LMFDB - Embedded newform 2000.2.c.j.1249.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2000,2,Mod(1249,2000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2000.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.9700804043$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.309760000.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 $ x^8 - 4x^7 + 8x^6 - 2x^5 - x^4 - 2x^3 + 18x^2 + 6x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$-0.748606 - 0.748606i$ of defining polynomial
Character	$\chi$	$=$	2000.1249
Dual form			2000.2.c.j.1249.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.543362i q^{3} +2.11525i q^{7} +2.70476 q^{9} +4.99442 q^{11} -6.32279i q^{13} +3.75836i q^{17} +1.90770 q^{19} +1.14934 q^{21} +9.35131i q^{23} -3.09975i q^{27} -7.22705 q^{29} -0.994424 q^{31} -2.71378i q^{33} -6.37984i q^{37} -3.43556 q^{39} +4.18247 q^{41} +1.45664i q^{43} -2.78501i q^{47} +2.52573 q^{49} +2.04215 q^{51} -1.52786i q^{53} -1.03657i q^{57} +1.47771 q^{59} +8.79148 q^{61} +5.72123i q^{63} +12.0811i q^{67} +5.08115 q^{69} +12.3798 q^{71} -9.09362i q^{73} +10.5644i q^{77} -8.91328 q^{79} +6.42999 q^{81} +12.1964i q^{83} +3.92690i q^{87} -3.08328 q^{89} +13.3743 q^{91} +0.540332i q^{93} +10.1382i q^{97} +13.5087 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9} + 16 q^{21} - 20 q^{29} + 32 q^{31} - 48 q^{39} + 4 q^{41} - 4 q^{49} + 16 q^{59} + 12 q^{61} - 24 q^{69} + 48 q^{71} - 96 q^{79} + 32 q^{81} - 20 q^{89} + 16 q^{91} + 16 q^{99}+O(q^{100}) $ 8 * q - 12 * q^9 + 16 * q^21 - 20 * q^29 + 32 * q^31 - 48 * q^39 + 4 * q^41 - 4 * q^49 + 16 * q^59 + 12 * q^61 - 24 * q^69 + 48 * q^71 - 96 * q^79 + 32 * q^81 - 20 * q^89 + 16 * q^91 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times$.

$n$	$501$	$751$	$1377$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 0.543362i	− 0.313710i	−0.987622	−	0.156855i	$-0.949864\pi$
$3$	− 0.543362i	− 0.313710i	0.987622	−	0.156855i	$-0.0501356\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.11525i	0.799488i	0.916627	+	0.399744i	$0.130901\pi$
$7$	2.11525i	0.799488i	−0.916627	+	0.399744i	$0.869099\pi$
$8$	0	0
$9$	2.70476	0.901586
$10$	0	0
$11$	4.99442	1.50588	0.752938	−	0.658092i	$-0.228636\pi$
$11$	4.99442	1.50588	0.752938	+	0.658092i	$0.228636\pi$
$12$	0	0
$13$	− 6.32279i	− 1.75363i	−0.480831	−	0.876813i	$-0.659665\pi$
$13$	− 6.32279i	− 1.75363i	0.480831	−	0.876813i	$-0.340335\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.75836i	0.911535i	0.890099	+	0.455768i	$0.150635\pi$
$17$	3.75836i	0.911535i	−0.890099	+	0.455768i	$0.849365\pi$
$18$	0	0
$19$	1.90770	0.437656	0.218828	−	0.975763i	$-0.429777\pi$
$19$	1.90770	0.437656	0.218828	+	0.975763i	$0.429777\pi$
$20$	0	0
$21$	1.14934	0.250807
$22$	0	0
$23$	9.35131i	1.94988i	0.222460	+	0.974942i	$0.428591\pi$
$23$	9.35131i	1.94988i	−0.222460	+	0.974942i	$0.571409\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.09975i	− 0.596547i
$28$	0	0
$29$	−7.22705	−1.34203	−0.671014	−	0.741444i	$-0.734141\pi$
$29$	−7.22705	−1.34203	−0.671014	+	0.741444i	$0.734141\pi$
$30$	0	0
$31$	−0.994424	−0.178604	−0.0893019	−	0.996005i	$-0.528464\pi$
$31$	−0.994424	−0.178604	−0.0893019	+	0.996005i	$0.528464\pi$
$32$	0	0
$33$	− 2.71378i	− 0.472408i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.37984i	− 1.04884i	−0.851460	−	0.524419i	$-0.824282\pi$
$37$	− 6.37984i	− 1.04884i	0.851460	−	0.524419i	$-0.175718\pi$
$38$	0	0
$39$	−3.43556	−0.550131
$40$	0	0
$41$	4.18247	0.653192	0.326596	−	0.945164i	$-0.394098\pi$
$41$	4.18247	0.653192	0.326596	+	0.945164i	$0.394098\pi$
$42$	0	0
$43$	1.45664i	0.222135i	0.993813	+	0.111068i	$0.0354270\pi$
$43$	1.45664i	0.222135i	−0.993813	+	0.111068i	$0.964573\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.78501i	− 0.406235i	−0.979154	−	0.203117i	$-0.934893\pi$
$47$	− 2.78501i	− 0.406235i	0.979154	−	0.203117i	$-0.0651073\pi$
$48$	0	0
$49$	2.52573	0.360819
$50$	0	0
$51$	2.04215	0.285958
$52$	0	0
$53$	− 1.52786i	− 0.209868i	−0.994479	−	0.104934i	$-0.966537\pi$
$53$	− 1.52786i	− 0.209868i	0.994479	−	0.104934i	$-0.0334632\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.03657i	− 0.137297i
$58$	0	0
$59$	1.47771	0.192382	0.0961908	−	0.995363i	$-0.469334\pi$
$59$	1.47771	0.192382	0.0961908	+	0.995363i	$0.469334\pi$
$60$	0	0
$61$	8.79148	1.12563	0.562817	−	0.826582i	$-0.309718\pi$
$61$	8.79148	1.12563	0.562817	+	0.826582i	$0.309718\pi$
$62$	0	0
$63$	5.72123i	0.720807i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0811i	1.47595i	0.674830	+	0.737974i	$0.264217\pi$
$67$	12.0811i	1.47595i	−0.674830	+	0.737974i	$0.735783\pi$
$68$	0	0
$69$	5.08115	0.611698
$70$	0	0
$71$	12.3798	1.46922	0.734608	−	0.678492i	$-0.237366\pi$
$71$	12.3798	1.46922	0.734608	+	0.678492i	$0.237366\pi$
