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Properties

Label 63.2.f.a.22.3
Level $63$
Weight $2$
Character 63.22
Analytic conductor $0.503$
Analytic rank $0$
Dimension $6$
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(22,63)] 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("63.22"); 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([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 22.3
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 63.22
Dual form 63.2.f.a.43.3

$q$-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+(0.439693 + 0.761570i) q^{2} +(-0.592396 + 1.62760i) q^{3} +(0.613341 - 1.06234i) q^{4} +(-0.673648 + 1.16679i) q^{5} +(-1.50000 + 0.264490i) q^{6} +(-0.500000 - 0.866025i) q^{7} +2.83750 q^{8} +(-2.29813 - 1.92836i) q^{9} -1.18479 q^{10} +(-0.826352 - 1.43128i) q^{11} +(1.36571 + 1.62760i) q^{12} +(1.68479 - 2.91815i) q^{13} +(0.439693 - 0.761570i) q^{14} +(-1.50000 - 1.78763i) q^{15} +(0.0209445 + 0.0362770i) q^{16} +0.467911 q^{17} +(0.458111 - 2.59808i) q^{18} -3.22668 q^{19} +(0.826352 + 1.43128i) q^{20} +(1.70574 - 0.300767i) q^{21} +(0.726682 - 1.25865i) q^{22} +(-4.47178 + 7.74535i) q^{23} +(-1.68092 + 4.61830i) q^{24} +(1.59240 + 2.75811i) q^{25} +2.96316 q^{26} +(4.50000 - 2.59808i) q^{27} -1.22668 q^{28} +(-3.13429 - 5.42874i) q^{29} +(0.701867 - 1.92836i) q^{30} +(-4.61721 + 7.99724i) q^{31} +(2.81908 - 4.88279i) q^{32} +(2.81908 - 0.497079i) q^{33} +(0.205737 + 0.356347i) q^{34} +1.34730 q^{35} +(-3.45811 + 1.25865i) q^{36} +9.23442 q^{37} +(-1.41875 - 2.45734i) q^{38} +(3.75150 + 4.47086i) q^{39} +(-1.91147 + 3.31077i) q^{40} +(-1.70574 + 2.95442i) q^{41} +(0.979055 + 1.16679i) q^{42} +(2.20574 + 3.82045i) q^{43} -2.02734 q^{44} +(3.79813 - 1.38241i) q^{45} -7.86484 q^{46} +(-4.67752 - 8.10170i) q^{47} +(-0.0714517 + 0.0125989i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-1.40033 + 2.42544i) q^{50} +(-0.277189 + 0.761570i) q^{51} +(-2.06670 - 3.57964i) q^{52} -0.573978 q^{53} +(3.95723 + 2.28471i) q^{54} +2.22668 q^{55} +(-1.41875 - 2.45734i) q^{56} +(1.91147 - 5.25173i) q^{57} +(2.75624 - 4.77396i) q^{58} +(5.19846 - 9.00400i) q^{59} +(-2.81908 + 0.497079i) q^{60} +(-3.81908 - 6.61484i) q^{61} -8.12061 q^{62} +(-0.520945 + 2.95442i) q^{63} +5.04189 q^{64} +(2.26991 + 3.93161i) q^{65} +(1.61809 + 1.92836i) q^{66} +(-0.298133 + 0.516382i) q^{67} +(0.286989 - 0.497079i) q^{68} +(-9.95723 - 11.8666i) q^{69} +(0.592396 + 1.02606i) q^{70} -0.554378 q^{71} +(-6.52094 - 5.47172i) q^{72} +2.04963 q^{73} +(4.06031 + 7.03266i) q^{74} +(-5.43242 + 0.957882i) q^{75} +(-1.97906 + 3.42782i) q^{76} +(-0.826352 + 1.43128i) q^{77} +(-1.75537 + 4.82283i) q^{78} +(1.20187 + 2.08169i) q^{79} -0.0564370 q^{80} +(1.56283 + 8.86327i) q^{81} -3.00000 q^{82} +(7.52481 + 13.0334i) q^{83} +(0.726682 - 1.99654i) q^{84} +(-0.315207 + 0.545955i) q^{85} +(-1.93969 + 3.35965i) q^{86} +(10.6925 - 