LMFDB - Embedded newform 6975.2.a.bp.1.2

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,-4,0,6,0,0,2,-9,0,0,2,0,4,-2,0,4,-19] 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:	5.5.144209.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - x^{2} + 6x - 1 $ x^5 - 6x^3 - x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 775)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.93413$ 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)

$f(q)$	$=$	$q-2.23959 q^{2} +3.01578 q^{4} +3.67497 q^{7} -2.27494 q^{8} -5.02619 q^{11} -2.67247 q^{13} -8.23043 q^{14} -0.936618 q^{16} -3.30547 q^{17} +2.88050 q^{19} +11.2566 q^{22} -6.01040 q^{23} +5.98526 q^{26} +11.0829 q^{28} +5.13050 q^{29} -1.00000 q^{31} +6.64753 q^{32} +7.40291 q^{34} -7.90540 q^{37} -6.45115 q^{38} +9.55651 q^{41} +5.25000 q^{43} -15.1579 q^{44} +13.4609 q^{46} +1.06691 q^{47} +6.50538 q^{49} -8.05960 q^{52} +0.521408 q^{53} -8.36034 q^{56} -11.4902 q^{58} +8.09515 q^{59} +0.598687 q^{61} +2.23959 q^{62} -13.0145 q^{64} +12.4705 q^{67} -9.96858 q^{68} +8.80048 q^{71} +9.87423 q^{73} +17.7049 q^{74} +8.68697 q^{76} -18.4711 q^{77} -2.40291 q^{79} -21.4027 q^{82} +2.91400 q^{83} -11.7579 q^{86} +11.4343 q^{88} +7.65918 q^{89} -9.82125 q^{91} -18.1261 q^{92} -2.38946 q^{94} -14.5224 q^{97} -14.5694 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 6 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} + 4 q^{13} - 2 q^{14} + 4 q^{16} - 19 q^{17} + 8 q^{19} - 10 q^{22} - 12 q^{23} + 16 q^{26} + 6 q^{28} - 6 q^{29} - 5 q^{31} - 7 q^{32} + 31 q^{34}+ \cdots - 10 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 6 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 + 4 * q^13 - 2 * q^14 + 4 * q^16 - 19 * q^17 + 8 * q^19 - 10 * q^22 - 12 * q^23 + 16 * q^26 + 6 * q^28 - 6 * q^29 - 5 * q^31 - 7 * q^32 + 31 * q^34 + 16 * q^37 - 14 * q^38 + 2 * q^41 + q^43 + 4 * q^44 - 11 * q^46 - 8 * q^47 - 9 * q^49 - 26 * q^52 - 3 * q^53 + 9 * q^56 + 5 * q^58 + 4 * q^59 - 5 * q^61 + 4 * q^62 - 5 * q^64 + 13 * q^67 - 30 * q^68 - 6 * q^71 + 19 * q^73 - 4 * q^74 - 5 * q^76 - 23 * q^77 - 6 * q^79 - 27 * q^82 - 18 * q^83 + 7 * q^86 - q^88 + 31 * q^89 + 8 * q^91 + 16 * q^92 - 14 * q^94 - q^97 - 10 * 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 $

$2$	−2.23959	−1.58363	−0.791816	−	0.610759i	$-0.790864\pi$
$2$	−2.23959	−1.58363	−0.791816	+	0.610759i	$0.790864\pi$
$3$	0	0
$3$	0	0
$4$	3.01578	1.50789
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.67497	1.38901	0.694503	−	0.719489i	$-0.255624\pi$
$7$	3.67497	1.38901	0.694503	+	0.719489i	$0.255624\pi$
$8$	−2.27494	−0.804314
$9$	0	0
$10$	0	0
$11$	−5.02619	−1.51545	−0.757726	−	0.652573i	$-0.773690\pi$
$11$	−5.02619	−1.51545	−0.757726	+	0.652573i	$0.773690\pi$
$12$	0	0
$13$	−2.67247	−0.741211	−0.370605	−	0.928790i	$-0.620850\pi$
$13$	−2.67247	−0.741211	−0.370605	+	0.928790i	$0.620850\pi$
$14$	−8.23043	−2.19968
$15$	0	0
$16$	−0.936618	−0.234155
$17$	−3.30547	−0.801694	−0.400847	−	0.916145i	$-0.631284\pi$
$17$	−3.30547	−0.801694	−0.400847	+	0.916145i	$0.631284\pi$
$18$	0	0
$19$	2.88050	0.660832	0.330416	−	0.943835i	$-0.392811\pi$
$19$	2.88050	0.660832	0.330416	+	0.943835i	$0.392811\pi$
$20$	0	0
$21$	0	0
$22$	11.2566	2.39992
$23$	−6.01040	−1.25326	−0.626628	−	0.779319i	$-0.715565\pi$
$23$	−6.01040	−1.25326	−0.626628	+	0.779319i	$0.715565\pi$
$24$	0	0
$25$	0	0
$26$	5.98526	1.17381
$27$	0	0
$28$	11.0829	2.09447
$29$	5.13050	0.952710	0.476355	−	0.879253i	$-0.341958\pi$
$29$	5.13050	0.952710	0.476355	+	0.879253i	$0.341958\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	6.64753	1.17513
$33$	0	0
$34$	7.40291	1.26959
$35$	0	0
$36$	0	0
$37$	−7.90540	−1.29964	−0.649820	−	0.760088i	$-0.725156\pi$
$37$	−7.90540	−1.29964	−0.649820	+	0.760088i	$0.725156\pi$
$38$	−6.45115	−1.04652
$39$	0	0
$40$	0	0
$41$	9.55651	1.49248	0.746238	−	0.665679i	$-0.231858\pi$
$41$	9.55651	1.49248	0.746238	+	0.665679i	$0.231858\pi$
$42$	0	0
$43$	5.25000	0.800617	0.400309	−	0.916380i	$-0.368903\pi$
$43$	5.25000	0.800617	0.400309	+	0.916380i	$0.368903\pi$
$44$	−15.1579	−2.28514
$45$	0	0
$46$	13.4609	1.98470
$47$	1.06691	0.155625	0.0778127	−	0.996968i	$-0.475206\pi$
$47$	1.06691	0.155625	0.0778127	+	0.996968i	$0.475206\pi$
$48$	0	0
$49$	6.50538	0.929339
$50$	0	0
$51$	0	0
$52$	−8.05960	−1.11767
$53$	0.521408	0.0716210	0.0358105	−	0.999359i	$-0.488599\pi$
$53$	0.521408	0.0716210	0.0358105	+	0.999359i	$0.488599\pi$
$54$	0	0
$55$	0	0
$56$	−8.36034	−1.11720
$57$	0	0
$58$	−11.4902	−1.50874
$59$	8.09515	1.05390	0.526950	−	0.849897i	$-0.323336\pi$
$59$	8.09515	1.05390	0.526950	+	0.849897i	$0.323336\pi$
$60$	0	0
$61$	0.598687	0.0766541	0.0383270	−	0.999265i	$-0.487797\pi$
$61$	0.598687	0.0766541	0.0383270	+	0.999265i	$0.487797\pi$
