Properties

Label 2880.2.f.d
Level $2880$
Weight $2$
Character orbit 2880.f
Analytic conductor $22.997$
Analytic rank $1$
Dimension $2$
CM discriminant -4
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1440)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + i ) q^{5} +O(q^{10})\) \( q + ( -2 + i ) q^{5} + 4 i q^{13} -2 i q^{17} + ( 3 - 4 i ) q^{25} -4 q^{29} + 12 i q^{37} -8 q^{41} + 7 q^{49} -14 i q^{53} -10 q^{61} + ( -4 - 8 i ) q^{65} -16 i q^{73} + ( 2 + 4 i ) q^{85} -16 q^{89} -8 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + O(q^{10}) \) \( 2q - 4q^{5} + 6q^{25} - 8q^{29} - 16q^{41} + 14q^{49} - 20q^{61} - 8q^{65} + 4q^{85} - 32q^{89} + O(q^{100}) \)

Character values

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

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

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 −2.00000 1.00000i 0 0 0 0 0
1729.2 0 0 0 −2.00000 + 1.00000i 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d 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.f.d 2
3.b odd 2 1 2880.2.f.s 2
4.b odd 2 1 CM 2880.2.f.d 2
5.b even 2 1 inner 2880.2.f.d 2
8.b even 2 1 1440.2.f.f yes 2
8.d odd 2 1 1440.2.f.f yes 2
12.b even 2 1 2880.2.f.s 2
15.d odd 2 1 2880.2.f.s 2
20.d odd 2 1 inner 2880.2.f.d 2
24.f even 2 1 1440.2.f.a 2
24.h odd 2 1 1440.2.f.a 2
40.e odd 2 1 1440.2.f.f yes 2
40.f even 2 1 1440.2.f.f yes 2
40.i odd 4 1 7200.2.a.x 1
40.i odd 4 1 7200.2.a.bc 1
40.k even 4 1 7200.2.a.x 1
40.k even 4 1 7200.2.a.bc 1
60.h even 2 1 2880.2.f.s 2
120.i odd 2 1 1440.2.f.a 2
120.m even 2 1 1440.2.f.a 2
120.q odd 4 1 7200.2.a.w 1
120.q odd 4 1 7200.2.a.bd 1
120.w even 4 1 7200.2.a.w 1
120.w even 4 1 7200.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.f.a 2 24.f even 2 1
1440.2.f.a 2 24.h odd 2 1
1440.2.f.a 2 120.i odd 2 1
1440.2.f.a 2 120.m even 2 1
1440.2.f.f yes 2 8.b even 2 1
1440.2.f.f yes 2 8.d odd 2 1
1440.2.f.f yes 2 40.e odd 2 1
1440.2.f.f yes 2 40.f even 2 1
2880.2.f.d 2 1.a even 1 1 trivial
2880.2.f.d 2 4.b odd 2 1 CM
2880.2.f.d 2 5.b even 2 1 inner
2880.2.f.d 2 20.d odd 2 1 inner
2880.2.f.s 2 3.b odd 2 1
2880.2.f.s 2 12.b even 2 1
2880.2.f.s 2 15.d odd 2 1
2880.2.f.s 2 60.h even 2 1
7200.2.a.w 1 120.q odd 4 1
7200.2.a.w 1 120.w even 4 1
7200.2.a.x 1 40.i odd 4 1
7200.2.a.x 1 40.k even 4 1
7200.2.a.bc 1 40.i odd 4 1
7200.2.a.bc 1 40.k even 4 1
7200.2.a.bd 1 120.q odd 4 1
7200.2.a.bd 1 120.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 \(S_{2}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7} \)
\( T_{11} \)
\( T_{13}^{2} + 16 \)
\( T_{17}^{2} + 4 \)
\( T_{19} \)
\( T_{29} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 144 + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 196 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 256 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 16 + T )^{2} \)
$97$ \( 64 + T^{2} \)
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