LMFDB - Embedded newform 3549.1.dd.a.1283.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,1,Mod(263,3549)]

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, 32]))

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

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

chi := DirichletCharacter("3549.263");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3549.dd (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:	$D_{39}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{39} - \cdots)$

Embedding invariants

Embedding label			1283.1
Root			$-0.845190 - 0.534466i$ of defining polynomial
Character	$\chi$	$=$	3549.1283
Dual form			3549.1.dd.a.2993.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.748511 - 0.663123i) q^{3} +(0.278217 - 0.960518i) q^{4} +(0.987050 - 0.160411i) q^{7} +(0.120537 + 0.992709i) q^{9} +O(q^{10})$	\(q+(-0.748511 - 0.663123i) q^{3} +(0.278217 - 0.960518i) q^{4} +(0.987050 - 0.160411i) q^{7} +(0.120537 + 0.992709i) q^{9} +(-0.845190 + 0.534466i) q^{12} +(-0.500000 + 0.866025i) q^{13} +(-0.845190 - 0.534466i) q^{16} +1.59889 q^{19} +(-0.845190 - 0.534466i) q^{21} +(0.428693 - 0.903450i) q^{25} +(0.568065 - 0.822984i) q^{27} +(0.120537 - 0.992709i) q^{28} +(1.93559 + 0.156257i) q^{31} +(0.987050 + 0.160411i) q^{36} +(-1.96770 - 0.158849i) q^{37} +(0.948536 - 0.316668i) q^{39} +(0.846282 - 1.78350i) q^{43} +(0.278217 + 0.960518i) q^{48} +(0.948536 - 0.316668i) q^{49} +(0.692724 + 0.721202i) q^{52} +(-1.19678 - 1.06026i) q^{57} +(-0.197315 + 0.520276i) q^{61} +(0.278217 + 0.960518i) q^{63} +(-0.748511 + 0.663123i) q^{64} +(-1.10312 - 0.271894i) q^{67} +(1.55512 - 0.662573i) q^{73} +(-0.919979 + 0.391967i) q^{75} +(0.444838 - 1.53576i) q^{76} +(-1.03702 + 1.07966i) q^{79} +(-0.970942 + 0.239316i) q^{81} +(-0.748511 + 0.663123i) q^{84} +(-0.354605 + 0.935016i) q^{91} +(-1.34519 - 1.40049i) q^{93} +(0.845190 + 0.534466i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{19} + q^{21} + q^{25} - 2 q^{27} - 2 q^{28} + 2 q^{31} + q^{36} - q^{37} + q^{39} - q^{43} + q^{48} + q^{49} + q^{52} + 2 q^{57} + 2 q^{61} + q^{63} - 2 q^{64} - 4 q^{67} - q^{73} + q^{75} - q^{76} + 2 q^{79} - 2 q^{81} - 2 q^{84} - 2 q^{91} - 11 q^{93} - q^{97}+O(q^{100}) $ 24 * q - 2 * q^3 + q^4 + q^7 - 2 * q^9 + q^12 - 12 * q^13 + q^16 + 2 * q^19 + q^21 + q^25 - 2 * q^27 - 2 * q^28 + 2 * q^31 + q^36 - q^37 + q^39 - q^43 + q^48 + q^49 + q^52 + 2 * q^57 + 2 * q^61 + q^63 - 2 * q^64 - 4 * q^67 - q^73 + q^75 - q^76 + 2 * q^79 - 2 * q^81 - 2 * q^84 - 2 * q^91 - 11 * q^93 - q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1184$	$1522$	$3382$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{39}\right)$

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		0.799443	−	0.600742i	$-0.205128\pi$
$2$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$3$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$3$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$4$	0.278217	−	0.960518i	0.278217	−	0.960518i
$5$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$5$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$6$		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		0.120537	−	0.992709i	$-0.461538\pi$
$11$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$12$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$13$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$13$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$14$		0			0
$15$		0			0
$16$	−0.845190	−	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
$19$	1.59889			1.59889			0.799443	−	0.600742i	$-0.205128\pi$
$19$	1.59889			1.59889			0.799443	+	0.600742i	$0.205128\pi$
$20$		0			0
$21$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$22$		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
$25$	0.428693	−	0.903450i	0.428693	−	0.903450i
$26$		0			0
$27$	0.568065	−	0.822984i	0.568065	−	0.822984i
$28$	0.120537	−	0.992709i	0.120537	−	0.992709i
$29$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$29$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$30$		0			0
$31$	1.93559	+	0.156257i	1.93559	+	0.156257i	0.987050	−	0.160411i	$-0.0512821\pi$
$31$	1.93559	+	0.156257i	1.93559	+	0.156257i	0.948536	+	0.316668i	$0.102564\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.987050	+	0.160411i	0.987050	+	0.160411i
$37$	−1.96770	−	0.158849i	−1.96770	−	0.158849i	−0.970942	−	0.239316i	$-0.923077\pi$
$37$	−1.96770	−	0.158849i	−1.96770	−	0.158849i	−0.996757	+	0.0804666i	$0.974359\pi$
$38$		0			0
$39$	0.948536	−	0.316668i	0.948536	−	0.316668i
$40$		0			0
$41$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$41$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$42$		0			0
$43$	0.846282	−	1.78350i	0.846282	−	1.78350i	0.278217	−	0.960518i	$-0.410256\pi$
