LMFDB - Embedded newform 2704.1.cm.a.1759.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,1,Mod(95,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(78))

chi = DirichletCharacter(H, H._module([39, 0, 37]))

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

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

chi := DirichletCharacter("2704.95");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2704.cm (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:	$1.34947179416$
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:	$\Z[a_1, \ldots, a_{9}]$
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			1759.1
Root			$0.799443 + 0.600742i$ of defining polynomial
Character	$\chi$	$=$	2704.1759
Dual form			2704.1.cm.a.2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.558358 + 1.06386i) q^{5} +(0.799443 - 0.600742i) q^{9} +(0.632445 + 0.774605i) q^{13} +(0.876221 - 1.07318i) q^{17} +(-0.251974 - 0.365047i) q^{25} +(-0.368039 - 0.156807i) q^{29} +(-0.965727 + 0.458243i) q^{37} +(0.248247 + 0.743589i) q^{41} +(0.192732 + 1.18593i) q^{45} +(0.200026 + 0.979791i) q^{49} +(0.448536 + 1.18269i) q^{53} +(1.87251 - 0.304312i) q^{61} +(-1.17720 + 0.240328i) q^{65} +(1.71942 + 0.208776i) q^{73} +(0.278217 - 0.960518i) q^{81} +(0.652466 + 1.53139i) q^{85} +(-0.804924 - 0.464723i) q^{89} +(-0.879714 + 1.39116i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{9} - q^{13} + q^{17} + 4 q^{25} + q^{29} + 3 q^{37} - 3 q^{41} - 3 q^{45} - q^{49} - 11 q^{53} - q^{61} - 3 q^{65} + q^{81} + 10 q^{85}+O(q^{100}) $ 24 * q + q^9 - q^13 + q^17 + 4 * q^25 + q^29 + 3 * q^37 - 3 * q^41 - 3 * q^45 - q^49 - 11 * q^53 - q^61 - 3 * q^65 + q^81 + 10 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$e\left(\frac{49}{78}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$3$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$4$		0			0
$5$	−0.558358	+	1.06386i	−0.558358	+	1.06386i	0.428693	+	0.903450i	$0.358974\pi$
$5$	−0.558358	+	1.06386i	−0.558358	+	1.06386i	−0.987050	+	0.160411i	$0.948718\pi$
$6$		0			0
$7$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$7$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$8$		0			0
$9$	0.799443	−	0.600742i	0.799443	−	0.600742i
$10$		0			0
$11$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$11$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$12$		0			0
$13$	0.632445	+	0.774605i	0.632445	+	0.774605i
$13$	0.632445	+	0.774605i	0.632445	+	0.774605i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.876221	−	1.07318i	0.876221	−	1.07318i	−0.120537	−	0.992709i	$-0.538462\pi$
$17$	0.876221	−	1.07318i	0.876221	−	1.07318i	0.996757	−	0.0804666i	$-0.0256410\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.251974	−	0.365047i	−0.251974	−	0.365047i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.368039	−	0.156807i	−0.368039	−	0.156807i	0.200026	−	0.979791i	$-0.435897\pi$
$29$	−0.368039	−	0.156807i	−0.368039	−	0.156807i	−0.568065	+	0.822984i	$0.692308\pi$
$30$		0			0
$31$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$31$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.965727	+	0.458243i	−0.965727	+	0.458243i	−0.845190	−	0.534466i	$-0.820513\pi$
$37$	−0.965727	+	0.458243i	−0.965727	+	0.458243i	−0.120537	+	0.992709i	$0.538462\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.248247	+	0.743589i	0.248247	+	0.743589i	0.996757	+	0.0804666i	$0.0256410\pi$
$41$	0.248247	+	0.743589i	0.248247	+	0.743589i	−0.748511	+	0.663123i	$0.769231\pi$
$42$		0			0
$43$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$43$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$44$		0			0
$45$	0.192732	+	1.18593i	0.192732	+	1.18593i
$46$		0			0
$47$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$47$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$48$		0			0
$49$	0.200026	+	0.979791i	0.200026	+	0.979791i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.448536	+	1.18269i	0.448536	+	1.18269i	0.948536	+	0.316668i	$0.102564\pi$
$53$	0.448536	+	1.18269i	0.448536	+	1.18269i	−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$59$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$60$		0			0
