LMFDB - Embedded newform 8037.2.a.q.1.20

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:	$23$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+2.11032 q^{2} +2.45347 q^{4} -3.83102 q^{5} +4.31069 q^{7} +0.956959 q^{8} +O(q^{10})$	\(q+2.11032 q^{2} +2.45347 q^{4} -3.83102 q^{5} +4.31069 q^{7} +0.956959 q^{8} -8.08470 q^{10} -2.76575 q^{11} +3.24197 q^{13} +9.09695 q^{14} -2.88744 q^{16} +0.177222 q^{17} -1.00000 q^{19} -9.39928 q^{20} -5.83662 q^{22} -5.19445 q^{23} +9.67673 q^{25} +6.84160 q^{26} +10.5761 q^{28} +5.03429 q^{29} +3.49646 q^{31} -8.00735 q^{32} +0.373997 q^{34} -16.5143 q^{35} +11.2740 q^{37} -2.11032 q^{38} -3.66613 q^{40} -0.266938 q^{41} -1.82212 q^{43} -6.78567 q^{44} -10.9620 q^{46} -1.00000 q^{47} +11.5820 q^{49} +20.4210 q^{50} +7.95406 q^{52} +3.63372 q^{53} +10.5956 q^{55} +4.12515 q^{56} +10.6240 q^{58} +5.46233 q^{59} +3.95547 q^{61} +7.37866 q^{62} -11.1232 q^{64} -12.4201 q^{65} -12.8454 q^{67} +0.434809 q^{68} -34.8506 q^{70} +11.3561 q^{71} -2.40126 q^{73} +23.7918 q^{74} -2.45347 q^{76} -11.9223 q^{77} +2.23110 q^{79} +11.0618 q^{80} -0.563326 q^{82} -0.0436954 q^{83} -0.678943 q^{85} -3.84526 q^{86} -2.64671 q^{88} -16.6468 q^{89} +13.9751 q^{91} -12.7444 q^{92} -2.11032 q^{94} +3.83102 q^{95} +9.83417 q^{97} +24.4419 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8	$ 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8} - 5 q^{10} - 12 q^{11} + 9 q^{13} - 3 q^{14} + 60 q^{16} - 16 q^{17} - 23 q^{19} - 25 q^{20} + 3 q^{22} - 12 q^{23} + 54 q^{25} - 5 q^{26} + 8 q^{28} - 27 q^{29} + 10 q^{31} + 34 q^{32} + 6 q^{35} + 15 q^{37} + 2 q^{38} + 3 q^{40} - 10 q^{41} + 24 q^{43} - 39 q^{44} + 43 q^{46} - 23 q^{47} + 78 q^{49} + 32 q^{50} + 38 q^{52} + 2 q^{53} + 5 q^{55} - 58 q^{56} - 11 q^{58} + 51 q^{59} + 48 q^{61} + 22 q^{62} + 125 q^{64} - 15 q^{65} + 26 q^{67} - 26 q^{68} + 86 q^{70} - 24 q^{71} + 53 q^{73} - 26 q^{74} - 30 q^{76} - 18 q^{77} + 29 q^{79} - 5 q^{80} + 47 q^{82} + 22 q^{83} - 5 q^{85} + 28 q^{86} + 62 q^{88} - 38 q^{89} + 15 q^{91} - 15 q^{92} + 2 q^{94} + 9 q^{95} + 33 q^{97} + 30 q^{98}+O(q^{100}) $ 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8 - 5 * q^10 - 12 * q^11 + 9 * q^13 - 3 * q^14 + 60 * q^16 - 16 * q^17 - 23 * q^19 - 25 * q^20 + 3 * q^22 - 12 * q^23 + 54 * q^25 - 5 * q^26 + 8 * q^28 - 27 * q^29 + 10 * q^31 + 34 * q^32 + 6 * q^35 + 15 * q^37 + 2 * q^38 + 3 * q^40 - 10 * q^41 + 24 * q^43 - 39 * q^44 + 43 * q^46 - 23 * q^47 + 78 * q^49 + 32 * q^50 + 38 * q^52 + 2 * q^53 + 5 * q^55 - 58 * q^56 - 11 * q^58 + 51 * q^59 + 48 * q^61 + 22 * q^62 + 125 * q^64 - 15 * q^65 + 26 * q^67 - 26 * q^68 + 86 * q^70 - 24 * q^71 + 53 * q^73 - 26 * q^74 - 30 * q^76 - 18 * q^77 + 29 * q^79 - 5 * q^80 + 47 * q^82 + 22 * q^83 - 5 * q^85 + 28 * q^86 + 62 * q^88 - 38 * q^89 + 15 * q^91 - 15 * q^92 + 2 * q^94 + 9 * q^95 + 33 * q^97 + 30 * 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 $

$2$	2.11032	1.49222	0.746112	−	0.665820i	$-0.231918\pi$
$2$	2.11032	1.49222	0.746112	+	0.665820i	$0.231918\pi$
$3$	0	0
$3$	0	0
$4$	2.45347	1.22673
$5$	−3.83102	−1.71329	−0.856643	−	0.515910i	$-0.827454\pi$
$5$	−3.83102	−1.71329	−0.856643	+	0.515910i	$0.827454\pi$
$6$	0	0
$7$	4.31069	1.62929	0.814644	−	0.579962i	$-0.196932\pi$
$7$	4.31069	1.62929	0.814644	+	0.579962i	$0.196932\pi$
$8$	0.956959	0.338336
$9$	0	0
$10$	−8.08470	−2.55661
$11$	−2.76575	−0.833904	−0.416952	−	0.908928i	$-0.636902\pi$
$11$	−2.76575	−0.833904	−0.416952	+	0.908928i	$0.636902\pi$
$12$	0	0
$13$	3.24197	0.899160	0.449580	−	0.893240i	$-0.351574\pi$
$13$	3.24197	0.899160	0.449580	+	0.893240i	$0.351574\pi$
$14$	9.09695	2.43126
$15$	0	0
$16$	−2.88744	−0.721860
$17$	0.177222	0.0429827	0.0214914	−	0.999769i	$-0.493159\pi$
$17$	0.177222	0.0429827	0.0214914	+	0.999769i	$0.493159\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−9.39928	−2.10174
$21$	0	0
$22$	−5.83662	−1.24437
$23$	−5.19445	−1.08312	−0.541559	−	0.840663i	$-0.682166\pi$
$23$	−5.19445	−1.08312	−0.541559	+	0.840663i	$0.682166\pi$
$24$	0	0
$25$	9.67673	1.93535
$26$	6.84160	1.34175
$27$	0	0
$28$	10.5761	1.99870
$29$	5.03429	0.934844	0.467422	−	0.884034i	$-0.345183\pi$
$29$	5.03429	0.934844	0.467422	+	0.884034i	$0.345183\pi$
$30$	0	0
$31$	3.49646	0.627983	0.313992	−	0.949426i	$-0.398334\pi$
$31$	3.49646	0.627983	0.313992	+	0.949426i	$0.398334\pi$
$32$	−8.00735	−1.41551
$33$	0	0
$34$	0.373997	0.0641399
$35$	−16.5143	−2.79143
$36$	0	0
$37$	11.2740	1.85343	0.926717	−	0.375761i	$-0.122619\pi$
$37$	11.2740	1.85343	0.926717	+	0.375761i	$0.122619\pi$
$38$	−2.11032	−0.342340
$39$	0	0
$40$	−3.66613	−0.579666
$41$	−0.266938	−0.0416887	−0.0208444	−	0.999783i	$-0.506635\pi$
$41$	−0.266938	−0.0416887	−0.0208444	+	0.999783i	$0.506635\pi$
$42$	0	0
$43$	−1.82212	−0.277870	−0.138935	−	0.990301i	$-0.544368\pi$
$43$	−1.82212	−0.277870	−0.138935	+	0.990301i	$0.544368\pi$
$44$	−6.78567	−1.02298
$45$	0	0
$46$	−10.9620	−1.61625
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	11.5820	1.65458
$50$	20.4210	2.88797
$51$	0	0
$52$	7.95406	1.10303
$53$	3.63372	0.499129	0.249565	−	0.968358i	$-0.419712\pi$
$53$	3.63372	0.499129	0.249565	+	0.968358i	$0.419712\pi$
