LMFDB - Embedded newform 2500.1.t.a.471.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2500,1,Mod(11,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, base_ring=CyclotomicField(250))

chi = DirichletCharacter(H, H._module([125, 238]))

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

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

chi := DirichletCharacter("2500.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2500 = 2^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2500.t (of order $250$, degree $100$, 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:	$1.24766253158$
Analytic rank:	$0$
Dimension:	$100$
Coefficient field:	$\Q(\zeta_{250})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{100} - x^{75} + x^{50} - x^{25} + 1 $ x^100 - x^75 + x^50 - x^25 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{125}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{125} - \cdots)$

Embedding invariants

Embedding label			471.1
Root			$0.947098 + 0.320944i$ of defining polynomial
Character	$\chi$	$=$	2500.471
Dual form			2500.1.t.a.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.236499 - 0.971632i) q^{2} +(-0.888136 + 0.459580i) q^{4} +(0.994951 + 0.100362i) q^{5} +(0.656586 + 0.754251i) q^{8} +(-0.675333 + 0.737513i) q^{9} +O(q^{10})$	\(q+(-0.236499 - 0.971632i) q^{2} +(-0.888136 + 0.459580i) q^{4} +(0.994951 + 0.100362i) q^{5} +(0.656586 + 0.754251i) q^{8} +(-0.675333 + 0.737513i) q^{9} +(-0.137790 - 0.990461i) q^{10} +(-1.95909 + 0.399294i) q^{13} +(0.577573 - 0.816339i) q^{16} +(0.878701 + 1.75167i) q^{17} +(0.876307 + 0.481754i) q^{18} +(-0.929776 + 0.368125i) q^{20} +(0.979855 + 0.199710i) q^{25} +(0.851290 + 1.80908i) q^{26} +(-1.06348 + 1.58621i) q^{29} +(-0.929776 - 0.368125i) q^{32} +(1.49417 - 1.26804i) q^{34} +(0.260842 - 0.965382i) q^{36} +(-0.00346296 - 0.275559i) q^{37} +(0.577573 + 0.816339i) q^{40} +(0.669845 - 1.25556i) q^{41} +(-0.745941 + 0.666012i) q^{45} +(-0.637424 - 0.770513i) q^{49} +(-0.0376902 - 0.999289i) q^{50} +(1.55643 - 1.25499i) q^{52} +(-0.624858 + 1.77043i) q^{53} +(1.79273 + 0.658175i) q^{58} +(0.813304 + 1.52446i) q^{61} +(-0.137790 + 0.990461i) q^{64} +(-1.98927 + 0.200660i) q^{65} +(-1.58544 - 1.15189i) q^{68} +(-0.999684 - 0.0251301i) q^{72} +(0.468030 + 1.22693i) q^{73} +(-0.266923 + 0.0685341i) q^{74} +(0.656586 - 0.754251i) q^{80} +(-0.0878512 - 0.996134i) q^{81} +(-1.37836 - 0.353903i) q^{82} +(0.698464 + 1.83101i) q^{85} +(1.55250 - 1.25182i) q^{89} +(0.823533 + 0.567269i) q^{90} +(-1.46154 + 0.536583i) q^{97} +(-0.597905 + 0.801567i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 100 q+O(q^{10}) $ 100 * q	$ 100 q+O(q^{100}) $ 100 * q

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times$.

