Properties

Label 5040.2.f.e
Level $5040$
Weight $2$
Character orbit 5040.f
Analytic conductor $40.245$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(881,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("5040.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.f (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: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 2520)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + (\beta_{7} + \beta_{3} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (\beta_{7} + \beta_{3} - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_1 - 1) q^{11} + ( - \beta_{6} + 2 \beta_{3}) q^{13} + (\beta_{7} - \beta_{5} - \beta_{4} + 2) q^{17} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_1 - 1) q^{19} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_1 + 1) q^{23} + q^{25} + (\beta_{7} + \beta_{5} + 2 \beta_{3} - \beta_1 - 1) q^{29} + ( - \beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{31} + ( - \beta_{7} - \beta_{3} + 1) q^{35} + (\beta_{4} - 2 \beta_{2} - 3) q^{37} + ( - 4 \beta_{4} - \beta_{2} - 2) q^{41} + (\beta_{4} + \beta_{2} + 3) q^{43} + (\beta_{7} - \beta_{5} + \beta_{4} + 2) q^{47} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{49} + (3 \beta_{6} - 4 \beta_{3} + 3 \beta_1) q^{53} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_1 + 1) q^{55} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{2} + 5) q^{59} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - 5 \beta_{3} + 3 \beta_1 + 1) q^{61} + (\beta_{6} - 2 \beta_{3}) q^{65} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 2) q^{67} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} + 7 \beta_{3} - 4 \beta_1 - 2) q^{71} + (\beta_{6} + 4 \beta_{3} - 2 \beta_1) q^{73} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 3 \beta_1 - 4) q^{77} + (3 \beta_{7} - 3 \beta_{5} + 2 \beta_{2} - 1) q^{79} + ( - 2 \beta_{4} + 3 \beta_{2} - 2) q^{83} + ( - \beta_{7} + \beta_{5} + \beta_{4} - 2) q^{85} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{2}) q^{89} + (\beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 4) q^{91} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{95} + ( - 3 \beta_{7} - \beta_{6} - 3 \beta_{5} + 2 \beta_1 + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 4 q^{7} + 16 q^{17} + 8 q^{25} + 4 q^{35} - 24 q^{37} - 16 q^{41} + 24 q^{43} + 16 q^{47} + 40 q^{59} + 16 q^{67} - 32 q^{77} - 8 q^{79} - 16 q^{83} - 16 q^{85} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 7\nu^{3} - 6\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 7\nu^{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 11\nu^{3} + 26\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 19\nu^{2} - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 12\nu^{5} - 20\nu^{4} + 41\nu^{3} - 46\nu^{2} + 46\nu - 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 12\nu^{5} - 41\nu^{3} - 38\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 12\nu^{5} + 20\nu^{4} + 41\nu^{3} + 46\nu^{2} + 46\nu + 12 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - 2\beta_{4} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - 5\beta_{5} + 2\beta_{3} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 7\beta_{5} + 14\beta_{4} + 4\beta_{2} + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29\beta_{7} + 29\beta_{6} + 29\beta_{5} - 14\beta_{3} - 22\beta _1 - 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 51\beta_{7} - 51\beta_{5} - 94\beta_{4} - 40\beta_{2} - 217 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -181\beta_{7} - 189\beta_{6} - 181\beta_{5} + 86\beta_{3} + 182\beta _1 + 181 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
2.63640i
2.63640i
1.65222i
1.65222i
0.222191i
0.222191i
2.06644i
2.06644i
0 0 0 −1.00000 0 −2.63640 0.222191i 0 0 0
881.2 0 0 0 −1.00000 0 −2.63640 + 0.222191i 0 0 0
881.3 0 0 0 −1.00000 0 −1.65222 2.06644i 0 0 0
881.4 0 0 0 −1.00000 0 −1.65222 + 2.06644i 0 0 0
881.5 0 0 0 −1.00000 0 0.222191 2.63640i 0 0 0
881.6 0 0 0 −1.00000 0 0.222191 + 2.63640i 0 0 0
881.7 0 0 0 −1.00000 0 2.06644 1.65222i 0 0 0
881.8 0 0 0 −1.00000 0 2.06644 + 1.65222i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5040.2.f.e 8
3.b odd 2 1 5040.2.f.h 8
4.b odd 2 1 2520.2.f.b 8
7.b odd 2 1 5040.2.f.h 8
12.b even 2 1 2520.2.f.d yes 8
21.c even 2 1 inner 5040.2.f.e 8
28.d even 2 1 2520.2.f.d yes 8
84.h odd 2 1 2520.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.2.f.b 8 4.b odd 2 1
2520.2.f.b 8 84.h odd 2 1
2520.2.f.d yes 8 12.b even 2 1
2520.2.f.d yes 8 28.d even 2 1
5040.2.f.e 8 1.a even 1 1 trivial
5040.2.f.e 8 21.c even 2 1 inner
5040.2.f.h 8 3.b odd 2 1
5040.2.f.h 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5040, [\chi])\):

\( T_{11}^{8} + 84T_{11}^{6} + 2548T_{11}^{4} + 32480T_{11}^{2} + 141376 \) Copy content Toggle raw display
\( T_{17}^{4} - 8T_{17}^{3} - 2T_{17}^{2} + 104T_{17} - 136 \) Copy content Toggle raw display
\( T_{41}^{4} + 8T_{41}^{3} - 152T_{41}^{2} - 672T_{41} + 6544 \) Copy content Toggle raw display
\( T_{47}^{4} - 8T_{47}^{3} - 26T_{47}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + 8 T^{6} + 20 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 84 T^{6} + 2548 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$13$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 8 T^{3} - 2 T^{2} + 104 T - 136)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 68 T^{6} + 628 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{8} + 148 T^{6} + 7684 T^{4} + \cdots + 1290496 \) Copy content Toggle raw display
$29$ \( T^{8} + 52 T^{6} + 708 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{8} + 160 T^{6} + 5344 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} - 38 T^{2} - 480 T + 368)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} - 152 T^{2} - 672 T + 6544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{3} + 34 T^{2} + 24 T - 136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{3} - 26 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 284 T^{6} + 23748 T^{4} + \cdots + 135424 \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{3} + 84 T^{2} - 96 T + 32)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 260 T^{6} + 11332 T^{4} + \cdots + 541696 \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} - 66 T^{2} - 64 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 536 T^{6} + \cdots + 123921424 \) Copy content Toggle raw display
$73$ \( T^{8} + 224 T^{6} + 8160 T^{4} + \cdots + 430336 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} - 196 T^{2} - 256 T + 8704)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} - 216 T^{2} - 704 T + 5248)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 232 T^{2} + 384 T + 656)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 520 T^{6} + 59344 T^{4} + \cdots + 2262016 \) Copy content Toggle raw display
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