# Properties

 Label 819.2.fy.a Level $819$ Weight $2$ Character orbit 819.fy Analytic conductor $6.540$ Analytic rank $0$ Dimension $336$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.fy (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$336$$ Relative dimension: $$84$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 12 q^{6} + 36 q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$336 q - 12 q^{6} + 36 q^{8} - 12 q^{11} + 4 q^{15} + 168 q^{16} - 4 q^{18} - 4 q^{21} + 8 q^{24} - 96 q^{26} + 24 q^{27} - 120 q^{30} - 72 q^{32} + 36 q^{33} + 72 q^{38} + 8 q^{39} - 48 q^{41} - 76 q^{45} + 24 q^{48} - 24 q^{50} - 36 q^{52} + 72 q^{54} + 32 q^{57} + 36 q^{58} - 72 q^{59} + 4 q^{60} + 36 q^{62} - 8 q^{63} - 72 q^{65} - 20 q^{66} - 48 q^{71} - 176 q^{72} - 60 q^{75} - 124 q^{78} - 12 q^{79} - 96 q^{80} - 20 q^{81} - 156 q^{83} - 12 q^{84} - 108 q^{85} - 24 q^{87} + 60 q^{89} - 72 q^{92} + 20 q^{93} + 96 q^{94} + 88 q^{96} - 96 q^{97} + 108 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
50.1 −0.727236 2.71408i 1.68275 + 0.410318i −5.10531 + 2.94755i −0.0876862 0.327249i −0.110119 4.86551i −0.707107 + 0.707107i 7.73897 + 7.73897i 2.66328 + 1.38092i −0.824413 + 0.475975i
50.2 −0.721163 2.69142i −1.27046 + 1.17726i −4.99160 + 2.88190i −0.822532 3.06973i 4.08470 + 2.57033i 0.707107 0.707107i 7.41566 + 7.41566i 0.228118 2.99131i −7.66875 + 4.42756i
50.3 −0.708020 2.64237i 0.160527 + 1.72460i −4.74876 + 2.74170i 0.977651 + 3.64864i 4.44336 1.64522i 0.707107 0.707107i 6.73810 + 6.73810i −2.94846 + 0.553689i 8.94886 5.16663i
50.4 −0.693142 2.58684i −1.33631 1.10194i −4.47926 + 2.58610i 0.766927 + 2.86221i −1.92428 + 4.22063i −0.707107 + 0.707107i 6.00720 + 6.00720i 0.571467 + 2.94507i 6.87250 3.96784i
50.5 −0.676409 2.52439i 1.29284 1.15263i −4.18297 + 2.41504i −0.237212 0.885285i −3.78419 2.48398i 0.707107 0.707107i 5.22994 + 5.22994i 0.342872 2.98034i −2.07435 + 1.19763i
50.6 −0.663488 2.47617i −0.692898 1.58742i −3.95915 + 2.28582i −0.816615 3.04765i −3.47098 + 2.76896i −0.707107 + 0.707107i 4.66155 + 4.66155i −2.03979 + 2.19984i −7.00468 + 4.04416i
50.7 −0.634582 2.36829i −1.61523 0.625330i −3.47406 + 2.00575i 0.0143461 + 0.0535404i −0.455971 + 4.22216i 0.707107 0.707107i 3.48736 + 3.48736i 2.21792 + 2.02010i 0.117696 0.0679516i
50.8 −0.629766 2.35032i −0.786534 + 1.54317i −3.39535 + 1.96031i −0.0503052 0.187742i 4.12227 + 0.876773i −0.707107 + 0.707107i 3.30451 + 3.30451i −1.76273 2.42751i −0.409572 + 0.236467i
50.9 −0.606137 2.26213i 1.35078 1.08415i −3.01779 + 1.74232i 0.760590 + 2.83856i −3.27125 2.39851i −0.707107 + 0.707107i 2.45857 + 2.45857i 0.649233 2.92891i 5.96018 3.44111i
50.10 −0.603210 2.25121i 0.907564 + 1.47524i −2.97204 + 1.71591i −1.06637 3.97976i 2.77362 2.93300i 0.707107 0.707107i 2.35963 + 2.35963i −1.35266 + 2.67775i −8.31604 + 4.80127i
50.11 −0.585079 2.18354i 0.305712 + 1.70486i −2.69350 + 1.55509i 0.0226908 + 0.0846831i 3.54377 1.66501i −0.707107 + 0.707107i 1.77459 + 1.77459i −2.81308 + 1.04239i 0.171633 0.0990926i
50.12 −0.580718 2.16727i 0.116460 1.72813i −2.62776 + 1.51714i 0.952172 + 3.55355i −3.81295 + 0.751156i 0.707107 0.707107i 1.64094 + 1.64094i −2.97287 0.402516i 7.14856 4.12722i
50.13 −0.572772 2.13762i 1.16473 + 1.28195i −2.50928 + 1.44873i 0.0776352 + 0.289738i 2.07319 3.22402i 0.707107 0.707107i 1.40439 + 1.40439i −0.286797 + 2.98626i 0.574882 0.331908i
50.14 −0.563544 2.10318i −1.71440 + 0.246659i −2.37372 + 1.37047i −0.775066 2.89259i 1.48491 + 3.46668i −0.707107 + 0.707107i 1.14076 + 1.14076i 2.87832 0.845744i −5.64684 + 3.26020i
50.15 −0.548397 2.04664i −1.71570 + 0.237418i −2.15597 + 1.24475i 0.530257 + 1.97895i 1.42680 + 3.38123i 0.707107 0.707107i 0.733384 + 0.733384i 2.88726 0.814679i 3.75941 2.17049i
50.16 −0.541434 2.02066i 1.72494 + 0.156795i −2.05786 + 1.18811i −1.05055 3.92071i −0.617111 3.57041i −0.707107 + 0.707107i 0.556508 + 0.556508i 2.95083 + 0.540924i −7.35361 + 4.24561i
50.17 −0.503059 1.87744i 1.45910 + 0.933282i −1.53967 + 0.888931i 0.750845 + 2.80219i 1.01817 3.20888i −0.707107 + 0.707107i −0.305303 0.305303i 1.25797 + 2.72351i 4.88324 2.81934i
50.18 −0.459010 1.71305i 1.54753 0.777908i −0.991796 + 0.572614i −0.538351 2.00915i −2.04293 2.29393i 0.707107 0.707107i −1.07192 1.07192i 1.78972 2.40768i −3.19467 + 1.84444i
50.19 −0.452193 1.68761i −1.18434 + 1.26385i −0.911485 + 0.526246i 0.371355 + 1.38591i 2.66844 + 1.42720i 0.707107 0.707107i −1.17056 1.17056i −0.194657 2.99368i 2.17095 1.25340i
50.20 −0.445095 1.66112i −0.117775 1.72804i −0.829146 + 0.478708i 0.109473 + 0.408558i −2.81806 + 0.964781i −0.707107 + 0.707107i −1.26780 1.26780i −2.97226 + 0.407041i 0.629936 0.363694i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 743.84 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.bc even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.fy.a yes 336
9.d odd 6 1 819.2.ew.a 336
13.f odd 12 1 819.2.ew.a 336
117.bc even 12 1 inner 819.2.fy.a yes 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.ew.a 336 9.d odd 6 1
819.2.ew.a 336 13.f odd 12 1
819.2.fy.a yes 336 1.a even 1 1 trivial
819.2.fy.a yes 336 117.bc even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(819, [\chi])$$.