LMFDB - Embedded newform 900.2.bj.c.703.1

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,-2] 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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			703.1
Root			$-0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	900.703
Dual form			900.2.bj.c.667.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.642040 + 1.26007i) q^{2} +(-1.17557 + 1.61803i) q^{4} +(-1.98459 - 1.03025i) q^{5} +(-2.79360 - 0.442463i) q^{8} +(0.0240055 - 3.16219i) q^{10} +(2.21972 - 4.35645i) q^{13} +(-1.23607 - 3.80423i) q^{16} +(1.25711 - 7.93710i) q^{17} +(4.00000 - 2.00000i) q^{20} +(2.87718 + 4.08924i) q^{25} +6.91460 q^{26} +(-4.05242 + 5.57768i) q^{29} +(4.00000 - 4.00000i) q^{32} +(10.8084 - 3.51188i) q^{34} +(10.3357 + 5.26632i) q^{37} +(5.08831 + 3.75621i) q^{40} +(-3.95309 - 12.1663i) q^{41} -7.00000i q^{49} +(-3.30548 + 6.25090i) q^{50} +(4.43945 + 8.71290i) q^{52} +(-0.960638 - 6.06523i) q^{53} +(-9.63011 - 1.52526i) q^{58} +(4.81636 - 14.8232i) q^{61} +(7.60845 + 2.47214i) q^{64} +(-8.89346 + 6.35889i) q^{65} +(11.3647 + 11.3647i) q^{68} +(-3.05007 + 1.55409i) q^{73} +16.4050i q^{74} +(-1.46621 + 8.82328i) q^{80} +(12.7925 - 12.7925i) q^{82} +(-10.6720 + 14.4567i) q^{85} +(-11.5332 - 3.74737i) q^{89} +(-15.1115 + 2.39343i) q^{97} +(8.82051 - 4.49428i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 2 q^{5} - 4 q^{8} + 6 q^{10} - 10 q^{13} + 8 q^{16} + 6 q^{17} + 32 q^{20} + 6 q^{25} + 20 q^{26} + 32 q^{32} + 50 q^{34} + 50 q^{37} + 4 q^{40} - 20 q^{41} + 2 q^{50} - 20 q^{52} - 52 q^{53}+ \cdots - 14 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 - 2 * q^5 - 4 * q^8 + 6 * q^10 - 10 * q^13 + 8 * q^16 + 6 * q^17 + 32 * q^20 + 6 * q^25 + 20 * q^26 + 32 * q^32 + 50 * q^34 + 50 * q^37 + 4 * q^40 - 20 * q^41 + 2 * q^50 - 20 * q^52 - 52 * q^53 - 20 * q^58 + 24 * q^61 - 12 * q^68 + 10 * q^73 + 8 * q^80 + 20 * q^82 - 52 * q^85 - 80 * q^89 + 10 * q^97 - 14 * q^98

Character values

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

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$e\left(\frac{7}{20}\right)$

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 $

$2$	0.642040	+	1.26007i	0.453990	+	0.891007i
$2$	0.642040	+	1.26007i	0.453990	+	0.891007i
$3$		0			0
$3$		0			0
$4$	−1.17557	+	1.61803i	−0.587785	+	0.809017i
$5$	−1.98459	−	1.03025i	−0.887535	−	0.460741i
$5$	−1.98459	−	1.03025i	−0.887535	−	0.460741i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−2.79360	−	0.442463i	−0.987688	−	0.156434i
$9$		0			0
$10$	0.0240055	−	3.16219i	0.00759122	−	0.999971i
$11$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$11$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$12$		0			0
$13$	2.21972	−	4.35645i	0.615640	−	1.20826i	−0.347098	−	0.937829i	$-0.612833\pi$
$13$	2.21972	−	4.35645i	0.615640	−	1.20826i	0.962739	−	0.270434i	$-0.0871670\pi$
$14$		0			0
$15$		0			0
$16$	−1.23607	−	3.80423i	−0.309017	−	0.951057i
$17$	1.25711	−	7.93710i	0.304895	−	1.92503i	−0.0691254	−	0.997608i	$-0.522021\pi$
$17$	1.25711	−	7.93710i	0.304895	−	1.92503i	0.374020	−	0.927421i	$-0.377979\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	4.00000	−	2.00000i	0.894427	−	0.447214i
$21$		0			0
$22$		0			0
$23$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$23$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$24$		0			0
$25$	2.87718	+	4.08924i	0.575435	+	0.817848i
$26$	6.91460			1.35606
$27$		0			0
$28$		0			0
$29$	−4.05242	+	5.57768i	−0.752516	+	1.03575i	0.245284	+	0.969451i	$0.421119\pi$
$29$	−4.05242	+	5.57768i	−0.752516	+	1.03575i	−0.997800	+	0.0662984i	$0.978881\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		0			0
$34$	10.8084	−	3.51188i	1.85363	−	0.602282i
$35$		0			0
$36$		0			0
$37$	10.3357	+	5.26632i	1.69918	+	0.865777i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	10.3357	+	5.26632i	1.69918	+	0.865777i	0.712789	+	0.701378i	$0.247432\pi$
$38$		0			0
$39$		0			0
$40$	5.08831	+	3.75621i	0.804532	+	0.593910i
$41$	−3.95309	−	12.1663i	−0.617368	−	1.90006i	−0.352819	−	0.935692i	$-0.614777\pi$
$41$	−3.95309	−	12.1663i	−0.617368	−	1.90006i	−0.264550	−	0.964372i	$-0.585223\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$47$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$	−3.30548	+	6.25090i	−0.467465	+	0.884011i
$51$		0			0
$52$	4.43945	+	8.71290i	0.615640	+	1.20826i
$53$	−0.960638	−	6.06523i	−0.131954	−	0.833124i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−0.960638	−	6.06523i	−0.131954	−	0.833124i	0.829570	−	0.558403i	$-0.188586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−9.63011	−	1.52526i	−1.26450	−	0.200276i
$59$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$59$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$	4.81636	−	14.8232i	0.616671	−	1.89792i	0.245213	−	0.969469i	$-0.421142\pi$
