LMFDB - Embedded newform 935.1.y.a.274.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [935,1,Mod(219,935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(935, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 4, 3]))

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

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

chi := DirichletCharacter("935.219");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 935 = 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	935.y (of order $8$, degree $4$, 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:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{32})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1 $ x^16 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			274.4
Root			$-0.555570 - 0.831470i$ of defining polynomial
Character	$\chi$	$=$	935.274
Dual form			935.1.y.a.604.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.17588 + 1.17588i) q^{2} +1.76537i q^{4} +(0.382683 + 0.923880i) q^{5} +(-0.425215 + 1.02656i) q^{7} +(-0.899976 + 0.899976i) q^{8} +(0.707107 - 0.707107i) q^{9} +O(q^{10})$	\(q+(1.17588 + 1.17588i) q^{2} +1.76537i q^{4} +(0.382683 + 0.923880i) q^{5} +(-0.425215 + 1.02656i) q^{7} +(-0.899976 + 0.899976i) q^{8} +(0.707107 - 0.707107i) q^{9} +(-0.636379 + 1.53636i) q^{10} +(-0.923880 - 0.382683i) q^{11} -1.96157i q^{13} +(-1.70711 + 0.707107i) q^{14} -0.351153 q^{16} +(-0.980785 - 0.195090i) q^{17} +1.66294 q^{18} +(-1.63099 + 0.675577i) q^{20} +(-0.636379 - 1.53636i) q^{22} +(-0.707107 + 0.707107i) q^{25} +(2.30656 - 2.30656i) q^{26} +(-1.81225 - 0.750661i) q^{28} +(1.30656 - 0.541196i) q^{31} +(0.487064 + 0.487064i) q^{32} +(-0.923880 - 1.38268i) q^{34} -1.11114 q^{35} +(1.24830 + 1.24830i) q^{36} +(-1.17588 - 0.487064i) q^{40} +(-1.17588 + 1.17588i) q^{43} +(0.675577 - 1.63099i) q^{44} +(0.923880 + 0.382683i) q^{45} +(-0.165911 - 0.165911i) q^{49} -1.66294 q^{50} +3.46289 q^{52} -1.00000i q^{55} +(-0.541196 - 1.30656i) q^{56} +(2.17273 + 0.899976i) q^{62} +(0.425215 + 1.02656i) q^{63} +1.49661i q^{64} +(1.81225 - 0.750661i) q^{65} +(0.344406 - 1.73145i) q^{68} +(-1.30656 - 1.30656i) q^{70} +(-1.70711 + 0.707107i) q^{71} +1.27276i q^{72} +(0.149316 + 0.360480i) q^{73} +(0.785695 - 0.785695i) q^{77} +(-0.134381 - 0.324423i) q^{80} -1.00000i q^{81} +(-0.275899 - 0.275899i) q^{83} +(-0.195090 - 0.980785i) q^{85} -2.76537 q^{86} +(1.17588 - 0.487064i) q^{88} -1.84776i q^{89} +(0.636379 + 1.53636i) q^{90} +(2.01367 + 0.834089i) q^{91} -0.390181i q^{98} +(-0.923880 + 0.382683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{14} - 16 q^{16} + 16 q^{26} + 16 q^{44} - 16 q^{71} - 16 q^{80} - 32 q^{86}+O(q^{100}) $ 16 * q - 16 * q^14 - 16 * q^16 + 16 * q^26 + 16 * q^44 - 16 * q^71 - 16 * q^80 - 32 * q^86

Display 100 coefficients 10 coefficients

Character values

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

$n$	$496$	$562$	$596$
$\chi(n)$	$e\left(\frac{7}{8}\right)$	$-1$	$-1$

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 $

$2$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$2$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$3$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$3$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$4$			1.76537i			1.76537i
$5$	0.382683	+	0.923880i	0.382683	+	0.923880i
$5$	0.382683	+	0.923880i	0.382683	+	0.923880i
$6$		0			0
$7$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$7$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$8$	−0.899976	+	0.899976i	−0.899976	+	0.899976i
$9$	0.707107	−	0.707107i	0.707107	−	0.707107i
$10$	−0.636379	+	1.53636i	−0.636379	+	1.53636i
$11$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$11$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$12$		0			0
$13$		−	1.96157i		−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$13$		−	1.96157i		−	1.96157i	0.195090	−	0.980785i	$-0.437500\pi$
$14$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$15$		0			0
$16$	−0.351153			−0.351153
$17$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$17$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$18$	1.66294			1.66294
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$	−1.63099	+	0.675577i	−1.63099	+	0.675577i
$21$		0			0
$22$	−0.636379	−	1.53636i	−0.636379	−	1.53636i
$23$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$23$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$24$		0			0
$25$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$26$	2.30656	−	2.30656i	2.30656	−	2.30656i
$27$		0			0
$28$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$29$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$29$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$30$		0			0
$31$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$31$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$32$	0.487064	+	0.487064i	0.487064	+	0.487064i
$33$		0			0
$34$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$35$	−1.11114			−1.11114
$36$	1.24830	+	1.24830i	1.24830	+	1.24830i