$62$	2.23959	0.284429
$63$	0	0
$64$	−13.0145	−1.62682
$65$	0	0
$66$	0	0
$67$	12.4705	1.52351	0.761755	−	0.647865i	$-0.224338\pi$
$67$	12.4705	1.52351	0.761755	+	0.647865i	$0.224338\pi$
$68$	−9.96858	−1.20887
$69$	0	0
$70$	0	0
$71$	8.80048	1.04442	0.522212	−	0.852815i	$-0.325107\pi$
$71$	8.80048	1.04442	0.522212	+	0.852815i	$0.325107\pi$
$72$	0	0
$73$	9.87423	1.15569	0.577845	−	0.816146i	$-0.303894\pi$
$73$	9.87423	1.15569	0.577845	+	0.816146i	$0.303894\pi$
$74$	17.7049	2.05815
$75$	0	0
$76$	8.68697	0.996464
$77$	−18.4711	−2.10497
$78$	0	0
$79$	−2.40291	−0.270349	−0.135174	−	0.990822i	$-0.543159\pi$
$79$	−2.40291	−0.270349	−0.135174	+	0.990822i	$0.543159\pi$
$80$	0	0
$81$	0	0
$82$	−21.4027	−2.36353
$83$	2.91400	0.319853	0.159927	−	0.987129i	$-0.448874\pi$
$83$	2.91400	0.319853	0.159927	+	0.987129i	$0.448874\pi$
$84$	0	0
$85$	0	0
$86$	−11.7579	−1.26788
$87$	0	0
$88$	11.4343	1.21890
$89$	7.65918	0.811872	0.405936	−	0.913902i	$-0.366946\pi$
$89$	7.65918	0.811872	0.405936	+	0.913902i	$0.366946\pi$
$90$	0	0
$91$	−9.82125	−1.02955
$92$	−18.1261	−1.88977
$93$	0	0
$94$	−2.38946	−0.246454
$95$	0	0
$96$	0	0
$97$	−14.5224	−1.47452	−0.737261	−	0.675608i	$-0.763881\pi$
$97$	−14.5224	−1.47452	−0.737261	+	0.675608i	$0.763881\pi$
$98$	−14.5694	−1.47173
$99$	0	0
$100$	0	0
$101$	14.6832	1.46103	0.730516	−	0.682896i	$-0.239280\pi$
$101$	14.6832	1.46103	0.730516	+	0.682896i	$0.239280\pi$
$102$	0	0
$103$	−16.2414	−1.60031	−0.800156	−	0.599792i	$-0.795250\pi$
$103$	−16.2414	−1.60031	−0.800156	+	0.599792i	$0.795250\pi$
$104$	6.07972	0.596166
$105$	0	0
$106$	−1.16774	−0.113421
$107$	−7.72859	−0.747151	−0.373576	−	0.927600i	$-0.621868\pi$
$107$	−7.72859	−0.747151	−0.373576	+	0.927600i	$0.621868\pi$
$108$	0	0
$109$	−6.60865	−0.632994	−0.316497	−	0.948594i	$-0.602507\pi$
$109$	−6.60865	−0.632994	−0.316497	+	0.948594i	$0.602507\pi$
$110$	0	0
$111$	0	0
$112$	−3.44204	−0.325242
$113$	−11.3537	−1.06807	−0.534034	−	0.845463i	$-0.679324\pi$
$113$	−11.3537	−1.06807	−0.534034	+	0.845463i	$0.679324\pi$
$114$	0	0
$115$	0	0
$116$	15.4725	1.43658
$117$	0	0
$118$	−18.1299	−1.66899
$119$	−12.1475	−1.11356
$120$	0	0
$121$	14.2626	1.29660
$122$	−1.34082	−0.121392
$123$	0	0
$124$	−3.01578	−0.270825
$125$	0	0
$126$	0	0
$127$	−11.1669	−0.990901	−0.495451	−	0.868636i	$-0.664997\pi$
$127$	−11.1669	−0.990901	−0.495451	+	0.868636i	$0.664997\pi$
$128$	15.8522	1.40115
$129$	0	0
$130$	0	0
$131$	1.93468	0.169034	0.0845170	−	0.996422i	$-0.473065\pi$
$131$	1.93468	0.169034	0.0845170	+	0.996422i	$0.473065\pi$
$132$	0	0
$133$	10.5857	0.917901
$134$	−27.9288	−2.41268
$135$	0	0
$136$	7.51975	0.644813
$137$	−16.5372	−1.41287	−0.706434	−	0.707779i	$-0.749697\pi$
$137$	−16.5372	−1.41287	−0.706434	+	0.707779i	$0.749697\pi$
$138$	0	0
$139$	−22.3954	−1.89956	−0.949778	−	0.312924i	$-0.898691\pi$
$139$	−22.3954	−1.89956	−0.949778	+	0.312924i	$0.898691\pi$
$140$	0	0
$141$	0	0
$142$	−19.7095	−1.65399
$143$	13.4324	1.12327
$144$	0	0
$145$	0	0
$146$	−22.1143	−1.83019
$147$	0	0
$148$	−23.8410	−1.95972
$149$	4.86954	0.398928	0.199464	−	0.979905i	$-0.436080\pi$
$149$	4.86954	0.398928	0.199464	+	0.979905i	$0.436080\pi$
$150$	0	0
$151$	19.5409	1.59022	0.795109	−	0.606466i	$-0.207414\pi$
$151$	19.5409	1.59022	0.795109	+	0.606466i	$0.207414\pi$
$152$	−6.55297	−0.531516
$153$	0	0
$154$	41.3677	3.33350
$155$	0	0
$156$	0	0
$157$	−14.9019	−1.18930	−0.594649	−	0.803985i	$-0.702709\pi$
$157$	−14.9019	−1.18930	−0.594649	+	0.803985i	$0.702709\pi$
$158$	5.38154	0.428133
$159$	0	0
$160$	0	0
$161$	−22.0880	−1.74078
$162$	0	0
$163$	−10.7995	−0.845878	−0.422939	−	0.906158i	$-0.639002\pi$
$163$	−10.7995	−0.845878	−0.422939	+	0.906158i	$0.639002\pi$
$164$	28.8204	2.25049
$165$	0	0
$166$	−6.52618	−0.506530
$167$	−16.2405	−1.25673	−0.628364	−	0.777920i	$-0.716275\pi$
$167$	−16.2405	−1.25673	−0.628364	+	0.777920i	$0.716275\pi$
$168$	0	0
$169$	−5.85789	−0.450607
$170$	0	0
$171$	0	0
$172$	15.8329	1.20724
$173$	3.97252	0.302025	0.151013	−	0.988532i	$-0.451747\pi$
$173$	3.97252	0.302025	0.151013	+	0.988532i	$0.451747\pi$
$174$	0	0
$175$	0	0
$176$	4.70762	0.354850
$177$	0	0
$178$	−17.1535	−1.28571
$179$	−24.8049	−1.85401	−0.927003	−	0.375053i	$-0.877624\pi$
$179$	−24.8049	−1.85401	−0.927003	+	0.375053i	$0.877624\pi$
$180$	0	0
$181$	−15.9359	−1.18451	−0.592253	−	0.805752i	$-0.701761\pi$
$181$	−15.9359	−1.18451	−0.592253	+	0.805752i	$0.701761\pi$
$182$	21.9956	1.63042
$183$	0	0
$184$	13.6733	1.00801
$185$	0	0
$186$	0	0
$187$	16.6139	1.21493
$188$	3.21758	0.234666
$189$	0	0
$190$	0	0
$191$	−22.5120	−1.62891	−0.814456	−	0.580226i	$-0.802964\pi$
$191$	−22.5120	−1.62891	−0.814456	+	0.580226i	$0.802964\pi$
$192$	0	0
$193$	19.0305	1.36984	0.684922	−	0.728616i	$-0.259836\pi$
$193$	19.0305	1.36984	0.684922	+	0.728616i	$0.259836\pi$
