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Properties

Label 1218.2.p.b
Level $1218$
Weight $2$
Character orbit 1218.p
Analytic conductor $9.726$
Analytic rank $0$
Dimension $44$
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] = [1218,2,Mod(289,1218)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1218, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) 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("1218.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1218 = 2 \cdot 3 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1218.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] 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: \(9.72577896619\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 22 q^{4} - 44 q^{6} - 8 q^{7} + 22 q^{9} - 16 q^{13} - 22 q^{16} - 4 q^{23} - 22 q^{24} - 22 q^{25} - 4 q^{28} + 4 q^{29} - 16 q^{34} - 10 q^{35} + 44 q^{36} + 4 q^{38} + 8 q^{42} - 52 q^{49} + 8 q^{51}+ \cdots + 22 q^{96}+O(q^{100}) \) 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 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.38285 − 2.39516i −1.00000 −1.23344 + 2.34065i − 1.00000i 0.500000 − 0.866025i −2.39516 + 1.38285i
289.2 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.10904 + 1.92091i −1.00000 1.47129 − 2.19893i − 1.00000i 0.500000 − 0.866025i 1.92091 − 1.10904i
289.3 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 0.197710 − 0.342443i −1.00000 −2.52690 − 0.784063i − 1.00000i 0.500000 − 0.866025i −0.342443 + 0.197710i
289.4 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.09965 + 1.90465i −1.00000 −2.59884 − 0.496020i − 1.00000i 0.500000 − 0.866025i 1.90465 − 1.09965i
289.5 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.83597 − 3.17999i −1.00000 2.42090 + 1.06736i − 1.00000i 0.500000 − 0.866025i −3.17999 + 1.83597i
289.6 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.47473 + 2.55431i −1.00000 1.71941 + 2.01087i − 1.00000i 0.500000 − 0.866025i 2.55431 − 1.47473i
289.7 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 0.110464 − 0.191328i −1.00000 −1.52713 − 2.16053i − 1.00000i 0.500000 − 0.866025i −0.191328 + 0.110464i
289.8 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.79012 + 3.10059i −1.00000 −1.03878 + 2.43330i − 1.00000i 0.500000 − 0.866025i 3.10059 − 1.79012i
289.9 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −0.270202 + 0.468004i −1.00000 1.40998 − 2.23874i − 1.00000i 0.500000 − 0.866025i 0.468004 − 0.270202i
289.10 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 0.469195 − 0.812670i −1.00000 0.642299 + 2.56660i − 1.00000i 0.500000 − 0.866025i −0.812670 + 0.469195i
289.11 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.74756 − 3.02686i −1.00000 −0.738796 − 2.54051i − 1.00000i 0.500000 − 0.866025i −3.02686 + 1.74756i
289.12 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.09965 + 1.90465i −1.00000 −2.59884 − 0.496020i 1.00000i 0.500000 − 0.866025i −1.90465 + 1.09965i
289.13 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.270202 + 0.468004i −1.00000 1.40998 − 2.23874i 1.00000i 0.500000 − 0.866025i −0.468004 + 0.270202i
289.14 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.83597 − 3.17999i −1.00000 2.42090 + 1.06736i 1.00000i 0.500000 − 0.866025i 3.17999 − 1.83597i
289.15 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.74756 − 3.02686i −1.00000 −0.738796 − 2.54051i 1.00000i 0.500000 − 0.866025i 3.02686 − 1.74756i
289.16 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.38285 − 2.39516i −1.00000 −1.23344 + 2.34065i 1.00000i 0.500000 − 0.866025i 2.39516 − 1.38285i
289.17 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.79012 + 3.10059i −1.00000 −1.03878 + 2.43330i 1.00000i 0.500000 − 0.866025i −3.10059 + 1.79012i
289.18 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.110464 − 0.191328i −1.00000 −1.52713 − 2.16053i 1.00000i 0.500000 − 0.866025i 0.191328 − 0.110464i
289.19 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.10904 + 1.92091i −1.00000 1.47129 − 2.19893i 1.00000i 0.500000 − 0.866025i −1.92091 + 1.10904i
289.20 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.47473 + 2.55431i −1.00000 1.71941 + 2.01087i 1.00000i 0.500000 − 0.866025i −2.55431 + 1.47473i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.22
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1218.2.p.b ✓ 44
7.c even 3 1 inner 1218.2.p.b ✓ 44
29.b even 2 1 inner 1218.2.p.b ✓ 44
203.j even 6 1 inner 1218.2.p.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1218.2.p.b ✓ 44 1.a even 1 1 trivial
1218.2.p.b ✓ 44 7.c even 3 1 inner
1218.2.p.b ✓ 44 29.b even 2 1 inner
1218.2.p.b ✓ 44 203.j even 6 1 inner