Properties

Label 3360.1.ft.a
Level $3360$
Weight $1$
Character orbit 3360.ft
Analytic conductor $1.677$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3360.ft (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{5} q^{3} -\zeta_{24}^{11} q^{5} + \zeta_{24}^{8} q^{7} + \zeta_{24}^{10} q^{9} +O(q^{10})\) \( q + \zeta_{24}^{5} q^{3} -\zeta_{24}^{11} q^{5} + \zeta_{24}^{8} q^{7} + \zeta_{24}^{10} q^{9} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{11} + \zeta_{24}^{4} q^{15} -\zeta_{24} q^{21} -\zeta_{24}^{10} q^{25} -\zeta_{24}^{3} q^{27} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{29} + \zeta_{24}^{2} q^{31} + ( -\zeta_{24}^{6} + \zeta_{24}^{8} ) q^{33} + \zeta_{24}^{7} q^{35} + \zeta_{24}^{9} q^{45} -\zeta_{24}^{4} q^{49} + ( -\zeta_{24} + \zeta_{24}^{9} ) q^{53} + ( -1 + \zeta_{24}^{2} ) q^{55} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{59} -\zeta_{24}^{6} q^{63} + ( -\zeta_{24}^{2} - \zeta_{24}^{8} ) q^{73} + \zeta_{24}^{3} q^{75} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{77} + ( 1 - \zeta_{24}^{8} ) q^{79} -\zeta_{24}^{8} q^{81} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{83} + ( -1 + \zeta_{24}^{10} ) q^{87} + \zeta_{24}^{7} q^{93} + ( \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{97} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{7} + O(q^{10}) \) \( 8q - 4q^{7} + 4q^{15} - 4q^{33} - 4q^{49} - 8q^{55} + 4q^{73} + 12q^{79} + 4q^{81} - 8q^{87} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(\zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
0 −0.965926 + 0.258819i 0 −0.258819 0.965926i 0 −0.500000 + 0.866025i 0 0.866025 0.500000i 0
17.2 0 0.965926 0.258819i 0 0.258819 + 0.965926i 0 −0.500000 + 0.866025i 0 0.866025 0.500000i 0
593.1 0 −0.965926 0.258819i 0 −0.258819 + 0.965926i 0 −0.500000 0.866025i 0 0.866025 + 0.500000i 0
593.2 0 0.965926 + 0.258819i 0 0.258819 0.965926i 0 −0.500000 0.866025i 0 0.866025 + 0.500000i 0
1937.1 0 −0.258819 + 0.965926i 0 −0.965926 0.258819i 0 −0.500000 0.866025i 0 −0.866025 0.500000i 0
1937.2 0 0.258819 0.965926i 0 0.965926 + 0.258819i 0 −0.500000 0.866025i 0 −0.866025 0.500000i 0
2033.1 0 −0.258819 0.965926i 0 −0.965926 + 0.258819i 0 −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0
2033.2 0 0.258819 + 0.965926i 0 0.965926 0.258819i 0 −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2033.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
35.k even 12 1 inner
105.w odd 12 1 inner
280.bv even 12 1 inner
840.dh odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.ft.a 8
3.b odd 2 1 inner 3360.1.ft.a 8
4.b odd 2 1 840.1.dh.b yes 8
5.c odd 4 1 3360.1.ft.b 8
7.d odd 6 1 3360.1.ft.b 8
8.b even 2 1 inner 3360.1.ft.a 8
8.d odd 2 1 840.1.dh.b yes 8
12.b even 2 1 840.1.dh.b yes 8
15.e even 4 1 3360.1.ft.b 8
20.e even 4 1 840.1.dh.a 8
21.g even 6 1 3360.1.ft.b 8
24.f even 2 1 840.1.dh.b yes 8
24.h odd 2 1 CM 3360.1.ft.a 8
28.f even 6 1 840.1.dh.a 8
35.k even 12 1 inner 3360.1.ft.a 8
40.i odd 4 1 3360.1.ft.b 8
40.k even 4 1 840.1.dh.a 8
56.j odd 6 1 3360.1.ft.b 8
56.m even 6 1 840.1.dh.a 8
60.l odd 4 1 840.1.dh.a 8
84.j odd 6 1 840.1.dh.a 8
105.w odd 12 1 inner 3360.1.ft.a 8
120.q odd 4 1 840.1.dh.a 8
120.w even 4 1 3360.1.ft.b 8
140.x odd 12 1 840.1.dh.b yes 8
168.ba even 6 1 3360.1.ft.b 8
168.be odd 6 1 840.1.dh.a 8
280.bp odd 12 1 840.1.dh.b yes 8
280.bv even 12 1 inner 3360.1.ft.a 8
420.br even 12 1 840.1.dh.b yes 8
840.dh odd 12 1 inner 3360.1.ft.a 8
840.dk even 12 1 840.1.dh.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.dh.a 8 20.e even 4 1
840.1.dh.a 8 28.f even 6 1
840.1.dh.a 8 40.k even 4 1
840.1.dh.a 8 56.m even 6 1
840.1.dh.a 8 60.l odd 4 1
840.1.dh.a 8 84.j odd 6 1
840.1.dh.a 8 120.q odd 4 1
840.1.dh.a 8 168.be odd 6 1
840.1.dh.b yes 8 4.b odd 2 1
840.1.dh.b yes 8 8.d odd 2 1
840.1.dh.b yes 8 12.b even 2 1
840.1.dh.b yes 8 24.f even 2 1
840.1.dh.b yes 8 140.x odd 12 1
840.1.dh.b yes 8 280.bp odd 12 1
840.1.dh.b yes 8 420.br even 12 1
840.1.dh.b yes 8 840.dk even 12 1
3360.1.ft.a 8 1.a even 1 1 trivial
3360.1.ft.a 8 3.b odd 2 1 inner
3360.1.ft.a 8 8.b even 2 1 inner
3360.1.ft.a 8 24.h odd 2 1 CM
3360.1.ft.a 8 35.k even 12 1 inner
3360.1.ft.a 8 105.w odd 12 1 inner
3360.1.ft.a 8 280.bv even 12 1 inner
3360.1.ft.a 8 840.dh odd 12 1 inner
3360.1.ft.b 8 5.c odd 4 1
3360.1.ft.b 8 7.d odd 6 1
3360.1.ft.b 8 15.e even 4 1
3360.1.ft.b 8 21.g even 6 1
3360.1.ft.b 8 40.i odd 4 1
3360.1.ft.b 8 56.j odd 6 1
3360.1.ft.b 8 120.w even 4 1
3360.1.ft.b 8 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{73}^{4} - 2 T_{73}^{3} + 2 T_{73}^{2} - 4 T_{73} + 4 \) acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 1 - T^{4} + T^{8} \)
$5$ \( 1 - T^{4} + T^{8} \)
$7$ \( ( 1 + T + T^{2} )^{4} \)
$11$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( 81 - 9 T^{4} + T^{8} \)
$59$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$79$ \( ( 3 - 3 T + T^{2} )^{4} \)
$83$ \( ( 9 + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( ( 1 + 2 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
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