LMFDB - Embedded newform 2385.1.ca.a.694.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2385,1,Mod(19,2385)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2385, base_ring=CyclotomicField(52))

chi = DirichletCharacter(H, H._module([0, 26, 37]))

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

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

chi := DirichletCharacter("2385.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2385 = 3^{2} \cdot 5 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2385.ca (of order $52$, degree $24$, 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:	$1.19027005513$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$2$ over $\Q(\zeta_{52})$
Coefficient field:	$\Q(\zeta_{104})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} - x^{44} + x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{52}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{52} - \cdots)$

Embedding invariants

Embedding label			694.2
Root			$0.787183 + 0.616719i$ of defining polynomial
Character	$\chi$	$=$	2385.694
Dual form			2385.1.ca.a.244.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.556340 + 1.78536i) q^{2} +(-2.05501 + 1.41847i) q^{4} +(0.998176 - 0.0603785i) q^{5} +(-2.20370 - 1.72649i) q^{8} +O(q^{10})$	\(q+(0.556340 + 1.78536i) q^{2} +(-2.05501 + 1.41847i) q^{4} +(0.998176 - 0.0603785i) q^{5} +(-2.20370 - 1.72649i) q^{8} +(0.663123 + 1.74851i) q^{10} +(0.970942 - 2.56016i) q^{16} +(-0.0989396 + 0.814841i) q^{17} +(0.186505 + 1.01773i) q^{19} +(-1.96561 + 1.53996i) q^{20} +(-1.37312 + 1.37312i) q^{23} +(0.992709 - 0.120537i) q^{25} +(-0.509195 + 0.307819i) q^{31} +(2.31661 + 0.140129i) q^{32} +(-1.50983 + 0.276686i) q^{34} +(-1.71325 + 0.899182i) q^{38} +(-2.30393 - 1.59029i) q^{40} +(-3.21543 - 1.68759i) q^{46} +(1.30452 + 1.47250i) q^{47} +(-0.568065 - 0.822984i) q^{49} +(0.767485 + 1.70528i) q^{50} +(0.787183 - 0.616719i) q^{53} +(0.970942 - 1.23932i) q^{61} +(-0.832854 - 0.737844i) q^{62} +(0.383375 + 1.55542i) q^{64} +(-0.952506 - 1.81485i) q^{68} +(-1.82689 - 1.82689i) q^{76} +(0.593921 - 1.90596i) q^{79} +(0.814592 - 2.61412i) q^{80} +(1.16387 + 1.16387i) q^{83} +(-0.0495602 + 0.819328i) q^{85} +(0.874043 - 4.76950i) q^{92} +(-1.90318 + 3.14825i) q^{94} +(0.247614 + 1.00461i) q^{95} +(1.15328 - 1.47206i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 4 q^{16} - 4 q^{19} - 4 q^{31} + 4 q^{49} + 4 q^{61} + 4 q^{76} + 4 q^{79} - 4 q^{85}+O(q^{100}) $ 48 * q + 4 * q^16 - 4 * q^19 - 4 * q^31 + 4 * q^49 + 4 * q^61 + 4 * q^76 + 4 * q^79 - 4 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2385\mathbb{Z}\right)^\times$.

$n$	$1432$	$1486$	$1856$
$\chi(n)$	$-1$	$e\left(\frac{47}{52}\right)$	$1$

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 $

$2$	0.556340	+	1.78536i	0.556340	+	1.78536i	0.616719	+	0.787183i	$0.288462\pi$
$2$	0.556340	+	1.78536i	0.556340	+	1.78536i	−0.0603785	+	0.998176i	$0.519231\pi$
$3$		0			0
$3$		0			0
$4$	−2.05501	+	1.41847i	−2.05501	+	1.41847i
$5$	0.998176	−	0.0603785i	0.998176	−	0.0603785i
$5$	0.998176	−	0.0603785i	0.998176	−	0.0603785i
$6$		0			0
$7$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$7$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$8$	−2.20370	−	1.72649i	−2.20370	−	1.72649i
$9$		0			0
$10$	0.663123	+	1.74851i	0.663123	+	1.74851i
$11$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$11$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$12$		0			0
$13$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$13$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$14$		0			0
$15$		0			0
$16$	0.970942	−	2.56016i	0.970942	−	2.56016i
$17$	−0.0989396	+	0.814841i	−0.0989396	+	0.814841i	0.855781	+	0.517338i	$0.173077\pi$
$17$	−0.0989396	+	0.814841i	−0.0989396	+	0.814841i	−0.954721	+	0.297503i	$0.903846\pi$
$18$		0			0
$19$	0.186505	+	1.01773i	0.186505	+	1.01773i	0.935016	+	0.354605i	$0.115385\pi$
$19$	0.186505	+	1.01773i	0.186505	+	1.01773i	−0.748511	+	0.663123i	$0.769231\pi$
$20$	−1.96561	+	1.53996i	−1.96561	+	1.53996i
$21$		0			0
$22$		0			0
$23$	−1.37312	+	1.37312i	−1.37312	+	1.37312i	−0.517338	+	0.855781i	$0.673077\pi$
$23$	−1.37312	+	1.37312i	−1.37312	+	1.37312i	−0.855781	+	0.517338i	$0.826923\pi$
$24$		0			0
$25$	0.992709	−	0.120537i	0.992709	−	0.120537i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$29$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$30$		0			0
$31$	−0.509195	+	0.307819i	−0.509195	+	0.307819i	−0.748511	−	0.663123i	$-0.769231\pi$
$31$	−0.509195	+	0.307819i	−0.509195	+	0.307819i	0.239316	+	0.970942i	$0.423077\pi$
$32$	2.31661	+	0.140129i	2.31661	+	0.140129i
$33$		0			0
$34$	−1.50983	+	0.276686i	−1.50983	+	0.276686i
$35$		0			0
$36$		0			0
$37$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$37$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$38$	−1.71325	+	0.899182i	−1.71325	+	0.899182i
$39$		0			0
$40$	−2.30393	−	1.59029i	−2.30393	−	1.59029i
$41$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$41$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$42$		0			0
$43$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$43$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$44$		0			0
$45$		0			0
$46$	−3.21543	−	1.68759i	−3.21543	−	1.68759i
$47$	1.30452	+	1.47250i	1.30452	+	1.47250i	0.787183	+	0.616719i	$0.211538\pi$
