LMFDB - Embedded newform 416.2.i.d.321.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(289,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("416.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.i (of order $3$, degree $2$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 121 $ x^4 + 11*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			321.1
Root			$-1.65831 + 2.87228i$ of defining polynomial
Character	$\chi$	$=$	416.321
Dual form			416.2.i.d.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.65831 + 2.87228i) q^{3} +(-1.65831 - 2.87228i) q^{7} +(-4.00000 - 6.92820i) q^{9} +O(q^{10})$	$q+(-1.65831 + 2.87228i) q^{3} +(-1.65831 - 2.87228i) q^{7} +(-4.00000 - 6.92820i) q^{9} +(-1.65831 + 2.87228i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{17} +(-1.65831 - 2.87228i) q^{19} +11.0000 q^{21} +(1.65831 - 2.87228i) q^{23} -5.00000 q^{25} +16.5831 q^{27} +(2.50000 - 4.33013i) q^{29} +(-5.50000 - 9.52628i) q^{33} +(-4.50000 + 7.79423i) q^{37} +(11.6082 + 2.87228i) q^{39} +(1.50000 - 2.59808i) q^{41} +(-4.97494 - 8.61684i) q^{43} -6.63325 q^{47} +(-2.00000 + 3.46410i) q^{49} -9.94987 q^{51} -8.00000 q^{53} +11.0000 q^{57} +(-1.65831 - 2.87228i) q^{59} +(4.50000 + 7.79423i) q^{61} +(-13.2665 + 22.9783i) q^{63} +(-4.97494 + 8.61684i) q^{67} +(5.50000 + 9.52628i) q^{69} +(-4.97494 - 8.61684i) q^{71} -4.00000 q^{73} +(8.29156 - 14.3614i) q^{75} +11.0000 q^{77} +6.63325 q^{79} +(-15.5000 + 26.8468i) q^{81} -13.2665 q^{83} +(8.29156 + 14.3614i) q^{87} +(0.500000 - 0.866025i) q^{89} +(-8.29156 + 8.61684i) q^{91} +(-3.50000 - 6.06218i) q^{97} +26.5330 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{9}+O(q^{10}) $ 4 * q - 16 * q^9	$ 4 q - 16 q^{9} - 4 q^{13} + 6 q^{17} + 44 q^{21} - 20 q^{25} + 10 q^{29} - 22 q^{33} - 18 q^{37} + 6 q^{41} - 8 q^{49} - 32 q^{53} + 44 q^{57} + 18 q^{61} + 22 q^{69} - 16 q^{73} + 44 q^{77} - 62 q^{81} + 2 q^{89} - 14 q^{97}+O(q^{100}) $ 4 * q - 16 * q^9 - 4 * q^13 + 6 * q^17 + 44 * q^21 - 20 * q^25 + 10 * q^29 - 22 * q^33 - 18 * q^37 + 6 * q^41 - 8 * q^49 - 32 * q^53 + 44 * q^57 + 18 * q^61 + 22 * q^69 - 16 * q^73 + 44 * q^77 - 62 * q^81 + 2 * q^89 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.65831	+	2.87228i	−0.957427	+	1.65831i	−0.228714	+	0.973494i	$0.573452\pi$
$3$	−1.65831	+	2.87228i	−0.957427	+	1.65831i	−0.728714	+	0.684819i	$0.759881\pi$
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−1.65831	−	2.87228i	−0.626783	−	1.08562i	−0.988193	−	0.153213i	$-0.951038\pi$
$7$	−1.65831	−	2.87228i	−0.626783	−	1.08562i	0.361410	−	0.932407i	$-0.382295\pi$
$8$		0			0
$9$	−4.00000	−	6.92820i	−1.33333	−	2.30940i
$10$		0			0
$11$	−1.65831	+	2.87228i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$11$	−1.65831	+	2.87228i	−0.500000	+	0.866025i	−1.00000			$\pi$
$12$		0			0
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−1.65831	−	2.87228i	−0.380443	−	0.658947i	0.610683	−	0.791875i	$-0.290895\pi$
$19$	−1.65831	−	2.87228i	−0.380443	−	0.658947i	−0.991126	+	0.132929i	$0.957562\pi$
$20$		0			0
$21$	11.0000			2.40040
$22$		0			0
$23$	1.65831	−	2.87228i	0.345782	−	0.598912i	−0.639713	−	0.768613i	$-0.720947\pi$
$23$	1.65831	−	2.87228i	0.345782	−	0.598912i	0.985496	+	0.169701i	$0.0542803\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$	16.5831			3.19142
$28$		0			0
$29$	2.50000	−	4.33013i	0.464238	−	0.804084i	−0.534928	−	0.844897i	$-0.679661\pi$
$29$	2.50000	−	4.33013i	0.464238	−	0.804084i	0.999167	+	0.0408130i	$0.0129948\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	−5.50000	−	9.52628i	−0.957427	−	1.65831i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.50000	+	7.79423i	−0.739795	+	1.28136i	0.212792	+	0.977098i	$0.431744\pi$
$37$	−4.50000	+	7.79423i	−0.739795	+	1.28136i	−0.952587	+	0.304266i	$0.901589\pi$
$38$		0			0
$39$	11.6082	+	2.87228i	1.85880	+	0.459933i
$40$		0			0
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	$-0.758066\pi$
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	0.959058	+	0.283211i	$0.0913998\pi$
$42$		0			0
$43$	−4.97494	−	8.61684i	−0.758671	−	1.31406i	−0.943529	−	0.331291i	$-0.892516\pi$
$43$	−4.97494	−	8.61684i	−0.758671	−	1.31406i	0.184858	−	0.982765i	$-0.440818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.63325			−0.967559			−0.483779	−	0.875190i	$-0.660736\pi$
$47$	−6.63325			−0.967559			−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		0			0
$51$	−9.94987			−1.39326
$52$		0			0
$53$	−8.00000			−1.09888			−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000			−1.09888			−0.549442	+	0.835532i	$0.685160\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	11.0000			1.45699
