LMFDB - Embedded newform 8001.2.a.ba.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.308449 q^{2} -1.90486 q^{4} +1.85496 q^{5} +1.00000 q^{7} +1.20445 q^{8} +O(q^{10})$	\(q-0.308449 q^{2} -1.90486 q^{4} +1.85496 q^{5} +1.00000 q^{7} +1.20445 q^{8} -0.572160 q^{10} -1.15845 q^{11} -6.94841 q^{13} -0.308449 q^{14} +3.43821 q^{16} -5.72916 q^{17} +0.736412 q^{19} -3.53344 q^{20} +0.357322 q^{22} -8.44637 q^{23} -1.55912 q^{25} +2.14323 q^{26} -1.90486 q^{28} -2.24001 q^{29} -2.94811 q^{31} -3.46941 q^{32} +1.76715 q^{34} +1.85496 q^{35} -3.40391 q^{37} -0.227145 q^{38} +2.23420 q^{40} +8.37907 q^{41} -9.36793 q^{43} +2.20668 q^{44} +2.60527 q^{46} +3.88765 q^{47} +1.00000 q^{49} +0.480908 q^{50} +13.2358 q^{52} +5.19832 q^{53} -2.14888 q^{55} +1.20445 q^{56} +0.690927 q^{58} +9.94087 q^{59} +1.25028 q^{61} +0.909339 q^{62} -5.80628 q^{64} -12.8890 q^{65} +8.40607 q^{67} +10.9133 q^{68} -0.572160 q^{70} +3.31234 q^{71} +4.14472 q^{73} +1.04993 q^{74} -1.40276 q^{76} -1.15845 q^{77} +6.66640 q^{79} +6.37774 q^{80} -2.58451 q^{82} +10.6307 q^{83} -10.6274 q^{85} +2.88953 q^{86} -1.39529 q^{88} +0.0324518 q^{89} -6.94841 q^{91} +16.0891 q^{92} -1.19914 q^{94} +1.36602 q^{95} +14.0672 q^{97} -0.308449 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

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 $

$2$	−0.308449	−0.218106	−0.109053	−	0.994036i	$-0.534782\pi$
$2$	−0.308449	−0.218106	−0.109053	+	0.994036i	$0.534782\pi$
$3$	0	0
$3$	0	0
$4$	−1.90486	−0.952430
$5$	1.85496	0.829564	0.414782	−	0.909921i	$-0.363858\pi$
$5$	1.85496	0.829564	0.414782	+	0.909921i	$0.363858\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.20445	0.425837
$9$	0	0
$10$	−0.572160	−0.180933
$11$	−1.15845	−0.349285	−0.174643	−	0.984632i	$-0.555877\pi$
$11$	−1.15845	−0.349285	−0.174643	+	0.984632i	$0.555877\pi$
$12$	0	0
$13$	−6.94841	−1.92714	−0.963572	−	0.267450i	$-0.913819\pi$
$13$	−6.94841	−1.92714	−0.963572	+	0.267450i	$0.913819\pi$
$14$	−0.308449	−0.0824363
$15$	0	0
$16$	3.43821	0.859552
$17$	−5.72916	−1.38953	−0.694763	−	0.719239i	$-0.744491\pi$
$17$	−5.72916	−1.38953	−0.694763	+	0.719239i	$0.744491\pi$
$18$	0	0
$19$	0.736412	0.168945	0.0844723	−	0.996426i	$-0.473080\pi$
$19$	0.736412	0.168945	0.0844723	+	0.996426i	$0.473080\pi$
$20$	−3.53344	−0.790101
$21$	0	0
$22$	0.357322	0.0761812
$23$	−8.44637	−1.76119	−0.880594	−	0.473871i	$-0.842856\pi$
$23$	−8.44637	−1.76119	−0.880594	+	0.473871i	$0.842856\pi$
$24$	0	0
$25$	−1.55912	−0.311824
$26$	2.14323	0.420322
$27$	0	0
$28$	−1.90486	−0.359985
$29$	−2.24001	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$29$	−2.24001	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$30$	0	0
$31$	−2.94811	−0.529495	−0.264748	−	0.964318i	$-0.585289\pi$
$31$	−2.94811	−0.529495	−0.264748	+	0.964318i	$0.585289\pi$
$32$	−3.46941	−0.613310
$33$	0	0
$34$	1.76715	0.303064
$35$	1.85496	0.313546
$36$	0	0
$37$	−3.40391	−0.559599	−0.279800	−	0.960058i	$-0.590268\pi$
$37$	−3.40391	−0.559599	−0.279800	+	0.960058i	$0.590268\pi$
$38$	−0.227145	−0.0368478
$39$	0	0
$40$	2.23420	0.353259
$41$	8.37907	1.30859	0.654296	−	0.756239i	$-0.272965\pi$
$41$	8.37907	1.30859	0.654296	+	0.756239i	$0.272965\pi$
$42$	0	0
$43$	−9.36793	−1.42860	−0.714298	−	0.699841i	$-0.753254\pi$
$43$	−9.36793	−1.42860	−0.714298	+	0.699841i	$0.753254\pi$
$44$	2.20668	0.332670
$45$	0	0
$46$	2.60527	0.384126
$47$	3.88765	0.567072	0.283536	−	0.958962i	$-0.408492\pi$
$47$	3.88765	0.567072	0.283536	+	0.958962i	$0.408492\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.480908	0.0680107
$51$	0	0
$52$	13.2358	1.83547
$53$	5.19832	0.714044	0.357022	−	0.934096i	$-0.383792\pi$
$53$	5.19832	0.714044	0.357022	+	0.934096i	$0.383792\pi$
$54$	0	0
$55$	−2.14888	−0.289754
$56$	1.20445	0.160951
$57$	0	0
$58$	0.690927	0.0907232
$59$	9.94087	1.29419	0.647096	−	0.762409i	$-0.275983\pi$
$59$	9.94087	1.29419	0.647096	+	0.762409i	$0.275983\pi$
$60$	0	0
$61$	1.25028	0.160083	0.0800413	−	0.996792i	$-0.474495\pi$
$61$	1.25028	0.160083	0.0800413	+	0.996792i	$0.474495\pi$
$62$	0.909339	0.115486
$63$	0	0
$64$	−5.80628	−0.725785
$65$	−12.8890	−1.59869
$66$	0	0
$67$	8.40607	1.02696	0.513482	−	0.858100i	$-0.328355\pi$
$67$	8.40607	1.02696	0.513482	+	0.858100i	$0.328355\pi$
$68$	10.9133	1.32343
$69$	0	0
$70$	−0.572160	−0.0683862
$71$	3.31234	0.393102	0.196551	−	0.980494i	$-0.437026\pi$
$71$	3.31234	0.393102	0.196551	+	0.980494i	$0.437026\pi$
$72$	0	0
$73$	4.14472	0.485103	0.242551	−	0.970139i	$-0.422016\pi$
$73$	4.14472	0.485103	0.242551	+	0.970139i	$0.422016\pi$
$74$	1.04993	0.122052
$75$	0	0
$76$	−1.40276	−0.160908
$77$	−1.15845	−0.132017
$78$	0	0
$79$	6.66640	0.750028	0.375014	−	0.927019i	$-0.377638\pi$
$79$	6.66640	0.750028	0.375014	+	0.927019i	$0.377638\pi$
$80$	6.37774	0.713053
$81$	0	0
$82$	−2.58451	−0.285412
