LMFDB - Embedded newform 3525.1.l.d.1268.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,1,Mod(1268,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("3525.1268");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3525.l (of order $4$, degree $2$, 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:	$1.75920416953$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{120})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 $ x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{30}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{30} - \cdots)$

Embedding invariants

Embedding label			1268.12
Root			$0.838671 + 0.544639i$ of defining polynomial
Character	$\chi$	$=$	3525.1268
Dual form			3525.1.l.d.1832.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.946294 + 0.946294i) q^{2} +(0.933580 - 0.358368i) q^{3} +0.790943i q^{4} +(1.22256 + 0.544320i) q^{6} +(-0.294032 + 0.294032i) q^{7} +(0.197829 - 0.197829i) q^{8} +(0.743145 - 0.669131i) q^{9} +O(q^{10})$	\(q+(0.946294 + 0.946294i) q^{2} +(0.933580 - 0.358368i) q^{3} +0.790943i q^{4} +(1.22256 + 0.544320i) q^{6} +(-0.294032 + 0.294032i) q^{7} +(0.197829 - 0.197829i) q^{8} +(0.743145 - 0.669131i) q^{9} +(0.283449 + 0.738409i) q^{12} -0.556480 q^{14} +1.16535 q^{16} +(-0.147826 - 0.147826i) q^{17} +(1.33643 + 0.0700392i) q^{18} +(-0.169131 + 0.379874i) q^{21} +(0.113794 - 0.255585i) q^{24} +(0.453990 - 0.891007i) q^{27} +(-0.232562 - 0.232562i) q^{28} +(0.904936 + 0.904936i) q^{32} -0.279773i q^{34} +(0.529244 + 0.587785i) q^{36} +(-0.831254 + 0.831254i) q^{37} +(-0.519519 + 0.199425i) q^{42} +(-0.707107 - 0.707107i) q^{47} +(1.08795 - 0.417625i) q^{48} +0.827091i q^{49} +(-0.190983 - 0.0850311i) q^{51} +(0.437016 - 0.437016i) q^{53} +(1.27276 - 0.413545i) q^{54} +0.116336i q^{56} -0.415823 q^{59} -0.618034 q^{61} +(-0.0217625 + 0.415254i) q^{63} +0.547318i q^{64} +(0.116922 - 0.116922i) q^{68} +0.813473i q^{71} +(0.0146422 - 0.279389i) q^{72} -1.57322 q^{74} +1.61803i q^{79} +(0.104528 - 0.994522i) q^{81} +(-0.707107 + 0.707107i) q^{83} +(-0.300458 - 0.133773i) q^{84} +1.90211 q^{89} -1.33826i q^{94} +(1.16913 + 0.520530i) q^{96} +(-1.34500 + 1.34500i) q^{97} +(-0.782671 + 0.782671i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{6}+O(q^{10}) $ 32 * q - 4 * q^6	$ 32 q - 4 q^{6} - 48 q^{16} + 12 q^{21} - 24 q^{51} + 16 q^{61} - 4 q^{81} + 20 q^{96}+O(q^{100}) $ 32 * q - 4 * q^6 - 48 * q^16 + 12 * q^21 - 24 * q^51 + 16 * q^61 - 4 * q^81 + 20 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$.

