LMFDB - Embedded newform 3960.1.b.a.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.653400.2

Embedding invariants

Embedding label			1979.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3960.1979
Dual form			3960.1.b.a.1979.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{5} -1.41421i q^{7} -1.00000 q^{8} -1.00000i q^{10} +(0.707107 + 0.707107i) q^{11} +1.41421i q^{13} +1.41421i q^{14} +1.00000 q^{16} -2.00000i q^{19} +1.00000i q^{20} +(-0.707107 - 0.707107i) q^{22} -2.00000i q^{23} -1.00000 q^{25} -1.41421i q^{26} -1.41421i q^{28} -1.00000 q^{32} +1.41421 q^{35} +1.41421 q^{37} +2.00000i q^{38} -1.00000i q^{40} +1.41421 q^{41} +(0.707107 + 0.707107i) q^{44} +2.00000i q^{46} -1.00000 q^{49} +1.00000 q^{50} +1.41421i q^{52} +(-0.707107 + 0.707107i) q^{55} +1.41421i q^{56} +1.41421i q^{59} +1.00000 q^{64} -1.41421 q^{65} -1.41421 q^{70} -1.41421 q^{74} -2.00000i q^{76} +(1.00000 - 1.00000i) q^{77} +1.00000i q^{80} -1.41421 q^{82} +(-0.707107 - 0.707107i) q^{88} -1.41421i q^{89} +2.00000 q^{91} -2.00000i q^{92} +2.00000 q^{95} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} - 4 q^{25} - 4 q^{32} - 4 q^{49} + 4 q^{50} + 4 q^{64} + 4 q^{77} + 8 q^{91} + 8 q^{95} + 4 q^{98}+O(q^{100})$ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 - 4 * q^25 - 4 * q^32 - 4 * q^49 + 4 * q^50 + 4 * q^64 + 4 * q^77 + 8 * q^91 + 8 * q^95 + 4 * q^98

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−1.00000			−1.00000
$2$	−1.00000			−1.00000
$3$		0			0
$3$		0			0
$4$	1.00000			1.00000
$5$			1.00000i			1.00000i
$5$			1.00000i			1.00000i
$6$		0			0
$7$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$7$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$8$	−1.00000			−1.00000
$9$		0			0
$10$		−	1.00000i		−	1.00000i
$11$	0.707107	+	0.707107i	0.707107	+	0.707107i
$11$	0.707107	+	0.707107i	0.707107	+	0.707107i
$12$		0			0
$13$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$13$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$14$			1.41421i			1.41421i
$15$		0			0
$16$	1.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$19$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$20$			1.00000i			1.00000i
$21$		0			0
$22$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$23$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$23$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$24$		0			0
$25$	−1.00000			−1.00000
$26$		−	1.41421i		−	1.41421i
$27$		0			0
$28$		−	1.41421i		−	1.41421i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$		0			0
$35$	1.41421			1.41421
$36$		0			0
$37$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$37$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$38$			2.00000i			2.00000i
$39$		0			0
$40$		−	1.00000i		−	1.00000i
$41$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$41$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	0.707107	+	0.707107i	0.707107	+	0.707107i
$45$		0			0
$46$			2.00000i			2.00000i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$	1.00000			1.00000
$51$		0			0
$52$			1.41421i			1.41421i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$56$			1.41421i			1.41421i
$57$		0			0
$58$		0			0
$59$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$59$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$	−1.41421			−1.41421
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$	−1.41421			−1.41421
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	−1.41421			−1.41421
$75$		0			0
$76$		−	2.00000i		−	2.00000i
$77$	1.00000	−	1.00000i	1.00000	−	1.00000i
$78$		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
$82$	−1.41421			−1.41421
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$89$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$89$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$90$		0			0
$91$	2.00000			2.00000
$92$		−	2.00000i		−	2.00000i
$93$		0			0
$94$		0			0
$95$	2.00000			2.00000
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	1.00000			1.00000
$99$		0			0
$100$	−1.00000			−1.00000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$103$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$104$		−	1.41421i		−	1.41421i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.707107	−	0.707107i	0.707107	−	0.707107i
$111$		0			0
$112$		−	1.41421i		−	1.41421i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	2.00000			2.00000
$116$		0			0
$117$		0			0
$118$		−	1.41421i		−	1.41421i
$119$		0			0
$120$		0			0
$121$			1.00000i			1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	1.00000i
$126$		0			0
$127$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$127$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$128$	−1.00000			−1.00000
$129$		0			0
$130$	1.41421			1.41421
$131$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$131$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$	−2.82843			−2.82843
$134$		0			0
$135$		0			0
$136$		0			0
$137$		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			$\pi$
$140$	1.41421			1.41421
$141$		0			0
$142$		0			0
$143$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.41421			1.41421
$149$		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	$0.5\pi$
$152$			2.00000i			2.00000i
$153$		0			0
$154$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$155$		0			0
$156$		0			0
$157$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$157$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		−	1.00000i		−	1.00000i