$83$	10.6307	1.16688	0.583438	−	0.812158i	$-0.301707\pi$
$83$	10.6307	1.16688	0.583438	+	0.812158i	$0.301707\pi$
$84$	0	0
$85$	−10.6274	−1.15270
$86$	2.88953	0.311586
$87$	0	0
$88$	−1.39529	−0.148738
$89$	0.0324518	0.00343988	0.00171994	−	0.999999i	$-0.499453\pi$
$89$	0.0324518	0.00343988	0.00171994	+	0.999999i	$0.499453\pi$
$90$	0	0
$91$	−6.94841	−0.728392
$92$	16.0891	1.67741
$93$	0	0
$94$	−1.19914	−0.123682
$95$	1.36602	0.140150
$96$	0	0
$97$	14.0672	1.42830	0.714152	−	0.699990i	$-0.246812\pi$
$97$	14.0672	1.42830	0.714152	+	0.699990i	$0.246812\pi$
$98$	−0.308449	−0.0311580
$99$	0	0
$100$	2.96990	0.296990
$101$	10.9152	1.08610	0.543052	−	0.839699i	$-0.317269\pi$
$101$	10.9152	1.08610	0.543052	+	0.839699i	$0.317269\pi$
$102$	0	0
$103$	2.52392	0.248689	0.124345	−	0.992239i	$-0.460317\pi$
$103$	2.52392	0.248689	0.124345	+	0.992239i	$0.460317\pi$
$104$	−8.36901	−0.820649
$105$	0	0
$106$	−1.60342	−0.155737
$107$	−16.2638	−1.57228	−0.786142	−	0.618046i	$-0.787924\pi$
$107$	−16.2638	−1.57228	−0.786142	+	0.618046i	$0.787924\pi$
$108$	0	0
$109$	18.0713	1.73091	0.865457	−	0.500984i	$-0.167028\pi$
$109$	18.0713	1.73091	0.865457	+	0.500984i	$0.167028\pi$
$110$	0.662818	0.0631972
$111$	0	0
$112$	3.43821	0.324880
$113$	12.6917	1.19394	0.596968	−	0.802265i	$-0.296372\pi$
$113$	12.6917	1.19394	0.596968	+	0.802265i	$0.296372\pi$
$114$	0	0
$115$	−15.6677	−1.46102
$116$	4.26690	0.396172
$117$	0	0
$118$	−3.06625	−0.282271
$119$	−5.72916	−0.525192
$120$	0	0
$121$	−9.65800	−0.878000
$122$	−0.385649	−0.0349150
$123$	0	0
$124$	5.61573	0.504307
$125$	−12.1669	−1.08824
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.72975	0.771609
$129$	0	0
$130$	3.97561	0.348684
$131$	0.438946	0.0383509	0.0191754	−	0.999816i	$-0.493896\pi$
$131$	0.438946	0.0383509	0.0191754	+	0.999816i	$0.493896\pi$
$132$	0	0
$133$	0.736412	0.0638550
$134$	−2.59284	−0.223987
$135$	0	0
$136$	−6.90048	−0.591711
$137$	−7.68794	−0.656825	−0.328413	−	0.944534i	$-0.606514\pi$
$137$	−7.68794	−0.656825	−0.328413	+	0.944534i	$0.606514\pi$
$138$	0	0
$139$	9.57505	0.812145	0.406072	−	0.913841i	$-0.366898\pi$
$139$	9.57505	0.812145	0.406072	+	0.913841i	$0.366898\pi$
$140$	−3.53344	−0.298630
$141$	0	0
$142$	−1.02169	−0.0857380
$143$	8.04938	0.673123
$144$	0	0
$145$	−4.15513	−0.345065
$146$	−1.27843	−0.105804
$147$	0	0
$148$	6.48397	0.532979
$149$	14.6864	1.20315	0.601577	−	0.798815i	$-0.294539\pi$
$149$	14.6864	1.20315	0.601577	+	0.798815i	$0.294539\pi$
$150$	0	0
$151$	14.3026	1.16393	0.581965	−	0.813214i	$-0.302284\pi$
$151$	14.3026	1.16393	0.581965	+	0.813214i	$0.302284\pi$
$152$	0.886970	0.0719428
$153$	0	0
$154$	0.357322	0.0287938
$155$	−5.46862	−0.439250
$156$	0	0
$157$	−15.4234	−1.23092	−0.615462	−	0.788166i	$-0.711031\pi$
$157$	−15.4234	−1.23092	−0.615462	+	0.788166i	$0.711031\pi$
$158$	−2.05624	−0.163586
$159$	0	0
$160$	−6.43561	−0.508780
$161$	−8.44637	−0.665667
$162$	0	0
$163$	−4.12792	−0.323323	−0.161662	−	0.986846i	$-0.551685\pi$
$163$	−4.12792	−0.323323	−0.161662	+	0.986846i	$0.551685\pi$
$164$	−15.9610	−1.24634
$165$	0	0
$166$	−3.27904	−0.254503
$167$	−9.56284	−0.739995	−0.369998	−	0.929033i	$-0.620641\pi$
$167$	−9.56284	−0.739995	−0.369998	+	0.929033i	$0.620641\pi$
$168$	0	0
$169$	35.2805	2.71388
$170$	3.27800	0.251411
$171$	0	0
$172$	17.8446	1.36064
$173$	−8.27300	−0.628985	−0.314492	−	0.949260i	$-0.601834\pi$
$173$	−8.27300	−0.628985	−0.314492	+	0.949260i	$0.601834\pi$
$174$	0	0
$175$	−1.55912	−0.117858
$176$	−3.98299	−0.300229
$177$	0	0
$178$	−0.0100097	−0.000750259 0
$179$	4.18392	0.312721	0.156360	−	0.987700i	$-0.450024\pi$
$179$	4.18392	0.312721	0.156360	+	0.987700i	$0.450024\pi$
$180$	0	0
$181$	−11.4008	−0.847418	−0.423709	−	0.905798i	$-0.639272\pi$
$181$	−11.4008	−0.847418	−0.423709	+	0.905798i	$0.639272\pi$
$182$	2.14323	0.158867
$183$	0	0
$184$	−10.1732	−0.749979
$185$	−6.31412	−0.464223
$186$	0	0
$187$	6.63694	0.485341
$188$	−7.40543	−0.540096
$189$	0	0
$190$	−0.421346	−0.0305676
$191$	−22.5708	−1.63317	−0.816584	−	0.577227i	$-0.804135\pi$
$191$	−22.5708	−1.63317	−0.816584	+	0.577227i	$0.804135\pi$
$192$	0	0
$193$	20.2943	1.46081	0.730407	−	0.683013i	$-0.239331\pi$
$193$	20.2943	1.46081	0.730407	+	0.683013i	$0.239331\pi$
$194$	−4.33900	−0.311522
$195$	0	0
$196$	−1.90486	−0.136061
$197$	−5.25249	−0.374225	−0.187112	−	0.982339i	$-0.559913\pi$
$197$	−5.25249	−0.374225	−0.187112	+	0.982339i	$0.559913\pi$
$198$	0	0
$199$	14.8645	1.05372	0.526860	−	0.849952i	$-0.323369\pi$
$199$	14.8645	1.05372	0.526860	+	0.849952i	$0.323369\pi$
$200$	−1.87788	−0.132786
$201$	0	0
$202$	−3.36678	−0.236886
$203$	−2.24001	−0.157218
$204$	0	0
$205$	15.5429	1.08556
$206$	−0.778500	−0.0542407
$207$	0	0
$208$	−23.8901	−1.65648
$209$	−0.853095	−0.0590098
$210$	0	0
$211$	5.14691	0.354328	0.177164	−	0.984181i	$-0.443308\pi$
