LMFDB - Embedded newform 8037.2.a.m.1.6

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 $ x^12 - x^11 - 15x^10 + 14x^9 + 84x^8 - 76x^7 - 213x^6 + 196x^5 + 225x^4 - 229x^3 - 49x^2 + 83x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.332508$ 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)

$f(q)$	$=$	$q+0.332508 q^{2} -1.88944 q^{4} +1.32217 q^{5} +1.64871 q^{7} -1.29327 q^{8} +O(q^{10})$	\(q+0.332508 q^{2} -1.88944 q^{4} +1.32217 q^{5} +1.64871 q^{7} -1.29327 q^{8} +0.439633 q^{10} +2.44033 q^{11} +3.38397 q^{13} +0.548211 q^{14} +3.34886 q^{16} -4.00898 q^{17} +1.00000 q^{19} -2.49816 q^{20} +0.811429 q^{22} -0.324166 q^{23} -3.25186 q^{25} +1.12520 q^{26} -3.11515 q^{28} -1.08715 q^{29} -8.83217 q^{31} +3.70006 q^{32} -1.33302 q^{34} +2.17988 q^{35} -9.92982 q^{37} +0.332508 q^{38} -1.70992 q^{40} -10.9158 q^{41} +1.05406 q^{43} -4.61085 q^{44} -0.107788 q^{46} -1.00000 q^{47} -4.28174 q^{49} -1.08127 q^{50} -6.39380 q^{52} -6.87937 q^{53} +3.22654 q^{55} -2.13223 q^{56} -0.361488 q^{58} +7.79484 q^{59} -9.96268 q^{61} -2.93677 q^{62} -5.46741 q^{64} +4.47419 q^{65} -6.45240 q^{67} +7.57473 q^{68} +0.724829 q^{70} +2.04338 q^{71} +11.8359 q^{73} -3.30174 q^{74} -1.88944 q^{76} +4.02341 q^{77} +6.05262 q^{79} +4.42776 q^{80} -3.62958 q^{82} +16.4227 q^{83} -5.30056 q^{85} +0.350483 q^{86} -3.15600 q^{88} +1.78097 q^{89} +5.57920 q^{91} +0.612492 q^{92} -0.332508 q^{94} +1.32217 q^{95} -15.1752 q^{97} -1.42371 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.332508	0.235119	0.117559	−	0.993066i	$-0.462493\pi$
$2$	0.332508	0.235119	0.117559	+	0.993066i	$0.462493\pi$
$3$	0	0
$3$	0	0
$4$	−1.88944	−0.944719
$5$	1.32217	0.591293	0.295647	−	0.955297i	$-0.404465\pi$
$5$	1.32217	0.591293	0.295647	+	0.955297i	$0.404465\pi$
$6$	0	0
$7$	1.64871	0.623156	0.311578	−	0.950221i	$-0.399143\pi$
$7$	1.64871	0.623156	0.311578	+	0.950221i	$0.399143\pi$
$8$	−1.29327	−0.457240
$9$	0	0
$10$	0.439633	0.139024
$11$	2.44033	0.735787	0.367894	−	0.929868i	$-0.380079\pi$
$11$	2.44033	0.735787	0.367894	+	0.929868i	$0.380079\pi$
$12$	0	0
$13$	3.38397	0.938545	0.469272	−	0.883054i	$-0.344516\pi$
$13$	3.38397	0.938545	0.469272	+	0.883054i	$0.344516\pi$
$14$	0.548211	0.146515
$15$	0	0
$16$	3.34886	0.837214
$17$	−4.00898	−0.972321	−0.486161	−	0.873870i	$-0.661603\pi$
$17$	−4.00898	−0.972321	−0.486161	+	0.873870i	$0.661603\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.49816	−0.558606
$21$	0	0
$22$	0.811429	0.172997
$23$	−0.324166	−0.0675933	−0.0337966	−	0.999429i	$-0.510760\pi$
$23$	−0.324166	−0.0675933	−0.0337966	+	0.999429i	$0.510760\pi$
$24$	0	0
$25$	−3.25186	−0.650372
$26$	1.12520	0.220669
$27$	0	0
$28$	−3.11515	−0.588707
$29$	−1.08715	−0.201880	−0.100940	−	0.994893i	$-0.532185\pi$
$29$	−1.08715	−0.201880	−0.100940	+	0.994893i	$0.532185\pi$
$30$	0	0
$31$	−8.83217	−1.58630	−0.793152	−	0.609024i	$-0.791561\pi$
$31$	−8.83217	−1.58630	−0.793152	+	0.609024i	$0.791561\pi$
$32$	3.70006	0.654084
$33$	0	0
$34$	−1.33302	−0.228611
$35$	2.17988	0.368468
$36$	0	0
$37$	−9.92982	−1.63245	−0.816226	−	0.577733i	$-0.803938\pi$
$37$	−9.92982	−1.63245	−0.816226	+	0.577733i	$0.803938\pi$
$38$	0.332508	0.0539399
$39$	0	0
$40$	−1.70992	−0.270363
$41$	−10.9158	−1.70476	−0.852379	−	0.522924i	$-0.824841\pi$
$41$	−10.9158	−1.70476	−0.852379	+	0.522924i	$0.824841\pi$
$42$	0	0
$43$	1.05406	0.160743	0.0803713	−	0.996765i	$-0.474389\pi$
$43$	1.05406	0.160743	0.0803713	+	0.996765i	$0.474389\pi$
$44$	−4.61085	−0.695112
$45$	0	0
$46$	−0.107788	−0.0158924
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−4.28174	−0.611677
$50$	−1.08127	−0.152915
$51$	0	0
$52$	−6.39380	−0.886661
$53$	−6.87937	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$53$	−6.87937	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$54$	0	0
$55$	3.22654	0.435066
$56$	−2.13223	−0.284931
$57$	0	0
$58$	−0.361488	−0.0474656
$59$	7.79484	1.01480	0.507401	−	0.861710i	$-0.330606\pi$
$59$	7.79484	1.01480	0.507401	+	0.861710i	$0.330606\pi$
$60$	0	0
$61$	−9.96268	−1.27559	−0.637795	−	0.770206i	$-0.720153\pi$
$61$	−9.96268	−1.27559	−0.637795	+	0.770206i	$0.720153\pi$
$62$	−2.93677	−0.372970
$63$	0	0
$64$	−5.46741	−0.683427
$65$	4.47419	0.554955
$66$	0	0
$67$	−6.45240	−0.788286	−0.394143	−	0.919049i	$-0.628959\pi$
$67$	−6.45240	−0.788286	−0.394143	+	0.919049i	$0.628959\pi$
$68$	7.57473	0.918570
$69$	0	0
$70$	0.724829	0.0866336
$71$	2.04338	0.242505	0.121252	−	0.992622i	$-0.461309\pi$
$71$	2.04338	0.242505	0.121252	+	0.992622i	$0.461309\pi$
$72$	0	0
$73$	11.8359	1.38528	0.692642	−	0.721282i	$-0.256447\pi$
$73$	11.8359	1.38528	0.692642	+	0.721282i	$0.256447\pi$
$74$	−3.30174	−0.383820
$75$	0	0
$76$	−1.88944	−0.216733
$77$	4.02341	0.458510
$78$	0	0
$79$	6.05262	0.680973	0.340487	−	0.940249i	$-0.389408\pi$
$79$	6.05262	0.680973	0.340487	+	0.940249i	$0.389408\pi$
