Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,4,Mod(9,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 10]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.9");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.j (of order \(22\), degree \(10\), 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: | \(13.5704393013\) |
Analytic rank: | \(0\) |
Dimension: | \(360\) |
Relative dimension: | \(36\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −0.563465 | − | 1.91899i | −7.34353 | + | 6.36321i | −3.36501 | + | 2.16256i | −7.26047 | − | 8.50209i | 16.3487 | + | 10.5067i | −18.7911 | − | 8.58160i | 6.04600 | + | 5.23889i | 9.59456 | − | 66.7316i | −12.2244 | + | 18.7234i |
9.2 | −0.563465 | − | 1.91899i | −6.85739 | + | 5.94196i | −3.36501 | + | 2.16256i | 11.1057 | − | 1.29006i | 15.2664 | + | 9.81115i | 9.56919 | + | 4.37010i | 6.04600 | + | 5.23889i | 7.87438 | − | 54.7675i | −8.73326 | − | 20.5847i |
9.3 | −0.563465 | − | 1.91899i | −6.12185 | + | 5.30461i | −3.36501 | + | 2.16256i | −10.8097 | + | 2.85485i | 13.6289 | + | 8.75878i | 32.7383 | + | 14.9511i | 6.04600 | + | 5.23889i | 5.49562 | − | 38.2229i | 11.5693 | + | 19.1351i |
9.4 | −0.563465 | − | 1.91899i | −4.91203 | + | 4.25629i | −3.36501 | + | 2.16256i | 6.66177 | + | 8.97891i | 10.9355 | + | 7.02783i | −6.13831 | − | 2.80327i | 6.04600 | + | 5.23889i | 2.16945 | − | 15.0889i | 13.4767 | − | 17.8432i |
9.5 | −0.563465 | − | 1.91899i | −4.12395 | + | 3.57342i | −3.36501 | + | 2.16256i | 9.20079 | − | 6.35181i | 9.18104 | + | 5.90030i | −12.3150 | − | 5.62406i | 6.04600 | + | 5.23889i | 0.395105 | − | 2.74801i | −17.3734 | − | 14.0772i |
9.6 | −0.563465 | − | 1.91899i | −3.19029 | + | 2.76440i | −3.36501 | + | 2.16256i | −9.29353 | + | 6.21533i | 7.10247 | + | 4.56448i | −18.3244 | − | 8.36846i | 6.04600 | + | 5.23889i | −1.30647 | + | 9.08669i | 17.1637 | + | 14.3320i |
9.7 | −0.563465 | − | 1.91899i | −3.14614 | + | 2.72615i | −3.36501 | + | 2.16256i | −3.36640 | − | 10.6615i | 7.00418 | + | 4.50131i | 17.7210 | + | 8.09292i | 6.04600 | + | 5.23889i | −1.37617 | + | 9.57148i | −18.5624 | + | 12.4675i |
9.8 | −0.563465 | − | 1.91899i | −2.50934 | + | 2.17435i | −3.36501 | + | 2.16256i | 0.326731 | + | 11.1756i | 5.58647 | + | 3.59021i | 12.0046 | + | 5.48233i | 6.04600 | + | 5.23889i | −2.27354 | + | 15.8128i | 21.2617 | − | 6.92403i |
9.9 | −0.563465 | − | 1.91899i | 0.694762 | − | 0.602015i | −3.36501 | + | 2.16256i | 4.50572 | − | 10.2322i | −1.54673 | − | 0.994024i | −24.3344 | − | 11.1132i | 6.04600 | + | 5.23889i | −3.72223 | + | 25.8887i | −22.1743 | − | 2.88090i |
9.10 | −0.563465 | − | 1.91899i | 0.729490 | − | 0.632107i | −3.36501 | + | 2.16256i | −6.36706 | − | 9.19024i | −1.62405 | − | 1.04371i | 6.54228 | + | 2.98776i | 6.04600 | + | 5.23889i | −3.70990 | + | 25.8029i | −14.0483 | + | 17.3967i |
