Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,4,Mod(4,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.e (of order \(11\), 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: | \(4.07113179040\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | −2.04147 | − | 4.47020i | −2.87848 | + | 0.845198i | −10.5762 | + | 12.2056i | 16.4166 | + | 10.5503i | 9.65454 | + | 11.1419i | 0.635011 | + | 4.41660i | 38.4307 | + | 11.2843i | 7.57128 | − | 4.86577i | 13.6479 | − | 94.9235i |
4.2 | −1.25209 | − | 2.74169i | −2.87848 | + | 0.845198i | −0.710260 | + | 0.819684i | −9.32369 | − | 5.99197i | 5.92138 | + | 6.83364i | 2.45322 | + | 17.0626i | −19.9992 | − | 5.87229i | 7.57128 | − | 4.86577i | −4.75405 | + | 33.0652i |
4.3 | −0.464261 | − | 1.01659i | −2.87848 | + | 0.845198i | 4.42097 | − | 5.10207i | 6.11072 | + | 3.92712i | 2.19559 | + | 2.53384i | −1.36340 | − | 9.48266i | −15.8177 | − | 4.64450i | 7.57128 | − | 4.86577i | 1.15530 | − | 8.03531i |
4.4 | 0.700410 | + | 1.53368i | −2.87848 | + | 0.845198i | 3.37727 | − | 3.89758i | −14.3333 | − | 9.21148i | −3.31238 | − | 3.82269i | −3.63633 | − | 25.2912i | 21.2851 | + | 6.24988i | 7.57128 | − | 4.86577i | 4.08828 | − | 28.4346i |
4.5 | 0.718480 | + | 1.57325i | −2.87848 | + | 0.845198i | 3.27998 | − | 3.78529i | 5.04169 | + | 3.24010i | −3.39784 | − | 3.92132i | 4.06727 | + | 28.2885i | 21.5877 | + | 6.33873i | 7.57128 | − | 4.86577i | −1.47514 | + | 10.2598i |
4.6 | 2.08583 | + | 4.56734i | −2.87848 | + | 0.845198i | −11.2710 | + | 13.0074i | −5.38817 | − | 3.46277i | −9.86434 | − | 11.3841i | 1.35014 | + | 9.39044i | −44.3773 | − | 13.0304i | 7.57128 | − | 4.86577i | 4.57681 | − | 31.8324i |
13.1 | −3.15951 | − | 3.64626i | 2.52376 | + | 1.62192i | −2.17425 | + | 15.1222i | 0.670517 | − | 1.46823i | −2.05988 | − | 14.3268i | 29.0868 | + | 8.54065i | 29.5387 | − | 18.9834i | 3.73874 | + | 8.18669i | −7.47204 | + | 2.19399i |
13.2 | −2.24664 | − | 2.59277i | 2.52376 | + | 1.62192i | −0.536504 | + | 3.73147i | −5.18705 | + | 11.3581i | −1.46473 | − | 10.1874i | −25.5228 | − | 7.49416i | −12.2087 | + | 7.84605i | 3.73874 | + | 8.18669i | 41.1022 | − | 12.0687i |
13.3 | −0.456676 | − | 0.527032i | 2.52376 | + | 1.62192i | 1.06931 | − | 7.43721i | 3.32762 | − | 7.28646i | −0.297736 | − | 2.07080i | −13.2035 | − | 3.87690i | −9.10125 | + | 5.84902i | 3.73874 | + | 8.18669i | −5.35984 | + | 1.57379i |
13.4 | 0.280926 | + | 0.324206i | 2.52376 | + | 1.62192i | 1.11233 | − | 7.73642i | −9.21494 | + | 20.1779i | 0.183153 | + | 1.27386i | 29.8306 | + | 8.75906i | 5.70777 | − | 3.66816i | 3.73874 | + | 8.18669i | −9.13053 | + | 2.68096i |
13.5 | 1.38600 | + | 1.59953i | 2.52376 | + | 1.62192i | 0.501022 | − | 3.48468i | 6.21106 | − | 13.6003i | 0.903619 | + | 6.28481i | 10.6195 | + | 3.11816i | 20.5122 | − | 13.1824i | 3.73874 | + | 8.18669i | 30.3627 | − | 8.91528i |