$58$		0			0
$59$	−1.65831	−	2.87228i	−0.215894	−	0.373939i	0.737655	−	0.675178i	$-0.235933\pi$
$59$	−1.65831	−	2.87228i	−0.215894	−	0.373939i	−0.953549	+	0.301239i	$0.902600\pi$
$60$		0			0
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	0.995914	+	0.0903080i	$0.0287851\pi$
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	−0.419748	+	0.907641i	$0.637882\pi$
$62$		0			0
$63$	−13.2665	+	22.9783i	−1.67142	+	2.89499i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.97494	+	8.61684i	−0.607785	+	1.05272i	0.383819	+	0.923408i	$0.374609\pi$
$67$	−4.97494	+	8.61684i	−0.607785	+	1.05272i	−0.991605	+	0.129307i	$0.958725\pi$
$68$		0			0
$69$	5.50000	+	9.52628i	0.662122	+	1.14683i
$70$		0			0
$71$	−4.97494	−	8.61684i	−0.590416	−	1.02263i	−0.994176	−	0.107766i	$-0.965630\pi$
$71$	−4.97494	−	8.61684i	−0.590416	−	1.02263i	0.403760	−	0.914865i	$-0.367703\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$	8.29156	−	14.3614i	0.957427	−	1.65831i
$76$		0			0
$77$	11.0000			1.25357
$78$		0			0
$79$	6.63325			0.746299			0.373149	−	0.927771i	$-0.378278\pi$
$79$	6.63325			0.746299			0.373149	+	0.927771i	$0.378278\pi$
$80$		0			0
$81$	−15.5000	+	26.8468i	−1.72222	+	2.98298i
$82$		0			0
$83$	−13.2665			−1.45619			−0.728094	−	0.685478i	$-0.759593\pi$
$83$	−13.2665			−1.45619			−0.728094	+	0.685478i	$0.759593\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	8.29156	+	14.3614i	0.888949	+	1.53970i
$88$		0			0
$89$	0.500000	−	0.866025i	0.0529999	−	0.0917985i	−0.838308	−	0.545197i	$-0.816455\pi$
$89$	0.500000	−	0.866025i	0.0529999	−	0.0917985i	0.891308	+	0.453398i	$0.149788\pi$
$90$		0			0
$91$	−8.29156	+	8.61684i	−0.869192	+	0.903291i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.50000	−	6.06218i	−0.355371	−	0.615521i	0.631810	−	0.775123i	$-0.282312\pi$
$97$	−3.50000	−	6.06218i	−0.355371	−	0.615521i	−0.987181	+	0.159602i	$0.948979\pi$
$98$		0			0
$99$	26.5330			2.66667
$100$		0			0
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	−0.203317	−	0.979113i	$-0.565172\pi$
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	0.949595	−	0.313478i	$-0.101494\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.65831	+	2.87228i	−0.160315	+	0.277674i	−0.934982	−	0.354696i	$-0.884584\pi$
$107$	−1.65831	+	2.87228i	−0.160315	+	0.277674i	0.774667	+	0.632370i	$0.217918\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$	−14.9248	−	25.8505i	−1.41660	−	2.45362i
$112$		0			0
$113$	5.50000	+	9.52628i	0.517396	+	0.896157i	0.999796	+	0.0202056i	$0.00643208\pi$
$113$	5.50000	+	9.52628i	0.517396	+	0.896157i	−0.482399	+	0.875951i	$0.660235\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−20.0000	+	20.7846i	−1.84900	+	1.92154i
$118$		0			0
$119$	4.97494	−	8.61684i	0.456052	−	0.789905i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	4.97494	+	8.61684i	0.448575	+	0.776955i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.97494	−	8.61684i	0.441454	−	0.764621i	−0.556343	−	0.830952i	$-0.687796\pi$
$127$	4.97494	−	8.61684i	0.441454	−	0.764621i	0.997798	+	0.0663312i	$0.0211294\pi$
$128$		0			0
$129$	33.0000			2.90549
$130$		0			0
$131$	13.2665			1.15910			0.579550	−	0.814937i	$-0.303228\pi$
$131$	13.2665			1.15910			0.579550	+	0.814937i	$0.303228\pi$
$132$		0			0
$133$	−5.50000	+	9.52628i	−0.476910	+	0.826033i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.50000	+	7.79423i	0.384461	+	0.665906i	0.991694	−	0.128618i	$-0.0410540\pi$
$137$	4.50000	+	7.79423i	0.384461	+	0.665906i	−0.607233	+	0.794524i	$0.707721\pi$
$138$		0			0
$139$	−1.65831	−	2.87228i	−0.140656	−	0.243624i	0.787088	−	0.616841i	$-0.211588\pi$
$139$	−1.65831	−	2.87228i	−0.140656	−	0.243624i	−0.927744	+	0.373217i	$0.878255\pi$
$140$		0			0
$141$	11.0000	−	19.0526i	0.926367	−	1.60451i
$142$		0			0
$143$	11.6082	+	2.87228i	0.970725	+	0.240192i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.63325	−	11.4891i	−0.547101	−	0.947607i
$148$		0			0
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	0.885779	−	0.464107i	$-0.153625\pi$
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	−0.844818	+	0.535054i	$0.820291\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	12.0000	−	20.7846i	0.970143	−	1.68034i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−8.00000			−0.638470			−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000			−0.638470			−0.319235	+	0.947676i	$0.603426\pi$
$158$		0			0
$159$	13.2665	−	22.9783i	1.05210	−	1.82229i
$160$		0			0
$161$	−11.0000			−0.866921
$162$		0			0
$163$	8.29156	+	14.3614i	0.649445	+	1.12487i	0.983256	+	0.182232i	$0.0583322\pi$
