LMFDB - Embedded newform 3549.1.ds.a.2669.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,1,Mod(179,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, base_ring=CyclotomicField(78))

chi = DirichletCharacter(H, H._module([39, 52, 5]))

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

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

chi := DirichletCharacter("3549.179");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3549.ds (of order $78$, degree $24$, 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:	$24$
Coefficient field:	$\Q(\zeta_{39})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 $ x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{78}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{78} - \cdots)$

Embedding invariants

Embedding label			2669.1
Root			$0.948536 + 0.316668i$ of defining polynomial
Character	$\chi$	$=$	3549.2669
Dual form			3549.1.ds.a.3182.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.120537 + 0.992709i) q^{3} +(0.845190 - 0.534466i) q^{4} +(0.948536 + 0.316668i) q^{7} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})$	\(q+(-0.120537 + 0.992709i) q^{3} +(0.845190 - 0.534466i) q^{4} +(0.948536 + 0.316668i) q^{7} +(-0.970942 - 0.239316i) q^{9} +(0.428693 + 0.903450i) q^{12} +(0.500000 - 0.866025i) q^{13} +(0.428693 - 0.903450i) q^{16} -1.92104i q^{19} +(-0.428693 + 0.903450i) q^{21} +(0.632445 - 0.774605i) q^{25} +(0.354605 - 0.935016i) q^{27} +(0.970942 - 0.239316i) q^{28} +(-0.149094 - 0.917410i) q^{31} +(-0.948536 + 0.316668i) q^{36} +(0.101594 + 0.625134i) q^{37} +(0.799443 + 0.600742i) q^{39} +(-1.19979 + 1.46948i) q^{43} +(0.845190 + 0.534466i) q^{48} +(0.799443 + 0.600742i) q^{49} +(-0.0402659 - 0.999189i) q^{52} +(1.90703 + 0.231555i) q^{57} +(-1.26527 + 1.12093i) q^{61} +(-0.845190 - 0.534466i) q^{63} +(-0.120537 - 0.992709i) q^{64} +(0.869047 + 1.65583i) q^{67} +(-1.30314 + 1.25168i) q^{73} +(0.692724 + 0.721202i) q^{75} +(-1.02673 - 1.62364i) q^{76} +(0.00970705 - 0.240878i) q^{79} +(0.885456 + 0.464723i) q^{81} +(0.120537 + 0.992709i) q^{84} +(0.748511 - 0.663123i) q^{91} +(0.928693 - 0.0374250i) q^{93} +(-1.56482 - 0.742517i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} - q^{4} + q^{7} - 2 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 - q^4 + q^7 - 2 * q^9	$ 24 q + 2 q^{3} - q^{4} + q^{7} - 2 q^{9} + q^{12} + 12 q^{13} + q^{16} - q^{21} - q^{25} + 2 q^{27} + 2 q^{28} - q^{36} + 3 q^{37} + q^{39} - q^{43} - q^{48} + q^{49} + q^{52} - 2 q^{61} + q^{63} + 2 q^{64} - 3 q^{73} + q^{75} - 3 q^{76} - 2 q^{79} - 2 q^{81} - 2 q^{84} + 2 q^{91} + 13 q^{93} + 3 q^{97}+O(q^{100}) $ 24 * q + 2 * q^3 - q^4 + q^7 - 2 * q^9 + q^12 + 12 * q^13 + q^16 - q^21 - q^25 + 2 * q^27 + 2 * q^28 - q^36 + 3 * q^37 + q^39 - q^43 - q^48 + q^49 + q^52 - 2 * q^61 + q^63 + 2 * q^64 - 3 * q^73 + q^75 - 3 * q^76 - 2 * q^79 - 2 * q^81 - 2 * q^84 + 2 * q^91 + 13 * q^93 + 3 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1184$	$1522$	$3382$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{19}{78}\right)$

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		0.960518	−	0.278217i	$-0.0897436\pi$
$2$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$3$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$3$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$4$	0.845190	−	0.534466i	0.845190	−	0.534466i
$5$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$5$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$6$		0			0
$7$	0.948536	+	0.316668i	0.948536	+	0.316668i
$7$	0.948536	+	0.316668i	0.948536	+	0.316668i
$8$		0			0
$9$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$10$		0			0
$11$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$11$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$12$	0.428693	+	0.903450i	0.428693	+	0.903450i
$13$	0.500000	−	0.866025i	0.500000	−	0.866025i
$13$	0.500000	−	0.866025i	0.500000	−	0.866025i
$14$		0			0
$15$		0			0
$16$	0.428693	−	0.903450i	0.428693	−	0.903450i
$17$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$17$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$18$		0			0
$19$		−	1.92104i		−	1.92104i	−0.278217	−	0.960518i	$-0.589744\pi$
$19$		−	1.92104i		−	1.92104i	0.278217	−	0.960518i	$-0.410256\pi$
$20$		0			0
$21$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	0.632445	−	0.774605i	0.632445	−	0.774605i
$26$		0			0
$27$	0.354605	−	0.935016i	0.354605	−	0.935016i
$28$	0.970942	−	0.239316i	0.970942	−	0.239316i
$29$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$29$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$30$		0			0
$31$	−0.149094	−	0.917410i	−0.149094	−	0.917410i	−0.948536	−	0.316668i	$-0.897436\pi$
$31$	−0.149094	−	0.917410i	−0.149094	−	0.917410i	0.799443	−	0.600742i	$-0.205128\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$37$	0.101594	+	0.625134i	0.101594	+	0.625134i	0.987050	+	0.160411i	$0.0512821\pi$
$37$	0.101594	+	0.625134i	0.101594	+	0.625134i	−0.885456	+	0.464723i	$0.846154\pi$
$38$		0			0
$39$	0.799443	+	0.600742i	0.799443	+	0.600742i
$40$		0			0
$41$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$41$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$42$		0			0