$54$	0	0
$55$	10.5956	1.42872
$56$	4.12515	0.551247
$57$	0	0
$58$	10.6240	1.39500
$59$	5.46233	0.711135	0.355568	−	0.934651i	$-0.384288\pi$
$59$	5.46233	0.711135	0.355568	+	0.934651i	$0.384288\pi$
$60$	0	0
$61$	3.95547	0.506446	0.253223	−	0.967408i	$-0.418509\pi$
$61$	3.95547	0.506446	0.253223	+	0.967408i	$0.418509\pi$
$62$	7.37866	0.937091
$63$	0	0
$64$	−11.1232	−1.39040
$65$	−12.4201	−1.54052
$66$	0	0
$67$	−12.8454	−1.56932	−0.784658	−	0.619929i	$-0.787161\pi$
$67$	−12.8454	−1.56932	−0.784658	+	0.619929i	$0.787161\pi$
$68$	0.434809	0.0527283
$69$	0	0
$70$	−34.8506	−4.16545
$71$	11.3561	1.34772	0.673859	−	0.738860i	$-0.264636\pi$
$71$	11.3561	1.34772	0.673859	+	0.738860i	$0.264636\pi$
$72$	0	0
$73$	−2.40126	−0.281047	−0.140523	−	0.990077i	$-0.544879\pi$
$73$	−2.40126	−0.281047	−0.140523	+	0.990077i	$0.544879\pi$
$74$	23.7918	2.76574
$75$	0	0
$76$	−2.45347	−0.281432
$77$	−11.9223	−1.35867
$78$	0	0
$79$	2.23110	0.251018	0.125509	−	0.992092i	$-0.459944\pi$
$79$	2.23110	0.251018	0.125509	+	0.992092i	$0.459944\pi$
$80$	11.0618	1.23675
$81$	0	0
$82$	−0.563326	−0.0622089
$83$	−0.0436954	−0.00479620	−0.00239810	−	0.999997i	$-0.500763\pi$
$83$	−0.0436954	−0.00479620	−0.00239810	+	0.999997i	$0.500763\pi$
$84$	0	0
$85$	−0.678943	−0.0736417
$86$	−3.84526	−0.414645
$87$	0	0
$88$	−2.64671	−0.282140
$89$	−16.6468	−1.76456	−0.882279	−	0.470726i	$-0.843992\pi$
$89$	−16.6468	−1.76456	−0.882279	+	0.470726i	$0.843992\pi$
$90$	0	0
$91$	13.9751	1.46499
$92$	−12.7444	−1.32870
$93$	0	0
$94$	−2.11032	−0.217663
$95$	3.83102	0.393055
$96$	0	0
$97$	9.83417	0.998509	0.499254	−	0.866455i	$-0.333607\pi$
$97$	9.83417	0.998509	0.499254	+	0.866455i	$0.333607\pi$
$98$	24.4419	2.46900
$99$	0	0
$100$	23.7415	2.37415
$101$	17.1552	1.70701	0.853505	−	0.521084i	$-0.174472\pi$
$101$	17.1552	1.70701	0.853505	+	0.521084i	$0.174472\pi$
$102$	0	0
$103$	−2.16722	−0.213542	−0.106771	−	0.994284i	$-0.534051\pi$
$103$	−2.16722	−0.213542	−0.106771	+	0.994284i	$0.534051\pi$
$104$	3.10243	0.304218
$105$	0	0
$106$	7.66832	0.744813
$107$	19.2939	1.86521	0.932606	−	0.360897i	$-0.117529\pi$
$107$	19.2939	1.86521	0.932606	+	0.360897i	$0.117529\pi$
$108$	0	0
$109$	−13.2377	−1.26794	−0.633971	−	0.773357i	$-0.718576\pi$
$109$	−13.2377	−1.26794	−0.633971	+	0.773357i	$0.718576\pi$
$110$	22.3602	2.13196
$111$	0	0
$112$	−12.4469	−1.17612
$113$	19.5573	1.83979	0.919896	−	0.392162i	$-0.128273\pi$
$113$	19.5573	1.83979	0.919896	+	0.392162i	$0.128273\pi$
$114$	0	0
$115$	19.9001	1.85569
$116$	12.3515	1.14680
$117$	0	0
$118$	11.5273	1.06117
$119$	0.763951	0.0700313
$120$	0	0
$121$	−3.35064	−0.304604
$122$	8.34732	0.755731
$123$	0	0
$124$	8.57845	0.770367
$125$	−17.9167	−1.60251
$126$	0	0
$127$	14.7181	1.30602	0.653012	−	0.757348i	$-0.273505\pi$
$127$	14.7181	1.30602	0.653012	+	0.757348i	$0.273505\pi$
$128$	−7.45889	−0.659279
$129$	0	0
$130$	−26.2103	−2.29880
$131$	16.5011	1.44171	0.720855	−	0.693086i	$-0.243749\pi$
$131$	16.5011	1.44171	0.720855	+	0.693086i	$0.243749\pi$
$132$	0	0
$133$	−4.31069	−0.373784
$134$	−27.1080	−2.34177
$135$	0	0
$136$	0.169595	0.0145426
$137$	7.30221	0.623870	0.311935	−	0.950103i	$-0.399023\pi$
$137$	7.30221	0.623870	0.311935	+	0.950103i	$0.399023\pi$
$138$	0	0
$139$	9.04600	0.767272	0.383636	−	0.923484i	$-0.374672\pi$
$139$	9.04600	0.767272	0.383636	+	0.923484i	$0.374672\pi$
$140$	−40.5174	−3.42434
$141$	0	0
$142$	23.9650	2.01110
$143$	−8.96646	−0.749813
$144$	0	0
$145$	−19.2865	−1.60165
$146$	−5.06744	−0.419385
$147$	0	0
$148$	27.6604	2.27367
$149$	5.53714	0.453620	0.226810	−	0.973939i	$-0.427170\pi$
$149$	5.53714	0.453620	0.226810	+	0.973939i	$0.427170\pi$
$150$	0	0
$151$	2.40299	0.195552	0.0977762	−	0.995208i	$-0.468827\pi$
$151$	2.40299	0.195552	0.0977762	+	0.995208i	$0.468827\pi$
$152$	−0.956959	−0.0776196
$153$	0	0
$154$	−25.1599	−2.02744
$155$	−13.3950	−1.07591
$156$	0	0
$157$	−7.85589	−0.626968	−0.313484	−	0.949593i	$-0.601496\pi$
$157$	−7.85589	−0.626968	−0.313484	+	0.949593i	$0.601496\pi$
$158$	4.70833	0.374575
$159$	0	0
$160$	30.6763	2.42518
$161$	−22.3917	−1.76471
$162$	0	0
$163$	−6.46407	−0.506305	−0.253152	−	0.967426i	$-0.581467\pi$
$163$	−6.46407	−0.506305	−0.253152	+	0.967426i	$0.581467\pi$
$164$	−0.654923	−0.0511409
$165$	0	0
$166$	−0.0922115	−0.00715700
$167$	9.77926	0.756741	0.378371	−	0.925654i	$-0.376484\pi$
$167$	9.77926	0.756741	0.378371	+	0.925654i	$0.376484\pi$
$168$	0	0
$169$	−2.48964	−0.191511
$170$	−1.43279	−0.109890
$171$	0	0
$172$	−4.47050	−0.340873
$173$	9.24278	0.702716	0.351358	−	0.936241i	$-0.385720\pi$
$173$	9.24278	0.702716	0.351358	+	0.936241i	$0.385720\pi$
$174$	0	0
$175$	41.7134	3.15324
$176$	7.98592	0.601962
$177$	0	0
$178$	−35.1302	−2.63312
$179$	2.47917	0.185302	0.0926510	−	0.995699i	$-0.470466\pi$
$179$	2.47917	0.185302	0.0926510	+	0.995699i	$0.470466\pi$
$180$	0	0
$181$	−0.635684	−0.0472500	−0.0236250	−	0.999721i	$-0.507521\pi$
$181$	−0.635684	−0.0472500	−0.0236250	+	0.999721i	$0.507521\pi$
$182$	29.4920	2.18609
$183$	0	0
$184$	−4.97088	−0.366458
$185$	−43.1909	−3.17546
$186$	0	0
$187$	−0.490152	−0.0358435
$188$	−2.45347	−0.178937
$189$	0	0
$190$	8.08470	0.586526
$191$	−6.59486	−0.477187	−0.238594	−	0.971119i	$-0.576686\pi$