$72$	0	0
$73$	− 9.09362i	− 1.06433i	−0.846642	−	0.532164i	$-0.821379\pi$
$73$	− 9.09362i	− 1.06433i	0.846642	−	0.532164i	$-0.178621\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.5644i	1.20393i
$78$	0	0
$79$	−8.91328	−1.00282	−0.501411	−	0.865209i	$-0.667186\pi$
$79$	−8.91328	−1.00282	−0.501411	+	0.865209i	$0.667186\pi$
$80$	0	0
$81$	6.42999	0.714443
$82$	0	0
$83$	12.1964i	1.33873i	0.742935	+	0.669364i	$0.233433\pi$
$83$	12.1964i	1.33873i	−0.742935	+	0.669364i	$0.766567\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.92690i	0.421008i
$88$	0	0
$89$	−3.08328	−0.326827	−0.163413	−	0.986558i	$-0.552250\pi$
$89$	−3.08328	−0.326827	−0.163413	+	0.986558i	$0.552250\pi$
$90$	0	0
$91$	13.3743	1.40200
$92$	0	0
$93$	0.540332i	0.0560298i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.1382i	1.02938i	0.857377	+	0.514689i	$0.172093\pi$
$97$	10.1382i	1.02938i	−0.857377	+	0.514689i	$0.827907\pi$
$98$	0	0
$99$	13.5087	1.35768
$100$	0	0
$101$	11.5202	1.14630	0.573149	−	0.819451i	$-0.305722\pi$
$101$	11.5202	1.14630	0.573149	+	0.819451i	$0.305722\pi$
$102$	0	0
$103$	− 6.20639i	− 0.611534i	−0.952106	−	0.305767i	$-0.901087\pi$
$103$	− 6.20639i	− 0.611534i	0.952106	−	0.305767i	$-0.0989128\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.7168i	− 1.22938i	−0.788769	−	0.614690i	$-0.789281\pi$
$107$	− 12.7168i	− 1.22938i	0.788769	−	0.614690i	$-0.210719\pi$
$108$	0	0
$109$	−0.381966	−0.0365857	−0.0182929	−	0.999833i	$-0.505823\pi$
$109$	−0.381966	−0.0365857	−0.0182929	+	0.999833i	$0.505823\pi$
$110$	0	0
$111$	−3.46656	−0.329031
$112$	0	0
$113$	8.31164i	0.781893i	0.920413	+	0.390947i	$0.127852\pi$
$113$	8.31164i	0.781893i	−0.920413	+	0.390947i	$0.872148\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 17.1016i	− 1.58105i
$118$	0	0
$119$	−7.94985	−0.728761
$120$	0	0
$121$	13.9443	1.26766
$122$	0	0
$123$	− 2.27259i	− 0.204913i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.3017i	− 0.914130i	−0.889433	−	0.457065i	$-0.848901\pi$
$127$	− 10.3017i	− 0.914130i	0.889433	−	0.457065i	$-0.151099\pi$
$128$	0	0
$129$	0.791482	0.0696861
$130$	0	0
$131$	−3.08672	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$131$	−3.08672	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$132$	0	0
$133$	4.03526i	0.349901i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.1053i	− 1.37596i	−0.725728	−	0.687982i	$-0.758497\pi$
$137$	− 16.1053i	− 1.37596i	0.725728	−	0.687982i	$-0.241503\pi$
$138$	0	0
$139$	15.5477	1.31874	0.659370	−	0.751819i	$-0.270823\pi$
$139$	15.5477	1.31874	0.659370	+	0.751819i	$0.270823\pi$
$140$	0	0
$141$	−1.51327	−0.127440
$142$	0	0
$143$	− 31.5787i	− 2.64074i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.37239i	− 0.113193i
$148$	0	0
$149$	15.2636	1.25044	0.625222	−	0.780447i	$-0.285008\pi$
$149$	15.2636	1.25044	0.625222	+	0.780447i	$0.285008\pi$
$150$	0	0
$151$	16.1313	1.31275	0.656373	−	0.754436i	$-0.272090\pi$
$151$	16.1313	1.31275	0.656373	+	0.754436i	$0.272090\pi$
$152$	0	0
$153$	10.1654i	0.821827i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 0.241644i	− 0.0192853i	−0.999954	−	0.00964264i	$-0.996931\pi$
$157$	− 0.241644i	− 0.0192853i	0.999954	−	0.00964264i	$-0.00306940\pi$
$158$	0	0
$159$	−0.830183	−0.0658378
$160$	0	0
$161$	−19.7803	−1.55891
$162$	0	0
$163$	0.314742i	0.0246525i	0.999924	+	0.0123263i	$0.00392367\pi$
$163$	0.314742i	0.0246525i	−0.999924	+	0.0123263i	$0.996076\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.71813i	− 0.597247i	−0.954371	−	0.298623i	$-0.903473\pi$
$167$	− 7.71813i	− 0.597247i	0.954371	−	0.298623i	$-0.0965274\pi$
$168$	0	0
$169$	−26.9777	−2.07521
$170$	0	0
$171$	5.15987	0.394585
$172$	0	0
$173$	− 13.7472i	− 1.04518i	−0.852584	−	0.522590i	$-0.824966\pi$
$173$	− 13.7472i	− 1.04518i	0.852584	−	0.522590i	$-0.175034\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 0.802932i	− 0.0603521i
$178$	0	0
$179$	1.29311	0.0966518	0.0483259	−	0.998832i	$-0.484611\pi$
$179$	1.29311	0.0966518	0.0483259	+	0.998832i	$0.484611\pi$
$180$	0	0
$181$	−18.4589	−1.37204	−0.686018	−	0.727585i	$-0.740643\pi$
$181$	−18.4589	−1.37204	−0.686018	+	0.727585i	$0.740643\pi$
$182$	0	0
$183$	− 4.77696i	− 0.353123i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.7708i	1.37266i
$188$	0	0
$189$	6.55673	0.476932
$190$	0	0
$191$	−5.90770	−0.427466	−0.213733	−	0.976892i	$-0.568562\pi$
$191$	−5.90770	−0.427466	−0.213733	+	0.976892i	$0.568562\pi$
$192$	0	0
$193$	20.8520i	1.50096i	0.660894	+	0.750479i	$0.270177\pi$
$193$	20.8520i	1.50096i	−0.660894	+	0.750479i	$0.729823\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4735i	1.38743i	0.720252	+	0.693713i	$0.244026\pi$
$197$	19.4735i	1.38743i	−0.720252	+	0.693713i	$0.755974\pi$
$198$	0	0
