LMFDB - Embedded newform 6012.2.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.a (trivial)

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:	yes
Analytic conductor:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 $ x^10 - 4x^9 - 26x^8 + 82x^7 + 211x^6 - 340x^5 - 593x^4 + 192x^3 + 423x^2 + 126*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.07621$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.98604 q^{5} +1.07621 q^{7} +O(q^{10})$	$q+3.98604 q^{5} +1.07621 q^{7} +2.72051 q^{11} +1.35927 q^{13} +5.62926 q^{17} +7.96667 q^{19} -0.725481 q^{23} +10.8885 q^{25} +0.488099 q^{29} +4.05410 q^{31} +4.28980 q^{35} -6.63379 q^{37} +1.03121 q^{41} +2.77062 q^{43} -2.73500 q^{47} -5.84178 q^{49} -6.80477 q^{53} +10.8441 q^{55} -8.67320 q^{59} -7.54493 q^{61} +5.41809 q^{65} -0.660487 q^{67} -9.99008 q^{71} -0.870311 q^{73} +2.92783 q^{77} -2.47131 q^{79} +4.15135 q^{83} +22.4385 q^{85} -12.1744 q^{89} +1.46285 q^{91} +31.7555 q^{95} -16.2394 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) $ 10 * q + 6 * q^5 + 4 * q^7	$ 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) $ 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

Display 100 coefficients 10 coefficients

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
$5$	3.98604	1.78261	0.891305	−	0.453404i	$-0.149790\pi$
$5$	3.98604	1.78261	0.891305	+	0.453404i	$0.149790\pi$
$6$	0	0
$7$	1.07621	0.406768	0.203384	−	0.979099i	$-0.434806\pi$
$7$	1.07621	0.406768	0.203384	+	0.979099i	$0.434806\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.72051	0.820266	0.410133	−	0.912026i	$-0.365482\pi$
$11$	2.72051	0.820266	0.410133	+	0.912026i	$0.365482\pi$
$12$	0	0
$13$	1.35927	0.376993	0.188496	−	0.982074i	$-0.439639\pi$
$13$	1.35927	0.376993	0.188496	+	0.982074i	$0.439639\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.62926	1.36530	0.682649	−	0.730747i	$-0.260828\pi$
$17$	5.62926	1.36530	0.682649	+	0.730747i	$0.260828\pi$
$18$	0	0
$19$	7.96667	1.82768	0.913840	−	0.406074i	$-0.133103\pi$
$19$	7.96667	1.82768	0.913840	+	0.406074i	$0.133103\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.725481	−0.151273	−0.0756366	−	0.997135i	$-0.524099\pi$
$23$	−0.725481	−0.151273	−0.0756366	+	0.997135i	$0.524099\pi$
$24$	0	0
$25$	10.8885	2.17770
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.488099	0.0906377	0.0453189	−	0.998973i	$-0.485570\pi$
$29$	0.488099	0.0906377	0.0453189	+	0.998973i	$0.485570\pi$
$30$	0	0
$31$	4.05410	0.728137	0.364069	−	0.931372i	$-0.381387\pi$
$31$	4.05410	0.728137	0.364069	+	0.931372i	$0.381387\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.28980	0.725108
$36$	0	0
$37$	−6.63379	−1.09059	−0.545294	−	0.838245i	$-0.683582\pi$
$37$	−6.63379	−1.09059	−0.545294	+	0.838245i	$0.683582\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.03121	0.161048	0.0805240	−	0.996753i	$-0.474341\pi$
$41$	1.03121	0.161048	0.0805240	+	0.996753i	$0.474341\pi$
$42$	0	0
$43$	2.77062	0.422516	0.211258	−	0.977430i	$-0.432244\pi$
$43$	2.77062	0.422516	0.211258	+	0.977430i	$0.432244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.73500	−0.398941	−0.199471	−	0.979904i	$-0.563922\pi$
$47$	−2.73500	−0.398941	−0.199471	+	0.979904i	$0.563922\pi$
$48$	0	0
$49$	−5.84178	−0.834540
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.80477	−0.934708	−0.467354	−	0.884070i	$-0.654793\pi$
$53$	−6.80477	−0.934708	−0.467354	+	0.884070i	$0.654793\pi$
$54$	0	0
$55$	10.8441	1.46221
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.67320	−1.12915	−0.564577	−	0.825380i	$-0.690961\pi$
$59$	−8.67320	−1.12915	−0.564577	+	0.825380i	$0.690961\pi$
$60$	0	0
$61$	−7.54493	−0.966029	−0.483014	−	0.875612i	$-0.660458\pi$
$61$	−7.54493	−0.966029	−0.483014	+	0.875612i	$0.660458\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.41809	0.672031
$66$	0	0
$67$	−0.660487	−0.0806913	−0.0403456	−	0.999186i	$-0.512846\pi$
$67$	−0.660487	−0.0806913	−0.0403456	+	0.999186i	$0.512846\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.99008	−1.18560	−0.592802	−	0.805348i	$-0.701978\pi$
$71$	−9.99008	−1.18560	−0.592802	+	0.805348i	$0.701978\pi$
$72$	0	0
$73$	−0.870311	−0.101862	−0.0509311	−	0.998702i	$-0.516219\pi$
$73$	−0.870311	−0.101862	−0.0509311	+	0.998702i	$0.516219\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.92783	0.333658
$78$	0	0
$79$	−2.47131	−0.278044	−0.139022	−	0.990289i	$-0.544396\pi$
