LMFDB - Embedded newform 36.7.g.a.5.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,7,Mod(5,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("36.5");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 36 = 2^{2} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	36.g (of order $6$, degree $2$, 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:	$8.28194701031$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 59 x^{10} + 602 x^{9} + 26655 x^{8} + 692184 x^{7} - 4209870 x^{6} + \cdots + 166668145981081 $ x^12 - 2x^11 + 59x^10 + 602x^9 + 26655x^8 + 692184x^7 - 4209870x^6 - 34154736x^5 + 2261179815x^4 - 21421205158x^3 + 1095551670899x^2 - 17316244703642*x + 166668145981081
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{15} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.4
Root			$-16.7265 - 5.11172i$ of defining polynomial
Character	$\chi$	$=$	36.5
Dual form			36.7.g.a.29.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.85375 + 25.5071i) q^{3} +(51.3356 + 29.6386i) q^{5} +(76.8738 + 133.149i) q^{7} +(-572.222 + 451.667i) q^{9} +O(q^{10})$	\(q+(8.85375 + 25.5071i) q^{3} +(51.3356 + 29.6386i) q^{5} +(76.8738 + 133.149i) q^{7} +(-572.222 + 451.667i) q^{9} +(-587.230 + 339.038i) q^{11} +(-826.070 + 1430.80i) q^{13} +(-301.482 + 1571.84i) q^{15} +6420.23i q^{17} +2857.79 q^{19} +(-2715.63 + 3139.70i) q^{21} +(6075.95 + 3507.95i) q^{23} +(-6055.60 - 10488.6i) q^{25} +(-16587.0 - 10596.8i) q^{27} +(8027.87 - 4634.89i) q^{29} +(26119.0 - 45239.4i) q^{31} +(-13847.1 - 11976.8i) q^{33} +9113.74i q^{35} +58462.5 q^{37} +(-43809.2 - 8402.73i) q^{39} +(102770. + 59334.4i) q^{41} +(-44827.4 - 77643.4i) q^{43} +(-42762.2 + 6226.71i) q^{45} +(-58769.9 + 33930.8i) q^{47} +(47005.3 - 81415.6i) q^{49} +(-163761. + 56843.1i) q^{51} +221631. i q^{53} -40194.5 q^{55} +(25302.1 + 72893.8i) q^{57} +(145784. + 84168.7i) q^{59} +(106190. + 183926. i) q^{61} +(-104128. - 41469.6i) q^{63} +(-84813.7 + 48967.2i) q^{65} +(54371.1 - 94173.5i) q^{67} +(-35682.6 + 186038. i) q^{69} -408102. i q^{71} +39555.6 q^{73} +(213919. - 247324. i) q^{75} +(-90285.3 - 52126.2i) q^{77} +(-209166. - 362286. i) q^{79} +(123435. - 516907. i) q^{81} +(-22035.7 + 12722.3i) q^{83} +(-190287. + 329586. i) q^{85} +(189299. + 163731. i) q^{87} -1.26291e6i q^{89} -254013. q^{91} +(1.38518e6 + 265680. i) q^{93} +(146706. + 84700.9i) q^{95} +(435032. + 753498. i) q^{97} +(182894. - 459237. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} - 216 q^{5} - 120 q^{7} + 174 q^{9}+O(q^{10}) $ 12 * q + 6 * q^3 - 216 * q^5 - 120 * q^7 + 174 * q^9	$ 12 q + 6 q^{3} - 216 q^{5} - 120 q^{7} + 174 q^{9} - 2214 q^{11} - 840 q^{13} - 5040 q^{15} - 12900 q^{19} + 5892 q^{21} - 8208 q^{23} + 24078 q^{25} + 40176 q^{27} - 29268 q^{29} + 3120 q^{31} + 2718 q^{33} + 12768 q^{37} + 223812 q^{39} - 16578 q^{41} + 71430 q^{43} - 61524 q^{45} - 329508 q^{47} - 238914 q^{49} - 153198 q^{51} + 5400 q^{55} + 652854 q^{57} + 428058 q^{59} - 93576 q^{61} - 493392 q^{63} - 1426464 q^{65} + 104334 q^{67} - 860256 q^{69} + 221820 q^{73} + 2598978 q^{75} + 3461184 q^{77} + 123468 q^{79} - 1125882 q^{81} - 2901420 q^{83} + 377568 q^{85} - 4083948 q^{87} + 91488 q^{91} + 6111144 q^{93} + 7249716 q^{95} - 1033482 q^{97} - 3503484 q^{99}+O(q^{100}) $ 12 * q + 6 * q^3 - 216 * q^5 - 120 * q^7 + 174 * q^9 - 2214 * q^11 - 840 * q^13 - 5040 * q^15 - 12900 * q^19 + 5892 * q^21 - 8208 * q^23 + 24078 * q^25 + 40176 * q^27 - 29268 * q^29 + 3120 * q^31 + 2718 * q^33 + 12768 * q^37 + 223812 * q^39 - 16578 * q^41 + 71430 * q^43 - 61524 * q^45 - 329508 * q^47 - 238914 * q^49 - 153198 * q^51 + 5400 * q^55 + 652854 * q^57 + 428058 * q^59 - 93576 * q^61 - 493392 * q^63 - 1426464 * q^65 + 104334 * q^67 - 860256 * q^69 + 221820 * q^73 + 2598978 * q^75 + 3461184 * q^77 + 123468 * q^79 - 1125882 * q^81 - 2901420 * q^83 + 377568 * q^85 - 4083948 * q^87 + 91488 * q^91 + 6111144 * q^93 + 7249716 * q^95 - 1033482 * q^97 - 3503484 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$.

$n$	$19$	$29$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	8.85375	+	25.5071i	0.327917	+	0.944707i
$3$	8.85375	+	25.5071i	0.327917	+	0.944707i
$4$		0			0
$5$	51.3356	+	29.6386i	0.410685	+	0.237109i	0.691084	−	0.722774i	$-0.257133\pi$
$5$	51.3356	+	29.6386i	0.410685	+	0.237109i	−0.280399	+	0.959884i	$0.590467\pi$
$6$		0			0
$7$	76.8738	+	133.149i	0.224122	+	0.388190i	0.956056	−	0.293185i	$-0.0947153\pi$
$7$	76.8738	+	133.149i	0.224122	+	0.388190i	−0.731934	+	0.681376i	$0.761382\pi$
$8$		0			0
$9$	−572.222	+	451.667i	−0.784941	+	0.619570i
$10$		0			0
$11$	−587.230	+	339.038i	−0.441195	+	0.254724i	−0.704104	−	0.710097i	$-0.748651\pi$
$11$	−587.230	+	339.038i	−0.441195	+	0.254724i	0.262909	+	0.964821i	$0.415318\pi$
$12$		0			0
$13$	−826.070	+	1430.80i	−0.375999	+	0.651250i	−0.990476	−	0.137685i	$-0.956034\pi$
$13$	−826.070	+	1430.80i	−0.375999	+	0.651250i	0.614477	+	0.788935i	$0.289367\pi$
$14$		0			0
$15$	−301.482	+	1571.84i	−0.0893280	+	0.465729i
$16$		0			0
$17$			6420.23i			1.30678i	0.757020	+	0.653392i	$0.226655\pi$
$17$			6420.23i			1.30678i	−0.757020	+	0.653392i	$0.773345\pi$
$18$		0			0
$19$	2857.79			0.416648			0.208324	−	0.978060i	$-0.433199\pi$
$19$	2857.79			0.416648			0.208324	+	0.978060i	$0.433199\pi$
$20$		0			0
$21$	−2715.63	+	3139.70i	−0.293233	+	0.339024i
$22$		0			0
$23$	6075.95	+	3507.95i	0.499379	+	0.288317i	0.728457	−	0.685091i	$-0.240238\pi$
$23$	6075.95	+	3507.95i	0.499379	+	0.288317i	−0.229078	+	0.973408i	$0.573571\pi$
$24$		0			0
$25$	−6055.60	−	10488.6i	−0.387559	−	0.671271i
$26$		0			0
$27$	−16587.0	−	10596.8i	−0.842708	−	0.538372i
$28$		0			0
$29$	8027.87	−	4634.89i	0.329160	−	0.190040i	−0.326308	−	0.945263i	$-0.605805\pi$
$29$	8027.87	−	4634.89i	0.329160	−	0.190040i	0.655468	+	0.755223i	$0.272471\pi$
$30$		0			0
$31$	26119.0	−	45239.4i	0.876740	−	1.51856i	0.0218431	−	0.999761i	$-0.493047\pi$
$31$	26119.0	−	45239.4i	0.876740	−	1.51856i	0.854897	−	0.518797i	$-0.173620\pi$
$32$		0			0
$33$	−13847.1	−	11976.8i	−0.385315	−	0.333271i
$34$		0			0
$35$			9113.74i			0.212565i
$36$		0			0
$37$	58462.5			1.15418			0.577088	−	0.816682i	$-0.304189\pi$
$37$	58462.5			1.15418			0.577088	+	0.816682i	$0.304189\pi$
$38$		0			0
$39$	−43809.2	−	8402.73i	−0.738536	−	0.141653i
$40$		0			0
$41$	102770.	+	59334.4i	1.49113	+	0.860904i	0.999948	−	0.0101531i	$-0.00323189\pi$
$41$	102770.	+	59334.4i	1.49113	+	0.860904i	0.491181	+	0.871057i	$0.336565\pi$
$42$		0			0
$43$	−44827.4	−	77643.4i	−0.563817	−	0.976560i	−0.997159	−	0.0753303i	$-0.975999\pi$
$43$	−44827.4	−	77643.4i	−0.563817	−	0.976560i	0.433341	−	0.901230i	$-0.357334\pi$
$44$		0			0
$45$	−42762.2	+	6226.71i	−0.469269	+	0.0683316i
$46$		0			0
$47$	−58769.9	+	33930.8i	−0.566058	+	0.326814i	−0.755573	−	0.655064i	$-0.772642\pi$
$47$	−58769.9	+	33930.8i	−0.566058	+	0.326814i	0.189515	+	0.981878i	$0.439308\pi$
$48$		0			0
$49$	47005.3	−	81415.6i	0.399539	−	0.692021i
$50$		0			0
$51$	−163761.	+	56843.1i	−1.23453	+	0.428516i
$52$		0			0
$53$			221631.i			1.48869i	0.667797	+	0.744343i	$0.267237\pi$
$53$			221631.i			1.48869i	−0.667797	+	0.744343i	$0.732763\pi$
$54$		0			0
