Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [140,2,Mod(31,140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(140, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("140.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 140 = 2^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 140.o (of order \(6\), 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: | \(1.11790562830\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −1.39328 | + | 0.242400i | −0.406021 | + | 0.703249i | 1.88248 | − | 0.675465i | 0.866025 | − | 0.500000i | 0.395235 | − | 1.07825i | 0.336411 | − | 2.62428i | −2.45910 | + | 1.39743i | 1.17029 | + | 2.02701i | −1.08542 | + | 0.906567i |
31.2 | −1.27092 | + | 0.620297i | −0.331177 | + | 0.573616i | 1.23046 | − | 1.57669i | −0.866025 | + | 0.500000i | 0.0650866 | − | 0.934447i | 1.68748 | + | 2.03775i | −0.585797 | + | 2.76710i | 1.28064 | + | 2.21814i | 0.790498 | − | 1.17265i |
31.3 | −1.26796 | − | 0.626319i | −1.49907 | + | 2.59647i | 1.21545 | + | 1.58830i | −0.866025 | + | 0.500000i | 3.52698 | − | 2.35332i | −2.06101 | − | 1.65899i | −0.546365 | − | 2.77516i | −2.99443 | − | 5.18651i | 1.41125 | − | 0.0915727i |
31.4 | −0.950668 | + | 1.04701i | 1.36859 | − | 2.37047i | −0.192463 | − | 1.99072i | −0.866025 | + | 0.500000i | 1.18083 | + | 3.68646i | −1.02440 | − | 2.43939i | 2.26727 | + | 1.69100i | −2.24609 | − | 3.89033i | 0.299797 | − | 1.38207i |
31.5 | −0.894275 | − | 1.09557i | 0.895374 | − | 1.55083i | −0.400544 | + | 1.95948i | 0.866025 | − | 0.500000i | −2.49976 | + | 0.405928i | 0.644798 | − | 2.56598i | 2.50494 | − | 1.31349i | −0.103389 | − | 0.179074i | −1.32225 | − | 0.501653i |
31.6 | −0.501653 | − | 1.32225i | −0.895374 | + | 1.55083i | −1.49669 | + | 1.32662i | 0.866025 | − | 0.500000i | 2.49976 | + | 0.405928i | −0.644798 | + | 2.56598i | 2.50494 | + | 1.31349i | −0.103389 | − | 0.179074i | −1.09557 | − | 0.894275i |
31.7 | −0.397222 | + | 1.35728i | 0.556469 | − | 0.963833i | −1.68443 | − | 1.07828i | 0.866025 | − | 0.500000i | 1.08715 | + | 1.13814i | 2.32410 | + | 1.26433i | 2.13263 | − | 1.85793i | 0.880685 | + | 1.52539i | 0.334637 | + | 1.37405i |
31.8 | 0.0915727 | − | 1.41125i | 1.49907 | − | 2.59647i | −1.98323 | − | 0.258463i | −0.866025 | + | 0.500000i | −3.52698 | − | 2.35332i | 2.06101 | + | 1.65899i | −0.546365 | + | 2.77516i | −2.99443 | − | 5.18651i | 0.626319 | + | 1.26796i |
31.9 | 0.288532 | + | 1.38447i | −0.450639 | + | 0.780530i | −1.83350 | + | 0.798926i | −0.866025 | + | 0.500000i | −1.21064 | − | 0.398687i | −2.29962 | + | 1.30833i | −1.63511 | − | 2.30790i | 1.09385 | + | 1.89460i | −0.942109 | − | 1.05472i |
31.10 | 0.569639 | + | 1.29442i | −1.51353 | + | 2.62152i | −1.35102 | + | 1.47470i | 0.866025 | − | 0.500000i | −4.25550 | − | 0.465823i | 2.57616 | − | 0.602834i | −2.67847 | − | 0.908739i | −3.08156 | − | 5.33743i | 1.14053 | + | 0.836177i |
31.11 | 0.836177 | + | 1.14053i | 1.51353 | − | 2.62152i | −0.601615 | + | 1.90737i | 0.866025 | − | 0.500000i | 4.25550 | − | 0.465823i | −2.57616 | + | 0.602834i | −2.67847 | + | 0.908739i | −3.08156 | − | 5.33743i | 1.29442 | + | 0.569639i |
31.12 | 0.906567 | − | 1.08542i | 0.406021 | − | 0.703249i | −0.356272 | − | 1.96801i | 0.866025 | − | 0.500000i | −0.395235 | − | 1.07825i | −0.336411 | + | 2.62428i | −2.45910 | − | 1.39743i | 1.17029 | + | 2.02701i | 0.242400 | − | 1.39328i |
31.13 | 1.05472 | + | 0.942109i | 0.450639 | − | 0.780530i | 0.224860 | + | 1.98732i | −0.866025 | + | 0.500000i | 1.21064 | − | 0.398687i | 2.29962 | − | 1.30833i | −1.63511 | + | 2.30790i | 1.09385 | + | 1.89460i | −1.38447 | − | 0.288532i |
31.14 | 1.17265 | − | 0.790498i | 0.331177 | − | 0.573616i | 0.750225 | − | 1.85396i | −0.866025 | + | 0.500000i | −0.0650866 | − | 0.934447i | −1.68748 | − | 2.03775i | −0.585797 | − | 2.76710i | 1.28064 | + | 2.21814i | −0.620297 | + | 1.27092i |
