LMFDB - Embedded newform 400.2.bi.a.287.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,2,Mod(47,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 0, 17])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.47"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	400.bi (of order $20$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$3.19401608085$
Analytic rank:	$0$
Dimension:	$8$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			287.1
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	400.287
Dual form			400.2.bi.a.223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.56909 - 1.59310i) q^{5} +(1.76336 + 2.42705i) q^{9} +(0.552226 - 3.48662i) q^{13} +(2.87515 - 1.46496i) q^{17} +(-0.0759100 - 4.99942i) q^{25} +(2.51249 - 0.816356i) q^{29} +(11.3973 + 1.80516i) q^{37} +(-9.69421 + 7.04325i) q^{41} +(6.63339 + 0.999068i) q^{45} +7.00000i q^{49} +(-8.83869 - 4.50354i) q^{53} +(-4.69421 - 3.41054i) q^{61} +(-4.68802 - 6.35056i) q^{65} +(-16.7710 + 2.65626i) q^{73} +(-2.78115 + 8.55951i) q^{81} +(2.17754 - 6.87904i) q^{85} +(-11.0634 + 15.2274i) q^{89} +(-2.30885 + 4.53138i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} + 10 q^{13} + 10 q^{17} - 6 q^{25} + 50 q^{37} - 16 q^{41} - 6 q^{45} - 30 q^{53} + 24 q^{61} - 50 q^{65} - 10 q^{73} + 18 q^{81} - 60 q^{85} - 50 q^{89} - 10 q^{97}+O(q^{100}) $ 8 * q - 4 * q^5 + 10 * q^13 + 10 * q^17 - 6 * q^25 + 50 * q^37 - 16 * q^41 - 6 * q^45 - 30 * q^53 + 24 * q^61 - 50 * q^65 - 10 * q^73 + 18 * q^81 - 60 * q^85 - 50 * q^89 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{9}{20}\right)$	$-1$

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			0		−0.891007	−	0.453990i	$-0.850000\pi$
$3$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$4$		0			0
$5$	1.56909	−	1.59310i	0.701719	−	0.712454i
$5$	1.56909	−	1.59310i	0.701719	−	0.712454i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$	1.76336	+	2.42705i	0.587785	+	0.809017i
$10$		0			0
$11$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$11$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$12$		0			0
$13$	0.552226	−	3.48662i	0.153160	−	0.967013i	−0.784669	−	0.619915i	$-0.787167\pi$
$13$	0.552226	−	3.48662i	0.153160	−	0.967013i	0.937829	−	0.347098i	$-0.112833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.87515	−	1.46496i	0.697325	−	0.355305i	−0.0691254	−	0.997608i	$-0.522021\pi$
$17$	2.87515	−	1.46496i	0.697325	−	0.355305i	0.766451	+	0.642303i	$0.222021\pi$
$18$		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		0.987688	−	0.156434i	$-0.0500000\pi$
$23$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$24$		0			0
$25$	−0.0759100	−	4.99942i	−0.0151820	−	0.999885i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.51249	−	0.816356i	0.466557	−	0.151594i	−0.0662984	−	0.997800i	$-0.521119\pi$
$29$	2.51249	−	0.816356i	0.466557	−	0.151594i	0.532855	+	0.846206i	$0.321119\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	11.3973	+	1.80516i	1.87371	+	0.296766i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	11.3973	+	1.80516i	1.87371	+	0.296766i	0.887314	+	0.461165i	$0.152568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.69421	+	7.04325i	−1.51398	+	1.09997i	−0.549609	+	0.835422i	$0.685223\pi$
$41$	−9.69421	+	7.04325i	−1.51398	+	1.09997i	−0.964372	+	0.264550i	$0.914777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	6.63339	+	0.999068i	0.988847	+	0.148932i
$46$		0			0
$47$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$47$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.83869	−	4.50354i	−1.21409	−	0.618608i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−8.83869	−	4.50354i	−1.21409	−	0.618608i	−0.939366	+	0.342916i	$0.888586\pi$
$54$		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
$61$	−4.69421	−	3.41054i	−0.601032	−	0.436675i	0.245213	−	0.969469i	$-0.421142\pi$
$61$	−4.69421	−	3.41054i	−0.601032	−	0.436675i	−0.846245	+	0.532794i	$0.821142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.68802	−	6.35056i	−0.581478	−	0.787691i
$66$		0			0
$67$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$67$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$		0			0
$73$	−16.7710	+	2.65626i	−1.96290	+	0.310892i	−0.963891	+	0.266296i	$0.914200\pi$