$80$	4.42776	0.495039
$81$	0	0
$82$	−3.62958	−0.400820
$83$	16.4227	1.80262	0.901311	−	0.433173i	$-0.142606\pi$
$83$	16.4227	1.80262	0.901311	+	0.433173i	$0.142606\pi$
$84$	0	0
$85$	−5.30056	−0.574927
$86$	0.350483	0.0377936
$87$	0	0
$88$	−3.15600	−0.336431
$89$	1.78097	0.188783	0.0943914	−	0.995535i	$-0.469909\pi$
$89$	1.78097	0.188783	0.0943914	+	0.995535i	$0.469909\pi$
$90$	0	0
$91$	5.57920	0.584859
$92$	0.612492	0.0638567
$93$	0	0
$94$	−0.332508	−0.0342956
$95$	1.32217	0.135652
$96$	0	0
$97$	−15.1752	−1.54081	−0.770404	−	0.637556i	$-0.779945\pi$
$97$	−15.1752	−1.54081	−0.770404	+	0.637556i	$0.779945\pi$
$98$	−1.42371	−0.143817
$99$	0	0
$100$	6.14419	0.614419
$101$	4.03486	0.401484	0.200742	−	0.979644i	$-0.435665\pi$
$101$	4.03486	0.401484	0.200742	+	0.979644i	$0.435665\pi$
$102$	0	0
$103$	8.68186	0.855449	0.427724	−	0.903909i	$-0.359315\pi$
$103$	8.68186	0.855449	0.427724	+	0.903909i	$0.359315\pi$
$104$	−4.37638	−0.429140
$105$	0	0
$106$	−2.28744	−0.222176
$107$	17.1897	1.66179	0.830893	−	0.556432i	$-0.187830\pi$
$107$	17.1897	1.66179	0.830893	+	0.556432i	$0.187830\pi$
$108$	0	0
$109$	6.67591	0.639437	0.319718	−	0.947513i	$-0.396412\pi$
$109$	6.67591	0.639437	0.319718	+	0.947513i	$0.396412\pi$
$110$	1.07285	0.102292
$111$	0	0
$112$	5.52131	0.521714
$113$	−11.0768	−1.04202	−0.521008	−	0.853552i	$-0.674444\pi$
$113$	−11.0768	−1.04202	−0.521008	+	0.853552i	$0.674444\pi$
$114$	0	0
$115$	−0.428603	−0.0399674
$116$	2.05411	0.190720
$117$	0	0
$118$	2.59185	0.238599
$119$	−6.60967	−0.605907
$120$	0	0
$121$	−5.04479	−0.458617
$122$	−3.31267	−0.299915
$123$	0	0
$124$	16.6878	1.49861
$125$	−10.9104	−0.975854
$126$	0	0
$127$	12.8845	1.14331	0.571657	−	0.820493i	$-0.306301\pi$
$127$	12.8845	1.14331	0.571657	+	0.820493i	$0.306301\pi$
$128$	−9.21807	−0.814770
$129$	0	0
$130$	1.48770	0.130480
$131$	−11.6787	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$131$	−11.6787	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$132$	0	0
$133$	1.64871	0.142962
$134$	−2.14547	−0.185341
$135$	0	0
$136$	5.18469	0.444584
$137$	2.48303	0.212139	0.106070	−	0.994359i	$-0.466173\pi$
$137$	2.48303	0.212139	0.106070	+	0.994359i	$0.466173\pi$
$138$	0	0
$139$	−3.70637	−0.314370	−0.157185	−	0.987569i	$-0.550242\pi$
$139$	−3.70637	−0.314370	−0.157185	+	0.987569i	$0.550242\pi$
$140$	−4.11876	−0.348099
$141$	0	0
$142$	0.679440	0.0570173
$143$	8.25801	0.690569
$144$	0	0
$145$	−1.43741	−0.119370
$146$	3.93552	0.325706
$147$	0	0
$148$	18.7618	1.54221
$149$	−11.8886	−0.973953	−0.486976	−	0.873415i	$-0.661900\pi$
$149$	−11.8886	−0.973953	−0.486976	+	0.873415i	$0.661900\pi$
$150$	0	0
$151$	−14.3716	−1.16955	−0.584773	−	0.811197i	$-0.698816\pi$
$151$	−14.3716	−1.16955	−0.584773	+	0.811197i	$0.698816\pi$
$152$	−1.29327	−0.104898
$153$	0	0
$154$	1.33782	0.107804
$155$	−11.6776	−0.937971
$156$	0	0
$157$	−23.7132	−1.89252	−0.946259	−	0.323409i	$-0.895171\pi$
$157$	−23.7132	−1.89252	−0.946259	+	0.323409i	$0.895171\pi$
$158$	2.01254	0.160109
$159$	0	0
$160$	4.89211	0.386756
$161$	−0.534457	−0.0421211
$162$	0	0
$163$	−19.2735	−1.50962	−0.754809	−	0.655944i	$-0.772271\pi$
$163$	−19.2735	−1.50962	−0.754809	+	0.655944i	$0.772271\pi$
$164$	20.6247	1.61052
$165$	0	0
$166$	5.46066	0.423830
$167$	4.99831	0.386781	0.193390	−	0.981122i	$-0.438052\pi$
$167$	4.99831	0.386781	0.193390	+	0.981122i	$0.438052\pi$
$168$	0	0
$169$	−1.54874	−0.119134
$170$	−1.76248	−0.135176
$171$	0	0
$172$	−1.99158	−0.151857
$173$	−3.63313	−0.276222	−0.138111	−	0.990417i	$-0.544103\pi$
$173$	−3.63313	−0.276222	−0.138111	+	0.990417i	$0.544103\pi$
$174$	0	0
$175$	−5.36139	−0.405283
$176$	8.17231	0.616011
$177$	0	0
$178$	0.592187	0.0443863
$179$	−4.53330	−0.338835	−0.169417	−	0.985544i	$-0.554189\pi$
$179$	−4.53330	−0.338835	−0.169417	+	0.985544i	$0.554189\pi$
$180$	0	0
$181$	17.5178	1.30208	0.651042	−	0.759042i	$-0.274332\pi$
$181$	17.5178	1.30208	0.651042	+	0.759042i	$0.274332\pi$
$182$	1.85513	0.137511
$183$	0	0
$184$	0.419234	0.0309063
$185$	−13.1289	−0.965258
$186$	0	0
$187$	−9.78324	−0.715422
$188$	1.88944	0.137801
$189$	0	0
$190$	0.439633	0.0318943
$191$	12.8617	0.930639	0.465319	−	0.885143i	$-0.345939\pi$
$191$	12.8617	0.930639	0.465319	+	0.885143i	$0.345939\pi$
$192$	0	0
$193$	−1.68810	−0.121512	−0.0607561	−	0.998153i	$-0.519351\pi$
$193$	−1.68810	−0.121512	−0.0607561	+	0.998153i	$0.519351\pi$
$194$	−5.04587	−0.362272
$195$	0	0
$196$	8.09008	0.577863
$197$	8.31060	0.592106	0.296053	−	0.955172i	$-0.404329\pi$
$197$	8.31060	0.592106	0.296053	+	0.955172i	$0.404329\pi$
$198$	0	0
$199$	−21.2119	−1.50367	−0.751837	−	0.659349i	$-0.770832\pi$
$199$	−21.2119	−1.50367	−0.751837	+	0.659349i	$0.770832\pi$
$200$	4.20553	0.297376
$201$	0	0
$202$	1.34162	0.0943963
$203$	−1.79241	−0.125802
$204$	0	0
$205$	−14.4325	−1.00801
$206$	2.88679	0.201132
$207$	0	0
