Properties

Label 507.4.b.h
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 13^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 3 q^{3} + (\beta_1 - 6) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + 3 \beta_{2} q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3} + 5 \beta_{2}) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 3 q^{3} + (\beta_1 - 6) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + 3 \beta_{2} q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3} + 5 \beta_{2}) q^{8} + 9 q^{9} + ( - \beta_{4} - 3 \beta_1 + 17) q^{10} + ( - \beta_{7} - \beta_{5} + \beta_{3} + 4 \beta_{2}) q^{11} + ( - 3 \beta_1 + 18) q^{12} + (\beta_{6} + \beta_{4} - 5 \beta_1 + 12) q^{14} + ( - 3 \beta_{5} - 3 \beta_{2}) q^{15} + ( - 2 \beta_{4} - 7 \beta_1 + 34) q^{16} + (2 \beta_{6} + \beta_{4} + 3 \beta_1 - 27) q^{17} - 9 \beta_{2} q^{18} + (3 \beta_{7} + 7 \beta_{5} - \beta_{3} - 6 \beta_{2}) q^{19} + (8 \beta_{7} - 9 \beta_{5} - 2 \beta_{3} - 23 \beta_{2}) q^{20} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{21} + (\beta_{4} - 8 \beta_1 + 59) q^{22} + ( - 2 \beta_{6} + \beta_{4} - 25) q^{23} + (3 \beta_{7} - 6 \beta_{5} - 3 \beta_{3} - 15 \beta_{2}) q^{24} + (2 \beta_{4} + 7 \beta_1 + 11) q^{25} - 27 q^{27} + ( - 5 \beta_{7} - \beta_{5} + \beta_{3} - 48 \beta_{2}) q^{28} + ( - 4 \beta_{6} + 2 \beta_{4} - 3 \beta_1 + 52) q^{29} + (3 \beta_{4} + 9 \beta_1 - 51) q^{30} + ( - \beta_{7} + 7 \beta_{5} - 53 \beta_{2}) q^{31} + (9 \beta_{7} - 20 \beta_{5} + 3 \beta_{3} - 29 \beta_{2}) q^{32} + (3 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} - 12 \beta_{2}) q^{33} + ( - 18 \beta_{7} + 13 \beta_{5} + 37 \beta_{2}) q^{34} + (6 \beta_{6} + \beta_{4} + 8 \beta_1 + 15) q^{35} + (9 \beta_1 - 54) q^{36} + ( - \beta_{7} - 2 \beta_{5} - \beta_{3} + 21 \beta_{2}) q^{37} + (2 \beta_{6} - 5 \beta_{4} + 2 \beta_1 - 85) q^{38} + (6 \beta_{6} + 7 \beta_{4} + 41 \beta_1 - 273) q^{40} + ( - 21 \beta_{7} - 3 \beta_{3} - 43 \beta_{2}) q^{41} + ( - 3 \beta_{6} - 3 \beta_{4} + 15 \beta_1 - 36) q^{42} + (6 \beta_{6} + 5 \beta_{4} - 3 \beta_1 - 114) q^{43} + ( - 5 \beta_{7} - 13 \beta_{5} - \beta_{3} - 90 \beta_{2}) q^{44} + (9 \beta_{5} + 9 \beta_{2}) q^{45} + (5 \beta_{7} + 15 \beta_{5} + \beta_{3} + 22 \beta_{2}) q^{46} + ( - 3 \beta_{7} - 47 \beta_{5} + 3 \beta_{3} - 8 \beta_{2}) q^{47} + (6 \beta_{4} + 21 \beta_1 - 102) q^{48} + ( - 2 \beta_{4} - 31 \beta_1 - 252) q^{49} + ( - 17 \beta_{7} + 36 \beta_{5} + 5 \beta_{3} + 24 \beta_{2}) q^{50} + ( - 6 \beta_{6} - 3 \beta_{4} - 9 \beta_1 + 81) q^{51} + ( - 2 \beta_{4} - 25 \beta_1 + 78) q^{53} + 27 \beta_{2} q^{54} + (6 \beta_{6} + 3 \beta_{4} + 26 \beta_1 + 35) q^{55} + (4 \beta_{6} + 5 \beta_{4} - 4 \beta_1 - 541) q^{56} + ( - 9 \beta_{7} - 21 \beta_{5} + 3 \beta_{3} + 18 \beta_{2}) q^{57} + (13 \beta_{7} + 24 \beta_{5} - \beta_{3} - 79 \beta_{2}) q^{58} + ( - 8 \beta_{7} + 24 \beta_{5} + 14 \beta_{3} + 10 \beta_{2}) q^{59} + ( - 24 \beta_{7} + 27 \beta_{5} + 6 \beta_{3} + 69 \beta_{2}) q^{60} + ( - 6 \beta_{6} - 3 \beta_{4} + 22 \beta_1 - 240) q^{61} + ( - \beta_{6} - 8 \beta_{4} + 37 \beta_1 - 713) q^{62} + (9 \beta_{3} + 9 \beta_{2}) q^{63} + (12 \beta_{6} + 16 \beta_{4} + 19 \beta_1 - 272) q^{64} + ( - 3 \beta_{4} + 24 \beta_1 - 177) q^{66} + ( - 26 \beta_{7} + 14 \beta_{5} + \beta_{3} - 129 \beta_{2}) q^{67} + ( - 2 \beta_{6} - 23 \beta_{4} - 75 \beta_1 + 