Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,2,Mod(99,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.99");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 722.e (of order \(9\), degree \(6\), not 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: | \(5.76519902594\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
99.1 | 0.939693 | + | 0.342020i | −0.485104 | − | 2.75116i | 0.766044 | + | 0.642788i | −1.79605 | + | 1.50706i | 0.485104 | − | 2.75116i | 0.642040 | − | 1.11205i | 0.500000 | + | 0.866025i | −4.51450 | + | 1.64314i | −2.20318 | + | 0.801892i |
99.2 | 0.939693 | + | 0.342020i | −0.222978 | − | 1.26457i | 0.766044 | + | 0.642788i | 2.83108 | − | 2.37556i | 0.222978 | − | 1.26457i | 0.221232 | − | 0.383185i | 0.500000 | + | 0.866025i | 1.26966 | − | 0.462117i | 3.47284 | − | 1.26401i |
99.3 | 0.939693 | + | 0.342020i | −0.0768330 | − | 0.435741i | 0.766044 | + | 0.642788i | −0.682922 | + | 0.573040i | 0.0768330 | − | 0.435741i | −1.26007 | + | 2.18251i | 0.500000 | + | 0.866025i | 2.63511 | − | 0.959102i | −0.837728 | + | 0.304908i |
99.4 | 0.939693 | + | 0.342020i | 0.437619 | + | 2.48186i | 0.766044 | + | 0.642788i | −1.88420 | + | 1.58103i | −0.437619 | + | 2.48186i | 1.39680 | − | 2.41933i | 0.500000 | + | 0.866025i | −3.14904 | + | 1.14616i | −2.31131 | + | 0.841250i |
245.1 | −0.173648 | + | 0.984808i | −2.14003 | − | 1.79569i | −0.939693 | − | 0.342020i | 2.20318 | − | 0.801892i | 2.14003 | − | 1.79569i | 0.642040 | + | 1.11205i | 0.500000 | − | 0.866025i | 0.834245 | + | 4.73124i | 0.407131 | + | 2.30896i |
245.2 | −0.173648 | + | 0.984808i | −0.983662 | − | 0.825390i | −0.939693 | − | 0.342020i | −3.47284 | + | 1.26401i | 0.983662 | − | 0.825390i | 0.221232 | + | 0.383185i | 0.500000 | − | 0.866025i | −0.234623 | − | 1.33061i | −0.641755 | − | 3.63957i |
245.3 | −0.173648 | + | 0.984808i | −0.338947 | − | 0.284410i | −0.939693 | − | 0.342020i | 0.837728 | − | 0.304908i | 0.338947 | − | 0.284410i | −1.26007 | − | 2.18251i | 0.500000 | − | 0.866025i | −0.486949 | − | 2.76162i | 0.154806 | + | 0.877948i |
245.4 | −0.173648 | + | 0.984808i | 1.93054 | + | 1.61992i | −0.939693 | − | 0.342020i | 2.31131 | − | 0.841250i | −1.93054 | + | 1.61992i | 1.39680 | + | 2.41933i | 0.500000 | − | 0.866025i | 0.581920 | + | 3.30023i | 0.427114 | + | 2.42228i |
389.1 | −0.173648 | − | 0.984808i | −2.14003 | + | 1.79569i | −0.939693 | + | 0.342020i | 2.20318 | + | 0.801892i | 2.14003 | + | 1.79569i | 0.642040 | − | 1.11205i | 0.500000 | + | 0.866025i | 0.834245 | − | 4.73124i | 0.407131 | − | 2.30896i |
389.2 | −0.173648 | − | 0.984808i | −0.983662 | + | 0.825390i | −0.939693 | + | 0.342020i | −3.47284 | − | 1.26401i | 0.983662 | + | 0.825390i | 0.221232 | − | 0.383185i | 0.500000 | + | 0.866025i | −0.234623 | + | 1.33061i | −0.641755 | + | 3.63957i |
389.3 | −0.173648 | − | 0.984808i | −0.338947 | + | 0.284410i | −0.939693 | + | 0.342020i | 0.837728 | + | 0.304908i | 0.338947 | + | 0.284410i | −1.26007 | + | 2.18251i | 0.500000 | + | 0.866025i | −0.486949 | + | 2.76162i | 0.154806 | − | 0.877948i |
389.4 | −0.173648 | − | 0.984808i | 1.93054 | − | 1.61992i | −0.939693 | + | 0.342020i | 2.31131 | + | 0.841250i | −1.93054 | − | 1.61992i | 1.39680 | − | 2.41933i | 0.500000 | + | 0.866025i | 0.581920 | − | 3.30023i | 0.427114 | − | 2.42228i |
415.1 | −0.766044 | − | 0.642788i | −2.36816 | + | 0.861941i | 0.173648 | + | 0.984808i | −0.427114 | + | 2.42228i | 2.36816 | + | 0.861941i | 1.39680 | + | 2.41933i | 0.500000 | − | 0.866025i | 2.56712 | − | 2.15407i | 1.88420 | − | 1.58103i |
