Properties

Label 507.4.a.d
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
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

\(f(q)\) \(=\) \( q + q^{2} + 3 q^{3} - 7 q^{4} - 7 q^{5} + 3 q^{6} + 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 3 q^{3} - 7 q^{4} - 7 q^{5} + 3 q^{6} + 10 q^{7} - 15 q^{8} + 9 q^{9} - 7 q^{10} + 22 q^{11} - 21 q^{12} + 10 q^{14} - 21 q^{15} + 41 q^{16} + 37 q^{17} + 9 q^{18} - 30 q^{19} + 49 q^{20} + 30 q^{21} + 22 q^{22} - 162 q^{23} - 45 q^{24} - 76 q^{25} + 27 q^{27} - 70 q^{28} - 113 q^{29} - 21 q^{30} - 196 q^{31} + 161 q^{32} + 66 q^{33} + 37 q^{34} - 70 q^{35} - 63 q^{36} - 13 q^{37} - 30 q^{38} + 105 q^{40} - 285 q^{41} + 30 q^{42} - 246 q^{43} - 154 q^{44} - 63 q^{45} - 162 q^{46} + 462 q^{47} + 123 q^{48} - 243 q^{49} - 76 q^{50} + 111 q^{51} - 537 q^{53} + 27 q^{54} - 154 q^{55} - 150 q^{56} - 90 q^{57} - 113 q^{58} - 576 q^{59} + 147 q^{60} - 635 q^{61} - 196 q^{62} + 90 q^{63} - 167 q^{64} + 66 q^{66} - 202 q^{67} - 259 q^{68} - 486 q^{69} - 70 q^{70} + 1086 q^{71} - 135 q^{72} + 805 q^{73} - 13 q^{74} - 228 q^{75} + 210 q^{76} + 220 q^{77} + 884 q^{79} - 287 q^{80} + 81 q^{81} - 285 q^{82} - 518 q^{83} - 210 q^{84} - 259 q^{85} - 246 q^{86} - 339 q^{87} - 330 q^{88} - 194 q^{89} - 63 q^{90} + 1134 q^{92} - 588 q^{93} + 462 q^{94} + 210 q^{95} + 483 q^{96} + 1202 q^{97} - 243 q^{98} + 198 q^{99}+O(q^{100}) \) 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.

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 3.00000 −7.00000 −7.00000 3.00000 10.0000 −15.0000 9.00000 −7.00000
\(n\): 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 507.4.a.d 1
3.b odd 2 1 1521.4.a.e 1
13.b even 2 1 507.4.a.b 1
13.d odd 4 2 507.4.b.d 2
13.e even 6 2 39.4.e.b 2
39.d odd 2 1 1521.4.a.h 1
39.h odd 6 2 117.4.g.a 2
52.i odd 6 2 624.4.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 13.e even 6 2
117.4.g.a 2 39.h odd 6 2
507.4.a.b 1 13.b even 2 1
507.4.a.d 1 1.a even 1 1 trivial
507.4.b.d 2 13.d odd 4 2
624.4.q.c 2 52.i odd 6 2
1521.4.a.e 1 3.b odd 2 1
1521.4.a.h 1 39.d odd 2 1

Hecke kernels

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

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 7 \) Copy content Toggle raw display
$7$ \( T - 10 \) Copy content Toggle raw display
$11$ \( T - 22 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 37 \) Copy content Toggle raw display
$19$ \( T + 30 \) Copy content Toggle raw display
$23$ \( T + 162 \) Copy content Toggle raw display
$29$ \( T + 113 \) Copy content Toggle raw display
$31$ \( T + 196 \) Copy content Toggle raw display
$37$ \( T + 13 \) Copy content Toggle raw display
$41$ \( T + 285 \) Copy content Toggle raw display
$43$ \( T + 246 \) Copy content Toggle raw display
$47$ \( T - 462 \) Copy content Toggle raw display
$53$ \( T + 537 \) Copy content Toggle raw display
$59$ \( T + 576 \) Copy content Toggle raw display
$61$ \( T + 635 \) Copy content Toggle raw display
$67$ \( T + 202 \) Copy content Toggle raw display
$71$ \( T - 1086 \) Copy content Toggle raw display
$73$ \( T - 805 \) Copy content Toggle raw display
$79$ \( T - 884 \) Copy content Toggle raw display
$83$ \( T + 518 \) Copy content Toggle raw display
$89$ \( T + 194 \) Copy content Toggle raw display
$97$ \( T - 1202 \) Copy content Toggle raw display
show more
show less