LMFDB - Embedded newform 976.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 122)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47283 q^{3} -3.58774 q^{5} -3.75698 q^{7} -0.830760 q^{9} +O(q^{10})$	$q-1.47283 q^{3} -3.58774 q^{5} -3.75698 q^{7} -0.830760 q^{9} +2.64207 q^{11} -0.642074 q^{13} +5.28415 q^{15} -6.22982 q^{17} -1.11491 q^{19} +5.53341 q^{21} -3.39905 q^{23} +7.87189 q^{25} +5.64207 q^{27} +6.06058 q^{29} -0.188687 q^{31} -3.89134 q^{33} +13.4791 q^{35} -7.93246 q^{37} +0.945668 q^{39} +11.2904 q^{41} +4.45963 q^{43} +2.98055 q^{45} -5.17548 q^{47} +7.11491 q^{49} +9.17548 q^{51} +12.0800 q^{53} -9.47908 q^{55} +1.64207 q^{57} +11.5877 q^{59} -1.00000 q^{61} +3.12115 q^{63} +2.30359 q^{65} -12.1560 q^{67} +5.00624 q^{69} -15.3447 q^{71} -2.93246 q^{73} -11.5940 q^{75} -9.92622 q^{77} +10.7632 q^{79} -5.81756 q^{81} +0.316798 q^{83} +22.3510 q^{85} -8.92622 q^{87} -15.4053 q^{89} +2.41226 q^{91} +0.277904 q^{93} +4.00000 q^{95} +0.756981 q^{97} -2.19493 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9	$ 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} + 7 q^{11} - q^{13} + 14 q^{15} - 6 q^{17} + 3 q^{19} - 6 q^{21} - 2 q^{23} + 10 q^{25} + 16 q^{27} + q^{29} + 3 q^{31} + 10 q^{33} + 7 q^{35} + 7 q^{37} - 8 q^{39} + 4 q^{41} - 12 q^{43} + 17 q^{45} + 8 q^{47} + 15 q^{49} + 4 q^{51} + 11 q^{53} + 5 q^{55} + 4 q^{57} + 23 q^{59} - 3 q^{61} - 25 q^{63} - 3 q^{65} - 21 q^{67} - 13 q^{69} - 27 q^{71} + 22 q^{73} + 5 q^{75} - 27 q^{77} - 3 q^{79} + 7 q^{81} + 11 q^{83} + 20 q^{85} - 24 q^{87} - 10 q^{89} + 19 q^{91} + 27 q^{93} + 12 q^{95} - 5 q^{97} + 25 q^{99}+O(q^{100}) $ 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9 + 7 * q^11 - q^13 + 14 * q^15 - 6 * q^17 + 3 * q^19 - 6 * q^21 - 2 * q^23 + 10 * q^25 + 16 * q^27 + q^29 + 3 * q^31 + 10 * q^33 + 7 * q^35 + 7 * q^37 - 8 * q^39 + 4 * q^41 - 12 * q^43 + 17 * q^45 + 8 * q^47 + 15 * q^49 + 4 * q^51 + 11 * q^53 + 5 * q^55 + 4 * q^57 + 23 * q^59 - 3 * q^61 - 25 * q^63 - 3 * q^65 - 21 * q^67 - 13 * q^69 - 27 * q^71 + 22 * q^73 + 5 * q^75 - 27 * q^77 - 3 * q^79 + 7 * q^81 + 11 * q^83 + 20 * q^85 - 24 * q^87 - 10 * q^89 + 19 * q^91 + 27 * q^93 + 12 * q^95 - 5 * q^97 + 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.47283	−0.850341	−0.425171	−	0.905113i	$-0.639786\pi$
$3$	−1.47283	−0.850341	−0.425171	+	0.905113i	$0.639786\pi$
$4$	0	0
$5$	−3.58774	−1.60449	−0.802243	−	0.596997i	$-0.796360\pi$
$5$	−3.58774	−1.60449	−0.802243	+	0.596997i	$0.796360\pi$
$6$	0	0
$7$	−3.75698	−1.42001	−0.710003	−	0.704199i	$-0.751306\pi$
$7$	−3.75698	−1.42001	−0.710003	+	0.704199i	$0.751306\pi$
$8$	0	0
$9$	−0.830760	−0.276920
$10$	0	0
$11$	2.64207	0.796615	0.398308	−	0.917252i	$-0.369598\pi$
$11$	2.64207	0.796615	0.398308	+	0.917252i	$0.369598\pi$
$12$	0	0
$13$	−0.642074	−0.178079	−0.0890396	−	0.996028i	$-0.528380\pi$
$13$	−0.642074	−0.178079	−0.0890396	+	0.996028i	$0.528380\pi$
$14$	0	0
$15$	5.28415	1.36436
$16$	0	0
$17$	−6.22982	−1.51095	−0.755476	−	0.655176i	$-0.772594\pi$
$17$	−6.22982	−1.51095	−0.755476	+	0.655176i	$0.772594\pi$
$18$	0	0
$19$	−1.11491	−0.255777	−0.127889	−	0.991789i	$-0.540820\pi$
$19$	−1.11491	−0.255777	−0.127889	+	0.991789i	$0.540820\pi$
$20$	0	0
$21$	5.53341	1.20749
$22$	0	0
$23$	−3.39905	−0.708752	−0.354376	−	0.935103i	$-0.615307\pi$
$23$	−3.39905	−0.708752	−0.354376	+	0.935103i	$0.615307\pi$
$24$	0	0
$25$	7.87189	1.57438
$26$	0	0
$27$	5.64207	1.08582
$28$	0	0
$29$	6.06058	1.12542	0.562710	−	0.826654i	$-0.309759\pi$
$29$	6.06058	1.12542	0.562710	+	0.826654i	$0.309759\pi$
$30$	0	0
$31$	−0.188687	−0.0338891	−0.0169446	−	0.999856i	$-0.505394\pi$
$31$	−0.188687	−0.0338891	−0.0169446	+	0.999856i	$0.505394\pi$
$32$	0	0
$33$	−3.89134	−0.677395
$34$	0	0
$35$	13.4791	2.27838
$36$	0	0
$37$	−7.93246	−1.30409	−0.652045	−	0.758181i	$-0.726088\pi$
$37$	−7.93246	−1.30409	−0.652045	+	0.758181i	$0.726088\pi$
$38$	0	0
$39$	0.945668	0.151428
$40$	0	0
$41$	11.2904	1.76326	0.881631	−	0.471939i	$-0.156446\pi$
$41$	11.2904	1.76326	0.881631	+	0.471939i	$0.156446\pi$
$42$	0	0
$43$	4.45963	0.680087	0.340044	−	0.940410i	$-0.389558\pi$
$43$	4.45963	0.680087	0.340044	+	0.940410i	$0.389558\pi$
$44$	0	0
$45$	2.98055	0.444315
$46$	0	0
$47$	−5.17548	−0.754922	−0.377461	−	0.926026i	$-0.623203\pi$
$47$	−5.17548	−0.754922	−0.377461	+	0.926026i	$0.623203\pi$
$48$	0	0
$49$	7.11491	1.01642
$50$	0	0
$51$	9.17548	1.28482
$52$	0	0
$53$	12.0800	1.65932	0.829659	−	0.558270i	$-0.188535\pi$
$53$	12.0800	1.65932	0.829659	+	0.558270i	$0.188535\pi$
$54$	0	0
$55$	−9.47908	−1.27816
$56$	0	0
$57$	1.64207	0.217498
$58$	0	0
$59$	11.5877	1.50860	0.754298	−	0.656533i	$-0.227978\pi$
$59$	11.5877	1.50860	0.754298	+	0.656533i	$0.227978\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	3.12115	0.393228
$64$	0	0
