Properties

Label 507.4.a.c
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
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] = [507,4,Mod(1,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("507.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.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: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 8 q^{4} + 12 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 8 q^{4} + 12 q^{5} - 2 q^{7} + 9 q^{9} + 36 q^{11} + 24 q^{12} - 36 q^{15} + 64 q^{16} - 78 q^{17} - 74 q^{19} - 96 q^{20} + 6 q^{21} - 96 q^{23} + 19 q^{25} - 27 q^{27} + 16 q^{28} + 18 q^{29} + 214 q^{31} - 108 q^{33} - 24 q^{35} - 72 q^{36} + 286 q^{37} + 384 q^{41} + 524 q^{43} - 288 q^{44} + 108 q^{45} - 300 q^{47} - 192 q^{48} - 339 q^{49} + 234 q^{51} + 558 q^{53} + 432 q^{55} + 222 q^{57} - 576 q^{59} + 288 q^{60} + 74 q^{61} - 18 q^{63} - 512 q^{64} - 38 q^{67} + 624 q^{68} + 288 q^{69} + 456 q^{71} + 682 q^{73} - 57 q^{75} + 592 q^{76} - 72 q^{77} + 704 q^{79} + 768 q^{80} + 81 q^{81} + 888 q^{83} - 48 q^{84} - 936 q^{85} - 54 q^{87} + 1020 q^{89} + 768 q^{92} - 642 q^{93} - 888 q^{95} - 110 q^{97} + 324 q^{99}+O(q^{100}) \) 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 −8.00000 12.0000 0 −2.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(13\) \( +1 \)

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 507.4.a.c 1
3.b odd 2 1 1521.4.a.f 1
13.b even 2 1 39.4.a.a 1
13.d odd 4 2 507.4.b.b 2
39.d odd 2 1 117.4.a.a 1
52.b odd 2 1 624.4.a.g 1
65.d even 2 1 975.4.a.e 1
91.b odd 2 1 1911.4.a.f 1
104.e even 2 1 2496.4.a.o 1
104.h odd 2 1 2496.4.a.f 1
156.h even 2 1 1872.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 13.b even 2 1
117.4.a.a 1 39.d odd 2 1
507.4.a.c 1 1.a even 1 1 trivial
507.4.b.b 2 13.d odd 4 2
624.4.a.g 1 52.b odd 2 1
975.4.a.e 1 65.d even 2 1
1521.4.a.f 1 3.b odd 2 1
1872.4.a.m 1 156.h even 2 1
1911.4.a.f 1 91.b odd 2 1
2496.4.a.f 1 104.h odd 2 1
2496.4.a.o 1 104.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} - 12 \) 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
$5$ \( T - 12 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 36 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 78 \) Copy content Toggle raw display
$19$ \( T + 74 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T - 18 \) Copy content Toggle raw display
$31$ \( T - 214 \) Copy content Toggle raw display
$37$ \( T - 286 \) Copy content Toggle raw display
$41$ \( T - 384 \) Copy content Toggle raw display
$43$ \( T - 524 \) Copy content Toggle raw display
$47$ \( T + 300 \) Copy content Toggle raw display
$53$ \( T - 558 \) Copy content Toggle raw display
$59$ \( T + 576 \) Copy content Toggle raw display
$61$ \( T - 74 \) Copy content Toggle raw display
$67$ \( T + 38 \) Copy content Toggle raw display
$71$ \( T - 456 \) Copy content Toggle raw display
$73$ \( T - 682 \) Copy content Toggle raw display
$79$ \( T - 704 \) Copy content Toggle raw display
$83$ \( T - 888 \) Copy content Toggle raw display
$89$ \( T - 1020 \) Copy content Toggle raw display
$97$ \( T + 110 \) Copy content Toggle raw display
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