LMFDB - Embedded newform 4005.2.a.x.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.30924$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30924 q^{2} -0.285886 q^{4} +1.00000 q^{5} +4.28141 q^{7} +2.99278 q^{8} +O(q^{10})$	\(q-1.30924 q^{2} -0.285886 q^{4} +1.00000 q^{5} +4.28141 q^{7} +2.99278 q^{8} -1.30924 q^{10} -3.71314 q^{11} +1.17965 q^{13} -5.60540 q^{14} -3.34650 q^{16} -7.72282 q^{17} -0.216462 q^{19} -0.285886 q^{20} +4.86139 q^{22} +1.27959 q^{23} +1.00000 q^{25} -1.54444 q^{26} -1.22400 q^{28} +5.32898 q^{29} -3.94926 q^{31} -1.60418 q^{32} +10.1110 q^{34} +4.28141 q^{35} +0.356617 q^{37} +0.283401 q^{38} +2.99278 q^{40} +11.1264 q^{41} +3.34534 q^{43} +1.06154 q^{44} -1.67530 q^{46} -2.32065 q^{47} +11.3304 q^{49} -1.30924 q^{50} -0.337245 q^{52} -5.58573 q^{53} -3.71314 q^{55} +12.8133 q^{56} -6.97692 q^{58} +1.34310 q^{59} +3.59677 q^{61} +5.17053 q^{62} +8.79326 q^{64} +1.17965 q^{65} +8.60212 q^{67} +2.20785 q^{68} -5.60540 q^{70} +7.59580 q^{71} +3.56548 q^{73} -0.466898 q^{74} +0.0618836 q^{76} -15.8975 q^{77} +14.7905 q^{79} -3.34650 q^{80} -14.5672 q^{82} -9.17254 q^{83} -7.72282 q^{85} -4.37986 q^{86} -11.1126 q^{88} -1.00000 q^{89} +5.05055 q^{91} -0.365818 q^{92} +3.03829 q^{94} -0.216462 q^{95} +7.73120 q^{97} -14.8343 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8	$ 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) $ 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8 + 5 * q^10 + 2 * q^11 + 8 * q^13 + 4 * q^14 + 33 * q^16 + 10 * q^17 + 32 * q^19 + 21 * q^20 + 8 * q^22 + 15 * q^23 + 17 * q^25 - 15 * q^26 + 24 * q^28 + q^29 + 18 * q^31 + 25 * q^32 + 14 * q^34 + 12 * q^35 + 12 * q^37 + 22 * q^38 + 15 * q^40 - 7 * q^41 + 28 * q^43 - 14 * q^44 + 4 * q^46 + 26 * q^47 + 41 * q^49 + 5 * q^50 + 10 * q^52 + 12 * q^53 + 2 * q^55 + 13 * q^56 + 16 * q^58 - 23 * q^59 + 26 * q^61 + 10 * q^62 + 59 * q^64 + 8 * q^65 + 31 * q^67 - q^68 + 4 * q^70 - 2 * q^71 + 33 * q^73 - 10 * q^74 + 66 * q^76 + 12 * q^77 + 33 * q^79 + 33 * q^80 + 30 * q^82 + 13 * q^83 + 10 * q^85 - 20 * q^86 + 12 * q^88 - 17 * q^89 + 40 * q^91 + 16 * q^92 + 38 * q^94 + 32 * q^95 + 45 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.30924	−0.925774	−0.462887	−	0.886417i	$-0.653186\pi$
$2$	−1.30924	−0.925774	−0.462887	+	0.886417i	$0.653186\pi$
$3$	0	0
$3$	0	0
$4$	−0.285886	−0.142943
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.28141	1.61822	0.809110	−	0.587657i	$-0.199950\pi$
$7$	4.28141	1.61822	0.809110	+	0.587657i	$0.199950\pi$
$8$	2.99278	1.05811
$9$	0	0
$10$	−1.30924	−0.414019
$11$	−3.71314	−1.11955	−0.559777	−	0.828644i	$-0.689113\pi$
$11$	−3.71314	−1.11955	−0.559777	+	0.828644i	$0.689113\pi$
$12$	0	0
$13$	1.17965	0.327175	0.163588	−	0.986529i	$-0.447693\pi$
$13$	1.17965	0.327175	0.163588	+	0.986529i	$0.447693\pi$
$14$	−5.60540	−1.49811
$15$	0	0
$16$	−3.34650	−0.836624
$17$	−7.72282	−1.87306	−0.936529	−	0.350589i	$-0.885981\pi$
$17$	−7.72282	−1.87306	−0.936529	+	0.350589i	$0.885981\pi$
$18$	0	0
$19$	−0.216462	−0.0496598	−0.0248299	−	0.999692i	$-0.507904\pi$
$19$	−0.216462	−0.0496598	−0.0248299	+	0.999692i	$0.507904\pi$
$20$	−0.285886	−0.0639261
$21$	0	0
$22$	4.86139	1.03645
$23$	1.27959	0.266814	0.133407	−	0.991061i	$-0.457408\pi$
$23$	1.27959	0.266814	0.133407	+	0.991061i	$0.457408\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.54444	−0.302890
$27$	0	0
$28$	−1.22400	−0.231313
$29$	5.32898	0.989566	0.494783	−	0.869017i	$-0.335248\pi$
$29$	5.32898	0.989566	0.494783	+	0.869017i	$0.335248\pi$
$30$	0	0
$31$	−3.94926	−0.709308	−0.354654	−	0.934998i	$-0.615401\pi$
$31$	−3.94926	−0.709308	−0.354654	+	0.934998i	$0.615401\pi$
$32$	−1.60418	−0.283582
$33$	0	0
$34$	10.1110	1.73403
$35$	4.28141	0.723690
$36$	0	0
$37$	0.356617	0.0586275	0.0293138	−	0.999570i	$-0.490668\pi$
$37$	0.356617	0.0586275	0.0293138	+	0.999570i	$0.490668\pi$
$38$	0.283401	0.0459738
$39$	0	0
$40$	2.99278	0.473200
$41$	11.1264	1.73766	0.868828	−	0.495113i	$-0.164873\pi$
$41$	11.1264	1.73766	0.868828	+	0.495113i	$0.164873\pi$
$42$	0	0
$43$	3.34534	0.510160	0.255080	−	0.966920i	$-0.417898\pi$
$43$	3.34534	0.510160	0.255080	+	0.966920i	$0.417898\pi$
$44$	1.06154	0.160032
$45$	0	0
$46$	−1.67530	−0.247009
$47$	−2.32065	−0.338501	−0.169251	−	0.985573i	$-0.554135\pi$
$47$	−2.32065	−0.338501	−0.169251	+	0.985573i	$0.554135\pi$
$48$	0	0
$49$	11.3304	1.61864
$50$	−1.30924	−0.185155
$51$	0	0
$52$	−0.337245	−0.0467675
$53$	−5.58573	−0.767259	−0.383630	−	0.923487i	$-0.625326\pi$
$53$	−5.58573	−0.767259	−0.383630	+	0.923487i	$0.625326\pi$
$54$	0	0
$55$	−3.71314	−0.500679
$56$	12.8133	1.71225
$57$	0	0
$58$	−6.97692	−0.916114
$59$	1.34310	0.174857	0.0874287	−	0.996171i	$-0.472135\pi$
$59$	1.34310	0.174857	0.0874287	+	0.996171i	$0.472135\pi$
$60$	0	0
$61$	3.59677	0.460519	0.230260	−	0.973129i	$-0.426042\pi$
$61$	3.59677	0.460519	0.230260	+	0.973129i	$0.426042\pi$
$62$	5.17053	0.656658
$63$	0	0
$64$	8.79326	1.09916
$65$	1.17965	0.146317
$66$	0	0
$67$	8.60212	1.05092	0.525458	−	0.850820i	$-0.323894\pi$
