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Properties

Label 792.2.b.a
Level $792$
Weight $2$
Character orbit 792.b
Analytic conductor $6.324$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [792,2,Mod(593,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("792.593"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 792.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.32415184009\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12781568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 17x^{2} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{4} + \beta_{2}) q^{7} + ( - \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{3}) q^{17} + (\beta_{4} - \beta_1) q^{19} + ( - 2 \beta_{4} - \beta_{2} + \beta_1) q^{23}+ \cdots + ( - 3 \beta_{5} + \beta_{3} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{11} - 4 q^{17} - 26 q^{25} + 8 q^{29} - 12 q^{31} - 16 q^{35} + 12 q^{37} - 4 q^{41} - 14 q^{49} + 16 q^{55} - 56 q^{65} + 32 q^{67} + 40 q^{77} + 44 q^{83} + 40 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} + 17x^{2} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 7\nu^{2} + 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{4} - 5\beta_{2} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} + 7\beta_{3} + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -23\beta_{4} + 27\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/792\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\) \(397\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
− 1.59038i
− 0.810603i
− 2.19399i
2.19399i
0.810603i
1.59038i
0 0 0 − 3.92933i 0 2.07986i 0 0 0
593.2 0 0 0 − 3.52039i 0 − 4.72761i 0 0 0
593.3 0 0 0 − 0.408946i 0 1.15061i 0 0 0
593.4 0 0 0 0.408946i 0 − 1.15061i 0 0 0
593.5 0 0 0 3.52039i 0 4.72761i 0 0 0
593.6 0 0 0 3.92933i 0 − 2.07986i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 792.2.b.a ✓ 6
3.b odd 2 1 792.2.b.b yes 6
4.b odd 2 1 1584.2.b.g 6
8.b even 2 1 6336.2.b.y 6
8.d odd 2 1 6336.2.b.w 6
11.b odd 2 1 792.2.b.b yes 6
12.b even 2 1 1584.2.b.f 6
24.f even 2 1 6336.2.b.z 6
24.h odd 2 1 6336.2.b.x 6
33.d even 2 1 inner 792.2.b.a ✓ 6
44.c even 2 1 1584.2.b.f 6
88.b odd 2 1 6336.2.b.x 6
88.g even 2 1 6336.2.b.z 6
132.d odd 2 1 1584.2.b.g 6
264.m even 2 1 6336.2.b.y 6
264.p odd 2 1 6336.2.b.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.b.a ✓ 6 1.a even 1 1 trivial
792.2.b.a ✓ 6 33.d even 2 1 inner
792.2.b.b yes 6 3.b odd 2 1
792.2.b.b yes 6 11.b odd 2 1
1584.2.b.f 6 12.b even 2 1
1584.2.b.f 6 44.c even 2 1
1584.2.b.g 6 4.b odd 2 1
1584.2.b.g 6 132.d odd 2 1
6336.2.b.w 6 8.d odd 2 1
6336.2.b.w 6 264.p odd 2 1
6336.2.b.x 6 24.h odd 2 1
6336.2.b.x 6 88.b odd 2 1
6336.2.b.y 6 8.b even 2 1
6336.2.b.y 6 264.m even 2 1
6336.2.b.z 6 24.f even 2 1
6336.2.b.z 6 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{3} + 2T_{17}^{2} - 16T_{17} - 16 \) acting on \(S_{2}^{\mathrm{new}}(792, [\chi])\). 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
$5$ \( T^{6} + 28 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( T^{6} + 28 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots + 1331 \) Copy content Toggle raw display
$13$ \( T^{6} + 34 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 16 T - 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 30 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$23$ \( T^{6} + 82 T^{4} + \cdots + 15488 \) Copy content Toggle raw display
$29$ \( (T^{3} - 4 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} - 16 T - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 6 T^{2} - 16 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} + \cdots + 304)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 110 T^{4} + \cdots + 2888 \) Copy content Toggle raw display
$47$ \( T^{6} + 82 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$53$ \( T^{6} + 52 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$59$ \( T^{6} + 160 T^{4} + \cdots + 8192 \) Copy content Toggle raw display
$61$ \( T^{6} + 210 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$67$ \( (T^{3} - 16 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 34 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$73$ \( T^{6} + 456 T^{4} + \cdots + 3442688 \) Copy content Toggle raw display
$79$ \( T^{6} + 388 T^{4} + \cdots + 2048288 \) Copy content Toggle raw display
$83$ \( (T^{3} - 22 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{3} \) Copy content Toggle raw display
$97$ \( (T^{3} - 8 T^{2} + \cdots + 848)^{2} \) Copy content Toggle raw display
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