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Properties

Label 1728.2.r.e.1441.6
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 1441.6
Root \(-0.180407 + 0.673288i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1441
Dual form 1728.2.r.e.289.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{5} +(2.17731 + 3.77121i) q^{7} +(-5.05328 + 2.91751i) q^{11} +(-3.48133 - 2.00994i) q^{13} -1.70739 q^{17} -2.00000i q^{19} +(3.32123 - 5.75253i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(4.06108 - 2.34467i) q^{29} +(2.43071 - 4.21012i) q^{31} -7.54241i q^{35} -5.75194i q^{37} +(-0.646305 + 1.11943i) q^{41} +(3.06782 - 1.77121i) q^{43} +(-5.30669 - 9.19145i) q^{47} +(-5.98133 + 10.3600i) q^{49} +0.506816i q^{53} +10.1066 q^{55} +(1.62152 + 0.936184i) q^{59} +(1.06108 - 0.612617i) q^{61} +(3.48133 + 6.02983i) q^{65} +(1.36811 + 0.789879i) q^{67} +9.88549 q^{71} -7.96265 q^{73} +(-22.0051 - 12.7046i) q^{77} +(1.03339 + 1.78988i) q^{79} +(1.33577 - 0.771205i) q^{83} +(2.56108 + 1.47864i) q^{85} -3.96265 q^{89} -17.5051i q^{91} +(-1.73205 + 3.00000i) q^{95} +(-3.06108 - 5.30195i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} + 6 q^{13} - 12 q^{25} + 18 q^{29} - 18 q^{41} - 24 q^{49} - 18 q^{61} - 6 q^{65} - 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) 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
\(7\) 2.17731 + 3.77121i 0.822944 + 1.42538i 0.903480 + 0.428630i \(0.141004\pi\)
−0.0805357 + 0.996752i \(0.525663\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.05328 + 2.91751i −1.52362 + 0.879663i −0.524011 + 0.851712i \(0.675565\pi\)
−0.999609 + 0.0279511i \(0.991102\pi\)
\(12\) 0 0
\(13\) −3.48133 − 2.00994i −0.965546 − 0.557458i −0.0676707 − 0.997708i \(-0.521557\pi\)
−0.897876 + 0.440249i \(0.854890\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.70739 −0.414103 −0.207051 − 0.978330i \(-0.566387\pi\)
−0.207051 + 0.978330i \(0.566387\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.32123 − 5.75253i 0.692523 − 1.19949i −0.278485 − 0.960441i \(-0.589832\pi\)
0.971008 − 0.239045i \(-0.0768344\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.06108 − 2.34467i 0.754124 − 0.435394i −0.0730578 − 0.997328i \(-0.523276\pi\)
0.827182 + 0.561934i \(0.189942\pi\)
\(30\) 0 0
\(31\) 2.43071 − 4.21012i 0.436569 − 0.756160i −0.560853 − 0.827915i \(-0.689527\pi\)
0.997422 + 0.0717553i \(0.0228601\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 7.54241i − 1.27490i
\(36\) 0 0
\(37\) − 5.75194i − 0.945613i −0.881166 − 0.472807i \(-0.843241\pi\)
0.881166 − 0.472807i \(-0.156759\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.646305 + 1.11943i −0.100936 + 0.174826i −0.912071 − 0.410033i \(-0.865517\pi\)
0.811135 + 0.584860i \(0.198850\pi\)
\(42\) 0 0
\(43\) 3.06782 − 1.77121i 0.467838 − 0.270106i −0.247496 − 0.968889i \(-0.579608\pi\)
0.715334 + 0.698783i \(0.246274\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.30669 − 9.19145i −0.774060 − 1.34071i −0.935322 − 0.353799i \(-0.884890\pi\)
0.161262 − 0.986912i \(-0.448444\pi\)
\(48\) 0 0
\(49\) −5.98133 + 10.3600i −0.854475 + 1.47999i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.506816i 0.0696166i 0.999394 + 0.0348083i \(0.0110821\pi\)
−0.999394 + 0.0348083i \(0.988918\pi\)
\(54\) 0 0
\(55\) 10.1066 1.36277
