Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [129,2,Mod(5,129)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(129, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("129.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 129 = 3 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 129.n (of order \(42\), degree \(12\), 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: | \(1.03007018607\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −0.608139 | + | 2.66443i | −0.739274 | − | 1.56636i | −4.92742 | − | 2.37292i | −0.999992 | − | 2.54794i | 4.62303 | − | 1.01718i | −0.457900 | + | 0.264368i | 5.91110 | − | 7.41228i | −1.90695 | + | 2.31593i | 7.39694 | − | 1.11491i |
5.2 | −0.560235 | + | 2.45455i | −0.790747 | + | 1.54101i | −3.90902 | − | 1.88248i | 1.36058 | + | 3.46671i | −3.33949 | − | 2.80426i | 2.34383 | − | 1.35321i | 3.67113 | − | 4.60345i | −1.74944 | − | 2.43710i | −9.27146 | + | 1.39745i |
5.3 | −0.437293 | + | 1.91590i | 1.65136 | − | 0.522508i | −1.67753 | − | 0.807855i | 0.258500 | + | 0.658648i | 0.278949 | + | 3.39233i | −0.427704 | + | 0.246935i | −0.169191 | + | 0.212159i | 2.45397 | − | 1.72570i | −1.37495 | + | 0.207240i |
5.4 | −0.306235 | + | 1.34170i | −1.69962 | − | 0.333615i | 0.0955536 | + | 0.0460162i | 0.503180 | + | 1.28208i | 0.968094 | − | 2.17822i | −3.19967 | + | 1.84733i | −1.80710 | + | 2.26604i | 2.77740 | + | 1.13404i | −1.87426 | + | 0.282499i |
5.5 | −0.234198 | + | 1.02609i | 0.968186 | + | 1.43618i | 0.803929 | + | 0.387152i | −0.485964 | − | 1.23822i | −1.70040 | + | 0.657093i | 2.66652 | − | 1.53951i | −1.89795 | + | 2.37995i | −1.12523 | + | 2.78098i | 1.38433 | − | 0.208654i |
5.6 | −0.0804380 | + | 0.352422i | 0.736839 | − | 1.56750i | 1.68421 | + | 0.811071i | −1.30036 | − | 3.31326i | 0.493153 | + | 0.385765i | −0.812758 | + | 0.469246i | −0.872078 | + | 1.09355i | −1.91414 | − | 2.31000i | 1.27226 | − | 0.191763i |
5.7 | 0.0804380 | − | 0.352422i | 0.274201 | − | 1.71021i | 1.68421 | + | 0.811071i | 1.30036 | + | 3.31326i | −0.580659 | − | 0.234200i | −0.812758 | + | 0.469246i | 0.872078 | − | 1.09355i | −2.84963 | − | 0.937883i | 1.27226 | − | 0.191763i |
5.8 | 0.234198 | − | 1.02609i | −1.60898 | + | 0.641230i | 0.803929 | + | 0.387152i | 0.485964 | + | 1.23822i | 0.281139 | + | 1.80113i | 2.66652 | − | 1.53951i | 1.89795 | − | 2.37995i | 2.17765 | − | 2.06346i | 1.38433 | − | 0.208654i |
5.9 | 0.306235 | − | 1.34170i | 1.59222 | + | 0.681783i | 0.0955536 | + | 0.0460162i | −0.503180 | − | 1.28208i | 1.40234 | − | 1.92750i | −3.19967 | + | 1.84733i | 1.80710 | − | 2.26604i | 2.07034 | + | 2.17110i | −1.87426 | + | 0.282499i |
5.10 | 0.437293 | − | 1.91590i | −1.07008 | − | 1.36196i | −1.67753 | − | 0.807855i | −0.258500 | − | 0.658648i | −3.07732 | + | 1.45459i | −0.427704 | + | 0.246935i | 0.169191 | − | 0.212159i | −0.709870 | + | 2.91480i | −1.37495 | + | 0.207240i |
5.11 | 0.560235 | − | 2.45455i | −0.214738 | + | 1.71869i | −3.90902 | − | 1.88248i | −1.36058 | − | 3.46671i | 4.09830 | + | 1.48995i | 2.34383 | − | 1.35321i | −3.67113 | + | 4.60345i | −2.90778 | − | 0.738134i | −9.27146 | + | 1.39745i |
