LMFDB - Embedded newform 8037.2.a.w.1.25

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			1.25
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)

$f(q)$	$=$	$q+1.41986 q^{2} +0.0160129 q^{4} +3.53530 q^{5} -3.17409 q^{7} -2.81699 q^{8} +O(q^{10})$	\(q+1.41986 q^{2} +0.0160129 q^{4} +3.53530 q^{5} -3.17409 q^{7} -2.81699 q^{8} +5.01964 q^{10} +3.68649 q^{11} -5.98876 q^{13} -4.50678 q^{14} -4.03177 q^{16} +2.25108 q^{17} -1.00000 q^{19} +0.0566105 q^{20} +5.23432 q^{22} +1.00617 q^{23} +7.49834 q^{25} -8.50323 q^{26} -0.0508265 q^{28} -1.40193 q^{29} +2.93770 q^{31} -0.0905807 q^{32} +3.19623 q^{34} -11.2214 q^{35} +3.21614 q^{37} -1.41986 q^{38} -9.95891 q^{40} +4.29305 q^{41} +6.16907 q^{43} +0.0590316 q^{44} +1.42863 q^{46} +1.00000 q^{47} +3.07485 q^{49} +10.6466 q^{50} -0.0958977 q^{52} +1.50966 q^{53} +13.0329 q^{55} +8.94138 q^{56} -1.99054 q^{58} +3.82125 q^{59} +0.924079 q^{61} +4.17113 q^{62} +7.93493 q^{64} -21.1721 q^{65} +6.84623 q^{67} +0.0360464 q^{68} -15.9328 q^{70} -2.44002 q^{71} +7.69026 q^{73} +4.56648 q^{74} -0.0160129 q^{76} -11.7013 q^{77} +8.43491 q^{79} -14.2535 q^{80} +6.09555 q^{82} +4.61907 q^{83} +7.95824 q^{85} +8.75923 q^{86} -10.3848 q^{88} +8.43920 q^{89} +19.0089 q^{91} +0.0161118 q^{92} +1.41986 q^{94} -3.53530 q^{95} +5.67971 q^{97} +4.36587 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * 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 $

$2$	1.41986	1.00400	0.501998	−	0.864869i	$-0.332599\pi$
$2$	1.41986	1.00400	0.501998	+	0.864869i	$0.332599\pi$
$3$	0	0
$3$	0	0
$4$	0.0160129	0.00800647
$5$	3.53530	1.58103	0.790517	−	0.612440i	$-0.209812\pi$
$5$	3.53530	1.58103	0.790517	+	0.612440i	$0.209812\pi$
$6$	0	0
$7$	−3.17409	−1.19969	−0.599847	−	0.800115i	$-0.704772\pi$
$7$	−3.17409	−1.19969	−0.599847	+	0.800115i	$0.704772\pi$
$8$	−2.81699	−0.995957
$9$	0	0
$10$	5.01964	1.58735
$11$	3.68649	1.11152	0.555760	−	0.831343i	$-0.312427\pi$
$11$	3.68649	1.11152	0.555760	+	0.831343i	$0.312427\pi$
$12$	0	0
$13$	−5.98876	−1.66098	−0.830492	−	0.557030i	$-0.811941\pi$
$13$	−5.98876	−1.66098	−0.830492	+	0.557030i	$0.811941\pi$
$14$	−4.50678	−1.20449
$15$	0	0
$16$	−4.03177	−1.00794
$17$	2.25108	0.545967	0.272984	−	0.962019i	$-0.411990\pi$
$17$	2.25108	0.545967	0.272984	+	0.962019i	$0.411990\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.0566105	0.0126585
$21$	0	0
$22$	5.23432	1.11596
$23$	1.00617	0.209802	0.104901	−	0.994483i	$-0.466547\pi$
$23$	1.00617	0.209802	0.104901	+	0.994483i	$0.466547\pi$
$24$	0	0
$25$	7.49834	1.49967
$26$	−8.50323	−1.66762
$27$	0	0
$28$	−0.0508265	−0.00960531
$29$	−1.40193	−0.260331	−0.130166	−	0.991492i	$-0.541551\pi$
$29$	−1.40193	−0.260331	−0.130166	+	0.991492i	$0.541551\pi$
$30$	0	0
$31$	2.93770	0.527626	0.263813	−	0.964574i	$-0.415020\pi$
$31$	2.93770	0.527626	0.263813	+	0.964574i	$0.415020\pi$
$32$	−0.0905807	−0.0160126
$33$	0	0
$34$	3.19623	0.548148
$35$	−11.2214	−1.89676
$36$	0	0
$37$	3.21614	0.528730	0.264365	−	0.964423i	$-0.414838\pi$
$37$	3.21614	0.528730	0.264365	+	0.964423i	$0.414838\pi$
$38$	−1.41986	−0.230332
$39$	0	0
$40$	−9.95891	−1.57464
$41$	4.29305	0.670462	0.335231	−	0.942136i	$-0.391186\pi$
$41$	4.29305	0.670462	0.335231	+	0.942136i	$0.391186\pi$
$42$	0	0
$43$	6.16907	0.940774	0.470387	−	0.882460i	$-0.344114\pi$
$43$	6.16907	0.940774	0.470387	+	0.882460i	$0.344114\pi$
$44$	0.0590316	0.00889935
$45$	0	0
$46$	1.42863	0.210640
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	3.07485	0.439264
$50$	10.6466	1.50566
$51$	0	0
$52$	−0.0958977	−0.0132986
$53$	1.50966	0.207367	0.103684	−	0.994610i	$-0.466937\pi$
$53$	1.50966	0.207367	0.103684	+	0.994610i	$0.466937\pi$
$54$	0	0
$55$	13.0329	1.75735
$56$	8.94138	1.19484
$57$	0	0
$58$	−1.99054	−0.261371
$59$	3.82125	0.497485	0.248742	−	0.968570i	$-0.419983\pi$
$59$	3.82125	0.497485	0.248742	+	0.968570i	$0.419983\pi$
$60$	0	0
$61$	0.924079	0.118316	0.0591581	−	0.998249i	$-0.481158\pi$
$61$	0.924079	0.118316	0.0591581	+	0.998249i	$0.481158\pi$
$62$	4.17113	0.529734
$63$	0	0
$64$	7.93493	0.991866
$65$	−21.1721	−2.62607
$66$	0	0
$67$	6.84623	0.836400	0.418200	−	0.908355i	$-0.362661\pi$
$67$	6.84623	0.836400	0.418200	+	0.908355i	$0.362661\pi$
$68$	0.0360464	0.00437127
$69$	0	0
$70$	−15.9328	−1.90433
$71$	−2.44002	−0.289578	−0.144789	−	0.989463i	$-0.546250\pi$
$71$	−2.44002	−0.289578	−0.144789	+	0.989463i	$0.546250\pi$
$72$	0	0
$73$	7.69026	0.900077	0.450039	−	0.893009i	$-0.351410\pi$
$73$	7.69026	0.900077	0.450039	+	0.893009i	$0.351410\pi$
$74$	4.56648	0.530842
$75$	0	0
$76$	−0.0160129	−0.00183681
$77$	−11.7013	−1.33348
$78$	0	0
$79$	8.43491	0.949001	0.474501	−	0.880255i	$-0.342629\pi$
$79$	8.43491	0.949001	0.474501	+	0.880255i	$0.342629\pi$
$80$	−14.2535	−1.59359
$81$	0	0
$82$	6.09555	0.673140
$83$	4.61907	0.507009	0.253505	−	0.967334i	$-0.418417\pi$
