LMFDB - Embedded newform 93.2.p.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [93,2,Mod(11,93)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(93, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 23]))

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

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

chi := DirichletCharacter("93.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 93 = 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	93.p (of order $30$, degree $8$, 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:	$0.742608738798$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

Embedding invariants

Embedding label			11.1
Root			$0.913545 + 0.406737i$ of defining polynomial
Character	$\chi$	$=$	93.11
Dual form			93.2.p.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28716 + 1.15897i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(0.473579 + 4.50581i) q^{7} +(0.313585 - 2.98357i) q^{9} +O(q^{10})$	$q+(-1.28716 + 1.15897i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(0.473579 + 4.50581i) q^{7} +(0.313585 - 2.98357i) q^{9} +(0.720227 - 3.38840i) q^{12} +(-0.262977 - 1.23721i) q^{13} +(1.23607 - 3.80423i) q^{16} +(8.52685 + 1.81244i) q^{19} +(-5.83166 - 5.25085i) q^{21} +(-2.50000 + 4.33013i) q^{25} +(3.05422 + 4.20378i) q^{27} +(-6.06316 - 6.73382i) q^{28} +(-4.32380 - 3.50781i) q^{31} +(3.00000 + 5.19615i) q^{36} +(7.66945 + 4.42796i) q^{37} +(1.77238 + 1.28771i) q^{39} +(0.209710 - 0.986609i) q^{43} +(2.81795 + 6.32923i) q^{48} +(-13.2310 + 2.81233i) q^{49} +(1.87993 + 1.69270i) q^{52} +(-13.0760 + 7.54944i) q^{57} -13.8340i q^{61} +13.5919 q^{63} +(2.47214 + 7.60845i) q^{64} +(5.45406 + 9.44671i) q^{67} +(6.79380 - 15.2591i) q^{73} +(-1.80057 - 8.47101i) q^{75} +(-15.9274 + 7.09132i) q^{76} +(-6.65573 - 14.9490i) q^{79} +(-8.80333 - 1.87121i) q^{81} +(15.6086 + 1.64053i) q^{84} +(5.45008 - 1.77084i) q^{91} +(9.63089 - 0.496015i) q^{93} +(12.3326 - 8.96014i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9} + 6 q^{12} + 9 q^{13} - 8 q^{16} + 7 q^{19} - 33 q^{21} - 20 q^{25} - 18 q^{28} - 4 q^{31} + 24 q^{36} + 9 q^{37} + 27 q^{39} + 44 q^{43} + 12 q^{48} + 9 q^{49} + 18 q^{52} - 21 q^{57} - 24 q^{63} - 16 q^{64} + 16 q^{67} + 3 q^{73} - 15 q^{75} - 66 q^{76} - 35 q^{79} + 9 q^{81} + 24 q^{84} - 5 q^{91} + 15 q^{93} + 15 q^{97}+O(q^{100}) $ 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9 + 6 * q^12 + 9 * q^13 - 8 * q^16 + 7 * q^19 - 33 * q^21 - 20 * q^25 - 18 * q^28 - 4 * q^31 + 24 * q^36 + 9 * q^37 + 27 * q^39 + 44 * q^43 + 12 * q^48 + 9 * q^49 + 18 * q^52 - 21 * q^57 - 24 * q^63 - 16 * q^64 + 16 * q^67 + 3 * q^73 - 15 * q^75 - 66 * q^76 - 35 * q^79 + 9 * q^81 + 24 * q^84 - 5 * q^91 + 15 * q^93 + 15 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/93\mathbb{Z}\right)^\times$.

$n$	$32$	$34$
$\chi(n)$	$-1$	$e\left(\frac{23}{30}\right)$

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		−0.309017	−	0.951057i	$-0.600000\pi$
$2$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$3$	−1.28716	+	1.15897i	−0.743145	+	0.669131i
$3$	−1.28716	+	1.15897i	−0.743145	+	0.669131i
$4$	−1.61803	+	1.17557i	−0.809017	+	0.587785i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	0.473579	+	4.50581i	0.178996	+	1.70303i	0.603300	+	0.797514i	$0.293852\pi$
$7$	0.473579	+	4.50581i	0.178996	+	1.70303i	−0.424304	+	0.905520i	$0.639481\pi$
$8$		0			0
$9$	0.313585	−	2.98357i	0.104528	−	0.994522i
$10$		0			0
$11$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$11$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$12$	0.720227	−	3.38840i	0.207912	−	0.978148i
$13$	−0.262977	−	1.23721i	−0.0729367	−	0.343140i	0.926515	−	0.376258i	$-0.122790\pi$
$13$	−0.262977	−	1.23721i	−0.0729367	−	0.343140i	−0.999451	+	0.0331183i	$0.989456\pi$
$14$		0			0
$15$		0			0
$16$	1.23607	−	3.80423i	0.309017	−	0.951057i
$17$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$17$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$18$		0			0
$19$	8.52685	+	1.81244i	1.95619	+	0.415802i	0.980226	+	0.197884i	$0.0634068\pi$
$19$	8.52685	+	1.81244i	1.95619	+	0.415802i	0.975967	+	0.217918i	$0.0699265\pi$
$20$		0			0
$21$	−5.83166	−	5.25085i	−1.27257	−	1.14583i
$22$		0			0
$23$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$23$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$	3.05422	+	4.20378i	0.587785	+	0.809017i
$28$	−6.06316	−	6.73382i	−1.14583	−	1.27257i
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$	−4.32380	−	3.50781i	−0.776578	−	0.630022i
$31$	−4.32380	−	3.50781i	−0.776578	−	0.630022i
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.00000	+	5.19615i	0.500000	+	0.866025i
$37$	7.66945	+	4.42796i	1.26085	+	0.727952i	0.973239	−	0.229795i	$-0.0738055\pi$
$37$	7.66945	+	4.42796i	1.26085	+	0.727952i	0.287611	+	0.957747i	$0.407139\pi$
$38$		0			0
$39$	1.77238	+	1.28771i	0.283808	+	0.206199i
$40$		0			0
$41$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$41$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$42$		0			0
$43$	0.209710	−	0.986609i	0.0319805	−	0.150456i	−0.959260	−	0.282524i	$-0.908828\pi$