$n$	$1251$	$1877$
$\chi(n)$	$-1$	$e\left(\frac{53}{125}\right)$

Coefficient data

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

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

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

$2$	−0.236499	−	0.971632i	−0.236499	−	0.971632i
$2$	−0.236499	−	0.971632i	−0.236499	−	0.971632i
$3$		0			0		−0.402906	−	0.915241i	$-0.632000\pi$
$3$		0			0		0.402906	+	0.915241i	$0.368000\pi$
$4$	−0.888136	+	0.459580i	−0.888136	+	0.459580i
$5$	0.994951	+	0.100362i	0.994951	+	0.100362i
$5$	0.994951	+	0.100362i	0.994951	+	0.100362i
$6$		0			0
$7$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$7$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$8$	0.656586	+	0.754251i	0.656586	+	0.754251i
$9$	−0.675333	+	0.737513i	−0.675333	+	0.737513i
$10$	−0.137790	−	0.990461i	−0.137790	−	0.990461i
$11$		0			0		0.850994	−	0.525175i	$-0.176000\pi$
$11$		0			0		−0.850994	+	0.525175i	$0.824000\pi$
$12$		0			0
$13$	−1.95909	+	0.399294i	−1.95909	+	0.399294i	−0.974527	+	0.224271i	$0.928000\pi$
$13$	−1.95909	+	0.399294i	−1.95909	+	0.399294i	−0.984564	+	0.175023i	$0.944000\pi$
$14$		0			0
$15$		0			0
$16$	0.577573	−	0.816339i	0.577573	−	0.816339i
$17$	0.878701	+	1.75167i	0.878701	+	1.75167i	0.617860	+	0.786288i	$0.288000\pi$
$17$	0.878701	+	1.75167i	0.878701	+	1.75167i	0.260842	+	0.965382i	$0.416000\pi$
$18$	0.876307	+	0.481754i	0.876307	+	0.481754i
$19$		0			0		−0.745941	−	0.666012i	$-0.768000\pi$
$19$		0			0		0.745941	+	0.666012i	$0.232000\pi$
$20$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$21$		0			0
$22$		0			0
$23$		0			0		0.693653	−	0.720309i	$-0.256000\pi$
$23$		0			0		−0.693653	+	0.720309i	$0.744000\pi$
$24$		0			0
$25$	0.979855	+	0.199710i	0.979855	+	0.199710i
$26$	0.851290	+	1.80908i	0.851290	+	1.80908i
$27$		0			0
$28$		0			0
$29$	−1.06348	+	1.58621i	−1.06348	+	1.58621i	−0.285019	+	0.958522i	$0.592000\pi$
$29$	−1.06348	+	1.58621i	−1.06348	+	1.58621i	−0.778462	+	0.627691i	$0.784000\pi$
$30$		0			0
$31$		0			0		−0.448383	−	0.893841i	$-0.648000\pi$
$31$		0			0		0.448383	+	0.893841i	$0.352000\pi$
$32$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$33$		0			0
$34$	1.49417	−	1.26804i	1.49417	−	1.26804i
$35$		0			0
$36$	0.260842	−	0.965382i	0.260842	−	0.965382i
$37$	−0.00346296	−	0.275559i	−0.00346296	−	0.275559i	−0.992115	−	0.125333i	$-0.960000\pi$
$37$	−0.00346296	−	0.275559i	−0.00346296	−	0.275559i	0.988652	−	0.150226i	$-0.0480000\pi$
$38$		0			0
$39$		0			0
$40$	0.577573	+	0.816339i	0.577573	+	0.816339i
$41$	0.669845	−	1.25556i	0.669845	−	1.25556i	−0.285019	−	0.958522i	$-0.592000\pi$
$41$	0.669845	−	1.25556i	0.669845	−	1.25556i	0.954865	−	0.297042i	$-0.0960000\pi$
$42$		0			0
$43$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$43$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$44$		0			0
$45$	−0.745941	+	0.666012i	−0.745941	+	0.666012i
$46$		0			0
$47$		0			0		−0.920232	−	0.391374i	$-0.872000\pi$
$47$		0			0		0.920232	+	0.391374i	$0.128000\pi$
$48$		0			0
$49$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$50$	−0.0376902	−	0.999289i	−0.0376902	−	0.999289i
$51$		0			0
$52$	1.55643	−	1.25499i	1.55643	−	1.25499i
$53$	−0.624858	+	1.77043i	−0.624858	+	1.77043i	0.0125660	+	0.999921i	$0.496000\pi$
$53$	−0.624858	+	1.77043i	−0.624858	+	1.77043i	−0.637424	+	0.770513i	$0.720000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.79273	+	0.658175i	1.79273	+	0.658175i
$59$		0			0		0.837528	−	0.546394i	$-0.184000\pi$
$59$		0			0		−0.837528	+	0.546394i	$0.816000\pi$
$60$		0			0
$61$	0.813304	+	1.52446i	0.813304	+	1.52446i	0.850994	+	0.525175i	$0.176000\pi$
$61$	0.813304	+	1.52446i	0.813304	+	1.52446i	−0.0376902	+	0.999289i	$0.512000\pi$
$62$		0			0
$63$		0			0
$64$	−0.137790	+	0.990461i	−0.137790	+	0.990461i
$65$	−1.98927	+	0.200660i	−1.98927	+	0.200660i
$66$		0			0
$67$		0			0		0.0376902	−	0.999289i	$-0.488000\pi$
$67$		0			0		−0.0376902	+	0.999289i	$0.512000\pi$
$68$	−1.58544	−	1.15189i	−1.58544	−	1.15189i
$69$		0			0
$70$		0			0
$71$		0			0		−0.514440	−	0.857527i	$-0.672000\pi$
$71$		0			0		0.514440	+	0.857527i	$0.328000\pi$
$72$	−0.999684	−	0.0251301i	−0.999684	−	0.0251301i
$73$	0.468030	+	1.22693i	0.468030	+	1.22693i	0.938734	+	0.344643i	$0.112000\pi$
$73$	0.468030	+	1.22693i	0.468030	+	1.22693i	−0.470704	+	0.882291i	$0.656000\pi$
$74$	−0.266923	+	0.0685341i	−0.266923	+	0.0685341i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.402906	−	0.915241i	$-0.632000\pi$
$79$		0			0		0.402906	+	0.915241i	$0.368000\pi$
$80$	0.656586	−	0.754251i	0.656586	−	0.754251i
$81$	−0.0878512	−	0.996134i	−0.0878512	−	0.996134i
$82$	−1.37836	−	0.353903i	−1.37836	−	0.353903i
$83$		0			0		−0.994951	−	0.100362i	$-0.968000\pi$
$83$		0			0		0.994951	+	0.100362i	$0.0320000\pi$
$84$		0			0
$85$	0.698464	+	1.83101i	0.698464	+	1.83101i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.55250	−	1.25182i	1.55250	−	1.25182i	0.728969	−	0.684547i	$-0.240000\pi$
$89$	1.55250	−	1.25182i	1.55250	−	1.25182i	0.823533	−	0.567269i	$-0.192000\pi$
$90$	0.823533	+	0.567269i	0.823533	+	0.567269i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.46154	+	0.536583i	−1.46154	+	0.536583i	−0.947098	−	0.320944i	$-0.896000\pi$
$97$	−1.46154	+	0.536583i	−1.46154	+	0.536583i	−0.514440	+	0.857527i	$0.672000\pi$
$98$	−0.597905	+	0.801567i	−0.597905	+	0.801567i
$99$		0			0
$100$	−0.962028	+	0.272952i	−0.962028	+	0.272952i
$101$	−0.133570	−	0.700198i	−0.133570	−	0.700198i	−0.984564	−	0.175023i	$-0.944000\pi$
$101$	−0.133570	−	0.700198i	−0.133570	−	0.700198i	0.850994	−	0.525175i	$-0.176000\pi$
$102$		0			0
$103$		0			0		0.888136	−	0.459580i	$-0.152000\pi$
$103$		0			0		−0.888136	+	0.459580i	$0.848000\pi$
$104$	−1.58748	−	1.21548i	−1.58748	−	1.21548i
$105$		0			0
$106$	1.86799	+	0.188426i	1.86799	+	0.188426i
$107$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$107$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$108$		0			0
$109$	0.974450	−	0.330212i	0.974450	−	0.330212i	0.212007	−	0.977268i	$-0.432000\pi$
$109$	0.974450	−	0.330212i	0.974450	−	0.330212i	0.762443	+	0.647056i	$0.224000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.396149	−	0.258443i	0.396149	−	0.258443i	−0.332820	−	0.942991i	$-0.608000\pi$
$113$	0.396149	−	0.258443i	0.396149	−	0.258443i	0.728969	+	0.684547i	$0.240000\pi$
$114$		0			0
$115$		0			0
$116$	0.215525	−	1.89753i	0.215525	−	1.89753i
$117$	1.02855	−	1.71451i	1.02855	−	1.71451i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.448383	−	0.893841i	0.448383	−	0.893841i
$122$	1.28887	−	1.15077i	1.28887	−	1.15077i
$123$		0			0
$124$		0			0
$125$	0.954865	+	0.297042i	0.954865	+	0.297042i
$126$		0			0
$127$		0			0		0.850994	−	0.525175i	$-0.176000\pi$
$127$		0			0		−0.850994	+	0.525175i	$0.824000\pi$
$128$	0.994951	−	0.100362i	0.994951	−	0.100362i
$129$		0			0
$130$	0.665429	+	1.88539i	0.665429	+	1.88539i
