LMFDB - Embedded newform 6975.2.a.cg.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16,0,0,4,0,0,0,0,0,8,0,0,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.116450197504.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 58x^{4} - 62x^{2} + 1 $ x^8 - 14x^6 + 58x^4 - 62*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.27109$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.978865 q^{2} -1.04182 q^{4} -1.38432 q^{7} +2.97753 q^{8} -2.53620 q^{11} -4.49692 q^{13} +1.35507 q^{14} -0.830955 q^{16} +4.15867 q^{17} -5.63702 q^{19} +2.48260 q^{22} -7.13620 q^{23} +4.40188 q^{26} +1.44222 q^{28} +5.51373 q^{29} -1.00000 q^{31} -5.14168 q^{32} -4.07077 q^{34} -5.88124 q^{37} +5.51788 q^{38} -3.95640 q^{41} +7.65750 q^{43} +2.64227 q^{44} +6.98538 q^{46} +3.58020 q^{47} -5.08365 q^{49} +4.68500 q^{52} -10.8535 q^{53} -4.12187 q^{56} -5.39720 q^{58} -7.07107 q^{59} -10.5747 q^{61} +0.978865 q^{62} +6.69491 q^{64} +12.1255 q^{67} -4.33260 q^{68} +4.53901 q^{71} +4.68500 q^{73} +5.75694 q^{74} +5.87278 q^{76} +3.51092 q^{77} +0.0707729 q^{79} +3.87278 q^{82} -17.2810 q^{83} -7.49566 q^{86} -7.55162 q^{88} +11.6752 q^{89} +6.22519 q^{91} +7.43467 q^{92} -3.50453 q^{94} +16.9564 q^{97} +4.97620 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 4 q^{7} + 8 q^{13} + 12 q^{16} + 28 q^{22} + 36 q^{28} - 8 q^{31} - 28 q^{34} + 12 q^{37} + 52 q^{43} - 16 q^{46} + 8 q^{49} + 56 q^{52} + 16 q^{58} - 28 q^{61} + 48 q^{64} + 24 q^{67}+ \cdots + 44 q^{97}+O(q^{100}) $ 8 * q + 16 * q^4 + 4 * q^7 + 8 * q^13 + 12 * q^16 + 28 * q^22 + 36 * q^28 - 8 * q^31 - 28 * q^34 + 12 * q^37 + 52 * q^43 - 16 * q^46 + 8 * q^49 + 56 * q^52 + 16 * q^58 - 28 * q^61 + 48 * q^64 + 24 * q^67 + 56 * q^73 + 4 * q^76 - 4 * q^79 - 12 * q^82 + 52 * q^88 - 8 * q^91 - 64 * q^94 + 44 * q^97

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 $

$2$	−0.978865	−0.692162	−0.346081	−	0.938205i	$-0.612488\pi$
$2$	−0.978865	−0.692162	−0.346081	+	0.938205i	$0.612488\pi$
$3$	0	0
$3$	0	0
$4$	−1.04182	−0.520912
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.38432	−0.523225	−0.261613	−	0.965173i	$-0.584254\pi$
$7$	−1.38432	−0.523225	−0.261613	+	0.965173i	$0.584254\pi$
$8$	2.97753	1.05272
$9$	0	0
$10$	0	0
$11$	−2.53620	−0.764693	−0.382346	−	0.924019i	$-0.624884\pi$
$11$	−2.53620	−0.764693	−0.382346	+	0.924019i	$0.624884\pi$
$12$	0	0
$13$	−4.49692	−1.24722	−0.623611	−	0.781735i	$-0.714335\pi$
$13$	−4.49692	−1.24722	−0.623611	+	0.781735i	$0.714335\pi$
$14$	1.35507	0.362157
$15$	0	0
$16$	−0.830955	−0.207739
$17$	4.15867	1.00863	0.504313	−	0.863521i	$-0.331746\pi$
$17$	4.15867	1.00863	0.504313	+	0.863521i	$0.331746\pi$
$18$	0	0
$19$	−5.63702	−1.29322	−0.646610	−	0.762821i	$-0.723814\pi$
$19$	−5.63702	−1.29322	−0.646610	+	0.762821i	$0.723814\pi$
$20$	0	0
$21$	0	0
$22$	2.48260	0.529291
$23$	−7.13620	−1.48800	−0.744000	−	0.668179i	$-0.767074\pi$
$23$	−7.13620	−1.48800	−0.744000	+	0.668179i	$0.767074\pi$
$24$	0	0
$25$	0	0
$26$	4.40188	0.863279
$27$	0	0
$28$	1.44222	0.272554
$29$	5.51373	1.02387	0.511937	−	0.859023i	$-0.328928\pi$
$29$	5.51373	1.02387	0.511937	+	0.859023i	$0.328928\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.14168	−0.908928
$33$	0	0
$34$	−4.07077	−0.698132
$35$	0	0
$36$	0	0
$37$	−5.88124	−0.966871	−0.483435	−	0.875380i	$-0.660611\pi$
$37$	−5.88124	−0.966871	−0.483435	+	0.875380i	$0.660611\pi$
$38$	5.51788	0.895118
$39$	0	0
$40$	0	0
$41$	−3.95640	−0.617886	−0.308943	−	0.951081i	$-0.599975\pi$
$41$	−3.95640	−0.617886	−0.308943	+	0.951081i	$0.599975\pi$
$42$	0	0
$43$	7.65750	1.16776	0.583879	−	0.811841i	$-0.301534\pi$
$43$	7.65750	1.16776	0.583879	+	0.811841i	$0.301534\pi$
$44$	2.64227	0.398338
$45$	0	0
$46$	6.98538	1.02994
$47$	3.58020	0.522226	0.261113	−	0.965308i	$-0.415911\pi$
$47$	3.58020	0.522226	0.261113	+	0.965308i	$0.415911\pi$
$48$	0	0
$49$	−5.08365	−0.726235
$50$	0	0
$51$	0	0
$52$	4.68500	0.649693
$53$	−10.8535	−1.49085	−0.745424	−	0.666591i	$-0.767753\pi$
$53$	−10.8535	−1.49085	−0.745424	+	0.666591i	$0.767753\pi$
$54$	0	0
$55$	0	0
$56$	−4.12187	−0.550808
$57$	0	0
$58$	−5.39720	−0.708687
$59$	−7.07107	−0.920575	−0.460287	−	0.887770i	$-0.652254\pi$
$59$	−7.07107	−0.920575	−0.460287	+	0.887770i	$0.652254\pi$
$60$	0	0
$61$	−10.5747	−1.35395	−0.676976	−	0.736005i	$-0.736710\pi$
$61$	−10.5747	−1.35395	−0.676976	+	0.736005i	$0.736710\pi$
$62$	0.978865	0.124316
$63$	0	0
$64$	6.69491	0.836864
$65$	0	0
$66$	0	0
$67$	12.1255	1.48136	0.740681	−	0.671857i	$-0.234503\pi$
$67$	12.1255	1.48136	0.740681	+	0.671857i	$0.234503\pi$
$68$	−4.33260	−0.525405
$69$	0	0
$70$	0	0
$71$	4.53901	0.538682	0.269341	−	0.963045i	$-0.413194\pi$