$61$	1.87251	−	0.304312i	1.87251	−	0.304312i	0.885456	−	0.464723i	$-0.153846\pi$
$61$	1.87251	−	0.304312i	1.87251	−	0.304312i	0.987050	+	0.160411i	$0.0512821\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.17720	+	0.240328i	−1.17720	+	0.240328i
$66$		0			0
$67$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$67$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$71$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$72$		0			0
$73$	1.71942	+	0.208776i	1.71942	+	0.208776i	0.919979	−	0.391967i	$-0.128205\pi$
$73$	1.71942	+	0.208776i	1.71942	+	0.208776i	0.799443	+	0.600742i	$0.205128\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$79$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$80$		0			0
$81$	0.278217	−	0.960518i	0.278217	−	0.960518i
$82$		0			0
$83$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$83$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$84$		0			0
$85$	0.652466	+	1.53139i	0.652466	+	1.53139i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.804924	−	0.464723i	−0.804924	−	0.464723i	0.0402659	−	0.999189i	$-0.487179\pi$
$89$	−0.804924	−	0.464723i	−0.804924	−	0.464723i	−0.845190	+	0.534466i	$0.820513\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.879714	+	1.39116i	−0.879714	+	1.39116i	0.0402659	+	0.999189i	$0.487179\pi$
$97$	−0.879714	+	1.39116i	−0.879714	+	1.39116i	−0.919979	+	0.391967i	$0.871795\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.854605	+	0.0689908i	−0.854605	+	0.0689908i	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−0.854605	+	0.0689908i	−0.854605	+	0.0689908i	−0.354605	+	0.935016i	$0.615385\pi$
$102$		0			0
$103$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$103$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$107$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$108$		0			0
$109$	0.393906	−	0.271894i	0.393906	−	0.271894i	−0.354605	−	0.935016i	$-0.615385\pi$
$109$	0.393906	−	0.271894i	0.393906	−	0.271894i	0.748511	+	0.663123i	$0.230769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.0763874	+	0.0255019i	0.0763874	+	0.0255019i	0.354605	−	0.935016i	$-0.384615\pi$
$113$	0.0763874	+	0.0255019i	0.0763874	+	0.0255019i	−0.278217	+	0.960518i	$0.589744\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.970942	+	0.239316i	0.970942	+	0.239316i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.663673	+	0.0805846i	−0.663673	+	0.0805846i
$126$		0			0
$127$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$127$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$131$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.80544	−	0.856690i	−1.80544	−	0.856690i	−0.919979	−	0.391967i	$-0.871795\pi$
$137$	−1.80544	−	0.856690i	−1.80544	−	0.856690i	−0.885456	−	0.464723i	$-0.846154\pi$
$138$		0			0
$139$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$139$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.372318	−	0.303988i	0.372318	−	0.303988i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.154579	−	0.0447744i	0.154579	−	0.0447744i	−0.200026	−	0.979791i	$-0.564103\pi$
$149$	0.154579	−	0.0447744i	0.154579	−	0.0447744i	0.354605	+	0.935016i	$0.384615\pi$
$150$		0			0
$151$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$151$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$152$		0			0
$153$	0.0557864	−	1.38433i	0.0557864	−	1.38433i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.76517	+	0.926432i	1.76517	+	0.926432i	0.919979	+	0.391967i	$0.128205\pi$
$157$	1.76517	+	0.926432i	1.76517	+	0.926432i	0.845190	+	0.534466i	$0.179487\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$163$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$167$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$168$		0			0
$169$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.03702	−	1.07966i	−1.03702	−	1.07966i	−0.996757	−	0.0804666i	$-0.974359\pi$
$173$	−1.03702	−	1.07966i	−1.03702	−	1.07966i	−0.0402659	−	0.999189i	$-0.512821\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$179$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$180$		0			0
