Properties

Label 7524.2.a.r
Level $7524$
Weight $2$
Character orbit 7524.a
Self dual yes
Analytic conductor $60.079$
Analytic rank $0$
Dimension $6$
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] = [7524,2,Mod(1,7524)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7524, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7524.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7524.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: \(60.0794424808\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 836)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + \beta_{4} q^{7} - q^{11} + \beta_{3} q^{13} + (\beta_{5} + \beta_{4} + 2 \beta_1 - 3) q^{17} + q^{19} + (\beta_{5} + \beta_{2} + 2 \beta_1) q^{23} + (\beta_{2} + 2 \beta_1 + 4) q^{25} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{29} + (\beta_{5} + \beta_{3} - \beta_{2} - 4) q^{31} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 4) q^{35} + (2 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 + 3) q^{37} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{41} + ( - \beta_{2} - 2 \beta_1 + 2) q^{43} + ( - 2 \beta_{3} - 2) q^{47} + ( - \beta_{2} + 2 \beta_1 + 7) q^{49} + (2 \beta_{2} + 2 \beta_1 - 2) q^{53} - \beta_{5} q^{55} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 3) q^{59} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3) q^{61} + (\beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_1 - 1) q^{65} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{67} + ( - 3 \beta_1 + 6) q^{71} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 4) q^{73} - \beta_{4} q^{77} + ( - 3 \beta_{5} - \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 1) q^{79} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{83} + ( - \beta_{5} - \beta_{4} - 4 \beta_{2} + 1) q^{85} + (2 \beta_{5} - \beta_{4} + \beta_{2} + 3) q^{89} + (4 \beta_{5} + \beta_{4} + \beta_{2} + 7 \beta_1 + 1) q^{91} + \beta_{5} q^{95} + ( - \beta_{4} + \beta_{2} - 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{7} - 6 q^{11} + 2 q^{13} - 12 q^{17} + 6 q^{19} + 2 q^{23} + 26 q^{25} - 4 q^{29} - 20 q^{31} - 20 q^{35} + 22 q^{37} + 2 q^{41} + 10 q^{43} - 16 q^{47} + 48 q^{49} - 12 q^{53} + 14 q^{59} + 12 q^{61} - 10 q^{65} - 12 q^{67} + 30 q^{71} + 24 q^{73} - 2 q^{77} + 14 q^{83} + 12 q^{85} + 14 q^{89} + 20 q^{91} - 46 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} - 12\nu^{3} - 5\nu^{2} + 28\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 15\nu^{3} - 25\nu^{2} - 43\nu + 48 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} + \nu^{4} + 24\nu^{3} - 20\nu^{2} - 50\nu + 42 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} + 7\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{3} + 10\beta_{2} - 2\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{5} + 12\beta_{4} - 11\beta_{3} - 5\beta_{2} + 58\beta _1 - 46 \) Copy content Toggle raw display

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
2.81471
1.24102
−1.64714
0.658537
−3.25580
2.18868
0 0 0 −4.18951 0 4.08740 0 0 0
1.2 0 0 0 −2.83235 0 −4.46564 0 0 0
1.3 0 0 0 −1.84900 0 3.60453 0 0 0
1.4 0 0 0 2.39807 0 4.45908 0 0 0
1.5 0 0 0 2.84405 0 −1.37411 0 0 0
1.6 0 0 0 3.62873 0 −4.31127 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)
\(19\) \(-1\)

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 7524.2.a.r 6
3.b odd 2 1 836.2.a.d 6
12.b even 2 1 3344.2.a.x 6
33.d even 2 1 9196.2.a.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.a.d 6 3.b odd 2 1
3344.2.a.x 6 12.b even 2 1
7524.2.a.r 6 1.a even 1 1 trivial
9196.2.a.k 6 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7524))\):

\( T_{5}^{6} - 28T_{5}^{4} + 6T_{5}^{3} + 228T_{5}^{2} - 48T_{5} - 543 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} - 43T_{7}^{4} + 78T_{7}^{3} + 547T_{7}^{2} - 760T_{7} - 1738 \) Copy content Toggle raw display
\( T_{17}^{6} + 12T_{17}^{5} - 18T_{17}^{4} - 480T_{17}^{3} - 96T_{17}^{2} + 4032T_{17} - 2592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 28 T^{4} + 6 T^{3} + 228 T^{2} + \cdots - 543 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} - 43 T^{4} + \cdots - 1738 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} - 49 T^{4} + \cdots + 2462 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} - 18 T^{4} + \cdots - 2592 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} - 75 T^{4} + \cdots - 1584 \) Copy content Toggle raw display
$29$ \( T^{6} + 4 T^{5} - 69 T^{4} + \cdots - 2682 \) Copy content Toggle raw display
$31$ \( T^{6} + 20 T^{5} + 88 T^{4} + \cdots - 7593 \) Copy content Toggle raw display
$37$ \( T^{6} - 22 T^{5} + 67 T^{4} + \cdots + 91824 \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} - 159 T^{4} + \cdots + 24912 \) Copy content Toggle raw display
$43$ \( T^{6} - 10 T^{5} - 25 T^{4} + \cdots + 4900 \) Copy content Toggle raw display
$47$ \( T^{6} + 16 T^{5} - 96 T^{4} + \cdots + 328896 \) Copy content Toggle raw display
$53$ \( T^{6} + 12 T^{5} - 96 T^{4} + \cdots - 31104 \) Copy content Toggle raw display
$59$ \( T^{6} - 14 T^{5} - 81 T^{4} + \cdots + 31536 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} - 214 T^{4} + \cdots + 109728 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{5} - 184 T^{4} + \cdots + 253887 \) Copy content Toggle raw display
$71$ \( T^{6} - 30 T^{5} + 252 T^{4} + \cdots + 2187 \) Copy content Toggle raw display
$73$ \( T^{6} - 24 T^{5} - 52 T^{4} + \cdots - 217728 \) Copy content Toggle raw display
$79$ \( T^{6} - 370 T^{4} - 1184 T^{3} + \cdots + 428832 \) Copy content Toggle raw display
$83$ \( T^{6} - 14 T^{5} - 135 T^{4} + \cdots - 177642 \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} - 173 T^{4} + \cdots + 19776 \) Copy content Toggle raw display
$97$ \( T^{6} + 46 T^{5} + 791 T^{4} + \cdots + 752 \) Copy content Toggle raw display
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