LMFDB - Embedded newform 4000.2.c.e.1249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.268960000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 5x^{6} + 13x^{4} - 20x^{2} + 16 $ x^8 - 5x^6 + 13x^4 - 20*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$-1.38004 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	4000.1249
Dual form			4000.2.c.e.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +O(q^{10})$	$q+0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +5.52016 q^{11} +5.52016i q^{13} +3.41164i q^{17} +3.41164 q^{19} -1.00000 q^{21} +2.38197i q^{23} +3.47214i q^{27} -1.09017 q^{29} -5.52016 q^{31} +3.41164i q^{33} -8.93180i q^{37} -3.41164 q^{39} +5.85410 q^{41} +12.3262i q^{43} +1.09017i q^{47} +4.38197 q^{49} -2.10851 q^{51} -11.0403i q^{53} +2.10851i q^{57} -12.3434 q^{59} +1.14590 q^{61} +4.23607i q^{63} -10.4721i q^{67} -1.47214 q^{69} -3.41164 q^{71} -12.3434i q^{73} +8.93180i q^{77} +8.93180 q^{79} +5.70820 q^{81} +12.5623i q^{83} -0.673762i q^{87} -3.14590 q^{89} -8.93180 q^{91} -3.41164i q^{93} -12.3434i q^{97} +14.4520 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9}+O(q^{10}) $ 8 * q + 12 * q^9	$ 8 q + 12 q^{9} - 8 q^{21} + 36 q^{29} + 20 q^{41} + 44 q^{49} + 36 q^{61} + 24 q^{69} - 8 q^{81} - 52 q^{89}+O(q^{100}) $ 8 * q + 12 * q^9 - 8 * q^21 + 36 * q^29 + 20 * q^41 + 44 * q^49 + 36 * q^61 + 24 * q^69 - 8 * q^81 - 52 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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	0
$2$	0	0
$3$	0.618034i	0.356822i	0.983956	+	0.178411i	$0.0570957\pi$
$3$	0.618034i	0.356822i	−0.983956	+	0.178411i	$0.942904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.61803i	0.611559i	0.952102	+	0.305780i	$0.0989171\pi$
$7$	1.61803i	0.611559i	−0.952102	+	0.305780i	$0.901083\pi$
$8$	0	0
$9$	2.61803	0.872678
$10$	0	0
$11$	5.52016	1.66439	0.832195	−	0.554483i	$-0.187084\pi$
$11$	5.52016	1.66439	0.832195	+	0.554483i	$0.187084\pi$
$12$	0	0
$13$	5.52016i	1.53102i	0.643426	+	0.765508i	$0.277512\pi$
$13$	5.52016i	1.53102i	−0.643426	+	0.765508i	$0.722488\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.41164i	0.827445i	0.910403	+	0.413723i	$0.135772\pi$
$17$	3.41164i	0.827445i	−0.910403	+	0.413723i	$0.864228\pi$
$18$	0	0
$19$	3.41164	0.782685	0.391342	−	0.920245i	$-0.372011\pi$
$19$	3.41164	0.782685	0.391342	+	0.920245i	$0.372011\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.38197i	0.496674i	0.968674	+	0.248337i	$0.0798841\pi$
$23$	2.38197i	0.496674i	−0.968674	+	0.248337i	$0.920116\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.47214i	0.668213i
$28$	0	0
$29$	−1.09017	−0.202439	−0.101220	−	0.994864i	$-0.532275\pi$
$29$	−1.09017	−0.202439	−0.101220	+	0.994864i	$0.532275\pi$
$30$	0	0
$31$	−5.52016	−0.991450	−0.495725	−	0.868480i	$-0.665098\pi$
$31$	−5.52016	−0.991450	−0.495725	+	0.868480i	$0.665098\pi$
$32$	0	0
$33$	3.41164i	0.593891i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.93180i	− 1.46838i	−0.678944	−	0.734190i	$-0.737562\pi$
$37$	− 8.93180i	− 1.46838i	0.678944	−	0.734190i	$-0.262438\pi$
$38$	0	0
$39$	−3.41164	−0.546300
$40$	0	0
$41$	5.85410	0.914257	0.457129	−	0.889401i	$-0.348878\pi$
$41$	5.85410	0.914257	0.457129	+	0.889401i	$0.348878\pi$
$42$	0	0
$43$	12.3262i	1.87973i	0.341541	+	0.939867i	$0.389051\pi$
$43$	12.3262i	1.87973i	−0.341541	+	0.939867i	$0.610949\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.09017i	0.159018i	0.996834	+	0.0795088i	$0.0253352\pi$
$47$	1.09017i	0.159018i	−0.996834	+	0.0795088i	$0.974665\pi$
$48$	0	0
$49$	4.38197	0.625995
$50$	0	0
$51$	−2.10851	−0.295251
$52$	0	0
$53$	− 11.0403i	− 1.51650i	−0.651962	−	0.758252i	$-0.726054\pi$
$53$	− 11.0403i	− 1.51650i	0.651962	−	0.758252i	$-0.273946\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.10851i	0.279279i
$58$	0	0
$59$	−12.3434	−1.60698	−0.803490	−	0.595318i	$-0.797026\pi$
$59$	−12.3434	−1.60698	−0.803490	+	0.595318i	$0.797026\pi$
$60$	0	0
$61$	1.14590	0.146717	0.0733586	−	0.997306i	$-0.476628\pi$
$61$	1.14590	0.146717	0.0733586	+	0.997306i	$0.476628\pi$
$62$	0	0
$63$	4.23607i	0.533694i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.4721i	− 1.27938i	−0.768635	−	0.639688i	$-0.779064\pi$
$67$	− 10.4721i	− 1.27938i	0.768635	−	0.639688i	$-0.220936\pi$
$68$	0	0
$69$	−1.47214	−0.177224
$70$	0	0
$71$	−3.41164	−0.404888	−0.202444	−	0.979294i	$-0.564888\pi$
$71$	−3.41164	−0.404888	−0.202444	+	0.979294i	$0.564888\pi$
$72$	0	0
$73$	− 12.3434i	− 1.44469i	−0.691532	−	0.722346i	$-0.743064\pi$
$73$	− 12.3434i	− 1.44469i	0.691532	−	0.722346i	$-0.256936\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.93180i	1.01787i
$78$	0	0