$71$	4.53901	0.538682	0.269341	+	0.963045i	$0.413194\pi$
$72$	0	0
$73$	4.68500	0.548338	0.274169	−	0.961682i	$-0.411597\pi$
$73$	4.68500	0.548338	0.274169	+	0.961682i	$0.411597\pi$
$74$	5.75694	0.669231
$75$	0	0
$76$	5.87278	0.673654
$77$	3.51092	0.400107
$78$	0	0
$79$	0.0707729	0.00796257	0.00398129	−	0.999992i	$-0.498733\pi$
$79$	0.0707729	0.00796257	0.00398129	+	0.999992i	$0.498733\pi$
$80$	0	0
$81$	0	0
$82$	3.87278	0.427677
$83$	−17.2810	−1.89684	−0.948418	−	0.317023i	$-0.897317\pi$
$83$	−17.2810	−1.89684	−0.948418	+	0.317023i	$0.897317\pi$
$84$	0	0
$85$	0	0
$86$	−7.49566	−0.808277
$87$	0	0
$88$	−7.55162	−0.805005
$89$	11.6752	1.23757	0.618785	−	0.785560i	$-0.287625\pi$
$89$	11.6752	1.23757	0.618785	+	0.785560i	$0.287625\pi$
$90$	0	0
$91$	6.22519	0.652578
$92$	7.43467	0.775117
$93$	0	0
$94$	−3.50453	−0.361465
$95$	0	0
$96$	0	0
$97$	16.9564	1.72166	0.860832	−	0.508889i	$-0.169944\pi$
$97$	16.9564	1.72166	0.860832	+	0.508889i	$0.169944\pi$
$98$	4.97620	0.502672
$99$	0	0
$100$	0	0
$101$	−8.31734	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$101$	−8.31734	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$102$	0	0
$103$	−14.4779	−1.42655	−0.713274	−	0.700885i	$-0.752789\pi$
$103$	−14.4779	−1.42655	−0.713274	+	0.700885i	$0.752789\pi$
$104$	−13.3897	−1.31297
$105$	0	0
$106$	10.6241	1.03191
$107$	5.98341	0.578438	0.289219	−	0.957263i	$-0.406604\pi$
$107$	5.98341	0.578438	0.289219	+	0.957263i	$0.406604\pi$
$108$	0	0
$109$	−13.8941	−1.33082	−0.665408	−	0.746480i	$-0.731742\pi$
$109$	−13.8941	−1.33082	−0.665408	+	0.746480i	$0.731742\pi$
$110$	0	0
$111$	0	0
$112$	1.15031	0.108694
$113$	4.87427	0.458533	0.229267	−	0.973364i	$-0.426367\pi$
$113$	4.87427	0.458533	0.229267	+	0.973364i	$0.426367\pi$
$114$	0	0
$115$	0	0
$116$	−5.74434	−0.533349
$117$	0	0
$118$	6.92162	0.637187
$119$	−5.75694	−0.527738
$120$	0	0
$121$	−4.56769	−0.415245
$122$	10.3512	0.937154
$123$	0	0
$124$	1.04182	0.0935586
$125$	0	0
$126$	0	0
$127$	9.95202	0.883099	0.441549	−	0.897237i	$-0.354429\pi$
$127$	9.95202	0.883099	0.441549	+	0.897237i	$0.354429\pi$
$128$	3.72994	0.329683
$129$	0	0
$130$	0	0
$131$	−9.08979	−0.794178	−0.397089	−	0.917780i	$-0.629980\pi$
$131$	−9.08979	−0.794178	−0.397089	+	0.917780i	$0.629980\pi$
$132$	0	0
$133$	7.80346	0.676645
$134$	−11.8692	−1.02534
$135$	0	0
$136$	12.3826	1.06180
$137$	14.2356	1.21623	0.608115	−	0.793849i	$-0.291926\pi$
$137$	14.2356	1.21623	0.608115	+	0.793849i	$0.291926\pi$
$138$	0	0
$139$	17.0646	1.44740	0.723701	−	0.690114i	$-0.242440\pi$
$139$	17.0646	1.44740	0.723701	+	0.690114i	$0.242440\pi$
$140$	0	0
$141$	0	0
$142$	−4.44308	−0.372855
$143$	11.4051	0.953741
$144$	0	0
$145$	0	0
$146$	−4.58598	−0.379538
$147$	0	0
$148$	6.12722	0.503654
$149$	−23.3095	−1.90959	−0.954794	−	0.297269i	$-0.903924\pi$
$149$	−23.3095	−1.90959	−0.954794	+	0.297269i	$0.903924\pi$
$150$	0	0
$151$	2.63702	0.214597	0.107299	−	0.994227i	$-0.465780\pi$
$151$	2.63702	0.214597	0.107299	+	0.994227i	$0.465780\pi$
$152$	−16.7844	−1.36140
$153$	0	0
$154$	−3.43672	−0.276938
$155$	0	0
$156$	0	0
$157$	7.07222	0.564425	0.282212	−	0.959352i	$-0.408932\pi$
$157$	7.07222	0.564425	0.282212	+	0.959352i	$0.408932\pi$
$158$	−0.0692771	−0.00551139
$159$	0	0
$160$	0	0
$161$	9.87881	0.778560
$162$	0	0
$163$	−0.741148	−0.0580512	−0.0290256	−	0.999579i	$-0.509240\pi$
$163$	−0.741148	−0.0580512	−0.0290256	+	0.999579i	$0.509240\pi$
$164$	4.12187	0.321864
$165$	0	0
$166$	16.9158	1.31292
$167$	1.75132	0.135521	0.0677605	−	0.997702i	$-0.478415\pi$
$167$	1.75132	0.135521	0.0677605	+	0.997702i	$0.478415\pi$
$168$	0	0
$169$	7.22229	0.555561
$170$	0	0
$171$	0	0
$172$	−7.97777	−0.608299
$173$	6.20242	0.471561	0.235781	−	0.971806i	$-0.424235\pi$
$173$	6.20242	0.471561	0.235781	+	0.971806i	$0.424235\pi$
$174$	0	0
$175$	0	0
$176$	2.10747	0.158856
$177$	0	0
$178$	−11.4285	−0.856599
$179$	−20.8301	−1.55692	−0.778459	−	0.627696i	$-0.783998\pi$
$179$	−20.8301	−1.55692	−0.778459	+	0.627696i	$0.783998\pi$
$180$	0	0
$181$	0.762788	0.0566976	0.0283488	−	0.999598i	$-0.490975\pi$
$181$	0.762788	0.0566976	0.0283488	+	0.999598i	$0.490975\pi$
$182$	−6.09362	−0.451689
$183$	0	0
$184$	−21.2483	−1.56644
$185$	0	0
$186$	0	0
$187$	−10.5472	−0.771288
$188$	−3.72994	−0.272034
$189$	0	0
$190$	0	0
$191$	−5.38074	−0.389337	−0.194668	−	0.980869i	$-0.562363\pi$
$191$	−5.38074	−0.389337	−0.194668	+	0.980869i	$0.562363\pi$
$192$	0	0
$193$	22.2117	1.59884	0.799418	−	0.600776i	$-0.205142\pi$
$193$	22.2117	1.59884	0.799418	+	0.600776i	$0.205142\pi$
$194$	−16.5980	−1.19167
$195$	0	0
$196$	5.29627	0.378305
$197$	−21.4537	−1.52851	−0.764256	−	0.644913i	$-0.776894\pi$
$197$	−21.4537	−1.52851	−0.764256	+	0.644913i	$0.776894\pi$
