LMFDB - Embedded newform 8281.2.a.cs.1.13

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			1.13
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	$-0.300971\pi$
$2$	1.65552	1.17063	0.585314	+	0.810807i	$0.300971\pi$
$3$	−0.494977	−0.285775	−0.142888	−	0.989739i	$-0.545639\pi$
$3$	−0.494977	−0.285775	−0.142888	+	0.989739i	$0.545639\pi$
$4$	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	$-0.372766\pi$
$11$	2.58138	0.778315	0.389157	+	0.921171i	$0.372766\pi$
$12$	−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	$-0.589004\pi$
$17$	−2.27584	−0.551972	−0.275986	+	0.961162i	$0.589004\pi$
$18$	−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	$-0.339055\pi$
$37$	5.89241	0.968707	0.484353	+	0.874872i	$0.339055\pi$
$38$	−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	$-0.709461\pi$
$61$	−9.55301	−1.22314	−0.611569	+	0.791191i	$0.709461\pi$
$62$	−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	$-0.0757332\pi$
$67$	15.9095	1.94366	0.971830	+	0.235685i	$0.0757332\pi$
$68$	−1.68581	−0.204434
$69$	3.54862	0.427204
$70$	0	0
$71$	4.79396	0.568938	0.284469	−	0.958685i	$-0.408183\pi$
$71$	4.79396	0.568938	0.284469	+	0.958685i	$0.408183\pi$
$72$	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	$-0.165352\pi$
$101$	17.4483	1.73617	0.868083	+	0.496419i	$0.165352\pi$
$102$	1.86492	0.184655
$103$	6.55698	0.646078	0.323039	−	0.946386i	$-0.395295\pi$
$103$	6.55698	0.646078	0.323039	+	0.946386i	$0.395295\pi$
$104$	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	$-0.659267\pi$
$109$	−10.0171	−0.959466	−0.479733	+	0.877414i	$0.659267\pi$
$110$	−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	$-0.529738\pi$
$131$	−2.13546	−0.186576	−0.0932878	+	0.995639i	$0.529738\pi$
$132$	−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	$-0.494163\pi$
$137$	0.429210	0.0366699	0.0183349	+	0.999832i	$0.494163\pi$
$138$	5.87480	0.500096
$139$	19.4858	1.65276	0.826381	−	0.563112i	$-0.190396\pi$
$139$	19.4858	1.65276	0.826381	+	0.563112i	$0.190396\pi$
$140$	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	$-0.112642\pi$
$149$	22.9004	1.87607	0.938037	+	0.346534i	$0.112642\pi$
$150$	−2.67078	−0.218068
$151$	11.2207	0.913131	0.456565	−	0.889690i	$-0.349079\pi$
$151$	11.2207	0.913131	0.456565	+	0.889690i	$0.349079\pi$
$152$	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	$-0.451700\pi$
$157$	3.78799	0.302315	0.151157	+	0.988510i	$0.451700\pi$
$158$	−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	$-0.778799\pi$
$163$	−19.6130	−1.53621	−0.768103	+	0.640326i	$0.778799\pi$
$164$	−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	$-0.573301\pi$
$173$	−6.00439	−0.456505	−0.228253	+	0.973602i	$0.573301\pi$
$174$	−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	$-0.233265\pi$
$181$	19.9999	1.48658	0.743289	+	0.668970i	$0.233265\pi$
$182$	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	$-0.352325\pi$
$193$	12.4329	0.894942	0.447471	+	0.894298i	$0.352325\pi$
$194$	−23.2405	−1.66857
$195$	0	0
$196$	0	0
$197$	18.9805	1.35231	0.676154	−	0.736761i	$-0.263646\pi$
$197$	18.9805	1.35231	0.676154	+	0.736761i	$0.263646\pi$
$198$	−11.7735	−0.836709
$199$	−12.8188	−0.908701	−0.454350	−	0.890823i	$-0.650129\pi$
$199$	−12.8188	−0.908701	−0.454350	+	0.890823i	$0.650129\pi$
$200$	−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	$-0.779814\pi$
$239$	−23.8122	−1.54028	−0.770140	+	0.637875i	$0.779814\pi$
$240$	−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	$-0.422710\pi$
$251$	7.61843	0.480871	0.240435	+	0.970665i	$0.422710\pi$
$252$	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	$-0.381568\pi$
$257$	11.6560	0.727082	0.363541	+	0.931578i	$0.381568\pi$
$258$	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	$-0.317040\pi$
$269$	17.8332	1.08731	0.543656	+	0.839308i	$0.317040\pi$
$270$	−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	$-0.515791\pi$
$281$	−1.66255	−0.0991794	−0.0495897	+	0.998770i	$0.515791\pi$
$282$	−1.10795	−0.0659776
$283$	13.7957	0.820066	0.410033	−	0.912071i	$-0.365517\pi$
$283$	13.7957	0.820066	0.410033	+	0.912071i	$0.365517\pi$
$284$	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	$-0.661522\pi$
