Properties

Label 507.2.k.h
Level $507$
Weight $2$
Character orbit 507.k
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.56070144.2
Defining polynomial: \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( 2 \beta_{2} + \beta_{3} ) q^{3} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{7} ) q^{6} + ( \beta_{5} + \beta_{6} ) q^{8} + ( -3 - 3 \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( 2 \beta_{2} + \beta_{3} ) q^{3} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{7} ) q^{6} + ( \beta_{5} + \beta_{6} ) q^{8} + ( -3 - 3 \beta_{4} ) q^{9} + ( -6 - \beta_{3} - 3 \beta_{4} ) q^{10} + ( \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{11} + ( -2 + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} ) q^{12} + ( 2 \beta_{1} - 3 \beta_{5} - \beta_{7} ) q^{15} + ( 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{16} + ( -3 \beta_{5} + 3 \beta_{6} ) q^{18} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} + ( 7 - 5 \beta_{2} - 10 \beta_{3} + 7 \beta_{4} ) q^{22} + 3 \beta_{7} q^{24} + ( -4 - 5 \beta_{2} - 5 \beta_{3} - 8 \beta_{4} ) q^{25} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{27} + ( 1 - 9 \beta_{2} - \beta_{4} ) q^{30} -\beta_{6} q^{32} + ( 3 \beta_{1} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{33} + ( -6 + 6 \beta_{3} - 3 \beta_{4} ) q^{36} + ( -9 + \beta_{2} - \beta_{3} ) q^{40} + ( -2 \beta_{1} + 5 \beta_{5} + \beta_{7} ) q^{41} + 4 \beta_{2} q^{43} + ( 3 \beta_{1} + 5 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} ) q^{44} + ( -3 \beta_{1} + 3 \beta_{6} - 3 \beta_{7} ) q^{45} + ( \beta_{1} - 3 \beta_{5} - 3 \beta_{6} ) q^{47} + ( -6 - \beta_{2} - 2 \beta_{3} - 6 \beta_{4} ) q^{48} -7 \beta_{3} q^{49} + ( -4 \beta_{5} + 8 \beta_{6} - 5 \beta_{7} ) q^{50} + ( -6 \beta_{1} + 3 \beta_{7} ) q^{54} + ( 4 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} ) q^{55} + ( -5 \beta_{1} + 6 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} ) q^{59} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} ) q^{60} + ( 8 \beta_{2} + 16 \beta_{3} ) q^{61} + ( 5 - 2 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} ) q^{64} + ( 15 + 7 \beta_{2} - 7 \beta_{3} ) q^{66} + ( -\beta_{1} - 5 \beta_{6} - \beta_{7} ) q^{71} + ( -6 \beta_{5} + 3 \beta_{6} ) q^{72} + ( 10 + 12 \beta_{3} + 5 \beta_{4} ) q^{75} + ( -6 \beta_{2} + 6 \beta_{3} ) q^{79} + ( 6 \beta_{1} - 7 \beta_{5} - 3 \beta_{7} ) q^{80} + 9 \beta_{4} q^{81} + ( 1 + 17 \beta_{2} - \beta_{4} ) q^{82} + ( \beta_{1} - 7 \beta_{5} + 7 \beta_{6} - 2 \beta_{7} ) q^{83} + 4 \beta_{1} q^{86} + ( 14 - 15 \beta_{3} + 7 \beta_{4} ) q^{88} + ( 3 \beta_{5} - 6 \beta_{6} + 7 \beta_{7} ) q^{89} + ( 9 + 3 \beta_{2} + 3 \beta_{3} + 18 \beta_{4} ) q^{90} + ( -22 \beta_{2} - 11 \beta_{3} + 5 \beta_{4} ) q^{94} + ( \beta_{1} - 2 \beta_{7} ) q^{96} + ( 7 \beta_{1} - 