$79$	−2.47131	−0.278044	−0.139022	+	0.990289i	$0.544396\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.15135	0.455670	0.227835	−	0.973700i	$-0.426835\pi$
$83$	4.15135	0.455670	0.227835	+	0.973700i	$0.426835\pi$
$84$	0	0
$85$	22.4385	2.43379
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.1744	−1.29049	−0.645243	−	0.763977i	$-0.723244\pi$
$89$	−12.1744	−1.29049	−0.645243	+	0.763977i	$0.723244\pi$
$90$	0	0
$91$	1.46285	0.153349
$92$	0	0
$93$	0	0
$94$	0	0
$95$	31.7555	3.25804
$96$	0	0
$97$	−16.2394	−1.64887	−0.824433	−	0.565960i	$-0.808506\pi$
$97$	−16.2394	−1.64887	−0.824433	+	0.565960i	$0.808506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.36422	0.533760	0.266880	−	0.963730i	$-0.414007\pi$
$101$	5.36422	0.533760	0.266880	+	0.963730i	$0.414007\pi$
$102$	0	0
$103$	10.1112	0.996290	0.498145	−	0.867094i	$-0.334015\pi$
$103$	10.1112	0.996290	0.498145	+	0.867094i	$0.334015\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.6142	1.02612	0.513059	−	0.858354i	$-0.328512\pi$
$107$	10.6142	1.02612	0.513059	+	0.858354i	$0.328512\pi$
$108$	0	0
$109$	−0.269857	−0.0258476	−0.0129238	−	0.999916i	$-0.504114\pi$
$109$	−0.269857	−0.0258476	−0.0129238	+	0.999916i	$0.504114\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.5720	−1.84118	−0.920592	−	0.390527i	$-0.872293\pi$
$113$	−19.5720	−1.84118	−0.920592	+	0.390527i	$0.872293\pi$
$114$	0	0
$115$	−2.89179	−0.269661
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.05825	0.555359
$120$	0	0
$121$	−3.59880	−0.327164
$122$	0	0
$123$	0	0
$124$	0	0
$125$	23.4718	2.09938
$126$	0	0
$127$	−12.6521	−1.12269	−0.561345	−	0.827582i	$-0.689716\pi$
$127$	−12.6521	−1.12269	−0.561345	+	0.827582i	$0.689716\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.43691	0.125544	0.0627719	−	0.998028i	$-0.480006\pi$
$131$	1.43691	0.125544	0.0627719	+	0.998028i	$0.480006\pi$
$132$	0	0
$133$	8.57378	0.743441
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.2628	1.30399	0.651993	−	0.758225i	$-0.273933\pi$
$137$	15.2628	1.30399	0.651993	+	0.758225i	$0.273933\pi$
$138$	0	0
$139$	−10.8843	−0.923191	−0.461596	−	0.887090i	$-0.652723\pi$
$139$	−10.8843	−0.923191	−0.461596	+	0.887090i	$0.652723\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.69791	0.309234
$144$	0	0
$145$	1.94558	0.161572
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.22665	0.182414	0.0912071	−	0.995832i	$-0.470927\pi$
$149$	2.22665	0.182414	0.0912071	+	0.995832i	$0.470927\pi$
$150$	0	0
$151$	14.8224	1.20623	0.603114	−	0.797655i	$-0.293926\pi$
$151$	14.8224	1.20623	0.603114	+	0.797655i	$0.293926\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.1598	1.29799
$156$	0	0
$157$	−20.4688	−1.63359	−0.816795	−	0.576928i	$-0.804251\pi$
$157$	−20.4688	−1.63359	−0.816795	+	0.576928i	$0.804251\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.780767	−0.0615330
$162$	0	0
$163$	−15.2582	−1.19512	−0.597559	−	0.801825i	$-0.703863\pi$
$163$	−15.2582	−1.19512	−0.597559	+	0.801825i	$0.703863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.1524	−0.857876
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.5794	1.41256	0.706281	−	0.707931i	$-0.250371\pi$
$173$	18.5794	1.41256	0.706281	+	0.707931i	$0.250371\pi$
$174$	0	0
$175$	11.7183	0.885818
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.37738	−0.551411	−0.275706	−	0.961242i	$-0.588911\pi$
$179$	−7.37738	−0.551411	−0.275706	+	0.961242i	$0.588911\pi$
$180$	0	0
$181$	−16.0510	−1.19306	−0.596532	−	0.802589i	$-0.703455\pi$
$181$	−16.0510	−1.19306	−0.596532	+	0.802589i	$0.703455\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−26.4425	−1.94409
$186$	0	0
$187$	15.3145	1.11991
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.2943	1.03430	0.517151	−	0.855894i	$-0.326993\pi$
$191$	14.2943	1.03430	0.517151	+	0.855894i	$0.326993\pi$
$192$	0	0
$193$	−4.14936	−0.298677	−0.149339	−	0.988786i	$-0.547714\pi$
$193$	−4.14936	−0.298677	−0.149339	+	0.988786i	$0.547714\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.14806	0.366784	0.183392	−	0.983040i	$-0.441292\pi$
$197$	5.14806	0.366784	0.183392	+	0.983040i	$0.441292\pi$
$198$	0	0
$199$	11.6316	0.824539	0.412269	−	0.911062i	$-0.364736\pi$
$199$	11.6316	0.824539	0.412269	+	0.911062i	$0.364736\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.525295	0.0368685
$204$	0	0
$205$	4.11044	0.287086
$206$	0	0
$207$	0	0
$208$	0	0