$37$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$37$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$38$		0			0
$39$		0			0
$40$	−1.17588	−	0.487064i	−1.17588	−	0.487064i
$41$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$41$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$42$		0			0
$43$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.195090	+	0.980785i	$0.562500\pi$
$43$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$44$	0.675577	−	1.63099i	0.675577	−	1.63099i
$45$	0.923880	+	0.382683i	0.923880	+	0.382683i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−0.165911	−	0.165911i	−0.165911	−	0.165911i
$50$	−1.66294			−1.66294
$51$		0			0
$52$	3.46289			3.46289
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		−	1.00000i		−	1.00000i
$56$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$61$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$62$	2.17273	+	0.899976i	2.17273	+	0.899976i
$63$	0.425215	+	1.02656i	0.425215	+	1.02656i
$64$			1.49661i			1.49661i
$65$	1.81225	−	0.750661i	1.81225	−	0.750661i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	0.344406	−	1.73145i	0.344406	−	1.73145i
$69$		0			0
$70$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$71$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$71$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$72$			1.27276i			1.27276i
$73$	0.149316	+	0.360480i	0.149316	+	0.360480i	0.980785	−	0.195090i	$-0.0625000\pi$
$73$	0.149316	+	0.360480i	0.149316	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.785695	−	0.785695i	0.785695	−	0.785695i
$78$		0			0
$79$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$79$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$80$	−0.134381	−	0.324423i	−0.134381	−	0.324423i
$81$		−	1.00000i		−	1.00000i
$82$		0			0
$83$	−0.275899	−	0.275899i	−0.275899	−	0.275899i	0.555570	−	0.831470i	$-0.312500\pi$
$83$	−0.275899	−	0.275899i	−0.275899	−	0.275899i	−0.831470	+	0.555570i	$0.812500\pi$
$84$		0			0
$85$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$86$	−2.76537			−2.76537
$87$		0			0
$88$	1.17588	−	0.487064i	1.17588	−	0.487064i
$89$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$89$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$90$	0.636379	+	1.53636i	0.636379	+	1.53636i
$91$	2.01367	+	0.834089i	2.01367	+	0.834089i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$97$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$98$		−	0.390181i		−	0.390181i
$99$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$100$	−1.24830	−	1.24830i	−1.24830	−	1.24830i
$101$		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			$\pi$
$104$	1.76537	+	1.76537i	1.76537	+	1.76537i
$105$		0			0
$106$		0			0
$107$	−0.149316	−	0.360480i	−0.149316	−	0.360480i	0.831470	−	0.555570i	$-0.187500\pi$
$107$	−0.149316	−	0.360480i	−0.149316	−	0.360480i	−0.980785	+	0.195090i	$0.937500\pi$
$108$		0			0
$109$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$109$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$110$	1.17588	−	1.17588i	1.17588	−	1.17588i
$111$		0			0
$112$	0.149316	−	0.360480i	0.149316	−	0.360480i
$113$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$113$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$118$		0			0
$119$	0.617317	−	0.923880i	0.617317	−	0.923880i
$120$		0			0
$121$	0.707107	+	0.707107i	0.707107	+	0.707107i
$122$		0			0
$123$		0			0
$124$	0.955410	+	2.30656i	0.955410	+	2.30656i
$125$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$126$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$127$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$127$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$128$	−1.27276	+	1.27276i	−1.27276	+	1.27276i
$129$		0			0
$130$	3.01367	+	1.24830i	3.01367	+	1.24830i
$131$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$131$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.05826	−	0.707107i	1.05826	−	0.707107i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$139$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$140$		−	1.96157i		−	1.96157i
$141$		0			0
$142$	−2.83881	−	1.17588i	−2.83881	−	1.17588i
$143$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$144$	−0.248303	+	0.248303i	−0.248303	+	0.248303i
$145$		0			0
$146$	−0.248303	+	0.599456i	−0.248303	+	0.599456i
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$154$	1.84776			1.84776
$155$	1.00000	+	1.00000i	1.00000	+	1.00000i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$	−0.263597	+	0.636379i	−0.263597	+	0.636379i
$161$		0			0
$162$	1.17588	−	1.17588i	1.17588	−	1.17588i
$163$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$163$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$164$		0			0
$165$		0			0
$166$		−	0.648847i		−	0.648847i
$167$	1.02656	−	0.425215i	1.02656	−	0.425215i	0.195090	−	0.980785i	$-0.437500\pi$
$167$	1.02656	−	0.425215i	1.02656	−	0.425215i	0.831470	+	0.555570i	$0.187500\pi$
$168$		0			0
$169$	−2.84776			−2.84776
$170$	0.923880	−	1.38268i	0.923880	−	1.38268i
