LMFDB - Embedded newform 8037.2.a.k.1.2

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 9x^{5} + 14x^{4} + 23x^{3} - 19x^{2} - 14x + 5 $ x^7 - 2x^6 - 9x^5 + 14x^4 + 23x^3 - 19x^2 - 14x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.37483$ of defining polynomial
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)

$f(q)$	$=$	$q-1.37483 q^{2} -0.109849 q^{4} -0.874519 q^{5} -1.13950 q^{7} +2.90068 q^{8} +O(q^{10})$	\(q-1.37483 q^{2} -0.109849 q^{4} -0.874519 q^{5} -1.13950 q^{7} +2.90068 q^{8} +1.20231 q^{10} +0.0912702 q^{11} -6.53290 q^{13} +1.56661 q^{14} -3.76823 q^{16} -3.65033 q^{17} -1.00000 q^{19} +0.0960655 q^{20} -0.125481 q^{22} +5.00000 q^{23} -4.23522 q^{25} +8.98162 q^{26} +0.125173 q^{28} -2.53290 q^{29} -7.09603 q^{31} -0.620686 q^{32} +5.01858 q^{34} +0.996512 q^{35} -0.716135 q^{37} +1.37483 q^{38} -2.53670 q^{40} +3.80822 q^{41} -0.158190 q^{43} -0.0100260 q^{44} -6.87414 q^{46} -1.00000 q^{47} -5.70155 q^{49} +5.82269 q^{50} +0.717636 q^{52} -3.56909 q^{53} -0.0798176 q^{55} -3.30532 q^{56} +3.48231 q^{58} -4.62038 q^{59} +2.26467 q^{61} +9.75581 q^{62} +8.38980 q^{64} +5.71315 q^{65} -9.51421 q^{67} +0.400987 q^{68} -1.37003 q^{70} -4.94113 q^{71} -9.66573 q^{73} +0.984562 q^{74} +0.109849 q^{76} -0.104002 q^{77} -6.90971 q^{79} +3.29539 q^{80} -5.23564 q^{82} -7.33833 q^{83} +3.19229 q^{85} +0.217485 q^{86} +0.264746 q^{88} +7.76413 q^{89} +7.44423 q^{91} -0.549247 q^{92} +1.37483 q^{94} +0.874519 q^{95} +7.70638 q^{97} +7.83864 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8	$ 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8} - 2 q^{10} + 9 q^{11} - 2 q^{13} + 6 q^{14} + 14 q^{16} + 6 q^{17} - 7 q^{19} + 25 q^{20} - 13 q^{22} + 35 q^{23} + 3 q^{25} + 23 q^{26} - 9 q^{28} + 26 q^{29} - 11 q^{31} + 21 q^{32} + 18 q^{34} + 17 q^{35} - 3 q^{37} - 2 q^{38} + 14 q^{40} + 20 q^{41} - 5 q^{43} - 18 q^{44} + 10 q^{46} - 7 q^{47} - 10 q^{49} - 17 q^{50} + 15 q^{52} - 3 q^{53} - 3 q^{55} + 3 q^{56} + 31 q^{58} + 15 q^{59} - 9 q^{61} + 4 q^{62} + 4 q^{64} + 13 q^{65} - 4 q^{67} - 5 q^{68} + 18 q^{70} + 4 q^{71} - 8 q^{73} - 28 q^{74} - 8 q^{76} + 11 q^{77} - 11 q^{79} + 12 q^{80} - 19 q^{82} + 4 q^{83} - 6 q^{85} + 36 q^{86} - 7 q^{88} + 23 q^{89} + 19 q^{91} + 40 q^{92} - 2 q^{94} - 6 q^{95} - 7 q^{97}+O(q^{100}) $ 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8 - 2 * q^10 + 9 * q^11 - 2 * q^13 + 6 * q^14 + 14 * q^16 + 6 * q^17 - 7 * q^19 + 25 * q^20 - 13 * q^22 + 35 * q^23 + 3 * q^25 + 23 * q^26 - 9 * q^28 + 26 * q^29 - 11 * q^31 + 21 * q^32 + 18 * q^34 + 17 * q^35 - 3 * q^37 - 2 * q^38 + 14 * q^40 + 20 * q^41 - 5 * q^43 - 18 * q^44 + 10 * q^46 - 7 * q^47 - 10 * q^49 - 17 * q^50 + 15 * q^52 - 3 * q^53 - 3 * q^55 + 3 * q^56 + 31 * q^58 + 15 * q^59 - 9 * q^61 + 4 * q^62 + 4 * q^64 + 13 * q^65 - 4 * q^67 - 5 * q^68 + 18 * q^70 + 4 * q^71 - 8 * q^73 - 28 * q^74 - 8 * q^76 + 11 * q^77 - 11 * q^79 + 12 * q^80 - 19 * q^82 + 4 * q^83 - 6 * q^85 + 36 * q^86 - 7 * q^88 + 23 * q^89 + 19 * q^91 + 40 * q^92 - 2 * q^94 - 6 * q^95 - 7 * q^97

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 $

$2$	−1.37483	−0.972150	−0.486075	−	0.873917i	$-0.661572\pi$
$2$	−1.37483	−0.972150	−0.486075	+	0.873917i	$0.661572\pi$
$3$	0	0
$3$	0	0
$4$	−0.109849	−0.0549247
$5$	−0.874519	−0.391097	−0.195548	−	0.980694i	$-0.562649\pi$
$5$	−0.874519	−0.391097	−0.195548	+	0.980694i	$0.562649\pi$
$6$	0	0
$7$	−1.13950	−0.430689	−0.215345	−	0.976538i	$-0.569088\pi$
$7$	−1.13950	−0.430689	−0.215345	+	0.976538i	$0.569088\pi$
$8$	2.90068	1.02554
$9$	0	0
$10$	1.20231	0.380205
$11$	0.0912702	0.0275190	0.0137595	−	0.999905i	$-0.495620\pi$
$11$	0.0912702	0.0275190	0.0137595	+	0.999905i	$0.495620\pi$
$12$	0	0
$13$	−6.53290	−1.81190	−0.905951	−	0.423383i	$-0.860842\pi$
$13$	−6.53290	−1.81190	−0.905951	+	0.423383i	$0.860842\pi$
$14$	1.56661	0.418695
$15$	0	0
$16$	−3.76823	−0.942059
$17$	−3.65033	−0.885336	−0.442668	−	0.896686i	$-0.645968\pi$
$17$	−3.65033	−0.885336	−0.442668	+	0.896686i	$0.645968\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.0960655	0.0214809
$21$	0	0
$22$	−0.125481	−0.0267526
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	−4.23522	−0.847043
$26$	8.98162	1.76144
$27$	0	0
$28$	0.125173	0.0236555
$29$	−2.53290	−0.470348	−0.235174	−	0.971953i	$-0.575566\pi$
$29$	−2.53290	−0.470348	−0.235174	+	0.971953i	$0.575566\pi$
$30$	0	0
$31$	−7.09603	−1.27448	−0.637242	−	0.770664i	$-0.719925\pi$
$31$	−7.09603	−1.27448	−0.637242	+	0.770664i	$0.719925\pi$
$32$	−0.620686	−0.109723
$33$	0	0
$34$	5.01858	0.860679
$35$	0.996512	0.168441
$36$	0	0
$37$	−0.716135	−0.117732	−0.0588659	−	0.998266i	$-0.518748\pi$
$37$	−0.716135	−0.117732	−0.0588659	+	0.998266i	$0.518748\pi$
$38$	1.37483	0.223026
$39$	0	0
$40$	−2.53670	−0.401087
$41$	3.80822	0.594743	0.297372	−	0.954762i	$-0.403890\pi$
$41$	3.80822	0.594743	0.297372	+	0.954762i	$0.403890\pi$
$42$	0	0
$43$	−0.158190	−0.0241238	−0.0120619	−	0.999927i	$-0.503840\pi$
$43$	−0.158190	−0.0241238	−0.0120619	+	0.999927i	$0.503840\pi$
$44$	−0.0100260	−0.00151147
$45$	0	0
$46$	−6.87414	−1.01354
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.70155	−0.814507
$50$	5.82269	0.823453
$51$	0	0
$52$	0.717636	0.0995182