$208$	11.3324	0.785762
$209$	2.44033	0.168801
$210$	0	0
$211$	−12.8107	−0.881927	−0.440963	−	0.897525i	$-0.645363\pi$
$211$	−12.8107	−0.881927	−0.440963	+	0.897525i	$0.645363\pi$
$212$	12.9981	0.892717
$213$	0	0
$214$	5.71569	0.390717
$215$	1.39365	0.0950460
$216$	0	0
$217$	−14.5617	−0.988514
$218$	2.21979	0.150343
$219$	0	0
$220$	−6.09634	−0.411015
$221$	−13.5663	−0.912567
$222$	0	0
$223$	−8.95190	−0.599464	−0.299732	−	0.954023i	$-0.596897\pi$
$223$	−8.95190	−0.599464	−0.299732	+	0.954023i	$0.596897\pi$
$224$	6.10034	0.407596
$225$	0	0
$226$	−3.68312	−0.244997
$227$	−1.55304	−0.103079	−0.0515393	−	0.998671i	$-0.516413\pi$
$227$	−1.55304	−0.103079	−0.0515393	+	0.998671i	$0.516413\pi$
$228$	0	0
$229$	−1.54975	−0.102410	−0.0512050	−	0.998688i	$-0.516306\pi$
$229$	−1.54975	−0.102410	−0.0512050	+	0.998688i	$0.516306\pi$
$230$	−0.142514	−0.00939709
$231$	0	0
$232$	1.40598	0.0923073
$233$	−12.2911	−0.805214	−0.402607	−	0.915373i	$-0.631896\pi$
$233$	−12.2911	−0.805214	−0.402607	+	0.915373i	$0.631896\pi$
$234$	0	0
$235$	−1.32217	−0.0862490
$236$	−14.7279	−0.958703
$237$	0	0
$238$	−2.19777	−0.142460
$239$	−26.1582	−1.69203	−0.846017	−	0.533156i	$-0.821006\pi$
$239$	−26.1582	−1.69203	−0.846017	+	0.533156i	$0.821006\pi$
$240$	0	0
$241$	13.8359	0.891246	0.445623	−	0.895221i	$-0.352982\pi$
$241$	13.8359	0.891246	0.445623	+	0.895221i	$0.352982\pi$
$242$	−1.67743	−0.107829
$243$	0	0
$244$	18.8239	1.20508
$245$	−5.66120	−0.361681
$246$	0	0
$247$	3.38397	0.215317
$248$	11.4224	0.725321
$249$	0	0
$250$	−3.62779	−0.229441
$251$	14.8578	0.937815	0.468907	−	0.883247i	$-0.344648\pi$
$251$	14.8578	0.937815	0.468907	+	0.883247i	$0.344648\pi$
$252$	0	0
$253$	−0.791072	−0.0497343
$254$	4.28419	0.268814
$255$	0	0
$256$	7.86974	0.491859
$257$	26.4944	1.65268	0.826338	−	0.563174i	$-0.190420\pi$
$257$	26.4944	1.65268	0.826338	+	0.563174i	$0.190420\pi$
$258$	0	0
$259$	−16.3714	−1.01727
$260$	−8.45371	−0.524277
$261$	0	0
$262$	−3.88327	−0.239909
$263$	28.1656	1.73677	0.868384	−	0.495892i	$-0.165159\pi$
$263$	28.1656	1.73677	0.868384	+	0.495892i	$0.165159\pi$
$264$	0	0
$265$	−9.09571	−0.558745
$266$	0.548211	0.0336129
$267$	0	0
$268$	12.1914	0.744709
$269$	9.30234	0.567173	0.283587	−	0.958947i	$-0.408476\pi$
$269$	9.30234	0.567173	0.283587	+	0.958947i	$0.408476\pi$
$270$	0	0
$271$	−18.1852	−1.10467	−0.552336	−	0.833622i	$-0.686263\pi$
$271$	−18.1852	−1.10467	−0.552336	+	0.833622i	$0.686263\pi$
$272$	−13.4255	−0.814041
$273$	0	0
$274$	0.825626	0.0498779
$275$	−7.93562	−0.478536
$276$	0	0
$277$	−4.12913	−0.248096	−0.124048	−	0.992276i	$-0.539588\pi$
$277$	−4.12913	−0.248096	−0.124048	+	0.992276i	$0.539588\pi$
$278$	−1.23240	−0.0739143
$279$	0	0
$280$	−2.81918	−0.168478
$281$	−22.6958	−1.35392	−0.676959	−	0.736021i	$-0.736702\pi$
$281$	−22.6958	−1.35392	−0.676959	+	0.736021i	$0.736702\pi$
$282$	0	0
$283$	−18.5347	−1.10177	−0.550886	−	0.834581i	$-0.685710\pi$
$283$	−18.5347	−1.10177	−0.550886	+	0.834581i	$0.685710\pi$
$284$	−3.86084	−0.229099
$285$	0	0
$286$	2.74585	0.162366
$287$	−17.9970	−1.06233
$288$	0	0
$289$	−0.928061	−0.0545918
$290$	−0.477949	−0.0280661
$291$	0	0
$292$	−22.3631	−1.30870
$293$	19.2511	1.12466	0.562330	−	0.826913i	$-0.309905\pi$
$293$	19.2511	1.12466	0.562330	+	0.826913i	$0.309905\pi$
$294$	0	0
$295$	10.3061	0.600046
$296$	12.8419	0.746422
$297$	0	0
$298$	−3.95306	−0.228994
$299$	−1.09697	−0.0634393
$300$	0	0
$301$	1.73784	0.100168
$302$	−4.77868	−0.274982
$303$	0	0
$304$	3.34886	0.192070
$305$	−13.1724	−0.754248
$306$	0	0
$307$	−4.53549	−0.258854	−0.129427	−	0.991589i	$-0.541314\pi$
$307$	−4.53549	−0.258854	−0.129427	+	0.991589i	$0.541314\pi$
$308$	−7.60198	−0.433163
$309$	0	0
$310$	−3.88291	−0.220534
$311$	1.67918	0.0952177	0.0476089	−	0.998866i	$-0.484840\pi$
$311$	1.67918	0.0952177	0.0476089	+	0.998866i	$0.484840\pi$
$312$	0	0
$313$	−12.3525	−0.698207	−0.349103	−	0.937084i	$-0.613514\pi$
$313$	−12.3525	−0.698207	−0.349103	+	0.937084i	$0.613514\pi$
$314$	−7.88482	−0.444966
$315$	0	0
$316$	−11.4361	−0.643328
$317$	28.1674	1.58204	0.791018	−	0.611792i	$-0.209551\pi$
$317$	28.1674	1.58204	0.791018	+	0.611792i	$0.209551\pi$
$318$	0	0
$319$	−2.65302	−0.148540
$320$	−7.22886	−0.404106
$321$	0	0
$322$	−0.177711	−0.00990346
$323$	−4.00898	−0.223066
$324$	0	0
$325$	−11.0042	−0.610403
$326$	−6.40860	−0.354939
$327$	0	0
$328$	14.1170	0.779483
$329$	−1.64871	−0.0908966
$330$	0	0
$331$	16.5839	0.911533	0.455767	−	0.890099i	$-0.349365\pi$
$331$	16.5839	0.911533	0.455767	+	0.890099i	$0.349365\pi$
$332$	−31.0296	−1.70297
$333$	0	0
$334$	1.66198	0.0909393
$335$	−8.53118	−0.466108
$336$	0	0
$337$	7.12703	0.388234	0.194117	−	0.980978i	$-0.437816\pi$
$337$	7.12703	0.388234	0.194117	+	0.980978i	$0.437816\pi$
$338$	−0.514970	−0.0280107
$339$	0	0
$340$	10.0151	0.543145
$341$	−21.5534	−1.16718
$342$	0	0