$131$		0			0		0.285019	−	0.958522i	$-0.408000\pi$
$131$		0			0		−0.285019	+	0.958522i	$0.592000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.744257	+	1.81288i	−0.744257	+	1.81288i
$137$	0.151124	−	0.396169i	0.151124	−	0.396169i	−0.837528	−	0.546394i	$-0.816000\pi$
$137$	0.151124	−	0.396169i	0.151124	−	0.396169i	0.988652	+	0.150226i	$0.0480000\pi$
$138$		0			0
$139$		0			0		−0.675333	−	0.737513i	$-0.736000\pi$
$139$		0			0		0.675333	+	0.737513i	$0.264000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.212007	+	0.977268i	0.212007	+	0.977268i
$145$	−1.21731	+	1.47147i	−1.21731	+	1.47147i
$146$	1.08144	−	0.744921i	1.08144	−	0.744921i
$147$		0			0
$148$	0.129717	+	0.243142i	0.129717	+	0.243142i
$149$	1.08287	−	0.595312i	1.08287	−	0.595312i	0.162637	−	0.986686i	$-0.448000\pi$
$149$	1.08287	−	0.595312i	1.08287	−	0.595312i	0.920232	+	0.391374i	$0.128000\pi$
$150$		0			0
$151$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$151$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$152$		0			0
$153$	−1.88530	−	0.534907i	−1.88530	−	0.534907i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.0957483	−	1.52188i	0.0957483	−	1.52188i	−0.597905	−	0.801567i	$-0.704000\pi$
$157$	0.0957483	−	1.52188i	0.0957483	−	1.52188i	0.693653	−	0.720309i	$-0.256000\pi$
$158$		0			0
$159$		0			0
$160$	−0.888136	−	0.459580i	−0.888136	−	0.459580i
$161$		0			0
$162$	−0.947098	+	0.320944i	−0.947098	+	0.320944i
$163$		0			0		−0.693653	−	0.720309i	$-0.744000\pi$
$163$		0			0		0.693653	+	0.720309i	$0.256000\pi$
$164$	−0.0178824	+	1.42296i	−0.0178824	+	1.42296i
$165$		0			0
$166$		0			0
$167$		0			0		0.863923	−	0.503623i	$-0.168000\pi$
$167$		0			0		−0.863923	+	0.503623i	$0.832000\pi$
$168$		0			0
$169$	2.75837	−	1.17313i	2.75837	−	1.17313i
$170$	1.61389	−	1.11168i	1.61389	−	1.11168i
$171$		0			0
$172$		0			0
$173$	0.257047	−	0.453960i	0.257047	−	0.453960i	−0.711536	−	0.702650i	$-0.752000\pi$
$173$	0.257047	−	0.453960i	0.257047	−	0.453960i	0.968583	+	0.248690i	$0.0800000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.58347	−	1.21241i	−1.58347	−	1.21241i
$179$		0			0		0.656586	−	0.754251i	$-0.272000\pi$
$179$		0			0		−0.656586	+	0.754251i	$0.728000\pi$
$180$	0.356412	−	0.934329i	0.356412	−	0.934329i
$181$	−0.469300	+	0.597232i	−0.469300	+	0.597232i	−0.962028	−	0.272952i	$-0.912000\pi$
$181$	−0.469300	+	0.597232i	−0.469300	+	0.597232i	0.492727	+	0.870184i	$0.336000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0242101	−	0.274515i	0.0242101	−	0.274515i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.260842	−	0.965382i	$-0.584000\pi$
$191$		0			0		0.260842	+	0.965382i	$0.416000\pi$
$192$		0			0
$193$	−0.651916	+	0.473645i	−0.651916	+	0.473645i	−0.863923	−	0.503623i	$-0.832000\pi$
$193$	−0.651916	+	0.473645i	−0.651916	+	0.473645i	0.212007	+	0.977268i	$0.432000\pi$
$194$	0.867013	+	1.29318i	0.867013	+	1.29318i
$195$		0			0
$196$	0.920232	+	0.391374i	0.920232	+	0.391374i
$197$	0.506272	+	1.70260i	0.506272	+	1.70260i	0.693653	+	0.720309i	$0.256000\pi$
$197$	0.506272	+	1.70260i	0.506272	+	1.70260i	−0.187381	+	0.982287i	$0.560000\pi$
$198$		0			0
$199$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$199$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$200$	0.492727	+	0.870184i	0.492727	+	0.870184i
$201$		0			0
$202$	−0.648745	+	0.295377i	−0.648745	+	0.295377i
$203$		0			0
$204$		0			0
$205$	0.792474	−	1.18200i	0.792474	−	1.18200i
$206$		0			0
$207$		0			0
$208$	−0.805558	+	1.82990i	−0.805558	+	1.82990i
$209$		0			0
$210$		0			0
$211$		0			0		0.954865	−	0.297042i	$-0.0960000\pi$
$211$		0			0		−0.954865	+	0.297042i	$0.904000\pi$
$212$	−0.258697	−	1.85956i	−0.258697	−	1.85956i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.551301	−	0.868711i	−0.551301	−	0.868711i
$219$		0			0
$220$		0			0
$221$	−2.42089	−	3.08082i	−2.42089	−	3.08082i
$222$		0			0
$223$		0			0		−0.577573	−	0.816339i	$-0.696000\pi$
$223$		0			0		0.577573	+	0.816339i	$0.304000\pi$
$224$		0			0
$225$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$226$	−0.344801	−	0.323789i	−0.344801	−	0.323789i
$227$		0			0		0.137790	−	0.990461i	$-0.456000\pi$
$227$		0			0		−0.137790	+	0.990461i	$0.544000\pi$
$228$		0			0
$229$	0.0214849	+	0.0722537i	0.0214849	+	0.0722537i	0.968583	−	0.248690i	$-0.0800000\pi$
$229$	0.0214849	+	0.0722537i	0.0214849	+	0.0722537i	−0.947098	+	0.320944i	$0.896000\pi$
$230$		0			0
$231$		0			0
$232$	−1.89467	+	0.239353i	−1.89467	+	0.239353i
$233$	0.321583	−	1.95098i	0.321583	−	1.95098i	0.0125660	−	0.999921i	$-0.496000\pi$
$233$	0.321583	−	1.95098i	0.321583	−	1.95098i	0.309017	−	0.951057i	$-0.400000\pi$
$234$	−1.90913	−	0.593896i	−1.90913	−	0.593896i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.762443	−	0.647056i	$-0.776000\pi$
$239$		0			0		0.762443	+	0.647056i	$0.224000\pi$
$240$		0			0
$241$	0.886323	+	1.56530i	0.886323	+	1.56530i	0.823533	+	0.567269i	$0.192000\pi$
$241$	0.886323	+	1.56530i	0.886323	+	1.56530i	0.0627905	+	0.998027i	$0.480000\pi$
$242$	−0.974527	−	0.224271i	−0.974527	−	0.224271i
$243$		0			0
$244$	−1.42294	−	0.980154i	−1.42294	−	0.980154i
$245$	−0.556876	−	0.830596i	−0.556876	−	0.830596i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$251$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.332820	−	0.942991i	−0.332820	−	0.942991i
$257$	0.0329233	+	0.172590i	0.0329233	+	0.172590i	0.994951	−	0.100362i	$-0.0320000\pi$
$257$	0.0329233	+	0.172590i	0.0329233	+	0.172590i	−0.962028	+	0.272952i	$0.912000\pi$
$258$		0			0
$259$		0			0
$260$	1.67453	−	1.09244i	1.67453	−	1.09244i
$261$	−0.451649	−	1.85555i	−0.451649	−	1.85555i
$262$		0			0
$263$		0			0		−0.910106	−	0.414376i	$-0.864000\pi$
$263$		0			0		0.910106	+	0.414376i	$0.136000\pi$
$264$		0			0
$265$	−0.799387	+	1.69878i	−0.799387	+	1.69878i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.601089	+	1.36543i	−0.601089	+	1.36543i	0.309017	+	0.951057i	$0.400000\pi$
$269$	−0.601089	+	1.36543i	−0.601089	+	1.36543i	−0.910106	+	0.414376i	$0.864000\pi$
$270$		0			0
$271$		0			0		0.332820	−	0.942991i	$-0.392000\pi$
$271$		0			0		−0.332820	+	0.942991i	$0.608000\pi$
$272$	1.93747	+	0.294399i	1.93747	+	0.294399i
$273$		0			0
$274$	−0.420671	−	0.0531431i	−0.420671	−	0.0531431i
$275$		0			0
$276$		0			0
$277$	−1.14184	+	0.355205i	−1.14184	+	0.355205i	−0.809017	−	0.587785i	$-0.800000\pi$
$277$	−1.14184	+	0.355205i	−1.14184	+	0.355205i	−0.332820	+	0.942991i	$0.608000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.10111	+	0.167314i	−1.10111	+	0.167314i	−0.675333	−	0.737513i	$-0.736000\pi$
$281$	−1.10111	+	0.167314i	−1.10111	+	0.167314i	−0.425779	+	0.904827i	$0.640000\pi$
$282$		0			0
$283$		0			0		−0.332820	−	0.942991i	$-0.608000\pi$
$283$		0			0		0.332820	+	0.942991i	$0.392000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.899405	−	0.437116i	0.899405	−	0.437116i
$289$	−1.69833	+	2.27682i	−1.69833	+	2.27682i