$83$	4.61907	0.507009	0.253505	+	0.967334i	$0.418417\pi$
$84$	0	0
$85$	7.95824	0.863192
$86$	8.75923	0.944532
$87$	0	0
$88$	−10.3848	−1.10703
$89$	8.43920	0.894554	0.447277	−	0.894396i	$-0.352394\pi$
$89$	8.43920	0.894554	0.447277	+	0.894396i	$0.352394\pi$
$90$	0	0
$91$	19.0089	1.99267
$92$	0.0161118	0.00167977
$93$	0	0
$94$	1.41986	0.146448
$95$	−3.53530	−0.362714
$96$	0	0
$97$	5.67971	0.576687	0.288343	−	0.957527i	$-0.406896\pi$
$97$	5.67971	0.576687	0.288343	+	0.957527i	$0.406896\pi$
$98$	4.36587	0.441019
$99$	0	0
$100$	0.120070	0.0120070
$101$	−12.8511	−1.27873	−0.639367	−	0.768902i	$-0.720803\pi$
$101$	−12.8511	−1.27873	−0.639367	+	0.768902i	$0.720803\pi$
$102$	0	0
$103$	2.77751	0.273676	0.136838	−	0.990593i	$-0.456306\pi$
$103$	2.77751	0.273676	0.136838	+	0.990593i	$0.456306\pi$
$104$	16.8703	1.65427
$105$	0	0
$106$	2.14351	0.208196
$107$	−4.96050	−0.479549	−0.239775	−	0.970829i	$-0.577074\pi$
$107$	−4.96050	−0.479549	−0.239775	+	0.970829i	$0.577074\pi$
$108$	0	0
$109$	4.85696	0.465212	0.232606	−	0.972571i	$-0.425275\pi$
$109$	4.85696	0.465212	0.232606	+	0.972571i	$0.425275\pi$
$110$	18.5049	1.76437
$111$	0	0
$112$	12.7972	1.20922
$113$	17.2822	1.62578	0.812888	−	0.582420i	$-0.197894\pi$
$113$	17.2822	1.62578	0.812888	+	0.582420i	$0.197894\pi$
$114$	0	0
$115$	3.55713	0.331704
$116$	−0.0224489	−0.00208433
$117$	0	0
$118$	5.42566	0.499472
$119$	−7.14513	−0.654993
$120$	0	0
$121$	2.59024	0.235477
$122$	1.31207	0.118789
$123$	0	0
$124$	0.0470411	0.00422442
$125$	8.83239	0.789993
$126$	0	0
$127$	−1.78084	−0.158024	−0.0790120	−	0.996874i	$-0.525177\pi$
$127$	−1.78084	−0.158024	−0.0790120	+	0.996874i	$0.525177\pi$
$128$	11.4477	1.01184
$129$	0	0
$130$	−30.0615	−2.63656
$131$	10.6090	0.926913	0.463457	−	0.886120i	$-0.346609\pi$
$131$	10.6090	0.926913	0.463457	+	0.886120i	$0.346609\pi$
$132$	0	0
$133$	3.17409	0.275229
$134$	9.72072	0.839742
$135$	0	0
$136$	−6.34127	−0.543760
$137$	6.14976	0.525409	0.262705	−	0.964876i	$-0.415386\pi$
$137$	6.14976	0.525409	0.262705	+	0.964876i	$0.415386\pi$
$138$	0	0
$139$	−9.24522	−0.784169	−0.392084	−	0.919929i	$-0.628246\pi$
$139$	−9.24522	−0.784169	−0.392084	+	0.919929i	$0.628246\pi$
$140$	−0.179687	−0.0151863
$141$	0	0
$142$	−3.46450	−0.290735
$143$	−22.0775	−1.84622
$144$	0	0
$145$	−4.95623	−0.411592
$146$	10.9191	0.903673
$147$	0	0
$148$	0.0514998	0.00423326
$149$	−10.4876	−0.859174	−0.429587	−	0.903025i	$-0.641341\pi$
$149$	−10.4876	−0.859174	−0.429587	+	0.903025i	$0.641341\pi$
$150$	0	0
$151$	−7.64750	−0.622344	−0.311172	−	0.950354i	$-0.600722\pi$
$151$	−7.64750	−0.622344	−0.311172	+	0.950354i	$0.600722\pi$
$152$	2.81699	0.228488
$153$	0	0
$154$	−16.6142	−1.33881
$155$	10.3856	0.834194
$156$	0	0
$157$	−1.85060	−0.147694	−0.0738471	−	0.997270i	$-0.523528\pi$
$157$	−1.85060	−0.147694	−0.0738471	+	0.997270i	$0.523528\pi$
$158$	11.9764	0.952793
$159$	0	0
$160$	−0.320230	−0.0253164
$161$	−3.19369	−0.251698
$162$	0	0
$163$	13.3110	1.04260	0.521301	−	0.853373i	$-0.325447\pi$
$163$	13.3110	1.04260	0.521301	+	0.853373i	$0.325447\pi$
$164$	0.0687444	0.00536803
$165$	0	0
$166$	6.55845	0.509035
$167$	−20.9056	−1.61772	−0.808861	−	0.587999i	$-0.799916\pi$
$167$	−20.9056	−1.61772	−0.808861	+	0.587999i	$0.799916\pi$
$168$	0	0
$169$	22.8653	1.75887
$170$	11.2996	0.866641
$171$	0	0
$172$	0.0987849	0.00753228
$173$	16.6763	1.26787	0.633937	−	0.773384i	$-0.281438\pi$
$173$	16.6763	1.26787	0.633937	+	0.773384i	$0.281438\pi$
$174$	0	0
$175$	−23.8004	−1.79914
$176$	−14.8631	−1.12035
$177$	0	0
$178$	11.9825	0.898127
$179$	3.43446	0.256704	0.128352	−	0.991729i	$-0.459031\pi$
$179$	3.43446	0.256704	0.128352	+	0.991729i	$0.459031\pi$
$180$	0	0
$181$	−3.10824	−0.231034	−0.115517	−	0.993306i	$-0.536852\pi$
$181$	−3.10824	−0.231034	−0.115517	+	0.993306i	$0.536852\pi$
$182$	26.9900	2.00063
$183$	0	0
$184$	−2.83438	−0.208954
$185$	11.3700	0.835939
$186$	0	0
$187$	8.29859	0.606853
$188$	0.0160129	0.00116786
$189$	0	0
$190$	−5.01964	−0.364163
$191$	−11.6702	−0.844428	−0.422214	−	0.906496i	$-0.638747\pi$
$191$	−11.6702	−0.844428	−0.422214	+	0.906496i	$0.638747\pi$
$192$	0	0
$193$	7.25789	0.522435	0.261217	−	0.965280i	$-0.415876\pi$
$193$	7.25789	0.522435	0.261217	+	0.965280i	$0.415876\pi$
$194$	8.06441	0.578991
$195$	0	0
$196$	0.0492374	0.00351696
$197$	−3.74077	−0.266519	−0.133260	−	0.991081i	$-0.542544\pi$
$197$	−3.74077	−0.266519	−0.133260	+	0.991081i	$0.542544\pi$
$198$	0	0
$199$	2.47498	0.175447	0.0877234	−	0.996145i	$-0.472041\pi$
$199$	2.47498	0.175447	0.0877234	+	0.996145i	$0.472041\pi$
$200$	−21.1228	−1.49360
$201$	0	0
$202$	−18.2468	−1.28384
$203$	4.44984	0.312317
$204$	0	0
$205$	15.1772	1.06002
$206$	3.94369	0.274770
$207$	0	0
$208$	24.1453	1.67418
$209$	−3.68649	−0.255000
$210$	0	0
$211$	19.1332	1.31719	0.658593	−	0.752500i	$-0.271152\pi$