$199$	−23.1989	−1.64452	−0.822262	−	0.569109i	$-0.807288\pi$
$199$	−23.1989	−1.64452	−0.822262	+	0.569109i	$0.807288\pi$
$200$	0	0
$201$	6.56444	0.463020
$202$	0	0
$203$	− 15.2870i	− 1.07294i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.2930i	1.75799i
$208$	0	0
$209$	9.52786	0.659056
$210$	0	0
$211$	−23.7323	−1.63380	−0.816900	−	0.576780i	$-0.804309\pi$
$211$	−23.7323	−1.63380	−0.816900	+	0.576780i	$0.804309\pi$
$212$	0	0
$213$	− 6.72673i	− 0.460908i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.10345i	− 0.142792i
$218$	0	0
$219$	−4.94112	−0.333890
$220$	0	0
$221$	23.7633	1.59849
$222$	0	0
$223$	10.6314i	0.711931i	0.934499	+	0.355966i	$0.115848\pi$
$223$	10.6314i	0.711931i	−0.934499	+	0.355966i	$0.884152\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 25.0243i	− 1.66092i	−0.557079	−	0.830459i	$-0.688078\pi$
$227$	− 25.0243i	− 1.66092i	0.557079	−	0.830459i	$-0.311922\pi$
$228$	0	0
$229$	−11.8962	−0.786126	−0.393063	−	0.919511i	$-0.628585\pi$
$229$	−11.8962	−0.786126	−0.393063	+	0.919511i	$0.628585\pi$
$230$	0	0
$231$	5.74031	0.377685
$232$	0	0
$233$	16.0347i	1.05047i	0.850957	+	0.525235i	$0.176023\pi$
$233$	16.0347i	1.05047i	−0.850957	+	0.525235i	$0.823977\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.84313i	0.314595i
$238$	0	0
$239$	−11.0867	−0.717141	−0.358570	−	0.933503i	$-0.616736\pi$
$239$	−11.0867	−0.717141	−0.358570	+	0.933503i	$0.616736\pi$
$240$	0	0
$241$	6.56473	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$241$	6.56473	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$242$	0	0
$243$	− 12.7931i	− 0.820675i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 12.0620i	− 0.767486i
$248$	0	0
$249$	6.62706	0.419973
$250$	0	0
$251$	−1.22918	−0.0775849	−0.0387924	−	0.999247i	$-0.512351\pi$
$251$	−1.22918	−0.0775849	−0.0387924	+	0.999247i	$0.512351\pi$
$252$	0	0
$253$	46.7044i	2.93628i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.52786i	0.344819i	0.985025	+	0.172409i	$0.0551552\pi$
$257$	5.52786i	0.344819i	−0.985025	+	0.172409i	$0.944845\pi$
$258$	0	0
$259$	13.4949	0.838534
$260$	0	0
$261$	−19.5474	−1.20995
$262$	0	0
$263$	6.91631i	0.426478i	0.977000	+	0.213239i	$0.0684012\pi$
$263$	6.91631i	0.426478i	−0.977000	+	0.213239i	$0.931599\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.67534i	0.102529i
$268$	0	0
$269$	−2.94427	−0.179515	−0.0897577	−	0.995964i	$-0.528609\pi$
$269$	−2.94427	−0.179515	−0.0897577	+	0.995964i	$0.528609\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	− 7.26706i	− 0.439823i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.81408i	0.169082i	0.996420	+	0.0845410i	$0.0269424\pi$
$277$	2.81408i	0.169082i	−0.996420	+	0.0845410i	$0.973058\pi$
$278$	0	0
$279$	−2.68968	−0.161027
$280$	0	0
$281$	−2.59393	−0.154741	−0.0773705	−	0.997002i	$-0.524652\pi$
$281$	−2.59393	−0.154741	−0.0773705	+	0.997002i	$0.524652\pi$
$282$	0	0
$283$	− 17.6090i	− 1.04675i	−0.852103	−	0.523374i	$-0.824673\pi$
$283$	− 17.6090i	− 1.04675i	0.852103	−	0.523374i	$-0.175327\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.84695i	0.522219i
$288$	0	0
$289$	2.87476	0.169103
$290$	0	0
$291$	5.50871	0.322926
$292$	0	0
$293$	17.7392i	1.03634i	0.855279	+	0.518168i	$0.173386\pi$
$293$	17.7392i	1.03634i	−0.855279	+	0.518168i	$0.826614\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 15.4815i	− 0.898325i
$298$	0	0
$299$	59.1264	3.41937
$300$	0	0
$301$	−3.08115	−0.177594
$302$	0	0
$303$	− 6.25962i	− 0.359606i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.4770i	0.712102i	0.934466	+	0.356051i	$0.115877\pi$
$307$	12.4770i	0.712102i	−0.934466	+	0.356051i	$0.884123\pi$
$308$	0	0
$309$	−3.37232	−0.191844
$310$	0	0
$311$	−20.2262	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$311$	−20.2262	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$312$	0	0
$313$	− 12.6456i	− 0.714771i	−0.933957	−	0.357385i	$-0.883668\pi$
$313$	− 12.6456i	− 0.714771i	0.933957	−	0.357385i	$-0.116332\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.67852i	0.375103i	0.982255	+	0.187552i	$0.0600552\pi$
$317$	6.67852i	0.375103i	−0.982255	+	0.187552i	$0.939945\pi$
$318$	0	0
$319$	−36.0949	−2.02093
$320$	0	0
$321$	−6.90983	−0.385669
$322$	0	0
$323$	7.16982i	0.398939i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.207546i	0.0114773i
$328$	0	0
$329$	5.89097	0.324780
$330$	0	0
$331$	−27.2628	−1.49850	−0.749250	−	0.662288i	$-0.769586\pi$
$331$	−27.2628	−1.49850	−0.749250	+	0.662288i	$0.769586\pi$
$332$	0	0
$333$	− 17.2559i	− 0.945618i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 29.3835i	− 1.60062i	−0.599587	−	0.800310i	$-0.704668\pi$
$337$	− 29.3835i	− 1.60062i	0.599587	−	0.800310i	$-0.295332\pi$
$338$	0	0
$339$	4.51623	0.245288
$340$	0	0
$341$	−4.96658	−0.268955
$342$	0	0
