LMFDB - Embedded newform 2925.1.er.a.2638.3

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			2638.3
Root			$0.233445 - 0.972370i$ of defining polynomial
Character	$\chi$	$=$	2925.2638
Dual form			2925.1.er.a.1702.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.163474 + 1.03213i) q^{2} +(-0.0875153 + 0.0284354i) q^{4} +(0.0784591 + 0.996917i) q^{5} +(0.430763 + 0.845420i) q^{8} +O(q^{10})$	\(q+(0.163474 + 1.03213i) q^{2} +(-0.0875153 + 0.0284354i) q^{4} +(0.0784591 + 0.996917i) q^{5} +(0.430763 + 0.845420i) q^{8} +(-1.01612 + 0.243950i) q^{10} +(-0.763472 + 1.05083i) q^{11} +(-0.987688 - 0.156434i) q^{13} +(-0.876612 + 0.636896i) q^{16} +(-0.0352141 - 0.0850145i) q^{20} +(-1.20940 - 0.616221i) q^{22} +(-0.987688 + 0.156434i) q^{25} -1.04500i q^{26} +(-0.129733 - 0.129733i) q^{32} +(-0.809017 + 0.495766i) q^{40} +(-0.274431 - 0.377723i) q^{41} +(0.221232 + 0.221232i) q^{43} +(0.0369347 - 0.113673i) q^{44} +(-0.211964 + 0.416003i) q^{47} -1.00000i q^{49} +(-0.322922 - 0.993851i) q^{50} +(0.0908861 - 0.0143949i) q^{52} +(-1.10749 - 0.678671i) q^{55} +(1.61305 - 1.17195i) q^{59} +(1.44168 + 1.04744i) q^{61} +(-0.524202 + 0.721502i) q^{64} +(0.0784591 - 0.996917i) q^{65} +(-1.44638 + 0.469957i) q^{71} +(1.34500 - 0.437016i) q^{79} +(-0.703710 - 0.823939i) q^{80} +(0.344997 - 0.344997i) q^{82} +(0.774181 + 1.51942i) q^{83} +(-0.192175 + 0.264506i) q^{86} +(-1.21727 - 0.192796i) q^{88} +(1.49487 + 1.08609i) q^{89} +(-0.464020 - 0.150769i) q^{94} +(1.03213 - 0.163474i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$e\left(\frac{19}{20}\right)$	$-1$

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 $

$2$	0.163474	+	1.03213i	0.163474	+	1.03213i	0.923880	+	0.382683i	$0.125000\pi$
$2$	0.163474	+	1.03213i	0.163474	+	1.03213i	−0.760406	+	0.649448i	$0.775000\pi$
$3$		0			0
$3$		0			0
$4$	−0.0875153	+	0.0284354i	−0.0875153	+	0.0284354i
$5$	0.0784591	+	0.996917i	0.0784591	+	0.996917i
$5$	0.0784591	+	0.996917i	0.0784591	+	0.996917i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	0.430763	+	0.845420i	0.430763	+	0.845420i
$9$		0			0
$10$	−1.01612	+	0.243950i	−1.01612	+	0.243950i
$11$	−0.763472	+	1.05083i	−0.763472	+	1.05083i	0.233445	+	0.972370i	$0.425000\pi$
$11$	−0.763472	+	1.05083i	−0.763472	+	1.05083i	−0.996917	+	0.0784591i	$0.975000\pi$
$12$		0			0
$13$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$13$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$14$		0			0
$15$		0			0
$16$	−0.876612	+	0.636896i	−0.876612	+	0.636896i
$17$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$17$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$18$		0			0
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$	−0.0352141	−	0.0850145i	−0.0352141	−	0.0850145i
$21$		0			0
$22$	−1.20940	−	0.616221i	−1.20940	−	0.616221i
$23$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$23$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$24$		0			0
$25$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$26$		−	1.04500i		−	1.04500i
$27$		0			0
$28$		0			0
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	−0.129733	−	0.129733i	−0.129733	−	0.129733i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$37$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$38$		0			0
$39$		0			0
$40$	−0.809017	+	0.495766i	−0.809017	+	0.495766i
$41$	−0.274431	−	0.377723i	−0.274431	−	0.377723i	0.649448	−	0.760406i	$-0.275000\pi$
$41$	−0.274431	−	0.377723i	−0.274431	−	0.377723i	−0.923880	+	0.382683i	$0.875000\pi$
$42$		0			0
$43$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$43$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$44$	0.0369347	−	0.113673i	0.0369347	−	0.113673i
$45$		0			0
$46$		0			0
$47$	−0.211964	+	0.416003i	−0.211964	+	0.416003i	−0.972370	−	0.233445i	$-0.925000\pi$
$47$	−0.211964	+	0.416003i	−0.211964	+	0.416003i	0.760406	+	0.649448i	$0.225000\pi$
$48$		0			0
$49$		−	1.00000i		−	1.00000i
$50$	−0.322922	−	0.993851i	−0.322922	−	0.993851i
$51$		0			0
$52$	0.0908861	−	0.0143949i	0.0908861	−	0.0143949i
$53$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$53$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$54$		0			0
$55$	−1.10749	−	0.678671i	−1.10749	−	0.678671i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.61305	−	1.17195i	1.61305	−	1.17195i	0.760406	−	0.649448i	$-0.225000\pi$
$59$	1.61305	−	1.17195i	1.61305	−	1.17195i	0.852640	−	0.522499i	$-0.175000\pi$
$60$		0			0
$61$	1.44168	+	1.04744i	1.44168	+	1.04744i	0.987688	+	0.156434i	$0.0500000\pi$
$61$	1.44168	+	1.04744i	1.44168	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$62$		0			0
$63$		0			0
$64$	−0.524202	+	0.721502i	−0.524202	+	0.721502i
$65$	0.0784591	−	0.996917i	0.0784591	−	0.996917i
$66$		0			0
