LMFDB - Embedded newform 961.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [961,2,Mod(1,961)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(961, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("961.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$7.67362363425$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36147 q^{2} +3.57653 q^{4} +3.93800 q^{5} -0.784934 q^{7} -3.72294 q^{8} -3.00000 q^{9} -9.29947 q^{10} +1.85360 q^{14} +1.63853 q^{16} +7.08441 q^{18} +8.66094 q^{19} +14.0844 q^{20} +10.5079 q^{25} -2.80734 q^{28} +3.57653 q^{32} -3.09107 q^{35} -10.7296 q^{36} -20.4525 q^{38} -14.6609 q^{40} +2.36814 q^{41} -11.8140 q^{45} +8.00000 q^{47} -6.38388 q^{49} -24.8140 q^{50} +2.92226 q^{56} +10.2308 q^{59} +2.35480 q^{63} -11.7229 q^{64} -12.0000 q^{67} +7.29947 q^{70} +16.5369 q^{71} +11.1688 q^{72} +30.9762 q^{76} +6.45254 q^{80} +9.00000 q^{81} -5.59228 q^{82} +27.8984 q^{90} -18.8918 q^{94} +34.1068 q^{95} +10.2441 q^{97} +15.0753 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 3 q^{8} - 9 q^{9} - 9 q^{10} + 15 q^{14} + 12 q^{16} + 21 q^{20} + 15 q^{25} + 27 q^{28} + 6 q^{32} + 12 q^{35} - 18 q^{36} - 33 q^{38} - 18 q^{40} + 24 q^{47} + 21 q^{49} - 39 q^{50} + 30 q^{56}+ \cdots + 57 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 + 3 * q^8 - 9 * q^9 - 9 * q^10 + 15 * q^14 + 12 * q^16 + 21 * q^20 + 15 * q^25 + 27 * q^28 + 6 * q^32 + 12 * q^35 - 18 * q^36 - 33 * q^38 - 18 * q^40 + 24 * q^47 + 21 * q^49 - 39 * q^50 + 30 * q^56 - 21 * q^64 - 36 * q^67 + 3 * q^70 - 9 * q^72 + 15 * q^76 - 9 * q^80 + 27 * q^81 + 21 * q^82 + 27 * q^90 + 48 * q^95 + 57 * 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 $

$2$	−2.36147	−1.66981	−0.834905	−	0.550394i	$-0.814478\pi$
$2$	−2.36147	−1.66981	−0.834905	+	0.550394i	$0.814478\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	3.57653	1.78827
$5$	3.93800	1.76113	0.880564	−	0.473927i	$-0.157164\pi$
$5$	3.93800	1.76113	0.880564	+	0.473927i	$0.157164\pi$
$6$	0	0
$7$	−0.784934	−0.296677	−0.148339	−	0.988937i	$-0.547393\pi$
$7$	−0.784934	−0.296677	−0.148339	+	0.988937i	$0.547393\pi$
$8$	−3.72294	−1.31626
$9$	−3.00000	−1.00000
$10$	−9.29947	−2.94075
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.85360	0.495395
$15$	0	0
$16$	1.63853	0.409633
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	7.08441	1.66981
$19$	8.66094	1.98696	0.993478	−	0.114023i	$-0.0363739\pi$
$19$	8.66094	1.98696	0.993478	+	0.114023i	$0.0363739\pi$
$20$	14.0844	3.14937
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	10.5079	2.10157
$26$	0	0
$27$	0	0
$28$	−2.80734	−0.530538
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	3.57653	0.632248
$33$	0	0
$34$	0	0
$35$	−3.09107	−0.522487
$36$	−10.7296	−1.78827
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−20.4525	−3.31784
$39$	0	0
$40$	−14.6609	−2.31810
$41$	2.36814	0.369841	0.184920	−	0.982754i	$-0.440797\pi$
$41$	2.36814	0.369841	0.184920	+	0.982754i	$0.440797\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−11.8140	−1.76113
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.38388	−0.911983
$50$	−24.8140	−3.50923
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	2.92226	0.390503
$57$	0	0
$58$	0	0
$59$	10.2308	1.33194	0.665969	−	0.745979i	$-0.268018\pi$
$59$	10.2308	1.33194	0.665969	+	0.745979i	$0.268018\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	2.35480	0.296677
$64$	−11.7229	−1.46537
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	7.29947	0.872454
$71$	16.5369	1.96257	0.981287	−	0.192549i	$-0.0616755\pi$
$71$	16.5369	1.96257	0.981287	+	0.192549i	$0.0616755\pi$