$191$	−6.59486	−0.477187	−0.238594	+	0.971119i	$0.576686\pi$
$192$	0	0
$193$	−21.4627	−1.54492	−0.772460	−	0.635063i	$-0.780974\pi$
$193$	−21.4627	−1.54492	−0.772460	+	0.635063i	$0.780974\pi$
$194$	20.7533	1.49000
$195$	0	0
$196$	28.4162	2.02973
$197$	15.2239	1.08466	0.542328	−	0.840167i	$-0.317543\pi$
$197$	15.2239	1.08466	0.542328	+	0.840167i	$0.317543\pi$
$198$	0	0
$199$	11.5525	0.818937	0.409469	−	0.912324i	$-0.365714\pi$
$199$	11.5525	0.818937	0.409469	+	0.912324i	$0.365714\pi$
$200$	9.26023	0.654797
$201$	0	0
$202$	36.2031	2.54724
$203$	21.7013	1.52313
$204$	0	0
$205$	1.02265	0.0714246
$206$	−4.57353	−0.318653
$207$	0	0
$208$	−9.36098	−0.649067
$209$	2.76575	0.191311
$210$	0	0
$211$	13.4252	0.924228	0.462114	−	0.886820i	$-0.347091\pi$
$211$	13.4252	0.924228	0.462114	+	0.886820i	$0.347091\pi$
$212$	8.91520	0.612298
$213$	0	0
$214$	40.7164	2.78331
$215$	6.98057	0.476071
$216$	0	0
$217$	15.0722	1.02316
$218$	−27.9358	−1.89205
$219$	0	0
$220$	25.9960	1.75265
$221$	0.574549	0.0386484
$222$	0	0
$223$	−6.80933	−0.455987	−0.227993	−	0.973663i	$-0.573216\pi$
$223$	−6.80933	−0.455987	−0.227993	+	0.973663i	$0.573216\pi$
$224$	−34.5172	−2.30628
$225$	0	0
$226$	41.2721	2.74538
$227$	−9.97429	−0.662017	−0.331008	−	0.943628i	$-0.607389\pi$
$227$	−9.97429	−0.662017	−0.331008	+	0.943628i	$0.607389\pi$
$228$	0	0
$229$	15.4584	1.02152	0.510761	−	0.859722i	$-0.329364\pi$
$229$	15.4584	1.02152	0.510761	+	0.859722i	$0.329364\pi$
$230$	41.9956	2.76911
$231$	0	0
$232$	4.81761	0.316292
$233$	−26.6465	−1.74567	−0.872834	−	0.488017i	$-0.837720\pi$
$233$	−26.6465	−1.74567	−0.872834	+	0.488017i	$0.837720\pi$
$234$	0	0
$235$	3.83102	0.249908
$236$	13.4016	0.872373
$237$	0	0
$238$	1.61218	0.104502
$239$	19.0415	1.23170	0.615848	−	0.787865i	$-0.288814\pi$
$239$	19.0415	1.23170	0.615848	+	0.787865i	$0.288814\pi$
$240$	0	0
$241$	−11.7881	−0.759341	−0.379670	−	0.925122i	$-0.623963\pi$
$241$	−11.7881	−0.759341	−0.379670	+	0.925122i	$0.623963\pi$
$242$	−7.07094	−0.454537
$243$	0	0
$244$	9.70461	0.621274
$245$	−44.3711	−2.83476
$246$	0	0
$247$	−3.24197	−0.206281
$248$	3.34597	0.212469
$249$	0	0
$250$	−37.8099	−2.39131
$251$	7.12582	0.449778	0.224889	−	0.974384i	$-0.427798\pi$
$251$	7.12582	0.449778	0.224889	+	0.974384i	$0.427798\pi$
$252$	0	0
$253$	14.3665	0.903217
$254$	31.0600	1.94888
$255$	0	0
$256$	6.50576	0.406610
$257$	31.6167	1.97220	0.986098	−	0.166165i	$-0.0531384\pi$
$257$	31.6167	1.97220	0.986098	+	0.166165i	$0.0531384\pi$
$258$	0	0
$259$	48.5987	3.01978
$260$	−30.4722	−1.88980
$261$	0	0
$262$	34.8227	2.15135
$263$	24.4848	1.50980	0.754898	−	0.655842i	$-0.227686\pi$
$263$	24.4848	1.50980	0.754898	+	0.655842i	$0.227686\pi$
$264$	0	0
$265$	−13.9208	−0.855151
$266$	−9.09695	−0.557770
$267$	0	0
$268$	−31.5157	−1.92513
$269$	−24.8011	−1.51215	−0.756074	−	0.654487i	$-0.772885\pi$
$269$	−24.8011	−1.51215	−0.756074	+	0.654487i	$0.772885\pi$
$270$	0	0
$271$	−22.4551	−1.36405	−0.682024	−	0.731330i	$-0.738900\pi$
$271$	−22.4551	−1.36405	−0.682024	+	0.731330i	$0.738900\pi$
$272$	−0.511719	−0.0310275
$273$	0	0
$274$	15.4100	0.930954
$275$	−26.7634	−1.61389
$276$	0	0
$277$	19.7357	1.18580	0.592902	−	0.805275i	$-0.297982\pi$
$277$	19.7357	1.18580	0.592902	+	0.805275i	$0.297982\pi$
$278$	19.0900	1.14494
$279$	0	0
$280$	−15.8036	−0.944443
$281$	−20.5939	−1.22853	−0.614265	−	0.789100i	$-0.710547\pi$
$281$	−20.5939	−1.22853	−0.614265	+	0.789100i	$0.710547\pi$
$282$	0	0
$283$	−21.1450	−1.25694	−0.628469	−	0.777835i	$-0.716318\pi$
$283$	−21.1450	−1.25694	−0.628469	+	0.777835i	$0.716318\pi$
$284$	27.8617	1.65329
$285$	0	0
$286$	−18.9221	−1.11889
$287$	−1.15069	−0.0679229
$288$	0	0
$289$	−16.9686	−0.998152
$290$	−40.7007	−2.39003
$291$	0	0
$292$	−5.89142	−0.344769
$293$	−6.17446	−0.360716	−0.180358	−	0.983601i	$-0.557726\pi$
$293$	−6.17446	−0.360716	−0.180358	+	0.983601i	$0.557726\pi$
$294$	0	0
$295$	−20.9263	−1.21838
$296$	10.7887	0.627083
$297$	0	0
$298$	11.6852	0.676903
$299$	−16.8402	−0.973896
$300$	0	0
$301$	−7.85458	−0.452731
$302$	5.07108	0.291808
$303$	0	0
$304$	2.88744	0.165606
$305$	−15.1535	−0.867687
$306$	0	0
$307$	8.05970	0.459991	0.229996	−	0.973192i	$-0.426129\pi$
$307$	8.05970	0.459991	0.229996	+	0.973192i	$0.426129\pi$
$308$	−29.2509	−1.66672
$309$	0	0
$310$	−28.2678	−1.60550
$311$	−10.9020	−0.618193	−0.309097	−	0.951031i	$-0.600027\pi$
$311$	−10.9020	−0.618193	−0.309097	+	0.951031i	$0.600027\pi$
$312$	0	0
$313$	11.8057	0.667299	0.333650	−	0.942697i	$-0.391720\pi$
$313$	11.8057	0.667299	0.333650	+	0.942697i	$0.391720\pi$
$314$	−16.5785	−0.935577
$315$	0	0
$316$	5.47392	0.307932
$317$	−1.75290	−0.0984525	−0.0492263	−	0.998788i	$-0.515676\pi$
$317$	−1.75290	−0.0984525	−0.0492263	+	0.998788i	$0.515676\pi$
$318$	0	0
$319$	−13.9236	−0.779571
$320$	42.6133	2.38215
$321$	0	0
$322$	−47.2537	−2.63334
$323$	−0.177222	−0.00986092
$324$	0	0
$325$	31.3717	1.74019
$326$	−13.6413	−0.755520
$327$	0	0
$328$	−0.255449	−0.0141048
$329$	−4.31069	−0.237656
$330$	0	0
$331$	−22.8188	−1.25424	−0.627118	−	0.778924i	$-0.715766\pi$
$331$	−22.8188	−1.25424	−0.627118	+	0.778924i	$0.715766\pi$
$332$	−0.107205	−0.00588365
$333$	0	0
$334$	20.6374	1.12923