$163$	8.29156	+	14.3614i	0.649445	+	1.12487i	−0.333810	+	0.942640i	$0.608335\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.65831	−	2.87228i	0.128324	−	0.222264i	−0.794703	−	0.606998i	$-0.792374\pi$
$167$	1.65831	−	2.87228i	0.128324	−	0.222264i	0.923027	+	0.384734i	$0.125707\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$	−13.2665	+	22.9783i	−1.01451	+	1.75719i
$172$		0			0
$173$	−3.50000	−	6.06218i	−0.266100	−	0.460899i	0.701751	−	0.712422i	$-0.252402\pi$
$173$	−3.50000	−	6.06218i	−0.266100	−	0.460899i	−0.967851	+	0.251523i	$0.919068\pi$
$174$		0			0
$175$	8.29156	+	14.3614i	0.626783	+	1.08562i
$176$		0			0
$177$	11.0000			0.826811
$178$		0			0
$179$	−1.65831	+	2.87228i	−0.123948	+	0.214684i	−0.921321	−	0.388802i	$-0.872889\pi$
$179$	−1.65831	+	2.87228i	−0.123948	+	0.214684i	0.797373	+	0.603487i	$0.206222\pi$
$180$		0			0
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$		0			0
$183$	−29.8496			−2.20655
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−9.94987			−0.727607
$188$		0			0
$189$	−27.5000	−	47.6314i	−2.00033	−	3.46467i
$190$		0			0
$191$	11.6082	+	20.1060i	0.839939	+	1.45482i	0.889945	+	0.456068i	$0.150743\pi$
$191$	11.6082	+	20.1060i	0.839939	+	1.45482i	−0.0500060	+	0.998749i	$0.515924\pi$
$192$		0			0
$193$	−6.50000	+	11.2583i	−0.467880	+	0.810392i	−0.999326	−	0.0366998i	$-0.988315\pi$
$193$	−6.50000	+	11.2583i	−0.467880	+	0.810392i	0.531446	+	0.847092i	$0.321649\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.50000	−	11.2583i	0.463106	−	0.802123i	−0.536008	−	0.844213i	$-0.680068\pi$
$197$	6.50000	−	11.2583i	0.463106	−	0.802123i	0.999114	+	0.0420901i	$0.0134016\pi$
$198$		0			0
$199$	−4.97494	−	8.61684i	−0.352664	−	0.610832i	0.634051	−	0.773291i	$-0.281391\pi$
$199$	−4.97494	−	8.61684i	−0.352664	−	0.610832i	−0.986715	+	0.162459i	$0.948057\pi$
$200$		0			0
$201$	−16.5000	−	28.5788i	−1.16382	−	2.01580i
$202$		0			0
$203$	−16.5831			−1.16391
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−26.5330			−1.84417
$208$		0			0
$209$	11.0000			0.760886
$210$		0			0
$211$	−1.65831	+	2.87228i	−0.114163	+	0.197736i	−0.917445	−	0.397863i	$-0.869752\pi$
$211$	−1.65831	+	2.87228i	−0.114163	+	0.197736i	0.803282	+	0.595599i	$0.203085\pi$
$212$		0			0
$213$	33.0000			2.26112
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	6.63325	−	11.4891i	0.448233	−	0.776363i
$220$		0			0
$221$	7.50000	−	7.79423i	0.504505	−	0.524297i
$222$		0			0
$223$	1.65831	−	2.87228i	0.111049	−	0.192342i	−0.805145	−	0.593079i	$-0.797912\pi$
$223$	1.65831	−	2.87228i	0.111049	−	0.192342i	0.916193	+	0.400736i	$0.131246\pi$
$224$		0			0
$225$	20.0000	+	34.6410i	1.33333	+	2.30940i
$226$		0			0
$227$	−1.65831	−	2.87228i	−0.110066	−	0.190640i	0.805731	−	0.592282i	$-0.201773\pi$
$227$	−1.65831	−	2.87228i	−0.110066	−	0.190640i	−0.915797	+	0.401642i	$0.868440\pi$
$228$		0			0
$229$	8.00000			0.528655			0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000			0.528655			0.264327	+	0.964433i	$0.414850\pi$
$230$		0			0
$231$	−18.2414	+	31.5951i	−1.20020	+	2.07880i
$232$		0			0
$233$	4.00000			0.262049			0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000			0.262049			0.131024	+	0.991379i	$0.458173\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−11.0000	+	19.0526i	−0.714527	+	1.23760i
$238$		0			0
$239$	26.5330			1.71628			0.858138	−	0.513418i	$-0.171621\pi$
$239$	26.5330			1.71628			0.858138	+	0.513418i	$0.171621\pi$
$240$		0			0
$241$	−6.50000	−	11.2583i	−0.418702	−	0.725213i	0.577107	−	0.816668i	$-0.304181\pi$
$241$	−6.50000	−	11.2583i	−0.418702	−	0.725213i	−0.995809	+	0.0914555i	$0.970848\pi$
$242$		0			0
$243$	−26.5330	−	45.9565i	−1.70209	−	2.94811i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.29156	+	8.61684i	−0.527579	+	0.548277i
$248$		0			0
$249$	22.0000	−	38.1051i	1.39419	−	2.41481i
$250$		0			0
$251$	8.29156	+	14.3614i	0.523359	+	0.906484i	0.999630	+	0.0271858i	$0.00865456\pi$
$251$	8.29156	+	14.3614i	0.523359	+	0.906484i	−0.476272	+	0.879298i	$0.658012\pi$
$252$		0			0
$253$	5.50000	+	9.52628i	0.345782	+	0.598912i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.50000	+	11.2583i	−0.405459	+	0.702275i	−0.994375	−	0.105919i	$-0.966222\pi$
$257$	−6.50000	+	11.2583i	−0.405459	+	0.702275i	0.588916	+	0.808194i	$0.299555\pi$
$258$		0			0
$259$	29.8496			1.85477
$260$		0			0
$261$	−40.0000			−2.47594
$262$		0			0
$263$	−11.6082	+	20.1060i	−0.715791	+	1.23979i	0.246862	+	0.969051i	$0.420601\pi$
$263$	−11.6082	+	20.1060i	−0.715791	+	1.23979i	−0.962653	+	0.270737i	$0.912733\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.65831	+	2.87228i	0.101487	+	0.175781i
$268$		0			0
$269$	−14.5000	−	25.1147i	−0.884081	−	1.53127i	−0.846764	−	0.531969i	$-0.821452\pi$
