Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,7,Mod(335,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.335");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.o (of order \(2\), degree \(1\), 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: | \(77.2981720963\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
335.1 | 0 | −20.0330 | − | 18.1019i | 0 | −138.405 | 0 | 272.716 | + | 208.027i | 0 | 73.6404 | + | 725.271i | 0 | ||||||||||||
335.2 | 0 | −20.0330 | − | 18.1019i | 0 | 138.405 | 0 | 272.716 | − | 208.027i | 0 | 73.6404 | + | 725.271i | 0 | ||||||||||||
335.3 | 0 | −20.0330 | + | 18.1019i | 0 | −138.405 | 0 | 272.716 | − | 208.027i | 0 | 73.6404 | − | 725.271i | 0 | ||||||||||||
335.4 | 0 | −20.0330 | + | 18.1019i | 0 | 138.405 | 0 | 272.716 | + | 208.027i | 0 | 73.6404 | − | 725.271i | 0 | ||||||||||||
335.5 | 0 | −18.7134 | − | 19.4630i | 0 | −214.780 | 0 | −147.649 | − | 309.595i | 0 | −28.6171 | + | 728.438i | 0 | ||||||||||||
335.6 | 0 | −18.7134 | − | 19.4630i | 0 | 214.780 | 0 | −147.649 | + | 309.595i | 0 | −28.6171 | + | 728.438i | 0 | ||||||||||||
335.7 | 0 | −18.7134 | + | 19.4630i | 0 | −214.780 | 0 | −147.649 | + | 309.595i | 0 | −28.6171 | − | 728.438i | 0 | ||||||||||||
335.8 | 0 | −18.7134 | + | 19.4630i | 0 | 214.780 | 0 | −147.649 | − | 309.595i | 0 | −28.6171 | − | 728.438i | 0 | ||||||||||||
335.9 | 0 | −5.43032 | − | 26.4483i | 0 | −57.7046 | 0 | −251.052 | + | 233.713i | 0 | −670.023 | + | 287.245i | 0 | ||||||||||||
335.10 | 0 | −5.43032 | − | 26.4483i | 0 | 57.7046 | 0 | −251.052 | − | 233.713i | 0 | −670.023 | + | 287.245i | 0 | ||||||||||||
335.11 | 0 | −5.43032 | + | 26.4483i | 0 | −57.7046 | 0 | −251.052 | − | 233.713i | 0 | −670.023 | − | 287.245i | 0 | ||||||||||||
335.12 | 0 | −5.43032 | + | 26.4483i | 0 | 57.7046 | 0 | −251.052 | + | 233.713i | 0 | −670.023 | − | 287.245i | 0 | ||||||||||||
335.13 | 0 | 5.43032 | − | 26.4483i | 0 | −57.7046 | 0 | 251.052 | + | 233.713i | 0 | −670.023 | − | 287.245i | 0 | ||||||||||||
335.14 | 0 | 5.43032 | − | 26.4483i | 0 | 57.7046 | 0 | 251.052 | − | 233.713i | 0 | −670.023 | − | 287.245i | 0 | ||||||||||||
335.15 | 0 | 5.43032 | + | 26.4483i | 0 | −57.7046 | 0 | 251.052 | − | 233.713i | 0 | −670.023 | + | 287.245i | 0 | ||||||||||||
335.16 | 0 | 5.43032 | + | 26.4483i | 0 | 57.7046 | 0 | 251.052 | + | 233.713i | 0 | −670.023 | + | 287.245i | 0 | ||||||||||||
335.17 | 0 | 18.7134 | − | 19.4630i | 0 | −214.780 | 0 | 147.649 | − | 309.595i | 0 | −28.6171 | − | 728.438i | 0 | ||||||||||||
335.18 | 0 | 18.7134 | − | 19.4630i | 0 | 214.780 | 0 | 147.649 | + | 309.595i | 0 | −28.6171 | − | 728.438i | 0 | ||||||||||||
335.19 | 0 | 18.7134 | + | 19.4630i | 0 | −214.780 | 0 | 147.649 | + | 309.595i | 0 | −28.6171 | + | 728.438i | 0 | ||||||||||||
335.20 | 0 | 18.7134 | + | 19.4630i | 0 | 214.780 | 0 | 147.649 | − | 309.595i | 0 | −28.6171 | + | 728.438i | 0 | ||||||||||||
See all 24 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.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
84.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 336.7.o.e | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
7.b | odd | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
12.b | even | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
21.c | even | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
28.d | even | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
84.h | odd | 2 | 1 | inner | 336.7.o.e | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.7.o.e | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
336.7.o.e | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
336.7.o.e | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
336.7.o.e | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
336.7.o.e | ✓ | 24 | 12.b | even | 2 | 1 | inner |
336.7.o.e | ✓ | 24 | 21.c | even | 2 | 1 | inner |
336.7.o.e | ✓ | 24 | 28.d | even | 2 | 1 | inner |
336.7.o.e | ✓ | 24 | 84.h | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\):
\( T_{5}^{6} - 68616T_{5}^{4} + 1101059100T_{5}^{2} - 2942456220000 \) |
\( T_{19}^{6} - 95297796T_{19}^{4} + 2223702800731356T_{19}^{2} - 1827590823378704673520 \) |