LMFDB - Embedded newform 935.1.j.a.659.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [935,1,Mod(659,935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(935, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

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

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

chi := DirichletCharacter("935.659");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 935 = 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	935.j (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:	$0.466625786812$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.44174768466125.1

Embedding invariants

Embedding label			659.3
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	935.659
Dual form			935.1.j.a.769.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367i q^{2} +0.414214 q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.541196 + 0.541196i) q^{7} +1.08239i q^{8} +1.00000i q^{9} +O(q^{10})$	\(q+0.765367i q^{2} +0.414214 q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.541196 + 0.541196i) q^{7} +1.08239i q^{8} +1.00000i q^{9} +(0.541196 - 0.541196i) q^{10} +(0.707107 - 0.707107i) q^{11} +1.84776 q^{13} +(-0.414214 - 0.414214i) q^{14} -0.414214 q^{16} +(-0.923880 + 0.382683i) q^{17} -0.765367 q^{18} +(-0.292893 - 0.292893i) q^{20} +(0.541196 + 0.541196i) q^{22} +1.00000i q^{25} +1.41421i q^{26} +(-0.224171 + 0.224171i) q^{28} +0.765367i q^{32} +(-0.292893 - 0.707107i) q^{34} +0.765367 q^{35} +0.414214i q^{36} +(0.765367 - 0.765367i) q^{40} -0.765367i q^{43} +(0.292893 - 0.292893i) q^{44} +(0.707107 - 0.707107i) q^{45} +0.414214i q^{49} -0.765367 q^{50} +0.765367 q^{52} -1.00000 q^{55} +(-0.585786 - 0.585786i) q^{56} -2.00000i q^{59} +(-0.541196 - 0.541196i) q^{63} -1.00000 q^{64} +(-1.30656 - 1.30656i) q^{65} +(-0.382683 + 0.158513i) q^{68} +0.585786i q^{70} +(1.00000 + 1.00000i) q^{71} -1.08239 q^{72} +(-1.30656 - 1.30656i) q^{73} +0.765367i q^{77} +(0.292893 + 0.292893i) q^{80} -1.00000 q^{81} -1.84776i q^{83} +(0.923880 + 0.382683i) q^{85} +0.585786 q^{86} +(0.765367 + 0.765367i) q^{88} -1.41421 q^{89} +(0.541196 + 0.541196i) q^{90} +(-1.00000 + 1.00000i) q^{91} -0.317025 q^{98} +(0.707107 + 0.707107i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{14} + 8 q^{16} - 8 q^{20} - 8 q^{34} + 8 q^{44} - 8 q^{55} - 16 q^{56} - 8 q^{64} + 8 q^{71} + 8 q^{80} - 8 q^{81} + 16 q^{86} - 8 q^{91}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^14 + 8 * q^16 - 8 * q^20 - 8 * q^34 + 8 * q^44 - 8 * q^55 - 16 * q^56 - 8 * q^64 + 8 * q^71 + 8 * q^80 - 8 * q^81 + 16 * q^86 - 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/935\mathbb{Z}\right)^\times$.

