LMFDB - Newform orbit 2880.2.o.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2879,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2880.2879");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.o (of order $2$, degree $1$, 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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{5} - \beta_{9} q^{7}+O(q^{10}) $ q - b2 * q^5 - b9 * q^7	$ q - \beta_{2} q^{5} - \beta_{9} q^{7} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{11} + (\beta_{8} - \beta_{7}) q^{13} - \beta_{6} q^{17} - 2 \beta_{7} q^{19} + ( - \beta_{11} + 2 \beta_{5} + \cdots + \beta_{2}) q^{23}+ \cdots + (\beta_{10} + 2 \beta_{8} + \cdots + \beta_1) q^{97}+O(q^{100}) $ q - b2 * q^5 - b9 * q^7 + (-b4 - b3 + b2) * q^11 + (b8 - b7) * q^13 - b6 * q^17 - 2b7 q^19 + (-b11 + 2b5 + b3 + b2) q^23 + (b9 + b8 + b7 + b1 - 1) * q^25 + (2b11 + b3 + b2) q^29 + (-b10 + b8 - 2b7 - b1) q^31 + (-b11 + 2b5 + b4 - 2b3) * q^35 + (-b10 - b8 - b7 - b1) * q^37 + (-2b11 + b5) q^41 + (2b10 + 2b9 - 2b1 - 4) q^43 + (-2b5 + 2b3 + 2b2) q^47 + (-b10 - 2b9 + b1 - 1) q^49 + (3b6 + 2b4 - 2b3 + 2b2) * q^53 + (-b10 - b9 - b8 - 2b7 - b1 - 4) q^55 + (-2b6 + b4 - b3 + b2) q^59 + (2b10 - 2b1 + 2) * q^61 + (-2b6 - b5 + 2b4 + b3 - b2) * q^65 + (2b10 - 2b1 + 4) * q^67 + (-2b6 - 4b4 + 2b3 - 2b2) * q^71 + (b10 - 4b8 + b1) q^73 + (b6 - 2b4 + b3 - b2) q^77 + (-3b10 - b8 - 2b7 - 3b1) q^79 + (-2b11 - 4b5 - 2b3 - 2b2) * q^83 + (-b10 - b9 - 3b8 + b7 - 2) q^85 + (-2b11 - b5) q^89 + (-b10 + 3b8 - 4b7 - b1) * q^91 + (-2b11 - 2b6 - 2b5 + 4b4 + 2b3 - 2b2) * q^95 + (b10 + 2b8 + 4b7 + b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 8 q^{25} - 64 q^{43} - 4 q^{49} - 48 q^{55} + 8 q^{61} + 32 q^{67} - 20 q^{85}+O(q^{100}) $ 12 * q - 8 * q^25 - 64 * q^43 - 4 * q^49 - 48 * q^55 + 8 * q^61 + 32 * q^67 - 20 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4*x^9 - 3*x^8 + 4*x^7 + 8*x^6 + 8*x^5 - 12*x^4 - 32*x^3 + 64 :