$198$	0	0
$199$	−12.4560	−0.882979	−0.441490	−	0.897266i	$-0.645550\pi$
$199$	−12.4560	−0.882979	−0.441490	+	0.897266i	$0.645550\pi$
$200$	0	0
$201$	0	0
$202$	8.14155	0.572837
$203$	−7.63279	−0.535717
$204$	0	0
$205$	0	0
$206$	14.1719	0.987402
$207$	0	0
$208$	3.73674	0.259096
$209$	14.2966	0.988916
$210$	0	0
$211$	25.7900	1.77546	0.887728	−	0.460368i	$-0.152283\pi$
$211$	25.7900	1.77546	0.887728	+	0.460368i	$0.152283\pi$
$212$	11.3075	0.776600
$213$	0	0
$214$	−5.85694	−0.400372
$215$	0	0
$216$	0	0
$217$	1.38432	0.0939740
$218$	13.6005	0.921140
$219$	0	0
$220$	0	0
$221$	−18.7012	−1.25798
$222$	0	0
$223$	−0.764238	−0.0511771	−0.0255886	−	0.999673i	$-0.508146\pi$
$223$	−0.764238	−0.0511771	−0.0255886	+	0.999673i	$0.508146\pi$
$224$	7.11774	0.475574
$225$	0	0
$226$	−4.77125	−0.317379
$227$	−24.5684	−1.63066	−0.815330	−	0.578997i	$-0.803444\pi$
$227$	−24.5684	−1.63066	−0.815330	+	0.578997i	$0.803444\pi$
$228$	0	0
$229$	−30.0629	−1.98661	−0.993305	−	0.115521i	$-0.963146\pi$
$229$	−30.0629	−1.98661	−0.993305	+	0.115521i	$0.963146\pi$
$230$	0	0
$231$	0	0
$232$	16.4173	1.07785
$233$	3.04681	0.199603	0.0998016	−	0.995007i	$-0.468179\pi$
$233$	3.04681	0.199603	0.0998016	+	0.995007i	$0.468179\pi$
$234$	0	0
$235$	0	0
$236$	7.36681	0.479538
$237$	0	0
$238$	5.63527	0.365280
$239$	20.1975	1.30647	0.653235	−	0.757155i	$-0.273411\pi$
$239$	20.1975	1.30647	0.653235	+	0.757155i	$0.273411\pi$
$240$	0	0
$241$	13.4185	0.864364	0.432182	−	0.901786i	$-0.357744\pi$
$241$	13.4185	0.864364	0.432182	+	0.901786i	$0.357744\pi$
$242$	4.47115	0.287417
$243$	0	0
$244$	11.0170	0.705290
$245$	0	0
$246$	0	0
$247$	25.3492	1.61293
$248$	−2.97753	−0.189074
$249$	0	0
$250$	0	0
$251$	−18.7746	−1.18504	−0.592522	−	0.805554i	$-0.701868\pi$
$251$	−18.7746	−1.18504	−0.592522	+	0.805554i	$0.701868\pi$
$252$	0	0
$253$	18.0988	1.13786
$254$	−9.74168	−0.611247
$255$	0	0
$256$	−17.0409	−1.06506
$257$	10.1179	0.631136	0.315568	−	0.948903i	$-0.397805\pi$
$257$	10.1179	0.631136	0.315568	+	0.948903i	$0.397805\pi$
$258$	0	0
$259$	8.14155	0.505891
$260$	0	0
$261$	0	0
$262$	8.89767	0.549700
$263$	2.31388	0.142680	0.0713400	−	0.997452i	$-0.477272\pi$
$263$	2.31388	0.142680	0.0713400	+	0.997452i	$0.477272\pi$
$264$	0	0
$265$	0	0
$266$	−7.63853	−0.468348
$267$	0	0
$268$	−12.6326	−0.771659
$269$	20.0986	1.22543	0.612717	−	0.790302i	$-0.290076\pi$
$269$	20.0986	1.22543	0.612717	+	0.790302i	$0.290076\pi$
$270$	0	0
$271$	8.93739	0.542908	0.271454	−	0.962451i	$-0.412496\pi$
$271$	8.93739	0.542908	0.271454	+	0.962451i	$0.412496\pi$
$272$	−3.45567	−0.209531
$273$	0	0
$274$	−13.9347	−0.841828
$275$	0	0
$276$	0	0
$277$	13.3290	0.800864	0.400432	−	0.916326i	$-0.368860\pi$
$277$	13.3290	0.800864	0.400432	+	0.916326i	$0.368860\pi$
$278$	−16.7039	−1.00184
$279$	0	0
$280$	0	0
$281$	31.8659	1.90096	0.950480	−	0.310786i	$-0.100592\pi$
$281$	31.8659	1.90096	0.950480	+	0.310786i	$0.100592\pi$
$282$	0	0
$283$	−8.75518	−0.520441	−0.260221	−	0.965549i	$-0.583795\pi$
$283$	−8.75518	−0.520441	−0.260221	+	0.965549i	$0.583795\pi$
$284$	−4.72885	−0.280606
$285$	0	0
$286$	−11.1640	−0.660143
$287$	5.47694	0.323293
$288$	0	0
$289$	0.294517	0.0173245
$290$	0	0
$291$	0	0
$292$	−4.88094	−0.285636
$293$	−4.36094	−0.254769	−0.127384	−	0.991853i	$-0.540658\pi$
$293$	−4.36094	−0.254769	−0.127384	+	0.991853i	$0.540658\pi$
$294$	0	0
$295$	0	0
$296$	−17.5116	−1.01784
$297$	0	0
$298$	22.8168	1.32174
$299$	32.0909	1.85587
$300$	0	0
$301$	−10.6005	−0.611000
$302$	−2.58128	−0.148536
$303$	0	0
$304$	4.68411	0.268652
$305$	0	0
$306$	0	0
$307$	3.67884	0.209963	0.104981	−	0.994474i	$-0.466522\pi$
$307$	3.67884	0.209963	0.104981	+	0.994474i	$0.466522\pi$
$308$	−3.65776	−0.208420
$309$	0	0
$310$	0	0
$311$	−19.5312	−1.10751	−0.553755	−	0.832679i	$-0.686806\pi$
$311$	−19.5312	−1.10751	−0.553755	+	0.832679i	$0.686806\pi$
$312$	0	0
$313$	27.4498	1.55155	0.775777	−	0.631007i	$-0.217358\pi$
$313$	27.4498	1.55155	0.775777	+	0.631007i	$0.217358\pi$
$314$	−6.92275	−0.390673
$315$	0	0
$316$	−0.0737329	−0.00414780
$317$	28.7699	1.61588	0.807938	−	0.589268i	$-0.200584\pi$
$317$	28.7699	1.61588	0.807938	+	0.589268i	$0.200584\pi$
$318$	0	0
$319$	−13.9839	−0.782950
$320$	0	0
$321$	0	0
$322$	−9.67002	−0.538889
$323$	−23.4425	−1.30437
$324$	0	0
$325$	0	0
$326$	0.725484	0.0401808
$327$	0	0
$328$	−11.7803	−0.650459
$329$	−4.95615	−0.273242
$330$	0	0
$331$	−8.31441	−0.457001	−0.228501	−	0.973544i	$-0.573382\pi$
$331$	−8.31441	−0.457001	−0.228501	+	0.973544i	$0.573382\pi$
$332$	18.0038	0.988085
$333$	0	0
$334$	−1.71430	−0.0938025
$335$	0	0
$336$	0	0
$337$	1.91049	0.104071	0.0520356	−	0.998645i	$-0.483429\pi$
$337$	1.91049	0.104071	0.0520356	+	0.998645i	$0.483429\pi$