415.2 | −0.766044 | − | 0.642788i | 0.415780 | − | 0.151331i | 0.173648 | + | 0.984808i | −0.154806 | + | 0.877948i | −0.415780 | − | 0.151331i | −1.26007 | − | 2.18251i | 0.500000 | − | 0.866025i | −2.14816 | + | 1.80252i | 0.682922 | − | 0.573040i |
415.3 | −0.766044 | − | 0.642788i | 1.20664 | − | 0.439181i | 0.173648 | + | 0.984808i | 0.641755 | − | 3.63957i | −1.20664 | − | 0.439181i | 0.221232 | + | 0.383185i | 0.500000 | − | 0.866025i | −1.03503 | + | 0.868497i | −2.83108 | + | 2.37556i |
415.4 | −0.766044 | − | 0.642788i | 2.62513 | − | 0.955469i | 0.173648 | + | 0.984808i | −0.407131 | + | 2.30896i | −2.62513 | − | 0.955469i | 0.642040 | + | 1.11205i | 0.500000 | − | 0.866025i | 3.68025 | − | 3.08810i | 1.79605 | − | 1.50706i |
423.1 | 0.939693 | − | 0.342020i | −0.485104 | + | 2.75116i | 0.766044 | − | 0.642788i | −1.79605 | − | 1.50706i | 0.485104 | + | 2.75116i | 0.642040 | + | 1.11205i | 0.500000 | − | 0.866025i | −4.51450 | − | 1.64314i | −2.20318 | − | 0.801892i |
423.2 | 0.939693 | − | 0.342020i | −0.222978 | + | 1.26457i | 0.766044 | − | 0.642788i | 2.83108 | + | 2.37556i | 0.222978 | + | 1.26457i | 0.221232 | + | 0.383185i | 0.500000 | − | 0.866025i | 1.26966 | + | 0.462117i | 3.47284 | + | 1.26401i |
423.3 | 0.939693 | − | 0.342020i | −0.0768330 | + | 0.435741i | 0.766044 | − | 0.642788i | −0.682922 | − | 0.573040i | 0.0768330 | + | 0.435741i | −1.26007 | − | 2.18251i | 0.500000 | − | 0.866025i | 2.63511 | + | 0.959102i | −0.837728 | − | 0.304908i |
423.4 | 0.939693 | − | 0.342020i | 0.437619 | − | 2.48186i | 0.766044 | − | 0.642788i | −1.88420 | − | 1.58103i | −0.437619 | − | 2.48186i | 1.39680 | + | 2.41933i | 0.500000 | − | 0.866025i | −3.14904 | − | 1.14616i | −2.31131 | − | 0.841250i |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 2 | inner |
19.e | even | 9 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 722.2.e.s | 24 | |
19.b | odd | 2 | 1 | 722.2.e.r | 24 | ||
19.c | even | 3 | 2 | inner | 722.2.e.s | 24 | |
19.d | odd | 6 | 2 | 722.2.e.r | 24 | ||
19.e | even | 9 | 1 | 722.2.a.m | ✓ | 4 | |
19.e | even | 9 | 2 | 722.2.c.n | 8 | ||
19.e | even | 9 | 3 | inner | 722.2.e.s | 24 | |
19.f | odd | 18 | 1 | 722.2.a.n | yes | 4 | |
19.f | odd | 18 | 2 | 722.2.c.m | 8 | ||
19.f | odd | 18 | 3 | 722.2.e.r | 24 | ||
57.j | even | 18 | 1 | 6498.2.a.bx | 4 | ||
57.l | odd | 18 | 1 | 6498.2.a.ca | 4 | ||
76.k | even | 18 | 1 | 5776.2.a.bt | 4 | ||
76.l | odd | 18 | 1 | 5776.2.a.bv | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
722.2.a.m | ✓ | 4 | 19.e | even | 9 | 1 | |
722.2.a.n | yes | 4 | 19.f | odd | 18 | 1 | |
722.2.c.m | 8 | 19.f | odd | 18 | 2 | ||
722.2.c.n | 8 | 19.e | even | 9 | 2 | ||
722.2.e.r | 24 | 19.b | odd | 2 | 1 | ||
722.2.e.r | 24 | 19.d | odd | 6 | 2 | ||
722.2.e.r | 24 | 19.f | odd | 18 | 3 | ||
722.2.e.s | 24 | 1.a | even | 1 | 1 | trivial | |
722.2.e.s | 24 | 19.c | even | 3 | 2 | inner | |
722.2.e.s | 24 | 19.e | even | 9 | 3 | inner | |
5776.2.a.bt | 4 | 76.k | even | 18 | 1 | ||
5776.2.a.bv | 4 | 76.l | odd | 18 | 1 | ||
6498.2.a.bx | 4 | 57.j | even | 18 | 1 | ||
6498.2.a.ca | 4 | 57.l | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):
\( T_{3}^{24} - 8 T_{3}^{21} + 400 T_{3}^{18} + 1152 T_{3}^{15} + 119104 T_{3}^{12} - 259072 T_{3}^{9} + 568320 T_{3}^{6} - 49152 T_{3}^{3} + 4096 \) |
\( T_{5}^{24} + 22 T_{5}^{21} + 1710 T_{5}^{18} - 48048 T_{5}^{15} + 1278099 T_{5}^{12} - 12617792 T_{5}^{9} + 102640310 T_{5}^{6} - 72280142 T_{5}^{3} + 47045881 \) |
\( T_{7}^{8} - 2T_{7}^{7} + 10T_{7}^{6} - 12T_{7}^{5} + 64T_{7}^{4} - 88T_{7}^{3} + 120T_{7}^{2} - 48T_{7} + 16 \) |
\( T_{13}^{24} - 642 T_{13}^{21} + 281870 T_{13}^{18} - 67527952 T_{13}^{15} + 11801523939 T_{13}^{12} - 1049930053408 T_{13}^{9} + 64940441655990 T_{13}^{6} - 1829557198438 T_{13}^{3} + \cdots + 51520374361 \) |