$290$	1.71762	+	0.834773i	1.71762	+	0.834773i
$291$		0			0
$292$	−0.979549	−	0.874588i	−0.979549	−	0.874588i
$293$	−0.0233672	+	0.00925174i	−0.0233672	+	0.00925174i	−0.379779	−	0.925077i	$-0.624000\pi$
$293$	−0.0233672	+	0.00925174i	−0.0233672	+	0.00925174i	0.356412	+	0.934329i	$0.384000\pi$
$294$		0			0
$295$		0			0
$296$	0.205567	−	0.183540i	0.205567	−	0.183540i
$297$		0			0
$298$	−0.834522	−	0.911359i	−0.834522	−	0.911359i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.656200	+	1.59839i	0.656200	+	1.59839i
$306$	−0.0738618	+	1.95832i	−0.0738618	+	1.95832i
$307$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$307$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.984564	−	0.175023i	$-0.0560000\pi$
$311$		0			0		−0.984564	+	0.175023i	$0.944000\pi$
$312$		0			0
$313$	0.469268	−	1.92794i	0.469268	−	1.92794i	0.112856	−	0.993611i	$-0.464000\pi$
$313$	0.469268	−	1.92794i	0.469268	−	1.92794i	0.356412	−	0.934329i	$-0.384000\pi$
$314$	−1.50135	+	0.266890i	−1.50135	+	0.266890i
$315$		0			0
$316$		0			0
$317$	−0.596778	−	0.890111i	−0.596778	−	0.890111i	0.402906	−	0.915241i	$-0.368000\pi$
$317$	−0.596778	−	0.890111i	−0.596778	−	0.890111i	−0.999684	+	0.0251301i	$0.992000\pi$
$318$		0			0
$319$		0			0
$320$	−0.236499	+	0.971632i	−0.236499	+	0.971632i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.535827	+	0.844328i	0.535827	+	0.844328i
$325$	−1.99937			−1.99937
$326$		0			0
$327$		0			0
$328$	1.38682	−	0.319153i	1.38682	−	0.319153i
$329$		0			0
$330$		0			0
$331$		0			0		−0.974527	−	0.224271i	$-0.928000\pi$
$331$		0			0		0.974527	+	0.224271i	$0.0720000\pi$
$332$		0			0
$333$	0.205567	+	0.183540i	0.205567	+	0.183540i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.14660	+	0.557256i	−1.14660	+	0.557256i	−0.910106	−	0.414376i	$-0.864000\pi$
$337$	−1.14660	+	0.557256i	−1.14660	+	0.557256i	−0.236499	+	0.971632i	$0.576000\pi$
$338$	−1.79220	−	2.40268i	−1.79220	−	2.40268i
$339$		0			0
$340$	−1.46183	−	1.30519i	−1.46183	−	1.30519i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.501873	−	0.142394i	−0.501873	−	0.142394i
$347$		0			0		0.556876	−	0.830596i	$-0.312000\pi$
$347$		0			0		−0.556876	+	0.830596i	$0.688000\pi$
$348$		0			0
$349$	0.933985	−	0.117990i	0.933985	−	0.117990i	0.356412	−	0.934329i	$-0.384000\pi$
$349$	0.933985	−	0.117990i	0.933985	−	0.117990i	0.577573	+	0.816339i	$0.304000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.370510	−	0.0562989i	−0.370510	−	0.0562989i	−0.0376902	−	0.999289i	$-0.512000\pi$
$353$	−0.370510	−	0.0562989i	−0.370510	−	0.0562989i	−0.332820	+	0.942991i	$0.608000\pi$
$354$		0			0
$355$		0			0
$356$	−0.803523	+	1.82528i	−0.803523	+	1.82528i
$357$		0			0
$358$		0			0
$359$		0			0		−0.597905	−	0.801567i	$-0.704000\pi$
$359$		0			0		0.597905	+	0.801567i	$0.296000\pi$
$360$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$361$	0.112856	+	0.993611i	0.112856	+	0.993611i
$362$	0.691278	+	0.314742i	0.691278	+	0.314742i
$363$		0			0
$364$		0			0
$365$	0.342530	+	1.26771i	0.342530	+	1.26771i
$366$		0			0
$367$		0			0		−0.0878512	−	0.996134i	$-0.528000\pi$
$367$		0			0		0.0878512	+	0.996134i	$0.472000\pi$
$368$		0			0
$369$	0.473626	+	1.34194i	0.473626	+	1.34194i
$370$	−0.272453	+	0.0413993i	−0.272453	+	0.0413993i
$371$		0			0
$372$		0			0
$373$	−0.649264	−	0.551006i	−0.649264	−	0.551006i	0.260842	−	0.965382i	$-0.416000\pi$
$373$	−0.649264	−	0.551006i	−0.649264	−	0.551006i	−0.910106	+	0.414376i	$0.864000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.45009	−	3.53218i	1.45009	−	3.53218i
$378$		0			0
$379$		0			0		−0.162637	−	0.986686i	$-0.552000\pi$
$379$		0			0		0.162637	+	0.986686i	$0.448000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.974527	−	0.224271i	$-0.928000\pi$
$383$		0			0		0.974527	+	0.224271i	$0.0720000\pi$
$384$		0			0
$385$		0			0
$386$	0.614386	+	0.521406i	0.614386	+	0.521406i
$387$		0			0
$388$	1.05144	−	1.14825i	1.05144	−	1.14825i
$389$	0.441861	−	0.780352i	0.441861	−	0.780352i	−0.556876	−	0.830596i	$-0.688000\pi$
$389$	0.441861	−	0.780352i	0.441861	−	0.780352i	0.998737	+	0.0502443i	$0.0160000\pi$
$390$		0			0
$391$		0			0
$392$	0.162637	−	0.986686i	0.162637	−	0.986686i
$393$		0			0
$394$	1.53456	−	0.894572i	1.53456	−	0.894572i
$395$		0			0
$396$		0			0
$397$	0.315055	+	1.91137i	0.315055	+	1.91137i	0.402906	+	0.915241i	$0.368000\pi$
$397$	0.315055	+	1.91137i	0.315055	+	1.91137i	−0.0878512	+	0.996134i	$0.528000\pi$
$398$		0			0
$399$		0			0
$400$	0.728969	−	0.684547i	0.728969	−	0.684547i
$401$	1.45058	−	1.36218i	1.45058	−	1.36218i	0.656586	−	0.754251i	$-0.272000\pi$
$401$	1.45058	−	1.36218i	1.45058	−	1.36218i	0.793990	−	0.607930i	$-0.208000\pi$
$402$		0			0
$403$		0			0
$404$	0.440425	+	0.560485i	0.440425	+	0.560485i
$405$	0.0125660	−	0.999921i	0.0125660	−	0.999921i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.526870	−	1.28337i	−0.526870	−	1.28337i	−0.929776	−	0.368125i	$-0.880000\pi$
$409$	−0.526870	−	1.28337i	−0.526870	−	1.28337i	0.402906	−	0.915241i	$-0.368000\pi$
$410$	−1.33589	−	0.490451i	−1.33589	−	0.490451i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.96851	+	0.349936i	1.96851	+	0.349936i
$417$		0			0
$418$		0			0
$419$		0			0		−0.656586	−	0.754251i	$-0.728000\pi$
$419$		0			0		0.656586	+	0.754251i	$0.272000\pi$
$420$		0			0
$421$	−0.355124	+	1.63698i	−0.355124	+	1.63698i	0.356412	+	0.934329i	$0.384000\pi$
$421$	−0.355124	+	1.63698i	−0.355124	+	1.63698i	−0.711536	+	0.702650i	$0.752000\pi$
$422$		0			0
$423$		0			0
$424$	−1.74563	+	0.691142i	−1.74563	+	0.691142i
$425$	0.511174	+	1.89187i	0.511174	+	1.89187i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.556876	−	0.830596i	$-0.688000\pi$
$431$		0			0		0.556876	+	0.830596i	$0.312000\pi$
$432$		0			0
$433$	−0.312923	+	0.0887843i	−0.312923	+	0.0887843i	−0.425779	−	0.904827i	$-0.640000\pi$
$433$	−0.312923	+	0.0887843i	−0.312923	+	0.0887843i	0.112856	+	0.993611i	$0.464000\pi$
$434$		0			0
$435$		0			0
$436$	−0.713685	+	0.741111i	−0.713685	+	0.741111i
$437$		0			0
$438$		0			0
$439$		0			0		−0.899405	−	0.437116i	$-0.856000\pi$
$439$		0			0		0.899405	+	0.437116i	$0.144000\pi$
$440$		0			0
$441$	0.998737	+	0.0502443i	0.998737	+	0.0502443i
$442$	−2.42089	+	3.08082i	−2.42089	+	3.08082i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	1.67030	−	1.08968i	1.67030	−	1.08968i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.106869	−	1.69863i	0.106869	−	1.69863i	−0.470704	−	0.882291i	$-0.656000\pi$
$449$	0.106869	−	1.69863i	0.106869	−	1.69863i	0.577573	−	0.816339i	$-0.304000\pi$
$450$	0.762443	+	0.647056i	0.762443	+	0.647056i
$451$		0			0
$452$	−0.233059	+	0.411595i	−0.233059	+	0.411595i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.18532	+	1.43281i	1.18532	+	1.43281i	0.876307	+	0.481754i	$0.160000\pi$
$457$	1.18532	+	1.43281i	1.18532	+	1.43281i	0.309017	+	0.951057i	$0.400000\pi$
$458$	0.0651229	−	0.0379633i	0.0651229	−	0.0379633i
