LMFDB - Embedded newform 900.2.bj.a.487.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(127,900)] 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("900.127"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.bj (of order $20$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,-2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$7.18653618192$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			487.1
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	900.487
Dual form			900.2.bj.a.523.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.221232 - 1.39680i) q^{2} +(-1.90211 + 0.618034i) q^{4} +(-0.333023 - 2.21113i) q^{5} +(1.28408 + 2.52015i) q^{8} +(-3.01484 + 0.954339i) q^{10} +(1.12038 - 7.07383i) q^{13} +(3.23607 - 2.35114i) q^{16} +(-0.645000 + 0.328644i) q^{17} +(2.00000 + 4.00000i) q^{20} +(-4.77819 + 1.47271i) q^{25} -10.1286 q^{26} +(-8.66785 + 2.81636i) q^{29} +(-4.00000 - 4.00000i) q^{32} +(0.601745 + 0.828232i) q^{34} +(-12.0154 - 1.90305i) q^{37} +(5.14475 - 3.67853i) q^{40} +(5.69421 - 4.13708i) q^{41} +7.00000i q^{49} +(3.11418 + 6.34838i) q^{50} +(2.24077 + 14.1477i) q^{52} +(2.48755 + 1.26747i) q^{53} +(5.85150 + 11.4842i) q^{58} +(-10.6942 - 7.76980i) q^{61} +(-4.70228 + 6.47214i) q^{64} +(-16.0143 - 0.121571i) q^{65} +(1.02375 - 1.02375i) q^{68} +(11.3902 - 1.80403i) q^{73} +17.2041i q^{74} +(-6.27636 - 6.37238i) q^{80} +(-7.03843 - 7.03843i) q^{82} +(0.941475 + 1.31673i) q^{85} +(-4.15354 + 5.71685i) q^{89} +(4.86556 - 9.54921i) q^{97} +(9.77762 - 1.54862i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{10} + 2 q^{13} + 8 q^{16} + 6 q^{17} + 16 q^{20} - 6 q^{25} + 4 q^{26} - 32 q^{32} + 50 q^{34} - 46 q^{37} + 4 q^{40} - 16 q^{41} - 14 q^{50} + 4 q^{52} - 2 q^{53}+ \cdots + 14 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 - 6 * q^10 + 2 * q^13 + 8 * q^16 + 6 * q^17 + 16 * q^20 - 6 * q^25 + 4 * q^26 - 32 * q^32 + 50 * q^34 - 46 * q^37 + 4 * q^40 - 16 * q^41 - 14 * q^50 + 4 * q^52 - 2 * q^53 - 8 * q^58 - 24 * q^61 - 78 * q^65 - 12 * q^68 + 22 * q^73 + 16 * q^80 - 16 * q^82 - 52 * q^85 + 50 * q^89 - 26 * q^97 + 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$e\left(\frac{9}{20}\right)$

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 $

$2$	−0.221232	−	1.39680i	−0.156434	−	0.987688i
$2$	−0.221232	−	1.39680i	−0.156434	−	0.987688i
$3$		0			0
$3$		0			0
$4$	−1.90211	+	0.618034i	−0.951057	+	0.309017i
$5$	−0.333023	−	2.21113i	−0.148932	−	0.988847i
$5$	−0.333023	−	2.21113i	−0.148932	−	0.988847i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	1.28408	+	2.52015i	0.453990	+	0.891007i
$9$		0			0
$10$	−3.01484	+	0.954339i	−0.953375	+	0.301788i
$11$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$11$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$12$		0			0
$13$	1.12038	−	7.07383i	0.310739	−	1.96193i	0.0403050	−	0.999187i	$-0.487167\pi$
$13$	1.12038	−	7.07383i	0.310739	−	1.96193i	0.270434	−	0.962739i	$-0.412833\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	−	2.35114i	0.809017	−	0.587785i
$17$	−0.645000	+	0.328644i	−0.156436	+	0.0797079i	−0.530456	−	0.847713i	$-0.677979\pi$
$17$	−0.645000	+	0.328644i	−0.156436	+	0.0797079i	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	$0.100000\pi$
$20$	2.00000	+	4.00000i	0.447214	+	0.894427i
$21$		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
$25$	−4.77819	+	1.47271i	−0.955638	+	0.294542i
$26$	−10.1286			−1.98638
$27$		0			0
$28$		0			0
$29$	−8.66785	+	2.81636i	−1.60958	+	0.522984i	−0.969451	−	0.245284i	$-0.921119\pi$
$29$	−8.66785	+	2.81636i	−1.60958	+	0.522984i	−0.640129	+	0.768268i	$0.721119\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$	0.601745	+	0.828232i	0.103198	+	0.142041i
$35$		0			0
$36$		0			0
$37$	−12.0154	−	1.90305i	−1.97531	−	0.312859i	−0.988918	−	0.148460i	$-0.952568\pi$
$37$	−12.0154	−	1.90305i	−1.97531	−	0.312859i	−0.986394	−	0.164399i	$-0.947432\pi$
$38$		0			0
$39$		0			0
$40$	5.14475	−	3.67853i	0.813456	−	0.581627i
$41$	5.69421	−	4.13708i	0.889286	−	0.646104i	−0.0464057	−	0.998923i	$-0.514777\pi$
$41$	5.69421	−	4.13708i	0.889286	−	0.646104i	0.935692	+	0.352819i	$0.114777\pi$
$42$		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
$45$		0			0
$46$		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
$50$	3.11418	+	6.34838i	0.440411	+	0.897796i
$51$		0			0
$52$	2.24077	+	14.1477i	0.310739	+	1.96193i
$53$	2.48755	+	1.26747i	0.341691	+	0.174100i	0.616412	−	0.787424i	$-0.288586\pi$