$65$	2.30359	0.285726
$66$	0	0
$67$	−12.1560	−1.48510	−0.742548	−	0.669793i	$-0.766383\pi$
$67$	−12.1560	−1.48510	−0.742548	+	0.669793i	$0.766383\pi$
$68$	0	0
$69$	5.00624	0.602681
$70$	0	0
$71$	−15.3447	−1.82108	−0.910542	−	0.413417i	$-0.864335\pi$
$71$	−15.3447	−1.82108	−0.910542	+	0.413417i	$0.864335\pi$
$72$	0	0
$73$	−2.93246	−0.343219	−0.171609	−	0.985165i	$-0.554897\pi$
$73$	−2.93246	−0.343219	−0.171609	+	0.985165i	$0.554897\pi$
$74$	0	0
$75$	−11.5940	−1.33876
$76$	0	0
$77$	−9.92622	−1.13120
$78$	0	0
$79$	10.7632	1.21096	0.605479	−	0.795862i	$-0.292982\pi$
$79$	10.7632	1.21096	0.605479	+	0.795862i	$0.292982\pi$
$80$	0	0
$81$	−5.81756	−0.646395
$82$	0	0
$83$	0.316798	0.0347731	0.0173865	−	0.999849i	$-0.494465\pi$
$83$	0.316798	0.0347731	0.0173865	+	0.999849i	$0.494465\pi$
$84$	0	0
$85$	22.3510	2.42430
$86$	0	0
$87$	−8.92622	−0.956991
$88$	0	0
$89$	−15.4053	−1.63296	−0.816479	−	0.577375i	$-0.804077\pi$
$89$	−15.4053	−1.63296	−0.816479	+	0.577375i	$0.804077\pi$
$90$	0	0
$91$	2.41226	0.252873
$92$	0	0
$93$	0.277904	0.0288173
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	0.756981	0.0768598	0.0384299	−	0.999261i	$-0.487764\pi$
$97$	0.756981	0.0768598	0.0384299	+	0.999261i	$0.487764\pi$
$98$	0	0
$99$	−2.19493	−0.220599
$100$	0	0
$101$	−9.36417	−0.931770	−0.465885	−	0.884845i	$-0.654264\pi$
$101$	−9.36417	−0.931770	−0.465885	+	0.884845i	$0.654264\pi$
$102$	0	0
$103$	−7.66152	−0.754912	−0.377456	−	0.926028i	$-0.623201\pi$
$103$	−7.66152	−0.754912	−0.377456	+	0.926028i	$0.623201\pi$
$104$	0	0
$105$	−19.8524	−1.93740
$106$	0	0
$107$	16.2904	1.57485	0.787426	−	0.616410i	$-0.211413\pi$
$107$	16.2904	1.57485	0.787426	+	0.616410i	$0.211413\pi$
$108$	0	0
$109$	0.642074	0.0614995	0.0307498	−	0.999527i	$-0.490211\pi$
$109$	0.642074	0.0614995	0.0307498	+	0.999527i	$0.490211\pi$
$110$	0	0
$111$	11.6832	1.10892
$112$	0	0
$113$	9.98680	0.939479	0.469739	−	0.882805i	$-0.344348\pi$
$113$	9.98680	0.939479	0.469739	+	0.882805i	$0.344348\pi$
$114$	0	0
$115$	12.1949	1.13718
$116$	0	0
$117$	0.533409	0.0493137
$118$	0	0
$119$	23.4053	2.14556
$120$	0	0
$121$	−4.01945	−0.365404
$122$	0	0
$123$	−16.6289	−1.49937
$124$	0	0
$125$	−10.3036	−0.921581
$126$	0	0
$127$	10.1212	0.898107	0.449053	−	0.893505i	$-0.351761\pi$
$127$	10.1212	0.898107	0.449053	+	0.893505i	$0.351761\pi$
$128$	0	0
$129$	−6.56829	−0.578306
$130$	0	0
$131$	2.10866	0.184235	0.0921174	−	0.995748i	$-0.470636\pi$
$131$	2.10866	0.184235	0.0921174	+	0.995748i	$0.470636\pi$
$132$	0	0
$133$	4.18869	0.363205
$134$	0	0
$135$	−20.2423	−1.74218
$136$	0	0
$137$	−6.60095	−0.563957	−0.281978	−	0.959421i	$-0.590991\pi$
$137$	−6.60095	−0.563957	−0.281978	+	0.959421i	$0.590991\pi$
$138$	0	0
$139$	2.07378	0.175896	0.0879478	−	0.996125i	$-0.471969\pi$
$139$	2.07378	0.175896	0.0879478	+	0.996125i	$0.471969\pi$
$140$	0	0
$141$	7.62263	0.641941
$142$	0	0
$143$	−1.69641	−0.141861
$144$	0	0
$145$	−21.7438	−1.80572
$146$	0	0
$147$	−10.4791	−0.864300
$148$	0	0
$149$	13.5613	1.11099	0.555494	−	0.831521i	$-0.312529\pi$
$149$	13.5613	1.11099	0.555494	+	0.831521i	$0.312529\pi$
$150$	0	0
$151$	−9.20813	−0.749347	−0.374674	−	0.927157i	$-0.622245\pi$
$151$	−9.20813	−0.749347	−0.374674	+	0.927157i	$0.622245\pi$
$152$	0	0
$153$	5.17548	0.418413
$154$	0	0
$155$	0.676959	0.0543746
$156$	0	0
$157$	−9.11491	−0.727449	−0.363724	−	0.931507i	$-0.618495\pi$
$157$	−9.11491	−0.727449	−0.363724	+	0.931507i	$0.618495\pi$
$158$	0	0
$159$	−17.7919	−1.41099
$160$	0	0
$161$	12.7702	1.00643
$162$	0	0
$163$	5.93246	0.464666	0.232333	−	0.972636i	$-0.425364\pi$
$163$	5.93246	0.464666	0.232333	+	0.972636i	$0.425364\pi$
$164$	0	0
$165$	13.9611	1.08687
$166$	0	0
$167$	15.1491	1.17227	0.586135	−	0.810213i	$-0.300649\pi$
$167$	15.1491	1.17227	0.586135	+	0.810213i	$0.300649\pi$
$168$	0	0
$169$	−12.5877	−0.968288
$170$	0	0
$171$	0.926221	0.0708299
$172$	0	0
$173$	−12.2904	−0.934421	−0.467211	−	0.884146i	$-0.654741\pi$
$173$	−12.2904	−0.934421	−0.467211	+	0.884146i	$0.654741\pi$
$174$	0	0
$175$	−29.5745	−2.23562
$176$	0	0
$177$	−17.0668	−1.28282
$178$	0	0
$179$	5.09546	0.380853	0.190426	−	0.981701i	$-0.439013\pi$
$179$	5.09546	0.380853	0.190426	+	0.981701i	$0.439013\pi$
$180$	0	0
$181$	−2.08698	−0.155124	−0.0775621	−	0.996988i	$-0.524714\pi$
$181$	−2.08698	−0.155124	−0.0775621	+	0.996988i	$0.524714\pi$
$182$	0	0
$183$	1.47283	0.108875
$184$	0	0
$185$	28.4596	2.09239
$186$	0	0
$187$	−16.4596	−1.20365
$188$	0	0
$189$	−21.1972	−1.54187
$190$	0	0
$191$	6.11491	0.442459	0.221230	−	0.975222i	$-0.428993\pi$
$191$	6.11491	0.442459	0.221230	+	0.975222i	$0.428993\pi$
$192$	0	0
$193$	25.5529	1.83933	0.919667	−	0.392698i	$-0.128458\pi$
$193$	25.5529	1.83933	0.919667	+	0.392698i	$0.128458\pi$
$194$	0	0
$195$	−3.39281	−0.242964
$196$	0	0
$197$	5.84396	0.416365	0.208183	−	0.978090i	$-0.433245\pi$
