Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,4,Mod(59,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.59");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 105.p (of order \(6\), degree \(2\), 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: | \(6.19520055060\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 | −2.65960 | − | 4.60657i | 1.84211 | + | 4.85867i | −10.1470 | + | 17.5751i | 0.975062 | − | 11.1377i | 17.4825 | − | 21.4079i | 7.48937 | − | 16.9384i | 65.3942 | −20.2133 | + | 17.9004i | −53.9000 | + | 25.1303i | ||
59.2 | −2.65960 | − | 4.60657i | 5.12878 | − | 0.834020i | −10.1470 | + | 17.5751i | 10.1331 | + | 4.72444i | −17.4825 | − | 21.4079i | −7.48937 | + | 16.9384i | 65.3942 | 25.6088 | − | 8.55501i | −5.18656 | − | 59.2440i | ||
59.3 | −2.57122 | − | 4.45348i | −4.78955 | + | 2.01500i | −9.22230 | + | 15.9735i | −9.15758 | + | 6.41394i | 21.2887 | + | 16.1492i | −17.8760 | − | 4.84251i | 53.7107 | 18.8796 | − | 19.3018i | 52.1105 | + | 24.2914i | ||
59.4 | −2.57122 | − | 4.45348i | −0.649738 | − | 5.15537i | −9.22230 | + | 15.9735i | −10.1334 | + | 4.72373i | −21.2887 | + | 16.1492i | 17.8760 | + | 4.84251i | 53.7107 | −26.1557 | + | 6.69928i | 47.0922 | + | 32.9833i | ||
59.5 | −2.26674 | − | 3.92610i | −5.16785 | + | 0.541618i | −6.27618 | + | 10.8707i | 11.1613 | + | 0.652852i | 13.8406 | + | 19.0618i | 18.5202 | + | 0.0273336i | 20.6380 | 26.4133 | − | 5.59799i | −22.7365 | − | 45.3001i | ||
59.6 | −2.26674 | − | 3.92610i | −2.11487 | − | 4.74630i | −6.27618 | + | 10.8707i | 5.01525 | − | 9.99236i | −13.8406 | + | 19.0618i | −18.5202 | − | 0.0273336i | 20.6380 | −18.0547 | + | 20.0756i | −50.5993 | + | 2.95968i | ||
59.7 | −1.89916 | − | 3.28944i | 4.02025 | + | 3.29205i | −3.21362 | + | 5.56616i | −5.19426 | + | 9.90049i | 3.19390 | − | 19.4765i | 6.82394 | − | 17.2173i | −5.97382 | 5.32484 | + | 26.4697i | 42.4318 | − | 1.71639i | ||
59.8 | −1.89916 | − | 3.28944i | 4.86112 | + | 1.83562i | −3.21362 | + | 5.56616i | −11.1712 | − | 0.451881i | −3.19390 | − | 19.4765i | −6.82394 | + | 17.2173i | −5.97382 | 20.2610 | + | 17.8463i | 19.7295 | + | 37.6052i | ||
59.9 | −1.82858 | − | 3.16720i | −2.45762 | + | 4.57822i | −2.68744 | + | 4.65479i | −6.67931 | − | 8.96587i | 18.9941 | − | 0.587877i | 2.93091 | + | 18.2869i | −9.60048 | −14.9202 | − | 22.5031i | −16.1830 | + | 37.5496i | ||
59.10 | −1.82858 | − | 3.16720i | 2.73604 | − | 4.41747i | −2.68744 | + | 4.65479i | 4.42501 | + | 10.2674i | −18.9941 | − | 0.587877i | −2.93091 | − | 18.2869i | −9.60048 | −12.0281 | − | 24.1728i | 24.4274 | − | 32.7897i | ||
59.11 | −1.76958 | − | 3.06500i | −0.764269 | + | 5.13964i | −2.26281 | + | 3.91930i | 9.87152 | + | 5.24910i | 17.1054 | − | 6.75251i | −18.4952 | − | 0.962729i | −12.2964 | −25.8318 | − | 7.85614i | −1.37994 | − | 39.5449i | ||
59.12 | −1.76958 | − | 3.06500i | 4.06892 | − | 3.23170i | −2.26281 | + | 3.91930i | 0.389908 | − | 11.1735i | −17.1054 | − | 6.75251i | 18.4952 | + | 0.962729i | −12.2964 | 6.11228 | − | 26.2991i | −34.9368 | + | 18.5774i | ||
