LMFDB - Embedded newform 8281.2.a.cs.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,0,0,20,0,0,0,0,16,0,0,0,0,0,0,28,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 42x^{14} + 641x^{12} - 4448x^{10} + 14076x^{8} - 17900x^{6} + 6960x^{4} - 416x^{2} + 4 $ x^16 - 42x^14 + 641x^12 - 4448x^10 + 14076x^8 - 17900x^6 + 6960x^4 - 416*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.06973$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.65552 q^{2} +0.494977 q^{3} +0.740740 q^{4} -2.87389 q^{5} -0.819443 q^{6} +2.08473 q^{8} -2.75500 q^{9} +4.75778 q^{10} -2.58138 q^{11} +0.366649 q^{12} -1.42251 q^{15} -4.93278 q^{16} +2.27584 q^{17} +4.56095 q^{18} -4.39571 q^{19} -2.12881 q^{20} +4.27352 q^{22} -7.16926 q^{23} +1.03189 q^{24} +3.25926 q^{25} -2.84859 q^{27} +6.19574 q^{29} +2.35499 q^{30} -6.85404 q^{31} +3.99686 q^{32} -1.27772 q^{33} -3.76769 q^{34} -2.04074 q^{36} -5.89241 q^{37} +7.27717 q^{38} -5.99128 q^{40} -3.98014 q^{41} -1.31425 q^{43} -1.91213 q^{44} +7.91757 q^{45} +11.8688 q^{46} +1.35208 q^{47} -2.44161 q^{48} -5.39576 q^{50} +1.12649 q^{51} -8.28778 q^{53} +4.71589 q^{54} +7.41860 q^{55} -2.17577 q^{57} -10.2572 q^{58} -9.61085 q^{59} -1.05371 q^{60} +9.55301 q^{61} +11.3470 q^{62} +3.24870 q^{64} +2.11529 q^{66} -15.9095 q^{67} +1.68581 q^{68} -3.54862 q^{69} -4.79396 q^{71} -5.74342 q^{72} +5.70507 q^{73} +9.75500 q^{74} +1.61326 q^{75} -3.25607 q^{76} -2.01426 q^{79} +14.1763 q^{80} +6.85501 q^{81} +6.58920 q^{82} +3.11798 q^{83} -6.54052 q^{85} +2.17577 q^{86} +3.06675 q^{87} -5.38147 q^{88} -13.0584 q^{89} -13.1077 q^{90} -5.31056 q^{92} -3.39259 q^{93} -2.23839 q^{94} +12.6328 q^{95} +1.97835 q^{96} -14.0382 q^{97} +7.11169 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} + 36 q^{23} + 44 q^{25} + 36 q^{29} - 52 q^{36} + 36 q^{43} + 72 q^{51} + 12 q^{53} + 164 q^{64} + 96 q^{74} + 36 q^{79} + 16 q^{81} - 136 q^{88} + 24 q^{92}+ \cdots + 84 q^{95}+O(q^{100}) $ 16 * q + 20 * q^4 + 16 * q^9 + 28 * q^16 - 8 * q^22 + 36 * q^23 + 44 * q^25 + 36 * q^29 - 52 * q^36 + 36 * q^43 + 72 * q^51 + 12 * q^53 + 164 * q^64 + 96 * q^74 + 36 * q^79 + 16 * q^81 - 136 * q^88 + 24 * q^92 + 84 * q^95

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 $

$2$	−1.65552	−1.17063	−0.585314	−	0.810807i	$-0.699029\pi$
$2$	−1.65552	−1.17063	−0.585314	+	0.810807i	$0.699029\pi$
$3$	0.494977	0.285775	0.142888	−	0.989739i	$-0.454361\pi$
$3$	0.494977	0.285775	0.142888	+	0.989739i	$0.454361\pi$
$4$	0.740740	0.370370
$5$	−2.87389	−1.28524	−0.642622	−	0.766183i	$-0.722153\pi$
$5$	−2.87389	−1.28524	−0.642622	+	0.766183i	$0.722153\pi$
$6$	−0.819443	−0.334536
$7$	0	0
$7$	0	0
$8$	2.08473	0.737063
$9$	−2.75500	−0.918333
$10$	4.75778	1.50454
$11$	−2.58138	−0.778315	−0.389157	−	0.921171i	$-0.627234\pi$
$11$	−2.58138	−0.778315	−0.389157	+	0.921171i	$0.627234\pi$
$12$	0.366649	0.105842
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.42251	−0.367291
$16$	−4.93278	−1.23320
$17$	2.27584	0.551972	0.275986	−	0.961162i	$-0.410996\pi$
$17$	2.27584	0.551972	0.275986	+	0.961162i	$0.410996\pi$
$18$	4.56095	1.07503
$19$	−4.39571	−1.00844	−0.504222	−	0.863574i	$-0.668221\pi$
$19$	−4.39571	−1.00844	−0.504222	+	0.863574i	$0.668221\pi$
$20$	−2.12881	−0.476016
$21$	0	0
$22$	4.27352	0.911117
$23$	−7.16926	−1.49489	−0.747447	−	0.664321i	$-0.768721\pi$
$23$	−7.16926	−1.49489	−0.747447	+	0.664321i	$0.768721\pi$
$24$	1.03189	0.210634
$25$	3.25926	0.651852
$26$	0	0
$27$	−2.84859	−0.548212
$28$	0	0
$29$	6.19574	1.15052	0.575260	−	0.817971i	$-0.304901\pi$
$29$	6.19574	1.15052	0.575260	+	0.817971i	$0.304901\pi$
$30$	2.35499	0.429961
$31$	−6.85404	−1.23102	−0.615511	−	0.788129i	$-0.711050\pi$
$31$	−6.85404	−1.23102	−0.615511	+	0.788129i	$0.711050\pi$
$32$	3.99686	0.706551
$33$	−1.27772	−0.222423
$34$	−3.76769	−0.646154
$35$	0	0
$36$	−2.04074	−0.340123
$37$	−5.89241	−0.968707	−0.484353	−	0.874872i	$-0.660945\pi$
$37$	−5.89241	−0.968707	−0.484353	+	0.874872i	$0.660945\pi$
$38$	7.27717	1.18051
$39$	0	0
$40$	−5.99128	−0.947305
$41$	−3.98014	−0.621594	−0.310797	−	0.950476i	$-0.600596\pi$
$41$	−3.98014	−0.621594	−0.310797	+	0.950476i	$0.600596\pi$
$42$	0	0
$43$	−1.31425	−0.200422	−0.100211	−	0.994966i	$-0.531952\pi$
$43$	−1.31425	−0.200422	−0.100211	+	0.994966i	$0.531952\pi$
$44$	−1.91213	−0.288264
$45$	7.91757	1.18028
$46$	11.8688	1.74997
$47$	1.35208	0.197221	0.0986106	−	0.995126i	$-0.468560\pi$
$47$	1.35208	0.197221	0.0986106	+	0.995126i	$0.468560\pi$
$48$	−2.44161	−0.352417
$49$	0	0
$50$	−5.39576	−0.763076
$51$	1.12649	0.157740
$52$	0	0
$53$	−8.28778	−1.13841	−0.569207	−	0.822194i	$-0.692750\pi$
$53$	−8.28778	−1.13841	−0.569207	+	0.822194i	$0.692750\pi$
$54$	4.71589	0.641752
$55$	7.41860	1.00032
$56$	0	0
$57$	−2.17577	−0.288188
$58$	−10.2572	−1.34683
$59$	−9.61085	−1.25123	−0.625613	−	0.780133i	$-0.715151\pi$
$59$	−9.61085	−1.25123	−0.625613	+	0.780133i	$0.715151\pi$
$60$	−1.05371	−0.136033
$61$	9.55301	1.22314	0.611569	−	0.791191i	$-0.290539\pi$
$61$	9.55301	1.22314	0.611569	+	0.791191i	$0.290539\pi$
$62$	11.3470	1.44107
$63$	0	0
$64$	3.24870	0.406087
$65$	0	0
$66$	2.11529	0.260375
$67$	−15.9095	−1.94366	−0.971830	−	0.235685i	$-0.924267\pi$
$67$	−15.9095	−1.94366	−0.971830	+	0.235685i	$0.924267\pi$
$68$	1.68581	0.204434
$69$	−3.54862	−0.427204
$70$	0	0
$71$	−4.79396	−0.568938	−0.284469	−	0.958685i	$-0.591817\pi$
