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Properties

Label 117.4.ba.a
Level $117$
Weight $4$
Character orbit 117.ba
Analytic conductor $6.903$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,4,Mod(71,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.71");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.ba (of order \(12\), degree \(4\), 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: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{7} + 72 q^{10} + 180 q^{13} + 304 q^{16} - 56 q^{19} - 432 q^{22} + 344 q^{28} - 136 q^{31} + 48 q^{34} - 2020 q^{37} - 1776 q^{40} - 840 q^{43} + 336 q^{46} + 5976 q^{49} + 2336 q^{52} + 2016 q^{55} - 2616 q^{58} - 276 q^{61} + 1600 q^{67} - 4440 q^{70} - 4196 q^{73} - 9776 q^{76} - 5112 q^{82} - 6600 q^{85} + 16848 q^{88} + 12520 q^{91} + 6552 q^{94} + 452 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −4.80615 + 1.28780i 0 14.5125 − 8.37877i 13.7475 + 13.7475i 0 3.89700 − 14.5438i −30.8121 + 30.8121i 0 −83.7764 − 48.3683i
71.2 −4.78440 + 1.28198i 0 14.3188 − 8.26696i −3.63780 − 3.63780i 0 2.21462 − 8.26507i −29.8894 + 29.8894i 0 22.0682 + 12.7411i
71.3 −4.15264 + 1.11270i 0 9.07815 − 5.24127i −13.4564 − 13.4564i 0 −1.13368 + 4.23095i −7.54679 + 7.54679i 0 70.8526 + 40.9068i
71.4 −2.54603 + 0.682207i 0 −0.911329 + 0.526156i 12.8112 + 12.8112i 0 3.35243 − 12.5114i 16.8719 − 16.8719i 0 −41.3576 − 23.8778i
71.5 −2.25514 + 0.604263i 0 −2.20768 + 1.27460i −5.63980 − 5.63980i 0 −6.04890 + 22.5748i 17.4154 − 17.4154i 0 16.1265 + 9.31062i
71.6 −1.66521 + 0.446191i 0 −4.35438 + 2.51400i 2.52869 + 2.52869i 0 −3.79091 + 14.1479i 15.8813 − 15.8813i 0 −5.33907 − 3.08251i
71.7 −0.893421 + 0.239391i 0 −6.18731 + 3.57225i 0.663107 + 0.663107i 0 7.43765 − 27.7577i 9.90494 − 9.90494i 0 −0.751176 − 0.433692i
71.8 0.893421 − 0.239391i 0 −6.18731 + 3.57225i −0.663107 − 0.663107i 0 7.43765 − 27.7577i −9.90494 + 9.90494i 0 −0.751176 − 0.433692i
71.9 1.66521 − 0.446191i 0 −4.35438 + 2.51400i −2.52869 − 2.52869i 0 −3.79091 + 14.1479i −15.8813 + 15.8813i 0 −5.33907 − 3.08251i
71.10 2.25514 − 0.604263i 0 −2.20768 + 1.27460i 5.63980 + 5.63980i 0 −6.04890 + 22.5748i −17.4154 + 17.4154i 0 16.1265 + 9.31062i
71.11 2.54603 − 0.682207i 0 −0.911329 + 0.526156i −12.8112 − 12.8112i 0 3.35243 − 12.5114i −16.8719 + 16.8719i 0 −41.3576 − 23.8778i
71.12 4.15264 − 1.11270i 0 9.07815 − 5.24127i 13.4564 + 13.4564i 0 −1.13368 + 4.23095i 7.54679 − 7.54679i 0 70.8526 + 40.9068i
71.13 4.78440 − 1.28198i 0 14.3188 − 8.26696i 3.63780 + 3.63780i 0 2.21462 − 8.26507i 29.8894 − 29.8894i 0 22.0682 + 12.7411i
71.14 4.80615 − 1.28780i 0 14.5125 − 8.37877i −13.7475 − 13.7475i 0 3.89700 − 14.5438i 30.8121 − 30.8121i 0 −83.7764 − 48.3683i
80.1 −1.38181 + 5.15699i 0 −17.7570 − 10.2520i −3.17239 − 3.17239i 0 −22.8022 + 6.10984i 47.2049 − 47.2049i 0 20.7436 − 11.9763i
80.2 −1.11928 + 4.17719i 0 −9.26796 − 5.35086i 6.96987 + 6.96987i 0 −18.1013 + 4.85023i 8.26162 − 8.26162i 0 −36.9157 + 21.3133i
80.3 −1.07432 + 4.00943i 0 −7.99316 − 4.61485i −13.4826 − 13.4826i 0 22.6125 − 6.05901i 3.60932 − 3.60932i 0 68.5422 − 39.5729i
80.4 −0.894170 + 3.33709i 0 −3.40842 − 1.96785i 5.11575 + 5.11575i 0 35.3495 − 9.47186i −9.92874 + 9.92874i 0 −21.6461 + 12.4974i
80.5 −0.655246 + 2.44541i 0 1.37752 + 0.795309i −6.42816 − 6.42816i 0 −7.98757 + 2.14026i −17.1688 + 17.1688i 0 19.9315 − 11.5075i
80.6 −0.279542 + 1.04327i 0 5.91794 + 3.41673i 5.15692 + 5.15692i 0 10.5702 − 2.83228i −11.3287 + 11.3287i 0 −6.82162 + 3.93846i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.ba.a ✓ 56
3.b odd 2 1 inner 117.4.ba.a ✓ 56
13.f odd 12 1 inner 117.4.ba.a ✓ 56
39.k even 12 1 inner 117.4.ba.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.ba.a ✓ 56 1.a even 1 1 trivial
117.4.ba.a ✓ 56 3.b odd 2 1 inner
117.4.ba.a ✓ 56 13.f odd 12 1 inner
117.4.ba.a ✓ 56 39.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(117, [\chi])\).