Properties

Label 507.4.a.h
Level $507$
Weight $4$
Character orbit 507.a
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + (2 \beta_{2} - 2) q^{5} + (3 \beta_1 - 3) q^{6} + ( - 6 \beta_1 - 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + (2 \beta_{2} - 2) q^{5} + (3 \beta_1 - 3) q^{6} + ( - 6 \beta_1 - 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + 9 q^{9} + ( - 4 \beta_{2} + 6 \beta_1 - 2) q^{10} + ( - 6 \beta_{2} - 2 \beta_1 + 8) q^{11} + (3 \beta_{2} + 9) q^{12} + ( - 6 \beta_{2} - 14 \beta_1 - 52) q^{14} + (6 \beta_{2} - 6) q^{15} + ( - 5 \beta_{2} - 6 \beta_1 - 33) q^{16} + ( - 8 \beta_1 - 46) q^{17} + (9 \beta_1 - 9) q^{18} + ( - 16 \beta_{2} + 6 \beta_1 - 28) q^{19} + ( - 2 \beta_{2} - 12 \beta_1 + 86) q^{20} + ( - 18 \beta_1 - 24) q^{21} + (10 \beta_{2} - 18 \beta_1 - 16) q^{22} + ( - 8 \beta_{2} + 32 \beta_1 - 24) q^{23} + ( - 6 \beta_{2} - 3 \beta_1 + 9) q^{24} + ( - 20 \beta_{2} - 24 \beta_1 + 63) q^{25} + 27 q^{27} + ( - 2 \beta_{2} - 42 \beta_1 - 12) q^{28} + (8 \beta_{2} + 20 \beta_1 - 10) q^{29} + ( - 12 \beta_{2} + 18 \beta_1 - 6) q^{30} + (4 \beta_{2} + 54 \beta_1 - 120) q^{31} + (20 \beta_{2} - 51 \beta_1 - 41) q^{32} + ( - 18 \beta_{2} - 6 \beta_1 + 24) q^{33} + ( - 8 \beta_{2} - 54 \beta_1 - 34) q^{34} + ( - 4 \beta_{2} - 36 \beta_1 + 40) q^{35} + (9 \beta_{2} + 27) q^{36} + (28 \beta_{2} + 48 \beta_1 - 150) q^{37} + (38 \beta_{2} - 86 \beta_1 + 120) q^{38} + (24 \beta_{2} + 18 \beta_1 - 186) q^{40} + ( - 34 \beta_{2} + 4 \beta_1 - 150) q^{41} + ( - 18 \beta_{2} - 42 \beta_1 - 156) q^{42} + (4 \beta_{2} - 60 \beta_1 - 68) q^{43} + (10 \beta_{2} + 22 \beta_1 - 248) q^{44} + (18 \beta_{2} - 18) q^{45} + (48 \beta_{2} - 24 \beta_1 + 360) q^{46} + (42 \beta_{2} - 54 \beta_1 + 12) q^{47} + ( - 15 \beta_{2} - 18 \beta_1 - 99) q^{48} + (36 \beta_{2} + 168 \beta_1 + 81) q^{49} + (16 \beta_{2} - 41 \beta_1 - 263) q^{50} + ( - 24 \beta_1 - 138) q^{51} + ( - 12 \beta_{2} - 108 \beta_1 - 186) q^{53} + (27 \beta_1 - 27) q^{54} + (68 \beta_{2} + 60 \beta_1 - 560) q^{55} + (10 \beta_{2} + 50 \beta_1 + 12) q^{56} + ( - 48 \beta_{2} + 18 \beta_1 - 84) q^{57} + (4 \beta_{2} + 42 \beta_1 + 194) q^{58} + ( - 2 \beta_{2} - 82 \beta_1 + 624) q^{59} + ( - 6 \beta_{2} - 36 \beta_1 + 258) q^{60} + ( - 28 \beta_{2} + 24 \beta_1 + 78) q^{61} + (46 \beta_{2} - 50 \beta_1 + 652) q^{62} + ( - 54 \beta_1 - 72) q^{63} + ( - 51 \beta_{2} + 36 \beta_1 - 245) q^{64} + (30 \beta_{2} - 54 \beta_1 - 48) q^{66} + (76 \beta_{2} - 42 \beta_1 - 36) q^{67} + ( - 38 \beta_{2} - 56 \beta_1 - 122) q^{68} + ( - 24 \beta_{2} + 96 \beta_1 - 72) q^{69} + ( - 28 \beta_{2} - 12 \beta_1 - 392) q^{70} + ( - 14 \beta_{2} + 134 \beta_1 + 276) q^{71} + ( - 18 \beta_{2} - 9 \beta_1 + 27) q^{72} + ( - 12 \beta_{2} + 240 \beta_1 - 2) q^{73} + ( - 8 \beta_{2} + 10 \beta_1 + 574) q^{74} + ( - 60 \beta_{2} - 72 \beta_1 + 189) q^{75} + ( - 34 \beta_{2} + 138 \beta_1 - 832) q^{76} + (24 \beta_{2} + 136 \beta_1 - 16) q^{77} + ( - 24 \beta_{2} + 48 \beta_1 - 16) q^{79} + ( - 14 \beta_{2} + 24 \beta_1 - 370) q^{80} + 81 q^{81} + (72 \beta_{2} - 282 \beta_1 + 258) q^{82} + (10 \beta_{2} - 30 \beta_1 + 272) q^{83} + ( - 6 \beta_{2} - 126 \beta_1 - 36) q^{84} + ( - 76 \beta_{2} - 48 \beta_1 + 124) q^{85} + ( - 68 \beta_{2} - 112 \beta_1 - 540) q^{86} + (24 \beta_{2} + 60 \beta_1 - 30) q^{87} + ( - 78 \beta_{2} - 42 \beta_1 + 576) q^{88} + ( - 30 \beta_{2} - 116 \beta_1 - 430) q^{89} + ( - 36 \beta_{2} + 54 \beta_1 - 18) q^{90} + ( - 56 \beta_{2} + 272 \beta_1 - 504) q^{92} + (12 \beta_{2} + 162 \beta_1 - 360) q^{93} + ( - 138 \beta_{2} + 126 \beta_1 - 636) q^{94} + (60 \beta_{2} + 228 \beta_1 - 1440) q^{95} + (60 \beta_{2} - 153 \beta_1 - 123) q^{96} + (4 \beta_{2} + 48 \beta_1 - 1098) q^{97} + (96 \beta_{2} + 393 \beta_1 + 1527) q^{98} + ( - 54 \beta_{2} - 18 \beta_1 + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 4 q^{10} + 16 q^{11} + 30 q^{12} - 176 q^{14} - 12 q^{15} - 110 q^{16} - 146 q^{17} - 18 q^{18} - 94 q^{19} + 244 q^{20} - 90 q^{21} - 56 q^{22} - 48 q^{23} + 18 q^{24} + 145 q^{25} + 81 q^{27} - 80 q^{28} - 2 q^{29} - 12 q^{30} - 302 q^{31} - 154 q^{32} + 48 q^{33} - 164 q^{34} + 80 q^{35} + 90 q^{36} - 374 q^{37} + 312 q^{38} - 516 q^{40} - 480 q^{41} - 528 q^{42} - 260 q^{43} - 712 q^{44} - 36 q^{45} + 1104 q^{46} + 24 q^{47} - 330 q^{48} + 447 q^{49} - 814 q^{50} - 438 q^{51} - 678 q^{53} - 54 q^{54} - 1552 q^{55} + 96 q^{56} - 282 q^{57} + 628 q^{58} + 1788 q^{59} + 732 q^{60} + 230 q^{61} + 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} - 460 q^{68} - 144 q^{69} - 1216 q^{70} + 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 435 q^{75} - 2392 q^{76} + 112 q^{77} - 24 q^{79} - 1100 q^{80} + 243 q^{81} + 564 q^{82} + 796 q^{83} - 240 q^{84} + 248 q^{85} - 1800 q^{86} - 6 q^{87} + 1608 q^{88} - 1436 q^{89} - 36 q^{90} - 1296 q^{92} - 906 q^{93} - 1920 q^{94} - 4032 q^{95} - 462 q^{96} - 3242 q^{97} + 5070 q^{98} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 16x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 10 \) 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
−3.20905
−0.526440
4.73549
−4.20905 3.00000 9.71610 11.4322 −12.6271 11.2543 −7.22315 9.00000 −48.1187
1.2 −1.52644 3.00000 −5.66998 −19.3400 −4.57932 −4.84136 20.8664 9.00000 29.5213
1.3 3.73549 3.00000 5.95388 3.90776 11.2065 −36.4129 −7.64325 9.00000 14.5974
\(n\): 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.h 3
