LMFDB - Embedded newform 3549.1.ea.a.215.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("3549.110");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.77118172983$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{156})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{42} - x^{40} + x^{36} + x^{34} - x^{30} - x^{28} + x^{24} - x^{20} - x^{18} + \cdots + 1 $ x^48 + x^46 - x^42 - x^40 + x^36 + x^34 - x^30 - x^28 + x^24 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{156}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{156} - \cdots)$

Embedding invariants

Embedding label			215.1
Root			$-0.999189 + 0.0402659i$ of defining polynomial
Character	$\chi$	$=$	3549.215
Dual form			3549.1.ea.a.1172.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.992709 - 0.120537i) q^{3} +(-0.999189 - 0.0402659i) q^{4} +(-0.200026 + 0.979791i) q^{7} +(0.970942 - 0.239316i) q^{9} +O(q^{10})$	\(q+(0.992709 - 0.120537i) q^{3} +(-0.999189 - 0.0402659i) q^{4} +(-0.200026 + 0.979791i) q^{7} +(0.970942 - 0.239316i) q^{9} +(-0.996757 + 0.0804666i) q^{12} +(0.866025 + 0.500000i) q^{13} +(0.996757 + 0.0804666i) q^{16} +(-1.41393 + 1.41393i) q^{19} +(-0.0804666 + 0.996757i) q^{21} +(-0.160411 - 0.987050i) q^{25} +(0.935016 - 0.354605i) q^{27} +(0.239316 - 0.970942i) q^{28} +(-0.0598112 - 0.591992i) q^{31} +(-0.979791 + 0.200026i) q^{36} +(0.167722 + 1.66006i) q^{37} +(0.919979 + 0.391967i) q^{39} +(0.314339 + 1.93421i) q^{43} +(0.999189 - 0.0402659i) q^{48} +(-0.919979 - 0.391967i) q^{49} +(-0.845190 - 0.534466i) q^{52} +(-1.23319 + 1.57405i) q^{57} +(0.0534025 - 0.0602790i) q^{61} +(0.0402659 + 0.999189i) q^{63} +(-0.992709 - 0.120537i) q^{64} +(0.542586 + 1.74122i) q^{67} +(0.625186 - 1.13504i) q^{73} +(-0.278217 - 0.960518i) q^{75} +(1.46971 - 1.35585i) q^{76} +(1.67806 - 1.06114i) q^{79} +(0.885456 - 0.464723i) q^{81} +(0.120537 - 0.992709i) q^{84} +(-0.663123 + 0.748511i) q^{91} +(-0.130732 - 0.580467i) q^{93} +(-0.394291 + 0.335386i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 48 * q + 2 * q^7 + 4 * q^9	$ 48 q + 2 q^{7} + 4 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{19} - 2 q^{31} - 2 q^{37} - 2 q^{39} + 6 q^{43} + 2 q^{49} + 2 q^{52} + 2 q^{57} - 2 q^{63} + 4 q^{67} - 4 q^{73} - 2 q^{75} + 4 q^{76} - 4 q^{81} - 4 q^{84} + 2 q^{93} + 2 q^{97}+O(q^{100}) $ 48 * q + 2 * q^7 + 4 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^19 - 2 * q^31 - 2 * q^37 - 2 * q^39 + 6 * q^43 + 2 * q^49 + 2 * q^52 + 2 * q^57 - 2 * q^63 + 4 * q^67 - 4 * q^73 - 2 * q^75 + 4 * q^76 - 4 * q^81 - 4 * q^84 + 2 * q^93 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1184$	$1522$	$3382$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{131}{156}\right)$

Coefficient data

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

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

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

$2$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$2$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$3$	0.992709	−	0.120537i	0.992709	−	0.120537i
$3$	0.992709	−	0.120537i	0.992709	−	0.120537i
$4$	−0.999189	−	0.0402659i	−0.999189	−	0.0402659i
$5$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$5$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$6$		0			0
$7$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$7$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$8$		0			0
$9$	0.970942	−	0.239316i	0.970942	−	0.239316i
$10$		0			0
$11$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$11$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$12$	−0.996757	+	0.0804666i	−0.996757	+	0.0804666i
$13$	0.866025	+	0.500000i	0.866025	+	0.500000i
$13$	0.866025	+	0.500000i	0.866025	+	0.500000i
$14$		0			0
$15$		0			0
$16$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$17$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$17$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$18$		0			0
$19$	−1.41393	+	1.41393i	−1.41393	+	1.41393i	−0.692724	+	0.721202i	$0.743590\pi$
$19$	−1.41393	+	1.41393i	−1.41393	+	1.41393i	−0.721202	+	0.692724i	$0.756410\pi$
$20$		0			0
$21$	−0.0804666	+	0.996757i	−0.0804666	+	0.996757i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−0.160411	−	0.987050i	−0.160411	−	0.987050i
$26$		0			0
$27$	0.935016	−	0.354605i	0.935016	−	0.354605i
$28$	0.239316	−	0.970942i	0.239316	−	0.970942i
$29$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$29$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$30$		0			0
$31$	−0.0598112	−	0.591992i	−0.0598112	−	0.591992i	−0.979791	−	0.200026i	$-0.935897\pi$
$31$	−0.0598112	−	0.591992i	−0.0598112	−	0.591992i	0.919979	−	0.391967i	$-0.128205\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.979791	+	0.200026i	−0.979791	+	0.200026i
$37$	0.167722	+	1.66006i	0.167722	+	1.66006i	0.632445	+	0.774605i	$0.282051\pi$
$37$	0.167722	+	1.66006i	0.167722	+	1.66006i	−0.464723	+	0.885456i	$0.653846\pi$
$38$		0			0
$39$	0.919979	+	0.391967i	0.919979	+	0.391967i
$40$		0			0
$41$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$41$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$42$		0			0
$43$	0.314339	+	1.93421i	0.314339	+	1.93421i	0.354605	+	0.935016i	$0.384615\pi$
