Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [429,2,Mod(116,429)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(429, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 9, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("429.116");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 429.y (of order \(10\), degree \(4\), 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: | \(3.42558224671\) |
Analytic rank: | \(0\) |
Dimension: | \(176\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
116.1 | −2.50188 | + | 0.812912i | −0.234485 | + | 1.71611i | 3.98057 | − | 2.89205i | 0.944274 | − | 2.90618i | −0.808388 | − | 4.48411i | −2.69270 | + | 1.95636i | −4.51544 | + | 6.21497i | −2.89003 | − | 0.804801i | 8.03853i | ||
116.2 | −2.50188 | + | 0.812912i | 1.55965 | − | 0.753314i | 3.98057 | − | 2.89205i | 0.944274 | − | 2.90618i | −3.28969 | + | 3.15256i | 2.69270 | − | 1.95636i | −4.51544 | + | 6.21497i | 1.86504 | − | 2.34982i | 8.03853i | ||
116.3 | −2.12303 | + | 0.689813i | −1.71440 | + | 0.246677i | 2.41336 | − | 1.75341i | −0.370527 | + | 1.14036i | 3.46954 | − | 1.70631i | −1.00158 | + | 0.727691i | −1.28990 | + | 1.77539i | 2.87830 | − | 0.845805i | − | 2.67661i | |
116.4 | −2.12303 | + | 0.689813i | −0.295173 | − | 1.70671i | 2.41336 | − | 1.75341i | −0.370527 | + | 1.14036i | 1.80397 | + | 3.41978i | 1.00158 | − | 0.727691i | −1.28990 | + | 1.77539i | −2.82575 | + | 1.00755i | − | 2.67661i | |
116.5 | −2.06588 | + | 0.671246i | −1.48049 | + | 0.898973i | 2.19926 | − | 1.59786i | 0.803773 | − | 2.47376i | 2.45508 | − | 2.85094i | 3.30516 | − | 2.40134i | −0.917289 | + | 1.26254i | 1.38370 | − | 2.66184i | 5.65002i | ||
116.6 | −2.06588 | + | 0.671246i | 0.397478 | − | 1.68583i | 2.19926 | − | 1.59786i | 0.803773 | − | 2.47376i | 0.310462 | + | 3.74952i | −3.30516 | + | 2.40134i | −0.917289 | + | 1.26254i | −2.68402 | − | 1.34016i | 5.65002i | ||
116.7 | −2.03433 | + | 0.660995i | −0.778983 | + | 1.54699i | 2.08356 | − | 1.51379i | −0.766534 | + | 2.35915i | 0.562157 | − | 3.66200i | −2.59152 | + | 1.88285i | −0.723470 | + | 0.995771i | −1.78637 | − | 2.41016i | − | 5.30597i | |
116.8 | −2.03433 | + | 0.660995i | 1.23056 | − | 1.21890i | 2.08356 | − | 1.51379i | −0.766534 | + | 2.35915i | −1.69767 | + | 3.29305i | 2.59152 | − | 1.88285i | −0.723470 | + | 0.995771i | 0.0285456 | − | 2.99986i | − | 5.30597i | |
116.9 | −1.39922 | + | 0.454633i | 0.606125 | + | 1.62253i | 0.133079 | − | 0.0966876i | 0.724806 | − | 2.23072i | −1.58576 | − | 1.99471i | −0.197187 | + | 0.143265i | 1.58728 | − | 2.18470i | −2.26523 | + | 1.96691i | 3.45079i | ||
116.10 | −1.39922 | + | 0.454633i | 1.73042 | + | 0.0750685i | 0.133079 | − | 0.0966876i | 0.724806 | − | 2.23072i | −2.45536 | + | 0.681670i | 0.197187 | − | 0.143265i | 1.58728 | − | 2.18470i | 2.98873 | + | 0.259801i | 3.45079i | ||
116.11 | −1.39675 | + | 0.453832i | −0.128417 | + | 1.72728i | 0.126919 | − | 0.0922120i | −0.121706 | + | 0.374573i | −0.604530 | − | 2.47087i | 2.86530 | − | 2.08177i | 1.59106 | − | 2.18990i | −2.96702 | − | 0.443626i | − | 0.578420i | |
116.12 | −1.39675 | + | 0.453832i | 1.60306 | − | 0.655892i | 0.126919 | − | 0.0922120i | −0.121706 | + | 0.374573i | −1.94141 | + | 1.64364i | −2.86530 | + | 2.08177i | 1.59106 | − | 2.18990i | 2.13961 | − | 2.10287i | − | 0.578420i | |
