Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,15,Mod(2,11)]
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([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.2");
S:= CuspForms(chi, 15);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 11 \) |
Weight: | \( k \) | \(=\) | \( 15 \) |
Character orbit: | \([\chi]\) | \(=\) | 11.d (of order \(10\), 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: | \(13.6761864967\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | −236.091 | − | 76.7107i | 1404.04 | − | 1020.09i | 36599.6 | + | 26591.2i | 17129.5 | + | 52719.2i | −409733. | + | 133130.i | 347419. | − | 478181.i | −4.21038e6 | − | 5.79510e6i | −547289. | + | 1.68438e6i | − | 1.37606e7i | |
2.2 | −199.758 | − | 64.9052i | −3456.81 | + | 2511.52i | 22435.6 | + | 16300.4i | −9471.22 | − | 29149.4i | 853535. | − | 277330.i | −55107.8 | + | 75849.3i | −1.40098e6 | − | 1.92828e6i | 4.16378e6 | − | 1.28148e7i | 6.43755e6i | ||
2.3 | −157.861 | − | 51.2920i | 79.9689 | − | 58.1008i | 9034.19 | + | 6563.72i | −14721.0 | − | 45306.7i | −15604.0 | + | 5070.06i | −565560. | + | 778426.i | 509000. | + | 700579.i | −1.47500e6 | + | 4.53958e6i | 7.90722e6i | ||
2.4 | −119.417 | − | 38.8010i | 3112.87 | − | 2261.63i | −499.976 | − | 363.254i | −39265.4 | − | 120847.i | −459484. | + | 149295.i | 577116. | − | 794332.i | 1.25481e6 | + | 1.72710e6i | 3.09696e6 | − | 9.53146e6i | 1.59547e7i | ||
2.5 | −116.568 | − | 37.8754i | −829.952 | + | 602.996i | −1101.28 | − | 800.129i | 25059.8 | + | 77126.2i | 119585. | − | 38855.5i | 411520. | − | 566408.i | 1.27843e6 | + | 1.75960e6i | −1.15280e6 | + | 3.54796e6i | − | 9.93962e6i | |
2.6 | −57.1439 | − | 18.5672i | 2553.58 | − | 1855.29i | −10334.3 | − | 7508.27i | 43338.6 | + | 133382.i | −180369. | + | 58605.5i | −281229. | + | 387078.i | 1.02976e6 | + | 1.41735e6i | 1.60068e6 | − | 4.92639e6i | − | 8.42667e6i | |
2.7 | −13.9048 | − | 4.51795i | −1467.63 | + | 1066.30i | −13082.0 | − | 9504.63i | −19976.8 | − | 61482.1i | 25224.6 | − | 8195.98i | 351170. | − | 483344.i | 279760. | + | 385057.i | −461067. | + | 1.41902e6i | 945153.i | ||
2.8 | 47.9694 | + | 15.5862i | −2902.88 | + | 2109.07i | −11196.8 | − | 8134.95i | 33214.8 | + | 102224.i | −172122. | + | 55925.8i | −499247. | + | 687155.i | −896042. | − | 1.23330e6i | 2.50055e6 | − | 7.69589e6i | 5.42134e6i | ||
2.9 | 53.1516 | + | 17.2700i | 1186.53 | − | 862.066i | −10728.1 | − | 7794.42i | −13864.2 | − | 42669.5i | 77954.0 | − | 25328.8i | −759822. | + | 1.04581e6i | −973812. | − | 1.34034e6i | −813318. | + | 2.50313e6i | − | 2.50739e6i | |
2.10 | 103.867 | + | 33.7486i | 1680.48 | − | 1220.94i | −3605.45 | − | 2619.51i | 2385.75 | + | 7342.59i | 215753. | − | 70102.3i | 607900. | − | 836702.i | −1.33783e6 | − | 1.84137e6i | −144697. | + | 445330.i | 843172.i | ||
2.11 | 178.910 | + | 58.1314i | −2296.87 | + | 1668.77i | 15374.6 | + | 11170.3i | −40945.0 | − | 126016.i | −507941. | + | 165040.i | −7104.29 | + | 9778.22i | 289708. | + | 398749.i | 1.01278e6 | − | 3.11702e6i | − | 2.49256e7i | |
2.12 | 186.716 | + | 60.6678i | −624.250 | + | 453.544i | 17927.4 | + | 13025.0i | 24181.1 | + | 74421.8i | −144073. | + | 46812.2i | 68223.2 | − | 93901.2i | 666478. | + | 917329.i | −1.29403e6 | + | 3.98262e6i | 1.53628e7i | ||
2.13 | 226.579 | + | 73.6200i | 2991.54 | − | 2173.48i | 32663.3 | + | 23731.3i | −18469.4 | − | 56842.9i | 837832. | − | 272228.i | −545797. | + | 751225.i | 3.35941e6 | + | 4.62383e6i | 2.74727e6 | − | 8.45522e6i | − | 1.42391e7i | |
6.1 | −236.091 | + | 76.7107i | 1404.04 | + | 1020.09i | 36599.6 | − | 26591.2i | 17129.5 | − | 52719.2i | −409733. | − | 133130.i | 347419. | + | 478181.i | −4.21038e6 | + | 5.79510e6i | −547289. | − | 1.68438e6i | 1.37606e7i | ||
6.2 | −199.758 | + | 64.9052i | −3456.81 | − | 2511.52i | 22435.6 | − | 16300.4i | −9471.22 | + | 29149.4i | 853535. | + | 277330.i | −55107.8 | − | 75849.3i | −1.40098e6 | + | 1.92828e6i | 4.16378e6 | + | 1.28148e7i | − | 6.43755e6i | |
6.3 | −157.861 | + | 51.2920i | 79.9689 | + | 58.1008i | 9034.19 | − | 6563.72i | −14721.0 | + | 45306.7i | −15604.0 | − | 5070.06i | −565560. | − | 778426.i | 509000. | − | 700579.i | −1.47500e6 | − | 4.53958e6i | − | 7.90722e6i | |
6.4 | −119.417 | + | 38.8010i | 3112.87 | + | 2261.63i | −499.976 | + | 363.254i | −39265.4 | + | 120847.i | −459484. | − | 149295.i | 577116. | + | 794332.i | 1.25481e6 | − | 1.72710e6i | 3.09696e6 | + | 9.53146e6i | − | 1.59547e7i | |
6.5 | −116.568 | + | 37.8754i | −829.952 | − | 602.996i | −1101.28 | + | 800.129i | 25059.8 | − | 77126.2i | 119585. | + | 38855.5i | 411520. | + | 566408.i | 1.27843e6 | − | 1.75960e6i | −1.15280e6 | − | 3.54796e6i | 9.93962e6i | ||
6.6 | −57.1439 | + | 18.5672i | 2553.58 | + | 1855.29i | −10334.3 | + | 7508.27i | 43338.6 | − | 133382.i | −180369. | − | 58605.5i | −281229. | − | 387078.i | 1.02976e6 | − | 1.41735e6i | 1.60068e6 | + | 4.92639e6i | 8.42667e6i | ||
6.7 | −13.9048 | + | 4.51795i | −1467.63 | − | 1066.30i | −13082.0 | + | 9504.63i | −19976.8 | + | 61482.1i | 25224.6 | + | 8195.98i | 351170. | + | 483344.i | 279760. | − | 385057.i | −461067. | − | 1.41902e6i | − | 945153.i | |
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 11.15.d.a | ✓ | 52 |
11.d | odd | 10 | 1 | inner | 11.15.d.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.15.d.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
11.15.d.a | ✓ | 52 | 11.d | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(11, [\chi])\).