Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,2,Mod(76,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([14, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.76");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.l (of order \(9\), degree \(6\), 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: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(66\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 | −2.03285 | + | 1.70577i | −1.72224 | + | 0.184133i | 0.875557 | − | 4.96553i | 0 | 3.18696 | − | 3.31205i | −0.859447 | − | 4.87417i | 4.03645 | + | 6.99134i | 2.93219 | − | 0.634241i | 0 | ||||
76.2 | −1.70314 | + | 1.42910i | 0.320463 | + | 1.70215i | 0.511048 | − | 2.89830i | 0 | −2.97833 | − | 2.44102i | 0.653660 | + | 3.70709i | 1.04829 | + | 1.81569i | −2.79461 | + | 1.09095i | 0 | ||||
76.3 | −1.02568 | + | 0.860644i | −0.817526 | − | 1.52697i | −0.0359939 | + | 0.204131i | 0 | 2.15270 | + | 0.862581i | −0.0601126 | − | 0.340915i | −1.47769 | − | 2.55944i | −1.66330 | + | 2.49668i | 0 | ||||
76.4 | −0.672508 | + | 0.564301i | −1.45527 | + | 0.939248i | −0.213465 | + | 1.21062i | 0 | 0.448662 | − | 1.45286i | 0.0883715 | + | 0.501180i | −1.41750 | − | 2.45517i | 1.23563 | − | 2.73372i | 0 | ||||
76.5 | −0.211903 | + | 0.177808i | 0.483303 | + | 1.66326i | −0.334009 | + | 1.89426i | 0 | −0.398153 | − | 0.266514i | −0.293371 | − | 1.66379i | −0.542656 | − | 0.939908i | −2.53284 | + | 1.60771i | 0 | ||||
76.6 | −0.194784 | + | 0.163444i | 1.72511 | − | 0.154950i | −0.336069 | + | 1.90594i | 0 | −0.310698 | + | 0.312139i | 0.449901 | + | 2.55151i | −0.500326 | − | 0.866590i | 2.95198 | − | 0.534610i | 0 | ||||
76.7 | 0.594907 | − | 0.499186i | 0.900618 | − | 1.47949i | −0.242569 | + | 1.37568i | 0 | −0.202756 | − | 1.32973i | −0.855088 | − | 4.84944i | 1.31901 | + | 2.28459i | −1.37777 | − | 2.66491i | 0 | ||||
76.8 | 1.01600 | − | 0.852522i | −0.196018 | − | 1.72092i | −0.0418420 | + | 0.237298i | 0 | −1.66628 | − | 1.58134i | 0.769922 | + | 4.36644i | 1.48608 | + | 2.57396i | −2.92315 | + | 0.674663i | 0 | ||||
76.9 | 1.23677 | − | 1.03778i | −1.21085 | + | 1.23848i | 0.105334 | − | 0.597379i | 0 | −0.212279 | + | 2.78832i | −0.302296 | − | 1.71440i | 1.12482 | + | 1.94825i | −0.0676836 | − | 2.99924i | 0 | ||||
76.10 | 1.55121 | − | 1.30162i | 1.45868 | + | 0.933937i | 0.364742 | − | 2.06856i | 0 | 3.47836 | − | 0.449921i | −0.0652816 | − | 0.370230i | −0.101721 | − | 0.176186i | 1.25552 | + | 2.72464i | 0 | ||||
76.11 | 1.84958 | − | 1.55198i | −1.53931 | − | 0.794060i | 0.665000 | − | 3.77140i | 0 | −4.07944 | + | 0.920299i | −0.0583477 | − | 0.330906i | −2.20872 | − | 3.82562i | 1.73894 | + | 2.44461i | 0 | ||||