$61$	4.81636	−	14.8232i	0.616671	−	1.89792i	0.371458	−	0.928450i	$-0.378858\pi$
$62$		0			0
$63$		0			0
$64$	7.60845	+	2.47214i	0.951057	+	0.309017i
$65$	−8.89346	+	6.35889i	−1.10310	+	0.788724i
$66$		0			0
$67$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$67$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$68$	11.3647	+	11.3647i	1.37817	+	1.37817i
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$		0			0
$73$	−3.05007	+	1.55409i	−0.356984	+	0.181893i	−0.623280	−	0.781999i	$-0.714200\pi$
$73$	−3.05007	+	1.55409i	−0.356984	+	0.181893i	0.266296	+	0.963891i	$0.414200\pi$
$74$			16.4050i			1.90704i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$	−1.46621	+	8.82328i	−0.163928	+	0.986472i
$81$		0			0
$82$	12.7925	−	12.7925i	1.41269	−	1.41269i
$83$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$83$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$84$		0			0
$85$	−10.6720	+	14.4567i	−1.15754	+	1.56805i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.5332	−	3.74737i	−1.22252	−	0.397220i	−0.374519	−	0.927219i	$-0.622192\pi$
$89$	−11.5332	−	3.74737i	−1.22252	−	0.397220i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−15.1115	+	2.39343i	−1.53434	+	0.243016i	−0.865698	−	0.500567i	$-0.833125\pi$
$97$	−15.1115	+	2.39343i	−1.53434	+	0.243016i	−0.668644	+	0.743583i	$0.733125\pi$
$98$	8.82051	−	4.49428i	0.891007	−	0.453990i
$99$		0			0
$100$	−9.99885	−	0.151820i	−0.999885	−	0.0151820i
$101$	4.27823			0.425699			0.212850	−	0.977085i	$-0.431726\pi$
$101$	4.27823			0.425699			0.212850	+	0.977085i	$0.431726\pi$
$102$		0			0
$103$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$103$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$104$	−8.12860	+	11.1881i	−0.797075	+	1.09708i
$105$		0			0
$106$	7.02587	−	5.10459i	0.682413	−	0.495802i
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	6.56381	−	2.13271i	0.628699	−	0.204277i	0.0227005	−	0.999742i	$-0.492774\pi$
$109$	6.56381	−	2.13271i	0.628699	−	0.204277i	0.605999	+	0.795465i	$0.292774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.96084	+	17.5866i	−0.842965	+	1.65441i	−0.0903879	+	0.995907i	$0.528811\pi$
$113$	−8.96084	+	17.5866i	−0.842965	+	1.65441i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$	−4.26097	−	13.1139i	−0.395621	−	1.21760i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.89919	−	6.46564i	−0.809017	−	0.587785i
$122$	21.7706	−	3.44813i	1.97102	−	0.312179i
$123$		0			0
$124$		0			0
$125$	−1.49707	−	11.0797i	−0.133902	−	0.990995i
$126$		0			0
$127$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$127$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$128$	1.76985	+	11.1744i	0.156434	+	0.987688i
$129$		0			0
$130$	−13.7226	−	7.12376i	−1.20355	−	0.624795i
$131$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$131$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−7.02375	+	21.6169i	−0.602282	+	1.85363i
$137$	−3.97384	−	2.02477i	−0.339508	−	0.172988i	0.275921	−	0.961180i	$-0.411017\pi$
$137$	−3.97384	−	2.02477i	−0.339508	−	0.172988i	−0.615429	+	0.788192i	$0.711017\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	13.7888	−	6.89440i	1.14510	−	0.572548i
$146$	−3.91654	−	2.84553i	−0.324135	−	0.235498i
$147$		0			0
$148$	−20.6715	+	10.5326i	−1.69918	+	0.865777i
$149$			7.95135i			0.651400i	0.945473	+	0.325700i	$0.105600\pi$
$149$			7.95135i			0.651400i	−0.945473	+	0.325700i	$0.894400\pi$
$150$		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
$157$	−17.7130	+	17.7130i	−1.41365	+	1.41365i	−0.686955	+	0.726700i	$0.741053\pi$
$157$	−17.7130	+	17.7130i	−1.41365	+	1.41365i	−0.726700	+	0.686955i	$0.758947\pi$
$158$		0			0
$159$		0			0
$160$	−12.0593	+	3.81736i	−0.953375	+	0.301788i
$161$		0			0
$162$		0			0
$163$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$163$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$164$	24.3327	+	7.90617i	1.90006	+	0.617368i
$165$		0			0
$166$		0			0
$167$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$167$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$168$		0			0
$169$	−6.41029	−	8.82301i	−0.493100	−	0.678693i
$170$	−25.0684	−	4.16576i	−1.92266	−	0.319499i
$171$		0			0
$172$		0			0
$173$	13.6928	−	6.97683i	1.04104	−	0.530438i	0.152057	−	0.988372i	$-0.451410\pi$
$173$	13.6928	−	6.97683i	1.04104	−	0.530438i	0.888986	+	0.457933i	$0.151410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−2.68282	−	16.9386i	−0.201086	−	1.26961i
$179$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$179$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$180$		0			0
$181$	10.8884	−	7.91090i	0.809330	−	0.588012i	−0.104306	−	0.994545i	$-0.533262\pi$
$181$	10.8884	−	7.91090i	0.809330	−	0.588012i	0.913636	+	0.406533i	$0.133262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.0865	−	21.0998i	−1.10918	−	1.55129i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$	15.2130	+	15.2130i	1.09506	+	1.09506i	0.994980	+	0.100076i	$0.0319087\pi$
