LMFDB - Embedded newform 3380.1.cs.a.1047.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(7,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 39, 107]))

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

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

chi := DirichletCharacter("3380.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3380 = 2^{2} \cdot 5 \cdot 13^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3380.cs (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:	$1.68683974270$
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:	$\Z[a_1, a_2]$
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			1047.1
Root			$-0.316668 + 0.948536i$ of defining polynomial
Character	$\chi$	$=$	3380.1047
Dual form			3380.1.cs.a.3083.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.600742 - 0.799443i) q^{2} +(-0.278217 - 0.960518i) q^{4} +(-0.632445 + 0.774605i) q^{5} +(-0.935016 - 0.354605i) q^{8} +(0.391967 + 0.919979i) q^{9} +(0.239316 + 0.970942i) q^{10} +(-0.987050 - 0.160411i) q^{13} +(-0.845190 + 0.534466i) q^{16} +(-1.02399 + 1.42140i) q^{17} +(0.970942 + 0.239316i) q^{18} +(0.919979 + 0.391967i) q^{20} +(-0.200026 - 0.979791i) q^{25} +(-0.721202 + 0.692724i) q^{26} +(-1.13965 + 1.51660i) q^{29} +(-0.0804666 + 0.996757i) q^{32} +(0.521177 + 1.67252i) q^{34} +(0.774605 - 0.632445i) q^{36} +(1.99190 - 0.160803i) q^{37} +(0.866025 - 0.500000i) q^{40} +(-1.09182 + 1.65196i) q^{41} +(-0.960518 - 0.278217i) q^{45} +(0.948536 - 0.316668i) q^{49} +(-0.903450 - 0.428693i) q^{50} +(0.120537 + 0.992709i) q^{52} +(-0.666000 - 1.47979i) q^{53} +(0.527799 + 1.82217i) q^{58} +(-1.23933 + 1.51790i) q^{61} +(0.748511 + 0.663123i) q^{64} +(0.748511 - 0.663123i) q^{65} +(1.65017 + 0.588099i) q^{68} +(-0.0402659 - 0.999189i) q^{72} +(-0.992709 + 0.120537i) q^{73} +(1.06806 - 1.68901i) q^{74} +(0.120537 - 0.992709i) q^{80} +(-0.692724 + 0.721202i) q^{81} +(0.664749 + 1.86525i) q^{82} +(-0.453409 - 1.69214i) q^{85} +(0.574732 - 0.153999i) q^{89} +(-0.799443 + 0.600742i) q^{90} +(1.64463 - 0.0662764i) q^{97} +(0.316668 - 0.948536i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74}+ \cdots - 2 q^{90}+O(q^{100})$ 48 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^18 - 2 * q^20 + 2 * q^25 + 2 * q^34 - 2 * q^41 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 + 20 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$ .

$n$	$677$	$1691$	$1861$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{71}{156}\right)$

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

)

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

$2$	0.600742	−	0.799443i	0.600742	−	0.799443i
$2$	0.600742	−	0.799443i	0.600742	−	0.799443i
$3$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$3$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$4$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$5$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$5$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$6$		0			0
$7$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$7$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$8$	−0.935016	−	0.354605i	−0.935016	−	0.354605i
$9$	0.391967	+	0.919979i	0.391967	+	0.919979i
$10$	0.239316	+	0.970942i	0.239316	+	0.970942i
$11$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$11$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$12$		0			0
$13$	−0.987050	−	0.160411i	−0.987050	−	0.160411i
$13$	−0.987050	−	0.160411i	−0.987050	−	0.160411i
$14$		0			0
$15$		0			0
$16$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$17$	−1.02399	+	1.42140i	−1.02399	+	1.42140i	−0.120537	+	0.992709i	$0.538462\pi$
$17$	−1.02399	+	1.42140i	−1.02399	+	1.42140i	−0.903450	+	0.428693i	$0.858974\pi$
$18$	0.970942	+	0.239316i	0.970942	+	0.239316i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$	0.919979	+	0.391967i	0.919979	+	0.391967i
$21$		0			0
$22$		0			0
$23$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$23$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$26$	−0.721202	+	0.692724i	−0.721202	+	0.692724i
$27$		0			0
$28$		0			0
$29$	−1.13965	+	1.51660i	−1.13965	+	1.51660i	−0.316668	+	0.948536i	$0.602564\pi$
$29$	−1.13965	+	1.51660i	−1.13965	+	1.51660i	−0.822984	+	0.568065i	$0.807692\pi$
$30$		0			0
$31$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$31$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$32$	−0.0804666	+	0.996757i	−0.0804666	+	0.996757i
$33$		0			0
$34$	0.521177	+	1.67252i	0.521177	+	1.67252i
$35$		0			0
$36$	0.774605	−	0.632445i	0.774605	−	0.632445i
$37$	1.99190	−	0.160803i	1.99190	−	0.160803i	0.992709	−	0.120537i	$-0.0384615\pi$
$37$	1.99190	−	0.160803i	1.99190	−	0.160803i	0.999189	−	0.0402659i	$-0.0128205\pi$
$38$		0			0
$39$		0			0
$40$	0.866025	−	0.500000i	0.866025	−	0.500000i
$41$	−1.09182	+	1.65196i	−1.09182	+	1.65196i	−0.428693	+	0.903450i	$0.641026\pi$
$41$	−1.09182	+	1.65196i	−1.09182	+	1.65196i	−0.663123	+	0.748511i	$0.730769\pi$
$42$		0			0
$43$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$43$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$44$		0			0
$45$	−0.960518	−	0.278217i	−0.960518	−	0.278217i
$46$		0			0
$47$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$47$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$48$		0			0
$49$	0.948536	−	0.316668i	0.948536	−	0.316668i
$50$	−0.903450	−	0.428693i	−0.903450	−	0.428693i
$51$		0			0
$52$	0.120537	+	0.992709i	0.120537	+	0.992709i
