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Properties

Label 117.4.r.a
Level $117$
Weight $4$
Character orbit 117.r
Analytic conductor $6.903$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,4,Mod(43,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.43");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.r (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: \(6.90322347067\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 3 q^{2} - q^{3} + 153 q^{4} - 3 q^{6} - 17 q^{9} - 10 q^{10} - 3 q^{11} - 101 q^{12} - 13 q^{13} + 126 q^{14} - 84 q^{15} - 551 q^{16} - 138 q^{17} + 168 q^{18} - 96 q^{19} + 249 q^{21} - 31 q^{22}+ \cdots + 2331 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −4.66466 + 2.69314i −5.17990 + 0.410710i 10.5060 − 18.1970i −8.96369 + 5.17519i 23.0563 − 15.8660i − 22.5801i 70.0867i 26.6626 − 4.25487i 27.8750 − 48.2810i
43.2 −4.65052 + 2.68498i 1.15725 + 5.06565i 10.4182 − 18.0449i 10.1662 − 5.86946i −18.9830 − 20.4507i − 8.42000i 68.9311i −24.3215 + 11.7244i −31.5187 + 54.5920i
43.3 −4.60227 + 2.65712i 4.46766 − 2.65330i 10.1206 − 17.5294i −6.42468 + 3.70929i −13.5113 + 24.0823i 9.45757i 65.0530i 12.9200 − 23.7081i 19.7121 − 34.1423i
43.4 −4.03088 + 2.32723i 0.353584 − 5.18411i 6.83199 − 11.8334i 16.7449 − 9.66766i 10.6394 + 21.7194i − 10.0132i 26.3627i −26.7500 − 3.66603i −44.9977 + 77.9384i
43.5 −3.77028 + 2.17677i −4.76526 − 2.07178i 5.47669 − 9.48590i 7.24527 − 4.18306i 22.4762 − 2.56172i 21.7451i 12.8576i 18.4155 + 19.7451i −18.2111 + 31.5426i
43.6 −3.59142 + 2.07351i −1.69070 + 4.91340i 4.59887 − 7.96548i −15.6155 + 9.01561i −4.11595 − 21.1518i 28.4248i 4.96706i −21.2830 − 16.6142i 37.3879 − 64.7577i
43.7 −3.59109 + 2.07331i 4.58787 + 2.43956i 4.59727 − 7.96270i 4.17848 − 2.41245i −21.5334 + 0.751409i 15.5822i 4.95329i 15.0971 + 22.3848i −10.0035 + 17.3266i
43.8 −3.44134 + 1.98686i −0.951787 − 5.10824i 3.89522 − 6.74672i −9.30367 + 5.37147i 13.4248 + 15.6881i − 16.8418i − 0.832741i −25.1882 + 9.72391i 21.3447 − 36.9702i
43.9 −3.39094 + 1.95776i −3.85298 + 3.48634i 3.66566 − 6.34911i 5.36534 − 3.09768i 6.23981 − 19.3652i 7.97770i − 2.61824i 2.69089 − 26.8656i −12.1290 + 21.0081i
43.10 −3.17897 + 1.83538i 4.08181 + 3.21540i 2.73723 − 4.74102i −15.6805 + 9.05312i −18.8774 − 2.72998i − 31.1968i − 9.27065i 6.32242 + 26.2493i 33.2318 − 57.5592i
43.11 −2.45287 + 1.41616i 3.66210 − 3.68633i 0.0110460 − 0.0191322i −5.22947 + 3.01924i −3.76221 + 14.2282i 11.1486i − 22.5961i −0.178006 − 26.9994i 8.55148 − 14.8116i
43.12 −2.40967 + 1.39123i −1.40720 + 5.00198i −0.128983 + 0.223404i 7.52439 − 4.34421i −3.56799 − 14.0109i − 26.2259i − 22.9774i −23.0396 − 14.0776i −12.0876 + 20.9363i
43.13 −1.99949 + 1.15440i −5.03465 − 1.28542i −1.33470 + 2.31177i 4.82576 − 2.78615i 11.5506 − 3.24185i − 19.2427i − 24.6336i 23.6954 + 12.9432i −6.43269 + 11.1418i
43.14 −1.96456 + 1.13424i 5.09919 − 0.999155i −1.42699 + 2.47162i 13.8137 − 7.97537i −8.88439 + 7.74661i − 9.23062i − 24.6221i 25.0034 − 10.1898i −18.0920 + 31.3362i
43.15 −1.89505 + 1.09411i −3.65737 − 3.69102i −1.60586 + 2.78143i −12.2426 + 7.06827i 10.9693 + 2.99312i 17.6600i − 24.5336i −0.247313 + 26.9989i 15.4669 − 26.7895i
43.16 −1.05904 + 0.611437i 3.21920 + 4.07882i −3.25229 + 5.63313i −0.324345 + 0.187261i −5.90320 − 2.35129i 12.1253i − 17.7373i −6.27348 + 26.2611i 0.228996 − 0.396633i
43.17 −0.744609 + 0.429900i 0.340197 − 5.18500i −3.63037 + 6.28799i 9.13119 − 5.27190i 1.97572 + 4.00705i 33.8282i − 13.1212i −26.7685 − 3.52784i −4.53278 + 7.85101i
43.18 −0.507395 + 0.292945i −4.95248 + 1.57256i −3.82837 + 6.63093i −12.8183 + 7.40065i 2.05219 − 2.24871i − 2.87637i − 9.17312i 22.0541 − 15.5762i 4.33596 − 7.51011i
43.19 −0.434777 + 0.251018i −0.0491927 + 5.19592i −3.87398 + 6.70993i −5.67184 + 3.27464i −1.28288 − 2.27141i 5.82604i − 7.90605i −26.9952 − 0.511203i 1.64399 − 2.84747i
43.20 −0.379429 + 0.219063i −4.38499 + 2.78780i −3.90402 + 6.76197i 14.8736 − 8.58728i 1.05309 − 2.01836i 27.3751i − 6.92592i 11.4563 − 24.4490i −3.76232 + 6.51652i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.r.a yes 80
3.b odd 2 1 351.4.r.a 80
9.c even 3 1 117.4.l.a ✓ 80
9.d odd 6 1 351.4.l.a 80
13.e even 6 1 117.4.l.a ✓ 80
39.h odd 6 1 351.4.l.a 80
117.m odd 6 1 351.4.r.a 80
117.r even 6 1 inner 117.4.r.a yes 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.l.a ✓ 80 9.c even 3 1
117.4.l.a ✓ 80 13.e even 6 1
117.4.r.a yes 80 1.a even 1 1 trivial
117.4.r.a yes 80 117.r even 6 1 inner
351.4.l.a 80 9.d odd 6 1
351.4.l.a 80 39.h odd 6 1
351.4.r.a 80 3.b odd 2 1
351.4.r.a 80 117.m odd 6 1

Hecke kernels

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