$79$	8.93180	1.00491	0.502453	−	0.864604i	$-0.332431\pi$
$79$	8.93180	1.00491	0.502453	+	0.864604i	$0.332431\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	12.5623i	1.37889i	0.724337	+	0.689446i	$0.242146\pi$
$83$	12.5623i	1.37889i	−0.724337	+	0.689446i	$0.757854\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.673762i	− 0.0722349i
$88$	0	0
$89$	−3.14590	−0.333465	−0.166732	−	0.986002i	$-0.553322\pi$
$89$	−3.14590	−0.333465	−0.166732	+	0.986002i	$0.553322\pi$
$90$	0	0
$91$	−8.93180	−0.936307
$92$	0	0
$93$	− 3.41164i	− 0.353771i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.3434i	− 1.25329i	−0.779306	−	0.626644i	$-0.784428\pi$
$97$	− 12.3434i	− 1.25329i	0.779306	−	0.626644i	$-0.215572\pi$
$98$	0	0
$99$	14.4520	1.45248
$100$	0	0
$101$	−14.0902	−1.40202	−0.701012	−	0.713149i	$-0.747268\pi$
$101$	−14.0902	−1.40202	−0.701012	+	0.713149i	$0.747268\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.1459i	− 0.980841i	−0.871486	−	0.490420i	$-0.836843\pi$
$107$	− 10.1459i	− 0.980841i	0.871486	−	0.490420i	$-0.163157\pi$
$108$	0	0
$109$	−6.32624	−0.605944	−0.302972	−	0.953000i	$-0.597979\pi$
$109$	−6.32624	−0.605944	−0.302972	+	0.953000i	$0.597979\pi$
$110$	0	0
$111$	5.52016	0.523950
$112$	0	0
$113$	5.52016i	0.519293i	0.965704	+	0.259646i	$0.0836060\pi$
$113$	5.52016i	0.519293i	−0.965704	+	0.259646i	$0.916394\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.4520i	1.33608i
$118$	0	0
$119$	−5.52016	−0.506032
$120$	0	0
$121$	19.4721	1.77019
$122$	0	0
$123$	3.61803i	0.326227i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.09017i	0.806622i	0.915063	+	0.403311i	$0.132141\pi$
$127$	9.09017i	0.806622i	−0.915063	+	0.403311i	$0.867859\pi$
$128$	0	0
$129$	−7.61803	−0.670730
$130$	0	0
$131$	2.10851	0.184222	0.0921108	−	0.995749i	$-0.470639\pi$
$131$	2.10851	0.184222	0.0921108	+	0.995749i	$0.470639\pi$
$132$	0	0
$133$	5.52016i	0.478658i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.41164i	0.291476i	0.989323	+	0.145738i	$0.0465557\pi$
$137$	3.41164i	0.291476i	−0.989323	+	0.145738i	$0.953444\pi$
$138$	0	0
$139$	−2.10851	−0.178842	−0.0894208	−	0.995994i	$-0.528502\pi$
$139$	−2.10851	−0.178842	−0.0894208	+	0.995994i	$0.528502\pi$
$140$	0	0
$141$	−0.673762	−0.0567410
$142$	0	0
$143$	30.4721i	2.54821i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.70820i	0.223369i
$148$	0	0
$149$	9.14590	0.749261	0.374631	−	0.927174i	$-0.377770\pi$
$149$	9.14590	0.749261	0.374631	+	0.927174i	$0.377770\pi$
$150$	0	0
$151$	19.9721	1.62531	0.812654	−	0.582747i	$-0.198022\pi$
$151$	19.9721	1.62531	0.812654	+	0.582747i	$0.198022\pi$
$152$	0	0
$153$	8.93180i	0.722093i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.62867i	− 0.608834i	−0.952539	−	0.304417i	$-0.901538\pi$
$157$	− 7.62867i	− 0.608834i	0.952539	−	0.304417i	$-0.0984616\pi$
$158$	0	0
$159$	6.82329	0.541122
$160$	0	0
$161$	−3.85410	−0.303746
$162$	0	0
$163$	18.2705i	1.43106i	0.698584	+	0.715528i	$0.253814\pi$
$163$	18.2705i	1.43106i	−0.698584	+	0.715528i	$0.746186\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.6180i	− 1.13118i	−0.824687	−	0.565589i	$-0.808649\pi$
$167$	− 14.6180i	− 1.13118i	0.824687	−	0.565589i	$-0.191351\pi$
$168$	0	0
$169$	−17.4721	−1.34401
$170$	0	0
$171$	8.93180	0.683032
$172$	0	0
$173$	10.2349i	0.778148i	0.921207	+	0.389074i	$0.127205\pi$
$173$	10.2349i	0.778148i	−0.921207	+	0.389074i	$0.872795\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 7.62867i	− 0.573406i
$178$	0	0
$179$	−12.3434	−0.922593	−0.461296	−	0.887246i	$-0.652615\pi$
$179$	−12.3434	−0.922593	−0.461296	+	0.887246i	$0.652615\pi$
$180$	0	0
$181$	−21.0902	−1.56762	−0.783810	−	0.621001i	$-0.786726\pi$
$181$	−21.0902	−1.56762	−0.783810	+	0.621001i	$0.786726\pi$
$182$	0	0
$183$	0.708204i	0.0523519i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.8328i	1.37719i
$188$	0	0
$189$	−5.61803	−0.408652
$190$	0	0
$191$	10.2349	0.740574	0.370287	−	0.928917i	$-0.379259\pi$
$191$	10.2349	0.740574	0.370287	+	0.928917i	$0.379259\pi$
$192$	0	0
$193$	8.93180i	0.642925i	0.946922	+	0.321463i	$0.104174\pi$
$193$	8.93180i	0.642925i	−0.946922	+	0.321463i	$0.895826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.41164i	− 0.243070i	−0.992587	−	0.121535i	$-0.961218\pi$
$197$	− 3.41164i	− 0.243070i	0.992587	−	0.121535i	$-0.0387816\pi$
$198$	0	0
$199$	−14.4520	−1.02447	−0.512236	−	0.858845i	$-0.671183\pi$
$199$	−14.4520	−1.02447	−0.512236	+	0.858845i	$0.671183\pi$
$200$	0	0
$201$	6.47214	0.456509
$202$	0	0
$203$	− 1.76393i	− 0.123804i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.23607i	0.433437i
$208$	0	0
$209$	18.8328	1.30269
$210$	0	0
$211$	−13.1488	−0.905203	−0.452601	−	0.891713i	$-0.649504\pi$