$343$	20.1493i	1.08796i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.5007i	− 0.832119i	−0.909337	−	0.416059i	$-0.863411\pi$
$347$	− 15.5007i	− 0.832119i	0.909337	−	0.416059i	$-0.136589\pi$
$348$	0	0
$349$	−5.98540	−0.320391	−0.160196	−	0.987085i	$-0.551213\pi$
$349$	−5.98540	−0.320391	−0.160196	+	0.987085i	$0.551213\pi$
$350$	0	0
$351$	−19.5991	−1.04612
$352$	0	0
$353$	− 22.8841i	− 1.21800i	−0.793171	−	0.608998i	$-0.791572\pi$
$353$	− 22.8841i	− 1.21800i	0.793171	−	0.608998i	$-0.208428\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.31964i	0.228620i
$358$	0	0
$359$	11.3525	0.599161	0.299580	−	0.954071i	$-0.403153\pi$
$359$	11.3525	0.599161	0.299580	+	0.954071i	$0.403153\pi$
$360$	0	0
$361$	−15.3607	−0.808457
$362$	0	0
$363$	− 7.57679i	− 0.397678i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.8123i	0.929794i	0.885365	+	0.464897i	$0.153909\pi$
$367$	17.8123i	0.929794i	−0.885365	+	0.464897i	$0.846091\pi$
$368$	0	0
$369$	11.3126	0.588909
$370$	0	0
$371$	3.23181	0.167787
$372$	0	0
$373$	− 13.0366i	− 0.675008i	−0.941324	−	0.337504i	$-0.890417\pi$
$373$	− 13.0366i	− 0.675008i	0.941324	−	0.337504i	$-0.109583\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.6951i	2.35342i
$378$	0	0
$379$	−31.2800	−1.60675	−0.803373	−	0.595476i	$-0.796964\pi$
$379$	−31.2800	−1.60675	−0.803373	+	0.595476i	$0.796964\pi$
$380$	0	0
$381$	−5.59756	−0.286772
$382$	0	0
$383$	13.3291i	0.681084i	0.940229	+	0.340542i	$0.110611\pi$
$383$	13.3291i	0.681084i	−0.940229	+	0.340542i	$0.889389\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.93985i	0.200274i
$388$	0	0
$389$	−2.66950	−0.135349	−0.0676746	−	0.997707i	$-0.521558\pi$
$389$	−2.66950	−0.135349	−0.0676746	+	0.997707i	$0.521558\pi$
$390$	0	0
$391$	−35.1456	−1.77739
$392$	0	0
$393$	1.67721i	0.0846040i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 24.9572i	− 1.25257i	−0.779595	−	0.626284i	$-0.784575\pi$
$397$	− 24.9572i	− 1.25257i	0.779595	−	0.626284i	$-0.215425\pi$
$398$	0	0
$399$	2.19260	0.109768
$400$	0	0
$401$	−28.3039	−1.41343	−0.706716	−	0.707498i	$-0.749824\pi$
$401$	−28.3039	−1.41343	−0.706716	+	0.707498i	$0.749824\pi$
$402$	0	0
$403$	6.28754i	0.313204i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 31.8636i	− 1.57942i
$408$	0	0
$409$	14.5647	0.720180	0.360090	−	0.932918i	$-0.382746\pi$
$409$	14.5647	0.720180	0.360090	+	0.932918i	$0.382746\pi$
$410$	0	0
$411$	−8.75098	−0.431654
$412$	0	0
$413$	3.12572i	0.153807i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.44803i	− 0.413702i
$418$	0	0
$419$	28.4610	1.39041	0.695205	−	0.718811i	$-0.255314\pi$
$419$	28.4610	1.39041	0.695205	+	0.718811i	$0.255314\pi$
$420$	0	0
$421$	−20.5443	−1.00127	−0.500633	−	0.865660i	$-0.666899\pi$
$421$	−20.5443	−1.00127	−0.500633	+	0.865660i	$0.666899\pi$
$422$	0	0
$423$	− 7.53277i	− 0.366256i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.5961i	0.899931i
$428$	0	0
$429$	−17.1587	−0.828428
$430$	0	0
$431$	−28.9753	−1.39569	−0.697845	−	0.716249i	$-0.745857\pi$
$431$	−28.9753	−1.39569	−0.697845	+	0.716249i	$0.745857\pi$
$432$	0	0
$433$	− 20.9002i	− 1.00440i	−0.864752	−	0.502199i	$-0.832524\pi$
$433$	− 20.9002i	− 1.00440i	0.864752	−	0.502199i	$-0.167476\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.8395i	0.853379i
$438$	0	0
$439$	−30.2546	−1.44397	−0.721987	−	0.691907i	$-0.756771\pi$
$439$	−30.2546	−1.44397	−0.721987	+	0.691907i	$0.756771\pi$
$440$	0	0
$441$	6.83150	0.325309
$442$	0	0
$443$	6.09346i	0.289509i	0.989468	+	0.144754i	$0.0462392\pi$
$443$	6.09346i	0.289509i	−0.989468	+	0.144754i	$0.953761\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.29367i	− 0.392277i
$448$	0	0
$449$	19.3135	0.911459	0.455730	−	0.890118i	$-0.349378\pi$
$449$	19.3135	0.911459	0.455730	+	0.890118i	$0.349378\pi$
$450$	0	0
$451$	20.8890	0.983626
$452$	0	0
$453$	− 8.76513i	− 0.411822i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.1289i	0.614143i	0.951686	+	0.307071i	$0.0993490\pi$
$457$	13.1289i	0.614143i	−0.951686	+	0.307071i	$0.900651\pi$
$458$	0	0
$459$	11.6500	0.543773
$460$	0	0
$461$	−6.28409	−0.292679	−0.146340	−	0.989234i	$-0.546749\pi$
$461$	−6.28409	−0.292679	−0.146340	+	0.989234i	$0.546749\pi$
$462$	0	0
$463$	0.705730i	0.0327981i	0.999866	+	0.0163990i	$0.00522021\pi$
$463$	0.705730i	0.0327981i	−0.999866	+	0.0163990i	$0.994780\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.1190i	0.468251i	0.972206	+	0.234126i	$0.0752227\pi$
$467$	10.1190i	0.468251i	−0.972206	+	0.234126i	$0.924777\pi$
$468$	0	0
$469$	−25.5546	−1.18000
$470$	0	0
$471$	−0.131300	−0.00604999
$472$	0	0
$473$	7.27507i	0.334508i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.13250i	− 0.189214i
$478$	0	0
$479$	−29.0533	−1.32748	−0.663739	−	0.747964i	$-0.731031\pi$
