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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,26,Mod(7,18)] 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("18.7"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), 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: \(71.2794203914\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 49152 q^{2} - 97956 q^{3} - 201326592 q^{4} + 49854096 q^{5} - 6381453312 q^{6} + 34065485256 q^{7} - 1649267441664 q^{8} - 1581889193388 q^{9} + 408404754432 q^{10} - 2177291699892 q^{11} - 24495003795456 q^{12}+ \cdots + 33\!\cdots\!08 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 2048.00 − 3547.24i −918460. − 60989.5i −8.38861e6 − 1.45295e7i −3.54629e8 − 6.14236e8i −2.09735e9 + 3.13309e9i −7.86828e9 + 1.36283e10i −6.87195e10 8.39849e11 + 1.12033e11i −2.90512e12
7.2 2048.00 − 3547.24i −862905. + 320443.i −8.38861e6 − 1.45295e7i 2.85522e8 + 4.94539e8i −6.30542e8 + 3.71720e9i 2.29208e10 − 3.97000e10i −6.87195e10 6.41921e11 − 5.53023e11i 2.33900e12
7.3 2048.00 − 3547.24i −763669. − 513904.i −8.38861e6 − 1.45295e7i 2.55901e8 + 4.43233e8i −3.38694e9 + 1.65644e9i −3.30211e10 + 5.71942e10i −6.87195e10 3.19093e11 + 7.84906e11i 2.09634e12
7.4 2048.00 − 3547.24i −528144. − 753891.i −8.38861e6 − 1.45295e7i 1.05809e8 + 1.83266e8i −3.75587e9 + 3.29485e8i 1.92956e10 − 3.34210e10i −6.87195e10 −2.89416e11 + 7.96327e11i 8.66784e11
7.5 2048.00 − 3547.24i −343285. + 854075.i −8.38861e6 − 1.45295e7i −2.15554e8 − 3.73350e8i 2.32656e9 + 2.96686e9i −2.22521e9 + 3.85417e9i −6.87195e10 −6.11599e11 − 5.86383e11i −1.76582e12
7.6 2048.00 − 3547.24i 141934. + 909474.i −8.38861e6 − 1.45295e7i 3.33118e8 + 5.76976e8i 3.51680e9 + 1.35913e9i −1.03215e10 + 1.78773e10i −6.87195e10 −8.06998e11 + 2.58170e11i 2.72890e12
7.7 2048.00 − 3547.24i 253827. − 884794.i −8.38861e6 − 1.45295e7i −1.95819e8 − 3.39168e8i −2.61874e9 − 2.71244e9i −2.03084e10 + 3.51752e10i −6.87195e10 −7.18432e11 − 4.49170e11i −1.60415e12
7.8 2048.00 − 3547.24i 315565. − 864701.i −8.38861e6 − 1.45295e7i −3.75091e8 − 6.49677e8i −2.42102e9 − 2.89029e9i 2.37877e10 − 4.12015e10i −6.87195e10 −6.48126e11 − 5.45739e11i −3.07275e12
7.9 2048.00 − 3547.24i 335494. − 857165.i −8.38861e6 − 1.45295e7i 5.12811e8 + 8.88214e8i −2.35348e9 − 2.94555e9i 4.05434e9 − 7.02232e9i −6.87195e10 −6.22176e11 − 5.75148e11i 4.20094e12
7.10 2048.00 − 3547.24i 667013. + 634336.i −8.38861e6 − 1.45295e7i 1.71048e8 + 2.96264e8i 3.61618e9 − 1.06694e9i 3.61628e7 − 6.26357e7i −6.87195e10 4.25250e10 + 8.46221e11i 1.40122e12
7.11 2048.00 − 3547.24i 751952. + 530901.i −8.38861e6 − 1.45295e7i −4.49094e8 − 7.77854e8i 3.42323e9 − 1.58007e9i 3.51328e10 − 6.08519e10i −6.87195e10 2.83576e11 + 7.98425e11i −3.67898e12
7.12 2048.00 − 3547.24i 901700. − 185002.i −8.38861e6 − 1.45295e7i −4.90937e7 − 8.50327e7i 1.19044e9 − 3.57743e9i −1.44503e10 + 2.50287e10i −6.87195e10 7.78837e11 − 3.33633e11i −4.02175e11
13.1 2048.00 + 3547.24i −918460. + 60989.5i −8.38861e6 + 1.45295e7i −3.54629e8 + 6.14236e8i −2.09735e9 − 3.13309e9i −7.86828e9 − 1.36283e10i −6.87195e10 8.39849e11 − 1.12033e11i −2.90512e12
13.2 2048.00 + 3547.24i −862905. − 320443.i −8.38861e6 + 1.45295e7i 2.85522e8 − 4.94539e8i −6.30542e8 − 3.71720e9i 2.29208e10 + 3.97000e10i −6.87195e10 6.41921e11 + 5.53023e11i 2.33900e12
13.3 2048.00 + 3547.24i −763669. + 513904.i −8.38861e6 + 1.45295e7i 2.55901e8 − 4.43233e8i −3.38694e9 − 1.65644e9i −3.30211e10 − 5.71942e10i −6.87195e10 3.19093e11 − 7.84906e11i 2.09634e12
13.4 2048.00 + 3547.24i −528144. + 753891.i −8.38861e6 + 1.45295e7i 1.05809e8 − 1.83266e8i −3.75587e9 − 3.29485e8i 1.92956e10 + 3.34210e10i −6.87195e10 −2.89416e11 − 7.96327e11i 8.66784e11
13.5 2048.00 + 3547.24i −343285. − 854075.i −8.38861e6 + 1.45295e7i −2.15554e8 + 3.73350e8i 2.32656e9 − 2.96686e9i −2.22521e9 − 3.85417e9i −6.87195e10 −6.11599e11 + 5.86383e11i −1.76582e12
13.6 2048.00 + 3547.24i 141934. − 909474.i −8.38861e6 + 1.45295e7i 3.33118e8 − 5.76976e8i 3.51680e9 − 1.35913e9i −1.03215e10 − 1.78773e10i −6.87195e10 −8.06998e11 − 2.58170e11i 2.72890e12
13.7 2048.00 + 3547.24i 253827. + 884794.i −8.38861e6 + 1.45295e7i −1.95819e8 + 3.39168e8i −2.61874e9 + 2.71244e9i −2.03084e10 − 3.51752e10i −6.87195e10 −7.18432e11 + 4.49170e11i −1.60415e12
13.8 2048.00 + 3547.24i 315565. + 864701.i −8.38861e6 + 1.45295e7i −3.75091e8 + 6.49677e8i −2.42102e9 + 2.89029e9i 2.37877e10 + 4.12015e10i −6.87195e10 −6.48126e11 + 5.45739e11i −3.07275e12
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.26.c.a ✓ 24
3.b odd 2 1 54.26.c.a 24
9.c even 3 1 inner 18.26.c.a ✓ 24
9.d odd 6 1 54.26.c.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.26.c.a ✓ 24 1.a even 1 1 trivial
18.26.c.a ✓ 24 9.c even 3 1 inner
54.26.c.a 24 3.b odd 2 1
54.26.c.a 24 9.d odd 6 1