LMFDB - Embedded newform 2925.1.er.a.298.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,1,Mod(298,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 11, 10]))

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

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

chi := DirichletCharacter("2925.298");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2925.er (of order $20$, degree $8$, 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:	$1.45976516195$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$4$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{80})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} - x^{24} + x^{16} - x^{8} + 1 $ x^32 - x^24 + x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

Embedding invariants

Embedding label			298.2
Root			$-0.522499 + 0.852640i$ of defining polynomial
Character	$\chi$	$=$	2925.298
Dual form			2925.1.er.a.2287.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.461143 - 0.0730378i) q^{2} +(-0.743739 - 0.241655i) q^{4} +(-0.760406 + 0.649448i) q^{5} +(0.741322 + 0.377723i) q^{8} +O(q^{10})$	\(q+(-0.461143 - 0.0730378i) q^{2} +(-0.743739 - 0.241655i) q^{4} +(-0.760406 + 0.649448i) q^{5} +(0.741322 + 0.377723i) q^{8} +(0.398090 - 0.243950i) q^{10} +(-1.17195 - 1.61305i) q^{11} +(0.156434 + 0.987688i) q^{13} +(0.318395 + 0.231327i) q^{16} +(0.722486 - 0.299263i) q^{20} +(0.422621 + 0.829441i) q^{22} +(0.156434 - 0.987688i) q^{25} -0.466891i q^{26} +(-0.718246 - 0.718246i) q^{32} +(-0.809017 + 0.194228i) q^{40} +(-0.614234 + 0.845420i) q^{41} +(1.39680 + 1.39680i) q^{43} +(0.481821 + 1.48289i) q^{44} +(0.931099 - 0.474419i) q^{47} -1.00000i q^{49} +(-0.144277 + 0.444039i) q^{50} +(0.122334 - 0.772385i) q^{52} +(1.93874 + 0.465451i) q^{55} +(1.05083 + 0.763472i) q^{59} +(0.734572 - 0.533698i) q^{61} +(0.0474275 + 0.0652784i) q^{64} +(-0.760406 - 0.649448i) q^{65} +(0.149238 + 0.0484904i) q^{71} +(1.34500 + 0.437016i) q^{79} +(-0.392344 + 0.0308782i) q^{80} +(0.344997 - 0.344997i) q^{82} +(1.73278 + 0.882893i) q^{83} +(-0.542106 - 0.746144i) q^{86} +(-0.259506 - 1.63846i) q^{88} +(-0.619195 + 0.449871i) q^{89} +(-0.464020 + 0.150769i) q^{94} +(-0.0730378 + 0.461143i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{10} + 8 q^{16} - 8 q^{22} - 8 q^{40} + 8 q^{43} + 8 q^{52} + 40 q^{64} - 32 q^{82} + 24 q^{88} - 40 q^{94}+O(q^{100}) $ 32 * q + 8 * q^10 + 8 * q^16 - 8 * q^22 - 8 * q^40 + 8 * q^43 + 8 * q^52 + 40 * q^64 - 32 * q^82 + 24 * q^88 - 40 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$e\left(\frac{11}{20}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.461143	−	0.0730378i	−0.461143	−	0.0730378i	−0.0784591	−	0.996917i	$-0.525000\pi$
$2$	−0.461143	−	0.0730378i	−0.461143	−	0.0730378i	−0.382683	+	0.923880i	$0.625000\pi$
$3$		0			0
$3$		0			0
$4$	−0.743739	−	0.241655i	−0.743739	−	0.241655i
$5$	−0.760406	+	0.649448i	−0.760406	+	0.649448i
$5$	−0.760406	+	0.649448i	−0.760406	+	0.649448i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	0.741322	+	0.377723i	0.741322	+	0.377723i
$9$		0			0
$10$	0.398090	−	0.243950i	0.398090	−	0.243950i
$11$	−1.17195	−	1.61305i	−1.17195	−	1.61305i	−0.649448	−	0.760406i	$-0.725000\pi$
$11$	−1.17195	−	1.61305i	−1.17195	−	1.61305i	−0.522499	−	0.852640i	$-0.675000\pi$
$12$		0			0
$13$	0.156434	+	0.987688i	0.156434	+	0.987688i
$13$	0.156434	+	0.987688i	0.156434	+	0.987688i
$14$		0			0
$15$		0			0
$16$	0.318395	+	0.231327i	0.318395	+	0.231327i
$17$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$17$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	0.722486	−	0.299263i	0.722486	−	0.299263i
$21$		0			0
$22$	0.422621	+	0.829441i	0.422621	+	0.829441i
$23$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$23$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$24$		0			0
$25$	0.156434	−	0.987688i	0.156434	−	0.987688i
$26$		−	0.466891i		−	0.466891i
$27$		0			0
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$31$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$32$	−0.718246	−	0.718246i	−0.718246	−	0.718246i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$37$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$38$		0			0
$39$		0			0
$40$	−0.809017	+	0.194228i	−0.809017	+	0.194228i
$41$	−0.614234	+	0.845420i	−0.614234	+	0.845420i	−0.996917	−	0.0784591i	$-0.975000\pi$
$41$	−0.614234	+	0.845420i	−0.614234	+	0.845420i	0.382683	+	0.923880i	$0.375000\pi$
$42$		0			0
$43$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$43$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$44$	0.481821	+	1.48289i	0.481821	+	1.48289i
$45$		0			0
$46$		0			0
$47$	0.931099	−	0.474419i	0.931099	−	0.474419i	0.0784591	−	0.996917i	$-0.475000\pi$
$47$	0.931099	−	0.474419i	0.931099	−	0.474419i	0.852640	+	0.522499i	$0.175000\pi$
$48$		0			0
$49$		−	1.00000i		−	1.00000i
