LMFDB - Embedded newform 6975.2.a.cj.1.6

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 = [11,3,0,15,0,0,-8,9,0,0,0,0,-14,14,0,27,12] 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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 $ x^11 - 3x^10 - 14x^9 + 44x^8 + 61x^7 - 211x^6 - 83x^5 + 369x^4 + 10x^3 - 168x^2 - 31x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.104188$ 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)

$f(q)$	$=$	$q+0.104188 q^{2} -1.98914 q^{4} +3.88714 q^{7} -0.415622 q^{8} +5.28458 q^{11} -0.694025 q^{13} +0.404994 q^{14} +3.93499 q^{16} +1.67629 q^{17} +3.99063 q^{19} +0.550591 q^{22} +2.82890 q^{23} -0.0723093 q^{26} -7.73208 q^{28} +8.18906 q^{29} +1.00000 q^{31} +1.24122 q^{32} +0.174650 q^{34} +9.62461 q^{37} +0.415777 q^{38} -7.41833 q^{41} -2.34101 q^{43} -10.5118 q^{44} +0.294738 q^{46} -0.567889 q^{47} +8.10984 q^{49} +1.38052 q^{52} +11.3489 q^{53} -1.61558 q^{56} +0.853204 q^{58} -4.57276 q^{59} +4.98804 q^{61} +0.104188 q^{62} -7.74065 q^{64} -13.7584 q^{67} -3.33439 q^{68} -10.0186 q^{71} -8.46096 q^{73} +1.00277 q^{74} -7.93794 q^{76} +20.5419 q^{77} -0.437300 q^{79} -0.772903 q^{82} +13.9850 q^{83} -0.243905 q^{86} -2.19639 q^{88} +2.97354 q^{89} -2.69777 q^{91} -5.62709 q^{92} -0.0591674 q^{94} -9.79254 q^{97} +0.844950 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{2} + 15 q^{4} - 8 q^{7} + 9 q^{8} - 14 q^{13} + 14 q^{14} + 27 q^{16} + 12 q^{17} + 12 q^{19} - 10 q^{22} + 12 q^{23} + 6 q^{26} - 22 q^{28} + 8 q^{29} + 11 q^{31} + 21 q^{32} + 2 q^{34} - 16 q^{37}+ \cdots + 17 q^{98}+O(q^{100}) $ 11 * q + 3 * q^2 + 15 * q^4 - 8 * q^7 + 9 * q^8 - 14 * q^13 + 14 * q^14 + 27 * q^16 + 12 * q^17 + 12 * q^19 - 10 * q^22 + 12 * q^23 + 6 * q^26 - 22 * q^28 + 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 - 16 * q^37 + 26 * q^38 + 8 * q^41 - 28 * q^43 + 26 * q^44 + 8 * q^46 + 14 * q^47 + 23 * q^49 - 18 * q^52 + 26 * q^53 + 42 * q^56 - 6 * q^58 + 6 * q^59 + 14 * q^61 + 3 * q^62 + 29 * q^64 - 6 * q^67 + 54 * q^68 + 18 * q^71 - 26 * q^73 + 40 * q^74 + 10 * q^76 + 2 * q^77 + 8 * q^79 + 30 * q^82 + 48 * q^83 + 12 * q^86 - 40 * q^88 + 14 * q^89 - 4 * q^91 - 30 * q^92 - 4 * q^94 - 16 * q^97 + 17 * q^98

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 $

$2$	0.104188	0.0736723	0.0368361	−	0.999321i	$-0.488272\pi$
$2$	0.104188	0.0736723	0.0368361	+	0.999321i	$0.488272\pi$
$3$	0	0
$3$	0	0
$4$	−1.98914	−0.994572
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.88714	1.46920	0.734600	−	0.678501i	$-0.237370\pi$
$7$	3.88714	1.46920	0.734600	+	0.678501i	$0.237370\pi$
$8$	−0.415622	−0.146945
$9$	0	0
$10$	0	0
$11$	5.28458	1.59336	0.796680	−	0.604401i	$-0.206588\pi$
$11$	5.28458	1.59336	0.796680	+	0.604401i	$0.206588\pi$
$12$	0	0
$13$	−0.694025	−0.192488	−0.0962440	−	0.995358i	$-0.530683\pi$
$13$	−0.694025	−0.192488	−0.0962440	+	0.995358i	$0.530683\pi$
$14$	0.404994	0.108239
$15$	0	0
$16$	3.93499	0.983747
$17$	1.67629	0.406560	0.203280	−	0.979121i	$-0.434840\pi$
$17$	1.67629	0.406560	0.203280	+	0.979121i	$0.434840\pi$
$18$	0	0
$19$	3.99063	0.915513	0.457756	−	0.889078i	$-0.348653\pi$
$19$	3.99063	0.915513	0.457756	+	0.889078i	$0.348653\pi$
$20$	0	0
$21$	0	0
$22$	0.550591	0.117386
$23$	2.82890	0.589867	0.294933	−	0.955518i	$-0.404703\pi$
$23$	2.82890	0.589867	0.294933	+	0.955518i	$0.404703\pi$
$24$	0	0
$25$	0	0
$26$	−0.0723093	−0.0141810
$27$	0	0
$28$	−7.73208	−1.46123
$29$	8.18906	1.52067	0.760335	−	0.649531i	$-0.225035\pi$
$29$	8.18906	1.52067	0.760335	+	0.649531i	$0.225035\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.24122	0.219419
$33$	0	0
$34$	0.174650	0.0299522
$35$	0	0
$36$	0	0
$37$	9.62461	1.58228	0.791138	−	0.611638i	$-0.209489\pi$
$37$	9.62461	1.58228	0.791138	+	0.611638i	$0.209489\pi$
$38$	0.415777	0.0674479
$39$	0	0
$40$	0	0
$41$	−7.41833	−1.15855	−0.579274	−	0.815133i	$-0.696664\pi$
$41$	−7.41833	−1.15855	−0.579274	+	0.815133i	$0.696664\pi$
$42$	0	0
$43$	−2.34101	−0.357000	−0.178500	−	0.983940i	$-0.557124\pi$
$43$	−2.34101	−0.357000	−0.178500	+	0.983940i	$0.557124\pi$
$44$	−10.5118	−1.58471
$45$	0	0
$46$	0.294738	0.0434568
$47$	−0.567889	−0.0828352	−0.0414176	−	0.999142i	$-0.513187\pi$
$47$	−0.567889	−0.0828352	−0.0414176	+	0.999142i	$0.513187\pi$
$48$	0	0
$49$	8.10984	1.15855
$50$	0	0
$51$	0	0
$52$	1.38052	0.191443
$53$	11.3489	1.55890	0.779449	−	0.626466i	$-0.215499\pi$
$53$	11.3489	1.55890	0.779449	+	0.626466i	$0.215499\pi$
$54$	0	0
$55$	0	0
$56$	−1.61558	−0.215891
$57$	0	0
$58$	0.853204	0.112031
$59$	−4.57276	−0.595322	−0.297661	−	0.954672i	$-0.596207\pi$
$59$	−4.57276	−0.595322	−0.297661	+	0.954672i	$0.596207\pi$
$60$	0	0
$61$	4.98804	0.638653	0.319326	−	0.947645i	$-0.396543\pi$
$61$	4.98804	0.638653	0.319326	+	0.947645i	$0.396543\pi$
$62$	0.104188	0.0132319
$63$	0	0