1.88538i) q^{87} +(-2.34477 - 4.06126i) q^{88} +9.08647 q^{89} +(2.72281 + 2.28471i) q^{90} -3.36959 q^{91} +(5.48545 + 9.50108i) q^{92} +(-10.2811 - 12.2525i) q^{93} +(4.11334 - 7.12452i) q^{94} +(2.17365 - 3.76487i) q^{95} +(6.27719 + 7.48086i) q^{96} +(0.949493 + 1.64457i) q^{97} -0.879385 q^{98} +(-0.860967 + 4.88279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} + 12 q^{8} - 6 q^{11} + 18 q^{12} + 3 q^{13} - 3 q^{14} - 9 q^{15} - 3 q^{16} + 12 q^{17} + 9 q^{18} - 6 q^{19} + 6 q^{20} - 9 q^{22} - 12 q^{23} - 27 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.439693 + 0.761570i 0.310910 + 0.538511i 0.978560 − 0.205964i \(-0.0660330\pi\)
−0.667650 + 0.744475i \(0.732700\pi\)
\(3\) −0.592396 + 1.62760i −0.342020 + 0.939693i
\(4\) 0.613341 − 1.06234i 0.306670 − 0.531169i
\(5\) −0.673648 + 1.16679i −0.301265 + 0.521806i −0.976423 − 0.215867i \(-0.930742\pi\)
0.675158 + 0.737673i \(0.264075\pi\)
\(6\) −1.50000 + 0.264490i −0.612372 + 0.107978i
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 2.83750 1.00321
\(9\) −2.29813 − 1.92836i −0.766044 − 0.642788i
\(10\) −1.18479 −0.374664
\(11\) −0.826352 − 1.43128i −0.249154 − 0.431548i 0.714137 − 0.700006i \(-0.246819\pi\)
−0.963291 + 0.268458i \(0.913486\pi\)
\(12\) 1.36571 + 1.62760i 0.394248 + 0.469846i
\(13\) 1.68479 − 2.91815i 0.467277 − 0.809348i −0.532024 − 0.846729i \(-0.678568\pi\)
0.999301 + 0.0373813i \(0.0119016\pi\)
\(14\) 0.439693 − 0.761570i 0.117513 − 0.203538i
\(15\) −1.50000 − 1.78763i −0.387298 − 0.461564i
\(16\) 0.0209445 + 0.0362770i 0.00523613 + 0.00906925i
\(17\) 0.467911 0.113485 0.0567426 − 0.998389i \(-0.481929\pi\)
0.0567426 + 0.998389i \(0.481929\pi\)
\(18\) 0.458111 − 2.59808i 0.107978 − 0.612372i
\(19\) −3.22668 −0.740252 −0.370126 − 0.928982i \(-0.620685\pi\)
−0.370126 + 0.928982i \(0.620685\pi\)
\(20\) 0.826352 + 1.43128i 0.184778 + 0.320045i
\(21\) 1.70574 − 0.300767i 0.372222 − 0.0656328i
\(22\) 0.726682 − 1.25865i 0.154929 − 0.268345i
\(23\) −4.47178 + 7.74535i −0.932431 + 1.61502i −0.153279 + 0.988183i \(0.548983\pi\)
−0.779152 + 0.626835i \(0.784350\pi\)
\(24\) −1.68092 + 4.61830i −0.343117 + 0.942706i
\(25\) 1.59240 + 2.75811i 0.318479 + 0.551622i
\(26\) 2.96316 0.581124
\(27\) 4.50000 − 2.59808i 0.866025 − 0.500000i
\(28\) −1.22668 −0.231821
\(29\) −3.13429 − 5.42874i −0.582022 − 1.00809i −0.995239 − 0.0974595i \(-0.968928\pi\)
0.413217 − 0.910632i \(-0.364405\pi\)
\(30\) 0.701867 − 1.92836i 0.128143 − 0.352069i
\(31\) −4.61721 + 7.99724i −0.829276 + 1.43635i 0.0693317 + 0.997594i \(0.477913\pi\)
−0.898607 + 0.438754i \(0.855420\pi\)
\(32\) 2.81908 − 4.88279i 0.498347 − 0.863163i
\(33\) 2.81908 − 0.497079i 0.490738 − 0.0865304i
\(34\) 0.205737 + 0.356347i 0.0352836 + 0.0611130i
\(35\) 1.34730 0.227735
\(36\) −3.45811 + 1.25865i −0.576352 + 0.209775i
\(37\) 9.23442 1.51813 0.759065 − 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) −1.41875 − 2.45734i −0.230151 − 0.398634i