$67$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$67$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.44638	+	0.469957i	−1.44638	+	0.469957i	−0.923880	−	0.382683i	$-0.875000\pi$
$71$	−1.44638	+	0.469957i	−1.44638	+	0.469957i	−0.522499	+	0.852640i	$0.675000\pi$
$72$		0			0
$73$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$73$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$79$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$80$	−0.703710	−	0.823939i	−0.703710	−	0.823939i
$81$		0			0
$82$	0.344997	−	0.344997i	0.344997	−	0.344997i
$83$	0.774181	+	1.51942i	0.774181	+	1.51942i	0.852640	+	0.522499i	$0.175000\pi$
$83$	0.774181	+	1.51942i	0.774181	+	1.51942i	−0.0784591	+	0.996917i	$0.525000\pi$
$84$		0			0
$85$		0			0
$86$	−0.192175	+	0.264506i	−0.192175	+	0.264506i
$87$		0			0
$88$	−1.21727	−	0.192796i	−1.21727	−	0.192796i
$89$	1.49487	+	1.08609i	1.49487	+	1.08609i	0.972370	+	0.233445i	$0.0750000\pi$
$89$	1.49487	+	1.08609i	1.49487	+	1.08609i	0.522499	+	0.852640i	$0.325000\pi$
$90$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$97$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$98$	1.03213	−	0.163474i	1.03213	−	0.163474i
$99$		0			0
$100$	0.0819895	−	0.0417758i	0.0819895	−	0.0417758i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−0.642040	+	1.26007i	−0.642040	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$103$	−0.642040	+	1.26007i	−0.642040	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$104$	−0.293207	−	0.902398i	−0.293207	−	0.902398i
$105$		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	$-0.700000\pi$
$109$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$110$	0.519433	−	1.25402i	0.519433	−	1.25402i
$111$		0			0
$112$		0			0
$113$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$113$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.47329	+	1.47329i	1.47329	+	1.47329i
$119$		0			0
$120$		0			0
$121$	−0.212335	−	0.653500i	−0.212335	−	0.653500i
$122$	−0.845420	+	1.65923i	−0.845420	+	1.65923i
$123$		0			0
$124$		0			0
$125$	−0.233445	−	0.972370i	−0.233445	−	0.972370i
$126$		0			0
$127$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.891007	−	0.453990i	$-0.850000\pi$
$127$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$128$	−0.993851	−	0.506393i	−0.993851	−	0.506393i
$129$		0			0
$130$	1.04178	−	0.0819895i	1.04178	−	0.0819895i
$131$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$131$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.82501	−	0.289053i	−1.82501	−	0.289053i	−0.852640	−	0.522499i	$-0.825000\pi$
$137$	−1.82501	−	0.289053i	−1.82501	−	0.289053i	−0.972370	+	0.233445i	$0.925000\pi$
$138$		0			0
$139$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$139$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$140$		0			0
$141$		0			0
$142$	−0.721502	−	1.41603i	−0.721502	−	1.41603i
$143$	0.918458	−	0.918458i	0.918458	−	0.918458i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$149$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$150$		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
$157$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$157$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$158$	0.670929	+	1.31677i	0.670929	+	1.31677i
$159$		0			0
$160$	0.119155	−	0.139512i	0.119155	−	0.139512i
$161$		0			0
$162$		0			0
$163$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$163$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$164$	0.0347576	+	0.0252529i	0.0347576	+	0.0252529i
$165$		0			0
$166$	−1.44168	+	1.04744i	−1.44168	+	1.04744i
$167$	−0.139815	+	0.0712394i	−0.139815	+	0.0712394i	−0.522499	−	0.852640i	$-0.675000\pi$
$167$	−0.139815	+	0.0712394i	−0.139815	+	0.0712394i	0.382683	+	0.923880i	$0.375000\pi$
$168$		0			0
$169$	0.951057	+	0.309017i	0.951057	+	0.309017i
$170$		0			0
$171$		0			0
$172$	−0.0256520	−	0.0130703i	−0.0256520	−	0.0130703i
$173$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$173$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$174$		0			0
$175$		0			0
$176$		−	1.40742i		−	1.40742i
$177$		0			0
$178$	−0.876612	+	1.72045i	−0.876612	+	1.72045i
$179$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$179$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$180$		0			0
$181$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$181$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.00672087	−	0.0424339i	0.00672087	−	0.0424339i
$189$		0			0
$190$		0			0
$191$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$191$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$192$		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
$196$	0.0284354	+	0.0875153i	0.0284354	+	0.0875153i
$197$	0.347469	−	0.681947i	0.347469	−	0.681947i	−0.649448	−	0.760406i	$-0.725000\pi$
