LMFDB - Embedded newform 2597.1.bf.a.521.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2597,1,Mod(68,2597)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2597, base_ring=CyclotomicField(78))

chi = DirichletCharacter(H, H._module([65, 18]))

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

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

chi := DirichletCharacter("2597.68");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2597 = 7^{2} \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2597.bf (of order $78$, degree $24$, not 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:	$1.29607183781$
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:	no (minimal twist has level 371)
Projective image:	$D_{13}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{13} - \cdots)$

Embedding invariants

Embedding label			521.1
Root			$0.428693 + 0.903450i$ of defining polynomial
Character	$\chi$	$=$	2597.521
Dual form			2597.1.bf.a.2273.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.00970705 - 0.240878i) q^{2} +(0.938829 - 0.0757901i) q^{4} +(-0.0564276 - 0.464723i) q^{8} +(-0.200026 + 0.979791i) q^{9} +O(q^{10})$	$q+(-0.00970705 - 0.240878i) q^{2} +(0.938829 - 0.0757901i) q^{4} +(-0.0564276 - 0.464723i) q^{8} +(-0.200026 + 0.979791i) q^{9} +(-0.416498 + 1.43792i) q^{11} +(0.818293 - 0.132986i) q^{16} +(0.237952 + 0.0386709i) q^{18} +(0.350405 + 0.0863671i) q^{22} +(0.970942 + 1.68172i) q^{23} +(-0.919979 - 0.391967i) q^{25} +(-0.234068 - 0.0576926i) q^{29} +(-0.133616 - 0.654493i) q^{32} +(-0.113532 + 0.935016i) q^{36} +(1.12142 - 0.182248i) q^{37} +(0.251489 + 0.663123i) q^{43} +(-0.282040 + 1.38152i) q^{44} +(0.395664 - 0.250203i) q^{46} +(-0.0854858 + 0.225408i) q^{50} +(-0.200026 - 0.979791i) q^{53} +(-0.0116248 + 0.0569419i) q^{58} +(0.648582 - 0.159861i) q^{64} +(1.49217 - 0.120460i) q^{67} +(-0.627974 - 1.65583i) q^{71} +(0.466618 + 0.0376693i) q^{72} +(-0.0547851 - 0.268355i) q^{74} +(-1.49676 - 0.946492i) q^{79} +(-0.919979 - 0.391967i) q^{81} +(0.157290 - 0.0670152i) q^{86} +(0.691735 + 0.112418i) q^{88} +(1.03901 + 1.50526i) q^{92} +(-1.32555 - 0.695701i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{2} + 3 q^{4} + 18 q^{8} + q^{9}+O(q^{10}) $ 24 * q + 2 * q^2 + 3 * q^4 + 18 * q^8 + q^9	$ 24 q + 2 q^{2} + 3 q^{4} + 18 q^{8} + q^{9} + 2 q^{11} + 5 q^{16} + 2 q^{18} - 8 q^{22} + 2 q^{23} + q^{25} - 4 q^{29} + 6 q^{32} - 6 q^{36} + 2 q^{37} + 22 q^{43} - 7 q^{44} + 4 q^{46} - 4 q^{50} + q^{53} + 4 q^{58} + 12 q^{64} + 2 q^{67} - 4 q^{71} + 4 q^{72} - 9 q^{74} - 11 q^{79} + q^{81} + 4 q^{86} - 5 q^{88} - 12 q^{92} - 4 q^{99}+O(q^{100}) $ 24 * q + 2 * q^2 + 3 * q^4 + 18 * q^8 + q^9 + 2 * q^11 + 5 * q^16 + 2 * q^18 - 8 * q^22 + 2 * q^23 + q^25 - 4 * q^29 + 6 * q^32 - 6 * q^36 + 2 * q^37 + 22 * q^43 - 7 * q^44 + 4 * q^46 - 4 * q^50 + q^53 + 4 * q^58 + 12 * q^64 + 2 * q^67 - 4 * q^71 + 4 * q^72 - 9 * q^74 - 11 * q^79 + q^81 + 4 * q^86 - 5 * q^88 - 12 * q^92 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2597\mathbb{Z}\right)^\times$.