$n$	$1552$	$2026$	$2351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$	0.946294	+	0.946294i	0.946294	+	0.946294i	0.998630	−	0.0523360i	$-0.0166667\pi$
$2$	0.946294	+	0.946294i	0.946294	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$3$	0.933580	−	0.358368i	0.933580	−	0.358368i
$3$	0.933580	−	0.358368i	0.933580	−	0.358368i
$4$			0.790943i			0.790943i
$5$		0			0
$5$		0			0
$6$	1.22256	+	0.544320i	1.22256	+	0.544320i
$7$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$7$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$8$	0.197829	−	0.197829i	0.197829	−	0.197829i
$9$	0.743145	−	0.669131i	0.743145	−	0.669131i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	0.283449	+	0.738409i	0.283449	+	0.738409i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	−0.556480			−0.556480
$15$		0			0
$16$	1.16535			1.16535
$17$	−0.147826	−	0.147826i	−0.147826	−	0.147826i	0.629320	−	0.777146i	$-0.283333\pi$
$17$	−0.147826	−	0.147826i	−0.147826	−	0.147826i	−0.777146	+	0.629320i	$0.783333\pi$
$18$	1.33643	+	0.0700392i	1.33643	+	0.0700392i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−0.169131	+	0.379874i	−0.169131	+	0.379874i
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	0.113794	−	0.255585i	0.113794	−	0.255585i
$25$		0			0
$26$		0			0
$27$	0.453990	−	0.891007i	0.453990	−	0.891007i
$28$	−0.232562	−	0.232562i	−0.232562	−	0.232562i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.904936	+	0.904936i	0.904936	+	0.904936i
$33$		0			0
$34$		−	0.279773i		−	0.279773i
$35$		0			0
$36$	0.529244	+	0.587785i	0.529244	+	0.587785i
$37$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$37$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−0.519519	+	0.199425i	−0.519519	+	0.199425i
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$47$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$48$	1.08795	−	0.417625i	1.08795	−	0.417625i
$49$			0.827091i			0.827091i
$50$		0			0
$51$	−0.190983	−	0.0850311i	−0.190983	−	0.0850311i
$52$		0			0
$53$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$53$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$54$	1.27276	−	0.413545i	1.27276	−	0.413545i
$55$		0			0
$56$			0.116336i			0.116336i
$57$		0			0
$58$		0			0
$59$	−0.415823			−0.415823			−0.207912	−	0.978148i	$-0.566667\pi$
$59$	−0.415823			−0.415823			−0.207912	+	0.978148i	$0.566667\pi$
$60$		0			0
$61$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$61$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$62$		0			0
$63$	−0.0217625	+	0.415254i	−0.0217625	+	0.415254i
$64$			0.547318i			0.547318i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	0.116922	−	0.116922i	0.116922	−	0.116922i
$69$		0			0
$70$		0			0
$71$			0.813473i			0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$71$			0.813473i			0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$72$	0.0146422	−	0.279389i	0.0146422	−	0.279389i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	−1.57322			−1.57322
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$79$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$80$		0			0
$81$	0.104528	−	0.994522i	0.104528	−	0.994522i
$82$		0			0
$83$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$83$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$84$	−0.300458	−	0.133773i	−0.300458	−	0.133773i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$89$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		−	1.33826i		−	1.33826i
$95$		0			0
$96$	1.16913	+	0.520530i	1.16913	+	0.520530i
$97$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$97$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$98$	−0.782671	+	0.782671i	−0.782671	+	0.782671i
$99$		0			0
$100$		0			0
$101$		−	1.98904i		−	1.98904i	−0.104528	−	0.994522i	$-0.533333\pi$
$101$		−	1.98904i		−	1.98904i	0.104528	−	0.994522i	$-0.466667\pi$
$102$	−0.100262	−	0.261190i	−0.100262	−	0.261190i
$103$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.777146	−	0.629320i	$-0.783333\pi$
$103$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.629320	−	0.777146i	$-0.716667\pi$
$104$		0			0
$105$		0			0
$106$	0.827091			0.827091
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	0.704735	+	0.359081i	0.704735	+	0.359081i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−0.478148	+	1.07394i	−0.478148	+	1.07394i
$112$	−0.342650	+	0.342650i	−0.342650	+	0.342650i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.393491	−	0.393491i	−0.393491	−	0.393491i
$119$	0.0869308			0.0869308
$120$		0			0
$121$	−1.00000			−1.00000
