LMFDB - Embedded newform 324.2.i.a.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,2,Mod(37,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("324.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	324.i (of order $9$, degree $6$, not 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:	$2.58715302549$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$3$ over $\Q(\zeta_{9})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 $ x^18 - 6x^17 + 24x^16 - 66x^15 + 153x^14 - 315x^13 + 651x^12 - 1350x^11 + 2700x^10 - 4941x^9 + 8100x^8 - 12150x^7 + 17577x^6 - 25515x^5 + 37179x^4 - 48114x^3 + 52488x^2 - 39366*x + 19683
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			37.3
Root			$0.472963 + 1.66622i$ of defining polynomial
Character	$\chi$	$=$	324.37
Dual form			324.2.i.a.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.94709 - 1.43662i) q^{5} +(0.610312 + 3.46125i) q^{7} +O(q^{10})$	$q+(3.94709 - 1.43662i) q^{5} +(0.610312 + 3.46125i) q^{7} +(-1.73646 - 0.632019i) q^{11} +(-1.78502 - 1.49781i) q^{13} +(0.799928 - 1.38552i) q^{17} +(2.31046 + 4.00184i) q^{19} +(0.308317 - 1.74855i) q^{23} +(9.68544 - 8.12705i) q^{25} +(-0.882314 + 0.740350i) q^{29} +(0.322800 - 1.83069i) q^{31} +(7.38148 + 12.7851i) q^{35} +(-4.38364 + 7.59269i) q^{37} +(-2.98440 - 2.50421i) q^{41} +(-2.41848 - 0.880255i) q^{43} +(-1.29725 - 7.35705i) q^{47} +(-5.02994 + 1.83075i) q^{49} -8.02417 q^{53} -7.76194 q^{55} +(1.15006 - 0.418589i) q^{59} +(-0.754920 - 4.28136i) q^{61} +(-9.19743 - 3.34759i) q^{65} +(-4.86356 - 4.08101i) q^{67} +(-0.871328 + 1.50918i) q^{71} +(-1.37908 - 2.38864i) q^{73} +(1.12780 - 6.39605i) q^{77} +(-7.63735 + 6.40849i) q^{79} +(-8.65194 + 7.25984i) q^{83} +(1.16692 - 6.61795i) q^{85} +(2.71167 + 4.69675i) q^{89} +(4.09488 - 7.09254i) q^{91} +(14.8688 + 12.4764i) q^{95} +(11.3643 + 4.13626i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 3 q^{5}+O(q^{10}) $ 18 * q - 3 * q^5	$ 18 q - 3 q^{5} - 3 q^{11} + 12 q^{17} + 30 q^{23} + 9 q^{25} + 24 q^{29} + 9 q^{31} + 21 q^{35} - 21 q^{41} - 9 q^{43} - 45 q^{47} - 18 q^{49} - 66 q^{53} - 60 q^{59} - 18 q^{61} - 33 q^{65} - 27 q^{67} + 12 q^{71} + 9 q^{73} + 75 q^{77} - 36 q^{79} + 45 q^{83} - 36 q^{85} + 48 q^{89} + 9 q^{91} - 6 q^{95} - 27 q^{97}+O(q^{100}) $ 18 * q - 3 * q^5 - 3 * q^11 + 12 * q^17 + 30 * q^23 + 9 * q^25 + 24 * q^29 + 9 * q^31 + 21 * q^35 - 21 * q^41 - 9 * q^43 - 45 * q^47 - 18 * q^49 - 66 * q^53 - 60 * q^59 - 18 * q^61 - 33 * q^65 - 27 * q^67 + 12 * q^71 + 9 * q^73 + 75 * q^77 - 36 * q^79 + 45 * q^83 - 36 * q^85 + 48 * q^89 + 9 * q^91 - 6 * q^95 - 27 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{7}{9}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.94709	−	1.43662i	1.76519	−	0.642478i	0.765195	−	0.643799i	$-0.222643\pi$
$5$	3.94709	−	1.43662i	1.76519	−	0.642478i	0.999999	+	0.00132064i	$0.000420373\pi$
$6$		0			0
$7$	0.610312	+	3.46125i	0.230676	+	1.30823i	0.851531	+	0.524305i	$0.175675\pi$
$7$	0.610312	+	3.46125i	0.230676	+	1.30823i	−0.620854	+	0.783926i	$0.713214\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.73646	−	0.632019i	−0.523562	−	0.190561i	0.0666996	−	0.997773i	$-0.478753\pi$
$11$	−1.73646	−	0.632019i	−0.523562	−	0.190561i	−0.590261	+	0.807212i	$0.700975\pi$
$12$		0			0
$13$	−1.78502	−	1.49781i	−0.495075	−	0.415418i	0.360766	−	0.932656i	$-0.382515\pi$
$13$	−1.78502	−	1.49781i	−0.495075	−	0.415418i	−0.855841	+	0.517239i	$0.826960\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.799928	−	1.38552i	0.194011	−	0.336037i	−0.752565	−	0.658518i	$-0.771184\pi$
$17$	0.799928	−	1.38552i	0.194011	−	0.336037i	0.946576	+	0.322481i	$0.104517\pi$
$18$		0			0
$19$	2.31046	+	4.00184i	0.530056	+	0.918085i	0.999385	+	0.0350612i	$0.0111626\pi$
$19$	2.31046	+	4.00184i	0.530056	+	0.918085i	−0.469329	+	0.883024i	$0.655504\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.308317	−	1.74855i	0.0642885	−	0.364598i	−0.935644	−	0.352946i	$-0.885180\pi$
$23$	0.308317	−	1.74855i	0.0642885	−	0.364598i	0.999932	−	0.0116518i	$-0.00370898\pi$
$24$		0			0
$25$	9.68544	−	8.12705i	1.93709	−	1.62541i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.882314	+	0.740350i	−0.163842	+	0.137479i	−0.721022	−	0.692912i	$-0.756328\pi$
$29$	−0.882314	+	0.740350i	−0.163842	+	0.137479i	0.557181	+	0.830391i	$0.311883\pi$
$30$		0			0
$31$	0.322800	−	1.83069i	0.0579766	−	0.328802i	−0.942000	−	0.335612i	$-0.891057\pi$
$31$	0.322800	−	1.83069i	0.0579766	−	0.328802i	0.999977	+	0.00681062i	$0.00216791\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.38148	+	12.7851i	1.24770	+	2.16108i
$36$		0			0
$37$	−4.38364	+	7.59269i	−0.720666	+	1.24823i	0.240067	+	0.970756i	$0.422830\pi$
$37$	−4.38364	+	7.59269i	−0.720666	+	1.24823i	−0.960733	+	0.277474i	$0.910503\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.98440	−	2.50421i	−0.466085	−	0.391092i	0.379279	−	0.925282i	$-0.376172\pi$
$41$	−2.98440	−	2.50421i	−0.466085	−	0.391092i	−0.845364	+	0.534191i	$0.820616\pi$
$42$		0			0
$43$	−2.41848	−	0.880255i	−0.368815	−	0.134238i	0.150962	−	0.988540i	$-0.451763\pi$
$43$	−2.41848	−	0.880255i	−0.368815	−	0.134238i	−0.519777	+	0.854302i	$0.673985\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.29725	−	7.35705i	−0.189223	−	1.07314i	−0.920409	−	0.390958i	$-0.872144\pi$
$47$	−1.29725	−	7.35705i	−0.189223	−	1.07314i	0.731186	−	0.682178i	$-0.238967\pi$
$48$		0			0
$49$	−5.02994	+	1.83075i	−0.718563	+	0.261535i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.02417			−1.10220			−0.551102	−	0.834438i	$-0.685793\pi$
$53$	−8.02417			−1.10220			−0.551102	+	0.834438i	$0.685793\pi$
$54$		0			0
$55$	−7.76194			−1.04662
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.15006	−	0.418589i	0.149726	−	0.0544957i	−0.266070	−	0.963954i	$-0.585725\pi$
$59$	1.15006	−	0.418589i	0.149726	−	0.0544957i	0.415796	+	0.909458i	$0.363503\pi$
$60$		0			0
$61$	−0.754920	−	4.28136i	−0.0966575	−	0.548172i	−0.994227	−	0.107299i	$-0.965780\pi$
$61$	−0.754920	−	4.28136i	−0.0966575	−	0.548172i	0.897569	−	0.440873i	$-0.145331\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.19743	−	3.34759i	−1.14080	−	0.415218i
$66$		0			0
$67$	−4.86356	−	4.08101i	−0.594178	−	0.498574i	0.295390	−	0.955377i	$-0.404550\pi$
