LMFDB - Embedded newform 3549.1.ea.a.2126.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,1,Mod(110,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 130, 113]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3549.110");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3549.ea (of order $156$, degree $48$, 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:	$1.77118172983$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{156})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{42} - x^{40} + x^{36} + x^{34} - x^{30} - x^{28} + x^{24} - x^{20} - x^{18} + \cdots + 1 $ x^48 + x^46 - x^42 - x^40 + x^36 + x^34 - x^30 - x^28 + x^24 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{156}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{156} - \cdots)$

Embedding invariants

Embedding label			2126.1
Root			$0.960518 - 0.278217i$ of defining polynomial
Character	$\chi$	$=$	3549.2126
Dual form			3549.1.ea.a.626.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.663123 + 0.748511i) q^{3} +(0.960518 + 0.278217i) q^{4} +(0.987050 - 0.160411i) q^{7} +(-0.120537 - 0.992709i) q^{9} +O(q^{10})$	\(q+(-0.663123 + 0.748511i) q^{3} +(0.960518 + 0.278217i) q^{4} +(0.987050 - 0.160411i) q^{7} +(-0.120537 - 0.992709i) q^{9} +(-0.845190 + 0.534466i) q^{12} +(0.866025 + 0.500000i) q^{13} +(0.845190 + 0.534466i) q^{16} +(-0.198700 + 0.198700i) q^{19} +(-0.534466 + 0.845190i) q^{21} +(-0.903450 - 0.428693i) q^{25} +(0.822984 + 0.568065i) q^{27} +(0.992709 + 0.120537i) q^{28} +(-0.788125 + 0.670382i) q^{31} +(0.160411 - 0.987050i) q^{36} +(1.23607 - 1.05141i) q^{37} +(-0.948536 + 0.316668i) q^{39} +(-0.289847 - 0.137534i) q^{43} +(-0.960518 + 0.278217i) q^{48} +(0.948536 - 0.316668i) q^{49} +(0.692724 + 0.721202i) q^{52} +(-0.0169667 - 0.280492i) q^{57} +(0.520276 + 0.197315i) q^{61} +(-0.278217 - 0.960518i) q^{63} +(0.663123 + 0.748511i) q^{64} +(-0.186505 - 0.308518i) q^{67} +(0.407653 + 0.164066i) q^{73} +(0.919979 - 0.391967i) q^{75} +(-0.246137 + 0.135573i) q^{76} +(0.918722 - 0.956491i) q^{79} +(-0.970942 + 0.239316i) q^{81} +(-0.748511 + 0.663123i) q^{84} +(0.935016 + 0.354605i) q^{91} +(0.0208353 - 1.03447i) q^{93} +(-0.504989 + 0.113733i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 48 * q + 2 * q^7 + 4 * q^9	$ 48 q + 2 q^{7} + 4 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{19} - 2 q^{31} - 2 q^{37} - 2 q^{39} + 6 q^{43} + 2 q^{49} + 2 q^{52} + 2 q^{57} - 2 q^{63} + 4 q^{67} - 4 q^{73} - 2 q^{75} + 4 q^{76} - 4 q^{81} - 4 q^{84} + 2 q^{93} + 2 q^{97}+O(q^{100}) $ 48 * q + 2 * q^7 + 4 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^19 - 2 * q^31 - 2 * q^37 - 2 * q^39 + 6 * q^43 + 2 * q^49 + 2 * q^52 + 2 * q^57 - 2 * q^63 + 4 * q^67 - 4 * q^73 - 2 * q^75 + 4 * q^76 - 4 * q^81 - 4 * q^84 + 2 * q^93 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3549\mathbb{Z}\right)^\times$.

$n$	$1184$	$1522$	$3382$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{59}{156}\right)$

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		−0.990080	−	0.140502i	$-0.955128\pi$
$2$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$3$	−0.663123	+	0.748511i	−0.663123	+	0.748511i
$3$	−0.663123	+	0.748511i	−0.663123	+	0.748511i
$4$	0.960518	+	0.278217i	0.960518	+	0.278217i
$5$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$5$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$6$		0			0
$7$	0.987050	−	0.160411i	0.987050	−	0.160411i
$7$	0.987050	−	0.160411i	0.987050	−	0.160411i
$8$		0			0
$9$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$10$		0			0
$11$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$11$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$12$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$13$	0.866025	+	0.500000i	0.866025	+	0.500000i
$13$	0.866025	+	0.500000i	0.866025	+	0.500000i
$14$		0			0
$15$		0			0
$16$	0.845190	+	0.534466i	0.845190	+	0.534466i
$17$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$17$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$18$		0			0
$19$	−0.198700	+	0.198700i	−0.198700	+	0.198700i	−0.799443	−	0.600742i	$-0.794872\pi$
$19$	−0.198700	+	0.198700i	−0.198700	+	0.198700i	0.600742	+	0.799443i	$0.294872\pi$
$20$		0			0
$21$	−0.534466	+	0.845190i	−0.534466	+	0.845190i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−0.903450	−	0.428693i	−0.903450	−	0.428693i
$26$		0			0
$27$	0.822984	+	0.568065i	0.822984	+	0.568065i
$28$	0.992709	+	0.120537i	0.992709	+	0.120537i
$29$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$29$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$30$		0			0
$31$	−0.788125	+	0.670382i	−0.788125	+	0.670382i	−0.948536	−	0.316668i	$-0.897436\pi$
$31$	−0.788125	+	0.670382i	−0.788125	+	0.670382i	0.160411	+	0.987050i	$0.448718\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.160411	−	0.987050i	0.160411	−	0.987050i
$37$	1.23607	−	1.05141i	1.23607	−	1.05141i	0.239316	−	0.970942i	$-0.423077\pi$
$37$	1.23607	−	1.05141i	1.23607	−	1.05141i	0.996757	−	0.0804666i	$-0.0256410\pi$
$38$		0			0
$39$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$40$		0			0
$41$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$41$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$42$		0			0
$43$	−0.289847	−	0.137534i	−0.289847	−	0.137534i	0.278217	−	0.960518i	$-0.410256\pi$
