LMFDB - Embedded newform 22.9.b.a.21.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [22,9,Mod(21,22)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("22.21");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 22 = 2 \cdot 11 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	22.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.96232942134$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7944x^{6} + 15215349x^{4} + 1757611988x^{2} + 38177252100 $ x^8 + 7944x^6 + 15215349x^4 + 1757611988*x^2 + 38177252100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 11^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			21.6
Root			$53.7858i$ of defining polynomial
Character	$\chi$	$=$	22.21
Dual form			22.9.b.a.21.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} -1.48475 q^{3} -128.000 q^{4} -214.559 q^{5} -16.7981i q^{6} -2223.81i q^{7} -1448.15i q^{8} -6558.80 q^{9} +O(q^{10})$	\(q+11.3137i q^{2} -1.48475 q^{3} -128.000 q^{4} -214.559 q^{5} -16.7981i q^{6} -2223.81i q^{7} -1448.15i q^{8} -6558.80 q^{9} -2427.46i q^{10} +(14356.2 - 2873.81i) q^{11} +190.048 q^{12} -45674.7i q^{13} +25159.5 q^{14} +318.567 q^{15} +16384.0 q^{16} -57872.1i q^{17} -74204.3i q^{18} +7002.19i q^{19} +27463.6 q^{20} +3301.81i q^{21} +(32513.5 + 162422. i) q^{22} -370074. q^{23} +2150.15i q^{24} -344589. q^{25} +516751. q^{26} +19479.7 q^{27} +284647. i q^{28} +976934. i q^{29} +3604.18i q^{30} -98983.8 q^{31} +185364. i q^{32} +(-21315.4 + 4266.90i) q^{33} +654748. q^{34} +477138. i q^{35} +839526. q^{36} -1.22373e6 q^{37} -79220.8 q^{38} +67815.7i q^{39} +310715. i q^{40} -3.00097e6i q^{41} -37355.7 q^{42} -3.17189e6i q^{43} +(-1.83759e6 + 367848. i) q^{44} +1.40725e6 q^{45} -4.18690e6i q^{46} -2.24545e6 q^{47} -24326.2 q^{48} +819484. q^{49} -3.89858e6i q^{50} +85925.7i q^{51} +5.84637e6i q^{52} +1.29564e7 q^{53} +220387. i q^{54} +(-3.08025e6 + 616602. i) q^{55} -3.22042e6 q^{56} -10396.5i q^{57} -1.10527e7 q^{58} +1.29244e7 q^{59} -40776.6 q^{60} +5.39065e6i q^{61} -1.11987e6i q^{62} +1.45855e7i q^{63} -2.09715e6 q^{64} +9.79992e6i q^{65} +(-48274.5 - 241156. i) q^{66} -1.86222e7 q^{67} +7.40762e6i q^{68} +549468. q^{69} -5.39820e6 q^{70} -5.69111e6 q^{71} +9.49815e6i q^{72} +1.85193e7i q^{73} -1.38449e7i q^{74} +511630. q^{75} -896281. i q^{76} +(-6.39080e6 - 3.19254e7i) q^{77} -767247. q^{78} +841589. i q^{79} -3.51533e6 q^{80} +4.30033e7 q^{81} +3.39521e7 q^{82} -8.26457e7i q^{83} -422631. i q^{84} +1.24170e7i q^{85} +3.58858e7 q^{86} -1.45051e6i q^{87} +(-4.16172e6 - 2.07900e7i) q^{88} -1.71143e7 q^{89} +1.59212e7i q^{90} -1.01572e8 q^{91} +4.73694e7 q^{92} +146967. q^{93} -2.54043e7i q^{94} -1.50238e6i q^{95} -275220. i q^{96} +1.06451e8 q^{97} +9.27140e6i q^{98} +(-9.41593e7 + 1.88487e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 182 q^{3} - 1024 q^{4} - 1410 q^{5} + 44582 q^{9}+O(q^{10}) $ 8 * q + 182 * q^3 - 1024 * q^4 - 1410 * q^5 + 44582 * q^9	$ 8 q + 182 q^{3} - 1024 q^{4} - 1410 q^{5} + 44582 q^{9} + 5808 q^{11} - 23296 q^{12} + 12288 q^{14} + 87958 q^{15} + 131072 q^{16} + 180480 q^{20} + 359040 q^{22} - 802026 q^{23} + 999558 q^{25} - 1321728 q^{26} - 1561354 q^{27} + 196726 q^{31} - 4286722 q^{33} - 701952 q^{34} - 5706496 q^{36} + 8627998 q^{37} + 9288960 q^{38} + 4366848 q^{42} - 743424 q^{44} + 1146988 q^{45} - 14335392 q^{47} + 2981888 q^{48} - 6714712 q^{49} + 55946352 q^{53} - 10078442 q^{55} - 1572864 q^{56} - 23226624 q^{58} + 21793110 q^{59} - 11258624 q^{60} - 16777216 q^{64} - 44242176 q^{66} - 113809034 q^{67} - 171636914 q^{69} + 137817600 q^{70} + 16741974 q^{71} + 346496844 q^{75} - 137074080 q^{77} - 57993216 q^{78} - 23101440 q^{80} + 85282724 q^{81} + 47480832 q^{82} + 49839360 q^{86} - 45957120 q^{88} + 42055422 q^{89} - 146801952 q^{91} + 102659328 q^{92} + 253251118 q^{93} + 100034782 q^{97} - 333541978 q^{99}+O(q^{100}) $ 8 * q + 182 * q^3 - 1024 * q^4 - 1410 * q^5 + 44582 * q^9 + 5808 * q^11 - 23296 * q^12 + 12288 * q^14 + 87958 * q^15 + 131072 * q^16 + 180480 * q^20 + 359040 * q^22 - 802026 * q^23 + 999558 * q^25 - 1321728 * q^26 - 1561354 * q^27 + 196726 * q^31 - 4286722 * q^33 - 701952 * q^34 - 5706496 * q^36 + 8627998 * q^37 + 9288960 * q^38 + 4366848 * q^42 - 743424 * q^44 + 1146988 * q^45 - 14335392 * q^47 + 2981888 * q^48 - 6714712 * q^49 + 55946352 * q^53 - 10078442 * q^55 - 1572864 * q^56 - 23226624 * q^58 + 21793110 * q^59 - 11258624 * q^60 - 16777216 * q^64 - 44242176 * q^66 - 113809034 * q^67 - 171636914 * q^69 + 137817600 * q^70 + 16741974 * q^71 + 346496844 * q^75 - 137074080 * q^77 - 57993216 * q^78 - 23101440 * q^80 + 85282724 * q^81 + 47480832 * q^82 + 49839360 * q^86 - 45957120 * q^88 + 42055422 * q^89 - 146801952 * q^91 + 102659328 * q^92 + 253251118 * q^93 + 100034782 * q^97 - 333541978 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/22\mathbb{Z}\right)^\times$.

$n$	$13$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			11.3137i			0.707107i
$2$			11.3137i			0.707107i
$3$	−1.48475			−0.0183303			−0.00916514	−	0.999958i	$-0.502917\pi$
$3$	−1.48475			−0.0183303			−0.00916514	+	0.999958i	$0.502917\pi$
$4$	−128.000			−0.500000
$5$	−214.559			−0.343294			−0.171647	−	0.985158i	$-0.554909\pi$
$5$	−214.559			−0.343294			−0.171647	+	0.985158i	$0.554909\pi$
$6$		−	16.7981i		−	0.0129615i
$7$		−	2223.81i		−	0.926200i	−0.886306	−	0.463100i	$-0.846737\pi$
$7$		−	2223.81i		−	0.926200i	0.886306	−	0.463100i	$-0.153263\pi$
$8$		−	1448.15i		−	0.353553i
$9$	−6558.80			−0.999664
$10$		−	2427.46i		−	0.242746i
$11$	14356.2	−	2873.81i	0.980547	−	0.196285i
$11$	14356.2	−	2873.81i	0.980547	−	0.196285i
$12$	190.048			0.00916514
$13$		−	45674.7i		−	1.59920i	−0.600534	−	0.799600i	$-0.705045\pi$
$13$		−	45674.7i		−	1.59920i	0.600534	−	0.799600i	$-0.294955\pi$
$14$	25159.5			0.654923
$15$	318.567			0.00629269
$16$	16384.0			0.250000
$17$		−	57872.1i		−	0.692904i	−0.938068	−	0.346452i	$-0.887386\pi$
$17$		−	57872.1i		−	0.692904i	0.938068	−	0.346452i	$-0.112614\pi$
$18$		−	74204.3i		−	0.706869i
$19$			7002.19i			0.0537304i	0.999639	+	0.0268652i	$0.00855248\pi$
$19$			7002.19i			0.0537304i	−0.999639	+	0.0268652i	$0.991448\pi$
$20$	27463.6			0.171647
$21$			3301.81i			0.0169775i
$22$	32513.5	+	162422.i	0.138795	+	0.693351i
$23$	−370074.			−1.32244			−0.661221	−	0.750191i	$-0.729962\pi$
$23$	−370074.			−1.32244			−0.661221	+	0.750191i	$0.729962\pi$
$24$			2150.15i			0.00648074i
$25$	−344589.			−0.882149
$26$	516751.			1.13080
$27$	19479.7			0.0366544
$28$			284647.i			0.463100i
$29$			976934.i			1.38125i	0.723212	+	0.690626i	$0.242665\pi$
$29$			976934.i			1.38125i	−0.723212	+	0.690626i	$0.757335\pi$
$30$			3604.18i			0.00444960i
$31$	−98983.8			−0.107181			−0.0535904	−	0.998563i	$-0.517067\pi$
$31$	−98983.8			−0.107181			−0.0535904	+	0.998563i	$0.517067\pi$
$32$			185364.i			0.176777i
$33$	−21315.4	+	4266.90i	−0.0179737	+	0.00359796i
$34$	654748.			0.489957
$35$			477138.i			0.317959i
$36$	839526.			0.499832
$37$	−1.22373e6			−0.652946			−0.326473	−	0.945206i	$-0.605860\pi$
$37$	−1.22373e6			−0.652946			−0.326473	+	0.945206i	$0.605860\pi$
$38$	−79220.8			−0.0379931
$39$			67815.7i			0.0293138i
$40$			310715.i			0.121373i
$41$		−	3.00097e6i		−	1.06200i	−0.847371	−	0.531002i	$-0.821816\pi$
$41$		−	3.00097e6i		−	1.06200i	0.847371	−	0.531002i	$-0.178184\pi$
$42$	−37355.7			−0.0120049
$43$		−	3.17189e6i		−	0.927778i	−0.885893	−	0.463889i	$-0.846454\pi$
$43$		−	3.17189e6i		−	0.927778i	0.885893	−	0.463889i	$-0.153546\pi$
$44$	−1.83759e6	+	367848.i	−0.490273	+	0.0981426i
