↑

Properties

Label 187.3.o.a
Level $187$
Weight $3$
Character orbit 187.o
Analytic conductor $5.095$
Analytic rank $0$
Dimension $272$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [187,3,Mod(13,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 187.o (of order \(20\), degree \(8\), 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: \(5.09538094354\)
Analytic rank: \(0\)
Dimension: \(272\)
Relative dimension: \(34\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q - 14 q^{3} - 140 q^{4} - 2 q^{5} - 50 q^{6} - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 272 q - 14 q^{3} - 140 q^{4} - 2 q^{5} - 50 q^{6} - 40 q^{7} - 14 q^{11} - 12 q^{12} - 80 q^{13} + 18 q^{14} - 204 q^{16} + 70 q^{17} - 20 q^{18} + 106 q^{20} - 80 q^{22} + 112 q^{23} + 450 q^{24} - 200 q^{27} - 10 q^{28} + 150 q^{29} - 20 q^{30} - 62 q^{31} - 440 q^{33} - 192 q^{34} - 20 q^{35} + 150 q^{37} + 416 q^{38} - 330 q^{39} + 360 q^{40} - 280 q^{41} - 112 q^{44} - 212 q^{45} - 290 q^{46} - 8 q^{47} - 34 q^{48} + 960 q^{50} + 630 q^{51} + 140 q^{52} - 716 q^{55} + 900 q^{56} - 550 q^{57} + 174 q^{58} + 420 q^{61} - 740 q^{62} + 920 q^{63} - 324 q^{64} - 368 q^{67} - 450 q^{68} - 96 q^{69} - 382 q^{71} - 500 q^{72} - 10 q^{73} - 660 q^{74} - 514 q^{75} - 1032 q^{78} + 640 q^{79} + 1096 q^{80} - 640 q^{81} + 390 q^{82} - 660 q^{84} + 760 q^{85} - 12 q^{86} + 456 q^{88} + 240 q^{89} - 780 q^{90} + 482 q^{91} + 394 q^{92} - 200 q^{95} + 540 q^{96} - 764 q^{97} - 172 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.17810 + 3.62581i −3.43440 − 0.543956i −8.52252 − 6.19197i 1.67141 − 3.28033i 6.01834 − 11.8117i 0.487466 + 3.07774i 20.1541 − 14.6428i 2.93973 + 0.955176i 9.92478 + 9.92478i
13.2 −1.16781 + 3.59416i 2.73433 + 0.433076i −8.31813 − 6.04348i 2.27141 − 4.45789i −4.74974 + 9.32188i −1.53477 − 9.69014i 19.2058 − 13.9538i −1.27048 − 0.412805i 13.3698 + 13.3698i
13.3 −1.04166 + 3.20591i 5.00182 + 0.792210i −5.95674 − 4.32783i −3.03955 + 5.96546i −7.74997 + 15.2102i 0.705295 + 4.45306i 9.17112 − 6.66321i 15.8311 + 5.14383i −15.9586 − 15.9586i
13.4 −1.03284 + 3.17875i −2.44733 − 0.387619i −5.80164 − 4.21514i −3.80650 + 7.47068i 3.75984 − 7.37910i 0.527319 + 3.32936i 8.57499 − 6.23010i −2.72034 − 0.883891i −19.8159 − 19.8159i
13.5 −1.01941 + 3.13743i 1.68724 + 0.267232i −5.56820 − 4.04553i 0.683950 − 1.34233i −2.55841 + 5.02117i 1.52138 + 9.60562i 7.69344 − 5.58961i −5.78415 − 1.87938i 3.51423 + 3.51423i
13.6 −0.891706 + 2.74439i 0.471614 + 0.0746963i −3.50046 − 2.54324i −2.51282 + 4.93168i −0.625536 + 1.22768i −1.53113 − 9.66720i 0.762947 − 0.554313i −8.34267 − 2.71070i −11.2937 − 11.2937i
13.7 −0.887624 + 2.73182i −3.75198 − 0.594255i −3.43892 − 2.49852i 0.668014 − 1.31105i 4.95375 − 9.72227i −0.613939 − 3.87626i 0.582676 − 0.423339i 5.16470 + 1.67811i 2.98862 + 2.98862i