$43$	−1.19979	+	1.46948i	−1.19979	+	1.46948i	−0.354605	+	0.935016i	$0.615385\pi$
$43$	−1.19979	+	1.46948i	−1.19979	+	1.46948i	−0.845190	+	0.534466i	$0.820513\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$47$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$48$	0.845190	+	0.534466i	0.845190	+	0.534466i
$49$	0.799443	+	0.600742i	0.799443	+	0.600742i
$50$		0			0
$51$		0			0
$52$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$53$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$53$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.90703	+	0.231555i	1.90703	+	0.231555i
$58$		0			0
$59$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$59$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$60$		0			0
$61$	−1.26527	+	1.12093i	−1.26527	+	1.12093i	−0.278217	+	0.960518i	$0.589744\pi$
$61$	−1.26527	+	1.12093i	−1.26527	+	1.12093i	−0.987050	+	0.160411i	$0.948718\pi$
$62$		0			0
$63$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$64$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$65$		0			0
$66$		0			0
$67$	0.869047	+	1.65583i	0.869047	+	1.65583i	0.748511	+	0.663123i	$0.230769\pi$
$67$	0.869047	+	1.65583i	0.869047	+	1.65583i	0.120537	+	0.992709i	$0.461538\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$71$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$72$		0			0
$73$	−1.30314	+	1.25168i	−1.30314	+	1.25168i	−0.354605	+	0.935016i	$0.615385\pi$
$73$	−1.30314	+	1.25168i	−1.30314	+	1.25168i	−0.948536	+	0.316668i	$0.897436\pi$
$74$		0			0
$75$	0.692724	+	0.721202i	0.692724	+	0.721202i
$76$	−1.02673	−	1.62364i	−1.02673	−	1.62364i
$77$		0			0
$78$		0			0
$79$	0.00970705	−	0.240878i	0.00970705	−	0.240878i	−0.987050	−	0.160411i	$-0.948718\pi$
$79$	0.00970705	−	0.240878i	0.00970705	−	0.240878i	0.996757	−	0.0804666i	$-0.0256410\pi$
$80$		0			0
$81$	0.885456	+	0.464723i	0.885456	+	0.464723i
$82$		0			0
$83$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$83$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$84$	0.120537	+	0.992709i	0.120537	+	0.992709i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	0.748511	−	0.663123i	0.748511	−	0.663123i
$92$		0			0
$93$	0.928693	−	0.0374250i	0.928693	−	0.0374250i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.56482	−	0.742517i	−1.56482	−	0.742517i	−0.568065	−	0.822984i	$-0.692308\pi$
$97$	−1.56482	−	0.742517i	−1.56482	−	0.742517i	−0.996757	+	0.0804666i	$0.974359\pi$
$98$		0			0
$99$		0			0
$100$	0.120537	−	0.992709i	0.120537	−	0.992709i
$101$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$101$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$102$		0			0
$103$	0.685430	+	0.515067i	0.685430	+	0.515067i	0.885456	−	0.464723i	$-0.153846\pi$
$103$	0.685430	+	0.515067i	0.685430	+	0.515067i	−0.200026	+	0.979791i	$0.564103\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$107$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$108$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$109$	0.124660	+	0.101781i	0.124660	+	0.101781i	0.692724	−	0.721202i	$-0.256410\pi$
$109$	0.124660	+	0.101781i	0.124660	+	0.101781i	−0.568065	+	0.822984i	$0.692308\pi$
$110$		0			0
$111$	−0.632822	+	0.0255019i	−0.632822	+	0.0255019i
$112$	0.692724	−	0.721202i	0.692724	−	0.721202i
$113$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$113$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$122$		0			0
$123$		0			0
$124$	−0.616337	−	0.695701i	−0.616337	−	0.695701i
$125$		0			0
$126$		0			0
$127$	−0.0802707	+	1.99190i	−0.0802707	+	1.99190i	0.0402659	+	0.999189i	$0.487179\pi$
$127$	−0.0802707	+	1.99190i	−0.0802707	+	1.99190i	−0.120537	+	0.992709i	$0.538462\pi$
$128$		0			0
$129$	−1.31415	−	1.36817i	−1.31415	−	1.36817i
$130$		0			0
$131$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$131$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$132$		0			0
$133$	0.608331	−	1.82217i	0.608331	−	1.82217i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$137$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$138$		0			0
$139$	−0.304033	+	0.640736i	−0.304033	+	0.640736i	−0.996757	−	0.0804666i	$-0.974359\pi$
$139$	−0.304033	+	0.640736i	−0.304033	+	0.640736i	0.692724	+	0.721202i	$0.256410\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$145$		0			0
$146$		0			0
$147$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$148$	0.419979	+	0.474059i	0.419979	+	0.474059i
$149$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$149$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$150$		0			0
$151$	1.86852	−	0.0752986i	1.86852	−	0.0752986i	0.919979	−	0.391967i	$-0.128205\pi$
$151$	1.86852	−	0.0752986i	1.86852	−	0.0752986i	0.948536	+	0.316668i	$0.102564\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$157$	0.0802707	+	0.00648012i	0.0802707	+	0.00648012i	0.120537	−	0.992709i	$-0.461538\pi$
$157$	0.0802707	+	0.00648012i	0.0802707	+	0.00648012i	−0.0402659	+	0.999189i	$0.512821\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.44854	+	0.549357i	1.44854	+	0.549357i	0.948536	−	0.316668i	$-0.102564\pi$
$163$	1.44854	+	0.549357i	1.44854	+	0.549357i	0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$167$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	−0.459734	+	1.86521i	−0.459734	+	1.86521i