$53$	−3.56909	−0.490252	−0.245126	−	0.969491i	$-0.578829\pi$
$53$	−3.56909	−0.490252	−0.245126	+	0.969491i	$0.578829\pi$
$54$	0	0
$55$	−0.0798176	−0.0107626
$56$	−3.30532	−0.441691
$57$	0	0
$58$	3.48231	0.457249
$59$	−4.62038	−0.601522	−0.300761	−	0.953700i	$-0.597241\pi$
$59$	−4.62038	−0.601522	−0.300761	+	0.953700i	$0.597241\pi$
$60$	0	0
$61$	2.26467	0.289961	0.144981	−	0.989434i	$-0.453688\pi$
$61$	2.26467	0.289961	0.144981	+	0.989434i	$0.453688\pi$
$62$	9.75581	1.23899
$63$	0	0
$64$	8.38980	1.04873
$65$	5.71315	0.708629
$66$	0	0
$67$	−9.51421	−1.16235	−0.581173	−	0.813780i	$-0.697406\pi$
$67$	−9.51421	−1.16235	−0.581173	+	0.813780i	$0.697406\pi$
$68$	0.400987	0.0486268
$69$	0	0
$70$	−1.37003	−0.163750
$71$	−4.94113	−0.586404	−0.293202	−	0.956050i	$-0.594721\pi$
$71$	−4.94113	−0.586404	−0.293202	+	0.956050i	$0.594721\pi$
$72$	0	0
$73$	−9.66573	−1.13129	−0.565644	−	0.824649i	$-0.691372\pi$
$73$	−9.66573	−1.13129	−0.565644	+	0.824649i	$0.691372\pi$
$74$	0.984562	0.114453
$75$	0	0
$76$	0.109849	0.0126006
$77$	−0.104002	−0.0118521
$78$	0	0
$79$	−6.90971	−0.777403	−0.388701	−	0.921364i	$-0.627076\pi$
$79$	−6.90971	−0.777403	−0.388701	+	0.921364i	$0.627076\pi$
$80$	3.29539	0.368436
$81$	0	0
$82$	−5.23564	−0.578180
$83$	−7.33833	−0.805487	−0.402743	−	0.915313i	$-0.631943\pi$
$83$	−7.33833	−0.805487	−0.402743	+	0.915313i	$0.631943\pi$
$84$	0	0
$85$	3.19229	0.346252
$86$	0.217485	0.0234520
$87$	0	0
$88$	0.264746	0.0282220
$89$	7.76413	0.822996	0.411498	−	0.911411i	$-0.365006\pi$
$89$	7.76413	0.822996	0.411498	+	0.911411i	$0.365006\pi$
$90$	0	0
$91$	7.44423	0.780367
$92$	−0.549247	−0.0572630
$93$	0	0
$94$	1.37483	0.141803
$95$	0.874519	0.0897238
$96$	0	0
$97$	7.70638	0.782464	0.391232	−	0.920292i	$-0.372049\pi$
$97$	7.70638	0.782464	0.391232	+	0.920292i	$0.372049\pi$
$98$	7.83864	0.791822
$99$	0	0
$100$	0.465236	0.0465236
$101$	4.92979	0.490532	0.245266	−	0.969456i	$-0.421125\pi$
$101$	4.92979	0.490532	0.245266	+	0.969456i	$0.421125\pi$
$102$	0	0
$103$	6.95526	0.685322	0.342661	−	0.939459i	$-0.388672\pi$
$103$	6.95526	0.685322	0.342661	+	0.939459i	$0.388672\pi$
$104$	−18.9499	−1.85819
$105$	0	0
$106$	4.90688	0.476599
$107$	5.90192	0.570560	0.285280	−	0.958444i	$-0.407913\pi$
$107$	5.90192	0.570560	0.285280	+	0.958444i	$0.407913\pi$
$108$	0	0
$109$	−3.04780	−0.291926	−0.145963	−	0.989290i	$-0.546628\pi$
$109$	−3.04780	−0.291926	−0.145963	+	0.989290i	$0.546628\pi$
$110$	0.109735	0.0104629
$111$	0	0
$112$	4.29389	0.405735
$113$	8.51692	0.801204	0.400602	−	0.916252i	$-0.368801\pi$
$113$	8.51692	0.801204	0.400602	+	0.916252i	$0.368801\pi$
$114$	0	0
$115$	−4.37260	−0.407747
$116$	0.278238	0.0258338
$117$	0	0
$118$	6.35222	0.584770
$119$	4.15955	0.381305
$120$	0	0
$121$	−10.9917	−0.999243
$122$	−3.11353	−0.281886
$123$	0	0
$124$	0.779495	0.0700007
$125$	8.07637	0.722373
$126$	0	0
$127$	15.8038	1.40236	0.701182	−	0.712982i	$-0.252656\pi$
$127$	15.8038	1.40236	0.701182	+	0.712982i	$0.252656\pi$
$128$	−10.2932	−0.909796
$129$	0	0
$130$	−7.85460	−0.688894
$131$	−3.50329	−0.306084	−0.153042	−	0.988220i	$-0.548907\pi$
$131$	−3.50329	−0.306084	−0.153042	+	0.988220i	$0.548907\pi$
$132$	0	0
$133$	1.13950	0.0988069
$134$	13.0804	1.12997
$135$	0	0
$136$	−10.5884	−0.907952
$137$	−16.3660	−1.39825	−0.699123	−	0.715002i	$-0.746426\pi$
$137$	−16.3660	−1.39825	−0.699123	+	0.715002i	$0.746426\pi$
$138$	0	0
$139$	−10.0260	−0.850398	−0.425199	−	0.905100i	$-0.639796\pi$
$139$	−10.0260	−0.850398	−0.425199	+	0.905100i	$0.639796\pi$
$140$	−0.109466	−0.00925159
$141$	0	0
$142$	6.79320	0.570073
$143$	−0.596260	−0.0498617
$144$	0	0
$145$	2.21507	0.183952
$146$	13.2887	1.09978
$147$	0	0
$148$	0.0786671	0.00646639
$149$	−6.76049	−0.553841	−0.276921	−	0.960893i	$-0.589314\pi$
$149$	−6.76049	−0.553841	−0.276921	+	0.960893i	$0.589314\pi$
$150$	0	0
$151$	14.5178	1.18144	0.590721	−	0.806876i	$-0.298843\pi$
$151$	14.5178	1.18144	0.590721	+	0.806876i	$0.298843\pi$
$152$	−2.90068	−0.235276
$153$	0	0
$154$	0.142985	0.0115221
$155$	6.20561	0.498447
$156$	0	0
$157$	11.2657	0.899099	0.449550	−	0.893255i	$-0.351584\pi$
$157$	11.2657	0.899099	0.449550	+	0.893255i	$0.351584\pi$
$158$	9.49965	0.755752
$159$	0	0
$160$	0.542802	0.0429123
$161$	−5.69749	−0.449025
$162$	0	0
$163$	−20.0624	−1.57141	−0.785706	−	0.618600i	$-0.787700\pi$
$163$	−20.0624	−1.57141	−0.785706	+	0.618600i	$0.787700\pi$
$164$	−0.418330	−0.0326661
$165$	0	0
$166$	10.0889	0.783054
$167$	−11.8538	−0.917276	−0.458638	−	0.888623i	$-0.651663\pi$
$167$	−11.8538	−0.917276	−0.458638	+	0.888623i	$0.651663\pi$
$168$	0	0
$169$	29.6788	2.28299
$170$	−4.38884	−0.336609
$171$	0	0
$172$	0.0173771	0.00132499
$173$	7.00736	0.532760	0.266380	−	0.963868i	$-0.414172\pi$
$173$	7.00736	0.532760	0.266380	+	0.963868i	$0.414172\pi$
$174$	0	0
$175$	4.82602	0.364813
$176$	−0.343928	−0.0259245
$177$	0	0
$178$	−10.6743	−0.800076
$179$	−9.52206	−0.711712	−0.355856	−	0.934541i	$-0.615811\pi$
$179$	−9.52206	−0.711712	−0.355856	+	0.934541i	$0.615811\pi$
$180$	0	0
$181$	−5.51796	−0.410147	−0.205073	−	0.978747i	$-0.565743\pi$
$181$	−5.51796	−0.410147	−0.205073	+	0.978747i	$0.565743\pi$
$182$	−10.2345	−0.758633
$183$	0	0
$184$	14.5034	1.06920
$185$	0.626274	0.0460446
$186$	0	0
$187$	−0.333167	−0.0243636
$188$	0.109849	0.00801160
