LMFDB - Embedded newform 8037.2.a.x.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.146324 q^{2} -1.97859 q^{4} -1.83482 q^{5} -1.25452 q^{7} -0.582164 q^{8} +O(q^{10})$	\(q+0.146324 q^{2} -1.97859 q^{4} -1.83482 q^{5} -1.25452 q^{7} -0.582164 q^{8} -0.268479 q^{10} +4.86721 q^{11} +6.26308 q^{13} -0.183567 q^{14} +3.87199 q^{16} +6.31596 q^{17} +1.00000 q^{19} +3.63036 q^{20} +0.712191 q^{22} +0.426324 q^{23} -1.63343 q^{25} +0.916441 q^{26} +2.48219 q^{28} -3.05144 q^{29} +8.77465 q^{31} +1.73089 q^{32} +0.924178 q^{34} +2.30183 q^{35} +1.53907 q^{37} +0.146324 q^{38} +1.06817 q^{40} +4.20462 q^{41} -0.679839 q^{43} -9.63021 q^{44} +0.0623815 q^{46} -1.00000 q^{47} -5.42617 q^{49} -0.239010 q^{50} -12.3921 q^{52} -2.93855 q^{53} -8.93046 q^{55} +0.730339 q^{56} -0.446500 q^{58} +8.94412 q^{59} +5.09550 q^{61} +1.28394 q^{62} -7.49072 q^{64} -11.4916 q^{65} -9.67239 q^{67} -12.4967 q^{68} +0.336813 q^{70} -6.13376 q^{71} -1.44466 q^{73} +0.225203 q^{74} -1.97859 q^{76} -6.10603 q^{77} +16.7816 q^{79} -7.10442 q^{80} +0.615237 q^{82} -3.57940 q^{83} -11.5887 q^{85} -0.0994768 q^{86} -2.83351 q^{88} -4.91265 q^{89} -7.85719 q^{91} -0.843520 q^{92} -0.146324 q^{94} -1.83482 q^{95} +0.401628 q^{97} -0.793980 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * q^98

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 $

$2$	0.146324	0.103467	0.0517334	−	0.998661i	$-0.483525\pi$
$2$	0.146324	0.103467	0.0517334	+	0.998661i	$0.483525\pi$
$3$	0	0
$3$	0	0
$4$	−1.97859	−0.989295
$5$	−1.83482	−0.820557	−0.410279	−	0.911960i	$-0.634569\pi$
$5$	−1.83482	−0.820557	−0.410279	+	0.911960i	$0.634569\pi$
$6$	0	0
$7$	−1.25452	−0.474166	−0.237083	−	0.971489i	$-0.576191\pi$
$7$	−1.25452	−0.474166	−0.237083	+	0.971489i	$0.576191\pi$
$8$	−0.582164	−0.205826
$9$	0	0
$10$	−0.268479	−0.0849005
$11$	4.86721	1.46752	0.733760	−	0.679409i	$-0.237764\pi$
$11$	4.86721	1.46752	0.733760	+	0.679409i	$0.237764\pi$
$12$	0	0
$13$	6.26308	1.73707	0.868533	−	0.495631i	$-0.165063\pi$
$13$	6.26308	1.73707	0.868533	+	0.495631i	$0.165063\pi$
$14$	−0.183567	−0.0490604
$15$	0	0
$16$	3.87199	0.967998
$17$	6.31596	1.53185	0.765923	−	0.642933i	$-0.222282\pi$
$17$	6.31596	1.53185	0.765923	+	0.642933i	$0.222282\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.63036	0.811773
$21$	0	0
$22$	0.712191	0.151840
$23$	0.426324	0.0888947	0.0444474	−	0.999012i	$-0.485847\pi$
$23$	0.426324	0.0888947	0.0444474	+	0.999012i	$0.485847\pi$
$24$	0	0
$25$	−1.63343	−0.326686
$26$	0.916441	0.179729
$27$	0	0
$28$	2.48219	0.469090
$29$	−3.05144	−0.566638	−0.283319	−	0.959026i	$-0.591436\pi$
$29$	−3.05144	−0.566638	−0.283319	+	0.959026i	$0.591436\pi$
$30$	0	0
$31$	8.77465	1.57597	0.787987	−	0.615692i	$-0.211123\pi$
$31$	8.77465	1.57597	0.787987	+	0.615692i	$0.211123\pi$
$32$	1.73089	0.305982
$33$	0	0
$34$	0.924178	0.158495
$35$	2.30183	0.389080
$36$	0	0
$37$	1.53907	0.253021	0.126511	−	0.991965i	$-0.459622\pi$
$37$	1.53907	0.253021	0.126511	+	0.991965i	$0.459622\pi$
$38$	0.146324	0.0237369
$39$	0	0
$40$	1.06817	0.168892
$41$	4.20462	0.656651	0.328325	−	0.944565i	$-0.393516\pi$
$41$	4.20462	0.656651	0.328325	+	0.944565i	$0.393516\pi$
$42$	0	0
$43$	−0.679839	−0.103674	−0.0518372	−	0.998656i	$-0.516508\pi$
$43$	−0.679839	−0.103674	−0.0518372	+	0.998656i	$0.516508\pi$
$44$	−9.63021	−1.45181
$45$	0	0
$46$	0.0623815	0.00919766
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.42617	−0.775167
$50$	−0.239010	−0.0338012
$51$	0	0
$52$	−12.3921	−1.71847
$53$	−2.93855	−0.403641	−0.201821	−	0.979422i	$-0.564686\pi$
$53$	−2.93855	−0.403641	−0.201821	+	0.979422i	$0.564686\pi$
$54$	0	0
$55$	−8.93046	−1.20418
$56$	0.730339	0.0975956
$57$	0	0
$58$	−0.446500	−0.0586283
$59$	8.94412	1.16443	0.582213	−	0.813036i	$-0.302187\pi$
$59$	8.94412	1.16443	0.582213	+	0.813036i	$0.302187\pi$
$60$	0	0
$61$	5.09550	0.652412	0.326206	−	0.945299i	$-0.394230\pi$
$61$	5.09550	0.652412	0.326206	+	0.945299i	$0.394230\pi$
$62$	1.28394	0.163061
$63$	0	0
$64$	−7.49072	−0.936339
$65$	−11.4916	−1.42536
$66$	0	0
$67$	−9.67239	−1.18167	−0.590835	−	0.806792i	$-0.701202\pi$
$67$	−9.67239	−1.18167	−0.590835	+	0.806792i	$0.701202\pi$
$68$	−12.4967	−1.51545
$69$	0	0
$70$	0.336813	0.0402569
$71$	−6.13376	−0.727943	−0.363972	−	0.931410i	$-0.618579\pi$
$71$	−6.13376	−0.727943	−0.363972	+	0.931410i	$0.618579\pi$
$72$	0	0
$73$	−1.44466	−0.169085	−0.0845426	−	0.996420i	$-0.526943\pi$
$73$	−1.44466	−0.169085	−0.0845426	+	0.996420i	$0.526943\pi$
$74$	0.225203	0.0261793
$75$	0	0
$76$	−1.97859	−0.226960
$77$	−6.10603	−0.695847
$78$	0	0
$79$	16.7816	1.88807	0.944037	−	0.329841i	$-0.106995\pi$
$79$	16.7816	1.88807	0.944037	+	0.329841i	$0.106995\pi$
$80$	−7.10442	−0.794298
$81$	0	0
$82$	0.615237	0.0679416
$83$	−3.57940	−0.392890	−0.196445	−	0.980515i	$-0.562940\pi$
$83$	−3.57940	−0.392890	−0.196445	+	0.980515i	$0.562940\pi$
$84$	0	0
$85$	−11.5887	−1.25697
$86$	−0.0994768	−0.0107269
$87$	0	0
$88$	−2.83351	−0.302054
$89$	−4.91265	−0.520740	−0.260370	−	0.965509i	$-0.583844\pi$
