Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [232,3,Mod(115,232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(232, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("232.115");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 232.b (of order \(2\), degree \(1\), 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: | \(6.32154213316\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
115.1 | −1.99351 | − | 0.160993i | − | 4.31476i | 3.94816 | + | 0.641884i | 9.09838i | −0.694647 | + | 8.60151i | − | 8.77758i | −7.76736 | − | 1.91523i | −9.61712 | 1.46478 | − | 18.1377i | ||||||
115.2 | −1.99351 | + | 0.160993i | 4.31476i | 3.94816 | − | 0.641884i | − | 9.09838i | −0.694647 | − | 8.60151i | 8.77758i | −7.76736 | + | 1.91523i | −9.61712 | 1.46478 | + | 18.1377i | |||||||
115.3 | −1.92780 | − | 0.532521i | − | 4.75153i | 3.43284 | + | 2.05319i | − | 5.77072i | −2.53029 | + | 9.16001i | 4.89588i | −5.52447 | − | 5.78621i | −13.5770 | −3.07303 | + | 11.1248i | ||||||
115.4 | −1.92780 | + | 0.532521i | 4.75153i | 3.43284 | − | 2.05319i | 5.77072i | −2.53029 | − | 9.16001i | − | 4.89588i | −5.52447 | + | 5.78621i | −13.5770 | −3.07303 | − | 11.1248i | |||||||
115.5 | −1.85899 | − | 0.737665i | 4.63447i | 2.91170 | + | 2.74263i | − | 1.58058i | 3.41869 | − | 8.61545i | − | 8.48084i | −3.38969 | − | 7.24638i | −12.4783 | −1.16594 | + | 2.93828i | ||||||
115.6 | −1.85899 | + | 0.737665i | − | 4.63447i | 2.91170 | − | 2.74263i | 1.58058i | 3.41869 | + | 8.61545i | 8.48084i | −3.38969 | + | 7.24638i | −12.4783 | −1.16594 | − | 2.93828i | |||||||
115.7 | −1.78690 | − | 0.898321i | 3.19007i | 2.38604 | + | 3.21042i | 3.89874i | 2.86570 | − | 5.70034i | 12.5367i | −1.37963 | − | 7.88014i | −1.17652 | 3.50232 | − | 6.96667i | ||||||||
115.8 | −1.78690 | + | 0.898321i | − | 3.19007i | 2.38604 | − | 3.21042i | − | 3.89874i | 2.86570 | + | 5.70034i | − | 12.5367i | −1.37963 | + | 7.88014i | −1.17652 | 3.50232 | + | 6.96667i | |||||
115.9 | −1.75422 | − | 0.960583i | − | 0.800506i | 2.15456 | + | 3.37014i | − | 9.11788i | −0.768952 | + | 1.40426i | − | 7.90968i | −0.542273 | − | 7.98160i | 8.35919 | −8.75848 | + | 15.9948i | |||||
115.10 | −1.75422 | + | 0.960583i | 0.800506i | 2.15456 | − | 3.37014i | 9.11788i | −0.768952 | − | 1.40426i | 7.90968i | −0.542273 | + | 7.98160i | 8.35919 | −8.75848 | − | 15.9948i | ||||||||
115.11 | −1.64262 | − | 1.14096i | 1.23647i | 1.39641 | + | 3.74834i | 6.52788i | 1.41077 | − | 2.03106i | − | 8.84035i | 1.98293 | − | 7.75035i | 7.47113 | 7.44806 | − | 10.7228i | |||||||
115.12 | −1.64262 | + | 1.14096i | − | 1.23647i | 1.39641 | − | 3.74834i | − | 6.52788i | 1.41077 | + | 2.03106i | 8.84035i | 1.98293 | + | 7.75035i | 7.47113 | 7.44806 | + | 10.7228i | ||||||
