LMFDB - Embedded newform 3525.1.l.d.1268.9

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			1268.9
Root			$0.933580 + 0.358368i$ of defining polynomial
Character	$\chi$	$=$	3525.1268
Dual form			3525.1.l.d.1832.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.147826 + 0.147826i) q^{2} +(-0.998630 - 0.0523360i) q^{3} -0.956295i q^{4} +(-0.139886 - 0.155360i) q^{6} +(-0.575212 + 0.575212i) q^{7} +(0.289190 - 0.289190i) q^{8} +(0.994522 + 0.104528i) q^{9} +O(q^{10})$	\(q+(0.147826 + 0.147826i) q^{2} +(-0.998630 - 0.0523360i) q^{3} -0.956295i q^{4} +(-0.139886 - 0.155360i) q^{6} +(-0.575212 + 0.575212i) q^{7} +(0.289190 - 0.289190i) q^{8} +(0.994522 + 0.104528i) q^{9} +(-0.0500486 + 0.954985i) q^{12} -0.170062 q^{14} -0.870796 q^{16} +(1.38331 + 1.38331i) q^{17} +(0.131564 + 0.162468i) q^{18} +(0.604528 - 0.544320i) q^{21} +(-0.303929 + 0.273659i) q^{24} +(-0.987688 - 0.156434i) q^{27} +(0.550073 + 0.550073i) q^{28} +(-0.417916 - 0.417916i) q^{32} +0.408977i q^{34} +(0.0999601 - 0.951057i) q^{36} +(-1.34500 + 1.34500i) q^{37} +(0.169829 + 0.00890037i) q^{42} +(0.707107 + 0.707107i) q^{47} +(0.869602 + 0.0455739i) q^{48} +0.338261i q^{49} +(-1.30902 - 1.45381i) q^{51} +(1.14412 - 1.14412i) q^{53} +(-0.122881 - 0.169131i) q^{54} +0.332692i q^{56} +0.813473 q^{59} +1.61803 q^{61} +(-0.632187 + 0.511935i) q^{63} +0.747238i q^{64} +(1.32285 - 1.32285i) q^{68} +1.48629i q^{71} +(0.317835 - 0.257378i) q^{72} -0.397650 q^{74} -0.618034i q^{79} +(0.978148 + 0.207912i) q^{81} +(0.707107 - 0.707107i) q^{83} +(-0.520530 - 0.578108i) q^{84} +1.17557 q^{89} +0.209057i q^{94} +(0.395472 + 0.439216i) q^{96} +(0.831254 - 0.831254i) q^{97} +(-0.0500037 + 0.0500037i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.147826	+	0.147826i	0.147826	+	0.147826i	0.777146	−	0.629320i	$-0.216667\pi$
$2$	0.147826	+	0.147826i	0.147826	+	0.147826i	−0.629320	+	0.777146i	$0.716667\pi$
$3$	−0.998630	−	0.0523360i	−0.998630	−	0.0523360i
$3$	−0.998630	−	0.0523360i	−0.998630	−	0.0523360i
$4$		−	0.956295i		−	0.956295i
$5$		0			0
$5$		0			0
$6$	−0.139886	−	0.155360i	−0.139886	−	0.155360i
$7$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$7$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$8$	0.289190	−	0.289190i	0.289190	−	0.289190i
$9$	0.994522	+	0.104528i	0.994522	+	0.104528i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.0500486	+	0.954985i	−0.0500486	+	0.954985i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	−0.170062			−0.170062
$15$		0			0
$16$	−0.870796			−0.870796
$17$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$17$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.544639	+	0.838671i	$0.316667\pi$
$18$	0.131564	+	0.162468i	0.131564	+	0.162468i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	0.604528	−	0.544320i	0.604528	−	0.544320i
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	−0.303929	+	0.273659i	−0.303929	+	0.273659i
$25$		0			0
$26$		0			0
$27$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$28$	0.550073	+	0.550073i	0.550073	+	0.550073i
$29$		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			$\pi$
$32$	−0.417916	−	0.417916i	−0.417916	−	0.417916i
$33$		0			0
$34$			0.408977i			0.408977i
$35$		0			0
$36$	0.0999601	−	0.951057i	0.0999601	−	0.951057i
$37$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$37$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	0.169829	+	0.00890037i	0.169829	+	0.00890037i
$43$		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
$48$	0.869602	+	0.0455739i	0.869602	+	0.0455739i
$49$			0.338261i			0.338261i
$50$		0			0
$51$	−1.30902	−	1.45381i	−1.30902	−	1.45381i
$52$		0			0
$53$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$53$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$54$	−0.122881	−	0.169131i	−0.122881	−	0.169131i
$55$		0			0
$56$			0.332692i			0.332692i
$57$		0			0
$58$		0			0
$59$	0.813473			0.813473			0.406737	−	0.913545i	$-0.366667\pi$
$59$	0.813473			0.813473			0.406737	+	0.913545i	$0.366667\pi$
$60$		0			0
$61$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$61$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$	−0.632187	+	0.511935i	−0.632187	+	0.511935i
