LMFDB - Embedded newform 6975.2.a.bv.1.1

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 = [5,3,0,3,0,0,-8,9,0,0,0,0,-6,-16,0,3,8] 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:	$1$
Dimension:	$5$
Coefficient field:	5.5.126032.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 6x - 2 $ x^5 - 6x^3 + 6x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.33253$ 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)

$f(q)$	$=$	$q-1.39276 q^{2} -0.0602300 q^{4} +0.392756 q^{7} +2.86940 q^{8} +2.96404 q^{11} -2.92963 q^{13} -0.547014 q^{14} -3.87591 q^{16} -0.739663 q^{17} -3.50091 q^{19} -4.12818 q^{22} +7.61557 q^{23} +4.08026 q^{26} -0.0236557 q^{28} -0.883778 q^{29} -1.00000 q^{31} -0.340595 q^{32} +1.03017 q^{34} +1.01775 q^{37} +4.87591 q^{38} +0.566932 q^{41} -4.98562 q^{43} -0.178524 q^{44} -10.6066 q^{46} +2.22210 q^{47} -6.84574 q^{49} +0.176452 q^{52} -7.23391 q^{53} +1.12697 q^{56} +1.23089 q^{58} +4.56679 q^{59} -2.23789 q^{61} +1.39276 q^{62} +8.22619 q^{64} -3.28668 q^{67} +0.0445499 q^{68} -9.31452 q^{71} +15.6910 q^{73} -1.41748 q^{74} +0.210860 q^{76} +1.16414 q^{77} -10.7258 q^{79} -0.789599 q^{82} +5.97573 q^{83} +6.94375 q^{86} +8.50500 q^{88} +2.02861 q^{89} -1.15063 q^{91} -0.458686 q^{92} -3.09484 q^{94} -2.14846 q^{97} +9.53445 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 3 q^{4} - 8 q^{7} + 9 q^{8} - 6 q^{13} - 16 q^{14} + 3 q^{16} + 8 q^{17} - 4 q^{19} - 6 q^{22} + 4 q^{23} + 6 q^{26} - 18 q^{28} - 4 q^{29} - 5 q^{31} + q^{32} + 6 q^{34} + 4 q^{37} + 2 q^{38}+ \cdots + 29 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 3 * q^4 - 8 * q^7 + 9 * q^8 - 6 * q^13 - 16 * q^14 + 3 * q^16 + 8 * q^17 - 4 * q^19 - 6 * q^22 + 4 * q^23 + 6 * q^26 - 18 * q^28 - 4 * q^29 - 5 * q^31 + q^32 + 6 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^43 - 6 * q^44 - 16 * q^46 + 18 * q^47 - 11 * q^49 - 2 * q^52 + 2 * q^53 - 18 * q^56 - 14 * q^58 - 10 * q^59 - 14 * q^61 - 3 * q^62 + 11 * q^64 - 18 * q^67 + 6 * q^68 - 6 * q^71 - 10 * q^73 + 4 * q^74 - 8 * q^76 + 6 * q^77 - 24 * q^79 - 30 * q^82 + 48 * q^83 - 4 * q^86 - 8 * q^88 - 10 * q^89 - 26 * q^92 + 12 * q^94 - 20 * q^97 + 29 * 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 $

$2$	−1.39276	−0.984827	−0.492414	−	0.870361i	$-0.663885\pi$
$2$	−1.39276	−0.984827	−0.492414	+	0.870361i	$0.663885\pi$
$3$	0	0
$3$	0	0
$4$	−0.0602300	−0.0301150
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.392756	0.148448	0.0742240	−	0.997242i	$-0.476352\pi$
$7$	0.392756	0.148448	0.0742240	+	0.997242i	$0.476352\pi$
$8$	2.86940	1.01449
$9$	0	0
$10$	0	0
$11$	2.96404	0.893691	0.446845	−	0.894611i	$-0.352547\pi$
$11$	2.96404	0.893691	0.446845	+	0.894611i	$0.352547\pi$
$12$	0	0
$13$	−2.92963	−0.812533	−0.406266	−	0.913755i	$-0.633169\pi$
$13$	−2.92963	−0.812533	−0.406266	+	0.913755i	$0.633169\pi$
$14$	−0.547014	−0.146196
$15$	0	0
$16$	−3.87591	−0.968978
$17$	−0.739663	−0.179395	−0.0896973	−	0.995969i	$-0.528590\pi$
$17$	−0.739663	−0.179395	−0.0896973	+	0.995969i	$0.528590\pi$
$18$	0	0
$19$	−3.50091	−0.803164	−0.401582	−	0.915823i	$-0.631539\pi$
$19$	−3.50091	−0.803164	−0.401582	+	0.915823i	$0.631539\pi$
$20$	0	0
$21$	0	0
$22$	−4.12818	−0.880131
$23$	7.61557	1.58796	0.793979	−	0.607946i	$-0.208006\pi$
$23$	7.61557	1.58796	0.793979	+	0.607946i	$0.208006\pi$
$24$	0	0
$25$	0	0
$26$	4.08026	0.800204
$27$	0	0
$28$	−0.0236557	−0.00447051
$29$	−0.883778	−0.164114	−0.0820568	−	0.996628i	$-0.526149\pi$
$29$	−0.883778	−0.164114	−0.0820568	+	0.996628i	$0.526149\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−0.340595	−0.0602093
$33$	0	0
$34$	1.03017	0.176673
$35$	0	0
$36$	0	0
$37$	1.01775	0.167317	0.0836587	−	0.996494i	$-0.473339\pi$
$37$	1.01775	0.167317	0.0836587	+	0.996494i	$0.473339\pi$
$38$	4.87591	0.790977
$39$	0	0
$40$	0	0
$41$	0.566932	0.0885400	0.0442700	−	0.999020i	$-0.485904\pi$
$41$	0.566932	0.0885400	0.0442700	+	0.999020i	$0.485904\pi$
$42$	0	0
$43$	−4.98562	−0.760300	−0.380150	−	0.924925i	$-0.624128\pi$
$43$	−4.98562	−0.760300	−0.380150	+	0.924925i	$0.624128\pi$
$44$	−0.178524	−0.0269135
$45$	0	0
$46$	−10.6066	−1.56386
$47$	2.22210	0.324126	0.162063	−	0.986780i	$-0.448185\pi$
$47$	2.22210	0.324126	0.162063	+	0.986780i	$0.448185\pi$
$48$	0	0
$49$	−6.84574	−0.977963
$50$	0	0
$51$	0	0
$52$	0.176452	0.0244694
$53$	−7.23391	−0.993654	−0.496827	−	0.867850i	$-0.665502\pi$
$53$	−7.23391	−0.993654	−0.496827	+	0.867850i	$0.665502\pi$
$54$	0	0
$55$	0	0
$56$	1.12697	0.150598
$57$	0	0
$58$	1.23089	0.161624
$59$	4.56679	0.594545	0.297272	−	0.954793i	$-0.403923\pi$
$59$	4.56679	0.594545	0.297272	+	0.954793i	$0.403923\pi$
$60$	0	0
$61$	−2.23789	−0.286532	−0.143266	−	0.989684i	$-0.545760\pi$
$61$	−2.23789	−0.286532	−0.143266	+	0.989684i	$0.545760\pi$
$62$	1.39276	0.176880
$63$	0	0
$64$	8.22619	1.02827