$73$	−16.7710	+	2.65626i	−1.96290	+	0.310892i	−0.999005	+	0.0445966i	$0.985800\pi$
$74$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$83$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$84$		0			0
$85$	2.17754	−	6.87904i	0.236188	−	0.746137i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.0634	+	15.2274i	−1.17271	+	1.61410i	−0.529999	+	0.847998i	$0.677808\pi$
$89$	−11.0634	+	15.2274i	−1.17271	+	1.61410i	−0.642716	+	0.766105i	$0.722192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.30885	+	4.53138i	−0.234429	+	0.460092i	−0.978011	−	0.208552i	$-0.933125\pi$
$97$	−2.30885	+	4.53138i	−0.234429	+	0.460092i	0.743583	+	0.668644i	$0.233125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.3737			−1.33074			−0.665368	−	0.746515i	$-0.731726\pi$
$101$	−13.3737			−1.33074			−0.665368	+	0.746515i	$0.731726\pi$
$102$		0			0
$103$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$103$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	12.2702	+	16.8884i	1.17527	+	1.61762i	0.605999	+	0.795465i	$0.292774\pi$
$109$	12.2702	+	16.8884i	1.17527	+	1.61762i	0.569269	+	0.822152i	$0.307226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.81474	−	17.7716i	0.264788	−	1.67181i	−0.393716	−	0.919232i	$-0.628811\pi$
$113$	2.81474	−	17.7716i	0.264788	−	1.67181i	0.658505	−	0.752577i	$-0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.43597	−	4.80786i	0.872355	−	0.444487i
$118$		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
$125$	−8.08367	−	7.72362i	−0.723026	−	0.690821i
$126$		0			0
$127$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$127$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$131$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.82453	−	0.447362i	−0.241316	−	0.0382207i	0.0346048	−	0.999401i	$-0.488983\pi$
$137$	−2.82453	−	0.447362i	−0.241316	−	0.0382207i	−0.275921	+	0.961180i	$0.588983\pi$
$138$		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
$145$	2.64178	−	5.28357i	0.219388	−	0.438777i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	14.6949i		−	1.20385i	−0.798552	−	0.601926i	$-0.794400\pi$
$149$		−	14.6949i		−	1.20385i	0.798552	−	0.601926i	$-0.205600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	8.62544	+	4.39488i	0.697325	+	0.355305i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.0374	+	14.0374i	1.12031	+	1.12031i	0.991695	+	0.128615i	$0.0410530\pi$
$157$	14.0374	+	14.0374i	1.12031	+	1.12031i	0.128615	+	0.991695i	$0.458947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$163$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$167$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$168$		0			0
$169$	0.512197	+	0.166423i	0.0393997	+	0.0128018i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−25.9818	+	4.11511i	−1.97536	+	0.312866i	−0.986986	+	0.160809i	$0.948590\pi$
$173$	−25.9818	+	4.11511i	−1.97536	+	0.312866i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		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
$181$	0.867287	−	2.66924i	0.0644650	−	0.198403i	−0.913636	−	0.406533i	$-0.866738\pi$
$181$	0.867287	−	2.66924i	0.0644650	−	0.198403i	0.978101	+	0.208130i	$0.0667377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.7592	−	15.3246i	1.52625	−	1.12668i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$		0			0
$193$	16.5250	−	16.5250i	1.18949	−	1.18949i	0.212287	−	0.977207i	$-0.431909\pi$
$193$	16.5250	−	16.5250i	1.18949	−	1.18949i	0.977207	−	0.212287i	$-0.0680913\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.9140	−	21.4200i	0.777592	−	1.52611i	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	10.9140	−	21.4200i	0.777592	−	1.52611i	0.848839	−	0.528651i	$-0.177302\pi$
$198$		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
$205$	−3.99051	+	26.4953i	−0.278709	+	1.85051i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		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
$221$	−3.52002	−	10.8335i	−0.236782	−	0.728741i
$222$		0			0
$223$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$223$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$224$		0			0
$225$	12.0000	−	9.00000i	0.800000	−	0.600000i
$226$		0			0
$227$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$227$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$228$		0			0
