Properties

Label 9660.2.a.x
Level $9660$
Weight $2$
Character orbit 9660.a
Self dual yes
Analytic conductor $77.135$
Analytic rank $1$
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] = [9660,2,Mod(1,9660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9660, 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("9660.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9660.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: \(77.1354883526\)
Analytic rank: \(1\)
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: \( x^{7} - x^{6} - 20x^{5} + 16x^{4} + 110x^{3} - 56x^{2} - 120x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} - q^{7} + q^{9} + ( - \beta_{4} + 1) q^{11} + (\beta_{6} - 1) q^{13} + q^{15} + (\beta_{5} - \beta_{3} - 1) q^{17} + (\beta_{3} - 1) q^{19} + q^{21} + q^{23} + q^{25} - q^{27} + ( - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{29} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{31} + (\beta_{4} - 1) q^{33} + q^{35} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{37} + ( - \beta_{6} + 1) q^{39} + ( - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 1) q^{41} + (\beta_{3} - \beta_{2} - 1) q^{43} - q^{45} + (\beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{47} + q^{49} + ( - \beta_{5} + \beta_{3} + 1) q^{51} - 3 \beta_1 q^{53} + (\beta_{4} - 1) q^{55} + ( - \beta_{3} + 1) q^{57} + ( - \beta_{6} - \beta_{5} + 2 \beta_1 + 3) q^{59} + ( - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{61} - q^{63} + ( - \beta_{6} + 1) q^{65} + ( - \beta_{5} - \beta_{3} + 1) q^{67} - q^{69} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{71} + (\beta_{6} + \beta_{5} + \beta_{2} - 2 \beta_1 - 1) q^{73} - q^{75} + (\beta_{4} - 1) q^{77} + (\beta_{2} - \beta_1 + 2) q^{79} + q^{81} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} + 3 \beta_1 - 1) q^{83} + ( - \beta_{5} + \beta_{3} + 1) q^{85} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{87} + (\beta_{6} + \beta_{5} - \beta_{4} + 2) q^{89} + ( - \beta_{6} + 1) q^{91} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{93} + ( - \beta_{3} + 1) q^{95} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1) q^{97} + ( - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{3} - 7 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{3} - 7 q^{5} - 7 q^{7} + 7 q^{9} + 4 q^{11} - 5 q^{13} + 7 q^{15} - 5 q^{17} - 6 q^{19} + 7 q^{21} + 7 q^{23} + 7 q^{25} - 7 q^{27} + 6 q^{29} - 2 q^{31} - 4 q^{33} + 7 q^{35} + 13 q^{37} + 5 q^{39} - 6 q^{41} - 5 q^{43} - 7 q^{45} - 5 q^{47} + 7 q^{49} + 5 q^{51} - 3 q^{53} - 4 q^{55} + 6 q^{57} + 18 q^{59} + 8 q^{61} - 7 q^{63} + 5 q^{65} + 3 q^{67} - 7 q^{69} + 12 q^{71} - 5 q^{73} - 7 q^{75} - 4 q^{77} + 12 q^{79} + 7 q^{81} - 11 q^{83} + 5 q^{85} - 6 q^{87} + 16 q^{89} + 5 q^{91} + 2 q^{93} + 6 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 20x^{5} + 16x^{4} + 110x^{3} - 56x^{2} - 120x - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{5} - 18\nu^{4} - 20\nu^{3} + 86\nu^{2} + 84\nu - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 5\nu^{5} + 62\nu^{4} - 84\nu^{3} - 338\nu^{2} + 324\nu + 256 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 19\nu^{4} - 32\nu^{3} - 94\nu^{2} + 118\nu + 64 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{6} - 7\nu^{5} - 94\nu^{4} + 124\nu^{3} + 462\nu^{2} - 532\nu - 320 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + 3\beta_{4} + 11\beta_{2} + 2\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17\beta_{6} + 19\beta_{5} - \beta_{4} - 12\beta_{3} - \beta_{2} + 88\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 21\beta_{6} - 17\beta_{5} + 55\beta_{4} + 8\beta_{3} + 113\beta_{2} + 44\beta _1 + 522 \) 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
−0.380978
3.40934
−3.27215
−2.68731
−0.590446
1.65584
2.86569
0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 0
1.6 0 −1.00000 0 −1.00000 0 −1.00000 0 1.00000 0
1.7 0 −1.00000 0 −1.00000 0 −1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

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 9660.2.a.x 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9660.2.a.x 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{7} - 4T_{11}^{6} - 48T_{11}^{5} + 161T_{11}^{4} + 718T_{11}^{3} - 2086T_{11}^{2} - 3370T_{11} + 8972 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9660))\). 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
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 4 T^{6} - 48 T^{5} + \cdots + 8972 \) Copy content Toggle raw display
$13$ \( T^{7} + 5 T^{6} - 49 T^{5} - 292 T^{4} + \cdots + 864 \) Copy content Toggle raw display
$17$ \( T^{7} + 5 T^{6} - 77 T^{5} + \cdots + 2528 \) Copy content Toggle raw display
$19$ \( T^{7} + 6 T^{6} - 56 T^{5} - 325 T^{4} + \cdots + 216 \) Copy content Toggle raw display
$23$ \( (T - 1)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 6 T^{6} - 128 T^{5} + \cdots + 30528 \) Copy content Toggle raw display
$31$ \( T^{7} + 2 T^{6} - 114 T^{5} + \cdots + 14592 \) Copy content Toggle raw display
$37$ \( T^{7} - 13 T^{6} - 143 T^{5} + \cdots - 939384 \) Copy content Toggle raw display
$41$ \( T^{7} + 6 T^{6} - 184 T^{5} + \cdots - 715632 \) Copy content Toggle raw display
$43$ \( T^{7} + 5 T^{6} - 69 T^{5} - 548 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$47$ \( T^{7} + 5 T^{6} - 200 T^{5} + \cdots + 6912 \) Copy content Toggle raw display
$53$ \( T^{7} + 3 T^{6} - 180 T^{5} + \cdots + 69984 \) Copy content Toggle raw display
$59$ \( T^{7} - 18 T^{6} - 32 T^{5} + \cdots + 17904 \) Copy content Toggle raw display
$61$ \( T^{7} - 8 T^{6} - 48 T^{5} + \cdots - 3256 \) Copy content Toggle raw display
$67$ \( T^{7} - 3 T^{6} - 185 T^{5} + \cdots - 185344 \) Copy content Toggle raw display
$71$ \( T^{7} - 12 T^{6} - 136 T^{5} + \cdots - 349952 \) Copy content Toggle raw display
$73$ \( T^{7} + 5 T^{6} - 285 T^{5} + \cdots - 4147632 \) Copy content Toggle raw display
$79$ \( T^{7} - 12 T^{6} - 20 T^{5} + \cdots + 10368 \) Copy content Toggle raw display
$83$ \( T^{7} + 11 T^{6} - 393 T^{5} + \cdots - 46336 \) Copy content Toggle raw display
$89$ \( T^{7} - 16 T^{6} + 20 T^{5} + \cdots - 16384 \) Copy content Toggle raw display
$97$ \( T^{7} - 2 T^{6} - 460 T^{5} + \cdots - 366336 \) Copy content Toggle raw display
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