$171$		0			0
$172$	−2.07585	−	2.07585i	−2.07585	−	2.07585i
$173$	−1.02656	+	0.425215i	−1.02656	+	0.425215i	−0.831470	−	0.555570i	$-0.812500\pi$
$173$	−1.02656	+	0.425215i	−1.02656	+	0.425215i	−0.195090	+	0.980785i	$0.562500\pi$
$174$		0			0
$175$	−0.425215	−	1.02656i	−0.425215	−	1.02656i
$176$	0.324423	+	0.134381i	0.324423	+	0.134381i
$177$		0			0
$178$	2.17273	−	2.17273i	2.17273	−	2.17273i
$179$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$179$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$180$	−0.675577	+	1.63099i	−0.675577	+	1.63099i
$181$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$181$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$182$	1.38704	+	3.34861i	1.38704	+	3.34861i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.831470	+	0.555570i	0.831470	+	0.555570i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$191$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$192$		0			0
$193$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.980785	−	0.195090i	$-0.0625000\pi$
$193$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.555570	+	0.831470i	$0.312500\pi$
$194$		0			0
$195$		0			0
$196$	0.292893	−	0.292893i	0.292893	−	0.292893i
$197$	0.149316	−	0.360480i	0.149316	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$197$	0.149316	−	0.360480i	0.149316	−	0.360480i	0.980785	+	0.195090i	$0.0625000\pi$
$198$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$199$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$199$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$200$		−	1.27276i		−	1.27276i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			0.688812i			0.688812i
$209$		0			0
$210$		0			0
$211$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$211$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$212$		0			0
$213$		0			0
$214$	0.248303	−	0.599456i	0.248303	−	0.599456i
$215$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$216$		0			0
$217$			1.57139i			1.57139i
$218$		0			0
$219$		0			0
$220$	1.76537			1.76537
$221$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$225$			1.00000i			1.00000i
$226$		0			0
$227$	−0.360480	−	0.149316i	−0.360480	−	0.149316i	0.195090	−	0.980785i	$-0.437500\pi$
$227$	−0.360480	−	0.149316i	−0.360480	−	0.149316i	−0.555570	+	0.831470i	$0.687500\pi$
$228$		0			0
$229$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$229$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.750661	+	1.81225i	0.750661	+	1.81225i	0.555570	+	0.831470i	$0.312500\pi$
$233$	0.750661	+	1.81225i	0.750661	+	1.81225i	0.195090	+	0.980785i	$0.437500\pi$
$234$		−	3.26197i		−	3.26197i
$235$		0			0
$236$		0			0
$237$		0			0
$238$	1.81225	−	0.360480i	1.81225	−	0.360480i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$241$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$242$			1.66294i			1.66294i
$243$		0			0
$244$		0			0
$245$	0.0897902	−	0.216773i	0.0897902	−	0.216773i
$246$		0			0
$247$		0			0
$248$	−0.688812	+	1.66294i	−0.688812	+	1.66294i
$249$		0			0
$250$	−0.636379	−	1.53636i	−0.636379	−	1.53636i
$251$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$251$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$252$	−1.81225	+	0.750661i	−1.81225	+	0.750661i
$253$		0			0
$254$	3.26197			3.26197
$255$		0			0
$256$	−1.49661			−1.49661
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$	1.32519	+	3.19929i	1.32519	+	3.19929i
$261$		0			0
$262$		0			0
$263$	−0.785695	+	0.785695i	−0.785695	+	0.785695i	−0.980785	−	0.195090i	$-0.937500\pi$
$263$	−0.785695	+	0.785695i	−0.785695	+	0.785695i	0.195090	+	0.980785i	$0.437500\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$269$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	0.344406	+	0.0685066i	0.344406	+	0.0685066i
$273$		0			0
$274$		0			0
$275$	0.923880	−	0.382683i	0.923880	−	0.382683i
$276$		0			0
$277$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.555570	−	0.831470i	$-0.687500\pi$
$277$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.195090	−	0.980785i	$-0.562500\pi$
$278$		0			0
$279$	0.541196	−	1.30656i	0.541196	−	1.30656i
$280$	1.00000	−	1.00000i	1.00000	−	1.00000i
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$	1.81225	+	0.750661i	1.81225	+	0.750661i	0.980785	+	0.195090i	$0.0625000\pi$
$283$	1.81225	+	0.750661i	1.81225	+	0.750661i	0.831470	+	0.555570i	$0.187500\pi$
$284$	−1.24830	−	3.01367i	−1.24830	−	3.01367i
$285$		0			0
$286$	−3.01367	+	1.24830i	−3.01367	+	1.24830i
$287$		0			0
$288$	0.688812			0.688812
$289$	0.923880	+	0.382683i	0.923880	+	0.382683i
$290$		0			0
$291$		0			0
$292$	−0.636379	+	0.263597i	−0.636379	+	0.263597i
$293$		−	1.96157i		−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$293$		−	1.96157i		−	1.96157i	0.195090	−	0.980785i	$-0.437500\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	−1.63099	−	0.324423i	−1.63099	−	0.324423i
$307$	1.11114			1.11114			0.555570	−	0.831470i	$-0.312500\pi$
$307$	1.11114			1.11114			0.555570	+	0.831470i	$0.312500\pi$
$308$	1.38704	+	1.38704i	1.38704	+	1.38704i
$309$		0			0
