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Properties

Label 9680.2.a.cj
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + q^{5} + ( - \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + q^{5} + ( - \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{9} + ( - 2 \beta_{3} + \beta_{2} - \beta_1) q^{13} + (\beta_1 - 1) q^{15} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{19} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{21} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{23} + q^{25} + ( - \beta_{3} + \beta_{2} - 2) q^{27} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{29} + (2 \beta_{3} - 4 \beta_{2} - 6) q^{31} + ( - \beta_{3} + \beta_1 + 1) q^{35} + ( - 3 \beta_{2} + \beta_1 - 4) q^{37} + (2 \beta_{3} - 8 \beta_{2} - 2 \beta_1 - 6) q^{39} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 7) q^{43} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{45} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{47} + ( - 7 \beta_{2} + \beta_1 - 1) q^{49} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{51} + (3 \beta_{3} + \beta_1 - 1) q^{53} + (\beta_{3} - 7 \beta_{2} - 10) q^{57} + (\beta_{3} - 8 \beta_{2} - 5) q^{59} + ( - 2 \beta_{3} + 6 \beta_{2} + \cdots + 2) q^{61}+ \cdots + (6 \beta_{2} + 3 \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 7 q^{9} - 7 q^{13} - 3 q^{15} - q^{17} - 4 q^{19} + 16 q^{21} - 13 q^{23} + 4 q^{25} - 12 q^{27} + 4 q^{29} - 12 q^{31} + 3 q^{35} - 9 q^{37} - 6 q^{39} - 19 q^{43} + 7 q^{45} - 3 q^{47} + 11 q^{49} - 10 q^{51} + 3 q^{53} - 24 q^{57} - 2 q^{59} - 10 q^{61} - 29 q^{63} - 7 q^{65} + 13 q^{67} + 30 q^{69} - 16 q^{71} - q^{73} - 3 q^{75} - 2 q^{79} - 8 q^{81} - 27 q^{83} - q^{85} - 10 q^{87} + 16 q^{89} + 16 q^{91} - 4 q^{93} - 4 q^{95} - 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} + 2\nu - 11 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \) 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
−1.88301
−1.48718
1.26498
3.10522
0 −2.88301 0 1.00000 0 −3.92979 0 5.31175 0
1.2 0 −2.48718 0 1.00000 0 0.431946 0 3.18609 0
1.3 0 0.264977 0 1.00000 0 4.31175 0 −2.92979 0
1.4 0 2.10522 0 1.00000 0 2.18609 0 1.43195 0
\(n\): e.g. 2-40 or 990-1000
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.cj 4
4.b odd 2 1 1210.2.a.u 4
11.b odd 2 1 9680.2.a.ci 4
11.c even 5 2 880.2.bo.g 8
20.d odd 2 1 6050.2.a.dh 4
44.c even 2 1 1210.2.a.v 4
44.h odd 10 2 110.2.g.c ✓ 8
132.o even 10 2 990.2.n.j 8
220.g even 2 1 6050.2.a.cy 4
220.n odd 10 2 550.2.h.l 8
220.v even 20 4 550.2.ba.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.g.c ✓ 8 44.h odd 10 2
550.2.h.l 8 220.n odd 10 2
550.2.ba.f 16 220.v even 20 4
880.2.bo.g 8 11.c even 5 2
990.2.n.j 8 132.o even 10 2
1210.2.a.u 4 4.b odd 2 1
1210.2.a.v 4 44.c even 2 1
6050.2.a.cy 4 220.g even 2 1
6050.2.a.dh 4 20.d odd 2 1
9680.2.a.ci 4 11.b odd 2 1
9680.2.a.cj 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{4} + 3T_{3}^{3} - 5T_{3}^{2} - 14T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 15T_{7}^{2} + 44T_{7} - 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 7T_{13}^{3} - 21T_{13}^{2} - 132T_{13} + 176 \) Copy content Toggle raw display
\( T_{17}^{4} + T_{17}^{3} - 33T_{17}^{2} + 74T_{17} - 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} + \cdots + 176 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + \cdots - 44 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots - 149 \) Copy content Toggle raw display
$23$ \( T^{4} + 13 T^{3} + \cdots - 484 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots - 1136 \) Copy content Toggle raw display
$37$ \( T^{4} + 9 T^{3} + \cdots - 44 \) Copy content Toggle raw display
$41$ \( T^{4} - 59 T^{2} + \cdots - 271 \) Copy content Toggle raw display
$43$ \( T^{4} + 19 T^{3} + \cdots - 44 \) Copy content Toggle raw display
$47$ \( T^{4} + 3 T^{3} + \cdots - 244 \) Copy content Toggle raw display
$53$ \( T^{4} - 3 T^{3} + \cdots + 1516 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + \cdots + 5651 \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + \cdots - 2816 \) Copy content Toggle raw display
$67$ \( T^{4} - 13 T^{3} + \cdots - 604 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} + \cdots - 5296 \) Copy content Toggle raw display
$73$ \( T^{4} + T^{3} + \cdots + 4076 \) Copy content Toggle raw display
$79$ \( T^{4} + 2 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$83$ \( T^{4} + 27 T^{3} + \cdots - 1324 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{4} + 13 T^{3} + \cdots - 2384 \) Copy content Toggle raw display
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