LMFDB - Embedded newform 19.7.b.b.18.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [19,7,Mod(18,19)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(19, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("19.18");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	19.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:	$4.37102758878$
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} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 $ x^8 + 483x^6 + 75582x^4 + 4242376*x^2 + 71047680
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3}\cdot 29 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			18.5
Root			$5.43437i$ of defining polynomial
Character	$\chi$	$=$	19.18
Dual form			19.7.b.b.18.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.43437i q^{2} +33.7437i q^{3} +34.4677 q^{4} -51.7597 q^{5} -183.376 q^{6} -313.488 q^{7} +535.109i q^{8} -409.636 q^{9} +O(q^{10})$	\(q+5.43437i q^{2} +33.7437i q^{3} +34.4677 q^{4} -51.7597 q^{5} -183.376 q^{6} -313.488 q^{7} +535.109i q^{8} -409.636 q^{9} -281.281i q^{10} +460.660 q^{11} +1163.07i q^{12} +177.130i q^{13} -1703.61i q^{14} -1746.56i q^{15} -702.050 q^{16} +6127.64 q^{17} -2226.11i q^{18} +(5407.00 + 4220.22i) q^{19} -1784.04 q^{20} -10578.2i q^{21} +2503.40i q^{22} +10910.4 q^{23} -18056.6 q^{24} -12945.9 q^{25} -962.592 q^{26} +10776.5i q^{27} -10805.2 q^{28} +11145.5i q^{29} +9491.47 q^{30} -15953.5i q^{31} +30431.8i q^{32} +15544.4i q^{33} +33299.8i q^{34} +16226.1 q^{35} -14119.2 q^{36} -76514.9i q^{37} +(-22934.2 + 29383.6i) q^{38} -5977.04 q^{39} -27697.1i q^{40} -125282. i q^{41} +57486.1 q^{42} -20603.1 q^{43} +15877.9 q^{44} +21202.7 q^{45} +59290.9i q^{46} +150655. q^{47} -23689.8i q^{48} -19374.2 q^{49} -70352.9i q^{50} +206769. i q^{51} +6105.27i q^{52} +113881. i q^{53} -58563.5 q^{54} -23843.6 q^{55} -167750. i q^{56} +(-142406. + 182452. i) q^{57} -60568.7 q^{58} -231084. i q^{59} -60200.0i q^{60} -314764. q^{61} +86697.2 q^{62} +128416. q^{63} -210309. q^{64} -9168.22i q^{65} -84473.8 q^{66} +379771. i q^{67} +211205. q^{68} +368156. i q^{69} +88178.4i q^{70} -218789. i q^{71} -219200. i q^{72} +379291. q^{73} +415810. q^{74} -436843. i q^{75} +(186367. + 145461. i) q^{76} -144411. q^{77} -32481.4i q^{78} -331511. i q^{79} +36337.9 q^{80} -662264. q^{81} +680827. q^{82} +191832. q^{83} -364607. i q^{84} -317165. q^{85} -111965. i q^{86} -376090. q^{87} +246503. i q^{88} -268881. i q^{89} +115223. i q^{90} -55528.3i q^{91} +376054. q^{92} +538330. q^{93} +818714. i q^{94} +(-279865. - 218437. i) q^{95} -1.02688e6 q^{96} +1.70707e6i q^{97} -105286. i q^{98} -188703. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 454 q^{4} + 108 q^{5} - 358 q^{6} - 140 q^{7} - 1052 q^{9}+O(q^{10}) $ 8 * q - 454 * q^4 + 108 * q^5 - 358 * q^6 - 140 * q^7 - 1052 * q^9	$ 8 q - 454 q^{4} + 108 q^{5} - 358 q^{6} - 140 q^{7} - 1052 q^{9} - 2024 q^{11} + 11546 q^{16} + 6008 q^{17} + 20552 q^{19} - 10732 q^{20} - 50252 q^{23} + 64310 q^{24} + 78492 q^{25} - 37522 q^{26} - 135818 q^{28} - 187696 q^{30} + 210800 q^{35} + 35052 q^{36} + 103318 q^{38} + 43724 q^{39} - 429970 q^{42} + 260800 q^{43} + 693512 q^{44} - 191012 q^{45} - 100248 q^{47} - 301872 q^{49} + 390202 q^{54} - 52480 q^{55} - 186860 q^{57} - 405186 q^{58} - 54548 q^{61} - 1461908 q^{62} - 137408 q^{63} - 858058 q^{64} + 1539556 q^{66} + 1243910 q^{68} + 479968 q^{73} + 2645844 q^{74} - 2569288 q^{76} - 1755300 q^{77} + 2344672 q^{80} - 4279648 q^{81} + 1847172 q^{82} + 483040 q^{83} + 2111780 q^{85} + 2802652 q^{87} + 3905498 q^{92} + 1507528 q^{93} - 2383888 q^{95} - 8462238 q^{96} + 528224 q^{99}+O(q^{100}) $ 8 * q - 454 * q^4 + 108 * q^5 - 358 * q^6 - 140 * q^7 - 1052 * q^9 - 2024 * q^11 + 11546 * q^16 + 6008 * q^17 + 20552 * q^19 - 10732 * q^20 - 50252 * q^23 + 64310 * q^24 + 78492 * q^25 - 37522 * q^26 - 135818 * q^28 - 187696 * q^30 + 210800 * q^35 + 35052 * q^36 + 103318 * q^38 + 43724 * q^39 - 429970 * q^42 + 260800 * q^43 + 693512 * q^44 - 191012 * q^45 - 100248 * q^47 - 301872 * q^49 + 390202 * q^54 - 52480 * q^55 - 186860 * q^57 - 405186 * q^58 - 54548 * q^61 - 1461908 * q^62 - 137408 * q^63 - 858058 * q^64 + 1539556 * q^66 + 1243910 * q^68 + 479968 * q^73 + 2645844 * q^74 - 2569288 * q^76 - 1755300 * q^77 + 2344672 * q^80 - 4279648 * q^81 + 1847172 * q^82 + 483040 * q^83 + 2111780 * q^85 + 2802652 * q^87 + 3905498 * q^92 + 1507528 * q^93 - 2383888 * q^95 - 8462238 * q^96 + 528224 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/19\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$			5.43437i			0.679296i	0.940553	+	0.339648i	$0.110308\pi$
$2$			5.43437i			0.679296i	−0.940553	+	0.339648i	$0.889692\pi$
$3$			33.7437i			1.24977i	0.780718	+	0.624883i	$0.214853\pi$
$3$			33.7437i			1.24977i	−0.780718	+	0.624883i	$0.785147\pi$
$4$	34.4677			0.538557
$5$	−51.7597			−0.414078			−0.207039	−	0.978333i	$-0.566383\pi$
$5$	−51.7597			−0.414078			−0.207039	+	0.978333i	$0.566383\pi$
$6$	−183.376			−0.848961
$7$	−313.488			−0.913960			−0.456980	−	0.889477i	$-0.651069\pi$
$7$	−313.488			−0.913960			−0.456980	+	0.889477i	$0.651069\pi$
$8$			535.109i			1.04514i
$9$	−409.636			−0.561915
$10$		−	281.281i		−	0.281281i
$11$	460.660			0.346101			0.173050	−	0.984913i	$-0.444638\pi$
$11$	460.660			0.346101			0.173050	+	0.984913i	$0.444638\pi$
$12$			1163.07i			0.673071i
$13$			177.130i			0.0806238i	0.999187	+	0.0403119i	$0.0128352\pi$
$13$			177.130i			0.0806238i	−0.999187	+	0.0403119i	$0.987165\pi$
$14$		−	1703.61i		−	0.620849i
$15$		−	1746.56i		−	0.517500i
$16$	−702.050			−0.171399
$17$	6127.64			1.24723			0.623615	−	0.781732i	$-0.285663\pi$
$17$	6127.64			1.24723			0.623615	+	0.781732i	$0.285663\pi$
$18$		−	2226.11i		−	0.381707i
$19$	5407.00	+	4220.22i	0.788308	+	0.615281i
$19$	5407.00	+	4220.22i	0.788308	+	0.615281i
$20$	−1784.04			−0.223005
$21$		−	10578.2i		−	1.14224i
$22$			2503.40i			0.235105i
$23$	10910.4			0.896717			0.448358	−	0.893854i	$-0.352009\pi$
$23$	10910.4			0.896717			0.448358	+	0.893854i	$0.352009\pi$
$24$	−18056.6			−1.30617
$25$	−12945.9			−0.828540
$26$	−962.592			−0.0547674
$27$			10776.5i			0.547503i
$28$	−10805.2			−0.492220
$29$			11145.5i			0.456988i	0.973545	+	0.228494i	$0.0733802\pi$
$29$			11145.5i			0.456988i	−0.973545	+	0.228494i	$0.926620\pi$
$30$	9491.47			0.351536
$31$		−	15953.5i		−	0.535514i	−0.963486	−	0.267757i	$-0.913718\pi$
$31$		−	15953.5i		−	0.535514i	0.963486	−	0.267757i	$-0.0862824\pi$
$32$			30431.8i			0.928705i
$33$			15544.4i			0.432545i
$34$			33299.8i			0.847238i
$35$	16226.1			0.378450
$36$	−14119.2			−0.302624
$37$		−	76514.9i		−	1.51057i	−0.655397	−	0.755285i	$-0.727498\pi$
$37$		−	76514.9i		−	1.51057i	0.655397	−	0.755285i	$-0.272502\pi$
$38$	−22934.2	+	29383.6i	−0.417958	+	0.535494i
$39$	−5977.04			−0.100761
$40$		−	27697.1i		−	0.432767i
$41$		−	125282.i		−	1.81776i	−0.417058	−	0.908880i	$-0.636939\pi$
$41$		−	125282.i		−	1.81776i	0.417058	−	0.908880i	$-0.363061\pi$
$42$	57486.1			0.775916
$43$	−20603.1			−0.259136			−0.129568	−	0.991571i	$-0.541359\pi$
$43$	−20603.1			−0.259136			−0.129568	+	0.991571i	$0.541359\pi$
$44$	15877.9			0.186395
$45$	21202.7			0.232677
