# Properties

 Label 539.2.r.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10})$$ 276 * q + 3 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^6 + 6 * q^8 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9} + 3 q^{10} + 23 q^{11} - 32 q^{12} - 27 q^{13} - 23 q^{14} + 6 q^{15} + 70 q^{16} - 3 q^{17} - 20 q^{18} - 61 q^{19} - 12 q^{20} - 23 q^{21} - 6 q^{23} + 6 q^{24} - 11 q^{25} + 9 q^{26} - 12 q^{27} - 3 q^{28} - 24 q^{29} - 26 q^{30} - 75 q^{31} + 137 q^{32} + 3 q^{33} - 58 q^{34} + 15 q^{35} - 36 q^{36} - 29 q^{37} - 6 q^{39} - 53 q^{40} - 25 q^{41} + 4 q^{42} + 20 q^{43} + 6 q^{44} + 3 q^{45} + 52 q^{46} + 25 q^{47} + 288 q^{48} + 84 q^{49} - 218 q^{50} - 160 q^{51} + 65 q^{52} + 5 q^{53} + 66 q^{54} - 12 q^{55} - 178 q^{56} + 23 q^{57} + 5 q^{58} + 28 q^{59} - 54 q^{60} - 65 q^{61} - 92 q^{62} + 85 q^{63} - 50 q^{64} + 201 q^{65} - 38 q^{66} - 16 q^{67} - 91 q^{68} - 42 q^{69} - 255 q^{70} - 63 q^{71} - 165 q^{72} + 6 q^{73} - 166 q^{74} - 117 q^{75} - 87 q^{76} + 33 q^{78} + 2 q^{79} - 48 q^{80} + 5 q^{81} + 53 q^{82} + 37 q^{83} - 84 q^{84} + 26 q^{85} + 67 q^{86} - 59 q^{87} - 3 q^{88} - 155 q^{89} + 9 q^{90} + 47 q^{91} - 19 q^{92} + 81 q^{93} + 126 q^{94} - 37 q^{95} + 395 q^{96} + 302 q^{97} + 90 q^{98} + 268 q^{99}+O(q^{100})$$ 276 * q + 3 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^6 + 6 * q^8 + 6 * q^9 + 3 * q^10 + 23 * q^11 - 32 * q^12 - 27 * q^13 - 23 * q^14 + 6 * q^15 + 70 * q^16 - 3 * q^17 - 20 * q^18 - 61 * q^19 - 12 * q^20 - 23 * q^21 - 6 * q^23 + 6 * q^24 - 11 * q^25 + 9 * q^26 - 12 * q^27 - 3 * q^28 - 24 * q^29 - 26 * q^30 - 75 * q^31 + 137 * q^32 + 3 * q^33 - 58 * q^34 + 15 * q^35 - 36 * q^36 - 29 * q^37 - 6 * q^39 - 53 * q^40 - 25 * q^41 + 4 * q^42 + 20 * q^43 + 6 * q^44 + 3 * q^45 + 52 * q^46 + 25 * q^47 + 288 * q^48 + 84 * q^49 - 218 * q^50 - 160 * q^51 + 65 * q^52 + 5 * q^53 + 66 * q^54 - 12 * q^55 - 178 * q^56 + 23 * q^57 + 5 * q^58 + 28 * q^59 - 54 * q^60 - 65 * q^61 - 92 * q^62 + 85 * q^63 - 50 * q^64 + 201 * q^65 - 38 * q^66 - 16 * q^67 - 91 * q^68 - 42 * q^69 - 255 * q^70 - 63 * q^71 - 165 * q^72 + 6 * q^73 - 166 * q^74 - 117 * q^75 - 87 * q^76 + 33 * q^78 + 2 * q^79 - 48 * q^80 + 5 * q^81 + 53 * q^82 + 37 * q^83 - 84 * q^84 + 26 * q^85 + 67 * q^86 - 59 * q^87 - 3 * q^88 - 155 * q^89 + 9 * q^90 + 47 * q^91 - 19 * q^92 + 81 * q^93 + 126 * q^94 - 37 * q^95 + 395 * q^96 + 302 * q^97 + 90 * 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 −0.209677 + 2.79795i 2.93495 0.905312i −5.80688 0.875247i 0.632033 + 0.586441i 1.91762 + 8.40165i 2.01885 1.71005i 2.41777 10.5929i 5.31562 3.62413i −1.77335 + 1.64543i
23.2 −0.186858 + 2.49345i −1.31214 + 0.404741i −4.20469 0.633755i 2.05474 + 1.90652i −0.764016 3.34737i 1.19228 + 2.36188i 1.25312 5.49026i −0.920824 + 0.627807i −5.13775 + 4.76714i
23.3 −0.185221 + 2.47160i 0.579946 0.178890i −4.09683 0.617498i −0.257511 0.238935i 0.334725 + 1.46653i −1.95317 1.78470i 1.18198 5.17858i −2.17438 + 1.48247i 0.638248 0.592207i
23.4 −0.166566 + 2.22267i −2.75461 + 0.849686i −2.93486 0.442359i −3.17205 2.94323i −1.42975 6.26413i −2.43449 + 1.03599i 0.480110 2.10350i 4.38722 2.99115i 7.07019 6.56017i
23.5 −0.149048 + 1.98891i 2.21799 0.684159i −1.95589 0.294803i 1.33474 + 1.23846i 1.03014 + 4.51335i −1.59553 + 2.11052i −0.00977309 + 0.0428187i 1.97268 1.34495i −2.66212 + 2.47009i
