Properties

Label 2880.2.d.i
Level $2880$
Weight $2$
Character orbit 2880.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 960)
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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} +O(q^{10})\) \( q -\beta_{4} q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + \beta_{1} q^{11} -\beta_{2} q^{13} + \beta_{1} q^{17} + \beta_{3} q^{19} + \beta_{6} q^{25} + \beta_{7} q^{29} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{31} + ( -5 + \beta_{6} ) q^{35} + ( -\beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{37} -6 q^{41} + ( 2 - \beta_{3} - 2 \beta_{6} ) q^{43} + ( \beta_{4} - \beta_{5} + \beta_{7} ) q^{47} + ( -3 + \beta_{3} + 2 \beta_{6} ) q^{49} + ( -2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{53} + ( 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{55} + ( \beta_{1} - \beta_{3} ) q^{59} + ( 3 \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{61} + ( -2 + \beta_{1} + \beta_{3} ) q^{65} + ( -6 - \beta_{3} - 2 \beta_{6} ) q^{67} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{71} -3 \beta_{3} q^{73} + ( 6 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{79} + ( -2 - \beta_{3} - 2 \beta_{6} ) q^{83} + ( 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{85} + 2 q^{89} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{91} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{95} + 2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{35} - 48q^{41} + 16q^{43} - 24q^{49} - 16q^{65} - 48q^{67} - 16q^{83} + 16q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 15 \nu^{3} + 42 \nu \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 3 \nu^{5} - 5 \nu^{3} - 4 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/10\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} - \nu^{6} + 5 \nu^{5} + 5 \nu^{4} + 15 \nu^{3} + 15 \nu^{2} + 30 \nu + 8 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} + \nu^{6} + 5 \nu^{5} - 5 \nu^{4} + 15 \nu^{3} - 15 \nu^{2} + 30 \nu - 8 \)\()/20\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} + 10 \nu^{6} + 5 \nu^{5} + 30 \nu^{4} - 5 \nu^{3} + 10 \nu^{2} + 16 \nu + 60 \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{6} + 15 \nu^{4} + 45 \nu^{2} + 96 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - \beta_{3} - 6\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 2\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{7} + 15 \beta_{5} - 15 \beta_{4} - 36\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(20 \beta_{5} + 20 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - 14 \beta_{1}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.228425 + 1.39564i
−0.228425 1.39564i
−1.09445 0.895644i
−1.09445 + 0.895644i
1.09445 + 0.895644i
1.09445 0.895644i
0.228425 1.39564i
0.228425 + 1.39564i
0 0 0 −2.18890 0.456850i 0 0.913701i 0 0 0
289.2 0 0 0 −2.18890 + 0.456850i 0 0.913701i 0 0 0
289.3 0 0 0 −0.456850 2.18890i 0 4.37780i 0 0 0
289.4 0 0 0 −0.456850 + 2.18890i 0 4.37780i 0 0 0
289.5 0 0 0 0.456850 2.18890i 0 4.37780i 0 0 0
289.6 0 0 0 0.456850 + 2.18890i 0 4.37780i 0 0 0
289.7 0 0 0 2.18890 0.456850i 0 0.913701i 0 0 0
289.8 0 0 0 2.18890 + 0.456850i 0 0.913701i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.d.i 8
3.b odd 2 1 960.2.d.f yes 8
4.b odd 2 1 2880.2.d.j 8
5.b even 2 1 2880.2.d.j 8
8.b even 2 1 2880.2.d.j 8
8.d odd 2 1 inner 2880.2.d.i 8
12.b even 2 1 960.2.d.e 8
15.d odd 2 1 960.2.d.e 8
15.e even 4 1 4800.2.k.q 8
15.e even 4 1 4800.2.k.r 8
20.d odd 2 1 inner 2880.2.d.i 8
24.f even 2 1 960.2.d.f yes 8
24.h odd 2 1 960.2.d.e 8
40.e odd 2 1 2880.2.d.j 8
40.f even 2 1 inner 2880.2.d.i 8
48.i odd 4 1 3840.2.f.i 8
48.i odd 4 1 3840.2.f.k 8
48.k even 4 1 3840.2.f.i 8
48.k even 4 1 3840.2.f.k 8
60.h even 2 1 960.2.d.f yes 8
60.l odd 4 1 4800.2.k.q 8
60.l odd 4 1 4800.2.k.r 8
120.i odd 2 1 960.2.d.f yes 8
120.m even 2 1 960.2.d.e 8
120.q odd 4 1 4800.2.k.q 8
120.q odd 4 1 4800.2.k.r 8
120.w even 4 1 4800.2.k.q 8
120.w even 4 1 4800.2.k.r 8
240.t even 4 1 3840.2.f.i 8
240.t even 4 1 3840.2.f.k 8
240.bm odd 4 1 3840.2.f.i 8
240.bm odd 4 1 3840.2.f.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.e 8 12.b even 2 1
960.2.d.e 8 15.d odd 2 1
960.2.d.e 8 24.h odd 2 1
960.2.d.e 8 120.m even 2 1
960.2.d.f yes 8 3.b odd 2 1
960.2.d.f yes 8 24.f even 2 1
960.2.d.f yes 8 60.h even 2 1
960.2.d.f yes 8 120.i odd 2 1
2880.2.d.i 8 1.a even 1 1 trivial
2880.2.d.i 8 8.d odd 2 1 inner
2880.2.d.i 8 20.d odd 2 1 inner
2880.2.d.i 8 40.f even 2 1 inner
2880.2.d.j 8 4.b odd 2 1
2880.2.d.j 8 5.b even 2 1
2880.2.d.j 8 8.b even 2 1
2880.2.d.j 8 40.e odd 2 1
3840.2.f.i 8 48.i odd 4 1
3840.2.f.i 8 48.k even 4 1
3840.2.f.i 8 240.t even 4 1
3840.2.f.i 8 240.bm odd 4 1
3840.2.f.k 8 48.i odd 4 1
3840.2.f.k 8 48.k even 4 1
3840.2.f.k 8 240.t even 4 1
3840.2.f.k 8 240.bm odd 4 1
4800.2.k.q 8 15.e even 4 1
4800.2.k.q 8 60.l odd 4 1
4800.2.k.q 8 120.q odd 4 1
4800.2.k.q 8 120.w even 4 1
4800.2.k.r 8 15.e even 4 1
4800.2.k.r 8 60.l odd 4 1
4800.2.k.r 8 120.q odd 4 1
4800.2.k.r 8 120.w even 4 1