$\beta_{1}$	$=$	$ ( - \nu^{11} + 12 \nu^{9} + 16 \nu^{8} + 11 \nu^{7} - 28 \nu^{6} - 44 \nu^{5} + 52 \nu^{4} + 100 \nu^{3} + \cdots - 288 ) / 80 $ (-v^11 + 12v^9 + 16v^8 + 11v^7 - 28v^6 - 44v^5 + 52v^4 + 100v^3 + 56v^2 - 96*v - 288) / 80
$\beta_{2}$	$=$	$ ( - 3 \nu^{11} + \nu^{10} + 22 \nu^{9} + 2 \nu^{8} - 19 \nu^{7} - 67 \nu^{6} - 22 \nu^{5} + 110 \nu^{4} + \cdots - 352 ) / 160 $ (-3v^11 + v^10 + 22v^9 + 2v^8 - 19v^7 - 67v^6 - 22v^5 + 110v^4 + 172v^3 - 40v^2 - 272v - 352) / 160
$\beta_{3}$	$=$	$ ( - 4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + \cdots - 32 ) / 160 $ (-4v^11 + 5v^10 + 8v^9 - 6v^8 + 4v^7 - 27v^6 + 4v^5 + 18v^4 + 60v^3 + 104v^2 + 16*v - 32) / 160
$\beta_{4}$	$=$	$ ( -\nu^{11} + 4\nu^{10} + \nu^{9} + 2\nu^{8} - 7\nu^{7} - 20\nu^{6} + \nu^{5} + 18\nu^{4} + 18\nu^{3} - 16\nu^{2} - 72\nu ) / 80 $ (-v^11 + 4v^10 + v^9 + 2v^8 - 7v^7 - 20v^6 + v^5 + 18v^4 + 18v^3 - 16v^2 - 72v) / 80
$\beta_{5}$	$=$	$ ( -\nu^{10} - \nu^{9} + 2\nu^{7} + 3\nu^{6} - 5\nu^{5} - 4\nu^{4} - 2\nu^{3} + 8\nu^{2} + 24\nu ) / 16 $ (-v^10 - v^9 + 2v^7 + 3v^6 - 5v^5 - 4v^4 - 2v^3 + 8v^2 + 24*v) / 16
$\beta_{6}$	$=$	$ ( - 3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} + 16 \nu^{8} + 29 \nu^{7} - 60 \nu^{6} - 102 \nu^{5} + \cdots - 320 ) / 160 $ (-3v^11 - 8v^10 + 18v^9 + 16v^8 + 29v^7 - 60v^6 - 102v^5 + 4v^4 + 24v^3 + 352v^2 + 64*v - 320) / 160
$\beta_{7}$	$=$	$ ( - \nu^{11} - 12 \nu^{10} - 20 \nu^{9} - 12 \nu^{8} + 35 \nu^{7} + 48 \nu^{6} + 36 \nu^{5} + \cdots + 448 ) / 160 $ (-v^11 - 12v^10 - 20v^9 - 12v^8 + 35v^7 + 48v^6 + 36v^5 - 136v^4 - 164v^3 - 48v^2 + 432v + 448) / 160
$\beta_{8}$	$=$	$ ( -3\nu^{11} - \nu^{10} + 4\nu^{8} + 5\nu^{7} - \nu^{6} - 12\nu^{5} - 8\nu^{4} - 12\nu^{3} + 36\nu^{2} + 16\nu + 64 ) / 40 $ (-3v^11 - v^10 + 4v^8 + 5v^7 - v^6 - 12v^5 - 8v^4 - 12v^3 + 36v^2 + 16v + 64) / 40
$\beta_{9}$	$=$	$ ( - 7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} + \cdots - 320 ) / 160 $ (-7v^11 - 12v^10 + 12v^9 + 44v^8 + 21v^7 - 40v^6 - 108v^5 - 24v^4 + 196v^3 + 208v^2 + 176*v - 320) / 160
$\beta_{10}$	$=$	$ ( 2 \nu^{11} + \nu^{10} + 2 \nu^{9} - 3 \nu^{8} - 4 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + 5 \nu^{4} + \cdots - 72 ) / 20 $ (2v^11 + v^10 + 2v^9 - 3v^8 - 4v^7 + 3v^6 - 2v^5 + 5v^4 + 2v^3 - 30v^2 - 12v - 72) / 20
$\beta_{11}$	$=$	$ ( - 11 \nu^{11} - 22 \nu^{10} - 30 \nu^{9} + 28 \nu^{8} + 85 \nu^{7} + 38 \nu^{6} - 94 \nu^{5} + \cdots + 448 ) / 160 $ (-11v^11 - 22v^10 - 30v^9 + 28v^8 + 85v^7 + 38v^6 - 94v^5 - 256v^4 - 64v^3 + 352v^2 + 352*v + 448) / 160

