Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,16,Mod(4,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.4");
S:= CuspForms(chi, 16);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 17 \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 17.c (of order \(4\), degree \(2\), 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: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | − | 348.759i | 5015.99 | − | 5015.99i | −88865.1 | −20803.9 | + | 20803.9i | −1.74937e6 | − | 1.74937e6i | −1.51898e6 | − | 1.51898e6i | 1.95644e7i | − | 3.59714e7i | 7.25557e6 | + | 7.25557e6i | ||||||
4.2 | − | 338.865i | −2552.04 | + | 2552.04i | −82061.8 | −182025. | + | 182025.i | 864799. | + | 864799.i | −103726. | − | 103726.i | 1.67040e7i | 1.32307e6i | 6.16820e7 | + | 6.16820e7i | |||||||
4.3 | − | 302.124i | −334.184 | + | 334.184i | −58511.1 | 109767. | − | 109767.i | 100965. | + | 100965.i | 1.54065e6 | + | 1.54065e6i | 7.77762e6i | 1.41255e7i | −3.31634e7 | − | 3.31634e7i | |||||||
4.4 | − | 250.655i | −4190.42 | + | 4190.42i | −30059.9 | 114890. | − | 114890.i | 1.05035e6 | + | 1.05035e6i | −2.34763e6 | − | 2.34763e6i | − | 678803.i | − | 2.07703e7i | −2.87976e7 | − | 2.87976e7i | |||||
4.5 | − | 235.953i | 1501.32 | − | 1501.32i | −22905.8 | 44470.0 | − | 44470.0i | −354242. | − | 354242.i | −175577. | − | 175577.i | − | 2.32702e6i | 9.84096e6i | −1.04928e7 | − | 1.04928e7i | ||||||
4.6 | − | 191.501i | 1757.52 | − | 1757.52i | −3904.64 | −212618. | + | 212618.i | −336567. | − | 336567.i | −620986. | − | 620986.i | − | 5.52736e6i | 8.17116e6i | 4.07165e7 | + | 4.07165e7i | ||||||
4.7 | − | 155.544i | −3969.60 | + | 3969.60i | 8574.15 | −108667. | + | 108667.i | 617447. | + | 617447.i | 2.10978e6 | + | 2.10978e6i | − | 6.43051e6i | − | 1.71666e7i | 1.69025e7 | + | 1.69025e7i | |||||
4.8 | − | 133.029i | 4742.62 | − | 4742.62i | 15071.4 | −14498.4 | + | 14498.4i | −630905. | − | 630905.i | 2.54959e6 | + | 2.54959e6i | − | 6.36401e6i | − | 3.06360e7i | 1.92870e6 | + | 1.92870e6i | |||||
4.9 | − | 80.4188i | 3327.44 | − | 3327.44i | 26300.8 | 211555. | − | 211555.i | −267589. | − | 267589.i | −2.55424e6 | − | 2.55424e6i | − | 4.75024e6i | − | 7.79484e6i | −1.70130e7 | − | 1.70130e7i | |||||
4.10 | − | 66.3760i | −1952.69 | + | 1952.69i | 28362.2 | −33212.7 | + | 33212.7i | 129612. | + | 129612.i | −1.64518e6 | − | 1.64518e6i | − | 4.05758e6i | 6.72291e6i | 2.20452e6 | + | 2.20452e6i | ||||||
4.11 | − | 53.4016i | −2453.73 | + | 2453.73i | 29916.3 | 225279. | − | 225279.i | 131033. | + | 131033.i | 2.15818e6 | + | 2.15818e6i | − | 3.34744e6i | 2.30736e6i | −1.20303e7 | − | 1.20303e7i | ||||||
4.12 | 31.6064i | −152.613 | + | 152.613i | 31769.0 | −69379.4 | + | 69379.4i | −4823.54 | − | 4823.54i | −162757. | − | 162757.i | 2.03978e6i | 1.43023e7i | −2.19284e6 | − | 2.19284e6i | ||||||||
4.13 | 39.3257i | 2394.27 | − | 2394.27i | 31221.5 | 13542.0 | − | 13542.0i | 94156.4 | + | 94156.4i | 1.69479e6 | + | 1.69479e6i | 2.51643e6i | 2.88386e6i | 532547. | + | 532547.i | ||||||||
4.14 | 119.498i | 4421.91 | − | 4421.91i | 18488.2 | −204259. | + | 204259.i | 528410. | + | 528410.i | −2.13755e6 | − | 2.13755e6i | 6.12502e6i | − | 2.47577e7i | −2.44085e7 | − | 2.44085e7i | |||||||
4.15 | 138.706i | −4846.03 | + | 4846.03i | 13528.7 | −100507. | + | 100507.i | −672173. | − | 672173.i | −478607. | − | 478607.i | 6.42162e6i | − | 3.26192e7i | −1.39409e7 | − | 1.39409e7i | |||||||
4.16 | 167.241i | −2780.51 | + | 2780.51i | 4798.53 | 136750. | − | 136750.i | −465015. | − | 465015.i | −526443. | − | 526443.i | 6.28266e6i | − | 1.11360e6i | 2.28702e7 | + | 2.28702e7i | |||||||
4.17 | 184.257i | 3098.34 | − | 3098.34i | −1182.81 | 93660.4 | − | 93660.4i | 570892. | + | 570892.i | −101141. | − | 101141.i | 5.81981e6i | − | 4.85050e6i | 1.72576e7 | + | 1.72576e7i | |||||||
4.18 | 224.704i | −688.326 | + | 688.326i | −17723.8 | −200306. | + | 200306.i | −154669. | − | 154669.i | 2.77180e6 | + | 2.77180e6i | 3.38049e6i | 1.34013e7i | −4.50094e7 | − | 4.50094e7i | ||||||||
4.19 | 274.663i | 931.417 | − | 931.417i | −42671.9 | 129422. | − | 129422.i | 255826. | + | 255826.i | 151712. | + | 151712.i | − | 2.72024e6i | 1.26138e7i | 3.55476e7 | + | 3.55476e7i | |||||||
4.20 | 300.568i | −890.573 | + | 890.573i | −57572.8 | −102815. | + | 102815.i | −267677. | − | 267677.i | −2.55244e6 | − | 2.55244e6i | − | 7.45553e6i | 1.27627e7i | −3.09027e7 | − | 3.09027e7i | |||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.c | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 17.16.c.a | ✓ | 44 |
17.c | even | 4 | 1 | inner | 17.16.c.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.16.c.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
17.16.c.a | ✓ | 44 | 17.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(17, [\chi])\).