$197$	0.347469	−	0.681947i	0.347469	−	0.681947i	0.996917	+	0.0784591i	$0.0250000\pi$
$198$		0			0
$199$			1.97538i			1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$199$			1.97538i			1.97538i	−0.156434	+	0.987688i	$0.550000\pi$
$200$	−0.557713	−	0.767626i	−0.557713	−	0.767626i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.355026	−	0.303221i	0.355026	−	0.303221i
$206$	−1.40552	−	0.456680i	−1.40552	−	0.456680i
$207$		0			0
$208$	0.965451	−	0.491922i	0.965451	−	0.491922i
$209$		0			0
$210$		0			0
$211$	0.253116	+	0.183900i	0.253116	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$211$	0.253116	+	0.183900i	0.253116	+	0.183900i	−0.453990	+	0.891007i	$0.650000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.203192	+	0.237907i	−0.203192	+	0.237907i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.116221	+	0.0279021i	0.116221	+	0.0279021i
$221$		0			0
$222$		0			0
$223$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$223$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.289053	+	1.82501i	0.289053	+	1.82501i	0.522499	+	0.852640i	$0.325000\pi$
$227$	0.289053	+	1.82501i	0.289053	+	1.82501i	−0.233445	+	0.972370i	$0.575000\pi$
$228$		0			0
$229$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$229$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$233$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$234$		0			0
$235$	−0.431351	−	0.178671i	−0.431351	−	0.178671i
$236$	−0.107841	+	0.148431i	−0.107841	+	0.148431i
$237$		0			0
$238$		0			0
$239$	−0.619195	−	0.449871i	−0.619195	−	0.449871i	0.233445	−	0.972370i	$-0.425000\pi$
$239$	−0.619195	−	0.449871i	−0.619195	−	0.449871i	−0.852640	+	0.522499i	$0.825000\pi$
$240$		0			0
$241$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$241$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$242$	0.639787	−	0.325988i	0.639787	−	0.325988i
$243$		0			0
$244$	−0.155953	−	0.0506723i	−0.155953	−	0.0506723i
$245$	0.996917	−	0.0784591i	0.996917	−	0.0784591i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.965451	−	0.399903i	0.965451	−	0.399903i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−0.522499	−	1.60809i	−0.522499	−	1.60809i
$255$		0			0
$256$	0.0846061	−	0.260391i	0.0846061	−	0.260391i
$257$		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
$260$	0.0214814	+	0.0894765i	0.0214814	+	0.0894765i
$261$		0			0
$262$		0			0
$263$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$263$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$269$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$		−	1.93090i		−	1.93090i
$275$	0.589686	−	1.15732i	0.589686	−	1.15732i
$276$		0			0
$277$	1.95106	−	0.309017i	1.95106	−	0.309017i	0.951057	−	0.309017i	$-0.100000\pi$
$277$	1.95106	−	0.309017i	1.95106	−	0.309017i	1.00000			$0$
$278$	1.77106	+	0.902398i	1.77106	+	0.902398i
$279$		0			0
$280$		0			0
$281$	−0.149238	−	0.0484904i	−0.149238	−	0.0484904i	0.233445	−	0.972370i	$-0.425000\pi$
$281$	−0.149238	−	0.0484904i	−0.149238	−	0.0484904i	−0.382683	+	0.923880i	$0.625000\pi$
$282$		0			0
$283$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	−0.707107	−	0.707107i	$-0.750000\pi$
$283$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$284$	0.113217	−	0.0822568i	0.113217	−	0.0822568i
$285$		0			0
$286$	1.09811	+	0.797826i	1.09811	+	0.797826i
$287$		0			0
$288$		0			0
$289$	0.587785	−	0.809017i	0.587785	−	0.809017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.37514	−	1.37514i	1.37514	−	1.37514i	0.522499	−	0.852640i	$-0.325000\pi$
$293$	1.37514	−	1.37514i	1.37514	−	1.37514i	0.852640	−	0.522499i	$-0.175000\pi$
$294$		0			0
$295$	1.29489	+	1.51612i	1.29489	+	1.51612i
$296$		0			0
$297$		0			0
$298$	0.125117	+	0.789959i	0.125117	+	0.789959i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.931099	+	1.51942i	−0.931099	+	1.51942i
$306$		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	$-0.300000\pi$
$311$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$312$		0			0
$313$	1.59811	+	0.253116i	1.59811	+	0.253116i	0.891007	−	0.453990i	$-0.150000\pi$
$313$	1.59811	+	0.253116i	1.59811	+	0.253116i	0.707107	+	0.707107i	$0.250000\pi$
$314$	0.767626	+	0.557713i	0.767626	+	0.557713i
$315$		0			0
$316$	−0.105281	+	0.0764912i	−0.105281	+	0.0764912i
$317$	−1.51942	+	0.774181i	−1.51942	+	0.774181i	−0.996917	−	0.0784591i	$-0.975000\pi$
$317$	−1.51942	+	0.774181i	−1.51942	+	0.774181i	−0.522499	+	0.852640i	$0.675000\pi$
$318$		0			0
$319$		0			0
$320$	−0.760406	−	0.465977i	−0.760406	−	0.465977i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.00000			1.00000
$326$		0			0
$327$		0			0
$328$	0.201119	−	0.394719i	0.201119	−	0.394719i
$329$		0			0
$330$		0			0
$331$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$331$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$332$	−0.110958	−	0.110958i	−0.110958	−	0.110958i
