Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [725,2,Mod(149,725)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(725, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([7, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("725.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 725.p (of order \(14\), degree \(6\), not 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: | \(5.78915414654\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{14})\) |
Twist minimal: | no (minimal twist has level 29) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 | −0.504621 | + | 2.21089i | 2.55926 | − | 1.23248i | −2.83146 | − | 1.36356i | 0 | 1.43341 | + | 6.28018i | −0.677709 | − | 1.40728i | 1.61565 | − | 2.02596i | 3.16036 | − | 3.96297i | 0 | ||||
149.2 | −0.122359 | + | 0.536089i | −1.77743 | + | 0.855966i | 1.52952 | + | 0.736577i | 0 | −0.241390 | − | 1.05760i | 1.94112 | + | 4.03077i | −1.26771 | + | 1.58965i | 0.556117 | − | 0.697349i | 0 | ||||
149.3 | 0.122359 | − | 0.536089i | 1.77743 | − | 0.855966i | 1.52952 | + | 0.736577i | 0 | −0.241390 | − | 1.05760i | −1.94112 | − | 4.03077i | 1.26771 | − | 1.58965i | 0.556117 | − | 0.697349i | 0 | ||||
149.4 | 0.504621 | − | 2.21089i | −2.55926 | + | 1.23248i | −2.83146 | − | 1.36356i | 0 | 1.43341 | + | 6.28018i | 0.677709 | + | 1.40728i | −1.61565 | + | 2.02596i | 3.16036 | − | 3.96297i | 0 | ||||
274.1 | −2.34472 | − | 1.12916i | −0.273923 | + | 0.343489i | 2.97573 | + | 3.73144i | 0 | 1.03013 | − | 0.496082i | 0.0587399 | + | 0.0468435i | −1.60566 | − | 7.03485i | 0.624612 | + | 2.73660i | 0 | ||||
274.2 | −0.154074 | − | 0.0741982i | 0.701005 | − | 0.879032i | −1.22875 | − | 1.54080i | 0 | −0.173229 | + | 0.0834229i | 2.28763 | + | 1.82432i | 0.151100 | + | 0.662012i | 0.386273 | + | 1.69237i | 0 | ||||
274.3 | 0.154074 | + | 0.0741982i | −0.701005 | + | 0.879032i | −1.22875 | − | 1.54080i | 0 | −0.173229 | + | 0.0834229i | −2.28763 | − | 1.82432i | −0.151100 | − | 0.662012i | 0.386273 | + | 1.69237i | 0 | ||||
274.4 | 2.34472 | + | 1.12916i | 0.273923 | − | 0.343489i | 2.97573 | + | 3.73144i | 0 | 1.03013 | − | 0.496082i | −0.0587399 | − | 0.0468435i | 1.60566 | + | 7.03485i | 0.624612 | + | 2.73660i | 0 | ||||
299.1 | −2.34472 | + | 1.12916i | −0.273923 | − | 0.343489i | 2.97573 | − | 3.73144i | 0 | 1.03013 | + | 0.496082i | 0.0587399 | − | 0.0468435i | −1.60566 | + | 7.03485i | 0.624612 | − | 2.73660i | 0 | ||||
299.2 | −0.154074 | + | 0.0741982i | 0.701005 | + | 0.879032i | −1.22875 | + | 1.54080i | 0 | −0.173229 | − | 0.0834229i | 2.28763 | − | 1.82432i | 0.151100 | − | 0.662012i | 0.386273 | − | 1.69237i | 0 | ||||
299.3 | 0.154074 | − | 0.0741982i | −0.701005 | − | 0.879032i | −1.22875 | + | 1.54080i | 0 | −0.173229 | − | 0.0834229i | −2.28763 | + | 1.82432i | −0.151100 | + | 0.662012i | 0.386273 | − | 1.69237i | 0 | ||||
