Properties

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

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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

\(f(q)\) \(=\) \( q + 3 q^{2} + 3 q^{3} + q^{4} - 9 q^{5} + 9 q^{6} + 2 q^{7} - 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + 3 q^{3} + q^{4} - 9 q^{5} + 9 q^{6} + 2 q^{7} - 21 q^{8} + 9 q^{9} - 27 q^{10} + 30 q^{11} + 3 q^{12} + 6 q^{14} - 27 q^{15} - 71 q^{16} - 111 q^{17} + 27 q^{18} - 46 q^{19} - 9 q^{20} + 6 q^{21} + 90 q^{22} - 6 q^{23} - 63 q^{24} - 44 q^{25} + 27 q^{27} + 2 q^{28} - 105 q^{29} - 81 q^{30} - 100 q^{31} - 45 q^{32} + 90 q^{33} - 333 q^{34} - 18 q^{35} + 9 q^{36} + 17 q^{37} - 138 q^{38} + 189 q^{40} - 231 q^{41} + 18 q^{42} - 514 q^{43} + 30 q^{44} - 81 q^{45} - 18 q^{46} - 162 q^{47} - 213 q^{48} - 339 q^{49} - 132 q^{50} - 333 q^{51} + 639 q^{53} + 81 q^{54} - 270 q^{55} - 42 q^{56} - 138 q^{57} - 315 q^{58} + 600 q^{59} - 27 q^{60} + 233 q^{61} - 300 q^{62} + 18 q^{63} + 433 q^{64} + 270 q^{66} + 926 q^{67} - 111 q^{68} - 18 q^{69} - 54 q^{70} - 930 q^{71} - 189 q^{72} - 253 q^{73} + 51 q^{74} - 132 q^{75} - 46 q^{76} + 60 q^{77} - 1324 q^{79} + 639 q^{80} + 81 q^{81} - 693 q^{82} + 810 q^{83} + 6 q^{84} + 999 q^{85} - 1542 q^{86} - 315 q^{87} - 630 q^{88} + 498 q^{89} - 243 q^{90} - 6 q^{92} - 300 q^{93} - 486 q^{94} + 414 q^{95} - 135 q^{96} + 1358 q^{97} - 1017 q^{98} + 270 q^{99}+O(q^{100}) \) 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
3.00000 3.00000 1.00000 −9.00000 9.00000 2.00000 −21.0000 9.00000 −27.0000
\(n\): 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.e 1
3.b odd 2 1 1521.4.a.c 1
13.b even 2 1 507.4.a.a 1
13.c even 3 2 39.4.e.a 2
13.d odd 4 2 507.4.b.c 2
39.d odd 2 1 1521.4.a.j 1
39.i odd 6 2 117.4.g.b 2
52.j odd 6 2 624.4.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 13.c even 3 2
117.4.g.b 2 39.i odd 6 2
507.4.a.a 1 13.b even 2 1
507.4.a.e 1 1.a even 1 1 trivial
507.4.b.c 2 13.d odd 4 2
624.4.q.b 2 52.j odd 6 2
1521.4.a.c 1 3.b odd 2 1
1521.4.a.j 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 9 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 111 \) Copy content Toggle raw display
$19$ \( T + 46 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 105 \) Copy content Toggle raw display
$31$ \( T + 100 \) Copy content Toggle raw display
$37$ \( T - 17 \) Copy content Toggle raw display
$41$ \( T + 231 \) Copy content Toggle raw display
$43$ \( T + 514 \) Copy content Toggle raw display
$47$ \( T + 162 \) Copy content Toggle raw display
$53$ \( T - 639 \) Copy content Toggle raw display
$59$ \( T - 600 \) Copy content Toggle raw display
$61$ \( T - 233 \) Copy content Toggle raw display
$67$ \( T - 926 \) Copy content Toggle raw display
$71$ \( T + 930 \) Copy content Toggle raw display
$73$ \( T + 253 \) Copy content Toggle raw display
$79$ \( T + 1324 \) Copy content Toggle raw display
$83$ \( T - 810 \) Copy content Toggle raw display
$89$ \( T - 498 \) Copy content Toggle raw display
$97$ \( T - 1358 \) Copy content Toggle raw display
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