115.13 | −1.63691 | − | 1.14914i | − | 3.33686i | 1.35897 | + | 3.76207i | 0.938063i | −3.83451 | + | 5.46215i | 5.06710i | 2.09861 | − | 7.71983i | −2.13463 | 1.07796 | − | 1.53553i | |||||||
115.14 | −1.63691 | + | 1.14914i | 3.33686i | 1.35897 | − | 3.76207i | − | 0.938063i | −3.83451 | − | 5.46215i | − | 5.06710i | 2.09861 | + | 7.71983i | −2.13463 | 1.07796 | + | 1.53553i | ||||||
115.15 | −1.23306 | − | 1.57467i | − | 2.41553i | −0.959149 | + | 3.88330i | 3.79912i | −3.80365 | + | 2.97848i | 1.38473i | 7.29759 | − | 3.27799i | 3.16523 | 5.98235 | − | 4.68452i | |||||||
115.16 | −1.23306 | + | 1.57467i | 2.41553i | −0.959149 | − | 3.88330i | − | 3.79912i | −3.80365 | − | 2.97848i | − | 1.38473i | 7.29759 | + | 3.27799i | 3.16523 | 5.98235 | + | 4.68452i | ||||||
115.17 | −1.02899 | − | 1.71499i | 0.366308i | −1.88236 | + | 3.52941i | − | 4.93433i | 0.628213 | − | 0.376927i | 6.63118i | 7.98982 | − | 0.403509i | 8.86582 | −8.46232 | + | 5.07738i | |||||||
115.18 | −1.02899 | + | 1.71499i | − | 0.366308i | −1.88236 | − | 3.52941i | 4.93433i | 0.628213 | + | 0.376927i | − | 6.63118i | 7.98982 | + | 0.403509i | 8.86582 | −8.46232 | − | 5.07738i | ||||||
115.19 | −0.951200 | − | 1.75932i | 3.37464i | −2.19044 | + | 3.34694i | − | 4.09203i | 5.93709 | − | 3.20996i | 0.0802561i | 7.97189 | + | 0.670088i | −2.38822 | −7.19920 | + | 3.89234i | |||||||
115.20 | −0.951200 | + | 1.75932i | − | 3.37464i | −2.19044 | − | 3.34694i | 4.09203i | 5.93709 | + | 3.20996i | − | 0.0802561i | 7.97189 | − | 0.670088i | −2.38822 | −7.19920 | − | 3.89234i | ||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
29.b | even | 2 | 1 | inner |
232.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 232.3.b.c | ✓ | 56 |
4.b | odd | 2 | 1 | 928.3.b.c | 56 | ||
8.b | even | 2 | 1 | 928.3.b.c | 56 | ||
8.d | odd | 2 | 1 | inner | 232.3.b.c | ✓ | 56 |
29.b | even | 2 | 1 | inner | 232.3.b.c | ✓ | 56 |
116.d | odd | 2 | 1 | 928.3.b.c | 56 | ||
232.b | odd | 2 | 1 | inner | 232.3.b.c | ✓ | 56 |
232.g | even | 2 | 1 | 928.3.b.c | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
232.3.b.c | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
232.3.b.c | ✓ | 56 | 8.d | odd | 2 | 1 | inner |
232.3.b.c | ✓ | 56 | 29.b | even | 2 | 1 | inner |
232.3.b.c | ✓ | 56 | 232.b | odd | 2 | 1 | inner |
928.3.b.c | 56 | 4.b | odd | 2 | 1 | ||
928.3.b.c | 56 | 8.b | even | 2 | 1 | ||
928.3.b.c | 56 | 116.d | odd | 2 | 1 | ||
928.3.b.c | 56 | 232.g | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(232, [\chi])\):
\( T_{3}^{28} + 172 T_{3}^{26} + 13002 T_{3}^{24} + 569432 T_{3}^{22} + 16036759 T_{3}^{20} + 304789316 T_{3}^{18} + 3991348352 T_{3}^{16} + 36115653072 T_{3}^{14} + 222992195775 T_{3}^{12} + \cdots + 89548674228 \) |
\( T_{31}^{28} - 12300 T_{31}^{26} + 67923754 T_{31}^{24} - 222275550672 T_{31}^{22} + 479106557841303 T_{31}^{20} + \cdots + 59\!\cdots\!00 \) |