59.13 | −1.29175 | − | 2.23737i | −4.90178 | − | 1.72412i | 0.662782 | − | 1.14797i | −9.00517 | − | 6.62623i | 2.47436 | + | 13.1942i | 1.81286 | − | 18.4313i | −24.0925 | 21.0548 | + | 16.9025i | −3.19292 | + | 28.7073i | ||
59.14 | −1.29175 | − | 2.23737i | −3.94402 | − | 3.38300i | 0.662782 | − | 1.14797i | 1.23589 | + | 11.1118i | −2.47436 | + | 13.1942i | −1.81286 | + | 18.4313i | −24.0925 | 4.11059 | + | 26.6853i | 23.2648 | − | 17.1188i | ||
59.15 | −0.878214 | − | 1.52111i | 3.00499 | + | 4.23911i | 2.45748 | − | 4.25648i | 11.1091 | − | 1.25980i | 3.80914 | − | 8.29376i | 15.1670 | + | 10.6284i | −22.6842 | −8.94012 | + | 25.4769i | −11.6725 | − | 15.7919i | ||
59.16 | −0.878214 | − | 1.52111i | 5.17367 | + | 0.482839i | 2.45748 | − | 4.25648i | 6.64558 | − | 8.99090i | −3.80914 | − | 8.29376i | −15.1670 | − | 10.6284i | −22.6842 | 26.5337 | + | 4.99610i | −19.5124 | − | 2.21274i | ||
59.17 | −0.850362 | − | 1.47287i | −3.03689 | + | 4.21632i | 2.55377 | − | 4.42326i | −6.34140 | + | 9.20796i | 8.79254 | + | 0.887546i | 17.5269 | − | 5.98407i | −22.2923 | −8.55463 | − | 25.6089i | 18.9546 | + | 1.50996i | ||
59.18 | −0.850362 | − | 1.47287i | 2.13299 | − | 4.73818i | 2.55377 | − | 4.42326i | −11.1450 | + | 0.887834i | −8.79254 | + | 0.887546i | −17.5269 | + | 5.98407i | −22.2923 | −17.9007 | − | 20.2130i | 10.7850 | + | 15.6602i | ||
59.19 | −0.388071 | − | 0.672158i | −5.13489 | + | 0.795580i | 3.69880 | − | 6.40651i | 5.93654 | − | 9.47404i | 2.52746 | + | 3.14272i | −10.2225 | + | 15.4435i | −11.9507 | 25.7341 | − | 8.17043i | −8.67186 | − | 0.313697i | ||
59.20 | −0.388071 | − | 0.672158i | −1.87845 | − | 4.84473i | 3.69880 | − | 6.40651i | 11.1730 | − | 0.404175i | −2.52746 | + | 3.14272i | 10.2225 | − | 15.4435i | −11.9507 | −19.9428 | + | 18.2012i | −4.60760 | − | 7.35320i | ||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
15.d | odd | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
105.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 105.4.p.a | ✓ | 88 |
3.b | odd | 2 | 1 | inner | 105.4.p.a | ✓ | 88 |
5.b | even | 2 | 1 | inner | 105.4.p.a | ✓ | 88 |
7.d | odd | 6 | 1 | inner | 105.4.p.a | ✓ | 88 |
15.d | odd | 2 | 1 | inner | 105.4.p.a | ✓ | 88 |
21.g | even | 6 | 1 | inner | 105.4.p.a | ✓ | 88 |
35.i | odd | 6 | 1 | inner | 105.4.p.a | ✓ | 88 |
105.p | even | 6 | 1 | inner | 105.4.p.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.4.p.a | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
105.4.p.a | ✓ | 88 | 3.b | odd | 2 | 1 | inner |
105.4.p.a | ✓ | 88 | 5.b | even | 2 | 1 | inner |
105.4.p.a | ✓ | 88 | 7.d | odd | 6 | 1 | inner |
105.4.p.a | ✓ | 88 | 15.d | odd | 2 | 1 | inner |
105.4.p.a | ✓ | 88 | 21.g | even | 6 | 1 | inner |
105.4.p.a | ✓ | 88 | 35.i | odd | 6 | 1 | inner |
105.4.p.a | ✓ | 88 | 105.p | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(105, [\chi])\).