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 = [6,1] 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: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 43.3
Root \(0.500000 + 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 63.43
Dual form 63.2.f.b.22.3

$q$-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+(1.23025 - 2.13086i) q^{2} +(-1.73025 + 0.0789082i) q^{3} +(-2.02704 - 3.51094i) q^{4} +(1.29679 + 2.24611i) q^{5} +(-1.96050 + 3.78400i) q^{6} +(0.500000 - 0.866025i) q^{7} -5.05408 q^{8} +(2.98755 - 0.273062i) q^{9} +6.38151 q^{10} +(-2.25729 + 3.90975i) q^{11} +(3.78434 + 5.91486i) q^{12} +(-0.500000 - 0.866025i) q^{13} +(-1.23025 - 2.13086i) q^{14} +(-2.42101 - 3.78400i) q^{15} +(-2.16372 + 3.74766i) q^{16} -0.945916 q^{17} +(3.09358 - 6.70198i) q^{18} -4.05408 q^{19} +(5.25729 - 9.10590i) q^{20} +(-0.796790 + 1.53790i) q^{21} +(5.55408 + 9.61996i) q^{22} +(0.136673 + 0.236725i) q^{23} +(8.74484 - 0.398809i) q^{24} +(-0.863327 + 1.49533i) q^{25} -2.46050 q^{26} +(-5.14766 + 0.708209i) q^{27} -4.05408 q^{28} +(-1.23025 + 2.13086i) q^{29} +(-11.0416 + 0.503554i) q^{30} +(-1.16372 - 2.01561i) q^{31} +(0.269748 + 0.467216i) q^{32} +(3.59718 - 6.94297i) q^{33} +(-1.16372 + 2.01561i) q^{34} +2.59358 q^{35} +(-7.01459 - 9.93559i) q^{36} +1.78074 q^{37} +(-4.98755 + 8.63868i) q^{38} +(0.933463 + 1.45899i) q^{39} +(-6.55408 - 11.3520i) q^{40} +(3.20321 + 5.54812i) q^{41} +(2.29679 + 3.58985i) q^{42} +(5.21780 - 9.03749i) q^{43} +18.3025 q^{44} +(4.48755 + 6.35624i) q^{45} +0.672570 q^{46} +(6.08113 - 10.5328i) q^{47} +(3.44805 - 6.65514i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(2.12422 + 3.67926i) q^{50} +(1.63667 - 0.0746406i) q^{51} +(-2.02704 + 3.51094i) q^{52} -6.27335 q^{53} +(-4.82383 + 11.8402i) q^{54} -11.7089 q^{55} +(-2.52704 + 4.37697i) q^{56} +(7.01459 - 0.319901i) q^{57} +(3.02704 + 5.24299i) q^{58} +(1.36333 + 2.36135i) q^{59} +(-8.37792 + 16.1704i) q^{60} +(1.13667 - 1.96878i) q^{61} -5.72665 q^{62} +(1.25729 - 2.72382i) q^{63} -7.32743 q^{64} +(1.29679 - 2.24611i) q^{65} +(-10.3691 - 16.2067i) q^{66} +(7.90856 + 13.6980i) q^{67} +(1.91741 + 3.32105i) q^{68} +(-0.255158 - 0.398809i) q^{69} +(3.19076 - 5.52655i) q^{70} +3.27335 q^{71} +(-15.0993 + 1.38008i) q^{72} -1.50739 q^{73} +(2.19076 - 3.79450i) q^{74} +(1.37578 - 2.65542i) q^{75} +(8.21780 + 14.2336i) q^{76} +(2.25729 + 3.90975i) q^{77} +(4.25729 - 0.194154i) q^{78} +(-7.35447 + 12.7383i) q^{79} -11.2235 q^{80} +(8.85087 - 1.63157i) q^{81} +15.7630 q^{82} +(0.472958 - 0.819187i) q^{83} +(7.01459 - 0.319901i) q^{84} +(-1.22665 - 2.12463i) q^{85} +(-12.8384 - 22.2368i) q^{86} +(1.96050 - 3.78400i) q^{87} +(11.4086 - 19.7602i) q^{88} -14.3566 q^{89} +(19.0651 - 1.74255i) q^{90} -1.00000 q^{91} +(0.554084 - 0.959702i) q^{92} +(2.17257 + 3.39569i) q^{93} +(-14.9626 - 25.9161i) q^{94} +(-5.25729 - 9.10590i) q^{95} +(-0.503599 - 0.787117i) q^{96} +(5.74484 - 9.95036i) q^{97} -2.46050 q^{98} +(-5.67617 + 12.2969i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 4 q^{3} - 3 q^{4} + 5 q^{5} + q^{6} + 3 q^{7} - 12 q^{8} - 4 q^{9} + 2 q^{11} - 2 q^{12} - 3 q^{13} - q^{14} + 11 q^{15} - 3 q^{16} - 24 q^{17} + 13 q^{18} - 6 q^{19} + 16 q^{20} - 2 q^{21}+ \cdots - 46 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.23025 − 2.13086i 0.869920 − 1.50675i 0.00784213 − 0.999969i \(-0.497504\pi\)
0.862078 − 0.506776i \(-0.169163\pi\)
\(3\) −1.73025 + 0.0789082i −0.998962 + 0.0455577i
\(4\) −2.02704 − 3.51094i −1.01352 − 1.75547i
\(5\) 1.29679 + 2.24611i 0.579942 + 1.00449i 0.995485 + 0.0949156i \(0.0302581\pi\)
−0.415543 + 0.909573i \(0.636409\pi\)
\(6\) −1.96050 + 3.78400i −0.800373 + 1.54481i
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) −5.05408 −1.78689
\(9\) 2.98755 − 0.273062i 0.995849 − 0.0910208i
\(10\) 6.38151 2.01801
\(11\) −2.25729 + 3.90975i −0.680600 + 1.17883i 0.294198 + 0.955744i \(0.404947\pi\)
−0.974798 + 0.223089i \(0.928386\pi\)
\(12\) 3.78434 + 5.91486i 1.09244 + 1.70747i
\(13\) −0.500000 − 0.866025i −0.138675 − 0.240192i 0.788320 − 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) −1.23025 − 2.13086i −0.328799 − 0.569496i
\(15\) −2.42101 − 3.78400i −0.625102 − 0.977025i
\(16\) −2.16372 + 3.74766i −0.540929 + 0.936916i
\(17\) −0.945916 −0.229418 −0.114709 − 0.993399i \(-0.536594\pi\)
−0.114709 + 0.993399i \(0.536594\pi\)
\(18\) 3.09358 − 6.70198i 0.729164 − 1.57967i
\(19\) −4.05408 −0.930071 −0.465035 − 0.885292i \(-0.653958\pi\)
−0.465035 + 0.885292i \(0.653958\pi\)
\(20\) 5.25729 − 9.10590i 1.17557 − 2.03614i
\(21\) −0.796790 + 1.53790i −0.173874 + 0.335597i
\(22\) 5.55408 + 9.61996i 1.18413 + 2.05098i
\(23\) 0.136673 + 0.236725i 0.0284983 + 0.0493605i 0.879923 − 0.475117i \(-0.157594\pi\)
−0.851425 + 0.524477i \(0.824261\pi\)
\(24\) 8.74484 − 0.398809i 1.78503 − 0.0814065i
\(25\) −0.863327 + 1.49533i −0.172665 + 0.299065i
\(26\) −2.46050 −0.482545
\(27\) −5.14766 + 0.708209i −0.990668 + 0.136295i
\(28\) −4.05408 −0.766150
\(29\) −1.23025 + 2.13086i −0.228452 + 0.395691i −0.957350 − 0.288932i \(-0.906700\pi\)
0.728897 + 0.684623i \(0.240033\pi\)
\(30\) −11.0416 + 0.503554i −2.01592 + 0.0919360i
\(31\) −1.16372 − 2.01561i −0.209009 − 0.362015i 0.742393 − 0.669964i \(-0.233691\pi\)
−0.951403 + 0.307949i \(0.900357\pi\)
\(32\) 0.269748 + 0.467216i 0.0476851 + 0.0825930i
\(33\) 3.59718 − 6.94297i 0.626188 − 1.20862i
\(34\) −1.16372 + 2.01561i −0.199576 + 0.345675i
\(35\) 2.59358 0.438395
\(36\) −7.01459 − 9.93559i −1.16910 − 1.65593i
\(37\) 1.78074 0.292752 0.146376 − 0.989229i \(-0.453239\pi\)
0.146376 + 0.989229i \(0.453239\pi\)
\(38\) −4.98755 + 8.63868i −0.809087 + 1.40138i
\(39\) 0.933463 + 1.45899i 0.149474 + 0.233625i