$43$	0.846282	−	1.78350i	0.846282	−	1.78350i	0.568065	−	0.822984i	$-0.307692\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$47$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$48$	0.278217	+	0.960518i	0.278217	+	0.960518i
$49$	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
$57$	−1.19678	−	1.06026i	−1.19678	−	1.06026i
$58$		0			0
$59$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$59$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$60$		0			0
$61$	−0.197315	+	0.520276i	−0.197315	+	0.520276i	−0.996757	−	0.0804666i	$-0.974359\pi$
$61$	−0.197315	+	0.520276i	−0.197315	+	0.520276i	0.799443	+	0.600742i	$0.205128\pi$
$62$		0			0
$63$	0.278217	+	0.960518i	0.278217	+	0.960518i
$64$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$65$		0			0
$66$		0			0
$67$	−1.10312	−	0.271894i	−1.10312	−	0.271894i	−0.354605	−	0.935016i	$-0.615385\pi$
$67$	−1.10312	−	0.271894i	−1.10312	−	0.271894i	−0.748511	+	0.663123i	$0.769231\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$71$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$72$		0			0
$73$	1.55512	−	0.662573i	1.55512	−	0.662573i	0.568065	−	0.822984i	$-0.307692\pi$
$73$	1.55512	−	0.662573i	1.55512	−	0.662573i	0.987050	+	0.160411i	$0.0512821\pi$
$74$		0			0
$75$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$76$	0.444838	−	1.53576i	0.444838	−	1.53576i
$77$		0			0
$78$		0			0
$79$	−1.03702	+	1.07966i	−1.03702	+	1.07966i	−0.0402659	+	0.999189i	$0.512821\pi$
$79$	−1.03702	+	1.07966i	−1.03702	+	1.07966i	−0.996757	+	0.0804666i	$0.974359\pi$
$80$		0			0
$81$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$82$		0			0
$83$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$83$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$84$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$92$		0			0
$93$	−1.34519	−	1.40049i	−1.34519	−	1.40049i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.845190	+	0.534466i	0.845190	+	0.534466i	0.885456	−	0.464723i	$-0.153846\pi$
$97$	0.845190	+	0.534466i	0.845190	+	0.534466i	−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
$109$	−0.0345234	+	0.0727566i	−0.0345234	+	0.0727566i	−0.919979	−	0.391967i	$-0.871795\pi$
$109$	−0.0345234	+	0.0727566i	−0.0345234	+	0.0727566i	0.885456	+	0.464723i	$0.153846\pi$
$110$		0			0
$111$	1.36751	+	1.42373i	1.36751	+	1.42373i
$112$	−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
$117$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$122$		0			0
$123$		0			0
$124$	0.688601	−	1.81569i	0.688601	−	1.81569i
$125$		0			0
$126$		0			0
$127$	−0.0557864	+	0.0580798i	−0.0557864	+	0.0580798i	−0.748511	−	0.663123i	$-0.769231\pi$
$127$	−0.0557864	+	0.0580798i	−0.0557864	+	0.0580798i	0.692724	+	0.721202i	$0.256410\pi$
$128$		0			0
$129$	−1.81613	+	0.773781i	−1.81613	+	0.773781i
$130$		0			0
$131$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$131$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$132$		0			0
$133$	1.57818	−	0.256479i	1.57818	−	0.256479i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$137$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$138$		0			0
$139$	−0.960245	−	0.607222i	−0.960245	−	0.607222i	−0.0402659	−	0.999189i	$-0.512821\pi$
$139$	−0.960245	−	0.607222i	−0.960245	−	0.607222i	−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
$147$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$148$	−0.700026	+	1.84582i	−0.700026	+	1.84582i
$149$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$149$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$150$		0			0
$151$	0.787025	+	0.819379i	0.787025	+	0.819379i	0.987050	−	0.160411i	$-0.0512821\pi$
$151$	0.787025	+	0.819379i	0.787025	+	0.819379i	−0.200026	+	0.979791i	$0.564103\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$157$	−0.0557864	−	1.38433i	−0.0557864	−	1.38433i	−0.748511	−	0.663123i	$-0.769231\pi$
$157$	−0.0557864	−	1.38433i	−0.0557864	−	1.38433i	0.692724	−	0.721202i	$-0.256410\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.487050	−	0.705614i	0.487050	−	0.705614i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	0.487050	−	0.705614i	0.487050	−	0.705614i	0.987050	+	0.160411i	$0.0512821\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$167$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	0.192724	+	1.58723i	0.192724	+	1.58723i
$172$	−1.47764	−	1.30907i	−1.47764	−	1.30907i
$173$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$173$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$174$		0			0
$175$	0.278217	−	0.960518i	0.278217	−	0.960518i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$179$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$180$		0			0
$181$	0.759177	+	0.398447i	0.759177	+	0.398447i	0.799443	−	0.600742i	$-0.205128\pi$
$181$	0.759177	+	0.398447i	0.759177	+	0.398447i	−0.0402659	+	0.999189i	$0.512821\pi$
