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Properties

Label 9792.2.a.dr
Level $9792$
Weight $2$
Character orbit 9792.a
Self dual yes
Analytic conductor $78.190$
Analytic rank $0$
Dimension $6$
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] = [9792,2,Mod(1,9792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9792, 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("9792.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9792.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: \(78.1895136592\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.102503232.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 15x^{4} - 16x^{3} + 27x^{2} + 42x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 4896)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{5} - \beta_{2} q^{7} + \beta_{5} q^{11} + ( - \beta_{4} - 1) q^{13} - q^{17} - \beta_1 q^{19} + ( - \beta_{5} - \beta_{2}) q^{23} + (\beta_{4} - 2 \beta_{3} + 2) q^{25} + ( - \beta_{4} - \beta_{3} + 2) q^{29} - \beta_{2} q^{31} + ( - \beta_{5} - 2 \beta_{2} - \beta_1) q^{35} + (\beta_{4} - 3 \beta_{3} - 2) q^{37} + ( - \beta_{4} + 2 \beta_{3} + 1) q^{41} + (\beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{43} + ( - 2 \beta_{2} - 2 \beta_1) q^{47} + ( - 2 \beta_{4} + 5) q^{49} + (2 \beta_{4} - 2 \beta_{3} + 2) q^{53} + (3 \beta_{5} + 2 \beta_{2}) q^{55} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{59} + (\beta_{4} - \beta_{3} - 4) q^{61} + ( - \beta_{4} + 4 \beta_{3} + 1) q^{65} + (\beta_{5} + 2 \beta_{2} + \beta_1) q^{67} + (2 \beta_{5} + \beta_{2} + 2 \beta_1) q^{71} + 6 q^{73} + ( - 4 \beta_{4} + 4 \beta_{3} + 4) q^{77} + ( - 2 \beta_{5} - \beta_{2}) q^{79} + (\beta_{5} + 2 \beta_{2} - \beta_1) q^{83} + (\beta_{3} - 1) q^{85} + (2 \beta_{4} + 4) q^{89} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{91} + (\beta_{5} + 2 \beta_1) q^{95} + (2 \beta_{4} - 2 \beta_{3} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{5} - 6 q^{13} - 6 q^{17} + 12 q^{25} + 12 q^{29} - 12 q^{37} + 6 q^{41} + 30 q^{49} + 12 q^{53} - 24 q^{61} + 6 q^{65} + 36 q^{73} + 24 q^{77} - 6 q^{85} + 24 q^{89} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 15x^{4} - 16x^{3} + 27x^{2} + 42x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 11\nu^{3} - 27\nu^{2} - 11\nu + 41 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + 4\nu^{4} + 38\nu^{3} - \nu^{2} - 58\nu - 32 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 14\nu^{3} - 2\nu^{2} + 28\nu + 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{5} + 3\nu^{4} + 26\nu^{3} - 7\nu^{2} - 50\nu - 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{5} - 21\nu^{4} - 227\nu^{3} + 9\nu^{2} + 427\nu + 163 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{3} + 2\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 2\beta_{4} - 4\beta_{3} + 4\beta_{2} - \beta _1 + 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{5} + 3\beta_{4} - 15\beta_{3} + 9\beta_{2} + 4\beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 59\beta_{5} + 38\beta_{4} - 84\beta_{3} + 68\beta_{2} - \beta _1 + 228 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 261\beta_{5} + 126\beta_{4} - 452\beta_{3} + 272\beta_{2} + 81\beta _1 + 660 ) / 4 \) 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
−2.69534
−0.410146
1.60704
−1.38358
4.07891
−1.19689
0 0 0 −2.10548 0 −2.17070 0 0 0
1.2 0 0 0 −2.10548 0 2.17070 0 0 0
1.3 0 0 0 1.22346 0 −4.88878 0 0 0
1.4 0 0 0 1.22346 0 4.88878 0 0 0
1.5 0 0 0 3.88202 0 −2.71807 0 0 0
1.6 0 0 0 3.88202 0 2.71807 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
\(2\) \(-1\)
\(3\) \(1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9792.2.a.dr 6
3.b odd 2 1 9792.2.a.dq 6
4.b odd 2 1 inner 9792.2.a.dr 6
8.b even 2 1 4896.2.a.bk ✓ 6
8.d odd 2 1 4896.2.a.bk ✓ 6
12.b even 2 1 9792.2.a.dq 6
24.f even 2 1 4896.2.a.bl yes 6
24.h odd 2 1 4896.2.a.bl yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4896.2.a.bk ✓ 6 8.b even 2 1
4896.2.a.bk ✓ 6 8.d odd 2 1
4896.2.a.bl yes 6 24.f even 2 1
4896.2.a.bl yes 6 24.h odd 2 1
9792.2.a.dq 6 3.b odd 2 1
9792.2.a.dq 6 12.b even 2 1
9792.2.a.dr 6 1.a even 1 1 trivial
9792.2.a.dr 6 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9792))\):

\( T_{5}^{3} - 3T_{5}^{2} - 6T_{5} + 10 \) Copy content Toggle raw display
\( T_{7}^{6} - 36T_{7}^{4} + 324T_{7}^{2} - 832 \) Copy content Toggle raw display
\( T_{11}^{6} - 69T_{11}^{4} + 1200T_{11}^{2} - 832 \) Copy content Toggle raw display
\( T_{13}^{3} + 3T_{13}^{2} - 24T_{13} - 76 \) Copy content Toggle raw display
\( T_{19}^{6} - 69T_{19}^{4} + 264T_{19}^{2} - 208 \) Copy content Toggle raw display
\( T_{23}^{6} - 81T_{23}^{4} + 168T_{23}^{2} - 52 \) 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
$5$ \( (T^{3} - 3 T^{2} - 6 T + 10)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 36 T^{4} + \cdots - 832 \) Copy content Toggle raw display
$11$ \( T^{6} - 69 T^{4} + \cdots - 832 \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 24 T - 76)^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 69 T^{4} + \cdots - 208 \) Copy content Toggle raw display
$23$ \( T^{6} - 81 T^{4} + \cdots - 52 \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} + \cdots + 100)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 36 T^{4} + \cdots - 832 \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} + \cdots - 500)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 3 T^{2} + \cdots + 164)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 309 T^{4} + \cdots - 851968 \) Copy content Toggle raw display
$47$ \( T^{6} - 324 T^{4} + \cdots - 269568 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} + \cdots + 360)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 300 T^{4} + \cdots - 140608 \) Copy content Toggle raw display
$61$ \( (T^{3} + 12 T^{2} + \cdots - 40)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 168 T^{4} + \cdots - 83200 \) Copy content Toggle raw display
$71$ \( T^{6} - 420 T^{4} + \cdots - 300352 \) Copy content Toggle raw display
$73$ \( (T - 6)^{6} \) Copy content Toggle raw display
$79$ \( T^{6} - 264 T^{4} + \cdots - 1872 \) Copy content Toggle raw display
$83$ \( T^{6} - 300 T^{4} + \cdots - 140608 \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} + \cdots + 768)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 18 T^{2} + \cdots + 632)^{2} \) Copy content Toggle raw display
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