Properties

Label 9680.2.a.cw
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.27433728.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9x^{4} + 15x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 605)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + q^{5} - \beta_1 q^{7} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + q^{5} - \beta_1 q^{7} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{13} + (\beta_{3} - 1) q^{15} + (2 \beta_{4} - 2 \beta_1) q^{17} + ( - \beta_{5} + \beta_{4}) q^{19} + ( - \beta_{4} + 2 \beta_1) q^{21} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{23} + q^{25} + (3 \beta_{3} + 3 \beta_{2} - 5) q^{27} + ( - 2 \beta_{5} + 2 \beta_1) q^{29} + ( - 2 \beta_{3} + \beta_{2}) q^{31} - \beta_1 q^{35} + 2 \beta_{3} q^{37} + (\beta_{5} - \beta_{4} - 4 \beta_1) q^{39} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{41} + (2 \beta_{5} - \beta_1) q^{43} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{45} + ( - \beta_{3} - 7) q^{47} + (\beta_{2} - 4) q^{49} + ( - 2 \beta_{5} - 4 \beta_{4} + 6 \beta_1) q^{51} + (4 \beta_{3} + 3 \beta_{2} + 4) q^{53} + ( - \beta_{5} + 3 \beta_{4}) q^{57} + \beta_{2} q^{59} - \beta_{5} q^{61} + (\beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{63} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{65} + ( - \beta_{3} - 5) q^{67} + (2 \beta_{3} + 3 \beta_{2} - 4) q^{69} - \beta_{2} q^{71} + ( - 4 \beta_{4} + 4 \beta_1) q^{73} + (\beta_{3} - 1) q^{75} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{79} + ( - 8 \beta_{3} - 3 \beta_{2} + 5) q^{81} + ( - \beta_{5} - 3 \beta_{4} + 6 \beta_1) q^{83} + (2 \beta_{4} - 2 \beta_1) q^{85} + (10 \beta_{4} - 6 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_{2} - 5) q^{89} + ( - 3 \beta_{2} - 6) q^{91} + (\beta_{2} - 10) q^{93} + ( - \beta_{5} + \beta_{4}) q^{95} + ( - 4 \beta_{3} - \beta_{2} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 6 q^{5} + 12 q^{9} - 6 q^{15} - 12 q^{23} + 6 q^{25} - 30 q^{27} + 12 q^{45} - 42 q^{47} - 24 q^{49} + 24 q^{53} - 30 q^{67} - 24 q^{69} - 6 q^{75} + 30 q^{81} - 30 q^{89} - 36 q^{91} - 60 q^{93} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} + 15x^{2} - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 8\nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 10\nu^{3} - 19\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} + 8\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 10\beta_{4} + 31\beta_1 \) 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
1.37268
−1.37268
2.62383
−2.62383
0.480901
−0.480901
0 −3.26180 0 1.00000 0 −1.37268 0 7.63935 0
1.2 0 −3.26180 0 1.00000 0 1.37268 0 7.63935 0
1.3 0 −1.33988 0 1.00000 0 −2.62383 0 −1.20473 0
1.4 0 −1.33988 0 1.00000 0 2.62383 0 −1.20473 0
1.5 0 1.60168 0 1.00000 0 −0.480901 0 −0.434624 0
1.6 0 1.60168 0 1.00000 0 0.480901 0 −0.434624 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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.cw 6
4.b odd 2 1 605.2.a.m 6
11.b odd 2 1 inner 9680.2.a.cw 6
12.b even 2 1 5445.2.a.bx 6
20.d odd 2 1 3025.2.a.bg 6
44.c even 2 1 605.2.a.m 6
44.g even 10 4 605.2.g.q 24
44.h odd 10 4 605.2.g.q 24
132.d odd 2 1 5445.2.a.bx 6
220.g even 2 1 3025.2.a.bg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 4.b odd 2 1
605.2.a.m 6 44.c even 2 1
605.2.g.q 24 44.g even 10 4
605.2.g.q 24 44.h odd 10 4
3025.2.a.bg 6 20.d odd 2 1
3025.2.a.bg 6 220.g even 2 1
5445.2.a.bx 6 12.b even 2 1
5445.2.a.bx 6 132.d odd 2 1
9680.2.a.cw 6 1.a even 1 1 trivial
9680.2.a.cw 6 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{3} + 3T_{3}^{2} - 3T_{3} - 7 \) Copy content Toggle raw display
\( T_{7}^{6} - 9T_{7}^{4} + 15T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{13}^{6} - 72T_{13}^{4} + 1296T_{13}^{2} - 3888 \) Copy content Toggle raw display
\( T_{17}^{6} - 48T_{17}^{4} + 384T_{17}^{2} - 768 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} + 3 T^{2} - 3 T - 7)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 72 T^{4} + \cdots - 3888 \) Copy content Toggle raw display
$17$ \( T^{6} - 48 T^{4} + \cdots - 768 \) Copy content Toggle raw display
$19$ \( T^{6} - 36 T^{4} + \cdots - 48 \) Copy content Toggle raw display
$23$ \( (T^{3} + 6 T^{2} - 12 T - 84)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 144 T^{4} + \cdots - 6912 \) Copy content Toggle raw display
$31$ \( (T^{3} - 48 T + 124)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 24 T - 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 105 T^{4} + \cdots - 7203 \) Copy content Toggle raw display
$43$ \( T^{6} - 129 T^{4} + \cdots - 43923 \) Copy content Toggle raw display
$47$ \( (T^{3} + 21 T^{2} + \cdots + 303)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 12 T^{2} + \cdots + 732)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 12 T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 33 T^{4} + \cdots - 1083 \) Copy content Toggle raw display
$67$ \( (T^{3} + 15 T^{2} + \cdots + 97)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 192 T^{4} + \cdots - 49152 \) Copy content Toggle raw display
$79$ \( T^{6} - 276 T^{4} + \cdots - 101568 \) Copy content Toggle raw display
$83$ \( T^{6} - 312 T^{4} + \cdots - 40368 \) Copy content Toggle raw display
$89$ \( (T^{3} + 15 T^{2} + \cdots - 147)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 12 T^{2} + \cdots + 76)^{2} \) Copy content Toggle raw display
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