$197$	5.84396	0.416365	0.208183	+	0.978090i	$0.433245\pi$
$198$	0	0
$199$	−2.08226	−0.147607	−0.0738036	−	0.997273i	$-0.523514\pi$
$199$	−2.08226	−0.147607	−0.0738036	+	0.997273i	$0.523514\pi$
$200$	0	0
$201$	17.9038	1.26284
$202$	0	0
$203$	−22.7695	−1.59810
$204$	0	0
$205$	−40.5070	−2.82913
$206$	0	0
$207$	2.82380	0.196268
$208$	0	0
$209$	−2.94567	−0.203756
$210$	0	0
$211$	−1.77018	−0.121865	−0.0609323	−	0.998142i	$-0.519407\pi$
$211$	−1.77018	−0.121865	−0.0609323	+	0.998142i	$0.519407\pi$
$212$	0	0
$213$	22.6002	1.54854
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	0.708892	0.0481227
$218$	0	0
$219$	4.31903	0.291853
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−20.5070	−1.37325	−0.686625	−	0.727012i	$-0.740909\pi$
$223$	−20.5070	−1.37325	−0.686625	+	0.727012i	$0.740909\pi$
$224$	0	0
$225$	−6.53965	−0.435977
$226$	0	0
$227$	−10.3859	−0.689333	−0.344667	−	0.938725i	$-0.612008\pi$
$227$	−10.3859	−0.689333	−0.344667	+	0.938725i	$0.612008\pi$
$228$	0	0
$229$	−12.4247	−0.821050	−0.410525	−	0.911849i	$-0.634655\pi$
$229$	−12.4247	−0.821050	−0.410525	+	0.911849i	$0.634655\pi$
$230$	0	0
$231$	14.6197	0.961904
$232$	0	0
$233$	17.4876	1.14565	0.572824	−	0.819678i	$-0.305848\pi$
$233$	17.4876	1.14565	0.572824	+	0.819678i	$0.305848\pi$
$234$	0	0
$235$	18.5683	1.21126
$236$	0	0
$237$	−15.8524	−1.02973
$238$	0	0
$239$	12.0125	0.777023	0.388512	−	0.921444i	$-0.372989\pi$
$239$	12.0125	0.777023	0.388512	+	0.921444i	$0.372989\pi$
$240$	0	0
$241$	−3.56606	−0.229710	−0.114855	−	0.993382i	$-0.536640\pi$
$241$	−3.56606	−0.229710	−0.114855	+	0.993382i	$0.536640\pi$
$242$	0	0
$243$	−8.35793	−0.536161
$244$	0	0
$245$	−25.5264	−1.63082
$246$	0	0
$247$	0.715853	0.0455486
$248$	0	0
$249$	−0.466591	−0.0295690
$250$	0	0
$251$	−2.19092	−0.138290	−0.0691449	−	0.997607i	$-0.522027\pi$
$251$	−2.19092	−0.138290	−0.0691449	+	0.997607i	$0.522027\pi$
$252$	0	0
$253$	−8.98055	−0.564603
$254$	0	0
$255$	−32.9193	−2.06148
$256$	0	0
$257$	6.85868	0.427833	0.213917	−	0.976852i	$-0.431378\pi$
$257$	6.85868	0.427833	0.213917	+	0.976852i	$0.431378\pi$
$258$	0	0
$259$	29.8021	1.85181
$260$	0	0
$261$	−5.03489	−0.311652
$262$	0	0
$263$	15.4876	0.955004	0.477502	−	0.878631i	$-0.341542\pi$
$263$	15.4876	0.955004	0.477502	+	0.878631i	$0.341542\pi$
$264$	0	0
$265$	−43.3400	−2.66235
$266$	0	0
$267$	22.6894	1.38857
$268$	0	0
$269$	6.25622	0.381449	0.190724	−	0.981644i	$-0.438916\pi$
$269$	6.25622	0.381449	0.190724	+	0.981644i	$0.438916\pi$
$270$	0	0
$271$	21.0279	1.27736	0.638678	−	0.769474i	$-0.279482\pi$
$271$	21.0279	1.27736	0.638678	+	0.769474i	$0.279482\pi$
$272$	0	0
$273$	−3.55286	−0.215029
$274$	0	0
$275$	20.7981	1.25417
$276$	0	0
$277$	6.06058	0.364145	0.182072	−	0.983285i	$-0.441719\pi$
$277$	6.06058	0.364145	0.182072	+	0.983285i	$0.441719\pi$
$278$	0	0
$279$	0.156753	0.00938458
$280$	0	0
$281$	20.4596	1.22052	0.610260	−	0.792201i	$-0.291065\pi$
$281$	20.4596	1.22052	0.610260	+	0.792201i	$0.291065\pi$
$282$	0	0
$283$	−23.3642	−1.38886	−0.694428	−	0.719562i	$-0.744342\pi$
$283$	−23.3642	−1.38886	−0.694428	+	0.719562i	$0.744342\pi$
$284$	0	0
$285$	−5.89134	−0.348973
$286$	0	0
$287$	−42.4178	−2.50384
$288$	0	0
$289$	21.8106	1.28298
$290$	0	0
$291$	−1.11491	−0.0653570
$292$	0	0
$293$	−2.45963	−0.143693	−0.0718466	−	0.997416i	$-0.522889\pi$
$293$	−2.45963	−0.143693	−0.0718466	+	0.997416i	$0.522889\pi$
$294$	0	0
$295$	−41.5738	−2.42052
$296$	0	0
$297$	14.9068	0.864979
$298$	0	0
$299$	2.18244	0.126214
$300$	0	0
$301$	−16.7547	−0.965728
$302$	0	0
$303$	13.7919	0.792322
$304$	0	0
$305$	3.58774	0.205433
$306$	0	0
$307$	−11.0194	−0.628913	−0.314457	−	0.949272i	$-0.601822\pi$
$307$	−11.0194	−0.628913	−0.314457	+	0.949272i	$0.601822\pi$
$308$	0	0
$309$	11.2841	0.641933
$310$	0	0
$311$	9.50076	0.538739	0.269369	−	0.963037i	$-0.413185\pi$
$311$	9.50076	0.538739	0.269369	+	0.963037i	$0.413185\pi$
$312$	0	0
$313$	0.568295	0.0321219	0.0160610	−	0.999871i	$-0.494887\pi$
$313$	0.568295	0.0321219	0.0160610	+	0.999871i	$0.494887\pi$
$314$	0	0
$315$	−11.1979	−0.630929
$316$	0	0
$317$	17.0668	0.958568	0.479284	−	0.877660i	$-0.340896\pi$
$317$	17.0668	0.958568	0.479284	+	0.877660i	$0.340896\pi$
$318$	0	0
$319$	16.0125	0.896527
$320$	0	0
$321$	−23.9930	−1.33916
$322$	0	0
$323$	6.94567	0.386467
$324$	0	0
$325$	−5.05433	−0.280364
$326$	0	0
$327$	−0.945668	−0.0522955
$328$	0	0
$329$	19.4442	1.07199
$330$	0	0
$331$	−0.680967	−0.0374294	−0.0187147	−	0.999825i	$-0.505957\pi$
$331$	−0.680967	−0.0374294	−0.0187147	+	0.999825i	$0.505957\pi$
$332$	0	0
$333$	6.58998	0.361128
$334$	0	0
$335$	43.6127	2.38282
$336$	0	0
$337$	21.3230	1.16154	0.580770	−	0.814068i	$-0.302752\pi$
$337$	21.3230	1.16154	0.580770	+	0.814068i	$0.302752\pi$
$338$	0	0
$339$	−14.7089	−0.798877
$340$	0	0
$341$	−0.498524	−0.0269966
$342$	0	0
