Properties

Label 504.2.ch.a
Level 504
Weight 2
Character orbit 504.ch
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.ch (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: 8.0.4857532416.2
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 2 - 2 \beta_{3} ) q^{4} -\beta_{5} q^{5} + ( 3 - \beta_{3} ) q^{7} + 2 \beta_{4} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 2 - 2 \beta_{3} ) q^{4} -\beta_{5} q^{5} + ( 3 - \beta_{3} ) q^{7} + 2 \beta_{4} q^{8} + ( -\beta_{1} + \beta_{7} ) q^{10} + ( \beta_{5} - \beta_{6} ) q^{11} + \beta_{1} q^{13} + ( 2 \beta_{2} + \beta_{4} ) q^{14} -4 \beta_{3} q^{16} + ( -\beta_{2} - \beta_{4} ) q^{17} -\beta_{7} q^{19} + 2 \beta_{6} q^{20} + ( \beta_{1} - 2 \beta_{7} ) q^{22} -\beta_{2} q^{23} + ( 6 - 6 \beta_{3} ) q^{25} + 2 \beta_{5} q^{26} + ( 4 - 6 \beta_{3} ) q^{28} + ( 2 \beta_{5} + \beta_{6} ) q^{29} + ( 3 + 3 \beta_{3} ) q^{31} + ( -4 \beta_{2} + 4 \beta_{4} ) q^{32} + ( -2 + 4 \beta_{3} ) q^{34} + ( -2 \beta_{5} + \beta_{6} ) q^{35} + ( -2 \beta_{1} + \beta_{7} ) q^{37} + ( -2 \beta_{5} - 2 \beta_{6} ) q^{38} + 2 \beta_{7} q^{40} + ( -8 \beta_{2} + 4 \beta_{4} ) q^{41} + ( -\beta_{1} + 2 \beta_{7} ) q^{43} + ( -2 \beta_{5} - 4 \beta_{6} ) q^{44} + ( -2 + 2 \beta_{3} ) q^{46} + ( 3 \beta_{2} - 6 \beta_{4} ) q^{47} + ( 8 - 5 \beta_{3} ) q^{49} + 6 \beta_{4} q^{50} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{52} + ( -\beta_{5} + \beta_{6} ) q^{53} + ( -11 + 22 \beta_{3} ) q^{55} + ( -2 \beta_{2} + 6 \beta_{4} ) q^{56} + ( 2 \beta_{1} - \beta_{7} ) q^{58} + ( \beta_{5} + \beta_{6} ) q^{59} -\beta_{7} q^{61} + ( 6 \beta_{2} - 3 \beta_{4} ) q^{62} -8 q^{64} -11 \beta_{2} q^{65} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{68} + ( -2 \beta_{1} + 3 \beta_{7} ) q^{70} + \beta_{4} q^{71} + ( -2 - 2 \beta_{3} ) q^{73} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{74} -2 \beta_{1} q^{76} + ( \beta_{5} - 4 \beta_{6} ) q^{77} + 7 \beta_{3} q^{79} + ( 4 \beta_{5} + 4 \beta_{6} ) q^{80} + ( -16 + 8 \beta_{3} ) q^{82} -\beta_{6} q^{83} + ( \beta_{1} - 2 \beta_{7} ) q^{85} + ( 2 \beta_{5} + 4 \beta_{6} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{7} ) q^{88} + ( 4 \beta_{2} - 8 \beta_{4} ) q^{89} + ( 3 \beta_{1} - \beta_{7} ) q^{91} -2 \beta_{4} q^{92} + ( 6 + 6 \beta_{3} ) q^{94} + ( 11 \beta_{2} - 11 \beta_{4} ) q^{95} + ( 1 - 2 \beta_{3} ) q^{97} + ( 3 \beta_{2} + 5 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + 20q^{7} + O(q^{10}) \) \( 8q + 8q^{4} + 20q^{7} - 16q^{16} + 24q^{25} + 8q^{28} + 36q^{31} - 8q^{46} + 44q^{49} - 64q^{64} - 24q^{73} + 28q^{79} - 96q^{82} + 72q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 7 x^{6} - 2 x^{5} + 98 x^{4} - 98 x^{3} + 67 x^{2} - 30 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 33 \nu^{7} + 61 \nu^{6} - 282 \nu^{5} - 294 \nu^{4} + 388 \nu^{3} - 312 \nu^{2} + 153 \nu + 44177 \)\()/9430\)
\(\beta_{2}\)\(=\)\((\)\( -3823 \nu^{7} + 12079 \nu^{6} + 22382 \nu^{5} - 30236 \nu^{4} - 414148 \nu^{3} + 781972 \nu^{2} - 298053 \nu + 135243 \)\()/84870\)
\(\beta_{3}\)\(=\)\((\)\( -1508 \nu^{7} + 2499 \nu^{6} + 11172 \nu^{5} + 7434 \nu^{4} - 143178 \nu^{3} + 100842 \nu^{2} - 96148 \nu + 42843 \)\()/28290\)
\(\beta_{4}\)\(=\)\((\)\( 209 \nu^{7} - 227 \nu^{6} - 1786 \nu^{5} - 1862 \nu^{4} + 19784 \nu^{3} - 1976 \nu^{2} + 969 \nu + 261 \)\()/2070\)
\(\beta_{5}\)\(=\)\((\)\( -9068 \nu^{7} + 27959 \nu^{6} + 51772 \nu^{5} - 65806 \nu^{4} - 976178 \nu^{3} + 1837142 \nu^{2} - 700428 \nu + 317583 \)\()/84870\)
\(\beta_{6}\)\(=\)\((\)\( -88 \nu^{7} + 97 \nu^{6} + 752 \nu^{5} + 784 \nu^{4} - 8278 \nu^{3} + 832 \nu^{2} - 408 \nu - 108 \)\()/369\)
\(\beta_{7}\)\(=\)\((\)\( -7023 \nu^{7} + 11879 \nu^{6} + 51442 \nu^{5} + 32564 \nu^{4} - 668948 \nu^{3} + 471032 \nu^{2} - 450053 \nu + 200643 \)\()/28290\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{7} + \beta_{5} + 9 \beta_{3} - 2 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{6} + 19 \beta_{4} - 3 \beta_{1} + 14\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-16 \beta_{7} + 17 \beta_{6} + 17 \beta_{5} + 40 \beta_{4} + 75 \beta_{3} - 40 \beta_{2} + 16 \beta_{1} - 75\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-45 \beta_{7} - 61 \beta_{5} + 211 \beta_{3} + 143 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(113 \beta_{6} + 265 \beta_{4} + 55 \beta_{1} - 258\)