$43$	0.314339	+	1.93421i	0.314339	+	1.93421i	−0.0402659	+	0.999189i	$0.512821\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$47$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$48$	0.999189	−	0.0402659i	0.999189	−	0.0402659i
$49$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$50$		0			0
$51$		0			0
$52$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$53$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$53$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.23319	+	1.57405i	−1.23319	+	1.57405i
$58$		0			0
$59$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$59$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$60$		0			0
$61$	0.0534025	−	0.0602790i	0.0534025	−	0.0602790i	−0.721202	−	0.692724i	$-0.756410\pi$
$61$	0.0534025	−	0.0602790i	0.0534025	−	0.0602790i	0.774605	+	0.632445i	$0.217949\pi$
$62$		0			0
$63$	0.0402659	+	0.999189i	0.0402659	+	0.999189i
$64$	−0.992709	−	0.120537i	−0.992709	−	0.120537i
$65$		0			0
$66$		0			0
$67$	0.542586	+	1.74122i	0.542586	+	1.74122i	0.663123	+	0.748511i	$0.269231\pi$
$67$	0.542586	+	1.74122i	0.542586	+	1.74122i	−0.120537	+	0.992709i	$0.538462\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$71$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$72$		0			0
$73$	0.625186	−	1.13504i	0.625186	−	1.13504i	−0.354605	−	0.935016i	$-0.615385\pi$
$73$	0.625186	−	1.13504i	0.625186	−	1.13504i	0.979791	−	0.200026i	$-0.0641026\pi$
$74$		0			0
$75$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$76$	1.46971	−	1.35585i	1.46971	−	1.35585i
$77$		0			0
$78$		0			0
$79$	1.67806	−	1.06114i	1.67806	−	1.06114i	0.774605	−	0.632445i	$-0.217949\pi$
$79$	1.67806	−	1.06114i	1.67806	−	1.06114i	0.903450	−	0.428693i	$-0.141026\pi$
$80$		0			0
$81$	0.885456	−	0.464723i	0.885456	−	0.464723i
$82$		0			0
$83$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$83$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$84$	0.120537	−	0.992709i	0.120537	−	0.992709i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$89$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$90$		0			0
$91$	−0.663123	+	0.748511i	−0.663123	+	0.748511i
$92$		0			0
$93$	−0.130732	−	0.580467i	−0.130732	−	0.580467i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.394291	+	0.335386i	−0.394291	+	0.335386i	−0.822984	−	0.568065i	$-0.807692\pi$
$97$	−0.394291	+	0.335386i	−0.394291	+	0.335386i	0.428693	+	0.903450i	$0.358974\pi$
$98$		0			0
$99$		0			0
$100$	0.120537	+	0.992709i	0.120537	+	0.992709i
$101$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$101$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$102$		0			0
$103$	1.83399	+	0.781391i	1.83399	+	0.781391i	0.948536	+	0.316668i	$0.102564\pi$
$103$	1.83399	+	0.781391i	1.83399	+	0.781391i	0.885456	+	0.464723i	$0.153846\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$107$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$108$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$109$	−0.544766	−	0.392453i	−0.544766	−	0.392453i	0.278217	−	0.960518i	$-0.410256\pi$
$109$	−0.544766	−	0.392453i	−0.544766	−	0.392453i	−0.822984	+	0.568065i	$0.807692\pi$
$110$		0			0
$111$	0.366598	+	1.62774i	0.366598	+	1.62774i
$112$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$113$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$113$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.960518	+	0.278217i	0.960518	+	0.278217i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.464723	−	0.885456i	0.464723	−	0.885456i
$122$		0			0
$123$		0			0
$124$	0.0359256	+	0.593921i	0.0359256	+	0.593921i
$125$		0			0
$126$		0			0
$127$	−0.458243	−	0.724653i	−0.458243	−	0.724653i	0.534466	−	0.845190i	$-0.320513\pi$
$127$	−0.458243	−	0.724653i	−0.458243	−	0.724653i	−0.992709	+	0.120537i	$0.961538\pi$
$128$		0			0
$129$	0.545190	+	1.88221i	0.545190	+	1.88221i
$130$		0			0
$131$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$131$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$132$		0			0
$133$	−1.10253	−	1.66817i	−1.10253	−	1.66817i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$137$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$138$		0			0
$139$	−0.150475	+	1.86397i	−0.150475	+	1.86397i	0.278217	+	0.960518i	$0.410256\pi$
$139$	−0.150475	+	1.86397i	−0.150475	+	1.86397i	−0.428693	+	0.903450i	$0.641026\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.987050	−	0.160411i	0.987050	−	0.160411i
$145$		0			0
$146$		0			0
$147$	−0.960518	−	0.278217i	−0.960518	−	0.278217i
$148$	−0.100742	−	1.66547i	−0.100742	−	1.66547i
$149$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$149$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$150$		0			0
$151$	0.800768	−	0.180348i	0.800768	−	0.180348i	0.200026	−	0.979791i	$-0.435897\pi$
$151$	0.800768	−	0.180348i	0.800768	−	0.180348i	0.600742	+	0.799443i	$0.294872\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.903450	−	0.428693i	−0.903450	−	0.428693i
$157$	−0.965727	−	0.458243i	−0.965727	−	0.458243i	−0.120537	−	0.992709i	$-0.538462\pi$
$157$	−0.965727	−	0.458243i	−0.965727	−	0.458243i	−0.845190	+	0.534466i	$0.820513\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.06605	−	0.479791i	−1.06605	−	0.479791i	−0.200026	−	0.979791i	$-0.564103\pi$
$163$	−1.06605	−	0.479791i	−1.06605	−	0.479791i	−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$167$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$168$		0			0
