Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.3");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 11 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
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: | \(8.45177498616\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) 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 | −66.2723 | + | 48.1496i | 155.409 | + | 478.301i | 1440.76 | − | 4434.21i | −3908.19 | − | 2839.47i | −33329.3 | − | 24215.2i | 15024.9 | − | 46241.9i | 66180.4 | + | 203682.i | −61304.7 | + | 44540.5i | 395724. | ||
3.2 | −55.1898 | + | 40.0977i | −247.284 | − | 761.061i | 805.219 | − | 2478.21i | −3143.04 | − | 2283.56i | 44164.4 | + | 32087.3i | 15890.9 | − | 48907.1i | 11757.6 | + | 36186.1i | −374750. | + | 272272.i | 265029. | ||
3.3 | −49.0498 | + | 35.6368i | −8.11730 | − | 24.9825i | 503.038 | − | 1548.19i | 10386.7 | + | 7546.38i | 1288.45 | + | 936.111i | −13532.7 | + | 41649.2i | −7871.40 | − | 24225.7i | 142757. | − | 103719.i | −778394. | ||
3.4 | −32.2868 | + | 23.4578i | −12.6213 | − | 38.8444i | −140.693 | + | 433.009i | −5225.04 | − | 3796.22i | 1318.70 | + | 958.095i | −10421.9 | + | 32075.4i | −30871.8 | − | 95013.6i | 141965. | − | 103144.i | 257751. | ||
3.5 | −6.97513 | + | 5.06773i | 187.776 | + | 577.916i | −609.896 | + | 1877.07i | −462.160 | − | 335.779i | −4238.49 | − | 3079.44i | 5696.63 | − | 17532.4i | −10714.8 | − | 32976.7i | −155412. | + | 112914.i | 4925.26 | ||
3.6 | 4.91043 | − | 3.56764i | −108.526 | − | 334.010i | −621.482 | + | 1912.73i | 6091.05 | + | 4425.40i | −1724.54 | − | 1252.95i | 20002.0 | − | 61559.7i | 7613.44 | + | 23431.7i | 43530.5 | − | 31626.8i | 45697.9 | ||
3.7 | 21.2132 | − | 15.4123i | −143.663 | − | 442.150i | −420.407 | + | 1293.88i | −5027.95 | − | 3653.02i | −9862.09 | − | 7165.22i | −13499.1 | + | 41546.0i | 27617.8 | + | 84998.8i | −31542.8 | + | 22917.2i | −162960. | ||
3.8 | 44.3233 | − | 32.2028i | 122.608 | + | 377.349i | 294.671 | − | 906.903i | 3995.51 | + | 2902.91i | 17586.1 | + | 12777.0i | −12513.3 | + | 38512.0i | 18528.6 | + | 57025.2i | 15955.6 | − | 11592.4i | 270576. | ||
3.9 | 56.7091 | − | 41.2016i | 40.8374 | + | 125.685i | 885.488 | − | 2725.25i | −8463.91 | − | 6149.39i | 7494.26 | + | 5444.90i | 23115.4 | − | 71141.9i | −17707.8 | − | 54499.0i | 129186. | − | 93859.1i | −733345. | ||
3.10 | 66.9203 | − | 48.6204i | −196.291 | − | 604.123i | 1481.51 | − | 4559.62i | 7813.25 | + | 5676.66i | −42508.6 | − | 30884.3i | −7835.02 | + | 24113.7i | −70198.2 | − | 216048.i | −183119. | + | 133044.i | 798867. | ||
4.1 | −66.2723 | − | 48.1496i | 155.409 | − | 478.301i | 1440.76 | + | 4434.21i | −3908.19 | + | 2839.47i | −33329.3 | + | 24215.2i | 15024.9 | + | 46241.9i | 66180.4 | − | 203682.i | −61304.7 | − | 44540.5i | 395724. | ||
