Properties

Label 504.2.cj.a
Level 504
Weight 2
Character orbit 504.cj
Analytic conductor 4.024
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 504.cj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{4} + \beta_{5} ) q^{2} + \beta_{6} q^{4} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( 1 + 3 \beta_{2} ) q^{7} + ( \beta_{4} - 2 \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( \beta_{4} + \beta_{5} ) q^{2} + \beta_{6} q^{4} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( 1 + 3 \beta_{2} ) q^{7} + ( \beta_{4} - 2 \beta_{7} ) q^{8} + ( 2 + 2 \beta_{2} + \beta_{6} ) q^{10} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{11} + ( \beta_{4} - 2 \beta_{5} ) q^{14} + ( -\beta_{1} - 4 \beta_{2} ) q^{16} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{17} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 3 \beta_{4} - 2 \beta_{7} ) q^{20} + ( -2 - \beta_{1} - \beta_{6} ) q^{22} + ( -4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{6} ) q^{28} + ( -\beta_{4} - \beta_{7} ) q^{29} + ( 1 + \beta_{2} ) q^{31} + ( 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{32} + ( 4 - 2 \beta_{1} - 2 \beta_{6} ) q^{34} + ( -2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{35} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -8 \beta_{3} + 2 \beta_{5} + 8 \beta_{7} ) q^{38} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{40} + ( 6 \beta_{4} - 6 \beta_{7} ) q^{41} + ( 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{44} + ( -8 - 8 \beta_{2} + 4 \beta_{6} ) q^{46} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -8 - 3 \beta_{2} ) q^{49} + ( 5 \beta_{3} + 5 \beta_{5} - 5 \beta_{7} ) q^{53} -5 q^{55} + ( 6 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} ) q^{56} + ( \beta_{1} - 2 \beta_{2} ) q^{58} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{59} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{61} + \beta_{4} q^{62} + ( 4 + 3 \beta_{1} + 3 \beta_{6} ) q^{64} + ( 2 + 2 \beta_{2} - 4 \beta_{6} ) q^{67} + ( 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -4 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} ) q^{70} + ( -6 \beta_{4} + 6 \beta_{7} ) q^{71} + ( 10 + 10 \beta_{2} ) q^{73} + ( -8 \beta_{3} + 2 \beta_{5} + 8 \beta_{7} ) q^{74} + ( -16 + 2 \beta_{1} + 2 \beta_{6} ) q^{76} + ( -\beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{77} -13 \beta_{2} q^{79} + ( 6 \beta_{3} + \beta_{5} - 6 \beta_{7} ) q^{80} + ( -6 \beta_{1} - 12 \beta_{2} ) q^{82} + ( 5 \beta_{4} + 5 \beta_{7} ) q^{83} + ( 2 - 4 \beta_{1} - 4 \beta_{6} ) q^{85} + ( 4 + 4 \beta_{2} - 3 \beta_{6} ) q^{88} + ( 8 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{89} + ( -4 \beta_{4} - 8 \beta_{7} ) q^{92} + ( 4 + 4 \beta_{2} - 2 \beta_{6} ) q^{94} + ( -10 \beta_{3} + 10 \beta_{5} + 10 \beta_{7} ) q^{95} - q^{97} + ( -8 \beta_{4} - 5 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} - 4q^{7} + O(q^{10}) \) \( 8q + 2q^{4} - 4q^{7} + 10q^{10} + 14q^{16} - 20q^{22} - 10q^{28} + 4q^{31} + 24q^{34} + 10q^{40} - 24q^{46} - 52q^{49} - 40q^{55} + 10q^{58} + 44q^{64} - 50q^{70} + 40q^{73} - 120q^{76} + 52q^{79} + 36q^{82} + 10q^{88} + 12q^{94} - 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} + 8 \nu^{4} - 16 \nu^{2} + 19 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 1 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 12 \nu^{3} + 11 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{5} - 6 \nu^{3} - 3 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + 16 \nu^{5} - 40 \nu^{3} + 23 \nu \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{6} + 16 \nu^{4} - 40 \nu^{2} - 3 \)\()/8\)
\(\beta_{7}\)\(=\)\( -\nu^{7} + 3 \nu^{5} - 8 \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} - 5 \beta_{2} + \beta_{1} - 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 4 \beta_{5} - \beta_{4} - 4 \beta_{3}\)\()/3\)
\(\nu^{4}\)\(=\)\(\beta_{6} - 3 \beta_{2} + 2 \beta_{1} - 4\)