$269$	−14.5000	−	25.1147i	−0.884081	−	1.53127i	−0.0373168	−	0.999303i	$-0.511881\pi$
$270$		0			0
$271$	14.9248	−	25.8505i	0.906618	−	1.57031i	0.0878869	−	0.996130i	$-0.471989\pi$
$271$	14.9248	−	25.8505i	0.906618	−	1.57031i	0.818731	−	0.574178i	$-0.194678\pi$
$272$		0			0
$273$	−11.0000	−	38.1051i	−0.665750	−	2.30623i
$274$		0			0
$275$	8.29156	−	14.3614i	0.500000	−	0.866025i
$276$		0			0
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	0.781094	−	0.624413i	$-0.214662\pi$
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$	11.6082	−	20.1060i	0.690035	−	1.19518i	−0.281791	−	0.959476i	$-0.590928\pi$
$283$	11.6082	−	20.1060i	0.690035	−	1.19518i	0.971826	−	0.235700i	$-0.0757383\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.94987			−0.587323
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$	23.2164			1.36097
$292$		0			0
$293$	−3.50000	−	6.06218i	−0.204472	−	0.354156i	0.745492	−	0.666514i	$-0.232214\pi$
$293$	−3.50000	−	6.06218i	−0.204472	−	0.354156i	−0.949964	+	0.312358i	$0.898881\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−27.5000	+	47.6314i	−1.59571	+	2.76385i
$298$		0			0
$299$	−11.6082	−	2.87228i	−0.671319	−	0.166108i
$300$		0			0
$301$	−16.5000	+	28.5788i	−0.951044	+	1.64726i
$302$		0			0
$303$	24.8747	+	43.0842i	1.42901	+	2.47512i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	13.2665			0.757159			0.378580	−	0.925569i	$-0.376413\pi$
$307$	13.2665			0.757159			0.378580	+	0.925569i	$0.376413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	26.5330			1.50455			0.752274	−	0.658850i	$-0.228957\pi$
$311$	26.5330			1.50455			0.752274	+	0.658850i	$0.228957\pi$
$312$		0			0
$313$	4.00000			0.226093			0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000			0.226093			0.113047	+	0.993590i	$0.463939\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$	8.29156	+	14.3614i	0.464238	+	0.804084i
$320$		0			0
$321$	−5.50000	−	9.52628i	−0.306980	−	0.531705i
$322$		0			0
$323$	4.97494	−	8.61684i	0.276813	−	0.479454i
$324$		0			0
$325$	5.00000	+	17.3205i	0.277350	+	0.960769i
$326$		0			0
$327$	26.5330	−	45.9565i	1.46728	−	2.54140i
$328$		0			0
$329$	11.0000	+	19.0526i	0.606450	+	1.05040i
$330$		0			0
$331$	−4.97494	−	8.61684i	−0.273447	−	0.473625i	0.696295	−	0.717756i	$-0.254831\pi$
$331$	−4.97494	−	8.61684i	−0.273447	−	0.473625i	−0.969742	+	0.244131i	$0.921497\pi$
$332$		0			0
$333$	72.0000			3.94558
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12.0000			−0.653682			−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000			−0.653682			−0.326841	+	0.945079i	$0.605984\pi$
$338$		0			0
$339$	−36.4829			−1.98148
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−9.94987			−0.537243
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.29156	+	14.3614i	0.445114	+	0.770961i	0.998060	−	0.0622565i	$-0.0198297\pi$
$347$	8.29156	+	14.3614i	0.445114	+	0.770961i	−0.552946	+	0.833217i	$0.686496\pi$
$348$		0			0
$349$	−0.500000	+	0.866025i	−0.0267644	+	0.0463573i	−0.879097	−	0.476642i	$-0.841854\pi$
$349$	−0.500000	+	0.866025i	−0.0267644	+	0.0463573i	0.852333	+	0.523000i	$0.175187\pi$
$350$		0			0
$351$	−16.5831	−	57.4456i	−0.885142	−	3.06622i
$352$		0			0
$353$	−15.5000	+	26.8468i	−0.824982	+	1.42891i	0.0769515	+	0.997035i	$0.475481\pi$
$353$	−15.5000	+	26.8468i	−0.824982	+	1.42891i	−0.901933	+	0.431875i	$0.857852\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	16.5000	+	28.5788i	0.873273	+	1.51255i
$358$		0			0
$359$	−26.5330			−1.40036			−0.700179	−	0.713967i	$-0.746897\pi$
$359$	−26.5330			−1.40036			−0.700179	+	0.713967i	$0.746897\pi$
$360$		0			0
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.6082	+	20.1060i	−0.605942	+	1.04952i	0.385959	+	0.922516i	$0.373870\pi$
$367$	−11.6082	+	20.1060i	−0.605942	+	1.04952i	−0.991902	+	0.127007i	$0.959463\pi$
$368$		0			0
$369$	−24.0000			−1.24939
$370$		0			0
$371$	13.2665	+	22.9783i	0.688762	+	1.19297i
$372$		0			0
$373$	−6.50000	−	11.2583i	−0.336557	−	0.582934i	0.647225	−	0.762299i	$-0.275929\pi$
$373$	−6.50000	−	11.2583i	−0.336557	−	0.582934i	−0.983783	+	0.179364i	$0.942596\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−17.5000	−	4.33013i	−0.901296	−	0.223013i
$378$		0			0
$379$	−14.9248	+	25.8505i	−0.766636	+	1.32785i	0.172741	+	0.984967i	$0.444738\pi$
$379$	−14.9248	+	25.8505i	−0.766636	+	1.32785i	−0.939377	+	0.342885i	$0.888596\pi$
$380$		0			0
$381$	16.5000	+	28.5788i	0.845321	+	1.46414i
$382$		0			0
$383$	−1.65831	−	2.87228i	−0.0847358	−	0.146767i	0.820543	−	0.571585i	$-0.193671\pi$