$67$	−4.86356	−	4.08101i	−0.594178	−	0.498574i	−0.889568	+	0.456802i	$0.848995\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.871328	+	1.50918i	−0.103408	+	0.179107i	−0.913087	−	0.407766i	$-0.866308\pi$
$71$	−0.871328	+	1.50918i	−0.103408	+	0.179107i	0.809679	+	0.586873i	$0.199641\pi$
$72$		0			0
$73$	−1.37908	−	2.38864i	−0.161409	−	0.279569i	0.773965	−	0.633228i	$-0.218271\pi$
$73$	−1.37908	−	2.38864i	−0.161409	−	0.279569i	−0.935374	+	0.353659i	$0.884937\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.12780	−	6.39605i	0.128524	−	0.728897i
$78$		0			0
$79$	−7.63735	+	6.40849i	−0.859268	+	0.721012i	−0.961810	−	0.273717i	$-0.911747\pi$
$79$	−7.63735	+	6.40849i	−0.859268	+	0.721012i	0.102542	+	0.994729i	$0.467302\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.65194	+	7.25984i	−0.949674	+	0.796871i	−0.979243	−	0.202692i	$-0.935031\pi$
$83$	−8.65194	+	7.25984i	−0.949674	+	0.796871i	0.0295686	+	0.999563i	$0.490587\pi$
$84$		0			0
$85$	1.16692	−	6.61795i	0.126571	−	0.717818i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.71167	+	4.69675i	0.287436	+	0.497854i	0.973197	−	0.229973i	$-0.0738637\pi$
$89$	2.71167	+	4.69675i	0.287436	+	0.497854i	−0.685761	+	0.727827i	$0.740530\pi$
$90$		0			0
$91$	4.09488	−	7.09254i	0.429260	−	0.743500i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.8688	+	12.4764i	1.52550	+	1.28005i
$96$		0			0
$97$	11.3643	+	4.13626i	1.15387	+	0.419974i	0.846903	−	0.531747i	$-0.178464\pi$
$97$	11.3643	+	4.13626i	1.15387	+	0.419974i	0.306966	+	0.951721i	$0.400686\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.45951	+	13.9486i	0.244731	+	1.38794i	0.821117	+	0.570760i	$0.193351\pi$
$101$	2.45951	+	13.9486i	0.244731	+	1.38794i	−0.576386	+	0.817178i	$0.695538\pi$
$102$		0			0
$103$	−1.41828	+	0.516213i	−0.139748	+	0.0508640i	−0.410947	−	0.911659i	$-0.634802\pi$
$103$	−1.41828	+	0.516213i	−0.139748	+	0.0508640i	0.271199	+	0.962523i	$0.412580\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.4114			−1.87658			−0.938288	−	0.345856i	$-0.887589\pi$
$107$	−19.4114			−1.87658			−0.938288	+	0.345856i	$0.887589\pi$
$108$		0			0
$109$	15.2590			1.46155			0.730775	−	0.682619i	$-0.239159\pi$
$109$	15.2590			1.46155			0.730775	+	0.682619i	$0.239159\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.6990	+	3.89411i	−1.00647	+	0.366327i	−0.792078	−	0.610420i	$-0.791001\pi$
$113$	−10.6990	+	3.89411i	−1.00647	+	0.366327i	−0.214397	+	0.976747i	$0.568778\pi$
$114$		0			0
$115$	−1.29506	−	7.34463i	−0.120765	−	0.684890i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.28382	+	1.92315i	0.484367	+	0.176295i
$120$		0			0
$121$	−5.81065	−	4.87572i	−0.528241	−	0.443247i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	16.0528	−	27.8042i	1.43581	−	2.48689i
$126$		0			0
$127$	0.804999	+	1.39430i	0.0714321	+	0.123724i	0.899529	−	0.436861i	$-0.143910\pi$
$127$	0.804999	+	1.39430i	0.0714321	+	0.123724i	−0.828097	+	0.560585i	$0.810576\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.914583	−	5.18686i	0.0799075	−	0.453178i	−0.918432	−	0.395578i	$-0.870544\pi$
$131$	0.914583	−	5.18686i	0.0799075	−	0.453178i	0.998340	−	0.0575995i	$-0.0183446\pi$
$132$		0			0
$133$	−12.4413	+	10.4395i	−1.07879	+	0.905216i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.9771	−	10.0500i	1.02328	−	0.858630i	0.0332397	−	0.999447i	$-0.489418\pi$
$137$	11.9771	−	10.0500i	1.02328	−	0.858630i	0.990036	+	0.140818i	$0.0449731\pi$
$138$		0			0
$139$	−2.12790	+	12.0679i	−0.180486	+	1.02359i	0.751133	+	0.660151i	$0.229508\pi$
$139$	−2.12790	+	12.0679i	−0.180486	+	1.02359i	−0.931619	+	0.363436i	$0.881603\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.15297	+	3.72905i	0.180040	+	0.311839i
$144$		0			0
$145$	−2.41897	+	4.18978i	−0.200885	+	0.347943i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.0411	−	13.4601i	−1.31414	−	1.10269i	−0.987513	−	0.157540i	$-0.949644\pi$
$149$	−16.0411	−	13.4601i	−1.31414	−	1.10269i	−0.326626	−	0.945154i	$-0.605912\pi$
$150$		0			0
$151$	9.06208	+	3.29833i	0.737462	+	0.268414i	0.683320	−	0.730119i	$-0.260535\pi$
$151$	9.06208	+	3.29833i	0.737462	+	0.268414i	0.0541419	+	0.998533i	$0.482758\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.35589	−	7.68965i	−0.108908	−	0.617647i
$156$		0			0
$157$	11.0515	−	4.02243i	0.882008	−	0.321025i	0.138988	−	0.990294i	$-0.455615\pi$
$157$	11.0515	−	4.02243i	0.882008	−	0.321025i	0.743020	+	0.669269i	$0.233393\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.24034			0.491808
$162$		0			0
$163$	−14.3539			−1.12429			−0.562143	−	0.827040i	$-0.690023\pi$
$163$	−14.3539			−1.12429			−0.562143	+	0.827040i	$0.690023\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.95006	−	2.89359i	0.615194	−	0.223912i	−0.0155803	−	0.999879i	$-0.504960\pi$
$167$	7.95006	−	2.89359i	0.615194	−	0.223912i	0.630774	+	0.775966i	$0.282737\pi$
$168$		0			0
$169$	−1.31456	−	7.45526i	−0.101120	−	0.573482i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	23.5461	+	8.57008i	1.79018	+	0.651571i	0.999211	+	0.0397102i	$0.0126435\pi$
$173$	23.5461	+	8.57008i	1.79018	+	0.651571i	0.790965	+	0.611861i	$0.209579\pi$
$174$		0			0
$175$	34.0409	+	28.5637i	2.57325	+	2.15921i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.84160	−	8.38590i	0.361878	−	0.626792i	−0.626392	−	0.779508i	$-0.715469\pi$
$179$	4.84160	−	8.38590i	0.361878	−	0.626792i	0.988270	+	0.152717i	$0.0488023\pi$
$180$		0			0
$181$	0.302082	+	0.523221i	0.0224535	+	0.0388907i	0.877034	−	0.480429i	$-0.159519\pi$
$181$	0.302082	+	0.523221i	0.0224535	+	0.0388907i	−0.854580	+	0.519319i	$0.826186\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.39479	+	36.2667i	−0.470155	+	2.66638i
$186$		0			0
$187$	−2.26471	+	1.90032i	−0.165612	+	0.138965i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.8632	−	13.3108i	1.14782	−	0.963138i	0.148157	−	0.988964i	$-0.452666\pi$
$191$	15.8632	−	13.3108i	1.14782	−	0.963138i	0.999666	+	0.0258260i	$0.00822160\pi$
$192$		0			0
$193$	−1.93360	+	10.9660i	−0.139184	+	0.789350i	0.832671	+	0.553768i	$0.186811\pi$
$193$	−1.93360	+	10.9660i	−0.139184	+	0.789350i	−0.971855	+	0.235582i	$0.924300\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.31196	+	10.9326i	0.449709	+	0.778918i	0.998367	−	0.0571286i	$-0.0181945\pi$
$197$	6.31196	+	10.9326i	0.449709	+	0.778918i	−0.548658	+	0.836047i	$0.684861\pi$
