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Properties

Label 18.21.d.a
Level $18$
Weight $21$
Character orbit 18.d
Analytic conductor $45.632$
Analytic rank $0$
Dimension $40$
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] = [18,21,Mod(5,18)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 21, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18.5"); S:= CuspForms(chi, 21); N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.6324777185\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 18846 q^{3} + 10485760 q^{4} + 29763918 q^{5} + 102690816 q^{6} + 133479866 q^{7} + 967310598 q^{9} - 35793208728 q^{11} + 26559381504 q^{12} + 39827158550 q^{13} + 187564400640 q^{14} - 965812538286 q^{15}+ \cdots + 52\!\cdots\!22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −627.069 + 362.039i −29056.0 − 51405.6i 262144. − 454047.i 1.35661e7 + 7.83237e6i 3.68309e7 + 2.17155e7i −1.44553e8 − 2.50374e8i 3.79625e8i −1.79828e9 + 2.98728e9i −1.13425e10
5.2 −627.069 + 362.039i 57522.4 − 13339.9i 262144. − 454047.i 8.06063e6 + 4.65381e6i −3.12410e7 + 2.91904e7i −1.01532e8 − 1.75858e8i 3.79625e8i 3.13088e9 − 1.53468e9i −6.73943e9
5.3 −627.069 + 362.039i −3428.09 + 58949.4i 262144. − 454047.i 2.28544e6 + 1.31950e6i −1.91923e7 − 3.82065e7i −2.28670e8 − 3.96069e8i 3.79625e8i −3.46328e9 − 4.04167e8i −1.91084e9
5.4 −627.069 + 362.039i −21083.2 − 55156.9i 262144. − 454047.i −667393. − 385320.i 3.31896e7 + 2.69543e7i 2.18628e8 + 3.78675e8i 3.79625e8i −2.59778e9 + 2.32577e9i 5.58002e8
5.5 −627.069 + 362.039i −39906.5 + 43523.0i 262144. − 454047.i −5.93247e6 − 3.42511e6i 9.26713e6 − 4.17397e7i 1.73104e8 + 2.99825e8i 3.79625e8i −3.01725e8 − 3.47371e9i 4.96010e9
5.6 −627.069 + 362.039i 27305.7 − 52356.3i 262144. − 454047.i −7.64335e6 − 4.41289e6i 1.83243e6 + 4.27168e7i −7.20445e7 − 1.24785e8i 3.79625e8i −1.99558e9 − 2.85925e9i 6.39055e9
5.7 −627.069 + 362.039i −56191.2 + 18147.7i 262144. − 454047.i 7.36161e6 + 4.25023e6i 2.86656e7 − 3.17232e7i −1.38813e7 − 2.40432e7i 3.79625e8i 2.82811e9 − 2.03948e9i −6.15499e9
5.8 −627.069 + 362.039i 53724.5 + 24504.3i 262144. − 454047.i −8.59905e6 − 4.96467e6i −4.25605e7 + 4.08445e6i 1.75216e7 + 3.03482e7i 3.79625e8i 2.28586e9 + 2.63296e9i 7.18960e9
5.9 −627.069 + 362.039i 28178.4 + 51891.8i 262144. − 454047.i 1.35635e7 + 7.83088e6i −3.64567e7 − 2.23381e7i 2.49396e8 + 4.31967e8i 3.79625e8i −1.89874e9 + 2.92446e9i −1.13403e10
5.10 −627.069 + 362.039i −55730.0 − 19518.0i 262144. − 454047.i −1.45540e7 − 8.40275e6i 4.20128e7 − 7.93729e6i −1.01988e8 − 1.76649e8i 3.79625e8i 2.72488e9 + 2.17547e9i 1.21685e10
5.11 627.069 − 362.039i 37919.5 + 45264.8i 262144. − 454047.i 1.57165e7 + 9.07392e6i 4.01657e7 + 1.46558e7i −1.29309e8 − 2.23969e8i − 3.79625e8i −6.11012e8 + 3.43283e9i 1.31404e10
5.12 627.069 − 362.039i 50872.9 − 29978.9i 262144. − 454047.i −1.42842e7 − 8.24698e6i 2.10473e7 − 3.72168e7i −5.08606e6 − 8.80931e6i − 3.79625e8i 1.68932e9 − 3.05022e9i −1.19429e10
5.13 627.069 − 362.039i −58768.8 − 5745.70i 262144. − 454047.i 1.51936e7 + 8.77204e6i −3.89323e7 + 1.76736e7i 2.43968e8 + 4.22564e8i − 3.79625e8i 3.42076e9 + 6.75336e8i 1.27033e10
5.14 627.069 − 362.039i −40381.2 − 43083.0i 262144. − 454047.i −1.17375e7 − 6.77667e6i −4.09195e7 − 1.23965e7i 3.57156e7 + 6.18612e7i − 3.79625e8i −2.25502e8 + 3.47948e9i −9.81366e9
5.15 627.069 − 362.039i −2376.61 + 59001.2i 262144. − 454047.i −6.58251e6 − 3.80041e6i 1.98704e7 + 3.78582e7i 7.46524e7 + 1.29302e8i − 3.79625e8i −3.47549e9 − 2.80445e8i −5.50358e9
5.16 627.069 − 362.039i −20088.3 + 55527.0i 262144. − 454047.i 2.60680e6 + 1.50504e6i 7.50614e6 + 4.20920e7i 3.31729e7 + 5.74572e7i − 3.79625e8i −2.67970e9 − 2.23089e9i 2.17953e9
5.17 627.069 − 362.039i −56785.7 + 16191.7i 262144. − 454047.i −2.17822e6 − 1.25760e6i −2.97465e7 + 3.07119e7i −1.62510e8 − 2.81476e8i − 3.79625e8i 2.96244e9 − 1.83892e9i −1.82119e9
5.18 627.069 − 362.039i 54030.9 + 23821.1i 262144. − 454047.i −964142. − 556647.i 4.25053e7 − 4.62377e6i 1.11151e8 + 1.92519e8i − 3.79625e8i 2.35189e9 + 2.57415e9i −8.06111e8
5.19 627.069 − 362.039i 47482.1 − 35103.2i 262144. − 454047.i 5.33096e6 + 3.07783e6i 1.70658e7 − 3.92025e7i −2.63403e8 − 4.56228e8i − 3.79625e8i 1.02231e9 − 3.33355e9i 4.45718e9
5.20 627.069 − 362.039i 17336.1 − 56446.8i 262144. − 454047.i 4.33970e6 + 2.50553e6i −9.56498e6 − 4.16724e7i 1.32408e8 + 2.29337e8i − 3.79625e8i −2.88570e9 − 1.95714e9i 3.62839e9
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.21.d.a ✓ 40
3.b odd 2 1 54.21.d.a 40
9.c even 3 1 54.21.d.a 40
9.d odd 6 1 inner 18.21.d.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.21.d.a ✓ 40 1.a even 1 1 trivial
18.21.d.a ✓ 40 9.d odd 6 1 inner
54.21.d.a 40 3.b odd 2 1
54.21.d.a 40 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{21}^{\mathrm{new}}(18, [\chi])\).