$479$	−29.0533	−1.32748	−0.663739	+	0.747964i	$0.731031\pi$
$480$	0	0
$481$	−40.3384	−1.83927
$482$	0	0
$483$	10.7479i	0.489045i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.59798i	0.344297i	0.985071	+	0.172149i	$0.0550709\pi$
$487$	7.59798i	0.344297i	−0.985071	+	0.172149i	$0.944929\pi$
$488$	0	0
$489$	0.171019	0.00773375
$490$	0	0
$491$	11.6822	0.527208	0.263604	−	0.964631i	$-0.415089\pi$
$491$	11.6822	0.527208	0.263604	+	0.964631i	$0.415089\pi$
$492$	0	0
$493$	− 27.1618i	− 1.22331i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.1864i	1.17462i
$498$	0	0
$499$	4.21754	0.188803	0.0944015	−	0.995534i	$-0.469906\pi$
$499$	4.21754	0.188803	0.0944015	+	0.995534i	$0.469906\pi$
$500$	0	0
$501$	−4.19374	−0.187362
$502$	0	0
$503$	8.12640i	0.362338i	0.983452	+	0.181169i	$0.0579881\pi$
$503$	8.12640i	0.362338i	−0.983452	+	0.181169i	$0.942012\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.6587i	0.651014i
$508$	0	0
$509$	20.8190	0.922787	0.461394	−	0.887196i	$-0.347350\pi$
$509$	20.8190	0.922787	0.461394	+	0.887196i	$0.347350\pi$
$510$	0	0
$511$	19.2352	0.850917
$512$	0	0
$513$	− 5.91339i	− 0.261083i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13.9095i	− 0.611739i
$518$	0	0
$519$	−7.46971	−0.327884
$520$	0	0
$521$	15.7390	0.689539	0.344769	−	0.938687i	$-0.387957\pi$
$521$	15.7390	0.689539	0.344769	+	0.938687i	$0.387957\pi$
$522$	0	0
$523$	39.6698i	1.73464i	0.497753	+	0.867319i	$0.334159\pi$
$523$	39.6698i	1.73464i	−0.497753	+	0.867319i	$0.665841\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.73740i	− 0.162804i
$528$	0	0
$529$	−64.4471	−2.80205
$530$	0	0
$531$	3.99685	0.173449
$532$	0	0
$533$	− 26.4449i	− 1.14546i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.702628i	− 0.0303206i
$538$	0	0
$539$	12.6146	0.543349
$540$	0	0
$541$	10.1162	0.434930	0.217465	−	0.976068i	$-0.430221\pi$
$541$	10.1162	0.434930	0.217465	+	0.976068i	$0.430221\pi$
$542$	0	0
$543$	10.0298i	0.430422i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.91575i	0.167425i	0.996490	+	0.0837127i	$0.0266778\pi$
$547$	3.91575i	0.167425i	−0.996490	+	0.0837127i	$0.973322\pi$
$548$	0	0
$549$	23.7788	1.01486
$550$	0	0
$551$	−13.7870	−0.587348
$552$	0	0
$553$	− 18.8538i	− 0.801744i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 25.9095i	− 1.09782i	−0.835881	−	0.548910i	$-0.815043\pi$
$557$	− 25.9095i	− 1.09782i	0.835881	−	0.548910i	$-0.184957\pi$
$558$	0	0
$559$	9.21002	0.389542
$560$	0	0
$561$	10.1994	0.430617
$562$	0	0
$563$	30.7156i	1.29451i	0.762275	+	0.647254i	$0.224083\pi$
$563$	30.7156i	1.29451i	−0.762275	+	0.647254i	$0.775917\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	13.6010i	0.571189i
$568$	0	0
$569$	12.3448	0.517519	0.258760	−	0.965942i	$-0.416686\pi$
$569$	12.3448	0.517519	0.258760	+	0.965942i	$0.416686\pi$
$570$	0	0
$571$	−33.9368	−1.42021	−0.710104	−	0.704096i	$-0.751352\pi$
$571$	−33.9368	−1.42021	−0.710104	+	0.704096i	$0.751352\pi$
$572$	0	0
$573$	3.21002i	0.134100i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 45.2207i	− 1.88256i	−0.337626	−	0.941280i	$-0.609624\pi$
$577$	− 45.2207i	− 1.88256i	0.337626	−	0.941280i	$-0.390376\pi$
$578$	0	0
$579$	11.3302	0.470866
$580$	0	0
$581$	−25.7984	−1.07030
$582$	0	0
$583$	− 7.63080i	− 0.316035i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.6710i	− 1.55485i	−0.628976	−	0.777424i	$-0.716526\pi$
$587$	− 37.6710i	− 1.55485i	0.628976	−	0.777424i	$-0.283474\pi$
$588$	0	0
$589$	−1.89706	−0.0781671
$590$	0	0
$591$	10.5811	0.435250
$592$	0	0
$593$	− 6.19524i	− 0.254408i	−0.991877	−	0.127204i	$-0.959400\pi$
$593$	− 6.19524i	− 0.254408i	0.991877	−	0.127204i	$-0.0406003\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.6054i	0.515904i
$598$	0	0
$599$	−4.03100	−0.164702	−0.0823510	−	0.996603i	$-0.526243\pi$
$599$	−4.03100	−0.164702	−0.0823510	+	0.996603i	$0.526243\pi$
$600$	0	0
$601$	42.2631	1.72395	0.861974	−	0.506953i	$-0.169228\pi$
$601$	42.2631	1.72395	0.861974	+	0.506953i	$0.169228\pi$
$602$	0	0
$603$	32.6766i	1.33069i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 30.9920i	− 1.25793i	−0.777436	−	0.628963i	$-0.783480\pi$
$607$	− 30.9920i	− 1.25793i	0.777436	−	0.628963i	$-0.216520\pi$
$608$	0	0
$609$	−8.30636	−0.336591
$610$	0	0
$611$	−17.6090	−0.712384
$612$	0	0
$613$	6.63443i	0.267962i	0.990984	+	0.133981i	$0.0427761\pi$
$613$	6.63443i	0.267962i	−0.990984	+	0.133981i	$0.957224\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.2862i	0.856951i	0.903553	+	0.428475i	$0.140949\pi$
$617$	21.2862i	0.856951i	−0.903553	+	0.428475i	$0.859051\pi$
$618$	0	0
$619$	17.4244	0.700346	0.350173	−	0.936685i	$-0.386123\pi$
$619$	17.4244	0.700346	0.350173	+	0.936685i	$0.386123\pi$
$620$	0	0
$621$	28.9867	1.16320
$622$	0	0
