Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [429,2,Mod(25,429)]
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([0, 8, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("429.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 429.bb (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: | \(56\) |
Relative dimension: | \(14\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −1.45613 | − | 2.00419i | 0.309017 | + | 0.951057i | −1.27843 | + | 3.93462i | −2.20631 | + | 3.03673i | 1.45613 | − | 2.00419i | −2.63274 | − | 0.855429i | 5.03515 | − | 1.63602i | −0.809017 | + | 0.587785i | 9.29887 | ||
25.2 | −1.38149 | − | 1.90146i | 0.309017 | + | 0.951057i | −1.08900 | + | 3.35161i | 0.382487 | − | 0.526448i | 1.38149 | − | 1.90146i | 0.0445990 | + | 0.0144911i | 3.40681 | − | 1.10694i | −0.809017 | + | 0.587785i | −1.52942 | ||
25.3 | −1.22454 | − | 1.68544i | 0.309017 | + | 0.951057i | −0.723159 | + | 2.22565i | 1.88056 | − | 2.58836i | 1.22454 | − | 1.68544i | 3.04968 | + | 0.990901i | 0.674037 | − | 0.219008i | −0.809017 | + | 0.587785i | −6.66533 | ||
25.4 | −0.897576 | − | 1.23541i | 0.309017 | + | 0.951057i | −0.102555 | + | 0.315631i | 0.900087 | − | 1.23886i | 0.897576 | − | 1.23541i | −2.72966 | − | 0.886920i | −2.42263 | + | 0.787161i | −0.809017 | + | 0.587785i | −2.33840 | ||
25.5 | −0.817916 | − | 1.12577i | 0.309017 | + | 0.951057i | 0.0196739 | − | 0.0605500i | −1.89008 | + | 2.60147i | 0.817916 | − | 1.12577i | 2.89956 | + | 0.942124i | −2.73109 | + | 0.887385i | −0.809017 | + | 0.587785i | 4.47458 | ||
25.6 | −0.665766 | − | 0.916348i | 0.309017 | + | 0.951057i | 0.221584 | − | 0.681966i | −1.07016 | + | 1.47295i | 0.665766 | − | 0.916348i | −1.70616 | − | 0.554365i | −2.92691 | + | 0.951009i | −0.809017 | + | 0.587785i | 2.06221 | ||
25.7 | −0.150436 | − | 0.207057i | 0.309017 | + | 0.951057i | 0.597792 | − | 1.83982i | 0.694723 | − | 0.956204i | 0.150436 | − | 0.207057i | −2.82596 | − | 0.918210i | −0.957697 | + | 0.311175i | −0.809017 | + | 0.587785i | −0.302500 | ||
25.8 | 0.150436 | + | 0.207057i | 0.309017 | + | 0.951057i | 0.597792 | − | 1.83982i | −0.694723 | + | 0.956204i | −0.150436 | + | 0.207057i | 2.82596 | + | 0.918210i | 0.957697 | − | 0.311175i | −0.809017 | + | 0.587785i | −0.302500 | ||
25.9 | 0.665766 | + | 0.916348i | 0.309017 | + | 0.951057i | 0.221584 | − | 0.681966i | 1.07016 | − | 1.47295i | −0.665766 | + | 0.916348i | 1.70616 | + | 0.554365i | 2.92691 | − | 0.951009i | −0.809017 | + | 0.587785i | 2.06221 | ||
25.10 | 0.817916 | + | 1.12577i | 0.309017 | + | 0.951057i | 0.0196739 | − | 0.0605500i | 1.89008 | − | 2.60147i | −0.817916 | + | 1.12577i | −2.89956 | − | 0.942124i | 2.73109 | − | 0.887385i | −0.809017 | + | 0.587785i | 4.47458 | ||
25.11 | 0.897576 | + | 1.23541i | 0.309017 | + | 0.951057i | −0.102555 | + | 0.315631i | −0.900087 | + | 1.23886i | −0.897576 | + | 1.23541i | 2.72966 | + | 0.886920i | 2.42263 | − | 0.787161i | −0.809017 | + | 0.587785i | −2.33840 | ||