$50$	−0.144277	+	0.444039i	−0.144277	+	0.444039i
$51$		0			0
$52$	0.122334	−	0.772385i	0.122334	−	0.772385i
$53$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$53$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$54$		0			0
$55$	1.93874	+	0.465451i	1.93874	+	0.465451i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.05083	+	0.763472i	1.05083	+	0.763472i	0.972370	−	0.233445i	$-0.0750000\pi$
$59$	1.05083	+	0.763472i	1.05083	+	0.763472i	0.0784591	+	0.996917i	$0.475000\pi$
$60$		0			0
$61$	0.734572	−	0.533698i	0.734572	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$61$	0.734572	−	0.533698i	0.734572	−	0.533698i	0.891007	+	0.453990i	$0.150000\pi$
$62$		0			0
$63$		0			0
$64$	0.0474275	+	0.0652784i	0.0474275	+	0.0652784i
$65$	−0.760406	−	0.649448i	−0.760406	−	0.649448i
$66$		0			0
$67$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$67$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.149238	+	0.0484904i	0.149238	+	0.0484904i	0.382683	−	0.923880i	$-0.375000\pi$
$71$	0.149238	+	0.0484904i	0.149238	+	0.0484904i	−0.233445	+	0.972370i	$0.575000\pi$
$72$		0			0
$73$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$73$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$79$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$80$	−0.392344	+	0.0308782i	−0.392344	+	0.0308782i
$81$		0			0
$82$	0.344997	−	0.344997i	0.344997	−	0.344997i
$83$	1.73278	+	0.882893i	1.73278	+	0.882893i	0.972370	+	0.233445i	$0.0750000\pi$
$83$	1.73278	+	0.882893i	1.73278	+	0.882893i	0.760406	+	0.649448i	$0.225000\pi$
$84$		0			0
$85$		0			0
$86$	−0.542106	−	0.746144i	−0.542106	−	0.746144i
$87$		0			0
$88$	−0.259506	−	1.63846i	−0.259506	−	1.63846i
$89$	−0.619195	+	0.449871i	−0.619195	+	0.449871i	−0.852640	−	0.522499i	$-0.825000\pi$
$89$	−0.619195	+	0.449871i	−0.619195	+	0.449871i	0.233445	+	0.972370i	$0.425000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.464020	+	0.150769i	−0.464020	+	0.150769i
$95$		0			0
$96$		0			0
$97$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$97$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$98$	−0.0730378	+	0.461143i	−0.0730378	+	0.461143i
$99$		0			0
$100$	−0.355026	+	0.696779i	−0.355026	+	0.696779i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	1.26007	−	0.642040i	1.26007	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$103$	1.26007	−	0.642040i	1.26007	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$104$	−0.257104	+	0.791284i	−0.257104	+	0.791284i
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$109$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$110$	−0.860042	−	0.356241i	−0.860042	−	0.356241i
$111$		0			0
$112$		0			0
$113$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$113$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.428820	−	0.428820i	−0.428820	−	0.428820i
$119$		0			0
$120$		0			0
$121$	−0.919442	+	2.82975i	−0.919442	+	2.82975i
$122$	−0.377723	+	0.192459i	−0.377723	+	0.192459i
$123$		0			0
$124$		0			0
$125$	0.522499	+	0.852640i	0.522499	+	0.852640i
$126$		0			0
$127$	0.253116	−	1.59811i	0.253116	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$127$	0.253116	−	1.59811i	0.253116	−	1.59811i	0.707107	−	0.707107i	$-0.250000\pi$
$128$	0.444039	+	0.871477i	0.444039	+	0.871477i
$129$		0			0
$130$	0.303221	+	0.355026i	0.303221	+	0.355026i
$131$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$131$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.119730	−	0.755944i	−0.119730	−	0.755944i	−0.972370	−	0.233445i	$-0.925000\pi$
$137$	−0.119730	−	0.755944i	−0.119730	−	0.755944i	0.852640	−	0.522499i	$-0.175000\pi$
$138$		0			0
$139$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$139$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$140$		0			0
$141$		0			0
$142$	−0.0652784	−	0.0332610i	−0.0652784	−	0.0332610i
$143$	1.40985	−	1.40985i	1.40985	−	1.40985i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$149$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$157$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$158$	−0.588317	−	0.299762i	−0.588317	−	0.299762i
$159$		0			0
$160$	1.01262	+	0.0796951i	1.01262	+	0.0796951i
$161$		0			0
$162$		0			0
$163$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$163$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$164$	0.661130	−	0.480339i	0.661130	−	0.480339i
$165$		0			0
$166$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$167$	0.690434	−	1.35505i	0.690434	−	1.35505i	−0.233445	−	0.972370i	$-0.575000\pi$
$167$	0.690434	−	1.35505i	0.690434	−	1.35505i	0.923880	−	0.382683i	$-0.125000\pi$
$168$		0			0
$169$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$170$		0			0
$171$		0			0
$172$	−0.701311	−	1.37640i	−0.701311	−	1.37640i
$173$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$173$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$174$		0			0