$229$	28.3108	−	9.19874i	1.87083	−	0.607870i	0.879604	−	0.475706i	$-0.157808\pi$
$229$	28.3108	−	9.19874i	1.87083	−	0.607870i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.8149	+	27.1133i	0.905047	+	1.77625i	0.524097	+	0.851658i	$0.324403\pi$
$233$	13.8149	+	27.1133i	0.905047	+	1.77625i	0.380949	+	0.924596i	$0.375597\pi$
$234$		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
$241$	19.5019	−	14.1690i	1.25623	−	0.912704i	0.257663	−	0.966235i	$-0.417048\pi$
$241$	19.5019	−	14.1690i	1.25623	−	0.912704i	0.998566	+	0.0535313i	$0.0170477\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.1517	+	10.9836i	0.712454	+	0.701719i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.5327	−	11.5327i	−0.719390	−	0.719390i	0.249090	−	0.968480i	$-0.419869\pi$
$257$	−11.5327	−	11.5327i	−0.719390	−	0.719390i	−0.968480	+	0.249090i	$0.919869\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.41175	+	4.65841i	0.396877	+	0.288348i
$262$		0			0
$263$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$263$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$264$		0			0
$265$	−21.0433	+	7.01442i	−1.29268	+	0.430893i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−25.7314	−	8.36063i	−1.56887	−	0.509757i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−25.7314	−	8.36063i	−1.56887	−	0.509757i	−0.959159	+	0.282867i	$0.908714\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.09594	+	13.2333i	0.125933	+	0.795110i	0.967112	+	0.254353i	$0.0818624\pi$
$277$	2.09594	+	13.2333i	0.125933	+	0.795110i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.116802	+	0.359479i	−0.00696781	+	0.0214447i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−0.116802	+	0.359479i	−0.00696781	+	0.0214447i	0.947512	+	0.319720i	$0.103589\pi$
$282$		0			0
$283$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$283$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.87199	+	5.32934i	−0.227764	+	0.313490i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.967335	+	0.967335i	−0.0565123	+	0.0565123i	−0.734798	−	0.678286i	$-0.762723\pi$
$293$	−0.967335	+	0.967335i	−0.0565123	+	0.0565123i	0.678286	+	0.734798i	$0.262723\pi$
$294$		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
$305$	−12.7990	+	2.12688i	−0.732866	+	0.121785i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$		0			0
$313$	5.53079	−	34.9201i	0.312619	−	1.97380i	0.116834	−	0.993152i	$-0.462726\pi$
$313$	5.53079	−	34.9201i	0.312619	−	1.97380i	0.195785	−	0.980647i	$-0.437274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.5018	+	16.0510i	−1.76932	+	0.901513i	−0.830569	+	0.556916i	$0.811985\pi$
$317$	−31.5018	+	16.0510i	−1.76932	+	0.901513i	−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
$325$	−17.4730	−	2.49614i	−0.969227	−	0.138461i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$331$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$332$		0			0
$333$	15.7163	+	30.8450i	0.861249	+	1.69030i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−34.9201	−	5.53079i	−1.90222	−	0.301282i	−0.908929	−	0.416951i	$-0.863099\pi$
$337$	−34.9201	−	5.53079i	−1.90222	−	0.301282i	−0.993288	+	0.115670i	$0.963099\pi$
$338$		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		0.453990	−	0.891007i	$-0.350000\pi$
$347$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$348$		0			0
$349$		−	35.0025i		−	1.87364i	−0.349813	−	0.936819i	$-0.613755\pi$
$349$		−	35.0025i		−	1.87364i	0.349813	−	0.936819i	$-0.386245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	31.5018	+	16.0510i	1.67667	+	0.854308i	0.992112	+	0.125353i	$0.0400062\pi$
$353$	31.5018	+	16.0510i	1.67667	+	0.854308i	0.684561	+	0.728955i	$0.259994\pi$
$354$		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
$365$	−22.0835	+	30.8857i	−1.15590	+	1.61663i
$366$		0			0
$367$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$367$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$368$		0			0
$369$	−34.1887	−	11.1086i	−1.77979	−	0.578289i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−34.9201	+	5.53079i	−1.80809	+	0.286374i	−0.967048	−	0.254593i	$-0.918058\pi$
$373$	−34.9201	+	5.53079i	−1.80809	+	0.286374i	−0.841044	+	0.540967i	$0.818058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.45886	−	9.21089i	−0.0751352	−	0.474385i