$53$	−0.666000	−	1.47979i	−0.666000	−	1.47979i	−0.866025	−	0.500000i	$-0.833333\pi$
$53$	−0.666000	−	1.47979i	−0.666000	−	1.47979i	0.200026	−	0.979791i	$-0.435897\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.527799	+	1.82217i	0.527799	+	1.82217i
$59$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$59$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$60$		0			0
$61$	−1.23933	+	1.51790i	−1.23933	+	1.51790i	−0.464723	+	0.885456i	$0.653846\pi$
$61$	−1.23933	+	1.51790i	−1.23933	+	1.51790i	−0.774605	+	0.632445i	$0.782051\pi$
$62$		0			0
$63$		0			0
$64$	0.748511	+	0.663123i	0.748511	+	0.663123i
$65$	0.748511	−	0.663123i	0.748511	−	0.663123i
$66$		0			0
$67$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$67$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$68$	1.65017	+	0.588099i	1.65017	+	0.588099i
$69$		0			0
$70$		0			0
$71$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$71$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$72$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$73$	−0.992709	+	0.120537i	−0.992709	+	0.120537i	−0.600742	−	0.799443i	$-0.705128\pi$
$73$	−0.992709	+	0.120537i	−0.992709	+	0.120537i	−0.391967	+	0.919979i	$0.628205\pi$
$74$	1.06806	−	1.68901i	1.06806	−	1.68901i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$79$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$80$	0.120537	−	0.992709i	0.120537	−	0.992709i
$81$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$82$	0.664749	+	1.86525i	0.664749	+	1.86525i
$83$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$83$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$84$		0			0
$85$	−0.453409	−	1.69214i	−0.453409	−	1.69214i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.574732	−	0.153999i	0.574732	−	0.153999i	0.0402659	−	0.999189i	$-0.487179\pi$
$89$	0.574732	−	0.153999i	0.574732	−	0.153999i	0.534466	+	0.845190i	$0.320513\pi$
$90$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.64463	−	0.0662764i	1.64463	−	0.0662764i	0.799443	−	0.600742i	$-0.205128\pi$
$97$	1.64463	−	0.0662764i	1.64463	−	0.0662764i	0.845190	+	0.534466i	$0.179487\pi$
$98$	0.316668	−	0.948536i	0.316668	−	0.948536i
$99$		0			0
$100$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$101$	−0.145395	+	0.0689908i	−0.145395	+	0.0689908i	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−0.145395	+	0.0689908i	−0.145395	+	0.0689908i	0.354605	+	0.935016i	$0.384615\pi$
$102$		0			0
$103$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$103$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$104$	0.866025	+	0.500000i	0.866025	+	0.500000i
$105$		0			0
$106$	−1.58310	−	0.356544i	−1.58310	−	0.356544i
$107$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$107$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$108$		0			0
$109$	0.186505	+	1.01773i	0.186505	+	1.01773i	0.935016	+	0.354605i	$0.115385\pi$
$109$	0.186505	+	1.01773i	0.186505	+	1.01773i	−0.748511	+	0.663123i	$0.769231\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.242292	−	0.366598i	−0.242292	−	0.366598i	0.692724	−	0.721202i	$-0.256410\pi$
$113$	−0.242292	−	0.366598i	−0.242292	−	0.366598i	−0.935016	+	0.354605i	$0.884615\pi$
$114$		0			0
$115$		0			0
$116$	1.77379	+	0.672711i	1.77379	+	0.672711i
$117$	−0.239316	−	0.970942i	−0.239316	−	0.970942i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.721202	−	0.692724i	0.721202	−	0.692724i
$122$	0.468959	+	1.90264i	0.468959	+	1.90264i
$123$		0			0
$124$		0			0
$125$	0.885456	+	0.464723i	0.885456	+	0.464723i
$126$		0			0
$127$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$127$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$128$	0.979791	−	0.200026i	0.979791	−	0.200026i
$129$		0			0
$130$	−0.0804666	−	0.996757i	−0.0804666	−	0.996757i
$131$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$131$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.46148	−	0.965923i	1.46148	−	0.965923i
$137$	1.68490	+	0.136019i	1.68490	+	0.136019i	0.885456	−	0.464723i	$-0.153846\pi$
$137$	1.68490	+	0.136019i	1.68490	+	0.136019i	0.799443	+	0.600742i	$0.205128\pi$
$138$		0			0
$139$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$139$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$145$	−0.453999	−	1.84195i	−0.453999	−	1.84195i
$146$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$147$		0			0
$148$	−0.708635	−	1.86852i	−0.708635	−	1.86852i
$149$	0.671273	+	0.0135202i	0.671273	+	0.0135202i	0.354605	−	0.935016i	$-0.384615\pi$
$149$	0.671273	+	0.0135202i	0.671273	+	0.0135202i	0.316668	+	0.948536i	$0.397436\pi$
$150$		0			0
$151$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$151$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$152$		0			0
$153$	−1.70903	−	0.384905i	−1.70903	−	0.384905i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.79863	−	0.560476i	−1.79863	−	0.560476i	−0.799443	−	0.600742i	$-0.794872\pi$
$157$	−1.79863	−	0.560476i	−1.79863	−	0.560476i	−0.999189	+	0.0402659i	$0.987179\pi$
$158$		0			0
$159$		0			0
$160$	−0.721202	−	0.692724i	−0.721202	−	0.692724i
$161$		0			0
$162$	0.160411	+	0.987050i	0.160411	+	0.987050i
$163$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$163$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$164$	1.89050	+	0.589104i	1.89050	+	0.589104i
$165$		0			0
$166$		0			0
$167$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$167$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$168$		0			0
$169$	0.948536	+	0.316668i	0.948536	+	0.316668i
