Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,5,Mod(2,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.2");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 19 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 19.f (of order \(18\), degree \(6\), 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: | \(1.96402929859\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −7.01187 | − | 1.23638i | −2.17032 | + | 2.58649i | 32.6027 | + | 11.8664i | 26.1793 | − | 9.52849i | 18.4159 | − | 15.4528i | 22.6666 | + | 39.2597i | −115.276 | − | 66.5547i | 12.0859 | + | 68.5424i | −195.347 | + | 34.4449i |
2.2 | −3.73687 | − | 0.658911i | 0.577174 | − | 0.687849i | −1.50506 | − | 0.547798i | −39.5293 | + | 14.3875i | −2.61005 | + | 2.19009i | −34.7004 | − | 60.1028i | 57.8416 | + | 33.3949i | 13.9255 | + | 78.9754i | 157.196 | − | 27.7179i |
2.3 | −1.06853 | − | 0.188411i | 6.31184 | − | 7.52216i | −13.9288 | − | 5.06968i | 33.8670 | − | 12.3266i | −8.16166 | + | 6.84845i | −8.52110 | − | 14.7590i | 28.9626 | + | 16.7216i | −2.67805 | − | 15.1880i | −38.5105 | + | 6.79043i |
2.4 | −0.301706 | − | 0.0531989i | −9.51563 | + | 11.3403i | −14.9469 | − | 5.44022i | −0.554323 | + | 0.201757i | 3.47421 | − | 2.91521i | 33.9928 | + | 58.8772i | 8.46520 | + | 4.88738i | −23.9894 | − | 136.050i | 0.177976 | − | 0.0313819i |
2.5 | 4.43376 | + | 0.781791i | 6.29743 | − | 7.50499i | 4.01191 | + | 1.46022i | −26.3200 | + | 9.57971i | 33.7886 | − | 28.3520i | 37.8762 | + | 65.6035i | −45.7374 | − | 26.4065i | −2.60170 | − | 14.7550i | −124.186 | + | 21.8973i |
2.6 | 5.74553 | + | 1.01309i | −4.05101 | + | 4.82781i | 16.9497 | + | 6.16919i | 13.0485 | − | 4.74927i | −28.1662 | + | 23.6343i | −38.0385 | − | 65.8845i | 10.2945 | + | 5.94354i | 7.16848 | + | 40.6545i | 79.7821 | − | 14.0677i |
3.1 | −5.01717 | + | 5.97923i | 4.42557 | + | 12.1592i | −7.80084 | − | 44.2408i | −1.37942 | + | 7.82309i | −94.9062 | − | 34.5430i | −2.12885 | − | 3.68728i | 195.510 | + | 112.878i | −66.2097 | + | 55.5565i | −39.8553 | − | 47.4977i |
3.2 | −3.41023 | + | 4.06416i | −4.67230 | − | 12.8370i | −2.10931 | − | 11.9625i | 3.16499 | − | 17.9496i | 68.1054 | + | 24.7883i | −22.5528 | − | 39.0626i | −17.7027 | − | 10.2207i | −80.9096 | + | 67.8913i | 62.1565 | + | 74.0752i |
3.3 | −1.35909 | + | 1.61970i | 0.767327 | + | 2.10821i | 2.00207 | + | 11.3543i | −3.09216 | + | 17.5365i | −4.45754 | − | 1.62241i | 14.2495 | + | 24.6809i | −50.4090 | − | 29.1037i | 58.1938 | − | 48.8304i | −24.2013 | − | 28.8420i |
3.4 | 1.68606 | − | 2.00936i | 4.63257 | + | 12.7279i | 1.58361 | + | 8.98112i | 4.60964 | − | 26.1426i | 33.3857 | + | 12.1514i | −34.5022 | − | 59.7596i | 57.0623 | + | 32.9449i | −78.4888 | + | 65.8599i | −44.7579 | − | 53.3404i |
3.5 | 2.50472 | − | 2.98501i | −3.27675 | − | 9.00281i | 0.141711 | + | 0.803685i | 2.69721 | − | 15.2966i | −35.0808 | − | 12.7684i | 18.9504 | + | 32.8230i | 56.7476 | + | 32.7632i | −8.26383 | + | 6.93418i | −38.9048 | − | 46.3649i |
3.6 | 4.76936 | − | 5.68391i | 1.42559 | + | 3.91678i | −6.78159 | − | 38.4603i | −7.62340 | + | 43.2344i | 29.0618 | + | 10.5776i | −9.09343 | − | 15.7503i | −148.137 | − | 85.5267i | 48.7408 | − | 40.8984i | 209.382 | + | 249.531i |
10.1 | −7.01187 | + | 1.23638i | −2.17032 | − | 2.58649i | 32.6027 | − | 11.8664i | 26.1793 | + | 9.52849i | 18.4159 | + | 15.4528i | 22.6666 | − | 39.2597i | −115.276 | + | 66.5547i | 12.0859 | − | 68.5424i | −195.347 | − | 34.4449i |
10.2 | −3.73687 | + | 0.658911i | 0.577174 | + | 0.687849i | −1.50506 | + | 0.547798i | −39.5293 | − | 14.3875i | −2.61005 | − | 2.19009i | −34.7004 | + | 60.1028i | 57.8416 | − | 33.3949i | 13.9255 | − | 78.9754i | 157.196 | + | 27.7179i |
10.3 | −1.06853 | + | 0.188411i | 6.31184 | + | 7.52216i | −13.9288 | + | 5.06968i | 33.8670 | + | 12.3266i | −8.16166 | − | 6.84845i | −8.52110 | + | 14.7590i | 28.9626 | − | 16.7216i | −2.67805 | + | 15.1880i | −38.5105 | − | 6.79043i |
10.4 | −0.301706 | + | 0.0531989i | −9.51563 | − | 11.3403i | −14.9469 | + | 5.44022i | −0.554323 | − | 0.201757i | 3.47421 | + | 2.91521i | 33.9928 | − | 58.8772i | 8.46520 | − | 4.88738i | −23.9894 | + | 136.050i | 0.177976 | + | 0.0313819i |
10.5 | 4.43376 | − | 0.781791i | 6.29743 | + | 7.50499i | 4.01191 | − | 1.46022i | −26.3200 | − | 9.57971i | 33.7886 | + | 28.3520i | 37.8762 | − | 65.6035i | −45.7374 | + | 26.4065i | −2.60170 | + | 14.7550i | −124.186 | − | 21.8973i |
10.6 | 5.74553 | − | 1.01309i | −4.05101 | − | 4.82781i | 16.9497 | − | 6.16919i | 13.0485 | + | 4.74927i | −28.1662 | − | 23.6343i | −38.0385 | + | 65.8845i | 10.2945 | − | 5.94354i | 7.16848 | − | 40.6545i | 79.7821 | + | 14.0677i |
13.1 | −5.01717 | − | 5.97923i | 4.42557 | − | 12.1592i | −7.80084 | + | 44.2408i | −1.37942 | − | 7.82309i | −94.9062 | + | 34.5430i | −2.12885 | + | 3.68728i | 195.510 | − | 112.878i | −66.2097 | − | 55.5565i | −39.8553 | + | 47.4977i |
13.2 | −3.41023 | − | 4.06416i | −4.67230 | + | 12.8370i | −2.10931 | + | 11.9625i | 3.16499 | + | 17.9496i | 68.1054 | − | 24.7883i | −22.5528 | + | 39.0626i | −17.7027 | + | 10.2207i | −80.9096 | − | 67.8913i | 62.1565 | − | 74.0752i |
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 19.5.f.a | ✓ | 36 |
19.f | odd | 18 | 1 | inner | 19.5.f.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.5.f.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
19.5.f.a | ✓ | 36 | 19.f | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(19, [\chi])\).