\(39\) 3.75150 + 4.47086i 0.600720 + 0.715910i
\(40\) −1.91147 + 3.31077i −0.302231 + 0.523479i
\(41\) −1.70574 + 2.95442i −0.266391 + 0.461403i −0.967927 − 0.251231i \(-0.919165\pi\)
0.701536 + 0.712634i \(0.252498\pi\)
\(42\) 0.979055 + 1.16679i 0.151072 + 0.180040i
\(43\) 2.20574 + 3.82045i 0.336372 + 0.582613i 0.983747 − 0.179558i \(-0.0574668\pi\)
−0.647376 + 0.762171i \(0.724133\pi\)
\(44\) −2.02734 −0.305633
\(45\) 3.79813 − 1.38241i 0.566192 − 0.206077i
\(46\) −7.86484 −1.15961
\(47\) −4.67752 − 8.10170i −0.682286 − 1.18175i −0.974281 − 0.225335i \(-0.927652\pi\)
0.291995 − 0.956420i \(-0.405681\pi\)
\(48\) −0.0714517 + 0.0125989i −0.0103132 + 0.00181849i
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) −1.40033 + 2.42544i −0.198037 + 0.343009i
\(51\) −0.277189 + 0.761570i −0.0388142 + 0.106641i
\(52\) −2.06670 − 3.57964i −0.286600 − 0.496406i
\(53\) −0.573978 −0.0788419 −0.0394210 − 0.999223i \(-0.512551\pi\)
−0.0394210 + 0.999223i \(0.512551\pi\)
\(54\) 3.95723 + 2.28471i 0.538511 + 0.310910i
\(55\) 2.22668 0.300246
\(56\) −1.41875 − 2.45734i −0.189588 − 0.328376i
\(57\) 1.91147 − 5.25173i 0.253181 − 0.695609i
\(58\) 2.75624 − 4.77396i 0.361913 − 0.626851i
\(59\) 5.19846 − 9.00400i 0.676782 − 1.17222i −0.299162 − 0.954202i \(-0.596707\pi\)
0.975945 − 0.218019i \(-0.0699595\pi\)
\(60\) −2.81908 + 0.497079i −0.363941 + 0.0641727i
\(61\) −3.81908 − 6.61484i −0.488983 − 0.846943i 0.510937 − 0.859618i \(-0.329299\pi\)
−0.999920 + 0.0126752i \(0.995965\pi\)
\(62\) −8.12061 −1.03132
\(63\) −0.520945 + 2.95442i −0.0656328 + 0.372222i
\(64\) 5.04189 0.630236
\(65\) 2.26991 + 3.93161i 0.281548 + 0.487656i
\(66\) 1.61809 + 1.92836i 0.199173 + 0.237365i
\(67\) −0.298133 + 0.516382i −0.0364228 + 0.0630861i −0.883662 − 0.468125i \(-0.844930\pi\)
0.847239 + 0.531211i \(0.178263\pi\)
\(68\) 0.286989 − 0.497079i 0.0348025 − 0.0602797i
\(69\) −9.95723 − 11.8666i −1.19871 − 1.42857i
\(70\) 0.592396 + 1.02606i 0.0708049 + 0.122638i
\(71\) −0.554378 −0.0657925 −0.0328963 − 0.999459i \(-0.510473\pi\)
−0.0328963 + 0.999459i \(0.510473\pi\)
\(72\) −6.52094 − 5.47172i −0.768501 − 0.644849i
\(73\) 2.04963 0.239891 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 4.06031 + 7.03266i 0.472001 + 0.817530i
\(75\) −5.43242 + 0.957882i −0.627282 + 0.110607i
\(76\) −1.97906 + 3.42782i −0.227013 + 0.393198i
\(77\) −0.826352 + 1.43128i −0.0941715 + 0.163110i
\(78\) −1.75537 + 4.82283i −0.198756 + 0.546078i
\(79\) 1.20187 + 2.08169i 0.135221 + 0.234209i 0.925682 − 0.378303i \(-0.123492\pi\)
−0.790461 + 0.612512i \(0.790159\pi\)
\(80\) −0.0564370 −0.00630985
\(81\) 1.56283 + 8.86327i 0.173648 + 0.984808i
\(82\) −3.00000 −0.331295
\(83\) 7.52481 + 13.0334i 0.825956 + 1.43060i 0.901187 + 0.433431i \(0.142697\pi\)
−0.0752309 + 0.997166i \(0.523969\pi\)
\(84\) 0.726682 − 1.99654i 0.0792875 − 0.217841i
\(85\) −0.315207 + 0.545955i −0.0341891 + 0.0592172i