\(\nu^{7}\)\(=\)\((\)\(-546 \beta_{7} - 365 \beta_{6} - 365 \beta_{5} - 856 \beta_{4} + 2561 \beta_{3} + 856 \beta_{2} + 546 \beta_{1} - 2561\)\()/2\)

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
2.91089 + 1.10325i
0.0386042 0.555062i
0.461396 0.310963i
−2.41089 1.96928i
2.91089 1.10325i
0.0386042 + 0.555062i
0.461396 + 0.310963i
−2.41089 + 1.96928i
−1.22474 + 0.707107i 0 1.00000 1.73205i −2.87228 + 1.65831i 0 2.50000 0.866025i 2.82843i 0 2.34521 4.06202i
269.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 2.87228 1.65831i 0 2.50000 0.866025i 2.82843i 0 −2.34521 + 4.06202i
269.3 1.22474 0.707107i 0 1.00000 1.73205i −2.87228 + 1.65831i 0 2.50000 0.866025i 2.82843i 0 −2.34521 + 4.06202i
269.4 1.22474 0.707107i 0 1.00000 1.73205i 2.87228 1.65831i 0 2.50000 0.866025i 2.82843i 0 2.34521 4.06202i
341.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −2.87228 1.65831i 0 2.50000 + 0.866025i 2.82843i 0 2.34521 + 4.06202i
341.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 2.87228 + 1.65831i 0 2.50000 + 0.866025i 2.82843i 0 −2.34521 4.06202i
341.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −2.87228 1.65831i 0 2.50000 + 0.866025i 2.82843i 0 −2.34521 4.06202i
341.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.87228 + 1.65831i 0 2.50000 + 0.866025i 2.82843i 0 2.34521 + 4.06202i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.ch.a 8
3.b odd 2 1 inner 504.2.ch.a 8
4.b odd 2 1 2016.2.cp.a 8
7.d odd 6 1 inner 504.2.ch.a 8
8.b even 2 1 inner 504.2.ch.a 8
8.d odd 2 1 2016.2.cp.a 8
12.b even 2 1 2016.2.cp.a 8
21.g even 6 1 inner 504.2.ch.a 8
24.f even 2 1 2016.2.cp.a 8
24.h odd 2 1 inner 504.2.ch.a 8
28.f even 6 1 2016.2.cp.a 8
56.j odd 6 1 inner 504.2.ch.a 8
56.m even 6 1 2016.2.cp.a 8
84.j odd 6 1 2016.2.cp.a 8
168.ba even 6 1 inner 504.2.ch.a 8
168.be odd 6 1 2016.2.cp.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.ch.a 8 1.a even 1 1 trivial
504.2.ch.a 8 3.b odd 2 1 inner
504.2.ch.a 8 7.d odd 6 1 inner
504.2.ch.a 8 8.b even 2 1 inner
504.2.ch.a 8 21.g even 6 1 inner
504.2.ch.a 8 24.h odd 2 1 inner
504.2.ch.a 8 56.j odd 6 1 inner
504.2.ch.a 8 168.ba even 6 1 inner
2016.2.cp.a 8 4.b odd 2 1
2016.2.cp.a 8 8.d odd 2 1
2016.2.cp.a 8 12.b even 2 1
2016.2.cp.a 8 24.f even 2 1
2016.2.cp.a 8 28.f even 6 1
2016.2.cp.a 8 56.m even 6 1
2016.2.cp.a 8 84.j odd 6 1
2016.2.cp.a 8 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 11 T_{5}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ 1
$5$ \( ( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{4}( 1 - 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 4 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 16 T^{2} - 105 T^{4} - 5776 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 44 T^{2} + 1407 T^{4} + 23276 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 25 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 8 T^{2} - 1305 T^{4} + 10952 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 14 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 20 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 73 T^{2} + 2520 T^{4} - 205057 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} )^{2}( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 100 T^{2} + 6279 T^{4} - 372100 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 67 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 140 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 155 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 82 T^{2} - 1197 T^{4} - 649522 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 191 T^{2} + 9409 T^{4} )^{4} \)
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