$65$	0	0
$66$	0	0
$67$	−3.28668	−0.401531	−0.200766	−	0.979639i	$-0.564343\pi$
$67$	−3.28668	−0.401531	−0.200766	+	0.979639i	$0.564343\pi$
$68$	0.0445499	0.00540247
$69$	0	0
$70$	0	0
$71$	−9.31452	−1.10543	−0.552715	−	0.833370i	$-0.686408\pi$
$71$	−9.31452	−1.10543	−0.552715	+	0.833370i	$0.686408\pi$
$72$	0	0
$73$	15.6910	1.83649	0.918246	−	0.396011i	$-0.129606\pi$
$73$	15.6910	1.83649	0.918246	+	0.396011i	$0.129606\pi$
$74$	−1.41748	−0.164779
$75$	0	0
$76$	0.210860	0.0241873
$77$	1.16414	0.132667
$78$	0	0
$79$	−10.7258	−1.20675	−0.603373	−	0.797459i	$-0.706177\pi$
$79$	−10.7258	−1.20675	−0.603373	+	0.797459i	$0.706177\pi$
$80$	0	0
$81$	0	0
$82$	−0.789599	−0.0871966
$83$	5.97573	0.655922	0.327961	−	0.944691i	$-0.393639\pi$
$83$	5.97573	0.655922	0.327961	+	0.944691i	$0.393639\pi$
$84$	0	0
$85$	0	0
$86$	6.94375	0.748764
$87$	0	0
$88$	8.50500	0.906636
$89$	2.02861	0.215033	0.107516	−	0.994203i	$-0.465710\pi$
$89$	2.02861	0.215033	0.107516	+	0.994203i	$0.465710\pi$
$90$	0	0
$91$	−1.15063	−0.120619
$92$	−0.458686	−0.0478214
$93$	0	0
$94$	−3.09484	−0.319209
$95$	0	0
$96$	0	0
$97$	−2.14846	−0.218143	−0.109072	−	0.994034i	$-0.534788\pi$
$97$	−2.14846	−0.218143	−0.109072	+	0.994034i	$0.534788\pi$
$98$	9.53445	0.963125
$99$	0	0
$100$	0	0
$101$	−10.1754	−1.01249	−0.506246	−	0.862389i	$-0.668967\pi$
$101$	−10.1754	−1.01249	−0.506246	+	0.862389i	$0.668967\pi$
$102$	0	0
$103$	−17.4957	−1.72391	−0.861953	−	0.506989i	$-0.830758\pi$
$103$	−17.4957	−1.72391	−0.861953	+	0.506989i	$0.830758\pi$
$104$	−8.40627	−0.824303
$105$	0	0
$106$	10.0751	0.978578
$107$	5.35074	0.517275	0.258638	−	0.965974i	$-0.416726\pi$
$107$	5.35074	0.517275	0.258638	+	0.965974i	$0.416726\pi$
$108$	0	0
$109$	−5.41010	−0.518194	−0.259097	−	0.965851i	$-0.583425\pi$
$109$	−5.41010	−0.518194	−0.259097	+	0.965851i	$0.583425\pi$
$110$	0	0
$111$	0	0
$112$	−1.52229	−0.143843
$113$	−0.847152	−0.0796934	−0.0398467	−	0.999206i	$-0.512687\pi$
$113$	−0.847152	−0.0796934	−0.0398467	+	0.999206i	$0.512687\pi$
$114$	0	0
$115$	0	0
$116$	0.0532300	0.00494228
$117$	0	0
$118$	−6.36042	−0.585524
$119$	−0.290507	−0.0266307
$120$	0	0
$121$	−2.21449	−0.201317
$122$	3.11683	0.282185
$123$	0	0
$124$	0.0602300	0.00540882
$125$	0	0
$126$	0	0
$127$	0.440271	0.0390677	0.0195339	−	0.999809i	$-0.493782\pi$
$127$	0.440271	0.0390677	0.0195339	+	0.999809i	$0.493782\pi$
$128$	−10.7759	−0.952463
$129$	0	0
$130$	0	0
$131$	−20.9706	−1.83221	−0.916106	−	0.400935i	$-0.868685\pi$
$131$	−20.9706	−1.83221	−0.916106	+	0.400935i	$0.868685\pi$
$132$	0	0
$133$	−1.37500	−0.119228
$134$	4.57754	0.395439
$135$	0	0
$136$	−2.12239	−0.181993
$137$	10.7649	0.919706	0.459853	−	0.887995i	$-0.347902\pi$
$137$	10.7649	0.919706	0.459853	+	0.887995i	$0.347902\pi$
$138$	0	0
$139$	11.2882	0.957456	0.478728	−	0.877963i	$-0.341098\pi$
$139$	11.2882	0.957456	0.478728	+	0.877963i	$0.341098\pi$
$140$	0	0
$141$	0	0
$142$	12.9729	1.08866
$143$	−8.68353	−0.726153
$144$	0	0
$145$	0	0
$146$	−21.8537	−1.80863
$147$	0	0
$148$	−0.0612993	−0.00503877
$149$	−3.86637	−0.316746	−0.158373	−	0.987379i	$-0.550625\pi$
$149$	−3.86637	−0.316746	−0.158373	+	0.987379i	$0.550625\pi$
$150$	0	0
$151$	16.8876	1.37429	0.687147	−	0.726518i	$-0.258863\pi$
$151$	16.8876	1.37429	0.687147	+	0.726518i	$0.258863\pi$
$152$	−10.0455	−0.814798
$153$	0	0
$154$	−1.62137	−0.130654
$155$	0	0
$156$	0	0
$157$	12.2006	0.973716	0.486858	−	0.873481i	$-0.338143\pi$
$157$	12.2006	0.973716	0.486858	+	0.873481i	$0.338143\pi$
$158$	14.9384	1.18844
$159$	0	0
$160$	0	0
$161$	2.99106	0.235729
$162$	0	0
$163$	−10.4423	−0.817901	−0.408951	−	0.912556i	$-0.634105\pi$
$163$	−10.4423	−0.817901	−0.408951	+	0.912556i	$0.634105\pi$
$164$	−0.0341464	−0.00266638
$165$	0	0
$166$	−8.32274	−0.645970
$167$	9.62985	0.745180	0.372590	−	0.927996i	$-0.378470\pi$
$167$	9.62985	0.745180	0.372590	+	0.927996i	$0.378470\pi$
$168$	0	0
$169$	−4.41728	−0.339791
$170$	0	0
$171$	0	0
$172$	0.300284	0.0228964
$173$	25.7568	1.95825	0.979125	−	0.203259i	$-0.0651533\pi$
$173$	25.7568	1.95825	0.979125	+	0.203259i	$0.0651533\pi$
$174$	0	0
$175$	0	0
$176$	−11.4883	−0.865967
$177$	0	0
$178$	−2.82537	−0.211770
$179$	−22.5410	−1.68479	−0.842396	−	0.538860i	$-0.818855\pi$
$179$	−22.5410	−1.68479	−0.842396	+	0.538860i	$0.818855\pi$
$180$	0	0
$181$	6.49425	0.482714	0.241357	−	0.970436i	$-0.422408\pi$
$181$	6.49425	0.482714	0.241357	+	0.970436i	$0.422408\pi$
$182$	1.60255	0.118789
$183$	0	0
$184$	21.8521	1.61096
$185$	0	0
$186$	0	0
$187$	−2.19239	−0.160323
$188$	−0.133837	−0.00976107
$189$	0	0
$190$	0	0
$191$	15.6289	1.13087	0.565435	−	0.824793i	$-0.308708\pi$
$191$	15.6289	1.13087	0.565435	+	0.824793i	$0.308708\pi$
$192$	0	0
$193$	15.2744	1.09947	0.549737	−	0.835338i	$-0.314728\pi$
$193$	15.2744	1.09947	0.549737	+	0.835338i	$0.314728\pi$
$194$	2.99229	0.214834
