Properties

Label 507.4.a.j
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5054412.1
Defining polynomial: \( x^{4} - 29x^{2} + 48 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{3} + 7) q^{4} - \beta_{2} q^{5} - 3 \beta_1 q^{6} - 6 \beta_1 q^{7} + (2 \beta_{2} + 9 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{3} + 7) q^{4} - \beta_{2} q^{5} - 3 \beta_1 q^{6} - 6 \beta_1 q^{7} + (2 \beta_{2} + 9 \beta_1) q^{8} + 9 q^{9} + ( - 2 \beta_{3} - 6) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{11} + ( - 3 \beta_{3} - 21) q^{12} + ( - 6 \beta_{3} - 90) q^{14} + 3 \beta_{2} q^{15} + (5 \beta_{3} + 91) q^{16} - 54 q^{17} + 9 \beta_1 q^{18} + (6 \beta_{2} + 6 \beta_1) q^{19} + (4 \beta_{2} - 26 \beta_1) q^{20} + 18 \beta_1 q^{21} + ( - 4 \beta_{3} - 36) q^{22} + ( - 12 \beta_{3} - 36) q^{23} + ( - 6 \beta_{2} - 27 \beta_1) q^{24} + ( - 8 \beta_{3} + 7) q^{25} - 27 q^{27} + ( - 12 \beta_{2} - 102 \beta_1) q^{28} + ( - 12 \beta_{3} - 18) q^{29} + (6 \beta_{3} + 18) q^{30} + (6 \beta_{2} - 18 \beta_1) q^{31} + ( - 6 \beta_{2} + 69 \beta_1) q^{32} + (3 \beta_{2} + 6 \beta_1) q^{33} - 54 \beta_1 q^{34} + (12 \beta_{3} + 36) q^{35} + (9 \beta_{3} + 63) q^{36} + (24 \beta_{2} - 48 \beta_1) q^{37} + (18 \beta_{3} + 126) q^{38} + ( - 2 \beta_{3} - 318) q^{40} + (13 \beta_{2} - 4 \beta_1) q^{41} + (18 \beta_{3} + 270) q^{42} + (12 \beta_{3} - 260) q^{43} - 60 \beta_1 q^{44} - 9 \beta_{2} q^{45} + ( - 24 \beta_{2} - 156 \beta_1) q^{46} + ( - 21 \beta_{2} + 122 \beta_1) q^{47} + ( - 15 \beta_{3} - 273) q^{48} + (36 \beta_{3} + 197) q^{49} + ( - 16 \beta_{2} - 73 \beta_1) q^{50} + 162 q^{51} + (36 \beta_{3} - 198) q^{53} - 27 \beta_1 q^{54} + ( - 4 \beta_{3} + 144) q^{55} + ( - 78 \beta_{3} - 882) q^{56} + ( - 18 \beta_{2} - 18 \beta_1) q^{57} + ( - 24 \beta_{2} - 138 \beta_1) q^{58} + ( - 15 \beta_{2} - 34 \beta_1) q^{59} + ( - 12 \beta_{2} + 78 \beta_1) q^{60} + (8 \beta_{3} - 566) q^{61} + ( - 6 \beta_{3} - 234) q^{62} - 54 \beta_1 q^{63} + (17 \beta_{3} + 271) q^{64} + (12 \beta_{3} + 108) q^{66} + (12 \beta_{2} + 150 \beta_1) q^{67} + ( - 54 \beta_{3} - 378) q^{68} + (36 \beta_{3} + 108) q^{69} + (24 \beta_{2} + 156 \beta_1) q^{70} + (23 \beta_{2} + 6 \beta_1) q^{71} + (18 \beta_{2} + 81 \beta_1) q^{72} + 54 \beta_{2} q^{73} - 576 q^{74} + (24 \beta_{3} - 21) q^{75} + ( - 12 \beta_{2} + 258 \beta_1) q^{76} + (24 \beta_{3} + 216) q^{77} + (32 \beta_{3} + 88) q^{79} + ( - 36 \beta_{2} - 130 \beta_1) q^{80} + 81 q^{81} + (22 \beta_{3} + 18) q^{82} + ( - 25 \beta_{2} + 138 \beta_1) q^{83} + (36 \beta_{2} + 306 \beta_1) q^{84} + 54 \beta_{2} q^{85} + (24 \beta_{2} - 140 \beta_1) q^{86} + (36 \beta_{3} + 54) q^{87} + ( - 28 \beta_{3} - 612) q^{88} + (15 \beta_{2} - 4 \beta_1) q^{89} + ( - 18 \beta_{3} - 54) q^{90} + ( - 108 \beta_{3} - 2196) q^{92} + ( - 18 \beta_{2} + 54 \beta_1) q^{93} + (80 \beta_{3} + 1704) q^{94} + (36 \beta_{3} - 828) q^{95} + (18 \beta_{2} - 207 \beta_1) q^{96} + (54 \beta_{2} - 336 \beta_1) q^{97} + (72 \beta_{2} + 557 \beta_1) q^{98} + ( - 9 \beta_{2} - 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9} - 20 q^{10} - 78 q^{12} - 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} + 44 q^{25} - 108 q^{27} - 48 q^{29} + 60 q^{30} + 120 q^{35} + 234 q^{36} + 468 q^{38} - 1268 q^{40} + 1044 q^{42} - 1064 q^{43} - 