$211$	−13.1488	−0.905203	−0.452601	+	0.891713i	$0.649504\pi$
$212$	0	0
$213$	− 2.10851i	− 0.144473i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.93180i	− 0.606330i
$218$	0	0
$219$	7.62867	0.515498
$220$	0	0
$221$	−18.8328	−1.26683
$222$	0	0
$223$	18.3262i	1.22722i	0.789611	+	0.613608i	$0.210282\pi$
$223$	18.3262i	1.22722i	−0.789611	+	0.613608i	$0.789718\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.85410i	− 0.189433i	−0.995504	−	0.0947167i	$-0.969805\pi$
$227$	− 2.85410i	− 0.189433i	0.995504	−	0.0947167i	$-0.0301945\pi$
$228$	0	0
$229$	25.3262	1.67360	0.836802	−	0.547505i	$-0.184422\pi$
$229$	25.3262	1.67360	0.836802	+	0.547505i	$0.184422\pi$
$230$	0	0
$231$	−5.52016	−0.363200
$232$	0	0
$233$	− 3.41164i	− 0.223504i	−0.993736	−	0.111752i	$-0.964354\pi$
$233$	− 3.41164i	− 0.223504i	0.993736	−	0.111752i	$-0.0356463\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.52016i	0.358573i
$238$	0	0
$239$	8.93180	0.577750	0.288875	−	0.957367i	$-0.406719\pi$
$239$	8.93180	0.577750	0.288875	+	0.957367i	$0.406719\pi$
$240$	0	0
$241$	14.5623	0.938041	0.469020	−	0.883187i	$-0.344607\pi$
$241$	14.5623	0.938041	0.469020	+	0.883187i	$0.344607\pi$
$242$	0	0
$243$	13.9443i	0.894525i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.8328i	1.19830i
$248$	0	0
$249$	−7.76393	−0.492019
$250$	0	0
$251$	−24.6869	−1.55822	−0.779111	−	0.626885i	$-0.784329\pi$
$251$	−24.6869	−1.55822	−0.779111	+	0.626885i	$0.784329\pi$
$252$	0	0
$253$	13.1488i	0.826660i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.8636i	1.11430i	0.830412	+	0.557151i	$0.188105\pi$
$257$	17.8636i	1.11430i	−0.830412	+	0.557151i	$0.811895\pi$
$258$	0	0
$259$	14.4520	0.898001
$260$	0	0
$261$	−2.85410	−0.176664
$262$	0	0
$263$	18.3262i	1.13004i	0.825076	+	0.565022i	$0.191132\pi$
$263$	18.3262i	1.13004i	−0.825076	+	0.565022i	$0.808868\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.94427i	− 0.118988i
$268$	0	0
$269$	−3.52786	−0.215098	−0.107549	−	0.994200i	$-0.534300\pi$
$269$	−3.52786	−0.215098	−0.107549	+	0.994200i	$0.534300\pi$
$270$	0	0
$271$	−11.0403	−0.670651	−0.335326	−	0.942102i	$-0.608846\pi$
$271$	−11.0403	−0.670651	−0.335326	+	0.942102i	$0.608846\pi$
$272$	0	0
$273$	− 5.52016i	− 0.334095i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 3.41164i	− 0.204986i	−0.994734	−	0.102493i	$-0.967318\pi$
$277$	− 3.41164i	− 0.204986i	0.994734	−	0.102493i	$-0.0326819\pi$
$278$	0	0
$279$	−14.4520	−0.865216
$280$	0	0
$281$	22.5623	1.34595	0.672977	−	0.739663i	$-0.265015\pi$
$281$	22.5623	1.34595	0.672977	+	0.739663i	$0.265015\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.47214i	0.559123i
$288$	0	0
$289$	5.36068	0.315334
$290$	0	0
$291$	7.62867	0.447201
$292$	0	0
$293$	− 1.30313i	− 0.0761298i	−0.999275	−	0.0380649i	$-0.987881\pi$
$293$	− 1.30313i	− 0.0761298i	0.999275	−	0.0380649i	$-0.0121194\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	19.1667i	1.11217i
$298$	0	0
$299$	−13.1488	−0.760416
$300$	0	0
$301$	−19.9443	−1.14957
$302$	0	0
$303$	− 8.70820i	− 0.500273i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.6180i	− 0.891368i	−0.895190	−	0.445684i	$-0.852960\pi$
$307$	− 15.6180i	− 0.891368i	0.895190	−	0.445684i	$-0.147040\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	0	0
$311$	−23.3838	−1.32597	−0.662986	−	0.748632i	$-0.730711\pi$
$311$	−23.3838	−1.32597	−0.662986	+	0.748632i	$0.730711\pi$
$312$	0	0
$313$	− 22.0806i	− 1.24807i	−0.781396	−	0.624035i	$-0.785492\pi$
$313$	− 22.0806i	− 1.24807i	0.781396	−	0.624035i	$-0.214508\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 33.6187i	− 1.88821i	−0.329639	−	0.944107i	$-0.606927\pi$
$317$	− 33.6187i	− 1.88821i	0.329639	−	0.944107i	$-0.393073\pi$
$318$	0	0
$319$	−6.01791	−0.336938
$320$	0	0
$321$	6.27051	0.349986
$322$	0	0
$323$	11.6393i	0.647629i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.90983i	− 0.216214i
$328$	0	0
$329$	−1.76393	−0.0972487
$330$	0	0
$331$	26.7954	1.47281	0.736404	−	0.676542i	$-0.236522\pi$
$331$	26.7954	1.47281	0.736404	+	0.676542i	$0.236522\pi$
$332$	0	0
$333$	− 23.3838i	− 1.28142i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.1488i	0.716262i	0.933671	+	0.358131i	$0.116586\pi$
$337$	13.1488i	0.716262i	−0.933671	+	0.358131i	$0.883414\pi$
$338$	0	0
$339$	−3.41164	−0.185295
$340$	0	0
$341$	−30.4721	−1.65016
$342$	0	0
$343$	18.4164i	0.994393i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.67376i	− 0.358266i	−0.983825	−	0.179133i	$-0.942671\pi$
$347$	− 6.67376i	− 0.358266i	0.983825	−	0.179133i	$-0.0573293\pi$
$348$	0	0
$349$	8.38197	0.448676	0.224338	−	0.974511i	$-0.427978\pi$
$349$	8.38197	0.448676	0.224338	+	0.974511i	$0.427978\pi$
$350$	0	0