$193$	15.2130	+	15.2130i	1.09506	+	1.09506i	0.100076	+	0.994980i	$0.468091\pi$
$194$	−12.7181	−	17.5049i	−0.913105	−	1.25678i
$195$		0			0
$196$	11.3262	+	8.22899i	0.809017	+	0.587785i
$197$	27.0058	−	4.27729i	1.92408	−	0.304745i	0.926623	−	0.375992i	$-0.122698\pi$
$197$	27.0058	−	4.27729i	1.92408	−	0.304745i	0.997459	+	0.0712470i	$0.0226979\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−6.22835	−	12.6968i	−0.440411	−	0.897796i
$201$		0			0
$202$	2.74679	+	5.39088i	0.193264	+	0.379301i
$203$		0			0
$204$		0			0
$205$	−4.68912	+	28.2178i	−0.327502	+	1.97082i
$206$		0			0
$207$		0			0
$208$	−19.3167	−	3.05946i	−1.33937	−	0.212135i
$209$		0			0
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$	10.9430	+	5.57576i	0.751572	+	0.382945i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	6.90160	+	6.90160i	0.467436	+	0.467436i
$219$		0			0
$220$		0			0
$221$	−31.7871	−	23.0947i	−2.13823	−	1.55352i
$222$		0			0
$223$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$223$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$224$		0			0
$225$		0			0
$226$	−27.9137			−1.85679
$227$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$227$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$228$		0			0
$229$	−8.46263	+	11.6478i	−0.559226	+	0.769709i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−8.46263	+	11.6478i	−0.559226	+	0.769709i	0.432001	+	0.901873i	$0.357808\pi$
$230$		0			0
$231$		0			0
$232$	13.7888	−	13.7888i	0.905279	−	0.905279i
$233$	27.8359	+	4.40877i	1.82359	+	0.288828i	0.971930	−	0.235269i	$-0.0755972\pi$
$233$	27.8359	+	4.40877i	1.82359	+	0.288828i	0.851658	+	0.524097i	$0.175597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	−9.58071	−	29.4864i	−0.617148	−	1.89939i	−0.359485	−	0.933151i	$-0.617048\pi$
$241$	−9.58071	−	29.4864i	−0.617148	−	1.89939i	−0.257663	−	0.966235i	$-0.582952\pi$
$242$	2.43355	−	15.3648i	0.156434	−	0.987688i
$243$		0			0
$244$	18.3225	+	25.2188i	1.17298	+	1.61447i
$245$	−7.21174	+	13.8921i	−0.460741	+	0.887535i
$246$		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
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	4.93651	−	4.93651i	0.307931	−	0.307931i	−0.536175	−	0.844107i	$-0.680131\pi$
$257$	4.93651	−	4.93651i	0.307931	−	0.307931i	0.844107	+	0.536175i	$0.180131\pi$
$258$		0			0
$259$		0			0
$260$	0.165989	−	21.8653i	0.0102942	−	1.35603i
$261$		0			0
$262$		0			0
$263$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$263$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$264$		0			0
$265$	−4.34223	+	13.0267i	−0.266741	+	0.800223i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.4933	+	25.4539i	1.12756	+	1.55195i	0.792624	+	0.609711i	$0.208714\pi$
$269$	18.4933	+	25.4539i	1.12756	+	1.55195i	0.334935	+	0.942241i	$0.391286\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$	−31.7484	+	5.02845i	−1.92503	+	0.304895i
$273$		0			0
$274$		−	6.30732i		−	0.381039i
$275$		0			0
$276$		0			0
$277$	13.0522	+	25.6163i	0.784228	+	1.53913i	0.841178	+	0.540758i	$0.181862\pi$
$277$	13.0522	+	25.6163i	0.784228	+	1.53913i	−0.0569502	+	0.998377i	$0.518138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.67182	−	6.30045i	0.517317	−	0.375853i	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	8.67182	−	6.30045i	0.517317	−	0.375853i	0.815592	+	0.578627i	$0.196411\pi$
$282$		0			0
$283$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$283$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−45.2492	−	14.7024i	−2.66172	−	0.864845i
$290$	17.5404	+	12.9484i	1.03001	+	0.760357i
$291$		0			0
$292$	1.07101	−	6.76207i	0.0626759	−	0.395720i
$293$	20.1372	+	20.1372i	1.17643	+	1.17643i	0.980648	+	0.195778i	$0.0627231\pi$
$293$	20.1372	+	20.1372i	1.17643	+	1.17643i	0.195778	+	0.980648i	$0.437277\pi$
$294$		0			0
$295$		0			0
$296$	−26.5438	−	19.2852i	−1.54283	−	1.12093i
$297$		0			0
$298$	−10.0193	+	5.10508i	−0.580401	+	0.295729i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−24.8301	+	24.4559i	−1.42177	+	1.40034i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	16.0510	−	31.5018i	0.907255	−	1.78059i	0.418016	−	0.908440i	$-0.362726\pi$
$313$	16.0510	−	31.5018i	0.907255	−	1.78059i	0.489240	−	0.872149i	$-0.337274\pi$
$314$	−33.6922	−	10.9473i	−1.90136	−	0.617790i
$315$		0			0
$316$		0			0
$317$	0.663695	−	4.19041i	0.0372768	−	0.235357i	−0.962014	−	0.272999i	$-0.911985\pi$
$317$	0.663695	−	4.19041i	0.0372768	−	0.235357i	0.999291	+	0.0376418i	$0.0119846\pi$
$318$		0			0
$319$		0			0
$320$	−12.5527	−	12.7448i	−0.701719	−	0.712454i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	24.2011	−	3.45730i	1.34244	−	0.191776i
$326$		0			0
$327$		0			0
$328$	5.66019	+	35.7371i	0.312532	+	1.97325i
$329$		0			0
