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Properties

Label 799.2.o.a
Level $799$
Weight $2$
Character orbit 799.o
Analytic conductor $6.380$
Analytic rank $0$
Dimension $3080$
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] = [799,2,Mod(4,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(92))
 
chi = DirichletCharacter(H, H._module([69, 72]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.o (of order \(92\), degree \(44\), 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: \(6.38004712150\)
Analytic rank: \(0\)
Dimension: \(3080\)
Relative dimension: \(70\) over \(\Q(\zeta_{92})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{92}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3080 q - 42 q^{3} + 52 q^{4} - 38 q^{5} - 42 q^{6} - 46 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3080 q - 42 q^{3} + 52 q^{4} - 38 q^{5} - 42 q^{6} - 46 q^{7} - 58 q^{10} - 50 q^{11} - 48 q^{12} - 84 q^{13} - 52 q^{14} - 244 q^{16} - 30 q^{17} - 100 q^{18} - 50 q^{20} - 84 q^{21} - 18 q^{22} - 58 q^{23} - 68 q^{24} - 30 q^{27} + 6 q^{28} - 54 q^{29} - 116 q^{30} - 38 q^{31} - 68 q^{33} - 18 q^{34} - 16 q^{35} - 76 q^{37} - 84 q^{38} - 62 q^{39} - 42 q^{40} - 46 q^{41} - 234 q^{44} + 116 q^{45} - 72 q^{46} + 20 q^{47} - 532 q^{48} - 156 q^{50} - 38 q^{51} + 244 q^{52} - 272 q^{54} - 52 q^{55} - 38 q^{56} - 34 q^{57} - 34 q^{58} - 36 q^{61} + 14 q^{62} - 58 q^{63} + 28 q^{64} - 34 q^{65} - 92 q^{67} - 84 q^{68} - 116 q^{69} - 58 q^{71} - 132 q^{72} - 66 q^{73} - 54 q^{74} - 134 q^{75} - 282 q^{78} + 118 q^{79} - 300 q^{80} + 40 q^{81} - 50 q^{82} - 664 q^{84} + 338 q^{85} - 164 q^{86} + 346 q^{88} + 68 q^{89} - 50 q^{90} - 14 q^{91} + 78 q^{92} - 448 q^{95} + 662 q^{96} - 206 q^{97} - 156 q^{98} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.738719 + 2.63652i 0.552492 + 1.74076i −4.69670 − 2.85613i 0.232826 − 2.26478i −4.99768 + 0.170726i 1.07197 + 1.41361i 6.99764 − 6.53534i −0.274080 + 0.193466i 5.79915 + 2.28689i
4.2 −0.736498 + 2.62860i −0.861966 − 2.71583i −4.65825 − 2.83275i −0.230384 + 2.24102i 7.77366 − 0.265556i 0.117056 + 0.154362i 6.88684 − 6.43186i −4.18185 + 2.95187i −5.72107 − 2.25610i
4.3 −0.732642 + 2.61483i 0.181404 + 0.571558i −4.59175 − 2.79230i −0.0215367 + 0.209495i −1.62743 + 0.0555948i −1.36934 − 1.80575i 6.69629 − 6.25390i 2.15714 − 1.52267i −0.532015 − 0.209799i
4.4 −0.717240 + 2.55986i −0.615889 − 1.94051i −4.32963 − 2.63291i 0.308401 − 2.99992i 5.40917 − 0.184783i −0.243529 − 0.321141i 5.95950 − 5.56578i −0.935334 + 0.660231i 7.45818 + 2.94112i
4.5 −0.691057 + 2.46642i 0.0266226 + 0.0838808i −3.89681 − 2.36970i −0.330769 + 3.21750i −0.225283 + 0.00769589i 0.762927 + 1.00607i 4.79366 − 4.47697i 2.44458 − 1.72557i −7.70712 − 3.03929i
4.6 −0.656890 + 2.34447i 0.903316 + 2.84611i −3.35620 − 2.04095i −0.0122623 + 0.119279i −7.26601 + 0.248214i 2.19891 + 2.89969i 3.43079 − 3.20413i −4.83347 + 3.41184i −0.271591 − 0.107102i
4.7 −0.630891 + 2.25168i 0.876627 + 2.76202i −2.96320 − 1.80196i −0.172581 + 1.67875i −6.77225 + 0.231347i −2.89067 − 3.81192i 2.50894 − 2.34319i −4.40939 + 3.11249i −3.67114 − 1.44771i