$43$	−0.289847	−	0.137534i	−0.289847	−	0.137534i	−0.568065	+	0.822984i	$0.692308\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$47$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$48$	−0.960518	+	0.278217i	−0.960518	+	0.278217i
$49$	0.948536	−	0.316668i	0.948536	−	0.316668i
$50$		0			0
$51$		0			0
$52$	0.692724	+	0.721202i	0.692724	+	0.721202i
$53$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$53$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.0169667	−	0.280492i	−0.0169667	−	0.280492i
$58$		0			0
$59$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$59$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$60$		0			0
$61$	0.520276	+	0.197315i	0.520276	+	0.197315i	0.600742	−	0.799443i	$-0.294872\pi$
$61$	0.520276	+	0.197315i	0.520276	+	0.197315i	−0.0804666	+	0.996757i	$0.525641\pi$
$62$		0			0
$63$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$64$	0.663123	+	0.748511i	0.663123	+	0.748511i
$65$		0			0
$66$		0			0
$67$	−0.186505	−	0.308518i	−0.186505	−	0.308518i	0.748511	−	0.663123i	$-0.230769\pi$
$67$	−0.186505	−	0.308518i	−0.186505	−	0.308518i	−0.935016	+	0.354605i	$0.884615\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$71$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$72$		0			0
$73$	0.407653	+	0.164066i	0.407653	+	0.164066i	0.568065	−	0.822984i	$-0.307692\pi$
$73$	0.407653	+	0.164066i	0.407653	+	0.164066i	−0.160411	+	0.987050i	$0.551282\pi$
$74$		0			0
$75$	0.919979	−	0.391967i	0.919979	−	0.391967i
$76$	−0.246137	+	0.135573i	−0.246137	+	0.135573i
$77$		0			0
$78$		0			0
$79$	0.918722	−	0.956491i	0.918722	−	0.956491i	−0.0804666	−	0.996757i	$-0.525641\pi$
$79$	0.918722	−	0.956491i	0.918722	−	0.956491i	0.999189	+	0.0402659i	$0.0128205\pi$
$80$		0			0
$81$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$82$		0			0
$83$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$83$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$84$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$89$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$90$		0			0
$91$	0.935016	+	0.354605i	0.935016	+	0.354605i
$92$		0			0
$93$	0.0208353	−	1.03447i	0.0208353	−	1.03447i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.504989	+	0.113733i	−0.504989	+	0.113733i	−0.464723	−	0.885456i	$-0.653846\pi$
$97$	−0.504989	+	0.113733i	−0.504989	+	0.113733i	−0.0402659	+	0.999189i	$0.512821\pi$
$98$		0			0
$99$		0			0
$100$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$101$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$101$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$102$		0			0
$103$	−1.60339	+	0.535289i	−1.60339	+	0.535289i	−0.970942	−	0.239316i	$-0.923077\pi$
$103$	−1.60339	+	0.535289i	−1.60339	+	0.535289i	−0.632445	+	0.774605i	$0.717949\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$107$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$108$	0.632445	+	0.774605i	0.632445	+	0.774605i
$109$	−1.38470	+	0.493489i	−1.38470	+	0.493489i	−0.919979	−	0.391967i	$-0.871795\pi$
$109$	−1.38470	+	0.493489i	−1.38470	+	0.493489i	−0.464723	+	0.885456i	$0.653846\pi$
$110$		0			0
$111$	−0.0326775	+	1.62243i	−0.0326775	+	1.62243i
$112$	0.919979	+	0.391967i	0.919979	+	0.391967i
$113$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$113$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.391967	−	0.919979i	0.391967	−	0.919979i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.239316	+	0.970942i	−0.239316	+	0.970942i
$122$		0			0
$123$		0			0
$124$	−0.943521	+	0.424644i	−0.943521	+	0.424644i
$125$		0			0
$126$		0			0
$127$	−0.0580798	−	0.0557864i	−0.0580798	−	0.0557864i	0.663123	−	0.748511i	$-0.269231\pi$
$127$	−0.0580798	−	0.0557864i	−0.0580798	−	0.0557864i	−0.721202	+	0.692724i	$0.756410\pi$
$128$		0			0
$129$	0.295150	−	0.125752i	0.295150	−	0.125752i
$130$		0			0
$131$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$131$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$132$		0			0
$133$	−0.164254	+	0.228001i	−0.164254	+	0.228001i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$137$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$138$		0			0
$139$	−0.879714	+	1.39116i	−0.879714	+	1.39116i	0.0402659	+	0.999189i	$0.487179\pi$
$139$	−0.879714	+	1.39116i	−0.879714	+	1.39116i	−0.919979	+	0.391967i	$0.871795\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.428693	−	0.903450i	0.428693	−	0.903450i
$145$		0			0
$146$		0			0
$147$	−0.391967	+	0.919979i	−0.391967	+	0.919979i
$148$	1.47979	−	0.666000i	1.47979	−	0.666000i
$149$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$149$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$150$		0			0
$151$	−1.96684	−	0.0396144i	−1.96684	−	0.0396144i	−0.979791	−	0.200026i	$-0.935897\pi$
$151$	−1.96684	−	0.0396144i	−1.96684	−	0.0396144i	−0.987050	+	0.160411i	$0.948718\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.999189	+	0.0402659i	−0.999189	+	0.0402659i
$157$	1.44124	−	0.0580798i	1.44124	−	0.0580798i	0.692724	−	0.721202i	$-0.256410\pi$
$157$	1.44124	−	0.0580798i	1.44124	−	0.0580798i	0.748511	+	0.663123i	$0.230769\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.121025	+	0.660411i	0.121025	+	0.660411i	0.987050	+	0.160411i	$0.0512821\pi$
$163$	0.121025	+	0.660411i	0.121025	+	0.660411i	−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$167$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$168$		0			0
