Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [129,3,Mod(22,129)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(129, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("129.22");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 129 = 3 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 129.k (of order \(14\), 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: | \(3.51499541025\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 | −1.41223 | + | 2.93252i | −0.751509 | − | 1.56052i | −4.11131 | − | 5.15543i | 5.65471 | − | 1.29065i | 5.63756 | 9.63817i | 8.23151 | − | 1.87879i | −1.87047 | + | 2.34549i | −4.20087 | + | 18.4052i | ||||
22.2 | −1.16061 | + | 2.41002i | −0.751509 | − | 1.56052i | −1.96725 | − | 2.46685i | −1.61798 | + | 0.369294i | 4.63311 | − | 9.66910i | −2.20305 | + | 0.502833i | −1.87047 | + | 2.34549i | 0.987833 | − | 4.32798i | |||
22.3 | −1.14117 | + | 2.36966i | 0.751509 | + | 1.56052i | −1.81907 | − | 2.28104i | 0.108463 | − | 0.0247561i | −4.55551 | 5.42580i | −2.77557 | + | 0.633505i | −1.87047 | + | 2.34549i | −0.0651116 | + | 0.285272i | ||||
22.4 | −0.826539 | + | 1.71632i | 0.751509 | + | 1.56052i | 0.231354 | + | 0.290109i | −8.96045 | + | 2.04516i | −3.29952 | − | 8.80602i | −8.11800 | + | 1.85288i | −1.87047 | + | 2.34549i | 3.89599 | − | 17.0694i | |||
22.5 | −0.352919 | + | 0.732845i | 0.751509 | + | 1.56052i | 2.08145 | + | 2.61006i | 5.98914 | − | 1.36698i | −1.40884 | 0.499685i | −5.81936 | + | 1.32823i | −1.87047 | + | 2.34549i | −1.11190 | + | 4.87155i | ||||
22.6 | −0.310075 | + | 0.643878i | −0.751509 | − | 1.56052i | 2.17553 | + | 2.72802i | −5.26232 | + | 1.20109i | 1.23781 | 3.62578i | −5.21803 | + | 1.19098i | −1.87047 | + | 2.34549i | 0.858360 | − | 3.76072i | ||||
22.7 | 0.212773 | − | 0.441828i | 0.751509 | + | 1.56052i | 2.34402 | + | 2.93931i | 1.18678 | − | 0.270874i | 0.849383 | − | 1.46169i | 3.70980 | − | 0.846737i | −1.87047 | + | 2.34549i | 0.132835 | − | 0.581986i | |||
22.8 | 0.422999 | − | 0.878366i | −0.751509 | − | 1.56052i | 1.90136 | + | 2.38423i | 5.11914 | − | 1.16841i | −1.68860 | 9.90706i | 6.70038 | − | 1.52932i | −1.87047 | + | 2.34549i | 1.13910 | − | 4.99071i | ||||
22.9 | 0.516112 | − | 1.07172i | −0.751509 | − | 1.56052i | 1.61175 | + | 2.02108i | 0.417208 | − | 0.0952250i | −2.06030 | − | 10.4524i | 7.63663 | − | 1.74301i | −1.87047 | + | 2.34549i | 0.113272 | − | 0.496275i | |||
22.10 | 0.641871 | − | 1.33286i | 0.751509 | + | 1.56052i | 1.12944 | + | 1.41628i | −8.38778 | + | 1.91446i | 2.56233 | 12.6855i | 8.38174 | − | 1.91308i | −1.87047 | + | 2.34549i | −2.83217 | + | 12.4086i | ||||
22.11 | 1.13308 | − | 2.35286i | 0.751509 | + | 1.56052i | −1.75812 | − | 2.20461i | 0.515964 | − | 0.117765i | 4.52321 | − | 8.20311i | 3.00479 | − | 0.685825i | −1.87047 | + | 2.34549i | 0.307541 | − | 1.34743i | |||
22.12 | 1.17892 | − | 2.44805i | 0.751509 | + | 1.56052i | −2.10913 | − | 2.64476i | 6.59618 | − | 1.50554i | 4.70620 | 4.23961i | 1.63502 | − | 0.373182i | −1.87047 | + | 2.34549i | 4.09073 | − | 17.9227i | ||||
22.13 | 1.32508 | − | 2.75156i | −0.751509 | − | 1.56052i | −3.32127 | − | 4.16474i | −8.70094 | + | 1.98593i | −5.28968 | − | 2.32386i | −3.95074 | + | 0.901730i | −1.87047 | + | 2.34549i | −6.06503 | + | 26.5726i | |||
22.14 | 1.46473 | − | 3.04154i | −0.751509 | − | 1.56052i | −4.61158 | − | 5.78274i | 5.64984 | − | 1.28954i | −5.84715 | − | 2.51254i | −11.1783 | + | 2.55137i | −1.87047 | + | 2.34549i | 4.35330 | − | 19.0730i | |||
70.1 | −3.76661 | + | 0.859705i | 1.68862 | + | 0.385418i | 9.84441 | − | 4.74082i | 3.59221 | − | 2.86470i | −6.69174 | − | 7.95256i | −20.9220 | + | 16.6848i | 2.70291 | + | 1.30165i | −11.0677 | + | 13.8784i | |||
70.2 | −3.21984 | + | 0.734907i | −1.68862 | − | 0.385418i | 6.22340 | − | 2.99703i | −2.14187 | + | 1.70808i | 5.72035 | − | 7.45032i | −7.50738 | + | 5.98693i | 2.70291 | + | 1.30165i | 5.64119 | − | 7.07383i | |||
70.3 | −2.69435 | + | 0.614968i | −1.68862 | − | 0.385418i | 3.27747 | − | 1.57835i | 1.94272 | − | 1.54927i | 4.78677 | 9.96218i | 0.782782 | − | 0.624248i | 2.70291 | + | 1.30165i | −4.28162 | + | 5.36899i | ||||
70.4 | −2.24438 | + | 0.512265i | 1.68862 | + | 0.385418i | 1.17095 | − | 0.563898i | 1.48452 | − | 1.18387i | −3.98735 | 5.24089i | 4.86023 | − | 3.87590i | 2.70291 | + | 1.30165i | −2.72538 | + | 3.41751i | ||||
70.5 | −2.16778 | + | 0.494783i | 1.68862 | + | 0.385418i | 0.850604 | − | 0.409629i | −7.32813 | + | 5.84399i | −3.85127 | − | 10.2781i | 5.31246 | − | 4.23655i | 2.70291 | + | 1.30165i | 12.9943 | − | 16.2943i | |||
70.6 | −1.21121 | + | 0.276451i | −1.68862 | − | 0.385418i | −2.21327 | + | 1.06585i | −0.564929 | + | 0.450516i | 2.15183 | − | 4.88887i | 6.27134 | − | 5.00123i | 2.70291 | + | 1.30165i | 0.559703 | − | 0.701845i | |||
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.f | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 129.3.k.a | ✓ | 84 |
3.b | odd | 2 | 1 | 387.3.w.d | 84 | ||
43.f | odd | 14 | 1 | inner | 129.3.k.a | ✓ | 84 |
129.j | even | 14 | 1 | 387.3.w.d | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
129.3.k.a | ✓ | 84 | 1.a | even | 1 | 1 | trivial |
129.3.k.a | ✓ | 84 | 43.f | odd | 14 | 1 | inner |
387.3.w.d | 84 | 3.b | odd | 2 | 1 | ||
387.3.w.d | 84 | 129.j | even | 14 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(129, [\chi])\).