$209$	21.6734	1.49918
$210$	0	0
$211$	5.03773	0.346812	0.173406	−	0.984850i	$-0.444523\pi$
$211$	5.03773	0.346812	0.173406	+	0.984850i	$0.444523\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.0438	0.753181
$216$	0	0
$217$	4.36305	0.296183
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.65167	0.514707
$222$	0	0
$223$	−9.94461	−0.665940	−0.332970	−	0.942937i	$-0.608051\pi$
$223$	−9.94461	−0.665940	−0.332970	+	0.942937i	$0.608051\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.1539	−1.53678	−0.768390	−	0.639981i	$-0.778942\pi$
$227$	−23.1539	−1.53678	−0.768390	+	0.639981i	$0.778942\pi$
$228$	0	0
$229$	3.96832	0.262234	0.131117	−	0.991367i	$-0.458144\pi$
$229$	3.96832	0.262234	0.131117	+	0.991367i	$0.458144\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.22268	−0.211125	−0.105562	−	0.994413i	$-0.533664\pi$
$233$	−3.22268	−0.211125	−0.105562	+	0.994413i	$0.533664\pi$
$234$	0	0
$235$	−10.9018	−0.711157
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.61958	0.557554	0.278777	−	0.960356i	$-0.410071\pi$
$239$	8.61958	0.557554	0.278777	+	0.960356i	$0.410071\pi$
$240$	0	0
$241$	−16.1857	−1.04261	−0.521305	−	0.853371i	$-0.674555\pi$
$241$	−16.1857	−1.04261	−0.521305	+	0.853371i	$0.674555\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−23.2856	−1.48766
$246$	0	0
$247$	10.8288	0.689022
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.3478	−1.59994	−0.799970	−	0.600040i	$-0.795152\pi$
$251$	−25.3478	−1.59994	−0.799970	+	0.600040i	$0.795152\pi$
$252$	0	0
$253$	−1.97368	−0.124084
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1203	1.13031	0.565157	−	0.824984i	$-0.308816\pi$
$257$	18.1203	1.13031	0.565157	+	0.824984i	$0.308816\pi$
$258$	0	0
$259$	−7.13932	−0.443616
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.8425	−1.22354	−0.611772	−	0.791034i	$-0.709543\pi$
$263$	−19.8425	−1.22354	−0.611772	+	0.791034i	$0.709543\pi$
$264$	0	0
$265$	−27.1241	−1.66622
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.92511	0.544174	0.272087	−	0.962273i	$-0.412286\pi$
$269$	8.92511	0.544174	0.272087	+	0.962273i	$0.412286\pi$
$270$	0	0
$271$	10.2587	0.623173	0.311587	−	0.950218i	$-0.399140\pi$
$271$	10.2587	0.623173	0.311587	+	0.950218i	$0.399140\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	29.6223	1.78629
$276$	0	0
$277$	−23.1891	−1.39330	−0.696648	−	0.717413i	$-0.745326\pi$
$277$	−23.1891	−1.39330	−0.696648	+	0.717413i	$0.745326\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−25.4496	−1.51820	−0.759099	−	0.650975i	$-0.774360\pi$
$281$	−25.4496	−1.51820	−0.759099	+	0.650975i	$0.774360\pi$
$282$	0	0
$283$	18.8323	1.11946	0.559732	−	0.828674i	$-0.310904\pi$
$283$	18.8323	1.11946	0.559732	+	0.828674i	$0.310904\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.10980	0.0655091
$288$	0	0
$289$	14.6886	0.864036
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.4422	1.66161	0.830806	−	0.556562i	$-0.187880\pi$
$293$	28.4422	1.66161	0.830806	+	0.556562i	$0.187880\pi$
$294$	0	0
$295$	−34.5717	−2.01284
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.986122	−0.0570289
$300$	0	0
$301$	2.98176	0.171866
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−30.0744	−1.72205
$306$	0	0
$307$	25.3562	1.44715	0.723577	−	0.690244i	$-0.242497\pi$
$307$	25.3562	1.44715	0.723577	+	0.690244i	$0.242497\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.7803	0.781408	0.390704	−	0.920516i	$-0.372232\pi$
$311$	13.7803	0.781408	0.390704	+	0.920516i	$0.372232\pi$
$312$	0	0
$313$	−0.0219175	−0.00123885	−0.000619425	−	1.00000i	$-0.500197\pi$
$313$	−0.0219175	−0.00123885	−0.000619425		1.00000i	$0.500197\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.7498	1.22159	0.610796	−	0.791788i	$-0.290850\pi$
$317$	21.7498	1.22159	0.610796	+	0.791788i	$0.290850\pi$
$318$	0	0
$319$	1.32788	0.0743471
$320$	0	0
$321$	0	0
$322$	0	0
$323$	44.8465	2.49533
$324$	0	0
$325$	14.8004	0.820977
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.94343	−0.162276
$330$	0	0
$331$	−35.3559	−1.94333	−0.971667	−	0.236353i	$-0.924048\pi$
$331$	−35.3559	−1.94333	−0.971667	+	0.236353i	$0.924048\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.63272	−0.143841
$336$	0	0
$337$	−16.5269	−0.900276	−0.450138	−	0.892959i	$-0.648625\pi$
$337$	−16.5269	−0.900276	−0.450138	+	0.892959i	$0.648625\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.0292	0.597266
$342$	0	0