$53$	2.48755	+	1.26747i	0.341691	+	0.174100i	−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	5.85150	+	11.4842i	0.768339	+	1.50795i
$59$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$59$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$60$		0			0
$61$	−10.6942	−	7.76980i	−1.36925	−	0.994821i	−0.997795	−	0.0663709i	$-0.978858\pi$
$61$	−10.6942	−	7.76980i	−1.36925	−	0.994821i	−0.371458	−	0.928450i	$-0.621142\pi$
$62$		0			0
$63$		0			0
$64$	−4.70228	+	6.47214i	−0.587785	+	0.809017i
$65$	−16.0143	−	0.121571i	−1.98632	−	0.0150791i
$66$		0			0
$67$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$67$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$68$	1.02375	−	1.02375i	0.124148	−	0.124148i
$69$		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
$73$	11.3902	−	1.80403i	1.33312	−	0.211145i	0.551121	−	0.834425i	$-0.314200\pi$
$73$	11.3902	−	1.80403i	1.33312	−	0.211145i	0.781999	+	0.623280i	$0.214200\pi$
$74$			17.2041i			1.99993i
$75$		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	$0.900000\pi$
$80$	−6.27636	−	6.37238i	−0.701719	−	0.712454i
$81$		0			0
$82$	−7.03843	−	7.03843i	−0.777264	−	0.777264i
$83$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$83$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$84$		0			0
$85$	0.941475	+	1.31673i	0.102117	+	0.142820i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.15354	+	5.71685i	−0.440274	+	0.605985i	−0.970273	−	0.242013i	$-0.922192\pi$
$89$	−4.15354	+	5.71685i	−0.440274	+	0.605985i	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
$97$	4.86556	−	9.54921i	0.494023	−	0.969575i	−0.500567	−	0.865698i	$-0.666875\pi$
$97$	4.86556	−	9.54921i	0.494023	−	0.969575i	0.994590	−	0.103877i	$-0.0331249\pi$
$98$	9.77762	−	1.54862i	0.987688	−	0.156434i
$99$		0			0
$100$	8.17848	−	5.75435i	0.817848	−	0.575435i
$101$	−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		−0.891007	−	0.453990i	$-0.850000\pi$
$103$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$104$	19.2657	−	6.25982i	1.88916	−	0.613826i
$105$		0			0
$106$	1.22008	−	3.75502i	0.118505	−	0.364720i
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−10.0905	−	13.8884i	−0.966497	−	1.33027i	−0.943797	−	0.330527i	$-0.892774\pi$
$109$	−10.0905	−	13.8884i	−0.966497	−	1.33027i	−0.0227005	−	0.999742i	$-0.507226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.12953	−	13.4453i	0.200329	−	1.26483i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	2.12953	−	13.4453i	0.200329	−	1.26483i	0.858834	−	0.512254i	$-0.171189\pi$
$114$		0			0
$115$		0			0
$116$	14.7466	−	10.7141i	1.36919	−	0.994775i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.39919	+	10.4616i	0.309017	+	0.951057i
$122$	−8.48697	+	16.6566i	−0.768374	+	1.50802i
$123$		0			0
$124$		0			0
$125$	4.84760	+	10.0748i	0.433583	+	0.901114i
$126$		0			0
$127$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$127$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$128$	10.0806	+	5.13632i	0.891007	+	0.453990i
$129$		0			0
$130$	3.37305	+	22.3956i	0.295836	+	1.96423i
$131$		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
$136$	−1.65646	−	1.20349i	−0.142041	−	0.103198i
$137$	16.9051	+	2.67751i	1.44430	+	0.228755i	0.828873	−	0.559437i	$-0.188983\pi$
$137$	16.9051	+	2.67751i	1.44430	+	0.228755i	0.615429	+	0.788192i	$0.288983\pi$
$138$		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
$145$	9.11392	+	18.2278i	0.756870	+	1.51374i
$146$	−5.03974	−	15.5107i	−0.417092	−	1.28368i
$147$		0			0
$148$	24.0307	−	3.80609i	1.97531	−	0.312859i
$149$		−	23.3474i		−	1.91269i	−0.292239	−	0.956345i	$-0.594400\pi$
$149$		−	23.3474i		−	1.91269i	0.292239	−	0.956345i	$-0.405600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.6922	+	16.6922i	1.33218	+	1.33218i	0.903414	+	0.428770i	$0.141053\pi$
$157$	16.6922	+	16.6922i	1.33218	+	1.33218i	0.428770	+	0.903414i	$0.358947\pi$
$158$		0			0
$159$		0			0
$160$	−7.51243	+	10.1766i	−0.593910	+	0.804532i
$161$		0			0
$162$		0			0
$163$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$163$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$164$	−8.27417	+	11.3884i	−0.646104	+	0.889286i
$165$		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
$169$	−36.4200	−	11.8336i	−2.80154	−	0.910275i
$170$	1.63093	−	1.60636i	0.125087	−	0.123202i
$171$		0			0
$172$		0			0
$173$	24.7457	−	3.91933i	1.88138	−	0.297981i	0.893009	−	0.450039i	$-0.148590\pi$
$173$	24.7457	−	3.91933i	1.88138	−	0.297981i	0.988372	+	0.152057i	$0.0485898\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	8.90421	+	4.53692i	0.667399	+	0.340057i