$172$	−0.228667	+	1.88324i	−0.228667	+	1.88324i
$173$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$173$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$174$		0			0
$175$	0.845190	−	0.534466i	0.845190	−	0.534466i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$179$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$180$		0			0
$181$	−0.718540	+	1.04098i	−0.718540	+	1.04098i	0.278217	+	0.960518i	$0.410256\pi$
$181$	−0.718540	+	1.04098i	−0.718540	+	1.04098i	−0.996757	+	0.0804666i	$0.974359\pi$
$182$		0			0
$183$	−0.960245	−	1.39116i	−0.960245	−	1.39116i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.632445	−	0.774605i	0.632445	−	0.774605i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	1.00000			1.00000
$193$	−1.29944	−	1.46677i	−1.29944	−	1.46677i	−0.799443	−	0.600742i	$-0.794872\pi$
$193$	−1.29944	−	1.46677i	−1.29944	−	1.46677i	−0.500000	−	0.866025i	$-0.666667\pi$
$194$		0			0
$195$		0			0
$196$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$197$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$197$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$198$		0			0
$199$	0.554631	−	0.0447744i	0.554631	−	0.0447744i	0.200026	−	0.979791i	$-0.435897\pi$
$199$	0.554631	−	0.0447744i	0.554631	−	0.0447744i	0.354605	+	0.935016i	$0.384615\pi$
$200$		0			0
$201$	−1.74851	+	0.663123i	−1.74851	+	0.663123i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$209$		0			0
$210$		0			0
$211$	−0.0509320	+	1.26386i	−0.0509320	+	1.26386i	0.748511	+	0.663123i	$0.230769\pi$
$211$	−0.0509320	+	1.26386i	−0.0509320	+	1.26386i	−0.799443	+	0.600742i	$0.794872\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.149094	−	0.917410i	0.149094	−	0.917410i
$218$		0			0
$219$	−1.08548	−	1.44451i	−1.08548	−	1.44451i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$223$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$224$		0			0
$225$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$226$		0			0
$227$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$227$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$228$	1.73556	−	0.823534i	1.73556	−	0.823534i
$229$	−0.0258155	+	0.158849i	−0.0258155	+	0.158849i	−0.996757	−	0.0804666i	$-0.974359\pi$
$229$	−0.0258155	+	0.158849i	−0.0258155	+	0.158849i	0.970942	+	0.239316i	$0.0769231\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$233$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.237952	+	0.0386709i	0.237952	+	0.0386709i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.0747894	+	0.926432i	0.0747894	+	0.926432i	0.919979	+	0.391967i	$0.128205\pi$
$241$	0.0747894	+	0.926432i	0.0747894	+	0.926432i	−0.845190	+	0.534466i	$0.820513\pi$
$242$		0			0
$243$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$244$	−0.470293	+	1.62364i	−0.470293	+	1.62364i
$245$		0			0
$246$		0			0
$247$	−1.66367	−	0.960518i	−1.66367	−	0.960518i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$251$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$252$	−1.00000			−1.00000
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$257$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$257$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$258$		0			0
$259$	−0.101594	+	0.625134i	−0.101594	+	0.625134i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$263$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.61950	+	0.935016i	1.61950	+	0.935016i
$269$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$269$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$270$		0			0
$271$	1.06806	−	1.68901i	1.06806	−	1.68901i	0.500000	−	0.866025i	$-0.333333\pi$
$271$	1.06806	−	1.68901i	1.06806	−	1.68901i	0.568065	−	0.822984i	$-0.307692\pi$
$272$		0			0
$273$	0.568065	+	0.822984i	0.568065	+	0.822984i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.368039	+	1.80277i	0.368039	+	1.80277i	0.568065	+	0.822984i	$0.307692\pi$
$277$	0.368039	+	1.80277i	0.368039	+	1.80277i	−0.200026	+	0.979791i	$0.564103\pi$
$278$		0			0
$279$	−0.0747894	+	0.926432i	−0.0747894	+	0.926432i
$280$		0			0
$281$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$281$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$282$		0			0
$283$	0.304033	−	0.801669i	0.304033	−	0.801669i	−0.692724	−	0.721202i	$-0.743590\pi$
$283$	0.304033	−	0.801669i	0.304033	−	0.801669i	0.996757	−	0.0804666i	$-0.0256410\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.799443	−	0.600742i	0.799443	−	0.600742i
$290$		0			0
$291$	0.925722	−	1.46391i	0.925722	−	1.46391i
$292$	−0.432420	+	1.75440i	−0.432420	+	1.75440i
$293$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$293$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.970942	+	0.239316i	0.970942	+	0.239316i
$301$	−1.60339	+	1.01392i	−1.60339	+	1.01392i
$302$		0			0
$303$		0			0
$304$	−1.73556	−	0.823534i	−1.73556	−	0.823534i
$305$		0			0
$306$		0			0
$307$	1.24006	+	0.470293i	1.24006	+	0.470293i	0.885456	−	0.464723i	$-0.153846\pi$
$307$	1.24006	+	0.470293i	1.24006	+	0.470293i	0.354605	+	0.935016i	$0.384615\pi$
$308$		0			0
$309$	−0.593932	+	0.618348i	−0.593932	+	0.618348i
$310$		0			0
$311$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$311$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$312$		0			0
