# Properties

 Label 731.2.e.b Level 731 Weight 2 Character orbit 731.e Analytic conductor 5.837 Analytic rank 0 Dimension 58 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.e (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$58$$ Relative dimension: $$29$$ over $$\Q(\zeta_{3})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$58q + 2q^{2} - 5q^{3} + 54q^{4} - 3q^{5} - 8q^{6} - 13q^{7} + 12q^{8} - 30q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$58q + 2q^{2} - 5q^{3} + 54q^{4} - 3q^{5} - 8q^{6} - 13q^{7} + 12q^{8} - 30q^{9} + 4q^{10} - 8q^{12} + 2q^{13} + 5q^{14} - 5q^{15} + 30q^{16} - 29q^{17} + 16q^{18} - 4q^{19} - 7q^{20} - 26q^{21} - 10q^{22} + 5q^{23} - 28q^{24} - 24q^{25} - 8q^{26} + 28q^{27} - 25q^{28} + 10q^{29} - 5q^{30} + 3q^{31} + 16q^{32} - 23q^{33} - q^{34} - 26q^{35} - 39q^{36} - 16q^{37} + q^{38} + 130q^{39} + 15q^{40} - 22q^{41} + 112q^{42} + 7q^{43} + 24q^{44} + 2q^{45} - 61q^{46} + 12q^{47} + 22q^{48} - 36q^{49} - 17q^{50} + 10q^{51} - 35q^{52} - 10q^{53} - 10q^{54} - 20q^{55} - 4q^{56} + 9q^{57} - 17q^{58} - 36q^{59} + 22q^{60} - 35q^{61} + 3q^{62} - 22q^{63} + 28q^{64} - 28q^{65} + 14q^{66} - 17q^{67} - 27q^{68} - 23q^{69} + 10q^{70} - 6q^{71} + 17q^{72} - 14q^{73} + 21q^{74} + 70q^{75} - 44q^{76} - 11q^{77} - 184q^{78} - 10q^{79} + 8q^{80} - 13q^{81} + 184q^{82} + 9q^{83} - 36q^{84} + 6q^{85} - 76q^{86} + 8q^{87} + 50q^{88} - 54q^{89} - 34q^{90} - 23q^{91} - 66q^{92} + 96q^{94} - 3q^{95} - 60q^{96} + 36q^{97} - 16q^{98} + 20q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1 −2.61335 1.52887 2.64809i 4.82960 −0.766598 + 1.32779i −3.99548 + 6.92038i −0.834694 1.44573i −7.39472 −3.17491 5.49911i 2.00339 3.46997i
307.2 −2.46945 −0.974328 + 1.68758i 4.09816 0.310056 0.537033i 2.40605 4.16740i 1.71828 + 2.97615i −5.18130 −0.398628 0.690445i −0.765666 + 1.32617i
307.3 −2.46652 0.742625 1.28626i 4.08373 0.428499 0.742181i −1.83170 + 3.17260i −2.43048 4.20972i −5.13956 0.397017 + 0.687654i −1.05690 + 1.83061i
307.4 −2.24624 −0.247428 + 0.428559i 3.04561 −0.237822 + 0.411920i 0.555785 0.962647i −0.154590 0.267757i −2.34870 1.37756 + 2.38600i 0.534206 0.925272i
307.5 −2.08718 −1.33964 + 2.32032i 2.35632 −2.10822 + 3.65155i 2.79606 4.84292i 0.688078 + 1.19179i −0.743705 −2.08925 3.61869i 4.40024 7.62144i
307.6 −1.72289 −1.55986 + 2.70176i 0.968349 1.14718 1.98697i 2.68747 4.65483i −2.28646 3.96026i 1.77742 −3.36633 5.83066i −1.97646 + 3.42332i
307.7 −1.53983 0.868079 1.50356i 0.371078 0.736400 1.27548i −1.33669 + 2.31522i 0.767588 + 1.32950i 2.50826 −0.00712137 0.0123346i −1.13393 + 1.96403i
307.8 −1.48275 −0.591829 + 1.02508i 0.198553 0.830949 1.43925i 0.877535 1.51994i −0.706420 1.22356i 2.67110 0.799477 + 1.38474i −1.23209 + 2.13404i
307.9 −1.32448 1.04438 1.80892i −0.245749 1.40548 2.43436i −1.38326 + 2.39588i 0.178508 + 0.309184i 2.97445 −0.681454 1.18031i −1.86153 + 3.22427i
307.10 −1.12544 1.19443 2.06882i −0.733389 −2.16193 + 3.74458i −1.34426 + 2.32833i −1.34379 2.32751i 3.07626 −1.35334 2.34406i 2.43312 4.21429i
307.11 −1.09472 0.124885 0.216307i −0.801596 −1.31747 + 2.28193i −0.136714 + 0.236795i 0.589522 + 1.02108i 3.06695 1.46881 + 2.54405i 1.44226 2.49806i
307.12 −0.517241 −1.48646 + 2.57463i −1.73246 0.969459 1.67915i 0.768860 1.33171i 1.00663 + 1.74354i 1.93058 −2.91915 5.05612i −0.501444 + 0.868526i
307.13 −0.482661 −0.0594428 + 0.102958i −1.76704 0.220118 0.381256i 0.0286908 0.0496939i −2.49924 4.32882i 1.81820 1.49293 + 2.58584i −0.106243 + 0.184018i
307.14 −0.267888 −0.791039 + 1.37012i −1.92824 −1.21931 + 2.11190i 0.211910 0.367038i 2.34329 + 4.05870i 1.05233 0.248515 + 0.430440i 0.326638 0.565753i
307.15 0.0823862 0.281246 0.487132i −1.99321 −0.854403 + 1.47987i 0.0231708 0.0401330i −0.768980 1.33191i −0.328986 1.34180 + 2.32407i −0.0703910 + 0.121921i
307.16 0.175605 1.59696 2.76601i −1.96916 1.26069 2.18359i 0.280434 0.485725i −0.781954 1.35438i −0.697005 −3.60054 6.23632i 0.221384 0.383449i
307.17 0.428243 −1.51618 + 2.62610i −1.81661 −1.16373 + 2.01564i −0.649294 + 1.12461i −0.814374 1.41054i −1.63444 −3.09761 5.36521i −0.498360 + 0.863184i
307.18 0.504225 0.395848 0.685630i −1.74576 1.62259 2.81041i 0.199597 0.345712i 1.77182 + 3.06888i −1.88870 1.18661 + 2.05527i 0.818151 1.41708i
307.19 1.34037 −0.719541 + 1.24628i −0.203418 2.16961 3.75787i −0.964449 + 1.67047i −1.07792 1.86701i −2.95339 0.464522 + 0.804575i 2.90807 5.03692i
307.20 1.35318 1.03357 1.79019i −0.168903 0.832416 1.44179i 1.39860 2.42245i −1.69298 2.93233i −2.93492 −0.636523 1.10249i 1.12641 1.95100i
See all 58 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 681.29 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 $$T_{2}^{29} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.