Properties

Label 3360.1.dm.a
Level $3360$
Weight $1$
Character orbit 3360.dm
Analytic conductor $1.677$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
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.dm (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.518616000.10

$q$-expansion

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

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

Character values

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

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1649.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 0.500000i 0 1.00000i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0
1649.2 0 0.866025 + 0.500000i 0 1.00000i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0
3089.1 0 −0.866025 + 0.500000i 0 1.00000i 0 −0.500000 0.866025i 0 0.500000 0.866025i 0
3089.2 0 0.866025 0.500000i 0 1.00000i 0 −0.500000 0.866025i 0 0.500000 0.866025i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner
280.bf even 6 1 inner
840.cg odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.dm.a 4
3.b odd 2 1 inner 3360.1.dm.a 4
4.b odd 2 1 840.1.cg.b yes 4
5.b even 2 1 3360.1.dm.b 4
7.c even 3 1 3360.1.dm.b 4
8.b even 2 1 inner 3360.1.dm.a 4
8.d odd 2 1 840.1.cg.b yes 4
12.b even 2 1 840.1.cg.b yes 4
15.d odd 2 1 3360.1.dm.b 4
20.d odd 2 1 840.1.cg.a 4
21.h odd 6 1 3360.1.dm.b 4
24.f even 2 1 840.1.cg.b yes 4
24.h odd 2 1 CM 3360.1.dm.a 4
28.g odd 6 1 840.1.cg.a 4
35.j even 6 1 inner 3360.1.dm.a 4
40.e odd 2 1 840.1.cg.a 4
40.f even 2 1 3360.1.dm.b 4
56.k odd 6 1 840.1.cg.a 4
56.p even 6 1 3360.1.dm.b 4
60.h even 2 1 840.1.cg.a 4
84.n even 6 1 840.1.cg.a 4
105.o odd 6 1 inner 3360.1.dm.a 4
120.i odd 2 1 3360.1.dm.b 4
120.m even 2 1 840.1.cg.a 4
140.p odd 6 1 840.1.cg.b yes 4
168.s odd 6 1 3360.1.dm.b 4
168.v even 6 1 840.1.cg.a 4
280.bf even 6 1 inner 3360.1.dm.a 4
280.bi odd 6 1 840.1.cg.b yes 4
420.ba even 6 1 840.1.cg.b yes 4
840.cg odd 6 1 inner 3360.1.dm.a 4
840.cv even 6 1 840.1.cg.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.cg.a 4 20.d odd 2 1
840.1.cg.a 4 28.g odd 6 1
840.1.cg.a 4 40.e odd 2 1
840.1.cg.a 4 56.k odd 6 1
840.1.cg.a 4 60.h even 2 1
840.1.cg.a 4 84.n even 6 1
840.1.cg.a 4 120.m even 2 1
840.1.cg.a 4 168.v even 6 1
840.1.cg.b yes 4 4.b odd 2 1
840.1.cg.b yes 4 8.d odd 2 1
840.1.cg.b yes 4 12.b even 2 1
840.1.cg.b yes 4 24.f even 2 1
840.1.cg.b yes 4 140.p odd 6 1
840.1.cg.b yes 4 280.bi odd 6 1
840.1.cg.b yes 4 420.ba even 6 1
840.1.cg.b yes 4 840.cv even 6 1
3360.1.dm.a 4 1.a even 1 1 trivial
3360.1.dm.a 4 3.b odd 2 1 inner
3360.1.dm.a 4 8.b even 2 1 inner
3360.1.dm.a 4 24.h odd 2 1 CM
3360.1.dm.a 4 35.j even 6 1 inner
3360.1.dm.a 4 105.o odd 6 1 inner
3360.1.dm.a 4 280.bf even 6 1 inner
3360.1.dm.a 4 840.cg odd 6 1 inner
3360.1.dm.b 4 5.b even 2 1
3360.1.dm.b 4 7.c even 3 1
3360.1.dm.b 4 15.d odd 2 1
3360.1.dm.b 4 21.h odd 6 1
3360.1.dm.b 4 40.f even 2 1
3360.1.dm.b 4 56.p even 6 1
3360.1.dm.b 4 120.i odd 2 1
3360.1.dm.b 4 168.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{193}^{2} + 3 T_{193} + 3 \) acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\).

Hecke characteristic polynomials

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