$313$	1.81613	+	0.295150i	1.81613	+	0.295150i	0.970942	−	0.239316i	$-0.0769231\pi$
$313$	1.81613	+	0.295150i	1.81613	+	0.295150i	0.845190	+	0.534466i	$0.179487\pi$
$314$		0			0
$315$		0			0
$316$	−0.120537	−	0.208776i	−0.120537	−	0.208776i
$317$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$317$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$325$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$326$		0			0
$327$	−0.116065	+	0.111482i	−0.116065	+	0.111482i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.212745	+	0.240139i	−0.212745	+	0.240139i	−0.845190	−	0.534466i	$-0.820513\pi$
$331$	−0.212745	+	0.240139i	−0.212745	+	0.240139i	0.632445	+	0.774605i	$0.282051\pi$
$332$		0			0
$333$	0.0509624	−	0.631282i	0.0509624	−	0.631282i
$334$		0			0
$335$		0			0
$336$	0.632445	+	0.774605i	0.632445	+	0.774605i
$337$	0.400051			0.400051			0.200026	−	0.979791i	$-0.435897\pi$
$337$	0.400051			0.400051			0.200026	+	0.979791i	$0.435897\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.568065	+	0.822984i	0.568065	+	0.822984i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$347$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$348$		0			0
$349$	−1.66367	−	0.481887i	−1.66367	−	0.481887i	−0.692724	−	0.721202i	$-0.743590\pi$
$349$	−1.66367	−	0.481887i	−1.66367	−	0.481887i	−0.970942	+	0.239316i	$0.923077\pi$
$350$		0			0
$351$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$352$		0			0
$353$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$353$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$359$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$360$		0			0
$361$	−2.69038			−2.69038
$362$		0			0
$363$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$364$	0.278217	−	0.960518i	0.278217	−	0.960518i
$365$		0			0
$366$		0			0
$367$	−1.55242	−	0.382638i	−1.55242	−	0.382638i	−0.632445	−	0.774605i	$-0.717949\pi$
$367$	−1.55242	−	0.382638i	−1.55242	−	0.382638i	−0.919979	+	0.391967i	$0.871795\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.764919	−	0.527986i	0.764919	−	0.527986i
$373$	1.71945	−	0.423807i	1.71945	−	0.423807i	0.748511	−	0.663123i	$-0.230769\pi$
$373$	1.71945	−	0.423807i	1.71945	−	0.423807i	0.970942	+	0.239316i	$0.0769231\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.287534	−	0.382638i	0.287534	−	0.382638i	−0.632445	−	0.774605i	$-0.717949\pi$
$379$	0.287534	−	0.382638i	0.287534	−	0.382638i	0.919979	+	0.391967i	$0.128205\pi$
$380$		0			0
$381$	−1.96770	−	0.319782i	−1.96770	−	0.319782i
$382$		0			0
$383$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$383$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.51660	−	1.13965i	1.51660	−	1.13965i
$388$	−1.71942	+	0.208776i	−1.71942	+	0.208776i
$389$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$389$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.879714	−	0.607222i	−0.879714	−	0.607222i	0.0402659	−	0.999189i	$-0.487179\pi$
$397$	−0.879714	−	0.607222i	−0.879714	−	0.607222i	−0.919979	+	0.391967i	$0.871795\pi$
$398$		0			0
$399$	1.73556	+	0.823534i	1.73556	+	0.823534i
$400$	−0.428693	−	0.903450i	−0.428693	−	0.903450i
$401$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$401$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$402$		0			0
$403$	−0.869047	−	0.329586i	−0.869047	−	0.329586i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.866514	−	1.15312i	0.866514	−	1.15312i	−0.120537	−	0.992709i	$-0.538462\pi$
$409$	0.866514	−	1.15312i	0.866514	−	1.15312i	0.987050	−	0.160411i	$-0.0512821\pi$
$410$		0			0
$411$		0			0
$412$	0.854605	+	0.0689908i	0.854605	+	0.0689908i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.599417	−	0.379048i	−0.599417	−	0.379048i
$418$		0			0
$419$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$419$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$420$		0			0
$421$	−1.63397	+	0.198399i	−1.63397	+	0.198399i	−0.885456	−	0.464723i	$-0.846154\pi$
$421$	−1.63397	+	0.198399i	−1.63397	+	0.198399i	−0.748511	+	0.663123i	$0.769231\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.55512	+	0.662573i	−1.55512	+	0.662573i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$431$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$432$	−0.692724	−	0.721202i	−0.692724	−	0.721202i
$433$	−1.93559	−	0.156257i	−1.93559	−	0.156257i	−0.948536	−	0.316668i	$-0.897436\pi$
$433$	−1.93559	−	0.156257i	−1.93559	−	0.156257i	−0.987050	+	0.160411i	$0.948718\pi$
$434$		0			0
$435$		0			0
$436$	0.159760	+	0.0193983i	0.159760	+	0.0193983i
$437$		0			0
$438$		0			0
$439$	−0.351915	+	0.431017i	−0.351915	+	0.431017i	−0.919979	−	0.391967i	$-0.871795\pi$
$439$	−0.351915	+	0.431017i	−0.351915	+	0.431017i	0.568065	+	0.822984i	$0.307692\pi$
$440$		0			0
$441$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$442$		0			0
$443$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$443$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$444$	−0.521225	+	0.359776i	−0.521225	+	0.359776i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.200026	−	0.979791i	0.200026	−	0.979791i
$449$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$449$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.150475	+	1.86397i	−0.150475	+	1.86397i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.988802	−	1.31586i	−0.988802	−	1.31586i	−0.948536	−	0.316668i	$-0.897436\pi$