$211$	5.14691	0.354328	0.177164	+	0.984181i	$0.443308\pi$
$212$	−9.90207	−0.680077
$213$	0	0
$214$	5.01655	0.342925
$215$	−17.3772	−1.18511
$216$	0	0
$217$	−2.94811	−0.200130
$218$	−5.57405	−0.377523
$219$	0	0
$220$	4.09331	0.275971
$221$	39.8086	2.67782
$222$	0	0
$223$	29.3376	1.96459	0.982295	−	0.187340i	$-0.0599867\pi$
$223$	29.3376	1.96459	0.982295	+	0.187340i	$0.0599867\pi$
$224$	−3.46941	−0.231810
$225$	0	0
$226$	−3.91474	−0.260405
$227$	1.07501	0.0713506	0.0356753	−	0.999363i	$-0.488642\pi$
$227$	1.07501	0.0713506	0.0356753	+	0.999363i	$0.488642\pi$
$228$	0	0
$229$	14.9717	0.989359	0.494680	−	0.869075i	$-0.335285\pi$
$229$	14.9717	0.989359	0.494680	+	0.869075i	$0.335285\pi$
$230$	4.83267	0.318657
$231$	0	0
$232$	−2.69797	−0.177131
$233$	−26.2411	−1.71911	−0.859555	−	0.511044i	$-0.829259\pi$
$233$	−26.2411	−1.71911	−0.859555	+	0.511044i	$0.829259\pi$
$234$	0	0
$235$	7.21144	0.470423
$236$	−18.9360	−1.23263
$237$	0	0
$238$	1.76715	0.114547
$239$	−13.0710	−0.845492	−0.422746	−	0.906248i	$-0.638934\pi$
$239$	−13.0710	−0.845492	−0.422746	+	0.906248i	$0.638934\pi$
$240$	0	0
$241$	5.91450	0.380986	0.190493	−	0.981689i	$-0.438991\pi$
$241$	5.91450	0.380986	0.190493	+	0.981689i	$0.438991\pi$
$242$	2.97900	0.191497
$243$	0	0
$244$	−2.38162	−0.152467
$245$	1.85496	0.118509
$246$	0	0
$247$	−5.11690	−0.325580
$248$	−3.55084	−0.225479
$249$	0	0
$250$	3.75287	0.237352
$251$	15.7215	0.992335	0.496168	−	0.868227i	$-0.334740\pi$
$251$	15.7215	0.992335	0.496168	+	0.868227i	$0.334740\pi$
$252$	0	0
$253$	9.78468	0.615157
$254$	0.308449	0.0193538
$255$	0	0
$256$	8.91989	0.557493
$257$	1.15010	0.0717411	0.0358705	−	0.999356i	$-0.488580\pi$
$257$	1.15010	0.0717411	0.0358705	+	0.999356i	$0.488580\pi$
$258$	0	0
$259$	−3.40391	−0.211509
$260$	24.5518	1.52264
$261$	0	0
$262$	−0.135392	−0.00836456
$263$	13.1067	0.808191	0.404096	−	0.914717i	$-0.367586\pi$
$263$	13.1067	0.808191	0.404096	+	0.914717i	$0.367586\pi$
$264$	0	0
$265$	9.64269	0.592345
$266$	−0.227145	−0.0139272
$267$	0	0
$268$	−16.0124	−0.978112
$269$	2.96535	0.180800	0.0904002	−	0.995906i	$-0.471185\pi$
$269$	2.96535	0.180800	0.0904002	+	0.995906i	$0.471185\pi$
$270$	0	0
$271$	−21.2724	−1.29220	−0.646102	−	0.763251i	$-0.723602\pi$
$271$	−21.2724	−1.29220	−0.646102	+	0.763251i	$0.723602\pi$
$272$	−19.6981	−1.19437
$273$	0	0
$274$	2.37133	0.143258
$275$	1.80616	0.108915
$276$	0	0
$277$	14.9174	0.896302	0.448151	−	0.893958i	$-0.352083\pi$
$277$	14.9174	0.896302	0.448151	+	0.893958i	$0.352083\pi$
$278$	−2.95341	−0.177134
$279$	0	0
$280$	2.23420	0.133519
$281$	6.47139	0.386051	0.193025	−	0.981194i	$-0.438170\pi$
$281$	6.47139	0.386051	0.193025	+	0.981194i	$0.438170\pi$
$282$	0	0
$283$	−12.0825	−0.718230	−0.359115	−	0.933293i	$-0.616921\pi$
$283$	−12.0825	−0.718230	−0.359115	+	0.933293i	$0.616921\pi$
$284$	−6.30954	−0.374402
$285$	0	0
$286$	−2.48282	−0.146812
$287$	8.37907	0.494601
$288$	0	0
$289$	15.8233	0.930783
$290$	1.28164	0.0752607
$291$	0	0
$292$	−7.89511	−0.462026
$293$	−21.7215	−1.26898	−0.634492	−	0.772929i	$-0.718791\pi$
$293$	−21.7215	−1.26898	−0.634492	+	0.772929i	$0.718791\pi$
$294$	0	0
$295$	18.4399	1.07361
$296$	−4.09983	−0.238298
$297$	0	0
$298$	−4.52999	−0.262415
$299$	58.6889	3.39406
$300$	0	0
$301$	−9.36793	−0.539959
$302$	−4.41162	−0.253860
$303$	0	0
$304$	2.53194	0.145217
$305$	2.31923	0.132799
$306$	0	0
$307$	−28.4780	−1.62533	−0.812664	−	0.582732i	$-0.801984\pi$
$307$	−28.4780	−1.62533	−0.812664	+	0.582732i	$0.801984\pi$
$308$	2.20668	0.125737
$309$	0	0
$310$	1.68679	0.0958031
$311$	−21.6623	−1.22836	−0.614179	−	0.789167i	$-0.710513\pi$
$311$	−21.6623	−1.22836	−0.614179	+	0.789167i	$0.710513\pi$
$312$	0	0
$313$	2.94234	0.166311	0.0831554	−	0.996537i	$-0.473500\pi$
$313$	2.94234	0.166311	0.0831554	+	0.996537i	$0.473500\pi$
$314$	4.75734	0.268472
$315$	0	0
$316$	−12.6985	−0.714349
$317$	−11.0071	−0.618218	−0.309109	−	0.951027i	$-0.600031\pi$
$317$	−11.0071	−0.618218	−0.309109	+	0.951027i	$0.600031\pi$
$318$	0	0
$319$	2.59493	0.145288
$320$	−10.7704	−0.602085
$321$	0	0
$322$	2.60527	0.145186
$323$	−4.21903	−0.234753
$324$	0	0
$325$	10.8334	0.600929
$326$	1.27325	0.0705188
$327$	0	0
$328$	10.0922	0.557246
$329$	3.88765	0.214333
$330$	0	0
$331$	−7.39001	−0.406192	−0.203096	−	0.979159i	$-0.565100\pi$
$331$	−7.39001	−0.406192	−0.203096	+	0.979159i	$0.565100\pi$
$332$	−20.2501	−1.11137
$333$	0	0
$334$	2.94965	0.161397
$335$	15.5929	0.851933
$336$	0	0
$337$	−29.4047	−1.60177	−0.800887	−	0.598816i	$-0.795638\pi$
$337$	−29.4047	−1.60177	−0.800887	+	0.598816i	$0.795638\pi$
$338$	−10.8822	−0.591914
$339$	0	0
$340$	20.2437	1.09787
$341$	3.41523	0.184945
$342$	0	0
$343$	1.00000	0.0539949
$344$	−11.2832	−0.608349
$345$	0	0
$346$	2.55180	0.137185
$347$	−10.3132	−0.553641	−0.276821	−	0.960922i	$-0.589281\pi$
