Properties

Label 624.4.a.h
Level $624$
Weight $4$
Character orbit 624.a
Self dual yes
Analytic conductor $36.817$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} - 6 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 6 q^{5} + 4 q^{7} + 9 q^{9} - 36 q^{11} + 13 q^{13} - 18 q^{15} + 66 q^{17} - 56 q^{19} + 12 q^{21} - 96 q^{23} - 89 q^{25} + 27 q^{27} + 222 q^{29} - 260 q^{31} - 108 q^{33} - 24 q^{35} - 106 q^{37} + 39 q^{39} - 90 q^{41} - 44 q^{43} - 54 q^{45} - 168 q^{47} - 327 q^{49} + 198 q^{51} + 30 q^{53} + 216 q^{55} - 168 q^{57} - 348 q^{59} - 346 q^{61} + 36 q^{63} - 78 q^{65} + 256 q^{67} - 288 q^{69} + 168 q^{71} - 814 q^{73} - 267 q^{75} - 144 q^{77} - 200 q^{79} + 81 q^{81} - 1236 q^{83} - 396 q^{85} + 666 q^{87} + 318 q^{89} + 52 q^{91} - 780 q^{93} + 336 q^{95} - 502 q^{97} - 324 q^{99}+O(q^{100}) \) 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 3.00000 0 −6.00000 0 4.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

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 624.4.a.h 1
3.b odd 2 1 1872.4.a.j 1
4.b odd 2 1 156.4.a.a 1
8.b even 2 1 2496.4.a.e 1
8.d odd 2 1 2496.4.a.n 1
12.b even 2 1 468.4.a.b 1
52.b odd 2 1 2028.4.a.a 1
52.f even 4 2 2028.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.a 1 4.b odd 2 1
468.4.a.b 1 12.b even 2 1
624.4.a.h 1 1.a even 1 1 trivial
1872.4.a.j 1 3.b odd 2 1
2028.4.a.a 1 52.b odd 2 1
2028.4.b.a 2 52.f even 4 2
2496.4.a.e 1 8.b even 2 1
2496.4.a.n 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(624))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 36 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 66 \) Copy content Toggle raw display
$19$ \( T + 56 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T - 222 \) Copy content Toggle raw display
$31$ \( T + 260 \) Copy content Toggle raw display
$37$ \( T + 106 \) Copy content Toggle raw display
$41$ \( T + 90 \) Copy content Toggle raw display
$43$ \( T + 44 \) Copy content Toggle raw display
$47$ \( T + 168 \) Copy content Toggle raw display
$53$ \( T - 30 \) Copy content Toggle raw display
$59$ \( T + 348 \) Copy content Toggle raw display
$61$ \( T + 346 \) Copy content Toggle raw display
$67$ \( T - 256 \) Copy content Toggle raw display
$71$ \( T - 168 \) Copy content Toggle raw display
$73$ \( T + 814 \) Copy content Toggle raw display
$79$ \( T + 200 \) Copy content Toggle raw display
$83$ \( T + 1236 \) Copy content Toggle raw display
$89$ \( T - 318 \) Copy content Toggle raw display
$97$ \( T + 502 \) Copy content Toggle raw display
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