$64$	−7.74065	−0.967582
$65$	0	0
$66$	0	0
$67$	−13.7584	−1.68086	−0.840431	−	0.541919i	$-0.817698\pi$
$67$	−13.7584	−1.68086	−0.840431	+	0.541919i	$0.817698\pi$
$68$	−3.33439	−0.404354
$69$	0	0
$70$	0	0
$71$	−10.0186	−1.18899	−0.594497	−	0.804098i	$-0.702649\pi$
$71$	−10.0186	−1.18899	−0.594497	+	0.804098i	$0.702649\pi$
$72$	0	0
$73$	−8.46096	−0.990281	−0.495141	−	0.868813i	$-0.664883\pi$
$73$	−8.46096	−0.990281	−0.495141	+	0.868813i	$0.664883\pi$
$74$	1.00277	0.116570
$75$	0	0
$76$	−7.93794	−0.910544
$77$	20.5419	2.34096
$78$	0	0
$79$	−0.437300	−0.0492001	−0.0246000	−	0.999697i	$-0.507831\pi$
$79$	−0.437300	−0.0492001	−0.0246000	+	0.999697i	$0.507831\pi$
$80$	0	0
$81$	0	0
$82$	−0.772903	−0.0853528
$83$	13.9850	1.53505	0.767526	−	0.641017i	$-0.221487\pi$
$83$	13.9850	1.53505	0.767526	+	0.641017i	$0.221487\pi$
$84$	0	0
$85$	0	0
$86$	−0.243905	−0.0263010
$87$	0	0
$88$	−2.19639	−0.234136
$89$	2.97354	0.315195	0.157597	−	0.987503i	$-0.449625\pi$
$89$	2.97354	0.315195	0.157597	+	0.987503i	$0.449625\pi$
$90$	0	0
$91$	−2.69777	−0.282803
$92$	−5.62709	−0.586665
$93$	0	0
$94$	−0.0591674	−0.00610265
$95$	0	0
$96$	0	0
$97$	−9.79254	−0.994282	−0.497141	−	0.867670i	$-0.665617\pi$
$97$	−9.79254	−0.994282	−0.497141	+	0.867670i	$0.665617\pi$
$98$	0.844950	0.0853529
$99$	0	0
$100$	0	0
$101$	7.93874	0.789934	0.394967	−	0.918695i	$-0.370756\pi$
$101$	7.93874	0.789934	0.394967	+	0.918695i	$0.370756\pi$
$102$	0	0
$103$	8.25376	0.813267	0.406634	−	0.913591i	$-0.366703\pi$
$103$	8.25376	0.813267	0.406634	+	0.913591i	$0.366703\pi$
$104$	0.288452	0.0282851
$105$	0	0
$106$	1.18243	0.114848
$107$	7.95648	0.769182	0.384591	−	0.923087i	$-0.374342\pi$
$107$	7.95648	0.769182	0.384591	+	0.923087i	$0.374342\pi$
$108$	0	0
$109$	−19.1086	−1.83027	−0.915134	−	0.403151i	$-0.867915\pi$
$109$	−19.1086	−1.83027	−0.915134	+	0.403151i	$0.867915\pi$
$110$	0	0
$111$	0	0
$112$	15.2958	1.44532
$113$	−15.6174	−1.46916	−0.734580	−	0.678523i	$-0.762621\pi$
$113$	−15.6174	−1.46916	−0.734580	+	0.678523i	$0.762621\pi$
$114$	0	0
$115$	0	0
$116$	−16.2892	−1.51242
$117$	0	0
$118$	−0.476428	−0.0438587
$119$	6.51598	0.597319
$120$	0	0
$121$	16.9268	1.53880
$122$	0.519695	0.0470510
$123$	0	0
$124$	−1.98914	−0.178630
$125$	0	0
$126$	0	0
$127$	−20.9367	−1.85783	−0.928916	−	0.370289i	$-0.879259\pi$
$127$	−20.9367	−1.85783	−0.928916	+	0.370289i	$0.879259\pi$
$128$	−3.28893	−0.290703
$129$	0	0
$130$	0	0
$131$	11.7903	1.03013	0.515063	−	0.857152i	$-0.327769\pi$
$131$	11.7903	1.03013	0.515063	+	0.857152i	$0.327769\pi$
$132$	0	0
$133$	15.5121	1.34507
$134$	−1.43347	−0.123833
$135$	0	0
$136$	−0.696704	−0.0597419
$137$	1.14623	0.0979292	0.0489646	−	0.998801i	$-0.484408\pi$
$137$	1.14623	0.0979292	0.0489646	+	0.998801i	$0.484408\pi$
$138$	0	0
$139$	6.08625	0.516229	0.258115	−	0.966114i	$-0.416899\pi$
$139$	6.08625	0.516229	0.258115	+	0.966114i	$0.416899\pi$
$140$	0	0
$141$	0	0
$142$	−1.04383	−0.0875959
$143$	−3.66763	−0.306703
$144$	0	0
$145$	0	0
$146$	−0.881534	−0.0729562
$147$	0	0
$148$	−19.1447	−1.57369
$149$	−1.80203	−0.147628	−0.0738139	−	0.997272i	$-0.523517\pi$
$149$	−1.80203	−0.147628	−0.0738139	+	0.997272i	$0.523517\pi$
$150$	0	0
$151$	−9.26966	−0.754354	−0.377177	−	0.926141i	$-0.623105\pi$
$151$	−9.26966	−0.754354	−0.377177	+	0.926141i	$0.623105\pi$
$152$	−1.65859	−0.134530
$153$	0	0
$154$	2.14022	0.172464
$155$	0	0
$156$	0	0
$157$	−5.63336	−0.449591	−0.224796	−	0.974406i	$-0.572171\pi$
$157$	−5.63336	−0.449591	−0.224796	+	0.974406i	$0.572171\pi$
$158$	−0.0455615	−0.00362468
$159$	0	0
$160$	0	0
$161$	10.9963	0.866632
$162$	0	0
$163$	−21.1252	−1.65466	−0.827328	−	0.561719i	$-0.810140\pi$
$163$	−21.1252	−1.65466	−0.827328	+	0.561719i	$0.810140\pi$
$164$	14.7561	1.15226
$165$	0	0
$166$	1.45707	0.113091
$167$	3.85688	0.298454	0.149227	−	0.988803i	$-0.452321\pi$
$167$	3.85688	0.298454	0.149227	+	0.988803i	$0.452321\pi$
$168$	0	0
$169$	−12.5183	−0.962948
$170$	0	0
$171$	0	0
$172$	4.65660	0.355062
$173$	5.72231	0.435059	0.217530	−	0.976054i	$-0.430200\pi$
$173$	5.72231	0.435059	0.217530	+	0.976054i	$0.430200\pi$
$174$	0	0
$175$	0	0
$176$	20.7947	1.56746
$177$	0	0
$178$	0.309808	0.0232211
$179$	12.5993	0.941719	0.470859	−	0.882208i	$-0.343944\pi$
$179$	12.5993	0.941719	0.470859	+	0.882208i	$0.343944\pi$
$180$	0	0
$181$	−9.58112	−0.712159	−0.356079	−	0.934456i	$-0.615887\pi$
$181$	−9.58112	−0.712159	−0.356079	+	0.934456i	$0.615887\pi$
$182$	−0.281076	−0.0208347
$183$	0	0
$184$	−1.17575	−0.0866777
$185$	0	0
$186$	0	0
$187$	8.85849	0.647797
$188$	1.12961	0.0823856
$189$	0	0
$190$	0	0
$191$	8.50009	0.615045	0.307522	−	0.951541i	$-0.400500\pi$
$191$	8.50009	0.615045	0.307522	+	0.951541i	$0.400500\pi$
$192$	0	0
$193$	−15.9422	−1.14754	−0.573772	−	0.819015i	$-0.694520\pi$
$193$	−15.9422	−1.14754	−0.573772	+	0.819015i	$0.694520\pi$
