Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [241,2,Mod(2,241)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(241, base_ring=CyclotomicField(24))
chi = DirichletCharacter(H, H._module([19]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("241.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 241 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 241.m (of order \(24\), degree \(8\), 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: | \(160\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{24})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{24}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −0.704088 | − | 2.62769i | 2.52559 | + | 0.676731i | −4.67697 | + | 2.70025i | 0.313348 | + | 0.313348i | − | 7.11296i | 2.71787 | − | 2.08550i | 6.54122 | + | 6.54122i | 3.32258 | + | 1.91829i | 0.602757 | − | 1.04401i | |
2.2 | −0.667750 | − | 2.49208i | −2.07790 | − | 0.556771i | −4.03251 | + | 2.32817i | 2.64848 | + | 2.64848i | 5.55006i | −0.592030 | + | 0.454281i | 4.84603 | + | 4.84603i | 1.40958 | + | 0.813823i | 4.83169 | − | 8.36872i | ||
2.3 | −0.601890 | − | 2.24628i | 0.155005 | + | 0.0415335i | −2.95147 | + | 1.70403i | −0.474194 | − | 0.474194i | − | 0.373184i | −2.57684 | + | 1.97728i | 2.31542 | + | 2.31542i | −2.57577 | − | 1.48712i | −0.779762 | + | 1.35059i | |
2.4 | −0.542269 | − | 2.02378i | −3.00395 | − | 0.804907i | −2.06956 | + | 1.19486i | −2.34995 | − | 2.34995i | 6.51580i | −1.24675 | + | 0.956667i | 0.577378 | + | 0.577378i | 5.77778 | + | 3.33581i | −3.48147 | + | 6.03008i | ||
2.5 | −0.427399 | − | 1.59507i | 1.86568 | + | 0.499906i | −0.629538 | + | 0.363464i | 1.72756 | + | 1.72756i | − | 3.18955i | −0.628487 | + | 0.482255i | −1.48653 | − | 1.48653i | 0.632762 | + | 0.365326i | 2.01722 | − | 3.49393i | |
2.6 | −0.420927 | − | 1.57092i | −1.37342 | − | 0.368007i | −0.558561 | + | 0.322485i | −0.0180347 | − | 0.0180347i | 2.31244i | 3.75903 | − | 2.88441i | −1.55828 | − | 1.55828i | −0.847224 | − | 0.489145i | −0.0207398 | + | 0.0359224i | ||
2.7 | −0.362641 | − | 1.35340i | 2.40372 | + | 0.644075i | 0.0318802 | − | 0.0184060i | −2.75079 | − | 2.75079i | − | 3.48675i | 1.31419 | − | 1.00841i | −2.01798 | − | 2.01798i | 2.76496 | + | 1.59635i | −2.72536 | + | 4.72046i | |
2.8 | −0.222155 | − | 0.829092i | 0.311636 | + | 0.0835027i | 1.09401 | − | 0.631627i | −1.02800 | − | 1.02800i | − | 0.276926i | −0.302572 | + | 0.232172i | −1.98059 | − | 1.98059i | −2.50793 | − | 1.44796i | −0.623932 | + | 1.08068i | |
2.9 | −0.104391 | − | 0.389594i | −2.83228 | − | 0.758906i | 1.59117 | − | 0.918660i | 2.04617 | + | 2.04617i | 1.18266i | 0.0933977 | − | 0.0716666i | −1.09441 | − | 1.09441i | 4.84778 | + | 2.79887i | 0.583573 | − | 1.01078i | ||