$67$	8.60212	1.05092	0.525458	+	0.850820i	$0.323894\pi$
$68$	2.20785	0.267741
$69$	0	0
$70$	−5.60540	−0.669973
$71$	7.59580	0.901455	0.450728	−	0.892662i	$-0.351165\pi$
$71$	7.59580	0.901455	0.450728	+	0.892662i	$0.351165\pi$
$72$	0	0
$73$	3.56548	0.417307	0.208654	−	0.977990i	$-0.433092\pi$
$73$	3.56548	0.417307	0.208654	+	0.977990i	$0.433092\pi$
$74$	−0.466898	−0.0542758
$75$	0	0
$76$	0.0618836	0.00709854
$77$	−15.8975	−1.81168
$78$	0	0
$79$	14.7905	1.66407	0.832033	−	0.554726i	$-0.187177\pi$
$79$	14.7905	1.66407	0.832033	+	0.554726i	$0.187177\pi$
$80$	−3.34650	−0.374150
$81$	0	0
$82$	−14.5672	−1.60868
$83$	−9.17254	−1.00682	−0.503408	−	0.864049i	$-0.667921\pi$
$83$	−9.17254	−1.00682	−0.503408	+	0.864049i	$0.667921\pi$
$84$	0	0
$85$	−7.72282	−0.837657
$86$	−4.37986	−0.472293
$87$	0	0
$88$	−11.1126	−1.18461
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	5.05055	0.529442
$92$	−0.365818	−0.0381392
$93$	0	0
$94$	3.03829	0.313376
$95$	−0.216462	−0.0222086
$96$	0	0
$97$	7.73120	0.784984	0.392492	−	0.919755i	$-0.371613\pi$
$97$	7.73120	0.784984	0.392492	+	0.919755i	$0.371613\pi$
$98$	−14.8343	−1.49849
$99$	0	0
$100$	−0.285886	−0.0285886
$101$	−0.160949	−0.0160150	−0.00800750	−	0.999968i	$-0.502549\pi$
$101$	−0.160949	−0.0160150	−0.00800750	+	0.999968i	$0.502549\pi$
$102$	0	0
$103$	5.55740	0.547586	0.273793	−	0.961789i	$-0.411722\pi$
$103$	5.55740	0.547586	0.273793	+	0.961789i	$0.411722\pi$
$104$	3.53042	0.346186
$105$	0	0
$106$	7.31307	0.710308
$107$	9.26187	0.895379	0.447689	−	0.894189i	$-0.352247\pi$
$107$	9.26187	0.895379	0.447689	+	0.894189i	$0.352247\pi$
$108$	0	0
$109$	−2.54908	−0.244158	−0.122079	−	0.992520i	$-0.538956\pi$
$109$	−2.54908	−0.244158	−0.122079	+	0.992520i	$0.538956\pi$
$110$	4.86139	0.463516
$111$	0	0
$112$	−14.3277	−1.35384
$113$	−4.32315	−0.406688	−0.203344	−	0.979107i	$-0.565181\pi$
$113$	−4.32315	−0.406688	−0.203344	+	0.979107i	$0.565181\pi$
$114$	0	0
$115$	1.27959	0.119323
$116$	−1.52348	−0.141452
$117$	0	0
$118$	−1.75845	−0.161878
$119$	−33.0645	−3.03102
$120$	0	0
$121$	2.78739	0.253399
$122$	−4.70904	−0.426336
$123$	0	0
$124$	1.12904	0.101391
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.5962	1.73888	0.869441	−	0.494037i	$-0.164479\pi$
$127$	19.5962	1.73888	0.869441	+	0.494037i	$0.164479\pi$
$128$	−8.30413	−0.733988
$129$	0	0
$130$	−1.54444	−0.135457
$131$	−10.1540	−0.887157	−0.443578	−	0.896236i	$-0.646291\pi$
$131$	−10.1540	−0.887157	−0.443578	+	0.896236i	$0.646291\pi$
$132$	0	0
$133$	−0.926763	−0.0803605
$134$	−11.2622	−0.972910
$135$	0	0
$136$	−23.1127	−1.98190
$137$	−2.14694	−0.183425	−0.0917126	−	0.995786i	$-0.529234\pi$
$137$	−2.14694	−0.183425	−0.0917126	+	0.995786i	$0.529234\pi$
$138$	0	0
$139$	3.93489	0.333753	0.166877	−	0.985978i	$-0.446632\pi$
$139$	3.93489	0.333753	0.166877	+	0.985978i	$0.446632\pi$
$140$	−1.22400	−0.103447
$141$	0	0
$142$	−9.94473	−0.834544
$143$	−4.38019	−0.366290
$144$	0	0
$145$	5.32898	0.442547
$146$	−4.66807	−0.386332
$147$	0	0
$148$	−0.101952	−0.00838041
$149$	−18.6498	−1.52785	−0.763924	−	0.645306i	$-0.776730\pi$
$149$	−18.6498	−1.52785	−0.763924	+	0.645306i	$0.776730\pi$
$150$	0	0
$151$	−19.2951	−1.57021	−0.785105	−	0.619362i	$-0.787391\pi$
$151$	−19.2951	−1.57021	−0.785105	+	0.619362i	$0.787391\pi$
$152$	−0.647823	−0.0525454
$153$	0	0
$154$	20.8136	1.67721
$155$	−3.94926	−0.317212
$156$	0	0
$157$	10.1947	0.813625	0.406813	−	0.913512i	$-0.366640\pi$
$157$	10.1947	0.813625	0.406813	+	0.913512i	$0.366640\pi$
$158$	−19.3644	−1.54055
$159$	0	0
$160$	−1.60418	−0.126822
$161$	5.47846	0.431763
$162$	0	0
$163$	5.59553	0.438276	0.219138	−	0.975694i	$-0.429676\pi$
$163$	5.59553	0.438276	0.219138	+	0.975694i	$0.429676\pi$
$164$	−3.18090	−0.248386
$165$	0	0
$166$	12.0091	0.932084
$167$	3.84648	0.297650	0.148825	−	0.988864i	$-0.452451\pi$
$167$	3.84648	0.297650	0.148825	+	0.988864i	$0.452451\pi$
$168$	0	0
$169$	−11.6084	−0.892956
$170$	10.1110	0.775481
$171$	0	0
$172$	−0.956388	−0.0729239
$173$	7.78812	0.592120	0.296060	−	0.955169i	$-0.404327\pi$
$173$	7.78812	0.592120	0.296060	+	0.955169i	$0.404327\pi$
$174$	0	0
$175$	4.28141	0.323644
$176$	12.4260	0.936645
$177$	0	0
$178$	1.30924	0.0981318
$179$	20.4862	1.53121	0.765607	−	0.643309i	$-0.222439\pi$
$179$	20.4862	1.53121	0.765607	+	0.643309i	$0.222439\pi$
$180$	0	0
$181$	15.3218	1.13886	0.569431	−	0.822039i	$-0.307163\pi$
$181$	15.3218	1.13886	0.569431	+	0.822039i	$0.307163\pi$
$182$	−6.61239	−0.490143
$183$	0	0
$184$	3.82954	0.282317
$185$	0.356617	0.0262190
$186$	0	0
$187$	28.6759	2.09699
$188$	0.663442	0.0483864
$189$	0	0
$190$	0.283401	0.0205601
$191$	−3.96467	−0.286873	−0.143437	−	0.989660i	$-0.545815\pi$
$191$	−3.96467	−0.286873	−0.143437	+	0.989660i	$0.545815\pi$
$192$	0	0
$193$	−8.23872	−0.593036	−0.296518	−	0.955027i	$-0.595825\pi$
$193$	−8.23872	−0.593036	−0.296518	+	0.955027i	$0.595825\pi$
$194$	−10.1220	−0.726717
$195$	0	0
$196$	−3.23922	−0.231373
$197$	8.05142	0.573640	0.286820	−	0.957985i	$-0.407402\pi$