$189$	0	0
$190$	−1.20231	−0.0872250
$191$	−9.02295	−0.652878	−0.326439	−	0.945218i	$-0.605849\pi$
$191$	−9.02295	−0.652878	−0.326439	+	0.945218i	$0.605849\pi$
$192$	0	0
$193$	−26.2903	−1.89242	−0.946208	−	0.323559i	$-0.895120\pi$
$193$	−26.2903	−1.89242	−0.946208	+	0.323559i	$0.895120\pi$
$194$	−10.5949	−0.760672
$195$	0	0
$196$	0.626312	0.0447366
$197$	16.0277	1.14193	0.570963	−	0.820976i	$-0.306570\pi$
$197$	16.0277	1.14193	0.570963	+	0.820976i	$0.306570\pi$
$198$	0	0
$199$	1.74397	0.123627	0.0618135	−	0.998088i	$-0.480312\pi$
$199$	1.74397	0.123627	0.0618135	+	0.998088i	$0.480312\pi$
$200$	−12.2850	−0.868681
$201$	0	0
$202$	−6.77761	−0.476871
$203$	2.88624	0.202574
$204$	0	0
$205$	−3.33036	−0.232602
$206$	−9.56229	−0.666236
$207$	0	0
$208$	24.6175	1.70692
$209$	−0.0912702	−0.00631329
$210$	0	0
$211$	−7.74969	−0.533510	−0.266755	−	0.963764i	$-0.585952\pi$
$211$	−7.74969	−0.533510	−0.266755	+	0.963764i	$0.585952\pi$
$212$	0.392063	0.0269270
$213$	0	0
$214$	−8.11412	−0.554670
$215$	0.138341	0.00943475
$216$	0	0
$217$	8.08590	0.548907
$218$	4.19020	0.283796
$219$	0	0
$220$	0.00876792	0.000591133 0
$221$	23.8473	1.60414
$222$	0	0
$223$	−1.83532	−0.122902	−0.0614511	−	0.998110i	$-0.519573\pi$
$223$	−1.83532	−0.122902	−0.0614511	+	0.998110i	$0.519573\pi$
$224$	0.707270	0.0472565
$225$	0	0
$226$	−11.7093	−0.778890
$227$	10.9472	0.726592	0.363296	−	0.931674i	$-0.381651\pi$
$227$	10.9472	0.726592	0.363296	+	0.931674i	$0.381651\pi$
$228$	0	0
$229$	3.59332	0.237453	0.118727	−	0.992927i	$-0.462119\pi$
$229$	3.59332	0.237453	0.118727	+	0.992927i	$0.462119\pi$
$230$	6.01156	0.396391
$231$	0	0
$232$	−7.34714	−0.482363
$233$	−2.90957	−0.190613	−0.0953063	−	0.995448i	$-0.530383\pi$
$233$	−2.90957	−0.190613	−0.0953063	+	0.995448i	$0.530383\pi$
$234$	0	0
$235$	0.874519	0.0570473
$236$	0.507546	0.0330384
$237$	0	0
$238$	−5.71866	−0.370685
$239$	−11.0730	−0.716254	−0.358127	−	0.933673i	$-0.616584\pi$
$239$	−11.0730	−0.716254	−0.358127	+	0.933673i	$0.616584\pi$
$240$	0	0
$241$	18.3651	1.18300	0.591500	−	0.806305i	$-0.298536\pi$
$241$	18.3651	1.18300	0.591500	+	0.806305i	$0.298536\pi$
$242$	15.1116	0.971414
$243$	0	0
$244$	−0.248773	−0.0159260
$245$	4.98611	0.318551
$246$	0	0
$247$	6.53290	0.415679
$248$	−20.5833	−1.30704
$249$	0	0
$250$	−11.1036	−0.702255
$251$	21.2669	1.34236	0.671179	−	0.741295i	$-0.265788\pi$
$251$	21.2669	1.34236	0.671179	+	0.741295i	$0.265788\pi$
$252$	0	0
$253$	0.456351	0.0286906
$254$	−21.7276	−1.36331
$255$	0	0
$256$	−2.62829	−0.164268
$257$	−9.37955	−0.585080	−0.292540	−	0.956253i	$-0.594501\pi$
$257$	−9.37955	−0.585080	−0.292540	+	0.956253i	$0.594501\pi$
$258$	0	0
$259$	0.816034	0.0507059
$260$	−0.627586	−0.0389213
$261$	0	0
$262$	4.81642	0.297559
$263$	−8.01358	−0.494139	−0.247069	−	0.968998i	$-0.579468\pi$
$263$	−8.01358	−0.494139	−0.247069	+	0.968998i	$0.579468\pi$
$264$	0	0
$265$	3.12124	0.191736
$266$	−1.56661	−0.0960551
$267$	0	0
$268$	1.04513	0.0638415
$269$	−17.0152	−1.03744	−0.518718	−	0.854946i	$-0.673590\pi$
$269$	−17.0152	−1.03744	−0.518718	+	0.854946i	$0.673590\pi$
$270$	0	0
$271$	−16.2102	−0.984698	−0.492349	−	0.870398i	$-0.663862\pi$
$271$	−16.2102	−0.984698	−0.492349	+	0.870398i	$0.663862\pi$
$272$	13.7553	0.834038
$273$	0	0
$274$	22.5005	1.35930
$275$	−0.386549	−0.0233098
$276$	0	0
$277$	9.59580	0.576556	0.288278	−	0.957547i	$-0.406917\pi$
$277$	9.59580	0.576556	0.288278	+	0.957547i	$0.406917\pi$
$278$	13.7841	0.826714
$279$	0	0
$280$	2.89056	0.172744
$281$	−20.2739	−1.20944	−0.604721	−	0.796438i	$-0.706715\pi$
$281$	−20.2739	−1.20944	−0.604721	+	0.796438i	$0.706715\pi$
$282$	0	0
$283$	−17.0131	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$283$	−17.0131	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$284$	0.542781	0.0322081
$285$	0	0
$286$	0.819754	0.0484731
$287$	−4.33945	−0.256150
$288$	0	0
$289$	−3.67506	−0.216180
$290$	−3.04534	−0.178829
$291$	0	0
$292$	1.06178	0.0621357
$293$	6.19040	0.361647	0.180824	−	0.983516i	$-0.442124\pi$
$293$	6.19040	0.361647	0.180824	+	0.983516i	$0.442124\pi$
$294$	0	0
$295$	4.04061	0.235253
$296$	−2.07728	−0.120739
$297$	0	0
$298$	9.29451	0.538417
$299$	−32.6645	−1.88904
$300$	0	0
$301$	0.180258	0.0103899
$302$	−19.9595	−1.14854
$303$	0	0
$304$	3.76823	0.216123
$305$	−1.98050	−0.113403
$306$	0	0
$307$	11.7299	0.669459	0.334730	−	0.942314i	$-0.391355\pi$
$307$	11.7299	0.669459	0.334730	+	0.942314i	$0.391355\pi$
$308$	0.0114246	0.000650976 0
$309$	0	0
$310$	−8.53165	−0.484565
$311$	−22.2285	−1.26046	−0.630232	−	0.776407i	$-0.717040\pi$
$311$	−22.2285	−1.26046	−0.630232	+	0.776407i	$0.717040\pi$
$312$	0	0
$313$	−25.0955	−1.41848	−0.709240	−	0.704967i	$-0.750962\pi$
$313$	−25.0955	−1.41848	−0.709240	+	0.704967i	$0.750962\pi$
$314$	−15.4884	−0.874059
$315$	0	0
$316$	0.759028	0.0426986
$317$	−16.5653	−0.930401	−0.465201	−	0.885205i	$-0.654018\pi$
$317$	−16.5653	−0.930401	−0.465201	+	0.885205i	$0.654018\pi$
$318$	0	0
$319$	−0.231179	−0.0129435
$320$	−7.33705	−0.410153
$321$	0	0
$322$	7.83306	0.436519
$323$	3.65033	0.203110
$324$	0	0
$325$	27.6683	1.53476
$326$	27.5824	1.52765
$327$	0	0
$328$	11.0464	0.609936
$329$	1.13950	0.0628225
$330$	0	0
$331$	−21.0978	−1.15964	−0.579820	−	0.814745i	$-0.696877\pi$
