LMFDB - Embedded newform 6975.2.a.cf.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 = [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:	$1$
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			1.1
Root			$-2.61861$ 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)

$f(q)$	$=$	$q-2.72741 q^{2} +5.43874 q^{4} -3.85713 q^{7} -9.37885 q^{8} +2.51759 q^{11} -1.31707 q^{13} +10.5200 q^{14} +14.7024 q^{16} +3.65872 q^{17} -2.96781 q^{19} -6.86649 q^{22} +5.72013 q^{23} +3.59219 q^{26} -20.9780 q^{28} +6.86125 q^{29} -1.00000 q^{31} -21.3418 q^{32} -9.97880 q^{34} -5.17421 q^{37} +8.09443 q^{38} -6.65144 q^{41} -6.41839 q^{43} +13.6925 q^{44} -15.6011 q^{46} +6.59594 q^{47} +7.87748 q^{49} -7.16322 q^{52} -9.83502 q^{53} +36.1755 q^{56} -18.7134 q^{58} -7.07107 q^{59} -12.9560 q^{61} +2.72741 q^{62} +28.8029 q^{64} +7.31623 q^{67} +19.8988 q^{68} -5.36703 q^{71} -7.16322 q^{73} +14.1122 q^{74} -16.1412 q^{76} -9.71069 q^{77} +5.97880 q^{79} +18.1412 q^{82} -4.35024 q^{83} +17.5056 q^{86} -23.6121 q^{88} +0.353102 q^{89} +5.08012 q^{91} +31.1103 q^{92} -17.9898 q^{94} +18.0187 q^{97} -21.4851 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.72741	−1.92857	−0.964284	−	0.264872i	$-0.914670\pi$
$2$	−2.72741	−1.92857	−0.964284	+	0.264872i	$0.914670\pi$
$3$	0	0
$3$	0	0
$4$	5.43874	2.71937
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.85713	−1.45786	−0.728930	−	0.684588i	$-0.759982\pi$
$7$	−3.85713	−1.45786	−0.728930	+	0.684588i	$0.759982\pi$
$8$	−9.37885	−3.31592
$9$	0	0
$10$	0	0
$11$	2.51759	0.759082	0.379541	−	0.925175i	$-0.376082\pi$
$11$	2.51759	0.759082	0.379541	+	0.925175i	$0.376082\pi$
$12$	0	0
$13$	−1.31707	−0.365290	−0.182645	−	0.983179i	$-0.558466\pi$
$13$	−1.31707	−0.365290	−0.182645	+	0.983179i	$0.558466\pi$
$14$	10.5200	2.81158
$15$	0	0
$16$	14.7024	3.67561
$17$	3.65872	0.887369	0.443685	−	0.896183i	$-0.353671\pi$
$17$	3.65872	0.887369	0.443685	+	0.896183i	$0.353671\pi$
$18$	0	0
$19$	−2.96781	−0.680863	−0.340432	−	0.940269i	$-0.610573\pi$
$19$	−2.96781	−0.680863	−0.340432	+	0.940269i	$0.610573\pi$
$20$	0	0
$21$	0	0
$22$	−6.86649	−1.46394
$23$	5.72013	1.19273	0.596365	−	0.802714i	$-0.296611\pi$
$23$	5.72013	1.19273	0.596365	+	0.802714i	$0.296611\pi$
$24$	0	0
$25$	0	0
$26$	3.59219	0.704486
$27$	0	0
$28$	−20.9780	−3.96446
$29$	6.86125	1.27410	0.637051	−	0.770821i	$-0.280154\pi$
$29$	6.86125	1.27410	0.637051	+	0.770821i	$0.280154\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−21.3418	−3.77273
$33$	0	0
$34$	−9.97880	−1.71135
$35$	0	0
$36$	0	0
$37$	−5.17421	−0.850634	−0.425317	−	0.905044i	$-0.639837\pi$
$37$	−5.17421	−0.850634	−0.425317	+	0.905044i	$0.639837\pi$
$38$	8.09443	1.31309
$39$	0	0
$40$	0	0
$41$	−6.65144	−1.03878	−0.519390	−	0.854537i	$-0.673841\pi$
$41$	−6.65144	−1.03878	−0.519390	+	0.854537i	$0.673841\pi$
$42$	0	0
$43$	−6.41839	−0.978796	−0.489398	−	0.872061i	$-0.662783\pi$
$43$	−6.41839	−0.978796	−0.489398	+	0.872061i	$0.662783\pi$
$44$	13.6925	2.06423
$45$	0	0
$46$	−15.6011	−2.30026
$47$	6.59594	0.962116	0.481058	−	0.876689i	$-0.340253\pi$
$47$	6.59594	0.962116	0.481058	+	0.876689i	$0.340253\pi$
$48$	0	0
$49$	7.87748	1.12535
$50$	0	0
$51$	0	0
$52$	−7.16322	−0.993359
$53$	−9.83502	−1.35094	−0.675472	−	0.737385i	$-0.736060\pi$
$53$	−9.83502	−1.35094	−0.675472	+	0.737385i	$0.736060\pi$
$54$	0	0
$55$	0	0
$56$	36.1755	4.83415
$57$	0	0
$58$	−18.7134	−2.45719
$59$	−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
$61$	−12.9560	−1.65884	−0.829422	−	0.558623i	$-0.811330\pi$
$61$	−12.9560	−1.65884	−0.829422	+	0.558623i	$0.811330\pi$
$62$	2.72741	0.346381
$63$	0	0
$64$	28.8029	3.60036
$65$	0	0
$66$	0	0
$67$	7.31623	0.893819	0.446910	−	0.894579i	$-0.352525\pi$
$67$	7.31623	0.893819	0.446910	+	0.894579i	$0.352525\pi$
$68$	19.8988	2.41309
$69$	0	0
$70$	0	0
$71$	−5.36703	−0.636949	−0.318474	−	0.947931i	$-0.603171\pi$
$71$	−5.36703	−0.636949	−0.318474	+	0.947931i	$0.603171\pi$
$72$	0	0
$73$	−7.16322	−0.838391	−0.419196	−	0.907896i	$-0.637688\pi$
$73$	−7.16322	−0.838391	−0.419196	+	0.907896i	$0.637688\pi$
$74$	14.1122	1.64051
$75$	0	0
$76$	−16.1412	−1.85152
$77$	−9.71069	−1.10664
$78$	0	0
$79$	5.97880	0.672668	0.336334	−	0.941743i	$-0.390813\pi$
$79$	5.97880	0.672668	0.336334	+	0.941743i	$0.390813\pi$
$80$	0	0
$81$	0	0
$82$	18.1412	2.00336
$83$	−4.35024	−0.477501	−0.238750	−	0.971081i	$-0.576738\pi$
$83$	−4.35024	−0.477501	−0.238750	+	0.971081i	$0.576738\pi$
$84$	0	0
$85$	0	0
$86$	17.5056	1.88767
$87$	0	0
$88$	−23.6121	−2.51706