$89$	−4.91265	−0.520740	−0.260370	+	0.965509i	$0.583844\pi$
$90$	0	0
$91$	−7.85719	−0.823657
$92$	−0.843520	−0.0879431
$93$	0	0
$94$	−0.146324	−0.0150922
$95$	−1.83482	−0.188249
$96$	0	0
$97$	0.401628	0.0407792	0.0203896	−	0.999792i	$-0.493509\pi$
$97$	0.401628	0.0407792	0.0203896	+	0.999792i	$0.493509\pi$
$98$	−0.793980	−0.0802041
$99$	0	0
$100$	3.23189	0.323189
$101$	−8.12736	−0.808702	−0.404351	−	0.914604i	$-0.632503\pi$
$101$	−8.12736	−0.808702	−0.404351	+	0.914604i	$0.632503\pi$
$102$	0	0
$103$	−6.70609	−0.660771	−0.330385	−	0.943846i	$-0.607179\pi$
$103$	−6.70609	−0.660771	−0.330385	+	0.943846i	$0.607179\pi$
$104$	−3.64614	−0.357534
$105$	0	0
$106$	−0.429982	−0.0417635
$107$	17.9047	1.73091	0.865455	−	0.500987i	$-0.167030\pi$
$107$	17.9047	1.73091	0.865455	+	0.500987i	$0.167030\pi$
$108$	0	0
$109$	18.2116	1.74435	0.872177	−	0.489191i	$-0.162708\pi$
$109$	18.2116	1.74435	0.872177	+	0.489191i	$0.162708\pi$
$110$	−1.30674	−0.124593
$111$	0	0
$112$	−4.85751	−0.458992
$113$	13.7474	1.29324	0.646621	−	0.762811i	$-0.276181\pi$
$113$	13.7474	1.29324	0.646621	+	0.762811i	$0.276181\pi$
$114$	0	0
$115$	−0.782229	−0.0729432
$116$	6.03755	0.560572
$117$	0	0
$118$	1.30874	0.120479
$119$	−7.92353	−0.726349
$120$	0	0
$121$	12.6897	1.15361
$122$	0.745595	0.0675030
$123$	0	0
$124$	−17.3614	−1.55910
$125$	12.1712	1.08862
$126$	0	0
$127$	−11.5502	−1.02491	−0.512457	−	0.858713i	$-0.671265\pi$
$127$	−11.5502	−1.02491	−0.512457	+	0.858713i	$0.671265\pi$
$128$	−4.55786	−0.402862
$129$	0	0
$130$	−1.68151	−0.147478
$131$	−4.92285	−0.430112	−0.215056	−	0.976602i	$-0.568993\pi$
$131$	−4.92285	−0.430112	−0.215056	+	0.976602i	$0.568993\pi$
$132$	0	0
$133$	−1.25452	−0.108781
$134$	−1.41531	−0.122264
$135$	0	0
$136$	−3.67693	−0.315294
$137$	15.3541	1.31179	0.655894	−	0.754853i	$-0.272292\pi$
$137$	15.3541	1.31179	0.655894	+	0.754853i	$0.272292\pi$
$138$	0	0
$139$	−16.1236	−1.36758	−0.683792	−	0.729677i	$-0.739670\pi$
$139$	−16.1236	−1.36758	−0.683792	+	0.729677i	$0.739670\pi$
$140$	−4.55437	−0.384915
$141$	0	0
$142$	−0.897517	−0.0753180
$143$	30.4837	2.54918
$144$	0	0
$145$	5.59885	0.464959
$146$	−0.211389	−0.0174947
$147$	0	0
$148$	−3.04518	−0.250312
$149$	−9.17258	−0.751447	−0.375723	−	0.926732i	$-0.622606\pi$
$149$	−9.17258	−0.751447	−0.375723	+	0.926732i	$0.622606\pi$
$150$	0	0
$151$	−13.7066	−1.11543	−0.557713	−	0.830034i	$-0.688321\pi$
$151$	−13.7066	−1.11543	−0.557713	+	0.830034i	$0.688321\pi$
$152$	−0.582164	−0.0472197
$153$	0	0
$154$	−0.893461	−0.0719971
$155$	−16.0999	−1.29318
$156$	0	0
$157$	−14.1840	−1.13200	−0.566002	−	0.824404i	$-0.691511\pi$
$157$	−14.1840	−1.13200	−0.566002	+	0.824404i	$0.691511\pi$
$158$	2.45555	0.195353
$159$	0	0
$160$	−3.17588	−0.251076
$161$	−0.534834	−0.0421508
$162$	0	0
$163$	7.80785	0.611558	0.305779	−	0.952103i	$-0.401083\pi$
$163$	7.80785	0.611558	0.305779	+	0.952103i	$0.401083\pi$
$164$	−8.31921	−0.649621
$165$	0	0
$166$	−0.523753	−0.0406511
$167$	9.73357	0.753206	0.376603	−	0.926375i	$-0.377092\pi$
$167$	9.73357	0.753206	0.376603	+	0.926375i	$0.377092\pi$
$168$	0	0
$169$	26.2262	2.01740
$170$	−1.69570	−0.130054
$171$	0	0
$172$	1.34512	0.102565
$173$	11.6076	0.882510	0.441255	−	0.897382i	$-0.354533\pi$
$173$	11.6076	0.882510	0.441255	+	0.897382i	$0.354533\pi$
$174$	0	0
$175$	2.04918	0.154903
$176$	18.8458	1.42056
$177$	0	0
$178$	−0.718839	−0.0538793
$179$	−10.9060	−0.815156	−0.407578	−	0.913170i	$-0.633627\pi$
$179$	−10.9060	−0.815156	−0.407578	+	0.913170i	$0.633627\pi$
$180$	0	0
$181$	−9.30995	−0.692003	−0.346002	−	0.938234i	$-0.612461\pi$
$181$	−9.30995	−0.692003	−0.346002	+	0.938234i	$0.612461\pi$
$182$	−1.14970	−0.0852212
$183$	0	0
$184$	−0.248191	−0.0182968
$185$	−2.82391	−0.207618
$186$	0	0
$187$	30.7411	2.24801
$188$	1.97859	0.144303
$189$	0	0
$190$	−0.268479	−0.0194775
$191$	−9.84362	−0.712260	−0.356130	−	0.934436i	$-0.615904\pi$
$191$	−9.84362	−0.712260	−0.356130	+	0.934436i	$0.615904\pi$
$192$	0	0
$193$	−7.45256	−0.536447	−0.268224	−	0.963357i	$-0.586437\pi$
$193$	−7.45256	−0.536447	−0.268224	+	0.963357i	$0.586437\pi$
$194$	0.0587679	0.00421929
$195$	0	0
$196$	10.7362	0.766868
$197$	−12.8372	−0.914610	−0.457305	−	0.889310i	$-0.651185\pi$
$197$	−12.8372	−0.914610	−0.457305	+	0.889310i	$0.651185\pi$
$198$	0	0
$199$	−19.2113	−1.36185	−0.680926	−	0.732352i	$-0.738422\pi$
$199$	−19.2113	−1.36185	−0.680926	+	0.732352i	$0.738422\pi$
$200$	0.950924	0.0672404
$201$	0	0
$202$	−1.18923	−0.0836739
$203$	3.82811	0.268680
$204$	0	0
$205$	−7.71472	−0.538819
$206$	−0.981263	−0.0683679
$207$	0	0
$208$	24.2506	1.68148
$209$	4.86721	0.336672
$210$	0	0
$211$	4.71866	0.324846	0.162423	−	0.986721i	$-0.448069\pi$
$211$	4.71866	0.324846	0.162423	+	0.986721i	$0.448069\pi$
$212$	5.81419	0.399320
$213$	0	0
$214$	2.61989	0.179092
$215$	1.24738	0.0850708
$216$	0	0
$217$	−11.0080	−0.747273
$218$	2.66480	0.180483
$219$	0	0
$220$	17.6697	1.19129
$221$	39.5574	2.66092
$222$	0	0
$223$	13.7654	0.921799	0.460899	−	0.887452i	$-0.347527\pi$