$182$		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
$189$	0.428693	−	0.903450i	0.428693	−	0.903450i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	1.00000			1.00000
$193$	0.448536	−	1.18269i	0.448536	−	1.18269i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	0.448536	−	1.18269i	0.448536	−	1.18269i	0.948536	−	0.316668i	$-0.102564\pi$
$194$		0			0
$195$		0			0
$196$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$197$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$197$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$198$		0			0
$199$	−0.0643806	+	1.59759i	−0.0643806	+	1.59759i	0.568065	+	0.822984i	$0.307692\pi$
$199$	−0.0643806	+	1.59759i	−0.0643806	+	1.59759i	−0.632445	+	0.774605i	$0.717949\pi$
$200$		0			0
$201$	0.645395	+	0.935016i	0.645395	+	0.935016i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.885456	−	0.464723i	0.885456	−	0.464723i
$209$		0			0
$210$		0			0
$211$	0.593932	−	0.618348i	0.593932	−	0.618348i	−0.354605	−	0.935016i	$-0.615385\pi$
$211$	0.593932	−	0.618348i	0.593932	−	0.618348i	0.948536	+	0.316668i	$0.102564\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.93559	−	0.156257i	1.93559	−	0.156257i
$218$		0			0
$219$	−1.60339	−	0.535289i	−1.60339	−	0.535289i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.69038	+	1.06893i	−1.69038	+	1.06893i	−0.845190	+	0.534466i	$0.820513\pi$
$223$	−1.69038	+	1.06893i	−1.69038	+	1.06893i	−0.845190	+	0.534466i	$0.820513\pi$
$224$		0			0
$225$	0.948536	+	0.316668i	0.948536	+	0.316668i
$226$		0			0
$227$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$227$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$228$	−1.35136	+	0.854550i	−1.35136	+	0.854550i
$229$	0.0802707	−	0.00648012i	0.0802707	−	0.00648012i	−0.0402659	−	0.999189i	$-0.512821\pi$
$229$	0.0802707	−	0.00648012i	0.0802707	−	0.00648012i	0.120537	+	0.992709i	$0.461538\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$233$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.49217	−	0.120460i	1.49217	−	0.120460i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	0.0781918	−	1.94031i	0.0781918	−	1.94031i	−0.200026	−	0.979791i	$-0.564103\pi$
$241$	0.0781918	−	1.94031i	0.0781918	−	1.94031i	0.278217	−	0.960518i	$-0.410256\pi$
$242$		0			0
$243$	0.885456	+	0.464723i	0.885456	+	0.464723i
$244$	0.444838	+	0.334274i	0.444838	+	0.334274i
$245$		0			0
$246$		0			0
$247$	−0.799443	+	1.38468i	−0.799443	+	1.38468i
$248$		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	$0.0256410\pi$
$252$	1.00000			1.00000
$253$		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
$259$	−1.96770	+	0.158849i	−1.96770	+	0.158849i
$260$		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
$268$	−0.568065	+	0.983917i	−0.568065	+	0.983917i
$269$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$269$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$270$		0			0
$271$	0.385456	+	1.33075i	0.385456	+	1.33075i	0.885456	+	0.464723i	$0.153846\pi$
$271$	0.385456	+	1.33075i	0.385456	+	1.33075i	−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$	0.885456	−	0.464723i	0.885456	−	0.464723i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.253011	+	0.309882i	0.253011	+	0.309882i	0.885456	−	0.464723i	$-0.153846\pi$
$277$	0.253011	+	0.309882i	0.253011	+	0.309882i	−0.632445	+	0.774605i	$0.717949\pi$
$278$		0			0
$279$	0.0781918	+	1.94031i	0.0781918	+	1.94031i
$280$		0			0
$281$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$281$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$282$		0			0
$283$	−0.960245	+	1.39116i	−0.960245	+	1.39116i	−0.0402659	+	0.999189i	$0.512821\pi$
$283$	−0.960245	+	1.39116i	−0.960245	+	1.39116i	−0.919979	+	0.391967i	$0.871795\pi$
$284$		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
$291$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$292$	−0.203753	−	1.67806i	−0.203753	−	1.67806i
$293$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$293$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.120537	+	0.992709i	0.120537	+	0.992709i
$301$	0.549229	−	1.89616i	0.549229	−	1.89616i
$302$		0			0
$303$		0			0
$304$	−1.35136	−	0.854550i	−1.35136	−	0.854550i
$305$		0			0
$306$		0			0
$307$	−0.402877	+	0.583668i	−0.402877	+	0.583668i	−0.970942	−	0.239316i	$-0.923077\pi$
$307$	−0.402877	+	0.583668i	−0.402877	+	0.583668i	0.568065	+	0.822984i	$0.307692\pi$
$308$		0			0
$309$	1.55512	+	0.662573i	1.55512	+	0.662573i
$310$		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
$313$	0.398754	−	0.0321908i	0.398754	−	0.0321908i	0.120537	−	0.992709i	$-0.461538\pi$
$313$	0.398754	−	0.0321908i	0.398754	−	0.0321908i	0.278217	+	0.960518i	$0.410256\pi$
$314$		0			0
$315$		0			0
$316$	0.748511	+	1.29646i	0.748511	+	1.29646i
