Properties

Label 504.2.s.h
Level 504
Weight 2
Character orbit 504.s
Analytic conductor 4.024
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 4 \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 4 - 4 \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} -3 q^{13} -4 \zeta_{6} q^{17} + ( -7 + 7 \zeta_{6} ) q^{19} + ( 4 - 4 \zeta_{6} ) q^{23} -11 \zeta_{6} q^{25} + 8 q^{29} + 5 \zeta_{6} q^{31} + ( 8 - 12 \zeta_{6} ) q^{35} + ( -3 + 3 \zeta_{6} ) q^{37} -8 q^{41} + 11 q^{43} + ( 4 - 4 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{53} + 12 \zeta_{6} q^{59} + ( 2 - 2 \zeta_{6} ) q^{61} + ( -12 + 12 \zeta_{6} ) q^{65} + 3 \zeta_{6} q^{67} -12 q^{71} -\zeta_{6} q^{73} + ( -1 + \zeta_{6} ) q^{79} -12 q^{83} -16 q^{85} + ( 8 - 8 \zeta_{6} ) q^{89} + ( -9 + 3 \zeta_{6} ) q^{91} + 28 \zeta_{6} q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + 5q^{7} + O(q^{10}) \) \( 2q + 4q^{5} + 5q^{7} - 6q^{13} - 4q^{17} - 7q^{19} + 4q^{23} - 11q^{25} + 16q^{29} + 5q^{31} + 4q^{35} - 3q^{37} - 16q^{41} + 22q^{43} + 4q^{47} + 11q^{49} + 4q^{53} + 12q^{59} + 2q^{61} - 12q^{65} + 3q^{67} - 24q^{71} - q^{73} - q^{79} - 24q^{83} - 32q^{85} + 8q^{89} - 15q^{91} + 28q^{95} - 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\) \(1\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 2.00000 3.46410i 0 2.50000 0.866025i 0 0 0
361.1 0 0 0 2.00000 + 3.46410i 0 2.50000 + 0.866025i 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 504.2.s.h yes 2
3.b odd 2 1 504.2.s.a 2
4.b odd 2 1 1008.2.s.q 2
7.b odd 2 1 3528.2.s.b 2
7.c even 3 1 inner 504.2.s.h yes 2
7.c even 3 1 3528.2.a.a 1
7.d odd 6 1 3528.2.a.ba 1
7.d odd 6 1 3528.2.s.b 2
12.b even 2 1 1008.2.s.a 2
21.c even 2 1 3528.2.s.bb 2
21.g even 6 1 3528.2.a.c 1
21.g even 6 1 3528.2.s.bb 2
21.h odd 6 1 504.2.s.a 2
21.h odd 6 1 3528.2.a.z 1
28.f even 6 1 7056.2.a.cc 1
28.g odd 6 1 1008.2.s.q 2
28.g odd 6 1 7056.2.a.b 1
84.j odd 6 1 7056.2.a.d 1
84.n even 6 1 1008.2.s.a 2
84.n even 6 1 7056.2.a.cb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.a 2 3.b odd 2 1
504.2.s.a 2 21.h odd 6 1
504.2.s.h yes 2 1.a even 1 1 trivial
504.2.s.h yes 2 7.c even 3 1 inner
1008.2.s.a 2 12.b even 2 1
1008.2.s.a 2 84.n even 6 1
1008.2.s.q 2 4.b odd 2 1
1008.2.s.q 2 28.g odd 6 1
3528.2.a.a 1 7.c even 3 1
3528.2.a.c 1 21.g even 6 1
3528.2.a.z 1 21.h odd 6 1
3528.2.a.ba 1 7.d odd 6 1
3528.2.s.b 2 7.b odd 2 1
3528.2.s.b 2 7.d odd 6 1
3528.2.s.bb 2 21.c even 2 1
3528.2.s.bb 2 21.g even 6 1
7056.2.a.b 1 28.g odd 6 1
7056.2.a.d 1 84.j odd 6 1
7056.2.a.cb 1 84.n even 6 1
7056.2.a.cc 1 28.f even 6 1

Hecke kernels

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

\( T_{5}^{2} - 4 T_{5} + 16 \)
\( T_{11} \)
\( T_{13} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 11 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 4 T - 31 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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