LMFDB - Embedded newform 800.2.bg.b.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(129,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(10))

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("800.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.bg (of order $10$, degree $4$, 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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			609.2
Root			$-0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	800.609
Dual form			800.2.bg.b.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.21113 - 0.333023i) q^{5} +(-2.42705 + 1.76336i) q^{9} +O(q^{10})$	$q+(2.21113 - 0.333023i) q^{5} +(-2.42705 + 1.76336i) q^{9} +(2.62756 + 3.61653i) q^{13} +(5.03733 + 1.63673i) q^{17} +(4.77819 - 1.47271i) q^{25} +(1.77346 + 5.45814i) q^{29} +(-5.01536 - 6.90305i) q^{37} +(8.65537 - 6.28849i) q^{41} +(-4.77929 + 4.70727i) q^{45} +7.00000 q^{49} +(-11.4875 + 3.73253i) q^{53} +(6.73305 + 4.89185i) q^{61} +(7.01426 + 7.12157i) q^{65} +(-0.447932 + 0.616525i) q^{73} +(2.78115 - 8.55951i) q^{81} +(11.6833 + 1.94147i) q^{85} +(14.1535 + 10.2831i) q^{89} +(-3.90694 + 1.26944i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 8 * q - 2 * q^5 - 6 * q^9	$ 8 q - 2 q^{5} - 6 q^{9} + 6 q^{25} + 20 q^{29} + 10 q^{37} + 20 q^{41} - 6 q^{45} + 56 q^{49} - 70 q^{53} - 20 q^{61} + 6 q^{65} - 18 q^{81} + 38 q^{85} + 30 q^{89}+O(q^{100}) $ 8 * q - 2 * q^5 - 6 * q^9 + 6 * q^25 + 20 * q^29 + 10 * q^37 + 20 * q^41 - 6 * q^45 + 56 * q^49 - 70 * q^53 - 20 * q^61 + 6 * q^65 - 18 * q^81 + 38 * q^85 + 30 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{7}{10}\right)$

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			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$		0			0
$5$	2.21113	−	0.333023i	0.988847	−	0.148932i
$5$	2.21113	−	0.333023i	0.988847	−	0.148932i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−2.42705	+	1.76336i	−0.809017	+	0.587785i
$10$		0			0
$11$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$11$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$12$		0			0
$13$	2.62756	+	3.61653i	0.728754	+	1.00304i	0.999187	+	0.0403050i	$0.0128330\pi$
$13$	2.62756	+	3.61653i	0.728754	+	1.00304i	−0.270434	+	0.962739i	$0.587167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.03733	+	1.63673i	1.22173	+	0.396965i	0.847713	−	0.530456i	$-0.177979\pi$
$17$	5.03733	+	1.63673i	1.22173	+	0.396965i	0.374020	+	0.927421i	$0.377979\pi$
$18$		0			0
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$23$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$24$		0			0
$25$	4.77819	−	1.47271i	0.955638	−	0.294542i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.77346	+	5.45814i	0.329323	+	1.01355i	0.969451	+	0.245284i	$0.0788811\pi$
$29$	1.77346	+	5.45814i	0.329323	+	1.01355i	−0.640129	+	0.768268i	$0.721119\pi$
$30$		0			0
$31$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$31$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.01536	−	6.90305i	−0.824519	−	1.13485i	−0.988918	−	0.148460i	$-0.952568\pi$
$37$	−5.01536	−	6.90305i	−0.824519	−	1.13485i	0.164399	−	0.986394i	$-0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.65537	−	6.28849i	1.35174	−	0.982097i	0.352819	−	0.935692i	$-0.385223\pi$
$41$	8.65537	−	6.28849i	1.35174	−	0.982097i	0.998923	−	0.0464057i	$-0.0147767\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−4.77929	+	4.70727i	−0.712454	+	0.701719i
$46$		0			0
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.4875	+	3.73253i	−1.57794	+	0.512703i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−11.4875	+	3.73253i	−1.57794	+	0.512703i	−0.616412	+	0.787424i	$0.711414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$59$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$60$		0			0
$61$	6.73305	+	4.89185i	0.862079	+	0.626337i	0.928450	−	0.371458i	$-0.121142\pi$
$61$	6.73305	+	4.89185i	0.862079	+	0.626337i	−0.0663709	+	0.997795i	$0.521142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.01426	+	7.12157i	0.870012	+	0.883322i
$66$		0			0
$67$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$67$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$71$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$72$		0			0