$317$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$317$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$325$	0.568065	+	0.822984i	0.568065	+	0.822984i
$326$		0			0
$327$	0.0740877	−	0.0315658i	0.0740877	−	0.0315658i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.706910	+	1.86397i	0.706910	+	1.86397i	0.428693	+	0.903450i	$0.358974\pi$
$331$	0.706910	+	1.86397i	0.706910	+	1.86397i	0.278217	+	0.960518i	$0.410256\pi$
$332$		0			0
$333$	−0.0794890	−	1.97250i	−0.0794890	−	1.97250i
$334$		0			0
$335$		0			0
$336$	0.428693	+	0.903450i	0.428693	+	0.903450i
$337$	−1.26489			−1.26489			−0.632445	−	0.774605i	$-0.717949\pi$
$337$	−1.26489			−1.26489			−0.632445	+	0.774605i	$0.717949\pi$
$338$		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
$349$	−0.799443	−	0.600742i	−0.799443	−	0.600742i	0.120537	−	0.992709i	$-0.461538\pi$
$349$	−0.799443	−	0.600742i	−0.799443	−	0.600742i	−0.919979	+	0.391967i	$0.871795\pi$
$350$		0			0
$351$	0.428693	+	0.903450i	0.428693	+	0.903450i
$352$		0			0
$353$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$353$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$359$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$360$		0			0
$361$	1.55643			1.55643
$362$		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
$367$	0.228667	+	1.88324i	0.228667	+	1.88324i	0.428693	+	0.903450i	$0.358974\pi$
$367$	0.228667	+	1.88324i	0.228667	+	1.88324i	−0.200026	+	0.979791i	$0.564103\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.71945	+	0.902438i	−1.71945	+	0.902438i
$373$	−0.234068	+	1.92773i	−0.234068	+	1.92773i	0.120537	+	0.992709i	$0.461538\pi$
$373$	−0.234068	+	1.92773i	−0.234068	+	1.92773i	−0.354605	+	0.935016i	$0.615385\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.228667	−	0.0763402i	0.228667	−	0.0763402i	−0.200026	−	0.979791i	$-0.564103\pi$
$379$	0.228667	−	0.0763402i	0.228667	−	0.0763402i	0.428693	+	0.903450i	$0.358974\pi$
$380$		0			0
$381$	0.0802707	−	0.00648012i	0.0802707	−	0.00648012i
$382$		0			0
$383$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$383$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.87251	+	0.625134i	1.87251	+	0.625134i
$388$	0.748511	−	0.663123i	0.748511	−	0.663123i
$389$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$389$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.492699	+	0.258588i	0.492699	+	0.258588i	0.692724	−	0.721202i	$-0.256410\pi$
$397$	0.492699	+	0.258588i	0.492699	+	0.258588i	−0.200026	+	0.979791i	$0.564103\pi$
$398$		0			0
$399$	−1.35136	−	0.854550i	−1.35136	−	0.854550i
$400$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$401$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$401$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$402$		0			0
$403$	−1.10312	+	1.59814i	−1.10312	+	1.59814i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.74527	+	0.582656i	−1.74527	+	0.582656i	−0.996757	−	0.0804666i	$-0.974359\pi$
$409$	−1.74527	+	0.582656i	−1.74527	+	0.582656i	−0.748511	+	0.663123i	$0.769231\pi$
$410$		0			0
$411$		0			0
$412$	0.0680647	+	1.68901i	0.0680647	+	1.68901i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.316091	+	1.09127i	0.316091	+	1.09127i
$418$		0			0
$419$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$419$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$420$		0			0
$421$	−1.32555	+	1.17433i	−1.32555	+	1.17433i	−0.354605	+	0.935016i	$0.615385\pi$
$421$	−1.32555	+	1.17433i	−1.32555	+	1.17433i	−0.970942	+	0.239316i	$0.923077\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.111301	+	0.545190i	−0.111301	+	0.545190i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$431$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$432$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$433$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i	−0.996757	−	0.0804666i	$-0.974359\pi$
$433$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i	0.987050	−	0.160411i	$-0.0512821\pi$
$434$		0			0
$435$		0			0
$436$	0.0602790	+	0.0534025i	0.0602790	+	0.0534025i
$437$		0			0
$438$		0			0
$439$	0.685430	−	1.44451i	0.685430	−	1.44451i	−0.200026	−	0.979791i	$-0.564103\pi$
$439$	0.685430	−	1.44451i	0.685430	−	1.44451i	0.885456	−	0.464723i	$-0.153846\pi$
$440$		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	$0.435897\pi$
$444$	1.74798	−	0.917410i	1.74798	−	0.917410i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$449$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$449$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.0457473	−	1.13521i	−0.0457473	−	1.13521i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.67977	+	0.560791i	1.67977	+	0.560791i	0.987050	−	0.160411i	$-0.0512821\pi$
$457$	1.67977	+	0.560791i	1.67977	+	0.560791i	0.692724	+	0.721202i	$0.256410\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$461$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$462$		0			0
