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(226,1323)] 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("1323.226"); 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, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,2,0,6,-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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 226.3
Root \(0.500000 - 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 1323.226
Dual form 1323.2.h.d.802.3

$q$-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)
 
\(f(q)\) \(=\) \(q+2.46050 q^{2} +4.05408 q^{4} +(-1.29679 - 2.24611i) q^{5} +5.05408 q^{8} +(-3.19076 - 5.52655i) q^{10} +(2.25729 - 3.90975i) q^{11} +(-0.500000 + 0.866025i) q^{13} +4.32743 q^{16} +(-0.472958 - 0.819187i) q^{17} +(2.02704 - 3.51094i) q^{19} +(-5.25729 - 9.10590i) q^{20} +(5.55408 - 9.61996i) q^{22} +(-0.136673 - 0.236725i) q^{23} +(-0.863327 + 1.49533i) q^{25} +(-1.23025 + 2.13086i) q^{26} +(1.23025 + 2.13086i) q^{29} +2.32743 q^{31} +0.539495 q^{32} +(-1.16372 - 2.01561i) q^{34} +(-0.890369 + 1.54216i) q^{37} +(4.98755 - 8.63868i) q^{38} +(-6.55408 - 11.3520i) q^{40} +(-3.20321 + 5.54812i) q^{41} +(5.21780 + 9.03749i) q^{43} +(9.15126 - 15.8505i) q^{44} +(-0.336285 - 0.582462i) q^{46} +12.1623 q^{47} +(-2.12422 + 3.67926i) q^{50} +(-2.02704 + 3.51094i) q^{52} +(-3.13667 - 5.43288i) q^{53} -11.7089 q^{55} +(3.02704 + 5.24299i) q^{58} +2.72665 q^{59} -2.27335 q^{61} +5.72665 q^{62} -7.32743 q^{64} +2.59358 q^{65} -15.8171 q^{67} +(-1.91741 - 3.32105i) q^{68} -3.27335 q^{71} +(0.753696 + 1.30544i) q^{73} +(-2.19076 + 3.79450i) q^{74} +(8.21780 - 14.2336i) q^{76} +14.7089 q^{79} +(-5.61177 - 9.71987i) q^{80} +(-7.88151 + 13.6512i) q^{82} +(-0.472958 - 0.819187i) q^{83} +(-1.22665 + 2.12463i) q^{85} +(12.8384 + 22.2368i) q^{86} +(11.4086 - 19.7602i) q^{88} +(-7.17830 + 12.4332i) q^{89} +(-0.554084 - 0.959702i) q^{92} +29.9253 q^{94} -10.5146 q^{95} +(5.74484 + 9.95036i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 12 q^{8} - 2 q^{11} - 3 q^{13} + 6 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} + 15 q^{22} - 6 q^{25} - q^{26} + q^{29} - 6 q^{31} + 16 q^{32} + 3 q^{34} + 3 q^{37} + 8 q^{38}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.46050 1.73984 0.869920 − 0.493193i \(-0.164170\pi\)
0.869920 + 0.493193i \(0.164170\pi\)
\(3\) 0 0
\(4\) 4.05408 2.02704
\(5\) −1.29679 − 2.24611i −0.579942 − 1.00449i −0.995485 − 0.0949156i \(-0.969742\pi\)
0.415543 − 0.909573i \(-0.363591\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 5.05408 1.78689
\(9\) 0 0
\(10\) −3.19076 − 5.52655i −1.00901 − 1.74765i