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$		0			0
$171$	−1.03447	+	1.71122i	−1.03447	+	1.71122i
$172$	−0.236201	−	1.94529i	−0.236201	−	1.94529i
$173$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$173$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$174$		0			0
$175$	0.999189	+	0.0402659i	0.999189	+	0.0402659i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$179$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$180$		0			0
$181$	−1.12142	−	1.62465i	−1.12142	−	1.62465i	−0.692724	−	0.721202i	$-0.743590\pi$
$181$	−1.12142	−	1.62465i	−1.12142	−	1.62465i	−0.428693	−	0.903450i	$-0.641026\pi$
$182$		0			0
$183$	0.0457473	−	0.0662764i	0.0457473	−	0.0662764i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.160411	+	0.987050i	0.160411	+	0.987050i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.00000			−1.00000
$193$	−0.891967	+	0.0539540i	−0.891967	+	0.0539540i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−0.891967	+	0.0539540i	−0.891967	+	0.0539540i	−0.391967	+	0.919979i	$0.628205\pi$
$194$		0			0
$195$		0			0
$196$	0.903450	+	0.428693i	0.903450	+	0.428693i
$197$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$197$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$198$		0			0
$199$	−0.618348	−	1.30314i	−0.618348	−	1.30314i	−0.935016	−	0.354605i	$-0.884615\pi$
$199$	−0.618348	−	1.30314i	−0.618348	−	1.30314i	0.316668	−	0.948536i	$-0.397436\pi$
$200$		0			0
$201$	0.748511	+	1.66312i	0.748511	+	1.66312i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.822984	+	0.568065i	0.822984	+	0.568065i
$209$		0			0
$210$		0			0
$211$	−0.271156	+	0.171469i	−0.271156	+	0.171469i	−0.663123	−	0.748511i	$-0.730769\pi$
$211$	−0.271156	+	0.171469i	−0.271156	+	0.171469i	0.391967	+	0.919979i	$0.371795\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.591992	+	0.0598112i	0.591992	+	0.0598112i
$218$		0			0
$219$	0.483813	−	1.20212i	0.483813	−	1.20212i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.916291	−	1.07722i	0.916291	−	1.07722i	−0.0804666	−	0.996757i	$-0.525641\pi$
$223$	0.916291	−	1.07722i	0.916291	−	1.07722i	0.996757	−	0.0804666i	$-0.0256410\pi$
$224$		0			0
$225$	−0.391967	−	0.919979i	−0.391967	−	0.919979i
$226$		0			0
$227$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$227$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$228$	1.29557	−	1.52312i	1.29557	−	1.52312i
$229$	0.189377	−	1.87439i	0.189377	−	1.87439i	−0.239316	−	0.970942i	$-0.576923\pi$
$229$	0.189377	−	1.87439i	0.189377	−	1.87439i	0.428693	−	0.903450i	$-0.358974\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$233$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.53791	−	1.25567i	1.53791	−	1.25567i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	0.641008	−	1.79863i	0.641008	−	1.79863i	0.0402659	−	0.999189i	$-0.487179\pi$
$241$	0.641008	−	1.79863i	0.641008	−	1.79863i	0.600742	−	0.799443i	$-0.294872\pi$
$242$		0			0
$243$	0.822984	−	0.568065i	0.822984	−	0.568065i
$244$	−0.0557864	+	0.0580798i	−0.0557864	+	0.0580798i
$245$		0			0
$246$		0			0
$247$	−1.93146	+	0.517533i	−1.93146	+	0.517533i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$251$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$252$		−	1.00000i		−	1.00000i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.987050	+	0.160411i	0.987050	+	0.160411i
$257$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$257$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$258$		0			0
$259$	−1.66006	−	0.167722i	−1.66006	−	0.167722i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$263$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.472034	−	1.76166i	−0.472034	−	1.76166i
$269$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$269$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$270$		0			0
$271$	−0.297961	+	0.322984i	−0.297961	+	0.322984i	−0.866025	−	0.500000i	$-0.833333\pi$
$271$	−0.297961	+	0.322984i	−0.297961	+	0.322984i	0.568065	+	0.822984i	$0.307692\pi$
$272$		0			0
$273$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.380472	−	1.13965i	−0.380472	−	1.13965i	−0.948536	−	0.316668i	$-0.897436\pi$
$277$	−0.380472	−	1.13965i	−0.380472	−	1.13965i	0.568065	−	0.822984i	$-0.307692\pi$
$278$		0			0
$279$	−0.199746	−	0.560476i	−0.199746	−	0.560476i
$280$		0			0
$281$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$281$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$282$		0			0
$283$	0.0570677	+	0.150475i	0.0570677	+	0.150475i	0.960518	−	0.278217i	$-0.0897436\pi$
$283$	0.0570677	+	0.150475i	0.0570677	+	0.150475i	−0.903450	+	0.428693i	$0.858974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$290$		0			0
$291$	−0.350990	+	0.380467i	−0.350990	+	0.380467i
$292$	−0.670382	+	1.10895i	−0.670382	+	1.10895i
$293$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$293$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.239316	+	0.970942i	0.239316	+	0.970942i
$301$	−1.95799	−	0.0789044i	−1.95799	−	0.0789044i
$302$		0			0
$303$		0			0
$304$	−1.52312	+	1.29557i	−1.52312	+	1.29557i
$305$		0			0
$306$		0			0
$307$	0.0495602	−	0.110118i	0.0495602	−	0.110118i	−0.885456	−	0.464723i	$-0.846154\pi$
$307$	0.0495602	−	0.110118i	0.0495602	−	0.110118i	0.935016	+	0.354605i	$0.115385\pi$
$308$		0			0
