Properties

Label 2880.2.w.l
Level $2880$
Weight $2$
Character orbit 2880.w
Analytic conductor $22.997$
Analytic rank $0$
Dimension $4$
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.w (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 - 2 \zeta_{8}^{2} ) q^{7} +O(q^{10})\) \( q + ( \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 - 2 \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( 3 + 3 \zeta_{8}^{2} ) q^{13} + 4 \zeta_{8} q^{17} -8 \zeta_{8}^{2} q^{19} + 4 \zeta_{8}^{3} q^{23} + ( -4 - 3 \zeta_{8}^{2} ) q^{25} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{29} + 4 q^{31} + ( 6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( 3 - 3 \zeta_{8}^{2} ) q^{37} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{41} -\zeta_{8}^{2} q^{49} + 4 \zeta_{8}^{3} q^{53} + ( 6 + 2 \zeta_{8}^{2} ) q^{55} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + ( -3 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{65} + ( 4 - 4 \zeta_{8}^{2} ) q^{67} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( 1 + \zeta_{8}^{2} ) q^{73} -8 \zeta_{8} q^{77} -12 \zeta_{8}^{2} q^{79} -12 \zeta_{8}^{3} q^{83} + ( -8 + 4 \zeta_{8}^{2} ) q^{85} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} + 12 q^{91} + ( 16 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} + ( -3 + 3 \zeta_{8}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} + O(q^{10}) \) \( 4q + 8q^{7} + 12q^{13} - 16q^{25} + 16q^{31} + 12q^{37} + 24q^{55} + 16q^{67} + 4q^{73} - 32q^{85} + 48q^{91} - 12q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 −0.707107 2.12132i 0 2.00000 + 2.00000i 0 0 0
2177.2 0 0 0 0.707107 + 2.12132i 0 2.00000 + 2.00000i 0 0 0
2753.1 0 0 0 −0.707107 + 2.12132i 0 2.00000 2.00000i 0 0 0
2753.2 0 0 0 0.707107 2.12132i 0 2.00000 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e 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.w.l 4
3.b odd 2 1 inner 2880.2.w.l 4
4.b odd 2 1 2880.2.w.c 4
5.c odd 4 1 inner 2880.2.w.l 4
8.b even 2 1 90.2.f.a 4
8.d odd 2 1 720.2.w.a 4
12.b even 2 1 2880.2.w.c 4
15.e even 4 1 inner 2880.2.w.l 4
20.e even 4 1 2880.2.w.c 4
24.f even 2 1 720.2.w.a 4
24.h odd 2 1 90.2.f.a 4
40.e odd 2 1 3600.2.w.g 4
40.f even 2 1 450.2.f.b 4
40.i odd 4 1 90.2.f.a 4
40.i odd 4 1 450.2.f.b 4
40.k even 4 1 720.2.w.a 4
40.k even 4 1 3600.2.w.g 4
60.l odd 4 1 2880.2.w.c 4
72.j odd 6 2 810.2.m.c 8
72.n even 6 2 810.2.m.c 8
120.i odd 2 1 450.2.f.b 4
120.m even 2 1 3600.2.w.g 4
120.q odd 4 1 720.2.w.a 4
120.q odd 4 1 3600.2.w.g 4
120.w even 4 1 90.2.f.a 4
120.w even 4 1 450.2.f.b 4
360.br even 12 2 810.2.m.c 8
360.bu odd 12 2 810.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.f.a 4 8.b even 2 1
90.2.f.a 4 24.h odd 2 1
90.2.f.a 4 40.i odd 4 1
90.2.f.a 4 120.w even 4 1
450.2.f.b 4 40.f even 2 1
450.2.f.b 4 40.i odd 4 1
450.2.f.b 4 120.i odd 2 1
450.2.f.b 4 120.w even 4 1
720.2.w.a 4 8.d odd 2 1
720.2.w.a 4 24.f even 2 1
720.2.w.a 4 40.k even 4 1
720.2.w.a 4 120.q odd 4 1
810.2.m.c 8 72.j odd 6 2
810.2.m.c 8 72.n even 6 2
810.2.m.c 8 360.br even 12 2
810.2.m.c 8 360.bu odd 12 2
2880.2.w.c 4 4.b odd 2 1
2880.2.w.c 4 12.b even 2 1
2880.2.w.c 4 20.e even 4 1
2880.2.w.c 4 60.l odd 4 1
2880.2.w.l 4 1.a even 1 1 trivial
2880.2.w.l 4 3.b odd 2 1 inner
2880.2.w.l 4 5.c odd 4 1 inner
2880.2.w.l 4 15.e even 4 1 inner
3600.2.w.g 4 40.e odd 2 1
3600.2.w.g 4 40.k even 4 1
3600.2.w.g 4 120.m even 2 1
3600.2.w.g 4 120.q odd 4 1

Hecke kernels

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

\( T_{7}^{2} - 4 T_{7} + 8 \)
\( T_{11}^{2} + 8 \)
\( T_{13}^{2} - 6 T_{13} + 18 \)
\( T_{31} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( ( 8 - 4 T + T^{2} )^{2} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( 18 - 6 T + T^{2} )^{2} \)
$17$ \( 256 + T^{4} \)
$19$ \( ( 64 + T^{2} )^{2} \)
$23$ \( 256 + T^{4} \)
$29$ \( ( -2 + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 18 - 6 T + T^{2} )^{2} \)
$41$ \( ( 98 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 256 + T^{4} \)
$59$ \( ( -8 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 32 - 8 T + T^{2} )^{2} \)
$71$ \( ( 32 + T^{2} )^{2} \)
$73$ \( ( 2 - 2 T + T^{2} )^{2} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( -98 + T^{2} )^{2} \)
$97$ \( ( 18 + 6 T + T^{2} )^{2} \)
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