$197$	8.05142	0.573640	0.286820	+	0.957985i	$0.407402\pi$
$198$	0	0
$199$	−20.4588	−1.45029	−0.725143	−	0.688599i	$-0.758226\pi$
$199$	−20.4588	−1.45029	−0.725143	+	0.688599i	$0.758226\pi$
$200$	2.99278	0.211621
$201$	0	0
$202$	0.210721	0.0148263
$203$	22.8155	1.60134
$204$	0	0
$205$	11.1264	0.777104
$206$	−7.27597	−0.506941
$207$	0	0
$208$	−3.94769	−0.273723
$209$	0.803754	0.0555968
$210$	0	0
$211$	5.94093	0.408990	0.204495	−	0.978868i	$-0.434445\pi$
$211$	5.94093	0.408990	0.204495	+	0.978868i	$0.434445\pi$
$212$	1.59688	0.109674
$213$	0	0
$214$	−12.1260	−0.828918
$215$	3.34534	0.228151
$216$	0	0
$217$	−16.9084	−1.14782
$218$	3.33737	0.226035
$219$	0	0
$220$	1.06154	0.0715687
$221$	−9.11021	−0.612819
$222$	0	0
$223$	−8.70415	−0.582873	−0.291437	−	0.956590i	$-0.594133\pi$
$223$	−8.70415	−0.582873	−0.291437	+	0.956590i	$0.594133\pi$
$224$	−6.86816	−0.458898
$225$	0	0
$226$	5.66005	0.376501
$227$	−1.00295	−0.0665679	−0.0332840	−	0.999446i	$-0.510597\pi$
$227$	−1.00295	−0.0665679	−0.0332840	+	0.999446i	$0.510597\pi$
$228$	0	0
$229$	13.2838	0.877821	0.438911	−	0.898531i	$-0.355364\pi$
$229$	13.2838	0.877821	0.438911	+	0.898531i	$0.355364\pi$
$230$	−1.67530	−0.110466
$231$	0	0
$232$	15.9484	1.04707
$233$	7.50825	0.491882	0.245941	−	0.969285i	$-0.420903\pi$
$233$	7.50825	0.491882	0.245941	+	0.969285i	$0.420903\pi$
$234$	0	0
$235$	−2.32065	−0.151382
$236$	−0.383975	−0.0249947
$237$	0	0
$238$	43.2895	2.80604
$239$	21.6195	1.39845	0.699224	−	0.714902i	$-0.253529\pi$
$239$	21.6195	1.39845	0.699224	+	0.714902i	$0.253529\pi$
$240$	0	0
$241$	18.0531	1.16290	0.581451	−	0.813582i	$-0.302485\pi$
$241$	18.0531	1.16290	0.581451	+	0.813582i	$0.302485\pi$
$242$	−3.64937	−0.234590
$243$	0	0
$244$	−1.02827	−0.0658281
$245$	11.3304	0.723876
$246$	0	0
$247$	−0.255349	−0.0162475
$248$	−11.8193	−0.750523
$249$	0	0
$250$	−1.30924	−0.0828037
$251$	25.5948	1.61553	0.807763	−	0.589507i	$-0.200678\pi$
$251$	25.5948	1.61553	0.807763	+	0.589507i	$0.200678\pi$
$252$	0	0
$253$	−4.75131	−0.298712
$254$	−25.6562	−1.60981
$255$	0	0
$256$	−6.71440	−0.419650
$257$	−1.78189	−0.111151	−0.0555756	−	0.998454i	$-0.517699\pi$
$257$	−1.78189	−0.111151	−0.0555756	+	0.998454i	$0.517699\pi$
$258$	0	0
$259$	1.52682	0.0948723
$260$	−0.337245	−0.0209151
$261$	0	0
$262$	13.2940	0.821307
$263$	−4.17388	−0.257372	−0.128686	−	0.991685i	$-0.541076\pi$
$263$	−4.17388	−0.257372	−0.128686	+	0.991685i	$0.541076\pi$
$264$	0	0
$265$	−5.58573	−0.343129
$266$	1.21336	0.0743957
$267$	0	0
$268$	−2.45923	−0.150221
$269$	22.3934	1.36535	0.682676	−	0.730721i	$-0.260816\pi$
$269$	22.3934	1.36535	0.682676	+	0.730721i	$0.260816\pi$
$270$	0	0
$271$	−16.4111	−0.996903	−0.498452	−	0.866918i	$-0.666098\pi$
$271$	−16.4111	−0.996903	−0.498452	+	0.866918i	$0.666098\pi$
$272$	25.8444	1.56705
$273$	0	0
$274$	2.81086	0.169810
$275$	−3.71314	−0.223911
$276$	0	0
$277$	22.0755	1.32639	0.663195	−	0.748446i	$-0.269200\pi$
$277$	22.0755	1.32639	0.663195	+	0.748446i	$0.269200\pi$
$278$	−5.15173	−0.308980
$279$	0	0
$280$	12.8133	0.765741
$281$	3.80934	0.227246	0.113623	−	0.993524i	$-0.463754\pi$
$281$	3.80934	0.227246	0.113623	+	0.993524i	$0.463754\pi$
$282$	0	0
$283$	0.870512	0.0517465	0.0258733	−	0.999665i	$-0.491763\pi$
$283$	0.870512	0.0517465	0.0258733	+	0.999665i	$0.491763\pi$
$284$	−2.17153	−0.128857
$285$	0	0
$286$	5.73473	0.339102
$287$	47.6368	2.81191
$288$	0	0
$289$	42.6419	2.50835
$290$	−6.97692	−0.409699
$291$	0	0
$292$	−1.01932	−0.0596512
$293$	14.1589	0.827174	0.413587	−	0.910465i	$-0.364276\pi$
$293$	14.1589	0.827174	0.413587	+	0.910465i	$0.364276\pi$
$294$	0	0
$295$	1.34310	0.0781986
$296$	1.06728	0.0620342
$297$	0	0
$298$	24.4170	1.41444
$299$	1.50947	0.0872948
$300$	0	0
$301$	14.3228	0.825551
$302$	25.2619	1.45366
$303$	0	0
$304$	0.724390	0.0415466
$305$	3.59677	0.205950
$306$	0	0
$307$	−0.0273873	−0.00156308	−0.000781539	−	1.00000i	$-0.500249\pi$
$307$	−0.0273873	−0.00156308	−0.000781539		1.00000i	$0.500249\pi$
$308$	4.54486	0.258968
$309$	0	0
$310$	5.17053	0.293667
$311$	−1.23518	−0.0700406	−0.0350203	−	0.999387i	$-0.511150\pi$
$311$	−1.23518	−0.0700406	−0.0350203	+	0.999387i	$0.511150\pi$
$312$	0	0
$313$	16.5256	0.934084	0.467042	−	0.884235i	$-0.345320\pi$
$313$	16.5256	0.934084	0.467042	+	0.884235i	$0.345320\pi$
$314$	−13.3473	−0.753233
$315$	0	0
$316$	−4.22841	−0.237867
$317$	23.9548	1.34543	0.672717	−	0.739900i	$-0.265127\pi$
$317$	23.9548	1.34543	0.672717	+	0.739900i	$0.265127\pi$
$318$	0	0
$319$	−19.7872	−1.10787
$320$	8.79326	0.491558
$321$	0	0
$322$	−7.17263	−0.399715
$323$	1.67170	0.0930158
$324$	0	0
$325$	1.17965	0.0654351
$326$	−7.32590	−0.405744
$327$	0	0
$328$	33.2989	1.83863
$329$	−9.93564	−0.547770
$330$	0	0
$331$	−22.8272	−1.25470	−0.627349	−	0.778738i	$-0.715860\pi$
$331$	−22.8272	−1.25470	−0.627349	+	0.778738i	$0.715860\pi$
$332$	2.62230	0.143918
$333$	0	0
$334$	−5.03598	−0.275556
$335$	8.60212	0.469984
$336$	0	0
$337$	−19.9043	−1.08425	−0.542127	−	0.840297i	$-0.682381\pi$
$337$	−19.9043	−1.08425	−0.542127	+	0.840297i	$0.682381\pi$