151.1 | −2.03285 | − | 1.70577i | −1.72224 | − | 0.184133i | 0.875557 | + | 4.96553i | 0 | 3.18696 | + | 3.31205i | −0.859447 | + | 4.87417i | 4.03645 | − | 6.99134i | 2.93219 | + | 0.634241i | 0 | ||||
151.2 | −1.70314 | − | 1.42910i | 0.320463 | − | 1.70215i | 0.511048 | + | 2.89830i | 0 | −2.97833 | + | 2.44102i | 0.653660 | − | 3.70709i | 1.04829 | − | 1.81569i | −2.79461 | − | 1.09095i | 0 | ||||
151.3 | −1.02568 | − | 0.860644i | −0.817526 | + | 1.52697i | −0.0359939 | − | 0.204131i | 0 | 2.15270 | − | 0.862581i | −0.0601126 | + | 0.340915i | −1.47769 | + | 2.55944i | −1.66330 | − | 2.49668i | 0 | ||||
151.4 | −0.672508 | − | 0.564301i | −1.45527 | − | 0.939248i | −0.213465 | − | 1.21062i | 0 | 0.448662 | + | 1.45286i | 0.0883715 | − | 0.501180i | −1.41750 | + | 2.45517i | 1.23563 | + | 2.73372i | 0 | ||||
151.5 | −0.211903 | − | 0.177808i | 0.483303 | − | 1.66326i | −0.334009 | − | 1.89426i | 0 | −0.398153 | + | 0.266514i | −0.293371 | + | 1.66379i | −0.542656 | + | 0.939908i | −2.53284 | − | 1.60771i | 0 | ||||
151.6 | −0.194784 | − | 0.163444i | 1.72511 | + | 0.154950i | −0.336069 | − | 1.90594i | 0 | −0.310698 | − | 0.312139i | 0.449901 | − | 2.55151i | −0.500326 | + | 0.866590i | 2.95198 | + | 0.534610i | 0 | ||||
151.7 | 0.594907 | + | 0.499186i | 0.900618 | + | 1.47949i | −0.242569 | − | 1.37568i | 0 | −0.202756 | + | 1.32973i | −0.855088 | + | 4.84944i | 1.31901 | − | 2.28459i | −1.37777 | + | 2.66491i | 0 | ||||
151.8 | 1.01600 | + | 0.852522i | −0.196018 | + | 1.72092i | −0.0418420 | − | 0.237298i | 0 | −1.66628 | + | 1.58134i | 0.769922 | − | 4.36644i | 1.48608 | − | 2.57396i | −2.92315 | − | 0.674663i | 0 | ||||
151.9 | 1.23677 | + | 1.03778i | −1.21085 | − | 1.23848i | 0.105334 | + | 0.597379i | 0 | −0.212279 | − | 2.78832i | −0.302296 | + | 1.71440i | 1.12482 | − | 1.94825i | −0.0676836 | + | 2.99924i | 0 | ||||
See all 66 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.l.g | yes | 66 |
5.b | even | 2 | 1 | 675.2.l.f | ✓ | 66 | |
5.c | odd | 4 | 2 | 675.2.u.e | 132 | ||
27.e | even | 9 | 1 | inner | 675.2.l.g | yes | 66 |
135.p | even | 18 | 1 | 675.2.l.f | ✓ | 66 | |
135.r | odd | 36 | 2 | 675.2.u.e | 132 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
675.2.l.f | ✓ | 66 | 5.b | even | 2 | 1 | |
675.2.l.f | ✓ | 66 | 135.p | even | 18 | 1 | |
675.2.l.g | yes | 66 | 1.a | even | 1 | 1 | trivial |
675.2.l.g | yes | 66 | 27.e | even | 9 | 1 | inner |
675.2.u.e | 132 | 5.c | odd | 4 | 2 | ||
675.2.u.e | 132 | 135.r | odd | 36 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} - 6 T_{2}^{65} + 18 T_{2}^{64} - 44 T_{2}^{63} + 84 T_{2}^{62} - 99 T_{2}^{61} + 527 T_{2}^{60} + \cdots + 23409 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).