$347$	−10.3132	−0.553641	−0.276821	+	0.960922i	$0.589281\pi$
$348$	0	0
$349$	21.8417	1.16916	0.584581	−	0.811336i	$-0.301259\pi$
$349$	21.8417	1.16916	0.584581	+	0.811336i	$0.301259\pi$
$350$	0.480908	0.0257056
$351$	0	0
$352$	4.01913	0.214220
$353$	−28.8655	−1.53636	−0.768178	−	0.640236i	$-0.778836\pi$
$353$	−28.8655	−1.53636	−0.768178	+	0.640236i	$0.778836\pi$
$354$	0	0
$355$	6.14426	0.326103
$356$	−0.0618161	−0.00327624
$357$	0	0
$358$	−1.29052	−0.0682063
$359$	34.0930	1.79936	0.899679	−	0.436551i	$-0.143800\pi$
$359$	34.0930	1.79936	0.899679	+	0.436551i	$0.143800\pi$
$360$	0	0
$361$	−18.4577	−0.971458
$362$	3.51657	0.184827
$363$	0	0
$364$	13.2358	0.693742
$365$	7.68829	0.402424
$366$	0	0
$367$	0.697650	0.0364170	0.0182085	−	0.999834i	$-0.494204\pi$
$367$	0.697650	0.0364170	0.0182085	+	0.999834i	$0.494204\pi$
$368$	−29.0404	−1.51383
$369$	0	0
$370$	1.94758	0.101250
$371$	5.19832	0.269883
$372$	0	0
$373$	19.9367	1.03229	0.516143	−	0.856503i	$-0.327367\pi$
$373$	19.9367	1.03229	0.516143	+	0.856503i	$0.327367\pi$
$374$	−2.04715	−0.105856
$375$	0	0
$376$	4.68247	0.241480
$377$	15.5645	0.801613
$378$	0	0
$379$	−15.3724	−0.789628	−0.394814	−	0.918761i	$-0.629191\pi$
$379$	−15.3724	−0.789628	−0.394814	+	0.918761i	$0.629191\pi$
$380$	−2.60207	−0.133483
$381$	0	0
$382$	6.96194	0.356204
$383$	13.7610	0.703156	0.351578	−	0.936159i	$-0.385645\pi$
$383$	13.7610	0.703156	0.351578	+	0.936159i	$0.385645\pi$
$384$	0	0
$385$	−2.14888	−0.109517
$386$	−6.25974	−0.318612
$387$	0	0
$388$	−26.7960	−1.36036
$389$	−11.8096	−0.598773	−0.299386	−	0.954132i	$-0.596782\pi$
$389$	−11.8096	−0.598773	−0.299386	+	0.954132i	$0.596782\pi$
$390$	0	0
$391$	48.3906	2.44722
$392$	1.20445	0.0608338
$393$	0	0
$394$	1.62012	0.0816206
$395$	12.3659	0.622196
$396$	0	0
$397$	−8.16562	−0.409821	−0.204910	−	0.978781i	$-0.565690\pi$
$397$	−8.16562	−0.409821	−0.204910	+	0.978781i	$0.565690\pi$
$398$	−4.58495	−0.229823
$399$	0	0
$400$	−5.36058	−0.268029
$401$	−12.6585	−0.632137	−0.316069	−	0.948736i	$-0.602363\pi$
$401$	−12.6585	−0.632137	−0.316069	+	0.948736i	$0.602363\pi$
$402$	0	0
$403$	20.4847	1.02041
$404$	−20.7920	−1.03444
$405$	0	0
$406$	0.690927	0.0342902
$407$	3.94325	0.195460
$408$	0	0
$409$	−2.51318	−0.124269	−0.0621345	−	0.998068i	$-0.519791\pi$
$409$	−2.51318	−0.124269	−0.0621345	+	0.998068i	$0.519791\pi$
$410$	−4.79417	−0.236767
$411$	0	0
$412$	−4.80772	−0.236859
$413$	9.94087	0.489158
$414$	0	0
$415$	19.7196	0.967998
$416$	24.1069	1.18194
$417$	0	0
$418$	0.263136	0.0128704
$419$	8.37210	0.409004	0.204502	−	0.978866i	$-0.434443\pi$
$419$	8.37210	0.409004	0.204502	+	0.978866i	$0.434443\pi$
$420$	0	0
$421$	14.0608	0.685280	0.342640	−	0.939467i	$-0.388679\pi$
$421$	14.0608	0.685280	0.342640	+	0.939467i	$0.388679\pi$
$422$	−1.58756	−0.0772810
$423$	0	0
$424$	6.26111	0.304066
$425$	8.93245	0.433288
$426$	0	0
$427$	1.25028	0.0605055
$428$	30.9803	1.49749
$429$	0	0
$430$	5.35996	0.258480
$431$	24.5595	1.18299	0.591494	−	0.806309i	$-0.298538\pi$
$431$	24.5595	1.18299	0.591494	+	0.806309i	$0.298538\pi$
$432$	0	0
$433$	13.9330	0.669576	0.334788	−	0.942293i	$-0.391335\pi$
$433$	13.9330	0.669576	0.334788	+	0.942293i	$0.391335\pi$
$434$	0.909339	0.0436497
$435$	0	0
$436$	−34.4232	−1.64857
$437$	−6.22001	−0.297543
$438$	0	0
$439$	−8.72309	−0.416330	−0.208165	−	0.978094i	$-0.566749\pi$
$439$	−8.72309	−0.416330	−0.208165	+	0.978094i	$0.566749\pi$
$440$	−2.58821	−0.123388
$441$	0	0
$442$	−12.2789	−0.584048
$443$	−24.5674	−1.16723	−0.583616	−	0.812029i	$-0.698363\pi$
$443$	−24.5674	−1.16723	−0.583616	+	0.812029i	$0.698363\pi$
$444$	0	0
$445$	0.0601968	0.00285360
$446$	−9.04914	−0.428489
$447$	0	0
$448$	−5.80628	−0.274321
$449$	6.39621	0.301856	0.150928	−	0.988545i	$-0.451774\pi$
$449$	6.39621	0.301856	0.150928	+	0.988545i	$0.451774\pi$
$450$	0	0
$451$	−9.70672	−0.457072
$452$	−24.1760	−1.13714
$453$	0	0
$454$	−0.331584	−0.0155620
$455$	−12.8890	−0.604247
$456$	0	0
$457$	30.9294	1.44681	0.723407	−	0.690421i	$-0.242575\pi$
$457$	30.9294	1.44681	0.723407	+	0.690421i	$0.242575\pi$
$458$	−4.61801	−0.215785
$459$	0	0
$460$	29.8447	1.39152
$461$	−3.53338	−0.164566	−0.0822829	−	0.996609i	$-0.526221\pi$
$461$	−3.53338	−0.164566	−0.0822829	+	0.996609i	$0.526221\pi$
$462$	0	0
$463$	−17.7249	−0.823747	−0.411873	−	0.911241i	$-0.635125\pi$
$463$	−17.7249	−0.823747	−0.411873	+	0.911241i	$0.635125\pi$
$464$	−7.70162	−0.357539
$465$	0	0
$466$	8.09402	0.374948
$467$	21.4959	0.994711	0.497355	−	0.867547i	$-0.334305\pi$
$467$	21.4959	0.994711	0.497355	+	0.867547i	$0.334305\pi$
$468$	0	0
$469$	8.40607	0.388156
$470$	−2.22436	−0.102602
$471$	0	0
$472$	11.9733	0.551114
$473$	10.8523	0.498988
$474$	0	0
$475$	−1.14815	−0.0526809
$476$	10.9133	0.500208
$477$	0	0
$478$	4.03173	0.184407
$479$	29.7836	1.36085	0.680424	−	0.732819i	$-0.261796\pi$