$310$			2.35175i			2.35175i
$311$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$311$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$312$		0			0
$313$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$313$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$314$		0			0
$315$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$316$		0			0
$317$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$317$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$318$		0			0
$319$		0			0
$320$	−1.38268	+	0.572726i	−1.38268	+	0.572726i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.76537			1.76537
$325$	1.38704	+	1.38704i	1.38704	+	1.38704i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$331$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$332$	0.487064	−	0.487064i	0.487064	−	0.487064i
$333$		0			0
$334$	1.70711	+	0.707107i	1.70711	+	0.707107i
$335$		0			0
$336$		0			0
$337$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$337$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$338$	−3.34861	−	3.34861i	−3.34861	−	3.34861i
$339$		0			0
$340$	1.73145	−	0.344406i	1.73145	−	0.344406i
$341$	−1.41421			−1.41421
$342$		0			0
$343$	−0.785695	+	0.325446i	−0.785695	+	0.325446i
$344$		−	2.11652i		−	2.11652i
$345$		0			0
$346$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$347$	−0.636379	+	1.53636i	−0.636379	+	1.53636i	0.195090	+	0.980785i	$0.437500\pi$
$347$	−0.636379	+	1.53636i	−0.636379	+	1.53636i	−0.831470	+	0.555570i	$0.812500\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	0.707107	−	1.70711i	0.707107	−	1.70711i
$351$		0			0
$352$	−0.263597	−	0.636379i	−0.263597	−	0.636379i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$356$	3.26197			3.26197
$357$		0			0
$358$	−3.07271			−3.07271
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	−1.17588	+	0.487064i	−1.17588	+	0.487064i
$361$			1.00000i			1.00000i
$362$	−0.899976	−	2.17273i	−0.899976	−	2.17273i
$363$		0			0
$364$	−1.47247	+	3.55487i	−1.47247	+	3.55487i
$365$	−0.275899	+	0.275899i	−0.275899	+	0.275899i
$366$		0			0
$367$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$367$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.66294			1.66294			0.831470	−	0.555570i	$-0.187500\pi$
$373$	1.66294			1.66294			0.831470	+	0.555570i	$0.187500\pi$
$374$	0.324423	+	1.63099i	0.324423	+	1.63099i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$379$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$380$		0			0
$381$		0			0
$382$	−2.17273	+	2.17273i	−2.17273	+	2.17273i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$	1.02656	+	0.425215i	1.02656	+	0.425215i
$386$	1.05826	+	2.55487i	1.05826	+	2.55487i
$387$			1.66294i			1.66294i
$388$		0			0
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	0.298631			0.298631
$393$		0			0
$394$	0.599456	−	0.248303i	0.599456	−	0.248303i
$395$		0			0
$396$	−0.675577	−	1.63099i	−0.675577	−	1.63099i
$397$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$397$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$398$	0.487064	−	1.17588i	0.487064	−	1.17588i
$399$		0			0
$400$	0.248303	−	0.248303i	0.248303	−	0.248303i
$401$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$401$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$402$		0			0
$403$	−1.06159	−	2.56292i	−1.06159	−	2.56292i
$404$		0			0
$405$	0.923880	−	0.382683i	0.923880	−	0.382683i
$406$		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
$413$		0			0
$414$		0			0
$415$	0.149316	−	0.360480i	0.149316	−	0.360480i
$416$	0.955410	−	0.955410i	0.955410	−	0.955410i
$417$		0			0
$418$		0			0
$419$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$419$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$420$		0			0
$421$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$421$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.831470	−	0.555570i	0.831470	−	0.555570i
$426$		0			0
$427$		0			0
$428$	0.636379	−	0.263597i	0.636379	−	0.263597i
$429$		0			0
$430$	−1.05826	−	2.55487i	−1.05826	−	2.55487i
$431$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$431$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$	−1.84776	+	1.84776i	−1.84776	+	1.84776i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$439$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$440$	0.899976	+	0.899976i	0.899976	+	0.899976i
$441$	−0.234633			−0.234633
$442$	−2.71223	+	1.81225i	−2.71223	+	1.81225i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	1.70711	−	0.707107i	1.70711	−	0.707107i
$446$		0			0
$447$		0			0
$448$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$449$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$449$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$450$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.248303	−	0.599456i	−0.248303	−	0.599456i
$455$			2.17958i			2.17958i
$456$		0			0
$457$	0.785695	+	0.785695i	0.785695	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$457$	0.785695	+	0.785695i	0.785695	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$458$	−2.35175			−2.35175