$333$		0			0
$334$	−0.0963845	−	0.132662i	−0.0963845	−	0.132662i
$335$		0			0
$336$		0			0
$337$	0.183900	−	1.16110i	0.183900	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$337$	0.183900	−	1.16110i	0.183900	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$338$	−0.163474	+	1.03213i	−0.163474	+	1.03213i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−0.0917353	+	0.282332i	−0.0917353	+	0.282332i
$345$		0			0
$346$		0			0
$347$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$347$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	0.235375	−	0.0372798i	0.235375	−	0.0372798i
$353$	−1.73278	−	0.882893i	−1.73278	−	0.882893i	−0.972370	−	0.233445i	$-0.925000\pi$
$353$	−1.73278	−	0.882893i	−1.73278	−	0.882893i	−0.760406	−	0.649448i	$-0.775000\pi$
$354$		0			0
$355$	−0.581990	−	1.40505i	−0.581990	−	1.40505i
$356$	−0.161707	−	0.0525418i	−0.161707	−	0.0525418i
$357$		0			0
$358$		0			0
$359$	1.57333	−	1.14309i	1.57333	−	1.14309i	0.649448	−	0.760406i	$-0.275000\pi$
$359$	1.57333	−	1.14309i	1.57333	−	1.14309i	0.923880	−	0.382683i	$-0.125000\pi$
$360$		0			0
$361$	0.809017	+	0.587785i	0.809017	+	0.587785i
$362$	1.96323	+	0.310945i	1.96323	+	0.310945i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.896802	+	1.76007i	0.896802	+	1.76007i	0.587785	+	0.809017i	$0.300000\pi$
$367$	0.896802	+	1.76007i	0.896802	+	1.76007i	0.309017	+	0.951057i	$0.400000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.297556	−	1.87869i	−0.297556	−	1.87869i	−0.453990	−	0.891007i	$-0.650000\pi$
$373$	−0.297556	−	1.87869i	−0.297556	−	1.87869i	0.156434	−	0.987688i	$-0.450000\pi$
$374$		0			0
$375$		0			0
$376$	−0.443003			−0.443003
$377$		0			0
$378$		0			0
$379$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$379$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.347469	+	0.681947i	0.347469	+	0.681947i	0.996917	−	0.0784591i	$-0.0250000\pi$
$383$	0.347469	+	0.681947i	0.347469	+	0.681947i	−0.649448	+	0.760406i	$0.725000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$389$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$390$		0			0
$391$		0			0
$392$	0.845420	−	0.430763i	0.845420	−	0.430763i
$393$		0			0
$394$	0.760661	+	0.247154i	0.760661	+	0.247154i
$395$	0.541196	+	1.30656i	0.541196	+	1.30656i
$396$		0			0
$397$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$397$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$398$	−2.03885	+	0.322922i	−2.03885	+	0.322922i
$399$		0			0
$400$	0.766187	−	0.766187i	0.766187	−	0.766187i
$401$			1.52081i			1.52081i	0.649448	+	0.760406i	$0.275000\pi$
$401$			1.52081i			1.52081i	−0.649448	+	0.760406i	$0.725000\pi$
$402$		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	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$	0.371002	+	0.316865i	0.371002	+	0.316865i
$411$		0			0
$412$	0.0203575	−	0.128532i	0.0203575	−	0.128532i
$413$		0			0
$414$		0			0
$415$	−1.45399	+	0.891007i	−1.45399	+	0.891007i
$416$	0.107841	+	0.148431i	0.107841	+	0.148431i
$417$		0			0
$418$		0			0
$419$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$419$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$	−0.148431	+	0.291312i	−0.148431	+	0.291312i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−0.278768	−	0.170829i	−0.278768	−	0.170829i
$431$	−0.993851	−	0.322922i	−0.993851	−	0.322922i	−0.233445	−	0.972370i	$-0.575000\pi$
$431$	−0.993851	−	0.322922i	−0.993851	−	0.322922i	−0.760406	+	0.649448i	$0.775000\pi$
$432$		0			0
$433$	1.69480	−	0.863541i	1.69480	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$433$	1.69480	−	0.863541i	1.69480	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$434$		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			$1.00000\pi$
$440$	0.0966962	−	1.22864i	0.0966962	−	1.22864i
$441$		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
$445$	−0.965451	+	1.57547i	−0.965451	+	1.57547i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.99383			−1.99383			−0.996917	−	0.0784591i	$-0.975000\pi$
$449$	−1.99383			−1.99383			−0.996917	+	0.0784591i	$0.975000\pi$
$450$		0			0
$451$	0.606443			0.606443
$452$		0			0
$453$		0			0
$454$	−1.83640	+	0.596682i	−1.83640	+	0.596682i
$455$		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
$461$	0.0922342	−	0.126949i	0.0922342	−	0.126949i	−0.760406	−	0.649448i	$-0.775000\pi$
$461$	0.0922342	−	0.126949i	0.0922342	−	0.126949i	0.852640	+	0.522499i	$0.175000\pi$
$462$		0			0
$463$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$463$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$467$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$468$		0			0
$469$		0			0
$470$	0.113898	−	0.474419i	0.113898	−	0.474419i
$471$		0			0
$472$	1.68563	+	0.858871i	1.68563	+	0.858871i