$198$		0			0
$199$	10.5243	−	18.2286i	0.746049	−	1.29219i	−0.203654	−	0.979043i	$-0.565282\pi$
$199$	10.5243	−	18.2286i	0.746049	−	1.29219i	0.949703	−	0.313152i	$-0.101385\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.10102	−	2.60207i	−0.217649	−	0.182629i
$204$		0			0
$205$	−15.3773	−	5.59688i	−1.07400	−	0.390903i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.48278	−	8.40928i	−0.102566	−	0.581682i
$210$		0			0
$211$	20.8317	−	7.58212i	1.43411	−	0.521975i	0.496006	−	0.868319i	$-0.334799\pi$
$211$	20.8317	−	7.58212i	1.43411	−	0.521975i	0.938108	+	0.346344i	$0.112577\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.8106			−0.737275
$216$		0			0
$217$	6.53349			0.443522
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.50312	+	1.27503i	−0.235646	+	0.0857680i
$222$		0			0
$223$	2.39430	+	13.5787i	0.160334	+	0.909299i	0.953746	+	0.300614i	$0.0971916\pi$
$223$	2.39430	+	13.5787i	0.160334	+	0.909299i	−0.793412	+	0.608685i	$0.791697\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.73632	−	3.17976i	−0.579850	−	0.211048i	0.0354095	−	0.999373i	$-0.488726\pi$
$227$	−8.73632	−	3.17976i	−0.579850	−	0.211048i	−0.615259	+	0.788325i	$0.710949\pi$
$228$		0			0
$229$	−0.113444	−	0.0951912i	−0.00749662	−	0.00629041i	0.639032	−	0.769180i	$-0.279335\pi$
$229$	−0.113444	−	0.0951912i	−0.00749662	−	0.00629041i	−0.646528	+	0.762890i	$0.723780\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.824403	+	1.42791i	−0.0540084	+	0.0935454i	−0.891766	−	0.452498i	$-0.850533\pi$
$233$	−0.824403	+	1.42791i	−0.0540084	+	0.0935454i	0.837757	+	0.546043i	$0.183866\pi$
$234$		0			0
$235$	−15.6897	−	27.1753i	−1.02348	−	1.77272i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.11203	−	11.9779i	0.136616	−	0.774787i	−0.837105	−	0.547042i	$-0.815754\pi$
$239$	2.11203	−	11.9779i	0.136616	−	0.774787i	0.973721	−	0.227745i	$-0.0731351\pi$
$240$		0			0
$241$	0.743637	−	0.623986i	0.0479019	−	0.0401945i	−0.618522	−	0.785767i	$-0.712268\pi$
$241$	0.743637	−	0.623986i	0.0479019	−	0.0401945i	0.666424	+	0.745573i	$0.267824\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.2235	+	14.4523i	−1.10037	+	0.923322i
$246$		0			0
$247$	1.86977	−	10.6040i	0.118971	−	0.674716i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.1814	−	21.0988i	−0.768884	−	1.33175i	−0.938168	−	0.346180i	$-0.887479\pi$
$251$	−12.1814	−	21.0988i	−0.768884	−	1.33175i	0.169284	−	0.985567i	$-0.445855\pi$
$252$		0			0
$253$	−1.64050	+	2.84142i	−0.103137	+	0.178639i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.702943	+	0.589839i	0.0438484	+	0.0367931i	0.664448	−	0.747334i	$-0.268667\pi$
$257$	0.702943	+	0.589839i	0.0438484	+	0.0367931i	−0.620600	+	0.784127i	$0.713111\pi$
$258$		0			0
$259$	−28.9556	−	10.5390i	−1.79921	−	0.654860i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.07212	+	23.0941i	0.251097	+	1.42404i	0.805895	+	0.592059i	$0.201685\pi$
$263$	4.07212	+	23.0941i	0.251097	+	1.42404i	−0.554797	+	0.831986i	$0.687204\pi$
$264$		0			0
$265$	−31.6722	+	11.5277i	−1.94561	+	0.708142i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−16.5865			−1.01130			−0.505649	−	0.862739i	$-0.668747\pi$
$269$	−16.5865			−1.01130			−0.505649	+	0.862739i	$0.668747\pi$
$270$		0			0
$271$	17.5443			1.06574			0.532869	−	0.846198i	$-0.321114\pi$
$271$	17.5443			1.06574			0.532869	+	0.846198i	$0.321114\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.9548	+	7.99089i	−1.32392	+	0.481869i
$276$		0			0
$277$	4.20802	+	23.8648i	0.252835	+	1.43390i	0.801569	+	0.597902i	$0.203999\pi$
$277$	4.20802	+	23.8648i	0.252835	+	1.43390i	−0.548734	+	0.835997i	$0.684890\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.8477	+	7.22397i	1.18401	+	0.430946i	0.857618	−	0.514287i	$-0.171943\pi$
$281$	19.8477	+	7.22397i	1.18401	+	0.430946i	0.326397	+	0.945233i	$0.394165\pi$
$282$		0			0
$283$	−19.7119	−	16.5402i	−1.17175	−	0.983214i	−0.171751	−	0.985140i	$-0.554942\pi$
$283$	−19.7119	−	16.5402i	−1.17175	−	0.983214i	−0.999998	+	0.00192598i	$0.999387\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.84628	−	11.8581i	0.404123	−	0.699962i
$288$		0			0
$289$	7.22023	+	12.5058i	0.424720	+	0.735636i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.17783	+	18.0224i	−0.185651	+	1.05288i	0.739465	+	0.673195i	$0.235078\pi$
$293$	−3.17783	+	18.0224i	−0.185651	+	1.05288i	−0.925116	+	0.379684i	$0.876033\pi$
$294$		0			0
$295$	3.93806	−	3.30442i	0.229282	−	0.192391i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.16935	+	2.65940i	−0.183288	+	0.153797i
$300$		0			0
$301$	1.57076	−	8.90821i	0.0905369	−	0.513460i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.13045	−	15.8144i	−0.522808	−	0.905530i
$306$		0			0
$307$	6.26334	−	10.8484i	0.357468	−	0.619152i	−0.630069	−	0.776539i	$-0.716974\pi$
$307$	6.26334	−	10.8484i	0.357468	−	0.619152i	0.987537	+	0.157387i	$0.0503069\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.00776	+	5.04111i	0.340669	+	0.285855i	0.797030	−	0.603939i	$-0.206403\pi$
$311$	6.00776	+	5.04111i	0.340669	+	0.285855i	−0.456361	+	0.889795i	$0.650848\pi$
$312$		0			0
$313$	−6.04147	−	2.19892i	−0.341484	−	0.124290i	0.165585	−	0.986196i	$-0.447049\pi$
$313$	−6.04147	−	2.19892i	−0.341484	−	0.124290i	−0.507069	+	0.861905i	$0.669271\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.31089	+	18.7770i	0.185958	+	1.05462i	0.924718	+	0.380653i	$0.124301\pi$
$317$	3.31089	+	18.7770i	0.185958	+	1.05462i	−0.738760	+	0.673969i	$0.764588\pi$
$318$		0			0
$319$	2.00002	−	0.727946i	0.111979	−	0.0407572i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.39281			0.411347
$324$		0			0
$325$	−29.4615			−1.63423
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.6729	−	8.98020i	1.36026	−	0.495094i
$330$		0			0
$331$	2.61923	+	14.8544i	0.143966	+	0.816472i	0.968191	+	0.250210i	$0.0804998\pi$
$331$	2.61923	+	14.8544i	0.143966	+	0.816472i	−0.824225	+	0.566262i	$0.808389\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−25.0598	−	9.12102i	−1.36916	−	0.498334i
$336$		0			0
$337$	−2.78707	−	2.33863i	−0.151821	−	0.127393i	0.563713	−	0.825971i	$-0.309372\pi$
$337$	−2.78707	−	2.33863i	−0.151821	−	0.127393i	−0.715534	+	0.698577i	$0.753817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.71756	+	2.97490i	−0.0930111	+	0.161100i
