# Properties

 Label 273.2.y.b Level $273$ Weight $2$ Character orbit 273.y Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.y (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 28q^{4} - 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 28q^{4} - 12q^{7} - 6q^{12} + 12q^{13} - 9q^{15} - 8q^{16} + 36q^{18} + 12q^{21} - 18q^{22} - 24q^{24} + 16q^{25} + 6q^{28} + 44q^{30} - 36q^{31} - 30q^{33} + 24q^{34} - 4q^{36} - 36q^{37} - 27q^{39} - 54q^{40} - 24q^{42} - 12q^{43} - 3q^{45} - 24q^{46} + 54q^{48} + 8q^{49} - 14q^{51} - 6q^{52} + 9q^{54} - 42q^{55} + 9q^{60} + 6q^{63} - 28q^{64} - 9q^{66} + 39q^{69} + 84q^{70} + 12q^{73} + 51q^{75} - 18q^{76} - 11q^{78} + 4q^{79} + 30q^{84} - 30q^{85} + 3q^{87} + 60q^{88} + 4q^{91} - 18q^{93} + 24q^{96} + 12q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 −1.30618 + 2.26238i −1.71658 + 0.230971i −2.41223 4.17810i 0.648305 0.374299i 1.71963 4.18524i −2.44285 1.01610i 7.37851 2.89330 0.792960i 1.95561i
101.2 −1.30520 + 2.26068i 1.39879 1.02146i −2.40711 4.16923i −1.65792 + 0.957199i 0.483493 + 4.49543i 2.51605 0.818240i 7.34624 0.913228 2.85762i 4.99736i
101.3 −1.18406 + 2.05084i −0.641138 1.60902i −1.80398 3.12458i 0.0190083 0.0109745i 4.05899 + 0.590294i 0.134389 + 2.64234i 3.80781 −2.17788 + 2.06321i 0.0519775i
101.4 −1.14404 + 1.98153i −0.240189 + 1.71532i −1.61764 2.80184i −3.02328 + 1.74549i −3.12417 2.43833i 0.229018 + 2.63582i 2.82644 −2.88462 0.824000i 7.98764i
101.5 −1.06779 + 1.84947i 1.37855 1.04862i −1.28036 2.21764i 3.07271 1.77403i 0.467389 + 3.66929i −2.57290 0.616608i 1.19745 0.800790 2.89115i 7.57717i
101.6 −0.985417 + 1.70679i −0.524054 + 1.65087i −0.942093 1.63175i −0.0906165 + 0.0523174i −2.30128 2.52124i 0.412313 2.61343i −0.228249 −2.45074 1.73029i 0.206218i
101.7 −0.884751 + 1.53243i −1.73022 0.0796811i −0.565569 0.979594i 1.27320 0.735080i 1.65292 2.58094i 2.56619 + 0.643937i −1.53745 2.98730 + 0.275731i 2.60145i
101.8 −0.799109 + 1.38410i 1.59038 + 0.686061i −0.277151 0.480040i 1.21627 0.702213i −2.22047 + 1.65301i 2.53934 0.742795i −2.31054 2.05864 + 2.18220i 2.24458i
101.9 −0.760690 + 1.31755i 0.347458 1.69684i −0.157299 0.272450i −1.84777 + 1.06681i 1.97137 + 1.74857i −2.06030 1.65987i −2.56414 −2.75855 1.17916i 3.24606i
101.10 −0.638569 + 1.10603i 1.72991 0.0860586i 0.184460 + 0.319495i −2.29307 + 1.32390i −1.00948 + 1.96829i −0.551092 + 2.58772i −3.02544 2.98519 0.297747i 3.38161i
101.11 −0.636825 + 1.10301i 0.696603 + 1.58579i 0.188907 + 0.327197i 1.93046 1.11455i −2.19277 0.241511i −2.30379 + 1.30098i −3.02850 −2.02949 + 2.20934i 2.83909i
101.12 −0.491530 + 0.851355i −0.901093 1.47920i 0.516796 + 0.895118i 1.05876 0.611275i 1.70224 0.0400792i 0.826182 2.51345i −2.98220 −1.37606 + 2.66579i 1.20184i
