LMFDB - Embedded newform 3960.1.b.c.1979.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,1,Mod(1979,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3960.1979");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3960.b (of order $2$, degree $1$, 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:	$1.97629745003$
Analytic rank:	$0$
Dimension:	$8$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.2483965440000.8

Embedding invariants

Embedding label			1979.3
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3960.1979
Dual form			3960.1.b.c.1979.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.382683 - 0.923880i) q^{2} +(-0.707107 + 0.707107i) q^{4} -1.00000 q^{5} -0.765367i q^{7} +(0.923880 + 0.382683i) q^{8} +(0.382683 + 0.923880i) q^{10} -1.00000i q^{11} +1.84776i q^{13} +(-0.707107 + 0.292893i) q^{14} -1.00000i q^{16} -0.765367i q^{17} +(0.707107 - 0.707107i) q^{20} +(-0.923880 + 0.382683i) q^{22} +1.00000 q^{25} +(1.70711 - 0.707107i) q^{26} +(0.541196 + 0.541196i) q^{28} -1.41421i q^{31} +(-0.923880 + 0.382683i) q^{32} +(-0.707107 + 0.292893i) q^{34} +0.765367i q^{35} +(-0.923880 - 0.382683i) q^{40} -0.765367 q^{43} +(0.707107 + 0.707107i) q^{44} +0.414214 q^{49} +(-0.382683 - 0.923880i) q^{50} +(-1.30656 - 1.30656i) q^{52} +1.00000i q^{55} +(0.292893 - 0.707107i) q^{56} -1.41421i q^{59} +(-1.30656 + 0.541196i) q^{62} +(0.707107 + 0.707107i) q^{64} -1.84776i q^{65} +(0.541196 + 0.541196i) q^{68} +(0.707107 - 0.292893i) q^{70} -1.41421 q^{71} -1.84776 q^{73} -0.765367 q^{77} +1.00000i q^{80} -1.84776i q^{83} +0.765367i q^{85} +(0.292893 + 0.707107i) q^{86} +(0.382683 - 0.923880i) q^{88} -2.00000i q^{89} +1.41421 q^{91} +(-0.158513 - 0.382683i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 8 q^{25} + 8 q^{26} - 8 q^{49} + 8 q^{56} + 8 q^{86}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^25 + 8 * q^26 - 8 * q^49 + 8 * q^56 + 8 * q^86

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$2$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$3$		0			0
$3$		0			0
$4$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$5$	−1.00000			−1.00000
$5$	−1.00000			−1.00000
$6$		0			0
$7$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$7$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$8$	0.923880	+	0.382683i	0.923880	+	0.382683i
$9$		0			0
$10$	0.382683	+	0.923880i	0.382683	+	0.923880i
$11$		−	1.00000i		−	1.00000i
$11$		−	1.00000i		−	1.00000i
$12$		0			0
$13$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$13$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$14$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$15$		0			0
$16$		−	1.00000i		−	1.00000i
$17$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$17$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.707107	−	0.707107i	0.707107	−	0.707107i
$21$		0			0
$22$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	1.00000			1.00000
$26$	1.70711	−	0.707107i	1.70711	−	0.707107i
$27$		0			0
$28$	0.541196	+	0.541196i	0.541196	+	0.541196i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$31$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$32$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$33$		0			0
$34$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$35$			0.765367i			0.765367i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$43$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$44$	0.707107	+	0.707107i	0.707107	+	0.707107i
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	0.414214			0.414214
$50$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$51$		0			0
$52$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			1.00000i			1.00000i
$56$	0.292893	−	0.707107i	0.292893	−	0.707107i
$57$		0			0
$58$		0			0
$59$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$		−	1.84776i		−	1.84776i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	0.541196	+	0.541196i	0.541196	+	0.541196i
$69$		0			0
$70$	0.707107	−	0.292893i	0.707107	−	0.292893i
$71$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$71$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$72$		0			0
$73$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$73$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.765367			−0.765367
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$			1.00000i			1.00000i
$81$		0			0
$82$		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
$85$			0.765367i			0.765367i
$86$	0.292893	+	0.707107i	0.292893	+	0.707107i
$87$		0			0
$88$	0.382683	−	0.923880i	0.382683	−	0.923880i
$89$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$89$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$90$		0			0
$91$	1.41421			1.41421
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$99$		0			0
$100$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$105$		0			0
$106$		0			0
$107$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$107$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.923880	−	0.382683i	0.923880	−	0.382683i
$111$		0			0
$112$	−0.765367			−0.765367
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$119$	−0.585786			−0.585786
$120$		0			0
$121$	−1.00000			−1.00000
$122$		0			0
$123$		0			0
$124$	1.00000	+	1.00000i	1.00000	+	1.00000i
$125$	−1.00000			−1.00000
$126$		0			0
$127$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$127$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$128$	0.382683	−	0.923880i	0.382683	−	0.923880i
$129$		0			0