$342$		0			0
$343$	2.89475	+	5.01386i	0.156302	+	0.270723i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.96153	−	11.1244i	0.105300	−	0.597187i	−0.885800	−	0.464068i	$-0.846389\pi$
$347$	1.96153	−	11.1244i	0.105300	−	0.597187i	0.991100	−	0.133120i	$-0.0424994\pi$
$348$		0			0
$349$	20.4174	−	17.1322i	1.09292	−	0.917065i	0.0959872	−	0.995383i	$-0.469399\pi$
$349$	20.4174	−	17.1322i	1.09292	−	0.917065i	0.996928	+	0.0783174i	$0.0249548\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.46470	+	1.22903i	−0.0779579	+	0.0654144i	−0.680934	−	0.732345i	$-0.738426\pi$
$353$	−1.46470	+	1.22903i	−0.0779579	+	0.0654144i	0.602976	+	0.797759i	$0.293981\pi$
$354$		0			0
$355$	−1.27108	+	7.20866i	−0.0674620	+	0.382596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	13.2372	+	22.9275i	0.698631	+	1.21006i	0.968941	+	0.247291i	$0.0795405\pi$
$359$	13.2372	+	22.9275i	0.698631	+	1.21006i	−0.270310	+	0.962773i	$0.587126\pi$
$360$		0			0
$361$	−1.17647	+	2.03771i	−0.0619197	+	0.107248i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.87495	−	7.44697i	−0.464536	−	0.389792i
$366$		0			0
$367$	−11.7457	−	4.27510i	−0.613123	−	0.223159i	0.0167461	−	0.999860i	$-0.494669\pi$
$367$	−11.7457	−	4.27510i	−0.613123	−	0.223159i	−0.629869	+	0.776701i	$0.716892\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.89725	−	27.7737i	−0.254253	−	1.44194i
$372$		0			0
$373$	3.01015	−	1.09561i	0.155860	−	0.0567284i	−0.262912	−	0.964820i	$-0.584683\pi$
$373$	3.01015	−	1.09561i	0.155860	−	0.0567284i	0.418772	+	0.908091i	$0.362461\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.68385			0.138225
$378$		0			0
$379$	10.7650			0.552963			0.276481	−	0.961019i	$-0.410832\pi$
$379$	10.7650			0.552963			0.276481	+	0.961019i	$0.410832\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−28.3359	+	10.3134i	−1.44789	+	0.526990i	−0.942002	−	0.335606i	$-0.891059\pi$
$383$	−28.3359	+	10.3134i	−1.44789	+	0.526990i	−0.505892	+	0.862597i	$0.668837\pi$
$384$		0			0
$385$	−4.73720	−	26.8660i	−0.241430	−	1.36922i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.1771	−	6.25195i	−0.870913	−	0.316986i	−0.132376	−	0.991200i	$-0.542261\pi$
$389$	−17.1771	−	6.25195i	−0.870913	−	0.316986i	−0.738537	+	0.674213i	$0.764483\pi$
$390$		0			0
$391$	−2.17601	−	1.82589i	−0.110046	−	0.0923393i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20.9387	+	36.2669i	−1.05354	+	1.82479i
$396$		0			0
$397$	−4.90869	−	8.50210i	−0.246360	−	0.426708i	0.716153	−	0.697943i	$-0.245901\pi$
$397$	−4.90869	−	8.50210i	−0.246360	−	0.426708i	−0.962513	+	0.271235i	$0.912568\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.00593	−	22.7188i	0.200047	−	1.13452i	−0.705000	−	0.709208i	$-0.749053\pi$
$401$	4.00593	−	22.7188i	0.200047	−	1.13452i	0.905046	−	0.425313i	$-0.139836\pi$
$402$		0			0
$403$	−3.31823	+	2.78433i	−0.165293	+	0.138697i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.4107	−	10.4138i	0.615177	−	0.516195i
$408$		0			0
$409$	−3.26154	+	18.4971i	−0.161273	+	0.914623i	0.791552	+	0.611102i	$0.209273\pi$
$409$	−3.26154	+	18.4971i	−0.161273	+	0.914623i	−0.952825	+	0.303521i	$0.901838\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.15074	+	3.72519i	0.105831	+	0.183305i
$414$		0			0
$415$	−23.7204	+	41.0849i	−1.16439	+	2.01678i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.0404	+	20.1723i	1.17445	+	0.985483i	1.00000		0.000817323i	$0.000260162\pi$
$419$	24.0404	+	20.1723i	1.17445	+	0.985483i	0.174453	+	0.984665i	$0.444184\pi$
$420$		0			0
$421$	19.5621	+	7.12002i	0.953398	+	0.347008i	0.771443	−	0.636299i	$-0.219535\pi$
$421$	19.5621	+	7.12002i	0.953398	+	0.347008i	0.181955	+	0.983307i	$0.441758\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.51250	−	19.9204i	−0.170381	−	0.966280i
$426$		0			0
$427$	14.3581	−	5.22593i	0.694839	−	0.252901i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	30.8928			1.48806			0.744028	−	0.668149i	$-0.232913\pi$
$431$	30.8928			1.48806			0.744028	+	0.668149i	$0.232913\pi$
$432$		0			0
$433$	−15.5840			−0.748917			−0.374459	−	0.927244i	$-0.622171\pi$
$433$	−15.5840			−0.748917			−0.374459	+	0.927244i	$0.622171\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.70977	−	2.80613i	0.368808	−	0.134235i
$438$		0			0
$439$	−1.93807	−	10.9914i	−0.0924992	−	0.524589i	−0.995485	−	0.0949187i	$-0.969741\pi$
$439$	−1.93807	−	10.9914i	−0.0924992	−	0.524589i	0.902986	−	0.429670i	$-0.141370\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.6033	+	4.95121i	0.646315	+	0.235239i	0.644317	−	0.764759i	$-0.277142\pi$
$443$	13.6033	+	4.95121i	0.646315	+	0.235239i	0.00199787	+	0.999998i	$0.499364\pi$
$444$		0			0
$445$	17.4507	+	14.6428i	0.827241	+	0.694138i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.6936	+	32.3783i	−0.882207	+	1.52803i	−0.0333252	+	0.999445i	$0.510610\pi$
$449$	−18.6936	+	32.3783i	−0.882207	+	1.52803i	−0.848882	+	0.528583i	$0.822724\pi$
$450$		0			0
$451$	3.59958	+	6.23465i	0.169497	+	0.293578i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.97355	−	33.8777i	0.280044	−	1.58821i
$456$		0			0
$457$	10.5050	−	8.81472i	0.491402	−	0.412335i	−0.363126	−	0.931740i	$-0.618291\pi$
$457$	10.5050	−	8.81472i	0.491402	−	0.412335i	0.854528	+	0.519405i	$0.173846\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.09478	+	3.43593i	−0.190713	+	0.160027i	−0.733145	−	0.680073i	$-0.761948\pi$
$461$	−4.09478	+	3.43593i	−0.190713	+	0.160027i	0.542432	+	0.840100i	$0.317504\pi$
$462$		0			0
$463$	6.31705	−	35.8258i	0.293578	−	1.66496i	−0.379348	−	0.925254i	$-0.623852\pi$
$463$	6.31705	−	35.8258i	0.293578	−	1.66496i	0.672926	−	0.739710i	$-0.265037\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.61809	+	4.53466i	0.121151	+	0.209839i	0.920222	−	0.391397i	$-0.128008\pi$
$467$	2.61809	+	4.53466i	0.121151	+	0.209839i	−0.799071	+	0.601237i	$0.794675\pi$
$468$		0			0
$469$	11.1571	−	19.3247i	0.515188	−	0.892331i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.64325	+	3.05705i	0.167517	+	0.140563i
$474$		0			0
$475$	54.9010	+	19.9823i	2.51903	+	0.916852i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.03109	−	11.5189i	−0.0928027	−	0.526310i	−0.995399	−	0.0958213i	$-0.969452\pi$
$479$	−2.03109	−	11.5189i	−0.0928027	−	0.526310i	0.902596	−	0.430489i	$-0.141659\pi$
