Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,2,Mod(17,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([5, 27]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.n (of order \(30\), 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: | \(0.614848095564\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −1.95877 | − | 1.76368i | 0.650787 | + | 1.46169i | 0.517138 | + | 4.92024i | 0.446428 | + | 2.10028i | 1.30322 | − | 4.01090i | −1.30623 | + | 2.30082i | 4.56624 | − | 6.28489i | 0.294372 | − | 0.326934i | 2.82977 | − | 4.90131i |
17.2 | −1.33360 | − | 1.20078i | −1.17499 | − | 2.63908i | 0.127565 | + | 1.21370i | 0.418270 | + | 1.96780i | −1.60198 | + | 4.93040i | −1.17364 | − | 2.37120i | −0.822345 | + | 1.13186i | −3.57672 | + | 3.97235i | 1.80510 | − | 3.12652i |
17.3 | −1.11268 | − | 1.00186i | 0.245339 | + | 0.551041i | 0.0252713 | + | 0.240440i | −0.491349 | − | 2.31161i | 0.279082 | − | 0.858925i | 2.00635 | − | 1.72470i | −1.54736 | + | 2.12976i | 1.76394 | − | 1.95905i | −1.76920 | + | 3.06434i |
17.4 | −0.0202070 | − | 0.0181945i | −0.500742 | − | 1.12469i | −0.208980 | − | 1.98831i | −0.240558 | − | 1.13174i | −0.0103446 | + | 0.0318373i | −0.296416 | + | 2.62909i | −0.0639186 | + | 0.0879764i | 0.993218 | − | 1.10308i | −0.0157304 | + | 0.0272459i |
17.5 | 0.386517 | + | 0.348022i | 0.460952 | + | 1.03532i | −0.180780 | − | 1.72001i | 0.678628 | + | 3.19269i | −0.182146 | + | 0.560588i | −1.97073 | − | 1.76528i | 1.14015 | − | 1.56929i | 1.14799 | − | 1.27497i | −0.848825 | + | 1.47021i |
17.6 | 1.75157 | + | 1.57712i | −0.821231 | − | 1.84451i | 0.371633 | + | 3.53585i | −0.00617973 | − | 0.0290734i | 1.47058 | − | 4.52598i | −2.21185 | + | 1.45180i | −2.15474 | + | 2.96574i | −0.720422 | + | 0.800109i | 0.0350280 | − | 0.0606703i |
19.1 | −2.06490 | + | 0.217030i | −0.138863 | − | 0.125032i | 2.26043 | − | 0.480470i | −1.04356 | − | 2.34387i | 0.313874 | + | 0.228043i | 2.62996 | − | 0.288618i | −0.613987 | + | 0.199497i | −0.309936 | − | 2.94884i | 2.66354 | + | 4.61339i |
19.2 | −1.30277 | + | 0.136927i | 2.02980 | + | 1.82764i | −0.277833 | + | 0.0590552i | 0.166444 | + | 0.373838i | −2.89461 | − | 2.10306i | −2.50634 | + | 0.847492i | 2.84553 | − | 0.924570i | 0.466233 | + | 4.43591i | −0.268026 | − | 0.464235i |
19.3 | −0.476751 | + | 0.0501086i | −1.66384 | − | 1.49813i | −1.73151 | + | 0.368045i | −0.547175 | − | 1.22898i | 0.868306 | + | 0.630861i | −1.63770 | − | 2.07796i | 1.71889 | − | 0.558501i | 0.210388 | + | 2.00171i | 0.322449 | + | 0.558497i |
19.4 | 1.11135 | − | 0.116808i | 1.15989 | + | 1.04437i | −0.734836 | + | 0.156194i | −1.25246 | − | 2.81307i | 1.41104 | + | 1.02518i | 1.16071 | + | 2.37755i | −2.92398 | + | 0.950058i | −0.0589482 | − | 0.560855i | −1.72051 | − | 2.98001i |
