Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,3,Mod(13,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 7, 6]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.13");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 160.v (of order \(8\), degree \(4\), 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: | \(4.35968422976\) |
Analytic rank: | \(0\) |
Dimension: | \(184\) |
Relative dimension: | \(46\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −1.99562 | − | 0.132347i | −4.49249 | + | 1.86085i | 3.96497 | + | 0.528226i | 4.34634 | + | 2.47171i | 9.21157 | − | 3.11898i | − | 9.26439i | −7.84265 | − | 1.57889i | 10.3558 | − | 10.3558i | −8.34650 | − | 5.50781i | |
13.2 | −1.98630 | + | 0.233717i | 0.152407 | − | 0.0631292i | 3.89075 | − | 0.928464i | 1.13682 | − | 4.86905i | −0.287972 | + | 0.161014i | − | 5.77129i | −7.51119 | + | 2.75354i | −6.34472 | + | 6.34472i | −1.12009 | + | 9.93707i | |
13.3 | −1.98201 | + | 0.267624i | 1.91566 | − | 0.793494i | 3.85675 | − | 1.06087i | −0.189284 | + | 4.99642i | −3.58451 | + | 2.08539i | 3.33475i | −7.36023 | + | 3.13482i | −3.32383 | + | 3.32383i | −0.961999 | − | 9.95362i | ||
13.4 | −1.97439 | − | 0.319035i | −3.58463 | + | 1.48480i | 3.79643 | + | 1.25980i | −4.99639 | + | 0.190010i | 7.55117 | − | 1.78796i | 8.87581i | −7.09372 | − | 3.69853i | 4.28100 | − | 4.28100i | 9.92544 | + | 1.21887i | ||
13.5 | −1.96655 | − | 0.364277i | 4.81555 | − | 1.99467i | 3.73461 | + | 1.43273i | −4.99995 | + | 0.0217470i | −10.1966 | + | 2.16841i | − | 5.72496i | −6.82236 | − | 4.17796i | 12.8469 | − | 12.8469i | 9.84056 | + | 1.77860i | |
13.6 | −1.86795 | − | 0.714680i | −0.137258 | + | 0.0568543i | 2.97847 | + | 2.66997i | 4.82878 | − | 1.29725i | 0.297024 | − | 0.00810515i | 10.6903i | −3.65545 | − | 7.11602i | −6.34835 | + | 6.34835i | −9.94704 | − | 1.02785i | ||
13.7 | −1.70391 | + | 1.04723i | 5.07893 | − | 2.10376i | 1.80660 | − | 3.56878i | 2.76666 | − | 4.16480i | −6.45089 | + | 8.90345i | 13.0758i | 0.659077 | + | 7.97280i | 15.0058 | − | 15.0058i | −0.352614 | + | 9.99378i | ||
13.8 | −1.66520 | − | 1.10776i | 3.97729 | − | 1.64745i | 1.54576 | + | 3.68926i | 4.80144 | + | 1.39504i | −8.44794 | − | 1.66254i | − | 4.01011i | 1.51281 | − | 7.85566i | 6.74079 | − | 6.74079i | −6.44998 | − | 7.64184i | |
13.9 | −1.66483 | + | 1.10831i | 1.14849 | − | 0.475719i | 1.54331 | − | 3.69028i | −4.93972 | − | 0.774033i | −1.38479 | + | 2.06487i | − | 0.575911i | 1.52062 | + | 7.85415i | −5.27125 | + | 5.27125i | 9.08166 | − | 4.18610i | |
13.10 | −1.60071 | + | 1.19905i | −2.59734 | + | 1.07585i | 1.12457 | − | 3.83867i | 0.271832 | + | 4.99261i | 2.86760 | − | 4.83647i | 5.11582i | 2.80264 | + | 7.49301i | −0.775245 | + | 0.775245i | −6.42150 | − | 7.66579i | ||