$343$	−18.6004	−1.00433
$344$	−1.36318	−0.0734979
$345$	0	0
$346$	−1.20805	−0.0649449
$347$	−12.7356	−0.683681	−0.341841	−	0.939758i	$-0.611050\pi$
$347$	−12.7356	−0.683681	−0.341841	+	0.939758i	$0.611050\pi$
$348$	0	0
$349$	−27.7139	−1.48349	−0.741746	−	0.670681i	$-0.766002\pi$
$349$	−27.7139	−1.48349	−0.741746	+	0.670681i	$0.766002\pi$
$350$	−1.78270	−0.0952896
$351$	0	0
$352$	9.02936	0.481267
$353$	−9.80556	−0.521897	−0.260949	−	0.965353i	$-0.584035\pi$
$353$	−9.80556	−0.521897	−0.260949	+	0.965353i	$0.584035\pi$
$354$	0	0
$355$	2.70170	0.143391
$356$	−3.36504	−0.178347
$357$	0	0
$358$	−1.50736	−0.0796663
$359$	7.32485	0.386591	0.193295	−	0.981141i	$-0.438082\pi$
$359$	7.32485	0.386591	0.193295	+	0.981141i	$0.438082\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	5.82479	0.306144
$363$	0	0
$364$	−10.5416	−0.552528
$365$	15.6491	0.819109
$366$	0	0
$367$	−26.0038	−1.35739	−0.678695	−	0.734421i	$-0.737454\pi$
$367$	−26.0038	−1.35739	−0.678695	+	0.734421i	$0.737454\pi$
$368$	−1.08558	−0.0565900
$369$	0	0
$370$	−4.36547	−0.226950
$371$	−11.3421	−0.588854
$372$	0	0
$373$	0.312012	0.0161554	0.00807768	−	0.999967i	$-0.497429\pi$
$373$	0.312012	0.0161554	0.00807768	+	0.999967i	$0.497429\pi$
$374$	−3.25300	−0.168209
$375$	0	0
$376$	1.29327	0.0666952
$377$	−3.67890	−0.189473
$378$	0	0
$379$	20.9102	1.07408	0.537042	−	0.843556i	$-0.319542\pi$
$379$	20.9102	1.07408	0.537042	+	0.843556i	$0.319542\pi$
$380$	−2.49816	−0.128153
$381$	0	0
$382$	4.27661	0.218810
$383$	7.17012	0.366376	0.183188	−	0.983078i	$-0.441358\pi$
$383$	7.17012	0.366376	0.183188	+	0.983078i	$0.441358\pi$
$384$	0	0
$385$	5.31964	0.271114
$386$	−0.561307	−0.0285698
$387$	0	0
$388$	28.6726	1.45563
$389$	9.97273	0.505638	0.252819	−	0.967514i	$-0.418642\pi$
$389$	9.97273	0.505638	0.252819	+	0.967514i	$0.418642\pi$
$390$	0	0
$391$	1.29958	0.0657223
$392$	5.53744	0.279683
$393$	0	0
$394$	2.76334	0.139215
$395$	8.00261	0.402655
$396$	0	0
$397$	28.3725	1.42398	0.711988	−	0.702191i	$-0.247795\pi$
$397$	28.3725	1.42398	0.711988	+	0.702191i	$0.247795\pi$
$398$	−7.05314	−0.353542
$399$	0	0
$400$	−10.8900	−0.544501
$401$	−36.5033	−1.82289	−0.911445	−	0.411422i	$-0.865032\pi$
$401$	−36.5033	−1.82289	−0.911445	+	0.411422i	$0.865032\pi$
$402$	0	0
$403$	−29.8878	−1.48882
$404$	−7.62363	−0.379290
$405$	0	0
$406$	−0.595990	−0.0295785
$407$	−24.2320	−1.20114
$408$	0	0
$409$	26.3260	1.30174	0.650868	−	0.759191i	$-0.274405\pi$
$409$	26.3260	1.30174	0.650868	+	0.759191i	$0.274405\pi$
$410$	−4.79893	−0.237002
$411$	0	0
$412$	−16.4038	−0.808159
$413$	12.8515	0.632379
$414$	0	0
$415$	21.7136	1.06588
$416$	12.5209	0.613887
$417$	0	0
$418$	0.811429	0.0396883
$419$	8.18126	0.399681	0.199840	−	0.979828i	$-0.435958\pi$
$419$	8.18126	0.399681	0.199840	+	0.979828i	$0.435958\pi$
$420$	0	0
$421$	6.83502	0.333119	0.166559	−	0.986031i	$-0.446734\pi$
$421$	6.83502	0.333119	0.166559	+	0.986031i	$0.446734\pi$
$422$	−4.25967	−0.207357
$423$	0	0
$424$	8.89688	0.432071
$425$	13.0367	0.632371
$426$	0	0
$427$	−16.4256	−0.794892
$428$	−32.4788	−1.56992
$429$	0	0
$430$	0.463399	0.0223471
$431$	−38.6082	−1.85969	−0.929846	−	0.367950i	$-0.880060\pi$
$431$	−38.6082	−1.85969	−0.929846	+	0.367950i	$0.880060\pi$
$432$	0	0
$433$	−2.77435	−0.133327	−0.0666633	−	0.997776i	$-0.521235\pi$
$433$	−2.77435	−0.133327	−0.0666633	+	0.997776i	$0.521235\pi$
$434$	−4.84189	−0.232418
$435$	0	0
$436$	−12.6137	−0.604088
$437$	−0.324166	−0.0155070
$438$	0	0
$439$	7.05276	0.336610	0.168305	−	0.985735i	$-0.446171\pi$
$439$	7.05276	0.336610	0.168305	+	0.985735i	$0.446171\pi$
$440$	−4.17278	−0.198929
$441$	0	0
$442$	−4.51089	−0.214561
$443$	−0.592635	−0.0281569	−0.0140785	−	0.999901i	$-0.504481\pi$
$443$	−0.592635	−0.0281569	−0.0140785	+	0.999901i	$0.504481\pi$
$444$	0	0
$445$	2.35475	0.111626
$446$	−2.97658	−0.140945
$447$	0	0
$448$	−9.01420	−0.425881
$449$	−8.79097	−0.414871	−0.207436	−	0.978249i	$-0.566512\pi$
$449$	−8.79097	−0.414871	−0.207436	+	0.978249i	$0.566512\pi$
$450$	0	0
$451$	−26.6381	−1.25434
$452$	20.9289	0.984413
$453$	0	0
$454$	−0.516396	−0.0242357
$455$	7.37667	0.345823
$456$	0	0
$457$	12.6963	0.593908	0.296954	−	0.954892i	$-0.404029\pi$
$457$	12.6963	0.593908	0.296954	+	0.954892i	$0.404029\pi$
$458$	−0.515303	−0.0240785
$459$	0	0
$460$	0.809819	0.0377580
$461$	8.08904	0.376744	0.188372	−	0.982098i	$-0.439679\pi$
$461$	8.08904	0.376744	0.188372	+	0.982098i	$0.439679\pi$
$462$	0	0
$463$	−13.3725	−0.621473	−0.310736	−	0.950496i	$-0.600576\pi$
$463$	−13.3725	−0.621473	−0.310736	+	0.950496i	$0.600576\pi$
$464$	−3.64072	−0.169016
$465$	0	0
$466$	−4.08687	−0.189321
$467$	34.4756	1.59534	0.797669	−	0.603095i	$-0.206066\pi$
$467$	34.4756	1.59534	0.797669	+	0.603095i	$0.206066\pi$
$468$	0	0
$469$	−10.6382	−0.491225
$470$	−0.439633	−0.0202787
$471$	0	0
$472$	−10.0808	−0.464008
$473$	2.57225	0.118272