$179$		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
$185$	−0.206497	+	27.2013i	−0.0151819	+	1.99988i
$186$		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
$193$	1.11604	−	1.11604i	0.0803343	−	0.0803343i	−0.665798	−	0.746132i	$-0.731909\pi$
$193$	1.11604	−	1.11604i	0.0803343	−	0.0803343i	0.746132	+	0.665798i	$0.231909\pi$
$194$	−14.4148	−	4.68364i	−1.03492	−	0.336266i
$195$		0			0
$196$	−4.32624	−	13.3148i	−0.309017	−	0.951057i
$197$	11.7385	−	23.0380i	0.836330	−	1.64139i	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	11.7385	−	23.0380i	0.836330	−	1.64139i	0.765083	−	0.643932i	$-0.222698\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−9.84703	−	10.1507i	−0.696290	−	0.717761i
$201$		0			0
$202$	2.24277	+	14.1603i	0.157801	+	0.996317i
$203$		0			0
$204$		0			0
$205$	−11.0439	−	11.2129i	−0.771342	−	0.783142i
$206$		0			0
$207$		0			0
$208$	−13.0059	−	25.5256i	−0.901798	−	1.76988i
$209$		0			0
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$	−5.51494	−	0.873480i	−0.378767	−	0.0599909i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−17.1670	+	17.1670i	−1.16270	+	1.16270i
$219$		0			0
$220$		0			0
$221$	1.60212	+	4.93083i	0.107770	+	0.331683i
$222$		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
$225$		0			0
$226$	−19.2516			−1.28060
$227$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$227$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$228$		0			0
$229$	25.9597	−	8.43481i	1.71546	−	0.557388i	0.724236	−	0.689552i	$-0.242192\pi$
$229$	25.9597	−	8.43481i	1.71546	−	0.557388i	0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$	−18.2278	−	18.2278i	−1.19672	−	1.19672i
$233$	−7.21950	−	14.1691i	−0.472965	−	0.928247i	−0.997063	−	0.0765885i	$-0.975597\pi$
$233$	−7.21950	−	14.1691i	−0.472965	−	0.928247i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		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
$241$	9.02978	−	6.56052i	0.581659	−	0.422600i	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	9.02978	−	6.56052i	0.581659	−	0.422600i	0.839322	+	0.543635i	$0.182952\pi$
$242$	13.8608	−	7.06243i	0.891007	−	0.453990i
$243$		0			0
$244$	25.1436	+	8.16965i	1.60965	+	0.523008i
$245$	15.4779	−	2.33116i	0.988847	−	0.148932i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	13.0000	−	9.00000i	0.822192	−	0.569210i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	−9.01256	−	9.01256i	−0.562188	−	0.562188i	0.367740	−	0.929928i	$-0.380131\pi$
$257$	−9.01256	−	9.01256i	−0.562188	−	0.562188i	−0.929928	+	0.367740i	$0.880131\pi$
$258$		0			0
$259$		0			0
$260$	30.5361	−	9.66611i	1.89377	−	0.599467i
$261$		0			0
$262$		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
$265$	1.97413	−	5.92239i	0.121270	−	0.363809i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.4490	−	3.39507i	−0.637084	−	0.207001i	−0.0273737	−	0.999625i	$-0.508714\pi$
$269$	−10.4490	−	3.39507i	−0.637084	−	0.207001i	−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$	−1.31458	+	2.58000i	−0.0797079	+	0.156436i
$273$		0			0
$274$		−	24.2055i		−	1.46231i
$275$		0			0
$276$		0			0
$277$	3.46636	+	21.8857i	0.208273	+	1.31499i	0.841178	+	0.540758i	$0.181862\pi$
$277$	3.46636	+	21.8857i	0.208273	+	1.31499i	−0.632905	+	0.774229i	$0.718138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.99465	+	18.4496i	−0.357611	+	1.10061i	0.596869	+	0.802339i	$0.296411\pi$
$281$	−5.99465	+	18.4496i	−0.357611	+	1.10061i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		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
$289$	−9.68433	+	13.3293i	−0.569667	+	0.784079i
$290$	23.4444	−	16.7629i	1.37670	−	0.984353i
$291$		0			0
$292$	−20.5505	+	10.4710i	−1.20262	+	0.612768i
$293$	6.55454	−	6.55454i	0.382921	−	0.382921i	−0.489233	−	0.872153i	$-0.662723\pi$
$293$	6.55454	−	6.55454i	0.382921	−	0.382921i	0.872153	+	0.489233i	$0.162723\pi$
$294$		0			0
$295$		0			0
$296$	−10.6327	−	32.7241i	−0.618014	−	1.90205i
$297$		0			0
$298$	−32.6117	+	5.16518i	−1.88914	+	0.299211i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.6186	+	26.2338i	−0.779800	+	1.50214i
$306$		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
$313$	0.221232	−	1.39680i	0.0125048	−	0.0789519i	−0.980647	−	0.195785i	$-0.937274\pi$
$313$	0.221232	−	1.39680i	0.0125048	−	0.0789519i	0.993152	+	0.116834i	$0.0372744\pi$
$314$	19.6229	−	27.0086i	1.10738	−	1.52418i
$315$		0			0
$316$		0			0
$317$	−3.78022	+	1.92612i	−0.212318	+	0.108182i	−0.556916	−	0.830569i	$-0.688015\pi$