$338$	15.1982	0.826675
$339$	0	0
$340$	2.20785	0.119737
$341$	14.6641	0.794108
$342$	0	0
$343$	18.5404	1.00109
$344$	10.0119	0.539804
$345$	0	0
$346$	−10.1965	−0.548169
$347$	−33.4631	−1.79639	−0.898196	−	0.439595i	$-0.855122\pi$
$347$	−33.4631	−1.79639	−0.898196	+	0.439595i	$0.855122\pi$
$348$	0	0
$349$	−33.7346	−1.80577	−0.902885	−	0.429883i	$-0.858555\pi$
$349$	−33.7346	−1.80577	−0.902885	+	0.429883i	$0.858555\pi$
$350$	−5.60540	−0.299621
$351$	0	0
$352$	5.95655	0.317485
$353$	−26.7136	−1.42182	−0.710910	−	0.703283i	$-0.751717\pi$
$353$	−26.7136	−1.42182	−0.710910	+	0.703283i	$0.751717\pi$
$354$	0	0
$355$	7.59580	0.403143
$356$	0.285886	0.0151519
$357$	0	0
$358$	−26.8214	−1.41756
$359$	8.72203	0.460331	0.230166	−	0.973151i	$-0.426073\pi$
$359$	8.72203	0.460331	0.230166	+	0.973151i	$0.426073\pi$
$360$	0	0
$361$	−18.9531	−0.997534
$362$	−20.0600	−1.05433
$363$	0	0
$364$	−1.44388	−0.0756801
$365$	3.56548	0.186625
$366$	0	0
$367$	18.8586	0.984410	0.492205	−	0.870479i	$-0.336191\pi$
$367$	18.8586	0.984410	0.492205	+	0.870479i	$0.336191\pi$
$368$	−4.28215	−0.223223
$369$	0	0
$370$	−0.466898	−0.0242729
$371$	−23.9148	−1.24159
$372$	0	0
$373$	15.4920	0.802145	0.401073	−	0.916046i	$-0.368638\pi$
$373$	15.4920	0.802145	0.401073	+	0.916046i	$0.368638\pi$
$374$	−37.5437	−1.94134
$375$	0	0
$376$	−6.94518	−0.358171
$377$	6.28631	0.323762
$378$	0	0
$379$	24.3260	1.24954	0.624770	−	0.780809i	$-0.285193\pi$
$379$	24.3260	1.24954	0.624770	+	0.780809i	$0.285193\pi$
$380$	0.0618836	0.00317456
$381$	0	0
$382$	5.19071	0.265580
$383$	18.3571	0.938003	0.469001	−	0.883197i	$-0.344614\pi$
$383$	18.3571	0.938003	0.469001	+	0.883197i	$0.344614\pi$
$384$	0	0
$385$	−15.8975	−0.810209
$386$	10.7865	0.549017
$387$	0	0
$388$	−2.21024	−0.112208
$389$	−3.18435	−0.161453	−0.0807264	−	0.996736i	$-0.525724\pi$
$389$	−3.18435	−0.161453	−0.0807264	+	0.996736i	$0.525724\pi$
$390$	0	0
$391$	−9.88207	−0.499758
$392$	33.9095	1.71269
$393$	0	0
$394$	−10.5413	−0.531061
$395$	14.7905	0.744193
$396$	0	0
$397$	3.74826	0.188120	0.0940598	−	0.995567i	$-0.470016\pi$
$397$	3.74826	0.188120	0.0940598	+	0.995567i	$0.470016\pi$
$398$	26.7855	1.34264
$399$	0	0
$400$	−3.34650	−0.167325
$401$	−32.7548	−1.63570	−0.817849	−	0.575433i	$-0.804834\pi$
$401$	−32.7548	−1.63570	−0.817849	+	0.575433i	$0.804834\pi$
$402$	0	0
$403$	−4.65873	−0.232068
$404$	0.0460131	0.00228924
$405$	0	0
$406$	−29.8710	−1.48247
$407$	−1.32417	−0.0656367
$408$	0	0
$409$	21.7470	1.07532	0.537661	−	0.843161i	$-0.319308\pi$
$409$	21.7470	1.07532	0.537661	+	0.843161i	$0.319308\pi$
$410$	−14.5672	−0.719422
$411$	0	0
$412$	−1.58878	−0.0782737
$413$	5.75038	0.282958
$414$	0	0
$415$	−9.17254	−0.450262
$416$	−1.89237	−0.0927811
$417$	0	0
$418$	−1.05231	−0.0514701
$419$	−10.8946	−0.532235	−0.266118	−	0.963941i	$-0.585741\pi$
$419$	−10.8946	−0.532235	−0.266118	+	0.963941i	$0.585741\pi$
$420$	0	0
$421$	22.8674	1.11449	0.557245	−	0.830348i	$-0.311859\pi$
$421$	22.8674	1.11449	0.557245	+	0.830348i	$0.311859\pi$
$422$	−7.77811	−0.378632
$423$	0	0
$424$	−16.7169	−0.811842
$425$	−7.72282	−0.374612
$426$	0	0
$427$	15.3992	0.745221
$428$	−2.64784	−0.127988
$429$	0	0
$430$	−4.37986	−0.211216
$431$	0.458869	0.0221030	0.0110515	−	0.999939i	$-0.496482\pi$
$431$	0.458869	0.0221030	0.0110515	+	0.999939i	$0.496482\pi$
$432$	0	0
$433$	−10.8831	−0.523009	−0.261505	−	0.965202i	$-0.584219\pi$
$433$	−10.8831	−0.523009	−0.261505	+	0.965202i	$0.584219\pi$
$434$	22.1372	1.06262
$435$	0	0
$436$	0.728748	0.0349007
$437$	−0.276984	−0.0132499
$438$	0	0
$439$	−11.7622	−0.561380	−0.280690	−	0.959799i	$-0.590563\pi$
$439$	−11.7622	−0.561380	−0.280690	+	0.959799i	$0.590563\pi$
$440$	−11.1126	−0.529772
$441$	0	0
$442$	11.9275	0.567331
$443$	−27.8113	−1.32135	−0.660677	−	0.750670i	$-0.729731\pi$
$443$	−27.8113	−1.32135	−0.660677	+	0.750670i	$0.729731\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	11.3958	0.539609
$447$	0	0
$448$	37.6475	1.77868
$449$	21.7033	1.02424	0.512121	−	0.858913i	$-0.328860\pi$
$449$	21.7033	1.02424	0.512121	+	0.858913i	$0.328860\pi$
$450$	0	0
$451$	−41.3140	−1.94540
$452$	1.23593	0.0581332
$453$	0	0
$454$	1.31310	0.0616268
$455$	5.05055	0.236774
$456$	0	0
$457$	20.2724	0.948302	0.474151	−	0.880444i	$-0.342755\pi$
$457$	20.2724	0.948302	0.474151	+	0.880444i	$0.342755\pi$
$458$	−17.3918	−0.812664
$459$	0	0
$460$	−0.365818	−0.0170564
$461$	34.1046	1.58841	0.794206	−	0.607649i	$-0.207887\pi$
$461$	34.1046	1.58841	0.794206	+	0.607649i	$0.207887\pi$
$462$	0	0
$463$	−0.293496	−0.0136399	−0.00681995	−	0.999977i	$-0.502171\pi$
$463$	−0.293496	−0.0136399	−0.00681995	+	0.999977i	$0.502171\pi$
$464$	−17.8334	−0.827895
$465$	0	0
$466$	−9.83011	−0.455371
$467$	33.8874	1.56812	0.784061	−	0.620684i	$-0.213145\pi$
$467$	33.8874	1.56812	0.784061	+	0.620684i	$0.213145\pi$
$468$	0	0
$469$	36.8292	1.70061
$470$	3.03829	0.140146
$471$	0	0
$472$	4.01961	0.185018
$473$	−12.4217	−0.571151
$474$	0	0
$475$	−0.216462	−0.00993197
$476$	9.45270	0.433264
$477$	0	0
$478$	−28.3051	−1.29465