$55$	−40194.5			−0.241590
$56$		0			0
$57$	25302.1	+	72893.8i	0.136626	+	0.393610i
$58$		0			0
$59$	145784.	+	84168.7i	0.709831	+	0.409821i	0.810999	−	0.585048i	$-0.198924\pi$
$59$	145784.	+	84168.7i	0.709831	+	0.409821i	−0.101167	+	0.994869i	$0.532258\pi$
$60$		0			0
$61$	106190.	+	183926.i	0.467836	+	0.810316i	0.999324	−	0.0367498i	$-0.0117005\pi$
$61$	106190.	+	183926.i	0.467836	+	0.810316i	−0.531488	+	0.847066i	$0.678367\pi$
$62$		0			0
$63$	−104128.	−	41469.6i	−0.416434	−	0.165847i
$64$		0			0
$65$	−84813.7	+	48967.2i	−0.308834	+	0.178306i
$66$		0			0
$67$	54371.1	−	94173.5i	0.180777	−	0.313115i	−0.761368	−	0.648320i	$-0.775472\pi$
$67$	54371.1	−	94173.5i	0.180777	−	0.313115i	0.942145	+	0.335204i	$0.108805\pi$
$68$		0			0
$69$	−35682.6	+	186038.i	−0.108620	+	0.566311i
$70$		0			0
$71$		−	408102.i		−	1.14023i	−0.821564	−	0.570117i	$-0.806898\pi$
$71$		−	408102.i		−	1.14023i	0.821564	−	0.570117i	$-0.193102\pi$
$72$		0			0
$73$	39555.6			0.101681			0.0508404	−	0.998707i	$-0.483810\pi$
$73$	39555.6			0.101681			0.0508404	+	0.998707i	$0.483810\pi$
$74$		0			0
$75$	213919.	−	247324.i	0.507067	−	0.586250i
$76$		0			0
$77$	−90285.3	−	52126.2i	−0.197763	−	0.114178i
$78$		0			0
$79$	−209166.	−	362286.i	−0.424238	−	0.734803i	0.572111	−	0.820177i	$-0.306125\pi$
$79$	−209166.	−	362286.i	−0.424238	−	0.734803i	−0.996349	+	0.0853740i	$0.972791\pi$
$80$		0			0
$81$	123435.	−	516907.i	0.232265	−	0.972652i
$82$		0			0
$83$	−22035.7	+	12722.3i	−0.0385383	+	0.0222501i	−0.519145	−	0.854686i	$-0.673750\pi$
$83$	−22035.7	+	12722.3i	−0.0385383	+	0.0222501i	0.480607	+	0.876936i	$0.340416\pi$
$84$		0			0
$85$	−190287.	+	329586.i	−0.309850	+	0.536677i
$86$		0			0
$87$	189299.	+	163731.i	0.287469	+	0.248642i
$88$		0			0
$89$		−	1.26291e6i		−	1.79145i	−0.444611	−	0.895724i	$-0.646658\pi$
$89$		−	1.26291e6i		−	1.79145i	0.444611	−	0.895724i	$-0.353342\pi$
$90$		0			0
$91$	−254013.			−0.337079
$92$		0			0
$93$	1.38518e6	+	265680.i	1.72209	+	0.330301i
$94$		0			0
$95$	146706.	+	84700.9i	0.171111	+	0.0987910i
$96$		0			0
$97$	435032.	+	753498.i	0.476658	+	0.825595i	0.999642	−	0.0267470i	$-0.00851484\pi$
$97$	435032.	+	753498.i	0.476658	+	0.825595i	−0.522985	+	0.852342i	$0.675182\pi$
$98$		0			0
$99$	182894.	−	459237.i	0.188493	−	0.473295i
$100$		0			0
$101$	−1.23317e6	+	711969.i	−1.19690	+	0.691030i	−0.959863	−	0.280471i	$-0.909509\pi$
$101$	−1.23317e6	+	711969.i	−1.19690	+	0.691030i	−0.237037	+	0.971501i	$0.576176\pi$
$102$		0			0
$103$	−812565.	+	1.40740e6i	−0.743612	+	1.28797i	0.207228	+	0.978293i	$0.433556\pi$
$103$	−812565.	+	1.40740e6i	−0.743612	+	1.28797i	−0.950840	+	0.309681i	$0.899778\pi$
$104$		0			0
$105$	−232465.	+	80690.8i	−0.200812	+	0.0697038i
$106$		0			0
$107$		−	1.10780e6i		−	0.904292i	−0.891944	−	0.452146i	$-0.850659\pi$
$107$		−	1.10780e6i		−	0.904292i	0.891944	−	0.452146i	$-0.149341\pi$
$108$		0			0
$109$	−513511.			−0.396525			−0.198262	−	0.980149i	$-0.563530\pi$
$109$	−513511.			−0.396525			−0.198262	+	0.980149i	$0.563530\pi$
$110$		0			0
$111$	517612.	+	1.49121e6i	0.378474	+	1.09036i
$112$		0			0
$113$	−424592.	−	245138.i	−0.294264	−	0.169893i	0.345599	−	0.938382i	$-0.387676\pi$
$113$	−424592.	−	245138.i	−0.294264	−	0.169893i	−0.639863	+	0.768489i	$0.721009\pi$
$114$		0			0
$115$	207942.	+	360166.i	0.136725	+	0.236815i
$116$		0			0
$117$	−173547.	−	1.19184e6i	−0.108358	−	0.744151i
$118$		0			0
$119$	−854849.	+	493547.i	−0.507281	+	0.292879i
$120$		0			0
$121$	−655887.	+	1.13603e6i	−0.370231	+	0.641260i
$122$		0			0
$123$	−603545.	+	3.14670e6i	−0.324335	+	1.69099i
$124$		0			0
$125$		−	1.64413e6i		−	0.841793i
$126$		0			0
$127$	−114175.			−0.0557389			−0.0278694	−	0.999612i	$-0.508872\pi$
$127$	−114175.			−0.0557389			−0.0278694	+	0.999612i	$0.508872\pi$
$128$		0			0
$129$	1.58356e6	−	1.83085e6i	0.737678	−	0.852872i
$130$		0			0
$131$	3.84454e6	+	2.21965e6i	1.71014	+	0.987348i	0.934354	+	0.356347i	$0.115978\pi$
$131$	3.84454e6	+	2.21965e6i	1.71014	+	0.987348i	0.775782	+	0.631001i	$0.217355\pi$
$132$		0			0
$133$	219689.	+	380512.i	0.0933799	+	0.161739i
$134$		0			0
$135$	−537431.	−	1.03561e6i	−0.218435	−	0.420915i
$136$		0			0
$137$	−2.46358e6	+	1.42235e6i	−0.958085	+	0.553151i	−0.895583	−	0.444894i	$-0.853241\pi$
$137$	−2.46358e6	+	1.42235e6i	−0.958085	+	0.553151i	−0.0625020	+	0.998045i	$0.519908\pi$
$138$		0			0
$139$	2.02940e6	−	3.51502e6i	0.755654	−	1.30883i	−0.189395	−	0.981901i	$-0.560653\pi$
$139$	2.02940e6	−	3.51502e6i	0.755654	−	1.30883i	0.945049	−	0.326929i	$-0.106014\pi$
$140$		0			0
$141$	−1.38581e6	−	1.19863e6i	−0.494363	−	0.427591i
$142$		0			0
$143$		−	1.12028e6i		−	0.383104i
$144$		0			0
$145$	549488.			0.180241
$146$		0			0
$147$	2.49285e6	+	478135.i	0.784773	+	0.150521i
$148$		0			0
$149$	−506083.	−	292187.i	−0.152990	−	0.0883289i	0.421551	−	0.906805i	$-0.361486\pi$
$149$	−506083.	−	292187.i	−0.152990	−	0.0883289i	−0.574541	+	0.818476i	$0.694819\pi$
$150$		0			0
$151$	−2.13100e6	−	3.69100e6i	−0.618946	−	1.07205i	−0.989678	−	0.143306i	$-0.954227\pi$
$151$	−2.13100e6	−	3.69100e6i	−0.618946	−	1.07205i	0.370733	−	0.928740i	$-0.379107\pi$
$152$		0			0
$153$	−2.89980e6	−	3.67380e6i	−0.809644	−	1.02575i
$154$		0			0
$155$	2.68167e6	−	1.54826e6i	0.720128	−	0.415766i
$156$		0			0
$157$	2.14089e6	−	3.70813e6i	0.553217	−	0.958201i	−0.444822	−	0.895619i	$-0.646733\pi$
$157$	2.14089e6	−	3.70813e6i	0.553217	−	0.958201i	0.998040	−	0.0625818i	$-0.0199334\pi$
$158$		0			0
$159$	−5.65316e6	+	1.96227e6i	−1.40637	+	0.488165i
$160$		0			0
$161$			1.07868e6i			0.258472i
$162$		0			0
$163$	1.85229e6			0.427707			0.213853	−	0.976866i	$-0.431399\pi$
$163$	1.85229e6			0.427707			0.213853	+	0.976866i	$0.431399\pi$
$164$		0			0
$165$	−355872.	−	1.02524e6i	−0.0792213	−	0.228231i
$166$		0			0
$167$	7.77994e6	+	4.49175e6i	1.67042	+	0.964420i	0.967400	+	0.253253i	$0.0815005\pi$
$167$	7.77994e6	+	4.49175e6i	1.67042	+	0.964420i	0.703023	+	0.711167i	$0.251833\pi$
$168$		0			0
$169$	1.04862e6	+	1.81626e6i	0.217249	+	0.376287i
$170$		0			0
$171$	−1.63529e6	+	1.29077e6i	−0.327044	+	0.258143i
$172$		0			0
$173$	−4.75707e6	+	2.74649e6i	−0.918757	+	0.530445i	−0.883238	−	0.468924i	$-0.844642\pi$
$173$	−4.75707e6	+	2.74649e6i	−0.918757	+	0.530445i	−0.0355189	+	0.999369i	$0.511308\pi$
$174$		0			0
$175$	931034.	−	1.61260e6i	0.173721	−	0.300893i
$176$		0			0
$177$	−856158.	+	4.46374e6i	−0.154395	+	0.804970i
$178$		0			0
$179$			2.48807e6i			0.433814i	0.976192	+	0.216907i	$0.0695969\pi$
$179$			2.48807e6i			0.433814i	−0.976192	+	0.216907i	$0.930403\pi$
$180$		0			0
$181$	5.06731e6			0.854558			0.427279	−	0.904120i	$-0.359472\pi$
$181$	5.06731e6			0.854558			0.427279	+	0.904120i	$0.359472\pi$
$182$		0			0
$183$	−3.75124e6	+	4.33703e6i	−0.612099	+	0.707684i
$184$		0			0
$185$	3.00121e6	+	1.73275e6i	0.474003	+	0.273666i
$186$		0			0
$187$	−2.17670e6	−	3.77015e6i	−0.332869	−	0.576546i
$188$		0			0
$189$	135846.	−	3.02316e6i	0.0201215	−	0.447792i
$190$		0			0
$191$	−4.69888e6	+	2.71290e6i	−0.674364	+	0.389344i	−0.797728	−	0.603017i	$-0.793965\pi$
$191$	−4.69888e6	+	2.71290e6i	−0.674364	+	0.389344i	0.123364	+	0.992361i	$0.460632\pi$
$192$		0			0
$193$	−2.81775e6	+	4.88049e6i	−0.391950	+	0.678877i	−0.992707	−	0.120555i	$-0.961533\pi$
