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Properties

Label 126.16.k.a
Level $126$
Weight $16$
Character orbit 126.k
Analytic conductor $179.794$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,16,Mod(17,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.17");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 126.k (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(179.793816426\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 655360 q^{4} + 3791920 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 655360 q^{4} + 3791920 q^{7} + 102707712 q^{10} - 10737418240 q^{16} - 5231980980 q^{19} - 76084435968 q^{22} - 166454663812 q^{25} + 69552373760 q^{28} - 983657992368 q^{31} + 1236827709140 q^{37} + 1682763153408 q^{40} - 20620202500312 q^{43} - 5919750832128 q^{46} - 24566041364224 q^{49} + 4760594743296 q^{52} + 8623175483904 q^{58} + 98122426115520 q^{61} - 351843720888320 q^{64} + 74357461762756 q^{67} + 45826556232192 q^{70} + 715181167398996 q^{73} - 191003605272536 q^{79} - 11\!\cdots\!64 q^{82}+ \cdots + 608503530304512 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −110.851 + 64.0000i 0 8192.00 − 14189.0i 146320. + 253434.i 0 377282. + 2.14598e6i 2.09715e6i 0 −3.24395e7 − 1.87289e7i
17.2 −110.851 + 64.0000i 0 8192.00 − 14189.0i 133299. + 230881.i 0 1.11200e6 + 1.87377e6i 2.09715e6i 0 −2.95528e7 − 1.70623e7i
17.3 −110.851 + 64.0000i 0 8192.00 − 14189.0i 109363. + 189423.i 0 −2.16641e6 + 232865.i 2.09715e6i 0 −2.42461e7 − 1.39985e7i
17.4 −110.851 + 64.0000i 0 8192.00 − 14189.0i 103181. + 178715.i 0 1.56649e6 − 1.51449e6i 2.09715e6i 0 −2.28755e7 − 1.32072e7i
17.5 −110.851 + 64.0000i 0 8192.00 − 14189.0i −62040.1 − 107457.i 0 1.92348e6 + 1.02361e6i 2.09715e6i 0 1.37544e7 + 7.94113e6i
17.6 −110.851 + 64.0000i 0 8192.00 − 14189.0i 61647.7 + 106777.i 0 −624733. − 2.08741e6i 2.09715e6i 0 −1.36674e7 − 7.89091e6i
17.7 −110.851 + 64.0000i 0 8192.00 − 14189.0i −62366.9 − 108023.i 0 −1.38221e6 − 1.68436e6i 2.09715e6i 0 1.38269e7 + 7.98296e6i
17.8 −110.851 + 64.0000i 0 8192.00 − 14189.0i 61146.5 + 105909.i 0 319430. + 2.15535e6i 2.09715e6i 0 −1.35563e7 − 7.82675e6i
17.9 −110.851 + 64.0000i 0 8192.00 − 14189.0i 57865.5 + 100226.i 0 1.77691e6 − 1.26101e6i 2.09715e6i 0 −1.28289e7 − 7.40678e6i
17.10 −110.851 + 64.0000i 0 8192.00 − 14189.0i 55779.7 + 96613.3i 0 2.00300e6 − 857649.i 2.09715e6i 0 −1.23665e7 − 7.13980e6i
17.11 −110.851 + 64.0000i 0 8192.00 − 14189.0i −20615.2 − 35706.6i 0 −2.17696e6 + 91747.2i 2.09715e6i 0 4.57045e6 + 2.63875e6i
17.12 −110.851 + 64.0000i 0 8192.00 − 14189.0i −39931.7 − 69163.8i 0 2.16388e6 − 255285.i 2.09715e6i 0 8.85297e6 + 5.11126e6i
17.13 −110.851 + 64.0000i 0 8192.00 − 14189.0i −47896.7 − 82959.5i 0 −946305. − 1.96267e6i 2.09715e6i 0 1.06188e7 + 6.13078e6i
17.14 −110.851 + 64.0000i 0 8192.00 − 14189.0i 67456.1 + 116837.i 0 −2.02081e6 + 814796.i 2.09715e6i 0 −1.49552e7 − 8.63439e6i
17.15 −110.851 + 64.0000i 0 8192.00 − 14189.0i −77338.2 − 133954.i 0 −663635. + 2.07537e6i 2.09715e6i 0 1.71461e7 + 9.89929e6i
17.16 −110.851 + 64.0000i 0 8192.00 − 14189.0i −92338.7 − 159935.i 0 333681. − 2.15319e6i 2.09715e6i 0 2.04717e7 + 1.18194e7i
17.17 −110.851 + 64.0000i 0 8192.00 − 14189.0i −103813. − 179809.i 0 −1.56294e6 + 1.51815e6i 2.09715e6i 0 2.30156e7 + 1.32881e7i
17.18 −110.851 + 64.0000i 0 8192.00 − 14189.0i −111390. − 192933.i 0 −1.03310e6 + 1.91840e6i 2.09715e6i 0 2.46954e7 + 1.42579e7i
17.19 −110.851 + 64.0000i 0 8192.00 − 14189.0i −141845. − 245683.i 0 2.03396e6 + 781390.i 2.09715e6i 0 3.14474e7 + 1.81561e7i
17.20 −110.851 + 64.0000i 0 8192.00 − 14189.0i −152301. − 263794.i 0 −85034.5 − 2.17723e6i 2.09715e6i 0 3.37656e7 + 1.94946e7i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.16.k.a ✓ 80
3.b odd 2 1 inner 126.16.k.a ✓ 80
7.d odd 6 1 inner 126.16.k.a ✓ 80
21.g even 6 1 inner 126.16.k.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.16.k.a ✓ 80 1.a even 1 1 trivial
126.16.k.a ✓ 80 3.b odd 2 1 inner
126.16.k.a ✓ 80 7.d odd 6 1 inner
126.16.k.a ✓ 80 21.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(126, [\chi])\).