$194$	−1.02027	−0.0732510
$195$	0	0
$196$	−16.1316	−1.15226
$197$	6.87178	0.489594	0.244797	−	0.969574i	$-0.421279\pi$
$197$	6.87178	0.489594	0.244797	+	0.969574i	$0.421279\pi$
$198$	0	0
$199$	17.4295	1.23554	0.617772	−	0.786357i	$-0.288035\pi$
$199$	17.4295	1.23554	0.617772	+	0.786357i	$0.288035\pi$
$200$	0	0
$201$	0	0
$202$	0.827124	0.0581962
$203$	31.8320	2.23417
$204$	0	0
$205$	0	0
$206$	0.859945	0.0599152
$207$	0	0
$208$	−2.73098	−0.189359
$209$	21.0888	1.45874
$210$	0	0
$211$	6.07786	0.418417	0.209209	−	0.977871i	$-0.432911\pi$
$211$	6.07786	0.418417	0.209209	+	0.977871i	$0.432911\pi$
$212$	−22.5747	−1.55044
$213$	0	0
$214$	0.828972	0.0566674
$215$	0	0
$216$	0	0
$217$	3.88714	0.263876
$218$	−1.99089	−0.134840
$219$	0	0
$220$	0	0
$221$	−1.16339	−0.0782580
$222$	0	0
$223$	5.05639	0.338601	0.169301	−	0.985564i	$-0.445849\pi$
$223$	5.05639	0.338601	0.169301	+	0.985564i	$0.445849\pi$
$224$	4.82481	0.322371
$225$	0	0
$226$	−1.62715	−0.108236
$227$	−16.0132	−1.06283	−0.531416	−	0.847111i	$-0.678340\pi$
$227$	−16.0132	−1.06283	−0.531416	+	0.847111i	$0.678340\pi$
$228$	0	0
$229$	19.9345	1.31731	0.658655	−	0.752445i	$-0.271126\pi$
$229$	19.9345	1.31731	0.658655	+	0.752445i	$0.271126\pi$
$230$	0	0
$231$	0	0
$232$	−3.40355	−0.223454
$233$	−5.23419	−0.342903	−0.171452	−	0.985193i	$-0.554846\pi$
$233$	−5.23419	−0.342903	−0.171452	+	0.985193i	$0.554846\pi$
$234$	0	0
$235$	0	0
$236$	9.09588	0.592091
$237$	0	0
$238$	0.678888	0.0440058
$239$	−5.03499	−0.325687	−0.162843	−	0.986652i	$-0.552067\pi$
$239$	−5.03499	−0.325687	−0.162843	+	0.986652i	$0.552067\pi$
$240$	0	0
$241$	−7.25850	−0.467561	−0.233780	−	0.972289i	$-0.575110\pi$
$241$	−7.25850	−0.467561	−0.233780	+	0.972289i	$0.575110\pi$
$242$	1.76357	0.113367
$243$	0	0
$244$	−9.92193	−0.635186
$245$	0	0
$246$	0	0
$247$	−2.76960	−0.176225
$248$	−0.415622	−0.0263920
$249$	0	0
$250$	0	0
$251$	−14.3668	−0.906825	−0.453412	−	0.891301i	$-0.649794\pi$
$251$	−14.3668	−0.906825	−0.453412	+	0.891301i	$0.649794\pi$
$252$	0	0
$253$	14.9495	0.939870
$254$	−2.18136	−0.136871
$255$	0	0
$256$	15.1386	0.946165
$257$	−24.2390	−1.51199	−0.755994	−	0.654579i	$-0.772846\pi$
$257$	−24.2390	−1.51199	−0.755994	+	0.654579i	$0.772846\pi$
$258$	0	0
$259$	37.4122	2.32468
$260$	0	0
$261$	0	0
$262$	1.22841	0.0758917
$263$	29.4904	1.81845	0.909227	−	0.416300i	$-0.136674\pi$
$263$	29.4904	1.81845	0.909227	+	0.416300i	$0.136674\pi$
$264$	0	0
$265$	0	0
$266$	1.61618	0.0990944
$267$	0	0
$268$	27.3675	1.67174
$269$	27.1024	1.65246	0.826231	−	0.563332i	$-0.190481\pi$
$269$	27.1024	1.65246	0.826231	+	0.563332i	$0.190481\pi$
$270$	0	0
$271$	−3.26492	−0.198330	−0.0991650	−	0.995071i	$-0.531617\pi$
$271$	−3.26492	−0.198330	−0.0991650	+	0.995071i	$0.531617\pi$
$272$	6.59619	0.399952
$273$	0	0
$274$	0.119424	0.00721466
$275$	0	0
$276$	0	0
$277$	2.38699	0.143420	0.0717101	−	0.997426i	$-0.477154\pi$
$277$	2.38699	0.143420	0.0717101	+	0.997426i	$0.477154\pi$
$278$	0.634117	0.0380318
$279$	0	0
$280$	0	0
$281$	−3.30607	−0.197224	−0.0986118	−	0.995126i	$-0.531440\pi$
$281$	−3.30607	−0.197224	−0.0986118	+	0.995126i	$0.531440\pi$
$282$	0	0
$283$	−12.4622	−0.740799	−0.370400	−	0.928873i	$-0.620779\pi$
$283$	−12.4622	−0.740799	−0.370400	+	0.928873i	$0.620779\pi$
$284$	19.9285	1.18254
$285$	0	0
$286$	−0.382124	−0.0225955
$287$	−28.8361	−1.70214
$288$	0	0
$289$	−14.1900	−0.834709
$290$	0	0
$291$	0	0
$292$	16.8301	0.984906
$293$	−23.5587	−1.37631	−0.688157	−	0.725562i	$-0.741580\pi$
$293$	−23.5587	−1.37631	−0.688157	+	0.725562i	$0.741580\pi$
$294$	0	0
$295$	0	0
$296$	−4.00020	−0.232507
$297$	0	0
$298$	−0.187750	−0.0108761
$299$	−1.96333	−0.113542
$300$	0	0
$301$	−9.09981	−0.524505
$302$	−0.965790	−0.0555750
$303$	0	0
$304$	15.7031	0.900633
$305$	0	0
$306$	0	0
$307$	−15.7904	−0.901207	−0.450603	−	0.892724i	$-0.648791\pi$
$307$	−15.7904	−0.901207	−0.450603	+	0.892724i	$0.648791\pi$
$308$	−40.8608	−2.32826
$309$	0	0
$310$	0	0
$311$	0.116768	0.00662133	0.00331066	−	0.999995i	$-0.498946\pi$
$311$	0.116768	0.00662133	0.00331066	+	0.999995i	$0.498946\pi$
$312$	0	0
$313$	28.7756	1.62649	0.813247	−	0.581919i	$-0.197698\pi$
$313$	28.7756	1.62649	0.813247	+	0.581919i	$0.197698\pi$
$314$	−0.586930	−0.0331224
$315$	0	0
$316$	0.869853	0.0489331
$317$	3.27767	0.184092	0.0920462	−	0.995755i	$-0.470659\pi$
$317$	3.27767	0.184092	0.0920462	+	0.995755i	$0.470659\pi$
$318$	0	0
$319$	43.2757	2.42297
$320$	0	0
$321$	0	0
$322$	1.14569	0.0638467
$323$	6.68946	0.372211
$324$	0	0
$325$	0	0
$326$	−2.20100	−0.121902
$327$	0	0
$328$	3.08322	0.170242
$329$	−2.20746	−0.121701
$330$	0	0
$331$	16.2926	0.895521	0.447761	−	0.894153i	$-0.352222\pi$
$331$	16.2926	0.895521	0.447761	+	0.894153i	$0.352222\pi$
$332$	−27.8182	−1.52672
$333$	0	0
$334$	0.401842	0.0219878
$335$	0	0
$336$	0	0
$337$	−27.7285	−1.51047	−0.755235	−	0.655454i	$-0.772477\pi$