$223$	13.7654	0.921799	0.460899	+	0.887452i	$0.347527\pi$
$224$	−2.17145	−0.145086
$225$	0	0
$226$	2.01157	0.133808
$227$	15.6922	1.04153	0.520764	−	0.853701i	$-0.325647\pi$
$227$	15.6922	1.04153	0.520764	+	0.853701i	$0.325647\pi$
$228$	0	0
$229$	2.66038	0.175803	0.0879014	−	0.996129i	$-0.471984\pi$
$229$	2.66038	0.175803	0.0879014	+	0.996129i	$0.471984\pi$
$230$	−0.114459	−0.00754720
$231$	0	0
$232$	1.77644	0.116629
$233$	−10.7639	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$233$	−10.7639	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$234$	0	0
$235$	1.83482	0.119691
$236$	−17.6967	−1.15196
$237$	0	0
$238$	−1.15940	−0.0751530
$239$	2.54032	0.164319	0.0821597	−	0.996619i	$-0.473818\pi$
$239$	2.54032	0.164319	0.0821597	+	0.996619i	$0.473818\pi$
$240$	0	0
$241$	8.08024	0.520494	0.260247	−	0.965542i	$-0.416196\pi$
$241$	8.08024	0.520494	0.260247	+	0.965542i	$0.416196\pi$
$242$	1.85682	0.119361
$243$	0	0
$244$	−10.0819	−0.645428
$245$	9.95605	0.636069
$246$	0	0
$247$	6.26308	0.398510
$248$	−5.10829	−0.324376
$249$	0	0
$250$	1.78094	0.112636
$251$	27.6053	1.74243	0.871216	−	0.490899i	$-0.163332\pi$
$251$	27.6053	1.74243	0.871216	+	0.490899i	$0.163332\pi$
$252$	0	0
$253$	2.07501	0.130455
$254$	−1.69007	−0.106045
$255$	0	0
$256$	14.3145	0.894657
$257$	−13.9867	−0.872467	−0.436234	−	0.899833i	$-0.643688\pi$
$257$	−13.9867	−0.872467	−0.436234	+	0.899833i	$0.643688\pi$
$258$	0	0
$259$	−1.93080	−0.119974
$260$	22.7372	1.41010
$261$	0	0
$262$	−0.720333	−0.0445023
$263$	−11.2869	−0.695983	−0.347991	−	0.937498i	$-0.613136\pi$
$263$	−11.2869	−0.695983	−0.347991	+	0.937498i	$0.613136\pi$
$264$	0	0
$265$	5.39172	0.331211
$266$	−0.183567	−0.0112552
$267$	0	0
$268$	19.1377	1.16902
$269$	−14.9401	−0.910916	−0.455458	−	0.890257i	$-0.650524\pi$
$269$	−14.9401	−0.910916	−0.455458	+	0.890257i	$0.650524\pi$
$270$	0	0
$271$	−25.7263	−1.56276	−0.781381	−	0.624055i	$-0.785484\pi$
$271$	−25.7263	−1.56276	−0.781381	+	0.624055i	$0.785484\pi$
$272$	24.4554	1.48282
$273$	0	0
$274$	2.24668	0.135727
$275$	−7.95024	−0.479418
$276$	0	0
$277$	−8.35177	−0.501809	−0.250905	−	0.968012i	$-0.580728\pi$
$277$	−8.35177	−0.501809	−0.250905	+	0.968012i	$0.580728\pi$
$278$	−2.35927	−0.141500
$279$	0	0
$280$	−1.34004	−0.0800828
$281$	29.4696	1.75801	0.879004	−	0.476814i	$-0.158209\pi$
$281$	29.4696	1.75801	0.879004	+	0.476814i	$0.158209\pi$
$282$	0	0
$283$	20.5474	1.22141	0.610707	−	0.791857i	$-0.290885\pi$
$283$	20.5474	1.22141	0.610707	+	0.791857i	$0.290885\pi$
$284$	12.1362	0.720150
$285$	0	0
$286$	4.46051	0.263755
$287$	−5.27479	−0.311361
$288$	0	0
$289$	22.8914	1.34655
$290$	0.819247	0.0481078
$291$	0	0
$292$	2.85840	0.167275
$293$	23.3456	1.36387	0.681933	−	0.731415i	$-0.261140\pi$
$293$	23.3456	1.36387	0.681933	+	0.731415i	$0.261140\pi$
$294$	0	0
$295$	−16.4109	−0.955478
$296$	−0.895989	−0.0520783
$297$	0	0
$298$	−1.34217	−0.0777498
$299$	2.67010	0.154416
$300$	0	0
$301$	0.852874	0.0491588
$302$	−2.00560	−0.115410
$303$	0	0
$304$	3.87199	0.222074
$305$	−9.34934	−0.535342
$306$	0	0
$307$	−4.57302	−0.260996	−0.130498	−	0.991449i	$-0.541658\pi$
$307$	−4.57302	−0.260996	−0.130498	+	0.991449i	$0.541658\pi$
$308$	12.0813	0.688398
$309$	0	0
$310$	−2.35581	−0.133801
$311$	25.7104	1.45790	0.728952	−	0.684564i	$-0.240008\pi$
$311$	25.7104	1.45790	0.728952	+	0.684564i	$0.240008\pi$
$312$	0	0
$313$	24.8487	1.40453	0.702266	−	0.711914i	$-0.252172\pi$
$313$	24.8487	1.40453	0.702266	+	0.711914i	$0.252172\pi$
$314$	−2.07546	−0.117125
$315$	0	0
$316$	−33.2038	−1.86786
$317$	28.9977	1.62867	0.814335	−	0.580395i	$-0.197102\pi$
$317$	28.9977	1.62867	0.814335	+	0.580395i	$0.197102\pi$
$318$	0	0
$319$	−14.8520	−0.831552
$320$	13.7441	0.768320
$321$	0	0
$322$	−0.0782592	−0.00436121
$323$	6.31596	0.351430
$324$	0	0
$325$	−10.2303	−0.567475
$326$	1.14248	0.0632760
$327$	0	0
$328$	−2.44778	−0.135156
$329$	1.25452	0.0691642
$330$	0	0
$331$	22.5291	1.23831	0.619157	−	0.785267i	$-0.287475\pi$
$331$	22.5291	1.23831	0.619157	+	0.785267i	$0.287475\pi$
$332$	7.08217	0.388684
$333$	0	0
$334$	1.42426	0.0779319
$335$	17.7471	0.969628
$336$	0	0
$337$	−5.20323	−0.283438	−0.141719	−	0.989907i	$-0.545263\pi$
$337$	−5.20323	−0.283438	−0.141719	+	0.989907i	$0.545263\pi$
$338$	3.83753	0.208734
$339$	0	0
$340$	22.9292	1.24351
$341$	42.7081	2.31277
$342$	0	0
$343$	15.5889	0.841723
$344$	0.395777	0.0213389
$345$	0	0
$346$	1.69847	0.0913106
$347$	27.8582	1.49550	0.747752	−	0.663978i	$-0.231133\pi$
$347$	27.8582	1.49550	0.747752	+	0.663978i	$0.231133\pi$
$348$	0	0
$349$	−12.4740	−0.667717	−0.333859	−	0.942623i	$-0.608351\pi$
$349$	−12.4740	−0.667717	−0.333859	+	0.942623i	$0.608351\pi$
$350$	0.299844	0.0160273
$351$	0	0
$352$	8.42463	0.449034
$353$	−20.1347	−1.07166	−0.535832	−	0.844325i	$-0.680002\pi$
$353$	−20.1347	−1.07166	−0.535832	+	0.844325i	$0.680002\pi$
$354$	0	0
$355$	11.2544	0.597319
$356$	9.72011	0.515165
$357$	0	0
$358$	−1.59582	−0.0843416
$359$	28.2621	1.49162	0.745808	−	0.666161i	$-0.232064\pi$