1062 q^{48} + 716 q^{49} + 648 q^{51} - 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} + 1050 q^{64} + 408 q^{66} - 1404 q^{68} + 360 q^{69} - 2304 q^{74} - 132 q^{75} + 816 q^{77} + 288 q^{79} + 324 q^{81} + 28 q^{82} + 144 q^{87} - 2392 q^{88} - 180 q^{90} - 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 29x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 25\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 25\beta_1 \) 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
−5.21898
−1.32750
1.32750
5.21898
−5.21898 −3.00000 19.2377 5.83936 15.6569 31.3139 −58.6495 9.00000 −30.4755
1.2 −1.32750 −3.00000 −6.23774 −15.4241 3.98251 7.96501 18.9006 9.00000 20.4755
1.3 1.32750 −3.00000 −6.23774 15.4241 −3.98251 −7.96501 −18.9006 9.00000 20.4755
1.4 5.21898 −3.00000 19.2377 −5.83936 −15.6569 −31.3139 58.6495 9.00000 −30.4755
\(n\): 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.j 4
3.b odd 2 1 1521.4.a.x 4
13.b even 2 1 inner 507.4.a.j 4
13.d odd 4 2 39.4.b.a 4
39.d odd 2 1 1521.4.a.x 4
39.f even 4 2 117.4.b.d 4
52.f even 4 2 624.4.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.a 4 13.d odd 4 2
117.4.b.d 4 39.f even 4 2
507.4.a.j 4 1.a even 1 1 trivial
507.4.a.j 4 13.b even 2 1 inner
624.4.c.e 4 52.f even 4 2
1521.4.a.x 4 3.b odd 2 1
1521.4.a.x 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{4} - 29T_{2}^{2} + 48 \) Copy content Toggle raw display
\( T_{5}^{4} - 272T_{5}^{2} + 8112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 29T^{2} + 48 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 272T^{2} + 8112 \) Copy content Toggle raw display
$7$ \( T^{4} - 1044 T^{2} + 62208 \) Copy content Toggle raw display
$11$ \( T^{4} - 428 T^{2} + 43200 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T + 54)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 11556 T^{2} + \cdots + 31492800 \) Copy content Toggle raw display
$23$ \( (T^{2} + 60 T - 22464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T - 23220)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 17028 T^{2} + \cdots + 47044800 \) Copy content Toggle raw display
$37$ \( T^{4} - 200448 T^{2} + \cdots + 2293235712 \) Copy content Toggle raw display
$41$ \( T^{4} - 45392 T^{2} + \cdots + 128314800 \) Copy content Toggle raw display
$43$ \( (T^{2} + 532 T + 47392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 500348 T^{2} + \cdots + 62387841792 \) Copy content Toggle raw display
$53$ \( (T^{2} + 432 T - 163620)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 104924 T^{2} + \cdots + 2436066048 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1140 T + 314516)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 727668 T^{2} + \cdots + 143327232 \) Copy content Toggle raw display
$71$ \( T^{4} - 147692 T^{2} + \cdots + 3298756800 \) Copy content Toggle raw display
$73$ \( T^{4} - 793152 T^{2} + \cdots + 68976790272 \) Copy content Toggle raw display
$79$ \( (T^{2} - 144 T - 160960)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 653276 T^{2} + \cdots + 106682723328 \) Copy content Toggle raw display
$89$ \( T^{4} - 60464 T^{2} + \cdots + 249304368 \) Copy content Toggle raw display
$97$ \( T^{4} - 3704256 T^{2} + \cdots + 3383532000000 \) Copy content Toggle raw display
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