$479$	−4.47505	−0.204470	−0.102235	−	0.994760i	$-0.532599\pi$
$479$	−4.47505	−0.204470	−0.102235	+	0.994760i	$0.532599\pi$
$480$	0	0
$481$	0.420683	0.0191815
$482$	−23.6359	−1.07658
$483$	0	0
$484$	−0.796877	−0.0362217
$485$	7.73120	0.351056
$486$	0	0
$487$	−30.5341	−1.38363	−0.691816	−	0.722074i	$-0.743189\pi$
$487$	−30.5341	−1.38363	−0.691816	+	0.722074i	$0.743189\pi$
$488$	10.7643	0.487278
$489$	0	0
$490$	−14.8343	−0.670145
$491$	−40.9988	−1.85025	−0.925124	−	0.379664i	$-0.876040\pi$
$491$	−40.9988	−1.85025	−0.925124	+	0.379664i	$0.876040\pi$
$492$	0	0
$493$	−41.1547	−1.85352
$494$	0.334314	0.0150415
$495$	0	0
$496$	13.2162	0.593424
$497$	32.5207	1.45875
$498$	0	0
$499$	−33.7211	−1.50956	−0.754781	−	0.655976i	$-0.772257\pi$
$499$	−33.7211	−1.50956	−0.754781	+	0.655976i	$0.772257\pi$
$500$	−0.285886	−0.0127852
$501$	0	0
$502$	−33.5097	−1.49561
$503$	−1.84488	−0.0822591	−0.0411296	−	0.999154i	$-0.513096\pi$
$503$	−1.84488	−0.0822591	−0.0411296	+	0.999154i	$0.513096\pi$
$504$	0	0
$505$	−0.160949	−0.00716213
$506$	6.22061	0.276540
$507$	0	0
$508$	−5.60229	−0.248561
$509$	−8.65691	−0.383711	−0.191855	−	0.981423i	$-0.561450\pi$
$509$	−8.65691	−0.383711	−0.191855	+	0.981423i	$0.561450\pi$
$510$	0	0
$511$	15.2653	0.675295
$512$	25.3990	1.12249
$513$	0	0
$514$	2.33292	0.102901
$515$	5.55740	0.244888
$516$	0	0
$517$	8.61689	0.378970
$518$	−1.99898	−0.0878302
$519$	0	0
$520$	3.53042	0.154819
$521$	27.2745	1.19492	0.597459	−	0.801899i	$-0.296177\pi$
$521$	27.2745	1.19492	0.597459	+	0.801899i	$0.296177\pi$
$522$	0	0
$523$	−28.2048	−1.23331	−0.616655	−	0.787233i	$-0.711513\pi$
$523$	−28.2048	−1.23331	−0.616655	+	0.787233i	$0.711513\pi$
$524$	2.90288	0.126813
$525$	0	0
$526$	5.46461	0.238268
$527$	30.4994	1.32858
$528$	0	0
$529$	−21.3626	−0.928810
$530$	7.31307	0.317660
$531$	0	0
$532$	0.264949	0.0114870
$533$	13.1253	0.568519
$534$	0	0
$535$	9.26187	0.400425
$536$	25.7442	1.11198
$537$	0	0
$538$	−29.3184	−1.26401
$539$	−42.0715	−1.81215
$540$	0	0
$541$	−30.1932	−1.29811	−0.649054	−	0.760743i	$-0.724835\pi$
$541$	−30.1932	−1.29811	−0.649054	+	0.760743i	$0.724835\pi$
$542$	21.4861	0.922907
$543$	0	0
$544$	12.3888	0.531166
$545$	−2.54908	−0.109191
$546$	0	0
$547$	43.3588	1.85389	0.926945	−	0.375197i	$-0.122425\pi$
$547$	43.3588	1.85389	0.926945	+	0.375197i	$0.122425\pi$
$548$	0.613780	0.0262194
$549$	0	0
$550$	4.86139	0.207291
$551$	−1.15352	−0.0491417
$552$	0	0
$553$	63.3243	2.69282
$554$	−28.9022	−1.22794
$555$	0	0
$556$	−1.12493	−0.0477077
$557$	−8.27467	−0.350609	−0.175304	−	0.984514i	$-0.556091\pi$
$557$	−8.27467	−0.350609	−0.175304	+	0.984514i	$0.556091\pi$
$558$	0	0
$559$	3.94633	0.166912
$560$	−14.3277	−0.605456
$561$	0	0
$562$	−4.98735	−0.210379
$563$	11.1946	0.471798	0.235899	−	0.971778i	$-0.424197\pi$
$563$	11.1946	0.471798	0.235899	+	0.971778i	$0.424197\pi$
$564$	0	0
$565$	−4.32315	−0.181876
$566$	−1.13971	−0.0479056
$567$	0	0
$568$	22.7325	0.953836
$569$	−8.48034	−0.355514	−0.177757	−	0.984074i	$-0.556884\pi$
$569$	−8.48034	−0.355514	−0.177757	+	0.984074i	$0.556884\pi$
$570$	0	0
$571$	27.9358	1.16908	0.584539	−	0.811366i	$-0.301276\pi$
$571$	27.9358	1.16908	0.584539	+	0.811366i	$0.301276\pi$
$572$	1.25224	0.0523587
$573$	0	0
$574$	−62.3681	−2.60319
$575$	1.27959	0.0533627
$576$	0	0
$577$	21.4559	0.893221	0.446610	−	0.894729i	$-0.352631\pi$
$577$	21.4559	0.893221	0.446610	+	0.894729i	$0.352631\pi$
$578$	−55.8286	−2.32216
$579$	0	0
$580$	−1.52348	−0.0632591
$581$	−39.2714	−1.62925
$582$	0	0
$583$	20.7406	0.858988
$584$	10.6707	0.441556
$585$	0	0
$586$	−18.5375	−0.765776
$587$	9.13301	0.376959	0.188480	−	0.982077i	$-0.439644\pi$
$587$	9.13301	0.376959	0.188480	+	0.982077i	$0.439644\pi$
$588$	0	0
$589$	0.854865	0.0352241
$590$	−1.75845	−0.0723942
$591$	0	0
$592$	−1.19342	−0.0490492
$593$	−13.6118	−0.558968	−0.279484	−	0.960150i	$-0.590163\pi$
$593$	−13.6118	−0.558968	−0.279484	+	0.960150i	$0.590163\pi$
$594$	0	0
$595$	−33.0645	−1.35551
$596$	5.33171	0.218395
$597$	0	0
$598$	−1.97626	−0.0808153
$599$	42.4939	1.73625	0.868126	−	0.496344i	$-0.165324\pi$
$599$	42.4939	1.73625	0.868126	+	0.496344i	$0.165324\pi$
$600$	0	0
$601$	−16.4683	−0.671757	−0.335879	−	0.941905i	$-0.609033\pi$
$601$	−16.4683	−0.671757	−0.335879	+	0.941905i	$0.609033\pi$
$602$	−18.7520	−0.764273
$603$	0	0
$604$	5.51620	0.224451
$605$	2.78739	0.113324
$606$	0	0
$607$	45.0289	1.82766	0.913832	−	0.406092i	$-0.133109\pi$
$607$	45.0289	1.82766	0.913832	+	0.406092i	$0.133109\pi$
$608$	0.347245	0.0140826
$609$	0	0
$610$	−4.70904	−0.190663
$611$	−2.73755	−0.110749
$612$	0	0
$613$	−23.1320	−0.934292	−0.467146	−	0.884180i	$-0.654718\pi$
$613$	−23.1320	−0.934292	−0.467146	+	0.884180i	$0.654718\pi$
$614$	0.0358566	0.00144706
$615$	0	0
$616$	−47.5775	−1.91695
$617$	9.74256	0.392221	0.196110	−	0.980582i	$-0.437169\pi$
$617$	9.74256	0.392221	0.196110	+	0.980582i	$0.437169\pi$
$618$	0	0
$619$	−27.1769	−1.09233	−0.546165	−	0.837678i	$-0.683913\pi$
$619$	−27.1769	−1.09233	−0.546165	+	0.837678i	$0.683913\pi$
