Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(25,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([24, 22]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bl (of order \(15\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −0.285584 | + | 2.71715i | −0.403470 | + | 0.293138i | −5.34504 | − | 1.13612i | −1.62469 | − | 2.81404i | −0.681275 | − | 1.18000i | −0.0580480 | + | 0.552290i | 2.92494 | − | 9.00203i | −0.850193 | + | 2.61662i | 8.11015 | − | 3.61087i |
25.2 | −0.281438 | + | 2.67771i | −1.08221 | + | 0.786270i | −5.13461 | − | 1.09140i | 1.29436 | + | 2.24190i | −1.80083 | − | 3.11912i | −0.0373564 | + | 0.355422i | 2.70348 | − | 8.32047i | −0.374098 | + | 1.15136i | −6.36743 | + | 2.83496i |
25.3 | −0.278667 | + | 2.65134i | 1.36335 | − | 0.990534i | −4.99567 | − | 1.06186i | −0.0346755 | − | 0.0600598i | 2.24632 | + | 3.89075i | 0.0117499 | − | 0.111793i | 2.55985 | − | 7.87840i | −0.0494776 | + | 0.152276i | 0.168902 | − | 0.0752001i |
25.4 | −0.264807 | + | 2.51947i | 1.12049 | − | 0.814085i | −4.32131 | − | 0.918523i | 1.77724 | + | 3.07827i | 1.75435 | + | 3.03862i | −0.466240 | + | 4.43597i | 1.89281 | − | 5.82546i | −0.334284 | + | 1.02882i | −8.22622 | + | 3.66255i |
25.5 | −0.259044 | + | 2.46464i | 2.52872 | − | 1.83722i | −4.05106 | − | 0.861079i | −1.07529 | − | 1.86245i | 3.87305 | + | 6.70831i | 0.0234834 | − | 0.223430i | 1.64003 | − | 5.04750i | 2.09199 | − | 6.43848i | 4.86882 | − | 2.16774i |
25.6 | −0.257677 | + | 2.45163i | −2.00543 | + | 1.45703i | −3.98780 | − | 0.847633i | 0.396793 | + | 0.687265i | −3.05534 | − | 5.29201i | 0.406827 | − | 3.87070i | 1.58211 | − | 4.86923i | 0.971759 | − | 2.99077i | −1.78716 | + | 0.795697i |
25.7 | −0.233053 | + | 2.21735i | −2.47998 | + | 1.80181i | −2.90603 | − | 0.617697i | −1.93612 | − | 3.35346i | −3.41728 | − | 5.91890i | 0.0513781 | − | 0.488830i | 0.668963 | − | 2.05886i | 1.97673 | − | 6.08374i | 7.88702 | − | 3.51153i |
25.8 | −0.230219 | + | 2.19039i | −0.668805 | + | 0.485916i | −2.78852 | − | 0.592718i | −0.140529 | − | 0.243404i | −0.910374 | − | 1.57681i | −0.0854680 | + | 0.813173i | 0.579063 | − | 1.78217i | −0.715864 | + | 2.20320i | 0.565503 | − | 0.251778i |
25.9 | −0.220803 | + | 2.10080i | 1.10241 | − | 0.800945i | −2.40832 | − | 0.511903i | −0.922080 | − | 1.59709i | 1.43921 | + | 2.49279i | 0.240891 | − | 2.29192i | 0.301652 | − | 0.928389i | −0.353265 | + | 1.08724i | 3.55877 | − | 1.58446i |
25.10 | −0.219890 | + | 2.09211i | −1.49834 | + | 1.08861i | −2.37228 | − | 0.504243i | −0.827887 | − | 1.43394i | −1.94802 | − | 3.37407i | −0.393963 | + | 3.74831i | 0.276454 | − | 0.850839i | 0.132907 | − | 0.409047i | 3.18201 | − | 1.41672i |
25.11 | −0.216483 | + | 2.05970i | 0.649806 | − | 0.472112i | −2.23921 | − | 0.475960i | 1.63567 | + | 2.83307i | 0.831737 | + | 1.44061i | 0.530941 | − | 5.05157i | 0.185110 | − | 0.569710i | −0.727693 | + | 2.23961i | −6.18937 | + | 2.75568i |