$359$	28.2621	1.49162	0.745808	+	0.666161i	$0.232064\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.36227	−0.0715994
$363$	0	0
$364$	15.5462	0.814840
$365$	2.65070	0.138744
$366$	0	0
$367$	3.03246	0.158293	0.0791466	−	0.996863i	$-0.474780\pi$
$367$	3.03246	0.158293	0.0791466	+	0.996863i	$0.474780\pi$
$368$	1.65072	0.0860499
$369$	0	0
$370$	−0.413207	−0.0214816
$371$	3.68649	0.191393
$372$	0	0
$373$	−5.57204	−0.288509	−0.144255	−	0.989541i	$-0.546078\pi$
$373$	−5.57204	−0.288509	−0.144255	+	0.989541i	$0.546078\pi$
$374$	4.49817	0.232595
$375$	0	0
$376$	0.582164	0.0300228
$377$	−19.1114	−0.984288
$378$	0	0
$379$	−18.6864	−0.959854	−0.479927	−	0.877308i	$-0.659337\pi$
$379$	−18.6864	−0.959854	−0.479927	+	0.877308i	$0.659337\pi$
$380$	3.63036	0.186233
$381$	0	0
$382$	−1.44036	−0.0736953
$383$	27.3187	1.39592	0.697960	−	0.716137i	$-0.254091\pi$
$383$	27.3187	1.39592	0.697960	+	0.716137i	$0.254091\pi$
$384$	0	0
$385$	11.2035	0.570982
$386$	−1.09049	−0.0555045
$387$	0	0
$388$	−0.794657	−0.0403426
$389$	15.1841	0.769866	0.384933	−	0.922945i	$-0.374225\pi$
$389$	15.1841	0.769866	0.384933	+	0.922945i	$0.374225\pi$
$390$	0	0
$391$	2.69265	0.136173
$392$	3.15892	0.159550
$393$	0	0
$394$	−1.87839	−0.0946318
$395$	−30.7912	−1.54927
$396$	0	0
$397$	−23.0611	−1.15741	−0.578703	−	0.815538i	$-0.696441\pi$
$397$	−23.0611	−1.15741	−0.578703	+	0.815538i	$0.696441\pi$
$398$	−2.81108	−0.140907
$399$	0	0
$400$	−6.32463	−0.316231
$401$	13.4118	0.669751	0.334875	−	0.942262i	$-0.391306\pi$
$401$	13.4118	0.669751	0.334875	+	0.942262i	$0.391306\pi$
$402$	0	0
$403$	54.9564	2.73757
$404$	16.0807	0.800045
$405$	0	0
$406$	0.560145	0.0277995
$407$	7.49096	0.371313
$408$	0	0
$409$	−15.0315	−0.743259	−0.371629	−	0.928381i	$-0.621201\pi$
$409$	−15.0315	−0.743259	−0.371629	+	0.928381i	$0.621201\pi$
$410$	−1.12885	−0.0557499
$411$	0	0
$412$	13.2686	0.653697
$413$	−11.2206	−0.552131
$414$	0	0
$415$	6.56756	0.322389
$416$	10.8407	0.531511
$417$	0	0
$418$	0.712191	0.0348344
$419$	−30.7774	−1.50357	−0.751786	−	0.659407i	$-0.770807\pi$
$419$	−30.7774	−1.50357	−0.751786	+	0.659407i	$0.770807\pi$
$420$	0	0
$421$	16.6911	0.813473	0.406736	−	0.913546i	$-0.366667\pi$
$421$	16.6911	0.813473	0.406736	+	0.913546i	$0.366667\pi$
$422$	0.690455	0.0336108
$423$	0	0
$424$	1.71072	0.0830799
$425$	−10.3167	−0.500432
$426$	0	0
$427$	−6.39243	−0.309351
$428$	−35.4260	−1.71238
$429$	0	0
$430$	0.182522	0.00880201
$431$	7.85213	0.378224	0.189112	−	0.981956i	$-0.439439\pi$
$431$	7.85213	0.378224	0.189112	+	0.981956i	$0.439439\pi$
$432$	0	0
$433$	−18.2186	−0.875531	−0.437765	−	0.899089i	$-0.644230\pi$
$433$	−18.2186	−0.875531	−0.437765	+	0.899089i	$0.644230\pi$
$434$	−1.61074	−0.0773179
$435$	0	0
$436$	−36.0332	−1.72568
$437$	0.426324	0.0203938
$438$	0	0
$439$	−33.7986	−1.61312	−0.806559	−	0.591154i	$-0.798673\pi$
$439$	−33.7986	−1.61312	−0.806559	+	0.591154i	$0.798673\pi$
$440$	5.19899	0.247852
$441$	0	0
$442$	5.78820	0.275317
$443$	17.0632	0.810699	0.405349	−	0.914162i	$-0.367150\pi$
$443$	17.0632	0.810699	0.405349	+	0.914162i	$0.367150\pi$
$444$	0	0
$445$	9.01383	0.427297
$446$	2.01421	0.0953756
$447$	0	0
$448$	9.39729	0.443980
$449$	−26.5493	−1.25294	−0.626470	−	0.779446i	$-0.715501\pi$
$449$	−26.5493	−1.25294	−0.626470	+	0.779446i	$0.715501\pi$
$450$	0	0
$451$	20.4647	0.963647
$452$	−27.2004	−1.27940
$453$	0	0
$454$	2.29615	0.107764
$455$	14.4165	0.675858
$456$	0	0
$457$	13.6857	0.640190	0.320095	−	0.947385i	$-0.396285\pi$
$457$	13.6857	0.640190	0.320095	+	0.947385i	$0.396285\pi$
$458$	0.389278	0.0181898
$459$	0	0
$460$	1.54771	0.0721623
$461$	−2.86611	−0.133488	−0.0667440	−	0.997770i	$-0.521261\pi$
$461$	−2.86611	−0.133488	−0.0667440	+	0.997770i	$0.521261\pi$
$462$	0	0
$463$	−19.0748	−0.886480	−0.443240	−	0.896403i	$-0.646171\pi$
$463$	−19.0748	−0.886480	−0.443240	+	0.896403i	$0.646171\pi$
$464$	−11.8152	−0.548505
$465$	0	0
$466$	−1.57502	−0.0729612
$467$	−14.5731	−0.674365	−0.337182	−	0.941439i	$-0.609474\pi$
$467$	−14.5731	−0.674365	−0.337182	+	0.941439i	$0.609474\pi$
$468$	0	0
$469$	12.1342	0.560308
$470$	0.268479	0.0123840
$471$	0	0
$472$	−5.20695	−0.239669
$473$	−3.30892	−0.152144
$474$	0	0
$475$	−1.63343	−0.0749469
$476$	15.6774	0.718573
$477$	0	0
$478$	0.371710	0.0170016
$479$	38.3763	1.75346	0.876729	−	0.480985i	$-0.159721\pi$
$479$	38.3763	1.75346	0.876729	+	0.480985i	$0.159721\pi$
$480$	0	0
$481$	9.63930	0.439514
$482$	1.18234	0.0538539
$483$	0	0
$484$	−25.1078	−1.14126
$485$	−0.736916	−0.0334616
$486$	0	0
$487$	−21.2666	−0.963681	−0.481840	−	0.876259i	$-0.660032\pi$
$487$	−21.2666	−0.963681	−0.481840	+	0.876259i	$0.660032\pi$
$488$	−2.96642	−0.134283
$489$	0	0
$490$	1.45681	0.0658120
$491$	−15.5358	−0.701122	−0.350561	−	0.936540i	$-0.614009\pi$
$491$	−15.5358	−0.701122	−0.350561	+	0.936540i	$0.614009\pi$
$492$	0	0
$493$	−19.2728	−0.868002
$494$	0.916441	0.0412326
$495$	0	0
$496$	33.9754	1.52554
$497$	7.69495	0.345166
$498$	0	0
$499$	−21.6343	−0.968485	−0.484243	−	0.874934i	$-0.660905\pi$