$181$	−1.74798	−	0.917410i	−1.74798	−	0.917410i	−0.948536	−	0.316668i	$-0.897436\pi$
$181$	−1.74798	−	0.917410i	−1.74798	−	0.917410i	−0.799443	−	0.600742i	$-0.794872\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0517137	−	1.28326i	0.0517137	−	1.28326i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−0.124660	+	0.101781i	−0.124660	+	0.101781i	−0.692724	−	0.721202i	$-0.743590\pi$
$193$	−0.124660	+	0.101781i	−0.124660	+	0.101781i	0.568065	+	0.822984i	$0.307692\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.106718	−	1.32194i	−0.106718	−	1.32194i	−0.799443	−	0.600742i	$-0.794872\pi$
$197$	−0.106718	−	1.32194i	−0.106718	−	1.32194i	0.692724	−	0.721202i	$-0.256410\pi$
$198$		0			0
$199$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$199$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.929686	−	0.151089i	−0.929686	−	0.151089i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$211$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.38545			1.38545
$222$		0			0
$223$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$223$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$224$		0			0
$225$	−0.420738	−	0.140463i	−0.420738	−	0.140463i
$226$		0			0
$227$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$227$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$228$		0			0
$229$	1.53901	−	1.06230i	1.53901	−	1.06230i	0.568065	−	0.822984i	$-0.307692\pi$
$229$	1.53901	−	1.06230i	1.53901	−	1.06230i	0.970942	−	0.239316i	$-0.0769231\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.850405	+	0.753393i	−0.850405	+	0.753393i	−0.970942	−	0.239316i	$-0.923077\pi$
$233$	−0.850405	+	0.753393i	−0.850405	+	0.753393i	0.120537	+	0.992709i	$0.461538\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.338496	−	0.535289i	0.338496	−	0.535289i	−0.632445	−	0.774605i	$-0.717949\pi$
$241$	0.338496	−	0.535289i	0.338496	−	0.535289i	0.970942	+	0.239316i	$0.0769231\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.15405	−	0.334274i	−1.15405	−	0.334274i
$246$		0			0
$247$		0			0
$248$		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	$0.0256410\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.549229	−	1.89616i	0.549229	−	1.89616i	0.120537	−	0.992709i	$-0.461538\pi$
$257$	0.549229	−	1.89616i	0.549229	−	1.89616i	0.428693	−	0.903450i	$-0.358974\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.388427	+	0.0957386i	−0.388427	+	0.0957386i
$262$		0			0
$263$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$263$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$264$		0			0
$265$	−1.50867	−	0.183185i	−1.50867	−	0.183185i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.354228	−	1.73512i	0.354228	−	1.73512i	−0.278217	−	0.960518i	$-0.589744\pi$
$269$	0.354228	−	1.73512i	0.354228	−	1.73512i	0.632445	−	0.774605i	$-0.282051\pi$
$270$		0			0
$271$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$271$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.987050	+	0.160411i	−0.987050	+	0.160411i	−0.632445	−	0.774605i	$-0.717949\pi$
$277$	−0.987050	+	0.160411i	−0.987050	+	0.160411i	−0.354605	+	0.935016i	$0.615385\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.02732	−	1.15960i	1.02732	−	1.15960i	0.0402659	−	0.999189i	$-0.487179\pi$
$281$	1.02732	−	1.15960i	1.02732	−	1.15960i	0.987050	−	0.160411i	$-0.0512821\pi$
$282$		0			0
$283$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$283$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.183917	−	0.900886i	−0.183917	−	0.900886i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.289847	+	1.78350i	0.289847	+	1.78350i	0.568065	+	0.822984i	$0.307692\pi$
$293$	0.289847	+	1.78350i	0.289847	+	1.78350i	−0.278217	+	0.960518i	$0.589744\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.721783	+	2.16200i	−0.721783	+	2.16200i
$306$		0			0
$307$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$307$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$311$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$312$		0			0
