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Properties

Label 368.2.x.a
Level $368$
Weight $2$
Character orbit 368.x
Analytic conductor $2.938$
Analytic rank $0$
Dimension $920$
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] = [368,2,Mod(11,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 11, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.x (of order \(44\), degree \(20\), 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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(920\)
Relative dimension: \(46\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 18 q^{2} - 18 q^{3} - 22 q^{4} - 22 q^{5} - 12 q^{6} - 44 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 920 q - 18 q^{2} - 18 q^{3} - 22 q^{4} - 22 q^{5} - 12 q^{6} - 44 q^{7} - 18 q^{8} - 22 q^{10} - 22 q^{11} - 12 q^{12} - 18 q^{13} - 22 q^{14} - 6 q^{16} - 44 q^{17} - 46 q^{18} - 22 q^{19} - 22 q^{20} - 22 q^{21} - 48 q^{23} - 48 q^{24} + 2 q^{26} - 42 q^{27} - 22 q^{28} - 2 q^{29} - 22 q^{30} + 2 q^{32} - 44 q^{33} - 66 q^{34} - 38 q^{35} - 60 q^{36} - 22 q^{37} + 88 q^{38} - 12 q^{39} - 22 q^{40} - 242 q^{42} - 22 q^{43} - 22 q^{44} + 170 q^{46} - 78 q^{48} + 32 q^{49} - 184 q^{50} - 22 q^{51} + 50 q^{52} - 22 q^{53} + 102 q^{54} - 36 q^{55} - 22 q^{56} - 58 q^{58} - 30 q^{59} - 22 q^{60} - 22 q^{61} - 20 q^{62} - 64 q^{64} - 44 q^{65} - 22 q^{66} - 22 q^{67} + 2 q^{69} - 32 q^{70} - 36 q^{71} + 44 q^{72} - 22 q^{74} - 10 q^{75} - 22 q^{76} + 26 q^{77} - 56 q^{78} - 22 q^{80} + 24 q^{81} - 88 q^{82} - 22 q^{83} - 22 q^{84} - 6 q^{85} - 22 q^{86} - 36 q^{87} - 22 q^{88} - 22 q^{90} - 26 q^{92} - 292 q^{93} + 82 q^{94} - 46 q^{96} - 44 q^{97} + 66 q^{98} - 22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.40754 − 0.137230i 0.973926 − 0.729072i 1.96234 + 0.386313i −0.350311 + 0.641546i −1.47089 + 0.892546i −1.07732 + 0.933507i −2.70905 − 0.813042i −0.428212 + 1.45836i 0.581115 − 0.854929i
11.2 −1.39928 + 0.204983i −1.15383 + 0.863745i 1.91596 − 0.573657i −0.0600777 + 0.110024i 1.43747 − 1.44514i 2.40328 − 2.08245i −2.56338 + 1.19545i −0.259934 + 0.885253i 0.0615123 − 0.166269i
11.3 −1.39536 − 0.230161i 2.22730 − 1.66733i 1.89405 + 0.642314i 1.54484 − 2.82916i −3.49163 + 1.81389i −2.61306 + 2.26423i −2.49505 − 1.33220i 1.33565 − 4.54881i −2.80677 + 3.59214i
11.4 −1.38579 + 0.282117i −2.29370 + 1.71704i 1.84082 − 0.781909i 1.59597 − 2.92281i 2.69418 − 3.02655i −1.57126 + 1.36150i −2.33040 + 1.60289i 1.46763 − 4.99830i −1.38711 + 4.50064i
11.5 −1.36059 + 0.385738i 2.68981 − 2.01357i 1.70241 − 1.04966i −1.61466 + 2.95703i −2.88302 + 3.77720i 1.86470 − 1.61577i −1.91139 + 2.08485i 2.33543 − 7.95375i 1.05625 − 4.64614i
11.6 −1.32112 − 0.504624i −1.58855 + 1.18918i 1.49071 + 1.33334i −1.20872 + 2.21361i 2.69875 − 0.769421i −3.42067 + 2.96403i −1.29657 − 2.51374i 0.264164 − 0.899660i 2.71390 − 2.31449i
11.7 −1.27334 − 0.615319i 0.285694 − 0.213868i 1.24276 + 1.56701i −1.75226 + 3.20903i −0.495381 + 0.0965323i 2.33298 − 2.02154i −0.618242 − 2.76003i −0.809316 + 2.75628i 4.20579 − 3.00797i
