Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(79,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.79");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.q (of order \(3\), degree \(2\), 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: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
79.1 | −1.30642 | − | 2.26278i | −1.71591 | − | 0.235914i | −2.41345 | + | 4.18023i | −0.409745 | 1.70787 | + | 4.19093i | 2.62346 | + | 0.342704i | 7.38625 | 2.88869 | + | 0.809612i | 0.535298 | + | 0.927164i | ||||
79.2 | −1.28061 | − | 2.21808i | 1.63464 | − | 0.572663i | −2.27991 | + | 3.94892i | 3.88979 | −3.36355 | − | 2.89241i | 1.21756 | − | 2.34895i | 6.55624 | 2.34411 | − | 1.87220i | −4.98129 | − | 8.62785i | ||||
79.3 | −1.23220 | − | 2.13423i | 1.15780 | − | 1.28822i | −2.03662 | + | 3.52753i | −1.72333 | −4.17599 | − | 0.883668i | −2.48312 | − | 0.913305i | 5.10930 | −0.319007 | − | 2.98299i | 2.12348 | + | 3.67797i | ||||
79.4 | −1.23208 | − | 2.13403i | −1.12814 | − | 1.31427i | −2.03606 | + | 3.52656i | 2.20096 | −1.41473 | + | 4.02678i | −2.44344 | + | 1.01469i | 5.10606 | −0.454599 | + | 2.96536i | −2.71177 | − | 4.69692i | ||||
79.5 | −1.16734 | − | 2.02190i | −0.0130928 | + | 1.73200i | −1.72537 | + | 2.98844i | −1.75883 | 3.51721 | − | 1.99537i | −0.457521 | + | 2.60589i | 3.38704 | −2.99966 | − | 0.0453535i | 2.05316 | + | 3.55617i | ||||
79.6 | −1.11130 | − | 1.92482i | 1.65382 | + | 0.514665i | −1.46996 | + | 2.54604i | −2.34763 | −0.847245 | − | 3.75525i | −0.424108 | − | 2.61154i | 2.08904 | 2.47024 | + | 1.70233i | 2.60891 | + | 4.51876i | ||||
79.7 | −1.10741 | − | 1.91809i | −1.45241 | + | 0.943673i | −1.45271 | + | 2.51616i | 1.29021 | 3.41845 | + | 1.74081i | 0.190765 | − | 2.63887i | 2.00533 | 1.21896 | − | 2.74119i | −1.42879 | − | 2.47473i | ||||
79.8 | −0.965178 | − | 1.67174i | −0.461029 | − | 1.66957i | −0.863138 | + | 1.49500i | −3.28745 | −2.34610 | + | 2.38215i | 2.50497 | + | 0.851542i | −0.528383 | −2.57490 | + | 1.53944i | 3.17297 | + | 5.49575i | ||||
79.9 | −0.936148 | − | 1.62146i | 1.61108 | + | 0.635949i | −0.752744 | + | 1.30379i | 3.05727 | −0.477042 | − | 3.20763i | −0.901056 | + | 2.48759i | −0.925871 | 2.19114 | + | 2.04913i | −2.86206 | − | 4.95723i | ||||
79.10 | −0.871435 | − | 1.50937i | −0.149918 | + | 1.72555i | −0.518798 | + | 0.898585i | 3.07299 | 2.73514 | − | 1.27742i | −2.63585 | − | 0.228635i | −1.67734 | −2.95505 | − | 0.517383i | −2.67791 | − | 4.63828i | ||||
79.11 | −0.797019 | − | 1.38048i | 1.23146 | + | 1.21799i | −0.270480 | + | 0.468484i | −2.27464 | 0.699905 | − | 2.67077i | 1.37143 | − | 2.26256i | −2.32577 | 0.0330102 | + | 2.99982i | 1.81293 | + | 3.14009i | ||||