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.62152 + 0.936184i 0.211104 + 0.121881i 0.601824 − 0.798629i \(-0.294441\pi\)
−0.390721 + 0.920509i \(0.627774\pi\)
\(60\) 0 0
\(61\) 1.06108 − 0.612617i 0.135858 − 0.0784376i −0.430531 − 0.902576i \(-0.641674\pi\)
0.566388 + 0.824138i \(0.308340\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.48133 + 6.02983i 0.431805 + 0.747909i
\(66\) 0 0
\(67\) 1.36811 + 0.789879i 0.167141 + 0.0964990i 0.581237 − 0.813734i \(-0.302569\pi\)
−0.414096 + 0.910233i \(0.635902\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.88549 1.17319 0.586596 − 0.809880i \(-0.300468\pi\)
0.586596 + 0.809880i \(0.300468\pi\)
\(72\) 0 0
\(73\) −7.96265 −0.931958 −0.465979 − 0.884796i \(-0.654298\pi\)
−0.465979 + 0.884796i \(0.654298\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −22.0051 − 12.7046i −2.50771 − 1.44783i
\(78\) 0 0
\(79\) 1.03339 + 1.78988i 0.116265 + 0.201377i 0.918285 − 0.395921i \(-0.129574\pi\)
−0.802020 + 0.597298i \(0.796241\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.33577 − 0.771205i 0.146619 − 0.0846508i −0.424896 − 0.905242i \(-0.639689\pi\)
0.571515 + 0.820592i \(0.306356\pi\)
\(84\) 0 0
\(85\) 2.56108 + 1.47864i 0.277789 + 0.160381i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.96265 −0.420040 −0.210020 − 0.977697i \(-0.567353\pi\)
−0.210020 + 0.977697i \(0.567353\pi\)
\(90\) 0 0
\(91\) − 17.5051i − 1.83503i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.73205 + 3.00000i −0.177705 + 0.307794i
\(96\) 0 0
\(97\) −3.06108 − 5.30195i −0.310806 − 0.538332i 0.667731 − 0.744403i \(-0.267266\pi\)
−0.978537 + 0.206071i \(0.933932\pi\)
\(98\) 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.1441.6 12
3.2 odd 2 576.2.r.f.481.4 yes 12
4.3 odd 2 inner 1728.2.r.e.1441.1 12
8.3 odd 2 1728.2.r.f.1441.1 12
8.5 even 2 1728.2.r.f.1441.6 12
9.2 odd 6 576.2.r.e.97.3 ✓ 12
9.4 even 3 5184.2.d.r.2593.7 12
9.5 odd 6 5184.2.d.q.2593.1 12
9.7 even 3 1728.2.r.f.289.6 12
12.11 even 2 576.2.r.f.481.3 yes 12
24.5 odd 2 576.2.r.e.481.3 yes 12
24.11 even 2 576.2.r.e.481.4 yes 12
36.7 odd 6 1728.2.r.f.289.1 12
36.11 even 6 576.2.r.e.97.4 yes 12
36.23 even 6 5184.2.d.q.2593.6 12
36.31 odd 6 5184.2.d.r.2593.12 12
72.5 odd 6 5184.2.d.q.2593.7 12
72.11 even 6 576.2.r.f.97.3 yes 12
72.13 even 6 5184.2.d.r.2593.1 12
72.29 odd 6 576.2.r.f.97.4 yes 12
72.43 odd 6 inner 1728.2.r.e.289.1 12
72.59 even 6 5184.2.d.q.2593.12 12
72.61 even 6 inner 1728.2.r.e.289.6 12
72.67 odd 6 5184.2.d.r.2593.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.e.97.3 ✓ 12 9.2 odd 6
576.2.r.e.97.4 yes 12 36.11 even 6
576.2.r.e.481.3 yes 12 24.5 odd 2
576.2.r.e.481.4 yes 12 24.11 even 2
576.2.r.f.97.3 yes 12 72.11 even 6
576.2.r.f.97.4 yes 12 72.29 odd 6
576.2.r.f.481.3 yes 12 12.11 even 2
576.2.r.f.481.4 yes 12 3.2 odd 2
1728.2.r.e.289.1 12 72.43 odd 6 inner
1728.2.r.e.289.6 12 72.61 even 6 inner
1728.2.r.e.1441.1 12 4.3 odd 2 inner
1728.2.r.e.1441.6 12 1.1 even 1 trivial
1728.2.r.f.289.1 12 36.7 odd 6
1728.2.r.f.289.6 12 9.7 even 3
1728.2.r.f.1441.1 12 8.3 odd 2
1728.2.r.f.1441.6 12 8.5 even 2
5184.2.d.q.2593.1 12 9.5 odd 6
5184.2.d.q.2593.6 12 36.23 even 6
5184.2.d.q.2593.7 12 72.5 odd 6
5184.2.d.q.2593.12 12 72.59 even 6
5184.2.d.r.2593.1 12 72.13 even 6
5184.2.d.r.2593.6 12 72.67 odd 6
5184.2.d.r.2593.7 12 9.4 even 3
5184.2.d.r.2593.12 12 36.31 odd 6