# Properties

 Label 539.2.r.a Level $539$ Weight $2$ Character orbit 539.r Analytic conductor $4.304$ Analytic rank $0$ Dimension $276$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 539.r (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.30393666895$$ Analytic rank: $$0$$ Dimension: $$276$$ Relative dimension: $$23$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10})$$ 276 * q - q^3 + 24 * q^4 - 2 * q^5 - 12 * q^6 - 2 * q^7 + 6 * q^8 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9} - 9 q^{10} - 23 q^{11} - 44 q^{12} + 43 q^{13} - 47 q^{14} + 6 q^{15} - 34 q^{16} - 3 q^{17} + 10 q^{18} + 59 q^{19} - 28 q^{20} + 25 q^{21} + 6 q^{23} + 2 q^{24} + 53 q^{25} + q^{26} - 4 q^{27} - 13 q^{28} - 28 q^{29} - 6 q^{30} - 3 q^{31} - 253 q^{32} + q^{33} - 50 q^{34} - 9 q^{35} - 76 q^{36} - 17 q^{37} + 8 q^{38} + 10 q^{39} - 59 q^{40} + 73 q^{41} - 12 q^{42} - 38 q^{44} - 9 q^{45} + 74 q^{46} - 87 q^{47} - 160 q^{48} - 60 q^{49} + 30 q^{50} + 128 q^{51} + 49 q^{52} + 21 q^{53} + 76 q^{54} - 4 q^{55} + 102 q^{56} - 41 q^{57} + 17 q^{58} + 36 q^{59} + 10 q^{60} + 11 q^{61} + 72 q^{62} - 201 q^{63} - 42 q^{64} - 215 q^{65} - 34 q^{66} - 8 q^{67} + 75 q^{68} + 6 q^{69} + 97 q^{70} + q^{71} - 95 q^{72} - 20 q^{73} + 154 q^{74} - 101 q^{75} - 27 q^{76} + 2 q^{77} - 87 q^{78} - 14 q^{79} - 124 q^{80} - 203 q^{81} - 15 q^{82} + 29 q^{83} - 96 q^{84} - 74 q^{85} + 63 q^{86} + 65 q^{87} + 3 q^{88} + 141 q^{89} + 491 q^{90} + 71 q^{91} + 41 q^{92} + 121 q^{93} + 116 q^{94} - 7 q^{95} - 141 q^{96} - 298 q^{97} - 102 q^{98} - 268 q^{99}+O(q^{100})$$ 276 * q - q^3 + 24 * q^4 - 2 * q^5 - 12 * q^6 - 2 * q^7 + 6 * q^8 + 6 * q^9 - 9 * q^10 - 23 * q^11 - 44 * q^12 + 43 * q^13 - 47 * q^14 + 6 * q^15 - 34 * q^16 - 3 * q^17 + 10 * q^18 + 59 * q^19 - 28 * q^20 + 25 * q^21 + 6 * q^23 + 2 * q^24 + 53 * q^25 + q^26 - 4 * q^27 - 13 * q^28 - 28 * q^29 - 6 * q^30 - 3 * q^31 - 253 * q^32 + q^33 - 50 * q^34 - 9 * q^35 - 76 * q^36 - 17 * q^37 + 8 * q^38 + 10 * q^39 - 59 * q^40 + 73 * q^41 - 12 * q^42 - 38 * q^44 - 9 * q^45 + 74 * q^46 - 87 * q^47 - 160 * q^48 - 60 * q^49 + 30 * q^50 + 128 * q^51 + 49 * q^52 + 21 * q^53 + 76 * q^54 - 4 * q^55 + 102 * q^56 - 41 * q^57 + 17 * q^58 + 36 * q^59 + 10 * q^60 + 11 * q^61 + 72 * q^62 - 201 * q^63 - 42 * q^64 - 215 * q^65 - 34 * q^66 - 8 * q^67 + 75 * q^68 + 6 * q^69 + 97 * q^70 + q^71 - 95 * q^72 - 20 * q^73 + 154 * q^74 - 101 * q^75 - 27 * q^76 + 2 * q^77 - 87 * q^78 - 14 * q^79 - 124 * q^80 - 203 * q^81 - 15 * q^82 + 29 * q^83 - 96 * q^84 - 74 * q^85 + 63 * q^86 + 65 * q^87 + 3 * q^88 + 141 * q^89 + 491 * q^90 + 71 * q^91 + 41 * q^92 + 121 * q^93 + 116 * q^94 - 7 * q^95 - 141 * q^96 - 298 * q^97 - 102 * q^98 - 268 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −0.207378 + 2.76727i 0.432433 0.133388i −5.63711 0.849658i −2.28382 2.11908i 0.279443 + 1.22432i −0.165781 + 2.64055i 2.28524 10.0123i −2.30951 + 1.57460i 6.33768 5.88051i
23.2 −0.194832 + 2.59986i −2.33478 + 0.720186i −4.74363 0.714987i 0.904184 + 0.838960i −1.41749 6.21042i −0.823302 2.51439i 1.62279 7.10990i 2.45384 1.67300i −2.35734 + 2.18729i
23.3 −0.158928 + 2.12075i 0.357452 0.110259i −2.49465 0.376009i −1.46094 1.35556i 0.177023 + 0.775588i 2.63653 0.220744i 0.247422 1.08403i −2.36310 + 1.61114i 3.10698 2.88285i
23.4 −0.156649 + 2.09034i 1.86160 0.574229i −2.36732 0.356817i 1.22307 + 1.13484i 0.908715 + 3.98134i 2.06946 + 1.64844i 0.183810 0.805326i 0.657114 0.448012i −2.56379 + 2.37885i