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$		0			0
$171$	0.221202	+	0.173301i	0.221202	+	0.173301i
$172$	−0.240139	−	0.212745i	−0.240139	−	0.212745i
$173$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$173$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$174$		0			0
$175$	−0.960518	−	0.278217i	−0.960518	−	0.278217i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$179$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$180$		0			0
$181$	−0.759177	−	0.398447i	−0.759177	−	0.398447i	0.0402659	−	0.999189i	$-0.487179\pi$
$181$	−0.759177	−	0.398447i	−0.759177	−	0.398447i	−0.799443	+	0.600742i	$0.794872\pi$
$182$		0			0
$183$	−0.492699	+	0.258588i	−0.492699	+	0.258588i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.903450	+	0.428693i	0.903450	+	0.428693i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.00000			−1.00000
$193$	−0.816668	−	1.81456i	−0.816668	−	1.81456i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−0.816668	−	1.81456i	−0.816668	−	1.81456i	−0.316668	−	0.948536i	$-0.602564\pi$
$194$		0			0
$195$		0			0
$196$	0.999189	−	0.0402659i	0.999189	−	0.0402659i
$197$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$197$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$198$		0			0
$199$	−0.0483789	+	1.20051i	−0.0483789	+	1.20051i	0.774605	+	0.632445i	$0.217949\pi$
$199$	−0.0483789	+	1.20051i	−0.0483789	+	1.20051i	−0.822984	+	0.568065i	$0.807692\pi$
$200$		0			0
$201$	0.354605	+	0.0649838i	0.354605	+	0.0649838i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.464723	+	0.885456i	0.464723	+	0.885456i
$209$		0			0
$210$		0			0
$211$	1.25168	−	1.30314i	1.25168	−	1.30314i	0.316668	−	0.948536i	$-0.397436\pi$
$211$	1.25168	−	1.30314i	1.25168	−	1.30314i	0.935016	−	0.354605i	$-0.115385\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.670382	+	0.788125i	−0.670382	+	0.788125i
$218$		0			0
$219$	−0.393130	+	0.196337i	−0.393130	+	0.196337i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.310724	−	1.37966i	0.310724	−	1.37966i	−0.534466	−	0.845190i	$-0.679487\pi$
$223$	0.310724	−	1.37966i	0.310724	−	1.37966i	0.845190	−	0.534466i	$-0.179487\pi$
$224$		0			0
$225$	−0.316668	+	0.948536i	−0.316668	+	0.948536i
$226$		0			0
$227$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$227$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$228$	0.0617411	−	0.274138i	0.0617411	−	0.274138i
$229$	−1.03297	−	0.878652i	−1.03297	−	0.878652i	−0.0402659	−	0.999189i	$-0.512821\pi$
$229$	−1.03297	−	0.878652i	−1.03297	−	0.878652i	−0.992709	+	0.120537i	$0.961538\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$233$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.106718	+	1.32194i	0.106718	+	1.32194i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−1.25801	+	1.16054i	−1.25801	+	1.16054i	−0.278217	+	0.960518i	$0.589744\pi$
$241$	−1.25801	+	1.16054i	−1.25801	+	1.16054i	−0.979791	+	0.200026i	$0.935897\pi$
$242$		0			0
$243$	0.464723	−	0.885456i	0.464723	−	0.885456i
$244$	0.444838	+	0.334274i	0.444838	+	0.334274i
$245$		0			0
$246$		0			0
$247$	−0.271430	+	0.0727294i	−0.271430	+	0.0727294i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$251$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$252$		−	1.00000i		−	1.00000i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.428693	+	0.903450i	0.428693	+	0.903450i
$257$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$257$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$258$		0			0
$259$	1.05141	−	1.23607i	1.05141	−	1.23607i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$263$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.0933069	−	0.348226i	−0.0933069	−	0.348226i
$269$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$269$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$270$		0			0
$271$	0.0194306	−	0.0352768i	0.0194306	−	0.0352768i	−0.866025	−	0.500000i	$-0.833333\pi$
$271$	0.0194306	−	0.0352768i	0.0194306	−	0.0352768i	0.885456	+	0.464723i	$0.153846\pi$
$272$		0			0
$273$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.51790	−	1.23933i	1.51790	−	1.23933i	0.632445	−	0.774605i	$-0.282051\pi$
$277$	1.51790	−	1.23933i	1.51790	−	1.23933i	0.885456	−	0.464723i	$-0.153846\pi$
$278$		0			0
$279$	0.760492	+	0.701573i	0.760492	+	0.701573i
$280$		0			0
$281$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$281$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$282$		0			0
$283$	−0.607222	+	0.879714i	−0.607222	+	0.879714i	−0.999189	−	0.0402659i	$-0.987179\pi$
$283$	−0.607222	+	0.879714i	−0.607222	+	0.879714i	0.391967	+	0.919979i	$0.371795\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.948536	+	0.316668i	0.948536	+	0.316668i
$290$		0			0
$291$	0.249739	−	0.453409i	0.249739	−	0.453409i
$292$	0.345912	+	0.271005i	0.345912	+	0.271005i
$293$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$293$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.992709	−	0.120537i	0.992709	−	0.120537i
$301$	−0.308156	−	0.0892584i	−0.308156	−	0.0892584i
$302$		0			0
$303$		0			0
$304$	−0.274138	+	0.0617411i	−0.274138	+	0.0617411i
$305$		0			0
$306$		0			0
$307$	1.79393	−	0.328749i	1.79393	−	0.328749i	0.822984	−	0.568065i	$-0.192308\pi$
$307$	1.79393	−	0.328749i	1.79393	−	0.328749i	0.970942	+	0.239316i	$0.0769231\pi$
