# Properties

 Label 63.3.n.b Level $63$ Weight $3$ Character orbit 63.n Analytic conductor $1.717$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 63.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.71662566547$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 6 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 3 q^{7} + 20 q^{9}+O(q^{10})$$ 22 * q - 6 * q^2 + 8 * q^3 + 12 * q^4 - 8 * q^6 + 3 * q^7 + 20 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 6 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 3 q^{7} + 20 q^{9} + 25 q^{10} - 20 q^{12} - 18 q^{13} - 90 q^{14} + 53 q^{15} + 12 q^{16} + 6 q^{17} - 56 q^{18} + 3 q^{19} - 39 q^{20} - 2 q^{21} - 59 q^{22} + 15 q^{24} - 114 q^{25} - 3 q^{26} - 97 q^{27} + 34 q^{28} - 63 q^{29} - 20 q^{30} - 29 q^{31} + 246 q^{32} + 77 q^{33} - 99 q^{34} - 27 q^{35} + 76 q^{36} - 20 q^{37} + 200 q^{39} + 210 q^{40} - 51 q^{41} + 80 q^{42} + 65 q^{43} + 54 q^{44} + 71 q^{45} + 75 q^{46} + 261 q^{47} - 113 q^{48} - 131 q^{49} + 63 q^{50} - 78 q^{51} + 92 q^{52} - 63 q^{53} - 485 q^{54} - 100 q^{55} + 153 q^{56} + 224 q^{57} - 80 q^{58} - 102 q^{59} + 103 q^{60} + 78 q^{61} + 421 q^{63} + 106 q^{64} - 225 q^{65} - 401 q^{66} - 132 q^{67} - 297 q^{69} + 179 q^{70} - 66 q^{72} + q^{73} - 245 q^{75} + 233 q^{76} - 447 q^{77} - 440 q^{78} + 140 q^{79} + 96 q^{80} + 104 q^{81} - 157 q^{82} + 255 q^{83} - 316 q^{84} + 102 q^{85} - 136 q^{87} - 816 q^{88} - 720 q^{89} + 418 q^{90} - 70 q^{91} - 1239 q^{92} + 210 q^{93} + 261 q^{94} + 642 q^{95} + 539 q^{96} + 178 q^{97} + 483 q^{98} - 103 q^{99}+O(q^{100})$$ 22 * q - 6 * q^2 + 8 * q^3 + 12 * q^4 - 8 * q^6 + 3 * q^7 + 20 * q^9 + 25 * q^10 - 20 * q^12 - 18 * q^13 - 90 * q^14 + 53 * q^15 + 12 * q^16 + 6 * q^17 - 56 * q^18 + 3 * q^19 - 39 * q^20 - 2 * q^21 - 59 * q^22 + 15 * q^24 - 114 * q^25 - 3 * q^26 - 97 * q^27 + 34 * q^28 - 63 * q^29 - 20 * q^30 - 29 * q^31 + 246 * q^32 + 77 * q^33 - 99 * q^34 - 27 * q^35 + 76 * q^36 - 20 * q^37 + 200 * q^39 + 210 * q^40 - 51 * q^41 + 80 * q^42 + 65 * q^43 + 54 * q^44 + 71 * q^45 + 75 * q^46 + 261 * q^47 - 113 * q^48 - 131 * q^49 + 63 * q^50 - 78 * q^51 + 92 * q^52 - 63 * q^53 - 485 * q^54 - 100 * q^55 + 153 * q^56 + 224 * q^57 - 80 * q^58 - 102 * q^59 + 103 * q^60 + 78 * q^61 + 421 * q^63 + 106 * q^64 - 225 * q^65 - 401 * q^66 - 132 * q^67 - 297 * q^69 + 179 * q^70 - 66 * q^72 + q^73 - 245 * q^75 + 233 * q^76 - 447 * q^77 - 440 * q^78 + 140 * q^79 + 96 * q^80 + 104 * q^81 - 157 * q^82 + 255 * q^83 - 316 * q^84 + 102 * q^85 - 136 * q^87 - 816 * q^88 - 720 * q^89 + 418 * q^90 - 70 * q^91 - 1239 * q^92 + 210 * q^93 + 261 * q^94 + 642 * q^95 + 539 * q^96 + 178 * q^97 + 483 * q^98 - 103 * 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 −3.19625 + 1.84536i 2.96329 + 0.467854i 4.81069 8.33236i 5.83234i −10.3348 + 3.97296i 3.24949 6.20007i 20.7469i 8.56223 + 2.77278i 10.7628 + 18.6416i
2.2 −2.79169 + 1.61178i −0.476488 2.96192i 3.19568 5.53509i 5.53294i 6.10417 + 7.50076i 3.31972 + 6.16275i 7.70873i −8.54592 + 2.82264i −8.91790 15.4463i
2.3 −2.09020 + 1.20678i −2.95029 + 0.543839i 0.912615 1.58070i 6.85138i 5.51041 4.69707i 5.41224 + 4.43934i 5.24892i 8.40848 3.20897i 8.26808 + 14.3207i
2.4 −1.26920 + 0.732774i 1.95453 + 2.27592i −0.926086 + 1.60403i 1.15270i −4.14843 1.45637i −2.90907 + 6.36689i 8.57663i −1.35961 + 8.89671i 0.844669 + 1.46301i
2.5 −1.11793 + 0.645440i −0.401800 2.97297i −1.16681 + 2.02098i 4.83636i 2.36806 + 3.06425i −1.74042 6.78019i 8.17595i −8.67711 + 2.38908i 3.12158 + 5.40673i