$343$	−0.431705	−0.0233099
$344$	0	0
$345$	−17.9611	−0.966993
$346$	0	0
$347$	3.78267	0.203064	0.101532	−	0.994832i	$-0.467626\pi$
$347$	3.78267	0.203064	0.101532	+	0.994832i	$0.467626\pi$
$348$	0	0
$349$	25.4178	1.36058	0.680291	−	0.732942i	$-0.261853\pi$
$349$	25.4178	1.36058	0.680291	+	0.732942i	$0.261853\pi$
$350$	0	0
$351$	−3.62263	−0.193362
$352$	0	0
$353$	7.22758	0.384685	0.192343	−	0.981328i	$-0.438391\pi$
$353$	7.22758	0.384685	0.192343	+	0.981328i	$0.438391\pi$
$354$	0	0
$355$	55.0529	2.92190
$356$	0	0
$357$	−34.4721	−1.82446
$358$	0	0
$359$	−28.2640	−1.49172	−0.745858	−	0.666105i	$-0.767960\pi$
$359$	−28.2640	−1.49172	−0.745858	+	0.666105i	$0.767960\pi$
$360$	0	0
$361$	−17.7570	−0.934578
$362$	0	0
$363$	5.91998	0.310718
$364$	0	0
$365$	10.5209	0.550690
$366$	0	0
$367$	0.568295	0.0296647	0.0148324	−	0.999890i	$-0.495279\pi$
$367$	0.568295	0.0296647	0.0148324	+	0.999890i	$0.495279\pi$
$368$	0	0
$369$	−9.37961	−0.488283
$370$	0	0
$371$	−45.3844	−2.35624
$372$	0	0
$373$	−4.53965	−0.235054	−0.117527	−	0.993070i	$-0.537497\pi$
$373$	−4.53965	−0.235054	−0.117527	+	0.993070i	$0.537497\pi$
$374$	0	0
$375$	15.1755	0.783659
$376$	0	0
$377$	−3.89134	−0.200414
$378$	0	0
$379$	30.6414	1.57394	0.786970	−	0.616991i	$-0.211648\pi$
$379$	30.6414	1.57394	0.786970	+	0.616991i	$0.211648\pi$
$380$	0	0
$381$	−14.9068	−0.763697
$382$	0	0
$383$	2.35168	0.120165	0.0600827	−	0.998193i	$-0.480864\pi$
$383$	2.35168	0.120165	0.0600827	+	0.998193i	$0.480864\pi$
$384$	0	0
$385$	35.6127	1.81499
$386$	0	0
$387$	−3.70488	−0.188330
$388$	0	0
$389$	14.8587	0.753365	0.376683	−	0.926342i	$-0.377065\pi$
$389$	14.8587	0.753365	0.376683	+	0.926342i	$0.377065\pi$
$390$	0	0
$391$	21.1755	1.07089
$392$	0	0
$393$	−3.10571	−0.156662
$394$	0	0
$395$	−38.6157	−1.94297
$396$	0	0
$397$	35.1708	1.76517	0.882585	−	0.470153i	$-0.155801\pi$
$397$	35.1708	1.76517	0.882585	+	0.470153i	$0.155801\pi$
$398$	0	0
$399$	−6.16924	−0.308848
$400$	0	0
$401$	11.4053	0.569553	0.284777	−	0.958594i	$-0.408081\pi$
$401$	11.4053	0.569553	0.284777	+	0.958594i	$0.408081\pi$
$402$	0	0
$403$	0.121151	0.00603495
$404$	0	0
$405$	20.8719	1.03713
$406$	0	0
$407$	−20.9582	−1.03886
$408$	0	0
$409$	−3.59622	−0.177822	−0.0889108	−	0.996040i	$-0.528339\pi$
$409$	−3.59622	−0.177822	−0.0889108	+	0.996040i	$0.528339\pi$
$410$	0	0
$411$	9.72210	0.479556
$412$	0	0
$413$	−43.5349	−2.14221
$414$	0	0
$415$	−1.13659	−0.0557930
$416$	0	0
$417$	−3.05433	−0.149571
$418$	0	0
$419$	39.3914	1.92439	0.962197	−	0.272353i	$-0.0878019\pi$
$419$	39.3914	1.92439	0.962197	+	0.272353i	$0.0878019\pi$
$420$	0	0
$421$	−13.9130	−0.678079	−0.339039	−	0.940772i	$-0.610102\pi$
$421$	−13.9130	−0.678079	−0.339039	+	0.940772i	$0.610102\pi$
$422$	0	0
$423$	4.29959	0.209053
$424$	0	0
$425$	−49.0404	−2.37881
$426$	0	0
$427$	3.75698	0.181813
$428$	0	0
$429$	2.49852	0.120630
$430$	0	0
$431$	−15.5529	−0.749155	−0.374577	−	0.927196i	$-0.622212\pi$
$431$	−15.5529	−0.749155	−0.374577	+	0.927196i	$0.622212\pi$
$432$	0	0
$433$	−1.80908	−0.0869388	−0.0434694	−	0.999055i	$-0.513841\pi$
$433$	−1.80908	−0.0869388	−0.0434694	+	0.999055i	$0.513841\pi$
$434$	0	0
$435$	32.0250	1.53548
$436$	0	0
$437$	3.78963	0.181283
$438$	0	0
$439$	−15.0668	−0.719100	−0.359550	−	0.933126i	$-0.617070\pi$
$439$	−15.0668	−0.719100	−0.359550	+	0.933126i	$0.617070\pi$
$440$	0	0
$441$	−5.91078	−0.281466
$442$	0	0
$443$	30.2974	1.43947	0.719735	−	0.694249i	$-0.244263\pi$
$443$	30.2974	1.43947	0.719735	+	0.694249i	$0.244263\pi$
$444$	0	0
$445$	55.2702	2.62006
$446$	0	0
$447$	−19.9736	−0.944719
$448$	0	0
$449$	8.93942	0.421878	0.210939	−	0.977499i	$-0.432348\pi$
$449$	8.93942	0.421878	0.210939	+	0.977499i	$0.432348\pi$
$450$	0	0
$451$	29.8300	1.40464
$452$	0	0
$453$	13.5621	0.637201
$454$	0	0
$455$	−8.65456	−0.405732
$456$	0	0
$457$	−24.4596	−1.14417	−0.572087	−	0.820193i	$-0.693866\pi$
$457$	−24.4596	−1.14417	−0.572087	+	0.820193i	$0.693866\pi$
$458$	0	0
$459$	−35.1491	−1.64062
$460$	0	0
$461$	−22.0823	−1.02847	−0.514237	−	0.857648i	$-0.671925\pi$
$461$	−22.0823	−1.02847	−0.514237	+	0.857648i	$0.671925\pi$
$462$	0	0
$463$	−22.0947	−1.02683	−0.513415	−	0.858141i	$-0.671620\pi$
$463$	−22.0947	−1.02683	−0.513415	+	0.858141i	$0.671620\pi$
$464$	0	0
$465$	−0.997048	−0.0462370
$466$	0	0
$467$	36.3594	1.68251	0.841257	−	0.540635i	$-0.181816\pi$
$467$	36.3594	1.68251	0.841257	+	0.540635i	$0.181816\pi$
$468$	0	0
$469$	45.6700	2.10884
$470$	0	0
$471$	13.4247	0.618580
$472$	0	0
$473$	11.7827	0.541768
$474$	0	0
$475$	−8.77643	−0.402690
$476$	0	0
$477$	−10.0356	−0.459499
$478$	0	0
$479$	19.4317	0.887857	0.443929	−	0.896062i	$-0.353584\pi$
$479$	19.4317	0.887857	0.443929	+	0.896062i	$0.353584\pi$
$480$	0	0
$481$	5.09323	0.232231
$482$	0	0
$483$	−18.8084	−0.855810
$484$	0	0
$485$	−2.71585	−0.123321
$486$	0	0
