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Properties

Label 3360.1.dm.b
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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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3360,1,Mod(1649,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 3, 4])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1649"); S:= CuspForms(chi, 1); N := Newforms(S);
 
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

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} - \zeta_{12} q^{5} - \zeta_{12}^{4} q^{7} + \zeta_{12}^{2} q^{9} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{11} + \zeta_{12}^{2} q^{15} + \zeta_{12}^{5} q^{21} + \zeta_{12}^{2} q^{25} + \cdots + ( - \zeta_{12}^{5} + \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7} + 2 q^{9} + 2 q^{15} + 2 q^{25} + 2 q^{31} - 6 q^{33} - 2 q^{49} - 6 q^{55} + 4 q^{63} - 2 q^{79} - 2 q^{81} - 6 q^{87}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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 −0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0
1649.2 0 0.866025 + 0.500000i 0 0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0
3089.1 0 −0.866025 + 0.500000i 0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0.500000 − 0.866025i 0
3089.2 0 0.866025 − 0.500000i 0 0.866025 − 0.500000i 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.b 4
3.b odd 2 1 inner 3360.1.dm.b 4
4.b odd 2 1 840.1.cg.a ✓ 4
5.b even 2 1 3360.1.dm.a 4
7.c even 3 1 3360.1.dm.a 4
8.b even 2 1 inner 3360.1.dm.b 4
8.d odd 2 1 840.1.cg.a ✓ 4
12.b even 2 1 840.1.cg.a ✓ 4
15.d odd 2 1 3360.1.dm.a 4
20.d odd 2 1 840.1.cg.b yes 4
21.h odd 6 1 3360.1.dm.a 4
24.f even 2 1 840.1.cg.a ✓ 4
24.h odd 2 1 CM 3360.1.dm.b 4
28.g odd 6 1 840.1.cg.b yes 4
35.j even 6 1 inner 3360.1.dm.b 4
40.e odd 2 1 840.1.cg.b yes 4
40.f even 2 1 3360.1.dm.a 4
56.k odd 6 1 840.1.cg.b yes 4
56.p even 6 1 3360.1.dm.a 4
60.h even 2 1 840.1.cg.b yes 4
84.n even 6 1 840.1.cg.b yes 4
105.o odd 6 1 inner 3360.1.dm.b 4
120.i odd 2 1 3360.1.dm.a 4
120.m even 2 1 840.1.cg.b yes 4
140.p odd 6 1 840.1.cg.a ✓ 4
168.s odd 6 1 3360.1.dm.a 4
168.v even 6 1 840.1.cg.b yes 4
280.bf even 6 1 inner 3360.1.dm.b 4
280.bi odd 6 1 840.1.cg.a ✓ 4
420.ba even 6 1 840.1.cg.a ✓ 4
840.cg odd 6 1 inner 3360.1.dm.b 4
840.cv even 6 1 840.1.cg.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.cg.a ✓ 4 4.b odd 2 1
840.1.cg.a ✓ 4 8.d odd 2 1
840.1.cg.a ✓ 4 12.b even 2 1
840.1.cg.a ✓ 4 24.f even 2 1
840.1.cg.a ✓ 4 140.p odd 6 1
840.1.cg.a ✓ 4 280.bi odd 6 1
840.1.cg.a ✓ 4 420.ba even 6 1
840.1.cg.a ✓ 4 840.cv even 6 1
840.1.cg.b yes 4 20.d odd 2 1
840.1.cg.b yes 4 28.g odd 6 1
840.1.cg.b yes 4 40.e odd 2 1
840.1.cg.b yes 4 56.k odd 6 1
840.1.cg.b yes 4 60.h even 2 1
840.1.cg.b yes 4 84.n even 6 1
840.1.cg.b yes 4 120.m even 2 1
840.1.cg.b yes 4 168.v even 6 1
3360.1.dm.a 4 5.b even 2 1
3360.1.dm.a 4 7.c even 3 1
3360.1.dm.a 4 15.d odd 2 1
3360.1.dm.a 4 21.h odd 6 1
3360.1.dm.a 4 40.f even 2 1
3360.1.dm.a 4 56.p even 6 1
3360.1.dm.a 4 120.i odd 2 1
3360.1.dm.a 4 168.s odd 6 1
3360.1.dm.b 4 1.a even 1 1 trivial
3360.1.dm.b 4 3.b odd 2 1 inner
3360.1.dm.b 4 8.b even 2 1 inner
3360.1.dm.b 4 24.h odd 2 1 CM
3360.1.dm.b 4 35.j even 6 1 inner
3360.1.dm.b 4 105.o odd 6 1 inner
3360.1.dm.b 4 280.bf even 6 1 inner
3360.1.dm.b 4 840.cg odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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