$71$	−4.79396	−0.568938	−0.284469	+	0.958685i	$0.591817\pi$
$72$	−5.74342	−0.676869
$73$	5.70507	0.667728	0.333864	−	0.942621i	$-0.391647\pi$
$73$	5.70507	0.667728	0.333864	+	0.942621i	$0.391647\pi$
$74$	9.75500	1.13400
$75$	1.61326	0.186283
$76$	−3.25607	−0.373497
$77$	0	0
$78$	0	0
$79$	−2.01426	−0.226622	−0.113311	−	0.993560i	$-0.536146\pi$
$79$	−2.01426	−0.226622	−0.113311	+	0.993560i	$0.536146\pi$
$80$	14.1763	1.58496
$81$	6.85501	0.761667
$82$	6.58920	0.727655
$83$	3.11798	0.342243	0.171122	−	0.985250i	$-0.445261\pi$
$83$	3.11798	0.342243	0.171122	+	0.985250i	$0.445261\pi$
$84$	0	0
$85$	−6.54052	−0.709419
$86$	2.17577	0.234620
$87$	3.06675	0.328790
$88$	−5.38147	−0.573667
$89$	−13.0584	−1.38418	−0.692092	−	0.721810i	$-0.743311\pi$
$89$	−13.0584	−1.38418	−0.692092	+	0.721810i	$0.743311\pi$
$90$	−13.1077	−1.38167
$91$	0	0
$92$	−5.31056	−0.553664
$93$	−3.39259	−0.351795
$94$	−2.23839	−0.230873
$95$	12.6328	1.29610
$96$	1.97835	0.201915
$97$	−14.0382	−1.42536	−0.712682	−	0.701488i	$-0.752520\pi$
$97$	−14.0382	−1.42536	−0.712682	+	0.701488i	$0.752520\pi$
$98$	0	0
$99$	7.11169	0.714752
$100$	2.41426	0.241426
$101$	−17.4483	−1.73617	−0.868083	−	0.496419i	$-0.834648\pi$
$101$	−17.4483	−1.73617	−0.868083	+	0.496419i	$0.834648\pi$
$102$	−1.86492	−0.184655
$103$	−6.55698	−0.646078	−0.323039	−	0.946386i	$-0.604705\pi$
$103$	−6.55698	−0.646078	−0.323039	+	0.946386i	$0.604705\pi$
$104$	0	0
$105$	0	0
$106$	13.7206	1.33266
$107$	15.6020	1.50831	0.754153	−	0.656699i	$-0.228048\pi$
$107$	15.6020	1.50831	0.754153	+	0.656699i	$0.228048\pi$
$108$	−2.11006	−0.203041
$109$	10.0171	0.959466	0.479733	−	0.877414i	$-0.340733\pi$
$109$	10.0171	0.959466	0.479733	+	0.877414i	$0.340733\pi$
$110$	−12.2816	−1.17101
$111$	−2.91661	−0.276832
$112$	0	0
$113$	−3.95074	−0.371654	−0.185827	−	0.982582i	$-0.559496\pi$
$113$	−3.95074	−0.371654	−0.185827	+	0.982582i	$0.559496\pi$
$114$	3.60203	0.337361
$115$	20.6037	1.92130
$116$	4.58943	0.426118
$117$	0	0
$118$	15.9109	1.46472
$119$	0	0
$120$	−2.96555	−0.270716
$121$	−4.33649	−0.394226
$122$	−15.8152	−1.43184
$123$	−1.97008	−0.177636
$124$	−5.07706	−0.455933
$125$	5.00270	0.447455
$126$	0	0
$127$	2.37353	0.210616	0.105308	−	0.994440i	$-0.466417\pi$
$127$	2.37353	0.210616	0.105308	+	0.994440i	$0.466417\pi$
$128$	−13.3720	−1.18193
$129$	−0.650526	−0.0572756
$130$	0	0
$131$	2.13546	0.186576	0.0932878	−	0.995639i	$-0.470262\pi$
$131$	2.13546	0.186576	0.0932878	+	0.995639i	$0.470262\pi$
$132$	−0.946460	−0.0823788
$133$	0	0
$134$	26.3385	2.27530
$135$	8.18655	0.704586
$136$	4.74451	0.406838
$137$	−0.429210	−0.0366699	−0.0183349	−	0.999832i	$-0.505837\pi$
$137$	−0.429210	−0.0366699	−0.0183349	+	0.999832i	$0.505837\pi$
$138$	5.87480	0.500096
$139$	−19.4858	−1.65276	−0.826381	−	0.563112i	$-0.809604\pi$
$139$	−19.4858	−1.65276	−0.826381	+	0.563112i	$0.809604\pi$
$140$	0	0
$141$	0.669248	0.0563609
$142$	7.93648	0.666015
$143$	0	0
$144$	13.5898	1.13248
$145$	−17.8059	−1.47870
$146$	−9.44485	−0.781661
$147$	0	0
$148$	−4.36475	−0.358780
$149$	−22.9004	−1.87607	−0.938037	−	0.346534i	$-0.887358\pi$
$149$	−22.9004	−1.87607	−0.938037	+	0.346534i	$0.887358\pi$
$150$	−2.67078	−0.218068
$151$	−11.2207	−0.913131	−0.456565	−	0.889690i	$-0.650921\pi$
$151$	−11.2207	−0.913131	−0.456565	+	0.889690i	$0.650921\pi$
$152$	−9.16385	−0.743286
$153$	−6.26993	−0.506894
$154$	0	0
$155$	19.6978	1.58216
$156$	0	0
$157$	−3.78799	−0.302315	−0.151157	−	0.988510i	$-0.548300\pi$
$157$	−3.78799	−0.302315	−0.151157	+	0.988510i	$0.548300\pi$
$158$	3.33464	0.265290
$159$	−4.10226	−0.325330
$160$	−11.4865	−0.908091
$161$	0	0
$162$	−11.3486	−0.891629
$163$	19.6130	1.53621	0.768103	−	0.640326i	$-0.221201\pi$
$163$	19.6130	1.53621	0.768103	+	0.640326i	$0.221201\pi$
$164$	−2.94825	−0.230220
$165$	3.67204	0.285868
$166$	−5.16188	−0.400639
$167$	−10.2922	−0.796435	−0.398217	−	0.917291i	$-0.630371\pi$
$167$	−10.2922	−0.796435	−0.398217	+	0.917291i	$0.630371\pi$
$168$	0	0
$169$	0	0
$170$	10.8279	0.830466
$171$	12.1102	0.926087
$172$	−0.973521	−0.0742303
$173$	6.00439	0.456505	0.228253	−	0.973602i	$-0.426699\pi$
$173$	6.00439	0.456505	0.228253	+	0.973602i	$0.426699\pi$
$174$	−5.07706	−0.384891
$175$	0	0
$176$	12.7334	0.959815
$177$	−4.75715	−0.357569
$178$	21.6184	1.62036
$179$	−7.75130	−0.579359	−0.289680	−	0.957124i	$-0.593549\pi$
$179$	−7.75130	−0.579359	−0.289680	+	0.957124i	$0.593549\pi$
$180$	5.86486	0.437141
$181$	−19.9999	−1.48658	−0.743289	−	0.668970i	$-0.766735\pi$
$181$	−19.9999	−1.48658	−0.743289	+	0.668970i	$0.766735\pi$
$182$	0	0
$183$	4.72852	0.349542
$184$	−14.9460	−1.10183
$185$	16.9342	1.24502
$186$	5.61649	0.411821
$187$	−5.87480	−0.429608
$188$	1.00154	0.0730448
$189$	0	0
$190$	−20.9138	−1.51725
$191$	20.4835	1.48214	0.741068	−	0.671430i	$-0.234320\pi$
$191$	20.4835	1.48214	0.741068	+	0.671430i	$0.234320\pi$
$192$	1.60803	0.116050
$193$	−12.4329	−0.894942	−0.447471	−	0.894298i	$-0.647675\pi$
$193$	−12.4329	−0.894942	−0.447471	+	0.894298i	$0.647675\pi$
$194$	23.2405	1.66857
$195$	0	0
$196$	0	0
$197$	−18.9805	−1.35231	−0.676154	−	0.736761i	$-0.736354\pi$
$197$	−18.9805	−1.35231	−0.676154	+	0.736761i	$0.736354\pi$
$198$	−11.7735	−0.836709
$199$	12.8188	0.908701	0.454350	−	0.890823i	$-0.349871\pi$
$199$	12.8188	0.908701	0.454350	+	0.890823i	$0.349871\pi$
$200$	6.79467	0.480456
