Properties

Label 462.2.a.g
Level $462$
Weight $2$
Character orbit 462.a
Self dual yes
Analytic conductor $3.689$
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] = [462,2,Mod(1,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, 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("462.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.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: \(3.68908857338\)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} - 2 q^{13} - q^{14} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + 2 q^{20} - q^{21} + q^{22} + q^{24} - q^{25} - 2 q^{26} + q^{27} - q^{28} - 2 q^{29} + 2 q^{30} + 4 q^{31} + q^{32} + q^{33} - 2 q^{34} - 2 q^{35} + q^{36} - 2 q^{37} - 2 q^{39} + 2 q^{40} - 10 q^{41} - q^{42} + 4 q^{43} + q^{44} + 2 q^{45} + 4 q^{47} + q^{48} + q^{49} - q^{50} - 2 q^{51} - 2 q^{52} - 2 q^{53} + q^{54} + 2 q^{55} - q^{56} - 2 q^{58} - 12 q^{59} + 2 q^{60} - 2 q^{61} + 4 q^{62} - q^{63} + q^{64} - 4 q^{65} + q^{66} + 12 q^{67} - 2 q^{68} - 2 q^{70} + 8 q^{71} + q^{72} + 6 q^{73} - 2 q^{74} - q^{75} - q^{77} - 2 q^{78} - 8 q^{79} + 2 q^{80} + q^{81} - 10 q^{82} - 8 q^{83} - q^{84} - 4 q^{85} + 4 q^{86} - 2 q^{87} + q^{88} - 14 q^{89} + 2 q^{90} + 2 q^{91} + 4 q^{93} + 4 q^{94} + q^{96} - 14 q^{97} + q^{98} + q^{99}+O(q^{100}) \) 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 1.00000 1.00000 2.00000 1.00000 −1.00000 1.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 462.2.a.g 1
3.b odd 2 1 1386.2.a.a 1
4.b odd 2 1 3696.2.a.m 1
7.b odd 2 1 3234.2.a.p 1
11.b odd 2 1 5082.2.a.n 1
21.c even 2 1 9702.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.g 1 1.a even 1 1 trivial
1386.2.a.a 1 3.b odd 2 1
3234.2.a.p 1 7.b odd 2 1
3696.2.a.m 1 4.b odd 2 1
5082.2.a.n 1 11.b odd 2 1
9702.2.a.r 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(462))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 8 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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