$457$	−0.988802	−	1.31586i	−0.988802	−	1.31586i	−0.0402659	−	0.999189i	$-0.512821\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$461$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$462$		0			0
$463$	0.345190	−	1.40049i	0.345190	−	1.40049i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	0.345190	−	1.40049i	0.345190	−	1.40049i	0.845190	−	0.534466i	$-0.179487\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$467$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$468$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$469$	0.299974	+	1.84582i	0.299974	+	1.84582i
$470$		0			0
$471$	−0.0161084	+	0.0789044i	−0.0161084	+	0.0789044i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.48804	−	1.21495i	−1.48804	−	1.21495i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$479$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$480$		0			0
$481$	0.592179	+	0.224584i	0.592179	+	0.224584i
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	1.99676	+	0.0804666i	1.99676	+	0.0804666i	1.00000			$0$
$487$	1.99676	+	0.0804666i	1.99676	+	0.0804666i	0.996757	+	0.0804666i	$0.0256410\pi$
$488$		0			0
$489$	−0.719954	+	1.37176i	−0.719954	+	1.37176i
$490$		0			0
$491$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$491$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.892750	−	0.258588i	−0.892750	−	0.258588i
$497$		0			0
$498$		0			0
$499$	−0.572188	−	1.71391i	−0.572188	−	1.71391i	−0.692724	−	0.721202i	$-0.743590\pi$
$499$	−0.572188	−	1.71391i	−0.572188	−	1.71391i	0.120537	−	0.992709i	$-0.461538\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$503$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.919979	−	0.391967i	0.919979	−	0.391967i
$508$	0.996757	+	1.72643i	0.996757	+	1.72643i
$509$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$509$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$510$		0			0
$511$	−1.63245	+	0.774605i	−1.63245	+	0.774605i
$512$		0			0
$513$	−1.79620	−	0.681209i	−1.79620	−	0.681209i
$514$		0			0
$515$		0			0
$516$	−1.84195	−	0.453999i	−1.84195	−	0.453999i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$521$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$522$		0			0
$523$	0.593932	−	1.25168i	0.593932	−	1.25168i	−0.354605	−	0.935016i	$-0.615385\pi$
$523$	0.593932	−	1.25168i	0.593932	−	1.25168i	0.948536	−	0.316668i	$-0.102564\pi$
$524$		0			0
$525$	0.428693	+	0.903450i	0.428693	+	0.903450i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.459734	−	1.86521i	−0.459734	−	1.86521i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.0258155	+	0.319782i	0.0258155	+	0.319782i	0.996757	+	0.0804666i	$0.0256410\pi$
$541$	0.0258155	+	0.319782i	0.0258155	+	0.319782i	−0.970942	+	0.239316i	$0.923077\pi$
$542$		0			0
$543$	−0.946784	−	0.838778i	−0.946784	−	0.838778i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.0781918	−	0.0192725i	0.0781918	−	0.0192725i	−0.200026	−	0.979791i	$-0.564103\pi$
$547$	0.0781918	−	0.0192725i	0.0781918	−	0.0192725i	0.278217	+	0.960518i	$0.410256\pi$
$548$		0			0
$549$	1.49676	−	0.785559i	1.49676	−	0.785559i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.0854858	−	0.225408i	0.0854858	−	0.225408i
$554$		0			0
$555$		0			0
$556$	0.0854858	+	0.704039i	0.0854858	+	0.704039i
$557$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$557$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$558$		0			0
$559$	0.672711	+	1.77379i	0.672711	+	1.77379i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$563$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.692724	+	0.721202i	0.692724	+	0.721202i
$568$		0			0
$569$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$569$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$570$		0			0
$571$	−1.27822	+	0.960518i	−1.27822	+	0.960518i	−0.278217	+	0.960518i	$0.589744\pi$
$571$	−1.27822	+	0.960518i	−1.27822	+	0.960518i	−1.00000			$1.00000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$577$	1.34166	−	0.774605i	1.34166	−	0.774605i	0.354605	−	0.935016i	$-0.384615\pi$
$577$	1.34166	−	0.774605i	1.34166	−	0.774605i	0.987050	+	0.160411i	$0.0512821\pi$
$578$		0			0
$579$	1.61270	−	1.11317i	1.61270	−	1.11317i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$589$	−1.76238	+	0.286414i	−1.76238	+	0.286414i
$590$		0			0
$591$		0			0
$592$	0.608331	+	0.176205i	0.608331	+	0.176205i
$593$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$593$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.0224054	+	0.555984i	−0.0224054	+	0.555984i
$598$		0			0
$599$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$599$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$600$		0			0
$601$	−0.527799	+	1.82217i	−0.527799	+	1.82217i	0.0402659	+	0.999189i	$0.487179\pi$
$601$	−0.527799	+	1.82217i	−0.527799	+	1.82217i	−0.568065	+	0.822984i	$0.692308\pi$
$602$		0			0
$603$	−0.447528	−	1.81569i	−0.447528	−	1.81569i
$604$	1.53901	−	1.06230i	1.53901	−	1.06230i
$605$		0			0
$606$		0			0
$607$	−0.0482209	+	0.397135i	−0.0482209	+	0.397135i	0.948536	+	0.316668i	$0.102564\pi$
$607$	−0.0482209	+	0.397135i	−0.0482209	+	0.397135i	−0.996757	+	0.0804666i	$0.974359\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.61270	+	1.11317i	−1.61270	+	1.11317i	−0.692724	+	0.721202i	$0.743590\pi$
