Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3584,2,Mod(897,3584)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3584, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3584.897");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3584 = 2^{9} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3584.m (of order \(4\), 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: | \(28.6183840844\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
897.1 | 0 | −2.34057 | + | 2.34057i | 0 | 1.01838 | + | 1.01838i | 0 | − | 1.00000i | 0 | − | 7.95654i | 0 | ||||||||||||
897.2 | 0 | −1.63240 | + | 1.63240i | 0 | 0.462322 | + | 0.462322i | 0 | − | 1.00000i | 0 | − | 2.32947i | 0 | ||||||||||||
897.3 | 0 | −1.53441 | + | 1.53441i | 0 | −1.58168 | − | 1.58168i | 0 | − | 1.00000i | 0 | − | 1.70884i | 0 | ||||||||||||
897.4 | 0 | −1.07038 | + | 1.07038i | 0 | 2.97612 | + | 2.97612i | 0 | − | 1.00000i | 0 | 0.708572i | 0 | |||||||||||||
897.5 | 0 | −0.482987 | + | 0.482987i | 0 | 2.27930 | + | 2.27930i | 0 | − | 1.00000i | 0 | 2.53345i | 0 | |||||||||||||
897.6 | 0 | −0.303325 | + | 0.303325i | 0 | −2.39032 | − | 2.39032i | 0 | − | 1.00000i | 0 | 2.81599i | 0 | |||||||||||||
897.7 | 0 | 0.125826 | − | 0.125826i | 0 | −1.32428 | − | 1.32428i | 0 | − | 1.00000i | 0 | 2.96834i | 0 | |||||||||||||
897.8 | 0 | 0.714770 | − | 0.714770i | 0 | 0.214515 | + | 0.214515i | 0 | − | 1.00000i | 0 | 1.97821i | 0 | |||||||||||||
897.9 | 0 | 1.19058 | − | 1.19058i | 0 | 0.370923 | + | 0.370923i | 0 | − | 1.00000i | 0 | 0.165048i | 0 | |||||||||||||
897.10 | 0 | 1.22968 | − | 1.22968i | 0 | 1.60657 | + | 1.60657i | 0 | − | 1.00000i | 0 | − | 0.0242347i | 0 | ||||||||||||
897.11 | 0 | 1.77137 | − | 1.77137i | 0 | −1.80277 | − | 1.80277i | 0 | − | 1.00000i | 0 | − | 3.27553i | 0 | ||||||||||||
897.12 | 0 | 2.33184 | − | 2.33184i | 0 | 2.17092 | + | 2.17092i | 0 | − | 1.00000i | 0 | − | 7.87499i | 0 | ||||||||||||
2689.1 | 0 | −2.34057 | − | 2.34057i | 0 | 1.01838 | − | 1.01838i | 0 | 1.00000i | 0 | 7.95654i | 0 | ||||||||||||||
2689.2 | 0 | −1.63240 | − | 1.63240i | 0 | 0.462322 | − | 0.462322i | 0 | 1.00000i | 0 | 2.32947i | 0 | ||||||||||||||
2689.3 | 0 | −1.53441 | − | 1.53441i | 0 | −1.58168 | + | 1.58168i | 0 | 1.00000i | 0 | 1.70884i | 0 | ||||||||||||||
2689.4 | 0 | −1.07038 | − | 1.07038i | 0 | 2.97612 | − | 2.97612i | 0 | 1.00000i | 0 | − | 0.708572i | 0 | |||||||||||||
2689.5 | 0 | −0.482987 | − | 0.482987i | 0 | 2.27930 | − | 2.27930i | 0 | 1.00000i | 0 | − | 2.53345i | 0 | |||||||||||||
