Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,2,Mod(26,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("465.26");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 465.o (of order \(6\), 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: | \(3.71304369399\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 | − | 2.75560i | 0.965315 | + | 1.43811i | −5.59331 | 0.866025 | − | 0.500000i | 3.96286 | − | 2.66002i | −2.20986 | + | 3.82759i | 9.90170i | −1.13633 | + | 2.77646i | −1.37780 | − | 2.38642i | |||||
26.2 | − | 2.64726i | −1.47729 | + | 0.904227i | −5.00797 | −0.866025 | + | 0.500000i | 2.39372 | + | 3.91076i | −0.0311239 | + | 0.0539082i | 7.96287i | 1.36475 | − | 2.67160i | 1.32363 | + | 2.29259i | |||||
26.3 | − | 2.62586i | −0.134987 | − | 1.72678i | −4.89514 | 0.866025 | − | 0.500000i | −4.53429 | + | 0.354457i | 2.14649 | − | 3.71783i | 7.60223i | −2.96356 | + | 0.466187i | −1.31293 | − | 2.27406i | |||||
26.4 | − | 2.38701i | 0.715863 | + | 1.57719i | −3.69781 | −0.866025 | + | 0.500000i | 3.76478 | − | 1.70877i | 1.50161 | − | 2.60087i | 4.05268i | −1.97508 | + | 2.25811i | 1.19350 | + | 2.06721i | |||||
26.5 | − | 2.29159i | −1.47595 | + | 0.906412i | −3.25139 | 0.866025 | − | 0.500000i | 2.07713 | + | 3.38227i | 0.405889 | − | 0.703021i | 2.86768i | 1.35683 | − | 2.67563i | −1.14580 | − | 1.98458i | |||||
26.6 | − | 2.07473i | 1.45124 | − | 0.945463i | −2.30452 | 0.866025 | − | 0.500000i | −1.96158 | − | 3.01094i | −0.753936 | + | 1.30586i | 0.631788i | 1.21220 | − | 2.74419i | −1.03737 | − | 1.79677i | |||||
26.7 | − | 2.01133i | −1.26968 | − | 1.17810i | −2.04544 | −0.866025 | + | 0.500000i | −2.36955 | + | 2.55374i | −0.941394 | + | 1.63054i | 0.0913974i | 0.224160 | + | 2.99161i | 1.00566 | + | 1.74186i | |||||
26.8 | − | 1.78703i | −0.483456 | + | 1.66321i | −1.19348 | −0.866025 | + | 0.500000i | 2.97221 | + | 0.863952i | −1.94940 | + | 3.37646i | − | 1.44127i | −2.53254 | − | 1.60818i | 0.893516 | + | 1.54761i | ||||
26.9 | − | 1.78039i | −0.284115 | − | 1.70859i | −1.16979 | −0.866025 | + | 0.500000i | −3.04196 | + | 0.505835i | 1.86834 | − | 3.23606i | − | 1.47810i | −2.83856 | + | 0.970870i | 0.890195 | + | 1.54186i | ||||
26.10 | − | 1.64741i | 1.01103 | − | 1.40635i | −0.713974 | −0.866025 | + | 0.500000i | −2.31685 | − | 1.66558i | −0.179912 | + | 0.311616i | − | 2.11862i | −0.955654 | − | 2.84372i | 0.823707 | + | 1.42670i | ||||
26.11 | − | 1.59574i | −1.69863 | − | 0.338601i | −0.546377 | 0.866025 | − | 0.500000i | −0.540317 | + | 2.71057i | 1.58858 | − | 2.75151i | − | 2.31960i | 2.77070 | + | 1.15032i | −0.797869 | − | 1.38195i | ||||
26.12 | − | 1.57069i | 1.37717 | + | 1.05043i | −0.467081 | 0.866025 | − | 0.500000i | 1.64991 | − | 2.16311i | 0.571783 | − | 0.990357i | − | 2.40775i | 0.793189 | + | 2.89324i | −0.785347 | − | 1.36026i | ||||
26.13 | − | 1.47744i | 1.61791 | + | 0.618351i | −0.182823 | −0.866025 | + | 0.500000i | 0.913576 | − | 2.39037i | −2.25563 | + | 3.90686i | − | 2.68477i | 2.23528 | + | 2.00088i | 0.738719 | + | 1.27950i | ||||
26.14 | − | 0.929497i | 1.71887 | + | 0.213270i | 1.13603 | −0.866025 | + | 0.500000i | 0.198233 | − | 1.59769i | 2.04756 | − | 3.54648i | − | 2.91494i | 2.90903 | + | 0.733165i | 0.464749 | + | 0.804968i | ||||
26.15 | − | 0.874240i | −1.69245 | + | 0.368254i | 1.23570 | 0.866025 | − | 0.500000i | 0.321943 | + | 1.47961i | −2.57080 | + | 4.45276i | − | 2.82878i | 2.72878 | − | 1.24650i | −0.437120 | − | 0.757114i | ||||
26.16 | − | 0.873617i | −0.976350 | + | 1.43064i | 1.23679 | −0.866025 | + | 0.500000i | 1.24984 | + | 0.852956i | 1.44167 | − | 2.49704i | − | 2.82772i | −1.09348 | − | 2.79362i | 0.436809 | + | 0.756575i | ||||
26.17 | − | 0.831589i | −0.465689 | − | 1.66827i | 1.30846 | 0.866025 | − | 0.500000i | −1.38732 | + | 0.387262i | 0.0353422 | − | 0.0612145i | − | 2.75128i | −2.56627 | + | 1.55379i | −0.415794 | − | 0.720177i | ||||
26.18 | − | 0.778487i | −1.71064 | − | 0.271514i | 1.39396 | −0.866025 | + | 0.500000i | −0.211370 | + | 1.33171i | −0.567828 | + | 0.983507i | − | 2.64215i | 2.85256 | + | 0.928925i | 0.389244 | + | 0.674190i | ||||
26.19 | − | 0.459345i | −1.16522 | + | 1.28150i | 1.78900 | 0.866025 | − | 0.500000i | 0.588653 | + | 0.535240i | 1.20094 | − | 2.08009i | − | 1.74046i | −0.284505 | − | 2.98648i | −0.229673 | − | 0.397805i | ||||
26.20 | − | 0.151885i | 0.0908737 | + | 1.72967i | 1.97693 | 0.866025 | − | 0.500000i | 0.262711 | − | 0.0138024i | −0.275801 | + | 0.477701i | − | 0.604038i | −2.98348 | + | 0.314362i | −0.0759427 | − | 0.131537i | ||||
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
31.e | odd | 6 | 1 | inner |
93.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 465.2.o.a | ✓ | 84 |
3.b | odd | 2 | 1 | inner | 465.2.o.a | ✓ | 84 |
31.e | odd | 6 | 1 | inner | 465.2.o.a | ✓ | 84 |
93.g | even | 6 | 1 | inner | 465.2.o.a | ✓ | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
465.2.o.a | ✓ | 84 | 1.a | even | 1 | 1 | trivial |
465.2.o.a | ✓ | 84 | 3.b | odd | 2 | 1 | inner |
465.2.o.a | ✓ | 84 | 31.e | odd | 6 | 1 | inner |
465.2.o.a | ✓ | 84 | 93.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).