$43$	0.209710	−	0.986609i	0.0319805	−	0.150456i	0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$	2.81795	+	6.32923i	0.406737	+	0.913545i
$49$	−13.2310	+	2.81233i	−1.89014	+	0.401761i
$50$		0			0
$51$		0			0
$52$	1.87993	+	1.69270i	0.260700	+	0.234735i
$53$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$53$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−13.0760	+	7.54944i	−1.73196	+	0.999948i
$58$		0			0
$59$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$59$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$60$		0			0
$61$		−	13.8340i		−	1.77127i	−0.464386	−	0.885633i	$-0.653725\pi$
$61$		−	13.8340i		−	1.77127i	0.464386	−	0.885633i	$-0.346275\pi$
$62$		0			0
$63$	13.5919			1.71242
$64$	2.47214	+	7.60845i	0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	5.45406	+	9.44671i	0.666320	+	1.15410i	0.978926	+	0.204217i	$0.0654647\pi$
$67$	5.45406	+	9.44671i	0.666320	+	1.15410i	−0.312606	+	0.949883i	$0.601202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$71$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$72$		0			0
$73$	6.79380	−	15.2591i	0.795154	−	1.78595i	0.204795	−	0.978805i	$-0.434347\pi$
$73$	6.79380	−	15.2591i	0.795154	−	1.78595i	0.590359	−	0.807141i	$-0.298986\pi$
$74$		0			0
$75$	−1.80057	−	8.47101i	−0.207912	−	0.978148i
$76$	−15.9274	+	7.09132i	−1.82700	+	0.813431i
$77$		0			0
$78$		0			0
$79$	−6.65573	−	14.9490i	−0.748828	−	1.68189i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−6.65573	−	14.9490i	−0.748828	−	1.68189i	−0.0175207	−	0.999847i	$-0.505577\pi$
$80$		0			0
$81$	−8.80333	−	1.87121i	−0.978148	−	0.207912i
$82$		0			0
$83$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$83$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$84$	15.6086	+	1.64053i	1.70303	+	0.178996i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$89$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$90$		0			0
$91$	5.45008	−	1.77084i	0.571324	−	0.185634i
$92$		0			0
$93$	9.63089	−	0.496015i	0.998676	−	0.0514344i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.3326	−	8.96014i	1.25218	−	0.909764i	0.253837	−	0.967247i	$-0.418307\pi$
$97$	12.3326	−	8.96014i	1.25218	−	0.909764i	0.998346	+	0.0574829i	$0.0183075\pi$
$98$		0			0
$99$		0			0
$100$	−1.04528	−	9.94522i	−0.104528	−	0.994522i
$101$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$101$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$102$		0			0
$103$	−7.23215	+	8.03212i	−0.712605	+	0.791428i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−7.23215	+	8.03212i	−0.712605	+	0.791428i	0.272724	+	0.962092i	$0.412075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$107$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$108$	−9.88367	−	3.21140i	−0.951057	−	0.309017i
$109$	−1.93990	+	5.97041i	−0.185809	+	0.571861i	−0.999961	−	0.00878995i	$-0.997202\pi$
$109$	−1.93990	+	5.97041i	−0.185809	+	0.571861i	0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$	−15.0037	+	3.18914i	−1.42409	+	0.302700i
$112$	17.7265	+	3.76788i	1.67499	+	0.356031i
$113$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$113$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.77376	+	0.396638i	−0.348884	+	0.0366692i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.36044	−	8.17459i	−0.669131	−	0.743145i
$122$		0			0
$123$		0			0
$124$	11.1197	+	0.592827i	0.998582	+	0.0532375i
$125$		0			0
$126$		0			0
$127$	−13.1006	+	11.7958i	−1.16249	+	1.04671i	−0.164309	+	0.986409i	$0.552539\pi$
$127$	−13.1006	+	11.7958i	−1.16249	+	1.04671i	−0.998180	+	0.0603008i	$0.980794\pi$
$128$		0			0
$129$	0.873517	+	1.51298i	0.0769089	+	0.133210i
$130$		0			0
$131$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$131$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$132$		0			0
$133$	−4.12835	+	39.2786i	−0.357973	+	3.40589i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$137$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$138$		0			0
$139$	9.38177	+	3.04832i	0.795751	+	0.258555i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	9.38177	+	3.04832i	0.795751	+	0.258555i	0.117200	+	0.993108i	$0.462608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−10.9625	−	4.88084i	−0.913545	−	0.406737i
$145$		0			0
$146$		0			0
$147$	13.7710	−	18.9542i	1.13582	−	1.56332i
$148$	−17.6148	+	1.85139i	−1.44793	+	0.152183i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	14.4304	+	19.8617i	1.17433	+	1.61633i	0.624330	+	0.781161i	$0.285372\pi$
$151$	14.4304	+	19.8617i	1.17433	+	1.61633i	0.550000	+	0.835165i	$0.314628\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.38157			−0.350806
$157$	−1.33983	−	4.12357i	−0.106930	−	0.329096i	0.883249	−	0.468905i	$-0.155351\pi$
$157$	−1.33983	−	4.12357i	−0.106930	−	0.329096i	−0.990179	+	0.139808i	$0.955351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.25197	+	1.63615i	0.176388	+	0.128153i	0.672476	−	0.740119i	$-0.265231\pi$
$163$	2.25197	+	1.63615i	0.176388	+	0.128153i	−0.496088	+	0.868272i	$0.665231\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$167$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$168$		0			0