$343$	−13.8204	−0.746232
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.9071	1.12235	0.561176	−	0.827697i	$-0.310349\pi$
$347$	20.9071	1.12235	0.561176	+	0.827697i	$0.310349\pi$
$348$	0	0
$349$	−25.4915	−1.36453	−0.682264	−	0.731105i	$-0.739005\pi$
$349$	−25.4915	−1.36453	−0.682264	+	0.731105i	$0.739005\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.8059	0.575143	0.287571	−	0.957759i	$-0.407152\pi$
$353$	10.8059	0.575143	0.287571	+	0.957759i	$0.407152\pi$
$354$	0	0
$355$	−39.8208	−2.11347
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.6799	0.933109	0.466555	−	0.884492i	$-0.345495\pi$
$359$	17.6799	0.933109	0.466555	+	0.884492i	$0.345495\pi$
$360$	0	0
$361$	44.4679	2.34041
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.46909	−0.181581
$366$	0	0
$367$	−2.22310	−0.116045	−0.0580224	−	0.998315i	$-0.518479\pi$
$367$	−2.22310	−0.116045	−0.0580224	+	0.998315i	$0.518479\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.32334	−0.380209
$372$	0	0
$373$	−7.93331	−0.410771	−0.205385	−	0.978681i	$-0.565845\pi$
$373$	−7.93331	−0.410771	−0.205385	+	0.978681i	$0.565845\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.663457	0.0341698
$378$	0	0
$379$	−24.8635	−1.27715	−0.638576	−	0.769559i	$-0.720476\pi$
$379$	−24.8635	−1.27715	−0.638576	+	0.769559i	$0.720476\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.1513	1.38737	0.693684	−	0.720280i	$-0.255987\pi$
$383$	27.1513	1.38737	0.693684	+	0.720280i	$0.255987\pi$
$384$	0	0
$385$	11.6705	0.594782
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.00443	0.152331	0.0761654	−	0.997095i	$-0.475732\pi$
$389$	3.00443	0.152331	0.0761654	+	0.997095i	$0.475732\pi$
$390$	0	0
$391$	−4.08392	−0.206533
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.85072	−0.495644
$396$	0	0
$397$	4.22555	0.212074	0.106037	−	0.994362i	$-0.466184\pi$
$397$	4.22555	0.212074	0.106037	+	0.994362i	$0.466184\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.46684	−0.422814	−0.211407	−	0.977398i	$-0.567805\pi$
$401$	−8.46684	−0.422814	−0.211407	+	0.977398i	$0.567805\pi$
$402$	0	0
$403$	5.51060	0.274503
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.0473	−0.894572
$408$	0	0
$409$	19.7391	0.976038	0.488019	−	0.872833i	$-0.337720\pi$
$409$	19.7391	0.976038	0.488019	+	0.872833i	$0.337720\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.33415	−0.459303
$414$	0	0
$415$	16.5474	0.812281
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.0264	1.07606	0.538030	−	0.842926i	$-0.319169\pi$
$419$	22.0264	1.07606	0.538030	+	0.842926i	$0.319169\pi$
$420$	0	0
$421$	−31.1074	−1.51608	−0.758041	−	0.652207i	$-0.773843\pi$
$421$	−31.1074	−1.51608	−0.758041	+	0.652207i	$0.773843\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	61.2942	2.97321
$426$	0	0
$427$	−8.11990	−0.392949
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.5605	0.990362	0.495181	−	0.868790i	$-0.335102\pi$
$431$	20.5605	0.990362	0.495181	+	0.868790i	$0.335102\pi$
$432$	0	0
$433$	26.6369	1.28009	0.640045	−	0.768337i	$-0.278916\pi$
$433$	26.6369	1.28009	0.640045	+	0.768337i	$0.278916\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.77967	−0.276479
$438$	0	0
$439$	10.6578	0.508667	0.254334	−	0.967117i	$-0.418144\pi$
$439$	10.6578	0.508667	0.254334	+	0.967117i	$0.418144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.9274	−1.13683	−0.568413	−	0.822743i	$-0.692443\pi$
$443$	−23.9274	−1.13683	−0.568413	+	0.822743i	$0.692443\pi$
$444$	0	0
$445$	−48.5277	−2.30043
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−29.3373	−1.38451	−0.692257	−	0.721651i	$-0.743383\pi$
$449$	−29.3373	−1.38451	−0.692257	+	0.721651i	$0.743383\pi$
$450$	0	0
$451$	2.80542	0.132102
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.83098	0.273361
$456$	0	0
$457$	25.2616	1.18169	0.590845	−	0.806785i	$-0.298795\pi$
$457$	25.2616	1.18169	0.590845	+	0.806785i	$0.298795\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.0384	−1.49218	−0.746090	−	0.665846i	$-0.768071\pi$
$461$	−32.0384	−1.49218	−0.746090	+	0.665846i	$0.768071\pi$
$462$	0	0
$463$	29.0212	1.34873	0.674365	−	0.738398i	$-0.264417\pi$
$463$	29.0212	1.34873	0.674365	+	0.738398i	$0.264417\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.3713	0.618751	0.309375	−	0.950940i	$-0.399880\pi$
$467$	13.3713	0.618751	0.309375	+	0.950940i	$0.399880\pi$
$468$	0	0
$469$	−0.710820	−0.0328226