13.11 | −1.53272 | − | 1.28482i | −0.518083 | + | 0.214597i | 0.698458 | + | 3.93855i | −1.24748 | + | 4.84188i | 1.06980 | + | 0.336728i | − | 7.03161i | 3.98980 | − | 6.93408i | −6.14160 | + | 6.14160i | 8.13300 | − | 5.81845i | |
13.12 | −1.50904 | + | 1.31255i | −4.42876 | + | 1.83445i | 0.554433 | − | 3.96139i | −0.125524 | − | 4.99842i | 4.27539 | − | 8.58123i | 2.19283i | 4.36285 | + | 6.70563i | 9.88475 | − | 9.88475i | 6.75010 | + | 7.37809i | ||
13.13 | −1.44356 | − | 1.38425i | −1.90642 | + | 0.789667i | 0.167724 | + | 3.99648i | −3.37146 | − | 3.69233i | 3.84513 | + | 1.49903i | − | 6.30545i | 5.29000 | − | 6.00133i | −3.35308 | + | 3.35308i | −0.244192 | + | 9.99702i | |
13.14 | −1.42228 | + | 1.40610i | 2.71037 | − | 1.12267i | 0.0457753 | − | 3.99974i | 4.68901 | + | 1.73586i | −2.27632 | + | 5.40780i | − | 12.7706i | 5.55892 | + | 5.75312i | −0.278249 | + | 0.278249i | −9.10988 | + | 4.12433i | |
13.15 | −1.01850 | − | 1.72124i | −3.74118 | + | 1.54965i | −1.92531 | + | 3.50616i | 4.07056 | − | 2.90354i | 6.47770 | + | 4.86113i | 2.79047i | 7.99587 | − | 0.257117i | 5.23103 | − | 5.23103i | −9.14355 | − | 4.04914i | ||
13.16 | −1.01554 | − | 1.72298i | 2.65551 | − | 1.09995i | −1.93734 | + | 3.49953i | −4.54176 | + | 2.09103i | −4.59197 | − | 3.45835i | 12.6857i | 7.99709 | − | 0.215915i | −0.522125 | + | 0.522125i | 8.21517 | + | 5.70185i | ||
13.17 | −0.756816 | + | 1.85128i | −3.28804 | + | 1.36195i | −2.85446 | − | 2.80215i | −3.29788 | + | 3.75819i | −0.0329077 | − | 7.11781i | − | 11.5572i | 7.34786 | − | 3.16368i | 2.59232 | − | 2.59232i | −4.46157 | − | 8.94955i | |
13.18 | −0.710424 | + | 1.86957i | 1.17810 | − | 0.487984i | −2.99060 | − | 2.65638i | −4.10083 | − | 2.86063i | 0.0753728 | + | 2.54922i | 3.86601i | 7.09088 | − | 3.70398i | −5.21417 | + | 5.21417i | 8.26148 | − | 5.63454i | ||
13.19 | −0.702991 | − | 1.87238i | 3.43142 | − | 1.42134i | −3.01161 | + | 2.63253i | 1.08056 | − | 4.88184i | −5.07355 | − | 5.42573i | − | 2.20595i | 7.04623 | + | 3.78822i | 3.39049 | − | 3.39049i | −9.90028 | + | 1.40868i | |
13.20 | −0.669438 | + | 1.88464i | −0.764258 | + | 0.316566i | −3.10371 | − | 2.52329i | 4.97587 | − | 0.490617i | −0.0849887 | − | 1.65227i | 4.84631i | 6.83323 | − | 4.16017i | −5.88009 | + | 5.88009i | −2.40640 | + | 9.70614i | ||
See next 80 embeddings (of 184 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
160.v | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 160.3.v.a | ✓ | 184 |
5.c | odd | 4 | 1 | 160.3.bb.a | yes | 184 | |
32.g | even | 8 | 1 | 160.3.bb.a | yes | 184 | |
160.v | odd | 8 | 1 | inner | 160.3.v.a | ✓ | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
160.3.v.a | ✓ | 184 | 1.a | even | 1 | 1 | trivial |
160.3.v.a | ✓ | 184 | 160.v | odd | 8 | 1 | inner |
160.3.bb.a | yes | 184 | 5.c | odd | 4 | 1 | |
160.3.bb.a | yes | 184 | 32.g | even | 8 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(160, [\chi])\).