$46$			59290.9i			0.609136i
$47$	150655.			1.45107			0.725537	−	0.688183i	$-0.241591\pi$
$47$	150655.			1.45107			0.725537	+	0.688183i	$0.241591\pi$
$48$		−	23689.8i		−	0.214209i
$49$	−19374.2			−0.164678
$50$		−	70352.9i		−	0.562823i
$51$			206769.i			1.55874i
$52$			6105.27i			0.0434205i
$53$			113881.i			0.764935i	0.923969	+	0.382467i	$0.124926\pi$
$53$			113881.i			0.764935i	−0.923969	+	0.382467i	$0.875074\pi$
$54$	−58563.5			−0.371917
$55$	−23843.6			−0.143313
$56$		−	167750.i		−	0.955212i
$57$	−142406.	+	182452.i	−0.768958	+	0.985200i
$58$	−60568.7			−0.310430
$59$		−	231084.i		−	1.12516i	−0.826743	−	0.562580i	$-0.809809\pi$
$59$		−	231084.i		−	1.12516i	0.826743	−	0.562580i	$-0.190191\pi$
$60$		−	60200.0i		−	0.278704i
$61$	−314764.			−1.38674			−0.693370	−	0.720582i	$-0.743875\pi$
$61$	−314764.			−1.38674			−0.693370	+	0.720582i	$0.743875\pi$
$62$	86697.2			0.363772
$63$	128416.			0.513568
$64$	−210309.			−0.802264
$65$		−	9168.22i		−	0.0333845i
$66$	−84473.8			−0.293826
$67$			379771.i			1.26269i	0.775501	+	0.631346i	$0.217497\pi$
$67$			379771.i			1.26269i	−0.775501	+	0.631346i	$0.782503\pi$
$68$	211205.			0.671704
$69$			368156.i			1.12069i
$70$			88178.4i			0.257080i
$71$		−	218789.i		−	0.611295i	−0.952145	−	0.305648i	$-0.901127\pi$
$71$		−	218789.i		−	0.611295i	0.952145	−	0.305648i	$-0.0988729\pi$
$72$		−	219200.i		−	0.587278i
$73$	379291.			0.974999			0.487499	−	0.873123i	$-0.337909\pi$
$73$	379291.			0.974999			0.487499	+	0.873123i	$0.337909\pi$
$74$	415810.			1.02612
$75$		−	436843.i		−	1.03548i
$76$	186367.	+	145461.i	0.424549	+	0.331364i
$77$	−144411.			−0.316322
$78$		−	32481.4i		−	0.0684465i
$79$		−	331511.i		−	0.672383i	−0.941794	−	0.336192i	$-0.890861\pi$
$79$		−	331511.i		−	0.672383i	0.941794	−	0.336192i	$-0.109139\pi$
$80$	36337.9			0.0709725
$81$	−662264.			−1.24617
$82$	680827.			1.23480
$83$	191832.			0.335496			0.167748	−	0.985830i	$-0.446351\pi$
$83$	191832.			0.335496			0.167748	+	0.985830i	$0.446351\pi$
$84$		−	364607.i		−	0.615159i
$85$	−317165.			−0.516450
$86$		−	111965.i		−	0.176030i
$87$	−376090.			−0.571129
$88$			246503.i			0.361722i
$89$		−	268881.i		−	0.381408i	−0.981648	−	0.190704i	$-0.938923\pi$
$89$		−	268881.i		−	0.381408i	0.981648	−	0.190704i	$-0.0610771\pi$
$90$			115223.i			0.158056i
$91$		−	55528.3i		−	0.0736869i
$92$	376054.			0.482933
$93$	538330.			0.669267
$94$			818714.i			0.985708i
$95$	−279865.	−	218437.i	−0.326421	−	0.254774i
$96$	−1.02688e6			−1.16066
$97$			1.70707e6i			1.87041i	0.354109	+	0.935204i	$0.384784\pi$
$97$			1.70707e6i			1.87041i	−0.354109	+	0.935204i	$0.615216\pi$
$98$		−	105286.i		−	0.111865i
$99$	−188703.			−0.194479
$100$	−446216.			−0.446216
$101$	712272.			0.691324			0.345662	−	0.938359i	$-0.387654\pi$
$101$	712272.			0.691324			0.345662	+	0.938359i	$0.387654\pi$
$102$	−1.12366e6			−1.05885
$103$		−	1.46263e6i		−	1.33852i	−0.743030	−	0.669259i	$-0.766612\pi$
$103$		−	1.46263e6i		−	1.33852i	0.743030	−	0.669259i	$-0.233388\pi$
$104$	−94784.2			−0.0842628
$105$			547527.i			0.472974i
$106$	−618872.			−0.519617
$107$			681335.i			0.556173i	0.960556	+	0.278086i	$0.0897001\pi$
$107$			681335.i			0.556173i	−0.960556	+	0.278086i	$0.910300\pi$
$108$			371441.i			0.294862i
$109$		−	750959.i		−	0.579878i	−0.957045	−	0.289939i	$-0.906365\pi$
$109$		−	750959.i		−	0.579878i	0.957045	−	0.289939i	$-0.0936350\pi$
$110$		−	129575.i		−	0.0973516i
$111$	2.58189e6			1.88786
$112$	220084.			0.156652
$113$			828261.i			0.574027i	0.957927	+	0.287013i	$0.0926624\pi$
$113$			828261.i			0.574027i	−0.957927	+	0.287013i	$0.907338\pi$
$114$	−991512.	−	773884.i	−0.669242	−	0.522350i
$115$	−564717.			−0.371310
$116$			384159.i			0.246114i
$117$		−	72559.1i		−	0.0453038i
$118$	1.25580e6			0.764316
$119$	−1.92094e6			−1.13992
$120$	934603.			0.540858
$121$	−1.55935e6			−0.880214
$122$		−	1.71054e6i		−	0.942006i
$123$	4.22747e6			2.27177
$124$		−	549880.i		−	0.288405i
$125$	1.47882e6			0.757158
$126$			697860.i			0.348865i
$127$		−	3.56446e6i		−	1.74013i	−0.492935	−	0.870066i	$-0.664076\pi$
$127$		−	3.56446e6i		−	1.74013i	0.492935	−	0.870066i	$-0.335924\pi$
$128$			804740.i			0.383730i
$129$		−	695225.i		−	0.323859i
$130$	49823.5			0.0226780
$131$	1.28645e6			0.572242			0.286121	−	0.958194i	$-0.407634\pi$
$131$	1.28645e6			0.572242			0.286121	+	0.958194i	$0.407634\pi$
$132$			535778.i			0.232950i
$133$	−1.69503e6	−	1.32299e6i	−0.720481	−	0.562342i
$134$	−2.06382e6			−0.857742
$135$		−	557789.i		−	0.226709i
$136$			3.27896e6i			1.30352i
$137$	−3.79392e6			−1.47546			−0.737728	−	0.675098i	$-0.764101\pi$
$137$	−3.79392e6			−1.47546			−0.737728	+	0.675098i	$0.764101\pi$
$138$	−2.00069e6			−0.761277
$139$	3.53764e6			1.31725			0.658626	−	0.752470i	$-0.271138\pi$
$139$	3.53764e6			1.31725			0.658626	+	0.752470i	$0.271138\pi$
$140$	559274.			0.203817
$141$			5.08365e6i			1.81350i
$142$	1.18898e6			0.415250
$143$			81596.9i			0.0279040i
$144$	287585.			0.0963117
$145$		−	576887.i		−	0.189229i
$146$			2.06121e6i			0.662313i
$147$		−	653755.i		−	0.205808i
$148$		−	2.63729e6i		−	0.813528i
$149$	644521.			0.194840			0.0974200	−	0.995243i	$-0.468941\pi$
$149$	644521.			0.194840			0.0974200	+	0.995243i	$0.468941\pi$
$150$	2.37397e6			0.703398
$151$			1.00793e6i			0.292752i	0.989229	+	0.146376i	$0.0467609\pi$
$151$			1.00793e6i			0.292752i	−0.989229	+	0.146376i	$0.953239\pi$
$152$	−2.25828e6	+	2.89334e6i	−0.643052	+	0.823888i
$153$	−2.51010e6			−0.700837
$154$		−	784785.i		−	0.214876i
$155$			825749.i			0.221744i
$156$	−206014.			−0.0542655
$157$	−6.99464e6			−1.80745			−0.903725	−	0.428113i	$-0.859178\pi$
$157$	−6.99464e6			−1.80745			−0.903725	+	0.428113i	$0.859178\pi$
$158$	1.80155e6			0.456747
$159$	−3.84277e6			−0.955990
$160$		−	1.57514e6i		−	0.384556i
$161$	−3.42027e6			−0.819563
$162$		−	3.59899e6i		−	0.846516i
$163$	3.86384e6			0.892187			0.446094	−	0.894986i	$-0.352815\pi$
$163$	3.86384e6			0.892187			0.446094	+	0.894986i	$0.352815\pi$
$164$		−	4.31817e6i		−	0.978968i
$165$		−	804572.i		−	0.179107i
$166$			1.04249e6i			0.227901i
$167$			5.91450e6i			1.26990i	0.772554	+	0.634949i	$0.218979\pi$
$167$			5.91450e6i			1.26990i	−0.772554	+	0.634949i	$0.781021\pi$
$168$	5.66052e6			1.19379
$169$	4.79543e6			0.993500
$170$		−	1.72359e6i		−	0.350822i
$171$	−2.21490e6	−	1.72875e6i	−0.442962	−	0.345736i
$172$	−710141.			−0.139559
$173$		−	3.74467e6i		−	0.723228i	−0.932328	−	0.361614i	$-0.882226\pi$
$173$		−	3.74467e6i		−	0.723228i	0.932328	−	0.361614i	$-0.117774\pi$
$174$		−	2.04381e6i		−	0.387965i
$175$	4.05840e6			0.757252
$176$	−323406.			−0.0593213
$177$	7.79763e6			1.40619
$178$	1.46120e6			0.259089
$179$			1.09184e7i			1.90371i	0.306552	+	0.951854i	$0.400825\pi$
$179$			1.09184e7i			1.90371i	−0.306552	+	0.951854i	$0.599175\pi$
$180$	730806.			0.125310
$181$			44238.1i			0.00746038i	0.999993	+	0.00373019i	$0.00118736\pi$
$181$			44238.1i			0.00746038i	−0.999993	+	0.00373019i	$0.998813\pi$
$182$	301761.			0.0500552
$183$		−	1.06213e7i		−	1.73310i
$184$			5.83823e6i			0.937190i
$185$			3.96039e6i			0.625493i
$186$			2.92548e6i			0.454631i
$187$	2.82276e6			0.431667
$188$	5.19272e6			0.781486
$189$		−	3.37831e6i		−	0.500396i
$190$	1.18707e6	−	1.52089e6i	0.173067	−	0.221736i
