Properties

Label 9310.2.a.u
Level $9310$
Weight $2$
Character orbit 9310.a
Self dual yes
Analytic conductor $74.341$
Analytic rank $0$
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] = [9310,2,Mod(1,9310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9310 = 2 \cdot 5 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9310.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: \(74.3407242818\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + 3 q^{3} + q^{4} + q^{5} + 3 q^{6} + q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 3 q^{3} + q^{4} + q^{5} + 3 q^{6} + q^{8} + 6 q^{9} + q^{10} - 4 q^{11} + 3 q^{12} + q^{13} + 3 q^{15} + q^{16} + 3 q^{17} + 6 q^{18} - q^{19} + q^{20} - 4 q^{22} + 7 q^{23} + 3 q^{24} + q^{25} + q^{26} + 9 q^{27} - 3 q^{29} + 3 q^{30} + 2 q^{31} + q^{32} - 12 q^{33} + 3 q^{34} + 6 q^{36} - 2 q^{37} - q^{38} + 3 q^{39} + q^{40} + 6 q^{41} + 6 q^{43} - 4 q^{44} + 6 q^{45} + 7 q^{46} + 3 q^{48} + q^{50} + 9 q^{51} + q^{52} - 13 q^{53} + 9 q^{54} - 4 q^{55} - 3 q^{57} - 3 q^{58} + 9 q^{59} + 3 q^{60} + 12 q^{61} + 2 q^{62} + q^{64} + q^{65} - 12 q^{66} - 3 q^{67} + 3 q^{68} + 21 q^{69} + 6 q^{72} - 11 q^{73} - 2 q^{74} + 3 q^{75} - q^{76} + 3 q^{78} - 2 q^{79} + q^{80} + 9 q^{81} + 6 q^{82} + 10 q^{83} + 3 q^{85} + 6 q^{86} - 9 q^{87} - 4 q^{88} - 2 q^{89} + 6 q^{90} + 7 q^{92} + 6 q^{93} - q^{95} + 3 q^{96} + 2 q^{97} - 24 q^{99}+O(q^{100}) \) 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 3.00000 1.00000 1.00000 3.00000 0 1.00000 6.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(19\) \(1\)

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 9310.2.a.u 1
7.b odd 2 1 190.2.a.b 1
21.c even 2 1 1710.2.a.g 1
28.d even 2 1 1520.2.a.j 1
35.c odd 2 1 950.2.a.c 1
35.f even 4 2 950.2.b.a 2
56.e even 2 1 6080.2.a.b 1
56.h odd 2 1 6080.2.a.x 1
105.g even 2 1 8550.2.a.bm 1
133.c even 2 1 3610.2.a.e 1
140.c even 2 1 7600.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.b 1 7.b odd 2 1
950.2.a.c 1 35.c odd 2 1
950.2.b.a 2 35.f even 4 2
1520.2.a.j 1 28.d even 2 1
1710.2.a.g 1 21.c even 2 1
3610.2.a.e 1 133.c even 2 1
6080.2.a.b 1 56.e even 2 1
6080.2.a.x 1 56.h odd 2 1
7600.2.a.a 1 140.c even 2 1
8550.2.a.bm 1 105.g even 2 1
9310.2.a.u 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9310))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 7 \) Copy content Toggle raw display
$29$ \( T + 3 \) Copy content Toggle raw display
$31$ \( T - 2 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 13 \) Copy content Toggle raw display
$59$ \( T - 9 \) Copy content Toggle raw display
$61$ \( T - 12 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T + 2 \) Copy content Toggle raw display
$83$ \( T - 10 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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