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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,2,Mod(575,1728)] 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("1728.575"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1151
Dual form 1728.2.s.e.575.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{5} +(-2.59808 + 1.50000i) q^{7} +(-2.59808 - 4.50000i) q^{11} +(0.500000 - 0.866025i) q^{13} +3.46410i q^{17} -6.00000i q^{19} +(-2.59808 + 4.50000i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-7.50000 + 4.33013i) q^{29} +(-2.59808 - 1.50000i) q^{31} -5.19615 q^{35} +4.00000 q^{37} +(-4.50000 - 2.59808i) q^{41} +(2.59808 - 1.50000i) q^{43} +(-2.59808 - 4.50000i) q^{47} +(1.00000 - 1.73205i) q^{49} -10.3923i q^{53} -9.00000i q^{55} +(-2.59808 + 4.50000i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(1.50000 - 0.866025i) q^{65} +(-7.79423 - 4.50000i) q^{67} -10.3923 q^{71} +4.00000 q^{73} +(13.5000 + 7.79423i) q^{77} +(12.9904 - 7.50000i) q^{79} +(2.59808 + 4.50000i) q^{83} +(-3.00000 + 5.19615i) q^{85} -3.46410i q^{89} +3.00000i q^{91} +(5.19615 - 9.00000i) q^{95} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} + 2 q^{13} - 4 q^{25} - 30 q^{29} + 16 q^{37} - 18 q^{41} + 4 q^{49} - 14 q^{61} + 6 q^{65} + 16 q^{73} + 54 q^{77} - 12 q^{85} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 − 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) −2.59808 + 1.50000i −0.981981 + 0.566947i −0.902867 − 0.429919i \(-0.858542\pi\)
−0.0791130 + 0.996866i \(0.525209\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.59808 − 4.50000i −0.783349 − 1.35680i −0.929980 − 0.367610i \(-0.880176\pi\)
0.146631 − 0.989191i \(-0.453157\pi\)
\(12\) 0 0
\(13\) 0.500000 − 0.866025i 0.138675 − 0.240192i −0.788320 − 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) − 6.00000i − 1.37649i −0.725476 − 0.688247i \(-0.758380\pi\)
0.725476 − 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.59808 + 4.50000i −0.541736 + 0.938315i 0.457068 + 0.889432i \(0.348900\pi\)
−0.998805 + 0.0488832i \(0.984434\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.50000 + 4.33013i −1.39272 + 0.804084i −0.993615 − 0.112823i \(-0.964011\pi\)
−0.399100 + 0.916907i \(0.630677\pi\)
\(30\) 0 0
\(31\) −2.59808 − 1.50000i −0.466628 − 0.269408i 0.248199 − 0.968709i \(-0.420161\pi\)
−0.714827 + 0.699301i \(0.753495\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.19615 −0.878310
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.50000 − 2.59808i −0.702782 − 0.405751i 0.105601 − 0.994409i \(-0.466323\pi\)
−0.808383 + 0.588657i \(0.799657\pi\)
\(42\) 0 0
\(43\) 2.59808 − 1.50000i 0.396203 − 0.228748i −0.288641 − 0.957437i \(-0.593204\pi\)
0.684844 + 0.728689i \(0.259870\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.59808 − 4.50000i −0.378968 − 0.656392i 0.611944 − 0.790901i \(-0.290388\pi\)
−0.990912 + 0.134509i \(0.957054\pi\)
\(48\) 0 0
\(49\) 1.00000 − 1.73205i 0.142857 − 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 10.3923i − 1.42749i −0.700404 − 0.713746i \(-0.746997\pi\)