$335$	49.2110	2.68868
$336$	0	0
$337$	21.3653	1.16384	0.581921	−	0.813246i	$-0.302301\pi$
$337$	21.3653	1.16384	0.581921	+	0.813246i	$0.302301\pi$
$338$	−5.25395	−0.285777
$339$	0	0
$340$	−1.66576	−0.0903387
$341$	−9.67033	−0.523678
$342$	0	0
$343$	19.7518	1.06650
$344$	−1.74369	−0.0940135
$345$	0	0
$346$	19.5053	1.04861
$347$	−0.0216134	−0.00116027	−0.000580135	−	1.00000i	$-0.500185\pi$
$347$	−0.0216134	−0.00116027	−0.000580135		1.00000i	$0.500185\pi$
$348$	0	0
$349$	10.9619	0.586778	0.293389	−	0.955993i	$-0.405217\pi$
$349$	10.9619	0.586778	0.293389	+	0.955993i	$0.405217\pi$
$350$	88.0287	4.70533
$351$	0	0
$352$	22.1463	1.18040
$353$	−20.4111	−1.08637	−0.543187	−	0.839612i	$-0.682782\pi$
$353$	−20.4111	−1.08637	−0.543187	+	0.839612i	$0.682782\pi$
$354$	0	0
$355$	−43.5054	−2.30902
$356$	−40.8424	−2.16464
$357$	0	0
$358$	5.23185	0.276512
$359$	−10.4886	−0.553567	−0.276783	−	0.960932i	$-0.589268\pi$
$359$	−10.4886	−0.553567	−0.276783	+	0.960932i	$0.589268\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.34150	−0.0705076
$363$	0	0
$364$	34.2875	1.79715
$365$	9.19930	0.481513
$366$	0	0
$367$	−14.7751	−0.771252	−0.385626	−	0.922655i	$-0.626014\pi$
$367$	−14.7751	−0.771252	−0.385626	+	0.922655i	$0.626014\pi$
$368$	14.9987	0.781859
$369$	0	0
$370$	−91.1468	−4.73850
$371$	15.6638	0.813225
$372$	0	0
$373$	−1.03691	−0.0536894	−0.0268447	−	0.999640i	$-0.508546\pi$
$373$	−1.03691	−0.0536894	−0.0268447	+	0.999640i	$0.508546\pi$
$374$	−1.03438	−0.0534865
$375$	0	0
$376$	−0.956959	−0.0493514
$377$	16.3210	0.840575
$378$	0	0
$379$	−2.39878	−0.123217	−0.0616086	−	0.998100i	$-0.519623\pi$
$379$	−2.39878	−0.123217	−0.0616086	+	0.998100i	$0.519623\pi$
$380$	9.39928	0.482173
$381$	0	0
$382$	−13.9173	−0.712070
$383$	21.9474	1.12146	0.560729	−	0.828000i	$-0.310521\pi$
$383$	21.9474	1.12146	0.560729	+	0.828000i	$0.310521\pi$
$384$	0	0
$385$	45.6745	2.32779
$386$	−45.2933	−2.30537
$387$	0	0
$388$	24.1278	1.22490
$389$	20.0659	1.01738	0.508690	−	0.860950i	$-0.330130\pi$
$389$	20.0659	1.01738	0.508690	+	0.860950i	$0.330130\pi$
$390$	0	0
$391$	−0.920573	−0.0465554
$392$	11.0835	0.559803
$393$	0	0
$394$	32.1273	1.61855
$395$	−8.54738	−0.430065
$396$	0	0
$397$	14.9735	0.751497	0.375748	−	0.926722i	$-0.377386\pi$
$397$	14.9735	0.751497	0.375748	+	0.926722i	$0.377386\pi$
$398$	24.3796	1.22204
$399$	0	0
$400$	−27.9410	−1.39705
$401$	−16.7806	−0.837986	−0.418993	−	0.907990i	$-0.637617\pi$
$401$	−16.7806	−0.837986	−0.418993	+	0.907990i	$0.637617\pi$
$402$	0	0
$403$	11.3354	0.564657
$404$	42.0898	2.09405
$405$	0	0
$406$	45.7967	2.27285
$407$	−31.1810	−1.54559
$408$	0	0
$409$	13.7678	0.680772	0.340386	−	0.940286i	$-0.389442\pi$
$409$	13.7678	0.680772	0.340386	+	0.940286i	$0.389442\pi$
$410$	2.15811	0.106582
$411$	0	0
$412$	−5.31719	−0.261959
$413$	23.5464	1.15864
$414$	0	0
$415$	0.167398	0.00821725
$416$	−25.9596	−1.27277
$417$	0	0
$418$	5.83662	0.285478
$419$	38.8585	1.89836	0.949180	−	0.314733i	$-0.101915\pi$
$419$	38.8585	1.89836	0.949180	+	0.314733i	$0.101915\pi$
$420$	0	0
$421$	1.53237	0.0746829	0.0373415	−	0.999303i	$-0.488111\pi$
$421$	1.53237	0.0746829	0.0373415	+	0.999303i	$0.488111\pi$
$422$	28.3315	1.37916
$423$	0	0
$424$	3.47732	0.168873
$425$	1.71493	0.0831865
$426$	0	0
$427$	17.0508	0.825146
$428$	47.3369	2.28812
$429$	0	0
$430$	14.7313	0.710405
$431$	−15.9042	−0.766078	−0.383039	−	0.923732i	$-0.625123\pi$
$431$	−15.9042	−0.766078	−0.383039	+	0.923732i	$0.625123\pi$
$432$	0	0
$433$	−34.5794	−1.66178	−0.830889	−	0.556438i	$-0.812168\pi$
$433$	−34.5794	−1.66178	−0.830889	+	0.556438i	$0.812168\pi$
$434$	31.8071	1.52679
$435$	0	0
$436$	−32.4783	−1.55543
$437$	5.19445	0.248484
$438$	0	0
$439$	20.8679	0.995970	0.497985	−	0.867186i	$-0.334073\pi$
$439$	20.8679	0.995970	0.497985	+	0.867186i	$0.334073\pi$
$440$	10.1396	0.483386
$441$	0	0
$442$	1.21249	0.0576720
$443$	29.4994	1.40156	0.700778	−	0.713379i	$-0.252836\pi$
$443$	29.4994	1.40156	0.700778	+	0.713379i	$0.252836\pi$
$444$	0	0
$445$	63.7743	3.02319
$446$	−14.3699	−0.680434
$447$	0	0
$448$	−47.9487	−2.26536
$449$	−13.3009	−0.627709	−0.313855	−	0.949471i	$-0.601620\pi$
$449$	−13.3009	−0.627709	−0.313855	+	0.949471i	$0.601620\pi$
$450$	0	0
$451$	0.738283	0.0347644
$452$	47.9831	2.25693
$453$	0	0
$454$	−21.0490	−0.987877
$455$	−53.5390	−2.50995
$456$	0	0
$457$	−19.5133	−0.912795	−0.456397	−	0.889776i	$-0.650860\pi$
$457$	−19.5133	−0.912795	−0.456397	+	0.889776i	$0.650860\pi$
$458$	32.6223	1.52434
$459$	0	0
$460$	48.8241	2.27644
$461$	−7.63314	−0.355511	−0.177755	−	0.984075i	$-0.556884\pi$
$461$	−7.63314	−0.355511	−0.177755	+	0.984075i	$0.556884\pi$
$462$	0	0
$463$	−1.57750	−0.0733127	−0.0366563	−	0.999328i	$-0.511671\pi$
$463$	−1.57750	−0.0733127	−0.0366563	+	0.999328i	$0.511671\pi$
$464$	−14.5362	−0.674826
$465$	0	0
$466$	−56.2327	−2.60493
$467$	−33.0768	−1.53061	−0.765306	−	0.643666i	$-0.777412\pi$
$467$	−33.0768	−1.53061	−0.765306	+	0.643666i	$0.777412\pi$
$468$	0	0
$469$	−55.3725	−2.55687
$470$	8.08470	0.372919
$471$	0	0
$472$	5.22723	0.240603
$473$	5.03951	0.231717
$474$	0	0
$475$	−9.67673	−0.443999
$476$	1.87433	0.0859096
$477$	0	0
$478$	40.1838	1.83797
