# Properties

 Label 273.2.bv.b Level $273$ Weight $2$ Character orbit 273.bv Analytic conductor $2.180$ Analytic rank $0$ Dimension $128$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128 q + 2 q^{3} - 4 q^{6} - 16 q^{7} + 8 q^{9}+O(q^{10})$$ 128 * q + 2 * q^3 - 4 * q^6 - 16 * q^7 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$128 q + 2 q^{3} - 4 q^{6} - 16 q^{7} + 8 q^{9} - 12 q^{10} + 48 q^{12} - 16 q^{13} - 6 q^{15} - 64 q^{16} - 2 q^{18} - 4 q^{19} - 6 q^{21} - 8 q^{22} + 2 q^{24} - 40 q^{27} + 68 q^{28} + 18 q^{30} + 20 q^{31} - 16 q^{33} - 48 q^{34} - 60 q^{36} - 8 q^{37} + 4 q^{39} + 44 q^{40} + 2 q^{42} - 144 q^{43} - 2 q^{45} - 24 q^{46} - 64 q^{48} - 60 q^{49} - 36 q^{51} + 48 q^{52} + 14 q^{54} - 16 q^{55} + 40 q^{57} + 44 q^{58} - 58 q^{60} + 20 q^{61} + 14 q^{63} - 34 q^{66} - 84 q^{67} - 54 q^{69} - 104 q^{70} + 46 q^{72} - 48 q^{73} + 144 q^{76} + 82 q^{78} - 24 q^{79} + 24 q^{81} + 36 q^{82} + 184 q^{84} + 56 q^{85} + 4 q^{87} + 132 q^{88} + 24 q^{91} + 16 q^{93} - 16 q^{94} - 90 q^{96} + 52 q^{97} - 10 q^{99}+O(q^{100})$$ 128 * q + 2 * q^3 - 4 * q^6 - 16 * q^7 + 8 * q^9 - 12 * q^10 + 48 * q^12 - 16 * q^13 - 6 * q^15 - 64 * q^16 - 2 * q^18 - 4 * q^19 - 6 * q^21 - 8 * q^22 + 2 * q^24 - 40 * q^27 + 68 * q^28 + 18 * q^30 + 20 * q^31 - 16 * q^33 - 48 * q^34 - 60 * q^36 - 8 * q^37 + 4 * q^39 + 44 * q^40 + 2 * q^42 - 144 * q^43 - 2 * q^45 - 24 * q^46 - 64 * q^48 - 60 * q^49 - 36 * q^51 + 48 * q^52 + 14 * q^54 - 16 * q^55 + 40 * q^57 + 44 * q^58 - 58 * q^60 + 20 * q^61 + 14 * q^63 - 34 * q^66 - 84 * q^67 - 54 * q^69 - 104 * q^70 + 46 * q^72 - 48 * q^73 + 144 * q^76 + 82 * q^78 - 24 * q^79 + 24 * q^81 + 36 * q^82 + 184 * q^84 + 56 * q^85 + 4 * q^87 + 132 * q^88 + 24 * q^91 + 16 * q^93 - 16 * q^94 - 90 * q^96 + 52 * q^97 - 10 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1.83790 1.83790i 1.29065 + 1.15509i 4.75574i 2.20098 0.589752i −0.249144 4.49501i −2.64509 0.0592679i 5.06476 5.06476i 0.331543 + 2.98162i −5.12909 2.96128i
2.2 −1.81876 1.81876i −0.267828 1.71122i 4.61580i 2.15482 0.577382i −2.62519 + 3.59942i 1.30976 2.29881i 4.75753 4.75753i −2.85654 + 0.916623i −4.96923 2.86898i
2.3 −1.73689 1.73689i 1.00077 1.41367i 4.03355i −2.06624 + 0.553646i −4.19360 + 0.717172i −1.93609 + 1.80321i 3.53204 3.53204i −0.996937 2.82951i 4.55044 + 2.62720i
2.4 −1.52398 1.52398i −1.72942 0.0953716i 2.64502i −2.04381 + 0.547638i 2.49026 + 2.78095i −2.26018 1.37535i 0.982996 0.982996i 2.98181 + 0.329876i 3.94931 + 2.28014i
2.5 −1.48334 1.48334i 1.73149 0.0439971i 2.40062i −0.600039 + 0.160780i −2.63366 2.50313i 2.54988 + 0.705761i 0.594253 0.594253i 2.99613 0.152361i 1.12856 + 0.651572i
2.6 −1.44872 1.44872i −0.783380 + 1.54477i 2.19760i 0.629445 0.168659i 3.37285 1.10304i −0.719902 + 2.54593i 0.286268 0.286268i −1.77263 2.42029i −1.15623 0.667551i
2.7 −1.32954 1.32954i −1.62246 0.606325i 1.53537i 3.58803 0.961410i 1.35099 + 2.96326i 0.951973 + 2.46855i −0.617751 + 0.617751i 2.26474 + 1.96747i −6.04868 3.49220i
2.8 −1.21487 1.21487i 0.917597 + 1.46902i 0.951801i −2.79534 + 0.749010i 0.669904 2.89942i −0.107896 2.64355i −1.27342 + 1.27342i −1.31603 + 2.69593i 4.30591 + 2.48602i