$122$	−0.584842	−	0.584842i	−0.584842	−	0.584842i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.413545	+	0.372358i	−0.413545	+	0.372358i
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	0.387012	−	0.387012i	0.387012	−	0.387012i
$129$		0			0
$130$		0			0
$131$		−	1.48629i		−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$131$		−	1.48629i		−	1.48629i	0.669131	−	0.743145i	$-0.266667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.0584884			−0.0584884
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$142$	−0.769785	+	0.769785i	−0.769785	+	0.769785i
$143$		0			0
$144$	0.866025	−	0.779773i	0.866025	−	0.779773i
$145$		0			0
$146$		0			0
$147$	0.296403	+	0.772156i	0.296403	+	0.772156i
$148$	−0.657474	−	0.657474i	−0.657474	−	0.657474i
$149$	0.813473			0.813473			0.406737	−	0.913545i	$-0.366667\pi$
$149$	0.813473			0.813473			0.406737	+	0.913545i	$0.366667\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−0.208770	−	0.0109412i	−0.208770	−	0.0109412i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.0523360	+	0.998630i	$0.516667\pi$
$157$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.998630	+	0.0523360i	$0.983333\pi$
$158$	−1.53114	+	1.53114i	−1.53114	+	1.53114i
$159$	0.251377	−	0.564602i	0.251377	−	0.564602i
$160$		0			0
$161$		0			0
$162$	1.04002	−	0.842195i	1.04002	−	0.842195i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	−1.33826			−1.33826
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$	0.0416911	+	0.108609i	0.0416911	+	0.108609i
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.38331	+	1.38331i	−1.38331	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$173$	−1.38331	+	1.38331i	−1.38331	+	1.38331i	−0.838671	+	0.544639i	$0.816667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.388205	+	0.149018i	−0.388205	+	0.149018i
$178$	1.79996	+	1.79996i	1.79996	+	1.79996i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−0.576984	+	0.221484i	−0.576984	+	0.221484i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.559281	−	0.559281i	0.559281	−	0.559281i
$189$	0.128496	+	0.395472i	0.128496	+	0.395472i
$190$		0			0
$191$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$192$	0.196141	+	0.510966i	0.196141	+	0.510966i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$	−2.54552			−2.54552
$195$		0			0
$196$	−0.654182			−0.654182
$197$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$197$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$	1.88222	−	1.88222i	1.88222	−	1.88222i
$203$		0			0
$204$	0.0672548	−	0.151057i	0.0672548	−	0.151057i
$205$		0			0
$206$		−	2.66186i		−	2.66186i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.345655	+	0.345655i	0.345655	+	0.345655i
$213$	0.291523	+	0.759443i	0.291523	+	0.759443i
$214$		0			0
$215$		0			0
$216$	−0.0864545	−	0.266080i	−0.0864545	−	0.266080i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−1.46873	+	0.563792i	−1.46873	+	0.563792i
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$	−0.532159			−0.532159
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		−	0.328893i		−	0.328893i
$237$	0.579852	+	1.51056i	0.579852	+	1.51056i
$238$	0.0822620	+	0.0822620i	0.0822620	+	0.0822620i
$239$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$239$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$241$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$242$	−0.946294	−	0.946294i	−0.946294	−	0.946294i
$243$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$244$		−	0.488830i		−	0.488830i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.406737	+	0.913545i	−0.406737	+	0.913545i
$250$		0			0
$251$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$251$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$252$	−0.328442	−	0.0172129i	−0.328442	−	0.0172129i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.27977			1.27977
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		−	0.488830i		−	0.488830i
$260$		0			0
$261$		0			0
$262$	1.40647	−	1.40647i	1.40647	−	1.40647i
$263$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$263$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.77578	−	0.681656i	1.77578	−	0.681656i
$268$		0			0
$269$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$269$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$271$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$272$	−0.172269	−	0.172269i	−0.172269	−	0.172269i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$277$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−0.479590	−	1.24937i	−0.479590	−	1.24937i
$283$	1.05097	+	1.05097i	1.05097	+	1.05097i	0.998630	+	0.0523360i	$0.0166667\pi$
