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Properties

Label 63.2.f.a.43.2
Level $63$
Weight $2$
Character 63.43
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 43.2
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 63.43
Dual form 63.2.f.a.22.2

$q$-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+(-0.673648 + 1.16679i) q^{2} +(1.70574 + 0.300767i) q^{3} +(0.0923963 + 0.160035i) q^{4} +(-1.26604 - 2.19285i) q^{5} +(-1.50000 + 1.78763i) q^{6} +(-0.500000 + 0.866025i) q^{7} -2.94356 q^{8} +(2.81908 + 1.02606i) q^{9} +3.41147 q^{10} +(-0.233956 + 0.405223i) q^{11} +(0.109470 + 0.300767i) q^{12} +(-2.91147 - 5.04282i) q^{13} +(-0.673648 - 1.16679i) q^{14} +(-1.50000 - 4.12122i) q^{15} +(1.79813 - 3.11446i) q^{16} +3.87939 q^{17} +(-3.09627 + 2.59808i) q^{18} -2.18479 q^{19} +(0.233956 - 0.405223i) q^{20} +(-1.11334 + 1.32683i) q^{21} +(-0.315207 - 0.545955i) q^{22} +(0.0530334 + 0.0918566i) q^{23} +(-5.02094 - 0.885328i) q^{24} +(-0.705737 + 1.22237i) q^{25} +7.84524 q^{26} +(4.50000 + 2.59808i) q^{27} -0.184793 q^{28} +(-4.39053 + 7.60462i) q^{29} +(5.81908 + 1.02606i) q^{30} +(3.84002 + 6.65111i) q^{31} +(-0.520945 - 0.902302i) q^{32} +(-0.520945 + 0.620838i) q^{33} +(-2.61334 + 4.52644i) q^{34} +2.53209 q^{35} +(0.0962667 + 0.545955i) q^{36} -7.68004 q^{37} +(1.47178 - 2.54920i) q^{38} +(-3.44949 - 9.47740i) q^{39} +(3.72668 + 6.45480i) q^{40} +(1.11334 + 1.92836i) q^{41} +(-0.798133 - 2.19285i) q^{42} +(-0.613341 + 1.06234i) q^{43} -0.0864665 q^{44} +(-1.31908 - 7.48086i) q^{45} -0.142903 q^{46} +(2.66637 - 4.61830i) q^{47} +(4.00387 - 4.77163i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-0.950837 - 1.64690i) q^{50} +(6.61721 + 1.16679i) q^{51} +(0.538019 - 0.931876i) q^{52} -0.716881 q^{53} +(-6.06283 + 3.50038i) q^{54} +1.18479 q^{55} +(1.47178 - 2.54920i) q^{56} +(-3.72668 - 0.657115i) q^{57} +(-5.91534 - 10.2457i) q^{58} +(-0.368241 - 0.637812i) q^{59} +(0.520945 - 0.620838i) q^{60} +(-0.479055 + 0.829748i) q^{61} -10.3473 q^{62} +(-2.29813 + 1.92836i) q^{63} +8.59627 q^{64} +(-7.37211 + 12.7689i) q^{65} +(-0.373455 - 1.02606i) q^{66} +(4.81908 + 8.34689i) q^{67} +(0.358441 + 0.620838i) q^{68} +(0.0628336 + 0.172634i) q^{69} +(-1.70574 + 2.95442i) q^{70} +13.2344 q^{71} +(-8.29813 - 3.02027i) q^{72} -10.2686 q^{73} +(5.17365 - 8.96102i) q^{74} +(-1.57145 + 1.87278i) q^{75} +(-0.201867 - 0.349643i) q^{76} +(-0.233956 - 0.405223i) q^{77} +(13.3819 + 2.35959i) q^{78} +(6.31908 - 10.9450i) q^{79} -9.10607 q^{80} +(6.89440 + 5.78509i) q^{81} -3.00000 q^{82} +(1.36571 - 2.36549i) q^{83} +(-0.315207 - 0.0555796i) q^{84} +(-4.91147 - 8.50692i) q^{85} +(-0.826352 - 1.43128i) q^{86} +(-9.77631 + 11.6510i) q^{87} +(0.688663 - 1.19280i) q^{88} -8.11381 q^{89} +(9.61721 + 3.50038i) q^{90} +5.82295 q^{91} +(-0.00980018 + 0.0169744i) q^{92} +(4.54963 + 12.5000i) q^{93} +(3.59240 + 6.22221i) q^{94} +(2.76604 + 4.79093i) q^{95} +(-0.617211 - 1.69577i) q^{96} +(6.80200 - 11.7814i) q^{97} +1.34730 q^{98} +(-1.07532 + 0.902302i) 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 \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.673648 + 1.16679i −0.476341 + 0.825047i −0.999633 − 0.0271067i \(-0.991371\pi\)