$313$	−0.136945	−	0.198399i	−0.136945	−	0.198399i	0.748511	−	0.663123i	$-0.230769\pi$
$313$	−0.136945	−	0.198399i	−0.136945	−	0.198399i	−0.885456	+	0.464723i	$0.846154\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.44854	−	0.549357i	−1.44854	−	0.549357i	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.44854	−	0.549357i	−1.44854	−	0.549357i	−0.948536	+	0.316668i	$0.897436\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.123408	−	0.426052i	0.123408	−	0.426052i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$331$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$332$		0			0
$333$	−0.496757	+	0.946492i	−0.496757	+	0.946492i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.857385			−0.857385			−0.428693	−	0.903450i	$-0.641026\pi$
$337$	−0.857385			−0.857385			−0.428693	+	0.903450i	$0.641026\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$347$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$348$		0			0
$349$	0.558358	−	0.743039i	0.558358	−	0.743039i	−0.428693	−	0.903450i	$-0.641026\pi$
$349$	0.558358	−	0.743039i	0.558358	−	0.743039i	0.987050	+	0.160411i	$0.0512821\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.0514636	+	0.316668i	−0.0514636	+	0.316668i	0.948536	+	0.316668i	$0.102564\pi$
$353$	−0.0514636	+	0.316668i	−0.0514636	+	0.316668i	−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$359$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.18216	+	1.71266i	−1.18216	+	1.71266i
$366$		0			0
$367$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$367$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$368$		0			0
$369$	0.645164	+	0.445325i	0.645164	+	0.445325i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.69272	+	0.721202i	−1.69272	+	0.721202i	−0.692724	+	0.721202i	$0.743590\pi$
$373$	−1.69272	+	0.721202i	−1.69272	+	0.721202i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.111301	−	0.384257i	−0.111301	−	0.384257i
$378$		0			0
$379$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$379$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$383$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.599417	+	1.58053i	−0.599417	+	1.58053i	0.200026	+	0.979791i	$0.435897\pi$
$389$	−0.599417	+	1.58053i	−0.599417	+	1.58053i	−0.799443	+	0.600742i	$0.794872\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.98381	−	0.0799447i	1.98381	−	0.0799447i	0.987050	−	0.160411i	$-0.0512821\pi$
$397$	1.98381	−	0.0799447i	1.98381	−	0.0799447i	0.996757	+	0.0804666i	$0.0256410\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.80544	−	0.0727566i	−1.80544	−	0.0727566i	−0.885456	−	0.464723i	$-0.846154\pi$
$401$	−1.80544	−	0.0727566i	−1.80544	−	0.0727566i	−0.919979	+	0.391967i	$0.871795\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.866514	+	0.832298i	0.866514	+	0.832298i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.69705	−	0.346455i	−1.69705	−	0.346455i	−0.748511	−	0.663123i	$-0.769231\pi$
$409$	−1.69705	−	0.346455i	−1.69705	−	0.346455i	−0.948536	+	0.316668i	$0.897436\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		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
$421$	−0.419979	−	0.474059i	−0.419979	−	0.474059i	0.500000	−	0.866025i	$-0.333333\pi$
$421$	−0.419979	−	0.474059i	−0.419979	−	0.474059i	−0.919979	+	0.391967i	$0.871795\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.612544	−	0.0494497i	−0.612544	−	0.0494497i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$431$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$432$		0			0
$433$	0.0680647	+	1.68901i	0.0680647	+	1.68901i	0.568065	+	0.822984i	$0.307692\pi$
$433$	0.0680647	+	1.68901i	0.0680647	+	1.68901i	−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$439$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$440$		0			0
$441$	0.748511	+	0.663123i	0.748511	+	0.663123i
$442$		0			0
$443$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$443$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$444$		0			0
$445$	0.943837	−	0.596846i	0.943837	−	0.596846i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.43189	+	1.37535i	−1.43189	+	1.37535i	−0.632445	+	0.774605i	$0.717949\pi$