2.6 −0.444866 + 0.256844i 2.83879 0.970187i −1.86806 + 3.23558i 7.02462i −1.01370 + 1.16073i 5.34652 4.51827i 3.97395i 7.11748 5.50832i −1.80423 3.12502i
2.7 0.0664669 0.0383747i −2.92647 0.660129i −1.99705 + 3.45900i 4.07697i −0.219846 + 0.0684257i −6.97461 + 0.595686i 0.613543i 8.12846 + 3.86370i 0.156452 + 0.270983i
2.8 1.11318 0.642694i 2.34935 1.86563i −1.17389 + 2.03324i 7.87519i 1.41622 3.58669i 0.417718 + 6.98753i 8.15936i 2.03887 8.76601i −5.06133 8.76649i
2.9 1.86624 1.07747i 2.05320 + 2.18732i 0.321900 0.557548i 1.87862i 6.18854 + 1.86979i −3.79886 5.87951i 7.23243i −0.568739 + 8.98201i −2.02416 3.50595i
2.10 2.37724 1.37250i −2.35466 1.85892i 1.76751 3.06142i 2.68504i −8.14895 1.18734i 6.00002 3.60552i 1.27635i 2.08880 + 8.75425i −3.68521 6.38297i
2.11 2.48702 1.43588i 0.950543 2.84543i 2.12350 3.67801i 7.54889i −1.72168 8.44150i −6.82274 + 1.56534i 0.709334i −7.19294 5.40941i 10.8393 + 18.7742i
32.1 −3.19625 1.84536i 2.96329 0.467854i 4.81069 + 8.33236i 5.83234i −10.3348 3.97296i 3.24949 + 6.20007i 20.7469i 8.56223 2.77278i 10.7628 18.6416i
32.2 −2.79169 1.61178i −0.476488 + 2.96192i 3.19568 + 5.53509i 5.53294i 6.10417 7.50076i 3.31972 6.16275i 7.70873i −8.54592 2.82264i −8.91790 + 15.4463i
32.3 −2.09020 1.20678i −2.95029 0.543839i 0.912615 + 1.58070i 6.85138i 5.51041 + 4.69707i 5.41224 4.43934i 5.24892i 8.40848 + 3.20897i 8.26808 14.3207i
32.4 −1.26920 0.732774i 1.95453 2.27592i −0.926086 1.60403i 1.15270i −4.14843 + 1.45637i −2.90907 6.36689i 8.57663i −1.35961 8.89671i 0.844669 1.46301i
32.5 −1.11793 0.645440i −0.401800 + 2.97297i −1.16681 2.02098i 4.83636i 2.36806 3.06425i −1.74042 + 6.78019i 8.17595i −8.67711 2.38908i 3.12158 5.40673i
32.6 −0.444866 0.256844i 2.83879 + 0.970187i −1.86806 3.23558i 7.02462i −1.01370 1.16073i 5.34652 + 4.51827i 3.97395i 7.11748 + 5.50832i −1.80423 + 3.12502i
32.7 0.0664669 + 0.0383747i −2.92647 + 0.660129i −1.99705 3.45900i 4.07697i −0.219846 0.0684257i −6.97461 0.595686i 0.613543i 8.12846 3.86370i 0.156452 0.270983i
32.8 1.11318 + 0.642694i 2.34935 + 1.86563i −1.17389 2.03324i 7.87519i 1.41622 + 3.58669i 0.417718 6.98753i 8.15936i 2.03887 + 8.76601i −5.06133 + 8.76649i
32.9 1.86624 + 1.07747i 2.05320 2.18732i 0.321900 + 0.557548i 1.87862i 6.18854 1.86979i −3.79886 + 5.87951i 7.23243i −0.568739 8.98201i −2.02416 + 3.50595i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.n.b yes 22
3.b odd 2 1 189.3.n.b 22
7.b odd 2 1 441.3.n.f 22
7.c even 3 1 63.3.j.b 22
7.c even 3 1 441.3.r.g 22
7.d odd 6 1 441.3.j.f 22
7.d odd 6 1 441.3.r.f 22
9.c even 3 1 189.3.j.b 22
9.d odd 6 1 63.3.j.b 22
21.h odd 6 1 189.3.j.b 22
63.g even 3 1 189.3.n.b 22
63.i even 6 1 441.3.r.f 22
63.j odd 6 1 441.3.r.g 22
63.n odd 6 1 inner 63.3.n.b yes 22
63.o even 6 1 441.3.j.f 22
63.s even 6 1 441.3.n.f 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.b 22 7.c even 3 1
63.3.j.b 22 9.d odd 6 1
63.3.n.b yes 22 1.a even 1 1 trivial
63.3.n.b yes 22 63.n odd 6 1 inner
189.3.j.b 22 9.c even 3 1
189.3.j.b 22 21.h odd 6 1
189.3.n.b 22 3.b odd 2 1
189.3.n.b 22 63.g even 3 1
441.3.j.f 22 7.d odd 6 1
441.3.j.f 22 63.o even 6 1
441.3.n.f 22 7.b odd 2 1
441.3.n.f 22 63.s even 6 1
441.3.r.f 22 7.d odd 6 1
441.3.r.f 22 63.i even 6 1
441.3.r.g 22 7.c even 3 1
441.3.r.g 22 63.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{22} + 6 T_{2}^{21} - 10 T_{2}^{20} - 132 T_{2}^{19} + 63 T_{2}^{18} + 1884 T_{2}^{17} + 887 T_{2}^{16} - 15735 T_{2}^{15} - 12503 T_{2}^{14} + 93906 T_{2}^{13} + 112326 T_{2}^{12} - 336231 T_{2}^{11} - 485559 T_{2}^{10} + \cdots + 2187$$ acting on $$S_{3}^{\mathrm{new}}(63, [\chi])$$.