$73$	−0.447932	+	0.616525i	−0.0524265	+	0.0721588i	−0.834425	−	0.551121i	$-0.814200\pi$
$73$	−0.447932	+	0.616525i	−0.0524265	+	0.0721588i	0.781999	+	0.623280i	$0.214200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$79$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$80$		0			0
$81$	2.78115	−	8.55951i	0.309017	−	0.951057i
$82$		0			0
$83$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$83$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$84$		0			0
$85$	11.6833	+	1.94147i	1.26723	+	0.210582i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.1535	+	10.2831i	1.50027	+	1.09001i	0.970273	+	0.242013i	$0.0778077\pi$
$89$	14.1535	+	10.2831i	1.50027	+	1.09001i	0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.90694	+	1.26944i	−0.396690	+	0.128892i	−0.500567	−	0.865698i	$-0.666875\pi$
$97$	−3.90694	+	1.26944i	−0.396690	+	0.128892i	0.103877	+	0.994590i	$0.466875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.1377			−1.00874			−0.504368	−	0.863489i	$-0.668274\pi$
$101$	−10.1377			−1.00874			−0.504368	+	0.863489i	$0.668274\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−13.8884	+	10.0905i	−1.33027	+	0.966497i	−0.330527	+	0.943797i	$0.607226\pi$
$109$	−13.8884	+	10.0905i	−1.33027	+	0.966497i	−0.999742	+	0.0227005i	$0.992774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.4453	−	17.1295i	−1.17076	−	1.61141i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−12.4453	−	17.1295i	−1.17076	−	1.61141i	−0.512254	−	0.858834i	$-0.671189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−12.7544	−	4.14417i	−1.17915	−	0.383129i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.39919	−	10.4616i	−0.309017	−	0.951057i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.0748	−	4.84760i	0.901114	−	0.433583i
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$131$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.7515	−	18.9273i	−1.17487	−	1.61706i	−0.615429	−	0.788192i	$-0.711017\pi$
$137$	−13.7515	−	18.9273i	−1.17487	−	1.61706i	−0.559437	−	0.828873i	$-0.688983\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	5.73903	+	11.4781i	0.476600	+	0.953201i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−23.3474			−1.91269			−0.956345	−	0.292239i	$-0.905600\pi$
$149$	−23.3474			−1.91269			−0.956345	+	0.292239i	$0.905600\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−15.1120	+	4.91019i	−1.22173	+	0.396965i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	22.6395i		−	1.80683i	−0.428770	−	0.903414i	$-0.641053\pi$
$157$		−	22.6395i		−	1.80683i	0.428770	−	0.903414i	$-0.358947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$163$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	−2.15797	+	6.64154i	−0.165997	+	0.510887i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.08067	+	9.74570i	−0.538333	+	0.740952i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−7.08067	+	9.74570i	−0.538333	+	0.740952i	0.450039	+	0.893009i	$0.351410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$179$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$180$		0			0
$181$	8.13271	−	25.0299i	0.604500	−	1.86046i	0.104306	−	0.994545i	$-0.466738\pi$
$181$	8.13271	−	25.0299i	0.604500	−	1.86046i	0.500193	−	0.865914i	$-0.333262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.3885	−	13.5933i	−0.984340	−	0.999400i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$191$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$192$		0			0
$193$			20.7312i			1.49226i	0.665798	+	0.746132i	$0.268091\pi$
$193$			20.7312i			1.49226i	−0.665798	+	0.746132i	$0.731909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.0380	+	3.26155i	−0.715179	+	0.232376i	−0.643932	−	0.765083i	$-0.722698\pi$
$197$	−10.0380	+	3.26155i	−0.715179	+	0.232376i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	17.0439	−	16.7871i	1.19040	−	1.17246i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$211$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.31662	+	22.5182i	0.492169	+	1.51474i
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$		0			0
$227$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$227$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$228$		0			0
