# Properties

 Label 819.2.fm.g Level $819$ Weight $2$ Character orbit 819.fm 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.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 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{2} - 12q^{4} + 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{2} - 12q^{4} + 16q^{8} + 4q^{11} + 32q^{14} + 12q^{16} + 4q^{22} + 12q^{23} + 24q^{28} - 4q^{29} - 4q^{32} + 20q^{35} + 4q^{37} - 48q^{43} - 24q^{44} + 84q^{46} + 24q^{49} + 44q^{50} - 72q^{53} - 60q^{56} - 16q^{58} - 4q^{65} - 56q^{67} + 56q^{70} - 84q^{71} + 24q^{74} - 80q^{79} + 36q^{85} + 48q^{86} - 228q^{88} - 48q^{91} - 24q^{92} + 84q^{95} + 32q^{98} + O(q^{100})$$

## 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 −0.612835 2.28713i 0 −3.12335 + 1.80327i −2.25527 2.25527i 0 −2.60590 0.457468i 2.68981 + 2.68981i 0 −3.77599 + 6.54020i
370.2 −0.612835 2.28713i 0 −3.12335 + 1.80327i 2.25527 + 2.25527i 0 2.48551 + 0.906771i 2.68981 + 2.68981i 0 3.77599 6.54020i
370.3 −0.218036 0.813723i 0 1.11745 0.645157i −1.01306 1.01306i 0 2.43848 + 1.02656i −1.96000 1.96000i 0 −0.603466 + 1.04523i
370.4 −0.218036 0.813723i 0 1.11745 0.645157i 1.01306 + 1.01306i 0 −2.62506 0.330214i −1.96000 1.96000i 0 0.603466 1.04523i
370.5 0.357317 + 1.33353i 0 0.0814361 0.0470171i −2.02925 2.02925i 0 −2.32989 + 1.25364i 2.04421 + 2.04421i 0 1.98097 3.43114i
370.6 0.357317 + 1.33353i 0 0.0814361 0.0470171i 2.02925 + 2.02925i 0 1.39092 + 2.25063i 2.04421 + 2.04421i 0 −1.98097 + 3.43114i
370.7 0.607529 + 2.26733i 0 −3.03963 + 1.75493i −1.12678 1.12678i 0 −0.437995 2.60925i −2.50608 2.50608i 0 1.87023 3.23933i
370.8 0.607529 + 2.26733i 0 −3.03963 + 1.75493i 1.12678 + 1.12678i 0 1.68394 2.04067i −2.50608 2.50608i 0 −1.87023 + 3.23933i
496.1 −1.61897 0.433802i 0 0.700831 + 0.404625i −1.42145 1.42145i 0 −1.11484 2.39940i 1.41124 + 1.41124i 0 1.68466 + 2.91792i
496.2 −1.61897 0.433802i 0 0.700831 + 0.404625i 1.42145 + 1.42145i 0 0.234216 + 2.63536i 1.41124 + 1.41124i 0 −1.68466 2.91792i
496.3 0.112775 + 0.0302180i 0 −1.72025 0.993184i −1.24841 1.24841i 0 −2.63771 0.206123i −0.329103 0.329103i 0 −0.103066 0.178515i
496.4 0.112775 + 0.0302180i 0 −1.72025 0.993184i 1.24841 + 1.24841i 0 −2.18126 + 1.49736i −0.329103 0.329103i 0 0.103066 + 0.178515i
496.5 0.892257 + 0.239080i 0 −0.993087 0.573359i −2.80028 2.80028i 0 2.62900 + 0.297264i −2.05537 2.05537i 0 −1.82908 3.16806i
496.6 0.892257 + 0.239080i 0 −0.993087 0.573359i 2.80028 + 2.80028i 0 2.12815 1.57194i −2.05537 2.05537i 0 1.82908 + 3.16806i
496.7 2.47996 + 0.664504i 0 3.97660 + 2.29589i −0.281686 0.281686i 0 1.14426 + 2.38551i 4.70528 + 4.70528i 0 −0.511390 0.885754i
496.8 2.47996 + 0.664504i 0 3.97660 + 2.29589i 0.281686 + 0.281686i 0 −0.201802 2.63804i 4.70528 + 4.70528i 0 0.511390 + 0.885754i
622.1 −0.612835 + 2.28713i 0 −3.12335 1.80327i −2.25527 + 2.25527i 0 −2.60590 + 0.457468i 2.68981 2.68981i 0 −3.77599 6.54020i
622.2 −0.612835 + 2.28713i 0 −3.12335 1.80327i 2.25527 2.25527i 0 2.48551 0.906771i 2.68981 2.68981i 0 3.77599 + 6.54020i
622.3 −0.218036 + 0.813723i 0 1.11745 + 0.645157i −1.01306 + 1.01306i 0 2.43848 1.02656i −1.96000 + 1.96000i 0 −0.603466 1.04523i
622.4 −0.218036 + 0.813723i 0 1.11745 + 0.645157i 1.01306 1.01306i 0 −2.62506 + 0.330214i −1.96000 + 1.96000i 0 0.603466 + 1.04523i
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
7.b odd 2 1 inner
13.f odd 12 1 inner
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.g 32
3.b odd 2 1 91.2.bc.a 32
7.b odd 2 1 inner 819.2.fm.g 32
13.f odd 12 1 inner 819.2.fm.g 32
21.c even 2 1 91.2.bc.a 32
21.g even 6 1 637.2.x.b 32
21.g even 6 1 637.2.bb.b 32
21.h odd 6 1 637.2.x.b 32
21.h odd 6 1 637.2.bb.b 32
39.k even 12 1 91.2.bc.a 32
91.bc even 12 1 inner 819.2.fm.g 32
273.bs odd 12 1 637.2.x.b 32
273.bv even 12 1 637.2.x.b 32
273.bw even 12 1 637.2.bb.b 32
273.ca odd 12 1 91.2.bc.a 32
273.ch odd 12 1 637.2.bb.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bc.a 32 3.b odd 2 1
91.2.bc.a 32 21.c even 2 1
91.2.bc.a 32 39.k even 12 1
91.2.bc.a 32 273.ca odd 12 1
637.2.x.b 32 21.g even 6 1
637.2.x.b 32 21.h odd 6 1
637.2.x.b 32 273.bs odd 12 1
637.2.x.b 32 273.bv even 12 1
637.2.bb.b 32 21.g even 6 1
637.2.bb.b 32 21.h odd 6 1
637.2.bb.b 32 273.bw even 12 1
637.2.bb.b 32 273.ch odd 12 1
819.2.fm.g 32 1.a even 1 1 trivial
819.2.fm.g 32 7.b odd 2 1 inner
819.2.fm.g 32 13.f odd 12 1 inner
819.2.fm.g 32 91.bc 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 $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{16} - \cdots$$ $$T_{5}^{32} + \cdots$$ $$26\!\cdots\!13$$$$T_{19}^{16} -$$$$22\!\cdots\!40$$$$T_{19}^{14} -$$$$84\!\cdots\!44$$$$T_{19}^{12} +$$$$97\!\cdots\!76$$$$T_{19}^{10} +$$$$45\!\cdots\!76$$$$T_{19}^{8} -$$$$21\!\cdots\!36$$$$T_{19}^{6} +$$$$33\!\cdots\!40$$$$T_{19}^{4} -$$$$21\!\cdots\!28$$$$T_{19}^{2} +$$$$55\!\cdots\!76$$">$$T_{19}^{32} + \cdots$$