$473$	−0.401381	+	0.0635725i	−0.401381	+	0.0635725i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.363104	−	0.712633i	0.363104	−	0.712633i
$479$	−0.0484904	−	0.149238i	−0.0484904	−	0.149238i	0.923880	−	0.382683i	$-0.125000\pi$
$479$	−0.0484904	−	0.149238i	−0.0484904	−	0.149238i	−0.972370	+	0.233445i	$0.925000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.0371651	+	0.0511534i	0.0371651	+	0.0511534i
$485$		0			0
$486$		0			0
$487$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$487$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$488$	−0.264506	+	1.67002i	−0.264506	+	1.67002i
$489$		0			0
$490$	0.243950	+	1.01612i	0.243950	+	1.01612i
$491$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$491$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$492$		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	$0.5\pi$
$500$	0.0480798	+	0.0784591i	0.0480798	+	0.0784591i
$501$		0			0
$502$		0			0
$503$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$503$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.132662	−	0.0675946i	0.132662	−	0.0675946i
$509$	0.845420	−	0.614234i	0.845420	−	0.614234i	−0.0784591	−	0.996917i	$-0.525000\pi$
$509$	0.845420	−	0.614234i	0.845420	−	0.614234i	0.923880	+	0.382683i	$0.125000\pi$
$510$		0			0
$511$		0			0
$512$	−0.819104	−	0.129733i	−0.819104	−	0.129733i
$513$		0			0
$514$		0			0
$515$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$516$		0			0
$517$	−0.275319	−	0.540344i	−0.275319	−	0.540344i
$518$		0			0
$519$		0			0
$520$	0.876612	−	0.363104i	0.876612	−	0.363104i
$521$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$521$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$522$		0			0
$523$	−0.278768	−	1.76007i	−0.278768	−	1.76007i	−0.587785	−	0.809017i	$-0.700000\pi$
$523$	−0.278768	−	1.76007i	−0.278768	−	1.76007i	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
$533$	0.211964	+	0.416003i	0.211964	+	0.416003i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.05083	+	0.763472i	1.05083	+	0.763472i
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.809017	+	0.412215i	0.809017	+	0.412215i	0.809017	−	0.587785i	$-0.200000\pi$
$547$	0.809017	+	0.412215i	0.809017	+	0.412215i			1.00000i	$0.5\pi$
$548$	0.167936	−	0.0265984i	0.167936	−	0.0265984i
$549$		0			0
$550$	1.29091	+	0.419442i	1.29091	+	0.419442i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.637892	+	1.96323i	0.637892	+	1.96323i
$555$		0			0
$556$	−0.0540874	+	0.166464i	−0.0540874	+	0.166464i
$557$	0.110958	+	0.110958i	0.110958	+	0.110958i	0.760406	−	0.649448i	$-0.225000\pi$
$557$	0.110958	+	0.110958i	0.110958	+	0.110958i	−0.649448	+	0.760406i	$0.725000\pi$
$558$		0			0
$559$	−0.183900	−	0.253116i	−0.183900	−	0.253116i
$560$		0			0
$561$		0			0
$562$	0.0256520	−	0.161960i	0.0256520	−	0.161960i
$563$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$563$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$564$		0			0
$565$		0			0
$566$	−0.379617	−	0.522499i	−0.379617	−	0.522499i
$567$		0			0
$568$	−1.02036	−	1.02036i	−1.02036	−	1.02036i
$569$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$569$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$570$		0			0
$571$	−0.610425	−	1.87869i	−0.610425	−	1.87869i	−0.453990	−	0.891007i	$-0.650000\pi$
$571$	−0.610425	−	1.87869i	−0.610425	−	1.87869i	−0.156434	−	0.987688i	$-0.550000\pi$
$572$	−0.0542624	+	0.106496i	−0.0542624	+	0.106496i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$577$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$578$	0.931099	+	0.474419i	0.931099	+	0.474419i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.64412	+	1.19453i	1.64412	+	1.19453i
$587$	−0.154986	−	0.0245474i	−0.154986	−	0.0245474i	0.0784591	−	0.996917i	$-0.475000\pi$
$587$	−0.154986	−	0.0245474i	−0.154986	−	0.0245474i	−0.233445	+	0.972370i	$0.575000\pi$
$588$		0			0
$589$		0			0
$590$	−1.35316	+	1.58434i	−1.35316	+	1.58434i
$591$		0			0
$592$		0			0
$593$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$593$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$594$		0			0
$595$		0			0
$596$	−0.0669813	+	0.0217635i	−0.0669813	+	0.0217635i
$597$		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	$-0.750000\pi$
$601$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.634826	−	0.262954i	0.634826	−	0.262954i
$606$		0			0
$607$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$607$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$608$		0			0
$609$		0			0
$610$	−1.72045	−	0.712633i	−1.72045	−	0.712633i
$611$	0.274431	−	0.377723i	0.274431	−	0.377723i
$612$		0			0
$613$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$613$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.416003	+	0.211964i	−0.416003	+	0.211964i	−0.649448	−	0.760406i	$-0.725000\pi$
