Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,2,Mod(98,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 9, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("465.98");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 465.be (of order \(12\), 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: | \(3.71304369399\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
98.1 | −1.93622 | − | 1.93622i | −1.72994 | − | 0.0855625i | 5.49791i | 1.23362 | − | 1.86499i | 3.18387 | + | 3.51521i | −0.576519 | − | 2.15160i | 6.77273 | − | 6.77273i | 2.98536 | + | 0.296035i | −5.99959 | + | 1.22248i | ||
98.2 | −1.87901 | − | 1.87901i | −0.840793 | + | 1.51429i | 5.06133i | 0.913745 | + | 2.04085i | 4.42521 | − | 1.26550i | 0.793040 | + | 2.95967i | 5.75226 | − | 5.75226i | −1.58614 | − | 2.54640i | 2.11784 | − | 5.55170i | ||
98.3 | −1.84495 | − | 1.84495i | 1.68449 | − | 0.403121i | 4.80768i | −1.30163 | + | 1.81817i | −3.85153 | − | 2.36405i | −0.819803 | − | 3.05955i | 5.18002 | − | 5.18002i | 2.67499 | − | 1.35810i | 5.75588 | − | 0.952991i | ||
98.4 | −1.84242 | − | 1.84242i | −1.10472 | − | 1.33402i | 4.78904i | −1.47481 | + | 1.68076i | −0.422471 | + | 4.49318i | 0.651939 | + | 2.43307i | 5.13859 | − | 5.13859i | −0.559206 | + | 2.94742i | 5.81388 | − | 0.379441i | ||
98.5 | −1.84102 | − | 1.84102i | 0.834923 | + | 1.51753i | 4.77870i | −1.66280 | − | 1.49502i | 1.25670 | − | 4.33091i | 0.308327 | + | 1.15069i | 5.11564 | − | 5.11564i | −1.60581 | + | 2.53404i | 0.308888 | + | 5.81361i | ||
98.6 | −1.68480 | − | 1.68480i | 0.167725 | − | 1.72391i | 3.67707i | −0.970904 | − | 2.01429i | −3.18702 | + | 2.62186i | −1.13309 | − | 4.22876i | 2.82552 | − | 2.82552i | −2.94374 | − | 0.578284i | −1.75788 | + | 5.02943i | ||
98.7 | −1.62658 | − | 1.62658i | 1.72162 | − | 0.189762i | 3.29154i | 1.58899 | − | 1.57325i | −3.10903 | − | 2.49170i | 0.419558 | + | 1.56581i | 2.10079 | − | 2.10079i | 2.92798 | − | 0.653398i | −5.14364 | − | 0.0256022i | ||
98.8 | −1.46359 | − | 1.46359i | −1.13704 | − | 1.30657i | 2.28419i | 2.22776 | − | 0.192591i | −0.248128 | + | 3.57645i | 0.281899 | + | 1.05206i | 0.415932 | − | 0.415932i | −0.414276 | + | 2.97126i | −3.54240 | − | 2.97865i | ||
98.9 | −1.44260 | − | 1.44260i | −1.11708 | + | 1.32368i | 2.16217i | −2.23322 | − | 0.112833i | 3.52104 | − | 0.298036i | −0.231092 | − | 0.862448i | 0.233953 | − | 0.233953i | −0.504254 | − | 2.95732i | 3.05886 | + | 3.38441i | ||
98.10 | −1.40988 | − | 1.40988i | −0.596389 | + | 1.62614i | 1.97552i | 2.23540 | − | 0.0546927i | 3.13349 | − | 1.45182i | −0.419910 | − | 1.56713i | −0.0345080 | + | 0.0345080i | −2.28864 | − | 1.93962i | −3.22875 | − | 3.07453i | ||
98.11 | −1.38421 | − | 1.38421i | 1.36494 | + | 1.06628i | 1.83208i | 0.940181 | + | 2.02881i | −0.413413 | − | 3.36531i | 0.112718 | + | 0.420667i | −0.232438 | + | 0.232438i | 0.726114 | + | 2.91080i | 1.50689 | − | 4.10971i | ||
98.12 | −1.37486 | − | 1.37486i | −1.73040 | + | 0.0756319i | 1.78046i | 0.524907 | + | 2.17359i | 2.48303 | + | 2.27507i | −1.05912 | − | 3.95270i | −0.301832 | + | 0.301832i | 2.98856 | − | 0.261747i | 2.26670 | − | 3.71004i | ||