$191$	−1.09262e7			−1.56808			−0.784042	−	0.620708i	$-0.786845\pi$
$191$	−1.09262e7			−1.56808			−0.784042	+	0.620708i	$0.786845\pi$
$192$		−	7.09659e6i		−	1.00264i
$193$		−	1.58192e6i		−	0.220046i	−0.993929	−	0.110023i	$-0.964908\pi$
$193$		−	1.58192e6i		−	0.220046i	0.993929	−	0.110023i	$-0.0350924\pi$
$194$	−9.27685e6			−1.27056
$195$	309370.			0.0417228
$196$	−667782.			−0.0886883
$197$	−2.68965e6			−0.351801			−0.175901	−	0.984408i	$-0.556284\pi$
$197$	−2.68965e6			−0.351801			−0.175901	+	0.984408i	$0.556284\pi$
$198$		−	1.02548e6i		−	0.132109i
$199$	967242.			0.122737			0.0613686	−	0.998115i	$-0.480453\pi$
$199$	967242.			0.122737			0.0613686	+	0.998115i	$0.480453\pi$
$200$		−	6.92749e6i		−	0.865936i
$201$	−1.28149e7			−1.57807
$202$			3.87075e6i			0.469613i
$203$		−	3.49398e6i		−	0.417669i
$204$			7.12685e6i			0.839473i
$205$			6.48455e6i			0.752694i
$206$	7.94849e6			0.909249
$207$	−4.46928e6			−0.503879
$208$		−	124354.i		−	0.0138188i
$209$	2.49079e6	+	1.94408e6i	0.272834	+	0.212949i
$210$	−2.97546e6			−0.321290
$211$			3.68691e6i			0.392478i	0.980556	+	0.196239i	$0.0628729\pi$
$211$			3.68691e6i			0.392478i	−0.980556	+	0.196239i	$0.937127\pi$
$212$			3.92522e6i			0.411961i
$213$	7.38275e6			0.763976
$214$	−3.70263e6			−0.377806
$215$	1.06641e6			0.107302
$216$	−5.76661e6			−0.572215
$217$			5.00123e6i			0.489438i
$218$	4.08099e6			0.393909
$219$			1.27987e7i			1.21852i
$220$	−821834.			−0.0771820
$221$			1.08539e6i			0.100556i
$222$			1.40310e7i			1.28241i
$223$		−	1.00457e7i		−	0.905872i	−0.891543	−	0.452936i	$-0.850377\pi$
$223$		−	1.00457e7i		−	0.905872i	0.891543	−	0.452936i	$-0.149623\pi$
$224$		−	9.54001e6i		−	0.848799i
$225$	5.30312e6			0.465569
$226$	−4.50108e6			−0.389934
$227$		−	9.74935e6i		−	0.833486i	−0.909024	−	0.416743i	$-0.863172\pi$
$227$		−	9.74935e6i		−	0.833486i	0.909024	−	0.416743i	$-0.136828\pi$
$228$	−4.90839e6	+	6.28870e6i	−0.414128	+	0.530587i
$229$	1.25492e7			1.04499			0.522493	−	0.852644i	$-0.325002\pi$
$229$	1.25492e7			1.04499			0.522493	+	0.852644i	$0.325002\pi$
$230$		−	3.06888e6i		−	0.252230i
$231$		−	4.87298e6i		−	0.395329i
$232$	−5.96406e6			−0.477615
$233$	5.50788e6			0.435428			0.217714	−	0.976013i	$-0.430140\pi$
$233$	5.50788e6			0.435428			0.217714	+	0.976013i	$0.430140\pi$
$234$	394313.			0.0307747
$235$	−7.79785e6			−0.600857
$236$		−	7.96493e6i		−	0.605963i
$237$	1.11864e7			0.840322
$238$		−	1.04391e7i		−	0.774341i
$239$	1.03934e7			0.761314			0.380657	−	0.924716i	$-0.375698\pi$
$239$	1.03934e7			0.761314			0.380657	+	0.924716i	$0.375698\pi$
$240$			1.22617e6i			0.0886990i
$241$		−	2.34911e7i		−	1.67823i	−0.543952	−	0.839116i	$-0.683073\pi$
$241$		−	2.34911e7i		−	1.67823i	0.543952	−	0.839116i	$-0.316927\pi$
$242$		−	8.47410e6i		−	0.597926i
$243$		−	1.44912e7i		−	1.00991i
$244$	−1.08492e7			−0.746839
$245$	1.00280e6			0.0681893
$246$			2.29736e7i			1.54321i
$247$	−747529.	+	957745.i	−0.0496063	+	0.0635563i
$248$	8.53687e6			0.559685
$249$			6.47312e6i			0.419291i
$250$			8.03647e6i			0.514334i
$251$	200237.			0.0126626			0.00633132	−	0.999980i	$-0.497985\pi$
$251$	200237.			0.0126626			0.00633132	+	0.999980i	$0.497985\pi$
$252$	4.42620e6			0.276586
$253$	5.02596e6			0.310354
$254$	1.93706e7			1.18206
$255$		−	1.07023e7i		−	0.645441i
$256$	−1.78330e7			−1.06293
$257$			1.33812e7i			0.788309i	0.919044	+	0.394154i	$0.128962\pi$
$257$			1.33812e7i			0.788309i	−0.919044	+	0.394154i	$0.871038\pi$
$258$	3.77810e6			0.219996
$259$			2.39865e7i			1.38060i
$260$		−	316007.i		−	0.0179795i
$261$		−	4.56560e6i		−	0.256789i
$262$			6.99105e6i			0.388721i
$263$	−2.16755e7			−1.19152			−0.595760	−	0.803162i	$-0.703149\pi$
$263$	−2.16755e7			−1.19152			−0.595760	+	0.803162i	$0.703149\pi$
$264$	−8.31794e6			−0.452068
$265$		−	5.89446e6i		−	0.316743i
$266$	7.18960e6	−	9.21142e6i	0.381997	−	0.489420i
$267$	9.07304e6			0.476671
$268$			1.30898e7i			0.680032i
$269$		−	9.47444e6i		−	0.486740i	−0.969934	−	0.243370i	$-0.921747\pi$
$269$		−	9.47444e6i		−	0.486740i	0.969934	−	0.243370i	$-0.0782529\pi$
$270$	3.03123e6			0.154002
$271$	−9.62807e6			−0.483762			−0.241881	−	0.970306i	$-0.577764\pi$
$271$	−9.62807e6			−0.483762			−0.241881	+	0.970306i	$0.577764\pi$
$272$	−4.30191e6			−0.213774
$273$	1.87373e6			0.0920914
$274$		−	2.06175e7i		−	1.00227i
$275$	−5.96367e6			−0.286758
$276$			1.26895e7i			0.603554i
$277$	−1.47609e7			−0.694501			−0.347251	−	0.937772i	$-0.612885\pi$
$277$	−1.47609e7			−0.694501			−0.347251	+	0.937772i	$0.612885\pi$
$278$			1.92248e7i			0.894804i
$279$			6.53513e6i			0.300914i
$280$			8.68272e6i			0.395532i
$281$			2.57329e6i			0.115976i	0.998317	+	0.0579882i	$0.0184686\pi$
$281$			2.57329e6i			0.115976i	−0.998317	+	0.0579882i	$0.981531\pi$
$282$	−2.76264e7			−1.23190
$283$	1.68438e6			0.0743156			0.0371578	−	0.999309i	$-0.488170\pi$
$283$	1.68438e6			0.0743156			0.0371578	+	0.999309i	$0.488170\pi$
$284$		−	7.54115e6i		−	0.329217i
$285$	7.37087e6	−	9.44367e6i	0.318408	−	0.407949i
$286$	−443428.			−0.0189550
$287$			3.92744e7i			1.66136i
$288$		−	1.24660e7i		−	0.521854i
$289$	1.34104e7			0.555581
$290$	3.13502e6			0.128542
$291$	−5.76029e7			−2.33757
$292$	1.30733e7			0.525093
$293$		−	3.83169e7i		−	1.52331i	−0.647985	−	0.761653i	$-0.724388\pi$
$293$		−	3.83169e7i		−	1.52331i	0.647985	−	0.761653i	$-0.275612\pi$
$294$	3.55275e6			0.139805
$295$			1.19609e7i			0.465904i
$296$	4.09438e7			1.57875
$297$			4.96431e6i			0.189491i
$298$			3.50256e6i			0.132354i
$299$			1.93256e6i			0.0722967i
$300$		−	1.50570e7i		−	0.557666i
$301$	6.45883e6			0.236840
$302$	−5.47747e6			−0.198865
$303$			2.40347e7i			0.863993i
$304$	−3.79598e6	−	2.96280e6i	−0.135115	−	0.105459i
$305$	1.62921e7			0.574218
$306$		−	1.36408e7i		−	0.476076i
$307$		−	1.27696e7i		−	0.441328i	−0.975350	−	0.220664i	$-0.929178\pi$
$307$		−	1.27696e7i		−	0.441328i	0.975350	−	0.220664i	$-0.0708225\pi$
$308$	−4.97753e6			−0.170358
$309$	4.93547e7			1.67283
$310$	−4.48742e6			−0.150630
$311$	2.29592e7			0.763264			0.381632	−	0.924314i	$-0.375362\pi$
$311$	2.29592e7			0.763264			0.381632	+	0.924314i	$0.375362\pi$
$312$		−	3.19837e6i		−	0.105309i
$313$	−2.79625e7			−0.911892			−0.455946	−	0.890008i	$-0.650699\pi$
$313$	−2.79625e7			−0.911892			−0.455946	+	0.890008i	$0.650699\pi$
$314$		−	3.80114e7i		−	1.22779i
$315$	−6.64678e6			−0.212657
$316$		−	1.14264e7i		−	0.362117i
$317$		−	1.15462e7i		−	0.362462i	−0.983441	−	0.181231i	$-0.941992\pi$
$317$		−	1.15462e7i		−	0.362462i	0.983441	−	0.181231i	$-0.0580081\pi$
$318$		−	2.08830e7i		−	0.649400i
$319$			5.13428e6i			0.158164i
$320$	1.08855e7			0.332200
$321$	−2.29908e7			−0.695086
$322$		−	1.85870e7i		−	0.556726i
$323$	3.31321e7	+	2.58599e7i	0.983200	+	0.767397i
$324$	−2.28267e7			−0.671132
$325$		−	2.29312e6i		−	0.0668000i
$326$			2.09975e7i			0.606059i
$327$	2.53401e7			0.724712
$328$	6.70395e7			1.89981
$329$	−4.72285e7			−1.32622
$330$	4.37234e6			0.121667
$331$			3.61243e7i			0.996128i	0.867140	+	0.498064i	$0.165956\pi$
$331$			3.61243e7i			0.996128i	−0.867140	+	0.498064i	$0.834044\pi$
$332$	6.61201e6			0.180684
$333$			3.13433e7i			0.848813i
$334$	−3.21416e7			−0.862636
$335$		−	1.96568e7i		−	0.522853i