$449$	−1.43189	+	1.37535i	−1.43189	+	1.37535i	−0.799443	+	0.600742i	$0.794872\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.51790	−	0.309882i	1.51790	−	0.309882i	0.632445	−	0.774605i	$-0.282051\pi$
$457$	1.51790	−	0.309882i	1.51790	−	0.309882i	0.885456	+	0.464723i	$0.153846\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.20051	−	1.59759i	−1.20051	−	1.59759i	−0.632445	−	0.774605i	$-0.717949\pi$
$461$	−1.20051	−	1.59759i	−1.20051	−	1.59759i	−0.568065	−	0.822984i	$-0.692308\pi$
$462$		0			0
$463$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$463$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$467$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.06907	+	0.676041i	1.06907	+	0.676041i
$478$		0			0
$479$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$479$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$480$		0			0
$481$	−0.965727	−	0.458243i	−0.965727	−	0.458243i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.988802	−	1.71266i	−0.988802	−	1.71266i
$486$		0			0
$487$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$487$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$488$		0			0
$489$		0			0
$490$		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
$493$	−0.490765	+	0.257573i	−0.490765	+	0.257573i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$499$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$500$		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
$505$	0.403779	−	0.947703i	0.403779	−	0.947703i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.768090	−	1.80277i	0.768090	−	1.80277i	0.200026	−	0.979791i	$-0.435897\pi$
$509$	0.768090	−	1.80277i	0.768090	−	1.80277i	0.568065	−	0.822984i	$-0.307692\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.62920	−	0.855072i	1.62920	−	0.855072i	0.632445	−	0.774605i	$-0.282051\pi$
$521$	1.62920	−	0.855072i	1.62920	−	0.855072i	0.996757	−	0.0804666i	$-0.0256410\pi$
$522$		0			0
$523$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$523$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$524$		0			0
$525$		0			0
$526$		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			0
$533$	−0.418986	+	0.662573i	−0.418986	+	0.662573i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.149094	−	0.284074i	−0.149094	−	0.284074i	0.799443	−	0.600742i	$-0.205128\pi$
$541$	−0.149094	−	0.284074i	−0.149094	−	0.284074i	−0.948536	+	0.316668i	$0.897436\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.0693168	+	0.570875i	0.0693168	+	0.570875i
$546$		0			0
$547$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$547$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$548$		0			0
$549$	1.31415	−	1.36817i	1.31415	−	1.36817i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.69705	+	0.346455i	−1.69705	+	0.346455i	−0.948536	−	0.316668i	$-0.897436\pi$
$557$	−1.69705	+	0.346455i	−1.69705	+	0.346455i	−0.748511	+	0.663123i	$0.769231\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$563$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$564$		0			0
$565$	−0.0697820	+	0.0670265i	−0.0697820	+	0.0670265i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.64126	−	1.03787i	1.64126	−	1.03787i	0.692724	−	0.721202i	$-0.256410\pi$
$569$	1.64126	−	1.03787i	1.64126	−	1.03787i	0.948536	−	0.316668i	$-0.102564\pi$
$570$		0			0
$571$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$571$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	1.92104i		−	1.92104i	−0.278217	−	0.960518i	$-0.589744\pi$
$577$		−	1.92104i		−	1.92104i	0.278217	−	0.960518i	$-0.410256\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.796732	+	0.899324i	−0.796732	+	0.899324i
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.27388	+	1.43792i	1.27388	+	1.43792i	0.845190	+	0.534466i	$0.179487\pi$
$593$	1.27388	+	1.43792i	1.27388	+	1.43792i	0.428693	+	0.903450i	$0.358974\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$599$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$600$		0			0
$601$	−1.51660	−	1.13965i	−1.51660	−	1.13965i	−0.948536	−	0.316668i	$-0.897436\pi$