3.b odd 2 1 1521.4.a.u 3
13.b even 2 1 39.4.a.c 3
13.d odd 4 2 507.4.b.g 6
39.d odd 2 1 117.4.a.f 3
52.b odd 2 1 624.4.a.t 3
65.d even 2 1 975.4.a.l 3
91.b odd 2 1 1911.4.a.k 3
104.e even 2 1 2496.4.a.bl 3
104.h odd 2 1 2496.4.a.bp 3
156.h even 2 1 1872.4.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 13.b even 2 1
117.4.a.f 3 39.d odd 2 1
507.4.a.h 3 1.a even 1 1 trivial
507.4.b.g 6 13.d odd 4 2
624.4.a.t 3 52.b odd 2 1
975.4.a.l 3 65.d even 2 1
1521.4.a.u 3 3.b odd 2 1
1872.4.a.bk 3 156.h even 2 1
1911.4.a.k 3 91.b odd 2 1
2496.4.a.bl 3 104.e even 2 1
2496.4.a.bp 3 104.h odd 2 1

Hecke kernels

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

\( T_{2}^{3} + 2T_{2}^{2} - 15T_{2} - 24 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} - 252T_{5} + 864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 15 T - 24 \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 252 T + 864 \) Copy content Toggle raw display
$7$ \( T^{3} + 30 T^{2} - 288 T - 1984 \) Copy content Toggle raw display
$11$ \( T^{3} - 16 T^{2} - 2256 T - 30336 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 146 T^{2} + 6060 T + 71256 \) Copy content Toggle raw display
$19$ \( T^{3} + 94 T^{2} - 14432 T - 779616 \) Copy content Toggle raw display
$23$ \( T^{3} + 48 T^{2} - 20928 T + 534528 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 10116 T - 199176 \) Copy content Toggle raw display
$31$ \( T^{3} + 302 T^{2} - 17536 T - 7197248 \) Copy content Toggle raw display
$37$ \( T^{3} + 374 T^{2} - 36964 T - 7758104 \) Copy content Toggle raw display
$41$ \( T^{3} + 480 T^{2} + \cdots - 12919824 \) Copy content Toggle raw display
$43$ \( T^{3} + 260 T^{2} - 38096 T - 3663168 \) Copy content Toggle raw display
$47$ \( T^{3} - 24 T^{2} - 168480 T - 18102528 \) Copy content Toggle raw display
$53$ \( T^{3} + 678 T^{2} - 42228 T - 1471608 \) Copy content Toggle raw display
$59$ \( T^{3} - 1788 T^{2} + \cdots - 137423808 \) Copy content Toggle raw display
$61$ \( T^{3} - 230 T^{2} - 44452 T + 6279512 \) Copy content Toggle raw display
$67$ \( T^{3} + 74 T^{2} - 409216 T - 4260896 \) Copy content Toggle raw display
$71$ \( T^{3} - 948 T^{2} + \cdots + 70464384 \) Copy content Toggle raw display
$73$ \( T^{3} - 222 T^{2} + \cdots - 22780552 \) Copy content Toggle raw display
$79$ \( T^{3} + 24 T^{2} - 78336 T + 7757824 \) Copy content Toggle raw display
$83$ \( T^{3} - 796 T^{2} + \cdots - 13963968 \) Copy content Toggle raw display
$89$ \( T^{3} + 1436 T^{2} + \cdots + 30129888 \) Copy content Toggle raw display
$97$ \( T^{3} + 3242 T^{2} + \cdots + 1218481048 \) Copy content Toggle raw display
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