$383$	−1.65831	−	2.87228i	−0.0847358	−	0.146767i	−0.905279	+	0.424818i	$0.860338\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−39.7995	+	68.9348i	−2.02312	+	3.50415i
$388$		0			0
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$	9.94987			0.503187
$392$		0			0
$393$	−22.0000	+	38.1051i	−1.10975	+	1.92215i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.50000	−	4.33013i	−0.125471	−	0.217323i	0.796446	−	0.604710i	$-0.206711\pi$
$397$	−2.50000	−	4.33013i	−0.125471	−	0.217323i	−0.921917	+	0.387387i	$0.873378\pi$
$398$		0			0
$399$	−18.2414	−	31.5951i	−0.913214	−	1.58173i
$400$		0			0
$401$	5.50000	−	9.52628i	0.274657	−	0.475720i	−0.695392	−	0.718631i	$-0.744769\pi$
$401$	5.50000	−	9.52628i	0.274657	−	0.475720i	0.970049	+	0.242911i	$0.0781024\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.9248	−	25.8505i	−0.739795	−	1.28136i
$408$		0			0
$409$	−6.50000	−	11.2583i	−0.321404	−	0.556689i	0.659374	−	0.751815i	$-0.270822\pi$
$409$	−6.50000	−	11.2583i	−0.321404	−	0.556689i	−0.980778	+	0.195127i	$0.937488\pi$
$410$		0			0
$411$	−29.8496			−1.47237
$412$		0			0
$413$	−5.50000	+	9.52628i	−0.270637	+	0.468758i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	11.0000			0.538672
$418$		0			0
$419$	8.29156	−	14.3614i	0.405069	−	0.701601i	−0.589260	−	0.807943i	$-0.700581\pi$
$419$	8.29156	−	14.3614i	0.405069	−	0.701601i	0.994330	+	0.106343i	$0.0339141\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	26.5330	+	45.9565i	1.29008	+	2.23448i
$424$		0			0
$425$	−7.50000	−	12.9904i	−0.363803	−	0.630126i
$426$		0			0
$427$	14.9248	−	25.8505i	0.722262	−	1.25099i
$428$		0			0
$429$	−27.5000	+	28.5788i	−1.32771	+	1.37980i
$430$		0			0
$431$	−8.29156	+	14.3614i	−0.399390	+	0.691765i	−0.993651	−	0.112508i	$-0.964112\pi$
$431$	−8.29156	+	14.3614i	−0.399390	+	0.691765i	0.594260	+	0.804273i	$0.297445\pi$
$432$		0			0
$433$	−10.5000	−	18.1865i	−0.504598	−	0.873989i	−0.999986	−	0.00531724i	$-0.998307\pi$
$433$	−10.5000	−	18.1865i	−0.504598	−	0.873989i	0.495388	−	0.868672i	$-0.335026\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.0000			−0.526201
$438$		0			0
$439$	−8.29156	+	14.3614i	−0.395735	+	0.685433i	−0.993195	−	0.116466i	$-0.962843\pi$
$439$	−8.29156	+	14.3614i	−0.395735	+	0.685433i	0.597460	+	0.801899i	$0.296177\pi$
$440$		0			0
$441$	32.0000			1.52381
$442$		0			0
$443$	−13.2665			−0.630310			−0.315155	−	0.949040i	$-0.602057\pi$
$443$	−13.2665			−0.630310			−0.315155	+	0.949040i	$0.602057\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−3.31662			−0.156871
$448$		0			0
$449$	1.50000	+	2.59808i	0.0707894	+	0.122611i	0.899247	−	0.437440i	$-0.144115\pi$
$449$	1.50000	+	2.59808i	0.0707894	+	0.122611i	−0.828458	+	0.560051i	$0.810782\pi$
$450$		0			0
$451$	4.97494	+	8.61684i	0.234261	+	0.405751i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	−0.854094	−	0.520119i	$-0.825888\pi$
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	0.877483	+	0.479608i	$0.159221\pi$
$458$		0			0
$459$	24.8747	+	43.0842i	1.16105	+	2.01100i
$460$		0			0
$461$	1.50000	+	2.59808i	0.0698620	+	0.121004i	0.898840	−	0.438276i	$-0.144411\pi$
$461$	1.50000	+	2.59808i	0.0698620	+	0.121004i	−0.828978	+	0.559281i	$0.811077\pi$
$462$		0			0
$463$	−26.5330			−1.23309			−0.616547	−	0.787318i	$-0.711469\pi$
$463$	−26.5330			−1.23309			−0.616547	+	0.787318i	$0.711469\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.8997			0.920851			0.460425	−	0.887698i	$-0.347697\pi$
$467$	19.8997			0.920851			0.460425	+	0.887698i	$0.347697\pi$
$468$		0			0
$469$	33.0000			1.52380
$470$		0			0
$471$	13.2665	−	22.9783i	0.611288	−	1.05878i
$472$		0			0
$473$	33.0000			1.51734
$474$		0			0
$475$	8.29156	+	14.3614i	0.380443	+	0.658947i
$476$		0			0
$477$	32.0000	+	55.4256i	1.46518	+	2.53777i
$478$		0			0
$479$	4.97494	−	8.61684i	0.227311	−	0.393714i	−0.729700	−	0.683768i	$-0.760340\pi$
$479$	4.97494	−	8.61684i	0.227311	−	0.393714i	0.957010	+	0.290054i	$0.0936734\pi$
$480$		0			0
$481$	31.5000	+	7.79423i	1.43628	+	0.355386i
$482$		0			0
$483$	18.2414	−	31.5951i	0.830014	−	1.43763i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.97494	−	8.61684i	−0.225436	−	0.390466i	0.731014	−	0.682362i	$-0.239047\pi$
$487$	−4.97494	−	8.61684i	−0.225436	−	0.390466i	−0.956450	+	0.291896i	$0.905714\pi$
$488$		0			0
$489$	−55.0000			−2.48719
$490$		0			0
$491$	8.29156	−	14.3614i	0.374193	−	0.648121i	−0.616013	−	0.787736i	$-0.711253\pi$
$491$	8.29156	−	14.3614i	0.374193	−	0.648121i	0.990206	+	0.139615i	$0.0445864\pi$
$492$		0			0
$493$	15.0000			0.675566
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.5000	+	28.5788i	−0.740126	+	1.28194i
$498$		0			0
$499$	−13.2665			−0.593890			−0.296945	−	0.954895i	$-0.595968\pi$