$336$			7.42646e6i			0.195778i
$337$			3.86365e6i			0.100950i	0.998725	+	0.0504752i	$0.0160736\pi$
$337$			3.86365e6i			0.100950i	−0.998725	+	0.0504752i	$0.983926\pi$
$338$			2.60601e7i			0.674880i
$339$	−2.79486e7			−0.717399
$340$	−1.09319e7			−0.278138
$341$		−	7.34914e6i		−	0.185342i
$342$	9.39468e6	−	1.20366e7i	0.234857	−	0.300902i
$343$	4.29551e7			1.06447
$344$		−	1.10249e7i		−	0.270832i
$345$		−	1.90556e7i		−	0.464051i
$346$	2.03499e7			0.491286
$347$	−3.99827e7			−0.956939			−0.478469	−	0.878104i	$-0.658808\pi$
$347$	−3.99827e7			−0.956939			−0.478469	+	0.878104i	$0.658808\pi$
$348$	−1.29629e7			−0.307585
$349$	1.63712e7			0.385127			0.192563	−	0.981285i	$-0.438320\pi$
$349$	1.63712e7			0.385127			0.192563	+	0.981285i	$0.438320\pi$
$350$			2.20548e7i			0.514398i
$351$	−1.90885e6			−0.0441418
$352$			1.40187e7i			0.321425i
$353$	−6.15479e7			−1.39923			−0.699615	−	0.714520i	$-0.746645\pi$
$353$	−6.15479e7			−1.39923			−0.699615	+	0.714520i	$0.746645\pi$
$354$			4.23752e7i			0.955217i
$355$			1.13245e7i			0.253124i
$356$		−	9.26770e6i		−	0.205410i
$357$		−	6.48197e7i		−	1.42463i
$358$	−5.93346e7			−1.29318
$359$	−7.15774e7			−1.54701			−0.773504	−	0.633792i	$-0.781498\pi$
$359$	−7.15774e7			−1.54701			−0.773504	+	0.633792i	$0.781498\pi$
$360$			1.13457e7i			0.243179i
$361$	1.14254e7	+	4.56374e7i	0.242858	+	0.970062i
$362$	−240406.			−0.00506780
$363$		−	5.26183e7i		−	1.10006i
$364$		−	1.91393e6i		−	0.0396846i
$365$	−1.96320e7			−0.403725
$366$	5.77199e7			1.17729
$367$	5.78244e6			0.116980			0.0584902	−	0.998288i	$-0.481371\pi$
$367$	5.78244e6			0.116980			0.0584902	+	0.998288i	$0.481371\pi$
$368$	−7.65961e6			−0.153696
$369$			5.13200e7i			1.02143i
$370$	−2.15222e7			−0.424895
$371$		−	3.57004e7i		−	0.699120i
$372$	1.85550e7			0.360439
$373$		−	7.08862e7i		−	1.36595i	−0.730441	−	0.682976i	$-0.760686\pi$
$373$		−	7.08862e7i		−	1.36595i	0.730441	−	0.682976i	$-0.239314\pi$
$374$			1.53399e7i			0.293229i
$375$			4.99009e7i			0.946270i
$376$			8.06168e7i			1.51657i
$377$	−1.97421e6			−0.0368441
$378$	1.83590e7			0.339917
$379$		−	9.66939e7i		−	1.77616i	−0.459692	−	0.888079i	$-0.652040\pi$
$379$		−	9.66939e7i		−	1.77616i	0.459692	−	0.888079i	$-0.347960\pi$
$380$	−9.64629e6	−	7.52902e6i	−0.175796	−	0.137211i
$381$	1.20278e8			2.17476
$382$		−	5.93770e7i		−	1.06519i
$383$			8.74413e7i			1.55640i	0.628019	+	0.778198i	$0.283866\pi$
$383$			8.74413e7i			1.55640i	−0.628019	+	0.778198i	$0.716134\pi$
$384$	−2.71549e7			−0.479573
$385$	7.47470e6			0.130982
$386$	8.59674e6			0.149476
$387$	8.43978e6			0.145612
$388$			5.88388e7i			1.00732i
$389$	1.24502e7			0.211509			0.105754	−	0.994392i	$-0.466274\pi$
$389$	1.24502e7			0.211509			0.105754	+	0.994392i	$0.466274\pi$
$390$			1.68123e6i			0.0283422i
$391$	6.68547e7			1.11841
$392$		−	1.03673e7i		−	0.172110i
$393$			4.34096e7i			0.715168i
$394$		−	1.46165e7i		−	0.238977i
$395$			1.71589e7i			0.278419i
$396$	−6.50415e6			−0.104738
$397$	−9.66655e6			−0.154490			−0.0772449	−	0.997012i	$-0.524612\pi$
$397$	−9.66655e6			−0.154490			−0.0772449	+	0.997012i	$0.524612\pi$
$398$			5.25635e6i			0.0833748i
$399$	4.46425e7	−	5.71966e7i	0.702797	−	0.900433i
$400$	9.08869e6			0.142011
$401$		−	1.73149e7i		−	0.268526i	−0.990946	−	0.134263i	$-0.957133\pi$
$401$		−	1.73149e7i		−	0.268526i	0.990946	−	0.134263i	$-0.0428667\pi$
$402$		−	6.96408e7i		−	1.07198i
$403$	2.82585e6			0.0431752
$404$	2.45503e7			0.372317
$405$	3.42786e7			0.516010
$406$	1.89876e7			0.283721
$407$		−	3.52474e7i		−	0.522809i
$408$	−1.10644e8			−1.62910
$409$		−	9.98449e7i		−	1.45934i	−0.683801	−	0.729669i	$-0.739674\pi$
$409$		−	9.98449e7i		−	1.45934i	0.683801	−	0.729669i	$-0.260326\pi$
$410$	−3.52394e7			−0.511302
$411$		−	1.28021e8i		−	1.84398i
$412$		−	5.04136e7i		−	0.720868i
$413$			7.24422e7i			1.02835i
$414$		−	2.42877e7i		−	0.342283i
$415$	−9.92918e6			−0.138921
$416$	−5.39040e6			−0.0748757
$417$			1.19373e8i			1.64626i
$418$	−1.05649e7	+	1.35359e7i	−0.144656	+	0.185335i
$419$	3.42446e7			0.465533			0.232766	−	0.972533i	$-0.425222\pi$
$419$	3.42446e7			0.465533			0.232766	+	0.972533i	$0.425222\pi$
$420$			1.88720e7i			0.254724i
$421$			9.97444e6i			0.133673i	0.997764	+	0.0668363i	$0.0212905\pi$
$421$			9.97444e6i			0.133673i	−0.997764	+	0.0668363i	$0.978709\pi$
$422$	−2.00360e7			−0.266609
$423$	−6.17137e7			−0.815381
$424$	−6.09389e7			−0.799461
$425$	−7.93280e7			−1.03338
$426$			4.01206e7i			0.518966i
$427$	9.86746e7			1.26742
$428$			2.34840e7i			0.299531i
$429$	−2.75338e6			−0.0348734
$430$			5.79527e6i			0.0728900i
$431$			1.17535e7i			0.146803i	0.997302	+	0.0734014i	$0.0233854\pi$
$431$			1.17535e7i			0.146803i	−0.997302	+	0.0734014i	$0.976615\pi$
$432$		−	7.56565e6i		−	0.0938415i
$433$			8.93309e7i			1.10037i	0.835043	+	0.550184i	$0.185442\pi$
$433$			8.93309e7i			1.10037i	−0.835043	+	0.550184i	$0.814558\pi$
$434$	−2.71785e7			−0.332473
$435$	1.94663e7			0.236492
$436$		−	2.58838e7i		−	0.312297i
$437$	5.89923e7	+	4.60440e7i	0.706889	+	0.551733i
$438$	−6.95527e7			−0.827736
$439$			6.34878e7i			0.750406i	0.926943	+	0.375203i	$0.122427\pi$
$439$			6.34878e7i			0.750406i	−0.926943	+	0.375203i	$0.877573\pi$
$440$		−	1.27589e7i		−	0.149781i
$441$	7.93636e6			0.0925349
$442$	−5.89841e6			−0.0683075
$443$	−3.65478e7			−0.420388			−0.210194	−	0.977660i	$-0.567410\pi$
$443$	−3.65478e7			−0.420388			−0.210194	+	0.977660i	$0.567410\pi$
$444$	8.89919e7			1.01672
$445$			1.39172e7i			0.157933i
$446$	5.45922e7			0.615355
$447$			2.17485e7i			0.243505i
$448$	6.59293e7			0.733237
$449$			5.68613e7i			0.628170i	0.949395	+	0.314085i	$0.101698\pi$
$449$			5.68613e7i			0.628170i	−0.949395	+	0.314085i	$0.898302\pi$
$450$			2.88191e7i			0.316259i
$451$		−	5.77123e7i		−	0.629128i
$452$			2.85482e7i			0.309146i
$453$	−3.40113e7			−0.365872
$454$	5.29815e7			0.566183
$455$			2.87413e6i			0.0305121i
$456$	−9.76319e7	−	7.62026e7i	−1.02967	−	0.803665i
$457$	−1.53349e8			−1.60669			−0.803347	−	0.595512i	$-0.796949\pi$
$457$	−1.53349e8			−1.60669			−0.803347	+	0.595512i	$0.796949\pi$
$458$			6.81971e7i			0.709855i
$459$			6.60345e7i			0.682862i
$460$	−1.94645e7			−0.199972
$461$	−2.18233e7			−0.222750			−0.111375	−	0.993778i	$-0.535525\pi$
$461$	−2.18233e7			−0.222750			−0.111375	+	0.993778i	$0.535525\pi$
$462$	2.64815e7			0.268545
$463$	−4.91907e7			−0.495610			−0.247805	−	0.968810i	$-0.579709\pi$
$463$	−4.91907e7			−0.495610			−0.247805	+	0.968810i	$0.579709\pi$
$464$		−	7.82469e6i		−	0.0783273i
$465$	−2.78638e7			−0.277129
$466$			2.99318e7i			0.295785i
$467$	−8.46888e7			−0.831525			−0.415763	−	0.909473i	$-0.636485\pi$
$467$	−8.46888e7			−0.831525			−0.415763	+	0.909473i	$0.636485\pi$
$468$		−	2.50094e6i		−	0.0243987i
$469$		−	1.19054e8i		−	1.15405i
$470$		−	4.23764e7i		−	0.408160i
$471$		−	2.36025e8i		−	2.25889i
$472$	1.23655e8			1.17594
$473$	−9.49102e6			−0.0896870
$474$			6.07911e7i			0.570827i
$475$	−6.99987e7	−	5.46346e7i	−0.653144	−	0.509785i
$476$	−6.62104e7			−0.613911
$477$		−	4.66499e7i		−	0.429829i
$478$			5.64816e7i			0.517158i
$479$	1.59447e8			1.45080			0.725402	−	0.688325i	$-0.241654\pi$
$479$	1.59447e8			1.45080			0.725402	+	0.688325i	$0.241654\pi$
$480$	5.31511e7			0.480605
