# Properties

 Label 432.2.v.a Level $432$ Weight $2$ Character orbit 432.v Analytic conductor $3.450$ Analytic rank $0$ Dimension $88$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q + 6q^{2} - 2q^{4} + 6q^{5} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q + 6q^{2} - 2q^{4} + 6q^{5} - 4q^{7} - 8q^{10} + 6q^{11} - 2q^{13} + 6q^{14} - 2q^{16} - 8q^{19} + 48q^{20} - 2q^{22} + 12q^{23} + 8q^{28} + 6q^{29} + 6q^{32} + 2q^{34} - 8q^{37} + 6q^{38} - 2q^{40} - 2q^{43} - 40q^{46} - 24q^{49} - 72q^{50} - 2q^{52} - 16q^{55} - 36q^{56} + 16q^{58} + 42q^{59} - 2q^{61} - 44q^{64} + 12q^{65} - 2q^{67} - 96q^{68} - 16q^{70} - 78q^{74} - 14q^{76} + 6q^{77} - 36q^{82} - 54q^{83} + 8q^{85} - 54q^{86} + 22q^{88} + 20q^{91} - 108q^{92} + 6q^{94} - 4q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.41416 0.0121383i 0 1.99971 + 0.0343310i −0.323102 + 1.20583i 0 0.140266 0.242948i −2.82749 0.0728225i 0 0.471556 1.70132i
35.2 −1.24823 + 0.664769i 0 1.11616 1.65957i 0.521033 1.94452i 0 −0.322227 + 0.558114i −0.290001 + 2.81352i 0 0.642288 + 2.77358i
35.3 −1.20346 + 0.742754i 0 0.896632 1.78775i 0.206232 0.769670i 0 −2.17574 + 3.76849i 0.248799 + 2.81746i 0 0.323483 + 1.07945i
35.4 −1.19684 0.753382i 0 0.864831 + 1.80335i −0.546024 + 2.03779i 0 −0.0638076 + 0.110518i 0.323550 2.80986i 0 2.18873 2.02753i
35.5 −1.14165 0.834650i 0 0.606718 + 1.90575i 1.02195 3.81396i 0 1.46715 2.54117i 0.897978 2.68210i 0 −4.35003 + 3.50123i
35.6 −1.00831 + 0.991617i 0 0.0333930 1.99972i −0.473330 + 1.76649i 0 1.40613 2.43549i 1.94929 + 2.04946i 0 −1.27442 2.25054i
35.7 −0.714386 1.22051i 0 −0.979306 + 1.74383i 0.247419 0.923380i 0 −1.93471 + 3.35102i 2.82798 0.0505136i 0 −1.30375 + 0.357671i
35.8 −0.417977 1.35103i 0 −1.65059 + 1.12940i −0.0776974 + 0.289971i 0 −0.374023 + 0.647827i 2.21577 + 1.75794i 0 0.424236 0.0162292i
35.9 −0.378412 + 1.36265i 0 −1.71361 1.03128i −1.00059 + 3.73424i 0 −1.68236 + 2.91393i 2.05372 1.94479i 0 −4.70981 2.77653i
35.10 −0.254908 + 1.39105i 0 −1.87004 0.709181i 0.0458174 0.170993i 0 1.17432 2.03397i 1.46320 2.42055i 0 0.226181 + 0.107322i
35.11 −0.0381759 + 1.41370i 0 −1.99709 0.107938i 0.642590 2.39818i 0 −1.93190 + 3.34616i 0.228833 2.81916i 0 3.36577 + 0.999982i
35.12 0.0112605 1.41417i 0 −1.99975 0.0318485i 0.759273 2.83365i 0 1.41719 2.45465i −0.0675573 + 2.82762i 0 −3.99870 1.10565i
35.13 0.503111 + 1.32170i 0 −1.49376 + 1.32992i 0.746784 2.78704i 0 1.16672 2.02082i −2.50928 1.30520i 0 4.05933 0.415168i
35.14 0.639562 1.26133i 0 −1.18192 1.61340i −0.997280 + 3.72190i 0 −0.481387 + 0.833787i −2.79095 + 0.458925i 0 4.05673 + 3.63829i
35.15 0.725676 1.21383i 0 −0.946789 1.76170i −0.178044 + 0.664471i 0 0.645693 1.11837i −2.82548 0.129179i 0 0.677355 + 0.698307i
35.16 0.845208 + 1.13385i 0 −0.571246 + 1.91668i −0.310357 + 1.15827i 0 0.356047 0.616691i −2.65606 + 0.972288i 0 −1.57562 + 0.627079i
35.17 1.09161 0.899102i 0 0.383232 1.96294i 0.726024 2.70956i 0 −0.00424642 + 0.00735502i −1.34654 2.48733i 0 −1.64363 3.61055i
35.18 1.25625 + 0.649489i 0 1.15633 + 1.63184i 0.315990 1.17929i 0 1.93802 3.35676i 0.392771 + 2.80102i 0 1.16290 1.27625i
35.19 1.26595 + 0.630383i 0 1.20523 + 1.59606i −0.282421 + 1.05401i 0 −1.93586 + 3.35301i 0.519632 + 2.78028i 0 −1.02196 + 1.15629i
35.20 1.35029 0.420387i 0 1.64655 1.13529i −0.619079 + 2.31044i 0 2.51270 4.35213i 1.74605 2.22515i 0 0.135344 + 3.38000i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 395.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
16.f odd 4 1 inner
144.u 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.v.a 88
3.b odd 2 1 144.2.u.a 88
4.b odd 2 1 1728.2.z.a 88
9.c even 3 1 144.2.u.a 88
9.d odd 6 1 inner 432.2.v.a 88
12.b even 2 1 576.2.y.a 88
16.e even 4 1 1728.2.z.a 88
16.f odd 4 1 inner 432.2.v.a 88
36.f odd 6 1 576.2.y.a 88
36.h even 6 1 1728.2.z.a 88
48.i odd 4 1 576.2.y.a 88
48.k even 4 1 144.2.u.a 88
144.u even 12 1 inner 432.2.v.a 88
144.v odd 12 1 144.2.u.a 88
144.w odd 12 1 1728.2.z.a 88
144.x even 12 1 576.2.y.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.u.a 88 3.b odd 2 1
144.2.u.a 88 9.c even 3 1
144.2.u.a 88 48.k even 4 1
144.2.u.a 88 144.v odd 12 1
432.2.v.a 88 1.a even 1 1 trivial
432.2.v.a 88 9.d odd 6 1 inner
432.2.v.a 88 16.f odd 4 1 inner
432.2.v.a 88 144.u even 12 1 inner
576.2.y.a 88 12.b even 2 1
576.2.y.a 88 36.f odd 6 1
576.2.y.a 88 48.i odd 4 1
576.2.y.a 88 144.x even 12 1
1728.2.z.a 88 4.b odd 2 1
1728.2.z.a 88 16.e even 4 1
1728.2.z.a 88 36.h even 6 1
1728.2.z.a 88 144.w odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(432, [\chi])$$.