Properties

Label 2880.2.bl.a
Level $2880$
Weight $2$
Character orbit 2880.bl
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.bl (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.4
Defining polynomial: \(x^{8} + 17 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 720)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -1 + \beta_{6} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -1 + \beta_{6} ) q^{7} + ( \beta_{3} - \beta_{5} ) q^{11} + ( -\beta_{4} + \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{5} ) q^{17} + ( -2 - 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{19} + ( -\beta_{3} + \beta_{7} ) q^{23} -\beta_{2} q^{25} + 4 \beta_{5} q^{29} -4 \beta_{2} q^{31} + ( -\beta_{1} - \beta_{7} ) q^{35} + ( -1 - \beta_{2} ) q^{37} + ( -4 \beta_{1} + \beta_{3} + 4 \beta_{5} + \beta_{7} ) q^{41} + ( 6 - 6 \beta_{2} ) q^{43} + ( -\beta_{1} + \beta_{5} ) q^{47} + ( 9 - 2 \beta_{6} ) q^{49} + 2 \beta_{1} q^{53} + ( 1 - \beta_{6} ) q^{55} + ( -\beta_{3} - 9 \beta_{5} ) q^{59} + ( -3 + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{61} + ( \beta_{3} - \beta_{7} ) q^{65} + ( -7 - 7 \beta_{2} + \beta_{4} + \beta_{6} ) q^{67} + ( 3 \beta_{1} + \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{71} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{73} + ( -2 \beta_{3} + 16 \beta_{5} ) q^{77} + ( -4 \beta_{2} + 2 \beta_{4} ) q^{79} -2 \beta_{1} q^{83} + ( -1 - \beta_{2} ) q^{85} + ( -8 \beta_{1} - \beta_{3} + 8 \beta_{5} - \beta_{7} ) q^{89} + ( 15 - 15 \beta_{2} + \beta_{4} - \beta_{6} ) q^{91} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{95} + ( 2 + 4 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} - 16q^{19} - 8q^{37} + 48q^{43} + 72q^{49} + 8q^{55} - 24q^{61} - 56q^{67} - 8q^{85} + 120q^{91} + 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 17 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/28\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 33 \nu^{2} \)\()/112\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 61 \nu \)\()/28\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} + 17 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 13 \nu^{3} \)\()/448\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - \nu^{2} \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{7} - 251 \nu^{3} \)\()/448\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 7 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 11 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(7 \beta_{4} - 17\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{3} + 61 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-33 \beta_{6} - 7 \beta_{2}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} - 251 \beta_{5}\)\()/2\)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−1.01575 + 1.72286i
1.72286 1.01575i
1.01575 1.72286i
−1.72286 + 1.01575i
−1.01575 1.72286i
1.72286 + 1.01575i
1.01575 + 1.72286i
−1.72286 1.01575i
0 0 0 −0.707107 0.707107i 0 −4.87298 0 0 0
431.2 0 0 0 −0.707107 0.707107i 0 2.87298 0 0 0
431.3 0 0 0 0.707107 + 0.707107i 0 −4.87298 0 0 0
431.4 0 0 0 0.707107 + 0.707107i 0 2.87298 0 0 0
1871.1 0 0 0 −0.707107 + 0.707107i 0 −4.87298 0 0 0
1871.2 0 0 0 −0.707107 + 0.707107i 0 2.87298 0 0 0
1871.3 0 0 0 0.707107 0.707107i 0 −4.87298 0 0 0
1871.4 0 0 0 0.707107 0.707107i 0 2.87298 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1871.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.bl.a 8
3.b odd 2 1 inner 2880.2.bl.a 8
4.b odd 2 1 720.2.bl.a 8
12.b even 2 1 720.2.bl.a 8
16.e even 4 1 720.2.bl.a 8
16.f odd 4 1 inner 2880.2.bl.a 8
48.i odd 4 1 720.2.bl.a 8
48.k even 4 1 inner 2880.2.bl.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bl.a 8 4.b odd 2 1
720.2.bl.a 8 12.b even 2 1
720.2.bl.a 8 16.e even 4 1
720.2.bl.a 8 48.i odd 4 1
2880.2.bl.a 8 1.a even 1 1 trivial
2880.2.bl.a 8 3.b odd 2 1 inner
2880.2.bl.a 8 16.f odd 4 1 inner
2880.2.bl.a 8 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 2 T_{7} - 14 \) acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + T^{4} )^{2} \)
$7$ \( ( -14 + 2 T + T^{2} )^{4} \)
$11$ \( 38416 + 632 T^{4} + T^{8} \)
$13$ \( ( 900 + T^{4} )^{2} \)
$17$ \( ( 2 + T^{2} )^{4} \)
$19$ \( ( 484 - 176 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$23$ \( ( 30 + T^{2} )^{4} \)
$29$ \( ( 256 + T^{4} )^{2} \)
$31$ \( ( 16 + T^{2} )^{4} \)
$37$ \( ( 2 + 2 T + T^{2} )^{4} \)
$41$ \( ( 4 - 124 T^{2} + T^{4} )^{2} \)
$43$ \( ( 72 - 12 T + T^{2} )^{4} \)
$47$ \( ( -2 + T^{2} )^{4} \)
$53$ \( ( 16 + T^{4} )^{2} \)
$59$ \( 18974736 + 28152 T^{4} + T^{8} \)
$61$ \( ( 10404 - 1224 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$67$ \( ( 4624 + 1904 T + 392 T^{2} + 28 T^{3} + T^{4} )^{2} \)
$71$ \( ( 144 + 96 T^{2} + T^{4} )^{2} \)
$73$ \( ( 3136 + 128 T^{2} + T^{4} )^{2} \)
$79$ \( ( 1936 + 152 T^{2} + T^{4} )^{2} \)
$83$ \( ( 16 + T^{4} )^{2} \)
$89$ \( ( 9604 - 316 T^{2} + T^{4} )^{2} \)
$97$ \( ( -236 - 4 T + T^{2} )^{4} \)
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