$193$	−2.81775e6	+	4.88049e6i	−0.391950	+	0.678877i	0.600757	+	0.799432i	$0.294866\pi$
$194$		0			0
$195$	−1.99993e6	−	1.72981e6i	−0.269719	−	0.233289i
$196$		0			0
$197$			1.52261e6i			0.199154i	0.995030	+	0.0995771i	$0.0317490\pi$
$197$			1.52261e6i			0.199154i	−0.995030	+	0.0995771i	$0.968251\pi$
$198$		0			0
$199$	−9.65115e6			−1.22467			−0.612336	−	0.790597i	$-0.709770\pi$
$199$	−9.65115e6			−1.22467			−0.612336	+	0.790597i	$0.709770\pi$
$200$		0			0
$201$	2.88348e6	+	553059.i	0.355082	+	0.0681057i
$202$		0			0
$203$	1.23427e6	+	712604.i	0.147544	+	0.0851844i
$204$		0			0
$205$	3.51718e6	+	6.09194e6i	0.408257	+	0.707121i
$206$		0			0
$207$	−5.06121e6	+	736977.i	−0.570616	+	0.0830889i
$208$		0			0
$209$	−1.67818e6	+	968897.i	−0.183823	+	0.106130i
$210$		0			0
$211$	1.73941e6	−	3.01275e6i	0.185163	−	0.320712i	−0.758468	−	0.651710i	$-0.774052\pi$
$211$	1.73941e6	−	3.01275e6i	0.185163	−	0.320712i	0.943631	+	0.330998i	$0.107385\pi$
$212$		0			0
$213$	1.04095e7	−	3.61324e6i	1.07719	−	0.373902i
$214$		0			0
$215$		−	5.31450e6i		−	0.534745i
$216$		0			0
$217$	8.03146e6			0.785987
$218$		0			0
$219$	350215.	+	1.00895e6i	0.0333429	+	0.0960586i
$220$		0			0
$221$	−9.18603e6	−	5.30356e6i	−0.851042	−	0.491350i
$222$		0			0
$223$	−4.00207e6	−	6.93179e6i	−0.360886	−	0.625073i	0.627221	−	0.778841i	$-0.284192\pi$
$223$	−4.00207e6	−	6.93179e6i	−0.360886	−	0.625073i	−0.988107	+	0.153769i	$0.950859\pi$
$224$		0			0
$225$	8.20251e6	+	3.26670e6i	0.720110	+	0.286789i
$226$		0			0
$227$	1.70106e7	−	9.82108e6i	1.45426	−	0.839618i	0.455542	−	0.890214i	$-0.349445\pi$
$227$	1.70106e7	−	9.82108e6i	1.45426	−	0.839618i	0.998719	+	0.0505959i	$0.0161121\pi$
$228$		0			0
$229$	−5.27389e6	+	9.13465e6i	−0.439162	+	0.760651i	−0.997625	−	0.0688783i	$-0.978058\pi$
$229$	−5.27389e6	+	9.13465e6i	−0.439162	+	0.760651i	0.558463	+	0.829529i	$0.311391\pi$
$230$		0			0
$231$	530224.	−	2.76443e6i	0.0430154	−	0.224269i
$232$		0			0
$233$		−	5.28004e6i		−	0.417416i	−0.977978	−	0.208708i	$-0.933074\pi$
$233$		−	5.28004e6i		−	0.417416i	0.977978	−	0.208708i	$-0.0669259\pi$
$234$		0			0
$235$	−4.02265e6			−0.309962
$236$		0			0
$237$	7.38896e6	−	8.54281e6i	0.555058	−	0.641735i
$238$		0			0
$239$	−1.74830e7	−	1.00938e7i	−1.28063	−	0.739371i	−0.303665	−	0.952779i	$-0.598210\pi$
$239$	−1.74830e7	−	1.00938e7i	−1.28063	−	0.739371i	−0.976963	+	0.213407i	$0.931544\pi$
$240$		0			0
$241$	3.20399e6	+	5.54946e6i	0.228897	+	0.396461i	0.957481	−	0.288495i	$-0.0931549\pi$
$241$	3.20399e6	+	5.54946e6i	0.228897	+	0.396461i	−0.728585	+	0.684956i	$0.759822\pi$
$242$		0			0
$243$	1.42777e7	−	1.42810e6i	0.995035	−	0.0995266i
$244$		0			0
$245$	4.82610e6	−	2.78635e6i	0.328169	−	0.189469i
$246$		0			0
$247$	−2.36073e6	+	4.08891e6i	−0.156659	+	0.271342i
$248$		0			0
$249$	−519608.	−	449426.i	−0.0336572	−	0.0291112i
$250$		0			0
$251$		−	1.55033e7i		−	0.980400i	−0.871610	−	0.490200i	$-0.836924\pi$
$251$		−	1.55033e7i		−	0.980400i	0.871610	−	0.490200i	$-0.163076\pi$
$252$		0			0
$253$	−4.75731e6			−0.293765
$254$		0			0
$255$	−1.00915e7	−	1.93558e6i	−0.608607	−	0.116732i
$256$		0			0
$257$	−1.88422e7	−	1.08786e7i	−1.11002	−	0.640873i	−0.171189	−	0.985238i	$-0.554761\pi$
$257$	−1.88422e7	−	1.08786e7i	−1.11002	−	0.640873i	−0.938836	+	0.344365i	$0.888094\pi$
$258$		0			0
$259$	4.49423e6	+	7.78424e6i	0.258676	+	0.448040i
$260$		0			0
$261$	−2.50030e6	+	6.27811e6i	−0.140628	+	0.353108i
$262$		0			0
$263$	−6.28543e6	+	3.62890e6i	−0.345516	+	0.199484i	−0.662708	−	0.748878i	$-0.730593\pi$
$263$	−6.28543e6	+	3.62890e6i	−0.345516	+	0.199484i	0.317193	+	0.948361i	$0.397260\pi$
$264$		0			0
$265$	−6.56885e6	+	1.13776e7i	−0.352981	+	0.611381i
$266$		0			0
$267$	3.22133e7	−	1.11815e7i	1.69239	−	0.587446i
$268$		0			0
$269$			2.87490e7i			1.47695i	0.674280	+	0.738476i	$0.264454\pi$
$269$			2.87490e7i			1.47695i	−0.674280	+	0.738476i	$0.735546\pi$
$270$		0			0
$271$	−1.61888e7			−0.813405			−0.406703	−	0.913561i	$-0.633321\pi$
$271$	−1.61888e7			−0.813405			−0.406703	+	0.913561i	$0.633321\pi$
$272$		0			0
$273$	−2.24896e6	−	6.47912e6i	−0.110534	−	0.318440i
$274$		0			0
$275$	7.11207e6	+	4.10615e6i	0.341978	+	0.197441i
$276$		0			0
$277$	7.67531e6	+	1.32940e7i	0.361124	+	0.625485i	0.988146	−	0.153516i	$-0.0490597\pi$
$277$	7.67531e6	+	1.32940e7i	0.361124	+	0.625485i	−0.627022	+	0.779002i	$0.715726\pi$
$278$		0			0
$279$	5.48727e6	+	3.76840e7i	0.252664	+	1.73518i
$280$		0			0
$281$	1.63531e7	−	9.44146e6i	0.737023	−	0.425520i	−0.0839630	−	0.996469i	$-0.526758\pi$
$281$	1.63531e7	−	9.44146e6i	0.737023	−	0.425520i	0.820986	+	0.570948i	$0.193424\pi$
$282$		0			0
$283$	−105962.	+	183532.i	−0.00467511	+	0.00809752i	−0.868353	−	0.495946i	$-0.834821\pi$
$283$	−105962.	+	183532.i	−0.00467511	+	0.00809752i	0.863678	+	0.504043i	$0.168155\pi$
$284$		0			0
$285$	−861572.	+	4.49197e6i	−0.0372183	+	0.194045i
$286$		0			0
$287$			1.82450e7i			0.771790i
$288$		0			0
$289$	−1.70818e7			−0.707683
$290$		0			0
$291$	−1.53679e7	+	1.77677e7i	−0.623641	+	0.721028i
$292$		0			0
$293$	−1.29442e7	−	7.47332e6i	−0.514602	−	0.297106i	0.220121	−	0.975473i	$-0.429355\pi$
$293$	−1.29442e7	−	7.47332e6i	−0.514602	−	0.297106i	−0.734723	+	0.678367i	$0.762688\pi$
$294$		0			0
$295$	4.98929e6	+	8.64171e6i	0.194345	+	0.336615i
$296$		0			0
$297$	1.33331e7	+	599123.i	0.508934	+	0.0228690i
$298$		0			0
$299$	−1.00383e7	+	5.79562e6i	−0.375532	+	0.216814i
$300$		0			0
$301$	6.89211e6	−	1.19375e7i	0.252728	−	0.437737i
$302$		0			0
$303$	−2.90784e7	−	2.51509e7i	−1.04530	−	0.904118i
$304$		0			0
$305$			1.25893e7i			0.443713i
$306$		0			0
$307$	4.02227e7			1.39013			0.695066	−	0.718946i	$-0.255375\pi$
$307$	4.02227e7			1.39013			0.695066	+	0.718946i	$0.255375\pi$
$308$		0			0
$309$	−4.30930e7	−	8.26535e6i	−1.46060	−	0.280147i
$310$		0			0
$311$	481733.	+	278128.i	0.0160149	+	0.00924622i	0.507986	−	0.861365i	$-0.330390\pi$
$311$	481733.	+	278128.i	0.0160149	+	0.00924622i	−0.491971	+	0.870611i	$0.663723\pi$
$312$		0			0
$313$	−2.50885e7	−	4.34546e7i	−0.818166	−	1.41711i	−0.907032	−	0.421062i	$-0.861657\pi$
$313$	−2.50885e7	−	4.34546e7i	−0.818166	−	1.41711i	0.0888653	−	0.996044i	$-0.471676\pi$
$314$		0			0
$315$	−4.11637e6	−	5.21508e6i	−0.131699	−	0.166851i
$316$		0			0
$317$	4.04985e7	−	2.33818e7i	1.27134	−	0.734008i	0.296100	−	0.955157i	$-0.404314\pi$
$317$	4.04985e7	−	2.33818e7i	1.27134	−	0.734008i	0.975240	+	0.221149i	$0.0709807\pi$
$318$		0			0
$319$	−3.14281e6	+	5.44350e6i	−0.0968157	+	0.167690i
$320$		0			0
$321$	2.82567e7	−	9.80816e6i	0.854291	−	0.296533i
$322$		0			0
$323$			1.83476e7i			0.544469i
$324$		0			0
$325$	2.00094e7			0.582887
$326$		0			0
$327$	−4.54650e6	−	1.30982e7i	−0.130027	−	0.374600i
$328$		0			0
$329$	−9.03572e6	−	5.21678e6i	−0.253732	−	0.146492i
$330$		0			0
$331$	−1.72379e6	−	2.98570e6i	−0.0475337	−	0.0823307i	0.841280	−	0.540600i	$-0.181803\pi$
$331$	−1.72379e6	−	2.98570e6i	−0.0475337	−	0.0823307i	−0.888813	+	0.458269i	$0.848469\pi$
$332$		0			0
$333$	−3.34535e7	+	2.64056e7i	−0.905960	+	0.715093i
$334$		0			0
$335$	5.58235e6	−	3.22297e6i	0.148485	−	0.0857279i
$336$		0			0