$479$	29.8714	1.36486	0.682429	−	0.730952i	$-0.260923\pi$
$479$	29.8714	1.36486	0.682429	+	0.730952i	$0.260923\pi$
$480$	0	0
$481$	36.5499	1.66653
$482$	−24.8768	−1.13311
$483$	0	0
$484$	−8.22069	−0.373668
$485$	−37.6749	−1.71073
$486$	0	0
$487$	−11.4542	−0.519038	−0.259519	−	0.965738i	$-0.583564\pi$
$487$	−11.4542	−0.519038	−0.259519	+	0.965738i	$0.583564\pi$
$488$	3.78522	0.171349
$489$	0	0
$490$	−93.6373	−4.23010
$491$	5.58108	0.251870	0.125935	−	0.992038i	$-0.459807\pi$
$491$	5.58108	0.251870	0.125935	+	0.992038i	$0.459807\pi$
$492$	0	0
$493$	0.892189	0.0401822
$494$	−6.84160	−0.307818
$495$	0	0
$496$	−10.0958	−0.453316
$497$	48.9525	2.19582
$498$	0	0
$499$	−12.3567	−0.553160	−0.276580	−	0.960991i	$-0.589201\pi$
$499$	−12.3567	−0.553160	−0.276580	+	0.960991i	$0.589201\pi$
$500$	−43.9579	−1.96586
$501$	0	0
$502$	15.0378	0.671169
$503$	19.3692	0.863629	0.431815	−	0.901962i	$-0.357873\pi$
$503$	19.3692	0.863629	0.431815	+	0.901962i	$0.357873\pi$
$504$	0	0
$505$	−65.7221	−2.92460
$506$	30.3180	1.34780
$507$	0	0
$508$	36.1105	1.60214
$509$	−12.0812	−0.535490	−0.267745	−	0.963490i	$-0.586279\pi$
$509$	−12.0812	−0.535490	−0.267745	+	0.963490i	$0.586279\pi$
$510$	0	0
$511$	−10.3511	−0.457906
$512$	28.6470	1.26603
$513$	0	0
$514$	66.7215	2.94296
$515$	8.30265	0.365858
$516$	0	0
$517$	2.76575	0.121637
$518$	102.559	4.50618
$519$	0	0
$520$	−11.8855	−0.521213
$521$	−12.3006	−0.538898	−0.269449	−	0.963015i	$-0.586842\pi$
$521$	−12.3006	−0.538898	−0.269449	+	0.963015i	$0.586842\pi$
$522$	0	0
$523$	−6.00558	−0.262606	−0.131303	−	0.991342i	$-0.541916\pi$
$523$	−6.00558	−0.262606	−0.131303	+	0.991342i	$0.541916\pi$
$524$	40.4849	1.76859
$525$	0	0
$526$	51.6708	2.25295
$527$	0.619651	0.0269924
$528$	0	0
$529$	3.98232	0.173144
$530$	−29.3775	−1.27608
$531$	0	0
$532$	−10.5761	−0.458533
$533$	−0.865404	−0.0374848
$534$	0	0
$535$	−73.9153	−3.19564
$536$	−12.2925	−0.530956
$537$	0	0
$538$	−52.3383	−2.25646
$539$	−32.0330	−1.37976
$540$	0	0
$541$	22.0726	0.948977	0.474489	−	0.880262i	$-0.342633\pi$
$541$	22.0726	0.948977	0.474489	+	0.880262i	$0.342633\pi$
$542$	−47.3875	−2.03547
$543$	0	0
$544$	−1.41908	−0.0608426
$545$	50.7139	2.17235
$546$	0	0
$547$	−8.46907	−0.362111	−0.181056	−	0.983473i	$-0.557951\pi$
$547$	−8.46907	−0.362111	−0.181056	+	0.983473i	$0.557951\pi$
$548$	17.9157	0.765322
$549$	0	0
$550$	−56.4794	−2.40829
$551$	−5.03429	−0.214468
$552$	0	0
$553$	9.61756	0.408980
$554$	41.6487	1.76948
$555$	0	0
$556$	22.1941	0.941238
$557$	−29.9399	−1.26859	−0.634296	−	0.773090i	$-0.718710\pi$
$557$	−29.9399	−1.26859	−0.634296	+	0.773090i	$0.718710\pi$
$558$	0	0
$559$	−5.90724	−0.249850
$560$	47.6842	2.01502
$561$	0	0
$562$	−43.4598	−1.83324
$563$	35.1139	1.47987	0.739937	−	0.672676i	$-0.234855\pi$
$563$	35.1139	1.47987	0.739937	+	0.672676i	$0.234855\pi$
$564$	0	0
$565$	−74.9243	−3.15209
$566$	−44.6227	−1.87563
$567$	0	0
$568$	10.8673	0.455981
$569$	−30.1532	−1.26409	−0.632044	−	0.774933i	$-0.717784\pi$
$569$	−30.1532	−1.26409	−0.632044	+	0.774933i	$0.717784\pi$
$570$	0	0
$571$	25.2185	1.05536	0.527680	−	0.849443i	$-0.323062\pi$
$571$	25.2185	1.05536	0.527680	+	0.849443i	$0.323062\pi$
$572$	−21.9989	−0.919821
$573$	0	0
$574$	−2.42832	−0.101356
$575$	−50.2653	−2.09621
$576$	0	0
$577$	22.9873	0.956975	0.478488	−	0.878094i	$-0.341185\pi$
$577$	22.9873	0.956975	0.478488	+	0.878094i	$0.341185\pi$
$578$	−35.8092	−1.48947
$579$	0	0
$580$	−47.3187	−1.96480
$581$	−0.188357	−0.00781438
$582$	0	0
$583$	−10.0499	−0.416226
$584$	−2.29791	−0.0950882
$585$	0	0
$586$	−13.0301	−0.538269
$587$	18.9459	0.781982	0.390991	−	0.920395i	$-0.372132\pi$
$587$	18.9459	0.781982	0.390991	+	0.920395i	$0.372132\pi$
$588$	0	0
$589$	−3.49646	−0.144069
$590$	−44.1613	−1.81809
$591$	0	0
$592$	−32.5530	−1.33792
$593$	−38.3465	−1.57470	−0.787351	−	0.616505i	$-0.788548\pi$
$593$	−38.3465	−1.57470	−0.787351	+	0.616505i	$0.788548\pi$
$594$	0	0
$595$	−2.92671	−0.119984
$596$	13.5852	0.556471
$597$	0	0
$598$	−35.5384	−1.45327
$599$	−28.3962	−1.16024	−0.580118	−	0.814532i	$-0.696994\pi$
$599$	−28.3962	−1.16024	−0.580118	+	0.814532i	$0.696994\pi$
$600$	0	0
$601$	−11.3476	−0.462878	−0.231439	−	0.972849i	$-0.574343\pi$
$601$	−11.3476	−0.462878	−0.231439	+	0.972849i	$0.574343\pi$
$602$	−16.5757	−0.675575
$603$	0	0
$604$	5.89565	0.239891
$605$	12.8364	0.521873
$606$	0	0
$607$	3.61797	0.146849	0.0734245	−	0.997301i	$-0.476607\pi$
$607$	3.61797	0.146849	0.0734245	+	0.997301i	$0.476607\pi$
$608$	8.00735	0.324741
$609$	0	0
$610$	−31.9788	−1.29478
$611$	−3.24197	−0.131156
$612$	0	0
$613$	38.5361	1.55646	0.778228	−	0.627981i	$-0.216119\pi$
$613$	38.5361	1.55646	0.778228	+	0.627981i	$0.216119\pi$
$614$	17.0086	0.686410
$615$	0	0
$616$	−11.4091	−0.459687
$617$	−25.2688	−1.01728	−0.508642	−	0.860978i	$-0.669852\pi$
$617$	−25.2688	−1.01728	−0.508642	+	0.860978i	$0.669852\pi$
$618$	0	0
$619$	27.8944	1.12117	0.560585	−	0.828097i	$-0.310576\pi$
$619$	27.8944	1.12117	0.560585	+	0.828097i	$0.310576\pi$
$620$	−32.8642	−1.31986
$621$	0	0
$622$	−23.0067	−0.922483
$623$	−71.7592	−2.87497
$624$	0	0
$625$	20.2555	0.810218
$626$	24.9139	0.995760
$627$	0	0
$628$	−19.2742	−0.769122
$629$	1.99800	0.0796657
$630$	0	0
