# Properties

 Label 273.2.bt.b Level $273$ Weight $2$ Character orbit 273.bt Analytic conductor $2.180$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bt (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 2q^{7} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 2q^{7} + 20q^{9} - 8q^{11} + 24q^{12} - 18q^{14} - 64q^{16} + 8q^{17} - 14q^{19} - 14q^{20} - 8q^{21} + 4q^{22} + 18q^{24} - 24q^{25} - 10q^{26} - 2q^{28} + 8q^{29} - 8q^{31} + 10q^{32} - 8q^{33} + 24q^{34} - 22q^{35} + 12q^{37} + 8q^{38} - 24q^{39} - 30q^{40} + 2q^{41} + 6q^{42} - 66q^{43} + 28q^{44} + 40q^{46} + 10q^{47} - 24q^{48} + 38q^{49} - 20q^{50} + 40q^{52} - 8q^{53} + 42q^{55} + 20q^{56} - 14q^{57} - 48q^{58} - 26q^{59} + 14q^{60} - 12q^{61} - 24q^{62} - 4q^{63} - 44q^{65} - 18q^{66} + 46q^{67} - 4q^{69} + 32q^{70} - 6q^{71} - 18q^{72} + 10q^{73} + 40q^{74} + 48q^{75} + 64q^{76} - 24q^{77} + 8q^{78} + 34q^{80} - 20q^{81} + 24q^{82} + 12q^{83} + 20q^{84} + 2q^{85} + 12q^{86} - 84q^{88} - 16q^{89} + 26q^{91} + 236q^{92} + 22q^{93} + 30q^{94} + 26q^{96} + 62q^{97} - 14q^{98} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1 −1.75053 + 1.75053i −0.866025 + 0.500000i 4.12868i −0.286571 + 1.06950i 0.640737 2.39126i 1.02891 2.43749i 3.72630 + 3.72630i 0.500000 0.866025i −1.37053 2.37383i
136.2 −1.59737 + 1.59737i −0.866025 + 0.500000i 3.10321i 0.609466 2.27456i 0.584679 2.18205i 1.40839 + 2.23974i 1.76223 + 1.76223i 0.500000 0.866025i 2.65977 + 4.60686i
136.3 −1.03264 + 1.03264i −0.866025 + 0.500000i 0.132693i 0.456824 1.70489i 0.377973 1.41061i −2.58707 + 0.554121i −1.92826 1.92826i 0.500000 0.866025i 1.28880 + 2.23227i
136.4 −0.783932 + 0.783932i −0.866025 + 0.500000i 0.770902i −0.994915 + 3.71307i 0.286939 1.07087i 0.0389254 + 2.64546i −2.17220 2.17220i 0.500000 0.866025i −2.13085 3.69074i
136.5 −0.326747 + 0.326747i −0.866025 + 0.500000i 1.78647i −0.562238 + 2.09830i 0.119598 0.446344i 2.25993 1.37576i −1.23722 1.23722i 0.500000 0.866025i −0.501904 0.869322i
136.6 −0.184060 + 0.184060i −0.866025 + 0.500000i 1.93224i 0.900201 3.35960i 0.0673706 0.251431i 0.939961 2.47315i −0.723769 0.723769i 0.500000 0.866025i 0.452676 + 0.784058i
136.7 1.07562 1.07562i −0.866025 + 0.500000i 0.313900i −0.962166 + 3.59085i −0.393703 + 1.46932i −2.21486 1.44721i 1.81360 + 1.81360i 0.500000 0.866025i 2.82746 + 4.89730i
136.8 1.09526 1.09526i −0.866025 + 0.500000i 0.399185i 0.128983 0.481371i −0.400893 + 1.49615i 2.51963 + 0.807124i 1.75331 + 1.75331i 0.500000 0.866025i −0.385956 0.668496i
136.9 1.57115 1.57115i −0.866025 + 0.500000i 2.93701i 0.922649 3.44337i −0.575080 + 2.14623i −2.64488 + 0.0678683i −1.47218 1.47218i 0.500000 0.866025i −3.96043 6.85967i
136.10 1.93326 1.93326i −0.866025 + 0.500000i 5.47495i −0.212233 + 0.792066i −0.707621 + 2.64088i 2.21517 1.44673i −6.71797 6.71797i 0.500000 0.866025i 1.12096 + 1.94157i
145.1 −1.96855 + 1.96855i 0.866025 + 0.500000i 5.75037i −1.75415 + 0.470023i −2.68909 + 0.720539i 2.27503 + 1.35065i 7.38279 + 7.38279i 0.500000 + 0.866025i 2.52787 4.37840i
145.2 −1.65256 + 1.65256i 0.866025 + 0.500000i 3.46192i 2.69306 0.721604i −2.25744 + 0.604879i −2.54381 0.727358i 2.41591 + 2.41591i 0.500000 + 0.866025i −3.25795 + 5.64294i
145.3 −1.22934 + 1.22934i 0.866025 + 0.500000i 1.02253i −1.95691 + 0.524353i −1.67930 + 0.449968i −1.82011 1.92021i −1.20163 1.20163i 0.500000 + 0.866025i 1.76110 3.05031i
145.4 −0.507981 + 0.507981i 0.866025 + 0.500000i 1.48391i 1.10096 0.295002i −0.693915 + 0.185934i 0.718657 + 2.54628i −1.76976 1.76976i 0.500000 + 0.866025i −0.409414 + 0.709125i
145.5 −0.240784 + 0.240784i 0.866025 + 0.500000i 1.88405i 2.06336 0.552877i −0.328916 + 0.0881329i 1.35855 2.27032i −0.935214 0.935214i 0.500000 + 0.866025i −0.363700 + 0.629948i
145.6 0.465913 0.465913i 0.866025 + 0.500000i 1.56585i −3.81958 + 1.02345i 0.636449 0.170536i −2.61148 + 0.424440i 1.66138 + 1.66138i 0.500000 + 0.866025i −1.30275 + 2.25643i
145.7 0.837153 0.837153i 0.866025 + 0.500000i 0.598351i −1.58368 + 0.424345i 1.14357 0.306419i 2.46703 + 0.955901i 2.17522 + 2.17522i 0.500000 + 0.866025i −0.970539 + 1.68102i
145.8 0.884731 0.884731i 0.866025 + 0.500000i 0.434503i 3.68041 0.986163i 1.20856 0.323834i −2.53212 0.767050i 2.15388 + 2.15388i 0.500000 + 0.866025i 2.38368 4.12866i
145.9 1.55234 1.55234i 0.866025 + 0.500000i 2.81950i −0.926472 + 0.248247i 2.12053 0.568195i 0.619609 2.57218i −1.27214 1.27214i 0.500000 + 0.866025i −1.05283 + 1.82356i
145.10 1.85908 1.85908i 0.866025 + 0.500000i 4.91234i 0.502992 0.134776i 2.53955 0.680470i −1.89546 + 1.84587i −5.41427 5.41427i 0.500000 + 0.866025i 0.684542 1.18566i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.10 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.ba even 12 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.