$337$	3.53568e7	−	6.12397e7i	0.923810	−	1.60009i	0.130347	−	0.991468i	$-0.458391\pi$
$337$	3.53568e7	−	6.12397e7i	0.923810	−	1.60009i	0.793463	−	0.608618i	$-0.208276\pi$
$338$		0			0
$339$	2.49353e6	−	1.30005e7i	0.0640052	−	0.333704i
$340$		0			0
$341$			3.54213e7i			0.893307i
$342$		0			0
$343$	3.25422e7			0.806425
$344$		0			0
$345$	−7.34571e6	+	8.49280e6i	−0.178886	+	0.206821i
$346$		0			0
$347$	3.19802e7	+	1.84638e7i	0.765408	+	0.441908i	0.831234	−	0.555923i	$-0.187635\pi$
$347$	3.19802e7	+	1.84638e7i	0.765408	+	0.441908i	−0.0658262	+	0.997831i	$0.520968\pi$
$348$		0			0
$349$	−2.02452e7	−	3.50657e7i	−0.476261	−	0.824909i	0.523369	−	0.852106i	$-0.324675\pi$
$349$	−2.02452e7	−	3.50657e7i	−0.476261	−	0.824909i	−0.999630	+	0.0271976i	$0.991342\pi$
$350$		0			0
$351$	2.88638e7	−	1.49789e7i	0.667472	−	0.346386i
$352$		0			0
$353$	3.74286e7	−	2.16094e7i	0.850901	−	0.491268i	−0.0100535	−	0.999949i	$-0.503200\pi$
$353$	3.74286e7	−	2.16094e7i	0.850901	−	0.491268i	0.860955	+	0.508681i	$0.169867\pi$
$354$		0			0
$355$	1.20956e7	−	2.09502e7i	0.270360	−	0.468277i
$356$		0			0
$357$	−2.01576e7	−	1.74350e7i	−0.443030	−	0.383192i
$358$		0			0
$359$			3.23781e7i			0.699791i	0.936789	+	0.349896i	$0.113783\pi$
$359$			3.23781e7i			0.699791i	−0.936789	+	0.349896i	$0.886217\pi$
$360$		0			0
$361$	−3.88789e7			−0.826405
$362$		0			0
$363$	−3.47839e7	−	6.67164e6i	−0.727207	−	0.139480i
$364$		0			0
$365$	2.03061e6	+	1.17237e6i	0.0417588	+	0.0241095i
$366$		0			0
$367$	9.80140e6	+	1.69765e7i	0.198285	+	0.343440i	0.947972	−	0.318352i	$-0.103130\pi$
$367$	9.80140e6	+	1.69765e7i	0.198285	+	0.343440i	−0.749687	+	0.661792i	$0.769796\pi$
$368$		0			0
$369$	−8.56067e7	+	1.24654e7i	−1.70384	+	0.248101i
$370$		0			0
$371$	−2.95100e7	+	1.70376e7i	−0.577894	+	0.333647i
$372$		0			0
$373$	−905558.	+	1.56847e6i	−0.0174498	+	0.0302239i	−0.874618	−	0.484812i	$-0.838888\pi$
$373$	−905558.	+	1.56847e6i	−0.0174498	+	0.0302239i	0.857169	+	0.515036i	$0.172221\pi$
$374$		0			0
$375$	4.19369e7	−	1.45567e7i	0.795247	−	0.276038i
$376$		0			0
$377$			1.53150e7i			0.285820i
$378$		0			0
$379$	−1.03932e7			−0.190912			−0.0954559	−	0.995434i	$-0.530431\pi$
$379$	−1.03932e7			−0.190912			−0.0954559	+	0.995434i	$0.530431\pi$
$380$		0			0
$381$	−1.01087e6	−	2.91226e6i	−0.0182777	−	0.0526569i
$382$		0			0
$383$	−1.77788e7	−	1.02646e7i	−0.316450	−	0.182703i	0.333359	−	0.942800i	$-0.391818\pi$
$383$	−1.77788e7	−	1.02646e7i	−0.316450	−	0.182703i	−0.649809	+	0.760097i	$0.725151\pi$
$384$		0			0
$385$	−3.08990e6	−	5.35187e6i	−0.0541455	−	0.0937828i
$386$		0			0
$387$	6.07202e7	+	2.41822e7i	1.04761	+	0.417218i
$388$		0			0
$389$	−2.02842e7	+	1.17111e7i	−0.344595	+	0.198952i	−0.662302	−	0.749237i	$-0.730421\pi$
$389$	−2.02842e7	+	1.17111e7i	−0.344595	+	0.198952i	0.317707	+	0.948189i	$0.397087\pi$
$390$		0			0
$391$	−2.25218e7	+	3.90090e7i	−0.376768	+	0.652581i
$392$		0			0
$393$	−2.25781e7	+	1.17715e8i	−0.371971	+	1.93934i
$394$		0			0
$395$		−	2.47976e7i		−	0.402363i
$396$		0			0
$397$	2.83340e7			0.452831			0.226415	−	0.974031i	$-0.427299\pi$
$397$	2.83340e7			0.452831			0.226415	+	0.974031i	$0.427299\pi$
$398$		0			0
$399$	−7.76069e6	+	8.97259e6i	−0.122175	+	0.141253i
$400$		0			0
$401$	2.35430e7	+	1.35925e7i	0.365113	+	0.210798i	0.671322	−	0.741166i	$-0.265727\pi$
$401$	2.35430e7	+	1.35925e7i	0.365113	+	0.210798i	−0.306208	+	0.951965i	$0.599060\pi$
$402$		0			0
$403$	4.31522e7	+	7.47418e7i	0.659307	+	1.14195i
$404$		0			0
$405$	2.16571e7	−	2.28773e7i	0.326013	−	0.344382i
$406$		0			0
$407$	−3.43309e7	+	1.98210e7i	−0.509216	+	0.293996i
$408$		0			0
$409$	−3.49859e7	+	6.05973e7i	−0.511355	+	0.885693i	0.488558	+	0.872531i	$0.337523\pi$
$409$	−3.49859e7	+	6.05973e7i	−0.511355	+	0.885693i	−0.999913	+	0.0131620i	$0.995810\pi$
$410$		0			0
$411$	−5.80918e7	−	5.02455e7i	−0.836737	−	0.723722i
$412$		0			0
$413$			2.58815e7i			0.367400i
$414$		0			0
$415$	−1.50829e6			−0.0211028
$416$		0			0
$417$	1.07626e8	+	2.06429e7i	1.48425	+	0.284683i
$418$		0			0
$419$	−5.73935e7	−	3.31362e7i	−0.780227	−	0.450464i	0.0562839	−	0.998415i	$-0.482075\pi$
$419$	−5.73935e7	−	3.31362e7i	−0.780227	−	0.450464i	−0.836511	+	0.547951i	$0.815408\pi$
$420$		0			0
$421$	1.27092e6	+	2.20129e6i	0.0170322	+	0.0295006i	0.874416	−	0.485177i	$-0.161245\pi$
$421$	1.27092e6	+	2.20129e6i	0.0170322	+	0.0295006i	−0.857384	+	0.514678i	$0.827912\pi$
$422$		0			0
$423$	1.83040e7	−	4.59603e7i	0.241838	−	0.607242i
$424$		0			0
$425$	6.73393e7	−	3.88783e7i	0.877206	−	0.506455i
$426$		0			0
$427$	−1.63264e7	+	2.82782e7i	−0.209705	+	0.363219i
$428$		0			0
$429$	2.85750e7	−	9.91864e6i	0.361921	−	0.125626i
$430$		0			0
$431$		−	1.56169e8i		−	1.95057i	−0.220946	−	0.975286i	$-0.570914\pi$
$431$		−	1.56169e8i		−	1.95057i	0.220946	−	0.975286i	$-0.429086\pi$
$432$		0			0
$433$	−8.12049e7			−1.00027			−0.500136	−	0.865947i	$-0.666717\pi$
$433$	−8.12049e7			−1.00027			−0.500136	+	0.865947i	$0.666717\pi$
$434$		0			0
$435$	4.86503e6	+	1.40158e7i	0.0591041	+	0.170275i
$436$		0			0
$437$	1.73638e7	+	1.00250e7i	0.208065	+	0.120127i
$438$		0			0
$439$	−1.01429e7	−	1.75680e7i	−0.119886	−	0.207649i	0.799836	−	0.600218i	$-0.204920\pi$
$439$	−1.01429e7	−	1.75680e7i	−0.119886	−	0.207649i	−0.919722	+	0.392569i	$0.871586\pi$
$440$		0			0
$441$	9.87524e6	+	6.78186e7i	0.115142	+	0.790738i
$442$		0			0
$443$	−1.20858e8	+	6.97776e7i	−1.39016	+	0.802611i	−0.993333	−	0.115281i	$-0.963223\pi$
$443$	−1.20858e8	+	6.97776e7i	−1.39016	+	0.802611i	−0.396830	+	0.917892i	$0.629890\pi$
$444$		0			0
$445$	3.74311e7	−	6.48325e7i	0.424769	−	0.735721i
$446$		0			0
$447$	2.97211e6	−	1.54957e7i	0.0332768	−	0.173495i
$448$		0			0
$449$		−	1.19418e8i		−	1.31926i	−0.751588	−	0.659632i	$-0.770712\pi$
$449$		−	1.19418e8i		−	1.31926i	0.751588	−	0.659632i	$-0.229288\pi$
$450$		0			0
$451$	−8.04663e7			−0.877172
$452$		0			0
$453$	7.52793e7	−	8.70348e7i	0.809806	−	0.936264i
$454$		0			0
$455$	−1.30399e7	−	7.52859e6i	−0.138433	−	0.0799244i
$456$		0			0
$457$	1.62162e7	+	2.80872e7i	0.169902	+	0.294279i	0.938385	−	0.345591i	$-0.112321\pi$
$457$	1.62162e7	+	2.80872e7i	0.169902	+	0.294279i	−0.768483	+	0.639870i	$0.778988\pi$
$458$		0			0
$459$	6.80337e7	−	1.06492e8i	0.703535	−	1.10124i
$460$		0			0
$461$	3.14592e7	−	1.81630e7i	0.321104	−	0.185389i	−0.330781	−	0.943708i	$-0.607312\pi$
$461$	3.14592e7	−	1.81630e7i	0.321104	−	0.185389i	0.651884	+	0.758318i	$0.273979\pi$
$462$		0			0
$463$	−2.22272e7	+	3.84986e7i	−0.223945	+	0.387884i	−0.956002	−	0.293359i	$-0.905227\pi$
$463$	−2.22272e7	+	3.84986e7i	−0.223945	+	0.387884i	0.732057	+	0.681243i	$0.238560\pi$
$464$		0			0
$465$	6.32345e7	+	5.46936e7i	0.628919	+	0.543973i
$466$		0			0
$467$		−	1.52522e8i		−	1.49755i	−0.662822	−	0.748777i	$-0.730641\pi$
$467$		−	1.52522e8i		−	1.49755i	0.662822	−	0.748777i	$-0.269359\pi$
$468$		0			0
$469$	1.67189e7			0.162065
$470$		0			0
$471$	1.13539e8	+	2.17770e7i	1.08663	+	0.208418i
$472$		0			0
$473$	5.26480e7	+	3.03964e7i	0.497507	+	0.287236i
$474$		0			0
$475$	−1.73056e7	−	2.99742e7i	−0.161475	−	0.279684i
$476$		0			0
$477$	−1.00103e8	−	1.26822e8i	−0.922346	−	1.16853i
$478$		0			0
$479$	−9.57151e7	+	5.52612e7i	−0.870912	+	0.502821i	−0.867651	−	0.497174i	$-0.834371\pi$