$45$	1.40725e6			0.343179
$46$		−	4.18690e6i		−	0.935108i
$47$	−2.24545e6			−0.460162			−0.230081	−	0.973171i	$-0.573899\pi$
$47$	−2.24545e6			−0.460162			−0.230081	+	0.973171i	$0.573899\pi$
$48$	−24326.2			−0.00458257
$49$	819484.			0.142153
$50$		−	3.89858e6i		−	0.623774i
$51$			85925.7i			0.0127011i
$52$			5.84637e6i			0.799600i
$53$	1.29564e7			1.64203			0.821015	−	0.570907i	$-0.193408\pi$
$53$	1.29564e7			1.64203			0.821015	+	0.570907i	$0.193408\pi$
$54$			220387.i			0.0259186i
$55$	−3.08025e6	+	616602.i	−0.336616	+	0.0673836i
$56$	−3.22042e6			−0.327461
$57$		−	10396.5i		−	0.000984893i
$58$	−1.10527e7			−0.976693
$59$	1.29244e7			1.06661			0.533303	−	0.845924i	$-0.320951\pi$
$59$	1.29244e7			1.06661			0.533303	+	0.845924i	$0.320951\pi$
$60$	−40776.6			−0.00314634
$61$			5.39065e6i			0.389333i	0.980869	+	0.194667i	$0.0623625\pi$
$61$			5.39065e6i			0.389333i	−0.980869	+	0.194667i	$0.937637\pi$
$62$		−	1.11987e6i		−	0.0757883i
$63$			1.45855e7i			0.925889i
$64$	−2.09715e6			−0.125000
$65$			9.79992e6i			0.548996i
$66$	−48274.5	−	241156.i	−0.00254414	−	0.0127093i
$67$	−1.86222e7			−0.924128			−0.462064	−	0.886847i	$-0.652891\pi$
$67$	−1.86222e7			−0.924128			−0.462064	+	0.886847i	$0.652891\pi$
$68$			7.40762e6i			0.346452i
$69$	549468.			0.0242408
$70$	−5.39820e6			−0.224831
$71$	−5.69111e6			−0.223956			−0.111978	−	0.993711i	$-0.535719\pi$
$71$	−5.69111e6			−0.223956			−0.111978	+	0.993711i	$0.535719\pi$
$72$			9.49815e6i			0.353435i
$73$			1.85193e7i			0.652129i	0.945348	+	0.326065i	$0.105723\pi$
$73$			1.85193e7i			0.652129i	−0.945348	+	0.326065i	$0.894277\pi$
$74$		−	1.38449e7i		−	0.461703i
$75$	511630.			0.0161700
$76$		−	896281.i		−	0.0268652i
$77$	−6.39080e6	−	3.19254e7i	−0.181799	−	0.908183i
$78$	−767247.			−0.0207280
$79$			841589.i			0.0216069i	0.999942	+	0.0108034i	$0.00343890\pi$
$79$			841589.i			0.0216069i	−0.999942	+	0.0108034i	$0.996561\pi$
$80$	−3.51533e6			−0.0858236
$81$	4.30033e7			0.998992
$82$	3.39521e7			0.750950
$83$		−	8.26457e7i		−	1.74144i	−0.491781	−	0.870719i	$-0.663654\pi$
$83$		−	8.26457e7i		−	1.74144i	0.491781	−	0.870719i	$-0.336346\pi$
$84$		−	422631.i		−	0.00848876i
$85$			1.24170e7i			0.237870i
$86$	3.58858e7			0.656038
$87$		−	1.45051e6i		−	0.0253188i
$88$	−4.16172e6	−	2.07900e7i	−0.0693973	−	0.346676i
$89$	−1.71143e7			−0.272771			−0.136386	−	0.990656i	$-0.543549\pi$
$89$	−1.71143e7			−0.272771			−0.136386	+	0.990656i	$0.543549\pi$
$90$			1.59212e7i			0.242664i
$91$	−1.01572e8			−1.48118
$92$	4.73694e7			0.661221
$93$	146967.			0.00196466
$94$		−	2.54043e7i		−	0.325384i
$95$		−	1.50238e6i		−	0.0184453i
$96$		−	275220.i		−	0.00324037i
$97$	1.06451e8			1.20244			0.601218	−	0.799085i	$-0.294682\pi$
$97$	1.06451e8			1.20244			0.601218	+	0.799085i	$0.294682\pi$
$98$			9.27140e6i			0.100517i
$99$	−9.41593e7	+	1.88487e7i	−0.980217	+	0.196219i
$100$	4.41074e7			0.441074
$101$		−	1.27766e8i		−	1.22781i	−0.789381	−	0.613903i	$-0.789599\pi$
$101$		−	1.27766e8i		−	1.22781i	0.789381	−	0.613903i	$-0.210401\pi$
$102$	−972139.			−0.00898106
$103$	−1.25586e8			−1.11582			−0.557909	−	0.829902i	$-0.688396\pi$
$103$	−1.25586e8			−1.11582			−0.557909	+	0.829902i	$0.688396\pi$
$104$	−6.61441e7			−0.565402
$105$		−	708432.i		−	0.00582829i
$106$			1.46585e8i			1.16109i
$107$			1.42601e8i			1.08790i	0.839118	+	0.543949i	$0.183072\pi$
$107$			1.42601e8i			1.08790i	−0.839118	+	0.543949i	$0.816928\pi$
$108$	−2.49340e6			−0.0183272
$109$			1.50269e8i			1.06455i	0.846573	+	0.532273i	$0.178662\pi$
$109$			1.50269e8i			1.06455i	−0.846573	+	0.532273i	$0.821338\pi$
$110$	−6.97606e6	−	3.48490e7i	−0.0476474	−	0.238024i
$111$	1.81693e6			0.0119687
$112$		−	3.64349e7i		−	0.231550i
$113$	1.24886e8			0.765951			0.382975	−	0.923759i	$-0.374899\pi$
$113$	1.24886e8			0.765951			0.382975	+	0.923759i	$0.374899\pi$
$114$	117623.			0.000696424
$115$	7.94026e7			0.453987
$116$		−	1.25048e8i		−	0.690626i
$117$			2.99571e8i			1.59866i
$118$			1.46223e8i			0.754204i
$119$	−1.28696e8			−0.641768
$120$		−	461335.i		−	0.00222480i
$121$	1.97841e8	−	8.25139e7i	0.922944	−	0.384934i
$122$	−6.09882e7			−0.275300
$123$			4.45570e6i			0.0194668i
$124$	1.26699e7			0.0535904
$125$	1.57747e8			0.646131
$126$	−1.65016e8			−0.654702
$127$		−	2.41828e8i		−	0.929592i	−0.885418	−	0.464796i	$-0.846128\pi$
$127$		−	2.41828e8i		−	0.929592i	0.885418	−	0.464796i	$-0.153872\pi$
$128$		−	2.37266e7i		−	0.0883883i
$129$			4.70947e6i			0.0170064i
$130$	−1.10873e8			−0.388199
$131$		−	4.20730e8i		−	1.42862i	−0.699827	−	0.714312i	$-0.746740\pi$
$131$		−	4.20730e8i		−	1.42862i	0.699827	−	0.714312i	$-0.253260\pi$
$132$	2.72837e6	−	546163.i	0.00898685	−	0.00179898i
$133$	1.55715e7			0.0497651
$134$		−	2.10686e8i		−	0.653457i
$135$	−4.17954e6			−0.0125833
$136$	−8.38077e7			−0.244979
$137$	1.48404e8			0.421273			0.210637	−	0.977564i	$-0.432446\pi$
$137$	1.48404e8			0.421273			0.210637	+	0.977564i	$0.432446\pi$
$138$			6.21652e6i			0.0171408i
$139$			6.43984e8i			1.72511i	0.505966	+	0.862554i	$0.331136\pi$
$139$			6.43984e8i			1.72511i	−0.505966	+	0.862554i	$0.668864\pi$
$140$		−	6.10736e7i		−	0.158980i
$141$	3.33393e6			0.00843491
$142$		−	6.43875e7i		−	0.158361i
$143$	−1.31261e8	−	6.55715e8i	−0.313899	−	1.56809i
$144$	−1.07459e8			−0.249916
$145$		−	2.09610e8i		−	0.474176i
$146$	−2.09522e8			−0.461125
$147$	−1.21673e6			−0.00260571
$148$	1.56637e8			0.326473
$149$		−	8.74734e8i		−	1.77473i	−0.461072	−	0.887363i	$-0.652535\pi$
$149$		−	8.74734e8i		−	1.77473i	0.461072	−	0.887363i	$-0.347465\pi$
$150$			5.78844e6i			0.0114339i
$151$			7.38306e8i			1.42013i	0.704136	+	0.710066i	$0.251335\pi$
$151$			7.38306e8i			1.42013i	−0.704136	+	0.710066i	$0.748665\pi$
$152$	1.01403e7			0.0189965
$153$			3.79571e8i			0.692672i
$154$	3.61195e8	−	7.23037e7i	0.642182	−	0.128552i
$155$	2.12379e7			0.0367946
$156$		−	8.68041e6i		−	0.0146569i
$157$	8.59515e8			1.41467			0.707334	−	0.706879i	$-0.249898\pi$
$157$	8.59515e8			1.41467			0.707334	+	0.706879i	$0.249898\pi$
$158$	−9.52149e6			−0.0152784
$159$	−1.92371e7			−0.0300989
$160$		−	3.97715e7i		−	0.0606864i
$161$			8.22972e8i			1.22485i
$162$			4.86527e8i			0.706394i
$163$	−1.05456e9			−1.49390			−0.746950	−	0.664880i	$-0.768483\pi$
$163$	−1.05456e9			−1.49390			−0.746950	+	0.664880i	$0.768483\pi$
$164$			3.84124e8i			0.531002i
$165$	4.57341e6	−	915502.i	0.00617027	−	0.00123516i
$166$	9.35030e8			1.23138
$167$		−	4.11363e8i		−	0.528883i	−0.964402	−	0.264441i	$-0.914812\pi$
$167$		−	4.11363e8i		−	0.528883i	0.964402	−	0.264441i	$-0.0851876\pi$
$168$	4.78152e6			0.00600246
$169$	−1.27045e9			−1.55744
$170$	−1.40482e8			−0.168200
$171$		−	4.59260e7i		−	0.0537123i
$172$			4.06002e8i			0.463889i
$173$			6.61324e8i			0.738294i	0.929371	+	0.369147i	$0.120350\pi$
$173$			6.61324e8i			0.738294i	−0.929371	+	0.369147i	$0.879650\pi$
$174$	1.64106e7			0.0179031
$175$			7.66300e8i			0.817047i
$176$	2.35212e8	−	4.70845e7i	0.245137	−	0.0490713i
$177$	−1.91896e7			−0.0195512
$178$		−	1.93626e8i		−	0.192879i
$179$	4.25911e8			0.414865			0.207432	−	0.978249i	$-0.433489\pi$
$179$	4.25911e8			0.414865			0.207432	+	0.978249i	$0.433489\pi$
$180$	−1.80128e8			−0.171590
$181$	−1.10833e9			−1.03265			−0.516326	−	0.856392i	$-0.672701\pi$
$181$	−1.10833e9			−1.03265			−0.516326	+	0.856392i	$0.672701\pi$
$182$		−	1.14915e9i		−	1.04735i
$183$		−	8.00378e6i		−	0.00713659i
$184$			5.35924e8i			0.467554i
$185$	2.62562e8			0.224153
$186$			1.66274e6i			0.00138922i
$187$	−1.66313e8	−	8.30822e8i	−0.136007	−	0.679425i
