# Properties

 Label 731.2.y.a Level 731 Weight 2 Character orbit 731.y Analytic conductor 5.837 Analytic rank 0 Dimension 768 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.y (of order $$28$$ and degree $$12$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$768$$ Relative dimension: $$64$$ over $$\Q(\zeta_{28})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$768q - 14q^{3} + 104q^{4} - 6q^{5} - 36q^{6} - 28q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$768q - 14q^{3} + 104q^{4} - 6q^{5} - 36q^{6} - 28q^{7} - 18q^{10} - 4q^{11} + 34q^{12} + 4q^{13} + 26q^{14} - 152q^{16} - 10q^{17} - 24q^{18} + 22q^{20} - 20q^{21} + 12q^{22} - 24q^{23} - 100q^{24} + 10q^{27} - 42q^{28} - 18q^{29} + 48q^{30} + 4q^{31} + 36q^{33} - 4q^{34} - 60q^{35} + 40q^{37} + 12q^{38} + 34q^{39} + 6q^{40} - 48q^{41} - 104q^{44} - 52q^{45} - 74q^{46} + 20q^{47} + 94q^{48} - 344q^{50} + 80q^{51} + 12q^{52} + 60q^{54} - 32q^{55} - 50q^{56} + 38q^{57} + 112q^{58} - 12q^{61} + 10q^{62} + 52q^{63} + 144q^{64} - 10q^{65} - 20q^{67} - 54q^{68} - 12q^{69} + 14q^{71} - 208q^{72} - 176q^{73} + 46q^{74} - 116q^{75} + 140q^{78} - 168q^{79} + 32q^{80} + 116q^{81} + 186q^{82} - 60q^{84} - 184q^{85} + 176q^{86} + 24q^{88} - 4q^{89} - 58q^{90} - 152q^{91} - 136q^{92} + 70q^{95} - 332q^{96} + 72q^{97} + 104q^{98} - 56q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.19597 1.75123i −2.11738 0.238571i 1.31045 + 5.74146i −0.0617234 + 0.0215980i 4.23192 + 4.23192i −3.05915 3.05915i 4.73957 9.84181i 1.50159 + 0.342729i 0.173366 + 0.0606634i
4.2 −2.09603 1.67153i −0.439959 0.0495715i 1.15430 + 5.05731i −3.58688 + 1.25510i 0.839309 + 0.839309i 2.20062 + 2.20062i 3.70758 7.69887i −2.73368 0.623944i 9.61617 + 3.36484i
4.3 −2.01642 1.60804i −0.987529 0.111268i 1.03511 + 4.53513i 3.97010 1.38920i 1.81235 + 1.81235i 2.87738 + 2.87738i 2.96740 6.16187i −1.96195 0.447803i −10.2393 3.58289i
4.4 −2.00227 1.59676i 3.12141 + 0.351699i 1.01442 + 4.44445i −2.71628 + 0.950466i −5.68835 5.68835i −0.896379 0.896379i 2.84322 5.90401i 6.69474 + 1.52803i 6.95640 + 2.43415i
4.5 −1.99507 1.59101i 1.16029 + 0.130733i 1.00393 + 4.39849i −0.0820838 + 0.0287223i −2.10685 2.10685i 0.499882 + 0.499882i 2.78079 5.77437i −1.59561 0.364187i 0.209460 + 0.0732932i
4.6 −1.98792 1.58532i 1.88514 + 0.212405i 0.993571 + 4.35312i 0.785952 0.275017i −3.41079 3.41079i −0.526144 0.526144i 2.71950 5.64710i 0.583864 + 0.133263i −1.99840 0.699271i
4.7 −1.84201 1.46896i −3.11411 0.350875i 0.790139 + 3.46182i −0.601633 + 0.210521i 5.22081 + 5.22081i 2.55945 + 2.55945i 1.58534 3.29200i 6.64976 + 1.51776i 1.41746 + 0.495992i