11.8 −1.23557 + 0.688020i 0.164054 − 0.122809i 1.05326 − 1.70019i −0.205675 + 0.376665i −0.118205 + 0.264611i −0.650756 + 0.563883i −0.131605 + 2.82536i −0.833366 + 2.83818i −0.00502800 − 0.606904i
11.9 −1.19560 + 0.755342i 1.29396 − 0.968643i 0.858917 − 1.80617i 1.40833 − 2.57917i −0.815396 + 2.13549i 1.80740 − 1.56612i 0.337357 + 2.80824i −0.109146 + 0.371717i 0.264353 + 4.14743i
11.10 −1.18486 + 0.772074i −1.65860 + 1.24161i 0.807803 − 1.82960i −1.76181 + 3.22651i 1.00659 − 2.75170i −1.59739 + 1.38415i 0.455454 + 2.79152i 0.364148 − 1.24017i −0.403603 − 5.18321i
11.11 −1.15065 − 0.822189i −0.399833 + 0.299311i 0.648011 + 1.89211i 1.96982 − 3.60745i 0.706160 − 0.0156656i 2.66543 − 2.30961i 0.810035 − 2.70995i −0.774918 + 2.63913i −5.23258 + 2.53137i
11.12 −1.11238 − 0.873270i −2.51073 + 1.87951i 0.474801 + 1.94282i −0.246034 + 0.450578i 4.43421 + 0.101806i 2.53122 − 2.19332i 1.16845 − 2.57580i 1.92601 − 6.55938i 0.667160 − 0.286362i
11.13 −1.07561 − 0.918190i 1.63479 − 1.22379i 0.313854 + 1.97522i −0.208917 + 0.382602i −2.88206 − 0.184735i 0.383100 − 0.331958i 1.47605 − 2.41274i 0.329683 − 1.12280i 0.576014 − 0.219704i
11.14 −0.932238 − 1.06345i −0.232979 + 0.174406i −0.261866 + 1.98278i 1.14787 − 2.10218i 0.402665 + 0.0851746i −3.14997 + 2.72946i 2.35272 − 1.56994i −0.821336 + 2.79721i −3.30566 + 0.739016i
11.15 −0.795222 + 1.16945i −2.41514 + 1.80795i −0.735244 − 1.85995i −0.465829 + 0.853103i −0.193743 − 4.26212i 1.81839 − 1.57565i 2.75981 + 0.619240i 1.71902 − 5.85446i −0.627226 − 1.22317i
11.16 −0.749707 + 1.19914i −0.622900 + 0.466297i −0.875879 − 1.79801i 1.52012 − 2.78390i −0.0921637 − 1.09653i −3.28820 + 2.84924i 2.81272 + 0.297677i −0.674626 + 2.29757i 2.19864 + 3.90995i
11.17 −0.696332 + 1.23090i 1.49397 − 1.11837i −1.03024 − 1.71423i −0.899031 + 1.64645i 0.336308 + 2.61769i −2.63017 + 2.27906i 2.82745 − 0.0744536i 0.135990 − 0.463140i −1.40060 − 2.25310i
11.18 −0.609714 − 1.27603i −1.39048 + 1.04090i −1.25650 + 1.55602i −0.313563 + 0.574247i 2.17602 + 1.13964i 0.0844719 − 0.0731954i 2.75164 + 0.654598i 0.00476860 − 0.0162404i 0.923939 + 0.0499886i
11.19 −0.496428 − 1.32422i 2.48154 − 1.85765i −1.50712 + 1.31476i 1.03377 − 1.89320i −3.69185 − 2.36391i 1.52012 − 1.31719i 2.48921 + 1.34308i 1.86194 − 6.34120i −3.02021 − 0.429098i
11.20 −0.407811 + 1.35414i 1.91330 − 1.43228i −1.66738 − 1.10447i −0.120335 + 0.220378i 1.15924 + 3.17498i 1.85359 − 1.60614i 2.17557 − 1.80745i 0.764104 − 2.60230i −0.249348 − 0.252823i
See next 80 embeddings (of 920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
23.d odd 22 1 inner
368.x even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.x.a ✓ 920
16.f odd 4 1 inner 368.2.x.a ✓ 920
23.d odd 22 1 inner 368.2.x.a ✓ 920
368.x even 44 1 inner 368.2.x.a ✓ 920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.x.a ✓ 920 1.a even 1 1 trivial
368.2.x.a ✓ 920 16.f odd 4 1 inner
368.2.x.a ✓ 920 23.d odd 22 1 inner
368.2.x.a ✓ 920 368.x even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(368, [\chi])\).