$337$	−27.7285	−1.51047	−0.755235	+	0.655454i	$0.772477\pi$
$338$	−1.30426	−0.0709426
$339$	0	0
$340$	0	0
$341$	5.28458	0.286176
$342$	0	0
$343$	4.31409	0.232939
$344$	0.972974	0.0524593
$345$	0	0
$346$	0.596198	0.0320518
$347$	12.0273	0.645657	0.322829	−	0.946457i	$-0.395366\pi$
$347$	12.0273	0.645657	0.322829	+	0.946457i	$0.395366\pi$
$348$	0	0
$349$	24.5657	1.31497	0.657487	−	0.753466i	$-0.271620\pi$
$349$	24.5657	1.31497	0.657487	+	0.753466i	$0.271620\pi$
$350$	0	0
$351$	0	0
$352$	6.55934	0.349614
$353$	−26.6809	−1.42008	−0.710041	−	0.704160i	$-0.751324\pi$
$353$	−26.6809	−1.42008	−0.710041	+	0.704160i	$0.751324\pi$
$354$	0	0
$355$	0	0
$356$	−5.91481	−0.313484
$357$	0	0
$358$	1.31270	0.0693785
$359$	34.0862	1.79900	0.899500	−	0.436921i	$-0.143931\pi$
$359$	34.0862	1.79900	0.899500	+	0.436921i	$0.143931\pi$
$360$	0	0
$361$	−3.07489	−0.161837
$362$	−0.998240	−0.0524664
$363$	0	0
$364$	5.36626	0.281268
$365$	0	0
$366$	0	0
$367$	10.2307	0.534036	0.267018	−	0.963692i	$-0.413962\pi$
$367$	10.2307	0.534036	0.267018	+	0.963692i	$0.413962\pi$
$368$	11.1317	0.580279
$369$	0	0
$370$	0	0
$371$	44.1149	2.29033
$372$	0	0
$373$	9.79049	0.506932	0.253466	−	0.967344i	$-0.418429\pi$
$373$	9.79049	0.506932	0.253466	+	0.967344i	$0.418429\pi$
$374$	0.922951	0.0477247
$375$	0	0
$376$	0.236027	0.0121722
$377$	−5.68341	−0.292711
$378$	0	0
$379$	36.9424	1.89760	0.948801	−	0.315874i	$-0.102298\pi$
$379$	36.9424	1.89760	0.948801	+	0.315874i	$0.102298\pi$
$380$	0	0
$381$	0	0
$382$	0.885610	0.0453117
$383$	10.2672	0.524628	0.262314	−	0.964983i	$-0.415514\pi$
$383$	10.2672	0.524628	0.262314	+	0.964983i	$0.415514\pi$
$384$	0	0
$385$	0	0
$386$	−1.66099	−0.0845421
$387$	0	0
$388$	19.4788	0.988885
$389$	−13.9806	−0.708845	−0.354423	−	0.935085i	$-0.615323\pi$
$389$	−13.9806	−0.708845	−0.354423	+	0.935085i	$0.615323\pi$
$390$	0	0
$391$	4.74206	0.239816
$392$	−3.37063	−0.170242
$393$	0	0
$394$	0.715959	0.0360695
$395$	0	0
$396$	0	0
$397$	−5.61552	−0.281835	−0.140918	−	0.990021i	$-0.545005\pi$
$397$	−5.61552	−0.281835	−0.140918	+	0.990021i	$0.545005\pi$
$398$	1.81595	0.0910253
$399$	0	0
$400$	0	0
$401$	8.17499	0.408240	0.204120	−	0.978946i	$-0.434567\pi$
$401$	8.17499	0.408240	0.204120	+	0.978946i	$0.434567\pi$
$402$	0	0
$403$	−0.694025	−0.0345718
$404$	−15.7913	−0.785647
$405$	0	0
$406$	3.31652	0.164596
$407$	50.8620	2.52114
$408$	0	0
$409$	5.53871	0.273872	0.136936	−	0.990580i	$-0.456275\pi$
$409$	5.53871	0.273872	0.136936	+	0.990580i	$0.456275\pi$
$410$	0	0
$411$	0	0
$412$	−16.4179	−0.808853
$413$	−17.7749	−0.874648
$414$	0	0
$415$	0	0
$416$	−0.861440	−0.0422356
$417$	0	0
$418$	2.19720	0.107469
$419$	−0.697378	−0.0340692	−0.0170346	−	0.999855i	$-0.505423\pi$
$419$	−0.697378	−0.0340692	−0.0170346	+	0.999855i	$0.505423\pi$
$420$	0	0
$421$	39.4502	1.92268	0.961342	−	0.275357i	$-0.0887959\pi$
$421$	39.4502	1.92268	0.961342	+	0.275357i	$0.0887959\pi$
$422$	0.633242	0.0308257
$423$	0	0
$424$	−4.71687	−0.229072
$425$	0	0
$426$	0	0
$427$	19.3892	0.938309
$428$	−15.8266	−0.765007
$429$	0	0
$430$	0	0
$431$	−12.1326	−0.584406	−0.292203	−	0.956356i	$-0.594388\pi$
$431$	−12.1326	−0.584406	−0.292203	+	0.956356i	$0.594388\pi$
$432$	0	0
$433$	−15.4117	−0.740640	−0.370320	−	0.928904i	$-0.620752\pi$
$433$	−15.4117	−0.740640	−0.370320	+	0.928904i	$0.620752\pi$
$434$	0.404994	0.0194403
$435$	0	0
$436$	38.0097	1.82033
$437$	11.2891	0.540030
$438$	0	0
$439$	−30.1477	−1.43887	−0.719437	−	0.694558i	$-0.755600\pi$
$439$	−30.1477	−1.43887	−0.719437	+	0.694558i	$0.755600\pi$
$440$	0	0
$441$	0	0
$442$	−0.121211	−0.00576544
$443$	−10.7906	−0.512677	−0.256339	−	0.966587i	$-0.582516\pi$
$443$	−10.7906	−0.512677	−0.256339	+	0.966587i	$0.582516\pi$
$444$	0	0
$445$	0	0
$446$	0.526817	0.0249455
$447$	0	0
$448$	−30.0890	−1.42157
$449$	24.2640	1.14509	0.572545	−	0.819873i	$-0.305956\pi$
$449$	24.2640	1.14509	0.572545	+	0.819873i	$0.305956\pi$
$450$	0	0
$451$	−39.2027	−1.84598
$452$	31.0652	1.46119
$453$	0	0
$454$	−1.66839	−0.0783012
$455$	0	0
$456$	0	0
$457$	26.7751	1.25249	0.626244	−	0.779627i	$-0.284591\pi$
$457$	26.7751	1.25249	0.626244	+	0.779627i	$0.284591\pi$
$458$	2.07694	0.0970492
$459$	0	0
$460$	0	0
$461$	−22.8736	−1.06533	−0.532664	−	0.846327i	$-0.678809\pi$
$461$	−22.8736	−1.06533	−0.532664	+	0.846327i	$0.678809\pi$
$462$	0	0
$463$	20.5343	0.954310	0.477155	−	0.878819i	$-0.341668\pi$
$463$	20.5343	0.954310	0.477155	+	0.878819i	$0.341668\pi$
$464$	32.2238	1.49595
$465$	0	0
$466$	−0.545341	−0.0252625
$467$	29.6109	1.37023	0.685114	−	0.728436i	$-0.259752\pi$
$467$	29.6109	1.37023	0.685114	+	0.728436i	$0.259752\pi$
$468$	0	0
$469$	−53.4810	−2.46952
$470$	0	0
$471$	0	0
$472$	1.90054	0.0874794
$473$	−12.3712	−0.568830
$474$	0	0
$475$	0	0
$476$	−12.9612	−0.594077
$477$	0	0
$478$	−0.524588	−0.0239941
