Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,5,Mod(31,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.31");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 70.j (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: | \(7.23589741587\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) 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 | −1.41421 | − | 2.44949i | −9.27207 | − | 5.35323i | −4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 30.2824i | −2.45035 | + | 48.9387i | 22.6274 | 16.8142 | + | 29.1230i | 27.3861 | + | 15.8114i | ||||
31.2 | −1.41421 | − | 2.44949i | −1.47336 | − | 0.850642i | −4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 4.81196i | −47.5891 | + | 11.6737i | 22.6274 | −39.0528 | − | 67.6415i | −27.3861 | − | 15.8114i | ||||
31.3 | −1.41421 | − | 2.44949i | −0.950229 | − | 0.548615i | −4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 3.10343i | 46.7485 | + | 14.6824i | 22.6274 | −39.8980 | − | 69.1054i | −27.3861 | − | 15.8114i | ||||
31.4 | −1.41421 | − | 2.44949i | 4.31893 | + | 2.49353i | −4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | − | 14.1056i | −26.3103 | − | 41.3373i | 22.6274 | −28.0646 | − | 48.6093i | 27.3861 | + | 15.8114i | |||
31.5 | −1.41421 | − | 2.44949i | 10.1550 | + | 5.86302i | −4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | − | 33.1663i | 33.0212 | + | 36.2022i | 22.6274 | 28.2500 | + | 48.9305i | 27.3861 | + | 15.8114i | |||
31.6 | −1.41421 | − | 2.44949i | 14.7070 | + | 8.49107i | −4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | − | 48.0327i | 7.09468 | − | 48.4837i | 22.6274 | 103.696 | + | 179.607i | −27.3861 | − | 15.8114i | |||
31.7 | 1.41421 | + | 2.44949i | −12.1079 | − | 6.99053i | −4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | − | 39.5444i | 46.3256 | + | 15.9668i | −22.6274 | 57.2349 | + | 99.1338i | 27.3861 | + | 15.8114i | |||
31.8 | 1.41421 | + | 2.44949i | −7.38100 | − | 4.26142i | −4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | − | 24.1063i | 44.3801 | − | 20.7703i | −22.6274 | −4.18054 | − | 7.24091i | −27.3861 | − | 15.8114i | |||
31.9 | 1.41421 | + | 2.44949i | −5.53636 | − | 3.19642i | −4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | − | 18.0817i | −46.3868 | − | 15.7883i | −22.6274 | −20.0658 | − | 34.7551i | 27.3861 | + | 15.8114i | |||
31.10 | 1.41421 | + | 2.44949i | 0.181996 | + | 0.105076i | −4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 0.594397i | −17.7284 | + | 45.6804i | −22.6274 | −40.4779 | − | 70.1098i | −27.3861 | − | 15.8114i | ||||
31.11 | 1.41421 | + | 2.44949i | 10.4879 | + | 6.05522i | −4.00000 | + | 6.92820i | 9.68246 | − | 5.59017i | 34.2535i | 47.6639 | + | 11.3646i | −22.6274 | 32.8313 | + | 56.8655i | 27.3861 | + | 15.8114i | ||||
31.12 | 1.41421 | + | 2.44949i | 14.8701 | + | 8.58525i | −4.00000 | + | 6.92820i | −9.68246 | + | 5.59017i | 48.5655i | −46.7691 | + | 14.6168i | −22.6274 | 106.913 | + | 185.179i | −27.3861 | − | 15.8114i | ||||
61.1 | −1.41421 | + | 2.44949i | −9.27207 | + | 5.35323i | −4.00000 | − | 6.92820i | −9.68246 | − | 5.59017i | − | 30.2824i | −2.45035 | − | 48.9387i | 22.6274 | 16.8142 | − | 29.1230i | 27.3861 | − | 15.8114i | |||
61.2 | −1.41421 | + | 2.44949i | −1.47336 | + | 0.850642i | −4.00000 | − | 6.92820i | 9.68246 | + | 5.59017i | − | 4.81196i | −47.5891 | − | 11.6737i | 22.6274 | −39.0528 | + | 67.6415i | −27.3861 | + | 15.8114i | |||
61.3 | −1.41421 | + | 2.44949i | −0.950229 | + | 0.548615i | −4.00000 | − | 6.92820i | 9.68246 | + | 5.59017i | − | 3.10343i | 46.7485 | − | 14.6824i | 22.6274 | −39.8980 | + | 69.1054i | −27.3861 | + | 15.8114i | |||
61.4 | −1.41421 | + | 2.44949i | 4.31893 | − | 2.49353i | −4.00000 | − | 6.92820i | −9.68246 | − | 5.59017i | 14.1056i | −26.3103 | + | 41.3373i | 22.6274 | −28.0646 | + | 48.6093i | 27.3861 | − | 15.8114i | ||||
61.5 | −1.41421 | + | 2.44949i | 10.1550 | − | 5.86302i | −4.00000 | − | 6.92820i | −9.68246 | − | 5.59017i | 33.1663i | 33.0212 | − | 36.2022i | 22.6274 | 28.2500 | − | 48.9305i | 27.3861 | − | 15.8114i | ||||
61.6 | −1.41421 | + | 2.44949i | 14.7070 | − | 8.49107i | −4.00000 | − | 6.92820i | 9.68246 | + | 5.59017i | 48.0327i | 7.09468 | + | 48.4837i | 22.6274 | 103.696 | − | 179.607i | −27.3861 | + | 15.8114i | ||||
61.7 | 1.41421 | − | 2.44949i | −12.1079 | + | 6.99053i | −4.00000 | − | 6.92820i | 9.68246 | + | 5.59017i | 39.5444i | 46.3256 | − | 15.9668i | −22.6274 | 57.2349 | − | 99.1338i | 27.3861 | − | 15.8114i | ||||
61.8 | 1.41421 | − | 2.44949i | −7.38100 | + | 4.26142i | −4.00000 | − | 6.92820i | −9.68246 | − | 5.59017i | 24.1063i | 44.3801 | + | 20.7703i | −22.6274 | −4.18054 | + | 7.24091i | −27.3861 | + | 15.8114i | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 70.5.j.a | ✓ | 24 |
5.b | even | 2 | 1 | 350.5.k.d | 24 | ||
5.c | odd | 4 | 2 | 350.5.i.c | 48 | ||
7.c | even | 3 | 1 | 490.5.b.c | 24 | ||
7.d | odd | 6 | 1 | inner | 70.5.j.a | ✓ | 24 |
7.d | odd | 6 | 1 | 490.5.b.c | 24 | ||
35.i | odd | 6 | 1 | 350.5.k.d | 24 | ||
35.k | even | 12 | 2 | 350.5.i.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
70.5.j.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
70.5.j.a | ✓ | 24 | 7.d | odd | 6 | 1 | inner |
350.5.i.c | 48 | 5.c | odd | 4 | 2 | ||
350.5.i.c | 48 | 35.k | even | 12 | 2 | ||
350.5.k.d | 24 | 5.b | even | 2 | 1 | ||
350.5.k.d | 24 | 35.i | odd | 6 | 1 | ||
490.5.b.c | 24 | 7.c | even | 3 | 1 | ||
490.5.b.c | 24 | 7.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(70, [\chi])\).