$194$	32.5242	2.33510
$195$	0	0
$196$	19.6188	1.40134
$197$	1.27674	0.0909642	0.0454821	−	0.998965i	$-0.485518\pi$
$197$	1.27674	0.0909642	0.0454821	+	0.998965i	$0.485518\pi$
$198$	0	0
$199$	9.21392	0.653157	0.326579	−	0.945170i	$-0.394104\pi$
$199$	9.21392	0.653157	0.326579	+	0.945170i	$0.394104\pi$
$200$	0	0
$201$	0	0
$202$	−32.8844	−2.31374
$203$	18.8544	1.32332
$204$	0	0
$205$	0	0
$206$	36.3741	2.53431
$207$	0	0
$208$	2.50309	0.173558
$209$	−14.4779	−1.00146
$210$	0	0
$211$	2.86821	0.197456	0.0987279	−	0.995114i	$-0.468523\pi$
$211$	2.86821	0.197456	0.0987279	+	0.995114i	$0.468523\pi$
$212$	1.57245	0.107997
$213$	0	0
$214$	17.3089	1.18321
$215$	0	0
$216$	0	0
$217$	−3.67497	−0.249473
$218$	14.8007	1.00243
$219$	0	0
$220$	0	0
$221$	8.83378	0.594224
$222$	0	0
$223$	9.10462	0.609691	0.304845	−	0.952402i	$-0.401395\pi$
$223$	9.10462	0.609691	0.304845	+	0.952402i	$0.401395\pi$
$224$	24.4294	1.63226
$225$	0	0
$226$	25.4277	1.69143
$227$	−4.62499	−0.306971	−0.153486	−	0.988151i	$-0.549050\pi$
$227$	−4.62499	−0.306971	−0.153486	+	0.988151i	$0.549050\pi$
$228$	0	0
$229$	26.1553	1.72839	0.864196	−	0.503156i	$-0.167828\pi$
$229$	26.1553	1.72839	0.864196	+	0.503156i	$0.167828\pi$
$230$	0	0
$231$	0	0
$232$	−11.6716	−0.766278
$233$	4.76330	0.312054	0.156027	−	0.987753i	$-0.450131\pi$
$233$	4.76330	0.312054	0.156027	+	0.987753i	$0.450131\pi$
$234$	0	0
$235$	0	0
$236$	24.4132	1.58917
$237$	0	0
$238$	27.2054	1.76347
$239$	−0.674275	−0.0436152	−0.0218076	−	0.999762i	$-0.506942\pi$
$239$	−0.674275	−0.0436152	−0.0218076	+	0.999762i	$0.506942\pi$
$240$	0	0
$241$	4.14684	0.267122	0.133561	−	0.991041i	$-0.457359\pi$
$241$	4.14684	0.267122	0.133561	+	0.991041i	$0.457359\pi$
$242$	−31.9423	−2.05333
$243$	0	0
$244$	1.80551	0.115586
$245$	0	0
$246$	0	0
$247$	−7.69806	−0.489816
$248$	2.27494	0.144459
$249$	0	0
$250$	0	0
$251$	−23.0876	−1.45728	−0.728638	−	0.684899i	$-0.759846\pi$
$251$	−23.0876	−1.45728	−0.728638	+	0.684899i	$0.759846\pi$
$252$	0	0
$253$	30.2094	1.89925
$254$	25.0093	1.56922
$255$	0	0
$256$	−9.47347	−0.592092
$257$	−27.3951	−1.70886	−0.854429	−	0.519568i	$-0.826093\pi$
$257$	−27.3951	−1.70886	−0.854429	+	0.519568i	$0.826093\pi$
$258$	0	0
$259$	−29.0521	−1.80521
$260$	0	0
$261$	0	0
$262$	−4.33291	−0.267688
$263$	13.1328	0.809802	0.404901	−	0.914361i	$-0.367306\pi$
$263$	13.1328	0.809802	0.404901	+	0.914361i	$0.367306\pi$
$264$	0	0
$265$	0	0
$266$	−23.7078	−1.45362
$267$	0	0
$268$	37.6082	2.29729
$269$	−23.5415	−1.43535	−0.717676	−	0.696377i	$-0.754794\pi$
$269$	−23.5415	−1.43535	−0.717676	+	0.696377i	$0.754794\pi$
$270$	0	0
$271$	21.2089	1.28835	0.644173	−	0.764880i	$-0.277202\pi$
$271$	21.2089	1.28835	0.644173	+	0.764880i	$0.277202\pi$
$272$	3.09596	0.187720
$273$	0	0
$274$	37.0366	2.23746
$275$	0	0
$276$	0	0
$277$	−3.16271	−0.190029	−0.0950144	−	0.995476i	$-0.530290\pi$
$277$	−3.16271	−0.190029	−0.0950144	+	0.995476i	$0.530290\pi$
$278$	50.1567	3.00820
$279$	0	0
$280$	0	0
$281$	−21.9912	−1.31189	−0.655943	−	0.754810i	$-0.727729\pi$
$281$	−21.9912	−1.31189	−0.655943	+	0.754810i	$0.727729\pi$
$282$	0	0
$283$	15.6629	0.931065	0.465532	−	0.885031i	$-0.345863\pi$
$283$	15.6629	0.931065	0.465532	+	0.885031i	$0.345863\pi$
$284$	26.5403	1.57488
$285$	0	0
$286$	−30.0830	−1.77885
$287$	35.1198	2.07306
$288$	0	0
$289$	−6.07387	−0.357287
$290$	0	0
$291$	0	0
$292$	29.7785	1.74266
$293$	−29.1815	−1.70480	−0.852401	−	0.522889i	$-0.824854\pi$
$293$	−29.1815	−1.70480	−0.852401	+	0.522889i	$0.824854\pi$
$294$	0	0
$295$	0	0
$296$	17.9843	1.04532
$297$	0	0
$298$	−10.9058	−0.631756
$299$	16.0626	0.928927
$300$	0	0
$301$	19.2936	1.11206
$302$	−43.7638	−2.51832
$303$	0	0
$304$	−2.69793	−0.154737
$305$	0	0
$306$	0	0
$307$	−19.3160	−1.10243	−0.551213	−	0.834365i	$-0.685835\pi$
$307$	−19.3160	−1.10243	−0.551213	+	0.834365i	$0.685835\pi$
$308$	−55.7047	−3.17407
$309$	0	0
$310$	0	0
$311$	−6.15914	−0.349253	−0.174626	−	0.984635i	$-0.555872\pi$
$311$	−6.15914	−0.349253	−0.174626	+	0.984635i	$0.555872\pi$
$312$	0	0
$313$	−20.1839	−1.14086	−0.570430	−	0.821346i	$-0.693223\pi$
$313$	−20.1839	−1.14086	−0.570430	+	0.821346i	$0.693223\pi$
$314$	33.3741	1.88341
$315$	0	0
$316$	−7.24666	−0.407656
$317$	25.2088	1.41587	0.707933	−	0.706280i	$-0.249628\pi$
$317$	25.2088	1.41587	0.707933	+	0.706280i	$0.249628\pi$
$318$	0	0
$319$	−25.7869	−1.44379
$320$	0	0
$321$	0	0
$322$	49.4682	2.75676
$323$	−9.52141	−0.529785
$324$	0	0
$325$	0	0
$326$	24.1864	1.33956
$327$	0	0
$328$	−21.7405	−1.20042
$329$	3.92087	0.216165
$330$	0	0
$331$	−17.6922	−0.972454	−0.486227	−	0.873833i	$-0.661627\pi$
$331$	−17.6922	−0.972454	−0.486227	+	0.873833i	$0.661627\pi$
$332$	8.78800	0.482304
$333$	0	0
$334$	36.3721	1.99019
$335$	0	0
$336$	0	0
