Properties

Label 3211.2.a.a
Level $3211$
Weight $2$
Character orbit 3211.a
Self dual yes
Analytic conductor $25.640$
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] = [3211,2,Mod(1,3211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3211, 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("3211.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3211 = 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3211.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: \(25.6399640890\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} - 2 q^{4} - 3 q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} - 2 q^{4} - 3 q^{5} + q^{7} + q^{9} - 3 q^{11} + 4 q^{12} + 6 q^{15} + 4 q^{16} - 3 q^{17} - q^{19} + 6 q^{20} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 2 q^{28} + 6 q^{29} + 4 q^{31} + 6 q^{33} - 3 q^{35} - 2 q^{36} - 2 q^{37} + 6 q^{41} - q^{43} + 6 q^{44} - 3 q^{45} + 3 q^{47} - 8 q^{48} - 6 q^{49} + 6 q^{51} + 12 q^{53} + 9 q^{55} + 2 q^{57} + 6 q^{59} - 12 q^{60} - q^{61} + q^{63} - 8 q^{64} + 4 q^{67} + 6 q^{68} - 6 q^{71} + 7 q^{73} - 8 q^{75} + 2 q^{76} - 3 q^{77} + 8 q^{79} - 12 q^{80} - 11 q^{81} - 12 q^{83} + 4 q^{84} + 9 q^{85} - 12 q^{87} - 12 q^{89} - 8 q^{93} + 3 q^{95} - 8 q^{97} - 3 q^{99}+O(q^{100}) \) 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
0 −2.00000 −2.00000 −3.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)
\(19\) \( +1 \)

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 3211.2.a.a 1
13.b even 2 1 19.2.a.a 1
39.d odd 2 1 171.2.a.b 1
52.b odd 2 1 304.2.a.f 1
65.d even 2 1 475.2.a.b 1
65.h odd 4 2 475.2.b.a 2
91.b odd 2 1 931.2.a.a 1
91.r even 6 2 931.2.f.c 2
91.s odd 6 2 931.2.f.b 2
104.e even 2 1 1216.2.a.o 1
104.h odd 2 1 1216.2.a.b 1
143.d odd 2 1 2299.2.a.b 1
156.h even 2 1 2736.2.a.c 1
195.e odd 2 1 4275.2.a.i 1
221.b even 2 1 5491.2.a.b 1
247.d odd 2 1 361.2.a.b 1
247.n odd 6 2 361.2.c.a 2
247.q even 6 2 361.2.c.c 2
247.bl odd 18 6 361.2.e.e 6
247.bn even 18 6 361.2.e.d 6
260.g odd 2 1 7600.2.a.c 1
273.g even 2 1 8379.2.a.j 1
741.d even 2 1 3249.2.a.d 1
988.g even 2 1 5776.2.a.c 1
1235.e odd 2 1 9025.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 13.b even 2 1
171.2.a.b 1 39.d odd 2 1
304.2.a.f 1 52.b odd 2 1
361.2.a.b 1 247.d odd 2 1
361.2.c.a 2 247.n odd 6 2
361.2.c.c 2 247.q even 6 2
361.2.e.d 6 247.bn even 18 6
361.2.e.e 6 247.bl odd 18 6
475.2.a.b 1 65.d even 2 1
475.2.b.a 2 65.h odd 4 2
931.2.a.a 1 91.b odd 2 1
931.2.f.b 2 91.s odd 6 2
931.2.f.c 2 91.r even 6 2
1216.2.a.b 1 104.h odd 2 1
1216.2.a.o 1 104.e even 2 1
2299.2.a.b 1 143.d odd 2 1
2736.2.a.c 1 156.h even 2 1
3211.2.a.a 1 1.a even 1 1 trivial
3249.2.a.d 1 741.d even 2 1
4275.2.a.i 1 195.e odd 2 1
5491.2.a.b 1 221.b even 2 1
5776.2.a.c 1 988.g even 2 1
7600.2.a.c 1 260.g odd 2 1
8379.2.a.j 1 273.g even 2 1
9025.2.a.d 1 1235.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\). Copy content Toggle raw display

Hecke characteristic polynomials

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