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−1.24830	+	3.01367i	−1.24830	+	3.01367i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	2.44863	−	2.44863i	2.44863	−	2.44863i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.53636	−	0.636379i	1.53636	−	0.636379i
$474$		0			0
$475$		0			0
$476$	1.63099	+	1.08979i	1.63099	+	1.08979i
$477$		0			0
$478$		0			0
$479$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$479$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.24830	+	1.24830i	−1.24830	+	1.24830i
$485$		0			0
$486$		0			0
$487$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$487$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$488$		0			0
$489$		0			0
$490$	0.360480	−	0.149316i	0.360480	−	0.149316i
$491$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$496$	−0.458804	+	0.190043i	−0.458804	+	0.190043i
$497$		−	2.05312i		−	2.05312i
$498$		0			0
$499$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$499$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$500$	0.675577	−	1.63099i	0.675577	−	1.63099i
$501$		0			0
$502$	−1.66294	+	1.66294i	−1.66294	+	1.66294i
$503$	−0.636379	+	1.53636i	−0.636379	+	1.53636i	0.195090	+	0.980785i	$0.437500\pi$
$503$	−0.636379	+	1.53636i	−0.636379	+	1.53636i	−0.831470	+	0.555570i	$0.812500\pi$
$504$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.44863	+	2.44863i	2.44863	+	2.44863i
$509$	2.00000			2.00000			1.00000			$0$
$509$	2.00000			2.00000			1.00000			$0$
$510$		0			0
$511$	−0.433546			−0.433546
$512$	−0.487064	−	0.487064i	−0.487064	−	0.487064i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.955410	+	2.30656i	−0.955410	+	2.30656i
$521$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$521$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$522$		0			0
$523$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$523$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$524$		0			0
$525$		0			0
$526$	−1.84776			−1.84776
$527$	−1.38704	+	0.275899i	−1.38704	+	0.275899i
$528$		0			0
$529$	0.707107	+	0.707107i	0.707107	+	0.707107i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.275899	−	0.275899i	0.275899	−	0.275899i
$536$		0			0
$537$		0			0
$538$	1.17588	+	0.487064i	1.17588	+	0.487064i
$539$	0.0897902	+	0.216773i	0.0897902	+	0.216773i
$540$		0			0
$541$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$541$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$542$		0			0
$543$		0			0
$544$	−0.382683	−	0.572726i	−0.382683	−	0.572726i
$545$		0			0
$546$		0			0
$547$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.831470	−	0.555570i	$-0.187500\pi$
$547$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.980785	−	0.195090i	$-0.0625000\pi$
$548$		0			0
$549$		0			0
$550$	1.53636	+	0.636379i	1.53636	+	0.636379i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.24830	−	3.01367i	1.24830	−	3.01367i
$555$		0			0
$556$		0			0
$557$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$557$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$558$	2.17273	−	0.899976i	2.17273	−	0.899976i
$559$	2.30656	+	2.30656i	2.30656	+	2.30656i
$560$	0.390181			0.390181
$561$		0			0
$562$		0			0
$563$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$563$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$564$		0			0
$565$		0			0
$566$	1.24830	+	3.01367i	1.24830	+	3.01367i
$567$	1.02656	+	0.425215i	1.02656	+	0.425215i
$568$	0.899976	−	2.17273i	0.899976	−	2.17273i
$569$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$569$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$570$		0			0
$571$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$571$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$572$	−3.19929	−	1.32519i	−3.19929	−	1.32519i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.05826	+	1.05826i	1.05826	+	1.05826i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.636379	+	1.53636i	0.636379	+	1.53636i
$579$		0			0
$580$		0			0
$581$	0.400544	−	0.165911i	0.400544	−	0.165911i
$582$		0			0
$583$		0			0
$584$	−0.458804	−	0.190043i	−0.458804	−	0.190043i
$585$	0.750661	−	1.81225i	0.750661	−	1.81225i
$586$	2.30656	−	2.30656i	2.30656	−	2.30656i
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$593$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$594$		0			0
$595$	1.08979	+	0.216773i	1.08979	+	0.216773i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$599$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$600$		0			0
$601$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$601$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$602$	1.17588	−	2.83881i	1.17588	−	2.83881i
$603$		0			0
$604$		0			0
$605$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$606$		0			0
$607$	0.149316	+	0.360480i	0.149316	+	0.360480i	0.980785	−	0.195090i	$-0.0625000\pi$
$607$	0.149316	+	0.360480i	0.149316	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.980785	−	1.46785i	−0.980785	−	1.46785i
$613$	−0.390181			−0.390181			−0.195090	−	0.980785i	$-0.562500\pi$
