Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,13,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, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.2");
S:= CuspForms(chi, 13);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 11 \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
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: | \(10.0539319900\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(11\) 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 | −102.224 | − | 33.2144i | −276.412 | + | 200.825i | 6032.72 | + | 4383.03i | −6925.60 | − | 21314.8i | 34926.2 | − | 11348.2i | 128352. | − | 176662.i | −212330. | − | 292248.i | −128151. | + | 394409.i | 2.40891e6i | ||
2.2 | −92.8176 | − | 30.1583i | 691.311 | − | 502.267i | 4391.85 | + | 3190.87i | 1083.89 | + | 3335.88i | −79313.3 | + | 25770.5i | −69230.1 | + | 95287.1i | −76445.3 | − | 105218.i | 61414.8 | − | 189015.i | − | 342317.i | |
2.3 | −79.4376 | − | 25.8108i | −641.631 | + | 466.172i | 2330.39 | + | 1693.13i | 6734.78 | + | 20727.5i | 63001.9 | − | 20470.5i | −24839.2 | + | 34188.3i | 59673.9 | + | 82134.1i | 30149.4 | − | 92790.4i | − | 1.82037e6i | |
2.4 | −28.7412 | − | 9.33860i | −796.380 | + | 578.604i | −2574.88 | − | 1870.76i | −7517.68 | − | 23137.0i | 28292.3 | − | 9192.72i | −91007.2 | + | 125261.i | 129293. | + | 177956.i | 135214. | − | 416146.i | 735192.i | ||
2.5 | −21.1209 | − | 6.86260i | 669.622 | − | 486.509i | −2914.74 | − | 2117.68i | −1648.51 | − | 5073.61i | −17481.7 | + | 5680.16i | 14639.8 | − | 20149.9i | 100496. | + | 138321.i | 47478.3 | − | 146123.i | 118472.i | ||
2.6 | −8.21322 | − | 2.66864i | −83.6942 | + | 60.8074i | −3253.40 | − | 2363.73i | 3579.35 | + | 11016.1i | 849.672 | − | 276.075i | 61235.8 | − | 84283.8i | 41204.4 | + | 56713.0i | −160917. | + | 495252.i | − | 100030.i | |
2.7 | 56.5486 | + | 18.3737i | −199.859 | + | 145.206i | −453.588 | − | 329.551i | 1735.17 | + | 5340.30i | −13969.7 | + | 4539.04i | −101553. | + | 139776.i | −162745. | − | 224000.i | −145365. | + | 447389.i | 333868.i | ||
2.8 | 59.3643 | + | 19.2886i | −999.365 | + | 726.081i | −161.661 | − | 117.454i | 875.230 | + | 2693.68i | −73331.7 | + | 23826.9i | 110980. | − | 152751.i | −157610. | − | 216932.i | 307312. | − | 945809.i | 176791.i | ||
2.9 | 73.1050 | + | 23.7533i | 491.809 | − | 357.320i | 1466.40 | + | 1065.40i | −9397.34 | − | 28922.0i | 44441.2 | − | 14439.8i | 26666.1 | − | 36702.7i | −103169. | − | 142000.i | −50026.1 | + | 153965.i | − | 2.33757e6i | |
2.10 | 75.1176 | + | 24.4072i | 1006.77 | − | 731.464i | 1733.20 | + | 1259.25i | 7624.07 | + | 23464.5i | 93479.4 | − | 30373.3i | 11948.6 | − | 16445.8i | −90698.2 | − | 124835.i | 314329. | − | 967407.i | 1.94868e6i | ||
2.11 | 116.921 | + | 37.9900i | −222.926 | + | 161.965i | 8913.56 | + | 6476.08i | 698.010 | + | 2148.25i | −32217.8 | + | 10468.2i | 9655.01 | − | 13289.0i | 500175. | + | 688432.i | −140761. | + | 433218.i | 277694.i | ||
6.1 | −102.224 | + | 33.2144i | −276.412 | − | 200.825i | 6032.72 | − | 4383.03i | −6925.60 | + | 21314.8i | 34926.2 | + | 11348.2i | 128352. | + | 176662.i | −212330. | + | 292248.i | −128151. | − | 394409.i | − | 2.40891e6i | |
6.2 | −92.8176 | + | 30.1583i | 691.311 | + | 502.267i | 4391.85 | − | 3190.87i | 1083.89 | − | 3335.88i | −79313.3 | − | 25770.5i | −69230.1 | − | 95287.1i | −76445.3 | + | 105218.i | 61414.8 | + | 189015.i | 342317.i | ||
6.3 | −79.4376 | + | 25.8108i | −641.631 | − | 466.172i | 2330.39 | − | 1693.13i | 6734.78 | − | 20727.5i | 63001.9 | + | 20470.5i | −24839.2 | − | 34188.3i | 59673.9 | − | 82134.1i | 30149.4 | + | 92790.4i | 1.82037e6i | ||
6.4 | −28.7412 | + | 9.33860i | −796.380 | − | 578.604i | −2574.88 | + | 1870.76i | −7517.68 | + | 23137.0i | 28292.3 | + | 9192.72i | −91007.2 | − | 125261.i | 129293. | − | 177956.i | 135214. | + | 416146.i | − | 735192.i | |
6.5 | −21.1209 | + | 6.86260i | 669.622 | + | 486.509i | −2914.74 | + | 2117.68i | −1648.51 | + | 5073.61i | −17481.7 | − | 5680.16i | 14639.8 | + | 20149.9i | 100496. | − | 138321.i | 47478.3 | + | 146123.i | − | 118472.i | |
6.6 | −8.21322 | + | 2.66864i | −83.6942 | − | 60.8074i | −3253.40 | + | 2363.73i | 3579.35 | − | 11016.1i | 849.672 | + | 276.075i | 61235.8 | + | 84283.8i | 41204.4 | − | 56713.0i | −160917. | − | 495252.i | 100030.i | ||
6.7 | 56.5486 | − | 18.3737i | −199.859 | − | 145.206i | −453.588 | + | 329.551i | 1735.17 | − | 5340.30i | −13969.7 | − | 4539.04i | −101553. | − | 139776.i | −162745. | + | 224000.i | −145365. | − | 447389.i | − | 333868.i | |
6.8 | 59.3643 | − | 19.2886i | −999.365 | − | 726.081i | −161.661 | + | 117.454i | 875.230 | − | 2693.68i | −73331.7 | − | 23826.9i | 110980. | + | 152751.i | −157610. | + | 216932.i | 307312. | + | 945809.i | − | 176791.i | |
6.9 | 73.1050 | − | 23.7533i | 491.809 | + | 357.320i | 1466.40 | − | 1065.40i | −9397.34 | + | 28922.0i | 44441.2 | + | 14439.8i | 26666.1 | + | 36702.7i | −103169. | + | 142000.i | −50026.1 | − | 153965.i | 2.33757e6i | ||
See all 44 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 | 11.13.d.a | ✓ | 44 |
11.d | odd | 10 | 1 | inner | 11.13.d.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.13.d.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
11.13.d.a | ✓ | 44 | 11.d | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(11, [\chi])\).