Properties

Label 277.2.g.a
Level $277$
Weight $2$
Character orbit 277.g
Analytic conductor $2.212$
Analytic rank $0$
Dimension $462$
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] = [277,2,Mod(16,277)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(277, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("277.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 277 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 277.g (of order \(23\), degree \(22\), 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: \(2.21185613599\)
Analytic rank: \(0\)
Dimension: \(462\)
Relative dimension: \(21\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 462 q - 18 q^{2} - 19 q^{3} - 36 q^{4} - 15 q^{5} - 11 q^{6} - 11 q^{7} - 2 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 462 q - 18 q^{2} - 19 q^{3} - 36 q^{4} - 15 q^{5} - 11 q^{6} - 11 q^{7} - 2 q^{8} - 36 q^{9} - q^{10} - 11 q^{11} + 19 q^{12} - 93 q^{13} + 11 q^{14} - 9 q^{15} + 4 q^{16} + q^{17} + 34 q^{18} - q^{19} - 74 q^{20} - 56 q^{21} + 39 q^{22} - 79 q^{23} - 116 q^{24} - 6 q^{25} + 21 q^{26} - 121 q^{27} + 45 q^{28} - 81 q^{29} + 45 q^{30} + 11 q^{31} - 94 q^{32} + 17 q^{33} + 25 q^{34} + 8 q^{35} + 14 q^{36} - 15 q^{37} - 67 q^{38} + 43 q^{39} + 59 q^{40} + 25 q^{41} + 39 q^{42} + 41 q^{43} - 118 q^{44} - 61 q^{45} - 134 q^{46} - 45 q^{47} + 117 q^{48} + 24 q^{49} + 82 q^{50} + 31 q^{51} - 145 q^{52} - 118 q^{53} + 95 q^{54} - 24 q^{55} - 19 q^{56} + 47 q^{57} - 168 q^{58} + 51 q^{59} + 88 q^{60} - 185 q^{61} + 23 q^{62} + 10 q^{63} + 138 q^{64} - 83 q^{65} + 117 q^{66} + 43 q^{67} - 6 q^{68} + 5 q^{69} - 47 q^{70} + 63 q^{71} - 146 q^{72} + 73 q^{73} + 117 q^{74} - 459 q^{75} - 24 q^{76} + 87 q^{77} + 175 q^{78} - 27 q^{79} + 161 q^{80} + 20 q^{81} + 63 q^{82} + 87 q^{83} - 282 q^{84} + 127 q^{85} + 51 q^{86} - 173 q^{87} + 191 q^{88} + 52 q^{89} - 25 q^{90} + 7 q^{91} - 145 q^{92} - 155 q^{93} + 51 q^{94} + 96 q^{95} - 41 q^{96} + 69 q^{97} - 415 q^{98} - 123 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.26253 1.37587i 0.191052 2.79307i 2.30588 + 4.45014i −2.28039 0.990512i −4.27518 + 6.05655i 3.28175 1.42546i 0.544304 7.95744i −4.79271 0.658743i 3.79662 + 5.37858i
16.2 −2.13644 1.29920i −0.142346 + 2.08102i 1.95633 + 3.77554i 1.88109 + 0.817073i 3.00777 4.26104i 1.95720 0.850132i 0.384323 5.61861i −1.33832 0.183948i −2.95730 4.18953i
16.3 −2.04323 1.24251i −0.0903377 + 1.32069i 1.71080 + 3.30170i −1.31876 0.572817i 1.82556 2.58622i −1.60050 + 0.695194i 0.280465 4.10025i 1.23600 + 0.169884i 1.98279 + 2.80897i
16.4 −1.63953 0.997023i 0.0265916 0.388755i 0.773888 + 1.49354i −3.28121 1.42523i −0.431196 + 0.610865i −1.93539 + 0.840657i −0.0416251 + 0.608538i 2.82163 + 0.387825i 3.95867 + 5.60815i
16.5 −1.61130 0.979854i 0.106997 1.56423i 0.716048 + 1.38191i 1.44267 + 0.626638i −1.70513 + 2.41561i 2.54898 1.10718i −0.0570881 + 0.834598i 0.536677 + 0.0737645i −1.71056 2.42331i
16.6 −1.29591 0.788063i 0.228574 3.34164i 0.138222 + 0.266756i 1.36313 + 0.592092i −2.92964 + 4.15035i −3.34717 + 1.45388i −0.175912 + 2.57175i −8.14226 1.11913i −1.29990 1.84154i