$487$	16.8804	0.764922	0.382461	−	0.923972i	$-0.375077\pi$
$487$	16.8804	0.764922	0.382461	+	0.923972i	$0.375077\pi$
$488$	0	0
$489$	−8.73753	−0.395125
$490$	0	0
$491$	5.33224	0.240640	0.120320	−	0.992735i	$-0.461608\pi$
$491$	5.33224	0.240640	0.120320	+	0.992735i	$0.461608\pi$
$492$	0	0
$493$	−37.7563	−1.70046
$494$	0	0
$495$	7.87484	0.353948
$496$	0	0
$497$	57.6498	2.58595
$498$	0	0
$499$	−13.0070	−0.582272	−0.291136	−	0.956682i	$-0.594033\pi$
$499$	−13.0070	−0.582272	−0.291136	+	0.956682i	$0.594033\pi$
$500$	0	0
$501$	−22.3121	−0.996830
$502$	0	0
$503$	0.486038	0.0216713	0.0108357	−	0.999941i	$-0.496551\pi$
$503$	0.486038	0.0216713	0.0108357	+	0.999941i	$0.496551\pi$
$504$	0	0
$505$	33.5962	1.49501
$506$	0	0
$507$	18.5397	0.823375
$508$	0	0
$509$	36.0970	1.59997	0.799985	−	0.600020i	$-0.204841\pi$
$509$	36.0970	1.59997	0.799985	+	0.600020i	$0.204841\pi$
$510$	0	0
$511$	11.0172	0.487373
$512$	0	0
$513$	−6.29039	−0.277728
$514$	0	0
$515$	27.4876	1.21125
$516$	0	0
$517$	−13.6740	−0.601382
$518$	0	0
$519$	18.1017	0.794577
$520$	0	0
$521$	15.7049	0.688043	0.344022	−	0.938962i	$-0.388211\pi$
$521$	15.7049	0.688043	0.344022	+	0.938962i	$0.388211\pi$
$522$	0	0
$523$	24.9930	1.09287	0.546435	−	0.837502i	$-0.315985\pi$
$523$	24.9930	1.09287	0.546435	+	0.837502i	$0.315985\pi$
$524$	0	0
$525$	43.5584	1.90104
$526$	0	0
$527$	1.17548	0.0512048
$528$	0	0
$529$	−11.4464	−0.497671
$530$	0	0
$531$	−9.62664	−0.417760
$532$	0	0
$533$	−7.24926	−0.314000
$534$	0	0
$535$	−58.4457	−2.52683
$536$	0	0
$537$	−7.50477	−0.323855
$538$	0	0
$539$	18.7981	0.809692
$540$	0	0
$541$	5.12187	0.220206	0.110103	−	0.993920i	$-0.464882\pi$
$541$	5.12187	0.220206	0.110103	+	0.993920i	$0.464882\pi$
$542$	0	0
$543$	3.07378	0.131908
$544$	0	0
$545$	−2.30359	−0.0986751
$546$	0	0
$547$	−17.1406	−0.732879	−0.366440	−	0.930442i	$-0.619423\pi$
$547$	−17.1406	−0.732879	−0.366440	+	0.930442i	$0.619423\pi$
$548$	0	0
$549$	0.830760	0.0354560
$550$	0	0
$551$	−6.75698	−0.287857
$552$	0	0
$553$	−40.4372	−1.71957
$554$	0	0
$555$	−41.9163	−1.77925
$556$	0	0
$557$	12.4985	0.529579	0.264790	−	0.964306i	$-0.414697\pi$
$557$	12.4985	0.529579	0.264790	+	0.964306i	$0.414697\pi$
$558$	0	0
$559$	−2.86341	−0.121109
$560$	0	0
$561$	24.2423	1.02351
$562$	0	0
$563$	−21.7438	−0.916391	−0.458195	−	0.888851i	$-0.651504\pi$
$563$	−21.7438	−0.916391	−0.458195	+	0.888851i	$0.651504\pi$
$564$	0	0
$565$	−35.8300	−1.50738
$566$	0	0
$567$	21.8565	0.917885
$568$	0	0
$569$	−38.1421	−1.59900	−0.799500	−	0.600666i	$-0.794902\pi$
$569$	−38.1421	−1.59900	−0.799500	+	0.600666i	$0.794902\pi$
$570$	0	0
$571$	−21.1204	−0.883863	−0.441931	−	0.897049i	$-0.645707\pi$
$571$	−21.1204	−0.883863	−0.441931	+	0.897049i	$0.645707\pi$
$572$	0	0
$573$	−9.00624	−0.376241
$574$	0	0
$575$	−26.7570	−1.11584
$576$	0	0
$577$	2.63360	0.109638	0.0548190	−	0.998496i	$-0.482542\pi$
$577$	2.63360	0.109638	0.0548190	+	0.998496i	$0.482542\pi$
$578$	0	0
$579$	−37.6351	−1.56406
$580$	0	0
$581$	−1.19020	−0.0493780
$582$	0	0
$583$	31.9163	1.32184
$584$	0	0
$585$	−1.91373	−0.0791232
$586$	0	0
$587$	−32.7842	−1.35315	−0.676574	−	0.736375i	$-0.736536\pi$
$587$	−32.7842	−1.35315	−0.676574	+	0.736375i	$0.736536\pi$
$588$	0	0
$589$	0.210368	0.00866807
$590$	0	0
$591$	−8.60719	−0.354052
$592$	0	0
$593$	−16.4721	−0.676429	−0.338214	−	0.941069i	$-0.609823\pi$
$593$	−16.4721	−0.676429	−0.338214	+	0.941069i	$0.609823\pi$
$594$	0	0
$595$	−83.9722	−3.44252
$596$	0	0
$597$	3.06682	0.125517
$598$	0	0
$599$	22.9108	0.936109	0.468055	−	0.883700i	$-0.344955\pi$
$599$	22.9108	0.936109	0.468055	+	0.883700i	$0.344955\pi$
$600$	0	0
$601$	11.6678	0.475938	0.237969	−	0.971273i	$-0.423518\pi$
$601$	11.6678	0.475938	0.237969	+	0.971273i	$0.423518\pi$
$602$	0	0
$603$	10.0988	0.411253
$604$	0	0
$605$	14.4207	0.586286
$606$	0	0
$607$	−8.98456	−0.364672	−0.182336	−	0.983236i	$-0.558366\pi$
$607$	−8.98456	−0.364672	−0.182336	+	0.983236i	$0.558366\pi$
$608$	0	0
$609$	33.5356	1.35893
$610$	0	0
$611$	3.32304	0.134436
$612$	0	0
$613$	19.8524	0.801833	0.400916	−	0.916115i	$-0.368692\pi$
$613$	19.8524	0.801833	0.400916	+	0.916115i	$0.368692\pi$
$614$	0	0
$615$	59.6601	2.40573
$616$	0	0
$617$	−10.8804	−0.438027	−0.219014	−	0.975722i	$-0.570284\pi$
$617$	−10.8804	−0.438027	−0.219014	+	0.975722i	$0.570284\pi$
$618$	0	0
$619$	38.0272	1.52844	0.764221	−	0.644954i	$-0.223124\pi$
$619$	38.0272	1.52844	0.764221	+	0.644954i	$0.223124\pi$
$620$	0	0
$621$	−19.1777	−0.769575
$622$	0	0
$623$	57.8774	2.31881
$624$	0	0
$625$	−2.39281	−0.0957125
$626$	0	0
$627$	4.33848	0.173262
$628$	0	0
$629$	49.4178	1.97042
$630$	0	0
$631$	30.5940	1.21793	0.608964	−	0.793198i	$-0.291586\pi$
$631$	30.5940	1.21793	0.608964	+	0.793198i	$0.291586\pi$
$632$	0	0
$633$	2.60719	0.103626
$634$	0	0
$635$	−36.3121	−1.44100
$636$	0	0
$637$	−4.56829	−0.181002
