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Properties

Label 63.2.i.b.38.5
Level $63$
Weight $2$
Character 63.38
Analytic conductor $0.503$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,2,Mod(5,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 38.5
Root \(0.827154 + 1.43267i\) of defining polynomial
Character \(\chi\) \(=\) 63.38
Dual form 63.2.i.b.5.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.09548i q^{2} +(-1.72861 + 0.109097i) q^{3} -2.39104 q^{4} +(-1.04492 + 1.80985i) q^{5} +(-0.228612 - 3.62227i) q^{6} +(2.60068 - 0.486271i) q^{7} -0.819421i q^{8} +(2.97620 - 0.377174i) q^{9} +(-3.79250 - 2.18960i) q^{10} +(2.79620 - 1.61439i) q^{11} +(4.13318 - 0.260856i) q^{12} +(-2.68740 + 1.55157i) q^{13} +(1.01897 + 5.44968i) q^{14} +(1.60880 - 3.24252i) q^{15} -3.06500 q^{16} +(0.816304 - 1.41388i) q^{17} +(0.790361 + 6.23656i) q^{18} +(4.79094 - 2.76605i) q^{19} +(2.49844 - 4.32742i) q^{20} +(-4.44252 + 1.12430i) q^{21} +(3.38292 + 5.85939i) q^{22} +(-1.00527 - 0.580391i) q^{23} +(0.0893966 + 1.41646i) q^{24} +(0.316304 + 0.547854i) q^{25} +(-3.25129 - 5.63139i) q^{26} +(-5.10354 + 0.976682i) q^{27} +(-6.21834 + 1.16270i) q^{28} +(-7.05749 - 4.07464i) q^{29} +(6.79464 + 3.37122i) q^{30} -5.96849i q^{31} -8.06150i q^{32} +(-4.65742 + 3.09571i) q^{33} +(2.96276 + 1.71055i) q^{34} +(-1.83741 + 5.21495i) q^{35} +(-7.11621 + 0.901839i) q^{36} +(2.82656 + 4.89575i) q^{37} +(5.79620 + 10.0393i) q^{38} +(4.47620 - 2.97525i) q^{39} +(1.48303 + 0.856225i) q^{40} +(1.35369 + 2.34465i) q^{41} +(-2.35595 - 9.30921i) q^{42} +(-0.974903 + 1.68858i) q^{43} +(-6.68583 + 3.86007i) q^{44} +(-2.42725 + 5.78057i) q^{45} +(1.21620 - 2.10652i) q^{46} +8.13518 q^{47} +(5.29820 - 0.334384i) q^{48} +(6.52708 - 2.52927i) q^{49} +(-1.14802 + 0.662809i) q^{50} +(-1.25682 + 2.53311i) q^{51} +(6.42568 - 3.70987i) q^{52} +(-5.27766 - 3.04706i) q^{53} +(-2.04662 - 10.6944i) q^{54} +6.74759i q^{55} +(-0.398461 - 2.13105i) q^{56} +(-7.97990 + 5.30410i) q^{57} +(8.53834 - 14.7888i) q^{58} -3.96206 q^{59} +(-3.84672 + 7.75300i) q^{60} +4.79219i q^{61} +12.5068 q^{62} +(7.55673 - 2.42815i) q^{63} +10.7627 q^{64} -6.48504i q^{65} +(-6.48700 - 9.75954i) q^{66} -0.673961 q^{67} +(-1.95182 + 3.38065i) q^{68} +(1.80103 + 0.893598i) q^{69} +(-10.9278 - 3.85027i) q^{70} -7.01535i q^{71} +(-0.309064 - 2.43876i) q^{72} +(-2.96276 - 1.71055i) q^{73} +(-10.2590 + 5.92301i) q^{74} +(-0.606536 - 0.912519i) q^{75} +(-11.4553 + 6.61374i) q^{76} +(6.48700 - 5.55822i) q^{77} +(6.23458 + 9.37978i) q^{78} -14.1595 q^{79} +(3.20267 - 5.54718i) q^{80} +(8.71548 - 2.24509i) q^{81} +(-4.91318 + 2.83662i) q^{82} +(-1.54535 + 2.67662i) q^{83} +(10.6222 - 2.68825i) q^{84} +(1.70594 + 2.95477i) q^{85} +(-3.53839 - 2.04289i) q^{86} +(12.6442 + 6.27352i) q^{87} +(-1.32286 - 2.29127i) q^{88} +(2.45766 + 4.25679i) q^{89} +(-12.1131 - 5.08625i) q^{90} +(-6.23458 + 5.34194i) q^{91} +(2.40363 + 1.38774i) q^{92} +(0.651146 + 10.3172i) q^{93} +17.0471i q^{94} +11.5611i q^{95} +(0.879488 + 13.9352i) q^{96} +(-2.07939 - 1.20054i) q^{97} +(5.30004 + 13.6774i) q^{98} +(7.71314 - 5.85939i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} - 8 q^{4} + 12 q^{6} - 6 q^{7} + 3 q^{9} - 15 q^{10} - 12 q^{11} - 12 q^{12} - 6 q^{13} + 12 q^{14} - 3 q^{15} + 12 q^{16} + 12 q^{17} + 24 q^{18} + 3 q^{19} + 3 q^{20} - 9 q^{21} + 5 q^{22}+ \cdots + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.09548i 1.48173i 0.671655 + 0.740865i \(0.265584\pi\)
−0.671655 + 0.740865i \(0.734416\pi\)
\(3\) −1.72861 + 0.109097i −0.998014 + 0.0629874i
\(4\) −2.39104 −1.19552
\(5\) −1.04492 + 1.80985i −0.467300 + 0.809388i −0.999302 − 0.0373553i \(-0.988107\pi\)
0.532002 + 0.846743i \(0.321440\pi\)
\(6\) −0.228612 − 3.62227i −0.0933303 − 1.47879i
\(7\) 2.60068 − 0.486271i 0.982965 − 0.183793i
\(8\) − 0.819421i − 0.289709i
\(9\) 2.97620 − 0.377174i 0.992065 − 0.125725i
\(10\) −3.79250 − 2.18960i −1.19929 − 0.692412i
\(11\) 2.79620 − 1.61439i 0.843086 − 0.486756i −0.0152257 − 0.999884i \(-0.504847\pi\)
0.858312 + 0.513128i \(0.171513\pi\)
\(12\) 4.13318 − 0.260856i 1.19315 − 0.0753028i
\(13\) −2.68740 + 1.55157i −0.745350 + 0.430328i −0.824011 − 0.566573i \(-0.808269\pi\)
0.0786612 + 0.996901i \(0.474935\pi\)
\(14\) 1.01897 + 5.44968i 0.272332 + 1.45649i
\(15\) 1.60880 − 3.24252i 0.415391 − 0.837215i
\(16\) −3.06500 −0.766251
\(17\) 0.816304 − 1.41388i 0.197983 − 0.342916i −0.749891 − 0.661561i \(-0.769894\pi\)
0.947874 + 0.318645i \(0.103228\pi\)
\(18\) 0.790361 + 6.23656i 0.186290 + 1.46997i
\(19\) 4.79094 − 2.76605i 1.09912 − 0.634575i 0.163127 − 0.986605i \(-0.447842\pi\)
0.935989 + 0.352030i \(0.114509\pi\)
\(20\) 2.49844 − 4.32742i 0.558667 − 0.967640i
\(21\) −4.44252 + 1.12430i −0.969436 + 0.245343i
