Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [370,2,Mod(17,370)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([9, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("370.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 370.ba (of order \(36\), degree \(12\), 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: | \(2.95446487479\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | −0.173648 | − | 0.984808i | −2.77903 | + | 1.94590i | −0.939693 | + | 0.342020i | −0.881644 | + | 2.05492i | 2.39891 | + | 2.39891i | 0.205490 | + | 2.34876i | 0.500000 | + | 0.866025i | 2.91044 | − | 7.99636i | 2.17680 | + | 0.511416i |
17.2 | −0.173648 | − | 0.984808i | −1.47946 | + | 1.03593i | −0.939693 | + | 0.342020i | 1.60255 | − | 1.55943i | 1.27710 | + | 1.27710i | −0.151990 | − | 1.73726i | 0.500000 | + | 0.866025i | 0.0895920 | − | 0.246152i | −1.81402 | − | 1.30741i |
17.3 | −0.173648 | − | 0.984808i | −1.28340 | + | 0.898643i | −0.939693 | + | 0.342020i | −2.22534 | − | 0.218791i | 1.10785 | + | 1.10785i | −0.0448989 | − | 0.513197i | 0.500000 | + | 0.866025i | −0.186516 | + | 0.512448i | 0.170958 | + | 2.22952i |
17.4 | −0.173648 | − | 0.984808i | −0.809902 | + | 0.567099i | −0.939693 | + | 0.342020i | 0.168535 | + | 2.22971i | 0.699122 | + | 0.699122i | −0.288350 | − | 3.29585i | 0.500000 | + | 0.866025i | −0.691721 | + | 1.90049i | 2.16657 | − | 0.553159i |
17.5 | −0.173648 | − | 0.984808i | −0.124693 | + | 0.0873112i | −0.939693 | + | 0.342020i | 2.15046 | + | 0.612816i | 0.107638 | + | 0.107638i | 0.230601 | + | 2.63579i | 0.500000 | + | 0.866025i | −1.01814 | + | 2.79730i | 0.230083 | − | 2.22420i |
17.6 | −0.173648 | − | 0.984808i | 0.810570 | − | 0.567567i | −0.939693 | + | 0.342020i | −2.22842 | + | 0.184728i | −0.699698 | − | 0.699698i | 0.406884 | + | 4.65071i | 0.500000 | + | 0.866025i | −0.691169 | + | 1.89897i | 0.568884 | + | 2.16249i |
17.7 | −0.173648 | − | 0.984808i | 1.21087 | − | 0.847863i | −0.939693 | + | 0.342020i | −0.415110 | − | 2.19720i | −1.04525 | − | 1.04525i | −0.203881 | − | 2.33037i | 0.500000 | + | 0.866025i | −0.278717 | + | 0.765769i | −2.09174 | + | 0.790343i |
17.8 | −0.173648 | − | 0.984808i | 1.32542 | − | 0.928067i | −0.939693 | + | 0.342020i | 0.101493 | + | 2.23376i | −1.14412 | − | 1.14412i | 0.111534 | + | 1.27484i | 0.500000 | + | 0.866025i | −0.130639 | + | 0.358928i | 2.18220 | − | 0.487840i |
17.9 | −0.173648 | − | 0.984808i | 2.30212 | − | 1.61196i | −0.939693 | + | 0.342020i | −2.13489 | + | 0.665008i | −1.98723 | − | 1.98723i | −0.396149 | − | 4.52800i | 0.500000 | + | 0.866025i | 1.67527 | − | 4.60277i | 1.02563 | + | 1.98698i |
17.10 | −0.173648 | − | 0.984808i | 2.60236 | − | 1.82219i | −0.939693 | + | 0.342020i | 1.31462 | − | 1.80881i | −2.24641 | − | 2.24641i | 0.299132 | + | 3.41909i | 0.500000 | + | 0.866025i | 2.42584 | − | 6.66494i | −2.00961 | − | 0.980548i |
87.1 | −0.173648 | + | 0.984808i | −1.92441 | + | 2.74835i | −0.939693 | − | 0.342020i | 2.17626 | − | 0.513687i | −2.37242 | − | 2.37242i | −3.50361 | − | 0.306526i | 0.500000 | − | 0.866025i | −2.82399 | − | 7.75885i | 0.127979 | + | 2.23240i |