$72$	11.1688	1.31626
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	30.9762	3.55321
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	6.45254	0.721416
$81$	9.00000	1.00000
$82$	−5.59228	−0.617564
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	27.8984	2.94075
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−18.8918	−1.94854
$95$	34.1068	3.49929
$96$	0	0
$97$	10.2441	1.04014	0.520068	−	0.854125i	$-0.325907\pi$
$97$	10.2441	1.04014	0.520068	+	0.854125i	$0.325907\pi$
$98$	15.0753	1.52284
$99$	0	0
$100$	37.5818	3.75818
$101$	7.07774	0.704261	0.352131	−	0.935951i	$-0.385457\pi$
$101$	7.07774	0.704261	0.352131	+	0.935951i	$0.385457\pi$
$102$	0	0
$103$	−14.9671	−1.47475	−0.737375	−	0.675483i	$-0.763935\pi$
$103$	−14.9671	−1.47475	−0.737375	+	0.675483i	$0.763935\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.52120	0.533755	0.266878	−	0.963730i	$-0.414008\pi$
$107$	5.52120	0.533755	0.266878	+	0.963730i	$0.414008\pi$
$108$	0	0
$109$	14.9537	1.43231	0.716155	−	0.697942i	$-0.245901\pi$
$109$	14.9537	1.43231	0.716155	+	0.697942i	$0.245901\pi$
$110$	0	0
$111$	0	0
$112$	−1.28614	−0.121529
$113$	−21.2599	−1.99996	−0.999981	−	0.00618051i	$-0.998033\pi$
$113$	−21.2599	−1.99996	−0.999981	+	0.00618051i	$0.998033\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−24.1597	−2.22408
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	21.6900	1.94001
$126$	−5.56079	−0.495395
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	20.5303	1.81464
$129$	0	0
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	−6.79827	−0.589485
$134$	28.3376	2.44800
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−11.0553	−0.934346
$141$	0	0
$142$	−39.0515	−3.27713
$143$	0	0
$144$	−4.91559	−0.409633
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−32.2441	−2.61535
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−22.8298	−1.82201	−0.911006	−	0.412392i	$-0.864693\pi$
$157$	−22.8298	−1.82201	−0.911006	+	0.412392i	$0.864693\pi$
$158$	0	0
$159$	0	0
$160$	14.0844	1.11347
$161$	0	0
$162$	−21.2532	−1.66981
$163$	24.4130	1.91217	0.956085	−	0.293089i	$-0.0946833\pi$
$163$	24.4130	1.91217	0.956085	+	0.293089i	$0.0946833\pi$
$164$	8.46972	0.661374
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−25.9828	−1.98696
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−8.24799	−0.623489
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−42.2532	−3.14937
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	28.6123	2.08676
$189$	0	0
$190$	−80.5422	−5.84314
$191$	−13.3972	−0.969388	−0.484694	−	0.874684i	$-0.661069\pi$
$191$	−13.3972	−0.969388	−0.484694	+	0.874684i	$0.661069\pi$
$192$	0	0
$193$	−18.1201	−1.30432	−0.652158	−	0.758083i	$-0.726136\pi$
$193$	−18.1201	−1.30432	−0.652158	+	0.758083i	$0.726136\pi$
$194$	−24.1912	−1.73683
$195$	0	0
$196$	−22.8322	−1.63087
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−39.1201	−2.76621
$201$	0	0
$202$	−16.7139	−1.17598
$203$	0	0
$204$	0	0
$205$	9.32573	0.651337
$206$	35.3443	2.46255
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	22.8431	1.57258	0.786291	−	0.617856i	$-0.211998\pi$
$211$	22.8431	1.57258	0.786291	+	0.617856i	$0.211998\pi$
$212$	0	0
$213$	0	0
$214$	−13.0382	−0.891270
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−35.3128	−2.39168
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−2.80734	−0.187574
$225$	−31.5236	−2.10157
$226$	50.2046	3.33956
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.64761	−0.566524	−0.283262	−	0.959043i	$-0.591417\pi$
$233$	−8.64761	−0.566524	−0.283262	+	0.959043i	$0.591417\pi$
$234$	0	0