16.7 −1.20024 0.729879i −0.171528 + 2.50765i −0.0122887 0.0237160i −3.60151 1.56436i 2.03616 2.88458i 3.47311 1.50859i −0.194286 + 2.84036i −3.28684 0.451766i 3.18087 + 4.50627i
16.8 −1.09201 0.664069i −0.00466840 + 0.0682496i −0.168621 0.325424i 0.936249 + 0.406670i 0.0504204 0.0714294i 1.89133 0.821521i −0.206405 + 3.01754i 2.96742 + 0.407863i −0.752341 1.06582i
16.9 −0.965710 0.587261i −0.0593620 + 0.867841i −0.332411 0.641523i 3.35404 + 1.45687i 0.566976 0.803221i −4.23678 + 1.84029i −0.209992 + 3.06997i 2.22243 + 0.305466i −2.38347 3.37661i
16.10 −0.331159 0.201382i 0.0927188 1.35550i −0.851019 1.64239i −2.09811 0.911339i −0.303679 + 0.430215i −1.62492 + 0.705800i −0.101825 + 1.48863i 1.14327 + 0.157139i 0.511282 + 0.724321i
16.11 −0.0387400 0.0235583i −0.207250 + 3.02989i −0.919184 1.77395i 3.60660 + 1.56657i 0.0794080 0.112496i 4.01897 1.74568i −0.0123703 + 0.180847i −6.16522 0.847390i −0.102814 0.145654i
16.12 0.0563172 + 0.0342472i 0.163040 2.38357i −0.918131 1.77191i 1.84624 + 0.801934i 0.0908125 0.128652i 3.29494 1.43119i 0.0179727 0.262751i −2.68275 0.368735i 0.0765109 + 0.108391i
16.13 0.143933 + 0.0875274i −0.0735688 + 1.07554i −0.907074 1.75057i −0.165823 0.0720272i −0.104728 + 0.148366i −0.376419 + 0.163502i 0.0456574 0.667487i 1.82069 + 0.250248i −0.0175630 0.0248811i
16.14 0.296974 + 0.180594i −0.178174 + 2.60481i −0.864551 1.66851i −1.64277 0.713557i −0.523326 + 0.741384i −2.71762 + 1.18043i 0.0920117 1.34516i −3.78124 0.519719i −0.358997 0.508583i
16.15 0.722881 + 0.439593i 0.142533 2.08375i −0.590816 1.14022i 2.38739 + 1.03699i 1.01904 1.44365i −1.63366 + 0.709598i 0.189618 2.77212i −1.34965 0.185505i 1.26994 + 1.79910i
16.16 0.948416 + 0.576744i −0.00917760 + 0.134172i −0.353272 0.681783i −3.16508 1.37479i −0.0860870 + 0.121958i 3.88742 1.68854i 0.209666 3.06521i 2.95414 + 0.406037i −2.20891 3.12931i
16.17 1.19118 + 0.724372i −0.0329619 + 0.481886i −0.0259378 0.0500577i 2.50640 + 1.08868i −0.388328 + 0.550136i −0.0270097 + 0.0117320i 0.195643 2.86019i 2.74093 + 0.376732i 2.19696 + 3.11239i
16.18 1.33922 + 0.814396i 0.180529 2.63925i 0.210133 + 0.405538i −1.94291 0.843923i 2.39116 3.38750i −1.25200 + 0.543822i 0.165071 2.41326i −3.96098 0.544424i −1.91469 2.71249i
16.19 1.78257 + 1.08401i −0.170303 + 2.48974i 1.08237 + 2.08887i −0.215365 0.0935460i −3.00247 + 4.25353i 0.626806 0.272260i −0.0502065 + 0.733994i −3.19774 0.439519i −0.282499 0.400209i
16.20 2.00704 + 1.22051i 0.0304939 0.445804i 1.61844 + 3.12346i 0.912837 + 0.396501i 0.605311 0.857530i −4.02881 + 1.74996i −0.243317 + 3.55716i 2.77425 + 0.381311i 1.34817 + 1.90992i
See next 80 embeddings (of 462 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
277.g even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 277.2.g.a 462
277.g even 23 1 inner 277.2.g.a 462
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
277.2.g.a 462 1.a even 1 1 trivial
277.2.g.a 462 277.g even 23 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(277, [\chi])\).