# Properties

 Label 547.2.j.a Level 547 Weight 2 Character orbit 547.j Analytic conductor 4.368 Analytic rank 0 Dimension 1080 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 547.j (of order $$39$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.36781699056$$ Analytic rank: $$0$$ Dimension: $$1080$$ Relative dimension: $$45$$ over $$\Q(\zeta_{39})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1080q - 25q^{2} - 74q^{3} + 21q^{4} - 27q^{5} - 75q^{6} - 28q^{7} + 4q^{8} + 1010q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1080q - 25q^{2} - 74q^{3} + 21q^{4} - 27q^{5} - 75q^{6} - 28q^{7} + 4q^{8} + 1010q^{9} - 16q^{10} - 14q^{11} - 69q^{12} - 10q^{13} - 15q^{14} - 97q^{15} + 39q^{16} - 22q^{17} - 145q^{18} - 24q^{19} - 51q^{20} - 77q^{21} - 19q^{22} - 53q^{23} - 45q^{24} + 40q^{25} - 68q^{26} - 278q^{27} - 50q^{28} + 39q^{29} + 14q^{30} - 2q^{31} - 45q^{32} - 105q^{33} - 98q^{34} - 18q^{35} - 172q^{36} - 16q^{37} - 37q^{38} + 30q^{39} - 33q^{40} + 62q^{41} + 170q^{42} + 12q^{43} - 22q^{44} - 128q^{45} + 79q^{46} - 58q^{47} - 27q^{48} + 17q^{49} - 40q^{50} - 39q^{51} + 97q^{52} + 17q^{53} - 278q^{54} + 9q^{55} + 47q^{56} - 174q^{57} - 134q^{58} - 126q^{59} - 414q^{60} - 79q^{61} + 15q^{62} - 398q^{63} - 308q^{64} + 47q^{65} - 142q^{66} + 22q^{67} + 7q^{68} - 77q^{69} + 396q^{70} - 22q^{71} - 172q^{72} + 9q^{73} - 43q^{74} - 143q^{75} + 46q^{76} + 56q^{77} - 33q^{78} + 72q^{79} + 131q^{80} + 944q^{81} + 74q^{82} + 41q^{83} - 582q^{84} - 66q^{85} + 20q^{86} - 182q^{87} - 459q^{88} - 92q^{89} - 265q^{90} + 65q^{91} + 86q^{92} - 28q^{93} - 17q^{94} + 37q^{95} - 70q^{96} + 91q^{97} + 139q^{98} - 237q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.56764 1.09397i −1.22916 4.01056 + 4.17543i 1.76583 0.286975i 3.15604 + 1.34466i 3.76080 + 2.82606i −3.75049 9.88922i −1.48916 −4.84795 1.19491i
11.2 −2.53602 1.08050i 2.30650 3.87848 + 4.03793i −3.23891 + 0.526374i −5.84934 2.49217i −0.0846384 0.0636016i −3.51793 9.27602i 2.31995 8.78269 + 2.16474i
11.3 −2.52033 1.07381i −3.03640 3.81355 + 3.97033i −0.333001 + 0.0541179i 7.65274 + 3.26053i −2.25223 1.69244i −3.40511 8.97854i 6.21972 0.897385 + 0.221186i
11.4 −2.20012 0.937384i 0.493253 2.57640 + 2.68231i 0.125352 0.0203717i −1.08522 0.462368i −1.95219 1.46697i −1.45796 3.84432i −2.75670 −0.294885 0.0726827i
11.5 −2.12444 0.905139i −0.527263 2.30852 + 2.40342i −3.39503 + 0.551746i 1.12014 + 0.477247i 0.0185396 + 0.0139316i −1.09115 2.87714i −2.72199 7.71194 + 1.90082i
11.6 −2.10961 0.898822i −0.232269 2.25714 + 2.34993i 2.70427 0.439486i 0.489997 + 0.208768i 0.511767 + 0.384568i −1.02322 2.69801i −2.94605 −6.09997 1.50351i
