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Properties

Label 840.1.cg.a
Level $840$
Weight $1$
Character orbit 840.cg
Analytic conductor $0.419$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,1,Mod(149,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 3, 2])) 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("840.149"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 840.cg (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,-4,-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: \(0.419214610612\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} - q^{6} - \zeta_{12}^{2} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} + q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{11} + \cdots + (\zeta_{12}^{5} - \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{6} - 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{15} - 2 q^{16} - 2 q^{24} + 2 q^{25} + 2 q^{28} - 2 q^{31} - 6 q^{33} + 4 q^{36} + 2 q^{40} + 2 q^{42} - 2 q^{49} - 2 q^{54} + 6 q^{55} - 6 q^{58}+ \cdots + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −0.866025 + 0.500000i −1.00000 −0.500000 − 0.866025i − 1.00000i 0.500000 − 0.866025i 1.00000
149.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.866025 − 0.500000i −1.00000 −0.500000 − 0.866025i 1.00000i 0.500000 − 0.866025i 1.00000
389.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i −0.866025 − 0.500000i −1.00000 −0.500000 + 0.866025i 1.00000i 0.500000 + 0.866025i 1.00000
389.2 0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i − 1.00000i 0.500000 + 0.866025i 1.00000
\(n\): 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 840.1.cg.a ✓ 4
3.b odd 2 1 inner 840.1.cg.a ✓ 4
4.b odd 2 1 3360.1.dm.b 4
5.b even 2 1 840.1.cg.b yes 4
7.c even 3 1 840.1.cg.b yes 4
8.b even 2 1 inner 840.1.cg.a ✓ 4
8.d odd 2 1 3360.1.dm.b 4
12.b even 2 1 3360.1.dm.b 4
15.d odd 2 1 840.1.cg.b yes 4
20.d odd 2 1 3360.1.dm.a 4
21.h odd 6 1 840.1.cg.b yes 4
24.f even 2 1 3360.1.dm.b 4
24.h odd 2 1 CM 840.1.cg.a ✓ 4
28.g odd 6 1 3360.1.dm.a 4
35.j even 6 1 inner 840.1.cg.a ✓ 4
40.e odd 2 1 3360.1.dm.a 4
40.f even 2 1 840.1.cg.b yes 4
56.k odd 6 1 3360.1.dm.a 4
56.p even 6 1 840.1.cg.b yes 4
60.h even 2 1 3360.1.dm.a 4
84.n even 6 1 3360.1.dm.a 4
105.o odd 6 1 inner 840.1.cg.a ✓ 4
120.i odd 2 1 840.1.cg.b yes 4
120.m even 2 1 3360.1.dm.a 4
140.p odd 6 1 3360.1.dm.b 4
168.s odd 6 1 840.1.cg.b yes 4
168.v even 6 1 3360.1.dm.a 4
280.bf even 6 1 inner 840.1.cg.a ✓ 4
280.bi odd 6 1 3360.1.dm.b 4
420.ba even 6 1 3360.1.dm.b 4
840.cg odd 6 1 inner 840.1.cg.a ✓ 4
840.cv even 6 1 3360.1.dm.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.cg.a ✓ 4 1.a even 1 1 trivial
840.1.cg.a ✓ 4 3.b odd 2 1 inner
840.1.cg.a ✓ 4 8.b even 2 1 inner
840.1.cg.a ✓ 4 24.h odd 2 1 CM
840.1.cg.a ✓ 4 35.j even 6 1 inner
840.1.cg.a ✓ 4 105.o odd 6 1 inner
840.1.cg.a ✓ 4 280.bf even 6 1 inner
840.1.cg.a ✓ 4 840.cg odd 6 1 inner
840.1.cg.b yes 4 5.b even 2 1
840.1.cg.b yes 4 7.c even 3 1
840.1.cg.b yes 4 15.d odd 2 1
840.1.cg.b yes 4 21.h odd 6 1
840.1.cg.b yes 4 40.f even 2 1
840.1.cg.b yes 4 56.p even 6 1
840.1.cg.b yes 4 120.i odd 2 1
840.1.cg.b yes 4 168.s odd 6 1
3360.1.dm.a 4 20.d odd 2 1
3360.1.dm.a 4 28.g odd 6 1
3360.1.dm.a 4 40.e odd 2 1
3360.1.dm.a 4 56.k odd 6 1
3360.1.dm.a 4 60.h even 2 1
3360.1.dm.a 4 84.n even 6 1
3360.1.dm.a 4 120.m even 2 1
3360.1.dm.a 4 168.v even 6 1
3360.1.dm.b 4 4.b odd 2 1
3360.1.dm.b 4 8.d odd 2 1
3360.1.dm.b 4 12.b even 2 1
3360.1.dm.b 4 24.f even 2 1
3360.1.dm.b 4 140.p odd 6 1
3360.1.dm.b 4 280.bi odd 6 1
3360.1.dm.b 4 420.ba even 6 1
3360.1.dm.b 4 840.cv even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{193}^{2} - 3T_{193} + 3 \) acting on \(S_{1}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) 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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