# Properties

 Label 819.2.fn.e Level $819$ Weight $2$ Character orbit 819.fn Analytic conductor $6.540$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.fn (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 91) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{2} + 6q^{5} - 6q^{7} + 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{2} + 6q^{5} - 6q^{7} + 16q^{8} + 10q^{11} - 28q^{14} + 12q^{16} + 12q^{19} - 8q^{22} - 24q^{26} - 6q^{28} - 16q^{29} + 24q^{31} - 4q^{32} - 28q^{35} - 8q^{37} - 132q^{40} + 42q^{44} + 12q^{46} - 30q^{47} - 88q^{50} + 36q^{52} + 12q^{53} + 26q^{58} + 54q^{59} - 48q^{61} + 8q^{65} + 16q^{67} + 48q^{68} + 50q^{70} + 36q^{71} + 66q^{73} - 12q^{74} - 32q^{79} - 138q^{80} - 84q^{85} - 42q^{86} + 60q^{89} - 48q^{92} - 72q^{94} + 86q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1 −0.697597 + 2.60347i 0 −4.55935 2.63234i −2.44137 0.654162i 0 0.722189 + 2.54528i 6.22207 6.22207i 0 3.40618 5.89968i
73.2 −0.493585 + 1.84208i 0 −1.41759 0.818448i 3.30509 + 0.885596i 0 −1.12943 + 2.39257i −0.489646 + 0.489646i 0 −3.26268 + 5.65113i
73.3 −0.423548 + 1.58070i 0 −0.587174 0.339005i 2.77274 + 0.742954i 0 1.92960 1.81015i −1.52975 + 1.52975i 0 −2.34878 + 4.06820i
73.4 −0.200025 + 0.746505i 0 1.21479 + 0.701360i −1.76272 0.472319i 0 −2.63734 + 0.210751i −1.85952 + 1.85952i 0 0.705177 1.22140i
73.5 0.186083 0.694471i 0 1.28439 + 0.741542i 1.87130 + 0.501414i 0 −0.783278 2.52715i 1.77076 1.77076i 0 0.696434 1.20626i
73.6 0.211401 0.788958i 0 1.15429 + 0.666428i −3.03793 0.814012i 0 2.28951 1.32595i 1.92491 1.92491i 0 −1.28444 + 2.22472i
73.7 0.411280 1.53492i 0 −0.454770 0.262561i 0.0769162 + 0.0206096i 0 1.52171 + 2.16435i 1.65723 1.65723i 0 0.0632682 0.109584i
73.8 0.639966 2.38839i 0 −3.56278 2.05697i 1.58199 + 0.423894i 0 −2.54693 + 0.716327i −3.69606 + 3.69606i 0 2.02484 3.50713i
460.1 −0.697597 2.60347i 0 −4.55935 + 2.63234i −2.44137 + 0.654162i 0 0.722189 2.54528i 6.22207 + 6.22207i 0 3.40618 + 5.89968i
460.2 −0.493585 1.84208i 0 −1.41759 + 0.818448i 3.30509 0.885596i 0 −1.12943 2.39257i −0.489646 0.489646i 0 −3.26268 5.65113i
460.3 −0.423548 1.58070i 0 −0.587174 + 0.339005i 2.77274 0.742954i 0 1.92960 + 1.81015i −1.52975 1.52975i 0 −2.34878 4.06820i
460.4 −0.200025 0.746505i 0 1.21479 0.701360i −1.76272 + 0.472319i 0 −2.63734 0.210751i −1.85952 1.85952i 0 0.705177 + 1.22140i
460.5 0.186083 + 0.694471i 0 1.28439 0.741542i 1.87130 0.501414i 0 −0.783278 + 2.52715i 1.77076 + 1.77076i 0 0.696434 + 1.20626i