$169$	10.4146	−	4.63686i	0.801120	−	0.356682i
$170$		0			0
$171$	8.08142	−	24.8721i	0.618002	−	1.90201i
$172$	0.820510	+	1.84290i	0.0625633	+	0.140520i
$173$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$173$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$174$		0			0
$175$	−20.6947	−	9.21385i	−1.56437	−	0.696502i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$179$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$180$		0			0
$181$	12.7184	−	7.34295i	0.945348	−	0.545797i	0.0537152	−	0.998556i	$-0.482894\pi$
$181$	12.7184	−	7.34295i	0.945348	−	0.545797i	0.891633	+	0.452759i	$0.149560\pi$
$182$		0			0
$183$	16.0332	+	17.8067i	1.18521	+	1.31631i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−17.4950	+	15.7525i	−1.27257	+	1.14583i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−12.0000	−	6.92820i	−0.866025	−	0.500000i
$193$	1.36921	+	13.0272i	0.0985578	+	0.937715i	0.926346	+	0.376674i	$0.122932\pi$
$193$	1.36921	+	13.0272i	0.0985578	+	0.937715i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		0			0
$195$		0			0
$196$	18.1021	−	20.1044i	1.29300	−	1.43603i
$197$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$197$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$198$		0			0
$199$	−5.31404	−	25.0006i	−0.376702	−	1.77224i	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−5.31404	−	25.0006i	−0.376702	−	1.77224i	0.225847	−	0.974163i	$-0.427485\pi$
$200$		0			0
$201$	−17.9687	−	5.83839i	−1.26742	−	0.411808i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−5.03168	−	0.528851i	−0.348884	−	0.0366692i
$209$		0			0
$210$		0			0
$211$	−3.65711	+	6.33430i	−0.251766	+	0.436071i	−0.964012	−	0.265859i	$-0.914345\pi$
$211$	−3.65711	+	6.33430i	−0.251766	+	0.436071i	0.712246	+	0.701930i	$0.247678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	13.7579	−	21.1434i	0.933944	−	1.43531i
$218$		0			0
$219$	8.94010	+	27.5148i	0.604116	+	1.85928i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−25.0566	−	14.4665i	−1.67792	−	0.968745i	−0.962987	−	0.269549i	$-0.913126\pi$
$223$	−25.0566	−	14.4665i	−1.67792	−	0.968745i	−0.714929	−	0.699197i	$-0.753541\pi$
$224$		0			0
$225$	12.1353	+	8.81678i	0.809017	+	0.587785i
$226$		0			0
$227$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$227$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$228$	12.2825	−	27.5870i	0.813431	−	1.82700i
$229$	2.08733	−	9.82009i	0.137934	−	0.648930i	−0.853798	−	0.520605i	$-0.825707\pi$
$229$	2.08733	−	9.82009i	0.137934	−	0.648930i	0.991732	−	0.128325i	$-0.0409601\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$233$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	25.8924	+	11.5281i	1.68189	+	0.748828i
$238$		0			0
$239$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$239$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$240$		0			0
$241$	1.72256	−	0.181049i	0.110960	−	0.0116624i	−0.0488857	−	0.998804i	$-0.515567\pi$
$241$	1.72256	−	0.181049i	0.110960	−	0.0116624i	0.159846	+	0.987142i	$0.448900\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$	16.2629	+	22.3839i	1.04112	+	1.43298i
$245$		0			0
$246$		0			0
$247$		−	11.0261i		−	0.701575i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$251$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$252$	−21.9921	+	15.9782i	−1.38537	+	1.00653i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	−	9.40456i	−0.809017	−	0.587785i
$257$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$257$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$258$		0			0
$259$	−16.3194	+	36.6541i	−1.01404	+	2.27757i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$263$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−19.9301	−	8.87347i	−1.21743	−	0.542033i
$269$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$269$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$270$		0			0
$271$	−19.0429	+	26.2103i	−1.15677	+	1.59216i	−0.434296	+	0.900770i	$0.643003\pi$
$271$	−19.0429	+	26.2103i	−1.15677	+	1.59216i	−0.722479	+	0.691393i	$0.756997\pi$
$272$		0			0
$273$	−4.96281	+	8.59584i	−0.300363	+	0.520244i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.5309	+	3.74663i	−0.692827	+	0.225113i	−0.634203	−	0.773167i	$-0.718672\pi$
$277$	−11.5309	+	3.74663i	−0.692827	+	0.225113i	−0.0586248	+	0.998280i	$0.518672\pi$
$278$		0			0
$279$	−11.8217	+	11.8003i	−0.707745	+	0.706468i
$280$		0			0
$281$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$281$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$	27.0708	−	19.6681i	1.60919	−	1.16915i	0.743242	−	0.669023i	$-0.233287\pi$
$283$	27.0708	−	19.6681i	1.60919	−	1.16915i	0.865953	−	0.500125i	$-0.166713\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−11.3752	+	12.6335i	−0.669131	+	0.743145i
$290$		0			0
$291$	−5.48954	+	25.8262i	−0.321802	+	1.51396i
$292$	6.94558	+	32.6764i	0.406459	+	1.91224i
$293$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$293$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	12.8716	+	11.5897i	0.743145	+	0.669131i
$301$	4.54478	+	0.477676i	0.261957	+	0.0275328i