$338$	−7.06965	−0.384538
$339$	0	0
$340$	0	0
$341$	2.53620	0.137343
$342$	0	0
$343$	16.7277	0.903210
$344$	22.8005	1.22932
$345$	0	0
$346$	−6.07133	−0.326397
$347$	−2.82638	−0.151728	−0.0758639	−	0.997118i	$-0.524171\pi$
$347$	−2.82638	−0.151728	−0.0758639	+	0.997118i	$0.524171\pi$
$348$	0	0
$349$	−8.88971	−0.475855	−0.237928	−	0.971283i	$-0.576468\pi$
$349$	−8.88971	−0.475855	−0.237928	+	0.971283i	$0.576468\pi$
$350$	0	0
$351$	0	0
$352$	13.0403	0.695051
$353$	−12.5370	−0.667277	−0.333638	−	0.942701i	$-0.608276\pi$
$353$	−12.5370	−0.667277	−0.333638	+	0.942701i	$0.608276\pi$
$354$	0	0
$355$	0	0
$356$	−12.1635	−0.644665
$357$	0	0
$358$	20.3899	1.07764
$359$	−36.5337	−1.92818	−0.964089	−	0.265580i	$-0.914436\pi$
$359$	−36.5337	−1.92818	−0.964089	+	0.265580i	$0.914436\pi$
$360$	0	0
$361$	12.7760	0.672419
$362$	−0.746666	−0.0392439
$363$	0	0
$364$	−6.48556	−0.339935
$365$	0	0
$366$	0	0
$367$	−6.21528	−0.324435	−0.162217	−	0.986755i	$-0.551865\pi$
$367$	−6.21528	−0.324435	−0.162217	+	0.986755i	$0.551865\pi$
$368$	5.92986	0.309115
$369$	0	0
$370$	0	0
$371$	15.0248	0.780049
$372$	0	0
$373$	33.2100	1.71955	0.859774	−	0.510675i	$-0.170605\pi$
$373$	33.2100	1.71955	0.859774	+	0.510675i	$0.170605\pi$
$374$	10.3243	0.533856
$375$	0	0
$376$	10.6602	0.549756
$377$	−24.7948	−1.27700
$378$	0	0
$379$	4.50249	0.231277	0.115639	−	0.993291i	$-0.463109\pi$
$379$	4.50249	0.231277	0.115639	+	0.993291i	$0.463109\pi$
$380$	0	0
$381$	0	0
$382$	5.26702	0.269484
$383$	11.6738	0.596503	0.298252	−	0.954487i	$-0.403597\pi$
$383$	11.6738	0.596503	0.298252	+	0.954487i	$0.403597\pi$
$384$	0	0
$385$	0	0
$386$	−21.7423	−1.10665
$387$	0	0
$388$	−17.6656	−0.896836
$389$	−16.5184	−0.837517	−0.418759	−	0.908098i	$-0.637535\pi$
$389$	−16.5184	−0.837517	−0.418759	+	0.908098i	$0.637535\pi$
$390$	0	0
$391$	−29.6771	−1.50084
$392$	−15.1367	−0.764521
$393$	0	0
$394$	21.0003	1.05798
$395$	0	0
$396$	0	0
$397$	21.9433	1.10130	0.550650	−	0.834736i	$-0.314380\pi$
$397$	21.9433	1.10130	0.550650	+	0.834736i	$0.314380\pi$
$398$	12.1927	0.611164
$399$	0	0
$400$	0	0
$401$	−7.01996	−0.350560	−0.175280	−	0.984519i	$-0.556083\pi$
$401$	−7.01996	−0.350560	−0.175280	+	0.984519i	$0.556083\pi$
$402$	0	0
$403$	4.49692	0.224008
$404$	8.66520	0.431110
$405$	0	0
$406$	7.47147	0.370803
$407$	14.9160	0.739359
$408$	0	0
$409$	11.6095	0.574054	0.287027	−	0.957923i	$-0.407333\pi$
$409$	11.6095	0.574054	0.287027	+	0.957923i	$0.407333\pi$
$410$	0	0
$411$	0	0
$412$	15.0834	0.743106
$413$	9.78865	0.481668
$414$	0	0
$415$	0	0
$416$	23.1217	1.13363
$417$	0	0
$418$	−13.9944	−0.684490
$419$	3.37379	0.164820	0.0824101	−	0.996599i	$-0.473738\pi$
$419$	3.37379	0.164820	0.0824101	+	0.996599i	$0.473738\pi$
$420$	0	0
$421$	30.1307	1.46848	0.734241	−	0.678889i	$-0.237538\pi$
$421$	30.1307	1.46848	0.734241	+	0.678889i	$0.237538\pi$
$422$	−25.2449	−1.22890
$423$	0	0
$424$	−32.3168	−1.56944
$425$	0	0
$426$	0	0
$427$	14.6388	0.708422
$428$	−6.23365	−0.301315
$429$	0	0
$430$	0	0
$431$	−18.7905	−0.905108	−0.452554	−	0.891737i	$-0.649487\pi$
$431$	−18.7905	−0.905108	−0.452554	+	0.891737i	$0.649487\pi$
$432$	0	0
$433$	15.6932	0.754165	0.377083	−	0.926180i	$-0.376927\pi$
$433$	15.6932	0.754165	0.377083	+	0.926180i	$0.376927\pi$
$434$	−1.35507	−0.0650452
$435$	0	0
$436$	14.4752	0.693238
$437$	40.2269	1.92431
$438$	0	0
$439$	−5.67623	−0.270912	−0.135456	−	0.990783i	$-0.543250\pi$
$439$	−5.67623	−0.270912	−0.135456	+	0.990783i	$0.543250\pi$
$440$	0	0
$441$	0	0
$442$	18.3059	0.870725
$443$	31.0173	1.47368	0.736838	−	0.676069i	$-0.236318\pi$
$443$	31.0173	1.47368	0.736838	+	0.676069i	$0.236318\pi$
$444$	0	0
$445$	0	0
$446$	0.748085	0.0354229
$447$	0	0
$448$	−9.26793	−0.437868
$449$	−14.5835	−0.688237	−0.344118	−	0.938926i	$-0.611822\pi$
$449$	−14.5835	−0.688237	−0.344118	+	0.938926i	$0.611822\pi$
$450$	0	0
$451$	10.0342	0.472493
$452$	−5.07813	−0.238855
$453$	0	0
$454$	24.0491	1.12868
$455$	0	0
$456$	0	0
$457$	37.9535	1.77539	0.887694	−	0.460434i	$-0.152306\pi$
$457$	37.9535	1.77539	0.887694	+	0.460434i	$0.152306\pi$
$458$	29.4275	1.37506
$459$	0	0
$460$	0	0
$461$	30.1983	1.40648	0.703238	−	0.710954i	$-0.251737\pi$
$461$	30.1983	1.40648	0.703238	+	0.710954i	$0.251737\pi$
$462$	0	0
$463$	−11.5007	−0.534485	−0.267242	−	0.963629i	$-0.586112\pi$
$463$	−11.5007	−0.534485	−0.267242	+	0.963629i	$0.586112\pi$
$464$	−4.58166	−0.212698
$465$	0	0
$466$	−2.98242	−0.138158
$467$	6.48845	0.300250	0.150125	−	0.988667i	$-0.452032\pi$
$467$	6.48845	0.300250	0.150125	+	0.988667i	$0.452032\pi$
$468$	0	0
$469$	−16.7856	−0.775086
$470$	0	0
$471$	0	0
$472$	−21.0543	−0.969105
$473$	−19.4209	−0.892976
$474$	0	0
$475$	0	0
$476$	5.99772	0.274905
