Properties

Label 241.2.s.a
Level $241$
Weight $2$
Character orbit 241.s
Analytic conductor $1.924$
Analytic rank $0$
Dimension $640$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [241,2,Mod(3,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(120))
 
chi = DirichletCharacter(H, H._module([91]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.s (of order \(120\), 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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(20\) over \(\Q(\zeta_{120})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{120}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 32 q^{2} - 32 q^{3} - 24 q^{4} - 32 q^{5} - 40 q^{6} - 48 q^{7} - 64 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 32 q^{2} - 32 q^{3} - 24 q^{4} - 32 q^{5} - 40 q^{6} - 48 q^{7} - 64 q^{8} - 4 q^{9} - 44 q^{10} - 32 q^{11} - 76 q^{12} - 32 q^{13} - 96 q^{14} - 80 q^{15} + 360 q^{16} + 44 q^{17} - 48 q^{18} - 44 q^{19} - 104 q^{20} - 40 q^{21} - 56 q^{22} - 60 q^{23} - 108 q^{24} - 40 q^{25} - 60 q^{26} - 44 q^{27} - 72 q^{28} - 16 q^{29} - 84 q^{30} + 176 q^{31} - 104 q^{32} - 20 q^{33} + 12 q^{34} + 32 q^{35} + 256 q^{36} - 4 q^{37} + 20 q^{38} - 8 q^{39} - 120 q^{40} - 20 q^{41} + 24 q^{42} - 40 q^{43} - 4 q^{44} - 12 q^{45} + 40 q^{46} - 48 q^{47} - 264 q^{48} + 116 q^{49} + 356 q^{50} - 96 q^{51} - 152 q^{52} - 8 q^{53} - 28 q^{54} - 76 q^{55} - 40 q^{56} - 112 q^{57} - 16 q^{58} - 60 q^{59} - 96 q^{60} - 48 q^{61} + 28 q^{62} + 72 q^{63} + 228 q^{65} + 108 q^{66} - 12 q^{67} - 48 q^{68} + 188 q^{69} - 68 q^{70} - 40 q^{71} - 388 q^{72} - 100 q^{73} - 224 q^{74} - 120 q^{75} - 60 q^{76} - 200 q^{78} - 40 q^{79} + 148 q^{80} + 88 q^{81} - 16 q^{82} + 56 q^{83} + 304 q^{84} + 20 q^{85} - 132 q^{86} - 80 q^{87} - 164 q^{88} + 24 q^{89} + 92 q^{90} + 88 q^{91} + 188 q^{92} + 16 q^{93} + 12 q^{94} + 272 q^{95} + 168 q^{96} + 152 q^{97} - 104 q^{98} + 252 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.63760 0.706742i 0.0388054 + 0.00203370i 4.72539 + 2.72821i −1.46248 + 2.87029i −0.100916 0.0327895i −2.01129 + 1.90865i −6.67384 6.67384i −2.98206 0.313428i 5.88600 6.53707i
3.2 −2.45155 0.656890i 1.72096 + 0.0901919i 3.84653 + 2.22079i 1.44750 2.84088i −4.15978 1.35159i 1.62669 1.54367i −4.38182 4.38182i −0.0299826 0.00315130i −5.41476 + 6.01370i
3.3 −2.20393 0.590541i −3.02856 0.158720i 2.77652 + 1.60302i −0.162719 + 0.319354i 6.58101 + 2.13830i 1.66562 1.58062i −1.94583 1.94583i 6.16342 + 0.647802i 0.547213 0.607742i
3.4 −1.85977 0.498323i −1.14995 0.0602663i 1.47836 + 0.853531i 0.593949 1.16569i 2.10861 + 0.685128i −2.05081 + 1.94614i 0.398819 + 0.398819i −1.66481 0.174979i −1.68550 + 1.87193i
3.5 −1.83755 0.492369i 2.81865 + 0.147719i 1.40210 + 0.809501i −0.634709 + 1.24569i −5.10666 1.65926i −1.18204 + 1.12171i 0.512511 + 0.512511i 4.93940 + 0.519152i 1.77964 1.97649i
3.6 −1.66779 0.446883i 0.776835 + 0.0407122i 0.849764 + 0.490612i −1.75457 + 3.44354i −1.27740 0.415054i 2.74176 2.60184i 1.24383 + 1.24383i −2.38175 0.250332i 4.46511 4.95901i
