# Properties

 Label 432.2.y.e Level $432$ Weight $2$ Character orbit 432.y Analytic conductor $3.450$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 4q^{2} - 2q^{4} - 4q^{5} + 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 4q^{2} - 2q^{4} - 4q^{5} + 8q^{8} - 20q^{10} + 2q^{11} - 16q^{13} + 4q^{14} - 10q^{16} + 16q^{17} + 28q^{19} - 12q^{20} - 8q^{22} + 4q^{26} - 16q^{28} - 4q^{29} + 28q^{31} + 46q^{32} - 14q^{34} + 16q^{35} + 16q^{37} - 2q^{38} - 10q^{40} - 10q^{43} - 60q^{44} + 20q^{46} + 56q^{47} + 4q^{49} + 36q^{50} + 6q^{52} + 8q^{53} - 52q^{56} - 14q^{58} + 14q^{59} - 32q^{61} - 16q^{62} - 44q^{64} + 64q^{65} - 18q^{67} - 16q^{68} + 14q^{70} - 38q^{74} + 10q^{76} + 36q^{77} + 44q^{79} - 144q^{80} - 88q^{82} - 20q^{83} - 8q^{85} - 76q^{86} - 42q^{88} - 80q^{91} + 68q^{92} + 20q^{94} - 48q^{95} + 40q^{97} - 88q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −1.41364 0.0402314i 0 1.99676 + 0.113746i 2.21121 0.592492i 0 2.67054 1.54184i −2.81813 0.241128i 0 −3.14970 + 0.748611i
37.2 −1.30266 0.550520i 0 1.39386 + 1.43428i −2.73721 + 0.733432i 0 −1.14487 + 0.660988i −1.02612 2.63573i 0 3.96942 + 0.551472i
37.3 −1.11640 0.868135i 0 0.492685 + 1.93837i 2.41406 0.646846i 0 2.82197 1.62927i 1.13273 2.59170i 0 −3.25660 1.37359i
37.4 −1.10904 + 0.877514i 0 0.459938 1.94640i 1.98415 0.531653i 0 −1.54969 + 0.894715i 1.19790 + 2.56223i 0 −1.73397 + 2.33075i
37.5 −1.05869 0.937649i 0 0.241628 + 1.98535i 0.491749 0.131764i 0 −2.40518 + 1.38863i 1.60575 2.32842i 0 −0.644156 0.321592i
37.6 −0.902916 + 1.08846i 0 −0.369486 1.96557i −1.21694 + 0.326078i 0 −0.707732 + 0.408609i 2.47306 + 1.37258i 0 0.743871 1.61901i
37.7 −0.297230 1.38263i 0 −1.82331 + 0.821916i 0.0112878 0.00302457i 0 1.05753 0.610563i 1.67835 + 2.27665i 0 −0.00753693 0.0147079i
37.8 −0.268956 1.38840i 0 −1.85533 + 0.746838i −2.97932 + 0.798307i 0 1.78208 1.02889i 1.53591 + 2.37507i 0 1.90968 + 3.92179i
37.9 0.0882790 + 1.41146i 0 −1.98441 + 0.249204i −2.90072 + 0.777246i 0 −1.04527 + 0.603486i −0.526922 2.77891i 0 −1.35312 4.02562i
37.10 0.174715 1.40338i 0 −1.93895 0.490384i −0.174734 + 0.0468197i 0 −4.04791 + 2.33706i −1.02696 + 2.63540i 0 0.0351772 + 0.253398i
37.11 0.704761 + 1.22610i 0 −1.00662 + 1.72821i −2.53632 + 0.679606i 0 0.614293 0.354662i −2.82838 0.0162452i 0 −2.62076 2.63082i
37.12 0.751042 1.19831i 0 −0.871873 1.79995i 1.60558 0.430214i 0 3.62762 2.09441i −2.81171 0.307071i 0 0.690330 2.24708i
37.13 0.812713 + 1.15737i 0 −0.678997 + 1.88121i 2.69708 0.722679i 0 −2.89314 + 1.67035i −2.72908 + 0.743037i 0 3.02835 + 2.53418i
37.14 1.18142 0.777339i 0 0.791487 1.83672i −3.30105 + 0.884514i 0 2.63210 1.51965i −0.492680 2.78519i 0 −3.21235 + 3.61102i
37.15 1.20757 0.736049i 0 0.916464 1.77767i 4.10876 1.10094i 0 −1.63313 + 0.942891i −0.201751 2.82122i 0 4.15129 4.35372i
37.16 1.35510 + 0.404605i 0 1.67259 + 1.09656i −3.32531 + 0.891015i 0 −3.95817 + 2.28525i 1.82285 + 2.16269i 0 −4.86664 0.138025i
37.17 1.38700 0.276075i 0 1.84757 0.765835i 0.0691269 0.0185225i 0 1.28192 0.740118i 2.35115 1.57228i 0 0.0907658 0.0447750i
37.18 1.40500 + 0.161164i 0 1.94805 + 0.452871i 0.846545 0.226831i 0 −0.567074 + 0.327400i 2.66403 + 0.950239i 0 1.22595 0.182265i