$229$	4.04032	+	12.4348i	0.266992	+	0.821716i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.04032	+	12.4348i	0.266992	+	0.821716i	−0.724236	+	0.689552i	$0.757808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.2195	−	9.16907i	−1.84872	−	0.600686i	−0.997063	−	0.0765885i	$-0.975597\pi$
$233$	−28.2195	−	9.16907i	−1.84872	−	0.600686i	−0.851658	−	0.524097i	$-0.824403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$239$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	23.4395	−	17.0298i	1.50987	−	1.09698i	0.543635	−	0.839322i	$-0.317048\pi$
$241$	23.4395	−	17.0298i	1.50987	−	1.09698i	0.966235	−	0.257663i	$-0.0829523\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.4779	−	2.33116i	0.988847	−	0.148932i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.7907i			0.735481i	0.929928	+	0.367740i	$0.119869\pi$
$257$			11.7907i			0.735481i	−0.929928	+	0.367740i	$0.880131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−13.9289	−	10.1199i	−0.862178	−	0.626409i
$262$		0			0
$263$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$263$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$264$		0			0
$265$	−24.1574	+	12.0787i	−1.48398	+	0.741990i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.39507	+	10.4490i	−0.207001	+	0.637084i	0.792624	+	0.609711i	$0.208714\pi$
$269$	−3.39507	+	10.4490i	−0.207001	+	0.637084i	−0.999625	+	0.0273737i	$0.991286\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	19.5336	−	26.8857i	1.17366	−	1.61541i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	19.5336	−	26.8857i	1.17366	−	1.61541i	0.632905	−	0.774229i	$-0.281862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.44965	−	26.0053i	0.504064	−	1.55135i	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	8.44965	−	26.0053i	0.504064	−	1.55135i	0.802339	−	0.596869i	$-0.203589\pi$
$282$		0			0
$283$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$283$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.94255	+	6.49715i	0.526033	+	0.382185i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			16.7486i			0.978465i	0.872153	+	0.489233i	$0.162723\pi$
$293$			16.7486i			0.978465i	−0.872153	+	0.489233i	$0.837277\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.5167	+	8.57425i	0.945746	+	0.490960i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$311$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$312$		0			0
$313$	−14.1068	−	19.4164i	−0.797366	−	1.09748i	−0.993152	−	0.116834i	$-0.962726\pi$
$313$	−14.1068	−	19.4164i	−0.797366	−	1.09748i	0.195785	−	0.980647i	$-0.437274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−26.6296	−	8.65248i	−1.49567	−	0.485971i	−0.556916	−	0.830569i	$-0.688015\pi$
$317$	−26.6296	−	8.65248i	−1.49567	−	0.485971i	−0.938751	+	0.344597i	$0.888015\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	17.8811	+	13.4108i	0.991864	+	0.743898i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$331$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$332$		0			0
$333$	24.3450	+	7.91019i	1.33410	+	0.433476i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.8091	+	25.8885i	1.02460	+	1.41024i	0.908929	+	0.416951i	$0.136901\pi$
$337$	18.8091	+	25.8885i	1.02460	+	1.41024i	0.115670	+	0.993288i	$0.463099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$347$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$	29.2504			1.56574			0.782870	−	0.622185i	$-0.213755\pi$
$349$	29.2504			1.56574			0.782870	+	0.622185i	$0.213755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.2169	+	4.94427i	−0.809914	+	0.263157i	−0.684561	−	0.728955i	$-0.740006\pi$
$353$	−15.2169	+	4.94427i	−0.809914	+	0.263157i	−0.125353	+	0.992112i	$0.540006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$359$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$360$		0			0
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.785119	+	1.51239i	−0.0410950	+	0.0791621i
$366$		0			0
$367$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$367$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$368$		0			0
$369$	−9.91817	+	30.5250i	−0.516319	+	1.58907i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	21.1603	−	29.1246i	1.09564	−	1.50802i	0.254593	−	0.967048i	$-0.418058\pi$
$373$	21.1603	−	29.1246i	1.09564	−	1.50802i	0.841044	−	0.540967i	$-0.181942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.0796	+	20.7553i	−0.776641	+	1.06895i
