Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,8,Mod(75,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.75");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.d (of order \(2\), degree \(1\), 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: | \(23.7412619368\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
75.1 | −11.3050 | − | 0.442790i | 16.6522 | 127.608 | + | 10.0115i | 341.116 | −188.254 | − | 7.37345i | − | 1382.96i | −1438.18 | − | 169.684i | −1909.70 | −3856.33 | − | 151.043i | |||||||
75.2 | −11.3050 | + | 0.442790i | 16.6522 | 127.608 | − | 10.0115i | 341.116 | −188.254 | + | 7.37345i | 1382.96i | −1438.18 | + | 169.684i | −1909.70 | −3856.33 | + | 151.043i | ||||||||
75.3 | −11.1927 | − | 1.65015i | −65.7104 | 122.554 | + | 36.9394i | −95.9690 | 735.478 | + | 108.432i | 851.669i | −1310.76 | − | 615.686i | 2130.86 | 1074.15 | + | 158.364i | ||||||||
75.4 | −11.1927 | + | 1.65015i | −65.7104 | 122.554 | − | 36.9394i | −95.9690 | 735.478 | − | 108.432i | − | 851.669i | −1310.76 | + | 615.686i | 2130.86 | 1074.15 | − | 158.364i | |||||||
75.5 | −10.9753 | − | 2.74630i | −8.30507 | 112.916 | + | 60.2831i | −360.155 | 91.1509 | + | 22.8082i | 965.147i | −1073.73 | − | 971.727i | −2118.03 | 3952.82 | + | 989.094i | ||||||||
75.6 | −10.9753 | + | 2.74630i | −8.30507 | 112.916 | − | 60.2831i | −360.155 | 91.1509 | − | 22.8082i | − | 965.147i | −1073.73 | + | 971.727i | −2118.03 | 3952.82 | − | 989.094i | |||||||
75.7 | −10.7294 | − | 3.58884i | 80.7260 | 102.240 | + | 77.0123i | 174.727 | −866.143 | − | 289.713i | 433.835i | −820.594 | − | 1193.22i | 4329.69 | −1874.71 | − | 627.067i | ||||||||
75.8 | −10.7294 | + | 3.58884i | 80.7260 | 102.240 | − | 77.0123i | 174.727 | −866.143 | + | 289.713i | − | 433.835i | −820.594 | + | 1193.22i | 4329.69 | −1874.71 | + | 627.067i | |||||||
75.9 | −10.6582 | − | 3.79499i | 60.8414 | 99.1961 | + | 80.8959i | −369.508 | −648.462 | − | 230.893i | − | 900.474i | −750.256 | − | 1238.66i | 1514.68 | 3938.30 | + | 1402.28i | |||||||
75.10 | −10.6582 | + | 3.79499i | 60.8414 | 99.1961 | − | 80.8959i | −369.508 | −648.462 | + | 230.893i | 900.474i | −750.256 | + | 1238.66i | 1514.68 | 3938.30 | − | 1402.28i | ||||||||
75.11 | −9.80435 | − | 5.64577i | −65.3605 | 64.2506 | + | 110.706i | 399.628 | 640.817 | + | 369.010i | − | 321.615i | −4.91377 | − | 1448.15i | 2084.99 | −3918.10 | − | 2256.21i | |||||||
75.12 | −9.80435 | + | 5.64577i | −65.3605 | 64.2506 | − | 110.706i | 399.628 | 640.817 | − | 369.010i | 321.615i | −4.91377 | + | 1448.15i | 2084.99 | −3918.10 | + | 2256.21i | ||||||||
75.13 | −9.59747 | − | 5.99070i | −1.36022 | 56.2230 | + | 114.991i | 59.9018 | 13.0547 | + | 8.14867i | − | 254.899i | 149.279 | − | 1440.44i | −2185.15 | −574.906 | − | 358.854i | |||||||
75.14 | −9.59747 | + | 5.99070i | −1.36022 | 56.2230 | − | 114.991i | 59.9018 | 13.0547 | − | 8.14867i | 254.899i | 149.279 | + | 1440.44i | −2185.15 | −574.906 | + | 358.854i | ||||||||
75.15 | −8.06300 | − | 7.93650i | 49.8474 | 2.02391 | + | 127.984i | 33.4044 | −401.919 | − | 395.614i | 865.446i | 999.426 | − | 1048.00i | 297.759 | −269.340 | − | 265.114i | ||||||||
75.16 | −8.06300 | + | 7.93650i | 49.8474 | 2.02391 | − | 127.984i | 33.4044 | −401.919 | + | 395.614i | − | 865.446i | 999.426 | + | 1048.00i | 297.759 | −269.340 | + | 265.114i | |||||||
75.17 | −7.73771 | − | 8.25396i | −29.6173 | −8.25557 | + | 127.733i | −224.326 | 229.170 | + | 244.460i | − | 1196.17i | 1118.19 | − | 920.224i | −1309.82 | 1735.77 | + | 1851.58i | |||||||
75.18 | −7.73771 | + | 8.25396i | −29.6173 | −8.25557 | − | 127.733i | −224.326 | 229.170 | − | 244.460i | 1196.17i | 1118.19 | + | 920.224i | −1309.82 | 1735.77 | − | 1851.58i | ||||||||
75.19 | −7.35548 | − | 8.59633i | −83.0232 | −19.7938 | + | 126.460i | −447.252 | 610.676 | + | 713.695i | 106.200i | 1232.69 | − | 760.022i | 4705.86 | 3289.75 | + | 3844.72i | ||||||||
75.20 | −7.35548 | + | 8.59633i | −83.0232 | −19.7938 | − | 126.460i | −447.252 | 610.676 | − | 713.695i | − | 106.200i | 1232.69 | + | 760.022i | 4705.86 | 3289.75 | − | 3844.72i | |||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
76.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.8.d.a | ✓ | 68 |
4.b | odd | 2 | 1 | inner | 76.8.d.a | ✓ | 68 |
19.b | odd | 2 | 1 | inner | 76.8.d.a | ✓ | 68 |
76.d | even | 2 | 1 | inner | 76.8.d.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.8.d.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
76.8.d.a | ✓ | 68 | 4.b | odd | 2 | 1 | inner |
76.8.d.a | ✓ | 68 | 19.b | odd | 2 | 1 | inner |
76.8.d.a | ✓ | 68 | 76.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(76, [\chi])\).