$613$	−0.390181			−0.390181			−0.195090	+	0.980785i	$0.562500\pi$
$614$	1.30656	+	1.30656i	1.30656	+	1.30656i
$615$		0			0
$616$			1.41421i			1.41421i
$617$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$617$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$618$		0			0
$619$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$619$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$620$	−1.76537	+	1.76537i	−1.76537	+	1.76537i
$621$		0			0
$622$	1.17588	−	2.83881i	1.17588	−	2.83881i
$623$	1.89684	+	0.785695i	1.89684	+	0.785695i
$624$		0			0
$625$		−	1.00000i		−	1.00000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−1.84776			−1.84776
$631$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$631$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.81225	+	0.750661i	1.81225	+	0.750661i
$636$		0			0
$637$	−0.325446	+	0.325446i	−0.325446	+	0.325446i
$638$		0			0
$639$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$640$	−1.66294	−	0.688812i	−1.66294	−	0.688812i
$641$	0.541196	+	1.30656i	0.541196	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$641$	0.541196	+	1.30656i	0.541196	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$642$		0			0
$643$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$643$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.899976	+	0.899976i	0.899976	+	0.899976i
$649$		0			0
$650$			3.26197i			3.26197i
$651$		0			0
$652$		0			0
$653$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$653$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.360480	+	0.149316i	0.360480	+	0.149316i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$661$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$662$	1.27276			1.27276
$663$		0			0
$664$	0.496606			0.496606
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.750661	+	1.81225i	0.750661	+	1.81225i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.636379	+	1.53636i	−0.636379	+	1.53636i	0.195090	+	0.980785i	$0.437500\pi$
$673$	−0.636379	+	1.53636i	−0.636379	+	1.53636i	−0.831470	+	0.555570i	$0.812500\pi$
$674$	−0.599456	−	0.248303i	−0.599456	−	0.248303i
$675$		0			0
$676$		−	5.02734i		−	5.02734i
$677$	−1.53636	+	0.636379i	−1.53636	+	0.636379i	−0.980785	−	0.195090i	$-0.937500\pi$
$677$	−1.53636	+	0.636379i	−1.53636	+	0.636379i	−0.555570	+	0.831470i	$0.687500\pi$
$678$		0			0
$679$		0			0
$680$	1.05826	+	0.707107i	1.05826	+	0.707107i
$681$		0			0
$682$	−1.66294	−	1.66294i	−1.66294	−	1.66294i
$683$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$683$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$684$		0			0
$685$		0			0
$686$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$687$		0			0
$688$	0.412913	−	0.412913i	0.412913	−	0.412913i
$689$		0			0
$690$		0			0
$691$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$691$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$692$	−0.750661	−	1.81225i	−0.750661	−	1.81225i
$693$		−	1.11114i		−	1.11114i
$694$	−2.55487	+	1.05826i	−2.55487	+	1.05826i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.81225	−	0.750661i	1.81225	−	0.750661i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	0.572726	−	1.38268i	0.572726	−	1.38268i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$709$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$710$		−	3.07271i		−	3.07271i
$711$		0			0
$712$	1.66294	+	1.66294i	1.66294	+	1.66294i
$713$		0			0
$714$		0			0
$715$	−1.96157			−1.96157
$716$	−2.30656	−	2.30656i	−2.30656	−	2.30656i
$717$		0			0
$718$		0			0
$719$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$719$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$720$	−0.324423	−	0.134381i	−0.324423	−	0.134381i
$721$		0			0
$722$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$723$		0			0
$724$	0.955410	−	2.30656i	0.955410	−	2.30656i
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	−2.56292	+	1.06159i	−2.56292	+	1.06159i
$729$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$730$	−0.648847			−0.648847
$731$	1.38268	−	0.923880i	1.38268	−	0.923880i
$732$		0			0
$733$	0.275899	+	0.275899i	0.275899	+	0.275899i	0.831470	−	0.555570i	$-0.187500\pi$
$733$	0.275899	+	0.275899i	0.275899	+	0.275899i	−0.555570	+	0.831470i	$0.687500\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.555570	−	0.831470i	$-0.687500\pi$
$743$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.195090	−	0.980785i	$-0.562500\pi$
$744$		0			0
$745$		0			0
$746$	1.95541	+	1.95541i	1.95541	+	1.95541i
$747$	−0.390181			−0.390181
$748$	−0.980785	+	1.46785i	−0.980785	+	1.46785i
$749$	0.433546			0.433546
$750$		0			0
$751$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$751$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	1.17588	−	2.83881i	1.17588	−	2.83881i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$	−3.26197			−3.26197
$765$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$	0.707107	+	1.70711i	0.707107	+	1.70711i
$771$		0			0
$772$	−1.12344	+	2.71223i	−1.12344	+	2.71223i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$	−1.95541	+	1.95541i	−1.95541	+	1.95541i
