Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1323,2,Mod(442,1323)] 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("1323.442"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-1,0,-3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.5642081874\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 883.1
Root \(0.500000 + 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 1323.883
Dual form 1323.2.f.c.442.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.23025 + 2.13086i) q^{2} +(-2.02704 - 3.51094i) q^{4} +(1.29679 + 2.24611i) q^{5} +5.05408 q^{8} -6.38151 q^{10} +(2.25729 - 3.90975i) q^{11} +(0.500000 + 0.866025i) q^{13} +(-2.16372 + 3.74766i) q^{16} -0.945916 q^{17} +4.05408 q^{19} +(5.25729 - 9.10590i) q^{20} +(5.55408 + 9.61996i) q^{22} +(-0.136673 - 0.236725i) q^{23} +(-0.863327 + 1.49533i) q^{25} -2.46050 q^{26} +(1.23025 - 2.13086i) q^{29} +(1.16372 + 2.01561i) q^{31} +(-0.269748 - 0.467216i) q^{32} +(1.16372 - 2.01561i) q^{34} +1.78074 q^{37} +(-4.98755 + 8.63868i) q^{38} +(6.55408 + 11.3520i) q^{40} +(3.20321 + 5.54812i) q^{41} +(5.21780 - 9.03749i) q^{43} -18.3025 q^{44} +0.672570 q^{46} +(6.08113 - 10.5328i) q^{47} +(-2.12422 - 3.67926i) q^{50} +(2.02704 - 3.51094i) q^{52} +6.27335 q^{53} +11.7089 q^{55} +(3.02704 + 5.24299i) q^{58} +(1.36333 + 2.36135i) q^{59} +(-1.13667 + 1.96878i) q^{61} -5.72665 q^{62} -7.32743 q^{64} +(-1.29679 + 2.24611i) q^{65} +(7.90856 + 13.6980i) q^{67} +(1.91741 + 3.32105i) q^{68} -3.27335 q^{71} +1.50739 q^{73} +(-2.19076 + 3.79450i) q^{74} +(-8.21780 - 14.2336i) q^{76} +(-7.35447 + 12.7383i) q^{79} -11.2235 q^{80} -15.7630 q^{82} +(0.472958 - 0.819187i) q^{83} +(-1.22665 - 2.12463i) q^{85} +(12.8384 + 22.2368i) q^{86} +(11.4086 - 19.7602i) q^{88} -14.3566 q^{89} +(-0.554084 + 0.959702i) q^{92} +(14.9626 + 25.9161i) q^{94} +(5.25729 + 9.10590i) q^{95} +(-5.74484 + 9.95036i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 3 q^{4} + 5 q^{5} + 12 q^{8} - 2 q^{11} + 3 q^{13} - 3 q^{16} - 24 q^{17} + 6 q^{19} + 16 q^{20} + 15 q^{22} - 6 q^{25} - 2 q^{26} + q^{29} - 3 q^{31} - 8 q^{32} - 3 q^{34} - 6 q^{37} - 8 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 \(0.502496\pi\)
−0.862078 + 0.506776i \(0.830837\pi\)
\(3\) 0 0
\(4\) −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 \(0.636409\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 5.05408 1.78689
\(9\) 0 0
\(10\) −6.38151 −2.01801
\(11\) 2.25729 − 3.90975i 0.680600 − 1.17883i −0.294198 − 0.955744i \(-0.595053\pi\)
0.974798 − 0.223089i \(-0.0716141\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 − 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.16372 + 3.74766i −0.540929 + 0.936916i