$617$	−0.416003	+	0.211964i	−0.416003	+	0.211964i	0.233445	+	0.972370i	$0.425000\pi$
$618$		0			0
$619$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$619$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.951057	−	0.309017i	0.951057	−	0.309017i
$626$			1.69084i			1.69084i
$627$		0			0
$628$	−0.0379316	+	0.0744449i	−0.0379316	+	0.0744449i
$629$		0			0
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$	0.948838	+	0.948838i	0.948838	+	0.948838i
$633$		0			0
$634$	−1.04744	−	1.44168i	−1.04744	−	1.44168i
$635$	−0.377723	−	1.57333i	−0.377723	−	1.57333i
$636$		0			0
$637$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$638$		0			0
$639$		0			0
$640$	0.426855	−	1.03052i	0.426855	−	1.03052i
$641$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$641$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$642$		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		0.453990	−	0.891007i	$-0.350000\pi$
$647$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$648$		0			0
$649$			2.58978i			2.58978i
$650$	0.163474	+	1.03213i	0.163474	+	1.03213i
$651$		0			0
$652$		0			0
$653$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$653$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$654$		0			0
$655$		0			0
$656$	0.481140	+	0.156332i	0.481140	+	0.156332i
$657$		0			0
$658$		0			0
$659$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$659$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$661$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$662$		0			0
$663$		0			0
$664$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.0102102	−	0.0102102i	0.0102102	−	0.0102102i
$669$		0			0
$670$		0			0
$671$	−2.20136	+	0.715266i	−2.20136	+	0.715266i
$672$		0			0
$673$	−0.221232	−	1.39680i	−0.221232	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$673$	−0.221232	−	1.39680i	−0.221232	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$674$	1.22847			1.22847
$675$		0			0
$676$	−0.0920190			−0.0920190
$677$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$677$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.589686	+	1.15732i	0.589686	+	1.15732i	0.972370	+	0.233445i	$0.0750000\pi$
$683$	0.589686	+	1.15732i	0.589686	+	1.15732i	−0.382683	+	0.923880i	$0.625000\pi$
$684$		0			0
$685$	0.144974	−	1.84206i	0.144974	−	1.84206i
$686$		0			0
$687$		0			0
$688$	−0.334836	−	0.0530328i	−0.334836	−	0.0530328i
$689$		0			0
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.62182	+	0.993851i	1.62182	+	0.993851i
$696$		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
$704$	−0.357962	−	1.10169i	−0.357962	−	1.10169i
$705$		0			0
$706$	0.627999	−	1.93278i	0.627999	−	1.93278i
$707$		0			0
$708$		0			0
$709$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$709$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$710$	1.35505	−	0.830378i	1.35505	−	0.830378i
$711$		0			0
$712$	−0.274265	+	1.73164i	−0.274265	+	1.73164i
$713$		0			0
$714$		0			0
$715$	0.987688	+	0.843566i	0.987688	+	0.843566i
$716$		0			0
$717$		0			0
$718$	1.43702	+	1.43702i	1.43702	+	1.43702i
$719$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$719$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$720$		0			0
$721$		0			0
$722$	−0.474419	+	0.931099i	−0.474419	+	0.931099i
$723$		0			0
$724$			0.175031i			0.175031i
$725$		0			0
$726$		0			0
$727$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$727$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$733$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$734$	−1.67002	+	1.21334i	−1.67002	+	1.21334i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$739$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.07538	−	1.07538i	1.07538	−	1.07538i	0.0784591	−	0.996917i	$-0.475000\pi$
$743$	1.07538	−	1.07538i	1.07538	−	1.07538i	0.996917	−	0.0784591i	$-0.0250000\pi$
$744$		0			0
$745$	0.0600500	+	0.763007i	0.0600500	+	0.763007i
$746$	1.89042	−	0.614234i	1.89042	−	0.614234i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$751$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$752$	−0.0791402	−	0.499672i	−0.0791402	−	0.499672i
$753$		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
$761$	1.14309	−	1.57333i	1.14309	−	1.57333i	0.382683	−	0.923880i	$-0.375000\pi$
$761$	1.14309	−	1.57333i	1.14309	−	1.57333i	0.760406	−	0.649448i	$-0.225000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.647057	+	0.470114i	−0.647057	+	0.470114i
$767$	−1.77652	+	0.905182i	−1.77652	+	0.905182i
$768$		0			0
$769$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$769$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.96929	+	0.311904i	−1.96929	+	0.311904i	−0.972370	+	0.233445i	$0.925000\pi$
$773$	−1.96929	+	0.311904i	−1.96929	+	0.311904i	−0.996917	+	0.0784591i	$0.975000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.610425	−	1.87869i	0.610425	−	1.87869i