$235$	31.5040	2.05510
$236$	36.5908	2.38186
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	25.9762	1.66981
$243$	0	0
$244$	0	0
$245$	−25.1397	−1.60612
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−51.2203	−3.23946
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	8.42203	0.530538
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−25.0357	−1.56473
$257$	−24.3996	−1.52201	−0.761003	−	0.648748i	$-0.775293\pi$
$257$	−24.3996	−1.52201	−0.761003	+	0.648748i	$0.775293\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−47.2294	−2.91784
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	16.0539	0.984328
$267$	0	0
$268$	−42.9184	−2.62166
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	11.5079	0.687727
$281$	0.771601	0.0460298	0.0230149	−	0.999735i	$-0.492673\pi$
$281$	0.771601	0.0460298	0.0230149	+	0.999735i	$0.492673\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	59.1450	3.50961
$285$	0	0
$286$	0	0
$287$	−1.85883	−0.109723
$288$	−10.7296	−0.632248
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	40.2890	2.34571
$296$	0	0
$297$	0	0
$298$	−23.6147	−1.36796
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	14.1912	0.813922
$305$	0	0
$306$	0	0
$307$	−13.3705	−0.763097	−0.381549	−	0.924349i	$-0.624609\pi$
$307$	−13.3705	−0.763097	−0.381549	+	0.924349i	$0.624609\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.2466	−1.20478	−0.602391	−	0.798201i	$-0.705785\pi$
$311$	−21.2466	−1.20478	−0.602391	+	0.798201i	$0.705785\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	53.9118	3.04242
$315$	9.27322	0.522487
$316$	0	0
$317$	16.5503	0.929556	0.464778	−	0.885427i	$-0.346134\pi$
$317$	16.5503	0.929556	0.464778	+	0.885427i	$0.346134\pi$
$318$	0	0
$319$	0	0
$320$	−46.1650	−2.58070
$321$	0	0
$322$	0	0
$323$	0	0
$324$	32.1888	1.78827
$325$	0	0
$326$	−57.6504	−3.19296
$327$	0	0
$328$	−8.81642	−0.486805
$329$	−6.27947	−0.346199
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−47.2560	−2.58187
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	30.6991	1.66981
$339$	0	0
$340$	0	0
$341$	0	0
$342$	61.3576	3.31784
$343$	10.5055	0.567242
$344$	0	0
$345$	0	0
$346$	−33.0606	−1.77735
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	19.4774	1.04111
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	65.1226	3.45635
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.7191	−1.62129	−0.810646	−	0.585537i	$-0.800884\pi$
$359$	−30.7191	−1.62129	−0.810646	+	0.585537i	$0.800884\pi$
$360$	43.9828	2.31810
$361$	56.0119	2.94799
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−7.10441	−0.369841
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−38.5818	−1.99769	−0.998844	−	0.0480672i	$-0.984694\pi$
$373$	−38.5818	−1.99769	−0.998844	+	0.0480672i	$0.984694\pi$
$374$	0	0
$375$	0	0
$376$	−29.7835	−1.53597
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	121.984	6.25766
$381$	0	0
$382$	31.6371	1.61869
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	42.7902	2.17796
$387$	0	0
$388$	36.6385	1.86004
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	23.7668	1.20040
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.0935	−0.908086	−0.454043	−	0.890980i	$-0.650019\pi$
$397$	−18.0935	−0.908086	−0.454043	+	0.890980i	$0.650019\pi$
$398$	0	0
$399$	0	0
$400$	17.2175	0.860874
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	25.3138	1.25941
$405$	35.4420	1.76113
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	−22.0224	−1.08761
$411$	0	0
$412$	−53.5303	−2.63725
$413$	−8.03051	−0.395156
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.38147	−0.116342	−0.0581712	−	0.998307i	$-0.518527\pi$