$161$	−2.82843			−2.82843
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	1.41421			1.41421
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.00000			2.00000			1.00000			$0$
$173$	2.00000			2.00000			1.00000			$0$
$174$		0			0
$175$			1.41421i			1.41421i
$176$	0.707107	+	0.707107i	0.707107	+	0.707107i
$177$		0			0
$178$			1.41421i			1.41421i
$179$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−2.00000			−2.00000
$183$		0			0
$184$			2.00000i			2.00000i
$185$			1.41421i			1.41421i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−2.00000			−2.00000
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−1.00000			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	1.00000			1.00000
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			1.41421i			1.41421i
$206$	−1.41421			−1.41421
$207$		0			0
$208$			1.41421i			1.41421i
$209$	1.41421	−	1.41421i	1.41421	−	1.41421i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$221$		0			0
$222$		0			0
$223$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$223$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$224$			1.41421i			1.41421i
$225$		0			0
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	−2.00000			−2.00000
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$			1.41421i			1.41421i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$241$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$242$		−	1.00000i		−	1.00000i
$243$		0			0
$244$		0			0
$245$		−	1.00000i		−	1.00000i
$246$		0			0
$247$	2.82843			2.82843
$248$		0			0
$249$		0			0
$250$			1.00000i			1.00000i
$251$			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
$253$	1.41421	−	1.41421i	1.41421	−	1.41421i
$254$		−	1.41421i		−	1.41421i
$255$		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
$259$		−	2.00000i		−	2.00000i
$260$	−1.41421			−1.41421
$261$		0			0
$262$	1.41421			1.41421
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$	2.82843			2.82843
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$276$		0			0
$277$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$277$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$278$		0			0
$279$		0			0
$280$	−1.41421			−1.41421
$281$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$281$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$	1.00000	−	1.00000i	1.00000	−	1.00000i
$287$		−	2.00000i		−	2.00000i
$288$		0			0
$289$	1.00000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−1.41421			−1.41421
$296$	−1.41421			−1.41421
$297$		0			0
$298$		0			0
$299$	2.82843			2.82843
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	2.00000i		−	2.00000i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	1.00000	−	1.00000i	1.00000	−	1.00000i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−1.41421			−1.41421
$315$		0			0
$316$		0			0
$317$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$317$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$318$		0			0
$319$		0			0
$320$			1.00000i			1.00000i
$321$		0			0
$322$	2.82843			2.82843
$323$		0			0
$324$		0			0
$325$		−	1.41421i		−	1.41421i
$326$		0			0
$327$		0			0
$328$	−1.41421			−1.41421
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	1.00000			1.00000
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−2.00000			−2.00000
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		−	1.41421i		−	1.41421i
$351$		0			0
$352$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		−	1.41421i		−	1.41421i
$357$		0			0
$358$			1.41421i			1.41421i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−3.00000			−3.00000
$362$		0			0
$363$		0			0
$364$	2.00000			2.00000
$365$		0			0
$366$		0			0
$367$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$367$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$368$		−	2.00000i		−	2.00000i
$369$		0			0
$370$		−	1.41421i		−	1.41421i
$371$		0			0
$372$		0			0
$373$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$373$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.00000			−2.00000			−1.00000			$\pi$
$379$	−2.00000			−2.00000			−1.00000			$\pi$
$380$	2.00000			2.00000
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	1.00000	+	1.00000i	1.00000	+	1.00000i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	1.00000			1.00000
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$397$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$401$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.00000	+	1.00000i	1.00000	+	1.00000i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		−	1.41421i		−	1.41421i
$411$		0			0
$412$	1.41421			1.41421
$413$	2.00000			2.00000
$414$		0			0
$415$		0			0
$416$		−	1.41421i		−	1.41421i
$417$		0			0
$418$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$419$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$419$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		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			0
$435$		0			0
$436$		0			0
$437$	−4.00000			−4.00000
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	0.707107	−	0.707107i	0.707107	−	0.707107i
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	1.41421			1.41421
$446$	−1.41421			−1.41421
$447$		0			0
$448$		−	1.41421i		−	1.41421i
$449$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$449$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$450$		0			0
