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Properties

Label 5040.2.f.f
Level $5040$
Weight $2$
Character orbit 5040.f
Analytic conductor $40.245$
Analytic rank $0$
Dimension $8$
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: 8.0.7442857984.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 630)
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.

\(f(q)\) \(=\) \( q - q^{5} + (\beta_{6} + 1) q^{7} + ( - \beta_{6} - \beta_{5}) q^{11} - \beta_1 q^{13} + \beta_{2} q^{17} + ( - \beta_{4} + \beta_1) q^{19} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{23} + q^{25} + (\beta_{6} + \beta_{5} - \beta_1) q^{29}+ \cdots + ( - \beta_{6} - \beta_{5} + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} + 4 q^{7} + 8 q^{25} - 4 q^{35} - 8 q^{37} - 8 q^{41} + 16 q^{43} + 40 q^{47} + 4 q^{49} - 32 q^{67} + 52 q^{77} - 8 q^{79} - 16 q^{83} - 8 q^{89} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 73\nu^{3} - 198\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 16\nu^{4} - 21\nu^{2} + 126 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 20\nu^{5} - 132\nu^{4} + 73\nu^{3} - 594\nu^{2} - 198\nu + 108 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 26\nu^{5} + 187\nu^{3} + 306\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 20\nu^{5} - 132\nu^{4} + 73\nu^{3} - 738\nu^{2} - 54\nu - 756 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 6\nu^{6} + 20\nu^{5} + 132\nu^{4} + 73\nu^{3} + 738\nu^{2} - 54\nu + 756 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 112\nu^{5} + 665\nu^{3} + 954\nu ) / 72 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + 2\beta_{3} - \beta _1 - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 13\beta_{6} - 13\beta_{5} - 2\beta_{4} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{6} + 13\beta_{5} - 34\beta_{3} + 8\beta_{2} + 17\beta _1 + 120 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -38\beta_{7} + 163\beta_{6} + 163\beta_{5} + 50\beta_{4} - 73\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 187\beta_{6} - 187\beta_{5} + 502\beta_{3} - 176\beta_{2} - 251\beta _1 - 1416 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 614\beta_{7} - 2113\beta_{6} - 2113\beta_{5} - 854\beta_{4} + 895\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.91681i
− 1.91681i
− 3.73923i
3.73923i
− 2.73923i
2.73923i
0.916813i
− 0.916813i
0 0 0 −1.00000 0 −2.27220 − 1.35539i 0 0 0
881.2 0 0 0 −1.00000 0 −2.27220 + 1.35539i 0 0 0
881.3 0 0 0 −1.00000 0 −0.0951965 − 2.64404i 0 0 0
881.4 0 0 0 −1.00000 0 −0.0951965 + 2.64404i 0 0 0
881.5 0 0 0 −1.00000 0 1.80230 − 1.93693i 0 0 0
881.6 0 0 0 −1.00000 0 1.80230 + 1.93693i 0 0 0
881.7 0 0 0 −1.00000 0 2.56510 − 0.648285i 0 0 0
881.8 0 0 0 −1.00000 0 2.56510 + 0.648285i 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.f 8
3.b odd 2 1 5040.2.f.i 8
4.b odd 2 1 630.2.b.a ✓ 8
7.b odd 2 1 5040.2.f.i 8
12.b even 2 1 630.2.b.b yes 8
20.d odd 2 1 3150.2.b.e 8
20.e even 4 1 3150.2.d.c 8
20.e even 4 1 3150.2.d.d 8
21.c even 2 1 inner 5040.2.f.f 8
28.d even 2 1 630.2.b.b yes 8
60.h even 2 1 3150.2.b.f 8
60.l odd 4 1 3150.2.d.a 8
60.l odd 4 1 3150.2.d.f 8
84.h odd 2 1 630.2.b.a ✓ 8
140.c even 2 1 3150.2.b.f 8
140.j odd 4 1 3150.2.d.a 8
140.j odd 4 1 3150.2.d.f 8
420.o odd 2 1 3150.2.b.e 8
420.w even 4 1 3150.2.d.c 8
420.w even 4 1 3150.2.d.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.b.a ✓ 8 4.b odd 2 1
630.2.b.a ✓ 8 84.h odd 2 1
630.2.b.b yes 8 12.b even 2 1
630.2.b.b yes 8 28.d even 2 1
3150.2.b.e 8 20.d odd 2 1
3150.2.b.e 8 420.o odd 2 1
3150.2.b.f 8 60.h even 2 1
3150.2.b.f 8 140.c even 2 1
3150.2.d.a 8 60.l odd 4 1
3150.2.d.a 8 140.j odd 4 1
3150.2.d.c 8 20.e even 4 1
3150.2.d.c 8 420.w even 4 1
3150.2.d.d 8 20.e even 4 1
3150.2.d.d 8 420.w even 4 1
3150.2.d.f 8 60.l odd 4 1
3150.2.d.f 8 140.j odd 4 1
5040.2.f.f 8 1.a even 1 1 trivial
5040.2.f.f 8 21.c even 2 1 inner
5040.2.f.i 8 3.b odd 2 1
5040.2.f.i 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 \(S_{2}^{\mathrm{new}}(5040, [\chi])\):

\( T_{11}^{8} + 52T_{11}^{6} + 820T_{11}^{4} + 4320T_{11}^{2} + 5184 \) Copy content Toggle raw display
\( T_{17}^{4} - 50T_{17}^{2} + 120T_{17} - 72 \) Copy content Toggle raw display
\( T_{41}^{4} + 4T_{41}^{3} - 22T_{41}^{2} - 24T_{41} + 72 \) Copy content Toggle raw display
\( T_{47}^{4} - 20T_{47}^{3} - 22T_{47}^{2} + 2424T_{47} - 12024 \) 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
$7$ \( T^{8} - 4 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 52 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$17$ \( (T^{4} - 50 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 108 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$23$ \( T^{8} + 104 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( T^{8} + 104 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$31$ \( T^{8} + 216 T^{6} + \cdots + 1327104 \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 124)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} - 26 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 20 T^{3} + \cdots - 12024)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 120 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$59$ \( (T^{4} - 194 T^{2} + \cdots + 5112)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 252 T^{6} + \cdots + 4981824 \) Copy content Toggle raw display
$67$ \( (T^{4} + 16 T^{3} + \cdots - 4896)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 108 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$73$ \( T^{8} + 352 T^{6} + \cdots + 5992704 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 14368)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots - 5832)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 568 T^{6} + \cdots + 199148544 \) Copy content Toggle raw display
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