$64$			0.747238i			0.747238i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	1.32285	−	1.32285i	1.32285	−	1.32285i
$69$		0			0
$70$		0			0
$71$			1.48629i			1.48629i	0.669131	+	0.743145i	$0.266667\pi$
$71$			1.48629i			1.48629i	−0.669131	+	0.743145i	$0.733333\pi$
$72$	0.317835	−	0.257378i	0.317835	−	0.257378i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	−0.397650			−0.397650
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$79$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$80$		0			0
$81$	0.978148	+	0.207912i	0.978148	+	0.207912i
$82$		0			0
$83$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$83$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$84$	−0.520530	−	0.578108i	−0.520530	−	0.578108i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$89$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$			0.209057i			0.209057i
$95$		0			0
$96$	0.395472	+	0.439216i	0.395472	+	0.439216i
$97$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$97$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$98$	−0.0500037	+	0.0500037i	−0.0500037	+	0.0500037i
$99$		0			0
$100$		0			0
$101$			0.415823i			0.415823i	0.978148	+	0.207912i	$0.0666667\pi$
$101$			0.415823i			0.415823i	−0.978148	+	0.207912i	$0.933333\pi$
$102$	0.0214042	−	0.408417i	0.0214042	−	0.408417i
$103$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	0.544639	−	0.838671i	$-0.316667\pi$
$103$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	−0.838671	+	0.544639i	$0.816667\pi$
$104$		0			0
$105$		0			0
$106$	0.338261			0.338261
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	−0.149598	+	0.944522i	−0.149598	+	0.944522i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	1.41355	−	1.27276i	1.41355	−	1.27276i
$112$	0.500893	−	0.500893i	0.500893	−	0.500893i
$113$		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
$118$	0.120252	+	0.120252i	0.120252	+	0.120252i
$119$	−1.59139			−1.59139
$120$		0			0
$121$	−1.00000			−1.00000
$122$	0.239187	+	0.239187i	0.239187	+	0.239187i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.169131	−	0.0177763i	−0.169131	−	0.0177763i
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−0.528377	+	0.528377i	−0.528377	+	0.528377i
$129$		0			0
$130$		0			0
$131$		−	1.98904i		−	1.98904i	−0.104528	−	0.994522i	$-0.533333\pi$
$131$		−	1.98904i		−	1.98904i	0.104528	−	0.994522i	$-0.466667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.800080			0.800080
$137$		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
$141$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$142$	−0.219712	+	0.219712i	−0.219712	+	0.219712i
$143$		0			0
$144$	−0.866025	−	0.0910229i	−0.866025	−	0.0910229i
$145$		0			0
$146$		0			0
$147$	0.0177032	−	0.337798i	0.0177032	−	0.337798i
$148$	1.28621	+	1.28621i	1.28621	+	1.28621i
$149$	1.48629			1.48629			0.743145	−	0.669131i	$-0.233333\pi$
$149$	1.48629			1.48629			0.743145	+	0.669131i	$0.233333\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	1.23114	+	1.52033i	1.23114	+	1.52033i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.629320	−	0.777146i	$-0.283333\pi$
$157$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.777146	−	0.629320i	$-0.216667\pi$
$158$	0.0913612	−	0.0913612i	0.0913612	−	0.0913612i
$159$	−1.20243	+	1.08268i	−1.20243	+	1.08268i
$160$		0			0
$161$		0			0
$162$	0.113861	+	0.175330i	0.113861	+	0.175330i
$163$		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
$166$	0.209057			0.209057
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$	0.0174117	−	0.332236i	0.0174117	−	0.332236i
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.29195	+	1.29195i	−1.29195	+	1.29195i	−0.358368	+	0.933580i	$0.616667\pi$
$173$	−1.29195	+	1.29195i	−1.29195	+	1.29195i	−0.933580	+	0.358368i	$0.883333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.812358	−	0.0425739i	−0.812358	−	0.0425739i
$178$	0.173779	+	0.173779i	0.173779	+	0.173779i
$179$		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
$183$	−1.61582	−	0.0846814i	−1.61582	−	0.0846814i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.676203	−	0.676203i	0.676203	−	0.676203i
$189$	0.658114	−	0.478148i	0.658114	−	0.478148i
$190$		0			0
$191$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$191$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.0391074	−	0.746214i	0.0391074	−	0.746214i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$	0.245761			0.245761
