# Properties

 Label 429.2.bb.a Level $429$ Weight $2$ Character orbit 429.bb Analytic conductor $3.426$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bb (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 14q^{3} + 8q^{4} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 14q^{3} + 8q^{4} - 14q^{9} + 40q^{10} - 52q^{12} + 5q^{13} - 6q^{14} + 16q^{16} - 14q^{17} - 36q^{22} + 28q^{23} + 36q^{25} + 6q^{26} - 14q^{27} + 34q^{29} - 10q^{30} + 18q^{35} + 8q^{36} - 18q^{38} + 5q^{39} + 40q^{40} - 6q^{42} + 48q^{43} - 14q^{48} - 14q^{49} - 4q^{51} + 11q^{52} + 10q^{53} + 8q^{55} - 68q^{56} + 10q^{61} - 132q^{62} - 52q^{64} - 42q^{65} + 14q^{66} + 50q^{68} - 2q^{69} + 10q^{74} + 6q^{75} - 88q^{77} + 26q^{78} + 48q^{79} - 14q^{81} - 30q^{82} - 36q^{87} - 10q^{88} - 10q^{90} - 11q^{91} - 126q^{92} - 34q^{94} + 36q^{95} + O(q^{100})$$

## 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.

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
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{56} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.