# Properties

 Label 429.2.m.a Level $429$ Weight $2$ Character orbit 429.m Analytic conductor $3.426$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 28q^{3} + 4q^{5} + 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 28q^{3} + 4q^{5} + 28q^{9} + 4q^{11} + 48q^{14} - 4q^{15} - 52q^{16} - 8q^{20} - 32q^{22} - 4q^{26} - 28q^{27} + 24q^{31} - 4q^{33} + 16q^{34} - 12q^{37} - 48q^{42} - 24q^{44} + 4q^{45} - 8q^{47} + 52q^{48} - 8q^{53} + 48q^{55} - 64q^{58} + 4q^{59} + 8q^{60} + 32q^{66} + 28q^{67} - 4q^{70} + 12q^{71} + 4q^{78} + 56q^{80} + 28q^{81} - 8q^{86} - 104q^{89} - 76q^{91} - 24q^{93} - 8q^{97} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.92734 1.92734i −1.00000 5.42931i −1.59813 + 1.59813i 1.92734 + 1.92734i −1.66792 + 1.66792i 6.60946 6.60946i 1.00000 6.16029
109.2 −1.67111 1.67111i −1.00000 3.58521i 2.12568 2.12568i 1.67111 + 1.67111i −1.37191 + 1.37191i 2.64907 2.64907i 1.00000 −7.10448
109.3 −1.12106 1.12106i −1.00000 0.513571i 0.850971 0.850971i 1.12106 + 1.12106i −0.675060 + 0.675060i −1.66638 + 1.66638i 1.00000 −1.90799
109.4 −1.06339 1.06339i −1.00000 0.261597i 0.340859 0.340859i 1.06339 + 1.06339i −0.593196 + 0.593196i −1.84860 + 1.84860i 1.00000 −0.724932
109.5 −0.735263 0.735263i −1.00000 0.918776i −2.75658 + 2.75658i 0.735263 + 0.735263i −0.0552347 + 0.0552347i −2.14607 + 2.14607i 1.00000 4.05362
109.6 −0.200229 0.200229i −1.00000 1.91982i −0.906673 + 0.906673i 0.200229 + 0.200229i 2.29691 2.29691i −0.784861 + 0.784861i 1.00000 0.363084
109.7 −0.156364 0.156364i −1.00000 1.95110i 2.94387 2.94387i 0.156364 + 0.156364i 3.04129 3.04129i −0.617812 + 0.617812i 1.00000 −0.920633
109.8 0.156364 + 0.156364i −1.00000 1.95110i 2.94387 2.94387i −0.156364 0.156364i −3.04129 + 3.04129i 0.617812 0.617812i 1.00000 0.920633
109.9 0.200229 + 0.200229i −1.00000 1.91982i −0.906673 + 0.906673i −0.200229 0.200229i −2.29691 + 2.29691i 0.784861 0.784861i 1.00000 −0.363084
109.10 0.735263 + 0.735263i −1.00000 0.918776i −2.75658 + 2.75658i −0.735263 0.735263i 0.0552347 0.0552347i 2.14607 2.14607i 1.00000 −4.05362
109.11 1.06339 + 1.06339i −1.00000 0.261597i 0.340859 0.340859i −1.06339 1.06339i 0.593196 0.593196i 1.84860 1.84860i 1.00000 0.724932
109.12 1.12106 + 1.12106i −1.00000 0.513571i 0.850971 0.850971i −1.12106 1.12106i 0.675060 0.675060i 1.66638 1.66638i 1.00000 1.90799
109.13 1.67111 + 1.67111i −1.00000 3.58521i 2.12568 2.12568i −1.67111 1.67111i 1.37191 1.37191i −2.64907 + 2.64907i 1.00000 7.10448
109.14 1.92734 + 1.92734i −1.00000 5.42931i −1.59813 + 1.59813i −1.92734 1.92734i 1.66792 1.66792i −6.60946 + 6.60946i 1.00000 −6.16029
307.1 −1.92734 + 1.92734i −1.00000 5.42931i −1.59813 1.59813i 1.92734 1.92734i −1.66792 1.66792i 6.60946 + 6.60946i 1.00000 6.16029
307.2 −1.67111 + 1.67111i −1.00000 3.58521i 2.12568 + 2.12568i 1.67111 1.67111i −1.37191 1.37191i 2.64907 + 2.64907i 1.00000 −7.10448
307.3 −1.12106 + 1.12106i −1.00000 0.513571i 0.850971 + 0.850971i 1.12106 1.12106i −0.675060 0.675060i −1.66638 1.66638i 1.00000 −1.90799
307.4 −1.06339 + 1.06339i −1.00000 0.261597i 0.340859 + 0.340859i 1.06339 1.06339i −0.593196 0.593196i −1.84860 1.84860i 1.00000 −0.724932
307.5 −0.735263 + 0.735263i −1.00000 0.918776i −2.75658 2.75658i 0.735263 0.735263i −0.0552347 0.0552347i −2.14607 2.14607i 1.00000 4.05362
307.6 −0.200229 + 0.200229i −1.00000 1.91982i −0.906673 0.906673i 0.200229 0.200229i 2.29691 + 2.29691i −0.784861 0.784861i 1.00000 0.363084
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.14 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.d odd 4 1 inner
143.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.m.a 28
11.b odd 2 1 inner 429.2.m.a 28
13.d odd 4 1 inner 429.2.m.a 28
143.g even 4 1 inner 429.2.m.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.m.a 28 1.a even 1 1 trivial
429.2.m.a 28 11.b odd 2 1 inner
429.2.m.a 28 13.d odd 4 1 inner
429.2.m.a 28 143.g even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 99 T_{2}^{24} + 2857 T_{2}^{20} + 25707 T_{2}^{16} + 82143 T_{2}^{12} + 65769 T_{2}^{8} + 575 T_{2}^{4} + 1$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.