$481$	1.35531e7			0.121788
$482$	1.27659e8			1.14002
$483$		−	1.15412e8i		−	1.02426i
$484$	−5.37473e7			−0.474046
$485$		−	8.83575e7i		−	0.774495i
$486$	7.87502e7			0.686030
$487$			1.05220e8i			0.910983i	0.890240	+	0.455492i	$0.150536\pi$
$487$			1.05220e8i			0.910983i	−0.890240	+	0.455492i	$0.849464\pi$
$488$		−	1.68433e8i		−	1.44933i
$489$			1.30380e8i			1.11503i
$490$			5.44959e6i			0.0463207i
$491$	1.76955e8			1.49492			0.747461	−	0.664306i	$-0.231273\pi$
$491$	1.76955e8			1.49492			0.747461	+	0.664306i	$0.231273\pi$
$492$	1.45711e8			1.22348
$493$			6.82955e7i			0.569969i
$494$	−5.20474e6	−	4.06235e6i	−0.0431736	−	0.0336974i
$495$	9.76722e6			0.0805296
$496$			1.12002e7i			0.0917865i
$497$			6.85878e7i			0.558699i
$498$	−3.51773e7			−0.284823
$499$	−3.81855e7			−0.307324			−0.153662	−	0.988123i	$-0.549107\pi$
$499$	−3.81855e7			−0.307324			−0.153662	+	0.988123i	$0.549107\pi$
$500$	5.09716e7			0.407773
$501$	−1.99577e8			−1.58708
$502$			1.08816e6i			0.00860167i
$503$	1.08285e8			0.850873			0.425436	−	0.904988i	$-0.360121\pi$
$503$	1.08285e8			0.850873			0.425436	+	0.904988i	$0.360121\pi$
$504$			6.87167e7i			0.536748i
$505$	−3.68670e7			−0.286262
$506$			2.73129e7i			0.210822i
$507$			1.61816e8i			1.24164i
$508$		−	1.22859e8i		−	0.937161i
$509$		−	6.17811e7i		−	0.468492i	−0.972177	−	0.234246i	$-0.924738\pi$
$509$		−	6.17811e7i		−	0.468492i	0.972177	−	0.234246i	$-0.0752621\pi$
$510$	5.81603e7			0.438446
$511$	−1.18903e8			−0.891110
$512$		−	4.54077e7i		−	0.338314i
$513$	−4.54792e7	+	5.82686e7i	−0.336869	+	0.431601i
$514$	−7.27185e7			−0.535495
$515$			7.57055e7i			0.554250i
$516$		−	2.39628e7i		−	0.174417i
$517$	6.94007e7			0.502218
$518$	−1.30352e8			−0.937836
$519$	1.26359e8			0.903866
$520$	4.90600e6			0.0348913
$521$		−	2.19732e8i		−	1.55375i	−0.629657	−	0.776873i	$-0.716805\pi$
$521$		−	2.19732e8i		−	1.55375i	0.629657	−	0.776873i	$-0.283195\pi$
$522$	2.48111e7			0.174436
$523$			1.28062e8i			0.895186i	0.894237	+	0.447593i	$0.147719\pi$
$523$			1.28062e8i			0.895186i	−0.894237	+	0.447593i	$0.852281\pi$
$524$	4.43410e7			0.308185
$525$			1.36945e8i			0.946388i
$526$		−	1.17792e8i		−	0.809395i
$527$		−	9.77573e7i		−	0.667909i
$528$		−	1.09129e7i		−	0.0741377i
$529$	−2.90001e7			−0.195899
$530$	3.20327e7			0.215162
$531$			9.46605e7i			0.632245i
$532$	−5.84238e7	−	4.56003e7i	−0.388020	−	0.302854i
$533$	2.21912e7			0.146555
$534$			4.93062e7i			0.323801i
$535$		−	3.52657e7i		−	0.230299i
$536$	−2.03219e8			−1.31968
$537$	−3.68427e8			−2.37919
$538$	5.14876e7			0.330640
$539$	−8.92490e6			−0.0569950
$540$		−	1.92257e7i		−	0.122096i
$541$	7.84005e7			0.495139			0.247569	−	0.968870i	$-0.420368\pi$
$541$	7.84005e7			0.495139			0.247569	+	0.968870i	$0.420368\pi$
$542$		−	5.23225e7i		−	0.328617i
$543$	−1.49276e6			−0.00932373
$544$			1.86475e8i			1.15831i
$545$			3.88694e7i			0.240115i
$546$			1.01825e7i			0.0625573i
$547$			1.44687e8i			0.884030i	0.897008	+	0.442015i	$0.145736\pi$
$547$			1.44687e8i			0.884030i	−0.897008	+	0.442015i	$0.854264\pi$
$548$	−1.30768e8			−0.794618
$549$	1.28939e8			0.779230
$550$		−	3.24088e7i		−	0.194794i
$551$	−4.70364e7	+	6.02637e7i	−0.281176	+	0.360247i
$552$	−1.97003e8			−1.17127
$553$			1.03925e8i			0.614531i
$554$		−	8.02161e7i		−	0.471772i
$555$	−1.33638e8			−0.781720
$556$	1.21934e8			0.709416
$557$	3.19606e8			1.84948			0.924739	−	0.380602i	$-0.124283\pi$
$557$	3.19606e8			1.84948			0.924739	+	0.380602i	$0.124283\pi$
$558$	−3.55143e7			−0.204409
$559$		−	3.64944e6i		−	0.0208925i
$560$	−1.13915e7			−0.0648660
$561$			9.52502e7i			0.539483i
$562$	−1.39842e7			−0.0787823
$563$		−	1.52462e8i		−	0.854351i	−0.904169	−	0.427175i	$-0.859509\pi$
$563$		−	1.52462e8i		−	0.854351i	0.904169	−	0.427175i	$-0.140491\pi$
$564$			1.75222e8i			0.976675i
$565$		−	4.28706e7i		−	0.237692i
$566$			9.15352e6i			0.0504823i
$567$	2.07612e8			1.13895
$568$	1.17076e8			0.638886
$569$		−	1.12723e8i		−	0.611892i	−0.952049	−	0.305946i	$-0.901027\pi$
$569$		−	1.12723e8i		−	0.611892i	0.952049	−	0.305946i	$-0.0989728\pi$
$570$	5.13204e7	+	4.00560e7i	0.277118	+	0.216293i
$571$	2.03701e8			1.09417			0.547085	−	0.837077i	$-0.315737\pi$
$571$	2.03701e8			1.09417			0.547085	+	0.837077i	$0.315737\pi$
$572$			2.81246e6i			0.0150279i
$573$		−	3.68690e8i		−	1.95974i
$574$	−2.13431e8			−1.12855
$575$	−1.41245e8			−0.742965
$576$	8.61501e7			0.450805
$577$	−3.35128e8			−1.74455			−0.872275	−	0.489016i	$-0.837356\pi$
$577$	−3.35128e8			−1.74455			−0.872275	+	0.489016i	$0.837356\pi$
$578$			7.28768e7i			0.377404i
$579$	5.33799e7			0.275006
$580$		−	1.98840e7i		−	0.101910i
$581$	−6.01371e7			−0.306630
$582$		−	3.13035e8i		−	1.58790i
$583$			5.24605e7i			0.264745i
$584$			2.02962e8i			1.01901i
$585$			3.75564e6i			0.0187593i
$586$	2.08228e8			1.03478
$587$	2.06651e8			1.02170			0.510850	−	0.859670i	$-0.329331\pi$
$587$	2.06651e8			1.02170			0.510850	+	0.859670i	$0.329331\pi$
$588$		−	2.25334e7i		−	0.110840i
$589$	6.73272e7	−	8.62606e7i	0.329492	−	0.422150i
$590$	−6.49997e7			−0.316486
$591$		−	9.07587e7i		−	0.439669i
$592$			5.37173e7i			0.258910i
$593$	2.33592e8			1.12019			0.560097	−	0.828427i	$-0.310764\pi$
$593$	2.33592e8			1.12019			0.560097	+	0.828427i	$0.310764\pi$
$594$	−2.69779e7			−0.128721
$595$	9.94274e7			0.472014
$596$	2.22151e7			0.104933
$597$			3.26383e7i			0.153393i
$598$	−1.05022e7			−0.0491109
$599$		−	1.58646e8i		−	0.738156i	−0.929398	−	0.369078i	$-0.879674\pi$
$599$		−	1.58646e8i		−	0.738156i	0.929398	−	0.369078i	$-0.120326\pi$
$600$	2.33759e8			1.08222
$601$			2.01145e8i			0.926587i	0.886205	+	0.463294i	$0.153333\pi$
$601$			2.01145e8i			0.926587i	−0.886205	+	0.463294i	$0.846667\pi$
$602$			3.50996e7i			0.160884i
$603$		−	1.55568e8i		−	0.709526i
$604$			3.47410e7i			0.157664i
$605$	8.07117e7			0.364477
$606$	−1.30613e8			−0.586907
$607$		−	3.28217e8i		−	1.46756i	−0.679388	−	0.733779i	$-0.737755\pi$
$607$		−	3.28217e8i		−	1.46756i	0.679388	−	0.733779i	$-0.262245\pi$
$608$	−1.28429e8	+	1.64545e8i	−0.571415	+	0.732105i
$609$	1.17900e8			0.521989
$610$			8.85371e7i			0.390064i
$611$			2.66856e7i			0.116991i
$612$	−8.65174e7			−0.377441
$613$	−1.04114e8			−0.451988			−0.225994	−	0.974129i	$-0.572563\pi$
$613$	−1.04114e8			−0.451988			−0.225994	+	0.974129i	$0.572563\pi$
$614$	6.93946e7			0.299793
$615$	−2.18813e8			−0.940691
$616$		−	7.72759e7i		−	0.330599i
$617$	1.55494e8			0.662002			0.331001	−	0.943630i	$-0.392614\pi$
$617$	1.55494e8			0.662002			0.331001	+	0.943630i	$0.392614\pi$
$618$			2.68211e8i			1.13635i
$619$	−2.47327e8			−1.04280			−0.521398	−	0.853314i	$-0.674589\pi$
$619$	−2.47327e8			−1.04280			−0.521398	+	0.853314i	$0.674589\pi$
$620$			2.84616e7i			0.119422i
$621$			1.17575e8i			0.490955i
$622$			1.24768e8i			0.518482i
$623$			8.42910e7i			0.348592i
$624$	4.19618e6			0.0172703
$625$	1.25737e8			0.515018
$626$		−	1.51959e8i		−	0.619444i
$627$	−6.56006e7	+	8.40484e7i	−0.266137	+	0.340978i
$628$	−2.41089e8			−0.973415
$629$		−	4.68856e8i		−	1.88403i
$630$		−	3.61211e7i		−	0.144457i
$631$	−1.10573e7			−0.0440111			−0.0220055	−	0.999758i	$-0.507005\pi$
$631$	−1.10573e7			−0.0440111			−0.0220055	+	0.999758i	$0.507005\pi$
$632$	1.77395e8			0.702732