$623$	− 6.52189i	− 0.261294i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 5.17708i	− 0.206753i
$628$	0	0
$629$	23.9777	0.956053
$630$	0	0
$631$	−11.5148	−0.458396	−0.229198	−	0.973380i	$-0.573610\pi$
$631$	−11.5148	−0.458396	−0.229198	+	0.973380i	$0.573610\pi$
$632$	0	0
$633$	12.8952i	0.512539i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.9697i	− 0.632742i
$638$	0	0
$639$	33.4845	1.32462
$640$	0	0
$641$	−45.2710	−1.78810	−0.894048	−	0.447970i	$-0.852147\pi$
$641$	−45.2710	−1.78810	−0.894048	+	0.447970i	$0.852147\pi$
$642$	0	0
$643$	− 23.6449i	− 0.932464i	−0.884662	−	0.466232i	$-0.845611\pi$
$643$	− 23.6449i	− 0.932464i	0.884662	−	0.466232i	$-0.154389\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.6264i	0.968165i	0.875022	+	0.484082i	$0.160846\pi$
$647$	24.6264i	0.968165i	−0.875022	+	0.484082i	$0.839154\pi$
$648$	0	0
$649$	7.38032	0.289703
$650$	0	0
$651$	−1.14294	−0.0447952
$652$	0	0
$653$	− 7.93000i	− 0.310325i	−0.987889	−	0.155163i	$-0.950410\pi$
$653$	− 7.93000i	− 0.310325i	0.987889	−	0.155163i	$-0.0495901\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 24.5960i	− 0.959582i
$658$	0	0
$659$	−20.3575	−0.793017	−0.396508	−	0.918031i	$-0.629778\pi$
$659$	−20.3575	−0.793017	−0.396508	+	0.918031i	$0.629778\pi$
$660$	0	0
$661$	−39.5617	−1.53877	−0.769385	−	0.638785i	$-0.779437\pi$
$661$	−39.5617	−1.53877	−0.769385	+	0.638785i	$0.779437\pi$
$662$	0	0
$663$	− 12.9121i	− 0.501463i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 67.5824i	− 2.61680i
$668$	0	0
$669$	5.77670	0.223340
$670$	0	0
$671$	43.9084	1.69506
$672$	0	0
$673$	− 1.46282i	− 0.0563874i	−0.999602	−	0.0281937i	$-0.991024\pi$
$673$	− 1.46282i	− 0.0563874i	0.999602	−	0.0281937i	$-0.00897552\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 13.0148i	− 0.500199i	−0.968220	−	0.250099i	$-0.919537\pi$
$677$	− 13.0148i	− 0.500199i	0.968220	−	0.250099i	$-0.0804633\pi$
$678$	0	0
$679$	−21.4448	−0.822975
$680$	0	0
$681$	−13.5972	−0.521047
$682$	0	0
$683$	2.11337i	0.0808660i	0.999182	+	0.0404330i	$0.0128737\pi$
$683$	2.11337i	0.0808660i	−0.999182	+	0.0404330i	$0.987126\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.46397i	0.246616i
$688$	0	0
$689$	−9.66037	−0.368031
$690$	0	0
$691$	6.92443	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$691$	6.92443	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$692$	0	0
$693$	28.5742i	1.08545i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.7192i	0.595408i
$698$	0	0
$699$	8.71267	0.329543
$700$	0	0
$701$	1.65310	0.0624369	0.0312185	−	0.999513i	$-0.490061\pi$
$701$	1.65310	0.0624369	0.0312185	+	0.999513i	$0.490061\pi$
$702$	0	0
$703$	− 12.1708i	− 0.459031i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.3680i	0.916452i
$708$	0	0
$709$	1.67982	0.0630870	0.0315435	−	0.999502i	$-0.489958\pi$
$709$	1.67982	0.0630870	0.0315435	+	0.999502i	$0.489958\pi$
$710$	0	0
$711$	−24.1083	−0.904130
$712$	0	0
$713$	− 9.29917i	− 0.348257i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.02410i	0.224974i
$718$	0	0
$719$	−50.2441	−1.87379	−0.936895	−	0.349611i	$-0.886314\pi$
$719$	−50.2441	−1.87379	−0.936895	+	0.349611i	$0.886314\pi$
$720$	0	0
$721$	13.1280	0.488914
$722$	0	0
$723$	− 3.56703i	− 0.132659i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.2324i	0.676203i	0.941110	+	0.338101i	$0.109785\pi$
$727$	18.2324i	0.676203i	−0.941110	+	0.338101i	$0.890215\pi$
$728$	0	0
$729$	12.3387	0.456989
$730$	0	0
$731$	−5.47456	−0.202484
$732$	0	0
$733$	51.5137i	1.90270i	0.308112	+	0.951350i	$0.400303\pi$
$733$	51.5137i	1.90270i	−0.308112	+	0.951350i	$0.599697\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.3384i	2.22259i
$738$	0	0
$739$	−13.3261	−0.490207	−0.245103	−	0.969497i	$-0.578822\pi$
$739$	−13.3261	−0.490207	−0.245103	+	0.969497i	$0.578822\pi$
$740$	0	0
$741$	−6.55403	−0.240768
$742$	0	0
$743$	23.9448i	0.878448i	0.898378	+	0.439224i	$0.144747\pi$
$743$	23.9448i	0.878448i	−0.898378	+	0.439224i	$0.855253\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.9883i	1.20698i
$748$	0	0
$749$	26.8992	0.982875
$750$	0	0
$751$	−0.0607894	−0.00221823	−0.00110912	−	0.999999i	$-0.500353\pi$
$751$	−0.0607894	−0.00221823	−0.00110912	+	0.999999i	$0.500353\pi$
$752$	0	0
$753$	0.667887i	0.0243392i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.3402i	1.06639i	0.845993	+	0.533194i	$0.179008\pi$
$757$	29.3402i	1.06639i	−0.845993	+	0.533194i	$0.820992\pi$
$758$	0	0
$759$	25.3774	0.921142
$760$	0	0
$761$	33.1931	1.20325	0.601625	−	0.798779i	$-0.294520\pi$
$761$	33.1931	1.20325	0.601625	+	0.798779i	$0.294520\pi$
$762$	0	0
$763$	− 0.807952i	− 0.0292498i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.34326i	− 0.337366i
$768$	0	0
$769$	−7.90557	−0.285082	−0.142541	−	0.989789i	$-0.545527\pi$
$769$	−7.90557	−0.285082	−0.142541	+	0.989789i	$0.545527\pi$