$47$	1.30452	+	1.47250i	1.30452	+	1.47250i	0.517338	+	0.855781i	$0.326923\pi$
$48$		0			0
$49$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$50$	0.767485	+	1.70528i	0.767485	+	1.70528i
$51$		0			0
$52$		0			0
$53$	0.787183	−	0.616719i	0.787183	−	0.616719i
$53$	0.787183	−	0.616719i	0.787183	−	0.616719i
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$59$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$60$		0			0
$61$	0.970942	−	1.23932i	0.970942	−	1.23932i		−	1.00000i	$-0.5\pi$
$61$	0.970942	−	1.23932i	0.970942	−	1.23932i	0.970942	−	0.239316i	$-0.0769231\pi$
$62$	−0.832854	−	0.737844i	−0.832854	−	0.737844i
$63$		0			0
$64$	0.383375	+	1.55542i	0.383375	+	1.55542i
$65$		0			0
$66$		0			0
$67$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$67$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$68$	−0.952506	−	1.81485i	−0.952506	−	1.81485i
$69$		0			0
$70$		0			0
$71$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$71$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$72$		0			0
$73$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$73$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$74$		0			0
$75$		0			0
$76$	−1.82689	−	1.82689i	−1.82689	−	1.82689i
$77$		0			0
$78$		0			0
$79$	0.593921	−	1.90596i	0.593921	−	1.90596i	0.239316	−	0.970942i	$-0.423077\pi$
$79$	0.593921	−	1.90596i	0.593921	−	1.90596i	0.354605	−	0.935016i	$-0.384615\pi$
$80$	0.814592	−	2.61412i	0.814592	−	2.61412i
$81$		0			0
$82$		0			0
$83$	1.16387	+	1.16387i	1.16387	+	1.16387i	0.983620	+	0.180255i	$0.0576923\pi$
$83$	1.16387	+	1.16387i	1.16387	+	1.16387i	0.180255	+	0.983620i	$0.442308\pi$
$84$		0			0
$85$	−0.0495602	+	0.819328i	−0.0495602	+	0.819328i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$89$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$90$		0			0
$91$		0			0
$92$	0.874043	−	4.76950i	0.874043	−	4.76950i
$93$		0			0
$94$	−1.90318	+	3.14825i	−1.90318	+	3.14825i
$95$	0.247614	+	1.00461i	0.247614	+	1.00461i
$96$		0			0
$97$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$97$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$98$	1.15328	−	1.47206i	1.15328	−	1.47206i
$99$		0			0
$100$	−1.86905	+	1.65583i	−1.86905	+	1.65583i
$101$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$101$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$102$		0			0
$103$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$103$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$104$		0			0
$105$		0			0
$106$	1.53901	+	1.06230i	1.53901	+	1.06230i
$107$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$107$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$108$		0			0
$109$	−0.506219	−	1.12477i	−0.506219	−	1.12477i	−0.970942	−	0.239316i	$-0.923077\pi$
$109$	−0.506219	−	1.12477i	−0.506219	−	1.12477i	0.464723	−	0.885456i	$-0.346154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.76768	−	0.927751i	−1.76768	−	0.927751i	−0.911900	−	0.410413i	$-0.865385\pi$
$113$	−1.76768	−	0.927751i	−1.76768	−	0.927751i	−0.855781	−	0.517338i	$-0.826923\pi$
$114$		0			0
$115$	−1.28771	+	1.45352i	−1.28771	+	1.45352i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.885456	−	0.464723i	0.885456	−	0.464723i
$122$	2.75280	+	1.04400i	2.75280	+	1.04400i
$123$		0			0
$124$	0.609768	−	1.35485i	0.609768	−	1.35485i
$125$	0.983620	−	0.180255i	0.983620	−	0.180255i
$126$		0			0
$127$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$127$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$128$	−0.577549	+	0.349141i	−0.577549	+	0.349141i
$129$		0			0
$130$		0			0
$131$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$131$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.62485	−	1.62485i	1.62485	−	1.62485i
$137$	0.587763	+	0.972278i	0.587763	+	0.972278i	0.998176	+	0.0603785i	$0.0192308\pi$
$137$	0.587763	+	0.972278i	0.587763	+	0.972278i	−0.410413	+	0.911900i	$0.634615\pi$
$138$		0			0
$139$	1.50308	−	1.17759i	1.50308	−	1.17759i	0.568065	−	0.822984i	$-0.307692\pi$
$139$	1.50308	−	1.17759i	1.50308	−	1.17759i	0.935016	−	0.354605i	$-0.115385\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$149$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$150$		0			0
$151$	1.34731	+	1.05555i	1.34731	+	1.05555i	0.992709	+	0.120537i	$0.0384615\pi$
$151$	1.34731	+	1.05555i	1.34731	+	1.05555i	0.354605	+	0.935016i	$0.384615\pi$
$152$	1.34610	−	2.56477i	1.34610	−	2.56477i
$153$		0			0
$154$		0			0
$155$	−0.489680	+	0.338002i	−0.489680	+	0.338002i
$156$		0			0
$157$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$157$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$158$	3.73324			3.73324
$159$		0			0
$160$	2.32085			2.32085
$161$		0			0
$162$		0			0
$163$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$163$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$164$		0			0
$165$		0			0
$166$	−1.43042	+	2.72545i	−1.43042	+	2.72545i
$167$	−1.39403	−	1.09215i	−1.39403	−	1.09215i	−0.983620	−	0.180255i	$-0.942308\pi$
$167$	−1.39403	−	1.09215i	−1.39403	−	1.09215i	−0.410413	−	0.911900i	$-0.634615\pi$
$168$		0			0
$169$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$170$	−1.49037	+	0.367342i	−1.49037	+	0.367342i
$171$		0			0
$172$		0			0
$173$	−0.237125	−	0.0434547i	−0.237125	−	0.0434547i	0.0603785	−	0.998176i	$-0.480769\pi$