$n$	$496$	$562$	$596$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

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 $

$2$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$2$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$	0.414214			0.414214
$5$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$5$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$6$		0			0
$7$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$7$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$8$			1.08239i			1.08239i
$9$			1.00000i			1.00000i
$10$	0.541196	−	0.541196i	0.541196	−	0.541196i
$11$	0.707107	−	0.707107i	0.707107	−	0.707107i
$11$	0.707107	−	0.707107i	0.707107	−	0.707107i
$12$		0			0
$13$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$13$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$14$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$15$		0			0
$16$	−0.414214			−0.414214
$17$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$17$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$18$	−0.765367			−0.765367
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.292893	−	0.292893i	−0.292893	−	0.292893i
$21$		0			0
$22$	0.541196	+	0.541196i	0.541196	+	0.541196i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$			1.00000i			1.00000i
$26$			1.41421i			1.41421i
$27$		0			0
$28$	−0.224171	+	0.224171i	−0.224171	+	0.224171i
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$			0.765367i			0.765367i
$33$		0			0
$34$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$35$	0.765367			0.765367
$36$			0.414214i			0.414214i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	0.765367	−	0.765367i	0.765367	−	0.765367i
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$43$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$44$	0.292893	−	0.292893i	0.292893	−	0.292893i
$45$	0.707107	−	0.707107i	0.707107	−	0.707107i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$			0.414214i			0.414214i
$50$	−0.765367			−0.765367
$51$		0			0
$52$	0.765367			0.765367
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−1.00000			−1.00000
$56$	−0.585786	−	0.585786i	−0.585786	−	0.585786i
$57$		0			0
$58$		0			0
$59$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$59$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$64$	−1.00000			−1.00000
$65$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−0.382683	+	0.158513i	−0.382683	+	0.158513i
$69$		0			0
$70$			0.585786i			0.585786i
$71$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$71$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$72$	−1.08239			−1.08239
$73$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$73$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.765367i			0.765367i
$78$		0			0
$79$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$79$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$80$	0.292893	+	0.292893i	0.292893	+	0.292893i
$81$	−1.00000			−1.00000
$82$		0			0
$83$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$83$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$84$		0			0
$85$	0.923880	+	0.382683i	0.923880	+	0.382683i
$86$	0.585786			0.585786
$87$		0			0
$88$	0.765367	+	0.765367i	0.765367	+	0.765367i
$89$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$89$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$90$	0.541196	+	0.541196i	0.541196	+	0.541196i
$91$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	−0.317025			−0.317025
$99$	0.707107	+	0.707107i	0.707107	+	0.707107i
$100$			0.414214i			0.414214i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			2.00000i			2.00000i
$105$		0			0
$106$		0			0
$107$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$107$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$108$		0			0
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		−	0.765367i		−	0.765367i
$111$		0			0
$112$	0.224171	−	0.224171i	0.224171	−	0.224171i
$113$		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
$117$			1.84776i			1.84776i
$118$	1.53073			1.53073
$119$	0.292893	−	0.707107i	0.292893	−	0.707107i
$120$		0			0
$121$		−	1.00000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.707107	−	0.707107i
$126$	0.414214	−	0.414214i	0.414214	−	0.414214i
$127$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$127$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$128$		0			0
$129$		0			0
$130$	1.00000	−	1.00000i	1.00000	−	1.00000i
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.414214	−	1.00000i	−0.414214	−	1.00000i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$	0.317025			0.317025
$141$		0			0
$142$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$143$	1.30656	−	1.30656i	1.30656	−	1.30656i
$144$		−	0.414214i		−	0.414214i
$145$		0			0
$146$	1.00000	−	1.00000i	1.00000	−	1.00000i
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$154$	−0.585786			−0.585786
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	0.541196	−	0.541196i	0.541196	−	0.541196i
$161$		0			0
$162$		−	0.765367i		−	0.765367i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	1.41421			1.41421
$167$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$167$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	2.41421			2.41421
$170$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$171$		0			0
$172$		−	0.317025i		−	0.317025i
$173$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$173$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$174$		0			0
$175$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$176$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$177$		0			0
$178$		−	1.08239i		−	1.08239i
