# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bd (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 28q^{3} + 4q^{5} - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 28q^{3} + 4q^{5} - 28q^{9} - 12q^{11} + 24q^{14} - 8q^{15} + 68q^{16} - 8q^{20} - 10q^{22} - 12q^{23} - 48q^{26} + 56q^{27} + 4q^{31} + 12q^{33} + 32q^{34} + 36q^{37} - 12q^{42} + 24q^{44} + 4q^{45} - 8q^{47} + 68q^{48} - 24q^{49} - 56q^{53} - 2q^{55} + 72q^{56} - 88q^{58} + 56q^{59} - 44q^{60} + 20q^{66} + 92q^{67} + 12q^{69} - 216q^{70} + 72q^{71} - 12q^{75} - 12q^{78} + 44q^{80} - 28q^{81} - 156q^{82} - 120q^{86} - 6q^{88} - 56q^{89} + 92q^{91} + 32q^{92} - 20q^{93} - 44q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1 −2.55489 0.684581i −0.500000 0.866025i 4.32676 + 2.49806i −2.81573 2.81573i 0.684581 + 2.55489i −2.71459 + 0.727372i −5.60366 5.60366i −0.500000 + 0.866025i 5.26630 + 9.12149i
76.2 −2.52014 0.675270i −0.500000 0.866025i 4.16307 + 2.40355i 0.961072 + 0.961072i 0.675270 + 2.52014i 1.60148 0.429116i −5.17874 5.17874i −0.500000 + 0.866025i −1.77305 3.07102i
76.3 −2.00245 0.536555i −0.500000 0.866025i 1.98987 + 1.14885i 1.79777 + 1.79777i 0.536555 + 2.00245i −3.46337 + 0.928006i −0.436407 0.436407i −0.500000 + 0.866025i −2.63535 4.56456i
76.4 −1.43745 0.385163i −0.500000 0.866025i 0.185859 + 0.107306i −1.55279 1.55279i 0.385163 + 1.43745i −2.05994 + 0.551959i 1.87874 + 1.87874i −0.500000 + 0.866025i 1.63398 + 2.83014i
76.5 −1.42470 0.381747i −0.500000 0.866025i 0.151984 + 0.0877479i −0.971227 0.971227i 0.381747 + 1.42470i 4.17142 1.11773i 1.90287 + 1.90287i −0.500000 + 0.866025i 1.01294 + 1.75447i
76.6 −0.567668 0.152106i −0.500000 0.866025i −1.43294 0.827308i 0.319197 + 0.319197i 0.152106 + 0.567668i 2.56095 0.686204i 1.51872 + 1.51872i −0.500000 + 0.866025i −0.132646 0.229750i
76.7 −0.390674 0.104681i −0.500000 0.866025i −1.59038 0.918208i 1.89569 + 1.89569i 0.104681 + 0.390674i −1.61789 + 0.433512i 1.09719 + 1.09719i −0.500000 + 0.866025i −0.542154 0.939038i
76.8 0.390674 + 0.104681i −0.500000 0.866025i −1.59038 0.918208i 1.89569 + 1.89569i −0.104681 0.390674i 1.61789 0.433512i −1.09719 1.09719i −0.500000 + 0.866025i 0.542154 + 0.939038i
76.9 0.567668 + 0.152106i −0.500000 0.866025i −1.43294 0.827308i 0.319197 + 0.319197i −0.152106 0.567668i −2.56095 + 0.686204i −1.51872 1.51872i −0.500000 + 0.866025i 0.132646 + 0.229750i
76.10 1.42470 + 0.381747i −0.500000 0.866025i 0.151984 + 0.0877479i −0.971227 0.971227i −0.381747 1.42470i −4.17142 + 1.11773i −1.90287 1.90287i −0.500000 + 0.866025i −1.01294 1.75447i
76.11 1.43745 + 0.385163i −0.500000 0.866025i 0.185859 + 0.107306i −1.55279 1.55279i −0.385163 1.43745i 2.05994 0.551959i −1.87874 1.87874i −0.500000 + 0.866025i −1.63398 2.83014i
76.12 2.00245 + 0.536555i −0.500000 0.866025i 1.98987 + 1.14885i 1.79777 + 1.79777i −0.536555 2.00245i 3.46337 0.928006i 0.436407 + 0.436407i −0.500000 + 0.866025i 2.63535 + 4.56456i
76.13 2.52014 + 0.675270i −0.500000 0.866025i 4.16307 + 2.40355i 0.961072 + 0.961072i −0.675270 2.52014i −1.60148 + 0.429116i 5.17874 + 5.17874i −0.500000 + 0.866025i 1.77305 + 3.07102i
76.14 2.55489 + 0.684581i −0.500000 0.866025i 4.32676 + 2.49806i −2.81573 2.81573i −0.684581 2.55489i 2.71459 0.727372i 5.60366 + 5.60366i −0.500000 + 0.866025i −5.26630 9.12149i
175.1 −2.55489 + 0.684581i −0.500000 + 0.866025i 4.32676 2.49806i −2.81573 + 2.81573i 0.684581 2.55489i −2.71459 0.727372i −5.60366 + 5.60366i −0.500000 0.866025i 5.26630 9.12149i
175.2 −2.52014 + 0.675270i −0.500000 + 0.866025i 4.16307 2.40355i 0.961072 0.961072i 0.675270 2.52014i 1.60148 + 0.429116i −5.17874 + 5.17874i −0.500000 0.866025i −1.77305 + 3.07102i
175.3 −2.00245 + 0.536555i −0.500000 + 0.866025i 1.98987 1.14885i 1.79777 1.79777i 0.536555 2.00245i −3.46337 0.928006i −0.436407 + 0.436407i −0.500000 0.866025i −2.63535 + 4.56456i
175.4 −1.43745 + 0.385163i −0.500000 + 0.866025i 0.185859 0.107306i −1.55279 + 1.55279i 0.385163 1.43745i −2.05994 0.551959i 1.87874 1.87874i −0.500000 0.866025i 1.63398 2.83014i
175.5 −1.42470 + 0.381747i −0.500000 + 0.866025i 0.151984 0.0877479i −0.971227 + 0.971227i 0.381747 1.42470i 4.17142 + 1.11773i 1.90287 1.90287i −0.500000 0.866025i 1.01294 1.75447i
175.6 −0.567668 + 0.152106i −0.500000 + 0.866025i −1.43294 + 0.827308i 0.319197 0.319197i 0.152106 0.567668i 2.56095 + 0.686204i 1.51872 1.51872i −0.500000 0.866025i −0.132646 + 0.229750i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 340.14 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.b odd 2 1 inner
13.f odd 12 1 inner
143.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bd.a 56
11.b odd 2 1 inner 429.2.bd.a 56
13.f odd 12 1 inner 429.2.bd.a 56
143.o even 12 1 inner 429.2.bd.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bd.a 56 1.a even 1 1 trivial
429.2.bd.a 56 11.b odd 2 1 inner
429.2.bd.a 56 13.f odd 12 1 inner
429.2.bd.a 56 143.o even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{56} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.