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Properties

Label 840.2.cc.c
Level $840$
Weight $2$
Character orbit 840.cc
Analytic conductor $6.707$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [840,2,Mod(289,840)] 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, 0, 0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.cc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{5} + 12 q^{9} - 8 q^{11} + 8 q^{19} + 12 q^{21} + 2 q^{25} - 24 q^{29} + 4 q^{31} + 30 q^{35} + 4 q^{39} + 32 q^{41} - 2 q^{45} - 12 q^{49} - 20 q^{51} + 12 q^{55} - 4 q^{59} + 32 q^{61} + 22 q^{65}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −0.866025 + 0.500000i 0 −2.17066 − 0.536884i 0 −0.933675 + 2.47553i 0 0.500000 − 0.866025i 0
289.2 0 −0.866025 + 0.500000i 0 −2.13417 + 0.667333i 0 1.27407 − 2.31878i 0 0.500000 − 0.866025i 0
289.3 0 −0.866025 + 0.500000i 0 0.0914143 − 2.23420i 0 −2.35614 + 1.20358i 0 0.500000 − 0.866025i 0
289.4 0 −0.866025 + 0.500000i 0 0.410517 + 2.19806i 0 −2.09709 − 1.61313i 0 0.500000 − 0.866025i 0
289.5 0 −0.866025 + 0.500000i 0 2.06974 − 0.846282i 0 2.64327 − 0.114495i 0 0.500000 − 0.866025i 0
289.6 0 −0.866025 + 0.500000i 0 2.23316 − 0.114055i 0 −0.262488 − 2.63270i 0 0.500000 − 0.866025i 0
289.7 0 0.866025 − 0.500000i 0 −2.10884 + 0.743513i 0 2.09709 + 1.61313i 0 0.500000 − 0.866025i 0
289.8 0 0.866025 − 0.500000i 0 −1.01780 − 1.99100i 0 0.262488 + 2.63270i 0 0.500000 − 0.866025i 0
289.9 0 0.866025 − 0.500000i 0 −0.301966 − 2.21558i 0 −2.64327 + 0.114495i 0 0.500000 − 0.866025i 0
289.10 0 0.866025 − 0.500000i 0 0.489156 + 2.18191i 0 −1.27407 + 2.31878i 0 0.500000 − 0.866025i 0
289.11 0 0.866025 − 0.500000i 0 1.55028 + 1.61140i 0 0.933675 − 2.47553i 0 0.500000 − 0.866025i 0
289.12 0 0.866025 − 0.500000i 0 1.88917 − 1.19627i 0 2.35614 − 1.20358i 0 0.500000 − 0.866025i 0
529.1 0 −0.866025 − 0.500000i 0 −2.17066 + 0.536884i 0 −0.933675 − 2.47553i 0 0.500000 + 0.866025i 0
529.2 0 −0.866025 − 0.500000i 0 −2.13417 − 0.667333i 0 1.27407 + 2.31878i 0 0.500000 + 0.866025i 0
529.3 0 −0.866025 − 0.500000i 0 0.0914143 + 2.23420i 0 −2.35614 − 1.20358i 0 0.500000 + 0.866025i 0
529.4 0 −0.866025 − 0.500000i 0 0.410517 − 2.19806i 0 −2.09709 + 1.61313i 0 0.500000 + 0.866025i 0
529.5 0 −0.866025 − 0.500000i 0 2.06974 + 0.846282i 0 2.64327 + 0.114495i 0 0.500000 + 0.866025i 0
529.6 0 −0.866025 − 0.500000i 0 2.23316 + 0.114055i 0 −0.262488 + 2.63270i 0 0.500000 + 0.866025i 0
529.7 0 0.866025 + 0.500000i 0 −2.10884 − 0.743513i 0 2.09709 − 1.61313i 0 0.500000 + 0.866025i 0
529.8 0 0.866025 + 0.500000i 0 −1.01780 + 1.99100i 0 0.262488 − 2.63270i 0 0.500000 + 0.866025i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.cc.c ✓ 24
4.b odd 2 1 1680.2.di.g 24
5.b even 2 1 inner 840.2.cc.c ✓ 24
7.c even 3 1 inner 840.2.cc.c ✓ 24
20.d odd 2 1 1680.2.di.g 24
28.g odd 6 1 1680.2.di.g 24
35.j even 6 1 inner 840.2.cc.c ✓ 24
140.p odd 6 1 1680.2.di.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.cc.c ✓ 24 1.a even 1 1 trivial
840.2.cc.c ✓ 24 5.b even 2 1 inner
840.2.cc.c ✓ 24 7.c even 3 1 inner
840.2.cc.c ✓ 24 35.j even 6 1 inner
1680.2.di.g 24 4.b odd 2 1
1680.2.di.g 24 20.d odd 2 1
1680.2.di.g 24 28.g odd 6 1
1680.2.di.g 24 140.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} + 4 T_{11}^{11} + 61 T_{11}^{10} - 8 T_{11}^{9} + 1725 T_{11}^{8} - 1492 T_{11}^{7} + \cdots + 422500 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display