Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(341,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.341");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.t (of order \(6\), degree \(2\), 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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 315) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
341.1 | − | 2.62557i | 0 | −4.89362 | 0.500000 | + | 0.866025i | 0 | 1.43974 | − | 2.21972i | 7.59741i | 0 | 2.27381 | − | 1.31279i | |||||||||||
341.2 | − | 2.34793i | 0 | −3.51278 | 0.500000 | + | 0.866025i | 0 | 2.22052 | + | 1.43851i | 3.55190i | 0 | 2.03337 | − | 1.17397i | |||||||||||
341.3 | − | 2.21457i | 0 | −2.90433 | 0.500000 | + | 0.866025i | 0 | −0.612213 | + | 2.57395i | 2.00271i | 0 | 1.91788 | − | 1.10729i | |||||||||||
341.4 | − | 1.93560i | 0 | −1.74654 | 0.500000 | + | 0.866025i | 0 | −2.29351 | + | 1.31902i | − | 0.490599i | 0 | 1.67628 | − | 0.967799i | ||||||||||
341.5 | − | 1.23655i | 0 | 0.470942 | 0.500000 | + | 0.866025i | 0 | 2.44060 | − | 1.02150i | − | 3.05545i | 0 | 1.07088 | − | 0.618275i | ||||||||||
341.6 | − | 1.09551i | 0 | 0.799850 | 0.500000 | + | 0.866025i | 0 | −2.64482 | + | 0.0701258i | − | 3.06727i | 0 | 0.948743 | − | 0.547757i | ||||||||||
341.7 | − | 0.524306i | 0 | 1.72510 | 0.500000 | + | 0.866025i | 0 | −1.53835 | − | 2.15255i | − | 1.95309i | 0 | 0.454062 | − | 0.262153i | ||||||||||
341.8 | − | 0.103991i | 0 | 1.98919 | 0.500000 | + | 0.866025i | 0 | −0.107447 | + | 2.64357i | − | 0.414839i | 0 | 0.0900587 | − | 0.0519954i | ||||||||||
341.9 | 0.396951i | 0 | 1.84243 | 0.500000 | + | 0.866025i | 0 | 2.43622 | + | 1.03190i | 1.52526i | 0 | −0.343770 | + | 0.198476i | ||||||||||||
341.10 | 0.465802i | 0 | 1.78303 | 0.500000 | + | 0.866025i | 0 | 1.73784 | − | 1.99497i | 1.76214i | 0 | −0.403396 | + | 0.232901i | ||||||||||||
341.11 | 0.645959i | 0 | 1.58274 | 0.500000 | + | 0.866025i | 0 | −2.64433 | − | 0.0866018i | 2.31430i | 0 | −0.559417 | + | 0.322980i | ||||||||||||
341.12 | 1.57681i | 0 | −0.486323 | 0.500000 | + | 0.866025i | 0 | −2.52230 | − | 0.798755i | 2.38678i | 0 | −1.36556 | + | 0.788404i | ||||||||||||
341.13 | 1.71080i | 0 | −0.926832 | 0.500000 | + | 0.866025i | 0 | 1.44091 | + | 2.21896i | 1.83597i | 0 | −1.48159 | + | 0.855399i | ||||||||||||
341.14 | 2.32447i | 0 | −3.40314 | 0.500000 | + | 0.866025i | 0 | −1.17052 | + | 2.37274i | − | 3.26155i | 0 | −2.01305 | + | 1.16223i | |||||||||||
341.15 | 2.44435i | 0 | −3.97484 | 0.500000 | + | 0.866025i | 0 | 0.510801 | − | 2.59597i | − | 4.82720i | 0 | −2.11687 | + | 1.22217i | |||||||||||
341.16 | 2.51890i | 0 | −4.34486 | 0.500000 | + | 0.866025i | 0 | 1.80687 | − | 1.93267i | − | 5.90648i | 0 | −2.18143 | + | 1.25945i | |||||||||||
521.1 | − | 2.51890i | 0 | −4.34486 | 0.500000 | − | 0.866025i | 0 | 1.80687 | + | 1.93267i | 5.90648i | 0 | −2.18143 | − | 1.25945i | |||||||||||
521.2 | − | 2.44435i | 0 | −3.97484 | 0.500000 | − | 0.866025i | 0 | 0.510801 | + | 2.59597i | 4.82720i | 0 | −2.11687 | − | 1.22217i | |||||||||||
521.3 | − | 2.32447i | 0 | −3.40314 | 0.500000 | − | 0.866025i | 0 | −1.17052 | − | 2.37274i | 3.26155i | 0 | −2.01305 | − | 1.16223i | |||||||||||
521.4 | − | 1.71080i | 0 | −0.926832 | 0.500000 | − | 0.866025i | 0 | 1.44091 | − | 2.21896i | − | 1.83597i | 0 | −1.48159 | − | 0.855399i | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.t.c | 32 | |
3.b | odd | 2 | 1 | 315.2.t.c | ✓ | 32 | |
7.d | odd | 6 | 1 | 945.2.be.c | 32 | ||
9.c | even | 3 | 1 | 315.2.be.c | yes | 32 | |
9.d | odd | 6 | 1 | 945.2.be.c | 32 | ||
21.g | even | 6 | 1 | 315.2.be.c | yes | 32 | |
63.i | even | 6 | 1 | inner | 945.2.t.c | 32 | |
63.t | odd | 6 | 1 | 315.2.t.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.t.c | ✓ | 32 | 3.b | odd | 2 | 1 | |
315.2.t.c | ✓ | 32 | 63.t | odd | 6 | 1 | |
315.2.be.c | yes | 32 | 9.c | even | 3 | 1 | |
315.2.be.c | yes | 32 | 21.g | even | 6 | 1 | |
945.2.t.c | 32 | 1.a | even | 1 | 1 | trivial | |
945.2.t.c | 32 | 63.i | even | 6 | 1 | inner | |
945.2.be.c | 32 | 7.d | odd | 6 | 1 | ||
945.2.be.c | 32 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 48 T_{2}^{30} + 1032 T_{2}^{28} + 13118 T_{2}^{26} + 109596 T_{2}^{24} + 632922 T_{2}^{22} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).