$337$	2.89468	0.157683	0.0788415	−	0.996887i	$-0.474878\pi$
$337$	2.89468	0.157683	0.0788415	+	0.996887i	$0.474878\pi$
$338$	13.1193	0.713595
$339$	0	0
$340$	0	0
$341$	5.02619	0.272183
$342$	0	0
$343$	−1.81773	−0.0981480
$344$	−11.9434	−0.643947
$345$	0	0
$346$	−8.89684	−0.478297
$347$	−6.89681	−0.370240	−0.185120	−	0.982716i	$-0.559267\pi$
$347$	−6.89681	−0.370240	−0.185120	+	0.982716i	$0.559267\pi$
$348$	0	0
$349$	−14.6691	−0.785220	−0.392610	−	0.919705i	$-0.628428\pi$
$349$	−14.6691	−0.785220	−0.392610	+	0.919705i	$0.628428\pi$
$350$	0	0
$351$	0	0
$352$	−33.4117	−1.78085
$353$	10.5568	0.561883	0.280942	−	0.959725i	$-0.409353\pi$
$353$	10.5568	0.561883	0.280942	+	0.959725i	$0.409353\pi$
$354$	0	0
$355$	0	0
$356$	23.0984	1.22421
$357$	0	0
$358$	55.5530	2.93607
$359$	−20.4230	−1.07788	−0.538942	−	0.842343i	$-0.681176\pi$
$359$	−20.4230	−1.07788	−0.538942	+	0.842343i	$0.681176\pi$
$360$	0	0
$361$	−10.7027	−0.563301
$362$	35.6899	1.87582
$363$	0	0
$364$	−29.6188	−1.55244
$365$	0	0
$366$	0	0
$367$	−15.0168	−0.783872	−0.391936	−	0.919993i	$-0.628194\pi$
$367$	−15.0168	−0.783872	−0.391936	+	0.919993i	$0.628194\pi$
$368$	5.62945	0.293456
$369$	0	0
$370$	0	0
$371$	1.91616	0.0994820
$372$	0	0
$373$	19.2471	0.996578	0.498289	−	0.867011i	$-0.333962\pi$
$373$	19.2471	0.996578	0.498289	+	0.867011i	$0.333962\pi$
$374$	−37.2084	−1.92400
$375$	0	0
$376$	−2.42717	−0.125172
$377$	−13.7111	−0.706159
$378$	0	0
$379$	3.24968	0.166925	0.0834624	−	0.996511i	$-0.473402\pi$
$379$	3.24968	0.166925	0.0834624	+	0.996511i	$0.473402\pi$
$380$	0	0
$381$	0	0
$382$	50.4177	2.57960
$383$	2.01897	0.103164	0.0515822	−	0.998669i	$-0.483574\pi$
$383$	2.01897	0.103164	0.0515822	+	0.998669i	$0.483574\pi$
$384$	0	0
$385$	0	0
$386$	−42.6206	−2.16933
$387$	0	0
$388$	−43.7963	−2.22342
$389$	15.2990	0.775689	0.387844	−	0.921725i	$-0.373220\pi$
$389$	15.2990	0.775689	0.387844	+	0.921725i	$0.373220\pi$
$390$	0	0
$391$	19.8672	1.00473
$392$	−14.7994	−0.747480
$393$	0	0
$394$	−2.85939	−0.144054
$395$	0	0
$396$	0	0
$397$	14.1815	0.711751	0.355875	−	0.934533i	$-0.384183\pi$
$397$	14.1815	0.711751	0.355875	+	0.934533i	$0.384183\pi$
$398$	−20.6354	−1.03436
$399$	0	0
$400$	0	0
$401$	6.59958	0.329567	0.164784	−	0.986330i	$-0.447307\pi$
$401$	6.59958	0.329567	0.164784	+	0.986330i	$0.447307\pi$
$402$	0	0
$403$	2.67247	0.133125
$404$	44.2813	2.20308
$405$	0	0
$406$	−42.2262	−2.09565
$407$	39.7340	1.96954
$408$	0	0
$409$	−29.1922	−1.44346	−0.721730	−	0.692175i	$-0.756653\pi$
$409$	−29.1922	−1.44346	−0.721730	+	0.692175i	$0.756653\pi$
$410$	0	0
$411$	0	0
$412$	−48.9805	−2.41310
$413$	29.7494	1.46387
$414$	0	0
$415$	0	0
$416$	−17.7653	−0.871018
$417$	0	0
$418$	32.4247	1.58594
$419$	28.0209	1.36891	0.684454	−	0.729056i	$-0.260040\pi$
$419$	28.0209	1.36891	0.684454	+	0.729056i	$0.260040\pi$
$420$	0	0
$421$	−30.7552	−1.49892	−0.749459	−	0.662051i	$-0.769686\pi$
$421$	−30.7552	−1.49892	−0.749459	+	0.662051i	$0.769686\pi$
$422$	−6.42363	−0.312697
$423$	0	0
$424$	−1.18617	−0.0576057
$425$	0	0
$426$	0	0
$427$	2.20016	0.106473
$428$	−23.3078	−1.12662
$429$	0	0
$430$	0	0
$431$	26.9481	1.29804	0.649021	−	0.760770i	$-0.275179\pi$
$431$	26.9481	1.29804	0.649021	+	0.760770i	$0.275179\pi$
$432$	0	0
$433$	−9.45517	−0.454386	−0.227193	−	0.973850i	$-0.572955\pi$
$433$	−9.45517	−0.454386	−0.227193	+	0.973850i	$0.572955\pi$
$434$	8.23043	0.395073
$435$	0	0
$436$	−19.9303	−0.954486
$437$	−17.3130	−0.828192
$438$	0	0
$439$	−20.3379	−0.970673	−0.485337	−	0.874327i	$-0.661303\pi$
$439$	−20.3379	−0.970673	−0.485337	+	0.874327i	$0.661303\pi$
$440$	0	0
$441$	0	0
$442$	−19.7841	−0.941033
$443$	−13.6821	−0.650056	−0.325028	−	0.945704i	$-0.605374\pi$
$443$	−13.6821	−0.650056	−0.325028	+	0.945704i	$0.605374\pi$
$444$	0	0
$445$	0	0
$446$	−20.3907	−0.965526
$447$	0	0
$448$	−47.8280	−2.25966
$449$	−41.1092	−1.94006	−0.970031	−	0.242980i	$-0.921875\pi$
$449$	−41.1092	−1.94006	−0.970031	+	0.242980i	$0.921875\pi$
$450$	0	0
$451$	−48.0328	−2.26178
$452$	−34.2403	−1.61053
$453$	0	0
$454$	10.3581	0.486130
$455$	0	0
$456$	0	0
$457$	−16.6907	−0.780760	−0.390380	−	0.920654i	$-0.627656\pi$
$457$	−16.6907	−0.780760	−0.390380	+	0.920654i	$0.627656\pi$
$458$	−58.5773	−2.73714
$459$	0	0
$460$	0	0
$461$	−2.70160	−0.125826	−0.0629129	−	0.998019i	$-0.520039\pi$
$461$	−2.70160	−0.125826	−0.0629129	+	0.998019i	$0.520039\pi$
$462$	0	0
$463$	13.6205	0.633000	0.316500	−	0.948593i	$-0.397492\pi$
$463$	13.6205	0.633000	0.316500	+	0.948593i	$0.397492\pi$
$464$	−4.80532	−0.223081
$465$	0	0
$466$	−10.6678	−0.494179
$467$	27.5703	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$467$	27.5703	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$468$	0	0
$469$	45.8286	2.11617
$470$	0	0
$471$	0	0
$472$	−18.4160	−0.847665
$473$	−26.3875	−1.21330