7 \beta_{7} ) q^{98} + ( 3 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{4} - 12q^{9} + O(q^{10}) \) \( 8q + 12q^{4} - 12q^{9} - 36q^{10} - 4q^{16} + 28q^{22} + 12q^{30} - 36q^{36} - 72q^{40} - 24q^{48} + 32q^{55} + 120q^{66} + 60q^{75} - 36q^{81} + 12q^{82} + 84q^{88} - 20q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 390 \nu^{2} + 335 \nu - 107 \)\()/37\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 427 \nu^{2} + 335 \nu - 181 \)\()/37\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} + 29 \nu^{6} - 89 \nu^{5} + 261 \nu^{4} - 373 \nu^{3} + 498 \nu^{2} - 294 \nu + 152 \)\()/37\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{7} - 28 \nu^{6} + 114 \nu^{5} - 215 \nu^{4} + 378 \nu^{3} - 366 \nu^{2} + 266 \nu - 97 \)\()/37\)
\(\beta_{5}\)\(=\)\((\)\( -21 \nu^{7} + 55 \nu^{6} - 216 \nu^{5} + 273 \nu^{4} - 428 \nu^{3} + 156 \nu^{2} - 97 \nu - 46 \)\()/37\)
\(\beta_{6}\)\(=\)\((\)\( -24 \nu^{7} + 84 \nu^{6} - 305 \nu^{5} + 534 \nu^{4} - 801 \nu^{3} + 617 \nu^{2} - 317 \nu - 5 \)\()/37\)
\(\beta_{7}\)\(=\)\((\)\( -24 \nu^{7} + 84 \nu^{6} - 305 \nu^{5} + 571 \nu^{4} - 875 \nu^{3} + 876 \nu^{2} - 539 \nu + 217 \)\()/37\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(-\beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - 4 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + 4 \beta_{2} - 8\)\()/2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{6} - 3 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} - 4 \beta_{1} + 3\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{7} + 13 \beta_{6} - 14 \beta_{5} + 15 \beta_{4} - 26 \beta_{3} - \beta_{2} - 11 \beta_{1} + 41\)\()/2\)
\(\nu^{6}\)\(=\)\(5 \beta_{7} + 16 \beta_{6} - 4 \beta_{5} + 30 \beta_{4} - 31 \beta_{3} - 40 \beta_{2} + 11 \beta_{1} + 10\)
\(\nu^{7}\)\(=\)\((\)\(-46 \beta_{7} - 25 \beta_{6} + 63 \beta_{5} - 14 \beta_{4} + 71 \beta_{3} - 83 \beta_{2} + 77 \beta_{1} - 167\)\()/2\)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.500000 0.564882i
0.500000 + 1.56488i
0.500000 2.19293i
0.500000 + 1.19293i
0.500000 + 2.19293i
0.500000 1.19293i
0.500000 + 0.564882i
0.500000 1.56488i
−0.389774 + 1.45466i −0.866025 1.50000i −0.232051 0.133975i 2.90931 + 2.90931i 2.51954 0.675108i 0 −1.84443 + 1.84443i −1.50000 + 2.59808i −5.36603 + 3.09808i
80.2 0.389774 1.45466i −0.866025 1.50000i −0.232051 0.133975i −2.90931 2.90931i −2.51954 + 0.675108i 0 1.84443 1.84443i −1.50000 + 2.59808i −5.36603 + 3.09808i
89.1 −2.31259 0.619657i 0.866025 + 1.50000i 3.23205 + 1.86603i 1.23931 1.23931i −1.07328 4.00552i 0 −2.93225 2.93225i −1.50000 + 2.59808i −3.63397 + 2.09808i
89.2 2.31259 + 0.619657i 0.866025 + 1.50000i 3.23205 + 1.86603i −1.23931 + 1.23931i 1.07328 + 4.00552i 0 2.93225 + 2.93225i −1.50000 + 2.59808i −3.63397 + 2.09808i
