# Properties

 Label 32.3.h.a Level $32$ Weight $3$ Character orbit 32.h Analytic conductor $0.872$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 32.h (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.871936845953$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{9} - 44q^{10} - 4q^{11} - 52q^{12} - 4q^{13} - 20q^{14} - 8q^{15} + 16q^{16} + 56q^{18} - 4q^{19} + 76q^{20} - 4q^{21} + 144q^{22} - 68q^{23} + 208q^{24} - 4q^{25} + 96q^{26} - 100q^{27} + 56q^{28} - 4q^{29} + 20q^{30} - 24q^{32} - 8q^{33} - 48q^{34} + 92q^{35} - 336q^{36} - 4q^{37} - 396q^{38} + 188q^{39} - 408q^{40} - 4q^{41} - 424q^{42} + 92q^{43} - 188q^{44} - 40q^{45} - 36q^{46} - 8q^{47} + 48q^{48} + 308q^{50} + 224q^{51} + 420q^{52} - 164q^{53} + 592q^{54} + 252q^{55} + 552q^{56} - 4q^{57} + 528q^{58} + 124q^{59} + 440q^{60} - 68q^{61} + 216q^{62} - 232q^{64} - 8q^{65} - 580q^{66} - 164q^{67} - 368q^{68} + 188q^{69} - 664q^{70} - 260q^{71} - 748q^{72} - 4q^{73} - 532q^{74} - 488q^{75} - 516q^{76} + 220q^{77} - 236q^{78} - 520q^{79} + 312q^{80} + 636q^{82} - 484q^{83} + 992q^{84} + 96q^{85} + 688q^{86} - 452q^{87} + 672q^{88} - 4q^{89} + 872q^{90} - 196q^{91} + 616q^{92} + 32q^{93} + 40q^{94} - 128q^{96} - 8q^{97} - 328q^{98} + 216q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.96275 0.384217i −1.10785 + 2.67458i 3.70475 + 1.50824i 2.95565 + 7.13556i 3.20204 4.82387i −4.18452 4.18452i −6.69200 4.38373i 0.437918 + 0.437918i −3.05958 15.1409i
3.2 −1.46783 + 1.35848i 2.10187 5.07436i 0.309042 3.98804i 1.74699 + 4.21761i 3.80826 + 10.3037i −0.392379 0.392379i 4.96407 + 6.27359i −14.9674 14.9674i −8.29385 3.81747i
3.3 −1.44490 1.38284i 0.936461 2.26082i 0.175499 + 3.99615i −3.18221 7.68254i −4.47945 + 1.97169i 3.67370 + 3.67370i 5.27246 6.01674i 2.12963 + 2.12963i −6.02574 + 15.5010i
3.4 −0.658450 + 1.88850i −1.31872 + 3.18367i −3.13289 2.48697i −0.659338 1.59178i −5.14406 4.58669i 9.54718 + 9.54718i 6.75950 4.27892i −2.03276 2.03276i 3.44023 0.197052i
3.5 0.682385 1.87999i 0.299792 0.723762i −3.06870 2.56575i 1.34740 + 3.25291i −1.15609 1.05749i 0.583225 + 0.583225i −6.91761 + 4.01829i 5.93000 + 5.93000i 7.03487 0.313357i
3.6 1.20513 + 1.59614i 0.527719 1.27403i −1.09531 + 3.84712i −0.642823 1.55191i 2.66949 0.693061i −4.95044 4.95044i −7.46052 + 2.88803i 5.01930 + 5.01930i 1.70238 2.89629i
3.7 1.93931 0.488972i −1.73217 + 4.18183i 3.52181 1.89653i −1.85856 4.48696i −1.31441 + 8.95683i −5.27676 5.27676i 5.90252 5.40002i −8.12333 8.12333i −5.79831 7.79280i
11.1 −1.96275 + 0.384217i −1.10785 2.67458i 3.70475 1.50824i 2.95565 7.13556i 3.20204 + 4.82387i −4.18452 + 4.18452i −6.69200 + 4.38373i 0.437918 0.437918i −3.05958 + 15.1409i
11.2 −1.46783 1.35848i 2.10187 + 5.07436i 0.309042 + 3.98804i 1.74699 4.21761i 3.80826 10.3037i −0.392379 + 0.392379i 4.96407 6.27359i −14.9674 + 14.9674i −8.29385 + 3.81747i