Hecke kernels

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

\( T_{7}^{4} + 20 T_{7}^{2} + 16 \)
\( T_{11}^{4} + 44 T_{11}^{2} + 400 \)
\( T_{13}^{4} - 20 T_{13}^{2} + 16 \)
\( T_{31}^{2} - 28 \)
\( T_{41} + 6 \)
\( T_{43}^{2} - 4 T_{43} - 80 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 625 - 34 T^{4} + T^{8} \)
$7$ \( ( 16 + 20 T^{2} + T^{4} )^{2} \)
$11$ \( ( 400 + 44 T^{2} + T^{4} )^{2} \)
$13$ \( ( 16 - 20 T^{2} + T^{4} )^{2} \)
$17$ \( ( 400 + 44 T^{2} + T^{4} )^{2} \)
$19$ \( ( 16 + T^{2} )^{4} \)
$23$ \( T^{8} \)
$29$ \( ( 400 + 68 T^{2} + T^{4} )^{2} \)
$31$ \( ( -28 + T^{2} )^{4} \)
$37$ \( ( 400 - 68 T^{2} + T^{4} )^{2} \)
$41$ \( ( 6 + T )^{8} \)
$43$ \( ( -80 - 4 T + T^{2} )^{4} \)
$47$ \( ( 48 + T^{2} )^{4} \)
$53$ \( ( 400 - 68 T^{2} + T^{4} )^{2} \)
$59$ \( ( 144 + 60 T^{2} + T^{4} )^{2} \)
$61$ \( ( 112 + T^{2} )^{4} \)
$67$ \( ( -48 + 12 T + T^{2} )^{4} \)
$71$ \( ( -48 + T^{2} )^{4} \)
$73$ \( ( 144 + T^{2} )^{4} \)
$79$ \( ( -28 + T^{2} )^{4} \)
$83$ \( ( -80 + 4 T + T^{2} )^{4} \)
$89$ \( ( -2 + T )^{8} \)
$97$ \( ( 6400 + 176 T^{2} + T^{4} )^{2} \)
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