$419$	−2.38147	−0.116342	−0.0581712	+	0.998307i	$0.518527\pi$
$420$	0	0
$421$	10.2175	0.497969	0.248985	−	0.968507i	$-0.419903\pi$
$421$	10.2175	0.497969	0.248985	+	0.968507i	$0.419903\pi$
$422$	−53.9432	−2.62592
$423$	−24.0000	−1.16692
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	19.7468	0.954497
$429$	0	0
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	53.4826	2.56135
$437$	0	0
$438$	0	0
$439$	−36.9986	−1.76585	−0.882923	−	0.469519i	$-0.844428\pi$
$439$	−36.9986	−1.76585	−0.882923	+	0.469519i	$0.844428\pi$
$440$	0	0
$441$	19.1516	0.911983
$442$	0	0
$443$	−41.7348	−1.98288	−0.991441	−	0.130556i	$-0.958324\pi$
$443$	−41.7348	−1.98288	−0.991441	+	0.130556i	$0.958324\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	9.20173	0.434741
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	74.4420	3.50923
$451$	0	0
$452$	−76.0367	−3.57647
$453$	0	0
$454$	−66.1211	−3.10322
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	20.4211	0.945987
$467$	−40.1650	−1.85861	−0.929307	−	0.369309i	$-0.879594\pi$
$467$	−40.1650	−1.85861	−0.929307	+	0.369309i	$0.879594\pi$
$468$	0	0
$469$	9.41921	0.434939
$470$	−74.3958	−3.43162
$471$	0	0
$472$	−38.0887	−1.75317
$473$	0	0
$474$	0	0
$475$	91.0081	4.17574
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.1492	−1.33186	−0.665931	−	0.746013i	$-0.731966\pi$
$479$	−29.1492	−1.33186	−0.665931	+	0.746013i	$0.731966\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−39.3419	−1.78827
$485$	40.3415	1.83181
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	59.3667	2.68191
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.9804	−0.582251
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	77.5751	3.46926
$501$	0	0
$502$	0	0
$503$	22.8164	1.01733	0.508667	−	0.860963i	$-0.330138\pi$
$503$	22.8164	1.01733	0.508667	+	0.860963i	$0.330138\pi$
$504$	−8.76678	−0.390503
$505$	27.8722	1.24029
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	18.0606	0.798172
$513$	0	0
$514$	57.6189	2.54146
$515$	−58.9404	−2.59722
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	71.5307	3.12483
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−30.6924	−1.33194
$532$	−24.3142	−1.05416
$533$	0	0
$534$	0	0
$535$	21.7425	0.940011
$536$	44.6753	1.92968
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	46.4578	1.99738	0.998688	−	0.0512107i	$-0.0163080\pi$
$541$	46.4578	1.99738	0.998688	+	0.0512107i	$0.0163080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	58.8879	2.52248
$546$	0	0
$547$	−38.5951	−1.65021	−0.825104	−	0.564981i	$-0.808883\pi$
$547$	−38.5951	−1.65021	−0.825104	+	0.564981i	$0.808883\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	−5.06482	−0.214028
$561$	0	0
$562$	−1.82211	−0.0768611
$563$	44.8746	1.89124	0.945619	−	0.325277i	$-0.105457\pi$
$563$	44.8746	1.89124	0.945619	+	0.325277i	$0.105457\pi$
$564$	0	0
$565$	−83.7215	−3.52219
$566$	−9.44588	−0.397040
$567$	−7.06441	−0.296677
$568$	−61.5660	−2.58325
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	4.38957	0.183217
$575$	0	0
$576$	35.1688	1.46537
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	40.1450	1.66981
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	61.3982	2.53634
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	−95.1411	−3.91690
$591$	0	0
$592$	0	0
$593$	−48.0276	−1.97226	−0.986129	−	0.165978i	$-0.946922\pi$
$593$	−48.0276	−1.97226	−0.986129	+	0.165978i	$0.946922\pi$
$594$	0	0
$595$	0	0
$596$	35.7653	1.46501
$597$	0	0
$598$	0	0
$599$	−38.5684	−1.57586	−0.787932	−	0.615763i	$-0.788848\pi$
