LMFDB - Embedded newform 1260.2.c.a.811.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(811,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1260.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1260.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.0611506547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1260.811
Dual form			1260.2.c.a.811.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +1.00000i q^{5} +(1.73205 - 2.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +(0.366025 + 1.36603i) q^{10} +3.73205i q^{11} +6.46410i q^{13} +(1.63397 - 3.36603i) q^{14} +(2.00000 - 3.46410i) q^{16} -0.464102i q^{17} +6.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(1.36603 + 5.09808i) q^{22} -5.46410i q^{23} -1.00000 q^{25} +(2.36603 + 8.83013i) q^{26} +(1.00000 - 5.19615i) q^{28} +5.92820 q^{29} -6.00000 q^{31} +(1.46410 - 5.46410i) q^{32} +(-0.169873 - 0.633975i) q^{34} +(2.00000 + 1.73205i) q^{35} -2.53590 q^{37} +(8.19615 - 2.19615i) q^{38} +(2.00000 + 2.00000i) q^{40} -3.46410i q^{41} -2.00000i q^{43} +(3.73205 + 6.46410i) q^{44} +(-2.00000 - 7.46410i) q^{46} +1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} +(-1.36603 + 0.366025i) q^{50} +(6.46410 + 11.1962i) q^{52} -2.00000 q^{53} -3.73205 q^{55} +(-0.535898 - 7.46410i) q^{56} +(8.09808 - 2.16987i) q^{58} -3.46410 q^{59} +2.53590i q^{61} +(-8.19615 + 2.19615i) q^{62} -8.00000i q^{64} -6.46410 q^{65} -3.46410i q^{67} +(-0.464102 - 0.803848i) q^{68} +(3.36603 + 1.63397i) q^{70} +0.535898i q^{71} -0.928203i q^{73} +(-3.46410 + 0.928203i) q^{74} +(10.3923 - 6.00000i) q^{76} +(7.46410 + 6.46410i) q^{77} +2.66025i q^{79} +(3.46410 + 2.00000i) q^{80} +(-1.26795 - 4.73205i) q^{82} -8.53590 q^{83} +0.464102 q^{85} +(-0.732051 - 2.73205i) q^{86} +(7.46410 + 7.46410i) q^{88} -9.46410i q^{89} +(12.9282 + 11.1962i) q^{91} +(-5.46410 - 9.46410i) q^{92} +(2.36603 - 0.633975i) q^{94} +6.00000i q^{95} +7.39230i q^{97} +(-3.90192 - 9.09808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{8} - 2 q^{10} + 10 q^{14} + 8 q^{16} + 24 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{25} + 6 q^{26} + 4 q^{28} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 18 q^{34} + 8 q^{35} - 24 q^{37} + 12 q^{38}+ \cdots - 26 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^8 - 2 * q^10 + 10 * q^14 + 8 * q^16 + 24 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^25 + 6 * q^26 + 4 * q^28 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 18 * q^34 + 8 * q^35 - 24 * q^37 + 12 * q^38 + 8 * q^40 + 8 * q^44 - 8 * q^46 - 4 * q^49 - 2 * q^50 + 12 * q^52 - 8 * q^53 - 8 * q^55 - 16 * q^56 + 22 * q^58 - 12 * q^62 - 12 * q^65 + 12 * q^68 + 10 * q^70 + 16 * q^77 - 12 * q^82 - 48 * q^83 - 12 * q^85 + 4 * q^86 + 16 * q^88 + 24 * q^91 - 8 * q^92 + 6 * q^94 - 26 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$3$		0			0
$3$		0			0
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$		0			0
$10$	0.366025	+	1.36603i	0.115747	+	0.431975i
$11$			3.73205i			1.12526i	0.826710	+	0.562628i	$0.190210\pi$
$11$			3.73205i			1.12526i	−0.826710	+	0.562628i	$0.809790\pi$
$12$		0			0
$13$			6.46410i			1.79282i	0.443227	+	0.896410i	$0.353834\pi$
$13$			6.46410i			1.79282i	−0.443227	+	0.896410i	$0.646166\pi$
$14$	1.63397	−	3.36603i	0.436698	−	0.899608i
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$		−	0.464102i		−	0.112561i	−0.998415	−	0.0562806i	$-0.982076\pi$
$17$		−	0.464102i		−	0.112561i	0.998415	−	0.0562806i	$-0.0179241\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$	1.00000	+	1.73205i	0.223607	+	0.387298i
$21$		0			0
$22$	1.36603	+	5.09808i	0.291238	+	1.08691i
$23$		−	5.46410i		−	1.13934i	−0.821872	−	0.569672i	$-0.807070\pi$
$23$		−	5.46410i		−	1.13934i	0.821872	−	0.569672i	$-0.192930\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	2.36603	+	8.83013i	0.464016	+	1.73173i
$27$		0			0
$28$	1.00000	−	5.19615i	0.188982	−	0.981981i
$29$	5.92820			1.10084			0.550420	−	0.834888i	$-0.314468\pi$
$29$	5.92820			1.10084			0.550420	+	0.834888i	$0.314468\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.46410	−	5.46410i	0.258819	−	0.965926i
$33$		0			0
$34$	−0.169873	−	0.633975i	−0.0291330	−	0.108726i
$35$	2.00000	+	1.73205i	0.338062	+	0.292770i
$36$		0			0
$37$	−2.53590			−0.416899			−0.208450	−	0.978033i	$-0.566842\pi$
$37$	−2.53590			−0.416899			−0.208450	+	0.978033i	$0.566842\pi$
$38$	8.19615	−	2.19615i	1.32959	−	0.356263i
$39$		0			0
$40$	2.00000	+	2.00000i	0.316228	+	0.316228i
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$		0			0
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	3.73205	+	6.46410i	0.562628	+	0.974500i
$45$		0			0
$46$	−2.00000	−	7.46410i	−0.294884	−	1.10052i
$47$	1.73205			0.252646			0.126323	−	0.991989i	$-0.459682\pi$
$47$	1.73205			0.252646			0.126323	+	0.991989i	$0.459682\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$	−1.36603	+	0.366025i	−0.193185	+	0.0517638i
$51$		0			0
$52$	6.46410	+	11.1962i	0.896410	+	1.55263i
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−3.73205			−0.503230
$56$	−0.535898	−	7.46410i	−0.0716124	−	0.997433i
$57$		0			0
$58$	8.09808	−	2.16987i	1.06333	−	0.284918i
$59$	−3.46410			−0.450988			−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410			−0.450988			−0.225494	+	0.974245i	$0.572400\pi$