$175$		0			0
$176$		−	0.784688i		−	0.784688i
$177$		0			0
$178$	0.318395	−	0.162230i	0.318395	−	0.162230i
$179$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$179$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$180$		0			0
$181$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$181$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.807140	+	0.127838i	−0.807140	+	0.127838i
$189$		0			0
$190$		0			0
$191$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$191$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−0.241655	+	0.743739i	−0.241655	+	0.743739i
$197$	1.64637	−	0.838865i	1.64637	−	0.838865i	0.649448	−	0.760406i	$-0.275000\pi$
$197$	1.64637	−	0.838865i	1.64637	−	0.838865i	0.996917	−	0.0784591i	$-0.0250000\pi$
$198$		0			0
$199$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$199$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$200$	0.489040	−	0.673106i	0.489040	−	0.673106i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.0819895	−	1.04178i	−0.0819895	−	1.04178i
$206$	−0.627967	+	0.204039i	−0.627967	+	0.204039i
$207$		0			0
$208$	−0.178671	+	0.350662i	−0.178671	+	0.350662i
$209$		0			0
$210$		0			0
$211$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.707107	+	0.707107i	$0.750000\pi$
$211$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.96929	−	0.154986i	−1.96929	−	0.154986i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−1.32944	−	0.814682i	−1.32944	−	0.814682i
$221$		0			0
$222$		0			0
$223$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$223$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.755944	+	0.119730i	0.755944	+	0.119730i	0.522499	−	0.852640i	$-0.325000\pi$
$227$	0.755944	+	0.119730i	0.755944	+	0.119730i	0.233445	+	0.972370i	$0.425000\pi$
$228$		0			0
$229$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$229$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$233$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$234$		0			0
$235$	−0.399903	+	0.965451i	−0.399903	+	0.965451i
$236$	−0.597045	−	0.821762i	−0.597045	−	0.821762i
$237$		0			0
$238$		0			0
$239$	−1.49487	+	1.08609i	−1.49487	+	1.08609i	−0.522499	+	0.852640i	$0.675000\pi$
$239$	−1.49487	+	1.08609i	−1.49487	+	1.08609i	−0.972370	+	0.233445i	$0.925000\pi$
$240$		0			0
$241$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$241$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$242$	0.630673	−	1.23776i	0.630673	−	1.23776i
$243$		0			0
$244$	−0.675301	+	0.219418i	−0.675301	+	0.219418i
$245$	0.649448	+	0.760406i	0.649448	+	0.760406i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.178671	−	0.431351i	−0.178671	−	0.431351i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−0.233445	+	0.718471i	−0.233445	+	0.718471i
$255$		0			0
$256$	−0.166049	−	0.511046i	−0.166049	−	0.511046i
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$	0.408601	+	0.666776i	0.408601	+	0.666776i
$261$		0			0
$262$		0			0
$263$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$263$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$269$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$		0			0
$273$		0			0
$274$			0.357343i			0.357343i
$275$	−1.77652	+	0.905182i	−1.77652	+	0.905182i
$276$		0			0
$277$	0.0489435	−	0.309017i	0.0489435	−	0.309017i	−0.951057	−	0.309017i	$-0.900000\pi$
$277$	0.0489435	−	0.309017i	0.0489435	−	0.309017i	1.00000			$0$
$278$	−0.403179	−	0.791284i	−0.403179	−	0.791284i
$279$		0			0
$280$		0			0
$281$	−1.44638	+	0.469957i	−1.44638	+	0.469957i	−0.923880	−	0.382683i	$-0.875000\pi$
$281$	−1.44638	+	0.469957i	−1.44638	+	0.469957i	−0.522499	+	0.852640i	$0.675000\pi$
$282$		0			0
$283$	−0.280582	+	0.550672i	−0.280582	+	0.550672i	−0.987688	−	0.156434i	$-0.950000\pi$
$283$	−0.280582	+	0.550672i	−0.280582	+	0.550672i	0.707107	+	0.707107i	$0.250000\pi$
$284$	−0.0992762	−	0.0721283i	−0.0992762	−	0.0721283i
$285$		0			0
$286$	−0.753116	+	0.547171i	−0.753116	+	0.547171i
$287$		0			0
$288$		0			0
$289$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.20582	−	1.20582i	1.20582	−	1.20582i	0.233445	−	0.972370i	$-0.425000\pi$
$293$	1.20582	−	1.20582i	1.20582	−	1.20582i	0.972370	−	0.233445i	$-0.0750000\pi$
$294$		0			0
$295$	−1.29489	+	0.101910i	−1.29489	+	0.101910i
$296$		0			0
$297$		0			0
$298$	−0.852080	−	0.134956i	−0.852080	−	0.134956i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.211964	+	0.882893i	−0.211964	+	0.882893i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$311$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$312$		0			0
$313$	−0.253116	−	1.59811i	−0.253116	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$313$	−0.253116	−	1.59811i	−0.253116	−	1.59811i	0.453990	−	0.891007i	$-0.350000\pi$
$314$	0.673106	−	0.489040i	0.673106	−	0.489040i
$315$		0			0
$316$	−0.894719	−	0.650051i	−0.894719	−	0.650051i
