Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [375,2,Mod(49,375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(375, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("375.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 375 = 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 375.i (of order \(10\), 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: | \(2.99439007580\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 75) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | −1.26995 | + | 1.74793i | 0.951057 | − | 0.309017i | −0.824463 | − | 2.53744i | 0 | −0.667650 | + | 2.05481i | 3.16056i | 1.37265 | + | 0.446002i | 0.809017 | − | 0.587785i | 0 | ||||||
49.2 | −1.18666 | + | 1.63330i | −0.951057 | + | 0.309017i | −0.641469 | − | 1.97424i | 0 | 0.623865 | − | 1.92006i | 1.01887i | 0.145612 | + | 0.0473123i | 0.809017 | − | 0.587785i | 0 | ||||||
49.3 | −0.0832830 | + | 0.114629i | −0.951057 | + | 0.309017i | 0.611830 | + | 1.88302i | 0 | 0.0437845 | − | 0.134755i | − | 0.858311i | −0.536314 | − | 0.174259i | 0.809017 | − | 0.587785i | 0 | |||||
49.4 | 0.0832830 | − | 0.114629i | 0.951057 | − | 0.309017i | 0.611830 | + | 1.88302i | 0 | 0.0437845 | − | 0.134755i | 0.858311i | 0.536314 | + | 0.174259i | 0.809017 | − | 0.587785i | 0 | ||||||
49.5 | 1.18666 | − | 1.63330i | 0.951057 | − | 0.309017i | −0.641469 | − | 1.97424i | 0 | 0.623865 | − | 1.92006i | − | 1.01887i | −0.145612 | − | 0.0473123i | 0.809017 | − | 0.587785i | 0 | |||||
49.6 | 1.26995 | − | 1.74793i | −0.951057 | + | 0.309017i | −0.824463 | − | 2.53744i | 0 | −0.667650 | + | 2.05481i | − | 3.16056i | −1.37265 | − | 0.446002i | 0.809017 | − | 0.587785i | 0 | |||||
199.1 | −1.26995 | − | 1.74793i | 0.951057 | + | 0.309017i | −0.824463 | + | 2.53744i | 0 | −0.667650 | − | 2.05481i | − | 3.16056i | 1.37265 | − | 0.446002i | 0.809017 | + | 0.587785i | 0 | |||||
199.2 | −1.18666 | − | 1.63330i | −0.951057 | − | 0.309017i | −0.641469 | + | 1.97424i | 0 | 0.623865 | + | 1.92006i | − | 1.01887i | 0.145612 | − | 0.0473123i | 0.809017 | + | 0.587785i | 0 | |||||
199.3 | −0.0832830 | − | 0.114629i | −0.951057 | − | 0.309017i | 0.611830 | − | 1.88302i | 0 | 0.0437845 | + | 0.134755i | 0.858311i | −0.536314 | + | 0.174259i | 0.809017 | + | 0.587785i | 0 | ||||||
199.4 | 0.0832830 | + | 0.114629i | 0.951057 | + | 0.309017i | 0.611830 | − | 1.88302i | 0 | 0.0437845 | + | 0.134755i | − | 0.858311i | 0.536314 | − | 0.174259i | 0.809017 | + | 0.587785i | 0 | |||||
199.5 | 1.18666 | + | 1.63330i | 0.951057 | + | 0.309017i | −0.641469 | + | 1.97424i | 0 | 0.623865 | + | 1.92006i | 1.01887i | −0.145612 | + | 0.0473123i | 0.809017 | + | 0.587785i | 0 | ||||||
199.6 | 1.26995 | + | 1.74793i | −0.951057 | − | 0.309017i | −0.824463 | + | 2.53744i | 0 | −0.667650 | − | 2.05481i | 3.16056i | −1.37265 | + | 0.446002i | 0.809017 | + | 0.587785i | 0 | ||||||
274.1 | −2.55553 | − | 0.830342i | 0.587785 | − | 0.809017i | 4.22323 | + | 3.06835i | 0 | −2.17386 | + | 1.57940i | 1.68704i | −5.08599 | − | 7.00026i | −0.309017 | − | 0.951057i | 0 | ||||||
