Properties

Label 507.2.a.b
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 3 q^{8} + q^{9} - q^{10} + 2 q^{11} + q^{12} + 2 q^{14} - q^{15} - q^{16} - 7 q^{17} - q^{18} + 6 q^{19} - q^{20} + 2 q^{21} - 2 q^{22} - 6 q^{23} - 3 q^{24} - 4 q^{25} - q^{27} + 2 q^{28} - q^{29} + q^{30} - 4 q^{31} - 5 q^{32} - 2 q^{33} + 7 q^{34} - 2 q^{35} - q^{36} - q^{37} - 6 q^{38} + 3 q^{40} - 9 q^{41} - 2 q^{42} + 6 q^{43} - 2 q^{44} + q^{45} + 6 q^{46} - 6 q^{47} + q^{48} - 3 q^{49} + 4 q^{50} + 7 q^{51} - 9 q^{53} + q^{54} + 2 q^{55} - 6 q^{56} - 6 q^{57} + q^{58} + q^{60} + q^{61} + 4 q^{62} - 2 q^{63} + 7 q^{64} + 2 q^{66} + 2 q^{67} + 7 q^{68} + 6 q^{69} + 2 q^{70} - 6 q^{71} + 3 q^{72} - 11 q^{73} + q^{74} + 4 q^{75} - 6 q^{76} - 4 q^{77} - 4 q^{79} - q^{80} + q^{81} + 9 q^{82} + 14 q^{83} - 2 q^{84} - 7 q^{85} - 6 q^{86} + q^{87} + 6 q^{88} + 14 q^{89} - q^{90} + 6 q^{92} + 4 q^{93} + 6 q^{94} + 6 q^{95} + 5 q^{96} + 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 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 1.00000 1.00000 −2.00000 3.00000 1.00000 −1.00000
\(n\): 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.b 1
3.b odd 2 1 1521.2.a.d 1
4.b odd 2 1 8112.2.a.bc 1
13.b even 2 1 507.2.a.c 1
13.c even 3 2 507.2.e.c 2
13.d odd 4 2 507.2.b.b 2
13.e even 6 2 39.2.e.a 2
13.f odd 12 4 507.2.j.d 4
39.d odd 2 1 1521.2.a.a 1
39.f even 4 2 1521.2.b.c 2
39.h odd 6 2 117.2.g.b 2
52.b odd 2 1 8112.2.a.w 1
52.i odd 6 2 624.2.q.c 2
65.l even 6 2 975.2.i.f 2
65.r odd 12 4 975.2.bb.d 4
156.r even 6 2 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.e even 6 2
117.2.g.b 2 39.h odd 6 2
507.2.a.b 1 1.a even 1 1 trivial
507.2.a.c 1 13.b even 2 1
507.2.b.b 2 13.d odd 4 2
507.2.e.c 2 13.c even 3 2
507.2.j.d 4 13.f odd 12 4
624.2.q.c 2 52.i odd 6 2
975.2.i.f 2 65.l even 6 2
975.2.bb.d 4 65.r odd 12 4
1521.2.a.a 1 39.d odd 2 1
1521.2.a.d 1 3.b odd 2 1
1521.2.b.c 2 39.f even 4 2
1872.2.t.j 2 156.r even 6 2
8112.2.a.w 1 52.b odd 2 1
8112.2.a.bc 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
\( T_{5} - 1 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 7 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T + 9 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 6 \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 14 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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