$317$	−0.882893	+	1.73278i	−0.882893	+	1.73278i	−0.233445	+	0.972370i	$0.575000\pi$
$317$	−0.882893	+	1.73278i	−0.882893	+	1.73278i	−0.649448	+	0.760406i	$0.725000\pi$
$318$		0			0
$319$		0			0
$320$	−0.0784591	−	0.0188364i	−0.0784591	−	0.0188364i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.00000			1.00000
$326$		0			0
$327$		0			0
$328$	−0.774680	+	0.394719i	−0.774680	+	0.394719i
$329$		0			0
$330$		0			0
$331$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$331$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$332$	−1.07538	−	1.07538i	−1.07538	−	1.07538i
$333$		0			0
$334$	−0.417359	+	0.574445i	−0.417359	+	0.574445i
$335$		0			0
$336$		0			0
$337$	1.16110	−	0.183900i	1.16110	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$337$	1.16110	−	0.183900i	1.16110	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$338$	0.461143	−	0.0730378i	0.461143	−	0.0730378i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	0.507877	+	1.56308i	0.507877	+	1.56308i
$345$		0			0
$346$		0			0
$347$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$347$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−0.316818	+	2.00031i	−0.316818	+	2.00031i
$353$	0.774181	+	1.51942i	0.774181	+	1.51942i	0.852640	+	0.522499i	$0.175000\pi$
$353$	0.774181	+	1.51942i	0.774181	+	1.51942i	−0.0784591	+	0.996917i	$0.525000\pi$
$354$		0			0
$355$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i
$356$	0.569233	−	0.184955i	0.569233	−	0.184955i
$357$		0			0
$358$		0			0
$359$	−1.37960	−	1.00234i	−1.37960	−	1.00234i	−0.996917	−	0.0784591i	$-0.975000\pi$
$359$	−1.37960	−	1.00234i	−1.37960	−	1.00234i	−0.382683	−	0.923880i	$-0.625000\pi$
$360$		0			0
$361$	0.809017	−	0.587785i	0.809017	−	0.587785i
$362$	0.138926	+	0.877145i	0.138926	+	0.877145i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.278768	−	0.142040i	−0.278768	−	0.142040i	0.309017	−	0.951057i	$-0.400000\pi$
$367$	−0.278768	−	0.142040i	−0.278768	−	0.142040i	−0.587785	+	0.809017i	$0.700000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.87869	−	0.297556i	−1.87869	−	0.297556i	−0.891007	−	0.453990i	$-0.850000\pi$
$373$	−1.87869	−	0.297556i	−1.87869	−	0.297556i	−0.987688	+	0.156434i	$0.950000\pi$
$374$		0			0
$375$		0			0
$376$	0.869443			0.869443
$377$		0			0
$378$		0			0
$379$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$379$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.64637	+	0.838865i	1.64637	+	0.838865i	0.996917	+	0.0784591i	$0.0250000\pi$
$383$	1.64637	+	0.838865i	1.64637	+	0.838865i	0.649448	+	0.760406i	$0.275000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$	0.377723	−	0.741322i	0.377723	−	0.741322i
$393$		0			0
$394$	−0.820478	+	0.266589i	−0.820478	+	0.266589i
$395$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$396$		0			0
$397$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$397$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$398$	−0.0228513	+	0.144277i	−0.0228513	+	0.144277i
$399$		0			0
$400$	0.278287	−	0.278287i	0.278287	−	0.278287i
$401$			0.156918i			0.156918i	0.996917	+	0.0784591i	$0.0250000\pi$
$401$			0.156918i			0.156918i	−0.996917	+	0.0784591i	$0.975000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$	−0.0382802	+	0.486395i	−0.0382802	+	0.486395i
$411$		0			0
$412$	−1.09232	+	0.173006i	−1.09232	+	0.173006i
$413$		0			0
$414$		0			0
$415$	−1.89101	+	0.453990i	−1.89101	+	0.453990i
$416$	0.597045	−	0.821762i	0.597045	−	0.821762i
$417$		0			0
$418$		0			0
$419$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$419$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$420$		0			0
$421$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$421$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$422$	0.821762	−	0.418709i	0.821762	−	0.418709i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	0.896802	+	0.215303i	0.896802	+	0.215303i
$431$	0.444039	−	0.144277i	0.444039	−	0.144277i	−0.0784591	−	0.996917i	$-0.525000\pi$
$431$	0.444039	−	0.144277i	0.444039	−	0.144277i	0.522499	+	0.852640i	$0.325000\pi$
$432$		0			0
$433$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$433$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$439$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	−1.00000			$\pi$
$440$	1.26142	+	1.07736i	1.26142	+	1.07736i
$441$		0			0
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	0.178671	−	0.744220i	0.178671	−	0.744220i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.29890			−1.29890			−0.649448	−	0.760406i	$-0.725000\pi$
$449$	−1.29890			−1.29890			−0.649448	+	0.760406i	$0.725000\pi$
$450$		0			0
$451$	2.08355			2.08355
$452$		0			0
$453$		0			0
$454$	−0.339853	−	0.110425i	−0.339853	−	0.110425i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.893911	+	1.23036i	0.893911	+	1.23036i	0.972370	+	0.233445i	$0.0750000\pi$
$461$	0.893911	+	1.23036i	0.893911	+	1.23036i	−0.0784591	+	0.996917i	$0.525000\pi$