$479$	29.7836	1.36085	0.680424	+	0.732819i	$0.261796\pi$
$480$	0	0
$481$	23.6518	1.07843
$482$	−1.82432	−0.0830954
$483$	0	0
$484$	18.3971	0.836233
$485$	26.0941	1.18487
$486$	0	0
$487$	−27.2175	−1.23334	−0.616671	−	0.787221i	$-0.711519\pi$
$487$	−27.2175	−1.23334	−0.616671	+	0.787221i	$0.711519\pi$
$488$	1.50590	0.0681691
$489$	0	0
$490$	−0.572160	−0.0258476
$491$	−19.9769	−0.901545	−0.450773	−	0.892639i	$-0.648851\pi$
$491$	−19.9769	−0.901545	−0.450773	+	0.892639i	$0.648851\pi$
$492$	0	0
$493$	12.8334	0.577986
$494$	1.57830	0.0710110
$495$	0	0
$496$	−10.1362	−0.455129
$497$	3.31234	0.148579
$498$	0	0
$499$	9.50451	0.425480	0.212740	−	0.977109i	$-0.431761\pi$
$499$	9.50451	0.425480	0.212740	+	0.977109i	$0.431761\pi$
$500$	23.1763	1.03647
$501$	0	0
$502$	−4.84929	−0.216434
$503$	−25.6635	−1.14428	−0.572140	−	0.820156i	$-0.693887\pi$
$503$	−25.6635	−1.14428	−0.572140	+	0.820156i	$0.693887\pi$
$504$	0	0
$505$	20.2473	0.900993
$506$	−3.01807	−0.134170
$507$	0	0
$508$	1.90486	0.0845145
$509$	−27.6041	−1.22353	−0.611764	−	0.791040i	$-0.709540\pi$
$509$	−27.6041	−1.22353	−0.611764	+	0.791040i	$0.709540\pi$
$510$	0	0
$511$	4.14472	0.183352
$512$	−20.2108	−0.893201
$513$	0	0
$514$	−0.354746	−0.0156472
$515$	4.68178	0.206304
$516$	0	0
$517$	−4.50364	−0.198070
$518$	1.04993	0.0461313
$519$	0	0
$520$	−15.5242	−0.680780
$521$	13.2701	0.581373	0.290687	−	0.956818i	$-0.406116\pi$
$521$	13.2701	0.581373	0.290687	+	0.956818i	$0.406116\pi$
$522$	0	0
$523$	−5.90570	−0.258238	−0.129119	−	0.991629i	$-0.541215\pi$
$523$	−5.90570	−0.258238	−0.129119	+	0.991629i	$0.541215\pi$
$524$	−0.836130	−0.0365265
$525$	0	0
$526$	−4.04273	−0.176271
$527$	16.8902	0.735748
$528$	0	0
$529$	48.3411	2.10179
$530$	−2.97427	−0.129194
$531$	0	0
$532$	−1.40276	−0.0608174
$533$	−58.2213	−2.52184
$534$	0	0
$535$	−30.1688	−1.30431
$536$	10.1247	0.437319
$537$	0	0
$538$	−0.914657	−0.0394337
$539$	−1.15845	−0.0498979
$540$	0	0
$541$	14.9524	0.642854	0.321427	−	0.946934i	$-0.395837\pi$
$541$	14.9524	0.642854	0.321427	+	0.946934i	$0.395837\pi$
$542$	6.56143	0.281837
$543$	0	0
$544$	19.8768	0.852211
$545$	33.5215	1.43590
$546$	0	0
$547$	17.0806	0.730315	0.365157	−	0.930946i	$-0.381015\pi$
$547$	17.0806	0.730315	0.365157	+	0.930946i	$0.381015\pi$
$548$	14.6444	0.625580
$549$	0	0
$550$	−0.557107	−0.0237551
$551$	−1.64957	−0.0702740
$552$	0	0
$553$	6.66640	0.283484
$554$	−4.60126	−0.195489
$555$	0	0
$556$	−18.2391	−0.773511
$557$	8.09048	0.342805	0.171402	−	0.985201i	$-0.445170\pi$
$557$	8.09048	0.342805	0.171402	+	0.985201i	$0.445170\pi$
$558$	0	0
$559$	65.0923	2.75311
$560$	6.37774	0.269509
$561$	0	0
$562$	−1.99609	−0.0842000
$563$	31.0284	1.30769	0.653846	−	0.756627i	$-0.273154\pi$
$563$	31.0284	1.30769	0.653846	+	0.756627i	$0.273154\pi$
$564$	0	0
$565$	23.5427	0.990447
$566$	3.72683	0.156650
$567$	0	0
$568$	3.98954	0.167397
$569$	−36.9076	−1.54725	−0.773624	−	0.633645i	$-0.781558\pi$
$569$	−36.9076	−1.54725	−0.773624	+	0.633645i	$0.781558\pi$
$570$	0	0
$571$	36.4196	1.52411	0.762057	−	0.647510i	$-0.224190\pi$
$571$	36.4196	1.52411	0.762057	+	0.647510i	$0.224190\pi$
$572$	−15.3329	−0.641102
$573$	0	0
$574$	−2.58451	−0.107875
$575$	13.1689	0.549181
$576$	0	0
$577$	−6.08794	−0.253444	−0.126722	−	0.991938i	$-0.540446\pi$
$577$	−6.08794	−0.253444	−0.126722	+	0.991938i	$0.540446\pi$
$578$	−4.88068	−0.203010
$579$	0	0
$580$	7.91494	0.328650
$581$	10.6307	0.441038
$582$	0	0
$583$	−6.02199	−0.249405
$584$	4.99210	0.206575
$585$	0	0
$586$	6.69997	0.276773
$587$	−44.6249	−1.84187	−0.920933	−	0.389722i	$-0.872571\pi$
$587$	−44.6249	−1.84187	−0.920933	+	0.389722i	$0.872571\pi$
$588$	0	0
$589$	−2.17102	−0.0894553
$590$	−5.68777	−0.234162
$591$	0	0
$592$	−11.7033	−0.481005
$593$	−3.10600	−0.127548	−0.0637741	−	0.997964i	$-0.520314\pi$
$593$	−3.10600	−0.127548	−0.0637741	+	0.997964i	$0.520314\pi$
$594$	0	0
$595$	−10.6274	−0.435680
$596$	−27.9755	−1.14592
$597$	0	0
$598$	−18.1025	−0.740266
$599$	30.7166	1.25505	0.627523	−	0.778598i	$-0.284069\pi$
$599$	30.7166	1.25505	0.627523	+	0.778598i	$0.284069\pi$
$600$	0	0
$601$	28.0795	1.14539	0.572693	−	0.819770i	$-0.305899\pi$
$601$	28.0795	1.14539	0.572693	+	0.819770i	$0.305899\pi$
$602$	2.88953	0.117768
$603$	0	0
$604$	−27.2445	−1.10856
$605$	−17.9152	−0.728357
$606$	0	0
$607$	−2.89475	−0.117494	−0.0587472	−	0.998273i	$-0.518711\pi$
$607$	−2.89475	−0.117494	−0.0587472	+	0.998273i	$0.518711\pi$
$608$	−2.55491	−0.103615
$609$	0	0
$610$	−0.715363	−0.0289642
$611$	−27.0130	−1.09283
$612$	0	0
$613$	17.2282	0.695841	0.347920	−	0.937524i	$-0.386888\pi$
$613$	17.2282	0.695841	0.347920	+	0.937524i	$0.386888\pi$
$614$	8.78401	0.354494
$615$	0	0
$616$	−1.39529	−0.0562179
$617$	43.2096	1.73955	0.869776	−	0.493447i	$-0.164263\pi$
$617$	43.2096	1.73955	0.869776	+	0.493447i	$0.164263\pi$
