Properties

Label 241.4.s.a
Level $241$
Weight $4$
Character orbit 241.s
Analytic conductor $14.219$
Analytic rank $0$
Dimension $1888$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [241,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(14.2194603114\)
Analytic rank: \(0\)
Dimension: \(1888\)
Relative dimension: \(59\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1888 q - 32 q^{2} - 32 q^{3} - 120 q^{4} - 32 q^{5} - 40 q^{6} + 16 q^{7} - 160 q^{8} + 140 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1888 q - 32 q^{2} - 32 q^{3} - 120 q^{4} - 32 q^{5} - 40 q^{6} + 16 q^{7} - 160 q^{8} + 140 q^{9} + 44 q^{10} - 32 q^{11} - 748 q^{12} - 32 q^{13} + 280 q^{14} + 944 q^{15} + 13816 q^{16} - 1392 q^{17} - 216 q^{18} - 24 q^{19} - 1088 q^{20} - 376 q^{21} + 472 q^{22} + 296 q^{23} - 2148 q^{24} - 40 q^{25} - 316 q^{26} + 412 q^{27} - 400 q^{28} + 444 q^{29} + 708 q^{30} + 3408 q^{31} - 1608 q^{32} - 1696 q^{33} + 744 q^{34} + 864 q^{35} + 10504 q^{36} - 1236 q^{37} - 1116 q^{38} - 1088 q^{39} - 1320 q^{40} + 1020 q^{41} - 3464 q^{42} + 236 q^{43} - 1592 q^{44} - 1380 q^{45} + 4584 q^{46} + 704 q^{47} + 1056 q^{48} - 2368 q^{49} - 12764 q^{50} - 360 q^{51} - 1104 q^{52} - 3080 q^{53} - 2628 q^{54} + 2152 q^{55} - 4136 q^{56} - 16 q^{57} + 1072 q^{58} + 1020 q^{59} + 1320 q^{60} + 536 q^{61} + 10276 q^{62} + 1396 q^{63} - 8456 q^{65} + 132 q^{66} + 1988 q^{67} + 1300 q^{68} + 26928 q^{69} - 492 q^{70} + 144 q^{71} - 13540 q^{72} + 1192 q^{73} + 9112 q^{74} + 12280 q^{75} - 14220 q^{76} + 12512 q^{77} - 12608 q^{78} - 5888 q^{79} - 13884 q^{80} - 16700 q^{81} - 6340 q^{82} - 196 q^{83} + 800 q^{84} + 724 q^{85} + 3092 q^{86} + 4888 q^{87} + 4688 q^{88} - 5128 q^{89} - 10828 q^{90} - 1368 q^{91} + 3244 q^{92} + 5320 q^{93} - 2132 q^{94} - 9856 q^{95} - 15768 q^{96} - 18688 q^{97} + 1424 q^{98} - 12872 q^{99}+O(q^{100}) \) 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 −5.45583 1.46189i −5.86831 0.307545i 20.7008 + 11.9516i 3.34193 6.55890i 31.5670 + 10.2567i 14.7378 13.9857i −63.5167 63.5167i 7.49043 + 0.787276i −27.8214 + 30.8988i
3.2 −5.40671 1.44872i 8.67744 + 0.454765i 20.2055 + 11.6657i 4.12759 8.10085i −46.2576 15.0300i −26.2321 + 24.8933i −60.6814 60.6814i 48.2391 + 5.07013i −34.0526 + 37.8193i
3.3 −5.05874 1.35549i 4.70412 + 0.246532i 16.8253 + 9.71410i −0.400189 + 0.785414i −23.4627 7.62351i 20.8543 19.7900i −42.3216 42.3216i −4.78414 0.502833i 3.08907 3.43076i
3.4 −4.89020 1.31033i −3.05889 0.160310i 15.2689 + 8.81552i −3.30014 + 6.47689i 14.7485 + 4.79209i −10.5852 + 10.0450i −34.4779 34.4779i −17.5210 1.84153i 24.6252 27.3490i
3.5 −4.59862 1.23220i 5.43488 + 0.284830i 12.7008 + 7.33280i −6.98441 + 13.7077i −24.6420 8.00666i 4.61565 4.38008i −22.4392 22.4392i 2.60470 + 0.273765i 49.0092 54.4302i
3.6 −4.47910 1.20017i −6.60498 0.346153i 11.6937 + 6.75136i −9.65223 + 18.9436i 29.1689 + 9.47755i 11.4575 10.8727i −18.0430 18.0430i 16.6539 + 1.75039i 65.9688 73.2657i
