Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [891,2,Mod(136,891)]
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([20, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("891.136");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 891 = 3^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 891.n (of order \(15\), 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 \(\Q(\zeta_{15})\) |
Twist minimal: | no (minimal twist has level 297) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
136.1 | −1.49193 | + | 0.664249i | 0 | 0.446358 | − | 0.495731i | −2.73558 | − | 1.21796i | 0 | 1.48588 | + | 0.315834i | 0.672677 | − | 2.07029i | 0 | 4.89031 | ||||||||
136.2 | −0.330935 | + | 0.147342i | 0 | −1.25045 | + | 1.38877i | 2.45353 | + | 1.09238i | 0 | 2.43963 | + | 0.518560i | 0.433081 | − | 1.33288i | 0 | −0.972914 | ||||||||
136.3 | 1.11141 | − | 0.494834i | 0 | −0.347879 | + | 0.386359i | 0.271292 | + | 0.120787i | 0 | −4.20378 | − | 0.893541i | −0.947351 | + | 2.91565i | 0 | 0.361288 | ||||||||
136.4 | 2.53854 | − | 1.13023i | 0 | 3.82850 | − | 4.25198i | 0.924300 | + | 0.411525i | 0 | 1.86094 | + | 0.395555i | 3.19570 | − | 9.83534i | 0 | 2.81149 | ||||||||
190.1 | −1.49193 | − | 0.664249i | 0 | 0.446358 | + | 0.495731i | −2.73558 | + | 1.21796i | 0 | 1.48588 | − | 0.315834i | 0.672677 | + | 2.07029i | 0 | 4.89031 | ||||||||
190.2 | −0.330935 | − | 0.147342i | 0 | −1.25045 | − | 1.38877i | 2.45353 | − | 1.09238i | 0 | 2.43963 | − | 0.518560i | 0.433081 | + | 1.33288i | 0 | −0.972914 | ||||||||
190.3 | 1.11141 | + | 0.494834i | 0 | −0.347879 | − | 0.386359i | 0.271292 | − | 0.120787i | 0 | −4.20378 | + | 0.893541i | −0.947351 | − | 2.91565i | 0 | 0.361288 | ||||||||
190.4 | 2.53854 | + | 1.13023i | 0 | 3.82850 | + | 4.25198i | 0.924300 | − | 0.411525i | 0 | 1.86094 | − | 0.395555i | 3.19570 | + | 9.83534i | 0 | 2.81149 | ||||||||
379.1 | −2.29796 | − | 0.488446i | 0 | 3.21494 | + | 1.43138i | 3.35871 | − | 0.713915i | 0 | 0.298019 | + | 2.83546i | −2.88741 | − | 2.09782i | 0 | −8.06687 | ||||||||
379.2 | −1.70989 | − | 0.363447i | 0 | 0.964524 | + | 0.429434i | −3.77644 | + | 0.802707i | 0 | −0.0917633 | − | 0.873070i | 1.33531 | + | 0.970162i | 0 | 6.74902 | ||||||||
379.3 | 0.260096 | + | 0.0552850i | 0 | −1.76250 | − | 0.784714i | 1.07651 | − | 0.228819i | 0 | 0.161379 | + | 1.53541i | −0.845280 | − | 0.614132i | 0 | 0.292645 | ||||||||
379.4 | 1.79145 | + | 0.380785i | 0 | 1.23721 | + | 0.550844i | −1.63692 | + | 0.347939i | 0 | −0.432236 | − | 4.11245i | −0.956730 | − | 0.695105i | 0 | −3.06496 | ||||||||
433.1 | −0.290461 | + | 2.76355i | 0 | −5.59657 | − | 1.18959i | −0.105759 | − | 1.00623i | 0 | −1.27303 | + | 1.41384i | 3.19570 | − | 9.83534i | 0 | 2.81149 | ||||||||
433.2 | −0.127169 | + | 1.20993i | 0 | 0.508536 | + | 0.108093i | −0.0310414 | − | 0.295339i | 0 | 2.87572 | − | 3.19381i | −0.947351 | + | 2.91565i | 0 | 0.361288 | ||||||||