$211$	19.1332	1.31719	0.658593	+	0.752500i	$0.271152\pi$
$212$	0.0241740	0.00166028
$213$	0	0
$214$	−7.04323	−0.481465
$215$	21.8095	1.48740
$216$	0	0
$217$	−9.32451	−0.632989
$218$	6.89622	0.467071
$219$	0	0
$220$	0.208694	0.0140702
$221$	−13.4812	−0.906843
$222$	0	0
$223$	24.7470	1.65718	0.828590	−	0.559855i	$-0.189143\pi$
$223$	24.7470	1.65718	0.828590	+	0.559855i	$0.189143\pi$
$224$	0.287511	0.0192102
$225$	0	0
$226$	24.5384	1.63227
$227$	26.6992	1.77208	0.886042	−	0.463604i	$-0.153444\pi$
$227$	26.6992	1.77208	0.886042	+	0.463604i	$0.153444\pi$
$228$	0	0
$229$	−6.48037	−0.428235	−0.214117	−	0.976808i	$-0.568688\pi$
$229$	−6.48037	−0.428235	−0.214117	+	0.976808i	$0.568688\pi$
$230$	5.05064	0.333029
$231$	0	0
$232$	3.94921	0.259278
$233$	8.46521	0.554574	0.277287	−	0.960787i	$-0.410565\pi$
$233$	8.46521	0.554574	0.277287	+	0.960787i	$0.410565\pi$
$234$	0	0
$235$	3.53530	0.230618
$236$	0.0611895	0.00398310
$237$	0	0
$238$	−10.1451	−0.657610
$239$	10.6556	0.689252	0.344626	−	0.938740i	$-0.388006\pi$
$239$	10.6556	0.689252	0.344626	+	0.938740i	$0.388006\pi$
$240$	0	0
$241$	18.3951	1.18493	0.592466	−	0.805595i	$-0.298154\pi$
$241$	18.3951	1.18493	0.592466	+	0.805595i	$0.298154\pi$
$242$	3.67779	0.236417
$243$	0	0
$244$	0.0147972	0.000947296 0
$245$	10.8705	0.694492
$246$	0	0
$247$	5.98876	0.381056
$248$	−8.27546	−0.525492
$249$	0	0
$250$	12.5408	0.793149
$251$	−28.2338	−1.78210	−0.891051	−	0.453903i	$-0.850031\pi$
$251$	−28.2338	−1.78210	−0.891051	+	0.453903i	$0.850031\pi$
$252$	0	0
$253$	3.70926	0.233199
$254$	−2.52855	−0.158655
$255$	0	0
$256$	0.384286	0.0240179
$257$	−2.29088	−0.142901	−0.0714505	−	0.997444i	$-0.522763\pi$
$257$	−2.29088	−0.142901	−0.0714505	+	0.997444i	$0.522763\pi$
$258$	0	0
$259$	−10.2083	−0.634313
$260$	−0.339027	−0.0210256
$261$	0	0
$262$	15.0633	0.930616
$263$	6.87016	0.423632	0.211816	−	0.977310i	$-0.432062\pi$
$263$	6.87016	0.423632	0.211816	+	0.977310i	$0.432062\pi$
$264$	0	0
$265$	5.33709	0.327855
$266$	4.50678	0.276328
$267$	0	0
$268$	0.109628	0.00669662
$269$	8.37213	0.510458	0.255229	−	0.966881i	$-0.417849\pi$
$269$	8.37213	0.510458	0.255229	+	0.966881i	$0.417849\pi$
$270$	0	0
$271$	−9.88690	−0.600587	−0.300293	−	0.953847i	$-0.597085\pi$
$271$	−9.88690	−0.600587	−0.300293	+	0.953847i	$0.597085\pi$
$272$	−9.07583	−0.550303
$273$	0	0
$274$	8.73182	0.527508
$275$	27.6426	1.66691
$276$	0	0
$277$	−21.7515	−1.30692	−0.653462	−	0.756959i	$-0.726684\pi$
$277$	−21.7515	−1.30692	−0.653462	+	0.756959i	$0.726684\pi$
$278$	−13.1269	−0.787302
$279$	0	0
$280$	31.6105	1.88909
$281$	−32.3244	−1.92831	−0.964157	−	0.265331i	$-0.914519\pi$
$281$	−32.3244	−1.92831	−0.964157	+	0.265331i	$0.914519\pi$
$282$	0	0
$283$	−25.5232	−1.51720	−0.758598	−	0.651559i	$-0.774115\pi$
$283$	−25.5232	−1.51720	−0.758598	+	0.651559i	$0.774115\pi$
$284$	−0.0390720	−0.00231849
$285$	0	0
$286$	−31.3471	−1.85359
$287$	−13.6265	−0.804349
$288$	0	0
$289$	−11.9326	−0.701920
$290$	−7.03717	−0.413237
$291$	0	0
$292$	0.123144	0.00720644
$293$	−10.4301	−0.609336	−0.304668	−	0.952459i	$-0.598545\pi$
$293$	−10.4301	−0.609336	−0.304668	+	0.952459i	$0.598545\pi$
$294$	0	0
$295$	13.5093	0.786540
$296$	−9.05983	−0.526592
$297$	0	0
$298$	−14.8909	−0.862607
$299$	−6.02574	−0.348477
$300$	0	0
$301$	−19.5812	−1.12864
$302$	−10.8584	−0.624831
$303$	0	0
$304$	4.03177	0.231238
$305$	3.26690	0.187062
$306$	0	0
$307$	12.2572	0.699556	0.349778	−	0.936833i	$-0.386257\pi$
$307$	12.2572	0.699556	0.349778	+	0.936833i	$0.386257\pi$
$308$	−0.187372	−0.0106765
$309$	0	0
$310$	14.7462	0.837527
$311$	−6.02845	−0.341842	−0.170921	−	0.985285i	$-0.554674\pi$
$311$	−6.02845	−0.341842	−0.170921	+	0.985285i	$0.554674\pi$
$312$	0	0
$313$	20.9097	1.18189	0.590943	−	0.806714i	$-0.298756\pi$
$313$	20.9097	1.18189	0.590943	+	0.806714i	$0.298756\pi$
$314$	−2.62761	−0.148284
$315$	0	0
$316$	0.135068	0.00759815
$317$	−18.6533	−1.04767	−0.523836	−	0.851819i	$-0.675500\pi$
$317$	−18.6533	−1.04767	−0.523836	+	0.851819i	$0.675500\pi$
$318$	0	0
$319$	−5.16819	−0.289363
$320$	28.0523	1.56817
$321$	0	0
$322$	−4.53460	−0.252703
$323$	−2.25108	−0.125253
$324$	0	0
$325$	−44.9058	−2.49093
$326$	18.8999	1.04677
$327$	0	0
$328$	−12.0935	−0.667751
$329$	−3.17409	−0.174993
$330$	0	0
$331$	−13.1641	−0.723565	−0.361783	−	0.932262i	$-0.617832\pi$
$331$	−13.1641	−0.723565	−0.361783	+	0.932262i	$0.617832\pi$
$332$	0.0739649	0.00405935
$333$	0	0
$334$	−29.6831	−1.62419
$335$	24.2035	1.32238
$336$	0	0
$337$	−28.8933	−1.57392	−0.786959	−	0.617005i	$-0.788346\pi$
$337$	−28.8933	−1.57392	−0.786959	+	0.617005i	$0.788346\pi$
$338$	32.4656	1.76590
$339$	0	0
$340$	0.127435	0.00691112
$341$	10.8298	0.586466
$342$	0	0
$343$	12.4588	0.672711
$344$	−17.3782	−0.936970
$345$	0	0
$346$	23.6781	1.27294
$347$	−1.38980	−0.0746084	−0.0373042	−	0.999304i	$-0.511877\pi$