$631$	−4.99601	−0.198888	−0.0994440	−	0.995043i	$-0.531706\pi$
$631$	−4.99601	−0.198888	−0.0994440	+	0.995043i	$0.531706\pi$
$632$	2.13507	0.0849284
$633$	0	0
$634$	−3.69918	−0.146913
$635$	−56.3855	−2.23759
$636$	0	0
$637$	37.5486	1.48773
$638$	−29.3832	−1.16329
$639$	0	0
$640$	28.5752	1.12953
$641$	−45.2587	−1.78761	−0.893805	−	0.448457i	$-0.851974\pi$
$641$	−45.2587	−1.78761	−0.893805	+	0.448457i	$0.851974\pi$
$642$	0	0
$643$	−24.6554	−0.972314	−0.486157	−	0.873872i	$-0.661602\pi$
$643$	−24.6554	−0.972314	−0.486157	+	0.873872i	$0.661602\pi$
$644$	−54.9372	−2.16483
$645$	0	0
$646$	−0.373997	−0.0147147
$647$	−3.89284	−0.153043	−0.0765216	−	0.997068i	$-0.524381\pi$
$647$	−3.89284	−0.153043	−0.0765216	+	0.997068i	$0.524381\pi$
$648$	0	0
$649$	−15.1074	−0.593019
$650$	66.2043	2.59675
$651$	0	0
$652$	−15.8594	−0.621101
$653$	−8.09247	−0.316683	−0.158341	−	0.987384i	$-0.550615\pi$
$653$	−8.09247	−0.316683	−0.158341	+	0.987384i	$0.550615\pi$
$654$	0	0
$655$	−63.2162	−2.47006
$656$	0.770767	0.0300934
$657$	0	0
$658$	−9.09695	−0.354636
$659$	−1.00164	−0.0390183	−0.0195091	−	0.999810i	$-0.506210\pi$
$659$	−1.00164	−0.0390183	−0.0195091	+	0.999810i	$0.506210\pi$
$660$	0	0
$661$	15.8507	0.616522	0.308261	−	0.951302i	$-0.400253\pi$
$661$	15.8507	0.616522	0.308261	+	0.951302i	$0.400253\pi$
$662$	−48.1551	−1.87160
$663$	0	0
$664$	−0.0418147	−0.00162273
$665$	16.5143	0.640399
$666$	0	0
$667$	−26.1504	−1.01255
$668$	23.9931	0.928320
$669$	0	0
$670$	103.851	4.01212
$671$	−10.9398	−0.422327
$672$	0	0
$673$	−14.5217	−0.559771	−0.279886	−	0.960033i	$-0.590297\pi$
$673$	−14.5217	−0.559771	−0.279886	+	0.960033i	$0.590297\pi$
$674$	45.0877	1.73671
$675$	0	0
$676$	−6.10825	−0.234933
$677$	47.8423	1.83873	0.919365	−	0.393405i	$-0.128703\pi$
$677$	47.8423	1.83873	0.919365	+	0.393405i	$0.128703\pi$
$678$	0	0
$679$	42.3921	1.62686
$680$	−0.649721	−0.0249156
$681$	0	0
$682$	−20.4075	−0.781444
$683$	−36.2940	−1.38875	−0.694376	−	0.719613i	$-0.744319\pi$
$683$	−36.2940	−1.38875	−0.694376	+	0.719613i	$0.744319\pi$
$684$	0	0
$685$	−27.9749	−1.06887
$686$	41.6827	1.59145
$687$	0	0
$688$	5.26125	0.200583
$689$	11.7804	0.448797
$690$	0	0
$691$	−3.93750	−0.149790	−0.0748948	−	0.997191i	$-0.523862\pi$
$691$	−3.93750	−0.149790	−0.0748948	+	0.997191i	$0.523862\pi$
$692$	22.6768	0.862044
$693$	0	0
$694$	−0.0456113	−0.00173138
$695$	−34.6554	−1.31456
$696$	0	0
$697$	−0.0473074	−0.00179190
$698$	23.1332	0.875604
$699$	0	0
$700$	102.342	3.86818
$701$	0.0569862	0.00215234	0.00107617	−	0.999999i	$-0.499657\pi$
$701$	0.0569862	0.00215234	0.00107617	+	0.999999i	$0.499657\pi$
$702$	0	0
$703$	−11.2740	−0.425207
$704$	30.7640	1.15946
$705$	0	0
$706$	−43.0740	−1.62111
$707$	73.9509	2.78121
$708$	0	0
$709$	35.2447	1.32364	0.661821	−	0.749662i	$-0.269784\pi$
$709$	35.2447	1.32364	0.661821	+	0.749662i	$0.269784\pi$
$710$	−91.8104	−3.44558
$711$	0	0
$712$	−15.9303	−0.597014
$713$	−18.1622	−0.680180
$714$	0	0
$715$	34.3507	1.28464
$716$	6.08256	0.227316
$717$	0	0
$718$	−22.1343	−0.826045
$719$	−0.711745	−0.0265436	−0.0132718	−	0.999912i	$-0.504225\pi$
$719$	−0.711745	−0.0265436	−0.0132718	+	0.999912i	$0.504225\pi$
$720$	0	0
$721$	−9.34219	−0.347921
$722$	2.11032	0.0785381
$723$	0	0
$724$	−1.55963	−0.0579631
$725$	48.7155	1.80925
$726$	0	0
$727$	27.2997	1.01249	0.506245	−	0.862390i	$-0.331033\pi$
$727$	27.2997	1.01249	0.506245	+	0.862390i	$0.331033\pi$
$728$	13.3736	0.495659
$729$	0	0
$730$	19.4135	0.718526
$731$	−0.322920	−0.0119436
$732$	0	0
$733$	49.8400	1.84088	0.920442	−	0.390880i	$-0.127829\pi$
$733$	49.8400	1.84088	0.920442	+	0.390880i	$0.127829\pi$
$734$	−31.1801	−1.15088
$735$	0	0
$736$	41.5938	1.53317
$737$	35.5271	1.30866
$738$	0	0
$739$	−6.68506	−0.245914	−0.122957	−	0.992412i	$-0.539238\pi$
$739$	−6.68506	−0.245914	−0.122957	+	0.992412i	$0.539238\pi$
$740$	−105.967	−3.89544
$741$	0	0
$742$	33.0557	1.21351
$743$	−47.1237	−1.72880	−0.864401	−	0.502803i	$-0.832302\pi$
$743$	−47.1237	−1.72880	−0.864401	+	0.502803i	$0.832302\pi$
$744$	0	0
$745$	−21.2129	−0.777181
$746$	−2.18823	−0.0801166
$747$	0	0
$748$	−1.20257	−0.0439704
$749$	83.1700	3.03897
$750$	0	0
$751$	34.4091	1.25561	0.627804	−	0.778372i	$-0.283954\pi$
$751$	34.4091	1.25561	0.627804	+	0.778372i	$0.283954\pi$
$752$	2.88744	0.105294
$753$	0	0
$754$	34.4426	1.25433
$755$	−9.20590	−0.335037
$756$	0	0
$757$	−34.6326	−1.25875	−0.629373	−	0.777104i	$-0.716688\pi$
$757$	−34.6326	−1.25875	−0.629373	+	0.777104i	$0.716688\pi$
$758$	−5.06221	−0.183868
$759$	0	0
$760$	3.66613	0.132985
$761$	28.1681	1.02109	0.510546	−	0.859850i	$-0.329443\pi$
$761$	28.1681	1.02109	0.510546	+	0.859850i	$0.329443\pi$
$762$	0	0
$763$	−57.0636	−2.06584
$764$	−16.1803	−0.585381
$765$	0	0
$766$	46.3160	1.67347
$767$	17.7087	0.639425
$768$	0	0
$769$	30.4487	1.09801	0.549003	−	0.835820i	$-0.315007\pi$
$769$	30.4487	1.09801	0.549003	+	0.835820i	$0.315007\pi$
$770$	96.3880	3.47358
$771$	0	0
$772$	−52.6581	−1.89521
$773$	−50.6983	−1.82349	−0.911745	−	0.410756i	$-0.865265\pi$
$773$	−50.6983	−1.82349	−0.911745	+	0.410756i	$0.865265\pi$
$774$	0	0
$775$	33.8343	1.21536
$776$	9.41090	0.337832
$777$	0	0
$778$	42.3455	1.51816
$779$	0.266938	0.00956405
$780$	0	0