0.523291 + 0.852154i \(0.324704\pi\)
\(3\) 1.70574 + 0.300767i 0.984808 + 0.173648i
\(4\) 0.0923963 + 0.160035i 0.0461981 + 0.0800175i
\(5\) −1.26604 − 2.19285i −0.566192 − 0.980674i −0.996938 − 0.0782003i \(-0.975083\pi\)
0.430745 − 0.902473i \(-0.358251\pi\)
\(6\) −1.50000 + 1.78763i −0.612372 + 0.729797i
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) −2.94356 −1.04071
\(9\) 2.81908 + 1.02606i 0.939693 + 0.342020i
\(10\) 3.41147 1.07880
\(11\) −0.233956 + 0.405223i −0.0705403 + 0.122179i −0.899138 − 0.437665i \(-0.855806\pi\)
0.828598 + 0.559844i \(0.189139\pi\)
\(12\) 0.109470 + 0.300767i 0.0316014 + 0.0868241i
\(13\) −2.91147 − 5.04282i −0.807498 − 1.39863i −0.914592 − 0.404378i \(-0.867488\pi\)
0.107094 − 0.994249i \(-0.465845\pi\)
\(14\) −0.673648 − 1.16679i −0.180040 − 0.311839i
\(15\) −1.50000 − 4.12122i −0.387298 − 1.06409i
\(16\) 1.79813 − 3.11446i 0.449533 − 0.778615i
\(17\) 3.87939 0.940889 0.470445 − 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) −3.09627 + 2.59808i −0.729797 + 0.612372i
\(19\) −2.18479 −0.501226 −0.250613 − 0.968087i \(-0.580632\pi\)
−0.250613 + 0.968087i \(0.580632\pi\)
\(20\) 0.233956 − 0.405223i 0.0523141 − 0.0906106i
\(21\) −1.11334 + 1.32683i −0.242951 + 0.289538i
\(22\) −0.315207 − 0.545955i −0.0672025 − 0.116398i
\(23\) 0.0530334 + 0.0918566i 0.0110582 + 0.0191534i 0.871502 − 0.490393i \(-0.163147\pi\)
−0.860443 + 0.509546i \(0.829813\pi\)
\(24\) −5.02094 − 0.885328i −1.02490 − 0.180717i
\(25\) −0.705737 + 1.22237i −0.141147 + 0.244474i
\(26\) 7.84524 1.53858
\(27\) 4.50000 + 2.59808i 0.866025 + 0.500000i
\(28\) −0.184793 −0.0349225
\(29\) −4.39053 + 7.60462i −0.815301 + 1.41214i 0.0938108 + 0.995590i \(0.470095\pi\)
−0.909112 + 0.416552i \(0.863238\pi\)
\(30\) 5.81908 + 1.02606i 1.06241 + 0.187332i
\(31\) 3.84002 + 6.65111i 0.689688 + 1.19458i 0.971939 + 0.235235i \(0.0755858\pi\)
−0.282250 + 0.959341i \(0.591081\pi\)
\(32\) −0.520945 − 0.902302i −0.0920909 − 0.159506i
\(33\) −0.520945 + 0.620838i −0.0906848 + 0.108074i
\(34\) −2.61334 + 4.52644i −0.448184 + 0.776278i
\(35\) 2.53209 0.428001
\(36\) 0.0962667 + 0.545955i 0.0160444 + 0.0909926i
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 1.47178 − 2.54920i 0.238754 − 0.413535i
\(39\) −3.44949 − 9.47740i −0.552361 − 1.51760i
\(40\) 3.72668 + 6.45480i 0.589240 + 1.02059i
\(41\) 1.11334 + 1.92836i 0.173875 + 0.301160i 0.939771 − 0.341804i \(-0.111038\pi\)