\(40\) −6.55408 − 11.3520i −1.03629 − 1.79491i
\(41\) 3.20321 + 5.54812i 0.500257 + 0.866471i 1.00000 0.000297253i \(9.46187e-5\pi\)
−0.499743 + 0.866174i \(0.666572\pi\)
\(42\) 2.29679 + 3.58985i 0.354402 + 0.553926i
\(43\) 5.21780 − 9.03749i 0.795707 − 1.37820i −0.126682 − 0.991943i \(-0.540433\pi\)
0.922389 − 0.386262i \(-0.126234\pi\)
\(44\) 18.3025 2.75921
\(45\) 4.48755 + 6.35624i 0.668964 + 0.947533i
\(46\) 0.672570 0.0991650
\(47\) 6.08113 − 10.5328i 0.887023 − 1.53637i 0.0436467 − 0.999047i \(-0.486102\pi\)
0.843377 − 0.537323i \(-0.180564\pi\)
\(48\) 3.44805 − 6.65514i 0.497683 − 0.960587i
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 2.12422 + 3.67926i 0.300410 + 0.520326i
\(51\) 1.63667 − 0.0746406i 0.229180 − 0.0104518i
\(52\) −2.02704 + 3.51094i −0.281100 + 0.486880i
\(53\) −6.27335 −0.861710 −0.430855 − 0.902421i \(-0.641788\pi\)
−0.430855 + 0.902421i \(0.641788\pi\)
\(54\) −4.82383 + 11.8402i −0.656440 + 1.61125i
\(55\) −11.7089 −1.57883
\(56\) −2.52704 + 4.37697i −0.337690 + 0.584897i
\(57\) 7.01459 − 0.319901i 0.929105 − 0.0423719i
\(58\) 3.02704 + 5.24299i 0.397470 + 0.688438i
\(59\) 1.36333 + 2.36135i 0.177490 + 0.307422i 0.941020 − 0.338350i \(-0.109869\pi\)
−0.763530 + 0.645772i \(0.776536\pi\)
\(60\) −8.37792 + 16.1704i −1.08158 + 2.08758i
\(61\) 1.13667 − 1.96878i 0.145536 − 0.252076i −0.784037 − 0.620714i \(-0.786843\pi\)
0.929573 + 0.368639i \(0.120176\pi\)
\(62\) −5.72665 −0.727286
\(63\) 1.25729 − 2.72382i 0.158404 − 0.343169i
\(64\) −7.32743 −0.915929
\(65\) 1.29679 − 2.24611i 0.160847 − 0.278595i
\(66\) −10.3691 − 16.2067i −1.27634 − 1.99491i
\(67\) 7.90856 + 13.6980i 0.966184 + 1.67348i 0.706400 + 0.707813i \(0.250318\pi\)
0.259784 + 0.965667i \(0.416349\pi\)
\(68\) 1.91741 + 3.32105i 0.232520 + 0.402737i
\(69\) −0.255158 − 0.398809i −0.0307175 − 0.0480110i
\(70\) 3.19076 − 5.52655i 0.381368 − 0.660550i
\(71\) 3.27335 0.388475 0.194237 − 0.980955i \(-0.437777\pi\)
0.194237 + 0.980955i \(0.437777\pi\)
\(72\) −15.0993 + 1.38008i −1.77947 + 0.162644i
\(73\) −1.50739 −0.176427 −0.0882134 − 0.996102i \(-0.528116\pi\)
−0.0882134 + 0.996102i \(0.528116\pi\)
\(74\) 2.19076 − 3.79450i 0.254670 − 0.441102i
\(75\) 1.37578 − 2.65542i 0.158861 − 0.306621i
\(76\) 8.21780 + 14.2336i 0.942646 + 1.63271i
\(77\) 2.25729 + 3.90975i 0.257243 + 0.445557i
\(78\) 4.25729 − 0.194154i 0.482044 − 0.0219836i
\(79\) −7.35447 + 12.7383i −0.827443 + 1.43317i 0.0725952 + 0.997361i \(0.476872\pi\)
−0.900038 + 0.435811i \(0.856461\pi\)
\(80\) −11.2235 −1.25483
\(81\) 8.85087 − 1.63157i 0.983430 − 0.181286i
\(82\) 15.7630 1.74074
\(83\) 0.472958 − 0.819187i 0.0519139 − 0.0899175i −0.838901 − 0.544285i \(-0.816801\pi\)
0.890815 + 0.454367i \(0.150135\pi\)
\(84\) 7.01459 − 0.319901i 0.765354 − 0.0349040i
\(85\) −1.22665 − 2.12463i −0.133049 − 0.230448i
\(86\) −12.8384 − 22.2368i −1.38440 − 2.39786i