$173$	−0.237125	−	0.0434547i	−0.237125	−	0.0434547i	−0.297503	+	0.954721i	$0.596154\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$179$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$180$		0			0
$181$	−1.01773	−	1.68353i	−1.01773	−	1.68353i	−0.663123	−	0.748511i	$-0.730769\pi$
$181$	−1.01773	−	1.68353i	−1.01773	−	1.68353i	−0.354605	−	0.935016i	$-0.615385\pi$
$182$		0			0
$183$		0			0
$184$	5.39663	−	0.655269i	5.39663	−	0.655269i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−4.76950	−	1.17558i	−4.76950	−	1.17558i
$189$		0			0
$190$	−1.65583	+	1.00099i	−1.65583	+	1.00099i
$191$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$191$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$192$		0			0
$193$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$193$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$194$		0			0
$195$		0			0
$196$	2.33476	+	0.885456i	2.33476	+	0.885456i
$197$	−1.61489	+	0.847562i	−1.61489	+	0.847562i	−0.616719	+	0.787183i	$0.711538\pi$
$197$	−1.61489	+	0.847562i	−1.61489	+	0.847562i	−0.998176	+	0.0603785i	$0.980769\pi$
$198$		0			0
$199$	−1.09148	−	0.753393i	−1.09148	−	0.753393i	−0.120537	−	0.992709i	$-0.538462\pi$
$199$	−1.09148	−	0.753393i	−1.09148	−	0.753393i	−0.970942	+	0.239316i	$0.923077\pi$
$200$	−2.39574	−	1.44828i	−2.39574	−	1.44828i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			1.13613i			1.13613i	0.822984	+	0.568065i	$0.192308\pi$
$211$			1.13613i			1.13613i	−0.822984	+	0.568065i	$0.807692\pi$
$212$	−0.742872	+	2.38396i	−0.742872	+	2.38396i
$213$		0			0
$214$	2.52488	−	0.786784i	2.52488	−	0.786784i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.72649	−	1.52954i	1.72649	−	1.52954i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$223$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$224$		0			0
$225$		0			0
$226$	0.672936	−	3.67209i	0.672936	−	3.67209i
$227$	−0.0561186	−	0.106925i	−0.0561186	−	0.106925i	0.855781	−	0.517338i	$-0.173077\pi$
$227$	−0.0561186	−	0.106925i	−0.0561186	−	0.106925i	−0.911900	+	0.410413i	$0.865385\pi$
$228$		0			0
$229$	1.63397	+	0.198399i	1.63397	+	0.198399i	0.885456	−	0.464723i	$-0.153846\pi$
$229$	1.63397	+	0.198399i	1.63397	+	0.198399i	0.748511	+	0.663123i	$0.230769\pi$
$230$	−3.31146	−	1.49037i	−3.31146	−	1.49037i
$231$		0			0
$232$		0			0
$233$	−0.119877	+	1.98180i	−0.119877	+	1.98180i	0.0603785	+	0.998176i	$0.480769\pi$
$233$	−0.119877	+	1.98180i	−0.119877	+	1.98180i	−0.180255	+	0.983620i	$0.557692\pi$
$234$		0			0
$235$	1.39105	+	1.39105i	1.39105	+	1.39105i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$239$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$240$		0			0
$241$	−0.475142	+	1.92773i	−0.475142	+	1.92773i	−0.120537	+	0.992709i	$0.538462\pi$
$241$	−0.475142	+	1.92773i	−0.475142	+	1.92773i	−0.354605	+	0.935016i	$0.615385\pi$
$242$	1.32231	+	1.32231i	1.32231	+	1.32231i
$243$		0			0
$244$	−0.237362	+	3.92406i	−0.237362	+	3.92406i
$245$	−0.616719	−	0.787183i	−0.616719	−	0.787183i
$246$		0			0
$247$		0			0
$248$	1.65356	+	0.200779i	1.65356	+	0.200779i
$249$		0			0
$250$	0.869047	+	1.65583i	0.869047	+	1.65583i
$251$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$251$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.254432	+	0.225408i	0.254432	+	0.225408i
$257$	1.09215	−	1.39403i	1.09215	−	1.39403i	0.180255	−	0.983620i	$-0.442308\pi$
$257$	1.09215	−	1.39403i	1.09215	−	1.39403i	0.911900	−	0.410413i	$-0.134615\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.230158	−	0.0717201i	0.230158	−	0.0717201i	−0.180255	−	0.983620i	$-0.557692\pi$
$263$	0.230158	−	0.0717201i	0.230158	−	0.0717201i	0.410413	+	0.911900i	$0.365385\pi$
$264$		0			0
$265$	0.748511	−	0.663123i	0.748511	−	0.663123i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$269$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$270$		0			0
$271$	−0.616337	−	0.695701i	−0.616337	−	0.695701i	0.354605	−	0.935016i	$-0.384615\pi$
$271$	−0.616337	−	0.695701i	−0.616337	−	0.695701i	−0.970942	+	0.239316i	$0.923077\pi$
$272$	1.99006	+	1.04446i	1.99006	+	1.04446i
$273$		0			0
$274$	−1.40887	+	1.59029i	−1.40887	+	1.59029i
$275$		0			0
$276$		0			0
$277$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$277$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$278$	2.93864	+	2.02840i	2.93864	+	2.02840i
$279$		0			0
$280$		0			0
$281$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$281$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$282$		0			0
$283$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$283$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.316765	+	0.0780756i	0.316765	+	0.0780756i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.357882	+	0.0434547i	−0.357882	+	0.0434547i	−0.297503	−	0.954721i	$-0.596154\pi$
$293$	−0.357882	+	0.0434547i	−0.357882	+	0.0434547i	−0.0603785	+	0.998176i	$0.519231\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−1.13498	+	2.99269i	−1.13498	+	2.99269i
$303$		0			0
$304$	2.78663	+	0.510670i	2.78663	+	0.510670i
$305$	0.894342	−	1.29568i	0.894342	−	1.29568i
$306$		0			0
$307$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$307$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$308$		0			0
$309$		0			0
$310$	−0.875884	−	0.686211i	−0.875884	−	0.686211i
$311$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$311$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$312$		0			0