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{11} + \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} ) / 4 $ (-b11 + b9 + b7 + b5 + b4 + b3 - b2) / 4
$\nu^{2}$	$=$	$ ( -\beta_{10} - \beta_{8} - 2\beta_{7} + \beta_{6} - 2\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 ) / 4 $ (-b10 - b8 - 2b7 + b6 - 2b4 + b3 - b2 - b1) / 4
$\nu^{3}$	$=$	$ ( \beta_{11} + \beta_{9} - 2\beta_{8} - \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 4 ) / 4 $ (b11 + b9 - 2b8 - b7 - 2b6 + b5 - b4 + 3*b3 + b2 + 4) / 4
$\nu^{4}$	$=$	$ ( -3\beta_{11} - 3\beta_{10} - 2\beta_{9} + 6\beta_{5} - \beta_{3} - \beta_{2} + 3\beta _1 + 2 ) / 4 $ (-3b11 - 3b10 - 2b9 + 6b5 - b3 - b2 + 3*b1 + 2) / 4
$\nu^{5}$	$=$	$ ( -\beta_{11} - 4\beta_{10} + \beta_{9} - 4\beta_{8} + \beta_{7} - 7\beta_{5} - 7\beta_{4} + \beta_{3} - \beta_{2} - 8 ) / 4 $ (-b11 - 4b10 + b9 - 4b8 + b7 - 7b5 - 7b4 + b3 - b2 - 8) / 4
$\nu^{6}$	$=$	$ ( \beta_{10} + 5\beta_{8} - 6\beta_{7} - 5\beta_{6} - 14\beta_{4} + 3\beta_{3} - 3\beta_{2} + \beta_1 ) / 4 $ (b10 + 5b8 - 6b7 - 5b6 - 14b4 + 3b3 - 3b2 + b1) / 4
$\nu^{7}$	$=$	$ ( 3 \beta_{11} - 9 \beta_{9} - 2 \beta_{8} + 9 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} + \cdots + 4 ) / 4 $ (3b11 - 9b9 - 2b8 + 9b7 - 2b6 - b5 + b4 + 5b3 - b2 + 16*b1 + 4) / 4
$\nu^{8}$	$=$	$ ( -5\beta_{11} - 5\beta_{10} + 10\beta_{9} - 14\beta_{5} - 15\beta_{3} - 15\beta_{2} + 5\beta _1 - 2 ) / 4 $ (-5b11 - 5b10 + 10b9 - 14b5 - 15b3 - 15b2 + 5*b1 - 2) / 4
$\nu^{9}$	$=$	$ ( - 3 \beta_{11} + 12 \beta_{10} + 7 \beta_{9} + 16 \beta_{8} + 7 \beta_{7} + 4 \beta_{6} - 33 \beta_{5} + \cdots + 32 ) / 4 $ (-3b11 + 12b10 + 7b9 + 16b8 + 7b7 + 4b6 - 33b5 - 33b4 + 7*b3 + b2 + 32) / 4
$\nu^{10}$	$=$	$ ( 15\beta_{10} + 11\beta_{8} - 2\beta_{7} - 11\beta_{6} + 38\beta_{4} + 37\beta_{3} - 37\beta_{2} + 15\beta_1 ) / 4 $ (15b10 + 11b8 - 2b7 - 11b6 + 38b4 + 37b3 - 37b2 + 15b1) / 4
$\nu^{11}$	$=$	$ ( \beta_{11} + \beta_{9} - 50 \beta_{8} - \beta_{7} + 22 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} + \cdots + 100 ) / 4 $ (b11 + b9 - 50b8 - b7 + 22b6 - 7b5 + 7b4 - 21b3 - 23b2 + 8*b1 + 100) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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} $

2879.1

−0.0912546	+	1.41127i
−0.0912546	−	1.41127i
−0.760198	+	1.19252i
−0.760198	−	1.19252i
−1.35818	+	0.394157i
−1.35818	−	0.394157i
−0.394157	−	1.35818i
−0.394157	+	1.35818i
1.19252	+	0.760198i
1.19252	−	0.760198i
1.41127	+	0.0912546i
1.41127	−	0.0912546i