$613$	−1.61270	+	1.11317i	−1.61270	+	1.11317i	−0.919979	+	0.391967i	$0.871795\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$617$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$618$		0			0
$619$	−0.248247	−	0.582656i	−0.248247	−	0.582656i	0.748511	−	0.663123i	$-0.230769\pi$
$619$	−0.248247	−	0.582656i	−0.248247	−	0.582656i	−0.996757	+	0.0804666i	$0.974359\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.885456	−	0.464723i	0.885456	−	0.464723i
$625$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$626$		0			0
$627$		0			0
$628$	0.0713074	−	0.0374250i	0.0713074	−	0.0374250i
$629$		0			0
$630$		0			0
$631$	0.200557	−	0.600742i	0.200557	−	0.600742i	−0.799443	−	0.600742i	$-0.794872\pi$
$631$	0.200557	−	0.600742i	0.200557	−	0.600742i	1.00000			$0$
$632$		0			0
$633$	−1.24851	−	0.202903i	−1.24851	−	0.202903i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.919979	−	0.391967i	0.919979	−	0.391967i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$641$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$642$		0			0
$643$	−1.17720	−	1.56657i	−1.17720	−	1.56657i	−0.748511	−	0.663123i	$-0.769231\pi$
$643$	−1.17720	−	1.56657i	−1.17720	−	1.56657i	−0.428693	−	0.903450i	$-0.641026\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$647$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	0.892750	+	0.258588i	0.892750	+	0.258588i
$652$	1.51790	−	0.309882i	1.51790	−	0.309882i
$653$		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
$657$	1.56482	−	0.903450i	1.56482	−	0.903450i
$658$		0			0
$659$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$659$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$660$		0			0
$661$	0.317391	−	0.358261i	0.317391	−	0.358261i	−0.568065	−	0.822984i	$-0.692308\pi$
$661$	0.317391	−	0.358261i	0.317391	−	0.358261i	0.885456	+	0.464723i	$0.153846\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.10759	+	0.832298i	1.10759	+	0.832298i	0.987050	−	0.160411i	$-0.0512821\pi$
$673$	1.10759	+	0.832298i	1.10759	+	0.832298i	0.120537	+	0.992709i	$0.461538\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−1.24916	−	1.19983i	−1.24916	−	1.19983i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$683$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$684$	0.608331	+	1.82217i	0.608331	+	1.82217i
$685$		0			0
$686$		0			0
$687$	−0.154579	−	0.0447744i	−0.154579	−	0.0447744i
$688$	0.813261	+	1.71391i	0.813261	+	1.71391i
$689$		0			0
$690$		0			0
$691$	1.04733	+	0.213814i	1.04733	+	0.213814i	0.692724	−	0.721202i	$-0.256410\pi$
$691$	1.04733	+	0.213814i	1.04733	+	0.213814i	0.354605	+	0.935016i	$0.384615\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.428693	−	0.903450i	0.428693	−	0.903450i
$701$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$701$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$702$		0			0
$703$	1.20091	−	0.195166i	1.20091	−	0.195166i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.53791	+	0.186737i	−1.53791	+	0.186737i	−0.845190	−	0.534466i	$-0.820513\pi$
$709$	−1.53791	+	0.186737i	−1.53791	+	0.186737i	−0.692724	+	0.721202i	$0.743590\pi$
$710$		0			0
$711$	−0.0670708	+	0.231555i	−0.0670708	+	0.231555i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$719$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$720$		0			0
$721$	0.487050	+	0.705614i	0.487050	+	0.705614i
$722$		0			0
$723$	−0.928693	−	0.0374250i	−0.928693	−	0.0374250i
$724$	−0.0509320	+	1.26386i	−0.0509320	+	1.26386i
$725$		0			0
$726$		0			0
$727$	−1.12054	+	0.992709i	−1.12054	+	0.992709i	−0.120537	+	0.992709i	$0.538462\pi$
$727$	−1.12054	+	0.992709i	−1.12054	+	0.992709i	−1.00000			$\pi$
$728$		0			0
$729$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$730$		0			0
$731$		0			0
$732$	−1.55512	−	0.662573i	−1.55512	−	0.662573i
$733$	0.796732	−	1.06026i	0.796732	−	1.06026i	−0.200026	−	0.979791i	$-0.564103\pi$
$733$	0.796732	−	1.06026i	0.796732	−	1.06026i	0.996757	−	0.0804666i	$-0.0256410\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.879714	−	0.607222i	0.879714	−	0.607222i	−0.0402659	−	0.999189i	$-0.512821\pi$
$739$	0.879714	−	0.607222i	0.879714	−	0.607222i	0.919979	+	0.391967i	$0.128205\pi$
$740$		0			0
$741$	1.15405	−	1.53576i	1.15405	−	1.53576i
$742$		0			0
$743$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$743$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.38096	−	1.43773i	−1.38096	−	1.43773i	−0.748511	−	0.663123i	$-0.769231\pi$
$751$	−1.38096	−	1.43773i	−1.38096	−	1.43773i	−0.632445	−	0.774605i	$-0.717949\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.120537	−	0.992709i	0.120537	−	0.992709i
$757$	−1.26527	+	0.800107i	−1.26527	+	0.800107i	−0.987050	−	0.160411i	$-0.948718\pi$
$757$	−1.26527	+	0.800107i	−1.26527	+	0.800107i	−0.278217	+	0.960518i	$0.589744\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$761$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$762$		0			0
$763$	0.0860133	+	0.136019i	0.0860133	+	0.136019i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.845190	−	0.534466i	0.845190	−	0.534466i
$769$	0.171469	+	0.271156i	0.171469	+	0.271156i	0.919979	−	0.391967i	$-0.128205\pi$
