LMFDB - Embedded newform 8037.2.a.x.1.30

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			1.30
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)

$f(q)$	$=$	$q+2.29036 q^{2} +3.24574 q^{4} -1.89255 q^{5} +0.151615 q^{7} +2.85320 q^{8} +O(q^{10})$	\(q+2.29036 q^{2} +3.24574 q^{4} -1.89255 q^{5} +0.151615 q^{7} +2.85320 q^{8} -4.33463 q^{10} -3.32731 q^{11} -4.37847 q^{13} +0.347253 q^{14} +0.0433649 q^{16} +2.98588 q^{17} +1.00000 q^{19} -6.14274 q^{20} -7.62073 q^{22} +8.70880 q^{23} -1.41824 q^{25} -10.0283 q^{26} +0.492104 q^{28} +1.78773 q^{29} +5.37099 q^{31} -5.60708 q^{32} +6.83873 q^{34} -0.286940 q^{35} +6.55746 q^{37} +2.29036 q^{38} -5.39983 q^{40} +4.82536 q^{41} +7.41239 q^{43} -10.7996 q^{44} +19.9463 q^{46} -1.00000 q^{47} -6.97701 q^{49} -3.24828 q^{50} -14.2114 q^{52} +6.90782 q^{53} +6.29711 q^{55} +0.432588 q^{56} +4.09455 q^{58} -10.5959 q^{59} +14.2796 q^{61} +12.3015 q^{62} -12.9290 q^{64} +8.28650 q^{65} -1.40346 q^{67} +9.69139 q^{68} -0.657195 q^{70} +0.0570003 q^{71} +4.42666 q^{73} +15.0189 q^{74} +3.24574 q^{76} -0.504470 q^{77} +14.3838 q^{79} -0.0820704 q^{80} +11.0518 q^{82} +10.1368 q^{83} -5.65093 q^{85} +16.9770 q^{86} -9.49348 q^{88} -2.88078 q^{89} -0.663843 q^{91} +28.2665 q^{92} -2.29036 q^{94} -1.89255 q^{95} -15.9115 q^{97} -15.9799 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.29036	1.61953	0.809764	−	0.586756i	$-0.199595\pi$
$2$	2.29036	1.61953	0.809764	+	0.586756i	$0.199595\pi$
$3$	0	0
$3$	0	0
$4$	3.24574	1.62287
$5$	−1.89255	−0.846376	−0.423188	−	0.906042i	$-0.639089\pi$
$5$	−1.89255	−0.846376	−0.423188	+	0.906042i	$0.639089\pi$
$6$	0	0
$7$	0.151615	0.0573051	0.0286526	−	0.999589i	$-0.490878\pi$
$7$	0.151615	0.0573051	0.0286526	+	0.999589i	$0.490878\pi$
$8$	2.85320	1.00876
$9$	0	0
$10$	−4.33463	−1.37073
$11$	−3.32731	−1.00322	−0.501611	−	0.865094i	$-0.667259\pi$
$11$	−3.32731	−1.00322	−0.501611	+	0.865094i	$0.667259\pi$
$12$	0	0
$13$	−4.37847	−1.21437	−0.607185	−	0.794560i	$-0.707701\pi$
$13$	−4.37847	−1.21437	−0.607185	+	0.794560i	$0.707701\pi$
$14$	0.347253	0.0928073
$15$	0	0
$16$	0.0433649	0.0108412
$17$	2.98588	0.724181	0.362091	−	0.932143i	$-0.382063\pi$
$17$	2.98588	0.724181	0.362091	+	0.932143i	$0.382063\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−6.14274	−1.37356
$21$	0	0
$22$	−7.62073	−1.62474
$23$	8.70880	1.81591	0.907955	−	0.419068i	$-0.137643\pi$
$23$	8.70880	1.81591	0.907955	+	0.419068i	$0.137643\pi$
$24$	0	0
$25$	−1.41824	−0.283648
$26$	−10.0283	−1.96671
$27$	0	0
$28$	0.492104	0.0929989
$29$	1.78773	0.331974	0.165987	−	0.986128i	$-0.446919\pi$
$29$	1.78773	0.331974	0.165987	+	0.986128i	$0.446919\pi$
$30$	0	0
$31$	5.37099	0.964659	0.482329	−	0.875990i	$-0.339791\pi$
$31$	5.37099	0.964659	0.482329	+	0.875990i	$0.339791\pi$
$32$	−5.60708	−0.991201
$33$	0	0
$34$	6.83873	1.17283
$35$	−0.286940	−0.0485017
$36$	0	0
$37$	6.55746	1.07804	0.539020	−	0.842293i	$-0.318795\pi$
$37$	6.55746	1.07804	0.539020	+	0.842293i	$0.318795\pi$
$38$	2.29036	0.371545
$39$	0	0
$40$	−5.39983	−0.853789
$41$	4.82536	0.753595	0.376798	−	0.926296i	$-0.377025\pi$
$41$	4.82536	0.753595	0.376798	+	0.926296i	$0.377025\pi$
$42$	0	0
$43$	7.41239	1.13038	0.565190	−	0.824961i	$-0.308803\pi$
$43$	7.41239	1.13038	0.565190	+	0.824961i	$0.308803\pi$
$44$	−10.7996	−1.62810
$45$	0	0
$46$	19.9463	2.94092
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.97701	−0.996716
$50$	−3.24828	−0.459376
$51$	0	0
$52$	−14.2114	−1.97077
$53$	6.90782	0.948861	0.474431	−	0.880293i	$-0.342654\pi$
$53$	6.90782	0.948861	0.474431	+	0.880293i	$0.342654\pi$
$54$	0	0
$55$	6.29711	0.849102
$56$	0.432588	0.0578070
$57$	0	0
$58$	4.09455	0.537641
$59$	−10.5959	−1.37947	−0.689736	−	0.724061i	$-0.742273\pi$
$59$	−10.5959	−1.37947	−0.689736	+	0.724061i	$0.742273\pi$
$60$	0	0
$61$	14.2796	1.82832	0.914160	−	0.405354i	$-0.132852\pi$
$61$	14.2796	1.82832	0.914160	+	0.405354i	$0.132852\pi$
$62$	12.3015	1.56229
$63$	0	0
$64$	−12.9290	−1.61612
$65$	8.28650	1.02781
$66$	0	0
$67$	−1.40346	−0.171460	−0.0857302	−	0.996318i	$-0.527322\pi$
$67$	−1.40346	−0.171460	−0.0857302	+	0.996318i	$0.527322\pi$
$68$	9.69139	1.17525
$69$	0	0
$70$	−0.657195	−0.0785498
$71$	0.0570003	0.00676470	0.00338235	−	0.999994i	$-0.498923\pi$
$71$	0.0570003	0.00676470	0.00338235	+	0.999994i	$0.498923\pi$
$72$	0	0
$73$	4.42666	0.518101	0.259050	−	0.965864i	$-0.416590\pi$
$73$	4.42666	0.518101	0.259050	+	0.965864i	$0.416590\pi$
$74$	15.0189	1.74592
$75$	0	0
$76$	3.24574	0.372312
$77$	−0.504470	−0.0574897
$78$	0	0
$79$	14.3838	1.61830	0.809152	−	0.587600i	$-0.199927\pi$
$79$	14.3838	1.61830	0.809152	+	0.587600i	$0.199927\pi$
$80$	−0.0820704	−0.00917575
$81$	0	0
$82$	11.0518	1.22047
$83$	10.1368	1.11266	0.556332	−	0.830960i	$-0.312208\pi$
$83$	10.1368	1.11266	0.556332	+	0.830960i	$0.312208\pi$
