Properties

Label 504.4.s.f
Level $504$
Weight $4$
Character orbit 504.s
Analytic conductor $29.737$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{505})\)
Defining polynomial: \(x^{4} - x^{3} + 127 x^{2} + 126 x + 15876\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + 4 \beta_{2} ) q^{5} + ( 9 - 17 \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + 4 \beta_{2} ) q^{5} + ( 9 - 17 \beta_{2} - \beta_{3} ) q^{7} + ( 5 - 5 \beta_{1} - 5 \beta_{3} ) q^{11} + ( -10 + \beta_{3} ) q^{13} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{17} + ( -7 \beta_{1} - 55 \beta_{2} ) q^{19} + ( -4 \beta_{1} + 76 \beta_{2} ) q^{23} + ( -26 + 9 \beta_{1} + 17 \beta_{2} + 9 \beta_{3} ) q^{25} + ( 45 + 5 \beta_{3} ) q^{29} + ( 181 - 2 \beta_{1} - 179 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 85 - 4 \beta_{1} + 90 \beta_{2} - 17 \beta_{3} ) q^{35} + ( -\beta_{1} + 27 \beta_{2} ) q^{37} + ( 76 + 18 \beta_{3} ) q^{41} + ( 188 + 27 \beta_{3} ) q^{43} + ( 36 \beta_{1} + 166 \beta_{2} ) q^{47} + ( -82 - 34 \beta_{1} + 17 \beta_{2} - 17 \beta_{3} ) q^{49} + ( 351 - 5 \beta_{1} - 346 \beta_{2} - 5 \beta_{3} ) q^{53} + ( 655 - 25 \beta_{3} ) q^{55} + ( -279 - 27 \beta_{1} + 306 \beta_{2} - 27 \beta_{3} ) q^{59} + ( 28 \beta_{1} + 566 \beta_{2} ) q^{61} + ( -14 \beta_{1} - 162 \beta_{2} ) q^{65} + ( -80 - 73 \beta_{1} + 153 \beta_{2} - 73 \beta_{3} ) q^{67} + ( -346 + 76 \beta_{3} ) q^{71} + ( 440 - 63 \beta_{1} - 377 \beta_{2} - 63 \beta_{3} ) q^{73} + ( 590 - 45 \beta_{1} - 630 \beta_{2} + 40 \beta_{3} ) q^{77} + ( -48 \beta_{1} + 425 \beta_{2} ) q^{79} + ( 175 - 67 \beta_{3} ) q^{83} + ( 504 - 16 \beta_{3} ) q^{85} + ( 22 \beta_{1} - 940 \beta_{2} ) q^{89} + ( -216 + 17 \beta_{1} + 153 \beta_{2} + 18 \beta_{3} ) q^{91} + ( 1192 - 90 \beta_{1} - 1102 \beta_{2} - 90 \beta_{3} ) q^{95} + ( 837 + 55 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 9q^{5} + O(q^{10}) \) \( 4q + 9q^{5} + 5q^{11} - 38q^{13} - 4q^{17} - 117q^{19} + 148q^{23} - 43q^{25} + 190q^{29} + 360q^{31} + 482q^{35} + 53q^{37} + 340q^{41} + 806q^{43} + 368q^{47} - 362q^{49} + 697q^{53} + 2570q^{55} - 585q^{59} + 1160q^{61} - 338q^{65} - 233q^{67} - 1232q^{71} + 817q^{73} + 1135q^{77} + 802q^{79} + 566q^{83} + 1984q^{85} - 1858q^{89} - 505q^{91} + 2294q^{95} + 3458q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 127 x^{2} + 126 x + 15876\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 127 \nu^{2} - 127 \nu + 15876 \)\()/16002\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 253 \)\()/127\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 126 \beta_{2} + \beta_{1} - 127\)