$781$	−31.4080	−1.12387
$782$	−1.94271	−0.0694711
$783$	0	0
$784$	−33.4424	−1.19437
$785$	30.0961	1.07418
$786$	0	0
$787$	24.2499	0.864415	0.432208	−	0.901774i	$-0.357735\pi$
$787$	24.2499	0.864415	0.432208	+	0.901774i	$0.357735\pi$
$788$	37.3513	1.33058
$789$	0	0
$790$	−18.0377	−0.641754
$791$	84.3053	2.99755
$792$	0	0
$793$	12.8235	0.455376
$794$	31.5989	1.12140
$795$	0	0
$796$	28.3437	1.00462
$797$	−2.28293	−0.0808655	−0.0404328	−	0.999182i	$-0.512874\pi$
$797$	−2.28293	−0.0808655	−0.0404328	+	0.999182i	$0.512874\pi$
$798$	0	0
$799$	−0.177222	−0.00626968
$800$	−77.4849	−2.73951
$801$	0	0
$802$	−35.4126	−1.25046
$803$	6.64129	0.234366
$804$	0	0
$805$	85.7830	3.02345
$806$	23.9214	0.842595
$807$	0	0
$808$	16.4169	0.577543
$809$	22.4154	0.788084	0.394042	−	0.919092i	$-0.371076\pi$
$809$	22.4154	0.788084	0.394042	+	0.919092i	$0.371076\pi$
$810$	0	0
$811$	−38.7341	−1.36014	−0.680069	−	0.733148i	$-0.738050\pi$
$811$	−38.7341	−1.36014	−0.680069	+	0.733148i	$0.738050\pi$
$812$	53.2433	1.86847
$813$	0	0
$814$	−65.8020	−2.30636
$815$	24.7640	0.867444
$816$	0	0
$817$	1.82212	0.0637478
$818$	29.0544	1.01586
$819$	0	0
$820$	2.50903	0.0876190
$821$	8.52123	0.297393	0.148696	−	0.988883i	$-0.452492\pi$
$821$	8.52123	0.297393	0.148696	+	0.988883i	$0.452492\pi$
$822$	0	0
$823$	−37.4367	−1.30496	−0.652480	−	0.757806i	$-0.726272\pi$
$823$	−37.4367	−1.30496	−0.652480	+	0.757806i	$0.726272\pi$
$824$	−2.07394	−0.0722490
$825$	0	0
$826$	49.6906	1.72896
$827$	−16.7763	−0.583368	−0.291684	−	0.956515i	$-0.594216\pi$
$827$	−16.7763	−0.583368	−0.291684	+	0.956515i	$0.594216\pi$
$828$	0	0
$829$	34.1990	1.18778	0.593891	−	0.804545i	$-0.297591\pi$
$829$	34.1990	1.18778	0.593891	+	0.804545i	$0.297591\pi$
$830$	0.353264	0.0122620
$831$	0	0
$832$	−36.0611	−1.25019
$833$	2.05260	0.0711183
$834$	0	0
$835$	−37.4645	−1.29651
$836$	6.78567	0.234687
$837$	0	0
$838$	82.0040	2.83278
$839$	−41.8013	−1.44314	−0.721570	−	0.692341i	$-0.756579\pi$
$839$	−41.8013	−1.44314	−0.721570	+	0.692341i	$0.756579\pi$
$840$	0	0
$841$	−3.65592	−0.126066
$842$	3.23379	0.111444
$843$	0	0
$844$	32.9382	1.13378
$845$	9.53788	0.328113
$846$	0	0
$847$	−14.4436	−0.496287
$848$	−10.4921	−0.360301
$849$	0	0
$850$	3.61906	0.124133
$851$	−58.5622	−2.00749
$852$	0	0
$853$	−12.4015	−0.424619	−0.212309	−	0.977202i	$-0.568098\pi$
$853$	−12.4015	−0.424619	−0.212309	+	0.977202i	$0.568098\pi$
$854$	35.9827	1.23130
$855$	0	0
$856$	18.4635	0.631068
$857$	19.9201	0.680459	0.340229	−	0.940343i	$-0.389495\pi$
$857$	19.9201	0.680459	0.340229	+	0.940343i	$0.389495\pi$
$858$	0	0
$859$	−40.7955	−1.39192	−0.695962	−	0.718078i	$-0.745022\pi$
$859$	−40.7955	−1.39192	−0.695962	+	0.718078i	$0.745022\pi$
$860$	17.1266	0.584012
$861$	0	0
$862$	−33.5630	−1.14316
$863$	53.6152	1.82508	0.912541	−	0.408985i	$-0.134117\pi$
$863$	53.6152	1.82508	0.912541	+	0.408985i	$0.134117\pi$
$864$	0	0
$865$	−35.4093	−1.20395
$866$	−72.9736	−2.47974
$867$	0	0
$868$	36.9790	1.25515
$869$	−6.17065	−0.209325
$870$	0	0
$871$	−41.6444	−1.41107
$872$	−12.6679	−0.428991
$873$	0	0
$874$	10.9620	0.370794
$875$	−77.2331	−2.61096
$876$	0	0
$877$	31.6183	1.06767	0.533837	−	0.845588i	$-0.320750\pi$
$877$	31.6183	1.06767	0.533837	+	0.845588i	$0.320750\pi$
$878$	44.0380	1.48621
$879$	0	0
$880$	−30.5942	−1.03133
$881$	−29.0370	−0.978281	−0.489141	−	0.872205i	$-0.662690\pi$
$881$	−29.0370	−0.978281	−0.489141	+	0.872205i	$0.662690\pi$
$882$	0	0
$883$	−29.1166	−0.979852	−0.489926	−	0.871764i	$-0.662976\pi$
$883$	−29.1166	−0.979852	−0.489926	+	0.871764i	$0.662976\pi$
$884$	1.40964	0.0474112
$885$	0	0
$886$	62.2532	2.09144
$887$	−9.67487	−0.324850	−0.162425	−	0.986721i	$-0.551932\pi$
$887$	−9.67487	−0.324850	−0.162425	+	0.986721i	$0.551932\pi$
$888$	0	0
$889$	63.4454	2.12789
$890$	134.584	4.51128
$891$	0	0
$892$	−16.7065	−0.559374
$893$	1.00000	0.0334637
$894$	0	0
$895$	−9.49776	−0.317475
$896$	−32.1530	−1.07416
$897$	0	0
$898$	−28.0693	−0.936683
$899$	17.6022	0.587066
$900$	0	0
$901$	0.643976	0.0214539
$902$	1.55802	0.0518763
$903$	0	0
$904$	18.7155	0.622468
$905$	2.43532	0.0809527
$906$	0	0
$907$	−42.4724	−1.41027	−0.705137	−	0.709071i	$-0.749115\pi$
$907$	−42.4724	−1.41027	−0.705137	+	0.709071i	$0.749115\pi$
$908$	−24.4716	−0.812117
$909$	0	0
$910$	−112.985	−3.74540
$911$	−11.0772	−0.367003	−0.183502	−	0.983019i	$-0.558743\pi$
$911$	−11.0772	−0.367003	−0.183502	+	0.983019i	$0.558743\pi$
$912$	0	0
$913$	0.120851	0.00399957
$914$	−41.1794	−1.36209
$915$	0	0
$916$	37.9268	1.25314
$917$	71.1312	2.34896
$918$	0	0
$919$	3.86290	0.127425	0.0637127	−	0.997968i	$-0.479706\pi$
$919$	3.86290	0.127425	0.0637127	+	0.997968i	$0.479706\pi$
$920$	19.0435	0.627847
$921$	0	0
$922$	−16.1084	−0.530502
$923$	36.8160	1.21181
$924$	0	0
$925$	109.095	3.58703
$926$	−3.32904	−0.109399
$927$	0	0
$928$	−40.3113	−1.32328
$929$	−59.9745	−1.96770	−0.983849	−	0.179000i	$-0.942714\pi$
$929$	−59.9745	−1.96770	−0.983849	+	0.179000i	$0.942714\pi$
$930$	0	0
$931$	−11.5820	−0.379586
$932$	−65.3762	−2.14147
$933$	0	0
$934$	−69.8028	−2.28402
$935$	1.87778	0.0614101
$936$	0	0
$937$	−45.1610	−1.47535	−0.737673	−	0.675159i	$-0.764075\pi$
$937$	−45.1610	−1.47535	−0.737673	+	0.675159i	$0.764075\pi$