$480$		0			0
$481$	19.1973	−	6.98724i	0.875320	−	0.318591i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	50.7982			2.30663
$486$		0			0
$487$	20.7362			0.939645			0.469823	−	0.882761i	$-0.344318\pi$
$487$	20.7362			0.939645			0.469823	+	0.882761i	$0.344318\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.28722	−	3.38027i	0.419126	−	0.152550i	−0.123843	−	0.992302i	$-0.539522\pi$
$491$	9.28722	−	3.38027i	0.419126	−	0.152550i	0.542970	+	0.839752i	$0.317300\pi$
$492$		0			0
$493$	0.319978	+	1.81469i	0.0144111	+	0.0817293i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.75545	−	2.09481i	−0.258167	−	0.0939652i
$498$		0			0
$499$	−15.5261	−	13.0279i	−0.695042	−	0.583210i	0.225316	−	0.974286i	$-0.427659\pi$
$499$	−15.5261	−	13.0279i	−0.695042	−	0.583210i	−0.920358	+	0.391076i	$0.872103\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.34704	−	2.33314i	0.0600614	−	0.104029i	−0.834431	−	0.551112i	$-0.814204\pi$
$503$	1.34704	−	2.33314i	0.0600614	−	0.104029i	0.894493	+	0.447083i	$0.147537\pi$
$504$		0			0
$505$	29.7468	+	51.5230i	1.32372	+	2.29274i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.409562	−	2.32274i	0.0181535	−	0.102954i	−0.974385	−	0.224888i	$-0.927798\pi$
$509$	0.409562	−	2.32274i	0.0181535	−	0.102954i	0.992538	+	0.121934i	$0.0389096\pi$
$510$		0			0
$511$	7.42602	−	6.23117i	0.328508	−	0.275651i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.85650	+	4.07508i	−0.214003	+	0.179570i
$516$		0			0
$517$	−2.39718	+	13.5951i	−0.105428	+	0.597912i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.5589	−	30.4129i	−0.769270	−	1.33241i	−0.937959	−	0.346745i	$-0.887287\pi$
$521$	−17.5589	−	30.4129i	−0.769270	−	1.33241i	0.168690	−	0.985669i	$-0.446046\pi$
$522$		0			0
$523$	−14.8599	+	25.7381i	−0.649777	+	1.12545i	0.333399	+	0.942786i	$0.391804\pi$
$523$	−14.8599	+	25.7381i	−0.649777	+	1.12545i	−0.983176	+	0.182661i	$0.941529\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.27823	−	1.91166i	−0.0992414	−	0.0832734i
$528$		0			0
$529$	18.6506	+	6.78825i	0.810894	+	0.295141i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.57638	+	8.94012i	0.0682808	+	0.387240i
$534$		0			0
$535$	−76.6188	+	27.8870i	−3.31252	+	1.20566i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.89135			0.426050
$540$		0			0
$541$	−4.06242			−0.174657			−0.0873286	−	0.996180i	$-0.527833\pi$
$541$	−4.06242			−0.174657			−0.0873286	+	0.996180i	$0.527833\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	60.2288	−	21.9215i	2.57992	−	0.939013i
$546$		0			0
$547$	−6.47203	−	36.7047i	−0.276724	−	1.56938i	−0.733432	−	0.679763i	$-0.762083\pi$
$547$	−6.47203	−	36.7047i	−0.276724	−	1.56938i	0.456708	−	0.889617i	$-0.349028\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.00131	−	1.82033i	−0.213063	−	0.0775486i
$552$		0			0
$553$	−26.8426	−	22.5236i	−1.14146	−	0.957801i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.81761	−	8.34434i	0.204128	−	0.353561i	−0.745726	−	0.666253i	$-0.767897\pi$
$557$	4.81761	−	8.34434i	0.204128	−	0.353561i	0.949855	+	0.312692i	$0.101231\pi$
$558$		0			0
$559$	2.99858	+	5.19370i	0.126827	+	0.219670i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.77978	+	15.7649i	−0.117154	+	0.664413i	0.868507	+	0.495677i	$0.165080\pi$
$563$	−2.77978	+	15.7649i	−0.117154	+	0.664413i	−0.985661	+	0.168737i	$0.946031\pi$
$564$		0			0
$565$	−36.6355	+	30.7408i	−1.54127	+	1.29328i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.7944	+	12.4139i	−0.620212	+	0.520419i	−0.897870	−	0.440260i	$-0.854886\pi$
$569$	−14.7944	+	12.4139i	−0.620212	+	0.520419i	0.277658	+	0.960680i	$0.410442\pi$
$570$		0			0
$571$	−1.61529	+	9.16077i	−0.0675978	+	0.383366i	0.932174	+	0.362010i	$0.117909\pi$
$571$	−1.61529	+	9.16077i	−0.0675978	+	0.383366i	−0.999772	+	0.0213560i	$0.993202\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.2244	−	19.4412i	−0.468089	−	0.810753i
$576$		0			0
$577$	−3.05082	+	5.28418i	−0.127007	+	0.219983i	−0.922516	−	0.385959i	$-0.873871\pi$
$577$	−3.05082	+	5.28418i	−0.127007	+	0.219983i	0.795508	+	0.605943i	$0.207204\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−30.4085	−	25.5158i	−1.26156	−	1.05857i
$582$		0			0
$583$	13.9336	+	5.07143i	0.577072	+	0.210037i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.95166	+	22.4110i	0.163102	+	0.925000i	0.950999	+	0.309193i	$0.100059\pi$
$587$	3.95166	+	22.4110i	0.163102	+	0.925000i	−0.787897	+	0.615807i	$0.788830\pi$
$588$		0			0
$589$	8.07194	−	2.93795i	0.332599	−	0.121056i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.4186			−0.674233			−0.337117	−	0.941463i	$-0.609452\pi$
$593$	−16.4186			−0.674233			−0.337117	+	0.941463i	$0.609452\pi$
$594$		0			0
$595$	23.6186			0.968268
$596$		0			0
$597$		0			0
$598$		0			0
$599$	31.1090	−	11.3227i	1.27108	−	0.462635i	0.383607	−	0.923496i	$-0.374682\pi$
$599$	31.1090	−	11.3227i	1.27108	−	0.462635i	0.887472	+	0.460861i	$0.152459\pi$
$600$		0			0
$601$	3.94541	+	22.3755i	0.160937	+	0.912716i	0.953156	+	0.302480i	$0.0978145\pi$
$601$	3.94541	+	22.3755i	0.160937	+	0.912716i	−0.792219	+	0.610237i	$0.791074\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−29.9398	−	10.8972i	−1.21722	−	0.443033i
$606$		0			0
$607$	−13.1172	−	11.0066i	−0.532410	−	0.446745i	0.336523	−	0.941675i	$-0.390749\pi$
$607$	−13.1172	−	11.0066i	−0.532410	−	0.446745i	−0.868933	+	0.494930i	$0.835194\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.70385	+	15.0755i	−0.352120	+	0.609890i
$612$		0			0
$613$	−20.8362	−	36.0893i	−0.841564	−	1.45763i	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−20.8362	−	36.0893i	−0.841564	−	1.45763i	0.0470074	−	0.998895i	$-0.485032\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.15015	+	17.8654i	−0.126820	+	0.719233i	0.853390	+	0.521273i	$0.174543\pi$
$617$	−3.15015	+	17.8654i	−0.126820	+	0.719233i	−0.980210	+	0.197960i	$0.936568\pi$
$618$		0			0
$619$	−6.11217	+	5.12872i	−0.245669	+	0.206141i	−0.757305	−	0.653062i	$-0.773484\pi$
$619$	−6.11217	+	5.12872i	−0.245669	+	0.206141i	0.511636	+	0.859202i	$0.329040\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.6017	+	12.2522i	−0.585003	+	0.490876i
$624$		0			0
$625$	12.4400	−	70.5510i	0.497602	−	2.82204i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.01319	+	12.1472i	0.279634	+	0.484340i
