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Properties

Label 1728.2.r.e.1441.5
Level $1728$
Weight $2$
Character 1728.1441
Analytic conductor $13.798$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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(289,1728)] 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("1728.289"); 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([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-18,0,0,0,0,0,0,0,6,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1441.5
Root \(2.17840 + 0.583700i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1441
Dual form 1728.2.r.e.289.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{5} +(1.80664 + 3.12920i) q^{7} +(-0.635828 + 0.367095i) q^{11} +(-0.527909 - 0.304788i) q^{13} +5.52420 q^{17} +2.00000i q^{19} +(2.36788 - 4.10129i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-6.78630 + 3.91807i) q^{29} +(-4.70951 + 8.15710i) q^{31} -6.25839i q^{35} -2.34163i q^{37} +(-4.26210 + 7.38217i) q^{41} +(8.88403 - 5.12920i) q^{43} +(5.88032 + 10.1850i) q^{47} +(-3.02791 + 5.24449i) q^{49} +13.0323i q^{53} +1.27166 q^{55} +(-1.04788 - 0.604996i) q^{59} +(-9.78630 + 5.65012i) q^{61} +(0.527909 + 0.914365i) q^{65} +(5.46826 + 3.15710i) q^{67} +2.63999 q^{71} -2.05582 q^{73} +(-2.29743 - 1.32642i) q^{77} +(1.24540 + 2.15710i) q^{79} +(10.6161 - 6.12920i) q^{83} +(-8.28630 - 4.78410i) q^{85} +1.94418 q^{89} -2.20257i q^{91} +(1.73205 - 3.00000i) q^{95} +(7.78630 + 13.4863i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} + 6 q^{13} - 12 q^{25} + 18 q^{29} - 18 q^{41} - 24 q^{49} - 18 q^{61} - 6 q^{65} - 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.50000 − 0.866025i −0.670820 − 0.387298i 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 1.80664 + 3.12920i 0.682846 + 1.18272i 0.974108 + 0.226081i \(0.0725915\pi\)
−0.291262 + 0.956643i \(0.594075\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.635828 + 0.367095i −0.191709 + 0.110683i −0.592783 − 0.805363i \(-0.701971\pi\)
0.401073 + 0.916046i \(0.368637\pi\)
\(12\) 0 0
\(13\) −0.527909 − 0.304788i −0.146416 − 0.0845331i 0.425003 − 0.905192i \(-0.360273\pi\)
−0.571418 + 0.820659i \(0.693607\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.52420 1.33982 0.669908 − 0.742444i \(-0.266334\pi\)
0.669908 + 0.742444i \(0.266334\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.36788 − 4.10129i 0.493737 − 0.855177i −0.506237 − 0.862394i \(-0.668964\pi\)
0.999974 + 0.00721700i \(0.00229726\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.78630 + 3.91807i −1.26018 + 0.727568i −0.973110 − 0.230341i \(-0.926016\pi\)
−0.287074 + 0.957908i \(0.592683\pi\)
\(30\) 0 0
\(31\) −4.70951 + 8.15710i −0.845852 + 1.46506i 0.0390267 + 0.999238i \(0.487574\pi\)
−0.884879 + 0.465821i \(0.845759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 6.25839i − 1.05786i
\(36\) 0 0
\(37\) − 2.34163i − 0.384961i −0.981301 − 0.192481i \(-0.938347\pi\)
0.981301 − 0.192481i \(-0.0616532\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.26210 + 7.38217i −0.665628 + 1.15290i 0.313486 + 0.949593i \(0.398503\pi\)
−0.979115 + 0.203309i \(0.934830\pi\)
\(42\) 0 0
\(43\) 8.88403 − 5.12920i 1.35480 − 0.782195i 0.365884 − 0.930661i \(-0.380767\pi\)
0.988918 + 0.148466i \(0.0474334\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.88032 + 10.1850i 0.857733 + 1.48564i 0.874086 + 0.485771i \(0.161461\pi\)
−0.0163535 + 0.999866i \(0.505206\pi\)
\(48\) 0 0