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.53751	0.346575
$474$	0	0
$475$	86.7451	3.98014
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.6121	1.67285	0.836426	−	0.548080i	$-0.184641\pi$
$479$	36.6121	1.67285	0.836426	+	0.548080i	$0.184641\pi$
$480$	0	0
$481$	−9.01709	−0.411144
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−64.7310	−2.93928
$486$	0	0
$487$	−0.394191	−0.0178625	−0.00893125	−	0.999960i	$-0.502843\pi$
$487$	−0.394191	−0.0178625	−0.00893125	+	0.999960i	$0.502843\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.55268	0.340848	0.170424	−	0.985371i	$-0.445486\pi$
$491$	7.55268	0.340848	0.170424	+	0.985371i	$0.445486\pi$
$492$	0	0
$493$	2.74764	0.123747
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.7514	−0.482266
$498$	0	0
$499$	−17.3569	−0.777003	−0.388501	−	0.921448i	$-0.627007\pi$
$499$	−17.3569	−0.777003	−0.388501	+	0.921448i	$0.627007\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.2283	0.678997	0.339498	−	0.940607i	$-0.389743\pi$
$503$	15.2283	0.678997	0.339498	+	0.940607i	$0.389743\pi$
$504$	0	0
$505$	21.3820	0.951487
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.0461	−1.11015	−0.555074	−	0.831801i	$-0.687310\pi$
$509$	−25.0461	−1.11015	−0.555074	+	0.831801i	$0.687310\pi$
$510$	0	0
$511$	−0.936634	−0.0414343
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.3038	1.77600
$516$	0	0
$517$	−7.44061	−0.327238
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.9541	0.698960	0.349480	−	0.936944i	$-0.386358\pi$
$521$	15.9541	0.698960	0.349480	+	0.936944i	$0.386358\pi$
$522$	0	0
$523$	17.0590	0.745939	0.372969	−	0.927844i	$-0.378340\pi$
$523$	17.0590	0.745939	0.372969	+	0.927844i	$0.378340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.8216	0.994124
$528$	0	0
$529$	−22.4737	−0.977116
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.40169	0.0607140
$534$	0	0
$535$	42.3088	1.82917
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−15.8926	−0.684545
$540$	0	0
$541$	19.7206	0.847854	0.423927	−	0.905696i	$-0.360651\pi$
$541$	19.7206	0.847854	0.423927	+	0.905696i	$0.360651\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.07566	−0.0460762
$546$	0	0
$547$	−3.86165	−0.165112	−0.0825560	−	0.996586i	$-0.526308\pi$
$547$	−3.86165	−0.165112	−0.0825560	+	0.996586i	$0.526308\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.88853	0.165657
$552$	0	0
$553$	−2.65964	−0.113099
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.0103933	0.000440379 0	0.000220190	−	1.00000i	$-0.499930\pi$
$557$	0.0103933	0.000440379 0	0.000220190		1.00000i	$0.499930\pi$
$558$	0	0
$559$	3.76601	0.159285
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.97601	−0.420439	−0.210219	−	0.977654i	$-0.567418\pi$
$563$	−9.97601	−0.420439	−0.210219	+	0.977654i	$0.567418\pi$
$564$	0	0
$565$	−78.0149	−3.28211
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.2399	0.764655	0.382327	−	0.924027i	$-0.375123\pi$
$569$	18.2399	0.764655	0.382327	+	0.924027i	$0.375123\pi$
$570$	0	0
$571$	3.28635	0.137530	0.0687648	−	0.997633i	$-0.478094\pi$
$571$	3.28635	0.137530	0.0687648	+	0.997633i	$0.478094\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.89939	−0.329427
$576$	0	0
$577$	−11.9215	−0.496299	−0.248149	−	0.968722i	$-0.579822\pi$
$577$	−11.9215	−0.496299	−0.248149	+	0.968722i	$0.579822\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.46771	0.185352
$582$	0	0
$583$	−18.5125	−0.766709
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.0199	1.65180	0.825898	−	0.563819i	$-0.190669\pi$
$587$	40.0199	1.65180	0.825898	+	0.563819i	$0.190669\pi$
$588$	0	0
$589$	32.2977	1.33080
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.85756	0.281606	0.140803	−	0.990038i	$-0.455032\pi$
$593$	6.85756	0.281606	0.140803	+	0.990038i	$0.455032\pi$
$594$	0	0
$595$	24.1484	0.989988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.71380	−0.151742	−0.0758708	−	0.997118i	$-0.524174\pi$
$599$	−3.71380	−0.151742	−0.0758708	+	0.997118i	$0.524174\pi$
$600$	0	0
$601$	10.8484	0.442517	0.221259	−	0.975215i	$-0.428983\pi$
$601$	10.8484	0.442517	0.221259	+	0.975215i	$0.428983\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−14.3450	−0.583205
$606$	0	0
$607$	5.55526	0.225481	0.112740	−	0.993624i	$-0.464037\pi$
$607$	5.55526	0.225481	0.112740	+	0.993624i	$0.464037\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.71760	−0.150398
$612$	0	0