$463$	−0.221783	−	1.82654i	−0.221783	−	1.82654i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−0.221783	−	1.82654i	−0.221783	−	1.82654i	0.278217	−	0.960518i	$-0.410256\pi$
$464$		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
$469$	−1.13245	−	0.0914204i	−1.13245	−	0.0914204i
$470$		0			0
$471$	−0.876221	+	1.07318i	−0.876221	+	1.07318i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.685430	−	1.44451i	0.685430	−	1.44451i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$479$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$480$		0			0
$481$	1.12142	−	1.62465i	1.12142	−	1.62465i
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	0.959734	−	0.999189i	0.959734	−	0.999189i	−0.0402659	−	0.999189i	$-0.512821\pi$
$487$	0.959734	−	0.999189i	0.959734	−	0.999189i	1.00000			$0$
$488$		0			0
$489$	−0.832471	+	0.205186i	−0.832471	+	0.205186i
$490$		0			0
$491$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$491$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.55242	−	1.16657i	−1.55242	−	1.16657i
$497$		0			0
$498$		0			0
$499$	−1.66849	−	0.271156i	−1.66849	−	0.271156i	−0.748511	−	0.663123i	$-0.769231\pi$
$499$	−1.66849	−	0.271156i	−1.66849	−	0.271156i	−0.919979	+	0.391967i	$0.871795\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$503$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$508$	0.0402659	+	0.0697427i	0.0402659	+	0.0697427i
$509$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$509$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$510$		0			0
$511$	1.42869	−	0.903450i	1.42869	−	0.903450i
$512$		0			0
$513$	0.908271	−	1.31586i	0.908271	−	1.31586i
$514$		0			0
$515$		0			0
$516$	0.237952	+	1.95971i	0.237952	+	1.95971i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$521$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$522$		0			0
$523$	1.55512	+	0.983395i	1.55512	+	0.983395i	0.987050	+	0.160411i	$0.0512821\pi$
$523$	1.55512	+	0.983395i	1.55512	+	0.983395i	0.568065	+	0.822984i	$0.307692\pi$
$524$		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
$532$	0.192724	−	1.58723i	0.192724	−	1.58723i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.0802707	−	1.99190i	0.0802707	−	1.99190i	−0.0402659	−	0.999189i	$-0.512821\pi$
$541$	0.0802707	−	1.99190i	0.0802707	−	1.99190i	0.120537	−	0.992709i	$-0.461538\pi$
$542$		0			0
$543$	−0.304033	−	0.801669i	−0.304033	−	0.801669i
$544$		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
$549$	−0.540266	−	0.133164i	−0.540266	−	0.133164i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.850405	+	1.23202i	−0.850405	+	1.23202i
$554$		0			0
$555$		0			0
$556$	−0.850405	+	0.753393i	−0.850405	+	0.753393i
$557$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$557$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$558$		0			0
$559$	1.12142	+	1.62465i	1.12142	+	1.62465i
$560$		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		−0.0402659	−	0.999189i	$-0.512821\pi$
$569$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$570$		0			0
$571$	1.79944	+	0.600742i	1.79944	+	0.600742i	1.00000			$0$
$571$	1.79944	+	0.600742i	1.79944	+	0.600742i	0.799443	+	0.600742i	$0.205128\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$577$	−0.428693	−	0.742517i	−0.428693	−	0.742517i	0.568065	−	0.822984i	$-0.307692\pi$
$577$	−0.428693	−	0.742517i	−0.428693	−	0.742517i	−0.996757	+	0.0804666i	$0.974359\pi$
$578$		0			0
$579$	−1.12001	+	0.587824i	−1.12001	+	0.587824i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$589$	3.09478	+	0.249837i	3.09478	+	0.249837i
$590$		0			0
$591$		0			0
$592$	1.57818	+	1.18593i	1.57818	+	1.18593i
$593$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$593$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.10759	−	1.15312i	1.10759	−	1.15312i
$598$		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
$601$	1.57818	+	1.18593i	1.57818	+	1.18593i	0.885456	+	0.464723i	$0.153846\pi$
$601$	1.57818	+	1.18593i	1.57818	+	1.18593i	0.692724	+	0.721202i	$0.256410\pi$
$602$		0			0
$603$	0.136945	−	1.12785i	0.136945	−	1.12785i
$604$	1.00599	−	0.527986i	1.00599	−	0.527986i
$605$		0			0
$606$		0			0
$607$	0.946784	+	0.838778i	0.946784	+	0.838778i	0.987050	−	0.160411i	$-0.0512821\pi$
$607$	0.946784	+	0.838778i	0.946784	+	0.838778i	−0.0402659	+	0.999189i	$0.512821\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.12001	+	0.587824i	−1.12001	+	0.587824i	−0.919979	−	0.391967i	$-0.871795\pi$
$613$	−1.12001	+	0.587824i	−1.12001	+	0.587824i	−0.200026	+	0.979791i	$0.564103\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$617$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$618$		0			0
$619$	−0.394871	+	1.93421i	−0.394871	+	1.93421i	−0.0402659	+	0.999189i	$0.512821\pi$
$619$	−0.394871	+	1.93421i	−0.394871	+	1.93421i	−0.354605	+	0.935016i	$0.615385\pi$