433.3 | 0.0378659 | − | 0.360270i | 0 | 1.82793 | + | 0.388540i | −0.280735 | − | 2.67101i | 0 | −1.66890 | + | 1.85350i | 0.433081 | − | 1.33288i | 0 | −0.972914 | ||||||||
433.4 | 0.170707 | − | 1.62417i | 0 | −0.652494 | − | 0.138692i | 0.313007 | + | 2.97806i | 0 | −1.01646 | + | 1.12890i | 0.672677 | − | 2.07029i | 0 | 4.89031 | ||||||||
460.1 | −1.22550 | − | 1.36105i | 0 | −0.141563 | + | 1.34688i | 1.11978 | − | 1.24365i | 0 | 3.77761 | − | 1.68190i | −0.956730 | + | 0.695105i | 0 | −3.06496 | ||||||||
460.2 | −0.177926 | − | 0.197607i | 0 | 0.201666 | − | 1.91872i | −0.736416 | + | 0.817873i | 0 | −1.41040 | + | 0.627949i | −0.845280 | + | 0.614132i | 0 | 0.292645 | ||||||||
460.3 | 1.16970 | + | 1.29908i | 0 | −0.110361 | + | 1.05002i | 2.58338 | − | 2.86914i | 0 | 0.801982 | − | 0.357065i | 1.33531 | − | 0.970162i | 0 | 6.74902 | ||||||||
460.4 | 1.57199 | + | 1.74587i | 0 | −0.367856 | + | 3.49991i | −2.29762 | + | 2.55177i | 0 | −2.60459 | + | 1.15964i | −2.88741 | + | 2.09782i | 0 | −8.06687 | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
99.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 891.2.n.i | 32 | |
3.b | odd | 2 | 1 | 891.2.n.f | 32 | ||
9.c | even | 3 | 1 | 297.2.f.a | ✓ | 16 | |
9.c | even | 3 | 1 | inner | 891.2.n.i | 32 | |
9.d | odd | 6 | 1 | 297.2.f.d | yes | 16 | |
9.d | odd | 6 | 1 | 891.2.n.f | 32 | ||
11.c | even | 5 | 1 | inner | 891.2.n.i | 32 | |
33.h | odd | 10 | 1 | 891.2.n.f | 32 | ||
99.m | even | 15 | 1 | 297.2.f.a | ✓ | 16 | |
99.m | even | 15 | 1 | inner | 891.2.n.i | 32 | |
99.m | even | 15 | 1 | 3267.2.a.bm | 8 | ||
99.n | odd | 30 | 1 | 297.2.f.d | yes | 16 | |
99.n | odd | 30 | 1 | 891.2.n.f | 32 | ||
99.n | odd | 30 | 1 | 3267.2.a.be | 8 | ||
99.o | odd | 30 | 1 | 3267.2.a.bf | 8 | ||
99.p | even | 30 | 1 | 3267.2.a.bl | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
297.2.f.a | ✓ | 16 | 9.c | even | 3 | 1 | |
297.2.f.a | ✓ | 16 | 99.m | even | 15 | 1 | |
297.2.f.d | yes | 16 | 9.d | odd | 6 | 1 | |
297.2.f.d | yes | 16 | 99.n | odd | 30 | 1 | |
891.2.n.f | 32 | 3.b | odd | 2 | 1 | ||
891.2.n.f | 32 | 9.d | odd | 6 | 1 | ||
891.2.n.f | 32 | 33.h | odd | 10 | 1 | ||
891.2.n.f | 32 | 99.n | odd | 30 | 1 | ||
891.2.n.i | 32 | 1.a | even | 1 | 1 | trivial | |
891.2.n.i | 32 | 9.c | even | 3 | 1 | inner | |
891.2.n.i | 32 | 11.c | even | 5 | 1 | inner | |
891.2.n.i | 32 | 99.m | even | 15 | 1 | inner | |
3267.2.a.be | 8 | 99.n | odd | 30 | 1 | ||
3267.2.a.bf | 8 | 99.o | odd | 30 | 1 | ||
3267.2.a.bl | 8 | 99.p | even | 30 | 1 | ||
3267.2.a.bm | 8 | 99.m | even | 15 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 2 T_{2}^{31} - 4 T_{2}^{30} + 16 T_{2}^{29} - 24 T_{2}^{28} + 80 T_{2}^{27} + 180 T_{2}^{26} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\).