\(87\) 1.96050 − 3.78400i 0.210188 − 0.405688i
\(88\) 11.4086 − 19.7602i 1.21616 − 2.10644i
\(89\) −14.3566 −1.52180 −0.760899 − 0.648871i \(-0.775242\pi\)
−0.760899 + 0.648871i \(0.775242\pi\)
\(90\) 19.0651 − 1.74255i 2.00964 − 0.183681i
\(91\) −1.00000 −0.104828
\(92\) 0.554084 − 0.959702i 0.0577673 − 0.100056i
\(93\) 2.17257 + 3.39569i 0.225285 + 0.352117i
\(94\) −14.9626 − 25.9161i −1.54328 − 2.67304i
\(95\) −5.25729 − 9.10590i −0.539387 − 0.934246i
\(96\) −0.503599 − 0.787117i −0.0513983 − 0.0803348i
\(97\) 5.74484 − 9.95036i 0.583300 − 1.01031i −0.411785 − 0.911281i \(-0.635094\pi\)
0.995085 − 0.0990246i \(-0.0315722\pi\)
\(98\) −2.46050 −0.248549
\(99\) −5.67617 + 12.2969i −0.570476 + 1.23589i
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.b.43.3 yes 6
3.2 odd 2 189.2.f.a.127.1 6
4.3 odd 2 1008.2.r.k.673.3 6
7.2 even 3 441.2.h.c.214.1 6
7.3 odd 6 441.2.g.d.79.3 6
7.4 even 3 441.2.g.e.79.3 6
7.5 odd 6 441.2.h.b.214.1 6
7.6 odd 2 441.2.f.d.295.3 6
9.2 odd 6 567.2.a.g.1.3 3
9.4 even 3 inner 63.2.f.b.22.3 ✓ 6
9.5 odd 6 189.2.f.a.64.1 6
9.7 even 3 567.2.a.d.1.1 3
12.11 even 2 3024.2.r.g.2017.2 6
21.2 odd 6 1323.2.h.d.802.3 6
21.5 even 6 1323.2.h.e.802.3 6
21.11 odd 6 1323.2.g.c.667.1 6
21.17 even 6 1323.2.g.b.667.1 6
21.20 even 2 1323.2.f.c.883.1 6
36.7 odd 6 9072.2.a.bq.1.2 3
36.11 even 6 9072.2.a.cd.1.2 3
36.23 even 6 3024.2.r.g.1009.2 6
36.31 odd 6 1008.2.r.k.337.3 6
63.4 even 3 441.2.h.c.373.1 6
63.5 even 6 1323.2.g.b.361.1 6
63.13 odd 6 441.2.f.d.148.3 6
63.20 even 6 3969.2.a.p.1.3 3
63.23 odd 6 1323.2.g.c.361.1 6
63.31 odd 6 441.2.h.b.373.1 6
63.32 odd 6 1323.2.h.d.226.3 6
63.34 odd 6 3969.2.a.m.1.1 3
63.40 odd 6 441.2.g.d.67.3 6
63.41 even 6 1323.2.f.c.442.1 6
63.58 even 3 441.2.g.e.67.3 6
63.59 even 6 1323.2.h.e.226.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 ✓ 6 9.4 even 3 inner
63.2.f.b.43.3 yes 6 1.1 even 1 trivial
189.2.f.a.64.1 6 9.5 odd 6
189.2.f.a.127.1 6 3.2 odd 2
441.2.f.d.148.3 6 63.13 odd 6
441.2.f.d.295.3 6 7.6 odd 2
441.2.g.d.67.3 6 63.40 odd 6
441.2.g.d.79.3 6 7.3 odd 6
441.2.g.e.67.3 6 63.58 even 3
441.2.g.e.79.3 6 7.4 even 3
441.2.h.b.214.1 6 7.5 odd 6
441.2.h.b.373.1 6 63.31 odd 6
441.2.h.c.214.1 6 7.2 even 3
441.2.h.c.373.1 6 63.4 even 3
567.2.a.d.1.1 3 9.7 even 3
567.2.a.g.1.3 3 9.2 odd 6
1008.2.r.k.337.3 6 36.31 odd 6
1008.2.r.k.673.3 6 4.3 odd 2
1323.2.f.c.442.1 6 63.41 even 6
1323.2.f.c.883.1 6 21.20 even 2
1323.2.g.b.361.1 6 63.5 even 6
1323.2.g.b.667.1 6 21.17 even 6
1323.2.g.c.361.1 6 63.23 odd 6
1323.2.g.c.667.1 6 21.11 odd 6
1323.2.h.d.226.3 6 63.32 odd 6
1323.2.h.d.802.3 6 21.2 odd 6
1323.2.h.e.226.3 6 63.59 even 6
1323.2.h.e.802.3 6 21.5 even 6
3024.2.r.g.1009.2 6 36.23 even 6
3024.2.r.g.2017.2 6 12.11 even 2
3969.2.a.m.1.1 3 63.34 odd 6
3969.2.a.p.1.3 3 63.20 even 6
9072.2.a.bq.1.2 3 36.7 odd 6
9072.2.a.cd.1.2 3 36.11 even 6