0.700404 − 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) − 9.00000i − 1.21356i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.59808 + 4.50000i −0.338241 + 0.585850i −0.984102 − 0.177605i \(-0.943165\pi\)
0.645861 + 0.763455i \(0.276498\pi\)
\(60\) 0 0
\(61\) −3.50000 − 6.06218i −0.448129 − 0.776182i 0.550135 − 0.835076i \(-0.314576\pi\)
−0.998264 + 0.0588933i \(0.981243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.50000 − 0.866025i 0.186052 − 0.107417i
\(66\) 0 0
\(67\) −7.79423 − 4.50000i −0.952217 − 0.549762i −0.0584478 − 0.998290i \(-0.518615\pi\)
−0.893769 + 0.448528i \(0.851948\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 − 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.5000 + 7.79423i 1.53847 + 0.888235i
\(78\) 0 0
\(79\) 12.9904 − 7.50000i 1.46153 − 0.843816i 0.462450 − 0.886646i \(-0.346971\pi\)
0.999082 + 0.0428296i \(0.0136373\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.59808 + 4.50000i 0.285176 + 0.493939i 0.972652 − 0.232268i \(-0.0746146\pi\)
−0.687476 + 0.726207i \(0.741281\pi\)
\(84\) 0 0
\(85\) −3.00000 + 5.19615i −0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 3.46410i − 0.367194i −0.983002 − 0.183597i \(-0.941226\pi\)
0.983002 − 0.183597i \(-0.0587741\pi\)
\(90\) 0 0
\(91\) 3.00000i 0.314485i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.19615 − 9.00000i 0.533114 − 0.923381i
\(96\) 0 0
\(97\) −0.500000 − 0.866025i −0.0507673 − 0.0879316i 0.839525 − 0.543321i \(-0.182833\pi\)
−0.890292 + 0.455389i \(0.849500\pi\)
\(98\) 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.s.e.1151.1 4
3.2 odd 2 576.2.s.e.383.1 4
4.3 odd 2 inner 1728.2.s.e.1151.2 4
8.3 odd 2 432.2.s.e.287.2 4
8.5 even 2 432.2.s.e.287.1 4
9.2 odd 6 inner 1728.2.s.e.575.2 4
9.4 even 3 5184.2.c.f.5183.1 4
9.5 odd 6 5184.2.c.f.5183.3 4
9.7 even 3 576.2.s.e.191.2 4
12.11 even 2 576.2.s.e.383.2 4
24.5 odd 2 144.2.s.e.95.2 yes 4
24.11 even 2 144.2.s.e.95.1 yes 4
36.7 odd 6 576.2.s.e.191.1 4
36.11 even 6 inner 1728.2.s.e.575.1 4
36.23 even 6 5184.2.c.f.5183.4 4
36.31 odd 6 5184.2.c.f.5183.2 4
72.5 odd 6 1296.2.c.f.1295.1 4
72.11 even 6 432.2.s.e.143.1 4
72.13 even 6 1296.2.c.f.1295.3 4
72.29 odd 6 432.2.s.e.143.2 4
72.43 odd 6 144.2.s.e.47.2 yes 4
72.59 even 6 1296.2.c.f.1295.2 4
72.61 even 6 144.2.s.e.47.1 ✓ 4
72.67 odd 6 1296.2.c.f.1295.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.s.e.47.1 ✓ 4 72.61 even 6
144.2.s.e.47.2 yes 4 72.43 odd 6
144.2.s.e.95.1 yes 4 24.11 even 2
144.2.s.e.95.2 yes 4 24.5 odd 2
432.2.s.e.143.1 4 72.11 even 6
432.2.s.e.143.2 4 72.29 odd 6
432.2.s.e.287.1 4 8.5 even 2
432.2.s.e.287.2 4 8.3 odd 2
576.2.s.e.191.1 4 36.7 odd 6
576.2.s.e.191.2 4 9.7 even 3
576.2.s.e.383.1 4 3.2 odd 2
576.2.s.e.383.2 4 12.11 even 2
1296.2.c.f.1295.1 4 72.5 odd 6
1296.2.c.f.1295.2 4 72.59 even 6
1296.2.c.f.1295.3 4 72.13 even 6
1296.2.c.f.1295.4 4 72.67 odd 6
1728.2.s.e.575.1 4 36.11 even 6 inner
1728.2.s.e.575.2 4 9.2 odd 6 inner
1728.2.s.e.1151.1 4 1.1 even 1 trivial
1728.2.s.e.1151.2 4 4.3 odd 2 inner
5184.2.c.f.5183.1 4 9.4 even 3
5184.2.c.f.5183.2 4 36.31 odd 6
5184.2.c.f.5183.3 4 9.5 odd 6
5184.2.c.f.5183.4 4 36.23 even 6