$620$		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
$628$	−1.34519	−	0.331560i	−1.34519	−	0.331560i
$629$		0			0
$630$		0			0
$631$	1.94854	−	0.316668i	1.94854	−	0.316668i	0.948536	−	0.316668i	$-0.102564\pi$
$631$	1.94854	−	0.316668i	1.94854	−	0.316668i	1.00000			$0$
$632$		0			0
$633$	−0.854605	+	0.0689908i	−0.854605	+	0.0689908i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$641$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$642$		0			0
$643$	−1.19979	−	0.400550i	−1.19979	−	0.400550i	−0.354605	−	0.935016i	$-0.615385\pi$
$643$	−1.19979	−	0.400550i	−1.19979	−	0.400550i	−0.845190	+	0.534466i	$0.820513\pi$
$644$		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
$651$	−1.55242	−	1.16657i	−1.55242	−	1.16657i
$652$	−0.542249	−	0.664135i	−0.542249	−	0.664135i
$653$		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
$657$	0.845190	+	1.46391i	0.845190	+	1.46391i
$658$		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
$661$	−0.0854858	−	0.225408i	−0.0854858	−	0.225408i	0.885456	−	0.464723i	$-0.153846\pi$
$661$	−0.0854858	−	0.225408i	−0.0854858	−	0.225408i	−0.970942	+	0.239316i	$0.923077\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.97410	+	0.320823i	1.97410	+	0.320823i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.74527	+	0.582656i	−1.74527	+	0.582656i	−0.996757	−	0.0804666i	$-0.974359\pi$
$673$	−1.74527	+	0.582656i	−1.74527	+	0.582656i	−0.748511	+	0.663123i	$0.769231\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	0.919979	+	0.391967i	0.919979	+	0.391967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$683$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$684$	1.57818	+	0.256479i	1.57818	+	0.256479i
$685$		0			0
$686$		0			0
$687$	−0.0643806	−	0.0483789i	−0.0643806	−	0.0483789i
$688$	−1.66849	+	1.05509i	−1.66849	+	1.05509i
$689$		0			0
$690$		0			0
$691$	−0.351915	+	0.431017i	−0.351915	+	0.431017i	−0.919979	−	0.391967i	$-0.871795\pi$
$691$	−0.351915	+	0.431017i	−0.351915	+	0.431017i	0.568065	+	0.822984i	$0.307692\pi$
$692$		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		−0.568065	−	0.822984i	$-0.692308\pi$
$701$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$702$		0			0
$703$	−3.14613	−	0.253982i	−3.14613	−	0.253982i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.641762	+	0.568552i	−0.641762	+	0.568552i	−0.919979	−	0.391967i	$-0.871795\pi$
$709$	−0.641762	+	0.568552i	−0.641762	+	0.568552i	0.278217	+	0.960518i	$0.410256\pi$
$710$		0			0
$711$	−1.19678	−	0.899324i	−1.19678	−	0.899324i
$712$		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
$723$	−1.34519	+	1.40049i	−1.34519	+	1.40049i
$724$	0.593932	−	0.618348i	0.593932	−	0.618348i
$725$		0			0
$726$		0			0
$727$	0.251489	−	0.663123i	0.251489	−	0.663123i	−0.748511	−	0.663123i	$-0.769231\pi$
$727$	0.251489	−	0.663123i	0.251489	−	0.663123i	1.00000			$0$
$728$		0			0
$729$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$730$		0			0
$731$		0			0
$732$	−0.111301	−	0.545190i	−0.111301	−	0.545190i
$733$	−0.672711	+	0.224584i	−0.672711	+	0.224584i	−0.632445	−	0.774605i	$-0.717949\pi$
$733$	−0.672711	+	0.224584i	−0.672711	+	0.224584i	−0.0402659	+	0.999189i	$0.512821\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.492699	−	0.258588i	0.492699	−	0.258588i	−0.200026	−	0.979791i	$-0.564103\pi$
$739$	0.492699	−	0.258588i	0.492699	−	0.258588i	0.692724	+	0.721202i	$0.256410\pi$
$740$		0			0
$741$	1.51660	−	0.506316i	1.51660	−	0.506316i
$742$		0			0
$743$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$743$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.0740877	−	0.0315658i	0.0740877	−	0.0315658i	−0.354605	−	0.935016i	$-0.615385\pi$
$751$	0.0740877	−	0.0315658i	0.0740877	−	0.0315658i	0.428693	+	0.903450i	$0.358974\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$757$	−0.197315	+	0.681209i	−0.197315	+	0.681209i	0.799443	+	0.600742i	$0.205128\pi$
$757$	−0.197315	+	0.681209i	−0.197315	+	0.681209i	−0.996757	+	0.0804666i	$0.974359\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$761$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$762$		0			0
$763$	−0.0224054	+	0.0773523i	−0.0224054	+	0.0773523i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.278217	−	0.960518i	0.278217	−	0.960518i
$769$	−0.554631	+	1.91481i	−0.554631	+	1.91481i	−0.200026	+	0.979791i	$0.564103\pi$
$769$	−0.554631	+	1.91481i	−0.554631	+	1.91481i	−0.354605	+	0.935016i	$0.615385\pi$
$770$		0			0
$771$		0			0
$772$	−1.01121	−	0.759873i	−1.01121	−	0.759873i
$773$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$773$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$774$		0			0
$775$	0.970942	−	1.68172i	0.970942	−	1.68172i
$776$		0			0
$777$	1.57818	+	1.18593i	1.57818	+	1.18593i
$778$		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