$633$	−1.24410e8			−0.490506
$634$	6.27464e7			0.246219
$635$			1.84495e8i			0.720550i
$636$	−1.32451e8			−0.514855
$637$		−	3.43175e6i		−	0.0132769i
$638$	−2.79016e7			−0.107440
$639$			8.96240e7i			0.343496i
$640$		−	4.16531e7i		−	0.158894i
$641$			1.52484e8i			0.578961i	0.957184	+	0.289481i	$0.0934826\pi$
$641$			1.52484e8i			0.578961i	−0.957184	+	0.289481i	$0.906517\pi$
$642$		−	1.24940e8i		−	0.472169i
$643$	1.39087e8			0.523184			0.261592	−	0.965179i	$-0.415753\pi$
$643$	1.39087e8			0.523184			0.261592	+	0.965179i	$0.415753\pi$
$644$	−1.17889e8			−0.441382
$645$			3.59846e7i			0.134103i
$646$	−1.40532e8	+	1.80052e8i	−0.521290	+	0.667884i
$647$	−4.55618e8			−1.68224			−0.841121	−	0.540847i	$-0.818104\pi$
$647$	−4.55618e8			−1.68224			−0.841121	+	0.540847i	$0.818104\pi$
$648$		−	3.54384e8i		−	1.30241i
$649$		−	1.06451e8i		−	0.389419i
$650$	1.24616e7			0.0453770
$651$	−1.68760e8			−0.611683
$652$	1.33177e8			0.480494
$653$	2.89552e8			1.03989			0.519945	−	0.854200i	$-0.325953\pi$
$653$	2.89552e8			1.03989			0.519945	+	0.854200i	$0.325953\pi$
$654$			1.37707e8i			0.492294i
$655$	−6.65864e7			−0.236953
$656$			8.79541e7i			0.311562i
$657$	−1.55371e8			−0.547867
$658$		−	2.56657e8i		−	0.900898i
$659$			4.78112e8i			1.67060i	0.549791	+	0.835302i	$0.314707\pi$
$659$			4.78112e8i			1.67060i	−0.549791	+	0.835302i	$0.685293\pi$
$660$		−	2.77317e7i		−	0.0964595i
$661$			2.72696e8i			0.944224i	0.881539	+	0.472112i	$0.156508\pi$
$661$			2.72696e8i			0.944224i	−0.881539	+	0.472112i	$0.843492\pi$
$662$	−1.96313e8			−0.676666
$663$	−3.66251e7			−0.125672
$664$			1.02651e8i			0.350639i
$665$	8.77343e7	+	6.84775e7i	0.298335	+	0.232853i
$666$	−1.70331e8			−0.576595
$667$			1.21601e8i			0.409789i
$668$			2.03859e8i			0.683913i
$669$	3.38980e8			1.13213
$670$	1.06823e8			0.355172
$671$	−1.44999e8			−0.479951
$672$	3.21915e8			1.06080
$673$			3.70225e8i			1.21456i	0.794487	+	0.607282i	$0.207740\pi$
$673$			3.70225e8i			1.21456i	−0.794487	+	0.607282i	$0.792260\pi$
$674$	−2.09965e7			−0.0685752
$675$		−	1.39512e8i		−	0.453628i
$676$	1.65287e8			0.535056
$677$		−	2.52922e6i		−	0.00815119i	−0.999992	−	0.00407559i	$-0.998703\pi$
$677$		−	2.52922e6i		−	0.00815119i	0.999992	−	0.00407559i	$-0.00129731\pi$
$678$		−	1.51883e8i		−	0.487326i
$679$		−	5.35147e8i		−	1.70948i
$680$		−	1.69718e8i		−	0.539760i
$681$	3.28979e8			1.04166
$682$	3.99379e7			0.125902
$683$		−	9.64579e7i		−	0.302744i	−0.988477	−	0.151372i	$-0.951631\pi$
$683$		−	9.64579e7i		−	0.302744i	0.988477	−	0.151372i	$-0.0483692\pi$
$684$	−7.63426e7	−	5.95861e7i	−0.238560	−	0.186199i
$685$	1.96372e8			0.610954
$686$			2.33434e8i			0.723089i
$687$			4.23457e8i			1.30599i
$688$	1.44644e7			0.0444156
$689$	−2.01718e7			−0.0616720
$690$	1.03555e8			0.315228
$691$	−1.13291e8			−0.343369			−0.171685	−	0.985152i	$-0.554921\pi$
$691$	−1.13291e8			−0.343369			−0.171685	+	0.985152i	$0.554921\pi$
$692$		−	1.29070e8i		−	0.389500i
$693$	5.91562e7			0.177746
$694$		−	2.17281e8i		−	0.650044i
$695$	−1.83107e8			−0.545445
$696$		−	2.01249e8i		−	0.596907i
$697$		−	7.67682e8i		−	2.26716i
$698$			8.89670e7i			0.261615i
$699$			1.85856e8i			0.544183i
$700$	1.39883e8			0.407823
$701$	−1.17038e7			−0.0339761			−0.0169881	−	0.999856i	$-0.505408\pi$
$701$	−1.17038e7			−0.0339761			−0.0169881	+	0.999856i	$0.505408\pi$
$702$		−	1.03734e7i		−	0.0299853i
$703$	3.22909e8	−	4.13716e8i	0.929426	−	1.19079i
$704$	−9.68808e7			−0.277664
$705$		−	2.63128e8i		−	0.750931i
$706$		−	3.34474e8i		−	0.950492i
$707$	−2.23289e8			−0.631842
$708$	2.68766e8			0.757312
$709$	−6.14873e8			−1.72523			−0.862615	−	0.505861i	$-0.831175\pi$
$709$	−6.14873e8			−1.72523			−0.862615	+	0.505861i	$0.831175\pi$
$710$	−6.15413e7			−0.171946
$711$			1.35799e8i			0.377823i
$712$	1.43881e8			0.398623
$713$		−	1.74058e8i		−	0.480204i
$714$	3.52254e8			0.967745
$715$		−	4.22343e6i		−	0.0115544i
$716$			3.76332e8i			1.02526i
$717$			3.50712e8i			0.951465i
$718$		−	3.88978e8i		−	1.05088i
$719$	−6.26651e8			−1.68593			−0.842964	−	0.537969i	$-0.819192\pi$
$719$	−6.26651e8			−1.68593			−0.842964	+	0.537969i	$0.819192\pi$
$720$	−1.48853e7			−0.0398805
$721$			4.58518e8i			1.22335i
$722$	−2.48010e8	+	6.20901e7i	−0.658959	+	0.164972i
$723$	7.92676e8			2.09740
$724$			1.52478e6i			0.00401784i
$725$		−	1.44289e8i		−	0.378633i
$726$	2.85947e8			0.747268
$727$	−3.38552e8			−0.881094			−0.440547	−	0.897730i	$-0.645215\pi$
$727$	−3.38552e8			−0.881094			−0.440547	+	0.897730i	$0.645215\pi$
$728$	2.97137e7			0.0770128
$729$	6.19454e6			0.0159892
$730$		−	1.06687e8i		−	0.274249i
$731$	−1.26248e8			−0.323202
$732$		−	3.66091e8i		−	0.933374i
$733$	6.16356e8			1.56502			0.782510	−	0.622638i	$-0.213939\pi$
$733$	6.16356e8			1.56502			0.782510	+	0.622638i	$0.213939\pi$
$734$			3.14239e7i			0.0794643i
$735$			3.38382e7i			0.0852207i
$736$			3.32022e8i			0.832785i
$737$			1.74945e8i			0.437019i
$738$	−2.78892e8			−0.693851
$739$	−3.03227e8			−0.751335			−0.375668	−	0.926754i	$-0.622587\pi$
$739$	−3.03227e8			−0.751335			−0.375668	+	0.926754i	$0.622587\pi$
$740$			1.36505e8i			0.336864i
$741$	−3.23178e7	−	2.52244e7i	−0.0794306	−	0.0619963i
$742$	1.94009e8			0.474909
$743$			5.99361e8i			1.46124i	0.682784	+	0.730621i	$0.260769\pi$
$743$			5.99361e8i			1.46124i	−0.682784	+	0.730621i	$0.739231\pi$
$744$			2.88065e8i			0.699475i
$745$	−3.33602e7			−0.0806789
$746$	3.85222e8			0.927885
$747$	−7.85814e7			−0.188520
$748$	9.72938e7			0.232477
$749$		−	2.13591e8i		−	0.508319i
$750$	−2.71180e8			−0.642797
$751$			6.18350e8i			1.45987i	0.683516	+	0.729936i	$0.260450\pi$
$751$			6.18350e8i			1.45987i	−0.683516	+	0.729936i	$0.739550\pi$
$752$	−1.05767e8			−0.248712
$753$			6.75675e6i			0.0158253i
$754$		−	1.07286e7i		−	0.0250281i
$755$		−	5.21702e7i		−	0.121222i
$756$		−	1.16442e8i		−	0.269492i
$757$	−2.74435e8			−0.632632			−0.316316	−	0.948654i	$-0.602446\pi$
$757$	−2.74435e8			−0.632632			−0.316316	+	0.948654i	$0.602446\pi$
$758$	5.25470e8			1.20654
$759$			1.69595e8i			0.387870i
$760$	1.16888e8	−	1.49758e8i	0.266274	−	0.341154i
$761$	−1.68374e8			−0.382051			−0.191026	−	0.981585i	$-0.561181\pi$
$761$	−1.68374e8			−0.382051			−0.191026	+	0.981585i	$0.561181\pi$
$762$			6.53634e8i			1.47730i
$763$			2.35417e8i			0.529985i
$764$	−3.76601e8			−0.844503
$765$	1.29922e8			0.290201
$766$	−4.75188e8			−1.05725
$767$	4.09321e7			0.0907147
$768$		−	6.01752e8i		−	1.32841i
$769$	9.93895e7			0.218555			0.109278	−	0.994011i	$-0.465146\pi$
$769$	9.93895e7			0.218555			0.109278	+	0.994011i	$0.465146\pi$
$770$			4.06202e7i			0.0889755i
$771$	−4.51532e8			−0.985202
$772$		−	5.45251e7i		−	0.118507i
$773$		−	7.96234e8i		−	1.72386i	−0.507027	−	0.861930i	$-0.669256\pi$
$773$		−	7.96234e8i		−	1.72386i	0.507027	−	0.861930i	$-0.330744\pi$
$774$			4.58649e7i			0.0989138i
$775$			2.06533e8i			0.443695i
$776$	−9.13470e8			−1.95483
$777$	−8.09394e8			−1.72543
$778$			6.76590e7i			0.143677i
$779$	5.28716e8	−	6.77399e8i	1.11843	−	1.43295i
$780$	1.06632e7			0.0224701
$781$		−	1.00787e8i		−	0.211570i
$782$			3.63313e8i			0.759732i
$783$	−1.20109e8			−0.250203
$784$	1.36016e7			0.0282256
$785$	3.62040e8			0.748425
$786$	−2.35904e8			−0.485811