31.15 | 1.37405 | + | 0.334637i | −0.556469 | + | 0.963833i | 1.77604 | + | 0.919616i | 0.866025 | − | 0.500000i | −1.08715 | + | 1.13814i | −2.32410 | − | 1.26433i | 2.13263 | + | 1.85793i | 0.880685 | + | 1.52539i | 1.35728 | − | 0.397222i |
31.16 | 1.38207 | − | 0.299797i | −1.36859 | + | 2.37047i | 1.82024 | − | 0.828682i | −0.866025 | + | 0.500000i | −1.18083 | + | 3.68646i | 1.02440 | + | 2.43939i | 2.26727 | − | 1.69100i | −2.24609 | − | 3.89033i | −1.04701 | + | 0.950668i |
131.1 | −1.39328 | − | 0.242400i | −0.406021 | − | 0.703249i | 1.88248 | + | 0.675465i | 0.866025 | + | 0.500000i | 0.395235 | + | 1.07825i | 0.336411 | + | 2.62428i | −2.45910 | − | 1.39743i | 1.17029 | − | 2.02701i | −1.08542 | − | 0.906567i |
131.2 | −1.27092 | − | 0.620297i | −0.331177 | − | 0.573616i | 1.23046 | + | 1.57669i | −0.866025 | − | 0.500000i | 0.0650866 | + | 0.934447i | 1.68748 | − | 2.03775i | −0.585797 | − | 2.76710i | 1.28064 | − | 2.21814i | 0.790498 | + | 1.17265i |
131.3 | −1.26796 | + | 0.626319i | −1.49907 | − | 2.59647i | 1.21545 | − | 1.58830i | −0.866025 | − | 0.500000i | 3.52698 | + | 2.35332i | −2.06101 | + | 1.65899i | −0.546365 | + | 2.77516i | −2.99443 | + | 5.18651i | 1.41125 | + | 0.0915727i |
131.4 | −0.950668 | − | 1.04701i | 1.36859 | + | 2.37047i | −0.192463 | + | 1.99072i | −0.866025 | − | 0.500000i | 1.18083 | − | 3.68646i | −1.02440 | + | 2.43939i | 2.26727 | − | 1.69100i | −2.24609 | + | 3.89033i | 0.299797 | + | 1.38207i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
28.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 140.2.o.a | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 140.2.o.a | ✓ | 32 |
5.b | even | 2 | 1 | 700.2.p.c | 32 | ||
5.c | odd | 4 | 1 | 700.2.t.c | 32 | ||
5.c | odd | 4 | 1 | 700.2.t.d | 32 | ||
7.b | odd | 2 | 1 | 980.2.o.f | 32 | ||
7.c | even | 3 | 1 | 980.2.g.a | 32 | ||
7.c | even | 3 | 1 | 980.2.o.f | 32 | ||
7.d | odd | 6 | 1 | inner | 140.2.o.a | ✓ | 32 |
7.d | odd | 6 | 1 | 980.2.g.a | 32 | ||
20.d | odd | 2 | 1 | 700.2.p.c | 32 | ||
20.e | even | 4 | 1 | 700.2.t.c | 32 | ||
20.e | even | 4 | 1 | 700.2.t.d | 32 | ||
28.d | even | 2 | 1 | 980.2.o.f | 32 | ||
28.f | even | 6 | 1 | inner | 140.2.o.a | ✓ | 32 |
28.f | even | 6 | 1 | 980.2.g.a | 32 | ||
28.g | odd | 6 | 1 | 980.2.g.a | 32 | ||
28.g | odd | 6 | 1 | 980.2.o.f | 32 | ||
35.i | odd | 6 | 1 | 700.2.p.c | 32 | ||
35.k | even | 12 | 1 | 700.2.t.c | 32 | ||
35.k | even | 12 | 1 | 700.2.t.d | 32 | ||
140.s | even | 6 | 1 | 700.2.p.c | 32 | ||
140.x | odd | 12 | 1 | 700.2.t.c | 32 | ||
140.x | odd | 12 | 1 | 700.2.t.d | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
140.2.o.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
140.2.o.a | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
140.2.o.a | ✓ | 32 | 7.d | odd | 6 | 1 | inner |
140.2.o.a | ✓ | 32 | 28.f | even | 6 | 1 | inner |
700.2.p.c | 32 | 5.b | even | 2 | 1 | ||
700.2.p.c | 32 | 20.d | odd | 2 | 1 | ||
700.2.p.c | 32 | 35.i | odd | 6 | 1 | ||
700.2.p.c | 32 | 140.s | even | 6 | 1 | ||
700.2.t.c | 32 | 5.c | odd | 4 | 1 | ||
700.2.t.c | 32 | 20.e | even | 4 | 1 | ||
700.2.t.c | 32 | 35.k | even | 12 | 1 | ||
700.2.t.c | 32 | 140.x | odd | 12 | 1 | ||
700.2.t.d | 32 | 5.c | odd | 4 | 1 | ||
700.2.t.d | 32 | 20.e | even | 4 | 1 | ||
700.2.t.d | 32 | 35.k | even | 12 | 1 | ||
700.2.t.d | 32 | 140.x | odd | 12 | 1 | ||
980.2.g.a | 32 | 7.c | even | 3 | 1 | ||
980.2.g.a | 32 | 7.d | odd | 6 | 1 | ||
980.2.g.a | 32 | 28.f | even | 6 | 1 | ||
980.2.g.a | 32 | 28.g | odd | 6 | 1 | ||
980.2.o.f | 32 | 7.b | odd | 2 | 1 | ||
980.2.o.f | 32 | 7.c | even | 3 | 1 | ||
980.2.o.f | 32 | 28.d | even | 2 | 1 | ||
980.2.o.f | 32 | 28.g | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(140, [\chi])\).