$188$	2.87417e8			0.230081
$189$		−	4.33190e7i		−	0.0339493i
$190$	1.69975e7			0.0130428
$191$	2.39139e8			0.179687			0.0898435	−	0.995956i	$-0.471363\pi$
$191$	2.39139e8			0.179687			0.0898435	+	0.995956i	$0.471363\pi$
$192$	3.11375e6			0.00229129
$193$		−	2.27649e9i		−	1.64073i	−0.571839	−	0.820366i	$-0.693770\pi$
$193$		−	2.27649e9i		−	1.64073i	0.571839	−	0.820366i	$-0.306230\pi$
$194$			1.20435e9i			0.850250i
$195$		−	1.45505e7i		−	0.0100633i
$196$	−1.04894e8			−0.0710765
$197$		−	2.15763e8i		−	0.143256i	−0.997431	−	0.0716280i	$-0.977181\pi$
$197$		−	2.15763e8i		−	0.143256i	0.997431	−	0.0716280i	$-0.0228194\pi$
$198$	−2.13249e8	−	1.06529e9i	−0.138748	−	0.693118i
$199$	−7.80596e8			−0.497753			−0.248876	−	0.968535i	$-0.580061\pi$
$199$	−7.80596e8			−0.497753			−0.248876	+	0.968535i	$0.580061\pi$
$200$			4.99019e8i			0.311887i
$201$	2.76494e7			0.0169395
$202$	1.44551e9			0.868190
$203$	2.17251e9			1.27932
$204$		−	1.09985e7i		−	0.00635057i
$205$			6.43885e8i			0.364580i
$206$		−	1.42085e9i		−	0.789003i
$207$	2.42724e9			1.32200
$208$		−	7.48335e8i		−	0.399800i
$209$	2.01230e7	+	1.00525e8i	0.0105465	+	0.0526851i
$210$	8.01499e6			0.00412122
$211$		−	1.63928e9i		−	0.827032i	−0.910497	−	0.413516i	$-0.864301\pi$
$211$		−	1.63928e9i		−	0.827032i	0.910497	−	0.413516i	$-0.135699\pi$
$212$	−1.65842e9			−0.821015
$213$	8.44989e6			0.00410519
$214$	−1.61335e9			−0.769261
$215$			6.80557e8i			0.318501i
$216$		−	2.82096e7i		−	0.0129593i
$217$			2.20121e8i			0.0992709i
$218$	−1.70010e9			−0.752748
$219$		−	2.74966e7i		−	0.0119537i
$220$	3.94272e8	−	7.89251e7i	0.168308	−	0.0336918i
$221$	−2.64329e9			−1.10809
$222$			2.05562e7i			0.00846315i
$223$	1.68253e9			0.680369			0.340184	−	0.940359i	$-0.389511\pi$
$223$	1.68253e9			0.680369			0.340184	+	0.940359i	$0.389511\pi$
$224$	4.12213e8			0.163731
$225$	2.26009e9			0.881853
$226$			1.41293e9i			0.541609i
$227$			2.27805e9i			0.857946i	0.903317	+	0.428973i	$0.141124\pi$
$227$			2.27805e9i			0.857946i	−0.903317	+	0.428973i	$0.858876\pi$
$228$			1.33076e6i			0.000492446i
$229$	−2.57364e9			−0.935849			−0.467924	−	0.883768i	$-0.654998\pi$
$229$	−2.57364e9			−0.935849			−0.467924	+	0.883768i	$0.654998\pi$
$230$			8.98338e8i			0.321017i
$231$	9.48876e6	+	4.74013e7i	0.00333244	+	0.0166473i
$232$	1.41475e9			0.488347
$233$			7.71723e8i			0.261841i	0.991393	+	0.130920i	$0.0417933\pi$
$233$			7.71723e8i			0.261841i	−0.991393	+	0.130920i	$0.958207\pi$
$234$	−3.38926e9			−1.13042
$235$	4.81781e8			0.157971
$236$	−1.65433e9			−0.533303
$237$		−	1.24955e6i		−	0.000396060i
$238$		−	1.45603e9i		−	0.453799i
$239$			4.70857e9i			1.44310i	0.692360	+	0.721552i	$0.256571\pi$
$239$			4.70857e9i			1.44310i	−0.692360	+	0.721552i	$0.743429\pi$
$240$	5.21941e6			0.00157317
$241$		−	7.22359e7i		−	0.0214134i	−0.999943	−	0.0107067i	$-0.996592\pi$
$241$		−	7.22359e7i		−	0.0214134i	0.999943	−	0.0107067i	$-0.00340811\pi$
$242$	9.33539e8	+	2.23832e9i	0.272189	+	0.652620i
$243$	−1.91655e8			−0.0549662
$244$		−	6.90003e8i		−	0.194667i
$245$	−1.75828e8			−0.0488003
$246$	−5.04105e7			−0.0137651
$247$	3.19823e8			0.0859255
$248$			1.43344e8i			0.0378942i
$249$			1.22709e8i			0.0319211i
$250$			1.78470e9i			0.456884i
$251$	6.89580e9			1.73736			0.868680	−	0.495374i	$-0.164969\pi$
$251$	6.89580e9			1.73736			0.868680	+	0.495374i	$0.164969\pi$
$252$		−	1.86694e9i		−	0.462945i
$253$	−5.31285e9	+	1.06352e9i	−1.29672	+	0.259576i
$254$	2.73598e9			0.657321
$255$		−	1.84361e7i		−	0.00436023i
$256$	2.68435e8			0.0625000
$257$	−2.02817e7			−0.00464914			−0.00232457	−	0.999997i	$-0.500740\pi$
$257$	−2.02817e7			−0.00464914			−0.00232457	+	0.999997i	$0.500740\pi$
$258$	−5.32816e7			−0.0120254
$259$			2.72133e9i			0.604759i
$260$		−	1.25439e9i		−	0.274498i
$261$		−	6.40751e9i		−	1.38079i
$262$	4.76001e9			1.01019
$263$		−	3.33895e9i		−	0.697890i	−0.937143	−	0.348945i	$-0.886540\pi$
$263$		−	3.33895e9i		−	0.697890i	0.937143	−	0.348945i	$-0.113460\pi$
$264$	6.17913e6	+	3.08680e7i	0.00127207	+	0.00635467i
$265$	−2.77991e9			−0.563700
$266$			1.76172e8i			0.0351892i
$267$	2.54105e7			0.00499998
$268$	2.38364e9			0.462064
$269$	−4.94466e8			−0.0944338			−0.0472169	−	0.998885i	$-0.515035\pi$
$269$	−4.94466e8			−0.0944338			−0.0472169	+	0.998885i	$0.515035\pi$
$270$		−	4.72861e7i		−	0.00889771i
$271$		−	8.40516e9i		−	1.55836i	−0.626797	−	0.779182i	$-0.715635\pi$
$271$		−	8.40516e9i		−	1.55836i	0.626797	−	0.779182i	$-0.284365\pi$
$272$		−	9.48176e8i		−	0.173226i
$273$	1.50809e8			0.0271504
$274$			1.67900e9i			0.297885i
$275$	−4.94699e9	+	9.90285e8i	−0.864988	+	0.173153i
$276$	−7.03319e7			−0.0121204
$277$			2.24724e8i			0.0381707i	0.999818	+	0.0190854i	$0.00607543\pi$
$277$			2.24724e8i			0.0381707i	−0.999818	+	0.0190854i	$0.993925\pi$
$278$	−7.28585e9			−1.21984
$279$	6.49214e8			0.107145
$280$	6.90969e8			0.112416
$281$			6.61301e9i			1.06065i	0.847793	+	0.530327i	$0.177931\pi$
$281$			6.61301e9i			1.06065i	−0.847793	+	0.530327i	$0.822069\pi$
$282$			3.77191e7i			0.00596438i
$283$		−	1.10649e10i		−	1.72504i	−0.506019	−	0.862522i	$-0.668883\pi$
$283$		−	1.10649e10i		−	1.72504i	0.506019	−	0.862522i	$-0.331117\pi$
$284$	7.28462e8			0.111978
$285$			2.23067e6i			0.000338108i
$286$	7.41857e9	−	1.48504e9i	1.10881	−	0.221960i
$287$	−6.67357e9			−0.983628
$288$		−	1.21576e9i		−	0.176717i
$289$	3.62658e9			0.519884
$290$	2.37147e9			0.335293
$291$	−1.58053e8			−0.0220410
$292$		−	2.37047e9i		−	0.326065i
$293$		−	6.14596e9i		−	0.833911i	−0.908927	−	0.416955i	$-0.863097\pi$
$293$		−	6.14596e9i		−	0.833911i	0.908927	−	0.416955i	$-0.136903\pi$
$294$		−	1.37657e7i		−	0.00184251i
$295$	−2.77306e9			−0.366160
$296$			1.77215e9i			0.230851i
$297$	2.79654e8	−	5.59809e7i	0.0359414	−	0.00719472i
$298$	9.89649e9			1.25492
$299$			1.69030e10i			2.11485i
$300$	−6.54887e7			−0.00808502
$301$	−7.05367e9			−0.859308
$302$	−8.35298e9			−1.00418
$303$			1.89701e8i			0.0225060i
$304$			1.14724e8i			0.0134326i
$305$		−	1.15661e9i		−	0.133656i
$306$	−4.29436e9			−0.489793
$307$			2.57642e9i			0.290044i	0.989428	+	0.145022i	$0.0463253\pi$
$307$			2.57642e9i			0.290044i	−0.989428	+	0.145022i	$0.953675\pi$
$308$	8.18023e8	+	4.08645e9i	0.0908997	+	0.454091i
$309$	1.86465e8			0.0204533
$310$			2.40279e8i			0.0260177i
$311$	1.10814e10			1.18455			0.592275	−	0.805736i	$-0.298230\pi$
$311$	1.10814e10			1.18455			0.592275	+	0.805736i	$0.298230\pi$
$312$	9.82076e7			0.0103640
$313$	1.63067e10			1.69898			0.849492	−	0.527601i	$-0.176909\pi$
$313$	1.63067e10			1.69898			0.849492	+	0.527601i	$0.176909\pi$
$314$			9.72430e9i			1.00032i
$315$		−	3.12945e9i		−	0.317853i
$316$		−	1.07723e8i		−	0.0108034i
$317$	−6.20152e9			−0.614131			−0.307066	−	0.951688i	$-0.599347\pi$
$317$	−6.20152e9			−0.614131			−0.307066	+	0.951688i	$0.599347\pi$
$318$		−	2.17643e8i		−	0.0212831i
$319$	2.80752e9	+	1.40250e10i	0.271119	+	1.35438i
$320$	4.49963e8			0.0429118
$321$		−	2.11728e8i		−	0.0199415i
$322$	−9.31087e9			−0.866097
$323$	4.05231e8			0.0372300
$324$	−5.50443e9			−0.499496
$325$			1.57390e10i			1.41073i
$326$		−	1.19310e10i		−	1.05635i
$327$		−	2.23113e8i		−	0.0195134i
$328$	−4.34587e9			−0.375475
$329$			4.99344e9i			0.426203i
$330$	1.03577e7	+	5.17422e7i	0.000873391	+	0.00436304i
$331$	−1.64035e10			−1.36655			−0.683275	−	0.730161i	$-0.739445\pi$
$331$	−1.64035e10			−1.36655			−0.683275	+	0.730161i	$0.739445\pi$
$332$			1.05787e10i			0.870719i
$333$	8.02617e9			0.652727
$334$	4.65404e9			0.373977
$335$	3.99556e9			0.317248
$336$			5.40968e7i			0.00424438i