25.12 | 1.22454 | + | 1.68544i | 0.309017 | + | 0.951057i | −0.723159 | + | 2.22565i | −1.88056 | + | 2.58836i | −1.22454 | + | 1.68544i | −3.04968 | − | 0.990901i | −0.674037 | + | 0.219008i | −0.809017 | + | 0.587785i | −6.66533 | ||
25.13 | 1.38149 | + | 1.90146i | 0.309017 | + | 0.951057i | −1.08900 | + | 3.35161i | −0.382487 | + | 0.526448i | −1.38149 | + | 1.90146i | −0.0445990 | − | 0.0144911i | −3.40681 | + | 1.10694i | −0.809017 | + | 0.587785i | −1.52942 | ||
25.14 | 1.45613 | + | 2.00419i | 0.309017 | + | 0.951057i | −1.27843 | + | 3.93462i | 2.20631 | − | 3.03673i | −1.45613 | + | 2.00419i | 2.63274 | + | 0.855429i | −5.03515 | + | 1.63602i | −0.809017 | + | 0.587785i | 9.29887 | ||
64.1 | −2.53943 | − | 0.825112i | −0.809017 | + | 0.587785i | 4.14988 | + | 3.01506i | −1.45528 | + | 0.472849i | 2.53943 | − | 0.825112i | −0.367708 | + | 0.506107i | −4.91166 | − | 6.76032i | 0.309017 | − | 0.951057i | 4.08574 | ||
64.2 | −2.00472 | − | 0.651372i | −0.809017 | + | 0.587785i | 1.97658 | + | 1.43607i | −2.60228 | + | 0.845530i | 2.00472 | − | 0.651372i | 1.43209 | − | 1.97110i | −0.549095 | − | 0.755765i | 0.309017 | − | 0.951057i | 5.76758 | ||
64.3 | −1.81881 | − | 0.590967i | −0.809017 | + | 0.587785i | 1.34080 | + | 0.974145i | 2.40393 | − | 0.781084i | 1.81881 | − | 0.590967i | 2.67159 | − | 3.67713i | 0.385208 | + | 0.530193i | 0.309017 | − | 0.951057i | −4.83389 | ||
64.4 | −1.36111 | − | 0.442251i | −0.809017 | + | 0.587785i | 0.0389958 | + | 0.0283321i | −1.28374 | + | 0.417113i | 1.36111 | − | 0.442251i | −2.73757 | + | 3.76794i | 1.64187 | + | 2.25985i | 0.309017 | − | 0.951057i | 1.93178 | ||
64.5 | −1.23441 | − | 0.401083i | −0.809017 | + | 0.587785i | −0.255144 | − | 0.185373i | 3.32808 | − | 1.08136i | 1.23441 | − | 0.401083i | −1.47987 | + | 2.03686i | 1.76641 | + | 2.43126i | 0.309017 | − | 0.951057i | −4.54192 | ||
64.6 | −0.605812 | − | 0.196840i | −0.809017 | + | 0.587785i | −1.28977 | − | 0.937075i | −4.01620 | + | 1.30494i | 0.605812 | − | 0.196840i | 0.707733 | − | 0.974111i | 1.34573 | + | 1.85224i | 0.309017 | − | 0.951057i | 2.68992 | ||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
13.b | even | 2 | 1 | inner |
143.n | 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.bb.a | ✓ | 56 |
11.c | even | 5 | 1 | inner | 429.2.bb.a | ✓ | 56 |
13.b | even | 2 | 1 | inner | 429.2.bb.a | ✓ | 56 |
143.n | even | 10 | 1 | inner | 429.2.bb.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
429.2.bb.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
429.2.bb.a | ✓ | 56 | 11.c | even | 5 | 1 | inner |
429.2.bb.a | ✓ | 56 | 13.b | even | 2 | 1 | inner |
429.2.bb.a | ✓ | 56 | 143.n | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 18 T_{2}^{54} + 213 T_{2}^{52} - 2056 T_{2}^{50} + 17785 T_{2}^{48} - 125036 T_{2}^{46} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).