$613$	22.2433	0.898397	0.449198	−	0.893432i	$-0.351710\pi$
$613$	22.2433	0.898397	0.449198	+	0.893432i	$0.351710\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.2686	−0.896500	−0.448250	−	0.893908i	$-0.647953\pi$
$617$	−22.2686	−0.896500	−0.448250	+	0.893908i	$0.647953\pi$
$618$	0	0
$619$	45.9940	1.84866	0.924328	−	0.381599i	$-0.124626\pi$
$619$	45.9940	1.84866	0.924328	+	0.381599i	$0.124626\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.1022	−0.524928
$624$	0	0
$625$	39.1169	1.56468
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.3433	−1.48898
$630$	0	0
$631$	4.11215	0.163702	0.0818511	−	0.996645i	$-0.473917\pi$
$631$	4.11215	0.163702	0.0818511	+	0.996645i	$0.473917\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−50.4316	−2.00132
$636$	0	0
$637$	−7.94054	−0.314616
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.6463	0.460002	0.230001	−	0.973190i	$-0.426127\pi$
$641$	11.6463	0.460002	0.230001	+	0.973190i	$0.426127\pi$
$642$	0	0
$643$	12.3732	0.487952	0.243976	−	0.969781i	$-0.421548\pi$
$643$	12.3732	0.487952	0.243976	+	0.969781i	$0.421548\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.1460	0.398879	0.199440	−	0.979910i	$-0.436088\pi$
$647$	10.1460	0.398879	0.199440	+	0.979910i	$0.436088\pi$
$648$	0	0
$649$	−23.5956	−0.926206
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.6366	1.12064	0.560318	−	0.828278i	$-0.310679\pi$
$653$	28.6366	1.12064	0.560318	+	0.828278i	$0.310679\pi$
$654$	0	0
$655$	5.72760	0.223796
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.28002	−0.283589	−0.141795	−	0.989896i	$-0.545287\pi$
$659$	−7.28002	−0.283589	−0.141795	+	0.989896i	$0.545287\pi$
$660$	0	0
$661$	27.7814	1.08057	0.540285	−	0.841482i	$-0.318317\pi$
$661$	27.7814	1.08057	0.540285	+	0.841482i	$0.318317\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	34.1754	1.32527
$666$	0	0
$667$	−0.354107	−0.0137111
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20.5261	−0.792401
$672$	0	0
$673$	8.14415	0.313934	0.156967	−	0.987604i	$-0.449828\pi$
$673$	8.14415	0.313934	0.156967	+	0.987604i	$0.449828\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.56085	0.290587	0.145293	−	0.989389i	$-0.453587\pi$
$677$	7.56085	0.290587	0.145293	+	0.989389i	$0.453587\pi$
$678$	0	0
$679$	−17.4770	−0.670705
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.0681	−0.691357	−0.345678	−	0.938353i	$-0.612351\pi$
$683$	−18.0681	−0.691357	−0.345678	+	0.938353i	$0.612351\pi$
$684$	0	0
$685$	60.8379	2.32450
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.24951	−0.352378
$690$	0	0
$691$	−20.0591	−0.763085	−0.381542	−	0.924351i	$-0.624607\pi$
$691$	−20.0591	−0.763085	−0.381542	+	0.924351i	$0.624607\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−43.3851	−1.64569
$696$	0	0
$697$	5.80496	0.219878
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.25563	0.236272	0.118136	−	0.992997i	$-0.462308\pi$
$701$	6.25563	0.236272	0.118136	+	0.992997i	$0.462308\pi$
$702$	0	0
$703$	−52.8492	−1.99325
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.77301	0.217116
$708$	0	0
$709$	−3.89100	−0.146130	−0.0730648	−	0.997327i	$-0.523278\pi$
$709$	−3.89100	−0.146130	−0.0730648	+	0.997327i	$0.523278\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.94117	−0.110148
$714$	0	0
$715$	14.7400	0.551245
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.0479	−1.00872	−0.504358	−	0.863495i	$-0.668271\pi$
$719$	−27.0479	−1.00872	−0.504358	+	0.863495i	$0.668271\pi$
$720$	0	0
$721$	10.8818	0.405259
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.31467	0.197382
$726$	0	0
$727$	−47.6195	−1.76611	−0.883055	−	0.469269i	$-0.844517\pi$
$727$	−47.6195	−1.76611	−0.883055	+	0.469269i	$0.844517\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.5966	0.576859
$732$	0	0
$733$	3.56964	0.131848	0.0659239	−	0.997825i	$-0.479001\pi$
$733$	3.56964	0.131848	0.0659239	+	0.997825i	$0.479001\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.79686	−0.0661883
$738$	0	0
$739$	−23.4393	−0.862227	−0.431113	−	0.902298i	$-0.641879\pi$
$739$	−23.4393	−0.862227	−0.431113	+	0.902298i	$0.641879\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.2193	1.25538	0.627692	−	0.778462i	$-0.284000\pi$
$743$	34.2193	1.25538	0.627692	+	0.778462i	$0.284000\pi$
$744$	0	0
$745$	8.87550	0.325173
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.4231	0.417391
$750$	0	0