$451$	1.00000	+	1.00000i	1.00000	+	1.00000i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.00000i			2.00000i
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$	2.00000			2.00000
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$463$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		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
$472$		−	1.41421i		−	1.41421i
$473$		0			0
$474$		0			0
$475$			2.00000i			2.00000i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$			2.00000i			2.00000i
$482$		−	2.00000i		−	2.00000i
$483$		0			0
$484$			1.00000i			1.00000i
$485$		0			0
$486$		0			0
$487$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$			1.00000i			1.00000i
$491$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$491$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$		0			0
$494$	−2.82843			−2.82843
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		−	1.00000i		−	1.00000i
$501$		0			0
$502$		−	1.41421i		−	1.41421i
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$		0			0
$506$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$507$		0			0
$508$			1.41421i			1.41421i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$			1.41421i			1.41421i
$516$		0			0
$517$		0			0
$518$			2.00000i			2.00000i
$519$		0			0
$520$	1.41421			1.41421
$521$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$521$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−1.41421			−1.41421
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.00000			−3.00000
$530$		0			0
$531$		0			0
$532$	−2.82843			−2.82843
$533$			2.00000i			2.00000i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$540$		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			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	0.707107	+	0.707107i	0.707107	+	0.707107i
$551$		0			0
$552$		0			0
$553$		0			0
$554$			1.41421i			1.41421i
$555$		0			0
$556$		0			0
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	1.41421			1.41421
$561$		0			0
$562$	1.41421			1.41421
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$569$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$571$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$572$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$573$		0			0
$574$			2.00000i			2.00000i
$575$			2.00000i			2.00000i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−1.00000			−1.00000
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$	1.41421			1.41421
$591$		0			0
$592$	1.41421			1.41421
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−2.82843			−2.82843
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$601$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.00000			−1.00000
$606$		0			0
$607$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$607$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$608$			2.00000i			2.00000i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$613$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	−2.00000			−2.00000			−1.00000			$\pi$
$619$	−2.00000			−2.00000			−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.00000			−2.00000
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$	1.41421			1.41421
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			2.00000i			2.00000i
$635$	−1.41421			−1.41421
$636$		0			0
$637$		−	1.41421i		−	1.41421i
$638$		0			0
$639$		0			0
$640$		−	1.00000i		−	1.00000i
$641$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−2.82843			−2.82843
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$650$			1.41421i			1.41421i
$651$		0			0
$652$		0			0
$653$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$653$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$654$		0			0
$655$		−	1.41421i		−	1.41421i
$656$	1.41421			1.41421
$657$		0			0
$658$		0			0
$659$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$659$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.82843i		−	2.82843i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	−1.00000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	2.00000			2.00000
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$			1.41421i			1.41421i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	2.82843i		−	2.82843i
$704$	0.707107	+	0.707107i	0.707107	+	0.707107i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$			1.41421i			1.41421i
$713$		0			0
$714$		0			0
$715$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$716$		−	1.41421i		−	1.41421i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		−	2.00000i		−	2.00000i
$722$	3.00000			3.00000
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$727$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$728$	−2.00000			−2.00000
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$733$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$734$	1.41421			1.41421
$735$		0			0
$736$			2.00000i			2.00000i
$737$		0			0
$738$		0			0
$739$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$739$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$740$			1.41421i			1.41421i
$741$		0			0
$742$		0			0
$743$	−2.00000			−2.00000			−1.00000			$\pi$
$743$	−2.00000			−2.00000			−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		−	1.41421i		−	1.41421i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		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