$782$		0			0
$783$		0			0
$784$	0.636896	+	0.876612i	0.636896	+	0.876612i
$785$	0.690434	+	0.589686i	0.690434	+	0.589686i
$786$		0			0
$787$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$787$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$788$	−0.0110174	+	0.0695612i	−0.0110174	+	0.0695612i
$789$		0			0
$790$	−1.26007	+	0.772174i	−1.26007	+	0.772174i
$791$		0			0
$792$		0			0
$793$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$794$		0			0
$795$		0			0
$796$	−0.0561707	−	0.172876i	−0.0561707	−	0.172876i
$797$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$797$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$798$		0			0
$799$		0			0
$800$	0.148431	+	0.107841i	0.148431	+	0.107841i
$801$		0			0
$802$	−1.56968	+	0.248613i	−1.56968	+	0.248613i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$809$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$810$		0			0
$811$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$811$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−0.0224480	+	0.0366318i	−0.0224480	+	0.0366318i
$821$	0.993851	−	0.322922i	0.993851	−	0.322922i	0.233445	−	0.972370i	$-0.425000\pi$
$821$	0.993851	−	0.322922i	0.993851	−	0.322922i	0.760406	+	0.649448i	$0.225000\pi$
$822$		0			0
$823$	0.278768	+	1.76007i	0.278768	+	1.76007i	0.587785	+	0.809017i	$0.300000\pi$
$823$	0.278768	+	1.76007i	0.278768	+	1.76007i	−0.309017	+	0.951057i	$0.600000\pi$
$824$	−1.34186			−1.34186
$825$		0			0
$826$		0			0
$827$	−0.237907	−	1.50209i	−0.237907	−	1.50209i	−0.760406	−	0.649448i	$-0.775000\pi$
$827$	−0.237907	−	1.50209i	−0.237907	−	1.50209i	0.522499	−	0.852640i	$-0.325000\pi$
$828$		0			0
$829$	0.297556	−	0.0966818i	0.297556	−	0.0966818i	−0.156434	−	0.987688i	$-0.550000\pi$
$829$	0.297556	−	0.0966818i	0.297556	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$830$	−1.15732	−	1.35505i	−1.15732	−	1.35505i
$831$		0			0
$832$	0.630616	−	0.630616i	0.630616	−	0.630616i
$833$		0			0
$834$		0			0
$835$	−0.0819895	−	0.133795i	−0.0819895	−	0.133795i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.126949	−	0.0922342i	−0.126949	−	0.0922342i	0.522499	−	0.852640i	$-0.325000\pi$
$839$	−0.126949	−	0.0922342i	−0.126949	−	0.0922342i	−0.649448	+	0.760406i	$0.725000\pi$
$840$		0			0
$841$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$	−0.0273808	−	0.00889656i	−0.0273808	−	0.00889656i
$845$	−0.233445	+	0.972370i	−0.233445	+	0.972370i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$853$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$854$		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
$859$	1.16110	+	1.59811i	1.16110	+	1.59811i	0.707107	+	0.707107i	$0.250000\pi$
$859$	1.16110	+	1.59811i	1.16110	+	1.59811i	0.453990	+	0.891007i	$0.350000\pi$
$860$	0.0110174	−	0.0265984i	0.0110174	−	0.0265984i
$861$		0			0
$862$	0.170829	−	1.07857i	0.170829	−	1.07857i
$863$	0.304224	−	1.92080i	0.304224	−	1.92080i	−0.0784591	−	0.996917i	$-0.525000\pi$
$863$	0.304224	−	1.92080i	0.304224	−	1.92080i	0.382683	−	0.923880i	$-0.375000\pi$
$864$		0			0
$865$		0			0
$866$	1.16834	+	1.60809i	1.16834	+	1.60809i
$867$		0			0
$868$		0			0
$869$	−0.567638	+	1.74701i	−0.567638	+	1.74701i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$877$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$878$	−1.09457	−	0.557713i	−1.09457	−	0.557713i
$879$		0			0
$880$	1.40308	−	0.110425i	1.40308	−	0.110425i
$881$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$881$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$882$		0			0
$883$	1.44168	−	0.734572i	1.44168	−	0.734572i	0.453990	−	0.891007i	$-0.350000\pi$
$883$	1.44168	−	0.734572i	1.44168	−	0.734572i	0.987688	+	0.156434i	$0.0500000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$887$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$888$		0			0
$889$		0			0
$890$	−1.78392	−	0.738925i	−1.78392	−	0.738925i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.325939	−	2.05790i	−0.325939	−	2.05790i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	0.0991373	+	0.625928i	0.0991373	+	0.625928i
$903$		0			0
$904$		0			0
$905$	1.84956	+	0.444039i	1.84956	+	0.444039i
$906$		0			0
$907$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$907$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$908$	−0.0771915	−	0.151497i	−0.0771915	−	0.151497i
$909$		0			0
$910$		0			0
$911$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$911$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$912$		0			0
$913$	−2.18771	−	0.346500i	−2.18771	−	0.346500i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.69480	−	0.550672i	−1.69480	−	0.550672i	−0.707107	−	0.707107i	$-0.750000\pi$
