Properties

Label 507.4.a.l
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1362828.1
Defining polynomial: \( x^{4} - 23x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{3} + 4) q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + 3 \beta_1 q^{6} + (2 \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{2} + 3 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{3} + 4) q^{4} + (\beta_{2} + 2 \beta_1) q^{5} + 3 \beta_1 q^{6} + (2 \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{2} + 3 \beta_1) q^{8} + 9 q^{9} + (4 \beta_{3} + 24) q^{10} + ( - 3 \beta_{2} + 8 \beta_1) q^{11} + (3 \beta_{3} + 12) q^{12} + (6 \beta_{3} + 24) q^{14} + (3 \beta_{2} + 6 \beta_1) q^{15} + ( - \beta_{3} + 4) q^{16} + ( - 12 \beta_{3} - 30) q^{17} + 9 \beta_1 q^{18} + (8 \beta_{2} - 10 \beta_1) q^{19} + 36 \beta_1 q^{20} + (6 \beta_{2} + 6 \beta_1) q^{21} + (2 \beta_{3} + 96) q^{22} + 72 q^{23} + (6 \beta_{2} + 9 \beta_1) q^{24} + (4 \beta_{3} + 7) q^{25} + 27 q^{27} + ( - 4 \beta_{2} + 50 \beta_1) q^{28} + ( - 12 \beta_{3} - 102) q^{29} + (12 \beta_{3} + 72) q^{30} + ( - 20 \beta_{2} - 38 \beta_1) q^{31} + ( - 18 \beta_{2} - 27 \beta_1) q^{32} + ( - 9 \beta_{2} + 24 \beta_1) q^{33} + ( - 24 \beta_{2} - 114 \beta_1) q^{34} + 216 q^{35} + (9 \beta_{3} + 36) q^{36} + ( - 4 \beta_{2} + 68 \beta_1) q^{37} + (6 \beta_{3} - 120) q^{38} + (4 \beta_{3} + 240) q^{40} + ( - 13 \beta_{2} + 66 \beta_1) q^{41} + (18 \beta_{3} + 72) q^{42} + (12 \beta_{3} + 328) q^{43} + (28 \beta_{2} + 46 \beta_1) q^{44} + (9 \beta_{2} + 18 \beta_1) q^{45} + 72 \beta_1 q^{46} + (\beta_{2} - 36 \beta_1) q^{47} + ( - 3 \beta_{3} + 12) q^{48} + ( - 12 \beta_{3} + 41) q^{49} + (8 \beta_{2} + 35 \beta_1) q^{50} + ( - 36 \beta_{3} - 90) q^{51} + ( - 48 \beta_{3} + 222) q^{53} + 27 \beta_1 q^{54} + (44 \beta_{3} - 60) q^{55} + ( - 6 \beta_{3} + 408) q^{56} + (24 \beta_{2} - 30 \beta_1) q^{57} + ( - 24 \beta_{2} - 186 \beta_1) q^{58} + ( - \beta_{2} + 52 \beta_1) q^{59} + 108 \beta_1 q^{60} + ( - 28 \beta_{3} + 58) q^{61} + ( - 78 \beta_{3} - 456) q^{62} + (18 \beta_{2} + 18 \beta_1) q^{63} + ( - 55 \beta_{3} - 356) q^{64} + (6 \beta_{3} + 288) q^{66} + (54 \beta_{2} - 54 \beta_1) q^{67} + ( - 66 \beta_{3} - 1128) q^{68} + 216 q^{69} + 216 \beta_1 q^{70} + (41 \beta_{2} + 168 \beta_1) q^{71} + (18 \beta_{2} + 27 \beta_1) q^{72} + (42 \beta_{2} - 156 \beta_1) q^{73} + (60 \beta_{3} + 816) q^{74} + (12 \beta_{3} + 21) q^{75} + ( - 52 \beta_{2} + 2 \beta_1) q^{76} + (84 \beta_{3} - 312) q^{77} + ( - 16 \beta_{3} + 1072) q^{79} + (8 \beta_{2} - 20 \beta_1) q^{80} + 81 q^{81} + (40 \beta_{3} + 792) q^{82} + ( - 63 \beta_{2} - 188 \beta_1) q^{83} + ( - 12 \beta_{2} + 150 \beta_1) q^{84} + (18 \beta_{2} - 396 \beta_1) q^{85} + (24 \beta_{2} + 412 \beta_1) q^{86} + ( - 36 \beta_{3} - 306) q^{87} + (86 \beta_{3} - 216) q^{88} + (25 \beta_{2} - 138 \beta_1) q^{89} + (36 \beta_{3} + 216) q^{90} + (72 \beta_{3} + 288) q^{92} + ( - 60 \beta_{2} - 114 \beta_1) q^{93} + ( - 34 \beta_{3} - 432) q^{94} + ( - 72 \beta_{3} + 432) q^{95} + ( - 54 \beta_{2} - 81 \beta_1) q^{96} + ( - 14 \beta_{2} - 68 \beta_1) q^{97} + ( - 24 \beta_{2} - 43 \beta_1) q^{98} + ( - 27 \beta_{2} + 72 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9} + 88 q^{10} + 42 q^{12} + 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} + 20 q^{25} + 108 q^{27} - 384 q^{29} + 