$474$	0	0
$475$	−3.25186	−0.149206
$476$	12.4886	0.572412
$477$	0	0
$478$	−8.69781	−0.397828
$479$	−0.545440	−0.0249218	−0.0124609	−	0.999922i	$-0.503967\pi$
$479$	−0.545440	−0.0249218	−0.0124609	+	0.999922i	$0.503967\pi$
$480$	0	0
$481$	−33.6022	−1.53213
$482$	4.60053	0.209549
$483$	0	0
$484$	9.53181	0.433264
$485$	−20.0642	−0.911069
$486$	0	0
$487$	−0.931551	−0.0422126	−0.0211063	−	0.999777i	$-0.506719\pi$
$487$	−0.931551	−0.0422126	−0.0211063	+	0.999777i	$0.506719\pi$
$488$	12.8844	0.583251
$489$	0	0
$490$	−1.88239	−0.0850378
$491$	24.3142	1.09729	0.548643	−	0.836057i	$-0.315145\pi$
$491$	24.3142	1.09729	0.548643	+	0.836057i	$0.315145\pi$
$492$	0	0
$493$	4.35838	0.196292
$494$	1.12520	0.0506250
$495$	0	0
$496$	−29.5777	−1.32808
$497$	3.36895	0.151118
$498$	0	0
$499$	10.6587	0.477148	0.238574	−	0.971124i	$-0.423320\pi$
$499$	10.6587	0.477148	0.238574	+	0.971124i	$0.423320\pi$
$500$	20.6145	0.921908
$501$	0	0
$502$	4.94033	0.220498
$503$	−41.8080	−1.86413	−0.932063	−	0.362296i	$-0.881993\pi$
$503$	−41.8080	−1.86413	−0.932063	+	0.362296i	$0.881993\pi$
$504$	0	0
$505$	5.33478	0.237395
$506$	−0.263038	−0.0116934
$507$	0	0
$508$	−24.3444	−1.08011
$509$	26.2136	1.16190	0.580948	−	0.813941i	$-0.302682\pi$
$509$	26.2136	1.16190	0.580948	+	0.813941i	$0.302682\pi$
$510$	0	0
$511$	19.5140	0.863247
$512$	21.0529	0.930415
$513$	0	0
$514$	8.80960	0.388575
$515$	11.4789	0.505821
$516$	0	0
$517$	−2.44033	−0.107326
$518$	−5.44363	−0.239179
$519$	0	0
$520$	−5.78633	−0.253747
$521$	−32.4015	−1.41953	−0.709767	−	0.704437i	$-0.751200\pi$
$521$	−32.4015	−1.41953	−0.709767	+	0.704437i	$0.751200\pi$
$522$	0	0
$523$	−10.0983	−0.441569	−0.220785	−	0.975323i	$-0.570862\pi$
$523$	−10.0983	−0.441569	−0.220785	+	0.975323i	$0.570862\pi$
$524$	22.0663	0.963969
$525$	0	0
$526$	9.36530	0.408346
$527$	35.4080	1.54240
$528$	0	0
$529$	−22.8949	−0.995431
$530$	−3.02440	−0.131371
$531$	0	0
$532$	−3.11515	−0.135059
$533$	−36.9387	−1.59999
$534$	0	0
$535$	22.7277	0.982603
$536$	8.34469	0.360436
$537$	0	0
$538$	3.09310	0.133353
$539$	−10.4489	−0.450064
$540$	0	0
$541$	18.2570	0.784932	0.392466	−	0.919767i	$-0.371622\pi$
$541$	18.2570	0.784932	0.392466	+	0.919767i	$0.371622\pi$
$542$	−6.04672	−0.259729
$543$	0	0
$544$	−14.8335	−0.635980
$545$	8.82671	0.378095
$546$	0	0
$547$	−22.6122	−0.966827	−0.483414	−	0.875392i	$-0.660603\pi$
$547$	−22.6122	−0.966827	−0.483414	+	0.875392i	$0.660603\pi$
$548$	−4.69153	−0.200412
$549$	0	0
$550$	−2.63865	−0.112513
$551$	−1.08715	−0.0463144
$552$	0	0
$553$	9.97905	0.424352
$554$	−1.37297	−0.0583319
$555$	0	0
$556$	7.00296	0.296992
$557$	11.0869	0.469766	0.234883	−	0.972024i	$-0.424529\pi$
$557$	11.0869	0.469766	0.234883	+	0.972024i	$0.424529\pi$
$558$	0	0
$559$	3.56691	0.150864
$560$	7.30012	0.308486
$561$	0	0
$562$	−7.54653	−0.318331
$563$	−27.8896	−1.17541	−0.587703	−	0.809077i	$-0.699968\pi$
$563$	−27.8896	−1.17541	−0.587703	+	0.809077i	$0.699968\pi$
$564$	0	0
$565$	−14.6454	−0.616137
$566$	−6.16292	−0.259047
$567$	0	0
$568$	−2.64264	−0.110883
$569$	−11.5325	−0.483469	−0.241734	−	0.970342i	$-0.577716\pi$
$569$	−11.5325	−0.483469	−0.241734	+	0.970342i	$0.577716\pi$
$570$	0	0
$571$	−19.6098	−0.820643	−0.410322	−	0.911941i	$-0.634584\pi$
$571$	−19.6098	−0.820643	−0.410322	+	0.911941i	$0.634584\pi$
$572$	−15.6030	−0.652394
$573$	0	0
$574$	−5.98415	−0.249774
$575$	1.05414	0.0439608
$576$	0	0
$577$	8.75552	0.364497	0.182248	−	0.983253i	$-0.441662\pi$
$577$	8.75552	0.364497	0.182248	+	0.983253i	$0.441662\pi$
$578$	−0.308587	−0.0128355
$579$	0	0
$580$	2.71589	0.112771
$581$	27.0763	1.12331
$582$	0	0
$583$	−16.7879	−0.695285
$584$	−15.3070	−0.633407
$585$	0	0
$586$	6.40113	0.264428
$587$	16.2672	0.671421	0.335711	−	0.941965i	$-0.391024\pi$
$587$	16.2672	0.671421	0.335711	+	0.941965i	$0.391024\pi$
$588$	0	0
$589$	−8.83217	−0.363923
$590$	3.42687	0.141082
$591$	0	0
$592$	−33.2535	−1.36671
$593$	−32.0414	−1.31578	−0.657892	−	0.753112i	$-0.728552\pi$
$593$	−32.0414	−1.31578	−0.657892	+	0.753112i	$0.728552\pi$
$594$	0	0
$595$	−8.73912	−0.358269
$596$	22.4628	0.920112
$597$	0	0
$598$	−0.364750	−0.0149158
$599$	−3.41156	−0.139393	−0.0696963	−	0.997568i	$-0.522203\pi$
$599$	−3.41156	−0.139393	−0.0696963	+	0.997568i	$0.522203\pi$
$600$	0	0
$601$	24.1186	0.983817	0.491908	−	0.870647i	$-0.336300\pi$
$601$	24.1186	0.983817	0.491908	+	0.870647i	$0.336300\pi$
$602$	0.577847	0.0235513
$603$	0	0
$604$	27.1543	1.10489
$605$	−6.67008	−0.271177
$606$	0	0
$607$	−32.5534	−1.32130	−0.660650	−	0.750694i	$-0.729719\pi$
$607$	−32.5534	−1.32130	−0.660650	+	0.750694i	$0.729719\pi$
$608$	3.70006	0.150057
$609$	0	0
$610$	−4.37992	−0.177338
$611$	−3.38397	−0.136901
$612$	0	0
$613$	26.5577	1.07266	0.536328	−	0.844010i	$-0.319811\pi$
$613$	26.5577	1.07266	0.536328	+	0.844010i	$0.319811\pi$
$614$	−1.50808	−0.0608613