$620$	1.12904	0.0453433
$621$	0	0
$622$	1.61715	0.0648418
$623$	−4.28141	−0.171531
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.6360	−0.864750
$627$	0	0
$628$	−2.91452	−0.116302
$629$	−2.75409	−0.109813
$630$	0	0
$631$	−6.15947	−0.245204	−0.122602	−	0.992456i	$-0.539124\pi$
$631$	−6.15947	−0.245204	−0.122602	+	0.992456i	$0.539124\pi$
$632$	44.2648	1.76076
$633$	0	0
$634$	−31.3626	−1.24557
$635$	19.5962	0.777652
$636$	0	0
$637$	13.3659	0.529578
$638$	25.9063	1.02564
$639$	0	0
$640$	−8.30413	−0.328250
$641$	−17.4220	−0.688128	−0.344064	−	0.938946i	$-0.611804\pi$
$641$	−17.4220	−0.688128	−0.344064	+	0.938946i	$0.611804\pi$
$642$	0	0
$643$	−16.4751	−0.649714	−0.324857	−	0.945763i	$-0.605316\pi$
$643$	−16.4751	−0.649714	−0.324857	+	0.945763i	$0.605316\pi$
$644$	−1.56622	−0.0617176
$645$	0	0
$646$	−2.18866	−0.0861116
$647$	45.1088	1.77341	0.886705	−	0.462336i	$-0.152989\pi$
$647$	45.1088	1.77341	0.886705	+	0.462336i	$0.152989\pi$
$648$	0	0
$649$	−4.98713	−0.195762
$650$	−1.54444	−0.0605781
$651$	0	0
$652$	−1.59969	−0.0626485
$653$	−27.9236	−1.09274	−0.546368	−	0.837545i	$-0.683990\pi$
$653$	−27.9236	−1.09274	−0.546368	+	0.837545i	$0.683990\pi$
$654$	0	0
$655$	−10.1540	−0.396749
$656$	−37.2346	−1.45377
$657$	0	0
$658$	13.0082	0.507111
$659$	44.8255	1.74615	0.873077	−	0.487583i	$-0.162121\pi$
$659$	44.8255	1.74615	0.873077	+	0.487583i	$0.162121\pi$
$660$	0	0
$661$	−25.4919	−0.991520	−0.495760	−	0.868460i	$-0.665110\pi$
$661$	−25.4919	−0.991520	−0.495760	+	0.868460i	$0.665110\pi$
$662$	29.8863	1.16157
$663$	0	0
$664$	−27.4514	−1.06532
$665$	−0.926763	−0.0359383
$666$	0	0
$667$	6.81892	0.264030
$668$	−1.09966	−0.0425470
$669$	0	0
$670$	−11.2622	−0.435099
$671$	−13.3553	−0.515576
$672$	0	0
$673$	−46.5784	−1.79547	−0.897734	−	0.440539i	$-0.854787\pi$
$673$	−46.5784	−1.79547	−0.897734	+	0.440539i	$0.854787\pi$
$674$	26.0595	1.00377
$675$	0	0
$676$	3.31869	0.127642
$677$	−35.2820	−1.35600	−0.677998	−	0.735063i	$-0.737152\pi$
$677$	−35.2820	−1.35600	−0.677998	+	0.735063i	$0.737152\pi$
$678$	0	0
$679$	33.1004	1.27028
$680$	−23.1127	−0.886331
$681$	0	0
$682$	−19.1989	−0.735164
$683$	4.13944	0.158391	0.0791956	−	0.996859i	$-0.474765\pi$
$683$	4.13944	0.158391	0.0791956	+	0.996859i	$0.474765\pi$
$684$	0	0
$685$	−2.14694	−0.0820302
$686$	−24.2739	−0.926780
$687$	0	0
$688$	−11.1952	−0.426812
$689$	−6.58920	−0.251028
$690$	0	0
$691$	2.07285	0.0788548	0.0394274	−	0.999222i	$-0.487447\pi$
$691$	2.07285	0.0788548	0.0394274	+	0.999222i	$0.487447\pi$
$692$	−2.22652	−0.0846395
$693$	0	0
$694$	43.8113	1.66305
$695$	3.93489	0.149259
$696$	0	0
$697$	−85.9274	−3.25473
$698$	44.1667	1.67173
$699$	0	0
$700$	−1.22400	−0.0462627
$701$	7.85939	0.296845	0.148423	−	0.988924i	$-0.452580\pi$
$701$	7.85939	0.296845	0.148423	+	0.988924i	$0.452580\pi$
$702$	0	0
$703$	−0.0771942	−0.00291143
$704$	−32.6506	−1.23056
$705$	0	0
$706$	34.9745	1.31628
$707$	−0.689087	−0.0259158
$708$	0	0
$709$	−4.99900	−0.187741	−0.0938706	−	0.995584i	$-0.529924\pi$
$709$	−4.99900	−0.187741	−0.0938706	+	0.995584i	$0.529924\pi$
$710$	−9.94473	−0.373219
$711$	0	0
$712$	−2.99278	−0.112159
$713$	−5.05344	−0.189253
$714$	0	0
$715$	−4.38019	−0.163810
$716$	−5.85673	−0.218876
$717$	0	0
$718$	−11.4192	−0.426162
$719$	−12.6136	−0.470409	−0.235204	−	0.971946i	$-0.575576\pi$
$719$	−12.6136	−0.470409	−0.235204	+	0.971946i	$0.575576\pi$
$720$	0	0
$721$	23.7935	0.886115
$722$	24.8142	0.923491
$723$	0	0
$724$	−4.38030	−0.162793
$725$	5.32898	0.197913
$726$	0	0
$727$	19.7779	0.733523	0.366761	−	0.930315i	$-0.380467\pi$
$727$	19.7779	0.733523	0.366761	+	0.930315i	$0.380467\pi$
$728$	15.1152	0.560206
$729$	0	0
$730$	−4.66807	−0.172773
$731$	−25.8355	−0.955560
$732$	0	0
$733$	−28.1273	−1.03891	−0.519453	−	0.854499i	$-0.673864\pi$
$733$	−28.1273	−1.03891	−0.519453	+	0.854499i	$0.673864\pi$
$734$	−24.6904	−0.911341
$735$	0	0
$736$	−2.05270	−0.0756636
$737$	−31.9408	−1.17656
$738$	0	0
$739$	51.3371	1.88847	0.944233	−	0.329278i	$-0.106805\pi$
$739$	51.3371	1.88847	0.944233	+	0.329278i	$0.106805\pi$
$740$	−0.101952	−0.00374783
$741$	0	0
$742$	31.3102	1.14944
$743$	48.5148	1.77984	0.889918	−	0.456120i	$-0.150761\pi$
$743$	48.5148	1.77984	0.889918	+	0.456120i	$0.150761\pi$
$744$	0	0
$745$	−18.6498	−0.683274
$746$	−20.2828	−0.742605
$747$	0	0
$748$	−8.19805	−0.299750
$749$	39.6538	1.44892
$750$	0	0
$751$	−51.4650	−1.87798	−0.938992	−	0.343939i	$-0.888239\pi$
$751$	−51.4650	−1.87798	−0.938992	+	0.343939i	$0.888239\pi$
$752$	7.76604	0.283198
$753$	0	0
$754$	−8.23030	−0.299730
$755$	−19.2951	−0.702220
$756$	0	0
$757$	23.1780	0.842418	0.421209	−	0.906964i	$-0.361606\pi$
$757$	23.1780	0.842418	0.421209	+	0.906964i	$0.361606\pi$
$758$	−31.8486	−1.15679
$759$	0	0
$760$	−0.647823	−0.0234990
$761$	21.7956	0.790089	0.395045	−	0.918662i	$-0.370729\pi$
$761$	21.7956	0.790089	0.395045	+	0.918662i	$0.370729\pi$
$762$	0	0
$763$	−10.9137	−0.395101
$764$	1.13344	0.0410066
$765$	0	0
$766$	−24.0339	−0.868378
$767$	1.58439	0.0572090
$768$	0	0
