# Properties

 Label 731.2.s.b Level 731 Weight 2 Character orbit 731.s Analytic conductor 5.837 Analytic rank 0 Dimension 496 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.s (of order $$16$$ and degree $$8$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$496$$ Relative dimension: $$62$$ over $$\Q(\zeta_{16})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$496q - 16q^{4} - 16q^{6} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$496q - 16q^{4} - 16q^{6} - 16q^{9} - 16q^{10} + 16q^{13} - 16q^{14} - 16q^{15} - 16q^{17} - 16q^{21} - 80q^{23} - 64q^{24} + 48q^{31} - 96q^{35} + 16q^{36} - 32q^{38} - 16q^{40} - 16q^{41} - 48q^{43} - 80q^{44} - 96q^{47} - 16q^{49} + 96q^{52} - 16q^{53} + 144q^{54} - 16q^{56} - 48q^{57} - 64q^{58} + 16q^{59} - 224q^{60} - 48q^{64} - 272q^{66} - 16q^{68} - 112q^{74} + 64q^{78} - 80q^{79} + 96q^{81} + 16q^{83} + 96q^{86} - 16q^{87} + 64q^{90} + 368q^{92} - 96q^{95} + 80q^{96} - 80q^{97} - 128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
214.1 −2.59137 1.07338i 1.43157 2.14250i 4.14883 + 4.14883i 0.218697 + 1.09946i −6.00945 + 4.01538i 0.509391 2.56088i −4.15111 10.0217i −1.39286 3.36265i 0.613418 3.08386i
214.2 −2.49245 1.03241i −1.39302 + 2.08480i 3.73222 + 3.73222i 0.435881 + 2.19132i 5.62439 3.75810i −0.139368 + 0.700650i −3.38438 8.17063i −1.25784 3.03670i 1.17592 5.91177i
214.3 −2.43313 1.00784i −0.227777 + 0.340892i 3.49018 + 3.49018i −0.210988 1.06071i 0.897773 0.599873i −0.282349 + 1.41946i −2.95886 7.14332i 1.08373 + 2.61634i −0.555659 + 2.79349i
214.4 −2.39134 0.990524i 1.43792 2.15200i 3.32313 + 3.32313i −0.286571 1.44069i −5.57015 + 3.72185i −0.790858 + 3.97591i −2.67404 6.45570i −1.41543 3.41716i −0.741749 + 3.72902i
214.5 −2.24727 0.930848i 0.466380 0.697987i 2.76951 + 2.76951i 0.751697 + 3.77904i −1.69780 + 1.13443i −0.800996 + 4.02688i −1.78414 4.30730i 0.878375 + 2.12058i 1.82845 9.19222i
214.6 −2.22769 0.922741i −0.269266 + 0.402985i 2.69695 + 2.69695i 0.606709 + 3.05013i 0.971692 0.649264i 0.588740 2.95980i −1.67391 4.04118i 1.05816 + 2.55462i 1.46292 7.35460i
214.7 −2.19579 0.909528i −1.68196 + 2.51724i 2.58006 + 2.58006i −0.588645 2.95932i 5.98274 3.99754i 0.916876 4.60945i −1.49959 3.62032i −2.35943 5.69616i −1.39904 + 7.03344i
214.8 −2.07104 0.857855i −0.0129146 + 0.0193280i 2.13910 + 2.13910i −0.719326 3.61630i 0.0433273 0.0289504i −0.318313 + 1.60027i −0.879419 2.12311i 1.14784 + 2.77114i −1.61250 + 8.10659i
214.9 −2.04983 0.849068i 0.990599 1.48254i 2.06668 + 2.06668i 0.292642 + 1.47121i −3.28933 + 2.19786i 0.459436 2.30974i −0.783452 1.89142i −0.0685766 0.165558i 0.649291 3.26421i
214.10 −2.03061 0.841105i 1.24216 1.85903i 2.00170 + 2.00170i −0.565245 2.84168i −4.08599 + 2.73017i 0.401952 2.02075i −0.698811 1.68708i −0.764972 1.84681i −1.24236 + 6.24577i
214.11 −1.93285 0.800613i 0.211222 0.316117i 1.68072 + 1.68072i −0.167026 0.839698i −0.661349 + 0.441899i 0.314174 1.57946i −0.301744 0.728474i 1.09274 + 2.63810i −0.349436 + 1.75673i
214.12 −1.90221 0.787922i −1.35593 + 2.02929i 1.58338 + 1.58338i −0.106216 0.533986i 4.17819 2.79178i −0.224543 + 1.12885i −0.188497 0.455072i −1.13143 2.73151i −0.218693 + 1.09945i
214.13 −1.67323 0.693076i −1.35744 + 2.03155i 0.905142 + 0.905142i 0.494506 + 2.48605i 3.67934 2.45845i 0.453283 2.27881i 0.498970 + 1.20462i −1.13651 2.74379i 0.895597 4.50247i
214.14 −1.55591 0.644478i −0.529491 + 0.792439i 0.591285 + 0.591285i −0.134545 0.676403i 1.33455 0.891717i −0.834241 + 4.19401i 0.750042 + 1.81076i 0.800452 + 1.93246i −0.226588 + 1.13913i
214.15 −1.44571 0.598831i −0.336754 + 0.503988i 0.317255 + 0.317255i 0.505403 + 2.54083i 0.788652 0.526960i 0.214808 1.07991i 0.928987 + 2.24277i 1.00745 + 2.43220i 0.790865 3.97595i
214.16 −1.35638 0.561829i −0.901163 + 1.34869i 0.109890 + 0.109890i −0.614380 3.08870i 1.98005 1.32302i 0.564221 2.83653i 1.03635 + 2.50196i 0.141192 + 0.340868i −0.901990 + 4.53461i
214.17 −1.34652 0.557745i 0.877192 1.31281i 0.0878103 + 0.0878103i 0.0560935 + 0.282001i −1.91337 + 1.27847i −0.596391 + 2.99826i 1.04623 + 2.52582i 0.194044 + 0.468464i 0.0817538 0.411004i
214.18 −1.30999 0.542617i 1.88023 2.81397i 0.00743503 + 0.00743503i −0.424086 2.13202i −3.99000 + 2.66603i 0.264470 1.32958i 1.07953 + 2.60621i −3.23509 7.81019i −0.601323 + 3.02305i
214.19 −1.25875 0.521390i 1.34853 2.01822i −0.101617 0.101617i 0.174697 + 0.878263i −2.74974 + 1.83732i −0.443684 + 2.23055i 1.11771 + 2.69839i −1.10663 2.67164i 0.238018 1.19660i
214.20 −1.16026 0.480596i −0.484485 + 0.725082i −0.298982 0.298982i 0.671308 + 3.37489i 0.910600 0.608443i −0.264895 + 1.33172i 1.16440 + 2.81111i 0.857031 + 2.06906i 0.843067 4.23838i
See next 80 embeddings (of 496 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 687.62 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 $$T_{2}^{496} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$.