$919$	−1.69480	−	0.550672i	−1.69480	−	0.550672i	−0.987688	+	0.156434i	$0.950000\pi$
$920$		0			0
$921$		0			0
$922$	0.146106	+	0.0744449i	0.146106	+	0.0744449i
$923$	1.50209	−	0.237907i	1.50209	−	0.237907i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.600958	−	1.84956i	−0.600958	−	1.84956i	−0.522499	−	0.852640i	$-0.675000\pi$
$929$	−0.600958	−	1.84956i	−0.600958	−	1.84956i	−0.0784591	−	0.996917i	$-0.525000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i	0.891007	+	0.453990i	$0.150000\pi$
$937$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i	−0.987688	+	0.156434i	$0.950000\pi$
$938$		0			0
$939$		0			0
$940$	0.0428304	+	0.00337082i	0.0428304	+	0.00337082i
$941$	−0.893911	−	1.23036i	−0.893911	−	1.23036i	−0.972370	−	0.233445i	$-0.925000\pi$
$941$	−0.893911	−	1.23036i	−0.893911	−	1.23036i	0.0784591	−	0.996917i	$-0.475000\pi$
$942$		0			0
$943$		0			0
$944$	−0.667607	+	2.05468i	−0.667607	+	2.05468i
$945$		0			0
$946$	−0.131230	−	0.403886i	−0.131230	−	0.403886i
$947$	0.905182	−	1.77652i	0.905182	−	1.77652i	0.382683	−	0.923880i	$-0.375000\pi$
$947$	0.905182	−	1.77652i	0.905182	−	1.77652i	0.522499	−	0.852640i	$-0.325000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$953$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$954$		0			0
$955$		0			0
$956$	0.0669813	+	0.0217635i	0.0669813	+	0.0217635i
$957$		0			0
$958$	0.146106	−	0.0744449i	0.146106	−	0.0744449i
$959$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$967$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$968$	0.461016	−	0.461016i	0.461016	−	0.461016i
$969$		0			0
$970$		0			0
$971$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$971$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.93090			−1.93090
$977$	−0.311904	−	1.96929i	−0.311904	−	1.96929i	−0.233445	−	0.972370i	$-0.575000\pi$
$977$	−0.311904	−	1.96929i	−0.311904	−	1.96929i	−0.0784591	−	0.996917i	$-0.525000\pi$
$978$		0			0
$979$	−2.28258	+	0.741655i	−2.28258	+	0.741655i
$980$	−0.0850145	+	0.0352141i	−0.0850145	+	0.0352141i
$981$		0			0
$982$		0			0
$983$	−0.882893	−	1.73278i	−0.882893	−	1.73278i	−0.649448	−	0.760406i	$-0.725000\pi$
$983$	−0.882893	−	1.73278i	−0.882893	−	1.73278i	−0.233445	−	0.972370i	$-0.575000\pi$
$984$		0			0
$985$	0.707107	+	0.292893i	0.707107	+	0.292893i
$986$		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
$995$	−1.96929	+	0.154986i	−1.96929	+	0.154986i
$996$		0			0
$997$	0.278768	+	0.142040i	0.278768	+	0.142040i	0.587785	−	0.809017i	$-0.300000\pi$
$997$	0.278768	+	0.142040i	0.278768	+	0.142040i	−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.2638.3	yes	32
3.2	odd	2	inner	2925.1.er.a.2638.2	yes	32
13.12	even	2	inner	2925.1.er.a.2638.2	yes	32
25.2	odd	20	inner	2925.1.er.a.1702.2	✓	32
39.38	odd	2	CM	2925.1.er.a.2638.3	yes	32
75.2	even	20	inner	2925.1.er.a.1702.3	yes	32
325.77	odd	20	inner	2925.1.er.a.1702.3	yes	32
975.77	even	20	inner	2925.1.er.a.1702.2	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2925.1.er.a.1702.2	✓	32	25.2	odd	20	inner
2925.1.er.a.1702.2	✓	32	975.77	even	20	inner
2925.1.er.a.1702.3	yes	32	75.2	even	20	inner
2925.1.er.a.1702.3	yes	32	325.77	odd	20	inner
2925.1.er.a.2638.2	yes	32	3.2	odd	2	inner
2925.1.er.a.2638.2	yes	32	13.12	even	2	inner
2925.1.er.a.2638.3	yes	32	1.1	even	1	trivial
2925.1.er.a.2638.3	yes	32	39.38	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.163474 + 1.03213i) q^{2} +(-0.0875153 + 0.0284354i) q^{4} +(0.0784591 + 0.996917i) q^{5} +(0.430763 + 0.845420i) q^{8} +O(q^{10})\)	\(q+(0.163474 + 1.03213i) q^{2} +(-0.0875153 + 0.0284354i) q^{4} +(0.0784591 + 0.996917i) q^{5} +(0.430763 + 0.845420i) q^{8} +(-1.01612 + 0.243950i) q^{10} +(-0.763472 + 1.05083i) q^{11} +(-0.987688 - 0.156434i) q^{13} +(-0.876612 + 0.636896i) q^{16} +(-0.0352141 - 0.0850145i) q^{20} +(-1.20940 - 0.616221i) q^{22} +(-0.987688 + 0.156434i) q^{25} -1.04500i q^{26} +(-0.129733 - 0.129733i) q^{32} +(-0.809017 + 0.495766i) q^{40} +(-0.274431 - 0.377723i) q^{41} +(0.221232 + 0.221232i) q^{43} +(0.0369347 - 0.113673i) q^{44} +(-0.211964 + 0.416003i) q^{47} -1.00000i q^{49} +(-0.322922 - 0.993851i) q^{50} +(0.0908861 - 0.0143949i) q^{52} +(-1.10749 - 0.678671i) q^{55} +(1.61305 - 1.17195i) q^{59} +(1.44168 + 1.04744i) q^{61} +(-0.524202 + 0.721502i) q^{64} +(0.0784591 - 0.996917i) q^{65} +(-1.44638 + 0.469957i) q^{71} +(1.34500 - 0.437016i) q^{79} +(-0.703710 - 0.823939i) q^{80} +(0.344997 - 0.344997i) q^{82} +(0.774181 + 1.51942i) q^{83} +(-0.192175 + 0.264506i) q^{86} +(-1.21727 - 0.192796i) q^{88} +(1.49487 + 1.08609i) q^{89} +(-0.464020 - 0.150769i) q^{94} +(1.03213 - 0.163474i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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Groups

Database

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