$195$		0			0
$196$	0.323478			0.323478
$197$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$197$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$	−0.0614693	+	0.0614693i	−0.0614693	+	0.0614693i
$203$		0			0
$204$	−1.39027	+	1.25181i	−1.39027	+	1.25181i
$205$		0			0
$206$		−	0.0869308i		−	0.0869308i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−1.09412	−	1.09412i	−1.09412	−	1.09412i
$213$	0.0777864	−	1.48425i	0.0777864	−	1.48425i
$214$		0			0
$215$		0			0
$216$	−0.330869	+	0.240391i	−0.330869	+	0.240391i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	0.397105	+	0.0208114i	0.397105	+	0.0208114i
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$	0.480781			0.480781
$225$		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
$236$		−	0.777921i		−	0.777921i
$237$	−0.0323454	+	0.617187i	−0.0323454	+	0.617187i
$238$	−0.235249	−	0.235249i	−0.235249	−	0.235249i
$239$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$239$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$241$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$242$	−0.147826	−	0.147826i	−0.147826	−	0.147826i
$243$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$244$		−	1.54732i		−	1.54732i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.743145	+	0.669131i	−0.743145	+	0.669131i
$250$		0			0
$251$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$251$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$252$	0.489561	+	0.604558i	0.489561	+	0.604558i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.591023			0.591023
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		−	1.54732i		−	1.54732i
$260$		0			0
$261$		0			0
$262$	0.294032	−	0.294032i	0.294032	−	0.294032i
$263$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$263$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.17396	−	0.0615246i	−1.17396	−	0.0615246i
$268$		0			0
$269$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$269$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$271$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$272$	−1.20458	−	1.20458i	−1.20458	−	1.20458i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$277$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	0.0109412	−	0.208770i	0.0109412	−	0.208770i
$283$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.777146	−	0.629320i	$-0.783333\pi$
$283$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.629320	−	0.777146i	$-0.716667\pi$
$284$	1.42133			1.42133
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.371943	−	0.459311i	−0.371943	−	0.459311i
$289$			2.82709i			2.82709i
$290$		0			0
$291$	−0.873619	+	0.786610i	−0.873619	+	0.786610i
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	0.0525521	−	0.0473181i	0.0525521	−	0.0473181i
$295$		0			0
$296$			0.777921i			0.777921i
$297$		0			0
$298$	0.219712	+	0.219712i	0.219712	+	0.219712i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	0.0217625	−	0.415254i	0.0217625	−	0.415254i
$304$		0			0
$305$		0			0
$306$	−0.0427497	+	0.406737i	−0.0427497	+	0.406737i
$307$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$307$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$308$		0			0
$309$	0.278240	+	0.309017i	0.278240	+	0.309017i
$310$		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	$0.250000\pi$
$314$	0.415823			0.415823
$315$		0			0
$316$	−0.591023			−0.591023
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$	−0.337798	−	0.0177032i	−0.337798	−	0.0177032i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.198825	−	0.935398i	0.198825	−	0.935398i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.813473			−0.813473
$330$		0			0
$331$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$331$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$332$	−0.676203	−	0.676203i	−0.676203	−	0.676203i
$333$	−1.47822	+	1.19704i	−1.47822	+	1.19704i
$334$		0			0
$335$		0			0
$336$	−0.526421	+	0.473991i	−0.526421	+	0.473991i
$337$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$337$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$338$	−0.147826	+	0.147826i	−0.147826	+	0.147826i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.769785	−	0.769785i	−0.769785	−	0.769785i
