Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,5,Mod(19,70)]
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([3, 5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.19");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 70.h (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: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −2.44949 | − | 1.41421i | −6.32716 | − | 10.9590i | 4.00000 | + | 6.92820i | −18.7989 | − | 16.4803i | 35.7918i | −21.6321 | − | 43.9665i | − | 22.6274i | −39.5658 | + | 68.5300i | 22.7412 | + | 66.9540i | |||
19.2 | −2.44949 | − | 1.41421i | −6.27030 | − | 10.8605i | 4.00000 | + | 6.92820i | 10.1380 | + | 22.8521i | 35.4702i | −32.1106 | + | 37.0123i | − | 22.6274i | −38.1334 | + | 66.0490i | 7.48477 | − | 70.3134i | |||
19.3 | −2.44949 | − | 1.41421i | −4.27603 | − | 7.40629i | 4.00000 | + | 6.92820i | 22.3515 | − | 11.1986i | 24.1889i | 48.9684 | − | 1.76024i | − | 22.6274i | 3.93122 | − | 6.80907i | −70.5871 | − | 4.17884i | |||
19.4 | −2.44949 | − | 1.41421i | 0.694226 | + | 1.20243i | 4.00000 | + | 6.92820i | −9.07375 | + | 23.2952i | − | 3.92713i | −24.8459 | − | 42.2337i | − | 22.6274i | 39.5361 | − | 68.4785i | 55.1705 | − | 44.2292i | ||
19.5 | −2.44949 | − | 1.41421i | 2.92307 | + | 5.06291i | 4.00000 | + | 6.92820i | −16.0876 | − | 19.1361i | − | 16.5354i | −17.3754 | + | 45.8159i | − | 22.6274i | 23.4113 | − | 40.5496i | 12.3439 | + | 69.6249i | ||
19.6 | −2.44949 | − | 1.41421i | 3.98422 | + | 6.90087i | 4.00000 | + | 6.92820i | 24.9999 | + | 0.0650489i | − | 22.5381i | −33.9906 | + | 35.2936i | − | 22.6274i | 8.75204 | − | 15.1590i | −61.1450 | − | 35.5146i | ||
19.7 | −2.44949 | − | 1.41421i | 5.75172 | + | 9.96228i | 4.00000 | + | 6.92820i | 1.98309 | − | 24.9212i | − | 32.5367i | 17.7931 | − | 45.6553i | − | 22.6274i | −25.6646 | + | 44.4524i | −40.1015 | + | 58.2398i | ||
19.8 | −2.44949 | − | 1.41421i | 8.41923 | + | 14.5825i | 4.00000 | + | 6.92820i | 2.88670 | + | 24.8328i | − | 47.6264i | 48.4962 | + | 7.00864i | − | 22.6274i | −101.267 | + | 175.399i | 28.0479 | − | 64.9101i | ||
19.9 | 2.44949 | + | 1.41421i | −8.41923 | − | 14.5825i | 4.00000 | + | 6.92820i | 22.9492 | − | 9.91643i | − | 47.6264i | −48.4962 | − | 7.00864i | 22.6274i | −101.267 | + | 175.399i | 70.2377 | + | 8.16483i | |||
19.10 | 2.44949 | + | 1.41421i | −5.75172 | − | 9.96228i | 4.00000 | + | 6.92820i | −20.5909 | + | 14.1780i | − | 32.5367i | −17.7931 | + | 45.6553i | 22.6274i | −25.6646 | + | 44.4524i | −70.4879 | + | 5.60903i | |||
19.11 | 2.44949 | + | 1.41421i | −3.98422 | − | 6.90087i | 4.00000 | + | 6.92820i | 12.5563 | + | 21.6180i | − | 22.5381i | 33.9906 | − | 35.2936i | 22.6274i | 8.75204 | − | 15.1590i | 0.183986 | + | 70.7104i | |||
19.12 | 2.44949 | + | 1.41421i | −2.92307 | − | 5.06291i | 4.00000 | + | 6.92820i | −24.6161 | − | 4.36422i | − | 16.5354i | 17.3754 | − | 45.8159i | 22.6274i | 23.4113 | − | 40.5496i | −54.1250 | − | 45.5026i | |||