$n$	$638$	$1326$
$\chi(n)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i	−0.996757	−	0.0804666i	$-0.974359\pi$
$2$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i	0.987050	−	0.160411i	$-0.0512821\pi$
$3$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$3$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$4$	0.938829	−	0.0757901i	0.938829	−	0.0757901i
$5$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$5$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−0.0564276	−	0.464723i	−0.0564276	−	0.464723i
$9$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$10$		0			0
$11$	−0.416498	+	1.43792i	−0.416498	+	1.43792i	0.428693	+	0.903450i	$0.358974\pi$
$11$	−0.416498	+	1.43792i	−0.416498	+	1.43792i	−0.845190	+	0.534466i	$0.820513\pi$
$12$		0			0
$13$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$13$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$14$		0			0
$15$		0			0
$16$	0.818293	−	0.132986i	0.818293	−	0.132986i
$17$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$17$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$18$	0.237952	+	0.0386709i	0.237952	+	0.0386709i
$19$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$19$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$20$		0			0
$21$		0			0
$22$	0.350405	+	0.0863671i	0.350405	+	0.0863671i
$23$	0.970942	+	1.68172i	0.970942	+	1.68172i	0.692724	+	0.721202i	$0.256410\pi$
$23$	0.970942	+	1.68172i	0.970942	+	1.68172i	0.278217	+	0.960518i	$0.410256\pi$
$24$		0			0
$25$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	0.120537	−	0.992709i	$-0.461538\pi$
$29$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	−0.354605	+	0.935016i	$0.615385\pi$
$30$		0			0
$31$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$31$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$32$	−0.133616	−	0.654493i	−0.133616	−	0.654493i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.113532	+	0.935016i	−0.113532	+	0.935016i
$37$	1.12142	−	0.182248i	1.12142	−	0.182248i	0.428693	−	0.903450i	$-0.358974\pi$
$37$	1.12142	−	0.182248i	1.12142	−	0.182248i	0.692724	+	0.721202i	$0.256410\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$41$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$42$		0			0
$43$	0.251489	+	0.663123i	0.251489	+	0.663123i	1.00000			$0$
$43$	0.251489	+	0.663123i	0.251489	+	0.663123i	−0.748511	+	0.663123i	$0.769231\pi$
$44$	−0.282040	+	1.38152i	−0.282040	+	1.38152i
$45$		0			0
$46$	0.395664	−	0.250203i	0.395664	−	0.250203i
$47$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$47$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$48$		0			0
$49$		0			0
$50$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i
$51$		0			0
$52$		0			0
$53$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$53$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.0116248	+	0.0569419i	−0.0116248	+	0.0569419i
$59$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$59$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$60$		0			0
$61$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$61$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$62$		0			0
$63$		0			0
$64$	0.648582	−	0.159861i	0.648582	−	0.159861i
$65$		0			0
$66$		0			0
$67$	1.49217	−	0.120460i	1.49217	−	0.120460i	0.692724	−	0.721202i	$-0.256410\pi$
$67$	1.49217	−	0.120460i	1.49217	−	0.120460i	0.799443	+	0.600742i	$0.205128\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.627974	−	1.65583i	−0.627974	−	1.65583i	−0.748511	−	0.663123i	$-0.769231\pi$
$71$	−0.627974	−	1.65583i	−0.627974	−	1.65583i	0.120537	−	0.992709i	$-0.461538\pi$
$72$	0.466618	+	0.0376693i	0.466618	+	0.0376693i
$73$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$73$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$74$	−0.0547851	−	0.268355i	−0.0547851	−	0.268355i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.49676	−	0.946492i	−1.49676	−	0.946492i	−0.996757	−	0.0804666i	$-0.974359\pi$
$79$	−1.49676	−	0.946492i	−1.49676	−	0.946492i	−0.500000	−	0.866025i	$-0.666667\pi$
$80$		0			0
$81$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	0.157290	−	0.0670152i	0.157290	−	0.0670152i
$87$		0			0
$88$	0.691735	+	0.112418i	0.691735	+	0.112418i
$89$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$89$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$90$		0			0
$91$		0			0
$92$	1.03901	+	1.50526i	1.03901	+	1.50526i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$97$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$98$		0			0
$99$	−1.32555	−	0.695701i	−1.32555	−	0.695701i
$100$	−0.893411	−	0.298264i	−0.893411	−	0.298264i
$101$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$101$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$102$		0			0
$103$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$103$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$104$		0			0
$105$		0			0
$106$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i
$107$	0.354605	+	0.614194i	0.354605	+	0.614194i	0.987050	−	0.160411i	$-0.0512821\pi$
$107$	0.354605	+	0.614194i	0.354605	+	0.614194i	−0.632445	+	0.774605i	$0.717949\pi$
$108$		0			0
$109$	−1.26489	−	1.54921i	−1.26489	−	1.54921i	−0.632445	−	0.774605i	$-0.717949\pi$
$109$	−1.26489	−	1.54921i	−1.26489	−	1.54921i	−0.632445	−	0.774605i	$-0.717949\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.885456	−	0.464723i	$-0.153846\pi$
$113$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.120537	+	0.992709i	$0.461538\pi$
$114$		0			0
$115$		0			0
$116$	−0.224123	−	0.0364235i	−0.224123	−	0.0364235i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.04894	−	0.663311i	−1.04894	−	0.663311i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.45352	−	1.28771i	1.45352	−	1.28771i	0.568065	−	0.822984i	$-0.307692\pi$
