Properties

Label 9680.2.a.da
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{3} - \beta_1) q^{9} + ( - \beta_{5} - \beta_{4}) q^{13} - \beta_1 q^{15} + (2 \beta_1 + 2) q^{17} + ( - \beta_{5} - \beta_{4} - 2) q^{19} + (2 \beta_{5} + \beta_{2} + 2 \beta_1) q^{21} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{23} + q^{25} + (\beta_{3} - 2 \beta_{2} + 2) q^{27} + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots + 2) q^{29}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} - 40 q^{51} + 8 q^{53} - 16 q^{57} - 4 q^{59} + 16 q^{61} - 28 q^{63} + 2 q^{67} + 8 q^{69} + 28 q^{71} + 16 q^{73} + 2 q^{75} - 10 q^{81} - 12 q^{83} + 8 q^{85} + 24 q^{87} + 18 q^{89} + 24 q^{91} + 20 q^{93} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + 7\nu^{3} - 10\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 7\nu^{3} + 4\nu^{2} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} + 8\beta_{4} + 9\beta_{3} + 7\beta_{2} + 9\beta _1 + 7 \) 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
2.27997
−2.08589
1.90131
−1.45825
−0.821728
0.184585
0 −2.19828 0 1.00000 0 −2.58485 0 1.83244 0
1.2 0 −1.35095 0 1.00000 0 3.00679 0 −1.17493 0
1.3 0 −0.614975 0 1.00000 0 −2.24598 0 −2.62181 0
1.4 0 0.873518 0 1.00000 0 3.64048 0 −2.23697 0
1.5 0 2.32476 0 1.00000 0 −4.78768 0 2.40452 0
1.6 0 2.96593 0 1.00000 0 −1.02876 0 5.79673 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
\(2\) \( +1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)

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 9680.2.a.da 6
4.b odd 2 1 4840.2.a.bd yes 6
11.b odd 2 1 9680.2.a.db 6
44.c even 2 1 4840.2.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.bc 6 44.c even 2 1
4840.2.a.bd yes 6 4.b odd 2 1
9680.2.a.da 6 1.a even 1 1 trivial
9680.2.a.db 6 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{6} - 2T_{3}^{5} - 9T_{3}^{4} + 12T_{3}^{3} + 23T_{3}^{2} - 10T_{3} - 11 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 21T_{7}^{4} - 84T_{7}^{3} + 71T_{7}^{2} + 440T_{7} + 313 \) Copy content Toggle raw display
\( T_{13}^{6} - 32T_{13}^{4} - 16T_{13}^{3} + 64T_{13}^{2} + 64T_{13} + 16 \) Copy content Toggle raw display
\( T_{17}^{6} - 8T_{17}^{5} - 16T_{17}^{4} + 192T_{17}^{3} - 1024T_{17} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots - 11 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots + 313 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 32 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{6} - 8 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + \cdots - 176 \) Copy content Toggle raw display
$23$ \( T^{6} - 8 T^{5} + \cdots - 9776 \) Copy content Toggle raw display
$29$ \( T^{6} - 16 T^{5} + \cdots - 5888 \) Copy content Toggle raw display
$31$ \( T^{6} - 4 T^{5} + \cdots - 368 \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + \cdots + 3328 \) Copy content Toggle raw display
$41$ \( T^{6} - 32 T^{5} + \cdots - 70187 \) Copy content Toggle raw display
$43$ \( T^{6} - 4 T^{5} + \cdots + 9097 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 277 \) Copy content Toggle raw display
$53$ \( T^{6} - 8 T^{5} + \cdots + 89296 \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} + \cdots - 221168 \) Copy content Toggle raw display
$61$ \( T^{6} - 16 T^{5} + \cdots + 252013 \) Copy content Toggle raw display
$67$ \( T^{6} - 2 T^{5} + \cdots - 1315739 \) Copy content Toggle raw display
$71$ \( T^{6} - 28 T^{5} + \cdots - 10736 \) Copy content Toggle raw display
$73$ \( T^{6} - 16 T^{5} + \cdots + 212992 \) Copy content Toggle raw display
$79$ \( T^{6} - 132 T^{4} + \cdots + 19008 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + \cdots + 927952 \) Copy content Toggle raw display
$89$ \( T^{6} - 18 T^{5} + \cdots + 75673 \) Copy content Toggle raw display
$97$ \( T^{6} - 240 T^{4} + \cdots + 103248 \) Copy content Toggle raw display
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