$60$		0			0
$61$			2.53590i			0.324689i	0.986734	+	0.162344i	$0.0519055\pi$
$61$			2.53590i			0.324689i	−0.986734	+	0.162344i	$0.948094\pi$
$62$	−8.19615	+	2.19615i	−1.04091	+	0.278912i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	−6.46410			−0.801773
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$	−0.464102	−	0.803848i	−0.0562806	−	0.0974808i
$69$		0			0
$70$	3.36603	+	1.63397i	0.402317	+	0.195297i
$71$			0.535898i			0.0635994i	0.999494	+	0.0317997i	$0.0101239\pi$
$71$			0.535898i			0.0635994i	−0.999494	+	0.0317997i	$0.989876\pi$
$72$		0			0
$73$		−	0.928203i		−	0.108638i	−0.998524	−	0.0543190i	$-0.982701\pi$
$73$		−	0.928203i		−	0.108638i	0.998524	−	0.0543190i	$-0.0172988\pi$
$74$	−3.46410	+	0.928203i	−0.402694	+	0.107901i
$75$		0			0
$76$	10.3923	−	6.00000i	1.19208	−	0.688247i
$77$	7.46410	+	6.46410i	0.850613	+	0.736653i
$78$		0			0
$79$			2.66025i			0.299302i	0.988739	+	0.149651i	$0.0478150\pi$
$79$			2.66025i			0.299302i	−0.988739	+	0.149651i	$0.952185\pi$
$80$	3.46410	+	2.00000i	0.387298	+	0.223607i
$81$		0			0
$82$	−1.26795	−	4.73205i	−0.140022	−	0.522568i
$83$	−8.53590			−0.936937			−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590			−0.936937			−0.468468	+	0.883480i	$0.655194\pi$
$84$		0			0
$85$	0.464102			0.0503389
$86$	−0.732051	−	2.73205i	−0.0789391	−	0.294605i
$87$		0			0
$88$	7.46410	+	7.46410i	0.795676	+	0.795676i
$89$		−	9.46410i		−	1.00319i	−0.865102	−	0.501596i	$-0.832746\pi$
$89$		−	9.46410i		−	1.00319i	0.865102	−	0.501596i	$-0.167254\pi$
$90$		0			0
$91$	12.9282	+	11.1962i	1.35524	+	1.17368i
$92$	−5.46410	−	9.46410i	−0.569672	−	0.986701i
$93$		0			0
$94$	2.36603	−	0.633975i	0.244037	−	0.0653895i
$95$			6.00000i			0.615587i
$96$		0			0
$97$			7.39230i			0.750575i	0.926908	+	0.375287i	$0.122456\pi$
$97$			7.39230i			0.750575i	−0.926908	+	0.375287i	$0.877544\pi$
$98$	−3.90192	−	9.09808i	−0.394154	−	0.919044i
$99$		0			0
$100$	−1.73205	+	1.00000i	−0.173205	+	0.100000i
$101$			8.53590i			0.849354i	0.905345	+	0.424677i	$0.139612\pi$
$101$			8.53590i			0.849354i	−0.905345	+	0.424677i	$0.860388\pi$
$102$		0			0
$103$	−17.1962			−1.69439			−0.847194	−	0.531284i	$-0.821710\pi$
$103$	−17.1962			−1.69439			−0.847194	+	0.531284i	$0.821710\pi$
$104$	12.9282	+	12.9282i	1.26771	+	1.26771i
$105$		0			0
$106$	−2.73205	+	0.732051i	−0.265360	+	0.0711031i
$107$			18.3923i			1.77805i	0.457857	+	0.889026i	$0.348617\pi$
$107$			18.3923i			1.77805i	−0.457857	+	0.889026i	$0.651383\pi$
$108$		0			0
$109$	15.9282			1.52565			0.762823	−	0.646608i	$-0.223813\pi$
$109$	15.9282			1.52565			0.762823	+	0.646608i	$0.223813\pi$
$110$	−5.09808	+	1.36603i	−0.486082	+	0.130245i
$111$		0			0
$112$	−3.46410	−	10.0000i	−0.327327	−	0.944911i
$113$	1.46410			0.137731			0.0688655	−	0.997626i	$-0.478062\pi$
$113$	1.46410			0.137731			0.0688655	+	0.997626i	$0.478062\pi$
$114$		0			0
$115$	5.46410			0.509530
$116$	10.2679	−	5.92820i	0.953355	−	0.550420i
$117$		0			0
$118$	−4.73205	+	1.26795i	−0.435621	+	0.116724i
$119$	−0.928203	−	0.803848i	−0.0850883	−	0.0736886i
$120$		0			0
$121$	−2.92820			−0.266200
$122$	0.928203	+	3.46410i	0.0840356	+	0.313625i
$123$		0			0
$124$	−10.3923	+	6.00000i	−0.933257	+	0.538816i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			8.53590i			0.757438i	0.925512	+	0.378719i	$0.123635\pi$
$127$			8.53590i			0.757438i	−0.925512	+	0.378719i	$0.876365\pi$
$128$	−2.92820	−	10.9282i	−0.258819	−	0.965926i
$129$		0			0
$130$	−8.83013	+	2.36603i	−0.774453	+	0.207514i
$131$	−9.46410			−0.826882			−0.413441	−	0.910531i	$-0.635673\pi$
$131$	−9.46410			−0.826882			−0.413441	+	0.910531i	$0.635673\pi$
$132$		0			0
$133$	10.3923	−	12.0000i	0.901127	−	1.04053i
$134$	−1.26795	−	4.73205i	−0.109534	−	0.408787i
$135$		0			0
$136$	−0.928203	−	0.928203i	−0.0795928	−	0.0795928i
$137$	0.392305			0.0335169			0.0167584	−	0.999860i	$-0.494665\pi$
$137$	0.392305			0.0335169			0.0167584	+	0.999860i	$0.494665\pi$
$138$		0			0
$139$	−6.92820			−0.587643			−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820			−0.587643			−0.293821	+	0.955860i	$0.594927\pi$
$140$	5.19615	+	1.00000i	0.439155	+	0.0845154i
$141$		0			0
$142$	0.196152	+	0.732051i	0.0164607	+	0.0614323i
$143$	−24.1244			−2.01738
$144$		0			0
$145$			5.92820i			0.492310i
$146$	−0.339746	−	1.26795i	−0.0281176	−	0.104936i
$147$		0			0
$148$	−4.39230	+	2.53590i	−0.361045	+	0.208450i
$149$	−16.9282			−1.38681			−0.693406	−	0.720547i	$-0.743891\pi$
$149$	−16.9282			−1.38681			−0.693406	+	0.720547i	$0.743891\pi$
$150$		0			0
$151$		−	4.80385i		−	0.390932i	−0.980711	−	0.195466i	$-0.937378\pi$
$151$		−	4.80385i		−	0.390932i	0.980711	−	0.195466i	$-0.0626219\pi$
$152$	12.0000	−	12.0000i	0.973329	−	0.973329i
$153$		0			0
$154$	12.5622	+	6.09808i	1.01229	+	0.491397i
$155$		−	6.00000i		−	0.481932i
$156$		0			0
$157$		−	19.8564i		−	1.58471i	−0.610058	−	0.792357i	$-0.708854\pi$
$157$		−	19.8564i		−	1.58471i	0.610058	−	0.792357i	$-0.291146\pi$
$158$	0.973721	+	3.63397i	0.0774650	+	0.289103i
$159$		0			0
$160$	5.46410	+	1.46410i	0.431975	+	0.115747i
$161$	−10.9282	−	9.46410i	−0.861263	−	0.745876i
$162$		0			0