$751$	14.9292	0.544775	0.272387	−	0.962188i	$-0.412187\pi$
$751$	14.9292	0.544775	0.272387	+	0.962188i	$0.412187\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	59.0826	2.15024
$756$	0	0
$757$	−40.5631	−1.47429	−0.737145	−	0.675734i	$-0.763827\pi$
$757$	−40.5631	−1.47429	−0.737145	+	0.675734i	$0.763827\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.7884	0.826080	0.413040	−	0.910713i	$-0.364467\pi$
$761$	22.7884	0.826080	0.413040	+	0.910713i	$0.364467\pi$
$762$	0	0
$763$	−0.290422	−0.0105140
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.7892	−0.425683
$768$	0	0
$769$	47.9787	1.73016	0.865079	−	0.501636i	$-0.167268\pi$
$769$	47.9787	1.73016	0.865079	+	0.501636i	$0.167268\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.63473	0.0587972	0.0293986	−	0.999568i	$-0.490641\pi$
$773$	1.63473	0.0587972	0.0293986	+	0.999568i	$0.490641\pi$
$774$	0	0
$775$	44.1430	1.58566
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.21532	0.294344
$780$	0	0
$781$	−27.1782	−0.972511
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−81.5895	−2.91206
$786$	0	0
$787$	−45.2538	−1.61312	−0.806561	−	0.591150i	$-0.798674\pi$
$787$	−45.2538	−1.61312	−0.806561	+	0.591150i	$0.798674\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−21.0636	−0.748934
$792$	0	0
$793$	−10.2556	−0.364186
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.5874	−0.375026	−0.187513	−	0.982262i	$-0.560043\pi$
$797$	−10.5874	−0.375026	−0.187513	+	0.982262i	$0.560043\pi$
$798$	0	0
$799$	−15.3961	−0.544673
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.36769	−0.0835541
$804$	0	0
$805$	−3.11217	−0.109689
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.6573	0.445006	0.222503	−	0.974932i	$-0.428577\pi$
$809$	12.6573	0.445006	0.222503	+	0.974932i	$0.428577\pi$
$810$	0	0
$811$	21.8496	0.767242	0.383621	−	0.923491i	$-0.374677\pi$
$811$	21.8496	0.767242	0.383621	+	0.923491i	$0.374677\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−60.8199	−2.13043
$816$	0	0
$817$	22.0726	0.772223
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.9492	−0.451929	−0.225965	−	0.974135i	$-0.572553\pi$
$821$	−12.9492	−0.451929	−0.225965	+	0.974135i	$0.572553\pi$
$822$	0	0
$823$	27.8726	0.971578	0.485789	−	0.874076i	$-0.338532\pi$
$823$	27.8726	0.971578	0.485789	+	0.874076i	$0.338532\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.13400	0.317620	0.158810	−	0.987309i	$-0.449234\pi$
$827$	9.13400	0.317620	0.158810	+	0.987309i	$0.449234\pi$
$828$	0	0
$829$	18.5297	0.643564	0.321782	−	0.946814i	$-0.395718\pi$
$829$	18.5297	0.643564	0.321782	+	0.946814i	$0.395718\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−32.8849	−1.13940
$834$	0	0
$835$	−3.98604	−0.137943
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.4815	1.70829	0.854146	−	0.520033i	$-0.174081\pi$
$839$	49.4815	1.70829	0.854146	+	0.520033i	$0.174081\pi$
$840$	0	0
$841$	−28.7618	−0.991785
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−44.4539	−1.52926
$846$	0	0
$847$	−3.87305	−0.133080
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.81268	0.164977
$852$	0	0
$853$	53.0577	1.81666	0.908330	−	0.418253i	$-0.137358\pi$
$853$	53.0577	1.81666	0.908330	+	0.418253i	$0.137358\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.8640	−0.576065	−0.288032	−	0.957621i	$-0.593001\pi$
$857$	−16.8640	−0.576065	−0.288032	+	0.957621i	$0.593001\pi$
$858$	0	0
$859$	−4.70157	−0.160415	−0.0802077	−	0.996778i	$-0.525558\pi$
$859$	−4.70157	−0.160415	−0.0802077	+	0.996778i	$0.525558\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.1921	−0.380985	−0.190492	−	0.981689i	$-0.561008\pi$
$863$	−11.1921	−0.380985	−0.190492	+	0.981689i	$0.561008\pi$
$864$	0	0
$865$	74.0580	2.51805
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.72323	−0.228070
$870$	0	0
$871$	−0.897778	−0.0304200
$872$	0	0
$873$	0	0
$874$	0	0
$875$	25.2605	0.853960
$876$	0	0
$877$	−4.91528	−0.165977	−0.0829886	−	0.996551i	$-0.526446\pi$
$877$	−4.91528	−0.165977	−0.0829886	+	0.996551i	$0.526446\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.40940	−0.317011	−0.158505	−	0.987358i	$-0.550668\pi$
$881$	−9.40940	−0.317011	−0.158505	+	0.987358i	$0.550668\pi$
$882$	0	0
$883$	−48.1886	−1.62167	−0.810837	−	0.585271i	$-0.800988\pi$
$883$	−48.1886	−1.62167	−0.810837	+	0.585271i	$0.800988\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.34460	0.0787240	0.0393620	−	0.999225i	$-0.487467\pi$