$459$		0			0
$460$		0			0
$461$	0.0123833	−	0.985377i	0.0123833	−	0.985377i	−0.863923	−	0.503623i	$-0.832000\pi$
$461$	0.0123833	−	0.985377i	0.0123833	−	0.985377i	0.876307	−	0.481754i	$-0.160000\pi$
$462$		0			0
$463$		0			0		0.947098	−	0.320944i	$-0.104000\pi$
$463$		0			0		−0.947098	+	0.320944i	$0.896000\pi$
$464$	0.680650	+	1.78432i	0.680650	+	1.78432i
$465$		0			0
$466$	−1.97169	+	0.148944i	−1.97169	+	0.148944i
$467$		0			0		0.793990	−	0.607930i	$-0.208000\pi$
$467$		0			0		−0.793990	+	0.607930i	$0.792000\pi$
$468$	−0.125541	+	1.99542i	−0.125541	+	1.99542i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.883731	−	1.65647i	−0.883731	−	1.65647i
$478$		0			0
$479$		0			0		0.823533	−	0.567269i	$-0.192000\pi$
$479$		0			0		−0.823533	+	0.567269i	$0.808000\pi$
$480$		0			0
$481$	0.116813	+	0.538462i	0.116813	+	0.538462i
$482$	1.31128	−	1.23137i	1.31128	−	1.23137i
$483$		0			0
$484$	0.0125660	+	0.999921i	0.0125660	+	0.999921i
$485$	−1.50801	+	0.387191i	−1.50801	+	0.387191i
$486$		0			0
$487$		0			0		0.112856	−	0.993611i	$-0.464000\pi$
$487$		0			0		−0.112856	+	0.993611i	$0.536000\pi$
$488$	−0.615825	+	1.61438i	−0.615825	+	1.61438i
$489$		0			0
$490$	−0.675333	+	0.737513i	−0.675333	+	0.737513i
$491$		0			0		0.979855	−	0.199710i	$-0.0640000\pi$
$491$		0			0		−0.979855	+	0.199710i	$0.936000\pi$
$492$		0			0
$493$	−3.71300	−	0.469062i	−3.71300	−	0.469062i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$499$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$500$	−0.984564	+	0.175023i	−0.984564	+	0.175023i
$501$		0			0
$502$		0			0
$503$		0			0		0.745941	−	0.666012i	$-0.232000\pi$
$503$		0			0		−0.745941	+	0.666012i	$0.768000\pi$
$504$		0			0
$505$	−0.0626224	−	0.710068i	−0.0626224	−	0.710068i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.152431	+	1.34204i	−0.152431	+	1.34204i	0.656586	+	0.754251i	$0.272000\pi$
$509$	−0.152431	+	1.34204i	−0.152431	+	1.34204i	−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$	−0.837528	+	0.546394i	−0.837528	+	0.546394i
$513$		0			0
$514$	0.159908	−	0.0728068i	0.159908	−	0.0728068i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.45748	−	1.36866i	−1.45748	−	1.36866i
$521$	−1.52768	−	1.16969i	−1.52768	−	1.16969i	−0.929776	−	0.368125i	$-0.880000\pi$
$521$	−1.52768	−	1.16969i	−1.52768	−	1.16969i	−0.597905	−	0.801567i	$-0.704000\pi$
$522$	−1.69610	+	0.877673i	−1.69610	+	0.877673i
$523$		0			0		−0.899405	−	0.437116i	$-0.856000\pi$
$523$		0			0		0.899405	+	0.437116i	$0.144000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.0376902	−	0.999289i	−0.0376902	−	0.999289i
$530$	1.83965	+	0.374949i	1.83965	+	0.374949i
$531$		0			0
$532$		0			0
$533$	−0.810949	+	2.72723i	−0.810949	+	2.72723i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.46885	+	0.261114i	1.46885	+	0.261114i
$539$		0			0
$540$		0			0
$541$	1.03894	−	0.504932i	1.03894	−	0.504932i	0.162637	−	0.986686i	$-0.448000\pi$
$541$	1.03894	−	0.504932i	1.03894	−	0.504932i	0.876307	+	0.481754i	$0.160000\pi$
$542$		0			0
$543$		0			0
$544$	−0.172163	−	1.95213i	−0.172163	−	1.95213i
$545$	1.00267	−	0.230747i	1.00267	−	0.230747i
$546$		0			0
$547$		0			0		0.656586	−	0.754251i	$-0.272000\pi$
$547$		0			0		−0.656586	+	0.754251i	$0.728000\pi$
$548$	0.0478527	+	0.421305i	0.0478527	+	0.421305i
$549$	−1.67356	−	0.429698i	−1.67356	−	0.429698i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.615172	+	1.02544i	0.615172	+	1.02544i
$555$		0			0
$556$		0			0
$557$	−1.37694	−	1.00040i	−1.37694	−	1.00040i	−0.997159	−	0.0753268i	$-0.976000\pi$
$557$	−1.37694	−	1.00040i	−1.37694	−	1.00040i	−0.379779	−	0.925077i	$-0.624000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.422979	+	1.03031i	0.422979	+	1.03031i
$563$		0			0		0.762443	−	0.647056i	$-0.224000\pi$
$563$		0			0		−0.762443	+	0.647056i	$0.776000\pi$
$564$		0			0
$565$	0.420087	−	0.217380i	0.420087	−	0.217380i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.75109	−	0.496830i	1.75109	−	0.496830i	0.762443	−	0.647056i	$-0.224000\pi$
$569$	1.75109	−	0.496830i	1.75109	−	0.496830i	0.988652	+	0.150226i	$0.0480000\pi$
$570$		0			0
$571$		0			0		0.0878512	−	0.996134i	$-0.472000\pi$
$571$		0			0		−0.0878512	+	0.996134i	$0.528000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$577$	0.123051	+	0.0250798i	0.123051	+	0.0250798i	0.260842	−	0.965382i	$-0.416000\pi$
$577$	0.123051	+	0.0250798i	0.123051	+	0.0250798i	−0.137790	+	0.990461i	$0.544000\pi$
$578$	2.61389	+	1.11168i	2.61389	+	1.11168i
$579$		0			0
$580$	0.404876	−	1.86632i	0.404876	−	1.86632i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.618115	+	1.15860i	−0.618115	+	1.15860i
$585$	1.19543	−	1.60263i	1.19543	−	1.60263i
$586$	0.0145156	+	0.0205163i	0.0145156	+	0.0205163i
$587$		0			0		0.470704	−	0.882291i	$-0.344000\pi$
$587$		0			0		−0.470704	+	0.882291i	$0.656000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.226950	−	0.156328i	−0.226950	−	0.156328i
$593$	−1.07403	−	0.425237i	−1.07403	−	0.425237i	−0.236499	−	0.971632i	$-0.576000\pi$
$593$	−1.07403	−	0.425237i	−1.07403	−	0.425237i	−0.837528	+	0.546394i	$0.816000\pi$
$594$		0			0
$595$		0			0
$596$	−0.688142	+	1.02638i	−0.688142	+	1.02638i
$597$		0			0
$598$		0			0
$599$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$599$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$600$		0			0
$601$	−0.676129	+	1.43685i	−0.676129	+	1.43685i	0.212007	+	0.977268i	$0.432000\pi$
$601$	−0.676129	+	1.43685i	−0.676129	+	1.43685i	−0.888136	+	0.459580i	$0.848000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.535827	−	0.844328i	0.535827	−	0.844328i
$606$		0			0
$607$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$607$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$608$		0			0
$609$		0			0
$610$	1.39786	−	1.01560i	1.39786	−	1.01560i
$611$		0			0
$612$	1.92023	−	0.391374i	1.92023	−	0.391374i
$613$	0.732851	+	0.590914i	0.732851	+	0.590914i	0.920232	−	0.391374i	$-0.128000\pi$
$613$	0.732851	+	0.590914i	0.732851	+	0.590914i	−0.187381	+	0.982287i	$0.560000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.588804	+	0.676387i	0.588804	+	0.676387i	0.968583	−	0.248690i	$-0.0800000\pi$
$617$	0.588804	+	0.676387i	0.588804	+	0.676387i	−0.379779	+	0.925077i	$0.624000\pi$
$618$		0			0
$619$		0			0		0.793990	−	0.607930i	$-0.208000\pi$
$619$		0			0		−0.793990	+	0.607930i	$0.792000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.920232	+	0.391374i	0.920232	+	0.391374i
$626$	−1.98423			−1.98423
$627$		0			0
$628$	0.614386	+	1.39564i	0.614386	+	1.39564i
$629$	0.479645	−	0.248200i	0.479645	−	0.248200i
$630$		0			0
$631$		0			0		0.793990	−	0.607930i	$-0.208000\pi$
$631$		0			0		−0.793990	+	0.607930i	$0.792000\pi$
$632$		0			0
$633$		0			0
$634$	−0.723723	+	0.790359i	−0.723723	+	0.790359i
$635$		0			0
$636$		0			0
$637$	1.55643	+	1.25499i	1.55643	+	1.25499i
$638$		0			0
$639$		0			0
$640$	1.00000			1.00000
