Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,2,Mod(49,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 425.n (of order \(8\), degree \(4\), 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: | \(3.39364208590\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{8})\) |
Twist minimal: | no (minimal twist has level 85) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | −1.86672 | + | 1.86672i | −1.95914 | + | 0.811501i | − | 4.96928i | 0 | 2.14231 | − | 5.17200i | 1.55444 | − | 3.75274i | 5.54282 | + | 5.54282i | 1.05836 | − | 1.05836i | 0 | |||||
49.2 | −1.09631 | + | 1.09631i | 1.05359 | − | 0.436412i | − | 0.403772i | 0 | −0.676617 | + | 1.63350i | 1.43180 | − | 3.45666i | −1.74995 | − | 1.74995i | −1.20172 | + | 1.20172i | 0 | |||||
49.3 | −0.680853 | + | 0.680853i | −2.44733 | + | 1.01372i | 1.07288i | 0 | 0.976080 | − | 2.35647i | −1.18426 | + | 2.85906i | −2.09218 | − | 2.09218i | 2.84049 | − | 2.84049i | 0 | ||||||
49.4 | 0.254738 | − | 0.254738i | 0.0501087 | − | 0.0207557i | 1.87022i | 0 | 0.00747733 | − | 0.0180519i | −0.114315 | + | 0.275980i | 0.985893 | + | 0.985893i | −2.11924 | + | 2.11924i | 0 | ||||||
49.5 | 0.528855 | − | 0.528855i | 2.84096 | − | 1.17676i | 1.44062i | 0 | 0.880118 | − | 2.12479i | −1.23707 | + | 2.98655i | 1.81959 | + | 1.81959i | 4.56494 | − | 4.56494i | 0 | ||||||
49.6 | 1.44607 | − | 1.44607i | −2.95240 | + | 1.22292i | − | 2.18224i | 0 | −2.50094 | + | 6.03781i | −0.450584 | + | 1.08781i | −0.263530 | − | 0.263530i | 5.09981 | − | 5.09981i | 0 | |||||
274.1 | −1.66305 | + | 1.66305i | 0.600564 | + | 1.44989i | − | 3.53144i | 0 | −3.41000 | − | 1.41247i | 3.30945 | + | 1.37082i | 2.54686 | + | 2.54686i | 0.379815 | − | 0.379815i | 0 | |||||
274.2 | −1.01710 | + | 1.01710i | 0.0420595 | + | 0.101541i | − | 0.0689897i | 0 | −0.146056 | − | 0.0604983i | −0.642174 | − | 0.265997i | −1.96403 | − | 1.96403i | 2.11278 | − | 2.11278i | 0 | |||||
274.3 | −0.213325 | + | 0.213325i | −0.406032 | − | 0.980249i | 1.90899i | 0 | 0.295728 | + | 0.122495i | −2.31879 | − | 0.960473i | −0.833883 | − | 0.833883i | 1.32529 | − | 1.32529i | 0 | ||||||
274.4 | 1.09994 | − | 1.09994i | −1.15110 | − | 2.77900i | − | 0.419729i | 0 | −4.32287 | − | 1.79059i | −3.19170 | − | 1.32205i | 1.73820 | + | 1.73820i | −4.27649 | + | 4.27649i | 0 | |||||
274.5 | 1.27691 | − | 1.27691i | −0.263254 | − | 0.635552i | − | 1.26102i | 0 | −1.14770 | − | 0.475393i | 4.01142 | + | 1.66158i | 0.943613 | + | 0.943613i | 1.78670 | − | 1.78670i | 0 | |||||
274.6 | 1.93083 | − | 1.93083i | 0.591976 | + | 1.42916i | − | 5.45623i | 0 | 3.90247 | + | 1.61646i | −1.16820 | − | 0.483886i | −6.67340 | − | 6.67340i | 0.429267 | − | 0.429267i | 0 | |||||
349.1 | −1.66305 | − | 1.66305i | 0.600564 | − | 1.44989i | 3.53144i | 0 | −3.41000 | + | 1.41247i | 3.30945 | − | 1.37082i | 2.54686 | − | 2.54686i | 0.379815 | + | 0.379815i | 0 | ||||||