$638$	0	0
$639$	12.7478	0.504295
$640$	0	0
$641$	−42.7842	−1.68987	−0.844937	−	0.534866i	$-0.820362\pi$
$641$	−42.7842	−1.68987	−0.844937	+	0.534866i	$0.820362\pi$
$642$	0	0
$643$	−29.0543	−1.14579	−0.572896	−	0.819628i	$-0.694180\pi$
$643$	−29.0543	−1.14579	−0.572896	+	0.819628i	$0.694180\pi$
$644$	0	0
$645$	23.5653	0.927884
$646$	0	0
$647$	38.9277	1.53041	0.765204	−	0.643788i	$-0.222638\pi$
$647$	38.9277	1.53041	0.765204	+	0.643788i	$0.222638\pi$
$648$	0	0
$649$	30.6157	1.20177
$650$	0	0
$651$	−1.04408	−0.0409207
$652$	0	0
$653$	−25.0451	−0.980092	−0.490046	−	0.871697i	$-0.663020\pi$
$653$	−25.0451	−0.980092	−0.490046	+	0.871697i	$0.663020\pi$
$654$	0	0
$655$	−7.56534	−0.295602
$656$	0	0
$657$	2.43617	0.0950442
$658$	0	0
$659$	−10.3921	−0.404819	−0.202409	−	0.979301i	$-0.564877\pi$
$659$	−10.3921	−0.404819	−0.202409	+	0.979301i	$0.564877\pi$
$660$	0	0
$661$	10.1887	0.396294	0.198147	−	0.980172i	$-0.436508\pi$
$661$	10.1887	0.396294	0.198147	+	0.980172i	$0.436508\pi$
$662$	0	0
$663$	−5.89134	−0.228801
$664$	0	0
$665$	−15.0279	−0.582758
$666$	0	0
$667$	−20.6002	−0.797644
$668$	0	0
$669$	30.2034	1.16773
$670$	0	0
$671$	−2.64207	−0.101996
$672$	0	0
$673$	30.9317	1.19233	0.596166	−	0.802861i	$-0.296690\pi$
$673$	30.9317	1.19233	0.596166	+	0.802861i	$0.296690\pi$
$674$	0	0
$675$	44.4138	1.70949
$676$	0	0
$677$	4.88037	0.187568	0.0937839	−	0.995593i	$-0.470104\pi$
$677$	4.88037	0.187568	0.0937839	+	0.995593i	$0.470104\pi$
$678$	0	0
$679$	−2.84396	−0.109141
$680$	0	0
$681$	15.2966	0.586168
$682$	0	0
$683$	1.14132	0.0436712	0.0218356	−	0.999762i	$-0.493049\pi$
$683$	1.14132	0.0436712	0.0218356	+	0.999762i	$0.493049\pi$
$684$	0	0
$685$	23.6825	0.904861
$686$	0	0
$687$	18.2996	0.698173
$688$	0	0
$689$	−7.75626	−0.295490
$690$	0	0
$691$	−8.05361	−0.306374	−0.153187	−	0.988197i	$-0.548954\pi$
$691$	−8.05361	−0.306374	−0.153187	+	0.988197i	$0.548954\pi$
$692$	0	0
$693$	8.24631	0.313251
$694$	0	0
$695$	−7.44018	−0.282222
$696$	0	0
$697$	−70.3370	−2.66421
$698$	0	0
$699$	−25.7563	−0.974191
$700$	0	0
$701$	−26.5132	−1.00139	−0.500696	−	0.865623i	$-0.666922\pi$
$701$	−26.5132	−1.00139	−0.500696	+	0.865623i	$0.666922\pi$
$702$	0	0
$703$	8.84396	0.333556
$704$	0	0
$705$	−27.3480	−1.02999
$706$	0	0
$707$	35.1810	1.32312
$708$	0	0
$709$	−10.1645	−0.381736	−0.190868	−	0.981616i	$-0.561130\pi$
$709$	−10.1645	−0.381736	−0.190868	+	0.981616i	$0.561130\pi$
$710$	0	0
$711$	−8.94166	−0.335338
$712$	0	0
$713$	0.641356	0.0240190
$714$	0	0
$715$	6.08627	0.227613
$716$	0	0
$717$	−17.6924	−0.660735
$718$	0	0
$719$	−26.6630	−0.994364	−0.497182	−	0.867646i	$-0.665632\pi$
$719$	−26.6630	−0.994364	−0.497182	+	0.867646i	$0.665632\pi$
$720$	0	0
$721$	28.7842	1.07198
$722$	0	0
$723$	5.25221	0.195332
$724$	0	0
$725$	47.7082	1.77184
$726$	0	0
$727$	5.89134	0.218498	0.109249	−	0.994014i	$-0.465155\pi$
$727$	5.89134	0.218498	0.109249	+	0.994014i	$0.465155\pi$
$728$	0	0
$729$	29.7625	1.10232
$730$	0	0
$731$	−27.7827	−1.02758
$732$	0	0
$733$	45.7772	1.69082	0.845410	−	0.534118i	$-0.179356\pi$
$733$	45.7772	1.69082	0.845410	+	0.534118i	$0.179356\pi$
$734$	0	0
$735$	37.5962	1.38676
$736$	0	0
$737$	−32.1171	−1.18305
$738$	0	0
$739$	6.91078	0.254217	0.127109	−	0.991889i	$-0.459430\pi$
$739$	6.91078	0.254217	0.127109	+	0.991889i	$0.459430\pi$
$740$	0	0
$741$	−1.05433	−0.0387319
$742$	0	0
$743$	12.7291	0.466984	0.233492	−	0.972359i	$-0.424985\pi$
$743$	12.7291	0.466984	0.233492	+	0.972359i	$0.424985\pi$
$744$	0	0
$745$	−48.6546	−1.78257
$746$	0	0
$747$	−0.263183	−0.00962937
$748$	0	0
$749$	−61.2027	−2.23630
$750$	0	0
$751$	38.8370	1.41718	0.708591	−	0.705619i	$-0.249331\pi$
$751$	38.8370	1.41718	0.708591	+	0.705619i	$0.249331\pi$
$752$	0	0
$753$	3.22686	0.117593
$754$	0	0
$755$	33.0364	1.20232
$756$	0	0
$757$	−30.5070	−1.10880	−0.554398	−	0.832252i	$-0.687051\pi$
$757$	−30.5070	−1.10880	−0.554398	+	0.832252i	$0.687051\pi$
$758$	0	0
$759$	13.2269	0.480105
$760$	0	0
$761$	−12.9068	−0.467870	−0.233935	−	0.972252i	$-0.575160\pi$
$761$	−12.9068	−0.467870	−0.233935	+	0.972252i	$0.575160\pi$
$762$	0	0
$763$	−2.41226	−0.0873296
$764$	0	0
$765$	−18.5683	−0.671338
$766$	0	0
$767$	−7.44018	−0.268649
$768$	0	0
$769$	27.6226	0.996097	0.498049	−	0.867149i	$-0.334050\pi$
$769$	27.6226	0.996097	0.498049	+	0.867149i	$0.334050\pi$
$770$	0	0
$771$	−10.1017	−0.363804
$772$	0	0
$773$	48.1336	1.73125	0.865623	−	0.500696i	$-0.166922\pi$
$773$	48.1336	1.73125	0.865623	+	0.500696i	$0.166922\pi$
$774$	0	0
$775$	−1.48532	−0.0533543
$776$	0	0
$777$	−43.8936	−1.57467
$778$	0	0
$779$	−12.5877	−0.451003
$780$	0	0
$781$	−40.5419	−1.45070
$782$	0	0
$783$	34.1942	1.22200
$784$	0	0
$785$	32.7019	1.16718
$786$	0	0
$787$	−48.4068	−1.72552	−0.862758	−	0.505617i	$-0.831265\pi$
$787$	−48.4068	−1.72552	−0.862758	+	0.505617i	$0.831265\pi$