$84$	0	0
$85$	−5.65093	−0.612930
$86$	16.9770	1.83068
$87$	0	0
$88$	−9.49348	−1.01201
$89$	−2.88078	−0.305362	−0.152681	−	0.988276i	$-0.548791\pi$
$89$	−2.88078	−0.305362	−0.152681	+	0.988276i	$0.548791\pi$
$90$	0	0
$91$	−0.663843	−0.0695896
$92$	28.2665	2.94699
$93$	0	0
$94$	−2.29036	−0.236232
$95$	−1.89255	−0.194172
$96$	0	0
$97$	−15.9115	−1.61557	−0.807783	−	0.589479i	$-0.799333\pi$
$97$	−15.9115	−1.61557	−0.807783	+	0.589479i	$0.799333\pi$
$98$	−15.9799	−1.61421
$99$	0	0
$100$	−4.60325	−0.460325
$101$	15.9190	1.58400	0.791998	−	0.610524i	$-0.209041\pi$
$101$	15.9190	1.58400	0.791998	+	0.610524i	$0.209041\pi$
$102$	0	0
$103$	15.5231	1.52953	0.764766	−	0.644308i	$-0.222855\pi$
$103$	15.5231	1.52953	0.764766	+	0.644308i	$0.222855\pi$
$104$	−12.4927	−1.22501
$105$	0	0
$106$	15.8214	1.53671
$107$	−5.73797	−0.554710	−0.277355	−	0.960767i	$-0.589458\pi$
$107$	−5.73797	−0.554710	−0.277355	+	0.960767i	$0.589458\pi$
$108$	0	0
$109$	5.95488	0.570374	0.285187	−	0.958472i	$-0.407944\pi$
$109$	5.95488	0.570374	0.285187	+	0.958472i	$0.407944\pi$
$110$	14.4226	1.37514
$111$	0	0
$112$	0.00657477	0.000621258 0
$113$	−4.23990	−0.398856	−0.199428	−	0.979912i	$-0.563908\pi$
$113$	−4.23990	−0.398856	−0.199428	+	0.979912i	$0.563908\pi$
$114$	0	0
$115$	−16.4819	−1.53694
$116$	5.80252	0.538751
$117$	0	0
$118$	−24.2685	−2.23409
$119$	0.452704	0.0414993
$120$	0	0
$121$	0.0709771	0.00645246
$122$	32.7055	2.96102
$123$	0	0
$124$	17.4329	1.56552
$125$	12.1469	1.08645
$126$	0	0
$127$	−21.6018	−1.91685	−0.958424	−	0.285348i	$-0.907891\pi$
$127$	−21.6018	−1.91685	−0.958424	+	0.285348i	$0.907891\pi$
$128$	−18.3978	−1.62615
$129$	0	0
$130$	18.9790	1.66457
$131$	16.2798	1.42237	0.711185	−	0.703005i	$-0.248159\pi$
$131$	16.2798	1.42237	0.711185	+	0.703005i	$0.248159\pi$
$132$	0	0
$133$	0.151615	0.0131467
$134$	−3.21444	−0.277685
$135$	0	0
$136$	8.51930	0.730524
$137$	5.29056	0.452003	0.226001	−	0.974127i	$-0.427435\pi$
$137$	5.29056	0.452003	0.226001	+	0.974127i	$0.427435\pi$
$138$	0	0
$139$	2.22833	0.189005	0.0945024	−	0.995525i	$-0.469874\pi$
$139$	2.22833	0.189005	0.0945024	+	0.995525i	$0.469874\pi$
$140$	−0.931333	−0.0787120
$141$	0	0
$142$	0.130551	0.0109556
$143$	14.5685	1.21828
$144$	0	0
$145$	−3.38338	−0.280974
$146$	10.1386	0.839079
$147$	0	0
$148$	21.2838	1.74952
$149$	3.49025	0.285933	0.142966	−	0.989728i	$-0.454336\pi$
$149$	3.49025	0.285933	0.142966	+	0.989728i	$0.454336\pi$
$150$	0	0
$151$	−3.48150	−0.283321	−0.141660	−	0.989915i	$-0.545244\pi$
$151$	−3.48150	−0.283321	−0.141660	+	0.989915i	$0.545244\pi$
$152$	2.85320	0.231425
$153$	0	0
$154$	−1.15542	−0.0931062
$155$	−10.1649	−0.816464
$156$	0	0
$157$	1.22154	0.0974896	0.0487448	−	0.998811i	$-0.484478\pi$
$157$	1.22154	0.0974896	0.0487448	+	0.998811i	$0.484478\pi$
$158$	32.9440	2.62089
$159$	0	0
$160$	10.6117	0.838928
$161$	1.32039	0.104061
$162$	0	0
$163$	10.8327	0.848484	0.424242	−	0.905549i	$-0.360541\pi$
$163$	10.8327	0.848484	0.424242	+	0.905549i	$0.360541\pi$
$164$	15.6619	1.22299
$165$	0	0
$166$	23.2170	1.80199
$167$	2.98860	0.231265	0.115633	−	0.993292i	$-0.463111\pi$
$167$	2.98860	0.231265	0.115633	+	0.993292i	$0.463111\pi$
$168$	0	0
$169$	6.17103	0.474694
$170$	−12.9427	−0.992657
$171$	0	0
$172$	24.0587	1.83446
$173$	9.64094	0.732987	0.366494	−	0.930421i	$-0.380558\pi$
$173$	9.64094	0.732987	0.366494	+	0.930421i	$0.380558\pi$
$174$	0	0
$175$	−0.215027	−0.0162545
$176$	−0.144288	−0.0108761
$177$	0	0
$178$	−6.59802	−0.494542
$179$	−11.9636	−0.894201	−0.447101	−	0.894484i	$-0.647543\pi$
$179$	−11.9636	−0.894201	−0.447101	+	0.894484i	$0.647543\pi$
$180$	0	0
$181$	−13.7638	−1.02306	−0.511529	−	0.859266i	$-0.670921\pi$
$181$	−13.7638	−1.02306	−0.511529	+	0.859266i	$0.670921\pi$
$182$	−1.52044	−0.112702
$183$	0	0
$184$	24.8479	1.83181
$185$	−12.4103	−0.912426
$186$	0	0
$187$	−9.93493	−0.726514
$188$	−3.24574	−0.236720
$189$	0	0
$190$	−4.33463	−0.314467
$191$	−13.9168	−1.00699	−0.503494	−	0.863999i	$-0.667952\pi$
$191$	−13.9168	−1.00699	−0.503494	+	0.863999i	$0.667952\pi$
$192$	0	0
$193$	−9.02809	−0.649856	−0.324928	−	0.945739i	$-0.605340\pi$
$193$	−9.02809	−0.649856	−0.324928	+	0.945739i	$0.605340\pi$
$194$	−36.4430	−2.61646
$195$	0	0
$196$	−22.6456	−1.61754
$197$	5.23367	0.372883	0.186442	−	0.982466i	$-0.440304\pi$
$197$	5.23367	0.372883	0.186442	+	0.982466i	$0.440304\pi$
$198$	0	0
$199$	6.75268	0.478685	0.239342	−	0.970935i	$-0.423068\pi$
$199$	6.75268	0.478685	0.239342	+	0.970935i	$0.423068\pi$
$200$	−4.04652	−0.286132
$201$	0	0
$202$	36.4601	2.56533
$203$	0.271047	0.0190238
$204$	0	0
$205$	−9.13226	−0.637825
$206$	35.5534	2.47712
$207$	0	0
$208$	−0.189872	−0.0131653
$209$	−3.32731	−0.230155
$210$	0	0
$211$	−7.50596	−0.516731	−0.258366	−	0.966047i	$-0.583184\pi$
