LMFDB - Newform orbit 960.2.w.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(127,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("960.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	960.w (of order $4$ , degree $2$ , not 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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64$ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$ .

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-\beta_{5}$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

127.1

	−	3.16053i
		2.16053i
		0.692297i
	−	1.69230i
		3.16053i
	−	2.16053i
	−	0.692297i
		1.69230i

−0.707107

−

0.707107i

−2.23483

0.0743018i

3.16053

−

3.16053i

1.00000i

127.2

−0.707107

−

0.707107i

1.52773

1.63280i

−2.16053

2.16053i

1.00000i

127.3

0.707107

0.707107i

−0.489528

2.18183i

−0.692297

0.692297i

1.00000i

127.4

0.707107

0.707107i

1.19663

−

1.88893i

1.69230

−

1.69230i

1.00000i

703.1

−0.707107

0.707107i

−2.23483

−

0.0743018i

3.16053

3.16053i

−

1.00000i

703.2

−0.707107

0.707107i

1.52773

−

1.63280i

−2.16053

−

2.16053i

−

1.00000i

703.3

0.707107

−

0.707107i

−0.489528

−

2.18183i

−0.692297

−

0.692297i

−

1.00000i

703.4

0.707107

−

0.707107i

1.19663

1.88893i

1.69230

1.69230i

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.2.w.f		8
4.b	odd	2	1		960.2.w.e		8
5.c	odd	4	1		960.2.w.e		8
8.b	even	2	1		480.2.w.d	yes	8
8.d	odd	2	1		480.2.w.c	✓	8
20.e	even	4	1	inner	960.2.w.f		8
24.f	even	2	1		1440.2.x.q		8
24.h	odd	2	1		1440.2.x.r		8
40.e	odd	2	1		2400.2.w.j		8
40.f	even	2	1		2400.2.w.i		8
40.i	odd	4	1		480.2.w.c	✓	8
40.i	odd	4	1		2400.2.w.j		8
40.k	even	4	1		480.2.w.d	yes	8
40.k	even	4	1		2400.2.w.i		8
120.q	odd	4	1		1440.2.x.r		8
120.w	even	4	1		1440.2.x.q		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
480.2.w.c	✓	8	8.d	odd	2	1
480.2.w.c	✓	8	40.i	odd	4	1
480.2.w.d	yes	8	8.b	even	2	1
480.2.w.d	yes	8	40.k	even	4	1
960.2.w.e		8	4.b	odd	2	1
960.2.w.e		8	5.c	odd	4	1
960.2.w.f		8	1.a	even	1	1	trivial
960.2.w.f		8	20.e	even	4	1	inner
1440.2.x.q		8	24.f	even	2	1
1440.2.x.q		8	120.w	even	4	1
1440.2.x.r		8	24.h	odd	2	1
1440.2.x.r		8	120.q	odd	4	1
2400.2.w.i		8	40.f	even	2	1
2400.2.w.i		8	40.k	even	4	1
2400.2.w.j		8	40.e	odd	2	1
2400.2.w.j		8	40.i	odd	4	1