\(17\) −0.945916 −0.229418 −0.114709 − 0.993399i \(-0.536594\pi\)
−0.114709 + 0.993399i \(0.536594\pi\)
\(18\) 0 0
\(19\) 4.05408 0.930071 0.465035 − 0.885292i \(-0.346042\pi\)
0.465035 + 0.885292i \(0.346042\pi\)
\(20\) 5.25729 − 9.10590i 1.17557 − 2.03614i
\(21\) 0 0
\(22\) 5.55408 + 9.61996i 1.18413 + 2.05098i
\(23\) −0.136673 − 0.236725i −0.0284983 − 0.0493605i 0.851425 − 0.524477i \(-0.175739\pi\)
−0.879923 + 0.475117i \(0.842406\pi\)
\(24\) 0 0
\(25\) −0.863327 + 1.49533i −0.172665 + 0.299065i
\(26\) −2.46050 −0.482545
\(27\) 0 0
\(28\) 0 0
\(29\) 1.23025 − 2.13086i 0.228452 − 0.395691i −0.728897 − 0.684623i \(-0.759967\pi\)
0.957350 + 0.288932i \(0.0933002\pi\)
\(30\) 0 0
\(31\) 1.16372 + 2.01561i 0.209009 + 0.362015i 0.951403 − 0.307949i \(-0.0996427\pi\)
−0.742393 + 0.669964i \(0.766309\pi\)
\(32\) −0.269748 − 0.467216i −0.0476851 − 0.0825930i
\(33\) 0 0
\(34\) 1.16372 − 2.01561i 0.199576 − 0.345675i
\(35\) 0 0
\(36\) 0 0
\(37\) 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 0
\(40\) 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 \(0.666572\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) 0.672570 0.0991650
\(47\) 6.08113 − 10.5328i 0.887023 − 1.53637i 0.0436467 − 0.999047i \(-0.486102\pi\)
0.843377 − 0.537323i \(-0.180564\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.12422 − 3.67926i −0.300410 − 0.520326i
\(51\) 0 0
\(52\) 2.02704 − 3.51094i 0.281100 − 0.486880i
\(53\) 6.27335 0.861710 0.430855 − 0.902421i \(-0.358212\pi\)
0.430855 + 0.902421i \(0.358212\pi\)
\(54\) 0 0
\(55\) 11.7089 1.57883
\(56\) 0 0
\(57\) 0 0
\(58\) 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 \(0.776536\pi\)
\(60\) 0 0
\(61\) −1.13667 + 1.96878i −0.145536 + 0.252076i −0.929573 − 0.368639i \(-0.879824\pi\)
0.784037 + 0.620714i \(0.213157\pi\)
\(62\) −5.72665 −0.727286
\(63\) 0 0
\(64\) −7.32743 −0.915929
\(65\) −1.29679 + 2.24611i −0.160847 + 0.278595i
\(66\) 0 0
\(67\) 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 0
\(70\) 0 0
\(71\) −3.27335 −0.388475 −0.194237 − 0.980955i \(-0.562223\pi\)
−0.194237 + 0.980955i \(0.562223\pi\)
\(72\) 0 0
\(73\) 1.50739 0.176427 0.0882134 − 0.996102i \(-0.471884\pi\)
0.0882134 + 0.996102i \(0.471884\pi\)
\(74\) −2.19076 + 3.79450i −0.254670 + 0.441102i
\(75\) 0 0
\(76\) −8.21780 − 14.2336i −0.942646 − 1.63271i
\(77\) 0 0
\(78\) 0 0
\(79\) −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
\(81\) 0 0
\(82\) −15.7630 −1.74074
\(83\) 0.472958 − 0.819187i 0.0519139 − 0.0899175i −0.838901 − 0.544285i \(-0.816801\pi\)
0.890815 + 0.454367i \(0.150135\pi\)
\(84\) 0 0
\(85\) −1.22665 − 2.12463i −0.133049 − 0.230448i
\(86\) 12.8384 + 22.2368i 1.38440 + 2.39786i