$769$	0.171469	+	0.271156i	0.171469	+	0.271156i	−0.748511	+	0.663123i	$0.769231\pi$
$770$		0			0
$771$		0			0
$772$	−1.88221	−	0.545190i	−1.88221	−	0.545190i
$773$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$773$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$774$		0			0
$775$	−0.804924	−	0.464723i	−0.804924	−	0.464723i
$776$		0			0
$777$	−0.608331	−	0.176205i	−0.608331	−	0.176205i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.885456	−	0.464723i	0.885456	−	0.464723i
$785$		0			0
$786$		0			0
$787$	0.965727	+	1.52717i	0.965727	+	1.52717i	0.845190	+	0.534466i	$0.179487\pi$
$787$	0.965727	+	1.52717i	0.965727	+	1.52717i	0.120537	+	0.992709i	$0.461538\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.338119	+	1.65622i	0.338119	+	1.65622i
$794$		0			0
$795$		0			0
$796$	0.444838	−	0.334274i	0.444838	−	0.334274i
$797$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$797$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.12341	+	1.49498i	−1.12341	+	1.49498i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$809$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$810$		0			0
$811$	−1.58098	+	1.09127i	−1.58098	+	1.09127i	−0.632445	+	0.774605i	$0.717949\pi$
$811$	−1.58098	+	1.09127i	−1.58098	+	1.09127i	−0.948536	+	0.316668i	$0.897436\pi$
$812$		0			0
$813$	1.54795	+	1.26386i	1.54795	+	1.26386i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.82293	+	2.30485i	2.82293	+	2.30485i
$818$		0			0
$819$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$820$		0			0
$821$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$821$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$822$		0			0
$823$	0.970942	−	1.68172i	0.970942	−	1.68172i	0.278217	−	0.960518i	$-0.410256\pi$
$823$	0.970942	−	1.68172i	0.970942	−	1.68172i	0.692724	−	0.721202i	$-0.256410\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$827$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$828$		0			0
$829$	1.13245	+	1.64063i	1.13245	+	1.64063i	0.632445	+	0.774605i	$0.282051\pi$
$829$	1.13245	+	1.64063i	1.13245	+	1.64063i	0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$	−1.83399	+	0.148055i	−1.83399	+	0.148055i
$832$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.910663	−	0.185913i	−0.910663	−	0.185913i
$838$		0			0
$839$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$839$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$840$		0			0
$841$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$842$		0			0
$843$		0			0
$844$	0.632445	+	1.09543i	0.632445	+	1.09543i
$845$		0			0
$846$		0			0
$847$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$848$		0			0
$849$	0.759177	+	0.398447i	0.759177	+	0.398447i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.09148	−	1.23202i	1.09148	−	1.23202i	0.120537	−	0.992709i	$-0.461538\pi$
$853$	1.09148	−	1.23202i	1.09148	−	1.23202i	0.970942	−	0.239316i	$-0.0769231\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$857$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$858$		0			0
$859$	1.59370	−	0.128657i	1.59370	−	0.128657i	0.748511	−	0.663123i	$-0.230769\pi$
$859$	1.59370	−	0.128657i	1.59370	−	0.128657i	0.845190	+	0.534466i	$0.179487\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$863$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.500000	+	0.866025i	0.500000	+	0.866025i
$868$	−0.364312	−	0.855072i	−0.364312	−	0.855072i
$869$		0			0
$870$		0			0
$871$	1.86852	+	0.0752986i	1.86852	+	0.0752986i
$872$		0			0
$873$	1.34166	+	1.09543i	1.34166	+	1.09543i
$874$		0			0
$875$		0			0
$876$	−1.68948	−	0.640736i	−1.68948	−	0.640736i
$877$	1.24006	−	0.470293i	1.24006	−	0.470293i	0.354605	−	0.935016i	$-0.384615\pi$
$877$	1.24006	−	0.470293i	1.24006	−	0.470293i	0.885456	+	0.464723i	$0.153846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$881$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$882$		0			0
$883$	1.84195	+	0.453999i	1.84195	+	0.453999i	0.996757	−	0.0804666i	$-0.0256410\pi$
$883$	1.84195	+	0.453999i	1.84195	+	0.453999i	0.845190	+	0.534466i	$0.179487\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$887$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$888$		0			0
$889$	−0.706910	+	1.86397i	−0.706910	+	1.86397i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$901$		0			0
$902$		0			0
$903$	−0.813261	−	1.71391i	−0.813261	−	1.71391i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.12142	+	1.62465i	−1.12142	+	1.62465i	−0.428693	+	0.903450i	$0.641026\pi$
$907$	−1.12142	+	1.62465i	−1.12142	+	1.62465i	−0.692724	+	0.721202i	$0.743590\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$911$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$912$	1.02673	−	1.62364i	1.02673	−	1.62364i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0630804	+	0.148055i	0.0630804	+	0.148055i
$917$		0			0
$918$		0			0
$919$	−1.12142	+	1.62465i	−1.12142	+	1.62465i	−0.428693	+	0.903450i	$0.641026\pi$
$919$	−1.12142	+	1.62465i	−1.12142	+	1.62465i	−0.692724	+	0.721202i	$0.743590\pi$
$920$		0			0
$921$	−0.616337	+	1.17433i	−0.616337	+	1.17433i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.548485	+	0.316668i	0.548485	+	0.316668i
$926$		0			0
$927$	−0.542249	−	0.664135i	−0.542249	−	0.664135i
$928$		0			0