$127$	1.45352	−	1.28771i	1.45352	−	1.28771i	0.885456	−	0.464723i	$-0.153846\pi$
$128$	−0.230650	−	0.796297i	−0.230650	−	0.796297i
$129$		0			0
$130$		0			0
$131$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$131$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$132$		0			0
$133$		0			0
$134$	−0.0435007	−	0.358261i	−0.0435007	−	0.358261i
$135$		0			0
$136$		0			0
$137$	−0.491287	+	0.511484i	−0.491287	+	0.511484i	−0.919979	−	0.391967i	$-0.871795\pi$
$137$	−0.491287	+	0.511484i	−0.491287	+	0.511484i	0.428693	+	0.903450i	$0.358974\pi$
$138$		0			0
$139$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$139$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$140$		0			0
$141$		0			0
$142$	−0.392757	+	0.167338i	−0.392757	+	0.167338i
$143$		0			0
$144$	−0.0333816	+	0.828356i	−0.0333816	+	0.828356i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.03901	−	0.256092i	1.03901	−	0.256092i
$149$	−1.12001	+	1.37176i	−1.12001	+	1.37176i	−0.200026	+	0.979791i	$0.564103\pi$
$149$	−1.12001	+	1.37176i	−1.12001	+	1.37176i	−0.919979	+	0.391967i	$0.871795\pi$
$150$		0			0
$151$	0.652458	+	0.277987i	0.652458	+	0.277987i	0.692724	−	0.721202i	$-0.256410\pi$
$151$	0.652458	+	0.277987i	0.652458	+	0.277987i	−0.0402659	+	0.999189i	$0.512821\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$157$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$158$	−0.213460	+	0.369723i	−0.213460	+	0.369723i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i
$163$	0.759177	−	1.59993i	0.759177	−	1.59993i	−0.0402659	−	0.999189i	$-0.512821\pi$
$163$	0.759177	−	1.59993i	0.759177	−	1.59993i	0.799443	−	0.600742i	$-0.205128\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$167$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$168$		0			0
$169$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$170$		0			0
$171$		0			0
$172$	0.286364	+	0.603499i	0.286364	+	0.603499i
$173$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$173$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$174$		0			0
$175$		0			0
$176$	−0.149595	+	1.23202i	−0.149595	+	1.23202i
$177$		0			0
$178$		0			0
$179$	0.908271	+	0.682521i	0.908271	+	0.682521i	0.948536	−	0.316668i	$-0.102564\pi$
$179$	0.908271	+	0.682521i	0.908271	+	0.682521i	−0.0402659	+	0.999189i	$0.512821\pi$
$180$		0			0
$181$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$181$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$182$		0			0
$183$		0			0
$184$	0.726747	−	0.546115i	0.726747	−	0.546115i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.89707	+	0.633336i	1.89707	+	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$191$	1.89707	+	0.633336i	1.89707	+	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$192$		0			0
$193$	0.487050	+	1.02644i	0.487050	+	1.02644i	0.987050	+	0.160411i	$0.0512821\pi$
$193$	0.487050	+	1.02644i	0.487050	+	1.02644i	−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.71945	+	0.902438i	−1.71945	+	0.902438i	−0.748511	+	0.663123i	$0.769231\pi$
$197$	−1.71945	+	0.902438i	−1.71945	+	0.902438i	−0.970942	+	0.239316i	$0.923077\pi$
$198$	−0.154712	+	0.326048i	−0.154712	+	0.326048i
$199$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$199$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$200$	−0.130244	+	0.449654i	−0.130244	+	0.449654i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.84195	+	0.614932i	−1.84195	+	0.614932i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.94188			−1.94188			−0.970942	−	0.239316i	$-0.923077\pi$
$211$	−1.94188			−1.94188			−0.970942	+	0.239316i	$0.923077\pi$
$212$	−0.262048	−	0.904696i	−0.262048	−	0.904696i
$213$		0			0
$214$	0.144503	−	0.0913785i	0.144503	−	0.0913785i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.360892	+	0.319722i	−0.360892	+	0.319722i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$223$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$224$		0			0
$225$	0.568065	−	0.822984i	0.568065	−	0.822984i
$226$	0.117415	−	0.247447i	0.117415	−	0.247447i
$227$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$227$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$228$		0			0
$229$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$229$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$230$		0			0
$231$		0			0
$232$	−0.0136032	+	0.112032i	−0.0136032	+	0.112032i
$233$	−1.41998	−	0.474059i	−1.41998	−	0.474059i	−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.41998	−	0.474059i	−1.41998	−	0.474059i	−0.919979	+	0.391967i	$0.871795\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.120537	−	0.992709i	$-0.461538\pi$
$239$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.885456	+	0.464723i	$0.153846\pi$
$240$		0			0
$241$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$241$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$242$	−0.149595	+	0.259106i	−0.149595	+	0.259106i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$251$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$252$		0			0
$253$	−2.82257	+	0.695701i	−2.82257	+	0.695701i
$254$	−0.324289	−	0.337621i	−0.324289	−	0.337621i
$255$		0			0
$256$	0.444044	−	0.148244i	0.444044	−	0.148244i
$257$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$257$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.103346	−	0.217798i	0.103346	−	0.217798i
$262$		0			0
$263$	1.26527	−	0.800107i	1.26527	−	0.800107i	0.278217	−	0.960518i	$-0.410256\pi$
$263$	1.26527	−	0.800107i	1.26527	−	0.800107i	0.987050	+	0.160411i	$0.0512821\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.39176	−	0.226183i	1.39176	−	0.226183i
$269$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$269$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$270$		0			0
$271$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$271$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$272$		0			0