$170$	−1.62515	−	0.654068i	−1.62515	−	0.654068i
$171$		0			0
$172$		0			0
$173$	0.963158	−	1.74864i	0.963158	−	1.74864i	0.428693	−	0.903450i	$-0.358974\pi$
$173$	0.963158	−	1.74864i	0.963158	−	1.74864i	0.534466	−	0.845190i	$-0.320513\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.222152	−	0.551979i	0.222152	−	0.551979i
$179$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$179$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$180$			1.00000i			1.00000i
$181$	−0.587824	−	1.12001i	−0.587824	−	1.12001i	−0.979791	−	0.200026i	$-0.935897\pi$
$181$	−0.587824	−	1.12001i	−0.587824	−	1.12001i	0.391967	−	0.919979i	$-0.371795\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.13521	+	1.64463i	−1.13521	+	1.64463i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−0.289847	+	1.78350i	−0.289847	+	1.78350i	0.278217	+	0.960518i	$0.410256\pi$
$193$	−0.289847	+	1.78350i	−0.289847	+	1.78350i	−0.568065	+	0.822984i	$0.692308\pi$
$194$	0.935016	−	1.35460i	0.935016	−	1.35460i
$195$		0			0
$196$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$197$	−1.19820	+	0.568552i	−1.19820	+	0.568552i	−0.919979	−	0.391967i	$-0.871795\pi$
$197$	−1.19820	+	0.568552i	−1.19820	+	0.568552i	−0.278217	+	0.960518i	$0.589744\pi$
$198$		0			0
$199$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$199$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$200$	−0.160411	+	0.987050i	−0.160411	+	0.987050i
$201$		0			0
$202$	−0.0321908	+	0.157681i	−0.0321908	+	0.157681i
$203$		0			0
$204$		0			0
$205$	−0.589104	−	1.89050i	−0.589104	−	1.89050i
$206$		0			0
$207$		0			0
$208$	0.919979	−	0.391967i	0.919979	−	0.391967i
$209$		0			0
$210$		0			0
$211$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$211$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$212$	−1.23607	+	1.05141i	−1.23607	+	1.05141i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.925657	+	0.462291i	0.925657	+	0.462291i
$219$		0			0
$220$		0			0
$221$	1.23874	−	1.23874i	1.23874	−	1.23874i
$222$		0			0
$223$		0			0		0.0402659	−	0.999189i	$-0.487179\pi$
$223$		0			0		−0.0402659	+	0.999189i	$0.512821\pi$
$224$		0			0
$225$	0.822984	−	0.568065i	0.822984	−	0.568065i
$226$	−0.438629	−	0.0265322i	−0.438629	−	0.0265322i
$227$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$227$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$228$		0			0
$229$	−0.807380	+	0.147958i	−0.807380	+	0.147958i	−0.568065	−	0.822984i	$-0.692308\pi$
$229$	−0.807380	+	0.147958i	−0.807380	+	0.147958i	−0.239316	+	0.970942i	$0.576923\pi$
$230$		0			0
$231$		0			0
$232$	1.60339	−	1.01392i	1.60339	−	1.01392i
$233$	0.0217671	+	0.359852i	0.0217671	+	0.359852i	0.992709	+	0.120537i	$0.0384615\pi$
$233$	0.0217671	+	0.359852i	0.0217671	+	0.359852i	−0.970942	+	0.239316i	$0.923077\pi$
$234$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	0.810531	+	0.747735i	0.810531	+	0.747735i	0.970942	−	0.239316i	$-0.0769231\pi$
$241$	0.810531	+	0.747735i	0.810531	+	0.747735i	−0.160411	+	0.987050i	$0.551282\pi$
$242$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$243$		0			0
$244$	1.80277	+	0.768090i	1.80277	+	0.768090i
$245$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.903450	−	0.428693i	0.903450	−	0.428693i
$251$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$251$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.428693	−	0.903450i	0.428693	−	0.903450i
$257$	0.00404843	+	0.201003i	0.00404843	+	0.201003i	0.996757	+	0.0804666i	$0.0256410\pi$
$257$	0.00404843	+	0.201003i	0.00404843	+	0.201003i	−0.992709	+	0.120537i	$0.961538\pi$
$258$		0			0
$259$		0			0
$260$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$261$	−1.84195	−	0.453999i	−1.84195	−	0.453999i
$262$		0			0
$263$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$263$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$264$		0			0
$265$	1.56746	+	0.420000i	1.56746	+	0.420000i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.294326	+	0.881614i	−0.294326	+	0.881614i	0.692724	+	0.721202i	$0.256410\pi$
$269$	−0.294326	+	0.881614i	−0.294326	+	0.881614i	−0.987050	+	0.160411i	$0.948718\pi$
$270$		0			0
$271$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$271$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$272$	0.105773	−	1.74864i	0.105773	−	1.74864i
$273$		0			0
$274$	1.12093	−	1.26527i	1.12093	−	1.26527i
$275$		0			0
$276$		0			0
$277$	−0.194194	−	1.92207i	−0.194194	−	1.92207i	−0.354605	−	0.935016i	$-0.615385\pi$
$277$	−0.194194	−	1.92207i	−0.194194	−	1.92207i	0.160411	−	0.987050i	$-0.448718\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.0705851	−	1.16691i	0.0705851	−	1.16691i	−0.774605	−	0.632445i	$-0.782051\pi$
$281$	0.0705851	−	1.16691i	0.0705851	−	1.16691i	0.845190	−	0.534466i	$-0.179487\pi$
$282$		0			0
$283$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$283$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$289$	−0.655164	−	1.96246i	−0.655164	−	1.96246i
$290$	−1.74527	−	0.743589i	−1.74527	−	0.743589i
$291$		0			0
$292$	0.391967	+	0.919979i	0.391967	+	0.919979i
$293$	−1.54419	−	1.26079i	−1.54419	−	1.26079i	−0.822984	−	0.568065i	$-0.807692\pi$
$293$	−1.54419	−	1.26079i	−1.54419	−	1.26079i	−0.721202	−	0.692724i	$-0.756410\pi$
$294$		0			0
$295$		0			0
$296$	−1.91948	−	0.555984i	−1.91948	−	0.555984i
$297$		0			0
$298$	0.414071	−	0.528522i	0.414071	−	0.528522i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.391967	−	1.91998i	−0.391967	−	1.91998i
$306$	−1.33440	+	1.13504i	−1.33440	+	1.13504i