2689.6 | 0 | −0.303325 | − | 0.303325i | 0 | −2.39032 | + | 2.39032i | 0 | 1.00000i | 0 | − | 2.81599i | 0 | |||||||||||||
2689.7 | 0 | 0.125826 | + | 0.125826i | 0 | −1.32428 | + | 1.32428i | 0 | 1.00000i | 0 | − | 2.96834i | 0 | |||||||||||||
2689.8 | 0 | 0.714770 | + | 0.714770i | 0 | 0.214515 | − | 0.214515i | 0 | 1.00000i | 0 | − | 1.97821i | 0 | |||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3584.2.m.bn | yes | 24 |
4.b | odd | 2 | 1 | 3584.2.m.bm | yes | 24 | |
8.b | even | 2 | 1 | 3584.2.m.bk | ✓ | 24 | |
8.d | odd | 2 | 1 | 3584.2.m.bl | yes | 24 | |
16.e | even | 4 | 1 | 3584.2.m.bk | ✓ | 24 | |
16.e | even | 4 | 1 | inner | 3584.2.m.bn | yes | 24 |
16.f | odd | 4 | 1 | 3584.2.m.bl | yes | 24 | |
16.f | odd | 4 | 1 | 3584.2.m.bm | yes | 24 | |
32.g | even | 8 | 1 | 7168.2.a.bg | 12 | ||
32.g | even | 8 | 1 | 7168.2.a.bk | 12 | ||
32.h | odd | 8 | 1 | 7168.2.a.bh | 12 | ||
32.h | odd | 8 | 1 | 7168.2.a.bl | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3584.2.m.bk | ✓ | 24 | 8.b | even | 2 | 1 | |
3584.2.m.bk | ✓ | 24 | 16.e | even | 4 | 1 | |
3584.2.m.bl | yes | 24 | 8.d | odd | 2 | 1 | |
3584.2.m.bl | yes | 24 | 16.f | odd | 4 | 1 | |
3584.2.m.bm | yes | 24 | 4.b | odd | 2 | 1 | |
3584.2.m.bm | yes | 24 | 16.f | odd | 4 | 1 | |
3584.2.m.bn | yes | 24 | 1.a | even | 1 | 1 | trivial |
3584.2.m.bn | yes | 24 | 16.e | even | 4 | 1 | inner |
7168.2.a.bg | 12 | 32.g | even | 8 | 1 | ||
7168.2.a.bh | 12 | 32.h | odd | 8 | 1 | ||
7168.2.a.bk | 12 | 32.g | even | 8 | 1 | ||
7168.2.a.bl | 12 | 32.h | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):
\( T_{3}^{24} + 176 T_{3}^{20} + 7632 T_{3}^{16} + 64 T_{3}^{15} - 1408 T_{3}^{13} + 106048 T_{3}^{12} - 9728 T_{3}^{11} + 20480 T_{3}^{9} + 430656 T_{3}^{8} - 22016 T_{3}^{7} + 2048 T_{3}^{6} + 105472 T_{3}^{5} + \cdots + 1024 \) |
\( T_{5}^{24} - 8 T_{5}^{23} + 32 T_{5}^{22} - 48 T_{5}^{21} + 248 T_{5}^{20} - 1696 T_{5}^{19} + 6784 T_{5}^{18} - 9984 T_{5}^{17} + 20224 T_{5}^{16} - 101952 T_{5}^{15} + 403456 T_{5}^{14} - 568448 T_{5}^{13} + 728512 T_{5}^{12} + \cdots + 262144 \) |
\( T_{11}^{24} + 1584 T_{11}^{20} + 128 T_{11}^{19} - 6144 T_{11}^{17} + 766016 T_{11}^{16} - 5120 T_{11}^{15} + 8192 T_{11}^{14} - 3514368 T_{11}^{13} + 107331584 T_{11}^{12} - 54820864 T_{11}^{11} + 5242880 T_{11}^{10} + \cdots + 67108864 \) |
\( T_{13}^{24} + 8 T_{13}^{23} + 32 T_{13}^{22} + 48 T_{13}^{21} + 1720 T_{13}^{20} + 13664 T_{13}^{19} + 55424 T_{13}^{18} + 78016 T_{13}^{17} + 834048 T_{13}^{16} + 6411968 T_{13}^{15} + 26374144 T_{13}^{14} + \cdots + 157856825344 \) |