Properties

Label 2880.2.b.a
Level $2880$
Weight $2$
Character orbit 2880.b
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q - q^{5} + \beta_{3} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + \beta_{2} q^{13} + ( \beta_{2} - \beta_{4} ) q^{17} + \beta_{5} q^{19} + q^{25} + ( -2 + \beta_{6} ) q^{29} + ( \beta_{1} - 2 \beta_{3} ) q^{31} -\beta_{3} q^{35} + ( -\beta_{2} + 2 \beta_{4} ) q^{37} + ( -2 \beta_{2} + \beta_{4} ) q^{41} + ( \beta_{5} - \beta_{7} ) q^{43} + \beta_{7} q^{47} -3 q^{49} + ( -4 - \beta_{6} ) q^{53} + ( \beta_{1} - \beta_{3} ) q^{55} + ( -5 \beta_{1} - \beta_{3} ) q^{59} + 2 \beta_{4} q^{61} -\beta_{2} q^{65} + ( -\beta_{5} - \beta_{7} ) q^{67} + ( -\beta_{5} - \beta_{7} ) q^{71} -2 q^{73} + ( -10 - \beta_{6} ) q^{77} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{83} + ( -\beta_{2} + \beta_{4} ) q^{85} + ( 2 \beta_{2} + \beta_{4} ) q^{89} + ( \beta_{5} - 2 \beta_{7} ) q^{91} -\beta_{5} q^{95} + ( 6 + \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{25} - 16q^{29} - 24q^{49} - 32q^{53} - 16q^{73} - 80q^{77} + 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} + 16 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} + 6 \nu^{6} - \nu^{5} - 13 \nu^{3} + 36 \nu^{2} - 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{7} - \nu^{5} - 29 \nu^{3} - 11 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{7} + 2 \nu^{5} + 4 \nu^{4} - 26 \nu^{3} + 10 \nu + 14 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{7} + 2 \nu^{5} - 58 \nu^{3} + 22 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{7} - 4 \nu^{5} + 4 \nu^{4} + 52 \nu^{3} - 20 \nu + 14 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 6 \beta_{3}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - 3 \beta_{2} + 9 \beta_{1}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3}\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} + 2 \beta_{5} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{7} - 15 \beta_{6} + 11 \beta_{5} - 22 \beta_{4} + 30 \beta_{3}\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{4} + 12 \beta_{2} - 27 \beta_{1}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{7} + 39 \beta_{6} - 29 \beta_{5} - 58 \beta_{4} + 78 \beta_{3}\)\()/24\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
1.14412 + 1.14412i
0.437016 + 0.437016i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
−1.14412 1.14412i
−0.437016 0.437016i
0.437016 0.437016i
1.14412 1.14412i
0 0 0 −1.00000 0 3.16228i 0 0 0
2591.2 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.3 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.4 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.5 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.6 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.7 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.8 0 0 0 −1.00000 0 3.16228i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.b.a 8
3.b odd 2 1 2880.2.b.d yes 8
4.b odd 2 1 inner 2880.2.b.a 8
8.b even 2 1 2880.2.b.d yes 8
8.d odd 2 1 2880.2.b.d yes 8
12.b even 2 1 2880.2.b.d yes 8
24.f even 2 1 inner 2880.2.b.a 8
24.h odd 2 1 inner 2880.2.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.2.b.a 8 1.a even 1 1 trivial
2880.2.b.a 8 4.b odd 2 1 inner
2880.2.b.a 8 24.f even 2 1 inner
2880.2.b.a 8 24.h odd 2 1 inner
2880.2.b.d yes 8 3.b odd 2 1
2880.2.b.d yes 8 8.b even 2 1
2880.2.b.d yes 8 8.d odd 2 1
2880.2.b.d yes 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{2} + 10 \)
\( T_{29}^{2} + 4 T_{29} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + T )^{8} \)
$7$ \( ( 10 + T^{2} )^{4} \)
$11$ \( ( 36 + 28 T^{2} + T^{4} )^{2} \)
$13$ \( ( 324 + 44 T^{2} + T^{4} )^{2} \)
$17$ \( ( 144 + 56 T^{2} + T^{4} )^{2} \)
$19$ \( ( 144 - 56 T^{2} + T^{4} )^{2} \)
$23$ \( T^{8} \)
$29$ \( ( -36 + 4 T + T^{2} )^{4} \)
$31$ \( ( 1296 + 88 T^{2} + T^{4} )^{2} \)
$37$ \( ( 900 + 140 T^{2} + T^{4} )^{2} \)
$41$ \( ( 6084 + 164 T^{2} + T^{4} )^{2} \)
$43$ \( ( -72 + T^{2} )^{4} \)
$47$ \( ( 144 - 104 T^{2} + T^{4} )^{2} \)
$53$ \( ( -24 + 8 T + T^{2} )^{4} \)
$59$ \( ( 8100 + 220 T^{2} + T^{4} )^{2} \)
$61$ \( ( 72 + T^{2} )^{4} \)
$67$ \( ( 5184 - 176 T^{2} + T^{4} )^{2} \)
$71$ \( ( 5184 - 176 T^{2} + T^{4} )^{2} \)
$73$ \( ( 2 + T )^{8} \)
$79$ \( ( 16 + 152 T^{2} + T^{4} )^{2} \)
$83$ \( ( 576 + 112 T^{2} + T^{4} )^{2} \)
$89$ \( ( 900 + 260 T^{2} + T^{4} )^{2} \)
$97$ \( ( -4 - 12 T + T^{2} )^{4} \)
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