$615$	0	0
$616$	−5.20335	−0.209649
$617$	1.60216	0.0645005	0.0322503	−	0.999480i	$-0.489733\pi$
$617$	1.60216	0.0645005	0.0322503	+	0.999480i	$0.489733\pi$
$618$	0	0
$619$	23.8475	0.958512	0.479256	−	0.877675i	$-0.340907\pi$
$619$	23.8475	0.958512	0.479256	+	0.877675i	$0.340907\pi$
$620$	22.0642	0.886119
$621$	0	0
$622$	0.558341	0.0223874
$623$	2.93632	0.117641
$624$	0	0
$625$	1.83390	0.0733562
$626$	−4.10732	−0.164161
$627$	0	0
$628$	44.8046	1.78790
$629$	39.8085	1.58727
$630$	0	0
$631$	36.3737	1.44801	0.724007	−	0.689793i	$-0.242299\pi$
$631$	36.3737	1.44801	0.724007	+	0.689793i	$0.242299\pi$
$632$	−7.82767	−0.311368
$633$	0	0
$634$	9.36587	0.371966
$635$	17.0355	0.676034
$636$	0	0
$637$	−14.4893	−0.574086
$638$	−0.882149	−0.0349246
$639$	0	0
$640$	−12.1879	−0.481768
$641$	26.9679	1.06517	0.532584	−	0.846377i	$-0.321221\pi$
$641$	26.9679	1.06517	0.532584	+	0.846377i	$0.321221\pi$
$642$	0	0
$643$	−10.0717	−0.397189	−0.198594	−	0.980082i	$-0.563638\pi$
$643$	−10.0717	−0.397189	−0.198594	+	0.980082i	$0.563638\pi$
$644$	1.00982	0.0397926
$645$	0	0
$646$	−1.33302	−0.0524469
$647$	21.5511	0.847261	0.423630	−	0.905835i	$-0.360756\pi$
$647$	21.5511	0.847261	0.423630	+	0.905835i	$0.360756\pi$
$648$	0	0
$649$	19.0220	0.746678
$650$	−3.65898	−0.143517
$651$	0	0
$652$	36.4161	1.42617
$653$	−6.85949	−0.268433	−0.134216	−	0.990952i	$-0.542852\pi$
$653$	−6.85949	−0.268433	−0.134216	+	0.990952i	$0.542852\pi$
$654$	0	0
$655$	−15.4413	−0.603341
$656$	−36.5554	−1.42725
$657$	0	0
$658$	−0.548211	−0.0213715
$659$	8.93826	0.348185	0.174093	−	0.984729i	$-0.444301\pi$
$659$	8.93826	0.348185	0.174093	+	0.984729i	$0.444301\pi$
$660$	0	0
$661$	−14.6153	−0.568468	−0.284234	−	0.958755i	$-0.591739\pi$
$661$	−14.6153	−0.568468	−0.284234	+	0.958755i	$0.591739\pi$
$662$	5.51428	0.214318
$663$	0	0
$664$	−21.2389	−0.824230
$665$	2.17988	0.0845323
$666$	0	0
$667$	0.352419	0.0136457
$668$	−9.44400	−0.365399
$669$	0	0
$670$	−2.83669	−0.109591
$671$	−24.3122	−0.938564
$672$	0	0
$673$	−29.8703	−1.15142	−0.575708	−	0.817655i	$-0.695274\pi$
$673$	−29.8703	−1.15142	−0.575708	+	0.817655i	$0.695274\pi$
$674$	2.36979	0.0912810
$675$	0	0
$676$	2.92626	0.112548
$677$	2.42622	0.0932473	0.0466236	−	0.998913i	$-0.485154\pi$
$677$	2.42622	0.0932473	0.0466236	+	0.998913i	$0.485154\pi$
$678$	0	0
$679$	−25.0196	−0.960163
$680$	6.85505	0.262879
$681$	0	0
$682$	−7.16668	−0.274426
$683$	1.18454	0.0453251	0.0226625	−	0.999743i	$-0.492786\pi$
$683$	1.18454	0.0453251	0.0226625	+	0.999743i	$0.492786\pi$
$684$	0	0
$685$	3.28299	0.125437
$686$	−6.18477	−0.236136
$687$	0	0
$688$	3.52989	0.134576
$689$	−23.2796	−0.886882
$690$	0	0
$691$	5.95641	0.226593	0.113296	−	0.993561i	$-0.463859\pi$
$691$	5.95641	0.226593	0.113296	+	0.993561i	$0.463859\pi$
$692$	6.86458	0.260952
$693$	0	0
$694$	−4.23468	−0.160746
$695$	−4.90046	−0.185885
$696$	0	0
$697$	43.7612	1.65757
$698$	−9.21509	−0.348796
$699$	0	0
$700$	10.1300	0.382879
$701$	−12.9742	−0.490030	−0.245015	−	0.969519i	$-0.578793\pi$
$701$	−12.9742	−0.490030	−0.245015	+	0.969519i	$0.578793\pi$
$702$	0	0
$703$	−9.92982	−0.374510
$704$	−13.3423	−0.502857
$705$	0	0
$706$	−3.26042	−0.122708
$707$	6.65234	0.250187
$708$	0	0
$709$	41.1655	1.54600	0.773001	−	0.634405i	$-0.218755\pi$
$709$	41.1655	1.54600	0.773001	+	0.634405i	$0.218755\pi$
$710$	0.898336	0.0337140
$711$	0	0
$712$	−2.30328	−0.0863189
$713$	2.86309	0.107223
$714$	0	0
$715$	10.9185	0.408329
$716$	8.56539	0.320104
$717$	0	0
$718$	2.43557	0.0908947
$719$	28.3678	1.05794	0.528970	−	0.848640i	$-0.322578\pi$
$719$	28.3678	1.05794	0.528970	+	0.848640i	$0.322578\pi$
$720$	0	0
$721$	14.3139	0.533078
$722$	0.332508	0.0123747
$723$	0	0
$724$	−33.0987	−1.23010
$725$	3.53528	0.131297
$726$	0	0
$727$	−8.00585	−0.296921	−0.148460	−	0.988918i	$-0.547432\pi$
$727$	−8.00585	−0.296921	−0.148460	+	0.988918i	$0.547432\pi$
$728$	−7.21541	−0.267421
$729$	0	0
$730$	5.20343	0.192588
$731$	−4.22571	−0.156293
$732$	0	0
$733$	−35.8913	−1.32568	−0.662838	−	0.748763i	$-0.730648\pi$
$733$	−35.8913	−1.32568	−0.662838	+	0.748763i	$0.730648\pi$
$734$	−8.64648	−0.319147
$735$	0	0
$736$	−1.19943	−0.0442117
$737$	−15.7460	−0.580011
$738$	0	0
$739$	−3.68602	−0.135592	−0.0677961	−	0.997699i	$-0.521597\pi$
$739$	−3.68602	−0.135592	−0.0677961	+	0.997699i	$0.521597\pi$
$740$	24.8063	0.911898
$741$	0	0
$742$	−3.77134	−0.138450
$743$	−31.6280	−1.16032	−0.580159	−	0.814504i	$-0.697009\pi$
$743$	−31.6280	−1.16032	−0.580159	+	0.814504i	$0.697009\pi$
$744$	0	0
$745$	−15.7188	−0.575892
$746$	0.103746	0.00379842
$747$	0	0
$748$	18.4848	0.675873
$749$	28.3408	1.03555
$750$	0	0
$751$	0.671939	0.0245194	0.0122597	−	0.999925i	$-0.496098\pi$
$751$	0.671939	0.0245194	0.0122597	+	0.999925i	$0.496098\pi$
$752$	−3.34886	−0.122120
$753$	0	0
$754$	−1.22326	−0.0445486
$755$	−19.0018	−0.691545
$756$	0	0