$211$	−7.50596	−0.516731	−0.258366	+	0.966047i	$0.583184\pi$
$212$	22.4210	1.53988
$213$	0	0
$214$	−13.1420	−0.898369
$215$	−14.0283	−0.956725
$216$	0	0
$217$	0.814324	0.0552799
$218$	13.6388	0.923737
$219$	0	0
$220$	20.4388	1.37798
$221$	−13.0736	−0.879424
$222$	0	0
$223$	−23.0115	−1.54096	−0.770482	−	0.637462i	$-0.779984\pi$
$223$	−23.0115	−1.54096	−0.770482	+	0.637462i	$0.779984\pi$
$224$	−0.850118	−0.0568009
$225$	0	0
$226$	−9.71089	−0.645959
$227$	14.8722	0.987102	0.493551	−	0.869717i	$-0.335699\pi$
$227$	14.8722	0.987102	0.493551	+	0.869717i	$0.335699\pi$
$228$	0	0
$229$	−22.7475	−1.50320	−0.751598	−	0.659621i	$-0.770717\pi$
$229$	−22.7475	−1.50320	−0.751598	+	0.659621i	$0.770717\pi$
$230$	−37.7494	−2.48912
$231$	0	0
$232$	5.10076	0.334881
$233$	15.2433	0.998623	0.499311	−	0.866423i	$-0.333586\pi$
$233$	15.2433	0.998623	0.499311	+	0.866423i	$0.333586\pi$
$234$	0	0
$235$	1.89255	0.123457
$236$	−34.3917	−2.23871
$237$	0	0
$238$	1.03685	0.0672093
$239$	−4.60838	−0.298091	−0.149046	−	0.988830i	$-0.547620\pi$
$239$	−4.60838	−0.298091	−0.149046	+	0.988830i	$0.547620\pi$
$240$	0	0
$241$	21.7319	1.39988	0.699938	−	0.714203i	$-0.253211\pi$
$241$	21.7319	1.39988	0.699938	+	0.714203i	$0.253211\pi$
$242$	0.162563	0.0104499
$243$	0	0
$244$	46.3480	2.96713
$245$	13.2044	0.843596
$246$	0	0
$247$	−4.37847	−0.278596
$248$	15.3245	0.973108
$249$	0	0
$250$	27.8207	1.75953
$251$	23.5297	1.48518	0.742590	−	0.669746i	$-0.233597\pi$
$251$	23.5297	1.48518	0.742590	+	0.669746i	$0.233597\pi$
$252$	0	0
$253$	−28.9768	−1.82176
$254$	−49.4758	−3.10439
$255$	0	0
$256$	−16.2796	−1.01748
$257$	21.4252	1.33647	0.668235	−	0.743951i	$-0.267050\pi$
$257$	21.4252	1.33647	0.668235	+	0.743951i	$0.267050\pi$
$258$	0	0
$259$	0.994210	0.0617772
$260$	26.8958	1.66801
$261$	0	0
$262$	37.2865	2.30357
$263$	6.19349	0.381907	0.190953	−	0.981599i	$-0.438842\pi$
$263$	6.19349	0.381907	0.190953	+	0.981599i	$0.438842\pi$
$264$	0	0
$265$	−13.0734	−0.803093
$266$	0.347253	0.0212914
$267$	0	0
$268$	−4.55529	−0.278258
$269$	32.0210	1.95235	0.976176	−	0.216978i	$-0.0696201\pi$
$269$	32.0210	1.95235	0.976176	+	0.216978i	$0.0696201\pi$
$270$	0	0
$271$	−31.4465	−1.91024	−0.955121	−	0.296217i	$-0.904275\pi$
$271$	−31.4465	−1.91024	−0.955121	+	0.296217i	$0.904275\pi$
$272$	0.129482	0.00785101
$273$	0	0
$274$	12.1173	0.732032
$275$	4.71892	0.284562
$276$	0	0
$277$	13.4340	0.807172	0.403586	−	0.914942i	$-0.367764\pi$
$277$	13.4340	0.807172	0.403586	+	0.914942i	$0.367764\pi$
$278$	5.10369	0.306099
$279$	0	0
$280$	−0.818696	−0.0489265
$281$	−23.6329	−1.40982	−0.704911	−	0.709296i	$-0.749013\pi$
$281$	−23.6329	−1.40982	−0.704911	+	0.709296i	$0.749013\pi$
$282$	0	0
$283$	−8.43962	−0.501683	−0.250842	−	0.968028i	$-0.580707\pi$
$283$	−8.43962	−0.501683	−0.250842	+	0.968028i	$0.580707\pi$
$284$	0.185008	0.0109782
$285$	0	0
$286$	33.3672	1.97304
$287$	0.731598	0.0431849
$288$	0	0
$289$	−8.08454	−0.475561
$290$	−7.74916	−0.455046
$291$	0	0
$292$	14.3678	0.840811
$293$	−16.2763	−0.950871	−0.475436	−	0.879750i	$-0.657710\pi$
$293$	−16.2763	−0.950871	−0.475436	+	0.879750i	$0.657710\pi$
$294$	0	0
$295$	20.0534	1.16755
$296$	18.7097	1.08748
$297$	0	0
$298$	7.99393	0.463076
$299$	−38.1312	−2.20519
$300$	0	0
$301$	1.12383	0.0647765
$302$	−7.97389	−0.458846
$303$	0	0
$304$	0.0433649	0.00248715
$305$	−27.0250	−1.54745
$306$	0	0
$307$	−18.9786	−1.08317	−0.541583	−	0.840647i	$-0.682175\pi$
$307$	−18.9786	−1.08317	−0.541583	+	0.840647i	$0.682175\pi$
$308$	−1.63738	−0.0932984
$309$	0	0
$310$	−23.2813	−1.32229
$311$	13.4767	0.764192	0.382096	−	0.924123i	$-0.375202\pi$
$311$	13.4767	0.764192	0.382096	+	0.924123i	$0.375202\pi$
$312$	0	0
$313$	−23.8778	−1.34965	−0.674826	−	0.737977i	$-0.735781\pi$
$313$	−23.8778	−1.34965	−0.674826	+	0.737977i	$0.735781\pi$
$314$	2.79777	0.157887
$315$	0	0
$316$	46.6861	2.62630
$317$	−0.115250	−0.00647309	−0.00323655	−	0.999995i	$-0.501030\pi$
$317$	−0.115250	−0.00647309	−0.00323655	+	0.999995i	$0.501030\pi$
$318$	0	0
$319$	−5.94834	−0.333043
$320$	24.4687	1.36784
$321$	0	0
$322$	3.02416	0.168530
$323$	2.98588	0.166139
$324$	0	0
$325$	6.20973	0.344454
$326$	24.8108	1.37414
$327$	0	0
$328$	13.7677	0.760195
$329$	−0.151615	−0.00835881
$330$	0	0
$331$	−2.76269	−0.151851	−0.0759256	−	0.997113i	$-0.524191\pi$
$331$	−2.76269	−0.151851	−0.0759256	+	0.997113i	$0.524191\pi$
$332$	32.9016	1.80571
$333$	0	0
$334$	6.84497	0.374540
$335$	2.65613	0.145120
$336$	0	0
$337$	4.01855	0.218904	0.109452	−	0.993992i	$-0.465090\pi$
$337$	4.01855	0.218904	0.109452	+	0.993992i	$0.465090\pi$
$338$	14.1339	0.768781
$339$	0	0
$340$	−18.3415	−0.994706
$341$	−17.8709	−0.967766
$342$	0	0
$343$	−2.11913	−0.114422
$344$	21.1490	1.14028
$345$	0	0
$346$	22.0812	1.18709
$347$	−28.7805	−1.54502	−0.772510	−	0.635003i	$-0.780999\pi$