$479$	−9.57151e7	+	5.52612e7i	−0.870912	+	0.502821i	−0.00326048	+	0.999995i	$0.501038\pi$
$480$		0			0
$481$	−4.82941e7	+	8.36478e7i	−0.433969	+	0.751656i
$482$		0			0
$483$	−2.75139e7	+	9.55034e6i	−0.244180	+	0.0847574i
$484$		0			0
$485$			5.15751e7i			0.452079i
$486$		0			0
$487$	1.83933e8			1.59247			0.796237	−	0.604985i	$-0.206821\pi$
$487$	1.83933e8			1.59247			0.796237	+	0.604985i	$0.206821\pi$
$488$		0			0
$489$	1.63997e7	+	4.72465e7i	0.140252	+	0.404058i
$490$		0			0
$491$	6.92505e6	+	3.99818e6i	0.0585031	+	0.0337768i	0.528966	−	0.848643i	$-0.322580\pi$
$491$	6.92505e6	+	3.99818e6i	0.0585031	+	0.0337768i	−0.470463	+	0.882420i	$0.655913\pi$
$492$		0			0
$493$	2.97571e7	+	5.15408e7i	0.248342	+	0.430140i
$494$		0			0
$495$	2.30002e7	−	1.81545e7i	0.189634	−	0.149682i
$496$		0			0
$497$	5.43385e7	−	3.13724e7i	0.442628	−	0.255551i
$498$		0			0
$499$	−2.06168e7	+	3.57094e7i	−0.165928	+	0.287396i	−0.936984	−	0.349371i	$-0.886395\pi$
$499$	−2.06168e7	+	3.57094e7i	−0.165928	+	0.287396i	0.771056	+	0.636767i	$0.219729\pi$
$500$		0			0
$501$	−4.56897e7	+	2.38212e8i	−0.363334	+	1.89431i
$502$		0			0
$503$		−	4.69138e7i		−	0.368635i	−0.982867	−	0.184317i	$-0.940993\pi$
$503$		−	4.69138e7i		−	0.368635i	0.982867	−	0.184317i	$-0.0590075\pi$
$504$		0			0
$505$	−8.44072e7			−0.655398
$506$		0			0
$507$	−3.70434e7	+	4.28280e7i	−0.284241	+	0.328628i
$508$		0			0
$509$	−4.78164e7	−	2.76068e7i	−0.362596	−	0.209345i	0.307623	−	0.951508i	$-0.400466\pi$
$509$	−4.78164e7	−	2.76068e7i	−0.362596	−	0.209345i	−0.670219	+	0.742163i	$0.733800\pi$
$510$		0			0
$511$	3.04079e6	+	5.26680e6i	0.0227889	+	0.0394715i
$512$		0			0
$513$	−4.74022e7	−	3.02833e7i	−0.351112	−	0.224311i
$514$		0			0
$515$	−8.34271e7	+	4.81666e7i	−0.610781	+	0.352634i
$516$		0			0
$517$	2.30076e7	−	3.98504e7i	0.166495	−	0.288377i
$518$		0			0
$519$	−1.12173e8	−	9.70221e7i	−0.802391	−	0.694014i
$520$		0			0
$521$			5.59371e7i			0.395537i	0.980249	+	0.197768i	$0.0633694\pi$
$521$			5.59371e7i			0.395537i	−0.980249	+	0.197768i	$0.936631\pi$
$522$		0			0
$523$	−1.43384e8			−1.00230			−0.501149	−	0.865361i	$-0.667089\pi$
$523$	−1.43384e8			−1.00230			−0.501149	+	0.865361i	$0.667089\pi$
$524$		0			0
$525$	4.93758e7	+	9.47041e6i	0.341222	+	0.0654472i
$526$		0			0
$527$	2.90447e8	+	1.67690e8i	1.98443	+	1.14571i
$528$		0			0
$529$	−4.94065e7	−	8.55746e7i	−0.333747	−	0.578067i
$530$		0			0
$531$	−1.21437e8	+	1.76828e7i	−0.811089	+	0.118105i
$532$		0			0
$533$	−1.69791e8	+	9.80287e7i	−1.12133	+	0.647398i
$534$		0			0
$535$	3.28336e7	−	5.68694e7i	0.214416	−	0.371379i
$536$		0			0
$537$	−6.34635e7	+	2.20288e7i	−0.409827	+	0.142255i
$538$		0			0
$539$			6.37463e7i			0.407088i
$540$		0			0
$541$	1.75550e8			1.10868			0.554342	−	0.832289i	$-0.312970\pi$
$541$	1.75550e8			1.10868			0.554342	+	0.832289i	$0.312970\pi$
$542$		0			0
$543$	4.48647e7	+	1.29252e8i	0.280224	+	0.807306i
$544$		0			0
$545$	−2.63614e7	−	1.52198e7i	−0.162847	−	0.0940197i
$546$		0			0
$547$	3.37262e6	+	5.84155e6i	0.0206066	+	0.0356916i	0.876145	−	0.482048i	$-0.160107\pi$
$547$	3.37262e6	+	5.84155e6i	0.0206066	+	0.0356916i	−0.855538	+	0.517740i	$0.826774\pi$
$548$		0			0
$549$	−1.43838e8	−	5.72842e7i	−0.869271	−	0.346193i
$550$		0			0
$551$	2.29419e7	−	1.32455e7i	0.137144	−	0.0791799i
$552$		0			0
$553$	3.21588e7	−	5.57007e7i	0.190162	−	0.329371i
$554$		0			0
$555$	−1.76254e7	+	9.18934e7i	−0.103100	+	0.537533i
$556$		0			0
$557$			2.14625e8i			1.24198i	0.783819	+	0.620990i	$0.213269\pi$
$557$			2.14625e8i			1.24198i	−0.783819	+	0.620990i	$0.786731\pi$
$558$		0			0
$559$	1.48122e8			0.847979
$560$		0			0
$561$	7.68936e7	−	8.89012e7i	0.435514	−	0.503523i
$562$		0			0
$563$	2.19704e8	+	1.26846e8i	1.23115	+	0.710807i	0.967270	−	0.253748i	$-0.0816634\pi$
$563$	2.19704e8	+	1.26846e8i	1.23115	+	0.710807i	0.263883	+	0.964555i	$0.414997\pi$
$564$		0			0
$565$	−1.45311e7	−	2.51687e7i	−0.0805664	−	0.139545i
$566$		0			0
$567$	7.83148e7	−	2.33013e7i	0.429630	−	0.127830i
$568$		0			0
$569$	9.51808e7	−	5.49527e7i	0.516669	−	0.298299i	−0.218901	−	0.975747i	$-0.570247\pi$
$569$	9.51808e7	−	5.49527e7i	0.516669	−	0.298299i	0.735571	+	0.677448i	$0.236914\pi$
$570$		0			0
$571$	6.39080e7	−	1.10692e8i	0.343279	−	0.594576i	−0.641761	−	0.766905i	$-0.721796\pi$
$571$	6.39080e7	−	1.10692e8i	0.343279	−	0.594576i	0.985040	+	0.172329i	$0.0551291\pi$
$572$		0			0
$573$	−1.10801e8	−	9.58353e7i	−0.588951	−	0.509403i
$574$		0			0
$575$		−	8.49710e7i		−	0.446958i
$576$		0			0
$577$	−5.97876e7			−0.311232			−0.155616	−	0.987818i	$-0.549736\pi$
$577$	−5.97876e7			−0.311232			−0.155616	+	0.987818i	$0.549736\pi$
$578$		0			0
$579$	−1.49435e8	−	2.86619e7i	−0.769867	−	0.147662i
$580$		0			0
$581$	−3.38793e6	−	1.95602e6i	−0.0172745	−	0.00997346i
$582$		0			0
$583$	−7.51413e7	−	1.30149e8i	−0.379204	−	0.656801i
$584$		0			0
$585$	2.64154e7	−	6.63276e7i	0.131944	−	0.331304i
$586$		0			0
$587$	−1.16069e8	+	6.70127e7i	−0.573857	+	0.331316i	−0.758688	−	0.651454i	$-0.774159\pi$
$587$	−1.16069e8	+	6.70127e7i	−0.573857	+	0.331316i	0.184831	+	0.982770i	$0.440826\pi$
$588$		0			0
$589$	7.46425e7	−	1.29285e8i	0.365292	−	0.632704i
$590$		0			0
$591$	−3.88373e7	+	1.34808e7i	−0.188142	+	0.0653060i
$592$		0			0
$593$		−	2.75560e8i		−	1.32146i	−0.750626	−	0.660728i	$-0.770248\pi$
$593$		−	2.75560e8i		−	1.32146i	0.750626	−	0.660728i	$-0.229752\pi$
$594$		0			0
$595$	−5.85123e7			−0.277777
$596$		0			0
$597$	−8.54489e7	−	2.46173e8i	−0.401591	−	1.15696i
$598$		0			0
$599$	2.55365e8	+	1.47435e8i	1.18817	+	0.685993i	0.957891	−	0.287131i	$-0.0927015\pi$
$599$	2.55365e8	+	1.47435e8i	1.18817	+	0.685993i	0.230283	+	0.973124i	$0.426035\pi$
$600$		0			0
$601$	−6.77957e7	−	1.17426e8i	−0.312305	−	0.540928i	0.666556	−	0.745455i	$-0.267768\pi$
$601$	−6.77957e7	−	1.17426e8i	−0.312305	−	0.540928i	−0.978861	+	0.204527i	$0.934434\pi$
$602$		0			0
$603$	1.14227e7	+	7.84458e7i	0.0520975	+	0.357781i
$604$		0			0
$605$	−6.73408e7	+	3.88792e7i	−0.304097	+	0.175570i
$606$		0			0
$607$	1.69952e8	−	2.94366e8i	0.759908	−	1.31620i	−0.182989	−	0.983115i	$-0.558577\pi$
$607$	1.69952e8	−	2.94366e8i	0.759908	−	1.31620i	0.942897	−	0.333084i	$-0.108089\pi$
$608$		0			0
$609$	−7.24855e6	+	3.77917e7i	−0.0320922	+	0.167319i
$610$		0			0
$611$		−	1.12117e8i		−	0.491527i
$612$		0			0
$613$	3.34467e8			1.45202			0.726008	−	0.687686i	$-0.241373\pi$
$613$	3.34467e8			1.45202			0.726008	+	0.687686i	$0.241373\pi$
$614$		0			0
$615$	−1.24247e8	+	1.43650e8i	−0.534148	+	0.617559i
$616$		0			0
$617$	−2.98933e8	−	1.72589e8i	−1.27268	−	0.734780i	−0.297186	−	0.954820i	$-0.596048\pi$
$617$	−2.98933e8	−	1.72589e8i	−1.27268	−	0.734780i	−0.975491	+	0.220039i	$0.929381\pi$
$618$		0			0
$619$	5.73101e7	+	9.92640e7i	0.241635	+	0.418523i	0.961180	−	0.275922i	$-0.0889831\pi$
$619$	5.73101e7	+	9.92640e7i	0.241635	+	0.418523i	−0.719545	+	0.694445i	$0.755650\pi$
$620$		0			0
$621$	−6.36089e7	−	1.22572e8i	−0.265609	−	0.511818i
$622$		0			0
$623$	1.68156e8	−	9.70851e7i	0.695423	−	0.401503i
$624$		0			0
$625$	−4.58891e7	+	7.94822e7i	−0.187962	+	0.325559i
$626$		0			0
$627$	−3.95719e7	−	3.42271e7i	−0.160540	−	0.138857i
$628$		0			0
$629$			3.75342e8i			1.50826i