$283$	1.05097	+	1.05097i	1.05097	+	1.05097i	0.0523360	+	0.998630i	$0.483333\pi$
$284$	−0.643411			−0.643411
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.27802	+	0.0669781i	1.27802	+	0.0669781i
$289$		−	0.956295i		−	0.956295i
$290$		0			0
$291$	−0.773659	+	1.73767i	−0.773659	+	1.73767i
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−0.450202	+	1.01117i	−0.450202	+	1.01117i
$295$		0			0
$296$			0.328893i			0.328893i
$297$		0			0
$298$	0.769785	+	0.769785i	0.769785	+	0.769785i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−0.712810	−	1.85693i	−0.712810	−	1.85693i
$304$		0			0
$305$		0			0
$306$	−0.187205	−	0.207912i	−0.187205	−	0.207912i
$307$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$307$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$308$		0			0
$309$	−1.81708	−	0.809017i	−1.81708	−	0.809017i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$	−1.98904			−1.98904
$315$		0			0
$316$	−1.27977			−1.27977
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$	0.772156	−	0.296403i	0.772156	−	0.296403i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.786610	+	0.0826761i	0.786610	+	0.0826761i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.415823			0.415823
$330$		0			0
$331$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$331$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$332$	−0.559281	−	0.559281i	−0.559281	−	0.559281i
$333$	−0.0615246	+	1.17396i	−0.0615246	+	1.17396i
$334$		0			0
$335$		0			0
$336$	−0.197097	+	0.442686i	−0.197097	+	0.442686i
$337$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.0523360	−	0.998630i	$-0.483333\pi$
$337$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.998630	−	0.0523360i	$-0.0166667\pi$
$338$	−0.946294	+	0.946294i	−0.946294	+	0.946294i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.537222	−	0.537222i	−0.537222	−	0.537222i
$344$		0			0
$345$		0			0
$346$	−2.61803			−2.61803
$347$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$347$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.38331	−	1.38331i	1.38331	−	1.38331i	0.544639	−	0.838671i	$-0.316667\pi$
$353$	1.38331	−	1.38331i	1.38331	−	1.38331i	0.838671	−	0.544639i	$-0.183333\pi$
$354$	−0.508370	−	0.226341i	−0.508370	−	0.226341i
$355$		0			0
$356$			1.50446i			1.50446i
$357$	0.0811569	−	0.0311532i	0.0811569	−	0.0311532i
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$	−0.933580	+	0.358368i	−0.933580	+	0.358368i
$364$		0			0
$365$		0			0
$366$	−0.755585	−	0.336408i	−0.755585	−	0.336408i
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.256993i			0.256993i
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	−0.279773			−0.279773
$377$		0			0
$378$	−0.252637	+	0.495828i	−0.252637	+	0.495828i
$379$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$379$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$380$		0			0
$381$		0			0
$382$	1.63903	−	1.63903i	1.63903	−	1.63903i
$383$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$383$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$384$	0.222614	−	0.500000i	0.222614	−	0.500000i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.06382	−	1.06382i	−1.06382	−	1.06382i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.163623	+	0.163623i	0.163623	+	0.163623i
$393$	−0.532639	−	1.38757i	−0.532639	−	1.38757i
$394$		−	1.33826i		−	1.33826i
$395$		0			0
$396$		0			0
$397$	0.575212	−	0.575212i	0.575212	−	0.575212i	−0.358368	−	0.933580i	$-0.616667\pi$
$397$	0.575212	−	0.575212i	0.575212	−	0.575212i	0.933580	+	0.358368i	$0.116667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$401$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$		0			0
$404$	1.57322			1.57322
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.0546037	+	0.0209604i	−0.0546037	+	0.0209604i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	1.11243	−	1.11243i	1.11243	−	1.11243i
$413$	0.122265	−	0.122265i	0.122265	−	0.122265i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−0.998630	−	0.0523360i	−0.998630	−	0.0523360i
$424$		−	0.172909i		−	0.172909i
$425$		0			0
$426$	−0.442790	+	0.994522i	−0.442790	+	0.994522i
$427$	0.181721	−	0.181721i	0.181721	−	0.181721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			1.48629i			1.48629i	0.669131	+	0.743145i	$0.266667\pi$
$431$			1.48629i			1.48629i	−0.669131	+	0.743145i	$0.733333\pi$
$432$	0.529059	−	1.03834i	0.529059	−	1.03834i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$439$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	0.553432	+	0.614648i	0.553432	+	0.614648i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$	−0.849423	−	0.378188i	−0.849423	−	0.378188i