$311$	−17.1392	−0.971876	−0.485938	+	0.873993i	$0.661522\pi$
$312$	0	0
$313$	2.55266	0.144285	0.0721424	−	0.997394i	$-0.477016\pi$
$313$	2.55266	0.144285	0.0721424	+	0.997394i	$0.477016\pi$
$314$	6.27109	0.353898
$315$	0	0
$316$	−1.49204	−0.0839339
$317$	−27.2888	−1.53269	−0.766345	−	0.642430i	$-0.777927\pi$
$317$	−27.2888	−1.53269	−0.766345	+	0.642430i	$0.777927\pi$
$318$	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	$-0.414169\pi$
$331$	9.69307	0.532779	0.266390	+	0.963865i	$0.414169\pi$
$332$	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	$-0.563432\pi$
$359$	−7.50164	−0.395921	−0.197961	+	0.980210i	$0.563432\pi$
$360$	−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	$-0.231901\pi$
$367$	28.5883	1.49230	0.746149	+	0.665779i	$0.231901\pi$
$368$	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	$-0.485436\pi$
$379$	1.78091	0.0914791	0.0457395	+	0.998953i	$0.485436\pi$
$380$	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	$-0.684735\pi$
$401$	−21.9604	−1.09665	−0.548326	+	0.836265i	$0.684735\pi$
$402$	−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	$-0.513614\pi$
$419$	−1.75047	−0.0855162	−0.0427581	+	0.999085i	$0.513614\pi$
$420$	0	0
$421$	16.2204	0.790533	0.395267	−	0.918566i	$-0.370652\pi$
$421$	16.2204	0.790533	0.395267	+	0.918566i	$0.370652\pi$
$422$	−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	$-0.590764\pi$
$431$	−11.6797	−0.562590	−0.281295	+	0.959621i	$0.590764\pi$
$432$	−14.0515	−0.676052
$433$	31.6289	1.51999	0.759994	−	0.649930i	$-0.225202\pi$
$433$	31.6289	1.51999	0.759994	+	0.649930i	$0.225202\pi$
$434$	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	$-0.574476\pi$
$439$	−9.71539	−0.463690	−0.231845	+	0.972753i	$0.574476\pi$
$440$	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	$-0.397033\pi$
$449$	13.4710	0.635735	0.317867	+	0.948135i	$0.397033\pi$
$450$	−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	$-0.570222\pi$
$457$	−9.35582	−0.437647	−0.218823	+	0.975764i	$0.570222\pi$
$458$	−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	$-0.691008\pi$
$463$	−24.3018	−1.12940	−0.564700	+	0.825296i	$0.691008\pi$
$464$	−30.5623	−1.41882
$465$	−9.74994	−0.452143
$466$	13.4805	0.624473
$467$	−32.7935	−1.51750	−0.758751	−	0.651381i	$-0.774190\pi$
$467$	−32.7935	−1.51750	−0.758751	+	0.651381i	$0.774190\pi$
$468$	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	$-0.673821\pi$
$487$	−22.9216	−1.03867	−0.519337	+	0.854569i	$0.673821\pi$
$488$	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	$-0.439207\pi$
$499$	8.48087	0.379656	0.189828	+	0.981817i	$0.439207\pi$
$500$	3.70570	0.165724
$501$	5.09440	0.227601
$502$	12.6124	0.562921
$503$	9.36890	0.417739	0.208869	−	0.977944i	$-0.433022\pi$
$503$	9.36890	0.417739	0.208869	+	0.977944i	$0.433022\pi$
$504$	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	$-0.612102\pi$
$521$	−15.7470	−0.689889	−0.344944	+	0.938623i	$0.612102\pi$
$522$	−28.2585	−1.23684
$523$	−17.9404	−0.784477	−0.392238	−	0.919864i	$-0.628299\pi$
$523$	−17.9404	−0.784477	−0.392238	+	0.919864i	$0.628299\pi$
$524$	−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	$-0.435542\pi$
$541$	9.35582	0.402238	0.201119	+	0.979567i	$0.435542\pi$
$542$	−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	$-0.575931\pi$
$557$	−11.1532	−0.472578	−0.236289	+	0.971683i	$0.575931\pi$
$558$	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	$-0.367577\pi$
$563$	19.1778	0.808247	0.404124	+	0.914704i	$0.367577\pi$
$564$	−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	$-0.441095\pi$
$601$	9.02164	0.368000	0.184000	+	0.982926i	$0.441095\pi$
$602$	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	$-0.395901\pi$
$607$	15.8290	0.642479	0.321239	+	0.946998i	$0.395901\pi$
$608$	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	$-0.384786\pi$
$613$	17.5344	0.708209	0.354104	+	0.935206i	$0.384786\pi$
$614$	34.6029	1.39646
$615$	−5.66179	−0.228306
$616$	0	0
$617$	−7.01448	−0.282392	−0.141196	−	0.989982i	$-0.545095\pi$
$617$	−7.01448	−0.282392	−0.141196	+	0.989982i	$0.545095\pi$
$618$	−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	$-0.581665\pi$
