# Properties

 Label 430.2.j.a Level 430 Weight 2 Character orbit 430.j Analytic conductor 3.434 Analytic rank 0 Dimension 44 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 44q^{4} - 4q^{5} + 2q^{6} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 44q^{4} - 4q^{5} + 2q^{6} + 20q^{9} + 8q^{11} + 4q^{14} - 4q^{15} + 44q^{16} - 4q^{19} + 4q^{20} - 24q^{21} - 2q^{24} + 12q^{26} - 10q^{29} - 20q^{31} - 12q^{34} + 12q^{35} - 20q^{36} + 120q^{39} + 20q^{41} - 8q^{44} - 28q^{45} + 42q^{49} - 112q^{51} - 68q^{54} - 26q^{55} - 4q^{56} + 40q^{59} + 4q^{60} + 8q^{61} - 44q^{64} - 60q^{65} - 12q^{66} - 4q^{69} + 48q^{70} - 20q^{71} - 12q^{74} + 4q^{75} + 4q^{76} - 44q^{79} - 4q^{80} + 2q^{81} + 24q^{84} + 20q^{85} + 14q^{86} - 26q^{89} + 68q^{90} + 4q^{94} - 34q^{95} + 2q^{96} + 72q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 1.00000i −2.75803 1.59235i −1.00000 1.06925 + 1.96385i −1.59235 + 2.75803i 3.76835 2.17566i 1.00000i 3.57116 + 6.18543i 1.96385 1.06925i
49.2 1.00000i −2.19536 1.26749i −1.00000 0.795101 2.08993i −1.26749 + 2.19536i −2.46471 + 1.42300i 1.00000i 1.71308 + 2.96715i −2.08993 0.795101i
49.3 1.00000i −1.92330 1.11042i −1.00000 −2.17265 + 0.528758i −1.11042 + 1.92330i −1.64658 + 0.950651i 1.00000i 0.966065 + 1.67327i 0.528758 + 2.17265i
49.4 1.00000i −0.409934 0.236675i −1.00000 −1.79811 + 1.32921i −0.236675 + 0.409934i 2.61255 1.50836i 1.00000i −1.38797 2.40403i 1.32921 + 1.79811i
49.5 1.00000i −0.377570 0.217990i −1.00000 2.13668 0.659257i −0.217990 + 0.377570i 0.236341 0.136452i 1.00000i −1.40496 2.43346i −0.659257 2.13668i
49.6 1.00000i 0.0417606 + 0.0241105i −1.00000 0.704124 + 2.12231i 0.0241105 0.0417606i −3.51224 + 2.02779i 1.00000i −1.49884 2.59606i 2.12231 0.704124i
49.7 1.00000i 1.09428 + 0.631780i −1.00000 0.410590 2.19805i 0.631780 1.09428i 2.36788 1.36710i 1.00000i −0.701708 1.21539i −2.19805 0.410590i
49.8 1.00000i 1.13990 + 0.658124i −1.00000 −1.72456 1.42334i 0.658124 1.13990i −3.46526 + 2.00067i 1.00000i −0.633745 1.09768i −1.42334 + 1.72456i
49.9 1.00000i 1.66171 + 0.959389i −1.00000 −2.16132 0.573330i 0.959389 1.66171i 0.863309 0.498432i 1.00000i 0.340854 + 0.590377i −0.573330 + 2.16132i
49.10 1.00000i 2.23224 + 1.28878i −1.00000 −0.429629 + 2.19441i 1.28878 2.23224i 2.28550 1.31953i 1.00000i 1.82193 + 3.15567i 2.19441 + 0.429629i
49.11 1.00000i 2.36034 + 1.36274i −1.00000 2.17053 + 0.537418i 1.36274 2.36034i −2.77719 + 1.60341i 1.00000i 2.21413 + 3.83499i 0.537418 2.17053i
49.12 1.00000i −2.36034 1.36274i −1.00000 −1.55068 1.61102i 1.36274 2.36034i 2.77719 1.60341i 1.00000i 2.21413 + 3.83499i 1.61102 1.55068i
49.13 1.00000i −2.23224 1.28878i −1.00000 −1.68560 + 1.46927i 1.28878 2.23224i −2.28550 + 1.31953i 1.00000i 1.82193 + 3.15567i −1.46927 1.68560i
49.14 1.00000i −1.66171 0.959389i −1.00000 1.57718 + 1.58509i 0.959389 1.66171i −0.863309 + 0.498432i 1.00000i 0.340854 + 0.590377i −1.58509 + 1.57718i
49.15 1.00000i −1.13990 0.658124i −1.00000 2.09493 + 0.781842i 0.658124 1.13990i 3.46526 2.00067i 1.00000i −0.633745 1.09768i −0.781842 + 2.09493i
49.16 1.00000i −1.09428 0.631780i −1.00000 1.69827 1.45461i 0.631780 1.09428i −2.36788 + 1.36710i 1.00000i −0.701708 1.21539i 1.45461 + 1.69827i
49.17 1.00000i −0.0417606 0.0241105i −1.00000 −2.19004 + 0.451367i 0.0241105 0.0417606i 3.51224 2.02779i 1.00000i −1.49884 2.59606i −0.451367 2.19004i
49.18 1.00000i 0.377570 + 0.217990i −1.00000 −0.497404 2.18004i −0.217990 + 0.377570i −0.236341 + 0.136452i 1.00000i −1.40496 2.43346i 2.18004 0.497404i
49.19 1.00000i 0.409934 + 0.236675i −1.00000 −0.252079 + 2.22181i −0.236675 + 0.409934i −2.61255 + 1.50836i 1.00000i −1.38797 2.40403i −2.22181 0.252079i
49.20 1.00000i 1.92330 + 1.11042i −1.00000 0.628408 + 2.14595i −1.11042 + 1.92330i 1.64658 0.950651i 1.00000i 0.966065 + 1.67327i −2.14595 + 0.628408i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
43.c even 3 1 inner
215.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.j.a 44
5.b even 2 1 inner 430.2.j.a 44
43.c even 3 1 inner 430.2.j.a 44
215.i even 6 1 inner 430.2.j.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.j.a 44 1.a even 1 1 trivial
430.2.j.a 44 5.b even 2 1 inner
430.2.j.a 44 43.c even 3 1 inner
430.2.j.a 44 215.i even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database