$163$		−	20.7846i		−	1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$		−	20.7846i		−	1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$	−3.46410	−	6.00000i	−0.270501	−	0.468521i
$165$		0			0
$166$	−11.6603	+	3.12436i	−0.905011	+	0.242497i
$167$	−5.19615			−0.402090			−0.201045	−	0.979582i	$-0.564434\pi$
$167$	−5.19615			−0.402090			−0.201045	+	0.979582i	$0.564434\pi$
$168$		0			0
$169$	−28.7846			−2.21420
$170$	0.633975	−	0.169873i	0.0486236	−	0.0130287i
$171$		0			0
$172$	−2.00000	−	3.46410i	−0.152499	−	0.264135i
$173$		−	20.3205i		−	1.54494i	−0.635051	−	0.772470i	$-0.719021\pi$
$173$		−	20.3205i		−	1.54494i	0.635051	−	0.772470i	$-0.280979\pi$
$174$		0			0
$175$	−1.73205	+	2.00000i	−0.130931	+	0.151186i
$176$	12.9282	+	7.46410i	0.974500	+	0.562628i
$177$		0			0
$178$	−3.46410	−	12.9282i	−0.259645	−	0.969010i
$179$			14.3923i			1.07573i	0.843031	+	0.537866i	$0.180769\pi$
$179$			14.3923i			1.07573i	−0.843031	+	0.537866i	$0.819231\pi$
$180$		0			0
$181$			12.9282i			0.960946i	0.877010	+	0.480473i	$0.159535\pi$
$181$			12.9282i			0.960946i	−0.877010	+	0.480473i	$0.840465\pi$
$182$	21.7583	+	10.5622i	1.61283	+	0.782921i
$183$		0			0
$184$	−10.9282	−	10.9282i	−0.805638	−	0.805638i
$185$		−	2.53590i		−	0.186443i
$186$		0			0
$187$	1.73205			0.126660
$188$	3.00000	−	1.73205i	0.218797	−	0.126323i
$189$		0			0
$190$	2.19615	+	8.19615i	0.159326	+	0.594611i
$191$		−	3.19615i		−	0.231265i	−0.993292	−	0.115633i	$-0.963110\pi$
$191$		−	3.19615i		−	0.231265i	0.993292	−	0.115633i	$-0.0368896\pi$
$192$		0			0
$193$	−2.53590			−0.182538			−0.0912690	−	0.995826i	$-0.529092\pi$
$193$	−2.53590			−0.182538			−0.0912690	+	0.995826i	$0.529092\pi$
$194$	2.70577	+	10.0981i	0.194263	+	0.725000i
$195$		0			0
$196$	−8.66025	−	11.0000i	−0.618590	−	0.785714i
$197$	−21.3205			−1.51902			−0.759512	−	0.650494i	$-0.774562\pi$
$197$	−21.3205			−1.51902			−0.759512	+	0.650494i	$0.774562\pi$
$198$		0			0
$199$	−3.46410			−0.245564			−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410			−0.245564			−0.122782	+	0.992434i	$0.539182\pi$
$200$	−2.00000	+	2.00000i	−0.141421	+	0.141421i
$201$		0			0
$202$	3.12436	+	11.6603i	0.219829	+	0.820413i
$203$	10.2679	−	11.8564i	0.720669	−	0.832157i
$204$		0			0
$205$	3.46410			0.241943
$206$	−23.4904	+	6.29423i	−1.63665	+	0.438540i
$207$		0			0
$208$	22.3923	+	12.9282i	1.55263	+	0.896410i
$209$			22.3923i			1.54891i
$210$		0			0
$211$			7.19615i			0.495404i	0.968836	+	0.247702i	$0.0796753\pi$
$211$			7.19615i			0.495404i	−0.968836	+	0.247702i	$0.920325\pi$
$212$	−3.46410	+	2.00000i	−0.237915	+	0.137361i
$213$		0			0
$214$	6.73205	+	25.1244i	0.460194	+	1.71747i
$215$	2.00000			0.136399
$216$		0			0
$217$	−10.3923	+	12.0000i	−0.705476	+	0.814613i
$218$	21.7583	−	5.83013i	1.47366	−	0.394866i
$219$		0			0
$220$	−6.46410	+	3.73205i	−0.435810	+	0.251615i
$221$	3.00000			0.201802
$222$		0			0
$223$	−10.2679			−0.687593			−0.343796	−	0.939044i	$-0.611713\pi$
$223$	−10.2679			−0.687593			−0.343796	+	0.939044i	$0.611713\pi$
$224$	−8.39230	−	12.3923i	−0.560734	−	0.827996i
$225$		0			0
$226$	2.00000	−	0.535898i	0.133038	−	0.0356474i
$227$	−3.33975			−0.221667			−0.110833	−	0.993839i	$-0.535352\pi$
$227$	−3.33975			−0.221667			−0.110833	+	0.993839i	$0.535352\pi$
$228$		0			0
$229$		−	15.4641i		−	1.02190i	−0.859611	−	0.510948i	$-0.829294\pi$
$229$		−	15.4641i		−	1.02190i	0.859611	−	0.510948i	$-0.170706\pi$
$230$	7.46410	−	2.00000i	0.492168	−	0.131876i
$231$		0			0
$232$	11.8564	−	11.8564i	0.778411	−	0.778411i
$233$	22.9282			1.50208			0.751038	−	0.660259i	$-0.229553\pi$
$233$	22.9282			1.50208			0.751038	+	0.660259i	$0.229553\pi$
$234$		0			0
$235$			1.73205i			0.112987i
$236$	−6.00000	+	3.46410i	−0.390567	+	0.225494i
$237$		0			0
$238$	−1.56218	−	0.758330i	−0.101261	−	0.0491552i
$239$			27.9808i			1.80993i	0.425491	+	0.904963i	$0.360101\pi$
$239$			27.9808i			1.80993i	−0.425491	+	0.904963i	$0.639899\pi$
$240$		0			0
$241$		−	4.39230i		−	0.282933i	−0.989943	−	0.141467i	$-0.954818\pi$
$241$		−	4.39230i		−	0.282933i	0.989943	−	0.141467i	$-0.0451818\pi$
$242$	−4.00000	+	1.07180i	−0.257130	+	0.0688977i
$243$		0			0
$244$	2.53590	+	4.39230i	0.162344	+	0.281189i
$245$	6.92820	−	1.00000i	0.442627	−	0.0638877i
$246$		0			0
$247$			38.7846i			2.46781i
$248$	−12.0000	+	12.0000i	−0.762001	+	0.762001i
$249$		0			0
$250$	−0.366025	−	1.36603i	−0.0231495	−	0.0863950i
$251$	−1.85641			−0.117175			−0.0585877	−	0.998282i	$-0.518660\pi$
$251$	−1.85641			−0.117175			−0.0585877	+	0.998282i	$0.518660\pi$
$252$		0			0
$253$	20.3923			1.28205
$254$	3.12436	+	11.6603i	0.196040	+	0.731629i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$	−4.39230	+	5.07180i	−0.272925	+	0.315146i
$260$	−11.1962	+	6.46410i	−0.694356	+	0.400887i
$261$		0			0
$262$	−12.9282	+	3.46410i	−0.798707	+	0.214013i
$263$		−	4.53590i		−	0.279695i	−0.990173	−	0.139848i	$-0.955339\pi$
$263$		−	4.53590i		−	0.279695i	0.990173	−	0.139848i	$-0.0446613\pi$
$264$		0			0
$265$		−	2.00000i		−	0.122859i
$266$	9.80385	−	20.1962i	0.601112	−	1.23831i
$267$		0			0
$268$	−3.46410	−	6.00000i	−0.211604	−	0.366508i
$269$			12.0000i			0.731653i	0.930683	+	0.365826i	$0.119214\pi$
$269$			12.0000i			0.731653i	−0.930683	+	0.365826i	$0.880786\pi$
$270$		0			0
$271$	2.53590			0.154045			0.0770224	−	0.997029i	$-0.475459\pi$