$313$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$313$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$314$		0			0
$315$		0			0
$316$	1.48303	+	4.75922i	1.48303	+	4.75922i
$317$	1.23344			1.23344			0.616719	−	0.787183i	$-0.288462\pi$
$317$	1.23344			1.23344			0.616719	+	0.787183i	$0.288462\pi$
$318$		0			0
$319$		0			0
$320$	0.476590	+	1.52943i	0.476590	+	1.52943i
$321$		0			0
$322$		0			0
$323$	−0.847739	+	0.0512787i	−0.847739	+	0.0512787i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.12785	+	1.63397i	−1.12785	+	1.63397i	−0.464723	+	0.885456i	$0.653846\pi$
$331$	−1.12785	+	1.63397i	−1.12785	+	1.63397i	−0.663123	+	0.748511i	$0.730769\pi$
$332$	−4.04269	−	0.740851i	−4.04269	−	0.740851i
$333$		0			0
$334$	1.17433	−	3.09646i	1.17433	−	3.09646i
$335$		0			0
$336$		0			0
$337$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$337$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$338$	1.47206	−	1.15328i	1.47206	−	1.15328i
$339$		0			0
$340$	−1.06035	−	1.75403i	−1.06035	−	1.75403i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.0543397	−	0.447528i	−0.0543397	−	0.447528i
$347$	0.350034	+	0.0862757i	0.350034	+	0.0862757i	0.410413	−	0.911900i	$-0.365385\pi$
$347$	0.350034	+	0.0862757i	0.350034	+	0.0862757i	−0.0603785	+	0.998176i	$0.519231\pi$
$348$		0			0
$349$	−0.103342	+	0.0624722i	−0.103342	+	0.0624722i	−0.568065	−	0.822984i	$-0.692308\pi$
$349$	−0.103342	+	0.0624722i	−0.103342	+	0.0624722i	0.464723	+	0.885456i	$0.346154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.291069	+	0.646728i	−0.291069	+	0.646728i	−0.998176	−	0.0603785i	$-0.980769\pi$
$353$	−0.291069	+	0.646728i	−0.291069	+	0.646728i	0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$359$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$360$		0			0
$361$	−0.0659688	+	0.0250187i	−0.0659688	+	0.0250187i
$362$	2.43950	−	2.75362i	2.43950	−	2.75362i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$367$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$368$	2.18219	+	4.84863i	2.18219	+	4.84863i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$373$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$374$		0			0
$375$		0			0
$376$	−0.332519	−	5.49720i	−0.332519	−	5.49720i
$377$		0			0
$378$		0			0
$379$	−0.872172	+	1.11325i	−0.872172	+	1.11325i	0.120537	+	0.992709i	$0.461538\pi$
$379$	−0.872172	+	1.11325i	−0.872172	+	1.11325i	−0.992709	+	0.120537i	$0.961538\pi$
$380$	−1.93386	−	1.71325i	−1.93386	−	1.71325i
$381$		0			0
$382$		0			0
$383$	−0.247614	+	0.409604i	−0.247614	+	0.409604i	−0.954721	−	0.297503i	$-0.903846\pi$
$383$	−0.247614	+	0.409604i	−0.247614	+	0.409604i	0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$389$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$390$		0			0
$391$	−0.983018	−	1.25473i	−0.983018	−	1.25473i
$392$	−0.169028	+	2.79437i	−0.169028	+	2.79437i
$393$		0			0
$394$	−2.41163	−	2.41163i	−2.41163	−	2.41163i
$395$	0.477758	−	1.93834i	0.477758	−	1.93834i
$396$		0			0
$397$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$397$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$398$	0.737844	−	2.36782i	0.737844	−	2.36782i
$399$		0			0
$400$	0.655269	−	2.65853i	0.655269	−	2.65853i
$401$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$401$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.902438	−	1.71945i	−0.902438	−	1.71945i	−0.663123	−	0.748511i	$-0.730769\pi$
$409$	−0.902438	−	1.71945i	−0.902438	−	1.71945i	−0.239316	−	0.970942i	$-0.576923\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.23202	+	1.09148i	1.23202	+	1.09148i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$419$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$420$		0			0
$421$	1.74122	−	0.783659i	1.74122	−	0.783659i	0.748511	−	0.663123i	$-0.230769\pi$
$421$	1.74122	−	0.783659i	1.74122	−	0.783659i	0.992709	−	0.120537i	$-0.0384615\pi$
$422$	−2.02840	+	0.632075i	−2.02840	+	0.632075i
$423$		0			0
$424$	−2.79948			−2.79948
$425$			0.820826i			0.820826i
$426$		0			0
$427$		0			0
$428$	2.00602	+	2.90622i	2.00602	+	2.90622i
$429$		0			0
$430$		0			0
$431$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$431$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$432$		0			0
$433$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$433$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$434$		0			0
$435$		0			0
$436$	2.63574	+	1.59336i	2.63574	+	1.59336i
$437$	−1.65356	−	1.14137i	−1.65356	−	1.14137i
$438$		0			0
$439$	−1.56806	+	0.822984i	−1.56806	+	0.822984i	−0.568065	+	0.822984i	$0.692308\pi$
$439$	−1.56806	+	0.822984i	−1.56806	+	0.822984i	−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.697593	+	0.127839i	−0.697593	+	0.127839i	−0.517338	−	0.855781i	$-0.673077\pi$
$443$	−0.697593	+	0.127839i	−0.697593	+	0.127839i	−0.180255	+	0.983620i	$0.557692\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$449$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$450$		0			0
$451$		0			0
$452$	4.94859	−	0.600867i	4.94859	−	0.600867i
$453$		0			0
$454$	0.159679	−	0.159679i	0.159679	−	0.159679i
$455$		0			0
$456$		0			0
$457$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$457$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$458$	0.554827	+	3.02759i	0.554827	+	3.02759i
$459$		0			0