3.7 −1.21604 0.325838i −0.393654 0.0206305i −0.359457 0.207533i 1.13796 2.23338i 0.471979 + 0.153355i 0.846934 0.803710i 2.14991 + 2.14991i −2.82903 0.297343i −2.11153 + 2.34509i
3.8 −0.452936 0.121364i −2.00149 0.104894i −1.54163 0.890060i −1.24052 + 2.43467i 0.893818 + 0.290419i 1.64261 1.55878i 1.25338 + 1.25338i 1.01141 + 0.106303i 0.857359 0.952194i
3.9 −0.438186 0.117412i 3.29106 + 0.172477i −1.55383 0.897104i 1.47341 2.89172i −1.42185 0.461986i −1.24246 + 1.17905i 1.21708 + 1.21708i 7.81779 + 0.821683i −0.985147 + 1.09412i
3.10 −0.383790 0.102836i 0.677022 + 0.0354812i −1.59533 0.921065i −0.689366 + 1.35296i −0.256186 0.0832398i −2.93731 + 2.78740i 1.07946 + 1.07946i −2.52647 0.265542i 0.403705 0.448359i
3.11 −0.0899442 0.0241005i −3.33752 0.174912i −1.72454 0.995665i 1.23222 2.41837i 0.295975 + 0.0961681i −2.76536 + 2.62422i 0.262804 + 0.262804i 8.12487 + 0.853959i −0.169115 + 0.187821i
3.12 0.544032 + 0.145773i 1.49190 + 0.0781872i −1.45733 0.841390i 0.659390 1.29413i 0.800244 + 0.260015i 2.62383 2.48992i −1.46670 1.46670i −0.763912 0.0802904i 0.547378 0.607925i
3.13 0.945508 + 0.253348i −0.900870 0.0472126i −0.902251 0.520915i 1.19603 2.34733i −0.839819 0.272874i 0.242852 0.230458i −2.10543 2.10543i −2.17423 0.228520i 1.72555 1.91641i
3.14 0.946629 + 0.253649i 3.10227 + 0.162583i −0.900282 0.519778i −1.96330 + 3.85320i 2.89546 + 0.940793i 0.937006 0.889186i −2.10635 2.10635i 6.61410 + 0.695170i −2.83588 + 3.14956i
3.15 0.986895 + 0.264438i −1.75155 0.0917949i −0.828016 0.478055i −0.804333 + 1.57859i −1.70432 0.553768i −0.238777 + 0.226591i −2.13566 2.13566i 0.0759399 + 0.00798161i −1.21123 + 1.34521i
3.16 1.64096 + 0.439693i 2.06237 + 0.108084i 0.767353 + 0.443032i 0.619663 1.21616i 3.33674 + 1.08417i −3.03296 + 2.87817i −1.33813 1.33813i 1.25814 + 0.132236i 1.55157 1.72320i
3.17 2.12211 + 0.568619i 1.06388 + 0.0557554i 2.44799 + 1.41335i −0.373752 + 0.733529i 2.22596 + 0.723259i −0.199145 + 0.188981i 1.28427 + 1.28427i −1.85484 0.194952i −1.21024 + 1.34411i
3.18 2.23659 + 0.599293i −0.255295 0.0133794i 2.91114 + 1.68075i −1.11515 + 2.18860i −0.562973 0.182921i 2.86193 2.71587i 2.22917 + 2.22917i −2.91857 0.306754i −3.80573 + 4.22670i
3.19 2.45678 + 0.658292i −1.06146 0.0556289i 3.87036 + 2.23455i 1.85617 3.64294i −2.57116 0.835420i −0.417872 + 0.396546i 4.44065 + 4.44065i −1.85996 0.195489i 6.95831 7.72798i
3.20 2.45948 + 0.659015i −2.98534 0.156455i 3.88268 + 2.24166i −1.24067 + 2.43494i −7.23927 2.35218i −2.92640 + 2.77705i 4.47114 + 4.47114i 5.90421 + 0.620558i −4.65605 + 5.17107i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.s even 120 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.s.a 640
241.s even 120 1 inner 241.2.s.a 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.s.a 640 1.a even 1 1 trivial
241.2.s.a 640 241.s even 120 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).