$499$	−21.6343	−0.968485	−0.484243	+	0.874934i	$0.660905\pi$
$500$	−24.0817	−1.07697
$501$	0	0
$502$	4.03933	0.180284
$503$	15.6280	0.696819	0.348409	−	0.937342i	$-0.386722\pi$
$503$	15.6280	0.696819	0.348409	+	0.937342i	$0.386722\pi$
$504$	0	0
$505$	14.9122	0.663586
$506$	0.303624	0.0134977
$507$	0	0
$508$	22.8531	1.01394
$509$	−37.6537	−1.66897	−0.834485	−	0.551031i	$-0.814235\pi$
$509$	−37.6537	−1.66897	−0.834485	+	0.551031i	$0.814235\pi$
$510$	0	0
$511$	1.81237	0.0801744
$512$	11.2103	0.495429
$513$	0	0
$514$	−2.04659	−0.0902714
$515$	12.3045	0.542200
$516$	0	0
$517$	−4.86721	−0.214060
$518$	−0.282522	−0.0124133
$519$	0	0
$520$	6.69002	0.293377
$521$	7.42330	0.325221	0.162610	−	0.986690i	$-0.448009\pi$
$521$	7.42330	0.325221	0.162610	+	0.986690i	$0.448009\pi$
$522$	0	0
$523$	−25.3091	−1.10669	−0.553344	−	0.832953i	$-0.686649\pi$
$523$	−25.3091	−1.10669	−0.553344	+	0.832953i	$0.686649\pi$
$524$	9.74030	0.425507
$525$	0	0
$526$	−1.65155	−0.0720112
$527$	55.4204	2.41415
$528$	0	0
$529$	−22.8182	−0.992098
$530$	0.788940	0.0342694
$531$	0	0
$532$	2.48219	0.107617
$533$	26.3339	1.14065
$534$	0	0
$535$	−32.8519	−1.42031
$536$	5.63092	0.243219
$537$	0	0
$538$	−2.18610	−0.0942496
$539$	−26.4103	−1.13757
$540$	0	0
$541$	14.4326	0.620507	0.310254	−	0.950654i	$-0.399586\pi$
$541$	14.4326	0.620507	0.310254	+	0.950654i	$0.399586\pi$
$542$	−3.76438	−0.161694
$543$	0	0
$544$	10.9323	0.468717
$545$	−33.4150	−1.43134
$546$	0	0
$547$	−2.96127	−0.126615	−0.0633074	−	0.997994i	$-0.520165\pi$
$547$	−2.96127	−0.126615	−0.0633074	+	0.997994i	$0.520165\pi$
$548$	−30.3794	−1.29775
$549$	0	0
$550$	−1.16331	−0.0496038
$551$	−3.05144	−0.129996
$552$	0	0
$553$	−21.0529	−0.895259
$554$	−1.22207	−0.0519206
$555$	0	0
$556$	31.9019	1.35294
$557$	−12.5967	−0.533740	−0.266870	−	0.963733i	$-0.585989\pi$
$557$	−12.5967	−0.533740	−0.266870	+	0.963733i	$0.585989\pi$
$558$	0	0
$559$	−4.25788	−0.180089
$560$	8.91267	0.376629
$561$	0	0
$562$	4.31211	0.181896
$563$	10.3071	0.434394	0.217197	−	0.976128i	$-0.430309\pi$
$563$	10.3071	0.434394	0.217197	+	0.976128i	$0.430309\pi$
$564$	0	0
$565$	−25.2240	−1.06118
$566$	3.00658	0.126376
$567$	0	0
$568$	3.57085	0.149830
$569$	18.3661	0.769946	0.384973	−	0.922928i	$-0.374211\pi$
$569$	18.3661	0.769946	0.384973	+	0.922928i	$0.374211\pi$
$570$	0	0
$571$	2.04003	0.0853725	0.0426862	−	0.999089i	$-0.486408\pi$
$571$	2.04003	0.0853725	0.0426862	+	0.999089i	$0.486408\pi$
$572$	−60.3148	−2.52189
$573$	0	0
$574$	−0.771830	−0.0322156
$575$	−0.696370	−0.0290406
$576$	0	0
$577$	−27.5932	−1.14872	−0.574359	−	0.818603i	$-0.694749\pi$
$577$	−27.5932	−1.14872	−0.574359	+	0.818603i	$0.694749\pi$
$578$	3.34956	0.139323
$579$	0	0
$580$	−11.0778	−0.459981
$581$	4.49045	0.186295
$582$	0	0
$583$	−14.3026	−0.592352
$584$	0.841031	0.0348021
$585$	0	0
$586$	3.41603	0.141115
$587$	40.7346	1.68130	0.840648	−	0.541581i	$-0.182174\pi$
$587$	40.7346	1.68130	0.840648	+	0.541581i	$0.182174\pi$
$588$	0	0
$589$	8.77465	0.361553
$590$	−2.40131	−0.0988603
$591$	0	0
$592$	5.95926	0.244924
$593$	−31.8398	−1.30750	−0.653752	−	0.756709i	$-0.726806\pi$
$593$	−31.8398	−1.30750	−0.653752	+	0.756709i	$0.726806\pi$
$594$	0	0
$595$	14.5383	0.596011
$596$	18.1488	0.743402
$597$	0	0
$598$	0.390701	0.0159769
$599$	−20.0447	−0.819006	−0.409503	−	0.912309i	$-0.634298\pi$
$599$	−20.0447	−0.819006	−0.409503	+	0.912309i	$0.634298\pi$
$600$	0	0
$601$	−38.1963	−1.55806	−0.779029	−	0.626988i	$-0.784287\pi$
$601$	−38.1963	−1.55806	−0.779029	+	0.626988i	$0.784287\pi$
$602$	0.124796	0.00508631
$603$	0	0
$604$	27.1197	1.10348
$605$	−23.2834	−0.946605
$606$	0	0
$607$	−29.9483	−1.21556	−0.607781	−	0.794105i	$-0.707940\pi$
$607$	−29.9483	−1.21556	−0.607781	+	0.794105i	$0.707940\pi$
$608$	1.73089	0.0701970
$609$	0	0
$610$	−1.36803	−0.0553901
$611$	−6.26308	−0.253377
$612$	0	0
$613$	10.2200	0.412782	0.206391	−	0.978470i	$-0.433828\pi$
$613$	10.2200	0.412782	0.206391	+	0.978470i	$0.433828\pi$
$614$	−0.669144	−0.0270045
$615$	0	0
$616$	3.55471	0.143223
$617$	−21.0489	−0.847396	−0.423698	−	0.905804i	$-0.639268\pi$
$617$	−21.0489	−0.847396	−0.423698	+	0.905804i	$0.639268\pi$
$618$	0	0
$619$	−0.298668	−0.0120045	−0.00600225	−	0.999982i	$-0.501911\pi$
$619$	−0.298668	−0.0120045	−0.00600225	+	0.999982i	$0.501911\pi$
$620$	31.8551	1.27933
$621$	0	0
$622$	3.76206	0.150845
$623$	6.16304	0.246917
$624$	0	0
$625$	−14.1648	−0.566591
$626$	3.63597	0.145323
$627$	0	0
$628$	28.0643	1.11989
$629$	9.72069	0.387589
$630$	0	0
$631$	41.8441	1.66579	0.832893	−	0.553434i	$-0.186683\pi$
$631$	41.8441	1.66579	0.832893	+	0.553434i	$0.186683\pi$
$632$	−9.76962	−0.388615
$633$	0	0
$634$	4.24306	0.168513
$635$	21.1926	0.841001
$636$	0	0
$637$	−33.9845	−1.34652
$638$	−2.17321	−0.0860381
$639$	0	0
$640$	8.36286	0.330571
$641$	30.6845	1.21197	0.605983	−	0.795478i	$-0.292780\pi$
$641$	30.6845	1.21197	0.605983	+	0.795478i	$0.292780\pi$
$642$	0	0
$643$	−5.15022	−0.203105	−0.101552	−	0.994830i	$-0.532381\pi$
$643$	−5.15022	−0.203105	−0.101552	+	0.994830i	$0.532381\pi$