$630$		0			0
$631$	0.118628	−	0.205470i	0.00472251	−	0.00817962i	−0.863654	−	0.504084i	$-0.831830\pi$
$631$	0.118628	−	0.205470i	0.00472251	−	0.00817962i	0.868377	+	0.495905i	$0.165163\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.18049	+	4.34695i	0.205582	+	0.172503i
$636$		0			0
$637$	11.7207	+	4.26597i	0.464389	+	0.169024i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.15329	−	23.5545i	−0.164045	−	0.930345i	−0.950044	−	0.312117i	$-0.898962\pi$
$641$	−4.15329	−	23.5545i	−0.164045	−	0.930345i	0.785999	−	0.618228i	$-0.212149\pi$
$642$		0			0
$643$	24.2499	−	8.82623i	0.956321	−	0.348073i	0.183730	−	0.982977i	$-0.441183\pi$
$643$	24.2499	−	8.82623i	0.956321	−	0.348073i	0.772591	+	0.634904i	$0.218960\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.60899			−0.259826			−0.129913	−	0.991525i	$-0.541470\pi$
$647$	−6.60899			−0.259826			−0.129913	+	0.991525i	$0.541470\pi$
$648$		0			0
$649$	−2.26159			−0.0887753
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23.4279	+	8.52708i	−0.916807	+	0.333690i	−0.756967	−	0.653453i	$-0.773320\pi$
$653$	−23.4279	+	8.52708i	−0.916807	+	0.333690i	−0.159839	+	0.987143i	$0.551098\pi$
$654$		0			0
$655$	−3.84162	−	21.7869i	−0.150105	−	0.851285i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.25426	+	2.64034i	0.282586	+	0.102853i	0.479425	−	0.877583i	$-0.340845\pi$
$659$	7.25426	+	2.64034i	0.282586	+	0.102853i	−0.196839	+	0.980436i	$0.563068\pi$
$660$		0			0
$661$	13.6830	+	11.4814i	0.532208	+	0.446575i	0.868863	−	0.495053i	$-0.164851\pi$
$661$	13.6830	+	11.4814i	0.532208	+	0.446575i	−0.336655	+	0.941628i	$0.609296\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−34.1093	+	59.0790i	−1.32270	+	2.29098i
$666$		0			0
$667$	1.02251	+	1.77103i	0.0395916	+	0.0685747i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.39502	+	7.91153i	−0.0538540	+	0.305421i
$672$		0			0
$673$	3.74755	−	3.14457i	0.144458	−	0.121214i	−0.567695	−	0.823239i	$-0.692165\pi$
$673$	3.74755	−	3.14457i	0.144458	−	0.121214i	0.712153	+	0.702025i	$0.247720\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.9410	−	10.8588i	0.497363	−	0.417337i	−0.359294	−	0.933225i	$-0.616982\pi$
$677$	12.9410	−	10.8588i	0.497363	−	0.417337i	0.856656	+	0.515888i	$0.172538\pi$
$678$		0			0
$679$	−7.38089	+	41.8591i	−0.283252	+	1.60640i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.8279	−	29.1467i	−0.643900	−	1.11527i	−0.984554	−	0.175080i	$-0.943982\pi$
$683$	−16.8279	−	29.1467i	−0.643900	−	1.11527i	0.340654	−	0.940189i	$-0.389352\pi$
$684$		0			0
$685$	32.8368	−	56.8750i	1.25463	−	2.17308i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.3233	+	12.0187i	0.545674	+	0.457875i
$690$		0			0
$691$	−15.4088	−	5.60833i	−0.586177	−	0.213351i	0.0318704	−	0.999492i	$-0.489854\pi$
$691$	−15.4088	−	5.60833i	−0.586177	−	0.213351i	−0.618047	+	0.786141i	$0.712076\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.93805	+	50.6902i	0.339040	+	1.92279i
$696$		0			0
$697$	−5.85692	+	2.13174i	−0.221847	+	0.0807456i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	34.9743			1.32096			0.660481	−	0.750843i	$-0.270352\pi$
$701$	34.9743			1.32096			0.660481	+	0.750843i	$0.270352\pi$
$702$		0			0
$703$	−40.5129			−1.52797
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−46.7786	+	17.0260i	−1.75929	+	0.640329i
$708$		0			0
$709$	−4.64168	−	26.3243i	−0.174322	−	0.988629i	−0.938924	−	0.344126i	$-0.888175\pi$
$709$	−4.64168	−	26.3243i	−0.174322	−	0.988629i	0.764602	−	0.644503i	$-0.222936\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.10153	−	1.12886i	−0.116153	−	0.0422763i
$714$		0			0
$715$	13.8552	+	11.6259i	0.518155	+	0.434784i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.0563	+	22.6141i	−0.486916	+	0.843364i	−0.999887	−	0.0150424i	$-0.995212\pi$
$719$	−13.0563	+	22.6141i	−0.486916	+	0.843364i	0.512970	+	0.858406i	$0.328545\pi$
$720$		0			0
$721$	−2.65234	−	4.59399i	−0.0987783	−	0.171089i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.52874	+	14.3412i	−0.0939152	+	0.532619i
$726$		0			0
$727$	10.4715	−	8.78667i	0.388368	−	0.325880i	−0.427609	−	0.903964i	$-0.640644\pi$
$727$	10.4715	−	8.78667i	0.388368	−	0.325880i	0.815977	+	0.578084i	$0.196199\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.15422	+	2.64670i	−0.116663	+	0.0978918i
$732$		0			0
$733$	−2.00652	+	11.3795i	−0.0741125	+	0.420313i	0.925067	+	0.379805i	$0.124009\pi$
$733$	−2.00652	+	11.3795i	−0.0741125	+	0.420313i	−0.999179	+	0.0405082i	$0.987102\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.86609	+	10.1604i	0.216080	+	0.374262i
$738$		0			0
$739$	−25.2426	+	43.7215i	−0.928566	+	1.60832i	−0.142841	+	0.989746i	$0.545624\pi$
$739$	−25.2426	+	43.7215i	−0.928566	+	1.60832i	−0.785724	+	0.618577i	$0.787709\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.0885	−	13.4998i	−0.590228	−	0.495260i	0.298060	−	0.954547i	$-0.403661\pi$
$743$	−16.0885	−	13.4998i	−0.590228	−	0.495260i	−0.888288	+	0.459287i	$0.848105\pi$
$744$		0			0
$745$	−82.6529	−	30.0832i	−3.02817	−	1.10216i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.8470	−	67.1879i	−0.432882	−	2.45499i
$750$		0			0
$751$	10.5146	−	3.82701i	0.383684	−	0.139649i	−0.142975	−	0.989726i	$-0.545667\pi$
$751$	10.5146	−	3.82701i	0.383684	−	0.139649i	0.526659	+	0.850077i	$0.323445\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	40.5073			1.47421
$756$		0			0
$757$	−22.2619			−0.809123			−0.404561	−	0.914511i	$-0.632576\pi$
$757$	−22.2619			−0.809123			−0.404561	+	0.914511i	$0.632576\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.56792	+	2.02656i	−0.201837	+	0.0734626i	−0.440960	−	0.897527i	$-0.645362\pi$
$761$	−5.56792	+	2.02656i	−0.201837	+	0.0734626i	0.239124	+	0.970989i	$0.423140\pi$
$762$		0			0
$763$	9.31277	+	52.8153i	0.337145	+	1.91204i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.67985	−	0.975387i	−0.0967639	−	0.0352192i
$768$		0			0
$769$	1.73719	+	1.45768i	0.0626447	+	0.0525651i	0.673572	−	0.739121i	$-0.264759\pi$
$769$	1.73719	+	1.45768i	0.0626447	+	0.0525651i	−0.610928	+	0.791686i	$0.709203\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.63951	+	8.03586i	−0.166872	+	0.289030i	−0.937318	−	0.348474i	$-0.886700\pi$
