Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [668,2,Mod(667,668)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(668, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("668.667");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 668 = 2^{2} \cdot 167 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 668.b (of order \(2\), degree \(1\), 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: | \(5.33400685502\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
667.1 | −1.40042 | − | 0.197044i | 2.07563i | 1.92235 | + | 0.551888i | 0 | 0.408990 | − | 2.90675i | − | 4.81975i | −2.58335 | − | 1.15166i | −1.30824 | 0 | |||||||||
667.2 | −1.40042 | + | 0.197044i | − | 2.07563i | 1.92235 | − | 0.551888i | 0 | 0.408990 | + | 2.90675i | 4.81975i | −2.58335 | + | 1.15166i | −1.30824 | 0 | |||||||||
667.3 | −1.28464 | − | 0.591360i | − | 0.736748i | 1.30059 | + | 1.51937i | 0 | −0.435683 | + | 0.946454i | − | 5.23983i | −0.772291 | − | 2.72095i | 2.45720 | 0 | ||||||||
667.4 | −1.28464 | + | 0.591360i | 0.736748i | 1.30059 | − | 1.51937i | 0 | −0.435683 | − | 0.946454i | 5.23983i | −0.772291 | + | 2.72095i | 2.45720 | 0 | ||||||||||
667.5 | −1.07158 | − | 0.922888i | − | 3.45525i | 0.296557 | + | 1.97789i | 0 | −3.18880 | + | 3.70256i | 4.00920i | 1.50759 | − | 2.39315i | −8.93873 | 0 | |||||||||
667.6 | −1.07158 | + | 0.922888i | 3.45525i | 0.296557 | − | 1.97789i | 0 | −3.18880 | − | 3.70256i | − | 4.00920i | 1.50759 | + | 2.39315i | −8.93873 | 0 | |||||||||
667.7 | −0.760993 | − | 1.19201i | 3.04057i | −0.841781 | + | 1.81422i | 0 | 3.62439 | − | 2.31385i | 5.23541i | 2.80316 | − | 0.377198i | −6.24504 | 0 | ||||||||||
667.8 | −0.760993 | + | 1.19201i | − | 3.04057i | −0.841781 | − | 1.81422i | 0 | 3.62439 | + | 2.31385i | − | 5.23541i | 2.80316 | + | 0.377198i | −6.24504 | 0 | ||||||||
667.9 | −0.402518 | − | 1.35572i | 2.44978i | −1.67596 | + | 1.09140i | 0 | 3.32122 | − | 0.986080i | − | 2.87385i | 2.15424 | + | 1.83282i | −3.00142 | 0 | |||||||||
667.10 | −0.402518 | + | 1.35572i | − | 2.44978i | −1.67596 | − | 1.09140i | 0 | 3.32122 | + | 0.986080i | 2.87385i | 2.15424 | − | 1.83282i | −3.00142 | 0 | |||||||||
667.11 | 0.00426214 | − | 1.41421i | − | 3.24555i | −1.99996 | − | 0.0120551i | 0 | −4.58987 | − | 0.0138330i | − | 4.80685i | −0.0255725 | + | 2.82831i | −7.53357 | 0 | ||||||||
667.12 | 0.00426214 | + | 1.41421i | 3.24555i | −1.99996 | + | 0.0120551i | 0 | −4.58987 | + | 0.0138330i | 4.80685i | −0.0255725 | − | 2.82831i | −7.53357 | 0 | ||||||||||
667.13 | 0.394339 | − | 1.35812i | 0.246717i | −1.68899 | − | 1.07112i | 0 | 0.335073 | + | 0.0972902i | 1.50567i | −2.12075 | + | 1.87148i | 2.93913 | 0 | ||||||||||
667.14 | 0.394339 | + | 1.35812i | − | 0.246717i | −1.68899 | + | 1.07112i | 0 | 0.335073 | − | 0.0972902i | − | 1.50567i | −2.12075 | − | 1.87148i | 2.93913 | 0 | ||||||||
667.15 | 0.768164 | − | 1.18740i | 1.21020i | −0.819849 | − | 1.82424i | 0 | 1.43699 | + | 0.929628i | 3.98887i | −2.79588 | − | 0.427823i | 1.53543 | 0 | ||||||||||
667.16 | 0.768164 | + | 1.18740i | − | 1.21020i | −0.819849 | + | 1.82424i | 0 | 1.43699 | − | 0.929628i | − | 3.98887i | −2.79588 | + | 0.427823i | 1.53543 | 0 | ||||||||
667.17 | 1.06600 | − | 0.929330i | − | 2.77291i | 0.272692 | − | 1.98132i | 0 | −2.57695 | − | 2.95591i | − | 0.0155204i | −1.55061 | − | 2.36550i | −4.68903 | 0 | ||||||||
667.18 | 1.06600 | + | 0.929330i | 2.77291i | 0.272692 | + | 1.98132i | 0 | −2.57695 | + | 2.95591i | 0.0155204i | −1.55061 | + | 2.36550i | −4.68903 | 0 | ||||||||||
667.19 | 1.28818 | − | 0.583606i | 1.66053i | 1.31881 | − | 1.50358i | 0 | 0.969093 | + | 2.13906i | − | 2.84773i | 0.821364 | − | 2.70654i | 0.242651 | 0 | |||||||||
667.20 | 1.28818 | + | 0.583606i | − | 1.66053i | 1.31881 | + | 1.50358i | 0 | 0.969093 | − | 2.13906i | 2.84773i | 0.821364 | + | 2.70654i | 0.242651 | 0 | |||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
167.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-167}) \) |
4.b | odd | 2 | 1 | inner |
668.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 668.2.b.a | ✓ | 22 |
4.b | odd | 2 | 1 | inner | 668.2.b.a | ✓ | 22 |
167.b | odd | 2 | 1 | CM | 668.2.b.a | ✓ | 22 |
668.b | even | 2 | 1 | inner | 668.2.b.a | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
668.2.b.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
668.2.b.a | ✓ | 22 | 4.b | odd | 2 | 1 | inner |
668.2.b.a | ✓ | 22 | 167.b | odd | 2 | 1 | CM |
668.2.b.a | ✓ | 22 | 668.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 66 T_{3}^{20} + 1881 T_{3}^{18} + 30294 T_{3}^{16} + 302940 T_{3}^{14} + 1945944 T_{3}^{12} + \cdots + 353372 \) acting on \(S_{2}^{\mathrm{new}}(668, [\chi])\).