$273$		0			0
$274$	0.127974	+	0.113375i	0.127974	+	0.113375i
$275$	0.946784	−	1.15960i	0.946784	−	1.15960i
$276$		0			0
$277$	0.492699	−	1.70099i	0.492699	−	1.70099i	−0.200026	−	0.979791i	$-0.564103\pi$
$277$	0.492699	−	1.70099i	0.492699	−	1.70099i	0.692724	−	0.721202i	$-0.256410\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.251489	−	0.663123i	0.251489	−	0.663123i	−0.748511	−	0.663123i	$-0.769231\pi$
$281$	0.251489	−	0.663123i	0.251489	−	0.663123i	1.00000			$0$
$282$		0			0
$283$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$283$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$284$	−0.715056	−	1.50695i	−0.715056	−	1.50695i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.667993			0.667993
$289$	0.278217	+	0.960518i	0.278217	+	0.960518i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$293$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$294$		0			0
$295$		0			0
$296$	−0.147974	−	0.510865i	−0.147974	−	0.510865i
$297$		0			0
$298$	0.341298	+	0.256469i	0.341298	+	0.256469i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.0606274	−	0.159861i	0.0606274	−	0.159861i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$307$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$311$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$312$		0			0
$313$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$313$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$314$		0			0
$315$		0			0
$316$	−1.47693	−	0.775155i	−1.47693	−	0.775155i
$317$	−0.885456	−	1.53365i	−0.885456	−	1.53365i	−0.845190	−	0.534466i	$-0.820513\pi$
$317$	−0.885456	−	1.53365i	−0.885456	−	1.53365i	−0.0402659	−	0.999189i	$-0.512821\pi$
$318$		0			0
$319$	0.180446	−	0.312542i	0.180446	−	0.312542i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.893411	−	0.298264i	−0.893411	−	0.298264i
$325$		0			0
$326$	−0.392757	−	0.167338i	−0.392757	−	0.167338i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.76517	−	0.142499i	−1.76517	−	0.142499i	−0.845190	−	0.534466i	$-0.820513\pi$
$331$	−1.76517	−	0.142499i	−1.76517	−	0.142499i	−0.919979	+	0.391967i	$0.871795\pi$
$332$		0			0
$333$	−0.0457473	+	1.13521i	−0.0457473	+	1.13521i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.850405	+	1.23202i	−0.850405	+	1.23202i	0.120537	+	0.992709i	$0.461538\pi$
$337$	−0.850405	+	1.23202i	−0.850405	+	1.23202i	−0.970942	+	0.239316i	$0.923077\pi$
$338$	−0.221783	+	0.0944927i	−0.221783	+	0.0944927i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	0.293978	−	0.154291i	0.293978	−	0.154291i
$345$		0			0
$346$		0			0
$347$	−1.03702	+	1.07966i	−1.03702	+	1.07966i	−0.0402659	+	0.999189i	$0.512821\pi$
$347$	−1.03702	+	1.07966i	−1.03702	+	1.07966i	−0.996757	+	0.0804666i	$0.974359\pi$
$348$		0			0
$349$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$349$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$350$		0			0
$351$		0			0
$352$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$353$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$353$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.155588	−	0.225408i	0.155588	−	0.225408i
$359$	0.166997	+	0.173863i	0.166997	+	0.173863i	0.799443	−	0.600742i	$-0.205128\pi$
$359$	0.166997	+	0.173863i	0.166997	+	0.173863i	−0.632445	+	0.774605i	$0.717949\pi$
$360$		0			0
$361$	0.987050	+	0.160411i	0.987050	+	0.160411i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$367$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$368$	1.01816	+	1.24702i	1.01816	+	1.24702i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i	−0.996757	−	0.0804666i	$-0.974359\pi$
$373$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i	0.987050	−	0.160411i	$-0.0512821\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.180446	−	1.48611i	−0.180446	−	1.48611i	−0.748511	−	0.663123i	$-0.769231\pi$
$379$	−0.180446	−	1.48611i	−0.180446	−	1.48611i	0.568065	−	0.822984i	$-0.307692\pi$
$380$		0			0
$381$		0			0
$382$	0.134142	−	0.463111i	0.134142	−	0.463111i
$383$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$383$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$384$		0			0
$385$		0			0
$386$	0.242518	−	0.127283i	0.242518	−	0.127283i
$387$	−0.700026	+	0.113765i	−0.700026	+	0.113765i
$388$		0			0
$389$	−1.47764	−	0.240139i	−1.47764	−	0.240139i	−0.632445	−	0.774605i	$-0.717949\pi$
$389$	−1.47764	−	0.240139i	−1.47764	−	0.240139i	−0.845190	+	0.534466i	$0.820513\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	0.234068	+	0.405418i	0.234068	+	0.405418i
$395$		0			0
$396$	−1.29719	−	0.552681i	−1.29719	−	0.552681i
$397$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$397$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$398$		0			0
$399$		0			0
$400$	−0.804938	−	0.198399i	−0.804938	−	0.198399i
$401$	−0.568065	−	0.983917i	−0.568065	−	0.983917i	−0.996757	−	0.0804666i	$-0.974359\pi$
$401$	−0.568065	−	0.983917i	−0.568065	−	0.983917i	0.428693	−	0.903450i	$-0.358974\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.205010	+	1.68841i	−0.205010	+	1.68841i
$408$		0			0
$409$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$409$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.166003	+	0.437715i	0.166003	+	0.437715i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$419$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$420$		0			0
$421$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	0.568065	+	0.822984i	$0.307692\pi$
$421$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	−0.970942	+	0.239316i	$0.923077\pi$
$422$	0.0188500	+	0.467757i	0.0188500	+	0.467757i
$423$		0			0
$424$	−0.444044	+	0.148244i	−0.444044	+	0.148244i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.379463	+	0.549748i	0.379463	+	0.549748i