$337$		−	1.93188e10i		−	1.49782i	−0.662670	−	0.748911i	$-0.730577\pi$
$337$		−	1.93188e10i		−	1.49782i	0.662670	−	0.748911i	$-0.269423\pi$
$338$		−	1.43735e10i		−	1.10127i
$339$	−1.85425e8			−0.0140401
$340$		−	1.58937e9i		−	0.118935i
$341$	−1.42103e9	+	2.84461e8i	−0.105096	+	0.0210380i
$342$	5.19593e8			0.0379803
$343$		−	1.46422e10i		−	1.05786i
$344$	−4.59338e9			−0.328019
$345$	−1.17893e8			−0.00832171
$346$	−7.48202e9			−0.522053
$347$		−	9.04611e9i		−	0.623942i	−0.950092	−	0.311971i	$-0.899011\pi$
$347$		−	9.04611e9i		−	0.623942i	0.950092	−	0.311971i	$-0.100989\pi$
$348$			1.85665e8i			0.0126594i
$349$			9.59039e9i			0.646450i	0.946322	+	0.323225i	$0.104767\pi$
$349$			9.59039e9i			0.646450i	−0.946322	+	0.323225i	$0.895233\pi$
$350$	−8.66970e9			−0.577739
$351$		−	8.89728e8i		−	0.0586177i
$352$	5.32701e8	+	2.66112e9i	0.0346986	+	0.173338i
$353$	1.76246e10			1.13506			0.567531	−	0.823352i	$-0.307899\pi$
$353$	1.76246e10			1.13506			0.567531	+	0.823352i	$0.307899\pi$
$354$		−	2.17106e8i		−	0.0138248i
$355$	1.22108e9			0.0768830
$356$	2.19063e9			0.136386
$357$	1.91082e8			0.0117638
$358$			4.81863e9i			0.293354i
$359$		−	4.99806e9i		−	0.300901i	−0.988618	−	0.150451i	$-0.951928\pi$
$359$		−	4.99806e9i		−	0.300901i	0.988618	−	0.150451i	$-0.0480725\pi$
$360$		−	2.03791e9i		−	0.121332i
$361$	1.69345e10			0.997113
$362$		−	1.25393e10i		−	0.730195i
$363$	−2.93746e8	+	1.22513e8i	−0.0169178	+	0.00705594i
$364$	1.30012e10			0.740589
$365$		−	3.97349e9i		−	0.223872i
$366$	9.05525e7			0.00504633
$367$	−2.74387e10			−1.51251			−0.756256	−	0.654276i	$-0.772973\pi$
$367$	−2.74387e10			−1.51251			−0.756256	+	0.654276i	$0.772973\pi$
$368$	−6.06329e9			−0.330611
$369$			1.96827e10i			1.06165i
$370$			2.97054e9i			0.158500i
$371$		−	2.88125e10i		−	1.52085i
$372$	−1.88117e7			−0.000982328
$373$		−	2.19801e10i		−	1.13552i	−0.823194	−	0.567760i	$-0.807810\pi$
$373$		−	2.19801e10i		−	1.13552i	0.823194	−	0.567760i	$-0.192190\pi$
$374$	9.39968e9	−	1.88162e9i	0.480426	−	0.0961714i
$375$	−2.34215e8			−0.0118438
$376$			3.25175e9i			0.162692i
$377$	4.46212e10			2.20890
$378$	4.90099e8			0.0240058
$379$	−3.78267e9			−0.183334			−0.0916668	−	0.995790i	$-0.529219\pi$
$379$	−3.78267e9			−0.183334			−0.0916668	+	0.995790i	$0.529219\pi$
$380$			1.92305e8i			0.00922266i
$381$			3.59055e8i			0.0170397i
$382$			2.70555e9i			0.127058i
$383$	−2.61356e10			−1.21461			−0.607307	−	0.794467i	$-0.707750\pi$
$383$	−2.61356e10			−1.21461			−0.607307	+	0.794467i	$0.707750\pi$
$384$			3.52281e7i			0.00162018i
$385$	1.37120e9	+	6.84988e9i	0.0624107	+	0.311774i
$386$	2.57556e10			1.16017
$387$			2.08038e10i			0.927466i
$388$	−1.36257e10			−0.601218
$389$	−1.21794e10			−0.531899			−0.265949	−	0.963987i	$-0.585685\pi$
$389$	−1.21794e10			−0.531899			−0.265949	+	0.963987i	$0.585685\pi$
$390$	1.64620e8			0.00711580
$391$			2.14169e10i			0.916326i
$392$		−	1.18674e9i		−	0.0502587i
$393$			6.24680e8i			0.0261871i
$394$	2.44108e9			0.101297
$395$		−	1.80570e8i		−	0.00741751i
$396$	1.20524e10	−	2.41264e9i	0.490109	−	0.0981096i
$397$	−3.49985e10			−1.40892			−0.704461	−	0.709743i	$-0.748811\pi$
$397$	−3.49985e10			−1.40892			−0.704461	+	0.709743i	$0.748811\pi$
$398$		−	8.83143e9i		−	0.351964i
$399$	−2.31199e7			−0.000912208
$400$	−5.64575e9			−0.220537
$401$	−9.82998e9			−0.380167			−0.190084	−	0.981768i	$-0.560876\pi$
$401$	−9.82998e9			−0.380167			−0.190084	+	0.981768i	$0.560876\pi$
$402$			3.12817e8i			0.0119781i
$403$			4.52106e9i			0.171404i
$404$			1.63540e10i			0.613903i
$405$	−9.22675e9			−0.342948
$406$			2.45792e10i			0.904614i
$407$	−1.75680e10	+	3.51676e9i	−0.640244	+	0.128164i
$408$	1.24434e8			0.00449053
$409$			2.69330e10i			0.962479i	0.876589	+	0.481239i	$0.159813\pi$
$409$			2.69330e10i			0.962479i	−0.876589	+	0.481239i	$0.840187\pi$
$410$	−7.28472e9			−0.257797
$411$	−2.20344e8			−0.00772206
$412$	1.60751e10			0.557909
$413$		−	2.87415e10i		−	0.987891i
$414$			2.74611e10i			0.934794i
$415$			1.77324e10i			0.597826i
$416$	8.46644e9			0.282701
$417$		−	9.56158e8i		−	0.0316217i
$418$	−1.13731e9	+	2.27666e8i	−0.0372540	+	0.00745748i
$419$	−1.09066e10			−0.353861			−0.176930	−	0.984223i	$-0.556617\pi$
$419$	−1.09066e10			−0.353861			−0.176930	+	0.984223i	$0.556617\pi$
$420$			9.06793e7i			0.00291414i
$421$	−2.63005e10			−0.837213			−0.418606	−	0.908168i	$-0.637481\pi$
$421$	−2.63005e10			−0.837213			−0.418606	+	0.908168i	$0.637481\pi$
$422$	1.85463e10			0.584800
$423$	1.47274e10			0.460008
$424$		−	1.87629e10i		−	0.580545i
$425$			1.99421e10i			0.611245i
$426$			9.55996e7i			0.00290280i
$427$	1.19878e10			0.360601
$428$		−	1.82530e10i		−	0.543949i
$429$	1.94890e8	+	9.73575e8i	0.00575386	+	0.0287435i
$430$	−7.69962e9			−0.225214
$431$		−	5.47962e10i		−	1.58797i	−0.607939	−	0.793983i	$-0.708004\pi$
$431$		−	5.47962e10i		−	1.58797i	0.607939	−	0.793983i	$-0.291996\pi$
$432$	3.19155e8			0.00916361
$433$	5.73287e10			1.63087			0.815437	−	0.578846i	$-0.196497\pi$
$433$	5.73287e10			1.63087			0.815437	+	0.578846i	$0.196497\pi$
$434$	−2.49038e9			−0.0701952
$435$			3.11219e8i			0.00869179i
$436$		−	1.92345e10i		−	0.532273i
$437$		−	2.59133e9i		−	0.0710553i
$438$	3.11089e8			0.00845255
$439$			3.77222e10i			1.01564i	0.861463	+	0.507820i	$0.169548\pi$
$439$			3.77222e10i			1.01564i	−0.861463	+	0.507820i	$0.830452\pi$
$440$	8.92935e8	+	4.46068e9i	0.0238237	+	0.119012i
$441$	−5.37483e9			−0.142105
$442$		−	2.99054e10i		−	0.783539i
$443$	−6.18748e9			−0.160657			−0.0803284	−	0.996768i	$-0.525597\pi$
$443$	−6.18748e9			−0.160657			−0.0803284	+	0.996768i	$0.525597\pi$
$444$	−2.32567e8			−0.00598435
$445$	3.67203e9			0.0936409
$446$			1.90357e10i			0.481093i
$447$			1.29876e9i			0.0325312i
$448$			4.66366e9i			0.115775i
$449$	2.95583e10			0.727268			0.363634	−	0.931542i	$-0.381536\pi$
$449$	2.95583e10			0.727268			0.363634	+	0.931542i	$0.381536\pi$
$450$			2.55700e10i			0.623564i
$451$	−8.62421e9	−	4.30824e10i	−0.208455	−	1.04134i
$452$	−1.59854e10			−0.382975
$453$		−	1.09620e9i		−	0.0260314i
$454$	−2.57732e10			−0.606659
$455$	2.17931e10			0.508480
$456$	−1.50558e7			−0.000348212
$457$		−	6.48701e10i		−	1.48724i	−0.668605	−	0.743618i	$-0.733108\pi$
$457$		−	6.48701e10i		−	1.48724i	0.668605	−	0.743618i	$-0.266892\pi$
$458$		−	2.91174e10i		−	0.661745i
$459$		−	1.12733e9i		−	0.0253980i
$460$	−1.01635e10			−0.226994
$461$		−	4.54926e9i		−	0.100725i	−0.998731	−	0.0503625i	$-0.983962\pi$
$461$		−	4.54926e9i		−	0.100725i	0.998731	−	0.0503625i	$-0.0160377\pi$
$462$	−5.36285e8	+	1.07353e8i	−0.0117714	+	0.00235639i
$463$	3.63538e10			0.791090			0.395545	−	0.918447i	$-0.370556\pi$
$463$	3.63538e10			0.791090			0.395545	+	0.918447i	$0.370556\pi$
$464$			1.60061e10i			0.345313i
$465$	−3.15330e7			−0.000674455
$466$	−8.73104e9			−0.185149
$467$	2.65035e10			0.557232			0.278616	−	0.960403i	$-0.410124\pi$
$467$	2.65035e10			0.557232			0.278616	+	0.960403i	$0.410124\pi$
$468$		−	3.83451e10i		−	0.799331i
$469$			4.14122e10i			0.855928i
$470$			5.45072e9i			0.111702i
$471$	−1.27617e9			−0.0259313
$472$		−	1.87166e10i		−	0.377102i
$473$	−9.11541e9	−	4.55362e10i	−0.182109	−	0.909730i
$474$	1.41371e7			0.000280057
$475$		−	2.41288e9i		−	0.0473982i
$476$	1.64731e10			0.320884
$477$	−8.49784e10			−1.64148
$478$	−5.32714e10			−1.02043
$479$		−	5.38841e10i		−	1.02357i	−0.859113	−	0.511786i	$-0.828984\pi$
$479$		−	5.38841e10i		−	1.02357i	0.859113	−	0.511786i	$-0.171016\pi$
$480$			5.90508e7i			0.00111240i
$481$			5.58934e10i			1.04419i
$482$	8.17256e8			0.0151415
