Properties

Label 507.2.f.d
Level $507$
Weight $2$
Character orbit 507.f
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.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{4} q^{3} + ( -2 \beta_{2} - \beta_{5} ) q^{4} + \beta_{6} q^{5} + ( \beta_{1} - \beta_{6} ) q^{6} + ( \beta_{3} + \beta_{7} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{4} q^{3} + ( -2 \beta_{2} - \beta_{5} ) q^{4} + \beta_{6} q^{5} + ( \beta_{1} - \beta_{6} ) q^{6} + ( \beta_{3} + \beta_{7} ) q^{8} + 3 q^{9} + ( \beta_{2} - 3 \beta_{5} ) q^{10} + ( 2 \beta_{3} + \beta_{7} ) q^{11} + ( 3 \beta_{2} + 2 \beta_{5} ) q^{12} + ( 2 \beta_{1} + \beta_{6} ) q^{15} + ( 1 + 2 \beta_{4} ) q^{16} -3 \beta_{1} q^{18} + ( 2 \beta_{3} + \beta_{7} ) q^{20} + ( -7 + 5 \beta_{4} ) q^{22} -3 \beta_{3} q^{24} + ( -5 \beta_{2} + 4 \beta_{5} ) q^{25} + 3 \beta_{4} q^{27} + ( 9 \beta_{2} - \beta_{5} ) q^{30} -\beta_{1} q^{32} + ( -4 \beta_{3} - \beta_{7} ) q^{33} + ( -6 \beta_{2} - 3 \beta_{5} ) q^{36} + ( -9 - \beta_{4} ) q^{40} + ( -4 \beta_{1} - \beta_{6} ) q^{41} -4 \beta_{2} q^{43} + ( 8 \beta_{1} - 3 \beta_{6} ) q^{44} + 3 \beta_{6} q^{45} + ( -2 \beta_{3} - 3 \beta_{7} ) q^{47} + ( 6 + \beta_{4} ) q^{48} + 7 \beta_{2} q^{49} + ( \beta_{3} - 4 \beta_{7} ) q^{50} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{54} + ( -8 + 2 \beta_{4} ) q^{55} + ( 2 \beta_{3} - 3 \beta_{7} ) q^{59} + ( -4 \beta_{3} - \beta_{7} ) q^{60} -8 \beta_{4} q^{61} + ( -2 \beta_{2} - 5 \beta_{5} ) q^{64} + ( 15 - 7 \beta_{4} ) q^{66} + ( -6 \beta_{1} + \beta_{6} ) q^{71} + ( 3 \beta_{3} + 3 \beta_{7} ) q^{72} + ( -12 \beta_{2} + 5 \beta_{5} ) q^{75} + 6 \beta_{4} q^{79} + ( 4 \beta_{1} + 3 \beta_{6} ) q^{80} + 9 q^{81} + ( -17 \beta_{2} - \beta_{5} ) q^{82} + ( -6 \beta_{1} - \beta_{6} ) q^{83} + 4 \beta_{3} q^{86} + ( 15 \beta_{2} + 7 \beta_{5} ) q^{88} + ( -4 \beta_{3} + 3 \beta_{7} ) q^{89} + ( 3 \beta_{2} - 9 \beta_{5} ) q^{90} + ( 5 - 11 \beta_{4} ) q^{94} + ( \beta_{1} - \beta_{6} ) q^{96} -7 \beta_{3} q^{98} + ( 6 \beta_{3} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} + 8q^{16} - 56q^{22} - 72q^{40} + 48q^{48} - 64q^{55} + 120q^{66} + 72q^{81} + 40q^{94} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{7} + 8 \nu^{6} - 22 \nu^{5} + 146 \nu^{4} - 256 \nu^{3} + 390 \nu^{2} - 335 \nu + 107 \)\()/37\)
\(\beta_{2}\)\(=\)\((\)\( 6 \nu^{7} - 21 \nu^{6} + 67 \nu^{5} - 115 \nu^{4} + 117 \nu^{3} - 71 \nu^{2} - 41 \nu + 29 \)\()/37\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} + 29 \nu^{6} - 89 \nu^{5} + 261 \nu^{4} - 373 \nu^{3} + 461 \nu^{2} - 220 \nu + 41 \)\()/37\)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} + 11 \nu^{4} - 17 \nu^{3} + 25 \nu^{2} - 17 \nu + 9 \)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{7} - 56 \nu^{6} + 228 \nu^{5} - 430 \nu^{4} + 756 \nu^{3} - 732 \nu^{2} + 532 \nu - 157 \)\()/37\)
\(\beta_{6}\)\(=\)\((\)\( -42 \nu^{7} + 147 \nu^{6} - 543 \nu^{5} + 953 \nu^{4} - 1485 \nu^{3} + 1163 \nu^{2} - 749 \nu + 56 \)\()/37\)