$641$	0.842065	−	1.19017i	0.842065	−	1.19017i	−0.137790	−	0.990461i	$-0.544000\pi$
$641$	0.842065	−	1.19017i	0.842065	−	1.19017i	0.979855	−	0.199710i	$-0.0640000\pi$
$642$		0			0
$643$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$643$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.711536	−	0.702650i	$-0.752000\pi$
$647$		0			0		0.711536	+	0.702650i	$0.248000\pi$
$648$	0.693653	−	0.720309i	0.693653	−	0.720309i
$649$		0			0
$650$	0.472849	+	1.94265i	0.472849	+	1.94265i
$651$		0			0
$652$		0			0
$653$	0.0631332	+	1.67387i	0.0631332	+	1.67387i	0.577573	+	0.816339i	$0.304000\pi$
$653$	0.0631332	+	1.67387i	0.0631332	+	1.67387i	−0.514440	+	0.857527i	$0.672000\pi$
$654$		0			0
$655$		0			0
$656$	−0.638081	−	1.27200i	−0.638081	−	1.27200i
$657$	−1.22096	−	0.483411i	−1.22096	−	0.483411i
$658$		0			0
$659$		0			0		0.762443	−	0.647056i	$-0.224000\pi$
$659$		0			0		−0.762443	+	0.647056i	$0.776000\pi$
$660$		0			0
$661$	0.457154	−	1.69194i	0.457154	−	1.69194i	−0.236499	−	0.971632i	$-0.576000\pi$
$661$	0.457154	−	1.69194i	0.457154	−	1.69194i	0.693653	−	0.720309i	$-0.256000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.129717	−	0.243142i	0.129717	−	0.243142i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.13188	+	0.230694i	1.13188	+	0.230694i	0.728969	−	0.684547i	$-0.240000\pi$
$673$	1.13188	+	0.230694i	1.13188	+	0.230694i	0.402906	+	0.915241i	$0.368000\pi$
$674$	0.812619	+	0.982287i	0.812619	+	0.982287i
$675$		0			0
$676$	−1.91066	+	2.30959i	−1.91066	+	2.30959i
$677$	−0.0977601	+	0.0788261i	−0.0977601	+	0.0788261i	−0.675333	−	0.737513i	$-0.736000\pi$
$677$	−0.0977601	+	0.0788261i	−0.0977601	+	0.0788261i	0.577573	+	0.816339i	$0.304000\pi$
$678$		0			0
$679$		0			0
$680$	−0.922443	+	1.72904i	−0.922443	+	1.72904i
$681$		0			0
$682$		0			0
$683$		0			0		−0.938734	−	0.344643i	$-0.888000\pi$
$683$		0			0		0.938734	+	0.344643i	$0.112000\pi$
$684$		0			0
$685$	0.190121	−	0.379001i	0.190121	−	0.379001i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.517230	−	3.71794i	0.517230	−	3.71794i
$690$		0			0
$691$		0			0		0.356412	−	0.934329i	$-0.384000\pi$
$691$		0			0		−0.356412	+	0.934329i	$0.616000\pi$
$692$	−0.0196623	+	0.521312i	−0.0196623	+	0.521312i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.78793	+	0.0700830i	2.78793	+	0.0700830i
$698$	−0.335529	−	0.879585i	−0.335529	−	0.879585i
$699$		0			0
$700$		0			0
$701$	1.59532	+	0.409608i	1.59532	+	0.409608i	0.938734	−	0.344643i	$-0.112000\pi$
$701$	1.59532	+	0.409608i	1.59532	+	0.409608i	0.656586	+	0.754251i	$0.272000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.0329233	+	0.373314i	0.0329233	+	0.373314i
$707$		0			0
$708$		0			0
$709$	−0.512696	+	0.249173i	−0.512696	+	0.249173i	−0.675333	−	0.737513i	$-0.736000\pi$
$709$	−0.512696	+	0.249173i	−0.512696	+	0.249173i	0.162637	+	0.986686i	$0.448000\pi$
$710$		0			0
$711$		0			0
$712$	1.96353	+	0.349052i	1.96353	+	0.349052i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.162637	−	0.986686i	$-0.448000\pi$
$719$		0			0		−0.162637	+	0.986686i	$0.552000\pi$
$720$	0.112856	+	0.993611i	0.112856	+	0.993611i
$721$		0			0
$722$	0.938734	−	0.344643i	0.938734	−	0.344643i
$723$		0			0
$724$	0.142327	−	0.746104i	0.142327	−	0.746104i
$725$	−1.35884	+	1.34187i	−1.35884	+	1.34187i
$726$		0			0
$727$		0			0		−0.899405	−	0.437116i	$-0.856000\pi$
$727$		0			0		0.899405	+	0.437116i	$0.144000\pi$
$728$		0			0
$729$	0.793990	+	0.607930i	0.793990	+	0.607930i
$730$	1.15074	−	0.632625i	1.15074	−	0.632625i
$731$		0			0
$732$		0			0
$733$	1.44139	−	0.219019i	1.44139	−	0.219019i	0.617860	−	0.786288i	$-0.288000\pi$
$733$	1.44139	−	0.219019i	1.44139	−	0.219019i	0.823533	+	0.567269i	$0.192000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.19186	−	0.777558i	1.19186	−	0.777558i
$739$		0			0		−0.997159	−	0.0753268i	$-0.976000\pi$
$739$		0			0		0.997159	+	0.0753268i	$0.0240000\pi$
$740$	0.104660	+	0.254933i	0.104660	+	0.254933i
$741$		0			0
$742$		0			0
$743$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$743$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$744$		0			0
$745$	1.13715	−	0.483628i	1.13715	−	0.483628i
$746$	−0.381825	+	0.761158i	−0.381825	+	0.761158i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$		0			0
$754$	−3.77492	−	0.573599i	−3.77492	−	0.573599i
$755$		0			0
$756$		0			0
$757$	1.48012	+	0.186982i	1.48012	+	0.186982i	0.823533	−	0.567269i	$-0.192000\pi$
$757$	1.48012	+	0.186982i	1.48012	+	0.186982i	0.656586	+	0.754251i	$0.272000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.0667281	−	0.162538i	0.0667281	−	0.162538i	−0.888136	−	0.459580i	$-0.848000\pi$
$761$	0.0667281	−	0.162538i	0.0667281	−	0.162538i	0.954865	+	0.297042i	$0.0960000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.82209	−	0.721417i	−1.82209	−	0.721417i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.410693	+	1.89313i	0.410693	+	1.89313i	0.448383	+	0.893841i	$0.352000\pi$
$769$	0.410693	+	1.89313i	0.410693	+	1.89313i	−0.0376902	+	0.999289i	$0.512000\pi$
$770$		0			0
$771$		0			0
$772$	0.361313	−	0.720269i	0.361313	−	0.720269i
$773$	−0.866313	−	1.62382i	−0.866313	−	1.62382i	−0.778462	−	0.627691i	$-0.784000\pi$
$773$	−0.866313	−	1.62382i	−0.866313	−	1.62382i	−0.0878512	−	0.996134i	$-0.528000\pi$
$774$		0			0
$775$		0			0
$776$	−1.36434	−	0.750054i	−1.36434	−	0.750054i
$777$		0			0
$778$	−0.862714	−	0.244774i	−0.862714	−	0.244774i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.997159	+	0.0753268i	−0.997159	+	0.0753268i
$785$	0.248003	−	1.50458i	0.248003	−	1.50458i
$786$		0			0
$787$		0			0		0.947098	−	0.320944i	$-0.104000\pi$
$787$		0			0		−0.947098	+	0.320944i	$0.896000\pi$
$788$	−1.23212	−	1.27947i	−1.23212	−	1.27947i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.20205	−	2.66182i	−2.20205	−	2.66182i
$794$	1.78264	−	0.758156i	1.78264	−	0.758156i
$795$		0			0
$796$		0			0
$797$	−0.0787820	+	0.893299i	−0.0787820	+	0.893299i	0.850994	+	0.525175i	$0.176000\pi$
$797$	−0.0787820	+	0.893299i	−0.0787820	+	0.893299i	−0.929776	+	0.368125i	$0.880000\pi$
$798$		0			0
$799$		0			0
$800$	−0.837528	−	0.546394i	−0.837528	−	0.546394i
$801$	−0.125224	+	1.99038i	−0.125224	+	1.99038i
$802$	−1.66660	−	1.08727i	−1.66660	−	1.08727i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.440425	−	0.560485i	0.440425	−	0.560485i
$809$	−1.81791	−	0.0914553i	−1.81791	−	0.0914553i	−0.888136	−	0.459580i	$-0.848000\pi$
$809$	−1.81791	−	0.0914553i	−1.81791	−	0.0914553i	−0.929776	+	0.368125i	$0.880000\pi$
$810$	−0.974527	+	0.224271i	−0.974527	+	0.224271i
$811$		0			0		−0.899405	−	0.437116i	$-0.856000\pi$
$811$		0			0		0.899405	+	0.437116i	$0.144000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.12235	+	0.815438i	−1.12235	+	0.815438i
$819$		0			0
$820$	−0.160603	+	1.41398i	−0.160603	+	1.41398i
$821$	−0.783631	−	0.333278i	−0.783631	−	0.333278i	−0.0376902	−	0.999289i	$-0.512000\pi$