$347$	−28.7805	−1.54502	−0.772510	+	0.635003i	$0.780999\pi$
$348$	0	0
$349$	17.0823	0.914397	0.457199	−	0.889365i	$-0.348853\pi$
$349$	17.0823	0.914397	0.457199	+	0.889365i	$0.348853\pi$
$350$	−0.492488	−0.0263246
$351$	0	0
$352$	18.6565	0.994394
$353$	28.9253	1.53954	0.769770	−	0.638321i	$-0.220371\pi$
$353$	28.9253	1.53954	0.769770	+	0.638321i	$0.220371\pi$
$354$	0	0
$355$	−0.107876	−0.00572547
$356$	−9.35027	−0.495563
$357$	0	0
$358$	−27.4009	−1.44818
$359$	18.2046	0.960803	0.480401	−	0.877049i	$-0.340491\pi$
$359$	18.2046	0.960803	0.480401	+	0.877049i	$0.340491\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−31.5241	−1.65687
$363$	0	0
$364$	−2.15466	−0.112935
$365$	−8.37769	−0.438508
$366$	0	0
$367$	−1.57595	−0.0822638	−0.0411319	−	0.999154i	$-0.513096\pi$
$367$	−1.57595	−0.0822638	−0.0411319	+	0.999154i	$0.513096\pi$
$368$	0.377656	0.0196867
$369$	0	0
$370$	−28.4241	−1.47770
$371$	1.04733	0.0543746
$372$	0	0
$373$	24.0284	1.24414	0.622072	−	0.782960i	$-0.286291\pi$
$373$	24.0284	1.24414	0.622072	+	0.782960i	$0.286291\pi$
$374$	−22.7546	−1.17661
$375$	0	0
$376$	−2.85320	−0.147143
$377$	−7.82754	−0.403139
$378$	0	0
$379$	16.8798	0.867056	0.433528	−	0.901140i	$-0.357268\pi$
$379$	16.8798	0.867056	0.433528	+	0.901140i	$0.357268\pi$
$380$	−6.14274	−0.315116
$381$	0	0
$382$	−31.8746	−1.63085
$383$	−4.61378	−0.235753	−0.117877	−	0.993028i	$-0.537609\pi$
$383$	−4.61378	−0.235753	−0.117877	+	0.993028i	$0.537609\pi$
$384$	0	0
$385$	0.954737	0.0486579
$386$	−20.6776	−1.05246
$387$	0	0
$388$	−51.6446	−2.62186
$389$	8.29582	0.420615	0.210307	−	0.977635i	$-0.432553\pi$
$389$	8.29582	0.420615	0.210307	+	0.977635i	$0.432553\pi$
$390$	0	0
$391$	26.0034	1.31505
$392$	−19.9068	−1.00545
$393$	0	0
$394$	11.9870	0.603895
$395$	−27.2221	−1.36969
$396$	0	0
$397$	5.96008	0.299128	0.149564	−	0.988752i	$-0.452213\pi$
$397$	5.96008	0.299128	0.149564	+	0.988752i	$0.452213\pi$
$398$	15.4661	0.775244
$399$	0	0
$400$	−0.0615018	−0.00307509
$401$	−0.141286	−0.00705547	−0.00352774	−	0.999994i	$-0.501123\pi$
$401$	−0.141286	−0.00705547	−0.00352774	+	0.999994i	$0.501123\pi$
$402$	0	0
$403$	−23.5168	−1.17145
$404$	51.6689	2.57062
$405$	0	0
$406$	0.620796	0.0308096
$407$	−21.8187	−1.08151
$408$	0	0
$409$	−22.0224	−1.08894	−0.544469	−	0.838781i	$-0.683269\pi$
$409$	−22.0224	−1.08894	−0.544469	+	0.838781i	$0.683269\pi$
$410$	−20.9161	−1.03297
$411$	0	0
$412$	50.3839	2.48223
$413$	−1.60650	−0.0790508
$414$	0	0
$415$	−19.1845	−0.941731
$416$	24.5504	1.20368
$417$	0	0
$418$	−7.62073	−0.372742
$419$	10.1094	0.493874	0.246937	−	0.969032i	$-0.420576\pi$
$419$	10.1094	0.493874	0.246937	+	0.969032i	$0.420576\pi$
$420$	0	0
$421$	21.5055	1.04811	0.524057	−	0.851683i	$-0.324418\pi$
$421$	21.5055	1.04811	0.524057	+	0.851683i	$0.324418\pi$
$422$	−17.1913	−0.836861
$423$	0	0
$424$	19.7094	0.957172
$425$	−4.23469	−0.205413
$426$	0	0
$427$	2.16501	0.104772
$428$	−18.6240	−0.900224
$429$	0	0
$430$	−32.1300	−1.54944
$431$	16.9446	0.816193	0.408097	−	0.912939i	$-0.366193\pi$
$431$	16.9446	0.816193	0.408097	+	0.912939i	$0.366193\pi$
$432$	0	0
$433$	23.6726	1.13763	0.568817	−	0.822464i	$-0.307401\pi$
$433$	23.6726	1.13763	0.568817	+	0.822464i	$0.307401\pi$
$434$	1.86509	0.0895274
$435$	0	0
$436$	19.3280	0.925644
$437$	8.70880	0.416598
$438$	0	0
$439$	35.1935	1.67969	0.839847	−	0.542823i	$-0.182644\pi$
$439$	35.1935	1.67969	0.839847	+	0.542823i	$0.182644\pi$
$440$	17.9669	0.856539
$441$	0	0
$442$	−29.9432	−1.42425
$443$	−17.8260	−0.846938	−0.423469	−	0.905911i	$-0.639188\pi$
$443$	−17.8260	−0.846938	−0.423469	+	0.905911i	$0.639188\pi$
$444$	0	0
$445$	5.45203	0.258451
$446$	−52.7046	−2.49563
$447$	0	0
$448$	−1.96022	−0.0926119
$449$	−10.3284	−0.487425	−0.243713	−	0.969848i	$-0.578365\pi$
$449$	−10.3284	−0.487425	−0.243713	+	0.969848i	$0.578365\pi$
$450$	0	0
$451$	−16.0555	−0.756022
$452$	−13.7616	−0.647292
$453$	0	0
$454$	34.0627	1.59864
$455$	1.25636	0.0588990
$456$	0	0
$457$	−22.2643	−1.04148	−0.520739	−	0.853716i	$-0.674344\pi$
$457$	−22.2643	−1.04148	−0.520739	+	0.853716i	$0.674344\pi$
$458$	−52.0999	−2.43447
$459$	0	0
$460$	−53.4959	−2.49426
$461$	−23.6377	−1.10092	−0.550459	−	0.834862i	$-0.685547\pi$
$461$	−23.6377	−1.10092	−0.550459	+	0.834862i	$0.685547\pi$
$462$	0	0
$463$	−27.2149	−1.26478	−0.632391	−	0.774649i	$-0.717926\pi$
$463$	−27.2149	−1.26478	−0.632391	+	0.774649i	$0.717926\pi$
$464$	0.0775248	0.00359900
$465$	0	0
$466$	34.9127	1.61730
$467$	−15.6542	−0.724390	−0.362195	−	0.932102i	$-0.617972\pi$
$467$	−15.6542	−0.724390	−0.362195	+	0.932102i	$0.617972\pi$
$468$	0	0
$469$	−0.212786	−0.00982556
$470$	4.33463	0.199941
$471$	0	0
$472$	−30.2323	−1.39155
$473$	−24.6633	−1.13402
$474$	0	0
$475$	−1.41824	−0.0650733
$476$	1.46936	0.0673481
$477$	0	0
$478$	−10.5548	−0.482767
$479$	−0.142152	−0.00649511	−0.00324756	−	0.999995i	$-0.501034\pi$