349.2 | −1.01710 | − | 1.01710i | 0.0420595 | − | 0.101541i | 0.0689897i | 0 | −0.146056 | + | 0.0604983i | −0.642174 | + | 0.265997i | −1.96403 | + | 1.96403i | 2.11278 | + | 2.11278i | 0 | ||||||
349.3 | −0.213325 | − | 0.213325i | −0.406032 | + | 0.980249i | − | 1.90899i | 0 | 0.295728 | − | 0.122495i | −2.31879 | + | 0.960473i | −0.833883 | + | 0.833883i | 1.32529 | + | 1.32529i | 0 | |||||
349.4 | 1.09994 | + | 1.09994i | −1.15110 | + | 2.77900i | 0.419729i | 0 | −4.32287 | + | 1.79059i | −3.19170 | + | 1.32205i | 1.73820 | − | 1.73820i | −4.27649 | − | 4.27649i | 0 | ||||||
349.5 | 1.27691 | + | 1.27691i | −0.263254 | + | 0.635552i | 1.26102i | 0 | −1.14770 | + | 0.475393i | 4.01142 | − | 1.66158i | 0.943613 | − | 0.943613i | 1.78670 | + | 1.78670i | 0 | ||||||
349.6 | 1.93083 | + | 1.93083i | 0.591976 | − | 1.42916i | 5.45623i | 0 | 3.90247 | − | 1.61646i | −1.16820 | + | 0.483886i | −6.67340 | + | 6.67340i | 0.429267 | + | 0.429267i | 0 | ||||||
399.1 | −1.86672 | − | 1.86672i | −1.95914 | − | 0.811501i | 4.96928i | 0 | 2.14231 | + | 5.17200i | 1.55444 | + | 3.75274i | 5.54282 | − | 5.54282i | 1.05836 | + | 1.05836i | 0 | ||||||
399.2 | −1.09631 | − | 1.09631i | 1.05359 | + | 0.436412i | 0.403772i | 0 | −0.676617 | − | 1.63350i | 1.43180 | + | 3.45666i | −1.74995 | + | 1.74995i | −1.20172 | − | 1.20172i | 0 | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
85.m | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 425.2.n.c | 24 | |
5.b | even | 2 | 1 | 425.2.n.f | 24 | ||
5.c | odd | 4 | 1 | 85.2.l.a | ✓ | 24 | |
5.c | odd | 4 | 1 | 425.2.m.b | 24 | ||
15.e | even | 4 | 1 | 765.2.be.b | 24 | ||
17.d | even | 8 | 1 | 425.2.n.f | 24 | ||
85.k | odd | 8 | 1 | 85.2.l.a | ✓ | 24 | |
85.m | even | 8 | 1 | inner | 425.2.n.c | 24 | |
85.n | odd | 8 | 1 | 425.2.m.b | 24 | ||
85.o | even | 16 | 1 | 1445.2.a.p | 12 | ||
85.o | even | 16 | 1 | 1445.2.a.q | 12 | ||
85.o | even | 16 | 2 | 1445.2.d.j | 24 | ||
85.r | even | 16 | 1 | 7225.2.a.bq | 12 | ||
85.r | even | 16 | 1 | 7225.2.a.bs | 12 | ||
255.ba | even | 8 | 1 | 765.2.be.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
85.2.l.a | ✓ | 24 | 5.c | odd | 4 | 1 | |
85.2.l.a | ✓ | 24 | 85.k | odd | 8 | 1 | |
425.2.m.b | 24 | 5.c | odd | 4 | 1 | ||
425.2.m.b | 24 | 85.n | odd | 8 | 1 | ||
425.2.n.c | 24 | 1.a | even | 1 | 1 | trivial | |
425.2.n.c | 24 | 85.m | even | 8 | 1 | inner | |
425.2.n.f | 24 | 5.b | even | 2 | 1 | ||
425.2.n.f | 24 | 17.d | even | 8 | 1 | ||
765.2.be.b | 24 | 15.e | even | 4 | 1 | ||
765.2.be.b | 24 | 255.ba | even | 8 | 1 | ||
1445.2.a.p | 12 | 85.o | even | 16 | 1 | ||
1445.2.a.q | 12 | 85.o | even | 16 | 1 | ||
1445.2.d.j | 24 | 85.o | even | 16 | 2 | ||
7225.2.a.bq | 12 | 85.r | even | 16 | 1 | ||
7225.2.a.bs | 12 | 85.r | even | 16 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90 T_{2}^{20} - 8 T_{2}^{17} + 2327 T_{2}^{16} - 128 T_{2}^{15} + 640 T_{2}^{13} + \cdots + 289 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\).