$179$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$179$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$180$	0.292893	−	0.292893i	0.292893	−	0.292893i
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$191$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$193$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$194$		0			0
$195$		0			0
$196$			0.171573i			0.171573i
$197$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$197$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$198$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$199$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$199$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$200$	−1.08239			−1.08239
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.765367			−0.765367
$209$		0			0
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$		0			0
$213$		0			0
$214$	1.00000	−	1.00000i	1.00000	−	1.00000i
$215$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.414214			−0.414214
$221$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$225$	−1.00000			−1.00000
$226$		0			0
$227$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$227$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$233$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$234$	−1.41421			−1.41421
$235$		0			0
$236$		−	0.828427i		−	0.828427i
$237$		0			0
$238$	0.541196	+	0.224171i	0.541196	+	0.224171i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$	0.765367			0.765367
$243$		0			0
$244$		0			0
$245$	0.292893	−	0.292893i	0.292893	−	0.292893i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.541196	+	0.541196i	0.541196	+	0.541196i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−0.224171	−	0.224171i	−0.224171	−	0.224171i
$253$		0			0
$254$	1.41421			1.41421
$255$		0			0
$256$	−1.00000			−1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$261$		0			0
$262$		0			0
$263$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$263$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$269$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	0.382683	−	0.158513i	0.382683	−	0.158513i
$273$		0			0
$274$		0			0
$275$	0.707107	+	0.707107i	0.707107	+	0.707107i
$276$		0			0
$277$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$277$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$278$		0			0
$279$		0			0
$280$			0.828427i			0.828427i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$283$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$284$	0.414214	+	0.414214i	0.414214	+	0.414214i
$285$		0			0
$286$	1.00000	+	1.00000i	1.00000	+	1.00000i
$287$		0			0
$288$	−0.765367			−0.765367
$289$	0.707107	−	0.707107i	0.707107	−	0.707107i
$290$		0			0
$291$		0			0
$292$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$293$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$293$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$294$		0			0
$295$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.414214	+	0.414214i	0.414214	+	0.414214i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	0.707107	−	0.292893i	0.707107	−	0.292893i
$307$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$307$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$308$			0.317025i			0.317025i
$309$		0			0
$310$		0			0
$311$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$311$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$			0.765367i			0.765367i
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$	0.707107	+	0.707107i	0.707107	+	0.707107i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.414214			−0.414214
$325$			1.84776i			1.84776i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$331$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$332$		−	0.765367i		−	0.765367i
$333$		0			0
$334$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$335$		0			0
$336$		0			0
$337$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$337$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$338$			1.84776i			1.84776i
$339$		0			0
$340$	0.382683	+	0.158513i	0.382683	+	0.158513i
$341$		0			0
$342$		0			0
$343$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$344$	0.828427			0.828427
$345$		0			0
$346$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$347$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$347$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	0.414214	−	0.414214i	0.414214	−	0.414214i
$351$		0			0
$352$	0.541196	+	0.541196i	0.541196	+	0.541196i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		−	1.41421i		−	1.41421i
$356$	−0.585786			−0.585786
$357$		0			0
$358$	−1.08239			−1.08239
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	0.765367	+	0.765367i	0.765367	+	0.765367i
$361$	−1.00000			−1.00000
$362$		0			0
$363$		0			0
$364$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$365$			1.84776i			1.84776i
$366$		0			0
$367$		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			0
$372$		0			0
$373$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$373$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$374$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$379$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		−	1.08239i		−	1.08239i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	0.541196	−	0.541196i	0.541196	−	0.541196i
$386$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$387$	0.765367			0.765367
$388$		0			0
$389$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$389$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−0.448342			−0.448342
$393$		0			0
$394$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$395$		0			0
$396$	0.292893	+	0.292893i	0.292893	+	0.292893i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$399$		0			0