$499$	−13.2665			−0.593890			−0.296945	+	0.954895i	$0.595968\pi$
$500$		0			0
$501$	5.50000	+	9.52628i	0.245722	+	0.425603i
$502$		0			0
$503$	−1.65831	−	2.87228i	−0.0739405	−	0.128069i	0.826685	−	0.562666i	$-0.190224\pi$
$503$	−1.65831	−	2.87228i	−0.0739405	−	0.128069i	−0.900625	+	0.434597i	$0.856891\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.65831	−	43.0842i	−0.0736482	−	1.91344i
$508$		0			0
$509$	19.5000	−	33.7750i	0.864322	−	1.49705i	−0.00339621	−	0.999994i	$-0.501081\pi$
$509$	19.5000	−	33.7750i	0.864322	−	1.49705i	0.867719	−	0.497056i	$-0.165586\pi$
$510$		0			0
$511$	6.63325	+	11.4891i	0.293438	+	0.508249i
$512$		0			0
$513$	−27.5000	−	47.6314i	−1.21415	−	2.10298i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	11.0000	−	19.0526i	0.483779	−	0.837931i
$518$		0			0
$519$	23.2164			1.01909
$520$		0			0
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		0			0
$523$	8.29156	−	14.3614i	0.362565	−	0.627980i	−0.625817	−	0.779970i	$-0.715235\pi$
$523$	8.29156	−	14.3614i	0.362565	−	0.627980i	0.988382	+	0.151989i	$0.0485679\pi$
$524$		0			0
$525$	−55.0000			−2.40040
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6.00000	+	10.3923i	0.260870	+	0.451839i
$530$		0			0
$531$	−13.2665	+	22.9783i	−0.575717	+	0.997171i
$532$		0			0
$533$	−10.5000	−	2.59808i	−0.454805	−	0.112535i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−5.50000	−	9.52628i	−0.237343	−	0.411089i
$538$		0			0
$539$	−6.63325	−	11.4891i	−0.285714	−	0.494872i
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$	13.2665	−	22.9783i	0.569320	−	0.986091i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−13.2665			−0.567235			−0.283617	−	0.958938i	$-0.591535\pi$
$547$	−13.2665			−0.567235			−0.283617	+	0.958938i	$0.591535\pi$
$548$		0			0
$549$	36.0000	−	62.3538i	1.53644	−	2.66120i
$550$		0			0
$551$	−16.5831			−0.706465
$552$		0			0
$553$	−11.0000	−	19.0526i	−0.467768	−	0.810197i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.5000	−	32.0429i	0.783870	−	1.35770i	−0.145802	−	0.989314i	$-0.546576\pi$
$557$	18.5000	−	32.0429i	0.783870	−	1.35770i	0.929672	−	0.368389i	$-0.120091\pi$
$558$		0			0
$559$	−24.8747	+	25.8505i	−1.05209	+	1.09336i
$560$		0			0
$561$	16.5000	−	28.5788i	0.696631	−	1.20660i
$562$		0			0
$563$	−1.65831	−	2.87228i	−0.0698895	−	0.121052i	0.828963	−	0.559303i	$-0.188931\pi$
$563$	−1.65831	−	2.87228i	−0.0698895	−	0.121052i	−0.898853	+	0.438251i	$0.855598\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	102.815			4.31784
$568$		0			0
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	−0.979313	−	0.202350i	$-0.935142\pi$
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	0.664897	+	0.746935i	$0.268475\pi$
$570$		0			0
$571$	13.2665			0.555186			0.277593	−	0.960699i	$-0.410463\pi$
$571$	13.2665			0.555186			0.277593	+	0.960699i	$0.410463\pi$
$572$		0			0
$573$	−77.0000			−3.21672
$574$		0			0
$575$	−8.29156	+	14.3614i	−0.345782	+	0.598912i
$576$		0			0
$577$	20.0000			0.832611			0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000			0.832611			0.416305	+	0.909225i	$0.363325\pi$
$578$		0			0
$579$	−21.5581	−	37.3397i	−0.895922	−	1.55178i
$580$		0			0
$581$	22.0000	+	38.1051i	0.912714	+	1.58087i
$582$		0			0
$583$	13.2665	−	22.9783i	0.549442	−	0.951662i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.5581	−	37.3397i	0.889796	−	1.54117i	0.0496808	−	0.998765i	$-0.484180\pi$
$587$	21.5581	−	37.3397i	0.889796	−	1.54117i	0.840116	−	0.542407i	$-0.182487\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	21.5581	+	37.3397i	0.886780	+	1.53595i
$592$		0			0
$593$	−36.0000			−1.47834			−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000			−1.47834			−0.739171	+	0.673517i	$0.764783\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	33.0000			1.35060
$598$		0			0
$599$	−33.1662			−1.35514			−0.677568	−	0.735460i	$-0.736966\pi$
$599$	−33.1662			−1.35514			−0.677568	+	0.735460i	$0.736966\pi$
$600$		0			0
$601$	−14.5000	+	25.1147i	−0.591467	+	1.02445i	0.402568	+	0.915390i	$0.368118\pi$
$601$	−14.5000	+	25.1147i	−0.591467	+	1.02445i	−0.994035	+	0.109061i	$0.965216\pi$
$602$		0			0
$603$	79.5990			3.24152
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.97494	−	8.61684i	−0.201926	−	0.349747i	0.747223	−	0.664574i	$-0.231387\pi$
$607$	−4.97494	−	8.61684i	−0.201926	−	0.349747i	−0.949149	+	0.314827i	$0.898054\pi$
$608$		0			0
$609$	27.5000	−	47.6314i	1.11436	−	1.93012i
$610$		0			0
$611$	6.63325	+	22.9783i	0.268353	+	0.929601i
$612$		0			0
$613$	3.50000	−	6.06218i	0.141364	−	0.244849i	−0.786647	−	0.617403i	$-0.788185\pi$