$378$		0			0
$379$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$379$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$383$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.8884	−	17.3560i	−1.21119	−	0.879982i	−0.215852	−	0.976426i	$-0.569253\pi$
$389$	−23.8884	−	17.3560i	−1.21119	−	0.879982i	−0.995338	+	0.0964443i	$0.969253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.4127	−	3.70820i	0.572786	−	0.186109i	−0.00828030	−	0.999966i	$-0.502636\pi$
$397$	11.4127	−	3.70820i	0.572786	−	0.186109i	0.581066	+	0.813856i	$0.302636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−37.4242			−1.86888			−0.934438	−	0.356125i	$-0.884098\pi$
$401$	−37.4242			−1.86888			−0.934438	+	0.356125i	$0.884098\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	3.29898	−	19.8524i	0.163928	−	0.986472i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.9482	−	16.6728i	1.13471	−	0.824418i	0.148340	−	0.988936i	$-0.452607\pi$
$409$	22.9482	−	16.6728i	1.13471	−	0.824418i	0.986374	+	0.164518i	$0.0526069\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$419$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$420$		0			0
$421$	12.5858	+	38.7351i	0.613394	+	1.88783i	0.423015	+	0.906123i	$0.360972\pi$
$421$	12.5858	+	38.7351i	0.613394	+	1.88783i	0.190380	+	0.981711i	$0.439028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	26.4798	+	0.402062i	1.28446	+	0.0195029i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$431$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$432$		0			0
$433$	23.6999	+	7.70055i	1.13894	+	0.370065i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	23.6999	+	7.70055i	1.13894	+	0.370065i	0.321975	+	0.946748i	$0.395653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$439$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$440$		0			0
$441$	−16.9894	+	12.3435i	−0.809017	+	0.587785i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	34.7198	+	18.0239i	1.64588	+	0.854416i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.1852			0.575054			0.287527	−	0.957773i	$-0.407167\pi$
$449$	12.1852			0.575054			0.287527	+	0.957773i	$0.407167\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	34.3819	+	24.9799i	1.60132	+	1.16343i	0.884918	+	0.465746i	$0.154214\pi$
$461$	34.3819	+	24.9799i	1.60132	+	1.16343i	0.716406	+	0.697684i	$0.245786\pi$
$462$		0			0
$463$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$463$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$467$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	21.2991	−	29.3157i	0.975218	−	1.34227i
$478$		0			0
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		0			0
$481$	11.7869	−	36.2763i	0.537436	−	1.65406i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.21600	+	4.10800i	−0.373069	+	0.186535i
$486$		0			0
$487$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$487$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$491$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$492$		0			0
$493$			30.3971i			1.36902i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$	−22.4157	+	3.37607i	−0.997486	+	0.150233i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.3782	−	10.4463i	0.637301	−	0.463026i	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	14.3782	−	10.4463i	0.637301	−	0.463026i	0.858922	+	0.512107i	$0.171135\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.65489	+	29.7147i	0.422988	+	1.30182i	0.904907	+	0.425609i	$0.139940\pi$
$521$	9.65489	+	29.7147i	0.422988	+	1.30182i	−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	7.10739	+	21.8743i	0.309017	+	0.951057i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	45.4850	+	14.7790i	1.97017	+	0.640148i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.9788	+	13.0624i	−0.772969	+	0.561595i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−17.9788	+	13.0624i	−0.772969	+	0.561595i	0.129892	+	0.991528i	$0.458537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−27.3487	+	26.9366i	−1.17149	+	1.15384i
$546$		0			0
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$		0			0
$549$	−24.9675			−1.06559
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	0.316636i		−	0.0134163i	−0.999978	−	0.00670815i	$-0.997865\pi$