299.4 | 2.34472 | − | 1.12916i | 0.273923 | + | 0.343489i | 2.97573 | − | 3.73144i | 0 | 1.03013 | + | 0.496082i | −0.0587399 | + | 0.0468435i | 1.60566 | − | 7.03485i | 0.624612 | − | 2.73660i | 0 | ||||
324.1 | −0.965958 | − | 1.21127i | −0.653024 | − | 2.86109i | −0.0890656 | + | 0.390222i | 0 | −2.83476 | + | 3.55468i | −3.32768 | + | 0.759522i | −2.23300 | + | 1.07536i | −5.05647 | + | 2.43507i | 0 | ||||
324.2 | −0.725171 | − | 0.909335i | −0.219141 | − | 0.960118i | 0.144024 | − | 0.631009i | 0 | −0.714155 | + | 0.895521i | 1.48749 | − | 0.339509i | −2.77404 | + | 1.33591i | 1.82910 | − | 0.880850i | 0 | ||||
324.3 | 0.725171 | + | 0.909335i | 0.219141 | + | 0.960118i | 0.144024 | − | 0.631009i | 0 | −0.714155 | + | 0.895521i | −1.48749 | + | 0.339509i | 2.77404 | − | 1.33591i | 1.82910 | − | 0.880850i | 0 | ||||
324.4 | 0.965958 | + | 1.21127i | 0.653024 | + | 2.86109i | −0.0890656 | + | 0.390222i | 0 | −2.83476 | + | 3.55468i | 3.32768 | − | 0.759522i | 2.23300 | − | 1.07536i | −5.05647 | + | 2.43507i | 0 | ||||
399.1 | −0.504621 | − | 2.21089i | 2.55926 | + | 1.23248i | −2.83146 | + | 1.36356i | 0 | 1.43341 | − | 6.28018i | −0.677709 | + | 1.40728i | 1.61565 | + | 2.02596i | 3.16036 | + | 3.96297i | 0 | ||||
399.2 | −0.122359 | − | 0.536089i | −1.77743 | − | 0.855966i | 1.52952 | − | 0.736577i | 0 | −0.241390 | + | 1.05760i | 1.94112 | − | 4.03077i | −1.26771 | − | 1.58965i | 0.556117 | + | 0.697349i | 0 | ||||
399.3 | 0.122359 | + | 0.536089i | 1.77743 | + | 0.855966i | 1.52952 | − | 0.736577i | 0 | −0.241390 | + | 1.05760i | −1.94112 | + | 4.03077i | 1.26771 | + | 1.58965i | 0.556117 | + | 0.697349i | 0 | ||||
399.4 | 0.504621 | + | 2.21089i | −2.55926 | − | 1.23248i | −2.83146 | + | 1.36356i | 0 | 1.43341 | − | 6.28018i | 0.677709 | − | 1.40728i | −1.61565 | − | 2.02596i | 3.16036 | + | 3.96297i | 0 | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
29.e | even | 14 | 1 | inner |
145.l | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 725.2.p.a | 24 | |
5.b | even | 2 | 1 | inner | 725.2.p.a | 24 | |
5.c | odd | 4 | 1 | 29.2.e.a | ✓ | 12 | |
5.c | odd | 4 | 1 | 725.2.q.a | 12 | ||
15.e | even | 4 | 1 | 261.2.o.a | 12 | ||
20.e | even | 4 | 1 | 464.2.y.d | 12 | ||
29.e | even | 14 | 1 | inner | 725.2.p.a | 24 | |
145.e | even | 4 | 1 | 841.2.d.m | 24 | ||
145.h | odd | 4 | 1 | 841.2.e.i | 12 | ||
145.j | even | 4 | 1 | 841.2.d.m | 24 | ||
145.l | even | 14 | 1 | inner | 725.2.p.a | 24 | |
145.o | even | 28 | 1 | 841.2.a.k | 12 | ||
145.o | even | 28 | 2 | 841.2.d.k | 24 | ||
145.o | even | 28 | 2 | 841.2.d.l | 24 | ||
145.o | even | 28 | 1 | 841.2.d.m | 24 | ||
145.p | odd | 28 | 1 | 841.2.b.e | 12 | ||
