Properties

Label 9450.2.a.ba
Level $9450$
Weight $2$
Character orbit 9450.a
Self dual yes
Analytic conductor $75.459$
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] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, 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("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.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: \(75.4586299101\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{7} - q^{8} - 6 q^{11} + 5 q^{13} - q^{14} + q^{16} - 6 q^{17} + 2 q^{19} + 6 q^{22} - 5 q^{26} + q^{28} - 9 q^{29} + 5 q^{31} - q^{32} + 6 q^{34} - 4 q^{37} - 2 q^{38} + 3 q^{41} - 4 q^{43} - 6 q^{44} - 6 q^{47} + q^{49} + 5 q^{52} + 9 q^{53} - q^{56} + 9 q^{58} + 12 q^{59} - 7 q^{61} - 5 q^{62} + q^{64} + 5 q^{67} - 6 q^{68} + 9 q^{71} - 4 q^{73} + 4 q^{74} + 2 q^{76} - 6 q^{77} - 10 q^{79} - 3 q^{82} + 3 q^{83} + 4 q^{86} + 6 q^{88} + 9 q^{89} + 5 q^{91} + 6 q^{94} - 4 q^{97} - q^{98}+O(q^{100}) \) 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 0 1.00000 0 0 1.00000 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 9450.2.a.ba yes 1
3.b odd 2 1 9450.2.a.dz yes 1
5.b even 2 1 9450.2.a.ca yes 1
15.d odd 2 1 9450.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9450.2.a.y 1 15.d odd 2 1
9450.2.a.ba yes 1 1.a even 1 1 trivial
9450.2.a.ca yes 1 5.b even 2 1
9450.2.a.dz yes 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11} + 6 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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