$773$	−4.63951	+	8.03586i	−0.166872	+	0.289030i	0.770447	+	0.637504i	$0.220033\pi$
$774$		0			0
$775$	−11.7516	−	20.3544i	−0.422132	−	0.731153i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.12609	−	17.7290i	0.112004	−	0.635206i
$780$		0			0
$781$	2.46686	−	2.06994i	0.0882711	−	0.0740683i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	37.8427	−	31.7538i	1.35066	−	1.13334i
$786$		0			0
$787$	−4.00609	+	22.7197i	−0.142802	+	0.809868i	0.826304	+	0.563224i	$0.190439\pi$
$787$	−4.00609	+	22.7197i	−0.142802	+	0.809868i	−0.969106	+	0.246644i	$0.920672\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−20.0082	−	34.6552i	−0.711410	−	1.23220i
$792$		0			0
$793$	−5.06512	+	8.77304i	−0.179868	+	0.311540i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.5527	−	13.0503i	−0.550906	−	0.462265i	0.324341	−	0.945940i	$-0.394857\pi$
$797$	−15.5527	−	13.0503i	−0.550906	−	0.462265i	−0.875248	+	0.483675i	$0.839302\pi$
$798$		0			0
$799$	−11.2310	−	4.08775i	−0.397325	−	0.144614i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.885052	+	5.01938i	0.0312328	+	0.177130i
$804$		0			0
$805$	24.6312	−	8.96503i	0.868136	−	0.315976i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−35.2976			−1.24100			−0.620499	−	0.784207i	$-0.713070\pi$
$809$	−35.2976			−1.24100			−0.620499	+	0.784207i	$0.713070\pi$
$810$		0			0
$811$	−40.1846			−1.41107			−0.705536	−	0.708675i	$-0.749293\pi$
$811$	−40.1846			−1.41107			−0.705536	+	0.708675i	$0.749293\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−56.6563	+	20.6212i	−1.98458	+	0.722330i
$816$		0			0
$817$	−2.06517	−	11.7122i	−0.0722512	−	0.409757i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.7450	+	4.63881i	0.444804	+	0.161895i	0.554705	−	0.832047i	$-0.312831\pi$
$821$	12.7450	+	4.63881i	0.444804	+	0.161895i	−0.109901	+	0.993943i	$0.535053\pi$
$822$		0			0
$823$	16.3197	+	13.6939i	0.568869	+	0.477338i	0.881271	−	0.472612i	$-0.156689\pi$
$823$	16.3197	+	13.6939i	0.568869	+	0.477338i	−0.312401	+	0.949950i	$0.601133\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	26.2719	−	45.5043i	0.913564	−	1.58234i	0.104573	−	0.994517i	$-0.466652\pi$
$827$	26.2719	−	45.5043i	0.913564	−	1.58234i	0.808990	−	0.587822i	$-0.200014\pi$
$828$		0			0
$829$	−6.89163	−	11.9366i	−0.239356	−	0.414577i	0.721174	−	0.692754i	$-0.243603\pi$
$829$	−6.89163	−	11.9366i	−0.239356	−	0.414577i	−0.960530	+	0.278177i	$0.910270\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.48706	+	8.43352i	−0.0515235	+	0.292204i
$834$		0			0
$835$	27.2226	−	22.8425i	0.942078	−	0.790497i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.0984	+	16.0254i	−0.659349	+	0.553260i	−0.909892	−	0.414846i	$-0.863835\pi$
$839$	−19.0984	+	16.0254i	−0.659349	+	0.553260i	0.250543	+	0.968106i	$0.419391\pi$
$840$		0			0
$841$	−4.80544	+	27.2530i	−0.165705	+	0.939758i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.8991	−	27.5381i	−0.546946	−	0.947339i
$846$		0			0
$847$	13.3298	−	23.0878i	0.458016	−	0.793308i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.9246	+	10.0060i	0.408771	+	0.343000i
$852$		0			0
$853$	4.36081	+	1.58720i	0.149311	+	0.0543448i	0.415595	−	0.909550i	$-0.363574\pi$
$853$	4.36081	+	1.58720i	0.149311	+	0.0543448i	−0.266283	+	0.963895i	$0.585796\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.29397	+	35.6949i	0.214998	+	1.21931i	0.880910	+	0.473284i	$0.156932\pi$
$857$	6.29397	+	35.6949i	0.214998	+	1.21931i	−0.665912	+	0.746031i	$0.731957\pi$
$858$		0			0
$859$	−25.7284	+	9.36438i	−0.877842	+	0.319508i	−0.741339	−	0.671131i	$-0.765809\pi$
$859$	−25.7284	+	9.36438i	−0.877842	+	0.319508i	−0.136503	+	0.990640i	$0.543586\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.9476			0.713065			0.356532	−	0.934283i	$-0.383959\pi$
$863$	20.9476			0.713065			0.356532	+	0.934283i	$0.383959\pi$
$864$		0			0
$865$	105.251			3.57863
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.3122	−	6.30113i	0.587277	−	0.213751i
$870$		0			0
$871$	2.56897	+	14.5694i	0.0870463	+	0.493664i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	106.035	+	38.5935i	3.58463	+	1.30470i
$876$		0			0
$877$	13.9493	+	11.7049i	0.471035	+	0.395246i	0.847172	−	0.531319i	$-0.178303\pi$
$877$	13.9493	+	11.7049i	0.471035	+	0.395246i	−0.376137	+	0.926564i	$0.622748\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.0582	−	39.9380i	0.776851	−	1.34555i	−0.156897	−	0.987615i	$-0.550149\pi$
$881$	23.0582	−	39.9380i	0.776851	−	1.34555i	0.933748	−	0.357931i	$-0.116518\pi$
$882$		0			0
$883$	9.03494	+	15.6490i	0.304050	+	0.526630i	0.977049	−	0.213013i	$-0.0683277\pi$
$883$	9.03494	+	15.6490i	0.304050	+	0.526630i	−0.672999	+	0.739643i	$0.734994\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.58846	+	26.0224i	−0.154065	+	0.873748i	0.805570	+	0.592501i	$0.201859\pi$
$887$	−4.58846	+	26.0224i	−0.154065	+	0.873748i	−0.959635	+	0.281247i	$0.909252\pi$
$888$		0			0
$889$	−4.33472	+	3.63726i	−0.145382	+	0.121990i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26.4445	−	22.1896i	0.884931	−	0.742545i
$894$		0			0
$895$	7.06286	−	40.0555i	0.236086	−	1.33891i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.07054	+	1.85423i	0.0357045	+	0.0618420i
$900$		0			0
$901$	−6.41876	+	11.1176i	−0.213840	+	0.370381i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.94402	+	1.63122i	0.0646213	+	0.0542237i
$906$		0			0
$907$	−35.2748	−	12.8390i	−1.17128	−	0.426312i	−0.318167	−	0.948035i	$-0.603067\pi$
$907$	−35.2748	−	12.8390i	−1.17128	−	0.426312i	−0.853115	+	0.521723i	$0.825289\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.77993	+	15.7658i	0.0921031	+	0.522343i	0.995597	+	0.0937409i	$0.0298825\pi$
$911$	2.77993	+	15.7658i	0.0921031	+	0.522343i	−0.903494	+	0.428602i	$0.859006\pi$
$912$		0			0
$913$	19.6121	−	7.13822i	0.649065	−	0.236240i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.5112			0.611294
$918$		0			0
$919$	36.8859			1.21675			0.608377	−	0.793648i	$-0.291821\pi$
$919$	36.8859			1.21675			0.608377	+	0.793648i	$0.291821\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.81581	−	1.38884i	0.125599	−	0.0457143i
$924$		0			0
$925$	19.2486	+	109.165i	0.632892	+	3.58931i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−31.9483	−	11.6282i	−1.04819	−	0.381510i	−0.240211	−	0.970721i	$-0.577217\pi$