$201$	−7.87485	−0.555449
$202$	28.8859	2.03241
$203$	0	0
$204$	0.834435	0.0584221
$205$	11.4385	0.798900
$206$	10.8552	0.756317
$207$	19.7513	1.37281
$208$	0	0
$209$	11.3470	0.784887
$210$	0	0
$211$	−7.86926	−0.541743	−0.270871	−	0.962616i	$-0.587312\pi$
$211$	−7.86926	−0.541743	−0.270871	+	0.962616i	$0.587312\pi$
$212$	−6.13909	−0.421634
$213$	−2.37290	−0.162588
$214$	−25.8294	−1.76566
$215$	3.77703	0.257591
$216$	−5.93854	−0.404066
$217$	0	0
$218$	−16.5835	−1.12318
$219$	2.82388	0.190820
$220$	5.49526	0.370490
$221$	0	0
$222$	4.82850	0.324068
$223$	−7.77059	−0.520357	−0.260179	−	0.965560i	$-0.583781\pi$
$223$	−7.77059	−0.520357	−0.260179	+	0.965560i	$0.583781\pi$
$224$	0	0
$225$	−8.97925	−0.598617
$226$	6.54052	0.435069
$227$	25.5431	1.69536	0.847678	−	0.530511i	$-0.178000\pi$
$227$	25.5431	1.69536	0.847678	+	0.530511i	$0.178000\pi$
$228$	−1.61168	−0.106736
$229$	−12.8986	−0.852362	−0.426181	−	0.904638i	$-0.640141\pi$
$229$	−12.8986	−0.852362	−0.426181	+	0.904638i	$0.640141\pi$
$230$	−34.1098	−2.24913
$231$	0	0
$232$	12.9164	0.848005
$233$	8.14278	0.533451	0.266726	−	0.963772i	$-0.414058\pi$
$233$	8.14278	0.533451	0.266726	+	0.963772i	$0.414058\pi$
$234$	0	0
$235$	−3.88573	−0.253477
$236$	−7.11914	−0.463417
$237$	−0.997011	−0.0647628
$238$	0	0
$239$	23.8122	1.54028	0.770140	−	0.637875i	$-0.220186\pi$
$239$	23.8122	1.54028	0.770140	+	0.637875i	$0.220186\pi$
$240$	7.01694	0.452941
$241$	0.288546	0.0185869	0.00929346	−	0.999957i	$-0.497042\pi$
$241$	0.288546	0.0185869	0.00929346	+	0.999957i	$0.497042\pi$
$242$	7.17913	0.461492
$243$	11.9388	0.765877
$244$	7.07630	0.453013
$245$	0	0
$246$	3.26150	0.207946
$247$	0	0
$248$	−14.2888	−0.907340
$249$	1.54333	0.0978045
$250$	−8.28206	−0.523803
$251$	−7.61843	−0.480871	−0.240435	−	0.970665i	$-0.577290\pi$
$251$	−7.61843	−0.480871	−0.240435	+	0.970665i	$0.577290\pi$
$252$	0	0
$253$	18.5066	1.16350
$254$	−3.92942	−0.246554
$255$	−3.23741	−0.202734
$256$	15.6402	0.977511
$257$	−11.6560	−0.727082	−0.363541	−	0.931578i	$-0.618432\pi$
$257$	−11.6560	−0.727082	−0.363541	+	0.931578i	$0.618432\pi$
$258$	1.07696	0.0670484
$259$	0	0
$260$	0	0
$261$	−17.0693	−1.05656
$262$	−3.53529	−0.218411
$263$	−13.5528	−0.835700	−0.417850	−	0.908516i	$-0.637216\pi$
$263$	−13.5528	−0.835700	−0.417850	+	0.908516i	$0.637216\pi$
$264$	−2.66370	−0.163940
$265$	23.8182	1.46314
$266$	0	0
$267$	−6.46359	−0.395565
$268$	−11.7848	−0.719873
$269$	−17.8332	−1.08731	−0.543656	−	0.839308i	$-0.682960\pi$
$269$	−17.8332	−1.08731	−0.543656	+	0.839308i	$0.682960\pi$
$270$	−13.5530	−0.824808
$271$	−30.3743	−1.84511	−0.922555	−	0.385866i	$-0.873903\pi$
$271$	−30.3743	−1.84511	−0.922555	+	0.385866i	$0.873903\pi$
$272$	−11.2262	−0.680690
$273$	0	0
$274$	0.710564	0.0429268
$275$	−8.41338	−0.507346
$276$	−2.62860	−0.158223
$277$	5.21425	0.313294	0.156647	−	0.987655i	$-0.449932\pi$
$277$	5.21425	0.313294	0.156647	+	0.987655i	$0.449932\pi$
$278$	32.2591	1.93477
$279$	18.8829	1.13049
$280$	0	0
$281$	1.66255	0.0991794	0.0495897	−	0.998770i	$-0.484209\pi$
$281$	1.66255	0.0991794	0.0495897	+	0.998770i	$0.484209\pi$
$282$	−1.10795	−0.0659776
$283$	−13.7957	−0.820066	−0.410033	−	0.912071i	$-0.634483\pi$
$283$	−13.7957	−0.820066	−0.410033	+	0.912071i	$0.634483\pi$
$284$	−3.55107	−0.210718
$285$	6.25294	0.370392
$286$	0	0
$287$	0	0
$288$	−11.0113	−0.648849
$289$	−11.8206	−0.695327
$290$	29.4780	1.73101
$291$	−6.94859	−0.407333
$292$	4.22597	0.247306
$293$	−12.8576	−0.751149	−0.375574	−	0.926792i	$-0.622555\pi$
$293$	−12.8576	−0.751149	−0.375574	+	0.926792i	$0.622555\pi$
$294$	0	0
$295$	27.6206	1.60813
$296$	−12.2841	−0.713998
$297$	7.35329	0.426681
$298$	37.9120	2.19619
$299$	0	0
$300$	1.19500	0.0689936
$301$	0	0
$302$	18.5761	1.06894
$303$	−8.63648	−0.496153
$304$	21.6831	1.24361
$305$	−27.4543	−1.57203
$306$	10.3800	0.593384
$307$	20.9016	1.19292	0.596458	−	0.802644i	$-0.296574\pi$
$307$	20.9016	1.19292	0.596458	+	0.802644i	$0.296574\pi$
$308$	0	0
$309$	−3.24555	−0.184633
$310$	−32.6100	−1.85212
$311$	17.1392	0.971876	0.485938	−	0.873993i	$-0.338478\pi$
$311$	17.1392	0.971876	0.485938	+	0.873993i	$0.338478\pi$
$312$	0	0
$313$	−2.55266	−0.144285	−0.0721424	−	0.997394i	$-0.522984\pi$
$313$	−2.55266	−0.144285	−0.0721424	+	0.997394i	$0.522984\pi$
$314$	6.27109	0.353898
$315$	0	0
$316$	−1.49204	−0.0839339
$317$	27.2888	1.53269	0.766345	−	0.642430i	$-0.222073\pi$
$317$	27.2888	1.53269	0.766345	+	0.642430i	$0.222073\pi$
$318$	6.79136	0.380841
$319$	−15.9936	−0.895467
$320$	−9.33641	−0.521921
$321$	7.72264	0.431036
$322$	0	0
$323$	−10.0039	−0.556633
$324$	5.07778	0.282099
$325$	0	0
$326$	−32.4696	−1.79833
$327$	4.95824	0.274192
$328$	−8.29751	−0.458154
$329$	0	0
$330$	−6.07912	−0.334645
$331$	−9.69307	−0.532779	−0.266390	−	0.963865i	$-0.585831\pi$
$331$	−9.69307	−0.532779	−0.266390	+	0.963865i	$0.585831\pi$
$332$	2.30961	0.126757
$333$	16.2336	0.889595
$334$	17.0389	0.932329
$335$	45.7223	2.49808
$336$	0	0
$337$	−6.31370	−0.343929	−0.171965	−	0.985103i	$-0.555012\pi$
$337$	−6.31370	−0.343929	−0.171965	+	0.985103i	$0.555012\pi$
$338$	0	0
$339$	−1.95552	−0.106210
$340$	−4.84482	−0.262747
$341$	17.6929	0.958122
$342$	−20.0486	−1.08410
$343$	0	0
$344$	−2.73986	−0.147724
$345$	10.1984	0.549061
$346$	−9.94038	−0.534398
$347$	29.1200	1.56324	0.781622	−	0.623753i	$-0.214393\pi$