$89$	0.353102	0.0374287	0.0187144	−	0.999825i	$-0.494043\pi$
$89$	0.353102	0.0374287	0.0187144	+	0.999825i	$0.494043\pi$
$90$	0	0
$91$	5.08012	0.532542
$92$	31.1103	3.24347
$93$	0	0
$94$	−17.9898	−1.85551
$95$	0	0
$96$	0	0
$97$	18.0187	1.82952	0.914759	−	0.404000i	$-0.132381\pi$
$97$	18.0187	1.82952	0.914759	+	0.404000i	$0.132381\pi$
$98$	−21.4851	−2.17032
$99$	0	0
$100$	0	0
$101$	7.31743	0.728112	0.364056	−	0.931377i	$-0.381392\pi$
$101$	7.31743	0.728112	0.364056	+	0.931377i	$0.381392\pi$
$102$	0	0
$103$	15.9051	1.56717	0.783587	−	0.621282i	$-0.213388\pi$
$103$	15.9051	1.56717	0.783587	+	0.621282i	$0.213388\pi$
$104$	12.3526	1.21127
$105$	0	0
$106$	26.8241	2.60539
$107$	15.1100	1.46074	0.730369	−	0.683052i	$-0.239348\pi$
$107$	15.1100	1.46074	0.730369	+	0.683052i	$0.239348\pi$
$108$	0	0
$109$	16.0305	1.53544	0.767721	−	0.640784i	$-0.221390\pi$
$109$	16.0305	1.53544	0.767721	+	0.640784i	$0.221390\pi$
$110$	0	0
$111$	0	0
$112$	−56.7093	−5.35852
$113$	9.68071	0.910685	0.455342	−	0.890316i	$-0.349517\pi$
$113$	9.68071	0.910685	0.455342	+	0.890316i	$0.349517\pi$
$114$	0	0
$115$	0	0
$116$	37.3166	3.46476
$117$	0	0
$118$	19.2857	1.77539
$119$	−14.1122	−1.29366
$120$	0	0
$121$	−4.66173	−0.423794
$122$	35.3362	3.19919
$123$	0	0
$124$	−5.43874	−0.488413
$125$	0	0
$126$	0	0
$127$	−4.80460	−0.426339	−0.213170	−	0.977015i	$-0.568379\pi$
$127$	−4.80460	−0.426339	−0.213170	+	0.977015i	$0.568379\pi$
$128$	−35.8736	−3.17081
$129$	0	0
$130$	0	0
$131$	−15.2210	−1.32987	−0.664933	−	0.746903i	$-0.731540\pi$
$131$	−15.2210	−1.32987	−0.664933	+	0.746903i	$0.731540\pi$
$132$	0	0
$133$	11.4473	0.992603
$134$	−19.9543	−1.72379
$135$	0	0
$136$	−34.3145	−2.94245
$137$	21.0765	1.80069	0.900343	−	0.435181i	$-0.143316\pi$
$137$	21.0765	1.80069	0.900343	+	0.435181i	$0.143316\pi$
$138$	0	0
$139$	11.3447	0.962241	0.481121	−	0.876654i	$-0.340230\pi$
$139$	11.3447	0.962241	0.481121	+	0.876654i	$0.340230\pi$
$140$	0	0
$141$	0	0
$142$	14.6381	1.22840
$143$	−3.31585	−0.277285
$144$	0	0
$145$	0	0
$146$	19.5370	1.61689
$147$	0	0
$148$	−28.1412	−2.31319
$149$	16.8549	1.38080	0.690402	−	0.723426i	$-0.257434\pi$
$149$	16.8549	1.38080	0.690402	+	0.723426i	$0.257434\pi$
$150$	0	0
$151$	−0.0321861	−0.00261927	−0.00130963	−	0.999999i	$-0.500417\pi$
$151$	−0.0321861	−0.00261927	−0.00130963	+	0.999999i	$0.500417\pi$
$152$	27.8347	2.25769
$153$	0	0
$154$	26.4850	2.13422
$155$	0	0
$156$	0	0
$157$	−21.6515	−1.72798	−0.863990	−	0.503509i	$-0.832042\pi$
$157$	−21.6515	−1.72798	−0.863990	+	0.503509i	$0.832042\pi$
$158$	−16.3066	−1.29729
$159$	0	0
$160$	0	0
$161$	−22.0633	−1.73883
$162$	0	0
$163$	−13.4591	−1.05420	−0.527099	−	0.849804i	$-0.676720\pi$
$163$	−13.4591	−1.05420	−0.527099	+	0.849804i	$0.676720\pi$
$164$	−36.1755	−2.82483
$165$	0	0
$166$	11.8649	0.920892
$167$	−19.8110	−1.53302	−0.766512	−	0.642230i	$-0.778009\pi$
$167$	−19.8110	−1.53302	−0.766512	+	0.642230i	$0.778009\pi$
$168$	0	0
$169$	−11.2653	−0.866563
$170$	0	0
$171$	0	0
$172$	−34.9080	−2.66171
$173$	−11.0529	−0.840337	−0.420169	−	0.907446i	$-0.638029\pi$
$173$	−11.0529	−0.840337	−0.420169	+	0.907446i	$0.638029\pi$
$174$	0	0
$175$	0	0
$176$	37.0147	2.79009
$177$	0	0
$178$	−0.963052	−0.0721838
$179$	−14.2232	−1.06309	−0.531545	−	0.847030i	$-0.678388\pi$
$179$	−14.2232	−1.06309	−0.531545	+	0.847030i	$0.678388\pi$
$180$	0	0
$181$	11.4363	0.850051	0.425025	−	0.905181i	$-0.360265\pi$
$181$	11.4363	0.850051	0.425025	+	0.905181i	$0.360265\pi$
$182$	−13.8556	−1.02704
$183$	0	0
$184$	−53.6482	−3.95500
$185$	0	0
$186$	0	0
$187$	9.21115	0.673586
$188$	35.8736	2.61635
$189$	0	0
$190$	0	0
$191$	−0.864784	−0.0625736	−0.0312868	−	0.999510i	$-0.509961\pi$
$191$	−0.864784	−0.0625736	−0.0312868	+	0.999510i	$0.509961\pi$
$192$	0	0
$193$	−21.9238	−1.57811	−0.789055	−	0.614323i	$-0.789429\pi$
$193$	−21.9238	−1.57811	−0.789055	+	0.614323i	$0.789429\pi$
$194$	−49.1442	−3.52835
$195$	0	0
$196$	42.8436	3.06026
$197$	19.7658	1.40825	0.704126	−	0.710075i	$-0.251339\pi$
$197$	19.7658	1.40825	0.704126	+	0.710075i	$0.251339\pi$
$198$	0	0
$199$	−3.78177	−0.268083	−0.134041	−	0.990976i	$-0.542795\pi$
$199$	−3.78177	−0.268083	−0.134041	+	0.990976i	$0.542795\pi$
$200$	0	0
$201$	0	0
$202$	−19.9576	−1.40421
$203$	−26.4648	−1.85746
$204$	0	0
$205$	0	0
$206$	−43.3796	−3.02240
$207$	0	0
$208$	−19.3642	−1.34266
$209$	−7.47174	−0.516831
$210$	0	0
$211$	7.39641	0.509190	0.254595	−	0.967048i	$-0.418058\pi$
$211$	7.39641	0.509190	0.254595	+	0.967048i	$0.418058\pi$
$212$	−53.4902	−3.67372
$213$	0	0
$214$	−41.2111	−2.81713
