# Properties

 Label 429.2.j.a Level $429$ Weight $2$ Character orbit 429.j Analytic conductor $3.426$ Analytic rank $0$ Dimension $96$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96q + 12q^{6} - 16q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q + 12q^{6} - 16q^{7} + 16q^{13} - 16q^{15} - 120q^{16} - 28q^{18} - 24q^{19} + 24q^{24} - 24q^{27} + 56q^{28} + 48q^{31} - 16q^{34} - 16q^{37} + 80q^{40} + 52q^{42} + 4q^{45} - 56q^{46} + 28q^{48} + 4q^{54} + 4q^{57} + 48q^{58} + 4q^{60} - 96q^{61} - 36q^{63} + 20q^{66} - 16q^{67} + 48q^{70} - 16q^{72} - 16q^{73} - 88q^{76} + 80q^{78} + 16q^{79} + 32q^{81} + 52q^{84} - 8q^{85} - 48q^{87} - 16q^{91} - 36q^{93} - 16q^{94} - 108q^{96} - 48q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1 −1.95700 1.95700i 1.66851 + 0.464826i 5.65973i −0.288277 0.288277i −2.35562 4.17495i −2.20581 2.20581i 7.16211 7.16211i 2.56787 + 1.55114i 1.12832i
122.2 −1.89533 1.89533i −0.602141 + 1.62402i 5.18453i 1.30667 + 1.30667i 4.21929 1.93679i 3.41860 + 3.41860i 6.03572 6.03572i −2.27485 1.95577i 4.95311i
122.3 −1.84950 1.84950i −1.73089 0.0633404i 4.84133i 2.48780 + 2.48780i 3.08414 + 3.31844i −2.91775 2.91775i 5.25505 5.25505i 2.99198 + 0.219271i 9.20239i
122.4 −1.78236 1.78236i −1.03830 + 1.38634i 4.35365i −3.07379 3.07379i 4.32159 0.620342i −1.36462 1.36462i 4.19506 4.19506i −0.843882 2.87887i 10.9572i
122.5 −1.74176 1.74176i −0.954649 1.44521i 4.06743i −0.190803 0.190803i −0.854447 + 4.17998i 1.94400 + 1.94400i 3.60095 3.60095i −1.17729 + 2.75934i 0.664665i
122.6 −1.70776 1.70776i 0.636556 1.61084i 3.83287i 0.490801 + 0.490801i −3.83800 + 1.66383i −0.206749 0.206749i 3.13009 3.13009i −2.18959 2.05078i 1.67634i
122.7 −1.52670 1.52670i 1.68807 0.387848i 2.66165i 1.02510 + 1.02510i −3.16931 1.98505i 0.667364 + 0.667364i 1.01014 1.01014i 2.69915 1.30943i 3.13005i
122.8 −1.43577 1.43577i −1.73083 0.0650121i 2.12285i −1.59194 1.59194i 2.39173 + 2.57841i 1.33791 + 1.33791i 0.176380 0.176380i 2.99155 + 0.225050i 4.57130i
122.9 −1.43233 1.43233i 0.643764 1.60797i 2.10314i −2.03215 2.03215i −3.22523 + 1.38106i −3.19246 3.19246i 0.147729 0.147729i −2.17114 2.07031i 5.82141i
122.10 −1.37738 1.37738i 1.00936 + 1.40755i 1.79436i 2.82918 + 2.82918i 0.548456 3.32900i −0.289134 0.289134i −0.283251 + 0.283251i −0.962383 + 2.84145i 7.79370i
122.11 −1.28581 1.28581i 0.911426 + 1.47286i 1.30661i −1.56311 1.56311i 0.721891 3.06573i −0.788371 0.788371i −0.891573 + 0.891573i −1.33861 + 2.68480i 4.01973i
122.12 −1.28495 1.28495i −0.594101 + 1.62697i 1.30220i 0.959579 + 0.959579i 2.85397 1.32719i −1.57675 1.57675i −0.896640 + 0.896640i −2.29409 1.93317i 2.46603i
122.13 −1.16086 1.16086i −1.66333 + 0.483035i 0.695179i 0.163665 + 0.163665i 2.49163 + 1.37016i 0.204315 + 0.204315i −1.51471 + 1.51471i 2.53335 1.60690i 0.379984i
122.14 −0.972463 0.972463i 1.43733 0.966479i 0.108632i 1.52250 + 1.52250i −2.33762 0.457885i 2.37496 + 2.37496i −2.05057 + 2.05057i 1.13184 2.77830i 2.96115i
122.15 −0.943039 0.943039i −1.40657 1.01073i 0.221356i −0.0801747 0.0801747i 0.373288 + 2.27960i −3.37063 3.37063i −2.09482 + 2.09482i 0.956852 + 2.84331i 0.151216i
122.16 −0.786557 0.786557i −0.138475 1.72651i 0.762656i −2.59877 2.59877i −1.24908 + 1.46691i 1.66123 + 1.66123i −2.17299 + 2.17299i −2.96165 + 0.478155i 4.08816i
122.17 −0.727121 0.727121i 1.73181 + 0.0289503i 0.942589i −1.91531 1.91531i −1.23818 1.28029i −1.22176 1.22176i −2.13962 + 2.13962i 2.99832 + 0.100273i 2.78533i
122.18 −0.680371 0.680371i −0.131936 + 1.72702i 1.07419i −1.70906 1.70906i 1.26478 1.08525i 2.81165 + 2.81165i −2.09159 + 2.09159i −2.96519 0.455712i 2.32560i
122.19 −0.487825 0.487825i 0.738212 1.56686i 1.52405i 2.39011 + 2.39011i −1.12447 + 0.404234i −2.10889 2.10889i −1.71912 + 1.71912i −1.91009 2.31335i 2.33191i
122.20 −0.388797 0.388797i 1.29884 + 1.14586i 1.69767i 1.22711 + 1.22711i −0.0594784 0.950496i 3.02639 + 3.02639i −1.43765 + 1.43765i 0.373993 + 2.97660i 0.954196i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 320.48 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
13.d odd 4 1 inner
39.f 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.j.a 96
3.b odd 2 1 inner 429.2.j.a 96
13.d odd 4 1 inner 429.2.j.a 96
39.f even 4 1 inner 429.2.j.a 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.j.a 96 1.a even 1 1 trivial
429.2.j.a 96 3.b odd 2 1 inner
429.2.j.a 96 13.d odd 4 1 inner
429.2.j.a 96 39.f even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(429, [\chi])$$.