$770$	0	0
$771$	3.00363	0.108173
$772$	0	0
$773$	27.7151i	0.996843i	0.866935	+	0.498421i	$0.166087\pi$
$773$	27.7151i	0.996843i	−0.866935	+	0.498421i	$0.833913\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.33263i	− 0.263057i
$778$	0	0
$779$	7.97890	0.285874
$780$	0	0
$781$	61.8302	2.21246
$782$	0	0
$783$	22.4020i	0.800583i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.1468i	0.611217i	0.952157	+	0.305609i	$0.0988599\pi$
$787$	17.1468i	0.611217i	−0.952157	+	0.305609i	$0.901140\pi$
$788$	0	0
$789$	3.75806	0.133790
$790$	0	0
$791$	−17.5812	−0.625114
$792$	0	0
$793$	− 55.5867i	− 1.97394i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.0575i	− 0.745896i	−0.927852	−	0.372948i	$-0.878347\pi$
$797$	− 21.0575i	− 0.745896i	0.927852	−	0.372948i	$-0.121653\pi$
$798$	0	0
$799$	10.4670	0.370297
$800$	0	0
$801$	−8.33952	−0.294662
$802$	0	0
$803$	− 45.4174i	− 1.60274i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.59981i	0.0563158i
$808$	0	0
$809$	−38.8466	−1.36577	−0.682887	−	0.730524i	$-0.739276\pi$
$809$	−38.8466	−1.36577	−0.682887	+	0.730524i	$0.739276\pi$
$810$	0	0
$811$	−31.2820	−1.09846	−0.549229	−	0.835672i	$-0.685079\pi$
$811$	−31.2820	−1.09846	−0.549229	+	0.835672i	$0.685079\pi$
$812$	0	0
$813$	− 8.69379i	− 0.304905i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.77883i	0.0972189i
$818$	0	0
$819$	36.1741	1.26403
$820$	0	0
$821$	8.50658	0.296882	0.148441	−	0.988921i	$-0.452575\pi$
$821$	8.50658	0.296882	0.148441	+	0.988921i	$0.452575\pi$
$822$	0	0
$823$	− 44.6746i	− 1.55726i	−0.627483	−	0.778630i	$-0.715915\pi$
$823$	− 44.6746i	− 1.55726i	0.627483	−	0.778630i	$-0.284085\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.6536i	0.926836i	0.886140	+	0.463418i	$0.153377\pi$
$827$	26.6536i	0.926836i	−0.886140	+	0.463418i	$0.846623\pi$
$828$	0	0
$829$	−4.21967	−0.146555	−0.0732776	−	0.997312i	$-0.523346\pi$
$829$	−4.21967	−0.146555	−0.0732776	+	0.997312i	$0.523346\pi$
$830$	0	0
$831$	1.52907	0.0530427
$832$	0	0
$833$	9.49261i	0.328899i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.08246i	0.106546i
$838$	0	0
$839$	17.1897	0.593453	0.296726	−	0.954963i	$-0.404105\pi$
$839$	17.1897	0.593453	0.296726	+	0.954963i	$0.404105\pi$
$840$	0	0
$841$	23.2302	0.801041
$842$	0	0
$843$	1.40944i	0.0485438i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29.4956i	1.01348i
$848$	0	0
$849$	−9.56807	−0.328375
$850$	0	0
$851$	59.6599	2.04511
$852$	0	0
$853$	− 28.2243i	− 0.966381i	−0.875515	−	0.483191i	$-0.839478\pi$
$853$	− 28.2243i	− 0.966381i	0.875515	−	0.483191i	$-0.160522\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 51.2310i	− 1.75002i	−0.484108	−	0.875008i	$-0.660856\pi$
$857$	− 51.2310i	− 1.75002i	0.484108	−	0.875008i	$-0.339144\pi$
$858$	0	0
$859$	−53.1686	−1.81409	−0.907044	−	0.421036i	$-0.861667\pi$
$859$	−53.1686	−1.81409	−0.907044	+	0.421036i	$0.861667\pi$
$860$	0	0
$861$	4.80710	0.163825
$862$	0	0
$863$	− 26.8953i	− 0.915526i	−0.889074	−	0.457763i	$-0.848651\pi$
$863$	− 26.8953i	− 0.915526i	0.889074	−	0.457763i	$-0.151349\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.56203i	− 0.0530495i
$868$	0	0
$869$	−44.5167	−1.51012
$870$	0	0
$871$	76.3866	2.58826
$872$	0	0
$873$	27.4214i	0.928072i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.7008i	0.530178i	0.964224	+	0.265089i	$0.0854014\pi$
$877$	15.7008i	0.530178i	−0.964224	+	0.265089i	$0.914599\pi$
$878$	0	0
$879$	9.63881	0.325109
$880$	0	0
$881$	2.32804	0.0784336	0.0392168	−	0.999231i	$-0.487514\pi$
$881$	2.32804	0.0784336	0.0392168	+	0.999231i	$0.487514\pi$
$882$	0	0
$883$	− 45.0863i	− 1.51727i	−0.651514	−	0.758637i	$-0.725866\pi$
$883$	− 45.0863i	− 1.51727i	0.651514	−	0.758637i	$-0.274134\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 45.9907i	− 1.54422i	−0.635491	−	0.772108i	$-0.719202\pi$
$887$	− 45.9907i	− 1.54422i	0.635491	−	0.772108i	$-0.280798\pi$
$888$	0	0
$889$	21.7907	0.730836
$890$	0	0
$891$	32.1141	1.07586
$892$	0	0
$893$	− 5.31296i	− 0.177791i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 32.1270i	− 1.07269i
$898$	0	0
$899$	7.18675	0.239691
$900$	0	0
$901$	5.74226	0.191302
$902$	0	0
$903$	1.67418i	0.0557132i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41.9462i	− 1.39280i	−0.717653	−	0.696401i	$-0.754784\pi$
$907$	− 41.9462i	− 1.39280i	0.717653	−	0.696401i	$-0.245216\pi$
$908$	0	0
$909$	31.1592	1.03349
$910$	0	0
$911$	38.6141	1.27934	0.639671	−	0.768649i	$-0.279071\pi$
$911$	38.6141	1.27934	0.639671	+	0.768649i	$0.279071\pi$
$912$	0	0
$913$	60.9140i	2.01596i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.52918i	− 0.215613i
$918$	0	0
$919$	−46.7156	−1.54100	−0.770502	−	0.637437i	$-0.779995\pi$
$919$	−46.7156	−1.54100	−0.770502	+	0.637437i	$0.779995\pi$
