# Properties

 Label 429.2.p.b Level $429$ Weight $2$ Character orbit 429.p Analytic conductor $3.426$ Analytic rank $0$ Dimension $96$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$48$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$96q + 2q^{3} - 60q^{4} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q + 2q^{3} - 60q^{4} - 2q^{9} + 28q^{12} - 8q^{15} - 44q^{16} + 16q^{22} - 56q^{25} + 44q^{27} - 40q^{31} - 16q^{33} - 56q^{34} + 40q^{36} + 28q^{37} - 22q^{42} - 10q^{45} + 24q^{48} - 8q^{49} - 12q^{55} - 16q^{58} - 76q^{60} + 104q^{64} + 68q^{66} - 24q^{67} + 30q^{69} + 88q^{70} - 66q^{75} - 28q^{78} - 66q^{81} + 32q^{82} - 8q^{88} - 80q^{91} + 30q^{93} - 80q^{97} - 68q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
230.1 −1.38419 + 2.39749i −1.43448 0.970706i −2.83198 4.90513i 1.85441i 4.31285 2.09551i 1.32496 0.764967i 10.1432 1.11546 + 2.78491i −4.44593 2.56686i
230.2 −1.38419 + 2.39749i 1.55790 + 0.756943i −2.83198 4.90513i 1.85441i −3.97119 + 2.68729i −1.32496 + 0.764967i 10.1432 1.85408 + 2.35847i 4.44593 + 2.56686i
230.3 −1.23621 + 2.14118i −1.16044 + 1.28584i −2.05645 3.56187i 2.31580i −1.31867 4.07429i −3.20654 + 1.85129i 5.22398 −0.306758 2.98428i −4.95856 2.86283i
230.4 −1.23621 + 2.14118i −0.533348 + 1.64789i −2.05645 3.56187i 2.31580i −2.86910 3.17914i 3.20654 1.85129i 5.22398 −2.43108 1.75780i 4.95856 + 2.86283i
230.5 −1.18009 + 2.04397i 0.425511 1.67897i −1.78521 3.09207i 0.303518i 2.92962 + 2.85106i −1.36283 + 0.786828i 3.70646 −2.63788 1.42884i −0.620383 0.358178i
230.6 −1.18009 + 2.04397i 1.24128 1.20799i −1.78521 3.09207i 0.303518i 1.00428 + 3.96266i 1.36283 0.786828i 3.70646 0.0815279 2.99889i 0.620383 + 0.358178i
230.7 −1.15197 + 1.99526i −0.989927 1.42128i −1.65405 2.86490i 3.64772i 3.97620 0.337897i 1.03838 0.599507i 3.01377 −1.04009 + 2.81393i 7.27817 + 4.20205i
230.8 −1.15197 + 1.99526i 1.72583 + 0.146661i −1.65405 2.86490i 3.64772i −2.28072 + 3.27454i −1.03838 + 0.599507i 3.01377 2.95698 + 0.506224i −7.27817 4.20205i
230.9 −1.00971 + 1.74887i −1.56002 + 0.752553i −1.03904 1.79966i 1.23164i 0.259053 3.48814i 3.16404 1.82676i 0.157663 1.86733 2.34800i −2.15397 1.24360i
230.10 −1.00971 + 1.74887i 0.128280 + 1.72729i −1.03904 1.79966i 1.23164i −3.15034 1.51972i −3.16404 + 1.82676i 0.157663 −2.96709 + 0.443156i 2.15397 + 1.24360i
230.11 −0.964601 + 1.67074i −1.68588 0.397235i −0.860909 1.49114i 1.71348i 2.28988 2.43350i −3.58916 + 2.07221i −0.536667 2.68441 + 1.33939i 2.86278 + 1.65283i
230.12 −0.964601 + 1.67074i 1.18696 + 1.26140i −0.860909 1.49114i 1.71348i −3.25241 + 0.766347i 3.58916 2.07221i −0.536667 −0.182262 + 2.99446i −2.86278 1.65283i
230.13 −0.790522 + 1.36922i −1.05747 1.37177i −0.249851 0.432755i 3.89310i 2.71422 0.363499i −2.15619 + 1.24488i −2.37204 −0.763513 + 2.90121i −5.33053 3.07758i
230.14 −0.790522 + 1.36922i 1.71672 + 0.229910i −0.249851 0.432755i 3.89310i −1.67191 + 2.16883i 2.15619 1.24488i −2.37204 2.89428 + 0.789386i 5.33053 + 3.07758i
230.15 −0.642823 + 1.11340i −1.71725 + 0.225956i 0.173557 + 0.300610i 1.88239i 0.852307 2.05724i 0.680391 0.392824i −3.01756 2.89789 0.776047i 2.09586 + 1.21004i
230.16 −0.642823 + 1.11340i 0.662940 + 1.60016i 0.173557 + 0.300610i 1.88239i −2.20777 0.290500i −0.680391 + 0.392824i −3.01756 −2.12102 + 2.12162i −2.09586 1.21004i
230.17 −0.570464 + 0.988072i 0.0427205 1.73152i 0.349142 + 0.604732i 0.927976i 1.68650 + 1.02998i −1.26517 + 0.730447i −3.07855 −2.99635 0.147943i −0.916907 0.529376i
230.18 −0.570464 + 0.988072i 1.47818 0.902759i 0.349142 + 0.604732i 0.927976i 0.0487410 + 1.97554i 1.26517 0.730447i −3.07855 1.37005 2.66889i 0.916907 + 0.529376i
230.19 −0.400235 + 0.693228i −1.18152 + 1.26649i 0.679624 + 1.17714i 2.49177i −0.405079 1.32596i −0.533256 + 0.307876i −2.68898 −0.207999 2.99278i 1.72737 + 0.997295i
230.20 −0.400235 + 0.693228i −0.506051 + 1.65648i 0.679624 + 1.17714i 2.49177i −0.945776 1.01379i 0.533256 0.307876i −2.68898 −2.48782 1.67652i −1.72737 0.997295i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 263.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.c even 3 1 inner
33.d even 2 1 inner
39.i odd 6 1 inner
143.k odd 6 1 inner
429.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.p.b 96
3.b odd 2 1 inner 429.2.p.b 96
11.b odd 2 1 inner 429.2.p.b 96
13.c even 3 1 inner 429.2.p.b 96
33.d even 2 1 inner 429.2.p.b 96
39.i odd 6 1 inner 429.2.p.b 96
143.k odd 6 1 inner 429.2.p.b 96
429.p even 6 1 inner 429.2.p.b 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.p.b 96 1.a even 1 1 trivial
429.2.p.b 96 3.b odd 2 1 inner
429.2.p.b 96 11.b odd 2 1 inner
429.2.p.b 96 13.c even 3 1 inner
429.2.p.b 96 33.d even 2 1 inner
429.2.p.b 96 39.i odd 6 1 inner
429.2.p.b 96 143.k odd 6 1 inner
429.2.p.b 96 429.p even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.