$330$		0			0
$331$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$331$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−31.5018	−	16.0510i	−1.71601	−	0.874353i	−0.980417	−	0.196934i	$-0.936901\pi$
$337$	−31.5018	−	16.0510i	−1.71601	−	0.874353i	−0.735598	−	0.677419i	$-0.763099\pi$
$338$	7.00198	−	13.7422i	0.380858	−	0.747475i
$339$		0			0
$340$	−10.8457	−	34.2626i	−0.588193	−	1.85815i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	17.5826	+	12.7745i	0.945248	+	0.686763i
$347$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$347$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$348$		0			0
$349$		−	20.6352i		−	1.10458i	−0.833653	−	0.552288i	$-0.813755\pi$
$349$		−	20.6352i		−	1.10458i	0.833653	−	0.552288i	$-0.186245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.99109	+	12.5712i	0.105975	+	0.669099i	0.982291	+	0.187362i	$0.0599938\pi$
$353$	1.99109	+	12.5712i	0.105975	+	0.669099i	−0.876316	+	0.481736i	$0.840006\pi$
$354$		0			0
$355$		0			0
$356$	19.6215	−	14.2558i	1.03994	−	0.755558i
$357$		0			0
$358$		0			0
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	−5.87132	+	18.0701i	−0.309017	+	0.951057i
$362$	16.9591	+	8.64110i	0.891351	+	0.454166i
$363$		0			0
$364$		0			0
$365$	7.65424	+	0.0581067i	0.400641	+	0.00304144i
$366$		0			0
$367$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$367$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$368$		0			0
$369$		0			0
$370$	16.9012	−	32.5571i	0.878651	−	1.69256i
$371$		0			0
$372$		0			0
$373$	31.5018	−	16.0510i	1.63110	−	0.831089i	0.632712	−	0.774387i	$-0.281942\pi$
$373$	31.5018	−	16.0510i	1.63110	−	0.831089i	0.998391	−	0.0567016i	$-0.0180584\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.3036	+	30.0351i	0.788178	+	1.54689i
$378$		0			0
$379$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$379$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$383$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$384$		0			0
$385$		0			0
$386$	−9.40215	+	28.9369i	−0.478557	+	1.47285i
$387$		0			0
$388$	13.8920	−	27.2646i	0.705260	−	1.38415i
$389$	3.61816	+	1.17561i	0.183448	+	0.0596059i	0.399300	−	0.916820i	$-0.369253\pi$
$389$	3.61816	+	1.17561i	0.183448	+	0.0596059i	−0.215852	+	0.976426i	$0.569253\pi$
$390$		0			0
$391$		0			0
$392$	−3.09724	+	19.5552i	−0.156434	+	0.987688i
$393$		0			0
$394$	22.7285	+	31.2831i	1.14504	+	1.57602i
$395$		0			0
$396$		0			0
$397$	−34.9201	+	5.53079i	−1.75259	+	0.277583i	−0.948465	−	0.316881i	$-0.897364\pi$
$397$	−34.9201	+	5.53079i	−1.75259	+	0.277583i	−0.804122	+	0.594464i	$0.797364\pi$
$398$		0			0
$399$		0			0
$400$	12.0000	−	16.0000i	0.600000	−	0.800000i
$401$	33.5363			1.67472			0.837360	−	0.546652i	$-0.184098\pi$
$401$	33.5363			1.67472			0.837360	+	0.546652i	$0.184098\pi$
$402$		0			0
$403$		0			0
$404$	−5.02936	+	6.92232i	−0.250220	+	0.344398i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−34.4170	+	11.1828i	−1.70181	+	0.552952i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−34.4170	+	11.1828i	−1.70181	+	0.552952i	−0.712874	+	0.701292i	$0.752607\pi$
$410$	−38.5671	+	12.2083i	−1.90470	+	0.602927i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−8.54691	−	26.3047i	−0.419047	−	1.28969i
$417$		0			0
$418$		0			0
$419$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$419$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$	30.0826	+	21.8563i	1.46614	+	1.06521i	0.981711	+	0.190380i	$0.0609719\pi$
$421$	30.0826	+	21.8563i	1.46614	+	1.06521i	0.484427	+	0.874832i	$0.339028\pi$
$422$		0			0
$423$		0			0
$424$			17.3689i			0.843509i
$425$	36.0736	−	17.6958i	1.74983	−	0.858372i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$	29.4598	+	4.66597i	1.41575	+	0.224232i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	29.4598	+	4.66597i	1.41575	+	0.224232i	0.598778	+	0.800915i	$0.295653\pi$
$434$		0			0
$435$		0			0
$436$	−4.26543	+	13.1276i	−0.204277	+	0.628699i
$437$		0			0
$438$		0			0
$439$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$439$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$440$		0			0
$441$		0			0
$442$	8.69243	−	54.8818i	0.413457	−	2.61046i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	19.0279	+	19.3191i	0.902011	+	0.915811i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	0.954111i		−	0.0450273i	−0.999747	−	0.0225137i	$-0.992833\pi$
$449$		−	0.954111i		−	0.0450273i	0.999747	−	0.0225137i	$-0.00716692\pi$
$450$		0			0
$451$		0			0
$452$	−17.9217	−	35.1733i	−0.842965	−	1.65441i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	−20.1104	−	3.18518i	−0.939699	−	0.148834i
$459$		0			0
$460$		0			0
$461$	−13.0777	+	40.2492i	−0.609092	+	1.87459i	−0.143345	+	0.989673i	$0.545786\pi$
$461$	−13.0777	+	40.2492i	−0.609092	+	1.87459i	−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$463$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$464$	26.2278	+	8.52194i	1.21760	+	0.395621i