$378$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$383$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.00473	+	6.88842i	−0.253750	+	0.349257i	−0.916820	−	0.399300i	$-0.869253\pi$
$389$	−5.00473	+	6.88842i	−0.253750	+	0.349257i	0.663070	+	0.748557i	$0.269253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.0510	+	31.5018i	−0.805576	+	1.58103i	0.00828030	+	0.999966i	$0.497364\pi$
$397$	−16.0510	+	31.5018i	−0.805576	+	1.58103i	−0.813856	+	0.581066i	$0.802636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	38.6603			1.93060			0.965301	−	0.261138i	$-0.0840977\pi$
$401$	38.6603			1.93060			0.965301	+	0.261138i	$0.0840977\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	9.27224	+	17.8613i	0.460741	+	0.887535i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	10.9665	+	15.0941i	0.542258	+	0.746354i	0.988936	−	0.148340i	$-0.0473931\pi$
$409$	10.9665	+	15.0941i	0.542258	+	0.746354i	−0.446678	+	0.894695i	$0.647393\pi$
$410$		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
$421$	−12.4491	−	38.3143i	−0.606730	−	1.86732i	−0.484427	−	0.874832i	$-0.660972\pi$
$421$	−12.4491	−	38.3143i	−0.606730	−	1.86732i	−0.122304	−	0.992493i	$-0.539028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.54221	−	14.2629i	−0.365851	−	0.691851i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$431$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$432$		0			0
$433$	16.7159	+	32.8067i	0.803313	+	1.57659i	0.816968	+	0.576683i	$0.195653\pi$
$433$	16.7159	+	32.8067i	0.803313	+	1.57659i	−0.0136552	+	0.999907i	$0.504347\pi$
$434$		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		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	6.89932	+	41.5182i	0.327059	+	1.96815i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	34.8376i		−	1.64409i	−0.569422	−	0.822045i	$-0.692833\pi$
$449$		−	34.8376i		−	1.64409i	0.569422	−	0.822045i	$-0.307167\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.0000	−	25.0000i	−1.16945	−	1.16945i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−25.0000	−	25.0000i	−1.16945	−	1.16945i	−0.187112	−	0.982339i	$-0.559913\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.3608	−	11.1602i	−0.715422	−	0.519784i	0.169496	−	0.985531i	$-0.445786\pi$
$461$	−15.3608	−	11.1602i	−0.715422	−	0.519784i	−0.884918	+	0.465746i	$0.845786\pi$
$462$		0			0
$463$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$463$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$467$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$468$		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
$477$	−4.65544	−	29.3933i	−0.213158	−	1.34583i
$478$		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
$481$	12.5878	−	38.7412i	0.573954	−	1.76645i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.59612	+	10.7884i	0.163292	+	0.489875i
$486$		0			0
$487$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$487$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$	6.02784	−	6.02784i	0.271480	−	0.271480i
$494$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$503$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$504$		0			0
$505$	−20.9846	+	21.3057i	−0.933803	+	0.948089i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.4683	−	24.0431i	−0.774270	−	1.06569i	−0.995891	−	0.0905584i	$-0.971135\pi$
$509$	−17.4683	−	24.0431i	−0.774270	−	1.06569i	0.221621	−	0.975133i	$-0.428865\pi$
$510$		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
$521$	13.8565	+	42.6460i	0.607065	+	1.86835i	0.481919	+	0.876216i	$0.339940\pi$
$521$	13.8565	+	42.6460i	0.607065	+	1.86835i	0.125146	+	0.992138i	$0.460060\pi$
$522$		0			0
$523$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$523$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.8743	−	7.10739i	0.951057	−	0.309017i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.2037	+	37.6895i	0.831806	+	1.63251i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	36.9999	−	26.8820i	1.59075	−	1.15575i	0.687890	−	0.725815i	$-0.258537\pi$
$541$	36.9999	−	26.8820i	1.59075	−	1.15575i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	46.1579	+	6.95192i	1.97719	+	0.297788i
$546$		0			0
$547$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$547$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$548$		0			0