$460$	0.584473	−	4.81357i	0.584473	−	4.81357i
$461$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$461$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$462$		0			0
$463$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$463$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$464$		0			0
$465$		0			0
$466$	−3.60491	+	0.888530i	−3.60491	+	0.888530i
$467$	−0.677097	−	1.78536i	−0.677097	−	1.78536i	−0.616719	−	0.787183i	$-0.711538\pi$
$467$	−0.677097	−	1.78536i	−0.677097	−	1.78536i	−0.0603785	−	0.998176i	$-0.519231\pi$
$468$		0			0
$469$		0			0
$470$	−1.70962	+	3.25742i	−1.70962	+	3.25742i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.307819	+	0.987826i	0.307819	+	0.987826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$479$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$480$		0			0
$481$		0			0
$482$	−3.70602	+	0.224173i	−3.70602	+	0.224173i
$483$		0			0
$484$	−1.16042	+	2.21100i	−1.16042	+	2.21100i
$485$		0			0
$486$		0			0
$487$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$487$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$488$	−4.27934	+	1.05476i	−4.27934	+	1.05476i
$489$		0			0
$490$	1.06230	−	1.53901i	1.06230	−	1.53901i
$491$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$491$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.293668	+	1.60250i	0.293668	+	1.60250i
$497$		0			0
$498$		0			0
$499$	0.186505	+	0.308518i	0.186505	+	0.308518i	0.935016	−	0.354605i	$-0.115385\pi$
$499$	0.186505	+	0.308518i	0.186505	+	0.308518i	−0.748511	+	0.663123i	$0.769231\pi$
$500$	−1.76566	+	1.76566i	−1.76566	+	1.76566i
$501$		0			0
$502$		0			0
$503$	0.677097	+	0.210992i	0.677097	+	0.210992i	0.616719	−	0.787183i	$-0.288462\pi$
$503$	0.677097	+	0.210992i	0.677097	+	0.210992i	0.0603785	+	0.998176i	$0.480769\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$509$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$510$		0			0
$511$		0			0
$512$	−0.537862	+	1.19508i	−0.537862	+	1.19508i
$513$		0			0
$514$	3.09646	+	1.17433i	3.09646	+	1.17433i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$521$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$522$		0			0
$523$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$523$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$524$		0			0
$525$		0			0
$526$	0.256092	+	0.371013i	0.256092	+	0.371013i
$527$	−0.200444	−	0.445368i	−0.200444	−	0.445368i
$528$		0			0
$529$		−	2.77091i		−	2.77091i
$530$	1.60034	+	0.967439i	1.60034	+	0.967439i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.0853881	−	1.41163i	−0.0853881	−	1.41163i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.423807	+	1.71945i	0.423807	+	1.71945i	0.663123	+	0.748511i	$0.269231\pi$
$541$	0.423807	+	1.71945i	0.423807	+	1.71945i	−0.239316	+	0.970942i	$0.576923\pi$
$542$	0.899182	−	1.48743i	0.899182	−	1.48743i
$543$		0			0
$544$	−0.343388	+	1.87381i	−0.343388	+	1.87381i
$545$	−0.573207	−	1.09215i	−0.573207	−	1.09215i
$546$		0			0
$547$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$547$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$548$	−2.58700	−	1.16432i	−2.58700	−	1.16432i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.41847	+	4.55203i	−1.41847	+	4.55203i
$557$	−0.489680	+	1.57144i	−0.489680	+	1.57144i	0.297503	+	0.954721i	$0.403846\pi$
$557$	−0.489680	+	1.57144i	−0.489680	+	1.57144i	−0.787183	+	0.616719i	$0.788462\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.295181	+	0.376771i	0.295181	+	0.376771i	0.911900	−	0.410413i	$-0.134615\pi$
$563$	0.295181	+	0.376771i	0.295181	+	0.376771i	−0.616719	+	0.787183i	$0.711538\pi$
$564$		0			0
$565$	−1.82047	−	0.819328i	−1.82047	−	0.819328i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$569$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$570$		0			0
$571$	−0.424644	+	0.702447i	−0.424644	+	0.702447i	−0.992709	−	0.120537i	$-0.961538\pi$
$571$	−0.424644	+	0.702447i	−0.424644	+	0.702447i	0.568065	+	0.822984i	$0.307692\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.19760	+	1.52862i	−1.19760	+	1.52862i
$576$		0			0
$577$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$577$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$578$	0.0368363	+	0.608976i	0.0368363	+	0.608976i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.276686	−	0.614771i	−0.276686	−	0.614771i
$587$	−0.894342	−	1.29568i	−0.894342	−	1.29568i	−0.954721	−	0.297503i	$-0.903846\pi$
$587$	−0.894342	−	1.29568i	−0.894342	−	1.29568i	0.0603785	−	0.998176i	$-0.480769\pi$
$588$		0			0
$589$	−0.408244	−	0.460812i	−0.408244	−	0.460812i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.32231	−	0.501487i	1.32231	−	0.501487i	0.410413	−	0.911900i	$-0.365385\pi$
$593$	1.32231	−	0.501487i	1.32231	−	0.501487i	0.911900	+	0.410413i	$0.134615\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$599$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$600$		0			0
$601$	−0.0495602	+	0.110118i	−0.0495602	+	0.110118i	−0.935016	−	0.354605i	$-0.884615\pi$
$601$	−0.0495602	+	0.110118i	−0.0495602	+	0.110118i	0.885456	+	0.464723i	$0.153846\pi$
$602$		0			0
$603$		0			0
$604$	−4.26601	−	0.258046i	−4.26601	−	0.258046i
$605$	0.855781	−	0.517338i	0.855781	−	0.517338i
$606$		0			0
$607$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$607$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$608$	0.289448	+	2.38382i	0.289448	+	2.38382i