$601$	−1.51660	−	1.13965i	−1.51660	−	1.13965i	−0.568065	−	0.822984i	$-0.692308\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.17720	+	0.240328i	1.17720	+	0.240328i
$606$		0			0
$607$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$607$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.632822	+	0.0255019i	0.632822	+	0.0255019i	0.354605	−	0.935016i	$-0.384615\pi$
$613$	0.632822	+	0.0255019i	0.632822	+	0.0255019i	0.278217	+	0.960518i	$0.410256\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.95799	−	0.0789044i	1.95799	−	0.0789044i	0.970942	−	0.239316i	$-0.0769231\pi$
$617$	1.95799	−	0.0789044i	1.95799	−	0.0789044i	0.987050	+	0.160411i	$0.0512821\pi$
$618$		0			0
$619$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$619$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.442127	−	1.16579i	0.442127	−	1.16579i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.354415	+	1.43792i	−0.354415	+	1.43792i
$630$		0			0
$631$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$631$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.919979	+	0.391967i	−0.919979	+	0.391967i	−0.799443	−	0.600742i	$-0.794872\pi$
$641$	−0.919979	+	0.391967i	−0.919979	+	0.391967i	−0.120537	+	0.992709i	$0.538462\pi$
$642$		0			0
$643$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$643$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$647$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.354605	+	0.614194i	−0.354605	+	0.614194i	−0.987050	−	0.160411i	$-0.948718\pi$
$653$	−0.354605	+	0.614194i	−0.354605	+	0.614194i	0.632445	+	0.774605i	$0.282051\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.50000	−	0.866025i	1.50000	−	0.866025i
$658$		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
$661$	0.308156	−	1.89616i	0.308156	−	1.89616i	−0.120537	−	0.992709i	$-0.538462\pi$
$661$	0.308156	−	1.89616i	0.308156	−	1.89616i	0.428693	−	0.903450i	$-0.358974\pi$
$662$		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
$673$	−1.87251	+	0.625134i	−1.87251	+	0.625134i	−0.885456	+	0.464723i	$0.846154\pi$
$673$	−1.87251	+	0.625134i	−1.87251	+	0.625134i	−0.987050	+	0.160411i	$0.948718\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.94188			−1.94188			−0.970942	−	0.239316i	$-0.923077\pi$
$677$	−1.94188			−1.94188			−0.970942	+	0.239316i	$0.923077\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$683$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$684$		0			0
$685$	1.91948	−	1.44239i	1.91948	−	1.44239i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.632445	+	1.09543i	−0.632445	+	1.09543i
$690$		0			0
$691$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$691$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.01552	+	0.385136i	1.01552	+	0.385136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.402877	+	0.583668i	0.402877	+	0.583668i	0.970942	−	0.239316i	$-0.0769231\pi$
$701$	0.402877	+	0.583668i	0.402877	+	0.583668i	−0.568065	+	0.822984i	$0.692308\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.608331	+	1.82217i	−0.608331	+	1.82217i	−0.0402659	+	0.999189i	$0.512821\pi$
$709$	−0.608331	+	1.82217i	−0.608331	+	1.82217i	−0.568065	+	0.822984i	$0.692308\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$719$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.0354943	+	0.173863i	0.0354943	+	0.173863i
$726$		0			0
$727$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$727$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$728$		0			0
$729$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.106718	−	0.120460i	0.106718	−	0.120460i	−0.692724	−	0.721202i	$-0.743590\pi$
$733$	0.106718	−	0.120460i	0.106718	−	0.120460i	0.799443	+	0.600742i	$0.205128\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$739$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$743$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$744$		0			0
$745$	−0.0386767	+	0.189451i	−0.0386767	+	0.189451i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$751$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.416498	+	1.43792i	−0.416498	+	1.43792i	0.428693	+	0.903450i	$0.358974\pi$
