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Properties

Label 63.3.j.a.23.1
Level $63$
Weight $3$
Character 63.23
Analytic conductor $1.717$
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] = [63,3,Mod(11,63)] 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("63.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.63369648.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 23.1
Root \(-1.34153 - 2.32360i\) of defining polynomial
Character \(\chi\) \(=\) 63.23
Dual form 63.3.j.a.11.3

$q$-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-3.79027i q^{2} +3.00000 q^{3} -10.3661 q^{4} +(0.282467 - 0.163082i) q^{5} -11.3708i q^{6} +(-2.88187 - 6.37925i) q^{7} +24.1293i q^{8} +9.00000 q^{9} +(-0.618126 - 1.07063i) q^{10} +(13.1141 + 7.57144i) q^{11} -31.0984 q^{12} +(-3.96553 + 6.86850i) q^{13} +(-24.1791 + 10.9231i) q^{14} +(0.847401 - 0.489247i) q^{15} +49.9920 q^{16} +(6.90356 - 3.98577i) q^{17} -34.1124i q^{18} +(2.86612 - 4.96427i) q^{19} +(-2.92809 + 1.69053i) q^{20} +(-8.64562 - 19.1377i) q^{21} +(28.6978 - 49.7060i) q^{22} +(1.79822 - 1.03820i) q^{23} +72.3879i q^{24} +(-12.4468 + 21.5585i) q^{25} +(26.0334 + 15.0304i) q^{26} +27.0000 q^{27} +(29.8739 + 66.1281i) q^{28} +(-5.71753 + 3.30102i) q^{29} +(-1.85438 - 3.21188i) q^{30} -50.5944 q^{31} -92.9657i q^{32} +(39.3423 + 22.7143i) q^{33} +(-15.1071 - 26.1663i) q^{34} +(-1.85438 - 1.33194i) q^{35} -93.2951 q^{36} +(5.56197 - 9.63361i) q^{37} +(-18.8159 - 10.8634i) q^{38} +(-11.8966 + 20.6055i) q^{39} +(3.93507 + 6.81573i) q^{40} +(-12.8721 - 7.43168i) q^{41} +(-72.5372 + 32.7692i) q^{42} +(3.67905 + 6.37230i) q^{43} +(-135.943 - 78.4864i) q^{44} +(2.54220 - 1.46774i) q^{45} +(-3.93507 - 6.81573i) q^{46} +66.9565i q^{47} +149.976 q^{48} +(-32.3896 + 36.7684i) q^{49} +(81.7125 + 47.1767i) q^{50} +(20.7107 - 11.9573i) q^{51} +(41.1071 - 71.1997i) q^{52} +(22.0895 - 12.7534i) q^{53} -102.337i q^{54} +4.93908 q^{55} +(153.927 - 69.5376i) q^{56} +(8.59836 - 14.8928i) q^{57} +(12.5117 + 21.6710i) q^{58} +39.7826i q^{59} +(-8.78427 + 5.07160i) q^{60} -74.3186 q^{61} +191.766i q^{62} +(-25.9369 - 57.4132i) q^{63} -152.397 q^{64} +2.58683i q^{65} +(86.0933 - 149.118i) q^{66} +42.9525 q^{67} +(-71.5631 + 41.3170i) q^{68} +(5.39466 - 3.11461i) q^{69} +(-5.04843 + 7.02859i) q^{70} -72.5073i q^{71} +217.164i q^{72} +(-45.8030 - 79.3331i) q^{73} +(-36.5140 - 21.0813i) q^{74} +(-37.3404 + 64.6755i) q^{75} +(-29.7106 + 51.4602i) q^{76} +(10.5069 - 105.478i) q^{77} +(78.1003 + 45.0912i) q^{78} +23.0315 q^{79} +(14.1211 - 8.15282i) q^{80} +81.0000 q^{81} +(-28.1681 + 48.7885i) q^{82} +(97.9426 - 56.5472i) q^{83} +(89.6216 + 198.384i) q^{84} +(1.30002 - 2.25170i) q^{85} +(24.1527 - 13.9446i) q^{86} +(-17.1526 + 9.90306i) q^{87} +(-182.693 + 316.434i) q^{88} +(-63.4862 - 36.6538i) q^{89} +(-5.56314 - 9.63563i) q^{90} +(55.2440 + 5.50295i) q^{91} +(-18.6406 + 10.7621i) q^{92} -151.783 q^{93} +253.783 q^{94} -1.86966i q^{95} -278.897i q^{96} +(-57.2065 - 99.0846i) q^{97} +(139.362 + 122.765i) q^{98} +(118.027 + 68.1429i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{7} + 54 q^{9} - 19 q^{10} - 9 q^{11} - 78 q^{12} + 11 q^{13} - 24 q^{14} - 45 q^{15} + 94 q^{16} + 33 q^{17} - 19 q^{19} + 45 q^{20} - 6 q^{21} + 65 q^{22} + 15 q^{23}+ \cdots - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.79027i − 1.89513i −0.319558 − 0.947567i \(-0.603534\pi\)
0.319558 − 0.947567i \(-0.396466\pi\)
\(3\) 3.00000 1.00000
\(4\) −10.3661 −2.59153
\(5\) 0.282467 − 0.163082i 0.0564934 − 0.0326165i −0.471487 − 0.881873i \(-0.656283\pi\)
0.527981 + 0.849256i \(0.322949\pi\)
\(6\) − 11.3708i − 1.89513i
\(7\) −2.88187 − 6.37925i −0.411696 − 0.911321i
\(8\) 24.1293i 3.01616i
\(9\) 9.00000 1.00000
\(10\) −0.618126 − 1.07063i −0.0618126 − 0.107063i
\(11\) 13.1141 + 7.57144i 1.19219 + 0.688313i 0.958803 − 0.284071i \(-0.0916851\pi\)
0.233389 + 0.972383i \(0.425018\pi\)
\(12\) −31.0984 −2.59153
\(13\) −3.96553 + 6.86850i −0.305041 + 0.528346i −0.977270 − 0.211997i \(-0.932003\pi\)
0.672230 + 0.740343i \(0.265337\pi\)
\(14\) −24.1791 + 10.9231i −1.72708 + 0.780219i
\(15\) 0.847401 − 0.489247i 0.0564934 − 0.0326165i
\(16\) 49.9920 3.12450
\(17\) 6.90356 − 3.98577i 0.406092 − 0.234457i −0.283017 − 0.959115i \(-0.591335\pi\)
0.689109 + 0.724658i \(0.258002\pi\)
\(18\) − 34.1124i − 1.89513i
\(19\) 2.86612 − 4.96427i 0.150848 − 0.261277i −0.780691 − 0.624917i \(-0.785133\pi\)
0.931540 + 0.363640i \(0.118466\pi\)
\(20\) −2.92809 + 1.69053i −0.146404 + 0.0845266i
\(21\) −8.64562 − 19.1377i −0.411696 − 0.911321i
\(22\) 28.6978 − 49.7060i 1.30444 − 2.25936i
\(23\) 1.79822 − 1.03820i 0.0781835 − 0.0451393i −0.460399 − 0.887712i \(-0.652294\pi\)
0.538582 + 0.842573i \(0.318960\pi\)
\(24\) 72.3879i 3.01616i
\(25\) −12.4468 + 21.5585i −0.497872 + 0.862340i
\(26\) 26.0334 + 15.0304i 1.00129 + 0.578093i
\(27\) 27.0000 1.00000
\(28\) 29.8739 + 66.1281i 1.06692 + 2.36172i
\(29\) −5.71753 + 3.30102i −0.197156 + 0.113828i −0.595328 − 0.803482i \(-0.702978\pi\)
0.398172 + 0.917311i \(0.369645\pi\)
\(30\) −1.85438 − 3.21188i −0.0618126 − 0.107063i