$644$	1.05822	0.0416996
$645$	0	0
$646$	0.924178	0.0363613
$647$	−37.2260	−1.46351	−0.731753	−	0.681570i	$-0.761297\pi$
$647$	−37.2260	−1.46351	−0.731753	+	0.681570i	$0.761297\pi$
$648$	0	0
$649$	43.5329	1.70882
$650$	−1.49694	−0.0587148
$651$	0	0
$652$	−15.4485	−0.605011
$653$	17.3932	0.680647	0.340324	−	0.940308i	$-0.389463\pi$
$653$	17.3932	0.680647	0.340324	+	0.940308i	$0.389463\pi$
$654$	0	0
$655$	9.03256	0.352931
$656$	16.2802	0.635637
$657$	0	0
$658$	0.183567	0.00715620
$659$	−3.95132	−0.153922	−0.0769608	−	0.997034i	$-0.524522\pi$
$659$	−3.95132	−0.153922	−0.0769608	+	0.997034i	$0.524522\pi$
$660$	0	0
$661$	24.1686	0.940049	0.470025	−	0.882653i	$-0.344245\pi$
$661$	24.1686	0.940049	0.470025	+	0.882653i	$0.344245\pi$
$662$	3.29656	0.128124
$663$	0	0
$664$	2.08380	0.0808671
$665$	2.30183	0.0892611
$666$	0	0
$667$	−1.30090	−0.0503711
$668$	−19.2587	−0.745143
$669$	0	0
$670$	2.59683	0.100324
$671$	24.8009	0.957427
$672$	0	0
$673$	32.5719	1.25555	0.627777	−	0.778394i	$-0.283965\pi$
$673$	32.5719	1.25555	0.627777	+	0.778394i	$0.283965\pi$
$674$	−0.761359	−0.0293265
$675$	0	0
$676$	−51.8909	−1.99580
$677$	−17.8331	−0.685381	−0.342691	−	0.939448i	$-0.611338\pi$
$677$	−17.8331	−0.685381	−0.342691	+	0.939448i	$0.611338\pi$
$678$	0	0
$679$	−0.503852	−0.0193361
$680$	6.74650	0.258717
$681$	0	0
$682$	6.24923	0.239295
$683$	−7.43606	−0.284533	−0.142266	−	0.989828i	$-0.545439\pi$
$683$	−7.43606	−0.284533	−0.142266	+	0.989828i	$0.545439\pi$
$684$	0	0
$685$	−28.1720	−1.07640
$686$	2.28104	0.0870904
$687$	0	0
$688$	−2.63233	−0.100357
$689$	−18.4044	−0.701152
$690$	0	0
$691$	16.3347	0.621401	0.310700	−	0.950508i	$-0.399436\pi$
$691$	16.3347	0.621401	0.310700	+	0.950508i	$0.399436\pi$
$692$	−22.9667	−0.873063
$693$	0	0
$694$	4.07632	0.154735
$695$	29.5839	1.12218
$696$	0	0
$697$	26.5562	1.00589
$698$	−1.82525	−0.0690866
$699$	0	0
$700$	−4.05448	−0.153245
$701$	−11.7972	−0.445574	−0.222787	−	0.974867i	$-0.571516\pi$
$701$	−11.7972	−0.445574	−0.222787	+	0.974867i	$0.571516\pi$
$702$	0	0
$703$	1.53907	0.0580470
$704$	−36.4589	−1.37410
$705$	0	0
$706$	−2.94620	−0.110882
$707$	10.1960	0.383459
$708$	0	0
$709$	17.5291	0.658319	0.329159	−	0.944274i	$-0.393235\pi$
$709$	17.5291	0.658319	0.329159	+	0.944274i	$0.393235\pi$
$710$	1.64678	0.0618027
$711$	0	0
$712$	2.85997	0.107182
$713$	3.74085	0.140096
$714$	0	0
$715$	−55.9322	−2.09175
$716$	21.5786	0.806430
$717$	0	0
$718$	4.13543	0.154333
$719$	17.8080	0.664127	0.332064	−	0.943257i	$-0.392255\pi$
$719$	17.8080	0.664127	0.332064	+	0.943257i	$0.392255\pi$
$720$	0	0
$721$	8.41295	0.313315
$722$	0.146324	0.00544562
$723$	0	0
$724$	18.4206	0.684595
$725$	4.98431	0.185113
$726$	0	0
$727$	−10.1124	−0.375046	−0.187523	−	0.982260i	$-0.560046\pi$
$727$	−10.1124	−0.375046	−0.187523	+	0.982260i	$0.560046\pi$
$728$	4.57417	0.169530
$729$	0	0
$730$	0.387862	0.0143554
$731$	−4.29383	−0.158813
$732$	0	0
$733$	−47.0636	−1.73833	−0.869167	−	0.494519i	$-0.835344\pi$
$733$	−47.0636	−1.73833	−0.869167	+	0.494519i	$0.835344\pi$
$734$	0.443723	0.0163781
$735$	0	0
$736$	0.737922	0.0272002
$737$	−47.0776	−1.73412
$738$	0	0
$739$	−18.7202	−0.688635	−0.344318	−	0.938853i	$-0.611890\pi$
$739$	−18.7202	−0.688635	−0.344318	+	0.938853i	$0.611890\pi$
$740$	5.58736	0.205396
$741$	0	0
$742$	0.539422	0.0198028
$743$	50.8570	1.86576	0.932881	−	0.360186i	$-0.117287\pi$
$743$	50.8570	1.86576	0.932881	+	0.360186i	$0.117287\pi$
$744$	0	0
$745$	16.8300	0.616605
$746$	−0.815324	−0.0298511
$747$	0	0
$748$	−60.8240	−2.22395
$749$	−22.4618	−0.820738
$750$	0	0
$751$	38.4967	1.40476	0.702381	−	0.711801i	$-0.252120\pi$
$751$	38.4967	1.40476	0.702381	+	0.711801i	$0.252120\pi$
$752$	−3.87199	−0.141197
$753$	0	0
$754$	−2.79646	−0.101841
$755$	25.1491	0.915270
$756$	0	0
$757$	34.9104	1.26884	0.634421	−	0.772988i	$-0.281239\pi$
$757$	34.9104	1.26884	0.634421	+	0.772988i	$0.281239\pi$
$758$	−2.73427	−0.0993131
$759$	0	0
$760$	1.06817	0.0387465
$761$	33.6484	1.21975	0.609877	−	0.792496i	$-0.291219\pi$
$761$	33.6484	1.21975	0.609877	+	0.792496i	$0.291219\pi$
$762$	0	0
$763$	−22.8469	−0.827112
$764$	19.4765	0.704635
$765$	0	0
$766$	3.99739	0.144431
$767$	56.0178	2.02268
$768$	0	0
$769$	14.9609	0.539505	0.269753	−	0.962930i	$-0.413058\pi$
$769$	14.9609	0.539505	0.269753	+	0.962930i	$0.413058\pi$
$770$	1.63934	0.0590778
$771$	0	0
$772$	14.7456	0.530704
$773$	−41.0102	−1.47504	−0.737518	−	0.675328i	$-0.764002\pi$
$773$	−41.0102	−1.47504	−0.737518	+	0.675328i	$0.764002\pi$
$774$	0	0
$775$	−14.3328	−0.514848
$776$	−0.233813	−0.00839342
$777$	0	0
$778$	2.22181	0.0796556
$779$	4.20462	0.150646
$780$	0	0
$781$	−29.8543	−1.06827
$782$	0.393999	0.0140894
$783$	0	0
$784$	−21.0101	−0.750360
$785$	26.0251	0.928875
$786$	0	0
$787$	−7.96901	−0.284065	−0.142032	−	0.989862i	$-0.545364\pi$
$787$	−7.96901	−0.284065	−0.142032	+	0.989862i	$0.545364\pi$
$788$	25.3995	0.904819
$789$	0	0
$790$	−4.50549	−0.160298
$791$	−17.2464	−0.613211
$792$	0	0
$793$	31.9136	1.13328
$794$	−3.37440	−0.119753