$400$		−	0.414214i		−	0.414214i
$401$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$401$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.707107	+	0.707107i	0.707107	+	0.707107i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.08239	+	1.08239i	1.08239	+	1.08239i
$414$		0			0
$415$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$416$			1.41421i			1.41421i
$417$		0			0
$418$		0			0
$419$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$419$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$421$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$426$		0			0
$427$		0			0
$428$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$429$		0			0
$430$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$		−	1.08239i		−	1.08239i
$441$	−0.414214			−0.414214
$442$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	1.00000	+	1.00000i	1.00000	+	1.00000i
$446$		0			0
$447$		0			0
$448$	0.541196	−	0.541196i	0.541196	−	0.541196i
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$		−	0.765367i		−	0.765367i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	1.00000	+	1.00000i	1.00000	+	1.00000i
$455$	1.41421			1.41421
$456$		0			0
$457$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$457$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$			0.765367i			0.765367i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	2.16478			2.16478
$473$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$474$		0			0
$475$		0			0
$476$	0.121320	−	0.292893i	0.121320	−	0.292893i
$477$		0			0
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		−	0.414214i		−	0.414214i
$485$		0			0
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$	0.224171	+	0.224171i	0.224171	+	0.224171i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	1.00000i		−	1.00000i
$496$		0			0
$497$	−1.08239			−1.08239
$498$		0			0
$499$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$499$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$500$	0.292893	−	0.292893i	0.292893	−	0.292893i
$501$		0			0
$502$		0			0
$503$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$503$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$504$	0.585786	−	0.585786i	0.585786	−	0.585786i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		−	0.765367i		−	0.765367i
$509$	2.00000			2.00000			1.00000			$0$
$509$	2.00000			2.00000			1.00000			$0$
$510$		0			0
$511$	1.41421			1.41421
$512$		−	0.765367i		−	0.765367i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.41421	−	1.41421i	1.41421	−	1.41421i
$521$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$521$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$522$		0			0
$523$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$523$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$524$		0			0
$525$		0			0
$526$	−0.585786			−0.585786
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$	2.00000			2.00000
$532$		0			0
$533$		0			0
$534$		0			0
$535$			1.84776i			1.84776i
$536$		0			0
$537$		0			0
$538$	0.765367	−	0.765367i	0.765367	−	0.765367i
$539$	0.292893	+	0.292893i	0.292893	+	0.292893i
$540$		0			0
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$		0			0
$544$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$545$		0			0
$546$		0			0
$547$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$547$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$548$		0			0
$549$		0			0
$550$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$555$		0			0
$556$		0			0
$557$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$557$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$558$		0			0
$559$		−	1.41421i		−	1.41421i
$560$	−0.317025			−0.317025
$561$		0			0
$562$		0			0
$563$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$563$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$564$		0			0
$565$		0			0
$566$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$567$	0.541196	−	0.541196i	0.541196	−	0.541196i
$568$	−1.08239	+	1.08239i	−1.08239	+	1.08239i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$	0.541196	−	0.541196i	0.541196	−	0.541196i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	1.00000i		−	1.00000i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.541196	+	0.541196i	0.541196	+	0.541196i
$579$		0			0
$580$		0			0
$581$	1.00000	+	1.00000i	1.00000	+	1.00000i
$582$		0			0
$583$		0			0
$584$	1.41421	−	1.41421i	1.41421	−	1.41421i
$585$	1.30656	−	1.30656i	1.30656	−	1.30656i
$586$			1.41421i			1.41421i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$	−1.08239	−	1.08239i	−1.08239	−	1.08239i
$591$		0			0
$592$		0			0
$593$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$593$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$594$		0			0
$595$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$599$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$600$		0			0
$601$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$601$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$602$	−0.317025	+	0.317025i	−0.317025	+	0.317025i
$603$		0			0
$604$		0			0
$605$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$606$		0			0
$607$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$607$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$613$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$613$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$614$			0.585786i			0.585786i
$615$		0			0
$616$	−0.828427			−0.828427
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$619$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$620$		0			0
$621$		0			0
$622$	0.765367	−	0.765367i	0.765367	−	0.765367i
$623$	0.765367	−	0.765367i	0.765367	−	0.765367i
$624$		0			0