$630$		0			0
$631$	−3.72217e8			−1.48152			−0.740760	−	0.671769i	$-0.765535\pi$
$631$	−3.72217e8			−1.48152			−0.740760	+	0.671769i	$0.765535\pi$
$632$		0			0
$633$	9.22466e7	+	1.76931e7i	0.363697	+	0.0697580i
$634$		0			0
$635$	−5.86122e6	−	3.38398e6i	−0.0228911	−	0.0132162i
$636$		0			0
$637$	7.76594e7	+	1.34510e8i	0.300452	+	0.520399i
$638$		0			0
$639$	1.84326e8	+	2.33525e8i	0.706455	+	0.895016i
$640$		0			0
$641$	3.33506e8	−	1.92550e8i	1.26628	−	0.731087i	0.291997	−	0.956419i	$-0.405680\pi$
$641$	3.33506e8	−	1.92550e8i	1.26628	−	0.731087i	0.974282	+	0.225332i	$0.0723468\pi$
$642$		0			0
$643$	−1.12238e8	+	1.94403e8i	−0.422191	+	0.731256i	−0.996153	−	0.0876258i	$-0.972072\pi$
$643$	−1.12238e8	+	1.94403e8i	−0.422191	+	0.731256i	0.573963	+	0.818881i	$0.305405\pi$
$644$		0			0
$645$	1.35557e8	−	4.70532e7i	0.505177	−	0.175352i
$646$		0			0
$647$			1.62221e8i			0.598955i	0.954103	+	0.299477i	$0.0968123\pi$
$647$			1.62221e8i			0.598955i	−0.954103	+	0.299477i	$0.903188\pi$
$648$		0			0
$649$	−1.14145e8			−0.417565
$650$		0			0
$651$	7.11086e7	+	2.04859e8i	0.257738	+	0.742527i
$652$		0			0
$653$	−2.40053e8	−	1.38595e8i	−0.862119	−	0.497745i	0.00260213	−	0.999997i	$-0.499172\pi$
$653$	−2.40053e8	−	1.38595e8i	−0.862119	−	0.497745i	−0.864721	+	0.502252i	$0.832505\pi$
$654$		0			0
$655$	1.31575e8	+	2.27894e8i	0.468218	+	0.810978i
$656$		0			0
$657$	−2.26346e7	+	1.78659e7i	−0.0798135	+	0.0629984i
$658$		0			0
$659$	−3.40540e7	+	1.96611e7i	−0.118990	+	0.0686992i	−0.558314	−	0.829630i	$-0.688552\pi$
$659$	−3.40540e7	+	1.96611e7i	−0.118990	+	0.0686992i	0.439323	+	0.898329i	$0.355218\pi$
$660$		0			0
$661$	−1.39609e8	+	2.41810e8i	−0.483404	+	0.837280i	−0.999818	−	0.0190590i	$-0.993933\pi$
$661$	−1.39609e8	+	2.41810e8i	−0.483404	+	0.837280i	0.516415	+	0.856339i	$0.327266\pi$
$662$		0			0
$663$	5.39474e7	−	2.81265e8i	0.185110	−	0.965107i
$664$		0			0
$665$			2.60451e7i			0.0885649i
$666$		0			0
$667$	6.50359e7			0.219167
$668$		0			0
$669$	1.41376e8	−	1.63453e8i	0.472170	−	0.545903i
$670$		0			0
$671$	−1.24716e8	−	7.20047e7i	−0.412814	−	0.238338i
$672$		0			0
$673$	6.34740e7	+	1.09940e8i	0.208234	+	0.360671i	0.951158	−	0.308704i	$-0.0998952\pi$
$673$	6.34740e7	+	1.09940e8i	0.208234	+	0.360671i	−0.742925	+	0.669375i	$0.766562\pi$
$674$		0			0
$675$	−1.07010e7	+	2.38145e8i	−0.0347948	+	0.774336i
$676$		0			0
$677$	−4.77884e8	+	2.75907e8i	−1.54013	+	0.889193i	−0.541298	+	0.840831i	$0.682067\pi$
$677$	−4.77884e8	+	2.75907e8i	−1.54013	+	0.889193i	−0.998830	+	0.0483627i	$0.984600\pi$
$678$		0			0
$679$	−6.68852e7	+	1.15849e8i	−0.213659	+	0.370068i
$680$		0			0
$681$	4.01115e8	+	3.46938e8i	1.27007	+	1.09853i
$682$		0			0
$683$		−	1.16196e7i		−	0.0364695i	−0.999834	−	0.0182347i	$-0.994195\pi$
$683$		−	1.16196e7i		−	0.0364695i	0.999834	−	0.0182347i	$-0.00580462\pi$
$684$		0			0
$685$	−1.68626e8			−0.524628
$686$		0			0
$687$	−2.79692e8	−	5.36457e7i	−0.862601	−	0.165449i
$688$		0			0
$689$	−3.17109e8	−	1.83083e8i	−0.969506	−	0.559745i
$690$		0			0
$691$	−3.27200e8	−	5.66727e8i	−0.991698	−	1.71767i	−0.607212	−	0.794540i	$-0.707712\pi$
$691$	−3.27200e8	−	5.66727e8i	−0.991698	−	1.71767i	−0.384486	−	0.923131i	$-0.625621\pi$
$692$		0			0
$693$	7.52069e7	−	1.09511e7i	0.225974	−	0.0329046i
$694$		0			0
$695$	2.08361e8	−	1.20297e8i	0.620671	−	0.358345i
$696$		0			0
$697$	−3.80940e8	+	6.59808e8i	−1.12502	+	1.94858i
$698$		0			0
$699$	1.34678e8	−	4.67482e7i	0.394336	−	0.136878i
$700$		0			0
$701$		−	8.65379e7i		−	0.251219i	−0.992080	−	0.125609i	$-0.959911\pi$
$701$		−	8.65379e7i		−	0.251219i	0.992080	−	0.125609i	$-0.0400886\pi$
$702$		0			0
$703$	1.67073e8			0.480885
$704$		0			0
$705$	−3.56156e7	−	1.02606e8i	−0.101642	−	0.292823i
$706$		0			0
$707$	−1.89596e8	−	1.09464e8i	−0.536503	−	0.309750i
$708$		0			0
$709$	3.39991e8	+	5.88881e8i	0.953957	+	1.65230i	0.736738	+	0.676178i	$0.236365\pi$
$709$	3.39991e8	+	5.88881e8i	0.953957	+	1.65230i	0.217218	+	0.976123i	$0.430302\pi$
$710$		0			0
$711$	2.83322e8	+	1.12835e8i	0.788264	+	0.313931i
$712$		0			0
$713$	3.17395e8	−	1.83248e8i	0.875652	−	0.505558i
$714$		0			0
$715$	3.32034e7	−	5.75100e7i	0.0908375	−	0.157335i
$716$		0			0
$717$	1.02674e8	−	5.35310e8i	0.278549	−	1.45227i
$718$		0			0
$719$		−	1.62934e8i		−	0.438354i	−0.975685	−	0.219177i	$-0.929663\pi$
$719$		−	1.62934e8i		−	0.438354i	0.975685	−	0.219177i	$-0.0703372\pi$
$720$		0			0
$721$	−2.49860e8			−0.666639
$722$		0			0
$723$	−1.13183e8	+	1.30858e8i	−0.299480	+	0.346246i
$724$		0			0
$725$	−9.72272e7	−	5.61341e7i	−0.255137	−	0.147304i
$726$		0			0
$727$	−7.67295e7	−	1.32899e8i	−0.199691	−	0.345875i	0.748737	−	0.662867i	$-0.230661\pi$
$727$	−7.67295e7	−	1.32899e8i	−0.199691	−	0.345875i	−0.948428	+	0.316992i	$0.897327\pi$
$728$		0			0
$729$	1.62838e8	+	3.51537e8i	0.420312	+	0.907380i
$730$		0			0
$731$	4.98488e8	−	2.87802e8i	1.27615	−	0.736787i
$732$		0			0
$733$	1.74756e8	−	3.02686e8i	0.443732	−	0.768566i	−0.554231	−	0.832363i	$-0.686988\pi$
$733$	1.74756e8	−	3.02686e8i	0.443732	−	0.768566i	0.997963	+	0.0637970i	$0.0203210\pi$
$734$		0			0
$735$	1.13801e8	+	9.84300e7i	0.286604	+	0.247894i
$736$		0			0
$737$			7.37354e7i			0.184193i
$738$		0			0
$739$	5.10104e7			0.126394			0.0631968	−	0.998001i	$-0.479870\pi$
$739$	5.10104e7			0.126394			0.0631968	+	0.998001i	$0.479870\pi$
$740$		0			0
$741$	−1.25197e8	−	2.40132e7i	−0.307709	−	0.0590195i
$742$		0			0
$743$	−4.04849e8	−	2.33740e8i	−0.987021	−	0.569857i	−0.0826387	−	0.996580i	$-0.526335\pi$
$743$	−4.04849e8	−	2.33740e8i	−0.987021	−	0.569857i	−0.904383	+	0.426723i	$0.859668\pi$
$744$		0			0
$745$	−1.73201e7	−	2.99993e7i	−0.0418872	−	0.0725507i
$746$		0			0
$747$	6.86306e6	−	1.72328e7i	0.0164648	−	0.0413422i
$748$		0			0
$749$	1.47502e8	−	8.51605e7i	0.351038	−	0.202672i
$750$		0			0
$751$	−1.38607e8	+	2.40074e8i	−0.327239	+	0.566795i	−0.981963	−	0.189073i	$-0.939452\pi$
$751$	−1.38607e8	+	2.40074e8i	−0.327239	+	0.566795i	0.654724	+	0.755868i	$0.272785\pi$
$752$		0			0
$753$	3.95444e8	−	1.37263e8i	0.926191	−	0.321490i
$754$		0			0
$755$		−	2.52640e8i		−	0.587031i
$756$		0			0
$757$	6.09490e8			1.40501			0.702504	−	0.711680i	$-0.252065\pi$
$757$	6.09490e8			1.40501			0.702504	+	0.711680i	$0.252065\pi$
$758$		0			0
$759$	−4.21200e7	−	1.21345e8i	−0.0963304	−	0.277521i
$760$		0			0
$761$	−2.53043e8	−	1.46095e8i	−0.574171	−	0.331498i	0.184643	−	0.982806i	$-0.440887\pi$
$761$	−2.53043e8	−	1.46095e8i	−0.574171	−	0.331498i	−0.758813	+	0.651308i	$0.774221\pi$
$762$		0			0
$763$	−3.94756e7	−	6.83737e7i	−0.0888699	−	0.153927i
$764$		0			0
$765$	−3.99769e7	−	2.74543e8i	−0.0892946	−	0.613234i
$766$		0			0
$767$	−2.40856e8	+	1.39058e8i	−0.533792	+	0.308185i
$768$		0			0
$769$	3.28242e8	−	5.68532e8i	0.721797	−	1.25019i	−0.238482	−	0.971147i	$-0.576650\pi$
$769$	3.28242e8	−	5.68532e8i	0.721797	−	1.25019i	0.960279	−	0.279042i	$-0.0900168\pi$
$770$		0			0
$771$	1.10656e8	−	5.76926e8i	0.241441	−	1.25880i
$772$		0			0
$773$			4.93327e8i			1.06806i	0.845465	+	0.534031i	$0.179323\pi$
$773$			4.93327e8i			1.06806i	−0.845465	+	0.534031i	$0.820677\pi$
$774$		0			0
$775$	−6.32664e8			−1.35915
$776$		0			0
$777$	−1.58762e8	+	1.83554e8i	−0.338442	+	0.391293i