79.12 | −0.725750 | − | 1.25704i | −1.70576 | + | 0.300649i | −0.0534274 | + | 0.0925389i | 1.67650 | 1.61588 | + | 1.92600i | 0.491302 | + | 2.59974i | −2.74790 | 2.81922 | − | 1.02567i | −1.21672 | − | 2.10743i | ||||
79.13 | −0.712041 | − | 1.23329i | 1.33346 | − | 1.10539i | −0.0140047 | + | 0.0242568i | −0.0771886 | −2.31275 | − | 0.857454i | 2.64248 | − | 0.131501i | −2.80828 | 0.556210 | − | 2.94799i | 0.0549615 | + | 0.0951961i | ||||
79.14 | −0.691944 | − | 1.19848i | −1.54607 | − | 0.780820i | 0.0424270 | − | 0.0734858i | −2.75753 | 0.133993 | + | 2.39322i | −0.989701 | + | 2.45367i | −2.88520 | 1.78064 | + | 2.41440i | 1.90806 | + | 3.30485i | ||||
79.15 | −0.666890 | − | 1.15509i | −1.17774 | + | 1.27001i | 0.110515 | − | 0.191417i | −3.70643 | 2.25240 | + | 0.513440i | −2.40597 | − | 1.10059i | −2.96237 | −0.225845 | − | 2.99149i | 2.47178 | + | 4.28126i | ||||
79.16 | −0.601202 | − | 1.04131i | 0.371418 | + | 1.69176i | 0.277112 | − | 0.479971i | 2.56337 | 1.53835 | − | 1.40385i | 2.53790 | − | 0.747695i | −3.07121 | −2.72410 | + | 1.25670i | −1.54110 | − | 2.66927i | ||||
79.17 | −0.301506 | − | 0.522224i | −0.286088 | − | 1.70826i | 0.818188 | − | 1.41714i | 3.51406 | −0.805836 | + | 0.664453i | 1.42443 | + | 2.22957i | −2.19278 | −2.83631 | + | 0.977427i | −1.05951 | − | 1.83512i | ||||
79.18 | −0.276008 | − | 0.478060i | −0.0310116 | − | 1.73177i | 0.847639 | − | 1.46815i | −1.57072 | −0.819332 | + | 0.492809i | −2.63406 | − | 0.248479i | −2.03985 | −2.99808 | + | 0.107410i | 0.433531 | + | 0.750897i | ||||
79.19 | −0.275079 | − | 0.476450i | 1.69564 | − | 0.353295i | 0.848663 | − | 1.46993i | 1.36525 | −0.634761 | − | 0.710703i | −1.69022 | − | 2.03548i | −2.03411 | 2.75037 | − | 1.19812i | −0.375551 | − | 0.650473i | ||||
79.20 | −0.208396 | − | 0.360952i | −1.38547 | − | 1.03945i | 0.913143 | − | 1.58161i | 3.62518 | −0.0864652 | + | 0.716707i | −1.74876 | − | 1.98541i | −1.59476 | 0.839080 | + | 2.88027i | −0.755471 | − | 1.30851i | ||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.q.b | ✓ | 96 |
7.c | even | 3 | 1 | 819.2.r.a | yes | 96 | |
9.c | even | 3 | 1 | 819.2.r.a | yes | 96 | |
63.g | even | 3 | 1 | inner | 819.2.q.b | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.q.b | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
819.2.q.b | ✓ | 96 | 63.g | even | 3 | 1 | inner |
819.2.r.a | yes | 96 | 7.c | even | 3 | 1 | |
819.2.r.a | yes | 96 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{96} - 6 T_{2}^{95} + 90 T_{2}^{94} - 440 T_{2}^{93} + 3954 T_{2}^{92} - 16835 T_{2}^{91} + 115310 T_{2}^{90} - 440438 T_{2}^{89} + 2514469 T_{2}^{88} - 8770004 T_{2}^{87} + 43669833 T_{2}^{86} + \cdots + 502681 \)
acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).