Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,2,Mod(3,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.k (of order \(18\), degree \(6\), 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: | \(0.606863055362\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.35051 | + | 0.419681i | −1.23656 | + | 0.450071i | 1.64774 | − | 1.13356i | −0.503046 | + | 2.85292i | 1.48110 | − | 1.12678i | −2.71118 | + | 1.56530i | −1.74954 | + | 2.22241i | −0.971618 | + | 0.815285i | −0.517948 | − | 4.06400i |
3.2 | −1.32415 | − | 0.496616i | −1.09089 | + | 0.397053i | 1.50675 | + | 1.31519i | 0.615920 | − | 3.49306i | 1.64169 | + | 0.0159973i | 3.53555 | − | 2.04125i | −1.34202 | − | 2.48978i | −1.26573 | + | 1.06208i | −2.55028 | + | 4.31946i |
3.3 | −1.19877 | − | 0.750298i | 2.80443 | − | 1.02073i | 0.874105 | + | 1.79887i | −0.165124 | + | 0.936467i | −4.12772 | − | 0.880539i | −2.67390 | + | 1.54378i | 0.301838 | − | 2.81228i | 4.52481 | − | 3.79677i | 0.900576 | − | 0.998718i |
3.4 | −0.178792 | + | 1.40287i | 1.23656 | − | 0.450071i | −1.93607 | − | 0.501643i | −0.503046 | + | 2.85292i | 0.410302 | + | 1.81520i | 2.71118 | − | 1.56530i | 1.04989 | − | 2.62635i | −0.971618 | + | 0.815285i | −3.91232 | − | 1.21579i |
3.5 | 0.223323 | − | 1.39647i | 0.855656 | − | 0.311433i | −1.90025 | − | 0.623726i | −0.00805719 | + | 0.0456946i | −0.243820 | − | 1.26445i | 1.20959 | − | 0.698356i | −1.29538 | + | 2.51435i | −1.66298 | + | 1.39540i | 0.0620117 | + | 0.0214562i |
3.6 | 0.719007 | + | 1.21780i | 1.09089 | − | 0.397053i | −0.966058 | + | 1.75121i | 0.615920 | − | 3.49306i | 1.26789 | + | 1.04300i | −3.53555 | + | 2.04125i | −2.82722 | + | 0.0826699i | −1.26573 | + | 1.06208i | 4.69669 | − | 1.76147i |
3.7 | 0.947064 | + | 1.05027i | −2.80443 | + | 1.02073i | −0.206140 | + | 1.98935i | −0.165124 | + | 0.936467i | −3.72802 | − | 1.97872i | 2.67390 | − | 1.54378i | −2.28458 | + | 1.66754i | 4.52481 | − | 3.79677i | −1.13993 | + | 0.713469i |
3.8 | 1.33647 | − | 0.462424i | −0.855656 | + | 0.311433i | 1.57233 | − | 1.23604i | −0.00805719 | + | 0.0456946i | −0.999548 | + | 0.811899i | −1.20959 | + | 0.698356i | 1.52980 | − | 2.37901i | −1.66298 | + | 1.39540i | 0.0103621 | + | 0.0647955i |
15.1 | −1.37896 | + | 0.313814i | −0.00846870 | − | 0.0480284i | 1.80304 | − | 0.865472i | 0.579816 | − | 0.486524i | 0.0267499 | + | 0.0635714i | 2.62687 | + | 1.51662i | −2.21472 | + | 1.75927i | 2.81684 | − | 1.02525i | −0.646863 | + | 0.852849i |
15.2 | −1.06726 | − | 0.927872i | 0.361434 | + | 2.04979i | 0.278109 | + | 1.98057i | −2.99965 | + | 2.51700i | 1.51620 | − | 2.52304i | 0.0108543 | + | 0.00626673i | 1.54090 | − | 2.37184i | −1.25194 | + | 0.455671i | 5.53688 | + | 0.0969785i |
15.3 | −0.647187 | + | 1.25744i | 0.547993 | + | 3.10782i | −1.16230 | − | 1.62759i | 1.46633 | − | 1.23040i | −4.26255 | − | 1.32228i | −1.58907 | − | 0.917452i | 2.79882 | − | 0.408157i | −6.53918 | + | 2.38007i | 0.598159 | + | 2.64012i |
