Properties

Label 507.2.a.a
Level $507$
Weight $2$
Character orbit 507.a
Self dual yes
Analytic conductor $4.048$
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] = [507,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(4.04841538248\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} - 4 q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} + 2 q^{20} - 4 q^{21} + 4 q^{22} - 3 q^{24} - q^{25} - q^{27} - 4 q^{28} - 10 q^{29} - 2 q^{30} - 4 q^{31} - 5 q^{32} + 4 q^{33} - 2 q^{34} - 8 q^{35} - q^{36} + 2 q^{37} - 6 q^{40} - 6 q^{41} + 4 q^{42} - 12 q^{43} + 4 q^{44} - 2 q^{45} + q^{48} + 9 q^{49} + q^{50} - 2 q^{51} + 6 q^{53} + q^{54} + 8 q^{55} + 12 q^{56} + 10 q^{58} - 12 q^{59} - 2 q^{60} - 2 q^{61} + 4 q^{62} + 4 q^{63} + 7 q^{64} - 4 q^{66} + 8 q^{67} - 2 q^{68} + 8 q^{70} + 3 q^{72} - 2 q^{73} - 2 q^{74} + q^{75} - 16 q^{77} + 8 q^{79} + 2 q^{80} + q^{81} + 6 q^{82} - 4 q^{83} + 4 q^{84} - 4 q^{85} + 12 q^{86} + 10 q^{87} - 12 q^{88} + 2 q^{89} + 2 q^{90} + 4 q^{93} + 5 q^{96} - 10 q^{97} - 9 q^{98} - 4 q^{99}+O(q^{100}) \) 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 4.00000 3.00000 1.00000 2.00000
\(n\): 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 507.2.a.a 1
3.b odd 2 1 1521.2.a.e 1
4.b odd 2 1 8112.2.a.s 1
13.b even 2 1 39.2.a.a 1
13.c even 3 2 507.2.e.b 2
13.d odd 4 2 507.2.b.a 2
13.e even 6 2 507.2.e.a 2
13.f odd 12 4 507.2.j.e 4
39.d odd 2 1 117.2.a.a 1
39.f even 4 2 1521.2.b.b 2
52.b odd 2 1 624.2.a.i 1
65.d even 2 1 975.2.a.f 1
65.h odd 4 2 975.2.c.f 2
91.b odd 2 1 1911.2.a.f 1
104.e even 2 1 2496.2.a.q 1
104.h odd 2 1 2496.2.a.e 1
117.n odd 6 2 1053.2.e.d 2
117.t even 6 2 1053.2.e.b 2
143.d odd 2 1 4719.2.a.c 1
156.h even 2 1 1872.2.a.h 1
195.e odd 2 1 2925.2.a.p 1
195.s even 4 2 2925.2.c.e 2
273.g even 2 1 5733.2.a.e 1
312.b odd 2 1 7488.2.a.bl 1
312.h even 2 1 7488.2.a.by 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 13.b even 2 1
117.2.a.a 1 39.d odd 2 1
507.2.a.a 1 1.a even 1 1 trivial
507.2.b.a 2 13.d odd 4 2
507.2.e.a 2 13.e even 6 2
507.2.e.b 2 13.c even 3 2
507.2.j.e 4 13.f odd 12 4
624.2.a.i 1 52.b odd 2 1
975.2.a.f 1 65.d even 2 1
975.2.c.f 2 65.h odd 4 2
1053.2.e.b 2 117.t even 6 2
1053.2.e.d 2 117.n odd 6 2
1521.2.a.e 1 3.b odd 2 1
1521.2.b.b 2 39.f even 4 2
1872.2.a.h 1 156.h even 2 1
1911.2.a.f 1 91.b odd 2 1
2496.2.a.e 1 104.h odd 2 1
2496.2.a.q 1 104.e even 2 1
2925.2.a.p 1 195.e odd 2 1
2925.2.c.e 2 195.s even 4 2
4719.2.a.c 1 143.d odd 2 1
5733.2.a.e 1 273.g even 2 1
7488.2.a.bl 1 312.b odd 2 1
7488.2.a.by 1 312.h even 2 1
8112.2.a.s 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 2 \) 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
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T \) 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
$29$ \( T + 10 \) 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
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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