Properties

Label 3360.1.ds.b
Level $3360$
Weight $1$
Character orbit 3360.ds
Analytic conductor $1.677$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -20
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.ds (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: yes
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.

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

Character values

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

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(\zeta_{24}^{8}\) \(-1\)

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
0.258819 0.965926i
0.965926 + 0.258819i
−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 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0
1409.2 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0
1409.3 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0
1409.4 0 0.965926 0.258819i 0 −0.866025 0.500000i 0 0.258819 0.965926i 0 0.866025 0.500000i 0
3329.1 0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0
3329.2 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0
3329.3 0 0.258819 0.965926i 0 0.866025 0.500000i 0 0.965926 0.258819i 0 −0.866025 0.500000i 0
3329.4 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3329.4
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
21.h odd 6 1 inner
84.n even 6 1 inner
105.o odd 6 1 inner
420.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.ds.b yes 8
3.b odd 2 1 3360.1.ds.a 8
4.b odd 2 1 inner 3360.1.ds.b yes 8
5.b even 2 1 inner 3360.1.ds.b yes 8
7.c even 3 1 3360.1.ds.a 8
12.b even 2 1 3360.1.ds.a 8
15.d odd 2 1 3360.1.ds.a 8
20.d odd 2 1 CM 3360.1.ds.b yes 8
21.h odd 6 1 inner 3360.1.ds.b yes 8
28.g odd 6 1 3360.1.ds.a 8
35.j even 6 1 3360.1.ds.a 8
60.h even 2 1 3360.1.ds.a 8
84.n even 6 1 inner 3360.1.ds.b yes 8
105.o odd 6 1 inner 3360.1.ds.b yes 8
140.p odd 6 1 3360.1.ds.a 8
420.ba even 6 1 inner 3360.1.ds.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.1.ds.a 8 3.b odd 2 1
3360.1.ds.a 8 7.c even 3 1
3360.1.ds.a 8 12.b even 2 1
3360.1.ds.a 8 15.d odd 2 1
3360.1.ds.a 8 28.g odd 6 1
3360.1.ds.a 8 35.j even 6 1
3360.1.ds.a 8 60.h even 2 1
3360.1.ds.a 8 140.p odd 6 1
3360.1.ds.b yes 8 1.a even 1 1 trivial
3360.1.ds.b yes 8 4.b odd 2 1 inner
3360.1.ds.b yes 8 5.b even 2 1 inner
3360.1.ds.b yes 8 20.d odd 2 1 CM
3360.1.ds.b yes 8 21.h odd 6 1 inner
3360.1.ds.b yes 8 84.n even 6 1 inner
3360.1.ds.b yes 8 105.o odd 6 1 inner
3360.1.ds.b yes 8 420.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{89}^{2} - 3 T_{89} + 3 \) acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\).

Hecke characteristic polynomials

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