$483$		−	1.22191e9i		−	0.0224518i
$484$	−2.53237e10	+	1.05618e10i	−0.461472	+	0.192467i
$485$	−2.28400e10			−0.412789
$486$		−	2.16833e9i		−	0.0388670i
$487$	2.76214e10			0.491054			0.245527	−	0.969390i	$-0.421039\pi$
$487$	2.76214e10			0.491054			0.245527	+	0.969390i	$0.421039\pi$
$488$	7.80649e9			0.137650
$489$	1.56576e9			0.0273836
$490$		−	1.98926e9i		−	0.0345070i
$491$			1.05788e11i			1.82016i	0.414437	+	0.910078i	$0.363978\pi$
$491$			1.05788e11i			1.82016i	−0.414437	+	0.910078i	$0.636022\pi$
$492$		−	5.70329e8i		−	0.00973341i
$493$	5.65372e10			0.957076
$494$			3.61839e9i			0.0607585i
$495$	2.02027e10	−	4.04417e9i	0.336503	−	0.0673610i
$496$	−1.62175e9			−0.0267952
$497$			1.26559e10i			0.207428i
$498$	−1.38829e9			−0.0225716
$499$	5.86163e10			0.945401			0.472701	−	0.881223i	$-0.343279\pi$
$499$	5.86163e10			0.945401			0.472701	+	0.881223i	$0.343279\pi$
$500$	−2.01916e10			−0.323066
$501$			6.10773e8i			0.00969457i
$502$			7.80171e10i			1.22850i
$503$			6.88326e10i			1.07528i	0.843174	+	0.537641i	$0.180684\pi$
$503$			6.88326e10i			1.07528i	−0.843174	+	0.537641i	$0.819316\pi$
$504$	2.11221e10			0.327351
$505$			2.74133e10i			0.421499i
$506$	−1.20324e10	−	6.01080e10i	−0.183548	−	0.916917i
$507$	1.88631e9			0.0285483
$508$			3.09540e10i			0.464796i
$509$	−1.02998e11			−1.53446			−0.767231	−	0.641371i	$-0.778366\pi$
$509$	−1.02998e11			−1.53446			−0.767231	+	0.641371i	$0.778366\pi$
$510$	2.08581e8			0.00308315
$511$	4.11834e10			0.604002
$512$			3.03700e9i			0.0441942i
$513$			1.36400e8i			0.00196945i
$514$		−	2.29462e8i		−	0.00328744i
$515$	2.69457e10			0.383054
$516$		−	6.02812e8i		−	0.00850322i
$517$	−3.22360e10	+	6.45299e9i	−0.451211	+	0.0903231i
$518$	−3.07884e10			−0.427629
$519$		−	9.81902e8i		−	0.0135332i
$520$	1.41918e10			0.194099
$521$	1.67021e10			0.226683			0.113342	−	0.993556i	$-0.463845\pi$
$521$	1.67021e10			0.226683			0.113342	+	0.993556i	$0.463845\pi$
$522$	7.24927e10			0.976365
$523$			3.26342e9i			0.0436181i	0.999762	+	0.0218090i	$0.00694258\pi$
$523$			3.26342e9i			0.0436181i	−0.999762	+	0.0218090i	$0.993057\pi$
$524$			5.38534e10i			0.714312i
$525$		−	1.13777e9i		−	0.0149767i
$526$	3.77759e10			0.493483
$527$			5.72840e9i			0.0742661i
$528$	−3.49231e8	+	6.99089e7i	−0.00449343	+	0.000899491i
$529$	5.86435e10			0.748854
$530$		−	3.14511e10i		−	0.398596i
$531$	−8.47688e10			−1.06625
$532$	−1.99316e9			−0.0248825
$533$	−1.37068e11			−1.69835
$534$			2.87487e8i			0.00353552i
$535$		−	3.05964e10i		−	0.373470i
$536$			2.69678e10i			0.326729i
$537$	−6.32372e8			−0.00760459
$538$		−	5.59424e9i		−	0.0667748i
$539$	1.17647e10	−	2.35504e9i	0.139388	−	0.0279025i
$540$	5.34981e8			0.00629163
$541$		−	1.43330e11i		−	1.67320i	−0.547813	−	0.836601i	$-0.684540\pi$
$541$		−	1.43330e11i		−	1.67320i	0.547813	−	0.836601i	$-0.315460\pi$
$542$	9.50936e10			1.10193
$543$	1.64559e9			0.0189288
$544$	1.07274e10			0.122489
$545$		−	3.22417e10i		−	0.365453i
$546$			1.70621e9i			0.0191983i
$547$			7.17247e10i			0.801161i	0.916262	+	0.400580i	$0.131191\pi$
$547$			7.17247e10i			0.801161i	−0.916262	+	0.400580i	$0.868809\pi$
$548$	−1.89957e10			−0.210637
$549$		−	3.53562e10i		−	0.389203i
$550$	−1.12038e10	−	5.59688e10i	−0.122437	−	0.611639i
$551$	−6.84068e9			−0.0742152
$552$		−	7.95715e8i		−	0.00857040i
$553$	1.87153e9			0.0200123
$554$	−2.54246e9			−0.0269908
$555$	−3.89839e8			−0.00410879
$556$		−	8.24300e10i		−	0.862554i
$557$			5.62488e10i			0.584377i	0.956361	+	0.292188i	$0.0943834\pi$
$557$			5.62488e10i			0.584377i	−0.956361	+	0.292188i	$0.905617\pi$
$558$			7.34502e9i			0.0757628i
$559$	−1.44875e11			−1.48370
$560$			7.81743e9i			0.0794898i
$561$	2.46934e8	+	1.23357e9i	0.00249304	+	0.0124541i
$562$	−7.48176e10			−0.749996
$563$		−	2.61401e10i		−	0.260179i	−0.991502	−	0.130090i	$-0.958473\pi$
$563$		−	2.61401e10i		−	0.260179i	0.991502	−	0.130090i	$-0.0415265\pi$
$564$	−4.26743e8			−0.00421745
$565$	−2.67955e10			−0.262947
$566$	1.25185e11			1.21979
$567$		−	9.56311e10i		−	0.925267i
$568$			8.24161e9i			0.0791805i
$569$		−	2.86188e10i		−	0.273025i	−0.990638	−	0.136512i	$-0.956411\pi$
$569$		−	2.86188e10i		−	0.273025i	0.990638	−	0.136512i	$-0.0435893\pi$
$570$	−2.52371e7			−0.000239079
$571$		−	1.44604e11i		−	1.36031i	−0.733070	−	0.680153i	$-0.761913\pi$
$571$		−	1.44604e11i		−	1.36031i	0.733070	−	0.680153i	$-0.238087\pi$
$572$	1.68013e10	+	8.39315e10i	0.156950	+	0.784045i
$573$	−3.55062e8			−0.00329371
$574$		−	7.55029e10i		−	0.695530i
$575$	1.27523e11			1.16659
$576$	1.37548e10			0.124958
$577$	4.55399e10			0.410856			0.205428	−	0.978672i	$-0.434141\pi$
$577$	4.55399e10			0.410856			0.205428	+	0.978672i	$0.434141\pi$
$578$			4.10301e10i			0.367613i
$579$			3.38003e9i			0.0300751i
$580$			2.68301e10i			0.237088i
$581$	−1.83788e11			−1.61292
$582$		−	1.78817e9i		−	0.0155853i
$583$	1.86005e11	−	3.72343e10i	1.61009	−	0.322306i
$584$	2.68188e10			0.230562
$585$		−	6.42757e10i		−	0.548812i
$586$	6.95336e10			0.589664
$587$	1.11640e11			0.940304			0.470152	−	0.882585i	$-0.344199\pi$
$587$	1.11640e11			0.940304			0.470152	+	0.882585i	$0.344199\pi$
$588$	1.55742e8			0.00130285
$589$		−	6.93104e8i		−	0.00575887i
$590$		−	3.13736e10i		−	0.258914i
$591$			3.20355e8i			0.00262592i
$592$	−2.00495e10			−0.163237
$593$			2.27803e11i			1.84222i	0.389304	+	0.921110i	$0.372716\pi$
$593$			2.27803e11i			1.84222i	−0.389304	+	0.921110i	$0.627284\pi$
$594$	6.33351e8	+	3.16392e9i	0.00508743	+	0.0254144i
$595$	2.76129e10			0.220315
$596$			1.11966e11i			0.887363i
$597$	1.15899e9			0.00912395
$598$	−1.91236e11			−1.49542
$599$	−1.81591e11			−1.41054			−0.705271	−	0.708938i	$-0.749175\pi$
$599$	−1.81591e11			−1.41054			−0.705271	+	0.708938i	$0.749175\pi$
$600$		−	7.40920e8i		−	0.00571697i
$601$			7.32472e10i			0.561426i	0.959792	+	0.280713i	$0.0905710\pi$
$601$			7.32472e10i			0.561426i	−0.959792	+	0.280713i	$0.909429\pi$
$602$		−	7.98031e10i		−	0.607623i
$603$	1.22139e11			0.923817
$604$		−	9.45031e10i		−	0.710066i
$605$	−4.24486e10	+	1.77041e10i	−0.316842	+	0.132146i
$606$	−2.14622e9			−0.0159142
$607$			2.16842e11i			1.59731i	0.601789	+	0.798655i	$0.294455\pi$
$607$			2.16842e11i			1.59731i	−0.601789	+	0.798655i	$0.705545\pi$
$608$	−1.29795e9			−0.00949827
$609$	−3.22565e9			−0.0234502
$610$	1.30856e10			0.0945091
$611$			1.02560e11i			0.735891i
$612$		−	4.85851e10i		−	0.346336i
$613$		−	1.11757e11i		−	0.791467i	−0.918365	−	0.395734i	$-0.870490\pi$
$613$		−	1.11757e11i		−	0.791467i	0.918365	−	0.395734i	$-0.129510\pi$
$614$	−2.91489e10			−0.205092
$615$		−	9.56010e8i		−	0.00668285i
$616$	−4.62329e10	+	9.25487e9i	−0.321091	+	0.0642758i
$617$	2.07093e11			1.42897			0.714487	−	0.699649i	$-0.246660\pi$
$617$	2.07093e11			1.42897			0.714487	+	0.699649i	$0.246660\pi$
$618$			2.10961e9i			0.0144626i
$619$	−1.26597e11			−0.862306			−0.431153	−	0.902279i	$-0.641893\pi$
$619$	−1.26597e11			−0.862306			−0.431153	+	0.902279i	$0.641893\pi$
$620$	−2.71845e9			−0.0183973
$621$	−7.20891e9			−0.0484734
$622$			1.25372e11i			0.837604i
$623$			3.80589e10i			0.252641i
$624$			1.11109e9i			0.00732845i
$625$	1.00759e11			0.660336
$626$			1.84490e11i			1.20136i
$627$	−2.98777e7	−	1.49255e8i	−0.000193320	−	0.000965734i
$628$	−1.10018e11			−0.707334
$629$			7.08196e10i			0.452429i
$630$	3.54057e10			0.224756
$631$	−2.14244e11			−1.35142			−0.675710	−	0.737167i	$-0.736163\pi$
$631$	−2.14244e11			−1.35142			−0.675710	+	0.737167i	$0.736163\pi$
$632$	1.21875e9			0.00763918
$633$			2.43392e9i			0.0151597i
$634$		−	7.01622e10i		−	0.434256i
$635$			5.18864e10i			0.319124i