$271$	2.53590			0.154045			0.0770224	+	0.997029i	$0.475459\pi$
$272$	−1.60770	−	0.928203i	−0.0974808	−	0.0562806i
$273$		0			0
$274$	0.535898	−	0.143594i	0.0323748	−	0.00867480i
$275$		−	3.73205i		−	0.225051i
$276$		0			0
$277$	−24.7846			−1.48916			−0.744581	−	0.667532i	$-0.767351\pi$
$277$	−24.7846			−1.48916			−0.744581	+	0.667532i	$0.767351\pi$
$278$	−9.46410	+	2.53590i	−0.567619	+	0.152093i
$279$		0			0
$280$	7.46410	−	0.535898i	0.446065	−	0.0320261i
$281$	−5.92820			−0.353647			−0.176823	−	0.984243i	$-0.556582\pi$
$281$	−5.92820			−0.353647			−0.176823	+	0.984243i	$0.556582\pi$
$282$		0			0
$283$	−12.1244			−0.720718			−0.360359	−	0.932814i	$-0.617346\pi$
$283$	−12.1244			−0.720718			−0.360359	+	0.932814i	$0.617346\pi$
$284$	0.535898	+	0.928203i	0.0317997	+	0.0550787i
$285$		0			0
$286$	−32.9545	+	8.83013i	−1.94864	+	0.522136i
$287$	−6.92820	−	6.00000i	−0.408959	−	0.354169i
$288$		0			0
$289$	16.7846			0.987330
$290$	2.16987	+	8.09808i	0.127419	+	0.475535i
$291$		0			0
$292$	−0.928203	−	1.60770i	−0.0543190	−	0.0940832i
$293$			14.3205i			0.836613i	0.908306	+	0.418307i	$0.137376\pi$
$293$			14.3205i			0.836613i	−0.908306	+	0.418307i	$0.862624\pi$
$294$		0			0
$295$		−	3.46410i		−	0.201688i
$296$	−5.07180	+	5.07180i	−0.294792	+	0.294792i
$297$		0			0
$298$	−23.1244	+	6.19615i	−1.33956	+	0.358933i
$299$	35.3205			2.04264
$300$		0			0
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$	−1.75833	−	6.56218i	−0.101181	−	0.377611i
$303$		0			0
$304$	12.0000	−	20.7846i	0.688247	−	1.19208i
$305$	−2.53590			−0.145205
$306$		0			0
$307$	−1.73205			−0.0988534			−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	−1.73205			−0.0988534			−0.0494267	+	0.998778i	$0.515739\pi$
$308$	19.3923	+	3.73205i	1.10498	+	0.212653i
$309$		0			0
$310$	−2.19615	−	8.19615i	−0.124733	−	0.465510i
$311$	19.8564			1.12595			0.562977	−	0.826473i	$-0.309656\pi$
$311$	19.8564			1.12595			0.562977	+	0.826473i	$0.309656\pi$
$312$		0			0
$313$		−	24.4641i		−	1.38279i	−0.722476	−	0.691396i	$-0.756996\pi$
$313$		−	24.4641i		−	1.38279i	0.722476	−	0.691396i	$-0.243004\pi$
$314$	−7.26795	−	27.1244i	−0.410154	−	1.53072i
$315$		0			0
$316$	2.66025	+	4.60770i	0.149651	+	0.259203i
$317$	16.9282			0.950783			0.475391	−	0.879774i	$-0.342306\pi$
$317$	16.9282			0.950783			0.475391	+	0.879774i	$0.342306\pi$
$318$		0			0
$319$			22.1244i			1.23873i
$320$	8.00000			0.447214
$321$		0			0
$322$	−18.3923	−	8.92820i	−1.02496	−	0.497549i
$323$		−	2.78461i		−	0.154940i
$324$		0			0
$325$		−	6.46410i		−	0.358564i
$326$	−7.60770	−	28.3923i	−0.421351	−	1.57250i
$327$		0			0
$328$	−6.92820	−	6.92820i	−0.382546	−	0.382546i
$329$	3.00000	−	3.46410i	0.165395	−	0.190982i
$330$		0			0
$331$			5.60770i			0.308227i	0.988053	+	0.154113i	$0.0492521\pi$
$331$			5.60770i			0.308227i	−0.988053	+	0.154113i	$0.950748\pi$
$332$	−14.7846	+	8.53590i	−0.811411	+	0.468468i
$333$		0			0
$334$	−7.09808	+	1.90192i	−0.388389	+	0.104069i
$335$	3.46410			0.189264
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−39.3205	+	10.5359i	−2.13875	+	0.573077i
$339$		0			0
$340$	0.803848	−	0.464102i	0.0435948	−	0.0251694i
$341$		−	22.3923i		−	1.21261i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$	−4.00000	−	4.00000i	−0.215666	−	0.215666i
$345$		0			0
$346$	−7.43782	−	27.7583i	−0.399860	−	1.49230i
$347$		−	14.2487i		−	0.764911i	−0.923974	−	0.382455i	$-0.875079\pi$
$347$		−	14.2487i		−	0.764911i	0.923974	−	0.382455i	$-0.124921\pi$
$348$		0			0
$349$			29.3205i			1.56949i	0.619818	+	0.784745i	$0.287206\pi$
$349$			29.3205i			1.56949i	−0.619818	+	0.784745i	$0.712794\pi$
$350$	−1.63397	+	3.36603i	−0.0873396	+	0.179922i
$351$		0			0
$352$	20.3923	+	5.46410i	1.08691	+	0.291238i
$353$		−	13.3923i		−	0.712800i	−0.934333	−	0.356400i	$-0.884004\pi$
$353$		−	13.3923i		−	0.712800i	0.934333	−	0.356400i	$-0.115996\pi$
$354$		0			0
$355$	−0.535898			−0.0284425
$356$	−9.46410	−	16.3923i	−0.501596	−	0.868790i
$357$		0			0
$358$	5.26795	+	19.6603i	0.278420	+	1.03908i
$359$		−	9.32051i		−	0.491918i	−0.969280	−	0.245959i	$-0.920897\pi$
$359$		−	9.32051i		−	0.491918i	0.969280	−	0.245959i	$-0.0791028\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	4.73205	+	17.6603i	0.248711	+	0.928202i
$363$		0			0
$364$	33.5885	+	6.46410i	1.76051	+	0.338811i
$365$	0.928203			0.0485844
$366$		0			0
$367$	36.1244			1.88568			0.942838	−	0.333251i	$-0.108146\pi$
$367$	36.1244			1.88568			0.942838	+	0.333251i	$0.108146\pi$
$368$	−18.9282	−	10.9282i	−0.986701	−	0.569672i
$369$		0			0
$370$	−0.928203	−	3.46410i	−0.0482550	−	0.180090i
$371$	−3.46410	+	4.00000i	−0.179847	+	0.207670i
$372$		0			0
$373$	−24.3923			−1.26299			−0.631493	−	0.775382i	$-0.717557\pi$
$373$	−24.3923			−1.26299			−0.631493	+	0.775382i	$0.717557\pi$
$374$	2.36603	−	0.633975i	0.122344	−	0.0327820i
$375$		0			0
$376$	3.46410	−	3.46410i	0.178647	−	0.178647i
$377$			38.3205i			1.97361i
$378$		0			0
$379$		−	26.3923i		−	1.35568i	−0.735209	−	0.677841i	$-0.762916\pi$
$379$		−	26.3923i		−	1.35568i	0.735209	−	0.677841i	$-0.237084\pi$
$380$	6.00000	+	10.3923i	0.307794	+	0.533114i
$381$		0			0
$382$	−1.16987	−	4.36603i	−0.0598559	−	0.223385i
$383$	−20.5359			−1.04934			−0.524668	−	0.851307i	$-0.675810\pi$
$383$	−20.5359			−1.04934			−0.524668	+	0.851307i	$0.675810\pi$