$479$	−0.142152	−0.00649511	−0.00324756	+	0.999995i	$0.501034\pi$
$480$	0	0
$481$	−28.7117	−1.30914
$482$	49.7739	2.26714
$483$	0	0
$484$	0.230373	0.0104715
$485$	30.1133	1.36738
$486$	0	0
$487$	−37.7476	−1.71051	−0.855254	−	0.518209i	$-0.826599\pi$
$487$	−37.7476	−1.71051	−0.855254	+	0.518209i	$0.826599\pi$
$488$	40.7427	1.84433
$489$	0	0
$490$	30.2427	1.36623
$491$	−4.51484	−0.203752	−0.101876	−	0.994797i	$-0.532485\pi$
$491$	−4.51484	−0.203752	−0.101876	+	0.994797i	$0.532485\pi$
$492$	0	0
$493$	5.33795	0.240409
$494$	−10.0283	−0.451193
$495$	0	0
$496$	0.232913	0.0104581
$497$	0.00864211	0.000387652 0
$498$	0	0
$499$	−2.62589	−0.117551	−0.0587755	−	0.998271i	$-0.518720\pi$
$499$	−2.62589	−0.117551	−0.0587755	+	0.998271i	$0.518720\pi$
$500$	39.4256	1.76317
$501$	0	0
$502$	53.8914	2.40529
$503$	37.9328	1.69134	0.845670	−	0.533705i	$-0.179201\pi$
$503$	37.9328	1.69134	0.845670	+	0.533705i	$0.179201\pi$
$504$	0	0
$505$	−30.1275	−1.34066
$506$	−66.3674	−2.95039
$507$	0	0
$508$	−70.1138	−3.11080
$509$	−20.4141	−0.904841	−0.452420	−	0.891805i	$-0.649439\pi$
$509$	−20.4141	−0.904841	−0.452420	+	0.891805i	$0.649439\pi$
$510$	0	0
$511$	0.671148	0.0296898
$512$	−0.490608	−0.0216820
$513$	0	0
$514$	49.0715	2.16445
$515$	−29.3782	−1.29456
$516$	0	0
$517$	3.32731	0.146335
$518$	2.27710	0.100050
$519$	0	0
$520$	23.6430	1.03682
$521$	4.54121	0.198954	0.0994769	−	0.995040i	$-0.468283\pi$
$521$	4.54121	0.198954	0.0994769	+	0.995040i	$0.468283\pi$
$522$	0	0
$523$	−4.15431	−0.181655	−0.0908277	−	0.995867i	$-0.528951\pi$
$523$	−4.15431	−0.181655	−0.0908277	+	0.995867i	$0.528951\pi$
$524$	52.8399	2.30832
$525$	0	0
$526$	14.1853	0.618509
$527$	16.0371	0.698588
$528$	0	0
$529$	52.8432	2.29753
$530$	−29.9428	−1.30063
$531$	0	0
$532$	0.492104	0.0213354
$533$	−21.1277	−0.915143
$534$	0	0
$535$	10.8594	0.469493
$536$	−4.00436	−0.172962
$537$	0	0
$538$	73.3395	3.16189
$539$	23.2147	0.999927
$540$	0	0
$541$	−10.4392	−0.448817	−0.224408	−	0.974495i	$-0.572045\pi$
$541$	−10.4392	−0.448817	−0.224408	+	0.974495i	$0.572045\pi$
$542$	−72.0239	−3.09369
$543$	0	0
$544$	−16.7420	−0.717809
$545$	−11.2699	−0.482751
$546$	0	0
$547$	25.2431	1.07932	0.539658	−	0.841884i	$-0.318553\pi$
$547$	25.2431	1.07932	0.539658	+	0.841884i	$0.318553\pi$
$548$	17.1718	0.733543
$549$	0	0
$550$	10.8080	0.460856
$551$	1.78773	0.0761600
$552$	0	0
$553$	2.18080	0.0927371
$554$	30.7687	1.30724
$555$	0	0
$556$	7.23260	0.306731
$557$	15.4686	0.655424	0.327712	−	0.944778i	$-0.393723\pi$
$557$	15.4686	0.655424	0.327712	+	0.944778i	$0.393723\pi$
$558$	0	0
$559$	−32.4550	−1.37270
$560$	−0.0124431	−0.000525817 0
$561$	0	0
$562$	−54.1279	−2.28325
$563$	22.4712	0.947049	0.473524	−	0.880781i	$-0.342982\pi$
$563$	22.4712	0.947049	0.473524	+	0.880781i	$0.342982\pi$
$564$	0	0
$565$	8.02423	0.337582
$566$	−19.3298	−0.812490
$567$	0	0
$568$	0.162633	0.00682394
$569$	4.32946	0.181501	0.0907503	−	0.995874i	$-0.471073\pi$
$569$	4.32946	0.181501	0.0907503	+	0.995874i	$0.471073\pi$
$570$	0	0
$571$	17.4993	0.732323	0.366161	−	0.930551i	$-0.380672\pi$
$571$	17.4993	0.732323	0.366161	+	0.930551i	$0.380672\pi$
$572$	47.2857	1.97711
$573$	0	0
$574$	1.67562	0.0699391
$575$	−12.3512	−0.515079
$576$	0	0
$577$	−33.1977	−1.38204	−0.691019	−	0.722836i	$-0.742838\pi$
$577$	−33.1977	−1.38204	−0.691019	+	0.722836i	$0.742838\pi$
$578$	−18.5165	−0.770185
$579$	0	0
$580$	−10.9816	−0.455986
$581$	1.53690	0.0637613
$582$	0	0
$583$	−22.9844	−0.951918
$584$	12.6301	0.522639
$585$	0	0
$586$	−37.2786	−1.53996
$587$	−20.2035	−0.833887	−0.416944	−	0.908932i	$-0.636899\pi$
$587$	−20.2035	−0.833887	−0.416944	+	0.908932i	$0.636899\pi$
$588$	0	0
$589$	5.37099	0.221308
$590$	45.9294	1.89088
$591$	0	0
$592$	0.284363	0.0116873
$593$	−21.1705	−0.869367	−0.434683	−	0.900583i	$-0.643140\pi$
$593$	−21.1705	−0.869367	−0.434683	+	0.900583i	$0.643140\pi$
$594$	0	0
$595$	−0.856767	−0.0351240
$596$	11.3285	0.464032
$597$	0	0
$598$	−87.3342	−3.57136
$599$	29.6172	1.21013	0.605063	−	0.796177i	$-0.293148\pi$
$599$	29.6172	1.21013	0.605063	+	0.796177i	$0.293148\pi$
$600$	0	0
$601$	−38.6805	−1.57781	−0.788906	−	0.614514i	$-0.789352\pi$
$601$	−38.6805	−1.57781	−0.788906	+	0.614514i	$0.789352\pi$
$602$	2.57398	0.104907
$603$	0	0
$604$	−11.3001	−0.459793
$605$	−0.134328	−0.00546121
$606$	0	0
$607$	39.9353	1.62092	0.810462	−	0.585792i	$-0.199216\pi$
$607$	39.9353	1.62092	0.810462	+	0.585792i	$0.199216\pi$
$608$	−5.60708	−0.227397
$609$	0	0
$610$	−61.8969	−2.50613
$611$	4.37847	0.177134
$612$	0	0
$613$	14.2710	0.576401	0.288201	−	0.957570i	$-0.406943\pi$
$613$	14.2710	0.576401	0.288201	+	0.957570i	$0.406943\pi$
$614$	−43.4678	−1.75422
$615$	0	0
$616$	−1.43935	−0.0579932
$617$	40.0386	1.61189	0.805946	−	0.591989i	$-0.201657\pi$