$465$		0			0
$466$	12.3164	+	37.9058i	0.570544	+	1.75595i
$467$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$467$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$468$		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		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$	45.8849	−	33.3373i	2.09217	−	1.52005i
$482$	31.0038	−	31.0038i	1.41219	−	1.41219i
$483$		0			0
$484$	20.9232	−	6.79837i	0.951057	−	0.309017i
$485$	32.4559	+	10.8186i	1.47375	+	0.491250i
$486$		0			0
$487$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$487$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$488$	−20.0137	+	39.2792i	−0.905979	+	1.77808i
$489$		0			0
$490$	−22.1353	−	0.168039i	−0.999971	−	0.00759122i
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$	39.1763	+	39.1763i	1.76441	+	1.76441i
$494$		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	$0.5\pi$
$500$	19.6872	+	10.6026i	0.880437	+	0.474163i
$501$		0			0
$502$		0			0
$503$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$503$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$504$		0			0
$505$	−8.49052	−	4.40764i	−0.377823	−	0.196137i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−42.7373	+	13.8862i	−1.89430	+	0.615495i	−0.919165	+	0.393873i	$0.871135\pi$
$509$	−42.7373	+	13.8862i	−1.89430	+	0.615495i	−0.975133	+	0.221621i	$0.928865\pi$
$510$		0			0
$511$		0			0
$512$	−20.1612	−	10.2726i	−0.891007	−	0.453990i
$513$		0			0
$514$	9.38980	+	3.05093i	0.414166	+	0.134571i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	27.6584	−	13.8292i	1.21290	−	0.606450i
$521$	36.6420	+	26.6219i	1.60531	+	1.16633i	0.876216	+	0.481919i	$0.160060\pi$
$521$	36.6420	+	26.6219i	1.60531	+	1.16633i	0.729098	+	0.684410i	$0.239940\pi$
$522$		0			0
$523$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$523$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.5191	−	18.6074i	0.587785	−	0.809017i
$530$	−19.2025	+	2.89212i	−0.834101	+	0.125626i
$531$		0			0
$532$		0			0
$533$	−61.7768	−	9.78449i	−2.67585	−	0.423813i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−20.2003	+	39.6454i	−0.870898	+	1.70923i
$539$		0			0
$540$		0			0
$541$	−9.88850	−	30.4337i	−0.425140	−	1.30845i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−9.88850	−	30.4337i	−0.425140	−	1.30845i	0.477721	−	0.878512i	$-0.341463\pi$
$542$		0			0
$543$		0			0
$544$	−26.7199	−	36.7768i	−1.14561	−	1.57679i
$545$	−15.2237	−	2.52981i	−0.652111	−	0.108365i
$546$		0			0
$547$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$547$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$548$	7.94769	−	4.04955i	0.339508	−	0.172988i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−23.8984	+	32.8934i	−1.01535	+	1.39750i
$555$		0			0
$556$		0			0
$557$	−14.9528	+	14.9528i	−0.633572	+	0.633572i	−0.948962	−	0.315390i	$-0.897865\pi$
$557$	−14.9528	+	14.9528i	−0.633572	+	0.633572i	0.315390	+	0.948962i	$0.397865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	13.5067	+	6.88199i	0.569745	+	0.290299i
$563$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$563$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$564$		0			0
$565$	35.9022	−	25.6703i	1.51042	−	1.07996i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.7999	+	30.0049i	0.913898	+	1.25787i	0.965818	+	0.259221i	$0.0834659\pi$
$569$	21.7999	+	30.0049i	0.913898	+	1.25787i	−0.0519200	+	0.998651i	$0.516534\pi$
$570$		0			0
$571$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$571$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.0510	+	31.5018i	0.668211	+	1.31144i	0.937367	+	0.348342i	$0.113255\pi$
$577$	16.0510	+	31.5018i	0.668211	+	1.31144i	−0.269156	+	0.963097i	$0.586745\pi$
$578$	−10.5257	−	66.4568i	−0.437812	−	2.76424i
$579$		0			0
$580$	−5.05433	+	30.4156i	−0.209870	+	1.26294i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	9.20833	−	2.99197i	0.381043	−	0.123809i
$585$		0			0
$586$	−12.4455	+	38.3032i	−0.514117	+	1.58229i
$587$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$587$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	7.25860	−	45.8290i	0.298326	−	1.88356i
$593$	4.68632	+	4.68632i	0.192444	+	0.192444i	0.796751	−	0.604307i	$-0.206550\pi$
$593$	4.68632	+	4.68632i	0.192444	+	0.192444i	−0.604307	+	0.796751i	$0.706550\pi$
$594$		0			0
$595$		0			0
$596$	−12.8655	−	9.34737i	−0.526993	−	0.382883i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−32.9550			−1.34426			−0.672130	−	0.740433i	$-0.734621\pi$
$601$	−32.9550			−1.34426			−0.672130	+	0.740433i	$0.734621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.0000	+	22.0000i	0.447214	+	0.894427i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$	−46.7582	−	15.5861i	−1.89318	−	0.631061i
$611$		0			0
$612$		0			0