$477$	0	0
$478$	−19.7707	−0.904289
$479$	19.7285	0.901416	0.450708	−	0.892671i	$-0.351172\pi$
$479$	19.7285	0.901416	0.450708	+	0.892671i	$0.351172\pi$
$480$	0	0
$481$	26.4475	1.20590
$482$	−13.1349	−0.598280
$483$	0	0
$484$	4.75873	0.216306
$485$	0	0
$486$	0	0
$487$	34.8555	1.57945	0.789726	−	0.613460i	$-0.210223\pi$
$487$	34.8555	1.57945	0.789726	+	0.613460i	$0.210223\pi$
$488$	−31.4866	−1.42533
$489$	0	0
$490$	0	0
$491$	31.8901	1.43918	0.719590	−	0.694400i	$-0.244330\pi$
$491$	31.8901	1.43918	0.719590	+	0.694400i	$0.244330\pi$
$492$	0	0
$493$	22.9298	1.03271
$494$	−24.8135	−1.11641
$495$	0	0
$496$	0.830955	0.0373110
$497$	−6.28346	−0.281852
$498$	0	0
$499$	30.9251	1.38440	0.692198	−	0.721707i	$-0.256642\pi$
$499$	30.9251	1.38440	0.692198	+	0.721707i	$0.256642\pi$
$500$	0	0
$501$	0	0
$502$	18.3778	0.820242
$503$	−37.8819	−1.68907	−0.844536	−	0.535498i	$-0.820124\pi$
$503$	−37.8819	−1.68907	−0.844536	+	0.535498i	$0.820124\pi$
$504$	0	0
$505$	0	0
$506$	−17.7163	−0.787586
$507$	0	0
$508$	−10.3682	−0.460017
$509$	44.8753	1.98906	0.994531	−	0.104443i	$-0.0333060\pi$
$509$	44.8753	1.98906	0.994531	+	0.104443i	$0.0333060\pi$
$510$	0	0
$511$	−6.48556	−0.286904
$512$	9.22089	0.407510
$513$	0	0
$514$	−9.90403	−0.436848
$515$	0	0
$516$	0	0
$517$	−9.08009	−0.399342
$518$	−7.96947	−0.350158
$519$	0	0
$520$	0	0
$521$	38.9965	1.70847	0.854234	−	0.519888i	$-0.174026\pi$
$521$	38.9965	1.70847	0.854234	+	0.519888i	$0.174026\pi$
$522$	0	0
$523$	−9.60221	−0.419875	−0.209938	−	0.977715i	$-0.567326\pi$
$523$	−9.60221	−0.419875	−0.209938	+	0.977715i	$0.567326\pi$
$524$	9.46996	0.413697
$525$	0	0
$526$	−2.26497	−0.0987576
$527$	−4.15867	−0.181154
$528$	0	0
$529$	27.9254	1.21415
$530$	0	0
$531$	0	0
$532$	−8.12983	−0.352473
$533$	17.7916	0.770640
$534$	0	0
$535$	0	0
$536$	36.1040	1.55946
$537$	0	0
$538$	−19.6738	−0.848199
$539$	12.8931	0.555347
$540$	0	0
$541$	7.44459	0.320068	0.160034	−	0.987112i	$-0.448840\pi$
$541$	7.44459	0.320068	0.160034	+	0.987112i	$0.448840\pi$
$542$	−8.74850	−0.375780
$543$	0	0
$544$	−21.3825	−0.916768
$545$	0	0
$546$	0	0
$547$	15.5525	0.664976	0.332488	−	0.943108i	$-0.392112\pi$
$547$	15.5525	0.664976	0.332488	+	0.943108i	$0.392112\pi$
$548$	−14.8310	−0.633549
$549$	0	0
$550$	0	0
$551$	−31.0810	−1.32410
$552$	0	0
$553$	−0.0979726	−0.00416622
$554$	−13.0473	−0.554327
$555$	0	0
$556$	−17.7783	−0.753969
$557$	15.8083	0.669818	0.334909	−	0.942251i	$-0.391294\pi$
$557$	15.8083	0.669818	0.334909	+	0.942251i	$0.391294\pi$
$558$	0	0
$559$	−34.4352	−1.45645
$560$	0	0
$561$	0	0
$562$	−31.1924	−1.31577
$563$	18.6803	0.787281	0.393641	−	0.919264i	$-0.371215\pi$
$563$	18.6803	0.787281	0.393641	+	0.919264i	$0.371215\pi$
$564$	0	0
$565$	0	0
$566$	8.57014	0.360230
$567$	0	0
$568$	13.5151	0.567079
$569$	18.5542	0.777832	0.388916	−	0.921273i	$-0.372850\pi$
$569$	18.5542	0.777832	0.388916	+	0.921273i	$0.372850\pi$
$570$	0	0
$571$	−18.8406	−0.788454	−0.394227	−	0.919013i	$-0.628988\pi$
$571$	−18.8406	−0.788454	−0.394227	+	0.919013i	$0.628988\pi$
$572$	−11.8821	−0.496815
$573$	0	0
$574$	−5.36118	−0.223771
$575$	0	0
$576$	0	0
$577$	−21.0962	−0.878247	−0.439124	−	0.898427i	$-0.644711\pi$
$577$	−21.0962	−0.878247	−0.439124	+	0.898427i	$0.644711\pi$
$578$	−0.288292	−0.0119914
$579$	0	0
$580$	0	0
$581$	23.9225	0.992472
$582$	0	0
$583$	27.5267	1.14004
$584$	13.9497	0.577245
$585$	0	0
$586$	4.26877	0.176341
$587$	−17.5014	−0.722361	−0.361181	−	0.932496i	$-0.617626\pi$
$587$	−17.5014	−0.722361	−0.361181	+	0.932496i	$0.617626\pi$
$588$	0	0
$589$	5.63702	0.232269
$590$	0	0
$591$	0	0
$592$	4.88705	0.200856
$593$	7.13893	0.293161	0.146580	−	0.989199i	$-0.453173\pi$
$593$	7.13893	0.293161	0.146580	+	0.989199i	$0.453173\pi$
$594$	0	0
$595$	0	0
$596$	24.2844	0.994727
$597$	0	0
$598$	−31.4127	−1.28456
$599$	11.5961	0.473802	0.236901	−	0.971534i	$-0.423868\pi$
$599$	11.5961	0.473802	0.236901	+	0.971534i	$0.423868\pi$
$600$	0	0
$601$	−33.4764	−1.36553	−0.682766	−	0.730637i	$-0.739223\pi$
$601$	−33.4764	−1.36553	−0.682766	+	0.730637i	$0.739223\pi$
$602$	10.3764	0.422911
$603$	0	0
$604$	−2.74731	−0.111786
$605$	0	0
$606$	0	0
$607$	−16.6165	−0.674444	−0.337222	−	0.941425i	$-0.609487\pi$
$607$	−16.6165	−0.674444	−0.337222	+	0.941425i	$0.609487\pi$
$608$	28.9837	1.17544
$609$	0	0
$610$	0	0
$611$	−16.0999	−0.651331
$612$	0	0
$613$	−29.8943	−1.20742	−0.603710	−	0.797204i	$-0.706312\pi$
$613$	−29.8943	−1.20742	−0.603710	+	0.797204i	$0.706312\pi$
$614$	−3.60109	−0.145328
$615$	0	0
$616$	10.4539	0.421199
$617$	−11.7612	−0.473490	−0.236745	−	0.971572i	$-0.576081\pi$
$617$	−11.7612	−0.473490	−0.236745	+	0.971572i	$0.576081\pi$
$618$	0	0
$619$	26.6104	1.06956	0.534781	−	0.844991i	$-0.320394\pi$