$308$		0			0
$309$	0.662573	−	1.55512i	0.662573	−	1.55512i
$310$		0			0
$311$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$311$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$312$		0			0
$313$	−0.0321908	−	0.398754i	−0.0321908	−	0.398754i	−0.992709	−	0.120537i	$-0.961538\pi$
$313$	−0.0321908	−	0.398754i	−0.0321908	−	0.398754i	0.960518	−	0.278217i	$-0.0897436\pi$
$314$		0			0
$315$		0			0
$316$	1.14856	−	0.663123i	1.14856	−	0.663123i
$317$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$317$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.999189	−	0.0402659i	−0.999189	−	0.0402659i
$325$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$326$		0			0
$327$	0.548846	−	1.36371i	0.548846	−	1.36371i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.625233	+	1.38921i	−0.625233	+	1.38921i	0.278217	+	0.960518i	$0.410256\pi$
$331$	−0.625233	+	1.38921i	−0.625233	+	1.38921i	−0.903450	+	0.428693i	$0.858974\pi$
$332$		0			0
$333$	−1.19273	−	1.10033i	−1.19273	−	1.10033i
$334$		0			0
$335$		0			0
$336$	−0.903450	+	0.428693i	−0.903450	+	0.428693i
$337$		−	1.26489i		−	1.26489i	−0.774605	−	0.632445i	$-0.782051\pi$
$337$		−	1.26489i		−	1.26489i	0.774605	−	0.632445i	$-0.217949\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.885456	−	0.464723i	0.885456	−	0.464723i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$347$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$348$		0			0
$349$	−0.271430	−	1.91269i	−0.271430	−	1.91269i	−0.391967	−	0.919979i	$-0.628205\pi$
$349$	−0.271430	−	1.91269i	−0.271430	−	1.91269i	0.120537	−	0.992709i	$-0.461538\pi$
$350$		0			0
$351$	0.428693	+	0.903450i	0.428693	+	0.903450i
$352$		0			0
$353$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$353$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$359$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$360$		0			0
$361$			0.921036i			0.921036i
$362$		0			0
$363$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$364$	0.799443	+	0.600742i	0.799443	+	0.600742i
$365$		0			0
$366$		0			0
$367$	−0.628718	+	0.0763402i	−0.628718	+	0.0763402i	−0.428693	−	0.903450i	$-0.641026\pi$
$367$	−0.628718	+	0.0763402i	−0.628718	+	0.0763402i	−0.200026	+	0.979791i	$0.564103\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.307819	−	0.987826i	0.307819	−	0.987826i
$373$	0.0576926	−	0.475142i	0.0576926	−	0.475142i	−0.935016	−	0.354605i	$-0.884615\pi$
$373$	0.0576926	−	0.475142i	0.0576926	−	0.475142i	0.992709	−	0.120537i	$-0.0384615\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.40848	−	0.703425i	−1.40848	−	0.703425i	−0.428693	−	0.903450i	$-0.641026\pi$
$379$	−1.40848	−	0.703425i	−1.40848	−	0.703425i	−0.979791	+	0.200026i	$0.935897\pi$
$380$		0			0
$381$	0.0802707	−	0.00648012i	0.0802707	−	0.00648012i
$382$		0			0
$383$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$383$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.101594	+	0.304312i	−0.101594	+	0.304312i
$388$	−0.516694	−	0.0312542i	−0.516694	−	0.0312542i
$389$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$389$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.521177	−	1.67252i	−0.521177	−	1.67252i	−0.721202	−	0.692724i	$-0.756410\pi$
$397$	−0.521177	−	1.67252i	−0.521177	−	1.67252i	0.200026	−	0.979791i	$-0.435897\pi$
$398$		0			0
$399$	−0.0617411	−	0.274138i	−0.0617411	−	0.274138i
$400$	−0.534466	−	0.845190i	−0.534466	−	0.845190i
$401$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$401$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$402$		0			0
$403$	−1.01773	+	0.186505i	−1.01773	+	0.186505i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.333635	−	0.668044i	0.333635	−	0.668044i	−0.663123	−	0.748511i	$-0.730769\pi$
$409$	0.333635	−	0.668044i	0.333635	−	0.668044i	0.996757	+	0.0804666i	$0.0256410\pi$
$410$		0			0
$411$		0			0
$412$	−1.68901	+	0.0680647i	−1.68901	+	0.0680647i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.457937	−	1.58098i	−0.457937	−	1.58098i
$418$		0			0
$419$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$419$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$420$		0			0
$421$	0.593921	+	0.0359256i	0.593921	+	0.0359256i	0.354605	−	0.935016i	$-0.384615\pi$
$421$	0.593921	+	0.0359256i	0.593921	+	0.0359256i	0.239316	+	0.970942i	$0.423077\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.545190	+	0.111301i	0.545190	+	0.111301i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$431$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$432$	0.391967	+	0.919979i	0.391967	+	0.919979i
$433$	−0.0799447	−	1.98381i	−0.0799447	−	1.98381i	−0.160411	−	0.987050i	$-0.551282\pi$
$433$	−0.0799447	−	1.98381i	−0.0799447	−	1.98381i	0.0804666	−	0.996757i	$-0.474359\pi$
$434$		0			0
$435$		0			0
$436$	−1.46733	+	0.0887571i	−1.46733	+	0.0887571i
$437$		0			0
$438$		0			0
$439$	−0.685430	+	1.44451i	−0.685430	+	1.44451i	0.200026	+	0.979791i	$0.435897\pi$
$439$	−0.685430	+	1.44451i	−0.685430	+	1.44451i	−0.885456	+	0.464723i	$0.846154\pi$
$440$		0			0
$441$	−0.428693	−	0.903450i	−0.428693	−	0.903450i
$442$		0			0
$443$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$443$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$444$	−0.482775	+	1.54928i	−0.482775	+	1.54928i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.774605	+	0.632445i	0.774605	+	0.632445i
