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Properties

Label 840.2.dd.b
Level $840$
Weight $2$
Character orbit 840.dd
Analytic conductor $6.707$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(73,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 0, 9, 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("840.73"); 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.dd (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,0,4] 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{7} - 4 q^{11} - 16 q^{13} - 4 q^{15} + 4 q^{17} - 8 q^{19} + 48 q^{23} - 20 q^{25} + 24 q^{33} - 4 q^{37} + 12 q^{39} + 16 q^{43} + 4 q^{45} - 12 q^{47} + 12 q^{49} - 52 q^{53} + 56 q^{55} + 8 q^{57}+ \cdots - 44 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0 −0.258819 − 0.965926i 0 −2.21777 − 0.285502i 0 0.311708 − 2.62733i 0 −0.866025 + 0.500000i 0
73.2 0 −0.258819 − 0.965926i 0 −0.538951 + 2.17015i 0 2.45086 + 0.996627i 0 −0.866025 + 0.500000i 0
73.3 0 −0.258819 − 0.965926i 0 −0.519578 + 2.17487i 0 −2.60415 − 0.467324i 0 −0.866025 + 0.500000i 0
73.4 0 −0.258819 − 0.965926i 0 −0.394296 − 2.20103i 0 2.44511 + 1.01066i 0 −0.866025 + 0.500000i 0
73.5 0 −0.258819 − 0.965926i 0 1.53444 − 1.62649i 0 −1.53287 + 2.15646i 0 −0.866025 + 0.500000i 0
73.6 0 −0.258819 − 0.965926i 0 2.23605 + 0.00919595i 0 0.136441 − 2.64223i 0 −0.866025 + 0.500000i 0
73.7 0 0.258819 + 0.965926i 0 −2.12334 + 0.701022i 0 −2.28895 + 1.32691i 0 −0.866025 + 0.500000i 0
73.8 0 0.258819 + 0.965926i 0 −1.98539 − 1.02871i 0 2.63547 − 0.233051i 0 −0.866025 + 0.500000i 0
73.9 0 0.258819 + 0.965926i 0 −0.583460 + 2.15860i 0 0.939373 − 2.47337i 0 −0.866025 + 0.500000i 0
73.10 0 0.258819 + 0.965926i 0 −0.0101553 − 2.23604i 0 −2.07473 + 1.64180i 0 −0.866025 + 0.500000i 0
73.11 0 0.258819 + 0.965926i 0 0.829089 + 2.07668i 0 1.82816 + 1.91255i 0 −0.866025 + 0.500000i 0
73.12 0 0.258819 + 0.965926i 0 2.04130 − 0.912737i 0 −1.24643 − 2.33376i 0 −0.866025 + 0.500000i 0
313.1 0 −0.965926 − 0.258819i 0 −1.60125 − 1.56077i 0 −0.580301 + 2.58133i 0 0.866025 + 0.500000i 0
313.2 0 −0.965926 − 0.258819i 0 −0.654313 + 2.13819i 0 1.01438 + 2.44357i 0 0.866025 + 0.500000i 0
313.3 0 −0.965926 − 0.258819i 0 −0.263454 + 2.22049i 0 1.18366 − 2.36621i 0 0.866025 + 0.500000i 0
313.4 0 −0.965926 − 0.258819i 0 0.0956625 − 2.23402i 0 −1.96987 − 1.76624i 0 0.866025 + 0.500000i 0
313.5 0 −0.965926 − 0.258819i 0 1.42291 − 1.72491i 0 2.63771 − 0.206168i 0 0.866025 + 0.500000i 0
313.6 0 −0.965926 − 0.258819i 0 2.12529 + 0.695091i 0 −2.49269 + 0.886856i 0 0.866025 + 0.500000i 0
313.7 0 0.965926 + 0.258819i 0 −2.22330 + 0.238577i 0 −2.64040 − 0.168257i 0 0.866025 + 0.500000i 0
313.8 0 0.965926 + 0.258819i 0 −1.93782 − 1.11573i 0 2.10702 + 1.60015i 0 0.866025 + 0.500000i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.dd.b yes 48
5.c odd 4 1 840.2.dd.a ✓ 48
7.d odd 6 1 840.2.dd.a ✓ 48
35.k even 12 1 inner 840.2.dd.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.dd.a ✓ 48 5.c odd 4 1
840.2.dd.a ✓ 48 7.d odd 6 1
840.2.dd.b yes 48 1.a even 1 1 trivial
840.2.dd.b yes 48 35.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{48} + 16 T_{13}^{47} + 128 T_{13}^{46} + 456 T_{13}^{45} + 3786 T_{13}^{44} + \cdots + 1186042971136 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display