$479$	24.4122	1.11542	0.557711	−	0.830035i	$-0.311680\pi$
$479$	24.4122	1.11542	0.557711	+	0.830035i	$0.311680\pi$
$480$	0	0
$481$	−6.67972	−0.304569
$482$	−0.756251	−0.0344463
$483$	0	0
$484$	−33.6698	−1.53044
$485$	0	0
$486$	0	0
$487$	−7.54901	−0.342078	−0.171039	−	0.985264i	$-0.554712\pi$
$487$	−7.54901	−0.342078	−0.171039	+	0.985264i	$0.554712\pi$
$488$	−2.07314	−0.0938466
$489$	0	0
$490$	0	0
$491$	9.75807	0.440376	0.220188	−	0.975458i	$-0.429333\pi$
$491$	9.75807	0.440376	0.220188	+	0.975458i	$0.429333\pi$
$492$	0	0
$493$	13.7272	0.618244
$494$	−0.288559	−0.0129829
$495$	0	0
$496$	3.93499	0.176686
$497$	−38.9438	−1.74687
$498$	0	0
$499$	6.60183	0.295539	0.147769	−	0.989022i	$-0.452791\pi$
$499$	6.60183	0.295539	0.147769	+	0.989022i	$0.452791\pi$
$500$	0	0
$501$	0	0
$502$	−1.49685	−0.0668078
$503$	26.6040	1.18622	0.593108	−	0.805123i	$-0.297901\pi$
$503$	26.6040	1.18622	0.593108	+	0.805123i	$0.297901\pi$
$504$	0	0
$505$	0	0
$506$	1.55757	0.0692423
$507$	0	0
$508$	41.6462	1.84775
$509$	−23.6567	−1.04857	−0.524283	−	0.851544i	$-0.675667\pi$
$509$	−23.6567	−1.04857	−0.524283	+	0.851544i	$0.675667\pi$
$510$	0	0
$511$	−32.8889	−1.45492
$512$	8.15514	0.360409
$513$	0	0
$514$	−2.52542	−0.111392
$515$	0	0
$516$	0	0
$517$	−3.00106	−0.131986
$518$	3.89791	0.171264
$519$	0	0
$520$	0	0
$521$	−13.2996	−0.582667	−0.291334	−	0.956622i	$-0.594099\pi$
$521$	−13.2996	−0.582667	−0.291334	+	0.956622i	$0.594099\pi$
$522$	0	0
$523$	−36.8010	−1.60920	−0.804598	−	0.593820i	$-0.797619\pi$
$523$	−36.8010	−1.60920	−0.804598	+	0.593820i	$0.797619\pi$
$524$	−23.4527	−1.02453
$525$	0	0
$526$	3.07255	0.133970
$527$	1.67629	0.0730204
$528$	0	0
$529$	−14.9973	−0.652057
$530$	0	0
$531$	0	0
$532$	−30.8558	−1.33777
$533$	5.14850	0.223006
$534$	0	0
$535$	0	0
$536$	5.71832	0.246994
$537$	0	0
$538$	2.82375	0.121741
$539$	42.8571	1.84598
$540$	0	0
$541$	1.65543	0.0711725	0.0355862	−	0.999367i	$-0.488670\pi$
$541$	1.65543	0.0711725	0.0355862	+	0.999367i	$0.488670\pi$
$542$	−0.340167	−0.0146114
$543$	0	0
$544$	2.08065	0.0892073
$545$	0	0
$546$	0	0
$547$	13.5862	0.580905	0.290453	−	0.956889i	$-0.406194\pi$
$547$	13.5862	0.580905	0.290453	+	0.956889i	$0.406194\pi$
$548$	−2.28002	−0.0973977
$549$	0	0
$550$	0	0
$551$	32.6795	1.39219
$552$	0	0
$553$	−1.69985	−0.0722848
$554$	0.248696	0.0105661
$555$	0	0
$556$	−12.1064	−0.513428
$557$	2.25594	0.0955870	0.0477935	−	0.998857i	$-0.484781\pi$
$557$	2.25594	0.0955870	0.0477935	+	0.998857i	$0.484781\pi$
$558$	0	0
$559$	1.62472	0.0687182
$560$	0	0
$561$	0	0
$562$	−0.344454	−0.0145299
$563$	−10.1910	−0.429501	−0.214751	−	0.976669i	$-0.568894\pi$
$563$	−10.1910	−0.429501	−0.214751	+	0.976669i	$0.568894\pi$
$564$	0	0
$565$	0	0
$566$	−1.29841	−0.0545763
$567$	0	0
$568$	4.16397	0.174716
$569$	−19.2455	−0.806812	−0.403406	−	0.915021i	$-0.632174\pi$
$569$	−19.2455	−0.806812	−0.403406	+	0.915021i	$0.632174\pi$
$570$	0	0
$571$	24.8562	1.04020	0.520099	−	0.854106i	$-0.325895\pi$
$571$	24.8562	1.04020	0.520099	+	0.854106i	$0.325895\pi$
$572$	7.29545	0.305038
$573$	0	0
$574$	−3.00438	−0.125400
$575$	0	0
$576$	0	0
$577$	2.28450	0.0951048	0.0475524	−	0.998869i	$-0.484858\pi$
$577$	2.28450	0.0951048	0.0475524	+	0.998869i	$0.484858\pi$
$578$	−1.47844	−0.0614949
$579$	0	0
$580$	0	0
$581$	54.3616	2.25530
$582$	0	0
$583$	59.9744	2.48389
$584$	3.51656	0.145516
$585$	0	0
$586$	−2.45454	−0.101396
$587$	−4.05298	−0.167285	−0.0836423	−	0.996496i	$-0.526655\pi$
$587$	−4.05298	−0.167285	−0.0836423	+	0.996496i	$0.526655\pi$
$588$	0	0
$589$	3.99063	0.164431
$590$	0	0
$591$	0	0
$592$	37.8727	1.55656
$593$	0.981357	0.0402995	0.0201498	−	0.999797i	$-0.493586\pi$
$593$	0.981357	0.0402995	0.0201498	+	0.999797i	$0.493586\pi$
$594$	0	0
$595$	0	0
$596$	3.58449	0.146826
$597$	0	0
$598$	−0.204556	−0.00836491
$599$	−31.7074	−1.29553	−0.647765	−	0.761840i	$-0.724296\pi$
$599$	−31.7074	−1.29553	−0.647765	+	0.761840i	$0.724296\pi$
$600$	0	0
$601$	23.7961	0.970664	0.485332	−	0.874330i	$-0.338699\pi$
$601$	23.7961	0.970664	0.485332	+	0.874330i	$0.338699\pi$
$602$	−0.948094	−0.0386414
$603$	0	0
$604$	18.4387	0.750260
$605$	0	0
$606$	0	0
$607$	7.89451	0.320428	0.160214	−	0.987082i	$-0.448782\pi$
$607$	7.89451	0.320428	0.160214	+	0.987082i	$0.448782\pi$
$608$	4.95326	0.200881
$609$	0	0
$610$	0	0
$611$	0.394129	0.0159448
$612$	0	0
$613$	−19.8129	−0.800237	−0.400119	−	0.916463i	$-0.631031\pi$
$613$	−19.8129	−0.800237	−0.400119	+	0.916463i	$0.631031\pi$
$614$	−1.64518	−0.0663939
$615$	0	0
$616$	−8.53766	−0.343992
$617$	35.5080	1.42950	0.714748	−	0.699382i	$-0.246541\pi$
$617$	35.5080	1.42950	0.714748	+	0.699382i	$0.246541\pi$
$618$	0	0
$619$	3.32252	0.133543	0.0667717	−	0.997768i	$-0.478730\pi$
$619$	3.32252	0.133543	0.0667717	+	0.997768i	$0.478730\pi$
$620$	0	0
$621$	0	0
$622$	0.0121659	0.000487808 0