$613$	−6.33342	+	12.4300i	−0.255805	+	0.502045i	−0.982818	−	0.184579i	$-0.940908\pi$
$613$	−6.33342	+	12.4300i	−0.255805	+	0.502045i	0.727013	+	0.686624i	$0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.01435	−	25.3456i	0.161611	−	1.02037i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	4.01435	−	25.3456i	0.161611	−	1.02037i	0.926523	−	0.376239i	$-0.122783\pi$
$618$		0			0
$619$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−8.44373	+	23.5309i	−0.337749	+	0.941236i
$626$	50.0000			1.99840
$627$		0			0
$628$	−7.83738	−	49.4832i	−0.312745	−	1.97460i
$629$	54.7924	−	75.4153i	2.18472	−	3.00701i
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	5.70634	−	1.85410i	0.226628	−	0.0736358i
$635$		0			0
$636$		0			0
$637$	−30.4952	−	15.5381i	−1.20826	−	0.615640i
$638$		0			0
$639$		0			0
$640$	8.00000	−	24.0000i	0.316228	−	0.948683i
$641$	−15.4508	−	47.5528i	−0.610272	−	1.87822i	−0.455394	−	0.890290i	$-0.650502\pi$
$641$	−15.4508	−	47.5528i	−0.610272	−	1.87822i	−0.154878	−	0.987934i	$-0.549498\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$647$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$648$		0			0
$649$		0			0
$650$	19.8945	+	28.2754i	0.780327	+	1.10905i
$651$		0			0
$652$		0			0
$653$	−6.74886	−	42.6106i	−0.264103	−	1.66748i	−0.661584	−	0.749871i	$-0.730115\pi$
$653$	−6.74886	−	42.6106i	−0.264103	−	1.66748i	0.397481	−	0.917611i	$-0.369885\pi$
$654$		0			0
$655$		0			0
$656$	−41.3972	+	30.0769i	−1.61629	+	1.17430i
$657$		0			0
$658$		0			0
$659$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$659$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$660$		0			0
$661$	−3.70820	+	11.4127i	−0.144232	+	0.443902i	−0.996911	−	0.0785333i	$-0.974976\pi$
$661$	−3.70820	+	11.4127i	−0.144232	+	0.443902i	0.852679	+	0.522435i	$0.174976\pi$
$662$		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
$673$	−0.446032	+	0.227265i	−0.0171933	+	0.00876041i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−0.446032	+	0.227265i	−0.0171933	+	0.00876041i	0.445373	+	0.895345i	$0.353071\pi$
$674$		−	50.0000i		−	1.92593i
$675$		0			0
$676$	21.8117			0.838911
$677$	−17.3351	−	34.0220i	−0.666241	−	1.30757i	−0.938477	−	0.345341i	$-0.887763\pi$
$677$	−17.3351	−	34.0220i	−0.666241	−	1.30757i	0.272237	−	0.962230i	$-0.412237\pi$
$678$		0			0
$679$		0			0
$680$	36.2100	−	35.6644i	1.38859	−	1.36767i
$681$		0			0
$682$		0			0
$683$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$683$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$684$		0			0
$685$	5.80042	+	8.11239i	0.221623	+	0.309958i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−28.5552	−	9.27816i	−1.08787	−	0.353470i
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$	−4.80810	+	30.3572i	−0.182777	+	1.15401i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−101.535	+	16.0816i	−3.84591	+	0.609132i
$698$	26.0018	−	13.2486i	0.984184	−	0.501467i
$699$		0			0
$700$		0			0
$701$	22.4747			0.848856			0.424428	−	0.905462i	$-0.360475\pi$
$701$	22.4747			0.848856			0.424428	+	0.905462i	$0.360475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−14.5623	+	10.5801i	−0.548060	+	0.398189i
$707$		0			0
$708$		0			0
$709$	−17.0840	+	5.55093i	−0.641603	+	0.208470i	−0.611708	−	0.791083i	$-0.709517\pi$
$709$	−17.0840	+	5.55093i	−0.641603	+	0.208470i	−0.0298952	+	0.999553i	$0.509517\pi$
$710$		0			0
$711$		0			0
$712$	30.5612	+	15.5717i	1.14533	+	0.583574i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$719$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$720$		0			0
$721$		0			0
$722$	−26.5392	+	4.20340i	−0.987688	+	0.156434i
$723$		0			0
$724$			26.9176i			1.00039i
$725$	−34.4680	−	0.523353i	−1.28011	−	0.0194369i
$726$		0			0
$727$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$727$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$728$		0			0
$729$		0			0
$730$	4.84111	+	9.68221i	0.179177	+	0.358355i
$731$		0			0
$732$		0			0
$733$	34.9201	+	5.53079i	1.28980	+	0.204285i	0.763386	−	0.645943i	$-0.223536\pi$
$733$	34.9201	+	5.53079i	1.28980	+	0.204285i	0.526416	+	0.850227i	$0.323536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$	51.8755	+	0.393810i	1.90698	+	0.0144767i
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	8.19186	−	15.7801i	0.300127	−	0.578140i
$746$	40.4508	+	29.3893i	1.48101	+	1.07602i
$747$		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
$754$	−28.0209	+	38.5674i	−1.02046	+	1.40454i
$755$		0			0
$756$		0			0
$757$	18.3232	−	18.3232i	0.665970	−	0.665970i	−0.290811	−	0.956781i	$-0.593925\pi$
$757$	18.3232	−	18.3232i	0.665970	−	0.665970i	0.956781	+	0.290811i	$0.0939250\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.09785	+	9.53419i	−0.112297	+	0.345614i	−0.991374	−	0.131066i	$-0.958160\pi$