3.7 −4.45304 1.19319i −9.12845 0.478402i 11.4776 + 6.62662i 0.431636 0.847134i 40.0785 + 13.0223i −19.8889 + 18.8739i −17.1248 17.1248i 56.2477 + 5.91187i −2.93288 + 3.25730i
3.8 −4.42372 1.18533i −1.23361 0.0646508i 11.2361 + 6.48714i 5.75748 11.2997i 5.38051 + 1.74823i −9.77533 + 9.27645i −16.1086 16.1086i −25.3345 2.66276i −38.8633 + 43.1621i
3.9 −4.32461 1.15878i 0.352706 + 0.0184845i 10.4313 + 6.02252i 8.71129 17.0969i −1.50390 0.488645i 1.36770 1.29790i −12.8059 12.8059i −26.7280 2.80923i −57.4844 + 63.8429i
3.10 −4.30950 1.15473i 2.77153 + 0.145250i 10.3102 + 5.95259i −0.410992 + 0.806617i −11.7762 3.82631i −7.29875 + 6.92626i −12.3200 12.3200i −19.1918 2.01714i 2.70259 3.00153i
3.11 −3.73936 1.00196i −7.99140 0.418811i 6.05069 + 3.49336i 6.11073 11.9930i 29.4631 + 9.57313i 13.1400 12.4694i 2.77373 + 2.77373i 36.8349 + 3.87151i −34.8667 + 38.7234i
3.12 −3.69998 0.991406i 8.86257 + 0.464468i 5.77875 + 3.33636i 5.01594 9.84434i −32.3308 10.5049i 12.7714 12.1196i 3.59502 + 3.59502i 51.4774 + 5.41049i −28.3186 + 31.4510i
3.13 −3.56129 0.954244i 9.05328 + 0.474462i 4.84399 + 2.79668i −6.57630 + 12.9067i −31.7886 10.3287i −10.9085 + 10.3518i 6.27423 + 6.27423i 54.8846 + 5.76861i 35.7362 39.6891i
3.14 −3.19486 0.856060i −3.62401 0.189926i 2.54607 + 1.46998i −2.92117 + 5.73312i 11.4156 + 3.70916i 13.4020 12.7180i 11.8344 + 11.8344i −13.7547 1.44568i 14.2406 15.8158i
3.15 −2.94380 0.788790i −0.721247 0.0377990i 1.11559 + 0.644088i 0.0967234 0.189830i 2.09340 + 0.680186i 22.4174 21.2733i 14.4641 + 14.4641i −26.3333 2.76774i −0.434471 + 0.482529i
3.16 −2.80709 0.752159i 0.707572 + 0.0370823i 0.385833 + 0.222761i −6.46030 + 12.6791i −1.95833 0.636300i −21.5036 + 20.4062i 15.5240 + 15.5240i −26.3528 2.76979i 27.6713 30.7321i
3.17 −2.73130 0.731849i 5.39299 + 0.282635i −0.00381879 0.00220478i 5.62183 11.0335i −14.5230 4.71881i −9.28388 + 8.81007i 16.0044 + 16.0044i 2.15235 + 0.226221i −23.4297 + 26.0214i
3.18 −2.68145 0.718492i −6.29664 0.329993i −0.254269 0.146802i −3.49067 + 6.85084i 16.6470 + 5.40894i −10.0835 + 9.56891i 16.2800 + 16.2800i 12.6867 + 1.33343i 14.2823 15.8621i
3.19 −2.14885 0.575782i 3.61279 + 0.189338i −2.64218 1.52546i −0.925061 + 1.81553i −7.65432 2.48704i 3.61723 3.43262i 17.3838 + 17.3838i −13.8357 1.45419i 3.03317 3.36867i
3.20 −1.98415 0.531653i 5.99839 + 0.314362i −3.27399 1.89024i 4.80556 9.43143i −11.7346 3.81280i −19.1648 + 18.1867i 17.1112 + 17.1112i 9.02972 + 0.949062i −14.5492 + 16.1585i
See next 80 embeddings (of 1888 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.59
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 241.4.s.a 1888
241.s even 120 1 inner 241.4.s.a 1888
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.4.s.a 1888 1.a even 1 1 trivial
241.4.s.a 1888 241.s even 120 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(241, [\chi])\).