Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,2,Mod(1,1728)] 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("1728.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,3,0,1,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1728.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{5} +1.00000 q^{7} +3.00000 q^{11} +4.00000 q^{13} +2.00000 q^{19} -6.00000 q^{23} +4.00000 q^{25} +6.00000 q^{29} -5.00000 q^{31} +3.00000 q^{35} -2.00000 q^{37} +6.00000 q^{41} -10.0000 q^{43} +6.00000 q^{47} -6.00000 q^{49} +9.00000 q^{53} +9.00000 q^{55} -12.0000 q^{59} -8.00000 q^{61} +12.0000 q^{65} +14.0000 q^{67} -7.00000 q^{73} +3.00000 q^{77} -8.00000 q^{79} +3.00000 q^{83} +18.0000 q^{89} +4.00000 q^{91} +6.00000 q^{95} -1.00000 q^{97} +O(q^{100})\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 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 1728.2.a.z.1.1 1
3.2 odd 2 1728.2.a.d.1.1 1
4.3 odd 2 1728.2.a.y.1.1 1
8.3 odd 2 54.2.a.b.1.1 yes 1
8.5 even 2 432.2.a.b.1.1 1
12.11 even 2 1728.2.a.c.1.1 1
24.5 odd 2 432.2.a.g.1.1 1
24.11 even 2 54.2.a.a.1.1 ✓ 1
40.3 even 4 1350.2.c.k.649.1 2
40.19 odd 2 1350.2.a.h.1.1 1
40.27 even 4 1350.2.c.k.649.2 2
56.27 even 2 2646.2.a.bd.1.1 1
72.5 odd 6 1296.2.i.c.865.1 2
72.11 even 6 162.2.c.c.109.1 2
72.13 even 6 1296.2.i.o.865.1 2
72.29 odd 6 1296.2.i.c.433.1 2
72.43 odd 6 162.2.c.b.109.1 2
72.59 even 6 162.2.c.c.55.1 2
72.61 even 6 1296.2.i.o.433.1 2
72.67 odd 6 162.2.c.b.55.1 2
88.43 even 2 6534.2.a.b.1.1 1
104.51 odd 2 9126.2.a.r.1.1 1
120.59 even 2 1350.2.a.r.1.1 1
120.83 odd 4 1350.2.c.b.649.2 2
120.107 odd 4 1350.2.c.b.649.1 2
168.83 odd 2 2646.2.a.a.1.1 1
264.131 odd 2 6534.2.a.bc.1.1 1
312.155 even 2 9126.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.a.a.1.1 ✓ 1 24.11 even 2
54.2.a.b.1.1 yes 1 8.3 odd 2
162.2.c.b.55.1 2 72.67 odd 6
162.2.c.b.109.1 2 72.43 odd 6
162.2.c.c.55.1 2 72.59 even 6
162.2.c.c.109.1 2 72.11 even 6
432.2.a.b.1.1 1 8.5 even 2
432.2.a.g.1.1 1 24.5 odd 2
1296.2.i.c.433.1 2 72.29 odd 6
1296.2.i.c.865.1 2 72.5 odd 6
1296.2.i.o.433.1 2 72.61 even 6
1296.2.i.o.865.1 2 72.13 even 6
1350.2.a.h.1.1 1 40.19 odd 2
1350.2.a.r.1.1 1 120.59 even 2
1350.2.c.b.649.1 2 120.107 odd 4
1350.2.c.b.649.2 2 120.83 odd 4
1350.2.c.k.649.1 2 40.3 even 4
1350.2.c.k.649.2 2 40.27 even 4
1728.2.a.c.1.1 1 12.11 even 2
1728.2.a.d.1.1 1 3.2 odd 2
1728.2.a.y.1.1 1 4.3 odd 2
1728.2.a.z.1.1 1 1.1 even 1 trivial
2646.2.a.a.1.1 1 168.83 odd 2
2646.2.a.bd.1.1 1 56.27 even 2
6534.2.a.b.1.1 1 88.43 even 2
6534.2.a.bc.1.1 1 264.131 odd 2
9126.2.a.r.1.1 1 104.51 odd 2
9126.2.a.u.1.1 1 312.155 even 2