$549$		−	17.4071i		−	0.742916i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.3518	+	15.3518i	0.650478	+	0.650478i	0.953108	−	0.302630i	$-0.0978647\pi$
$557$	15.3518	+	15.3518i	0.650478	+	0.650478i	−0.302630	+	0.953108i	$0.597865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$563$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$564$		0			0
$565$	−23.8952	−	32.3694i	−1.00528	−	1.36179i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.35574	−	0.765425i	−0.0987576	−	0.0320883i	0.259221	−	0.965818i	$-0.416534\pi$
$569$	−2.35574	−	0.765425i	−0.0987576	−	0.0320883i	−0.357979	+	0.933730i	$0.616534\pi$
$570$		0			0
$571$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$571$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	5.53079	+	34.9201i	0.230250	+	1.45374i	0.783846	+	0.620956i	$0.213255\pi$
$577$	5.53079	+	34.9201i	0.230250	+	1.45374i	−0.553596	+	0.832785i	$0.686745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	7.14649	−	22.5764i	0.295471	−	0.933418i
$586$		0			0
$587$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$587$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	30.3566	−	30.3566i	1.24660	−	1.24660i	0.289383	−	0.957214i	$-0.406550\pi$
$593$	30.3566	−	30.3566i	1.24660	−	1.24660i	0.957214	−	0.289383i	$-0.0934500\pi$
$594$		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
$601$	24.3434			0.992987			0.496494	−	0.868040i	$-0.334621\pi$
$601$	24.3434			0.992987			0.496494	+	0.868040i	$0.334621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	+	11.0000i	0.894427	+	0.447214i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	6.13427	−	38.7303i	0.247761	−	1.56430i	−0.479252	−	0.877677i	$-0.659092\pi$
$613$	6.13427	−	38.7303i	0.247761	−	1.56430i	0.727013	−	0.686624i	$-0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.0332	−	5.11219i	0.403923	−	0.205809i	−0.240213	−	0.970720i	$-0.577217\pi$
$617$	10.0332	−	5.11219i	0.403923	−	0.205809i	0.644136	+	0.764911i	$0.277217\pi$
$618$		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
$625$	−24.9885	+	0.759012i	−0.999539	+	0.0303605i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	35.4135	−	11.5065i	1.41203	−	0.458795i
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	24.4063	+	3.86558i	0.967013	+	0.153160i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.47214	+	4.70228i	−0.255634	+	0.185729i	−0.708220	−	0.705992i	$-0.750502\pi$
$641$	−6.47214	+	4.70228i	−0.255634	+	0.185729i	0.452586	+	0.891721i	$0.350502\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$647$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	42.3456	+	21.5762i	1.65711	+	0.844341i	0.995528	+	0.0944693i	$0.0301154\pi$
$653$	42.3456	+	21.5762i	1.65711	+	0.844341i	0.661584	+	0.749871i	$0.269885\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−36.0201	−	36.0201i	−1.40528	−	1.40528i
$658$		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
$661$	9.70820	+	7.05342i	0.377605	+	0.274346i	0.760358	−	0.649505i	$-0.225024\pi$
$661$	9.70820	+	7.05342i	0.377605	+	0.274346i	−0.382752	+	0.923851i	$0.625024\pi$
$662$		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
$673$	−30.5201	+	4.83390i	−1.17646	+	0.186333i	−0.713896	−	0.700252i	$-0.753071\pi$
$673$	−30.5201	+	4.83390i	−1.17646	+	0.186333i	−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.53079	−	34.9201i	−0.212566	−	1.34209i	−0.831010	−	0.556258i	$-0.812237\pi$
$677$	−5.53079	−	34.9201i	−0.212566	−	1.34209i	0.618444	−	0.785829i	$-0.287763\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$683$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$684$		0			0
$685$	−5.14463	+	3.79780i	−0.196566	+	0.145106i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−20.5831	+	28.3301i	−0.784152	+	1.07929i
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−17.5542	+	34.4520i	−0.664912	+	1.30496i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.55832			−0.247704			−0.123852	−	0.992301i	$-0.539525\pi$
$701$	−6.55832			−0.247704			−0.123852	+	0.992301i	$0.539525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	24.7625	+	34.0826i	0.929974	+	1.28000i	0.959870	+	0.280447i	$0.0904826\pi$
$709$	24.7625	+	34.0826i	0.929974	+	1.28000i	−0.0298952	+	0.999553i	$0.509517\pi$
$710$		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
$725$	−4.27203	−	12.4990i	−0.158659	−	0.464202i