$347$	−1.38980	−0.0746084	−0.0373042	+	0.999304i	$0.511877\pi$
$348$	0	0
$349$	21.5453	1.15329	0.576647	−	0.816994i	$-0.304361\pi$
$349$	21.5453	1.15329	0.576647	+	0.816994i	$0.304361\pi$
$350$	−33.7933	−1.80633
$351$	0	0
$352$	−0.333925	−0.0177983
$353$	−3.34257	−0.177907	−0.0889536	−	0.996036i	$-0.528352\pi$
$353$	−3.34257	−0.177907	−0.0889536	+	0.996036i	$0.528352\pi$
$354$	0	0
$355$	−8.62622	−0.457832
$356$	0.135136	0.00716222
$357$	0	0
$358$	4.87647	0.257730
$359$	20.4009	1.07672	0.538359	−	0.842715i	$-0.319044\pi$
$359$	20.4009	1.07672	0.538359	+	0.842715i	$0.319044\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.41328	−0.231957
$363$	0	0
$364$	0.304388	0.0159543
$365$	27.1874	1.42305
$366$	0	0
$367$	−29.7991	−1.55550	−0.777751	−	0.628573i	$-0.783639\pi$
$367$	−29.7991	−1.55550	−0.777751	+	0.628573i	$0.783639\pi$
$368$	−4.05666	−0.211468
$369$	0	0
$370$	16.1439	0.839279
$371$	−4.79179	−0.248777
$372$	0	0
$373$	3.14297	0.162737	0.0813685	−	0.996684i	$-0.474071\pi$
$373$	3.14297	0.162737	0.0813685	+	0.996684i	$0.474071\pi$
$374$	11.7829	0.609278
$375$	0	0
$376$	−2.81699	−0.145275
$377$	8.39580	0.432406
$378$	0	0
$379$	19.0377	0.977902	0.488951	−	0.872311i	$-0.337380\pi$
$379$	19.0377	0.977902	0.488951	+	0.872311i	$0.337380\pi$
$380$	−0.0566105	−0.00290406
$381$	0	0
$382$	−16.5701	−0.847801
$383$	13.1372	0.671281	0.335640	−	0.941990i	$-0.391047\pi$
$383$	13.1372	0.671281	0.335640	+	0.941990i	$0.391047\pi$
$384$	0	0
$385$	−41.3675	−2.10828
$386$	10.3052	0.524522
$387$	0	0
$388$	0.0909488	0.00461723
$389$	−9.19674	−0.466293	−0.233147	−	0.972442i	$-0.574902\pi$
$389$	−9.19674	−0.466293	−0.233147	+	0.972442i	$0.574902\pi$
$390$	0	0
$391$	2.26498	0.114545
$392$	−8.66183	−0.437488
$393$	0	0
$394$	−5.31139	−0.267584
$395$	29.8199	1.50040
$396$	0	0
$397$	−33.3229	−1.67243	−0.836213	−	0.548404i	$-0.815235\pi$
$397$	−33.3229	−1.67243	−0.836213	+	0.548404i	$0.815235\pi$
$398$	3.51414	0.176148
$399$	0	0
$400$	−30.2316	−1.51158
$401$	21.7404	1.08566	0.542831	−	0.839842i	$-0.317352\pi$
$401$	21.7404	1.08566	0.542831	+	0.839842i	$0.317352\pi$
$402$	0	0
$403$	−17.5932	−0.876378
$404$	−0.205784	−0.0102381
$405$	0	0
$406$	6.31816	0.313565
$407$	11.8563	0.587693
$408$	0	0
$409$	−2.41905	−0.119614	−0.0598072	−	0.998210i	$-0.519049\pi$
$409$	−2.41905	−0.119614	−0.0598072	+	0.998210i	$0.519049\pi$
$410$	21.5496	1.06426
$411$	0	0
$412$	0.0444761	0.00219118
$413$	−12.1290	−0.596829
$414$	0	0
$415$	16.3298	0.801599
$416$	0.542466	0.0265966
$417$	0	0
$418$	−5.23432	−0.256019
$419$	14.2655	0.696916	0.348458	−	0.937324i	$-0.386705\pi$
$419$	14.2655	0.696916	0.348458	+	0.937324i	$0.386705\pi$
$420$	0	0
$421$	−1.31640	−0.0641574	−0.0320787	−	0.999485i	$-0.510213\pi$
$421$	−1.31640	−0.0641574	−0.0320787	+	0.999485i	$0.510213\pi$
$422$	27.1666	1.32245
$423$	0	0
$424$	−4.25269	−0.206529
$425$	16.8794	0.818769
$426$	0	0
$427$	−2.93311	−0.141943
$428$	−0.0794321	−0.00383950
$429$	0	0
$430$	30.9665	1.49334
$431$	−24.3946	−1.17505	−0.587523	−	0.809208i	$-0.699897\pi$
$431$	−24.3946	−1.17505	−0.587523	+	0.809208i	$0.699897\pi$
$432$	0	0
$433$	9.42511	0.452942	0.226471	−	0.974018i	$-0.427281\pi$
$433$	9.42511	0.452942	0.226471	+	0.974018i	$0.427281\pi$
$434$	−13.2395	−0.635518
$435$	0	0
$436$	0.0777742	0.00372471
$437$	−1.00617	−0.0481318
$438$	0	0
$439$	24.5020	1.16942	0.584708	−	0.811244i	$-0.301209\pi$
$439$	24.5020	1.16942	0.584708	+	0.811244i	$0.301209\pi$
$440$	−36.7135	−1.75025
$441$	0	0
$442$	−19.1414	−0.910466
$443$	−3.27246	−0.155479	−0.0777397	−	0.996974i	$-0.524770\pi$
$443$	−3.27246	−0.155479	−0.0777397	+	0.996974i	$0.524770\pi$
$444$	0	0
$445$	29.8351	1.41432
$446$	35.1373	1.66380
$447$	0	0
$448$	−25.1862	−1.18993
$449$	−39.0868	−1.84462	−0.922310	−	0.386450i	$-0.873701\pi$
$449$	−39.0868	−1.84462	−0.922310	+	0.386450i	$0.873701\pi$
$450$	0	0
$451$	15.8263	0.745232
$452$	0.276740	0.0130167
$453$	0	0
$454$	37.9092	1.77916
$455$	67.2021	3.15048
$456$	0	0
$457$	8.68262	0.406156	0.203078	−	0.979163i	$-0.434906\pi$
$457$	8.68262	0.406156	0.203078	+	0.979163i	$0.434906\pi$
$458$	−9.20124	−0.429946
$459$	0	0
$460$	0.0569601	0.00265578
$461$	8.95559	0.417103	0.208552	−	0.978011i	$-0.433125\pi$
$461$	8.95559	0.417103	0.208552	+	0.978011i	$0.433125\pi$
$462$	0	0
$463$	−15.2633	−0.709345	−0.354673	−	0.934991i	$-0.615408\pi$
$463$	−15.2633	−0.709345	−0.354673	+	0.934991i	$0.615408\pi$
$464$	5.65224	0.262399
$465$	0	0
$466$	12.0194	0.556790
$467$	14.2466	0.659253	0.329626	−	0.944111i	$-0.393077\pi$
$467$	14.2466	0.659253	0.329626	+	0.944111i	$0.393077\pi$
$468$	0	0
$469$	−21.7306	−1.00342
$470$	5.01964	0.231539
$471$	0	0
$472$	−10.7644	−0.495473
$473$	22.7422	1.04569
$474$	0	0
$475$	−7.49834	−0.344048
$476$	−0.114415	−0.00524418
$477$	0	0
$478$	15.1295	0.692005
