# Properties

 Label 731.2.u.b Level 731 Weight 2 Character orbit 731.u Analytic conductor 5.837 Analytic rank 0 Dimension 348 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.u (of order $$21$$ and degree $$12$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$348$$ Relative dimension: $$29$$ over $$\Q(\zeta_{21})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$348q + 6q^{2} - 3q^{3} - 54q^{4} + q^{5} - 12q^{6} + 49q^{7} - 2q^{8} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$348q + 6q^{2} - 3q^{3} - 54q^{4} + q^{5} - 12q^{6} + 49q^{7} - 2q^{8} + 8q^{9} - 4q^{10} - 16q^{11} - 12q^{12} - 9q^{13} - 3q^{14} - 14q^{15} - 30q^{16} - 29q^{17} - 204q^{18} - 8q^{19} + 61q^{20} + 26q^{21} + 85q^{22} - 79q^{23} - 12q^{24} + 22q^{25} - 14q^{26} + 12q^{27} - 15q^{28} + 54q^{29} - 46q^{30} - 52q^{31} + 33q^{32} + 39q^{33} + 3q^{34} + 32q^{35} - 161q^{36} + 68q^{37} - 29q^{38} + 61q^{39} + 107q^{40} + 63q^{41} - 56q^{42} - 45q^{43} - 74q^{44} + 144q^{45} - 22q^{46} - 70q^{47} + 168q^{48} - 119q^{49} + 97q^{50} - 6q^{51} + 27q^{52} + 10q^{53} - 34q^{54} - 78q^{55} - 44q^{56} - 40q^{57} - 38q^{58} - 47q^{59} - 72q^{60} - 76q^{61} - 41q^{62} + 63q^{63} - 112q^{64} - 31q^{65} - 598q^{66} - 39q^{67} - 20q^{68} - 17q^{69} - 72q^{70} - 37q^{71} + 245q^{72} + 14q^{73} + 77q^{74} - 121q^{75} + 152q^{76} + 59q^{77} + 220q^{78} + 4q^{79} - 42q^{80} - 109q^{81} + 148q^{82} + 59q^{83} + 390q^{84} + 2q^{85} + 12q^{86} - 78q^{87} + 344q^{88} - 103q^{89} - 75q^{90} + 148q^{91} + 28q^{92} + 140q^{93} + 24q^{94} - 302q^{95} + 22q^{96} + 57q^{97} - 72q^{98} - 128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
52.1 −1.74998 2.19441i 0.963898 + 0.145284i −1.30795 + 5.73049i 0.755186 + 0.514877i −1.36799 2.36943i −0.932904 + 1.61584i 9.80631 4.72247i −1.95873 0.604187i −0.191712 2.55821i
52.2 −1.67210 2.09675i 2.29142 + 0.345377i −1.15539 + 5.06209i −2.26489 1.54418i −3.10732 5.38204i 2.23836 3.87695i 7.71335 3.71455i 2.26462 + 0.698543i 0.549377 + 7.33093i
52.3 −1.50074 1.88187i −1.04625 0.157697i −0.844170 + 3.69855i −1.03583 0.706220i 1.27338 + 2.20557i −1.99658 + 3.45819i 3.88980 1.87323i −1.79695 0.554286i 0.225505 + 3.00916i
52.4 −1.43554 1.80010i −3.00731 0.453278i −0.734572 + 3.21837i 1.89413 + 1.29140i 3.50115 + 6.06416i −1.09737 + 1.90070i 2.69909 1.29981i 5.97171 + 1.84203i −0.394444 5.26349i
52.5 −1.32549 1.66211i 2.59933 + 0.391785i −0.560648 + 2.45636i 2.86311 + 1.95203i −2.79419 4.83967i 2.53065 4.38321i 0.995100 0.479215i 3.73628 + 1.15249i −0.550522 7.34620i
52.6 −1.15937 1.45380i −0.101371 0.0152793i −0.324361 + 1.42112i −3.42704 2.33652i 0.0953136 + 0.165088i 0.698039 1.20904i −0.908594 + 0.437556i −2.85668 0.881168i 0.576369 + 7.69111i
