Properties

Label 2880.2.f.o
Level $2880$
Weight $2$
Character orbit 2880.f
Analytic conductor $22.997$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{5} + 2 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{5} + 2 \beta q^{7} + 2 \beta q^{13} + 8 q^{19} - 2 \beta q^{23} + (2 \beta - 3) q^{25} + 6 q^{29} + 8 q^{31} + (2 \beta - 8) q^{35} + 2 \beta q^{37} - 6 q^{41} - 2 \beta q^{43} - 2 \beta q^{47} - 9 q^{49} + 6 \beta q^{53} + 6 q^{61} + (2 \beta - 8) q^{65} + 6 \beta q^{67} - 16 q^{71} - 8 q^{79} - 6 \beta q^{83} - 10 q^{89} - 16 q^{91} + (8 \beta + 8) q^{95} + 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 16 q^{19} - 6 q^{25} + 12 q^{29} + 16 q^{31} - 16 q^{35} - 12 q^{41} - 18 q^{49} + 12 q^{61} - 16 q^{65} - 32 q^{71} - 16 q^{79} - 20 q^{89} - 32 q^{91} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1729.1
1.00000i
1.00000i
0 0 0 1.00000 2.00000i 0 4.00000i 0 0 0
1729.2 0 0 0 1.00000 + 2.00000i 0 4.00000i 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
5.b 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.f.o 2
3.b odd 2 1 960.2.f.d 2
4.b odd 2 1 2880.2.f.m 2
5.b even 2 1 inner 2880.2.f.o 2
8.b even 2 1 1440.2.f.b 2
8.d odd 2 1 1440.2.f.d 2
12.b even 2 1 960.2.f.e 2
15.d odd 2 1 960.2.f.d 2
15.e even 4 1 4800.2.a.e 1
15.e even 4 1 4800.2.a.cp 1
20.d odd 2 1 2880.2.f.m 2
24.f even 2 1 480.2.f.d yes 2
24.h odd 2 1 480.2.f.c 2
40.e odd 2 1 1440.2.f.d 2
40.f even 2 1 1440.2.f.b 2
40.i odd 4 1 7200.2.a.a 1
40.i odd 4 1 7200.2.a.ca 1
40.k even 4 1 7200.2.a.c 1
40.k even 4 1 7200.2.a.by 1
48.i odd 4 1 3840.2.d.p 2
48.i odd 4 1 3840.2.d.q 2
48.k even 4 1 3840.2.d.a 2
48.k even 4 1 3840.2.d.bf 2
60.h even 2 1 960.2.f.e 2
60.l odd 4 1 4800.2.a.c 1
60.l odd 4 1 4800.2.a.cr 1
120.i odd 2 1 480.2.f.c 2
120.m even 2 1 480.2.f.d yes 2
120.q odd 4 1 2400.2.a.o 1
120.q odd 4 1 2400.2.a.t 1
120.w even 4 1 2400.2.a.p 1
120.w even 4 1 2400.2.a.s 1
240.t even 4 1 3840.2.d.a 2
240.t even 4 1 3840.2.d.bf 2
240.bm odd 4 1 3840.2.d.p 2
240.bm odd 4 1 3840.2.d.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.c 2 24.h odd 2 1
480.2.f.c 2 120.i odd 2 1
480.2.f.d yes 2 24.f even 2 1
480.2.f.d yes 2 120.m even 2 1
960.2.f.d 2 3.b odd 2 1
960.2.f.d 2 15.d odd 2 1
960.2.f.e 2 12.b even 2 1
960.2.f.e 2 60.h even 2 1
1440.2.f.b 2 8.b even 2 1
1440.2.f.b 2 40.f even 2 1
1440.2.f.d 2 8.d odd 2 1
1440.2.f.d 2 40.e odd 2 1
2400.2.a.o 1 120.q odd 4 1
2400.2.a.p 1 120.w even 4 1
2400.2.a.s 1 120.w even 4 1
2400.2.a.t 1 120.q odd 4 1
2880.2.f.m 2 4.b odd 2 1
2880.2.f.m 2 20.d odd 2 1
2880.2.f.o 2 1.a even 1 1 trivial
2880.2.f.o 2 5.b even 2 1 inner
3840.2.d.a 2 48.k even 4 1
3840.2.d.a 2 240.t even 4 1
3840.2.d.p 2 48.i odd 4 1
3840.2.d.p 2 240.bm odd 4 1
3840.2.d.q 2 48.i odd 4 1
3840.2.d.q 2 240.bm odd 4 1
3840.2.d.bf 2 48.k even 4 1
3840.2.d.bf 2 240.t even 4 1
4800.2.a.c 1 60.l odd 4 1
4800.2.a.e 1 15.e even 4 1
4800.2.a.cp 1 15.e even 4 1
4800.2.a.cr 1 60.l odd 4 1
7200.2.a.a 1 40.i odd 4 1
7200.2.a.c 1 40.k even 4 1
7200.2.a.by 1 40.k even 4 1
7200.2.a.ca 1 40.i odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} - 8 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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