\(11\) 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 \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.32743 1.08186
\(17\) −0.472958 − 0.819187i −0.114709 − 0.198682i 0.802954 − 0.596041i \(-0.203260\pi\)
−0.917663 + 0.397359i \(0.869927\pi\)
\(18\) 0 0
\(19\) 2.02704 − 3.51094i 0.465035 − 0.805465i −0.534168 − 0.845378i \(-0.679375\pi\)
0.999203 + 0.0399136i \(0.0127083\pi\)
\(20\) −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
\(26\) −1.23025 + 2.13086i −0.241272 + 0.417896i
\(27\) 0 0
\(28\) 0 0
\(29\) 1.23025 + 2.13086i 0.228452 + 0.395691i 0.957350 − 0.288932i \(-0.0933002\pi\)
−0.728897 + 0.684623i \(0.759967\pi\)
\(30\) 0 0
\(31\) 2.32743 0.418019 0.209009 − 0.977914i \(-0.432976\pi\)
0.209009 + 0.977914i \(0.432976\pi\)
\(32\) 0.539495 0.0953702
\(33\) 0 0
\(34\) −1.16372 − 2.01561i −0.199576 − 0.345675i
\(35\) 0 0
\(36\) 0 0
\(37\) −0.890369 + 1.54216i −0.146376 + 0.253530i −0.929885 − 0.367849i \(-0.880094\pi\)
0.783510 + 0.621380i \(0.213428\pi\)
\(38\) 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 0.499743 + 0.866174i \(0.333428\pi\)
−1.00000 0.000297253i \(0.999905\pi\)
\(42\) 0 0
\(43\) 5.21780 + 9.03749i 0.795707 + 1.37820i 0.922389 + 0.386262i \(0.126234\pi\)
−0.126682 + 0.991943i \(0.540433\pi\)
\(44\) 9.15126 − 15.8505i 1.37960 − 2.38955i
\(45\) 0 0
\(46\) −0.336285 − 0.582462i −0.0495825 − 0.0858794i
\(47\) 12.1623 1.77405 0.887023 − 0.461724i \(-0.152769\pi\)
0.887023 + 0.461724i \(0.152769\pi\)
\(48\) 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
\(53\) −3.13667 − 5.43288i −0.430855 − 0.746263i 0.566092 − 0.824342i \(-0.308455\pi\)
−0.996947 + 0.0780790i \(0.975121\pi\)
\(54\) 0 0
\(55\) −11.7089 −1.57883
\(56\) 0 0
\(57\) 0 0
\(58\) 3.02704 + 5.24299i 0.397470 + 0.688438i
\(59\) 2.72665 0.354980 0.177490 − 0.984123i \(-0.443202\pi\)
0.177490 + 0.984123i \(0.443202\pi\)
\(60\) 0 0
\(61\) −2.27335 −0.291072 −0.145536 − 0.989353i \(-0.546491\pi\)
−0.145536 + 0.989353i \(0.546491\pi\)
\(62\) 5.72665 0.727286
\(63\) 0 0
\(64\) −7.32743 −0.915929
\(65\) 2.59358 0.321694
\(66\) 0 0
\(67\) −15.8171 −1.93237 −0.966184 − 0.257854i \(-0.916985\pi\)
−0.966184 + 0.257854i \(0.916985\pi\)
\(68\) −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
\(73\) 0.753696 + 1.30544i 0.0882134 + 0.152790i 0.906756 − 0.421656i \(-0.138551\pi\)
−0.818543 + 0.574446i \(0.805218\pi\)
\(74\) −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
\(79\) 14.7089 1.65489 0.827443 − 0.561550i \(-0.189795\pi\)
0.827443 + 0.561550i \(0.189795\pi\)
\(80\) −5.61177 − 9.71987i −0.627415 − 1.08671i
\(81\) 0 0
\(82\) −7.88151 + 13.6512i −0.870368 + 1.50752i
\(83\) −0.472958 − 0.819187i −0.0519139 − 0.0899175i 0.838901 − 0.544285i \(-0.183199\pi\)