52.7 −1.04859 1.31489i 2.19207 + 0.330401i −0.184358 + 0.807723i 1.73827 + 1.18513i −1.86414 3.22879i −0.383582 + 0.664384i −1.77514 + 0.854860i 1.82928 + 0.564257i −0.264414 3.52837i
52.8 −1.02252 1.28220i −0.661263 0.0996693i −0.153446 + 0.672293i −0.0486121 0.0331432i 0.548358 + 0.949785i 0.748574 1.29657i −1.93625 + 0.932450i −2.43938 0.752450i 0.00721069 + 0.0962200i
52.9 −0.916857 1.14970i −1.68959 0.254664i −0.0361471 + 0.158371i 2.78996 + 1.90216i 1.25632 + 2.17601i 0.435372 0.754087i −2.43457 + 1.17243i −0.0768702 0.0237113i −0.371074 4.95163i
52.10 −0.636414 0.798038i −2.64174 0.398179i 0.213200 0.934089i −2.05536 1.40132i 1.36348 + 2.36162i −0.398801 + 0.690743i −2.72041 + 1.31008i 3.95355 + 1.21951i 0.189753 + 2.53208i
52.11 −0.614029 0.769968i 1.49724 + 0.225673i 0.229223 1.00429i 0.606327 + 0.413386i −0.745588 1.29140i 0.356455 0.617398i −2.68862 + 1.29477i −0.675919 0.208493i −0.0540078 0.720683i
52.12 −0.362567 0.454644i 1.29324 + 0.194925i 0.369795 1.62018i −2.61272 1.78132i −0.380264 0.658637i −1.55140 + 2.68710i −1.91853 + 0.923914i −1.23224 0.380097i 0.137417 + 1.83370i
52.13 −0.266678 0.334404i 3.15596 + 0.475684i 0.404333 1.77150i 2.29587 + 1.56530i −0.682553 1.18222i −2.60569 + 4.51320i −1.47094 + 0.708369i 6.86706 + 2.11821i −0.0888167 1.18518i
52.14 −0.251421 0.315272i 1.17665 + 0.177352i 0.408858 1.79132i −1.00912 0.688006i −0.239921 0.415555i 1.75229 3.03505i −1.39418 + 0.671401i −1.51366 0.466903i 0.0368048 + 0.491126i
52.15 −0.123719 0.155139i −1.15042 0.173398i 0.436280 1.91147i 1.32212 + 0.901405i 0.115429 + 0.199928i −1.26206 + 2.18595i −0.708080 + 0.340993i −1.57331 0.485303i −0.0237284 0.316634i
52.16 0.176668 + 0.221534i 3.18926 + 0.480704i 0.427176 1.87158i −2.99852 2.04436i 0.456948 + 0.791457i 0.233016 0.403596i 1.00067 0.481898i 7.07361 + 2.18192i −0.0768466 1.02545i
52.17 0.384419 + 0.482046i −0.444042 0.0669286i 0.360451 1.57924i 0.0789639 + 0.0538367i −0.138436 0.239777i −1.46965 + 2.54550i 2.01084 0.968367i −2.67402 0.824827i 0.00440346 + 0.0587601i
52.18 0.429469 + 0.538537i −2.21655 0.334092i 0.339463 1.48729i 2.91429 + 1.98693i −0.772020 1.33718i 1.74515 3.02268i 2.18795 1.05366i 1.93477 + 0.596798i 0.181562 + 2.42278i
52.19 0.443477 + 0.556103i 1.01721 + 0.153320i 0.332464 1.45662i 2.99169 + 2.03970i 0.365847 + 0.633666i 0.454177 0.786658i 2.23916 1.07832i −1.85551 0.572349i 0.192464 + 2.56825i
52.20 0.486709 + 0.610314i −1.78702 0.269350i 0.309445 1.35577i −3.20929 2.18806i −0.705372 1.22174i 2.22375 3.85164i 2.38468 1.14840i 0.254185 + 0.0784058i −0.226589 3.02362i
See next 80 embeddings (of 348 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 698.29 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 $$T_{2}^{348} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.