\(\nu^{3}\)\(=\)\(127 \beta_{3} - 253\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−5.36805 + 9.29774i
5.86805 10.1638i
−5.36805 9.29774i
5.86805 + 10.1638i
0 0 0 −3.36805 + 5.83364i 0 −11.2361 + 14.7224i 0 0 0
289.2 0 0 0 7.86805 13.6279i 0 11.2361 + 14.7224i 0 0 0
361.1 0 0 0 −3.36805 5.83364i 0 −11.2361 14.7224i 0 0 0
361.2 0 0 0 7.86805 + 13.6279i 0 11.2361 14.7224i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.4.s.f 4
3.b odd 2 1 168.4.q.d 4
7.c even 3 1 inner 504.4.s.f 4
12.b even 2 1 336.4.q.g 4
21.g even 6 1 1176.4.a.u 2
21.h odd 6 1 168.4.q.d 4
21.h odd 6 1 1176.4.a.r 2
84.j odd 6 1 2352.4.a.bo 2
84.n even 6 1 336.4.q.g 4
84.n even 6 1 2352.4.a.cc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 3.b odd 2 1
168.4.q.d 4 21.h odd 6 1
336.4.q.g 4 12.b even 2 1
336.4.q.g 4 84.n even 6 1
504.4.s.f 4 1.a even 1 1 trivial
504.4.s.f 4 7.c even 3 1 inner
1176.4.a.r 2 21.h odd 6 1
1176.4.a.u 2 21.g even 6 1
2352.4.a.bo 2 84.j odd 6 1
2352.4.a.cc 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 9 T_{5}^{3} + 187 T_{5}^{2} + 954 T_{5} + 11236 \) acting on \(S_{4}^{\mathrm{new}}(504, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 11236 + 954 T + 187 T^{2} - 9 T^{3} + T^{4} \)
$7$ \( 117649 + 181 T^{2} + T^{4} \)
$11$ \( 9922500 + 15750 T + 3175 T^{2} - 5 T^{3} + T^{4} \)
$13$ \( ( -36 + 19 T + T^{2} )^{2} \)
$17$ \( 4064256 - 8064 T + 2032 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 7639696 - 323388 T + 16453 T^{2} + 117 T^{3} + T^{4} \)
$23$ \( 11943936 - 511488 T + 18448 T^{2} - 148 T^{3} + T^{4} \)
$29$ \( ( -900 - 95 T + T^{2} )^{2} \)
$31$ \( 1017291025 - 11482200 T + 97705 T^{2} - 360 T^{3} + T^{4} \)
$37$ \( 331776 - 30528 T + 2233 T^{2} - 53 T^{3} + T^{4} \)
$41$ \( ( -33680 - 170 T + T^{2} )^{2} \)
$43$ \( ( -51434 - 403 T + T^{2} )^{2} \)
$47$ \( 16838695696 + 47753152 T + 265188 T^{2} - 368 T^{3} + T^{4} \)
$53$ \( 13993943616 - 82452312 T + 367513 T^{2} - 697 T^{3} + T^{4} \)
$59$ \( 41990400 - 3790800 T + 348705 T^{2} + 585 T^{3} + T^{4} \)
$61$ \( 56368256400 - 275407200 T + 1108180 T^{2} - 1160 T^{3} + T^{4} \)
$67$ \( 434563097796 - 153596862 T + 713503 T^{2} + 233 T^{3} + T^{4} \)
$71$ \( ( -634356 + 616 T + T^{2} )^{2} \)
$73$ \( 111698997796 + 273052838 T + 1001703 T^{2} - 817 T^{3} + T^{4} \)
$79$ \( 16920546241 + 104323358 T + 773283 T^{2} - 802 T^{3} + T^{4} \)
$83$ \( ( -546714 - 283 T + T^{2} )^{2} \)
$89$ \( 643101348096 + 1489997088 T + 2650228 T^{2} + 1858 T^{3} + T^{4} \)
$97$ \( ( 365454 - 1729 T + T^{2} )^{2} \)
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