$479$	−16.1860	−0.739559	−0.369779	−	0.929120i	$-0.620567\pi$
$479$	−16.1860	−0.739559	−0.369779	+	0.929120i	$0.620567\pi$
$480$	0	0
$481$	−19.2607	−0.878211
$482$	26.1185	1.18967
$483$	0	0
$484$	0.0414774	0.00188534
$485$	20.0795	0.911762
$486$	0	0
$487$	23.1998	1.05128	0.525641	−	0.850707i	$-0.323826\pi$
$487$	23.1998	1.05128	0.525641	+	0.850707i	$0.323826\pi$
$488$	−2.60312	−0.117838
$489$	0	0
$490$	15.4347	0.697266
$491$	30.9336	1.39601	0.698007	−	0.716091i	$-0.254070\pi$
$491$	30.9336	1.39601	0.698007	+	0.716091i	$0.254070\pi$
$492$	0	0
$493$	−3.15585	−0.142132
$494$	8.50323	0.382578
$495$	0	0
$496$	−11.8441	−0.531816
$497$	7.74486	0.347404
$498$	0	0
$499$	43.3379	1.94007	0.970035	−	0.242965i	$-0.0781201\pi$
$499$	43.3379	1.94007	0.970035	+	0.242965i	$0.0781201\pi$
$500$	0.141432	0.00632505
$501$	0	0
$502$	−40.0882	−1.78922
$503$	26.1588	1.16636	0.583182	−	0.812342i	$-0.301807\pi$
$503$	26.1588	1.16636	0.583182	+	0.812342i	$0.301807\pi$
$504$	0	0
$505$	−45.4325	−2.02172
$506$	5.26664	0.234131
$507$	0	0
$508$	−0.0285165	−0.00126521
$509$	38.4605	1.70473	0.852365	−	0.522947i	$-0.175167\pi$
$509$	38.4605	1.70473	0.852365	+	0.522947i	$0.175167\pi$
$510$	0	0
$511$	−24.4096	−1.07982
$512$	−22.3497	−0.987727
$513$	0	0
$514$	−3.25274	−0.143472
$515$	9.81934	0.432692
$516$	0	0
$517$	3.68649	0.162132
$518$	−14.4944	−0.636848
$519$	0	0
$520$	59.6415	2.61545
$521$	12.7139	0.557006	0.278503	−	0.960435i	$-0.410162\pi$
$521$	12.7139	0.557006	0.278503	+	0.960435i	$0.410162\pi$
$522$	0	0
$523$	35.6830	1.56031	0.780153	−	0.625588i	$-0.215141\pi$
$523$	35.6830	1.56031	0.780153	+	0.625588i	$0.215141\pi$
$524$	0.169881	0.00742130
$525$	0	0
$526$	9.75469	0.425325
$527$	6.61299	0.288066
$528$	0	0
$529$	−21.9876	−0.955983
$530$	7.57794	0.329165
$531$	0	0
$532$	0.0508265	0.00220361
$533$	−25.7101	−1.11363
$534$	0	0
$535$	−17.5368	−0.758184
$536$	−19.2858	−0.833019
$537$	0	0
$538$	11.8873	0.512497
$539$	11.3354	0.488251
$540$	0	0
$541$	−35.6406	−1.53231	−0.766154	−	0.642657i	$-0.777832\pi$
$541$	−35.6406	−1.53231	−0.766154	+	0.642657i	$0.777832\pi$
$542$	−14.0381	−0.602986
$543$	0	0
$544$	−0.203904	−0.00874233
$545$	17.1708	0.735516
$546$	0	0
$547$	−29.8487	−1.27624	−0.638120	−	0.769937i	$-0.720288\pi$
$547$	−29.8487	−1.27624	−0.638120	+	0.769937i	$0.720288\pi$
$548$	0.0984757	0.00420667
$549$	0	0
$550$	39.2487	1.67357
$551$	1.40193	0.0597240
$552$	0	0
$553$	−26.7732	−1.13851
$554$	−30.8842	−1.31215
$555$	0	0
$556$	−0.148043	−0.00627842
$557$	−38.7404	−1.64149	−0.820743	−	0.571298i	$-0.806440\pi$
$557$	−38.7404	−1.64149	−0.820743	+	0.571298i	$0.806440\pi$
$558$	0	0
$559$	−36.9451	−1.56261
$560$	45.2419	1.91182
$561$	0	0
$562$	−45.8963	−1.93602
$563$	9.52959	0.401624	0.200812	−	0.979630i	$-0.435642\pi$
$563$	9.52959	0.401624	0.200812	+	0.979630i	$0.435642\pi$
$564$	0	0
$565$	61.0979	2.57041
$566$	−36.2394	−1.52326
$567$	0	0
$568$	6.87353	0.288407
$569$	21.7248	0.910751	0.455375	−	0.890300i	$-0.349505\pi$
$569$	21.7248	0.910751	0.455375	+	0.890300i	$0.349505\pi$
$570$	0	0
$571$	3.94645	0.165154	0.0825768	−	0.996585i	$-0.473685\pi$
$571$	3.94645	0.165154	0.0825768	+	0.996585i	$0.473685\pi$
$572$	−0.353526	−0.0147817
$573$	0	0
$574$	−19.3478	−0.807562
$575$	7.54464	0.314633
$576$	0	0
$577$	−6.27016	−0.261030	−0.130515	−	0.991446i	$-0.541663\pi$
$577$	−6.27016	−0.261030	−0.130515	+	0.991446i	$0.541663\pi$
$578$	−16.9427	−0.704724
$579$	0	0
$580$	−0.0793637	−0.00329540
$581$	−14.6614	−0.608256
$582$	0	0
$583$	5.56534	0.230493
$584$	−21.6634	−0.896438
$585$	0	0
$586$	−14.8094	−0.611770
$587$	24.9768	1.03090	0.515451	−	0.856919i	$-0.327625\pi$
$587$	24.9768	1.03090	0.515451	+	0.856919i	$0.327625\pi$
$588$	0	0
$589$	−2.93770	−0.121046
$590$	19.1813	0.789682
$591$	0	0
$592$	−12.9667	−0.532929
$593$	−23.3868	−0.960380	−0.480190	−	0.877165i	$-0.659432\pi$
$593$	−23.3868	−0.960380	−0.480190	+	0.877165i	$0.659432\pi$
$594$	0	0
$595$	−25.2602	−1.03557
$596$	−0.167937	−0.00687895
$597$	0	0
$598$	−8.55573	−0.349870
$599$	−23.0434	−0.941528	−0.470764	−	0.882259i	$-0.656022\pi$
$599$	−23.0434	−0.941528	−0.470764	+	0.882259i	$0.656022\pi$
$600$	0	0
$601$	−5.51783	−0.225077	−0.112538	−	0.993647i	$-0.535898\pi$
$601$	−5.51783	−0.225077	−0.112538	+	0.993647i	$0.535898\pi$
$602$	−27.8026	−1.13315
$603$	0	0
$604$	−0.122459	−0.00498278
$605$	9.15729	0.372297
$606$	0	0
$607$	−12.1828	−0.494484	−0.247242	−	0.968954i	$-0.579524\pi$
$607$	−12.1828	−0.494484	−0.247242	+	0.968954i	$0.579524\pi$
$608$	0.0905807	0.00367353
$609$	0	0
$610$	4.63855	0.187809
$611$	−5.98876	−0.242279
$612$	0	0
$613$	−7.89456	−0.318858	−0.159429	−	0.987209i	$-0.550965\pi$
$613$	−7.89456	−0.318858	−0.159429	+	0.987209i	$0.550965\pi$
$614$	17.4036	0.702351
$615$	0	0
$616$	32.9624	1.32809
$617$	4.33694	0.174599	0.0872993	−	0.996182i	$-0.472176\pi$
