Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,4,Mod(4,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 43 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 43.e (of order \(7\), degree \(6\), 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: | \(2.53708213025\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −3.08835 | + | 3.87267i | −0.198541 | − | 0.248963i | −3.67950 | − | 16.1209i | −11.1889 | − | 5.38829i | 1.57731 | 3.53619 | 38.0923 | + | 18.3443i | 5.98550 | − | 26.2242i | 55.4224 | − | 26.6900i | ||||
4.2 | −2.32023 | + | 2.90947i | 3.61377 | + | 4.53152i | −1.30141 | − | 5.70184i | 13.3469 | + | 6.42753i | −21.5691 | −12.2090 | −7.21369 | − | 3.47393i | −1.46730 | + | 6.42866i | −49.6685 | + | 23.9191i | ||||
4.3 | −2.29664 | + | 2.87990i | −5.14704 | − | 6.45419i | −1.23909 | − | 5.42880i | 11.0752 | + | 5.33352i | 30.4083 | 28.5576 | −8.06984 | − | 3.88623i | −9.15641 | + | 40.1169i | −40.7957 | + | 19.6462i | ||||
4.4 | −0.976506 | + | 1.22450i | −3.38192 | − | 4.24079i | 1.23433 | + | 5.40796i | −7.32010 | − | 3.52517i | 8.49531 | −30.7370 | −19.1161 | − | 9.20584i | −0.538880 | + | 2.36099i | 11.4647 | − | 5.52111i | ||||
4.5 | −0.873702 | + | 1.09559i | 4.80787 | + | 6.02888i | 1.34321 | + | 5.88499i | −14.6785 | − | 7.06881i | −10.8058 | 16.8292 | −17.7214 | − | 8.53416i | −7.22369 | + | 31.6490i | 20.5692 | − | 9.90558i | ||||
4.6 | 0.208948 | − | 0.262012i | −0.805279 | − | 1.00979i | 1.75518 | + | 7.68993i | 8.19351 | + | 3.94579i | −0.432838 | 17.7083 | 4.79710 | + | 2.31016i | 5.63687 | − | 24.6967i | 2.74586 | − | 1.32234i | ||||
4.7 | 1.20289 | − | 1.50838i | 3.21845 | + | 4.03581i | 0.951908 | + | 4.17058i | 3.04073 | + | 1.46434i | 9.95900 | −15.9770 | 21.3417 | + | 10.2776i | 0.0787180 | − | 0.344886i | 5.86645 | − | 2.82513i | ||||
4.8 | 1.80704 | − | 2.26595i | −3.86902 | − | 4.85159i | −0.0889949 | − | 0.389912i | −19.3746 | − | 9.33033i | −17.9850 | 25.5918 | 19.8456 | + | 9.55715i | −2.56060 | + | 11.2187i | −56.1528 | + | 27.0418i | ||||
4.9 | 2.72209 | − | 3.41339i | −2.84196 | − | 3.56370i | −2.46131 | − | 10.7837i | 8.30189 | + | 3.99798i | −19.9004 | −12.2688 | −12.0406 | − | 5.79846i | 1.38482 | − | 6.06730i | 36.2452 | − | 17.4548i | ||||
4.10 | 2.96047 | − | 3.71232i | 4.10366 | + | 5.14583i | −3.23672 | − | 14.1810i | −6.52318 | − | 3.14140i | 31.2517 | 6.49418 | −28.0025 | − | 13.4853i | −3.63147 | + | 15.9105i | −30.9735 | + | 14.9161i | ||||
11.1 | −3.08835 | − | 3.87267i | −0.198541 | + | 0.248963i | −3.67950 | + | 16.1209i | −11.1889 | + | 5.38829i | 1.57731 | 3.53619 | 38.0923 | − | 18.3443i | 5.98550 | + | 26.2242i | 55.4224 | + | 26.6900i | ||||
11.2 | −2.32023 | − | 2.90947i | 3.61377 | − | 4.53152i | −1.30141 | + | 5.70184i | 13.3469 | − | 6.42753i | −21.5691 | −12.2090 | −7.21369 | + | 3.47393i | −1.46730 | − | 6.42866i | −49.6685 | − | 23.9191i | ||||
11.3 | −2.29664 | − | 2.87990i | −5.14704 | + | 6.45419i | −1.23909 | + | 5.42880i | 11.0752 | − | 5.33352i | 30.4083 | 28.5576 | −8.06984 | + | 3.88623i | −9.15641 | − | 40.1169i | −40.7957 | − | 19.6462i | ||||
11.4 | −0.976506 | − | 1.22450i | −3.38192 | + | 4.24079i | 1.23433 | − | 5.40796i | −7.32010 | + | 3.52517i | 8.49531 | −30.7370 | −19.1161 | + | 9.20584i | −0.538880 | − | 2.36099i | 11.4647 | + | 5.52111i | ||||
11.5 | −0.873702 | − | 1.09559i | 4.80787 | − | 6.02888i | 1.34321 | − | 5.88499i | −14.6785 | + | 7.06881i | −10.8058 | 16.8292 | −17.7214 | + | 8.53416i | −7.22369 | − | 31.6490i | 20.5692 | + | 9.90558i | ||||
11.6 | 0.208948 | + | 0.262012i | −0.805279 | + | 1.00979i | 1.75518 | − | 7.68993i | 8.19351 | − | 3.94579i | −0.432838 | 17.7083 | 4.79710 | − | 2.31016i | 5.63687 | + | 24.6967i | 2.74586 | + | 1.32234i | ||||
11.7 | 1.20289 | + | 1.50838i | 3.21845 | − | 4.03581i | 0.951908 | − | 4.17058i | 3.04073 | − | 1.46434i | 9.95900 | −15.9770 | 21.3417 | − | 10.2776i | 0.0787180 | + | 0.344886i | 5.86645 | + | 2.82513i | ||||
11.8 | 1.80704 | + | 2.26595i | −3.86902 | + | 4.85159i | −0.0889949 | + | 0.389912i | −19.3746 | + | 9.33033i | −17.9850 | 25.5918 | 19.8456 | − | 9.55715i | −2.56060 | − | 11.2187i | −56.1528 | − | 27.0418i | ||||
11.9 | 2.72209 | + | 3.41339i | −2.84196 | + | 3.56370i | −2.46131 | + | 10.7837i | 8.30189 | − | 3.99798i | −19.9004 | −12.2688 | −12.0406 | + | 5.79846i | 1.38482 | + | 6.06730i | 36.2452 | + | 17.4548i | ||||
11.10 | 2.96047 | + | 3.71232i | 4.10366 | − | 5.14583i | −3.23672 | + | 14.1810i | −6.52318 | + | 3.14140i | 31.2517 | 6.49418 | −28.0025 | + | 13.4853i | −3.63147 | − | 15.9105i | −30.9735 | − | 14.9161i | ||||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.e | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 43.4.e.a | ✓ | 60 |
43.e | even | 7 | 1 | inner | 43.4.e.a | ✓ | 60 |
43.e | even | 7 | 1 | 1849.4.a.h | 30 | ||
43.f | odd | 14 | 1 | 1849.4.a.g | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.4.e.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
43.4.e.a | ✓ | 60 | 43.e | even | 7 | 1 | inner |
1849.4.a.g | 30 | 43.f | odd | 14 | 1 | ||
1849.4.a.h | 30 | 43.e | even | 7 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(43, [\chi])\).