485) q^{68} + (6 \beta_{6} - 3 \beta_{4} + 75) q^{69} + ( - 43 \beta_{7} + 15 \beta_{5} + \beta_{3} + 22 \beta_{2}) q^{70} + (23 \beta_{7} - 49 \beta_{5} - 11 \beta_{3} + 76 \beta_{2}) q^{71} + ( - 9 \beta_{7} + 18 \beta_{5} + 9 \beta_{3} + 45 \beta_{2}) q^{72} + ( - 27 \beta_{7} - 24 \beta_{5} - 2 \beta_{3} - 194 \beta_{2}) q^{73} + ( - 2 \beta_{6} - 15 \beta_1 + 298) q^{74} + ( - 6 \beta_{4} - 21 \beta_1 - 33) q^{75} + (37 \beta_{7} + \beta_{5} - 3 \beta_{3} + 82 \beta_{2}) q^{76} + ( - 16 \beta_{6} - 18 \beta_1 - 610) q^{77} + (2 \beta_{4} + 9 \beta_1 - 191) q^{79} + ( - 42 \beta_{7} + 75 \beta_{5} + 12 \beta_{3} + 315 \beta_{2}) q^{80} + 81 q^{81} + ( - 24 \beta_{6} - 24 \beta_{4} + 13 \beta_1 - 428) q^{82} + ( - \beta_{7} - \beta_{5} - 5 \beta_{3} + 138 \beta_{2}) q^{83} + (15 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} + 144 \beta_{2}) q^{84} + (25 \beta_{7} - 50 \beta_{5} - 25 \beta_{3} - 147 \beta_{2}) q^{85} + ( - 52 \beta_{7} + 37 \beta_{5} - 14 \beta_{3} + 46 \beta_{2}) q^{86} + (12 \beta_{6} - 6 \beta_{4} + 9 \beta_1 - 156) q^{87} + ( - 6 \beta_{6} + 15 \beta_{4} + 46 \beta_1 - 785) q^{88} + ( - 32 \beta_{7} - 50 \beta_{5} - 28 \beta_{3} - 146 \beta_{2}) q^{89} + ( - 9 \beta_{4} - 27 \beta_1 + 153) q^{90} + ( - 10 \beta_{6} - \beta_{4} - 46 \beta_1 + 111) q^{92} + (3 \beta_{7} - 21 \beta_{5} + 159 \beta_{2}) q^{93} + (47 \beta_{4} + 84 \beta_1 - 235) q^{94} + ( - 6 \beta_{6} + \beta_{4} - 34 \beta_1 - 531) q^{95} + ( - 27 \beta_{7} + 60 \beta_{5} - 9 \beta_{3} + 87 \beta_{2}) q^{96} + (37 \beta_{7} + 63 \beta_{5} + 22 \beta_{3} - 71 \beta_{2}) q^{97} + (41 \beta_{7} - 84 \beta_{5} - 29 \beta_{3} + 49 \beta_{2}) q^{98} + ( - 9 \beta_{7} - 9 \beta_{5} + 9 \beta_{3} + 36 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 40\nu^{4} - 329\nu^{2} + 178 ) / 78 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{7} - 518\nu^{5} - 6739\nu^{3} - 31348\nu ) / 11856 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{7} + 1622\nu^{5} + 41575\nu^{3} + 349012\nu ) / 11856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 118\nu^{4} + 2357\nu^{2} + 5048 ) / 156 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} + 397\nu^{5} + 6242\nu^{3} + 19814\nu ) / 1482 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} - 239\nu^{4} - 3166\nu^{2} - 8704 ) / 78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 518\nu^{5} + 6739\nu^{3} + 19492\nu ) / 912 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - 13\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 2\beta_{4} + 9\beta _1 - 179 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{7} - 13\beta_{5} + 13\beta_{3} + 273\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} + 6\beta_{4} - 17\beta _1 + 291 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -698\beta_{7} + 663\beta_{5} - 377\beta_{3} - 6487\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1422\beta_{6} - 2462\beta_{4} + 4865\beta _1 - 90115 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 21016\beta_{7} - 23257\beta_{5} + 9789\beta_{3} + 161265\beta_{2} ) / 13 \) 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
4.33039i
5.22605i
1.36176i
2.46610i
2.46610i
1.36176i
5.22605i
4.33039i
5.33039i −3.00000 −20.4131 16.4131i 15.9912i 9.67968i 66.1667i 9.00000 87.4882
337.2 4.22605i −3.00000 −9.85953 5.85953i 12.6782i 24.1254i 7.85849i 9.00000 −24.7627