2.9 −0.879582 0.879582i 1.39839 1.02202i 0.452670i 2.08749 0.559342i −2.12894 0.331049i 0.755530 2.53558i −2.15733 + 2.15733i 0.910969 2.85835i −2.32811 1.34414i
2.10 −0.824043 0.824043i −0.772097 1.55044i 0.641906i −0.114643 + 0.0307184i −0.641388 + 1.91387i −2.54084 0.737668i −2.17704 + 2.17704i −1.80773 + 2.39418i 0.119784 + 0.0691572i
2.11 −0.748547 0.748547i −1.33611 + 1.10219i 0.879356i −2.02207 + 0.541811i 1.82518 + 0.175098i 2.61787 + 0.383108i −2.15533 + 2.15533i 0.570358 2.94528i 1.91918 + 1.10804i
2.12 −0.583259 0.583259i −0.386481 + 1.68838i 1.31962i 3.27322 0.877056i 1.21018 0.759346i −0.829143 2.51247i −1.93620 + 1.93620i −2.70127 1.30505i −2.42069 1.39758i
2.13 −0.554014 0.554014i 0.671048 + 1.59678i 1.38614i −1.68690 + 0.452004i 0.512866 1.25641i −1.01706 + 2.44246i −1.87597 + 1.87597i −2.09939 + 2.14303i 1.18499 + 0.684151i
2.14 −0.397102 0.397102i 1.57374 0.723421i 1.68462i −3.29155 + 0.881968i −0.912208 0.337664i −2.64283 + 0.124223i −1.46317 + 1.46317i 1.95332 2.27696i 1.65731 + 0.956850i
2.15 −0.363959 0.363959i −1.62227 0.606835i 1.73507i 0.585566 0.156902i 0.369576 + 0.811302i 2.22038 1.43871i −1.35941 + 1.35941i 2.26350 + 1.96890i −0.270228 0.156016i
2.16 −0.340464 0.340464i 1.62034 + 0.611966i 1.76817i 3.37589 0.904567i −0.343315 0.760019i −0.304456 + 2.62818i −1.28292 + 1.28292i 2.25100 + 1.98318i −1.45734 0.841396i
2.17 0.340464 + 0.340464i −1.34015 1.09727i 1.76817i −3.37589 + 0.904567i −0.0826902 0.829853i −0.304456 + 2.62818i 1.28292 1.28292i 0.591989 + 2.94101i −1.45734 0.841396i
2.18 0.363959 + 0.363959i 1.33667 + 1.10151i 1.73507i −0.585566 + 0.156902i 0.0855890 + 0.887396i 2.22038 1.43871i 1.35941 1.35941i 0.573364 + 2.94470i −0.270228 0.156016i
2.19 0.397102 + 0.397102i −0.160369 1.72461i 1.68462i 3.29155 0.881968i 0.621163 0.748529i −2.64283 + 0.124223i 1.46317 1.46317i −2.94856 + 0.553150i 1.65731 + 0.956850i
2.20 0.554014 + 0.554014i −1.71837 + 0.217243i 1.38614i 1.68690 0.452004i −1.07236 0.831647i −1.01706 + 2.44246i 1.87597 1.87597i 2.90561 0.746610i 1.18499 + 0.684151i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.32 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.x odd 12 1 inner
273.bv 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.bv.b 128
3.b odd 2 1 inner 273.2.bv.b 128
7.c even 3 1 273.2.bw.b yes 128
13.f odd 12 1 273.2.bw.b yes 128
21.h odd 6 1 273.2.bw.b yes 128
39.k even 12 1 273.2.bw.b yes 128
91.x odd 12 1 inner 273.2.bv.b 128
273.bv even 12 1 inner 273.2.bv.b 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bv.b 128 1.a even 1 1 trivial
273.2.bv.b 128 3.b odd 2 1 inner
273.2.bv.b 128 91.x odd 12 1 inner
273.2.bv.b 128 273.bv even 12 1 inner
273.2.bw.b yes 128 7.c even 3 1
273.2.bw.b yes 128 13.f odd 12 1
273.2.bw.b yes 128 21.h odd 6 1
273.2.bw.b yes 128 39.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{128} + 452 T_{2}^{124} + 94644 T_{2}^{120} + 12196318 T_{2}^{116} + 1083883810 T_{2}^{112} + 70551504126 T_{2}^{108} + 3487095853283 T_{2}^{104} + 133909455267320 T_{2}^{100} + \cdots + 269517889$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.