Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [891,2,Mod(107,891)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([5, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("891.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 891 = 3^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 891.u (of order \(30\), degree \(8\), 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: | \(7.11467082010\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{30})\) |
Twist minimal: | no (minimal twist has level 297) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −0.255082 | − | 2.42694i | 0 | −3.86870 | + | 0.822317i | −0.209653 | − | 0.0220354i | 0 | 0.423586 | − | 0.381399i | 1.47436 | + | 4.53760i | 0 | 0.514437i | ||||||||
107.2 | −0.0855419 | − | 0.813877i | 0 | 1.30122 | − | 0.276582i | −1.55568 | − | 0.163508i | 0 | −0.725520 | + | 0.653261i | −0.842187 | − | 2.59198i | 0 | 1.28012i | ||||||||
107.3 | 0.151369 | + | 1.44018i | 0 | −0.0949044 | + | 0.0201726i | 2.18492 | + | 0.229644i | 0 | −2.71309 | + | 2.44288i | 0.851564 | + | 2.62084i | 0 | 3.18143i | ||||||||
107.4 | 0.189255 | + | 1.80064i | 0 | −1.25021 | + | 0.265739i | −4.27136 | − | 0.448938i | 0 | 2.14141 | − | 1.92813i | 0.403879 | + | 1.24301i | 0 | − | 7.77615i | |||||||
134.1 | −2.61129 | + | 0.555046i | 0 | 4.68366 | − | 2.08530i | 0.739203 | − | 3.47767i | 0 | 0.173335 | + | 0.0182182i | −6.75339 | + | 4.90663i | 0 | 9.49150i | ||||||||
134.2 | −0.172007 | + | 0.0365613i | 0 | −1.79884 | + | 0.800896i | 0.384570 | − | 1.80926i | 0 | −3.33045 | − | 0.350045i | 0.564664 | − | 0.410252i | 0 | 0.325266i | ||||||||
134.3 | 0.775410 | − | 0.164818i | 0 | −1.25300 | + | 0.557869i | −0.689350 | + | 3.24314i | 0 | 4.74370 | + | 0.498583i | −2.16231 | + | 1.57101i | 0 | 2.62838i | ||||||||
134.4 | 2.00789 | − | 0.426789i | 0 | 2.02236 | − | 0.900414i | 0.370816 | − | 1.74455i | 0 | 0.305114 | + | 0.0320688i | 0.354978 | − | 0.257906i | 0 | − | 3.66112i | |||||||
215.1 | −1.78633 | + | 1.98392i | 0 | −0.535907 | − | 5.09881i | −2.64215 | + | 2.37901i | 0 | −0.0708899 | + | 0.159221i | 6.75339 | + | 4.90663i | 0 | − | 9.49150i | |||||||
215.2 | −0.117667 | + | 0.130682i | 0 | 0.205825 | + | 1.95829i | −1.37458 | + | 1.23768i | 0 | 1.36208 | − | 3.05928i | −0.564664 | − | 0.410252i | 0 | − | 0.325266i | |||||||
215.3 | 0.530442 | − | 0.589116i | 0 | 0.143369 | + | 1.36406i | 2.46396 | − | 2.21856i | 0 | −1.94006 | + | 4.35745i | 2.16231 | + | 1.57101i | 0 | − | 2.62838i | |||||||
215.4 | 1.37355 | − | 1.52548i | 0 | −0.231400 | − | 2.20162i | −1.32542 | + | 1.19341i | 0 | −0.124785 | + | 0.280271i | −0.354978 | − | 0.257906i | 0 | 3.66112i | ||||||||
