Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,16,Mod(16,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.16");
S:= CuspForms(chi, 16);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 17 \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 17.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: | \(24.2578958670\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −343.151 | 2662.06i | 84984.7 | − | 187278.i | − | 913491.i | − | 2.90764e6i | −1.79182e7 | 7.26232e6 | 6.42646e7i | |||||||||||||||
16.2 | −343.151 | − | 2662.06i | 84984.7 | 187278.i | 913491.i | 2.90764e6i | −1.79182e7 | 7.26232e6 | − | 6.42646e7i | ||||||||||||||||
16.3 | −271.837 | − | 6935.71i | 41127.3 | − | 305155.i | 1.88538e6i | 956533.i | −2.27238e6 | −3.37551e7 | 8.29523e7i | ||||||||||||||||
16.4 | −271.837 | 6935.71i | 41127.3 | 305155.i | − | 1.88538e6i | − | 956533.i | −2.27238e6 | −3.37551e7 | − | 8.29523e7i | |||||||||||||||
16.5 | −252.822 | 1625.59i | 31151.0 | 14176.6i | − | 410984.i | 2.61491e6i | 408809. | 1.17064e7 | − | 3.58415e6i | ||||||||||||||||
16.6 | −252.822 | − | 1625.59i | 31151.0 | − | 14176.6i | 410984.i | − | 2.61491e6i | 408809. | 1.17064e7 | 3.58415e6i | |||||||||||||||
16.7 | −202.905 | 5602.01i | 8402.47 | − | 284728.i | − | 1.13668e6i | 2.21629e6i | 4.94389e6 | −1.70336e7 | 5.77728e7i | ||||||||||||||||
16.8 | −202.905 | − | 5602.01i | 8402.47 | 284728.i | 1.13668e6i | − | 2.21629e6i | 4.94389e6 | −1.70336e7 | − | 5.77728e7i | |||||||||||||||
16.9 | −103.869 | 2983.33i | −21979.3 | 46006.2i | − | 309875.i | − | 1.62872e6i | 5.68653e6 | 5.44863e6 | − | 4.77860e6i | |||||||||||||||
16.10 | −103.869 | − | 2983.33i | −21979.3 | − | 46006.2i | 309875.i | 1.62872e6i | 5.68653e6 | 5.44863e6 | 4.77860e6i | ||||||||||||||||
16.11 | 12.5353 | − | 1940.16i | −32610.9 | − | 330469.i | − | 24320.6i | − | 2.72193e6i | −819546. | 1.05847e7 | − | 4.14255e6i | |||||||||||||
16.12 | 12.5353 | 1940.16i | −32610.9 | 330469.i | 24320.6i | 2.72193e6i | −819546. | 1.05847e7 | 4.14255e6i | ||||||||||||||||||
16.13 | 52.4616 | 5953.93i | −30015.8 | − | 78941.6i | 312353.i | − | 404894.i | −3.29374e6 | −2.11004e7 | − | 4.14140e6i | |||||||||||||||
16.14 | 52.4616 | − | 5953.93i | −30015.8 | 78941.6i | − | 312353.i | 404894.i | −3.29374e6 | −2.11004e7 | 4.14140e6i | ||||||||||||||||
16.15 | 158.526 | 159.452i | −7637.56 | − | 199360.i | 25277.2i | 4.18240e6i | −6.40533e6 | 1.43235e7 | − | 3.16037e7i | ||||||||||||||||
16.16 | 158.526 | − | 159.452i | −7637.56 | 199360.i | − | 25277.2i | − | 4.18240e6i | −6.40533e6 | 1.43235e7 | 3.16037e7i | |||||||||||||||
16.17 | 210.994 | 3417.77i | 11750.3 | − | 46569.8i | 721128.i | 204211.i | −4.43460e6 | 2.66774e6 | − | 9.82593e6i | ||||||||||||||||
16.18 | 210.994 | − | 3417.77i | 11750.3 | 46569.8i | − | 721128.i | − | 204211.i | −4.43460e6 | 2.66774e6 | 9.82593e6i | |||||||||||||||
16.19 | 284.859 | 6544.95i | 48376.7 | 188557.i | 1.86439e6i | 714240.i | 4.44627e6 | −2.84875e7 | 5.37121e7i | ||||||||||||||||||
16.20 | 284.859 | − | 6544.95i | 48376.7 | − | 188557.i | − | 1.86439e6i | − | 714240.i | 4.44627e6 | −2.84875e7 | − | 5.37121e7i | |||||||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 17.16.b.a | ✓ | 22 |
17.b | even | 2 | 1 | inner | 17.16.b.a | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.16.b.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
17.16.b.a | ✓ | 22 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(17, [\chi])\).