# Properties

 Label 547.2.m.a Level 547 Weight 2 Character orbit 547.m Analytic conductor 4.368 Analytic rank 0 Dimension 3168 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 547.m (of order $$91$$, degree $$72$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.36781699056$$ Analytic rank: $$0$$ Dimension: $$3168$$ Relative dimension: $$44$$ over $$\Q(\zeta_{91})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{91}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$3168q - 76q^{2} - 38q^{3} - 34q^{4} - 75q^{5} - 42q^{6} - 60q^{7} - 77q^{8} - 550q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$3168q - 76q^{2} - 38q^{3} - 34q^{4} - 75q^{5} - 42q^{6} - 60q^{7} - 77q^{8} - 550q^{9} - 91q^{10} - 61q^{11} - 31q^{12} - 72q^{13} - 21q^{14} - 46q^{15} + 24q^{16} - 91q^{17} + 9q^{18} - 89q^{19} - 108q^{20} - 81q^{21} - 50q^{22} - 86q^{23} - 33q^{24} - 15q^{25} + 137q^{26} + 163q^{27} - 47q^{28} - 12q^{29} + 29q^{30} - 53q^{31} - 122q^{32} - 21q^{33} - 80q^{34} - 75q^{35} - 8q^{36} - 49q^{37} - 47q^{38} + 128q^{39} - 246q^{40} - 686q^{41} - 466q^{42} - 9q^{43} - 43q^{44} + 156q^{45} - 91q^{46} - 72q^{47} + 21q^{48} - 8q^{49} - 84q^{50} + 93q^{51} + 4q^{52} - 32q^{53} - 71q^{54} - 34q^{55} - 263q^{56} - 61q^{57} - 36q^{58} - 23q^{59} + 87q^{60} + 38q^{61} + 26q^{62} + 133q^{63} + 343q^{64} + 143q^{65} - 65q^{66} - 50q^{67} - 213q^{68} - 140q^{69} + 372q^{70} - 75q^{71} + 147q^{72} - 18q^{73} - 85q^{74} + 97q^{75} - 47q^{76} + 11q^{77} + 55q^{78} - 37q^{79} - 142q^{80} - 532q^{81} - 94q^{82} - 279q^{83} + 395q^{84} - 75q^{85} - 147q^{86} + 84q^{87} + 345q^{88} + 8q^{89} + 278q^{90} - 38q^{91} - 78q^{92} - 83q^{93} - 540q^{94} + 35q^{95} + 123q^{96} - 735q^{97} - 90q^{98} + 351q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −1.43254 2.41657i −0.00191599 0.00839448i −2.82779 + 5.16906i 2.23230 + 0.0770965i −0.0175412 + 0.0166555i −4.60768 + 0.640360i 10.9271 0.377388i 2.70284 1.30162i −3.01155 5.50497i
10.2 −1.38950 2.34398i −0.559135 2.44973i −2.60363 + 4.75932i −2.63335 0.0909474i −4.96519 + 4.71450i 0.746366 0.103727i 9.32698 0.322123i −2.98564 + 1.43781i 3.44587 + 6.29889i
10.3 −1.36435 2.30154i 0.713317 + 3.12524i −2.47578 + 4.52561i −3.57878 0.123599i 6.21968 5.90565i −0.332062 + 0.0461487i 8.44582 0.291691i −6.55543 + 3.15693i 4.59823 + 8.40534i
10.4 −1.21694 2.05288i 0.474632 + 2.07950i −1.77350 + 3.24187i 2.57302 + 0.0888638i 3.69137 3.50500i 2.57249 0.357516i 4.04333 0.139643i −1.39614 + 0.672344i −2.94880 5.39026i
10.5 −1.19472 2.01540i −0.168315 0.737438i −1.67460 + 3.06109i −1.16754 0.0403228i −1.28514 + 1.22026i 4.11811 0.572320i 3.48700 0.120429i 2.18742 1.05341i 1.31362 + 2.40123i
10.6 −1.11993 1.88923i 0.187104 + 0.819754i −1.35508 + 2.47702i −1.48051 0.0511319i 1.33916 1.27155i −2.38045 + 0.330826i 1.80741 0.0624220i 2.06592 0.994893i 1.56147 + 2.85429i