$617$	4.33694	0.174599	0.0872993	+	0.996182i	$0.472176\pi$
$618$	0	0
$619$	5.05802	0.203299	0.101650	−	0.994820i	$-0.467588\pi$
$619$	5.05802	0.203299	0.101650	+	0.994820i	$0.467588\pi$
$620$	0.166305	0.00667895
$621$	0	0
$622$	−8.55958	−0.343208
$623$	−26.7868	−1.07319
$624$	0	0
$625$	−6.26658	−0.250663
$626$	29.6889	1.18661
$627$	0	0
$628$	−0.0296336	−0.00118251
$629$	7.23978	0.288669
$630$	0	0
$631$	8.12764	0.323556	0.161778	−	0.986827i	$-0.448277\pi$
$631$	8.12764	0.323556	0.161778	+	0.986827i	$0.448277\pi$
$632$	−23.7611	−0.945164
$633$	0	0
$634$	−26.4851	−1.05186
$635$	−6.29580	−0.249841
$636$	0	0
$637$	−18.4146	−0.729611
$638$	−7.33813	−0.290519
$639$	0	0
$640$	40.4710	1.59976
$641$	−1.72043	−0.0679527	−0.0339764	−	0.999423i	$-0.510817\pi$
$641$	−1.72043	−0.0679527	−0.0339764	+	0.999423i	$0.510817\pi$
$642$	0	0
$643$	−0.645402	−0.0254522	−0.0127261	−	0.999919i	$-0.504051\pi$
$643$	−0.645402	−0.0254522	−0.0127261	+	0.999919i	$0.504051\pi$
$644$	−0.0511403	−0.00201521
$645$	0	0
$646$	−3.19623	−0.125754
$647$	11.2571	0.442561	0.221281	−	0.975210i	$-0.428976\pi$
$647$	11.2571	0.442561	0.221281	+	0.975210i	$0.428976\pi$
$648$	0	0
$649$	14.0870	0.552964
$650$	−63.7601	−2.50088
$651$	0	0
$652$	0.213149	0.00834756
$653$	2.24427	0.0878250	0.0439125	−	0.999035i	$-0.486018\pi$
$653$	2.24427	0.0878250	0.0439125	+	0.999035i	$0.486018\pi$
$654$	0	0
$655$	37.5060	1.46548
$656$	−17.3086	−0.675787
$657$	0	0
$658$	−4.50678	−0.175692
$659$	−10.9961	−0.428346	−0.214173	−	0.976796i	$-0.568706\pi$
$659$	−10.9961	−0.428346	−0.214173	+	0.976796i	$0.568706\pi$
$660$	0	0
$661$	39.2824	1.52791	0.763955	−	0.645270i	$-0.223255\pi$
$661$	39.2824	1.52791	0.763955	+	0.645270i	$0.223255\pi$
$662$	−18.6913	−0.726456
$663$	0	0
$664$	−13.0119	−0.504959
$665$	11.2214	0.435146
$666$	0	0
$667$	−1.41058	−0.0546179
$668$	−0.334760	−0.0129522
$669$	0	0
$670$	34.3656	1.32766
$671$	3.40661	0.131511
$672$	0	0
$673$	−7.91111	−0.304951	−0.152475	−	0.988307i	$-0.548724\pi$
$673$	−7.91111	−0.304951	−0.152475	+	0.988307i	$0.548724\pi$
$674$	−41.0245	−1.58021
$675$	0	0
$676$	0.366140	0.0140823
$677$	23.5818	0.906323	0.453161	−	0.891428i	$-0.350296\pi$
$677$	23.5818	0.906323	0.453161	+	0.891428i	$0.350296\pi$
$678$	0	0
$679$	−18.0279	−0.691848
$680$	−22.4183	−0.859702
$681$	0	0
$682$	15.3768	0.588810
$683$	31.9957	1.22428	0.612141	−	0.790749i	$-0.290309\pi$
$683$	31.9957	1.22428	0.612141	+	0.790749i	$0.290309\pi$
$684$	0	0
$685$	21.7412	0.830690
$686$	17.6898	0.675399
$687$	0	0
$688$	−24.8723	−0.948246
$689$	−9.04098	−0.344434
$690$	0	0
$691$	−7.80727	−0.297002	−0.148501	−	0.988912i	$-0.547445\pi$
$691$	−7.80727	−0.297002	−0.148501	+	0.988912i	$0.547445\pi$
$692$	0.267036	0.0101512
$693$	0	0
$694$	−1.97333	−0.0749065
$695$	−32.6846	−1.23980
$696$	0	0
$697$	9.66400	0.366050
$698$	30.5914	1.15790
$699$	0	0
$700$	−0.381115	−0.0144048
$701$	4.29944	0.162388	0.0811938	−	0.996698i	$-0.474127\pi$
$701$	4.29944	0.162388	0.0811938	+	0.996698i	$0.474127\pi$
$702$	0	0
$703$	−3.21614	−0.121299
$704$	29.2521	1.10248
$705$	0	0
$706$	−4.74600	−0.178618
$707$	40.7906	1.53409
$708$	0	0
$709$	−9.22941	−0.346618	−0.173309	−	0.984868i	$-0.555446\pi$
$709$	−9.22941	−0.346618	−0.173309	+	0.984868i	$0.555446\pi$
$710$	−12.2481	−0.459661
$711$	0	0
$712$	−23.7732	−0.890937
$713$	2.95583	0.110697
$714$	0	0
$715$	−78.0507	−2.91893
$716$	0.0549959	0.00205529
$717$	0	0
$718$	28.9665	1.08102
$719$	36.1102	1.34668	0.673341	−	0.739332i	$-0.264859\pi$
$719$	36.1102	1.34668	0.673341	+	0.739332i	$0.264859\pi$
$720$	0	0
$721$	−8.81608	−0.328328
$722$	1.41986	0.0528419
$723$	0	0
$724$	−0.0497721	−0.00184976
$725$	−10.5121	−0.390410
$726$	0	0
$727$	−49.0198	−1.81804	−0.909021	−	0.416749i	$-0.863169\pi$
$727$	−49.0198	−1.81804	−0.909021	+	0.416749i	$0.863169\pi$
$728$	−53.5478	−1.98461
$729$	0	0
$730$	38.6024	1.42874
$731$	13.8871	0.513631
$732$	0	0
$733$	26.4121	0.975552	0.487776	−	0.872969i	$-0.337808\pi$
$733$	26.4121	0.975552	0.487776	+	0.872969i	$0.337808\pi$
$734$	−42.3107	−1.56172
$735$	0	0
$736$	−0.0911400	−0.00335946
$737$	25.2386	0.929676
$738$	0	0
$739$	33.4968	1.23220	0.616100	−	0.787668i	$-0.288712\pi$
$739$	33.4968	1.23220	0.616100	+	0.787668i	$0.288712\pi$
$740$	0.182067	0.00669292
$741$	0	0
$742$	−6.80368	−0.249771
$743$	−3.53605	−0.129725	−0.0648625	−	0.997894i	$-0.520661\pi$
$743$	−3.53605	−0.129725	−0.0648625	+	0.997894i	$0.520661\pi$
$744$	0	0
$745$	−37.0767	−1.35838
$746$	4.46259	0.163387
$747$	0	0
$748$	0.132885	0.00485875
$749$	15.7451	0.575312
$750$	0	0
$751$	33.8284	1.23442	0.617208	−	0.786800i	$-0.288264\pi$
$751$	33.8284	1.23442	0.617208	+	0.786800i	$0.288264\pi$
$752$	−4.03177	−0.147024
$753$	0	0
$754$	11.9209	0.434133
$755$	−27.0362	−0.983948
$756$	0	0
$757$	4.30012	0.156291	0.0781453	−	0.996942i	$-0.475100\pi$
$757$	4.30012	0.156291	0.0781453	+	0.996942i	$0.475100\pi$