$302$		0			0
$303$		0			0
$304$	17.4347	−	30.1978i	0.999948	−	1.73196i
$305$		0			0
$306$		0			0
$307$	−21.8662	−	24.2849i	−1.24797	−	1.38601i	−0.892325	−	0.451393i	$-0.850927\pi$
$307$	−21.8662	−	24.2849i	−1.24797	−	1.38601i	−0.355647	−	0.934620i	$-0.615739\pi$
$308$		0			0
$309$		−	18.7205i		−	1.06497i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	16.7454	−	15.0776i	0.946506	−	0.852238i	−0.0426523	−	0.999090i	$-0.513581\pi$
$313$	16.7454	−	15.0776i	0.946506	−	0.852238i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$		0			0
$316$	28.3428	+	16.3637i	1.59441	+	0.920532i
$317$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$317$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	16.4438	−	7.32126i	0.913545	−	0.406737i
$325$	6.01472	+	1.95430i	0.333636	+	0.108405i
$326$		0			0
$327$	−4.42254	−	9.93318i	−0.244567	−	0.549306i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.1588	−	12.7487i	−0.778239	−	0.700729i	0.180951	−	0.983492i	$-0.442082\pi$
$331$	−14.1588	−	12.7487i	−0.778239	−	0.700729i	−0.959190	+	0.282763i	$0.908749\pi$
$332$		0			0
$333$	15.6161	−	21.4938i	0.855759	−	1.17785i
$334$		0			0
$335$		0			0
$336$	−27.1837	+	15.6945i	−1.48299	+	0.856208i
$337$	−5.15760	−	7.09883i	−0.280952	−	0.386698i	0.645097	−	0.764101i	$-0.276817\pi$
$337$	−5.15760	−	7.09883i	−0.280952	−	0.386698i	−0.926049	+	0.377403i	$0.876817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−9.13744	−	28.1222i	−0.493375	−	1.51845i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−13.2480	−	9.62526i	−0.709151	−	0.515228i	0.173749	−	0.984790i	$-0.444412\pi$
$349$	−13.2480	−	9.62526i	−0.709151	−	0.515228i	−0.882900	+	0.469562i	$0.844412\pi$
$350$		0			0
$351$	4.39776	−	4.88421i	0.234735	−	0.260700i
$352$		0			0
$353$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$353$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$359$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$360$		0			0
$361$	52.0648	+	23.1807i	2.74025	+	1.22004i
$362$		0			0
$363$	18.9482	+	1.99153i	0.994522	+	0.104528i
$364$	−6.73667	+	9.27224i	−0.353098	+	0.485997i
$365$		0			0
$366$		0			0
$367$	−19.5000	+	11.2583i	−1.01789	+	0.587680i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−19.5000	+	11.2583i	−1.01789	+	0.587680i	−0.104399	+	0.994535i	$0.533292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−15.0000	+	12.1244i	−0.777714	+	0.628619i
$373$	−10.8623			−0.562430			−0.281215	−	0.959645i	$-0.590737\pi$
$373$	−10.8623			−0.562430			−0.281215	+	0.959645i	$0.590737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.87384	−	36.8571i	−0.198986	−	1.89322i	−0.404452	−	0.914559i	$-0.632538\pi$
$379$	−3.87384	−	36.8571i	−0.198986	−	1.89322i	0.205466	−	0.978664i	$-0.434129\pi$
$380$		0			0
$381$	3.19163	−	30.3663i	0.163512	−	1.55571i
$382$		0			0
$383$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$383$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.87785	−	0.935071i	−0.146289	−	0.0475323i
$388$	−9.42125	+	28.9956i	−0.478291	+	1.47203i
$389$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$389$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.64485	+	13.2413i	−0.383684	+	0.664560i	−0.991586	−	0.129452i	$-0.958678\pi$
$397$	−7.64485	+	13.2413i	−0.383684	+	0.664560i	0.607902	+	0.794012i	$0.292011\pi$
$398$		0			0
$399$	−40.2088	−	55.3427i	−2.01296	−	2.77060i
$400$	13.3826	+	14.8629i	0.669131	+	0.743145i
$401$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$401$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$402$		0			0
$403$	−3.20284	+	6.27192i	−0.159545	+	0.312427i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.5000	+	12.9904i	1.11255	+	0.642333i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	22.5000	+	12.9904i	1.11255	+	0.642333i	0.173064	+	0.984911i	$0.444633\pi$
$410$		0			0
$411$		0			0
$412$	2.25955	−	21.4981i	0.111320	−	1.05914i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−15.6088	+	6.94948i	−0.764365	+	0.340317i
$418$		0			0
$419$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$419$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$420$		0			0
$421$	25.5758	−	5.43631i	1.24649	−	0.264949i	0.463002	−	0.886357i	$-0.346772\pi$
$421$	25.5758	−	5.43631i	1.24649	−	0.264949i	0.783487	+	0.621408i	$0.213439\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	62.3334	−	6.55151i	3.01653	−	0.317050i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$431$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$432$	19.7673	−	6.42280i	0.951057	−	0.309017i
$433$			40.2450i			1.93405i	0.254678	+	0.967026i	$0.418030\pi$
$433$			40.2450i			1.93405i	−0.254678	+	0.967026i	$0.581970\pi$
$434$		0			0
$435$		0			0
$436$	−3.87981	−	11.9408i	−0.185809	−	0.571861i
$437$		0			0
$438$		0			0
$439$	10.4530	+	18.1052i	0.498897	+	0.864114i	0.999999	−	0.00127367i	$-0.000405421\pi$
$439$	10.4530	+	18.1052i	0.498897	+	0.864114i	−0.501103	+	0.865388i	$0.667072\pi$
$440$		0			0
$441$	4.24173	+	40.3574i	0.201987	+	1.92178i
$442$		0			0