$307$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$307$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$311$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$312$		0			0
$313$	−0.222333	−	1.21323i	−0.222333	−	1.21323i	−0.885456	−	0.464723i	$-0.846154\pi$
$313$	−0.222333	−	1.21323i	−0.222333	−	1.21323i	0.663123	−	0.748511i	$-0.269231\pi$
$314$	−1.52858	+	1.10120i	−1.52858	+	1.10120i
$315$		0			0
$316$		0			0
$317$	−1.84582	+	0.700026i	−1.84582	+	0.700026i	−0.866025	+	0.500000i	$0.833333\pi$
$317$	−1.84582	+	0.700026i	−1.84582	+	0.700026i	−0.979791	+	0.200026i	$0.935897\pi$
$318$		0			0
$319$		0			0
$320$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.885456	+	0.464723i	0.885456	+	0.464723i
$325$	0.0402659	+	0.999189i	0.0402659	+	0.999189i
$326$		0			0
$327$		0			0
$328$	1.60666	−	1.15745i	1.60666	−	1.15745i
$329$		0			0
$330$		0			0
$331$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$331$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$332$		0			0
$333$	0.928693	+	1.76948i	0.928693	+	1.76948i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.916291	+	0.916291i	−0.916291	+	0.916291i	−0.996757	−	0.0804666i	$-0.974359\pi$
$337$	−0.916291	+	0.916291i	−0.916291	+	0.916291i	0.0804666	+	0.996757i	$0.474359\pi$
$338$	0.822984	−	0.568065i	0.822984	−	0.568065i
$339$		0			0
$340$	−1.49919	+	0.906291i	−1.49919	+	0.906291i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.819328	−	1.82047i	−0.819328	−	1.82047i
$347$		0			0		−0.927686	−	0.373361i	$-0.878205\pi$
$347$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$348$		0			0
$349$	−0.712912	−	1.77136i	−0.712912	−	1.77136i	−0.632445	−	0.774605i	$-0.717949\pi$
$349$	−0.712912	−	1.77136i	−0.712912	−	1.77136i	−0.0804666	−	0.996757i	$-0.525641\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.979791	+	1.20003i	0.979791	+	1.20003i	0.979791	+	0.200026i	$0.0641026\pi$
$353$	0.979791	+	1.20003i	0.979791	+	1.20003i			1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$	−0.307819	−	0.509195i	−0.307819	−	0.509195i
$357$		0			0
$358$		0			0
$359$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$359$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$360$	0.799443	+	0.600742i	0.799443	+	0.600742i
$361$	0.866025	+	0.500000i	0.866025	+	0.500000i
$362$	−1.24851	−	0.202903i	−1.24851	−	0.202903i
$363$		0			0
$364$		0			0
$365$	0.534466	−	0.845190i	0.534466	−	0.845190i
$366$		0			0
$367$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$367$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$368$		0			0
$369$	−1.94773	−	0.356934i	−1.94773	−	0.356934i
$370$	0.632822	+	1.89553i	0.632822	+	1.89553i
$371$		0			0
$372$		0			0
$373$	−0.0394819	+	0.278217i	−0.0394819	+	0.278217i	0.960518	+	0.278217i	$0.0897436\pi$
$373$	−0.0394819	+	0.278217i	−0.0394819	+	0.278217i	−1.00000			$1.00000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.36817	−	1.31415i	1.36817	−	1.31415i
$378$		0			0
$379$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$379$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$383$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$384$		0			0
$385$		0			0
$386$	1.25168	+	1.30314i	1.25168	+	1.30314i
$387$		0			0
$388$	−0.521225	−	1.56126i	−0.521225	−	1.56126i
$389$	−0.0285570	−	0.0752986i	−0.0285570	−	0.0752986i	0.919979	−	0.391967i	$-0.128205\pi$
$389$	−0.0285570	−	0.0752986i	−0.0285570	−	0.0752986i	−0.948536	+	0.316668i	$0.897436\pi$
$390$		0			0
$391$		0			0
$392$	−0.999189	−	0.0402659i	−0.999189	−	0.0402659i
$393$		0			0
$394$	−0.265283	+	1.29944i	−0.265283	+	1.29944i
$395$		0			0
$396$		0			0
$397$	1.67806	+	1.06114i	1.67806	+	1.06114i	0.903450	+	0.428693i	$0.141026\pi$
$397$	1.67806	+	1.06114i	1.67806	+	1.06114i	0.774605	+	0.632445i	$0.217949\pi$
$398$		0			0
$399$		0			0
$400$	0.692724	+	0.721202i	0.692724	+	0.721202i
$401$	−0.334720	+	1.48620i	−0.334720	+	1.48620i	0.464723	+	0.885456i	$0.346154\pi$
$401$	−0.334720	+	1.48620i	−0.334720	+	1.48620i	−0.799443	+	0.600742i	$0.794872\pi$
$402$		0			0
$403$		0			0
$404$	0.106718	+	0.120460i	0.106718	+	0.120460i
$405$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.463097	+	0.231280i	−0.463097	+	0.231280i	−0.663123	−	0.748511i	$-0.730769\pi$
$409$	−0.463097	+	0.231280i	−0.463097	+	0.231280i	0.200026	+	0.979791i	$0.435897\pi$
$410$	−1.86525	−	0.664749i	−1.86525	−	0.664749i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.239316	−	0.970942i	0.239316	−	0.970942i
$417$		0			0
$418$		0			0
$419$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$419$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$420$		0			0
$421$	0.100742	+	1.66547i	0.100742	+	1.66547i	0.600742	+	0.799443i	$0.294872\pi$
$421$	0.100742	+	1.66547i	0.100742	+	1.66547i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	0.0979795	+	1.61980i	0.0979795	+	1.61980i
$425$	1.59750	+	0.718976i	1.59750	+	0.718976i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$431$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$432$		0			0
$433$	0.322984	+	1.43409i	0.322984	+	1.43409i	0.822984	+	0.568065i	$0.192308\pi$
$433$	0.322984	+	1.43409i	0.322984	+	1.43409i	−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$	0.925657	−	0.462291i	0.925657	−	0.462291i
$437$		0			0
$438$		0			0
$439$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$439$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$440$		0			0
$441$	0.663123	+	0.748511i	0.663123	+	0.748511i
$442$	−0.246137	−	1.73446i	−0.246137	−	1.73446i