$445$		0			0
$446$		0			0
$447$	0.759443	−	0.291523i	0.759443	−	0.291523i
$448$	−0.160929	−	0.160929i	−0.160929	−	0.160929i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$457$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$458$		0			0
$459$	−0.198825	+	0.0646021i	−0.198825	+	0.0646021i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.604528	+	1.35779i	−0.604528	+	1.35779i
$472$	−0.0822620	+	0.0822620i	−0.0822620	+	0.0822620i
$473$		0			0
$474$	−0.880728	+	1.97815i	−0.880728	+	1.97815i
$475$		0			0
$476$			0.0687573i			0.0687573i
$477$	0.0323454	−	0.617187i	0.0323454	−	0.617187i
$478$	−1.11243	−	1.11243i	−1.11243	−	1.11243i
$479$	1.48629			1.48629			0.743145	−	0.669131i	$-0.233333\pi$
$479$	1.48629			1.48629			0.743145	+	0.669131i	$0.233333\pi$
$480$		0			0
$481$		0			0
$482$	1.53114	+	1.53114i	1.53114	+	1.53114i
$483$		0			0
$484$		−	0.790943i		−	0.790943i
$485$		0			0
$486$	0.669131	−	1.15897i	0.669131	−	1.15897i
$487$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$487$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$488$	−0.122265	+	0.122265i	−0.122265	+	0.122265i
$489$		0			0
$490$		0			0
$491$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$491$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.239187	−	0.239187i	−0.239187	−	0.239187i
$498$	−1.24937	+	0.479590i	−1.24937	+	0.479590i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$	1.11243	−	1.11243i	1.11243	−	1.11243i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	0.0778440	+	0.0864545i	0.0778440	+	0.0864545i
$505$		0			0
$506$		0			0
$507$	0.358368	+	0.933580i	0.358368	+	0.933580i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	0.824028	+	0.824028i	0.824028	+	0.824028i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.462576	−	0.462576i	0.462576	−	0.462576i
$519$	−0.795697	+	1.78716i	−0.795697	+	1.78716i
$520$		0			0
$521$			0.415823i			0.415823i	0.978148	+	0.207912i	$0.0666667\pi$
$521$			0.415823i			0.415823i	−0.978148	+	0.207912i	$0.933333\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$	1.17557			1.17557
$525$		0			0
$526$	−0.827091			−0.827091
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$	−0.309017	+	0.278240i	−0.309017	+	0.278240i
$532$		0			0
$533$		0			0
$534$	2.32545	+	1.03536i	2.32545	+	1.03536i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.63903	+	1.63903i	1.63903	+	1.63903i
$539$		0			0
$540$		0			0
$541$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$541$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$542$	−1.72896	−	1.72896i	−1.72896	−	1.72896i
$543$		0			0
$544$		−	0.267545i		−	0.267545i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$	−0.459289	+	0.413545i	−0.459289	+	0.413545i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.475753	−	0.475753i	−0.475753	−	0.475753i
$554$	−2.66186			−2.66186
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$	0.321706	−	0.722562i	0.321706	−	0.722562i
$565$		0			0
$566$			1.98904i			1.98904i
$567$	0.261686	+	0.323155i	0.261686	+	0.323155i
$568$	0.160929	+	0.160929i	0.160929	+	0.160929i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$	−0.620711	−	1.61701i	−0.620711	−	1.61701i
$574$		0			0
$575$		0			0
$576$	0.366227	+	0.406737i	0.366227	+	0.406737i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	0.904936	−	0.904936i	0.904936	−	0.904936i
$579$		0			0
$580$		0			0
$581$		−	0.415823i		−	0.415823i
$582$	−2.37645	+	0.912234i	−2.37645	+	0.912234i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	−0.610731	+	0.234438i	−0.610731	+	0.234438i
$589$		0			0
$590$		0			0
$591$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$592$	−0.968703	+	0.968703i	−0.968703	+	0.968703i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$			0.643411i			0.643411i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$601$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	1.08268	−	2.43173i	1.08268	−	2.43173i
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.00865386	−	0.165126i	0.00865386	−	0.165126i
$613$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$613$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$614$	1.57322			1.57322
$615$		0			0
$616$		0			0
$617$	1.29195	+	1.29195i	1.29195	+	1.29195i	0.933580	+	0.358368i	$0.116667\pi$
$617$	1.29195	+	1.29195i	1.29195	+	1.29195i	0.358368	+	0.933580i	$0.383333\pi$
$618$	−0.953925	−	2.48506i	−0.953925	−	2.48506i