$449$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$449$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.33391	−	1.44593i	1.33391	−	1.44593i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.853136	−	1.70825i	−0.853136	−	1.70825i	−0.692724	−	0.721202i	$-0.743590\pi$
$457$	−0.853136	−	1.70825i	−0.853136	−	1.70825i	−0.160411	−	0.987050i	$-0.551282\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$461$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$462$		0			0
$463$	−1.46052	−	1.14424i	−1.46052	−	1.14424i	−0.960518	−	0.278217i	$-0.910256\pi$
$463$	−1.46052	−	1.14424i	−1.46052	−	1.14424i	−0.500000	−	0.866025i	$-0.666667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$467$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$468$	0.632445	−	0.774605i	0.632445	−	0.774605i
$469$	−0.233580	−	0.274605i	−0.233580	−	0.274605i
$470$		0			0
$471$	−0.912242	+	1.11729i	−0.912242	+	1.11729i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.264697	−	0.0943346i	0.264697	−	0.0943346i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$479$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$480$		0			0
$481$	1.59617	−	0.292510i	1.59617	−	0.292510i
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−1.99919	+	0.0402659i	−1.99919	+	0.0402659i	−0.999189	+	0.0402659i	$0.987179\pi$
$487$	−1.99919	+	0.0402659i	−1.99919	+	0.0402659i	−1.00000			$\pi$
$488$		0			0
$489$	−0.574579	−	0.347345i	−0.574579	−	0.347345i
$490$		0			0
$491$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$491$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.02441	+	0.145374i	−1.02441	+	0.145374i
$497$		0			0
$498$		0			0
$499$	1.14048	+	1.58310i	1.14048	+	1.58310i	0.748511	+	0.663123i	$0.230769\pi$
$499$	1.14048	+	1.58310i	1.14048	+	1.58310i	0.391967	+	0.919979i	$0.371795\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$503$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.979791	−	0.200026i	−0.979791	−	0.200026i
$508$	−0.0402659	−	0.0697427i	−0.0402659	−	0.0697427i
$509$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$509$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$510$		0			0
$511$	0.428693	+	0.0965496i	0.428693	+	0.0965496i
$512$		0			0
$513$	−0.276402	+	0.0506526i	−0.276402	+	0.0506526i
$514$		0			0
$515$		0			0
$516$	0.318483	−	0.0386709i	0.318483	−	0.0386709i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$521$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$522$		0			0
$523$	0.418986	−	0.662573i	0.418986	−	0.662573i	−0.568065	−	0.822984i	$-0.692308\pi$
$523$	0.418986	−	0.662573i	0.418986	−	0.662573i	0.987050	+	0.160411i	$0.0512821\pi$
$524$		0			0
$525$	0.845190	−	0.534466i	0.845190	−	0.534466i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.221202	+	0.173301i	−0.221202	+	0.173301i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.878652	+	0.952443i	0.878652	+	0.952443i	0.999189	−	0.0402659i	$-0.0128205\pi$
$541$	0.878652	+	0.952443i	0.878652	+	0.952443i	−0.120537	+	0.992709i	$0.538462\pi$
$542$		0			0
$543$	0.801669	−	0.304033i	0.801669	−	0.304033i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.166997	−	1.37535i	0.166997	−	1.37535i	−0.632445	−	0.774605i	$-0.717949\pi$
$547$	0.166997	−	1.37535i	0.166997	−	1.37535i	0.799443	−	0.600742i	$-0.205128\pi$
$548$		0			0
$549$	0.133164	−	0.540266i	0.133164	−	0.540266i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.753393	−	1.09148i	0.753393	−	1.09148i
$554$		0			0
$555$		0			0
$556$	−1.23202	+	1.09148i	−1.23202	+	1.09148i
$557$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$557$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$558$		0			0
$559$	−0.182248	−	0.264032i	−0.182248	−	0.264032i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$563$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$568$		0			0
$569$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$569$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$570$		0			0
$571$	−0.600742	+	1.79944i	−0.600742	+	1.79944i			1.00000i	$0.5\pi$
$571$	−0.600742	+	1.79944i	−0.600742	+	1.79944i	−0.600742	+	0.799443i	$0.705128\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.663123	−	0.748511i	0.663123	−	0.748511i
$577$	−1.81974	−	0.487598i	−1.81974	−	0.487598i	−0.822984	−	0.568065i	$-0.807692\pi$
$577$	−1.81974	−	0.487598i	−1.81974	−	0.487598i	−0.996757	+	0.0804666i	$0.974359\pi$
$578$		0			0
$579$	1.89977	+	0.591992i	1.89977	+	0.591992i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$587$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$588$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$589$	0.0233956	−	0.289806i	0.0233956	−	0.289806i
$590$		0			0
$591$		0			0
$592$	1.60666	−	0.228001i	1.60666	−	0.228001i
$593$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$593$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.866514	−	0.832298i	−0.866514	−	0.832298i
$598$		0			0
$599$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$599$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$600$		0			0
$601$	−1.18593	+	1.57818i	−1.18593	+	1.57818i	−0.464723	+	0.885456i	$0.653846\pi$
$601$	−1.18593	+	1.57818i	−1.18593	+	1.57818i	−0.721202	+	0.692724i	$0.756410\pi$
$602$		0			0
$603$	−0.283788	+	0.222333i	−0.283788	+	0.222333i
$604$	−1.87816	−	0.585260i	−1.87816	−	0.585260i
$605$		0			0
$606$		0			0