$443$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$443$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$444$		0			0
$445$	−0.244198	+	0.542586i	−0.244198	+	0.542586i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.759568	+	1.37902i	−0.759568	+	1.37902i	0.160411	+	0.987050i	$0.448718\pi$
$449$	−0.759568	+	1.37902i	−0.759568	+	1.37902i	−0.919979	+	0.391967i	$0.871795\pi$
$450$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$451$		0			0
$452$	−0.284714	+	0.334720i	−0.284714	+	0.334720i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.101594	+	0.304312i	0.101594	+	0.304312i	0.987050	−	0.160411i	$-0.0512821\pi$
$457$	0.101594	+	0.304312i	0.101594	+	0.304312i	−0.885456	+	0.464723i	$0.846154\pi$
$458$	−0.366744	+	0.734339i	−0.366744	+	0.734339i
$459$		0			0
$460$		0			0
$461$	1.81003	+	0.728476i	1.81003	+	0.728476i	0.987050	+	0.160411i	$0.0512821\pi$
$461$	1.81003	+	0.728476i	1.81003	+	0.728476i	0.822984	+	0.568065i	$0.192308\pi$
$462$		0			0
$463$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$463$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$464$	0.152651	−	1.89092i	0.152651	−	1.89092i
$465$		0			0
$466$	0.300758	+	0.198777i	0.300758	+	0.198777i
$467$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$467$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$468$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.10033	−	1.19273i	1.10033	−	1.19273i
$478$		0			0
$479$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$479$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$480$		0			0
$481$	−1.99190	−	0.160803i	−1.99190	−	0.160803i
$482$	1.08469	−	0.198777i	1.08469	−	0.198777i
$483$		0			0
$484$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$485$	−0.988802	+	1.31586i	−0.988802	+	1.31586i
$486$		0			0
$487$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$487$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$488$	1.69705	−	0.979791i	1.69705	−	0.979791i
$489$		0			0
$490$	0.534466	+	0.845190i	0.534466	+	0.845190i
$491$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$491$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$492$		0			0
$493$	−0.988710	−	3.17288i	−0.988710	−	3.17288i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$499$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$500$	0.200026	−	0.979791i	0.200026	−	0.979791i
$501$		0			0
$502$		0			0
$503$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$503$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$504$		0			0
$505$	0.0385138	−	0.156257i	0.0385138	−	0.156257i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.884733	−	0.125553i	0.884733	−	0.125553i	0.316668	−	0.948536i	$-0.397436\pi$
$509$	0.884733	−	0.125553i	0.884733	−	0.125553i	0.568065	+	0.822984i	$0.307692\pi$
$510$		0			0
$511$		0			0
$512$	−0.464723	−	0.885456i	−0.464723	−	0.885456i
$513$		0			0
$514$	0.163123	+	0.117515i	0.163123	+	0.117515i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$521$	1.06386	+	0.558358i	1.06386	+	0.558358i	0.903450	−	0.428693i	$-0.141026\pi$
$521$	1.06386	+	0.558358i	1.06386	+	0.558358i	0.160411	+	0.987050i	$0.448718\pi$
$522$	−1.46948	+	1.19979i	−1.46948	+	1.19979i
$523$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$523$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$	1.27741	−	1.00078i	1.27741	−	1.00078i
$531$		0			0
$532$		0			0
$533$	1.34267	−	1.45543i	1.34267	−	1.45543i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.527986	+	0.764919i	0.527986	+	0.764919i
$539$		0			0
$540$		0			0
$541$	1.89977	−	0.591992i	1.89977	−	0.591992i	0.919979	−	0.391967i	$-0.128205\pi$
$541$	1.89977	−	0.591992i	1.89977	−	0.591992i	0.979791	−	0.200026i	$-0.0641026\pi$
$542$		0			0
$543$		0			0
$544$	−1.33440	−	1.13504i	−1.33440	−	1.13504i
$545$	−0.906291	−	0.499189i	−0.906291	−	0.499189i
$546$		0			0
$547$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$547$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$548$	−0.338119	−	1.65622i	−0.338119	−	1.65622i
$549$	−1.88221	−	0.545190i	−1.88221	−	0.545190i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.65324	−	0.999420i	−1.65324	−	0.999420i
$555$		0			0
$556$		0			0
$557$	1.64291	−	0.548485i	1.64291	−	0.548485i	0.663123	−	0.748511i	$-0.269231\pi$
$557$	1.64291	−	0.548485i	1.64291	−	0.548485i	0.979791	+	0.200026i	$0.0641026\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.890475	−	0.757442i	−0.890475	−	0.757442i
$563$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$563$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$564$		0			0
$565$	0.437205	+	0.0441724i	0.437205	+	0.0441724i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.0192725	+	0.478243i	−0.0192725	+	0.478243i	0.960518	+	0.278217i	$0.0897436\pi$
$569$	−0.0192725	+	0.478243i	−0.0192725	+	0.478243i	−0.979791	+	0.200026i	$0.935897\pi$
$570$		0			0
$571$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$571$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.316668	+	0.948536i	−0.316668	+	0.948536i
$577$			1.38545i			1.38545i	0.721202	+	0.692724i	$0.243590\pi$
$577$			1.38545i			1.38545i	−0.721202	+	0.692724i	$0.756410\pi$
$578$	−1.96246	−	0.655164i	−1.96246	−	0.655164i
$579$		0			0
$580$	−1.64291	+	0.948536i	−1.64291	+	0.948536i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.970942	+	0.239316i	0.970942	+	0.239316i
$585$	0.903450	+	0.428693i	0.903450	+	0.428693i
$586$	−1.93559	+	0.477079i	−1.93559	+	0.477079i