11.3 −1.44490 + 1.38284i 0.936461 + 2.26082i 0.175499 3.99615i −3.18221 + 7.68254i −4.47945 1.97169i 3.67370 3.67370i 5.27246 + 6.01674i 2.12963 2.12963i −6.02574 15.5010i
11.4 −0.658450 1.88850i −1.31872 3.18367i −3.13289 + 2.48697i −0.659338 + 1.59178i −5.14406 + 4.58669i 9.54718 9.54718i 6.75950 + 4.27892i −2.03276 + 2.03276i 3.44023 + 0.197052i
11.5 0.682385 + 1.87999i 0.299792 + 0.723762i −3.06870 + 2.56575i 1.34740 3.25291i −1.15609 + 1.05749i 0.583225 0.583225i −6.91761 4.01829i 5.93000 5.93000i 7.03487 + 0.313357i
11.6 1.20513 1.59614i 0.527719 + 1.27403i −1.09531 3.84712i −0.642823 + 1.55191i 2.66949 + 0.693061i −4.95044 + 4.95044i −7.46052 2.88803i 5.01930 5.01930i 1.70238 + 2.89629i
11.7 1.93931 + 0.488972i −1.73217 4.18183i 3.52181 + 1.89653i −1.85856 + 4.48696i −1.31441 8.95683i −5.27676 + 5.27676i 5.90252 + 5.40002i −8.12333 + 8.12333i −5.79831 + 7.79280i
19.1 −1.98676 0.229757i 1.58190 + 0.655246i 3.89442 + 0.912943i 4.18866 1.73500i −2.99232 1.66527i 3.93197 + 3.93197i −7.52753 2.70857i −4.29089 4.29089i −8.72048 + 2.48465i
19.2 −1.62478 + 1.16623i −4.68670 1.94129i 1.27980 3.78974i −4.51028 + 1.86822i 9.87885 2.31161i −3.85317 3.85317i 2.34032 + 7.65003i 11.8326 + 11.8326i 5.14942 8.29547i
19.3 −0.360897 + 1.96717i 2.49683 + 1.03422i −3.73951 1.41989i −0.452310 + 0.187353i −2.93558 + 4.53843i 0.429965 + 0.429965i 4.14274 6.84381i −1.19943 1.19943i −0.205317 0.957385i
19.4 −0.108191 1.99707i 4.35131 + 1.80237i −3.97659 + 0.432130i −2.81639 + 1.16659i 3.12870 8.88489i −6.23443 6.23443i 1.29322 + 7.89478i 9.32143 + 9.32143i 2.63447 + 5.49832i
19.5 0.345994 1.96984i −3.70255 1.53365i −3.76058 1.36311i 7.20074 2.98264i −4.30210 + 6.76281i 4.26150 + 4.26150i −3.98625 + 6.93612i 4.99283 + 4.99283i −3.38393 15.2163i
19.6 1.61758 + 1.17620i −1.37292 0.568682i 1.23313 + 3.80518i 2.28872 0.948019i −1.55193 2.53471i −6.37744 6.37744i −2.48095 + 7.60558i −4.80245 4.80245i 4.81725 + 1.15849i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.7 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 32.3.h.a 28
3.b odd 2 1 288.3.u.a 28
4.b odd 2 1 128.3.h.a 28
8.b even 2 1 256.3.h.b 28
8.d odd 2 1 256.3.h.a 28
32.g even 8 1 128.3.h.a 28
32.g even 8 1 256.3.h.a 28
32.h odd 8 1 inner 32.3.h.a 28
32.h odd 8 1 256.3.h.b 28
96.o even 8 1 288.3.u.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.h.a 28 1.a even 1 1 trivial
32.3.h.a 28 32.h odd 8 1 inner
128.3.h.a 28 4.b odd 2 1
128.3.h.a 28 32.g even 8 1
256.3.h.a 28 8.d odd 2 1
256.3.h.a 28 32.g even 8 1
256.3.h.b 28 8.b even 2 1
256.3.h.b 28 32.h odd 8 1
288.3.u.a 28 3.b odd 2 1
288.3.u.a 28 96.o even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database