# Properties

 Label 819.2.gh.d Level $819$ Weight $2$ Character orbit 819.gh Analytic conductor $6.540$ Analytic rank $0$ Dimension $40$ 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.gh (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{7} - 4q^{11} + 18q^{14} + 32q^{16} - 4q^{17} + 14q^{19} - 14q^{20} + 4q^{22} - 12q^{23} + 24q^{25} + 32q^{26} + 16q^{28} - 8q^{29} + 14q^{31} + 26q^{32} - 24q^{34} - 26q^{35} + 36q^{37} + 8q^{38} - 30q^{40} + 2q^{41} - 66q^{43} + 32q^{44} - 26q^{46} + 4q^{47} - 14q^{49} + 20q^{50} + 2q^{52} + 8q^{53} - 42q^{55} - 46q^{56} + 24q^{58} - 14q^{59} - 24q^{62} - 28q^{65} - 44q^{67} + 18q^{68} - 4q^{70} + 6q^{71} + 14q^{73} + 20q^{74} - 64q^{76} - 24q^{77} - 20q^{80} + 48q^{82} + 12q^{83} + 2q^{85} + 60q^{86} + 2q^{89} - 14q^{91} - 236q^{92} - 24q^{95} - 62q^{97} + 88q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −2.39126 0.640737i 0 3.57554 + 2.06434i 1.06950 0.286571i 0 0.327682 2.62538i −3.72630 3.72630i 0 −2.74106
19.2 −2.18205 0.584679i 0 2.68745 + 1.55160i −2.27456 + 0.609466i 0 −2.33957 + 1.23548i −1.76223 1.76223i 0 5.31955
19.3 −1.41061 0.377973i 0 0.114915 + 0.0663464i −1.70489 + 0.456824i 0 1.96341 + 1.77342i 1.92826 + 1.92826i 0 2.57761
19.4 −1.07087 0.286939i 0 −0.667621 0.385451i 3.71307 0.994915i 0 −1.35644 + 2.27158i 2.17220 + 2.17220i 0 −4.26170
19.5 −0.446344 0.119598i 0 −1.54713 0.893237i 2.09830 0.562238i 0 −1.26927 2.32141i 1.23722 + 1.23722i 0 −1.00381
19.6 −0.251431 0.0673706i 0 −1.67337 0.966122i −3.35960 + 0.900201i 0 0.422545 2.61179i 0.723769 + 0.723769i 0 0.905353
19.7 1.46932 + 0.393703i 0 0.271846 + 0.156950i 3.59085 0.962166i 0 2.64173 0.145891i −1.81360 1.81360i 0 5.65491
19.8 1.49615 + 0.400893i 0 0.345704 + 0.199592i −0.481371 + 0.128983i 0 −2.58563 0.560827i −1.75331 1.75331i 0 −0.771912
19.9 2.14623 + 0.575080i 0 2.54352 + 1.46850i −3.44337 + 0.922649i 0 2.25660 + 1.38122i 1.47218 + 1.47218i 0 −7.92086
19.10 2.64088 + 0.707621i 0 4.74145 + 2.73748i 0.792066 0.212233i 0 −1.19502 2.36049i 6.71797 + 6.71797i 0 2.24193
262.1 −0.680470 2.53955i 0 −4.25421 + 2.45617i −0.134776 + 0.502992i 0 −2.56445 0.650845i 5.41427 + 5.41427i 0 1.36908
262.2 −0.568195 2.12053i 0 −2.44176 + 1.40975i 0.248247 0.926472i 0 1.82268 + 1.91776i 1.27214 + 1.27214i 0 −2.10567
262.3 −0.323834 1.20856i 0 0.376291 0.217252i −0.986163 + 3.68041i 0 −1.80936 + 1.93034i −2.15388 2.15388i 0 4.76737
262.4 −0.306419 1.14357i 0 0.518187 0.299176i 0.424345 1.58368i 0 1.65856 2.06135i −2.17522 2.17522i 0 −1.94108
262.5 −0.170536 0.636449i 0 1.35607 0.782925i 1.02345 3.81958i 0 −2.47383 + 0.938166i −1.66138 1.66138i 0 −2.60550
262.6 0.0881329 + 0.328916i 0 1.63163 0.942023i −0.552877 + 2.06336i 0 2.31170 + 1.28688i 0.935214 + 0.935214i 0 −0.727401
262.7 0.185934 + 0.693915i 0 1.28510 0.741955i −0.295002 + 1.10096i 0 −0.650764 2.56447i 1.76976 + 1.76976i 0 −0.818827
262.8 0.449968 + 1.67930i 0 −0.885540 + 0.511267i 0.524353 1.95691i 0 −0.616155 + 2.57300i 1.20163 + 1.20163i 0 3.52219
262.9 0.604879 + 2.25744i 0 −2.99811 + 1.73096i −0.721604 + 2.69306i 0 −1.83932 + 1.90181i −2.41591 2.41591i 0 −6.51591
262.10 0.720539 + 2.68909i 0 −4.97997 + 2.87519i 0.470023 1.75415i 0 1.29491 2.30721i −7.38279 7.38279i 0 5.05574
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 397.10 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.w 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.gh.d 40
3.b odd 2 1 273.2.cg.b yes 40
7.d odd 6 1 819.2.et.d 40
13.f odd 12 1 819.2.et.d 40
21.g even 6 1 273.2.bt.b 40
39.k even 12 1 273.2.bt.b 40
91.w even 12 1 inner 819.2.gh.d 40
273.ch odd 12 1 273.2.cg.b yes 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.b 40 21.g even 6 1
273.2.bt.b 40 39.k even 12 1
273.2.cg.b yes 40 3.b odd 2 1
273.2.cg.b yes 40 273.ch odd 12 1
819.2.et.d 40 7.d odd 6 1
819.2.et.d 40 13.f odd 12 1
819.2.gh.d 40 1.a even 1 1 trivial
819.2.gh.d 40 91.w even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.