# Properties

 Label 429.2.bd.b Level $429$ Weight $2$ Character orbit 429.bd 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.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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 28q^{3} - 4q^{5} - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 28q^{3} - 4q^{5} - 28q^{9} - 4q^{11} - 24q^{14} - 8q^{15} + 4q^{16} + 80q^{20} - 10q^{22} - 12q^{23} + 40q^{26} - 56q^{27} - 12q^{31} - 8q^{33} - 16q^{34} - 12q^{37} - 12q^{42} + 24q^{44} - 4q^{45} - 40q^{47} - 4q^{48} + 24q^{49} - 88q^{53} - 18q^{55} + 72q^{56} + 40q^{58} + 8q^{59} + 52q^{60} - 20q^{66} - 4q^{67} - 12q^{69} + 40q^{70} - 24q^{71} + 12q^{75} - 4q^{78} - 188q^{80} - 28q^{81} - 12q^{82} + 32q^{86} + 18q^{88} - 16q^{89} + 28q^{91} - 96q^{92} - 12q^{93} - 28q^{97} - 4q^{99} + O(q^{100})$$

## 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 −2.44393 0.654849i 0.500000 + 0.866025i 3.81192 + 2.20081i −0.748465 0.748465i −0.654849 2.44393i −0.270535 + 0.0724897i −4.29670 4.29670i −0.500000 + 0.866025i 1.33906 + 2.31933i
76.2 −2.02512 0.542629i 0.500000 + 0.866025i 2.07462 + 1.19778i 1.41467 + 1.41467i −0.542629 2.02512i −0.256787 + 0.0688060i −0.586415 0.586415i −0.500000 + 0.866025i −2.09724 3.63252i
76.3 −1.77380 0.475288i 0.500000 + 0.866025i 1.18842 + 0.686133i −0.611710 0.611710i −0.475288 1.77380i −4.35587 + 1.16715i 0.815121 + 0.815121i −0.500000 + 0.866025i 0.794312 + 1.37579i
76.4 −1.69511 0.454204i 0.500000 + 0.866025i 0.935048 + 0.539850i 2.20359 + 2.20359i −0.454204 1.69511i 4.59665 1.23167i 1.14201 + 1.14201i −0.500000 + 0.866025i −2.73445 4.73621i
76.5 −1.19562 0.320366i 0.500000 + 0.866025i −0.405173 0.233927i −2.87074 2.87074i −0.320366 1.19562i 2.15836 0.578331i 2.16000 + 2.16000i −0.500000 + 0.866025i 2.51263 + 4.35201i
76.6 −0.391582 0.104924i 0.500000 + 0.866025i −1.58972 0.917827i 2.00961 + 2.00961i −0.104924 0.391582i −3.95482 + 1.05969i 1.09952 + 1.09952i −0.500000 + 0.866025i −0.576070 0.997782i
76.7 −0.225205 0.0603434i 0.500000 + 0.866025i −1.68498 0.972821i −1.03093 1.03093i −0.0603434 0.225205i −1.55041 + 0.415430i 0.650483 + 0.650483i −0.500000 + 0.866025i 0.169960 + 0.294380i
76.8 0.225205 + 0.0603434i 0.500000 + 0.866025i −1.68498 0.972821i −1.03093 1.03093i 0.0603434 + 0.225205i 1.55041 0.415430i −0.650483 0.650483i −0.500000 + 0.866025i −0.169960 0.294380i
76.9 0.391582 + 0.104924i 0.500000 + 0.866025i −1.58972 0.917827i 2.00961 + 2.00961i 0.104924 + 0.391582i 3.95482 1.05969i −1.09952 1.09952i −0.500000 + 0.866025i 0.576070 + 0.997782i
76.10 1.19562 + 0.320366i 0.500000 + 0.866025i −0.405173 0.233927i −2.87074 2.87074i 0.320366 + 1.19562i −2.15836 + 0.578331i −2.16000 2.16000i −0.500000 + 0.866025i −2.51263 4.35201i
76.11 1.69511 + 0.454204i 0.500000 + 0.866025i 0.935048 + 0.539850i 2.20359 + 2.20359i 0.454204 + 1.69511i −4.59665 + 1.23167i −1.14201 1.14201i −0.500000 + 0.866025i 2.73445 + 4.73621i
76.12 1.77380 + 0.475288i 0.500000 + 0.866025i 1.18842 + 0.686133i −0.611710 0.611710i 0.475288 + 1.77380i 4.35587 1.16715i −0.815121 0.815121i −0.500000 + 0.866025i −0.794312 1.37579i
76.13 2.02512 + 0.542629i 0.500000 + 0.866025i 2.07462 + 1.19778i 1.41467 + 1.41467i 0.542629 + 2.02512i 0.256787 0.0688060i 0.586415 + 0.586415i −0.500000 + 0.866025i 2.09724 + 3.63252i
76.14 2.44393 + 0.654849i 0.500000 + 0.866025i 3.81192 + 2.20081i −0.748465 0.748465i 0.654849 + 2.44393i 0.270535 0.0724897i 4.29670 + 4.29670i −0.500000 + 0.866025i −1.33906 2.31933i
175.1 −2.44393 + 0.654849i 0.500000 0.866025i 3.81192 2.20081i −0.748465 + 0.748465i −0.654849 + 2.44393i −0.270535 0.0724897i −4.29670 + 4.29670i −0.500000 0.866025i 1.33906 2.31933i
175.2 −2.02512 + 0.542629i 0.500000 0.866025i 2.07462 1.19778i 1.41467 1.41467i −0.542629 + 2.02512i −0.256787 0.0688060i −0.586415 + 0.586415i −0.500000 0.866025i −2.09724 + 3.63252i
175.3 −1.77380 + 0.475288i 0.500000 0.866025i 1.18842 0.686133i −0.611710 + 0.611710i −0.475288 + 1.77380i −4.35587 1.16715i 0.815121 0.815121i −0.500000 0.866025i 0.794312 1.37579i
175.4 −1.69511 + 0.454204i 0.500000 0.866025i 0.935048 0.539850i 2.20359 2.20359i −0.454204 + 1.69511i 4.59665 + 1.23167i 1.14201 1.14201i −0.500000 0.866025i −2.73445 + 4.73621i
175.5 −1.19562 + 0.320366i 0.500000 0.866025i −0.405173 + 0.233927i −2.87074 + 2.87074i −0.320366 + 1.19562i 2.15836 + 0.578331i 2.16000 2.16000i −0.500000 0.866025i 2.51263 4.35201i
175.6 −0.391582 + 0.104924i 0.500000 0.866025i −1.58972 + 0.917827i 2.00961 2.00961i −0.104924 + 0.391582i −3.95482 1.05969i 1.09952 1.09952i −0.500000 0.866025i −0.576070 + 0.997782i
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.b 56
11.b odd 2 1 inner 429.2.bd.b 56
13.f odd 12 1 inner 429.2.bd.b 56
143.o even 12 1 inner 429.2.bd.b 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bd.b 56 1.a even 1 1 trivial
429.2.bd.b 56 11.b odd 2 1 inner
429.2.bd.b 56 13.f odd 12 1 inner
429.2.bd.b 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 $$S_{2}^{\mathrm{new}}(429, [\chi])$$.