$617$	40.0386	1.61189	0.805946	+	0.591989i	$0.201657\pi$
$618$	0	0
$619$	48.0886	1.93284	0.966422	−	0.256962i	$-0.0827215\pi$
$619$	48.0886	1.93284	0.966422	+	0.256962i	$0.0827215\pi$
$620$	−32.9926	−1.32502
$621$	0	0
$622$	30.8664	1.23763
$623$	−0.436770	−0.0174988
$624$	0	0
$625$	−15.8974	−0.635896
$626$	−54.6887	−2.18580
$627$	0	0
$628$	3.96481	0.158213
$629$	19.5798	0.780696
$630$	0	0
$631$	−37.8147	−1.50538	−0.752689	−	0.658376i	$-0.771244\pi$
$631$	−37.8147	−1.50538	−0.752689	+	0.658376i	$0.771244\pi$
$632$	41.0398	1.63248
$633$	0	0
$634$	−0.263964	−0.0104834
$635$	40.8825	1.62237
$636$	0	0
$637$	30.5487	1.21038
$638$	−13.6238	−0.539373
$639$	0	0
$640$	34.8188	1.37633
$641$	−24.7557	−0.977790	−0.488895	−	0.872343i	$-0.662600\pi$
$641$	−24.7557	−0.977790	−0.488895	+	0.872343i	$0.662600\pi$
$642$	0	0
$643$	0.927623	0.0365819	0.0182909	−	0.999833i	$-0.494177\pi$
$643$	0.927623	0.0365819	0.0182909	+	0.999833i	$0.494177\pi$
$644$	4.28563	0.168878
$645$	0	0
$646$	6.83873	0.269066
$647$	36.0116	1.41576	0.707880	−	0.706333i	$-0.249652\pi$
$647$	36.0116	1.41576	0.707880	+	0.706333i	$0.249652\pi$
$648$	0	0
$649$	35.2559	1.38392
$650$	14.2225	0.557853
$651$	0	0
$652$	35.1602	1.37698
$653$	20.4015	0.798373	0.399186	−	0.916870i	$-0.369293\pi$
$653$	20.4015	0.798373	0.399186	+	0.916870i	$0.369293\pi$
$654$	0	0
$655$	−30.8103	−1.20386
$656$	0.209251	0.00816989
$657$	0	0
$658$	−0.347253	−0.0135373
$659$	8.42570	0.328219	0.164109	−	0.986442i	$-0.447525\pi$
$659$	8.42570	0.328219	0.164109	+	0.986442i	$0.447525\pi$
$660$	0	0
$661$	40.5935	1.57891	0.789453	−	0.613811i	$-0.210364\pi$
$661$	40.5935	1.57891	0.789453	+	0.613811i	$0.210364\pi$
$662$	−6.32755	−0.245927
$663$	0	0
$664$	28.9225	1.12241
$665$	−0.286940	−0.0111270
$666$	0	0
$667$	15.5690	0.602834
$668$	9.70024	0.375314
$669$	0	0
$670$	6.08349	0.235026
$671$	−47.5127	−1.83421
$672$	0	0
$673$	13.8735	0.534785	0.267393	−	0.963588i	$-0.413838\pi$
$673$	13.8735	0.534785	0.267393	+	0.963588i	$0.413838\pi$
$674$	9.20392	0.354522
$675$	0	0
$676$	20.0296	0.770368
$677$	−2.64152	−0.101522	−0.0507610	−	0.998711i	$-0.516165\pi$
$677$	−2.64152	−0.101522	−0.0507610	+	0.998711i	$0.516165\pi$
$678$	0	0
$679$	−2.41242	−0.0925803
$680$	−16.1232	−0.618298
$681$	0	0
$682$	−40.9309	−1.56732
$683$	−37.1261	−1.42059	−0.710295	−	0.703904i	$-0.751438\pi$
$683$	−37.1261	−1.42059	−0.710295	+	0.703904i	$0.751438\pi$
$684$	0	0
$685$	−10.0127	−0.382564
$686$	−4.85356	−0.185310
$687$	0	0
$688$	0.321438	0.0122547
$689$	−30.2457	−1.15227
$690$	0	0
$691$	−36.6701	−1.39499	−0.697497	−	0.716587i	$-0.745703\pi$
$691$	−36.6701	−1.39499	−0.697497	+	0.716587i	$0.745703\pi$
$692$	31.2920	1.18954
$693$	0	0
$694$	−65.9177	−2.50220
$695$	−4.21724	−0.159969
$696$	0	0
$697$	14.4079	0.545740
$698$	39.1247	1.48089
$699$	0	0
$700$	−0.697922	−0.0263790
$701$	43.7473	1.65231	0.826157	−	0.563440i	$-0.190522\pi$
$701$	43.7473	1.65231	0.826157	+	0.563440i	$0.190522\pi$
$702$	0	0
$703$	6.55746	0.247319
$704$	43.0186	1.62132
$705$	0	0
$706$	66.2494	2.49333
$707$	2.41355	0.0907711
$708$	0	0
$709$	31.1353	1.16931	0.584656	−	0.811281i	$-0.301229\pi$
$709$	31.1353	1.16931	0.584656	+	0.811281i	$0.301229\pi$
$710$	−0.247075	−0.00927257
$711$	0	0
$712$	−8.21944	−0.308036
$713$	46.7749	1.75173
$714$	0	0
$715$	−27.5717	−1.03112
$716$	−38.8308	−1.45117
$717$	0	0
$718$	41.6951	1.55605
$719$	7.60710	0.283697	0.141849	−	0.989888i	$-0.454695\pi$
$719$	7.60710	0.283697	0.141849	+	0.989888i	$0.454695\pi$
$720$	0	0
$721$	2.35353	0.0876500
$722$	2.29036	0.0852383
$723$	0	0
$724$	−44.6739	−1.66029
$725$	−2.53544	−0.0941637
$726$	0	0
$727$	−5.09985	−0.189143	−0.0945715	−	0.995518i	$-0.530148\pi$
$727$	−5.09985	−0.189143	−0.0945715	+	0.995518i	$0.530148\pi$
$728$	−1.89408	−0.0701991
$729$	0	0
$730$	−19.1879	−0.710176
$731$	22.1325	0.818600
$732$	0	0
$733$	2.40980	0.0890080	0.0445040	−	0.999009i	$-0.485829\pi$
$733$	2.40980	0.0890080	0.0445040	+	0.999009i	$0.485829\pi$
$734$	−3.60949	−0.133229
$735$	0	0
$736$	−48.8309	−1.79993
$737$	4.66976	0.172013
$738$	0	0
$739$	34.9335	1.28505	0.642526	−	0.766264i	$-0.277887\pi$
$739$	34.9335	1.28505	0.642526	+	0.766264i	$0.277887\pi$
$740$	−40.2808	−1.48075
$741$	0	0
$742$	2.39876	0.0880612
$743$	14.2943	0.524408	0.262204	−	0.965012i	$-0.415551\pi$
$743$	14.2943	0.524408	0.262204	+	0.965012i	$0.415551\pi$
$744$	0	0
$745$	−6.60549	−0.242006
$746$	55.0337	2.01493
$747$	0	0
$748$	−32.2462	−1.17904
$749$	−0.869963	−0.0317877
$750$	0	0
$751$	−50.3503	−1.83731	−0.918655	−	0.395061i	$-0.870723\pi$
$751$	−50.3503	−1.83731	−0.918655	+	0.395061i	$0.870723\pi$
$752$	−0.0433649	−0.00158135
$753$	0	0
$754$	−17.9279	−0.652895
$755$	6.58893	0.239796
$756$	0	0
$757$	−50.9156	−1.85056	−0.925279	−	0.379288i	$-0.876169\pi$
$757$	−50.9156	−1.85056	−0.925279	+	0.379288i	$0.876169\pi$