$769$	−24.1579	−0.871155	−0.435578	−	0.900151i	$-0.643456\pi$
$769$	−24.1579	−0.871155	−0.435578	+	0.900151i	$0.643456\pi$
$770$	20.8136	0.750070
$771$	0	0
$772$	2.35534	0.0847704
$773$	12.8586	0.462491	0.231245	−	0.972895i	$-0.425720\pi$
$773$	12.8586	0.462491	0.231245	+	0.972895i	$0.425720\pi$
$774$	0	0
$775$	−3.94926	−0.141862
$776$	23.1377	0.830597
$777$	0	0
$778$	4.16908	0.149469
$779$	−2.40845	−0.0862918
$780$	0	0
$781$	−28.2042	−1.00923
$782$	12.9380	0.462662
$783$	0	0
$784$	−37.9173	−1.35419
$785$	10.1947	0.363864
$786$	0	0
$787$	−5.11804	−0.182438	−0.0912191	−	0.995831i	$-0.529076\pi$
$787$	−5.11804	−0.182438	−0.0912191	+	0.995831i	$0.529076\pi$
$788$	−2.30179	−0.0819979
$789$	0	0
$790$	−19.3644	−0.688954
$791$	−18.5092	−0.658110
$792$	0	0
$793$	4.24292	0.150671
$794$	−4.90737	−0.174156
$795$	0	0
$796$	5.84889	0.207308
$797$	3.36787	0.119296	0.0596480	−	0.998219i	$-0.481002\pi$
$797$	3.36787	0.119296	0.0596480	+	0.998219i	$0.481002\pi$
$798$	0	0
$799$	17.9219	0.634033
$800$	−1.60418	−0.0567164
$801$	0	0
$802$	42.8840	1.51429
$803$	−13.2391	−0.467198
$804$	0	0
$805$	5.47846	0.193090
$806$	6.09941	0.214842
$807$	0	0
$808$	−0.481684	−0.0169456
$809$	−10.2528	−0.360468	−0.180234	−	0.983624i	$-0.557685\pi$
$809$	−10.2528	−0.360468	−0.180234	+	0.983624i	$0.557685\pi$
$810$	0	0
$811$	15.9782	0.561071	0.280535	−	0.959844i	$-0.409488\pi$
$811$	15.9782	0.561071	0.280535	+	0.959844i	$0.409488\pi$
$812$	−6.52264	−0.228900
$813$	0	0
$814$	1.73366	0.0607647
$815$	5.59553	0.196003
$816$	0	0
$817$	−0.724140	−0.0253345
$818$	−28.4721	−0.995505
$819$	0	0
$820$	−3.18090	−0.111082
$821$	−37.3703	−1.30423	−0.652116	−	0.758119i	$-0.726118\pi$
$821$	−37.3703	−1.30423	−0.652116	+	0.758119i	$0.726118\pi$
$822$	0	0
$823$	−6.51386	−0.227059	−0.113529	−	0.993535i	$-0.536216\pi$
$823$	−6.51386	−0.227059	−0.113529	+	0.993535i	$0.536216\pi$
$824$	16.6320	0.579405
$825$	0	0
$826$	−7.52863	−0.261955
$827$	25.3352	0.880993	0.440496	−	0.897754i	$-0.354803\pi$
$827$	25.3352	0.880993	0.440496	+	0.897754i	$0.354803\pi$
$828$	0	0
$829$	55.3353	1.92187	0.960937	−	0.276768i	$-0.0892635\pi$
$829$	55.3353	1.92187	0.960937	+	0.276768i	$0.0892635\pi$
$830$	12.0091	0.416841
$831$	0	0
$832$	10.3729	0.359617
$833$	−87.5030	−3.03180
$834$	0	0
$835$	3.84648	0.133113
$836$	−0.229782	−0.00794719
$837$	0	0
$838$	14.2636	0.492729
$839$	35.9058	1.23961	0.619803	−	0.784757i	$-0.287212\pi$
$839$	35.9058	1.23961	0.619803	+	0.784757i	$0.287212\pi$
$840$	0	0
$841$	−0.602022	−0.0207594
$842$	−29.9390	−1.03176
$843$	0	0
$844$	−1.69843	−0.0584624
$845$	−11.6084	−0.399342
$846$	0	0
$847$	11.9340	0.410056
$848$	18.6926	0.641908
$849$	0	0
$850$	10.1110	0.346806
$851$	0.456325	0.0156426
$852$	0	0
$853$	24.6348	0.843479	0.421740	−	0.906717i	$-0.361420\pi$
$853$	24.6348	0.843479	0.421740	+	0.906717i	$0.361420\pi$
$854$	−20.1613	−0.689906
$855$	0	0
$856$	27.7187	0.947406
$857$	49.9535	1.70638	0.853190	−	0.521600i	$-0.174665\pi$
$857$	49.9535	1.70638	0.853190	+	0.521600i	$0.174665\pi$
$858$	0	0
$859$	−8.85780	−0.302224	−0.151112	−	0.988517i	$-0.548285\pi$
$859$	−8.85780	−0.302224	−0.151112	+	0.988517i	$0.548285\pi$
$860$	−0.956388	−0.0326126
$861$	0	0
$862$	−0.600771	−0.0204623
$863$	−18.6137	−0.633617	−0.316809	−	0.948489i	$-0.602611\pi$
$863$	−18.6137	−0.633617	−0.316809	+	0.948489i	$0.602611\pi$
$864$	0	0
$865$	7.78812	0.264804
$866$	14.2486	0.484188
$867$	0	0
$868$	4.83387	0.164072
$869$	−54.9193	−1.86301
$870$	0	0
$871$	10.1475	0.343834
$872$	−7.62884	−0.258345
$873$	0	0
$874$	0.362638	0.0122664
$875$	4.28141	0.144738
$876$	0	0
$877$	6.75783	0.228196	0.114098	−	0.993470i	$-0.463602\pi$
$877$	6.75783	0.228196	0.114098	+	0.993470i	$0.463602\pi$
$878$	15.3996	0.519710
$879$	0	0
$880$	12.4260	0.418880
$881$	−13.6931	−0.461334	−0.230667	−	0.973033i	$-0.574091\pi$
$881$	−13.6931	−0.461334	−0.230667	+	0.973033i	$0.574091\pi$
$882$	0	0
$883$	−25.9817	−0.874354	−0.437177	−	0.899376i	$-0.644022\pi$
$883$	−25.9817	−0.874354	−0.437177	+	0.899376i	$0.644022\pi$
$884$	2.60448	0.0875983
$885$	0	0
$886$	36.4117	1.22327
$887$	29.8478	1.00219	0.501095	−	0.865392i	$-0.332931\pi$
$887$	29.8478	1.00219	0.501095	+	0.865392i	$0.332931\pi$
$888$	0	0
$889$	83.8993	2.81389
$890$	1.30924	0.0438859
$891$	0	0
$892$	2.48840	0.0833177
$893$	0.502333	0.0168099
$894$	0	0
$895$	20.4862	0.684779
$896$	−35.5534	−1.18775
$897$	0	0
$898$	−28.4149	−0.948217
$899$	−21.0455	−0.701907
$900$	0	0
$901$	43.1376	1.43712
$902$	54.0900	1.80100
$903$	0	0
$904$	−12.9382	−0.430319
$905$	15.3218	0.509315
$906$	0	0
$907$	4.87267	0.161794	0.0808972	−	0.996722i	$-0.474221\pi$
$907$	4.87267	0.161794	0.0808972	+	0.996722i	$0.474221\pi$
$908$	0.286729	0.00951543
$909$	0	0
$910$	−6.61239	−0.219199
$911$	−49.8361	−1.65114	−0.825572	−	0.564297i	$-0.809147\pi$
$911$	−49.8361	−1.65114	−0.825572	+	0.564297i	$0.809147\pi$
$912$	0	0
$913$	34.0589	1.12718
$914$	−26.5414	−0.877913
$915$	0	0
$916$	−3.79767	−0.125479
$917$	−43.4733	−1.43561
$918$	0	0