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} + 24T_{7}^{5} + 132T_{7}^{4} - 320T_{7}^{3} + 512T_{7}^{2} + 1024T_{7} + 1024$ T7^8 - 4*T7^7 + 8*T7^6 + 24*T7^5 + 132*T7^4 - 320*T7^3 + 512*T7^2 + 1024*T7 + 1024 acting on $S_{2}^{\mathrm{new}}(960, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$(T^{4} + 1)^{2}$ (T^4 + 1)^2
$5$	$T^{8} + 2 T^{6} + \cdots + 625$ T^8 + 2T^6 + 16T^5 + 2T^4 + 80T^3 + 50*T^2 + 625
$7$	$T^{8} - 4 T^{7} + \cdots + 1024$ T^8 - 4T^7 + 8T^6 + 24T^5 + 132T^4 - 320T^3 + 512T^2 + 1024*T + 1024
$11$	$T^{8} + 36 T^{6} + \cdots + 1024$ T^8 + 36T^6 + 388T^4 + 1408*T^2 + 1024
$13$	$T^{8} - 12 T^{7} + \cdots + 141376$ T^8 - 12T^7 + 72T^6 - 168T^5 + 852T^4 - 7392T^3 + 41472T^2 - 108288*T + 141376
$17$	$T^{8} + 4 T^{7} + \cdots + 64$ T^8 + 4T^7 + 8T^6 + 8T^5 + 340T^4 + 1472T^3 + 3200T^2 + 640*T + 64
$19$	$(T^{4} + 4 T^{3} + \cdots + 128)^{2}$ (T^4 + 4T^3 - 36T^2 + 128)^2
$23$	$T^{8} + 192 T^{5} + \cdots + 4096$ T^8 + 192T^5 + 2832T^4 + 9984T^3 + 18432T^2 + 12288*T + 4096
$29$	$T^{8} + 196 T^{6} + \cdots + 2458624$ T^8 + 196T^6 + 12868T^4 + 319872*T^2 + 2458624
$31$	$(T^{2} + 16)^{4}$ (T^2 + 16)^4
$37$	$T^{8} + 12 T^{7} + \cdots + 4426816$ T^8 + 12T^7 + 72T^6 + 360T^5 + 9684T^4 + 117600T^3 + 778752T^2 + 2625792*T + 4426816
$41$	$(T^{4} - 4 T^{3} + \cdots + 128)^{2}$ (T^4 - 4T^3 - 36T^2 + 128)^2
$43$	$T^{8} - 24 T^{7} + \cdots + 10240000$ T^8 - 24T^7 + 288T^6 - 1472T^5 + 6464T^4 - 66560T^3 + 819200T^2 - 4096000*T + 10240000
$47$	$T^{8} - 24 T^{7} + \cdots + 4096$ T^8 - 24T^7 + 288T^6 - 1248T^5 + 2832T^4 - 1536*T^3 + 4096
$53$	$T^{8} + 4 T^{7} + \cdots + 8111104$ T^8 + 4T^7 + 8T^6 - 24T^5 + 14468T^4 + 65856T^3 + 147968T^2 - 1549312*T + 8111104
$59$	$(T^{4} - 8 T^{3} + \cdots - 2848)^{2}$ (T^4 - 8T^3 - 98T^2 + 1136*T - 2848)^2
$61$	$(T^{4} + 12 T^{3} + \cdots - 1600)^{2}$ (T^4 + 12T^3 - 84T^2 - 1120*T - 1600)^2
$67$	$T^{8} - 24 T^{7} + \cdots + 36192256$ T^8 - 24T^7 + 288T^6 - 1344T^5 + 13632T^4 - 236544T^3 + 2654208T^2 - 13860864*T + 36192256
$71$	$(T^{2} + 32)^{4}$ (T^2 + 32)^4
$73$	$T^{8} + 16 T^{7} + \cdots + 795664$ T^8 + 16T^7 + 128T^6 + 288T^5 + 3080T^4 + 45376T^3 + 373248T^2 + 770688*T + 795664
$79$	$(T^{4} - 8 T^{3} + \cdots + 2048)^{2}$ (T^4 - 8T^3 - 144T^2 + 2048)^2
$83$	$T^{8} + 16 T^{7} + \cdots + 6885376$ T^8 + 16T^7 + 128T^6 - 1344T^5 + 27152T^4 + 234496T^3 + 1179648T^2 - 4030464*T + 6885376
$89$	$T^{8} + 584 T^{6} + \cdots + 154157056$ T^8 + 584T^6 + 106000T^4 + 7056384*T^2 + 154157056
$97$	$T^{8} - 32 T^{7} + \cdots + 258064$ T^8 - 32T^7 + 512T^6 - 4032T^5 + 16392T^4 - 8320T^3 + 2048T^2 - 32512*T + 258064
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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 $\F_{q}$

