# Properties

 Label 576.2.bb.e Level $576$ Weight $2$ Character orbit 576.bb Analytic conductor $4.599$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.bb (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 2q^{3} + 4q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 2q^{3} + 4q^{5} + 2q^{11} - 16q^{13} + 20q^{15} - 16q^{17} - 28q^{19} - 16q^{21} - 8q^{27} + 4q^{29} - 28q^{31} - 32q^{33} + 16q^{35} + 16q^{37} + 10q^{43} + 40q^{45} + 56q^{47} + 4q^{49} + 54q^{51} - 8q^{53} + 14q^{59} - 32q^{61} + 108q^{63} - 64q^{65} + 18q^{67} + 32q^{69} - 86q^{75} - 36q^{77} - 44q^{79} - 44q^{81} - 20q^{83} - 8q^{85} + 80q^{91} - 4q^{93} - 48q^{95} + 40q^{97} - 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −1.71513 + 0.241506i 0 −1.98415 + 0.531653i 0 1.54969 0.894715i 0 2.88335 0.828427i 0
49.2 0 −1.51206 0.844796i 0 3.32531 0.891015i 0 3.95817 2.28525i 0 1.57264 + 2.55476i 0
49.3 0 −1.45211 + 0.944122i 0 −0.491749 + 0.131764i 0 2.40518 1.38863i 0 1.21727 2.74194i 0
49.4 0 −1.31071 1.13227i 0 −0.0112878 + 0.00302457i 0 −1.05753 + 0.610563i 0 0.435936 + 2.96816i 0
49.5 0 −1.08700 + 1.34849i 0 2.90072 0.777246i 0 1.04527 0.603486i 0 −0.636858 2.93162i 0
49.6 0 −1.05287 1.37530i 0 −4.10876 + 1.10094i 0 1.63313 0.942891i 0 −0.782911 + 2.89604i 0
49.7 0 −0.841300 + 1.51401i 0 −0.0691269 + 0.0185225i 0 −1.28192 + 0.740118i 0 −1.58443 2.54747i 0
49.8 0 0.162237 + 1.72444i 0 −2.21121 + 0.592492i 0 −2.67054 + 1.54184i 0 −2.94736 + 0.559536i 0
49.9 0 0.222657 + 1.71768i 0 2.97932 0.798307i 0 −1.78208 + 1.02889i 0 −2.90085 + 0.764906i 0
49.10 0 0.409248 1.68301i 0 3.30105 0.884514i 0 −2.63210 + 1.51965i 0 −2.66503 1.37753i 0
49.11 0 0.460673 + 1.66966i 0 −2.69708 + 0.722679i 0 2.89314 1.67035i 0 −2.57556 + 1.53834i 0
49.12 0 0.795173 1.53873i 0 −2.41406 + 0.646846i 0 −2.82197 + 1.62927i 0 −1.73540 2.44712i 0
49.13 0 1.15652 + 1.28937i 0 1.21694 0.326078i 0 0.707732 0.408609i 0 −0.324925 + 2.98235i 0
49.14 0 1.22967 1.21980i 0 0.174734 0.0468197i 0 4.04791 2.33706i 0 0.0241875 2.99990i 0
49.15 0 1.58205 0.705067i 0 2.53632 0.679606i 0 −0.614293 + 0.354662i 0 2.00576 2.23090i 0
49.16 0 1.64464 + 0.543291i 0 −1.60558 + 0.430214i 0 −3.62762 + 2.09441i 0 2.40967 + 1.78703i 0
49.17 0 1.68781 + 0.388965i 0 −0.846545 + 0.226831i 0 0.567074 0.327400i 0 2.69741 + 1.31300i 0
49.18 0 1.71859 + 0.215523i 0 2.73721 0.733432i 0 1.14487 0.660988i 0 2.90710 + 0.740791i 0
241.1 0 −1.72444 0.162237i 0 0.592492 2.21121i 0 2.67054 + 1.54184i 0 2.94736 + 0.559536i 0
241.2 0 −1.71768 0.222657i 0 −0.798307 + 2.97932i 0 1.78208 + 1.02889i 0 2.90085 + 0.764906i 0
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.18 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 576.2.bb.e 72
3.b odd 2 1 1728.2.bc.e 72
4.b odd 2 1 144.2.x.e 72
9.c even 3 1 inner 576.2.bb.e 72
9.d odd 6 1 1728.2.bc.e 72
12.b even 2 1 432.2.y.e 72
16.e even 4 1 inner 576.2.bb.e 72
16.f odd 4 1 144.2.x.e 72
36.f odd 6 1 144.2.x.e 72
36.h even 6 1 432.2.y.e 72
48.i odd 4 1 1728.2.bc.e 72
48.k even 4 1 432.2.y.e 72
144.u even 12 1 432.2.y.e 72
144.v odd 12 1 144.2.x.e 72
144.w odd 12 1 1728.2.bc.e 72
144.x even 12 1 inner 576.2.bb.e 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.e 72 4.b odd 2 1
144.2.x.e 72 16.f odd 4 1
144.2.x.e 72 36.f odd 6 1
144.2.x.e 72 144.v odd 12 1
432.2.y.e 72 12.b even 2 1
432.2.y.e 72 36.h even 6 1
432.2.y.e 72 48.k even 4 1
432.2.y.e 72 144.u even 12 1
576.2.bb.e 72 1.a even 1 1 trivial
576.2.bb.e 72 9.c even 3 1 inner
576.2.bb.e 72 16.e even 4 1 inner
576.2.bb.e 72 144.x even 12 1 inner
1728.2.bc.e 72 3.b odd 2 1
1728.2.bc.e 72 9.d odd 6 1
1728.2.bc.e 72 48.i odd 4 1
1728.2.bc.e 72 144.w 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 $$S_{2}^{\mathrm{new}}(576, [\chi])$$.