\(87\) 0 0
\(88\) 11.4086 − 19.7602i 1.21616 − 2.10644i
\(89\) −14.3566 −1.52180 −0.760899 − 0.648871i \(-0.775242\pi\)
−0.760899 + 0.648871i \(0.775242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.554084 + 0.959702i −0.0577673 + 0.100056i
\(93\) 0 0
\(94\) 14.9626 + 25.9161i 1.54328 + 2.67304i
\(95\) 5.25729 + 9.10590i 0.539387 + 0.934246i
\(96\) 0 0
\(97\) −5.74484 + 9.95036i −0.583300 + 1.01031i 0.411785 + 0.911281i \(0.364906\pi\)
−0.995085 + 0.0990246i \(0.968428\pi\)
\(98\) 0 0
\(99\) 0 0
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 1323.2.f.c.883.1 6
3.2 odd 2 441.2.f.d.295.3 6
7.2 even 3 1323.2.h.e.802.3 6
7.3 odd 6 1323.2.g.c.667.1 6
7.4 even 3 1323.2.g.b.667.1 6
7.5 odd 6 1323.2.h.d.802.3 6
7.6 odd 2 189.2.f.a.127.1 6
9.2 odd 6 3969.2.a.m.1.1 3
9.4 even 3 inner 1323.2.f.c.442.1 6
9.5 odd 6 441.2.f.d.148.3 6
9.7 even 3 3969.2.a.p.1.3 3
21.2 odd 6 441.2.h.b.214.1 6
21.5 even 6 441.2.h.c.214.1 6
21.11 odd 6 441.2.g.d.79.3 6
21.17 even 6 441.2.g.e.79.3 6
21.20 even 2 63.2.f.b.43.3 yes 6
28.27 even 2 3024.2.r.g.2017.2 6
63.4 even 3 1323.2.h.e.226.3 6
63.5 even 6 441.2.g.e.67.3 6
63.13 odd 6 189.2.f.a.64.1 6
63.20 even 6 567.2.a.d.1.1 3
63.23 odd 6 441.2.g.d.67.3 6
63.31 odd 6 1323.2.h.d.226.3 6
63.32 odd 6 441.2.h.b.373.1 6
63.34 odd 6 567.2.a.g.1.3 3
63.40 odd 6 1323.2.g.c.361.1 6
63.41 even 6 63.2.f.b.22.3 ✓ 6
63.58 even 3 1323.2.g.b.361.1 6
63.59 even 6 441.2.h.c.373.1 6
84.83 odd 2 1008.2.r.k.673.3 6
252.83 odd 6 9072.2.a.bq.1.2 3
252.139 even 6 3024.2.r.g.1009.2 6
252.167 odd 6 1008.2.r.k.337.3 6
252.223 even 6 9072.2.a.cd.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 ✓ 6 63.41 even 6
63.2.f.b.43.3 yes 6 21.20 even 2
189.2.f.a.64.1 6 63.13 odd 6
189.2.f.a.127.1 6 7.6 odd 2
441.2.f.d.148.3 6 9.5 odd 6
441.2.f.d.295.3 6 3.2 odd 2
441.2.g.d.67.3 6 63.23 odd 6
441.2.g.d.79.3 6 21.11 odd 6
441.2.g.e.67.3 6 63.5 even 6
441.2.g.e.79.3 6 21.17 even 6
441.2.h.b.214.1 6 21.2 odd 6
441.2.h.b.373.1 6 63.32 odd 6
441.2.h.c.214.1 6 21.5 even 6
441.2.h.c.373.1 6 63.59 even 6
567.2.a.d.1.1 3 63.20 even 6
567.2.a.g.1.3 3 63.34 odd 6
1008.2.r.k.337.3 6 252.167 odd 6
1008.2.r.k.673.3 6 84.83 odd 2
1323.2.f.c.442.1 6 9.4 even 3 inner
1323.2.f.c.883.1 6 1.1 even 1 trivial
1323.2.g.b.361.1 6 63.58 even 3
1323.2.g.b.667.1 6 7.4 even 3
1323.2.g.c.361.1 6 63.40 odd 6
1323.2.g.c.667.1 6 7.3 odd 6
1323.2.h.d.226.3 6 63.31 odd 6
1323.2.h.d.802.3 6 7.5 odd 6
1323.2.h.e.226.3 6 63.4 even 3
1323.2.h.e.802.3 6 7.2 even 3
3024.2.r.g.1009.2 6 252.139 even 6
3024.2.r.g.2017.2 6 28.27 even 2
3969.2.a.m.1.1 3 9.2 odd 6
3969.2.a.p.1.3 3 9.7 even 3
9072.2.a.bq.1.2 3 252.83 odd 6
9072.2.a.cd.1.2 3 252.223 even 6