# Properties

 Label 430.2.t.a Level 430 Weight 2 Character orbit 430.t Analytic conductor 3.434 Analytic rank 0 Dimension 264 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.t (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$264$$ Relative dimension: $$22$$ over $$\Q(\zeta_{42})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$264q + 44q^{4} + 4q^{5} - 2q^{6} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$264q + 44q^{4} + 4q^{5} - 2q^{6} - 6q^{9} - 8q^{11} + 10q^{14} + 32q^{15} - 44q^{16} + 4q^{19} - 4q^{20} + 24q^{21} + 2q^{24} + 28q^{25} - 12q^{26} - 46q^{29} - 36q^{31} + 12q^{34} - 68q^{35} - 134q^{36} - 64q^{39} - 20q^{41} + 8q^{44} - 70q^{45} + 112q^{49} - 28q^{50} - 28q^{51} + 68q^{54} - 30q^{55} + 4q^{56} - 40q^{59} - 4q^{60} + 20q^{61} + 44q^{64} + 18q^{65} - 44q^{66} + 32q^{69} - 48q^{70} + 20q^{71} + 40q^{74} + 122q^{75} + 52q^{76} + 16q^{79} + 4q^{80} - 16q^{81} - 24q^{84} + 120q^{85} - 14q^{86} - 142q^{89} - 68q^{90} - 4q^{94} - 22q^{95} - 2q^{96} - 268q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −0.781831 + 0.623490i −0.511545 + 3.39388i 0.222521 0.974928i 1.50535 + 1.65346i −1.71611 2.97238i 1.77402 + 1.02423i 0.433884 + 0.900969i −8.39001 2.58798i −2.20784 0.354160i
9.2 −0.781831 + 0.623490i −0.362457 + 2.40475i 0.222521 0.974928i −2.23136 + 0.145098i −1.21595 2.10610i −1.90024 1.09710i 0.433884 + 0.900969i −2.78472 0.858971i 1.65408 1.50467i
9.3 −0.781831 + 0.623490i −0.299194 + 1.98502i 0.222521 0.974928i 0.934065 2.03163i −1.00372 1.73850i 1.47449 + 0.851296i 0.433884 + 0.900969i −0.984076 0.303547i 0.536420 + 2.17077i
9.4 −0.781831 + 0.623490i −0.192290 + 1.27576i 0.222521 0.974928i 2.15300 0.603819i −0.645086 1.11732i −4.41993 2.55185i 0.433884 + 0.900969i 1.27612 + 0.393632i −1.30681 + 1.81446i
9.5 −0.781831 + 0.623490i −0.115718 + 0.767737i 0.222521 0.974928i −0.503717 + 2.17859i −0.388204 0.672390i 0.338318 + 0.195328i 0.433884 + 0.900969i 2.29069 + 0.706584i −0.964509 2.01736i
9.6 −0.781831 + 0.623490i 0.0612820 0.406579i 0.222521 0.974928i 2.16843 + 0.545829i 0.205586 + 0.356085i 1.79116 + 1.03412i 0.433884 + 0.900969i 2.70517 + 0.834434i −2.03566 + 0.925245i
9.7 −0.781831 + 0.623490i 0.0684663 0.454244i 0.222521 0.974928i −0.0829298 2.23453i 0.229688 + 0.397830i −1.76000 1.01613i 0.433884 + 0.900969i 2.66507 + 0.822065i 1.45804 + 1.69532i
9.8 −0.781831 + 0.623490i 0.225928 1.49894i 0.222521 0.974928i −2.22164 + 0.253613i 0.757933 + 1.31278i 3.55836 + 2.05442i 0.433884 + 0.900969i 0.670955 + 0.206962i 1.57882 1.58345i
9.9 −0.781831 + 0.623490i 0.287268 1.90590i 0.222521 0.974928i 0.339942 + 2.21008i 0.963713 + 1.66920i −2.79107 1.61143i 0.433884 + 0.900969i −0.683210 0.210742i −1.64374 1.51596i
9.10 −0.781831 + 0.623490i 0.320497 2.12636i 0.222521 0.974928i 1.14747 1.91919i 1.07519 + 1.86228i 3.36198 + 1.94104i 0.433884 + 0.900969i −1.55196 0.478717i 0.299466 + 2.21592i
9.11 −0.781831 + 0.623490i 0.408860 2.71261i 0.222521 0.974928i −2.17831 + 0.504950i 1.37162 + 2.37572i −2.50700 1.44741i 0.433884 + 0.900969i −4.32436 1.33389i 1.38824 1.75294i
9.12 0.781831 0.623490i −0.408860 + 2.71261i 0.222521 0.974928i −0.325780 2.21221i 1.37162 + 2.37572i 2.50700 + 1.44741i −0.433884 0.900969i −4.32436 1.33389i −1.63399 1.52645i
9.13 0.781831 0.623490i −0.320497 + 2.12636i 0.222521 0.974928i −1.36731 + 1.76931i 1.07519 + 1.86228i −3.36198 1.94104i −0.433884 0.900969i −1.55196 0.478717i 0.0341455 + 2.23581i
9.14 0.781831 0.623490i −0.287268 + 1.90590i 0.222521 0.974928i 2.18150 0.490989i 0.963713 + 1.66920i 2.79107 + 1.61143i −0.433884 0.900969i −0.683210 0.210742i 1.39944 1.74401i
9.15 0.781831 0.623490i −0.225928 + 1.49894i 0.222521 0.974928i −0.575574 2.16072i 0.757933 + 1.31278i −3.55836 2.05442i −0.433884 0.900969i 0.670955 + 0.206962i −1.79719 1.33046i
9.16 0.781831 0.623490i −0.0684663 + 0.454244i 0.222521 0.974928i −2.11036 + 0.739168i 0.229688 + 0.397830i 1.76000 + 1.01613i −0.433884 0.900969i 2.66507 + 0.822065i −1.18908 + 1.89369i
9.17 0.781831 0.623490i −0.0612820 + 0.406579i 0.222521 0.974928i 1.30031 + 1.81912i 0.205586 + 0.356085i −1.79116 1.03412i −0.433884 0.900969i 2.70517 + 0.834434i 2.15083 + 0.611511i
9.18 0.781831 0.623490i 0.115718 0.767737i 0.222521 0.974928i 1.84397 1.26483i −0.388204 0.672390i −0.338318 0.195328i −0.433884 0.900969i 2.29069 + 0.706584i 0.653065 2.13858i
9.19 0.781831 0.623490i 0.192290 1.27576i 0.222521 0.974928i 0.224499 + 2.22477i −0.645086 1.11732i 4.41993 + 2.55185i −0.433884 0.900969i 1.27612 + 0.393632i 1.56264 + 1.59942i
9.20 0.781831 0.623490i 0.299194 1.98502i 0.222521 0.974928i −1.54994 + 1.61173i −1.00372 1.73850i −1.47449 0.851296i −0.433884 0.900969i −0.984076 0.303547i −0.206891 + 2.22648i
See next 80 embeddings (of 264 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.22 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.g even 21 1 inner
215.u even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.t.a 264
5.b even 2 1 inner 430.2.t.a 264
43.g even 21 1 inner 430.2.t.a 264
215.u even 42 1 inner 430.2.t.a 264

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.t.a 264 1.a even 1 1 trivial
430.2.t.a 264 5.b even 2 1 inner
430.2.t.a 264 43.g even 21 1 inner
430.2.t.a 264 215.u even 42 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