$599$	−38.5684	−1.57586	−0.787932	+	0.615763i	$0.788848\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	36.0000	1.46603
$604$	0	0
$605$	−43.3180	−1.76113
$606$	0	0
$607$	−48.0000	−1.94826	−0.974130	−	0.225989i	$-0.927439\pi$
$607$	−48.0000	−1.94826	−0.974130	+	0.225989i	$0.927439\pi$
$608$	30.9762	1.25625
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	31.5741	1.27423
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	50.1731	2.01176
$623$	0	0
$624$	0	0
$625$	32.8760	1.31504
$626$	0	0
$627$	0	0
$628$	−81.6514	−3.25825
$629$	0	0
$630$	−21.8984	−0.872454
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	−39.0830	−1.55218
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−49.6108	−1.96257
$640$	80.8483	3.19581
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−33.5064	−1.31626
$649$	0	0
$650$	0	0
$651$	0	0
$652$	87.3138	3.41947
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	78.7601	3.07741
$656$	3.88026	0.151499
$657$	0	0
$658$	14.8288	0.578086
$659$	0.811601	0.0316155	0.0158077	−	0.999875i	$-0.494968\pi$
$659$	0.811601	0.0316155	0.0158077	+	0.999875i	$0.494968\pi$
$660$	0	0
$661$	−32.3023	−1.25641	−0.628207	−	0.778046i	$-0.716211\pi$
$661$	−32.3023	−1.25641	−0.628207	+	0.778046i	$0.716211\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−26.7716	−1.03816
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	111.594	4.31124
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	−46.4950	−1.78827
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−8.04098	−0.308584
$680$	0	0
$681$	0	0
$682$	0	0
$683$	37.0252	1.41673	0.708366	−	0.705846i	$-0.249433\pi$
$683$	37.0252	1.41673	0.708366	+	0.705846i	$0.249433\pi$
$684$	−92.9285	−3.55321
$685$	0	0
$686$	−24.8083	−0.947186
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−18.1335	−0.689830	−0.344915	−	0.938634i	$-0.612092\pi$
$691$	−18.1335	−0.689830	−0.344915	+	0.938634i	$0.612092\pi$
$692$	50.0715	1.90343
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	70.8441	2.68149
$699$	0	0
$700$	−29.4992	−1.11497
$701$	49.5975	1.87327	0.936636	−	0.350304i	$-0.113922\pi$
$701$	49.5975	1.87327	0.936636	+	0.350304i	$0.113922\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.55556	−0.208938
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	−153.785	−5.77144
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	72.5422	2.70725
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−19.3576	−0.721416
$721$	11.7482	0.437525
$722$	−132.270	−4.92259
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−52.7506	−1.95641	−0.978205	−	0.207640i	$-0.933422\pi$
$727$	−52.7506	−1.95641	−0.978205	+	0.207640i	$0.933422\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.4105	0.495330	0.247665	−	0.968846i	$-0.420337\pi$
$733$	13.4105	0.495330	0.247665	+	0.968846i	$0.420337\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	16.7768	0.617564
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	39.3800	1.44277
$746$	91.1096	3.33576
$747$	0	0
$748$	0	0
$749$	−4.33378	−0.158353
$750$	0	0
$751$	−24.3863	−0.889868	−0.444934	−	0.895563i	$-0.646773\pi$
$751$	−24.3863	−0.889868	−0.444934	+	0.895563i	$0.646773\pi$
$752$	13.1082	0.478009
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−47.2294	−1.71545
$759$	0	0
$760$	−126.978	−4.60596
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−11.7377	−0.424933
$764$	−47.9156	−1.73353
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−52.7639	−1.90272	−0.951358	−	0.308089i	$-0.900311\pi$
$769$	−52.7639	−1.90272	−0.951358	+	0.308089i	$0.900311\pi$
$770$	0	0
$771$	0	0
$772$	−64.8073	−2.33247