−0.765897 + 0.642964i \(0.777705\pi\)
\(42\) −0.798133 − 2.19285i −0.123155 − 0.338365i
\(43\) −0.613341 + 1.06234i −0.0935336 + 0.162005i −0.908996 − 0.416806i \(-0.863150\pi\)
0.815462 + 0.578811i \(0.196483\pi\)
\(44\) −0.0864665 −0.0130353
\(45\) −1.31908 − 7.48086i −0.196637 − 1.11518i
\(46\) −0.142903 −0.0210700
\(47\) 2.66637 − 4.61830i 0.388931 − 0.673648i −0.603375 − 0.797457i \(-0.706178\pi\)
0.992306 + 0.123810i \(0.0395112\pi\)
\(48\) 4.00387 − 4.77163i 0.577909 − 0.688725i
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) −0.950837 − 1.64690i −0.134469 − 0.232907i
\(51\) 6.61721 + 1.16679i 0.926595 + 0.163384i
\(52\) 0.538019 − 0.931876i 0.0746098 − 0.129228i
\(53\) −0.716881 −0.0984712 −0.0492356 − 0.998787i \(-0.515679\pi\)
−0.0492356 + 0.998787i \(0.515679\pi\)
\(54\) −6.06283 + 3.50038i −0.825047 + 0.476341i
\(55\) 1.18479 0.159757
\(56\) 1.47178 − 2.54920i 0.196675 − 0.340651i
\(57\) −3.72668 − 0.657115i −0.493611 − 0.0870369i
\(58\) −5.91534 − 10.2457i −0.776723 − 1.34532i
\(59\) −0.368241 − 0.637812i −0.0479409 − 0.0830360i 0.841059 − 0.540943i \(-0.181933\pi\)
−0.889000 + 0.457907i \(0.848599\pi\)
\(60\) 0.520945 − 0.620838i 0.0672537 − 0.0801498i
\(61\) −0.479055 + 0.829748i −0.0613368 + 0.106238i −0.895063 − 0.445939i \(-0.852870\pi\)
0.833726 + 0.552178i \(0.186203\pi\)
\(62\) −10.3473 −1.31411
\(63\) −2.29813 + 1.92836i −0.289538 + 0.242951i
\(64\) 8.59627 1.07453
\(65\) −7.37211 + 12.7689i −0.914398 + 1.58378i
\(66\) −0.373455 − 1.02606i −0.0459692 − 0.126299i
\(67\) 4.81908 + 8.34689i 0.588744 + 1.01973i 0.994397 + 0.105708i \(0.0337107\pi\)
−0.405653 + 0.914027i \(0.632956\pi\)
\(68\) 0.358441 + 0.620838i 0.0434673 + 0.0752876i
\(69\) 0.0628336 + 0.172634i 0.00756428 + 0.0207827i
\(70\) −1.70574 + 2.95442i −0.203875 + 0.353121i
\(71\) 13.2344 1.57064 0.785318 − 0.619092i \(-0.212499\pi\)
0.785318 + 0.619092i \(0.212499\pi\)
\(72\) −8.29813 − 3.02027i −0.977944 − 0.355943i
\(73\) −10.2686 −1.20185 −0.600923 − 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 5.17365 − 8.96102i 0.601424 − 1.04170i
\(75\) −1.57145 + 1.87278i −0.181456 + 0.216250i
\(76\) −0.201867 − 0.349643i −0.0231557 − 0.0401068i
\(77\) −0.233956 − 0.405223i −0.0266617 − 0.0461794i
\(78\) 13.3819 + 2.35959i 1.51520 + 0.267171i
\(79\) 6.31908 − 10.9450i 0.710952 − 1.23140i −0.253548 − 0.967323i \(-0.581598\pi\)
0.964500 − 0.264082i \(-0.0850689\pi\)
\(80\) −9.10607 −1.01809
\(81\) 6.89440 + 5.78509i 0.766044 + 0.642788i
\(82\) −3.00000 −0.331295
\(83\) 1.36571 − 2.36549i 0.149907 − 0.259646i −0.781286 − 0.624173i \(-0.785436\pi\)
0.931193 + 0.364527i \(0.118769\pi\)
\(84\) −0.315207 − 0.0555796i −0.0343920 − 0.00606423i
\(85\) −4.91147 − 8.50692i −0.532724 − 0.922705i
\(86\) −0.826352 − 1.43128i −0.0891078 − 0.154339i