$195$	0	0
$196$	0.412319	0.0294514
$197$	15.8270	1.12763	0.563815	−	0.825901i	$-0.309333\pi$
$197$	15.8270	1.12763	0.563815	+	0.825901i	$0.309333\pi$
$198$	0	0
$199$	−6.38186	−0.452398	−0.226199	−	0.974081i	$-0.572630\pi$
$199$	−6.38186	−0.452398	−0.226199	+	0.974081i	$0.572630\pi$
$200$	0	0
$201$	0	0
$202$	14.1719	0.997129
$203$	−0.347109	−0.0243623
$204$	0	0
$205$	0	0
$206$	24.3673	1.69775
$207$	0	0
$208$	11.3550	0.787326
$209$	−10.3768	−0.717780
$210$	0	0
$211$	−16.8427	−1.15950	−0.579751	−	0.814794i	$-0.696850\pi$
$211$	−16.8427	−1.15950	−0.579751	+	0.814794i	$0.696850\pi$
$212$	0.435699	0.0299239
$213$	0	0
$214$	−7.45227	−0.509427
$215$	0	0
$216$	0	0
$217$	−0.392756	−0.0266620
$218$	7.53495	0.510331
$219$	0	0
$220$	0	0
$221$	2.16694	0.145764
$222$	0	0
$223$	−21.3746	−1.43135	−0.715673	−	0.698436i	$-0.753880\pi$
$223$	−21.3746	−1.43135	−0.715673	+	0.698436i	$0.753880\pi$
$224$	−0.133771	−0.00893795
$225$	0	0
$226$	1.17988	0.0784842
$227$	21.7731	1.44513	0.722564	−	0.691304i	$-0.242963\pi$
$227$	21.7731	1.44513	0.722564	+	0.691304i	$0.242963\pi$
$228$	0	0
$229$	20.9196	1.38241	0.691203	−	0.722661i	$-0.257081\pi$
$229$	20.9196	1.38241	0.691203	+	0.722661i	$0.257081\pi$
$230$	0	0
$231$	0	0
$232$	−2.53591	−0.166491
$233$	−14.6529	−0.959943	−0.479972	−	0.877284i	$-0.659353\pi$
$233$	−14.6529	−0.959943	−0.479972	+	0.877284i	$0.659353\pi$
$234$	0	0
$235$	0	0
$236$	−0.275058	−0.0179047
$237$	0	0
$238$	0.404606	0.0262267
$239$	−19.2602	−1.24584	−0.622921	−	0.782285i	$-0.714054\pi$
$239$	−19.2602	−1.24584	−0.622921	+	0.782285i	$0.714054\pi$
$240$	0	0
$241$	0.395294	0.0254631	0.0127316	−	0.999919i	$-0.495947\pi$
$241$	0.395294	0.0254631	0.0127316	+	0.999919i	$0.495947\pi$
$242$	3.08424	0.198263
$243$	0	0
$244$	0.134788	0.00862892
$245$	0	0
$246$	0	0
$247$	10.2564	0.652597
$248$	−2.86940	−0.182207
$249$	0	0
$250$	0	0
$251$	21.6504	1.36656	0.683282	−	0.730155i	$-0.260552\pi$
$251$	21.6504	1.36656	0.683282	+	0.730155i	$0.260552\pi$
$252$	0	0
$253$	22.5728	1.41914
$254$	−0.613190	−0.0384750
$255$	0	0
$256$	−1.44420	−0.0902623
$257$	−21.1733	−1.32075	−0.660376	−	0.750935i	$-0.729603\pi$
$257$	−21.1733	−1.32075	−0.660376	+	0.750935i	$0.729603\pi$
$258$	0	0
$259$	0.399729	0.0248379
$260$	0	0
$261$	0	0
$262$	29.2070	1.80441
$263$	4.22941	0.260797	0.130398	−	0.991462i	$-0.458374\pi$
$263$	4.22941	0.260797	0.130398	+	0.991462i	$0.458374\pi$
$264$	0	0
$265$	0	0
$266$	1.91505	0.117419
$267$	0	0
$268$	0.197957	0.0120921
$269$	−25.7692	−1.57117	−0.785587	−	0.618751i	$-0.787639\pi$
$269$	−25.7692	−1.57117	−0.785587	+	0.618751i	$0.787639\pi$
$270$	0	0
$271$	17.1588	1.04232	0.521161	−	0.853458i	$-0.325499\pi$
$271$	17.1588	1.04232	0.521161	+	0.853458i	$0.325499\pi$
$272$	2.86687	0.173829
$273$	0	0
$274$	−14.9929	−0.905751
$275$	0	0
$276$	0	0
$277$	−6.16879	−0.370647	−0.185323	−	0.982678i	$-0.559333\pi$
$277$	−6.16879	−0.370647	−0.185323	+	0.982678i	$0.559333\pi$
$278$	−15.7218	−0.942929
$279$	0	0
$280$	0	0
$281$	−24.9164	−1.48638	−0.743192	−	0.669078i	$-0.766689\pi$
$281$	−24.9164	−1.48638	−0.743192	+	0.669078i	$0.766689\pi$
$282$	0	0
$283$	−25.1858	−1.49714	−0.748569	−	0.663057i	$-0.769259\pi$
$283$	−25.1858	−1.49714	−0.748569	+	0.663057i	$0.769259\pi$
$284$	0.561014	0.0332900
$285$	0	0
$286$	12.0940	0.715135
$287$	0.222666	0.0131436
$288$	0	0
$289$	−16.4529	−0.967818
$290$	0	0
$291$	0	0
$292$	−0.945069	−0.0553060
$293$	10.6574	0.622615	0.311307	−	0.950309i	$-0.399233\pi$
$293$	10.6574	0.622615	0.311307	+	0.950309i	$0.399233\pi$
$294$	0	0
$295$	0	0
$296$	2.92034	0.169741
$297$	0	0
$298$	5.38492	0.311940
$299$	−22.3108	−1.29027
$300$	0	0
$301$	−1.95813	−0.112865
$302$	−23.5203	−1.35344
$303$	0	0
$304$	13.5692	0.778248
$305$	0	0
$306$	0	0
$307$	27.0247	1.54238	0.771191	−	0.636603i	$-0.219661\pi$
$307$	27.0247	1.54238	0.771191	+	0.636603i	$0.219661\pi$
$308$	−0.0701164	−0.00399525
$309$	0	0
$310$	0	0
$311$	−33.9130	−1.92303	−0.961515	−	0.274752i	$-0.911404\pi$
$311$	−33.9130	−1.92303	−0.961515	+	0.274752i	$0.911404\pi$
$312$	0	0
$313$	−0.748537	−0.0423098	−0.0211549	−	0.999776i	$-0.506734\pi$
$313$	−0.748537	−0.0423098	−0.0211549	+	0.999776i	$0.506734\pi$
$314$	−16.9925	−0.958943
$315$	0	0
$316$	0.646015	0.0363412
$317$	25.4898	1.43165	0.715824	−	0.698281i	$-0.246051\pi$
$317$	25.4898	1.43165	0.715824	+	0.698281i	$0.246051\pi$
$318$	0	0
$319$	−2.61955	−0.146667
$320$	0	0
$321$	0	0
$322$	−4.16582	−0.232152
$323$	2.58949	0.144083
$324$	0	0
$325$	0	0
$326$	14.5435	0.805492
$327$	0	0
$328$	1.62675	0.0898225
$329$	0.872743	0.0481159
$330$	0	0
$331$	2.77571	0.152567	0.0762834	−	0.997086i	$-0.475695\pi$
$331$	2.77571	0.152567	0.0762834	+	0.997086i	$0.475695\pi$
$332$	−0.359919	−0.0197531
$333$	0	0
$334$	−13.4120	−0.733874
$335$	0	0
$336$	0	0
$337$	−2.73960	−0.149235	−0.0746176	−	0.997212i	$-0.523774\pi$