$351$	−19.1667	−1.02304
$352$	0	0
$353$	− 12.3434i	− 0.656975i	−0.944508	−	0.328488i	$-0.893461\pi$
$353$	− 12.3434i	− 0.656975i	0.944508	−	0.328488i	$-0.106539\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.41164i	− 0.180563i
$358$	0	0
$359$	34.4241	1.81683	0.908417	−	0.418066i	$-0.137292\pi$
$359$	34.4241	1.81683	0.908417	+	0.418066i	$0.137292\pi$
$360$	0	0
$361$	−7.36068	−0.387404
$362$	0	0
$363$	12.0344i	0.631644i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.9787i	− 0.729683i	−0.931070	−	0.364841i	$-0.881123\pi$
$367$	− 13.9787i	− 0.729683i	0.931070	−	0.364841i	$-0.118877\pi$
$368$	0	0
$369$	15.3262	0.797852
$370$	0	0
$371$	17.8636	0.927432
$372$	0	0
$373$	− 4.71478i	− 0.244122i	−0.992523	−	0.122061i	$-0.961050\pi$
$373$	− 4.71478i	− 0.244122i	0.992523	−	0.122061i	$-0.0389503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.01791i	− 0.309938i
$378$	0	0
$379$	17.8636	0.917592	0.458796	−	0.888542i	$-0.348281\pi$
$379$	17.8636	0.917592	0.458796	+	0.888542i	$0.348281\pi$
$380$	0	0
$381$	−5.61803	−0.287821
$382$	0	0
$383$	− 22.9098i	− 1.17064i	−0.810803	−	0.585319i	$-0.800969\pi$
$383$	− 22.9098i	− 1.17064i	0.810803	−	0.585319i	$-0.199031\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.2705i	1.64040i
$388$	0	0
$389$	−11.8541	−0.601027	−0.300513	−	0.953778i	$-0.597158\pi$
$389$	−11.8541	−0.601027	−0.300513	+	0.953778i	$0.597158\pi$
$390$	0	0
$391$	−8.12642	−0.410971
$392$	0	0
$393$	1.30313i	0.0657343i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.2071i	1.51605i	0.652226	+	0.758024i	$0.273835\pi$
$397$	30.2071i	1.51605i	−0.652226	+	0.758024i	$0.726165\pi$
$398$	0	0
$399$	−3.41164	−0.170796
$400$	0	0
$401$	−0.0901699	−0.00450287	−0.00225144	−	0.999997i	$-0.500717\pi$
$401$	−0.0901699	−0.00450287	−0.00225144	+	0.999997i	$0.500717\pi$
$402$	0	0
$403$	− 30.4721i	− 1.51793i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 49.3050i	− 2.44396i
$408$	0	0
$409$	7.50658	0.371176	0.185588	−	0.982628i	$-0.440581\pi$
$409$	7.50658	0.371176	0.185588	+	0.982628i	$0.440581\pi$
$410$	0	0
$411$	−2.10851	−0.104005
$412$	0	0
$413$	− 19.9721i	− 0.982764i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.30313i	− 0.0638147i
$418$	0	0
$419$	−15.2573	−0.745370	−0.372685	−	0.927958i	$-0.621563\pi$
$419$	−15.2573	−0.745370	−0.372685	+	0.927958i	$0.621563\pi$
$420$	0	0
$421$	−1.38197	−0.0673529	−0.0336765	−	0.999433i	$-0.510722\pi$
$421$	−1.38197	−0.0673529	−0.0336765	+	0.999433i	$0.510722\pi$
$422$	0	0
$423$	2.85410i	0.138771i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.85410i	0.0897263i
$428$	0	0
$429$	−18.8328	−0.909257
$430$	0	0
$431$	19.9721	0.962023	0.481012	−	0.876714i	$-0.340270\pi$
$431$	19.9721	0.962023	0.481012	+	0.876714i	$0.340270\pi$
$432$	0	0
$433$	− 8.93180i	− 0.429235i	−0.976698	−	0.214618i	$-0.931150\pi$
$433$	− 8.93180i	− 0.429235i	0.976698	−	0.214618i	$-0.0688505\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.12642i	0.388739i
$438$	0	0
$439$	10.2349	0.488487	0.244243	−	0.969714i	$-0.421460\pi$
$439$	10.2349	0.488487	0.244243	+	0.969714i	$0.421460\pi$
$440$	0	0
$441$	11.4721	0.546292
$442$	0	0
$443$	8.20163i	0.389671i	0.980836	+	0.194836i	$0.0624173\pi$
$443$	8.20163i	0.389671i	−0.980836	+	0.194836i	$0.937583\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.65248i	0.267353i
$448$	0	0
$449$	3.52786	0.166490	0.0832451	−	0.996529i	$-0.473472\pi$
$449$	3.52786	0.166490	0.0832451	+	0.996529i	$0.473472\pi$
$450$	0	0
$451$	32.3156	1.52168
$452$	0	0
$453$	12.3434i	0.579946i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 33.1209i	− 1.54933i	−0.632370	−	0.774666i	$-0.717918\pi$
$457$	− 33.1209i	− 1.54933i	0.632370	−	0.774666i	$-0.282082\pi$
$458$	0	0
$459$	−11.8457	−0.552910
$460$	0	0
$461$	−5.50658	−0.256467	−0.128233	−	0.991744i	$-0.540931\pi$
$461$	−5.50658	−0.256467	−0.128233	+	0.991744i	$0.540931\pi$
$462$	0	0
$463$	26.7426i	1.24284i	0.783479	+	0.621418i	$0.213443\pi$
$463$	26.7426i	1.24284i	−0.783479	+	0.621418i	$0.786557\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 26.4508i	− 1.22400i	−0.790858	−	0.612000i	$-0.790365\pi$
$467$	− 26.4508i	− 1.22400i	0.790858	−	0.612000i	$-0.209635\pi$
$468$	0	0
$469$	16.9443	0.782414
$470$	0	0
$471$	4.71478	0.217245
$472$	0	0
$473$	68.0428i	3.12861i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 28.9039i	− 1.32342i
$478$	0	0
$479$	−2.10851	−0.0963404	−0.0481702	−	0.998839i	$-0.515339\pi$
$479$	−2.10851	−0.0963404	−0.0481702	+	0.998839i	$0.515339\pi$
$480$	0	0
$481$	49.3050	2.24811
$482$	0	0
$483$	− 2.38197i	− 0.108383i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.2705i	1.23574i	0.786278	+	0.617872i	$0.212005\pi$