$929$	−31.9483	−	11.6282i	−1.04819	−	0.381510i	−0.807979	+	0.589211i	$0.799439\pi$
$930$		0			0
$931$	−18.9478	−	15.8991i	−0.620991	−	0.521073i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.20899	+	10.7543i	−0.203056	+	0.351703i
$936$		0			0
$937$	−6.09208	−	10.5518i	−0.199020	−	0.344712i	0.749191	−	0.662354i	$-0.230442\pi$
$937$	−6.09208	−	10.5518i	−0.199020	−	0.344712i	−0.948211	+	0.317642i	$0.897109\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.86766	+	10.5920i	−0.0608839	+	0.345290i	0.939115	+	0.343604i	$0.111648\pi$
$941$	−1.86766	+	10.5920i	−0.0608839	+	0.345290i	−0.999999	+	0.00168585i	$0.999463\pi$
$942$		0			0
$943$	−5.29887	+	4.44628i	−0.172555	+	0.144791i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−29.3920	+	24.6628i	−0.955110	+	0.801433i	−0.980151	−	0.198254i	$-0.936473\pi$
$947$	−29.3920	+	24.6628i	−0.955110	+	0.801433i	0.0250403	+	0.999686i	$0.492029\pi$
$948$		0			0
$949$	−1.11604	+	6.32937i	−0.0362282	+	0.205460i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.2733	−	19.5259i	−0.365177	−	0.632506i	0.623627	−	0.781722i	$-0.285658\pi$
$953$	−11.2733	−	19.5259i	−0.365177	−	0.632506i	−0.988805	+	0.149216i	$0.952325\pi$
$954$		0			0
$955$	43.4910	−	75.3286i	1.40734	−	2.43758i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	42.0954	+	35.3222i	1.35933	+	1.14061i
$960$		0			0
$961$	25.8832	+	9.42073i	0.834943	+	0.303895i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.12192	+	46.0617i	0.261454	+	1.48278i
$966$		0			0
$967$	27.5163	−	10.0151i	0.884865	−	0.322065i	0.140694	−	0.990053i	$-0.455067\pi$
$967$	27.5163	−	10.0151i	0.884865	−	0.322065i	0.744172	+	0.667988i	$0.232844\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−33.2319			−1.06646			−0.533232	−	0.845969i	$-0.679023\pi$
$971$	−33.2319			−1.06646			−0.533232	+	0.845969i	$0.679023\pi$
$972$		0			0
$973$	−43.0688			−1.38072
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.90284	−	2.14846i	0.188849	−	0.0687352i	−0.245865	−	0.969304i	$-0.579072\pi$
$977$	5.90284	−	2.14846i	0.188849	−	0.0687352i	0.434713	+	0.900569i	$0.356850\pi$
$978$		0			0
$979$	−1.74026	−	9.86953i	−0.0556191	−	0.315431i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.34031	+	0.487834i	0.0427493	+	0.0155595i	0.363306	−	0.931670i	$-0.381648\pi$
$983$	1.34031	+	0.487834i	0.0427493	+	0.0155595i	−0.320557	+	0.947229i	$0.603870\pi$
$984$		0			0
$985$	40.6200	+	34.0842i	1.29426	+	1.08601i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.28483	+	3.95744i	−0.0726533	+	0.125839i
$990$		0			0
$991$	5.58886	+	9.68018i	0.177536	+	0.307501i	0.941036	−	0.338307i	$-0.109854\pi$
$991$	5.58886	+	9.68018i	0.177536	+	0.307501i	−0.763500	+	0.645808i	$0.776521\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	15.3527	−	87.0697i	0.486714	−	2.76029i
$996$		0			0
$997$	−30.8832	+	25.9141i	−0.978082	+	0.820708i	−0.983799	−	0.179275i	$-0.942625\pi$
$997$	−30.8832	+	25.9141i	−0.978082	+	0.820708i	0.00571696	+	0.999984i	$0.498180\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.2.i.a.37.3		18
3.2	odd	2		108.2.i.a.49.2	✓	18
9.2	odd	6		972.2.i.a.433.3		18
9.4	even	3		972.2.i.b.757.1		18
9.5	odd	6		972.2.i.c.757.3		18
9.7	even	3		972.2.i.d.433.1		18
12.11	even	2		432.2.u.d.49.2		18
27.2	odd	18		972.2.i.a.541.3		18
27.4	even	9		2916.2.a.c.1.1		9
27.5	odd	18		2916.2.e.c.973.1		18
27.7	even	9		972.2.i.b.217.1		18
27.11	odd	18		108.2.i.a.97.2	yes	18
27.13	even	9		2916.2.e.d.1945.9		18
27.14	odd	18		2916.2.e.c.1945.1		18
27.16	even	9	inner	324.2.i.a.289.3		18
27.20	odd	18		972.2.i.c.217.3		18
27.22	even	9		2916.2.e.d.973.9		18
27.23	odd	18		2916.2.a.d.1.9		9
27.25	even	9		972.2.i.d.541.1		18
108.11	even	18		432.2.u.d.97.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.2.i.a.49.2	✓	18	3.2	odd	2
108.2.i.a.97.2	yes	18	27.11	odd	18
324.2.i.a.37.3		18	1.1	even	1	trivial
324.2.i.a.289.3		18	27.16	even	9	inner
432.2.u.d.49.2		18	12.11	even	2
432.2.u.d.97.2		18	108.11	even	18
972.2.i.a.433.3		18	9.2	odd	6
972.2.i.a.541.3		18	27.2	odd	18
972.2.i.b.217.1		18	27.7	even	9
972.2.i.b.757.1		18	9.4	even	3
972.2.i.c.217.3		18	27.20	odd	18
972.2.i.c.757.3		18	9.5	odd	6
972.2.i.d.433.1		18	9.7	even	3
972.2.i.d.541.1		18	27.25	even	9
2916.2.a.c.1.1		9	27.4	even	9
2916.2.a.d.1.9		9	27.23	odd	18
2916.2.e.c.973.1		18	27.5	odd	18
2916.2.e.c.1945.1		18	27.14	odd	18
2916.2.e.d.973.9		18	27.22	even	9
2916.2.e.d.1945.9		18	27.13	even	9

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.94709 - 1.43662i) q^{5} +(0.610312 + 3.46125i) q^{7} +O(q^{10})\)	\(q+(3.94709 - 1.43662i) q^{5} +(0.610312 + 3.46125i) q^{7} +(-1.73646 - 0.632019i) q^{11} +(-1.78502 - 1.49781i) q^{13} +(0.799928 - 1.38552i) q^{17} +(2.31046 + 4.00184i) q^{19} +(0.308317 - 1.74855i) q^{23} +(9.68544 - 8.12705i) q^{25} +(-0.882314 + 0.740350i) q^{29} +(0.322800 - 1.83069i) q^{31} +(7.38148 + 12.7851i) q^{35} +(-4.38364 + 7.59269i) q^{37} +(-2.98440 - 2.50421i) q^{41} +(-2.41848 - 0.880255i) q^{43} +(-1.29725 - 7.35705i) q^{47} +(-5.02994 + 1.83075i) q^{49} -8.02417 q^{53} -7.76194 q^{55} +(1.15006 - 0.418589i) q^{59} +(-0.754920 - 4.28136i) q^{61} +(-9.19743 - 3.34759i) q^{65} +(-4.86356 - 4.08101i) q^{67} +(-0.871328 + 1.50918i) q^{71} +(-1.37908 - 2.38864i) q^{73} +(1.12780 - 6.39605i) q^{77} +(-7.63735 + 6.40849i) q^{79} +(-8.65194 + 7.25984i) q^{83} +(1.16692 - 6.61795i) q^{85} +(2.71167 + 4.69675i) q^{89} +(4.09488 - 7.09254i) q^{91} +(14.8688 + 12.4764i) q^{95} +(11.3643 + 4.13626i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 3 q^{5}+O(q^{10}) \) 18 * q - 3 * q^5	\( 18 q - 3 q^{5} - 3 q^{11} + 12 q^{17} + 30 q^{23} + 9 q^{25} + 24 q^{29} + 9 q^{31} + 21 q^{35} - 21 q^{41} - 9 q^{43} - 45 q^{47} - 18 q^{49} - 66 q^{53} - 60 q^{59} - 18 q^{61} - 33 q^{65} - 27 q^{67} + 12 q^{71} + 9 q^{73} + 75 q^{77} - 36 q^{79} + 45 q^{83} - 36 q^{85} + 48 q^{89} + 9 q^{91} - 6 q^{95} - 27 q^{97}+O(q^{100}) \) 18 * q - 3 * q^5 - 3 * q^11 + 12 * q^17 + 30 * q^23 + 9 * q^25 + 24 * q^29 + 9 * q^31 + 21 * q^35 - 21 * q^41 - 9 * q^43 - 45 * q^47 - 18 * q^49 - 66 * q^53 - 60 * q^59 - 18 * q^61 - 33 * q^65 - 27 * q^67 + 12 * q^71 + 9 * q^73 + 75 * q^77 - 36 * q^79 + 45 * q^83 - 36 * q^85 + 48 * q^89 + 9 * q^91 - 6 * q^95 - 27 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more