$317$	−3.78022	+	1.92612i	−0.212318	+	0.108182i	0.344597	+	0.938751i	$0.388015\pi$
$318$		0			0
$319$		0			0
$320$	15.8767	+	8.24199i	0.887535	+	0.460741i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	5.06430	+	35.4501i	0.280917	+	1.96642i
$326$		0			0
$327$		0			0
$328$	17.7379	+	9.03790i	0.979410	+	0.499035i
$329$		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
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.77762	+	1.54862i	0.532621	+	0.0843588i	0.416951	−	0.908929i	$-0.363099\pi$
$337$	9.77762	+	1.54862i	0.532621	+	0.0843588i	0.115670	+	0.993288i	$0.463099\pi$
$338$	−8.47191	+	53.4895i	−0.460811	+	2.90945i
$339$		0			0
$340$	−2.60458	−	1.92271i	−0.141253	−	0.104274i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−10.9491	−	33.6978i	−0.588626	−	1.81160i
$347$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$347$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$348$		0			0
$349$			23.2468i			1.24437i	0.782870	+	0.622185i	$0.213755\pi$
$349$			23.2468i			1.24437i	−0.782870	+	0.622185i	$0.786245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.3407	−	5.77836i	−0.603603	−	0.307551i	0.125353	−	0.992112i	$-0.459994\pi$
$353$	−11.3407	−	5.77836i	−0.603603	−	0.307551i	−0.728955	+	0.684561i	$0.759994\pi$
$354$		0			0
$355$		0			0
$356$	4.36729	−	13.4411i	0.231466	−	0.712378i
$357$		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
$362$	−36.7611	−	5.82238i	−1.93212	−	0.306017i
$363$		0			0
$364$		0			0
$365$	−7.78213	−	24.5844i	−0.407335	−	1.28681i
$366$		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
$369$		0			0
$370$	38.0405	−	5.72935i	1.97763	−	0.297855i
$371$		0			0
$372$		0			0
$373$	15.3648	−	2.43355i	0.795560	−	0.126004i	0.254593	−	0.967048i	$-0.418058\pi$
$373$	15.3648	−	2.43355i	0.795560	−	0.126004i	0.540967	+	0.841044i	$0.318058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10.2111	+	64.4703i	0.525898	+	3.32039i
$378$		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
$386$	−1.80579	−	1.31198i	−0.0919123	−	0.0667782i
$387$		0			0
$388$	−3.35312	+	21.1708i	−0.170229	+	1.07478i
$389$	−17.3560	+	23.8884i	−0.879982	+	1.21119i	0.0964443	+	0.995338i	$0.469253\pi$
$389$	−17.3560	+	23.8884i	−0.879982	+	1.21119i	−0.976426	+	0.215852i	$0.930747\pi$
$390$		0			0
$391$		0			0
$392$	−17.6410	+	8.98855i	−0.891007	+	0.453990i
$393$		0			0
$394$	−34.7765	−	11.2996i	−1.75201	−	0.569264i
$395$		0			0
$396$		0			0
$397$	−8.34651	+	16.3810i	−0.418900	+	0.822137i	0.581066	+	0.813856i	$0.302636\pi$
$397$	−8.34651	+	16.3810i	−0.418900	+	0.822137i	−0.999966	+	0.00828030i	$0.997364\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	+	16.0000i	−0.600000	+	0.800000i
$401$	37.4242			1.86888			0.934438	−	0.356125i	$-0.115902\pi$
$401$	37.4242			1.86888			0.934438	+	0.356125i	$0.115902\pi$
$402$		0			0
$403$		0			0
$404$	19.2830	−	6.26543i	0.959365	−	0.311717i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−16.6728	−	22.9482i	−0.824418	−	1.13471i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−16.6728	−	22.9482i	−0.824418	−	1.13471i	0.164518	−	0.986374i	$-0.447393\pi$
$410$	−13.2189	+	17.9068i	−0.652836	+	0.884356i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−32.7768	+	23.8138i	−1.60702	+	1.16757i
$417$		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
$421$	1.55093	+	4.77327i	0.0755878	+	0.232635i	0.981711	−	0.190380i	$-0.0609719\pi$
$421$	1.55093	+	4.77327i	0.0755878	+	0.232635i	−0.906123	+	0.423015i	$0.860972\pi$
$422$		0			0
$423$		0			0
$424$			7.89652i			0.383489i
$425$	2.59794	−	2.52022i	0.126018	−	0.122249i
$426$		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
$433$	−2.70055	−	5.30013i	−0.129780	−	0.254708i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	−2.70055	−	5.30013i	−0.129780	−	0.254708i	−0.946748	+	0.321975i	$0.895653\pi$
$434$		0			0
$435$		0			0
$436$	27.7768	+	20.1811i	1.33027	+	0.966497i
$437$		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
$441$		0			0
$442$	6.53295	−	3.32870i	0.310741	−	0.158330i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	14.0239	+	7.28017i	0.664798	+	0.345113i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.1852i		−	0.575054i	−0.957773	−	0.287527i	$-0.907167\pi$
$449$		−	12.1852i		−	0.575054i	0.957773	−	0.287527i	$-0.0928331\pi$
$450$		0			0
$451$		0			0
$452$	4.25906	+	26.8907i	0.200329	+	1.26483i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	−17.5249	−	34.3945i	−0.818883	−	1.60715i
$459$		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		0.156434	−	0.987688i	$-0.450000\pi$
