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Properties

Label 840.2.da.a
Level $840$
Weight $2$
Character orbit 840.da
Analytic conductor $6.707$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(89,840)] 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, 3, 3, 5])) 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("840.89"); 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.da (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{3} - 3 q^{5} - q^{9} - 14 q^{15} + 5 q^{21} - 2 q^{23} + q^{25} + 6 q^{31} - 24 q^{33} + 4 q^{35} + 2 q^{39} - 3 q^{45} + 12 q^{51} + 6 q^{53} - 20 q^{57} + 18 q^{61} + 26 q^{63} - 10 q^{65}+ \cdots - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 0 −1.71363 + 0.251942i 0 1.98178 + 1.03564i 0 1.54613 − 2.14697i 0 2.87305 − 0.863469i 0
89.2 0 −1.69080 − 0.375748i 0 0.113055 − 2.23321i 0 −2.32178 + 1.26861i 0 2.71763 + 1.27063i 0
89.3 0 −1.65401 − 0.514040i 0 −0.955903 + 2.02145i 0 −1.96308 − 1.77379i 0 2.47153 + 1.70046i 0
89.4 0 −1.59188 − 0.682575i 0 −1.77181 − 1.36407i 0 2.55997 + 0.668258i 0 2.06818 + 2.17316i 0
89.5 0 −1.58278 + 0.703435i 0 −1.36472 + 1.77131i 0 1.08470 + 2.41318i 0 2.01036 − 2.22676i 0
89.6 0 −1.40058 + 1.01901i 0 0.851635 − 2.06754i 0 −1.08470 − 2.41318i 0 0.923253 − 2.85440i 0
89.7 0 −1.07500 + 1.35808i 0 1.88778 + 1.19845i 0 −1.54613 + 2.14697i 0 −0.688739 − 2.91987i 0
89.8 0 −0.898406 − 1.48083i 0 1.98193 − 1.03534i 0 2.34610 − 1.22303i 0 −1.38573 + 2.66078i 0
89.9 0 −0.684539 − 1.59104i 0 1.55313 + 1.60866i 0 −2.47935 + 0.923490i 0 −2.06281 + 2.17826i 0
89.10 0 −0.587907 − 1.62922i 0 −2.19487 − 0.427254i 0 −0.571108 − 2.58338i 0 −2.30873 + 1.91566i 0
89.11 0 −0.519994 + 1.65215i 0 −1.87749 + 1.21451i 0 2.32178 − 1.26861i 0 −2.45921 − 1.71822i 0
89.12 0 −0.381836 + 1.68944i 0 1.27267 − 1.83856i 0 1.96308 + 1.77379i 0 −2.70840 − 1.29018i 0
89.13 0 −0.204814 + 1.71990i 0 −2.06723 − 0.852393i 0 −2.55997 − 0.668258i 0 −2.91610 − 0.704519i 0
89.14 0 0.0583061 − 1.73107i 0 1.18105 − 1.89872i 0 −0.693447 + 2.55326i 0 −2.99320 − 0.201864i 0
89.15 0 0.395473 − 1.68630i 0 −2.06804 + 0.850426i 0 0.514896 + 2.59517i 0 −2.68720 − 1.33377i 0
89.16 0 0.833236 + 1.51846i 0 0.0943325 + 2.23408i 0 −2.34610 + 1.22303i 0 −1.61144 + 2.53047i 0
89.17 0 1.03561 + 1.38835i 0 2.16971 + 0.540722i 0 2.47935 − 0.923490i 0 −0.855021 + 2.87558i 0
89.18 0 1.11699 + 1.32375i 0 −1.46745 − 1.68719i 0 0.571108 + 2.58338i 0 −0.504648 + 2.95725i 0
89.19 0 1.27083 − 1.17686i 0 2.23332 + 0.110759i 0 −1.74016 − 1.99295i 0 0.230007 − 2.99117i 0
89.20 0 1.28768 − 1.15840i 0 −0.720646 − 2.11676i 0 2.62364 − 0.341357i 0 0.316218 − 2.98329i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p 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.da.a ✓ 48
3.b odd 2 1 840.2.da.b yes 48
5.b even 2 1 840.2.da.b yes 48
7.d odd 6 1 inner 840.2.da.a ✓ 48
15.d odd 2 1 inner 840.2.da.a ✓ 48
21.g even 6 1 840.2.da.b yes 48
35.i odd 6 1 840.2.da.b yes 48
105.p even 6 1 inner 840.2.da.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.da.a ✓ 48 1.a even 1 1 trivial
840.2.da.a ✓ 48 7.d odd 6 1 inner
840.2.da.a ✓ 48 15.d odd 2 1 inner
840.2.da.a ✓ 48 105.p even 6 1 inner
840.2.da.b yes 48 3.b odd 2 1
840.2.da.b yes 48 5.b even 2 1
840.2.da.b yes 48 21.g even 6 1
840.2.da.b yes 48 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{24} - 97 T_{17}^{22} + 5953 T_{17}^{20} + 1464 T_{17}^{19} - 220468 T_{17}^{18} + \cdots + 1695412326400 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display