$443$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$443$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$444$	20.5275	−	22.7981i	0.974190	−	1.08195i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−33.1115	+	14.7422i	−1.56437	+	0.696502i
$449$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$449$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−41.5934	−	8.84096i	−1.95423	−	0.415384i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.3073	+	23.8214i	−0.809599	+	1.11432i	0.181786	+	0.983338i	$0.441812\pi$
$457$	−17.3073	+	23.8214i	−0.809599	+	1.11432i	−0.991385	+	0.130980i	$0.958188\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$461$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$462$		0			0
$463$	−22.7051	+	7.37732i	−1.05519	+	0.342853i	−0.784705	−	0.619870i	$-0.787185\pi$
$463$	−22.7051	+	7.37732i	−1.05519	+	0.342853i	−0.270489	+	0.962723i	$0.587185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$467$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$468$	5.63980	−	5.07810i	0.260700	−	0.234735i
$469$	−39.9821	+	29.0487i	−1.84620	+	1.34134i
$470$		0			0
$471$	6.50366	+	3.75489i	0.299673	+	0.173016i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−29.1652	+	32.3912i	−1.33819	+	1.48621i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$479$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$480$		0			0
$481$	3.46143	−	10.6532i	0.157827	−	0.485743i
$482$		0			0
$483$		0			0
$484$	21.5192	+	4.57406i	0.978148	+	0.207912i
$485$		0			0
$486$		0			0
$487$	−30.2140	−	3.17562i	−1.36913	−	0.143901i	−0.608739	−	0.793370i	$-0.708325\pi$
$487$	−30.2140	−	3.17562i	−1.36913	−	0.143901i	−0.760388	+	0.649469i	$0.774991\pi$
$488$		0			0
$489$	−4.79491	+	0.503965i	−0.216833	+	0.0227901i
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−18.6890	+	12.1128i	−0.839162	+	0.543882i
$497$		0			0
$498$		0			0
$499$	32.7220	−	29.4630i	1.46484	−	1.31895i	0.619143	−	0.785278i	$-0.287480\pi$
$499$	32.7220	−	29.4630i	1.46484	−	1.31895i	0.845694	−	0.533667i	$-0.179187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$503$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8.03128	+	18.0385i	−0.356682	+	0.801120i
$508$	7.33037	−	34.4867i	0.325233	−	1.53010i
$509$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$509$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$510$		0			0
$511$	71.9721	+	23.3851i	3.18386	+	1.03450i
$512$		0			0
$513$	18.4238	+	41.3805i	0.813431	+	1.82700i
$514$		0			0
$515$		0			0
$516$	−3.19199	−	1.42117i	−0.140520	−	0.0625633i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−12.6518	−	17.4137i	−0.553224	−	0.761448i	0.437221	−	0.899354i	$-0.355963\pi$
$523$	−12.6518	−	17.4137i	−0.553224	−	0.761448i	−0.990445	+	0.137906i	$0.955963\pi$
$524$		0			0
$525$	37.3160	−	12.1247i	1.62860	−	0.529165i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.10739	−	21.8743i	−0.309017	−	0.951057i
$530$		0			0
$531$		0			0
$532$	−39.4950	−	68.4073i	−1.71233	−	2.96583i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	35.8157	−	15.9462i	1.53984	−	0.685579i	0.550989	−	0.834513i	$-0.314251\pi$
$541$	35.8157	−	15.9462i	1.53984	−	0.685579i	0.988847	+	0.148933i	$0.0475840\pi$
$542$		0			0
$543$	−7.86038	+	24.1918i	−0.337321	+	1.03817i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−25.8506	−	11.5094i	−1.10529	−	0.492108i	−0.228775	−	0.973479i	$-0.573472\pi$
$547$	−25.8506	−	11.5094i	−1.10529	−	0.492108i	−0.876517	+	0.481371i	$0.840139\pi$
$548$		0			0
$549$	−41.2747	−	4.33815i	−1.76156	−	0.185148i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	64.2053	−	37.0689i	2.73029	−	1.57633i
$554$		0			0
$555$		0			0
$556$	−18.7635	+	6.09664i	−0.795751	+	0.258555i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	−1.27579			−0.0539602
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.26221	−	40.5522i	0.178996	−	1.70303i
$568$		0			0
$569$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$569$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$570$		0			0
$571$	7.56239	+	35.5782i	0.316476	+	1.48890i	0.792718	+	0.609588i	$0.208665\pi$
$571$	7.56239	+	35.5782i	0.316476	+	1.48890i	−0.476242	+	0.879314i	$0.658001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	23.4755	−	4.98988i	0.978148	−	0.207912i
$577$	−1.01390	−	0.215512i	−0.0422094	−	0.00897189i	0.186759	−	0.982406i	$-0.440202\pi$
$577$	−1.01390	−	0.215512i	−0.0422094	−	0.00897189i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	−16.8604	−	15.1812i	−0.700696	−	0.630910i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$587$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$588$			46.8574i			1.93237i
$589$	−30.5107	−	37.7472i	−1.25717	−	1.55535i
$590$		0			0
$591$		0			0
$592$	26.3249	−	23.7031i	1.08195	−	0.974190i
$593$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$593$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	35.8149	+	26.0211i	1.46581	+	1.06497i
$598$		0			0
$599$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$599$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$600$		0			0
$601$	−2.47038	+	11.6222i	−0.100769	+	0.474081i	0.898607	+	0.438755i	$0.144580\pi$