$384$		0			0
$385$	−6.46410	+	7.46410i	−0.329441	+	0.380406i
$386$	−3.46410	+	0.928203i	−0.176318	+	0.0472443i
$387$		0			0
$388$	7.39230	+	12.8038i	0.375287	+	0.650017i
$389$	−6.85641			−0.347634			−0.173817	−	0.984778i	$-0.555610\pi$
$389$	−6.85641			−0.347634			−0.173817	+	0.984778i	$0.555610\pi$
$390$		0			0
$391$	−2.53590			−0.128246
$392$	−15.8564	−	11.8564i	−0.800869	−	0.598839i
$393$		0			0
$394$	−29.1244	+	7.80385i	−1.46726	+	0.393152i
$395$	−2.66025			−0.133852
$396$		0			0
$397$			5.53590i			0.277839i	0.990304	+	0.138919i	$0.0443629\pi$
$397$			5.53590i			0.277839i	−0.990304	+	0.138919i	$0.955637\pi$
$398$	−4.73205	+	1.26795i	−0.237196	+	0.0635566i
$399$		0			0
$400$	−2.00000	+	3.46410i	−0.100000	+	0.173205i
$401$	23.9282			1.19492			0.597459	−	0.801900i	$-0.296177\pi$
$401$	23.9282			1.19492			0.597459	+	0.801900i	$0.296177\pi$
$402$		0			0
$403$		−	38.7846i		−	1.93200i
$404$	8.53590	+	14.7846i	0.424677	+	0.735562i
$405$		0			0
$406$	9.68653	−	19.9545i	0.480735	−	0.990324i
$407$		−	9.46410i		−	0.469118i
$408$		0			0
$409$			31.8564i			1.57520i	0.616188	+	0.787599i	$0.288676\pi$
$409$			31.8564i			1.57520i	−0.616188	+	0.787599i	$0.711324\pi$
$410$	4.73205	−	1.26795i	0.233699	−	0.0626195i
$411$		0			0
$412$	−29.7846	+	17.1962i	−1.46738	+	0.847194i
$413$	−6.00000	+	6.92820i	−0.295241	+	0.340915i
$414$		0			0
$415$		−	8.53590i		−	0.419011i
$416$	35.3205	+	9.46410i	1.73173	+	0.464016i
$417$		0			0
$418$	8.19615	+	30.5885i	0.400887	+	1.49613i
$419$	−24.2487			−1.18463			−0.592314	−	0.805708i	$-0.701785\pi$
$419$	−24.2487			−1.18463			−0.592314	+	0.805708i	$0.701785\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$	2.63397	+	9.83013i	0.128220	+	0.478523i
$423$		0			0
$424$	−4.00000	+	4.00000i	−0.194257	+	0.194257i
$425$			0.464102i			0.0225122i
$426$		0			0
$427$	5.07180	+	4.39230i	0.245441	+	0.212559i
$428$	18.3923	+	31.8564i	0.889026	+	1.53984i
$429$		0			0
$430$	2.73205	−	0.732051i	0.131751	−	0.0353026i
$431$			17.5885i			0.847206i	0.905848	+	0.423603i	$0.139235\pi$
$431$			17.5885i			0.847206i	−0.905848	+	0.423603i	$0.860765\pi$
$432$		0			0
$433$		−	4.14359i		−	0.199128i	−0.995031	−	0.0995642i	$-0.968255\pi$
$433$		−	4.14359i		−	0.199128i	0.995031	−	0.0995642i	$-0.0317449\pi$
$434$	−9.80385	+	20.1962i	−0.470600	+	0.969446i
$435$		0			0
$436$	27.5885	−	15.9282i	1.32125	−	0.762823i
$437$		−	32.7846i		−	1.56830i
$438$		0			0
$439$	15.7128			0.749932			0.374966	−	0.927039i	$-0.377654\pi$
$439$	15.7128			0.749932			0.374966	+	0.927039i	$0.377654\pi$
$440$	−7.46410	+	7.46410i	−0.355837	+	0.355837i
$441$		0			0
$442$	4.09808	−	1.09808i	0.194926	−	0.0522302i
$443$		−	26.0000i		−	1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$		−	26.0000i		−	1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$		0			0
$445$	9.46410			0.448641
$446$	−14.0263	+	3.75833i	−0.664164	+	0.177962i
$447$		0			0
$448$	−16.0000	−	13.8564i	−0.755929	−	0.654654i
$449$	1.92820			0.0909975			0.0454988	−	0.998964i	$-0.485512\pi$
$449$	1.92820			0.0909975			0.0454988	+	0.998964i	$0.485512\pi$
$450$		0			0
$451$	12.9282			0.608765
$452$	2.53590	−	1.46410i	0.119279	−	0.0688655i
$453$		0			0
$454$	−4.56218	+	1.22243i	−0.214114	+	0.0573716i
$455$	−11.1962	+	12.9282i	−0.524884	+	0.606084i
$456$		0			0
$457$	27.4641			1.28472			0.642358	−	0.766405i	$-0.277956\pi$
$457$	27.4641			1.28472			0.642358	+	0.766405i	$0.277956\pi$
$458$	−5.66025	−	21.1244i	−0.264486	−	0.987076i
$459$		0			0
$460$	9.46410	−	5.46410i	0.441266	−	0.254765i
$461$			27.7128i			1.29071i	0.763881	+	0.645357i	$0.223291\pi$
$461$			27.7128i			1.29071i	−0.763881	+	0.645357i	$0.776709\pi$
$462$		0			0
$463$		−	4.39230i		−	0.204128i	−0.994778	−	0.102064i	$-0.967455\pi$
$463$		−	4.39230i		−	0.204128i	0.994778	−	0.102064i	$-0.0325446\pi$
$464$	11.8564	−	20.5359i	0.550420	−	0.953355i
$465$		0			0
$466$	31.3205	−	8.39230i	1.45089	−	0.388766i
$467$	22.5167			1.04195			0.520973	−	0.853573i	$-0.325569\pi$
$467$	22.5167			1.04195			0.520973	+	0.853573i	$0.325569\pi$
$468$		0			0
$469$	−6.92820	−	6.00000i	−0.319915	−	0.277054i
$470$	0.633975	+	2.36603i	0.0292431	+	0.109137i
$471$		0			0
$472$	−6.92820	+	6.92820i	−0.318896	+	0.318896i
$473$	7.46410			0.343200
$474$		0			0
$475$	−6.00000			−0.275299
$476$	−2.41154	−	0.464102i	−0.110533	−	0.0212721i
$477$		0			0
$478$	10.2417	+	38.2224i	0.468443	+	1.74825i
$479$	37.1769			1.69866			0.849328	−	0.527865i	$-0.177007\pi$
$479$	37.1769			1.69866			0.849328	+	0.527865i	$0.177007\pi$
$480$		0			0
$481$		−	16.3923i		−	0.747425i
$482$	−1.60770	−	6.00000i	−0.0732285	−	0.273293i
$483$		0			0
$484$	−5.07180	+	2.92820i	−0.230536	+	0.133100i
$485$	−7.39230			−0.335667
$486$		0			0
$487$		−	28.7846i		−	1.30436i	−0.758066	−	0.652178i	$-0.773856\pi$
$487$		−	28.7846i		−	1.30436i	0.758066	−	0.652178i	$-0.226144\pi$
$488$	5.07180	+	5.07180i	0.229589	+	0.229589i
$489$		0			0
$490$	9.09808	−	3.90192i	0.411009	−	0.176271i
$491$		−	34.1244i		−	1.54001i	−0.638037	−	0.770005i	$-0.720254\pi$
$491$		−	34.1244i		−	1.54001i	0.638037	−	0.770005i	$-0.279746\pi$
$492$		0			0
$493$		−	2.75129i		−	0.123912i
$494$	14.1962	+	52.9808i	0.638715	+	2.38372i
$495$		0			0
$496$	−12.0000	+	20.7846i	−0.538816	+	0.933257i
$497$	1.07180	+	0.928203i	0.0480767	+	0.0416356i