11.7 −2.10072 0.895035i 2.75344 2.22651 + 2.31804i 1.93817 0.314983i −5.78423 2.46443i 1.71600 + 1.28949i −0.983108 2.59224i 4.58145 −4.35347 1.07303i
11.8 −1.88116 0.801488i −2.40219 1.51094 + 1.57305i −2.47981 + 0.403009i 4.51890 + 1.92533i 0.855464 + 0.642839i −0.131350 0.346342i 2.77051 4.98793 + 1.22941i
11.9 −1.76793 0.753245i −2.36967 1.17275 + 1.22097i 4.05563 0.659104i 4.18940 + 1.78494i −3.52772 2.65091i 0.209233 + 0.551702i 2.61531 −7.66654 1.88963i
11.10 −1.56208 0.665541i 2.43313 0.611706 + 0.636853i −2.03141 + 0.330136i −3.80075 1.61935i −2.12230 1.59480i 0.672522 + 1.77329i 2.92013 3.39294 + 0.836285i
11.11 −1.54016 0.656201i −1.76098 0.556047 + 0.578906i 2.39819 0.389743i 2.71220 + 1.15556i 1.40974 + 1.05935i 0.710783 + 1.87418i 0.101057 −3.94934 0.973425i
11.12 −1.41644 0.603488i 0.684807 0.256650 + 0.267201i −2.74804 + 0.446599i −0.969987 0.413273i 3.82197 + 2.87203i 0.889652 + 2.34582i −2.53104 4.16194 + 1.02583i
11.13 −1.41327 0.602138i 1.23661 0.249311 + 0.259560i 0.850668 0.138247i −1.74767 0.744612i −2.23146 1.67683i 0.893433 + 2.35579i −1.47078 −1.28547 0.316839i
11.14 −1.19793 0.510392i 0.561439 −0.210904 0.219574i 0.714133 0.116058i −0.672566 0.286554i 2.29902 + 1.72760i 1.06406 + 2.80571i −2.68479 −0.914718 0.225458i
11.15 −0.985454 0.419863i −2.18181 −0.590614 0.614895i −1.42571 + 0.231701i 2.15008 + 0.916062i 0.947050 + 0.711662i 1.08354 + 2.85705i 1.76031 1.50225 + 0.370272i
11.16 −0.959893 0.408972i −1.71601 −0.631312 0.657265i −0.863810 + 0.140383i 1.64718 + 0.701799i −3.57814 2.68879i 1.07717 + 2.84026i −0.0553205 0.886578 + 0.218522i
11.17 −0.877457 0.373850i 3.24938 −0.755281 0.786331i −3.27067 + 0.531536i −2.85119 1.21478i 1.56237 + 1.17405i 1.04519 + 2.75593i 7.55848 3.06859 + 0.756339i
11.18 −0.849270 0.361840i 2.50274 −0.795117 0.827805i 3.63253 0.590344i −2.12551 0.905594i −1.20475 0.905312i 1.03044 + 2.71704i 3.26373 −3.29861 0.813035i
11.19 −0.399147 0.170061i −3.34382 −1.25505 1.30665i 0.514784 0.0836605i 1.33468 + 0.568652i 0.0677031 + 0.0508756i 0.586442 + 1.54632i 8.18115 −0.219702 0.0541516i
11.20 −0.264152 0.112545i −0.431166 −1.32834 1.38295i −4.08985 + 0.664666i 0.113894 + 0.0485255i −1.06831 0.802782i 0.398875 + 1.05175i −2.81410 1.15515 + 0.284719i
See next 80 embeddings (of 1080 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 521.45 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.j even 39 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.j.a 1080
547.j even 39 1 inner 547.2.j.a 1080

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.j.a 1080 1.a even 1 1 trivial
547.2.j.a 1080 547.j even 39 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(547, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database