$757$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$757$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$758$	2.00000			2.00000
$759$		0			0
$760$	−2.00000			−2.00000
$761$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$761$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000			−2.00000
$768$		0			0
$769$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$769$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$770$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$771$		0			0
$772$		0			0
$773$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$773$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	2.82843i		−	2.82843i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$			1.41421i			1.41421i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.41421			1.41421
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			1.00000
$801$		0			0
$802$		−	1.41421i		−	1.41421i
$803$		0			0
$804$		0			0
$805$		−	2.82843i		−	2.82843i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$809$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$			1.41421i			1.41421i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$823$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$824$	−1.41421			−1.41421
$825$		0			0
$826$	−2.00000			−2.00000
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$			1.41421i			1.41421i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	1.41421	−	1.41421i	1.41421	−	1.41421i
$837$		0			0
$838$		−	1.41421i		−	1.41421i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1.00000i		−	1.00000i
$846$		0			0
$847$	1.41421			1.41421
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2.82843i		−	2.82843i
$852$		0			0
$853$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	2.00000			2.00000			1.00000			$0$
$859$	2.00000			2.00000			1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$864$		0			0
$865$			2.00000i			2.00000i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	4.00000			4.00000
$875$	−1.41421			−1.41421
$876$		0			0
$877$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$877$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$881$			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			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	2.00000			2.00000
$890$	−1.41421			−1.41421
$891$		0			0
$892$	1.41421			1.41421
$893$		0			0
$894$		0			0
$895$	1.41421			1.41421
$896$			1.41421i			1.41421i
$897$		0			0
$898$			1.41421i			1.41421i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		0			0
$910$		−	2.00000i		−	2.00000i
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.00000i			2.00000i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−2.00000			−2.00000
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.41421			−1.41421
$926$	1.41421			1.41421
$927$		0			0
$928$		0			0
$929$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$929$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$930$		0			0
$931$			2.00000i			2.00000i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		−	2.82843i		−	2.82843i
$944$			1.41421i			1.41421i
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		−	2.00000i		−	2.00000i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		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
$962$		−	2.00000i		−	2.00000i
$963$		0			0
$964$			2.00000i			2.00000i
$965$		0			0
$966$		0			0
$967$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$967$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$968$		−	1.00000i		−	1.00000i
$969$		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
$974$	1.41421			1.41421
$975$		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
$980$		−	1.00000i		−	1.00000i
$981$		0			0
$982$	−1.41421			−1.41421
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.82843			2.82843
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$997$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.1.b.a.1979.3	yes	4
3.2	odd	2		3960.1.b.b.1979.1	yes	4
5.4	even	2		3960.1.b.b.1979.2	yes	4
8.3	odd	2		3960.1.b.b.1979.2	yes	4
11.10	odd	2		3960.1.b.b.1979.4	yes	4
15.14	odd	2	inner	3960.1.b.a.1979.4	yes	4
24.11	even	2	inner	3960.1.b.a.1979.4	yes	4
33.32	even	2	inner	3960.1.b.a.1979.2	yes	4
40.19	odd	2	CM	3960.1.b.a.1979.3	yes	4
55.54	odd	2	inner	3960.1.b.a.1979.1	✓	4
88.43	even	2	inner	3960.1.b.a.1979.1	✓	4
120.59	even	2		3960.1.b.b.1979.1	yes	4
165.164	even	2		3960.1.b.b.1979.3	yes	4
264.131	odd	2		3960.1.b.b.1979.3	yes	4
440.219	even	2		3960.1.b.b.1979.4	yes	4
1320.659	odd	2	inner	3960.1.b.a.1979.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.1.b.a.1979.1	✓	4	55.54	odd	2	inner
3960.1.b.a.1979.1	✓	4	88.43	even	2	inner
3960.1.b.a.1979.2	yes	4	33.32	even	2	inner
3960.1.b.a.1979.2	yes	4	1320.659	odd	2	inner
3960.1.b.a.1979.3	yes	4	1.1	even	1	trivial
3960.1.b.a.1979.3	yes	4	40.19	odd	2	CM
3960.1.b.a.1979.4	yes	4	15.14	odd	2	inner
3960.1.b.a.1979.4	yes	4	24.11	even	2	inner
3960.1.b.b.1979.1	yes	4	3.2	odd	2
3960.1.b.b.1979.1	yes	4	120.59	even	2
3960.1.b.b.1979.2	yes	4	5.4	even	2
3960.1.b.b.1979.2	yes	4	8.3	odd	2
3960.1.b.b.1979.3	yes	4	165.164	even	2
3960.1.b.b.1979.3	yes	4	264.131	odd	2
3960.1.b.b.1979.4	yes	4	11.10	odd	2
3960.1.b.b.1979.4	yes	4	440.219	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3960.1.b.a.1979.3

Level	$3960$
Weight	$1$
Character	3960.1979
Analytic conductor	$1.976$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{4}$
CM discriminant	-40
Inner twists	$8$