$607$	−1.02732	+	1.15960i	−1.02732	+	1.15960i	−0.0402659	+	0.999189i	$0.512821\pi$
$607$	−1.02732	+	1.15960i	−1.02732	+	1.15960i	−0.987050	+	0.160411i	$0.948718\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.191941	+	0.0598112i	0.191941	+	0.0598112i	0.391967	−	0.919979i	$-0.371795\pi$
$613$	0.191941	+	0.0598112i	0.191941	+	0.0598112i	−0.200026	+	0.979791i	$0.564103\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$617$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$618$		0			0
$619$	0.975282	−	0.644584i	0.975282	−	0.644584i	0.0402659	−	0.999189i	$-0.487179\pi$
$619$	0.975282	−	0.644584i	0.975282	−	0.644584i	0.935016	+	0.354605i	$0.115385\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$625$	0.632445	+	0.774605i	0.632445	+	0.774605i
$626$		0			0
$627$		0			0
$628$	1.40049	+	0.345190i	1.40049	+	0.345190i
$629$		0			0
$630$		0			0
$631$	−0.683332	+	0.948536i	−0.683332	+	0.948536i	0.316668	+	0.948536i	$0.397436\pi$
$631$	−0.683332	+	0.948536i	−0.683332	+	0.948536i	−1.00000			$\pi$
$632$		0			0
$633$	0.145395	+	1.80104i	0.145395	+	1.80104i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.979791	+	0.200026i	0.979791	+	0.200026i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$641$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$642$		0			0
$643$	0.179861	−	0.0898262i	0.179861	−	0.0898262i	−0.354605	−	0.935016i	$-0.615385\pi$
$643$	0.179861	−	0.0898262i	0.179861	−	0.0898262i	0.534466	+	0.845190i	$0.320513\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$647$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.145374	−	1.02441i	−0.145374	−	1.02441i
$652$	−0.0674914	+	0.668008i	−0.0674914	+	0.668008i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.113733	−	0.424457i	0.113733	−	0.424457i
$658$		0			0
$659$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$659$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$660$		0			0
$661$	0.506219	−	1.12477i	0.506219	−	1.12477i	−0.464723	−	0.885456i	$-0.653846\pi$
$661$	0.506219	−	1.12477i	0.506219	−	1.12477i	0.970942	−	0.239316i	$-0.0769231\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.826639	+	1.14746i	0.826639	+	1.14746i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.248247	+	0.743589i	0.248247	+	0.743589i	0.996757	+	0.0804666i	$0.0256410\pi$
$673$	0.248247	+	0.743589i	0.248247	+	0.743589i	−0.748511	+	0.663123i	$0.769231\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	0.239316	+	0.970942i	0.239316	+	0.970942i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−0.480206	+	0.193266i	−0.480206	+	0.193266i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$683$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$684$	0.164254	+	0.228001i	0.164254	+	0.228001i
$685$		0			0
$686$		0			0
$687$	1.34267	−	0.190538i	1.34267	−	0.190538i
$688$	−0.171469	−	0.271156i	−0.171469	−	0.271156i
$689$		0			0
$690$		0			0
$691$	−0.0969956	−	0.960031i	−0.0969956	−	0.960031i	−0.919979	−	0.391967i	$-0.871795\pi$
$691$	−0.0969956	−	0.960031i	−0.0969956	−	0.960031i	0.822984	−	0.568065i	$-0.192308\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$701$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$701$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$702$		0			0
$703$	−0.0366929	+	0.454524i	−0.0366929	+	0.454524i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.113749	−	1.88050i	0.113749	−	1.88050i	−0.278217	−	0.960518i	$-0.589744\pi$
$709$	0.113749	−	1.88050i	0.113749	−	1.88050i	0.391967	−	0.919979i	$-0.371795\pi$
$710$		0			0
$711$	−1.06026	−	0.796732i	−1.06026	−	0.796732i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$719$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$720$		0			0
$721$	−1.49676	+	0.785559i	−1.49676	+	0.785559i
$722$		0			0
$723$	−0.0344658	−	1.71122i	−0.0344658	−	1.71122i
$724$	−0.618348	−	0.593932i	−0.618348	−	0.593932i
$725$		0			0
$726$		0			0
$727$	0.663123	−	1.74851i	0.663123	−	1.74851i		−	1.00000i	$-0.5\pi$
$727$	0.663123	−	1.74851i	0.663123	−	1.74851i	0.663123	−	0.748511i	$-0.269231\pi$
$728$		0			0
$729$	0.354605	+	0.935016i	0.354605	+	0.935016i
$730$		0			0
$731$		0			0
$732$	−0.545190	+	0.111301i	−0.545190	+	0.111301i
$733$	−0.366744	+	0.734339i	−0.366744	+	0.734339i	−0.999189	−	0.0402659i	$-0.987179\pi$
$733$	−0.366744	+	0.734339i	−0.366744	+	0.734339i	0.632445	+	0.774605i	$0.282051\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.287066	−	0.921228i	0.287066	−	0.921228i	−0.692724	−	0.721202i	$-0.743590\pi$
$739$	0.287066	−	0.921228i	0.287066	−	0.921228i	0.979791	−	0.200026i	$-0.0641026\pi$
$740$		0			0
$741$	0.125553	−	0.251397i	0.125553	−	0.251397i
$742$		0			0
$743$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$743$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.783297	+	1.83847i	0.783297	+	1.83847i	0.428693	+	0.903450i	$0.358974\pi$
$751$	0.783297	+	1.83847i	0.783297	+	1.83847i	0.354605	+	0.935016i	$0.384615\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.748511	+	0.663123i	0.748511	+	0.663123i
$757$	0.520276	−	1.79620i	0.520276	−	1.79620i	−0.0804666	−	0.996757i	$-0.525641\pi$
$757$	0.520276	−	1.79620i	0.520276	−	1.79620i	0.600742	−	0.799443i	$-0.294872\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$761$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$762$		0			0