19.13 | 2.44949 | + | 1.41421i | −0.694226 | − | 1.20243i | 4.00000 | + | 6.92820i | 15.6374 | − | 19.5057i | − | 3.92713i | 24.8459 | + | 42.2337i | 22.6274i | 39.5361 | − | 68.4785i | 65.8888 | − | 25.6645i | |||
19.14 | 2.44949 | + | 1.41421i | 4.27603 | + | 7.40629i | 4.00000 | + | 6.92820i | 1.47744 | + | 24.9563i | 24.1889i | −48.9684 | + | 1.76024i | 22.6274i | 3.93122 | − | 6.80907i | −31.6746 | + | 63.2196i | ||||
19.15 | 2.44949 | + | 1.41421i | 6.27030 | + | 10.8605i | 4.00000 | + | 6.92820i | 24.8596 | − | 2.64627i | 35.4702i | 32.1106 | − | 37.0123i | 22.6274i | −38.1334 | + | 66.0490i | 64.6356 | + | 28.6747i | ||||
19.16 | 2.44949 | + | 1.41421i | 6.32716 | + | 10.9590i | 4.00000 | + | 6.92820i | −23.6718 | − | 8.04022i | 35.7918i | 21.6321 | + | 43.9665i | 22.6274i | −39.5658 | + | 68.5300i | −46.6133 | − | 53.1714i | ||||
59.1 | −2.44949 | + | 1.41421i | −6.32716 | + | 10.9590i | 4.00000 | − | 6.92820i | −18.7989 | + | 16.4803i | − | 35.7918i | −21.6321 | + | 43.9665i | 22.6274i | −39.5658 | − | 68.5300i | 22.7412 | − | 66.9540i | |||
59.2 | −2.44949 | + | 1.41421i | −6.27030 | + | 10.8605i | 4.00000 | − | 6.92820i | 10.1380 | − | 22.8521i | − | 35.4702i | −32.1106 | − | 37.0123i | 22.6274i | −38.1334 | − | 66.0490i | 7.48477 | + | 70.3134i | |||
59.3 | −2.44949 | + | 1.41421i | −4.27603 | + | 7.40629i | 4.00000 | − | 6.92820i | 22.3515 | + | 11.1986i | − | 24.1889i | 48.9684 | + | 1.76024i | 22.6274i | 3.93122 | + | 6.80907i | −70.5871 | + | 4.17884i | |||
59.4 | −2.44949 | + | 1.41421i | 0.694226 | − | 1.20243i | 4.00000 | − | 6.92820i | −9.07375 | − | 23.2952i | 3.92713i | −24.8459 | + | 42.2337i | 22.6274i | 39.5361 | + | 68.4785i | 55.1705 | + | 44.2292i | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
35.i | 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.h.a | ✓ | 32 |
5.b | even | 2 | 1 | inner | 70.5.h.a | ✓ | 32 |
5.c | odd | 4 | 2 | 350.5.k.e | 32 | ||
7.c | even | 3 | 1 | 490.5.d.a | 32 | ||
7.d | odd | 6 | 1 | inner | 70.5.h.a | ✓ | 32 |
7.d | odd | 6 | 1 | 490.5.d.a | 32 | ||
35.i | odd | 6 | 1 | inner | 70.5.h.a | ✓ | 32 |
35.i | odd | 6 | 1 | 490.5.d.a | 32 | ||
35.j | even | 6 | 1 | 490.5.d.a | 32 | ||
35.k | even | 12 | 2 | 350.5.k.e | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
70.5.h.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
70.5.h.a | ✓ | 32 | 5.b | even | 2 | 1 | inner |
70.5.h.a | ✓ | 32 | 7.d | odd | 6 | 1 | inner |
70.5.h.a | ✓ | 32 | 35.i | odd | 6 | 1 | inner |
350.5.k.e | 32 | 5.c | odd | 4 | 2 | ||
350.5.k.e | 32 | 35.k | even | 12 | 2 | ||
490.5.d.a | 32 | 7.c | even | 3 | 1 | ||
490.5.d.a | 32 | 7.d | odd | 6 | 1 | ||
490.5.d.a | 32 | 35.i | odd | 6 | 1 | ||
490.5.d.a | 32 | 35.j | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(70, [\chi])\).