$487$	27.2705i	1.23574i	−0.786278	+	0.617872i	$0.787995\pi$
$488$	0	0
$489$	−11.2918	−0.510633
$490$	0	0
$491$	17.0582	0.769827	0.384913	−	0.922953i	$-0.374231\pi$
$491$	17.0582	0.769827	0.384913	+	0.922953i	$0.374231\pi$
$492$	0	0
$493$	− 3.71927i	− 0.167508i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.52016i	− 0.247613i
$498$	0	0
$499$	−17.0582	−0.763631	−0.381815	−	0.924239i	$-0.624701\pi$
$499$	−17.0582	−0.763631	−0.381815	+	0.924239i	$0.624701\pi$
$500$	0	0
$501$	9.03444	0.403629
$502$	0	0
$503$	− 38.1591i	− 1.70143i	−0.525629	−	0.850714i	$-0.676170\pi$
$503$	− 38.1591i	− 1.70143i	0.525629	−	0.850714i	$-0.323830\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 10.7984i	− 0.479573i
$508$	0	0
$509$	−43.3050	−1.91946	−0.959729	−	0.280927i	$-0.909358\pi$
$509$	−43.3050	−1.91946	−0.959729	+	0.280927i	$0.909358\pi$
$510$	0	0
$511$	19.9721	0.883514
$512$	0	0
$513$	11.8457i	0.523000i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.01791i	0.264667i
$518$	0	0
$519$	−6.32554	−0.277660
$520$	0	0
$521$	7.09017	0.310626	0.155313	−	0.987865i	$-0.450361\pi$
$521$	7.09017	0.310626	0.155313	+	0.987865i	$0.450361\pi$
$522$	0	0
$523$	− 12.9787i	− 0.567520i	−0.958895	−	0.283760i	$-0.908418\pi$
$523$	− 12.9787i	− 0.567520i	0.958895	−	0.283760i	$-0.0915818\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 18.8328i	− 0.820370i
$528$	0	0
$529$	17.3262	0.753315
$530$	0	0
$531$	−32.3156	−1.40238
$532$	0	0
$533$	32.3156i	1.39974i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.62867i	− 0.329201i
$538$	0	0
$539$	24.1891	1.04190
$540$	0	0
$541$	−27.5623	−1.18500	−0.592498	−	0.805572i	$-0.701858\pi$
$541$	−27.5623	−1.18500	−0.592498	+	0.805572i	$0.701858\pi$
$542$	0	0
$543$	− 13.0344i	− 0.559361i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.9098i	0.765769i	0.923796	+	0.382885i	$0.125069\pi$
$547$	17.9098i	0.765769i	−0.923796	+	0.382885i	$0.874931\pi$
$548$	0	0
$549$	3.00000	0.128037
$550$	0	0
$551$	−3.71927	−0.158446
$552$	0	0
$553$	14.4520i	0.614560i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.7093i	1.25882i	0.777072	+	0.629412i	$0.216704\pi$
$557$	29.7093i	1.25882i	−0.777072	+	0.629412i	$0.783296\pi$
$558$	0	0
$559$	−68.0428	−2.87790
$560$	0	0
$561$	−11.6393	−0.491412
$562$	0	0
$563$	8.94427i	0.376956i	0.982077	+	0.188478i	$0.0603554\pi$
$563$	8.94427i	0.376956i	−0.982077	+	0.188478i	$0.939645\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.23607i	0.387878i
$568$	0	0
$569$	−15.0902	−0.632613	−0.316306	−	0.948657i	$-0.602443\pi$
$569$	−15.0902	−0.632613	−0.316306	+	0.948657i	$0.602443\pi$
$570$	0	0
$571$	−42.5505	−1.78068	−0.890341	−	0.455293i	$-0.849534\pi$
$571$	−42.5505	−1.78068	−0.890341	+	0.455293i	$0.849534\pi$
$572$	0	0
$573$	6.32554i	0.264253i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	35.7272i	1.48734i	0.668545	+	0.743672i	$0.266917\pi$
$577$	35.7272i	1.48734i	−0.668545	+	0.743672i	$0.733083\pi$
$578$	0	0
$579$	−5.52016	−0.229410
$580$	0	0
$581$	−20.3262	−0.843274
$582$	0	0
$583$	− 60.9443i	− 2.52405i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.7771i	1.47668i	0.674430	+	0.738339i	$0.264390\pi$
$587$	35.7771i	1.47668i	−0.674430	+	0.738339i	$0.735610\pi$
$588$	0	0
$589$	−18.8328	−0.775993
$590$	0	0
$591$	2.10851	0.0867326
$592$	0	0
$593$	− 33.6187i	− 1.38055i	−0.723545	−	0.690277i	$-0.757489\pi$
$593$	− 33.6187i	− 1.38055i	0.723545	−	0.690277i	$-0.242511\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.93180i	− 0.365554i
$598$	0	0
$599$	42.0527	1.71823	0.859114	−	0.511784i	$-0.171015\pi$
$599$	42.0527	1.71823	0.859114	+	0.511784i	$0.171015\pi$
$600$	0	0
$601$	6.43769	0.262599	0.131300	−	0.991343i	$-0.458085\pi$
$601$	6.43769	0.262599	0.131300	+	0.991343i	$0.458085\pi$
$602$	0	0
$603$	− 27.4164i	− 1.11648i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.36068i	− 0.176995i	−0.996076	−	0.0884973i	$-0.971794\pi$
$607$	− 4.36068i	− 0.176995i	0.996076	−	0.0884973i	$-0.0282065\pi$
$608$	0	0
$609$	1.09017	0.0441759
$610$	0	0
$611$	−6.01791	−0.243459
$612$	0	0
$613$	11.0403i	0.445914i	0.974828	+	0.222957i	$0.0715710\pi$
$613$	11.0403i	0.445914i	−0.974828	+	0.222957i	$0.928429\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.62867i	− 0.307119i	−0.988139	−	0.153559i	$-0.950926\pi$
$617$	− 7.62867i	− 0.307119i	0.988139	−	0.153559i	$-0.0490736\pi$
$618$	0	0
$619$	18.6690	0.750370	0.375185	−	0.926950i	$-0.377579\pi$
$619$	18.6690	0.750370	0.375185	+	0.926950i	$0.377579\pi$
$620$	0	0
$621$	−8.27051	−0.331884
$622$	0	0
$623$	− 5.09017i	− 0.203933i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.6393i	0.464830i
$628$	0	0
$629$	30.4721	1.21500
$630$	0	0