264 q^{30} + 864 q^{35} + 126 q^{36} - 492 q^{38} + 952 q^{40} + 252 q^{42} + 1288 q^{43} + 54 q^{48} + 188 q^{49} - 288 q^{51} + 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} - 1314 q^{64} + 1140 q^{66} - 4380 q^{68} + 864 q^{69} + 3144 q^{74} + 60 q^{75} - 1416 q^{77} + 4320 q^{79} + 324 q^{81} + 3088 q^{82} - 1152 q^{87} - 1036 q^{88} + 792 q^{90} + 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 23x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 19\beta_1 \) 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
−4.54739
−1.52356
1.52356
4.54739
−4.54739 3.00000 12.6788 −12.9118 −13.6422 −16.7289 −21.2762 9.00000 58.7151
1.2 −1.52356 3.00000 −5.67878 9.65841 −4.57067 22.3639 20.8404 9.00000 −14.7151
1.3 1.52356 3.00000 −5.67878 −9.65841 4.57067 −22.3639 −20.8404 9.00000 −14.7151
1.4 4.54739 3.00000 12.6788 12.9118 13.6422 16.7289 21.2762 9.00000 58.7151
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.a.l 4
3.b odd 2 1 1521.4.a.w 4
13.b even 2 1 inner 507.4.a.l 4
13.d odd 4 2 39.4.b.b 4
39.d odd 2 1 1521.4.a.w 4
39.f even 4 2 117.4.b.e 4
52.f even 4 2 624.4.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.b 4 13.d odd 4 2
117.4.b.e 4 39.f even 4 2
507.4.a.l 4 1.a even 1 1 trivial
507.4.a.l 4 13.b even 2 1 inner
624.4.c.c 4 52.f even 4 2
1521.4.a.w 4 3.b odd 2 1
1521.4.a.w 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{4} - 23T_{2}^{2} + 48 \) Copy content Toggle raw display
\( T_{5}^{4} - 260T_{5}^{2} + 15552 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 23T^{2} + 48 \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 260 T^{2} + 15552 \) Copy content Toggle raw display
$7$ \( T^{4} - 780 T^{2} + 139968 \) Copy content Toggle raw display
$11$ \( T^{4} - 3152 T^{2} + \cdots + 1572528 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 48 T - 11556)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 13884 T^{2} + \cdots + 3048192 \) Copy content Toggle raw display
$23$ \( (T - 72)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 192 T - 2916)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 100572 T^{2} + \cdots + 2389782528 \) Copy content Toggle raw display
$37$ \( T^{4} - 110256 T^{2} + \cdots + 2060577792 \) Copy content Toggle raw display
$41$ \( T^{4} - 133364 T^{2} + \cdots + 4431055872 \) Copy content Toggle raw display
$43$ \( (T^{2} - 644 T + 91552)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 30128 T^{2} + \cdots + 116663088 \) Copy content Toggle raw display
$53$ \( (T^{2} - 492 T - 133596)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 62576 T^{2} + \cdots + 457419312 \) Copy content Toggle raw display
$61$ \( (T^{2} - 144 T - 60868)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 591948 T^{2} + \cdots + 918330048 \) Copy content Toggle raw display
$71$ \( T^{4} - 917456 T^{2} + \cdots + 59484058032 \) Copy content Toggle raw display
$73$ \( T^{4} - 896400 T^{2} + \cdots + 179358354432 \) Copy content Toggle raw display
$79$ \( (T^{2} - 2160 T + 1144832)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 1464080 T^{2} + \cdots + 317023217328 \) Copy content Toggle raw display
$89$ \( T^{4} - 561812 T^{2} + \cdots + 78903164928 \) Copy content Toggle raw display
$97$ \( T^{4} - 137040 T^{2} + \cdots + 725594112 \) Copy content Toggle raw display
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