$601$	−2.47038	+	11.6222i	−0.100769	+	0.474081i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$	29.8952	−	13.3102i	1.21743	−	0.542033i
$604$	−46.6978	−	15.1730i	−1.90010	−	0.617382i
$605$		0			0
$606$		0			0
$607$	−0.0447920	+	0.00952083i	−0.00181805	+	0.000386439i	−0.208821	−	0.977954i	$-0.566962\pi$
$607$	−0.0447920	+	0.00952083i	−0.00181805	+	0.000386439i	0.207003	+	0.978340i	$0.433629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−40.0168	+	4.20594i	−1.61626	+	0.169876i	−0.869056	−	0.494714i	$-0.835273\pi$
$613$	−40.0168	+	4.20594i	−1.61626	+	0.169876i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$617$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$618$		0			0
$619$			47.8263i			1.92230i	0.276022	+	0.961151i	$0.410984\pi$
$619$			47.8263i			1.92230i	−0.276022	+	0.961151i	$0.589016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	7.08952	−	5.15084i	0.283808	−	0.206199i
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	7.01543	+	5.09701i	0.279946	+	0.203393i
$629$		0			0
$630$		0			0
$631$	−2.54004	+	5.70503i	−0.101117	+	0.227114i	−0.957019	−	0.290025i	$-0.906336\pi$
$631$	−2.54004	+	5.70503i	−0.101117	+	0.227114i	0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$	−2.63395	−	12.3918i	−0.104690	−	0.492528i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.95888	+	15.6299i	0.275721	+	0.619279i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$641$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$642$		0			0
$643$	21.4671	−	29.5469i	0.846580	−	1.16522i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	21.4671	−	29.5469i	0.846580	−	1.16522i	0.984606	−	0.174788i	$-0.0559241\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$647$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	6.79594	+	43.1600i	0.266354	+	1.69157i
$652$	−5.56718			−0.218028
$653$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$653$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−43.3962	−	25.0548i	−1.69305	−	0.977480i
$658$		0			0
$659$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$659$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$660$		0			0
$661$	−20.0521	+	22.2701i	−0.779935	+	0.866205i	−0.993860	−	0.110645i	$-0.964708\pi$
$661$	−20.0521	+	22.2701i	−0.779935	+	0.866205i	0.213925	+	0.976850i	$0.431375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	49.0182	−	10.4191i	1.89515	−	0.402827i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−43.0339	−	4.52305i	−1.65884	−	0.174351i	−0.771741	−	0.635937i	$-0.780614\pi$
$673$	−43.0339	−	4.52305i	−1.65884	−	0.174351i	−0.887095	+	0.461587i	$0.847280\pi$
$674$		0			0
$675$	−25.8384	+	2.71573i	−0.994522	+	0.104528i
$676$	−11.4002	+	19.7456i	−0.438468	+	0.759448i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	46.2131	+	51.3248i	1.77350	+	1.96967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	16.1628	+	49.7441i	0.618002	+	1.90201i
$685$		0			0
$686$		0			0
$687$	8.69444	+	15.0592i	0.331714	+	0.574545i
$688$	−3.49407	−	2.01730i	−0.133210	−	0.0769089i
$689$		0			0
$690$		0			0
$691$	3.86905	−	36.8116i	0.147186	−	1.40038i	−0.632671	−	0.774421i	$-0.718041\pi$
$691$	3.86905	−	36.8116i	0.147186	−	1.40038i	0.779857	−	0.625958i	$-0.215292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	44.3162	−	9.41970i	1.67499	−	0.356031i
$701$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$701$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$702$		0			0
$703$	57.3708	+	51.6569i	2.16378	+	1.94828i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−25.4518	−	35.0315i	−0.955864	−	1.31563i	−0.948873	−	0.315659i	$-0.897775\pi$
$709$	−25.4518	−	35.0315i	−0.955864	−	1.31563i	−0.00699141	−	0.999976i	$-0.502225\pi$
$710$		0			0
$711$	−46.6885	+	15.1700i	−1.75096	+	0.568920i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−39.6162	−	28.7828i	−1.47538	−	1.07193i
$722$		0			0
$723$	−2.00739	+	2.22943i	−0.0746557	+	0.0829136i
$724$	−11.9466	+	26.8325i	−0.443991	+	0.997221i
$725$		0			0
$726$		0			0
$727$	4.56773	−	2.03368i	0.169408	−	0.0754251i	−0.320281	−	0.947323i	$-0.603777\pi$
$727$	4.56773	−	2.03368i	0.169408	−	0.0754251i	0.489688	+	0.871898i	$0.337111\pi$
$728$		0			0
$729$	−8.34346	+	25.6785i	−0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$	−46.8753	−	9.96364i	−1.73256	−	0.368267i
$733$	6.39482	+	2.84716i	0.236198	+	0.105162i	0.521423	−	0.853299i	$-0.325402\pi$
$733$	6.39482	+	2.84716i	0.236198	+	0.105162i	−0.285225	+	0.958461i	$0.592068\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.4843	−	10.6719i	0.679955	−	0.392572i	−0.119883	−	0.992788i	$-0.538252\pi$
$739$	18.4843	−	10.6719i	0.679955	−	0.392572i	0.799838	+	0.600216i	$0.204919\pi$
$740$		0			0
$741$	12.7789	+	14.1924i	0.469446	+	0.521372i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.14981	+	10.9397i	0.0419573	+	0.399197i	0.995265	+	0.0971968i	$0.0309876\pi$
$751$	1.14981	+	10.9397i	0.0419573	+	0.399197i	−0.953308	+	0.302000i	$0.902346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	9.78924	−	46.0547i	0.356031	−	1.67499i
$757$	4.98007	+	23.4294i	0.181004	+	0.851556i	0.971119	+	0.238597i	$0.0766875\pi$