$619$	26.6104	1.06956	0.534781	+	0.844991i	$0.320394\pi$
$620$	0	0
$621$	0	0
$622$	19.1184	0.766577
$623$	−16.1623	−0.647528
$624$	0	0
$625$	0	0
$626$	−26.8696	−1.07393
$627$	0	0
$628$	−7.36801	−0.294016
$629$	−24.4581	−0.975210
$630$	0	0
$631$	2.66241	0.105989	0.0529945	−	0.998595i	$-0.483123\pi$
$631$	2.66241	0.105989	0.0529945	+	0.998595i	$0.483123\pi$
$632$	0.210729	0.00838234
$633$	0	0
$634$	−28.1618	−1.11845
$635$	0	0
$636$	0	0
$637$	22.8608	0.905776
$638$	13.6884	0.541928
$639$	0	0
$640$	0	0
$641$	−0.0519505	−0.00205192	−0.00102596	−	0.999999i	$-0.500327\pi$
$641$	−0.0519505	−0.00205192	−0.00102596	+	0.999999i	$0.500327\pi$
$642$	0	0
$643$	−28.8262	−1.13679	−0.568397	−	0.822754i	$-0.692436\pi$
$643$	−28.8262	−1.13679	−0.568397	+	0.822754i	$0.692436\pi$
$644$	−10.2920	−0.405561
$645$	0	0
$646$	22.9470	0.902838
$647$	−19.5187	−0.767361	−0.383680	−	0.923466i	$-0.625344\pi$
$647$	−19.5187	−0.767361	−0.383680	+	0.923466i	$0.625344\pi$
$648$	0	0
$649$	17.9336	0.703957
$650$	0	0
$651$	0	0
$652$	0.772146	0.0302396
$653$	−8.27640	−0.323880	−0.161940	−	0.986801i	$-0.551775\pi$
$653$	−8.27640	−0.323880	−0.161940	+	0.986801i	$0.551775\pi$
$654$	0	0
$655$	0	0
$656$	3.28759	0.128359
$657$	0	0
$658$	4.85140	0.189127
$659$	−35.0707	−1.36616	−0.683081	−	0.730343i	$-0.739360\pi$
$659$	−35.0707	−1.36616	−0.683081	+	0.730343i	$0.739360\pi$
$660$	0	0
$661$	6.77481	0.263509	0.131755	−	0.991282i	$-0.457939\pi$
$661$	6.77481	0.263509	0.131755	+	0.991282i	$0.457939\pi$
$662$	8.13868	0.316319
$663$	0	0
$664$	−51.4548	−1.99683
$665$	0	0
$666$	0	0
$667$	−39.3471	−1.52353
$668$	−1.82456	−0.0705945
$669$	0	0
$670$	0	0
$671$	26.8196	1.03536
$672$	0	0
$673$	−6.21933	−0.239738	−0.119869	−	0.992790i	$-0.538247\pi$
$673$	−6.21933	−0.239738	−0.119869	+	0.992790i	$0.538247\pi$
$674$	−1.87011	−0.0720341
$675$	0	0
$676$	−7.52436	−0.289398
$677$	27.9378	1.07374	0.536869	−	0.843666i	$-0.319607\pi$
$677$	27.9378	1.07374	0.536869	+	0.843666i	$0.319607\pi$
$678$	0	0
$679$	−23.4732	−0.900818
$680$	0	0
$681$	0	0
$682$	−2.48260	−0.0950635
$683$	25.9011	0.991079	0.495539	−	0.868586i	$-0.334970\pi$
$683$	25.9011	0.991079	0.495539	+	0.868586i	$0.334970\pi$
$684$	0	0
$685$	0	0
$686$	−16.3741	−0.625167
$687$	0	0
$688$	−6.36304	−0.242589
$689$	48.8075	1.85942
$690$	0	0
$691$	−37.2027	−1.41526	−0.707628	−	0.706585i	$-0.750235\pi$
$691$	−37.2027	−1.41526	−0.707628	+	0.706585i	$0.750235\pi$
$692$	−6.46183	−0.245642
$693$	0	0
$694$	2.76664	0.105020
$695$	0	0
$696$	0	0
$697$	−16.4533	−0.623215
$698$	8.70182	0.329369
$699$	0	0
$700$	0	0
$701$	26.1276	0.986825	0.493412	−	0.869795i	$-0.335749\pi$
$701$	26.1276	0.986825	0.493412	+	0.869795i	$0.335749\pi$
$702$	0	0
$703$	33.1527	1.25038
$704$	−16.9796	−0.639944
$705$	0	0
$706$	12.2720	0.461864
$707$	11.5139	0.433024
$708$	0	0
$709$	−24.6118	−0.924316	−0.462158	−	0.886797i	$-0.652925\pi$
$709$	−24.6118	−0.924316	−0.462158	+	0.886797i	$0.652925\pi$
$710$	0	0
$711$	0	0
$712$	34.7633	1.30281
$713$	7.13620	0.267253
$714$	0	0
$715$	0	0
$716$	21.7013	0.811017
$717$	0	0
$718$	35.7616	1.33461
$719$	21.3025	0.794450	0.397225	−	0.917721i	$-0.369973\pi$
$719$	21.3025	0.794450	0.397225	+	0.917721i	$0.369973\pi$
$720$	0	0
$721$	20.0421	0.746406
$722$	−12.5059	−0.465423
$723$	0	0
$724$	−0.794691	−0.0295345
$725$	0	0
$726$	0	0
$727$	33.5250	1.24337	0.621686	−	0.783266i	$-0.286448\pi$
$727$	33.5250	1.24337	0.621686	+	0.783266i	$0.286448\pi$
$728$	18.5357	0.686980
$729$	0	0
$730$	0	0
$731$	31.8450	1.17783
$732$	0	0
$733$	−8.06056	−0.297724	−0.148862	−	0.988858i	$-0.547561\pi$
$733$	−8.06056	−0.297724	−0.148862	+	0.988858i	$0.547561\pi$
$734$	6.08392	0.224561
$735$	0	0
$736$	36.6920	1.35249
$737$	−30.7526	−1.13279
$738$	0	0
$739$	31.8857	1.17293	0.586466	−	0.809974i	$-0.300519\pi$
$739$	31.8857	1.17293	0.586466	+	0.809974i	$0.300519\pi$
$740$	0	0
$741$	0	0
$742$	−14.7073	−0.539920
$743$	41.9865	1.54033	0.770167	−	0.637842i	$-0.220173\pi$
$743$	41.9865	1.54033	0.770167	+	0.637842i	$0.220173\pi$
$744$	0	0
$745$	0	0
$746$	−32.5081	−1.19021
$747$	0	0
$748$	10.9883	0.401773
$749$	−8.28297	−0.302653
$750$	0	0
$751$	21.8880	0.798703	0.399352	−	0.916798i	$-0.369235\pi$
$751$	21.8880	0.798703	0.399352	+	0.916798i	$0.369235\pi$
$752$	−2.97498	−0.108486
$753$	0	0
$754$	24.2708	0.883889
$755$	0	0
$756$	0	0
$757$	10.9541	0.398134	0.199067	−	0.979986i	$-0.436209\pi$
$757$	10.9541	0.398134	0.199067	+	0.979986i	$0.436209\pi$
$758$	−4.40732	−0.160081
$759$	0	0
$760$	0	0
$761$	−39.1067	−1.41762	−0.708809	−	0.705400i	$-0.750767\pi$
$761$	−39.1067	−1.41762	−0.708809	+	0.705400i	$0.750767\pi$
$762$	0	0
$763$	19.2340	0.696316
$764$	5.60579	0.202810