$636$	2.46234e9			0.0150494
$637$		−	3.74297e10i		−	0.227331i
$638$	−1.58675e11	+	3.17635e10i	−0.957693	+	0.191710i
$639$	3.73268e10			0.223881
$640$			5.09075e9i			0.0303432i
$641$	−2.07622e10			−0.122982			−0.0614910	−	0.998108i	$-0.519586\pi$
$641$	−2.07622e10			−0.122982			−0.0614910	+	0.998108i	$0.519586\pi$
$642$	2.39543e9			0.0141008
$643$	−6.78226e10			−0.396763			−0.198381	−	0.980125i	$-0.563568\pi$
$643$	−6.78226e10			−0.396763			−0.198381	+	0.980125i	$0.563568\pi$
$644$		−	1.05340e11i		−	0.612423i
$645$		−	1.01046e9i		−	0.00583821i
$646$			4.58467e9i			0.0263256i
$647$	−3.32740e10			−0.189884			−0.0949420	−	0.995483i	$-0.530267\pi$
$647$	−3.32740e10			−0.189884			−0.0949420	+	0.995483i	$0.530267\pi$
$648$		−	6.22755e10i		−	0.353197i
$649$	1.85546e11	−	3.71424e10i	1.04586	−	0.209359i
$650$	−1.78067e11			−0.997538
$651$		−	3.26825e8i		−	0.00181967i
$652$	1.34984e11			0.746950
$653$	1.71017e11			0.940562			0.470281	−	0.882517i	$-0.344153\pi$
$653$	1.71017e11			0.940562			0.470281	+	0.882517i	$0.344153\pi$
$654$	2.52424e9			0.0137981
$655$			9.02714e10i			0.490439i
$656$		−	4.91679e10i		−	0.265501i
$657$		−	1.21464e11i		−	0.651910i
$658$	−5.64943e10			−0.301371
$659$		−	6.18263e10i		−	0.327817i	−0.986476	−	0.163909i	$-0.947590\pi$
$659$		−	6.18263e10i		−	0.327817i	0.986476	−	0.163909i	$-0.0524102\pi$
$660$	−5.85396e8	+	1.17184e8i	−0.00308514	+	0.000617580i
$661$	2.37186e11			1.24246			0.621231	−	0.783627i	$-0.286633\pi$
$661$	2.37186e11			1.24246			0.621231	+	0.783627i	$0.286633\pi$
$662$		−	1.85585e11i		−	0.966297i
$663$	3.92463e9			0.0203116
$664$	−1.19684e11			−0.615691
$665$	−3.34101e9			−0.0170841
$666$			9.08058e10i			0.461548i
$667$		−	3.61537e11i		−	1.82663i
$668$			5.26545e10i			0.264441i
$669$	−2.49815e9			−0.0124714
$670$			4.52046e10i			0.224328i
$671$	1.54917e10	+	7.73892e10i	0.0764204	+	0.381760i
$672$	−6.12035e8			−0.00300123
$673$			2.26065e11i			1.10198i	0.834513	+	0.550989i	$0.185749\pi$
$673$			2.26065e11i			1.10198i	−0.834513	+	0.550989i	$0.814251\pi$
$674$	2.18567e11			1.05912
$675$	−6.71249e9			−0.0323347
$676$	1.62618e11			0.778719
$677$			1.91296e11i			0.910648i	0.890326	+	0.455324i	$0.150477\pi$
$677$			1.91296e11i			0.910648i	−0.890326	+	0.455324i	$0.849523\pi$
$678$		−	2.09785e9i		−	0.00992785i
$679$		−	2.36726e11i		−	1.11370i
$680$	1.79817e10			0.0840998
$681$		−	3.38234e9i		−	0.0157264i
$682$	−3.21831e9	−	1.60771e10i	−0.0148761	−	0.0743140i
$683$	−4.03653e10			−0.185492			−0.0927460	−	0.995690i	$-0.529564\pi$
$683$	−4.03653e10			−0.185492			−0.0927460	+	0.995690i	$0.529564\pi$
$684$			5.87852e9i			0.0268561i
$685$	−3.18415e10			−0.144621
$686$	1.65657e11			0.748022
$687$	3.82122e9			0.0171544
$688$		−	5.19682e10i		−	0.231944i
$689$		−	5.91780e11i		−	2.62593i
$690$		−	1.33381e9i		−	0.00588434i
$691$	−2.44285e11			−1.07148			−0.535741	−	0.844383i	$-0.679968\pi$
$691$	−2.44285e11			−1.07148			−0.535741	+	0.844383i	$0.679968\pi$
$692$		−	8.46494e10i		−	0.369147i
$693$	4.19160e10	+	2.09392e11i	0.181738	+	0.907878i
$694$	1.02345e11			0.441194
$695$		−	1.38173e11i		−	0.592220i
$696$	−2.10056e9			−0.00895153
$697$	−1.73672e11			−0.735867
$698$	−1.08503e11			−0.457109
$699$		−	1.14582e9i		−	0.00479962i
$700$		−	9.80864e10i		−	0.408523i
$701$		−	5.41695e8i		−	0.00224328i	−0.999999	−	0.00112164i	$-0.999643\pi$
$701$		−	5.41695e8i		−	0.00224328i	0.999999	−	0.00112164i	$-0.000357029\pi$
$702$	1.00661e10			0.0414490
$703$		−	8.56877e9i		−	0.0350830i
$704$	−3.01071e10	+	6.02682e9i	−0.122568	+	0.0245356i
$705$	−7.15325e8			−0.00289566
$706$			1.99399e11i			0.802611i
$707$	−2.84127e11			−1.13719
$708$	2.45627e9			0.00977560
$709$	9.08055e10			0.359358			0.179679	−	0.983725i	$-0.442494\pi$
$709$	9.08055e10			0.359358			0.179679	+	0.983725i	$0.442494\pi$
$710$			1.38149e10i			0.0543645i
$711$		−	5.51981e9i		−	0.0215996i
$712$			2.47841e10i			0.0964393i
$713$	3.66313e10			0.141741
$714$			2.16185e9i			0.00831826i
$715$	2.81631e10	+	1.40690e11i	0.107760	+	0.538316i
$716$	−5.45166e10			−0.207432
$717$		−	6.99107e9i		−	0.0264525i
$718$	5.65466e10			0.212769
$719$	4.11573e11			1.54004			0.770019	−	0.638021i	$-0.220247\pi$
$719$	4.11573e11			1.54004			0.770019	+	0.638021i	$0.220247\pi$
$720$	2.30564e10			0.0857948
$721$			2.79280e11i			1.03347i
$722$			1.91592e11i			0.705065i
$723$			1.07253e8i			0.000392513i
$724$	1.41866e11			0.516326
$725$		−	3.36641e11i		−	1.21847i
$726$	−1.38607e9	−	3.32335e9i	−0.00498931	−	0.0119627i
$727$	−4.64918e11			−1.66433			−0.832164	−	0.554530i	$-0.812898\pi$
$727$	−4.64918e11			−1.66433			−0.832164	+	0.554530i	$0.812898\pi$
$728$			1.47092e11i			0.523676i
$729$	−2.81860e11			−0.997985
$730$	4.49549e10			0.158302
$731$	−1.83564e11			−0.642861
$732$			1.02448e9i			0.00356830i
$733$			2.38640e11i			0.826663i	0.910581	+	0.413331i	$0.135635\pi$
$733$			2.38640e11i			0.826663i	−0.910581	+	0.413331i	$0.864365\pi$
$734$		−	3.10433e11i		−	1.06951i
$735$	2.61061e8			0.000894524
$736$		−	6.85983e10i		−	0.233777i
$737$	−2.67344e11	+	5.35167e10i	−0.906151	+	0.181393i
$738$	−2.22685e11			−0.750697
$739$		−	1.11881e11i		−	0.375127i	−0.982252	−	0.187563i	$-0.939941\pi$
$739$		−	1.11881e11i		−	0.375127i	0.982252	−	0.187563i	$-0.0600590\pi$
$740$	−3.36079e10			−0.112076
$741$	−4.74859e8			−0.00157504
$742$	3.25977e11			1.07540
$743$			4.17624e11i			1.37035i	0.728380	+	0.685173i	$0.240273\pi$
$743$			4.17624e11i			1.37035i	−0.728380	+	0.685173i	$0.759727\pi$
$744$		−	2.12830e8i		−	0.000694611i
$745$			1.87682e11i			0.609253i
$746$	2.48677e11			0.802933
$747$			5.42056e11i			1.74085i
$748$	2.12881e10	+	1.06345e11i	0.0680034	+	0.339713i
$749$	3.17118e11			1.00761
$750$		−	2.64984e9i		−	0.00837481i
$751$	−3.59909e11			−1.13144			−0.565721	−	0.824596i	$-0.691402\pi$
$751$	−3.59909e11			−1.13144			−0.565721	+	0.824596i	$0.691402\pi$
$752$	−3.67894e10			−0.115041
$753$	−1.02386e10			−0.0318463
$754$			5.04831e11i			1.56193i
$755$		−	1.58410e11i		−	0.487523i
$756$			5.54483e9i			0.0169747i
$757$	4.22853e11			1.28767			0.643837	−	0.765163i	$-0.277341\pi$
$757$	4.22853e11			1.28767			0.643837	+	0.765163i	$0.277341\pi$
$758$		−	4.27961e10i		−	0.129636i
$759$	7.88827e9	−	1.57907e9i	0.0237692	−	0.00475810i
$760$	−2.17568e9			−0.00652141
$761$			3.31553e11i			0.988587i	0.869295	+	0.494293i	$0.164573\pi$
$761$			3.31553e11i			0.988587i	−0.869295	+	0.494293i	$0.835427\pi$
$762$	−4.06225e9			−0.0120489
$763$	3.34170e11			0.985983
$764$	−3.06098e10			−0.0898435
$765$		−	8.14404e10i		−	0.237790i
$766$		−	2.95691e11i		−	0.858861i
$767$		−	5.90321e11i		−	1.70572i
$768$	−3.98560e8			−0.00114564
$769$		−	2.60313e11i		−	0.744372i	−0.928158	−	0.372186i	$-0.878608\pi$
$769$		−	2.60313e11i		−	0.744372i	0.928158	−	0.372186i	$-0.121392\pi$
$770$	−7.74975e10	+	1.55134e10i	−0.220458	+	0.0441310i
$771$	3.01134e7			8.52201e−5
$772$			2.91391e11i			0.820366i
$773$	3.71425e11			1.04029			0.520144	−	0.854078i	$-0.325878\pi$
$773$	3.71425e11			1.04029			0.520144	+	0.854078i	$0.325878\pi$
$774$	−2.35368e11			−0.655818
$775$	3.41088e10			0.0945495
$776$		−	1.54157e11i		−	0.425125i
$777$		−	4.04051e9i		−	0.0110854i
$778$		−	1.37795e11i		−	0.376109i
$779$	2.10134e10			0.0570618
$780$			1.86246e9i			0.00503163i
$781$	−8.17026e10	+	1.63552e10i	−0.219600	+	0.0439593i
$782$	−2.42305e11			−0.647940
$783$			1.90303e10i			0.0506290i
$784$	1.34264e10			0.0355382
$785$	−1.84417e11			−0.485648
$786$	−7.06745e9			−0.0185171
$787$		−	5.60469e10i		−	0.146101i	−0.997328	−	0.0730504i	$-0.976727\pi$