$757$	4.98007	+	23.4294i	0.181004	+	0.851556i	−0.790115	+	0.612959i	$0.789979\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$761$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$762$		0			0
$763$	−27.8202	−	5.91336i	−1.00716	−	0.214078i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.5610	−	2.89678i	0.994522	−	0.104528i
$769$	26.6655	−	46.1859i	0.961581	−	1.66551i	0.243049	−	0.970014i	$-0.421852\pi$
$769$	26.6655	−	46.1859i	0.961581	−	1.66551i	0.718532	−	0.695494i	$-0.244814\pi$
$770$		0			0
$771$		0			0
$772$	−17.5298	−	19.4688i	−0.630910	−	0.700696i
$773$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$773$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$774$		0			0
$775$	25.9988	−	9.95307i	0.933904	−	0.357525i
$776$		0			0
$777$	−21.4751	−	66.0935i	−0.770414	−	2.37109i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−5.65564	+	53.8098i	−0.201987	+	1.92178i
$785$		0			0
$786$		0			0
$787$	−9.72307	+	45.7434i	−0.346590	+	1.63058i	0.367147	+	0.930163i	$0.380335\pi$
$787$	−9.72307	+	45.7434i	−0.346590	+	1.63058i	−0.713737	+	0.700414i	$0.752999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−17.1156	+	3.63803i	−0.607792	+	0.129190i
$794$		0			0
$795$		0			0
$796$	37.9882	+	34.2048i	1.34646	+	1.21236i
$797$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$797$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	35.9374	−	11.6768i	1.26742	−	0.411808i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$809$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$810$		0			0
$811$	−9.50000	−	16.4545i	−0.333590	−	0.577795i	0.649623	−	0.760257i	$-0.274927\pi$
$811$	−9.50000	−	16.4545i	−0.333590	−	0.577795i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	−5.86557	−	55.8071i	−0.205714	−	1.95724i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.57633	−	8.03258i	0.125120	−	0.281024i
$818$		0			0
$819$	−3.57435	−	16.8160i	−0.124898	−	0.587598i
$820$		0			0
$821$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$821$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$822$		0			0
$823$	−23.1629	−	52.0248i	−0.807409	−	1.81347i	−0.515048	−	0.857161i	$-0.672226\pi$
$823$	−23.1629	−	52.0248i	−0.807409	−	1.81347i	−0.292361	−	0.956308i	$-0.594441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$827$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$828$		0			0
$829$	−29.5985	+	40.7389i	−1.02800	+	1.41492i	−0.121560	+	0.992584i	$0.538790\pi$
$829$	−29.5985	+	40.7389i	−1.02800	+	1.41492i	−0.906439	+	0.422336i	$0.861210\pi$
$830$		0			0
$831$	10.5000	−	18.1865i	0.364241	−	0.630884i
$832$	8.76313	−	5.05940i	0.303807	−	0.175403i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.54021	−	28.8899i	0.0532375	−	0.998582i
$838$		0			0
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$	−1.52909	−	14.5483i	−0.0526334	−	0.500773i
$845$		0			0
$846$		0			0
$847$	33.3474	−	37.0360i	1.14583	−	1.27257i
$848$		0			0
$849$	−12.0499	+	56.6904i	−0.413552	+	1.94561i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−7.10739	+	21.8743i	−0.243352	+	0.748962i	0.752551	+	0.658534i	$0.228823\pi$
$853$	−7.10739	+	21.8743i	−0.243352	+	0.748962i	−0.995903	+	0.0904274i	$0.971177\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$857$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$858$		0			0
$859$	28.4285	+	2.98795i	0.969967	+	0.101948i	0.576241	−	0.817280i	$-0.304519\pi$
$859$	28.4285	+	2.98795i	0.969967	+	0.101948i	0.393726	+	0.919228i	$0.371186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$	2.59491	+	50.3841i	0.0880771	+	1.71015i
$869$		0			0
$870$		0			0
$871$	10.2533	−	9.23208i	0.347419	−	0.312817i
$872$		0			0
$873$	−22.8658	−	39.6048i	−0.773892	−	1.34042i
$874$		0			0
$875$		0			0
$876$	−46.8110	−	34.0102i	−1.58160	−	1.14910i
$877$	−2.61321	+	24.8630i	−0.0882419	+	0.839565i	0.857464	+	0.514543i	$0.172038\pi$
$877$	−2.61321	+	24.8630i	−0.0882419	+	0.839565i	−0.945706	+	0.325022i	$0.894628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$881$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$882$		0			0
$883$	53.2017	+	17.2863i	1.79038	+	0.581729i	0.999541	−	0.0302962i	$-0.00964506\pi$
$883$	53.2017	+	17.2863i	1.79038	+	0.581729i	0.790838	+	0.612026i	$0.209645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$887$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$888$		0			0
$889$	−59.3538	−	53.4424i	−1.99066	−	1.79240i
$890$		0			0
$891$		0			0
$892$	57.5488	−	6.04862i	1.92688	−	0.202523i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−30.0000			−1.00000
$901$		0			0
$902$		0			0
$903$	−6.40350	+	4.65241i	−0.213095	+	0.154823i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	47.5950	+	34.5798i	1.58036	+	1.14820i	0.916275	+	0.400549i	$0.131180\pi$
$907$	47.5950	+	34.5798i	1.58036	+	1.14820i	0.664089	+	0.747653i	$0.268820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$911$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$912$	12.5569	+	59.0757i	0.415802	+	1.95619i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	8.16685	+	18.3430i	0.269840	+	0.606071i