$474$	0	0
$475$	0	0
$476$	−36.6342	−1.67913
$477$	0	0
$478$	1.51010	0.0690704
$479$	−5.41728	−0.247522	−0.123761	−	0.992312i	$-0.539496\pi$
$479$	−5.41728	−0.247522	−0.123761	+	0.992312i	$0.539496\pi$
$480$	0	0
$481$	21.1270	0.963307
$482$	−9.28724	−0.423022
$483$	0	0
$484$	43.0128	1.95513
$485$	0	0
$486$	0	0
$487$	1.23959	0.0561714	0.0280857	−	0.999606i	$-0.491059\pi$
$487$	1.23959	0.0561714	0.0280857	+	0.999606i	$0.491059\pi$
$488$	−1.36198	−0.0616539
$489$	0	0
$490$	0	0
$491$	−19.0450	−0.859487	−0.429743	−	0.902951i	$-0.641396\pi$
$491$	−19.0450	−0.859487	−0.429743	+	0.902951i	$0.641396\pi$
$492$	0	0
$493$	−16.9587	−0.763782
$494$	17.2405	0.775689
$495$	0	0
$496$	0.936618	0.0420554
$497$	32.3415	1.45071
$498$	0	0
$499$	3.81671	0.170859	0.0854296	−	0.996344i	$-0.472774\pi$
$499$	3.81671	0.170859	0.0854296	+	0.996344i	$0.472774\pi$
$500$	0	0
$501$	0	0
$502$	51.7068	2.30779
$503$	−30.4237	−1.35653	−0.678263	−	0.734819i	$-0.737267\pi$
$503$	−30.4237	−1.35653	−0.678263	+	0.734819i	$0.737267\pi$
$504$	0	0
$505$	0	0
$506$	−67.6568	−3.00771
$507$	0	0
$508$	−33.6769	−1.49417
$509$	−9.97990	−0.442351	−0.221176	−	0.975234i	$-0.570989\pi$
$509$	−9.97990	−0.442351	−0.221176	+	0.975234i	$0.570989\pi$
$510$	0	0
$511$	36.2875	1.60526
$512$	−10.4877	−0.463495
$513$	0	0
$514$	61.3539	2.70620
$515$	0	0
$516$	0	0
$517$	−5.36251	−0.235843
$518$	65.0649	2.85879
$519$	0	0
$520$	0	0
$521$	−8.27726	−0.362633	−0.181317	−	0.983425i	$-0.558036\pi$
$521$	−8.27726	−0.362633	−0.181317	+	0.983425i	$0.558036\pi$
$522$	0	0
$523$	−28.3888	−1.24136	−0.620678	−	0.784066i	$-0.713143\pi$
$523$	−28.3888	−1.24136	−0.620678	+	0.784066i	$0.713143\pi$
$524$	5.83459	0.254885
$525$	0	0
$526$	−29.4121	−1.28243
$527$	3.30547	0.143988
$528$	0	0
$529$	13.1250	0.570650
$530$	0	0
$531$	0	0
$532$	31.9243	1.38409
$533$	−25.5395	−1.10624
$534$	0	0
$535$	0	0
$536$	−28.3696	−1.22538
$537$	0	0
$538$	52.7235	2.27307
$539$	−32.6972	−1.40837
$540$	0	0
$541$	−3.10137	−0.133338	−0.0666691	−	0.997775i	$-0.521237\pi$
$541$	−3.10137	−0.133338	−0.0666691	+	0.997775i	$0.521237\pi$
$542$	−47.4992	−2.04027
$543$	0	0
$544$	−21.9732	−0.942093
$545$	0	0
$546$	0	0
$547$	35.4405	1.51533	0.757663	−	0.652646i	$-0.226341\pi$
$547$	35.4405	1.51533	0.757663	+	0.652646i	$0.226341\pi$
$548$	−49.8726	−2.13045
$549$	0	0
$550$	0	0
$551$	14.7784	0.629582
$552$	0	0
$553$	−8.83061	−0.375516
$554$	7.08319	0.300936
$555$	0	0
$556$	−67.5398	−2.86432
$557$	−11.6471	−0.493505	−0.246752	−	0.969079i	$-0.579363\pi$
$557$	−11.6471	−0.493505	−0.246752	+	0.969079i	$0.579363\pi$
$558$	0	0
$559$	−14.0305	−0.593426
$560$	0	0
$561$	0	0
$562$	49.2514	2.07755
$563$	31.6878	1.33548	0.667741	−	0.744394i	$-0.267261\pi$
$563$	31.6878	1.33548	0.667741	+	0.744394i	$0.267261\pi$
$564$	0	0
$565$	0	0
$566$	−35.0786	−1.47446
$567$	0	0
$568$	−20.0206	−0.840045
$569$	29.7931	1.24899	0.624495	−	0.781028i	$-0.285305\pi$
$569$	29.7931	1.24899	0.624495	+	0.781028i	$0.285305\pi$
$570$	0	0
$571$	35.6580	1.49224	0.746121	−	0.665810i	$-0.231914\pi$
$571$	35.6580	1.49224	0.746121	+	0.665810i	$0.231914\pi$
$572$	40.5091	1.69377
$573$	0	0
$574$	−78.6542	−3.28296
$575$	0	0
$576$	0	0
$577$	6.76735	0.281728	0.140864	−	0.990029i	$-0.455012\pi$
$577$	6.76735	0.281728	0.140864	+	0.990029i	$0.455012\pi$
$578$	13.6030	0.565811
$579$	0	0
$580$	0	0
$581$	10.7089	0.444278
$582$	0	0
$583$	−2.62070	−0.108538
$584$	−22.4633	−0.929538
$585$	0	0
$586$	65.3547	2.69978
$587$	−1.89702	−0.0782982	−0.0391491	−	0.999233i	$-0.512465\pi$
$587$	−1.89702	−0.0782982	−0.0391491	+	0.999233i	$0.512465\pi$
$588$	0	0
$589$	−2.88050	−0.118689
$590$	0	0
$591$	0	0
$592$	7.40434	0.304317
$593$	−24.2080	−0.994103	−0.497051	−	0.867721i	$-0.665584\pi$
$593$	−24.2080	−0.994103	−0.497051	+	0.867721i	$0.665584\pi$
$594$	0	0
$595$	0	0
$596$	14.6855	0.601541
$597$	0	0
$598$	−35.9738	−1.47108
$599$	−43.2713	−1.76802	−0.884009	−	0.467469i	$-0.845166\pi$
$599$	−43.2713	−1.76802	−0.884009	+	0.467469i	$0.845166\pi$
$600$	0	0
$601$	4.08692	0.166709	0.0833545	−	0.996520i	$-0.473437\pi$
$601$	4.08692	0.166709	0.0833545	+	0.996520i	$0.473437\pi$
$602$	−43.2098	−1.76110
$603$	0	0
$604$	58.9312	2.39788
$605$	0	0
$606$	0	0
$607$	0.641570	0.0260405	0.0130203	−	0.999915i	$-0.495855\pi$
$607$	0.641570	0.0260405	0.0130203	+	0.999915i	$0.495855\pi$
$608$	19.1482	0.776563
$609$	0	0
$610$	0	0
$611$	−2.85130	−0.115351
$612$	0	0
$613$	−16.9855	−0.686038	−0.343019	−	0.939328i	$-0.611450\pi$
$613$	−16.9855	−0.686038	−0.343019	+	0.939328i	$0.611450\pi$
$614$	43.2601	1.74584
$615$	0	0
$616$	42.0206	1.69306
$617$	−12.1444	−0.488915	−0.244457	−	0.969660i	$-0.578610\pi$
$617$	−12.1444	−0.488915	−0.244457	+	0.969660i	$0.578610\pi$
$618$	0	0
$619$	7.09579	0.285204	0.142602	−	0.989780i	$-0.454453\pi$
