# Properties

 Label 288.3.u.a Level $288$ Weight $3$ Character orbit 288.u Analytic conductor $7.847$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 288.u (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.84743161358$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{8})$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 4q^{2} - 4q^{4} + 4q^{5} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 4q^{2} - 4q^{4} + 4q^{5} - 4q^{7} + 4q^{8} - 44q^{10} + 4q^{11} - 4q^{13} + 20q^{14} + 16q^{16} - 4q^{19} - 76q^{20} + 144q^{22} + 68q^{23} - 4q^{25} - 96q^{26} + 56q^{28} + 4q^{29} + 24q^{32} - 48q^{34} - 92q^{35} - 4q^{37} + 396q^{38} - 408q^{40} + 4q^{41} + 92q^{43} + 188q^{44} - 36q^{46} + 8q^{47} - 308q^{50} + 420q^{52} + 164q^{53} + 252q^{55} - 552q^{56} + 528q^{58} - 124q^{59} - 68q^{61} - 216q^{62} - 232q^{64} + 8q^{65} - 164q^{67} + 368q^{68} - 664q^{70} + 260q^{71} - 4q^{73} + 532q^{74} - 516q^{76} - 220q^{77} - 520q^{79} - 312q^{80} + 636q^{82} + 484q^{83} + 96q^{85} - 688q^{86} + 672q^{88} + 4q^{89} - 196q^{91} - 616q^{92} + 40q^{94} - 8q^{97} + 328q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.82416 + 0.820030i 0 2.65510 2.99173i 7.60625 3.15061i 0 6.84161 + 6.84161i −2.39002 + 7.63465i 0 −11.2914 + 11.9846i
19.2 −1.61758 1.17620i 0 1.23313 + 3.80518i −2.28872 + 0.948019i 0 −6.37744 6.37744i 2.48095 7.60558i 0 4.81725 + 1.15849i
19.3 −0.345994 + 1.96984i 0 −3.76058 1.36311i −7.20074 + 2.98264i 0 4.26150 + 4.26150i 3.98625 6.93612i 0 −3.38393 15.2163i
19.4 0.108191 + 1.99707i 0 −3.97659 + 0.432130i 2.81639 1.16659i 0 −6.23443 6.23443i −1.29322 7.89478i 0 2.63447 + 5.49832i
19.5 0.360897 1.96717i 0 −3.73951 1.41989i 0.452310 0.187353i 0 0.429965 + 0.429965i −4.14274 + 6.84381i 0 −0.205317 0.957385i
19.6 1.62478 1.16623i 0 1.27980 3.78974i 4.51028 1.86822i 0 −3.85317 3.85317i −2.34032 7.65003i 0 5.14942 8.29547i
19.7 1.98676 + 0.229757i 0 3.89442 + 0.912943i −4.18866 + 1.73500i 0 3.93197 + 3.93197i 7.52753 + 2.70857i 0 −8.72048 + 2.48465i
91.1 −1.82416 0.820030i 0 2.65510 + 2.99173i 7.60625 + 3.15061i 0 6.84161 6.84161i −2.39002 7.63465i 0 −11.2914 11.9846i
91.2 −1.61758 + 1.17620i 0 1.23313 3.80518i −2.28872 0.948019i 0 −6.37744 + 6.37744i 2.48095 + 7.60558i 0 4.81725 1.15849i
91.3 −0.345994 1.96984i 0 −3.76058 + 1.36311i −7.20074 2.98264i 0 4.26150 4.26150i 3.98625 + 6.93612i 0 −3.38393 + 15.2163i
91.4 0.108191 1.99707i 0 −3.97659 0.432130i 2.81639 + 1.16659i 0 −6.23443 + 6.23443i −1.29322 + 7.89478i 0 2.63447 5.49832i
91.5 0.360897 + 1.96717i 0 −3.73951 + 1.41989i 0.452310 + 0.187353i 0 0.429965 0.429965i −4.14274 6.84381i 0 −0.205317 + 0.957385i
91.6 1.62478 + 1.16623i 0 1.27980 + 3.78974i 4.51028 + 1.86822i 0 −3.85317 + 3.85317i −2.34032 + 7.65003i 0 5.14942 + 8.29547i
91.7 1.98676 0.229757i 0 3.89442 0.912943i −4.18866 1.73500i 0 3.93197 3.93197i 7.52753 2.70857i 0 −8.72048 2.48465i
163.1 −1.93931 + 0.488972i 0 3.52181 1.89653i 1.85856 + 4.48696i 0 −5.27676 5.27676i −5.90252 + 5.40002i 0 −5.79831 7.79280i
163.2 −1.20513 1.59614i 0 −1.09531 + 3.84712i 0.642823 + 1.55191i 0 −4.95044 4.95044i 7.46052 2.88803i 0 1.70238 2.89629i
163.3 −0.682385 + 1.87999i 0 −3.06870 2.56575i −1.34740 3.25291i 0 0.583225 + 0.583225i 6.91761 4.01829i 0 7.03487 0.313357i
163.4 0.658450 1.88850i 0 −3.13289 2.48697i 0.659338 + 1.59178i 0 9.54718 + 9.54718i −6.75950 + 4.27892i 0 3.44023 0.197052i
163.5 1.44490 + 1.38284i 0 0.175499 + 3.99615i 3.18221 + 7.68254i 0 3.67370 + 3.67370i −5.27246 + 6.01674i 0 −6.02574 + 15.5010i
163.6 1.46783 1.35848i 0 0.309042 3.98804i −1.74699 4.21761i 0 −0.392379 0.392379i −4.96407 6.27359i 0 −8.29385 3.81747i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 235.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.u.a 28
3.b odd 2 1 32.3.h.a 28
12.b even 2 1 128.3.h.a 28
24.f even 2 1 256.3.h.a 28
24.h odd 2 1 256.3.h.b 28
32.h odd 8 1 inner 288.3.u.a 28
96.o even 8 1 32.3.h.a 28
96.o even 8 1 256.3.h.b 28
96.p odd 8 1 128.3.h.a 28
96.p odd 8 1 256.3.h.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.h.a 28 3.b odd 2 1
32.3.h.a 28 96.o even 8 1
128.3.h.a 28 12.b even 2 1
128.3.h.a 28 96.p odd 8 1
256.3.h.a 28 24.f even 2 1
256.3.h.a 28 96.p odd 8 1
256.3.h.b 28 24.h odd 2 1
256.3.h.b 28 96.o even 8 1
288.3.u.a 28 1.a even 1 1 trivial
288.3.u.a 28 32.h odd 8 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$37\!\cdots\!00$$$$T_{5}^{12} -$$$$73\!\cdots\!04$$$$T_{5}^{11} +$$$$78\!\cdots\!72$$$$T_{5}^{10} -$$$$10\!\cdots\!48$$$$T_{5}^{9} +$$$$23\!\cdots\!12$$$$T_{5}^{8} +$$$$45\!\cdots\!68$$$$T_{5}^{7} -$$$$78\!\cdots\!72$$$$T_{5}^{6} +$$$$16\!\cdots\!76$$$$T_{5}^{5} +$$$$10\!\cdots\!96$$$$T_{5}^{4} -$$$$21\!\cdots\!88$$$$T_{5}^{3} +$$$$37\!\cdots\!24$$$$T_{5}^{2} -$$$$24\!\cdots\!00$$$$T_{5} +$$$$58\!\cdots\!28$$">$$T_{5}^{28} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database