145.p | odd | 28 | 1 | 841.2.e.a | 12 | ||
145.p | odd | 28 | 1 | 841.2.e.e | 12 | ||
145.p | odd | 28 | 1 | 841.2.e.f | 12 | ||
145.p | odd | 28 | 1 | 841.2.e.h | 12 | ||
145.p | odd | 28 | 1 | 841.2.e.i | 12 | ||
145.q | odd | 28 | 1 | 29.2.e.a | ✓ | 12 | |
145.q | odd | 28 | 1 | 725.2.q.a | 12 | ||
145.q | odd | 28 | 1 | 841.2.b.e | 12 | ||
145.q | odd | 28 | 1 | 841.2.e.a | 12 | ||
145.q | odd | 28 | 1 | 841.2.e.e | 12 | ||
145.q | odd | 28 | 1 | 841.2.e.f | 12 | ||
145.q | odd | 28 | 1 | 841.2.e.h | 12 | ||
145.t | even | 28 | 1 | 841.2.a.k | 12 | ||
145.t | even | 28 | 2 | 841.2.d.k | 24 | ||
145.t | even | 28 | 2 | 841.2.d.l | 24 | ||
145.t | even | 28 | 1 | 841.2.d.m | 24 | ||
435.bc | odd | 28 | 1 | 7569.2.a.bp | 12 | ||
435.bg | even | 28 | 1 | 261.2.o.a | 12 | ||
435.bn | odd | 28 | 1 | 7569.2.a.bp | 12 | ||
580.bh | even | 28 | 1 | 464.2.y.d | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.2.e.a | ✓ | 12 | 5.c | odd | 4 | 1 | |
29.2.e.a | ✓ | 12 | 145.q | odd | 28 | 1 | |
261.2.o.a | 12 | 15.e | even | 4 | 1 | ||
261.2.o.a | 12 | 435.bg | even | 28 | 1 | ||
464.2.y.d | 12 | 20.e | even | 4 | 1 | ||
464.2.y.d | 12 | 580.bh | even | 28 | 1 | ||
725.2.p.a | 24 | 1.a | even | 1 | 1 | trivial | |
725.2.p.a | 24 | 5.b | even | 2 | 1 | inner | |
725.2.p.a | 24 | 29.e | even | 14 | 1 | inner | |
725.2.p.a | 24 | 145.l | even | 14 | 1 | inner | |
725.2.q.a | 12 | 5.c | odd | 4 | 1 | ||
725.2.q.a | 12 | 145.q | odd | 28 | 1 | ||
841.2.a.k | 12 | 145.o | even | 28 | 1 | ||
841.2.a.k | 12 | 145.t | even | 28 | 1 | ||
841.2.b.e | 12 | 145.p | odd | 28 | 1 | ||
841.2.b.e | 12 | 145.q | odd | 28 | 1 | ||
841.2.d.k | 24 | 145.o | even | 28 | 2 | ||
841.2.d.k | 24 | 145.t | even | 28 | 2 | ||
841.2.d.l | 24 | 145.o | even | 28 | 2 | ||
841.2.d.l | 24 | 145.t | even | 28 | 2 | ||
841.2.d.m | 24 | 145.e | even | 4 | 1 | ||
841.2.d.m | 24 | 145.j | even | 4 | 1 | ||
841.2.d.m | 24 | 145.o | even | 28 | 1 | ||
841.2.d.m | 24 | 145.t | even | 28 | 1 | ||
841.2.e.a | 12 | 145.p | odd | 28 | 1 | ||
841.2.e.a | 12 | 145.q | odd | 28 | 1 | ||
841.2.e.e | 12 | 145.p | odd | 28 | 1 | ||
841.2.e.e | 12 | 145.q | odd | 28 | 1 | ||
841.2.e.f | 12 | 145.p | odd | 28 | 1 | ||
841.2.e.f | 12 | 145.q | odd | 28 | 1 | ||
841.2.e.h | 12 | 145.p | odd | 28 | 1 | ||
841.2.e.h | 12 | 145.q | odd | 28 | 1 | ||
841.2.e.i | 12 | 145.h | odd | 4 | 1 | ||
841.2.e.i | 12 | 145.p | odd | 28 | 1 | ||
7569.2.a.bp | 12 | 435.bc | odd | 28 | 1 | ||
7569.2.a.bp | 12 | 435.bn | odd | 28 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} + 3 T_{2}^{22} + 5 T_{2}^{20} + 206 T_{2}^{18} + 1620 T_{2}^{16} + 4449 T_{2}^{14} + 12992 T_{2}^{12} + 14569 T_{2}^{10} + 18003 T_{2}^{8} + 7101 T_{2}^{6} + 902 T_{2}^{4} - 36 T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\).