$331$	−21.0978	−1.15964	−0.579820	+	0.814745i	$0.696877\pi$
$332$	0.806112	0.0442412
$333$	0	0
$334$	16.2970	0.891730
$335$	8.32036	0.454590
$336$	0	0
$337$	−15.3751	−0.837535	−0.418767	−	0.908094i	$-0.637538\pi$
$337$	−15.3751	−0.837535	−0.418767	+	0.908094i	$0.637538\pi$
$338$	−40.8033	−2.21941
$339$	0	0
$340$	−0.350671	−0.0190178
$341$	−0.647656	−0.0350725
$342$	0	0
$343$	14.4734	0.781489
$344$	−0.458860	−0.0247401
$345$	0	0
$346$	−9.63391	−0.517922
$347$	23.3727	1.25471	0.627356	−	0.778732i	$-0.284137\pi$
$347$	23.3727	1.25471	0.627356	+	0.778732i	$0.284137\pi$
$348$	0	0
$349$	6.86957	0.367720	0.183860	−	0.982952i	$-0.441141\pi$
$349$	6.86957	0.367720	0.183860	+	0.982952i	$0.441141\pi$
$350$	−6.63494	−0.354652
$351$	0	0
$352$	−0.0566502	−0.00301946
$353$	28.1629	1.49896	0.749481	−	0.662026i	$-0.230303\pi$
$353$	28.1629	1.49896	0.749481	+	0.662026i	$0.230303\pi$
$354$	0	0
$355$	4.32111	0.229341
$356$	−0.852886	−0.0452028
$357$	0	0
$358$	13.0912	0.691891
$359$	18.7264	0.988343	0.494172	−	0.869364i	$-0.335471\pi$
$359$	18.7264	0.988343	0.494172	+	0.869364i	$0.335471\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.58625	0.398724
$363$	0	0
$364$	−0.817744	−0.0428614
$365$	8.45287	0.442443
$366$	0	0
$367$	3.27387	0.170895	0.0854474	−	0.996343i	$-0.472768\pi$
$367$	3.27387	0.170895	0.0854474	+	0.996343i	$0.472768\pi$
$368$	−18.8412	−0.982164
$369$	0	0
$370$	−0.861019	−0.0447622
$371$	4.06697	0.211146
$372$	0	0
$373$	20.8508	1.07961	0.539807	−	0.841789i	$-0.318497\pi$
$373$	20.8508	1.07961	0.539807	+	0.841789i	$0.318497\pi$
$374$	0.458047	0.0236850
$375$	0	0
$376$	−2.90068	−0.149591
$377$	16.5472	0.852225
$378$	0	0
$379$	10.2520	0.526611	0.263305	−	0.964713i	$-0.415187\pi$
$379$	10.2520	0.526611	0.263305	+	0.964713i	$0.415187\pi$
$380$	−0.0960655	−0.00492805
$381$	0	0
$382$	12.4050	0.634695
$383$	−27.2809	−1.39399	−0.696995	−	0.717076i	$-0.745480\pi$
$383$	−27.2809	−1.39399	−0.696995	+	0.717076i	$0.745480\pi$
$384$	0	0
$385$	0.0909519	0.00463534
$386$	36.1446	1.83971
$387$	0	0
$388$	−0.846542	−0.0429766
$389$	28.2763	1.43366	0.716832	−	0.697245i	$-0.245591\pi$
$389$	28.2763	1.43366	0.716832	+	0.697245i	$0.245591\pi$
$390$	0	0
$391$	−18.2517	−0.923027
$392$	−16.5384	−0.835313
$393$	0	0
$394$	−22.0353	−1.11012
$395$	6.04267	0.304040
$396$	0	0
$397$	31.5348	1.58269	0.791344	−	0.611372i	$-0.209382\pi$
$397$	31.5348	1.58269	0.791344	+	0.611372i	$0.209382\pi$
$398$	−2.39766	−0.120184
$399$	0	0
$400$	15.9593	0.797964
$401$	19.3220	0.964893	0.482446	−	0.875926i	$-0.339748\pi$
$401$	19.3220	0.964893	0.482446	+	0.875926i	$0.339748\pi$
$402$	0	0
$403$	46.3577	2.30924
$404$	−0.541534	−0.0269423
$405$	0	0
$406$	−3.96808	−0.196932
$407$	−0.0653618	−0.00323987
$408$	0	0
$409$	−7.98064	−0.394617	−0.197309	−	0.980341i	$-0.563220\pi$
$409$	−7.98064	−0.394617	−0.197309	+	0.980341i	$0.563220\pi$
$410$	4.57867	0.226124
$411$	0	0
$412$	−0.764032	−0.0376411
$413$	5.26491	0.259069
$414$	0	0
$415$	6.41751	0.315023
$416$	4.05488	0.198807
$417$	0	0
$418$	0.125481	0.00613747
$419$	30.9606	1.51252	0.756261	−	0.654270i	$-0.227024\pi$
$419$	30.9606	1.51252	0.756261	+	0.654270i	$0.227024\pi$
$420$	0	0
$421$	−8.41051	−0.409903	−0.204952	−	0.978772i	$-0.565704\pi$
$421$	−8.41051	−0.409903	−0.204952	+	0.978772i	$0.565704\pi$
$422$	10.6545	0.518652
$423$	0	0
$424$	−10.3528	−0.502776
$425$	15.4600	0.749918
$426$	0	0
$427$	−2.58059	−0.124883
$428$	−0.648322	−0.0313378
$429$	0	0
$430$	−0.190194	−0.00917199
$431$	−17.2893	−0.832796	−0.416398	−	0.909182i	$-0.636708\pi$
$431$	−17.2893	−0.832796	−0.416398	+	0.909182i	$0.636708\pi$
$432$	0	0
$433$	33.6876	1.61892	0.809462	−	0.587172i	$-0.199759\pi$
$433$	33.6876	1.61892	0.809462	+	0.587172i	$0.199759\pi$
$434$	−11.1167	−0.533620
$435$	0	0
$436$	0.334799	0.0160340
$437$	−5.00000	−0.239182
$438$	0	0
$439$	1.20928	0.0577156	0.0288578	−	0.999584i	$-0.490813\pi$
$439$	1.20928	0.0577156	0.0288578	+	0.999584i	$0.490813\pi$
$440$	−0.231525	−0.0110375
$441$	0	0
$442$	−32.7859	−1.55947
$443$	−29.4943	−1.40132	−0.700659	−	0.713496i	$-0.747111\pi$
$443$	−29.4943	−1.40132	−0.700659	+	0.713496i	$0.747111\pi$
$444$	0	0
$445$	−6.78988	−0.321871
$446$	2.52325	0.119479
$447$	0	0
$448$	−9.56016	−0.451675
$449$	27.1431	1.28096	0.640480	−	0.767975i	$-0.278735\pi$
$449$	27.1431	1.28096	0.640480	+	0.767975i	$0.278735\pi$
$450$	0	0
$451$	0.347577	0.0163667
$452$	−0.935579	−0.0440059
$453$	0	0
$454$	−15.0505	−0.706356
$455$	−6.51012	−0.305199
$456$	0	0
$457$	19.4614	0.910367	0.455184	−	0.890398i	$-0.349574\pi$
$457$	19.4614	0.910367	0.455184	+	0.890398i	$0.349574\pi$
$458$	−4.94019	−0.230840
$459$	0	0
$460$	0.480327	0.0223954
$461$	34.0310	1.58498	0.792492	−	0.609883i	$-0.208783\pi$
$461$	34.0310	1.58498	0.792492	+	0.609883i	$0.208783\pi$
$462$	0	0
$463$	12.9591	0.602259	0.301129	−	0.953583i	$-0.402636\pi$
$463$	12.9591	0.602259	0.301129	+	0.953583i	$0.402636\pi$
$464$	9.54457	0.443096
$465$	0	0
$466$	4.00016	0.185304
$467$	21.4855	0.994229	0.497115	−	0.867685i	$-0.334393\pi$
$467$	21.4855	0.994229	0.497115	+	0.867685i	$0.334393\pi$
$468$	0	0
$469$	10.8414	0.500610
$470$	−1.20231	−0.0554586
$471$	0	0
$472$	−13.4022	−0.616888
$473$	−0.0144381	−0.000663864 0
$474$	0	0
$475$	4.23522	0.194325
$476$	−0.456924	−0.0209431