$795$	0	0
$796$	38.0112	1.34727
$797$	−0.424750	−0.0150454	−0.00752270	−	0.999972i	$-0.502395\pi$
$797$	−0.424750	−0.0150454	−0.00752270	+	0.999972i	$0.502395\pi$
$798$	0	0
$799$	−6.31596	−0.223443
$800$	−2.82729	−0.0999599
$801$	0	0
$802$	1.96246	0.0692970
$803$	−7.03149	−0.248136
$804$	0	0
$805$	0.981325	0.0345872
$806$	8.04145	0.283248
$807$	0	0
$808$	4.73145	0.166452
$809$	47.6410	1.67497	0.837484	−	0.546462i	$-0.184026\pi$
$809$	47.6410	1.67497	0.837484	+	0.546462i	$0.184026\pi$
$810$	0	0
$811$	34.2334	1.20210	0.601049	−	0.799212i	$-0.294750\pi$
$811$	34.2334	1.20210	0.601049	+	0.799212i	$0.294750\pi$
$812$	−7.57425	−0.265804
$813$	0	0
$814$	1.09611	0.0384186
$815$	−14.3260	−0.501818
$816$	0	0
$817$	−0.679839	−0.0237845
$818$	−2.19947	−0.0769026
$819$	0	0
$820$	15.2643	0.533051
$821$	−20.4007	−0.711991	−0.355995	−	0.934488i	$-0.615858\pi$
$821$	−20.4007	−0.711991	−0.355995	+	0.934488i	$0.615858\pi$
$822$	0	0
$823$	−32.2445	−1.12397	−0.561986	−	0.827147i	$-0.689963\pi$
$823$	−32.2445	−1.12397	−0.561986	+	0.827147i	$0.689963\pi$
$824$	3.90404	0.136004
$825$	0	0
$826$	−1.64185	−0.0571272
$827$	4.16860	0.144957	0.0724783	−	0.997370i	$-0.476909\pi$
$827$	4.16860	0.144957	0.0724783	+	0.997370i	$0.476909\pi$
$828$	0	0
$829$	−48.2070	−1.67430	−0.837150	−	0.546974i	$-0.815780\pi$
$829$	−48.2070	−1.67430	−0.837150	+	0.546974i	$0.815780\pi$
$830$	0.960994	0.0333566
$831$	0	0
$832$	−46.9150	−1.62648
$833$	−34.2715	−1.18744
$834$	0	0
$835$	−17.8594	−0.618049
$836$	−9.63021	−0.333068
$837$	0	0
$838$	−4.50347	−0.155570
$839$	−16.6660	−0.575373	−0.287687	−	0.957725i	$-0.592886\pi$
$839$	−16.6660	−0.575373	−0.287687	+	0.957725i	$0.592886\pi$
$840$	0	0
$841$	−19.6887	−0.678921
$842$	2.44231	0.0841674
$843$	0	0
$844$	−9.33630	−0.321369
$845$	−48.1204	−1.65539
$846$	0	0
$847$	−15.9196	−0.547003
$848$	−11.3781	−0.390724
$849$	0	0
$850$	−1.50958	−0.0517781
$851$	0.656141	0.0224922
$852$	0	0
$853$	41.4482	1.41916	0.709580	−	0.704625i	$-0.248885\pi$
$853$	41.4482	1.41916	0.709580	+	0.704625i	$0.248885\pi$
$854$	−0.935368	−0.0320076
$855$	0	0
$856$	−10.4234	−0.356266
$857$	40.0382	1.36768	0.683839	−	0.729633i	$-0.260309\pi$
$857$	40.0382	1.36768	0.683839	+	0.729633i	$0.260309\pi$
$858$	0	0
$859$	−3.41157	−0.116401	−0.0582006	−	0.998305i	$-0.518536\pi$
$859$	−3.41157	−0.116401	−0.0582006	+	0.998305i	$0.518536\pi$
$860$	−2.46806	−0.0841601
$861$	0	0
$862$	1.14896	0.0391336
$863$	52.6421	1.79196	0.895978	−	0.444098i	$-0.146476\pi$
$863$	52.6421	1.79196	0.895978	+	0.444098i	$0.146476\pi$
$864$	0	0
$865$	−21.2979	−0.724150
$866$	−2.66582	−0.0905884
$867$	0	0
$868$	21.7803	0.739273
$869$	81.6794	2.77078
$870$	0	0
$871$	−60.5790	−2.05264
$872$	−10.6021	−0.359033
$873$	0	0
$874$	0.0623815	0.00211009
$875$	−15.2690	−0.516187
$876$	0	0
$877$	15.1160	0.510430	0.255215	−	0.966884i	$-0.417854\pi$
$877$	15.1160	0.510430	0.255215	+	0.966884i	$0.417854\pi$
$878$	−4.94555	−0.166904
$879$	0	0
$880$	−34.5787	−1.16565
$881$	−51.4291	−1.73269	−0.866346	−	0.499445i	$-0.833537\pi$
$881$	−51.4291	−1.73269	−0.866346	+	0.499445i	$0.833537\pi$
$882$	0	0
$883$	−47.4099	−1.59547	−0.797734	−	0.603009i	$-0.793968\pi$
$883$	−47.4099	−1.59547	−0.797734	+	0.603009i	$0.793968\pi$
$884$	−78.2678	−2.63243
$885$	0	0
$886$	2.49676	0.0838804
$887$	50.9997	1.71240	0.856201	−	0.516643i	$-0.172819\pi$
$887$	50.9997	1.71240	0.856201	+	0.516643i	$0.172819\pi$
$888$	0	0
$889$	14.4900	0.485979
$890$	1.31894	0.0442110
$891$	0	0
$892$	−27.2361	−0.911930
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	20.0107	0.668882
$896$	5.71795	0.191023
$897$	0	0
$898$	−3.88481	−0.129638
$899$	−26.7753	−0.893007
$900$	0	0
$901$	−18.5598	−0.618316
$902$	2.99449	0.0997055
$903$	0	0
$904$	−8.00322	−0.266183
$905$	17.0821	0.567828
$906$	0	0
$907$	−45.6905	−1.51713	−0.758565	−	0.651598i	$-0.774099\pi$
$907$	−45.6905	−1.51713	−0.758565	+	0.651598i	$0.774099\pi$
$908$	−31.0484	−1.03038
$909$	0	0
$910$	2.10949	0.0699289
$911$	−24.0553	−0.796987	−0.398493	−	0.917171i	$-0.630467\pi$
$911$	−24.0553	−0.796987	−0.398493	+	0.917171i	$0.630467\pi$
$912$	0	0
$913$	−17.4217	−0.576574
$914$	2.00255	0.0662384
$915$	0	0
$916$	−5.26380	−0.173921
$917$	6.17584	0.203944
$918$	0	0
$919$	13.0273	0.429730	0.214865	−	0.976644i	$-0.431069\pi$
$919$	13.0273	0.429730	0.214865	+	0.976644i	$0.431069\pi$
$920$	0.455385	0.0150136
$921$	0	0
$922$	−0.419381	−0.0138116
$923$	−38.4162	−1.26449
$924$	0	0
$925$	−2.51396	−0.0826584
$926$	−2.79110	−0.0917212
$927$	0	0
$928$	−5.28172	−0.173381
$929$	50.7741	1.66584	0.832922	−	0.553390i	$-0.186666\pi$
$929$	50.7741	1.66584	0.832922	+	0.553390i	$0.186666\pi$
$930$	0	0
$931$	−5.42617	−0.177835
$932$	21.2973	0.697616
$933$	0	0
$934$	−2.13240	−0.0697744
$935$	−56.4045	−1.84462
$936$	0	0
$937$	6.15480	0.201069	0.100534	−	0.994934i	$-0.467945\pi$
$937$	6.15480	0.201069	0.100534	+	0.994934i	$0.467945\pi$
$938$	1.77553	0.0579733
$939$	0	0
$940$	−3.63036	−0.118409
$941$	39.6159	1.29144	0.645721	−	0.763574i	$-0.276557\pi$