$618$	0	0
$619$	17.4806	0.702606	0.351303	−	0.936262i	$-0.385739\pi$
$619$	17.4806	0.702606	0.351303	+	0.936262i	$0.385739\pi$
$620$	10.4170	0.418355
$621$	0	0
$622$	6.68171	0.267912
$623$	0.0324518	0.00130015
$624$	0	0
$625$	−14.7735	−0.590942
$626$	−0.907561	−0.0362734
$627$	0	0
$628$	29.3795	1.17237
$629$	19.5016	0.777578
$630$	0	0
$631$	35.4652	1.41185	0.705924	−	0.708288i	$-0.250532\pi$
$631$	35.4652	1.41185	0.705924	+	0.708288i	$0.250532\pi$
$632$	8.02933	0.319390
$633$	0	0
$634$	3.39511	0.134837
$635$	−1.85496	−0.0736119
$636$	0	0
$637$	−6.94841	−0.275306
$638$	−0.800404	−0.0316883
$639$	0	0
$640$	16.1934	0.640098
$641$	−27.8828	−1.10131	−0.550653	−	0.834734i	$-0.685621\pi$
$641$	−27.8828	−1.10131	−0.550653	+	0.834734i	$0.685621\pi$
$642$	0	0
$643$	2.92965	0.115534	0.0577670	−	0.998330i	$-0.481602\pi$
$643$	2.92965	0.115534	0.0577670	+	0.998330i	$0.481602\pi$
$644$	16.0891	0.634001
$645$	0	0
$646$	1.30135	0.0512010
$647$	−43.7801	−1.72117	−0.860587	−	0.509303i	$-0.829903\pi$
$647$	−43.7801	−1.72117	−0.860587	+	0.509303i	$0.829903\pi$
$648$	0	0
$649$	−11.5160	−0.452042
$650$	−3.34155	−0.131066
$651$	0	0
$652$	7.86310	0.307943
$653$	40.2893	1.57664	0.788320	−	0.615265i	$-0.210951\pi$
$653$	40.2893	1.57664	0.788320	+	0.615265i	$0.210951\pi$
$654$	0	0
$655$	0.814228	0.0318145
$656$	28.8090	1.12480
$657$	0	0
$658$	−1.19914	−0.0467474
$659$	−19.7014	−0.767458	−0.383729	−	0.923446i	$-0.625360\pi$
$659$	−19.7014	−0.767458	−0.383729	+	0.923446i	$0.625360\pi$
$660$	0	0
$661$	39.2335	1.52601	0.763003	−	0.646395i	$-0.223724\pi$
$661$	39.2335	1.52601	0.763003	+	0.646395i	$0.223724\pi$
$662$	2.27944	0.0885928
$663$	0	0
$664$	12.8042	0.496899
$665$	1.36602	0.0529718
$666$	0	0
$667$	18.9199	0.732583
$668$	18.2159	0.704793
$669$	0	0
$670$	−4.80962	−0.185812
$671$	−1.44839	−0.0559145
$672$	0	0
$673$	1.80350	0.0695199	0.0347600	−	0.999396i	$-0.488933\pi$
$673$	1.80350	0.0695199	0.0347600	+	0.999396i	$0.488933\pi$
$674$	9.06982	0.349357
$675$	0	0
$676$	−67.2043	−2.58478
$677$	10.4806	0.402802	0.201401	−	0.979509i	$-0.435451\pi$
$677$	10.4806	0.402802	0.201401	+	0.979509i	$0.435451\pi$
$678$	0	0
$679$	14.0672	0.539848
$680$	−12.8001	−0.490862
$681$	0	0
$682$	−1.05342	−0.0403376
$683$	−9.08766	−0.347730	−0.173865	−	0.984770i	$-0.555626\pi$
$683$	−9.08766	−0.347730	−0.173865	+	0.984770i	$0.555626\pi$
$684$	0	0
$685$	−14.2608	−0.544878
$686$	−0.308449	−0.0117766
$687$	0	0
$688$	−32.2089	−1.22795
$689$	−36.1201	−1.37607
$690$	0	0
$691$	−27.0320	−1.02835	−0.514173	−	0.857687i	$-0.671901\pi$
$691$	−27.0320	−1.02835	−0.514173	+	0.857687i	$0.671901\pi$
$692$	15.7589	0.599064
$693$	0	0
$694$	3.18109	0.120753
$695$	17.7613	0.673726
$696$	0	0
$697$	−48.0051	−1.81832
$698$	−6.73705	−0.255001
$699$	0	0
$700$	2.96990	0.112252
$701$	−20.6713	−0.780745	−0.390372	−	0.920657i	$-0.627654\pi$
$701$	−20.6713	−0.780745	−0.390372	+	0.920657i	$0.627654\pi$
$702$	0	0
$703$	−2.50668	−0.0945412
$704$	6.72628	0.253506
$705$	0	0
$706$	8.90352	0.335089
$707$	10.9152	0.410509
$708$	0	0
$709$	−36.1898	−1.35914	−0.679568	−	0.733613i	$-0.737833\pi$
$709$	−36.1898	−1.35914	−0.679568	+	0.733613i	$0.737833\pi$
$710$	−1.89519	−0.0711251
$711$	0	0
$712$	0.0390865	0.00146483
$713$	24.9008	0.932541
$714$	0	0
$715$	14.9313	0.558398
$716$	−7.96978	−0.297845
$717$	0	0
$718$	−10.5159	−0.392451
$719$	−13.3295	−0.497107	−0.248554	−	0.968618i	$-0.579955\pi$
$719$	−13.3295	−0.497107	−0.248554	+	0.968618i	$0.579955\pi$
$720$	0	0
$721$	2.52392	0.0939958
$722$	5.69325	0.211881
$723$	0	0
$724$	21.7170	0.807106
$725$	3.49244	0.129706
$726$	0	0
$727$	42.1450	1.56307	0.781537	−	0.623859i	$-0.214436\pi$
$727$	42.1450	1.56307	0.781537	+	0.623859i	$0.214436\pi$
$728$	−8.36901	−0.310176
$729$	0	0
$730$	−2.37144	−0.0877710
$731$	53.6704	1.98507
$732$	0	0
$733$	15.3186	0.565804	0.282902	−	0.959149i	$-0.408703\pi$
$733$	15.3186	0.565804	0.282902	+	0.959149i	$0.408703\pi$
$734$	−0.215189	−0.00794278
$735$	0	0
$736$	29.3039	1.08016
$737$	−9.73800	−0.358704
$738$	0	0
$739$	−25.2875	−0.930214	−0.465107	−	0.885254i	$-0.653984\pi$
$739$	−25.2875	−0.930214	−0.465107	+	0.885254i	$0.653984\pi$
$740$	12.0275	0.442140
$741$	0	0
$742$	−1.60342	−0.0588632
$743$	30.3812	1.11458	0.557289	−	0.830319i	$-0.311842\pi$
$743$	30.3812	1.11458	0.557289	+	0.830319i	$0.311842\pi$
$744$	0	0
$745$	27.2426	0.998093
$746$	−6.14946	−0.225148
$747$	0	0
$748$	−12.6424	−0.462253
$749$	−16.2638	−0.594267
$750$	0	0
$751$	32.7403	1.19471	0.597355	−	0.801977i	$-0.296218\pi$
$751$	32.7403	1.19471	0.597355	+	0.801977i	$0.296218\pi$
$752$	13.3666	0.487428
$753$	0	0
$754$	−4.80085	−0.174837
$755$	26.5308	0.965555
$756$	0	0
$757$	32.2648	1.17268	0.586342	−	0.810064i	$-0.300567\pi$
$757$	32.2648	1.17268	0.586342	+	0.810064i	$0.300567\pi$