$920$	0	0
$921$	6.77955	0.223394
$922$	0	0
$923$	− 78.2751i	− 2.57646i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.7868i	− 0.551350i
$928$	0	0
$929$	5.49036	0.180133	0.0900665	−	0.995936i	$-0.471292\pi$
$929$	5.49036	0.180133	0.0900665	+	0.995936i	$0.471292\pi$
$930$	0	0
$931$	4.81834	0.157915
$932$	0	0
$933$	10.9902i	0.359802i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.6902i	0.577912i	0.957342	+	0.288956i	$0.0933082\pi$
$937$	17.6902i	0.577912i	−0.957342	+	0.288956i	$0.906692\pi$
$938$	0	0
$939$	−6.87113	−0.224231
$940$	0	0
$941$	41.0472	1.33810	0.669050	−	0.743217i	$-0.266701\pi$
$941$	41.0472	1.33810	0.669050	+	0.743217i	$0.266701\pi$
$942$	0	0
$943$	39.1116i	1.27365i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26.6027i	− 0.864472i	−0.901760	−	0.432236i	$-0.857725\pi$
$947$	− 26.6027i	− 0.864472i	0.901760	−	0.432236i	$-0.142275\pi$
$948$	0	0
$949$	−57.4970	−1.86643
$950$	0	0
$951$	3.62886	0.117674
$952$	0	0
$953$	22.1029i	0.715984i	0.933725	+	0.357992i	$0.116539\pi$
$953$	22.1029i	0.715984i	−0.933725	+	0.357992i	$0.883461\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.6126i	0.633986i
$958$	0	0
$959$	34.0666	1.10007
$960$	0	0
$961$	−30.0111	−0.968101
$962$	0	0
$963$	− 34.3959i	− 1.10839i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.05390i	− 0.0338912i	−0.999856	−	0.0169456i	$-0.994606\pi$
$967$	− 1.05390i	− 0.0338912i	0.999856	−	0.0169456i	$-0.00539420\pi$
$968$	0	0
$969$	3.89581	0.125151
$970$	0	0
$971$	1.47456	0.0473210	0.0236605	−	0.999720i	$-0.492468\pi$
$971$	1.47456	0.0473210	0.0236605	+	0.999720i	$0.492468\pi$
$972$	0	0
$973$	32.8872i	1.05432i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 5.53670i	− 0.177135i	−0.996070	−	0.0885674i	$-0.971771\pi$
$977$	− 5.53670i	− 0.177135i	0.996070	−	0.0885674i	$-0.0282289\pi$
$978$	0	0
$979$	−15.3992	−0.492160
$980$	0	0
$981$	−1.03313	−0.0329852
$982$	0	0
$983$	− 4.18145i	− 0.133368i	−0.997774	−	0.0666838i	$-0.978758\pi$
$983$	− 4.18145i	− 0.133368i	0.997774	−	0.0666838i	$-0.0212419\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.20093i	− 0.101887i
$988$	0	0
$989$	−13.6215	−0.433138
$990$	0	0
$991$	21.5056	0.683147	0.341573	−	0.939855i	$-0.389040\pi$
$991$	21.5056	0.683147	0.341573	+	0.939855i	$0.389040\pi$
$992$	0	0
$993$	14.8136i	0.470094i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.3446i	− 0.834341i	−0.908828	−	0.417171i	$-0.863022\pi$
$997$	− 26.3446i	− 0.834341i	0.908828	−	0.417171i	$-0.136978\pi$
$998$	0	0
$999$	−19.7759	−0.625681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2000.2.c.j.1249.4		8
4.3	odd	2		1000.2.c.d.249.5		8
5.2	odd	4		2000.2.a.r.1.2		4
5.3	odd	4		2000.2.a.m.1.3		4
5.4	even	2	inner	2000.2.c.j.1249.5		8
20.3	even	4		1000.2.a.h.1.2	yes	4
20.7	even	4		1000.2.a.e.1.3	✓	4
20.19	odd	2		1000.2.c.d.249.4		8
40.3	even	4		8000.2.a.ba.1.3		4
40.13	odd	4		8000.2.a.bq.1.2		4
40.27	even	4		8000.2.a.br.1.2		4
40.37	odd	4		8000.2.a.bb.1.3		4
60.23	odd	4		9000.2.a.ba.1.1		4
60.47	odd	4		9000.2.a.r.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.e.1.3	✓	4	20.7	even	4
1000.2.a.h.1.2	yes	4	20.3	even	4
1000.2.c.d.249.4		8	20.19	odd	2
1000.2.c.d.249.5		8	4.3	odd	2
2000.2.a.m.1.3		4	5.3	odd	4
2000.2.a.r.1.2		4	5.2	odd	4
2000.2.c.j.1249.4		8	1.1	even	1	trivial
2000.2.c.j.1249.5		8	5.4	even	2	inner
8000.2.a.ba.1.3		4	40.3	even	4
8000.2.a.bb.1.3		4	40.37	odd	4
8000.2.a.bq.1.2		4	40.13	odd	4
8000.2.a.br.1.2		4	40.27	even	4
9000.2.a.r.1.4		4	60.47	odd	4
9000.2.a.ba.1.1		4	60.23	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.543362i q^{3} +2.11525i q^{7} +2.70476 q^{9} +4.99442 q^{11} -6.32279i q^{13} +3.75836i q^{17} +1.90770 q^{19} +1.14934 q^{21} +9.35131i q^{23} -3.09975i q^{27} -7.22705 q^{29} -0.994424 q^{31} -2.71378i q^{33} -6.37984i q^{37} -3.43556 q^{39} +4.18247 q^{41} +1.45664i q^{43} -2.78501i q^{47} +2.52573 q^{49} +2.04215 q^{51} -1.52786i q^{53} -1.03657i q^{57} +1.47771 q^{59} +8.79148 q^{61} +5.72123i q^{63} +12.0811i q^{67} +5.08115 q^{69} +12.3798 q^{71} -9.09362i q^{73} +10.5644i q^{77} -8.91328 q^{79} +6.42999 q^{81} +12.1964i q^{83} +3.92690i q^{87} -3.08328 q^{89} +13.3743 q^{91} +0.540332i q^{93} +10.1382i q^{97} +13.5087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9} + 16 q^{21} - 20 q^{29} + 32 q^{31} - 48 q^{39} + 4 q^{41} - 4 q^{49} + 16 q^{59} + 12 q^{61} - 24 q^{69} + 48 q^{71} - 96 q^{79} + 32 q^{81} - 20 q^{89} + 16 q^{91} + 16 q^{99}+O(q^{100}) \) 8 * q - 12 * q^9 + 16 * q^21 - 20 * q^29 + 32 * q^31 - 48 * q^39 + 4 * q^41 - 4 * q^49 + 16 * q^59 + 12 * q^61 - 24 * q^69 + 48 * q^71 - 96 * q^79 + 32 * q^81 - 20 * q^89 + 16 * q^91 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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