$821$	−0.783631	−	0.333278i	−0.783631	−	0.333278i	−0.745941	+	0.666012i	$0.768000\pi$
$822$		0			0
$823$		0			0		0.998737	−	0.0502443i	$-0.0160000\pi$
$823$		0			0		−0.998737	+	0.0502443i	$0.984000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.910106	−	0.414376i	$-0.136000\pi$
$827$		0			0		−0.910106	+	0.414376i	$0.864000\pi$
$828$		0			0
$829$	−0.120852	+	0.557080i	−0.120852	+	0.557080i	0.876307	+	0.481754i	$0.160000\pi$
$829$	−0.120852	+	0.557080i	−0.120852	+	0.557080i	−0.997159	+	0.0753268i	$0.976000\pi$
$830$		0			0
$831$		0			0
$832$	−0.125541	−	1.99542i	−0.125541	−	1.99542i
$833$	0.789580	−	1.79361i	0.789580	−	1.79361i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.998737	−	0.0502443i	$-0.0160000\pi$
$839$		0			0		−0.998737	+	0.0502443i	$0.984000\pi$
$840$		0			0
$841$	−1.00530	−	2.44874i	−1.00530	−	2.44874i
$842$	1.67453	−	0.0420943i	1.67453	−	0.0420943i
$843$		0			0
$844$		0			0
$845$	2.86218	−	0.890374i	2.86218	−	0.890374i
$846$		0			0
$847$		0			0
$848$	1.08437	+	1.53265i	1.08437	+	1.53265i
$849$		0			0
$850$	1.71731	−	0.944098i	1.71731	−	0.944098i
$851$		0			0
$852$		0			0
$853$	−0.0122596	−	0.0743767i	−0.0122596	−	0.0743767i	0.979855	−	0.199710i	$-0.0640000\pi$
$853$	−0.0122596	−	0.0743767i	−0.0122596	−	0.0743767i	−0.992115	+	0.125333i	$0.960000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.84489	−	0.233064i	1.84489	−	0.233064i	0.876307	−	0.481754i	$-0.160000\pi$
$857$	1.84489	−	0.233064i	1.84489	−	0.233064i	0.968583	+	0.248690i	$0.0800000\pi$
$858$		0			0
$859$		0			0		−0.954865	−	0.297042i	$-0.904000\pi$
$859$		0			0		0.954865	+	0.297042i	$0.0960000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.260842	−	0.965382i	$-0.584000\pi$
$863$		0			0		0.260842	+	0.965382i	$0.416000\pi$
$864$		0			0
$865$	0.301310	−	0.425870i	0.301310	−	0.425870i
$866$	0.160272	+	0.283048i	0.160272	+	0.283048i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.888873	+	0.518167i	0.888873	+	0.518167i
$873$	0.591287	−	1.44028i	0.591287	−	1.44028i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.40325	+	1.19088i	1.40325	+	1.19088i	0.954865	+	0.297042i	$0.0960000\pi$
$877$	1.40325	+	1.19088i	1.40325	+	1.19088i	0.448383	+	0.893841i	$0.352000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.124728	+	0.353398i	0.124728	+	0.353398i	0.988652	−	0.150226i	$-0.0480000\pi$
$881$	0.124728	+	0.353398i	0.124728	+	0.353398i	−0.863923	+	0.503623i	$0.832000\pi$
$882$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$883$		0			0		−0.0878512	−	0.996134i	$-0.528000\pi$
$883$		0			0		0.0878512	+	0.996134i	$0.472000\pi$
$884$	3.56596	+	1.62360i	3.56596	+	1.62360i
$885$		0			0
$886$		0			0
$887$		0			0		−0.998737	−	0.0502443i	$-0.984000\pi$
$887$		0			0		0.998737	+	0.0502443i	$0.0160000\pi$
$888$		0			0
$889$		0			0
$890$	−1.45380	−	1.36520i	−1.45380	−	1.36520i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.67572	+	0.297887i	−1.67572	+	0.297887i
$899$		0			0
$900$	0.448383	−	0.893841i	0.448383	−	0.893841i
$901$	−3.65028	+	0.461138i	−3.65028	+	0.461138i
$902$		0			0
$903$		0			0
$904$	0.455037	+	0.129106i	0.455037	+	0.129106i
$905$	−0.526870	+	0.547117i	−0.526870	+	0.547117i
$906$		0			0
$907$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$907$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$908$		0			0
$909$	0.606609	+	0.374357i	0.606609	+	0.374357i
$910$		0			0
$911$		0			0		−0.837528	−	0.546394i	$-0.816000\pi$
$911$		0			0		0.837528	+	0.546394i	$0.184000\pi$
$912$		0			0
$913$		0			0
$914$	1.11184	−	1.49056i	1.11184	−	1.49056i
$915$		0			0
$916$	−0.0522878	−	0.0542972i	−0.0522878	−	0.0542972i
$917$		0			0
$918$		0			0
$919$		0			0		−0.974527	−	0.224271i	$-0.928000\pi$
$919$		0			0		0.974527	+	0.224271i	$0.0720000\pi$
$920$		0			0
$921$		0			0
$922$	−0.960352	+	0.221009i	−0.960352	+	0.221009i
$923$		0			0
$924$		0			0
$925$	0.0516387	−	0.270699i	0.0516387	−	0.270699i
$926$		0			0
$927$		0			0
$928$	1.57272	−	1.08333i	1.57272	−	1.08333i
$929$	1.63239	−	0.375666i	1.63239	−	0.375666i	0.693653	−	0.720309i	$-0.256000\pi$
$929$	1.63239	−	0.375666i	1.63239	−	0.375666i	0.938734	+	0.344643i	$0.112000\pi$
$930$		0			0
$931$		0			0
$932$	0.611020	+	1.88053i	0.611020	+	1.88053i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.96851	−	0.349936i	1.96851	−	0.349936i
$937$	−0.425417	+	1.74778i	−0.425417	+	1.74778i	0.212007	+	0.977268i	$0.432000\pi$
$937$	−0.425417	+	1.74778i	−0.425417	+	1.74778i	−0.637424	+	0.770513i	$0.720000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.0195644	−	0.0157752i	−0.0195644	−	0.0157752i	0.617860	−	0.786288i	$-0.288000\pi$
$941$	−0.0195644	−	0.0157752i	−0.0195644	−	0.0157752i	−0.637424	+	0.770513i	$0.720000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.823533	−	0.567269i	$-0.192000\pi$
$947$		0			0		−0.823533	+	0.567269i	$0.808000\pi$
$948$		0			0
$949$	−1.40682	−	2.21679i	−1.40682	−	2.21679i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.77385	−	0.408220i	1.77385	−	0.408220i	0.793990	−	0.607930i	$-0.208000\pi$
$953$	1.77385	−	0.408220i	1.77385	−	0.408220i	0.979855	+	0.199710i	$0.0640000\pi$
$954$	−1.40048	+	1.25042i	−1.40048	+	1.25042i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.597905	+	0.801567i	−0.597905	+	0.801567i
$962$	0.495561	−	0.240845i	0.495561	−	0.240845i
$963$		0			0
$964$	−1.50655	−	0.982860i	−1.50655	−	0.982860i
$965$	−0.696161	+	0.405826i	−0.696161	+	0.405826i
$966$		0			0
$967$		0			0		−0.332820	−	0.942991i	$-0.608000\pi$
$967$		0			0		0.332820	+	0.942991i	$0.392000\pi$
$968$	0.968583	−	0.248690i	0.968583	−	0.248690i
$969$		0			0
$970$	0.732851	+	1.37366i	0.732851	+	1.37366i
$971$		0			0		−0.962028	−	0.272952i	$-0.912000\pi$
$971$		0			0		0.962028	+	0.272952i	$0.0880000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.71422	+	0.216557i	1.71422	+	0.216557i
$977$	1.91897	−	0.341129i	1.91897	−	0.341129i	0.920232	−	0.391374i	$-0.128000\pi$
$977$	1.91897	−	0.341129i	1.91897	−	0.341129i	0.998737	+	0.0502443i	$0.0160000\pi$
$978$		0			0
$979$		0			0
$980$	0.876307	+	0.481754i	0.876307	+	0.481754i
$981$	−0.414542	+	0.941672i	−0.414542	+	0.941672i
$982$		0			0
$983$		0			0		−0.888136	−	0.459580i	$-0.848000\pi$
$983$		0			0		0.888136	+	0.459580i	$0.152000\pi$
$984$		0			0
$985$	0.332840	+	1.74481i	0.332840	+	1.74481i
$986$	0.422367	+	3.71861i	0.422367	+	3.71861i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.910106	−	0.414376i	$-0.864000\pi$
$991$		0			0		0.910106	+	0.414376i	$0.136000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.410693	−	1.89313i	0.410693	−	1.89313i	−0.0376902	−	0.999289i	$-0.512000\pi$
$997$	0.410693	−	1.89313i	0.410693	−	1.89313i	0.448383	−	0.893841i	$-0.352000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2500.1.t.a.471.1	✓	100
4.3	odd	2	CM	2500.1.t.a.471.1	✓	100
625.556	even	125	inner	2500.1.t.a.2431.1	yes	100
2500.2431	odd	250	inner	2500.1.t.a.2431.1	yes	100