\(87\) −9.77631 + 11.6510i −1.04813 + 1.24911i
\(88\) 0.688663 − 1.19280i 0.0734117 − 0.127153i
\(89\) −8.11381 −0.860062 −0.430031 − 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(90\) 9.61721 + 3.50038i 1.01374 + 0.368972i
\(91\) 5.82295 0.610411
\(92\) −0.00980018 + 0.0169744i −0.00102174 + 0.00176970i
\(93\) 4.54963 + 12.5000i 0.471775 + 1.29619i
\(94\) 3.59240 + 6.22221i 0.370527 + 0.641772i
\(95\) 2.76604 + 4.79093i 0.283790 + 0.491539i
\(96\) −0.617211 − 1.69577i −0.0629939 − 0.173074i
\(97\) 6.80200 − 11.7814i 0.690639 − 1.19622i −0.280990 − 0.959711i \(-0.590663\pi\)
0.971629 − 0.236511i \(-0.0760039\pi\)
\(98\) 1.34730 0.136097
\(99\) −1.07532 + 0.902302i −0.108074 + 0.0906848i
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.43.2 yes 6
3.2 odd 2 189.2.f.b.127.2 6
4.3 odd 2 1008.2.r.h.673.1 6
7.2 even 3 441.2.h.d.214.2 6
7.3 odd 6 441.2.g.b.79.2 6
7.4 even 3 441.2.g.c.79.2 6
7.5 odd 6 441.2.h.e.214.2 6
7.6 odd 2 441.2.f.c.295.2 6
9.2 odd 6 567.2.a.c.1.2 3
9.4 even 3 inner 63.2.f.a.22.2 ✓ 6
9.5 odd 6 189.2.f.b.64.2 6
9.7 even 3 567.2.a.h.1.2 3
12.11 even 2 3024.2.r.k.2017.3 6
21.2 odd 6 1323.2.h.c.802.2 6
21.5 even 6 1323.2.h.b.802.2 6
21.11 odd 6 1323.2.g.d.667.2 6
21.17 even 6 1323.2.g.e.667.2 6
21.20 even 2 1323.2.f.d.883.2 6
36.7 odd 6 9072.2.a.ca.1.3 3
36.11 even 6 9072.2.a.bs.1.1 3
36.23 even 6 3024.2.r.k.1009.3 6
36.31 odd 6 1008.2.r.h.337.1 6
63.4 even 3 441.2.h.d.373.2 6
63.5 even 6 1323.2.g.e.361.2 6
63.13 odd 6 441.2.f.c.148.2 6
63.20 even 6 3969.2.a.l.1.2 3
63.23 odd 6 1323.2.g.d.361.2 6
63.31 odd 6 441.2.h.e.373.2 6
63.32 odd 6 1323.2.h.c.226.2 6
63.34 odd 6 3969.2.a.q.1.2 3
63.40 odd 6 441.2.g.b.67.2 6
63.41 even 6 1323.2.f.d.442.2 6
63.58 even 3 441.2.g.c.67.2 6
63.59 even 6 1323.2.h.b.226.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 9.4 even 3 inner
63.2.f.a.43.2 yes 6 1.1 even 1 trivial
189.2.f.b.64.2 6 9.5 odd 6
189.2.f.b.127.2 6 3.2 odd 2
441.2.f.c.148.2 6 63.13 odd 6
441.2.f.c.295.2 6 7.6 odd 2
441.2.g.b.67.2 6 63.40 odd 6
441.2.g.b.79.2 6 7.3 odd 6
441.2.g.c.67.2 6 63.58 even 3
441.2.g.c.79.2 6 7.4 even 3
441.2.h.d.214.2 6 7.2 even 3
441.2.h.d.373.2 6 63.4 even 3
441.2.h.e.214.2 6 7.5 odd 6
441.2.h.e.373.2 6 63.31 odd 6
567.2.a.c.1.2 3 9.2 odd 6
567.2.a.h.1.2 3 9.7 even 3
1008.2.r.h.337.1 6 36.31 odd 6
1008.2.r.h.673.1 6 4.3 odd 2
1323.2.f.d.442.2 6 63.41 even 6
1323.2.f.d.883.2 6 21.20 even 2
1323.2.g.d.361.2 6 63.23 odd 6
1323.2.g.d.667.2 6 21.11 odd 6
1323.2.g.e.361.2 6 63.5 even 6
1323.2.g.e.667.2 6 21.17 even 6
1323.2.h.b.226.2 6 63.59 even 6
1323.2.h.b.802.2 6 21.5 even 6
1323.2.h.c.226.2 6 63.32 odd 6
1323.2.h.c.802.2 6 21.2 odd 6
3024.2.r.k.1009.3 6 36.23 even 6
3024.2.r.k.2017.3 6 12.11 even 2
3969.2.a.l.1.2 3 63.20 even 6
3969.2.a.q.1.2 3 63.34 odd 6
9072.2.a.bs.1.1 3 36.11 even 6
9072.2.a.ca.1.3 3 36.7 odd 6