$765$	0	0
$766$	−11.4271	−0.412877
$767$	31.7980	1.14816
$768$	0	0
$769$	29.6460	1.06906	0.534532	−	0.845148i	$-0.320488\pi$
$769$	29.6460	1.06906	0.534532	+	0.845148i	$0.320488\pi$
$770$	0	0
$771$	0	0
$772$	−23.1407	−0.832852
$773$	48.0787	1.72927	0.864635	−	0.502400i	$-0.167550\pi$
$773$	48.0787	1.72927	0.864635	+	0.502400i	$0.167550\pi$
$774$	0	0
$775$	0	0
$776$	50.4883	1.81243
$777$	0	0
$778$	16.1693	0.579698
$779$	22.3023	0.799062
$780$	0	0
$781$	−11.5118	−0.411926
$782$	29.0499	1.03882
$783$	0	0
$784$	4.22428	0.150867
$785$	0	0
$786$	0	0
$787$	−25.7911	−0.919355	−0.459677	−	0.888086i	$-0.652035\pi$
$787$	−25.7911	−0.919355	−0.459677	+	0.888086i	$0.652035\pi$
$788$	22.3510	0.796220
$789$	0	0
$790$	0	0
$791$	−6.74757	−0.239916
$792$	0	0
$793$	47.5536	1.68868
$794$	−21.4795	−0.762278
$795$	0	0
$796$	12.9769	0.459954
$797$	−22.1228	−0.783629	−0.391815	−	0.920044i	$-0.628153\pi$
$797$	−22.1228	−0.783629	−0.391815	+	0.920044i	$0.628153\pi$
$798$	0	0
$799$	14.8889	0.526730
$800$	0	0
$801$	0	0
$802$	6.87159	0.242644
$803$	−11.8821	−0.419310
$804$	0	0
$805$	0	0
$806$	−4.40188	−0.155049
$807$	0	0
$808$	−24.7651	−0.871235
$809$	−13.9129	−0.489153	−0.244577	−	0.969630i	$-0.578649\pi$
$809$	−13.9129	−0.489153	−0.244577	+	0.969630i	$0.578649\pi$
$810$	0	0
$811$	−41.9060	−1.47152	−0.735760	−	0.677242i	$-0.763175\pi$
$811$	−41.9060	−1.47152	−0.735760	+	0.677242i	$0.763175\pi$
$812$	7.95203	0.279061
$813$	0	0
$814$	−14.6008	−0.511756
$815$	0	0
$816$	0	0
$817$	−43.1655	−1.51017
$818$	−11.3641	−0.397338
$819$	0	0
$820$	0	0
$821$	−45.4837	−1.58739	−0.793696	−	0.608315i	$-0.791846\pi$
$821$	−45.4837	−1.58739	−0.793696	+	0.608315i	$0.791846\pi$
$822$	0	0
$823$	−42.6844	−1.48788	−0.743942	−	0.668244i	$-0.767046\pi$
$823$	−42.6844	−1.48788	−0.743942	+	0.668244i	$0.767046\pi$
$824$	−43.1084	−1.50175
$825$	0	0
$826$	−9.58176	−0.333392
$827$	−16.4041	−0.570425	−0.285213	−	0.958464i	$-0.592064\pi$
$827$	−16.4041	−0.570425	−0.285213	+	0.958464i	$0.592064\pi$
$828$	0	0
$829$	34.6525	1.20353	0.601765	−	0.798673i	$-0.294464\pi$
$829$	34.6525	1.20353	0.601765	+	0.798673i	$0.294464\pi$
$830$	0	0
$831$	0	0
$832$	−30.1065	−1.04376
$833$	−21.1412	−0.732499
$834$	0	0
$835$	0	0
$836$	−14.8945	−0.515138
$837$	0	0
$838$	−3.30248	−0.114082
$839$	13.7651	0.475224	0.237612	−	0.971360i	$-0.423635\pi$
$839$	13.7651	0.475224	0.237612	+	0.971360i	$0.423635\pi$
$840$	0	0
$841$	1.40125	0.0483191
$842$	−29.4939	−1.01643
$843$	0	0
$844$	−26.8686	−0.924856
$845$	0	0
$846$	0	0
$847$	6.32317	0.217267
$848$	9.01880	0.309707
$849$	0	0
$850$	0	0
$851$	41.9697	1.43870
$852$	0	0
$853$	−7.97777	−0.273154	−0.136577	−	0.990629i	$-0.543610\pi$
$853$	−7.97777	−0.273154	−0.136577	+	0.990629i	$0.543610\pi$
$854$	−14.3294	−0.490343
$855$	0	0
$856$	17.8158	0.608931
$857$	−23.9717	−0.818858	−0.409429	−	0.912342i	$-0.634272\pi$
$857$	−23.9717	−0.818858	−0.409429	+	0.912342i	$0.634272\pi$
$858$	0	0
$859$	−7.16644	−0.244516	−0.122258	−	0.992498i	$-0.539013\pi$
$859$	−7.16644	−0.244516	−0.122258	+	0.992498i	$0.539013\pi$
$860$	0	0
$861$	0	0
$862$	18.3934	0.626481
$863$	−32.6266	−1.11062	−0.555311	−	0.831643i	$-0.687401\pi$
$863$	−32.6266	−1.11062	−0.555311	+	0.831643i	$0.687401\pi$
$864$	0	0
$865$	0	0
$866$	−15.3615	−0.522005
$867$	0	0
$868$	−1.44222	−0.0489522
$869$	−0.179494	−0.00608892
$870$	0	0
$871$	−54.5273	−1.84759
$872$	−41.3702	−1.40097
$873$	0	0
$874$	−39.3767	−1.33194
$875$	0	0
$876$	0	0
$877$	−6.39951	−0.216096	−0.108048	−	0.994146i	$-0.534460\pi$
$877$	−6.39951	−0.216096	−0.108048	+	0.994146i	$0.534460\pi$
$878$	5.55627	0.187515
$879$	0	0
$880$	0	0
$881$	4.98365	0.167903	0.0839517	−	0.996470i	$-0.473246\pi$
$881$	4.98365	0.167903	0.0839517	+	0.996470i	$0.473246\pi$
$882$	0	0
$883$	−0.276340	−0.00929957	−0.00464978	−	0.999989i	$-0.501480\pi$
$883$	−0.276340	−0.00929957	−0.00464978	+	0.999989i	$0.501480\pi$
$884$	19.4834	0.655296
$885$	0	0
$886$	−30.3617	−1.02002
$887$	−27.4923	−0.923103	−0.461551	−	0.887113i	$-0.652707\pi$
$887$	−27.4923	−0.923103	−0.461551	+	0.887113i	$0.652707\pi$
$888$	0	0
$889$	−13.7768	−0.462059
$890$	0	0
$891$	0	0
$892$	0.796201	0.0266588
$893$	−20.1816	−0.675353
$894$	0	0
$895$	0	0
$896$	−5.16344	−0.172498
$897$	0	0
$898$	14.2752	0.476371
$899$	−5.51373	−0.183893
$900$	0	0
$901$	−45.1362	−1.50371
$902$	−9.82214	−0.327041
$903$	0	0
$904$	14.5133	0.482706
$905$	0	0
$906$	0	0
$907$	−8.32231	−0.276338	−0.138169	−	0.990409i	$-0.544122\pi$
$907$	−8.32231	−0.276338	−0.138169	+	0.990409i	$0.544122\pi$
$908$	25.5959	0.849430
$909$	0	0
$910$	0	0
$911$	4.62993	0.153396	0.0766982	−	0.997054i	$-0.475562\pi$
$911$	4.62993	0.153396	0.0766982	+	0.997054i	$0.475562\pi$
$912$	0	0