$609$		0			0
$610$	2.81081	+	0.875884i	2.81081	+	0.875884i
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.269846	−	1.47250i	−0.269846	−	1.47250i	−0.787183	−	0.616719i	$-0.788462\pi$
$617$	−0.269846	−	1.47250i	−0.269846	−	1.47250i	0.517338	−	0.855781i	$-0.326923\pi$
$618$		0			0
$619$	−0.0576926	+	0.475142i	−0.0576926	+	0.475142i	0.935016	+	0.354605i	$0.115385\pi$
$619$	−0.0576926	+	0.475142i	−0.0576926	+	0.475142i	−0.992709	+	0.120537i	$0.961538\pi$
$620$	0.526852	−	1.38919i	0.526852	−	1.38919i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.970942	−	0.239316i	0.970942	−	0.239316i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.120537	+	0.00729113i	−0.120537	+	0.00729113i	−0.120537	−	0.992709i	$-0.538462\pi$
$631$	−0.120537	+	0.00729113i	−0.120537	+	0.00729113i			1.00000i	$0.5\pi$
$632$	−4.59945	+	3.17477i	−4.59945	+	3.17477i
$633$		0			0
$634$	0.686211	+	2.20213i	0.686211	+	2.20213i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.555415	+	0.383375i	−0.555415	+	0.383375i
$641$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$641$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$642$		0			0
$643$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$643$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$644$		0			0
$645$		0			0
$646$	−0.563182	−	1.48499i	−0.563182	−	1.48499i
$647$	1.93834	−	0.477758i	1.93834	−	0.477758i	0.954721	−	0.297503i	$-0.0961538\pi$
$647$	1.93834	−	0.477758i	1.93834	−	0.477758i	0.983620	−	0.180255i	$-0.0576923\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.189769	−	1.56289i	0.189769	−	1.56289i	−0.517338	−	0.855781i	$-0.673077\pi$
$653$	0.189769	−	1.56289i	0.189769	−	1.56289i	0.707107	−	0.707107i	$-0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−0.239316	+	0.0290582i	−0.239316	+	0.0290582i	−0.239316	−	0.970942i	$-0.576923\pi$
$661$	−0.239316	+	0.0290582i	−0.239316	+	0.0290582i			1.00000i	$0.5\pi$
$662$	−3.54468	−	1.10457i	−3.54468	−	1.10457i
$663$		0			0
$664$	−0.555415	−	4.57426i	−0.555415	−	4.57426i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	4.41394	+	0.266994i	4.41394	+	0.266994i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$673$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$674$		0			0
$675$		0			0
$676$	2.05501	+	1.41847i	2.05501	+	1.41847i
$677$	−0.409604	−	0.247614i	−0.409604	−	0.247614i	0.297503	−	0.954721i	$-0.403846\pi$
$677$	−0.409604	−	0.247614i	−0.409604	−	0.247614i	−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$	1.52378	−	1.71999i	1.52378	−	1.71999i
$681$		0			0
$682$		0			0
$683$	−1.20940	−	1.36513i	−1.20940	−	1.36513i	−0.911900	−	0.410413i	$-0.865385\pi$
$683$	−1.20940	−	1.36513i	−1.20940	−	1.36513i	−0.297503	−	0.954721i	$-0.596154\pi$
$684$		0			0
$685$	0.645395	+	0.935016i	0.645395	+	0.935016i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.50308	−	0.468379i	1.50308	−	0.468379i	0.568065	−	0.822984i	$-0.307692\pi$
$691$	1.50308	−	0.468379i	1.50308	−	0.468379i	0.935016	+	0.354605i	$0.115385\pi$
$692$	0.548932	−	0.247054i	0.548932	−	0.247054i
$693$		0			0
$694$	0.0407051	+	0.672936i	0.0407051	+	0.672936i
$695$	1.42924	−	1.26619i	1.42924	−	1.26619i
$696$		0			0
$697$		0			0
$698$	−0.169028	−	0.149746i	−0.169028	−	0.149746i
$699$		0			0
$700$		0			0
$701$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$701$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.31658	−	0.159861i	−1.31658	−	0.159861i
$707$		0			0
$708$		0			0
$709$	−0.872172	−	1.11325i	−0.872172	−	1.11325i	−0.992709	−	0.120537i	$-0.961538\pi$
$709$	−0.872172	−	1.11325i	−0.872172	−	1.11325i	0.120537	−	0.992709i	$-0.461538\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.276513	−	1.12186i	0.276513	−	1.12186i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$719$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$720$		0			0
$721$		0			0
$722$	−0.0813684	−	0.103859i	−0.0813684	−	0.103859i
$723$		0			0
$724$	4.47947	+	2.01605i	4.47947	+	2.01605i
$725$		0			0
$726$		0			0
$727$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$727$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$733$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$734$		0			0
$735$		0			0
$736$	−3.37340	+	2.98857i	−3.37340	+	2.98857i
$737$		0			0
$738$		0			0
$739$	−1.12477	+	0.506219i	−1.12477	+	0.506219i	−0.885456	−	0.464723i	$-0.846154\pi$
$739$	−1.12477	+	0.506219i	−1.12477	+	0.506219i	−0.239316	+	0.970942i	$0.576923\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			0.120757i			0.120757i	0.998176	+	0.0603785i	$0.0192308\pi$
$743$			0.120757i			0.120757i	−0.998176	+	0.0603785i	$0.980769\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.159861	+	0.180446i	−0.159861	+	0.180446i	−0.822984	−	0.568065i	$-0.807692\pi$
$751$	−0.159861	+	0.180446i	−0.159861	+	0.180446i	0.663123	+	0.748511i	$0.269231\pi$
$752$	5.03645	−	1.91008i	5.03645	−	1.91008i
$753$		0			0
$754$		0			0
$755$	1.40859	+	0.972278i	1.40859	+	0.972278i
$756$		0			0
$757$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$757$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$758$	−2.47277	−	0.937797i	−2.47277	−	0.937797i
$759$		0			0
$760$	1.18878	−	2.64137i	1.18878	−	2.64137i
$761$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$761$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.869047	−	0.214201i	−0.869047	−	0.214201i
