# Properties

 Label 432.3.x.a Level $432$ Weight $3$ Character orbit 432.x Analytic conductor $11.771$ Analytic rank $0$ Dimension $184$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 432.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7711474204$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$184q + 6q^{2} - 2q^{4} + 6q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q + 6q^{2} - 2q^{4} + 6q^{5} - 8q^{10} + 6q^{11} - 2q^{13} + 6q^{14} - 2q^{16} - 8q^{19} - 120q^{20} - 2q^{22} - 72q^{28} + 6q^{29} - 4q^{31} + 6q^{32} + 6q^{34} - 8q^{37} + 6q^{38} - 2q^{40} - 2q^{43} - 160q^{46} + 12q^{47} + 472q^{49} - 228q^{50} - 2q^{52} + 300q^{56} - 92q^{58} + 438q^{59} - 2q^{61} + 244q^{64} + 12q^{65} - 2q^{67} + 144q^{68} + 96q^{70} - 246q^{74} - 158q^{76} + 6q^{77} - 4q^{79} - 388q^{82} + 726q^{83} + 48q^{85} - 894q^{86} + 22q^{88} - 204q^{91} + 348q^{92} - 18q^{94} + 12q^{95} - 4q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1 −1.99910 0.0600540i 0 3.99279 + 0.240108i −7.92852 + 2.12444i 0 −8.42400 + 4.86360i −7.96755 0.719782i 0 15.9775 3.77083i
125.2 −1.94424 0.468982i 0 3.56011 + 1.82362i 0.517104 0.138558i 0 10.5540 6.09335i −6.06645 5.21518i 0 −1.07035 + 0.0268763i
125.3 −1.93835 + 0.492752i 0 3.51439 1.91025i −1.22916 + 0.329352i 0 0.837167 0.483338i −5.87084 + 5.43445i 0 2.22025 1.24407i
125.4 −1.93717 0.497372i 0 3.50524 + 1.92699i 5.08604 1.36280i 0 −9.00556 + 5.19936i −5.83182 5.47630i 0 −10.5303 + 0.110321i
125.5 −1.93332 + 0.512131i 0 3.47544 1.98023i 2.46048 0.659283i 0 2.84517 1.64266i −5.70500 + 5.60829i 0 −4.41925 + 2.53469i
125.6 −1.92351 0.547812i 0 3.39980 + 2.10745i 2.68051 0.718241i 0 −3.19237 + 1.84312i −5.38508 5.91615i 0 −5.54946 0.0868707i
125.7 −1.71077 + 1.03599i 0 1.85347 3.54467i 7.93365 2.12582i 0 −4.39189 + 2.53566i 0.501371 + 7.98427i 0 −11.3703 + 11.8559i
125.8 −1.65125 1.12844i 0 1.45327 + 3.72666i −7.93556 + 2.12633i 0 2.99388 1.72852i 1.80558 7.79358i 0 15.5030 + 5.44366i
125.9 −1.61311 + 1.18232i 0 1.20425 3.81442i −4.41854 + 1.18394i 0 1.46095 0.843480i 2.56726 + 7.57688i 0 5.72780 7.13395i
125.10 −1.43029 1.39796i 0 0.0914307 + 3.99895i −3.58315 + 0.960103i 0 2.87499 1.65987i 5.45960 5.84746i 0 6.46711 + 3.63587i
125.11 −1.41597 + 1.41245i 0 0.00995269 3.99999i −7.30894 + 1.95842i 0 −10.0693 + 5.81351i 5.63570 + 5.67793i 0 7.58307 13.0966i
125.12 −1.34550 1.47974i 0 −0.379247 + 3.98198i 7.96785 2.13498i 0 10.2471 5.91616i 6.40256 4.79658i 0 −13.8800 8.91771i
125.13 −1.31731 + 1.50489i 0 −0.529401 3.96481i 4.63766 1.24266i 0 6.36669 3.67581i 6.66400 + 4.42619i 0 −4.23916 + 8.61614i
125.14 −1.22012 1.58471i 0 −1.02259 + 3.86708i 0.432017 0.115759i 0 −7.23679 + 4.17816i 7.37588 3.09780i 0 −0.710557 0.543380i
125.15 −0.927252 + 1.77206i 0 −2.28041 3.28630i −5.06118 + 1.35614i 0 8.88571 5.13017i 7.93803 0.993796i 0 2.28983 10.2262i
125.16 −0.911412 1.78026i 0 −2.33866 + 3.24510i −4.51043 + 1.20857i 0 −7.69955 + 4.44534i 7.90861 + 1.20579i 0 6.26242 + 6.92824i
125.17 −0.904313 + 1.78388i 0 −2.36444 3.22637i 6.35235 1.70211i 0 1.38176 0.797761i 7.89363 1.30022i 0 −2.70816 + 12.8711i
125.18 −0.897969 1.78708i 0 −2.38730 + 3.20948i 3.11886 0.835695i 0 −0.108974 + 0.0629164i 7.87933 + 1.38428i 0 −4.29409 4.82321i
125.19 −0.672644 + 1.88349i 0 −3.09510 2.53384i 2.96132 0.793484i 0 −7.08251 + 4.08909i 6.85437 4.12523i 0 −0.497392 + 6.11137i
125.20 −0.266371 + 1.98218i 0 −3.85809 1.05599i −4.55983 + 1.22180i 0 5.29059 3.05452i 3.12086 7.36616i 0 −1.20723 9.36386i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
16.e even 4 1 inner
144.w odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.3.x.a 184
3.b odd 2 1 144.3.w.a 184
9.c even 3 1 144.3.w.a 184
9.d odd 6 1 inner 432.3.x.a 184
16.e even 4 1 inner 432.3.x.a 184
48.i odd 4 1 144.3.w.a 184
144.w odd 12 1 inner 432.3.x.a 184
144.x even 12 1 144.3.w.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.w.a 184 3.b odd 2 1
144.3.w.a 184 9.c even 3 1
144.3.w.a 184 48.i odd 4 1
144.3.w.a 184 144.x even 12 1
432.3.x.a 184 1.a even 1 1 trivial
432.3.x.a 184 9.d odd 6 1 inner
432.3.x.a 184 16.e even 4 1 inner
432.3.x.a 184 144.w odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(432, [\chi])$$.