$631$	5.52016	0.219754	0.109877	−	0.993945i	$-0.464954\pi$
$631$	5.52016	0.219754	0.109877	+	0.993945i	$0.464954\pi$
$632$	0	0
$633$	− 8.12642i	− 0.322996i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	24.1891i	0.958409i
$638$	0	0
$639$	−8.93180	−0.353337
$640$	0	0
$641$	41.2705	1.63009	0.815044	−	0.579400i	$-0.196713\pi$
$641$	41.2705	1.63009	0.815044	+	0.579400i	$0.196713\pi$
$642$	0	0
$643$	18.0902i	0.713407i	0.934218	+	0.356703i	$0.116099\pi$
$643$	18.0902i	0.713407i	−0.934218	+	0.356703i	$0.883901\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.05573i	− 0.120133i	−0.998194	−	0.0600665i	$-0.980869\pi$
$647$	− 3.05573i	− 0.120133i	0.998194	−	0.0600665i	$-0.0191313\pi$
$648$	0	0
$649$	−68.1378	−2.67464
$650$	0	0
$651$	5.52016	0.216352
$652$	0	0
$653$	− 4.71478i	− 0.184503i	−0.995736	−	0.0922517i	$-0.970594\pi$
$653$	− 4.71478i	− 0.184503i	0.995736	−	0.0922517i	$-0.0294065\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 32.3156i	− 1.26075i
$658$	0	0
$659$	25.4923	0.993038	0.496519	−	0.868026i	$-0.334611\pi$
$659$	25.4923	0.993038	0.496519	+	0.868026i	$0.334611\pi$
$660$	0	0
$661$	27.5623	1.07205	0.536025	−	0.844202i	$-0.319925\pi$
$661$	27.5623	1.07205	0.536025	+	0.844202i	$0.319925\pi$
$662$	0	0
$663$	− 11.6393i	− 0.452034i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.59675i	− 0.100546i
$668$	0	0
$669$	−11.3262	−0.437898
$670$	0	0
$671$	6.32554	0.244195
$672$	0	0
$673$	− 27.6008i	− 1.06393i	−0.846766	−	0.531966i	$-0.821453\pi$
$673$	− 27.6008i	− 1.06393i	0.846766	−	0.531966i	$-0.178547\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.8636i	0.686554i	0.939234	+	0.343277i	$0.111537\pi$
$677$	17.8636i	0.686554i	−0.939234	+	0.343277i	$0.888463\pi$
$678$	0	0
$679$	19.9721	0.766459
$680$	0	0
$681$	1.76393	0.0675940
$682$	0	0
$683$	− 48.6869i	− 1.86295i	−0.363801	−	0.931477i	$-0.618521\pi$
$683$	− 48.6869i	− 1.86295i	0.363801	−	0.931477i	$-0.381479\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.6525i	0.597179i
$688$	0	0
$689$	60.9443	2.32179
$690$	0	0
$691$	−33.6187	−1.27892	−0.639458	−	0.768826i	$-0.720841\pi$
$691$	−33.6187	−1.27892	−0.639458	+	0.768826i	$0.720841\pi$
$692$	0	0
$693$	23.3838i	0.888276i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.9721i	0.756498i
$698$	0	0
$699$	2.10851	0.0797513
$700$	0	0
$701$	26.9443	1.01767	0.508836	−	0.860864i	$-0.330076\pi$
$701$	26.9443	1.01767	0.508836	+	0.860864i	$0.330076\pi$
$702$	0	0
$703$	− 30.4721i	− 1.14928i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 22.7984i	− 0.857421i
$708$	0	0
$709$	36.3820	1.36635	0.683177	−	0.730253i	$-0.260598\pi$
$709$	36.3820	1.36635	0.683177	+	0.730253i	$0.260598\pi$
$710$	0	0
$711$	23.3838	0.876960
$712$	0	0
$713$	− 13.1488i	− 0.492427i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.52016i	0.206154i
$718$	0	0
$719$	29.4017	1.09650	0.548249	−	0.836315i	$-0.315295\pi$
$719$	29.4017	1.09650	0.548249	+	0.836315i	$0.315295\pi$
$720$	0	0
$721$	−12.9443	−0.482070
$722$	0	0
$723$	9.00000i	0.334714i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.9098i	1.29473i	0.762178	+	0.647367i	$0.224130\pi$
$727$	34.9098i	1.29473i	−0.762178	+	0.647367i	$0.775870\pi$
$728$	0	0
$729$	8.50658	0.315058
$730$	0	0
$731$	−42.0527	−1.55538
$732$	0	0
$733$	− 9.73718i	− 0.359651i	−0.983699	−	0.179826i	$-0.942447\pi$
$733$	− 9.73718i	− 0.359651i	0.983699	−	0.179826i	$-0.0575533\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 57.8078i	− 2.12938i
$738$	0	0
$739$	−37.8357	−1.39181	−0.695905	−	0.718134i	$-0.744996\pi$
$739$	−37.8357	−1.39181	−0.695905	+	0.718134i	$0.744996\pi$
$740$	0	0
$741$	−11.6393	−0.427581
$742$	0	0
$743$	10.1115i	0.370953i	0.982649	+	0.185477i	$0.0593829\pi$
$743$	10.1115i	0.370953i	−0.982649	+	0.185477i	$0.940617\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.8885i	1.20333i
$748$	0	0
$749$	16.4164	0.599842
$750$	0	0
$751$	−37.8357	−1.38065	−0.690323	−	0.723502i	$-0.742531\pi$
$751$	−37.8357	−1.38065	−0.690323	+	0.723502i	$0.742531\pi$
$752$	0	0
$753$	− 15.2573i	− 0.556008i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.91389i	0.105907i	0.998597	+	0.0529536i	$0.0168635\pi$
$757$	2.91389i	0.105907i	−0.998597	+	0.0529536i	$0.983136\pi$
$758$	0	0
$759$	−8.12642	−0.294970
$760$	0	0
$761$	0.854102	0.0309612	0.0154806	−	0.999880i	$-0.495072\pi$
$761$	0.854102	0.0309612	0.0154806	+	0.999880i	$0.495072\pi$
$762$	0	0
$763$	− 10.2361i	− 0.370571i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 68.1378i	− 2.46031i
$768$	0	0
$769$	−14.7984	−0.533643	−0.266822	−	0.963746i	$-0.585973\pi$
$769$	−14.7984	−0.533643	−0.266822	+	0.963746i	$0.585973\pi$
$770$	0	0
$771$	−11.0403	−0.397607
$772$	0	0