$767$		0			0
$768$		0			0
$769$	1.63406	+	0.509195i	1.63406	+	0.509195i	0.970942	−	0.239316i	$-0.0769231\pi$
$769$	1.63406	+	0.509195i	1.63406	+	0.509195i	0.663123	+	0.748511i	$0.269231\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.366901	−	0.606928i	−0.366901	−	0.606928i	0.616719	−	0.787183i	$-0.288462\pi$
$773$	−0.366901	−	0.606928i	−0.366901	−	0.606928i	−0.983620	+	0.180255i	$0.942308\pi$
$774$		0			0
$775$	−0.468379	+	0.366951i	−0.468379	+	0.366951i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	1.69325	−	2.45310i	1.69325	−	2.45310i
$783$		0			0
$784$	−2.65853	+	0.655269i	−2.65853	+	0.655269i
$785$		0			0
$786$		0			0
$787$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$787$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$788$	2.11638	−	4.03243i	2.11638	−	4.03243i
$789$		0			0
$790$	3.72643	−	0.225408i	3.72643	−	0.225408i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	3.31166			3.31166
$797$	0.590668	+	1.89552i	0.590668	+	1.89552i	0.410413	+	0.911900i	$0.365385\pi$
$797$	0.590668	+	1.89552i	0.590668	+	1.89552i	0.180255	+	0.983620i	$0.442308\pi$
$798$		0			0
$799$	−1.32892	+	0.917289i	−1.32892	+	0.917289i
$800$	2.31661	−	0.140129i	2.31661	−	0.140129i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$809$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$810$		0			0
$811$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	0.120537	+	0.992709i	$0.461538\pi$
$811$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	−0.748511	+	0.663123i	$0.769231\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.56778	−	2.56778i	2.56778	−	2.56778i
$819$		0			0
$820$		0			0
$821$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$821$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$822$		0			0
$823$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$823$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.49429	+	0.0903879i	1.49429	+	0.0903879i	0.787183	−	0.616719i	$-0.211538\pi$
$827$	1.49429	+	0.0903879i	1.49429	+	0.0903879i	0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	1.79393	−	0.328749i	1.79393	−	0.328749i	0.822984	−	0.568065i	$-0.192308\pi$
$829$	1.79393	−	0.328749i	1.79393	−	0.328749i	0.970942	+	0.239316i	$0.0769231\pi$
$830$	−1.26326	+	2.80684i	−1.26326	+	2.80684i
$831$		0			0
$832$		0			0
$833$	0.726805	−	0.381457i	0.726805	−	0.381457i
$834$		0			0
$835$	−1.45743	−	1.00599i	−1.45743	−	1.00599i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$839$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$840$		0			0
$841$	0.885456	+	0.464723i	0.885456	+	0.464723i
$842$	2.36782	+	2.67272i	2.36782	+	2.67272i
$843$		0			0
$844$	−1.61157	−	2.33476i	−1.61157	−	2.33476i
$845$	−0.410413	−	0.911900i	−0.410413	−	0.911900i
$846$		0			0
$847$		0			0
$848$	−0.814592	−	2.61412i	−0.814592	−	2.61412i
$849$		0			0
$850$	−1.46547	+	0.456658i	−1.46547	+	0.456658i
$851$		0			0
$852$		0			0
$853$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$853$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$854$		0			0
$855$		0			0
$856$	−2.44163	+	3.11651i	−2.44163	+	3.11651i
$857$	1.36513	+	1.20940i	1.36513	+	1.20940i	0.954721	+	0.297503i	$0.0961538\pi$
$857$	1.36513	+	1.20940i	1.36513	+	1.20940i	0.410413	+	0.911900i	$0.365385\pi$
$858$		0			0
$859$	−0.317391	−	1.28771i	−0.317391	−	1.28771i	−0.885456	−	0.464723i	$-0.846154\pi$
$859$	−0.317391	−	1.28771i	−0.317391	−	1.28771i	0.568065	−	0.822984i	$-0.307692\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.795403	−	1.51551i	−0.795403	−	1.51551i	−0.855781	−	0.517338i	$-0.826923\pi$
$863$	−0.795403	−	1.51551i	−0.795403	−	1.51551i	0.0603785	−	0.998176i	$-0.480769\pi$
$864$		0			0
$865$	−0.239316	−	0.0290582i	−0.239316	−	0.0290582i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.826353	+	3.35265i	−0.826353	+	3.35265i
$873$		0			0
$874$	1.11781	−	3.58718i	1.11781	−	3.58718i
$875$		0			0
$876$		0			0
$877$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$877$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$878$	−2.34170	−	2.34170i	−2.34170	−	2.34170i
$879$		0			0
$880$		0			0
$881$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$881$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$882$		0			0
$883$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$883$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$884$		0			0
$885$		0			0
$886$	−0.616337	−	1.17433i	−0.616337	−	1.17433i
$887$	0.239062	−	1.30452i	0.239062	−	1.30452i	−0.616719	−	0.787183i	$-0.711538\pi$
$887$	0.239062	−	1.30452i	0.239062	−	1.30452i	0.855781	−	0.517338i	$-0.173077\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.25530	+	1.60228i	−1.25530	+	1.60228i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	0.424644	+	0.702447i	0.424644	+	0.702447i
$902$		0			0
$903$		0			0
$904$	2.29369	+	5.09637i	2.29369	+	5.09637i
$905$	−1.11752	−	1.61901i	−1.11752	−	1.61901i
$906$		0			0
$907$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$907$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$908$	0.266994	+	0.140129i	0.266994	+	0.140129i
$909$		0			0
$910$		0			0
$911$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$911$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.63924	+	1.91002i	−3.63924	+	1.91002i
$917$		0			0