15.4 | −0.312491 | + | 1.37926i | −0.547993 | − | 3.10782i | −1.80470 | − | 0.862010i | 1.46633 | − | 1.23040i | 4.45773 | + | 0.215343i | 1.58907 | + | 0.917452i | 1.75289 | − | 2.21977i | −6.53918 | + | 2.38007i | 1.23882 | + | 2.40694i |
15.5 | −0.0238852 | − | 1.41401i | −0.306623 | − | 1.73895i | −1.99886 | + | 0.0675479i | −0.220151 | + | 0.184728i | −2.45157 | + | 0.475104i | 0.588321 | + | 0.339668i | 0.143257 | + | 2.82480i | −0.110838 | + | 0.0403418i | 0.266467 | + | 0.306884i |
15.6 | 0.854626 | + | 1.12677i | 0.00846870 | + | 0.0480284i | −0.539229 | + | 1.92594i | 0.579816 | − | 0.486524i | −0.0468794 | + | 0.0505886i | −2.62687 | − | 1.51662i | −2.63093 | + | 1.03837i | 2.81684 | − | 1.02525i | 1.04373 | + | 0.237525i |
15.7 | 0.927206 | − | 1.06784i | 0.306623 | + | 1.73895i | −0.280577 | − | 1.98022i | −0.220151 | + | 0.184728i | 2.14122 | + | 1.28494i | −0.588321 | − | 0.339668i | −2.37472 | − | 1.53646i | −0.110838 | + | 0.0403418i | −0.00686428 | + | 0.406368i |
15.8 | 1.41400 | − | 0.0247662i | −0.361434 | − | 2.04979i | 1.99877 | − | 0.0700386i | −2.99965 | + | 2.51700i | −0.561832 | − | 2.88945i | −0.0108543 | − | 0.00626673i | 2.82452 | − | 0.148536i | −1.25194 | + | 0.455671i | −4.17916 | + | 3.63332i |
51.1 | −1.35051 | − | 0.419681i | −1.23656 | − | 0.450071i | 1.64774 | + | 1.13356i | −0.503046 | − | 2.85292i | 1.48110 | + | 1.12678i | −2.71118 | − | 1.56530i | −1.74954 | − | 2.22241i | −0.971618 | − | 0.815285i | −0.517948 | + | 4.06400i |
51.2 | −1.32415 | + | 0.496616i | −1.09089 | − | 0.397053i | 1.50675 | − | 1.31519i | 0.615920 | + | 3.49306i | 1.64169 | − | 0.0159973i | 3.53555 | + | 2.04125i | −1.34202 | + | 2.48978i | −1.26573 | − | 1.06208i | −2.55028 | − | 4.31946i |
51.3 | −1.19877 | + | 0.750298i | 2.80443 | + | 1.02073i | 0.874105 | − | 1.79887i | −0.165124 | − | 0.936467i | −4.12772 | + | 0.880539i | −2.67390 | − | 1.54378i | 0.301838 | + | 2.81228i | 4.52481 | + | 3.79677i | 0.900576 | + | 0.998718i |
51.4 | −0.178792 | − | 1.40287i | 1.23656 | + | 0.450071i | −1.93607 | + | 0.501643i | −0.503046 | − | 2.85292i | 0.410302 | − | 1.81520i | 2.71118 | + | 1.56530i | 1.04989 | + | 2.62635i | −0.971618 | − | 0.815285i | −3.91232 | + | 1.21579i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.f | odd | 18 | 1 | inner |
76.k | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.2.k.a | ✓ | 48 |
3.b | odd | 2 | 1 | 684.2.cf.a | 48 | ||
4.b | odd | 2 | 1 | inner | 76.2.k.a | ✓ | 48 |
12.b | even | 2 | 1 | 684.2.cf.a | 48 | ||
19.f | odd | 18 | 1 | inner | 76.2.k.a | ✓ | 48 |
57.j | even | 18 | 1 | 684.2.cf.a | 48 | ||
76.k | even | 18 | 1 | inner | 76.2.k.a | ✓ | 48 |
228.u | odd | 18 | 1 | 684.2.cf.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.2.k.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
76.2.k.a | ✓ | 48 | 4.b | odd | 2 | 1 | inner |
76.2.k.a | ✓ | 48 | 19.f | odd | 18 | 1 | inner |
76.2.k.a | ✓ | 48 | 76.k | even | 18 | 1 | inner |
684.2.cf.a | 48 | 3.b | odd | 2 | 1 | ||
684.2.cf.a | 48 | 12.b | even | 2 | 1 | ||
684.2.cf.a | 48 | 57.j | even | 18 | 1 | ||
684.2.cf.a | 48 | 228.u | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).