$938$	−116.854	−3.81542
$939$	0	0
$940$	9.39928	0.306571
$941$	−35.9688	−1.17255	−0.586274	−	0.810113i	$-0.699406\pi$
$941$	−35.9688	−1.17255	−0.586274	+	0.810113i	$0.699406\pi$
$942$	0	0
$943$	1.38660	0.0451538
$944$	−15.7722	−0.513340
$945$	0	0
$946$	10.6350	0.345774
$947$	4.18872	0.136115	0.0680575	−	0.997681i	$-0.478320\pi$
$947$	4.18872	0.136115	0.0680575	+	0.997681i	$0.478320\pi$
$948$	0	0
$949$	−7.78482	−0.252706
$950$	−20.4210	−0.662546
$951$	0	0
$952$	0.731070	0.0236941
$953$	−21.5507	−0.698094	−0.349047	−	0.937105i	$-0.613495\pi$
$953$	−21.5507	−0.698094	−0.349047	+	0.937105i	$0.613495\pi$
$954$	0	0
$955$	25.2651	0.817558
$956$	46.7178	1.51096
$957$	0	0
$958$	63.0383	2.03667
$959$	31.4776	1.01646
$960$	0	0
$961$	−18.7748	−0.605637
$962$	77.1322	2.48684
$963$	0	0
$964$	−28.9218	−0.931508
$965$	82.2242	2.64689
$966$	0	0
$967$	1.02850	0.0330743	0.0165372	−	0.999863i	$-0.494736\pi$
$967$	1.02850	0.0330743	0.0165372	+	0.999863i	$0.494736\pi$
$968$	−3.20643	−0.103058
$969$	0	0
$970$	−79.5063	−2.55279
$971$	4.93837	0.158480	0.0792399	−	0.996856i	$-0.474751\pi$
$971$	4.93837	0.158480	0.0792399	+	0.996856i	$0.474751\pi$
$972$	0	0
$973$	38.9945	1.25011
$974$	−24.1720	−0.774522
$975$	0	0
$976$	−11.4212	−0.365583
$977$	−56.6315	−1.81180	−0.905901	−	0.423489i	$-0.860805\pi$
$977$	−56.6315	−1.81180	−0.905901	+	0.423489i	$0.860805\pi$
$978$	0	0
$979$	46.0409	1.47147
$980$	−108.863	−3.47750
$981$	0	0
$982$	11.7779	0.375847
$983$	4.35750	0.138983	0.0694914	−	0.997583i	$-0.477862\pi$
$983$	4.35750	0.138983	0.0694914	+	0.997583i	$0.477862\pi$
$984$	0	0
$985$	−58.3230	−1.85833
$986$	1.88281	0.0599608
$987$	0	0
$988$	−7.95406	−0.253052
$989$	9.46490	0.300966
$990$	0	0
$991$	−54.4825	−1.73069	−0.865347	−	0.501174i	$-0.832902\pi$
$991$	−54.4825	−1.73069	−0.865347	+	0.501174i	$0.832902\pi$
$992$	−27.9974	−0.888918
$993$	0	0
$994$	103.306	3.27665
$995$	−44.2580	−1.40307
$996$	0	0
$997$	34.4168	1.08999	0.544995	−	0.838439i	$-0.316532\pi$
$997$	34.4168	1.08999	0.544995	+	0.838439i	$0.316532\pi$
$998$	−26.0766	−0.825439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.q.1.20		23
3.2	odd	2		2679.2.a.m.1.4	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.m.1.4	✓	23	3.2	odd	2
8037.2.a.q.1.20		23	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)

\(f(q)\)	\(=\)	\(q+2.11032 q^{2} +2.45347 q^{4} -3.83102 q^{5} +4.31069 q^{7} +0.956959 q^{8} +O(q^{10})\)	\(q+2.11032 q^{2} +2.45347 q^{4} -3.83102 q^{5} +4.31069 q^{7} +0.956959 q^{8} -8.08470 q^{10} -2.76575 q^{11} +3.24197 q^{13} +9.09695 q^{14} -2.88744 q^{16} +0.177222 q^{17} -1.00000 q^{19} -9.39928 q^{20} -5.83662 q^{22} -5.19445 q^{23} +9.67673 q^{25} +6.84160 q^{26} +10.5761 q^{28} +5.03429 q^{29} +3.49646 q^{31} -8.00735 q^{32} +0.373997 q^{34} -16.5143 q^{35} +11.2740 q^{37} -2.11032 q^{38} -3.66613 q^{40} -0.266938 q^{41} -1.82212 q^{43} -6.78567 q^{44} -10.9620 q^{46} -1.00000 q^{47} +11.5820 q^{49} +20.4210 q^{50} +7.95406 q^{52} +3.63372 q^{53} +10.5956 q^{55} +4.12515 q^{56} +10.6240 q^{58} +5.46233 q^{59} +3.95547 q^{61} +7.37866 q^{62} -11.1232 q^{64} -12.4201 q^{65} -12.8454 q^{67} +0.434809 q^{68} -34.8506 q^{70} +11.3561 q^{71} -2.40126 q^{73} +23.7918 q^{74} -2.45347 q^{76} -11.9223 q^{77} +2.23110 q^{79} +11.0618 q^{80} -0.563326 q^{82} -0.0436954 q^{83} -0.678943 q^{85} -3.84526 q^{86} -2.64671 q^{88} -16.6468 q^{89} +13.9751 q^{91} -12.7444 q^{92} -2.11032 q^{94} +3.83102 q^{95} +9.83417 q^{97} +24.4419 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8	\( 23 q - 2 q^{2} + 30 q^{4} - 9 q^{5} + 5 q^{7} + 3 q^{8} - 5 q^{10} - 12 q^{11} + 9 q^{13} - 3 q^{14} + 60 q^{16} - 16 q^{17} - 23 q^{19} - 25 q^{20} + 3 q^{22} - 12 q^{23} + 54 q^{25} - 5 q^{26} + 8 q^{28} - 27 q^{29} + 10 q^{31} + 34 q^{32} + 6 q^{35} + 15 q^{37} + 2 q^{38} + 3 q^{40} - 10 q^{41} + 24 q^{43} - 39 q^{44} + 43 q^{46} - 23 q^{47} + 78 q^{49} + 32 q^{50} + 38 q^{52} + 2 q^{53} + 5 q^{55} - 58 q^{56} - 11 q^{58} + 51 q^{59} + 48 q^{61} + 22 q^{62} + 125 q^{64} - 15 q^{65} + 26 q^{67} - 26 q^{68} + 86 q^{70} - 24 q^{71} + 53 q^{73} - 26 q^{74} - 30 q^{76} - 18 q^{77} + 29 q^{79} - 5 q^{80} + 47 q^{82} + 22 q^{83} - 5 q^{85} + 28 q^{86} + 62 q^{88} - 38 q^{89} + 15 q^{91} - 15 q^{92} + 2 q^{94} + 9 q^{95} + 33 q^{97} + 30 q^{98}+O(q^{100}) \) 23 * q - 2 * q^2 + 30 * q^4 - 9 * q^5 + 5 * q^7 + 3 * q^8 - 5 * q^10 - 12 * q^11 + 9 * q^13 - 3 * q^14 + 60 * q^16 - 16 * q^17 - 23 * q^19 - 25 * q^20 + 3 * q^22 - 12 * q^23 + 54 * q^25 - 5 * q^26 + 8 * q^28 - 27 * q^29 + 10 * q^31 + 34 * q^32 + 6 * q^35 + 15 * q^37 + 2 * q^38 + 3 * q^40 - 10 * q^41 + 24 * q^43 - 39 * q^44 + 43 * q^46 - 23 * q^47 + 78 * q^49 + 32 * q^50 + 38 * q^52 + 2 * q^53 + 5 * q^55 - 58 * q^56 - 11 * q^58 + 51 * q^59 + 48 * q^61 + 22 * q^62 + 125 * q^64 - 15 * q^65 + 26 * q^67 - 26 * q^68 + 86 * q^70 - 24 * q^71 + 53 * q^73 - 26 * q^74 - 30 * q^76 - 18 * q^77 + 29 * q^79 - 5 * q^80 + 47 * q^82 + 22 * q^83 - 5 * q^85 + 28 * q^86 + 62 * q^88 - 38 * q^89 + 15 * q^91 - 15 * q^92 + 2 * q^94 + 9 * q^95 + 33 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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