$778$		0			0
$779$	2.93695e8	+	1.69565e8i	0.621276	+	0.358694i
$780$		0			0
$781$	1.38362e8	+	2.39650e8i	0.290445	+	0.503065i
$782$		0			0
$783$	−1.82273e8	−	8.19045e6i	−0.379698	−	0.0170617i
$784$		0			0
$785$	2.19808e8	−	1.26906e8i	0.454396	−	0.262346i
$786$		0			0
$787$	7.12144e7	−	1.23347e8i	0.146098	−	0.253049i	−0.783684	−	0.621160i	$-0.786662\pi$
$787$	7.12144e7	−	1.23347e8i	0.146098	−	0.253049i	0.929782	+	0.368111i	$0.119995\pi$
$788$		0			0
$789$	−1.48212e8	−	1.28194e8i	−0.301754	−	0.260997i
$790$		0			0
$791$		−	7.53789e7i		−	0.152307i
$792$		0			0
$793$	−3.50881e8			−0.703624
$794$		0			0
$795$	−3.48368e8	−	6.68178e7i	−0.693324	−	0.132981i
$796$		0			0
$797$	−5.39457e7	−	3.11455e7i	−0.106557	−	0.0615207i	0.445774	−	0.895145i	$-0.352928\pi$
$797$	−5.39457e7	−	3.11455e7i	−0.106557	−	0.0615207i	−0.552331	+	0.833625i	$0.686262\pi$
$798$		0			0
$799$	−2.17843e8	−	3.77316e8i	−0.427075	−	0.739716i
$800$		0			0
$801$	5.70417e8	+	7.22668e8i	1.10993	+	1.40618i
$802$		0			0
$803$	−2.32282e7	+	1.34108e7i	−0.0448611	+	0.0259006i
$804$		0			0
$805$	−3.19705e7	+	5.53746e7i	−0.0612861	+	0.106151i
$806$		0			0
$807$	−7.33304e8	+	2.54537e8i	−1.39529	+	0.484317i
$808$		0			0
$809$			6.87496e8i			1.29845i	0.760597	+	0.649224i	$0.224906\pi$
$809$			6.87496e8i			1.29845i	−0.760597	+	0.649224i	$0.775094\pi$
$810$		0			0
$811$	1.75961e8			0.329879			0.164939	−	0.986304i	$-0.447257\pi$
$811$	1.75961e8			0.329879			0.164939	+	0.986304i	$0.447257\pi$
$812$		0			0
$813$	−1.43332e8	−	4.12929e8i	−0.266729	−	0.768429i
$814$		0			0
$815$	9.50885e7	+	5.48994e7i	0.175653	+	0.101413i
$816$		0			0
$817$	−1.28107e8	−	2.21888e8i	−0.234913	−	0.406882i
$818$		0			0
$819$	1.45352e8	−	1.14729e8i	0.264587	−	0.208844i
$820$		0			0
$821$	2.05686e8	−	1.18753e8i	0.371686	−	0.214593i	−0.302509	−	0.953147i	$-0.597824\pi$
$821$	2.05686e8	−	1.18753e8i	0.371686	−	0.214593i	0.674195	+	0.738554i	$0.264491\pi$
$822$		0			0
$823$	1.78908e8	−	3.09878e8i	0.320945	−	0.555892i	−0.659739	−	0.751495i	$-0.729333\pi$
$823$	1.78908e8	−	3.09878e8i	0.320945	−	0.555892i	0.980683	+	0.195603i	$0.0626663\pi$
$824$		0			0
$825$	−4.17675e7	+	2.17763e8i	−0.0743835	+	0.387813i
$826$		0			0
$827$			8.43721e8i			1.49170i	0.666113	+	0.745851i	$0.267957\pi$
$827$			8.43721e8i			1.49170i	−0.666113	+	0.745851i	$0.732043\pi$
$828$		0			0
$829$	1.00864e9			1.77040			0.885199	−	0.465212i	$-0.154022\pi$
$829$	1.00864e9			1.77040			0.885199	+	0.465212i	$0.154022\pi$
$830$		0			0
$831$	−2.71136e8	+	3.13477e8i	−0.472482	+	0.546264i
$832$		0			0
$833$	5.22707e8	+	3.01785e8i	0.904322	+	0.522111i
$834$		0			0
$835$	2.66259e8	+	4.61173e8i	0.457345	+	0.792145i
$836$		0			0
$837$	−9.12627e8	+	4.73610e8i	−1.55638	+	0.807689i
$838$		0			0
$839$	−1.25987e8	+	7.27385e7i	−0.213324	+	0.123162i	−0.602855	−	0.797851i	$-0.705970\pi$
$839$	−1.25987e8	+	7.27385e7i	−0.213324	+	0.123162i	0.389531	+	0.921013i	$0.372637\pi$
$840$		0			0
$841$	−2.54447e8	+	4.40715e8i	−0.427769	+	0.740918i
$842$		0			0
$843$	3.85610e8	+	3.33527e8i	0.643674	+	0.556735i
$844$		0			0
$845$			1.24319e8i			0.206047i
$846$		0			0
$847$	−2.01682e8			−0.331908
$848$		0			0
$849$	−5.61953e6	−	1.07784e6i	−0.00918283	−	0.00176129i
$850$		0			0
$851$	3.55215e8	+	2.05083e8i	0.576371	+	0.332768i
$852$		0			0
$853$	−2.57698e8	−	4.46345e8i	−0.415206	−	0.719157i	0.580244	−	0.814442i	$-0.302957\pi$
$853$	−2.57698e8	−	4.46345e8i	−0.415206	−	0.719157i	−0.995450	+	0.0952852i	$0.969624\pi$
$854$		0			0
$855$	−1.22205e8	+	1.77946e7i	−0.195520	+	0.0284702i
$856$		0			0
$857$	−3.77473e8	+	2.17934e8i	−0.599714	+	0.346245i	−0.768929	−	0.639334i	$-0.779210\pi$
$857$	−3.77473e8	+	2.17934e8i	−0.599714	+	0.346245i	0.169215	+	0.985579i	$0.445877\pi$
$858$		0			0
$859$	1.45729e8	−	2.52410e8i	0.229915	−	0.398224i	−0.727868	−	0.685717i	$-0.759489\pi$
$859$	1.45729e8	−	2.52410e8i	0.229915	−	0.398224i	0.957783	+	0.287493i	$0.0928220\pi$
$860$		0			0
$861$	−4.65378e8	+	1.61537e8i	−0.729115	+	0.253083i
$862$		0			0
$863$			9.92214e8i			1.54374i	0.635783	+	0.771868i	$0.280677\pi$
$863$			9.92214e8i			1.54374i	−0.635783	+	0.771868i	$0.719323\pi$
$864$		0			0
$865$	−3.25609e8			−0.503093
$866$		0			0
$867$	−1.51238e8	−	4.35706e8i	−0.232061	−	0.668553i
$868$		0			0
$869$	2.45657e8	+	1.41830e8i	0.374344	+	0.216127i
$870$		0			0
$871$	8.98287e7	+	1.55588e8i	0.135944	+	0.235462i
$872$		0			0
$873$	−5.89265e8	−	2.34679e8i	−0.885662	−	0.352721i
$874$		0			0
$875$	2.18914e8	−	1.26390e8i	0.326776	−	0.188664i
$876$		0			0
$877$	5.93503e8	−	1.02798e9i	0.879881	−	1.52400i	0.0284101	−	0.999596i	$-0.490956\pi$
$877$	5.93503e8	−	1.02798e9i	0.879881	−	1.52400i	0.851471	−	0.524402i	$-0.175711\pi$
$878$		0			0
$879$	7.60181e7	−	3.96335e8i	0.111931	−	0.583574i
$880$		0			0
$881$		−	1.23545e9i		−	1.80675i	−0.428847	−	0.903377i	$-0.641080\pi$
$881$		−	1.23545e9i		−	1.80675i	0.428847	−	0.903377i	$-0.358920\pi$
$882$		0			0
$883$	8.66631e8			1.25879			0.629394	−	0.777087i	$-0.283303\pi$
$883$	8.66631e8			1.25879			0.629394	+	0.777087i	$0.283303\pi$
$884$		0			0
$885$	−1.76251e8	+	2.03774e8i	−0.254273	+	0.293980i
$886$		0			0
$887$	−7.04112e8	−	4.06519e8i	−1.00895	−	0.582519i	−0.0980680	−	0.995180i	$-0.531266\pi$
$887$	−7.04112e8	−	4.06519e8i	−1.00895	−	0.582519i	−0.910885	+	0.412660i	$0.864600\pi$
$888$		0			0
$889$	−8.77703e6	−	1.52023e7i	−0.0124923	−	0.0216373i
$890$		0			0
$891$	1.02766e8	+	3.45393e8i	0.145284	+	0.488293i
$892$		0			0
$893$	−1.67952e8	+	9.69670e7i	−0.235847	+	0.136166i
$894$		0			0
$895$	−7.37431e7	+	1.27727e8i	−0.102861	+	0.178161i
$896$		0			0
$897$	−2.36706e8	−	2.04735e8i	−0.327969	−	0.283671i
$898$		0			0
$899$		−	4.84235e8i		−	0.666464i
$900$		0			0
$901$	−1.42292e9			−1.94539
$902$		0			0
$903$	3.65511e8	+	7.01060e7i	0.496407	+	0.0952121i
$904$		0			0
$905$	2.60133e8	+	1.50188e8i	0.350954	+	0.202623i
$906$		0			0
$907$	−1.06694e7	−	1.84799e7i	−0.0142994	−	0.0247672i	0.858787	−	0.512332i	$-0.171218\pi$
$907$	−1.06694e7	−	1.84799e7i	−0.0142994	−	0.0247672i	−0.873087	+	0.487565i	$0.837885\pi$
$908$		0			0
$909$	3.84072e8	−	9.64385e8i	0.511354	−	1.28398i
$910$		0			0
$911$	1.96531e8	−	1.13467e8i	0.259942	−	0.150078i	−0.364366	−	0.931256i	$-0.618714\pi$
$911$	1.96531e8	−	1.13467e8i	0.259942	−	0.150078i	0.624308	+	0.781178i	$0.285381\pi$
$912$		0			0
$913$	8.62668e6	−	1.49419e7i	0.0113353	−	0.0196332i
$914$		0			0
$915$	−3.21116e8	+	1.11463e8i	−0.419178	+	0.145501i
$916$		0			0
$917$			6.82531e8i			0.885145i
$918$		0			0
$919$	1.26885e9			1.63479			0.817396	−	0.576077i	$-0.195417\pi$
$919$	1.26885e9			1.63479			0.817396	+	0.576077i	$0.195417\pi$
$920$		0			0
$921$	3.56122e8	+	1.02596e9i	0.455848	+	1.31327i
$922$		0			0
$923$	5.83911e8	+	3.37121e8i	0.742577	+	0.428727i
$924$		0			0
$925$	−3.54025e8	−	6.13190e8i	−0.447311	−	0.774765i
$926$		0			0
$927$	−1.70710e8	−	1.17236e9i	−0.214299	−	1.47170i
$928$		0			0
$929$	4.83075e8	−	2.78903e8i	0.602514	−	0.347862i	−0.167516	−	0.985869i	$-0.553575\pi$
$929$	4.83075e8	−	2.78903e8i	0.602514	−	0.347862i	0.770030	+	0.638008i	$0.220241\pi$
$930$		0			0
$931$	1.34331e8	−	2.32669e8i	0.166467	−	0.288329i
$932$		0			0
$933$	−2.82910e6	+	1.47501e7i	−0.00348340	+	0.0181614i