$757$	25.0904	0.911925	0.455963	−	0.889999i	$-0.349295\pi$
$757$	25.0904	0.911925	0.455963	+	0.889999i	$0.349295\pi$
$758$	6.95280	0.252537
$759$	0	0
$760$	−1.70992	−0.0620255
$761$	−21.8703	−0.792798	−0.396399	−	0.918078i	$-0.629740\pi$
$761$	−21.8703	−0.792798	−0.396399	+	0.918078i	$0.629740\pi$
$762$	0	0
$763$	11.0067	0.398469
$764$	−24.3014	−0.879192
$765$	0	0
$766$	2.38412	0.0861418
$767$	26.3775	0.952437
$768$	0	0
$769$	−17.6414	−0.636166	−0.318083	−	0.948063i	$-0.603039\pi$
$769$	−17.6414	−0.636166	−0.318083	+	0.948063i	$0.603039\pi$
$770$	1.76882	0.0637439
$771$	0	0
$772$	3.18957	0.114795
$773$	−31.9647	−1.14969	−0.574846	−	0.818262i	$-0.694938\pi$
$773$	−31.9647	−1.14969	−0.574846	+	0.818262i	$0.694938\pi$
$774$	0	0
$775$	28.7210	1.03169
$776$	19.6256	0.704518
$777$	0	0
$778$	3.31601	0.118885
$779$	−10.9158	−0.391099
$780$	0	0
$781$	4.98652	0.178432
$782$	0.432119	0.0154525
$783$	0	0
$784$	−14.3389	−0.512104
$785$	−31.3529	−1.11903
$786$	0	0
$787$	22.3551	0.796873	0.398436	−	0.917196i	$-0.369553\pi$
$787$	22.3551	0.796873	0.398436	+	0.917196i	$0.369553\pi$
$788$	−15.7024	−0.559374
$789$	0	0
$790$	2.66093	0.0946716
$791$	−18.2625	−0.649338
$792$	0	0
$793$	−33.7134	−1.19720
$794$	9.43409	0.334803
$795$	0	0
$796$	40.0787	1.42055
$797$	−2.30656	−0.0817024	−0.0408512	−	0.999165i	$-0.513007\pi$
$797$	−2.30656	−0.0817024	−0.0408512	+	0.999165i	$0.513007\pi$
$798$	0	0
$799$	4.00898	0.141828
$800$	−12.0321	−0.425398
$801$	0	0
$802$	−12.1376	−0.428595
$803$	28.8834	1.01927
$804$	0	0
$805$	−0.706644	−0.0249059
$806$	−9.93793	−0.350049
$807$	0	0
$808$	−5.21816	−0.183574
$809$	0.949289	0.0333752	0.0166876	−	0.999861i	$-0.494688\pi$
$809$	0.949289	0.0333752	0.0166876	+	0.999861i	$0.494688\pi$
$810$	0	0
$811$	−45.2275	−1.58815	−0.794075	−	0.607819i	$-0.792044\pi$
$811$	−45.2275	−1.58815	−0.794075	+	0.607819i	$0.792044\pi$
$812$	3.38665	0.118848
$813$	0	0
$814$	−8.05734	−0.282410
$815$	−25.4829	−0.892628
$816$	0	0
$817$	1.05406	0.0368769
$818$	8.75360	0.306062
$819$	0	0
$820$	27.2694	0.952289
$821$	6.35107	0.221654	0.110827	−	0.993840i	$-0.464650\pi$
$821$	6.35107	0.221654	0.110827	+	0.993840i	$0.464650\pi$
$822$	0	0
$823$	49.5351	1.72669	0.863343	−	0.504618i	$-0.168367\pi$
$823$	49.5351	1.72669	0.863343	+	0.504618i	$0.168367\pi$
$824$	−11.2280	−0.391145
$825$	0	0
$826$	4.27321	0.148684
$827$	16.5865	0.576768	0.288384	−	0.957515i	$-0.406882\pi$
$827$	16.5865	0.576768	0.288384	+	0.957515i	$0.406882\pi$
$828$	0	0
$829$	−8.51571	−0.295763	−0.147881	−	0.989005i	$-0.547245\pi$
$829$	−8.51571	−0.295763	−0.147881	+	0.989005i	$0.547245\pi$
$830$	7.21994	0.250608
$831$	0	0
$832$	−18.5016	−0.641426
$833$	17.1654	0.594746
$834$	0	0
$835$	6.60863	0.228701
$836$	−4.61085	−0.159470
$837$	0	0
$838$	2.72033	0.0939723
$839$	−37.3834	−1.29062	−0.645309	−	0.763922i	$-0.723271\pi$
$839$	−37.3834	−1.29062	−0.645309	+	0.763922i	$0.723271\pi$
$840$	0	0
$841$	−27.8181	−0.959245
$842$	2.27270	0.0783224
$843$	0	0
$844$	24.2051	0.833173
$845$	−2.04771	−0.0704432
$846$	0	0
$847$	−8.31741	−0.285790
$848$	−23.0380	−0.791129
$849$	0	0
$850$	4.33479	0.148682
$851$	3.21891	0.110343
$852$	0	0
$853$	0.190743	0.00653091	0.00326546	−	0.999995i	$-0.498961\pi$
$853$	0.190743	0.00653091	0.00326546	+	0.999995i	$0.498961\pi$
$854$	−5.46165	−0.186894
$855$	0	0
$856$	−22.2308	−0.759835
$857$	−30.7258	−1.04957	−0.524786	−	0.851234i	$-0.675855\pi$
$857$	−30.7258	−1.04957	−0.524786	+	0.851234i	$0.675855\pi$
$858$	0	0
$859$	38.8072	1.32408	0.662042	−	0.749466i	$-0.269690\pi$
$859$	38.8072	1.32408	0.662042	+	0.749466i	$0.269690\pi$
$860$	−2.63321	−0.0897918
$861$	0	0
$862$	−12.8375	−0.437248
$863$	−51.8601	−1.76534	−0.882668	−	0.469996i	$-0.844255\pi$
$863$	−51.8601	−1.76534	−0.882668	+	0.469996i	$0.844255\pi$
$864$	0	0
$865$	−4.80363	−0.163328
$866$	−0.922493	−0.0313476
$867$	0	0
$868$	27.5135	0.933869
$869$	14.7704	0.501051
$870$	0	0
$871$	−21.8347	−0.739842
$872$	−8.63375	−0.292376
$873$	0	0
$874$	−0.107788	−0.00364597
$875$	−17.9881	−0.608109
$876$	0	0
$877$	49.0936	1.65777	0.828887	−	0.559417i	$-0.188975\pi$
$877$	49.0936	1.65777	0.828887	+	0.559417i	$0.188975\pi$
$878$	2.34510	0.0791432
$879$	0	0
$880$	10.8052	0.364243
$881$	−27.1421	−0.914442	−0.457221	−	0.889353i	$-0.651155\pi$
$881$	−27.1421	−0.914442	−0.457221	+	0.889353i	$0.651155\pi$
$882$	0	0
$883$	−0.854391	−0.0287525	−0.0143763	−	0.999897i	$-0.504576\pi$
$883$	−0.854391	−0.0287525	−0.0143763	+	0.999897i	$0.504576\pi$
$884$	25.6326	0.862119
$885$	0	0
$886$	−0.197056	−0.00662021
$887$	20.8030	0.698495	0.349247	−	0.937031i	$-0.386437\pi$
$887$	20.8030	0.698495	0.349247	+	0.937031i	$0.386437\pi$
$888$	0	0
$889$	21.2428	0.712462
$890$	0.782974	0.0262453
$891$	0	0
$892$	16.9141	0.566325
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−5.99380	−0.200351
$896$	−15.1980	−0.507729