\(31\) −50.5944 −1.63208 −0.816038 − 0.577998i \(-0.803834\pi\)
−0.816038 + 0.577998i \(0.803834\pi\)
\(32\) − 92.9657i − 2.90518i
\(33\) 39.3423 + 22.7143i 1.19219 + 0.688313i
\(34\) −15.1071 − 26.1663i −0.444328 − 0.769598i
\(35\) −1.85438 − 1.33194i −0.0529822 − 0.0380556i
\(36\) −93.2951 −2.59153
\(37\) 5.56197 − 9.63361i 0.150323 − 0.260368i −0.781023 − 0.624502i \(-0.785302\pi\)
0.931346 + 0.364135i \(0.118635\pi\)
\(38\) −18.8159 − 10.8634i −0.495155 − 0.285878i
\(39\) −11.8966 + 20.6055i −0.305041 + 0.528346i
\(40\) 3.93507 + 6.81573i 0.0983766 + 0.170393i
\(41\) −12.8721 − 7.43168i −0.313953 − 0.181261i 0.334741 − 0.942310i \(-0.391351\pi\)
−0.648694 + 0.761049i \(0.724684\pi\)
\(42\) −72.5372 + 32.7692i −1.72708 + 0.780219i
\(43\) 3.67905 + 6.37230i 0.0855593 + 0.148193i 0.905629 − 0.424070i \(-0.139399\pi\)
−0.820070 + 0.572263i \(0.806066\pi\)
\(44\) −135.943 − 78.4864i −3.08960 − 1.78378i
\(45\) 2.54220 − 1.46774i 0.0564934 − 0.0326165i
\(46\) −3.93507 − 6.81573i −0.0855449 − 0.148168i
\(47\) 66.9565i 1.42461i 0.701872 + 0.712303i \(0.252348\pi\)
−0.701872 + 0.712303i \(0.747652\pi\)
\(48\) 149.976 3.12450
\(49\) −32.3896 + 36.7684i −0.661012 + 0.750375i
\(50\) 81.7125 + 47.1767i 1.63425 + 0.943534i
\(51\) 20.7107 − 11.9573i 0.406092 − 0.234457i
\(52\) 41.1071 − 71.1997i 0.790522 − 1.36922i
\(53\) 22.0895 − 12.7534i 0.416782 − 0.240629i −0.276917 − 0.960894i \(-0.589313\pi\)
0.693700 + 0.720264i \(0.255979\pi\)
\(54\) − 102.337i − 1.89513i
\(55\) 4.93908 0.0898014
\(56\) 153.927 − 69.5376i 2.74869 − 1.24174i
\(57\) 8.59836 − 14.8928i 0.150848 − 0.261277i
\(58\) 12.5117 + 21.6710i 0.215720 + 0.373637i
\(59\) 39.7826i 0.674281i 0.941454 + 0.337141i \(0.109460\pi\)
−0.941454 + 0.337141i \(0.890540\pi\)
\(60\) −8.78427 + 5.07160i −0.146404 + 0.0845266i
\(61\) −74.3186 −1.21834 −0.609169 − 0.793041i \(-0.708497\pi\)
−0.609169 + 0.793041i \(0.708497\pi\)
\(62\) 191.766i 3.09300i
\(63\) −25.9369 − 57.4132i −0.411696 − 0.911321i
\(64\) −152.397 −2.38120
\(65\) 2.58683i 0.0397974i
\(66\) 86.0933 − 149.118i 1.30444 − 2.25936i
\(67\) 42.9525 0.641081 0.320541 − 0.947235i \(-0.396135\pi\)
0.320541 + 0.947235i \(0.396135\pi\)
\(68\) −71.5631 + 41.3170i −1.05240 + 0.607603i
\(69\) 5.39466 − 3.11461i 0.0781835 − 0.0451393i
\(70\) −5.04843 + 7.02859i −0.0721204 + 0.100408i
\(71\) − 72.5073i − 1.02123i −0.859810 − 0.510615i \(-0.829418\pi\)
0.859810 − 0.510615i \(-0.170582\pi\)
\(72\) 217.164i 3.01616i
\(73\) −45.8030 − 79.3331i −0.627438 − 1.08675i −0.988064 − 0.154044i \(-0.950770\pi\)