$763$	−1.28761	+	0.709221i	−1.28761	+	0.709221i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.960518	−	0.278217i	−0.960518	−	0.278217i
$769$	−0.625186	−	1.13504i	−0.625186	−	1.13504i	−0.979791	−	0.200026i	$-0.935897\pi$
$769$	−0.625186	−	1.13504i	−0.625186	−	1.13504i	0.354605	−	0.935016i	$-0.384615\pi$
$770$		0			0
$771$		0			0
$772$	−0.279582	−	1.97013i	−0.279582	−	1.97013i
$773$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$773$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$774$		0			0
$775$	0.999420	−	0.267794i	0.999420	−	0.267794i
$776$		0			0
$777$	0.228001	+	1.60666i	0.228001	+	1.60666i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.970942	+	0.239316i	0.970942	+	0.239316i
$785$		0			0
$786$		0			0
$787$	−1.70903	+	0.941340i	−1.70903	+	0.941340i	−0.748511	+	0.663123i	$0.769231\pi$
$787$	−1.70903	+	0.941340i	−1.70903	+	0.941340i	−0.960518	+	0.278217i	$0.910256\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.351915	+	0.431017i	0.351915	+	0.431017i
$794$		0			0
$795$		0			0
$796$	−0.380472	+	1.13965i	−0.380472	+	1.13965i
$797$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$797$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.322525	+	0.161075i	0.322525	+	0.161075i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$809$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$810$		0			0
$811$	0.589104	−	1.89050i	0.589104	−	1.89050i	0.160411	−	0.987050i	$-0.448718\pi$
$811$	0.589104	−	1.89050i	0.589104	−	1.89050i	0.428693	−	0.903450i	$-0.358974\pi$
$812$		0			0
$813$	0.0135202	+	0.0379369i	0.0135202	+	0.0379369i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.0849209	−	0.0302647i	0.0849209	−	0.0302647i
$818$		0			0
$819$	0.239316	−	0.970942i	0.239316	−	0.970942i
$820$		0			0
$821$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$821$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$822$		0			0
$823$	1.71942	+	0.992709i	1.71942	+	0.992709i	0.919979	+	0.391967i	$0.128205\pi$
$823$	1.71942	+	0.992709i	1.71942	+	0.992709i	0.799443	+	0.600742i	$0.205128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$827$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$828$		0			0
$829$	1.76948	−	0.928693i	1.76948	−	0.928693i	0.866025	−	0.500000i	$-0.166667\pi$
$829$	1.76948	−	0.928693i	1.76948	−	0.928693i	0.903450	−	0.428693i	$-0.141026\pi$
$830$		0			0
$831$	−0.0789044	+	1.95799i	−0.0789044	+	1.95799i
$832$	0.200026	+	0.979791i	0.200026	+	0.979791i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.02943	+	0.104008i	−1.02943	+	0.104008i
$838$		0			0
$839$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$839$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$840$		0			0
$841$	0.692724	+	0.721202i	0.692724	+	0.721202i
$842$		0			0
$843$		0			0
$844$	1.56482	−	0.903450i	1.56482	−	0.903450i
$845$		0			0
$846$		0			0
$847$	−0.0804666	+	0.996757i	−0.0804666	+	0.996757i
$848$		0			0
$849$	−0.255812	−	1.03787i	−0.255812	−	1.03787i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.244198	−	0.542586i	0.244198	−	0.542586i	−0.748511	−	0.663123i	$-0.769231\pi$
$853$	0.244198	−	0.542586i	0.244198	−	0.542586i	0.992709	+	0.120537i	$0.0384615\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$857$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$858$		0			0
$859$	−1.89553	−	0.0763874i	−1.89553	−	0.0763874i	−0.935016	−	0.354605i	$-0.884615\pi$
$859$	−1.89553	−	0.0763874i	−1.89553	−	0.0763874i	−0.960518	+	0.278217i	$0.910256\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$863$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$868$	−0.863184	+	0.570496i	−0.863184	+	0.570496i
$869$		0			0
$870$		0			0
$871$	−0.00725961	−	0.360437i	−0.00725961	−	0.360437i
$872$		0			0
$873$	0.173773	+	0.487598i	0.173773	+	0.487598i
$874$		0			0
$875$		0			0
$876$	−0.432233	+	0.0792096i	−0.432233	+	0.0792096i
$877$	−0.147958	+	0.807380i	−0.147958	+	0.807380i	0.822984	+	0.568065i	$0.192308\pi$
$877$	−0.147958	+	0.807380i	−0.147958	+	0.807380i	−0.970942	+	0.239316i	$0.923077\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$881$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$882$		0			0
$883$	1.95971	−	0.237952i	1.95971	−	0.237952i	0.960518	−	0.278217i	$-0.0897436\pi$
$883$	1.95971	−	0.237952i	1.95971	−	0.237952i	0.999189	+	0.0402659i	$0.0128205\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$887$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$888$		0			0
$889$	−0.0662764	−	0.0457473i	−0.0662764	−	0.0457473i
$890$		0			0
$891$		0			0
$892$	0.682301	−	1.23874i	0.682301	−	1.23874i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$901$		0			0
$902$		0			0
$903$	0.271156	−	0.171469i	0.271156	−	0.171469i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.926432	+	1.76517i	−0.926432	+	1.76517i	−0.391967	+	0.919979i	$0.628205\pi$
$907$	−0.926432	+	1.76517i	−0.926432	+	1.76517i	−0.534466	+	0.845190i	$0.679487\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$911$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$912$	0.135573	−	0.246137i	0.135573	−	0.246137i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.747735	−	1.13135i	−0.747735	−	1.13135i
$917$		0			0
$918$		0			0
$919$	0.142499	+	0.0747894i	0.142499	+	0.0747894i	0.534466	−	0.845190i	$-0.320513\pi$
$919$	0.142499	+	0.0747894i	0.142499	+	0.0747894i	−0.391967	+	0.919979i	$0.628205\pi$
$920$		0			0