$897$	0	0
$898$	−2.92307	−0.0975440
$899$	9.60194	0.320242
$900$	0	0
$901$	27.5793	0.918799
$902$	−8.85738	−0.294919
$903$	0	0
$904$	14.3253	0.476451
$905$	23.1615	0.769914
$906$	0	0
$907$	45.1876	1.50043	0.750215	−	0.661194i	$-0.229950\pi$
$907$	45.1876	1.50043	0.750215	+	0.661194i	$0.229950\pi$
$908$	2.93437	0.0973803
$909$	0	0
$910$	2.45280	0.0813095
$911$	−0.296402	−0.00982024	−0.00491012	−	0.999988i	$-0.501563\pi$
$911$	−0.296402	−0.00982024	−0.00491012	+	0.999988i	$0.501563\pi$
$912$	0	0
$913$	40.0767	1.32635
$914$	4.22162	0.139639
$915$	0	0
$916$	2.92815	0.0967488
$917$	−19.2549	−0.635853
$918$	0	0
$919$	−36.0236	−1.18831	−0.594154	−	0.804351i	$-0.702513\pi$
$919$	−36.0236	−1.18831	−0.594154	+	0.804351i	$0.702513\pi$
$920$	0.554299	0.0182747
$921$	0	0
$922$	2.68967	0.0885795
$923$	6.91474	0.227601
$924$	0	0
$925$	32.2904	1.06170
$926$	−4.44646	−0.146120
$927$	0	0
$928$	−4.02254	−0.132046
$929$	6.40106	0.210012	0.105006	−	0.994472i	$-0.466514\pi$
$929$	6.40106	0.210012	0.105006	+	0.994472i	$0.466514\pi$
$930$	0	0
$931$	−4.28174	−0.140328
$932$	23.2232	0.760701
$933$	0	0
$934$	11.4634	0.375094
$935$	−12.9351	−0.423024
$936$	0	0
$937$	−49.4161	−1.61435	−0.807177	−	0.590309i	$-0.799006\pi$
$937$	−49.4161	−1.61435	−0.807177	+	0.590309i	$0.799006\pi$
$938$	−3.53727	−0.115496
$939$	0	0
$940$	2.49816	0.0814811
$941$	48.2293	1.57223	0.786114	−	0.618081i	$-0.212090\pi$
$941$	48.2293	1.57223	0.786114	+	0.618081i	$0.212090\pi$
$942$	0	0
$943$	3.53852	0.115230
$944$	26.1038	0.849606
$945$	0	0
$946$	0.855295	0.0278080
$947$	−52.9692	−1.72127	−0.860635	−	0.509223i	$-0.829933\pi$
$947$	−52.9692	−1.72127	−0.860635	+	0.509223i	$0.829933\pi$
$948$	0	0
$949$	40.0522	1.30015
$950$	−1.08127	−0.0350810
$951$	0	0
$952$	8.54808	0.277045
$953$	−15.8307	−0.512806	−0.256403	−	0.966570i	$-0.582537\pi$
$953$	−15.8307	−0.512806	−0.256403	+	0.966570i	$0.582537\pi$
$954$	0	0
$955$	17.0054	0.550280
$956$	49.4243	1.59850
$957$	0	0
$958$	−0.181363	−0.00585957
$959$	4.09381	0.132196
$960$	0	0
$961$	47.0072	1.51636
$962$	−11.1730	−0.360232
$963$	0	0
$964$	−26.1420	−0.841978
$965$	−2.23196	−0.0718494
$966$	0	0
$967$	−39.5222	−1.27095	−0.635473	−	0.772123i	$-0.719195\pi$
$967$	−39.5222	−1.27095	−0.635473	+	0.772123i	$0.719195\pi$
$968$	6.52426	0.209698
$969$	0	0
$970$	−6.67151	−0.214209
$971$	−31.7372	−1.01850	−0.509248	−	0.860620i	$-0.670076\pi$
$971$	−31.7372	−1.01850	−0.509248	+	0.860620i	$0.670076\pi$
$972$	0	0
$973$	−6.11075	−0.195902
$974$	−0.309748	−0.00992496
$975$	0	0
$976$	−33.3636	−1.06794
$977$	40.5924	1.29867	0.649333	−	0.760504i	$-0.275048\pi$
$977$	40.5924	1.29867	0.649333	+	0.760504i	$0.275048\pi$
$978$	0	0
$979$	4.34616	0.138904
$980$	10.6965	0.341687
$981$	0	0
$982$	8.08468	0.257992
$983$	31.4644	1.00356	0.501780	−	0.864996i	$-0.332679\pi$
$983$	31.4644	1.00356	0.501780	+	0.864996i	$0.332679\pi$
$984$	0	0
$985$	10.9880	0.350108
$986$	1.44920	0.0461518
$987$	0	0
$988$	−6.39380	−0.203414
$989$	−0.341690	−0.0108651
$990$	0	0
$991$	4.62710	0.146985	0.0734924	−	0.997296i	$-0.476586\pi$
$991$	4.62710	0.146985	0.0734924	+	0.997296i	$0.476586\pi$
$992$	−32.6795	−1.03758
$993$	0	0
$994$	1.12020	0.0355307
$995$	−28.0458	−0.889113
$996$	0	0
$997$	25.7084	0.814192	0.407096	−	0.913385i	$-0.366541\pi$
$997$	25.7084	0.814192	0.407096	+	0.913385i	$0.366541\pi$
$998$	3.54409	0.112186
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.6		12
3.2	odd	2		893.2.a.a.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.7	✓	12	3.2	odd	2
8037.2.a.m.1.6		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.332508 q^{2} -1.88944 q^{4} +1.32217 q^{5} +1.64871 q^{7} -1.29327 q^{8} +O(q^{10})\)	\(q+0.332508 q^{2} -1.88944 q^{4} +1.32217 q^{5} +1.64871 q^{7} -1.29327 q^{8} +0.439633 q^{10} +2.44033 q^{11} +3.38397 q^{13} +0.548211 q^{14} +3.34886 q^{16} -4.00898 q^{17} +1.00000 q^{19} -2.49816 q^{20} +0.811429 q^{22} -0.324166 q^{23} -3.25186 q^{25} +1.12520 q^{26} -3.11515 q^{28} -1.08715 q^{29} -8.83217 q^{31} +3.70006 q^{32} -1.33302 q^{34} +2.17988 q^{35} -9.92982 q^{37} +0.332508 q^{38} -1.70992 q^{40} -10.9158 q^{41} +1.05406 q^{43} -4.61085 q^{44} -0.107788 q^{46} -1.00000 q^{47} -4.28174 q^{49} -1.08127 q^{50} -6.39380 q^{52} -6.87937 q^{53} +3.22654 q^{55} -2.13223 q^{56} -0.361488 q^{58} +7.79484 q^{59} -9.96268 q^{61} -2.93677 q^{62} -5.46741 q^{64} +4.47419 q^{65} -6.45240 q^{67} +7.57473 q^{68} +0.724829 q^{70} +2.04338 q^{71} +11.8359 q^{73} -3.30174 q^{74} -1.88944 q^{76} +4.02341 q^{77} +6.05262 q^{79} +4.42776 q^{80} -3.62958 q^{82} +16.4227 q^{83} -5.30056 q^{85} +0.350483 q^{86} -3.15600 q^{88} +1.78097 q^{89} +5.57920 q^{91} +0.612492 q^{92} -0.332508 q^{94} +1.32217 q^{95} -15.1752 q^{97} -1.42371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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