188.1 −2.31259 + 0.619657i 0.866025 1.50000i 3.23205 1.86603i 1.23931 + 1.23931i −1.07328 + 4.00552i 0 −2.93225 + 2.93225i −1.50000 2.59808i −3.63397 2.09808i
188.2 2.31259 0.619657i 0.866025 1.50000i 3.23205 1.86603i −1.23931 1.23931i 1.07328 4.00552i 0 2.93225 2.93225i −1.50000 2.59808i −3.63397 2.09808i
488.1 −0.389774 1.45466i −0.866025 + 1.50000i −0.232051 + 0.133975i 2.90931 2.90931i 2.51954 + 0.675108i 0 −1.84443 1.84443i −1.50000 2.59808i −5.36603 3.09808i
488.2 0.389774 + 1.45466i −0.866025 + 1.50000i −0.232051 + 0.133975i −2.90931 + 2.90931i −2.51954 0.675108i 0 1.84443 + 1.84443i −1.50000 2.59808i −5.36603 3.09808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 488.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
13.f odd 12 2 inner
39.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.h 8
3.b odd 2 1 inner 507.2.k.h 8
13.b even 2 1 inner 507.2.k.h 8
13.c even 3 1 507.2.f.d 8
13.c even 3 1 507.2.k.g 8
13.d odd 4 2 507.2.k.g 8
13.e even 6 1 507.2.f.d 8
13.e even 6 1 507.2.k.g 8
13.f odd 12 2 507.2.f.d 8
13.f odd 12 2 inner 507.2.k.h 8
39.d odd 2 1 CM 507.2.k.h 8
39.f even 4 2 507.2.k.g 8
39.h odd 6 1 507.2.f.d 8
39.h odd 6 1 507.2.k.g 8
39.i odd 6 1 507.2.f.d 8
39.i odd 6 1 507.2.k.g 8
39.k even 12 2 507.2.f.d 8
39.k even 12 2 inner 507.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.f.d 8 13.c even 3 1
507.2.f.d 8 13.e even 6 1
507.2.f.d 8 13.f odd 12 2
507.2.f.d 8 39.h odd 6 1
507.2.f.d 8 39.i odd 6 1
507.2.f.d 8 39.k even 12 2
507.2.k.g 8 13.c even 3 1
507.2.k.g 8 13.d odd 4 2
507.2.k.g 8 13.e even 6 1
507.2.k.g 8 39.f even 4 2
507.2.k.g 8 39.h odd 6 1
507.2.k.g 8 39.i odd 6 1
507.2.k.h 8 1.a even 1 1 trivial
507.2.k.h 8 3.b odd 2 1 inner
507.2.k.h 8 13.b even 2 1 inner
507.2.k.h 8 13.f odd 12 2 inner
507.2.k.h 8 39.d odd 2 1 CM
507.2.k.h 8 39.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} - 6 T_{2}^{6} - T_{2}^{4} + 78 T_{2}^{2} + 169 \)
\( T_{5}^{8} + 296 T_{5}^{4} + 2704 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 169 + 78 T^{2} - T^{4} - 6 T^{6} + T^{8} \)
$3$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$5$ \( 2704 + 296 T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 2704 - 3744 T^{2} + 1676 T^{4} + 72 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( 39589264 - 453024 T^{2} - 4564 T^{4} + 72 T^{6} + T^{8} \)
$43$ \( ( 256 - 16 T^{2} + T^{4} )^{2} \)
$47$ \( 77228944 + 17768 T^{4} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( 2704 - 21216 T^{2} + 55436 T^{4} + 408 T^{6} + T^{8} \)
$61$ \( ( 36864 + 192 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( 39589264 - 2567136 T^{2} + 49196 T^{4} + 408 T^{6} + T^{8} \)
$73$ \( T^{8} \)
$79$ \( ( -108 + T^{2} )^{4} \)
$83$ \( 756690064 + 55208 T^{4} + T^{8} \)
$89$ \( 39589264 + 3473184 T^{2} + 95276 T^{4} - 552 T^{6} + T^{8} \)
$97$ \( T^{8} \)
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