$477$	0	0
$478$	15.2235	0.696306
$479$	1.13356	0.0517939	0.0258969	−	0.999665i	$-0.491756\pi$
$479$	1.13356	0.0517939	0.0258969	+	0.999665i	$0.491756\pi$
$480$	0	0
$481$	4.67844	0.213319
$482$	−25.2489	−1.15005
$483$	0	0
$484$	1.20743	0.0548831
$485$	−6.73938	−0.306019
$486$	0	0
$487$	−10.5526	−0.478182	−0.239091	−	0.970997i	$-0.576849\pi$
$487$	−10.5526	−0.478182	−0.239091	+	0.970997i	$0.576849\pi$
$488$	6.56908	0.297368
$489$	0	0
$490$	−6.85504	−0.309679
$491$	14.6811	0.662547	0.331273	−	0.943535i	$-0.392522\pi$
$491$	14.6811	0.662547	0.331273	+	0.943535i	$0.392522\pi$
$492$	0	0
$493$	9.24594	0.416416
$494$	−8.98162	−0.404102
$495$	0	0
$496$	26.7395	1.20064
$497$	5.63041	0.252558
$498$	0	0
$499$	27.8807	1.24811	0.624055	−	0.781381i	$-0.285484\pi$
$499$	27.8807	1.24811	0.624055	+	0.781381i	$0.285484\pi$
$500$	−0.887185	−0.0396761
$501$	0	0
$502$	−29.2384	−1.30497
$503$	−2.12761	−0.0948656	−0.0474328	−	0.998874i	$-0.515104\pi$
$503$	−2.12761	−0.0948656	−0.0474328	+	0.998874i	$0.515104\pi$
$504$	0	0
$505$	−4.31119	−0.191846
$506$	−0.627404	−0.0278915
$507$	0	0
$508$	−1.73604	−0.0770245
$509$	−24.0746	−1.06709	−0.533543	−	0.845773i	$-0.679140\pi$
$509$	−24.0746	−1.06709	−0.533543	+	0.845773i	$0.679140\pi$
$510$	0	0
$511$	11.0141	0.487234
$512$	24.1998	1.06949
$513$	0	0
$514$	12.8953	0.568786
$515$	−6.08251	−0.268027
$516$	0	0
$517$	−0.0912702	−0.00401406
$518$	−1.12191	−0.0492937
$519$	0	0
$520$	16.5720	0.726731
$521$	14.5657	0.638136	0.319068	−	0.947732i	$-0.396630\pi$
$521$	14.5657	0.638136	0.319068	+	0.947732i	$0.396630\pi$
$522$	0	0
$523$	28.1966	1.23295	0.616476	−	0.787374i	$-0.288560\pi$
$523$	28.1966	1.23295	0.616476	+	0.787374i	$0.288560\pi$
$524$	0.384834	0.0168116
$525$	0	0
$526$	11.0173	0.480377
$527$	25.9029	1.12835
$528$	0	0
$529$	2.00000	0.0869565
$530$	−4.29116	−0.186396
$531$	0	0
$532$	−0.125173	−0.00542694
$533$	−24.8787	−1.07762
$534$	0	0
$535$	−5.16134	−0.223144
$536$	−27.5977	−1.19204
$537$	0	0
$538$	23.3930	1.00854
$539$	−0.520381	−0.0224144
$540$	0	0
$541$	−2.42473	−0.104247	−0.0521236	−	0.998641i	$-0.516599\pi$
$541$	−2.42473	−0.104247	−0.0521236	+	0.998641i	$0.516599\pi$
$542$	22.2862	0.957274
$543$	0	0
$544$	2.26571	0.0971416
$545$	2.66536	0.114171
$546$	0	0
$547$	36.4468	1.55835	0.779176	−	0.626806i	$-0.215638\pi$
$547$	36.4468	1.55835	0.779176	+	0.626806i	$0.215638\pi$
$548$	1.79780	0.0767983
$549$	0	0
$550$	0.531438	0.0226606
$551$	2.53290	0.107905
$552$	0	0
$553$	7.87359	0.334819
$554$	−13.1926	−0.560498
$555$	0	0
$556$	1.10136	0.0467079
$557$	−30.4316	−1.28943	−0.644715	−	0.764423i	$-0.723024\pi$
$557$	−30.4316	−1.28943	−0.644715	+	0.764423i	$0.723024\pi$
$558$	0	0
$559$	1.03344	0.0437100
$560$	−3.75509	−0.158682
$561$	0	0
$562$	27.8732	1.17576
$563$	−8.38566	−0.353413	−0.176707	−	0.984264i	$-0.556544\pi$
$563$	−8.38566	−0.353413	−0.176707	+	0.984264i	$0.556544\pi$
$564$	0	0
$565$	−7.44821	−0.313348
$566$	23.3901	0.983160
$567$	0	0
$568$	−14.3326	−0.601384
$569$	37.3668	1.56650	0.783248	−	0.621710i	$-0.213562\pi$
$569$	37.3668	1.56650	0.783248	+	0.621710i	$0.213562\pi$
$570$	0	0
$571$	−28.6183	−1.19764	−0.598819	−	0.800884i	$-0.704363\pi$
$571$	−28.6183	−1.19764	−0.598819	+	0.800884i	$0.704363\pi$
$572$	0.0654988	0.00273864
$573$	0	0
$574$	5.96600	0.249016
$575$	−21.1761	−0.883104
$576$	0	0
$577$	0.345456	0.0143815	0.00719076	−	0.999974i	$-0.497711\pi$
$577$	0.345456	0.0143815	0.00719076	+	0.999974i	$0.497711\pi$
$578$	5.05258	0.210159
$579$	0	0
$580$	−0.243325	−0.0101035
$581$	8.36201	0.346915
$582$	0	0
$583$	−0.325752	−0.0134913
$584$	−28.0372	−1.16019
$585$	0	0
$586$	−8.51073	−0.351575
$587$	2.26791	0.0936069	0.0468034	−	0.998904i	$-0.485097\pi$
$587$	2.26791	0.0936069	0.0468034	+	0.998904i	$0.485097\pi$
$588$	0	0
$589$	7.09603	0.292387
$590$	−5.55514	−0.228702
$591$	0	0
$592$	2.69856	0.110910
$593$	−1.01720	−0.0417713	−0.0208857	−	0.999782i	$-0.506649\pi$
$593$	−1.01720	−0.0417713	−0.0208857	+	0.999782i	$0.506649\pi$
$594$	0	0
$595$	−3.63760	−0.149127
$596$	0.742637	0.0304196
$597$	0	0
$598$	44.9081	1.83643
$599$	−24.0890	−0.984250	−0.492125	−	0.870525i	$-0.663780\pi$
$599$	−24.0890	−0.984250	−0.492125	+	0.870525i	$0.663780\pi$
$600$	0	0
$601$	−5.49868	−0.224296	−0.112148	−	0.993692i	$-0.535773\pi$
$601$	−5.49868	−0.224296	−0.112148	+	0.993692i	$0.535773\pi$
$602$	−0.247823	−0.0101005
$603$	0	0
$604$	−1.59477	−0.0648904
$605$	9.61243	0.390801
$606$	0	0
$607$	28.8653	1.17161	0.585804	−	0.810453i	$-0.300779\pi$
$607$	28.8653	1.17161	0.585804	+	0.810453i	$0.300779\pi$
$608$	0.620686	0.0251721
$609$	0	0
$610$	2.72284	0.110245
$611$	6.53290	0.264293
$612$	0	0
$613$	3.25746	0.131568	0.0657839	−	0.997834i	$-0.479045\pi$
$613$	3.25746	0.131568	0.0657839	+	0.997834i	$0.479045\pi$
$614$	−16.1266	−0.650815
$615$	0	0
$616$	−0.301677	−0.0121549
$617$	41.9867	1.69032	0.845161	−	0.534512i	$-0.179505\pi$
$617$	41.9867	1.69032	0.845161	+	0.534512i	$0.179505\pi$
$618$	0	0
$619$	−39.9398	−1.60531	−0.802657	−	0.596440i	$-0.796581\pi$
$619$	−39.9398	−1.60531	−0.802657	+	0.596440i	$0.796581\pi$
$620$	−0.681683	−0.0273771
$621$	0	0
$622$	30.5604	1.22536
$623$	−8.84720	−0.354456
$624$	0	0
$625$	14.1131	0.564525
$626$	34.5020	1.37898
$627$	0	0
$628$	−1.23753	−0.0493828
