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Properties

Label 9680.2.a.dd
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: 6.6.25903625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{3} - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{3} - \beta_{2} + \beta_1) q^{9} + (\beta_{5} - 2 \beta_{2} - 1) q^{13} - \beta_1 q^{15} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{3} - 8 \beta_{2} + \cdots - 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} + q^{13} - 3 q^{15} + 6 q^{17} + 7 q^{19} + 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} + 10 q^{29} - q^{31} - 7 q^{35} + 3 q^{37} + q^{39} - 6 q^{41} - 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} - 15 q^{51} - 23 q^{53} + 9 q^{57} + 2 q^{59} + 6 q^{61} + 49 q^{63} - q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} + 10 q^{73} + 3 q^{75} + 22 q^{79} + 10 q^{81} - 10 q^{83} - 6 q^{85} + 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} - 7 q^{95} - 33 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} - 5\nu^{4} + 2\nu^{3} + 16\nu^{2} - 13\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 5\nu^{4} + 2\nu^{3} + 17\nu^{2} - 14\nu - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{5} + 9\nu^{4} - 31\nu^{2} + 14\nu + 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} - 9\nu^{4} + \nu^{3} + 29\nu^{2} - 18\nu - 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{3} - 2\beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} + 3\beta_{4} + 9\beta_{3} - 11\beta_{2} + 13\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{5} + 13\beta_{4} + 25\beta_{3} - 34\beta_{2} + 50\beta _1 + 34 \) 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
−1.71280
−1.68797
−0.220878
1.03795
2.29082
3.29288
0 −1.71280 0 −1.00000 0 3.70505 0 −0.0663151 0
1.2 0 −1.68797 0 −1.00000 0 −0.193967 0 −0.150741 0
1.3 0 −0.220878 0 −1.00000 0 −2.08772 0 −2.95121 0
1.4 0 1.03795 0 −1.00000 0 −2.60210 0 −1.92266 0
1.5 0 2.29082 0 −1.00000 0 4.66366 0 2.24785 0
1.6 0 3.29288 0 −1.00000 0 3.51508 0 7.84307 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.dd 6
4.b odd 2 1 4840.2.a.ba 6
11.b odd 2 1 9680.2.a.dc 6
11.d odd 10 2 880.2.bo.i 12
44.c even 2 1 4840.2.a.bb 6
44.g even 10 2 440.2.y.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.c ✓ 12 44.g even 10 2
880.2.bo.i 12 11.d odd 10 2
4840.2.a.ba 6 4.b odd 2 1
4840.2.a.bb 6 44.c even 2 1
9680.2.a.dc 6 11.b odd 2 1
9680.2.a.dd 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{6} - 3T_{3}^{5} - 7T_{3}^{4} + 17T_{3}^{3} + 16T_{3}^{2} - 20T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{6} - 7T_{7}^{5} - 5T_{7}^{4} + 93T_{7}^{3} - 13T_{7}^{2} - 336T_{7} - 64 \) Copy content Toggle raw display
\( T_{13}^{6} - T_{13}^{5} - 33T_{13}^{4} + 71T_{13}^{3} + 91T_{13}^{2} - 220T_{13} + 80 \) Copy content Toggle raw display
\( T_{17}^{6} - 6T_{17}^{5} - 30T_{17}^{4} + 197T_{17}^{3} + 79T_{17}^{2} - 825T_{17} - 605 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots - 5 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 7 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + \cdots + 80 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} + \cdots - 605 \) Copy content Toggle raw display
$19$ \( T^{6} - 7 T^{5} + \cdots - 9791 \) Copy content Toggle raw display
$23$ \( T^{6} - 9 T^{5} + \cdots + 5956 \) Copy content Toggle raw display
$29$ \( T^{6} - 10 T^{5} + \cdots + 1100 \) Copy content Toggle raw display
$31$ \( T^{6} + T^{5} + \cdots + 3596 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots - 1900 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 3775 \) Copy content Toggle raw display
$43$ \( T^{6} + 18 T^{5} + \cdots + 3751 \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots - 9284 \) Copy content Toggle raw display
$53$ \( T^{6} + 23 T^{5} + \cdots + 199804 \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} + \cdots + 11759 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 6620 \) Copy content Toggle raw display
$67$ \( T^{6} - 22 T^{5} + \cdots + 126475 \) Copy content Toggle raw display
$71$ \( T^{6} - 13 T^{5} + \cdots + 620 \) Copy content Toggle raw display
$73$ \( T^{6} - 10 T^{5} + \cdots - 832031 \) Copy content Toggle raw display
$79$ \( T^{6} - 22 T^{5} + \cdots + 6724 \) Copy content Toggle raw display
$83$ \( T^{6} + 10 T^{5} + \cdots - 1375 \) Copy content Toggle raw display
$89$ \( T^{6} + 25 T^{5} + \cdots - 22429 \) Copy content Toggle raw display
$97$ \( T^{6} + 33 T^{5} + \cdots - 164695 \) Copy content Toggle raw display
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