Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,10,Mod(32,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.32");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.e (of order \(4\), degree \(2\), 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: | \(38.6276877123\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 15) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 | −30.5207 | + | 30.5207i | 129.276 | − | 54.5047i | − | 1351.03i | 0 | −2282.07 | + | 5609.11i | 2749.16 | + | 2749.16i | 25607.7 | + | 25607.7i | 13741.5 | − | 14092.3i | 0 | |||||
32.2 | −24.1261 | + | 24.1261i | −138.640 | + | 21.4940i | − | 652.142i | 0 | 2826.28 | − | 3863.41i | −1781.62 | − | 1781.62i | 3381.08 | + | 3381.08i | 18759.0 | − | 5959.84i | 0 | |||||
32.3 | −24.0812 | + | 24.0812i | 17.1863 | + | 139.239i | − | 647.806i | 0 | −3766.92 | − | 2939.18i | 4153.03 | + | 4153.03i | 3270.36 | + | 3270.36i | −19092.3 | + | 4786.01i | 0 | |||||
32.4 | −20.3831 | + | 20.3831i | −65.1534 | − | 124.250i | − | 318.940i | 0 | 3860.62 | + | 1204.57i | −1024.24 | − | 1024.24i | −3935.17 | − | 3935.17i | −11193.1 | + | 16190.6i | 0 | |||||
32.5 | −13.8737 | + | 13.8737i | 128.756 | − | 55.7208i | 127.039i | 0 | −1013.28 | + | 2559.39i | −6797.28 | − | 6797.28i | −8865.86 | − | 8865.86i | 13473.4 | − | 14348.8i | 0 | ||||||
32.6 | −10.0225 | + | 10.0225i | 84.4162 | + | 112.058i | 311.097i | 0 | −1969.17 | − | 277.038i | −2707.78 | − | 2707.78i | −8249.53 | − | 8249.53i | −5430.82 | + | 18919.0i | 0 | ||||||
32.7 | −6.35582 | + | 6.35582i | −0.342183 | − | 140.296i | 431.207i | 0 | 893.869 | + | 889.519i | 7866.00 | + | 7866.00i | −5994.85 | − | 5994.85i | −19682.8 | + | 96.0135i | 0 | ||||||
32.8 | −4.02674 | + | 4.02674i | −130.186 | + | 52.2930i | 479.571i | 0 | 313.655 | − | 734.796i | −17.2800 | − | 17.2800i | −3992.80 | − | 3992.80i | 14213.9 | − | 13615.7i | 0 | ||||||
32.9 | 4.02674 | − | 4.02674i | −52.2930 | + | 130.186i | 479.571i | 0 | 313.655 | + | 734.796i | −17.2800 | − | 17.2800i | 3992.80 | + | 3992.80i | −14213.9 | − | 13615.7i | 0 | ||||||
32.10 | 6.35582 | − | 6.35582i | 140.296 | + | 0.342183i | 431.207i | 0 | 893.869 | − | 889.519i | 7866.00 | + | 7866.00i | 5994.85 | + | 5994.85i | 19682.8 | + | 96.0135i | 0 | ||||||
32.11 | 10.0225 | − | 10.0225i | −112.058 | − | 84.4162i | 311.097i | 0 | −1969.17 | + | 277.038i | −2707.78 | − | 2707.78i | 8249.53 | + | 8249.53i | 5430.82 | + | 18919.0i | 0 | ||||||
32.12 | 13.8737 | − | 13.8737i | 55.7208 | − | 128.756i | 127.039i | 0 | −1013.28 | − | 2559.39i | −6797.28 | − | 6797.28i | 8865.86 | + | 8865.86i | −13473.4 | − | 14348.8i | 0 | ||||||
32.13 | 20.3831 | − | 20.3831i | 124.250 | + | 65.1534i | − | 318.940i | 0 | 3860.62 | − | 1204.57i | −1024.24 | − | 1024.24i | 3935.17 | + | 3935.17i | 11193.1 | + | 16190.6i | 0 | |||||
32.14 | 24.0812 | − | 24.0812i | −139.239 | − | 17.1863i | − | 647.806i | 0 | −3766.92 | + | 2939.18i | 4153.03 | + | 4153.03i | −3270.36 | − | 3270.36i | 19092.3 | + | 4786.01i | 0 | |||||
32.15 | 24.1261 | − | 24.1261i | −21.4940 | + | 138.640i | − | 652.142i | 0 | 2826.28 | + | 3863.41i | −1781.62 | − | 1781.62i | −3381.08 | − | 3381.08i | −18759.0 | − | 5959.84i | 0 | |||||
32.16 | 30.5207 | − | 30.5207i | 54.5047 | − | 129.276i | − | 1351.03i | 0 | −2282.07 | − | 5609.11i | 2749.16 | + | 2749.16i | −25607.7 | − | 25607.7i | −13741.5 | − | 14092.3i | 0 | |||||
68.1 | −30.5207 | − | 30.5207i | 129.276 | + | 54.5047i | 1351.03i | 0 | −2282.07 | − | 5609.11i | 2749.16 | − | 2749.16i | 25607.7 | − | 25607.7i | 13741.5 | + | 14092.3i | 0 | ||||||
68.2 | −24.1261 | − | 24.1261i | −138.640 | − | 21.4940i | 652.142i | 0 | 2826.28 | + | 3863.41i | −1781.62 | + | 1781.62i | 3381.08 | − | 3381.08i | 18759.0 | + | 5959.84i | 0 | ||||||
68.3 | −24.0812 | − | 24.0812i | 17.1863 | − | 139.239i | 647.806i | 0 | −3766.92 | + | 2939.18i | 4153.03 | − | 4153.03i | 3270.36 | − | 3270.36i | −19092.3 | − | 4786.01i | 0 | ||||||
68.4 | −20.3831 | − | 20.3831i | −65.1534 | + | 124.250i | 318.940i | 0 | 3860.62 | − | 1204.57i | −1024.24 | + | 1024.24i | −3935.17 | + | 3935.17i | −11193.1 | − | 16190.6i | 0 | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.10.e.f | 32 | |
3.b | odd | 2 | 1 | inner | 75.10.e.f | 32 | |
5.b | even | 2 | 1 | 15.10.e.a | ✓ | 32 | |
5.c | odd | 4 | 1 | 15.10.e.a | ✓ | 32 | |
5.c | odd | 4 | 1 | inner | 75.10.e.f | 32 | |
15.d | odd | 2 | 1 | 15.10.e.a | ✓ | 32 | |
15.e | even | 4 | 1 | 15.10.e.a | ✓ | 32 | |
15.e | even | 4 | 1 | inner | 75.10.e.f | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
15.10.e.a | ✓ | 32 | 5.b | even | 2 | 1 | |
15.10.e.a | ✓ | 32 | 5.c | odd | 4 | 1 | |
15.10.e.a | ✓ | 32 | 15.d | odd | 2 | 1 | |
15.10.e.a | ✓ | 32 | 15.e | even | 4 | 1 | |
75.10.e.f | 32 | 1.a | even | 1 | 1 | trivial | |
75.10.e.f | 32 | 3.b | odd | 2 | 1 | inner | |
75.10.e.f | 32 | 5.c | odd | 4 | 1 | inner | |
75.10.e.f | 32 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\):
\( T_{2}^{32} + 7057841 T_{2}^{28} + 16809877421296 T_{2}^{24} + \cdots + 17\!\cdots\!00 \) |
\( T_{7}^{16} - 4880 T_{7}^{15} + 11907200 T_{7}^{14} + 298591416240 T_{7}^{13} + \cdots + 69\!\cdots\!00 \) |