$347$	29.1200	1.56324	0.781622	+	0.623753i	$0.214393\pi$
$348$	2.27166	0.121774
$349$	−20.3297	−1.08822	−0.544112	−	0.839013i	$-0.683133\pi$
$349$	−20.3297	−1.08822	−0.544112	+	0.839013i	$0.683133\pi$
$350$	0	0
$351$	0	0
$352$	−10.3174	−0.549919
$353$	14.1980	0.755683	0.377841	−	0.925870i	$-0.376666\pi$
$353$	14.1980	0.755683	0.377841	+	0.925870i	$0.376666\pi$
$354$	7.87555	0.418581
$355$	13.7773	0.731224
$356$	−9.67285	−0.512660
$357$	0	0
$358$	12.8324	0.678214
$359$	7.50164	0.395921	0.197961	−	0.980210i	$-0.436568\pi$
$359$	7.50164	0.395921	0.197961	+	0.980210i	$0.436568\pi$
$360$	16.5060	0.869941
$361$	0.322229	0.0169594
$362$	33.1101	1.74023
$363$	−2.14646	−0.112660
$364$	0	0
$365$	−16.3958	−0.858193
$366$	−7.82815	−0.409184
$367$	−28.5883	−1.49230	−0.746149	−	0.665779i	$-0.768099\pi$
$367$	−28.5883	−1.49230	−0.746149	+	0.665779i	$0.768099\pi$
$368$	35.3644	1.84350
$369$	10.9653	0.570830
$370$	−28.0348	−1.45746
$371$	0	0
$372$	−2.51303	−0.130294
$373$	−7.91204	−0.409670	−0.204835	−	0.978797i	$-0.565666\pi$
$373$	−7.91204	−0.409670	−0.204835	+	0.978797i	$0.565666\pi$
$374$	9.72584	0.502911
$375$	2.47622	0.127871
$376$	2.81872	0.145364
$377$	0	0
$378$	0	0
$379$	−1.78091	−0.0914791	−0.0457395	−	0.998953i	$-0.514564\pi$
$379$	−1.78091	−0.0914791	−0.0457395	+	0.998953i	$0.514564\pi$
$380$	9.35761	0.480035
$381$	1.17484	0.0601889
$382$	−33.9108	−1.73503
$383$	14.1997	0.725572	0.362786	−	0.931872i	$-0.381826\pi$
$383$	14.1997	0.725572	0.362786	+	0.931872i	$0.381826\pi$
$384$	−6.61883	−0.337766
$385$	0	0
$386$	20.5829	1.04764
$387$	3.62077	0.184054
$388$	−10.3987	−0.527912
$389$	−6.77499	−0.343506	−0.171753	−	0.985140i	$-0.554943\pi$
$389$	−6.77499	−0.343506	−0.171753	+	0.985140i	$0.554943\pi$
$390$	0	0
$391$	−16.3161	−0.825140
$392$	0	0
$393$	1.05700	0.0533187
$394$	31.4226	1.58305
$395$	5.78876	0.291264
$396$	5.26791	0.264723
$397$	29.0023	1.45558	0.727792	−	0.685798i	$-0.240547\pi$
$397$	29.0023	1.45558	0.727792	+	0.685798i	$0.240547\pi$
$398$	−21.2218	−1.06375
$399$	0	0
$400$	−16.0772	−0.803861
$401$	21.9604	1.09665	0.548326	−	0.836265i	$-0.315265\pi$
$401$	21.9604	1.09665	0.548326	+	0.836265i	$0.315265\pi$
$402$	13.0370	0.650224
$403$	0	0
$404$	−12.9246	−0.643024
$405$	−19.7006	−0.978928
$406$	0	0
$407$	15.2105	0.753959
$408$	2.34842	0.116264
$409$	−4.00290	−0.197931	−0.0989653	−	0.995091i	$-0.531553\pi$
$409$	−4.00290	−0.197931	−0.0989653	+	0.995091i	$0.531553\pi$
$410$	−18.9366	−0.935214
$411$	−0.212449	−0.0104793
$412$	−4.85702	−0.239288
$413$	0	0
$414$	−32.6986	−1.60705
$415$	−8.96075	−0.439866
$416$	0	0
$417$	−9.64501	−0.472318
$418$	−18.7851	−0.918811
$419$	1.75047	0.0855162	0.0427581	−	0.999085i	$-0.486386\pi$
$419$	1.75047	0.0855162	0.0427581	+	0.999085i	$0.486386\pi$
$420$	0	0
$421$	−16.2204	−0.790533	−0.395267	−	0.918566i	$-0.629348\pi$
$421$	−16.2204	−0.790533	−0.395267	+	0.918566i	$0.629348\pi$
$422$	13.0277	0.634179
$423$	−3.72498	−0.181115
$424$	−17.2778	−0.839082
$425$	7.41755	0.359804
$426$	3.92837	0.190330
$427$	0	0
$428$	11.5570	0.558631
$429$	0	0
$430$	−6.25294	−0.301543
$431$	11.6797	0.562590	0.281295	−	0.959621i	$-0.409236\pi$
$431$	11.6797	0.562590	0.281295	+	0.959621i	$0.409236\pi$
$432$	14.0515	0.676052
$433$	−31.6289	−1.51999	−0.759994	−	0.649930i	$-0.774798\pi$
$433$	−31.6289	−1.51999	−0.759994	+	0.649930i	$0.774798\pi$
$434$	0	0
$435$	−8.81351	−0.422575
$436$	7.42008	0.355357
$437$	31.5140	1.50752
$438$	−4.67498	−0.223379
$439$	9.71539	0.463690	0.231845	−	0.972753i	$-0.425524\pi$
$439$	9.71539	0.463690	0.231845	+	0.972753i	$0.425524\pi$
$440$	15.4658	0.737302
$441$	0	0
$442$	0	0
$443$	21.1163	1.00327	0.501633	−	0.865081i	$-0.332733\pi$
$443$	21.1163	1.00327	0.501633	+	0.865081i	$0.332733\pi$
$444$	−2.16045	−0.102530
$445$	37.5283	1.77901
$446$	12.8644	0.609145
$447$	−11.3352	−0.536135
$448$	0	0
$449$	−13.4710	−0.635735	−0.317867	−	0.948135i	$-0.602967\pi$
$449$	−13.4710	−0.635735	−0.317867	+	0.948135i	$0.602967\pi$
$450$	14.8653	0.700758
$451$	10.2743	0.483796
$452$	−2.92647	−0.137650
$453$	−5.55401	−0.260950
$454$	−42.2871	−1.98463
$455$	0	0
$456$	−4.53589	−0.212413
$457$	9.35582	0.437647	0.218823	−	0.975764i	$-0.429778\pi$
$457$	9.35582	0.437647	0.218823	+	0.975764i	$0.429778\pi$
$458$	21.3538	0.997798
$459$	−6.48294	−0.302598
$460$	15.2620	0.711593
$461$	8.14109	0.379168	0.189584	−	0.981864i	$-0.439286\pi$
$461$	8.14109	0.379168	0.189584	+	0.981864i	$0.439286\pi$
$462$	0	0
$463$	24.3018	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$463$	24.3018	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$464$	−30.5623	−1.41882
$465$	9.74994	0.452143
$466$	−13.4805	−0.624473
$467$	32.7935	1.51750	0.758751	−	0.651381i	$-0.225810\pi$
$467$	32.7935	1.51750	0.758751	+	0.651381i	$0.225810\pi$
$468$	0	0
$469$	0	0
$470$	6.43290	0.296728
$471$	−1.87497	−0.0863940
$472$	−20.0360	−0.922232
$473$	3.39259	0.155991
$474$	1.65057	0.0758132
$475$	−14.3267	−0.657356
$476$	0	0
$477$	22.8328	1.04544
$478$	−39.4215	−1.80310
$479$	2.48109	0.113364	0.0566819	−	0.998392i	$-0.481948\pi$
$479$	2.48109	0.113364	0.0566819	+	0.998392i	$0.481948\pi$
$480$	−5.68557	−0.259510
$481$	0	0
$482$	−0.477694	−0.0217584
$483$	0	0
$484$	−3.21221	−0.146009
$485$	40.3443	1.83194
$486$	−19.7650	−0.896557
$487$	22.9216	1.03867	0.519337	−	0.854569i	$-0.326179\pi$