$130$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.292893	−	0.707107i	0.292893	−	0.707107i
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$141$		0			0
$142$	0.541196	+	1.30656i	0.541196	+	1.30656i
$143$	1.84776			1.84776
$144$		0			0
$145$		0			0
$146$	0.707107	+	1.70711i	0.707107	+	1.70711i
$147$		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			0
$154$	0.292893	+	0.707107i	0.292893	+	0.707107i
$155$			1.41421i			1.41421i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$	0.923880	−	0.382683i	0.923880	−	0.382683i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$167$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$167$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	−2.41421			−2.41421
$170$	0.707107	−	0.292893i	0.707107	−	0.292893i
$171$		0			0
$172$	0.541196	−	0.541196i	0.541196	−	0.541196i
$173$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$173$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$174$		0			0
$175$		−	0.765367i		−	0.765367i
$176$	−1.00000			−1.00000
$177$		0			0
$178$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$181$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$182$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.765367			−0.765367
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$193$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$194$		0			0
$195$		0			0
$196$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$197$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$197$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$198$		0			0
$199$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$199$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$200$	0.923880	+	0.382683i	0.923880	+	0.382683i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.84776			1.84776
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	1.70711	−	0.707107i	1.70711	−	0.707107i
$215$	0.765367			0.765367
$216$		0			0
$217$	−1.08239			−1.08239
$218$		0			0
$219$		0			0
$220$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$221$	1.41421			1.41421
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	0.292893	+	0.707107i	0.292893	+	0.707107i
$225$		0			0
$226$		0			0
$227$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$227$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$228$		0			0
$229$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$229$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$233$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$234$		0			0
$235$		0			0
$236$	1.00000	+	1.00000i	1.00000	+	1.00000i
$237$		0			0
$238$	0.224171	+	0.541196i	0.224171	+	0.541196i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.382683	+	0.923880i	0.382683	+	0.923880i
$243$		0			0
$244$		0			0
$245$	−0.414214			−0.414214
$246$		0			0
$247$		0			0
$248$	0.541196	−	1.30656i	0.541196	−	1.30656i
$249$		0			0
$250$	0.382683	+	0.923880i	0.382683	+	0.923880i
$251$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$251$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$255$		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
$260$	1.30656	+	1.30656i	1.30656	+	1.30656i
$261$		0			0
$262$		0			0
$263$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$263$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$269$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−0.765367			−0.765367
$273$		0			0
$274$		0			0
$275$		−	1.00000i		−	1.00000i
$276$		0			0
$277$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$277$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$278$		0			0
$279$		0			0
$280$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$283$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$284$	1.00000	−	1.00000i	1.00000	−	1.00000i
$285$		0			0
$286$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$287$		0			0
$288$		0			0
$289$	0.414214			0.414214
$290$		0			0
$291$		0			0
$292$	1.30656	−	1.30656i	1.30656	−	1.30656i
$293$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$293$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$294$		0			0
$295$			1.41421i			1.41421i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$			0.585786i			0.585786i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$307$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$308$	0.541196	−	0.541196i	0.541196	−	0.541196i
$309$		0			0
$310$	1.30656	−	0.541196i	1.30656	−	0.541196i
$311$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$311$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		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			0
$325$			1.84776i			1.84776i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$331$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$332$	1.30656	+	1.30656i	1.30656	+	1.30656i
$333$		0			0
$334$	0.292893	+	0.707107i	0.292893	+	0.707107i
$335$		0			0
$336$		0			0
$337$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$337$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$338$	0.923880	+	2.23044i	0.923880	+	2.23044i
$339$		0			0
$340$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$341$	−1.41421			−1.41421
$342$		0			0
$343$		−	1.08239i		−	1.08239i
$344$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$345$		0			0
$346$	0.707107	+	1.70711i	0.707107	+	1.70711i
$347$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$347$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$351$		0			0
$352$	0.382683	+	0.923880i	0.382683	+	0.923880i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	1.41421			1.41421
