Properties

Label 3249.2.a.b
Level $3249$
Weight $2$
Character orbit 3249.a
Self dual yes
Analytic conductor $25.943$
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] = [3249,2,Mod(1,3249)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3249, 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("3249.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.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: \(25.9433956167\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{10} - q^{11} - 2 q^{13} + 10 q^{14} - 4 q^{16} + q^{17} + 6 q^{20} + 2 q^{22} + 4 q^{23} + 4 q^{25} + 4 q^{26} - 10 q^{28} - 2 q^{29} + 6 q^{31} + 8 q^{32} - 2 q^{34} - 15 q^{35} - q^{43} - 2 q^{44} - 8 q^{46} + 9 q^{47} + 18 q^{49} - 8 q^{50} - 4 q^{52} + 10 q^{53} - 3 q^{55} + 4 q^{58} - 8 q^{59} - q^{61} - 12 q^{62} - 8 q^{64} - 6 q^{65} - 8 q^{67} + 2 q^{68} + 30 q^{70} - 12 q^{71} - 11 q^{73} + 5 q^{77} - 16 q^{79} - 12 q^{80} - 12 q^{83} + 3 q^{85} + 2 q^{86} - 6 q^{89} + 10 q^{91} + 8 q^{92} - 18 q^{94} + 10 q^{97} - 36 q^{98}+O(q^{100}) \) 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
−2.00000 0 2.00000 3.00000 0 −5.00000 0 0 −6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(19\) \( -1 \)

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 3249.2.a.b 1
3.b odd 2 1 1083.2.a.e 1
19.b odd 2 1 171.2.a.d 1
57.d even 2 1 57.2.a.a 1
76.d even 2 1 2736.2.a.v 1
95.d odd 2 1 4275.2.a.b 1
133.c even 2 1 8379.2.a.p 1
228.b odd 2 1 912.2.a.g 1
285.b even 2 1 1425.2.a.j 1
285.j odd 4 2 1425.2.c.b 2
399.h odd 2 1 2793.2.a.b 1
456.l odd 2 1 3648.2.a.r 1
456.p even 2 1 3648.2.a.bh 1
627.b odd 2 1 6897.2.a.f 1
741.d even 2 1 9633.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.a 1 57.d even 2 1
171.2.a.d 1 19.b odd 2 1
912.2.a.g 1 228.b odd 2 1
1083.2.a.e 1 3.b odd 2 1
1425.2.a.j 1 285.b even 2 1
1425.2.c.b 2 285.j odd 4 2
2736.2.a.v 1 76.d even 2 1
2793.2.a.b 1 399.h odd 2 1
3249.2.a.b 1 1.a even 1 1 trivial
3648.2.a.r 1 456.l odd 2 1
3648.2.a.bh 1 456.p even 2 1
4275.2.a.b 1 95.d odd 2 1
6897.2.a.f 1 627.b odd 2 1
8379.2.a.p 1 133.c even 2 1
9633.2.a.o 1 741.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T + 5 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 6 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T - 9 \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T + 8 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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