Properties

Label 504.2.i.a
Level 504
Weight 2
Character orbit 504.i
Analytic conductor 4.024
Analytic rank 0
Dimension 8
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.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{2} - \beta_{5} ) q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{2} - \beta_{5} ) q^{7} + \beta_{3} q^{8} + ( -\beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{11} + ( \beta_{3} + \beta_{7} ) q^{14} + \beta_{4} q^{16} + ( -2 - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{22} + ( 3 \beta_{1} - \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{23} + 5 q^{25} + ( 4 + \beta_{4} ) q^{28} + ( -3 \beta_{1} + \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{29} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{32} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( -4 \beta_{2} - 4 \beta_{5} ) q^{43} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{6} + 3 \beta_{7} ) q^{44} + ( -6 + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{46} + 7 q^{49} + 5 \beta_{1} q^{50} + ( -5 \beta_{1} - \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( 3 \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{56} + ( -6 - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{58} + ( -\beta_{2} + 4 \beta_{5} ) q^{64} + ( -2 - 4 \beta_{4} ) q^{67} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 2 \beta_{3} - 2 \beta_{7} ) q^{74} + ( \beta_{1} - 3 \beta_{3} + 3 \beta_{6} + 2 \beta_{7} ) q^{77} -8 q^{79} + ( -4 \beta_{3} + 4 \beta_{7} ) q^{86} + ( 8 - 3 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{88} + ( -5 \beta_{1} + 3 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{92} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{16} - 20q^{22} + 40q^{25} + 28q^{28} - 44q^{46} + 56q^{49} - 52q^{58} - 64q^{79} + 68q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\( \nu^{4} \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + \nu^{2} \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{5} + \nu^{3} + 4 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - \nu^{3} \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 2 \beta_{6} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(4 \beta_{5} - \beta_{2}\)
\(\nu^{7}\)\(=\)\(-4 \beta_{7} - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
−1.28897 0.581861i
−1.28897 + 0.581861i
−0.581861 1.28897i
−0.581861 + 1.28897i
0.581861 1.28897i
0.581861 + 1.28897i
1.28897 0.581861i
1.28897 + 0.581861i
−1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 2.64575 −0.832353 2.70318i 0 0
125.2 −1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 2.64575 −0.832353 + 2.70318i 0 0
125.3 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 −2.64575 2.70318 + 0.832353i 0 0
125.4 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 −2.64575 2.70318 0.832353i 0 0
125.5 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 −2.64575 −2.70318 + 0.832353i 0 0
125.6 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 −2.64575 −2.70318 0.832353i 0 0
125.7 1.28897 0.581861i 0 1.32288 1.50000i 0 0 2.64575 0.832353 2.70318i 0 0
125.8 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 2.64575 0.832353 + 2.70318i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.8
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.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i 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.i.a 8
3.b odd 2 1 inner 504.2.i.a 8
4.b odd 2 1 2016.2.i.a 8
7.b odd 2 1 CM 504.2.i.a 8
8.b even 2 1 inner 504.2.i.a 8
8.d odd 2 1 2016.2.i.a 8
12.b even 2 1 2016.2.i.a 8
21.c even 2 1 inner 504.2.i.a 8
24.f even 2 1 2016.2.i.a 8
24.h odd 2 1 inner 504.2.i.a 8
28.d even 2 1 2016.2.i.a 8
56.e even 2 1 2016.2.i.a 8
56.h odd 2 1 inner 504.2.i.a 8
84.h odd 2 1 2016.2.i.a 8
168.e odd 2 1 2016.2.i.a 8
168.i even 2 1 inner 504.2.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.i.a 8 1.a even 1 1 trivial
504.2.i.a 8 3.b odd 2 1 inner
504.2.i.a 8 7.b odd 2 1 CM
504.2.i.a 8 8.b even 2 1 inner
504.2.i.a 8 21.c even 2 1 inner
504.2.i.a 8 24.h odd 2 1 inner
504.2.i.a 8 56.h odd 2 1 inner
504.2.i.a 8 168.i even 2 1 inner
2016.2.i.a 8 4.b odd 2 1
2016.2.i.a 8 8.d odd 2 1
2016.2.i.a 8 12.b even 2 1
2016.2.i.a 8 24.f even 2 1
2016.2.i.a 8 28.d even 2 1
2016.2.i.a 8 56.e even 2 1
2016.2.i.a 8 84.h odd 2 1
2016.2.i.a 8 168.e odd 2 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} + 16 T^{8} \)
$3$ \( \)
$5$ \( ( 1 - 5 T^{2} )^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 - 206 T^{4} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{8} \)
$17$ \( ( 1 + 17 T^{2} )^{8} \)
$19$ \( ( 1 + 19 T^{2} )^{8} \)
$23$ \( ( 1 - 734 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 1234 T^{4} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 - 38 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{8} \)
$43$ \( ( 1 + 58 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{8} \)
$53$ \( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 59 T^{2} )^{8} \)
$61$ \( ( 1 + 61 T^{2} )^{8} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 2914 T^{4} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{8} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{8} \)
$83$ \( ( 1 - 83 T^{2} )^{8} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{8} \)
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