$623$	11.5586	0.463084
$624$	0	0
$625$	0	0
$626$	2.99808	0.119827
$627$	0	0
$628$	11.2056	0.447151
$629$	16.1337	0.643291
$630$	0	0
$631$	4.05076	0.161258	0.0806290	−	0.996744i	$-0.474307\pi$
$631$	4.05076	0.161258	0.0806290	+	0.996744i	$0.474307\pi$
$632$	0.181752	0.00722969
$633$	0	0
$634$	0.341495	0.0135625
$635$	0	0
$636$	0	0
$637$	−5.62843	−0.223007
$638$	4.50882	0.178506
$639$	0	0
$640$	0	0
$641$	−15.6430	−0.617861	−0.308931	−	0.951085i	$-0.599971\pi$
$641$	−15.6430	−0.617861	−0.308931	+	0.951085i	$0.599971\pi$
$642$	0	0
$643$	2.82535	0.111421	0.0557105	−	0.998447i	$-0.482258\pi$
$643$	2.82535	0.111421	0.0557105	+	0.998447i	$0.482258\pi$
$644$	−21.8733	−0.861928
$645$	0	0
$646$	0.696963	0.0274216
$647$	−34.4255	−1.35341	−0.676704	−	0.736255i	$-0.736592\pi$
$647$	−34.4255	−1.35341	−0.676704	+	0.736255i	$0.736592\pi$
$648$	0	0
$649$	−24.1651	−0.948563
$650$	0	0
$651$	0	0
$652$	42.0212	1.64568
$653$	−13.2653	−0.519111	−0.259555	−	0.965728i	$-0.583576\pi$
$653$	−13.2653	−0.519111	−0.259555	+	0.965728i	$0.583576\pi$
$654$	0	0
$655$	0	0
$656$	−29.1910	−1.13972
$657$	0	0
$658$	−0.229992	−0.00896602
$659$	2.73817	0.106664	0.0533319	−	0.998577i	$-0.483016\pi$
$659$	2.73817	0.106664	0.0533319	+	0.998577i	$0.483016\pi$
$660$	0	0
$661$	12.7031	0.494093	0.247046	−	0.969004i	$-0.420540\pi$
$661$	12.7031	0.494093	0.247046	+	0.969004i	$0.420540\pi$
$662$	1.69750	0.0659751
$663$	0	0
$664$	−5.81247	−0.225568
$665$	0	0
$666$	0	0
$667$	23.1660	0.896992
$668$	−7.67189	−0.296834
$669$	0	0
$670$	0	0
$671$	26.3597	1.01760
$672$	0	0
$673$	−25.4661	−0.981647	−0.490823	−	0.871259i	$-0.663304\pi$
$673$	−25.4661	−0.981647	−0.490823	+	0.871259i	$0.663304\pi$
$674$	−2.88899	−0.111280
$675$	0	0
$676$	24.9008	0.957722
$677$	17.4203	0.669516	0.334758	−	0.942304i	$-0.391345\pi$
$677$	17.4203	0.669516	0.334758	+	0.942304i	$0.391345\pi$
$678$	0	0
$679$	−38.0649	−1.46080
$680$	0	0
$681$	0	0
$682$	0.550591	0.0210832
$683$	−8.91386	−0.341079	−0.170540	−	0.985351i	$-0.554551\pi$
$683$	−8.91386	−0.341079	−0.170540	+	0.985351i	$0.554551\pi$
$684$	0	0
$685$	0	0
$686$	0.449478	0.0171611
$687$	0	0
$688$	−9.21183	−0.351198
$689$	−7.87645	−0.300069
$690$	0	0
$691$	−22.7640	−0.865982	−0.432991	−	0.901398i	$-0.642542\pi$
$691$	−22.7640	−0.865982	−0.432991	+	0.901398i	$0.642542\pi$
$692$	−11.3825	−0.432698
$693$	0	0
$694$	1.25310	0.0475670
$695$	0	0
$696$	0	0
$697$	−12.4353	−0.471020
$698$	2.55946	0.0968770
$699$	0	0
$700$	0	0
$701$	41.1738	1.55511	0.777556	−	0.628814i	$-0.216459\pi$
$701$	41.1738	1.55511	0.777556	+	0.628814i	$0.216459\pi$
$702$	0	0
$703$	38.4082	1.44859
$704$	−40.9061	−1.54171
$705$	0	0
$706$	−2.77984	−0.104621
$707$	30.8590	1.16057
$708$	0	0
$709$	50.4738	1.89558	0.947792	−	0.318890i	$-0.103310\pi$
$709$	50.4738	1.89558	0.947792	+	0.318890i	$0.103310\pi$
$710$	0	0
$711$	0	0
$712$	−1.23587	−0.0463162
$713$	2.82890	0.105943
$714$	0	0
$715$	0	0
$716$	−25.0619	−0.936608
$717$	0	0
$718$	3.55138	0.132536
$719$	37.9121	1.41388	0.706942	−	0.707272i	$-0.250074\pi$
$719$	37.9121	1.41388	0.706942	+	0.707272i	$0.250074\pi$
$720$	0	0
$721$	32.0835	1.19485
$722$	−0.320368	−0.0119229
$723$	0	0
$724$	19.0582	0.708294
$725$	0	0
$726$	0	0
$727$	−32.3656	−1.20037	−0.600186	−	0.799860i	$-0.704907\pi$
$727$	−32.3656	−1.20037	−0.600186	+	0.799860i	$0.704907\pi$
$728$	1.12125	0.0415564
$729$	0	0
$730$	0	0
$731$	−3.92421	−0.145142
$732$	0	0
$733$	−44.9832	−1.66149	−0.830746	−	0.556652i	$-0.812086\pi$
$733$	−44.9832	−1.66149	−0.830746	+	0.556652i	$0.812086\pi$
$734$	1.06591	0.0393436
$735$	0	0
$736$	3.51130	0.129428
$737$	−72.7076	−2.67822
$738$	0	0
$739$	−10.4689	−0.385103	−0.192551	−	0.981287i	$-0.561676\pi$
$739$	−10.4689	−0.385103	−0.192551	+	0.981287i	$0.561676\pi$
$740$	0	0
$741$	0	0
$742$	4.59626	0.168734
$743$	−7.00170	−0.256867	−0.128434	−	0.991718i	$-0.540995\pi$
$743$	−7.00170	−0.256867	−0.128434	+	0.991718i	$0.540995\pi$
$744$	0	0
$745$	0	0
$746$	1.02005	0.0373468
$747$	0	0
$748$	−17.6208	−0.644281
$749$	30.9279	1.13008
$750$	0	0
$751$	−10.1665	−0.370982	−0.185491	−	0.982646i	$-0.559388\pi$
$751$	−10.1665	−0.370982	−0.185491	+	0.982646i	$0.559388\pi$
$752$	−2.23464	−0.0814888
$753$	0	0
$754$	−0.592145	−0.0215646
$755$	0	0
$756$	0	0
$757$	−45.9707	−1.67083	−0.835417	−	0.549616i	$-0.814774\pi$
$757$	−45.9707	−1.67083	−0.835417	+	0.549616i	$0.814774\pi$
$758$	3.84896	0.139801
$759$	0	0
$760$	0	0
$761$	−31.7560	−1.15115	−0.575577	−	0.817747i	$-0.695223\pi$
$761$	−31.7560	−1.15115	−0.575577	+	0.817747i	$0.695223\pi$
$762$	0	0
$763$	−74.2776	−2.68903
$764$	−16.9079	−0.611707
$765$	0	0
$766$	1.06972	0.0386505
$767$	3.17361	0.114592
$768$	0	0
$769$	23.1867	0.836135	0.418067	−	0.908416i	$-0.362708\pi$
$769$	23.1867	0.836135	0.418067	+	0.908416i	$0.362708\pi$
$770$	0	0
$771$	0	0
$772$	31.7113	1.14131