19.5 | 1.39661 | − | 0.146790i | 0.151995 | + | 0.136856i | −0.0273118 | + | 0.00580530i | 0.618266 | + | 1.38865i | 0.232367 | + | 0.168824i | −1.57500 | − | 2.12588i | −2.70844 | + | 0.880026i | −0.309213 | − | 2.94196i | 1.06732 | + | 1.84865i |
19.6 | 2.05902 | − | 0.216412i | −2.33449 | − | 2.10199i | 2.23645 | − | 0.475371i | 0.483200 | + | 1.08528i | −5.26167 | − | 3.82283i | 2.02173 | + | 1.70664i | 0.563952 | − | 0.183239i | 0.717924 | + | 6.83059i | 1.22979 | + | 2.13006i |
24.1 | −0.548010 | − | 2.57818i | 1.59126 | − | 0.167248i | −4.51962 | + | 2.01227i | −1.59568 | − | 1.43676i | −1.30322 | − | 4.01090i | 2.59186 | + | 0.531305i | 4.56624 | + | 6.28489i | −0.430319 | + | 0.0914672i | −2.82977 | + | 4.90131i |
24.2 | −0.373106 | − | 1.75533i | −2.87300 | + | 0.301965i | −1.11488 | + | 0.496374i | −1.49503 | − | 1.34613i | 1.60198 | + | 4.93040i | −1.89247 | + | 1.84893i | −0.822345 | − | 1.13186i | 5.22852 | − | 1.11136i | −1.80510 | + | 3.12652i |
24.3 | −0.311296 | − | 1.46453i | 0.599886 | − | 0.0630505i | −0.220863 | + | 0.0983344i | 1.75624 | + | 1.58133i | −0.279082 | − | 0.858925i | −2.26028 | − | 1.37519i | −1.54736 | − | 2.12976i | −2.57856 | + | 0.548089i | 1.76920 | − | 3.06434i |
24.4 | −0.00565338 | − | 0.0265970i | −1.22438 | + | 0.128687i | 1.82642 | − | 0.813173i | 0.859835 | + | 0.774198i | 0.0103446 | + | 0.0318373i | 2.59202 | − | 0.530526i | −0.0639186 | − | 0.0879764i | −1.45190 | + | 0.308612i | 0.0157304 | − | 0.0272459i |
24.5 | 0.108137 | + | 0.508745i | 1.12709 | − | 0.118461i | 1.57996 | − | 0.703445i | −2.42564 | − | 2.18406i | 0.182146 | + | 0.560588i | −1.06990 | + | 2.41978i | 1.14015 | + | 1.56929i | −1.67816 | + | 0.356703i | 0.848825 | − | 1.47021i |
24.6 | 0.490042 | + | 2.30547i | −2.00801 | + | 0.211051i | −3.24795 | + | 1.44608i | 0.0220884 | + | 0.0198885i | −1.47058 | − | 4.52598i | 2.06424 | + | 1.65497i | −2.15474 | − | 2.96574i | 1.05313 | − | 0.223849i | −0.0350280 | + | 0.0606703i |
40.1 | −0.842093 | − | 1.89137i | 0.653128 | + | 3.07273i | −1.52991 | + | 1.69913i | 1.18148 | − | 0.124179i | 5.26167 | − | 3.82283i | 2.63875 | + | 0.192354i | 0.563952 | + | 0.183239i | −6.27443 | + | 2.79356i | −1.22979 | − | 2.13006i |
40.2 | −0.571183 | − | 1.28290i | −0.0425239 | − | 0.200059i | 0.0186834 | − | 0.0207501i | 1.51174 | − | 0.158890i | −0.232367 | + | 0.168824i | −2.52376 | − | 0.794113i | −2.70844 | − | 0.880026i | 2.70242 | − | 1.20320i | −1.06732 | − | 1.84865i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
11.d | odd | 10 | 1 | inner |
77.n | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.2.n.a | ✓ | 48 |
3.b | odd | 2 | 1 | 693.2.cg.a | 48 | ||
7.b | odd | 2 | 1 | 539.2.s.d | 48 | ||