4.2 | −55.1898 | − | 40.0977i | −247.284 | + | 761.061i | 805.219 | + | 2478.21i | −3143.04 | + | 2283.56i | 44164.4 | − | 32087.3i | 15890.9 | + | 48907.1i | 11757.6 | − | 36186.1i | −374750. | − | 272272.i | 265029. | ||
4.3 | −49.0498 | − | 35.6368i | −8.11730 | + | 24.9825i | 503.038 | + | 1548.19i | 10386.7 | − | 7546.38i | 1288.45 | − | 936.111i | −13532.7 | − | 41649.2i | −7871.40 | + | 24225.7i | 142757. | + | 103719.i | −778394. | ||
4.4 | −32.2868 | − | 23.4578i | −12.6213 | + | 38.8444i | −140.693 | − | 433.009i | −5225.04 | + | 3796.22i | 1318.70 | − | 958.095i | −10421.9 | − | 32075.4i | −30871.8 | + | 95013.6i | 141965. | + | 103144.i | 257751. | ||
4.5 | −6.97513 | − | 5.06773i | 187.776 | − | 577.916i | −609.896 | − | 1877.07i | −462.160 | + | 335.779i | −4238.49 | + | 3079.44i | 5696.63 | + | 17532.4i | −10714.8 | + | 32976.7i | −155412. | − | 112914.i | 4925.26 | ||
4.6 | 4.91043 | + | 3.56764i | −108.526 | + | 334.010i | −621.482 | − | 1912.73i | 6091.05 | − | 4425.40i | −1724.54 | + | 1252.95i | 20002.0 | + | 61559.7i | 7613.44 | − | 23431.7i | 43530.5 | + | 31626.8i | 45697.9 | ||
4.7 | 21.2132 | + | 15.4123i | −143.663 | + | 442.150i | −420.407 | − | 1293.88i | −5027.95 | + | 3653.02i | −9862.09 | + | 7165.22i | −13499.1 | − | 41546.0i | 27617.8 | − | 84998.8i | −31542.8 | − | 22917.2i | −162960. | ||
4.8 | 44.3233 | + | 32.2028i | 122.608 | − | 377.349i | 294.671 | + | 906.903i | 3995.51 | − | 2902.91i | 17586.1 | − | 12777.0i | −12513.3 | − | 38512.0i | 18528.6 | − | 57025.2i | 15955.6 | + | 11592.4i | 270576. | ||
4.9 | 56.7091 | + | 41.2016i | 40.8374 | − | 125.685i | 885.488 | + | 2725.25i | −8463.91 | + | 6149.39i | 7494.26 | − | 5444.90i | 23115.4 | + | 71141.9i | −17707.8 | + | 54499.0i | 129186. | + | 93859.1i | −733345. | ||
4.10 | 66.9203 | + | 48.6204i | −196.291 | + | 604.123i | 1481.51 | + | 4559.62i | 7813.25 | − | 5676.66i | −42508.6 | + | 30884.3i | −7835.02 | − | 24113.7i | −70198.2 | + | 216048.i | −183119. | − | 133044.i | 798867. | ||
See all 40 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 | 11.12.c.a | ✓ | 40 |
3.b | odd | 2 | 1 | 99.12.f.a | 40 | ||
11.c | even | 5 | 1 | inner | 11.12.c.a | ✓ | 40 |
11.c | even | 5 | 1 | 121.12.a.j | 20 | ||
11.d | odd | 10 | 1 | 121.12.a.h | 20 | ||
33.h | odd | 10 | 1 | 99.12.f.a | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.12.c.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
11.12.c.a | ✓ | 40 | 11.c | even | 5 | 1 | inner |
99.12.f.a | 40 | 3.b | odd | 2 | 1 | ||
99.12.f.a | 40 | 33.h | odd | 10 | 1 | ||
121.12.a.h | 20 | 11.d | odd | 10 | 1 | ||
121.12.a.j | 20 | 11.c | even | 5 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(11, [\chi])\).