$429$		0			0
$430$		0			0
$431$	−1.49676	+	0.946492i	−1.49676	+	0.946492i	−0.500000	+	0.866025i	$0.666667\pi$
$431$	−1.49676	+	0.946492i	−1.49676	+	0.946492i	−0.996757	+	0.0804666i	$0.974359\pi$
$432$		0			0
$433$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$433$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$434$		0			0
$435$		0			0
$436$	−1.30493	−	1.35858i	−1.30493	−	1.35858i
$437$		0			0
$438$		0			0
$439$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$439$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.13245	−	0.0914204i	−1.13245	−	0.0914204i	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.13245	−	0.0914204i	−1.13245	−	0.0914204i	−0.632445	+	0.774605i	$0.717949\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.234068	−	1.92773i	−0.234068	−	1.92773i	−0.354605	−	0.935016i	$-0.615385\pi$
$449$	−0.234068	−	1.92773i	−0.234068	−	1.92773i	0.120537	−	0.992709i	$-0.461538\pi$
$450$	−0.203753	−	0.128845i	−0.203753	−	0.128845i
$451$		0			0
$452$	0.984472	+	0.419444i	0.984472	+	0.419444i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.221783	+	0.0944927i	−0.221783	+	0.0944927i	−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−0.221783	+	0.0944927i	−0.221783	+	0.0944927i	0.278217	+	0.960518i	$0.410256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$461$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$462$		0			0
$463$	0.645395	+	0.935016i	0.645395	+	0.935016i	1.00000			$0$
$463$	0.645395	+	0.935016i	0.645395	+	0.935016i	−0.354605	+	0.935016i	$0.615385\pi$
$464$	−0.199209	−	0.0160818i	−0.199209	−	0.0160818i
$465$		0			0
$466$	−0.100406	+	0.346643i	−0.100406	+	0.346643i
$467$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$467$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.05826	+	0.0854315i	−1.05826	+	0.0854315i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.00000			1.00000
$478$	−0.136945	−	0.237196i	−0.136945	−	0.237196i
$479$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$479$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.03505	−	0.543237i	−1.03505	−	0.543237i
$485$		0			0
$486$		0			0
$487$	1.74798	+	0.284074i	1.74798	+	0.284074i	0.948536	−	0.316668i	$-0.102564\pi$
$487$	1.74798	+	0.284074i	1.74798	+	0.284074i	0.799443	+	0.600742i	$0.205128\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.13613	+	1.64597i	1.13613	+	1.64597i	0.568065	+	0.822984i	$0.307692\pi$
$491$	1.13613	+	1.64597i	1.13613	+	1.64597i	0.568065	+	0.822984i	$0.307692\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.0670708	+	0.231555i	0.0670708	+	0.231555i	0.987050	−	0.160411i	$-0.0512821\pi$
$499$	0.0670708	+	0.231555i	0.0670708	+	0.231555i	−0.919979	+	0.391967i	$0.871795\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$503$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$504$		0			0
$505$		0			0
$506$	0.194978	+	0.673141i	0.194978	+	0.673141i
$507$		0			0
$508$	1.26701	−	1.31910i	1.26701	−	1.31910i
$509$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$509$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$510$		0			0
$511$		0			0
$512$	−0.333997	−	0.880676i	−0.333997	−	0.880676i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$521$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$522$	−0.0534659	−	0.0227797i	−0.0534659	−	0.0227797i
$523$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$523$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$524$		0			0
$525$		0			0
$526$	−0.205010	−	0.297008i	−0.205010	−	0.297008i
$527$		0			0
$528$		0			0
$529$	−1.38546	+	2.39968i	−1.38546	+	2.39968i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.140180	−	0.686647i	−0.140180	−	0.686647i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.491287	−	0.511484i	−0.491287	−	0.511484i	0.428693	−	0.903450i	$-0.358974\pi$
$541$	−0.491287	−	0.511484i	−0.491287	−	0.511484i	−0.919979	+	0.391967i	$0.871795\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.136945	−	1.12785i	0.136945	−	1.12785i	−0.748511	−	0.663123i	$-0.769231\pi$
$547$	0.136945	−	1.12785i	0.136945	−	1.12785i	0.885456	−	0.464723i	$-0.153846\pi$
$548$	−0.422469	+	0.517431i	−0.422469	+	0.517431i
$549$		0			0
$550$	−0.288513	−	0.216803i	−0.288513	−	0.216803i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.414514	−	0.102169i	−0.414514	−	0.102169i
$555$		0			0
$556$		0			0
$557$	0.0781918	−	1.94031i	0.0781918	−	1.94031i	−0.200026	−	0.979791i	$-0.564103\pi$
$557$	0.0781918	−	1.94031i	0.0781918	−	1.94031i	0.278217	−	0.960518i	$-0.410256\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.162173	−	0.0541412i	−0.162173	−	0.0541412i
$563$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$563$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−0.734068	+	0.385269i	−0.734068	+	0.385269i
$569$	0.759177	−	1.59993i	0.759177	−	1.59993i	−0.0402659	−	0.999189i	$-0.512821\pi$
$569$	0.759177	−	1.59993i	0.759177	−	1.59993i	0.799443	−	0.600742i	$-0.205128\pi$
$570$		0			0
$571$	−0.197315	+	0.681209i	−0.197315	+	0.681209i	0.799443	+	0.600742i	$0.205128\pi$
$571$	−0.197315	+	0.681209i	−0.197315	+	0.681209i	−0.996757	+	0.0804666i	$0.974359\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.234068	−	1.92773i	−0.234068	−	1.92773i
$576$	0.0268974	+	0.667451i	0.0268974	+	0.667451i
$577$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$577$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$578$	0.228667	−	0.0763402i	0.228667	−	0.0763402i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.49217	+	0.120460i	1.49217	+	0.120460i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$587$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.893411	−	0.298264i	0.893411	−	0.298264i
$593$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$593$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$594$		0			0