$934$		0			0
$935$		−	2.58058e8i		−	0.315705i
$936$		0			0
$937$	−1.39223e8			−0.169236			−0.0846182	−	0.996413i	$-0.526967\pi$
$937$	−1.39223e8			−0.169236			−0.0846182	+	0.996413i	$0.526967\pi$
$938$		0			0
$939$	8.86271e8	−	1.02467e9i	1.07046	−	1.23762i
$940$		0			0
$941$	3.52077e8	+	2.03272e8i	0.422541	+	0.243954i	0.696164	−	0.717883i	$-0.254889\pi$
$941$	3.52077e8	+	2.03272e8i	0.422541	+	0.243954i	−0.273623	+	0.961837i	$0.588222\pi$
$942$		0			0
$943$	4.16284e8	+	7.21025e8i	0.496426	+	0.859835i
$944$		0			0
$945$	9.65762e7	−	1.51170e8i	0.114439	−	0.179130i
$946$		0			0
$947$	−8.67192e8	+	5.00674e8i	−1.02109	+	0.589529i	−0.914420	−	0.404766i	$-0.867353\pi$
$947$	−8.67192e8	+	5.00674e8i	−1.02109	+	0.589529i	−0.106673	+	0.994294i	$0.534020\pi$
$948$		0			0
$949$	−3.26757e7	+	5.65959e7i	−0.0382319	+	0.0662196i
$950$		0			0
$951$	9.54967e8	+	8.25982e8i	1.11032	+	0.960349i
$952$		0			0
$953$		−	1.33438e9i		−	1.54170i	−0.637017	−	0.770849i	$-0.719832\pi$
$953$		−	1.33438e9i		−	1.54170i	0.637017	−	0.770849i	$-0.280168\pi$
$954$		0			0
$955$	−3.21627e8			−0.369268
$956$		0			0
$957$	−1.66673e8	−	3.19684e7i	−0.190165	−	0.0364741i
$958$		0			0
$959$	−3.78769e8	−	2.18682e8i	−0.429456	−	0.247946i
$960$		0			0
$961$	−9.20650e8	−	1.59461e9i	−1.03735	−	1.79674i
$962$		0			0
$963$	5.00355e8	+	6.33906e8i	0.560273	+	0.709816i
$964$		0			0
$965$	−2.89302e8	+	1.67029e8i	−0.321936	+	0.185870i
$966$		0			0
$967$	−8.04831e8	+	1.39401e9i	−0.890072	+	1.54165i	−0.0502847	+	0.998735i	$0.516013\pi$
$967$	−8.04831e8	+	1.39401e9i	−0.890072	+	1.54165i	−0.839787	+	0.542915i	$0.817320\pi$
$968$		0			0
$969$	−4.67995e8	+	1.62446e8i	−0.514363	+	0.178540i
$970$		0			0
$971$		−	1.52289e9i		−	1.66345i	−0.555185	−	0.831727i	$-0.687352\pi$
$971$		−	1.52289e9i		−	1.66345i	0.555185	−	0.831727i	$-0.312648\pi$
$972$		0			0
$973$	6.24030e8			0.677434
$974$		0			0
$975$	1.77158e8	+	5.10381e8i	0.191138	+	0.550657i
$976$		0			0
$977$	1.10021e8	+	6.35204e7i	0.117975	+	0.0681129i	0.557826	−	0.829958i	$-0.311636\pi$
$977$	1.10021e8	+	6.35204e7i	0.117975	+	0.0681129i	−0.439851	+	0.898071i	$0.644969\pi$
$978$		0			0
$979$	4.28176e8	+	7.41622e8i	0.456325	+	0.790377i
$980$		0			0
$981$	2.93842e8	−	2.31936e8i	0.311249	−	0.245675i
$982$		0			0
$983$	−7.66278e8	+	4.42411e8i	−0.806726	+	0.465763i	−0.845818	−	0.533472i	$-0.820887\pi$
$983$	−7.66278e8	+	4.42411e8i	−0.806726	+	0.465763i	0.0390918	+	0.999236i	$0.487554\pi$
$984$		0			0
$985$	−4.51280e7	+	7.81641e7i	−0.0472213	+	0.0817897i
$986$		0			0
$987$	5.30647e7	−	2.76663e8i	0.0551892	−	0.287740i
$988$		0			0
$989$		−	6.29009e8i		−	0.650232i
$990$		0			0
$991$	−1.07134e9			−1.10079			−0.550396	−	0.834904i	$-0.685523\pi$
$991$	−1.07134e9			−1.10079			−0.550396	+	0.834904i	$0.685523\pi$
$992$		0			0
$993$	6.08944e7	−	7.04036e7i	0.0621913	−	0.0719030i
$994$		0			0
$995$	−4.95448e8	−	2.86047e8i	−0.502955	−	0.290381i
$996$		0			0
$997$	7.94263e8	+	1.37570e9i	0.801455	+	1.38816i	0.918659	+	0.395052i	$0.129274\pi$
$997$	7.94263e8	+	1.37570e9i	0.801455	+	1.38816i	−0.117204	+	0.993108i	$0.537393\pi$
$998$		0			0
$999$	−9.69718e8	−	6.19513e8i	−0.972633	−	0.621375i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.7.g.a.5.4	✓	12
3.2	odd	2		108.7.g.a.17.3		12
4.3	odd	2		144.7.q.b.113.3		12
9.2	odd	6	inner	36.7.g.a.29.4	yes	12
9.4	even	3		324.7.c.b.161.5		12
9.5	odd	6		324.7.c.b.161.8		12
9.7	even	3		108.7.g.a.89.3		12
12.11	even	2		432.7.q.c.17.3		12
36.7	odd	6		432.7.q.c.305.3		12
36.11	even	6		144.7.q.b.65.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.7.g.a.5.4	✓	12	1.1	even	1	trivial
36.7.g.a.29.4	yes	12	9.2	odd	6	inner
108.7.g.a.17.3		12	3.2	odd	2
108.7.g.a.89.3		12	9.7	even	3
144.7.q.b.65.3		12	36.11	even	6
144.7.q.b.113.3		12	4.3	odd	2
324.7.c.b.161.5		12	9.4	even	3
324.7.c.b.161.8		12	9.5	odd	6
432.7.q.c.17.3		12	12.11	even	2
432.7.q.c.305.3		12	36.7	odd	6

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:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	36.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(8.85375 + 25.5071i) q^{3} +(51.3356 + 29.6386i) q^{5} +(76.8738 + 133.149i) q^{7} +(-572.222 + 451.667i) q^{9} +O(q^{10})\)	\(q+(8.85375 + 25.5071i) q^{3} +(51.3356 + 29.6386i) q^{5} +(76.8738 + 133.149i) q^{7} +(-572.222 + 451.667i) q^{9} +(-587.230 + 339.038i) q^{11} +(-826.070 + 1430.80i) q^{13} +(-301.482 + 1571.84i) q^{15} +6420.23i q^{17} +2857.79 q^{19} +(-2715.63 + 3139.70i) q^{21} +(6075.95 + 3507.95i) q^{23} +(-6055.60 - 10488.6i) q^{25} +(-16587.0 - 10596.8i) q^{27} +(8027.87 - 4634.89i) q^{29} +(26119.0 - 45239.4i) q^{31} +(-13847.1 - 11976.8i) q^{33} +9113.74i q^{35} +58462.5 q^{37} +(-43809.2 - 8402.73i) q^{39} +(102770. + 59334.4i) q^{41} +(-44827.4 - 77643.4i) q^{43} +(-42762.2 + 6226.71i) q^{45} +(-58769.9 + 33930.8i) q^{47} +(47005.3 - 81415.6i) q^{49} +(-163761. + 56843.1i) q^{51} +221631. i q^{53} -40194.5 q^{55} +(25302.1 + 72893.8i) q^{57} +(145784. + 84168.7i) q^{59} +(106190. + 183926. i) q^{61} +(-104128. - 41469.6i) q^{63} +(-84813.7 + 48967.2i) q^{65} +(54371.1 - 94173.5i) q^{67} +(-35682.6 + 186038. i) q^{69} -408102. i q^{71} +39555.6 q^{73} +(213919. - 247324. i) q^{75} +(-90285.3 - 52126.2i) q^{77} +(-209166. - 362286. i) q^{79} +(123435. - 516907. i) q^{81} +(-22035.7 + 12722.3i) q^{83} +(-190287. + 329586. i) q^{85} +(189299. + 163731. i) q^{87} -1.26291e6i q^{89} -254013. q^{91} +(1.38518e6 + 265680. i) q^{93} +(146706. + 84700.9i) q^{95} +(435032. + 753498. i) q^{97} +(182894. - 459237. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} - 216 q^{5} - 120 q^{7} + 174 q^{9}+O(q^{10}) \) 12 * q + 6 * q^3 - 216 * q^5 - 120 * q^7 + 174 * q^9	\( 12 q + 6 q^{3} - 216 q^{5} - 120 q^{7} + 174 q^{9} - 2214 q^{11} - 840 q^{13} - 5040 q^{15} - 12900 q^{19} + 5892 q^{21} - 8208 q^{23} + 24078 q^{25} + 40176 q^{27} - 29268 q^{29} + 3120 q^{31} + 2718 q^{33} + 12768 q^{37} + 223812 q^{39} - 16578 q^{41} + 71430 q^{43} - 61524 q^{45} - 329508 q^{47} - 238914 q^{49} - 153198 q^{51} + 5400 q^{55} + 652854 q^{57} + 428058 q^{59} - 93576 q^{61} - 493392 q^{63} - 1426464 q^{65} + 104334 q^{67} - 860256 q^{69} + 221820 q^{73} + 2598978 q^{75} + 3461184 q^{77} + 123468 q^{79} - 1125882 q^{81} - 2901420 q^{83} + 377568 q^{85} - 4083948 q^{87} + 91488 q^{91} + 6111144 q^{93} + 7249716 q^{95} - 1033482 q^{97} - 3503484 q^{99}+O(q^{100}) \) 12 * q + 6 * q^3 - 216 * q^5 - 120 * q^7 + 174 * q^9 - 2214 * q^11 - 840 * q^13 - 5040 * q^15 - 12900 * q^19 + 5892 * q^21 - 8208 * q^23 + 24078 * q^25 + 40176 * q^27 - 29268 * q^29 + 3120 * q^31 + 2718 * q^33 + 12768 * q^37 + 223812 * q^39 - 16578 * q^41 + 71430 * q^43 - 61524 * q^45 - 329508 * q^47 - 238914 * q^49 - 153198 * q^51 + 5400 * q^55 + 652854 * q^57 + 428058 * q^59 - 93576 * q^61 - 493392 * q^63 - 1426464 * q^65 + 104334 * q^67 - 860256 * q^69 + 221820 * q^73 + 2598978 * q^75 + 3461184 * q^77 + 123468 * q^79 - 1125882 * q^81 - 2901420 * q^83 + 377568 * q^85 - 4083948 * q^87 + 91488 * q^91 + 6111144 * q^93 + 7249716 * q^95 - 1033482 * q^97 - 3503484 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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