$344$		0			0
$345$		0			0
$346$	−0.381966			−0.381966
$347$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$347$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.358368	−	0.933580i	$-0.383333\pi$
$353$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$354$	−0.113794	−	0.126381i	−0.113794	−	0.126381i
$355$		0			0
$356$		−	1.12419i		−	1.12419i
$357$	1.58921	+	0.0832871i	1.58921	+	0.0832871i
$358$		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
$363$	0.998630	+	0.0523360i	0.998630	+	0.0523360i
$364$		0			0
$365$		0			0
$366$	−0.226341	−	0.251377i	−0.226341	−	0.251377i
$367$		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
$371$			1.31623i			1.31623i
$372$		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
$376$	0.408977			0.408977
$377$		0			0
$378$	0.167968	+	0.0266036i	0.167968	+	0.0266036i
$379$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$379$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$380$		0			0
$381$		0			0
$382$	−0.256041	+	0.256041i	−0.256041	+	0.256041i
$383$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$383$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$384$	0.555306	−	0.500000i	0.555306	−	0.500000i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.794924	−	0.794924i	−0.794924	−	0.794924i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.0978219	+	0.0978219i	0.0978219	+	0.0978219i
$393$	−0.104099	+	1.98632i	−0.104099	+	1.98632i
$394$			0.209057i			0.209057i
$395$		0			0
$396$		0			0
$397$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.0523360	+	0.998630i	$0.516667\pi$
$397$	−1.05097	+	1.05097i	−1.05097	+	1.05097i	−0.998630	+	0.0523360i	$0.983333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$401$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$402$		0			0
$403$		0			0
$404$	0.397650			0.397650
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.798983	−	0.0418729i	−0.798983	−	0.0418729i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	−0.281181	+	0.281181i	−0.281181	+	0.281181i
$413$	−0.467920	+	0.467920i	−0.467920	+	0.467920i
$414$		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
$423$	0.629320	+	0.777146i	0.629320	+	0.777146i
$424$		−	0.661739i		−	0.661739i
$425$		0			0
$426$	0.230909	−	0.207912i	0.230909	−	0.207912i
$427$	−0.930713	+	0.930713i	−0.930713	+	0.930713i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			1.98904i			1.98904i	0.104528	+	0.994522i	$0.466667\pi$
$431$			1.98904i			1.98904i	−0.104528	+	0.994522i	$0.533333\pi$
$432$	0.860075	+	0.136222i	0.860075	+	0.136222i
$433$		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
$441$	−0.0353579	+	0.336408i	−0.0353579	+	0.336408i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$	−1.21714	−	1.35177i	−1.21714	−	1.35177i
$445$		0			0
$446$		0			0
$447$	−1.48425	−	0.0777864i	−1.48425	−	0.0777864i
$448$	−0.429821	−	0.429821i	−0.429821	−	0.429821i
$449$		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
$457$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.0523360	−	0.998630i	$-0.483333\pi$
$457$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.998630	−	0.0523360i	$-0.0166667\pi$
$458$		0			0
$459$	−1.14988	−	1.58268i	−1.14988	−	1.58268i
$460$		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
$471$	−1.47815	+	1.33093i	−1.47815	+	1.33093i
$472$	0.235249	−	0.235249i	0.235249	−	0.235249i
$473$		0			0
$474$	−0.0960175	+	0.0864545i	−0.0960175	+	0.0864545i
$475$		0			0
$476$			1.52184i			1.52184i
$477$	1.25745	−	1.01826i	1.25745	−	1.01826i
$478$	0.281181	+	0.281181i	0.281181	+	0.281181i
$479$	1.98904			1.98904			0.994522	−	0.104528i	$-0.0333333\pi$
$479$	1.98904			1.98904			0.994522	+	0.104528i	$0.0333333\pi$
$480$		0			0
$481$		0			0
$482$	−0.0913612	−	0.0913612i	−0.0913612	−	0.0913612i
$483$		0			0
$484$			0.956295i			0.956295i
$485$		0			0
$486$	−0.104528	−	0.181049i	−0.104528	−	0.181049i
$487$	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	$-0.0833333\pi$
$488$	0.467920	−	0.467920i	0.467920	−	0.467920i
$489$		0			0
$490$		0			0
$491$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$491$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.854932	−	0.854932i	−0.854932	−	0.854932i
