Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,2,Mod(49,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("368.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 368.m (of order \(11\), degree \(10\), not 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: | \(2.93849479438\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{11})\) |
Twist minimal: | no (minimal twist has level 184) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | 0 | −1.05870 | + | 2.31822i | 0 | −0.536040 | + | 0.618623i | 0 | −1.79431 | + | 1.15313i | 0 | −2.28874 | − | 2.64134i | 0 | ||||||||||
49.2 | 0 | 0.246469 | − | 0.539692i | 0 | −1.62555 | + | 1.87599i | 0 | 3.01490 | − | 1.93756i | 0 | 1.73406 | + | 2.00121i | 0 | ||||||||||
49.3 | 0 | 0.693988 | − | 1.51962i | 0 | 2.16159 | − | 2.49461i | 0 | −1.24823 | + | 0.802191i | 0 | 0.136950 | + | 0.158049i | 0 | ||||||||||
81.1 | 0 | −0.369168 | + | 2.56762i | 0 | −3.43820 | + | 1.00955i | 0 | 2.84293 | + | 3.28091i | 0 | −3.57792 | − | 1.05057i | 0 | ||||||||||
81.2 | 0 | −0.0226391 | + | 0.157458i | 0 | 0.0359441 | − | 0.0105542i | 0 | −2.08148 | − | 2.40216i | 0 | 2.85420 | + | 0.838068i | 0 | ||||||||||
81.3 | 0 | 0.152362 | − | 1.05970i | 0 | 3.40226 | − | 0.998994i | 0 | 2.50777 | + | 2.89412i | 0 | 1.77873 | + | 0.522282i | 0 | ||||||||||
177.1 | 0 | −1.67298 | + | 1.07516i | 0 | 1.60166 | + | 3.50714i | 0 | 0.512781 | − | 0.150566i | 0 | 0.396642 | − | 0.868525i | 0 | ||||||||||
177.2 | 0 | −1.21882 | + | 0.783290i | 0 | −1.24491 | − | 2.72597i | 0 | −1.62369 | + | 0.476758i | 0 | −0.374258 | + | 0.819511i | 0 | ||||||||||
177.3 | 0 | 1.78999 | − | 1.15036i | 0 | −0.356751 | − | 0.781176i | 0 | 3.61448 | − | 1.06131i | 0 | 0.634502 | − | 1.38936i | 0 | ||||||||||
193.1 | 0 | −1.32137 | − | 1.52494i | 0 | 0.0973464 | + | 0.677059i | 0 | 0.899416 | − | 1.96945i | 0 | −0.152486 | + | 1.06057i | 0 | ||||||||||
193.2 | 0 | 0.946512 | + | 1.09233i | 0 | 0.287791 | + | 2.00163i | 0 | −2.03440 | + | 4.45471i | 0 | 0.129638 | − | 0.901650i | 0 | ||||||||||
193.3 | 0 | 1.63153 | + | 1.88288i | 0 | −0.385137 | − | 2.67869i | 0 | 1.44647 | − | 3.16732i | 0 | −0.456422 | + | 3.17448i | 0 | ||||||||||
209.1 | 0 | −0.369168 | − | 2.56762i | 0 | −3.43820 | − | 1.00955i | 0 | 2.84293 | − | 3.28091i | 0 | −3.57792 | + | 1.05057i | 0 | ||||||||||
209.2 | 0 | −0.0226391 | − | 0.157458i | 0 | 0.0359441 | + | 0.0105542i | 0 | −2.08148 | + | 2.40216i | 0 | 2.85420 | − | 0.838068i | 0 | ||||||||||
209.3 | 0 | 0.152362 | + | 1.05970i | 0 | 3.40226 | + | 0.998994i | 0 | 2.50777 | − | 2.89412i | 0 | 1.77873 | − | 0.522282i | 0 | ||||||||||
225.1 | 0 | −1.32137 | + | 1.52494i | 0 | 0.0973464 | − | 0.677059i | 0 | 0.899416 | + | 1.96945i | 0 | −0.152486 | − | 1.06057i | 0 | ||||||||||
225.2 | 0 | 0.946512 | − | 1.09233i | 0 | 0.287791 | − | 2.00163i | 0 | −2.03440 | − | 4.45471i | 0 | 0.129638 | + | 0.901650i | 0 | ||||||||||
225.3 | 0 | 1.63153 | − | 1.88288i | 0 | −0.385137 | + | 2.67869i | 0 | 1.44647 | + | 3.16732i | 0 | −0.456422 | − | 3.17448i | 0 | ||||||||||
257.1 | 0 | −3.00781 | + | 0.883172i | 0 | −2.52491 | − | 1.62266i | 0 | −0.101654 | − | 0.707018i | 0 | 5.74315 | − | 3.69090i | 0 | ||||||||||
257.2 | 0 | −0.817468 | + | 0.240030i | 0 | 2.57662 | + | 1.65589i | 0 | 0.181823 | + | 1.26461i | 0 | −1.91312 | + | 1.22949i | 0 | ||||||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 368.2.m.e | 30 | |
4.b | odd | 2 | 1 | 184.2.i.b | ✓ | 30 | |
23.c | even | 11 | 1 | inner | 368.2.m.e | 30 | |
23.c | even | 11 | 1 | 8464.2.a.cg | 15 | ||
23.d | odd | 22 | 1 | 8464.2.a.ch | 15 | ||
92.g | odd | 22 | 1 | 184.2.i.b | ✓ | 30 | |
92.g | odd | 22 | 1 | 4232.2.a.bb | 15 | ||
92.h | even | 22 | 1 | 4232.2.a.ba | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
184.2.i.b | ✓ | 30 | 4.b | odd | 2 | 1 | |
184.2.i.b | ✓ | 30 | 92.g | odd | 22 | 1 | |
368.2.m.e | 30 | 1.a | even | 1 | 1 | trivial | |
368.2.m.e | 30 | 23.c | even | 11 | 1 | inner | |
4232.2.a.ba | 15 | 92.h | even | 22 | 1 | ||
4232.2.a.bb | 15 | 92.g | odd | 22 | 1 | ||
8464.2.a.cg | 15 | 23.c | even | 11 | 1 | ||
8464.2.a.ch | 15 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + 2 T_{3}^{29} - 4 T_{3}^{28} - 17 T_{3}^{27} - 21 T_{3}^{26} + 99 T_{3}^{25} + 379 T_{3}^{24} + \cdots + 175561 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\).