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2500.1.t.a.471.1	✓	100	1.1	even	1	trivial
2500.1.t.a.471.1	✓	100	4.3	odd	2	CM
2500.1.t.a.2431.1	yes	100	625.556	even	125	inner
2500.1.t.a.2431.1	yes	100	2500.2431	odd	250	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2500.t (of order \(250\), degree \(100\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.236499 - 0.971632i) q^{2} +(-0.888136 + 0.459580i) q^{4} +(0.994951 + 0.100362i) q^{5} +(0.656586 + 0.754251i) q^{8} +(-0.675333 + 0.737513i) q^{9} +O(q^{10})\)	\(q+(-0.236499 - 0.971632i) q^{2} +(-0.888136 + 0.459580i) q^{4} +(0.994951 + 0.100362i) q^{5} +(0.656586 + 0.754251i) q^{8} +(-0.675333 + 0.737513i) q^{9} +(-0.137790 - 0.990461i) q^{10} +(-1.95909 + 0.399294i) q^{13} +(0.577573 - 0.816339i) q^{16} +(0.878701 + 1.75167i) q^{17} +(0.876307 + 0.481754i) q^{18} +(-0.929776 + 0.368125i) q^{20} +(0.979855 + 0.199710i) q^{25} +(0.851290 + 1.80908i) q^{26} +(-1.06348 + 1.58621i) q^{29} +(-0.929776 - 0.368125i) q^{32} +(1.49417 - 1.26804i) q^{34} +(0.260842 - 0.965382i) q^{36} +(-0.00346296 - 0.275559i) q^{37} +(0.577573 + 0.816339i) q^{40} +(0.669845 - 1.25556i) q^{41} +(-0.745941 + 0.666012i) q^{45} +(-0.637424 - 0.770513i) q^{49} +(-0.0376902 - 0.999289i) q^{50} +(1.55643 - 1.25499i) q^{52} +(-0.624858 + 1.77043i) q^{53} +(1.79273 + 0.658175i) q^{58} +(0.813304 + 1.52446i) q^{61} +(-0.137790 + 0.990461i) q^{64} +(-1.98927 + 0.200660i) q^{65} +(-1.58544 - 1.15189i) q^{68} +(-0.999684 - 0.0251301i) q^{72} +(0.468030 + 1.22693i) q^{73} +(-0.266923 + 0.0685341i) q^{74} +(0.656586 - 0.754251i) q^{80} +(-0.0878512 - 0.996134i) q^{81} +(-1.37836 - 0.353903i) q^{82} +(0.698464 + 1.83101i) q^{85} +(1.55250 - 1.25182i) q^{89} +(0.823533 + 0.567269i) q^{90} +(-1.46154 + 0.536583i) q^{97} +(-0.597905 + 0.801567i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q+O(q^{10}) \) 100 * q	\( 100 q+O(q^{100}) \) 100 * q

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