$787$		−	5.60469e10i		−	0.146101i	0.997328	−	0.0730504i	$-0.0232734\pi$
$788$			2.76177e10i			0.0716280i
$789$			4.95752e9i			0.0127925i
$790$	2.04292e9			0.00524497
$791$		−	2.77723e11i		−	0.709424i
$792$	2.72959e10	+	1.36357e11i	0.0693740	+	0.346559i
$793$	2.46216e11			0.622622
$794$		−	3.95962e11i		−	0.996258i
$795$	4.12749e9			0.0103328
$796$	9.99162e10			0.248876
$797$	−4.49341e11			−1.11363			−0.556817	−	0.830635i	$-0.687978\pi$
$797$	−4.49341e11			−1.11363			−0.556817	+	0.830635i	$0.687978\pi$
$798$		−	2.61572e8i		−	0.000645029i
$799$			1.29949e11i			0.318849i
$800$		−	6.38744e10i		−	0.155943i
$801$	1.12249e11			0.272680
$802$		−	1.11213e11i		−	0.268819i
$803$	5.32210e10	+	2.65867e11i	0.128003	+	0.639443i
$804$	−3.53912e9			−0.00846977
$805$		−	1.76576e11i		−	0.420483i
$806$	−5.11499e10			−0.121201
$807$	7.34160e8			0.00173100
$808$	−1.85025e11			−0.434095
$809$		−	5.62894e11i		−	1.31411i	−0.753842	−	0.657056i	$-0.771801\pi$
$809$		−	5.62894e11i		−	1.31411i	0.753842	−	0.657056i	$-0.228199\pi$
$810$		−	1.04389e11i		−	0.242501i
$811$			2.52391e11i			0.583432i	0.956505	+	0.291716i	$0.0942262\pi$
$811$			2.52391e11i			0.583432i	−0.956505	+	0.291716i	$0.905774\pi$
$812$	−2.78082e11			−0.639658
$813$			1.24796e10i			0.0285653i
$814$	−3.97876e10	−	1.98760e11i	−0.0906254	−	0.452721i
$815$	2.26266e11			0.512848
$816$			1.40781e9i			0.00317528i
$817$	2.22102e10			0.0498498
$818$	−3.04712e11			−0.680575
$819$	6.66189e11			1.48068
$820$		−	8.24172e10i		−	0.182290i
$821$		−	3.53163e11i		−	0.777326i	−0.921380	−	0.388663i	$-0.872937\pi$
$821$		−	3.53163e11i		−	0.777326i	0.921380	−	0.388663i	$-0.127063\pi$
$822$		−	2.49290e9i		−	0.00546032i
$823$	−1.00800e11			−0.219716			−0.109858	−	0.993947i	$-0.535040\pi$
$823$	−1.00800e11			−0.219716			−0.109858	+	0.993947i	$0.535040\pi$
$824$			1.81868e11i			0.394501i
$825$	7.34506e9	−	1.47033e9i	0.0158555	−	0.00317394i
$826$	3.25173e11			0.698544
$827$		−	3.04487e11i		−	0.650948i	−0.945551	−	0.325474i	$-0.894476\pi$
$827$		−	3.04487e11i		−	0.650948i	0.945551	−	0.325474i	$-0.105524\pi$
$828$	−3.10686e11			−0.660999
$829$	−6.39906e10			−0.135487			−0.0677436	−	0.997703i	$-0.521580\pi$
$829$	−6.39906e10			−0.135487			−0.0677436	+	0.997703i	$0.521580\pi$
$830$	−2.00619e11			−0.422727
$831$		−	3.33660e8i		−	0.000699681i
$832$			9.57868e10i			0.199900i
$833$		−	4.74252e10i		−	0.0984984i
$834$	1.08177e10			0.0223599
$835$			8.82616e10i			0.181562i
$836$	−2.57574e9	−	1.28672e10i	−0.00527324	−	0.0263426i
$837$	−1.92817e9			−0.00392865
$838$		−	1.23394e11i		−	0.250217i
$839$	3.60066e11			0.726666			0.363333	−	0.931659i	$-0.381639\pi$
$839$	3.60066e11			0.726666			0.363333	+	0.931659i	$0.381639\pi$
$840$	−1.02592e9			−0.00206061
$841$	−4.54153e11			−0.907859
$842$		−	2.97556e11i		−	0.591999i
$843$		−	9.81869e9i		−	0.0194421i
$844$			2.09827e11i			0.413516i
$845$	2.72586e11			0.534660
$846$			1.66622e11i			0.325275i
$847$	−1.83495e11	−	4.39961e11i	−0.356526	−	0.854831i
$848$	2.12278e11			0.410507
$849$			1.64286e10i			0.0316206i
$850$	−2.25619e11			−0.432215
$851$	4.52869e11			0.863484
$852$	−1.08159e9			−0.00205259
$853$			7.46577e11i			1.41019i	0.709112	+	0.705096i	$0.249096\pi$
$853$			7.46577e11i			1.41019i	−0.709112	+	0.705096i	$0.750904\pi$
$854$			1.35626e11i			0.254983i
$855$			9.85383e9i			0.0184391i
$856$	2.06509e11			0.384630
$857$			7.22932e11i			1.34021i	0.742265	+	0.670107i	$0.233752\pi$
$857$			7.22932e11i			1.34021i	−0.742265	+	0.670107i	$0.766248\pi$
$858$	−1.10147e10	+	2.20492e9i	−0.0203248	+	0.00406859i
$859$	−6.42576e11			−1.18019			−0.590095	−	0.807334i	$-0.700910\pi$
$859$	−6.42576e11			−1.18019			−0.590095	+	0.807334i	$0.700910\pi$
$860$		−	8.71113e10i		−	0.159250i
$861$	9.90861e9			0.0180302
$862$	6.19948e11			1.12286
$863$	9.48060e11			1.70920			0.854600	−	0.519288i	$-0.173803\pi$
$863$	9.48060e11			1.70920			0.854600	+	0.519288i	$0.173803\pi$
$864$			3.61082e9i			0.00647965i
$865$		−	1.41893e11i		−	0.253452i
$866$			6.48600e11i			1.15320i
$867$	−5.38458e9			−0.00952962
$868$		−	2.81755e10i		−	0.0496355i
$869$	2.41857e9	+	1.20820e10i	0.00424110	+	0.0211865i
$870$	−3.52104e9			−0.00614602
$871$			8.50565e11i			1.47786i
$872$	2.17613e11			0.376374
$873$	−6.98189e11			−1.20203
$874$	2.93175e10			0.0502437
$875$		−	3.50799e11i		−	0.598447i
$876$			3.51957e9i			0.00597686i
$877$			3.66385e11i			0.619354i	0.950842	+	0.309677i	$0.100221\pi$
$877$			3.66385e11i			0.619354i	−0.950842	+	0.309677i	$0.899779\pi$
$878$	−4.26779e11			−0.718165
$879$			9.12524e9i			0.0152858i
$880$	−5.04668e10	+	1.01024e10i	−0.0841541	+	0.0168459i
$881$	−7.13223e11			−1.18392			−0.591959	−	0.805968i	$-0.701645\pi$
$881$	−7.13223e11			−1.18392			−0.591959	+	0.805968i	$0.701645\pi$
$882$		−	6.08092e10i		−	0.100484i
$883$	−3.06503e11			−0.504187			−0.252094	−	0.967703i	$-0.581119\pi$
$883$	−3.06503e11			−0.504187			−0.252094	+	0.967703i	$0.581119\pi$
$884$	3.38341e11			0.554046
$885$	4.11731e9			0.00671182
$886$		−	7.00033e10i		−	0.113601i
$887$			9.61136e11i			1.55271i	0.630296	+	0.776355i	$0.282934\pi$
$887$			9.61136e11i			1.55271i	−0.630296	+	0.776355i	$0.717066\pi$
$888$		−	2.63120e9i		−	0.00423157i
$889$	−5.37780e11			−0.860988
$890$			4.15442e10i			0.0662141i
$891$	6.17364e11	−	1.23583e11i	0.979559	−	0.196087i
$892$	−2.15364e11			−0.340184
$893$		−	1.57230e10i		−	0.0247247i
$894$	−1.46938e10			−0.0230031
$895$	−9.13830e10			−0.142421
$896$	−5.27633e10			−0.0818653
$897$		−	2.50968e10i		−	0.0387658i
$898$			3.34414e11i			0.514256i
$899$		−	9.67006e10i		−	0.148044i
$900$	−2.89292e11			−0.440926
$901$		−	7.49814e11i		−	1.13777i
$902$	4.87422e11	−	9.75718e10i	0.736341	−	0.147400i
$903$	1.04730e10			0.0157514
$904$		−	1.80855e11i		−	0.270805i
$905$	2.37802e11			0.354504
$906$	1.24021e10			0.0184070
$907$	−2.93103e11			−0.433103			−0.216551	−	0.976271i	$-0.569481\pi$
$907$	−2.93103e11			−0.433103			−0.216551	+	0.976271i	$0.569481\pi$
$908$		−	2.91590e11i		−	0.428973i
$909$			8.37991e11i			1.22739i
$910$			2.46561e11i			0.359550i
$911$	−5.96689e11			−0.866313			−0.433156	−	0.901319i	$-0.642600\pi$
$911$	−5.96689e11			−0.866313			−0.433156	+	0.901319i	$0.642600\pi$
$912$		−	1.70337e8i		−	0.000246223i
$913$	−2.37508e11	−	1.18648e12i	−0.341818	−	1.70756i
$914$	7.33921e11			1.05163
$915$			1.71728e9i			0.00244995i
$916$	3.29426e11			0.467924
$917$	−9.35622e11			−1.32319
$918$	1.27543e10			0.0179591
$919$		−	9.74737e11i		−	1.36655i	−0.730161	−	0.683275i	$-0.760555\pi$
$919$		−	9.74737e11i		−	1.36655i	0.730161	−	0.683275i	$-0.239445\pi$
$920$		−	1.14987e11i		−	0.160509i
$921$		−	3.82535e9i		−	0.00531659i
$922$	5.14690e10			0.0712233
$923$			2.59940e11i			0.358151i
$924$	−1.21456e9	−	6.06737e9i	−0.00166622	−	0.00832363i
$925$	4.21683e11			0.575996
$926$			4.11296e11i			0.559385i
$927$	8.23695e11			1.11544
$928$	−1.81088e11			−0.244173
$929$	−6.94543e11			−0.932473			−0.466237	−	0.884660i	$-0.654390\pi$
$929$	−6.94543e11			−0.932473			−0.466237	+	0.884660i	$0.654390\pi$
$930$		−	3.56755e8i		−	0.000476912i
$931$			5.73818e9i			0.00763793i
$932$		−	9.87805e10i		−	0.130920i
$933$	−1.64532e10			−0.0217132
$934$			2.99853e11i			0.394023i
$935$	3.56840e10	+	1.78260e11i	0.0466904	+	0.233243i
$936$	4.33825e11			0.565212
$937$		−	1.17558e12i		−	1.52508i	−0.646939	−	0.762542i	$-0.723951\pi$
$937$		−	1.17558e12i		−	1.52508i	0.646939	−	0.762542i	$-0.276049\pi$
$938$	−4.68526e11			−0.605232
$939$	−2.42115e10			−0.0311429
$940$	−6.16679e10			−0.0789856
$941$		−	8.84870e11i		−	1.12855i	−0.825587	−	0.564275i	$-0.809156\pi$
