Properties

Label 624.4.a.d
Level $624$
Weight $4$
Character orbit 624.a
Self dual yes
Analytic conductor $36.817$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} + 10 q^{5} + 8 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 10 q^{5} + 8 q^{7} + 9 q^{9} - 40 q^{11} + 13 q^{13} - 30 q^{15} + 130 q^{17} + 20 q^{19} - 24 q^{21} - 25 q^{25} - 27 q^{27} - 18 q^{29} + 184 q^{31} + 120 q^{33} + 80 q^{35} - 74 q^{37} - 39 q^{39} - 362 q^{41} - 76 q^{43} + 90 q^{45} + 452 q^{47} - 279 q^{49} - 390 q^{51} + 382 q^{53} - 400 q^{55} - 60 q^{57} - 464 q^{59} + 358 q^{61} + 72 q^{63} + 130 q^{65} + 700 q^{67} + 748 q^{71} + 1058 q^{73} + 75 q^{75} - 320 q^{77} + 976 q^{79} + 81 q^{81} + 1008 q^{83} + 1300 q^{85} + 54 q^{87} - 386 q^{89} + 104 q^{91} - 552 q^{93} + 200 q^{95} - 614 q^{97} - 360 q^{99}+O(q^{100}) \) 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 10.0000 0 8.00000 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.d 1
3.b odd 2 1 1872.4.a.d 1
4.b odd 2 1 78.4.a.c 1
8.b even 2 1 2496.4.a.j 1
8.d odd 2 1 2496.4.a.a 1
12.b even 2 1 234.4.a.h 1
20.d odd 2 1 1950.4.a.l 1
52.b odd 2 1 1014.4.a.j 1
52.f even 4 2 1014.4.b.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.c 1 4.b odd 2 1
234.4.a.h 1 12.b even 2 1
624.4.a.d 1 1.a even 1 1 trivial
1014.4.a.j 1 52.b odd 2 1
1014.4.b.h 2 52.f even 4 2
1872.4.a.d 1 3.b odd 2 1
1950.4.a.l 1 20.d odd 2 1
2496.4.a.a 1 8.d odd 2 1
2496.4.a.j 1 8.b even 2 1

Hecke kernels

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

\( T_{5} - 10 \) Copy content Toggle raw display
\( T_{7} - 8 \) 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
$5$ \( T - 10 \) Copy content Toggle raw display
$7$ \( T - 8 \) Copy content Toggle raw display
$11$ \( T + 40 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 130 \) Copy content Toggle raw display
$19$ \( T - 20 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 18 \) Copy content Toggle raw display
$31$ \( T - 184 \) Copy content Toggle raw display
$37$ \( T + 74 \) Copy content Toggle raw display
$41$ \( T + 362 \) Copy content Toggle raw display
$43$ \( T + 76 \) Copy content Toggle raw display
$47$ \( T - 452 \) Copy content Toggle raw display
$53$ \( T - 382 \) Copy content Toggle raw display
$59$ \( T + 464 \) Copy content Toggle raw display
$61$ \( T - 358 \) Copy content Toggle raw display
$67$ \( T - 700 \) Copy content Toggle raw display
$71$ \( T - 748 \) Copy content Toggle raw display
$73$ \( T - 1058 \) Copy content Toggle raw display
$79$ \( T - 976 \) Copy content Toggle raw display
$83$ \( T - 1008 \) Copy content Toggle raw display
$89$ \( T + 386 \) Copy content Toggle raw display
$97$ \( T + 614 \) Copy content Toggle raw display
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