\(\beta_{7}\)\(=\)\((\)\( -42 \nu^{7} + 147 \nu^{6} - 543 \nu^{5} + 1027 \nu^{4} - 1633 \nu^{3} + 1681 \nu^{2} - 1193 \nu + 500 \)\()/37\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + \beta_{2} - 2 \beta_{1} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} + 2 \beta_{1} - 13\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{6} - 3 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 12 \beta_{2} + 10 \beta_{1} + 9\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{7} - \beta_{6} + 15 \beta_{5} - 25 \beta_{4} + 36 \beta_{3} + 27 \beta_{2} + 14 \beta_{1} + 67\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{7} + 12 \beta_{6} + 30 \beta_{5} + 9 \beta_{4} + 25 \beta_{3} + 71 \beta_{2} - 39 \beta_{1} - 10\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-23 \beta_{7} + 19 \beta_{6} - 7 \beta_{5} + 77 \beta_{4} - 67 \beta_{3} + 6 \beta_{2} - 73 \beta_{1} - 160\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.500000 2.19293i
0.500000 1.56488i
0.500000 + 0.564882i
0.500000 + 1.19293i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 0.564882i
0.500000 1.19293i
−1.69293 1.69293i −1.73205 3.73205i −1.23931 1.23931i 2.93225 + 2.93225i 0 2.93225 2.93225i 3.00000 4.19615i
239.2 −1.06488 1.06488i 1.73205 0.267949i 2.90931 + 2.90931i −1.84443 1.84443i 0 −1.84443 + 1.84443i 3.00000 6.19615i
239.3 1.06488 + 1.06488i 1.73205 0.267949i −2.90931 2.90931i 1.84443 + 1.84443i 0 1.84443 1.84443i 3.00000 6.19615i
239.4 1.69293 + 1.69293i −1.73205 3.73205i 1.23931 + 1.23931i −2.93225 2.93225i 0 −2.93225 + 2.93225i 3.00000 4.19615i
437.1 −1.69293 + 1.69293i −1.73205 3.73205i −1.23931 + 1.23931i 2.93225 2.93225i 0 2.93225 + 2.93225i 3.00000 4.19615i
437.2 −1.06488 + 1.06488i 1.73205 0.267949i 2.90931 2.90931i −1.84443 + 1.84443i 0 −1.84443 1.84443i 3.00000 6.19615i
437.3 1.06488 1.06488i 1.73205 0.267949i −2.90931 + 2.90931i 1.84443 1.84443i 0 1.84443 + 1.84443i 3.00000 6.19615i
437.4 1.69293 1.69293i −1.73205 3.73205i 1.23931 1.23931i −2.93225 + 2.93225i 0 −2.93225 2.93225i 3.00000 4.19615i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 437.4
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.d odd 4 2 inner
39.f even 4 2 inner

Twists

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

Hecke kernels

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

\( T_{2}^{8} + 38 T_{2}^{4} + 169 \)
\( T_{5}^{8} + 296 T_{5}^{4} + 2704 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 169 + 38 T^{4} + T^{8} \)
$3$ \( ( -3 + T^{2} )^{4} \)
$5$ \( 2704 + 296 T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 2704 + 1832 T^{4} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( 39589264 + 14312 T^{4} + T^{8} \)
$43$ \( ( 16 + T^{2} )^{4} \)
$47$ \( 77228944 + 17768 T^{4} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( 2704 + 55592 T^{4} + T^{8} \)
$61$ \( ( -192 + T^{2} )^{4} \)
$67$ \( T^{8} \)
$71$ \( 39589264 + 68072 T^{4} + T^{8} \)
$73$ \( T^{8} \)
$79$ \( ( -108 + T^{2} )^{4} \)
$83$ \( 756690064 + 55208 T^{4} + T^{8} \)
$89$ \( 39589264 + 114152 T^{4} + T^{8} \)
$97$ \( T^{8} \)
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