$498$		0			0
$499$			5.58846i			0.250174i	0.992146	+	0.125087i	$0.0399210\pi$
$499$			5.58846i			0.250174i	−0.992146	+	0.125087i	$0.960079\pi$
$500$	−1.00000	−	1.73205i	−0.0447214	−	0.0774597i
$501$		0			0
$502$	−2.53590	+	0.679492i	−0.113183	+	0.0303272i
$503$	−15.5885			−0.695055			−0.347527	−	0.937670i	$-0.612979\pi$
$503$	−15.5885			−0.695055			−0.347527	+	0.937670i	$0.612979\pi$
$504$		0			0
$505$	−8.53590			−0.379842
$506$	27.8564	−	7.46410i	1.23837	−	0.331820i
$507$		0			0
$508$	8.53590	+	14.7846i	0.378719	+	0.655961i
$509$			1.85641i			0.0822838i	0.999153	+	0.0411419i	$0.0130996\pi$
$509$			1.85641i			0.0822838i	−0.999153	+	0.0411419i	$0.986900\pi$
$510$		0			0
$511$	−1.85641	−	1.60770i	−0.0821226	−	0.0711202i
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	−2.19615	−	8.19615i	−0.0968681	−	0.361517i
$515$		−	17.1962i		−	0.757753i
$516$		0			0
$517$			6.46410i			0.284291i
$518$	−4.14359	+	8.53590i	−0.182059	+	0.375046i
$519$		0			0
$520$	−12.9282	+	12.9282i	−0.566939	+	0.566939i
$521$		−	34.3923i		−	1.50675i	−0.657589	−	0.753377i	$-0.728424\pi$
$521$		−	34.3923i		−	1.50675i	0.657589	−	0.753377i	$-0.271576\pi$
$522$		0			0
$523$	24.2487			1.06032			0.530161	−	0.847897i	$-0.322131\pi$
$523$	24.2487			1.06032			0.530161	+	0.847897i	$0.322131\pi$
$524$	−16.3923	+	9.46410i	−0.716101	+	0.413441i
$525$		0			0
$526$	−1.66025	−	6.19615i	−0.0723905	−	0.270165i
$527$			2.78461i			0.121300i
$528$		0			0
$529$	−6.85641			−0.298105
$530$	−0.732051	−	2.73205i	−0.0317983	−	0.118673i
$531$		0			0
$532$	6.00000	−	31.1769i	0.260133	−	1.35169i
$533$	22.3923			0.969918
$534$		0			0
$535$	−18.3923			−0.795169
$536$	−6.92820	−	6.92820i	−0.299253	−	0.299253i
$537$		0			0
$538$	4.39230	+	16.3923i	0.189366	+	0.706722i
$539$	25.8564	−	3.73205i	1.11371	−	0.160751i
$540$		0			0
$541$	−33.7846			−1.45251			−0.726257	−	0.687423i	$-0.758742\pi$
$541$	−33.7846			−1.45251			−0.726257	+	0.687423i	$0.758742\pi$
$542$	3.46410	−	0.928203i	0.148796	−	0.0398697i
$543$		0			0
$544$	−2.53590	−	0.679492i	−0.108726	−	0.0291330i
$545$			15.9282i			0.682289i
$546$		0			0
$547$		−	14.5359i		−	0.621510i	−0.950490	−	0.310755i	$-0.899418\pi$
$547$		−	14.5359i		−	0.621510i	0.950490	−	0.310755i	$-0.100582\pi$
$548$	0.679492	−	0.392305i	0.0290265	−	0.0167584i
$549$		0			0
$550$	−1.36603	−	5.09808i	−0.0582475	−	0.217383i
$551$	35.5692			1.51530
$552$		0			0
$553$	5.32051	+	4.60770i	0.226251	+	0.195939i
$554$	−33.8564	+	9.07180i	−1.43842	+	0.385424i
$555$		0			0
$556$	−12.0000	+	6.92820i	−0.508913	+	0.293821i
$557$	−5.85641			−0.248144			−0.124072	−	0.992273i	$-0.539595\pi$
$557$	−5.85641			−0.248144			−0.124072	+	0.992273i	$0.539595\pi$
$558$		0			0
$559$	12.9282			0.546805
$560$	10.0000	−	3.46410i	0.422577	−	0.146385i
$561$		0			0
$562$	−8.09808	+	2.16987i	−0.341597	+	0.0915306i
$563$	43.1769			1.81969			0.909845	−	0.414948i	$-0.136200\pi$
$563$	43.1769			1.81969			0.909845	+	0.414948i	$0.136200\pi$
$564$		0			0
$565$			1.46410i			0.0615952i
$566$	−16.5622	+	4.43782i	−0.696160	+	0.186536i
$567$		0			0
$568$	1.07180	+	1.07180i	0.0449716	+	0.0449716i
$569$	−20.9282			−0.877356			−0.438678	−	0.898644i	$-0.644553\pi$
$569$	−20.9282			−0.877356			−0.438678	+	0.898644i	$0.644553\pi$
$570$		0			0
$571$			41.3205i			1.72921i	0.502453	+	0.864605i	$0.332431\pi$
$571$			41.3205i			1.72921i	−0.502453	+	0.864605i	$0.667569\pi$
$572$	−41.7846	+	24.1244i	−1.74710	+	1.00869i
$573$		0			0
$574$	−11.6603	−	5.66025i	−0.486690	−	0.236254i
$575$			5.46410i			0.227869i
$576$		0			0
$577$		−	1.39230i		−	0.0579624i	−0.999580	−	0.0289812i	$-0.990774\pi$
$577$		−	1.39230i		−	0.0579624i	0.999580	−	0.0289812i	$-0.00922630\pi$
$578$	22.9282	−	6.14359i	0.953688	−	0.255540i
$579$		0			0
$580$	5.92820	+	10.2679i	0.246155	+	0.426353i
$581$	−14.7846	+	17.0718i	−0.613369	+	0.708257i
$582$		0			0
$583$		−	7.46410i		−	0.309132i
$584$	−1.85641	−	1.85641i	−0.0768186	−	0.0768186i
$585$		0			0
$586$	5.24167	+	19.5622i	0.216531	+	0.808106i
$587$	27.4641			1.13356			0.566782	−	0.823868i	$-0.308188\pi$
$587$	27.4641			1.13356			0.566782	+	0.823868i	$0.308188\pi$
$588$		0			0
$589$	−36.0000			−1.48335
$590$	−1.26795	−	4.73205i	−0.0522006	−	0.194815i
$591$		0			0
$592$	−5.07180	+	8.78461i	−0.208450	+	0.361045i
$593$		−	23.5359i		−	0.966504i	−0.875481	−	0.483252i	$-0.839456\pi$
$593$		−	23.5359i		−	0.966504i	0.875481	−	0.483252i	$-0.160544\pi$
$594$		0			0
$595$	0.803848	−	0.928203i	0.0329545	−	0.0380526i
$596$	−29.3205	+	16.9282i	−1.20101	+	0.693406i
$597$		0			0
$598$	48.2487	−	12.9282i	1.97304	−	0.528674i
$599$			14.1244i			0.577106i	0.957464	+	0.288553i	$0.0931741\pi$
$599$			14.1244i			0.577106i	−0.957464	+	0.288553i	$0.906826\pi$
$600$		0			0
$601$			26.7846i			1.09257i	0.837600	+	0.546284i	$0.183958\pi$
$601$			26.7846i			1.09257i	−0.837600	+	0.546284i	$0.816042\pi$
$602$	−6.73205	−	3.26795i	−0.274378	−	0.133192i
$603$		0			0
$604$	−4.80385	−	8.32051i	−0.195466	−	0.338557i
$605$		−	2.92820i		−	0.119048i
$606$		0			0
$607$	−18.8038			−0.763225			−0.381612	−	0.924322i	$-0.624631\pi$
$607$	−18.8038			−0.763225			−0.381612	+	0.924322i	$0.624631\pi$
$608$	8.78461	−	32.7846i	0.356263	−	1.32959i
$609$		0			0
$610$	−3.46410	+	0.928203i	−0.140257	+	0.0375819i