$757$	−0.416498	+	1.43792i	−0.416498	+	1.43792i	−0.845190	+	0.534466i	$0.820513\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.732990	+	1.72039i	0.732990	+	1.72039i	0.692724	+	0.721202i	$0.256410\pi$
$761$	0.732990	+	1.72039i	0.732990	+	1.72039i	0.0402659	+	0.999189i	$0.487179\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.44158	+	0.832298i	1.44158	+	0.832298i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.27388	+	0.368985i	1.27388	+	0.368985i	0.845190	−	0.534466i	$-0.179487\pi$
$769$	1.27388	+	0.368985i	1.27388	+	0.368985i	0.428693	+	0.903450i	$0.358974\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.255812	−	0.404534i	0.255812	−	0.404534i	−0.692724	−	0.721202i	$-0.743590\pi$
$773$	0.255812	−	0.404534i	0.255812	−	0.404534i	0.948536	+	0.316668i	$0.102564\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.97119	+	1.36062i	−1.97119	+	1.36062i
$786$		0			0
$787$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$787$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.41998	+	1.25799i	1.41998	+	1.25799i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.0670708	−	0.231555i	−0.0670708	−	0.231555i	0.919979	−	0.391967i	$-0.128205\pi$
$797$	−0.0670708	−	0.231555i	−0.0670708	−	0.231555i	−0.987050	+	0.160411i	$0.948718\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.922670	+	0.112032i	−0.922670	+	0.112032i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.96770	+	0.319782i	1.96770	+	0.319782i	0.996757	+	0.0804666i	$0.0256410\pi$
$809$	1.96770	+	0.319782i	1.96770	+	0.319782i	0.970942	+	0.239316i	$0.0769231\pi$
$810$		0			0
$811$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$811$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.27497	−	1.04098i	1.27497	−	1.04098i	0.278217	−	0.960518i	$-0.410256\pi$
$821$	1.27497	−	1.04098i	1.27497	−	1.04098i	0.996757	−	0.0804666i	$-0.0256410\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$827$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$828$		0			0
$829$	0.0509320	−	1.26386i	0.0509320	−	1.26386i	−0.748511	−	0.663123i	$-0.769231\pi$
$829$	0.0509320	−	1.26386i	0.0509320	−	1.26386i	0.799443	−	0.600742i	$-0.205128\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.22675	+	0.643850i	1.22675	+	0.643850i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$839$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$840$		0			0
$841$	−0.581860	−	0.605780i	−0.581860	−	0.605780i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.930676	−	0.759873i	−0.930676	−	0.759873i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.34867	+	0.511484i	−1.34867	+	0.511484i	−0.919979	−	0.391967i	$-0.871795\pi$
$853$	−1.34867	+	0.511484i	−1.34867	+	0.511484i	−0.428693	+	0.903450i	$0.641026\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.67977	+	0.881614i	1.67977	+	0.881614i	0.987050	+	0.160411i	$0.0512821\pi$
$857$	1.67977	+	0.881614i	1.67977	+	0.881614i	0.692724	+	0.721202i	$0.256410\pi$
$858$		0			0
$859$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$859$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$863$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$864$		0			0
$865$	1.72763	−	0.500415i	1.72763	−	0.500415i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.132445	+	1.64063i	0.132445	+	1.64063i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.708245	−	0.336066i	−0.708245	−	0.336066i	0.0402659	−	0.999189i	$-0.487179\pi$
$877$	−0.708245	−	0.336066i	−0.708245	−	0.336066i	−0.748511	+	0.663123i	$0.769231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.81613	−	0.295150i	−1.81613	−	0.295150i	−0.845190	−	0.534466i	$-0.820513\pi$
$881$	−1.81613	−	0.295150i	−1.81613	−	0.295150i	−0.970942	+	0.239316i	$0.923077\pi$
$882$		0			0
$883$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$883$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$887$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$888$		0			0
$889$		0			0
$890$		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			0