$337$	−2.73960	−0.149235	−0.0746176	+	0.997212i	$0.523774\pi$
$338$	6.15219	0.334635
$339$	0	0
$340$	0	0
$341$	−2.96404	−0.160512
$342$	0	0
$343$	−5.43800	−0.293625
$344$	−14.3057	−0.771313
$345$	0	0
$346$	−35.8729	−1.92854
$347$	−24.9035	−1.33689	−0.668444	−	0.743762i	$-0.733039\pi$
$347$	−24.9035	−1.33689	−0.668444	+	0.743762i	$0.733039\pi$
$348$	0	0
$349$	−20.9690	−1.12245	−0.561223	−	0.827665i	$-0.689669\pi$
$349$	−20.9690	−1.12245	−0.561223	+	0.827665i	$0.689669\pi$
$350$	0	0
$351$	0	0
$352$	−1.00954	−0.0538085
$353$	9.40420	0.500535	0.250268	−	0.968177i	$-0.419481\pi$
$353$	9.40420	0.500535	0.250268	+	0.968177i	$0.419481\pi$
$354$	0	0
$355$	0	0
$356$	−0.122184	−0.00647571
$357$	0	0
$358$	31.3941	1.65923
$359$	−14.8384	−0.783139	−0.391570	−	0.920149i	$-0.628068\pi$
$359$	−14.8384	−0.783139	−0.391570	+	0.920149i	$0.628068\pi$
$360$	0	0
$361$	−6.74364	−0.354928
$362$	−9.04491	−0.475390
$363$	0	0
$364$	0.0693025	0.00363244
$365$	0	0
$366$	0	0
$367$	17.6485	0.921246	0.460623	−	0.887596i	$-0.347626\pi$
$367$	17.6485	0.921246	0.460623	+	0.887596i	$0.347626\pi$
$368$	−29.5173	−1.53870
$369$	0	0
$370$	0	0
$371$	−2.84116	−0.147506
$372$	0	0
$373$	−7.54359	−0.390592	−0.195296	−	0.980744i	$-0.562567\pi$
$373$	−7.54359	−0.390592	−0.195296	+	0.980744i	$0.562567\pi$
$374$	3.05346	0.157891
$375$	0	0
$376$	6.37609	0.328821
$377$	2.58914	0.133348
$378$	0	0
$379$	−10.4641	−0.537505	−0.268753	−	0.963209i	$-0.586611\pi$
$379$	−10.4641	−0.537505	−0.268753	+	0.963209i	$0.586611\pi$
$380$	0	0
$381$	0	0
$382$	−21.7673	−1.11371
$383$	−32.1479	−1.64268	−0.821339	−	0.570440i	$-0.806773\pi$
$383$	−32.1479	−1.64268	−0.821339	+	0.570440i	$0.806773\pi$
$384$	0	0
$385$	0	0
$386$	−21.2735	−1.08279
$387$	0	0
$388$	0.129402	0.00656939
$389$	0.586235	0.0297233	0.0148616	−	0.999890i	$-0.495269\pi$
$389$	0.586235	0.0297233	0.0148616	+	0.999890i	$0.495269\pi$
$390$	0	0
$391$	−5.63296	−0.284871
$392$	−19.6432	−0.992129
$393$	0	0
$394$	−22.0432	−1.11052
$395$	0	0
$396$	0	0
$397$	−7.15104	−0.358900	−0.179450	−	0.983767i	$-0.557432\pi$
$397$	−7.15104	−0.358900	−0.179450	+	0.983767i	$0.557432\pi$
$398$	8.88837	0.445534
$399$	0	0
$400$	0	0
$401$	13.7057	0.684430	0.342215	−	0.939622i	$-0.388823\pi$
$401$	13.7057	0.684430	0.342215	+	0.939622i	$0.388823\pi$
$402$	0	0
$403$	2.92963	0.145935
$404$	0.612866	0.0304912
$405$	0	0
$406$	0.483439	0.0239927
$407$	3.01666	0.149530
$408$	0	0
$409$	−25.6183	−1.26674	−0.633371	−	0.773849i	$-0.718329\pi$
$409$	−25.6183	−1.26674	−0.633371	+	0.773849i	$0.718329\pi$
$410$	0	0
$411$	0	0
$412$	1.05377	0.0519154
$413$	1.79363	0.0882590
$414$	0	0
$415$	0	0
$416$	0.997818	0.0489221
$417$	0	0
$418$	14.4524	0.706889
$419$	−3.52044	−0.171985	−0.0859925	−	0.996296i	$-0.527406\pi$
$419$	−3.52044	−0.171985	−0.0859925	+	0.996296i	$0.527406\pi$
$420$	0	0
$421$	35.0272	1.70712	0.853559	−	0.520995i	$-0.174439\pi$
$421$	35.0272	1.70712	0.853559	+	0.520995i	$0.174439\pi$
$422$	23.4578	1.14191
$423$	0	0
$424$	−20.7570	−1.00805
$425$	0	0
$426$	0	0
$427$	−0.878944	−0.0425351
$428$	−0.322275	−0.0155778
$429$	0	0
$430$	0	0
$431$	30.0399	1.44697	0.723486	−	0.690339i	$-0.242539\pi$
$431$	30.0399	1.44697	0.723486	+	0.690339i	$0.242539\pi$
$432$	0	0
$433$	−19.6374	−0.943713	−0.471857	−	0.881675i	$-0.656416\pi$
$433$	−19.6374	−0.943713	−0.471857	+	0.881675i	$0.656416\pi$
$434$	0.547014	0.0262575
$435$	0	0
$436$	0.325851	0.0156054
$437$	−26.6614	−1.27539
$438$	0	0
$439$	8.15307	0.389125	0.194562	−	0.980890i	$-0.437671\pi$
$439$	8.15307	0.389125	0.194562	+	0.980890i	$0.437671\pi$
$440$	0	0
$441$	0	0
$442$	−3.01801	−0.143552
$443$	−6.41904	−0.304978	−0.152489	−	0.988305i	$-0.548729\pi$
$443$	−6.41904	−0.304978	−0.152489	+	0.988305i	$0.548729\pi$
$444$	0	0
$445$	0	0
$446$	29.7695	1.40963
$447$	0	0
$448$	3.23089	0.152645
$449$	10.4764	0.494414	0.247207	−	0.968963i	$-0.420487\pi$
$449$	10.4764	0.494414	0.247207	+	0.968963i	$0.420487\pi$
$450$	0	0
$451$	1.68041	0.0791273
$452$	0.0510240	0.00239997
$453$	0	0
$454$	−30.3246	−1.42320
$455$	0	0
$456$	0	0
$457$	−27.1616	−1.27057	−0.635283	−	0.772279i	$-0.719117\pi$
$457$	−27.1616	−1.27057	−0.635283	+	0.772279i	$0.719117\pi$
$458$	−29.1359	−1.36143
$459$	0	0
$460$	0	0
$461$	−20.3271	−0.946727	−0.473363	−	0.880867i	$-0.656960\pi$
$461$	−20.3271	−0.946727	−0.473363	+	0.880867i	$0.656960\pi$
$462$	0	0
$463$	−19.3354	−0.898593	−0.449296	−	0.893383i	$-0.648325\pi$
$463$	−19.3354	−0.898593	−0.449296	+	0.893383i	$0.648325\pi$
$464$	3.42545	0.159022
$465$	0	0
$466$	20.4079	0.945378
$467$	8.76168	0.405442	0.202721	−	0.979237i	$-0.435022\pi$
$467$	8.76168	0.405442	0.202721	+	0.979237i	$0.435022\pi$
$468$	0	0
$469$	−1.29086	−0.0596065
$470$	0	0
$471$	0	0
$472$	13.1039	0.603157
$473$	−14.7776	−0.679473
$474$	0	0
$475$	0	0
$476$	0.0174973	0.000801985 0
$477$	0	0