$595$		0			0
$596$	−0.947528	+	1.37273i	−0.947528	+	1.37273i
$597$		0			0
$598$		0			0
$599$	−0.718540	−	0.880052i	−0.718540	−	0.880052i	0.278217	−	0.960518i	$-0.410256\pi$
$599$	−0.718540	−	0.880052i	−0.718540	−	0.880052i	−0.996757	+	0.0804666i	$0.974359\pi$
$600$		0			0
$601$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$601$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$602$		0			0
$603$	−0.180446	+	1.48611i	−0.180446	+	1.48611i
$604$	0.633616	+	0.211532i	0.633616	+	0.211532i
$605$		0			0
$606$		0			0
$607$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$607$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.354605	−	0.614194i	0.354605	−	0.614194i	−0.632445	−	0.774605i	$-0.717949\pi$
$613$	0.354605	−	0.614194i	0.354605	−	0.614194i	0.987050	+	0.160411i	$0.0512821\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.402877	+	0.583668i	−0.402877	+	0.583668i	−0.970942	−	0.239316i	$-0.923077\pi$
$617$	−0.402877	+	0.583668i	−0.402877	+	0.583668i	0.568065	+	0.822984i	$0.307692\pi$
$618$		0			0
$619$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$619$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.692724	+	0.721202i	0.692724	+	0.721202i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.530851	+	0.470293i	0.530851	+	0.470293i	0.885456	−	0.464723i	$-0.153846\pi$
$631$	0.530851	+	0.470293i	0.530851	+	0.470293i	−0.354605	+	0.935016i	$0.615385\pi$
$632$	−0.355398	+	0.748986i	−0.355398	+	0.748986i
$633$		0			0
$634$	−0.360828	+	0.228174i	−0.360828	+	0.228174i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.0770360	−	0.0404316i	−0.0770360	−	0.0404316i
$639$	1.74798	−	0.284074i	1.74798	−	0.284074i
$640$		0			0
$641$	1.67977	−	0.560791i	1.67977	−	0.560791i	0.692724	−	0.721202i	$-0.256410\pi$
$641$	1.67977	−	0.560791i	1.67977	−	0.560791i	0.987050	+	0.160411i	$0.0512821\pi$
$642$		0			0
$643$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$643$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$647$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$648$	−0.130244	+	0.449654i	−0.130244	+	0.449654i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.591478	−	1.55960i	0.591478	−	1.55960i
$653$	−0.221783	+	0.0944927i	−0.221783	+	0.0944927i	−0.500000	−	0.866025i	$-0.666667\pi$
$653$	−0.221783	+	0.0944927i	−0.221783	+	0.0944927i	0.278217	+	0.960518i	$0.410256\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.94188			−1.94188			−0.970942	−	0.239316i	$-0.923077\pi$
$659$	−1.94188			−1.94188			−0.970942	+	0.239316i	$0.923077\pi$
$660$		0			0
$661$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$661$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$662$	−0.0171903	+	0.426573i	−0.0171903	+	0.426573i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.273891			0.273891
$667$	−0.130244	−	0.449654i	−0.130244	−	0.449654i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	0.120537	+	0.992709i	$0.461538\pi$
$673$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	−0.748511	+	0.663123i	$0.769231\pi$
$674$	0.305022	+	0.192884i	0.305022	+	0.192884i
$675$		0			0
$676$	−0.403779	−	0.850945i	−0.403779	−	0.850945i
$677$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$677$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.0482209	−	0.236201i	−0.0482209	−	0.236201i	0.948536	−	0.316668i	$-0.102564\pi$
$683$	−0.0482209	−	0.236201i	−0.0482209	−	0.236201i	−0.996757	+	0.0804666i	$0.974359\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.293978	+	0.509184i	0.293978	+	0.509184i
$689$		0			0
$690$		0			0
$691$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$691$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$692$		0			0
$693$		0			0
$694$	0.270132	+	0.239316i	0.270132	+	0.239316i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.45352	−	0.358261i	1.45352	−	0.358261i	0.568065	−	0.822984i	$-0.307692\pi$
$701$	1.45352	−	0.358261i	1.45352	−	0.358261i	0.885456	+	0.464723i	$0.153846\pi$
$702$		0			0
$703$		0			0
$704$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.652458	−	0.277987i	0.652458	−	0.277987i	−0.0402659	−	0.999189i	$-0.512821\pi$
$709$	0.652458	−	0.277987i	0.652458	−	0.277987i	0.692724	+	0.721202i	$0.256410\pi$
$710$		0			0
$711$	1.22675	−	1.27719i	1.22675	−	1.27719i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.904439	+	0.571933i	0.904439	+	0.571933i
$717$		0			0
$718$	0.0402586	−	0.0419137i	0.0402586	−	0.0419137i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	0.0290582	−	0.239316i	0.0290582	−	0.239316i
$723$		0			0
$724$		0			0
$725$	0.192724	+	0.144823i	0.192724	+	0.144823i
$726$		0			0
$727$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$727$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$728$		0			0
$729$	0.568065	−	0.822984i	0.568065	−	0.822984i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$733$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$734$		0			0
$735$		0			0
$736$	0.970942	−	0.860179i	0.970942	−	0.860179i
$737$	−0.448272	+	2.19578i	−0.448272	+	2.19578i
$738$		0			0
$739$	−0.718540	−	0.880052i	−0.718540	−	0.880052i	0.278217	−	0.960518i	$-0.410256\pi$
$739$	−0.718540	−	0.880052i	−0.718540	−	0.880052i	−0.996757	+	0.0804666i	$0.974359\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.77091			1.77091			0.885456	−	0.464723i	$-0.153846\pi$
$743$	1.77091			1.77091			0.885456	+	0.464723i	$0.153846\pi$
$744$		0			0
$745$		0			0
$746$	−0.0579279	+	0.00467642i	−0.0579279	+	0.00467642i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.0482209	+	0.236201i	−0.0482209	+	0.236201i	−0.996757	−	0.0804666i	$-0.974359\pi$
$751$	−0.0482209	+	0.236201i	−0.0482209	+	0.236201i	0.948536	+	0.316668i	$0.102564\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.120537	−	0.992709i	$-0.461538\pi$