−0.890815 + 0.454367i \(0.849865\pi\)
\(84\) 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
\(89\) −7.17830 + 12.4332i −0.760899 + 1.31792i 0.181489 + 0.983393i \(0.441908\pi\)
−0.942388 + 0.334522i \(0.891425\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.554084 − 0.959702i −0.0577673 − 0.100056i
\(93\) 0 0
\(94\) 29.9253 3.08656
\(95\) −10.5146 −1.07877
\(96\) 0 0
\(97\) 5.74484 + 9.95036i 0.583300 + 1.01031i 0.995085 + 0.0990246i \(0.0315722\pi\)
−0.411785 + 0.911281i \(0.635094\pi\)
\(98\) 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.h.d.226.3 6
3.2 odd 2 441.2.h.c.373.1 6
7.2 even 3 189.2.f.a.64.1 6
7.3 odd 6 1323.2.g.b.361.1 6
7.4 even 3 1323.2.g.c.361.1 6
7.5 odd 6 1323.2.f.c.442.1 6
7.6 odd 2 1323.2.h.e.226.3 6
9.2 odd 6 441.2.g.e.79.3 6
9.7 even 3 1323.2.g.c.667.1 6
21.2 odd 6 63.2.f.b.22.3 ✓ 6
21.5 even 6 441.2.f.d.148.3 6
21.11 odd 6 441.2.g.e.67.3 6
21.17 even 6 441.2.g.d.67.3 6
21.20 even 2 441.2.h.b.373.1 6
28.23 odd 6 3024.2.r.g.1009.2 6
63.2 odd 6 63.2.f.b.43.3 yes 6
63.5 even 6 3969.2.a.m.1.1 3
63.11 odd 6 441.2.h.c.214.1 6
63.16 even 3 189.2.f.a.127.1 6
63.20 even 6 441.2.g.d.79.3 6
63.23 odd 6 567.2.a.d.1.1 3
63.25 even 3 inner 1323.2.h.d.802.3 6
63.34 odd 6 1323.2.g.b.667.1 6
63.38 even 6 441.2.h.b.214.1 6
63.40 odd 6 3969.2.a.p.1.3 3
63.47 even 6 441.2.f.d.295.3 6
63.52 odd 6 1323.2.h.e.802.3 6
63.58 even 3 567.2.a.g.1.3 3
63.61 odd 6 1323.2.f.c.883.1 6
84.23 even 6 1008.2.r.k.337.3 6
252.23 even 6 9072.2.a.bq.1.2 3
252.79 odd 6 3024.2.r.g.2017.2 6
252.191 even 6 1008.2.r.k.673.3 6
252.247 odd 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 21.2 odd 6
63.2.f.b.43.3 yes 6 63.2 odd 6
189.2.f.a.64.1 6 7.2 even 3
189.2.f.a.127.1 6 63.16 even 3
441.2.f.d.148.3 6 21.5 even 6
441.2.f.d.295.3 6 63.47 even 6
441.2.g.d.67.3 6 21.17 even 6
441.2.g.d.79.3 6 63.20 even 6
441.2.g.e.67.3 6 21.11 odd 6
441.2.g.e.79.3 6 9.2 odd 6
441.2.h.b.214.1 6 63.38 even 6
441.2.h.b.373.1 6 21.20 even 2
441.2.h.c.214.1 6 63.11 odd 6
441.2.h.c.373.1 6 3.2 odd 2
567.2.a.d.1.1 3 63.23 odd 6
567.2.a.g.1.3 3 63.58 even 3
1008.2.r.k.337.3 6 84.23 even 6
1008.2.r.k.673.3 6 252.191 even 6
1323.2.f.c.442.1 6 7.5 odd 6
1323.2.f.c.883.1 6 63.61 odd 6
1323.2.g.b.361.1 6 7.3 odd 6
1323.2.g.b.667.1 6 63.34 odd 6
1323.2.g.c.361.1 6 7.4 even 3
1323.2.g.c.667.1 6 9.7 even 3
1323.2.h.d.226.3 6 1.1 even 1 trivial
1323.2.h.d.802.3 6 63.25 even 3 inner
1323.2.h.e.226.3 6 7.6 odd 2
1323.2.h.e.802.3 6 63.52 odd 6
3024.2.r.g.1009.2 6 28.23 odd 6
3024.2.r.g.2017.2 6 252.79 odd 6
3969.2.a.m.1.1 3 63.5 even 6
3969.2.a.p.1.3 3 63.40 odd 6
9072.2.a.bq.1.2 3 252.23 even 6
9072.2.a.cd.1.2 3 252.247 odd 6