\(22\) 3.38292 + 5.85939i 0.721241 + 1.24923i
\(23\) −1.00527 − 0.580391i −0.209612 − 0.121020i 0.391519 − 0.920170i \(-0.371950\pi\)
−0.601131 + 0.799150i \(0.705283\pi\)
\(24\) 0.0893966 + 1.41646i 0.0182480 + 0.289134i
\(25\) 0.316304 + 0.547854i 0.0632608 + 0.109571i
\(26\) −3.25129 − 5.63139i −0.637630 − 1.10441i
\(27\) −5.10354 + 0.976682i −0.982176 + 0.187963i
\(28\) −6.21834 + 1.16270i −1.17516 + 0.219729i
\(29\) −7.05749 − 4.07464i −1.31054 − 0.756643i −0.328357 − 0.944554i \(-0.606495\pi\)
−0.982186 + 0.187911i \(0.939828\pi\)
\(30\) 6.79464 + 3.37122i 1.24053 + 0.615497i
\(31\) − 5.96849i − 1.07197i −0.844227 − 0.535986i \(-0.819940\pi\)
0.844227 − 0.535986i \(-0.180060\pi\)
\(32\) − 8.06150i − 1.42508i
\(33\) −4.65742 + 3.09571i −0.810753 + 0.538893i
\(34\) 2.96276 + 1.71055i 0.508109 + 0.293357i
\(35\) −1.83741 + 5.21495i −0.310580 + 0.881487i
\(36\) −7.11621 + 0.901839i −1.18603 + 0.150306i
\(37\) 2.82656 + 4.89575i 0.464684 + 0.804857i 0.999187 − 0.0403097i \(-0.0128345\pi\)
−0.534503 + 0.845167i \(0.679501\pi\)
\(38\) 5.79620 + 10.0393i 0.940268 + 1.62859i
\(39\) 4.47620 − 2.97525i 0.716765 − 0.476421i
\(40\) 1.48303 + 0.856225i 0.234487 + 0.135381i
\(41\) 1.35369 + 2.34465i 0.211410 + 0.366173i 0.952156 − 0.305612i \(-0.0988611\pi\)
−0.740746 + 0.671785i \(0.765528\pi\)
\(42\) −2.35595 − 9.30921i −0.363531 − 1.43644i
\(43\) −0.974903 + 1.68858i −0.148671 + 0.257506i −0.930737 − 0.365690i \(-0.880833\pi\)
0.782065 + 0.623196i \(0.214166\pi\)
\(44\) −6.68583 + 3.86007i −1.00793 + 0.581927i
\(45\) −2.42725 + 5.78057i −0.361832 + 0.861717i
\(46\) 1.21620 − 2.10652i 0.179319 − 0.310589i
\(47\) 8.13518 1.18664 0.593319 − 0.804967i \(-0.297817\pi\)
0.593319 + 0.804967i \(0.297817\pi\)
\(48\) 5.29820 − 0.334384i 0.764729 − 0.0482641i
\(49\) 6.52708 − 2.52927i 0.932440 − 0.361325i
\(50\) −1.14802 + 0.662809i −0.162354 + 0.0937353i
\(51\) −1.25682 + 2.53311i −0.175990 + 0.354706i
\(52\) 6.42568 − 3.70987i 0.891082 − 0.514466i
\(53\) −5.27766 − 3.04706i −0.724943 − 0.418546i 0.0916264 − 0.995793i \(-0.470793\pi\)
−0.816569 + 0.577248i \(0.804127\pi\)
\(54\) −2.04662 − 10.6944i −0.278510 − 1.45532i
\(55\) 6.74759i 0.909845i
\(56\) −0.398461 − 2.13105i −0.0532466 − 0.284774i
\(57\) −7.97990 + 5.30410i −1.05696 + 0.702545i
\(58\) 8.53834 − 14.7888i 1.12114 − 1.94187i
\(59\) −3.96206 −0.515816 −0.257908 − 0.966170i \(-0.583033\pi\)