\(\nu^{5}\)\(=\)\((\)\(11 \beta_{7} - 2 \beta_{5} - 13 \beta_{4} - 13 \beta_{3}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-8 \beta_{6} + 8 \beta_{2} + 8 \beta_{1} - 23\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(34 \beta_{7} - 34 \beta_{5} - 29 \beta_{4} - 5 \beta_{3}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.40126 + 0.809017i
−0.535233 + 0.309017i
0.535233 0.309017i
1.40126 0.809017i
−1.40126 0.809017i
−0.535233 0.309017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−1.40126 + 0.190983i 0 1.92705 0.535233i −1.93649 1.11803i 0 −0.500000 + 2.59808i −2.59808 + 1.11803i 0 2.92705 + 1.19682i
37.2 −0.535233 1.30902i 0 −1.42705 + 1.40126i −1.93649 1.11803i 0 −0.500000 + 2.59808i 2.59808 + 1.11803i 0 −0.427051 + 3.13331i
37.3 0.535233 + 1.30902i 0 −1.42705 + 1.40126i 1.93649 + 1.11803i 0 −0.500000 + 2.59808i −2.59808 1.11803i 0 −0.427051 + 3.13331i
37.4 1.40126 0.190983i 0 1.92705 0.535233i 1.93649 + 1.11803i 0 −0.500000 + 2.59808i 2.59808 1.11803i 0 2.92705 + 1.19682i
109.1 −1.40126 0.190983i 0 1.92705 + 0.535233i −1.93649 + 1.11803i 0 −0.500000 2.59808i −2.59808 1.11803i 0 2.92705 1.19682i
109.2 −0.535233 + 1.30902i 0 −1.42705 1.40126i −1.93649 + 1.11803i 0 −0.500000 2.59808i 2.59808 1.11803i 0 −0.427051 3.13331i
109.3 0.535233 1.30902i 0 −1.42705 1.40126i 1.93649 1.11803i 0 −0.500000 2.59808i −2.59808 + 1.11803i 0 −0.427051 3.13331i
109.4 1.40126 + 0.190983i 0 1.92705 + 0.535233i 1.93649 1.11803i 0 −0.500000 2.59808i 2.59808 + 1.11803i 0 2.92705 1.19682i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.a 8
3.b odd 2 1 inner 504.2.cj.a 8
4.b odd 2 1 2016.2.cr.a 8
7.c even 3 1 inner 504.2.cj.a 8
8.b even 2 1 inner 504.2.cj.a 8
8.d odd 2 1 2016.2.cr.a 8
12.b even 2 1 2016.2.cr.a 8
21.h odd 6 1 inner 504.2.cj.a 8
24.f even 2 1 2016.2.cr.a 8
24.h odd 2 1 inner 504.2.cj.a 8
28.g odd 6 1 2016.2.cr.a 8
56.k odd 6 1 2016.2.cr.a 8
56.p even 6 1 inner 504.2.cj.a 8
84.n even 6 1 2016.2.cr.a 8
168.s odd 6 1 inner 504.2.cj.a 8
168.v even 6 1 2016.2.cr.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.a 8 1.a even 1 1 trivial
504.2.cj.a 8 3.b odd 2 1 inner
504.2.cj.a 8 7.c even 3 1 inner
504.2.cj.a 8 8.b even 2 1 inner
504.2.cj.a 8 21.h odd 6 1 inner
504.2.cj.a 8 24.h odd 2 1 inner
504.2.cj.a 8 56.p even 6 1 inner
504.2.cj.a 8 168.s odd 6 1 inner
2016.2.cr.a 8 4.b odd 2 1
2016.2.cr.a 8 8.d odd 2 1
2016.2.cr.a 8 12.b even 2 1
2016.2.cr.a 8 24.f even 2 1
2016.2.cr.a 8 28.g odd 6 1
2016.2.cr.a 8 56.k odd 6 1
2016.2.cr.a 8 84.n even 6 1
2016.2.cr.a 8 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 5 T_{5}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \)
$3$ \( \)
$5$ \( ( 1 + 5 T^{2} )^{4}( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + T + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 17 T^{2} + 168 T^{4} + 2057 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 13 T^{2} )^{8} \)
$17$ \( ( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2}( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 2 T^{2} - 525 T^{4} + 1058 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 53 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 14 T^{2} - 1173 T^{4} + 19166 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 26 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{8} \)
$47$ \( ( 1 - 82 T^{2} + 4515 T^{4} - 181138 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 19 T^{2} - 2448 T^{4} - 53371 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 113 T^{2} + 9288 T^{4} + 393353 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 62 T^{2} + 123 T^{4} + 230702 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 74 T^{2} + 987 T^{4} + 332186 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 34 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )^{4}( 1 + 7 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )^{4}( 1 + 4 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 41 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 14 T^{2} - 7725 T^{4} + 110894 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + T + 97 T^{2} )^{8} \)
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