Properties

Label 4608.2.d.q
Level $4608$
Weight $2$
Character orbit 4608.d
Analytic conductor $36.795$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} - \beta_{3} q^{7} + \beta_1 q^{11} + (\beta_{6} - 5) q^{25} + ( - \beta_{7} + \beta_{5}) q^{29} + (\beta_{4} - 2 \beta_{3}) q^{31} + ( - \beta_{2} - 3 \beta_1) q^{35} + ( - \beta_{6} + 7) q^{49} + (\beta_{7} + 3 \beta_{5}) q^{53} + (\beta_{4} - 3 \beta_{3}) q^{55} - \beta_{2} q^{59} - \beta_{6} q^{73} + ( - \beta_{7} - 4 \beta_{5}) q^{77} + ( - 2 \beta_{4} - \beta_{3}) q^{79} + 5 \beta_1 q^{83} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{25} + 56 q^{49} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -8\zeta_{24}^{5} - 8\zeta_{24}^{3} + 8\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} + 4\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{6} - 3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 4\zeta_{24}^{2} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -8\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 4\zeta_{24}^{7} + 4\zeta_{24}^{6} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 4\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} + \beta_{4} + 3\beta_{3} ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( 2\beta_{4} - 2\beta_{3} - \beta_{2} ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 4\beta_{5} - 2\beta_{4} + 2\beta_{3} - \beta_{2} ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{7} + 2\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 4\beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta_{2} ) / 32 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2305.1
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 0.258819i
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
0 0 0 4.44949i 0 −4.87832 0 0 0
2305.2 0 0 0 4.44949i 0 4.87832 0 0 0
2305.3 0 0 0 0.449490i 0 −2.04989 0 0 0
2305.4 0 0 0 0.449490i 0 2.04989 0 0 0
2305.5 0 0 0 0.449490i 0 −2.04989 0 0 0
2305.6 0 0 0 0.449490i 0 2.04989 0 0 0
2305.7 0 0 0 4.44949i 0 −4.87832 0 0 0
2305.8 0 0 0 4.44949i 0 4.87832 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2305.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.d.q 8
3.b odd 2 1 inner 4608.2.d.q 8
4.b odd 2 1 inner 4608.2.d.q 8
8.b even 2 1 inner 4608.2.d.q 8
8.d odd 2 1 inner 4608.2.d.q 8
12.b even 2 1 inner 4608.2.d.q 8
16.e even 4 1 4608.2.a.s 4
16.e even 4 1 4608.2.a.bc yes 4
16.f odd 4 1 4608.2.a.s 4
16.f odd 4 1 4608.2.a.bc yes 4
24.f even 2 1 inner 4608.2.d.q 8
24.h odd 2 1 CM 4608.2.d.q 8
48.i odd 4 1 4608.2.a.s 4
48.i odd 4 1 4608.2.a.bc yes 4
48.k even 4 1 4608.2.a.s 4
48.k even 4 1 4608.2.a.bc yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.a.s 4 16.e even 4 1
4608.2.a.s 4 16.f odd 4 1
4608.2.a.s 4 48.i odd 4 1
4608.2.a.s 4 48.k even 4 1
4608.2.a.bc yes 4 16.e even 4 1
4608.2.a.bc yes 4 16.f odd 4 1
4608.2.a.bc yes 4 48.i odd 4 1
4608.2.a.bc yes 4 48.k even 4 1
4608.2.d.q 8 1.a even 1 1 trivial
4608.2.d.q 8 3.b odd 2 1 inner
4608.2.d.q 8 4.b odd 2 1 inner
4608.2.d.q 8 8.b even 2 1 inner
4608.2.d.q 8 8.d odd 2 1 inner
4608.2.d.q 8 12.b even 2 1 inner
4608.2.d.q 8 24.f even 2 1 inner
4608.2.d.q 8 24.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4608, [\chi])\):

\( T_{5}^{4} + 20T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 28T_{7}^{2} + 100 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 20 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 28 T^{2} + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 124 T^{2} + 1444)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 212 T^{2} + 8836)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 316 T^{2} + 3364)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 300)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T + 2)^{8} \) Copy content Toggle raw display
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