Properties

Label 2880.1.cd.a
Level 2880
Weight 1
Character orbit 2880.cd
Analytic conductor 1.437
Analytic rank 0
Dimension 8
Projective image \(D_{12}\)
CM discriminant -20
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1440)
Projective image \(D_{12}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{24}^{11} q^{3} + \zeta_{24}^{2} q^{5} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{10} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{11} q^{3} + \zeta_{24}^{2} q^{5} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{10} q^{9} + \zeta_{24} q^{15} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} ) q^{21} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{23} + \zeta_{24}^{4} q^{25} -\zeta_{24}^{9} q^{27} + ( 1 - \zeta_{24}^{8} ) q^{29} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{35} -\zeta_{24}^{2} q^{41} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{43} + q^{45} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{47} + ( \zeta_{24}^{6} - \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{49} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{61} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{63} + ( -\zeta_{24} - \zeta_{24}^{3} ) q^{67} + ( \zeta_{24}^{6} + \zeta_{24}^{8} ) q^{69} + \zeta_{24}^{3} q^{75} -\zeta_{24}^{8} q^{81} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{83} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{87} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{21} + 4q^{25} + 12q^{29} + 8q^{45} + 4q^{49} - 4q^{69} + 4q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{8}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1409.1
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
0 −0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0.448288 0.258819i 0 0.866025 0.500000i 0
1409.2 0 −0.258819 0.965926i 0 −0.866025 0.500000i 0 −1.67303 + 0.965926i 0 −0.866025 + 0.500000i 0
1409.3 0 0.258819 + 0.965926i 0 −0.866025 0.500000i 0 1.67303 0.965926i 0 −0.866025 + 0.500000i 0
1409.4 0 0.965926 0.258819i 0 0.866025 + 0.500000i 0 −0.448288 + 0.258819i 0 0.866025 0.500000i 0
2369.1 0 −0.965926 0.258819i 0 0.866025 0.500000i 0 0.448288 + 0.258819i 0 0.866025 + 0.500000i 0
2369.2 0 −0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 −1.67303 0.965926i 0 −0.866025 0.500000i 0
2369.3 0 0.258819 0.965926i 0 −0.866025 + 0.500000i 0 1.67303 + 0.965926i 0 −0.866025 0.500000i 0
2369.4 0 0.965926 + 0.258819i 0 0.866025 0.500000i 0 −0.448288 0.258819i 0 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2369.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.cd.a 8
4.b odd 2 1 inner 2880.1.cd.a 8
5.b even 2 1 inner 2880.1.cd.a 8
8.b even 2 1 1440.1.cd.a 8
8.d odd 2 1 1440.1.cd.a 8
9.d odd 6 1 inner 2880.1.cd.a 8
20.d odd 2 1 CM 2880.1.cd.a 8
36.h even 6 1 inner 2880.1.cd.a 8
40.e odd 2 1 1440.1.cd.a 8
40.f even 2 1 1440.1.cd.a 8
45.h odd 6 1 inner 2880.1.cd.a 8
72.j odd 6 1 1440.1.cd.a 8
72.l even 6 1 1440.1.cd.a 8
180.n even 6 1 inner 2880.1.cd.a 8
360.bd even 6 1 1440.1.cd.a 8
360.bh odd 6 1 1440.1.cd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.1.cd.a 8 8.b even 2 1
1440.1.cd.a 8 8.d odd 2 1
1440.1.cd.a 8 40.e odd 2 1
1440.1.cd.a 8 40.f even 2 1
1440.1.cd.a 8 72.j odd 6 1
1440.1.cd.a 8 72.l even 6 1
1440.1.cd.a 8 360.bd even 6 1
1440.1.cd.a 8 360.bh odd 6 1
2880.1.cd.a 8 1.a even 1 1 trivial
2880.1.cd.a 8 4.b odd 2 1 inner
2880.1.cd.a 8 5.b even 2 1 inner
2880.1.cd.a 8 9.d odd 6 1 inner
2880.1.cd.a 8 20.d odd 2 1 CM
2880.1.cd.a 8 36.h even 6 1 inner
2880.1.cd.a 8 45.h odd 6 1 inner
2880.1.cd.a 8 180.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2880, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T^{4} + T^{8} \)
$5$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$7$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$11$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$13$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$17$ \( ( 1 + T^{2} )^{8} \)
$19$ \( ( 1 + T^{2} )^{8} \)
$23$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$29$ \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \)
$31$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$37$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$41$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$43$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$47$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$53$ \( ( 1 + T^{2} )^{8} \)
$59$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$61$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$67$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$71$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$73$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$79$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$83$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$89$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$97$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
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