$557$		−	0.316636i		−	0.0134163i	0.999978	−	0.00670815i	$-0.00213529\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$		0			0
$565$	−33.2228	−	33.7310i	−1.39769	−	1.41908i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.7654	−	42.3656i	0.577077	−	1.77606i	−0.0519200	−	0.998651i	$-0.516534\pi$
$569$	13.7654	−	42.3656i	0.577077	−	1.77606i	0.628997	−	0.777408i	$-0.283466\pi$
$570$		0			0
$571$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$571$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−28.2137	+	38.8328i	−1.17455	+	1.61663i	−0.553596	+	0.832785i	$0.686745\pi$
$577$	−28.2137	+	38.8328i	−1.17455	+	1.61663i	−0.620956	+	0.783846i	$0.713255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−29.5818	−	4.91578i	−1.22306	−	0.203243i
$586$		0			0
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	39.9824i		−	1.64188i	−0.571014	−	0.820940i	$-0.693450\pi$
$593$		−	39.9824i		−	1.64188i	0.571014	−	0.820940i	$-0.306550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−48.7409			−1.98818			−0.994091	−	0.108550i	$-0.965379\pi$
$601$	−48.7409			−1.98818			−0.994091	+	0.108550i	$0.965379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.0000	−	22.0000i	−0.447214	−	0.894427i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−5.37231	−	7.39435i	−0.216985	−	0.298655i	0.686624	−	0.727013i	$-0.259092\pi$
$613$	−5.37231	−	7.39435i	−0.216985	−	0.298655i	−0.903609	+	0.428358i	$0.859092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	43.7759	+	14.2236i	1.76235	+	0.572623i	0.997440	−	0.0715132i	$-0.0227828\pi$
$617$	43.7759	+	14.2236i	1.76235	+	0.572623i	0.764911	+	0.644136i	$0.222783\pi$
$618$		0			0
$619$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$619$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	20.6622	−	14.0738i	0.826489	−	0.562952i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13.9656	−	42.9817i	−0.556845	−	1.71379i
$630$		0			0
$631$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$631$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.3929	+	25.3157i	0.728754	+	1.00304i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.4508	−	29.3893i	1.59771	−	1.16081i	0.705992	−	0.708220i	$-0.250502\pi$
$641$	40.4508	−	29.3893i	1.59771	−	1.16081i	0.891721	−	0.452586i	$-0.149498\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−48.3890	+	15.7225i	−1.89361	+	0.615270i	−0.917611	+	0.397481i	$0.869885\pi$
$653$	−48.3890	+	15.7225i	−1.89361	+	0.615270i	−0.975996	+	0.217789i	$0.930115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	2.28620i		−	0.0891932i
$658$		0			0
$659$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$659$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$660$		0			0
$661$	−40.4508	−	29.3893i	−1.57336	−	1.14311i	−0.923851	−	0.382752i	$-0.874976\pi$
$661$	−40.4508	−	29.3893i	−1.57336	−	1.14311i	−0.649505	−	0.760358i	$-0.725024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	27.3053	−	37.5825i	1.05254	−	1.44870i	0.165957	−	0.986133i	$-0.446929\pi$
$673$	27.3053	−	37.5825i	1.05254	−	1.44870i	0.886585	−	0.462566i	$-0.153071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.5648	+	42.0689i	−1.17470	+	1.61684i	−0.556258	+	0.831010i	$0.687763\pi$
$677$	−30.5648	+	42.0689i	−1.17470	+	1.61684i	−0.618444	+	0.785829i	$0.712237\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$683$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$	−36.7095	−	37.2711i	−1.40260	−	1.42405i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−43.6830	−	31.7376i	−1.66419	−	1.20910i
$690$		0			0
$691$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$691$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	53.8925	−	17.5107i	2.04132	−	0.663267i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−46.3648			−1.75117			−0.875587	−	0.483061i	$-0.839525\pi$
$701$	−46.3648			−1.75117			−0.875587	+	0.483061i	$0.839525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−41.3545	+	30.0458i	−1.55310	+	1.12839i	−0.611708	+	0.791083i	$0.709517\pi$
$709$	−41.3545	+	30.0458i	−1.55310	+	1.12839i	−0.941393	+	0.337311i	$0.890483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$719$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.5122	+	23.4683i	0.613247	+	0.871589i
