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Properties

Label 8040.2.a.t
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8040,2,Mod(1,8040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8040.a (trivial)

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: yes
Analytic conductor: \(64.1997232251\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + (\beta_{3} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + (\beta_{3} + 1) q^{7} + q^{9} + ( - \beta_{6} - \beta_1) q^{11} + (\beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{13} - q^{15} - \beta_{2} q^{17} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + (\beta_{3} + 1) q^{21} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{23}+ \cdots + ( - \beta_{6} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{6} - 4\nu^{5} - 73\nu^{4} - 43\nu^{3} + 42\nu^{2} + 143\nu + 112 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{6} - 23\nu^{5} - 76\nu^{4} + 234\nu^{3} + 104\nu^{2} - 484\nu + 39 ) / 55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{6} - 29\nu^{5} - 213\nu^{4} + 197\nu^{3} + 497\nu^{2} - 242\nu - 123 ) / 55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -24\nu^{6} + 16\nu^{5} + 347\nu^{4} + 62\nu^{3} - 773\nu^{2} - 132\nu + 267 ) / 55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 10\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} - 4\beta_{5} + 4\beta_{4} + 2\beta_{3} + 13\beta_{2} + 23\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -16\beta_{6} - 33\beta_{5} + 30\beta_{4} - 11\beta_{3} + 25\beta_{2} + 134\beta _1 + 105 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{6} - 85\beta_{5} + 83\beta_{4} + 19\beta_{3} + 175\beta_{2} + 410\beta _1 + 558 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.70642
1.51692
3.95767
−1.05266
0.211675
−2.84959
0.922407
0 1.00000 0 −1.00000 0 −2.80016 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.274078 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 −0.212435 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 0.670274 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.61083 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 4.47701 0 1.00000 0
1.7 0 1.00000 0 −1.00000 0 4.52856 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(67\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8040.2.a.t ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.t ✓ 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{7} - 10T_{7}^{6} + 19T_{7}^{5} + 74T_{7}^{4} - 223T_{7}^{3} + 17T_{7}^{2} + 52T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{7} - 41T_{11}^{5} + 18T_{11}^{4} + 525T_{11}^{3} - 377T_{11}^{2} - 2000T_{11} + 1530 \) Copy content Toggle raw display
\( T_{13}^{7} + T_{13}^{6} - 55T_{13}^{5} - 69T_{13}^{4} + 971T_{13}^{3} + 1494T_{13}^{2} - 5580T_{13} - 10152 \) Copy content Toggle raw display
\( T_{17}^{7} + 2T_{17}^{6} - 76T_{17}^{5} + 51T_{17}^{4} + 1653T_{17}^{3} - 4766T_{17}^{2} + 1834T_{17} + 3332 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 10 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{7} - 41 T^{5} + \cdots + 1530 \) Copy content Toggle raw display
$13$ \( T^{7} + T^{6} + \cdots - 10152 \) Copy content Toggle raw display
$17$ \( T^{7} + 2 T^{6} + \cdots + 3332 \) Copy content Toggle raw display
$19$ \( T^{7} - 9 T^{6} + \cdots + 136 \) Copy content Toggle raw display
$23$ \( T^{7} - 2 T^{6} + \cdots + 6848 \) Copy content Toggle raw display
$29$ \( T^{7} + T^{6} + \cdots + 96 \) Copy content Toggle raw display
$31$ \( T^{7} - 9 T^{6} + \cdots - 3632 \) Copy content Toggle raw display
$37$ \( T^{7} - 23 T^{6} + \cdots - 260146 \) Copy content Toggle raw display
$41$ \( T^{7} - 5 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$43$ \( T^{7} + 3 T^{6} + \cdots - 12152 \) Copy content Toggle raw display
$47$ \( T^{7} - 11 T^{6} + \cdots + 59632 \) Copy content Toggle raw display
$53$ \( T^{7} - 13 T^{6} + \cdots - 11840 \) Copy content Toggle raw display
$59$ \( T^{7} - T^{6} + \cdots + 400 \) Copy content Toggle raw display
$61$ \( T^{7} - 4 T^{6} + \cdots + 44482 \) Copy content Toggle raw display
$67$ \( (T - 1)^{7} \) Copy content Toggle raw display
$71$ \( T^{7} - T^{6} + \cdots - 58012 \) Copy content Toggle raw display
$73$ \( T^{7} - 14 T^{6} + \cdots + 10256 \) Copy content Toggle raw display
$79$ \( T^{7} - 25 T^{6} + \cdots + 195772 \) Copy content Toggle raw display
$83$ \( T^{7} + 29 T^{6} + \cdots + 129586 \) Copy content Toggle raw display
$89$ \( T^{7} - 7 T^{6} + \cdots - 2086668 \) Copy content Toggle raw display
$97$ \( T^{7} - 38 T^{6} + \cdots - 18245468 \) Copy content Toggle raw display
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