25.12 | −0.202793 | + | 1.92945i | 1.75434 | − | 1.27461i | −1.72536 | − | 0.366736i | 0.246757 | + | 0.427396i | 2.10352 | + | 3.64340i | −0.474463 | + | 4.51421i | −0.141544 | + | 0.435629i | 0.526052 | − | 1.61902i | −0.874681 | + | 0.389433i |
25.13 | −0.201519 | + | 1.91733i | −2.58944 | + | 1.88134i | −1.67924 | − | 0.356933i | 1.74045 | + | 3.01455i | −3.08532 | − | 5.34392i | −0.219352 | + | 2.08699i | −0.168744 | + | 0.519340i | 2.23871 | − | 6.89003i | −6.13062 | + | 2.72953i |
25.14 | −0.199584 | + | 1.89891i | 2.76603 | − | 2.00964i | −1.60974 | − | 0.342160i | 1.60160 | + | 2.77406i | 3.26407 | + | 5.65354i | 0.186263 | − | 1.77217i | −0.209047 | + | 0.643381i | 2.68523 | − | 8.26429i | −5.58735 | + | 2.48765i |
25.15 | −0.190665 | + | 1.81405i | 0.321613 | − | 0.233666i | −1.29815 | − | 0.275930i | −1.21559 | − | 2.10547i | 0.362562 | + | 0.627976i | 0.219407 | − | 2.08752i | −0.379261 | + | 1.16724i | −0.878216 | + | 2.70287i | 4.05121 | − | 1.80371i |
25.16 | −0.174592 | + | 1.66113i | −1.40510 | + | 1.02086i | −0.772583 | − | 0.164218i | 0.317456 | + | 0.549850i | −1.45047 | − | 2.51229i | −0.373806 | + | 3.55653i | −0.624617 | + | 1.92237i | 0.00508971 | − | 0.0156645i | −0.968800 | + | 0.431337i |
25.17 | −0.161327 | + | 1.53493i | 0.869183 | − | 0.631498i | −0.373680 | − | 0.0794282i | 1.42302 | + | 2.46474i | 0.829081 | + | 1.43601i | −0.00370149 | + | 0.0352173i | −0.771661 | + | 2.37493i | −0.570362 | + | 1.75540i | −4.01277 | + | 1.78660i |
25.18 | −0.129542 | + | 1.23251i | −0.658544 | + | 0.478460i | 0.454007 | + | 0.0965023i | 1.33569 | + | 2.31348i | −0.504396 | − | 0.873639i | 0.0276602 | − | 0.263170i | −0.943678 | + | 2.90434i | −0.722295 | + | 2.22300i | −3.02441 | + | 1.34655i |
25.19 | −0.125849 | + | 1.19737i | 1.72539 | − | 1.25357i | 0.538429 | + | 0.114447i | −0.362674 | − | 0.628170i | 1.28385 | + | 2.22369i | −0.0567740 | + | 0.540169i | −0.948890 | + | 2.92038i | 0.478476 | − | 1.47260i | 0.797796 | − | 0.355202i |
25.20 | −0.115962 | + | 1.10330i | −1.60803 | + | 1.16830i | 0.752464 | + | 0.159941i | −0.568079 | − | 0.983942i | −1.10252 | − | 1.90962i | 0.388961 | − | 3.70072i | −0.949356 | + | 2.92182i | 0.293774 | − | 0.904143i | 1.15146 | − | 0.512664i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
671.bl | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.bl.a | ✓ | 480 |
11.c | even | 5 | 1 | 671.2.bp.a | yes | 480 | |
61.i | even | 15 | 1 | 671.2.bp.a | yes | 480 | |
671.bl | even | 15 | 1 | inner | 671.2.bl.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bl.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
671.2.bl.a | ✓ | 480 | 671.bl | even | 15 | 1 | inner |
671.2.bp.a | yes | 480 | 11.c | even | 5 | 1 | |
671.2.bp.a | yes | 480 | 61.i | even | 15 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).