$758$	4.74160	0.172223
$759$	0	0
$760$	1.64530	0.0596811
$761$	−12.1489	−0.440397	−0.220198	−	0.975455i	$-0.570671\pi$
$761$	−12.1489	−0.440397	−0.220198	+	0.975455i	$0.570671\pi$
$762$	0	0
$763$	18.0713	0.654224
$764$	42.9943	1.55548
$765$	0	0
$766$	−4.24457	−0.153363
$767$	−69.0733	−2.49409
$768$	0	0
$769$	21.0353	0.758552	0.379276	−	0.925284i	$-0.376173\pi$
$769$	21.0353	0.758552	0.379276	+	0.925284i	$0.376173\pi$
$770$	0.662818	0.0238863
$771$	0	0
$772$	−38.6577	−1.39132
$773$	−32.1763	−1.15730	−0.578651	−	0.815575i	$-0.696421\pi$
$773$	−32.1763	−1.15730	−0.578651	+	0.815575i	$0.696421\pi$
$774$	0	0
$775$	4.59645	0.165109
$776$	16.9432	0.608225
$777$	0	0
$778$	3.64267	0.130596
$779$	6.17045	0.221079
$780$	0	0
$781$	−3.83717	−0.137305
$782$	−14.9260	−0.533753
$783$	0	0
$784$	3.43821	0.122793
$785$	−28.6099	−1.02113
$786$	0	0
$787$	−5.25170	−0.187203	−0.0936015	−	0.995610i	$-0.529838\pi$
$787$	−5.25170	−0.187203	−0.0936015	+	0.995610i	$0.529838\pi$
$788$	10.0053	0.356423
$789$	0	0
$790$	−3.81425	−0.135705
$791$	12.6917	0.451266
$792$	0	0
$793$	−8.68750	−0.308502
$794$	2.51867	0.0893844
$795$	0	0
$796$	−28.3149	−1.00359
$797$	2.83114	0.100284	0.0501421	−	0.998742i	$-0.484033\pi$
$797$	2.83114	0.100284	0.0501421	+	0.998742i	$0.484033\pi$
$798$	0	0
$799$	−22.2730	−0.787962
$800$	5.40922	0.191245
$801$	0	0
$802$	3.90451	0.137873
$803$	−4.80144	−0.169439
$804$	0	0
$805$	−15.6677	−0.552213
$806$	−6.31846	−0.222558
$807$	0	0
$808$	13.1468	0.462503
$809$	−24.3007	−0.854368	−0.427184	−	0.904165i	$-0.640494\pi$
$809$	−24.3007	−0.854368	−0.427184	+	0.904165i	$0.640494\pi$
$810$	0	0
$811$	30.7464	1.07965	0.539826	−	0.841776i	$-0.318490\pi$
$811$	30.7464	1.07965	0.539826	+	0.841776i	$0.318490\pi$
$812$	4.26690	0.149739
$813$	0	0
$814$	−1.21629	−0.0426310
$815$	−7.65712	−0.268217
$816$	0	0
$817$	−6.89866	−0.241354
$818$	0.775188	0.0271038
$819$	0	0
$820$	−29.6070	−1.03392
$821$	38.9672	1.35996	0.679982	−	0.733229i	$-0.261988\pi$
$821$	38.9672	1.35996	0.679982	+	0.733229i	$0.261988\pi$
$822$	0	0
$823$	41.1089	1.43297	0.716483	−	0.697604i	$-0.245751\pi$
$823$	41.1089	1.43297	0.716483	+	0.697604i	$0.245751\pi$
$824$	3.03993	0.105901
$825$	0	0
$826$	−3.06625	−0.106688
$827$	−7.21565	−0.250913	−0.125456	−	0.992099i	$-0.540039\pi$
$827$	−7.21565	−0.250913	−0.125456	+	0.992099i	$0.540039\pi$
$828$	0	0
$829$	37.7023	1.30946	0.654728	−	0.755864i	$-0.272783\pi$
$829$	37.7023	1.30946	0.654728	+	0.755864i	$0.272783\pi$
$830$	−6.08249	−0.211126
$831$	0	0
$832$	40.3445	1.39869
$833$	−5.72916	−0.198504
$834$	0	0
$835$	−17.7387	−0.613873
$836$	1.62503	0.0562027
$837$	0	0
$838$	−2.58236	−0.0892062
$839$	−53.8472	−1.85901	−0.929506	−	0.368807i	$-0.879766\pi$
$839$	−53.8472	−1.85901	−0.929506	+	0.368807i	$0.879766\pi$
$840$	0	0
$841$	−23.9824	−0.826978
$842$	−4.33703	−0.149464
$843$	0	0
$844$	−9.80413	−0.337472
$845$	65.4439	2.25134
$846$	0	0
$847$	−9.65800	−0.331853
$848$	17.8729	0.613758
$849$	0	0
$850$	−2.75520	−0.0945026
$851$	28.7507	0.985560
$852$	0	0
$853$	−18.9652	−0.649357	−0.324679	−	0.945824i	$-0.605256\pi$
$853$	−18.9652	−0.649357	−0.324679	+	0.945824i	$0.605256\pi$
$854$	−0.385649	−0.0131966
$855$	0	0
$856$	−19.5889	−0.669536
$857$	51.1191	1.74620	0.873098	−	0.487544i	$-0.162107\pi$
$857$	51.1191	1.74620	0.873098	+	0.487544i	$0.162107\pi$
$858$	0	0
$859$	4.97281	0.169670	0.0848351	−	0.996395i	$-0.472964\pi$
$859$	4.97281	0.169670	0.0848351	+	0.996395i	$0.472964\pi$
$860$	33.1010	1.12874
$861$	0	0
$862$	−7.57534	−0.258017
$863$	17.5650	0.597918	0.298959	−	0.954266i	$-0.403361\pi$
$863$	17.5650	0.597918	0.298959	+	0.954266i	$0.403361\pi$
$864$	0	0
$865$	−15.3461	−0.521783
$866$	−4.29761	−0.146039
$867$	0	0
$868$	5.61573	0.190610
$869$	−7.72267	−0.261974
$870$	0	0
$871$	−58.4089	−1.97911
$872$	21.7659	0.737086
$873$	0	0
$874$	1.91855	0.0648960
$875$	−12.1669	−0.411317
$876$	0	0
$877$	−1.52760	−0.0515833	−0.0257916	−	0.999667i	$-0.508211\pi$
$877$	−1.52760	−0.0515833	−0.0257916	+	0.999667i	$0.508211\pi$
$878$	2.69062	0.0908041
$879$	0	0
$880$	−7.38828	−0.249059
$881$	−13.4649	−0.453644	−0.226822	−	0.973936i	$-0.572834\pi$
$881$	−13.4649	−0.453644	−0.226822	+	0.973936i	$0.572834\pi$
$882$	0	0
$883$	−4.25568	−0.143215	−0.0716075	−	0.997433i	$-0.522813\pi$
$883$	−4.25568	−0.143215	−0.0716075	+	0.997433i	$0.522813\pi$
$884$	−75.8298	−2.55043
$885$	0	0
$886$	7.57778	0.254581
$887$	22.9191	0.769548	0.384774	−	0.923011i	$-0.374279\pi$
$887$	22.9191	0.769548	0.384774	+	0.923011i	$0.374279\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−0.0185676	−0.000622388 0
$891$	0	0
$892$	−55.8840	−1.87113
$893$	2.86291	0.0958037
$894$	0	0
$895$	7.76101	0.259422
$896$	8.72975	0.291641
$897$	0	0
$898$	−1.97290	−0.0658366
$899$	6.60378	0.220248
$900$	0	0
$901$	−29.7820	−0.992184
$902$	2.99402	0.0996901