$478$	26.8248	1.22694
$479$	−13.8608	−0.633318	−0.316659	−	0.948539i	$-0.602561\pi$
$479$	−13.8608	−0.633318	−0.316659	+	0.948539i	$0.602561\pi$
$480$	0	0
$481$	−2.98164	−0.135951
$482$	−0.550548	−0.0250768
$483$	0	0
$484$	0.133379	0.00606267
$485$	0	0
$486$	0	0
$487$	−31.7390	−1.43823	−0.719116	−	0.694890i	$-0.755453\pi$
$487$	−31.7390	−1.43823	−0.719116	+	0.694890i	$0.755453\pi$
$488$	−6.42139	−0.290683
$489$	0	0
$490$	0	0
$491$	3.53863	0.159696	0.0798481	−	0.996807i	$-0.474556\pi$
$491$	3.53863	0.159696	0.0798481	+	0.996807i	$0.474556\pi$
$492$	0	0
$493$	0.653698	0.0294411
$494$	−14.2846	−0.642695
$495$	0	0
$496$	3.87591	0.174034
$497$	−3.65834	−0.164099
$498$	0	0
$499$	17.2529	0.772345	0.386172	−	0.922427i	$-0.373797\pi$
$499$	17.2529	0.772345	0.386172	+	0.922427i	$0.373797\pi$
$500$	0	0
$501$	0	0
$502$	−30.1538	−1.34583
$503$	−28.5735	−1.27403	−0.637015	−	0.770852i	$-0.719831\pi$
$503$	−28.5735	−1.27403	−0.637015	+	0.770852i	$0.719831\pi$
$504$	0	0
$505$	0	0
$506$	−31.4385	−1.39761
$507$	0	0
$508$	−0.0265175	−0.00117652
$509$	20.1430	0.892824	0.446412	−	0.894828i	$-0.352702\pi$
$509$	20.1430	0.892824	0.446412	+	0.894828i	$0.352702\pi$
$510$	0	0
$511$	6.16273	0.272623
$512$	23.5632	1.04136
$513$	0	0
$514$	29.4892	1.30071
$515$	0	0
$516$	0	0
$517$	6.58638	0.289669
$518$	−0.556725	−0.0244611
$519$	0	0
$520$	0	0
$521$	−34.7238	−1.52128	−0.760639	−	0.649175i	$-0.775114\pi$
$521$	−34.7238	−1.52128	−0.760639	+	0.649175i	$0.775114\pi$
$522$	0	0
$523$	−21.7416	−0.950694	−0.475347	−	0.879798i	$-0.657678\pi$
$523$	−21.7416	−0.950694	−0.475347	+	0.879798i	$0.657678\pi$
$524$	1.26306	0.0551771
$525$	0	0
$526$	−5.89054	−0.256840
$527$	0.739663	0.0322202
$528$	0	0
$529$	34.9970	1.52161
$530$	0	0
$531$	0	0
$532$	0.0828165	0.00359055
$533$	−1.66090	−0.0719416
$534$	0	0
$535$	0	0
$536$	−9.43078	−0.407348
$537$	0	0
$538$	35.8902	1.54734
$539$	−20.2910	−0.873997
$540$	0	0
$541$	−35.5278	−1.52746	−0.763729	−	0.645537i	$-0.776634\pi$
$541$	−35.5278	−1.52746	−0.763729	+	0.645537i	$0.776634\pi$
$542$	−23.8980	−1.02651
$543$	0	0
$544$	0.251926	0.0108012
$545$	0	0
$546$	0	0
$547$	−9.55722	−0.408637	−0.204319	−	0.978904i	$-0.565498\pi$
$547$	−9.55722	−0.408637	−0.204319	+	0.978904i	$0.565498\pi$
$548$	−0.648369	−0.0276970
$549$	0	0
$550$	0	0
$551$	3.09403	0.131810
$552$	0	0
$553$	−4.21262	−0.179139
$554$	8.59162	0.365023
$555$	0	0
$556$	−0.679891	−0.0288338
$557$	−1.32742	−0.0562447	−0.0281223	−	0.999604i	$-0.508953\pi$
$557$	−1.32742	−0.0562447	−0.0281223	+	0.999604i	$0.508953\pi$
$558$	0	0
$559$	14.6060	0.617769
$560$	0	0
$561$	0	0
$562$	34.7024	1.46383
$563$	8.91259	0.375621	0.187810	−	0.982205i	$-0.439861\pi$
$563$	8.91259	0.375621	0.187810	+	0.982205i	$0.439861\pi$
$564$	0	0
$565$	0	0
$566$	35.0776	1.47442
$567$	0	0
$568$	−26.7271	−1.12144
$569$	19.2832	0.808393	0.404197	−	0.914672i	$-0.367551\pi$
$569$	19.2832	0.808393	0.404197	+	0.914672i	$0.367551\pi$
$570$	0	0
$571$	−36.3948	−1.52307	−0.761537	−	0.648121i	$-0.775555\pi$
$571$	−36.3948	−1.52307	−0.761537	+	0.648121i	$0.775555\pi$
$572$	0.523009	0.0218681
$573$	0	0
$574$	−0.310120	−0.0129441
$575$	0	0
$576$	0	0
$577$	−31.1724	−1.29772	−0.648861	−	0.760907i	$-0.724754\pi$
$577$	−31.1724	−1.29772	−0.648861	+	0.760907i	$0.724754\pi$
$578$	22.9149	0.953133
$579$	0	0
$580$	0	0
$581$	2.34701	0.0973702
$582$	0	0
$583$	−21.4416	−0.888019
$584$	45.0237	1.86309
$585$	0	0
$586$	−14.8432	−0.613168
$587$	−5.11584	−0.211153	−0.105577	−	0.994411i	$-0.533669\pi$
$587$	−5.11584	−0.211153	−0.105577	+	0.994411i	$0.533669\pi$
$588$	0	0
$589$	3.50091	0.144252
$590$	0	0
$591$	0	0
$592$	−3.94472	−0.162127
$593$	−38.8933	−1.59716	−0.798578	−	0.601892i	$-0.794414\pi$
$593$	−38.8933	−1.59716	−0.798578	+	0.601892i	$0.794414\pi$
$594$	0	0
$595$	0	0
$596$	0.232872	0.00953880
$597$	0	0
$598$	31.0735	1.27069
$599$	34.5003	1.40964	0.704821	−	0.709385i	$-0.251027\pi$
$599$	34.5003	1.40964	0.704821	+	0.709385i	$0.251027\pi$
$600$	0	0
$601$	−41.3204	−1.68550	−0.842748	−	0.538309i	$-0.819063\pi$
$601$	−41.3204	−1.68550	−0.842748	+	0.538309i	$0.819063\pi$
$602$	2.72720	0.111152
$603$	0	0
$604$	−1.01714	−0.0413869
$605$	0	0
$606$	0	0
$607$	−26.0876	−1.05886	−0.529432	−	0.848352i	$-0.677595\pi$
$607$	−26.0876	−1.05886	−0.529432	+	0.848352i	$0.677595\pi$
$608$	1.19239	0.0483579
$609$	0	0
$610$	0	0
$611$	−6.50992	−0.263363
$612$	0	0
$613$	1.92244	0.0776468	0.0388234	−	0.999246i	$-0.487639\pi$
$613$	1.92244	0.0776468	0.0388234	+	0.999246i	$0.487639\pi$
$614$	−37.6389	−1.51898
$615$	0	0
$616$	3.34039	0.134588
$617$	−11.7444	−0.472811	−0.236406	−	0.971654i	$-0.575969\pi$
$617$	−11.7444	−0.472811	−0.236406	+	0.971654i	$0.575969\pi$
$618$	0	0
$619$	−46.2670	−1.85963	−0.929815	−	0.368028i	$-0.880033\pi$
$619$	−46.2670	−1.85963	−0.929815	+	0.368028i	$0.880033\pi$
$620$	0	0