$498$	−0.208770	−	0.0109412i	−0.208770	−	0.0109412i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$	−0.281181	+	0.281181i	−0.281181	+	0.281181i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	−0.0347758	+	0.330869i	−0.0347758	+	0.330869i
$505$		0			0
$506$		0			0
$507$	0.0523360	−	0.998630i	0.0523360	−	0.998630i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	0.615746	+	0.615746i	0.615746	+	0.615746i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.228733	−	0.228733i	0.228733	−	0.228733i
$519$	1.35779	−	1.22256i	1.35779	−	1.22256i
$520$		0			0
$521$		−	0.813473i		−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$521$		−	0.813473i		−	0.813473i	0.913545	−	0.406737i	$-0.133333\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$	−1.90211			−1.90211
$525$		0			0
$526$	−0.338261			−0.338261
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$	0.809017	+	0.0850311i	0.809017	+	0.0850311i
$532$		0			0
$533$		0			0
$534$	−0.164446	−	0.182636i	−0.164446	−	0.182636i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.256041	−	0.256041i	−0.256041	−	0.256041i
$539$		0			0
$540$		0			0
$541$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$541$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$542$	−0.197829	−	0.197829i	−0.197829	−	0.197829i
$543$		0			0
$544$		−	1.15622i		−	1.15622i
$545$		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
$549$	1.60917	+	0.169131i	1.60917	+	0.169131i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.355501	+	0.355501i	0.355501	+	0.355501i
$554$	−0.0869308			−0.0869308
$555$		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	$0.750000\pi$
$564$	−0.710666	+	0.639886i	−0.710666	+	0.639886i
$565$		0			0
$566$		−	0.415823i		−	0.415823i
$567$	−0.682236	+	0.443049i	−0.682236	+	0.443049i
$568$	0.429821	+	0.429821i	0.429821	+	0.429821i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$571$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$572$		0			0
$573$	0.0906485	−	1.72968i	0.0906485	−	1.72968i
$574$		0			0
$575$		0			0
$576$	−0.0781077	+	0.743145i	−0.0781077	+	0.743145i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−0.417916	+	0.417916i	−0.417916	+	0.417916i
$579$		0			0
$580$		0			0
$581$			0.813473i			0.813473i
$582$	−0.245424	−	0.0128621i	−0.245424	−	0.0128621i
$583$		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	$0.250000\pi$
$588$	−0.323034	−	0.0169295i	−0.323034	−	0.0169295i
$589$		0			0
$590$		0			0
$591$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$592$	1.17122	−	1.17122i	1.17122	−	1.17122i
$593$		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
$596$		−	1.42133i		−	1.42133i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$601$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.0646021	−	0.0581680i	0.0646021	−	0.0581680i
$607$		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
$612$	1.45388	−	1.17733i	1.45388	−	1.17733i
$613$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$613$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$614$	0.397650			0.397650
$615$		0			0
$616$		0			0
$617$	−0.946294	−	0.946294i	−0.946294	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$617$	−0.946294	−	0.946294i	−0.946294	−	0.946294i	−0.998630	+	0.0523360i	$0.983333\pi$
$618$	−0.00454960	+	0.0868116i	−0.00454960	+	0.0868116i
$619$		−	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
$623$	−0.676203	+	0.676203i	−0.676203	+	0.676203i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−1.34500	−	1.34500i	−1.34500	−	1.34500i
$629$	−3.72109			−3.72109
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.178730	−	0.178730i	−0.178730	−	0.178730i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.03536	+	1.14988i	1.03536	+	1.14988i
$637$		0			0
$638$		0			0
$639$	−0.155360	+	1.47815i	−0.155360	+	1.47815i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.777146	+	0.629320i	$0.216667\pi$
$643$	1.40647	+	1.40647i	1.40647	+	1.40647i	0.629320	+	0.777146i	$0.283333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$647$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$648$	0.342997	−	0.222745i	0.342997	−	0.222745i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.946294	+	0.946294i	−0.946294	+	0.946294i	−0.998630	−	0.0523360i	$-0.983333\pi$
