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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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2q^{7} + 3q^{8} + q^{9} - q^{10} + 2q^{11} + q^{12} + 2q^{14} - q^{15} - q^{16} - 7q^{17} - q^{18} + 6q^{19} - q^{20} + 2q^{21} - 2q^{22} - 6q^{23} - 3q^{24} - 4q^{25} - q^{27} + 2q^{28} - q^{29} + q^{30} - 4q^{31} - 5q^{32} - 2q^{33} + 7q^{34} - 2q^{35} - q^{36} - q^{37} - 6q^{38} + 3q^{40} - 9q^{41} - 2q^{42} + 6q^{43} - 2q^{44} + q^{45} + 6q^{46} - 6q^{47} + q^{48} - 3q^{49} + 4q^{50} + 7q^{51} - 9q^{53} + q^{54} + 2q^{55} - 6q^{56} - 6q^{57} + q^{58} + q^{60} + q^{61} + 4q^{62} - 2q^{63} + 7q^{64} + 2q^{66} + 2q^{67} + 7q^{68} + 6q^{69} + 2q^{70} - 6q^{71} + 3q^{72} - 11q^{73} + q^{74} + 4q^{75} - 6q^{76} - 4q^{77} - 4q^{79} - q^{80} + q^{81} + 9q^{82} + 14q^{83} - 2q^{84} - 7q^{85} - 6q^{86} + q^{87} + 6q^{88} + 14q^{89} - q^{90} + 6q^{92} + 4q^{93} + 6q^{94} + 6q^{95} + 5q^{96} + 2q^{97} + 3q^{98} + 2q^{99} + O(q^{100}) \)

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2} + 1 \)
\( T_{5} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( 2 + T \)
$11$ \( -2 + T \)
$13$ \( T \)
$17$ \( 7 + T \)
$19$ \( -6 + T \)
$23$ \( 6 + T \)
$29$ \( 1 + T \)
$31$ \( 4 + T \)
$37$ \( 1 + T \)
$41$ \( 9 + T \)
$43$ \( -6 + T \)
$47$ \( 6 + T \)
$53$ \( 9 + T \)
$59$ \( T \)
$61$ \( -1 + T \)
$67$ \( -2 + T \)
$71$ \( 6 + T \)
$73$ \( 11 + T \)
$79$ \( 4 + T \)
$83$ \( -14 + T \)
$89$ \( -14 + T \)
$97$ \( -2 + T \)
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