9.11 | −0.563465 | − | 1.91899i | 1.26527 | − | 1.09636i | −3.36501 | + | 2.16256i | 9.80304 | + | 5.37592i | −2.81684 | − | 1.81027i | 20.0538 | + | 9.15826i | 6.04600 | + | 5.23889i | −3.44360 | + | 23.9508i | 4.79264 | − | 21.8410i |
9.12 | −0.563465 | − | 1.91899i | 2.02304 | − | 1.75298i | −3.36501 | + | 2.16256i | −11.1730 | − | 0.405517i | −4.50385 | − | 2.89445i | 8.71258 | + | 3.97890i | 6.04600 | + | 5.23889i | −2.82272 | + | 19.6325i | 5.51741 | + | 21.6693i |
9.13 | −0.563465 | − | 1.91899i | 2.50028 | − | 2.16650i | −3.36501 | + | 2.16256i | 8.44207 | + | 7.33017i | −5.56631 | − | 3.57725i | −13.3837 | − | 6.11215i | 6.04600 | + | 5.23889i | −2.28485 | + | 15.8915i | 9.30968 | − | 20.3305i |
9.14 | −0.563465 | − | 1.91899i | 4.45429 | − | 3.85967i | −3.36501 | + | 2.16256i | 7.13469 | − | 8.60792i | −9.91648 | − | 6.37294i | 21.1421 | + | 9.65527i | 6.04600 | + | 5.23889i | 1.10120 | − | 7.65898i | −20.5386 | − | 8.84110i |
9.15 | −0.563465 | − | 1.91899i | 4.68867 | − | 4.06275i | −3.36501 | + | 2.16256i | −10.9539 | + | 2.23870i | −10.4383 | − | 6.70826i | 0.432083 | + | 0.197326i | 6.04600 | + | 5.23889i | 1.63513 | − | 11.3726i | 10.4682 | + | 19.7590i |
9.16 | −0.563465 | − | 1.91899i | 6.02364 | − | 5.21951i | −3.36501 | + | 2.16256i | 10.3489 | − | 4.23080i | −13.4103 | − | 8.61826i | −6.46040 | − | 2.95037i | 6.04600 | + | 5.23889i | 5.19840 | − | 36.1557i | −13.9501 | − | 17.4755i |
9.17 | −0.563465 | − | 1.91899i | 6.71164 | − | 5.81567i | −3.36501 | + | 2.16256i | −2.68913 | + | 10.8521i | −14.9420 | − | 9.60261i | 25.7501 | + | 11.7597i | 6.04600 | + | 5.23889i | 7.38159 | − | 51.3401i | 22.3403 | − | 0.954398i |
9.18 | −0.563465 | − | 1.91899i | 7.66317 | − | 6.64017i | −3.36501 | + | 2.16256i | −7.45417 | − | 8.33278i | −17.0603 | − | 10.9640i | −22.8813 | − | 10.4496i | 6.04600 | + | 5.23889i | 10.7897 | − | 75.0443i | −11.7903 | + | 18.9997i |
9.19 | 0.563465 | + | 1.91899i | −7.66317 | + | 6.64017i | −3.36501 | + | 2.16256i | 4.80461 | − | 10.0953i | −17.0603 | − | 10.9640i | 22.8813 | + | 10.4496i | −6.04600 | − | 5.23889i | 10.7897 | − | 75.0443i | 22.0800 | + | 3.53161i |
9.20 | 0.563465 | + | 1.91899i | −6.71164 | + | 5.81567i | −3.36501 | + | 2.16256i | 5.63759 | + | 9.65492i | −14.9420 | − | 9.60261i | −25.7501 | − | 11.7597i | −6.04600 | − | 5.23889i | 7.38159 | − | 51.3401i | −15.3511 | + | 16.2587i |
See next 80 embeddings (of 360 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
23.c | even | 11 | 1 | inner |
115.j | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.4.j.a | ✓ | 360 |
5.b | even | 2 | 1 | inner | 230.4.j.a | ✓ | 360 |
23.c | even | 11 | 1 | inner | 230.4.j.a | ✓ | 360 |
115.j | even | 22 | 1 | inner | 230.4.j.a | ✓ | 360 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.4.j.a | ✓ | 360 | 1.a | even | 1 | 1 | trivial |
230.4.j.a | ✓ | 360 | 5.b | even | 2 | 1 | inner |
230.4.j.a | ✓ | 360 | 23.c | even | 11 | 1 | inner |
230.4.j.a | ✓ | 360 | 115.j | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(230, [\chi])\).