$613$	3.50000	−	6.06218i	0.141364	−	0.244849i	0.928010	+	0.372554i	$0.121518\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.50000	+	14.7224i	0.342197	+	0.592703i	0.984840	−	0.173463i	$-0.0554956\pi$
$617$	8.50000	+	14.7224i	0.342197	+	0.592703i	−0.642643	+	0.766165i	$0.722162\pi$
$618$		0			0
$619$	−39.7995			−1.59968			−0.799838	−	0.600215i	$-0.795082\pi$
$619$	−39.7995			−1.59968			−0.799838	+	0.600215i	$0.795082\pi$
$620$		0			0
$621$	27.5000	−	47.6314i	1.10354	−	1.91138i
$622$		0			0
$623$	−3.31662			−0.132878
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−18.2414	+	31.5951i	−0.728493	+	1.26179i
$628$		0			0
$629$	−27.0000			−1.07656
$630$		0			0
$631$	−4.97494	−	8.61684i	−0.198049	−	0.343031i	0.749847	−	0.661612i	$-0.230127\pi$
$631$	−4.97494	−	8.61684i	−0.198049	−	0.343031i	−0.947896	+	0.318580i	$0.896794\pi$
$632$		0			0
$633$	−5.50000	−	9.52628i	−0.218605	−	0.378636i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.0000	+	3.46410i	0.554700	+	0.137253i
$638$		0			0
$639$	−39.7995	+	68.9348i	−1.57444	+	2.72702i
$640$		0			0
$641$	−10.5000	−	18.1865i	−0.414725	−	0.718325i	0.580674	−	0.814136i	$-0.302789\pi$
$641$	−10.5000	−	18.1865i	−0.414725	−	0.718325i	−0.995400	+	0.0958109i	$0.969456\pi$
$642$		0			0
$643$	−18.2414	−	31.5951i	−0.719372	−	1.24599i	−0.961249	−	0.275682i	$-0.911096\pi$
$643$	−18.2414	−	31.5951i	−0.719372	−	1.24599i	0.241877	−	0.970307i	$-0.422237\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.97494	−	8.61684i	0.195585	−	0.338763i	−0.751507	−	0.659725i	$-0.770673\pi$
$647$	4.97494	−	8.61684i	0.195585	−	0.338763i	0.947092	+	0.320962i	$0.104006\pi$
$648$		0			0
$649$	11.0000			0.431788
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	−0.681137	−	0.732156i	$-0.738514\pi$
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	0.974634	+	0.223803i	$0.0718474\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	16.0000	+	27.7128i	0.624219	+	1.08118i
$658$		0			0
$659$	21.5581	+	37.3397i	0.839783	+	1.45455i	0.890076	+	0.455812i	$0.150651\pi$
$659$	21.5581	+	37.3397i	0.839783	+	1.45455i	−0.0502932	+	0.998734i	$0.516016\pi$
$660$		0			0
$661$	−8.50000	+	14.7224i	−0.330612	+	0.572636i	−0.982632	−	0.185565i	$-0.940588\pi$
$661$	−8.50000	+	14.7224i	−0.330612	+	0.572636i	0.652020	+	0.758202i	$0.273922\pi$
$662$		0			0
$663$	9.94987	+	34.4674i	0.386421	+	1.33860i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.29156	−	14.3614i	−0.321051	−	0.556076i
$668$		0			0
$669$	5.50000	+	9.52628i	0.212642	+	0.368307i
$670$		0			0
$671$	−29.8496			−1.15233
$672$		0			0
$673$	24.5000	−	42.4352i	0.944406	−	1.63576i	0.187469	−	0.982271i	$-0.439972\pi$
$673$	24.5000	−	42.4352i	0.944406	−	1.63576i	0.756937	−	0.653488i	$-0.226695\pi$
$674$		0			0
$675$	−82.9156			−3.19142
$676$		0			0
$677$	−24.0000			−0.922395			−0.461197	−	0.887298i	$-0.652580\pi$
$677$	−24.0000			−0.922395			−0.461197	+	0.887298i	$0.652580\pi$
$678$		0			0
$679$	−11.6082	+	20.1060i	−0.445481	+	0.771596i
$680$		0			0
$681$	11.0000			0.421521
$682$		0			0
$683$	−14.9248	−	25.8505i	−0.571082	−	0.989143i	−0.996455	−	0.0841251i	$-0.973190\pi$
$683$	−14.9248	−	25.8505i	−0.571082	−	0.989143i	0.425373	−	0.905018i	$-0.360143\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−13.2665	+	22.9783i	−0.506149	+	0.876675i
$688$		0			0
$689$	8.00000	+	27.7128i	0.304776	+	1.05577i
$690$		0			0
$691$	11.6082	−	20.1060i	0.441596	−	0.764867i	−0.556212	−	0.831041i	$-0.687746\pi$
$691$	11.6082	−	20.1060i	0.441596	−	0.764867i	0.997808	+	0.0661734i	$0.0210790\pi$
$692$		0			0
$693$	−44.0000	−	76.2102i	−1.67142	−	2.89499i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	9.00000			0.340899
$698$		0			0
$699$	−6.63325	+	11.4891i	−0.250893	+	0.434559i
$700$		0			0
$701$	−32.0000			−1.20862			−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000			−1.20862			−0.604312	+	0.796748i	$0.706552\pi$
$702$		0			0
$703$	29.8496			1.12580
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−49.7494			−1.87102
$708$		0			0
$709$	−14.5000	−	25.1147i	−0.544559	−	0.943204i	−0.998635	−	0.0522406i	$-0.983364\pi$
$709$	−14.5000	−	25.1147i	−0.544559	−	0.943204i	0.454076	−	0.890963i	$-0.349970\pi$
$710$		0			0
$711$	−26.5330	−	45.9565i	−0.995065	−	1.72350i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−44.0000	+	76.2102i	−1.64321	+	2.84612i
$718$		0			0
$719$	−14.9248	−	25.8505i	−0.556602	−	0.964062i	−0.997777	−	0.0666413i	$-0.978772\pi$
$719$	−14.9248	−	25.8505i	−0.556602	−	0.964062i	0.441175	−	0.897421i	$-0.354562\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	43.1161			1.60351
$724$		0			0
$725$	−12.5000	+	21.6506i	−0.464238	+	0.804084i
$726$		0			0