$653$	−0.946294	+	0.946294i	−0.946294	+	0.946294i	0.0523360	+	0.998630i	$0.483333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.120252	−	0.120252i	−0.120252	−	0.120252i
$659$	−1.48629			−1.48629			−0.743145	−	0.669131i	$-0.766667\pi$
$659$	−1.48629			−1.48629			−0.743145	+	0.669131i	$0.766667\pi$
$660$		0			0
$661$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$661$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$662$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$663$		0			0
$664$		−	0.408977i		−	0.408977i
$665$		0			0
$666$	−0.395472	−	0.0415657i	−0.395472	−	0.0415657i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.480122	−	0.0251622i	−0.480122	−	0.0251622i
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$	−0.415823			−0.415823
$675$		0			0
$676$	0.956295			0.956295
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			0.956295i			0.956295i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.358368	−	0.933580i	$-0.383333\pi$
$683$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$684$		0			0
$685$		0			0
$686$		−	0.227588i		−	0.227588i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	1.23548	+	1.23548i	1.23548	+	1.23548i
$693$		0			0
$694$		−	0.129204i		−	0.129204i
$695$		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
$706$	0.381966			0.381966
$707$	−0.239187	−	0.239187i	−0.239187	−	0.239187i
$708$	−0.0407132	+	0.776854i	−0.0407132	+	0.776854i
$709$		−	0.209057i		−	0.209057i	−0.994522	−	0.104528i	$-0.966667\pi$
$709$		−	0.209057i		−	0.209057i	0.994522	−	0.104528i	$-0.0333333\pi$
$710$		0			0
$711$	0.0646021	−	0.614648i	0.0646021	−	0.614648i
$712$	0.339964	−	0.339964i	0.339964	−	0.339964i
$713$		0			0
$714$	0.222614	+	0.247238i	0.222614	+	0.247238i
$715$		0			0
$716$		0			0
$717$	−1.89951	−	0.0995489i	−1.89951	−	0.0995489i
$718$		0			0
$719$	−0.813473			−0.813473			−0.406737	−	0.913545i	$-0.633333\pi$
$719$	−0.813473			−0.813473			−0.406737	+	0.913545i	$0.633333\pi$
$720$		0			0
$721$	0.338261			0.338261
$722$	−0.147826	−	0.147826i	−0.147826	−	0.147826i
$723$	0.617187	+	0.0323454i	0.617187	+	0.0323454i
$724$		0			0
$725$		0			0
$726$	0.139886	+	0.155360i	0.139886	+	0.155360i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	0.951057	+	0.309017i	0.951057	+	0.309017i
$730$		0			0
$731$		0			0
$732$	−0.0809804	+	1.54520i	−0.0809804	+	1.54520i
$733$	1.05097	+	1.05097i	1.05097	+	1.05097i	0.998630	+	0.0523360i	$0.0166667\pi$
$733$	1.05097	+	1.05097i	1.05097	+	1.05097i	0.0523360	+	0.998630i	$0.483333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			0.209057i			0.209057i	0.994522	+	0.104528i	$0.0333333\pi$
$739$			0.209057i			0.209057i	−0.994522	+	0.104528i	$0.966667\pi$
$740$		0			0
$741$		0			0
$742$	−0.194572	+	0.194572i	−0.194572	+	0.194572i
$743$		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
$747$	0.777146	−	0.629320i	0.777146	−	0.629320i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−0.615746	−	0.615746i	−0.615746	−	0.615746i
$753$	0.0995489	−	1.89951i	0.0995489	−	1.89951i
$754$		0			0
$755$		0			0
$756$	−0.457250	−	0.629351i	−0.457250	−	0.629351i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	0.239187	−	0.239187i	0.239187	−	0.239187i
$759$		0			0
$760$		0			0
$761$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$762$		0			0
$763$		0			0
$764$	1.65635			1.65635
$765$		0			0
$766$	−0.129204			−0.129204
$767$		0			0
$768$	−0.590213	−	0.0309318i	−0.590213	−	0.0309318i
$769$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$769$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$773$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$774$		0			0
$775$		0			0
$776$		−	0.480781i		−	0.480781i
$777$	−0.0809804	+	1.54520i	−0.0809804	+	1.54520i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	0.294556i		−	0.294556i
$785$		0			0
$786$	−0.309017	+	0.278240i	−0.309017	+	0.278240i
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	0.676203	−	0.676203i	0.676203	−	0.676203i
$789$	1.20243	−	1.08268i	1.20243	−	1.08268i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.310719			−0.310719
$795$		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
$799$			1.95630i			1.95630i