$621$	0	0
$622$	47.2326	1.89385
$623$	0.796751	0.0319212
$624$	0	0
$625$	0	0
$626$	1.04253	0.0416679
$627$	0	0
$628$	−0.734844	−0.0293235
$629$	−0.752794	−0.0300158
$630$	0	0
$631$	35.4518	1.41131	0.705657	−	0.708554i	$-0.250652\pi$
$631$	35.4518	1.41131	0.705657	+	0.708554i	$0.250652\pi$
$632$	−30.7766	−1.22423
$633$	0	0
$634$	−35.5010	−1.40993
$635$	0	0
$636$	0	0
$637$	20.0555	0.794627
$638$	3.64840	0.144441
$639$	0	0
$640$	0	0
$641$	33.3069	1.31554	0.657772	−	0.753218i	$-0.271499\pi$
$641$	33.3069	1.31554	0.657772	+	0.753218i	$0.271499\pi$
$642$	0	0
$643$	3.17760	0.125312	0.0626562	−	0.998035i	$-0.480043\pi$
$643$	3.17760	0.125312	0.0626562	+	0.998035i	$0.480043\pi$
$644$	−0.180152	−0.00709898
$645$	0	0
$646$	−3.60653	−0.141897
$647$	−26.3964	−1.03775	−0.518874	−	0.854851i	$-0.673649\pi$
$647$	−26.3964	−1.03775	−0.518874	+	0.854851i	$0.673649\pi$
$648$	0	0
$649$	13.5361	0.531339
$650$	0	0
$651$	0	0
$652$	0.628938	0.0246311
$653$	−19.0749	−0.746459	−0.373229	−	0.927739i	$-0.621750\pi$
$653$	−19.0749	−0.746459	−0.373229	+	0.927739i	$0.621750\pi$
$654$	0	0
$655$	0	0
$656$	−2.19738	−0.0857933
$657$	0	0
$658$	−1.21552	−0.0473858
$659$	42.0839	1.63936	0.819678	−	0.572824i	$-0.194152\pi$
$659$	42.0839	1.63936	0.819678	+	0.572824i	$0.194152\pi$
$660$	0	0
$661$	20.6874	0.804646	0.402323	−	0.915498i	$-0.368203\pi$
$661$	20.6874	0.804646	0.402323	+	0.915498i	$0.368203\pi$
$662$	−3.86589	−0.150252
$663$	0	0
$664$	17.1468	0.665423
$665$	0	0
$666$	0	0
$667$	−6.73048	−0.260605
$668$	−0.580006	−0.0224411
$669$	0	0
$670$	0	0
$671$	−6.63318	−0.256071
$672$	0	0
$673$	42.3406	1.63211	0.816055	−	0.577974i	$-0.196157\pi$
$673$	42.3406	1.63211	0.816055	+	0.577974i	$0.196157\pi$
$674$	3.81559	0.146971
$675$	0	0
$676$	0.266053	0.0102328
$677$	17.0763	0.656297	0.328149	−	0.944626i	$-0.393575\pi$
$677$	17.0763	0.656297	0.328149	+	0.944626i	$0.393575\pi$
$678$	0	0
$679$	−0.843823	−0.0323829
$680$	0	0
$681$	0	0
$682$	4.12818	0.158076
$683$	10.9639	0.419523	0.209761	−	0.977753i	$-0.432731\pi$
$683$	10.9639	0.419523	0.209761	+	0.977753i	$0.432731\pi$
$684$	0	0
$685$	0	0
$686$	7.57381	0.289169
$687$	0	0
$688$	19.3238	0.736714
$689$	21.1927	0.807376
$690$	0	0
$691$	−11.7138	−0.445614	−0.222807	−	0.974863i	$-0.571522\pi$
$691$	−11.7138	−0.445614	−0.222807	+	0.974863i	$0.571522\pi$
$692$	−1.55133	−0.0589727
$693$	0	0
$694$	34.6845	1.31660
$695$	0	0
$696$	0	0
$697$	−0.419339	−0.0158836
$698$	29.2047	1.10542
$699$	0	0
$700$	0	0
$701$	−22.2196	−0.839223	−0.419612	−	0.907704i	$-0.637834\pi$
$701$	−22.2196	−0.839223	−0.419612	+	0.907704i	$0.637834\pi$
$702$	0	0
$703$	−3.56306	−0.134383
$704$	24.3827	0.918959
$705$	0	0
$706$	−13.0978	−0.492941
$707$	−3.99646	−0.150302
$708$	0	0
$709$	44.4034	1.66760	0.833801	−	0.552064i	$-0.186160\pi$
$709$	44.4034	1.66760	0.833801	+	0.552064i	$0.186160\pi$
$710$	0	0
$711$	0	0
$712$	5.82090	0.218148
$713$	−7.61557	−0.285206
$714$	0	0
$715$	0	0
$716$	1.35764	0.0507375
$717$	0	0
$718$	20.6662	0.771257
$719$	30.8358	1.14998	0.574991	−	0.818160i	$-0.305006\pi$
$719$	30.8358	1.14998	0.574991	+	0.818160i	$0.305006\pi$
$720$	0	0
$721$	−6.87156	−0.255910
$722$	9.39225	0.349543
$723$	0	0
$724$	−0.391149	−0.0145369
$725$	0	0
$726$	0	0
$727$	−28.4927	−1.05674	−0.528368	−	0.849016i	$-0.677196\pi$
$727$	−28.4927	−1.05674	−0.528368	+	0.849016i	$0.677196\pi$
$728$	−3.30162	−0.122366
$729$	0	0
$730$	0	0
$731$	3.68768	0.136394
$732$	0	0
$733$	−18.2652	−0.674642	−0.337321	−	0.941390i	$-0.609521\pi$
$733$	−18.2652	−0.674642	−0.337321	+	0.941390i	$0.609521\pi$
$734$	−24.5801	−0.907268
$735$	0	0
$736$	−2.59383	−0.0956099
$737$	−9.74183	−0.358845
$738$	0	0
$739$	8.64342	0.317953	0.158977	−	0.987282i	$-0.449181\pi$
$739$	8.64342	0.317953	0.158977	+	0.987282i	$0.449181\pi$
$740$	0	0
$741$	0	0
$742$	3.95705	0.145268
$743$	−15.5432	−0.570223	−0.285112	−	0.958494i	$-0.592031\pi$
$743$	−15.5432	−0.570223	−0.285112	+	0.958494i	$0.592031\pi$
$744$	0	0
$745$	0	0
$746$	10.5064	0.384666
$747$	0	0
$748$	0.132048	0.00482814
$749$	2.10154	0.0767884
$750$	0	0
$751$	15.1248	0.551913	0.275956	−	0.961170i	$-0.411005\pi$
$751$	15.1248	0.551913	0.275956	+	0.961170i	$0.411005\pi$
$752$	−8.61266	−0.314071
$753$	0	0
$754$	−3.60604	−0.131324
$755$	0	0
$756$	0	0
$757$	−45.4568	−1.65215	−0.826077	−	0.563557i	$-0.809433\pi$
$757$	−45.4568	−1.65215	−0.826077	+	0.563557i	$0.809433\pi$
$758$	14.5740	0.529350
$759$	0	0
$760$	0	0
$761$	−43.2430	−1.56756	−0.783778	−	0.621041i	$-0.786710\pi$
$761$	−43.2430	−1.56756	−0.783778	+	0.621041i	$0.786710\pi$
$762$	0	0
$763$	−2.12485	−0.0769248
$764$	−0.941332	−0.0340562
$765$	0	0
$766$	44.7741	1.61775
$767$	−13.3790	−0.483087
$768$	0	0
$769$	12.6357	0.455655	0.227828	−	0.973701i	$-0.426838\pi$
$769$	12.6357	0.455655	0.227828	+	0.973701i	$0.426838\pi$
$770$	0	0
$771$	0	0