$611$			11.1962i			0.452948i
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$	−2.36603	+	0.633975i	−0.0954850	+	0.0255851i
$615$		0			0
$616$	27.8564	−	2.00000i	1.12237	−	0.0805823i
$617$	9.07180			0.365217			0.182608	−	0.983186i	$-0.441546\pi$
$617$	9.07180			0.365217			0.182608	+	0.983186i	$0.441546\pi$
$618$		0			0
$619$	35.3205			1.41965			0.709826	−	0.704378i	$-0.248774\pi$
$619$	35.3205			1.41965			0.709826	+	0.704378i	$0.248774\pi$
$620$	−6.00000	−	10.3923i	−0.240966	−	0.417365i
$621$		0			0
$622$	27.1244	−	7.26795i	1.08759	−	0.291418i
$623$	−18.9282	−	16.3923i	−0.758342	−	0.656744i
$624$		0			0
$625$	1.00000			0.0400000
$626$	−8.95448	−	33.4186i	−0.357893	−	1.33568i
$627$		0			0
$628$	−19.8564	−	34.3923i	−0.792357	−	1.37240i
$629$			1.17691i			0.0469267i
$630$		0			0
$631$			26.9090i			1.07123i	0.844463	+	0.535614i	$0.179920\pi$
$631$			26.9090i			1.07123i	−0.844463	+	0.535614i	$0.820080\pi$
$632$	5.32051	+	5.32051i	0.211638	+	0.211638i
$633$		0			0
$634$	23.1244	−	6.19615i	0.918385	−	0.246081i
$635$	−8.53590			−0.338737
$636$		0			0
$637$	44.7846	−	6.46410i	1.77443	−	0.256117i
$638$	8.09808	+	30.2224i	0.320606	+	1.19652i
$639$		0			0
$640$	10.9282	−	2.92820i	0.431975	−	0.115747i
$641$	4.92820			0.194652			0.0973262	−	0.995253i	$-0.468971\pi$
$641$	4.92820			0.194652			0.0973262	+	0.995253i	$0.468971\pi$
$642$		0			0
$643$	−31.0526			−1.22459			−0.612297	−	0.790628i	$-0.709754\pi$
$643$	−31.0526			−1.22459			−0.612297	+	0.790628i	$0.709754\pi$
$644$	−28.3923	−	5.46410i	−1.11881	−	0.215316i
$645$		0			0
$646$	−1.01924	−	3.80385i	−0.0401014	−	0.149660i
$647$	−10.3923			−0.408564			−0.204282	−	0.978912i	$-0.565486\pi$
$647$	−10.3923			−0.408564			−0.204282	+	0.978912i	$0.565486\pi$
$648$		0			0
$649$		−	12.9282i		−	0.507476i
$650$	−2.36603	−	8.83013i	−0.0928032	−	0.346346i
$651$		0			0
$652$	−20.7846	−	36.0000i	−0.813988	−	1.40987i
$653$	−38.3923			−1.50241			−0.751203	−	0.660071i	$-0.770526\pi$
$653$	−38.3923			−1.50241			−0.751203	+	0.660071i	$0.770526\pi$
$654$		0			0
$655$		−	9.46410i		−	0.369793i
$656$	−12.0000	−	6.92820i	−0.468521	−	0.270501i
$657$		0			0
$658$	2.83013	−	5.83013i	0.110330	−	0.227282i
$659$		−	20.8038i		−	0.810403i	−0.914227	−	0.405201i	$-0.867201\pi$
$659$		−	20.8038i		−	0.810403i	0.914227	−	0.405201i	$-0.132799\pi$
$660$		0			0
$661$		−	15.7128i		−	0.611158i	−0.952167	−	0.305579i	$-0.901150\pi$
$661$		−	15.7128i		−	0.611158i	0.952167	−	0.305579i	$-0.0988499\pi$
$662$	2.05256	+	7.66025i	0.0797750	+	0.297724i
$663$		0			0
$664$	−17.0718	+	17.0718i	−0.662514	+	0.662514i
$665$	12.0000	+	10.3923i	0.465340	+	0.402996i
$666$		0			0
$667$		−	32.3923i		−	1.25424i
$668$	−9.00000	+	5.19615i	−0.348220	+	0.201045i
$669$		0			0
$670$	4.73205	−	1.26795i	0.182815	−	0.0489852i
$671$	−9.46410			−0.365358
$672$		0			0
$673$	49.1769			1.89563			0.947815	−	0.318820i	$-0.103286\pi$
$673$	49.1769			1.89563			0.947815	+	0.318820i	$0.103286\pi$
$674$		0			0
$675$		0			0
$676$	−49.8564	+	28.7846i	−1.91755	+	1.10710i
$677$			4.60770i			0.177088i	0.996072	+	0.0885441i	$0.0282214\pi$
$677$			4.60770i			0.177088i	−0.996072	+	0.0885441i	$0.971779\pi$
$678$		0			0
$679$	14.7846	+	12.8038i	0.567381	+	0.491367i
$680$	0.928203	−	0.928203i	0.0355950	−	0.0355950i
$681$		0			0
$682$	−8.19615	−	30.5885i	−0.313847	−	1.17129i
$683$		−	27.3205i		−	1.04539i	−0.852520	−	0.522695i	$-0.824927\pi$
$683$		−	27.3205i		−	1.04539i	0.852520	−	0.522695i	$-0.175073\pi$
$684$		0			0
$685$			0.392305i			0.0149892i
$686$	−24.9545	−	7.95448i	−0.952767	−	0.303704i
$687$		0			0
$688$	−6.92820	−	4.00000i	−0.264135	−	0.152499i
$689$		−	12.9282i		−	0.492525i
$690$		0			0
$691$	46.6410			1.77431			0.887154	−	0.461474i	$-0.152679\pi$
$691$	46.6410			1.77431			0.887154	+	0.461474i	$0.152679\pi$
$692$	−20.3205	−	35.1962i	−0.772470	−	1.33796i
$693$		0			0
$694$	−5.21539	−	19.4641i	−0.197974	−	0.738847i
$695$		−	6.92820i		−	0.262802i
$696$		0			0
$697$	−1.60770			−0.0608958
$698$	10.7321	+	40.0526i	0.406214	+	1.51601i
$699$		0			0
$700$	−1.00000	+	5.19615i	−0.0377964	+	0.196396i
$701$	−3.78461			−0.142943			−0.0714714	−	0.997443i	$-0.522769\pi$
$701$	−3.78461			−0.142943			−0.0714714	+	0.997443i	$0.522769\pi$
$702$		0			0
$703$	−15.2154			−0.573859
$704$	29.8564			1.12526
$705$		0			0
$706$	−4.90192	−	18.2942i	−0.184486	−	0.688512i
$707$	17.0718	+	14.7846i	0.642051	+	0.556032i
$708$		0			0
$709$	21.0000			0.788672			0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000			0.788672			0.394336	+	0.918966i	$0.370975\pi$
$710$	−0.732051	+	0.196152i	−0.0274734	+	0.00736147i
$711$		0			0
$712$	−18.9282	−	18.9282i	−0.709364	−	0.709364i
$713$			32.7846i			1.22779i
$714$		0			0
$715$		−	24.1244i		−	0.902200i
$716$	14.3923	+	24.9282i	0.537866	+	0.931611i
$717$		0			0
$718$	−3.41154	−	12.7321i	−0.127318	−	0.475156i
$719$	−45.4641			−1.69552			−0.847762	−	0.530376i	$-0.822051\pi$
$719$	−45.4641			−1.69552			−0.847762	+	0.530376i	$0.822051\pi$
$720$		0			0
$721$	−29.7846	+	34.3923i	−1.10924	+	1.28084i
$722$	23.2224	−	6.22243i	0.864249	−	0.231575i
$723$		0			0
$724$	12.9282	+	22.3923i	0.480473	+	0.832203i
$725$	−5.92820			−0.220168
$726$		0			0
$727$	−34.3923			−1.27554			−0.637770	−	0.770227i	$-0.720143\pi$
