Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,6,Mod(160,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.160");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 161.c (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: | \(25.8217949899\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
160.1 | −10.2721 | 25.0021i | 73.5163 | 51.3975 | − | 256.825i | 128.551 | − | 16.7787i | −426.461 | −382.105 | −527.961 | |||||||||||||||
160.2 | −10.2721 | 25.0021i | 73.5163 | −51.3975 | − | 256.825i | −128.551 | + | 16.7787i | −426.461 | −382.105 | 527.961 | |||||||||||||||
160.3 | −10.2721 | − | 25.0021i | 73.5163 | −51.3975 | 256.825i | −128.551 | − | 16.7787i | −426.461 | −382.105 | 527.961 | |||||||||||||||
160.4 | −10.2721 | − | 25.0021i | 73.5163 | 51.3975 | 256.825i | 128.551 | + | 16.7787i | −426.461 | −382.105 | −527.961 | |||||||||||||||
160.5 | −9.51992 | 10.0279i | 58.6288 | 107.196 | − | 95.4647i | −42.1636 | + | 122.594i | −253.504 | 142.441 | −1020.50 | |||||||||||||||
160.6 | −9.51992 | 10.0279i | 58.6288 | −107.196 | − | 95.4647i | 42.1636 | − | 122.594i | −253.504 | 142.441 | 1020.50 | |||||||||||||||
160.7 | −9.51992 | − | 10.0279i | 58.6288 | −107.196 | 95.4647i | 42.1636 | + | 122.594i | −253.504 | 142.441 | 1020.50 | |||||||||||||||
160.8 | −9.51992 | − | 10.0279i | 58.6288 | 107.196 | 95.4647i | −42.1636 | − | 122.594i | −253.504 | 142.441 | −1020.50 | |||||||||||||||
160.9 | −8.50063 | − | 9.26458i | 40.2607 | 8.77494 | 78.7548i | −115.888 | − | 58.1113i | −70.2213 | 157.168 | −74.5925 | |||||||||||||||
160.10 | −8.50063 | − | 9.26458i | 40.2607 | −8.77494 | 78.7548i | 115.888 | + | 58.1113i | −70.2213 | 157.168 | 74.5925 | |||||||||||||||
160.11 | −8.50063 | 9.26458i | 40.2607 | −8.77494 | − | 78.7548i | 115.888 | − | 58.1113i | −70.2213 | 157.168 | 74.5925 | |||||||||||||||
160.12 | −8.50063 | 9.26458i | 40.2607 | 8.77494 | − | 78.7548i | −115.888 | + | 58.1113i | −70.2213 | 157.168 | −74.5925 | |||||||||||||||
160.13 | −7.69648 | − | 23.6280i | 27.2358 | 52.2503 | 181.853i | −79.9898 | + | 102.023i | 36.6673 | −315.283 | −402.144 | |||||||||||||||
160.14 | −7.69648 | 23.6280i | 27.2358 | −52.2503 | − | 181.853i | 79.9898 | + | 102.023i | 36.6673 | −315.283 | 402.144 | |||||||||||||||
160.15 | −7.69648 | − | 23.6280i | 27.2358 | −52.2503 | 181.853i | 79.9898 | − | 102.023i | 36.6673 | −315.283 | 402.144 | |||||||||||||||
160.16 | −7.69648 | 23.6280i | 27.2358 | 52.2503 | − | 181.853i | −79.9898 | − | 102.023i | 36.6673 | −315.283 | −402.144 | |||||||||||||||
160.17 | −5.36240 | − | 14.2998i | −3.24467 | 33.8739 | 76.6815i | −9.98159 | + | 129.257i | 188.996 | 38.5144 | −181.646 | |||||||||||||||
160.18 | −5.36240 | − | 14.2998i | −3.24467 | −33.8739 | 76.6815i | 9.98159 | − | 129.257i | 188.996 | 38.5144 | 181.646 | |||||||||||||||
160.19 | −5.36240 | 14.2998i | −3.24467 | −33.8739 | − | 76.6815i | 9.98159 | + | 129.257i | 188.996 | 38.5144 | 181.646 | |||||||||||||||
160.20 | −5.36240 | 14.2998i | −3.24467 | 33.8739 | − | 76.6815i | −9.98159 | − | 129.257i | 188.996 | 38.5144 | −181.646 | |||||||||||||||
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
161.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 161.6.c.b | ✓ | 76 |
7.b | odd | 2 | 1 | inner | 161.6.c.b | ✓ | 76 |
23.b | odd | 2 | 1 | inner | 161.6.c.b | ✓ | 76 |
161.c | even | 2 | 1 | inner | 161.6.c.b | ✓ | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
161.6.c.b | ✓ | 76 | 1.a | even | 1 | 1 | trivial |
161.6.c.b | ✓ | 76 | 7.b | odd | 2 | 1 | inner |
161.6.c.b | ✓ | 76 | 23.b | odd | 2 | 1 | inner |
161.6.c.b | ✓ | 76 | 161.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{19} - 4 T_{2}^{18} - 428 T_{2}^{17} + 1545 T_{2}^{16} + 75709 T_{2}^{15} - 242562 T_{2}^{14} - 7182298 T_{2}^{13} + 19955241 T_{2}^{12} + 397008040 T_{2}^{11} - 926477020 T_{2}^{10} + \cdots - 2870938054656 \)
acting on \(S_{6}^{\mathrm{new}}(161, [\chi])\).