$787$	−0.470293	+	1.62364i	−0.470293	+	1.62364i	0.278217	+	0.960518i	$0.410256\pi$
$787$	−0.470293	+	1.62364i	−0.470293	+	1.62364i	−0.748511	+	0.663123i	$0.769231\pi$
$788$		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
$796$	1.51660	+	0.506316i	1.51660	+	0.506316i
$797$		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
$804$	1.07766	−	0.359776i	1.07766	−	0.359776i
$805$		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
$811$	1.41574	−	0.743039i	1.41574	−	0.743039i	0.428693	−	0.903450i	$-0.358974\pi$
$811$	1.41574	−	0.743039i	1.41574	−	0.743039i	0.987050	+	0.160411i	$0.0512821\pi$
$812$		0			0
$813$	0.593932	−	1.25168i	0.593932	−	1.25168i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.35311	−	2.85162i	1.35311	−	2.85162i
$818$		0			0
$819$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$820$		0			0
$821$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$821$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$822$		0			0
$823$	−0.120537	+	0.208776i	−0.120537	+	0.208776i	−0.919979	−	0.391967i	$-0.871795\pi$
$823$	−0.120537	+	0.208776i	−0.120537	+	0.208776i	0.799443	+	0.600742i	$0.205128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$827$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$828$		0			0
$829$	−0.0713074	+	0.0374250i	−0.0713074	+	0.0374250i	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−0.0713074	+	0.0374250i	−0.0713074	+	0.0374250i	0.428693	+	0.903450i	$0.358974\pi$
$830$		0			0
$831$	0.0161084	−	0.399727i	0.0161084	−	0.399727i
$832$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.22814	−	1.50419i	1.22814	−	1.50419i
$838$		0			0
$839$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$839$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$840$		0			0
$841$	0.692724	+	0.721202i	0.692724	+	0.721202i
$842$		0			0
$843$		0			0
$844$	−0.428693	−	0.742517i	−0.428693	−	0.742517i
$845$		0			0
$846$		0			0
$847$	−0.996757	−	0.0804666i	−0.996757	−	0.0804666i
$848$		0			0
$849$	1.64126	−	0.404534i	1.64126	−	0.404534i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.627974	−	1.65583i	−0.627974	−	1.65583i	−0.748511	−	0.663123i	$-0.769231\pi$
$853$	−0.627974	−	1.65583i	−0.627974	−	1.65583i	0.120537	−	0.992709i	$-0.461538\pi$
$854$		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
$859$	−0.0763874	+	1.89553i	−0.0763874	+	1.89553i	0.278217	+	0.960518i	$0.410256\pi$
$859$	−0.0763874	+	1.89553i	−0.0763874	+	1.89553i	−0.354605	+	0.935016i	$0.615385\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$863$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$868$	0.388427	−	1.90264i	0.388427	−	1.90264i
$869$		0			0
$870$		0			0
$871$	0.787025	−	0.819379i	0.787025	−	0.819379i
$872$		0			0
$873$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$874$		0			0
$875$		0			0
$876$	−0.960245	+	1.39116i	−0.960245	+	1.39116i
$877$	−0.402877	−	0.583668i	−0.402877	−	0.583668i	0.568065	−	0.822984i	$-0.307692\pi$
$877$	−0.402877	−	0.583668i	−0.402877	−	0.583668i	−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
$883$	0.237952	+	1.95971i	0.237952	+	1.95971i	0.278217	+	0.960518i	$0.410256\pi$
$883$	0.237952	+	1.95971i	0.237952	+	1.95971i	−0.0402659	+	0.999189i	$0.512821\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$887$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$888$		0			0
$889$	−0.0457473	+	0.0662764i	−0.0457473	+	0.0662764i
$890$		0			0
$891$		0			0
$892$	0.556435	+	1.92104i	0.556435	+	1.92104i
$893$		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
$903$	−1.66849	+	1.05509i	−1.66849	+	1.05509i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.76517	−	0.926432i	−1.76517	−	0.926432i	−0.919979	−	0.391967i	$-0.871795\pi$
$907$	−1.76517	−	0.926432i	−1.76517	−	0.926432i	−0.845190	−	0.534466i	$-0.820513\pi$
$908$		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	$0.230769\pi$
$912$	0.444838	+	1.53576i	0.444838	+	1.53576i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0161084	−	0.0789044i	0.0161084	−	0.0789044i
$917$		0			0
$918$		0			0
$919$	−1.76517	−	0.926432i	−1.76517	−	0.926432i	−0.919979	−	0.391967i	$-0.871795\pi$
$919$	−1.76517	−	0.926432i	−1.76517	−	0.926432i	−0.845190	−	0.534466i	$-0.820513\pi$
$920$		0			0
$921$	0.688601	−	0.169725i	0.688601	−	0.169725i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.987050	+	1.70962i	−0.987050	+	1.70962i
$926$		0			0
$927$	−0.724653	−	1.52717i	−0.724653	−	1.52717i
$928$		0			0
$929$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$929$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$930$		0			0
$931$	1.51660	−	0.506316i	1.51660	−	0.506316i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.78649	−	0.440331i	1.78649	−	0.440331i	0.799443	−	0.600742i	$-0.205128\pi$
$937$	1.78649	−	0.440331i	1.78649	−	0.440331i	0.987050	+	0.160411i	$0.0512821\pi$
