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Properties

Label 444.2.bb.b
Level $444$
Weight $2$
Character orbit 444.bb
Analytic conductor $3.545$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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] = [444,2,Mod(25,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.bb (of order \(18\), degree \(6\), 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{7} + 6 q^{11} - 12 q^{13} - 15 q^{17} + 12 q^{19} + 3 q^{21} + 18 q^{25} + 12 q^{27} - 9 q^{29} + 18 q^{33} + 12 q^{35} + 12 q^{37} + 15 q^{39} + 15 q^{41} - 15 q^{47} + 15 q^{49} + 9 q^{51}+ \cdots + 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −0.766044 − 0.642788i 0 −1.30455 − 3.58423i 0 1.22792 − 0.446926i 0 0.173648 + 0.984808i 0
25.2 0 −0.766044 − 0.642788i 0 −0.104810 − 0.287962i 0 −3.56341 + 1.29697i 0 0.173648 + 0.984808i 0
25.3 0 −0.766044 − 0.642788i 0 0.322459 + 0.885950i 0 2.90162 − 1.05610i 0 0.173648 + 0.984808i 0
25.4 0 −0.766044 − 0.642788i 0 1.08690 + 2.98625i 0 −3.88521 + 1.41410i 0 0.173648 + 0.984808i 0
169.1 0 0.939693 − 0.342020i 0 −3.49983 − 0.617115i 0 −0.659784 + 3.74182i 0 0.766044 − 0.642788i 0
169.2 0 0.939693 − 0.342020i 0 −1.69595 − 0.299042i 0 0.760862 − 4.31506i 0 0.766044 − 0.642788i 0
169.3 0 0.939693 − 0.342020i 0 0.993249 + 0.175137i 0 0.0807073 − 0.457714i 0 0.766044 − 0.642788i 0
169.4 0 0.939693 − 0.342020i 0 4.20253 + 0.741020i 0 −0.160840 + 0.912170i 0 0.766044 − 0.642788i 0
289.1 0 0.939693 + 0.342020i 0 −3.49983 + 0.617115i 0 −0.659784 − 3.74182i 0 0.766044 + 0.642788i 0
289.2 0 0.939693 + 0.342020i 0 −1.69595 + 0.299042i 0 0.760862 + 4.31506i 0 0.766044 + 0.642788i 0
289.3 0 0.939693 + 0.342020i 0 0.993249 − 0.175137i 0 0.0807073 + 0.457714i 0 0.766044 + 0.642788i 0
289.4 0 0.939693 + 0.342020i 0 4.20253 − 0.741020i 0 −0.160840 − 0.912170i 0 0.766044 + 0.642788i 0
337.1 0 −0.173648 − 0.984808i 0 −2.25617 − 2.68880i 0 0.880410 − 0.738752i 0 −0.939693 + 0.342020i 0
337.2 0 −0.173648 − 0.984808i 0 −0.379753 − 0.452572i 0 −2.95963 + 2.48342i 0 −0.939693 + 0.342020i 0
337.3 0 −0.173648 − 0.984808i 0 0.603708 + 0.719471i 0 −0.0528059 + 0.0443094i 0 −0.939693 + 0.342020i 0
337.4 0 −0.173648 − 0.984808i 0 2.03222 + 2.42190i 0 3.93016 − 3.29779i 0 −0.939693 + 0.342020i 0
361.1 0 −0.173648 + 0.984808i 0 −2.25617 + 2.68880i 0 0.880410 + 0.738752i 0 −0.939693 − 0.342020i 0
361.2 0 −0.173648 + 0.984808i 0 −0.379753 + 0.452572i 0 −2.95963 − 2.48342i 0 −0.939693 − 0.342020i 0
361.3 0 −0.173648 + 0.984808i 0 0.603708 − 0.719471i 0 −0.0528059 − 0.0443094i 0 −0.939693 − 0.342020i 0
361.4 0 −0.173648 + 0.984808i 0 2.03222 − 2.42190i 0 3.93016 + 3.29779i 0 −0.939693 − 0.342020i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 25.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.bb.b ✓ 24
3.b odd 2 1 1332.2.ct.e 24
37.h even 18 1 inner 444.2.bb.b ✓ 24
111.n odd 18 1 1332.2.ct.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.bb.b ✓ 24 1.a even 1 1 trivial
444.2.bb.b ✓ 24 37.h even 18 1 inner
1332.2.ct.e 24 3.b odd 2 1
1332.2.ct.e 24 111.n odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 9 T_{5}^{22} - 36 T_{5}^{21} - 60 T_{5}^{20} + 270 T_{5}^{19} - 1079 T_{5}^{18} + \cdots + 322624 \) acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\). Copy content Toggle raw display