Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.441000.2

$q$-expansion

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

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

Character values

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

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1553.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 1.00000i 0 1.00000i 0
1553.2 0 0.707107 0.707107i 0 −0.707107 0.707107i 0 1.00000i 0 1.00000i 0
2897.1 0 −0.707107 0.707107i 0 0.707107 0.707107i 0 1.00000i 0 1.00000i 0
2897.2 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 1.00000i 0 1.00000i 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.f even 4 1 inner
105.k odd 4 1 inner
280.s even 4 1 inner
840.bp odd 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{2} - 2 \)
\( T_{23} \)
\( T_{73}^{2} - 2 T_{73} + 2 \)

Hecke characteristic polynomials

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