$761$	−3.09785	+	9.53419i	−0.112297	+	0.345614i	0.879077	+	0.476680i	$0.158160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.1068	−	19.4164i	−0.508706	−	0.700174i	0.474995	−	0.879989i	$-0.342450\pi$
$769$	−14.1068	−	19.4164i	−0.508706	−	0.700174i	−0.983700	+	0.179815i	$0.942450\pi$
$770$		0			0
$771$		0			0
$772$	−42.4991	+	6.73120i	−1.52958	+	0.242261i
$773$	36.0542	−	18.3705i	1.29678	−	0.660741i	0.337001	−	0.941504i	$-0.390588\pi$
$773$	36.0542	−	18.3705i	1.29678	−	0.660741i	0.959777	+	0.280763i	$0.0905876\pi$
$774$		0			0
$775$		0			0
$776$	43.2746			1.55347
$777$		0			0
$778$	0.841645	+	5.31394i	0.0301744	+	0.190514i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.6296	+	8.65248i	−0.951057	+	0.309017i
$785$	53.4019	−	16.9042i	1.90600	−	0.603338i
$786$		0			0
$787$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$787$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$788$	−24.8264	+	48.7245i	−0.884403	+	1.73574i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−53.8857	−	53.8857i	−1.91354	−	1.91354i
$794$	−29.3893	−	40.4508i	−1.04299	−	1.43555i
$795$		0			0
$796$		0			0
$797$	−29.4961	+	4.67172i	−1.04480	+	0.165481i	−0.655164	−	0.755487i	$-0.727401\pi$
$797$	−29.4961	+	4.67172i	−1.04480	+	0.165481i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$	27.8657	+	4.84825i	0.985200	+	0.171412i
$801$		0			0
$802$	21.5316	+	42.2581i	0.760307	+	1.49219i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−11.9517	−	1.89296i	−0.420458	−	0.0665941i
$809$	−47.7135	+	15.5031i	−1.67752	+	0.545059i	−0.984428	−	0.175791i	$-0.943752\pi$
$809$	−47.7135	+	15.5031i	−1.67752	+	0.545059i	−0.693091	+	0.720850i	$0.743752\pi$
$810$		0			0
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−36.1882	−	36.1882i	−1.26529	−	1.26529i
$819$		0			0
$820$	−40.1450	−	40.7592i	−1.40193	−	1.42337i
$821$	−40.4508	−	29.3893i	−1.41174	−	1.02569i	−0.993066	−	0.117558i	$-0.962493\pi$
$821$	−40.4508	−	29.3893i	−1.41174	−	1.02569i	−0.418678	−	0.908135i	$-0.637507\pi$
$822$		0			0
$823$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$823$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$827$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$828$		0			0
$829$	−32.5884	+	44.8540i	−1.13184	+	1.55784i	−0.347314	+	0.937749i	$0.612906\pi$
$829$	−32.5884	+	44.8540i	−1.13184	+	1.55784i	−0.784526	+	0.620096i	$0.787094\pi$
$830$		0			0
$831$		0			0
$832$	27.6584	−	27.6584i	0.958882	−	0.958882i
$833$	−55.5597	−	8.79979i	−1.92503	−	0.304895i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	−5.72692	−	17.6256i	−0.197480	−	0.607781i
$842$	−8.22632	+	51.9389i	−0.283498	+	1.78993i
$843$		0			0
$844$		0			0
$845$	3.63189	+	24.1142i	0.124941	+	0.829555i
$846$		0			0
$847$		0			0
$848$	−21.8861	+	11.1515i	−0.751572	+	0.382945i
$849$		0			0
$850$	45.4587	+	34.0940i	1.55922	+	1.16941i
$851$		0			0
$852$		0			0
$853$	4.43661	+	28.0116i	0.151906	+	0.959100i	0.939411	+	0.342792i	$0.111372\pi$
$853$	4.43661	+	28.0116i	0.151906	+	0.959100i	−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	−0.136637	+	0.990621i	$0.543630\pi$
$857$	−33.0000	+	33.0000i	−1.12726	+	1.12726i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$859$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$863$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$864$		0			0
$865$	−34.3624	−	0.260860i	−1.16836	−	0.00886951i
$866$	13.0349	+	40.1172i	0.442943	+	1.36324i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−19.2803	+	3.05371i	−0.652915	+	0.103412i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	26.8595	+	52.7147i	0.906981	+	1.78005i	0.495297	+	0.868723i	$0.335059\pi$
$877$	26.8595	+	52.7147i	0.906981	+	1.78005i	0.411683	+	0.911327i	$0.364941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40.4508	+	29.3893i	−1.36282	+	0.990149i	−0.364564	+	0.931178i	$0.618782\pi$
$881$	−40.4508	+	29.3893i	−1.36282	+	0.990149i	−0.998260	+	0.0589711i	$0.981218\pi$
$882$		0			0
$883$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$883$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$884$	74.7360	−	24.2832i	2.51364	−	0.816733i
$885$		0			0
$886$		0			0
$887$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$887$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$888$		0			0
$889$		0			0
$890$	−12.1267	+	36.3802i	−0.406489	+	1.21947i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.20225	−	0.612577i	0.0401196	−	0.0204420i
$899$		0			0
$900$		0			0
$901$	−49.3480			−1.64402
$902$		0			0
$903$		0			0
$904$	32.8145	−	45.1653i	1.09139	−	1.50217i
$905$	−29.7592	+	4.48209i	−0.989230	+	0.148990i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$		0			0
$914$	47.5528	+	15.4508i	1.57291	+	0.511069i
$915$		0			0
$916$	−8.89814	−	27.3857i	−0.294003	−	0.904847i
$917$		0			0
$918$		0			0
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$	−59.1134	+	9.36264i	−1.94679	+	0.308342i
$923$		0			0
$924$		0			0