337.3 2.36176i −3.00000 2.42208 6.42208i 7.08529i 29.4938i 24.6145i 9.00000 −15.1674
337.4 1.46610i −3.00000 5.85055 9.85055i 4.39830i 29.9396i 20.3063i 9.00000 14.4419
337.5 1.46610i −3.00000 5.85055 9.85055i 4.39830i 29.9396i 20.3063i 9.00000 14.4419
337.6 2.36176i −3.00000 2.42208 6.42208i 7.08529i 29.4938i 24.6145i 9.00000 −15.1674
337.7 4.22605i −3.00000 −9.85953 5.85953i 12.6782i 24.1254i 7.85849i 9.00000 −24.7627
337.8 5.33039i −3.00000 −20.4131 16.4131i 15.9912i 9.67968i 66.1667i 9.00000 87.4882
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
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.h 8
13.b even 2 1 inner 507.4.b.h 8
13.d odd 4 1 507.4.a.i 4
13.d odd 4 1 507.4.a.m 4
13.f odd 12 2 39.4.e.c 8
39.f even 4 1 1521.4.a.v 4
39.f even 4 1 1521.4.a.bb 4
39.k even 12 2 117.4.g.e 8
52.l even 12 2 624.4.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 13.f odd 12 2
117.4.g.e 8 39.k even 12 2
507.4.a.i 4 13.d odd 4 1
507.4.a.m 4 13.d odd 4 1
507.4.b.h 8 1.a even 1 1 trivial
507.4.b.h 8 13.b even 2 1 inner
624.4.q.i 8 52.l even 12 2
1521.4.a.v 4 39.f even 4 1
1521.4.a.bb 4 39.f even 4 1

Hecke kernels

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

\( T_{2}^{8} + 54T_{2}^{6} + 877T_{2}^{4} + 4476T_{2}^{2} + 6084 \) Copy content Toggle raw display
\( T_{5}^{8} + 442T_{5}^{6} + 55249T_{5}^{4} + 2494440T_{5}^{2} + 37015056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 54 T^{6} + 877 T^{4} + \cdots + 6084 \) Copy content Toggle raw display
$3$ \( (T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 442 T^{6} + \cdots + 37015056 \) Copy content Toggle raw display
$7$ \( T^{8} + 2442 T^{6} + \cdots + 42523388944 \) Copy content Toggle raw display
$11$ \( T^{8} + 4112 T^{6} + \cdots + 751198464 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 98 T^{3} - 7255 T^{2} + \cdots + 22571952)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 27240 T^{6} + \cdots + 2959721107456 \) Copy content Toggle raw display
$23$ \( (T^{4} + 104 T^{3} - 6824 T^{2} + \cdots - 2571504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 194 T^{3} - 32795 T^{2} + \cdots - 274591068)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 162626 T^{6} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 738573457717264 \) Copy content Toggle raw display
$41$ \( T^{8} + 395346 T^{6} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{4} + 450 T^{3} - 41275 T^{2} + \cdots - 2362804828)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 878416 T^{6} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} - 262 T^{3} - 111719 T^{2} + \cdots + 744728256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 622536 T^{6} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{4} + 928 T^{3} + 133150 T^{2} + \cdots - 5230543711)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 1254458 T^{6} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} + 1593296 T^{6} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} + 2482356 T^{6} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( (T^{4} + 746 T^{3} + 184337 T^{2} + \cdots + 680937616)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 1114728 T^{6} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + 4252360 T^{6} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + 3671450 T^{6} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
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