$619$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$619$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.559281	+	0.559281i	−0.559281	+	0.559281i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$629$	0.245761			0.245761
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	0.320094	+	0.320094i	0.320094	+	0.320094i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.446568	+	0.198825i	0.446568	+	0.198825i
$637$		0			0
$638$		0			0
$639$	0.544320	+	0.604528i	0.544320	+	0.604528i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−1.05097	−	1.05097i	−1.05097	−	1.05097i	−0.998630	−	0.0523360i	$-0.983333\pi$
$643$	−1.05097	−	1.05097i	−1.05097	−	1.05097i	−0.0523360	−	0.998630i	$-0.516667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$647$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$648$	−0.176067	−	0.217424i	−0.176067	−	0.217424i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.358368	−	0.933580i	$-0.383333\pi$
$653$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.393491	+	0.393491i	0.393491	+	0.393491i
$659$	−0.813473			−0.813473			−0.406737	−	0.913545i	$-0.633333\pi$
$659$	−0.813473			−0.813473			−0.406737	+	0.913545i	$0.633333\pi$
$660$		0			0
$661$	0.209057			0.209057			0.104528	−	0.994522i	$-0.466667\pi$
$661$	0.209057			0.209057			0.104528	+	0.994522i	$0.466667\pi$
$662$	1.85123	+	1.85123i	1.85123	+	1.85123i
$663$		0			0
$664$			0.279773i			0.279773i
$665$		0			0
$666$	−1.16913	+	1.05269i	−1.16913	+	1.05269i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.496814	+	0.190709i	−0.496814	+	0.190709i
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$	1.98904			1.98904
$675$		0			0
$676$	−0.790943			−0.790943
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		−	0.790943i		−	0.790943i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.38331	−	1.38331i	1.38331	−	1.38331i	0.544639	−	0.838671i	$-0.316667\pi$
$683$	1.38331	−	1.38331i	1.38331	−	1.38331i	0.838671	−	0.544639i	$-0.183333\pi$
$684$		0			0
$685$		0			0
$686$		−	1.01674i		−	1.01674i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−1.09412	−	1.09412i	−1.09412	−	1.09412i
$693$		0			0
$694$		−	2.16535i		−	2.16535i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	2.61803			2.61803
$707$	0.584842	+	0.584842i	0.584842	+	0.584842i
$708$	−0.117865	−	0.307048i	−0.117865	−	0.307048i
$709$			1.33826i			1.33826i	0.743145	+	0.669131i	$0.233333\pi$
$709$			1.33826i			1.33826i	−0.743145	+	0.669131i	$0.766667\pi$
$710$		0			0
$711$	1.08268	+	1.20243i	1.08268	+	1.20243i
$712$	0.376294	−	0.376294i	0.376294	−	0.376294i
$713$		0			0
$714$	0.106278	+	0.0473181i	0.106278	+	0.0473181i
$715$		0			0
$716$		0			0
$717$	−1.09749	+	0.421287i	−1.09749	+	0.421287i
$718$		0			0
$719$	0.415823			0.415823			0.207912	−	0.978148i	$-0.433333\pi$
$719$	0.415823			0.415823			0.207912	+	0.978148i	$0.433333\pi$
$720$		0			0
$721$	0.827091			0.827091
$722$	−0.946294	−	0.946294i	−0.946294	−	0.946294i
$723$	1.51056	−	0.579852i	1.51056	−	0.579852i
$724$		0			0
$725$		0			0
$726$	−1.22256	−	0.544320i	−1.22256	−	0.544320i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$730$		0			0
$731$		0			0
$732$	−0.175181	−	0.456362i	−0.175181	−	0.456362i
$733$	−0.575212	−	0.575212i	−0.575212	−	0.575212i	0.358368	−	0.933580i	$-0.383333\pi$
$733$	−0.575212	−	0.575212i	−0.575212	−	0.575212i	−0.933580	+	0.358368i	$0.883333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	1.33826i		−	1.33826i	−0.743145	−	0.669131i	$-0.766667\pi$
$739$		−	1.33826i		−	1.33826i	0.743145	−	0.669131i	$-0.233333\pi$
$740$		0			0
$741$		0			0
$742$	−0.243191	+	0.243191i	−0.243191	+	0.243191i
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.0523360	+	0.998630i	−0.0523360	+	0.998630i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−0.824028	−	0.824028i	−0.824028	−	0.824028i
$753$	−0.421287	−	1.09749i	−0.421287	−	1.09749i
$754$		0			0
$755$		0			0
$756$	−0.312795	+	0.101633i	−0.312795	+	0.101633i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	−0.584842	+	0.584842i	−0.584842	+	0.584842i
$759$		0			0
$760$		0			0
$761$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$761$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$	1.36995			1.36995
$765$		0			0
$766$	−2.16535			−2.16535
$767$		0			0
$768$	1.19477	−	0.458630i	1.19477	−	0.458630i
$769$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$769$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$773$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$774$		0			0
$775$		0			0
$776$			0.532159i			0.532159i
$777$	−0.175181	−	0.456362i	−0.175181	−	0.456362i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			0.963852i			0.963852i