$775$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	1.84776			1.84776
$782$		0			0
$783$		0			0
$784$	0.0582601	+	0.0582601i	0.0582601	+	0.0582601i
$785$		0			0
$786$		0			0
$787$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	−0.831470	−	0.555570i	$-0.812500\pi$
$787$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	0.195090	−	0.980785i	$-0.437500\pi$
$788$	0.636379	+	0.263597i	0.636379	+	0.263597i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.487064	−	1.17588i	0.487064	−	1.17588i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.24830	−	0.517064i	1.24830	−	0.517064i
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$	−0.688812			−0.688812
$801$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$802$		0			0
$803$		−	0.390181i		−	0.390181i
$804$		0			0
$805$		0			0
$806$	1.76537	−	4.26197i	1.76537	−	4.26197i
$807$		0			0
$808$		0			0
$809$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$809$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$810$	1.53636	+	0.636379i	1.53636	+	0.636379i
$811$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$811$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	2.01367	−	0.834089i	2.01367	−	0.834089i
$820$		0			0
$821$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$821$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$822$		0			0
$823$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$823$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.831470	−	0.555570i	$-0.187500\pi$
$827$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.195090	+	0.980785i	$0.437500\pi$
$828$		0			0
$829$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$829$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$830$	0.599456	−	0.248303i	0.599456	−	0.248303i
$831$		0			0
$832$	2.93570			2.93570
$833$	0.130355	+	0.195090i	0.130355	+	0.195090i
$834$		0			0
$835$	0.785695	+	0.785695i	0.785695	+	0.785695i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$839$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$840$		0			0
$841$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$842$	2.17273	−	2.17273i	2.17273	−	2.17273i
$843$		0			0
$844$		0			0
$845$	−1.08979	−	2.63099i	−1.08979	−	2.63099i
$846$		0			0
$847$	−1.02656	+	0.425215i	−1.02656	+	0.425215i
$848$		0			0
$849$		0			0
$850$	1.63099	+	0.324423i	1.63099	+	0.324423i
$851$		0			0
$852$		0			0
$853$	−1.53636	+	0.636379i	−1.53636	+	0.636379i	−0.980785	−	0.195090i	$-0.937500\pi$
$853$	−1.53636	+	0.636379i	−1.53636	+	0.636379i	−0.555570	+	0.831470i	$0.687500\pi$
$854$		0			0
$855$		0			0
$856$	0.458804	+	0.190043i	0.458804	+	0.190043i
$857$	0.425215	−	1.02656i	0.425215	−	1.02656i	−0.555570	−	0.831470i	$-0.687500\pi$
$857$	0.425215	−	1.02656i	0.425215	−	1.02656i	0.980785	−	0.195090i	$-0.0625000\pi$
$858$		0			0
$859$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$859$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$860$	1.12344	−	2.71223i	1.12344	−	2.71223i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$866$		0			0
$867$		0			0
$868$	−2.77408			−2.77408
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.785695	−	0.785695i	0.785695	−	0.785695i
$876$		0			0
$877$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$877$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	0.831470	+	0.555570i	$0.187500\pi$
$878$		0			0
$879$		0			0
$880$			0.351153i			0.351153i
$881$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$881$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$882$	−0.275899	−	0.275899i	−0.275899	−	0.275899i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−3.39635	−	0.675577i	−3.39635	−	0.675577i
$885$		0			0
$886$		0			0
$887$	0.360480	−	0.149316i	0.360480	−	0.149316i	−0.195090	−	0.980785i	$-0.562500\pi$
$887$	0.360480	−	0.149316i	0.360480	−	0.149316i	0.555570	+	0.831470i	$0.312500\pi$
$888$		0			0
$889$	0.834089	+	2.01367i	0.834089	+	2.01367i
$890$	2.83881	+	1.17588i	2.83881	+	1.17588i
$891$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$896$	−0.765367	−	1.84776i	−0.765367	−	1.84776i
$897$		0			0
$898$	−2.17273	+	0.899976i	−2.17273	+	0.899976i
$899$		0			0
$900$	−1.76537			−1.76537
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.41421i		−	1.41421i
$906$		0			0
$907$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$907$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$908$	0.263597	−	0.636379i	0.263597	−	0.636379i
$909$		0			0
$910$	−2.56292	+	2.56292i	−2.56292	+	2.56292i
$911$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$911$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$912$		0			0
$913$	0.149316	+	0.360480i	0.149316	+	0.360480i
$914$			1.84776i			1.84776i
$915$		0			0
$916$	−1.76537	−	1.76537i	−1.76537	−	1.76537i
$917$		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
$923$	1.38704	+	3.34861i	1.38704	+	3.34861i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$929$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$930$		0			0
$931$		0			0