$773$	−35.2212	−1.26682	−0.633409	−	0.773817i	$-0.718345\pi$
$773$	−35.2212	−1.26682	−0.633409	+	0.773817i	$0.718345\pi$
$774$	0	0
$775$	0	0
$776$	4.07000	0.146104
$777$	0	0
$778$	−1.45662	−0.0522222
$779$	−29.6038	−1.06067
$780$	0	0
$781$	−52.9443	−1.89450
$782$	0.494067	0.0176678
$783$	0	0
$784$	31.9121	1.13972
$785$	0	0
$786$	0	0
$787$	2.92047	0.104104	0.0520518	−	0.998644i	$-0.483424\pi$
$787$	2.92047	0.104104	0.0520518	+	0.998644i	$0.483424\pi$
$788$	−13.6690	−0.486937
$789$	0	0
$790$	0	0
$791$	−60.7069	−2.15849
$792$	0	0
$793$	−3.46182	−0.122933
$794$	−0.585072	−0.0207634
$795$	0	0
$796$	−34.6698	−1.22884
$797$	−49.3613	−1.74847	−0.874233	−	0.485506i	$-0.838635\pi$
$797$	−49.3613	−1.74847	−0.874233	+	0.485506i	$0.838635\pi$
$798$	0	0
$799$	−0.951948	−0.0336775
$800$	0	0
$801$	0	0
$802$	0.851739	0.0300759
$803$	−44.7126	−1.57787
$804$	0	0
$805$	0	0
$806$	−0.0723093	−0.00254699
$807$	0	0
$808$	−3.29952	−0.116077
$809$	−34.3035	−1.20605	−0.603024	−	0.797723i	$-0.706038\pi$
$809$	−34.3035	−1.20605	−0.603024	+	0.797723i	$0.706038\pi$
$810$	0	0
$811$	−40.7695	−1.43161	−0.715806	−	0.698299i	$-0.753941\pi$
$811$	−40.7695	−1.43161	−0.715806	+	0.698299i	$0.753941\pi$
$812$	−63.3184	−2.22204
$813$	0	0
$814$	5.29923	0.185738
$815$	0	0
$816$	0	0
$817$	−9.34208	−0.326838
$818$	0.577069	0.0201768
$819$	0	0
$820$	0	0
$821$	38.5003	1.34367	0.671834	−	0.740701i	$-0.265507\pi$
$821$	38.5003	1.34367	0.671834	+	0.740701i	$0.265507\pi$
$822$	0	0
$823$	−18.9267	−0.659744	−0.329872	−	0.944026i	$-0.607006\pi$
$823$	−18.9267	−0.659744	−0.329872	+	0.944026i	$0.607006\pi$
$824$	−3.43045	−0.119505
$825$	0	0
$826$	−1.85194	−0.0644373
$827$	−2.10231	−0.0731046	−0.0365523	−	0.999332i	$-0.511638\pi$
$827$	−2.10231	−0.0731046	−0.0365523	+	0.999332i	$0.511638\pi$
$828$	0	0
$829$	−23.4452	−0.814286	−0.407143	−	0.913364i	$-0.633475\pi$
$829$	−23.4452	−0.814286	−0.407143	+	0.913364i	$0.633475\pi$
$830$	0	0
$831$	0	0
$832$	5.37221	0.186248
$833$	13.5945	0.471020
$834$	0	0
$835$	0	0
$836$	−41.9486	−1.45082
$837$	0	0
$838$	−0.0726587	−0.00250995
$839$	−19.3211	−0.667039	−0.333520	−	0.942743i	$-0.608236\pi$
$839$	−19.3211	−0.667039	−0.333520	+	0.942743i	$0.608236\pi$
$840$	0	0
$841$	38.0607	1.31244
$842$	4.11025	0.141648
$843$	0	0
$844$	−12.0897	−0.416146
$845$	0	0
$846$	0	0
$847$	65.7967	2.26080
$848$	44.6580	1.53356
$849$	0	0
$850$	0	0
$851$	27.2271	0.933332
$852$	0	0
$853$	6.48660	0.222097	0.111048	−	0.993815i	$-0.464579\pi$
$853$	6.48660	0.222097	0.111048	+	0.993815i	$0.464579\pi$
$854$	2.02013	0.0691273
$855$	0	0
$856$	−3.30689	−0.113027
$857$	18.2551	0.623581	0.311791	−	0.950151i	$-0.399071\pi$
$857$	18.2551	0.623581	0.311791	+	0.950151i	$0.399071\pi$
$858$	0	0
$859$	−20.5239	−0.700267	−0.350134	−	0.936700i	$-0.613864\pi$
$859$	−20.5239	−0.700267	−0.350134	+	0.936700i	$0.613864\pi$
$860$	0	0
$861$	0	0
$862$	−1.26407	−0.0430545
$863$	9.39936	0.319958	0.159979	−	0.987120i	$-0.448857\pi$
$863$	9.39936	0.319958	0.159979	+	0.987120i	$0.448857\pi$
$864$	0	0
$865$	0	0
$866$	−1.60572	−0.0545646
$867$	0	0
$868$	−7.73208	−0.262444
$869$	−2.31095	−0.0783935
$870$	0	0
$871$	9.54871	0.323546
$872$	7.94194	0.268948
$873$	0	0
$874$	1.17619	0.0397853
$875$	0	0
$876$	0	0
$877$	−24.5124	−0.827724	−0.413862	−	0.910340i	$-0.635820\pi$
$877$	−24.5124	−0.827724	−0.413862	+	0.910340i	$0.635820\pi$
$878$	−3.14104	−0.106005
$879$	0	0
$880$	0	0
$881$	−4.12141	−0.138854	−0.0694269	−	0.997587i	$-0.522117\pi$
$881$	−4.12141	−0.138854	−0.0694269	+	0.997587i	$0.522117\pi$
$882$	0	0
$883$	14.1187	0.475132	0.237566	−	0.971371i	$-0.423650\pi$
$883$	14.1187	0.475132	0.237566	+	0.971371i	$0.423650\pi$
$884$	2.31415	0.0778332
$885$	0	0
$886$	−1.12425	−0.0377701
$887$	52.5146	1.76327	0.881634	−	0.471934i	$-0.156444\pi$
$887$	52.5146	1.76327	0.881634	+	0.471934i	$0.156444\pi$
$888$	0	0
$889$	−81.3839	−2.72953
$890$	0	0
$891$	0	0
$892$	−10.0579	−0.336763
$893$	−2.26623	−0.0758367
$894$	0	0
$895$	0	0
$896$	−12.7845	−0.427101
$897$	0	0
$898$	2.52803	0.0843614
$899$	8.18906	0.273120
$900$	0	0
$901$	19.0241	0.633786
$902$	−4.08446	−0.135998
$903$	0	0
$904$	6.49093	0.215885
$905$	0	0
$906$	0	0
$907$	−22.4962	−0.746973	−0.373486	−	0.927636i	$-0.621838\pi$
$907$	−22.4962	−0.746973	−0.373486	+	0.927636i	$0.621838\pi$
$908$	31.8525	1.05706
$909$	0	0
$910$	0	0
$911$	−31.7221	−1.05100	−0.525500	−	0.850794i	$-0.676122\pi$
$911$	−31.7221	−1.05100	−0.525500	+	0.850794i	$0.676122\pi$
$912$	0	0
$913$	73.9048	2.44589
$914$	2.78966	0.0922736
$915$	0	0
$916$	−39.6526	−1.31016
$917$	45.8306	1.51346
$918$	0	0
$919$	−33.5939	−1.10816	−0.554080	−	0.832463i	$-0.686930\pi$
$919$	−33.5939	−1.10816	−0.554080	+	0.832463i	$0.686930\pi$
$920$	0	0
$921$	0	0
$922$	−2.38316	−0.0784851
$923$	6.95319	0.228867
$924$	0	0
$925$	0	0
$926$	2.13943	0.0703062
$927$	0	0