$309$	1.91481	+	0.554631i	1.91481	+	0.554631i
$310$		0			0
$311$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$311$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$312$		0			0
$313$	−1.23850	+	1.01121i	−1.23850	+	1.01121i	−0.239316	+	0.970942i	$0.576923\pi$
$313$	−1.23850	+	1.01121i	−1.23850	+	1.01121i	−0.999189	+	0.0402659i	$0.987179\pi$
$314$		0			0
$315$		0			0
$316$	−1.71942	+	0.992709i	−1.71942	+	0.992709i
$317$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$317$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.903450	+	0.428693i	−0.903450	+	0.428693i
$325$	0.354605	−	0.935016i	0.354605	−	0.935016i
$326$		0			0
$327$	−0.588099	−	0.323928i	−0.588099	−	0.323928i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.200677	−	0.0121387i	−0.200677	−	0.0121387i	−0.0402659	−	0.999189i	$-0.512821\pi$
$331$	−0.200677	−	0.0121387i	−0.200677	−	0.0121387i	−0.160411	+	0.987050i	$0.551282\pi$
$332$		0			0
$333$	0.560127	+	1.57168i	0.560127	+	1.57168i
$334$		0			0
$335$		0			0
$336$	−0.160411	+	0.987050i	−0.160411	+	0.987050i
$337$			1.89707i			1.89707i	0.316668	+	0.948536i	$0.397436\pi$
$337$			1.89707i			1.89707i	−0.316668	+	0.948536i	$0.602564\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.568065	−	0.822984i	0.568065	−	0.822984i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$347$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$348$		0			0
$349$	−1.93146	+	0.0389018i	−1.93146	+	0.0389018i	−0.970942	−	0.239316i	$-0.923077\pi$
$349$	−1.93146	+	0.0389018i	−1.93146	+	0.0389018i	−0.960518	+	0.278217i	$0.910256\pi$
$350$		0			0
$351$	0.987050	+	0.160411i	0.987050	+	0.160411i
$352$		0			0
$353$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$353$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$359$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$360$		0			0
$361$		−	2.99838i		−	2.99838i
$362$		0			0
$363$	0.354605	−	0.935016i	0.354605	−	0.935016i
$364$	0.692724	−	0.721202i	0.692724	−	0.721202i
$365$		0			0
$366$		0			0
$367$	−0.187607	−	0.761154i	−0.187607	−	0.761154i	−0.987050	−	0.160411i	$-0.948718\pi$
$367$	−0.187607	−	0.761154i	−0.187607	−	0.761154i	0.799443	−	0.600742i	$-0.205128\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.107253	+	0.585260i	0.107253	+	0.585260i
$373$	0.902438	+	0.222431i	0.902438	+	0.222431i	0.663123	−	0.748511i	$-0.269231\pi$
$373$	0.902438	+	0.222431i	0.902438	+	0.222431i	0.239316	+	0.970942i	$0.423077\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.386308	−	0.959854i	−0.386308	−	0.959854i	−0.987050	−	0.160411i	$-0.948718\pi$
$379$	−0.386308	−	0.959854i	−0.386308	−	0.959854i	0.600742	−	0.799443i	$-0.294872\pi$
$380$		0			0
$381$	−0.542249	−	0.664135i	−0.542249	−	0.664135i
$382$		0			0
$383$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$383$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.768090	+	1.80277i	0.768090	+	1.80277i
$388$	0.407476	−	0.319237i	0.407476	−	0.319237i
$389$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$389$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.264977	+	1.44593i	−0.264977	+	1.44593i	0.534466	+	0.845190i	$0.320513\pi$
$397$	−0.264977	+	1.44593i	−0.264977	+	1.44593i	−0.799443	+	0.600742i	$0.794872\pi$
$398$		0			0
$399$	−1.29557	−	1.52312i	−1.29557	−	1.52312i
$400$	−0.0804666	−	0.996757i	−0.0804666	−	0.996757i
$401$		0			0		0.761712	−	0.647915i	$-0.224359\pi$
$401$		0			0		−0.761712	+	0.647915i	$0.775641\pi$
$402$		0			0
$403$	0.244198	−	0.542586i	0.244198	−	0.542586i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.62515	−	0.654068i	1.62515	−	0.654068i	0.632445	−	0.774605i	$-0.282051\pi$
$409$	1.62515	−	0.654068i	1.62515	−	0.654068i	0.992709	+	0.120537i	$0.0384615\pi$
$410$		0			0
$411$		0			0
$412$	−1.80104	−	0.854605i	−1.80104	−	0.854605i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.0752986	+	1.86852i	0.0752986	+	1.86852i
$418$		0			0
$419$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$419$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$420$		0			0
$421$	0.283788	−	0.222333i	0.283788	−	0.222333i	−0.464723	−	0.885456i	$-0.653846\pi$
$421$	0.283788	−	0.222333i	0.283788	−	0.222333i	0.748511	+	0.663123i	$0.230769\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.0483789	+	0.0643806i	0.0483789	+	0.0643806i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$431$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$432$	0.960518	−	0.278217i	0.960518	−	0.278217i
$433$	0.205186	−	0.432420i	0.205186	−	0.432420i	−0.774605	−	0.632445i	$-0.782051\pi$
$433$	0.205186	−	0.432420i	0.205186	−	0.432420i	0.979791	+	0.200026i	$0.0641026\pi$
$434$		0			0
$435$		0			0
$436$	0.528522	+	0.414071i	0.528522	+	0.414071i
$437$		0			0
$438$		0			0
$439$	−1.36751	+	0.222242i	−1.36751	+	0.222242i	−0.799443	−	0.600742i	$-0.794872\pi$
$439$	−1.36751	+	0.222242i	−1.36751	+	0.222242i	−0.568065	+	0.822984i	$0.692308\pi$
$440$		0			0
$441$	−0.987050	−	0.160411i	−0.987050	−	0.160411i
$442$		0			0
$443$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$443$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$444$	−0.300758	−	1.64118i	−0.300758	−	1.64118i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.316668	−	0.948536i	0.316668	−	0.948536i
