Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,3,Mod(305,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.305");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.h (of order \(2\), degree \(1\), 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: | \(24.8502001097\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Twist minimal: | no (minimal twist has level 456) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
305.1 | 0 | −2.98372 | − | 0.312096i | 0 | 2.00477i | 0 | 9.74789 | 0 | 8.80519 | + | 1.86242i | 0 | ||||||||||||||
305.2 | 0 | −2.98372 | + | 0.312096i | 0 | − | 2.00477i | 0 | 9.74789 | 0 | 8.80519 | − | 1.86242i | 0 | |||||||||||||
305.3 | 0 | −2.96520 | − | 0.455642i | 0 | 8.19836i | 0 | −10.4735 | 0 | 8.58478 | + | 2.70214i | 0 | ||||||||||||||
305.4 | 0 | −2.96520 | + | 0.455642i | 0 | − | 8.19836i | 0 | −10.4735 | 0 | 8.58478 | − | 2.70214i | 0 | |||||||||||||
305.5 | 0 | −2.76466 | − | 1.16475i | 0 | 7.59780i | 0 | −5.98667 | 0 | 6.28671 | + | 6.44028i | 0 | ||||||||||||||
305.6 | 0 | −2.76466 | + | 1.16475i | 0 | − | 7.59780i | 0 | −5.98667 | 0 | 6.28671 | − | 6.44028i | 0 | |||||||||||||
305.7 | 0 | −2.72635 | − | 1.25180i | 0 | − | 0.528472i | 0 | −1.87136 | 0 | 5.86601 | + | 6.82568i | 0 | |||||||||||||
305.8 | 0 | −2.72635 | + | 1.25180i | 0 | 0.528472i | 0 | −1.87136 | 0 | 5.86601 | − | 6.82568i | 0 | ||||||||||||||
305.9 | 0 | −2.23280 | − | 2.00365i | 0 | − | 8.74486i | 0 | 7.20322 | 0 | 0.970748 | + | 8.94749i | 0 | |||||||||||||
305.10 | 0 | −2.23280 | + | 2.00365i | 0 | 8.74486i | 0 | 7.20322 | 0 | 0.970748 | − | 8.94749i | 0 | ||||||||||||||
305.11 | 0 | −1.89465 | − | 2.32600i | 0 | − | 1.91924i | 0 | −7.15920 | 0 | −1.82058 | + | 8.81394i | 0 | |||||||||||||
305.12 | 0 | −1.89465 | + | 2.32600i | 0 | 1.91924i | 0 | −7.15920 | 0 | −1.82058 | − | 8.81394i | 0 | ||||||||||||||
305.13 | 0 | −1.27146 | − | 2.71724i | 0 | 4.57275i | 0 | 6.79364 | 0 | −5.76676 | + | 6.90974i | 0 | ||||||||||||||
305.14 | 0 | −1.27146 | + | 2.71724i | 0 | − | 4.57275i | 0 | 6.79364 | 0 | −5.76676 | − | 6.90974i | 0 | |||||||||||||
305.15 | 0 | −0.922699 | − | 2.85458i | 0 | 8.24976i | 0 | 6.69140 | 0 | −7.29725 | + | 5.26784i | 0 | ||||||||||||||
305.16 | 0 | −0.922699 | + | 2.85458i | 0 | − | 8.24976i | 0 | 6.69140 | 0 | −7.29725 | − | 5.26784i | 0 | |||||||||||||
305.17 | 0 | 0.316327 | − | 2.98328i | 0 | − | 4.15317i | 0 | −12.8970 | 0 | −8.79987 | − | 1.88738i | 0 | |||||||||||||
305.18 | 0 | 0.316327 | + | 2.98328i | 0 | 4.15317i | 0 | −12.8970 | 0 | −8.79987 | + | 1.88738i | 0 | ||||||||||||||
305.19 | 0 | 0.423217 | − | 2.97000i | 0 | − | 0.567002i | 0 | −4.38953 | 0 | −8.64177 | − | 2.51391i | 0 | |||||||||||||
305.20 | 0 | 0.423217 | + | 2.97000i | 0 | 0.567002i | 0 | −4.38953 | 0 | −8.64177 | + | 2.51391i | 0 | ||||||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.3.h.d | 36 | |
3.b | odd | 2 | 1 | inner | 912.3.h.d | 36 | |
4.b | odd | 2 | 1 | 456.3.h.a | ✓ | 36 | |
12.b | even | 2 | 1 | 456.3.h.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
456.3.h.a | ✓ | 36 | 4.b | odd | 2 | 1 | |
456.3.h.a | ✓ | 36 | 12.b | even | 2 | 1 | |
912.3.h.d | 36 | 1.a | even | 1 | 1 | trivial | |
912.3.h.d | 36 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{36} + 520 T_{5}^{34} + 122068 T_{5}^{32} + 17126088 T_{5}^{30} + 1602774310 T_{5}^{28} + 105771193736 T_{5}^{26} + 5074684791836 T_{5}^{24} + 179915051470136 T_{5}^{22} + \cdots + 16\!\cdots\!00 \)
acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\).