$215$	0	0
$216$	0	0
$217$	3.85713	0.261839
$218$	−43.7217	−2.96120
$219$	0	0
$220$	0	0
$221$	−4.81879	−0.324147
$222$	0	0
$223$	20.1090	1.34660	0.673299	−	0.739371i	$-0.264877\pi$
$223$	20.1090	1.34660	0.673299	+	0.739371i	$0.264877\pi$
$224$	82.3182	5.50012
$225$	0	0
$226$	−26.4032	−1.75632
$227$	20.1854	1.33975	0.669876	−	0.742473i	$-0.266347\pi$
$227$	20.1854	1.33975	0.669876	+	0.742473i	$0.266347\pi$
$228$	0	0
$229$	17.1127	1.13084	0.565421	−	0.824803i	$-0.308714\pi$
$229$	17.1127	1.13084	0.565421	+	0.824803i	$0.308714\pi$
$230$	0	0
$231$	0	0
$232$	−64.3506	−4.22483
$233$	6.92778	0.453854	0.226927	−	0.973912i	$-0.427132\pi$
$233$	6.92778	0.453854	0.226927	+	0.973912i	$0.427132\pi$
$234$	0	0
$235$	0	0
$236$	−38.4577	−2.50338
$237$	0	0
$238$	38.4896	2.49491
$239$	−8.57614	−0.554744	−0.277372	−	0.960763i	$-0.589464\pi$
$239$	−8.57614	−0.554744	−0.277372	+	0.960763i	$0.589464\pi$
$240$	0	0
$241$	−18.6027	−1.19831	−0.599154	−	0.800634i	$-0.704496\pi$
$241$	−18.6027	−1.19831	−0.599154	+	0.800634i	$0.704496\pi$
$242$	12.7144	0.817315
$243$	0	0
$244$	−70.4642	−4.51101
$245$	0	0
$246$	0	0
$247$	3.90883	0.248713
$248$	9.37885	0.595557
$249$	0	0
$250$	0	0
$251$	26.4435	1.66910	0.834550	−	0.550932i	$-0.185728\pi$
$251$	26.4435	1.66910	0.834550	+	0.550932i	$0.185728\pi$
$252$	0	0
$253$	14.4009	0.905380
$254$	13.1041	0.822224
$255$	0	0
$256$	40.2360	2.51475
$257$	−0.143288	−0.00893805	−0.00446903	−	0.999990i	$-0.501423\pi$
$257$	−0.143288	−0.00893805	−0.00446903	+	0.999990i	$0.501423\pi$
$258$	0	0
$259$	19.9576	1.24011
$260$	0	0
$261$	0	0
$262$	41.5139	2.56473
$263$	−11.7489	−0.724467	−0.362233	−	0.932087i	$-0.617986\pi$
$263$	−11.7489	−0.724467	−0.362233	+	0.932087i	$0.617986\pi$
$264$	0	0
$265$	0	0
$266$	−31.2213	−1.91430
$267$	0	0
$268$	39.7911	2.43063
$269$	9.24579	0.563726	0.281863	−	0.959455i	$-0.409048\pi$
$269$	9.24579	0.563726	0.281863	+	0.959455i	$0.409048\pi$
$270$	0	0
$271$	−18.7965	−1.14181	−0.570903	−	0.821017i	$-0.693407\pi$
$271$	−18.7965	−1.14181	−0.570903	+	0.821017i	$0.693407\pi$
$272$	53.7920	3.26162
$273$	0	0
$274$	−57.4841	−3.47274
$275$	0	0
$276$	0	0
$277$	−15.4253	−0.926815	−0.463408	−	0.886145i	$-0.653373\pi$
$277$	−15.4253	−0.926815	−0.463408	+	0.886145i	$0.653373\pi$
$278$	−30.9415	−1.85575
$279$	0	0
$280$	0	0
$281$	−1.96570	−0.117264	−0.0586319	−	0.998280i	$-0.518674\pi$
$281$	−1.96570	−0.117264	−0.0586319	+	0.998280i	$0.518674\pi$
$282$	0	0
$283$	−0.870601	−0.0517518	−0.0258759	−	0.999665i	$-0.508237\pi$
$283$	−0.870601	−0.0517518	−0.0258759	+	0.999665i	$0.508237\pi$
$284$	−29.1899	−1.73210
$285$	0	0
$286$	9.04367	0.534763
$287$	25.6555	1.51440
$288$	0	0
$289$	−3.61379	−0.212576
$290$	0	0
$291$	0	0
$292$	−38.9589	−2.27990
$293$	−13.9689	−0.816070	−0.408035	−	0.912966i	$-0.633786\pi$
$293$	−13.9689	−0.816070	−0.408035	+	0.912966i	$0.633786\pi$
$294$	0	0
$295$	0	0
$296$	48.5281	2.82064
$297$	0	0
$298$	−45.9700	−2.66297
$299$	−7.53382	−0.435692
$300$	0	0
$301$	24.7566	1.42695
$302$	0.0877845	0.00505143
$303$	0	0
$304$	−43.6341	−2.50259
$305$	0	0
$306$	0	0
$307$	5.47093	0.312242	0.156121	−	0.987738i	$-0.450101\pi$
$307$	5.47093	0.312242	0.156121	+	0.987738i	$0.450101\pi$
$308$	−52.8139	−3.00935
$309$	0	0
$310$	0	0
$311$	14.9045	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$311$	14.9045	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$312$	0	0
$313$	−7.14763	−0.404008	−0.202004	−	0.979385i	$-0.564745\pi$
$313$	−7.14763	−0.404008	−0.202004	+	0.979385i	$0.564745\pi$
$314$	59.0525	3.33253
$315$	0	0
$316$	32.5172	1.82923
$317$	−17.3124	−0.972361	−0.486180	−	0.873858i	$-0.661610\pi$
$317$	−17.3124	−0.972361	−0.486180	+	0.873858i	$0.661610\pi$
$318$	0	0
$319$	17.2738	0.967149
$320$	0	0
$321$	0	0
$322$	60.1756	3.35345
$323$	−10.8584	−0.604177
$324$	0	0
$325$	0	0
$326$	36.7084	2.03309
$327$	0	0
$328$	62.3828	3.44452
$329$	−25.4414	−1.40263
$330$	0	0
$331$	12.1758	0.669245	0.334622	−	0.942352i	$-0.391391\pi$
$331$	12.1758	0.669245	0.334622	+	0.942352i	$0.391391\pi$
$332$	−23.6598	−1.29850
$333$	0	0
$334$	54.0327	2.95654
$335$	0	0
$336$	0	0
$337$	−36.0280	−1.96257	−0.981286	−	0.192558i	$-0.938322\pi$
$337$	−36.0280	−1.96257	−0.981286	+	0.192558i	$0.938322\pi$
$338$	30.7251	1.67123
$339$	0	0
$340$	0	0
$341$	−2.51759	−0.136335
$342$	0	0
$343$	−3.38458	−0.182750
$344$	60.1971	3.24561
$345$	0	0
$346$	30.1458	1.62065
$347$	−9.43665	−0.506586	−0.253293	−	0.967390i	$-0.581514\pi$
$347$	−9.43665	−0.506586	−0.253293	+	0.967390i	$0.581514\pi$
$348$	0	0