$462$		0			0
$463$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$463$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$467$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$468$		0			0
$469$		0			0
$470$	0.254927	−	0.416003i	0.254927	−	0.416003i
$471$		0			0
$472$	0.490622	+	0.962901i	0.490622	+	0.962901i
$473$	0.616129	−	3.89008i	0.616129	−	3.89008i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.768673	−	0.391658i	0.768673	−	0.391658i
$479$	0.469957	−	1.44638i	0.469957	−	1.44638i	−0.382683	−	0.923880i	$-0.625000\pi$
$479$	0.469957	−	1.44638i	0.469957	−	1.44638i	0.852640	−	0.522499i	$-0.175000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.36765	−	1.88241i	1.36765	−	1.88241i
$485$		0			0
$486$		0			0
$487$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$487$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$488$	0.746144	−	0.118178i	0.746144	−	0.118178i
$489$		0			0
$490$	−0.243950	−	0.398090i	−0.243950	−	0.398090i
$491$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$491$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.182557	−	0.760406i	−0.182557	−	0.760406i
$501$		0			0
$502$		0			0
$503$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$503$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−0.574445	+	1.12741i	−0.574445	+	1.12741i
$509$	0.377723	+	0.274431i	0.377723	+	0.274431i	0.760406	−	0.649448i	$-0.225000\pi$
$509$	0.377723	+	0.274431i	0.377723	+	0.274431i	−0.382683	+	0.923880i	$0.625000\pi$
$510$		0			0
$511$		0			0
$512$	−0.113759	−	0.718246i	−0.113759	−	0.718246i
$513$		0			0
$514$		0			0
$515$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$516$		0			0
$517$	−1.85646	−	0.945913i	−1.85646	−	0.945913i
$518$		0			0
$519$		0			0
$520$	−0.318395	−	0.768673i	−0.318395	−	0.768673i
$521$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$521$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$522$		0			0
$523$	0.896802	+	0.142040i	0.896802	+	0.142040i	0.587785	−	0.809017i	$-0.300000\pi$
$523$	0.896802	+	0.142040i	0.896802	+	0.142040i	0.309017	+	0.951057i	$0.400000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.931099	−	0.474419i	−0.931099	−	0.474419i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.61305	+	1.17195i	−1.61305	+	1.17195i
$540$		0			0
$541$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$541$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.809017	+	1.58779i	0.809017	+	1.58779i	0.809017	+	0.587785i	$0.200000\pi$
$547$	0.809017	+	1.58779i	0.809017	+	1.58779i			1.00000i	$0.500000\pi$
$548$	−0.0936302	+	0.591158i	−0.0936302	+	0.591158i
$549$		0			0
$550$	0.885341	−	0.287665i	0.885341	−	0.287665i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.0451398	+	0.138926i	−0.0451398	+	0.138926i
$555$		0			0
$556$	−0.459656	−	1.41467i	−0.459656	−	1.41467i
$557$	1.07538	+	1.07538i	1.07538	+	1.07538i	0.996917	+	0.0784591i	$0.0250000\pi$
$557$	1.07538	+	1.07538i	1.07538	+	1.07538i	0.0784591	+	0.996917i	$0.475000\pi$
$558$		0			0
$559$	−1.16110	+	1.59811i	−1.16110	+	1.59811i
$560$		0			0
$561$		0			0
$562$	0.701311	−	0.111077i	0.701311	−	0.111077i
$563$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$563$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$564$		0			0
$565$		0			0
$566$	0.169608	−	0.233445i	0.169608	−	0.233445i
$567$		0			0
$568$	0.0923176	+	0.0923176i	0.0923176	+	0.0923176i
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	0.0966818	−	0.297556i	0.0966818	−	0.297556i	−0.891007	−	0.453990i	$-0.850000\pi$
$571$	0.0966818	−	0.297556i	0.0966818	−	0.297556i	0.987688	+	0.156434i	$0.0500000\pi$
$572$	−1.38926	+	0.707864i	−1.38926	+	0.707864i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$577$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$578$	0.211964	+	0.416003i	0.211964	+	0.416003i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.644123	+	0.467983i	−0.644123	+	0.467983i
$587$	−0.237907	−	1.50209i	−0.237907	−	1.50209i	−0.760406	−	0.649448i	$-0.775000\pi$
$587$	−0.237907	−	1.50209i	−0.237907	−	1.50209i	0.522499	−	0.852640i	$-0.325000\pi$
$588$		0			0
$589$		0			0
$590$	0.604573	+	0.0475809i	0.604573	+	0.0475809i
$591$		0			0
$592$		0			0
$593$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$593$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$594$		0			0
$595$		0			0
$596$	−1.37425	−	0.446521i	−1.37425	−	0.446521i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$601$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.13863	−	2.74889i	−1.13863	−	2.74889i
$606$		0			0
$607$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$607$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$608$		0			0
$609$		0			0
$610$	0.162230	−	0.391658i	0.162230	−	0.391658i
$611$	0.614234	+	0.845420i	0.614234	+	0.845420i
$612$		0			0