$449$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$449$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.773191	−	0.275555i	0.773191	−	0.275555i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.82498	+	0.734492i	1.82498	+	0.734492i	0.979791	+	0.200026i	$0.0641026\pi$
$457$	1.82498	+	0.734492i	1.82498	+	0.734492i	0.845190	+	0.534466i	$0.179487\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$461$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$462$		0			0
$463$	0.499189	−	0.825759i	0.499189	−	0.825759i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	0.499189	−	0.825759i	0.499189	−	0.825759i	0.999189	+	0.0402659i	$0.0128205\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$467$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$468$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$469$	−1.81456	+	0.183332i	−1.81456	+	0.183332i
$470$		0			0
$471$	−1.01392	−	0.338496i	−1.01392	−	0.338496i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.62243	+	1.16881i	1.62243	+	1.16881i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$479$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$480$		0			0
$481$	−0.684779	+	1.52152i	−0.684779	+	1.52152i
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−1.90345	−	0.428693i	−1.90345	−	0.428693i	−0.903450	−	0.428693i	$-0.858974\pi$
$487$	−1.90345	−	0.428693i	−1.90345	−	0.428693i	−1.00000			$\pi$
$488$		0			0
$489$	−1.11611	−	0.347794i	−1.11611	−	0.347794i
$490$		0			0
$491$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$491$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.0119817	−	0.594885i	−0.0119817	−	0.594885i
$497$		0			0
$498$		0			0
$499$	0.839981	−	1.27093i	0.839981	−	1.27093i	−0.120537	−	0.992709i	$-0.538462\pi$
$499$	0.839981	−	1.27093i	0.839981	−	1.27093i	0.960518	−	0.278217i	$-0.0897436\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$503$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.600742	+	0.799443i	0.600742	+	0.799443i
$508$	0.428693	+	0.742517i	0.428693	+	0.742517i
$509$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$509$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$510$		0			0
$511$	0.987050	+	0.839589i	0.987050	+	0.839589i
$512$		0			0
$513$	−0.820659	+	1.82343i	−0.820659	+	1.82343i
$514$		0			0
$515$		0			0
$516$	−0.468959	−	1.90264i	−0.468959	−	1.90264i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$521$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$522$		0			0
$523$	0.154579	−	1.91481i	0.154579	−	1.91481i	−0.200026	−	0.979791i	$-0.564103\pi$
$523$	0.154579	−	1.91481i	0.154579	−	1.91481i	0.354605	−	0.935016i	$-0.384615\pi$
$524$		0			0
$525$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	1.03447	+	1.71122i	1.03447	+	1.71122i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.87439	+	0.668008i	1.87439	+	0.668008i	0.970942	+	0.239316i	$0.0769231\pi$
$541$	1.87439	+	0.668008i	1.87439	+	0.668008i	0.903450	+	0.428693i	$0.141026\pi$
$542$		0			0
$543$	−1.30907	−	1.47764i	−1.30907	−	1.47764i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.64126	+	0.404534i	1.64126	+	0.404534i	0.948536	−	0.316668i	$-0.102564\pi$
$547$	1.64126	+	0.404534i	1.64126	+	0.404534i	0.692724	+	0.721202i	$0.256410\pi$
$548$		0			0
$549$	0.0374250	−	0.0713074i	0.0374250	−	0.0713074i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.704039	+	1.85640i	0.704039	+	1.85640i
$554$		0			0
$555$		0			0
$556$	0.225408	−	1.85640i	0.225408	−	1.85640i
$557$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$557$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$558$		0			0
$559$	−0.694877	+	1.83224i	−0.694877	+	1.83224i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		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
$567$	0.278217	+	0.960518i	0.278217	+	0.960518i
$568$		0			0
$569$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$569$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$570$		0			0
$571$	0.721202	+	1.69272i	0.721202	+	1.69272i	0.721202	+	0.692724i	$0.243590\pi$
$571$	0.721202	+	1.69272i	0.721202	+	1.69272i			1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.992709	+	0.120537i	−0.992709	+	0.120537i
$577$	−1.56746	−	0.420000i	−1.56746	−	0.420000i	−0.632445	−	0.774605i	$-0.717949\pi$
$577$	−1.56746	−	0.420000i	−1.56746	−	0.420000i	−0.935016	+	0.354605i	$0.884615\pi$
$578$		0			0
$579$	−0.878960	+	0.161075i	−0.878960	+	0.161075i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$587$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$588$	0.948536	+	0.316668i	0.948536	+	0.316668i
$589$	0.921602	+	0.752465i	0.921602	+	0.752465i
$590$		0			0
$591$		0			0
$592$	0.0335989	+	1.66817i	0.0335989	+	1.66817i
$593$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$593$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.770916	−	1.21911i	−0.770916	−	1.21911i
$598$		0			0
$599$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$599$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$600$		0			0
$601$	−0.288518	−	0.277125i	−0.288518	−	0.277125i	0.534466	−	0.845190i	$-0.320513\pi$
$601$	−0.288518	−	0.277125i	−0.288518	−	0.277125i	−0.822984	+	0.568065i	$0.807692\pi$
$602$		0			0
$603$	0.943521	+	1.56077i	0.943521	+	1.56077i
$604$	−0.807380	+	0.147958i	−0.807380	+	0.147958i
$605$		0			0
$606$		0			0
$607$	0.628718	−	0.0763402i	0.628718	−	0.0763402i	0.200026	−	0.979791i	$-0.435897\pi$
