Properties

Label 504.2.p.b
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: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{3} q^{4} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{3} q^{4} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + 2 q^{11} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{14} -4 q^{16} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} + ( 4 - 8 \zeta_{12}^{2} ) q^{20} + ( -2 + 2 \zeta_{12}^{3} ) q^{22} -2 \zeta_{12}^{3} q^{23} + 7 q^{25} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( -2 - 4 \zeta_{12}^{2} ) q^{28} + 8 \zeta_{12}^{3} q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 4 - 4 \zeta_{12}^{3} ) q^{32} + ( -2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{34} + ( 10 - 8 \zeta_{12}^{2} ) q^{35} + 4 \zeta_{12}^{3} q^{37} + ( -4 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{38} + ( -4 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{40} + ( -6 + 12 \zeta_{12}^{2} ) q^{41} + 6 q^{43} -4 \zeta_{12}^{3} q^{44} + ( 2 + 2 \zeta_{12}^{3} ) q^{46} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -7 + 7 \zeta_{12}^{3} ) q^{50} + ( -4 + 8 \zeta_{12}^{2} ) q^{52} + 4 \zeta_{12}^{3} q^{53} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{55} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56} + ( -8 - 8 \zeta_{12}^{3} ) q^{58} + ( 4 - 8 \zeta_{12}^{2} ) q^{59} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{61} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} -12 q^{65} + 10 q^{67} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{68} + ( -10 + 8 \zeta_{12} + 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{70} + 2 \zeta_{12}^{3} q^{71} + ( -4 - 4 \zeta_{12}^{3} ) q^{74} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{76} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + 12 \zeta_{12}^{3} q^{79} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{80} + ( 6 - 12 \zeta_{12} - 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{82} + ( -4 + 8 \zeta_{12}^{2} ) q^{83} -12 \zeta_{12}^{3} q^{85} + ( -6 + 6 \zeta_{12}^{3} ) q^{86} + ( 4 + 4 \zeta_{12}^{3} ) q^{88} + ( -6 + 12 \zeta_{12}^{2} ) q^{89} + ( -10 + 8 \zeta_{12}^{2} ) q^{91} -4 q^{92} + ( 4 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{94} -24 \zeta_{12}^{3} q^{95} + ( 8 - 16 \zeta_{12}^{2} ) q^{97} + ( -3 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 8q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 8q^{8} + 8q^{11} + 8q^{14} - 16q^{16} - 8q^{22} + 28q^{25} - 16q^{28} + 16q^{32} + 24q^{35} + 24q^{43} + 8q^{46} - 4q^{49} - 28q^{50} + 16q^{56} - 32q^{58} - 48q^{65} + 40q^{67} - 24q^{70} - 16q^{74} - 24q^{86} + 16q^{88} - 24q^{91} - 16q^{92} + 4q^{98} + O(q^{100}) \)

Character values

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

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

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
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−1.00000 1.00000i 0 2.00000i −3.46410 0 −1.73205 + 2.00000i 2.00000 2.00000i 0 3.46410 + 3.46410i
307.2 −1.00000 1.00000i 0 2.00000i 3.46410 0 1.73205 + 2.00000i 2.00000 2.00000i 0 −3.46410 3.46410i
307.3 −1.00000 + 1.00000i 0 2.00000i −3.46410 0 −1.73205 2.00000i 2.00000 + 2.00000i 0 3.46410 3.46410i
307.4 −1.00000 + 1.00000i 0 2.00000i 3.46410 0 1.73205 2.00000i 2.00000 + 2.00000i 0 −3.46410 + 3.46410i
\(n\): 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.b 4
3.b odd 2 1 504.2.p.e yes 4
4.b odd 2 1 2016.2.p.b 4
7.b odd 2 1 inner 504.2.p.b 4
8.b even 2 1 2016.2.p.b 4
8.d odd 2 1 inner 504.2.p.b 4
12.b even 2 1 2016.2.p.f 4
21.c even 2 1 504.2.p.e yes 4
24.f even 2 1 504.2.p.e yes 4
24.h odd 2 1 2016.2.p.f 4
28.d even 2 1 2016.2.p.b 4
56.e even 2 1 inner 504.2.p.b 4
56.h odd 2 1 2016.2.p.b 4
84.h odd 2 1 2016.2.p.f 4
168.e odd 2 1 504.2.p.e yes 4
168.i even 2 1 2016.2.p.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.p.b 4 1.a even 1 1 trivial
504.2.p.b 4 7.b odd 2 1 inner
504.2.p.b 4 8.d odd 2 1 inner
504.2.p.b 4 56.e even 2 1 inner
504.2.p.e yes 4 3.b odd 2 1
504.2.p.e yes 4 21.c even 2 1
504.2.p.e yes 4 24.f even 2 1
504.2.p.e yes 4 168.e odd 2 1
2016.2.p.b 4 4.b odd 2 1
2016.2.p.b 4 8.b even 2 1
2016.2.p.b 4 28.d even 2 1
2016.2.p.b 4 56.h odd 2 1
2016.2.p.f 4 12.b even 2 1
2016.2.p.f 4 24.h odd 2 1
2016.2.p.f 4 84.h odd 2 1
2016.2.p.f 4 168.i even 2 1

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{11} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 2 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 42 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 6 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 50 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 58 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 26 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 6 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 70 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 14 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 10 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 118 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \)
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