# Properties

 Label 430.2.x.a Level 430 Weight 2 Character orbit 430.x Analytic conductor 3.434 Analytic rank 0 Dimension 528 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.x (of order $$84$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$528$$ Relative dimension: $$22$$ over $$\Q(\zeta_{84})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$528q - 4q^{6} - 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$528q - 4q^{6} - 12q^{7} + 40q^{13} + 88q^{16} - 4q^{17} + 16q^{21} - 12q^{23} + 8q^{25} - 12q^{28} + 36q^{30} - 72q^{31} - 124q^{33} + 40q^{35} - 268q^{36} - 44q^{38} + 56q^{41} - 168q^{43} - 24q^{46} - 72q^{47} - 24q^{50} + 280q^{51} + 16q^{52} - 132q^{53} - 24q^{55} + 8q^{56} - 20q^{57} - 8q^{60} - 72q^{61} + 48q^{62} - 84q^{65} - 72q^{66} - 40q^{67} - 32q^{68} - 24q^{71} + 192q^{73} + 48q^{76} - 148q^{77} - 40q^{78} + 56q^{82} - 28q^{83} - 20q^{86} - 216q^{87} + 8q^{90} - 48q^{91} + 12q^{92} + 108q^{93} - 4q^{95} + 4q^{96} - 120q^{97} + 96q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −0.330279 0.943883i −2.12618 2.47066i −0.781831 + 0.623490i −1.43957 + 1.71103i −1.62979 + 2.82287i 0.781116 2.91516i 0.846724 + 0.532032i −1.13642 + 7.53964i 2.09047 + 0.793673i
3.2 −0.330279 0.943883i −1.48270 1.72293i −0.781831 + 0.623490i 1.46328 + 1.69080i −1.13654 + 1.96854i −0.664636 + 2.48045i 0.846724 + 0.532032i −0.322955 + 2.14267i 1.11263 1.93960i
3.3 −0.330279 0.943883i −1.40744 1.63547i −0.781831 + 0.623490i 1.98609 1.02735i −1.07885 + 1.86862i 0.821541 3.06603i 0.846724 + 0.532032i −0.246762 + 1.63716i −1.62566 1.53533i
3.4 −0.330279 0.943883i −1.23630 1.43661i −0.781831 + 0.623490i 1.28939 1.82688i −0.947664 + 1.64140i −1.29317 + 4.82619i 0.846724 + 0.532032i −0.0882734 + 0.585656i −2.15022 0.613651i
3.5 −0.330279 0.943883i −0.777489 0.903458i −0.781831 + 0.623490i −2.03981 0.916062i −0.595971 + 1.03225i 0.237699 0.887105i 0.846724 + 0.532032i 0.235379 1.56164i −0.190949 + 2.22790i
3.6 −0.330279 0.943883i −0.0131640 0.0152969i −0.781831 + 0.623490i −1.62810 + 1.53273i −0.0100907 + 0.0174775i 0.140005 0.522506i 0.846724 + 0.532032i 0.447066 2.96609i 1.98445 + 1.03051i
3.7 −0.330279 0.943883i 0.658429 + 0.765108i −0.781831 + 0.623490i −0.824716 2.07842i 0.504708 0.874179i −1.09920 + 4.10225i 0.846724 + 0.532032i 0.295265 1.95895i −1.68940 + 1.46490i
3.8 −0.330279 0.943883i 0.671823 + 0.780672i −0.781831 + 0.623490i 0.417921 2.19667i 0.514974 0.891962i 0.647385 2.41607i 0.846724 + 0.532032i 0.289024 1.91755i −2.21143 + 0.331044i
3.9 −0.330279 0.943883i 1.23231 + 1.43197i −0.781831 + 0.623490i 1.04170 + 1.97860i 0.944603 1.63610i 1.31062 4.89129i 0.846724 + 0.532032i −0.0848203 + 0.562746i 1.52352 1.63673i
3.10 −0.330279 0.943883i 1.64699 + 1.91384i −0.781831 + 0.623490i 2.22442 + 0.227923i 1.26247 2.18667i −0.541490 + 2.02087i 0.846724 + 0.532032i −0.503069 + 3.33764i −0.519547 2.17487i
3.11 −0.330279 0.943883i 1.75583 + 2.04031i −0.781831 + 0.623490i −2.23481 + 0.0750623i 1.34590 2.33117i −0.285871 + 1.06689i 0.846724 + 0.532032i −0.632803 + 4.19837i 0.808960 + 2.08461i
3.12 0.330279 + 0.943883i −1.92406 2.23579i −0.781831 + 0.623490i −2.01247 0.974668i 1.47485 2.55452i −0.743303 + 2.77404i −0.846724 0.532032i −0.849656 + 5.63710i 0.255297 2.22145i
3.13 0.330279 + 0.943883i −1.35577 1.57544i −0.781831 + 0.623490i 1.79734 1.33025i 1.03925 1.80003i −0.398389 + 1.48681i −0.846724 0.532032i −0.196755 + 1.30538i 1.84922 + 1.25713i
3.14 0.330279 + 0.943883i −1.26939 1.47505i −0.781831 + 0.623490i −0.345539 + 2.20921i 0.973028 1.68533i −0.0286434 + 0.106899i −0.846724 0.532032i −0.117316 + 0.778339i −2.19936 + 0.403507i
3.15 0.330279 + 0.943883i −0.667495 0.775643i −0.781831 + 0.623490i −1.70213 + 1.45008i 0.511657 0.886215i 0.854652 3.18961i −0.846724 0.532032i 0.291054 1.93102i −1.93089 1.12768i
3.16 0.330279 + 0.943883i −0.515215 0.598691i −0.781831 + 0.623490i 2.22238 + 0.247005i 0.394930 0.684038i 0.301636 1.12572i −0.846724 0.532032i 0.354143 2.34958i 0.500863 + 2.17925i
3.17 0.330279 + 0.943883i 0.0217181 + 0.0252369i −0.781831 + 0.623490i −2.02196 0.954820i −0.0166477 + 0.0288346i −0.751874 + 2.80603i −0.846724 0.532032i 0.446962 2.96540i 0.233427 2.22385i
3.18 0.330279 + 0.943883i 0.447979 + 0.520562i −0.781831 + 0.623490i −0.398933 2.20019i −0.343391 + 0.594771i 0.739536 2.75998i −0.846724 0.532032i 0.376828 2.50009i 1.94497 1.10322i
3.19 0.330279 + 0.943883i 0.993378 + 1.15433i −0.781831 + 0.623490i 1.09418 + 1.95007i −0.761458 + 1.31888i −0.436975 + 1.63081i −0.846724 0.532032i 0.101458 0.673127i −1.47925 + 1.67685i
3.20 0.330279 + 0.943883i 1.27566 + 1.48234i −0.781831 + 0.623490i 1.46555 1.68884i −0.977833 + 1.69366i −1.09982 + 4.10459i −0.846724 0.532032i −0.122904 + 0.815417i 2.07811 + 0.825518i
See next 80 embeddings (of 528 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 417.22 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.c odd 4 1 inner
43.h odd 42 1 inner
215.x even 84 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.x.a 528
5.c odd 4 1 inner 430.2.x.a 528
43.h odd 42 1 inner 430.2.x.a 528
215.x even 84 1 inner 430.2.x.a 528

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.x.a 528 1.a even 1 1 trivial
430.2.x.a 528 5.c odd 4 1 inner
430.2.x.a 528 43.h odd 42 1 inner
430.2.x.a 528 215.x even 84 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