$788$	0	0
$789$	−22.8106	−0.812079
$790$	0	0
$791$	−37.5202	−1.33406
$792$	0	0
$793$	0.642074	0.0228007
$794$	0	0
$795$	63.8326	2.26391
$796$	0	0
$797$	−17.7742	−0.629594	−0.314797	−	0.949159i	$-0.601936\pi$
$797$	−17.7742	−0.629594	−0.314797	+	0.949159i	$0.601936\pi$
$798$	0	0
$799$	32.2423	1.14065
$800$	0	0
$801$	12.7981	0.452199
$802$	0	0
$803$	−7.74779	−0.273413
$804$	0	0
$805$	−45.8161	−1.61481
$806$	0	0
$807$	−9.21438	−0.324361
$808$	0	0
$809$	41.3532	1.45390	0.726951	−	0.686690i	$-0.240937\pi$
$809$	41.3532	1.45390	0.726951	+	0.686690i	$0.240937\pi$
$810$	0	0
$811$	16.4806	0.578712	0.289356	−	0.957222i	$-0.406559\pi$
$811$	16.4806	0.578712	0.289356	+	0.957222i	$0.406559\pi$
$812$	0	0
$813$	−30.9706	−1.08619
$814$	0	0
$815$	−21.2841	−0.745551
$816$	0	0
$817$	−4.97208	−0.173951
$818$	0	0
$819$	−2.00401	−0.0700257
$820$	0	0
$821$	0.360161	0.0125697	0.00628485	−	0.999980i	$-0.497999\pi$
$821$	0.360161	0.0125697	0.00628485	+	0.999980i	$0.497999\pi$
$822$	0	0
$823$	22.4504	0.782573	0.391286	−	0.920269i	$-0.372030\pi$
$823$	22.4504	0.782573	0.391286	+	0.920269i	$0.372030\pi$
$824$	0	0
$825$	−30.6322	−1.06647
$826$	0	0
$827$	−25.1079	−0.873089	−0.436544	−	0.899683i	$-0.643798\pi$
$827$	−25.1079	−0.873089	−0.436544	+	0.899683i	$0.643798\pi$
$828$	0	0
$829$	−32.2383	−1.11968	−0.559841	−	0.828600i	$-0.689138\pi$
$829$	−32.2383	−1.11968	−0.559841	+	0.828600i	$0.689138\pi$
$830$	0	0
$831$	−8.92622	−0.309647
$832$	0	0
$833$	−44.3246	−1.53575
$834$	0	0
$835$	−54.3510	−1.88089
$836$	0	0
$837$	−1.06458	−0.0367974
$838$	0	0
$839$	−9.95664	−0.343741	−0.171871	−	0.985120i	$-0.554981\pi$
$839$	−9.95664	−0.343741	−0.171871	+	0.985120i	$0.554981\pi$
$840$	0	0
$841$	7.73057	0.266572
$842$	0	0
$843$	−30.1336	−1.03786
$844$	0	0
$845$	45.1616	1.55360
$846$	0	0
$847$	15.1010	0.518876
$848$	0	0
$849$	34.4115	1.18100
$850$	0	0
$851$	26.9629	0.924276
$852$	0	0
$853$	22.2074	0.760367	0.380184	−	0.924911i	$-0.375861\pi$
$853$	22.2074	0.760367	0.380184	+	0.924911i	$0.375861\pi$
$854$	0	0
$855$	−3.32304	−0.113646
$856$	0	0
$857$	−4.32081	−0.147596	−0.0737980	−	0.997273i	$-0.523512\pi$
$857$	−4.32081	−0.147596	−0.0737980	+	0.997273i	$0.523512\pi$
$858$	0	0
$859$	−39.7415	−1.35596	−0.677982	−	0.735079i	$-0.737145\pi$
$859$	−39.7415	−1.35596	−0.677982	+	0.735079i	$0.737145\pi$
$860$	0	0
$861$	62.4744	2.12912
$862$	0	0
$863$	−8.49852	−0.289293	−0.144647	−	0.989483i	$-0.546205\pi$
$863$	−8.49852	−0.289293	−0.144647	+	0.989483i	$0.546205\pi$
$864$	0	0
$865$	44.0947	1.49927
$866$	0	0
$867$	−32.1234	−1.09097
$868$	0	0
$869$	28.4372	0.964667
$870$	0	0
$871$	7.80507	0.264465
$872$	0	0
$873$	−0.628870	−0.0212840
$874$	0	0
$875$	38.7104	1.30865
$876$	0	0
$877$	33.4784	1.13048	0.565242	−	0.824925i	$-0.308783\pi$
$877$	33.4784	1.13048	0.565242	+	0.824925i	$0.308783\pi$
$878$	0	0
$879$	3.62263	0.122188
$880$	0	0
$881$	50.4978	1.70131	0.850657	−	0.525721i	$-0.176204\pi$
$881$	50.4978	1.70131	0.850657	+	0.525721i	$0.176204\pi$
$882$	0	0
$883$	3.50995	0.118119	0.0590597	−	0.998254i	$-0.481190\pi$
$883$	3.50995	0.118119	0.0590597	+	0.998254i	$0.481190\pi$
$884$	0	0
$885$	61.2313	2.05827
$886$	0	0
$887$	−58.3510	−1.95923	−0.979617	−	0.200875i	$-0.935621\pi$
$887$	−58.3510	−1.95923	−0.979617	+	0.200875i	$0.935621\pi$
$888$	0	0
$889$	−38.0250	−1.27532
$890$	0	0
$891$	−15.3704	−0.514928
$892$	0	0
$893$	5.77018	0.193092
$894$	0	0
$895$	−18.2812	−0.611073
$896$	0	0
$897$	−3.21438	−0.107325
$898$	0	0
$899$	−1.14355	−0.0381395
$900$	0	0
$901$	−75.2563	−2.50715
$902$	0	0
$903$	24.6770	0.821198
$904$	0	0
$905$	7.48755	0.248895
$906$	0	0
$907$	−38.7717	−1.28739	−0.643697	−	0.765281i	$-0.722600\pi$
$907$	−38.7717	−1.28739	−0.643697	+	0.765281i	$0.722600\pi$
$908$	0	0
$909$	7.77938	0.258026
$910$	0	0
$911$	−48.0125	−1.59072	−0.795362	−	0.606134i	$-0.792719\pi$
$911$	−48.0125	−1.59072	−0.795362	+	0.606134i	$0.792719\pi$
$912$	0	0
$913$	0.837003	0.0277008
$914$	0	0
$915$	−5.28415	−0.174689
$916$	0	0
$917$	−7.92221	−0.261614
$918$	0	0
$919$	8.37737	0.276344	0.138172	−	0.990408i	$-0.455877\pi$
$919$	8.37737	0.276344	0.138172	+	0.990408i	$0.455877\pi$
$920$	0	0
$921$	16.2298	0.534791
$922$	0	0
$923$	9.85244	0.324297
$924$	0	0
$925$	−62.4435	−2.05313
$926$	0	0
$927$	6.36489	0.209050
$928$	0	0
$929$	−15.0364	−0.493328	−0.246664	−	0.969101i	$-0.579334\pi$
$929$	−15.0364	−0.493328	−0.246664	+	0.969101i	$0.579334\pi$
$930$	0	0
$931$	−7.93246	−0.259976
$932$	0	0
$933$	−13.9930	−0.458112
$934$	0	0
$935$	59.0529	1.93124
$936$	0	0
$937$	−40.4155	−1.32032	−0.660159	−	0.751126i	$-0.729511\pi$
$937$	−40.4155	−1.32032	−0.660159	+	0.751126i	$0.729511\pi$
$938$	0	0
$939$	−0.837003	−0.0273146
$940$	0	0
$941$	18.6770	0.608851	0.304426	−	0.952536i	$-0.401535\pi$
$941$	18.6770	0.608851	0.304426	+	0.952536i	$0.401535\pi$
$942$	0	0