5.12 | 0.608139 | − | 2.66443i | 1.49318 | − | 0.877737i | −4.92742 | − | 2.37292i | 0.999992 | + | 2.54794i | −1.43061 | − | 4.51225i | −0.457900 | + | 0.264368i | −5.91110 | + | 7.41228i | 1.45915 | − | 2.62123i | 7.39694 | − | 1.11491i |
20.1 | −1.60638 | − | 2.01433i | 1.34415 | + | 1.09236i | −1.03205 | + | 4.52170i | −0.0116054 | + | 0.154863i | 0.0411734 | − | 4.46231i | 2.00737 | − | 1.15896i | 6.12349 | − | 2.94892i | 0.613480 | + | 2.93660i | 0.330588 | − | 0.225391i |
20.2 | −1.50765 | − | 1.89054i | −1.59375 | − | 0.678193i | −0.856069 | + | 3.75068i | −0.198318 | + | 2.64637i | 1.12068 | + | 4.03553i | 0.0463085 | − | 0.0267363i | 4.02422 | − | 1.93796i | 2.08011 | + | 2.16175i | 5.30205 | − | 3.61487i |
20.3 | −1.22242 | − | 1.53287i | −1.02339 | + | 1.39738i | −0.410329 | + | 1.79777i | 0.256779 | − | 3.42648i | 3.39302 | − | 0.139460i | −1.51706 | + | 0.875874i | −0.275559 | + | 0.132702i | −0.905335 | − | 2.86013i | −5.56624 | + | 3.79500i |
20.4 | −0.732899 | − | 0.919027i | −0.297099 | + | 1.70638i | 0.137573 | − | 0.602748i | −0.292695 | + | 3.90574i | 1.78595 | − | 0.977562i | −2.04471 | + | 1.18051i | −2.77291 | + | 1.33536i | −2.82346 | − | 1.01393i | 3.80400 | − | 2.59352i |
20.5 | −0.606050 | − | 0.759962i | −0.431951 | − | 1.67732i | 0.234796 | − | 1.02871i | 0.0149759 | − | 0.199839i | −1.01292 | + | 1.34481i | 0.0942007 | − | 0.0543868i | −2.67561 | + | 1.28851i | −2.62684 | + | 1.44905i | −0.160946 | + | 0.109731i |
20.6 | −0.553934 | − | 0.694612i | 1.71160 | + | 0.265385i | 0.269400 | − | 1.18032i | 0.147069 | − | 1.96250i | −0.763774 | − | 1.33590i | −1.81332 | + | 1.04692i | −2.57001 | + | 1.23765i | 2.85914 | + | 0.908467i | −1.44464 | + | 0.984941i |
20.7 | 0.553934 | + | 0.694612i | 1.07418 | − | 1.35872i | 0.269400 | − | 1.18032i | −0.147069 | + | 1.96250i | 1.53881 | − | 0.00650406i | −1.81332 | + | 1.04692i | 2.57001 | − | 1.23765i | −0.692263 | − | 2.91904i | −1.44464 | + | 0.984941i |
20.8 | 0.606050 | + | 0.759962i | 0.824228 | + | 1.52337i | 0.234796 | − | 1.02871i | −0.0149759 | + | 0.199839i | −0.658179 | + | 1.54962i | 0.0942007 | − | 0.0543868i | 2.67561 | − | 1.28851i | −1.64130 | + | 2.51120i | −0.160946 | + | 0.109731i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
43.h | odd | 42 | 1 | inner |
129.n | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 129.2.n.b | ✓ | 144 |
3.b | odd | 2 | 1 | inner | 129.2.n.b | ✓ | 144 |
43.h | odd | 42 | 1 | inner | 129.2.n.b | ✓ | 144 |
129.n | even | 42 | 1 | inner | 129.2.n.b | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
129.2.n.b | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
129.2.n.b | ✓ | 144 | 3.b | odd | 2 | 1 | inner |
129.2.n.b | ✓ | 144 | 43.h | odd | 42 | 1 | inner |
129.2.n.b | ✓ | 144 | 129.n | even | 42 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} + 48 T_{2}^{142} + 1232 T_{2}^{140} + 22460 T_{2}^{138} + 326216 T_{2}^{136} + \cdots + 31\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(129, [\chi])\).