$758$	27.0310	0.981809
$759$	0	0
$760$	9.95891	0.361248
$761$	−26.7226	−0.968694	−0.484347	−	0.874876i	$-0.660943\pi$
$761$	−26.7226	−0.968694	−0.484347	+	0.874876i	$0.660943\pi$
$762$	0	0
$763$	−15.4164	−0.558112
$764$	−0.186875	−0.00676088
$765$	0	0
$766$	18.6531	0.673963
$767$	−22.8846	−0.826314
$768$	0	0
$769$	−20.7214	−0.747232	−0.373616	−	0.927584i	$-0.621882\pi$
$769$	−20.7214	−0.747232	−0.373616	+	0.927584i	$0.621882\pi$
$770$	−58.7362	−2.11671
$771$	0	0
$772$	0.116220	0.00418286
$773$	−9.46211	−0.340328	−0.170164	−	0.985416i	$-0.554430\pi$
$773$	−9.46211	−0.340328	−0.170164	+	0.985416i	$0.554430\pi$
$774$	0	0
$775$	22.0278	0.791264
$776$	−15.9997	−0.574355
$777$	0	0
$778$	−13.0581	−0.468156
$779$	−4.29305	−0.153814
$780$	0	0
$781$	−8.99514	−0.321871
$782$	3.21596	0.115003
$783$	0	0
$784$	−12.3971	−0.442753
$785$	−6.54244	−0.233510
$786$	0	0
$787$	−25.9896	−0.926429	−0.463215	−	0.886246i	$-0.653304\pi$
$787$	−25.9896	−0.926429	−0.463215	+	0.886246i	$0.653304\pi$
$788$	−0.0599008	−0.00213388
$789$	0	0
$790$	42.3402	1.50640
$791$	−54.8554	−1.95043
$792$	0	0
$793$	−5.53409	−0.196521
$794$	−47.3139	−1.67911
$795$	0	0
$796$	0.0396317	0.00140471
$797$	11.5108	0.407733	0.203867	−	0.978999i	$-0.434649\pi$
$797$	11.5108	0.407733	0.203867	+	0.978999i	$0.434649\pi$
$798$	0	0
$799$	2.25108	0.0796375
$800$	−0.679205	−0.0240135
$801$	0	0
$802$	30.8683	1.09000
$803$	28.3501	1.00045
$804$	0	0
$805$	−11.2906	−0.397943
$806$	−24.9799	−0.879879
$807$	0	0
$808$	36.2015	1.27356
$809$	−41.3057	−1.45223	−0.726115	−	0.687573i	$-0.758676\pi$
$809$	−41.3057	−1.45223	−0.726115	+	0.687573i	$0.758676\pi$
$810$	0	0
$811$	23.1902	0.814320	0.407160	−	0.913357i	$-0.366519\pi$
$811$	23.1902	0.814320	0.407160	+	0.913357i	$0.366519\pi$
$812$	0.0712550	0.00250056
$813$	0	0
$814$	16.8343	0.590041
$815$	47.0585	1.64839
$816$	0	0
$817$	−6.16907	−0.215828
$818$	−3.43472	−0.120092
$819$	0	0
$820$	0.243032	0.00848704
$821$	−4.34540	−0.151655	−0.0758277	−	0.997121i	$-0.524160\pi$
$821$	−4.34540	−0.151655	−0.0758277	+	0.997121i	$0.524160\pi$
$822$	0	0
$823$	24.9047	0.868124	0.434062	−	0.900883i	$-0.357080\pi$
$823$	24.9047	0.868124	0.434062	+	0.900883i	$0.357080\pi$
$824$	−7.82423	−0.272570
$825$	0	0
$826$	−17.2215	−0.599213
$827$	−5.56809	−0.193621	−0.0968107	−	0.995303i	$-0.530864\pi$
$827$	−5.56809	−0.193621	−0.0968107	+	0.995303i	$0.530864\pi$
$828$	0	0
$829$	5.15564	0.179063	0.0895314	−	0.995984i	$-0.471463\pi$
$829$	5.15564	0.179063	0.0895314	+	0.995984i	$0.471463\pi$
$830$	23.1861	0.804801
$831$	0	0
$832$	−47.5204	−1.64747
$833$	6.92173	0.239824
$834$	0	0
$835$	−73.9075	−2.55767
$836$	−0.0590316	−0.00204165
$837$	0	0
$838$	20.2551	0.699700
$839$	−34.3756	−1.18678	−0.593389	−	0.804916i	$-0.702210\pi$
$839$	−34.3756	−1.18678	−0.593389	+	0.804916i	$0.702210\pi$
$840$	0	0
$841$	−27.0346	−0.932228
$842$	−1.86911	−0.0644137
$843$	0	0
$844$	0.306379	0.0105460
$845$	80.8356	2.78083
$846$	0	0
$847$	−8.22167	−0.282500
$848$	−6.08659	−0.209014
$849$	0	0
$850$	23.9664	0.822041
$851$	3.23599	0.110928
$852$	0	0
$853$	11.8665	0.406301	0.203151	−	0.979147i	$-0.434882\pi$
$853$	11.8665	0.406301	0.203151	+	0.979147i	$0.434882\pi$
$854$	−4.16462	−0.142510
$855$	0	0
$856$	13.9737	0.477610
$857$	−8.77530	−0.299758	−0.149879	−	0.988704i	$-0.547888\pi$
$857$	−8.77530	−0.299758	−0.149879	+	0.988704i	$0.547888\pi$
$858$	0	0
$859$	−21.9065	−0.747442	−0.373721	−	0.927541i	$-0.621918\pi$
$859$	−21.9065	−0.747442	−0.373721	+	0.927541i	$0.621918\pi$
$860$	0.349234	0.0119088
$861$	0	0
$862$	−34.6370	−1.17974
$863$	−38.2778	−1.30299	−0.651496	−	0.758652i	$-0.725858\pi$
$863$	−38.2778	−1.30299	−0.651496	+	0.758652i	$0.725858\pi$
$864$	0	0
$865$	58.9557	2.00455
$866$	13.3824	0.454751
$867$	0	0
$868$	−0.149313	−0.00506801
$869$	31.0952	1.05483
$870$	0	0
$871$	−41.0005	−1.38925
$872$	−13.6820	−0.463331
$873$	0	0
$874$	−1.42863	−0.0483241
$875$	−28.0348	−0.947749
$876$	0	0
$877$	56.5971	1.91115	0.955574	−	0.294750i	$-0.0952365\pi$
$877$	56.5971	1.91115	0.955574	+	0.294750i	$0.0952365\pi$
$878$	34.7895	1.17409
$879$	0	0
$880$	−52.5455	−1.77131
$881$	−46.8644	−1.57890	−0.789451	−	0.613814i	$-0.789635\pi$
$881$	−46.8644	−1.57890	−0.789451	+	0.613814i	$0.789635\pi$
$882$	0	0
$883$	−50.8371	−1.71080	−0.855402	−	0.517965i	$-0.826690\pi$
$883$	−50.8371	−1.71080	−0.855402	+	0.517965i	$0.826690\pi$
$884$	−0.215873	−0.00726061
$885$	0	0
$886$	−4.64645	−0.156100
$887$	−34.0841	−1.14443	−0.572216	−	0.820103i	$-0.693916\pi$
$887$	−34.0841	−1.14443	−0.572216	+	0.820103i	$0.693916\pi$
$888$	0	0
$889$	5.65255	0.189580
$890$	42.3618	1.41997
$891$	0	0
$892$	0.396272	0.0132682
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	12.1419	0.405858
$896$	−36.3360	−1.21390
$897$	0	0
$898$	−55.4979	−1.85199
$899$	−4.11843	−0.137357