$349$	−8.79276	−0.470666	−0.235333	−	0.971915i	$-0.575618\pi$
$349$	−8.79276	−0.470666	−0.235333	+	0.971915i	$0.575618\pi$
$350$	0	0
$351$	0	0
$352$	−53.7300	−2.86382
$353$	2.62860	0.139906	0.0699531	−	0.997550i	$-0.477715\pi$
$353$	2.62860	0.139906	0.0699531	+	0.997550i	$0.477715\pi$
$354$	0	0
$355$	0	0
$356$	1.92043	0.101783
$357$	0	0
$358$	38.7923	2.05024
$359$	−13.6194	−0.718806	−0.359403	−	0.933182i	$-0.617020\pi$
$359$	−13.6194	−0.718806	−0.359403	+	0.933182i	$0.617020\pi$
$360$	0	0
$361$	−10.1921	−0.536425
$362$	−31.1913	−1.63938
$363$	0	0
$364$	27.6295	1.44818
$365$	0	0
$366$	0	0
$367$	−14.5596	−0.760003	−0.380002	−	0.924986i	$-0.624077\pi$
$367$	−14.5596	−0.760003	−0.380002	+	0.924986i	$0.624077\pi$
$368$	84.0998	4.38401
$369$	0	0
$370$	0	0
$371$	37.9350	1.96949
$372$	0	0
$373$	8.53361	0.441854	0.220927	−	0.975290i	$-0.429092\pi$
$373$	8.53361	0.441854	0.220927	+	0.975290i	$0.429092\pi$
$374$	−25.1226	−1.29906
$375$	0	0
$376$	−61.8623	−3.19030
$377$	−9.03677	−0.465417
$378$	0	0
$379$	−7.69555	−0.395294	−0.197647	−	0.980273i	$-0.563330\pi$
$379$	−7.69555	−0.395294	−0.197647	+	0.980273i	$0.563330\pi$
$380$	0	0
$381$	0	0
$382$	2.35862	0.120677
$383$	−24.0071	−1.22671	−0.613354	−	0.789808i	$-0.710180\pi$
$383$	−24.0071	−1.22671	−0.613354	+	0.789808i	$0.710180\pi$
$384$	0	0
$385$	0	0
$386$	59.7951	3.04349
$387$	0	0
$388$	97.9988	4.97514
$389$	−15.8417	−0.803208	−0.401604	−	0.915814i	$-0.631547\pi$
$389$	−15.8417	−0.803208	−0.401604	+	0.915814i	$0.631547\pi$
$390$	0	0
$391$	20.9283	1.05839
$392$	−73.8817	−3.73159
$393$	0	0
$394$	−53.9093	−2.71591
$395$	0	0
$396$	0	0
$397$	26.9470	1.35243	0.676217	−	0.736703i	$-0.263618\pi$
$397$	26.9470	1.35243	0.676217	+	0.736703i	$0.263618\pi$
$398$	10.3144	0.517015
$399$	0	0
$400$	0	0
$401$	−36.1967	−1.80758	−0.903789	−	0.427978i	$-0.859226\pi$
$401$	−36.1967	−1.80758	−0.903789	+	0.427978i	$0.859226\pi$
$402$	0	0
$403$	1.31707	0.0656080
$404$	39.7976	1.98001
$405$	0	0
$406$	72.1802	3.58224
$407$	−13.0265	−0.645702
$408$	0	0
$409$	5.22299	0.258260	0.129130	−	0.991628i	$-0.458782\pi$
$409$	5.22299	0.258260	0.129130	+	0.991628i	$0.458782\pi$
$410$	0	0
$411$	0	0
$412$	86.5036	4.26173
$413$	27.2741	1.34207
$414$	0	0
$415$	0	0
$416$	28.1087	1.37814
$417$	0	0
$418$	20.3785	0.996744
$419$	18.6699	0.912085	0.456042	−	0.889958i	$-0.349267\pi$
$419$	18.6699	0.912085	0.456042	+	0.889958i	$0.349267\pi$
$420$	0	0
$421$	−28.4794	−1.38800	−0.694000	−	0.719975i	$-0.744153\pi$
$421$	−28.4794	−1.38800	−0.694000	+	0.719975i	$0.744153\pi$
$422$	−20.1730	−0.982007
$423$	0	0
$424$	92.2412	4.47963
$425$	0	0
$426$	0	0
$427$	49.9729	2.41836
$428$	82.1794	3.97229
$429$	0	0
$430$	0	0
$431$	15.4441	0.743918	0.371959	−	0.928249i	$-0.378686\pi$
$431$	15.4441	0.743918	0.371959	+	0.928249i	$0.378686\pi$
$432$	0	0
$433$	3.65449	0.175624	0.0878119	−	0.996137i	$-0.472013\pi$
$433$	3.65449	0.175624	0.0878119	+	0.996137i	$0.472013\pi$
$434$	−10.5200	−0.504975
$435$	0	0
$436$	87.1857	4.17544
$437$	−16.9763	−0.812086
$438$	0	0
$439$	35.5884	1.69854	0.849272	−	0.527956i	$-0.177041\pi$
$439$	35.5884	1.69854	0.849272	+	0.527956i	$0.177041\pi$
$440$	0	0
$441$	0	0
$442$	13.1428	0.625140
$443$	1.94017	0.0921803	0.0460901	−	0.998937i	$-0.485324\pi$
$443$	1.94017	0.0921803	0.0460901	+	0.998937i	$0.485324\pi$
$444$	0	0
$445$	0	0
$446$	−54.8454	−2.59700
$447$	0	0
$448$	−111.097	−5.24882
$449$	−26.0386	−1.22884	−0.614418	−	0.788981i	$-0.710609\pi$
$449$	−26.0386	−1.22884	−0.614418	+	0.788981i	$0.710609\pi$
$450$	0	0
$451$	−16.7456	−0.788520
$452$	52.6509	2.47649
$453$	0	0
$454$	−55.0538	−2.58380
$455$	0	0
$456$	0	0
$457$	−41.4773	−1.94023	−0.970114	−	0.242651i	$-0.921983\pi$
$457$	−41.4773	−1.94023	−0.970114	+	0.242651i	$0.921983\pi$
$458$	−46.6734	−2.18090
$459$	0	0
$460$	0	0
$461$	−3.42995	−0.159749	−0.0798743	−	0.996805i	$-0.525452\pi$
$461$	−3.42995	−0.159749	−0.0798743	+	0.996805i	$0.525452\pi$
$462$	0	0
$463$	−42.1530	−1.95901	−0.979507	−	0.201411i	$-0.935447\pi$
$463$	−42.1530	−1.95901	−0.979507	+	0.201411i	$0.935447\pi$
$464$	100.877	4.68310
$465$	0	0
$466$	−18.8949	−0.875288
$467$	−19.0895	−0.883358	−0.441679	−	0.897173i	$-0.645617\pi$
$467$	−19.0895	−0.883358	−0.441679	+	0.897173i	$0.645617\pi$
$468$	0	0
$469$	−28.2197	−1.30306
$470$	0	0
$471$	0	0
$472$	66.3185	3.05255
$473$	−16.1589	−0.742987
$474$	0	0
$475$	0	0
$476$	−76.7524	−3.51794
$477$	0	0
$478$	23.3906	1.06986
$479$	−35.7326	−1.63266	−0.816332	−	0.577582i	$-0.803996\pi$
$479$	−35.7326	−1.63266	−0.816332	+	0.577582i	$0.803996\pi$
$480$	0	0
