Properties

Label 507.4.b.g
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $6$
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.158155776.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + 3 q^{3} + (\beta_1 - 3) q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + 3 \beta_{4} q^{6} + (6 \beta_{4} - 7 \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + 3 q^{3} + (\beta_1 - 3) q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + 3 \beta_{4} q^{6} + (6 \beta_{4} - 7 \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{8} + 9 q^{9} + ( - 3 \beta_{3} - 4 \beta_1 - 1) q^{10} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{2}) q^{11} + (3 \beta_1 - 9) q^{12} + ( - 7 \beta_{3} + 6 \beta_1 - 59) q^{14} + ( - 3 \beta_{5} + 3 \beta_{2}) q^{15} + ( - 3 \beta_{3} + 5 \beta_1 - 36) q^{16} + (4 \beta_{3} + 50) q^{17} + 9 \beta_{4} q^{18} + (8 \beta_{5} + 6 \beta_{4} + 11 \beta_{2}) q^{19} + ( - \beta_{5} + 12 \beta_{4} + 37 \beta_{2}) q^{20} + (18 \beta_{4} - 21 \beta_{2}) q^{21} + ( - 9 \beta_{3} - 10 \beta_1 - 25) q^{22} + ( - 16 \beta_{3} - 8 \beta_1 + 8) q^{23} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}) q^{24} + (12 \beta_{3} - 20 \beta_1 - 51) q^{25} + 27 q^{27} + (\beta_{5} - 42 \beta_{4} + 27 \beta_{2}) q^{28} + (10 \beta_{3} - 8 \beta_1) q^{29} + ( - 9 \beta_{3} - 12 \beta_1 - 3) q^{30} + ( - 2 \beta_{5} + 54 \beta_{4} + 33 \beta_{2}) q^{31} + ( - 10 \beta_{5} - 51 \beta_{4} + 46 \beta_{2}) q^{32} + ( - 9 \beta_{5} + 6 \beta_{4} + 9 \beta_{2}) q^{33} + ( - 4 \beta_{5} + 54 \beta_{4} - 44 \beta_{2}) q^{34} + ( - 18 \beta_{3} + 4 \beta_1 + 22) q^{35} + (9 \beta_1 - 27) q^{36} + (14 \beta_{5} - 48 \beta_{4} - 51 \beta_{2}) q^{37} + (43 \beta_{3} + 38 \beta_1 - 77) q^{38} + (9 \beta_{3} - 24 \beta_1 - 177) q^{40} + (17 \beta_{5} + 4 \beta_{4} + 73 \beta_{2}) q^{41} + ( - 21 \beta_{3} + 18 \beta_1 - 177) q^{42} + (30 \beta_{3} + 4 \beta_1 + 98) q^{43} + ( - 5 \beta_{5} + 22 \beta_{4} + 113 \beta_{2}) q^{44} + ( - 9 \beta_{5} + 9 \beta_{2}) q^{45} + (24 \beta_{5} + 24 \beta_{4} + 168 \beta_{2}) q^{46} + (21 \beta_{5} + 54 \beta_{4} - 21 \beta_{2}) q^{47} + ( - 9 \beta_{3} + 15 \beta_1 - 108) q^{48} + ( - 84 \beta_{3} + 36 \beta_1 - 165) q^{49} + (8 \beta_{5} + 41 \beta_{4} - 152 \beta_{2}) q^{50} + (12 \beta_{3} + 150) q^{51} + ( - 54 \beta_{3} + 12 \beta_1 - 240) q^{53} + 27 \beta_{4} q^{54} + (30 \beta_{3} - 68 \beta_1 - 530) q^{55} + ( - 25 \beta_{3} + 10 \beta_1 - 37) q^{56} + (24 \beta_{5} + 18 \beta_{4} + 33 \beta_{2}) q^{57} + ( - 2 \beta_{5} + 42 \beta_{4} - 118 \beta_{2}) q^{58} + ( - \beta_{5} + 82 \beta_{4} + 271 \beta_{2}) q^{59} + ( - 3 \beta_{5} + 36 \beta_{4} + 111 \beta_{2}) q^{60} + (12 \beta_{3} + 28 \beta_1 + 90) q^{61} + (25 \beta_{3} + 46 \beta_1 - 627) q^{62} + (54 \beta_{4} - 63 \beta_{2}) q^{63} + ( - 18 \beta_{3} - 51 \beta_1 + 227) q^{64} + ( - 27 \beta_{3} - 30 \beta_1 - 75) q^{66} + ( - 38 \beta_{5} - 42 \beta_{4} + 39 \beta_{2}) q^{67} + ( - 28 \beta_{3} + 38 \beta_1 - 150) q^{68} + ( - 48 \beta_{3} - 24 \beta_1 + 24) q^{69} + (14 \beta_{5} - 12 \beta_{4} + 202 \beta_{2}) q^{70} + (7 \beta_{5} + 134 \beta_{4} - 205 \beta_{2}) q^{71} + ( - 9 \beta_{5} + 9 \beta_{4} + 9 \beta_{2}) q^{72} + ( - 6 \beta_{5} - 240 \beta_{4} + 119 \beta_{2}) q^{73} + (5 \beta_{3} + 8 \beta_1 + 579) q^{74} + (36 \beta_{3} - 60 \beta_1 - 153) q^{75} + ( - 17 \beta_{5} - 138 \beta_{4} - 347 \beta_{2}) q^{76} + ( - 68 \beta_{3} + 24 \beta_1 - 52) q^{77} + (24 \beta_{3} + 24 \beta_1 + 8) q^{79} + (7 \beta_{5} + 24 \beta_{4} + 173 \beta_{2}) q^{80} + 81 q^{81} + (141 \beta_{3} + 72 \beta_1 - 117) q^{82} + ( - 5 \beta_{5} - 30 \beta_{4} - 121 \beta_{2}) q^{83} + (3 \beta_{5} - 126 \beta_{4} + 81 \beta_{2}) q^{84} + ( - 38 \beta_{5} + 48 \beta_{4} + 38 \beta_{2}) q^{85} + ( - 34 \beta_{5} + 112 \beta_{4} - 326 \beta_{2}) q^{86} + (30 \beta_{3} - 24 \beta_1) q^{87} + (21 \beta_{3} - 78 \beta_1 - 555) q^{88} + ( - 15 \beta_{5} + 116 \beta_{4} - 273 \beta_{2}) q^{89} + ( - 27 \beta_{3} - 36 \beta_1 - 9) q^{90} + (136 \beta_{3} + 56 \beta_1 - 368) q^{92} + ( - 6 \beta_{5} + 162 \beta_{4} + 99 \beta_{2}) q^{93} + (63 \beta_{3} + 138 \beta_1 - 573) q^{94} + ( - 114 \beta_{3} + 60 \beta_1 + 1326) q^{95} + ( - 30 \beta_{5} - 153 \beta_{4} + 138 \beta_{2}) q^{96} + ( - 2 \beta_{5} + 48 \beta_{4} + 525 \beta_{2}) q^{97} + (48 \beta_{5} - 393 \beta_{4} + 960 \beta_{2}) q^{98} + ( - 27 \beta_{5} + 18 \beta_{4} + 27 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{5} - 33\nu^{4} + 44\nu^{3} + 48\nu^{2} + 24\nu - 1523 ) / 227 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\nu^{5} - 138\nu^{4} + 184\nu^{3} + 1150\nu^{2} + 4434\nu + 1638 ) / 681 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 68\nu^{5} - 334\nu^{4} + 748\nu^{3} + 816\nu^{2} + 408\nu - 5007 ) / 681 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 152\nu^{5} - 346\nu^{4} + 310\nu^{3} + 3867\nu^{2} + 10446\nu + 3870 ) / 681 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 406\nu^{5} - 966\nu^{4} + 1288\nu^{3} + 9412\nu^{2} + 31038\nu + 11466 ) / 681 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 2\beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{5} - 18\beta_{4} + 9\beta_{3} - 49\beta_{2} - 26\beta _1 - 107 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{3} - 17\beta _1 - 92 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -197\beta_{5} + 210\beta_{4} + 105\beta_{3} + 881\beta_{2} - 394\beta _1 - 1867 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−0.376763 0.376763i
−1.42234 + 1.42234i
2.79911 + 2.79911i
2.79911 2.79911i
−1.42234 1.42234i
−0.376763 + 0.376763i
4.20905i 3.00000 −9.71610 11.4322i 12.6271i 11.2543i 7.22315i 9.00000 48.1187
337.2 3.73549i 3.00000 −5.95388 3.90776i 11.2065i 36.4129i 7.64325i 9.00000 −14.5974
337.3 1.52644i 3.00000 5.66998 19.3400i 4.57932i 4.84136i 20.8664i 9.00000 −29.5213
337.4 1.52644i 3.00000 5.66998 19.3400i 4.57932i 4.84136i 20.8664i 9.00000 −29.5213
337.5 3.73549i 3.00000 −5.95388 3.90776i 11.2065i 36.4129i 7.64325i 9.00000 −14.5974
337.6 4.20905i 3.00000 −9.71610 11.4322i 12.6271i 11.2543i 7.22315i 9.00000 48.1187
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
Significant digits:
Format:

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.b.g 6
13.b even 2 1 inner 507.4.b.g 6
13.d odd 4 1 39.4.a.c 3
13.d odd 4 1 507.4.a.h 3
39.f even 4 1 117.4.a.f 3
39.f even 4 1 1521.4.a.u 3
52.f even 4 1 624.4.a.t 3
65.g odd 4 1 975.4.a.l 3
91.i even 4 1 1911.4.a.k 3
104.j odd 4 1 2496.4.a.bl 3
104.m even 4 1 2496.4.a.bp 3
156.l odd 4 1 1872.4.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 13.d odd 4 1
117.4.a.f 3 39.f even 4 1
507.4.a.h 3 13.d odd 4 1
507.4.b.g 6 1.a even 1 1 trivial
507.4.b.g 6 13.b even 2 1 inner
624.4.a.t 3 52.f even 4 1
975.4.a.l 3 65.g odd 4 1
1521.4.a.u 3 39.f even 4 1
1872.4.a.bk 3 156.l odd 4 1
1911.4.a.k 3 91.i even 4 1
2496.4.a.bl 3 104.j odd 4 1
2496.4.a.bp 3 104.m even 4 1

Hecke kernels

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

\( T_{2}^{6} + 34T_{2}^{4} + 321T_{2}^{2} + 576 \) Copy content Toggle raw display
\( T_{5}^{6} + 520T_{5}^{4} + 56592T_{5}^{2} + 746496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 34 T^{4} + 321 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 520 T^{4} + 56592 T^{2} + \cdots + 746496 \) Copy content Toggle raw display
$7$ \( T^{6} + 1476 T^{4} + \cdots + 3936256 \) Copy content Toggle raw display
$11$ \( T^{6} + 4768 T^{4} + \cdots + 920272896 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 146 T^{2} + 6060 T - 71256)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 37700 T^{4} + \cdots + 607801107456 \) Copy content Toggle raw display
$23$ \( (T^{3} - 48 T^{2} - 20928 T - 534528)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 10116 T - 199176)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 126276 T^{4} + \cdots + 51800378773504 \) Copy content Toggle raw display
$37$ \( T^{6} + 213804 T^{4} + \cdots + 60188177674816 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 166921852190976 \) Copy content Toggle raw display
$43$ \( (T^{3} - 260 T^{2} - 38096 T + 3663168)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 327701519990784 \) Copy content Toggle raw display
$53$ \( (T^{3} + 678 T^{2} - 42228 T - 1471608)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 1284720 T^{4} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{3} - 230 T^{2} - 44452 T + 6279512)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 823908 T^{4} + \cdots + 18155234722816 \) Copy content Toggle raw display
$71$ \( T^{6} + 923856 T^{4} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 518953549424704 \) Copy content Toggle raw display
$79$ \( (T^{3} + 24 T^{2} - 78336 T + 7757824)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 194992402305024 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 907810150892544 \) Copy content Toggle raw display
$97$ \( T^{6} + 3579564 T^{4} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
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