13.6 | 2.90770 | + | 3.35566i | 2.52376 | + | 1.62192i | −1.66725 | + | 11.5960i | −1.16276 | + | 2.54608i | 1.89571 | + | 13.1850i | −5.53916 | − | 1.62644i | −13.8774 | + | 8.91848i | 3.73874 | + | 8.18669i | −11.9247 | + | 3.50142i |
16.1 | −3.15951 | + | 3.64626i | 2.52376 | − | 1.62192i | −2.17425 | − | 15.1222i | 0.670517 | + | 1.46823i | −2.05988 | + | 14.3268i | 29.0868 | − | 8.54065i | 29.5387 | + | 18.9834i | 3.73874 | − | 8.18669i | −7.47204 | − | 2.19399i |
16.2 | −2.24664 | + | 2.59277i | 2.52376 | − | 1.62192i | −0.536504 | − | 3.73147i | −5.18705 | − | 11.3581i | −1.46473 | + | 10.1874i | −25.5228 | + | 7.49416i | −12.2087 | − | 7.84605i | 3.73874 | − | 8.18669i | 41.1022 | + | 12.0687i |
16.3 | −0.456676 | + | 0.527032i | 2.52376 | − | 1.62192i | 1.06931 | + | 7.43721i | 3.32762 | + | 7.28646i | −0.297736 | + | 2.07080i | −13.2035 | + | 3.87690i | −9.10125 | − | 5.84902i | 3.73874 | − | 8.18669i | −5.35984 | − | 1.57379i |
16.4 | 0.280926 | − | 0.324206i | 2.52376 | − | 1.62192i | 1.11233 | + | 7.73642i | −9.21494 | − | 20.1779i | 0.183153 | − | 1.27386i | 29.8306 | − | 8.75906i | 5.70777 | + | 3.66816i | 3.73874 | − | 8.18669i | −9.13053 | − | 2.68096i |
16.5 | 1.38600 | − | 1.59953i | 2.52376 | − | 1.62192i | 0.501022 | + | 3.48468i | 6.21106 | + | 13.6003i | 0.903619 | − | 6.28481i | 10.6195 | − | 3.11816i | 20.5122 | + | 13.1824i | 3.73874 | − | 8.18669i | 30.3627 | + | 8.91528i |
16.6 | 2.90770 | − | 3.35566i | 2.52376 | − | 1.62192i | −1.66725 | − | 11.5960i | −1.16276 | − | 2.54608i | 1.89571 | − | 13.1850i | −5.53916 | + | 1.62644i | −13.8774 | − | 8.91848i | 3.73874 | − | 8.18669i | −11.9247 | − | 3.50142i |
25.1 | −3.93223 | − | 2.52709i | −0.426945 | − | 2.96946i | 5.75294 | + | 12.5972i | −6.57189 | − | 1.92968i | −5.82527 | + | 12.7556i | −12.7631 | + | 14.7294i | 3.89062 | − | 27.0599i | −8.63544 | + | 2.53559i | 20.9657 | + | 24.1957i |
25.2 | −3.39636 | − | 2.18271i | −0.426945 | − | 2.96946i | 3.44773 | + | 7.54946i | 16.0504 | + | 4.71284i | −5.03142 | + | 11.0173i | 23.9549 | − | 27.6454i | 0.172065 | − | 1.19674i | −8.63544 | + | 2.53559i | −44.2263 | − | 51.0399i |
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.4.e.a | ✓ | 60 |
3.b | odd | 2 | 1 | 207.4.i.c | 60 | ||
23.c | even | 11 | 1 | inner | 69.4.e.a | ✓ | 60 |
23.c | even | 11 | 1 | 1587.4.a.t | 30 | ||
23.d | odd | 22 | 1 | 1587.4.a.u | 30 | ||
69.h | odd | 22 | 1 | 207.4.i.c | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.4.e.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
69.4.e.a | ✓ | 60 | 23.c | even | 11 | 1 | inner |
207.4.i.c | 60 | 3.b | odd | 2 | 1 | ||
207.4.i.c | 60 | 69.h | odd | 22 | 1 | ||
1587.4.a.t | 30 | 23.c | even | 11 | 1 | ||
1587.4.a.u | 30 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{60} + 38 T_{2}^{58} - 64 T_{2}^{57} + 679 T_{2}^{56} - 3280 T_{2}^{55} + 17276 T_{2}^{54} - 51024 T_{2}^{53} + 648894 T_{2}^{52} - 696949 T_{2}^{51} + 17770628 T_{2}^{50} - 19361942 T_{2}^{49} + \cdots + 86\!\cdots\!64 \)
acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\).