Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,10,Mod(3,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.3");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 11 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 11.c (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: | \(5.66539419780\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −33.4202 | + | 24.2812i | −18.4920 | − | 56.9126i | 369.117 | − | 1136.03i | −314.303 | − | 228.354i | 1999.91 | + | 1453.02i | −1491.31 | + | 4589.77i | 8712.23 | + | 26813.5i | 13026.8 | − | 9464.52i | 16048.8 | ||
3.2 | −21.4033 | + | 15.5504i | 76.8224 | + | 236.435i | 58.0694 | − | 178.719i | 1489.97 | + | 1082.53i | −5320.92 | − | 3865.87i | −720.731 | + | 2218.18i | −2649.49 | − | 8154.29i | −34076.0 | + | 24757.6i | −48724.0 | ||
3.3 | −14.4842 | + | 10.5234i | −4.17901 | − | 12.8617i | −59.1658 | + | 182.094i | −591.288 | − | 429.596i | 195.879 | + | 142.314i | 2103.13 | − | 6472.78i | −3891.91 | − | 11978.1i | 15775.9 | − | 11461.9i | 13085.2 | ||
3.4 | −7.91186 | + | 5.74830i | −77.1605 | − | 237.476i | −128.662 | + | 395.981i | 690.317 | + | 501.544i | 1975.57 | + | 1435.33i | −2608.39 | + | 8027.81i | −2805.56 | − | 8634.62i | −34517.1 | + | 25078.1i | −8344.72 | ||
3.5 | 7.79135 | − | 5.66074i | 51.4829 | + | 158.448i | −129.556 | + | 398.731i | −1985.48 | − | 1442.54i | 1298.06 | + | 943.093i | −2299.51 | + | 7077.16i | 2771.43 | + | 8529.59i | −6531.46 | + | 4745.38i | −23635.4 | ||
3.6 | 12.1558 | − | 8.83167i | 13.4151 | + | 41.2873i | −88.4528 | + | 272.230i | 1436.78 | + | 1043.88i | 527.706 | + | 383.401i | 741.893 | − | 2283.31i | 3706.29 | + | 11406.8i | 14399.2 | − | 10461.6i | 26684.4 | ||
3.7 | 24.5942 | − | 17.8687i | −45.7664 | − | 140.854i | 127.366 | − | 391.991i | −601.873 | − | 437.286i | −3642.47 | − | 2646.41i | 430.583 | − | 1325.20i | 937.875 | + | 2886.48i | −1821.51 | + | 1323.41i | −22616.3 | ||
3.8 | 35.8136 | − | 26.0201i | 63.0538 | + | 194.060i | 447.352 | − | 1376.81i | 376.543 | + | 273.575i | 7307.64 | + | 5309.31i | −442.773 | + | 1362.72i | −12799.4 | − | 39392.6i | −17759.5 | + | 12903.0i | 20603.8 | ||
4.1 | −33.4202 | − | 24.2812i | −18.4920 | + | 56.9126i | 369.117 | + | 1136.03i | −314.303 | + | 228.354i | 1999.91 | − | 1453.02i | −1491.31 | − | 4589.77i | 8712.23 | − | 26813.5i | 13026.8 | + | 9464.52i | 16048.8 | ||
4.2 | −21.4033 | − | 15.5504i | 76.8224 | − | 236.435i | 58.0694 | + | 178.719i | 1489.97 | − | 1082.53i | −5320.92 | + | 3865.87i | −720.731 | − | 2218.18i | −2649.49 | + | 8154.29i | −34076.0 | − | 24757.6i | −48724.0 | ||
4.3 | −14.4842 | − | 10.5234i | −4.17901 | + | 12.8617i | −59.1658 | − | 182.094i | −591.288 | + | 429.596i | 195.879 | − | 142.314i | 2103.13 | + | 6472.78i | −3891.91 | + | 11978.1i | 15775.9 | + | 11461.9i | 13085.2 | ||