274.2 | −2.32085 | − | 0.754089i | −0.587785 | + | 0.809017i | 3.19965 | + | 2.32468i | 0 | 1.97423 | − | 1.43436i | 3.44028i | −2.80415 | − | 3.85959i | −0.309017 | − | 0.951057i | 0 | ||||||
274.3 | −0.234682 | − | 0.0762527i | −0.587785 | + | 0.809017i | −1.56877 | − | 1.13978i | 0 | 0.199632 | − | 0.145041i | 1.24676i | 0.571334 | + | 0.786373i | −0.309017 | − | 0.951057i | 0 | ||||||
274.4 | 0.234682 | + | 0.0762527i | 0.587785 | − | 0.809017i | −1.56877 | − | 1.13978i | 0 | 0.199632 | − | 0.145041i | − | 1.24676i | −0.571334 | − | 0.786373i | −0.309017 | − | 0.951057i | 0 | |||||
274.5 | 2.32085 | + | 0.754089i | 0.587785 | − | 0.809017i | 3.19965 | + | 2.32468i | 0 | 1.97423 | − | 1.43436i | − | 3.44028i | 2.80415 | + | 3.85959i | −0.309017 | − | 0.951057i | 0 | |||||
274.6 | 2.55553 | + | 0.830342i | −0.587785 | + | 0.809017i | 4.22323 | + | 3.06835i | 0 | −2.17386 | + | 1.57940i | − | 1.68704i | 5.08599 | + | 7.00026i | −0.309017 | − | 0.951057i | 0 | |||||
349.1 | −2.55553 | + | 0.830342i | 0.587785 | + | 0.809017i | 4.22323 | − | 3.06835i | 0 | −2.17386 | − | 1.57940i | − | 1.68704i | −5.08599 | + | 7.00026i | −0.309017 | + | 0.951057i | 0 | |||||
349.2 | −2.32085 | + | 0.754089i | −0.587785 | − | 0.809017i | 3.19965 | − | 2.32468i | 0 | 1.97423 | + | 1.43436i | − | 3.44028i | −2.80415 | + | 3.85959i | −0.309017 | + | 0.951057i | 0 | |||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 375.2.i.d | 24 | |
5.b | even | 2 | 1 | inner | 375.2.i.d | 24 | |
5.c | odd | 4 | 1 | 75.2.g.c | ✓ | 12 | |
5.c | odd | 4 | 1 | 375.2.g.c | 12 | ||
15.e | even | 4 | 1 | 225.2.h.d | 12 | ||
25.d | even | 5 | 1 | inner | 375.2.i.d | 24 | |
25.d | even | 5 | 1 | 1875.2.b.f | 12 | ||
25.e | even | 10 | 1 | inner | 375.2.i.d | 24 | |
25.e | even | 10 | 1 | 1875.2.b.f | 12 | ||
25.f | odd | 20 | 1 | 75.2.g.c | ✓ | 12 | |
25.f | odd | 20 | 1 | 375.2.g.c | 12 | ||
25.f | odd | 20 | 1 | 1875.2.a.j | 6 | ||
25.f | odd | 20 | 1 | 1875.2.a.k | 6 | ||
75.l | even | 20 | 1 | 225.2.h.d | 12 | ||
75.l | even | 20 | 1 | 5625.2.a.p | 6 | ||
75.l | even | 20 | 1 | 5625.2.a.q | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.2.g.c | ✓ | 12 | 5.c | odd | 4 | 1 | |
75.2.g.c | ✓ | 12 | 25.f | odd | 20 | 1 | |
225.2.h.d | 12 | 15.e | even | 4 | 1 | ||
225.2.h.d | 12 | 75.l | even | 20 | 1 | ||
375.2.g.c | 12 | 5.c | odd | 4 | 1 | ||
375.2.g.c | 12 | 25.f | odd | 20 | 1 | ||
375.2.i.d | 24 | 1.a | even | 1 | 1 | trivial | |
375.2.i.d | 24 | 5.b | even | 2 | 1 | inner | |
375.2.i.d | 24 | 25.d | even | 5 | 1 | inner | |
375.2.i.d | 24 | 25.e | even | 10 | 1 | inner | |
1875.2.a.j | 6 | 25.f | odd | 20 | 1 | ||
1875.2.a.k | 6 | 25.f | odd | 20 | 1 | ||
1875.2.b.f | 12 | 25.d | even | 5 | 1 | ||
1875.2.b.f | 12 | 25.e | even | 10 | 1 | ||
5625.2.a.p | 6 | 75.l | even | 20 | 1 | ||
5625.2.a.q | 6 | 75.l | even | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} - 16 T_{2}^{22} + 132 T_{2}^{20} - 717 T_{2}^{18} + 4268 T_{2}^{16} - 19372 T_{2}^{14} + 64293 T_{2}^{12} - 147172 T_{2}^{10} + 681588 T_{2}^{8} - 58037 T_{2}^{6} + 1932 T_{2}^{4} + 4 T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\).