23.5 −0.141924 + 1.89385i −1.56085 + 0.481458i −1.58885 0.239481i 0.329056 + 0.305319i −0.690286 3.02434i −2.09779 + 1.61223i −0.166167 + 0.728025i −0.274269 + 0.186993i −0.624929 + 0.579849i
23.6 −0.102537 + 1.36825i 2.76390 0.852550i 0.116054 + 0.0174923i 1.55705 + 1.44473i 0.883105 + 3.86914i −0.773931 2.53003i −0.646472 + 2.83238i 4.43358 3.02276i −2.13641 + 1.98230i
23.7 −0.0946098 + 1.26248i −1.31797 + 0.406540i 0.392756 + 0.0591984i −1.70882 1.58555i −0.388556 1.70237i 1.20230 2.35679i −0.675328 + 2.95880i −0.906946 + 0.618345i 2.16340 2.00734i
23.8 −0.0864495 + 1.15359i −0.305281 + 0.0941668i 0.654369 + 0.0986303i 1.98812 + 1.84471i −0.0822384 0.360310i −2.42047 1.06834i −0.685183 + 3.00198i −2.39439 + 1.63247i −2.29990 + 2.13400i
23.9 −0.0414087 + 0.552561i 2.45816 0.758241i 1.67405 + 0.252323i −0.520559 0.483008i 0.317185 + 1.38968i 0.112603 + 2.64335i −0.455347 + 1.99500i 2.98889 2.03779i 0.288447 0.267640i
23.10 −0.0405728 + 0.541406i −0.866143 + 0.267170i 1.68619 + 0.254152i 2.31163 + 2.14488i −0.109505 0.479775i 1.75943 + 1.97596i −0.447637 + 1.96122i −1.79989 + 1.22715i −1.25504 + 1.16451i
23.11 −0.0167368 + 0.223336i −1.83029 + 0.564569i 1.92806 + 0.290609i −2.99990 2.78350i −0.0954557 0.418219i 1.55827 + 2.13818i −0.196846 + 0.862438i 0.552494 0.376684i 0.671866 0.623400i
23.12 −0.0155962 + 0.208117i 1.34855 0.415972i 1.93459 + 0.291593i −0.884730 0.820909i 0.0655386 + 0.287144i 1.06549 2.42172i −0.183738 + 0.805010i −0.833166 + 0.568043i 0.184644 0.171324i
23.13 0.0247632 0.330442i −2.89943 + 0.894354i 1.86908 + 0.281719i −0.350504 0.325220i 0.223733 + 0.980240i −2.40487 + 1.10301i 0.286850 1.25677i 5.12808 3.49627i −0.116146 + 0.107768i
23.14 0.0301718 0.402615i −2.74875 + 0.847876i 1.81647 + 0.273789i 2.87831 + 2.67068i 0.258433 + 1.13227i 0.132111 2.64245i 0.344721 1.51032i 4.35800 2.97123i 1.16210 1.07827i
23.15 0.0745093 0.994258i 0.0218832 0.00675006i 0.994665 + 0.149922i 1.81047 + 1.67987i −0.00508080 0.0222604i −0.549266 + 2.58811i 0.666900 2.92188i −2.47828 + 1.68966i 1.80512 1.67490i
23.16 0.0868885 1.15945i 1.50123 0.463069i 0.640896 + 0.0965995i 2.05182 + 1.90381i −0.406464 1.78084i 1.75377 1.98099i 0.685137 3.00178i −0.439445 + 0.299609i 2.38564 2.21355i
23.17 0.0905072 1.20773i 2.16284 0.667146i 0.527231 + 0.0794673i −1.64013 1.52182i −0.609983 2.67251i −2.55293 + 0.694643i 0.682693 2.99107i 1.75406 1.19589i −1.98639 + 1.84310i
23.18 0.125657 1.67678i −1.11000 + 0.342390i −0.818137 0.123314i −0.636598 0.590676i 0.434633 + 1.90425i 2.63131 0.276014i 0.438754 1.92231i −1.36384 + 0.929853i −1.07043 + 0.993211i
23.19 0.143761 1.91836i −2.52291 + 0.778215i −1.68177 0.253486i −1.84824 1.71492i 1.13020 + 4.95172i −0.883304 2.49395i 0.128093 0.561211i 3.28074 2.23677i −3.55553 + 3.29905i
23.20 0.147156 1.96366i 1.94642 0.600392i −1.85663 0.279843i −2.52738 2.34506i −0.892536 3.91046i 2.61893 + 0.375741i 0.0536308 0.234972i 0.949372 0.647271i −4.97682 + 4.61781i
See next 80 embeddings (of 276 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.23 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.r.a 276
49.g even 21 1 inner 539.2.r.a 276

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.r.a 276 1.a even 1 1 trivial
539.2.r.a 276 49.g even 21 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{276} - 35 T_{2}^{274} - 2 T_{2}^{273} + 581 T_{2}^{272} + 159 T_{2}^{271} - 5343 T_{2}^{270} - 4308 T_{2}^{269} + 17132 T_{2}^{268} + 63229 T_{2}^{267} + 246460 T_{2}^{266} - 526200 T_{2}^{265} - 4083701 T_{2}^{264} + \cdots + 16\!\cdots\!49$$ acting on $$S_{2}^{\mathrm{new}}(539, [\chi])$$.