−2.20963

−

0.342849i

−2.64002

2879.2

−2.20963

0.342849i

−2.64002

2879.3

−1.24561

−

1.85700i

−0.864641

2879.4

−1.24561

1.85700i

−0.864641

2879.5

−0.256912

−

2.22126i

3.50466

2879.6

−0.256912

2.22126i

3.50466

2879.7

0.256912

−

2.22126i

3.50466

2879.8

0.256912

2.22126i

3.50466

2879.9

1.24561

−

1.85700i

−0.864641

2879.10

1.24561

1.85700i

−0.864641

2879.11

2.20963

−

0.342849i

−2.64002

2879.12

2.20963

0.342849i

−2.64002

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
20.d	odd	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2880.2.o.e		12
3.b	odd	2	1	inner	2880.2.o.e		12
4.b	odd	2	1		2880.2.o.f		12
5.b	even	2	1		2880.2.o.f		12
8.b	even	2	1		1440.2.o.b	yes	12
8.d	odd	2	1		1440.2.o.a	✓	12
12.b	even	2	1		2880.2.o.f		12
15.d	odd	2	1		2880.2.o.f		12
20.d	odd	2	1	inner	2880.2.o.e		12
24.f	even	2	1		1440.2.o.a	✓	12
24.h	odd	2	1		1440.2.o.b	yes	12
40.e	odd	2	1		1440.2.o.b	yes	12
40.f	even	2	1		1440.2.o.a	✓	12
40.i	odd	4	1		7200.2.h.l		12
40.i	odd	4	1		7200.2.h.m		12
40.k	even	4	1		7200.2.h.l		12
40.k	even	4	1		7200.2.h.m		12
60.h	even	2	1	inner	2880.2.o.e		12
120.i	odd	2	1		1440.2.o.a	✓	12
120.m	even	2	1		1440.2.o.b	yes	12
120.q	odd	4	1		7200.2.h.l		12
120.q	odd	4	1		7200.2.h.m		12
120.w	even	4	1		7200.2.h.l		12
120.w	even	4	1		7200.2.h.m		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1440.2.o.a	✓	12	8.d	odd	2	1
1440.2.o.a	✓	12	24.f	even	2	1
1440.2.o.a	✓	12	40.f	even	2	1
1440.2.o.a	✓	12	120.i	odd	2	1
1440.2.o.b	yes	12	8.b	even	2	1
1440.2.o.b	yes	12	24.h	odd	2	1
1440.2.o.b	yes	12	40.e	odd	2	1
1440.2.o.b	yes	12	120.m	even	2	1
2880.2.o.e		12	1.a	even	1	1	trivial
2880.2.o.e		12	3.b	odd	2	1	inner
2880.2.o.e		12	20.d	odd	2	1	inner
2880.2.o.e		12	60.h	even	2	1	inner
2880.2.o.f		12	4.b	odd	2	1
2880.2.o.f		12	5.b	even	2	1
2880.2.o.f		12	12.b	even	2	1
2880.2.o.f		12	15.d	odd	2	1
7200.2.h.l		12	40.i	odd	4	1
7200.2.h.l		12	40.k	even	4	1
7200.2.h.l		12	120.q	odd	4	1
7200.2.h.l		12	120.w	even	4	1
7200.2.h.m		12	40.i	odd	4	1
7200.2.h.m		12	40.k	even	4	1
7200.2.h.m		12	120.q	odd	4	1
7200.2.h.m		12	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2880, [\chi])$:

$ T_{7}^{3} - 10T_{7} - 8 $

$ T_{17}^{6} - 34T_{17}^{4} + 288T_{17}^{2} - 128 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 4 T^{10} + \cdots + 15625 $ T^12 + 4T^10 - 17T^8 - 152T^6 - 425T^4 + 2500*T^2 + 15625
$7$	$ (T^{3} - 10 T - 8)^{4} $ (T^3 - 10*T - 8)^4
$11$	$ (T^{6} - 28 T^{4} + \cdots - 512)^{2} $ (T^6 - 28T^4 + 228T^2 - 512)^2
$13$	$ (T^{6} + 32 T^{4} + \cdots + 16)^{2} $ (T^6 + 32T^4 + 52T^2 + 16)^2
$17$	$ (T^{6} - 34 T^{4} + \cdots - 128)^{2} $ (T^6 - 34T^4 + 288T^2 - 128)^2
$19$	$ (T^{6} + 80 T^{4} + \cdots + 4096)^{2} $ (T^6 + 80T^4 + 1600T^2 + 4096)^2
$23$	$ (T^{6} + 88 T^{4} + \cdots + 512)^{2} $ (T^6 + 88T^4 + 528T^2 + 512)^2
$29$	$ (T^{6} + 98 T^{4} + \cdots + 3200)^{2} $ (T^6 + 98T^4 + 2144T^2 + 3200)^2
$31$	$ (T^{6} + 88 T^{4} + \cdots + 6400)^{2} $ (T^6 + 88T^4 + 1424T^2 + 6400)^2
$37$	$ (T^{6} + 44 T^{4} + \cdots + 64)^{2} $ (T^6 + 44T^4 + 132T^2 + 64)^2
$41$	$ (T^{6} + 102 T^{4} + \cdots + 13448)^{2} $ (T^6 + 102T^4 + 2828T^2 + 13448)^2
$43$	$ (T^{3} + 16 T^{2} + \cdots - 512)^{4} $ (T^3 + 16T^2 + 8T - 512)^4
$47$	$ (T^{6} + 176 T^{4} + \cdots + 51200)^{2} $ (T^6 + 176T^4 + 5696T^2 + 51200)^2
$53$	$ (T^{6} - 306 T^{4} + \cdots - 991232)^{2} $ (T^6 - 306T^4 + 30464T^2 - 991232)^2
$59$	$ (T^{6} - 220 T^{4} + \cdots - 359552)^{2} $ (T^6 - 220T^4 + 15652T^2 - 359552)^2
$61$	$ (T^{3} - 2 T^{2} + \cdots + 328)^{4} $ (T^3 - 2T^2 - 100T + 328)^4
$67$	$ (T^{3} - 8 T^{2} + \cdots + 512)^{4} $ (T^3 - 8T^2 - 80T + 512)^4
$71$	$ (T^{6} - 256 T^{4} + \cdots - 204800)^{2} $ (T^6 - 256T^4 + 15616T^2 - 204800)^2
$73$	$ (T^{6} + 276 T^{4} + \cdots + 215296)^{2} $ (T^6 + 276T^4 + 15872T^2 + 215296)^2
$79$	$ (T^{6} + 344 T^{4} + \cdots + 215296)^{2} $ (T^6 + 344T^4 + 29584T^2 + 215296)^2
$83$	$ (T^{6} + 256 T^{4} + \cdots + 204800)^{2} $ (T^6 + 256T^4 + 15616T^2 + 204800)^2
$89$	$ (T^{6} + 118 T^{4} + \cdots + 200)^{2} $ (T^6 + 118T^4 + 1356T^2 + 200)^2
$97$	$ (T^{6} + 276 T^{4} + \cdots + 719104)^{2} $ (T^6 + 276T^4 + 24704T^2 + 719104)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{5} - \beta_{9} q^{7}+O(q^{10}) \) q - b2 * q^5 - b9 * q^7	\( q - \beta_{2} q^{5} - \beta_{9} q^{7} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{11} + (\beta_{8} - \beta_{7}) q^{13} - \beta_{6} q^{17} - 2 \beta_{7} q^{19} + ( - \beta_{11} + 2 \beta_{5} + \cdots + \beta_{2}) q^{23}+ \cdots + (\beta_{10} + 2 \beta_{8} + \cdots + \beta_1) q^{97}+O(q^{100}) \) q - b2 * q^5 - b9 * q^7 + (-b4 - b3 + b2) * q^11 + (b8 - b7) * q^13 - b6 * q^17 - 2b7 q^19 + (-b11 + 2b5 + b3 + b2) q^23 + (b9 + b8 + b7 + b1 - 1) * q^25 + (2b11 + b3 + b2) q^29 + (-b10 + b8 - 2b7 - b1) q^31 + (-b11 + 2b5 + b4 - 2b3) * q^35 + (-b10 - b8 - b7 - b1) * q^37 + (-2b11 + b5) q^41 + (2b10 + 2b9 - 2b1 - 4) q^43 + (-2b5 + 2b3 + 2b2) q^47 + (-b10 - 2b9 + b1 - 1) q^49 + (3b6 + 2b4 - 2b3 + 2b2) * q^53 + (-b10 - b9 - b8 - 2b7 - b1 - 4) q^55 + (-2b6 + b4 - b3 + b2) q^59 + (2b10 - 2b1 + 2) * q^61 + (-2b6 - b5 + 2b4 + b3 - b2) * q^65 + (2b10 - 2b1 + 4) * q^67 + (-2b6 - 4b4 + 2b3 - 2b2) * q^71 + (b10 - 4b8 + b1) q^73 + (b6 - 2b4 + b3 - b2) q^77 + (-3b10 - b8 - 2b7 - 3b1) q^79 + (-2b11 - 4b5 - 2b3 - 2b2) * q^83 + (-b10 - b9 - 3b8 + b7 - 2) q^85 + (-2b11 - b5) q^89 + (-b10 + 3b8 - 4b7 - b1) * q^91 + (-2b11 - 2b6 - 2b5 + 4b4 + 2b3 - 2b2) * q^95 + (b10 + 2b8 + 4b7 + b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 8 q^{25} - 64 q^{43} - 4 q^{49} - 48 q^{55} + 8 q^{61} + 32 q^{67} - 20 q^{85}+O(q^{100}) \) 12 * q - 8 * q^25 - 64 * q^43 - 4 * q^49 - 48 * q^55 + 8 * q^61 + 32 * q^67 - 20 * q^85