$619$	7.09579	0.285204	0.142602	+	0.989780i	$0.454453\pi$
$620$	0	0
$621$	0	0
$622$	13.7940	0.553088
$623$	28.1472	1.12770
$624$	0	0
$625$	0	0
$626$	45.2037	1.80670
$627$	0	0
$628$	−44.9408	−1.79333
$629$	26.1311	1.04191
$630$	0	0
$631$	−34.4872	−1.37291	−0.686457	−	0.727171i	$-0.740835\pi$
$631$	−34.4872	−1.37291	−0.686457	+	0.727171i	$0.740835\pi$
$632$	5.46648	0.217445
$633$	0	0
$634$	−56.4574	−2.24221
$635$	0	0
$636$	0	0
$637$	−17.3854	−0.688836
$638$	57.7521	2.28643
$639$	0	0
$640$	0	0
$641$	20.5382	0.811211	0.405605	−	0.914048i	$-0.367061\pi$
$641$	20.5382	0.811211	0.405605	+	0.914048i	$0.367061\pi$
$642$	0	0
$643$	−32.3455	−1.27558	−0.637792	−	0.770209i	$-0.720152\pi$
$643$	−32.3455	−1.27558	−0.637792	+	0.770209i	$0.720152\pi$
$644$	−66.6127	−2.62491
$645$	0	0
$646$	21.3241	0.838985
$647$	−23.1202	−0.908948	−0.454474	−	0.890760i	$-0.650173\pi$
$647$	−23.1202	−0.908948	−0.454474	+	0.890760i	$0.650173\pi$
$648$	0	0
$649$	−40.6877	−1.59713
$650$	0	0
$651$	0	0
$652$	−32.5688	−1.27549
$653$	7.05761	0.276186	0.138093	−	0.990419i	$-0.455903\pi$
$653$	7.05761	0.276186	0.138093	+	0.990419i	$0.455903\pi$
$654$	0	0
$655$	0	0
$656$	−8.95080	−0.349470
$657$	0	0
$658$	−8.78117	−0.342326
$659$	13.8853	0.540896	0.270448	−	0.962735i	$-0.412828\pi$
$659$	13.8853	0.540896	0.270448	+	0.962735i	$0.412828\pi$
$660$	0	0
$661$	−10.1962	−0.396584	−0.198292	−	0.980143i	$-0.563539\pi$
$661$	−10.1962	−0.396584	−0.198292	+	0.980143i	$0.563539\pi$
$662$	39.6234	1.54001
$663$	0	0
$664$	−6.62919	−0.257262
$665$	0	0
$666$	0	0
$667$	−30.8364	−1.19399
$668$	−48.9778	−1.89501
$669$	0	0
$670$	0	0
$671$	−3.00912	−0.116166
$672$	0	0
$673$	23.9562	0.923445	0.461722	−	0.887025i	$-0.347232\pi$
$673$	23.9562	0.923445	0.461722	+	0.887025i	$0.347232\pi$
$674$	−6.48290	−0.249712
$675$	0	0
$676$	−17.6661	−0.679466
$677$	−9.99786	−0.384249	−0.192124	−	0.981371i	$-0.561538\pi$
$677$	−9.99786	−0.384249	−0.192124	+	0.981371i	$0.561538\pi$
$678$	0	0
$679$	−53.3692	−2.04812
$680$	0	0
$681$	0	0
$682$	−11.2566	−0.431038
$683$	−37.6974	−1.44245	−0.721226	−	0.692700i	$-0.756421\pi$
$683$	−37.6974	−1.44245	−0.721226	+	0.692700i	$0.756421\pi$
$684$	0	0
$685$	0	0
$686$	4.07097	0.155430
$687$	0	0
$688$	−4.91724	−0.187468
$689$	−1.39345	−0.0530862
$690$	0	0
$691$	−32.1264	−1.22214	−0.611072	−	0.791575i	$-0.709261\pi$
$691$	−32.1264	−1.22214	−0.611072	+	0.791575i	$0.709261\pi$
$692$	11.9803	0.455422
$693$	0	0
$694$	15.4461	0.586324
$695$	0	0
$696$	0	0
$697$	−31.5887	−1.19651
$698$	32.8529	1.24350
$699$	0	0
$700$	0	0
$701$	27.8813	1.05306	0.526530	−	0.850156i	$-0.323493\pi$
$701$	27.8813	1.05306	0.526530	+	0.850156i	$0.323493\pi$
$702$	0	0
$703$	−22.7715	−0.858844
$704$	65.4135	2.46536
$705$	0	0
$706$	−23.6430	−0.889816
$707$	53.9602	2.02938
$708$	0	0
$709$	30.4246	1.14262	0.571310	−	0.820734i	$-0.306435\pi$
$709$	30.4246	1.14262	0.571310	+	0.820734i	$0.306435\pi$
$710$	0	0
$711$	0	0
$712$	−17.4242	−0.653000
$713$	6.01040	0.225091
$714$	0	0
$715$	0	0
$716$	−74.8063	−2.79564
$717$	0	0
$718$	45.7392	1.70697
$719$	−25.7714	−0.961111	−0.480556	−	0.876964i	$-0.659565\pi$
$719$	−25.7714	−0.961111	−0.480556	+	0.876964i	$0.659565\pi$
$720$	0	0
$721$	−59.6866	−2.22284
$722$	23.9697	0.892061
$723$	0	0
$724$	−48.0592	−1.78611
$725$	0	0
$726$	0	0
$727$	2.49622	0.0925795	0.0462898	−	0.998928i	$-0.485260\pi$
$727$	2.49622	0.0925795	0.0462898	+	0.998928i	$0.485260\pi$
$728$	22.3428	0.828078
$729$	0	0
$730$	0	0
$731$	−17.3537	−0.641850
$732$	0	0
$733$	45.2444	1.67114	0.835569	−	0.549385i	$-0.185138\pi$
$733$	45.2444	1.67114	0.835569	+	0.549385i	$0.185138\pi$
$734$	33.6316	1.24136
$735$	0	0
$736$	−39.9543	−1.47274
$737$	−62.6789	−2.30881
$738$	0	0
$739$	8.57134	0.315302	0.157651	−	0.987495i	$-0.449608\pi$
$739$	8.57134	0.315302	0.157651	+	0.987495i	$0.449608\pi$
$740$	0	0
$741$	0	0
$742$	−4.29142	−0.157543
$743$	5.94817	0.218217	0.109109	−	0.994030i	$-0.465200\pi$
$743$	5.94817	0.218217	0.109109	+	0.994030i	$0.465200\pi$
$744$	0	0
$745$	0	0
$746$	−43.1057	−1.57821
$747$	0	0
$748$	50.1039	1.83198
$749$	−28.4023	−1.03780
$750$	0	0
$751$	46.7694	1.70664	0.853319	−	0.521388i	$-0.174586\pi$
$751$	46.7694	1.70664	0.853319	+	0.521388i	$0.174586\pi$
$752$	−0.999291	−0.0364404
$753$	0	0
$754$	30.7074	1.11830
$755$	0	0
$756$	0	0
$757$	−45.3088	−1.64678	−0.823388	−	0.567479i	$-0.807919\pi$
$757$	−45.3088	−1.64678	−0.823388	+	0.567479i	$0.807919\pi$
$758$	−7.27796	−0.264348
$759$	0	0
$760$	0	0
$761$	34.2873	1.24291	0.621457	−	0.783448i	$-0.286541\pi$
$761$	34.2873	1.24291	0.621457	+	0.783448i	$0.286541\pi$
$762$	0	0
$763$	−24.2866	−0.879233
$764$	−67.8913	−2.45622
$765$	0	0
$766$	−4.52167	−0.163375
$767$	−21.6341	−0.781161
$768$	0	0
$769$	38.2808	1.38044	0.690220	−	0.723600i	$-0.257514\pi$
