Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [345,2,Mod(16,345)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("345.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 345 = 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 345.m (of order \(11\), degree \(10\), 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: | \(2.75483886973\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −1.52611 | + | 1.76122i | −0.841254 | + | 0.540641i | −0.488270 | − | 3.39599i | −0.415415 | − | 0.909632i | 0.331655 | − | 2.30671i | 1.12603 | − | 0.330631i | 2.80528 | + | 1.80285i | 0.415415 | − | 0.909632i | 2.23603 | + | 0.656559i |
16.2 | −0.758566 | + | 0.875431i | −0.841254 | + | 0.540641i | 0.0936713 | + | 0.651498i | −0.415415 | − | 0.909632i | 0.164852 | − | 1.14657i | −1.51957 | + | 0.446185i | −2.59035 | − | 1.66472i | 0.415415 | − | 0.909632i | 1.11144 | + | 0.326348i |
16.3 | 0.996473 | − | 1.14999i | −0.841254 | + | 0.540641i | −0.0448918 | − | 0.312230i | −0.415415 | − | 0.909632i | −0.216554 | + | 1.50617i | 2.33660 | − | 0.686088i | 2.15640 | + | 1.38584i | 0.415415 | − | 0.909632i | −1.46002 | − | 0.428700i |
31.1 | −0.353423 | + | 2.45811i | −0.415415 | − | 0.909632i | −3.99842 | − | 1.17404i | 0.654861 | + | 0.755750i | 2.38279 | − | 0.699652i | 2.47922 | + | 1.59330i | 2.23579 | − | 4.89569i | −0.654861 | + | 0.755750i | −2.08916 | + | 1.34262i |
31.2 | −0.139065 | + | 0.967216i | −0.415415 | − | 0.909632i | 1.00282 | + | 0.294454i | 0.654861 | + | 0.755750i | 0.937580 | − | 0.275298i | 1.72023 | + | 1.10553i | −1.23611 | + | 2.70671i | −0.654861 | + | 0.755750i | −0.822041 | + | 0.528294i |
31.3 | 0.101148 | − | 0.703501i | −0.415415 | − | 0.909632i | 1.43430 | + | 0.421149i | 0.654861 | + | 0.755750i | −0.681946 | + | 0.200237i | −3.66580 | − | 2.35586i | 1.03186 | − | 2.25945i | −0.654861 | + | 0.755750i | 0.597909 | − | 0.384253i |
121.1 | −0.883723 | + | 1.93508i | 0.959493 | + | 0.281733i | −1.65386 | − | 1.90866i | −0.841254 | + | 0.540641i | −1.39310 | + | 1.60773i | −0.350847 | + | 2.44020i | 1.07266 | − | 0.314961i | 0.841254 | + | 0.540641i | −0.302750 | − | 2.10567i |
121.2 | 0.0328990 | − | 0.0720388i | 0.959493 | + | 0.281733i | 1.30561 | + | 1.50676i | −0.841254 | + | 0.540641i | 0.0518621 | − | 0.0598520i | −0.274339 | + | 1.90807i | 0.303474 | − | 0.0891079i | 0.841254 | + | 0.540641i | 0.0112707 | + | 0.0783895i |
121.3 | 0.597726 | − | 1.30884i | 0.959493 | + | 0.281733i | −0.0460597 | − | 0.0531557i | −0.841254 | + | 0.540641i | 0.942257 | − | 1.08742i | 0.462869 | − | 3.21932i | 2.66406 | − | 0.782239i | 0.841254 | + | 0.540641i | 0.204772 | + | 1.42422i |
151.1 | −1.52611 | − | 1.76122i | −0.841254 | − | 0.540641i | −0.488270 | + | 3.39599i | −0.415415 | + | 0.909632i | 0.331655 | + | 2.30671i | 1.12603 | + | 0.330631i | 2.80528 | − | 1.80285i | 0.415415 | + | 0.909632i | 2.23603 | − | 0.656559i |