$727$	−34.3923			−1.27554			−0.637770	+	0.770227i	$0.720143\pi$
$728$	48.2487	−	3.46410i	1.78822	−	0.128388i
$729$		0			0
$730$	1.26795	−	0.339746i	0.0469289	−	0.0125746i
$731$	−0.928203			−0.0343308
$732$		0			0
$733$			18.4641i			0.681987i	0.940066	+	0.340994i	$0.110763\pi$
$733$			18.4641i			0.681987i	−0.940066	+	0.340994i	$0.889237\pi$
$734$	49.3468	−	13.2224i	1.82142	−	0.488049i
$735$		0			0
$736$	−29.8564	−	8.00000i	−1.10052	−	0.294884i
$737$	12.9282			0.476216
$738$		0			0
$739$			7.73205i			0.284428i	0.989836	+	0.142214i	$0.0454221\pi$
$739$			7.73205i			0.284428i	−0.989836	+	0.142214i	$0.954578\pi$
$740$	−2.53590	−	4.39230i	−0.0932215	−	0.161464i
$741$		0			0
$742$	−3.26795	+	6.73205i	−0.119970	+	0.247141i
$743$			9.60770i			0.352472i	0.984348	+	0.176236i	$0.0563922\pi$
$743$			9.60770i			0.352472i	−0.984348	+	0.176236i	$0.943608\pi$
$744$		0			0
$745$		−	16.9282i		−	0.620201i
$746$	−33.3205	+	8.92820i	−1.21995	+	0.326885i
$747$		0			0
$748$	3.00000	−	1.73205i	0.109691	−	0.0633300i
$749$	36.7846	+	31.8564i	1.34408	+	1.16401i
$750$		0			0
$751$		−	5.58846i		−	0.203926i	−0.994788	−	0.101963i	$-0.967488\pi$
$751$		−	5.58846i		−	0.203926i	0.994788	−	0.101963i	$-0.0325123\pi$
$752$	3.46410	−	6.00000i	0.126323	−	0.218797i
$753$		0			0
$754$	14.0263	+	52.3468i	0.510807	+	1.90636i
$755$	4.80385			0.174830
$756$		0			0
$757$	10.1436			0.368675			0.184338	−	0.982863i	$-0.440986\pi$
$757$	10.1436			0.368675			0.184338	+	0.982863i	$0.440986\pi$
$758$	−9.66025	−	36.0526i	−0.350876	−	1.30949i
$759$		0			0
$760$	12.0000	+	12.0000i	0.435286	+	0.435286i
$761$		−	6.24871i		−	0.226516i	−0.993566	−	0.113258i	$-0.963871\pi$
$761$		−	6.24871i		−	0.226516i	0.993566	−	0.113258i	$-0.0361286\pi$
$762$		0			0
$763$	27.5885	−	31.8564i	0.998769	−	1.15328i
$764$	−3.19615	−	5.53590i	−0.115633	−	0.200282i
$765$		0			0
$766$	−28.0526	+	7.51666i	−1.01358	+	0.271588i
$767$		−	22.3923i		−	0.808539i
$768$		0			0
$769$			18.0000i			0.649097i	0.945869	+	0.324548i	$0.105212\pi$
$769$			18.0000i			0.649097i	−0.945869	+	0.324548i	$0.894788\pi$
$770$	−6.09808	+	12.5622i	−0.219759	+	0.452709i
$771$		0			0
$772$	−4.39230	+	2.53590i	−0.158083	+	0.0912690i
$773$			12.4641i			0.448303i	0.974554	+	0.224151i	$0.0719610\pi$
$773$			12.4641i			0.448303i	−0.974554	+	0.224151i	$0.928039\pi$
$774$		0			0
$775$	6.00000			0.215526
$776$	14.7846	+	14.7846i	0.530737	+	0.530737i
$777$		0			0
$778$	−9.36603	+	2.50962i	−0.335788	+	0.0899742i
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	−2.00000			−0.0715656
$782$	−3.46410	+	0.928203i	−0.123876	+	0.0331925i
$783$		0			0
$784$	−26.0000	−	10.3923i	−0.928571	−	0.371154i
$785$	19.8564			0.708706
$786$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1260.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +1.00000i q^{5} +(1.73205 - 2.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +(0.366025 + 1.36603i) q^{10} +3.73205i q^{11} +6.46410i q^{13} +(1.63397 - 3.36603i) q^{14} +(2.00000 - 3.46410i) q^{16} -0.464102i q^{17} +6.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(1.36603 + 5.09808i) q^{22} -5.46410i q^{23} -1.00000 q^{25} +(2.36603 + 8.83013i) q^{26} +(1.00000 - 5.19615i) q^{28} +5.92820 q^{29} -6.00000 q^{31} +(1.46410 - 5.46410i) q^{32} +(-0.169873 - 0.633975i) q^{34} +(2.00000 + 1.73205i) q^{35} -2.53590 q^{37} +(8.19615 - 2.19615i) q^{38} +(2.00000 + 2.00000i) q^{40} -3.46410i q^{41} -2.00000i q^{43} +(3.73205 + 6.46410i) q^{44} +(-2.00000 - 7.46410i) q^{46} +1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} +(-1.36603 + 0.366025i) q^{50} +(6.46410 + 11.1962i) q^{52} -2.00000 q^{53} -3.73205 q^{55} +(-0.535898 - 7.46410i) q^{56} +(8.09808 - 2.16987i) q^{58} -3.46410 q^{59} +2.53590i q^{61} +(-8.19615 + 2.19615i) q^{62} -8.00000i q^{64} -6.46410 q^{65} -3.46410i q^{67} +(-0.464102 - 0.803848i) q^{68} +(3.36603 + 1.63397i) q^{70} +0.535898i q^{71} -0.928203i q^{73} +(-3.46410 + 0.928203i) q^{74} +(10.3923 - 6.00000i) q^{76} +(7.46410 + 6.46410i) q^{77} +2.66025i q^{79} +(3.46410 + 2.00000i) q^{80} +(-1.26795 - 4.73205i) q^{82} -8.53590 q^{83} +0.464102 q^{85} +(-0.732051 - 2.73205i) q^{86} +(7.46410 + 7.46410i) q^{88} -9.46410i q^{89} +(12.9282 + 11.1962i) q^{91} +(-5.46410 - 9.46410i) q^{92} +(2.36603 - 0.633975i) q^{94} +6.00000i q^{95} +7.39230i q^{97} +(-3.90192 - 9.09808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{8} - 2 q^{10} + 10 q^{14} + 8 q^{16} + 24 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{25} + 6 q^{26} + 4 q^{28} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 18 q^{34} + 8 q^{35} - 24 q^{37} + 12 q^{38}+ \cdots - 26 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^8 - 2 * q^10 + 10 * q^14 + 8 * q^16 + 24 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^25 + 6 * q^26 + 4 * q^28 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 18 * q^34 + 8 * q^35 - 24 * q^37 + 12 * q^38 + 8 * q^40 + 8 * q^44 - 8 * q^46 - 4 * q^49 - 2 * q^50 + 12 * q^52 - 8 * q^53 - 8 * q^55 - 16 * q^56 + 22 * q^58 - 12 * q^62 - 12 * q^65 + 12 * q^68 + 10 * q^70 + 16 * q^77 - 12 * q^82 - 48 * q^83 - 12 * q^85 + 4 * q^86 + 16 * q^88 + 24 * q^91 - 8 * q^92 + 6 * q^94 - 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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