$356$	1.41421	+	1.41421i	1.41421	+	1.41421i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$363$		0			0
$364$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$365$	1.84776			1.84776
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$373$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$374$	0.292893	+	0.707107i	0.292893	+	0.707107i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	0.765367			0.765367
$386$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$387$		0			0
$388$		0			0
$389$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$389$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	0.382683	+	0.158513i	0.382683	+	0.158513i
$393$		0			0
$394$	0.707107	+	1.70711i	0.707107	+	1.70711i
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$399$		0			0
$400$		−	1.00000i		−	1.00000i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	2.61313			2.61313
$404$		0			0
$405$		0			0
$406$		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			−1.08239
$414$		0			0
$415$			1.84776i			1.84776i
$416$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$421$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.765367i		−	0.765367i
$426$		0			0
$427$		0			0
$428$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$429$		0			0
$430$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	0.414214	+	1.00000i	0.414214	+	1.00000i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$441$		0			0
$442$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$			2.00000i			2.00000i
$446$		0			0
$447$		0			0
$448$	0.541196	−	0.541196i	0.541196	−	0.541196i
$449$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$449$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	0.707107	−	0.292893i	0.707107	−	0.292893i
$455$	−1.41421			−1.41421
$456$		0			0
$457$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$457$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$458$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	1.70711	−	0.707107i	1.70711	−	0.707107i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.541196	−	1.30656i	0.541196	−	1.30656i
$473$			0.765367i			0.765367i
$474$		0			0
$475$		0			0
$476$	0.414214	−	0.414214i	0.414214	−	0.414214i
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.707107	−	0.707107i	0.707107	−	0.707107i
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	0.158513	+	0.382683i	0.158513	+	0.382683i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.41421			−1.41421
$497$			1.08239i			1.08239i
$498$		0			0
$499$	−2.00000			−2.00000			−1.00000			$\pi$
$499$	−2.00000			−2.00000			−1.00000			$\pi$
$500$	0.707107	−	0.707107i	0.707107	−	0.707107i
$501$		0			0
$502$	1.30656	−	0.541196i	1.30656	−	0.541196i
$503$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$503$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.30656	+	1.30656i	1.30656	+	1.30656i
$509$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$509$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$			1.41421i			1.41421i
$512$	0.382683	+	0.923880i	0.382683	+	0.923880i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.707107	−	1.70711i	0.707107	−	1.70711i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$523$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$524$		0			0
$525$		0			0
$526$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$527$	−1.08239			−1.08239
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	1.84776i		−	1.84776i
$536$		0			0
$537$		0			0
$538$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$539$		−	0.414214i		−	0.414214i
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$	0.292893	+	0.707107i	0.292893	+	0.707107i
$545$		0			0
$546$		0			0
$547$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$547$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$548$		0			0
$549$		0			0
$550$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.707107	−	0.292893i	0.707107	−	0.292893i
$555$		0			0
$556$		0			0
$557$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$557$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$558$		0			0
$559$		−	1.41421i		−	1.41421i
$560$	0.765367			0.765367
$561$		0			0
$562$		0			0
$563$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$563$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$564$		0			0
$565$		0			0
$566$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$567$		0			0
$568$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$579$		0			0
$580$		0			0
$581$	−1.41421			−1.41421
$582$		0			0
$583$		0			0
$584$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$585$		0			0
$586$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$	1.30656	−	0.541196i	1.30656	−	0.541196i
$591$		0			0
$592$		0			0
$593$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$593$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$594$		0			0
$595$	0.585786			0.585786
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.00000			2.00000			1.00000			$0$
$599$	2.00000			2.00000			1.00000			$0$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	0.541196	−	0.224171i	0.541196	−	0.224171i
$603$		0			0
$604$		0			0
$605$	1.00000			1.00000
$606$		0			0
$607$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$607$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$613$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$614$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$615$		0			0