$727$	−26.5330			−0.984054			−0.492027	−	0.870580i	$-0.663744\pi$
$727$	−26.5330			−0.984054			−0.492027	+	0.870580i	$0.663744\pi$
$728$		0			0
$729$	83.0000			3.07407
$730$		0			0
$731$	14.9248	−	25.8505i	0.552014	−	0.956116i
$732$		0			0
$733$	18.0000			0.664845			0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000			0.664845			0.332423	+	0.943131i	$0.392134\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.5000	−	28.5788i	−0.607785	−	1.05272i
$738$		0			0
$739$	−4.97494	+	8.61684i	−0.183006	+	0.316976i	−0.942903	−	0.333068i	$-0.891916\pi$
$739$	−4.97494	+	8.61684i	−0.183006	+	0.316976i	0.759897	+	0.650044i	$0.225249\pi$
$740$		0			0
$741$	−11.0000	−	38.1051i	−0.404095	−	1.39983i
$742$		0			0
$743$	−21.5581	+	37.3397i	−0.790889	+	1.36986i	0.134529	+	0.990910i	$0.457048\pi$
$743$	−21.5581	+	37.3397i	−0.790889	+	1.36986i	−0.925417	+	0.378949i	$0.876285\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	53.0660	+	91.9130i	1.94158	+	3.36292i
$748$		0			0
$749$	11.0000			0.401931
$750$		0			0
$751$	14.9248	−	25.8505i	0.544614	−	0.943299i	−0.454017	−	0.890993i	$-0.650009\pi$
$751$	14.9248	−	25.8505i	0.544614	−	0.943299i	0.998631	−	0.0523063i	$-0.0166572\pi$
$752$		0			0
$753$	−55.0000			−2.00431
$754$		0			0
$755$		0			0
$756$		0			0
$757$	3.50000	−	6.06218i	0.127210	−	0.220334i	−0.795385	−	0.606105i	$-0.792731\pi$
$757$	3.50000	−	6.06218i	0.127210	−	0.220334i	0.922595	+	0.385771i	$0.126065\pi$
$758$		0			0
$759$	−36.4829			−1.32424
$760$		0			0
$761$	13.5000	+	23.3827i	0.489375	+	0.847622i	0.999925	−	0.0122260i	$-0.00389175\pi$
$761$	13.5000	+	23.3827i	0.489375	+	0.847622i	−0.510551	+	0.859848i	$0.670558\pi$
$762$		0			0
$763$	26.5330	+	45.9565i	0.960559	+	1.66374i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.29156	+	8.61684i	−0.299391	+	0.311136i
$768$		0			0
$769$	13.5000	−	23.3827i	0.486822	−	0.843201i	−0.513063	−	0.858351i	$-0.671489\pi$
$769$	13.5000	−	23.3827i	0.486822	−	0.843201i	0.999885	+	0.0151499i	$0.00482254\pi$
$770$		0			0
$771$	−21.5581	−	37.3397i	−0.776395	−	1.34475i
$772$		0			0
$773$	−10.5000	−	18.1865i	−0.377659	−	0.654124i	0.613062	−	0.790034i	$-0.289937\pi$
$773$	−10.5000	−	18.1865i	−0.377659	−	0.654124i	−0.990721	+	0.135910i	$0.956604\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−49.5000	+	85.7365i	−1.77580	+	3.07578i
$778$		0			0
$779$	−9.94987			−0.356491
$780$		0			0
$781$	33.0000			1.18083
$782$		0			0
$783$	41.4578	−	71.8070i	1.48158	−	2.56617i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−4.97494	−	8.61684i	−0.177337	−	0.307157i	0.763630	−	0.645654i	$-0.223415\pi$
$787$	−4.97494	−	8.61684i	−0.177337	−	0.307157i	−0.940968	+	0.338496i	$0.890082\pi$
$788$		0			0
$789$	−38.5000	−	66.6840i	−1.37064	−	2.37401i
$790$		0			0
$791$	18.2414	−	31.5951i	0.648591	−	1.12339i
$792$		0			0
$793$	22.5000	−	23.3827i	0.798998	−	0.830344i

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.65831 + 2.87228i) q^{3} +(-1.65831 - 2.87228i) q^{7} +(-4.00000 - 6.92820i) q^{9} +O(q^{10})\)	\(q+(-1.65831 + 2.87228i) q^{3} +(-1.65831 - 2.87228i) q^{7} +(-4.00000 - 6.92820i) q^{9} +(-1.65831 + 2.87228i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{17} +(-1.65831 - 2.87228i) q^{19} +11.0000 q^{21} +(1.65831 - 2.87228i) q^{23} -5.00000 q^{25} +16.5831 q^{27} +(2.50000 - 4.33013i) q^{29} +(-5.50000 - 9.52628i) q^{33} +(-4.50000 + 7.79423i) q^{37} +(11.6082 + 2.87228i) q^{39} +(1.50000 - 2.59808i) q^{41} +(-4.97494 - 8.61684i) q^{43} -6.63325 q^{47} +(-2.00000 + 3.46410i) q^{49} -9.94987 q^{51} -8.00000 q^{53} +11.0000 q^{57} +(-1.65831 - 2.87228i) q^{59} +(4.50000 + 7.79423i) q^{61} +(-13.2665 + 22.9783i) q^{63} +(-4.97494 + 8.61684i) q^{67} +(5.50000 + 9.52628i) q^{69} +(-4.97494 - 8.61684i) q^{71} -4.00000 q^{73} +(8.29156 - 14.3614i) q^{75} +11.0000 q^{77} +6.63325 q^{79} +(-15.5000 + 26.8468i) q^{81} -13.2665 q^{83} +(8.29156 + 14.3614i) q^{87} +(0.500000 - 0.866025i) q^{89} +(-8.29156 + 8.61684i) q^{91} +(-3.50000 - 6.06218i) q^{97} +26.5330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{9}+O(q^{10}) \) 4 * q - 16 * q^9	\( 4 q - 16 q^{9} - 4 q^{13} + 6 q^{17} + 44 q^{21} - 20 q^{25} + 10 q^{29} - 22 q^{33} - 18 q^{37} + 6 q^{41} - 8 q^{49} - 32 q^{53} + 44 q^{57} + 18 q^{61} + 22 q^{69} - 16 q^{73} + 44 q^{77} - 62 q^{81} + 2 q^{89} - 14 q^{97}+O(q^{100}) \) 4 * q - 16 * q^9 - 4 * q^13 + 6 * q^17 + 44 * q^21 - 20 * q^25 + 10 * q^29 - 22 * q^33 - 18 * q^37 + 6 * q^41 - 8 * q^49 - 32 * q^53 + 44 * q^57 + 18 * q^61 + 22 * q^69 - 16 * q^73 + 44 * q^77 - 62 * q^81 + 2 * q^89 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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