$481$	6.81480	0.310728
$482$	50.7372	2.31102
$483$	0	0
$484$	−25.3540	−1.15245
$485$	0	0
$486$	0	0
$487$	−1.40117	−0.0634932	−0.0317466	−	0.999496i	$-0.510107\pi$
$487$	−1.40117	−0.0634932	−0.0317466	+	0.999496i	$0.510107\pi$
$488$	121.512	5.50059
$489$	0	0
$490$	0	0
$491$	−20.8777	−0.942197	−0.471099	−	0.882081i	$-0.656142\pi$
$491$	−20.8777	−0.942197	−0.471099	+	0.882081i	$0.656142\pi$
$492$	0	0
$493$	25.1034	1.13060
$494$	−10.6610	−0.479659
$495$	0	0
$496$	−14.7024	−0.660159
$497$	20.7013	0.928582
$498$	0	0
$499$	−20.0648	−0.898224	−0.449112	−	0.893475i	$-0.648260\pi$
$499$	−20.0648	−0.898224	−0.449112	+	0.893475i	$0.648260\pi$
$500$	0	0
$501$	0	0
$502$	−72.1222	−3.21897
$503$	30.3967	1.35532	0.677662	−	0.735374i	$-0.262993\pi$
$503$	30.3967	1.35532	0.677662	+	0.735374i	$0.262993\pi$
$504$	0	0
$505$	0	0
$506$	−39.2772	−1.74609
$507$	0	0
$508$	−26.1310	−1.15937
$509$	−12.6100	−0.558929	−0.279464	−	0.960156i	$-0.590157\pi$
$509$	−12.6100	−0.558929	−0.279464	+	0.960156i	$0.590157\pi$
$510$	0	0
$511$	27.6295	1.22226
$512$	−37.9928	−1.67906
$513$	0	0
$514$	0.390804	0.0172376
$515$	0	0
$516$	0	0
$517$	16.6059	0.730326
$518$	−54.4325	−2.39163
$519$	0	0
$520$	0	0
$521$	42.3642	1.85601	0.928004	−	0.372571i	$-0.121524\pi$
$521$	42.3642	1.85601	0.928004	+	0.372571i	$0.121524\pi$
$522$	0	0
$523$	15.7008	0.686548	0.343274	−	0.939235i	$-0.388464\pi$
$523$	15.7008	0.686548	0.343274	+	0.939235i	$0.388464\pi$
$524$	−82.7831	−3.61640
$525$	0	0
$526$	32.0440	1.39718
$527$	−3.65872	−0.159376
$528$	0	0
$529$	9.71988	0.422603
$530$	0	0
$531$	0	0
$532$	62.2587	2.69926
$533$	8.76043	0.379456
$534$	0	0
$535$	0	0
$536$	−68.6178	−2.96384
$537$	0	0
$538$	−25.2170	−1.08718
$539$	19.8323	0.854237
$540$	0	0
$541$	−29.5306	−1.26962	−0.634811	−	0.772668i	$-0.718922\pi$
$541$	−29.5306	−1.26962	−0.634811	+	0.772668i	$0.718922\pi$
$542$	51.2657	2.20205
$543$	0	0
$544$	−78.0836	−3.34781
$545$	0	0
$546$	0	0
$547$	24.9520	1.06687	0.533435	−	0.845841i	$-0.320901\pi$
$547$	24.9520	1.06687	0.533435	+	0.845841i	$0.320901\pi$
$548$	114.630	4.89673
$549$	0	0
$550$	0	0
$551$	−20.3629	−0.867490
$552$	0	0
$553$	−23.0611	−0.980656
$554$	42.0710	1.78742
$555$	0	0
$556$	61.7007	2.61669
$557$	−39.2604	−1.66352	−0.831759	−	0.555137i	$-0.812666\pi$
$557$	−39.2604	−1.66352	−0.831759	+	0.555137i	$0.812666\pi$
$558$	0	0
$559$	8.45349	0.357544
$560$	0	0
$561$	0	0
$562$	5.36126	0.226151
$563$	16.6986	0.703761	0.351881	−	0.936045i	$-0.385542\pi$
$563$	16.6986	0.703761	0.351881	+	0.936045i	$0.385542\pi$
$564$	0	0
$565$	0	0
$566$	2.37448	0.0998069
$567$	0	0
$568$	50.3365	2.11207
$569$	−28.9524	−1.21375	−0.606874	−	0.794798i	$-0.707577\pi$
$569$	−28.9524	−1.21375	−0.606874	+	0.794798i	$0.707577\pi$
$570$	0	0
$571$	−37.7093	−1.57809	−0.789043	−	0.614338i	$-0.789423\pi$
$571$	−37.7093	−1.57809	−0.789043	+	0.614338i	$0.789423\pi$
$572$	−18.0341	−0.754042
$573$	0	0
$574$	−69.9729	−2.92061
$575$	0	0
$576$	0	0
$577$	−43.5185	−1.81170	−0.905848	−	0.423602i	$-0.860766\pi$
$577$	−43.5185	−1.81170	−0.905848	+	0.423602i	$0.860766\pi$
$578$	9.85628	0.409967
$579$	0	0
$580$	0	0
$581$	16.7795	0.696129
$582$	0	0
$583$	−24.7606	−1.02548
$584$	67.1827	2.78004
$585$	0	0
$586$	38.0988	1.57385
$587$	−1.84136	−0.0760012	−0.0380006	−	0.999278i	$-0.512099\pi$
$587$	−1.84136	−0.0760012	−0.0380006	+	0.999278i	$0.512099\pi$
$588$	0	0
$589$	2.96781	0.122287
$590$	0	0
$591$	0	0
$592$	−76.0734	−3.12660
$593$	−14.4185	−0.592096	−0.296048	−	0.955173i	$-0.595669\pi$
$593$	−14.4185	−0.592096	−0.296048	+	0.955173i	$0.595669\pi$
$594$	0	0
$595$	0	0
$596$	91.6692	3.75492
$597$	0	0
$598$	20.5478	0.840262
$599$	−26.0707	−1.06522	−0.532609	−	0.846361i	$-0.678789\pi$
$599$	−26.0707	−1.06522	−0.532609	+	0.846361i	$0.678789\pi$
$600$	0	0
$601$	−26.2323	−1.07004	−0.535020	−	0.844839i	$-0.679696\pi$
$601$	−26.2323	−1.07004	−0.535020	+	0.844839i	$0.679696\pi$
$602$	−67.5213	−2.75196
$603$	0	0
$604$	−0.175052	−0.00712276
$605$	0	0
$606$	0	0
$607$	12.5172	0.508059	0.254029	−	0.967196i	$-0.418244\pi$
$607$	12.5172	0.508059	0.254029	+	0.967196i	$0.418244\pi$
$608$	63.3385	2.56872
$609$	0	0
$610$	0	0
$611$	−8.68732	−0.351452
$612$	0	0
$613$	28.4584	1.14942	0.574712	−	0.818356i	$-0.305114\pi$
$613$	28.4584	1.14942	0.574712	+	0.818356i	$0.305114\pi$
$614$	−14.9214	−0.602180
$615$	0	0
$616$	91.0751	3.66952
$617$	20.5195	0.826086	0.413043	−	0.910712i	$-0.364466\pi$
$617$	20.5195	0.826086	0.413043	+	0.910712i	$0.364466\pi$
$618$	0	0
$619$	10.8831	0.437428	0.218714	−	0.975789i	$-0.429814\pi$
