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Properties

Label 187.2.t.a
Level $187$
Weight $2$
Character orbit 187.t
Analytic conductor $1.493$
Analytic rank $0$
Dimension $512$
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] = [187,2,Mod(6,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(80))
 
chi = DirichletCharacter(H, H._module([72, 75]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 187.t (of order \(80\), degree \(32\), 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: \(1.49320251780\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(16\) over \(\Q(\zeta_{80})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{80}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512 q - 40 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 40 q^{6} - 40 q^{7} - 40 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 512 q - 40 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 40 q^{6} - 40 q^{7} - 40 q^{8} - 24 q^{9} - 40 q^{11} - 16 q^{12} - 56 q^{14} - 24 q^{15} - 40 q^{17} - 80 q^{18} - 40 q^{19} - 24 q^{20} - 48 q^{22} - 48 q^{23} + 80 q^{24} - 40 q^{25} - 56 q^{26} + 48 q^{27} - 40 q^{28} - 40 q^{29} - 40 q^{30} + 40 q^{31} - 64 q^{34} - 80 q^{35} - 56 q^{36} - 56 q^{37} + 80 q^{38} - 40 q^{39} - 40 q^{40} + 80 q^{41} + 72 q^{42} + 32 q^{44} - 64 q^{45} - 40 q^{46} - 24 q^{47} - 72 q^{48} - 88 q^{49} - 40 q^{51} + 240 q^{52} - 24 q^{53} + 128 q^{55} - 64 q^{56} - 40 q^{57} + 88 q^{58} - 88 q^{59} - 152 q^{60} - 40 q^{61} - 40 q^{62} + 80 q^{63} - 88 q^{64} + 408 q^{66} - 40 q^{68} + 192 q^{69} - 168 q^{70} - 24 q^{71} - 40 q^{72} - 40 q^{73} - 40 q^{74} - 88 q^{75} + 192 q^{77} - 64 q^{78} - 40 q^{79} + 248 q^{80} - 40 q^{81} - 24 q^{82} - 40 q^{85} - 112 q^{86} + 128 q^{88} - 64 q^{89} - 40 q^{90} + 56 q^{91} + 136 q^{92} - 88 q^{93} + 440 q^{94} - 40 q^{95} - 40 q^{96} - 88 q^{97} + 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −2.62383 − 0.206500i 0.338932 + 0.156250i 4.86645 + 0.770770i 2.42091 + 3.07091i −0.857033 − 0.479962i −0.252119 + 0.683399i −7.49112 − 1.79846i −1.85788 − 2.17530i −5.71790 − 8.55745i
6.2 −2.59078 − 0.203899i −2.26719 − 1.04519i 4.69518 + 0.743644i −1.25029 − 1.58598i 5.66068 + 3.17014i 1.40072 − 3.79682i −6.95859 − 1.67061i 2.09940 + 2.45808i 2.91584 + 4.36387i
6.3 −2.09150 − 0.164604i 0.560652 + 0.258464i 2.37189 + 0.375671i −1.69373 − 2.14848i −1.13006 − 0.632864i −0.674528 + 1.82839i −0.818977 − 0.196619i −1.70082 − 1.99140i 3.18878 + 4.77235i
6.4 −1.75467 − 0.138096i 2.81986 + 1.29997i 1.08442 + 0.171756i 0.649663 + 0.824093i −4.76840 − 2.67043i 1.37212 − 3.71929i 1.54385 + 0.370644i 4.31333 + 5.05026i −1.02614 − 1.53573i
6.5 −1.67599 − 0.131903i −1.59616 − 0.735838i 0.816165 + 0.129268i 0.713695 + 0.905318i 2.57808 + 1.44380i −1.01149 + 2.74177i 1.91861 + 0.460617i 0.0579148 + 0.0678096i −1.07673 − 1.61144i
6.6 −1.06157 − 0.0835470i −2.09399 − 0.965344i −0.855435 − 0.135488i −0.686188 − 0.870426i 2.14226 + 1.19972i 0.248365 − 0.673223i 2.96763 + 0.712466i 1.50457 + 1.76163i 0.655712 + 0.981343i