$773$	− 50.9845i	− 1.83379i	−0.399132	−	0.916893i	$-0.630689\pi$
$773$	− 50.9845i	− 1.83379i	0.399132	−	0.916893i	$-0.369311\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.93180i	0.320427i
$778$	0	0
$779$	19.9721	0.715575
$780$	0	0
$781$	−18.8328	−0.673891
$782$	0	0
$783$	− 3.78522i	− 0.135273i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.5623i	1.19637i	0.801359	+	0.598184i	$0.204111\pi$
$787$	33.5623i	1.19637i	−0.801359	+	0.598184i	$0.795889\pi$
$788$	0	0
$789$	−11.3262	−0.403225
$790$	0	0
$791$	−8.93180	−0.317578
$792$	0	0
$793$	6.32554i	0.224626i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 52.2877i	− 1.85212i	−0.377371	−	0.926062i	$-0.623172\pi$
$797$	− 52.2877i	− 1.85212i	0.377371	−	0.926062i	$-0.376828\pi$
$798$	0	0
$799$	−3.71927	−0.131578
$800$	0	0
$801$	−8.23607	−0.291007
$802$	0	0
$803$	− 68.1378i	− 2.40453i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.18034i	− 0.0767516i
$808$	0	0
$809$	2.38197	0.0837455	0.0418727	−	0.999123i	$-0.486668\pi$
$809$	2.38197	0.0837455	0.0418727	+	0.999123i	$0.486668\pi$
$810$	0	0
$811$	52.2877	1.83607	0.918034	−	0.396501i	$-0.129776\pi$
$811$	52.2877	1.83607	0.918034	+	0.396501i	$0.129776\pi$
$812$	0	0
$813$	− 6.82329i	− 0.239303i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	42.0527i	1.47124i
$818$	0	0
$819$	−23.3838	−0.817095
$820$	0	0
$821$	24.5623	0.857230	0.428615	−	0.903487i	$-0.359002\pi$
$821$	24.5623	0.857230	0.428615	+	0.903487i	$0.359002\pi$
$822$	0	0
$823$	− 54.2492i	− 1.89101i	−0.325609	−	0.945505i	$-0.605569\pi$
$823$	− 54.2492i	− 1.89101i	0.325609	−	0.945505i	$-0.394431\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.8328i	− 1.62854i	−0.580489	−	0.814268i	$-0.697138\pi$
$827$	− 46.8328i	− 1.62854i	0.580489	−	0.814268i	$-0.302862\pi$
$828$	0	0
$829$	−7.85410	−0.272784	−0.136392	−	0.990655i	$-0.543551\pi$
$829$	−7.85410	−0.272784	−0.136392	+	0.990655i	$0.543551\pi$
$830$	0	0
$831$	2.10851	0.0731435
$832$	0	0
$833$	14.9497i	0.517977i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 19.1667i	− 0.662499i
$838$	0	0
$839$	−51.4823	−1.77737	−0.888683	−	0.458522i	$-0.848379\pi$
$839$	−51.4823	−1.77737	−0.888683	+	0.458522i	$0.848379\pi$
$840$	0	0
$841$	−27.8115	−0.959018
$842$	0	0
$843$	13.9443i	0.480266i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	31.5066i	1.08258i
$848$	0	0
$849$	2.47214	0.0848435
$850$	0	0
$851$	21.2752	0.729306
$852$	0	0
$853$	− 15.2573i	− 0.522401i	−0.965285	−	0.261201i	$-0.915882\pi$
$853$	− 15.2573i	− 0.522401i	0.965285	−	0.261201i	$-0.0841184\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.2474i	1.40898i	0.709712	+	0.704492i	$0.248825\pi$
$857$	41.2474i	1.40898i	−0.709712	+	0.704492i	$0.751175\pi$
$858$	0	0
$859$	−6.82329	−0.232808	−0.116404	−	0.993202i	$-0.537137\pi$
$859$	−6.82329	−0.232808	−0.116404	+	0.993202i	$0.537137\pi$
$860$	0	0
$861$	−5.85410	−0.199507
$862$	0	0
$863$	− 15.5066i	− 0.527850i	−0.964543	−	0.263925i	$-0.914983\pi$
$863$	− 15.5066i	− 0.527850i	0.964543	−	0.263925i	$-0.0850172\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.31308i	0.112518i
$868$	0	0
$869$	49.3050	1.67256
$870$	0	0
$871$	57.8078	1.95874
$872$	0	0
$873$	− 32.3156i	− 1.09372i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19.9721i	− 0.674410i	−0.941431	−	0.337205i	$-0.890518\pi$
$877$	− 19.9721i	− 0.674410i	0.941431	−	0.337205i	$-0.109482\pi$
$878$	0	0
$879$	0.805380	0.0271648
$880$	0	0
$881$	29.9787	1.01001	0.505004	−	0.863117i	$-0.331491\pi$
$881$	29.9787	1.01001	0.505004	+	0.863117i	$0.331491\pi$
$882$	0	0
$883$	− 38.8541i	− 1.30754i	−0.756691	−	0.653772i	$-0.773185\pi$
$883$	− 38.8541i	− 1.30754i	0.756691	−	0.653772i	$-0.226815\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.43769i	0.182580i	0.995824	+	0.0912899i	$0.0290990\pi$
$887$	5.43769i	0.182580i	−0.995824	+	0.0912899i	$0.970901\pi$
$888$	0	0
$889$	−14.7082	−0.493297
$890$	0	0
$891$	31.5102	1.05563
$892$	0	0
$893$	3.71927i	0.124461i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.12642i	− 0.271333i
$898$	0	0
$899$	6.01791	0.200709
$900$	0	0
$901$	37.6656	1.25482
$902$	0	0
$903$	− 12.3262i	− 0.410192i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27.7984i	− 0.923030i	−0.887132	−	0.461515i	$-0.847306\pi$
$907$	− 27.7984i	− 0.923030i	0.887132	−	0.461515i	$-0.152694\pi$
$908$	0	0
$909$	−36.8885	−1.22352
$910$	0	0
$911$	8.12642	0.269240	0.134620	−	0.990897i	$-0.457019\pi$
$911$	8.12642	0.269240	0.134620	+	0.990897i	$0.457019\pi$
$912$	0	0
$913$	69.3459i	2.29501i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.41164i	0.112662i
$918$	0	0
$919$	39.1389	1.29107	0.645536	−	0.763730i	$-0.276634\pi$
$919$	39.1389	1.29107	0.645536	+	0.763730i	$0.276634\pi$