13.8 −0.706837 + 2.17542i 3.73488 + 0.591547i −0.996766 − 0.724193i 3.58951 − 7.04481i −3.92682 + 7.70681i 0.0358166 + 0.226137i −5.12212 + 3.72144i 5.03991 + 1.63757i 12.7882 + 12.7882i
13.9 −0.664441 + 2.04494i −1.13657 − 0.180015i −0.504227 − 0.366342i 1.88560 − 3.70069i 1.12330 − 2.20461i −0.891394 − 5.62804i −5.87394 + 4.26767i −7.30012 − 2.37195i 6.31482 + 6.31482i
13.10 −0.590418 + 1.81712i −4.31224 − 0.682991i 0.282734 + 0.205418i 3.69818 − 7.25808i 3.78710 − 7.43261i 1.31956 + 8.33137i −6.72314 + 4.88465i 9.56940 + 3.10929i 11.0053 + 11.0053i
13.11 −0.543806 + 1.67366i −4.96351 − 0.786142i 0.730645 + 0.530844i −2.79333 + 5.48222i 4.01492 − 7.87973i 1.22509 + 7.73488i −6.98060 + 5.07170i 15.4589 + 5.02289i −7.65636 − 7.65636i
13.12 −0.505897 + 1.55699i 4.90827 + 0.777393i 1.06778 + 0.775790i −0.684951 + 1.34429i −3.69347 + 7.24884i −0.320216 − 2.02176i −7.04590 + 5.11915i 14.9272 + 4.85015i −1.74653 − 1.74653i
13.13 −0.395997 + 1.21875i 1.85084 + 0.293144i 1.90752 + 1.38590i −2.09562 + 4.11288i −1.09020 + 2.13963i 0.991321 + 6.25896i −6.59137 + 4.78891i −5.21983 − 1.69603i −4.18273 − 4.18273i
13.14 −0.368794 + 1.13503i −0.0783055 − 0.0124024i 2.08378 + 1.51396i −0.877672 + 1.72253i 0.0429557 − 0.0843052i 0.419213 + 2.64680i −6.34893 + 4.61277i −8.55353 − 2.77921i −1.63144 − 1.63144i
13.15 −0.242213 + 0.745456i −5.02974 − 0.796632i 2.73903 + 1.99002i −0.592599 + 1.16304i 1.81212 − 3.55649i −2.13594 − 13.4858i −4.68339 + 3.40269i 16.1041 + 5.23255i −0.723461 − 0.723461i
13.16 −0.0528626 + 0.162694i −1.05437 − 0.166996i 3.21239 + 2.33394i 3.07326 − 6.03161i 0.0829059 − 0.162712i −0.625313 − 3.94807i −1.10312 + 0.801462i −7.47570 − 2.42900i 0.818849 + 0.818849i
13.17 0.0489239 − 0.150572i 4.41908 + 0.699913i 3.21579 + 2.33641i −0.387550 + 0.760610i 0.321586 − 0.631148i −1.57233 − 9.92729i 1.02146 − 0.742137i 10.4789 + 3.40479i 0.0955663 + 0.0955663i
13.18 0.0630932 − 0.194181i −2.58667 − 0.409688i 3.20234 + 2.32664i −2.42611 + 4.76151i −0.242755 + 0.476433i −0.243412 − 1.53684i 1.31455 − 0.955080i −2.03649 − 0.661697i 0.771523 + 0.771523i
13.19 0.0662878 − 0.204013i 4.06248 + 0.643434i 3.19884 + 2.32409i 3.19409 − 6.26876i 0.400562 − 0.786147i 1.76228 + 11.1266i 1.38036 − 1.00289i 7.53024 + 2.44672i −1.06718 − 1.06718i
13.20 0.181146 − 0.557509i −1.46319 − 0.231747i 2.95807 + 2.14916i 1.12488 − 2.20770i −0.394252 + 0.773762i 1.80812 + 11.4160i 3.63100 − 2.63807i −6.47228 − 2.10297i −1.02705 − 1.02705i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
17.c even 4 1 inner
187.o odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.3.o.a ✓ 272
11.d odd 10 1 inner 187.3.o.a ✓ 272
17.c even 4 1 inner 187.3.o.a ✓ 272
187.o odd 20 1 inner 187.3.o.a ✓ 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.3.o.a ✓ 272 1.a even 1 1 trivial
187.3.o.a ✓ 272 11.d odd 10 1 inner
187.3.o.a ✓ 272 17.c even 4 1 inner
187.3.o.a ✓ 272 187.o odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(187, [\chi])\).