$487$	22.9216	1.03867	0.519337	+	0.854569i	$0.326179\pi$
$488$	19.9154	0.901529
$489$	9.70797	0.439010
$490$	0	0
$491$	22.8735	1.03227	0.516134	−	0.856508i	$-0.327371\pi$
$491$	22.8735	1.03227	0.516134	+	0.856508i	$0.327371\pi$
$492$	−1.45932	−0.0657910
$493$	14.1005	0.635055
$494$	0	0
$495$	−20.4382	−0.918631
$496$	33.8095	1.51809
$497$	0	0
$498$	−2.55501	−0.114493
$499$	−8.48087	−0.379656	−0.189828	−	0.981817i	$-0.560793\pi$
$499$	−8.48087	−0.379656	−0.189828	+	0.981817i	$0.560793\pi$
$500$	3.70570	0.165724
$501$	−5.09440	−0.227601
$502$	12.6124	0.562921
$503$	−9.36890	−0.417739	−0.208869	−	0.977944i	$-0.566978\pi$
$503$	−9.36890	−0.417739	−0.208869	+	0.977944i	$0.566978\pi$
$504$	0	0
$505$	50.1444	2.23140
$506$	−30.6380	−1.36202
$507$	0	0
$508$	1.75817	0.0780060
$509$	21.5402	0.954753	0.477376	−	0.878699i	$-0.341588\pi$
$509$	21.5402	0.954753	0.477376	+	0.878699i	$0.341588\pi$
$510$	5.35958	0.237326
$511$	0	0
$512$	0.851388	0.0376264
$513$	12.5216	0.552841
$514$	19.2967	0.851143
$515$	18.8441	0.830368
$516$	−0.481870	−0.0212132
$517$	−3.49023	−0.153500
$518$	0	0
$519$	2.97203	0.130458
$520$	0	0
$521$	15.7470	0.689889	0.344944	−	0.938623i	$-0.387898\pi$
$521$	15.7470	0.689889	0.344944	+	0.938623i	$0.387898\pi$
$522$	28.2585	1.23684
$523$	17.9404	0.784477	0.392238	−	0.919864i	$-0.371701\pi$
$523$	17.9404	0.784477	0.392238	+	0.919864i	$0.371701\pi$
$524$	1.58182	0.0691020
$525$	0	0
$526$	22.4369	0.978293
$527$	−15.5987	−0.679489
$528$	6.30273	0.274291
$529$	28.3983	1.23471
$530$	−39.4314	−1.71279
$531$	26.4779	1.14904
$532$	0	0
$533$	0	0
$534$	10.7006	0.463060
$535$	−44.8386	−1.93854
$536$	−33.1671	−1.43260
$537$	−3.83672	−0.165566
$538$	29.5232	1.27284
$539$	0	0
$540$	6.06410	0.260957
$541$	−9.35582	−0.402238	−0.201119	−	0.979567i	$-0.564458\pi$
$541$	−9.35582	−0.402238	−0.201119	+	0.979567i	$0.564458\pi$
$542$	50.2853	2.15994
$543$	−9.89947	−0.424827
$544$	9.09621	0.389997
$545$	−28.7881	−1.23315
$546$	0	0
$547$	−33.1634	−1.41796	−0.708981	−	0.705227i	$-0.750845\pi$
$547$	−33.1634	−1.41796	−0.708981	+	0.705227i	$0.750845\pi$
$548$	−0.317933	−0.0135814
$549$	−26.3185	−1.12325
$550$	13.9285	0.593914
$551$	−27.2347	−1.16024
$552$	−7.39790	−0.314876
$553$	0	0
$554$	−8.63228	−0.366750
$555$	8.38202	0.355797
$556$	−14.4339	−0.612133
$557$	11.1532	0.472578	0.236289	−	0.971683i	$-0.424069\pi$
$557$	11.1532	0.472578	0.236289	+	0.971683i	$0.424069\pi$
$558$	−31.2609	−1.32338
$559$	0	0
$560$	0	0
$561$	−2.90789	−0.122771
$562$	−2.75238	−0.116102
$563$	−19.1778	−0.808247	−0.404124	−	0.914704i	$-0.632423\pi$
$563$	−19.1778	−0.808247	−0.404124	+	0.914704i	$0.632423\pi$
$564$	0.495739	0.0208744
$565$	11.3540	0.477666
$566$	22.8389	0.959993
$567$	0	0
$568$	−9.99409	−0.419343
$569$	13.4672	0.564575	0.282288	−	0.959330i	$-0.408907\pi$
$569$	13.4672	0.564575	0.282288	+	0.959330i	$0.408907\pi$
$570$	−10.3519	−0.433591
$571$	25.1608	1.05295	0.526473	−	0.850192i	$-0.323514\pi$
$571$	25.1608	1.05295	0.526473	+	0.850192i	$0.323514\pi$
$572$	0	0
$573$	10.1389	0.423557
$574$	0	0
$575$	−23.3665	−0.974450
$576$	−8.95016	−0.372923
$577$	21.6484	0.901235	0.450617	−	0.892717i	$-0.351204\pi$
$577$	21.6484	0.901235	0.450617	+	0.892717i	$0.351204\pi$
$578$	19.5691	0.813969
$579$	−6.15401	−0.255752
$580$	−13.1895	−0.547666
$581$	0	0
$582$	11.5035	0.476836
$583$	21.3939	0.886044
$584$	11.8935	0.492157
$585$	0	0
$586$	21.2860	0.879316
$587$	−34.5576	−1.42634	−0.713172	−	0.700989i	$-0.752742\pi$
$587$	−34.5576	−1.42634	−0.713172	+	0.700989i	$0.752742\pi$
$588$	0	0
$589$	30.1283	1.24142
$590$	−45.7263	−1.88252
$591$	−9.39493	−0.386456
$592$	29.0660	1.19461
$593$	1.26839	0.0520865	0.0260432	−	0.999661i	$-0.491709\pi$
$593$	1.26839	0.0520865	0.0260432	+	0.999661i	$0.491709\pi$
$594$	−12.1735	−0.499485
$595$	0	0
$596$	−16.9632	−0.694842
$597$	6.34501	0.259684
$598$	0	0
$599$	17.5143	0.715614	0.357807	−	0.933796i	$-0.383525\pi$
$599$	17.5143	0.715614	0.357807	+	0.933796i	$0.383525\pi$
$600$	3.36320	0.137302
$601$	−9.02164	−0.368000	−0.184000	−	0.982926i	$-0.558905\pi$
$601$	−9.02164	−0.368000	−0.184000	+	0.982926i	$0.558905\pi$
$602$	0	0
$603$	43.8307	1.78493
$604$	−8.31165	−0.338196
$605$	12.4626	0.506677
$606$	14.2979	0.580811
$607$	−15.8290	−0.642479	−0.321239	−	0.946998i	$-0.604099\pi$
$607$	−15.8290	−0.642479	−0.321239	+	0.946998i	$0.604099\pi$
$608$	−17.5690	−0.712517
$609$	0	0
$610$	45.4511	1.84026
$611$	0	0
$612$	−4.64439	−0.187738
$613$	−17.5344	−0.708209	−0.354104	−	0.935206i	$-0.615214\pi$
$613$	−17.5344	−0.708209	−0.354104	+	0.935206i	$0.615214\pi$
$614$	−34.6029	−1.39646
$615$	5.66179	0.228306
$616$	0	0
$617$	7.01448	0.282392	0.141196	−	0.989982i	$-0.454905\pi$
$617$	7.01448	0.282392	0.141196	+	0.989982i	$0.454905\pi$
$618$	5.37307	0.216137
$619$	−38.8263	−1.56056	−0.780280	−	0.625430i	$-0.784923\pi$
$619$	−38.8263	−1.56056	−0.780280	+	0.625430i	$0.784923\pi$
$620$	14.5909	0.585985
$621$	20.4223	0.819518
$622$	−28.3743	−1.13771
$623$	0	0
$624$	0	0
$625$	−30.6735	−1.22694
$626$	4.22597	0.168904
$627$	5.61649	0.224301
$628$	−2.80592	−0.111968
$629$	−13.4102	−0.534699
$630$	0	0
$631$	12.7484	0.507506	0.253753	−	0.967269i	$-0.418335\pi$
$631$	12.7484	0.507506	0.253753	+	0.967269i	$0.418335\pi$
$632$	−4.19918	−0.167034
$633$	−3.89510	−0.154816
$634$	−45.1770	−1.79421
$635$	−6.82126	−0.270694
$636$	−3.03871	−0.120493
$637$	0	0