$607$	0.628718	−	0.0763402i	0.628718	−	0.0763402i	0.428693	+	0.903450i	$0.358974\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.75996	−	0.322525i	1.75996	−	0.322525i	0.799443	−	0.600742i	$-0.205128\pi$
$613$	1.75996	−	0.322525i	1.75996	−	0.322525i	0.960518	+	0.278217i	$0.0897436\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$617$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$618$		0			0
$619$	−1.09182	−	0.154940i	−1.09182	−	0.154940i	−0.428693	−	0.903450i	$-0.641026\pi$
$619$	−1.09182	−	0.154940i	−1.09182	−	0.154940i	−0.663123	+	0.748511i	$0.730769\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.885456	+	0.464723i	0.885456	+	0.464723i
$625$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$626$		0			0
$627$		0			0
$628$	0.946492	+	0.496757i	0.946492	+	0.496757i
$629$		0			0
$630$		0			0
$631$	−0.608033	−	0.919979i	−0.608033	−	0.919979i	0.391967	−	0.919979i	$-0.371795\pi$
$631$	−0.608033	−	0.919979i	−0.608033	−	0.919979i	−1.00000			$\pi$
$632$		0			0
$633$	−0.248511	+	0.202903i	−0.248511	+	0.202903i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.600742	−	0.799443i	−0.600742	−	0.799443i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$641$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$642$		0			0
$643$	−0.668044	+	1.65988i	−0.668044	+	1.65988i	0.0804666	+	0.996757i	$0.474359\pi$
$643$	−0.668044	+	1.65988i	−0.668044	+	1.65988i	−0.748511	+	0.663123i	$0.769231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$647$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	0.594885	−	0.0119817i	0.594885	−	0.0119817i
$652$	1.04587	+	0.522327i	1.04587	+	0.522327i
$653$		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
$657$	0.335386	−	1.25168i	0.335386	−	1.25168i
$658$		0			0
$659$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$659$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$660$		0			0
$661$	−1.70844	−	0.103342i	−1.70844	−	0.103342i	−0.822984	−	0.568065i	$-0.807692\pi$
$661$	−1.70844	−	0.103342i	−1.70844	−	0.103342i	−0.885456	+	0.464723i	$0.846154\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.779765	−	1.17982i	0.779765	−	1.17982i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.752982	−	1.76731i	0.752982	−	1.76731i	0.120537	−	0.992709i	$-0.461538\pi$
$673$	0.752982	−	1.76731i	0.752982	−	1.76731i	0.632445	−	0.774605i	$-0.282051\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	−0.464723	−	0.885456i	−0.464723	−	0.885456i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−0.249739	−	0.453409i	−0.249739	−	0.453409i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$683$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$684$	1.10253	−	1.66817i	1.10253	−	1.66817i
$685$		0			0
$686$		0			0
$687$	−0.0379369	−	1.88355i	−0.0379369	−	1.88355i
$688$	0.157681	+	1.95323i	0.157681	+	1.95323i
$689$		0			0
$690$		0			0
$691$	1.21323	−	0.605913i	1.21323	−	0.605913i	0.278217	−	0.960518i	$-0.410256\pi$
$691$	1.21323	−	0.605913i	1.21323	−	0.605913i	0.935016	+	0.354605i	$0.115385\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.996757	−	0.0804666i	−0.996757	−	0.0804666i
$701$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$701$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$702$		0			0
$703$	−2.58435	−	2.11006i	−2.58435	−	2.11006i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.00078	+	1.27741i	1.00078	+	1.27741i	0.960518	+	0.278217i	$0.0897436\pi$
$709$	1.00078	+	1.27741i	1.00078	+	1.27741i	0.0402659	+	0.999189i	$0.487179\pi$
$710$		0			0
$711$	1.37535	−	1.43189i	1.37535	−	1.43189i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$719$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$720$		0			0
$721$	−1.13245	+	1.64063i	−1.13245	+	1.64063i
$722$		0			0
$723$	0.419533	−	1.86278i	0.419533	−	1.86278i
$724$	1.05509	+	1.66849i	1.05509	+	1.66849i
$725$		0			0
$726$		0			0
$727$	−0.992709	−	0.879463i	−0.992709	−	0.879463i		−	1.00000i	$-0.5\pi$
$727$	−0.992709	−	0.879463i	−0.992709	−	0.879463i	−0.992709	+	0.120537i	$0.961538\pi$
$728$		0			0
$729$	0.748511	−	0.663123i	0.748511	−	0.663123i
$730$		0			0
$731$		0			0
$732$	−0.0483789	+	0.0643806i	−0.0483789	+	0.0643806i
$733$	−1.85199	+	0.745361i	−1.85199	+	0.745361i	−0.903450	+	0.428693i	$0.858974\pi$
$733$	−1.85199	+	0.745361i	−1.85199	+	0.745361i	−0.948536	+	0.316668i	$0.897436\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.244448	+	1.33391i	0.244448	+	1.33391i	0.845190	+	0.534466i	$0.179487\pi$
$739$	0.244448	+	1.33391i	0.244448	+	1.33391i	−0.600742	+	0.799443i	$0.705128\pi$
$740$		0			0
$741$	−1.85500	+	0.746571i	−1.85500	+	0.746571i
$742$		0			0
$743$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$743$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.73556	−	0.502711i	1.73556	−	0.502711i	0.748511	−	0.663123i	$-0.230769\pi$
$751$	1.73556	−	0.502711i	1.73556	−	0.502711i	0.987050	+	0.160411i	$0.0512821\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$757$	0.0534025	−	1.32517i	0.0534025	−	1.32517i	−0.721202	−	0.692724i	$-0.756410\pi$
$757$	0.0534025	−	1.32517i	0.0534025	−	1.32517i	0.774605	−	0.632445i	$-0.217949\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$761$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$762$		0			0