23.6 −0.125667 + 1.67691i −0.134497 + 0.0414869i −0.818569 0.123379i 2.54910 + 2.36522i −0.0526678 0.230753i 1.56302 2.13470i −0.438624 + 1.92174i −2.46235 + 1.67880i −4.28660 + 3.97739i
23.7 −0.113141 + 1.50976i 2.71968 0.838911i −0.288924 0.0435483i −3.07265 2.85100i 0.958850 + 4.20099i 2.64504 + 0.0614254i −0.575355 + 2.52080i 4.21419 2.87318i 4.65198 4.31641i
23.8 −0.0831158 + 1.10910i 0.364108 0.112312i 0.754460 + 0.113717i −0.972765 0.902594i 0.0943029 + 0.413168i −1.26372 + 2.32444i −0.683812 + 2.99598i −2.35876 + 1.60817i 1.08192 1.00388i
23.9 −0.0692750 + 0.924411i −2.17201 + 0.669976i 1.12792 + 0.170007i −0.162950 0.151196i −0.468867 2.05424i 0.230607 2.63568i −0.647849 + 2.83841i 1.79004 1.22043i 0.151056 0.140159i
23.10 −0.0264393 + 0.352807i −1.73623 + 0.535555i 1.85389 + 0.279429i −1.35577 1.25797i −0.143043 0.626713i −2.45984 0.974260i −0.305054 + 1.33653i 0.248947 0.169729i 0.479666 0.445065i
23.11 −0.0144543 + 0.192879i 0.602741 0.185921i 1.94067 + 0.292509i 0.198124 + 0.183832i 0.0271481 + 0.118943i 2.64493 0.0659952i −0.170550 + 0.747227i −2.14999 + 1.46584i −0.0383210 + 0.0355567i
23.12 0.00564876 0.0753775i 1.46521 0.451959i 1.97201 + 0.297233i 2.26902 + 2.10534i −0.0257909 0.112997i −2.64466 0.0760191i 0.0671843 0.294354i −0.536131 + 0.365528i 0.171513 0.159141i
23.13 0.00841924 0.112347i 2.88830 0.890922i 1.96511 + 0.296193i −1.46461 1.35896i −0.0757751 0.331992i −2.03943 1.68544i 0.0999604 0.437955i 5.06980 3.45653i −0.165006 + 0.153103i
23.14 0.0510088 0.680665i −2.04893 + 0.632011i 1.51696 + 0.228645i 0.755449 + 0.700954i 0.325674 + 1.42687i 2.63798 + 0.202627i 0.536782 2.35180i 1.31996 0.899934i 0.515649 0.478452i
23.15 0.0682458 0.910677i 0.181705 0.0560486i 1.15299 + 0.173785i −3.20937 2.97786i −0.0386415 0.169300i 0.419147 2.61234i 0.643374 2.81881i −2.44884 + 1.66959i −2.93090 + 2.71947i
23.16 0.0963063 1.28512i 2.53294 0.781307i 0.335407 + 0.0505544i −0.328585 0.304883i −0.760135 3.33037i 1.66067 + 2.05965i 0.670806 2.93899i 3.32661 2.26804i −0.423455 + 0.392909i
23.17 0.110040 1.46838i −1.06483 + 0.328455i −0.166383 0.0250782i 0.939178 + 0.871429i 0.365125 + 1.59972i −0.817308 2.51635i 0.600192 2.62961i −1.45275 + 0.990465i 1.38294 1.28318i
23.18 0.126531 1.68844i −2.90212 + 0.895185i −0.857151 0.129195i −0.837978 0.777530i 1.14426 + 5.01332i 0.435612 + 2.60964i 0.426939 1.87054i 5.14222 3.50590i −1.41884 + 1.31649i
23.19 0.130700 1.74407i −1.64058 + 0.506053i −1.04703 0.157814i 2.53334 + 2.35060i 0.668166 + 2.92743i −2.49524 + 0.879644i 0.366275 1.60475i −0.0432948 + 0.0295179i 4.43070 4.11109i
23.20 0.167941 2.24102i 1.19225 0.367760i −3.01630 0.454633i 2.18464 + 2.02705i −0.623930 2.73361i 1.44285 + 2.21770i −0.525258 + 2.30131i −1.19251 + 0.813036i 4.90954 4.55539i
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.b 276
49.g even 21 1 inner 539.2.r.b 276

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{276} - 33 T_{2}^{274} - 2 T_{2}^{273} + 409 T_{2}^{272} + 77 T_{2}^{271} - 891 T_{2}^{270} - 1176 T_{2}^{269} - 37616 T_{2}^{268} + 8117 T_{2}^{267} + 490364 T_{2}^{266} - 2814 T_{2}^{265} - 1544447 T_{2}^{264} + \cdots + 277701594113161$$ acting on $$S_{2}^{\mathrm{new}}(539, [\chi])$$.