$932$	−3.19929	+	1.32519i	−3.19929	+	1.32519i
$933$		0			0
$934$		0			0
$935$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$936$	2.49661			2.49661
$937$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	0.195090	−	0.980785i	$-0.437500\pi$
$937$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	−0.980785	+	0.195090i	$0.937500\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$941$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	2.55487	+	1.05826i	2.55487	+	1.05826i
$947$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$947$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$948$		0			0
$949$	0.707107	−	0.292893i	0.707107	−	0.292893i
$950$		0			0
$951$		0			0
$952$	0.275899	+	1.38704i	0.275899	+	1.38704i
$953$	1.11114			1.11114			0.555570	−	0.831470i	$-0.312500\pi$
$953$	1.11114			1.11114			0.555570	+	0.831470i	$0.312500\pi$
$954$		0			0
$955$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.707107	−	0.707107i	0.707107	−	0.707107i
$962$		0			0
$963$	−0.360480	−	0.149316i	−0.360480	−	0.149316i
$964$		0			0
$965$			1.66294i			1.66294i
$966$		0			0
$967$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	0.195090	−	0.980785i	$-0.437500\pi$
$967$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	−0.980785	+	0.195090i	$0.937500\pi$
$968$	−1.27276			−1.27276
$969$		0			0
$970$		0			0
$971$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$971$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$980$	0.382683	+	0.158513i	0.382683	+	0.158513i
$981$		0			0
$982$		0			0
$983$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$983$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$984$		0			0
$985$	0.390181			0.390181
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		−	1.66294i		−	1.66294i
$991$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$991$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$992$	0.899976	+	0.372782i	0.899976	+	0.372782i
$993$		0			0
$994$	2.41421	−	2.41421i	2.41421	−	2.41421i
$995$	0.541196	−	0.541196i	0.541196	−	0.541196i
$996$		0			0
$997$	1.81225	+	0.750661i	1.81225	+	0.750661i	0.980785	+	0.195090i	$0.0625000\pi$
$997$	1.81225	+	0.750661i	1.81225	+	0.750661i	0.831470	+	0.555570i	$0.187500\pi$
$998$	−0.487064	−	1.17588i	−0.487064	−	1.17588i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	935.1.y.a.274.4	yes	16
5.4	even	2	inner	935.1.y.a.274.1	✓	16
11.10	odd	2	inner	935.1.y.a.274.1	✓	16
17.9	even	8	inner	935.1.y.a.604.4	yes	16
55.54	odd	2	CM	935.1.y.a.274.4	yes	16
85.9	even	8	inner	935.1.y.a.604.1	yes	16
187.43	odd	8	inner	935.1.y.a.604.1	yes	16
935.604	odd	8	inner	935.1.y.a.604.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
935.1.y.a.274.1	✓	16	5.4	even	2	inner
935.1.y.a.274.1	✓	16	11.10	odd	2	inner
935.1.y.a.274.4	yes	16	1.1	even	1	trivial
935.1.y.a.274.4	yes	16	55.54	odd	2	CM
935.1.y.a.604.1	yes	16	85.9	even	8	inner
935.1.y.a.604.1	yes	16	187.43	odd	8	inner
935.1.y.a.604.4	yes	16	17.9	even	8	inner
935.1.y.a.604.4	yes	16	935.604	odd	8	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.y (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.17588 + 1.17588i) q^{2} +1.76537i q^{4} +(0.382683 + 0.923880i) q^{5} +(-0.425215 + 1.02656i) q^{7} +(-0.899976 + 0.899976i) q^{8} +(0.707107 - 0.707107i) q^{9} +O(q^{10})\)	\(q+(1.17588 + 1.17588i) q^{2} +1.76537i q^{4} +(0.382683 + 0.923880i) q^{5} +(-0.425215 + 1.02656i) q^{7} +(-0.899976 + 0.899976i) q^{8} +(0.707107 - 0.707107i) q^{9} +(-0.636379 + 1.53636i) q^{10} +(-0.923880 - 0.382683i) q^{11} -1.96157i q^{13} +(-1.70711 + 0.707107i) q^{14} -0.351153 q^{16} +(-0.980785 - 0.195090i) q^{17} +1.66294 q^{18} +(-1.63099 + 0.675577i) q^{20} +(-0.636379 - 1.53636i) q^{22} +(-0.707107 + 0.707107i) q^{25} +(2.30656 - 2.30656i) q^{26} +(-1.81225 - 0.750661i) q^{28} +(1.30656 - 0.541196i) q^{31} +(0.487064 + 0.487064i) q^{32} +(-0.923880 - 1.38268i) q^{34} -1.11114 q^{35} +(1.24830 + 1.24830i) q^{36} +(-1.17588 - 0.487064i) q^{40} +(-1.17588 + 1.17588i) q^{43} +(0.675577 - 1.63099i) q^{44} +(0.923880 + 0.382683i) q^{45} +(-0.165911 - 0.165911i) q^{49} -1.66294 q^{50} +3.46289 q^{52} -1.00000i q^{55} +(-0.541196 - 1.30656i) q^{56} +(2.17273 + 0.899976i) q^{62} +(0.425215 + 1.02656i) q^{63} +1.49661i q^{64} +(1.81225 - 0.750661i) q^{65} +(0.344406 - 1.73145i) q^{68} +(-1.30656 - 1.30656i) q^{70} +(-1.70711 + 0.707107i) q^{71} +1.27276i q^{72} +(0.149316 + 0.360480i) q^{73} +(0.785695 - 0.785695i) q^{77} +(-0.134381 - 0.324423i) q^{80} -1.00000i q^{81} +(-0.275899 - 0.275899i) q^{83} +(-0.195090 - 0.980785i) q^{85} -2.76537 q^{86} +(1.17588 - 0.487064i) q^{88} -1.84776i q^{89} +(0.636379 + 1.53636i) q^{90} +(2.01367 + 0.834089i) q^{91} -0.390181i q^{98} +(-0.923880 + 0.382683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{14} - 16 q^{16} + 16 q^{26} + 16 q^{44} - 16 q^{71} - 16 q^{80} - 32 q^{86}+O(q^{100}) \) 16 * q - 16 * q^14 - 16 * q^16 + 16 * q^26 + 16 * q^44 - 16 * q^71 - 16 * q^80 - 32 * q^86

Introduction

L-functions

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