116.13 | −1.15602 | + | 0.375612i | −1.73161 | − | 0.0391064i | −0.422745 | + | 0.307142i | −1.20121 | + | 3.69693i | 2.01646 | − | 0.605206i | 3.46009 | − | 2.51391i | 1.80225 | − | 2.48058i | 2.99694 | + | 0.135434i | − | 4.72490i | |
116.14 | −1.15602 | + | 0.375612i | −0.572289 | − | 1.63477i | −0.422745 | + | 0.307142i | −1.20121 | + | 3.69693i | 1.27562 | + | 1.67487i | −3.46009 | + | 2.51391i | 1.80225 | − | 2.48058i | −2.34497 | + | 1.87113i | − | 4.72490i | |
116.15 | −0.743149 | + | 0.241464i | −1.46911 | + | 0.917452i | −1.12407 | + | 0.816683i | 0.877069 | − | 2.69934i | 0.870236 | − | 1.03654i | −1.52431 | + | 1.10747i | 1.55673 | − | 2.14266i | 1.31656 | − | 2.69567i | 2.21779i | ||
116.16 | −0.743149 | + | 0.241464i | 0.418569 | − | 1.68071i | −1.12407 | + | 0.816683i | 0.877069 | − | 2.69934i | 0.0947722 | + | 1.35009i | 1.52431 | − | 1.10747i | 1.55673 | − | 2.14266i | −2.64960 | − | 1.40699i | 2.21779i | ||
116.17 | −0.687043 | + | 0.223234i | −1.63884 | − | 0.560531i | −1.19584 | + | 0.868828i | −0.0885211 | + | 0.272440i | 1.25108 | + | 0.0192637i | −1.47481 | + | 1.07151i | 1.47687 | − | 2.03274i | 2.37161 | + | 1.83724i | − | 0.206939i | |
116.18 | −0.687043 | + | 0.223234i | −1.03953 | − | 1.38542i | −1.19584 | + | 0.868828i | −0.0885211 | + | 0.272440i | 1.02347 | + | 0.719784i | 1.47481 | − | 1.07151i | 1.47687 | − | 2.03274i | −0.838768 | + | 2.88036i | − | 0.206939i | |
116.19 | −0.473784 | + | 0.153942i | 0.663585 | + | 1.59989i | −1.41726 | + | 1.02970i | 0.269770 | − | 0.830268i | −0.560687 | − | 0.655850i | −3.14053 | + | 2.28173i | 1.09859 | − | 1.51208i | −2.11931 | + | 2.12333i | 0.434897i | ||
116.20 | −0.473784 | + | 0.153942i | 1.72665 | + | 0.136713i | −1.41726 | + | 1.02970i | 0.269770 | − | 0.830268i | −0.839104 | + | 0.201031i | 3.14053 | − | 2.28173i | 1.09859 | − | 1.51208i | 2.96262 | + | 0.472112i | 0.434897i | ||
See next 80 embeddings (of 176 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
13.b | even | 2 | 1 | inner |
33.f | even | 10 | 1 | inner |
39.d | odd | 2 | 1 | inner |
143.l | odd | 10 | 1 | inner |
429.y | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 429.2.y.b | ✓ | 176 |
3.b | odd | 2 | 1 | inner | 429.2.y.b | ✓ | 176 |
11.d | odd | 10 | 1 | inner | 429.2.y.b | ✓ | 176 |
13.b | even | 2 | 1 | inner | 429.2.y.b | ✓ | 176 |
33.f | even | 10 | 1 | inner | 429.2.y.b | ✓ | 176 |
39.d | odd | 2 | 1 | inner | 429.2.y.b | ✓ | 176 |
143.l | odd | 10 | 1 | inner | 429.2.y.b | ✓ | 176 |
429.y | even | 10 | 1 | inner | 429.2.y.b | ✓ | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
429.2.y.b | ✓ | 176 | 1.a | even | 1 | 1 | trivial |
429.2.y.b | ✓ | 176 | 3.b | odd | 2 | 1 | inner |
429.2.y.b | ✓ | 176 | 11.d | odd | 10 | 1 | inner |
429.2.y.b | ✓ | 176 | 13.b | even | 2 | 1 | inner |
429.2.y.b | ✓ | 176 | 33.f | even | 10 | 1 | inner |
429.2.y.b | ✓ | 176 | 39.d | odd | 2 | 1 | inner |
429.2.y.b | ✓ | 176 | 143.l | odd | 10 | 1 | inner |
429.2.y.b | ✓ | 176 | 429.y | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} - 27 T_{2}^{86} + 434 T_{2}^{84} - 5386 T_{2}^{82} + 57065 T_{2}^{80} - 520748 T_{2}^{78} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).