$463$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$464$	−21.4281	+	29.4933i	−0.994775	+	1.36919i
$465$		0			0
$466$	−18.1942	+	13.2189i	−0.842830	+	0.612352i
$467$		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
$477$		0			0
$478$		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
$481$	−26.9236	+	82.8624i	−1.22761	+	3.77820i
$482$	−11.1614	−	11.1614i	−0.508389	−	0.508389i
$483$		0			0
$484$	−12.9313	−	17.7984i	−0.587785	−	0.809017i
$485$	−22.7349	−	7.57829i	−1.03234	−	0.344113i
$486$		0			0
$487$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$487$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$488$	5.84882	−	36.9280i	0.264764	−	1.67165i
$489$		0			0
$490$	−6.68037	−	21.1039i	−0.301788	−	0.953375i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$	4.66519	−	4.66519i	0.210110	−	0.210110i
$494$		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	$0.5\pi$
$500$	−15.4472	−	16.1673i	−0.690821	−	0.723026i
$501$		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
$505$	3.37607	+	22.4157i	0.150233	+	0.997486i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.3782	−	33.5537i	−1.08054	−	1.48724i	−0.858922	−	0.512107i	$-0.828865\pi$
$509$	−24.3782	−	33.5537i	−1.08054	−	1.48724i	−0.221621	−	0.975133i	$-0.571135\pi$
$510$		0			0
$511$		0			0
$512$	−22.3488	−	3.53971i	−0.987688	−	0.156434i
$513$		0			0
$514$	−10.5949	+	14.5826i	−0.467321	+	0.643212i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−20.2572	−	40.5144i	−0.888337	−	1.77667i
$521$	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		0.987688	−	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
$530$	−8.70915	−	1.44725i	−0.378301	−	0.0628644i
$531$		0			0
$532$		0			0
$533$	−22.8853	−	44.9150i	−0.991273	−	1.94548i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−2.43060	+	15.3462i	−0.104791	+	0.661623i
$539$		0			0
$540$		0			0
$541$	17.9788	−	13.0624i	0.772969	−	0.561595i	−0.129892	−	0.991528i	$-0.541463\pi$
$541$	17.9788	−	13.0624i	0.772969	−	0.561595i	0.902861	+	0.429934i	$0.141463\pi$
$542$		0			0
$543$		0			0
$544$	3.89458	+	1.26543i	0.166979	+	0.0542547i
$545$	−27.3487	+	26.9366i	−1.17149	+	1.15384i
$546$		0			0
$547$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$547$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$548$	−33.8102	+	5.35502i	−1.44430	+	0.228755i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	29.8032	−	9.68364i	1.26622	−	0.411419i
$555$		0			0
$556$		0			0
$557$	23.7586	+	23.7586i	1.00669	+	1.00669i	0.999978	+	0.00670815i	$0.00213529\pi$
$557$	23.7586	+	23.7586i	1.00669	+	1.00669i	0.00670815	+	0.999978i	$0.497865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	27.0967	+	4.29170i	1.14301	+	0.181034i
$563$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$563$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$564$		0			0
$565$	−30.4386	−	0.231072i	−1.28056	−	0.00972129i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−42.3656	−	13.7654i	−1.77606	−	0.577077i	−0.777408	−	0.628997i	$-0.783466\pi$
$569$	−42.3656	−	13.7654i	−1.77606	−	0.577077i	−0.998651	+	0.0519200i	$0.983466\pi$
$570$		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
$577$	−5.08833	−	32.1265i	−0.211830	−	1.33744i	−0.832785	−	0.553596i	$-0.813255\pi$
$577$	−5.08833	−	32.1265i	−0.211830	−	1.33744i	0.620956	−	0.783846i	$-0.286745\pi$
$578$	20.7609	+	10.5782i	0.863541	+	0.439996i
$579$		0			0
$580$	−28.6011	−	29.0387i	−1.18760	−	1.20577i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	19.1723	+	26.3884i	0.793356	+	1.09196i
$585$		0			0
$586$	−10.6055	−	7.70533i	−0.438108	−	0.318304i
$587$		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
$592$	−43.3568	+	22.0914i	−1.78196	+	0.907951i
$593$	−33.8963	+	33.8963i	−1.39195	+	1.39195i	−0.571014	+	0.820940i	$0.693450\pi$
$593$	−33.8963	+	33.8963i	−1.39195	+	1.39195i	−0.820940	+	0.571014i	$0.806550\pi$
$594$		0			0
$595$		0			0
$596$	14.4295	+	44.4093i	0.591054	+	1.81908i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−5.32225			−0.217099			−0.108550	−	0.994091i	$-0.534621\pi$
$601$	−5.32225			−0.217099			−0.108550	+	0.994091i	$0.534621\pi$
$602$		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
$610$	39.6563	+	13.2188i	1.60564	+	0.535212i
$611$		0			0
$612$		0			0
$613$	4.37231	−	27.6057i	0.176596	−	1.11498i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	4.37231	−	27.6057i	0.176596	−	1.11498i	0.903609	−	0.428358i	$-0.140908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	40.7759	−	20.7764i	1.64158	−	0.836424i	0.644136	−	0.764911i	$-0.277217\pi$