$629$	2.61413	0.104232
$630$	0	0
$631$	−15.4870	−0.616529	−0.308264	−	0.951301i	$-0.599748\pi$
$631$	−15.4870	−0.616529	−0.308264	+	0.951301i	$0.599748\pi$
$632$	−20.0428	−0.797261
$633$	0	0
$634$	22.7745	0.904489
$635$	−13.8208	−0.548460
$636$	0	0
$637$	37.2477	1.47581
$638$	0.317831	0.0125830
$639$	0	0
$640$	9.00157	0.355818
$641$	17.4745	0.690200	0.345100	−	0.938566i	$-0.387845\pi$
$641$	17.4745	0.690200	0.345100	+	0.938566i	$0.387845\pi$
$642$	0	0
$643$	22.9915	0.906697	0.453348	−	0.891333i	$-0.350229\pi$
$643$	22.9915	0.906697	0.453348	+	0.891333i	$0.350229\pi$
$644$	0.625866	0.0246626
$645$	0	0
$646$	−5.01858	−0.197453
$647$	45.9798	1.80765	0.903827	−	0.427899i	$-0.140746\pi$
$647$	45.9798	1.80765	0.903827	+	0.427899i	$0.140746\pi$
$648$	0	0
$649$	−0.421703	−0.0165533
$650$	−38.0391	−1.49202
$651$	0	0
$652$	2.20385	0.0863094
$653$	−6.02798	−0.235893	−0.117947	−	0.993020i	$-0.537631\pi$
$653$	−6.02798	−0.235893	−0.117947	+	0.993020i	$0.537631\pi$
$654$	0	0
$655$	3.06369	0.119708
$656$	−14.3502	−0.560283
$657$	0	0
$658$	−1.56661	−0.0610729
$659$	12.3526	0.481188	0.240594	−	0.970626i	$-0.422658\pi$
$659$	12.3526	0.481188	0.240594	+	0.970626i	$0.422658\pi$
$660$	0	0
$661$	−29.5698	−1.15013	−0.575066	−	0.818107i	$-0.695024\pi$
$661$	−29.5698	−1.15013	−0.575066	+	0.818107i	$0.695024\pi$
$662$	29.0058	1.12734
$663$	0	0
$664$	−21.2862	−0.826063
$665$	−0.996512	−0.0386431
$666$	0	0
$667$	−12.6645	−0.490372
$668$	1.30214	0.0503811
$669$	0	0
$670$	−11.4391	−0.441929
$671$	0.206697	0.00797945
$672$	0	0
$673$	12.6628	0.488115	0.244057	−	0.969761i	$-0.421521\pi$
$673$	12.6628	0.488115	0.244057	+	0.969761i	$0.421521\pi$
$674$	21.1381	0.814209
$675$	0	0
$676$	−3.26020	−0.125392
$677$	25.3835	0.975567	0.487783	−	0.872965i	$-0.337806\pi$
$677$	25.3835	0.975567	0.487783	+	0.872965i	$0.337806\pi$
$678$	0	0
$679$	−8.78140	−0.336999
$680$	9.25980	0.355097
$681$	0	0
$682$	0.890415	0.0340958
$683$	27.7298	1.06105	0.530526	−	0.847669i	$-0.321994\pi$
$683$	27.7298	1.06105	0.530526	+	0.847669i	$0.321994\pi$
$684$	0	0
$685$	14.3124	0.546850
$686$	−19.8984	−0.759724
$687$	0	0
$688$	0.596099	0.0227261
$689$	23.3165	0.888289
$690$	0	0
$691$	35.9561	1.36784	0.683918	−	0.729559i	$-0.260275\pi$
$691$	35.9561	1.36784	0.683918	+	0.729559i	$0.260275\pi$
$692$	−0.769755	−0.0292617
$693$	0	0
$694$	−32.1334	−1.21977
$695$	8.76797	0.332588
$696$	0	0
$697$	−13.9013	−0.526548
$698$	−9.44448	−0.357479
$699$	0	0
$700$	−0.530135	−0.0200372
$701$	−9.07457	−0.342742	−0.171371	−	0.985207i	$-0.554820\pi$
$701$	−9.07457	−0.342742	−0.171371	+	0.985207i	$0.554820\pi$
$702$	0	0
$703$	0.716135	0.0270095
$704$	0.765739	0.0288599
$705$	0	0
$706$	−38.7192	−1.45722
$707$	−5.61748	−0.211267
$708$	0	0
$709$	−1.91327	−0.0718543	−0.0359272	−	0.999354i	$-0.511438\pi$
$709$	−1.91327	−0.0718543	−0.0359272	+	0.999354i	$0.511438\pi$
$710$	−5.94079	−0.222954
$711$	0	0
$712$	22.5212	0.844019
$713$	−35.4801	−1.32874
$714$	0	0
$715$	0.521441	0.0195008
$716$	1.04599	0.0390906
$717$	0	0
$718$	−25.7456	−0.960818
$719$	−29.1719	−1.08793	−0.543964	−	0.839108i	$-0.683077\pi$
$719$	−29.1719	−1.08793	−0.543964	+	0.839108i	$0.683077\pi$
$720$	0	0
$721$	−7.92550	−0.295161
$722$	−1.37483	−0.0511658
$723$	0	0
$724$	0.606145	0.0225272
$725$	10.7274	0.398405
$726$	0	0
$727$	−53.1607	−1.97162	−0.985810	−	0.167866i	$-0.946312\pi$
$727$	−53.1607	−1.97162	−0.985810	+	0.167866i	$0.946312\pi$
$728$	21.5933	0.800301
$729$	0	0
$730$	−11.6212	−0.430121
$731$	0.577448	0.0213577
$732$	0	0
$733$	−5.33917	−0.197207	−0.0986033	−	0.995127i	$-0.531437\pi$
$733$	−5.33917	−0.197207	−0.0986033	+	0.995127i	$0.531437\pi$
$734$	−4.50101	−0.166135
$735$	0	0
$736$	−3.10343	−0.114394
$737$	−0.868364	−0.0319866
$738$	0	0
$739$	9.86650	0.362945	0.181472	−	0.983396i	$-0.441914\pi$
$739$	9.86650	0.362945	0.181472	+	0.983396i	$0.441914\pi$
$740$	−0.0687959	−0.00252899
$741$	0	0
$742$	−5.59138	−0.205266
$743$	−39.2147	−1.43865	−0.719324	−	0.694675i	$-0.755548\pi$
$743$	−39.2147	−1.43865	−0.719324	+	0.694675i	$0.755548\pi$
$744$	0	0
$745$	5.91218	0.216606
$746$	−28.6663	−1.04955
$747$	0	0
$748$	0.0365982	0.00133816
$749$	−6.72522	−0.245734
$750$	0	0
$751$	14.3603	0.524014	0.262007	−	0.965066i	$-0.415616\pi$
$751$	14.3603	0.524014	0.262007	+	0.965066i	$0.415616\pi$
$752$	3.76823	0.137413
$753$	0	0
$754$	−22.7496	−0.828490
$755$	−12.6961	−0.462059
$756$	0	0
$757$	−46.4760	−1.68920	−0.844600	−	0.535398i	$-0.820162\pi$
$757$	−46.4760	−1.68920	−0.844600	+	0.535398i	$0.820162\pi$
$758$	−14.0948	−0.511944
$759$	0	0
$760$	2.53670	0.0920158
$761$	10.1108	0.366516	0.183258	−	0.983065i	$-0.441336\pi$
$761$	10.1108	0.366516	0.183258	+	0.983065i	$0.441336\pi$
$762$	0	0
$763$	3.47296	0.125730
$764$	0.991167	0.0358592
$765$	0	0
$766$	37.5065	1.35517
$767$	30.1845	1.08990
$768$	0	0
$769$	−20.2018	−0.728494	−0.364247	−	0.931302i	$-0.618674\pi$
$769$	−20.2018	−0.728494	−0.364247	+	0.931302i	$0.618674\pi$
$770$	−0.125043	−0.00450624
$771$	0	0
$772$	2.88797	0.103940
$773$	1.24072	0.0446258	0.0223129	−	0.999751i	$-0.492897\pi$
$773$	1.24072	0.0446258	0.0223129	+	0.999751i	$0.492897\pi$
$774$	0	0
$775$	30.0532	1.07954
$776$	22.3537	0.802452
$777$	0	0
$778$	−38.8750	−1.39374
$779$	−3.80822	−0.136443
$780$	0	0