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	−38.1383	−1.36909
$777$	0	0
$778$	0	0
$779$	20.5103	0.734857
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−10.4602	−0.373578
$785$	−89.9036	−3.20880
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.6876	0.593343
$792$	0	0
$793$	0	0
$794$	42.7272	1.51633
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	37.5818	1.32872
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−26.3500	−0.926989
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	−83.6953	−2.94075
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	96.1383	3.36758
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	33.3538	1.16476
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	55.7215	1.94115
$825$	0	0
$826$	18.9638	0.659835
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	5.62376	0.194270
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−24.1282	−0.831514
$843$	0	0
$844$	81.6991	2.81220
$845$	−51.1940	−1.76113
$846$	56.6753	1.94854
$847$	8.63427	0.296677
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	−102.320	−3.49929
$856$	−20.5551	−0.702559
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	−94.4588	−3.21728
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	55.1320	1.87455
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−55.6719	−1.88529
$873$	−30.7324	−1.04014
$874$	0	0
$875$	−17.0252	−0.575558
$876$	0	0
$877$	57.4735	1.94074	0.970371	−	0.241618i	$-0.0776781\pi$
$877$	57.4735	1.94074	0.970371	+	0.241618i	$0.0776781\pi$
$878$	87.3710	2.94863
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−45.2260	−1.52284
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	98.5555	3.31104
$887$	−26.0095	−0.873313	−0.436657	−	0.899628i	$-0.643838\pi$
$887$	−26.0095	−0.873313	−0.436657	+	0.899628i	$0.643838\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	69.2875	2.31862
$894$	0	0
$895$	0	0
$896$	−16.1149	−0.538362
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−112.745	−3.75818
$901$	0	0
$902$	0	0
$903$	0	0
$904$	79.1492	2.63246
$905$	0	0
$906$	0	0
$907$	48.0143	1.59429	0.797144	−	0.603789i	$-0.206343\pi$
$907$	48.0143	1.59429	0.797144	+	0.603789i	$0.206343\pi$
$908$	100.143	3.32336
$909$	−21.2332	−0.704261
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.6987	−0.518416
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	44.9012	1.47475
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−55.2904	−1.81207
$932$	−30.9285	−1.01310
$933$	0	0
$934$	94.8483	3.10353
$935$	0	0
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	−22.2432	−0.726265
$939$	0	0
$940$	112.675	3.67506
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	16.7635	0.545605
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	−214.913	−6.97269
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−52.7583	−1.70722
$956$	0	0
$957$	0	0
$958$	68.8350	2.22396
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	−16.5636	−0.533755
$964$	0	0
$965$	−71.3572	−2.29707
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	40.9523	1.31626
$969$	0	0
$970$	−95.2651	−3.05878
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.6242	1.58762	0.793809	−	0.608167i	$-0.208095\pi$
$977$	49.6242	1.58762	0.793809	+	0.608167i	$0.208095\pi$
$978$	0	0
$979$	0	0
$980$	−89.9131	−2.87217
$981$	−44.8612	−1.43231
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	30.6528	0.972249
$995$	0	0
$996$	0	0
$997$	62.2098	1.97020	0.985102	−	0.171972i	$-0.0550138\pi$
$997$	62.2098	1.97020	0.985102	+	0.171972i	$0.0550138\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.2.a.g.1.1	✓	3
3.2	odd	2		8649.2.a.q.1.3		3
