Properties

Label 504.2.p.d
Level 504
Weight 2
Character orbit 504.p
Analytic conductor 4.024
Analytic rank 0
Dimension 4
CM discriminant -7
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.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( 2 - \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} - 2 \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( 2 - \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} - 2 \beta_{3} ) q^{8} + ( -\beta_{1} - 3 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( 2 - 3 \beta_{2} ) q^{16} + ( 8 - 2 \beta_{2} ) q^{22} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{23} -5 q^{25} + ( 2 + 3 \beta_{2} ) q^{28} + ( -\beta_{1} + \beta_{3} ) q^{29} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{32} + ( -4 + 8 \beta_{2} ) q^{37} -12 q^{43} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{44} -8 \beta_{2} q^{46} -7 q^{49} + 5 \beta_{3} q^{50} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{53} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{56} + 2 \beta_{2} q^{58} + ( -2 - 5 \beta_{2} ) q^{64} + 4 q^{67} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{71} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{74} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{77} + ( -6 + 12 \beta_{2} ) q^{79} + 12 \beta_{3} q^{86} + ( 12 - 10 \beta_{2} ) q^{88} + 8 \beta_{1} q^{92} + 7 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + O(q^{10}) \) \( 4q + 6q^{4} + 2q^{16} + 28q^{22} - 20q^{25} + 14q^{28} - 48q^{43} - 16q^{46} - 28q^{49} + 4q^{58} - 18q^{64} + 16q^{67} + 28q^{88} + O(q^{100}) \)

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

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

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
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i
−1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i −1.32288 2.50000i 0 0
307.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 0 0 2.64575i −1.32288 + 2.50000i 0 0
307.3 1.32288 0.500000i 0 1.50000 1.32288i 0 0 2.64575i 1.32288 2.50000i 0 0
307.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i 1.32288 + 2.50000i 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
56.e even 2 1 inner
168.e odd 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.d 4
3.b odd 2 1 inner 504.2.p.d 4
4.b odd 2 1 2016.2.p.d 4
7.b odd 2 1 CM 504.2.p.d 4
8.b even 2 1 2016.2.p.d 4
8.d odd 2 1 inner 504.2.p.d 4
12.b even 2 1 2016.2.p.d 4
21.c even 2 1 inner 504.2.p.d 4
24.f even 2 1 inner 504.2.p.d 4
24.h odd 2 1 2016.2.p.d 4
28.d even 2 1 2016.2.p.d 4
56.e even 2 1 inner 504.2.p.d 4
56.h odd 2 1 2016.2.p.d 4
84.h odd 2 1 2016.2.p.d 4
168.e odd 2 1 inner 504.2.p.d 4
168.i even 2 1 2016.2.p.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.p.d 4 1.a even 1 1 trivial
504.2.p.d 4 3.b odd 2 1 inner
504.2.p.d 4 7.b odd 2 1 CM
504.2.p.d 4 8.d odd 2 1 inner
504.2.p.d 4 21.c even 2 1 inner
504.2.p.d 4 24.f even 2 1 inner
504.2.p.d 4 56.e even 2 1 inner
504.2.p.d 4 168.e odd 2 1 inner
2016.2.p.d 4 4.b odd 2 1
2016.2.p.d 4 8.b even 2 1
2016.2.p.d 4 12.b even 2 1
2016.2.p.d 4 24.h odd 2 1
2016.2.p.d 4 28.d even 2 1
2016.2.p.d 4 56.h odd 2 1
2016.2.p.d 4 84.h odd 2 1
2016.2.p.d 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 \(S_{2}^{\mathrm{new}}(504, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} - 28 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ \( \)
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{2}( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 + 12 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 59 T^{2} )^{4} \)
$61$ \( ( 1 + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{4} \)
$89$ \( ( 1 - 89 T^{2} )^{4} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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