$616$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$619$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$620$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$621$		0			0
$622$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$623$	−1.53073			−1.53073
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1.84776i			1.84776i
$636$		0			0
$637$			0.765367i			0.765367i
$638$		0			0
$639$		0			0
$640$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−1.41421			−1.41421
$650$	1.70711	−	0.707107i	1.70711	−	0.707107i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$661$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$662$	0.541196	+	1.30656i	0.541196	+	1.30656i
$663$		0			0
$664$	0.707107	−	1.70711i	0.707107	−	1.70711i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.541196	−	0.541196i	0.541196	−	0.541196i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$673$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$674$	0.292893	+	0.707107i	0.292893	+	0.707107i
$675$		0			0
$676$	1.70711	−	1.70711i	1.70711	−	1.70711i
$677$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$677$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$		0			0
$680$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$681$		0			0
$682$	0.541196	+	1.30656i	0.541196	+	1.30656i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	−1.00000	+	0.414214i	−1.00000	+	0.414214i
$687$		0			0
$688$			0.765367i			0.765367i
$689$		0			0
$690$		0			0
$691$	2.00000			2.00000			1.00000			$0$
$691$	2.00000			2.00000			1.00000			$0$
$692$	1.30656	−	1.30656i	1.30656	−	1.30656i
$693$		0			0
$694$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.541196	+	0.541196i	0.541196	+	0.541196i
$701$		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
$709$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$709$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$710$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$711$		0			0
$712$	0.765367	−	1.84776i	0.765367	−	1.84776i
$713$		0			0
$714$		0			0
$715$	−1.84776			−1.84776
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$723$		0			0
$724$	1.00000	+	1.00000i	1.00000	+	1.00000i
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	1.30656	+	0.541196i	1.30656	+	0.541196i
$729$		0			0
$730$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$731$			0.585786i			0.585786i
$732$		0			0
$733$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$733$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$743$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$744$		0			0
$745$		0			0
$746$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$747$		0			0
$748$	0.541196	−	0.541196i	0.541196	−	0.541196i
$749$	1.41421			1.41421
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−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			0
$759$		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
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.61313			2.61313
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$771$		0			0
$772$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		−	1.41421i		−	1.41421i
$776$		0			0
$777$		0			0
$778$	0.541196	+	1.30656i	0.541196	+	1.30656i
$779$		0			0
$780$		0			0
$781$			1.41421i			1.41421i
$782$		0			0
$783$		0			0
$784$		−	0.414214i		−	0.414214i
$785$		0			0
$786$		0			0
$787$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$787$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$788$	1.30656	−	1.30656i	1.30656	−	1.30656i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.41421	+	1.41421i	1.41421	+	1.41421i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$801$		0			0
$802$		0			0
$803$			1.84776i			1.84776i
$804$		0			0
$805$		0			0
$806$	−1.00000	−	2.41421i	−1.00000	−	2.41421i
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$	0.414214	+	1.00000i	0.414214	+	1.00000i
$827$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$827$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$828$		0			0
$829$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$829$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$830$	1.70711	−	0.707107i	1.70711	−	0.707107i
$831$		0			0
$832$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$833$		−	0.317025i		−	0.317025i
$834$		0			0
$835$	0.765367			0.765367
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$839$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	1.30656	−	0.541196i	1.30656	−	0.541196i
$843$		0			0
$844$		0			0
$845$	2.41421			2.41421
$846$		0			0
$847$			0.765367i			0.765367i
$848$		0			0
$849$		0			0
$850$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$851$		0			0
$852$		0			0
$853$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$853$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$854$		0			0
$855$		0			0
$856$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$857$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$857$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$858$		0			0
$859$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$859$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$860$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	1.84776			1.84776
$866$		0			0
$867$		0			0
$868$	0.765367	−	0.765367i	0.765367	−	0.765367i
$869$		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
$877$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$877$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$878$		0			0