$900$	0	0
$901$	3.39836	0.113216
$902$	22.4712	0.748209
$903$	0	0
$904$	−48.6839	−1.61920
$905$	−10.9886	−0.365272
$906$	0	0
$907$	1.47502	0.0489773	0.0244887	−	0.999700i	$-0.492204\pi$
$907$	1.47502	0.0489773	0.0244887	+	0.999700i	$0.492204\pi$
$908$	0.427532	0.0141881
$909$	0	0
$910$	95.4178	3.16307
$911$	3.71537	0.123096	0.0615479	−	0.998104i	$-0.480396\pi$
$911$	3.71537	0.123096	0.0615479	+	0.998104i	$0.480396\pi$
$912$	0	0
$913$	17.0282	0.563551
$914$	12.3281	0.407779
$915$	0	0
$916$	−0.103770	−0.00342865
$917$	−33.6739	−1.11201
$918$	0	0
$919$	−46.7428	−1.54190	−0.770951	−	0.636894i	$-0.780219\pi$
$919$	−46.7428	−1.54190	−0.770951	+	0.636894i	$0.780219\pi$
$920$	−10.0204	−0.330363
$921$	0	0
$922$	12.7157	0.418770
$923$	14.6127	0.480984
$924$	0	0
$925$	24.1157	0.792919
$926$	−21.6718	−0.712179
$927$	0	0
$928$	0.126987	0.00416856
$929$	17.2096	0.564630	0.282315	−	0.959322i	$-0.408898\pi$
$929$	17.2096	0.564630	0.282315	+	0.959322i	$0.408898\pi$
$930$	0	0
$931$	−3.07485	−0.100774
$932$	0.135553	0.00444018
$933$	0	0
$934$	20.2282	0.661887
$935$	29.3380	0.959456
$936$	0	0
$937$	−4.43486	−0.144881	−0.0724403	−	0.997373i	$-0.523079\pi$
$937$	−4.43486	−0.144881	−0.0724403	+	0.997373i	$0.523079\pi$
$938$	−30.8544	−1.00743
$939$	0	0
$940$	0.0566105	0.00184643
$941$	6.15226	0.200558	0.100279	−	0.994959i	$-0.468026\pi$
$941$	6.15226	0.200558	0.100279	+	0.994959i	$0.468026\pi$
$942$	0	0
$943$	4.31956	0.140664
$944$	−15.4064	−0.501436
$945$	0	0
$946$	32.2909	1.04987
$947$	24.7464	0.804150	0.402075	−	0.915607i	$-0.368289\pi$
$947$	24.7464	0.804150	0.402075	+	0.915607i	$0.368289\pi$
$948$	0	0
$949$	−46.0552	−1.49501
$950$	−10.6466	−0.345422
$951$	0	0
$952$	20.1278	0.652345
$953$	−13.3249	−0.431636	−0.215818	−	0.976434i	$-0.569242\pi$
$953$	−13.3249	−0.431636	−0.215818	+	0.976434i	$0.569242\pi$
$954$	0	0
$955$	−41.2577	−1.33507
$956$	0.170627	0.00551847
$957$	0	0
$958$	−22.9820	−0.742513
$959$	−19.5199	−0.630330
$960$	0	0
$961$	−22.3699	−0.721611
$962$	−27.3475	−0.881720
$963$	0	0
$964$	0.294560	0.00948712
$965$	25.6588	0.825987
$966$	0	0
$967$	53.7677	1.72905	0.864527	−	0.502587i	$-0.167618\pi$
$967$	53.7677	1.72905	0.864527	+	0.502587i	$0.167618\pi$
$968$	−7.29669	−0.234525
$969$	0	0
$970$	28.5101	0.915404
$971$	41.1808	1.32155	0.660777	−	0.750582i	$-0.270227\pi$
$971$	41.1808	1.32155	0.660777	+	0.750582i	$0.270227\pi$
$972$	0	0
$973$	29.3451	0.940762
$974$	32.9405	1.05548
$975$	0	0
$976$	−3.72568	−0.119256
$977$	−35.0687	−1.12195	−0.560974	−	0.827833i	$-0.689573\pi$
$977$	−35.0687	−1.12195	−0.560974	+	0.827833i	$0.689573\pi$
$978$	0	0
$979$	31.1111	0.994314
$980$	0.174069	0.00556043
$981$	0	0
$982$	43.9215	1.40159
$983$	3.93358	0.125462	0.0627308	−	0.998030i	$-0.480019\pi$
$983$	3.93358	0.125462	0.0627308	+	0.998030i	$0.480019\pi$
$984$	0	0
$985$	−13.2248	−0.421376
$986$	−4.48087	−0.142700
$987$	0	0
$988$	0.0958977	0.00305091
$989$	6.20715	0.197376
$990$	0	0
$991$	44.2549	1.40580	0.702901	−	0.711288i	$-0.251888\pi$
$991$	44.2549	1.40580	0.702901	+	0.711288i	$0.251888\pi$
$992$	−0.266098	−0.00844864
$993$	0	0
$994$	10.9966	0.348792
$995$	8.74980	0.277387
$996$	0	0
$997$	−23.0403	−0.729692	−0.364846	−	0.931068i	$-0.618878\pi$
$997$	−23.0403	−0.729692	−0.364846	+	0.931068i	$0.618878\pi$
$998$	61.5339	1.94782
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.25	yes	34
3.2	odd	2		8037.2.a.v.1.10	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.10	✓	34	3.2	odd	2
8037.2.a.w.1.25	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)

\(f(q)\)	\(=\)	\(q+1.41986 q^{2} +0.0160129 q^{4} +3.53530 q^{5} -3.17409 q^{7} -2.81699 q^{8} +O(q^{10})\)	\(q+1.41986 q^{2} +0.0160129 q^{4} +3.53530 q^{5} -3.17409 q^{7} -2.81699 q^{8} +5.01964 q^{10} +3.68649 q^{11} -5.98876 q^{13} -4.50678 q^{14} -4.03177 q^{16} +2.25108 q^{17} -1.00000 q^{19} +0.0566105 q^{20} +5.23432 q^{22} +1.00617 q^{23} +7.49834 q^{25} -8.50323 q^{26} -0.0508265 q^{28} -1.40193 q^{29} +2.93770 q^{31} -0.0905807 q^{32} +3.19623 q^{34} -11.2214 q^{35} +3.21614 q^{37} -1.41986 q^{38} -9.95891 q^{40} +4.29305 q^{41} +6.16907 q^{43} +0.0590316 q^{44} +1.42863 q^{46} +1.00000 q^{47} +3.07485 q^{49} +10.6466 q^{50} -0.0958977 q^{52} +1.50966 q^{53} +13.0329 q^{55} +8.94138 q^{56} -1.99054 q^{58} +3.82125 q^{59} +0.924079 q^{61} +4.17113 q^{62} +7.93493 q^{64} -21.1721 q^{65} +6.84623 q^{67} +0.0360464 q^{68} -15.9328 q^{70} -2.44002 q^{71} +7.69026 q^{73} +4.56648 q^{74} -0.0160129 q^{76} -11.7013 q^{77} +8.43491 q^{79} -14.2535 q^{80} +6.09555 q^{82} +4.61907 q^{83} +7.95824 q^{85} +8.75923 q^{86} -10.3848 q^{88} +8.43920 q^{89} +19.0089 q^{91} +0.0161118 q^{92} +1.41986 q^{94} -3.53530 q^{95} +5.67971 q^{97} +4.36587 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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