98.13 | −1.33275 | − | 1.33275i | −1.70465 | − | 0.306863i | 1.55244i | −1.43453 | − | 1.71526i | 1.86290 | + | 2.68084i | 0.858496 | + | 3.20395i | −0.596486 | + | 0.596486i | 2.81167 | + | 1.04619i | −0.374142 | + | 4.19788i | ||
98.14 | −1.13894 | − | 1.13894i | 0.0753204 | − | 1.73041i | 0.594373i | −1.62755 | + | 1.53332i | −2.05662 | + | 1.88505i | −0.0981291 | − | 0.366223i | −1.60093 | + | 1.60093i | −2.98865 | − | 0.260671i | 3.60005 | + | 0.107326i | ||
98.15 | −1.07746 | − | 1.07746i | −0.0990226 | − | 1.72922i | 0.321843i | 0.838391 | − | 2.07294i | −1.75647 | + | 1.96986i | 0.299875 | + | 1.11915i | −1.80815 | + | 1.80815i | −2.98039 | + | 0.342463i | −3.13685 | + | 1.33018i | ||
98.16 | −1.04825 | − | 1.04825i | 1.14484 | − | 1.29975i | 0.197665i | 0.693500 | + | 2.12581i | −2.56254 | + | 0.162386i | 1.04385 | + | 3.89570i | −1.88930 | + | 1.88930i | −0.378694 | − | 2.97600i | 1.50142 | − | 2.95535i | ||
98.17 | −1.03032 | − | 1.03032i | 0.741075 | + | 1.56551i | 0.123114i | −2.05042 | + | 0.892066i | 0.849426 | − | 2.37651i | −0.433832 | − | 1.61908i | −1.93379 | + | 1.93379i | −1.90162 | + | 2.32031i | 3.03170 | + | 1.19347i | ||
98.18 | −0.989024 | − | 0.989024i | 0.175826 | + | 1.72310i | − | 0.0436644i | 1.18284 | − | 1.89760i | 1.53029 | − | 1.87809i | 1.22619 | + | 4.57620i | −2.02123 | + | 2.02123i | −2.93817 | + | 0.605932i | −3.04663 | + | 0.706916i | |
98.19 | −0.870093 | − | 0.870093i | 1.66082 | − | 0.491607i | − | 0.485876i | −1.68819 | − | 1.46629i | −1.87281 | − | 1.01732i | −0.0904276 | − | 0.337480i | −2.16294 | + | 2.16294i | 2.51665 | − | 1.63294i | 0.193071 | + | 2.74469i | |
98.20 | −0.816536 | − | 0.816536i | 1.28127 | − | 1.16548i | − | 0.666538i | 2.16826 | + | 0.546490i | −1.99786 | − | 0.0945498i | −1.26333 | − | 4.71479i | −2.17732 | + | 2.17732i | 0.283319 | − | 2.98659i | −1.32423 | − | 2.21669i | |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
31.c | even | 3 | 1 | inner |
93.h | odd | 6 | 1 | inner |
155.o | odd | 12 | 1 | inner |
465.be | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 465.2.be.a | ✓ | 240 |
3.b | odd | 2 | 1 | inner | 465.2.be.a | ✓ | 240 |
5.c | odd | 4 | 1 | inner | 465.2.be.a | ✓ | 240 |
15.e | even | 4 | 1 | inner | 465.2.be.a | ✓ | 240 |
31.c | even | 3 | 1 | inner | 465.2.be.a | ✓ | 240 |
93.h | odd | 6 | 1 | inner | 465.2.be.a | ✓ | 240 |
155.o | odd | 12 | 1 | inner | 465.2.be.a | ✓ | 240 |
465.be | even | 12 | 1 | inner | 465.2.be.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
465.2.be.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
465.2.be.a | ✓ | 240 | 3.b | odd | 2 | 1 | inner |
465.2.be.a | ✓ | 240 | 5.c | odd | 4 | 1 | inner |
465.2.be.a | ✓ | 240 | 15.e | even | 4 | 1 | inner |
465.2.be.a | ✓ | 240 | 31.c | even | 3 | 1 | inner |
465.2.be.a | ✓ | 240 | 93.h | odd | 6 | 1 | inner |
465.2.be.a | ✓ | 240 | 155.o | odd | 12 | 1 | inner |
465.2.be.a | ✓ | 240 | 465.be | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).