460.6 0.211401 + 0.788958i 0 1.15429 0.666428i −3.03793 + 0.814012i 0 2.28951 + 1.32595i 1.92491 + 1.92491i 0 −1.28444 2.22472i
460.7 0.411280 + 1.53492i 0 −0.454770 + 0.262561i 0.0769162 0.0206096i 0 1.52171 2.16435i 1.65723 + 1.65723i 0 0.0632682 + 0.109584i
460.8 0.639966 + 2.38839i 0 −3.56278 + 2.05697i 1.58199 0.423894i 0 −2.54693 0.716327i −3.69606 3.69606i 0 2.02484 + 3.50713i
577.1 −2.38839 0.639966i 0 3.56278 + 2.05697i 0.423894 1.58199i 0 −0.716327 2.54693i −3.69606 3.69606i 0 −2.02484 + 3.50713i
577.2 −1.53492 0.411280i 0 0.454770 + 0.262561i 0.0206096 0.0769162i 0 −2.16435 + 1.52171i 1.65723 + 1.65723i 0 −0.0632682 + 0.109584i
577.3 −0.788958 0.211401i 0 −1.15429 0.666428i −0.814012 + 3.03793i 0 1.32595 + 2.28951i 1.92491 + 1.92491i 0 1.28444 2.22472i
577.4 −0.694471 0.186083i 0 −1.28439 0.741542i 0.501414 1.87130i 0 2.52715 0.783278i 1.77076 + 1.77076i 0 −0.696434 + 1.20626i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 775.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
13.d odd 4 1 inner
91.bb 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.fn.e 32
3.b odd 2 1 91.2.bb.a 32
7.d odd 6 1 inner 819.2.fn.e 32
13.d odd 4 1 inner 819.2.fn.e 32
21.c even 2 1 637.2.bc.b 32
21.g even 6 1 91.2.bb.a 32
21.g even 6 1 637.2.i.a 32
21.h odd 6 1 637.2.i.a 32
21.h odd 6 1 637.2.bc.b 32
39.f even 4 1 91.2.bb.a 32
91.bb even 12 1 inner 819.2.fn.e 32
273.o odd 4 1 637.2.bc.b 32
273.cb odd 12 1 91.2.bb.a 32
273.cb odd 12 1 637.2.i.a 32
273.cd even 12 1 637.2.i.a 32
273.cd even 12 1 637.2.bc.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bb.a 32 3.b odd 2 1
91.2.bb.a 32 21.g even 6 1
91.2.bb.a 32 39.f even 4 1
91.2.bb.a 32 273.cb odd 12 1
637.2.i.a 32 21.g even 6 1
637.2.i.a 32 21.h odd 6 1
637.2.i.a 32 273.cb odd 12 1
637.2.i.a 32 273.cd even 12 1
637.2.bc.b 32 21.c even 2 1
637.2.bc.b 32 21.h odd 6 1
637.2.bc.b 32 273.o odd 4 1
637.2.bc.b 32 273.cd even 12 1
819.2.fn.e 32 1.a even 1 1 trivial
819.2.fn.e 32 7.d odd 6 1 inner
819.2.fn.e 32 13.d odd 4 1 inner
819.2.fn.e 32 91.bb even 12 1 inner

## Hecke kernels

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

 $$T_{2}^{32} - \cdots$$ $$11\!\cdots\!16$$$$T_{19}^{12} +$$$$35\!\cdots\!16$$$$T_{19}^{11} +$$$$13\!\cdots\!82$$$$T_{19}^{10} +$$$$11\!\cdots\!44$$$$T_{19}^{9} -$$$$12\!\cdots\!94$$$$T_{19}^{8} -$$$$57\!\cdots\!52$$$$T_{19}^{7} -$$$$71\!\cdots\!80$$$$T_{19}^{6} -$$$$12\!\cdots\!18$$$$T_{19}^{5} +$$$$22\!\cdots\!12$$$$T_{19}^{4} +$$$$77\!\cdots\!00$$$$T_{19}^{3} +$$$$97\!\cdots\!50$$$$T_{19}^{2} +$$$$81\!\cdots\!10$$$$T_{19} +$$$$33\!\cdots\!41$$">$$T_{19}^{32} - \cdots$$