$f(q)$	$=$	$q + \beta_{3} q^{3} + \beta_{4} q^{5} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{7} - \beta_{5} q^{9} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{11} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + \cdots + 2) q^{13}+ \cdots + (\beta_{4} - \beta_{2}) q^{99}+O(q^{100})$ q + b3 * q^3 + b4 * q^5 + (b6 + b5 + b3 + b2 + 1) * q^7 - b5 * q^9 + (-b7 - b6 - b3 - b1) * q^11 + (-b7 + 2b5 + b4 + 3b1 + 2) * q^13 + (-b7 + 1) * q^15 + (b6 + b3 - b2) * q^17 + (b7 - b6 - b4 - b3 + b2 + b1 - 2) * q^19 + (b4 - b2) * q^21 + (b7 + b6 + b4 - 3b3 - b2 + b1) q^23 + (b6 + b5 - 3b3 + b2) q^25 + b1 * q^27 + (b7 + b6 + 4b5 + 2b4 + b3 + 2b2 + b1) q^29 - 4b5 q^31 + (b6 + b5 + b3 + b2 + 1) * q^33 + (-b7 - b6 - b3 - 5b1 - 4) q^35 + (2b7 + b6 + 2b5 + 2b4 - 3b3 - b2 + 2b1 - 2) q^37 + (-b7 + b6 + 2b3 - 2b1 - 2) * q^39 + (-b7 + b6 + b4 + b3 - b2 - b1 + 2) * q^41 + (2b7 - 2b5 + 2b4 + 4b3 + 2b1 + 2) q^43 + (b6 + b3) * q^45 + (b7 - b6 + 2b5 - b4 - b3 - b2 - 3b1 + 2) * q^47 + (b7 + b6 + 3b5 + b4 + 5b3 + b2 + 5b1) q^49 + (-2b5 - b4 - b3 - b2 - b1) q^51 + (b6 + b3 + b2 + 8b1) q^53 + (b7 + 5b5 + b4 + 5b1 - 1) * q^55 + (b7 - b6 + 2b5 + b4 - b3 + b2 + b1 - 2) q^57 + (-2b7 + 2b6 + b4 - 2b3 - b2 + 2b1 + 4) * q^59 + (-b7 + b6 + b4 + 5b3 - b2 - 5b1 - 2) * q^61 + (-b7 - b5 - b4 - b1 + 1) * q^63 + (-b6 + 2b5 + 2b4 - 9b3 - 3b2) * q^65 + (2b7 + 2b5 - 2b4 + 6b1 + 2) * q^67 + (-b7 - b6 + 2b5 - b4 - b3 - b2 - b1) q^69 + (4b3 + 4b1) * q^71 + (-2b7 - b5 + 2b4 - 2b1 - 1) q^73 + (4b5 + b4 - b3 - b2) q^75 + (-b7 - b6 - 4b5 - b4 - 9b3 + b2 - b1 + 4) * q^77 + (2b7 - 2b6 - 2b4 + 2b3 + 2b2 - 2b1) * q^79 - q^81 + (-3b7 + b6 - 3b4 + 5b3 - b2 - 3b1) * q^83 + (-b7 - b4 - 5b1 + 6) q^85 + (-2b7 - b6 + b5 + 2b4 - b3 - b2 - 2b1 + 1) q^87 + (b7 + b6 + 2b5 + b4 + 9b3 + b2 + 9b1) q^89 + (4b7 + 4b6 + 4b5 + 2b4 + 2b2) q^91 + 4b1 q^93 + (b7 - b6 - 6b5 - b4 + 3b3 - b2 + 5b1 - 6) q^95 + (2b7 - 3b5 + 2b4 + 2b1 + 3) * q^97 + (b4 - b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{7} + 12 q^{13} + 4 q^{15} - 4 q^{17} - 8 q^{19} - 4 q^{25} + 4 q^{33} - 32 q^{35} - 12 q^{37} - 24 q^{39} + 8 q^{41} + 24 q^{43} - 4 q^{45} + 24 q^{47} - 4 q^{53} - 4 q^{55} - 8 q^{57} + 16 q^{59}+ \cdots + 32 q^{97}+O(q^{100})$ 8 * q + 4 * q^7 + 12 * q^13 + 4 * q^15 - 4 * q^17 - 8 * q^19 - 4 * q^25 + 4 * q^33 - 32 * q^35 - 12 * q^37 - 24 * q^39 + 8 * q^41 + 24 * q^43 - 4 * q^45 + 24 * q^47 - 4 * q^53 - 4 * q^55 - 8 * q^57 + 16 * q^59 - 24 * q^61 + 4 * q^63 + 4 * q^65 + 24 * q^67 - 16 * q^73 + 32 * q^77 + 16 * q^79 - 8 * q^81 - 16 * q^83 + 44 * q^85 + 4 * q^87 - 40 * q^95 + 32 * q^97