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.59759	+	1.20051i	−1.59759	+	1.20051i
$593$	0.956491	−	1.07966i	0.956491	−	1.07966i	−0.0402659	−	0.999189i	$-0.512821\pi$
$593$	0.956491	−	1.07966i	0.956491	−	1.07966i	0.996757	−	0.0804666i	$-0.0256410\pi$
$594$		0			0
$595$		0			0
$596$	−0.173773	−	0.648531i	−0.173773	−	0.648531i
$597$		0			0
$598$		0			0
$599$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$599$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$600$		0			0
$601$	0.368039	+	0.156807i	0.368039	+	0.156807i	0.568065	−	0.822984i	$-0.307692\pi$
$601$	0.368039	+	0.156807i	0.368039	+	0.156807i	−0.200026	+	0.979791i	$0.564103\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.0804666	+	0.996757i	0.0804666	+	0.996757i
$606$		0			0
$607$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$607$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$608$		0			0
$609$		0			0
$610$	−1.77038	−	0.840058i	−1.77038	−	0.840058i
$611$		0			0
$612$	0.105773	+	1.74864i	0.105773	+	1.74864i
$613$	0.338119	−	0.213814i	0.338119	−	0.213814i	−0.354605	−	0.935016i	$-0.615385\pi$
$613$	0.338119	−	0.213814i	0.338119	−	0.213814i	0.692724	+	0.721202i	$0.256410\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.338496	+	0.535289i	−0.338496	+	0.535289i	−0.970942	−	0.239316i	$-0.923077\pi$
$617$	−0.338496	+	0.535289i	−0.338496	+	0.535289i	0.632445	+	0.774605i	$0.282051\pi$
$618$		0			0
$619$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$619$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$626$	−1.10348	−	0.551098i	−1.10348	−	0.551098i
$627$		0			0
$628$	−0.0379369	+	1.88355i	−0.0379369	+	1.88355i
$629$	−1.81111	+	2.99595i	−1.81111	+	2.99595i
$630$		0			0
$631$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$631$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$632$		0			0
$633$		0			0
$634$	−0.549229	+	1.89616i	−0.549229	+	1.89616i
$635$		0			0
$636$		0			0
$637$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$638$		0			0
$639$		0			0
$640$	−0.464723	+	0.885456i	−0.464723	+	0.885456i
$641$	1.04052	+	1.38468i	1.04052	+	1.38468i	0.919979	+	0.391967i	$0.128205\pi$
$641$	1.04052	+	1.38468i	1.04052	+	1.38468i	0.120537	+	0.992709i	$0.461538\pi$
$642$		0			0
$643$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$643$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$647$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$648$	0.903450	−	0.428693i	0.903450	−	0.428693i
$649$		0			0
$650$	0.822984	+	0.568065i	0.822984	+	0.568065i
$651$		0			0
$652$		0			0
$653$	−0.472034	−	1.76166i	−0.472034	−	1.76166i	−0.632445	−	0.774605i	$-0.717949\pi$
$653$	−0.472034	−	1.76166i	−0.472034	−	1.76166i	0.160411	−	0.987050i	$-0.448718\pi$
$654$		0			0
$655$		0			0
$656$	0.0398746	−	1.97976i	0.0398746	−	1.97976i
$657$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$658$		0			0
$659$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$659$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$660$		0			0
$661$	0.0400701	+	0.00404843i	0.0400701	+	0.00404843i	0.120537	−	0.992709i	$-0.461538\pi$
$661$	0.0400701	+	0.00404843i	0.0400701	+	0.00404843i	−0.0804666	+	0.996757i	$0.525641\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.97250	+	0.320562i	1.97250	+	0.320562i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.09717	−	1.66006i	1.09717	−	1.66006i	0.464723	−	0.885456i	$-0.346154\pi$
$673$	1.09717	−	1.66006i	1.09717	−	1.66006i	0.632445	−	0.774605i	$-0.282051\pi$
$674$	0.182067	+	1.28298i	0.182067	+	1.28298i
$675$		0			0
$676$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$677$	1.21026	+	1.21026i	1.21026	+	1.21026i	0.970942	+	0.239316i	$0.0769231\pi$
$677$	1.21026	+	1.21026i	1.21026	+	1.21026i	0.239316	+	0.970942i	$0.423077\pi$
$678$		0			0
$679$		0			0
$680$	−0.176098	+	1.74296i	−0.176098	+	1.74296i
$681$		0			0
$682$		0			0
$683$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$683$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$684$		0			0
$685$	−1.17097	+	1.21911i	−1.17097	+	1.21911i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.420000	+	1.56746i	0.420000	+	1.56746i
$690$		0			0
$691$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$691$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$692$	−1.94757	−	0.438629i	−1.94757	−	0.438629i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.23010	−	3.24349i	−1.23010	−	3.24349i
$698$	−1.84438	−	0.494200i	−1.84438	−	0.494200i
$699$		0			0
$700$		0			0
$701$	1.53901	+	1.06230i	1.53901	+	1.06230i	0.970942	+	0.239316i	$0.0769231\pi$
$701$	1.53901	+	1.06230i	1.53901	+	1.06230i	0.568065	+	0.822984i	$0.307692\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.54795	−	0.0623804i	1.54795	−	0.0623804i
$707$		0			0
$708$		0			0
$709$	1.66817	−	1.10253i	1.66817	−	1.10253i	0.822984	−	0.568065i	$-0.192308\pi$
$709$	1.66817	−	1.10253i	1.66817	−	1.10253i	0.845190	−	0.534466i	$-0.179487\pi$
$710$		0			0
$711$		0			0
$712$	−0.591992	−	0.0598112i	−0.591992	−	0.0598112i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$719$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$720$	0.960518	−	0.278217i	0.960518	−	0.278217i
$721$		0			0
$722$	0.919979	−	0.391967i	0.919979	−	0.391967i
$723$		0			0
$724$	−0.912242	+	0.876221i	−0.912242	+	0.876221i
$725$	1.71391	+	0.813261i	1.71391	+	0.813261i
$726$		0			0
$727$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$727$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$728$		0			0
$729$	−0.935016	−	0.354605i	−0.935016	−	0.354605i