4.8 −1.76597 1.40831i 0.0746772 + 0.00841410i 0.690259 + 3.02422i 1.99849 0.699300i −0.120028 0.120028i −2.90895 2.90895i 1.08000 2.24264i −2.91928 0.666306i −4.51410 1.57955i
4.9 −1.69014 1.34784i −2.41916 0.272574i 0.594855 + 2.60623i −2.17264 + 0.760238i 3.72134 + 3.72134i −1.13454 1.13454i 0.631484 1.31129i 2.85326 + 0.651238i 4.69674 + 1.64346i
4.10 −1.59420 1.27133i 3.17551 + 0.357794i 0.480150 + 2.10368i 3.26776 1.14344i −4.60753 4.60753i 2.24837 + 2.24837i 0.139587 0.289855i 7.03107 + 1.60480i −6.66317 2.33155i
4.11 −1.59352 1.27079i −2.77311 0.312454i 0.479356 + 2.10019i 3.68453 1.28927i 4.02194 + 4.02194i −1.17641 1.17641i 0.136369 0.283173i 4.66772 + 1.06538i −7.50976 2.62778i
4.12 −1.57336 1.25472i −1.09566 0.123452i 0.456119 + 1.99839i 1.15086 0.402703i 1.56898 + 1.56898i −0.279078 0.279078i 0.0434659 0.0902579i −1.73955 0.397040i −2.31600 0.810403i
4.13 −1.52642 1.21728i −1.43515 0.161703i 0.403150 + 1.76631i −2.05291 + 0.718343i 1.99381 + 1.99381i −0.0817540 0.0817540i −0.159477 + 0.331157i −0.891265 0.203425i 4.00802 + 1.40247i
4.14 −1.48420 1.18361i 0.454616 + 0.0512230i 0.356876 + 1.56358i −2.90786 + 1.01751i −0.614114 0.614114i −3.31960 3.31960i −0.326349 + 0.677670i −2.72073 0.620989i 5.52018 + 1.93160i
4.15 −1.47162 1.17358i 1.46588 + 0.165165i 0.343343 + 1.50428i −1.99219 + 0.697097i −1.96339 1.96339i 3.08138 + 3.08138i −0.373253 + 0.775068i −0.803250 0.183337i 3.74985 + 1.31213i
4.16 −1.36039 1.08488i 2.64691 + 0.298235i 0.228670 + 1.00187i 2.58211 0.903520i −3.27729 3.27729i −2.06532 2.06532i −0.734101 + 1.52438i 3.99239 + 0.911237i −4.49290 1.57213i
4.17 −1.35561 1.08107i 1.45849 + 0.164333i 0.223944 + 0.981161i 0.881224 0.308354i −1.79950 1.79950i 1.96518 + 1.96518i −0.747499 + 1.55220i −0.824584 0.188206i −1.52795 0.534653i
4.18 −1.17476 0.936838i 2.38910 + 0.269187i 0.0573487 + 0.251261i −3.52574 + 1.23371i −2.55443 2.55443i −0.191598 0.191598i −1.13586 + 2.35864i 2.71054 + 0.618664i 5.29767 + 1.85374i
4.19 −0.974593 0.777212i −1.71701 0.193461i −0.0992688 0.434925i 1.61232 0.564176i 1.52303 + 1.52303i 0.159716 + 0.159716i −1.32300 + 2.74724i −0.0140846 0.00321473i −2.00984 0.703275i
4.20 −0.921645 0.734988i −0.106119 0.0119568i −0.135818 0.595060i 0.980725 0.343171i 0.0890163 + 0.0890163i 1.87363 + 1.87363i −1.33513 + 2.77243i −2.91367 0.665025i −1.15611 0.404539i
See next 80 embeddings (of 768 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 557.64 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

There are no other newforms in $$S_{2}^{\mathrm{new}}(731, [\chi])$$.