Properties

Label 9610.2.a.a
Level $9610$
Weight $2$
Character orbit 9610.a
Self dual yes
Analytic conductor $76.736$
Analytic rank $1$
Dimension $1$
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] = [9610,2,Mod(1,9610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9610 = 2 \cdot 5 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9610.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: \(76.7362363425\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 310)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} + q^{8} + q^{9} - q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} - q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24} + q^{25} + 4 q^{27} + 4 q^{29} + 2 q^{30} + q^{32} + 4 q^{33} - 2 q^{34} + q^{36} + 8 q^{37} - 4 q^{38} - q^{40} + 6 q^{41} - 2 q^{43} - 2 q^{44} - q^{45} + 4 q^{46} - 2 q^{48} - 7 q^{49} + q^{50} + 4 q^{51} - 8 q^{53} + 4 q^{54} + 2 q^{55} + 8 q^{57} + 4 q^{58} + 8 q^{59} + 2 q^{60} + q^{64} + 4 q^{66} + 4 q^{67} - 2 q^{68} - 8 q^{69} + q^{72} - 6 q^{73} + 8 q^{74} - 2 q^{75} - 4 q^{76} + 4 q^{79} - q^{80} - 11 q^{81} + 6 q^{82} - 6 q^{83} + 2 q^{85} - 2 q^{86} - 8 q^{87} - 2 q^{88} + 6 q^{89} - q^{90} + 4 q^{92} + 4 q^{95} - 2 q^{96} - 2 q^{97} - 7 q^{98} - 2 q^{99}+O(q^{100}) \) 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
1.00000 −2.00000 1.00000 −1.00000 −2.00000 0 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(31\) \( -1 \)

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 9610.2.a.a 1
31.b odd 2 1 310.2.a.b 1
93.c even 2 1 2790.2.a.h 1
124.d even 2 1 2480.2.a.c 1
155.c odd 2 1 1550.2.a.a 1
155.f even 4 2 1550.2.b.e 2
248.b even 2 1 9920.2.a.bg 1
248.g odd 2 1 9920.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.b 1 31.b odd 2 1
1550.2.a.a 1 155.c odd 2 1
1550.2.b.e 2 155.f even 4 2
2480.2.a.c 1 124.d even 2 1
2790.2.a.h 1 93.c even 2 1
9610.2.a.a 1 1.a even 1 1 trivial
9920.2.a.d 1 248.g odd 2 1
9920.2.a.bg 1 248.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9610))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 4 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 8 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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