87.2 | −0.173648 | + | 0.984808i | −1.51194 | + | 2.15928i | −0.939693 | − | 0.342020i | −1.73937 | − | 1.40520i | −1.86393 | − | 1.86393i | −0.0300439 | − | 0.00262850i | 0.500000 | − | 0.866025i | −1.35045 | − | 3.71033i | 1.68589 | − | 1.46893i |
87.3 | −0.173648 | + | 0.984808i | −0.880716 | + | 1.25779i | −0.939693 | − | 0.342020i | −1.51276 | + | 1.64668i | −1.08575 | − | 1.08575i | −0.477859 | − | 0.0418072i | 0.500000 | − | 0.866025i | 0.219679 | + | 0.603563i | −1.35898 | − | 1.77572i |
87.4 | −0.173648 | + | 0.984808i | −0.673462 | + | 0.961804i | −0.939693 | − | 0.342020i | −1.35496 | − | 1.77879i | −0.830246 | − | 0.830246i | 4.61019 | + | 0.403339i | 0.500000 | − | 0.866025i | 0.554545 | + | 1.52360i | 1.98705 | − | 1.02549i |
87.5 | −0.173648 | + | 0.984808i | −0.572741 | + | 0.817959i | −0.939693 | − | 0.342020i | 1.67937 | + | 1.47639i | −0.706077 | − | 0.706077i | 1.51235 | + | 0.132313i | 0.500000 | − | 0.866025i | 0.685035 | + | 1.88212i | −1.74558 | + | 1.39748i |
87.6 | −0.173648 | + | 0.984808i | −0.308015 | + | 0.439891i | −0.939693 | − | 0.342020i | 1.13091 | − | 1.92900i | −0.379722 | − | 0.379722i | −2.53970 | − | 0.222195i | 0.500000 | − | 0.866025i | 0.927429 | + | 2.54809i | 1.70331 | + | 1.44870i |
87.7 | −0.173648 | + | 0.984808i | 0.681558 | − | 0.973365i | −0.939693 | − | 0.342020i | −1.46658 | − | 1.68794i | 0.840226 | + | 0.840226i | −4.08083 | − | 0.357026i | 0.500000 | − | 0.866025i | 0.543142 | + | 1.49227i | 1.91696 | − | 1.15120i |
87.8 | −0.173648 | + | 0.984808i | 0.995829 | − | 1.42219i | −0.939693 | − | 0.342020i | 1.96796 | − | 1.06167i | 1.22766 | + | 1.22766i | 2.05370 | + | 0.179675i | 0.500000 | − | 0.866025i | −0.00489254 | − | 0.0134421i | 0.703804 | + | 2.12242i |
87.9 | −0.173648 | + | 0.984808i | 1.04818 | − | 1.49696i | −0.939693 | − | 0.342020i | −0.716196 | + | 2.11827i | 1.29220 | + | 1.29220i | 0.497240 | + | 0.0435029i | 0.500000 | − | 0.866025i | −0.116144 | − | 0.319104i | −1.96172 | − | 1.07315i |
87.10 | −0.173648 | + | 0.984808i | 1.90295 | − | 2.71770i | −0.939693 | − | 0.342020i | −2.02835 | − | 0.941159i | 2.34597 | + | 2.34597i | 1.44290 | + | 0.126237i | 0.500000 | − | 0.866025i | −2.73859 | − | 7.52423i | 1.27908 | − | 1.83411i |
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
185.z | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 370.2.ba.b | ✓ | 120 |
5.c | odd | 4 | 1 | 370.2.bd.b | yes | 120 | |
37.i | odd | 36 | 1 | 370.2.bd.b | yes | 120 | |
185.z | even | 36 | 1 | inner | 370.2.ba.b | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
370.2.ba.b | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
370.2.ba.b | ✓ | 120 | 185.z | even | 36 | 1 | inner |
370.2.bd.b | yes | 120 | 5.c | odd | 4 | 1 | |
370.2.bd.b | yes | 120 | 37.i | odd | 36 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 6 T_{3}^{119} + 18 T_{3}^{118} + 34 T_{3}^{117} + 69 T_{3}^{116} + 348 T_{3}^{115} + \cdots + 17\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).