31.2	even	5		961.2.d.m.531.1		12
31.3	odd	30		961.2.g.u.846.1		24
31.4	even	5		961.2.d.m.388.3		12
31.5	even	3		961.2.c.g.521.1		6
31.6	odd	6		961.2.c.g.439.1		6
31.7	even	15		961.2.g.u.235.3		24
31.8	even	5		961.2.d.m.374.3		12
31.9	even	15		961.2.g.u.732.3		24
31.10	even	15		961.2.g.u.844.1		24
31.11	odd	30		961.2.g.u.338.3		24
31.12	odd	30		961.2.g.u.547.1		24
31.13	odd	30		961.2.g.u.448.1		24
31.14	even	15		961.2.g.u.816.3		24
31.15	odd	10		961.2.d.m.628.1		12
31.16	even	5		961.2.d.m.628.1		12
31.17	odd	30		961.2.g.u.816.3		24
31.18	even	15		961.2.g.u.448.1		24
31.19	even	15		961.2.g.u.547.1		24
31.20	even	15		961.2.g.u.338.3		24
31.21	odd	30		961.2.g.u.844.1		24
31.22	odd	30		961.2.g.u.732.3		24
31.23	odd	10		961.2.d.m.374.3		12
31.24	odd	30		961.2.g.u.235.3		24
31.25	even	3		961.2.c.g.439.1		6
31.26	odd	6		961.2.c.g.521.1		6
31.27	odd	10		961.2.d.m.388.3		12
31.28	even	15		961.2.g.u.846.1		24
31.29	odd	10		961.2.d.m.531.1		12
31.30	odd	2	CM	961.2.a.g.1.1	✓	3
93.92	even	2		8649.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.g.1.1	✓	3	1.1	even	1	trivial
961.2.a.g.1.1	✓	3	31.30	odd	2	CM
961.2.c.g.439.1		6	31.6	odd	6
961.2.c.g.439.1		6	31.25	even	3
961.2.c.g.521.1		6	31.5	even	3
961.2.c.g.521.1		6	31.26	odd	6
961.2.d.m.374.3		12	31.8	even	5
961.2.d.m.374.3		12	31.23	odd	10
961.2.d.m.388.3		12	31.4	even	5
961.2.d.m.388.3		12	31.27	odd	10
961.2.d.m.531.1		12	31.2	even	5
961.2.d.m.531.1		12	31.29	odd	10
961.2.d.m.628.1		12	31.15	odd	10
961.2.d.m.628.1		12	31.16	even	5
961.2.g.u.235.3		24	31.7	even	15
961.2.g.u.235.3		24	31.24	odd	30
961.2.g.u.338.3		24	31.11	odd	30
961.2.g.u.338.3		24	31.20	even	15
961.2.g.u.448.1		24	31.13	odd	30
961.2.g.u.448.1		24	31.18	even	15
961.2.g.u.547.1		24	31.12	odd	30
961.2.g.u.547.1		24	31.19	even	15
961.2.g.u.732.3		24	31.9	even	15
961.2.g.u.732.3		24	31.22	odd	30
961.2.g.u.816.3		24	31.14	even	15
961.2.g.u.816.3		24	31.17	odd	30
961.2.g.u.844.1		24	31.10	even	15
961.2.g.u.844.1		24	31.21	odd	30
961.2.g.u.846.1		24	31.3	odd	30
961.2.g.u.846.1		24	31.28	even	15
8649.2.a.q.1.3		3	3.2	odd	2
8649.2.a.q.1.3		3	93.92	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36147 q^{2} +3.57653 q^{4} +3.93800 q^{5} -0.784934 q^{7} -3.72294 q^{8} -3.00000 q^{9} -9.29947 q^{10} +1.85360 q^{14} +1.63853 q^{16} +7.08441 q^{18} +8.66094 q^{19} +14.0844 q^{20} +10.5079 q^{25} -2.80734 q^{28} +3.57653 q^{32} -3.09107 q^{35} -10.7296 q^{36} -20.4525 q^{38} -14.6609 q^{40} +2.36814 q^{41} -11.8140 q^{45} +8.00000 q^{47} -6.38388 q^{49} -24.8140 q^{50} +2.92226 q^{56} +10.2308 q^{59} +2.35480 q^{63} -11.7229 q^{64} -12.0000 q^{67} +7.29947 q^{70} +16.5369 q^{71} +11.1688 q^{72} +30.9762 q^{76} +6.45254 q^{80} +9.00000 q^{81} -5.59228 q^{82} +27.8984 q^{90} -18.8918 q^{94} +34.1068 q^{95} +10.2441 q^{97} +15.0753 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 3 q^{8} - 9 q^{9} - 9 q^{10} + 15 q^{14} + 12 q^{16} + 21 q^{20} + 15 q^{25} + 27 q^{28} + 6 q^{32} + 12 q^{35} - 18 q^{36} - 33 q^{38} - 18 q^{40} + 24 q^{47} + 21 q^{49} - 39 q^{50} + 30 q^{56}+ \cdots + 57 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 + 3 * q^8 - 9 * q^9 - 9 * q^10 + 15 * q^14 + 12 * q^16 + 21 * q^20 + 15 * q^25 + 27 * q^28 + 6 * q^32 + 12 * q^35 - 18 * q^36 - 33 * q^38 - 18 * q^40 + 24 * q^47 + 21 * q^49 - 39 * q^50 + 30 * q^56 - 21 * q^64 - 36 * q^67 + 3 * q^70 - 9 * q^72 + 15 * q^76 - 9 * q^80 + 27 * q^81 + 21 * q^82 + 27 * q^90 + 48 * q^95 + 57 * q^98

Introduction

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Database

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