$903$	0	0
$904$	15.2865	0.508422
$905$	−21.1481	−0.702987
$906$	0	0
$907$	−30.2405	−1.00412	−0.502060	−	0.864833i	$-0.667424\pi$
$907$	−30.2405	−1.00412	−0.502060	+	0.864833i	$0.667424\pi$
$908$	−2.04773	−0.0679565
$909$	0	0
$910$	3.97561	0.131790
$911$	4.80322	0.159138	0.0795688	−	0.996829i	$-0.474646\pi$
$911$	4.80322	0.159138	0.0795688	+	0.996829i	$0.474646\pi$
$912$	0	0
$913$	−12.3152	−0.407572
$914$	−9.54012	−0.315559
$915$	0	0
$916$	−28.5190	−0.942295
$917$	0.438946	0.0144953
$918$	0	0
$919$	−40.4734	−1.33509	−0.667547	−	0.744568i	$-0.732656\pi$
$919$	−40.4734	−1.33509	−0.667547	+	0.744568i	$0.732656\pi$
$920$	−18.8709	−0.622155
$921$	0	0
$922$	1.08987	0.0358928
$923$	−23.0155	−0.757565
$924$	0	0
$925$	5.30710	0.174496
$926$	5.46723	0.179664
$927$	0	0
$928$	7.77150	0.255112
$929$	−32.5731	−1.06869	−0.534345	−	0.845266i	$-0.679442\pi$
$929$	−32.5731	−1.06869	−0.534345	+	0.845266i	$0.679442\pi$
$930$	0	0
$931$	0.736412	0.0241349
$932$	49.9855	1.63733
$933$	0	0
$934$	−6.63037	−0.216952
$935$	12.3113	0.402621
$936$	0	0
$937$	−10.6999	−0.349551	−0.174776	−	0.984608i	$-0.555920\pi$
$937$	−10.6999	−0.349551	−0.174776	+	0.984608i	$0.555920\pi$
$938$	−2.59284	−0.0846592
$939$	0	0
$940$	−13.7368	−0.448044
$941$	−7.17070	−0.233758	−0.116879	−	0.993146i	$-0.537289\pi$
$941$	−7.17070	−0.233758	−0.116879	+	0.993146i	$0.537289\pi$
$942$	0	0
$943$	−70.7727	−2.30468
$944$	34.1788	1.11243
$945$	0	0
$946$	−3.34737	−0.108832
$947$	−45.0916	−1.46528	−0.732641	−	0.680616i	$-0.761712\pi$
$947$	−45.0916	−1.46528	−0.732641	+	0.680616i	$0.761712\pi$
$948$	0	0
$949$	−28.7992	−0.934862
$950$	0.354147	0.0114900
$951$	0	0
$952$	−6.90048	−0.223646
$953$	−27.4772	−0.890072	−0.445036	−	0.895513i	$-0.646809\pi$
$953$	−27.4772	−0.890072	−0.445036	+	0.895513i	$0.646809\pi$
$954$	0	0
$955$	−41.8680	−1.35482
$956$	24.8984	0.805272
$957$	0	0
$958$	−9.18671	−0.296809
$959$	−7.68794	−0.248257
$960$	0	0
$961$	−22.3087	−0.719635
$962$	−7.29535	−0.235212
$963$	0	0
$964$	−11.2663	−0.362863
$965$	37.6451	1.21184
$966$	0	0
$967$	−20.3503	−0.654421	−0.327210	−	0.944952i	$-0.606109\pi$
$967$	−20.3503	−0.654421	−0.327210	+	0.944952i	$0.606109\pi$
$968$	−11.6326	−0.373885
$969$	0	0
$970$	−8.04867	−0.258427
$971$	42.2647	1.35634	0.678169	−	0.734906i	$-0.262774\pi$
$971$	42.2647	1.35634	0.678169	+	0.734906i	$0.262774\pi$
$972$	0	0
$973$	9.57505	0.306962
$974$	8.39519	0.268999
$975$	0	0
$976$	4.29874	0.137599
$977$	25.1668	0.805157	0.402579	−	0.915385i	$-0.368114\pi$
$977$	25.1668	0.805157	0.402579	+	0.915385i	$0.368114\pi$
$978$	0	0
$979$	−0.0375937	−0.00120150
$980$	−3.53344	−0.112872
$981$	0	0
$982$	6.16185	0.196632
$983$	31.6165	1.00841	0.504205	−	0.863584i	$-0.331786\pi$
$983$	31.6165	1.00841	0.504205	+	0.863584i	$0.331786\pi$
$984$	0	0
$985$	−9.74317	−0.310443
$986$	−3.95844	−0.126062
$987$	0	0
$988$	9.74697	0.310092
$989$	79.1250	2.51603
$990$	0	0
$991$	−17.6056	−0.559260	−0.279630	−	0.960108i	$-0.590212\pi$
$991$	−17.6056	−0.559260	−0.279630	+	0.960108i	$0.590212\pi$
$992$	10.2282	0.324745
$993$	0	0
$994$	−1.02169	−0.0324059
$995$	27.5732	0.874128
$996$	0	0
$997$	−50.4731	−1.59850	−0.799249	−	0.601000i	$-0.794769\pi$
$997$	−50.4731	−1.59850	−0.799249	+	0.601000i	$0.794769\pi$
$998$	−2.93165	−0.0927998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.18	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.23	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.18	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.23	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.308449 q^{2} -1.90486 q^{4} +1.85496 q^{5} +1.00000 q^{7} +1.20445 q^{8} +O(q^{10})\)	\(q-0.308449 q^{2} -1.90486 q^{4} +1.85496 q^{5} +1.00000 q^{7} +1.20445 q^{8} -0.572160 q^{10} -1.15845 q^{11} -6.94841 q^{13} -0.308449 q^{14} +3.43821 q^{16} -5.72916 q^{17} +0.736412 q^{19} -3.53344 q^{20} +0.357322 q^{22} -8.44637 q^{23} -1.55912 q^{25} +2.14323 q^{26} -1.90486 q^{28} -2.24001 q^{29} -2.94811 q^{31} -3.46941 q^{32} +1.76715 q^{34} +1.85496 q^{35} -3.40391 q^{37} -0.227145 q^{38} +2.23420 q^{40} +8.37907 q^{41} -9.36793 q^{43} +2.20668 q^{44} +2.60527 q^{46} +3.88765 q^{47} +1.00000 q^{49} +0.480908 q^{50} +13.2358 q^{52} +5.19832 q^{53} -2.14888 q^{55} +1.20445 q^{56} +0.690927 q^{58} +9.94087 q^{59} +1.25028 q^{61} +0.909339 q^{62} -5.80628 q^{64} -12.8890 q^{65} +8.40607 q^{67} +10.9133 q^{68} -0.572160 q^{70} +3.31234 q^{71} +4.14472 q^{73} +1.04993 q^{74} -1.40276 q^{76} -1.15845 q^{77} +6.66640 q^{79} +6.37774 q^{80} -2.58451 q^{82} +10.6307 q^{83} -10.6274 q^{85} +2.88953 q^{86} -1.39529 q^{88} +0.0324518 q^{89} -6.94841 q^{91} +16.0891 q^{92} -1.19914 q^{94} +1.36602 q^{95} +14.0672 q^{97} -0.308449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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