$619$	10.8831	0.437428	0.218714	+	0.975789i	$0.429814\pi$
$620$	0	0
$621$	0	0
$622$	−40.6505	−1.62994
$623$	−1.36196	−0.0545658
$624$	0	0
$625$	0	0
$626$	19.4945	0.779156
$627$	0	0
$628$	−117.757	−4.69902
$629$	−18.9310	−0.754827
$630$	0	0
$631$	32.8687	1.30848	0.654242	−	0.756286i	$-0.272988\pi$
$631$	32.8687	1.30848	0.654242	+	0.756286i	$0.272988\pi$
$632$	−56.0743	−2.23052
$633$	0	0
$634$	47.2179	1.87526
$635$	0	0
$636$	0	0
$637$	−10.3752	−0.411081
$638$	−47.1128	−1.86521
$639$	0	0
$640$	0	0
$641$	3.65196	0.144244	0.0721219	−	0.997396i	$-0.477023\pi$
$641$	3.65196	0.144244	0.0721219	+	0.997396i	$0.477023\pi$
$642$	0	0
$643$	−49.8011	−1.96396	−0.981981	−	0.188980i	$-0.939482\pi$
$643$	−49.8011	−1.96396	−0.981981	+	0.188980i	$0.939482\pi$
$644$	−119.997	−4.72853
$645$	0	0
$646$	29.6152	1.16520
$647$	29.9626	1.17795	0.588976	−	0.808151i	$-0.299531\pi$
$647$	29.9626	1.17795	0.588976	+	0.808151i	$0.299531\pi$
$648$	0	0
$649$	−17.8021	−0.698792
$650$	0	0
$651$	0	0
$652$	−73.2005	−2.86675
$653$	−24.8785	−0.973571	−0.486785	−	0.873522i	$-0.661831\pi$
$653$	−24.8785	−0.973571	−0.486785	+	0.873522i	$0.661831\pi$
$654$	0	0
$655$	0	0
$656$	−97.7923	−3.81815
$657$	0	0
$658$	69.3891	2.70507
$659$	−3.66468	−0.142756	−0.0713779	−	0.997449i	$-0.522740\pi$
$659$	−3.66468	−0.142756	−0.0713779	+	0.997449i	$0.522740\pi$
$660$	0	0
$661$	7.91988	0.308047	0.154024	−	0.988067i	$-0.450777\pi$
$661$	7.91988	0.308047	0.154024	+	0.988067i	$0.450777\pi$
$662$	−33.2084	−1.29068
$663$	0	0
$664$	40.8002	1.58336
$665$	0	0
$666$	0	0
$667$	39.2473	1.51966
$668$	−107.747	−4.16886
$669$	0	0
$670$	0	0
$671$	−32.6179	−1.25920
$672$	0	0
$673$	26.2307	1.01112	0.505559	−	0.862792i	$-0.331286\pi$
$673$	26.2307	1.01112	0.505559	+	0.862792i	$0.331286\pi$
$674$	98.2630	3.78495
$675$	0	0
$676$	−61.2692	−2.35651
$677$	42.4848	1.63282	0.816412	−	0.577470i	$-0.195960\pi$
$677$	42.4848	1.63282	0.816412	+	0.577470i	$0.195960\pi$
$678$	0	0
$679$	−69.5004	−2.66718
$680$	0	0
$681$	0	0
$682$	6.86649	0.262932
$683$	−32.8420	−1.25666	−0.628332	−	0.777945i	$-0.716262\pi$
$683$	−32.8420	−1.25666	−0.628332	+	0.777945i	$0.716262\pi$
$684$	0	0
$685$	0	0
$686$	9.23111	0.352445
$687$	0	0
$688$	−94.3660	−3.59767
$689$	12.9534	0.493487
$690$	0	0
$691$	−7.94420	−0.302212	−0.151106	−	0.988518i	$-0.548283\pi$
$691$	−7.94420	−0.302212	−0.151106	+	0.988518i	$0.548283\pi$
$692$	−60.1139	−2.28519
$693$	0	0
$694$	25.7376	0.976985
$695$	0	0
$696$	0	0
$697$	−24.3357	−0.921782
$698$	23.9814	0.907711
$699$	0	0
$700$	0	0
$701$	3.61986	0.136720	0.0683601	−	0.997661i	$-0.478223\pi$
$701$	3.61986	0.136720	0.0683601	+	0.997661i	$0.478223\pi$
$702$	0	0
$703$	15.3561	0.579166
$704$	72.5140	2.73297
$705$	0	0
$706$	−7.16925	−0.269818
$707$	−28.2243	−1.06148
$708$	0	0
$709$	−17.5558	−0.659322	−0.329661	−	0.944099i	$-0.606935\pi$
$709$	−17.5558	−0.659322	−0.329661	+	0.944099i	$0.606935\pi$
$710$	0	0
$711$	0	0
$712$	−3.31169	−0.124111
$713$	−5.72013	−0.214221
$714$	0	0
$715$	0	0
$716$	−77.3561	−2.89093
$717$	0	0
$718$	37.1457	1.38627
$719$	0.950263	0.0354388	0.0177194	−	0.999843i	$-0.494359\pi$
$719$	0.950263	0.0354388	0.0177194	+	0.999843i	$0.494359\pi$
$720$	0	0
$721$	−61.3480	−2.28472
$722$	27.7979	1.03453
$723$	0	0
$724$	62.1989	2.31160
$725$	0	0
$726$	0	0
$727$	10.6968	0.396723	0.198362	−	0.980129i	$-0.436438\pi$
$727$	10.6968	0.396723	0.198362	+	0.980129i	$0.436438\pi$
$728$	−47.6457	−1.76587
$729$	0	0
$730$	0	0
$731$	−23.4831	−0.868553
$732$	0	0
$733$	−38.4456	−1.42002	−0.710010	−	0.704192i	$-0.751309\pi$
$733$	−38.4456	−1.42002	−0.710010	+	0.704192i	$0.751309\pi$
$734$	39.7098	1.46572
$735$	0	0
$736$	−122.078	−4.49985
$737$	18.4193	0.678483
$738$	0	0
$739$	−8.99746	−0.330977	−0.165489	−	0.986212i	$-0.552920\pi$
$739$	−8.99746	−0.330977	−0.165489	+	0.986212i	$0.552920\pi$
$740$	0	0
$741$	0	0
$742$	−103.464	−3.79829
$743$	−6.87594	−0.252254	−0.126127	−	0.992014i	$-0.540255\pi$
$743$	−6.87594	−0.252254	−0.126127	+	0.992014i	$0.540255\pi$
$744$	0	0
$745$	0	0
$746$	−23.2746	−0.852144
$747$	0	0
$748$	50.0971	1.83173
$749$	−58.2813	−2.12955
$750$	0	0
$751$	−19.6646	−0.717573	−0.358786	−	0.933420i	$-0.616809\pi$
$751$	−19.6646	−0.717573	−0.358786	+	0.933420i	$0.616809\pi$
$752$	96.9763	3.53636
$753$	0	0
$754$	24.6469	0.897588
$755$	0	0
$756$	0	0
$757$	23.3515	0.848724	0.424362	−	0.905493i	$-0.360498\pi$
$757$	23.3515	0.848724	0.424362	+	0.905493i	$0.360498\pi$
$758$	20.9889	0.762351
$759$	0	0
$760$	0	0
$761$	−43.6186	−1.58117	−0.790586	−	0.612351i	$-0.790224\pi$