$785$		0			0
$786$	0.809017	−	1.81708i	0.809017	−	1.81708i
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	0.559281	−	0.559281i	0.559281	−	0.559281i
$789$	−0.251377	+	0.564602i	−0.251377	+	0.564602i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.08864			1.08864
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$			0.209057i			0.209057i
$800$		0			0
$801$	1.41355	−	1.27276i	1.41355	−	1.27276i
$802$	−1.11243	+	1.11243i	−1.11243	+	1.11243i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.61701	−	0.620711i	1.61701	−	0.620711i
$808$	−0.393491	−	0.393491i	−0.393491	−	0.393491i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$811$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$812$		0			0
$813$	−1.70574	+	0.654771i	−1.70574	+	0.654771i
$814$		0			0
$815$		0			0
$816$	−0.222562	−	0.0990912i	−0.222562	−	0.0990912i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$	−0.556480			−0.556480
$825$		0			0
$826$	0.231398			0.231398
$827$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$827$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.544639	+	0.838671i	$0.316667\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−0.809017	+	1.81708i	−0.809017	+	1.81708i
$832$		0			0
$833$	0.122265	−	0.122265i	0.122265	−	0.122265i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	−0.895472	−	0.994522i	−0.895472	−	0.994522i
$847$	0.294032	−	0.294032i	0.294032	−	0.294032i
$848$	0.509278	−	0.509278i	0.509278	−	0.509278i
$849$	1.35779	+	0.604528i	1.35779	+	0.604528i
$850$		0			0
$851$		0			0
$852$	−0.600676	+	0.230578i	−0.600676	+	0.230578i
$853$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.777146	+	0.629320i	$0.216667\pi$
$853$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.629320	+	0.777146i	$0.283333\pi$
$854$	0.343924			0.343924
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−1.40647	+	1.40647i	−1.40647	+	1.40647i
$863$	0.147826	−	0.147826i	0.147826	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$863$	0.147826	−	0.147826i	0.147826	−	0.147826i	0.777146	+	0.629320i	$0.216667\pi$
$864$	1.21714	−	0.395472i	1.21714	−	0.395472i
$865$		0			0
$866$		0			0
$867$	−0.342706	−	0.892778i	−0.342706	−	0.892778i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.0995489	+	1.89951i	−0.0995489	+	1.89951i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	−0.946294	+	0.946294i	−0.946294	+	0.946294i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−0.0579288	+	1.10535i	−0.0579288	+	1.10535i
$883$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$883$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$	0.117865	+	0.307048i	0.117865	+	0.307048i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.994522	+	0.442790i	0.994522	+	0.442790i
$895$		0			0
$896$			0.227588i			0.227588i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.129204			−0.129204
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$907$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$908$		0			0
$909$	−1.33093	−	1.47815i	−1.33093	−	1.47815i
$910$		0			0
$911$		−	0.415823i		−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$911$		−	0.415823i		−	0.415823i	0.978148	−	0.207912i	$-0.0666667\pi$
$912$		0			0
$913$		0			0
$914$	−1.08864			−1.08864
$915$		0			0
$916$		0			0
$917$	0.437016	+	0.437016i	0.437016	+	0.437016i
$918$	−0.249279	−	0.127014i	−0.249279	−	0.127014i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	0.478148	−	1.07394i	0.478148	−	1.07394i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.98632	−	0.104099i	−1.98632	−	0.104099i
$928$		0			0
$929$	−1.98904			−1.98904			−0.994522	−	0.104528i	$-0.966667\pi$
$929$	−1.98904			−1.98904			−0.994522	+	0.104528i	$0.966667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	0.415823i		−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$941$		−	0.415823i		−	0.415823i	0.978148	−	0.207912i	$-0.0666667\pi$
$942$	−1.85693	+	0.712810i	−1.85693	+	0.712810i
$943$		0			0
$944$	−0.484581			−0.484581
$945$		0			0
$946$		0			0
$947$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$948$	−1.19477	+	0.458630i	−1.19477	+	0.458630i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.0171974	−	0.0171974i	0.0171974	−	0.0171974i
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$	0.614648	−	0.553432i	0.614648	−	0.553432i
$955$		0			0
$956$		−	0.929809i		−	0.929809i
$957$		0			0
$958$	1.40647	+	1.40647i	1.40647	+	1.40647i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$			1.27977i			1.27977i
$965$		0			0
$966$		0			0