$638$	26.4776	1.04826
$639$	13.2073	0.522474
$640$	38.4297	1.51907
$641$	−9.71429	−0.383691	−0.191846	−	0.981425i	$-0.561447\pi$
$641$	−9.71429	−0.383691	−0.191846	+	0.981425i	$0.561447\pi$
$642$	−12.7850	−0.504583
$643$	−0.458279	−0.0180728	−0.00903639	−	0.999959i	$-0.502876\pi$
$643$	−0.458279	−0.0180728	−0.00903639	+	0.999959i	$0.502876\pi$
$644$	0	0
$645$	1.86954	0.0736131
$646$	16.5617	0.651610
$647$	−6.17722	−0.242852	−0.121426	−	0.992601i	$-0.538747\pi$
$647$	−6.17722	−0.242852	−0.121426	+	0.992601i	$0.538747\pi$
$648$	14.2908	0.561397
$649$	24.8093	0.973848
$650$	0	0
$651$	0	0
$652$	14.5281	0.568965
$653$	28.4385	1.11289	0.556443	−	0.830886i	$-0.312166\pi$
$653$	28.4385	1.11289	0.556443	+	0.830886i	$0.312166\pi$
$654$	−8.20846	−0.320976
$655$	−6.13707	−0.239795
$656$	19.6332	0.766547
$657$	−15.7175	−0.613196
$658$	0	0
$659$	−25.9818	−1.01211	−0.506054	−	0.862502i	$-0.668897\pi$
$659$	−25.9818	−1.01211	−0.506054	+	0.862502i	$0.668897\pi$
$660$	2.72002	0.105877
$661$	−37.2752	−1.44984	−0.724918	−	0.688835i	$-0.758122\pi$
$661$	−37.2752	−1.44984	−0.724918	+	0.688835i	$0.758122\pi$
$662$	16.0471	0.623686
$663$	0	0
$664$	6.50015	0.252255
$665$	0	0
$666$	−26.8750	−1.04139
$667$	−44.4189	−1.71991
$668$	−7.62385	−0.294976
$669$	−3.84626	−0.148705
$670$	−75.6941	−2.92432
$671$	−24.6599	−0.951986
$672$	0	0
$673$	−41.7078	−1.60772	−0.803859	−	0.594820i	$-0.797223\pi$
$673$	−41.7078	−1.60772	−0.803859	+	0.594820i	$0.797223\pi$
$674$	10.4525	0.402613
$675$	−9.28430	−0.357353
$676$	0	0
$677$	18.2069	0.699750	0.349875	−	0.936796i	$-0.386224\pi$
$677$	18.2069	0.699750	0.349875	+	0.936796i	$0.386224\pi$
$678$	3.23741	0.124332
$679$	0	0
$680$	−13.6352	−0.522886
$681$	12.6432	0.484490
$682$	−29.2908	−1.12160
$683$	27.3360	1.04598	0.522991	−	0.852338i	$-0.324816\pi$
$683$	27.3360	1.04598	0.522991	+	0.852338i	$0.324816\pi$
$684$	8.97048	0.342995
$685$	1.23350	0.0471297
$686$	0	0
$687$	−6.38449	−0.243584
$688$	6.48294	0.247160
$689$	0	0
$690$	−16.8836	−0.642746
$691$	−8.39581	−0.319392	−0.159696	−	0.987166i	$-0.551051\pi$
$691$	−8.39581	−0.319392	−0.159696	+	0.987166i	$0.551051\pi$
$692$	4.44769	0.169076
$693$	0	0
$694$	−48.2087	−1.82998
$695$	56.0000	2.12420
$696$	6.39334	0.242339
$697$	−9.05817	−0.343102
$698$	33.6562	1.27391
$699$	4.03049	0.152447
$700$	0	0
$701$	−27.4998	−1.03865	−0.519327	−	0.854576i	$-0.673817\pi$
$701$	−27.4998	−1.03865	−0.519327	+	0.854576i	$0.673817\pi$
$702$	0	0
$703$	25.9013	0.976887
$704$	−8.38612	−0.316064
$705$	−1.92335	−0.0724375
$706$	−23.5050	−0.884623
$707$	0	0
$708$	−3.52381	−0.132433
$709$	−28.9820	−1.08844	−0.544221	−	0.838942i	$-0.683175\pi$
$709$	−28.9820	−1.08844	−0.544221	+	0.838942i	$0.683175\pi$
$710$	−22.8086	−0.855991
$711$	5.54928	0.208114
$712$	−27.2231	−1.02023
$713$	49.1384	1.84025
$714$	0	0
$715$	0	0
$716$	−5.74170	−0.214577
$717$	11.7865	0.440174
$718$	−12.4191	−0.463476
$719$	−17.5522	−0.654588	−0.327294	−	0.944923i	$-0.606137\pi$
$719$	−17.5522	−0.654588	−0.327294	+	0.944923i	$0.606137\pi$
$720$	−39.0557	−1.45552
$721$	0	0
$722$	−0.533456	−0.0198532
$723$	0.142824	0.00531168
$724$	−14.8147	−0.550584
$725$	20.1935	0.749969
$726$	3.55350	0.131883
$727$	22.1403	0.821139	0.410569	−	0.911829i	$-0.365330\pi$
$727$	22.1403	0.821139	0.410569	+	0.911829i	$0.365330\pi$
$728$	0	0
$729$	−14.6556	−0.542799
$730$	27.1435	1.00463
$731$	−2.99103	−0.110627
$732$	3.50260	0.129460
$733$	−51.3414	−1.89634	−0.948169	−	0.317768i	$-0.897067\pi$
$733$	−51.3414	−1.89634	−0.948169	+	0.317768i	$0.897067\pi$
$734$	47.3285	1.74693
$735$	0	0
$736$	−28.6545	−1.05622
$737$	41.0685	1.51278
$738$	−18.1532	−0.668229
$739$	22.7787	0.837929	0.418965	−	0.908003i	$-0.362393\pi$
$739$	22.7787	0.837929	0.418965	+	0.908003i	$0.362393\pi$
$740$	12.5438	0.461120
$741$	0	0
$742$	0	0
$743$	27.6199	1.01328	0.506638	−	0.862159i	$-0.330888\pi$
$743$	27.6199	1.01328	0.506638	+	0.862159i	$0.330888\pi$
$744$	−7.07262	−0.259295
$745$	65.8133	2.41121
$746$	13.0985	0.479571
$747$	−8.59004	−0.314293
$748$	−4.35170	−0.159114
$749$	0	0
$750$	−4.09943	−0.149690
$751$	41.1172	1.50039	0.750194	−	0.661218i	$-0.229960\pi$
$751$	41.1172	1.50039	0.750194	+	0.661218i	$0.229960\pi$
$752$	−6.66952	−0.243212
$753$	−3.77095	−0.137421
$754$	0	0
$755$	32.2472	1.17360
$756$	0	0
$757$	14.0143	0.509357	0.254678	−	0.967026i	$-0.418030\pi$
$757$	14.0143	0.509357	0.254678	+	0.967026i	$0.418030\pi$
$758$	2.94832	0.107088
$759$	9.16033	0.332499
$760$	26.3359	0.955304
$761$	−34.1648	−1.23847	−0.619236	−	0.785205i	$-0.712558\pi$
$761$	−34.1648	−1.23847	−0.619236	+	0.785205i	$0.712558\pi$
$762$	−1.94497	−0.0704588
$763$	0	0
$764$	15.1730	0.548938
$765$	18.0191	0.651483
$766$	−23.5079	−0.849375
$767$	0	0
$768$	7.74153	0.279348
$769$	52.9061	1.90784	0.953921	−	0.300059i	$-0.0970062\pi$
$769$	52.9061	1.90784	0.953921	+	0.300059i	$0.0970062\pi$
$770$	0	0
$771$	−5.76946	−0.207782
$772$	−9.20957	−0.331460
$773$	−45.9133	−1.65139	−0.825694	−	0.564118i	$-0.809216\pi$
$773$	−45.9133	−1.65139	−0.825694	+	0.564118i	$0.809216\pi$
$774$	−5.99425	−0.215459
$775$	−22.3391	−0.802444
$776$	−29.2658	−1.05058
$777$	0	0
$778$	11.2161	0.402117
$779$	17.4955	0.626843
$780$	0	0
$781$	12.3750	0.442813
$782$	27.0116	0.965932
$783$	−17.6491	−0.630728
$784$	0	0
$785$	10.8863	0.388548
$786$	−1.74989	−0.0624163