$781$	−0.450978	−0.0161373
$782$	25.0929	0.897320
$783$	0	0
$784$	21.4848	0.767313
$785$	−9.85205	−0.351635
$786$	0	0
$787$	−30.8085	−1.09821	−0.549103	−	0.835755i	$-0.685031\pi$
$787$	−30.8085	−1.09821	−0.549103	+	0.835755i	$0.685031\pi$
$788$	−1.76063	−0.0627199
$789$	0	0
$790$	−8.30763	−0.295572
$791$	−9.70500	−0.345070
$792$	0	0
$793$	−14.7949	−0.525381
$794$	−43.3549	−1.53861
$795$	0	0
$796$	−0.191575	−0.00679018
$797$	35.6910	1.26424	0.632119	−	0.774871i	$-0.282185\pi$
$797$	35.6910	1.26424	0.632119	+	0.774871i	$0.282185\pi$
$798$	0	0
$799$	3.65033	0.129140
$800$	2.62874	0.0929400
$801$	0	0
$802$	−26.5644	−0.938020
$803$	−0.882194	−0.0311319
$804$	0	0
$805$	4.98256	0.175612
$806$	−63.7338	−2.24493
$807$	0	0
$808$	14.2997	0.503063
$809$	−0.211834	−0.00744768	−0.00372384	−	0.999993i	$-0.501185\pi$
$809$	−0.211834	−0.00744768	−0.00372384	+	0.999993i	$0.501185\pi$
$810$	0	0
$811$	−32.0044	−1.12383	−0.561913	−	0.827196i	$-0.689935\pi$
$811$	−32.0044	−1.12383	−0.561913	+	0.827196i	$0.689935\pi$
$812$	−0.317052	−0.0111263
$813$	0	0
$814$	0.0898612	0.00314963
$815$	17.5450	0.614574
$816$	0	0
$817$	0.158190	0.00553438
$818$	10.9720	0.383627
$819$	0	0
$820$	0.365838	0.0127756
$821$	−5.26528	−0.183760	−0.0918798	−	0.995770i	$-0.529288\pi$
$821$	−5.26528	−0.183760	−0.0918798	+	0.995770i	$0.529288\pi$
$822$	0	0
$823$	−10.6818	−0.372343	−0.186171	−	0.982517i	$-0.559608\pi$
$823$	−10.6818	−0.372343	−0.186171	+	0.982517i	$0.559608\pi$
$824$	20.1750	0.702829
$825$	0	0
$826$	−7.23834	−0.251854
$827$	−11.2972	−0.392843	−0.196422	−	0.980520i	$-0.562932\pi$
$827$	−11.2972	−0.392843	−0.196422	+	0.980520i	$0.562932\pi$
$828$	0	0
$829$	−21.2500	−0.738043	−0.369021	−	0.929421i	$-0.620307\pi$
$829$	−21.2500	−0.738043	−0.369021	+	0.929421i	$0.620307\pi$
$830$	−8.82297	−0.306250
$831$	0	0
$832$	−54.8098	−1.90019
$833$	20.8125	0.721112
$834$	0	0
$835$	10.3664	0.358744
$836$	0.0100260	0.000346756 0
$837$	0	0
$838$	−42.5654	−1.47040
$839$	−10.4734	−0.361583	−0.180792	−	0.983521i	$-0.557866\pi$
$839$	−10.4734	−0.361583	−0.180792	+	0.983521i	$0.557866\pi$
$840$	0	0
$841$	−22.5844	−0.778772
$842$	11.5630	0.398487
$843$	0	0
$844$	0.851299	0.0293029
$845$	−25.9547	−0.892869
$846$	0	0
$847$	12.5250	0.430363
$848$	13.4492	0.461846
$849$	0	0
$850$	−21.2548	−0.729033
$851$	−3.58068	−0.122744
$852$	0	0
$853$	−37.4334	−1.28170	−0.640848	−	0.767668i	$-0.721417\pi$
$853$	−37.4334	−1.28170	−0.640848	+	0.767668i	$0.721417\pi$
$854$	3.54786	0.121405
$855$	0	0
$856$	17.1196	0.585135
$857$	34.8892	1.19179	0.595896	−	0.803062i	$-0.296797\pi$
$857$	34.8892	1.19179	0.595896	+	0.803062i	$0.296797\pi$
$858$	0	0
$859$	−5.36125	−0.182923	−0.0914617	−	0.995809i	$-0.529154\pi$
$859$	−5.36125	−0.182923	−0.0914617	+	0.995809i	$0.529154\pi$
$860$	−0.0151966	−0.000518201 0
$861$	0	0
$862$	23.7698	0.809603
$863$	5.70332	0.194143	0.0970716	−	0.995277i	$-0.469052\pi$
$863$	5.70332	0.194143	0.0970716	+	0.995277i	$0.469052\pi$
$864$	0	0
$865$	−6.12807	−0.208361
$866$	−46.3147	−1.57384
$867$	0	0
$868$	−0.888232	−0.0301486
$869$	−0.630651	−0.0213934
$870$	0	0
$871$	62.1554	2.10606
$872$	−8.84069	−0.299384
$873$	0	0
$874$	6.87414	0.232521
$875$	−9.20300	−0.311118
$876$	0	0
$877$	−39.4858	−1.33334	−0.666671	−	0.745352i	$-0.732281\pi$
$877$	−39.4858	−1.33334	−0.666671	+	0.745352i	$0.732281\pi$
$878$	−1.66255	−0.0561083
$879$	0	0
$880$	0.300771	0.0101390
$881$	11.6456	0.392352	0.196176	−	0.980569i	$-0.437148\pi$
$881$	11.6456	0.392352	0.196176	+	0.980569i	$0.437148\pi$
$882$	0	0
$883$	55.1597	1.85627	0.928135	−	0.372244i	$-0.121411\pi$
$883$	55.1597	1.85627	0.928135	+	0.372244i	$0.121411\pi$
$884$	−2.61961	−0.0881071
$885$	0	0
$886$	40.5496	1.36229
$887$	17.2301	0.578531	0.289266	−	0.957249i	$-0.406589\pi$
$887$	17.2301	0.578531	0.289266	+	0.957249i	$0.406589\pi$
$888$	0	0
$889$	−18.0084	−0.603984
$890$	9.33491	0.312907
$891$	0	0
$892$	0.201609	0.00675037
$893$	1.00000	0.0334637
$894$	0	0
$895$	8.32723	0.278348
$896$	11.7290	0.391839
$897$	0	0
$898$	−37.3170	−1.24529
$899$	17.9736	0.599452
$900$	0	0
$901$	13.0284	0.434038
$902$	−0.477858	−0.0159109
$903$	0	0
$904$	24.7048	0.821671
$905$	4.82556	0.160407
$906$	0	0
$907$	29.9105	0.993160	0.496580	−	0.867991i	$-0.334589\pi$
$907$	29.9105	0.993160	0.496580	+	0.867991i	$0.334589\pi$
$908$	−1.20254	−0.0399079
$909$	0	0
$910$	8.95029	0.296699
$911$	30.4799	1.00984	0.504922	−	0.863165i	$-0.331521\pi$
$911$	30.4799	1.00984	0.504922	+	0.863165i	$0.331521\pi$
$912$	0	0
$913$	−0.669771	−0.0221662
$914$	−26.7561	−0.885014
$915$	0	0
$916$	−0.394724	−0.0130421
$917$	3.99199	0.131827
$918$	0	0
$919$	22.5879	0.745108	0.372554	−	0.928011i	$-0.378482\pi$
$919$	22.5879	0.745108	0.372554	+	0.928011i	$0.378482\pi$
$920$	−12.6835	−0.418163
$921$	0	0
$922$	−46.7868	−1.54084
$923$	32.2799	1.06251
$924$	0	0
$925$	3.03299	0.0997240
$926$	−17.8165	−0.585486
$927$	0	0
$928$	1.57214	0.0516080
$929$	−2.00085	−0.0656457	−0.0328229	−	0.999461i	$-0.510450\pi$
$929$	−2.00085	−0.0656457	−0.0328229	+	0.999461i	$0.510450\pi$
$930$	0	0
$931$	5.70155	0.186861
$932$	0.319615	0.0104693
$933$	0	0
$934$	−29.5388	−0.966540
$935$	0.291361	0.00952852
$936$	0	0
$937$	8.74720	0.285759	0.142879	−	0.989740i	$-0.454364\pi$
