Properties

Label 504.2.p.f
Level 504
Weight 2
Character orbit 504.p
Analytic conductor 4.024
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 504.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} -2 \beta_{1} q^{4} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} -2 \beta_{1} q^{4} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} ) q^{8} + ( -\beta_{2} + \beta_{3} ) q^{10} -2 q^{11} + \beta_{3} q^{13} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} -4 q^{16} + 2 \beta_{2} q^{17} -\beta_{2} q^{19} -2 \beta_{2} q^{20} + ( -2 + 2 \beta_{1} ) q^{22} -4 \beta_{1} q^{23} + q^{25} + ( -\beta_{2} + \beta_{3} ) q^{26} + ( 2 - 2 \beta_{2} ) q^{28} + 4 \beta_{1} q^{29} -2 \beta_{3} q^{31} + ( -4 + 4 \beta_{1} ) q^{32} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 6 + \beta_{2} ) q^{35} -8 \beta_{1} q^{37} + ( -\beta_{2} - \beta_{3} ) q^{38} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{40} -6 q^{43} + 4 \beta_{1} q^{44} + ( -4 - 4 \beta_{1} ) q^{46} -2 \beta_{3} q^{47} + ( 5 + 2 \beta_{2} ) q^{49} + ( 1 - \beta_{1} ) q^{50} -2 \beta_{2} q^{52} -4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{56} + ( 4 + 4 \beta_{1} ) q^{58} + \beta_{2} q^{59} + 3 \beta_{3} q^{61} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{62} + 8 \beta_{1} q^{64} + 6 q^{65} -2 q^{67} + 4 \beta_{3} q^{68} + ( 6 - 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{70} + 10 \beta_{1} q^{71} + 6 \beta_{2} q^{73} + ( -8 - 8 \beta_{1} ) q^{74} -2 \beta_{3} q^{76} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{77} + 6 \beta_{1} q^{79} -4 \beta_{3} q^{80} -\beta_{2} q^{83} + 12 \beta_{1} q^{85} + ( -6 + 6 \beta_{1} ) q^{86} + ( 4 + 4 \beta_{1} ) q^{88} + 6 \beta_{2} q^{89} + ( 6 + \beta_{2} ) q^{91} -8 q^{92} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{94} -6 \beta_{1} q^{95} -2 \beta_{2} q^{97} + ( 5 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{8} - 8q^{11} + 4q^{14} - 16q^{16} - 8q^{22} + 4q^{25} + 8q^{28} - 16q^{32} + 24q^{35} - 24q^{43} - 16q^{46} + 20q^{49} + 4q^{50} + 8q^{56} + 16q^{58} + 24q^{65} - 8q^{67} + 24q^{70} - 32q^{74} - 24q^{86} + 16q^{88} + 24q^{91} - 32q^{92} + 20q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 3 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.00000 1.00000i 0 2.00000i −2.44949 0 −2.44949 + 1.00000i −2.00000 2.00000i 0 −2.44949 + 2.44949i
307.2 1.00000 1.00000i 0 2.00000i 2.44949 0 2.44949 + 1.00000i −2.00000 2.00000i 0 2.44949 2.44949i
307.3 1.00000 + 1.00000i 0 2.00000i −2.44949 0 −2.44949 1.00000i −2.00000 + 2.00000i 0 −2.44949 2.44949i
307.4 1.00000 + 1.00000i 0 2.00000i 2.44949 0 2.44949 1.00000i −2.00000 + 2.00000i 0 2.44949 + 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.p.f 4
3.b odd 2 1 56.2.e.b 4
4.b odd 2 1 2016.2.p.e 4
7.b odd 2 1 inner 504.2.p.f 4
8.b even 2 1 2016.2.p.e 4
8.d odd 2 1 inner 504.2.p.f 4
12.b even 2 1 224.2.e.b 4
21.c even 2 1 56.2.e.b 4
21.g even 6 2 392.2.m.f 8
21.h odd 6 2 392.2.m.f 8
24.f even 2 1 56.2.e.b 4
24.h odd 2 1 224.2.e.b 4
28.d even 2 1 2016.2.p.e 4
48.i odd 4 1 1792.2.f.e 4
48.i odd 4 1 1792.2.f.f 4
48.k even 4 1 1792.2.f.e 4
48.k even 4 1 1792.2.f.f 4
56.e even 2 1 inner 504.2.p.f 4
56.h odd 2 1 2016.2.p.e 4
84.h odd 2 1 224.2.e.b 4
84.j odd 6 2 1568.2.q.e 8
84.n even 6 2 1568.2.q.e 8
168.e odd 2 1 56.2.e.b 4
168.i even 2 1 224.2.e.b 4
168.s odd 6 2 1568.2.q.e 8
168.v even 6 2 392.2.m.f 8
168.ba even 6 2 1568.2.q.e 8
168.be odd 6 2 392.2.m.f 8
336.v odd 4 1 1792.2.f.e 4
336.v odd 4 1 1792.2.f.f 4
336.y even 4 1 1792.2.f.e 4
336.y even 4 1 1792.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b 4 3.b odd 2 1
56.2.e.b 4 21.c even 2 1
56.2.e.b 4 24.f even 2 1
56.2.e.b 4 168.e odd 2 1
224.2.e.b 4 12.b even 2 1
224.2.e.b 4 24.h odd 2 1
224.2.e.b 4 84.h odd 2 1
224.2.e.b 4 168.i even 2 1
392.2.m.f 8 21.g even 6 2
392.2.m.f 8 21.h odd 6 2
392.2.m.f 8 168.v even 6 2
392.2.m.f 8 168.be odd 6 2
504.2.p.f 4 1.a even 1 1 trivial
504.2.p.f 4 7.b odd 2 1 inner
504.2.p.f 4 8.d odd 2 1 inner
504.2.p.f 4 56.e even 2 1 inner
1568.2.q.e 8 84.j odd 6 2
1568.2.q.e 8 84.n even 6 2
1568.2.q.e 8 168.s odd 6 2
1568.2.q.e 8 168.ba even 6 2
1792.2.f.e 4 48.i odd 4 1
1792.2.f.e 4 48.k even 4 1
1792.2.f.e 4 336.v odd 4 1
1792.2.f.e 4 336.y even 4 1
1792.2.f.f 4 48.i odd 4 1
1792.2.f.f 4 48.k even 4 1
1792.2.f.f 4 336.v odd 4 1
1792.2.f.f 4 336.y even 4 1
2016.2.p.e 4 4.b odd 2 1
2016.2.p.e 4 8.b even 2 1
2016.2.p.e 4 28.d even 2 1
2016.2.p.e 4 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\):

\( T_{5}^{2} - 6 \)
\( T_{11} + 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )^{2} \)
$3$ \( \)
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 + 6 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 112 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 68 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 2 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 42 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 70 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 38 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \)
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