$787$			3.17143e8i			0.650625i	0.945607	+	0.325313i	$0.105470\pi$
$787$			3.17143e8i			0.650625i	−0.945607	+	0.325313i	$0.894530\pi$
$788$	−9.27060e7			−0.189465
$789$		−	7.31410e8i		−	1.48912i
$790$	−9.32479e7			−0.189129
$791$		−	2.59650e8i		−	0.524637i
$792$		−	1.00977e8i		−	0.203257i
$793$		−	5.57542e7i		−	0.111804i
$794$		−	5.25316e7i		−	0.104944i
$795$	1.98901e8			0.395854
$796$	3.33386e7			0.0661010
$797$			7.89712e8i			1.55989i	0.625848	+	0.779945i	$0.284753\pi$
$797$			7.89712e8i			1.55989i	−0.625848	+	0.779945i	$0.715247\pi$
$798$	3.10827e8	+	2.42604e8i	0.611660	+	0.477407i
$799$	9.23158e8			1.80982
$800$		−	3.93968e8i		−	0.769469i
$801$			1.10143e8i			0.214319i
$802$	9.40954e7			0.182409
$803$	1.74724e8			0.337448
$804$	−4.41699e8			−0.849881
$805$	1.77032e8			0.339363
$806$			1.53567e7i			0.0293287i
$807$	3.19702e8			0.608311
$808$			3.81143e8i			0.722527i
$809$	−3.58300e8			−0.676708			−0.338354	−	0.941019i	$-0.609870\pi$
$809$	−3.58300e8			−0.676708			−0.338354	+	0.941019i	$0.609870\pi$
$810$			1.86282e8i			0.350523i
$811$		−	2.10199e8i		−	0.394064i	−0.980397	−	0.197032i	$-0.936870\pi$
$811$		−	2.10199e8i		−	0.394064i	0.980397	−	0.197032i	$-0.0631303\pi$
$812$		−	1.20429e8i		−	0.224939i
$813$		−	3.24887e8i		−	0.604589i
$814$	1.91547e8			0.355142
$815$	−1.99991e8			−0.369435
$816$		−	1.45162e8i		−	0.267167i
$817$	−1.11401e8	−	8.69495e7i	−0.204279	−	0.159441i
$818$	5.42594e8			0.991322
$819$			2.27464e7i			0.0414058i
$820$			2.23507e8i			0.405369i
$821$	8.92910e8			1.61353			0.806767	−	0.590869i	$-0.201215\pi$
$821$	8.92910e8			1.61353			0.806767	+	0.590869i	$0.201215\pi$
$822$	6.95712e8			1.25260
$823$	3.24504e7			0.0582131			0.0291066	−	0.999576i	$-0.490734\pi$
$823$	3.24504e7			0.0582131			0.0291066	+	0.999576i	$0.490734\pi$
$824$	7.82669e8			1.39893
$825$		−	2.01236e8i		−	0.358381i
$826$	−3.93677e8			−0.698555
$827$			1.76093e8i			0.311334i	0.987810	+	0.155667i	$0.0497527\pi$
$827$			1.76093e8i			0.311334i	−0.987810	+	0.155667i	$0.950247\pi$
$828$	−1.54046e8			−0.271368
$829$			6.29583e8i			1.10507i	0.833490	+	0.552534i	$0.186339\pi$
$829$			6.29583e8i			1.10507i	−0.833490	+	0.552534i	$0.813661\pi$
$830$		−	5.39588e7i		−	0.0943687i
$831$		−	4.98087e8i		−	0.867964i
$832$		−	3.72521e7i		−	0.0646816i
$833$	−1.18718e8			−0.205391
$834$	−6.48716e8			−1.11830
$835$		−	3.06133e8i		−	0.525836i
$836$	8.58517e7	+	6.70080e7i	0.146937	+	0.114685i
$837$	1.71923e8			0.293196
$838$			1.86098e8i			0.316235i
$839$			6.78394e7i			0.114867i	0.998349	+	0.0574336i	$0.0182917\pi$
$839$			6.78394e7i			0.114867i	−0.998349	+	0.0574336i	$0.981708\pi$
$840$	−2.92987e8			−0.494322
$841$	4.70601e8			0.791162
$842$	−5.42048e7			−0.0908032
$843$	−8.68323e7			−0.144943
$844$			1.27079e8i			0.211372i
$845$	−2.48210e8			−0.411386
$846$		−	3.35375e8i		−	0.553885i
$847$	4.88839e8			0.804480
$848$		−	7.99503e7i		−	0.131109i
$849$			5.68371e7i			0.0928771i
$850$		−	4.31097e8i		−	0.701970i
$851$		−	8.34805e8i		−	1.35455i
$852$	2.54466e8			0.411445
$853$	−6.22120e8			−1.00237			−0.501184	−	0.865341i	$-0.667102\pi$
$853$	−6.22120e8			−1.00237			−0.501184	+	0.865341i	$0.667102\pi$
$854$			5.36234e8i			0.860956i
$855$	1.14643e8	+	8.94798e7i	0.183421	+	0.143162i
$856$	−3.64589e8			−0.581276
$857$		−	2.10361e8i		−	0.334213i	−0.985939	−	0.167107i	$-0.946558\pi$
$857$		−	2.10361e8i		−	0.334213i	0.985939	−	0.167107i	$-0.0534424\pi$
$858$		−	1.49629e7i		−	0.0236894i
$859$	3.19350e8			0.503833			0.251917	−	0.967749i	$-0.418939\pi$
$859$	3.19350e8			0.503833			0.251917	+	0.967749i	$0.418939\pi$
$860$	3.67567e7			0.0577884
$861$	−1.32526e9			−2.07631
$862$	−6.38727e7			−0.0997225
$863$			1.58867e7i			0.0247173i	0.999924	+	0.0123586i	$0.00393398\pi$
$863$			1.58867e7i			0.0247173i	−0.999924	+	0.0123586i	$0.996066\pi$
$864$	−3.27949e8			−0.508469
$865$			1.93823e8i			0.299473i
$866$	−4.85457e8			−0.747476
$867$			4.52515e8i			0.694346i
$868$			1.72381e8i			0.263591i
$869$		−	1.52714e8i		−	0.232712i
$870$			1.05787e8i			0.160648i
$871$	−6.72691e7			−0.101803
$872$	4.01845e8			0.606051
$873$		−	6.99279e8i		−	1.05101i
$874$	−2.50220e8	+	3.20586e8i	−0.374790	+	0.480186i
$875$	−4.63594e8			−0.692012
$876$			4.41141e8i			0.656243i
$877$		−	9.80406e8i		−	1.45347i	−0.686916	−	0.726737i	$-0.741036\pi$
$877$		−	9.80406e8i		−	1.45347i	0.686916	−	0.726737i	$-0.258964\pi$
$878$	−3.45016e8			−0.509748
$879$	1.29295e9			1.90378
$880$	1.67394e7			0.0245636
$881$	5.62161e8			0.822115			0.411058	−	0.911609i	$-0.365159\pi$
$881$	5.62161e8			0.822115			0.411058	+	0.911609i	$0.365159\pi$
$882$			4.31291e7i			0.0628586i
$883$	−9.19482e8			−1.33555			−0.667776	−	0.744362i	$-0.732754\pi$
$883$	−9.19482e8			−1.33555			−0.667776	+	0.744362i	$0.732754\pi$
$884$			3.74109e7i			0.0541553i
$885$	−4.03603e8			−0.582271
$886$		−	1.98614e8i		−	0.285568i
$887$			8.30961e8i			1.19072i	0.803459	+	0.595360i	$0.202991\pi$
$887$			8.30961e8i			1.19072i	−0.803459	+	0.595360i	$0.797009\pi$
$888$			1.38160e9i			1.97307i
$889$			1.11742e9i			1.59041i
$890$	−7.56312e7			−0.107283
$891$	−3.05079e8			−0.431299
$892$		−	3.46253e8i		−	0.487864i
$893$	8.14591e8	+	6.35796e8i	1.14389	+	0.892819i
$894$	−1.18189e8			−0.165412
$895$		−	5.65134e8i		−	0.788283i
$896$		−	2.52277e8i		−	0.350714i
$897$	−6.52116e7			−0.0903540
$898$	−3.09005e8			−0.426714
$899$	1.77810e8			0.244724
$900$	1.82786e8			0.250736
$901$			6.97823e8i			0.954049i
$902$	3.13630e8			0.427364
$903$			2.17945e8i			0.295994i
$904$	−4.43210e8			−0.599936
$905$		−	2.28975e6i		−	0.00308918i
$906$		−	1.84830e8i		−	0.248535i
$907$		−	4.99166e7i		−	0.0668995i	−0.999440	−	0.0334498i	$-0.989351\pi$
$907$		−	4.99166e7i		−	0.0668995i	0.999440	−	0.0334498i	$-0.0106494\pi$
$908$		−	3.36037e8i		−	0.448880i
$909$	−2.91772e8			−0.388466
$910$	−1.56191e7			−0.0207267
$911$		−	7.57244e8i		−	1.00157i	−0.865572	−	0.500784i	$-0.833045\pi$
$911$		−	7.57244e8i		−	1.00157i	0.865572	−	0.500784i	$-0.166955\pi$
$912$	9.99759e7	−	1.28091e8i	0.131799	−	0.168862i
$913$	8.83694e7			0.116115
$914$		−	8.33356e8i		−	1.09142i
$915$			5.49754e8i			0.717638i
$916$	4.32542e8			0.562785
$917$	−4.03287e8			−0.523006
$918$	−3.58856e8			−0.463865
$919$	−6.93924e8			−0.894058			−0.447029	−	0.894520i	$-0.647518\pi$
$919$	−6.93924e8			−0.894058			−0.447029	+	0.894520i	$0.647518\pi$
$920$		−	3.02185e8i		−	0.388070i
$921$	4.30893e8			0.551557
$922$		−	1.18596e8i		−	0.151313i
$923$	3.87542e7			0.0492849
$924$		−	1.67960e8i		−	0.212907i
$925$			9.90557e8i			1.25157i
$926$		−	2.67320e8i		−	0.336666i
$927$			5.99148e8i			0.752133i
$928$	−3.39177e8			−0.424407
$929$	7.33299e8			0.914606			0.457303	−	0.889311i	$-0.348816\pi$
$929$	7.33299e8			0.914606			0.457303	+	0.889311i	$0.348816\pi$
$930$		−	1.51422e8i		−	0.188252i
$931$	−1.04756e8	−	8.17631e7i	−0.129817	−	0.101323i
$932$	1.89844e8			0.234503
$933$			7.74727e8i			0.953902i
$934$		−	4.60230e8i		−	0.564852i
$935$	−1.46105e8			−0.178744
$936$	3.88270e7			0.0473486
$937$	−1.23695e7			−0.0150360			−0.00751800	−	0.999972i	$-0.502393\pi$
$937$	−1.23695e7			−0.0150360			−0.00751800	+	0.999972i	$0.502393\pi$
$938$	6.46982e8			0.783941
$939$		−	9.43558e8i		−	1.13965i
$940$	−2.68774e8			−0.323596