$920$	0	0
$921$	9.65248	0.318060
$922$	0	0
$923$	− 18.8328i	− 0.619890i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	20.9443i	0.687900i
$928$	0	0
$929$	1.03444	0.0339389	0.0169695	−	0.999856i	$-0.494598\pi$
$929$	1.03444	0.0339389	0.0169695	+	0.999856i	$0.494598\pi$
$930$	0	0
$931$	14.9497	0.489957
$932$	0	0
$933$	− 14.4520i	− 0.473136i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0806i	0.721343i	0.932693	+	0.360671i	$0.117452\pi$
$937$	22.0806i	0.721343i	−0.932693	+	0.360671i	$0.882548\pi$
$938$	0	0
$939$	13.6466	0.445339
$940$	0	0
$941$	0.111456	0.00363337	0.00181668	−	0.999998i	$-0.499422\pi$
$941$	0.111456	0.00363337	0.00181668	+	0.999998i	$0.499422\pi$
$942$	0	0
$943$	13.9443i	0.454088i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 37.2705i	− 1.21113i	−0.795796	−	0.605564i	$-0.792947\pi$
$947$	− 37.2705i	− 1.21113i	0.795796	−	0.605564i	$-0.207053\pi$
$948$	0	0
$949$	68.1378	2.21185
$950$	0	0
$951$	20.7775	0.673756
$952$	0	0
$953$	− 53.5908i	− 1.73598i	−0.496585	−	0.867988i	$-0.665413\pi$
$953$	− 53.5908i	− 1.73598i	0.496585	−	0.867988i	$-0.334587\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.71927i	− 0.120227i
$958$	0	0
$959$	−5.52016	−0.178255
$960$	0	0
$961$	−0.527864	−0.0170279
$962$	0	0
$963$	− 26.5623i	− 0.855958i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 42.3951i	− 1.36334i	−0.731662	−	0.681668i	$-0.761255\pi$
$967$	− 42.3951i	− 1.36334i	0.731662	−	0.681668i	$-0.238745\pi$
$968$	0	0
$969$	−7.19350	−0.231088
$970$	0	0
$971$	−2.10851	−0.0676654	−0.0338327	−	0.999428i	$-0.510771\pi$
$971$	−2.10851	−0.0676654	−0.0338327	+	0.999428i	$0.510771\pi$
$972$	0	0
$973$	− 3.41164i	− 0.109372i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11.8457i	− 0.378977i	−0.981883	−	0.189489i	$-0.939317\pi$
$977$	− 11.8457i	− 0.378977i	0.981883	−	0.189489i	$-0.0606830\pi$
$978$	0	0
$979$	−17.3659	−0.555015
$980$	0	0
$981$	−16.5623	−0.528794
$982$	0	0
$983$	− 7.41641i	− 0.236547i	−0.992981	−	0.118273i	$-0.962264\pi$
$983$	− 7.41641i	− 0.236547i	0.992981	−	0.118273i	$-0.0377359\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.09017i	− 0.0347005i
$988$	0	0
$989$	−29.3607	−0.933615
$990$	0	0
$991$	−37.3380	−1.18608	−0.593040	−	0.805173i	$-0.702072\pi$
$991$	−37.3380	−1.18608	−0.593040	+	0.805173i	$0.702072\pi$
$992$	0	0
$993$	16.5605i	0.525531i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 28.0985i	− 0.889890i	−0.895558	−	0.444945i	$-0.853223\pi$
$997$	− 28.0985i	− 0.889890i	0.895558	−	0.444945i	$-0.146777\pi$
$998$	0	0
$999$	31.0124	0.981190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.c.e.1249.6		8
4.3	odd	2	inner	4000.2.c.e.1249.3		8
5.2	odd	4		4000.2.a.c.1.4	yes	4
5.3	odd	4		4000.2.a.h.1.2	yes	4
5.4	even	2	inner	4000.2.c.e.1249.4		8
20.3	even	4		4000.2.a.c.1.3	✓	4
20.7	even	4		4000.2.a.h.1.1	yes	4
20.19	odd	2	inner	4000.2.c.e.1249.5		8
40.3	even	4		8000.2.a.bp.1.2		4
40.13	odd	4		8000.2.a.bc.1.3		4
40.27	even	4		8000.2.a.bc.1.4		4
40.37	odd	4		8000.2.a.bp.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.c.1.3	✓	4	20.3	even	4
4000.2.a.c.1.4	yes	4	5.2	odd	4
4000.2.a.h.1.1	yes	4	20.7	even	4
4000.2.a.h.1.2	yes	4	5.3	odd	4
4000.2.c.e.1249.3		8	4.3	odd	2	inner
4000.2.c.e.1249.4		8	5.4	even	2	inner
4000.2.c.e.1249.5		8	20.19	odd	2	inner
4000.2.c.e.1249.6		8	1.1	even	1	trivial
8000.2.a.bc.1.3		4	40.13	odd	4
8000.2.a.bc.1.4		4	40.27	even	4
8000.2.a.bp.1.1		4	40.37	odd	4
8000.2.a.bp.1.2		4	40.3	even	4

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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +O(q^{10})\)	\(q+0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +5.52016 q^{11} +5.52016i q^{13} +3.41164i q^{17} +3.41164 q^{19} -1.00000 q^{21} +2.38197i q^{23} +3.47214i q^{27} -1.09017 q^{29} -5.52016 q^{31} +3.41164i q^{33} -8.93180i q^{37} -3.41164 q^{39} +5.85410 q^{41} +12.3262i q^{43} +1.09017i q^{47} +4.38197 q^{49} -2.10851 q^{51} -11.0403i q^{53} +2.10851i q^{57} -12.3434 q^{59} +1.14590 q^{61} +4.23607i q^{63} -10.4721i q^{67} -1.47214 q^{69} -3.41164 q^{71} -12.3434i q^{73} +8.93180i q^{77} +8.93180 q^{79} +5.70820 q^{81} +12.5623i q^{83} -0.673762i q^{87} -3.14590 q^{89} -8.93180 q^{91} -3.41164i q^{93} -12.3434i q^{97} +14.4520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9}+O(q^{10}) \) 8 * q + 12 * q^9	\( 8 q + 12 q^{9} - 8 q^{21} + 36 q^{29} + 20 q^{41} + 44 q^{49} + 36 q^{61} + 24 q^{69} - 8 q^{81} - 52 q^{89}+O(q^{100}) \) 8 * q + 12 * q^9 - 8 * q^21 + 36 * q^29 + 20 * q^41 + 44 * q^49 + 36 * q^61 + 24 * q^69 - 8 * q^81 - 52 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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