$761$	−43.6186	−1.58117	−0.790586	+	0.612351i	$0.790224\pi$
$762$	0	0
$763$	−61.8318	−2.23846
$764$	−4.70334	−0.170161
$765$	0	0
$766$	65.4772	2.36579
$767$	9.31311	0.336277
$768$	0	0
$769$	−4.18980	−0.151088	−0.0755439	−	0.997142i	$-0.524069\pi$
$769$	−4.18980	−0.151088	−0.0755439	+	0.997142i	$0.524069\pi$
$770$	0	0
$771$	0	0
$772$	−119.238	−4.29146
$773$	−16.6744	−0.599737	−0.299868	−	0.953981i	$-0.596943\pi$
$773$	−16.6744	−0.599737	−0.299868	+	0.953981i	$0.596943\pi$
$774$	0	0
$775$	0	0
$776$	−168.994	−6.06654
$777$	0	0
$778$	43.2068	1.54904
$779$	19.7402	0.707267
$780$	0	0
$781$	−13.5120	−0.483497
$782$	−57.0800	−2.04118
$783$	0	0
$784$	115.818	4.13636
$785$	0	0
$786$	0	0
$787$	−16.7155	−0.595845	−0.297922	−	0.954590i	$-0.596294\pi$
$787$	−16.7155	−0.595845	−0.297922	+	0.954590i	$0.596294\pi$
$788$	107.501	3.82956
$789$	0	0
$790$	0	0
$791$	−37.3398	−1.32765
$792$	0	0
$793$	17.0640	0.605959
$794$	−73.4955	−2.60826
$795$	0	0
$796$	−20.5681	−0.729016
$797$	34.7924	1.23241	0.616205	−	0.787585i	$-0.288669\pi$
$797$	34.7924	1.23241	0.616205	+	0.787585i	$0.288669\pi$
$798$	0	0
$799$	24.1327	0.853752
$800$	0	0
$801$	0	0
$802$	98.7232	3.48604
$803$	−18.0341	−0.636408
$804$	0	0
$805$	0	0
$806$	−3.59219	−0.126530
$807$	0	0
$808$	−68.6291	−2.41436
$809$	−34.6659	−1.21879	−0.609395	−	0.792867i	$-0.708588\pi$
$809$	−34.6659	−1.21879	−0.609395	+	0.792867i	$0.708588\pi$
$810$	0	0
$811$	1.84266	0.0647045	0.0323523	−	0.999477i	$-0.489700\pi$
$811$	1.84266	0.0647045	0.0323523	+	0.999477i	$0.489700\pi$
$812$	−143.935	−5.05113
$813$	0	0
$814$	35.5287	1.24528
$815$	0	0
$816$	0	0
$817$	19.0486	0.666426
$818$	−14.2452	−0.498072
$819$	0	0
$820$	0	0
$821$	−29.1013	−1.01564	−0.507821	−	0.861462i	$-0.669549\pi$
$821$	−29.1013	−1.01564	−0.507821	+	0.861462i	$0.669549\pi$
$822$	0	0
$823$	−20.4061	−0.711313	−0.355657	−	0.934617i	$-0.615743\pi$
$823$	−20.4061	−0.711313	−0.355657	+	0.934617i	$0.615743\pi$
$824$	−149.171	−5.19663
$825$	0	0
$826$	−74.3874	−2.58827
$827$	29.5430	1.02731	0.513655	−	0.857997i	$-0.328291\pi$
$827$	29.5430	1.02731	0.513655	+	0.857997i	$0.328291\pi$
$828$	0	0
$829$	45.2290	1.57087	0.785435	−	0.618945i	$-0.212440\pi$
$829$	45.2290	1.57087	0.785435	+	0.618945i	$0.212440\pi$
$830$	0	0
$831$	0	0
$832$	−37.9355	−1.31518
$833$	28.8215	0.998605
$834$	0	0
$835$	0	0
$836$	−40.6369	−1.40546
$837$	0	0
$838$	−50.9204	−1.75902
$839$	−24.5789	−0.848559	−0.424280	−	0.905531i	$-0.639473\pi$
$839$	−24.5789	−0.848559	−0.424280	+	0.905531i	$0.639473\pi$
$840$	0	0
$841$	18.0768	0.623338
$842$	77.6748	2.67685
$843$	0	0
$844$	40.2272	1.38468
$845$	0	0
$846$	0	0
$847$	17.9809	0.617832
$848$	−144.599	−4.96554
$849$	0	0
$850$	0	0
$851$	−29.5971	−1.01458
$852$	0	0
$853$	−34.9080	−1.19523	−0.597613	−	0.801785i	$-0.703884\pi$
$853$	−34.9080	−1.19523	−0.597613	+	0.801785i	$0.703884\pi$
$854$	−136.297	−4.66397
$855$	0	0
$856$	−141.714	−4.84370
$857$	4.42914	0.151297	0.0756483	−	0.997135i	$-0.475897\pi$
$857$	4.42914	0.151297	0.0756483	+	0.997135i	$0.475897\pi$
$858$	0	0
$859$	9.41507	0.321238	0.160619	−	0.987016i	$-0.448651\pi$
$859$	9.41507	0.321238	0.160619	+	0.987016i	$0.448651\pi$
$860$	0	0
$861$	0	0
$862$	−42.1225	−1.43470
$863$	−5.38232	−0.183216	−0.0916081	−	0.995795i	$-0.529201\pi$
$863$	−5.38232	−0.183216	−0.0916081	+	0.995795i	$0.529201\pi$
$864$	0	0
$865$	0	0
$866$	−9.96729	−0.338702
$867$	0	0
$868$	20.9780	0.712038
$869$	15.0522	0.510611
$870$	0	0
$871$	−9.63600	−0.326503
$872$	−150.348	−5.09141
$873$	0	0
$874$	46.3012	1.56616
$875$	0	0
$876$	0	0
$877$	−18.3806	−0.620669	−0.310334	−	0.950627i	$-0.600441\pi$
$877$	−18.3806	−0.620669	−0.310334	+	0.950627i	$0.600441\pi$
$878$	−97.0641	−3.27575
$879$	0	0
$880$	0	0
$881$	−33.9000	−1.14212	−0.571059	−	0.820909i	$-0.693467\pi$
$881$	−33.9000	−1.14212	−0.571059	+	0.820909i	$0.693467\pi$
$882$	0	0
$883$	−5.73155	−0.192882	−0.0964410	−	0.995339i	$-0.530746\pi$
$883$	−5.73155	−0.192882	−0.0964410	+	0.995339i	$0.530746\pi$
$884$	−26.2082	−0.881476
$885$	0	0
$886$	−5.29163	−0.177776
$887$	−39.4256	−1.32378	−0.661891	−	0.749600i	$-0.730246\pi$
$887$	−39.4256	−1.32378	−0.661891	+	0.749600i	$0.730246\pi$
$888$	0	0
$889$	18.5320	0.621543
$890$	0	0
$891$	0	0
$892$	109.368	3.66190
$893$	−19.5755	−0.655070
$894$	0	0
$895$	0	0
$896$	138.369	4.62259
$897$	0	0
$898$	71.0178	2.36989
$899$	−6.86125	−0.228836
$900$	0	0
$901$	−35.9836	−1.19879
$902$	45.6721	1.52071
$903$	0	0
$904$	−90.7939	−3.01976
$905$	0	0
$906$	0	0
$907$	−6.64103	−0.220512	−0.110256	−	0.993903i	$-0.535167\pi$