$917$		0			0
$918$		0			0
$919$	31.8129	+	14.1640i	1.04941	+	0.467228i	0.857661	−	0.514216i	$-0.171917\pi$
$919$	31.8129	+	14.1640i	1.04941	+	0.467228i	0.191750	+	0.981444i	$0.438584\pi$
$920$		0			0
$921$	56.2909	+	5.91641i	1.85485	+	0.194952i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−38.3473	+	22.1398i	−1.26085	+	0.727952i
$926$		0			0
$927$	21.6965	+	24.0964i	0.712605	+	0.791428i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−117.916			−3.86453
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−3.86290	−	36.7530i	−0.126195	−	1.20067i	−0.855986	−	0.516999i	$-0.827049\pi$
$937$	−3.86290	−	36.7530i	−0.126195	−	1.20067i	0.729791	−	0.683670i	$-0.239617\pi$
$938$		0			0
$939$	−4.07960	+	38.8148i	−0.133133	+	1.26667i
$940$		0			0
$941$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$941$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$947$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$948$	−55.4469	+	11.7856i	−1.80083	+	0.382779i
$949$	−20.6653	−	4.39256i	−0.670826	−	0.142588i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$953$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	6.39050	+	30.3342i	0.206145	+	0.978521i
$962$		0			0
$963$		0			0
$964$	−2.57433	+	2.31794i	−0.0829136	+	0.0746557i
$965$		0			0
$966$		0			0
$967$	−32.2326	−	18.6095i	−1.03653	−	0.598442i	−0.117683	−	0.993051i	$-0.537547\pi$
$967$	−32.2326	−	18.6095i	−1.03653	−	0.598442i	−0.918849	+	0.394609i	$0.870880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$971$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$972$	−12.6808	+	28.4815i	−0.406737	+	0.913545i
$973$	−9.29213	+	43.7160i	−0.297892	+	1.40147i
$974$		0			0
$975$	−10.0069	+	4.45536i	−0.320477	+	0.142686i
$976$	−52.6278	−	17.0998i	−1.68457	−	0.547351i
$977$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$977$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	17.2048	+	7.66006i	0.549306	+	0.244567i
$982$		0			0
$983$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$983$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	12.9620	+	17.8406i	0.412376	+	0.567586i
$989$		0			0
$990$		0			0
$991$		−	39.0517i		−	1.24052i	−0.784397	−	0.620259i	$-0.787027\pi$
$991$		−	39.0517i		−	1.24052i	0.784397	−	0.620259i	$-0.212973\pi$
$992$		0			0
$993$	33.0000			1.04722
$994$		0			0
$995$		0			0
$996$		0			0
$997$	31.1961	+	54.0332i	0.987990	+	1.71125i	0.627811	+	0.778366i	$0.283951\pi$
$997$	31.1961	+	54.0332i	0.987990	+	1.71125i	0.360180	+	0.932883i	$0.382715\pi$
$998$		0			0
$999$	4.81006	+	45.7646i	0.152183	+	1.44793i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.a.11.1	✓	8
3.2	odd	2	CM	93.2.p.a.11.1	✓	8
31.17	odd	30	inner	93.2.p.a.17.1	yes	8
93.17	even	30	inner	93.2.p.a.17.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.a.11.1	✓	8	1.1	even	1	trivial
93.2.p.a.11.1	✓	8	3.2	odd	2	CM
93.2.p.a.17.1	yes	8	31.17	odd	30	inner
93.2.p.a.17.1	yes	8	93.17	even	30	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.p (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.28716 + 1.15897i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(0.473579 + 4.50581i) q^{7} +(0.313585 - 2.98357i) q^{9} +O(q^{10})\)	\(q+(-1.28716 + 1.15897i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(0.473579 + 4.50581i) q^{7} +(0.313585 - 2.98357i) q^{9} +(0.720227 - 3.38840i) q^{12} +(-0.262977 - 1.23721i) q^{13} +(1.23607 - 3.80423i) q^{16} +(8.52685 + 1.81244i) q^{19} +(-5.83166 - 5.25085i) q^{21} +(-2.50000 + 4.33013i) q^{25} +(3.05422 + 4.20378i) q^{27} +(-6.06316 - 6.73382i) q^{28} +(-4.32380 - 3.50781i) q^{31} +(3.00000 + 5.19615i) q^{36} +(7.66945 + 4.42796i) q^{37} +(1.77238 + 1.28771i) q^{39} +(0.209710 - 0.986609i) q^{43} +(2.81795 + 6.32923i) q^{48} +(-13.2310 + 2.81233i) q^{49} +(1.87993 + 1.69270i) q^{52} +(-13.0760 + 7.54944i) q^{57} -13.8340i q^{61} +13.5919 q^{63} +(2.47214 + 7.60845i) q^{64} +(5.45406 + 9.44671i) q^{67} +(6.79380 - 15.2591i) q^{73} +(-1.80057 - 8.47101i) q^{75} +(-15.9274 + 7.09132i) q^{76} +(-6.65573 - 14.9490i) q^{79} +(-8.80333 - 1.87121i) q^{81} +(15.6086 + 1.64053i) q^{84} +(5.45008 - 1.77084i) q^{91} +(9.63089 - 0.496015i) q^{93} +(12.3326 - 8.96014i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} - 4 q^{4} - 4 q^{7} - 3 q^{9} + 6 q^{12} + 9 q^{13} - 8 q^{16} + 7 q^{19} - 33 q^{21} - 20 q^{25} - 18 q^{28} - 4 q^{31} + 24 q^{36} + 9 q^{37} + 27 q^{39} + 44 q^{43} + 12 q^{48} + 9 q^{49} + 18 q^{52} - 21 q^{57} - 24 q^{63} - 16 q^{64} + 16 q^{67} + 3 q^{73} - 15 q^{75} - 66 q^{76} - 35 q^{79} + 9 q^{81} + 24 q^{84} - 5 q^{91} + 15 q^{93} + 15 q^{97}+O(q^{100}) \) 8 * q + 3 * q^3 - 4 * q^4 - 4 * q^7 - 3 * q^9 + 6 * q^12 + 9 * q^13 - 8 * q^16 + 7 * q^19 - 33 * q^21 - 20 * q^25 - 18 * q^28 - 4 * q^31 + 24 * q^36 + 9 * q^37 + 27 * q^39 + 44 * q^43 + 12 * q^48 + 9 * q^49 + 18 * q^52 - 21 * q^57 - 24 * q^63 - 16 * q^64 + 16 * q^67 + 3 * q^73 - 15 * q^75 - 66 * q^76 - 35 * q^79 + 9 * q^81 + 24 * q^84 - 5 * q^91 + 15 * q^93 + 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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