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Properties

Label 9680.2.a.ck
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) 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 220)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{13} + \beta_1 q^{15} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + (4 \beta_{2} + \beta_1 + 5) q^{21} + ( - \beta_{3} + 5 \beta_{2} + 5) q^{23} + q^{25} + ( - \beta_{3} - \beta_{2} + \beta_1 - 4) q^{27} + ( - \beta_{2} + 3 \beta_1 - 1) q^{29} + ( - 3 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{31}+ \cdots + ( - 4 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} - 3 q^{7} + 4 q^{9} + 3 q^{13} + 2 q^{15} + q^{17} - 4 q^{19} + 14 q^{21} + 11 q^{23} + 4 q^{25} - 11 q^{27} + 4 q^{29} + q^{31} + 3 q^{35} - 17 q^{37} - 19 q^{39} + 9 q^{41} - 5 q^{43} - 4 q^{45} - q^{47} + 3 q^{49} - 3 q^{51} + 21 q^{53} + 17 q^{57} + 17 q^{59} + 28 q^{61} - 13 q^{63} - 3 q^{65} - 13 q^{67} - 8 q^{69} - 17 q^{71} + 13 q^{73} - 2 q^{75} - 22 q^{79} - 24 q^{81} + 21 q^{83} - q^{85} - 47 q^{87} - 4 q^{89} + 4 q^{91} + 7 q^{93} + 4 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 4 \) 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
2.70636
2.12233
−1.12233
−1.70636
0 −2.70636 0 −1.00000 0 −3.76095 0 4.32440 0
1.2 0 −2.12233 0 −1.00000 0 −0.306360 0 1.50430 0
1.3 0 1.12233 0 −1.00000 0 −2.31167 0 −1.74037 0
1.4 0 1.70636 0 −1.00000 0 3.37899 0 −0.0883282 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.ck 4
4.b odd 2 1 2420.2.a.n 4
11.b odd 2 1 9680.2.a.cl 4
11.c even 5 2 880.2.bo.f 8
44.c even 2 1 2420.2.a.m 4
44.h odd 10 2 220.2.m.a ✓ 8
132.o even 10 2 1980.2.z.c 8
220.n odd 10 2 1100.2.n.c 8
220.v even 20 4 1100.2.cb.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.m.a ✓ 8 44.h odd 10 2
880.2.bo.f 8 11.c even 5 2
1100.2.n.c 8 220.n odd 10 2
1100.2.cb.c 16 220.v even 20 4
1980.2.z.c 8 132.o even 10 2
2420.2.a.m 4 44.c even 2 1
2420.2.a.n 4 4.b odd 2 1
9680.2.a.ck 4 1.a even 1 1 trivial
9680.2.a.cl 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{4} + 2T_{3}^{3} - 6T_{3}^{2} - 7T_{3} + 11 \) Copy content Toggle raw display
\( T_{7}^{4} + 3T_{7}^{3} - 11T_{7}^{2} - 33T_{7} - 9 \) Copy content Toggle raw display
\( T_{13}^{4} - 3T_{13}^{3} - 21T_{13}^{2} + 93T_{13} - 89 \) Copy content Toggle raw display
\( T_{17}^{4} - T_{17}^{3} - 24T_{17}^{2} - 16T_{17} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + \cdots - 9 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} + \cdots - 89 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots + 11 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{4} - 11 T^{3} + \cdots - 99 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{4} - T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$37$ \( T^{4} + 17 T^{3} + \cdots - 1019 \) Copy content Toggle raw display
$41$ \( T^{4} - 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots - 145 \) Copy content Toggle raw display
$47$ \( T^{4} + T^{3} + \cdots + 101 \) Copy content Toggle raw display
$53$ \( T^{4} - 21 T^{3} + \cdots + 241 \) Copy content Toggle raw display
$59$ \( T^{4} - 17 T^{3} + \cdots - 1089 \) Copy content Toggle raw display
$61$ \( T^{4} - 28 T^{3} + \cdots - 16529 \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + \cdots + 881 \) Copy content Toggle raw display
$71$ \( T^{4} + 17 T^{3} + \cdots - 6599 \) Copy content Toggle raw display
$73$ \( T^{4} - 13 T^{3} + \cdots + 801 \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + \cdots - 1459 \) Copy content Toggle raw display
$83$ \( T^{4} - 21 T^{3} + \cdots - 2439 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots + 881 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + \cdots + 981 \) Copy content Toggle raw display
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