Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,4,Mod(95,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.95");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.bj (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: | \(19.8246417619\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
95.1 | 0 | −5.19375 | − | 0.158148i | 0 | 17.5520 | − | 10.1337i | 0 | 18.5198 | + | 0.135878i | 0 | 26.9500 | + | 1.64276i | 0 | ||||||||||
95.2 | 0 | −5.16981 | − | 0.522553i | 0 | −5.91262 | + | 3.41365i | 0 | −13.7402 | − | 12.4180i | 0 | 26.4539 | + | 5.40300i | 0 | ||||||||||
95.3 | 0 | −5.06455 | + | 1.16202i | 0 | −6.80588 | + | 3.92938i | 0 | 0.395544 | + | 18.5160i | 0 | 24.2994 | − | 11.7702i | 0 | ||||||||||
95.4 | 0 | −3.03745 | − | 4.21591i | 0 | 5.91262 | − | 3.41365i | 0 | −13.7402 | − | 12.4180i | 0 | −8.54780 | + | 25.6112i | 0 | ||||||||||
95.5 | 0 | −2.89222 | + | 4.31684i | 0 | −6.70049 | + | 3.86853i | 0 | 13.9129 | − | 12.2242i | 0 | −10.2702 | − | 24.9705i | 0 | ||||||||||
95.6 | 0 | −2.73383 | − | 4.41884i | 0 | −17.5520 | + | 10.1337i | 0 | 18.5198 | + | 0.135878i | 0 | −12.0523 | + | 24.1607i | 0 | ||||||||||
95.7 | 0 | −2.29238 | + | 4.66315i | 0 | 6.70049 | − | 3.86853i | 0 | −13.9129 | + | 12.2242i | 0 | −16.4900 | − | 21.3795i | 0 | ||||||||||
95.8 | 0 | −1.52594 | − | 4.96704i | 0 | 6.80588 | − | 3.92938i | 0 | 0.395544 | + | 18.5160i | 0 | −22.3430 | + | 15.1588i | 0 | ||||||||||
95.9 | 0 | 1.52594 | + | 4.96704i | 0 | 6.80588 | − | 3.92938i | 0 | −0.395544 | − | 18.5160i | 0 | −22.3430 | + | 15.1588i | 0 | ||||||||||
95.10 | 0 | 2.29238 | − | 4.66315i | 0 | 6.70049 | − | 3.86853i | 0 | 13.9129 | − | 12.2242i | 0 | −16.4900 | − | 21.3795i | 0 | ||||||||||
95.11 | 0 | 2.73383 | + | 4.41884i | 0 | −17.5520 | + | 10.1337i | 0 | −18.5198 | − | 0.135878i | 0 | −12.0523 | + | 24.1607i | 0 | ||||||||||
95.12 | 0 | 2.89222 | − | 4.31684i | 0 | −6.70049 | + | 3.86853i | 0 | −13.9129 | + | 12.2242i | 0 | −10.2702 | − | 24.9705i | 0 | ||||||||||
95.13 | 0 | 3.03745 | + | 4.21591i | 0 | 5.91262 | − | 3.41365i | 0 | 13.7402 | + | 12.4180i | 0 | −8.54780 | + | 25.6112i | 0 | ||||||||||
95.14 | 0 | 5.06455 | − | 1.16202i | 0 | −6.80588 | + | 3.92938i | 0 | −0.395544 | − | 18.5160i | 0 | 24.2994 | − | 11.7702i | 0 | ||||||||||
95.15 | 0 | 5.16981 | + | 0.522553i | 0 | −5.91262 | + | 3.41365i | 0 | 13.7402 | + | 12.4180i | 0 | 26.4539 | + | 5.40300i | 0 | ||||||||||
95.16 | 0 | 5.19375 | + | 0.158148i | 0 | 17.5520 | − | 10.1337i | 0 | −18.5198 | − | 0.135878i | 0 | 26.9500 | + | 1.64276i | 0 | ||||||||||
191.1 | 0 | −5.19375 | + | 0.158148i | 0 | 17.5520 | + | 10.1337i | 0 | 18.5198 | − | 0.135878i | 0 | 26.9500 | − | 1.64276i | 0 | ||||||||||
191.2 | 0 | −5.16981 | + | 0.522553i | 0 | −5.91262 | − | 3.41365i | 0 | −13.7402 | + | 12.4180i | 0 | 26.4539 | − | 5.40300i | 0 | ||||||||||
191.3 | 0 | −5.06455 | − | 1.16202i | 0 | −6.80588 | − | 3.92938i | 0 | 0.395544 | − | 18.5160i | 0 | 24.2994 | + | 11.7702i | 0 | ||||||||||
191.4 | 0 | −3.03745 | + | 4.21591i | 0 | 5.91262 | + | 3.41365i | 0 | −13.7402 | + | 12.4180i | 0 | −8.54780 | − | 25.6112i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
12.b | even | 2 | 1 | inner |
21.h | odd | 6 | 1 | inner |
28.g | odd | 6 | 1 | inner |
84.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 336.4.bj.g | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 336.4.bj.g | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 336.4.bj.g | ✓ | 32 |
7.c | even | 3 | 1 | inner | 336.4.bj.g | ✓ | 32 |
12.b | even | 2 | 1 | inner | 336.4.bj.g | ✓ | 32 |
21.h | odd | 6 | 1 | inner | 336.4.bj.g | ✓ | 32 |
28.g | odd | 6 | 1 | inner | 336.4.bj.g | ✓ | 32 |
84.n | even | 6 | 1 | inner | 336.4.bj.g | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.4.bj.g | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
336.4.bj.g | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
336.4.bj.g | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
336.4.bj.g | ✓ | 32 | 7.c | even | 3 | 1 | inner |
336.4.bj.g | ✓ | 32 | 12.b | even | 2 | 1 | inner |
336.4.bj.g | ✓ | 32 | 21.h | odd | 6 | 1 | inner |
336.4.bj.g | ✓ | 32 | 28.g | odd | 6 | 1 | inner |
336.4.bj.g | ✓ | 32 | 84.n | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):
\( T_{5}^{16} - 579 T_{5}^{14} + 256770 T_{5}^{12} - 37395467 T_{5}^{10} + 3759550482 T_{5}^{8} - 233452565091 T_{5}^{6} + 10602642000841 T_{5}^{4} - 284536107802800 T_{5}^{2} + \cdots + 50\!\cdots\!00 \) |
\( T_{13}^{4} + 5T_{13}^{3} - 3684T_{13}^{2} + 85684T_{13} - 180880 \) |
\( T_{19}^{16} - 19023 T_{19}^{14} + 243345570 T_{19}^{12} - 1748942749615 T_{19}^{10} + \cdots + 37\!\cdots\!00 \) |