$901$	1.66225	+	0.554942i	1.66225	+	0.554942i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.95200	−	1.34737i	1.95200	−	1.34737i
$906$		0			0
$907$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$907$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$908$		0			0
$909$	−0.641762	+	0.568552i	−0.641762	+	0.568552i
$910$		0			0
$911$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$911$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$919$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.410618	+	0.237070i	0.410618	+	0.237070i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.565375	+	1.32698i	0.565375	+	1.32698i	0.919979	+	0.391967i	$0.128205\pi$
$929$	0.565375	+	1.32698i	0.565375	+	1.32698i	−0.354605	+	0.935016i	$0.615385\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.970942	−	0.239316i	0.970942	−	0.239316i	0.278217	−	0.960518i	$-0.410256\pi$
$937$	0.970942	−	0.239316i	0.970942	−	0.239316i	0.692724	+	0.721202i	$0.256410\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.31658	−	0.159861i	−1.31658	−	0.159861i	−0.568065	−	0.822984i	$-0.692308\pi$
$941$	−1.31658	−	0.159861i	−1.31658	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$947$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$948$		0			0
$949$	0.925722	+	1.46391i	0.925722	+	1.46391i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.47764	+	0.240139i	−1.47764	+	0.240139i	−0.845190	−	0.534466i	$-0.820513\pi$
$953$	−1.47764	+	0.240139i	−1.47764	+	0.240139i	−0.632445	+	0.774605i	$0.717949\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.354605	+	0.935016i	0.354605	+	0.935016i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.0386767	−	0.189451i	−0.0386767	−	0.189451i
$966$		0			0
$967$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$967$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$971$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.145395	−	0.0689908i	0.145395	−	0.0689908i	−0.354605	−	0.935016i	$-0.615385\pi$
$977$	0.145395	−	0.0689908i	0.145395	−	0.0689908i	0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.151567	−	0.453999i	0.151567	−	0.453999i
$982$		0			0
$983$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$983$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$984$		0			0
$985$	1.46595	+	0.624584i	1.46595	+	0.624584i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.0509320	−	0.0623804i	0.0509320	−	0.0623804i	−0.748511	−	0.663123i	$-0.769231\pi$
$997$	0.0509320	−	0.0623804i	0.0509320	−	0.0623804i	0.799443	+	0.600742i	$0.205128\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.1.cm.a.1759.1	✓	24
4.3	odd	2	CM	2704.1.cm.a.1759.1	✓	24
169.49	even	78	inner	2704.1.cm.a.2415.1	yes	24
676.387	odd	78	inner	2704.1.cm.a.2415.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2704.1.cm.a.1759.1	✓	24	1.1	even	1	trivial
2704.1.cm.a.1759.1	✓	24	4.3	odd	2	CM
2704.1.cm.a.2415.1	yes	24	169.49	even	78	inner
2704.1.cm.a.2415.1	yes	24	676.387	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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2704.cm (of order \(78\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.558358 + 1.06386i) q^{5} +(0.799443 - 0.600742i) q^{9} +(0.632445 + 0.774605i) q^{13} +(0.876221 - 1.07318i) q^{17} +(-0.251974 - 0.365047i) q^{25} +(-0.368039 - 0.156807i) q^{29} +(-0.965727 + 0.458243i) q^{37} +(0.248247 + 0.743589i) q^{41} +(0.192732 + 1.18593i) q^{45} +(0.200026 + 0.979791i) q^{49} +(0.448536 + 1.18269i) q^{53} +(1.87251 - 0.304312i) q^{61} +(-1.17720 + 0.240328i) q^{65} +(1.71942 + 0.208776i) q^{73} +(0.278217 - 0.960518i) q^{81} +(0.652466 + 1.53139i) q^{85} +(-0.804924 - 0.464723i) q^{89} +(-0.879714 + 1.39116i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{9} - q^{13} + q^{17} + 4 q^{25} + q^{29} + 3 q^{37} - 3 q^{41} - 3 q^{45} - q^{49} - 11 q^{53} - q^{61} - 3 q^{65} + q^{81} + 10 q^{85}+O(q^{100}) \) 24 * q + q^9 - q^13 + q^17 + 4 * q^25 + q^29 + 3 * q^37 - 3 * q^41 - 3 * q^45 - q^49 - 11 * q^53 - q^61 - 3 * q^65 + q^81 + 10 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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