4.8 −0.604811 + 2.15860i 0.590926 + 1.86186i −2.58491 − 1.57192i 0.295239 − 2.87189i −4.37640 + 0.149502i −1.44731 − 1.90856i 1.67986 − 1.56888i −0.666402 + 0.470398i 6.02070 + 2.37425i
4.9 −0.602821 + 2.15150i −0.341430 − 1.07576i −2.55670 − 1.55476i −0.244991 + 2.38311i 2.52030 − 0.0860962i −1.91918 − 2.53082i 1.62041 − 1.51336i 1.41023 − 0.995451i −4.97957 − 1.96369i
4.10 −0.593950 + 2.11984i −0.819416 − 2.58177i −2.43209 − 1.47899i 0.102208 − 0.994210i 5.95961 − 0.203587i 0.395788 + 0.521925i 1.36192 − 1.27194i −3.54317 + 2.50104i 2.04686 + 0.807175i
4.11 −0.587788 + 2.09784i −0.383516 − 1.20836i −2.34661 − 1.42701i 0.322911 − 3.14106i 2.76037 − 0.0942971i 1.29111 + 1.70259i 1.18850 − 1.10998i 1.13786 − 0.803192i 6.39965 + 2.52370i
4.12 −0.568451 + 2.02883i 0.130190 + 0.410196i −2.08417 − 1.26741i 0.0313098 − 0.304560i −0.906224 + 0.0309576i 2.13216 + 2.81168i 0.676417 − 0.631730i 2.29960 − 1.62323i 0.600102 + 0.236650i
4.13 −0.542161 + 1.93500i 0.267486 + 0.842778i −1.74143 − 1.05899i 0.111375 − 1.08338i −1.77579 + 0.0606629i −1.15299 − 1.52045i 0.0560256 − 0.0523242i 1.81218 − 1.27918i 2.03595 + 0.802875i
4.14 −0.513043 + 1.83107i −0.806369 − 2.54066i −1.38078 − 0.839670i −0.387660 + 3.77090i 5.06583 − 0.173054i 2.50273 + 3.30035i −0.533603 + 0.498351i −3.35380 + 2.36737i −6.70590 − 2.64447i
4.15 −0.493775 + 1.76231i 0.529534 + 1.66843i −1.15307 − 0.701199i −0.368304 + 3.58262i −3.20175 + 0.109375i 1.86380 + 2.45779i −0.870028 + 0.812550i −0.0523275 + 0.0369368i −6.13182 − 2.41807i
4.16 −0.453881 + 1.61992i 0.0256952 + 0.0809588i −0.709304 − 0.431337i −0.0580530 + 0.564700i −0.142809 + 0.00487852i −1.26457 − 1.66759i −1.43831 + 1.34329i 2.44502 − 1.72588i −0.888422 − 0.350348i
4.17 −0.440926 + 1.57369i −0.647641 − 2.04055i −0.573235 − 0.348592i −0.0847652 + 0.824539i 3.49675 − 0.119453i −0.788524 − 1.03982i −1.58747 + 1.48259i −1.29349 + 0.913048i −1.26019 − 0.496955i
4.18 −0.431991 + 1.54180i −0.985729 − 3.10578i −0.481678 − 0.292915i 0.286480 − 2.78669i 5.21430 − 0.178126i −2.97193 − 3.91908i −1.68069 + 1.56966i −6.22327 + 4.39286i 4.17275 + 1.64552i
4.19 −0.401224 + 1.43199i 0.944788 + 2.97678i −0.180772 − 0.109930i 0.428343 − 4.16663i −4.64179 + 0.158568i 0.612873 + 0.808194i −1.94376 + 1.81534i −5.51769 + 3.89481i 5.79471 + 2.28514i
4.20 −0.396198 + 1.41405i 0.345254 + 1.08780i −0.133728 − 0.0813216i −0.258375 + 2.51330i −1.67500 + 0.0572197i 1.02182 + 1.34747i −1.97850 + 1.84779i 1.38679 − 0.978904i −3.45157 − 1.36112i
See next 80 embeddings (of 3080 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner
47.c even 23 1 inner
799.o even 92 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.o.a ✓ 3080
17.c even 4 1 inner 799.2.o.a ✓ 3080
47.c even 23 1 inner 799.2.o.a ✓ 3080
799.o even 92 1 inner 799.2.o.a ✓ 3080
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.o.a ✓ 3080 1.a even 1 1 trivial
799.2.o.a ✓ 3080 17.c even 4 1 inner
799.2.o.a ✓ 3080 47.c even 23 1 inner
799.2.o.a ✓ 3080 799.o even 92 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(799, [\chi])\).