$772$	−0.919976	−0.0331107
$773$	12.2788	0.441638	0.220819	−	0.975315i	$-0.429127\pi$
$773$	12.2788	0.441638	0.220819	+	0.975315i	$0.429127\pi$
$774$	0	0
$775$	0	0
$776$	−6.16480	−0.221303
$777$	0	0
$778$	−0.816482	−0.0292723
$779$	−1.98478	−0.0711121
$780$	0	0
$781$	−27.6086	−0.987912
$782$	7.84533	0.280549
$783$	0	0
$784$	26.5335	0.947625
$785$	0	0
$786$	0	0
$787$	−0.641485	−0.0228665	−0.0114332	−	0.999935i	$-0.503639\pi$
$787$	−0.641485	−0.0228665	−0.0114332	+	0.999935i	$0.503639\pi$
$788$	−0.953263	−0.0339586
$789$	0	0
$790$	0	0
$791$	−0.332724	−0.0118303
$792$	0	0
$793$	6.55618	0.232817
$794$	9.95965	0.353455
$795$	0	0
$796$	0.384380	0.0136240
$797$	7.40406	0.262265	0.131133	−	0.991365i	$-0.458139\pi$
$797$	7.40406	0.262265	0.131133	+	0.991365i	$0.458139\pi$
$798$	0	0
$799$	−1.64360	−0.0581465
$800$	0	0
$801$	0	0
$802$	−19.0887	−0.674045
$803$	46.5087	1.64126
$804$	0	0
$805$	0	0
$806$	−4.08026	−0.143721
$807$	0	0
$808$	−29.1973	−1.02716
$809$	−46.8134	−1.64587	−0.822936	−	0.568133i	$-0.807666\pi$
$809$	−46.8134	−1.64587	−0.822936	+	0.568133i	$0.807666\pi$
$810$	0	0
$811$	−43.7309	−1.53560	−0.767799	−	0.640691i	$-0.778648\pi$
$811$	−43.7309	−1.53560	−0.767799	+	0.640691i	$0.778648\pi$
$812$	0.0209064	0.000733671 0
$813$	0	0
$814$	−4.20147	−0.147261
$815$	0	0
$816$	0	0
$817$	17.4542	0.610645
$818$	35.6800	1.24752
$819$	0	0
$820$	0	0
$821$	−20.1707	−0.703964	−0.351982	−	0.936007i	$-0.614492\pi$
$821$	−20.1707	−0.703964	−0.351982	+	0.936007i	$0.614492\pi$
$822$	0	0
$823$	53.8348	1.87656	0.938281	−	0.345874i	$-0.112418\pi$
$823$	53.8348	1.87656	0.938281	+	0.345874i	$0.112418\pi$
$824$	−50.2022	−1.74888
$825$	0	0
$826$	−2.49809	−0.0869198
$827$	34.6261	1.20407	0.602034	−	0.798471i	$-0.294357\pi$
$827$	34.6261	1.20407	0.602034	+	0.798471i	$0.294357\pi$
$828$	0	0
$829$	−46.9067	−1.62914	−0.814568	−	0.580068i	$-0.803026\pi$
$829$	−46.9067	−1.62914	−0.814568	+	0.580068i	$0.803026\pi$
$830$	0	0
$831$	0	0
$832$	−24.0997	−0.835506
$833$	5.06354	0.175441
$834$	0	0
$835$	0	0
$836$	0.624996	0.0216159
$837$	0	0
$838$	4.90312	0.169375
$839$	−28.8006	−0.994308	−0.497154	−	0.867662i	$-0.665622\pi$
$839$	−28.8006	−0.994308	−0.497154	+	0.867662i	$0.665622\pi$
$840$	0	0
$841$	−28.2189	−0.973067
$842$	−48.7843	−1.68122
$843$	0	0
$844$	1.01444	0.0349184
$845$	0	0
$846$	0	0
$847$	−0.869754	−0.0298851
$848$	28.0380	0.962829
$849$	0	0
$850$	0	0
$851$	7.75077	0.265693
$852$	0	0
$853$	−14.2701	−0.488600	−0.244300	−	0.969700i	$-0.578558\pi$
$853$	−14.2701	−0.488600	−0.244300	+	0.969700i	$0.578558\pi$
$854$	1.22416	0.0418897
$855$	0	0
$856$	15.3534	0.524768
$857$	17.1765	0.586737	0.293369	−	0.955999i	$-0.405224\pi$
$857$	17.1765	0.586737	0.293369	+	0.955999i	$0.405224\pi$
$858$	0	0
$859$	−8.36778	−0.285505	−0.142753	−	0.989758i	$-0.545595\pi$
$859$	−8.36778	−0.285505	−0.142753	+	0.989758i	$0.545595\pi$
$860$	0	0
$861$	0	0
$862$	−41.8383	−1.42502
$863$	−32.3054	−1.09969	−0.549845	−	0.835267i	$-0.685313\pi$
$863$	−32.3054	−1.09969	−0.549845	+	0.835267i	$0.685313\pi$
$864$	0	0
$865$	0	0
$866$	27.3501	0.929394
$867$	0	0
$868$	0.0236557	0.000802928 0
$869$	−31.7917	−1.07846
$870$	0	0
$871$	9.62874	0.326257
$872$	−15.5237	−0.525700
$873$	0	0
$874$	37.1329	1.25604
$875$	0	0
$876$	0	0
$877$	−36.8711	−1.24505	−0.622523	−	0.782601i	$-0.713892\pi$
$877$	−36.8711	−1.24505	−0.622523	+	0.782601i	$0.713892\pi$
$878$	−11.3552	−0.383221
$879$	0	0
$880$	0	0
$881$	−12.0232	−0.405074	−0.202537	−	0.979275i	$-0.564919\pi$
$881$	−12.0232	−0.405074	−0.202537	+	0.979275i	$0.564919\pi$
$882$	0	0
$883$	−23.0776	−0.776623	−0.388312	−	0.921528i	$-0.626942\pi$
$883$	−23.0776	−0.776623	−0.388312	+	0.921528i	$0.626942\pi$
$884$	−0.130515	−0.00438968
$885$	0	0
$886$	8.94015	0.300350
$887$	3.86173	0.129664	0.0648320	−	0.997896i	$-0.479349\pi$
$887$	3.86173	0.129664	0.0648320	+	0.997896i	$0.479349\pi$
$888$	0	0
$889$	0.172919	0.00579952
$890$	0	0
$891$	0	0
$892$	1.28739	0.0431050
$893$	−7.77936	−0.260326
$894$	0	0
$895$	0	0
$896$	−4.23230	−0.141391
$897$	0	0
$898$	−14.5911	−0.486912
$899$	0.883778	0.0294757
$900$	0	0
$901$	5.35065	0.178256
$902$	−2.34040	−0.0779268
$903$	0	0
$904$	−2.43082	−0.0808477
$905$	0	0
$906$	0	0
$907$	22.9516	0.762095	0.381048	−	0.924555i	$-0.375563\pi$
$907$	22.9516	0.762095	0.381048	+	0.924555i	$0.375563\pi$
$908$	−1.31139	−0.0435201
$909$	0	0
$910$	0	0
$911$	−50.0710	−1.65893	−0.829463	−	0.558562i	$-0.811353\pi$
$911$	−50.0710	−1.65893	−0.829463	+	0.558562i	$0.811353\pi$
$912$	0	0
$913$	17.7123	0.586191
$914$	37.8295	1.25129
$915$	0	0
$916$	−1.25999	−0.0416312
$917$	−8.23635	−0.271988
$918$	0	0
$919$	−21.4914	−0.708937	−0.354469	−	0.935068i	$-0.615338\pi$
$919$	−21.4914	−0.708937	−0.354469	+	0.935068i	$0.615338\pi$
$920$	0	0
$921$	0	0
$922$	28.3107	0.932363
$923$	27.2881	0.898198
$924$	0	0
$925$	0	0