6.7 −0.877055 − 0.0690257i 2.14542 + 0.989054i −1.21092 − 0.191790i 0.214388 + 0.271950i −1.81338 − 1.01554i −1.45693 + 3.94919i 2.75972 + 0.662550i 1.67627 + 1.96266i −0.169259 − 0.253314i
6.8 −0.426567 − 0.0335715i −0.467735 − 0.215629i −1.79454 − 0.284228i 1.12229 + 1.42362i 0.192281 + 0.107683i 1.60101 − 4.33973i 1.58808 + 0.381264i −1.77606 − 2.07950i −0.430939 − 0.644946i
6.9 −0.213005 − 0.0167639i 1.15403 + 0.532014i −1.93029 − 0.305727i −2.42133 − 3.07145i −0.236896 − 0.132668i 0.704712 − 1.91021i 0.821557 + 0.197238i −0.899602 − 1.05330i 0.464268 + 0.694826i
6.10 0.540115 + 0.0425079i −1.09908 − 0.506683i −1.68546 − 0.266951i −0.995194 − 1.26240i −0.572092 − 0.320387i −0.583247 + 1.58096i −1.95262 − 0.468784i −0.997093 − 1.16745i −0.483857 − 0.724143i
6.11 0.629923 + 0.0495761i 1.56161 + 0.719913i −1.58103 − 0.250411i 2.34147 + 2.97014i 0.948006 + 0.530909i −0.248193 + 0.672757i −2.21234 − 0.531136i −0.0279848 − 0.0327660i 1.32770 + 1.98704i
6.12 0.659382 + 0.0518945i −2.80529 − 1.29326i −1.54329 − 0.244432i 2.38463 + 3.02489i −1.78264 − 0.998329i −0.787459 + 2.13450i −2.29122 − 0.550073i 4.24880 + 4.97470i 1.41541 + 2.11831i
6.13 1.51646 + 0.119348i 1.78375 + 0.822318i 0.310038 + 0.0491051i −0.164969 − 0.209262i 2.60684 + 1.45990i 0.694532 − 1.88261i −2.49394 − 0.598743i 0.557198 + 0.652395i −0.225194 − 0.337026i
6.14 1.93716 + 0.152458i −2.58853 − 1.19333i 1.75398 + 0.277804i −2.16149 − 2.74184i −4.83248 − 2.70632i 0.639914 − 1.73456i −0.423527 − 0.101680i 3.32813 + 3.89674i −3.76915 − 5.64093i
6.15 2.13560 + 0.168075i 0.194323 + 0.0895840i 2.55715 + 0.405012i 0.179483 + 0.227673i 0.399938 + 0.223976i −0.474986 + 1.28751i 1.22694 + 0.294561i −1.91861 − 2.24640i 0.345037 + 0.516384i
6.16 2.39250 + 0.188294i −1.45628 − 0.671356i 3.71324 + 0.588119i 1.43760 + 1.82358i −3.35775 − 1.88043i 0.0778254 − 0.210955i 4.10601 + 0.985766i −0.278301 − 0.325848i 3.09608 + 4.63362i
7.1 −1.38079 − 2.25324i 0.273940 + 0.347491i −2.26253 + 4.44046i 0.237290 + 0.219348i 0.404728 − 1.09706i 0.586843 + 4.95822i 7.86045 − 0.618631i 0.654629 − 2.72673i 0.166598 − 0.837543i
7.2 −1.18360 − 1.93146i −0.181018 − 0.229620i −1.42165 + 2.79015i 2.86138 + 2.64503i −0.229250 + 0.621408i −0.444980 − 3.75961i 2.55516 − 0.201096i 0.680378 − 2.83398i 1.72204 − 8.65730i
7.3 −0.987367 − 1.61124i 1.05876 + 1.34304i −0.713210 + 1.39975i −2.47426 − 2.28718i 1.11856 − 3.03199i −0.0495816 − 0.418913i −0.808223 + 0.0636085i 0.0175722 − 0.0731935i −1.24219 + 6.24491i
7.4 −0.917683 − 1.49752i −0.728333 − 0.923886i −0.492452 + 0.966491i −0.908277 − 0.839602i −0.715161 + 1.93853i −0.0328041 − 0.277160i −1.60259 + 0.126126i 0.377240 − 1.57132i −0.423813 + 2.13065i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
17.e odd 16 1 inner
187.t even 80 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.2.t.a ✓ 512
11.d odd 10 1 inner 187.2.t.a ✓ 512
17.e odd 16 1 inner 187.2.t.a ✓ 512
187.t even 80 1 inner 187.2.t.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.2.t.a ✓ 512 1.a even 1 1 trivial
187.2.t.a ✓ 512 11.d odd 10 1 inner
187.2.t.a ✓ 512 17.e odd 16 1 inner
187.2.t.a ✓ 512 187.t even 80 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(187, [\chi])\).