$769$	38.2808	1.38044	0.690220	+	0.723600i	$0.257514\pi$
$770$	0	0
$771$	0	0
$772$	57.3918	2.06558
$773$	3.17845	0.114321	0.0571604	−	0.998365i	$-0.481795\pi$
$773$	3.17845	0.114321	0.0571604	+	0.998365i	$0.481795\pi$
$774$	0	0
$775$	0	0
$776$	33.0375	1.18598
$777$	0	0
$778$	−34.2635	−1.22841
$779$	27.5275	0.986276
$780$	0	0
$781$	−44.2329	−1.58278
$782$	−44.4945	−1.59112
$783$	0	0
$784$	−6.09305	−0.217609
$785$	0	0
$786$	0	0
$787$	9.42277	0.335886	0.167943	−	0.985797i	$-0.446288\pi$
$787$	9.42277	0.335886	0.167943	+	0.985797i	$0.446288\pi$
$788$	3.85038	0.137164
$789$	0	0
$790$	0	0
$791$	−41.7245	−1.48355
$792$	0	0
$793$	−1.59998	−0.0568168
$794$	−31.7609	−1.12715
$795$	0	0
$796$	27.7872	0.984891
$797$	8.28600	0.293505	0.146753	−	0.989173i	$-0.453118\pi$
$797$	8.28600	0.293505	0.146753	+	0.989173i	$0.453118\pi$
$798$	0	0
$799$	−3.52665	−0.124764
$800$	0	0
$801$	0	0
$802$	−14.7804	−0.521914
$803$	−49.6297	−1.75139
$804$	0	0
$805$	0	0
$806$	−5.98526	−0.210822
$807$	0	0
$808$	−33.4034	−1.17513
$809$	41.8918	1.47284	0.736418	−	0.676526i	$-0.236516\pi$
$809$	41.8918	1.47284	0.736418	+	0.676526i	$0.236516\pi$
$810$	0	0
$811$	−33.5938	−1.17964	−0.589820	−	0.807535i	$-0.700801\pi$
$811$	−33.5938	−1.17964	−0.589820	+	0.807535i	$0.700801\pi$
$812$	56.8608	1.99542
$813$	0	0
$814$	−88.9881	−3.11903
$815$	0	0
$816$	0	0
$817$	15.1226	0.529074
$818$	65.3786	2.28591
$819$	0	0
$820$	0	0
$821$	30.9932	1.08167	0.540836	−	0.841128i	$-0.318108\pi$
$821$	30.9932	1.08167	0.540836	+	0.841128i	$0.318108\pi$
$822$	0	0
$823$	9.48741	0.330710	0.165355	−	0.986234i	$-0.447123\pi$
$823$	9.48741	0.330710	0.165355	+	0.986234i	$0.447123\pi$
$824$	36.9482	1.28715
$825$	0	0
$826$	−66.6266	−2.31824
$827$	4.97893	0.173134	0.0865671	−	0.996246i	$-0.472410\pi$
$827$	4.97893	0.173134	0.0865671	+	0.996246i	$0.472410\pi$
$828$	0	0
$829$	−28.0734	−0.975030	−0.487515	−	0.873115i	$-0.662096\pi$
$829$	−28.0734	−0.975030	−0.487515	+	0.873115i	$0.662096\pi$
$830$	0	0
$831$	0	0
$832$	34.7810	1.20581
$833$	−21.5033	−0.745046
$834$	0	0
$835$	0	0
$836$	−43.6623	−1.51009
$837$	0	0
$838$	−62.7554	−2.16785
$839$	−0.672364	−0.0232126	−0.0116063	−	0.999933i	$-0.503694\pi$
$839$	−0.672364	−0.0232126	−0.0116063	+	0.999933i	$0.503694\pi$
$840$	0	0
$841$	−2.67797	−0.0923438
$842$	68.8792	2.37373
$843$	0	0
$844$	8.64990	0.297742
$845$	0	0
$846$	0	0
$847$	52.4144	1.80098
$848$	−0.488361	−0.0167704
$849$	0	0
$850$	0	0
$851$	47.5146	1.62878
$852$	0	0
$853$	22.9459	0.785652	0.392826	−	0.919613i	$-0.371498\pi$
$853$	22.9459	0.785652	0.392826	+	0.919613i	$0.371498\pi$
$854$	−4.92746	−0.168614
$855$	0	0
$856$	17.5821	0.600944
$857$	30.7670	1.05098	0.525491	−	0.850799i	$-0.323882\pi$
$857$	30.7670	1.05098	0.525491	+	0.850799i	$0.323882\pi$
$858$	0	0
$859$	7.20222	0.245737	0.122868	−	0.992423i	$-0.460791\pi$
$859$	7.20222	0.245737	0.122868	+	0.992423i	$0.460791\pi$
$860$	0	0
$861$	0	0
$862$	−60.3527	−2.05562
$863$	−15.3436	−0.522303	−0.261152	−	0.965298i	$-0.584102\pi$
$863$	−15.3436	−0.522303	−0.261152	+	0.965298i	$0.584102\pi$
$864$	0	0
$865$	0	0
$866$	21.1757	0.719581
$867$	0	0
$868$	−11.0829	−0.376178
$869$	12.0775	0.409700
$870$	0	0
$871$	−33.3270	−1.12924
$872$	15.0343	0.509125
$873$	0	0
$874$	38.7740	1.31155
$875$	0	0
$876$	0	0
$877$	−2.70184	−0.0912348	−0.0456174	−	0.998959i	$-0.514526\pi$
$877$	−2.70184	−0.0912348	−0.0456174	+	0.998959i	$0.514526\pi$
$878$	45.5486	1.53719
$879$	0	0
$880$	0	0
$881$	−21.5020	−0.724420	−0.362210	−	0.932097i	$-0.617978\pi$
$881$	−21.5020	−0.724420	−0.362210	+	0.932097i	$0.617978\pi$
$882$	0	0
$883$	15.4417	0.519653	0.259827	−	0.965655i	$-0.416335\pi$
$883$	15.4417	0.519653	0.259827	+	0.965655i	$0.416335\pi$
$884$	26.6408	0.896026
$885$	0	0
$886$	30.6424	1.02945
$887$	−41.1908	−1.38305	−0.691526	−	0.722352i	$-0.743061\pi$
$887$	−41.1908	−1.38305	−0.691526	+	0.722352i	$0.743061\pi$
$888$	0	0
$889$	−41.0379	−1.37637
$890$	0	0
$891$	0	0
$892$	27.4576	0.919347
$893$	3.07325	0.102842
$894$	0	0
$895$	0	0
$896$	58.2564	1.94621
$897$	0	0
$898$	92.0679	3.07235
$899$	−5.13050	−0.171112
$900$	0	0
$901$	−1.72350	−0.0574181
$902$	107.574	3.58182
$903$	0	0
$904$	25.8290	0.859061
$905$	0	0
$906$	0	0
$907$	−36.5292	−1.21293	−0.606466	−	0.795109i	$-0.707413\pi$
$907$	−36.5292	−1.21293	−0.606466	+	0.795109i	$0.707413\pi$
$908$	−13.9480	−0.462880
$909$	0	0
$910$	0	0
$911$	32.2943	1.06996	0.534979	−	0.844865i	$-0.320320\pi$
$911$	32.2943	1.06996	0.534979	+	0.844865i	$0.320320\pi$
$912$	0	0
$913$	−14.6463	−0.484723
$914$	37.3805	1.23644
$915$	0	0
$916$	78.8787	2.60623
$917$	7.10990	0.234789
$918$	0	0
$919$	9.06037	0.298874	0.149437	−	0.988771i	$-0.452254\pi$
$919$	9.06037	0.298874	0.149437	+	0.988771i	$0.452254\pi$
$920$	0	0
$921$	0	0
$922$	6.05048	0.199262