$929$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$929$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$930$		0			0
$931$	1.15405	−	1.53576i	1.15405	−	1.53576i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.22675	−	0.643850i	−1.22675	−	0.643850i	−0.278217	−	0.960518i	$-0.589744\pi$
$937$	−1.22675	−	0.643850i	−1.22675	−	0.643850i	−0.948536	+	0.316668i	$0.897436\pi$
$938$		0			0
$939$	−0.511909	+	1.76731i	−0.511909	+	1.76731i
$940$		0			0
$941$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$941$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$947$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$948$	0.221783	−	0.0944927i	0.221783	−	0.0944927i
$949$	0.432420	+	1.75440i	0.432420	+	1.75440i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$953$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.129124	−	0.0431079i	0.129124	−	0.0431079i
$962$		0			0
$963$		0			0
$964$	0.558358	+	0.743039i	0.558358	+	0.743039i
$965$		0			0
$966$		0			0
$967$	−0.496757	−	0.946492i	−0.496757	−	0.946492i	−0.996757	−	0.0804666i	$-0.974359\pi$
$967$	−0.496757	−	0.946492i	−0.496757	−	0.946492i	0.500000	−	0.866025i	$-0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$971$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$972$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$973$	−0.491287	+	0.511484i	−0.491287	+	0.511484i
$974$		0			0
$975$	0.970942	−	0.239316i	0.970942	−	0.239316i
$976$	0.470293	+	1.62364i	0.470293	+	1.62364i
$977$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$977$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.0966793	−	0.128657i	−0.0966793	−	0.128657i
$982$		0			0
$983$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$983$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.91948	+	0.0773523i	−1.91948	+	0.0773523i
$989$		0			0
$990$		0			0
$991$	0.556435			0.556435			0.278217	−	0.960518i	$-0.410256\pi$
$991$	0.556435			0.556435			0.278217	+	0.960518i	$0.410256\pi$
$992$		0			0
$993$	−0.212745	−	0.240139i	−0.212745	−	0.240139i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.87251	−	0.625134i	1.87251	−	0.625134i	0.885456	−	0.464723i	$-0.153846\pi$
$997$	1.87251	−	0.625134i	1.87251	−	0.625134i	0.987050	−	0.160411i	$-0.0512821\pi$
$998$		0			0
$999$	0.620537	+	0.126683i	0.620537	+	0.126683i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.1.ds.a.2669.1	yes	24
3.2	odd	2	CM	3549.1.ds.a.2669.1	yes	24
7.4	even	3		3549.1.da.a.641.1	✓	24
21.11	odd	6		3549.1.da.a.641.1	✓	24
169.140	even	78		3549.1.da.a.1661.1	yes	24
507.140	odd	78		3549.1.da.a.1661.1	yes	24
1183.816	even	78	inner	3549.1.ds.a.3182.1	yes	24
3549.3182	odd	78	inner	3549.1.ds.a.3182.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3549.1.da.a.641.1	✓	24	7.4	even	3
3549.1.da.a.641.1	✓	24	21.11	odd	6
3549.1.da.a.1661.1	yes	24	169.140	even	78
3549.1.da.a.1661.1	yes	24	507.140	odd	78
3549.1.ds.a.2669.1	yes	24	1.1	even	1	trivial
3549.1.ds.a.2669.1	yes	24	3.2	odd	2	CM
3549.1.ds.a.3182.1	yes	24	1183.816	even	78	inner
3549.1.ds.a.3182.1	yes	24	3549.3182	odd	78	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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3549.ds (of order \(78\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.120537 + 0.992709i) q^{3} +(0.845190 - 0.534466i) q^{4} +(0.948536 + 0.316668i) q^{7} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)	\(q+(-0.120537 + 0.992709i) q^{3} +(0.845190 - 0.534466i) q^{4} +(0.948536 + 0.316668i) q^{7} +(-0.970942 - 0.239316i) q^{9} +(0.428693 + 0.903450i) q^{12} +(0.500000 - 0.866025i) q^{13} +(0.428693 - 0.903450i) q^{16} -1.92104i q^{19} +(-0.428693 + 0.903450i) q^{21} +(0.632445 - 0.774605i) q^{25} +(0.354605 - 0.935016i) q^{27} +(0.970942 - 0.239316i) q^{28} +(-0.149094 - 0.917410i) q^{31} +(-0.948536 + 0.316668i) q^{36} +(0.101594 + 0.625134i) q^{37} +(0.799443 + 0.600742i) q^{39} +(-1.19979 + 1.46948i) q^{43} +(0.845190 + 0.534466i) q^{48} +(0.799443 + 0.600742i) q^{49} +(-0.0402659 - 0.999189i) q^{52} +(1.90703 + 0.231555i) q^{57} +(-1.26527 + 1.12093i) q^{61} +(-0.845190 - 0.534466i) q^{63} +(-0.120537 - 0.992709i) q^{64} +(0.869047 + 1.65583i) q^{67} +(-1.30314 + 1.25168i) q^{73} +(0.692724 + 0.721202i) q^{75} +(-1.02673 - 1.62364i) q^{76} +(0.00970705 - 0.240878i) q^{79} +(0.885456 + 0.464723i) q^{81} +(0.120537 + 0.992709i) q^{84} +(0.748511 - 0.663123i) q^{91} +(0.928693 - 0.0374250i) q^{93} +(-1.56482 - 0.742517i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} - q^{4} + q^{7} - 2 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 - q^4 + q^7 - 2 * q^9	\( 24 q + 2 q^{3} - q^{4} + q^{7} - 2 q^{9} + q^{12} + 12 q^{13} + q^{16} - q^{21} - q^{25} + 2 q^{27} + 2 q^{28} - q^{36} + 3 q^{37} + q^{39} - q^{43} - q^{48} + q^{49} + q^{52} - 2 q^{61} + q^{63} + 2 q^{64} - 3 q^{73} + q^{75} - 3 q^{76} - 2 q^{79} - 2 q^{81} - 2 q^{84} + 2 q^{91} + 13 q^{93} + 3 q^{97}+O(q^{100}) \) 24 * q + 2 * q^3 - q^4 + q^7 - 2 * q^9 + q^12 + 12 * q^13 + q^16 - q^21 - q^25 + 2 * q^27 + 2 * q^28 - q^36 + 3 * q^37 + q^39 - q^43 - q^48 + q^49 + q^52 - 2 * q^61 + q^63 + 2 * q^64 - 3 * q^73 + q^75 - 3 * q^76 - 2 * q^79 - 2 * q^81 - 2 * q^84 + 2 * q^91 + 13 * q^93 + 3 * q^97

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