$726$		0			0
$727$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$727$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$728$		0			0
$729$	8.34346	+	25.6785i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−3.80423	−	1.23607i	−0.140512	−	0.0456552i	0.237917	−	0.971286i	$-0.423536\pi$
$733$	−3.80423	−	1.23607i	−0.140512	−	0.0456552i	−0.378429	+	0.925630i	$0.623536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$739$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−51.6241	+	7.77520i	−1.89136	+	0.284861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			33.1879i			1.20623i	0.797652	+	0.603117i	$0.206075\pi$
$757$			33.1879i			1.20623i	−0.797652	+	0.603117i	$0.793925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	40.2768	+	29.2628i	1.46003	+	1.06078i	0.983354	+	0.181700i	$0.0581600\pi$
$761$	40.2768	+	29.2628i	1.46003	+	1.06078i	0.476680	+	0.879077i	$0.341840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−31.7794	+	15.8897i	−1.14899	+	0.574493i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	15.4508	−	47.5528i	0.557172	−	1.71480i	−0.132966	−	0.991121i	$-0.542450\pi$
$769$	15.4508	−	47.5528i	0.557172	−	1.71480i	0.690138	−	0.723678i	$-0.257550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	26.9985	−	37.1603i	0.971070	−	1.33656i	0.0295658	−	0.999563i	$-0.490588\pi$
$773$	26.9985	−	37.1603i	0.971070	−	1.33656i	0.941504	−	0.337001i	$-0.109412\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.53946	−	50.0588i	−0.269095	−	1.78668i
$786$		0			0
$787$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$787$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			37.2039i			1.32115i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−35.1816	+	11.4312i	−1.24620	+	0.404914i	−0.856556	−	0.516054i	$-0.827401\pi$
$797$	−35.1816	+	11.4312i	−1.24620	+	0.404914i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−52.4842			−1.85444
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	33.1747	−	24.1028i	1.16636	−	0.847410i	0.175791	−	0.984428i	$-0.443752\pi$
$809$	33.1747	−	24.1028i	1.16636	−	0.847410i	0.990569	+	0.137018i	$0.0437518\pi$
$810$		0			0
$811$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$811$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.4508	+	47.5528i	0.539238	+	1.65961i	0.734309	+	0.678816i	$0.237507\pi$
$821$	15.4508	+	47.5528i	0.539238	+	1.65961i	−0.195070	+	0.980789i	$0.562493\pi$
$822$		0			0
$823$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$823$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$827$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$828$		0			0
$829$	−11.0344	−	33.9604i	−0.383240	−	1.17949i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−11.0344	−	33.9604i	−0.383240	−	1.17949i	0.554508	−	0.832178i	$-0.312906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	35.2613	+	11.4571i	1.22173	+	0.396965i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$		0			0
$841$	−3.18466	+	2.31379i	−0.109816	+	0.0797858i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.55976	+	15.4040i	−0.0880585	+	0.529912i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	52.1875	−	16.9568i	1.78687	−	0.580588i	0.787505	−	0.616308i	$-0.211372\pi$
$853$	52.1875	−	16.9568i	1.78687	−	0.580588i	0.999362	+	0.0357200i	$0.0113725\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.00000i			0.273275i	0.990621	+	0.136637i	$0.0436295\pi$
$857$			8.00000i			0.273275i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$859$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$863$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$	−12.4107	+	23.9070i	−0.421978	+	0.812864i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	7.24386	−	9.97032i	0.245168	−	0.337444i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	34.6026	−	47.6264i	1.16845	−	1.60823i	0.495297	−	0.868723i	$-0.335059\pi$
$877$	34.6026	−	47.6264i	1.16845	−	1.60823i	0.673150	−	0.739506i	$-0.264941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.4508	+	47.5528i	−0.520552	+	1.60210i	0.252394	+	0.967624i	$0.418782\pi$
$881$	−15.4508	+	47.5528i	−0.520552	+	1.60210i	−0.772947	+	0.634471i	$0.781218\pi$
$882$		0			0