$613$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$613$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.474419	−	0.931099i	0.474419	−	0.931099i	−0.522499	−	0.852640i	$-0.675000\pi$
$617$	0.474419	−	0.931099i	0.474419	−	0.931099i	0.996917	−	0.0784591i	$-0.0250000\pi$
$618$		0			0
$619$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$619$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$626$			0.755445i			0.755445i
$627$		0			0
$628$	1.24167	−	0.632662i	1.24167	−	0.632662i
$629$		0			0
$630$		0			0
$631$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$631$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$632$	0.832005	+	0.832005i	0.832005	+	0.832005i
$633$		0			0
$634$	0.533698	−	0.734572i	0.533698	−	0.734572i
$635$	0.845420	+	1.37960i	0.845420	+	1.37960i
$636$		0			0
$637$	0.987688	−	0.156434i	0.987688	−	0.156434i
$638$		0			0
$639$		0			0
$640$	−0.903629	−	0.374295i	−0.903629	−	0.374295i
$641$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$641$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$647$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$648$		0			0
$649$		−	2.58978i		−	2.58978i
$650$	−0.461143	−	0.0730378i	−0.461143	−	0.0730378i
$651$		0			0
$652$		0			0
$653$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$653$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$654$		0			0
$655$		0			0
$656$	−0.391138	+	0.127088i	−0.391138	+	0.127088i
$657$		0			0
$658$		0			0
$659$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$659$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$660$		0			0
$661$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$661$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$662$		0			0
$663$		0			0
$664$	0.951057	+	1.30902i	0.951057	+	1.30902i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.840958	+	0.840958i	−0.840958	+	0.840958i
$669$		0			0
$670$		0			0
$671$	−1.72176	−	0.559433i	−1.72176	−	0.559433i
$672$		0			0
$673$	−1.39680	−	0.221232i	−1.39680	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$673$	−1.39680	−	0.221232i	−1.39680	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$674$	−0.548863			−0.548863
$675$		0			0
$676$	0.782013			0.782013
$677$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$677$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.77652	−	0.905182i	−1.77652	−	0.905182i	−0.923880	−	0.382683i	$-0.875000\pi$
$683$	−1.77652	−	0.905182i	−1.77652	−	0.905182i	−0.852640	−	0.522499i	$-0.825000\pi$
$684$		0			0
$685$	0.581990	+	0.497066i	0.581990	+	0.497066i
$686$		0			0
$687$		0			0
$688$	0.121616	+	0.767853i	0.121616	+	0.767853i
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.84956	−	0.444039i	−1.84956	−	0.444039i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	0.0497145	−	0.153006i	0.0497145	−	0.153006i
$705$		0			0
$706$	−0.246033	−	0.757212i	−0.246033	−	0.757212i
$707$		0			0
$708$		0			0
$709$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$709$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$710$	0.0712394	−	0.0171031i	0.0712394	−	0.0171031i
$711$		0			0
$712$	−0.628949	+	0.0996158i	−0.628949	+	0.0996158i
$713$		0			0
$714$		0			0
$715$	−0.156434	+	1.98769i	−0.156434	+	1.98769i
$716$		0			0
$717$		0			0
$718$	0.562984	+	0.562984i	0.562984	+	0.562984i
$719$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$719$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$		0			0
$722$	−0.416003	+	0.211964i	−0.416003	+	0.211964i
$723$		0			0
$724$			1.48748i			1.48748i
$725$		0			0
$726$		0			0
$727$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i	0.891007	+	0.453990i	$0.150000\pi$
$727$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i	−0.987688	+	0.156434i	$0.950000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$733$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$734$	0.118178	+	0.0858611i	0.118178	+	0.0858611i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$739$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.110958	+	0.110958i	−0.110958	+	0.110958i	−0.760406	−	0.649448i	$-0.775000\pi$
$743$	−0.110958	+	0.110958i	−0.110958	+	0.110958i	0.649448	+	0.760406i	$0.275000\pi$
$744$		0			0
$745$	−1.40505	+	1.20002i	−1.40505	+	1.20002i
$746$	0.844613	+	0.274431i	0.844613	+	0.274431i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$751$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$752$	0.406203	+	0.0643362i	0.406203	+	0.0643362i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$757$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.00234	+	1.37960i	1.00234	+	1.37960i	0.923880	+	0.382683i	$0.125000\pi$
$761$	1.00234	+	1.37960i	1.00234	+	1.37960i	0.0784591	+	0.996917i	$0.475000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.697940	−	0.507083i	−0.697940	−	0.507083i
$767$	−0.589686	+	1.15732i	−0.589686	+	1.15732i
$768$		0			0