7.c | even | 3 | 1 | 539.2.m.a | 48 | ||
7.c | even | 3 | 1 | 539.2.s.d | 48 | ||
7.d | odd | 6 | 1 | inner | 77.2.n.a | ✓ | 48 |
7.d | odd | 6 | 1 | 539.2.m.a | 48 | ||
11.b | odd | 2 | 1 | 847.2.r.c | 48 | ||
11.c | even | 5 | 1 | 847.2.i.b | 48 | ||
11.c | even | 5 | 1 | 847.2.r.a | 48 | ||
11.c | even | 5 | 1 | 847.2.r.c | 48 | ||
11.c | even | 5 | 1 | 847.2.r.d | 48 | ||
11.d | odd | 10 | 1 | inner | 77.2.n.a | ✓ | 48 |
11.d | odd | 10 | 1 | 847.2.i.b | 48 | ||
11.d | odd | 10 | 1 | 847.2.r.a | 48 | ||
11.d | odd | 10 | 1 | 847.2.r.d | 48 | ||
21.g | even | 6 | 1 | 693.2.cg.a | 48 | ||
33.f | even | 10 | 1 | 693.2.cg.a | 48 | ||
77.i | even | 6 | 1 | 847.2.r.c | 48 | ||
77.l | even | 10 | 1 | 539.2.s.d | 48 | ||
77.n | even | 30 | 1 | inner | 77.2.n.a | ✓ | 48 |
77.n | even | 30 | 1 | 539.2.m.a | 48 | ||
77.n | even | 30 | 1 | 847.2.i.b | 48 | ||
77.n | even | 30 | 1 | 847.2.r.a | 48 | ||
77.n | even | 30 | 1 | 847.2.r.d | 48 | ||
77.o | odd | 30 | 1 | 539.2.m.a | 48 | ||
77.o | odd | 30 | 1 | 539.2.s.d | 48 | ||
77.p | odd | 30 | 1 | 847.2.i.b | 48 | ||
77.p | odd | 30 | 1 | 847.2.r.a | 48 | ||
77.p | odd | 30 | 1 | 847.2.r.c | 48 | ||
77.p | odd | 30 | 1 | 847.2.r.d | 48 | ||
231.bf | odd | 30 | 1 | 693.2.cg.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.2.n.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
77.2.n.a | ✓ | 48 | 7.d | odd | 6 | 1 | inner |
77.2.n.a | ✓ | 48 | 11.d | odd | 10 | 1 | inner |
77.2.n.a | ✓ | 48 | 77.n | even | 30 | 1 | inner |
539.2.m.a | 48 | 7.c | even | 3 | 1 | ||
539.2.m.a | 48 | 7.d | odd | 6 | 1 | ||
539.2.m.a | 48 | 77.n | even | 30 | 1 | ||
539.2.m.a | 48 | 77.o | odd | 30 | 1 | ||
539.2.s.d | 48 | 7.b | odd | 2 | 1 | ||
539.2.s.d | 48 | 7.c | even | 3 | 1 | ||
539.2.s.d | 48 | 77.l | even | 10 | 1 | ||
539.2.s.d | 48 | 77.o | odd | 30 | 1 | ||
693.2.cg.a | 48 | 3.b | odd | 2 | 1 | ||
693.2.cg.a | 48 | 21.g | even | 6 | 1 | ||
693.2.cg.a | 48 | 33.f | even | 10 | 1 | ||
693.2.cg.a | 48 | 231.bf | odd | 30 | 1 | ||
847.2.i.b | 48 | 11.c | even | 5 | 1 | ||
847.2.i.b | 48 | 11.d | odd | 10 | 1 | ||
847.2.i.b | 48 | 77.n | even | 30 | 1 | ||
847.2.i.b | 48 | 77.p | odd | 30 | 1 | ||
847.2.r.a | 48 | 11.c | even | 5 | 1 | ||
847.2.r.a | 48 | 11.d | odd | 10 | 1 | ||
847.2.r.a | 48 | 77.n | even | 30 | 1 | ||
847.2.r.a | 48 | 77.p | odd | 30 | 1 | ||
847.2.r.c | 48 | 11.b | odd | 2 | 1 | ||
847.2.r.c | 48 | 11.c | even | 5 | 1 | ||
847.2.r.c | 48 | 77.i | even | 6 | 1 | ||
847.2.r.c | 48 | 77.p | odd | 30 | 1 | ||
847.2.r.d | 48 | 11.c | even | 5 | 1 | ||
847.2.r.d | 48 | 11.d | odd | 10 | 1 | ||
847.2.r.d | 48 | 77.n | even | 30 | 1 | ||
847.2.r.d | 48 | 77.p | odd | 30 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(77, [\chi])\).