# Properties

 Label 731.2.m.b Level 731 Weight 2 Character orbit 731.m Analytic conductor 5.837 Analytic rank 0 Dimension 116 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.m (of order $$8$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$116$$ Relative dimension: $$29$$ over $$\Q(\zeta_{8})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$116q - 8q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$116q - 8q^{6} - 8q^{10} + 8q^{14} + 4q^{15} - 68q^{16} - 4q^{17} - 44q^{18} + 12q^{19} + 8q^{20} - 16q^{22} - 28q^{23} - 12q^{24} - 4q^{25} - 8q^{26} + 24q^{28} + 80q^{33} + 32q^{34} - 112q^{35} + 160q^{36} - 20q^{37} + 8q^{39} - 112q^{40} + 8q^{41} + 4q^{42} + 32q^{44} - 52q^{45} - 40q^{46} + 40q^{48} + 8q^{49} + 100q^{50} - 32q^{51} - 152q^{52} + 28q^{53} - 36q^{54} + 124q^{56} - 104q^{57} - 32q^{58} - 36q^{59} - 24q^{60} + 52q^{61} - 68q^{62} + 20q^{63} + 20q^{65} - 60q^{66} + 64q^{67} - 128q^{69} + 188q^{70} + 52q^{73} - 104q^{74} + 36q^{75} - 112q^{76} + 28q^{77} + 56q^{78} - 108q^{79} - 44q^{80} + 52q^{82} - 52q^{83} + 120q^{84} + 12q^{85} - 20q^{86} + 56q^{87} + 36q^{88} + 144q^{90} - 16q^{92} - 176q^{93} - 8q^{94} + 164q^{95} - 164q^{96} - 8q^{97} - 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
87.1 −1.95693 1.95693i −2.36371 + 0.979079i 5.65912i 0.741144 + 1.78928i 6.54158 + 2.70961i −1.36877 + 3.30450i 7.16062 7.16062i 2.50719 2.50719i 2.05112 4.95185i
87.2 −1.81709 1.81709i −2.81259 + 1.16501i 4.60365i −1.37650 3.32317i 7.22768 + 2.99380i 0.758828 1.83197i 4.73108 4.73108i 4.43209 4.43209i −3.53728 + 8.53975i
87.3 −1.75194 1.75194i 1.70725 0.707167i 4.13857i −1.12861 2.72471i −4.22991 1.75209i −1.06160 + 2.56293i 3.74664 3.74664i 0.293302 0.293302i −2.79626 + 6.75077i
87.4 −1.65901 1.65901i 1.85997 0.770425i 3.50461i −1.10929 2.67805i −4.36385 1.80756i 1.56474 3.77763i 2.49616 2.49616i 0.744616 0.744616i −2.60260 + 6.28322i
87.5 −1.60811 1.60811i −0.921811 + 0.381827i 3.17206i 0.244536 + 0.590361i 2.09640 + 0.868356i −1.44759 + 3.49480i 1.88481 1.88481i −1.41738 + 1.41738i 0.556127 1.34261i
87.6 −1.52705 1.52705i 0.458286 0.189828i 2.66378i 1.62594 + 3.92537i −0.989705 0.409949i −0.150783 + 0.364023i 1.01362 1.01362i −1.94733 + 1.94733i 3.51135 8.47714i
87.7 −1.35151 1.35151i 2.57265 1.06563i 1.65316i 1.19002 + 2.87295i −4.91716 2.03675i 1.13248 2.73404i −0.468760 + 0.468760i 3.36163 3.36163i 2.27450 5.49114i
87.8 −1.09522 1.09522i 2.12589 0.880571i 0.399034i −0.733376 1.77053i −3.29275 1.36390i −0.642781 + 1.55181i −1.75342 + 1.75342i 1.62267 1.62267i −1.13591 + 2.74234i
87.9 −1.00590 1.00590i 0.0816782 0.0338322i 0.0236750i 0.261308 + 0.630853i −0.116192 0.0481283i 0.939495 2.26814i −1.98799 + 1.98799i −2.11579 + 2.11579i 0.371726 0.897426i
87.10 −1.00313 1.00313i −1.37786 + 0.570727i 0.0125460i −0.979244 2.36410i 1.95469 + 0.809657i 1.38293 3.33870i −1.99368 + 1.99368i −0.548561 + 0.548561i −1.38920 + 3.35382i
87.11 −0.993125 0.993125i −1.41917 + 0.587838i 0.0274062i 0.967808 + 2.33649i 1.99321 + 0.825613i 0.386865 0.933975i −2.01347 + 2.01347i −0.452840 + 0.452840i 1.35928 3.28158i
87.12 −0.740178 0.740178i 2.21650 0.918106i 0.904273i 0.764492 + 1.84565i −2.32017 0.961046i −2.00667 + 4.84454i −2.14968 + 2.14968i 1.94865 1.94865i 0.800248 1.93197i
87.13 −0.482959 0.482959i −1.45483 + 0.602612i 1.53350i 1.15531 + 2.78915i 0.993661 + 0.411588i −1.39214 + 3.36091i −1.70653 + 1.70653i −0.367918 + 0.367918i 0.789081 1.90501i
87.14 −0.292872 0.292872i 2.84462 1.17828i 1.82845i 0.0154981 + 0.0374157i −1.17820 0.488025i −0.341796 + 0.825169i −1.12125 + 1.12125i 4.58219 4.58219i 0.00641907 0.0154970i
87.15 −0.0607158 0.0607158i −1.28452 + 0.532068i 1.99263i −0.430997 1.04052i 0.110296 + 0.0456861i −0.993514 + 2.39855i −0.242416 + 0.242416i −0.754412 + 0.754412i −0.0370077 + 0.0893444i
87.16 −0.0400748 0.0400748i 0.683480 0.283106i 1.99679i −0.594750 1.43585i −0.0387357 0.0160449i 1.79398 4.33104i −0.160170 + 0.160170i −1.73433 + 1.73433i −0.0337070 + 0.0813759i
87.17 0.0756963 + 0.0756963i 0.293333 0.121503i 1.98854i −0.559792 1.35146i 0.0314015 + 0.0130069i −1.02421 + 2.47265i 0.301918 0.301918i −2.05004 + 2.05004i 0.0599261 0.144674i
87.18 0.544996 + 0.544996i −1.90994 + 0.791125i 1.40596i 0.670095 + 1.61775i −1.47207 0.609752i 1.29182 3.11873i 1.85623 1.85623i 0.900688 0.900688i −0.516470 + 1.24687i
87.19 0.598059 + 0.598059i −1.51219 + 0.626370i 1.28465i −1.53826 3.71369i −1.27899 0.529773i −0.409939 + 0.989680i 1.96441 1.96441i −0.226937 + 0.226937i 1.30103 3.14098i
87.20 0.629885 + 0.629885i −0.623684 + 0.258339i 1.20649i 1.24172 + 2.99777i −0.555573 0.230126i 1.05484 2.54660i 2.01972 2.01972i −1.79908 + 1.79908i −1.10611 + 2.67039i
See next 80 embeddings (of 116 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 689.29 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{116} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.