$\beta_{1}$	$=$	$( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64$ (v^7 + 2v^6 + 18v^5 + 28v^4 + 89v^3 + 74v^2 + 104v - 16) / 64
$\beta_{2}$	$=$	$( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64$ (v^7 + 18v^5 + 8v^4 + 105v^3 + 72v^2 + 248*v + 64) / 64
$\beta_{3}$	$=$	$( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64$ (v^7 - 2v^6 + 18v^5 - 28v^4 + 89v^3 - 74v^2 + 104v + 16) / 64
$\beta_{4}$	$=$	$( \nu^{7} + 18\nu^{5} - 8\nu^{4} + 105\nu^{3} - 72\nu^{2} + 248\nu - 64 ) / 64$ (v^7 + 18v^5 - 8v^4 + 105v^3 - 72v^2 + 248*v - 64) / 64
$\beta_{5}$	$=$	$( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64$ (-3v^7 - 46v^5 - 179v^3 - 168v) / 64
$\beta_{6}$	$=$	$( \nu^{7} - 6\nu^{6} + 10\nu^{5} - 92\nu^{4} - 15\nu^{3} - 358\nu^{2} - 120\nu - 336 ) / 64$ (v^7 - 6v^6 + 10v^5 - 92v^4 - 15v^3 - 358v^2 - 120v - 336) / 64
$\beta_{7}$	$=$	$( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64$ (v^7 + 6v^6 + 10v^5 + 92v^4 - 15v^3 + 358v^2 - 120v + 336) / 64

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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	960.2.w.f

Level	$960$
Weight	$2$
Character orbit	960.w
Analytic conductor	$7.666$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

$\nu$	$=$	$( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 2$ (b7 + b6 + 2*b5 + b4 + b3 + b2 + b1) / 2
$\nu^{2}$	$=$	$( \beta_{7} - \beta_{6} + \beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta _1 - 10 ) / 2$ (b7 - b6 + b4 + 3b3 - b2 - 3b1 - 10) / 2
$\nu^{3}$	$=$	$( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} - 5\beta_{4} - 13\beta_{3} - 5\beta_{2} - 13\beta_1 ) / 2$ (-9b7 - 9b6 - 18b5 - 5b4 - 13b3 - 5b2 - 13*b1) / 2
$\nu^{4}$	$=$	$( -9\beta_{7} + 9\beta_{6} - 17\beta_{4} - 27\beta_{3} + 17\beta_{2} + 27\beta _1 + 74 ) / 2$ (-9b7 + 9b6 - 17b4 - 27b3 + 17b2 + 27b1 + 74) / 2
$\nu^{5}$	$=$	$( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} + 37\beta_{4} + 149\beta_{3} + 37\beta_{2} + 149\beta_1 ) / 2$ (81b7 + 81b6 + 178b5 + 37b4 + 149b3 + 37b2 + 149*b1) / 2
$\nu^{6}$	$=$	$( 89\beta_{7} - 89\beta_{6} + 201\beta_{4} + 235\beta_{3} - 201\beta_{2} - 235\beta _1 - 650 ) / 2$ (89b7 - 89b6 + 201b4 + 235b3 - 201b2 - 235b1 - 650) / 2
$\nu^{7}$	$=$	$( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} - 325\beta_{4} - 1565\beta_{3} - 325\beta_{2} - 1565\beta_1 ) / 2$ (-761b7 - 761b6 - 1810b5 - 325b4 - 1565b3 - 325b2 - 1565*b1) / 2

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 127.4
Significant digits:
Format: