Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,10,Mod(37,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.37");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.f (of order \(5\), degree \(4\), 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: | \(50.9885477802\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{5})\) |
Twist minimal: | no (minimal twist has level 11) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −35.8136 | − | 26.0201i | 0 | 447.352 | + | 1376.81i | −376.543 | + | 273.575i | 0 | −442.773 | − | 1362.72i | 12799.4 | − | 39392.6i | 0 | 20603.8 | ||||||||
37.2 | −24.5942 | − | 17.8687i | 0 | 127.366 | + | 391.991i | 601.873 | − | 437.286i | 0 | 430.583 | + | 1325.20i | −937.875 | + | 2886.48i | 0 | −22616.3 | ||||||||
37.3 | −12.1558 | − | 8.83167i | 0 | −88.4528 | − | 272.230i | −1436.78 | + | 1043.88i | 0 | 741.893 | + | 2283.31i | −3706.29 | + | 11406.8i | 0 | 26684.4 | ||||||||
37.4 | −7.79135 | − | 5.66074i | 0 | −129.556 | − | 398.731i | 1985.48 | − | 1442.54i | 0 | −2299.51 | − | 7077.16i | −2771.43 | + | 8529.59i | 0 | −23635.4 | ||||||||
37.5 | 7.91186 | + | 5.74830i | 0 | −128.662 | − | 395.981i | −690.317 | + | 501.544i | 0 | −2608.39 | − | 8027.81i | 2805.56 | − | 8634.62i | 0 | −8344.72 | ||||||||
37.6 | 14.4842 | + | 10.5234i | 0 | −59.1658 | − | 182.094i | 591.288 | − | 429.596i | 0 | 2103.13 | + | 6472.78i | 3891.91 | − | 11978.1i | 0 | 13085.2 | ||||||||
37.7 | 21.4033 | + | 15.5504i | 0 | 58.0694 | + | 178.719i | −1489.97 | + | 1082.53i | 0 | −720.731 | − | 2218.18i | 2649.49 | − | 8154.29i | 0 | −48724.0 | ||||||||
37.8 | 33.4202 | + | 24.2812i | 0 | 369.117 | + | 1136.03i | 314.303 | − | 228.354i | 0 | −1491.31 | − | 4589.77i | −8712.23 | + | 26813.5i | 0 | 16048.8 | ||||||||
64.1 | −11.8250 | + | 36.3935i | 0 | −770.441 | − | 559.758i | 276.050 | + | 849.593i | 0 | 42.1653 | + | 30.6349i | 13631.4 | − | 9903.80i | 0 | −34184.0 | ||||||||
64.2 | −6.35502 | + | 19.5587i | 0 | 72.0591 | + | 52.3540i | −411.773 | − | 1267.31i | 0 | 2149.58 | + | 1561.76i | −10000.4 | + | 7265.71i | 0 | 27403.7 | ||||||||
64.3 | −4.40341 | + | 13.5523i | 0 | 249.942 | + | 181.594i | −63.1636 | − | 194.398i | 0 | −1584.15 | − | 1150.95i | −9464.08 | + | 6876.06i | 0 | 2912.67 | ||||||||
64.4 | 1.04500 | − | 3.21619i | 0 | 404.965 | + | 294.224i | 656.025 | + | 2019.04i | 0 | −6768.61 | − | 4917.68i | 2770.23 | − | 2012.69i | 0 | 7179.16 | ||||||||
64.5 | 4.28817 | − | 13.1976i | 0 | 258.428 | + | 187.759i | 237.559 | + | 731.131i | 0 | 9125.19 | + | 6629.84i | 9334.16 | − | 6781.66i | 0 | 10667.9 | ||||||||
64.6 | 6.86703 | − | 21.1346i | 0 | 14.7033 | + | 10.6826i | −597.948 | − | 1840.30i | 0 | −7706.91 | − | 5599.40i | 9531.54 | − | 6925.07i | 0 | −43000.0 | ||||||||
64.7 | 10.8459 | − | 33.3802i | 0 | −582.388 | − | 423.130i | −311.221 | − | 957.840i | 0 | 7268.43 | + | 5280.82i | −5902.49 | + | 4288.41i | 0 | −35348.4 | ||||||||
64.8 | 13.1726 | − | 40.5409i | 0 | −1055.84 | − | 767.109i | 602.641 | + | 1854.74i | 0 | −3576.09 | − | 2598.18i | −27350.5 | + | 19871.3i | 0 | 83131.1 | ||||||||
82.1 | −11.8250 | − | 36.3935i | 0 | −770.441 | + | 559.758i | 276.050 | − | 849.593i | 0 | 42.1653 | − | 30.6349i | 13631.4 | + | 9903.80i | 0 | −34184.0 | ||||||||
82.2 | −6.35502 | − | 19.5587i | 0 | 72.0591 | − | 52.3540i | −411.773 | + | 1267.31i | 0 | 2149.58 | − | 1561.76i | −10000.4 | − | 7265.71i | 0 | 27403.7 | ||||||||
82.3 | −4.40341 | − | 13.5523i | 0 | 249.942 | − | 181.594i | −63.1636 | + | 194.398i | 0 | −1584.15 | + | 1150.95i | −9464.08 | − | 6876.06i | 0 | 2912.67 | ||||||||
82.4 | 1.04500 | + | 3.21619i | 0 | 404.965 | − | 294.224i | 656.025 | − | 2019.04i | 0 | −6768.61 | + | 4917.68i | 2770.23 | + | 2012.69i | 0 | 7179.16 | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.10.f.a | 32 | |
3.b | odd | 2 | 1 | 11.10.c.a | ✓ | 32 | |
11.c | even | 5 | 1 | inner | 99.10.f.a | 32 | |
33.f | even | 10 | 1 | 121.10.a.i | 16 | ||
33.h | odd | 10 | 1 | 11.10.c.a | ✓ | 32 | |
33.h | odd | 10 | 1 | 121.10.a.h | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.10.c.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
11.10.c.a | ✓ | 32 | 33.h | odd | 10 | 1 | |
99.10.f.a | 32 | 1.a | even | 1 | 1 | trivial | |
99.10.f.a | 32 | 11.c | even | 5 | 1 | inner | |
121.10.a.h | 16 | 33.h | odd | 10 | 1 | ||
121.10.a.i | 16 | 33.f | even | 10 | 1 |