$730$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$731$		0			0
$732$		0			0
$733$	0.641762	−	0.568552i	0.641762	−	0.568552i	−0.278217	−	0.960518i	$-0.589744\pi$
$733$	0.641762	−	0.568552i	0.641762	−	0.568552i	0.919979	+	0.391967i	$0.128205\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.45543	+	1.34267i	−1.45543	+	1.34267i
$739$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$739$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$740$	1.89553	+	0.632822i	1.89553	+	0.632822i
$741$		0			0
$742$		0			0
$743$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$743$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$744$		0			0
$745$	−0.435016	+	0.511421i	−0.435016	+	0.511421i
$746$	0.198700	+	0.198700i	0.198700	+	0.198700i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$751$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$752$		0			0
$753$		0			0
$754$	−0.228667	−	1.88324i	−0.228667	−	1.88324i
$755$		0			0
$756$		0			0
$757$	−0.120733	+	0.00243169i	−0.120733	+	0.00243169i	−0.0804666	−	0.996757i	$-0.525641\pi$
$757$	−0.120733	+	0.00243169i	−0.120733	+	0.00243169i	−0.0402659	+	0.999189i	$0.512821\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.256248	+	1.80571i	−0.256248	+	1.80571i	0.278217	+	0.960518i	$0.410256\pi$
$761$	−0.256248	+	1.80571i	−0.256248	+	1.80571i	−0.534466	+	0.845190i	$0.679487\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.37902	−	1.08039i	1.37902	−	1.08039i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.00243169	+	0.120733i	0.00243169	+	0.120733i	0.999189	+	0.0402659i	$0.0128205\pi$
$769$	0.00243169	+	0.120733i	0.00243169	+	0.120733i	−0.996757	+	0.0804666i	$0.974359\pi$
$770$		0			0
$771$		0			0
$772$	1.79373	−	0.217798i	1.79373	−	0.217798i
$773$	−0.0192725	−	0.478243i	−0.0192725	−	0.478243i	−0.979791	−	0.200026i	$-0.935897\pi$
$773$	−0.0192725	−	0.478243i	−0.0192725	−	0.478243i	0.960518	−	0.278217i	$-0.0897436\pi$
$774$		0			0
$775$		0			0
$776$	−1.56126	−	0.521225i	−1.56126	−	0.521225i
$777$		0			0
$778$	−0.0773523	−	0.0224054i	−0.0773523	−	0.0224054i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$785$	1.57168	−	1.03876i	1.57168	−	1.03876i
$786$		0			0
$787$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$787$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$788$	0.879463	+	0.992709i	0.879463	+	0.992709i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.46677	−	1.29944i	1.46677	−	1.29944i
$794$	1.85640	−	0.704039i	1.85640	−	0.704039i
$795$		0			0
$796$		0			0
$797$	−1.57405	−	0.0317031i	−1.57405	−	0.0317031i	−0.774605	−	0.632445i	$-0.782051\pi$
$797$	−1.57405	−	0.0317031i	−1.57405	−	0.0317031i	−0.799443	+	0.600742i	$0.794872\pi$
$798$		0			0
$799$		0			0
$800$	0.992709	−	0.120537i	0.992709	−	0.120537i
$801$	0.366951	+	0.468379i	0.366951	+	0.468379i
$802$	0.987050	+	1.16041i	0.987050	+	1.16041i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.160411	−	0.0129497i	0.160411	−	0.0129497i
$809$	−0.664135	+	0.542249i	−0.664135	+	0.542249i	−0.903450	−	0.428693i	$-0.858974\pi$
$809$	−0.664135	+	0.542249i	−0.664135	+	0.542249i	0.239316	+	0.970942i	$0.423077\pi$
$810$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$811$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$811$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.0933069	+	0.509159i	−0.0933069	+	0.509159i
$819$		0			0
$820$	−1.65196	+	1.09182i	−1.65196	+	1.09182i
$821$	−1.14990	−	1.59617i	−1.14990	−	1.59617i	−0.721202	−	0.692724i	$-0.756410\pi$
$821$	−1.14990	−	1.59617i	−1.14990	−	1.59617i	−0.428693	−	0.903450i	$-0.641026\pi$
$822$		0			0
$823$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$823$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$827$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$828$		0			0
$829$	0.271156	−	0.171469i	0.271156	−	0.171469i	−0.391967	−	0.919979i	$-0.628205\pi$
$829$	0.271156	−	0.171469i	0.271156	−	0.171469i	0.663123	+	0.748511i	$0.269231\pi$
$830$		0			0
$831$		0			0
$832$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$833$	−0.521177	+	1.67252i	−0.521177	+	1.67252i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$839$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$840$		0			0
$841$	−0.723055	−	2.49628i	−0.723055	−	2.49628i
$842$	1.39197	+	0.919979i	1.39197	+	0.919979i
$843$		0			0
$844$		0			0
$845$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$846$		0			0
$847$		0			0
$848$	1.35379	+	0.894750i	1.35379	+	0.894750i
$849$		0			0
$850$	1.53447	−	0.845190i	1.53447	−	0.845190i
$851$		0			0
$852$		0			0
$853$	−0.520276	−	0.197315i	−0.520276	−	0.197315i	0.0804666	−	0.996757i	$-0.474359\pi$
$853$	−0.520276	−	0.197315i	−0.520276	−	0.197315i	−0.600742	+	0.799443i	$0.705128\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.496387	−	1.59296i	0.496387	−	1.59296i	−0.278217	−	0.960518i	$-0.589744\pi$
$857$	0.496387	−	1.59296i	0.496387	−	1.59296i	0.774605	−	0.632445i	$-0.217949\pi$
$858$		0			0
$859$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$859$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$863$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$864$		0			0
$865$	0.745361	+	1.85199i	0.745361	+	1.85199i
$866$	1.34050	+	0.603311i	1.34050	+	0.603311i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.186505	−	1.01773i	0.186505	−	1.01773i
$873$	0.705614	+	1.48705i	0.705614	+	1.48705i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.0966793	+	1.19759i	−0.0966793	+	1.19759i	0.748511	+	0.663123i	$0.230769\pi$