−0.257908 + 0.966170i \(0.583033\pi\)
\(60\) −3.84672 + 7.75300i −0.496609 + 1.00091i
\(61\) 4.79219i 0.613577i 0.951778 + 0.306788i \(0.0992544\pi\)
−0.951778 + 0.306788i \(0.900746\pi\)
\(62\) 12.5068 1.58837
\(63\) 7.55673 − 2.42815i 0.952058 − 0.305918i
\(64\) 10.7627 1.34534
\(65\) − 6.48504i − 0.804370i
\(66\) −6.48700 − 9.75954i −0.798494 − 1.20132i
\(67\) −0.673961 −0.0823375 −0.0411687 − 0.999152i \(-0.513108\pi\)
−0.0411687 + 0.999152i \(0.513108\pi\)
\(68\) −1.95182 + 3.38065i −0.236693 + 0.409963i
\(69\) 1.80103 + 0.893598i 0.216819 + 0.107577i
\(70\) −10.9278 − 3.85027i −1.30612 − 0.460195i
\(71\) − 7.01535i − 0.832568i −0.909235 − 0.416284i \(-0.863332\pi\)
0.909235 − 0.416284i \(-0.136668\pi\)
\(72\) −0.309064 − 2.43876i −0.0364236 − 0.287410i
\(73\) −2.96276 − 1.71055i −0.346765 − 0.200205i 0.316495 − 0.948594i \(-0.397494\pi\)
−0.663259 + 0.748390i \(0.730827\pi\)
\(74\) −10.2590 + 5.92301i −1.19258 + 0.688536i
\(75\) −0.606536 − 0.912519i −0.0700367 − 0.105369i
\(76\) −11.4553 + 6.61374i −1.31402 + 0.758648i
\(77\) 6.48700 − 5.55822i 0.739262 − 0.633418i
\(78\) 6.23458 + 9.37978i 0.705927 + 1.06205i
\(79\) −14.1595 −1.59306 −0.796532 − 0.604596i \(-0.793335\pi\)
−0.796532 + 0.604596i \(0.793335\pi\)
\(80\) 3.20267 − 5.54718i 0.358069 − 0.620194i
\(81\) 8.71548 − 2.24509i 0.968387 − 0.249454i
\(82\) −4.91318 + 2.83662i −0.542570 + 0.313253i
\(83\) −1.54535 + 2.67662i −0.169624 + 0.293798i −0.938288 − 0.345856i \(-0.887589\pi\)
0.768664 + 0.639653i \(0.220922\pi\)
\(84\) 10.6222 − 2.68825i 1.15898 − 0.293312i
\(85\) 1.70594 + 2.95477i 0.185035 + 0.320490i
\(86\) −3.53839 − 2.04289i −0.381554 − 0.220291i
\(87\) 12.6442 + 6.27352i 1.35560 + 0.672592i
\(88\) −1.32286 − 2.29127i −0.141018 − 0.244250i
\(89\) 2.45766 + 4.25679i 0.260511 + 0.451219i 0.966378 − 0.257126i \(-0.0827756\pi\)
−0.705867 + 0.708345i \(0.749442\pi\)
\(90\) −12.1131 − 5.08625i −1.27683 − 0.536138i
\(91\) −6.23458 + 5.34194i −0.653562 + 0.559988i
\(92\) 2.40363 + 1.38774i 0.250596 + 0.144682i
\(93\) 0.651146 + 10.3172i 0.0675207 + 1.06984i
\(94\) 17.0471i 1.75828i
\(95\) 11.5611i 1.18615i
\(96\) 0.879488 + 13.9352i 0.0897624 + 1.42226i
\(97\) −2.07939 − 1.20054i −0.211130 − 0.121896i 0.390706 − 0.920515i \(-0.372231\pi\)
−0.601837 + 0.798619i \(0.705564\pi\)
\(98\) 5.30004 + 13.6774i 0.535385 + 1.38162i