$887$	2.34460	0.0787240	0.0393620	+	0.999225i	$0.487467\pi$
$888$	0	0
$889$	−13.6162	−0.456674
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.7889	−0.729137
$894$	0	0
$895$	−29.4065	−0.982951
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.97880	0.0659967
$900$	0	0
$901$	−38.3059	−1.27615
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−63.9801	−2.12677
$906$	0	0
$907$	−9.21191	−0.305876	−0.152938	−	0.988236i	$-0.548874\pi$
$907$	−9.21191	−0.305876	−0.152938	+	0.988236i	$0.548874\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42.5980	−1.41133	−0.705667	−	0.708544i	$-0.749352\pi$
$911$	−42.5980	−1.41133	−0.705667	+	0.708544i	$0.749352\pi$
$912$	0	0
$913$	11.2938	0.373770
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.54642	0.0510672
$918$	0	0
$919$	24.2536	0.800053	0.400026	−	0.916504i	$-0.369001\pi$
$919$	24.2536	0.800053	0.400026	+	0.916504i	$0.369001\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13.5792	−0.446964
$924$	0	0
$925$	−72.2320	−2.37497
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20.8743	−0.684865	−0.342432	−	0.939543i	$-0.611251\pi$
$929$	−20.8743	−0.684865	−0.342432	+	0.939543i	$0.611251\pi$
$930$	0	0
$931$	−46.5395	−1.52527
$932$	0	0
$933$	0	0
$934$	0	0
$935$	61.0442	1.99636
$936$	0	0
$937$	41.9088	1.36910	0.684551	−	0.728965i	$-0.259998\pi$
$937$	41.9088	1.36910	0.684551	+	0.728965i	$0.259998\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48.8746	1.59327	0.796634	−	0.604463i	$-0.206612\pi$
$941$	48.8746	1.59327	0.796634	+	0.604463i	$0.206612\pi$
$942$	0	0
$943$	−0.748123	−0.0243622
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.0372445	0.00121028	0.000605141	−	1.00000i	$-0.499807\pi$
$947$	0.0372445	0.00121028	0.000605141		1.00000i	$0.499807\pi$
$948$	0	0
$949$	−1.18299	−0.0384013
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.07310	−0.131941	−0.0659703	−	0.997822i	$-0.521014\pi$
$953$	−4.07310	−0.131941	−0.0659703	+	0.997822i	$0.521014\pi$
$954$	0	0
$955$	56.9778	1.84376
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.4259	0.530419
$960$	0	0
$961$	−14.5643	−0.469816
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.5395	−0.532425
$966$	0	0
$967$	34.2602	1.10173	0.550867	−	0.834593i	$-0.314297\pi$
$967$	34.2602	1.10173	0.550867	+	0.834593i	$0.314297\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10.3068	−0.330761	−0.165380	−	0.986230i	$-0.552885\pi$
$971$	−10.3068	−0.330761	−0.165380	+	0.986230i	$0.552885\pi$
$972$	0	0
$973$	−11.7137	−0.375524
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.1012	0.419144	0.209572	−	0.977793i	$-0.432793\pi$
$977$	13.1012	0.419144	0.209572	+	0.977793i	$0.432793\pi$
$978$	0	0
$979$	−33.1207	−1.05854
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.2773	−0.806221	−0.403110	−	0.915151i	$-0.632071\pi$
$983$	−25.2773	−0.806221	−0.403110	+	0.915151i	$0.632071\pi$
$984$	0	0
$985$	20.5203	0.653832
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.01003	−0.0639153
$990$	0	0
$991$	40.5855	1.28924	0.644620	−	0.764503i	$-0.277015\pi$
$991$	40.5855	1.28924	0.644620	+	0.764503i	$0.277015\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	46.3638	1.46983
$996$	0	0
$997$	−51.5866	−1.63376	−0.816881	−	0.576806i	$-0.804299\pi$
$997$	−51.5866	−1.63376	−0.816881	+	0.576806i	$0.804299\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.k.1.10	yes	10
3.2	odd	2		6012.2.a.j.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.1	✓	10	3.2	odd	2
6012.2.a.k.1.10	yes	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+3.98604 q^{5} +1.07621 q^{7} +O(q^{10})\)	\(q+3.98604 q^{5} +1.07621 q^{7} +2.72051 q^{11} +1.35927 q^{13} +5.62926 q^{17} +7.96667 q^{19} -0.725481 q^{23} +10.8885 q^{25} +0.488099 q^{29} +4.05410 q^{31} +4.28980 q^{35} -6.63379 q^{37} +1.03121 q^{41} +2.77062 q^{43} -2.73500 q^{47} -5.84178 q^{49} -6.80477 q^{53} +10.8441 q^{55} -8.67320 q^{59} -7.54493 q^{61} +5.41809 q^{65} -0.660487 q^{67} -9.99008 q^{71} -0.870311 q^{73} +2.92783 q^{77} -2.47131 q^{79} +4.15135 q^{83} +22.4385 q^{85} -12.1744 q^{89} +1.46285 q^{91} +31.7555 q^{95} -16.2394 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \) 10 * q + 6 * q^5 + 4 * q^7	\( 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) \) 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

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