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−0.585786			−0.585786
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$636$		0			0
$637$			0.765367i			0.765367i
$638$		0			0
$639$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$640$		0			0
$641$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$641$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		−	1.08239i		−	1.08239i
$649$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$650$	−1.41421			−1.41421
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.30656	−	1.30656i	1.30656	−	1.30656i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$661$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$662$	1.08239			1.08239
$663$		0			0
$664$	2.00000			2.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.224171	+	0.224171i	0.224171	+	0.224171i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$673$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$674$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$675$		0			0
$676$	1.00000			1.00000
$677$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$677$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$678$		0			0
$679$		0			0
$680$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$	0.585786	−	0.585786i	0.585786	−	0.585786i
$687$		0			0
$688$			0.317025i			0.317025i
$689$		0			0
$690$		0			0
$691$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$691$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$692$	0.224171	+	0.224171i	0.224171	+	0.224171i
$693$	−0.765367			−0.765367
$694$	0.414214	+	0.414214i	0.414214	+	0.414214i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.224171	−	0.224171i	−0.224171	−	0.224171i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$	1.08239			1.08239
$711$		0			0
$712$		−	1.53073i		−	1.53073i
$713$		0			0
$714$		0			0
$715$	−1.84776			−1.84776
$716$			0.585786i			0.585786i
$717$		0			0
$718$		0			0
$719$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$719$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$720$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$721$		0			0
$722$		−	0.765367i		−	0.765367i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	−1.08239	−	1.08239i	−1.08239	−	1.08239i
$729$		−	1.00000i		−	1.00000i
$730$	−1.41421			−1.41421
$731$	0.292893	+	0.707107i	0.292893	+	0.707107i
$732$		0			0
$733$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$733$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$743$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$744$		0			0
$745$		0			0
$746$		−	0.585786i		−	0.585786i
$747$	1.84776			1.84776
$748$	−0.158513	+	0.382683i	−0.158513	+	0.382683i
$749$	1.41421			1.41421
$750$		0			0
$751$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$751$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	0.765367	−	0.765367i	0.765367	−	0.765367i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	−0.585786			−0.585786
$765$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$766$		0			0
$767$		−	3.69552i		−	3.69552i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	0.414214	+	0.414214i	0.414214	+	0.414214i
$771$		0			0
$772$	−0.224171	+	0.224171i	−0.224171	+	0.224171i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$			0.585786i			0.585786i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.53073			−1.53073
$779$		0			0
$780$		0			0
$781$	1.41421			1.41421
$782$		0			0
$783$		0			0
$784$		−	0.171573i		−	0.171573i
$785$		0			0
$786$		0			0
$787$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$787$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$788$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.414214	+	0.414214i	0.414214	+	0.414214i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−0.765367			−0.765367
$801$		−	1.41421i		−	1.41421i
$802$	1.08239	+	1.08239i	1.08239	+	1.08239i
$803$	−1.84776			−1.84776
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$809$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$810$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$	−0.828427	+	0.828427i	−0.828427	+	0.828427i
$827$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$827$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$828$		0			0
$829$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$829$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$830$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$831$		0			0
$832$	−1.84776			−1.84776
$833$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$834$		0			0
$835$		−	0.765367i		−	0.765367i
$836$		0			0
$837$		0			0
$838$	−1.08239	−	1.08239i	−1.08239	−	1.08239i
$839$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$839$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		−	1.08239i		−	1.08239i
$843$		0			0
$844$		0			0
$845$	−1.70711	−	1.70711i	−1.70711	−	1.70711i
$846$		0			0
$847$	0.541196	+	0.541196i	0.541196	+	0.541196i
$848$		0			0
$849$		0			0
$850$	0.707107	−	0.292893i	0.707107	−	0.292893i
$851$		0			0
$852$		0			0
$853$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$853$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$854$		0			0
$855$		0			0
$856$	1.41421	−	1.41421i	1.41421	−	1.41421i
$857$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$857$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$858$		0			0
$859$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$859$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$860$	−0.224171	+	0.224171i	−0.224171	+	0.224171i
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		−	0.765367i		−	0.765367i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			0.765367i			0.765367i
$876$		0			0