$800$		0			0
$801$	1.16913	+	0.122881i	1.16913	+	0.122881i
$802$	0.281181	−	0.281181i	0.281181	−	0.281181i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.72968	+	0.0906485i	1.72968	+	0.0906485i
$808$	0.120252	+	0.120252i	0.120252	+	0.120252i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$811$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$812$		0			0
$813$	1.33643	+	0.0700392i	1.33643	+	0.0700392i
$814$		0			0
$815$		0			0
$816$	1.13989	+	1.26597i	1.13989	+	1.26597i
$817$		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	$0.750000\pi$
$824$	−0.170062			−0.170062
$825$		0			0
$826$	−0.138341			−0.138341
$827$	1.29195	+	1.29195i	1.29195	+	1.29195i	0.933580	+	0.358368i	$0.116667\pi$
$827$	1.29195	+	1.29195i	1.29195	+	1.29195i	0.358368	+	0.933580i	$0.383333\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	0.309017	−	0.278240i	0.309017	−	0.278240i
$832$		0			0
$833$	−0.467920	+	0.467920i	−0.467920	+	0.467920i
$834$		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
$846$	−0.0218524	+	0.207912i	−0.0218524	+	0.207912i
$847$	0.575212	−	0.575212i	0.575212	−	0.575212i
$848$	−0.996297	+	0.996297i	−0.996297	+	0.996297i
$849$	1.33093	+	1.47815i	1.33093	+	1.47815i
$850$		0			0
$851$		0			0
$852$	−1.41938	−	0.0743868i	−1.41938	−	0.0743868i
$853$	0.294032	+	0.294032i	0.294032	+	0.294032i	0.838671	−	0.544639i	$-0.183333\pi$
$853$	0.294032	+	0.294032i	0.294032	+	0.294032i	−0.544639	+	0.838671i	$0.683333\pi$
$854$	−0.275166			−0.275166
$855$		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
$862$	−0.294032	+	0.294032i	−0.294032	+	0.294032i
$863$	−1.38331	+	1.38331i	−1.38331	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$863$	−1.38331	+	1.38331i	−1.38331	+	1.38331i	−0.838671	+	0.544639i	$0.816667\pi$
$864$	0.347395	+	0.478148i	0.347395	+	0.478148i
$865$		0			0
$866$		0			0
$867$	0.147959	−	2.82322i	0.147959	−	2.82322i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.913590	−	0.739810i	0.913590	−	0.739810i
$874$		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	$0.750000\pi$
$878$	−0.147826	+	0.147826i	−0.147826	+	0.147826i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−0.0549565	+	0.0445029i	−0.0549565	+	0.0445029i
$883$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$883$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$884$		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	$0.250000\pi$
$888$	0.0407132	−	0.776854i	0.0407132	−	0.776854i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−0.207912	−	0.230909i	−0.207912	−	0.230909i
$895$		0			0
$896$		−	0.607858i		−	0.607858i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	3.16535			3.16535
$902$		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
$909$	−0.0434654	+	0.413545i	−0.0434654	+	0.413545i
$910$		0			0
$911$			0.813473i			0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$911$			0.813473i			0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$912$		0			0
$913$		0			0
$914$	0.310719			0.310719
$915$		0			0
$916$		0			0
$917$	1.14412	+	1.14412i	1.14412	+	1.14412i
$918$	0.0639781	−	0.403942i	0.0639781	−	0.403942i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−1.41355	+	1.27276i	−1.41355	+	1.27276i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.261686	−	0.323155i	−0.261686	−	0.323155i
$928$		0			0
$929$	0.415823			0.415823			0.207912	−	0.978148i	$-0.433333\pi$
$929$	0.415823			0.415823			0.207912	+	0.978148i	$0.433333\pi$
$930$		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
$941$			0.813473i			0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$941$			0.813473i			0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$942$	−0.415254	−	0.0217625i	−0.415254	−	0.0217625i
$943$		0			0
$944$	−0.708369			−0.708369
$945$		0			0
$946$		0			0
$947$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$947$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$948$	0.590213	+	0.0309318i	0.590213	+	0.0309318i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−0.460216	+	0.460216i	−0.460216	+	0.460216i
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$	0.336408	+	0.0353579i	0.336408	+	0.0353579i
$955$		0			0
$956$		−	1.81898i		−	1.81898i
$957$		0			0