$928$	10.1645	0.333665
$929$	−22.5894	−0.741134	−0.370567	−	0.928806i	$-0.620837\pi$
$929$	−22.5894	−0.741134	−0.370567	+	0.928806i	$0.620837\pi$
$930$	0	0
$931$	32.3633	1.06067
$932$	10.4116	0.341042
$933$	0	0
$934$	3.08511	0.100948
$935$	0	0
$936$	0	0
$937$	55.9174	1.82674	0.913371	−	0.407129i	$-0.133470\pi$
$937$	55.9174	1.82674	0.913371	+	0.407129i	$0.133470\pi$
$938$	−5.57209	−0.181935
$939$	0	0
$940$	0	0
$941$	−6.93479	−0.226068	−0.113034	−	0.993591i	$-0.536057\pi$
$941$	−6.93479	−0.226068	−0.113034	+	0.993591i	$0.536057\pi$
$942$	0	0
$943$	−20.9857	−0.683389
$944$	−17.9937	−0.585646
$945$	0	0
$946$	−1.28894	−0.0419070
$947$	−26.6200	−0.865032	−0.432516	−	0.901626i	$-0.642374\pi$
$947$	−26.6200	−0.865032	−0.432516	+	0.901626i	$0.642374\pi$
$948$	0	0
$949$	5.87212	0.190617
$950$	0	0
$951$	0	0
$952$	−2.70818	−0.0877728
$953$	14.8177	0.479992	0.239996	−	0.970774i	$-0.422854\pi$
$953$	14.8177	0.479992	0.239996	+	0.970774i	$0.422854\pi$
$954$	0	0
$955$	0	0
$956$	10.0153	0.323919
$957$	0	0
$958$	2.54347	0.0821757
$959$	4.45556	0.143878
$960$	0	0
$961$	1.00000	0.0322581
$962$	−0.695949	−0.0224383
$963$	0	0
$964$	14.4382	0.465023
$965$	0	0
$966$	0	0
$967$	−24.3047	−0.781585	−0.390793	−	0.920479i	$-0.627799\pi$
$967$	−24.3047	−0.781585	−0.390793	+	0.920479i	$0.627799\pi$
$968$	−7.03514	−0.226118
$969$	0	0
$970$	0	0
$971$	25.6481	0.823085	0.411543	−	0.911390i	$-0.364990\pi$
$971$	25.6481	0.823085	0.411543	+	0.911390i	$0.364990\pi$
$972$	0	0
$973$	23.6581	0.758444
$974$	−0.786518	−0.0252017
$975$	0	0
$976$	19.6279	0.628273
$977$	−3.68615	−0.117930	−0.0589652	−	0.998260i	$-0.518780\pi$
$977$	−3.68615	−0.117930	−0.0589652	+	0.998260i	$0.518780\pi$
$978$	0	0
$979$	15.7139	0.502219
$980$	0	0
$981$	0	0
$982$	1.01668	0.0324435
$983$	−36.2170	−1.15514	−0.577572	−	0.816340i	$-0.696000\pi$
$983$	−36.2170	−1.15514	−0.577572	+	0.816340i	$0.696000\pi$
$984$	0	0
$985$	0	0
$986$	1.43022	0.0455474
$987$	0	0
$988$	5.50913	0.175269
$989$	−6.62248	−0.210582
$990$	0	0
$991$	41.7558	1.32642	0.663209	−	0.748434i	$-0.269194\pi$
$991$	41.7558	1.32642	0.663209	+	0.748434i	$0.269194\pi$
$992$	1.24122	0.0394089
$993$	0	0
$994$	−4.05749	−0.128696
$995$	0	0
$996$	0	0
$997$	−26.0697	−0.825637	−0.412818	−	0.910813i	$-0.635456\pi$
$997$	−26.0697	−0.825637	−0.412818	+	0.910813i	$0.635456\pi$
$998$	0.687834	0.0217730
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cj.1.6		11
3.2	odd	2		2325.2.a.bc.1.6		11
5.2	odd	4		1395.2.c.h.559.12		22
5.3	odd	4		1395.2.c.h.559.11		22
5.4	even	2		6975.2.a.ci.1.6		11
15.2	even	4		465.2.c.b.94.11	✓	22
15.8	even	4		465.2.c.b.94.12	yes	22
15.14	odd	2		2325.2.a.bd.1.6		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.b.94.11	✓	22	15.2	even	4
465.2.c.b.94.12	yes	22	15.8	even	4
1395.2.c.h.559.11		22	5.3	odd	4
1395.2.c.h.559.12		22	5.2	odd	4
2325.2.a.bc.1.6		11	3.2	odd	2
2325.2.a.bd.1.6		11	15.14	odd	2
6975.2.a.ci.1.6		11	5.4	even	2
6975.2.a.cj.1.6		11	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.104188 q^{2} -1.98914 q^{4} +3.88714 q^{7} -0.415622 q^{8} +5.28458 q^{11} -0.694025 q^{13} +0.404994 q^{14} +3.93499 q^{16} +1.67629 q^{17} +3.99063 q^{19} +0.550591 q^{22} +2.82890 q^{23} -0.0723093 q^{26} -7.73208 q^{28} +8.18906 q^{29} +1.00000 q^{31} +1.24122 q^{32} +0.174650 q^{34} +9.62461 q^{37} +0.415777 q^{38} -7.41833 q^{41} -2.34101 q^{43} -10.5118 q^{44} +0.294738 q^{46} -0.567889 q^{47} +8.10984 q^{49} +1.38052 q^{52} +11.3489 q^{53} -1.61558 q^{56} +0.853204 q^{58} -4.57276 q^{59} +4.98804 q^{61} +0.104188 q^{62} -7.74065 q^{64} -13.7584 q^{67} -3.33439 q^{68} -10.0186 q^{71} -8.46096 q^{73} +1.00277 q^{74} -7.93794 q^{76} +20.5419 q^{77} -0.437300 q^{79} -0.772903 q^{82} +13.9850 q^{83} -0.243905 q^{86} -2.19639 q^{88} +2.97354 q^{89} -2.69777 q^{91} -5.62709 q^{92} -0.0591674 q^{94} -9.79254 q^{97} +0.844950 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{2} + 15 q^{4} - 8 q^{7} + 9 q^{8} - 14 q^{13} + 14 q^{14} + 27 q^{16} + 12 q^{17} + 12 q^{19} - 10 q^{22} + 12 q^{23} + 6 q^{26} - 22 q^{28} + 8 q^{29} + 11 q^{31} + 21 q^{32} + 2 q^{34} - 16 q^{37}+ \cdots + 17 q^{98}+O(q^{100}) \) 11 * q + 3 * q^2 + 15 * q^4 - 8 * q^7 + 9 * q^8 - 14 * q^13 + 14 * q^14 + 27 * q^16 + 12 * q^17 + 12 * q^19 - 10 * q^22 + 12 * q^23 + 6 * q^26 - 22 * q^28 + 8 * q^29 + 11 * q^31 + 21 * q^32 + 2 * q^34 - 16 * q^37 + 26 * q^38 + 8 * q^41 - 28 * q^43 + 26 * q^44 + 8 * q^46 + 14 * q^47 + 23 * q^49 - 18 * q^52 + 26 * q^53 + 42 * q^56 - 6 * q^58 + 6 * q^59 + 14 * q^61 + 3 * q^62 + 29 * q^64 - 6 * q^67 + 54 * q^68 + 18 * q^71 - 26 * q^73 + 40 * q^74 + 10 * q^76 + 2 * q^77 + 8 * q^79 + 30 * q^82 + 48 * q^83 + 12 * q^86 - 40 * q^88 + 14 * q^89 - 4 * q^91 - 30 * q^92 - 4 * q^94 - 16 * q^97 + 17 * q^98

Introduction

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