$726$		0			0
$727$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$727$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$728$		0			0
$729$	−25.6785	+	8.34346i	−0.951057	+	0.309017i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−16.0510	−	31.5018i	−0.592857	−	1.16355i	−0.971286	−	0.237917i	$-0.923536\pi$
$733$	−16.0510	−	31.5018i	−0.592857	−	1.16355i	0.378429	−	0.925630i	$-0.376464\pi$
$734$		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		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−23.4104	−	23.0576i	−0.857690	−	0.844766i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	26.9836	+	26.9836i	0.980734	+	0.980734i	0.999818	−	0.0190839i	$-0.00607496\pi$
$757$	26.9836	+	26.9836i	0.980734	+	0.980734i	−0.0190839	+	0.999818i	$0.506075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.2768	−	15.4585i	−0.771285	−	0.560371i	0.131066	−	0.991374i	$-0.458160\pi$
$761$	−21.2768	−	15.4585i	−0.771285	−	0.560371i	−0.902351	+	0.431003i	$0.858160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	20.5356	−	6.84519i	0.742465	−	0.247488i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−22.8254	−	7.41641i	−0.823103	−	0.267443i	−0.132966	−	0.991121i	$-0.542450\pi$
$769$	−22.8254	−	7.41641i	−0.823103	−	0.267443i	−0.690138	+	0.723678i	$0.742450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−49.6509	+	7.86393i	−1.78582	+	0.282846i	−0.959777	−	0.280763i	$-0.909412\pi$
$773$	−49.6509	+	7.86393i	−1.78582	+	0.282846i	−0.826042	+	0.563609i	$0.809412\pi$
$774$		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
$785$	44.3890	−	0.336976i	1.58431	−	0.0120272i
$786$		0			0
$787$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$787$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−14.4835	+	14.4835i	−0.514325	+	0.514325i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.6168	−	38.5001i	0.694862	−	1.36374i	−0.226106	−	0.974103i	$-0.572599\pi$
$797$	19.6168	−	38.5001i	0.694862	−	1.36374i	0.920967	−	0.389640i	$-0.127401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−56.4664			−1.99514
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.0845	+	41.4078i	1.05771	+	1.45582i	0.881924	+	0.471392i	$0.156248\pi$
$809$	30.0845	+	41.4078i	1.05771	+	1.45582i	0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$811$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$812$		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
$821$	8.65248	+	26.6296i	0.301973	+	0.929379i	0.980789	+	0.195070i	$0.0624935\pi$
$821$	8.65248	+	26.6296i	0.301973	+	0.929379i	−0.678816	+	0.734309i	$0.737507\pi$
$822$		0			0
$823$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$823$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$827$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$828$		0			0
$829$	2.21997	−	0.721311i	0.0771027	−	0.0250522i	−0.270212	−	0.962801i	$-0.587094\pi$
$829$	2.21997	−	0.721311i	0.0771027	−	0.0250522i	0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.2547	+	20.1260i	0.355305	+	0.697325i
$834$		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
$841$	−17.8153	+	12.9436i	−0.614322	+	0.446331i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.06881	−	0.554846i	0.0367682	−	0.0190873i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−51.0814	−	26.0273i	−1.74899	−	0.891156i	−0.961488	−	0.274848i	$-0.911372\pi$
$853$	−51.0814	−	26.0273i	−1.74899	−	0.891156i	−0.787505	−	0.616308i	$-0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.0000	+	25.0000i	0.853984	+	0.853984i	0.990621	−	0.136637i	$-0.0436295\pi$
$857$	25.0000	+	25.0000i	0.853984	+	0.853984i	−0.136637	+	0.990621i	$0.543630\pi$
$858$		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		0.156434	−	0.987688i	$-0.450000\pi$
$863$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$864$		0			0
$865$	−34.2120	+	47.8484i	−1.16324	+	1.62690i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−15.0692	+	2.38673i	−0.510016	+	0.0807786i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.44654	+	15.4468i	0.0826137	+	0.521602i	0.993941	+	0.109919i	$0.0350591\pi$
$877$	2.44654	+	15.4468i	0.0826137	+	0.521602i	−0.911327	+	0.411683i	$0.864941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.88854	+	30.4338i	−0.333154	+	1.02534i	0.634471	+	0.772947i	$0.281218\pi$