4.4 | −7.91186 | − | 5.74830i | −77.1605 | + | 237.476i | −128.662 | − | 395.981i | 690.317 | − | 501.544i | 1975.57 | − | 1435.33i | −2608.39 | − | 8027.81i | −2805.56 | + | 8634.62i | −34517.1 | − | 25078.1i | −8344.72 | ||
4.5 | 7.79135 | + | 5.66074i | 51.4829 | − | 158.448i | −129.556 | − | 398.731i | −1985.48 | + | 1442.54i | 1298.06 | − | 943.093i | −2299.51 | − | 7077.16i | 2771.43 | − | 8529.59i | −6531.46 | − | 4745.38i | −23635.4 | ||
4.6 | 12.1558 | + | 8.83167i | 13.4151 | − | 41.2873i | −88.4528 | − | 272.230i | 1436.78 | − | 1043.88i | 527.706 | − | 383.401i | 741.893 | + | 2283.31i | 3706.29 | − | 11406.8i | 14399.2 | + | 10461.6i | 26684.4 | ||
4.7 | 24.5942 | + | 17.8687i | −45.7664 | + | 140.854i | 127.366 | + | 391.991i | −601.873 | + | 437.286i | −3642.47 | + | 2646.41i | 430.583 | + | 1325.20i | 937.875 | − | 2886.48i | −1821.51 | − | 1323.41i | −22616.3 | ||
4.8 | 35.8136 | + | 26.0201i | 63.0538 | − | 194.060i | 447.352 | + | 1376.81i | 376.543 | − | 273.575i | 7307.64 | − | 5309.31i | −442.773 | − | 1362.72i | −12799.4 | + | 39392.6i | −17759.5 | − | 12903.0i | 20603.8 | ||
5.1 | −13.1726 | − | 40.5409i | 121.812 | + | 88.5013i | −1055.84 | + | 767.109i | −602.641 | + | 1854.74i | 1983.36 | − | 6104.15i | −3576.09 | + | 2598.18i | 27350.5 | + | 19871.3i | 923.206 | + | 2841.34i | 83131.1 | ||
5.2 | −10.8459 | − | 33.3802i | −181.934 | − | 132.183i | −582.388 | + | 423.130i | 311.221 | − | 957.840i | −2439.05 | + | 7506.62i | 7268.43 | − | 5280.82i | 5902.49 | + | 4288.41i | 9545.25 | + | 29377.3i | −35348.4 | ||
5.3 | −6.86703 | − | 21.1346i | 42.7536 | + | 31.0623i | 14.7033 | − | 10.6826i | 597.948 | − | 1840.30i | 362.897 | − | 1116.88i | −7706.91 | + | 5599.40i | −9531.54 | − | 6925.07i | −5219.38 | − | 16063.6i | −43000.0 | ||
5.4 | −4.28817 | − | 13.1976i | 55.2813 | + | 40.1642i | 258.428 | − | 187.759i | −237.559 | + | 731.131i | 293.017 | − | 901.813i | 9125.19 | − | 6629.84i | −9334.16 | − | 6781.66i | −4639.52 | − | 14279.0i | 10667.9 | ||
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 | 11.10.c.a | ✓ | 32 |
3.b | odd | 2 | 1 | 99.10.f.a | 32 | ||
11.c | even | 5 | 1 | inner | 11.10.c.a | ✓ | 32 |
11.c | even | 5 | 1 | 121.10.a.h | 16 | ||
11.d | odd | 10 | 1 | 121.10.a.i | 16 | ||
33.h | odd | 10 | 1 | 99.10.f.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.10.c.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
11.10.c.a | ✓ | 32 | 11.c | even | 5 | 1 | inner |
99.10.f.a | 32 | 3.b | odd | 2 | 1 | ||
99.10.f.a | 32 | 33.h | odd | 10 | 1 | ||
121.10.a.h | 16 | 11.c | even | 5 | 1 | ||
121.10.a.i | 16 | 11.d | odd | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(11, [\chi])\).