431.1 | −1.78633 | − | 1.98392i | 0 | −0.535907 | + | 5.09881i | −2.64215 | − | 2.37901i | 0 | −0.0708899 | − | 0.159221i | 6.75339 | − | 4.90663i | 0 | 9.49150i | ||||||||
431.2 | −0.117667 | − | 0.130682i | 0 | 0.205825 | − | 1.95829i | −1.37458 | − | 1.23768i | 0 | 1.36208 | + | 3.05928i | −0.564664 | + | 0.410252i | 0 | 0.325266i | ||||||||
431.3 | 0.530442 | + | 0.589116i | 0 | 0.143369 | − | 1.36406i | 2.46396 | + | 2.21856i | 0 | −1.94006 | − | 4.35745i | 2.16231 | − | 1.57101i | 0 | 2.62838i | ||||||||
431.4 | 1.37355 | + | 1.52548i | 0 | −0.231400 | + | 2.20162i | −1.32542 | − | 1.19341i | 0 | −0.124785 | − | 0.280271i | −0.354978 | + | 0.257906i | 0 | − | 3.66112i | |||||||
458.1 | −0.255082 | + | 2.42694i | 0 | −3.86870 | − | 0.822317i | −0.209653 | + | 0.0220354i | 0 | 0.423586 | + | 0.381399i | 1.47436 | − | 4.53760i | 0 | − | 0.514437i | |||||||
458.2 | −0.0855419 | + | 0.813877i | 0 | 1.30122 | + | 0.276582i | −1.55568 | + | 0.163508i | 0 | −0.725520 | − | 0.653261i | −0.842187 | + | 2.59198i | 0 | − | 1.28012i | |||||||
458.3 | 0.151369 | − | 1.44018i | 0 | −0.0949044 | − | 0.0201726i | 2.18492 | − | 0.229644i | 0 | −2.71309 | − | 2.44288i | 0.851564 | − | 2.62084i | 0 | − | 3.18143i | |||||||
458.4 | 0.189255 | − | 1.80064i | 0 | −1.25021 | − | 0.265739i | −4.27136 | + | 0.448938i | 0 | 2.14141 | + | 1.92813i | 0.403879 | − | 1.24301i | 0 | 7.77615i | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
11.d | odd | 10 | 1 | inner |
99.p | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 891.2.u.b | 32 | |
3.b | odd | 2 | 1 | 891.2.u.d | 32 | ||
9.c | even | 3 | 1 | 297.2.k.b | ✓ | 32 | |
9.c | even | 3 | 1 | 891.2.u.d | 32 | ||
9.d | odd | 6 | 1 | 297.2.k.b | ✓ | 32 | |
9.d | odd | 6 | 1 | inner | 891.2.u.b | 32 | |
11.d | odd | 10 | 1 | inner | 891.2.u.b | 32 | |
33.f | even | 10 | 1 | 891.2.u.d | 32 | ||
99.o | odd | 30 | 1 | 297.2.k.b | ✓ | 32 | |
99.o | odd | 30 | 1 | 891.2.u.d | 32 | ||
99.p | even | 30 | 1 | 297.2.k.b | ✓ | 32 | |
99.p | even | 30 | 1 | inner | 891.2.u.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
297.2.k.b | ✓ | 32 | 9.c | even | 3 | 1 | |
297.2.k.b | ✓ | 32 | 9.d | odd | 6 | 1 | |
297.2.k.b | ✓ | 32 | 99.o | odd | 30 | 1 | |
297.2.k.b | ✓ | 32 | 99.p | even | 30 | 1 | |
891.2.u.b | 32 | 1.a | even | 1 | 1 | trivial | |
891.2.u.b | 32 | 9.d | odd | 6 | 1 | inner | |
891.2.u.b | 32 | 11.d | odd | 10 | 1 | inner | |
891.2.u.b | 32 | 99.p | even | 30 | 1 | inner | |
891.2.u.d | 32 | 3.b | odd | 2 | 1 | ||
891.2.u.d | 32 | 9.c | even | 3 | 1 | ||
891.2.u.d | 32 | 33.f | even | 10 | 1 | ||
891.2.u.d | 32 | 99.o | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 6 T_{2}^{30} + 8 T_{2}^{28} + 72 T_{2}^{26} + 30 T_{2}^{25} - 842 T_{2}^{24} - 180 T_{2}^{23} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\).