181.1 −1.41328 0.0512684i 0 1.99474 + 0.144914i −0.430214 + 1.60558i 0 −3.62762 2.09441i −2.81171 0.307071i 0 0.690330 2.24708i
181.2 −1.30272 + 0.550382i 0 1.39416 1.43399i 0.0468197 0.174734i 0 4.04791 + 2.33706i −1.02696 + 2.63540i 0 0.0351772 + 0.253398i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 397.18 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.c even 3 1 inner
16.e even 4 1 inner
144.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.y.e 72
3.b odd 2 1 144.2.x.e 72
4.b odd 2 1 1728.2.bc.e 72
9.c even 3 1 inner 432.2.y.e 72
9.d odd 6 1 144.2.x.e 72
12.b even 2 1 576.2.bb.e 72
16.e even 4 1 inner 432.2.y.e 72
16.f odd 4 1 1728.2.bc.e 72
36.f odd 6 1 1728.2.bc.e 72
36.h even 6 1 576.2.bb.e 72
48.i odd 4 1 144.2.x.e 72
48.k even 4 1 576.2.bb.e 72
144.u even 12 1 576.2.bb.e 72
144.v odd 12 1 1728.2.bc.e 72
144.w odd 12 1 144.2.x.e 72
144.x even 12 1 inner 432.2.y.e 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.e 72 3.b odd 2 1
144.2.x.e 72 9.d odd 6 1
144.2.x.e 72 48.i odd 4 1
144.2.x.e 72 144.w odd 12 1
432.2.y.e 72 1.a even 1 1 trivial
432.2.y.e 72 9.c even 3 1 inner
432.2.y.e 72 16.e even 4 1 inner
432.2.y.e 72 144.x even 12 1 inner
576.2.bb.e 72 12.b even 2 1
576.2.bb.e 72 36.h even 6 1
576.2.bb.e 72 48.k even 4 1
576.2.bb.e 72 144.u even 12 1
1728.2.bc.e 72 4.b odd 2 1
1728.2.bc.e 72 16.f odd 4 1
1728.2.bc.e 72 36.f odd 6 1
1728.2.bc.e 72 144.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$56\!\cdots\!68$$$$T_{5}^{49} -$$$$23\!\cdots\!23$$$$T_{5}^{48} +$$$$28\!\cdots\!16$$$$T_{5}^{47} +$$$$30\!\cdots\!36$$$$T_{5}^{46} +$$$$23\!\cdots\!16$$$$T_{5}^{45} +$$$$12\!\cdots\!94$$$$T_{5}^{44} -$$$$12\!\cdots\!56$$$$T_{5}^{43} +$$$$17\!\cdots\!44$$$$T_{5}^{42} -$$$$72\!\cdots\!68$$$$T_{5}^{41} -$$$$46\!\cdots\!58$$$$T_{5}^{40} +$$$$48\!\cdots\!72$$$$T_{5}^{39} -$$$$69\!\cdots\!04$$$$T_{5}^{38} +$$$$18\!\cdots\!68$$$$T_{5}^{37} +$$$$12\!\cdots\!98$$$$T_{5}^{36} -$$$$15\!\cdots\!00$$$$T_{5}^{35} +$$$$49\!\cdots\!52$$$$T_{5}^{34} -$$$$43\!\cdots\!08$$$$T_{5}^{33} -$$$$20\!\cdots\!75$$$$T_{5}^{32} +$$$$37\!\cdots\!48$$$$T_{5}^{31} -$$$$14\!\cdots\!04$$$$T_{5}^{30} +$$$$78\!\cdots\!52$$$$T_{5}^{29} +$$$$16\!\cdots\!58$$$$T_{5}^{28} -$$$$53\!\cdots\!56$$$$T_{5}^{27} +$$$$47\!\cdots\!88$$$$T_{5}^{26} -$$$$98\!\cdots\!52$$$$T_{5}^{25} +$$$$47\!\cdots\!09$$$$T_{5}^{24} +$$$$12\!\cdots\!28$$$$T_{5}^{23} -$$$$14\!\cdots\!72$$$$T_{5}^{22} +$$$$25\!\cdots\!40$$$$T_{5}^{21} -$$$$63\!\cdots\!20$$$$T_{5}^{20} -$$$$28\!\cdots\!32$$$$T_{5}^{19} +$$$$30\!\cdots\!40$$$$T_{5}^{18} -$$$$37\!\cdots\!12$$$$T_{5}^{17} +$$$$39\!\cdots\!60$$$$T_{5}^{16} -$$$$18\!\cdots\!60$$$$T_{5}^{15} +$$$$70\!\cdots\!60$$$$T_{5}^{14} -$$$$36\!\cdots\!60$$$$T_{5}^{13} +$$$$52\!\cdots\!20$$$$T_{5}^{12} +$$$$17\!\cdots\!20$$$$T_{5}^{11} -$$$$12\!\cdots\!44$$$$T_{5}^{10} +$$$$52\!\cdots\!20$$$$T_{5}^{9} +$$$$68\!\cdots\!64$$$$T_{5}^{8} -$$$$12\!\cdots\!76$$$$T_{5}^{7} +$$$$63\!\cdots\!96$$$$T_{5}^{6} -$$$$65\!\cdots\!88$$$$T_{5}^{5} +$$$$45\!\cdots\!64$$$$T_{5}^{4} - 66961801216 T_{5}^{3} + 615022592 T_{5}^{2} - 8978432 T_{5} + 65536$$">$$T_{5}^{72} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.