$769$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$769$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.203192	−	1.28290i	0.203192	−	1.28290i	−0.649448	−	0.760406i	$-0.725000\pi$
$773$	0.203192	−	1.28290i	0.203192	−	1.28290i	0.852640	−	0.522499i	$-0.175000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$782$		0			0
$783$		0			0
$784$	0.231327	−	0.318395i	0.231327	−	0.318395i
$785$	0.139815	−	1.77652i	0.139815	−	1.77652i
$786$		0			0
$787$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$787$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$788$	−1.42718	+	0.226043i	−1.42718	+	0.226043i
$789$		0			0
$790$	0.642040	−	0.154140i	0.642040	−	0.154140i
$791$		0			0
$792$		0			0
$793$	0.642040	+	0.642040i	0.642040	+	0.642040i
$794$		0			0
$795$		0			0
$796$	−0.0756064	+	0.232693i	−0.0756064	+	0.232693i
$797$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$797$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$798$		0			0
$799$		0			0
$800$	−0.821762	+	0.597045i	−0.821762	+	0.597045i
$801$		0			0
$802$	0.0114610	−	0.0723617i	0.0114610	−	0.0723617i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$811$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−0.190772	+	0.794622i	−0.190772	+	0.794622i
$821$	−0.444039	−	0.144277i	−0.444039	−	0.144277i	0.0784591	−	0.996917i	$-0.475000\pi$
$821$	−0.444039	−	0.144277i	−0.444039	−	0.144277i	−0.522499	+	0.852640i	$0.675000\pi$
$822$		0			0
$823$	−0.896802	−	0.142040i	−0.896802	−	0.142040i	−0.309017	−	0.951057i	$-0.600000\pi$
$823$	−0.896802	−	0.142040i	−0.896802	−	0.142040i	−0.587785	+	0.809017i	$0.700000\pi$
$824$	1.17663			1.17663
$825$		0			0
$826$		0			0
$827$	0.154986	+	0.0245474i	0.154986	+	0.0245474i	0.233445	−	0.972370i	$-0.425000\pi$
$827$	0.154986	+	0.0245474i	0.154986	+	0.0245474i	−0.0784591	+	0.996917i	$0.525000\pi$
$828$		0			0
$829$	1.87869	+	0.610425i	1.87869	+	0.610425i	0.987688	+	0.156434i	$0.0500000\pi$
$829$	1.87869	+	0.610425i	1.87869	+	0.610425i	0.891007	+	0.453990i	$0.150000\pi$
$830$	0.905182	−	0.0712394i	0.905182	−	0.0712394i
$831$		0			0
$832$	−0.0570554	+	0.0570554i	−0.0570554	+	0.0570554i
$833$		0			0
$834$		0			0
$835$	0.355026	+	1.47879i	0.355026	+	1.47879i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.23036	−	0.893911i	1.23036	−	0.893911i	0.233445	−	0.972370i	$-0.425000\pi$
$839$	1.23036	−	0.893911i	1.23036	−	0.893911i	0.996917	+	0.0784591i	$0.0250000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$	1.46916	−	0.477360i	1.46916	−	0.477360i
$845$	0.522499	−	0.852640i	0.522499	−	0.852640i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$853$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$	0.183900	−	0.253116i	0.183900	−	0.253116i	−0.707107	−	0.707107i	$-0.750000\pi$
$859$	0.183900	−	0.253116i	0.183900	−	0.253116i	0.891007	+	0.453990i	$0.150000\pi$
$860$	1.42718	+	0.591158i	1.42718	+	0.591158i
$861$		0			0
$862$	−0.215303	+	0.0341007i	−0.215303	+	0.0341007i
$863$	1.68429	−	0.266765i	1.68429	−	0.266765i	0.760406	−	0.649448i	$-0.225000\pi$
$863$	1.68429	−	0.266765i	1.68429	−	0.266765i	0.923880	+	0.382683i	$0.125000\pi$
$864$		0			0
$865$		0			0
$866$	0.522000	−	0.718471i	0.522000	−	0.718471i
$867$		0			0
$868$		0			0
$869$	−0.871338	−	2.68170i	−0.871338	−	2.68170i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$877$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$878$	0.249179	+	0.489040i	0.249179	+	0.489040i
$879$		0			0
$880$	0.509614	+	0.596682i	0.509614	+	0.596682i
$881$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$881$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$882$		0			0
$883$	0.734572	−	1.44168i	0.734572	−	1.44168i	−0.156434	−	0.987688i	$-0.550000\pi$
$883$	0.734572	−	1.44168i	0.734572	−	1.44168i	0.891007	−	0.453990i	$-0.150000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$887$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$888$		0			0
$889$		0			0
$890$	−0.136749	+	0.330142i	−0.136749	+	0.330142i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.598976	+	0.0948685i	0.598976	+	0.0948685i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.960814	−	0.152178i	−0.960814	−	0.152178i
$903$		0			0
$904$		0			0
$905$	1.62182	+	0.993851i	1.62182	+	0.993851i
$906$		0			0
$907$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$907$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$908$	−0.533291	−	0.271726i	−0.533291	−	0.271726i
$909$		0			0
$910$		0			0
$911$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$911$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$912$		0			0
$913$	−0.606573	−	3.82975i	−0.606573	−	3.82975i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.156434	−	0.987688i	$-0.450000\pi$
$919$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$920$		0			0
$921$		0			0
$922$	−0.322357	−	0.632662i	−0.322357	−	0.632662i