$763$	0.493489	−	0.455256i	0.493489	−	0.455256i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.999189	+	0.0402659i	0.999189	+	0.0402659i
$769$	1.34925	+	1.46257i	1.34925	+	1.46257i	0.748511	+	0.663123i	$0.230769\pi$
$769$	1.34925	+	1.46257i	1.34925	+	1.46257i	0.600742	+	0.799443i	$0.294872\pi$
$770$		0			0
$771$		0			0
$772$	0.893416	−	0.0179944i	0.893416	−	0.0179944i
$773$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$773$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$774$		0			0
$775$	−0.574732	+	0.153999i	−0.574732	+	0.153999i
$776$		0			0
$777$	−1.66817	+	0.0335989i	−1.66817	+	0.0335989i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$785$		0			0
$786$		0			0
$787$	1.11973	−	1.03297i	1.11973	−	1.03297i	0.120537	−	0.992709i	$-0.461538\pi$
$787$	1.11973	−	1.03297i	1.11973	−	1.03297i	0.999189	−	0.0402659i	$-0.0128205\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0763874	−	0.0255019i	0.0763874	−	0.0255019i
$794$		0			0
$795$		0			0
$796$	0.565375	+	1.32698i	0.565375	+	1.32698i
$797$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$797$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.680937	−	1.69191i	−0.680937	−	1.69191i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$809$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$810$		0			0
$811$	0.00725961	+	0.0396144i	0.00725961	+	0.0396144i	0.987050	−	0.160411i	$-0.0512821\pi$
$811$	0.00725961	+	0.0396144i	0.00725961	+	0.0396144i	−0.979791	+	0.200026i	$0.935897\pi$
$812$		0			0
$813$	−0.256857	+	0.356544i	−0.256857	+	0.356544i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3.17928	−	2.29037i	−3.17928	−	2.29037i
$818$		0			0
$819$	−0.464723	+	0.885456i	−0.464723	+	0.885456i
$820$		0			0
$821$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$821$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$822$		0			0
$823$	0.414507	+	0.239316i	0.414507	+	0.239316i	0.692724	−	0.721202i	$-0.256410\pi$
$823$	0.414507	+	0.239316i	0.414507	+	0.239316i	−0.278217	+	0.960518i	$0.589744\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$827$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$828$		0			0
$829$	1.02644	−	1.48705i	1.02644	−	1.48705i	0.160411	−	0.987050i	$-0.448718\pi$
$829$	1.02644	−	1.48705i	1.02644	−	1.48705i	0.866025	−	0.500000i	$-0.166667\pi$
$830$		0			0
$831$	−0.515067	−	1.08548i	−0.515067	−	1.08548i
$832$	−0.799443	−	0.600742i	−0.799443	−	0.600742i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.265848	−	0.532313i	−0.265848	−	0.532313i
$838$		0			0
$839$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$839$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$840$		0			0
$841$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$842$		0			0
$843$		0			0
$844$	0.277840	−	0.160411i	0.277840	−	0.160411i
$845$		0			0
$846$		0			0
$847$	0.774605	+	0.632445i	0.774605	+	0.632445i
$848$		0			0
$849$	0.0747894	+	0.142499i	0.0747894	+	0.142499i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.359852	+	0.0217671i	0.359852	+	0.0217671i	0.239316	−	0.970942i	$-0.423077\pi$
$853$	0.359852	+	0.0217671i	0.359852	+	0.0217671i	0.120537	+	0.992709i	$0.461538\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$857$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$858$		0			0
$859$	1.66231	−	0.788777i	1.66231	−	0.788777i	0.663123	−	0.748511i	$-0.269231\pi$
$859$	1.66231	−	0.788777i	1.66231	−	0.788777i	0.999189	−	0.0402659i	$-0.0128205\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$863$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$868$	−0.589104	−	0.0835998i	−0.589104	−	0.0835998i
$869$		0			0
$870$		0			0
$871$	−0.400717	+	1.77923i	−0.400717	+	1.77923i
$872$		0			0
$873$	−0.302571	+	0.420000i	−0.302571	+	0.420000i
$874$		0			0
$875$		0			0
$876$	−0.531826	+	1.18167i	−0.531826	+	1.18167i
$877$	1.82047	−	0.819328i	1.82047	−	0.819328i	0.885456	−	0.464723i	$-0.153846\pi$
$877$	1.82047	−	0.819328i	1.82047	−	0.819328i	0.935016	−	0.354605i	$-0.115385\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$881$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$882$		0			0
$883$	−0.0957386	−	0.388427i	−0.0957386	−	0.388427i	0.903450	−	0.428693i	$-0.141026\pi$
$883$	−0.0957386	−	0.388427i	−0.0957386	−	0.388427i	−0.999189	+	0.0402659i	$0.987179\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$887$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$888$		0			0
$889$	0.801669	−	0.304033i	0.801669	−	0.304033i
$890$		0			0
$891$		0			0
$892$	−0.958923	+	1.03945i	−0.958923	+	1.03945i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.354605	+	0.935016i	0.354605	+	0.935016i
$901$		0			0
$902$		0			0
$903$	−1.95323	+	0.157681i	−1.95323	+	0.157681i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.04098	+	0.718540i	−1.04098	+	0.718540i	−0.960518	−	0.278217i	$-0.910256\pi$
$907$	−1.04098	+	0.718540i	−1.04098	+	0.718540i	−0.0804666	+	0.996757i	$0.525641\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$911$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$912$	−1.35585	+	1.46971i	−1.35585	+	1.46971i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.264697	+	1.86525i	−0.264697	+	1.86525i
$917$		0			0
$918$		0			0
$919$	−0.880052	−	1.27497i	−0.880052	−	1.27497i	−0.960518	−	0.278217i	$-0.910256\pi$
$919$	−0.880052	−	1.27497i	−0.880052	−	1.27497i	0.0804666	−	0.996757i	$-0.474359\pi$
