# Properties

 Label 819.2.fm.e Level $819$ Weight $2$ Character orbit 819.fm Analytic conductor $6.540$ Analytic rank $0$ Dimension $32$ 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.fm (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 273) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{2} + 6q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{2} + 6q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + 2q^{10} + 4q^{11} + 6q^{13} - 34q^{14} + 14q^{16} + 8q^{17} + 2q^{19} - 44q^{20} - 4q^{22} + 18q^{23} + 28q^{26} - 32q^{28} + 18q^{29} - 14q^{31} + 8q^{32} - 66q^{34} - 22q^{35} - 24q^{37} - 24q^{38} - 6q^{43} + 20q^{44} - 58q^{46} + 28q^{47} + 8q^{49} - 70q^{50} + 28q^{52} + 80q^{53} + 60q^{55} + 54q^{56} - 4q^{58} + 42q^{59} + 36q^{61} - 52q^{62} - 14q^{65} + 26q^{67} + 72q^{68} - 116q^{70} + 4q^{71} + 12q^{73} + 18q^{74} - 48q^{76} - 28q^{77} - 4q^{79} + 98q^{80} + 20q^{82} + 36q^{83} - 10q^{85} + 40q^{86} + 96q^{88} + 54q^{89} + 148q^{91} + 4q^{92} - 60q^{95} - 40q^{97} - 36q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
370.1 −0.687394 2.56539i 0 −4.37666 + 2.52687i 1.17771 + 1.17771i 0 −1.53750 2.15316i 5.73490 + 5.73490i 0 2.21174 3.83085i
370.2 −0.385482 1.43864i 0 −0.189037 + 0.109141i 1.07205 + 1.07205i 0 1.81743 1.92274i −1.87643 1.87643i 0 1.12904 1.95556i
370.3 −0.0578232 0.215799i 0 1.68883 0.975044i −1.08760 1.08760i 0 −0.725943 + 2.54421i −0.624019 0.624019i 0 −0.171815 + 0.297592i
370.4 −0.0473445 0.176692i 0 1.70307 0.983269i −2.80040 2.80040i 0 −0.467560 2.60411i −0.513062 0.513062i 0 −0.362225 + 0.627392i
370.5 0.189683 + 0.707908i 0 1.26690 0.731443i 1.23329 + 1.23329i 0 −2.64473 0.0736014i 1.79455 + 1.79455i 0 −0.639122 + 1.10699i
370.6 0.281068 + 1.04896i 0 0.710736 0.410344i 1.02734 + 1.02734i 0 2.23270 1.41953i 2.16598 + 2.16598i 0 −0.788887 + 1.36639i
370.7 0.500642 + 1.86842i 0 −1.50830 + 0.870817i −2.44068 2.44068i 0 2.02526 + 1.70244i 0.353388 + 0.353388i 0 3.33831 5.78213i
370.8 0.706652 + 2.63726i 0 −4.72373 + 2.72725i 2.18431 + 2.18431i 0 0.666364 + 2.56046i −6.66928 6.66928i 0 −4.21705 + 7.30414i
496.1 −2.37607 0.636667i 0 3.50833 + 2.02554i 0.498430 + 0.498430i 0 −2.62820 0.304236i −3.56765 3.56765i 0 −0.866973 1.50164i
496.2 −1.96303 0.525993i 0 1.84478 + 1.06508i 1.56698 + 1.56698i 0 2.60496 + 0.462799i −0.187059 0.187059i 0 −2.25182 3.90026i
496.3 −1.34112 0.359352i 0 −0.0625832 0.0361324i −2.52867 2.52867i 0 −0.324044 + 2.62583i 2.03448 + 2.03448i 0 2.48257 + 4.29993i
496.4 −0.604240 0.161906i 0 −1.39316 0.804341i −0.965431 0.965431i 0 2.12303 1.57884i 1.59624 + 1.59624i 0 0.427043 + 0.739660i
496.5 1.11595 + 0.299019i 0 −0.576113 0.332619i −0.549341 0.549341i 0 1.61214 + 2.09786i −2.17732 2.17732i 0 −0.448776 0.777302i
496.6 1.38083 + 0.369991i 0 0.0377371 + 0.0217876i −0.512287 0.512287i 0 −2.41005 1.09162i −1.97762 1.97762i 0 −0.517838 0.896921i
496.7 2.11902 + 0.567791i 0 2.43582 + 1.40632i 3.00219 + 3.00219i 0 −2.60148 + 0.481982i 1.26060 + 1.26060i 0 4.65710 + 8.06633i
496.8 2.16866 + 0.581092i 0 2.63339 + 1.52039i −1.87790 1.87790i 0 1.25762 2.32774i 1.65230 + 1.65230i 0 −2.98130 5.16377i
622.1 −0.687394 + 2.56539i 0 −4.37666 2.52687i 1.17771 1.17771i 0 −1.53750 + 2.15316i 5.73490 5.73490i 0 2.21174 + 3.83085i
622.2 −0.385482 + 1.43864i 0 −0.189037 0.109141i 1.07205 1.07205i 0 1.81743 + 1.92274i −1.87643 + 1.87643i 0 1.12904 + 1.95556i
622.3 −0.0578232 + 0.215799i 0 1.68883 + 0.975044i −1.08760 + 1.08760i 0 −0.725943 2.54421i −0.624019 + 0.624019i 0 −0.171815 0.297592i
622.4 −0.0473445 + 0.176692i 0 1.70307 + 0.983269i −2.80040 + 2.80040i 0 −0.467560 + 2.60411i −0.513062 + 0.513062i 0 −0.362225 0.627392i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 748.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.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.fm.e 32
3.b odd 2 1 273.2.by.d yes 32
7.b odd 2 1 819.2.fm.f 32
13.f odd 12 1 819.2.fm.f 32
21.c even 2 1 273.2.by.c 32
39.k even 12 1 273.2.by.c 32
91.bc even 12 1 inner 819.2.fm.e 32
273.ca odd 12 1 273.2.by.d yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.by.c 32 21.c even 2 1
273.2.by.c 32 39.k even 12 1
273.2.by.d yes 32 3.b odd 2 1
273.2.by.d yes 32 273.ca odd 12 1
819.2.fm.e 32 1.a even 1 1 trivial
819.2.fm.e 32 91.bc even 12 1 inner
819.2.fm.f 32 7.b odd 2 1
819.2.fm.f 32 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{32} - \cdots$$ $$T_{5}^{32} + \cdots$$ $$33\!\cdots\!16$$$$T_{19}^{15} +$$$$17\!\cdots\!36$$$$T_{19}^{14} -$$$$80\!\cdots\!40$$$$T_{19}^{13} +$$$$27\!\cdots\!53$$$$T_{19}^{12} -$$$$69\!\cdots\!46$$$$T_{19}^{11} +$$$$14\!\cdots\!58$$$$T_{19}^{10} -$$$$12\!\cdots\!08$$$$T_{19}^{9} -$$$$17\!\cdots\!71$$$$T_{19}^{8} +$$$$68\!\cdots\!02$$$$T_{19}^{7} -$$$$10\!\cdots\!22$$$$T_{19}^{6} +$$$$20\!\cdots\!76$$$$T_{19}^{5} +$$$$20\!\cdots\!48$$$$T_{19}^{4} -$$$$34\!\cdots\!32$$$$T_{19}^{3} +$$$$40\!\cdots\!84$$$$T_{19}^{2} -$$$$12\!\cdots\!48$$$$T_{19} +$$$$12\!\cdots\!36$$">$$T_{19}^{32} - \cdots$$