$926$	26.9295	0.884959
$927$	0	0
$928$	0.301011	0.00988117
$929$	13.8766	0.455277	0.227638	−	0.973746i	$-0.426900\pi$
$929$	13.8766	0.455277	0.227638	+	0.973746i	$0.426900\pi$
$930$	0	0
$931$	23.9663	0.785464
$932$	0.882545	0.0289087
$933$	0	0
$934$	−12.2029	−0.399290
$935$	0	0
$936$	0	0
$937$	−37.0301	−1.20972	−0.604860	−	0.796332i	$-0.706771\pi$
$937$	−37.0301	−1.20972	−0.604860	+	0.796332i	$0.706771\pi$
$938$	1.79786	0.0587021
$939$	0	0
$940$	0	0
$941$	8.66187	0.282369	0.141184	−	0.989983i	$-0.454909\pi$
$941$	8.66187	0.282369	0.141184	+	0.989983i	$0.454909\pi$
$942$	0	0
$943$	4.31752	0.140598
$944$	−17.7005	−0.576101
$945$	0	0
$946$	20.5815	0.669164
$947$	19.5007	0.633686	0.316843	−	0.948478i	$-0.397377\pi$
$947$	19.5007	0.633686	0.316843	+	0.948478i	$0.397377\pi$
$948$	0	0
$949$	−45.9688	−1.49221
$950$	0	0
$951$	0	0
$952$	−0.833581	−0.0270165
$953$	−22.4104	−0.725945	−0.362973	−	0.931800i	$-0.618238\pi$
$953$	−22.4104	−0.725945	−0.362973	+	0.931800i	$0.618238\pi$
$954$	0	0
$955$	0	0
$956$	1.16004	0.0375185
$957$	0	0
$958$	19.3048	0.623709
$959$	4.22797	0.136528
$960$	0	0
$961$	1.00000	0.0322581
$962$	4.15269	0.133888
$963$	0	0
$964$	−0.0238086	−0.000766822 0
$965$	0	0
$966$	0	0
$967$	−37.5051	−1.20608	−0.603041	−	0.797710i	$-0.706045\pi$
$967$	−37.5051	−1.20608	−0.603041	+	0.797710i	$0.706045\pi$
$968$	−6.35425	−0.204233
$969$	0	0
$970$	0	0
$971$	6.10577	0.195944	0.0979718	−	0.995189i	$-0.468765\pi$
$971$	6.10577	0.195944	0.0979718	+	0.995189i	$0.468765\pi$
$972$	0	0
$973$	4.43353	0.142132
$974$	44.2047	1.41641
$975$	0	0
$976$	8.67386	0.277643
$977$	−19.0911	−0.610780	−0.305390	−	0.952227i	$-0.598787\pi$
$977$	−19.0911	−0.610780	−0.305390	+	0.952227i	$0.598787\pi$
$978$	0	0
$979$	6.01289	0.192173
$980$	0	0
$981$	0	0
$982$	−4.92845	−0.157273
$983$	24.2343	0.772954	0.386477	−	0.922299i	$-0.373692\pi$
$983$	24.2343	0.772954	0.386477	+	0.922299i	$0.373692\pi$
$984$	0	0
$985$	0	0
$986$	−0.910442	−0.0289944
$987$	0	0
$988$	−0.617741	−0.0196530
$989$	−37.9684	−1.20732
$990$	0	0
$991$	11.3078	0.359205	0.179602	−	0.983739i	$-0.442519\pi$
$991$	11.3078	0.359205	0.179602	+	0.983739i	$0.442519\pi$
$992$	0.340595	0.0108139
$993$	0	0
$994$	5.09517	0.161609
$995$	0	0
$996$	0	0
$997$	−10.6447	−0.337121	−0.168560	−	0.985691i	$-0.553912\pi$
$997$	−10.6447	−0.337121	−0.168560	+	0.985691i	$0.553912\pi$
$998$	−24.0290	−0.760626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bv.1.1		5
3.2	odd	2		2325.2.a.w.1.5		5
5.2	odd	4		1395.2.c.f.559.3		10
5.3	odd	4		1395.2.c.f.559.8		10
5.4	even	2		6975.2.a.bs.1.5		5
15.2	even	4		465.2.c.a.94.8	yes	10
15.8	even	4		465.2.c.a.94.3	✓	10
15.14	odd	2		2325.2.a.x.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.a.94.3	✓	10	15.8	even	4
465.2.c.a.94.8	yes	10	15.2	even	4
1395.2.c.f.559.3		10	5.2	odd	4
1395.2.c.f.559.8		10	5.3	odd	4
2325.2.a.w.1.5		5	3.2	odd	2
2325.2.a.x.1.1		5	15.14	odd	2
6975.2.a.bs.1.5		5	5.4	even	2
6975.2.a.bv.1.1		5	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)

\(f(q)\)	\(=\)	\(q-1.39276 q^{2} -0.0602300 q^{4} +0.392756 q^{7} +2.86940 q^{8} +2.96404 q^{11} -2.92963 q^{13} -0.547014 q^{14} -3.87591 q^{16} -0.739663 q^{17} -3.50091 q^{19} -4.12818 q^{22} +7.61557 q^{23} +4.08026 q^{26} -0.0236557 q^{28} -0.883778 q^{29} -1.00000 q^{31} -0.340595 q^{32} +1.03017 q^{34} +1.01775 q^{37} +4.87591 q^{38} +0.566932 q^{41} -4.98562 q^{43} -0.178524 q^{44} -10.6066 q^{46} +2.22210 q^{47} -6.84574 q^{49} +0.176452 q^{52} -7.23391 q^{53} +1.12697 q^{56} +1.23089 q^{58} +4.56679 q^{59} -2.23789 q^{61} +1.39276 q^{62} +8.22619 q^{64} -3.28668 q^{67} +0.0445499 q^{68} -9.31452 q^{71} +15.6910 q^{73} -1.41748 q^{74} +0.210860 q^{76} +1.16414 q^{77} -10.7258 q^{79} -0.789599 q^{82} +5.97573 q^{83} +6.94375 q^{86} +8.50500 q^{88} +2.02861 q^{89} -1.15063 q^{91} -0.458686 q^{92} -3.09484 q^{94} -2.14846 q^{97} +9.53445 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 3 q^{4} - 8 q^{7} + 9 q^{8} - 6 q^{13} - 16 q^{14} + 3 q^{16} + 8 q^{17} - 4 q^{19} - 6 q^{22} + 4 q^{23} + 6 q^{26} - 18 q^{28} - 4 q^{29} - 5 q^{31} + q^{32} + 6 q^{34} + 4 q^{37} + 2 q^{38}+ \cdots + 29 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 3 * q^4 - 8 * q^7 + 9 * q^8 - 6 * q^13 - 16 * q^14 + 3 * q^16 + 8 * q^17 - 4 * q^19 - 6 * q^22 + 4 * q^23 + 6 * q^26 - 18 * q^28 - 4 * q^29 - 5 * q^31 + q^32 + 6 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^43 - 6 * q^44 - 16 * q^46 + 18 * q^47 - 11 * q^49 - 2 * q^52 + 2 * q^53 - 18 * q^56 - 14 * q^58 - 10 * q^59 - 14 * q^61 - 3 * q^62 + 11 * q^64 - 18 * q^67 + 6 * q^68 - 6 * q^71 - 10 * q^73 + 4 * q^74 - 8 * q^76 + 6 * q^77 - 24 * q^79 - 30 * q^82 + 48 * q^83 - 4 * q^86 - 8 * q^88 - 10 * q^89 - 26 * q^92 + 12 * q^94 - 20 * q^97 + 29 * q^98

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