$617$	40.7759	−	20.7764i	1.64158	−	0.836424i	0.997440	−	0.0715132i	$-0.0227828\pi$
$618$		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
$626$	−2.00000			−0.0799361
$627$		0			0
$628$	−42.0668	−	21.4341i	−1.67865	−	0.855315i
$629$	8.37534	−	2.72131i	0.333946	−	0.108506i
$630$		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
$634$	3.52671	+	4.85410i	0.140064	+	0.192781i
$635$		0			0
$636$		0			0
$637$	49.5168	+	7.84269i	1.96193	+	0.310739i
$638$		0			0
$639$		0			0
$640$	8.00000	−	24.0000i	0.316228	−	0.948683i
$641$	−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
$650$	48.3964	−	14.9165i	1.89826	−	0.585074i
$651$		0			0
$652$		0			0
$653$	35.0976	+	17.8831i	1.37348	+	0.699821i	0.975996	−	0.217789i	$-0.0698846\pi$
$653$	35.0976	+	17.8831i	1.37348	+	0.699821i	0.397481	+	0.917611i	$0.369885\pi$
$654$		0			0
$655$		0			0
$656$	8.69998	−	26.7758i	0.339677	−	1.04542i
$657$		0			0
$658$		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	0.382752	−	0.923851i	$-0.374976\pi$
$661$	−9.70820	−	7.05342i	−0.377605	−	0.274346i	−0.760358	+	0.649505i	$0.774976\pi$
$662$		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
$673$	16.3053	−	2.58250i	0.628523	−	0.0995482i	0.165957	−	0.986133i	$-0.446929\pi$
$673$	16.3053	−	2.58250i	0.628523	−	0.0995482i	0.462566	+	0.886585i	$0.346929\pi$
$674$		−	14.0000i		−	0.539260i
$675$		0			0
$676$	76.5885			2.94571
$677$	−5.97326	−	37.7137i	−0.229571	−	1.44945i	−0.785829	−	0.618444i	$-0.787763\pi$
$677$	−5.97326	−	37.7137i	−0.229571	−	1.44945i	0.556258	−	0.831010i	$-0.312237\pi$
$678$		0			0
$679$		0			0
$680$	−2.10944	+	4.06345i	−0.0808932	+	0.155826i
$681$		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
$685$	0.290532	−	38.2711i	0.0111007	−	1.46226i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.7529	−	16.1764i	0.447748	−	0.616273i
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$	−44.6468	+	22.7487i	−1.69722	+	0.864776i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.31314	+	4.53979i	−0.0876164	+	0.171957i
$698$	32.4711	−	5.14292i	1.22905	−	0.194662i
$699$		0			0
$700$		0			0
$701$	−25.5794			−0.966122			−0.483061	−	0.875587i	$-0.660475\pi$
$701$	−25.5794			−0.966122			−0.483061	+	0.875587i	$0.660475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−5.56231	+	17.1190i	−0.209340	+	0.644283i
$707$		0			0
$708$		0			0
$709$	8.77854	+	12.0826i	0.329685	+	0.453773i	0.941393	−	0.337311i	$-0.109517\pi$
$709$	8.77854	+	12.0826i	0.329685	+	0.453773i	−0.611708	+	0.791083i	$0.709517\pi$
$710$		0			0
$711$		0			0
$712$	−19.7408	−	3.12663i	−0.739817	−	0.117176i
$713$		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
$722$	12.1988	−	23.9414i	0.453990	−	0.891007i
$723$		0			0
$724$			52.6360i			1.95620i
$725$	37.2690	−	26.2223i	1.38414	−	0.973873i
$726$		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
$729$		0			0
$730$	−32.6179	+	16.3089i	−1.20724	+	0.603621i
$731$		0			0
$732$		0			0
$733$	−18.6191	−	36.5421i	−0.687714	−	1.34971i	−0.925630	−	0.378429i	$-0.876464\pi$
$733$	−18.6191	−	36.5421i	−0.687714	−	1.34971i	0.237917	−	0.971286i	$-0.423536\pi$
$734$		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	$0.700000\pi$
$740$	−16.4185	−	51.8675i	−0.603557	−	1.90669i
$741$		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
$745$	−51.6241	+	7.77520i	−1.89136	+	0.284861i
$746$	−6.79837	−	20.9232i	−0.248906	−	0.766054i
$747$		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
$754$	87.7932	−	28.5257i	3.19724	−	1.03885i
$755$		0			0
$756$		0			0
$757$	−38.5403	−	38.5403i	−1.40077	−	1.40077i	−0.797652	−	0.603117i	$-0.793925\pi$
$757$	−38.5403	−	38.5403i	−1.40077	−	1.40077i	−0.603117	−	0.797652i	$-0.706075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−40.2768	−	29.2628i	−1.46003	−	1.06078i	−0.983354	−	0.181700i	$-0.941840\pi$
$761$	−40.2768	−	29.2628i	−1.46003	−	1.06078i	−0.476680	−	0.879077i	$-0.658160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	22.8254	+	7.41641i	0.823103	+	0.267443i	0.690138	−	0.723678i	$-0.257550\pi$
$769$	22.8254	+	7.41641i	0.823103	+	0.267443i	0.132966	+	0.991121i	$0.457550\pi$
$770$		0			0
$771$		0			0
$772$	−1.43308	+	2.81259i	−0.0515778	+	0.101227i
$773$	10.1916	−	1.61419i	0.366566	−	0.0580584i	0.0295658	−	0.999563i	$-0.490588\pi$
$773$	10.1916	−	1.61419i	0.366566	−	0.0580584i	0.337001	+	0.941504i	$0.390588\pi$
$774$		0			0
$775$		0			0
$776$	30.3132			1.08818