\(86\) −1.93969 + 3.35965i −0.209162 + 0.362280i
\(87\) 10.6925 − 1.88538i 1.14636 − 0.202134i
\(88\) −2.34477 − 4.06126i −0.249953 − 0.432932i
\(89\) 9.08647 0.963164 0.481582 − 0.876401i \(-0.340062\pi\)
0.481582 + 0.876401i \(0.340062\pi\)
\(90\) 2.72281 + 2.28471i 0.287010 + 0.240830i
\(91\) −3.36959 −0.353228
\(92\) 5.48545 + 9.50108i 0.571898 + 0.990556i
\(93\) −10.2811 − 12.2525i −1.06610 − 1.27052i
\(94\) 4.11334 − 7.12452i 0.424259 − 0.734838i
\(95\) 2.17365 − 3.76487i 0.223012 − 0.386267i
\(96\) 6.27719 + 7.48086i 0.640663 + 0.763512i
\(97\) 0.949493 + 1.64457i 0.0964064 + 0.166981i 0.910195 − 0.414181i \(-0.135932\pi\)
−0.813788 + 0.581161i \(0.802598\pi\)
\(98\) −0.879385 −0.0888313
\(99\) −0.860967 + 4.88279i −0.0865304 + 0.490738i
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.f.a.22.3 ✓ 6
3.2 odd 2 189.2.f.b.64.1 6
4.3 odd 2 1008.2.r.h.337.2 6
7.2 even 3 441.2.g.c.67.3 6
7.3 odd 6 441.2.h.e.373.1 6
7.4 even 3 441.2.h.d.373.1 6
7.5 odd 6 441.2.g.b.67.3 6
7.6 odd 2 441.2.f.c.148.3 6
9.2 odd 6 189.2.f.b.127.1 6
9.4 even 3 567.2.a.h.1.1 3
9.5 odd 6 567.2.a.c.1.3 3
9.7 even 3 inner 63.2.f.a.43.3 yes 6
12.11 even 2 3024.2.r.k.1009.2 6
21.2 odd 6 1323.2.g.d.361.1 6
21.5 even 6 1323.2.g.e.361.1 6
21.11 odd 6 1323.2.h.c.226.3 6
21.17 even 6 1323.2.h.b.226.3 6
21.20 even 2 1323.2.f.d.442.1 6
36.7 odd 6 1008.2.r.h.673.2 6
36.11 even 6 3024.2.r.k.2017.2 6
36.23 even 6 9072.2.a.bs.1.2 3
36.31 odd 6 9072.2.a.ca.1.2 3
63.2 odd 6 1323.2.h.c.802.3 6
63.11 odd 6 1323.2.g.d.667.1 6
63.13 odd 6 3969.2.a.q.1.1 3
63.16 even 3 441.2.h.d.214.1 6
63.20 even 6 1323.2.f.d.883.1 6
63.25 even 3 441.2.g.c.79.3 6
63.34 odd 6 441.2.f.c.295.3 6
63.38 even 6 1323.2.g.e.667.1 6
63.41 even 6 3969.2.a.l.1.3 3
63.47 even 6 1323.2.h.b.802.3 6
63.52 odd 6 441.2.g.b.79.3 6
63.61 odd 6 441.2.h.e.214.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 1.1 even 1 trivial
63.2.f.a.43.3 yes 6 9.7 even 3 inner
189.2.f.b.64.1 6 3.2 odd 2
189.2.f.b.127.1 6 9.2 odd 6
441.2.f.c.148.3 6 7.6 odd 2
441.2.f.c.295.3 6 63.34 odd 6
441.2.g.b.67.3 6 7.5 odd 6
441.2.g.b.79.3 6 63.52 odd 6
441.2.g.c.67.3 6 7.2 even 3
441.2.g.c.79.3 6 63.25 even 3
441.2.h.d.214.1 6 63.16 even 3
441.2.h.d.373.1 6 7.4 even 3
441.2.h.e.214.1 6 63.61 odd 6
441.2.h.e.373.1 6 7.3 odd 6
567.2.a.c.1.3 3 9.5 odd 6
567.2.a.h.1.1 3 9.4 even 3
1008.2.r.h.337.2 6 4.3 odd 2
1008.2.r.h.673.2 6 36.7 odd 6
1323.2.f.d.442.1 6 21.20 even 2
1323.2.f.d.883.1 6 63.20 even 6
1323.2.g.d.361.1 6 21.2 odd 6
1323.2.g.d.667.1 6 63.11 odd 6
1323.2.g.e.361.1 6 21.5 even 6
1323.2.g.e.667.1 6 63.38 even 6
1323.2.h.b.226.3 6 21.17 even 6
1323.2.h.b.802.3 6 63.47 even 6
1323.2.h.c.226.3 6 21.11 odd 6
1323.2.h.c.802.3 6 63.2 odd 6
3024.2.r.k.1009.2 6 12.11 even 2
3024.2.r.k.2017.2 6 36.11 even 6
3969.2.a.l.1.3 3 63.41 even 6
3969.2.a.q.1.1 3 63.13 odd 6
9072.2.a.bs.1.2 3 36.23 even 6
9072.2.a.ca.1.2 3 36.31 odd 6