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	2880.2.o.e

Level	$2880$
Weight	$2$
Character orbit	2880.o
Analytic conductor	$22.997$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(22.9969157821\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{10} \)
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{11} + 12 \nu^{9} + 16 \nu^{8} + 11 \nu^{7} - 28 \nu^{6} - 44 \nu^{5} + 52 \nu^{4} + 100 \nu^{3} + \cdots - 288 ) / 80 \) (-v^11 + 12v^9 + 16v^8 + 11v^7 - 28v^6 - 44v^5 + 52v^4 + 100v^3 + 56v^2 - 96*v - 288) / 80
\(\beta_{2}\)	\(=\)	\( ( - 3 \nu^{11} + \nu^{10} + 22 \nu^{9} + 2 \nu^{8} - 19 \nu^{7} - 67 \nu^{6} - 22 \nu^{5} + 110 \nu^{4} + \cdots - 352 ) / 160 \) (-3v^11 + v^10 + 22v^9 + 2v^8 - 19v^7 - 67v^6 - 22v^5 + 110v^4 + 172v^3 - 40v^2 - 272v - 352) / 160
\(\beta_{3}\)	\(=\)	\( ( - 4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + \cdots - 32 ) / 160 \) (-4v^11 + 5v^10 + 8v^9 - 6v^8 + 4v^7 - 27v^6 + 4v^5 + 18v^4 + 60v^3 + 104v^2 + 16*v - 32) / 160
\(\beta_{4}\)	\(=\)	\( ( -\nu^{11} + 4\nu^{10} + \nu^{9} + 2\nu^{8} - 7\nu^{7} - 20\nu^{6} + \nu^{5} + 18\nu^{4} + 18\nu^{3} - 16\nu^{2} - 72\nu ) / 80 \) (-v^11 + 4v^10 + v^9 + 2v^8 - 7v^7 - 20v^6 + v^5 + 18v^4 + 18v^3 - 16v^2 - 72v) / 80
\(\beta_{5}\)	\(=\)	\( ( -\nu^{10} - \nu^{9} + 2\nu^{7} + 3\nu^{6} - 5\nu^{5} - 4\nu^{4} - 2\nu^{3} + 8\nu^{2} + 24\nu ) / 16 \) (-v^10 - v^9 + 2v^7 + 3v^6 - 5v^5 - 4v^4 - 2v^3 + 8v^2 + 24*v) / 16
\(\beta_{6}\)	\(=\)	\( ( - 3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} + 16 \nu^{8} + 29 \nu^{7} - 60 \nu^{6} - 102 \nu^{5} + \cdots - 320 ) / 160 \) (-3v^11 - 8v^10 + 18v^9 + 16v^8 + 29v^7 - 60v^6 - 102v^5 + 4v^4 + 24v^3 + 352v^2 + 64*v - 320) / 160
\(\beta_{7}\)	\(=\)	\( ( - \nu^{11} - 12 \nu^{10} - 20 \nu^{9} - 12 \nu^{8} + 35 \nu^{7} + 48 \nu^{6} + 36 \nu^{5} + \cdots + 448 ) / 160 \) (-v^11 - 12v^10 - 20v^9 - 12v^8 + 35v^7 + 48v^6 + 36v^5 - 136v^4 - 164v^3 - 48v^2 + 432v + 448) / 160
\(\beta_{8}\)	\(=\)	\( ( -3\nu^{11} - \nu^{10} + 4\nu^{8} + 5\nu^{7} - \nu^{6} - 12\nu^{5} - 8\nu^{4} - 12\nu^{3} + 36\nu^{2} + 16\nu + 64 ) / 40 \) (-3v^11 - v^10 + 4v^8 + 5v^7 - v^6 - 12v^5 - 8v^4 - 12v^3 + 36v^2 + 16v + 64) / 40
\(\beta_{9}\)	\(=\)	\( ( - 7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} + \cdots - 320 ) / 160 \) (-7v^11 - 12v^10 + 12v^9 + 44v^8 + 21v^7 - 40v^6 - 108v^5 - 24v^4 + 196v^3 + 208v^2 + 176*v - 320) / 160
\(\beta_{10}\)	\(=\)	\( ( 2 \nu^{11} + \nu^{10} + 2 \nu^{9} - 3 \nu^{8} - 4 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + 5 \nu^{4} + \cdots - 72 ) / 20 \) (2v^11 + v^10 + 2v^9 - 3v^8 - 4v^7 + 3v^6 - 2v^5 + 5v^4 + 2v^3 - 30v^2 - 12v - 72) / 20
\(\beta_{11}\)	\(=\)	\( ( - 11 \nu^{11} - 22 \nu^{10} - 30 \nu^{9} + 28 \nu^{8} + 85 \nu^{7} + 38 \nu^{6} - 94 \nu^{5} + \cdots + 448 ) / 160 \) (-11v^11 - 22v^10 - 30v^9 + 28v^8 + 85v^7 + 38v^6 - 94v^5 - 256v^4 - 64v^3 + 352v^2 + 352*v + 448) / 160