$758$	38.6608	1.40422
$759$	0	0
$760$	−5.39983	−0.195873
$761$	52.6201	1.90748	0.953738	−	0.300638i	$-0.0971996\pi$
$761$	52.6201	1.90748	0.953738	+	0.300638i	$0.0971996\pi$
$762$	0	0
$763$	0.902849	0.0326853
$764$	−45.1705	−1.63421
$765$	0	0
$766$	−10.5672	−0.381809
$767$	46.3940	1.67519
$768$	0	0
$769$	−24.5743	−0.886171	−0.443085	−	0.896479i	$-0.646116\pi$
$769$	−24.5743	−0.886171	−0.443085	+	0.896479i	$0.646116\pi$
$770$	2.18669	0.0788028
$771$	0	0
$772$	−29.3029	−1.05463
$773$	−1.64459	−0.0591519	−0.0295759	−	0.999563i	$-0.509416\pi$
$773$	−1.64459	−0.0591519	−0.0295759	+	0.999563i	$0.509416\pi$
$774$	0	0
$775$	−7.61736	−0.273624
$776$	−45.3987	−1.62972
$777$	0	0
$778$	19.0004	0.681198
$779$	4.82536	0.172887
$780$	0	0
$781$	−0.189658	−0.00678648
$782$	59.5571	2.12976
$783$	0	0
$784$	−0.302557	−0.0108056
$785$	−2.31183	−0.0825128
$786$	0	0
$787$	−29.3132	−1.04490	−0.522452	−	0.852669i	$-0.674983\pi$
$787$	−29.3132	−1.04490	−0.522452	+	0.852669i	$0.674983\pi$
$788$	16.9871	0.605142
$789$	0	0
$790$	−62.3484	−2.21826
$791$	−0.642833	−0.0228565
$792$	0	0
$793$	−62.5230	−2.22026
$794$	13.6507	0.484446
$795$	0	0
$796$	21.9175	0.776844
$797$	32.3042	1.14427	0.572136	−	0.820159i	$-0.306115\pi$
$797$	32.3042	1.14427	0.572136	+	0.820159i	$0.306115\pi$
$798$	0	0
$799$	−2.98588	−0.105633
$800$	7.95219	0.281152
$801$	0	0
$802$	−0.323595	−0.0114265
$803$	−14.7288	−0.519770
$804$	0	0
$805$	−2.49890	−0.0880746
$806$	−53.8618	−1.89720
$807$	0	0
$808$	45.4200	1.59787
$809$	31.7451	1.11610	0.558048	−	0.829808i	$-0.311550\pi$
$809$	31.7451	1.11610	0.558048	+	0.829808i	$0.311550\pi$
$810$	0	0
$811$	−35.8898	−1.26026	−0.630131	−	0.776489i	$-0.716999\pi$
$811$	−35.8898	−1.26026	−0.630131	+	0.776489i	$0.716999\pi$
$812$	0.879750	0.0308732
$813$	0	0
$814$	−49.9726	−1.75154
$815$	−20.5015	−0.718136
$816$	0	0
$817$	7.41239	0.259327
$818$	−50.4392	−1.76356
$819$	0	0
$820$	−29.6410	−1.03511
$821$	23.8301	0.831676	0.415838	−	0.909439i	$-0.363488\pi$
$821$	23.8301	0.831676	0.415838	+	0.909439i	$0.363488\pi$
$822$	0	0
$823$	−17.6426	−0.614983	−0.307491	−	0.951551i	$-0.599490\pi$
$823$	−17.6426	−0.614983	−0.307491	+	0.951551i	$0.599490\pi$
$824$	44.2904	1.54293
$825$	0	0
$826$	−3.67947	−0.128025
$827$	−24.4342	−0.849659	−0.424829	−	0.905273i	$-0.639666\pi$
$827$	−24.4342	−0.849659	−0.424829	+	0.905273i	$0.639666\pi$
$828$	0	0
$829$	−17.2184	−0.598020	−0.299010	−	0.954250i	$-0.596656\pi$
$829$	−17.2184	−0.598020	−0.299010	+	0.954250i	$0.596656\pi$
$830$	−43.9395	−1.52516
$831$	0	0
$832$	56.6091	1.96257
$833$	−20.8325	−0.721803
$834$	0	0
$835$	−5.65609	−0.195737
$836$	−10.7996	−0.373512
$837$	0	0
$838$	23.1540	0.799843
$839$	33.2033	1.14631	0.573153	−	0.819448i	$-0.305720\pi$
$839$	33.2033	1.14631	0.573153	+	0.819448i	$0.305720\pi$
$840$	0	0
$841$	−25.8040	−0.889793
$842$	49.2553	1.69745
$843$	0	0
$844$	−24.3624	−0.838589
$845$	−11.6790	−0.401770
$846$	0	0
$847$	0.0107612	0.000369759 0
$848$	0.299557	0.0102868
$849$	0	0
$850$	−9.69896	−0.332672
$851$	57.1076	1.95762
$852$	0	0
$853$	33.8439	1.15879	0.579396	−	0.815046i	$-0.303288\pi$
$853$	33.8439	1.15879	0.579396	+	0.815046i	$0.303288\pi$
$854$	4.95865	0.169681
$855$	0	0
$856$	−16.3716	−0.559569
$857$	−1.34022	−0.0457811	−0.0228906	−	0.999738i	$-0.507287\pi$
$857$	−1.34022	−0.0457811	−0.0228906	+	0.999738i	$0.507287\pi$
$858$	0	0
$859$	38.1222	1.30071	0.650356	−	0.759630i	$-0.274620\pi$
$859$	38.1222	1.30071	0.650356	+	0.759630i	$0.274620\pi$
$860$	−45.5324	−1.55264
$861$	0	0
$862$	38.8092	1.32185
$863$	46.9501	1.59820	0.799101	−	0.601197i	$-0.205309\pi$
$863$	46.9501	1.59820	0.799101	+	0.601197i	$0.205309\pi$
$864$	0	0
$865$	−18.2460	−0.620382
$866$	54.2188	1.84243
$867$	0	0
$868$	2.64309	0.0897122
$869$	−47.8593	−1.62352
$870$	0	0
$871$	6.14503	0.208216
$872$	16.9905	0.575369
$873$	0	0
$874$	19.9463	0.674693
$875$	1.84165	0.0622591
$876$	0	0
$877$	−22.5178	−0.760370	−0.380185	−	0.924910i	$-0.624140\pi$
$877$	−22.5178	−0.760370	−0.380185	+	0.924910i	$0.624140\pi$
$878$	80.6058	2.72031
$879$	0	0
$880$	0.273073	0.00920530
$881$	−42.3565	−1.42703	−0.713514	−	0.700641i	$-0.752897\pi$
$881$	−42.3565	−1.42703	−0.713514	+	0.700641i	$0.752897\pi$
$882$	0	0
$883$	44.8399	1.50898	0.754492	−	0.656309i	$-0.227883\pi$
$883$	44.8399	1.50898	0.754492	+	0.656309i	$0.227883\pi$
$884$	−42.4335	−1.42719
$885$	0	0
$886$	−40.8279	−1.37164
$887$	−49.0214	−1.64598	−0.822989	−	0.568057i	$-0.807695\pi$
$887$	−49.0214	−1.64598	−0.822989	+	0.568057i	$0.807695\pi$
$888$	0	0
$889$	−3.27516	−0.109845
$890$	12.4871	0.418569
$891$	0	0
$892$	−74.6894	−2.50079
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	22.6418	0.756830
$896$	−2.78938	−0.0931867
$897$	0	0
$898$	−23.6556	−0.789399
$899$	9.60190	0.320241
$900$	0	0
$901$	20.6259	0.687148