151.2 | −0.758566 | − | 0.875431i | −0.841254 | − | 0.540641i | 0.0936713 | − | 0.651498i | −0.415415 | + | 0.909632i | 0.164852 | + | 1.14657i | −1.51957 | − | 0.446185i | −2.59035 | + | 1.66472i | 0.415415 | + | 0.909632i | 1.11144 | − | 0.326348i |
151.3 | 0.996473 | + | 1.14999i | −0.841254 | − | 0.540641i | −0.0448918 | + | 0.312230i | −0.415415 | + | 0.909632i | −0.216554 | − | 1.50617i | 2.33660 | + | 0.686088i | 2.15640 | − | 1.38584i | 0.415415 | + | 0.909632i | −1.46002 | + | 0.428700i |
196.1 | −2.00240 | + | 1.28686i | 0.142315 | − | 0.989821i | 1.52275 | − | 3.33435i | 0.959493 | − | 0.281733i | 0.988793 | + | 2.16515i | 2.34831 | + | 2.71009i | 0.564215 | + | 3.92420i | −0.959493 | − | 0.281733i | −1.55873 | + | 1.79887i |
196.2 | −0.766261 | + | 0.492446i | 0.142315 | − | 0.989821i | −0.486177 | + | 1.06458i | 0.959493 | − | 0.281733i | 0.378383 | + | 0.828544i | −1.59175 | − | 1.83698i | −0.410966 | − | 2.85833i | −0.959493 | − | 0.281733i | −0.596484 | + | 0.688379i |
196.3 | 1.83027 | − | 1.17624i | 0.142315 | − | 0.989821i | 1.13552 | − | 2.48643i | 0.959493 | − | 0.281733i | −0.903797 | − | 1.97904i | −0.659426 | − | 0.761019i | −0.227095 | − | 1.57948i | −0.959493 | − | 0.281733i | 1.42475 | − | 1.64425i |
211.1 | −0.883723 | − | 1.93508i | 0.959493 | − | 0.281733i | −1.65386 | + | 1.90866i | −0.841254 | − | 0.540641i | −1.39310 | − | 1.60773i | −0.350847 | − | 2.44020i | 1.07266 | + | 0.314961i | 0.841254 | − | 0.540641i | −0.302750 | + | 2.10567i |
211.2 | 0.0328990 | + | 0.0720388i | 0.959493 | − | 0.281733i | 1.30561 | − | 1.50676i | −0.841254 | − | 0.540641i | 0.0518621 | + | 0.0598520i | −0.274339 | − | 1.90807i | 0.303474 | + | 0.0891079i | 0.841254 | − | 0.540641i | 0.0112707 | − | 0.0783895i |
211.3 | 0.597726 | + | 1.30884i | 0.959493 | − | 0.281733i | −0.0460597 | + | 0.0531557i | −0.841254 | − | 0.540641i | 0.942257 | + | 1.08742i | 0.462869 | + | 3.21932i | 2.66406 | + | 0.782239i | 0.841254 | − | 0.540641i | 0.204772 | − | 1.42422i |
256.1 | −0.353423 | − | 2.45811i | −0.415415 | + | 0.909632i | −3.99842 | + | 1.17404i | 0.654861 | − | 0.755750i | 2.38279 | + | 0.699652i | 2.47922 | − | 1.59330i | 2.23579 | + | 4.89569i | −0.654861 | − | 0.755750i | −2.08916 | − | 1.34262i |
256.2 | −0.139065 | − | 0.967216i | −0.415415 | + | 0.909632i | 1.00282 | − | 0.294454i | 0.654861 | − | 0.755750i | 0.937580 | + | 0.275298i | 1.72023 | − | 1.10553i | −1.23611 | − | 2.70671i | −0.654861 | − | 0.755750i | −0.822041 | − | 0.528294i |
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 | 345.2.m.a | ✓ | 30 |
23.c | even | 11 | 1 | inner | 345.2.m.a | ✓ | 30 |
23.c | even | 11 | 1 | 7935.2.a.bp | 15 | ||
23.d | odd | 22 | 1 | 7935.2.a.bq | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
345.2.m.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
345.2.m.a | ✓ | 30 | 23.c | even | 11 | 1 | inner |
7935.2.a.bp | 15 | 23.c | even | 11 | 1 | ||
7935.2.a.bq | 15 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 2 T_{2}^{28} - 22 T_{2}^{27} + 15 T_{2}^{26} - 33 T_{2}^{25} + 371 T_{2}^{24} - 330 T_{2}^{23} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\).