$913$	43.8281	1.45050
$914$	−37.1513	−1.22886
$915$	0	0
$916$	31.3202	1.03485
$917$	12.5832	0.415534
$918$	0	0
$919$	35.8072	1.18117	0.590585	−	0.806975i	$-0.298897\pi$
$919$	35.8072	1.18117	0.590585	+	0.806975i	$0.298897\pi$
$920$	0	0
$921$	0	0
$922$	−29.5601	−0.973510
$923$	−20.4116	−0.671855
$924$	0	0
$925$	0	0
$926$	11.2577	0.369950
$927$	0	0
$928$	−28.3498	−0.930629
$929$	11.3400	0.372052	0.186026	−	0.982545i	$-0.440439\pi$
$929$	11.3400	0.372052	0.186026	+	0.982545i	$0.440439\pi$
$930$	0	0
$931$	28.6566	0.939182
$932$	−3.17424	−0.103976
$933$	0	0
$934$	−6.35132	−0.207822
$935$	0	0
$936$	0	0
$937$	16.0561	0.524532	0.262266	−	0.964996i	$-0.415530\pi$
$937$	16.0561	0.524532	0.262266	+	0.964996i	$0.415530\pi$
$938$	16.4308	0.536485
$939$	0	0
$940$	0	0
$941$	13.8935	0.452914	0.226457	−	0.974021i	$-0.427286\pi$
$941$	13.8935	0.452914	0.226457	+	0.974021i	$0.427286\pi$
$942$	0	0
$943$	28.2337	0.919414
$944$	5.87574	0.191239
$945$	0	0
$946$	19.0105	0.618084
$947$	−31.6353	−1.02801	−0.514004	−	0.857788i	$-0.671838\pi$
$947$	−31.6353	−1.02801	−0.514004	+	0.857788i	$0.671838\pi$
$948$	0	0
$949$	−21.0681	−0.683899
$950$	0	0
$951$	0	0
$952$	−17.1415	−0.555559
$953$	17.3676	0.562592	0.281296	−	0.959621i	$-0.409236\pi$
$953$	17.3676	0.562592	0.281296	+	0.959621i	$0.409236\pi$
$954$	0	0
$955$	0	0
$956$	−21.0423	−0.680556
$957$	0	0
$958$	−19.3115	−0.623926
$959$	−19.7067	−0.636362
$960$	0	0
$961$	1.00000	0.0322581
$962$	−25.8885	−0.834679
$963$	0	0
$964$	−13.9798	−0.450258
$965$	0	0
$966$	0	0
$967$	−16.9854	−0.546213	−0.273106	−	0.961984i	$-0.588051\pi$
$967$	−16.9854	−0.546213	−0.273106	+	0.961984i	$0.588051\pi$
$968$	−13.6005	−0.437135
$969$	0	0
$970$	0	0
$971$	−6.62785	−0.212698	−0.106349	−	0.994329i	$-0.533916\pi$
$971$	−6.62785	−0.212698	−0.106349	+	0.994329i	$0.533916\pi$
$972$	0	0
$973$	−23.6229	−0.757317
$974$	−34.1188	−1.09324
$975$	0	0
$976$	8.78711	0.281268
$977$	11.9176	0.381277	0.190639	−	0.981660i	$-0.438944\pi$
$977$	11.9176	0.381277	0.190639	+	0.981660i	$0.438944\pi$
$978$	0	0
$979$	−29.6107	−0.946361
$980$	0	0
$981$	0	0
$982$	−31.2161	−0.996145
$983$	−9.07162	−0.289340	−0.144670	−	0.989480i	$-0.546212\pi$
$983$	−9.07162	−0.289340	−0.144670	+	0.989480i	$0.546212\pi$
$984$	0	0
$985$	0	0
$986$	−22.4452	−0.714799
$987$	0	0
$988$	−26.4094	−0.840196
$989$	−54.6455	−1.73762
$990$	0	0
$991$	37.8154	1.20125	0.600623	−	0.799532i	$-0.294919\pi$
$991$	37.8154	1.20125	0.600623	+	0.799532i	$0.294919\pi$
$992$	5.14168	0.163248
$993$	0	0
$994$	6.15066	0.195087
$995$	0	0
$996$	0	0
$997$	−12.3773	−0.391993	−0.195997	−	0.980605i	$-0.562794\pi$
$997$	−12.3773	−0.391993	−0.195997	+	0.980605i	$0.562794\pi$
$998$	−30.2715	−0.958226
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cg.1.4	yes	8
3.2	odd	2	inner	6975.2.a.cg.1.5	yes	8
5.4	even	2		6975.2.a.cf.1.5	yes	8
15.14	odd	2		6975.2.a.cf.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.cf.1.4	✓	8	15.14	odd	2
6975.2.a.cf.1.5	yes	8	5.4	even	2
6975.2.a.cg.1.4	yes	8	1.1	even	1	trivial
6975.2.a.cg.1.5	yes	8	3.2	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-0.978865 q^{2} -1.04182 q^{4} -1.38432 q^{7} +2.97753 q^{8} -2.53620 q^{11} -4.49692 q^{13} +1.35507 q^{14} -0.830955 q^{16} +4.15867 q^{17} -5.63702 q^{19} +2.48260 q^{22} -7.13620 q^{23} +4.40188 q^{26} +1.44222 q^{28} +5.51373 q^{29} -1.00000 q^{31} -5.14168 q^{32} -4.07077 q^{34} -5.88124 q^{37} +5.51788 q^{38} -3.95640 q^{41} +7.65750 q^{43} +2.64227 q^{44} +6.98538 q^{46} +3.58020 q^{47} -5.08365 q^{49} +4.68500 q^{52} -10.8535 q^{53} -4.12187 q^{56} -5.39720 q^{58} -7.07107 q^{59} -10.5747 q^{61} +0.978865 q^{62} +6.69491 q^{64} +12.1255 q^{67} -4.33260 q^{68} +4.53901 q^{71} +4.68500 q^{73} +5.75694 q^{74} +5.87278 q^{76} +3.51092 q^{77} +0.0707729 q^{79} +3.87278 q^{82} -17.2810 q^{83} -7.49566 q^{86} -7.55162 q^{88} +11.6752 q^{89} +6.22519 q^{91} +7.43467 q^{92} -3.50453 q^{94} +16.9564 q^{97} +4.97620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 4 q^{7} + 8 q^{13} + 12 q^{16} + 28 q^{22} + 36 q^{28} - 8 q^{31} - 28 q^{34} + 12 q^{37} + 52 q^{43} - 16 q^{46} + 8 q^{49} + 56 q^{52} + 16 q^{58} - 28 q^{61} + 48 q^{64} + 24 q^{67}+ \cdots + 44 q^{97}+O(q^{100}) \) 8 * q + 16 * q^4 + 4 * q^7 + 8 * q^13 + 12 * q^16 + 28 * q^22 + 36 * q^28 - 8 * q^31 - 28 * q^34 + 12 * q^37 + 52 * q^43 - 16 * q^46 + 8 * q^49 + 56 * q^52 + 16 * q^58 - 28 * q^61 + 48 * q^64 + 24 * q^67 + 56 * q^73 + 4 * q^76 - 4 * q^79 - 12 * q^82 + 52 * q^88 - 8 * q^91 - 64 * q^94 + 44 * q^97

Introduction

L-functions

Modular forms

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Database

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