$918$		0			0
$919$	−0.646140	+	1.43566i	−0.646140	+	1.43566i	0.239316	+	0.970942i	$0.423077\pi$
$919$	−0.646140	+	1.43566i	−0.646140	+	1.43566i	−0.885456	+	0.464723i	$0.846154\pi$
$920$	5.34722	−	0.979914i	5.34722	−	0.979914i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$929$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$930$		0			0
$931$	0.731626	−	0.731626i	0.731626	−	0.731626i
$932$	−2.56477	−	4.24265i	−2.56477	−	4.24265i
$933$		0			0
$934$	2.81081	−	2.20213i	2.81081	−	2.20213i
$935$		0			0
$936$		0			0
$937$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$937$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$938$		0			0
$939$		0			0
$940$	−4.83178	−	0.885456i	−4.83178	−	0.885456i
$941$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$941$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.573207	−	1.09215i	0.573207	−	1.09215i	−0.410413	−	0.911900i	$-0.634615\pi$
$947$	0.573207	−	1.09215i	0.573207	−	1.09215i	0.983620	−	0.180255i	$-0.0576923\pi$
$948$		0			0
$949$		0			0
$950$	−1.59237	+	1.09914i	−1.59237	+	1.09914i
$951$		0			0
$952$		0			0
$953$	−1.03468			−1.03468			−0.517338	−	0.855781i	$-0.673077\pi$
$953$	−1.03468			−1.03468			−0.517338	+	0.855781i	$0.673077\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.300196	+	0.571976i	−0.300196	+	0.571976i
$962$		0			0
$963$		0			0
$964$	−1.75800	−	4.63547i	−1.75800	−	4.63547i
$965$		0			0
$966$		0			0
$967$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$967$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$968$	−2.75362	−	0.504620i	−2.75362	−	0.504620i
$969$		0			0
$970$		0			0
$971$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$971$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−2.23012	−	3.68907i	−2.23012	−	3.68907i
$977$	−0.937797	+	0.937797i	−0.937797	+	0.937797i	−0.998176	−	0.0603785i	$-0.980769\pi$
$977$	−0.937797	+	0.937797i	−0.937797	+	0.937797i	0.0603785	+	0.998176i	$0.480769\pi$
$978$		0			0
$979$		0			0
$980$	2.38396	+	0.742872i	2.38396	+	0.742872i
$981$		0			0
$982$		0			0
$983$	−1.85396	−	0.456959i	−1.85396	−	0.456959i	−0.855781	−	0.517338i	$-0.826923\pi$
$983$	−1.85396	−	0.456959i	−1.85396	−	0.456959i	−0.998176	+	0.0603785i	$0.980769\pi$
$984$		0			0
$985$	−1.56077	+	0.943521i	−1.56077	+	0.943521i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.447528	−	0.169725i	−0.447528	−	0.169725i	0.120537	−	0.992709i	$-0.461538\pi$
$991$	−0.447528	−	0.169725i	−0.447528	−	0.169725i	−0.568065	+	0.822984i	$0.692308\pi$
$992$	−1.22274	+	0.641745i	−1.22274	+	0.641745i
$993$		0			0
$994$		0			0
$995$	−1.13498	−	0.686117i	−1.13498	−	0.686117i
$996$		0			0
$997$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$997$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$998$	−0.447054	+	0.504620i	−0.447054	+	0.504620i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2385.1.ca.a.694.2	yes	48
3.2	odd	2	inner	2385.1.ca.a.694.1	yes	48
5.4	even	2	inner	2385.1.ca.a.694.1	yes	48
15.14	odd	2	CM	2385.1.ca.a.694.2	yes	48
53.32	odd	52	inner	2385.1.ca.a.244.1	✓	48
159.32	even	52	inner	2385.1.ca.a.244.2	yes	48
265.244	odd	52	inner	2385.1.ca.a.244.2	yes	48
795.509	even	52	inner	2385.1.ca.a.244.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2385.1.ca.a.244.1	✓	48	53.32	odd	52	inner
2385.1.ca.a.244.1	✓	48	795.509	even	52	inner
2385.1.ca.a.244.2	yes	48	159.32	even	52	inner
2385.1.ca.a.244.2	yes	48	265.244	odd	52	inner
2385.1.ca.a.694.1	yes	48	3.2	odd	2	inner
2385.1.ca.a.694.1	yes	48	5.4	even	2	inner
2385.1.ca.a.694.2	yes	48	1.1	even	1	trivial
2385.1.ca.a.694.2	yes	48	15.14	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:	\( N \)	\(=\)	\( 2385 = 3^{2} \cdot 5 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2385.ca (of order \(52\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(0.556340 + 1.78536i) q^{2} +(-2.05501 + 1.41847i) q^{4} +(0.998176 - 0.0603785i) q^{5} +(-2.20370 - 1.72649i) q^{8} +O(q^{10})\)	\(q+(0.556340 + 1.78536i) q^{2} +(-2.05501 + 1.41847i) q^{4} +(0.998176 - 0.0603785i) q^{5} +(-2.20370 - 1.72649i) q^{8} +(0.663123 + 1.74851i) q^{10} +(0.970942 - 2.56016i) q^{16} +(-0.0989396 + 0.814841i) q^{17} +(0.186505 + 1.01773i) q^{19} +(-1.96561 + 1.53996i) q^{20} +(-1.37312 + 1.37312i) q^{23} +(0.992709 - 0.120537i) q^{25} +(-0.509195 + 0.307819i) q^{31} +(2.31661 + 0.140129i) q^{32} +(-1.50983 + 0.276686i) q^{34} +(-1.71325 + 0.899182i) q^{38} +(-2.30393 - 1.59029i) q^{40} +(-3.21543 - 1.68759i) q^{46} +(1.30452 + 1.47250i) q^{47} +(-0.568065 - 0.822984i) q^{49} +(0.767485 + 1.70528i) q^{50} +(0.787183 - 0.616719i) q^{53} +(0.970942 - 1.23932i) q^{61} +(-0.832854 - 0.737844i) q^{62} +(0.383375 + 1.55542i) q^{64} +(-0.952506 - 1.81485i) q^{68} +(-1.82689 - 1.82689i) q^{76} +(0.593921 - 1.90596i) q^{79} +(0.814592 - 2.61412i) q^{80} +(1.16387 + 1.16387i) q^{83} +(-0.0495602 + 0.819328i) q^{85} +(0.874043 - 4.76950i) q^{92} +(-1.90318 + 3.14825i) q^{94} +(0.247614 + 1.00461i) q^{95} +(1.15328 - 1.47206i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 4 q^{16} - 4 q^{19} - 4 q^{31} + 4 q^{49} + 4 q^{61} + 4 q^{76} + 4 q^{79} - 4 q^{85}+O(q^{100}) \) 48 * q + 4 * q^16 - 4 * q^19 - 4 * q^31 + 4 * q^49 + 4 * q^61 + 4 * q^76 + 4 * q^79 - 4 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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