$877$	−0.0966793	+	1.19759i	−0.0966793	+	1.19759i	−0.845190	+	0.534466i	$0.820513\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.23850	+	1.01121i	−1.23850	+	1.01121i	−0.239316	+	0.970942i	$0.576923\pi$
$881$	−1.23850	+	1.01121i	−1.23850	+	1.01121i	−0.999189	+	0.0402659i	$0.987179\pi$
$882$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$883$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$883$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$884$	−1.53447	−	0.845190i	−1.53447	−	0.845190i
$885$		0			0
$886$		0			0
$887$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$887$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$888$		0			0
$889$		0			0
$890$	0.287066	+	0.521177i	0.287066	+	0.521177i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.646140	+	1.43566i	0.646140	+	1.43566i
$899$		0			0
$900$	−0.774605	−	0.632445i	−0.774605	−	0.632445i
$901$	2.78535	+	0.568634i	2.78535	+	0.568634i
$902$		0			0
$903$		0			0
$904$	0.0965496	+	0.428693i	0.0965496	+	0.428693i
$905$	1.23933	+	0.253011i	1.23933	+	0.253011i
$906$		0			0
$907$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$907$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$908$		0			0
$909$	−0.120460	−	0.106718i	−0.120460	−	0.106718i
$910$		0			0
$911$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$911$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$912$		0			0
$913$		0			0
$914$	0.304312	+	0.101594i	0.304312	+	0.101594i
$915$		0			0
$916$	0.366744	+	0.734339i	0.366744	+	0.734339i
$917$		0			0
$918$		0			0
$919$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$919$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$920$		0			0
$921$		0			0
$922$	1.66974	−	1.00939i	1.66974	−	1.00939i
$923$		0			0
$924$		0			0
$925$	−0.555984	−	1.91948i	−0.555984	−	1.91948i
$926$		0			0
$927$		0			0
$928$	−1.41998	−	1.25799i	−1.41998	−	1.25799i
$929$	−0.135573	+	0.955347i	−0.135573	+	0.955347i	0.799443	+	0.600742i	$0.205128\pi$
$929$	−0.135573	+	0.955347i	−0.135573	+	0.955347i	−0.935016	+	0.354605i	$0.884615\pi$
$930$		0			0
$931$		0			0
$932$	0.339589	−	0.121025i	0.339589	−	0.121025i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$937$	−0.267794	−	0.442985i	−0.267794	−	0.442985i	0.692724	−	0.721202i	$-0.256410\pi$
$937$	−0.267794	−	0.442985i	−0.267794	−	0.442985i	−0.960518	+	0.278217i	$0.910256\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.0744731	+	0.0950579i	−0.0744731	+	0.0950579i	−0.822984	−	0.568065i	$-0.807692\pi$
$941$	−0.0744731	+	0.0950579i	−0.0744731	+	0.0950579i	0.748511	+	0.663123i	$0.230769\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$947$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$948$		0			0
$949$	0.999189	+	0.0402659i	0.999189	+	0.0402659i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.120145	+	0.0121387i	−0.120145	+	0.0121387i	−0.160411	−	0.987050i	$-0.551282\pi$
$953$	−0.120145	+	0.0121387i	−0.120145	+	0.0121387i	0.0402659	+	0.999189i	$0.487179\pi$
$954$	−0.292510	−	1.59617i	−0.292510	−	1.59617i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.935016	−	0.354605i	−0.935016	−	0.354605i
$962$	−1.32517	+	1.49581i	−1.32517	+	1.49581i
$963$		0			0
$964$	0.492709	−	0.986562i	0.492709	−	0.986562i
$965$	−1.19820	−	1.35248i	−1.19820	−	1.35248i
$966$		0			0
$967$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$967$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$968$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$969$		0			0
$970$	0.457937	+	1.58098i	0.457937	+	1.58098i
$971$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$971$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.236201	−	1.94529i	0.236201	−	1.94529i
$977$	−0.0689908	−	0.854605i	−0.0689908	−	0.854605i	−0.935016	−	0.354605i	$-0.884615\pi$
$977$	−0.0689908	−	0.854605i	−0.0689908	−	0.854605i	0.866025	−	0.500000i	$-0.166667\pi$
$978$		0			0
$979$		0			0
$980$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$981$	−0.863184	+	0.570496i	−0.863184	+	0.570496i
$982$		0			0
$983$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$983$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$984$		0			0
$985$	0.317391	−	1.28771i	0.317391	−	1.28771i
$986$	−3.13050	−	1.11567i	−3.13050	−	1.11567i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.58310	+	1.14048i	1.58310	+	1.14048i	0.919979	+	0.391967i	$0.128205\pi$
$997$	1.58310	+	1.14048i	1.58310	+	1.14048i	0.663123	+	0.748511i	$0.269231\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.cs.a.1047.1	✓	48
4.3	odd	2	CM	3380.1.cs.a.1047.1	✓	48
5.3	odd	4		3380.1.cz.a.1723.1	yes	48
20.3	even	4		3380.1.cz.a.1723.1	yes	48
169.41	odd	156		3380.1.cz.a.2407.1	yes	48
676.379	even	156		3380.1.cz.a.2407.1	yes	48
845.548	even	156	inner	3380.1.cs.a.3083.1	yes	48
3380.3083	odd	156	inner	3380.1.cs.a.3083.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.cs.a.1047.1	✓	48	1.1	even	1	trivial
3380.1.cs.a.1047.1	✓	48	4.3	odd	2	CM
3380.1.cs.a.3083.1	yes	48	845.548	even	156	inner
3380.1.cs.a.3083.1	yes	48	3380.3083	odd	156	inner
3380.1.cz.a.1723.1	yes	48	5.3	odd	4
3380.1.cz.a.1723.1	yes	48	20.3	even	4
3380.1.cz.a.2407.1	yes	48	169.41	odd	156
3380.1.cz.a.2407.1	yes	48	676.379	even	156

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3380.1.cs.a.1047.1

Level	$3380$
Weight	$1$
Character	3380.1047
Analytic conductor	$1.687$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{156}$
CM discriminant	-4
Inner twists	$4$