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 = [2,0,0,0,0,0,0,0,0,0,2] 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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+3.60555 q^{5} +3.60555 q^{7} +1.00000 q^{11} -4.00000 q^{13} +7.21110 q^{19} +6.00000 q^{23} +8.00000 q^{25} -7.21110 q^{29} -3.60555 q^{31} +13.0000 q^{35} -10.0000 q^{37} +7.21110 q^{41} -7.21110 q^{43} +10.0000 q^{47} +6.00000 q^{49} -3.60555 q^{53} +3.60555 q^{55} -4.00000 q^{59} -14.4222 q^{65} -7.21110 q^{67} -8.00000 q^{71} -3.00000 q^{73} +3.60555 q^{77} -14.4222 q^{79} +9.00000 q^{83} -7.21110 q^{89} -14.4222 q^{91} +26.0000 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{11} - 8 q^{13} + 12 q^{23} + 16 q^{25} + 26 q^{35} - 20 q^{37} + 20 q^{47} + 12 q^{49} - 8 q^{59} - 16 q^{71} - 6 q^{73} + 18 q^{83} + 52 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 3.60555 1.61245 0.806226 − 0.591608i \(-0.201507\pi\)
0.806226 + 0.591608i \(0.201507\pi\)
\(6\) 0 0
\(7\) 3.60555 1.36277 0.681385 − 0.731925i \(-0.261378\pi\)
0.681385 + 0.731925i \(0.261378\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 7.21110 1.65434 0.827170 − 0.561951i \(-0.189949\pi\)
0.827170 + 0.561951i \(0.189949\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 8.00000 1.60000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.21110 −1.33907 −0.669534 − 0.742781i \(-0.733506\pi\)
−0.669534 + 0.742781i \(0.733506\pi\)
\(30\) 0 0
\(31\) −3.60555 −0.647576 −0.323788 − 0.946130i \(-0.604956\pi\)
−0.323788 + 0.946130i \(0.604956\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13.0000 2.19740
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.21110 1.12619 0.563093 − 0.826394i \(-0.309611\pi\)
0.563093 + 0.826394i \(0.309611\pi\)
\(42\) 0 0
\(43\) −7.21110 −1.09968 −0.549841 − 0.835269i \(-0.685312\pi\)
−0.549841 + 0.835269i \(0.685312\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.60555 −0.495261 −0.247630 − 0.968855i \(-0.579652\pi\)
−0.247630 + 0.968855i \(0.579652\pi\)
\(54\) 0 0
\(55\) 3.60555 0.486172
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.4222 −1.78885
\(66\) 0 0
\(67\) −7.21110 −0.880976 −0.440488 − 0.897758i \(-0.645195\pi\)
−0.440488 + 0.897758i \(0.645195\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 − 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.60555 0.410891
\(78\) 0 0
\(79\) −14.4222 −1.62262 −0.811312 − 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.21110 −0.764375 −0.382188 − 0.924085i \(-0.624829\pi\)
−0.382188 + 0.924085i \(0.624829\pi\)
\(90\) 0 0
\(91\) −14.4222 −1.51186
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 26.0000 2.66754
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 − 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 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.bd.1.2 2
3.2 odd 2 1728.2.a.bc.1.1 2
4.3 odd 2 1728.2.a.bc.1.2 2
8.3 odd 2 864.2.a.n.1.1 yes 2
8.5 even 2 864.2.a.m.1.1 ✓ 2
12.11 even 2 inner 1728.2.a.bd.1.1 2
24.5 odd 2 864.2.a.n.1.2 yes 2
24.11 even 2 864.2.a.m.1.2 yes 2
72.5 odd 6 2592.2.i.bb.865.1 4
72.11 even 6 2592.2.i.bc.1729.1 4
72.13 even 6 2592.2.i.bc.865.2 4
72.29 odd 6 2592.2.i.bb.1729.1 4
72.43 odd 6 2592.2.i.bb.1729.2 4
72.59 even 6 2592.2.i.bc.865.1 4
72.61 even 6 2592.2.i.bc.1729.2 4
72.67 odd 6 2592.2.i.bb.865.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.m.1.1 ✓ 2 8.5 even 2
864.2.a.m.1.2 yes 2 24.11 even 2
864.2.a.n.1.1 yes 2 8.3 odd 2
864.2.a.n.1.2 yes 2 24.5 odd 2
1728.2.a.bc.1.1 2 3.2 odd 2
1728.2.a.bc.1.2 2 4.3 odd 2
1728.2.a.bd.1.1 2 12.11 even 2 inner
1728.2.a.bd.1.2 2 1.1 even 1 trivial
2592.2.i.bb.865.1 4 72.5 odd 6
2592.2.i.bb.865.2 4 72.67 odd 6
2592.2.i.bb.1729.1 4 72.29 odd 6
2592.2.i.bb.1729.2 4 72.43 odd 6
2592.2.i.bc.865.1 4 72.59 even 6
2592.2.i.bc.865.2 4 72.13 even 6
2592.2.i.bc.1729.1 4 72.11 even 6
2592.2.i.bc.1729.2 4 72.61 even 6