0.360626 − 0.932711i \(-0.382563\pi\)
\(74\) −36.5140 − 21.0813i −0.493432 − 0.284883i
\(75\) −37.3404 + 64.6755i −0.497872 + 0.862340i
\(76\) −29.7106 + 51.4602i −0.390928 + 0.677108i
\(77\) 10.5069 − 105.478i 0.136453 − 1.36985i
\(78\) 78.1003 + 45.0912i 1.00129 + 0.578093i
\(79\) 23.0315 0.291538 0.145769 − 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(80\) 14.1211 − 8.15282i 0.176514 − 0.101910i
\(81\) 81.0000 1.00000
\(82\) −28.1681 + 48.7885i −0.343513 + 0.594982i
\(83\) 97.9426 − 56.5472i 1.18003 − 0.681292i 0.224010 − 0.974587i \(-0.428085\pi\)
0.956022 + 0.293295i \(0.0947519\pi\)
\(84\) 89.6216 + 198.384i 1.06692 + 2.36172i
\(85\) 1.30002 − 2.25170i 0.0152943 − 0.0264906i
\(86\) 24.1527 − 13.9446i 0.280846 − 0.162146i
\(87\) −17.1526 + 9.90306i −0.197156 + 0.113828i
\(88\) −182.693 + 316.434i −2.07606 + 3.59585i
\(89\) −63.4862 − 36.6538i −0.713328 − 0.411840i 0.0989642 − 0.995091i \(-0.468447\pi\)
−0.812292 + 0.583251i \(0.801780\pi\)
\(90\) −5.56314 − 9.63563i −0.0618126 − 0.107063i
\(91\) 55.2440 + 5.50295i 0.607077 + 0.0604719i
\(92\) −18.6406 + 10.7621i −0.202615 + 0.116980i
\(93\) −151.783 −1.63208
\(94\) 253.783 2.69982
\(95\) − 1.86966i − 0.0196806i
\(96\) − 278.897i − 2.90518i
\(97\) −57.2065 − 99.0846i −0.589758 − 1.02149i −0.994264 − 0.106956i \(-0.965890\pi\)
0.404506 − 0.914536i \(-0.367444\pi\)
\(98\) 139.362 + 122.765i 1.42206 + 1.25271i
\(99\) 118.027 + 68.1429i 1.19219 + 0.688313i
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 63.3.j.a.23.1 yes 6
3.2 odd 2 189.3.j.a.44.3 6
7.2 even 3 441.3.r.b.50.1 6
7.3 odd 6 441.3.n.c.410.3 6
7.4 even 3 63.3.n.a.32.3 yes 6
7.5 odd 6 441.3.r.c.50.1 6
7.6 odd 2 441.3.j.c.275.1 6
9.2 odd 6 63.3.n.a.2.3 yes 6
9.7 even 3 189.3.n.a.170.1 6
21.11 odd 6 189.3.n.a.179.1 6
63.2 odd 6 441.3.r.b.344.1 6
63.11 odd 6 inner 63.3.j.a.11.3 ✓ 6
63.20 even 6 441.3.n.c.128.3 6
63.25 even 3 189.3.j.a.116.1 6
63.38 even 6 441.3.j.c.263.3 6
63.47 even 6 441.3.r.c.344.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.3.j.a.11.3 ✓ 6 63.11 odd 6 inner
63.3.j.a.23.1 yes 6 1.1 even 1 trivial
63.3.n.a.2.3 yes 6 9.2 odd 6
63.3.n.a.32.3 yes 6 7.4 even 3
189.3.j.a.44.3 6 3.2 odd 2
189.3.j.a.116.1 6 63.25 even 3
189.3.n.a.170.1 6 9.7 even 3
189.3.n.a.179.1 6 21.11 odd 6
441.3.j.c.263.3 6 63.38 even 6
441.3.j.c.275.1 6 7.6 odd 2
441.3.n.c.128.3 6 63.20 even 6
441.3.n.c.410.3 6 7.3 odd 6
441.3.r.b.50.1 6 7.2 even 3
441.3.r.b.344.1 6 63.2 odd 6
441.3.r.c.50.1 6 7.5 odd 6
441.3.r.c.344.1 6 63.47 even 6