$923$	−23.5190	−0.774139
$924$	0	0
$925$	0	0
$926$	−30.5045	−1.00244
$927$	0	0
$928$	34.1051	1.11956
$929$	23.6578	0.776188	0.388094	−	0.921620i	$-0.373134\pi$
$929$	23.6578	0.776188	0.388094	+	0.921620i	$0.373134\pi$
$930$	0	0
$931$	18.7387	0.614138
$932$	14.3651	0.470543
$933$	0	0
$934$	−61.7463	−2.02040
$935$	0	0
$936$	0	0
$937$	35.7122	1.16667	0.583334	−	0.812232i	$-0.301748\pi$
$937$	35.7122	1.16667	0.583334	+	0.812232i	$0.301748\pi$
$938$	−102.637	−3.35123
$939$	0	0
$940$	0	0
$941$	41.9221	1.36662	0.683311	−	0.730128i	$-0.260540\pi$
$941$	41.9221	1.36662	0.683311	+	0.730128i	$0.260540\pi$
$942$	0	0
$943$	−57.4385	−1.87045
$944$	−7.58207	−0.246775
$945$	0	0
$946$	59.0972	1.92142
$947$	−42.0893	−1.36772	−0.683859	−	0.729614i	$-0.739700\pi$
$947$	−42.0893	−1.36772	−0.683859	+	0.729614i	$0.739700\pi$
$948$	0	0
$949$	−26.3886	−0.856611
$950$	0	0
$951$	0	0
$952$	27.6348	0.895650
$953$	−34.6630	−1.12284	−0.561422	−	0.827530i	$-0.689746\pi$
$953$	−34.6630	−1.12284	−0.561422	+	0.827530i	$0.689746\pi$
$954$	0	0
$955$	0	0
$956$	−2.03347	−0.0657670
$957$	0	0
$958$	12.1325	0.391984
$959$	−60.7736	−1.96248
$960$	0	0
$961$	1.00000	0.0322581
$962$	−47.3158	−1.52552
$963$	0	0
$964$	12.5060	0.402790
$965$	0	0
$966$	0	0
$967$	−25.6742	−0.825628	−0.412814	−	0.910815i	$-0.635454\pi$
$967$	−25.6742	−0.825628	−0.412814	+	0.910815i	$0.635454\pi$
$968$	−32.4465	−1.04287
$969$	0	0
$970$	0	0
$971$	−22.5664	−0.724192	−0.362096	−	0.932141i	$-0.617939\pi$
$971$	−22.5664	−0.724192	−0.362096	+	0.932141i	$0.617939\pi$
$972$	0	0
$973$	−82.3025	−2.63850
$974$	−2.77619	−0.0889548
$975$	0	0
$976$	−0.560741	−0.0179489
$977$	−19.5130	−0.624275	−0.312138	−	0.950037i	$-0.601045\pi$
$977$	−19.5130	−0.624275	−0.312138	+	0.950037i	$0.601045\pi$
$978$	0	0
$979$	−38.4965	−1.23035
$980$	0	0
$981$	0	0
$982$	42.6530	1.36111
$983$	39.3717	1.25576	0.627882	−	0.778309i	$-0.283922\pi$
$983$	39.3717	1.25576	0.627882	+	0.778309i	$0.283922\pi$
$984$	0	0
$985$	0	0
$986$	37.9806	1.20955
$987$	0	0
$988$	−23.2157	−0.738589
$989$	−31.5546	−1.00338
$990$	0	0
$991$	28.1540	0.894342	0.447171	−	0.894448i	$-0.352432\pi$
$991$	28.1540	0.894342	0.447171	+	0.894448i	$0.352432\pi$
$992$	−6.64753	−0.211059
$993$	0	0
$994$	−72.4318	−2.29740
$995$	0	0
$996$	0	0
$997$	−38.7855	−1.22835	−0.614175	−	0.789170i	$-0.710511\pi$
$997$	−38.7855	−1.22835	−0.614175	+	0.789170i	$0.710511\pi$
$998$	−8.54787	−0.270578
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bp.1.2		5
3.2	odd	2		775.2.a.k.1.4	yes	5
5.4	even	2		6975.2.a.by.1.4		5
15.2	even	4		775.2.b.g.249.9		10
15.8	even	4		775.2.b.g.249.2		10
15.14	odd	2		775.2.a.h.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.h.1.2	✓	5	15.14	odd	2
775.2.a.k.1.4	yes	5	3.2	odd	2
775.2.b.g.249.2		10	15.8	even	4
775.2.b.g.249.9		10	15.2	even	4
6975.2.a.bp.1.2		5	1.1	even	1	trivial
6975.2.a.by.1.4		5	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)

\(f(q)\)	\(=\)	\(q-2.23959 q^{2} +3.01578 q^{4} +3.67497 q^{7} -2.27494 q^{8} -5.02619 q^{11} -2.67247 q^{13} -8.23043 q^{14} -0.936618 q^{16} -3.30547 q^{17} +2.88050 q^{19} +11.2566 q^{22} -6.01040 q^{23} +5.98526 q^{26} +11.0829 q^{28} +5.13050 q^{29} -1.00000 q^{31} +6.64753 q^{32} +7.40291 q^{34} -7.90540 q^{37} -6.45115 q^{38} +9.55651 q^{41} +5.25000 q^{43} -15.1579 q^{44} +13.4609 q^{46} +1.06691 q^{47} +6.50538 q^{49} -8.05960 q^{52} +0.521408 q^{53} -8.36034 q^{56} -11.4902 q^{58} +8.09515 q^{59} +0.598687 q^{61} +2.23959 q^{62} -13.0145 q^{64} +12.4705 q^{67} -9.96858 q^{68} +8.80048 q^{71} +9.87423 q^{73} +17.7049 q^{74} +8.68697 q^{76} -18.4711 q^{77} -2.40291 q^{79} -21.4027 q^{82} +2.91400 q^{83} -11.7579 q^{86} +11.4343 q^{88} +7.65918 q^{89} -9.82125 q^{91} -18.1261 q^{92} -2.38946 q^{94} -14.5224 q^{97} -14.5694 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 6 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} + 4 q^{13} - 2 q^{14} + 4 q^{16} - 19 q^{17} + 8 q^{19} - 10 q^{22} - 12 q^{23} + 16 q^{26} + 6 q^{28} - 6 q^{29} - 5 q^{31} - 7 q^{32} + 31 q^{34}+ \cdots - 10 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 6 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 + 4 * q^13 - 2 * q^14 + 4 * q^16 - 19 * q^17 + 8 * q^19 - 10 * q^22 - 12 * q^23 + 16 * q^26 + 6 * q^28 - 6 * q^29 - 5 * q^31 - 7 * q^32 + 31 * q^34 + 16 * q^37 - 14 * q^38 + 2 * q^41 + q^43 + 4 * q^44 - 11 * q^46 - 8 * q^47 - 9 * q^49 - 26 * q^52 - 3 * q^53 + 9 * q^56 + 5 * q^58 + 4 * q^59 - 5 * q^61 + 4 * q^62 - 5 * q^64 + 13 * q^67 - 30 * q^68 - 6 * q^71 + 19 * q^73 - 4 * q^74 - 5 * q^76 - 23 * q^77 - 6 * q^79 - 27 * q^82 - 18 * q^83 + 7 * q^86 - q^88 + 31 * q^89 + 8 * q^91 + 16 * q^92 - 14 * q^94 - q^97 - 10 * q^98

Introduction

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