$921$	−0.943521	+	1.56077i	−0.943521	+	1.56077i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.56746	+	0.420000i	−1.56746	+	0.420000i
$926$		0			0
$927$	0.724653	+	1.52717i	0.724653	+	1.52717i
$928$		0			0
$929$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$929$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$930$		0			0
$931$	−0.125553	+	0.251397i	−0.125553	+	0.251397i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.440331	+	1.78649i	0.440331	+	1.78649i	0.600742	+	0.799443i	$0.294872\pi$
$937$	0.440331	+	1.78649i	0.440331	+	1.78649i	−0.160411	+	0.987050i	$0.551282\pi$
$938$		0			0
$939$	0.319818	+	0.240328i	0.319818	+	0.240328i
$940$		0			0
$941$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$941$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$947$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$948$	−0.265283	+	1.29944i	−0.265283	+	1.29944i
$949$	0.271005	+	0.345912i	0.271005	+	0.345912i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$953$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0113176	−	0.0696400i	0.0113176	−	0.0696400i
$962$		0			0
$963$		0			0
$964$	−1.53122	+	0.764724i	−1.53122	+	0.764724i
$965$		0			0
$966$		0			0
$967$	0.906291	+	1.49919i	0.906291	+	1.49919i	0.866025	+	0.500000i	$0.166667\pi$
$967$	0.906291	+	1.49919i	0.906291	+	1.49919i	0.0402659	+	0.999189i	$0.487179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$971$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$972$	0.692724	−	0.721202i	0.692724	−	0.721202i
$973$	−0.645164	+	1.51426i	−0.645164	+	1.51426i
$974$		0			0
$975$	0.992709	+	0.120537i	0.992709	+	0.120537i
$976$	0.334274	+	0.444838i	0.334274	+	0.444838i
$977$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$977$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.656799	+	1.31512i	0.656799	+	1.31512i
$982$		0			0
$983$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$983$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.280948	−	0.00565861i	−0.280948	−	0.00565861i
$989$		0			0
$990$		0			0
$991$	−1.59889			−1.59889			−0.799443	−	0.600742i	$-0.794872\pi$
$991$	−1.59889			−1.59889			−0.799443	+	0.600742i	$0.794872\pi$
$992$		0			0
$993$	−0.625233	−	1.38921i	−0.625233	−	1.38921i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0258155	+	0.158849i	−0.0258155	+	0.158849i	−0.996757	−	0.0804666i	$-0.974359\pi$
$997$	−0.0258155	+	0.158849i	−0.0258155	+	0.158849i	0.970942	+	0.239316i	$0.0769231\pi$
$998$		0			0
$999$	1.61454	−	0.163123i	1.61454	−	0.163123i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.1.ea.a.2126.1	yes	48
3.2	odd	2	CM	3549.1.ea.a.2126.1	yes	48
7.3	odd	6		3549.1.ep.a.605.1	yes	48
21.17	even	6		3549.1.ep.a.605.1	yes	48
169.119	odd	156		3549.1.ep.a.2147.1	yes	48
507.119	even	156		3549.1.ep.a.2147.1	yes	48
1183.626	even	156	inner	3549.1.ea.a.626.1	✓	48
3549.626	odd	156	inner	3549.1.ea.a.626.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3549.1.ea.a.626.1	✓	48	1183.626	even	156	inner
3549.1.ea.a.626.1	✓	48	3549.626	odd	156	inner
3549.1.ea.a.2126.1	yes	48	1.1	even	1	trivial
3549.1.ea.a.2126.1	yes	48	3.2	odd	2	CM
3549.1.ep.a.605.1	yes	48	7.3	odd	6
3549.1.ep.a.605.1	yes	48	21.17	even	6
3549.1.ep.a.2147.1	yes	48	169.119	odd	156
3549.1.ep.a.2147.1	yes	48	507.119	even	156

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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3549.ea (of order \(156\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.663123 + 0.748511i) q^{3} +(0.960518 + 0.278217i) q^{4} +(0.987050 - 0.160411i) q^{7} +(-0.120537 - 0.992709i) q^{9} +O(q^{10})\)	\(q+(-0.663123 + 0.748511i) q^{3} +(0.960518 + 0.278217i) q^{4} +(0.987050 - 0.160411i) q^{7} +(-0.120537 - 0.992709i) q^{9} +(-0.845190 + 0.534466i) q^{12} +(0.866025 + 0.500000i) q^{13} +(0.845190 + 0.534466i) q^{16} +(-0.198700 + 0.198700i) q^{19} +(-0.534466 + 0.845190i) q^{21} +(-0.903450 - 0.428693i) q^{25} +(0.822984 + 0.568065i) q^{27} +(0.992709 + 0.120537i) q^{28} +(-0.788125 + 0.670382i) q^{31} +(0.160411 - 0.987050i) q^{36} +(1.23607 - 1.05141i) q^{37} +(-0.948536 + 0.316668i) q^{39} +(-0.289847 - 0.137534i) q^{43} +(-0.960518 + 0.278217i) q^{48} +(0.948536 - 0.316668i) q^{49} +(0.692724 + 0.721202i) q^{52} +(-0.0169667 - 0.280492i) q^{57} +(0.520276 + 0.197315i) q^{61} +(-0.278217 - 0.960518i) q^{63} +(0.663123 + 0.748511i) q^{64} +(-0.186505 - 0.308518i) q^{67} +(0.407653 + 0.164066i) q^{73} +(0.919979 - 0.391967i) q^{75} +(-0.246137 + 0.135573i) q^{76} +(0.918722 - 0.956491i) q^{79} +(-0.970942 + 0.239316i) q^{81} +(-0.748511 + 0.663123i) q^{84} +(0.935016 + 0.354605i) q^{91} +(0.0208353 - 1.03447i) q^{93} +(-0.504989 + 0.113733i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 48 * q + 2 * q^7 + 4 * q^9	\( 48 q + 2 q^{7} + 4 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{19} - 2 q^{31} - 2 q^{37} - 2 q^{39} + 6 q^{43} + 2 q^{49} + 2 q^{52} + 2 q^{57} - 2 q^{63} + 4 q^{67} - 4 q^{73} - 2 q^{75} + 4 q^{76} - 4 q^{81} - 4 q^{84} + 2 q^{93} + 2 q^{97}+O(q^{100}) \) 48 * q + 2 * q^7 + 4 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^19 - 2 * q^31 - 2 * q^37 - 2 * q^39 + 6 * q^43 + 2 * q^49 + 2 * q^52 + 2 * q^57 - 2 * q^63 + 4 * q^67 - 4 * q^73 - 2 * q^75 + 4 * q^76 - 4 * q^81 - 4 * q^84 + 2 * q^93 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more