$923$	−0.0245474	+	0.154986i	−0.0245474	+	0.154986i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.526961	−	1.62182i	0.526961	−	1.62182i	−0.233445	−	0.972370i	$-0.575000\pi$
$929$	0.526961	−	1.62182i	0.526961	−	1.62182i	0.760406	−	0.649448i	$-0.225000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$937$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$938$		0			0
$939$		0			0
$940$	0.530730	−	0.621405i	0.530730	−	0.621405i
$941$	0.0922342	−	0.126949i	0.0922342	−	0.126949i	−0.760406	−	0.649448i	$-0.775000\pi$
$941$	0.0922342	−	0.126949i	0.0922342	−	0.126949i	0.852640	+	0.522499i	$0.175000\pi$
$942$		0			0
$943$		0			0
$944$	0.157966	+	0.486171i	0.157966	+	0.486171i
$945$		0			0
$946$	−0.568246	+	1.74888i	−0.568246	+	1.74888i
$947$	1.15732	−	0.589686i	1.15732	−	0.589686i	0.233445	−	0.972370i	$-0.425000\pi$
$947$	1.15732	−	0.589686i	1.15732	−	0.589686i	0.923880	+	0.382683i	$0.125000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$953$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$954$		0			0
$955$		0			0
$956$	1.37425	−	0.446521i	1.37425	−	0.446521i
$957$		0			0
$958$	−0.322357	+	0.632662i	−0.322357	+	0.632662i
$959$		0			0
$960$		0			0
$961$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$967$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$968$	−1.75046	+	1.75046i	−1.75046	+	1.75046i
$969$		0			0
$970$		0			0
$971$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$971$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.357343			0.357343
$977$	1.28290	+	0.203192i	1.28290	+	0.203192i	0.760406	−	0.649448i	$-0.225000\pi$
$977$	1.28290	+	0.203192i	1.28290	+	0.203192i	0.522499	+	0.852640i	$0.325000\pi$
$978$		0			0
$979$	1.45133	+	0.471565i	1.45133	+	0.471565i
$980$	−0.299263	−	0.722486i	−0.299263	−	0.722486i
$981$		0			0
$982$		0			0
$983$	1.51942	+	0.774181i	1.51942	+	0.774181i	0.996917	−	0.0784591i	$-0.0250000\pi$
$983$	1.51942	+	0.774181i	1.51942	+	0.774181i	0.522499	+	0.852640i	$0.325000\pi$
$984$		0			0
$985$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$991$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.203192	+	0.237907i	0.203192	+	0.237907i
$996$		0			0
$997$	−0.896802	−	1.76007i	−0.896802	−	1.76007i	−0.587785	−	0.809017i	$-0.700000\pi$
$997$	−0.896802	−	1.76007i	−0.896802	−	1.76007i	−0.309017	−	0.951057i	$-0.600000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.1.er.a.298.2	✓	32
3.2	odd	2	inner	2925.1.er.a.298.3	yes	32
13.12	even	2	inner	2925.1.er.a.298.3	yes	32
25.12	odd	20	inner	2925.1.er.a.2287.3	yes	32
39.38	odd	2	CM	2925.1.er.a.298.2	✓	32
75.62	even	20	inner	2925.1.er.a.2287.2	yes	32
325.12	odd	20	inner	2925.1.er.a.2287.2	yes	32
975.662	even	20	inner	2925.1.er.a.2287.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2925.1.er.a.298.2	✓	32	1.1	even	1	trivial
2925.1.er.a.298.2	✓	32	39.38	odd	2	CM
2925.1.er.a.298.3	yes	32	3.2	odd	2	inner
2925.1.er.a.298.3	yes	32	13.12	even	2	inner
2925.1.er.a.2287.2	yes	32	75.62	even	20	inner
2925.1.er.a.2287.2	yes	32	325.12	odd	20	inner
2925.1.er.a.2287.3	yes	32	25.12	odd	20	inner
2925.1.er.a.2287.3	yes	32	975.662	even	20	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2925.er (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.461143 - 0.0730378i) q^{2} +(-0.743739 - 0.241655i) q^{4} +(-0.760406 + 0.649448i) q^{5} +(0.741322 + 0.377723i) q^{8} +O(q^{10})\)	\(q+(-0.461143 - 0.0730378i) q^{2} +(-0.743739 - 0.241655i) q^{4} +(-0.760406 + 0.649448i) q^{5} +(0.741322 + 0.377723i) q^{8} +(0.398090 - 0.243950i) q^{10} +(-1.17195 - 1.61305i) q^{11} +(0.156434 + 0.987688i) q^{13} +(0.318395 + 0.231327i) q^{16} +(0.722486 - 0.299263i) q^{20} +(0.422621 + 0.829441i) q^{22} +(0.156434 - 0.987688i) q^{25} -0.466891i q^{26} +(-0.718246 - 0.718246i) q^{32} +(-0.809017 + 0.194228i) q^{40} +(-0.614234 + 0.845420i) q^{41} +(1.39680 + 1.39680i) q^{43} +(0.481821 + 1.48289i) q^{44} +(0.931099 - 0.474419i) q^{47} -1.00000i q^{49} +(-0.144277 + 0.444039i) q^{50} +(0.122334 - 0.772385i) q^{52} +(1.93874 + 0.465451i) q^{55} +(1.05083 + 0.763472i) q^{59} +(0.734572 - 0.533698i) q^{61} +(0.0474275 + 0.0652784i) q^{64} +(-0.760406 - 0.649448i) q^{65} +(0.149238 + 0.0484904i) q^{71} +(1.34500 + 0.437016i) q^{79} +(-0.392344 + 0.0308782i) q^{80} +(0.344997 - 0.344997i) q^{82} +(1.73278 + 0.882893i) q^{83} +(-0.542106 - 0.746144i) q^{86} +(-0.259506 - 1.63846i) q^{88} +(-0.619195 + 0.449871i) q^{89} +(-0.464020 + 0.150769i) q^{94} +(-0.0730378 + 0.461143i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{10} + 8 q^{16} - 8 q^{22} - 8 q^{40} + 8 q^{43} + 8 q^{52} + 40 q^{64} - 32 q^{82} + 24 q^{88} - 40 q^{94}+O(q^{100}) \) 32 * q + 8 * q^10 + 8 * q^16 - 8 * q^22 - 8 * q^40 + 8 * q^43 + 8 * q^52 + 40 * q^64 - 32 * q^82 + 24 * q^88 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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