$958$	0.294032	+	0.294032i	0.294032	+	0.294032i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$			0.591023i			0.591023i
$965$		0			0
$966$		0			0
$967$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$967$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$968$	−0.289190	+	0.289190i	−0.289190	+	0.289190i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−0.247507	+	0.923710i	−0.247507	+	0.923710i
$973$		0			0
$974$	0.362097			0.362097
$975$		0			0
$976$	−1.40898			−1.40898
$977$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$977$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.173779	+	0.173779i	−0.173779	+	0.173779i
$983$		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
$987$	0.812358	+	0.0425739i	0.812358	+	0.0425739i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$991$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$992$		0			0
$993$	1.82459	+	0.0956226i	1.82459	+	0.0956226i
$994$		−	0.252762i		−	0.252762i
$995$		0			0
$996$	0.639886	+	0.710666i	0.639886	+	0.710666i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$	1.53884	−	1.11803i	1.53884	−	1.11803i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.1.l.d.1268.9	yes	32
3.2	odd	2	inner	3525.1.l.d.1268.7	✓	32
5.2	odd	4	inner	3525.1.l.d.1832.7	yes	32
5.3	odd	4	inner	3525.1.l.d.1832.10	yes	32
5.4	even	2	inner	3525.1.l.d.1268.8	yes	32
15.2	even	4	inner	3525.1.l.d.1832.9	yes	32
15.8	even	4	inner	3525.1.l.d.1832.8	yes	32
15.14	odd	2	inner	3525.1.l.d.1268.10	yes	32
47.46	odd	2	CM	3525.1.l.d.1268.9	yes	32
141.140	even	2	inner	3525.1.l.d.1268.7	✓	32
235.93	even	4	inner	3525.1.l.d.1832.10	yes	32
235.187	even	4	inner	3525.1.l.d.1832.7	yes	32
235.234	odd	2	inner	3525.1.l.d.1268.8	yes	32
705.422	odd	4	inner	3525.1.l.d.1832.9	yes	32
705.563	odd	4	inner	3525.1.l.d.1832.8	yes	32
705.704	even	2	inner	3525.1.l.d.1268.10	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3525.1.l.d.1268.7	✓	32	3.2	odd	2	inner
3525.1.l.d.1268.7	✓	32	141.140	even	2	inner
3525.1.l.d.1268.8	yes	32	5.4	even	2	inner
3525.1.l.d.1268.8	yes	32	235.234	odd	2	inner
3525.1.l.d.1268.9	yes	32	1.1	even	1	trivial
3525.1.l.d.1268.9	yes	32	47.46	odd	2	CM
3525.1.l.d.1268.10	yes	32	15.14	odd	2	inner
3525.1.l.d.1268.10	yes	32	705.704	even	2	inner
3525.1.l.d.1832.7	yes	32	5.2	odd	4	inner
3525.1.l.d.1832.7	yes	32	235.187	even	4	inner
3525.1.l.d.1832.8	yes	32	15.8	even	4	inner
3525.1.l.d.1832.8	yes	32	705.563	odd	4	inner
3525.1.l.d.1832.9	yes	32	15.2	even	4	inner
3525.1.l.d.1832.9	yes	32	705.422	odd	4	inner
3525.1.l.d.1832.10	yes	32	5.3	odd	4	inner
3525.1.l.d.1832.10	yes	32	235.93	even	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)

\(f(q)\)	\(=\)	\(q+(0.147826 + 0.147826i) q^{2} +(-0.998630 - 0.0523360i) q^{3} -0.956295i q^{4} +(-0.139886 - 0.155360i) q^{6} +(-0.575212 + 0.575212i) q^{7} +(0.289190 - 0.289190i) q^{8} +(0.994522 + 0.104528i) q^{9} +O(q^{10})\)	\(q+(0.147826 + 0.147826i) q^{2} +(-0.998630 - 0.0523360i) q^{3} -0.956295i q^{4} +(-0.139886 - 0.155360i) q^{6} +(-0.575212 + 0.575212i) q^{7} +(0.289190 - 0.289190i) q^{8} +(0.994522 + 0.104528i) q^{9} +(-0.0500486 + 0.954985i) q^{12} -0.170062 q^{14} -0.870796 q^{16} +(1.38331 + 1.38331i) q^{17} +(0.131564 + 0.162468i) q^{18} +(0.604528 - 0.544320i) q^{21} +(-0.303929 + 0.273659i) q^{24} +(-0.987688 - 0.156434i) q^{27} +(0.550073 + 0.550073i) q^{28} +(-0.417916 - 0.417916i) q^{32} +0.408977i q^{34} +(0.0999601 - 0.951057i) q^{36} +(-1.34500 + 1.34500i) q^{37} +(0.169829 + 0.00890037i) q^{42} +(0.707107 + 0.707107i) q^{47} +(0.869602 + 0.0455739i) q^{48} +0.338261i q^{49} +(-1.30902 - 1.45381i) q^{51} +(1.14412 - 1.14412i) q^{53} +(-0.122881 - 0.169131i) q^{54} +0.332692i q^{56} +0.813473 q^{59} +1.61803 q^{61} +(-0.632187 + 0.511935i) q^{63} +0.747238i q^{64} +(1.32285 - 1.32285i) q^{68} +1.48629i q^{71} +(0.317835 - 0.257378i) q^{72} -0.397650 q^{74} -0.618034i q^{79} +(0.978148 + 0.207912i) q^{81} +(0.707107 - 0.707107i) q^{83} +(-0.520530 - 0.578108i) q^{84} +1.17557 q^{89} +0.209057i q^{94} +(0.395472 + 0.439216i) q^{96} +(0.831254 - 0.831254i) q^{97} +(-0.0500037 + 0.0500037i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{6}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

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