# Properties

 Label 731.2.k.b Level 731 Weight 2 Character orbit 731.k Analytic conductor 5.837 Analytic rank 0 Dimension 180 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.k (of order $$7$$ and degree $$6$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$180$$ Relative dimension: $$30$$ over $$\Q(\zeta_{7})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$180q + 2q^{3} - 34q^{4} + 2q^{5} + 6q^{6} - 54q^{7} + 14q^{8} - 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$180q + 2q^{3} - 34q^{4} + 2q^{5} + 6q^{6} - 54q^{7} + 14q^{8} - 18q^{9} - 8q^{10} + 4q^{11} - 4q^{12} - 5q^{13} + 24q^{14} + 11q^{15} - 58q^{16} + 30q^{17} - 84q^{18} - 4q^{20} - 28q^{21} - 49q^{22} - 23q^{23} - 6q^{24} - 14q^{25} + 14q^{26} + 20q^{27} - 14q^{28} - 60q^{29} - 5q^{30} + 36q^{31} + 39q^{32} - 39q^{33} - 14q^{35} + 224q^{36} - 184q^{37} + 32q^{38} + 58q^{39} + 22q^{40} - 48q^{41} - 58q^{42} + 2q^{43} + 134q^{44} - 54q^{45} - 5q^{46} - 35q^{47} + 44q^{48} + 174q^{49} - 70q^{50} - 2q^{51} - 22q^{52} + 8q^{53} + 70q^{54} + 51q^{55} - 61q^{56} + 72q^{57} + 29q^{58} + 35q^{59} + 96q^{60} - 40q^{61} + 20q^{62} - 35q^{63} - 18q^{64} - 17q^{65} - 218q^{66} + 16q^{67} + 27q^{68} + 50q^{69} + 72q^{70} - 20q^{71} - 143q^{72} - 4q^{73} - 35q^{74} - 45q^{75} - 148q^{76} + 40q^{77} - 220q^{78} - 12q^{79} - 222q^{80} + 12q^{81} + 50q^{82} + 16q^{83} - 170q^{84} - 2q^{85} + 108q^{86} + 78q^{87} - 14q^{88} + 55q^{89} + 105q^{90} - 53q^{91} - 28q^{92} - 142q^{93} + 108q^{94} - 49q^{95} - 148q^{96} + 73q^{97} + 6q^{98} - 73q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −0.609444 + 2.67015i 0.133331 + 0.584162i −4.95634 2.38685i −0.259403 + 0.325281i −1.64106 1.45101 5.97861 7.49695i 2.37944 1.14588i −0.710458 0.890886i
35.2 −0.560314 + 2.45489i −0.473877 2.07619i −3.91062 1.88326i −2.10087 + 2.63441i 5.36236 2.59720 3.67444 4.60760i −1.38311 + 0.666071i −5.29005 6.63352i
35.3 −0.520342 + 2.27977i 0.631123 + 2.76513i −3.12464 1.50475i 0.571583 0.716742i −6.63225 0.491347 2.14042 2.68400i −4.54473 + 2.18863i 1.33659 + 1.67603i
35.4 −0.487718 + 2.13683i 0.382826 + 1.67727i −2.52625 1.21658i 2.58588 3.24259i −3.77076 −1.56444 1.09861 1.37762i 0.0362221 0.0174436i 5.66770 + 7.10707i
35.5 −0.464976 + 2.03719i 0.214156 + 0.938278i −2.13202 1.02673i −1.50470 + 1.88683i −2.01103 −4.68221 0.477312 0.598530i 1.86840 0.899776i −3.14419 3.94269i
35.6 −0.452186 + 1.98116i −0.537392 2.35447i −1.91857 0.923934i 0.431020 0.540482i 4.90757 −1.66996 0.164014 0.205667i −2.55183 + 1.22889i 0.875878 + 1.09832i
35.7 −0.437378 + 1.91628i −0.377133 1.65233i −1.67888 0.808507i 1.35653 1.70103i 3.33126 −1.53495 −0.167382 + 0.209890i 0.114954 0.0553592i 2.66633 + 3.34348i
35.8 −0.335627 + 1.47048i −0.201066 0.880929i −0.247722 0.119297i −1.49293 + 1.87208i 1.36287 3.95170 −1.62225 + 2.03423i 1.96730 0.947401i −2.25178 2.82365i
35.9 −0.317607 + 1.39153i 0.469666 + 2.05774i −0.0335331 0.0161487i −2.08668 + 2.61661i −3.01257 −1.13533 −1.74671 + 2.19030i −1.31080 + 0.631250i −2.97834 3.73472i
35.10 −0.263173 + 1.15304i 0.751127 + 3.29090i 0.541703 + 0.260871i 0.905581 1.13556i −3.99221 4.03940 −1.91815 + 2.40528i −7.56295 + 3.64213i 1.07102 + 1.34302i
35.11 −0.224709 + 0.984515i 0.253079 + 1.10881i 0.883161 + 0.425308i −0.00986560 + 0.0123711i −1.14851 1.32938 −1.87642 + 2.35296i 1.53749 0.740415i −0.00996262 0.0124927i
35.12 −0.212222 + 0.929804i −0.151920 0.665606i 0.982440 + 0.473118i 2.43514 3.05357i 0.651124 4.10878 −1.83767 + 2.30436i 2.28296 1.09941i 2.32243 + 2.91224i
35.13 −0.0957949 + 0.419705i −0.643935 2.82126i 1.63496 + 0.787356i 0.574144 0.719953i 1.24578 1.75614 −1.02390 + 1.28393i −4.84197 + 2.33177i 0.247168 + 0.309939i
35.14 −0.0946208 + 0.414561i −0.485779 2.12833i 1.63903 + 0.789315i −0.816059 + 1.02331i 0.928289 −4.53691 −1.01255 + 1.26970i −1.59092 + 0.766147i −0.347007 0.435132i
35.15 −0.0268087 + 0.117457i 0.561962 + 2.46211i 1.78886 + 0.861470i −1.64843 + 2.06707i −0.304257 −2.08745 −0.299375 + 0.375405i −3.04330 + 1.46558i −0.198599 0.249035i
35.16 0.0248321 0.108796i 0.413289 + 1.81074i 1.79072 + 0.862364i 1.10115 1.38080i 0.207265 −1.59919 0.277445 0.347905i −0.405063 + 0.195068i −0.122882 0.154090i
35.17 0.0787180 0.344886i −0.0523067 0.229171i 1.68919 + 0.813470i −0.573925 + 0.719679i −0.0831553 0.480778 0.854650 1.07170i 2.65312 1.27768i 0.203029 + 0.254590i
35.18 0.195877 0.858195i 0.00959066 + 0.0420194i 1.10381 + 0.531566i 1.96536 2.46448i 0.0379394 −5.12558 1.77007 2.21960i 2.70123 1.30085i −1.73003 2.16939i
35.19 0.205703 0.901243i −0.218156 0.955803i 1.03201 + 0.496991i 0.408806 0.512626i −0.906286 4.36242 1.81293 2.27334i 1.83694 0.884623i −0.377908 0.473882i
35.20 0.231641 1.01488i −0.681367 2.98526i 0.825605 + 0.397590i 2.03864 2.55637i −3.18753 −0.963834 1.89284 2.37354i −5.74463 + 2.76647i −2.12219 2.66115i
See next 80 embeddings (of 180 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 613.30 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 $$T_{2}^{180} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.