$883$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$883$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$887$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−63.9757			−2.13134
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.64696	−	58.0528i	0.320676	−	1.92974i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	24.6046	−	17.8763i	0.816085	−	0.592920i
$910$		0			0
$911$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$911$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$919$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−34.1305	−	25.5979i	−1.12221	−	0.841654i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.7654	−	57.7540i	−0.615674	−	1.89485i	−0.390877	−	0.920443i	$-0.627828\pi$
$929$	−18.7654	−	57.7540i	−0.615674	−	1.89485i	−0.224797	−	0.974406i	$-0.572172\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−35.9540	−	49.4865i	−1.17457	−	1.61665i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−35.9540	−	49.4865i	−1.17457	−	1.61665i	−0.553864	−	0.832607i	$-0.686847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−47.4721	+	34.4905i	−1.54754	+	1.12436i	−0.602172	+	0.798366i	$0.705698\pi$
$941$	−47.4721	+	34.4905i	−1.54754	+	1.12436i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$947$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$948$		0			0
$949$	−3.40665			−0.110584
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−28.5531	+	9.27747i	−0.924927	+	0.300527i	−0.732486	−	0.680782i	$-0.761640\pi$
$953$	−28.5531	+	9.27747i	−0.924927	+	0.300527i	−0.192440	+	0.981309i	$0.561640\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	25.0795	+	18.2213i	0.809017	+	0.587785i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.90396	+	45.8394i	0.222246	+	1.47562i
$966$		0			0
$967$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$967$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$971$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.2060	+	45.7041i	−1.06235	+	1.46220i	−0.184768	+	0.982782i	$0.559153\pi$
$977$	−33.2060	+	45.7041i	−1.06235	+	1.46220i	−0.877585	+	0.479421i	$0.840847\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	15.9147	−	48.9804i	0.508118	−	1.56383i
$982$		0			0
$983$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$983$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$984$		0			0
$985$	−21.1092	+	10.5546i	−0.672594	+	0.336297i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$991$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.4127	+	3.70820i	−0.361443	+	0.117440i	−0.484108	−	0.875008i	$-0.660856\pi$
$997$	−11.4127	+	3.70820i	−0.361443	+	0.117440i	0.122665	+	0.992448i	$0.460856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.bg.b.609.2	yes	8
4.3	odd	2	CM	800.2.bg.b.609.2	yes	8
25.14	even	10	inner	800.2.bg.b.289.2	✓	8
100.39	odd	10	inner	800.2.bg.b.289.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.bg.b.289.2	✓	8	25.14	even	10	inner
800.2.bg.b.289.2	✓	8	100.39	odd	10	inner
800.2.bg.b.609.2	yes	8	1.1	even	1	trivial
800.2.bg.b.609.2	yes	8	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.bg (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(2.21113 - 0.333023i) q^{5} +(-2.42705 + 1.76336i) q^{9} +O(q^{10})\)	\(q+(2.21113 - 0.333023i) q^{5} +(-2.42705 + 1.76336i) q^{9} +(2.62756 + 3.61653i) q^{13} +(5.03733 + 1.63673i) q^{17} +(4.77819 - 1.47271i) q^{25} +(1.77346 + 5.45814i) q^{29} +(-5.01536 - 6.90305i) q^{37} +(8.65537 - 6.28849i) q^{41} +(-4.77929 + 4.70727i) q^{45} +7.00000 q^{49} +(-11.4875 + 3.73253i) q^{53} +(6.73305 + 4.89185i) q^{61} +(7.01426 + 7.12157i) q^{65} +(-0.447932 + 0.616525i) q^{73} +(2.78115 - 8.55951i) q^{81} +(11.6833 + 1.94147i) q^{85} +(14.1535 + 10.2831i) q^{89} +(-3.90694 + 1.26944i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 8 * q - 2 * q^5 - 6 * q^9	\( 8 q - 2 q^{5} - 6 q^{9} + 6 q^{25} + 20 q^{29} + 10 q^{37} + 20 q^{41} - 6 q^{45} + 56 q^{49} - 70 q^{53} - 20 q^{61} + 6 q^{65} - 18 q^{81} + 38 q^{85} + 30 q^{89}+O(q^{100}) \) 8 * q - 2 * q^5 - 6 * q^9 + 6 * q^25 + 20 * q^29 + 10 * q^37 + 20 * q^41 - 6 * q^45 + 56 * q^49 - 70 * q^53 - 20 * q^61 + 6 * q^65 - 18 * q^81 + 38 * q^85 + 30 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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