\(49\) −3.02791 + 5.24449i −0.432558 + 0.749213i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.0323i 1.79012i 0.445942 + 0.895062i \(0.352869\pi\)
−0.445942 + 0.895062i \(0.647131\pi\)
\(54\) 0 0
\(55\) 1.27166 0.171470
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.04788 − 0.604996i −0.136423 − 0.0787637i 0.430235 − 0.902717i \(-0.358431\pi\)
−0.566658 + 0.823953i \(0.691764\pi\)
\(60\) 0 0
\(61\) −9.78630 + 5.65012i −1.25301 + 0.723424i −0.971706 − 0.236195i \(-0.924099\pi\)
−0.281302 + 0.959619i \(0.590766\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.527909 + 0.914365i 0.0654790 + 0.113413i
\(66\) 0 0
\(67\) 5.46826 + 3.15710i 0.668055 + 0.385702i 0.795339 − 0.606165i \(-0.207293\pi\)
−0.127284 + 0.991866i \(0.540626\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.63999 0.313309 0.156655 − 0.987653i \(-0.449929\pi\)
0.156655 + 0.987653i \(0.449929\pi\)
\(72\) 0 0
\(73\) −2.05582 −0.240615 −0.120308 − 0.992737i \(-0.538388\pi\)
−0.120308 + 0.992737i \(0.538388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.29743 − 1.32642i −0.261816 − 0.151160i
\(78\) 0 0
\(79\) 1.24540 + 2.15710i 0.140119 + 0.242693i 0.927541 − 0.373721i \(-0.121918\pi\)
−0.787422 + 0.616414i \(0.788585\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.6161 − 6.12920i 1.16527 − 0.672767i 0.212706 − 0.977116i \(-0.431772\pi\)
0.952560 + 0.304350i \(0.0984392\pi\)
\(84\) 0 0
\(85\) −8.28630 − 4.78410i −0.898775 − 0.518908i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.94418 0.206083 0.103041 − 0.994677i \(-0.467143\pi\)
0.103041 + 0.994677i \(0.467143\pi\)
\(90\) 0 0
\(91\) − 2.20257i − 0.230892i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.73205 − 3.00000i 0.177705 − 0.307794i
\(96\) 0 0
\(97\) 7.78630 + 13.4863i 0.790579 + 1.36932i 0.925609 + 0.378481i \(0.123554\pi\)
−0.135030 + 0.990842i \(0.543113\pi\)
\(98\) 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.r.e.1441.5 12
3.2 odd 2 576.2.r.f.481.6 yes 12
4.3 odd 2 inner 1728.2.r.e.1441.2 12
8.3 odd 2 1728.2.r.f.1441.2 12
8.5 even 2 1728.2.r.f.1441.5 12
9.2 odd 6 576.2.r.e.97.1 ✓ 12
9.4 even 3 5184.2.d.r.2593.8 12
9.5 odd 6 5184.2.d.q.2593.2 12
9.7 even 3 1728.2.r.f.289.5 12
12.11 even 2 576.2.r.f.481.1 yes 12
24.5 odd 2 576.2.r.e.481.1 yes 12
24.11 even 2 576.2.r.e.481.6 yes 12
36.7 odd 6 1728.2.r.f.289.2 12
36.11 even 6 576.2.r.e.97.6 yes 12
36.23 even 6 5184.2.d.q.2593.5 12
36.31 odd 6 5184.2.d.r.2593.11 12
72.5 odd 6 5184.2.d.q.2593.8 12
72.11 even 6 576.2.r.f.97.1 yes 12
72.13 even 6 5184.2.d.r.2593.2 12
72.29 odd 6 576.2.r.f.97.6 yes 12
72.43 odd 6 inner 1728.2.r.e.289.2 12
72.59 even 6 5184.2.d.q.2593.11 12
72.61 even 6 inner 1728.2.r.e.289.5 12
72.67 odd 6 5184.2.d.r.2593.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.e.97.1 ✓ 12 9.2 odd 6
576.2.r.e.97.6 yes 12 36.11 even 6
576.2.r.e.481.1 yes 12 24.5 odd 2
576.2.r.e.481.6 yes 12 24.11 even 2
576.2.r.f.97.1 yes 12 72.11 even 6
576.2.r.f.97.6 yes 12 72.29 odd 6
576.2.r.f.481.1 yes 12 12.11 even 2
576.2.r.f.481.6 yes 12 3.2 odd 2
1728.2.r.e.289.2 12 72.43 odd 6 inner
1728.2.r.e.289.5 12 72.61 even 6 inner
1728.2.r.e.1441.2 12 4.3 odd 2 inner
1728.2.r.e.1441.5 12 1.1 even 1 trivial
1728.2.r.f.289.2 12 36.7 odd 6
1728.2.r.f.289.5 12 9.7 even 3
1728.2.r.f.1441.2 12 8.3 odd 2
1728.2.r.f.1441.5 12 8.5 even 2
5184.2.d.q.2593.2 12 9.5 odd 6
5184.2.d.q.2593.5 12 36.23 even 6
5184.2.d.q.2593.8 12 72.5 odd 6
5184.2.d.q.2593.11 12 72.59 even 6
5184.2.d.r.2593.2 12 72.13 even 6
5184.2.d.r.2593.5 12 72.67 odd 6
5184.2.d.r.2593.8 12 9.4 even 3
5184.2.d.r.2593.11 12 36.31 odd 6