$941$	39.6159	1.29144	0.645721	+	0.763574i	$0.276557\pi$
$942$	0	0
$943$	1.79253	0.0583728
$944$	34.6316	1.12716
$945$	0	0
$946$	−0.484175	−0.0157419
$947$	−55.8032	−1.81336	−0.906680	−	0.421820i	$-0.861392\pi$
$947$	−55.8032	−1.81336	−0.906680	+	0.421820i	$0.861392\pi$
$948$	0	0
$949$	−9.04805	−0.293712
$950$	−0.239010	−0.00775452
$951$	0	0
$952$	4.61279	0.149501
$953$	−32.6056	−1.05620	−0.528100	−	0.849182i	$-0.677095\pi$
$953$	−32.6056	−1.05620	−0.528100	+	0.849182i	$0.677095\pi$
$954$	0	0
$955$	18.0613	0.584450
$956$	−5.02624	−0.162560
$957$	0	0
$958$	5.61538	0.181425
$959$	−19.2621	−0.622005
$960$	0	0
$961$	45.9945	1.48369
$962$	1.41046	0.0454752
$963$	0	0
$964$	−15.9875	−0.514922
$965$	13.6741	0.440186
$966$	0	0
$967$	−43.5158	−1.39938	−0.699688	−	0.714449i	$-0.746677\pi$
$967$	−43.5158	−1.39938	−0.699688	+	0.714449i	$0.746677\pi$
$968$	−7.38751	−0.237443
$969$	0	0
$970$	−0.107829	−0.00346217
$971$	34.1230	1.09506	0.547530	−	0.836786i	$-0.315568\pi$
$971$	34.1230	1.09506	0.547530	+	0.836786i	$0.315568\pi$
$972$	0	0
$973$	20.2274	0.648461
$974$	−3.11182	−0.0997090
$975$	0	0
$976$	19.7298	0.631534
$977$	−6.23854	−0.199589	−0.0997943	−	0.995008i	$-0.531818\pi$
$977$	−6.23854	−0.199589	−0.0997943	+	0.995008i	$0.531818\pi$
$978$	0	0
$979$	−23.9109	−0.764195
$980$	−19.6989	−0.629259
$981$	0	0
$982$	−2.27327	−0.0725429
$983$	−25.7659	−0.821805	−0.410903	−	0.911679i	$-0.634786\pi$
$983$	−25.7659	−0.821805	−0.410903	+	0.911679i	$0.634786\pi$
$984$	0	0
$985$	23.5539	0.750490
$986$	−2.82007	−0.0898095
$987$	0	0
$988$	−12.3921	−0.394244
$989$	−0.289832	−0.00921611
$990$	0	0
$991$	36.4081	1.15654	0.578271	−	0.815845i	$-0.303728\pi$
$991$	36.4081	1.15654	0.578271	+	0.815845i	$0.303728\pi$
$992$	15.1880	0.482219
$993$	0	0
$994$	1.12596	0.0357132
$995$	35.2493	1.11748
$996$	0	0
$997$	16.4208	0.520053	0.260026	−	0.965602i	$-0.416269\pi$
$997$	16.4208	0.520053	0.260026	+	0.965602i	$0.416269\pi$
$998$	−3.16563	−0.100206
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.x.1.17	yes	34
3.2	odd	2		8037.2.a.u.1.18	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.18	✓	34	3.2	odd	2
8037.2.a.x.1.17	yes	34	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.146324 q^{2} -1.97859 q^{4} -1.83482 q^{5} -1.25452 q^{7} -0.582164 q^{8} +O(q^{10})\)	\(q+0.146324 q^{2} -1.97859 q^{4} -1.83482 q^{5} -1.25452 q^{7} -0.582164 q^{8} -0.268479 q^{10} +4.86721 q^{11} +6.26308 q^{13} -0.183567 q^{14} +3.87199 q^{16} +6.31596 q^{17} +1.00000 q^{19} +3.63036 q^{20} +0.712191 q^{22} +0.426324 q^{23} -1.63343 q^{25} +0.916441 q^{26} +2.48219 q^{28} -3.05144 q^{29} +8.77465 q^{31} +1.73089 q^{32} +0.924178 q^{34} +2.30183 q^{35} +1.53907 q^{37} +0.146324 q^{38} +1.06817 q^{40} +4.20462 q^{41} -0.679839 q^{43} -9.63021 q^{44} +0.0623815 q^{46} -1.00000 q^{47} -5.42617 q^{49} -0.239010 q^{50} -12.3921 q^{52} -2.93855 q^{53} -8.93046 q^{55} +0.730339 q^{56} -0.446500 q^{58} +8.94412 q^{59} +5.09550 q^{61} +1.28394 q^{62} -7.49072 q^{64} -11.4916 q^{65} -9.67239 q^{67} -12.4967 q^{68} +0.336813 q^{70} -6.13376 q^{71} -1.44466 q^{73} +0.225203 q^{74} -1.97859 q^{76} -6.10603 q^{77} +16.7816 q^{79} -7.10442 q^{80} +0.615237 q^{82} -3.57940 q^{83} -11.5887 q^{85} -0.0994768 q^{86} -2.83351 q^{88} -4.91265 q^{89} -7.85719 q^{91} -0.843520 q^{92} -0.146324 q^{94} -1.83482 q^{95} +0.401628 q^{97} -0.793980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8} + 18 q^{11} - 6 q^{13} + 12 q^{14} + 21 q^{16} + 36 q^{17} + 34 q^{19} + 40 q^{20} + 12 q^{22} + 38 q^{23} + 32 q^{25} + 15 q^{26} + 28 q^{28} + 14 q^{29} - 6 q^{31} + 35 q^{32} + 10 q^{34} + 46 q^{35} - 2 q^{37} + 5 q^{38} + 31 q^{40} + 18 q^{41} - 6 q^{43} + 42 q^{44} - 14 q^{46} - 34 q^{47} + 44 q^{49} + 9 q^{50} + 2 q^{52} + 32 q^{53} + 8 q^{55} - 4 q^{56} + 8 q^{58} + 62 q^{59} - 10 q^{61} + 30 q^{62} - 37 q^{64} + 8 q^{65} + 92 q^{68} - 62 q^{70} + 4 q^{71} - 8 q^{73} + 34 q^{74} + 31 q^{76} + 52 q^{77} + 40 q^{79} + 48 q^{80} - 2 q^{82} + 110 q^{83} - 12 q^{85} + 16 q^{86} - 44 q^{88} + 2 q^{89} - 28 q^{91} + 60 q^{92} - 5 q^{94} + 14 q^{95} + 2 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 14 * q^5 + 15 * q^8 + 18 * q^11 - 6 * q^13 + 12 * q^14 + 21 * q^16 + 36 * q^17 + 34 * q^19 + 40 * q^20 + 12 * q^22 + 38 * q^23 + 32 * q^25 + 15 * q^26 + 28 * q^28 + 14 * q^29 - 6 * q^31 + 35 * q^32 + 10 * q^34 + 46 * q^35 - 2 * q^37 + 5 * q^38 + 31 * q^40 + 18 * q^41 - 6 * q^43 + 42 * q^44 - 14 * q^46 - 34 * q^47 + 44 * q^49 + 9 * q^50 + 2 * q^52 + 32 * q^53 + 8 * q^55 - 4 * q^56 + 8 * q^58 + 62 * q^59 - 10 * q^61 + 30 * q^62 - 37 * q^64 + 8 * q^65 + 92 * q^68 - 62 * q^70 + 4 * q^71 - 8 * q^73 + 34 * q^74 + 31 * q^76 + 52 * q^77 + 40 * q^79 + 48 * q^80 - 2 * q^82 + 110 * q^83 - 12 * q^85 + 16 * q^86 - 44 * q^88 + 2 * q^89 - 28 * q^91 + 60 * q^92 - 5 * q^94 + 14 * q^95 + 2 * q^97 + 45 * q^98

Introduction

L-functions

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Database

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