$937$	8.74720	0.285759	0.142879	+	0.989740i	$0.454364\pi$
$938$	−14.9051	−0.486668
$939$	0	0
$940$	−0.0960655	−0.00313331
$941$	−43.9664	−1.43326	−0.716632	−	0.697451i	$-0.754317\pi$
$941$	−43.9664	−1.43326	−0.716632	+	0.697451i	$0.754317\pi$
$942$	0	0
$943$	19.0411	0.620063
$944$	17.4107	0.566669
$945$	0	0
$946$	0.0198499	0.000645375 0
$947$	−15.3057	−0.497367	−0.248683	−	0.968585i	$-0.579998\pi$
$947$	−15.3057	−0.497367	−0.248683	+	0.968585i	$0.579998\pi$
$948$	0	0
$949$	63.1453	2.04978
$950$	−5.82269	−0.188913
$951$	0	0
$952$	12.0655	0.391045
$953$	22.4008	0.725633	0.362816	−	0.931861i	$-0.381815\pi$
$953$	22.4008	0.725633	0.362816	+	0.931861i	$0.381815\pi$
$954$	0	0
$955$	7.89075	0.255339
$956$	1.21636	0.0393400
$957$	0	0
$958$	−1.55845	−0.0503514
$959$	18.6491	0.602210
$960$	0	0
$961$	19.3536	0.624310
$962$	−6.43205	−0.207378
$963$	0	0
$964$	−2.01740	−0.0649760
$965$	22.9914	0.740118
$966$	0	0
$967$	7.55932	0.243091	0.121546	−	0.992586i	$-0.461215\pi$
$967$	7.55932	0.243091	0.121546	+	0.992586i	$0.461215\pi$
$968$	−31.8833	−1.02477
$969$	0	0
$970$	9.26548	0.297497
$971$	48.0156	1.54089	0.770447	−	0.637505i	$-0.220033\pi$
$971$	48.0156	1.54089	0.770447	+	0.637505i	$0.220033\pi$
$972$	0	0
$973$	11.4247	0.366258
$974$	14.5079	0.464864
$975$	0	0
$976$	−8.53381	−0.273161
$977$	−5.66222	−0.181151	−0.0905753	−	0.995890i	$-0.528871\pi$
$977$	−5.66222	−0.181151	−0.0905753	+	0.995890i	$0.528871\pi$
$978$	0	0
$979$	0.708634	0.0226480
$980$	−0.547722	−0.0174963
$981$	0	0
$982$	−20.1839	−0.644095
$983$	18.6532	0.594944	0.297472	−	0.954730i	$-0.403856\pi$
$983$	18.6532	0.594944	0.297472	+	0.954730i	$0.403856\pi$
$984$	0	0
$985$	−14.0165	−0.446603
$986$	−12.7116	−0.404819
$987$	0	0
$988$	−0.717636	−0.0228310
$989$	−0.790952	−0.0251508
$990$	0	0
$991$	8.69793	0.276299	0.138149	−	0.990411i	$-0.455885\pi$
$991$	8.69793	0.276299	0.138149	+	0.990411i	$0.455885\pi$
$992$	4.40441	0.139840
$993$	0	0
$994$	−7.74084	−0.245524
$995$	−1.52514	−0.0483502
$996$	0	0
$997$	−13.3584	−0.423064	−0.211532	−	0.977371i	$-0.567845\pi$
$997$	−13.3584	−0.423064	−0.211532	+	0.977371i	$0.567845\pi$
$998$	−38.3311	−1.21335
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.k.1.2		7
3.2	odd	2		2679.2.a.l.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.l.1.6	✓	7	3.2	odd	2
8037.2.a.k.1.2		7	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)

\(f(q)\)	\(=\)	\(q-1.37483 q^{2} -0.109849 q^{4} -0.874519 q^{5} -1.13950 q^{7} +2.90068 q^{8} +O(q^{10})\)	\(q-1.37483 q^{2} -0.109849 q^{4} -0.874519 q^{5} -1.13950 q^{7} +2.90068 q^{8} +1.20231 q^{10} +0.0912702 q^{11} -6.53290 q^{13} +1.56661 q^{14} -3.76823 q^{16} -3.65033 q^{17} -1.00000 q^{19} +0.0960655 q^{20} -0.125481 q^{22} +5.00000 q^{23} -4.23522 q^{25} +8.98162 q^{26} +0.125173 q^{28} -2.53290 q^{29} -7.09603 q^{31} -0.620686 q^{32} +5.01858 q^{34} +0.996512 q^{35} -0.716135 q^{37} +1.37483 q^{38} -2.53670 q^{40} +3.80822 q^{41} -0.158190 q^{43} -0.0100260 q^{44} -6.87414 q^{46} -1.00000 q^{47} -5.70155 q^{49} +5.82269 q^{50} +0.717636 q^{52} -3.56909 q^{53} -0.0798176 q^{55} -3.30532 q^{56} +3.48231 q^{58} -4.62038 q^{59} +2.26467 q^{61} +9.75581 q^{62} +8.38980 q^{64} +5.71315 q^{65} -9.51421 q^{67} +0.400987 q^{68} -1.37003 q^{70} -4.94113 q^{71} -9.66573 q^{73} +0.984562 q^{74} +0.109849 q^{76} -0.104002 q^{77} -6.90971 q^{79} +3.29539 q^{80} -5.23564 q^{82} -7.33833 q^{83} +3.19229 q^{85} +0.217485 q^{86} +0.264746 q^{88} +7.76413 q^{89} +7.44423 q^{91} -0.549247 q^{92} +1.37483 q^{94} +0.874519 q^{95} +7.70638 q^{97} +7.83864 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8	\( 7 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 7 q^{7} + 12 q^{8} - 2 q^{10} + 9 q^{11} - 2 q^{13} + 6 q^{14} + 14 q^{16} + 6 q^{17} - 7 q^{19} + 25 q^{20} - 13 q^{22} + 35 q^{23} + 3 q^{25} + 23 q^{26} - 9 q^{28} + 26 q^{29} - 11 q^{31} + 21 q^{32} + 18 q^{34} + 17 q^{35} - 3 q^{37} - 2 q^{38} + 14 q^{40} + 20 q^{41} - 5 q^{43} - 18 q^{44} + 10 q^{46} - 7 q^{47} - 10 q^{49} - 17 q^{50} + 15 q^{52} - 3 q^{53} - 3 q^{55} + 3 q^{56} + 31 q^{58} + 15 q^{59} - 9 q^{61} + 4 q^{62} + 4 q^{64} + 13 q^{65} - 4 q^{67} - 5 q^{68} + 18 q^{70} + 4 q^{71} - 8 q^{73} - 28 q^{74} - 8 q^{76} + 11 q^{77} - 11 q^{79} + 12 q^{80} - 19 q^{82} + 4 q^{83} - 6 q^{85} + 36 q^{86} - 7 q^{88} + 23 q^{89} + 19 q^{91} + 40 q^{92} - 2 q^{94} - 6 q^{95} - 7 q^{97}+O(q^{100}) \) 7 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 7 * q^7 + 12 * q^8 - 2 * q^10 + 9 * q^11 - 2 * q^13 + 6 * q^14 + 14 * q^16 + 6 * q^17 - 7 * q^19 + 25 * q^20 - 13 * q^22 + 35 * q^23 + 3 * q^25 + 23 * q^26 - 9 * q^28 + 26 * q^29 - 11 * q^31 + 21 * q^32 + 18 * q^34 + 17 * q^35 - 3 * q^37 - 2 * q^38 + 14 * q^40 + 20 * q^41 - 5 * q^43 - 18 * q^44 + 10 * q^46 - 7 * q^47 - 10 * q^49 - 17 * q^50 + 15 * q^52 - 3 * q^53 - 3 * q^55 + 3 * q^56 + 31 * q^58 + 15 * q^59 - 9 * q^61 + 4 * q^62 + 4 * q^64 + 13 * q^65 - 4 * q^67 - 5 * q^68 + 18 * q^70 + 4 * q^71 - 8 * q^73 - 28 * q^74 - 8 * q^76 + 11 * q^77 - 11 * q^79 + 12 * q^80 - 19 * q^82 + 4 * q^83 - 6 * q^85 + 36 * q^86 - 7 * q^88 + 23 * q^89 + 19 * q^91 + 40 * q^92 - 2 * q^94 - 6 * q^95 - 7 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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