$757$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.885456	+	0.464723i	$0.153846\pi$
$758$	−0.356219	+	0.0578911i	−0.356219	+	0.0578911i
$759$		0			0
$760$		0			0
$761$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$761$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$762$		0			0
$763$		0			0
$764$	1.82903	+	0.450815i	1.82903	+	0.450815i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$769$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$770$		0			0
$771$		0			0
$772$	0.535051	+	0.926735i	0.535051	+	0.926735i
$773$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$773$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$774$	0.0341987	+	0.167516i	0.0341987	+	0.167516i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.0435007	+	0.358261i	−0.0435007	+	0.358261i
$779$		0			0
$780$		0			0
$781$	2.64250	−	0.213324i	2.64250	−	0.213324i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$787$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$788$	−1.54588	+	0.977553i	−1.54588	+	0.977553i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.248511	+	0.655269i	−0.248511	+	0.655269i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$797$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$798$		0			0
$799$		0			0
$800$	−0.133616	+	0.654493i	−0.133616	+	0.654493i
$801$		0			0
$802$	−0.231490	+	0.146385i	−0.231490	+	0.146385i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.13245	+	0.0914204i	−1.13245	+	0.0914204i	−0.632445	−	0.774605i	$-0.717949\pi$
$809$	−1.13245	+	0.0914204i	−1.13245	+	0.0914204i	−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$811$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$812$		0			0
$813$		0			0
$814$	0.408690	+	0.0329929i	0.408690	+	0.0329929i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.26527	+	0.800107i	1.26527	+	0.800107i	0.987050	−	0.160411i	$-0.0512821\pi$
$821$	1.26527	+	0.800107i	1.26527	+	0.800107i	0.278217	+	0.960518i	$0.410256\pi$
$822$		0			0
$823$	0.652458	+	0.277987i	0.652458	+	0.277987i	0.692724	−	0.721202i	$-0.256410\pi$
$823$	0.652458	+	0.277987i	0.652458	+	0.277987i	−0.0402659	+	0.999189i	$0.512821\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.530851	−	0.470293i	0.530851	−	0.470293i	−0.354605	−	0.935016i	$-0.615385\pi$
$827$	0.530851	−	0.470293i	0.530851	−	0.470293i	0.885456	+	0.464723i	$0.153846\pi$
$828$	−1.68267	+	0.716918i	−1.68267	+	0.716918i
$829$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$829$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$839$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$840$		0			0
$841$	−0.833997	−	0.437715i	−0.833997	−	0.437715i
$842$	0.259795	+	0.0867324i	0.259795	+	0.0867324i
$843$		0			0
$844$	−1.82310	+	0.147176i	−1.82310	+	0.147176i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−0.293978	−	0.775155i	−0.293978	−	0.775155i
$849$		0			0
$850$		0			0
$851$	1.39532	+	1.70896i	1.39532	+	1.70896i
$852$		0			0
$853$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$853$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$854$		0			0
$855$		0			0
$856$	0.265421	−	0.199451i	0.265421	−	0.199451i
$857$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$857$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$858$		0			0
$859$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$859$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$860$		0			0
$861$		0			0
$862$	0.242518	+	0.351348i	0.242518	+	0.351348i
$863$	0.599417	+	0.379048i	0.599417	+	0.379048i	0.799443	−	0.600742i	$-0.205128\pi$
$863$	0.599417	+	0.379048i	0.599417	+	0.379048i	−0.200026	+	0.979791i	$0.564103\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.98437	−	1.75800i	1.98437	−	1.75800i
$870$		0			0
$871$		0			0
$872$	−0.648579	+	0.675242i	−0.648579	+	0.675242i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.316091	+	1.09127i	0.316091	+	1.09127i	0.948536	+	0.316668i	$0.102564\pi$
$877$	0.316091	+	1.09127i	0.316091	+	1.09127i	−0.632445	+	0.774605i	$0.717949\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$881$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$882$		0			0
$883$	0.688601	+	1.81569i	0.688601	+	1.81569i	0.568065	+	0.822984i	$0.307692\pi$
$883$	0.688601	+	1.81569i	0.688601	+	1.81569i	0.120537	+	0.992709i	$0.461538\pi$
$884$		0			0
$885$		0			0
$886$	−0.0110285	+	0.273668i	−0.0110285	+	0.273668i
$887$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$887$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.946784	−	1.15960i	0.946784	−	1.15960i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.462074	+	0.0750944i	−0.462074	+	0.0750944i
$899$		0			0
$900$	0.470942	−	0.815695i	0.470942	−	0.815695i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.188601	−	0.497301i	0.188601	−	0.497301i
$905$		0			0
$906$		0			0
$907$	0.299443	+	1.46677i	0.299443	+	1.46677i	0.799443	+	0.600742i	$0.205128\pi$
$907$	0.299443	+	1.46677i	0.299443	+	1.46677i	−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.688601	+	1.81569i	0.688601	+	1.81569i	0.568065	+	0.822984i	$0.307692\pi$
$911$	0.688601	+	1.81569i	0.688601	+	1.81569i	0.120537	+	0.992709i	$0.461538\pi$
$912$		0			0
$913$		0			0
$914$	0.0249141	+	0.0525053i	0.0249141	+	0.0525053i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.718540	+	0.880052i	−0.718540	+	0.880052i	−0.996757	−	0.0804666i	$-0.974359\pi$
$919$	−0.718540	+	0.880052i	−0.718540	+	0.880052i	0.278217	+	0.960518i	$0.410256\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.10312	−	0.271894i	−1.10312	−	0.271894i
$926$	0.218960	−	0.164538i	0.218960	−	0.164538i
$927$		0			0
$928$	−0.00648424	+	0.160905i	−0.00648424	+	0.160905i
$929$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$929$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$930$		0			0