$777$		0			0
$778$	37.2071	+	18.9580i	1.33394	+	0.679676i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	16.4580	+	22.6525i	0.587785	+	0.809017i
$785$	31.3498	−	42.4675i	1.11892	−	1.51573i
$786$		0			0
$787$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$787$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$788$	−8.08959	+	51.0757i	−0.288180	+	1.81950i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−66.9438	+	66.9438i	−2.37724	+	2.37724i
$794$	24.7275	+	8.03444i	0.877545	+	0.285132i
$795$		0			0
$796$		0			0
$797$	1.81840	−	3.56880i	0.0644109	−	0.126414i	−0.856556	−	0.516054i	$-0.827401\pi$
$797$	1.81840	−	3.56880i	0.0644109	−	0.126414i	0.920967	+	0.389640i	$0.127401\pi$
$798$		0			0
$799$		0			0
$800$	25.0036	+	13.2219i	0.884011	+	0.467465i
$801$		0			0
$802$	−8.27943	−	52.2742i	−0.292357	−	1.84587i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−13.0176	−	25.5484i	−0.457957	−	0.898790i
$809$	23.1747	+	31.8972i	0.814778	+	1.12145i	0.990569	+	0.137018i	$0.0437518\pi$
$809$	23.1747	+	31.8972i	0.814778	+	1.12145i	−0.175791	+	0.984428i	$0.556248\pi$
$810$		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
$818$	−28.3655	+	28.3655i	−0.991777	+	0.991777i
$819$		0			0
$820$	27.9368	+	14.5027i	0.975594	+	0.506455i
$821$	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
$829$	−33.9604	+	11.0344i	−1.17949	+	0.383240i	−0.832178	−	0.554508i	$-0.812906\pi$
$829$	−33.9604	+	11.0344i	−1.17949	+	0.383240i	−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$	40.5144	+	40.5144i	1.40458	+	1.40458i
$833$	−2.30051	−	4.51500i	−0.0797079	−	0.156436i
$834$		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
$841$	43.7383	−	31.7778i	1.50822	−	1.09578i
$842$	6.32421	−	3.22234i	0.217947	−	0.111049i
$843$		0			0
$844$		0			0
$845$	−14.0369	+	84.4702i	−0.482884	+	2.90586i
$846$		0			0
$847$		0			0
$848$	11.0299	−	1.74696i	0.378767	−	0.0599909i
$849$		0			0
$850$	−4.09500	−	3.07125i	−0.140457	−	0.105343i
$851$		0			0
$852$		0			0
$853$	−21.9568	−	11.1875i	−0.751785	−	0.383054i	0.0357200	−	0.999362i	$-0.488628\pi$
$853$	−21.9568	−	11.1875i	−0.751785	−	0.383054i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−33.0000	−	33.0000i	−1.12726	−	1.12726i	−0.990621	−	0.136637i	$-0.956370\pi$
$857$	−33.0000	−	33.0000i	−1.12726	−	1.12726i	−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
$865$	−16.9070	−	53.4107i	−0.574856	−	1.81602i
$866$	−6.80579	+	4.94470i	−0.231270	+	0.168028i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	22.0438	−	43.2634i	0.746498	−	1.46508i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.79176	+	36.5677i	0.195574	+	1.23480i	0.868723	+	0.495297i	$0.164941\pi$
$877$	5.79176	+	36.5677i	0.195574	+	1.23480i	−0.673150	+	0.739506i	$0.735059\pi$
$878$		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	$0.350000\pi$
$884$	−6.09484	−	8.38883i	−0.204992	−	0.282147i
$885$		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
$890$	7.06642	−	21.1993i	0.236867	−	0.710601i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−17.0203	+	2.69575i	−0.567974	+	0.0899582i
$899$		0			0
$900$		0			0
$901$	−2.02102			−0.0673298
$902$		0			0
$903$		0			0
$904$	36.6187	−	11.8981i	1.21792	−	0.395726i
$905$	−58.0528	−	9.64696i	−1.92974	−	0.320676i
$906$		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
$909$		0			0
$910$		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
$914$	−19.9847	+	27.5066i	−0.661035	+	0.909837i
$915$		0			0
$916$	−44.1653	+	32.0879i	−1.45926	+	1.06022i
$917$		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
$922$	27.2856	−	53.5510i	0.898604	−	1.76361i
$923$		0			0
$924$		0			0
$925$	60.2143	−	8.60205i	1.97983	−	0.282834i
$926$		0			0
$927$		0			0
$928$	45.9368	+	23.4060i	1.50795	+	0.768339i
$929$	57.7540	−	18.7654i	1.89485	−	0.615674i	0.920443	−	0.390877i	$-0.127828\pi$
$929$	57.7540	−	18.7654i	1.89485	−	0.615674i	0.974406	−	0.224797i	$-0.0721717\pi$
$930$		0			0
$931$		0			0
$932$	22.4893	+	22.4893i	0.736661	+	0.736661i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	44.4865	+	7.04597i	1.45331	+	0.230182i	0.832607	−	0.553864i	$-0.186847\pi$
$937$	44.4865	+	7.04597i	1.45331	+	0.230182i	0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	47.4721	−	34.4905i	1.54754	−	1.12436i	0.602172	−	0.798366i	$-0.294302\pi$
$941$	47.4721	−	34.4905i	1.54754	−	1.12436i	0.945373	−	0.325991i	$-0.105698\pi$
$942$		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