10.7 −1.11775 1.88556i −0.271817 1.19091i −1.34608 + 2.46058i 3.31408 + 0.114457i −1.94170 + 1.84367i 0.214643 0.0298303i 1.76284 0.0608827i 1.35853 0.654232i −3.48851 6.37683i
10.8 −1.10037 1.85624i −0.473626 2.07509i −1.27492 + 2.33049i 1.13411 + 0.0391684i −3.33070 + 3.16254i 0.250897 0.0348688i 1.41565 0.0488921i −1.37878 + 0.663987i −1.17524 2.14827i
10.9 −0.929069 1.56726i −0.617025 2.70336i −0.633268 + 1.15758i −0.691646 0.0238872i −3.66362 + 3.47865i −4.21969 + 0.586437i −1.23913 + 0.0427955i −4.22454 + 2.03443i 0.605149 + 1.10618i
10.10 −0.900824 1.51962i 0.668884 + 2.93057i −0.537874 + 0.983208i 1.50378 + 0.0519357i 3.85080 3.65638i −3.52663 + 0.490119i −1.55238 + 0.0536140i −5.43795 + 2.61878i −1.27572 2.33196i
10.11 −0.882718 1.48907i 0.304921 + 1.33595i −0.478271 + 0.874256i −3.64866 0.126013i 1.72016 1.63331i 2.07032 0.287726i −1.73602 + 0.0599566i 1.01113 0.486934i 3.03310 + 5.54435i
10.12 −0.807206 1.36169i 0.219615 + 0.962195i −0.242743 + 0.443723i −0.516600 0.0178417i 1.13294 1.07574i −2.71722 + 0.377630i −2.36389 + 0.0816410i 1.82532 0.879027i 0.392708 + 0.717851i
10.13 −0.671741 1.13317i −0.619707 2.71511i 0.127033 0.232210i −2.49729 0.0862482i −2.66041 + 2.52609i 1.97865 0.274985i −2.98152 + 0.102972i −4.28489 + 2.06350i 1.57980 + 2.88779i
10.14 −0.607931 1.02553i 0.696260 + 3.05052i 0.277745 0.507704i −0.411634 0.0142165i 2.70512 2.56854i 3.12156 0.433823i −3.07245 + 0.106112i −6.11797 + 2.94626i 0.235665 + 0.430785i
10.15 −0.575837 0.971390i −0.152308 0.667306i 0.347865 0.635880i 0.600736 + 0.0207475i −0.560510 + 0.532210i 0.619289 0.0860666i −3.07514 + 0.106205i 2.28081 1.09838i −0.325773 0.595497i
10.16 −0.572243 0.965327i −0.717695 3.14443i 0.355481 0.649802i 3.54444 + 0.122413i −2.62470 + 2.49219i 4.44174 0.617298i −3.07374 + 0.106157i −6.66943 + 3.21183i −1.91011 3.49159i
10.17 −0.449712 0.758628i −0.258554 1.13280i 0.586600 1.07228i 0.442256 + 0.0152741i −0.743097 + 0.705579i −1.14748 + 0.159473i −2.84002 + 0.0980850i 1.48653 0.715873i −0.187301 0.342377i
10.18 −0.432175 0.729044i 0.215157 + 0.942665i 0.615146 1.12446i 3.24535 + 0.112084i 0.594259 0.564255i 2.06386 0.286828i −2.77965 + 0.0959999i 1.86058 0.896009i −1.32084 2.41444i
10.19 −0.348818 0.588427i 0.465635 + 2.04008i 0.735303 1.34410i 2.83356 + 0.0978618i 1.03802 0.985608i −4.08637 + 0.567909i −2.41467 + 0.0833947i −1.24220 + 0.598212i −0.930811 1.70148i
10.20 −0.290706 0.490397i 0.429128 + 1.88013i 0.803896 1.46948i −2.18526 0.0754717i 0.797261 0.757008i −1.33797 + 0.185946i −2.09382 + 0.0723137i −0.647842 + 0.311984i 0.598256 + 1.09358i
See next 80 embeddings (of 3168 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 539.44 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.m even 91 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.m.a 3168
547.m even 91 1 inner 547.2.m.a 3168

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.m.a 3168 1.a even 1 1 trivial
547.2.m.a 3168 547.m even 91 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