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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 6 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 3 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 6 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 3 q^{7} + 20 q^{9} + 25 q^{10} - 20 q^{12} - 18 q^{13} - 90 q^{14} + 53 q^{15} + 12 q^{16} + 6 q^{17} - 56 q^{18} + 3 q^{19} - 39 q^{20} - 2 q^{21} - 59 q^{22} + 15 q^{24} - 114 q^{25} - 3 q^{26} - 97 q^{27} + 34 q^{28} - 63 q^{29} - 20 q^{30} - 29 q^{31} + 246 q^{32} + 77 q^{33} - 99 q^{34} - 27 q^{35} + 76 q^{36} - 20 q^{37} + 200 q^{39} + 210 q^{40} - 51 q^{41} + 80 q^{42} + 65 q^{43} + 54 q^{44} + 71 q^{45} + 75 q^{46} + 261 q^{47} - 113 q^{48} - 131 q^{49} + 63 q^{50} - 78 q^{51} + 92 q^{52} - 63 q^{53} - 485 q^{54} - 100 q^{55} + 153 q^{56} + 224 q^{57} - 80 q^{58} - 102 q^{59} + 103 q^{60} + 78 q^{61} + 421 q^{63} + 106 q^{64} - 225 q^{65} - 401 q^{66} - 132 q^{67} - 297 q^{69} + 179 q^{70} - 66 q^{72} + q^{73} - 245 q^{75} + 233 q^{76} - 447 q^{77} - 440 q^{78} + 140 q^{79} + 96 q^{80} + 104 q^{81} - 157 q^{82} + 255 q^{83} - 316 q^{84} + 102 q^{85} - 136 q^{87} - 816 q^{88} - 720 q^{89} + 418 q^{90} - 70 q^{91} - 1239 q^{92} + 210 q^{93} + 261 q^{94} + 642 q^{95} + 539 q^{96} + 178 q^{97} + 483 q^{98} - 103 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:

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 \(S_{3}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display