$631$	−12.7484	−0.507506	−0.253753	+	0.967269i	$0.581665\pi$
$632$	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	$-0.461253\pi$
$647$	6.17722	0.242852	0.121426	+	0.992601i	$0.461253\pi$
$648$	−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	$-0.613776\pi$
$677$	−18.2069	−0.699750	−0.349875	+	0.936796i	$0.613776\pi$
$678$	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	$-0.675184\pi$
$683$	−27.3360	−1.04598	−0.522991	+	0.852338i	$0.675184\pi$
$684$	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	$-0.316825\pi$
$709$	28.9820	1.08844	0.544221	+	0.838942i	$0.316825\pi$
$710$	−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	$-0.393863\pi$
$719$	17.5522	0.654588	0.327294	+	0.944923i	$0.393863\pi$
$720$	−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	$-0.634670\pi$
$727$	−22.1403	−0.821139	−0.410569	+	0.911829i	$0.634670\pi$
$728$	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	$-0.637607\pi$
$739$	−22.7787	−0.837929	−0.418965	+	0.908003i	$0.637607\pi$
$740$	−12.5438	−0.461120
$741$	0	0
$742$	0	0
$743$	−27.6199	−1.01328	−0.506638	−	0.862159i	$-0.669112\pi$
$743$	−27.6199	−1.01328	−0.506638	+	0.862159i	$0.669112\pi$
$744$	−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	$-0.406777\pi$
$797$	16.3006	0.577398	0.288699	+	0.957420i	$0.406777\pi$
$798$	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	$-0.747737\pi$
$821$	−40.2325	−1.40412	−0.702061	+	0.712117i	$0.747737\pi$
$822$	−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	$-0.795164\pi$
$827$	−46.0118	−1.59999	−0.799993	+	0.600009i	$0.795164\pi$
$828$	14.6306	0.508448
$829$	30.3865	1.05537	0.527683	−	0.849441i	$-0.323061\pi$
$829$	30.3865	1.05537	0.527683	+	0.849441i	$0.323061\pi$
$830$	−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	$-0.418363\pi$
$857$	14.8521	0.507338	0.253669	+	0.967291i	$0.418363\pi$
$858$	0	0
$859$	44.3568	1.51343	0.756717	−	0.653742i	$-0.226802\pi$
$859$	44.3568	1.51343	0.756717	+	0.653742i	$0.226802\pi$
$860$	2.79780	0.0954040
$861$	0	0
$862$	−19.3359	−0.658584
$863$	25.8451	0.879778	0.439889	−	0.898052i	$-0.355018\pi$
$863$	25.8451	0.879778	0.439889	+	0.898052i	$0.355018\pi$
$864$	−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	$-0.507379\pi$
$877$	−1.37284	−0.0463575	−0.0231788	+	0.999731i	$0.507379\pi$
$878$	−16.0840	−0.542809
$879$	6.36421	0.214660
$880$	36.5944	1.23360
$881$	16.2514	0.547524	0.273762	−	0.961797i	$-0.411732\pi$
$881$	16.2514	0.547524	0.273762	+	0.961797i	$0.411732\pi$
$882$	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	$-0.319865\pi$
$887$	31.9379	1.07237	0.536185	+	0.844100i	$0.319865\pi$
$888$	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	$-0.394895\pi$
$937$	19.8495	0.648456	0.324228	+	0.945979i	$0.394895\pi$
$938$	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	$-0.616961\pi$
$947$	−22.1095	−0.718462	−0.359231	+	0.933249i	$0.616961\pi$
$948$	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	$-0.374556\pi$
$967$	23.8805	0.767945	0.383973	+	0.923344i	$0.374556\pi$
$968$	9.04039	0.290569
$969$	−4.95171	−0.159072
$970$	66.7907	2.14452
$971$	−25.9377	−0.832381	−0.416190	−	0.909277i	$-0.636635\pi$
$971$	−25.9377	−0.832381	−0.416190	+	0.909277i	$0.636635\pi$
$972$	−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	$-0.757972\pi$
$977$	−45.2971	−1.44918	−0.724592	+	0.689178i	$0.757972\pi$
$978$	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	$-0.533611\pi$
$997$	−6.65574	−0.210789	−0.105395	+	0.994430i	$0.533611\pi$
$998$	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.1.13		16
7.6	odd	2	inner	8281.2.a.cs.1.14		16
13.5	odd	4		637.2.c.g.246.3	✓	16
13.8	odd	4		637.2.c.g.246.13	yes	16
13.12	even	2	inner	8281.2.a.cs.1.3		16
91.5	even	12		637.2.r.g.116.3		32
91.18	odd	12		637.2.r.g.324.14		32
91.31	even	12		637.2.r.g.324.13		32
91.34	even	4		637.2.c.g.246.14	yes	16
91.44	odd	12		637.2.r.g.116.4		32
91.47	even	12		637.2.r.g.116.13		32
91.60	odd	12		637.2.r.g.324.4		32
91.73	even	12		637.2.r.g.324.3		32
91.83	even	4		637.2.c.g.246.4	yes	16
91.86	odd	12		637.2.r.g.116.14		32
91.90	odd	2	inner	8281.2.a.cs.1.4		16

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