\(\nu\)	\(=\)	\( ( -\beta_{11} + \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) (-b11 + b9 + b7 + b5 + b4 + b3 - b2) / 4
\(\nu^{2}\)	\(=\)	\( ( -\beta_{10} - \beta_{8} - 2\beta_{7} + \beta_{6} - 2\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 ) / 4 \) (-b10 - b8 - 2b7 + b6 - 2b4 + b3 - b2 - b1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + \beta_{9} - 2\beta_{8} - \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 4 ) / 4 \) (b11 + b9 - 2b8 - b7 - 2b6 + b5 - b4 + 3*b3 + b2 + 4) / 4
\(\nu^{4}\)	\(=\)	\( ( -3\beta_{11} - 3\beta_{10} - 2\beta_{9} + 6\beta_{5} - \beta_{3} - \beta_{2} + 3\beta _1 + 2 ) / 4 \) (-3b11 - 3b10 - 2b9 + 6b5 - b3 - b2 + 3*b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( -\beta_{11} - 4\beta_{10} + \beta_{9} - 4\beta_{8} + \beta_{7} - 7\beta_{5} - 7\beta_{4} + \beta_{3} - \beta_{2} - 8 ) / 4 \) (-b11 - 4b10 + b9 - 4b8 + b7 - 7b5 - 7b4 + b3 - b2 - 8) / 4
\(\nu^{6}\)	\(=\)	\( ( \beta_{10} + 5\beta_{8} - 6\beta_{7} - 5\beta_{6} - 14\beta_{4} + 3\beta_{3} - 3\beta_{2} + \beta_1 ) / 4 \) (b10 + 5b8 - 6b7 - 5b6 - 14b4 + 3b3 - 3b2 + b1) / 4
\(\nu^{7}\)	\(=\)	\( ( 3 \beta_{11} - 9 \beta_{9} - 2 \beta_{8} + 9 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} + \cdots + 4 ) / 4 \) (3b11 - 9b9 - 2b8 + 9b7 - 2b6 - b5 + b4 + 5b3 - b2 + 16*b1 + 4) / 4
\(\nu^{8}\)	\(=\)	\( ( -5\beta_{11} - 5\beta_{10} + 10\beta_{9} - 14\beta_{5} - 15\beta_{3} - 15\beta_{2} + 5\beta _1 - 2 ) / 4 \) (-5b11 - 5b10 + 10b9 - 14b5 - 15b3 - 15b2 + 5*b1 - 2) / 4
\(\nu^{9}\)	\(=\)	\( ( - 3 \beta_{11} + 12 \beta_{10} + 7 \beta_{9} + 16 \beta_{8} + 7 \beta_{7} + 4 \beta_{6} - 33 \beta_{5} + \cdots + 32 ) / 4 \) (-3b11 + 12b10 + 7b9 + 16b8 + 7b7 + 4b6 - 33b5 - 33b4 + 7*b3 + b2 + 32) / 4
\(\nu^{10}\)	\(=\)	\( ( 15\beta_{10} + 11\beta_{8} - 2\beta_{7} - 11\beta_{6} + 38\beta_{4} + 37\beta_{3} - 37\beta_{2} + 15\beta_1 ) / 4 \) (15b10 + 11b8 - 2b7 - 11b6 + 38b4 + 37b3 - 37b2 + 15b1) / 4
\(\nu^{11}\)	\(=\)	\( ( \beta_{11} + \beta_{9} - 50 \beta_{8} - \beta_{7} + 22 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} + \cdots + 100 ) / 4 \) (b11 + b9 - 50b8 - b7 + 22b6 - 7b5 + 7b4 - 21b3 - 23b2 + 8*b1 + 100) / 4