$902$	−36.7728	−1.22440
$903$	0	0
$904$	−12.0973	−0.402349
$905$	26.0488	0.865892
$906$	0	0
$907$	19.8849	0.660268	0.330134	−	0.943934i	$-0.392906\pi$
$907$	19.8849	0.660268	0.330134	+	0.943934i	$0.392906\pi$
$908$	48.2713	1.60194
$909$	0	0
$910$	2.87751	0.0953885
$911$	−35.1663	−1.16511	−0.582556	−	0.812791i	$-0.697947\pi$
$911$	−35.1663	−1.16511	−0.582556	+	0.812791i	$0.697947\pi$
$912$	0	0
$913$	−33.7284	−1.11625
$914$	−50.9931	−1.68670
$915$	0	0
$916$	−73.8325	−2.43950
$917$	2.46826	0.0815091
$918$	0	0
$919$	25.6860	0.847302	0.423651	−	0.905825i	$-0.360748\pi$
$919$	25.6860	0.847302	0.423651	+	0.905825i	$0.360748\pi$
$920$	−47.0261	−1.55040
$921$	0	0
$922$	−54.1389	−1.78297
$923$	−0.249574	−0.00821484
$924$	0	0
$925$	−9.30005	−0.305784
$926$	−62.3318	−2.04835
$927$	0	0
$928$	−10.0240	−0.329053
$929$	22.8650	0.750175	0.375088	−	0.926989i	$-0.377613\pi$
$929$	22.8650	0.750175	0.375088	+	0.926989i	$0.377613\pi$
$930$	0	0
$931$	−6.97701	−0.228662
$932$	49.4759	1.62064
$933$	0	0
$934$	−35.8537	−1.17317
$935$	18.8024	0.614904
$936$	0	0
$937$	14.6197	0.477604	0.238802	−	0.971068i	$-0.423245\pi$
$937$	14.6197	0.477604	0.238802	+	0.971068i	$0.423245\pi$
$938$	−0.487357	−0.0159128
$939$	0	0
$940$	6.14274	0.200354
$941$	−18.6471	−0.607879	−0.303939	−	0.952691i	$-0.598302\pi$
$941$	−18.6471	−0.607879	−0.303939	+	0.952691i	$0.598302\pi$
$942$	0	0
$943$	42.0231	1.36846
$944$	−0.459491	−0.0149552
$945$	0	0
$946$	−56.4878	−1.83658
$947$	29.2474	0.950412	0.475206	−	0.879875i	$-0.342374\pi$
$947$	29.2474	0.950412	0.475206	+	0.879875i	$0.342374\pi$
$948$	0	0
$949$	−19.3820	−0.629166
$950$	−3.24828	−0.105388
$951$	0	0
$952$	1.29166	0.0418628
$953$	−60.0594	−1.94551	−0.972757	−	0.231826i	$-0.925530\pi$
$953$	−60.0594	−1.94551	−0.972757	+	0.231826i	$0.925530\pi$
$954$	0	0
$955$	26.3384	0.852290
$956$	−14.9576	−0.483764
$957$	0	0
$958$	−0.325580	−0.0105190
$959$	0.802129	0.0259021
$960$	0	0
$961$	−2.15242	−0.0694330
$962$	−65.7600	−2.12019
$963$	0	0
$964$	70.5363	2.27182
$965$	17.0861	0.550022
$966$	0	0
$967$	−31.6809	−1.01879	−0.509395	−	0.860533i	$-0.670131\pi$
$967$	−31.6809	−1.01879	−0.509395	+	0.860533i	$0.670131\pi$
$968$	0.202512	0.00650898
$969$	0	0
$970$	68.9704	2.21451
$971$	−2.71651	−0.0871769	−0.0435884	−	0.999050i	$-0.513879\pi$
$971$	−2.71651	−0.0871769	−0.0435884	+	0.999050i	$0.513879\pi$
$972$	0	0
$973$	0.337849	0.0108309
$974$	−86.4556	−2.77022
$975$	0	0
$976$	0.619235	0.0198212
$977$	−37.6024	−1.20301	−0.601504	−	0.798870i	$-0.705431\pi$
$977$	−37.6024	−1.20301	−0.601504	+	0.798870i	$0.705431\pi$
$978$	0	0
$979$	9.58524	0.306346
$980$	42.8580	1.36905
$981$	0	0
$982$	−10.3406	−0.329982
$983$	−9.75084	−0.311004	−0.155502	−	0.987836i	$-0.549699\pi$
$983$	−9.75084	−0.311004	−0.155502	+	0.987836i	$0.549699\pi$
$984$	0	0
$985$	−9.90500	−0.315599
$986$	12.2258	0.389350
$987$	0	0
$988$	−14.2114	−0.452125
$989$	64.5530	2.05267
$990$	0	0
$991$	−7.27874	−0.231217	−0.115608	−	0.993295i	$-0.536882\pi$
$991$	−7.27874	−0.231217	−0.115608	+	0.993295i	$0.536882\pi$
$992$	−30.1156	−0.956171
$993$	0	0
$994$	0.0197935	0.000627813 0
$995$	−12.7798	−0.405147
$996$	0	0
$997$	−40.1022	−1.27005	−0.635024	−	0.772492i	$-0.719010\pi$
$997$	−40.1022	−1.27005	−0.635024	+	0.772492i	$0.719010\pi$
$998$	−6.01423	−0.190377
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.x.1.30	yes	34
3.2	odd	2		8037.2.a.u.1.5	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.5	✓	34	3.2	odd	2
8037.2.a.x.1.30	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)

\(f(q)\)	\(=\)	\(q+2.29036 q^{2} +3.24574 q^{4} -1.89255 q^{5} +0.151615 q^{7} +2.85320 q^{8} +O(q^{10})\)	\(q+2.29036 q^{2} +3.24574 q^{4} -1.89255 q^{5} +0.151615 q^{7} +2.85320 q^{8} -4.33463 q^{10} -3.32731 q^{11} -4.37847 q^{13} +0.347253 q^{14} +0.0433649 q^{16} +2.98588 q^{17} +1.00000 q^{19} -6.14274 q^{20} -7.62073 q^{22} +8.70880 q^{23} -1.41824 q^{25} -10.0283 q^{26} +0.492104 q^{28} +1.78773 q^{29} +5.37099 q^{31} -5.60708 q^{32} +6.83873 q^{34} -0.286940 q^{35} +6.55746 q^{37} +2.29036 q^{38} -5.39983 q^{40} +4.82536 q^{41} +7.41239 q^{43} -10.7996 q^{44} +19.9463 q^{46} -1.00000 q^{47} -6.97701 q^{49} -3.24828 q^{50} -14.2114 q^{52} +6.90782 q^{53} +6.29711 q^{55} +0.432588 q^{56} +4.09455 q^{58} -10.5959 q^{59} +14.2796 q^{61} +12.3015 q^{62} -12.9290 q^{64} +8.28650 q^{65} -1.40346 q^{67} +9.69139 q^{68} -0.657195 q^{70} +0.0570003 q^{71} +4.42666 q^{73} +15.0189 q^{74} +3.24574 q^{76} -0.504470 q^{77} +14.3838 q^{79} -0.0820704 q^{80} +11.0518 q^{82} +10.1368 q^{83} -5.65093 q^{85} +16.9770 q^{86} -9.49348 q^{88} -2.88078 q^{89} -0.663843 q^{91} +28.2665 q^{92} -2.29036 q^{94} -1.89255 q^{95} -15.9115 q^{97} -15.9799 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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