$941$		−	8.84870e11i		−	1.12855i	0.825587	−	0.564275i	$-0.190844\pi$
$942$		−	1.44382e10i		−	0.0183362i
$943$			1.11058e12i			1.40444i
$944$	2.11754e11			0.266651
$945$			9.29448e9i			0.0116546i
$946$	5.15183e11	−	1.03129e11i	0.643276	−	0.128771i
$947$	7.56628e10			0.0940767			0.0470384	−	0.998893i	$-0.485022\pi$
$947$	7.56628e10			0.0940767			0.0470384	+	0.998893i	$0.485022\pi$
$948$			1.59943e8i			0.000198030i
$949$	8.45865e11			1.04288
$950$	2.72986e10			0.0335156
$951$	9.20773e9			0.0112572
$952$			1.86372e11i			0.226899i
$953$			6.10054e11i			0.739600i	0.929111	+	0.369800i	$0.120574\pi$
$953$			6.10054e11i			0.739600i	−0.929111	+	0.369800i	$0.879426\pi$
$954$		−	9.61421e11i		−	1.16070i
$955$	−5.13094e10			−0.0616855
$956$		−	6.02697e11i		−	0.721552i
$957$	−4.16848e9	−	2.08237e10i	−0.00496970	−	0.0248262i
$958$	6.09629e11			0.723775
$959$		−	3.30022e11i		−	0.390183i
$960$	−6.68084e8			−0.000786586
$961$	−8.43093e11			−0.988512
$962$	−6.32361e11			−0.738355
$963$		−	9.35293e11i		−	1.08753i
$964$			9.24620e9i			0.0107067i
$965$			4.88442e11i			0.563254i
$966$	1.38243e10			0.0158758
$967$		−	5.23213e11i		−	0.598374i	−0.954195	−	0.299187i	$-0.903285\pi$
$967$		−	5.23213e11i		−	0.598374i	0.954195	−	0.299187i	$-0.0967154\pi$
$968$	−1.19493e11	−	2.86505e11i	−0.136095	−	0.326310i
$969$	−6.01669e8			−0.000682437
$970$		−	2.58405e11i		−	0.291886i
$971$	1.19192e12			1.34082			0.670411	−	0.741990i	$-0.266118\pi$
$971$	1.19192e12			1.34082			0.670411	+	0.741990i	$0.266118\pi$
$972$	2.45319e10			0.0274831
$973$	1.43210e12			1.59779
$974$			3.12500e11i			0.347227i
$975$		−	2.33686e10i		−	0.0258591i
$976$			8.83204e10i			0.0973334i
$977$	1.60470e11			0.176123			0.0880616	−	0.996115i	$-0.471933\pi$
$977$	1.60470e11			0.176123			0.0880616	+	0.996115i	$0.471933\pi$
$978$			1.77146e10i			0.0193631i
$979$	−2.45696e11	+	4.91832e10i	−0.267465	+	0.0535410i
$980$	2.25059e10			0.0244002
$981$		−	9.85586e11i		−	1.06419i
$982$	−1.19685e12			−1.28704
$983$	1.09243e12			1.16999			0.584993	−	0.811039i	$-0.301097\pi$
$983$	1.09243e12			1.16999			0.584993	+	0.811039i	$0.301097\pi$
$984$	6.45254e9			0.00688256
$985$			4.62940e10i			0.0491790i
$986$			6.39645e11i			0.676755i
$987$		−	7.41402e9i		−	0.00781242i
$988$	−4.09374e10			−0.0429628
$989$			1.17383e12i			1.22693i
$990$	4.57545e10	+	2.28568e11i	0.0476314	+	0.237944i
$991$	−1.56847e12			−1.62623			−0.813114	−	0.582105i	$-0.802229\pi$
$991$	−1.56847e12			−1.62623			−0.813114	+	0.582105i	$0.802229\pi$
$992$		−	1.83480e10i		−	0.0189471i
$993$	2.43552e10			0.0250493
$994$	−1.43185e11			−0.146674
$995$	1.67484e11			0.170876
$996$		−	1.57067e10i		−	0.0159605i
$997$			6.53775e11i			0.661680i	0.943687	+	0.330840i	$0.107332\pi$
$997$			6.53775e11i			0.661680i	−0.943687	+	0.330840i	$0.892668\pi$
$998$			6.63168e11i			0.668500i
$999$	−2.38378e10			−0.0239334

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.b.a.21.6	yes	8
3.2	odd	2		198.9.d.a.109.2		8
4.3	odd	2		176.9.h.e.65.6		8
11.10	odd	2	inner	22.9.b.a.21.2	✓	8
33.32	even	2		198.9.d.a.109.6		8
44.43	even	2		176.9.h.e.65.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.b.a.21.2	✓	8	11.10	odd	2	inner
22.9.b.a.21.6	yes	8	1.1	even	1	trivial
176.9.h.e.65.5		8	44.43	even	2
176.9.h.e.65.6		8	4.3	odd	2
198.9.d.a.109.2		8	3.2	odd	2
198.9.d.a.109.6		8	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	22.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} -1.48475 q^{3} -128.000 q^{4} -214.559 q^{5} -16.7981i q^{6} -2223.81i q^{7} -1448.15i q^{8} -6558.80 q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} -1.48475 q^{3} -128.000 q^{4} -214.559 q^{5} -16.7981i q^{6} -2223.81i q^{7} -1448.15i q^{8} -6558.80 q^{9} -2427.46i q^{10} +(14356.2 - 2873.81i) q^{11} +190.048 q^{12} -45674.7i q^{13} +25159.5 q^{14} +318.567 q^{15} +16384.0 q^{16} -57872.1i q^{17} -74204.3i q^{18} +7002.19i q^{19} +27463.6 q^{20} +3301.81i q^{21} +(32513.5 + 162422. i) q^{22} -370074. q^{23} +2150.15i q^{24} -344589. q^{25} +516751. q^{26} +19479.7 q^{27} +284647. i q^{28} +976934. i q^{29} +3604.18i q^{30} -98983.8 q^{31} +185364. i q^{32} +(-21315.4 + 4266.90i) q^{33} +654748. q^{34} +477138. i q^{35} +839526. q^{36} -1.22373e6 q^{37} -79220.8 q^{38} +67815.7i q^{39} +310715. i q^{40} -3.00097e6i q^{41} -37355.7 q^{42} -3.17189e6i q^{43} +(-1.83759e6 + 367848. i) q^{44} +1.40725e6 q^{45} -4.18690e6i q^{46} -2.24545e6 q^{47} -24326.2 q^{48} +819484. q^{49} -3.89858e6i q^{50} +85925.7i q^{51} +5.84637e6i q^{52} +1.29564e7 q^{53} +220387. i q^{54} +(-3.08025e6 + 616602. i) q^{55} -3.22042e6 q^{56} -10396.5i q^{57} -1.10527e7 q^{58} +1.29244e7 q^{59} -40776.6 q^{60} +5.39065e6i q^{61} -1.11987e6i q^{62} +1.45855e7i q^{63} -2.09715e6 q^{64} +9.79992e6i q^{65} +(-48274.5 - 241156. i) q^{66} -1.86222e7 q^{67} +7.40762e6i q^{68} +549468. q^{69} -5.39820e6 q^{70} -5.69111e6 q^{71} +9.49815e6i q^{72} +1.85193e7i q^{73} -1.38449e7i q^{74} +511630. q^{75} -896281. i q^{76} +(-6.39080e6 - 3.19254e7i) q^{77} -767247. q^{78} +841589. i q^{79} -3.51533e6 q^{80} +4.30033e7 q^{81} +3.39521e7 q^{82} -8.26457e7i q^{83} -422631. i q^{84} +1.24170e7i q^{85} +3.58858e7 q^{86} -1.45051e6i q^{87} +(-4.16172e6 - 2.07900e7i) q^{88} -1.71143e7 q^{89} +1.59212e7i q^{90} -1.01572e8 q^{91} +4.73694e7 q^{92} +146967. q^{93} -2.54043e7i q^{94} -1.50238e6i q^{95} -275220. i q^{96} +1.06451e8 q^{97} +9.27140e6i q^{98} +(-9.41593e7 + 1.88487e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 182 q^{3} - 1024 q^{4} - 1410 q^{5} + 44582 q^{9}+O(q^{10}) \) 8 * q + 182 * q^3 - 1024 * q^4 - 1410 * q^5 + 44582 * q^9	\( 8 q + 182 q^{3} - 1024 q^{4} - 1410 q^{5} + 44582 q^{9} + 5808 q^{11} - 23296 q^{12} + 12288 q^{14} + 87958 q^{15} + 131072 q^{16} + 180480 q^{20} + 359040 q^{22} - 802026 q^{23} + 999558 q^{25} - 1321728 q^{26} - 1561354 q^{27} + 196726 q^{31} - 4286722 q^{33} - 701952 q^{34} - 5706496 q^{36} + 8627998 q^{37} + 9288960 q^{38} + 4366848 q^{42} - 743424 q^{44} + 1146988 q^{45} - 14335392 q^{47} + 2981888 q^{48} - 6714712 q^{49} + 55946352 q^{53} - 10078442 q^{55} - 1572864 q^{56} - 23226624 q^{58} + 21793110 q^{59} - 11258624 q^{60} - 16777216 q^{64} - 44242176 q^{66} - 113809034 q^{67} - 171636914 q^{69} + 137817600 q^{70} + 16741974 q^{71} + 346496844 q^{75} - 137074080 q^{77} - 57993216 q^{78} - 23101440 q^{80} + 85282724 q^{81} + 47480832 q^{82} + 49839360 q^{86} - 45957120 q^{88} + 42055422 q^{89} - 146801952 q^{91} + 102659328 q^{92} + 253251118 q^{93} + 100034782 q^{97} - 333541978 q^{99}+O(q^{100}) \) 8 * q + 182 * q^3 - 1024 * q^4 - 1410 * q^5 + 44582 * q^9 + 5808 * q^11 - 23296 * q^12 + 12288 * q^14 + 87958 * q^15 + 131072 * q^16 + 180480 * q^20 + 359040 * q^22 - 802026 * q^23 + 999558 * q^25 - 1321728 * q^26 - 1561354 * q^27 + 196726 * q^31 - 4286722 * q^33 - 701952 * q^34 - 5706496 * q^36 + 8627998 * q^37 + 9288960 * q^38 + 4366848 * q^42 - 743424 * q^44 + 1146988 * q^45 - 14335392 * q^47 + 2981888 * q^48 - 6714712 * q^49 + 55946352 * q^53 - 10078442 * q^55 - 1572864 * q^56 - 23226624 * q^58 + 21793110 * q^59 - 11258624 * q^60 - 16777216 * q^64 - 44242176 * q^66 - 113809034 * q^67 - 171636914 * q^69 + 137817600 * q^70 + 16741974 * q^71 + 346496844 * q^75 - 137074080 * q^77 - 57993216 * q^78 - 23101440 * q^80 + 85282724 * q^81 + 47480832 * q^82 + 49839360 * q^86 - 45957120 * q^88 + 42055422 * q^89 - 146801952 * q^91 + 102659328 * q^92 + 253251118 * q^93 + 100034782 * q^97 - 333541978 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more