$787$	−15.1872	−0.541364	−0.270682	−	0.962669i	$-0.587249\pi$
$787$	−15.1872	−0.541364	−0.270682	+	0.962669i	$0.587249\pi$
$788$	−14.0596	−0.500854
$789$	−6.70831	−0.238822
$790$	−9.58340	−0.340962
$791$	0	0
$792$	14.8259	0.526817
$793$	0	0
$794$	−48.0138	−1.70395
$795$	11.7894	0.418129
$796$	9.49540	0.336555
$797$	−16.3006	−0.577398	−0.288699	−	0.957420i	$-0.593223\pi$
$797$	−16.3006	−0.577398	−0.288699	+	0.957420i	$0.593223\pi$
$798$	0	0
$799$	3.07712	0.108861
$800$	13.0268	0.460567
$801$	35.9758	1.27114
$802$	−36.3559	−1.28377
$803$	−14.7269	−0.519703
$804$	−5.83322	−0.205722
$805$	0	0
$806$	0	0
$807$	−8.82704	−0.310727
$808$	−36.3749	−1.27966
$809$	10.3857	0.365142	0.182571	−	0.983193i	$-0.441558\pi$
$809$	10.3857	0.365142	0.182571	+	0.983193i	$0.441558\pi$
$810$	32.6146	1.14596
$811$	10.9705	0.385227	0.192614	−	0.981275i	$-0.438304\pi$
$811$	10.9705	0.385227	0.192614	+	0.981275i	$0.438304\pi$
$812$	0	0
$813$	−15.0346	−0.527286
$814$	−25.1813	−0.882605
$815$	−56.3656	−1.97440
$816$	−5.55672	−0.194524
$817$	5.77708	0.202114
$818$	6.62687	0.231703
$819$	0	0
$820$	8.47296	0.295888
$821$	40.2325	1.40412	0.702061	−	0.712117i	$-0.252263\pi$
$821$	40.2325	1.40412	0.702061	+	0.712117i	$0.252263\pi$
$822$	0.351713	0.0122674
$823$	−3.26499	−0.113811	−0.0569053	−	0.998380i	$-0.518123\pi$
$823$	−3.26499	−0.113811	−0.0569053	+	0.998380i	$0.518123\pi$
$824$	−13.6695	−0.476200
$825$	−4.16443	−0.144987
$826$	0	0
$827$	46.0118	1.59999	0.799993	−	0.600009i	$-0.204836\pi$
$827$	46.0118	1.59999	0.799993	+	0.600009i	$0.204836\pi$
$828$	14.6306	0.508448
$829$	−30.3865	−1.05537	−0.527683	−	0.849441i	$-0.676939\pi$
$829$	−30.3865	−1.05537	−0.527683	+	0.849441i	$0.676939\pi$
$830$	14.8347	0.514919
$831$	2.58093	0.0895315
$832$	0	0
$833$	0	0
$834$	15.9675	0.552909
$835$	29.5787	1.02361
$836$	8.40516	0.290699
$837$	19.5243	0.674860
$838$	−2.89794	−0.100108
$839$	−1.52007	−0.0524787	−0.0262394	−	0.999656i	$-0.508353\pi$
$839$	−1.52007	−0.0524787	−0.0262394	+	0.999656i	$0.508353\pi$
$840$	0	0
$841$	9.38720	0.323697
$842$	26.8531	0.925420
$843$	0.822923	0.0283430
$844$	−5.82908	−0.200645
$845$	0	0
$846$	6.16677	0.212018
$847$	0	0
$848$	40.8818	1.40389
$849$	−6.82853	−0.234354
$850$	−12.2799	−0.421197
$851$	42.2443	1.44811
$852$	−1.75770	−0.0602178
$853$	43.7939	1.49947	0.749737	−	0.661736i	$-0.230180\pi$
$853$	43.7939	1.49947	0.749737	+	0.661736i	$0.230180\pi$
$854$	0	0
$855$	−34.8033	−1.19025
$856$	32.5260	1.11172
$857$	−14.8521	−0.507338	−0.253669	−	0.967291i	$-0.581637\pi$
$857$	−14.8521	−0.507338	−0.253669	+	0.967291i	$0.581637\pi$
$858$	0	0
$859$	−44.3568	−1.51343	−0.756717	−	0.653742i	$-0.773198\pi$
$859$	−44.3568	−1.51343	−0.756717	+	0.653742i	$0.773198\pi$
$860$	2.79780	0.0954040
$861$	0	0
$862$	−19.3359	−0.658584
$863$	−25.8451	−0.879778	−0.439889	−	0.898052i	$-0.644982\pi$
$863$	−25.8451	−0.879778	−0.439889	+	0.898052i	$0.644982\pi$
$864$	−11.3854	−0.387340
$865$	−17.2560	−0.586721
$866$	52.3622	1.77934
$867$	−5.85090	−0.198707
$868$	0	0
$869$	5.19956	0.176383
$870$	14.5909	0.494678
$871$	0	0
$872$	20.8830	0.707187
$873$	38.6752	1.30896
$874$	−52.1719	−1.76474
$875$	0	0
$876$	2.09176	0.0706740
$877$	1.37284	0.0463575	0.0231788	−	0.999731i	$-0.492621\pi$
$877$	1.37284	0.0463575	0.0231788	+	0.999731i	$0.492621\pi$
$878$	−16.0840	−0.542809
$879$	−6.36421	−0.214660
$880$	−36.5944	−1.23360
$881$	−16.2514	−0.547524	−0.273762	−	0.961797i	$-0.588268\pi$
$881$	−16.2514	−0.547524	−0.273762	+	0.961797i	$0.588268\pi$
$882$	0	0
$883$	−0.619110	−0.0208347	−0.0104174	−	0.999946i	$-0.503316\pi$
$883$	−0.619110	−0.0208347	−0.0104174	+	0.999946i	$0.503316\pi$
$884$	0	0
$885$	13.6715	0.459564
$886$	−34.9584	−1.17445
$887$	−31.9379	−1.07237	−0.536185	−	0.844100i	$-0.680135\pi$
$887$	−31.9379	−1.07237	−0.536185	+	0.844100i	$0.680135\pi$
$888$	−6.08034	−0.204043
$889$	0	0
$890$	−62.1288	−2.08256
$891$	−17.6954	−0.592817
$892$	−5.75599	−0.192725
$893$	−5.94335	−0.198886
$894$	18.7656	0.627615
$895$	22.2764	0.744618
$896$	0	0
$897$	0	0
$898$	22.3015	0.744209
$899$	−42.4658	−1.41631
$900$	−6.65129	−0.221710
$901$	−18.8616	−0.628373
$902$	−17.0092	−0.566345
$903$	0	0
$904$	−8.23621	−0.273932
$905$	57.4775	1.91062
$906$	9.19476	0.305475
$907$	20.7037	0.687455	0.343728	−	0.939069i	$-0.388310\pi$
$907$	20.7037	0.687455	0.343728	+	0.939069i	$0.388310\pi$
$908$	18.9208	0.627909
$909$	48.0699	1.59438
$910$	0	0
$911$	39.4143	1.30585	0.652926	−	0.757421i	$-0.273541\pi$
$911$	39.4143	1.30585	0.652926	+	0.757421i	$0.273541\pi$
$912$	10.7326	0.355392
$913$	−8.04869	−0.266373
$914$	−15.4887	−0.512321
$915$	−13.5893	−0.449247
$916$	−9.55449	−0.315689
$917$	0	0
$918$	10.7326	0.354229
$919$	14.2185	0.469025	0.234512	−	0.972113i	$-0.424651\pi$
$919$	14.2185	0.469025	0.234512	+	0.972113i	$0.424651\pi$
$920$	42.9531	1.41612
$921$	10.3458	0.340906
$922$	−13.4777	−0.443865
$923$	0	0
$924$	0	0
$925$	−19.2049	−0.631454
$926$	−40.2320	−1.32211
$927$	18.0645	0.593315
$928$	24.7635	0.812902
$929$	−28.7478	−0.943185	−0.471593	−	0.881817i	$-0.656321\pi$
$929$	−28.7478	−0.943185	−0.471593	+	0.881817i	$0.656321\pi$
$930$	−16.1412	−0.529291
$931$	0	0
$932$	6.03168	0.197574
$933$	8.48352	0.277738
$934$	−54.2902	−1.77643
$935$	16.8836	0.552151
$936$	0	0
$937$	−19.8495	−0.648456	−0.324228	−	0.945979i	$-0.605105\pi$
$937$	−19.8495	−0.648456	−0.324228	+	0.945979i	$0.605105\pi$