$949$		−	82.5934i		−	2.68109i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−52.9060	−	26.9570i	−1.71379	−	0.873222i	−0.981309	−	0.192440i	$-0.938360\pi$
$953$	−52.9060	−	26.9570i	−1.71379	−	0.873222i	−0.732486	−	0.680782i	$-0.761640\pi$
$954$		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
$962$	121.699	+	19.2752i	3.92372	+	0.621457i
$963$		0			0
$964$	−13.1210	+	18.0596i	−0.422600	+	0.581659i
$965$	−2.83938	−	2.09604i	−0.0914028	−	0.0674740i
$966$		0			0
$967$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$967$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$968$	−22.0000	+	22.0000i	−0.707107	+	0.707107i
$969$		0			0
$970$	−5.55570	+	33.4327i	−0.178383	+	1.07346i
$971$		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
$976$	−52.8751			−1.69249
$977$	−9.20997	−	58.1495i	−0.294653	−	1.86037i	−0.479421	−	0.877585i	$-0.659153\pi$
$977$	−9.20997	−	58.1495i	−0.294653	−	1.86037i	0.184768	−	0.982782i	$-0.440847\pi$
$978$		0			0
$979$		0			0
$980$	−28.0000	+	14.0000i	−0.894427	+	0.447214i
$981$		0			0
$982$		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
$985$	−54.8492	−	18.2831i	−1.74764	−	0.582547i
$986$	−7.54844	−	5.48426i	−0.240391	−	0.174654i
$987$		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
$997$	23.7555	−	46.6227i	0.752343	−	1.47656i	−0.122665	−	0.992448i	$-0.539144\pi$
$997$	23.7555	−	46.6227i	0.752343	−	1.47656i	0.875008	−	0.484108i	$-0.160856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.2.bj.a.487.1		8
3.2	odd	2		100.2.l.a.87.1	yes	8
4.3	odd	2	CM	900.2.bj.a.487.1		8
12.11	even	2		100.2.l.a.87.1	yes	8
15.2	even	4		500.2.l.a.43.1		8
15.8	even	4		500.2.l.c.43.1		8
15.14	odd	2		500.2.l.b.207.1		8
25.23	odd	20	inner	900.2.bj.a.523.1		8
60.23	odd	4		500.2.l.c.43.1		8
60.47	odd	4		500.2.l.a.43.1		8
60.59	even	2		500.2.l.b.207.1		8
75.2	even	20		500.2.l.b.343.1		8
75.11	odd	10		500.2.l.c.407.1		8
75.14	odd	10		500.2.l.a.407.1		8
75.23	even	20		100.2.l.a.23.1	✓	8
100.23	even	20	inner	900.2.bj.a.523.1		8
300.11	even	10		500.2.l.c.407.1		8
300.23	odd	20		100.2.l.a.23.1	✓	8
300.227	odd	20		500.2.l.b.343.1		8
300.239	even	10		500.2.l.a.407.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.l.a.23.1	✓	8	75.23	even	20
100.2.l.a.23.1	✓	8	300.23	odd	20
100.2.l.a.87.1	yes	8	3.2	odd	2
100.2.l.a.87.1	yes	8	12.11	even	2
500.2.l.a.43.1		8	15.2	even	4
500.2.l.a.43.1		8	60.47	odd	4
500.2.l.a.407.1		8	75.14	odd	10
500.2.l.a.407.1		8	300.239	even	10
500.2.l.b.207.1		8	15.14	odd	2
500.2.l.b.207.1		8	60.59	even	2
500.2.l.b.343.1		8	75.2	even	20
500.2.l.b.343.1		8	300.227	odd	20
500.2.l.c.43.1		8	15.8	even	4
500.2.l.c.43.1		8	60.23	odd	4
500.2.l.c.407.1		8	75.11	odd	10
500.2.l.c.407.1		8	300.11	even	10
900.2.bj.a.487.1		8	1.1	even	1	trivial
900.2.bj.a.487.1		8	4.3	odd	2	CM
900.2.bj.a.523.1		8	25.23	odd	20	inner
900.2.bj.a.523.1		8	100.23	even	20	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bj (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.221232 - 1.39680i) q^{2} +(-1.90211 + 0.618034i) q^{4} +(-0.333023 - 2.21113i) q^{5} +(1.28408 + 2.52015i) q^{8} +(-3.01484 + 0.954339i) q^{10} +(1.12038 - 7.07383i) q^{13} +(3.23607 - 2.35114i) q^{16} +(-0.645000 + 0.328644i) q^{17} +(2.00000 + 4.00000i) q^{20} +(-4.77819 + 1.47271i) q^{25} -10.1286 q^{26} +(-8.66785 + 2.81636i) q^{29} +(-4.00000 - 4.00000i) q^{32} +(0.601745 + 0.828232i) q^{34} +(-12.0154 - 1.90305i) q^{37} +(5.14475 - 3.67853i) q^{40} +(5.69421 - 4.13708i) q^{41} +7.00000i q^{49} +(3.11418 + 6.34838i) q^{50} +(2.24077 + 14.1477i) q^{52} +(2.48755 + 1.26747i) q^{53} +(5.85150 + 11.4842i) q^{58} +(-10.6942 - 7.76980i) q^{61} +(-4.70228 + 6.47214i) q^{64} +(-16.0143 - 0.121571i) q^{65} +(1.02375 - 1.02375i) q^{68} +(11.3902 - 1.80403i) q^{73} +17.2041i q^{74} +(-6.27636 - 6.37238i) q^{80} +(-7.03843 - 7.03843i) q^{82} +(0.941475 + 1.31673i) q^{85} +(-4.15354 + 5.71685i) q^{89} +(4.86556 - 9.54921i) q^{97} +(9.77762 - 1.54862i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{10} + 2 q^{13} + 8 q^{16} + 6 q^{17} + 16 q^{20} - 6 q^{25} + 4 q^{26} - 32 q^{32} + 50 q^{34} - 46 q^{37} + 4 q^{40} - 16 q^{41} - 14 q^{50} + 4 q^{52} - 2 q^{53}+ \cdots + 14 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 - 6 * q^10 + 2 * q^13 + 8 * q^16 + 6 * q^17 + 16 * q^20 - 6 * q^25 + 4 * q^26 - 32 * q^32 + 50 * q^34 - 46 * q^37 + 4 * q^40 - 16 * q^41 - 14 * q^50 + 4 * q^52 - 2 * q^53 - 8 * q^58 - 24 * q^61 - 78 * q^65 - 12 * q^68 + 22 * q^73 + 16 * q^80 - 16 * q^82 - 52 * q^85 + 50 * q^89 - 26 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more