Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [46,4,Mod(3,46)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 46 = 2 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 46.c (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: | \(2.71408786026\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −0.284630 | − | 1.97964i | −2.19033 | + | 4.79615i | −3.83797 | + | 1.12693i | 13.4347 | − | 15.5044i | 10.1181 | + | 2.97094i | 23.9800 | − | 15.4110i | 3.32332 | + | 7.27706i | −0.524312 | − | 0.605089i | −34.5171 | − | 22.1828i |
3.2 | −0.284630 | − | 1.97964i | −1.97837 | + | 4.33203i | −3.83797 | + | 1.12693i | −7.35113 | + | 8.48366i | 9.13898 | + | 2.68345i | −14.4370 | + | 9.27813i | 3.32332 | + | 7.27706i | 2.82871 | + | 3.26450i | 18.8870 | + | 12.1379i |
3.3 | −0.284630 | − | 1.97964i | 2.88906 | − | 6.32617i | −3.83797 | + | 1.12693i | 3.38697 | − | 3.90877i | −13.3459 | − | 3.91870i | −8.19404 | + | 5.26599i | 3.32332 | + | 7.27706i | −13.9925 | − | 16.1482i | −8.70200 | − | 5.59244i |
9.1 | −1.91899 | + | 0.563465i | −4.45973 | − | 5.14680i | 3.36501 | − | 2.16256i | 1.49104 | + | 10.3704i | 11.4582 | + | 7.36374i | −12.8017 | + | 28.0319i | −5.23889 | + | 6.04600i | −2.75788 | + | 19.1815i | −8.70468 | − | 19.0606i |
9.2 | −1.91899 | + | 0.563465i | −1.12472 | − | 1.29799i | 3.36501 | − | 2.16256i | 0.231505 | + | 1.61016i | 2.88969 | + | 1.85709i | 13.5480 | − | 29.6661i | −5.23889 | + | 6.04600i | 3.42270 | − | 23.8054i | −1.35152 | − | 2.95942i |
9.3 | −1.91899 | + | 0.563465i | 5.41593 | + | 6.25032i | 3.36501 | − | 2.16256i | 0.0474277 | + | 0.329867i | −13.9149 | − | 8.94259i | −3.79922 | + | 8.31912i | −5.23889 | + | 6.04600i | −5.89167 | + | 40.9775i | −0.276882 | − | 0.606286i |
13.1 | −1.30972 | − | 1.51150i | −4.75375 | − | 3.05505i | −0.569259 | + | 3.95929i | −6.73761 | + | 14.7533i | 1.60838 | + | 11.1865i | 22.9137 | + | 6.72806i | 6.73003 | − | 4.32513i | 2.04859 | + | 4.48580i | 31.1240 | − | 9.13883i |
13.2 | −1.30972 | − | 1.51150i | 2.05539 | + | 1.32092i | −0.569259 | + | 3.95929i | 5.37705 | − | 11.7741i | −0.695419 | − | 4.83675i | 20.2076 | + | 5.93348i | 6.73003 | − | 4.32513i | −8.73641 | − | 19.1301i | −24.8390 | + | 7.29338i |
13.3 | −1.30972 | − | 1.51150i | 7.02888 | + | 4.51719i | −0.569259 | + | 3.95929i | −7.76566 | + | 17.0044i | −2.37815 | − | 16.5404i | −9.77224 | − | 2.86939i | 6.73003 | − | 4.32513i | 17.7839 | + | 38.9414i | 35.8730 | − | 10.5333i |
25.1 | 1.68251 | + | 1.08128i | −1.15980 | − | 8.06660i | 1.66166 | + | 3.63853i | −1.44423 | − | 0.424065i | 6.77089 | − | 14.8262i | 22.1855 | − | 25.6034i | −1.13852 | + | 7.91857i | −37.8186 | + | 11.1045i | −1.97140 | − | 2.27512i |
25.2 | 1.68251 | + | 1.08128i | 0.0135219 | + | 0.0940468i | 1.66166 | + | 3.63853i | 11.1631 | + | 3.27779i | −0.0789404 | + | 0.172855i | −6.36931 | + | 7.35057i | −1.13852 | + | 7.91857i | 25.8976 | − | 7.60424i | 15.2378 | + | 17.5854i |