$877$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$877$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$878$		0			0
$879$		0			0
$880$	0.414214			0.414214
$881$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$881$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$882$		−	0.317025i		−	0.317025i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$885$		0			0
$886$		0			0
$887$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$887$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$888$		0			0
$889$	1.00000	+	1.00000i	1.00000	+	1.00000i
$890$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$891$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	1.00000	−	1.00000i	1.00000	−	1.00000i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.414214			−0.414214
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	0.541196	−	0.541196i	0.541196	−	0.541196i
$909$		0			0
$910$			1.08239i			1.08239i
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$914$	0.585786			0.585786
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.84776	+	1.84776i	1.84776	+	1.84776i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$929$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	0.541196	+	0.541196i	0.541196	+	0.541196i
$933$		0			0
$934$		0			0
$935$	0.923880	−	0.382683i	0.923880	−	0.382683i
$936$	−2.00000			−2.00000
$937$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$937$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$			0.828427i			0.828427i
$945$		0			0
$946$	0.414214	−	0.414214i	0.414214	−	0.414214i
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$	−2.41421	−	2.41421i	−2.41421	−	2.41421i
$950$		0			0
$951$		0			0
$952$	0.765367	+	0.317025i	0.765367	+	0.317025i
$953$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$953$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$954$		0			0
$955$	1.00000	+	1.00000i	1.00000	+	1.00000i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	1.00000i		−	1.00000i
$962$		0			0
$963$	1.30656	−	1.30656i	1.30656	−	1.30656i
$964$		0			0
$965$	0.765367			0.765367
$966$		0			0
$967$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$967$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$968$	1.08239			1.08239
$969$		0			0
$970$		0			0
$971$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$971$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$980$	0.121320	−	0.121320i	0.121320	−	0.121320i
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	1.84776			1.84776
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.765367			0.765367
$991$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$991$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$		0			0
$994$		−	0.828427i		−	0.828427i
$995$		−	1.41421i		−	1.41421i
$996$		0			0
$997$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$997$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$998$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	935.1.j.a.659.3	yes	8
5.4	even	2	inner	935.1.j.a.659.2	✓	8
11.10	odd	2	inner	935.1.j.a.659.2	✓	8
17.4	even	4	inner	935.1.j.a.769.2	yes	8
55.54	odd	2	CM	935.1.j.a.659.3	yes	8
85.4	even	4	inner	935.1.j.a.769.3	yes	8
187.21	odd	4	inner	935.1.j.a.769.3	yes	8
935.769	odd	4	inner	935.1.j.a.769.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
935.1.j.a.659.2	✓	8	5.4	even	2	inner
935.1.j.a.659.2	✓	8	11.10	odd	2	inner
935.1.j.a.659.3	yes	8	1.1	even	1	trivial
935.1.j.a.659.3	yes	8	55.54	odd	2	CM
935.1.j.a.769.2	yes	8	17.4	even	4	inner
935.1.j.a.769.2	yes	8	935.769	odd	4	inner
935.1.j.a.769.3	yes	8	85.4	even	4	inner
935.1.j.a.769.3	yes	8	187.21	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:	\( N \)	\(=\)	\( 935 = 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	935.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+0.765367i q^{2} +0.414214 q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.541196 + 0.541196i) q^{7} +1.08239i q^{8} +1.00000i q^{9} +O(q^{10})\)	\(q+0.765367i q^{2} +0.414214 q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.541196 + 0.541196i) q^{7} +1.08239i q^{8} +1.00000i q^{9} +(0.541196 - 0.541196i) q^{10} +(0.707107 - 0.707107i) q^{11} +1.84776 q^{13} +(-0.414214 - 0.414214i) q^{14} -0.414214 q^{16} +(-0.923880 + 0.382683i) q^{17} -0.765367 q^{18} +(-0.292893 - 0.292893i) q^{20} +(0.541196 + 0.541196i) q^{22} +1.00000i q^{25} +1.41421i q^{26} +(-0.224171 + 0.224171i) q^{28} +0.765367i q^{32} +(-0.292893 - 0.707107i) q^{34} +0.765367 q^{35} +0.414214i q^{36} +(0.765367 - 0.765367i) q^{40} -0.765367i q^{43} +(0.292893 - 0.292893i) q^{44} +(0.707107 - 0.707107i) q^{45} +0.414214i q^{49} -0.765367 q^{50} +0.765367 q^{52} -1.00000 q^{55} +(-0.585786 - 0.585786i) q^{56} -2.00000i q^{59} +(-0.541196 - 0.541196i) q^{63} -1.00000 q^{64} +(-1.30656 - 1.30656i) q^{65} +(-0.382683 + 0.158513i) q^{68} +0.585786i q^{70} +(1.00000 + 1.00000i) q^{71} -1.08239 q^{72} +(-1.30656 - 1.30656i) q^{73} +0.765367i q^{77} +(0.292893 + 0.292893i) q^{80} -1.00000 q^{81} -1.84776i q^{83} +(0.923880 + 0.382683i) q^{85} +0.585786 q^{86} +(0.765367 + 0.765367i) q^{88} -1.41421 q^{89} +(0.541196 + 0.541196i) q^{90} +(-1.00000 + 1.00000i) q^{91} -0.317025 q^{98} +(0.707107 + 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{14} + 8 q^{16} - 8 q^{20} - 8 q^{34} + 8 q^{44} - 8 q^{55} - 16 q^{56} - 8 q^{64} + 8 q^{71} + 8 q^{80} - 8 q^{81} + 16 q^{86} - 8 q^{91}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^14 + 8 * q^16 - 8 * q^20 - 8 * q^34 + 8 * q^44 - 8 * q^55 - 16 * q^56 - 8 * q^64 + 8 * q^71 + 8 * q^80 - 8 * q^81 + 16 * q^86 - 8 * q^91

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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