$967$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.0523360	−	0.998630i	$-0.483333\pi$
$967$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.998630	−	0.0523360i	$-0.0166667\pi$
$968$	−0.197829	+	0.197829i	−0.197829	+	0.197829i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	0.763992	−	0.204711i	0.763992	−	0.204711i
$973$		0			0
$974$	2.31794			2.31794
$975$		0			0
$976$	−0.720227			−0.720227
$977$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$977$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1.79996	+	1.79996i	−1.79996	+	1.79996i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.388205	−	0.149018i	0.388205	−	0.149018i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$991$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$992$		0			0
$993$	1.82636	−	0.701074i	1.82636	−	0.701074i
$994$		−	0.452682i		−	0.452682i
$995$		0			0
$996$	−0.722562	−	0.321706i	−0.722562	−	0.321706i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$	0.363271	+	1.11803i	0.363271	+	1.11803i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.1.l.d.1268.12	yes	32
3.2	odd	2	inner	3525.1.l.d.1268.6	yes	32
5.2	odd	4	inner	3525.1.l.d.1832.6	yes	32
5.3	odd	4	inner	3525.1.l.d.1832.11	yes	32
5.4	even	2	inner	3525.1.l.d.1268.5	✓	32
15.2	even	4	inner	3525.1.l.d.1832.12	yes	32
15.8	even	4	inner	3525.1.l.d.1832.5	yes	32
15.14	odd	2	inner	3525.1.l.d.1268.11	yes	32
47.46	odd	2	CM	3525.1.l.d.1268.12	yes	32
141.140	even	2	inner	3525.1.l.d.1268.6	yes	32
235.93	even	4	inner	3525.1.l.d.1832.11	yes	32
235.187	even	4	inner	3525.1.l.d.1832.6	yes	32
235.234	odd	2	inner	3525.1.l.d.1268.5	✓	32
705.422	odd	4	inner	3525.1.l.d.1832.12	yes	32
705.563	odd	4	inner	3525.1.l.d.1832.5	yes	32
705.704	even	2	inner	3525.1.l.d.1268.11	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3525.1.l.d.1268.5	✓	32	5.4	even	2	inner
3525.1.l.d.1268.5	✓	32	235.234	odd	2	inner
3525.1.l.d.1268.6	yes	32	3.2	odd	2	inner
3525.1.l.d.1268.6	yes	32	141.140	even	2	inner
3525.1.l.d.1268.11	yes	32	15.14	odd	2	inner
3525.1.l.d.1268.11	yes	32	705.704	even	2	inner
3525.1.l.d.1268.12	yes	32	1.1	even	1	trivial
3525.1.l.d.1268.12	yes	32	47.46	odd	2	CM
3525.1.l.d.1832.5	yes	32	15.8	even	4	inner
3525.1.l.d.1832.5	yes	32	705.563	odd	4	inner
3525.1.l.d.1832.6	yes	32	5.2	odd	4	inner
3525.1.l.d.1832.6	yes	32	235.187	even	4	inner
3525.1.l.d.1832.11	yes	32	5.3	odd	4	inner
3525.1.l.d.1832.11	yes	32	235.93	even	4	inner
3525.1.l.d.1832.12	yes	32	15.2	even	4	inner
3525.1.l.d.1832.12	yes	32	705.422	odd	4	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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3525.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.946294 + 0.946294i) q^{2} +(0.933580 - 0.358368i) q^{3} +0.790943i q^{4} +(1.22256 + 0.544320i) q^{6} +(-0.294032 + 0.294032i) q^{7} +(0.197829 - 0.197829i) q^{8} +(0.743145 - 0.669131i) q^{9} +O(q^{10})\)	\(q+(0.946294 + 0.946294i) q^{2} +(0.933580 - 0.358368i) q^{3} +0.790943i q^{4} +(1.22256 + 0.544320i) q^{6} +(-0.294032 + 0.294032i) q^{7} +(0.197829 - 0.197829i) q^{8} +(0.743145 - 0.669131i) q^{9} +(0.283449 + 0.738409i) q^{12} -0.556480 q^{14} +1.16535 q^{16} +(-0.147826 - 0.147826i) q^{17} +(1.33643 + 0.0700392i) q^{18} +(-0.169131 + 0.379874i) q^{21} +(0.113794 - 0.255585i) q^{24} +(0.453990 - 0.891007i) q^{27} +(-0.232562 - 0.232562i) q^{28} +(0.904936 + 0.904936i) q^{32} -0.279773i q^{34} +(0.529244 + 0.587785i) q^{36} +(-0.831254 + 0.831254i) q^{37} +(-0.519519 + 0.199425i) q^{42} +(-0.707107 - 0.707107i) q^{47} +(1.08795 - 0.417625i) q^{48} +0.827091i q^{49} +(-0.190983 - 0.0850311i) q^{51} +(0.437016 - 0.437016i) q^{53} +(1.27276 - 0.413545i) q^{54} +0.116336i q^{56} -0.415823 q^{59} -0.618034 q^{61} +(-0.0217625 + 0.415254i) q^{63} +0.547318i q^{64} +(0.116922 - 0.116922i) q^{68} +0.813473i q^{71} +(0.0146422 - 0.279389i) q^{72} -1.57322 q^{74} +1.61803i q^{79} +(0.104528 - 0.994522i) q^{81} +(-0.707107 + 0.707107i) q^{83} +(-0.300458 - 0.133773i) q^{84} +1.90211 q^{89} -1.33826i q^{94} +(1.16913 + 0.520530i) q^{96} +(-1.34500 + 1.34500i) q^{97} +(-0.782671 + 0.782671i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{6}+O(q^{10}) \) 32 * q - 4 * q^6	\( 32 q - 4 q^{6} - 48 q^{16} + 12 q^{21} - 24 q^{51} + 16 q^{61} - 4 q^{81} + 20 q^{96}+O(q^{100}) \) 32 * q - 4 * q^6 - 48 * q^16 + 12 * q^21 - 24 * q^51 + 16 * q^61 - 4 * q^81 + 20 * q^96

Introduction

L-functions

Modular forms

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Representations

Groups

Database

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