$941$			6.92901e8i			0.831577i	0.909461	+	0.415788i	$0.136494\pi$
$941$			6.92901e8i			0.831577i	−0.909461	+	0.415788i	$0.863506\pi$
$942$	1.28265e9			1.53445
$943$		−	1.36687e9i		−	1.63002i
$944$			1.62233e8i			0.192851i
$945$			1.74860e8i			0.207203i
$946$		−	5.15777e7i		−	0.0609240i
$947$	−1.33520e9			−1.57216			−0.786079	−	0.618126i	$-0.787892\pi$
$947$	−1.33520e9			−1.57216			−0.786079	+	0.618126i	$0.787892\pi$
$948$	3.85569e8			0.452561
$949$			6.71840e7i			0.0786081i
$950$	2.96905e8	−	3.80398e8i	0.346295	−	0.443678i
$951$	3.89612e8			0.452992
$952$		−	1.02791e9i		−	1.19137i
$953$		−	1.20779e9i		−	1.39545i	−0.716368	−	0.697723i	$-0.754197\pi$
$953$		−	1.20779e9i		−	1.39545i	0.716368	−	0.697723i	$-0.245803\pi$
$954$	2.53513e8			0.291981
$955$	5.65537e8			0.649308
$956$	3.58236e8			0.410011
$957$	−1.73250e8			−0.197668
$958$			8.66492e8i			0.985526i
$959$	1.18935e9			1.34851
$960$			3.67318e8i			0.415172i
$961$	6.32990e8			0.713225
$962$			7.36526e7i			0.0827300i
$963$		−	2.79100e8i		−	0.312522i
$964$		−	8.09683e8i		−	0.903824i
$965$			8.18798e7i			0.0911161i
$966$	6.27193e8			0.695777
$967$	6.74016e7			0.0745402			0.0372701	−	0.999305i	$-0.488134\pi$
$967$	6.74016e7			0.0745402			0.0372701	+	0.999305i	$0.488134\pi$
$968$		−	8.34425e8i		−	0.919943i
$969$	−8.72610e8	+	1.11800e9i	−0.959067	+	1.22877i
$970$	4.80167e8			0.526111
$971$			4.59641e8i			0.502067i	0.967978	+	0.251033i	$0.0807704\pi$
$971$			4.59641e8i			0.502067i	−0.967978	+	0.251033i	$0.919230\pi$
$972$		−	4.99476e8i		−	0.543896i
$973$	−1.10901e9			−1.20392
$974$	−5.71803e8			−0.618827
$975$	7.73783e7			0.0834844
$976$	2.20980e8			0.237686
$977$		−	3.29203e8i		−	0.353004i	−0.984300	−	0.176502i	$-0.943522\pi$
$977$		−	3.29203e8i		−	0.353004i	0.984300	−	0.176502i	$-0.0564783\pi$
$978$	−7.08533e8			−0.757432
$979$		−	1.23863e8i		−	0.132006i
$980$	3.45642e7			0.0367239
$981$			3.07620e8i			0.325842i
$982$			9.61639e8i			1.01549i
$983$			8.54966e8i			0.900095i	0.893005	+	0.450047i	$0.148593\pi$
$983$			8.54966e8i			0.900095i	−0.893005	+	0.450047i	$0.851407\pi$
$984$			2.26216e9i			2.37431i
$985$	1.39216e8			0.145673
$986$	−3.71143e8			−0.387178
$987$		−	1.59366e9i		−	1.65747i
$988$	−2.57656e7	+	3.30112e7i	−0.0267158	+	0.0342287i
$989$	−2.24787e8			−0.232371
$990$			5.30786e7i			0.0547034i
$991$		−	6.98814e8i		−	0.718027i	−0.933332	−	0.359013i	$-0.883113\pi$
$991$		−	6.98814e8i		−	0.718027i	0.933332	−	0.359013i	$-0.116887\pi$
$992$	4.85494e8			0.497335
$993$	−1.21897e9			−1.24493
$994$	−3.72731e8			−0.379522
$995$	−5.00642e7			−0.0508227
$996$			2.23113e8i			0.225812i
$997$	−1.30205e9			−1.31384			−0.656920	−	0.753960i	$-0.728141\pi$
$997$	−1.30205e9			−1.31384			−0.656920	+	0.753960i	$0.728141\pi$
$998$		−	2.07514e8i		−	0.208764i
$999$	8.24563e8			0.827042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.b.b.18.5	yes	8
3.2	odd	2		171.7.c.d.37.4		8
4.3	odd	2		304.7.e.d.113.1		8
19.18	odd	2	inner	19.7.b.b.18.4	✓	8
57.56	even	2		171.7.c.d.37.5		8
76.75	even	2		304.7.e.d.113.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.b.b.18.4	✓	8	19.18	odd	2	inner
19.7.b.b.18.5	yes	8	1.1	even	1	trivial
171.7.c.d.37.4		8	3.2	odd	2
171.7.c.d.37.5		8	57.56	even	2
304.7.e.d.113.1		8	4.3	odd	2
304.7.e.d.113.8		8	76.75	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:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	19.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.43437i q^{2} +33.7437i q^{3} +34.4677 q^{4} -51.7597 q^{5} -183.376 q^{6} -313.488 q^{7} +535.109i q^{8} -409.636 q^{9} +O(q^{10})\)	\(q+5.43437i q^{2} +33.7437i q^{3} +34.4677 q^{4} -51.7597 q^{5} -183.376 q^{6} -313.488 q^{7} +535.109i q^{8} -409.636 q^{9} -281.281i q^{10} +460.660 q^{11} +1163.07i q^{12} +177.130i q^{13} -1703.61i q^{14} -1746.56i q^{15} -702.050 q^{16} +6127.64 q^{17} -2226.11i q^{18} +(5407.00 + 4220.22i) q^{19} -1784.04 q^{20} -10578.2i q^{21} +2503.40i q^{22} +10910.4 q^{23} -18056.6 q^{24} -12945.9 q^{25} -962.592 q^{26} +10776.5i q^{27} -10805.2 q^{28} +11145.5i q^{29} +9491.47 q^{30} -15953.5i q^{31} +30431.8i q^{32} +15544.4i q^{33} +33299.8i q^{34} +16226.1 q^{35} -14119.2 q^{36} -76514.9i q^{37} +(-22934.2 + 29383.6i) q^{38} -5977.04 q^{39} -27697.1i q^{40} -125282. i q^{41} +57486.1 q^{42} -20603.1 q^{43} +15877.9 q^{44} +21202.7 q^{45} +59290.9i q^{46} +150655. q^{47} -23689.8i q^{48} -19374.2 q^{49} -70352.9i q^{50} +206769. i q^{51} +6105.27i q^{52} +113881. i q^{53} -58563.5 q^{54} -23843.6 q^{55} -167750. i q^{56} +(-142406. + 182452. i) q^{57} -60568.7 q^{58} -231084. i q^{59} -60200.0i q^{60} -314764. q^{61} +86697.2 q^{62} +128416. q^{63} -210309. q^{64} -9168.22i q^{65} -84473.8 q^{66} +379771. i q^{67} +211205. q^{68} +368156. i q^{69} +88178.4i q^{70} -218789. i q^{71} -219200. i q^{72} +379291. q^{73} +415810. q^{74} -436843. i q^{75} +(186367. + 145461. i) q^{76} -144411. q^{77} -32481.4i q^{78} -331511. i q^{79} +36337.9 q^{80} -662264. q^{81} +680827. q^{82} +191832. q^{83} -364607. i q^{84} -317165. q^{85} -111965. i q^{86} -376090. q^{87} +246503. i q^{88} -268881. i q^{89} +115223. i q^{90} -55528.3i q^{91} +376054. q^{92} +538330. q^{93} +818714. i q^{94} +(-279865. - 218437. i) q^{95} -1.02688e6 q^{96} +1.70707e6i q^{97} -105286. i q^{98} -188703. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 454 q^{4} + 108 q^{5} - 358 q^{6} - 140 q^{7} - 1052 q^{9}+O(q^{10}) \) 8 * q - 454 * q^4 + 108 * q^5 - 358 * q^6 - 140 * q^7 - 1052 * q^9	\( 8 q - 454 q^{4} + 108 q^{5} - 358 q^{6} - 140 q^{7} - 1052 q^{9} - 2024 q^{11} + 11546 q^{16} + 6008 q^{17} + 20552 q^{19} - 10732 q^{20} - 50252 q^{23} + 64310 q^{24} + 78492 q^{25} - 37522 q^{26} - 135818 q^{28} - 187696 q^{30} + 210800 q^{35} + 35052 q^{36} + 103318 q^{38} + 43724 q^{39} - 429970 q^{42} + 260800 q^{43} + 693512 q^{44} - 191012 q^{45} - 100248 q^{47} - 301872 q^{49} + 390202 q^{54} - 52480 q^{55} - 186860 q^{57} - 405186 q^{58} - 54548 q^{61} - 1461908 q^{62} - 137408 q^{63} - 858058 q^{64} + 1539556 q^{66} + 1243910 q^{68} + 479968 q^{73} + 2645844 q^{74} - 2569288 q^{76} - 1755300 q^{77} + 2344672 q^{80} - 4279648 q^{81} + 1847172 q^{82} + 483040 q^{83} + 2111780 q^{85} + 2802652 q^{87} + 3905498 q^{92} + 1507528 q^{93} - 2383888 q^{95} - 8462238 q^{96} + 528224 q^{99}+O(q^{100}) \) 8 * q - 454 * q^4 + 108 * q^5 - 358 * q^6 - 140 * q^7 - 1052 * q^9 - 2024 * q^11 + 11546 * q^16 + 6008 * q^17 + 20552 * q^19 - 10732 * q^20 - 50252 * q^23 + 64310 * q^24 + 78492 * q^25 - 37522 * q^26 - 135818 * q^28 - 187696 * q^30 + 210800 * q^35 + 35052 * q^36 + 103318 * q^38 + 43724 * q^39 - 429970 * q^42 + 260800 * q^43 + 693512 * q^44 - 191012 * q^45 - 100248 * q^47 - 301872 * q^49 + 390202 * q^54 - 52480 * q^55 - 186860 * q^57 - 405186 * q^58 - 54548 * q^61 - 1461908 * q^62 - 137408 * q^63 - 858058 * q^64 + 1539556 * q^66 + 1243910 * q^68 + 479968 * q^73 + 2645844 * q^74 - 2569288 * q^76 - 1755300 * q^77 + 2344672 * q^80 - 4279648 * q^81 + 1847172 * q^82 + 483040 * q^83 + 2111780 * q^85 + 2802652 * q^87 + 3905498 * q^92 + 1507528 * q^93 - 2383888 * q^95 - 8462238 * q^96 + 528224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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