101.13 −0.366273 + 0.634404i −1.53905 + 0.794556i 0.731688 + 1.26732i −2.08780 + 1.20539i 0.0596442 1.26741i −2.55085 0.702242i −2.53708 1.73736 2.44573i 1.76601i
101.14 −0.209784 + 0.363356i 0.760555 1.55614i 0.911982 + 1.57960i 3.21070 1.85370i 0.405879 + 0.602804i 2.20704 + 1.45910i −1.60441 −1.84311 2.36705i 1.55550i
101.15 −0.183709 + 0.318193i −1.40902 1.00730i 0.932502 + 1.61514i 0.333187 0.192366i 0.579367 0.263290i −1.67893 + 2.04479i −1.42007 0.970675 + 2.83862i 0.141357i
101.16 −0.0381512 + 0.0660798i 1.25975 + 1.18871i 0.997089 + 1.72701i −3.58354 + 2.06896i −0.126611 + 0.0378937i −0.269802 2.63196i −0.304765 0.173957 + 2.99495i 0.315733i
101.17 0.0381512 0.0660798i −1.25975 + 1.18871i 0.997089 + 1.72701i 3.58354 2.06896i 0.0304884 + 0.128595i −0.269802 2.63196i 0.304765 0.173957 2.99495i 0.315733i
101.18 0.183709 0.318193i 1.40902 1.00730i 0.932502 + 1.61514i −0.333187 + 0.192366i −0.0616679 0.633392i −1.67893 + 2.04479i 1.42007 0.970675 2.83862i 0.141357i
101.19 0.209784 0.363356i −0.760555 1.55614i 0.911982 + 1.57960i −3.21070 + 1.85370i −0.724983 0.0500996i 2.20704 + 1.45910i 1.60441 −1.84311 + 2.36705i 1.55550i
101.20 0.366273 0.634404i 1.53905 + 0.794556i 0.731688 + 1.26732i 2.08780 1.20539i 1.06778 0.685356i −2.55085 0.702242i 2.53708 1.73736 + 2.44573i 1.76601i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 173.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.p odd 6 1 inner
273.y even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.y.b 64
3.b odd 2 1 inner 273.2.y.b 64
7.d odd 6 1 273.2.br.b yes 64
13.e even 6 1 273.2.br.b yes 64
21.g even 6 1 273.2.br.b yes 64
39.h odd 6 1 273.2.br.b yes 64
91.p odd 6 1 inner 273.2.y.b 64
273.y even 6 1 inner 273.2.y.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.y.b 64 1.a even 1 1 trivial
273.2.y.b 64 3.b odd 2 1 inner
273.2.y.b 64 91.p odd 6 1 inner
273.2.y.b 64 273.y even 6 1 inner
273.2.br.b yes 64 7.d odd 6 1
273.2.br.b yes 64 13.e even 6 1
273.2.br.b yes 64 21.g even 6 1
273.2.br.b yes 64 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$14\!\cdots\!03$$$$T_{2}^{38} +$$$$40\!\cdots\!03$$$$T_{2}^{36} +$$$$96\!\cdots\!94$$$$T_{2}^{34} +$$$$20\!\cdots\!68$$$$T_{2}^{32} +$$$$36\!\cdots\!12$$$$T_{2}^{30} +$$$$56\!\cdots\!51$$$$T_{2}^{28} +$$$$75\!\cdots\!96$$$$T_{2}^{26} +$$$$84\!\cdots\!72$$$$T_{2}^{24} +$$$$79\!\cdots\!78$$$$T_{2}^{22} +$$$$62\!\cdots\!95$$$$T_{2}^{20} +$$$$38\!\cdots\!15$$$$T_{2}^{18} +$$$$19\!\cdots\!75$$$$T_{2}^{16} +$$$$71\!\cdots\!34$$$$T_{2}^{14} +$$$$20\!\cdots\!04$$$$T_{2}^{12} + 383499973607 T_{2}^{10} + 52672506167 T_{2}^{8} + 4462331843 T_{2}^{6} + 239056184 T_{2}^{4} + 1381845 T_{2}^{2} + 7225$$">$$T_{2}^{64} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.