$931$		0			0
$932$	−1.36905	−	0.337440i	−1.36905	−	0.337440i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$937$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$941$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.0308511	+	0.254082i	0.0308511	+	0.254082i
$947$	−0.0457473	−	1.13521i	−0.0457473	−	1.13521i	−0.845190	−	0.534466i	$-0.820513\pi$
$947$	−0.0457473	−	1.13521i	−0.0457473	−	1.13521i	0.799443	−	0.600742i	$-0.205128\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.709210			−0.709210			−0.354605	−	0.935016i	$-0.615385\pi$
$953$	−0.709210			−0.709210			−0.354605	+	0.935016i	$0.615385\pi$
$954$	−0.00970705	−	0.240878i	−0.00970705	−	0.240878i
$955$		0			0
$956$	0.904439	−	0.571933i	0.904439	−	0.571933i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$962$		0			0
$963$	−0.672711	+	0.224584i	−0.672711	+	0.224584i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.136945	−	0.198399i	0.136945	−	0.198399i	−0.748511	−	0.663123i	$-0.769231\pi$
$967$	0.136945	−	0.198399i	0.136945	−	0.198399i	0.885456	+	0.464723i	$0.153846\pi$
$968$	−0.249067	+	0.524897i	−0.249067	+	0.524897i
$969$		0			0
$970$		0			0
$971$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$971$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$972$		0			0
$973$		0			0
$974$	0.0514595	−	0.423807i	0.0514595	−	0.423807i
$975$		0			0
$976$		0			0
$977$	0.748511	−	1.29646i	0.748511	−	1.29646i	−0.200026	−	0.979791i	$-0.564103\pi$
$977$	0.748511	−	1.29646i	0.748511	−	1.29646i	0.948536	−	0.316668i	$-0.102564\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.77091	−	0.929446i	1.77091	−	0.929446i
$982$	0.385449	−	0.289646i	0.385449	−	0.289646i
$983$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$983$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.871006	+	1.06679i	−0.871006	+	1.06679i
$990$		0			0
$991$	0.946784	+	1.15960i	0.946784	+	1.15960i	0.987050	+	0.160411i	$0.0512821\pi$
$991$	0.946784	+	1.15960i	0.946784	+	1.15960i	−0.0402659	+	0.999189i	$0.512821\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$997$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$998$	0.0551255	−	0.0184036i	0.0551255	−	0.0184036i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2597.1.bf.a.521.1		24
7.2	even	3	inner	2597.1.bf.a.362.1		24
7.3	odd	6		371.1.n.a.97.1	✓	12
7.4	even	3		371.1.n.a.97.1	✓	12
7.5	odd	6	inner	2597.1.bf.a.362.1		24
7.6	odd	2	CM	2597.1.bf.a.521.1		24
21.11	odd	6		3339.1.ck.a.2323.1		12
21.17	even	6		3339.1.ck.a.2323.1		12
53.47	even	13	inner	2597.1.bf.a.2432.1		24
371.47	odd	78	inner	2597.1.bf.a.2273.1		24
371.100	even	39	inner	2597.1.bf.a.2273.1		24
371.153	odd	26	inner	2597.1.bf.a.2432.1		24
371.206	odd	78		371.1.n.a.153.1	yes	12
371.312	even	39		371.1.n.a.153.1	yes	12
1113.206	even	78		3339.1.ck.a.2008.1		12
1113.683	odd	78		3339.1.ck.a.2008.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
371.1.n.a.97.1	✓	12	7.3	odd	6
371.1.n.a.97.1	✓	12	7.4	even	3
371.1.n.a.153.1	yes	12	371.206	odd	78
371.1.n.a.153.1	yes	12	371.312	even	39
2597.1.bf.a.362.1		24	7.2	even	3	inner
2597.1.bf.a.362.1		24	7.5	odd	6	inner
2597.1.bf.a.521.1		24	1.1	even	1	trivial
2597.1.bf.a.521.1		24	7.6	odd	2	CM
2597.1.bf.a.2273.1		24	371.47	odd	78	inner
2597.1.bf.a.2273.1		24	371.100	even	39	inner
2597.1.bf.a.2432.1		24	53.47	even	13	inner
2597.1.bf.a.2432.1		24	371.153	odd	26	inner
3339.1.ck.a.2008.1		12	1113.206	even	78
3339.1.ck.a.2008.1		12	1113.683	odd	78
3339.1.ck.a.2323.1		12	21.11	odd	6
3339.1.ck.a.2323.1		12	21.17	even	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:	\( N \)	\(=\)	\( 2597 = 7^{2} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2597.bf (of order \(78\), degree \(24\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.00970705 - 0.240878i) q^{2} +(0.938829 - 0.0757901i) q^{4} +(-0.0564276 - 0.464723i) q^{8} +(-0.200026 + 0.979791i) q^{9} +O(q^{10})\)	\(q+(-0.00970705 - 0.240878i) q^{2} +(0.938829 - 0.0757901i) q^{4} +(-0.0564276 - 0.464723i) q^{8} +(-0.200026 + 0.979791i) q^{9} +(-0.416498 + 1.43792i) q^{11} +(0.818293 - 0.132986i) q^{16} +(0.237952 + 0.0386709i) q^{18} +(0.350405 + 0.0863671i) q^{22} +(0.970942 + 1.68172i) q^{23} +(-0.919979 - 0.391967i) q^{25} +(-0.234068 - 0.0576926i) q^{29} +(-0.133616 - 0.654493i) q^{32} +(-0.113532 + 0.935016i) q^{36} +(1.12142 - 0.182248i) q^{37} +(0.251489 + 0.663123i) q^{43} +(-0.282040 + 1.38152i) q^{44} +(0.395664 - 0.250203i) q^{46} +(-0.0854858 + 0.225408i) q^{50} +(-0.200026 - 0.979791i) q^{53} +(-0.0116248 + 0.0569419i) q^{58} +(0.648582 - 0.159861i) q^{64} +(1.49217 - 0.120460i) q^{67} +(-0.627974 - 1.65583i) q^{71} +(0.466618 + 0.0376693i) q^{72} +(-0.0547851 - 0.268355i) q^{74} +(-1.49676 - 0.946492i) q^{79} +(-0.919979 - 0.391967i) q^{81} +(0.157290 - 0.0670152i) q^{86} +(0.691735 + 0.112418i) q^{88} +(1.03901 + 1.50526i) q^{92} +(-1.32555 - 0.695701i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{2} + 3 q^{4} + 18 q^{8} + q^{9}+O(q^{10}) \) 24 * q + 2 * q^2 + 3 * q^4 + 18 * q^8 + q^9	\( 24 q + 2 q^{2} + 3 q^{4} + 18 q^{8} + q^{9} + 2 q^{11} + 5 q^{16} + 2 q^{18} - 8 q^{22} + 2 q^{23} + q^{25} - 4 q^{29} + 6 q^{32} - 6 q^{36} + 2 q^{37} + 22 q^{43} - 7 q^{44} + 4 q^{46} - 4 q^{50} + q^{53} + 4 q^{58} + 12 q^{64} + 2 q^{67} - 4 q^{71} + 4 q^{72} - 9 q^{74} - 11 q^{79} + q^{81} + 4 q^{86} - 5 q^{88} - 12 q^{92} - 4 q^{99}+O(q^{100}) \) 24 * q + 2 * q^2 + 3 * q^4 + 18 * q^8 + q^9 + 2 * q^11 + 5 * q^16 + 2 * q^18 - 8 * q^22 + 2 * q^23 + q^25 - 4 * q^29 + 6 * q^32 - 6 * q^36 + 2 * q^37 + 22 * q^43 - 7 * q^44 + 4 * q^46 - 4 * q^50 + q^53 + 4 * q^58 + 12 * q^64 + 2 * q^67 - 4 * q^71 + 4 * q^72 - 9 * q^74 - 11 * q^79 + q^81 + 4 * q^86 - 5 * q^88 - 12 * q^92 - 4 * q^99

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