$925$	8.20248	+	57.4174i	0.269696	+	1.88787i
$926$		0			0
$927$		0			0
$928$	6.10104	+	38.5204i	0.200276	+	1.26450i
$929$	32.9802	−	45.3934i	1.08205	−	1.48931i	0.224797	−	0.974406i	$-0.427828\pi$
$929$	32.9802	−	45.3934i	1.08205	−	1.48931i	0.857249	−	0.514902i	$-0.172172\pi$
$930$		0			0
$931$		0			0
$932$	−39.8566	+	39.8566i	−1.30555	+	1.30555i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	49.5843	+	25.2645i	1.61985	+	0.825354i	0.999146	+	0.0413087i	$0.0131527\pi$
$937$	49.5843	+	25.2645i	1.61985	+	0.825354i	0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.9556	+	58.3393i	0.617935	+	1.90181i	0.325991	+	0.945373i	$0.394302\pi$
$941$	18.9556	+	58.3393i	0.617935	+	1.90181i	0.291944	+	0.956435i	$0.405698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$947$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$948$		0			0
$949$			16.7372i			0.543311i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.353119	−	2.22951i	−0.0114387	−	0.0722208i	0.981309	−	0.192440i	$-0.0616402\pi$
$953$	−0.353119	−	2.22951i	−0.0114387	−	0.0722208i	−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
$961$	9.57953	−	29.4828i	0.309017	−	0.951057i
$962$	71.4674	+	36.4145i	2.30420	+	1.17405i
$963$		0			0
$964$	58.9728	+	19.1614i	1.89939	+	0.617148i
$965$	−14.5184	−	45.8647i	−0.467363	−	1.47644i
$966$		0			0
$967$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$967$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$968$	22.0000	+	22.0000i	0.707107	+	0.707107i
$969$		0			0
$970$	7.20571	+	47.8429i	0.231361	+	1.53614i
$971$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−62.3442			−1.99559
$977$	27.2327	+	53.4472i	0.871252	+	1.70993i	0.686483	+	0.727145i	$0.259153\pi$
$977$	27.2327	+	53.4472i	0.871252	+	1.70993i	0.184768	+	0.982782i	$0.440847\pi$
$978$		0			0
$979$		0			0
$980$	−14.0000	−	28.0000i	−0.447214	−	0.894427i
$981$		0			0
$982$		0			0
$983$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$983$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$984$		0			0
$985$	−58.0020	−	19.3340i	−1.84810	−	0.616032i
$986$	−24.2123	+	74.5177i	−0.771075	+	2.37313i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$991$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	34.9201	−	5.53079i	1.10593	−	0.175162i	0.423345	−	0.905969i	$-0.360856\pi$
$997$	34.9201	−	5.53079i	1.10593	−	0.175162i	0.682585	+	0.730807i	$0.260856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.2.bj.c.703.1	yes	8
3.2	odd	2		900.2.bj.b.703.1	yes	8
4.3	odd	2	CM	900.2.bj.c.703.1	yes	8
12.11	even	2		900.2.bj.b.703.1	yes	8
25.17	odd	20	inner	900.2.bj.c.667.1	yes	8
75.17	even	20		900.2.bj.b.667.1	✓	8
100.67	even	20	inner	900.2.bj.c.667.1	yes	8
300.167	odd	20		900.2.bj.b.667.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.bj.b.667.1	✓	8	75.17	even	20
900.2.bj.b.667.1	✓	8	300.167	odd	20
900.2.bj.b.703.1	yes	8	3.2	odd	2
900.2.bj.b.703.1	yes	8	12.11	even	2
900.2.bj.c.667.1	yes	8	25.17	odd	20	inner
900.2.bj.c.667.1	yes	8	100.67	even	20	inner
900.2.bj.c.703.1	yes	8	1.1	even	1	trivial
900.2.bj.c.703.1	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bj (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.642040 + 1.26007i) q^{2} +(-1.17557 + 1.61803i) q^{4} +(-1.98459 - 1.03025i) q^{5} +(-2.79360 - 0.442463i) q^{8} +(0.0240055 - 3.16219i) q^{10} +(2.21972 - 4.35645i) q^{13} +(-1.23607 - 3.80423i) q^{16} +(1.25711 - 7.93710i) q^{17} +(4.00000 - 2.00000i) q^{20} +(2.87718 + 4.08924i) q^{25} +6.91460 q^{26} +(-4.05242 + 5.57768i) q^{29} +(4.00000 - 4.00000i) q^{32} +(10.8084 - 3.51188i) q^{34} +(10.3357 + 5.26632i) q^{37} +(5.08831 + 3.75621i) q^{40} +(-3.95309 - 12.1663i) q^{41} -7.00000i q^{49} +(-3.30548 + 6.25090i) q^{50} +(4.43945 + 8.71290i) q^{52} +(-0.960638 - 6.06523i) q^{53} +(-9.63011 - 1.52526i) q^{58} +(4.81636 - 14.8232i) q^{61} +(7.60845 + 2.47214i) q^{64} +(-8.89346 + 6.35889i) q^{65} +(11.3647 + 11.3647i) q^{68} +(-3.05007 + 1.55409i) q^{73} +16.4050i q^{74} +(-1.46621 + 8.82328i) q^{80} +(12.7925 - 12.7925i) q^{82} +(-10.6720 + 14.4567i) q^{85} +(-11.5332 - 3.74737i) q^{89} +(-15.1115 + 2.39343i) q^{97} +(8.82051 - 4.49428i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 2 q^{5} - 4 q^{8} + 6 q^{10} - 10 q^{13} + 8 q^{16} + 6 q^{17} + 32 q^{20} + 6 q^{25} + 20 q^{26} + 32 q^{32} + 50 q^{34} + 50 q^{37} + 4 q^{40} - 20 q^{41} + 2 q^{50} - 20 q^{52} - 52 q^{53}+ \cdots - 14 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 - 2 * q^5 - 4 * q^8 + 6 * q^10 - 10 * q^13 + 8 * q^16 + 6 * q^17 + 32 * q^20 + 6 * q^25 + 20 * q^26 + 32 * q^32 + 50 * q^34 + 50 * q^37 + 4 * q^40 - 20 * q^41 + 2 * q^50 - 20 * q^52 - 52 * q^53 - 20 * q^58 + 24 * q^61 - 12 * q^68 + 10 * q^73 + 8 * q^80 + 20 * q^82 - 52 * q^85 - 80 * q^89 + 10 * q^97 - 14 * q^98

Introduction

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