\(99\) 7.71314 − 5.85939i 0.775199 − 0.588891i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 63.2.i.b.38.5 yes 10
3.2 odd 2 189.2.i.b.143.1 10
4.3 odd 2 1008.2.ca.b.353.5 10
7.2 even 3 441.2.s.b.362.5 10
7.3 odd 6 441.2.o.c.146.1 10
7.4 even 3 441.2.o.d.146.1 10
7.5 odd 6 63.2.s.b.47.5 yes 10
7.6 odd 2 441.2.i.b.227.5 10
9.2 odd 6 567.2.p.c.80.1 10
9.4 even 3 189.2.s.b.17.1 10
9.5 odd 6 63.2.s.b.59.5 yes 10
9.7 even 3 567.2.p.d.80.5 10
12.11 even 2 3024.2.ca.b.2033.5 10
21.2 odd 6 1323.2.s.b.656.1 10
21.5 even 6 189.2.s.b.89.1 10
21.11 odd 6 1323.2.o.c.440.5 10
21.17 even 6 1323.2.o.d.440.5 10
21.20 even 2 1323.2.i.b.521.1 10
28.19 even 6 1008.2.df.b.929.3 10
36.23 even 6 1008.2.df.b.689.3 10
36.31 odd 6 3024.2.df.b.17.5 10
63.4 even 3 1323.2.o.d.881.5 10
63.5 even 6 inner 63.2.i.b.5.1 ✓ 10
63.13 odd 6 1323.2.s.b.962.1 10
63.23 odd 6 441.2.i.b.68.1 10
63.31 odd 6 1323.2.o.c.881.5 10
63.32 odd 6 441.2.o.c.293.1 10
63.40 odd 6 189.2.i.b.152.5 10
63.41 even 6 441.2.s.b.374.5 10
63.47 even 6 567.2.p.d.404.5 10
63.58 even 3 1323.2.i.b.1097.5 10
63.59 even 6 441.2.o.d.293.1 10
63.61 odd 6 567.2.p.c.404.1 10
84.47 odd 6 3024.2.df.b.1601.5 10
252.103 even 6 3024.2.ca.b.2609.5 10
252.131 odd 6 1008.2.ca.b.257.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.i.b.5.1 ✓ 10 63.5 even 6 inner
63.2.i.b.38.5 yes 10 1.1 even 1 trivial
63.2.s.b.47.5 yes 10 7.5 odd 6
63.2.s.b.59.5 yes 10 9.5 odd 6
189.2.i.b.143.1 10 3.2 odd 2
189.2.i.b.152.5 10 63.40 odd 6
189.2.s.b.17.1 10 9.4 even 3
189.2.s.b.89.1 10 21.5 even 6
441.2.i.b.68.1 10 63.23 odd 6
441.2.i.b.227.5 10 7.6 odd 2
441.2.o.c.146.1 10 7.3 odd 6
441.2.o.c.293.1 10 63.32 odd 6
441.2.o.d.146.1 10 7.4 even 3
441.2.o.d.293.1 10 63.59 even 6
441.2.s.b.362.5 10 7.2 even 3
441.2.s.b.374.5 10 63.41 even 6
567.2.p.c.80.1 10 9.2 odd 6
567.2.p.c.404.1 10 63.61 odd 6
567.2.p.d.80.5 10 9.7 even 3
567.2.p.d.404.5 10 63.47 even 6
1008.2.ca.b.257.5 10 252.131 odd 6
1008.2.ca.b.353.5 10 4.3 odd 2
1008.2.df.b.689.3 10 36.23 even 6
1008.2.df.b.929.3 10 28.19 even 6
1323.2.i.b.521.1 10 21.20 even 2
1323.2.i.b.1097.5 10 63.58 even 3
1323.2.o.c.440.5 10 21.11 odd 6
1323.2.o.c.881.5 10 63.31 odd 6
1323.2.o.d.440.5 10 21.17 even 6
1323.2.o.d.881.5 10 63.4 even 3
1323.2.s.b.656.1 10 21.2 odd 6
1323.2.s.b.962.1 10 63.13 odd 6
3024.2.ca.b.2033.5 10 12.11 even 2
3024.2.ca.b.2609.5 10 252.103 even 6
3024.2.df.b.17.5 10 36.31 odd 6
3024.2.df.b.1601.5 10 84.47 odd 6