25.3 | 1.68251 | + | 1.08128i | 1.01294 | + | 7.04516i | 1.66166 | + | 3.63853i | −7.67379 | − | 2.25323i | −5.91352 | + | 12.9488i | 4.19793 | − | 4.84467i | −1.13852 | + | 7.91857i | −22.7019 | + | 6.66587i | −10.4748 | − | 12.0886i |
27.1 | 0.830830 | + | 1.81926i | −5.88717 | + | 1.72863i | −2.61944 | + | 3.02300i | −11.4083 | − | 7.33168i | −8.03607 | − | 9.27412i | −0.0110854 | − | 0.0771003i | −7.67594 | − | 2.25386i | 8.95675 | − | 5.75615i | 3.85989 | − | 26.8461i |
27.2 | 0.830830 | + | 1.81926i | −0.858963 | + | 0.252214i | −2.61944 | + | 3.02300i | 9.50458 | + | 6.10822i | −1.17250 | − | 1.35313i | 3.59878 | + | 25.0301i | −7.67594 | − | 2.25386i | −22.0396 | + | 14.1640i | −3.21578 | + | 22.3662i |
27.3 | 0.830830 | + | 1.81926i | 6.99711 | − | 2.05454i | −2.61944 | + | 3.02300i | 2.74437 | + | 1.76370i | 9.55115 | + | 11.0226i | −1.74684 | − | 12.1495i | −7.67594 | − | 2.25386i | 22.0245 | − | 14.1543i | −0.928530 | + | 6.45807i |
29.1 | 0.830830 | − | 1.81926i | −5.88717 | − | 1.72863i | −2.61944 | − | 3.02300i | −11.4083 | + | 7.33168i | −8.03607 | + | 9.27412i | −0.0110854 | + | 0.0771003i | −7.67594 | + | 2.25386i | 8.95675 | + | 5.75615i | 3.85989 | + | 26.8461i |
29.2 | 0.830830 | − | 1.81926i | −0.858963 | − | 0.252214i | −2.61944 | − | 3.02300i | 9.50458 | − | 6.10822i | −1.17250 | + | 1.35313i | 3.59878 | − | 25.0301i | −7.67594 | + | 2.25386i | −22.0396 | − | 14.1640i | −3.21578 | − | 22.3662i |
29.3 | 0.830830 | − | 1.81926i | 6.99711 | + | 2.05454i | −2.61944 | − | 3.02300i | 2.74437 | − | 1.76370i | 9.55115 | − | 11.0226i | −1.74684 | + | 12.1495i | −7.67594 | + | 2.25386i | 22.0245 | + | 14.1543i | −0.928530 | − | 6.45807i |
31.1 | −0.284630 | + | 1.97964i | −2.19033 | − | 4.79615i | −3.83797 | − | 1.12693i | 13.4347 | + | 15.5044i | 10.1181 | − | 2.97094i | 23.9800 | + | 15.4110i | 3.32332 | − | 7.27706i | −0.524312 | + | 0.605089i | −34.5171 | + | 22.1828i |
31.2 | −0.284630 | + | 1.97964i | −1.97837 | − | 4.33203i | −3.83797 | − | 1.12693i | −7.35113 | − | 8.48366i | 9.13898 | − | 2.68345i | −14.4370 | − | 9.27813i | 3.32332 | − | 7.27706i | 2.82871 | − | 3.26450i | 18.8870 | − | 12.1379i |
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 | 46.4.c.a | ✓ | 30 |
23.c | even | 11 | 1 | inner | 46.4.c.a | ✓ | 30 |
23.c | even | 11 | 1 | 1058.4.a.v | 15 | ||
23.d | odd | 22 | 1 | 1058.4.a.w | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
46.4.c.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
46.4.c.a | ✓ | 30 | 23.c | even | 11 | 1 | inner |
1058.4.a.v | 15 | 23.c | even | 11 | 1 | ||
1058.4.a.w | 15 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{30} - 6 T_{3}^{29} + 90 T_{3}^{28} - 357 T_{3}^{27} + 4939 T_{3}^{26} - 81 T_{3}^{25} + \cdots + 18\!\cdots\!61 \)
acting on \(S_{4}^{\mathrm{new}}(46, [\chi])\).