$943$	−38.3767	−1.24972
$944$	0	0
$945$	76.0499	2.47390
$946$	0	0
$947$	−19.2882	−0.626781	−0.313390	−	0.949624i	$-0.601465\pi$
$947$	−19.2882	−0.626781	−0.313390	+	0.949624i	$0.601465\pi$
$948$	0	0
$949$	1.88286	0.0611202
$950$	0	0
$951$	−25.1366	−0.815110
$952$	0	0
$953$	−5.28415	−0.171170	−0.0855852	−	0.996331i	$-0.527276\pi$
$953$	−5.28415	−0.171170	−0.0855852	+	0.996331i	$0.527276\pi$
$954$	0	0
$955$	−21.9387	−0.709920
$956$	0	0
$957$	−23.5837	−0.762354
$958$	0	0
$959$	24.7996	0.800822
$960$	0	0
$961$	−30.9644	−0.998852
$962$	0	0
$963$	−13.5334	−0.436108
$964$	0	0
$965$	−91.6770	−2.95119
$966$	0	0
$967$	9.02346	0.290175	0.145087	−	0.989419i	$-0.453654\pi$
$967$	9.02346	0.290175	0.145087	+	0.989419i	$0.453654\pi$
$968$	0	0
$969$	−10.2298	−0.328629
$970$	0	0
$971$	30.5521	0.980465	0.490232	−	0.871592i	$-0.336912\pi$
$971$	30.5521	0.980465	0.490232	+	0.871592i	$0.336912\pi$
$972$	0	0
$973$	−7.79115	−0.249773
$974$	0	0
$975$	7.44419	0.238405
$976$	0	0
$977$	−18.7306	−0.599244	−0.299622	−	0.954058i	$-0.596861\pi$
$977$	−18.7306	−0.599244	−0.299622	+	0.954058i	$0.596861\pi$
$978$	0	0
$979$	−40.7019	−1.30084
$980$	0	0
$981$	−0.533409	−0.0170304
$982$	0	0
$983$	−6.83924	−0.218138	−0.109069	−	0.994034i	$-0.534787\pi$
$983$	−6.83924	−0.218138	−0.109069	+	0.994034i	$0.534787\pi$
$984$	0	0
$985$	−20.9666	−0.668052
$986$	0	0
$987$	−28.6381	−0.911560
$988$	0	0
$989$	−15.1585	−0.482013
$990$	0	0
$991$	−40.7408	−1.29417	−0.647087	−	0.762416i	$-0.724013\pi$
$991$	−40.7408	−1.29417	−0.647087	+	0.762416i	$0.724013\pi$
$992$	0	0
$993$	1.00295	0.0318277
$994$	0	0
$995$	7.47060	0.236834
$996$	0	0
$997$	11.7724	0.372836	0.186418	−	0.982470i	$-0.440312\pi$
$997$	11.7724	0.372836	0.186418	+	0.982470i	$0.440312\pi$
$998$	0	0
$999$	−44.7555	−1.41600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.g.1.1		3
3.2	odd	2		8784.2.a.bm.1.3		3
4.3	odd	2		122.2.a.c.1.3	✓	3
8.3	odd	2		3904.2.a.u.1.1		3
8.5	even	2		3904.2.a.t.1.3		3
12.11	even	2		1098.2.a.p.1.3		3
20.3	even	4		3050.2.b.k.1099.3		6
20.7	even	4		3050.2.b.k.1099.4		6
20.19	odd	2		3050.2.a.t.1.1		3
28.27	even	2		5978.2.a.q.1.1		3
244.243	odd	2		7442.2.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
122.2.a.c.1.3	✓	3	4.3	odd	2
976.2.a.g.1.1		3	1.1	even	1	trivial
1098.2.a.p.1.3		3	12.11	even	2
3050.2.a.t.1.1		3	20.19	odd	2
3050.2.b.k.1099.3		6	20.3	even	4
3050.2.b.k.1099.4		6	20.7	even	4
3904.2.a.t.1.3		3	8.5	even	2
3904.2.a.u.1.1		3	8.3	odd	2
5978.2.a.q.1.1		3	28.27	even	2
7442.2.a.j.1.3		3	244.243	odd	2
8784.2.a.bm.1.3		3	3.2	odd	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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q-1.47283 q^{3} -3.58774 q^{5} -3.75698 q^{7} -0.830760 q^{9} +O(q^{10})\)	\(q-1.47283 q^{3} -3.58774 q^{5} -3.75698 q^{7} -0.830760 q^{9} +2.64207 q^{11} -0.642074 q^{13} +5.28415 q^{15} -6.22982 q^{17} -1.11491 q^{19} +5.53341 q^{21} -3.39905 q^{23} +7.87189 q^{25} +5.64207 q^{27} +6.06058 q^{29} -0.188687 q^{31} -3.89134 q^{33} +13.4791 q^{35} -7.93246 q^{37} +0.945668 q^{39} +11.2904 q^{41} +4.45963 q^{43} +2.98055 q^{45} -5.17548 q^{47} +7.11491 q^{49} +9.17548 q^{51} +12.0800 q^{53} -9.47908 q^{55} +1.64207 q^{57} +11.5877 q^{59} -1.00000 q^{61} +3.12115 q^{63} +2.30359 q^{65} -12.1560 q^{67} +5.00624 q^{69} -15.3447 q^{71} -2.93246 q^{73} -11.5940 q^{75} -9.92622 q^{77} +10.7632 q^{79} -5.81756 q^{81} +0.316798 q^{83} +22.3510 q^{85} -8.92622 q^{87} -15.4053 q^{89} +2.41226 q^{91} +0.277904 q^{93} +4.00000 q^{95} +0.756981 q^{97} -2.19493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9	\( 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} + 7 q^{11} - q^{13} + 14 q^{15} - 6 q^{17} + 3 q^{19} - 6 q^{21} - 2 q^{23} + 10 q^{25} + 16 q^{27} + q^{29} + 3 q^{31} + 10 q^{33} + 7 q^{35} + 7 q^{37} - 8 q^{39} + 4 q^{41} - 12 q^{43} + 17 q^{45} + 8 q^{47} + 15 q^{49} + 4 q^{51} + 11 q^{53} + 5 q^{55} + 4 q^{57} + 23 q^{59} - 3 q^{61} - 25 q^{63} - 3 q^{65} - 21 q^{67} - 13 q^{69} - 27 q^{71} + 22 q^{73} + 5 q^{75} - 27 q^{77} - 3 q^{79} + 7 q^{81} + 11 q^{83} + 20 q^{85} - 24 q^{87} - 10 q^{89} + 19 q^{91} + 27 q^{93} + 12 q^{95} - 5 q^{97} + 25 q^{99}+O(q^{100}) \) 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9 + 7 * q^11 - q^13 + 14 * q^15 - 6 * q^17 + 3 * q^19 - 6 * q^21 - 2 * q^23 + 10 * q^25 + 16 * q^27 + q^29 + 3 * q^31 + 10 * q^33 + 7 * q^35 + 7 * q^37 - 8 * q^39 + 4 * q^41 - 12 * q^43 + 17 * q^45 + 8 * q^47 + 15 * q^49 + 4 * q^51 + 11 * q^53 + 5 * q^55 + 4 * q^57 + 23 * q^59 - 3 * q^61 - 25 * q^63 - 3 * q^65 - 21 * q^67 - 13 * q^69 - 27 * q^71 + 22 * q^73 + 5 * q^75 - 27 * q^77 - 3 * q^79 + 7 * q^81 + 11 * q^83 + 20 * q^85 - 24 * q^87 - 10 * q^89 + 19 * q^91 + 27 * q^93 + 12 * q^95 - 5 * q^97 + 25 * q^99

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