$879$		0			0
$880$	1.00000			1.00000
$881$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$881$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$885$		0			0
$886$		0			0
$887$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$887$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$888$		0			0
$889$	−1.41421			−1.41421
$890$	1.84776	−	0.765367i	1.84776	−	0.765367i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$897$		0			0
$898$	1.84776	−	0.765367i	1.84776	−	0.765367i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1.41421i			1.41421i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$909$		0			0
$910$	0.541196	+	1.30656i	0.541196	+	1.30656i
$911$	2.00000			2.00000			1.00000			$0$
$911$	2.00000			2.00000			1.00000			$0$
$912$		0			0
$913$	−1.84776			−1.84776
$914$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$915$		0			0
$916$	1.41421	+	1.41421i	1.41421	+	1.41421i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2.61313i		−	2.61313i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$930$		0			0
$931$		0			0
$932$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$933$		0			0
$934$		0			0
$935$	0.765367			0.765367
$936$		0			0
$937$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$937$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	−1.41421			−1.41421
$945$		0			0
$946$	0.707107	−	0.292893i	0.707107	−	0.292893i
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		−	3.41421i		−	3.41421i
$950$		0			0
$951$		0			0
$952$	−0.541196	−	0.224171i	−0.541196	−	0.224171i
$953$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$953$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.00000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.765367			−0.765367
$966$		0			0
$967$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$967$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$968$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		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
$979$	−2.00000			−2.00000
$980$	0.292893	−	0.292893i	0.292893	−	0.292893i
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	1.84776			1.84776
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$991$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$992$	0.541196	+	1.30656i	0.541196	+	1.30656i
$993$		0			0
$994$	1.00000	−	0.414214i	1.00000	−	0.414214i
$995$			2.00000i			2.00000i
$996$		0			0
$997$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$997$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$998$	0.765367	+	1.84776i	0.765367	+	1.84776i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.1.b.c.1979.3	✓	8
3.2	odd	2		3960.1.b.d.1979.6	yes	8
5.4	even	2	inner	3960.1.b.c.1979.6	yes	8
8.3	odd	2		3960.1.b.d.1979.5	yes	8
11.10	odd	2	inner	3960.1.b.c.1979.6	yes	8
15.14	odd	2		3960.1.b.d.1979.3	yes	8
24.11	even	2	inner	3960.1.b.c.1979.4	yes	8
33.32	even	2		3960.1.b.d.1979.3	yes	8
40.19	odd	2		3960.1.b.d.1979.4	yes	8
55.54	odd	2	CM	3960.1.b.c.1979.3	✓	8
88.43	even	2		3960.1.b.d.1979.4	yes	8
120.59	even	2	inner	3960.1.b.c.1979.5	yes	8
165.164	even	2		3960.1.b.d.1979.6	yes	8
264.131	odd	2	inner	3960.1.b.c.1979.5	yes	8
440.219	even	2		3960.1.b.d.1979.5	yes	8
1320.659	odd	2	inner	3960.1.b.c.1979.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.1.b.c.1979.3	✓	8	1.1	even	1	trivial
3960.1.b.c.1979.3	✓	8	55.54	odd	2	CM
3960.1.b.c.1979.4	yes	8	24.11	even	2	inner
3960.1.b.c.1979.4	yes	8	1320.659	odd	2	inner
3960.1.b.c.1979.5	yes	8	120.59	even	2	inner
3960.1.b.c.1979.5	yes	8	264.131	odd	2	inner
3960.1.b.c.1979.6	yes	8	5.4	even	2	inner
3960.1.b.c.1979.6	yes	8	11.10	odd	2	inner
3960.1.b.d.1979.3	yes	8	15.14	odd	2
3960.1.b.d.1979.3	yes	8	33.32	even	2
3960.1.b.d.1979.4	yes	8	40.19	odd	2
3960.1.b.d.1979.4	yes	8	88.43	even	2
3960.1.b.d.1979.5	yes	8	8.3	odd	2
3960.1.b.d.1979.5	yes	8	440.219	even	2
3960.1.b.d.1979.6	yes	8	3.2	odd	2
3960.1.b.d.1979.6	yes	8	165.164	even	2

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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3960.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 - 0.923880i) q^{2} +(-0.707107 + 0.707107i) q^{4} -1.00000 q^{5} -0.765367i q^{7} +(0.923880 + 0.382683i) q^{8} +(0.382683 + 0.923880i) q^{10} -1.00000i q^{11} +1.84776i q^{13} +(-0.707107 + 0.292893i) q^{14} -1.00000i q^{16} -0.765367i q^{17} +(0.707107 - 0.707107i) q^{20} +(-0.923880 + 0.382683i) q^{22} +1.00000 q^{25} +(1.70711 - 0.707107i) q^{26} +(0.541196 + 0.541196i) q^{28} -1.41421i q^{31} +(-0.923880 + 0.382683i) q^{32} +(-0.707107 + 0.292893i) q^{34} +0.765367i q^{35} +(-0.923880 - 0.382683i) q^{40} -0.765367 q^{43} +(0.707107 + 0.707107i) q^{44} +0.414214 q^{49} +(-0.382683 - 0.923880i) q^{50} +(-1.30656 - 1.30656i) q^{52} +1.00000i q^{55} +(0.292893 - 0.707107i) q^{56} -1.41421i q^{59} +(-1.30656 + 0.541196i) q^{62} +(0.707107 + 0.707107i) q^{64} -1.84776i q^{65} +(0.541196 + 0.541196i) q^{68} +(0.707107 - 0.292893i) q^{70} -1.41421 q^{71} -1.84776 q^{73} -0.765367 q^{77} +1.00000i q^{80} -1.84776i q^{83} +0.765367i q^{85} +(0.292893 + 0.707107i) q^{86} +(0.382683 - 0.923880i) q^{88} -2.00000i q^{89} +1.41421 q^{91} +(-0.158513 - 0.382683i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 8 q^{25} + 8 q^{26} - 8 q^{49} + 8 q^{56} + 8 q^{86}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^25 + 8 * q^26 - 8 * q^49 + 8 * q^56 + 8 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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