2.10 | −0.0692757 | − | 0.258540i | −1.49309 | − | 0.400072i | 1.67001 | − | 0.964179i | −2.09042 | − | 2.09042i | 0.413739i | −2.26261 | + | 1.73616i | −0.743500 | − | 0.743500i | −0.528818 | − | 0.305313i | −0.395643 | + | 0.685274i | ||
2.11 | 0.0371148 | + | 0.138514i | 0.407739 | + | 0.109253i | 1.71424 | − | 0.989718i | 2.06635 | + | 2.06635i | 0.0605326i | 0.858916 | − | 0.659069i | 0.403513 | + | 0.403513i | −2.44376 | − | 1.41091i | −0.209527 | + | 0.362912i | ||
2.12 | 0.103325 | + | 0.385613i | 2.49259 | + | 0.667887i | 1.59403 | − | 0.920313i | −0.673429 | − | 0.673429i | 1.03018i | −2.81988 | + | 2.16377i | 1.08416 | + | 1.08416i | 3.16885 | + | 1.82954i | 0.190101 | − | 0.329265i | ||
2.13 | 0.312380 | + | 1.16582i | −1.80359 | − | 0.483270i | 0.470498 | − | 0.271642i | 0.138268 | + | 0.138268i | − | 2.25362i | 1.25476 | − | 0.962814i | 2.17054 | + | 2.17054i | 0.421309 | + | 0.243243i | −0.118003 | + | 0.204388i | |
2.14 | 0.365174 | + | 1.36285i | 1.64953 | + | 0.441989i | 0.00804676 | − | 0.00464580i | −1.42518 | − | 1.42518i | 2.40946i | 3.20408 | − | 2.45858i | 2.00462 | + | 2.00462i | −0.0724922 | − | 0.0418534i | 1.42187 | − | 2.46275i | ||
2.15 | 0.393808 | + | 1.46971i | −2.34258 | − | 0.627694i | −0.272916 | + | 0.157568i | 0.386161 | + | 0.386161i | − | 3.69011i | −3.60979 | + | 2.76989i | 1.81275 | + | 1.81275i | 2.49563 | + | 1.44085i | −0.415472 | + | 0.719618i | |
2.16 | 0.456332 | + | 1.70305i | 1.41788 | + | 0.379919i | −0.960099 | + | 0.554313i | 0.286711 | + | 0.286711i | 2.58809i | −1.54437 | + | 1.18504i | 1.11129 | + | 1.11129i | −0.732035 | − | 0.422641i | −0.357449 | + | 0.619120i | ||
2.17 | 0.590754 | + | 2.20472i | 0.415803 | + | 0.111414i | −2.77976 | + | 1.60490i | 2.55729 | + | 2.55729i | 0.982547i | 1.45041 | − | 1.11294i | −1.95257 | − | 1.95257i | −2.43760 | − | 1.40735i | −4.12738 | + | 7.14883i | ||
2.18 | 0.610972 | + | 2.28018i | −2.28675 | − | 0.612733i | −3.09388 | + | 1.78626i | −2.05274 | − | 2.05274i | − | 5.58857i | 2.61515 | − | 2.00668i | −2.62485 | − | 2.62485i | 2.25571 | + | 1.30234i | 3.42644 | − | 5.93477i | |
2.19 | 0.655534 | + | 2.44649i | −0.436959 | − | 0.117083i | −3.82352 | + | 2.20751i | 0.510985 | + | 0.510985i | − | 1.14577i | −1.23402 | + | 0.946899i | −4.32520 | − | 4.32520i | −2.42085 | − | 1.39768i | −0.915150 | + | 1.58509i | |
2.20 | 0.722236 | + | 2.69542i | 3.10525 | + | 0.832050i | −5.01162 | + | 2.89346i | −1.52568 | − | 1.52568i | 8.97090i | 0.201148 | − | 0.154346i | −7.47229 | − | 7.47229i | 6.35222 | + | 3.66746i | 3.01045 | − | 5.21425i | ||
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
241.m | even | 24 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 241.2.m.a | ✓ | 160 |
241.m | even | 24 | 1 | inner | 241.2.m.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
241.2.m.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
241.2.m.a | ✓ | 160 | 241.m | even | 24 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).