$938$		0			0
$939$	−0.319818	−	0.240328i	−0.319818	−	0.240328i
$940$		0			0
$941$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$941$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$947$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$948$	0.299443	−	1.46677i	0.299443	−	1.46677i
$949$	−0.203753	+	1.67806i	−0.203753	+	1.67806i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$953$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.73503	+	0.444486i	2.73503	+	0.444486i
$962$		0			0
$963$		0			0
$964$	−1.84195	−	0.614932i	−1.84195	−	0.614932i
$965$		0			0
$966$		0			0
$967$	−0.540266	−	0.133164i	−0.540266	−	0.133164i	−0.0402659	−	0.999189i	$-0.512821\pi$
$967$	−0.540266	−	0.133164i	−0.540266	−	0.133164i	−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$971$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$972$	0.692724	−	0.721202i	0.692724	−	0.721202i
$973$	−1.04522	−	0.445325i	−1.04522	−	0.445325i
$974$		0			0
$975$	0.120537	−	0.992709i	0.120537	−	0.992709i
$976$	0.444838	−	0.334274i	0.444838	−	0.334274i
$977$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$977$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.0763874	−	0.0255019i	−0.0763874	−	0.0255019i
$982$		0			0
$983$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$983$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.10759	+	1.15312i	1.10759	+	1.15312i
$989$		0			0
$990$		0			0
$991$	1.59889			1.59889			0.799443	−	0.600742i	$-0.205128\pi$
$991$	1.59889			1.59889			0.799443	+	0.600742i	$0.205128\pi$
$992$		0			0
$993$	0.706910	−	1.86397i	0.706910	−	1.86397i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.96770	−	0.319782i	−1.96770	−	0.319782i	−0.996757	−	0.0804666i	$-0.974359\pi$
$997$	−1.96770	−	0.319782i	−1.96770	−	0.319782i	−0.970942	−	0.239316i	$-0.923077\pi$
$998$		0			0
$999$	−1.24851	+	1.52915i	−1.24851	+	1.52915i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.1.dd.a.1283.1	✓	24
3.2	odd	2	CM	3549.1.dd.a.1283.1	✓	24
7.4	even	3		3549.1.dx.a.2804.1	yes	24
21.11	odd	6		3549.1.dx.a.2804.1	yes	24
169.120	even	39		3549.1.dx.a.1472.1	yes	24
507.458	odd	78		3549.1.dx.a.1472.1	yes	24
1183.627	even	39	inner	3549.1.dd.a.2993.1	yes	24
3549.2993	odd	78	inner	3549.1.dd.a.2993.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3549.1.dd.a.1283.1	✓	24	1.1	even	1	trivial
3549.1.dd.a.1283.1	✓	24	3.2	odd	2	CM
3549.1.dd.a.2993.1	yes	24	1183.627	even	39	inner
3549.1.dd.a.2993.1	yes	24	3549.2993	odd	78	inner
3549.1.dx.a.1472.1	yes	24	169.120	even	39
3549.1.dx.a.1472.1	yes	24	507.458	odd	78
3549.1.dx.a.2804.1	yes	24	7.4	even	3
3549.1.dx.a.2804.1	yes	24	21.11	odd	6

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.dd (of order \(78\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.748511 - 0.663123i) q^{3} +(0.278217 - 0.960518i) q^{4} +(0.987050 - 0.160411i) q^{7} +(0.120537 + 0.992709i) q^{9} +O(q^{10})\)	\(q+(-0.748511 - 0.663123i) q^{3} +(0.278217 - 0.960518i) q^{4} +(0.987050 - 0.160411i) q^{7} +(0.120537 + 0.992709i) q^{9} +(-0.845190 + 0.534466i) q^{12} +(-0.500000 + 0.866025i) q^{13} +(-0.845190 - 0.534466i) q^{16} +1.59889 q^{19} +(-0.845190 - 0.534466i) q^{21} +(0.428693 - 0.903450i) q^{25} +(0.568065 - 0.822984i) q^{27} +(0.120537 - 0.992709i) q^{28} +(1.93559 + 0.156257i) q^{31} +(0.987050 + 0.160411i) q^{36} +(-1.96770 - 0.158849i) q^{37} +(0.948536 - 0.316668i) q^{39} +(0.846282 - 1.78350i) q^{43} +(0.278217 + 0.960518i) q^{48} +(0.948536 - 0.316668i) q^{49} +(0.692724 + 0.721202i) q^{52} +(-1.19678 - 1.06026i) q^{57} +(-0.197315 + 0.520276i) q^{61} +(0.278217 + 0.960518i) q^{63} +(-0.748511 + 0.663123i) q^{64} +(-1.10312 - 0.271894i) q^{67} +(1.55512 - 0.662573i) q^{73} +(-0.919979 + 0.391967i) q^{75} +(0.444838 - 1.53576i) q^{76} +(-1.03702 + 1.07966i) q^{79} +(-0.970942 + 0.239316i) q^{81} +(-0.748511 + 0.663123i) q^{84} +(-0.354605 + 0.935016i) q^{91} +(-1.34519 - 1.40049i) q^{93} +(0.845190 + 0.534466i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{3} + q^{4} + q^{7} - 2 q^{9}+O(q^{10}) \) 24 * q - 2 * q^3 + q^4 + q^7 - 2 * q^9	\( 24 q - 2 q^{3} + q^{4} + q^{7} - 2 q^{9} + q^{12} - 12 q^{13} + q^{16} + 2 q^{19} + q^{21} + q^{25} - 2 q^{27} - 2 q^{28} + 2 q^{31} + q^{36} - q^{37} + q^{39} - q^{43} + q^{48} + q^{49} + q^{52} + 2 q^{57} + 2 q^{61} + q^{63} - 2 q^{64} - 4 q^{67} - q^{73} + q^{75} - q^{76} + 2 q^{79} - 2 q^{81} - 2 q^{84} - 2 q^{91} - 11 q^{93} - q^{97}+O(q^{100}) \) 24 * q - 2 * q^3 + q^4 + q^7 - 2 * q^9 + q^12 - 12 * q^13 + q^16 + 2 * q^19 + q^21 + q^25 - 2 * q^27 - 2 * q^28 + 2 * q^31 + q^36 - q^37 + q^39 - q^43 + q^48 + q^49 + q^52 + 2 * q^57 + 2 * q^61 + q^63 - 2 * q^64 - 4 * q^67 - q^73 + q^75 - q^76 + 2 * q^79 - 2 * q^81 - 2 * q^84 - 2 * q^91 - 11 * q^93 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more