$920$		0			0
$921$	0.0359256	−	0.115289i	0.0359256	−	0.115289i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.61166	−	0.431843i	1.61166	−	0.431843i
$926$		0			0
$927$	1.96770	+	0.319782i	1.96770	+	0.319782i
$928$		0			0
$929$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$929$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$930$		0			0
$931$	1.85500	−	0.746571i	1.85500	−	0.746571i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.258588	+	0.492699i	0.258588	+	0.492699i	0.979791	−	0.200026i	$-0.0641026\pi$
$937$	0.258588	+	0.492699i	0.258588	+	0.492699i	−0.721202	+	0.692724i	$0.756410\pi$
$938$		0			0
$939$	−1.10759	+	1.15312i	−1.10759	+	1.15312i
$940$		0			0
$941$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$941$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.735006	−	0.678061i	$-0.762821\pi$
$947$		0			0		0.735006	+	0.678061i	$0.237179\pi$
$948$	−1.58723	+	1.19272i	−1.58723	+	1.19272i
$949$	1.10895	−	0.670382i	1.10895	−	0.670382i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$953$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.632913	−	0.129210i	0.632913	−	0.129210i
$962$		0			0
$963$		0			0
$964$	−0.712912	+	1.77136i	−0.712912	+	1.77136i
$965$		0			0
$966$		0			0
$967$	0.437333	+	1.40345i	0.437333	+	1.40345i	0.866025	+	0.500000i	$0.166667\pi$
$967$	0.437333	+	1.40345i	0.437333	+	1.40345i	−0.428693	+	0.903450i	$0.641026\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$971$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$972$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$973$	−1.79620	−	0.520276i	−1.79620	−	0.520276i
$974$		0			0
$975$	0.239316	−	0.970942i	0.239316	−	0.970942i
$976$	0.0580798	−	0.0557864i	0.0580798	−	0.0557864i
$977$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$977$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.622857	−	0.250678i	−0.622857	−	0.250678i
$982$		0			0
$983$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$983$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.95073	−	0.439341i	1.95073	−	0.439341i
$989$		0			0
$990$		0			0
$991$	−1.38545			−1.38545			−0.692724	−	0.721202i	$-0.743590\pi$
$991$	−1.38545			−1.38545			−0.692724	+	0.721202i	$0.743590\pi$
$992$		0			0
$993$	−0.200677	+	0.0121387i	−0.200677	+	0.0121387i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.51790	+	0.309882i	−1.51790	+	0.309882i	−0.885456	−	0.464723i	$-0.846154\pi$
$997$	−1.51790	+	0.309882i	−1.51790	+	0.309882i	−0.632445	+	0.774605i	$0.717949\pi$
$998$		0			0
$999$	0.745489	+	1.49271i	0.745489	+	1.49271i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.1.ea.a.215.1	✓	48
3.2	odd	2	CM	3549.1.ea.a.215.1	✓	48
7.3	odd	6		3549.1.ep.a.2243.1	yes	48
21.17	even	6		3549.1.ep.a.2243.1	yes	48
169.158	odd	156		3549.1.ep.a.2693.1	yes	48
507.158	even	156		3549.1.ep.a.2693.1	yes	48
1183.1172	even	156	inner	3549.1.ea.a.1172.1	yes	48
3549.1172	odd	156	inner	3549.1.ea.a.1172.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3549.1.ea.a.215.1	✓	48	1.1	even	1	trivial
3549.1.ea.a.215.1	✓	48	3.2	odd	2	CM
3549.1.ea.a.1172.1	yes	48	1183.1172	even	156	inner
3549.1.ea.a.1172.1	yes	48	3549.1172	odd	156	inner
3549.1.ep.a.2243.1	yes	48	7.3	odd	6
3549.1.ep.a.2243.1	yes	48	21.17	even	6
3549.1.ep.a.2693.1	yes	48	169.158	odd	156
3549.1.ep.a.2693.1	yes	48	507.158	even	156

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.992709 - 0.120537i) q^{3} +(-0.999189 - 0.0402659i) q^{4} +(-0.200026 + 0.979791i) q^{7} +(0.970942 - 0.239316i) q^{9} +O(q^{10})\)	\(q+(0.992709 - 0.120537i) q^{3} +(-0.999189 - 0.0402659i) q^{4} +(-0.200026 + 0.979791i) q^{7} +(0.970942 - 0.239316i) q^{9} +(-0.996757 + 0.0804666i) q^{12} +(0.866025 + 0.500000i) q^{13} +(0.996757 + 0.0804666i) q^{16} +(-1.41393 + 1.41393i) q^{19} +(-0.0804666 + 0.996757i) q^{21} +(-0.160411 - 0.987050i) q^{25} +(0.935016 - 0.354605i) q^{27} +(0.239316 - 0.970942i) q^{28} +(-0.0598112 - 0.591992i) q^{31} +(-0.979791 + 0.200026i) q^{36} +(0.167722 + 1.66006i) q^{37} +(0.919979 + 0.391967i) q^{39} +(0.314339 + 1.93421i) q^{43} +(0.999189 - 0.0402659i) q^{48} +(-0.919979 - 0.391967i) q^{49} +(-0.845190 - 0.534466i) q^{52} +(-1.23319 + 1.57405i) q^{57} +(0.0534025 - 0.0602790i) q^{61} +(0.0402659 + 0.999189i) q^{63} +(-0.992709 - 0.120537i) q^{64} +(0.542586 + 1.74122i) q^{67} +(0.625186 - 1.13504i) q^{73} +(-0.278217 - 0.960518i) q^{75} +(1.46971 - 1.35585i) q^{76} +(1.67806 - 1.06114i) q^{79} +(0.885456 - 0.464723i) q^{81} +(0.120537 - 0.992709i) q^{84} +(-0.663123 + 0.748511i) q^{91} +(-0.130732 - 0.580467i) q^{93} +(-0.394291 + 0.335386i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 48 * q + 2 * q^7 + 4 * q^9	\( 48 q + 2 q^{7} + 4 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{19} - 2 q^{31} - 2 q^{37} - 2 q^{39} + 6 q^{43} + 2 q^{49} + 2 q^{52} + 2 q^{57} - 2 q^{63} + 4 q^{67} - 4 q^{73} - 2 q^{75} + 4 q^{76} - 4 q^{81} - 4 q^{84} + 2 q^{93} + 2 q^{97}+O(q^{100}) \) 48 * q + 2 * q^7 + 4 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^19 - 2 * q^31 - 2 * q^37 - 2 * q^39 + 6 * q^43 + 2 * q^49 + 2 * q^52 + 2 * q^57 - 2 * q^63 + 4 * q^67 - 4 * q^73 - 2 * q^75 + 4 * q^76 - 4 * q^81 - 4 * q^84 + 2 * q^93 + 2 * q^97

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