$907$	−6.64103	−0.220512	−0.110256	+	0.993903i	$0.535167\pi$
$908$	109.783	3.64328
$909$	0	0
$910$	0	0
$911$	47.0118	1.55757	0.778785	−	0.627290i	$-0.215836\pi$
$911$	47.0118	1.55757	0.778785	+	0.627290i	$0.215836\pi$
$912$	0	0
$913$	−10.9521	−0.362462
$914$	113.125	3.74186
$915$	0	0
$916$	93.0718	3.07518
$917$	58.7095	1.93876
$918$	0	0
$919$	24.5583	0.810103	0.405052	−	0.914294i	$-0.367254\pi$
$919$	24.5583	0.810103	0.405052	+	0.914294i	$0.367254\pi$
$920$	0	0
$921$	0	0
$922$	9.35487	0.308086
$923$	7.06876	0.232671
$924$	0	0
$925$	0	0
$926$	114.968	3.77809
$927$	0	0
$928$	−146.432	−4.80685
$929$	4.66679	0.153112	0.0765562	−	0.997065i	$-0.475608\pi$
$929$	4.66679	0.153112	0.0765562	+	0.997065i	$0.475608\pi$
$930$	0	0
$931$	−23.3789	−0.766213
$932$	37.6784	1.23420
$933$	0	0
$934$	52.0649	1.70362
$935$	0	0
$936$	0	0
$937$	0.622308	0.0203299	0.0101650	−	0.999948i	$-0.496764\pi$
$937$	0.622308	0.0203299	0.0101650	+	0.999948i	$0.496764\pi$
$938$	76.9665	2.51304
$939$	0	0
$940$	0	0
$941$	−10.5055	−0.342468	−0.171234	−	0.985230i	$-0.554775\pi$
$941$	−10.5055	−0.342468	−0.171234	+	0.985230i	$0.554775\pi$
$942$	0	0
$943$	−38.0471	−1.23898
$944$	−103.962	−3.38367
$945$	0	0
$946$	44.0719	1.43290
$947$	42.2121	1.37171	0.685855	−	0.727738i	$-0.259428\pi$
$947$	42.2121	1.37171	0.685855	+	0.727738i	$0.259428\pi$
$948$	0	0
$949$	9.43447	0.306256
$950$	0	0
$951$	0	0
$952$	132.356	4.28968
$953$	33.3115	1.07907	0.539533	−	0.841964i	$-0.318601\pi$
$953$	33.3115	1.07907	0.539533	+	0.841964i	$0.318601\pi$
$954$	0	0
$955$	0	0
$956$	−46.6434	−1.50856
$957$	0	0
$958$	97.4573	3.14870
$959$	−81.2949	−2.62515
$960$	0	0
$961$	1.00000	0.0322581
$962$	−18.5867	−0.599260
$963$	0	0
$964$	−101.176	−3.25864
$965$	0	0
$966$	0	0
$967$	−5.60111	−0.180120	−0.0900598	−	0.995936i	$-0.528706\pi$
$967$	−5.60111	−0.180120	−0.0900598	+	0.995936i	$0.528706\pi$
$968$	43.7217	1.40527
$969$	0	0
$970$	0	0
$971$	−11.9500	−0.383493	−0.191746	−	0.981444i	$-0.561415\pi$
$971$	−11.9500	−0.383493	−0.191746	+	0.981444i	$0.561415\pi$
$972$	0	0
$973$	−43.7579	−1.40281
$974$	3.82157	0.122451
$975$	0	0
$976$	−190.484	−6.09726
$977$	17.8697	0.571702	0.285851	−	0.958274i	$-0.407724\pi$
$977$	17.8697	0.571702	0.285851	+	0.958274i	$0.407724\pi$
$978$	0	0
$979$	0.888967	0.0284115
$980$	0	0
$981$	0	0
$982$	56.9420	1.81709
$983$	2.40197	0.0766110	0.0383055	−	0.999266i	$-0.487804\pi$
$983$	2.40197	0.0766110	0.0383055	+	0.999266i	$0.487804\pi$
$984$	0	0
$985$	0	0
$986$	−68.4671	−2.18044
$987$	0	0
$988$	21.2591	0.676342
$989$	−36.7140	−1.16744
$990$	0	0
$991$	−51.1008	−1.62327	−0.811636	−	0.584164i	$-0.801423\pi$
$991$	−51.1008	−1.62327	−0.811636	+	0.584164i	$0.801423\pi$
$992$	21.3418	0.677603
$993$	0	0
$994$	−56.4610	−1.79083
$995$	0	0
$996$	0	0
$997$	4.84862	0.153557	0.0767787	−	0.997048i	$-0.475537\pi$
$997$	4.84862	0.153557	0.0767787	+	0.997048i	$0.475537\pi$
$998$	54.7249	1.73229
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cf.1.1	✓	8
3.2	odd	2	inner	6975.2.a.cf.1.8	yes	8
5.4	even	2		6975.2.a.cg.1.8	yes	8
15.14	odd	2		6975.2.a.cg.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.cf.1.1	✓	8	1.1	even	1	trivial
6975.2.a.cf.1.8	yes	8	3.2	odd	2	inner
6975.2.a.cg.1.1	yes	8	15.14	odd	2
6975.2.a.cg.1.8	yes	8	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-2.72741 q^{2} +5.43874 q^{4} -3.85713 q^{7} -9.37885 q^{8} +2.51759 q^{11} -1.31707 q^{13} +10.5200 q^{14} +14.7024 q^{16} +3.65872 q^{17} -2.96781 q^{19} -6.86649 q^{22} +5.72013 q^{23} +3.59219 q^{26} -20.9780 q^{28} +6.86125 q^{29} -1.00000 q^{31} -21.3418 q^{32} -9.97880 q^{34} -5.17421 q^{37} +8.09443 q^{38} -6.65144 q^{41} -6.41839 q^{43} +13.6925 q^{44} -15.6011 q^{46} +6.59594 q^{47} +7.87748 q^{49} -7.16322 q^{52} -9.83502 q^{53} +36.1755 q^{56} -18.7134 q^{58} -7.07107 q^{59} -12.9560 q^{61} +2.72741 q^{62} +28.8029 q^{64} +7.31623 q^{67} +19.8988 q^{68} -5.36703 q^{71} -7.16322 q^{73} +14.1122 q^{74} -16.1412 q^{76} -9.71069 q^{77} +5.97880 q^{79} +18.1412 q^{82} -4.35024 q^{83} +17.5056 q^{86} -23.6121 q^{88} +0.353102 q^{89} +5.08012 q^{91} +31.1103 q^{92} -17.9898 q^{94} +18.0187 q^{97} -21.4851 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 4 q^{7} - 8 q^{13} + 12 q^{16} - 28 q^{22} - 36 q^{28} - 8 q^{31} - 28 q^{34} - 12 q^{37} - 52 q^{43} - 16 q^{46} + 8 q^{49} - 56 q^{52} - 16 q^{58} - 28 q^{61} + 48 q^{64} - 24 q^{67}+ \cdots - 44 q^{97}+O(q^{100}) \) 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

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