$938$	0	0
$939$	−1.26351	−0.0412330
$940$	−2.87832	−0.0938804
$941$	45.2320	1.47452	0.737260	−	0.675609i	$-0.236119\pi$
$941$	45.2320	1.47452	0.737260	+	0.675609i	$0.236119\pi$
$942$	3.10404	0.101135
$943$	28.5347	0.929217
$944$	47.4083	1.54301
$945$	0	0
$946$	−5.61649	−0.182608
$947$	22.1095	0.718462	0.359231	−	0.933249i	$-0.383039\pi$
$947$	22.1095	0.718462	0.359231	+	0.933249i	$0.383039\pi$
$948$	−0.738526	−0.0239862
$949$	0	0
$950$	23.7182	0.769520
$951$	13.5073	0.438004
$952$	0	0
$953$	34.3502	1.11271	0.556356	−	0.830944i	$-0.312199\pi$
$953$	34.3502	1.11271	0.556356	+	0.830944i	$0.312199\pi$
$954$	−37.8001	−1.22382
$955$	−58.8674	−1.90491
$956$	17.6386	0.570474
$957$	−7.91644	−0.255902
$958$	−4.10749	−0.132707
$959$	0	0
$960$	−4.62131	−0.149152
$961$	15.9778	0.515413
$962$	0	0
$963$	−42.9836	−1.38513
$964$	0.213738	0.00688403
$965$	35.7309	1.15022
$966$	0	0
$967$	−23.8805	−0.767945	−0.383973	−	0.923344i	$-0.625444\pi$
$967$	−23.8805	−0.767945	−0.383973	+	0.923344i	$0.625444\pi$
$968$	−9.04039	−0.290569
$969$	−4.95171	−0.159072
$970$	−66.7907	−2.14452
$971$	25.9377	0.832381	0.416190	−	0.909277i	$-0.363365\pi$
$971$	25.9377	0.832381	0.416190	+	0.909277i	$0.363365\pi$
$972$	8.84358	0.283658
$973$	0	0
$974$	−37.9470	−1.21590
$975$	0	0
$976$	−47.1229	−1.50837
$977$	45.2971	1.44918	0.724592	−	0.689178i	$-0.242028\pi$
$977$	45.2971	1.44918	0.724592	+	0.689178i	$0.242028\pi$
$978$	−16.0717	−0.513917
$979$	33.7086	1.07733
$980$	0	0
$981$	−27.5971	−0.881109
$982$	−37.8675	−1.20840
$983$	2.03101	0.0647792	0.0323896	−	0.999475i	$-0.489688\pi$
$983$	2.03101	0.0647792	0.0323896	+	0.999475i	$0.489688\pi$
$984$	−4.10708	−0.130929
$985$	54.5480	1.73804
$986$	−23.3437	−0.743413
$987$	0	0
$988$	0	0
$989$	9.42224	0.299610
$990$	33.8359	1.07537
$991$	−26.9665	−0.856618	−0.428309	−	0.903632i	$-0.640890\pi$
$991$	−26.9665	−0.856618	−0.428309	+	0.903632i	$0.640890\pi$
$992$	−27.3946	−0.869780
$993$	−4.79785	−0.152255
$994$	0	0
$995$	−36.8399	−1.16790
$996$	1.14321	0.0362239
$997$	6.65574	0.210789	0.105395	−	0.994430i	$-0.466389\pi$
$997$	6.65574	0.210789	0.105395	+	0.994430i	$0.466389\pi$
$998$	14.0402	0.444436
$999$	16.7851	0.531056

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cs.1.4		16
7.6	odd	2	inner	8281.2.a.cs.1.3		16
13.5	odd	4		637.2.c.g.246.14	yes	16
13.8	odd	4		637.2.c.g.246.4	yes	16
13.12	even	2	inner	8281.2.a.cs.1.14		16
91.5	even	12		637.2.r.g.116.14		32
91.18	odd	12		637.2.r.g.324.3		32
91.31	even	12		637.2.r.g.324.4		32
91.34	even	4		637.2.c.g.246.3	✓	16
91.44	odd	12		637.2.r.g.116.13		32
91.47	even	12		637.2.r.g.116.4		32
91.60	odd	12		637.2.r.g.324.13		32
91.73	even	12		637.2.r.g.324.14		32
91.83	even	4		637.2.c.g.246.13	yes	16
91.86	odd	12		637.2.r.g.116.3		32
91.90	odd	2	inner	8281.2.a.cs.1.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.c.g.246.3	✓	16	91.34	even	4
637.2.c.g.246.4	yes	16	13.8	odd	4
637.2.c.g.246.13	yes	16	91.83	even	4
637.2.c.g.246.14	yes	16	13.5	odd	4
637.2.r.g.116.3		32	91.86	odd	12
637.2.r.g.116.4		32	91.47	even	12
637.2.r.g.116.13		32	91.44	odd	12
637.2.r.g.116.14		32	91.5	even	12
637.2.r.g.324.3		32	91.18	odd	12
637.2.r.g.324.4		32	91.31	even	12
637.2.r.g.324.13		32	91.60	odd	12
637.2.r.g.324.14		32	91.73	even	12
8281.2.a.cs.1.3		16	7.6	odd	2	inner
8281.2.a.cs.1.4		16	1.1	even	1	trivial
8281.2.a.cs.1.13		16	91.90	odd	2	inner
8281.2.a.cs.1.14		16	13.12	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65552 q^{2} +0.494977 q^{3} +0.740740 q^{4} -2.87389 q^{5} -0.819443 q^{6} +2.08473 q^{8} -2.75500 q^{9} +4.75778 q^{10} -2.58138 q^{11} +0.366649 q^{12} -1.42251 q^{15} -4.93278 q^{16} +2.27584 q^{17} +4.56095 q^{18} -4.39571 q^{19} -2.12881 q^{20} +4.27352 q^{22} -7.16926 q^{23} +1.03189 q^{24} +3.25926 q^{25} -2.84859 q^{27} +6.19574 q^{29} +2.35499 q^{30} -6.85404 q^{31} +3.99686 q^{32} -1.27772 q^{33} -3.76769 q^{34} -2.04074 q^{36} -5.89241 q^{37} +7.27717 q^{38} -5.99128 q^{40} -3.98014 q^{41} -1.31425 q^{43} -1.91213 q^{44} +7.91757 q^{45} +11.8688 q^{46} +1.35208 q^{47} -2.44161 q^{48} -5.39576 q^{50} +1.12649 q^{51} -8.28778 q^{53} +4.71589 q^{54} +7.41860 q^{55} -2.17577 q^{57} -10.2572 q^{58} -9.61085 q^{59} -1.05371 q^{60} +9.55301 q^{61} +11.3470 q^{62} +3.24870 q^{64} +2.11529 q^{66} -15.9095 q^{67} +1.68581 q^{68} -3.54862 q^{69} -4.79396 q^{71} -5.74342 q^{72} +5.70507 q^{73} +9.75500 q^{74} +1.61326 q^{75} -3.25607 q^{76} -2.01426 q^{79} +14.1763 q^{80} +6.85501 q^{81} +6.58920 q^{82} +3.11798 q^{83} -6.54052 q^{85} +2.17577 q^{86} +3.06675 q^{87} -5.38147 q^{88} -13.0584 q^{89} -13.1077 q^{90} -5.31056 q^{92} -3.39259 q^{93} -2.23839 q^{94} +12.6328 q^{95} +1.97835 q^{96} -14.0382 q^{97} +7.11169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} + 36 q^{23} + 44 q^{25} + 36 q^{29} - 52 q^{36} + 36 q^{43} + 72 q^{51} + 12 q^{53} + 164 q^{64} + 96 q^{74} + 36 q^{79} + 16 q^{81} - 136 q^{88} + 24 q^{92}+ \cdots + 84 q^{95}+O(q^{100}) \) 16 * q + 20 * q^4 + 16 * q^9 + 28 * q^16 - 8 * q^22 + 36 * q^23 + 44 * q^25 + 36 * q^29 - 52 * q^36 + 36 * q^43 + 72 * q^51 + 12 * q^53 + 164 * q^64 + 96 * q^74 + 36 * q^79 + 16 * q^81 - 136 * q^88 + 24 * q^92 + 84 * q^95

Introduction

L-functions

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Database

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