\(n\)	\(577\)	\(641\)	\(901\)	\(2431\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2879.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 4 T^{10} + \cdots + 15625 \) T^12 + 4T^10 - 17T^8 - 152T^6 - 425T^4 + 2500*T^2 + 15625
$7$	\( (T^{3} - 10 T - 8)^{4} \) (T^3 - 10*T - 8)^4
$11$	\( (T^{6} - 28 T^{4} + \cdots - 512)^{2} \) (T^6 - 28T^4 + 228T^2 - 512)^2
$13$	\( (T^{6} + 32 T^{4} + \cdots + 16)^{2} \) (T^6 + 32T^4 + 52T^2 + 16)^2
$17$	\( (T^{6} - 34 T^{4} + \cdots - 128)^{2} \) (T^6 - 34T^4 + 288T^2 - 128)^2
$19$	\( (T^{6} + 80 T^{4} + \cdots + 4096)^{2} \) (T^6 + 80T^4 + 1600T^2 + 4096)^2
$23$	\( (T^{6} + 88 T^{4} + \cdots + 512)^{2} \) (T^6 + 88T^4 + 528T^2 + 512)^2
$29$	\( (T^{6} + 98 T^{4} + \cdots + 3200)^{2} \) (T^6 + 98T^4 + 2144T^2 + 3200)^2
$31$	\( (T^{6} + 88 T^{4} + \cdots + 6400)^{2} \) (T^6 + 88T^4 + 1424T^2 + 6400)^2
$37$	\( (T^{6} + 44 T^{4} + \cdots + 64)^{2} \) (T^6 + 44T^4 + 132T^2 + 64)^2
$41$	\( (T^{6} + 102 T^{4} + \cdots + 13448)^{2} \) (T^6 + 102T^4 + 2828T^2 + 13448)^2
$43$	\( (T^{3} + 16 T^{2} + \cdots - 512)^{4} \) (T^3 + 16T^2 + 8T - 512)^4
$47$	\( (T^{6} + 176 T^{4} + \cdots + 51200)^{2} \) (T^6 + 176T^4 + 5696T^2 + 51200)^2
$53$	\( (T^{6} - 306 T^{4} + \cdots - 991232)^{2} \) (T^6 - 306T^4 + 30464T^2 - 991232)^2
$59$	\( (T^{6} - 220 T^{4} + \cdots - 359552)^{2} \) (T^6 - 220T^4 + 15652T^2 - 359552)^2
$61$	\( (T^{3} - 2 T^{2} + \cdots + 328)^{4} \) (T^3 - 2T^2 - 100T + 328)^4
$67$	\( (T^{3} - 8 T^{2} + \cdots + 512)^{4} \) (T^3 - 8T^2 - 80T + 512)^4
$71$	\( (T^{6} - 256 T^{4} + \cdots - 204800)^{2} \) (T^6 - 256T^4 + 15616T^2 - 204800)^2
$73$	\( (T^{6} + 276 T^{4} + \cdots + 215296)^{2} \) (T^6 + 276T^4 + 15872T^2 + 215296)^2
$79$	\( (T^{6} + 344 T^{4} + \cdots + 215296)^{2} \) (T^6 + 344T^4 + 29584T^2 + 215296)^2
$83$	\( (T^{6} + 256 T^{4} + \cdots + 204800)^{2} \) (T^6 + 256T^4 + 15616T^2 + 204800)^2
$89$	\( (T^{6} + 118 T^{4} + \cdots + 200)^{2} \) (T^6 + 118T^4 + 1356T^2 + 200)^2
$97$	\( (T^{6} + 276 T^{4} + \cdots + 719104)^{2} \) (T^6 + 276T^4 + 24704T^2 + 719104)^2
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