$881$	−9.88854	+	30.4338i	−0.333154	+	1.02534i	−0.967624	+	0.252394i	$0.918782\pi$
$882$		0			0
$883$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$883$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$887$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$888$		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
$901$	−32.0100			−1.06641
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.89150	−	5.56994i	−0.0961166	−	0.185151i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$	−23.5827	−	32.4587i	−0.782187	−	1.07659i
$910$		0			0
$911$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$911$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$912$		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
$925$	8.15958	−	57.1171i	0.268285	−	1.87800i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.0327	+	4.23457i	−0.427588	+	0.138932i	−0.514902	−	0.857249i	$-0.672172\pi$
$929$	−13.0327	+	4.23457i	−0.427588	+	0.138932i	0.0873137	+	0.996181i	$0.472172\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	29.6537	+	4.69668i	0.968743	+	0.153434i	0.620703	−	0.784046i	$-0.286847\pi$
$937$	29.6537	+	4.69668i	0.968743	+	0.153434i	0.348040	+	0.937480i	$0.386847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.4509	+	20.6708i	−0.927474	+	0.673849i	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−28.4509	+	20.6708i	−0.927474	+	0.673849i	0.0178992	+	0.999840i	$0.494302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$947$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$948$		0			0
$949$			59.9409i			1.94576i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−45.6581	−	23.2639i	−1.47901	−	0.753593i	−0.486263	−	0.873813i	$-0.661640\pi$
$953$	−45.6581	−	23.2639i	−1.47901	−	0.753593i	−0.992747	+	0.120219i	$0.961640\pi$
$954$		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
$965$	−0.396691	−	52.2551i	−0.0127699	−	1.68215i
$966$		0			0
$967$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$967$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.07366	+	50.9751i	0.258299	+	1.63084i	0.686483	+	0.727145i	$0.259153\pi$
$977$	8.07366	+	50.9751i	0.258299	+	1.63084i	−0.428184	+	0.903691i	$0.640847\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−19.3524	+	59.5606i	−0.617875	+	1.90162i
$982$		0			0
$983$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$983$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$984$		0			0
$985$	−16.9990	−	50.9970i	−0.541633	−	1.62490i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$991$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	16.0510	−	31.5018i	0.508340	−	0.997673i	−0.484108	−	0.875008i	$-0.660856\pi$
$997$	16.0510	−	31.5018i	0.508340	−	0.997673i	0.992448	−	0.122665i	$-0.0391441\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bi.a.287.1	yes	8
4.3	odd	2	CM	400.2.bi.a.287.1	yes	8
25.23	odd	20	inner	400.2.bi.a.223.1	✓	8
100.23	even	20	inner	400.2.bi.a.223.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bi.a.223.1	✓	8	25.23	odd	20	inner
400.2.bi.a.223.1	✓	8	100.23	even	20	inner
400.2.bi.a.287.1	yes	8	1.1	even	1	trivial
400.2.bi.a.287.1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bi (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(1.56909 - 1.59310i) q^{5} +(1.76336 + 2.42705i) q^{9} +(0.552226 - 3.48662i) q^{13} +(2.87515 - 1.46496i) q^{17} +(-0.0759100 - 4.99942i) q^{25} +(2.51249 - 0.816356i) q^{29} +(11.3973 + 1.80516i) q^{37} +(-9.69421 + 7.04325i) q^{41} +(6.63339 + 0.999068i) q^{45} +7.00000i q^{49} +(-8.83869 - 4.50354i) q^{53} +(-4.69421 - 3.41054i) q^{61} +(-4.68802 - 6.35056i) q^{65} +(-16.7710 + 2.65626i) q^{73} +(-2.78115 + 8.55951i) q^{81} +(2.17754 - 6.87904i) q^{85} +(-11.0634 + 15.2274i) q^{89} +(-2.30885 + 4.53138i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} + 10 q^{13} + 10 q^{17} - 6 q^{25} + 50 q^{37} - 16 q^{41} - 6 q^{45} - 30 q^{53} + 24 q^{61} - 50 q^{65} - 10 q^{73} + 18 q^{81} - 60 q^{85} - 50 q^{89} - 10 q^{97}+O(q^{100}) \) 8 * q - 4 * q^5 + 10 * q^13 + 10 * q^17 - 6 * q^25 + 50 * q^37 - 16 * q^41 - 6 * q^45 - 30 * q^53 + 24 * q^61 - 50 * q^65 - 10 * q^73 + 18 * q^81 - 60 * q^85 - 50 * q^89 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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