$919$	25.8784	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$919$	25.8784	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$920$	3.82954	0.126256
$921$	0	0
$922$	−44.6512	−1.47051
$923$	8.96037	0.294934
$924$	0	0
$925$	0.356617	0.0117255
$926$	0.384257	0.0126275
$927$	0	0
$928$	−8.54865	−0.280623
$929$	−20.6440	−0.677307	−0.338654	−	0.940911i	$-0.609971\pi$
$929$	−20.6440	−0.677307	−0.338654	+	0.940911i	$0.609971\pi$
$930$	0	0
$931$	−2.45261	−0.0803812
$932$	−2.14651	−0.0703111
$933$	0	0
$934$	−44.3668	−1.45173
$935$	28.6759	0.937802
$936$	0	0
$937$	−30.5034	−0.996503	−0.498251	−	0.867033i	$-0.666024\pi$
$937$	−30.5034	−0.996503	−0.498251	+	0.867033i	$0.666024\pi$
$938$	−48.2183	−1.57438
$939$	0	0
$940$	0.663442	0.0216391
$941$	−27.6519	−0.901427	−0.450714	−	0.892669i	$-0.648830\pi$
$941$	−27.6519	−0.901427	−0.450714	+	0.892669i	$0.648830\pi$
$942$	0	0
$943$	14.2373	0.463631
$944$	−4.49470	−0.146290
$945$	0	0
$946$	16.2630	0.528757
$947$	19.4143	0.630879	0.315439	−	0.948946i	$-0.397848\pi$
$947$	19.4143	0.630879	0.315439	+	0.948946i	$0.397848\pi$
$948$	0	0
$949$	4.20600	0.136533
$950$	0.283401	0.00919475
$951$	0	0
$952$	−98.9548	−3.20714
$953$	8.10614	0.262584	0.131292	−	0.991344i	$-0.458088\pi$
$953$	8.10614	0.262584	0.131292	+	0.991344i	$0.458088\pi$
$954$	0	0
$955$	−3.96467	−0.128294
$956$	−6.18072	−0.199899
$957$	0	0
$958$	5.85893	0.189293
$959$	−9.19191	−0.296822
$960$	0	0
$961$	−15.4034	−0.496883
$962$	−0.550776	−0.0177577
$963$	0	0
$964$	−5.16113	−0.166229
$965$	−8.23872	−0.265214
$966$	0	0
$967$	−28.8008	−0.926172	−0.463086	−	0.886313i	$-0.653258\pi$
$967$	−28.8008	−0.926172	−0.463086	+	0.886313i	$0.653258\pi$
$968$	8.34205	0.268123
$969$	0	0
$970$	−10.1220	−0.324998
$971$	−20.4678	−0.656843	−0.328421	−	0.944531i	$-0.606517\pi$
$971$	−20.4678	−0.656843	−0.328421	+	0.944531i	$0.606517\pi$
$972$	0	0
$973$	16.8469	0.540086
$974$	39.9765	1.28093
$975$	0	0
$976$	−12.0366	−0.385281
$977$	−2.84518	−0.0910253	−0.0455126	−	0.998964i	$-0.514492\pi$
$977$	−2.84518	−0.0910253	−0.0455126	+	0.998964i	$0.514492\pi$
$978$	0	0
$979$	3.71314	0.118672
$980$	−3.23922	−0.103473
$981$	0	0
$982$	53.6773	1.71291
$983$	32.2906	1.02991	0.514955	−	0.857217i	$-0.327809\pi$
$983$	32.2906	1.02991	0.514955	+	0.857217i	$0.327809\pi$
$984$	0	0
$985$	8.05142	0.256540
$986$	53.8815	1.71594
$987$	0	0
$988$	0.0730008	0.00232247
$989$	4.28068	0.136118
$990$	0	0
$991$	39.8964	1.26735	0.633676	−	0.773599i	$-0.281545\pi$
$991$	39.8964	1.26735	0.633676	+	0.773599i	$0.281545\pi$
$992$	6.33533	0.201147
$993$	0	0
$994$	−42.5775	−1.35047
$995$	−20.4588	−0.648587
$996$	0	0
$997$	−48.1591	−1.52521	−0.762607	−	0.646862i	$-0.776081\pi$
$997$	−48.1591	−1.52521	−0.762607	+	0.646862i	$0.776081\pi$
$998$	44.1490	1.39751
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.x.1.5	yes	17
3.2	odd	2		4005.2.a.w.1.13	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.13	✓	17	3.2	odd	2
4005.2.a.x.1.5	yes	17	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30924 q^{2} -0.285886 q^{4} +1.00000 q^{5} +4.28141 q^{7} +2.99278 q^{8} +O(q^{10})\)	\(q-1.30924 q^{2} -0.285886 q^{4} +1.00000 q^{5} +4.28141 q^{7} +2.99278 q^{8} -1.30924 q^{10} -3.71314 q^{11} +1.17965 q^{13} -5.60540 q^{14} -3.34650 q^{16} -7.72282 q^{17} -0.216462 q^{19} -0.285886 q^{20} +4.86139 q^{22} +1.27959 q^{23} +1.00000 q^{25} -1.54444 q^{26} -1.22400 q^{28} +5.32898 q^{29} -3.94926 q^{31} -1.60418 q^{32} +10.1110 q^{34} +4.28141 q^{35} +0.356617 q^{37} +0.283401 q^{38} +2.99278 q^{40} +11.1264 q^{41} +3.34534 q^{43} +1.06154 q^{44} -1.67530 q^{46} -2.32065 q^{47} +11.3304 q^{49} -1.30924 q^{50} -0.337245 q^{52} -5.58573 q^{53} -3.71314 q^{55} +12.8133 q^{56} -6.97692 q^{58} +1.34310 q^{59} +3.59677 q^{61} +5.17053 q^{62} +8.79326 q^{64} +1.17965 q^{65} +8.60212 q^{67} +2.20785 q^{68} -5.60540 q^{70} +7.59580 q^{71} +3.56548 q^{73} -0.466898 q^{74} +0.0618836 q^{76} -15.8975 q^{77} +14.7905 q^{79} -3.34650 q^{80} -14.5672 q^{82} -9.17254 q^{83} -7.72282 q^{85} -4.37986 q^{86} -11.1126 q^{88} -1.00000 q^{89} +5.05055 q^{91} -0.365818 q^{92} +3.03829 q^{94} -0.216462 q^{95} +7.73120 q^{97} -14.8343 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) \) 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8 + 5 * q^10 + 2 * q^11 + 8 * q^13 + 4 * q^14 + 33 * q^16 + 10 * q^17 + 32 * q^19 + 21 * q^20 + 8 * q^22 + 15 * q^23 + 17 * q^25 - 15 * q^26 + 24 * q^28 + q^29 + 18 * q^31 + 25 * q^32 + 14 * q^34 + 12 * q^35 + 12 * q^37 + 22 * q^38 + 15 * q^40 - 7 * q^41 + 28 * q^43 - 14 * q^44 + 4 * q^46 + 26 * q^47 + 41 * q^49 + 5 * q^50 + 10 * q^52 + 12 * q^53 + 2 * q^55 + 13 * q^56 + 16 * q^58 - 23 * q^59 + 26 * q^61 + 10 * q^62 + 59 * q^64 + 8 * q^65 + 31 * q^67 - q^68 + 4 * q^70 - 2 * q^71 + 33 * q^73 - 10 * q^74 + 66 * q^76 + 12 * q^77 + 33 * q^79 + 33 * q^80 + 30 * q^82 + 13 * q^83 + 10 * q^85 - 20 * q^86 + 12 * q^88 - 17 * q^89 + 40 * q^91 + 16 * q^92 + 38 * q^94 + 32 * q^95 + 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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