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Properties

Label 1728.2.c.f.1727.1
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
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(1727,1728)] 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("1728.1727"); 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([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.f.1727.8

$q$-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)
 
\(f(q)\) \(=\) \(q-3.86370i q^{5} -3.73205i q^{7} -1.03528 q^{11} -4.46410 q^{13} -1.79315i q^{17} -1.73205i q^{19} +8.76268 q^{23} -9.92820 q^{25} -7.72741i q^{29} +7.46410i q^{31} -14.4195 q^{35} -0.464102 q^{37} +7.72741i q^{41} -0.535898i q^{43} -4.62158 q^{47} -6.92820 q^{49} +3.58630i q^{53} +4.00000i q^{55} +12.3490 q^{59} -11.3923 q^{61} +17.2480i q^{65} +6.26795i q^{67} +11.3137 q^{71} -3.92820 q^{73} +3.86370i q^{77} -4.80385i q^{79} -2.07055 q^{83} -6.92820 q^{85} -1.79315i q^{89} +16.6603i q^{91} -6.69213 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{13} - 24 q^{25} + 24 q^{37} - 8 q^{61} + 24 q^{73} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) − 3.86370i − 1.72790i −0.503577 − 0.863950i \(-0.667983\pi\)
0.503577 − 0.863950i \(-0.332017\pi\)
\(6\) 0 0
\(7\) − 3.73205i − 1.41058i −0.708918 − 0.705291i \(-0.750816\pi\)
0.708918 − 0.705291i \(-0.249184\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.03528 −0.312148 −0.156074 − 0.987745i \(-0.549884\pi\)
−0.156074 + 0.987745i \(0.549884\pi\)
\(12\) 0 0
\(13\) −4.46410 −1.23812 −0.619060 − 0.785344i \(-0.712486\pi\)
−0.619060 + 0.785344i \(0.712486\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.79315i − 0.434903i −0.976071 − 0.217451i \(-0.930226\pi\)
0.976071 − 0.217451i \(-0.0697744\pi\)
\(18\) 0 0
\(19\) − 1.73205i − 0.397360i −0.980064 − 0.198680i \(-0.936335\pi\)
0.980064 − 0.198680i \(-0.0636654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.76268 1.82715 0.913573 − 0.406675i \(-0.133312\pi\)
0.913573 + 0.406675i \(0.133312\pi\)
\(24\) 0 0
\(25\) −9.92820 −1.98564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.72741i − 1.43494i −0.696588 − 0.717472i \(-0.745299\pi\)
0.696588 − 0.717472i \(-0.254701\pi\)
\(30\) 0 0
\(31\) 7.46410i 1.34059i 0.742094 + 0.670296i \(0.233833\pi\)
−0.742094 + 0.670296i \(0.766167\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −14.4195 −2.43735
\(36\) 0 0
\(37\) −0.464102 −0.0762978 −0.0381489 − 0.999272i \(-0.512146\pi\)
−0.0381489 + 0.999272i \(0.512146\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.72741i 1.20682i 0.797432 + 0.603409i \(0.206191\pi\)
−0.797432 + 0.603409i \(0.793809\pi\)
\(42\) 0 0
\(43\) − 0.535898i − 0.0817237i −0.999165 − 0.0408619i \(-0.986990\pi\)
0.999165 − 0.0408619i \(-0.0130104\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.62158 −0.674126 −0.337063 − 0.941482i \(-0.609434\pi\)
−0.337063 + 0.941482i \(0.609434\pi\)
\(48\) 0 0
\(49\) −6.92820 −0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.58630i 0.492616i 0.969192 + 0.246308i \(0.0792175\pi\)
−0.969192 + 0.246308i \(0.920782\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.3490 1.60770 0.803850 − 0.594831i \(-0.202781\pi\)
0.803850 + 0.594831i \(0.202781\pi\)
\(60\) 0 0
\(61\) −11.3923 −1.45864 −0.729318 − 0.684175i \(-0.760162\pi\)
−0.729318 + 0.684175i \(0.760162\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 17.2480i 2.13935i
\(66\) 0 0
\(67\) 6.26795i 0.765752i 0.923800 + 0.382876i \(0.125066\pi\)
−0.923800 + 0.382876i \(0.874934\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 − 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 0 0
\(73\) −3.92820 −0.459761 −0.229881 − 0.973219i \(-0.573834\pi\)
−0.229881 + 0.973219i \(0.573834\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.86370i 0.440310i
\(78\) 0 0
\(79\) − 4.80385i − 0.540475i −0.962794 − 0.270238i \(-0.912898\pi\)
0.962794 − 0.270238i \(-0.0871022\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.07055 −0.227273 −0.113636 − 0.993522i \(-0.536250\pi\)
−0.113636 + 0.993522i \(0.536250\pi\)
\(84\) 0 0
\(85\) −6.92820 −0.751469
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.79315i − 0.190074i −0.995474 − 0.0950368i \(-0.969703\pi\)
0.995474 − 0.0950368i \(-0.0302969\pi\)
\(90\) 0 0
\(91\) 16.6603i 1.74647i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.69213 −0.686598
\(96\) 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.c.f.1727.1 8
3.2 odd 2 inner 1728.2.c.f.1727.7 8
4.3 odd 2 inner 1728.2.c.f.1727.2 8
8.3 odd 2 864.2.c.b.863.8 yes 8
8.5 even 2 864.2.c.b.863.7 yes 8
12.11 even 2 inner 1728.2.c.f.1727.8 8
24.5 odd 2 864.2.c.b.863.1 ✓ 8
24.11 even 2 864.2.c.b.863.2 yes 8
72.5 odd 6 2592.2.s.g.1727.1 8
72.11 even 6 2592.2.s.g.863.4 8
72.13 even 6 2592.2.s.g.1727.4 8
72.29 odd 6 2592.2.s.c.863.4 8
72.43 odd 6 2592.2.s.g.863.1 8
72.59 even 6 2592.2.s.c.1727.1 8
72.61 even 6 2592.2.s.c.863.1 8
72.67 odd 6 2592.2.s.c.1727.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.b.863.1 ✓ 8 24.5 odd 2
864.2.c.b.863.2 yes 8 24.11 even 2
864.2.c.b.863.7 yes 8 8.5 even 2
864.2.c.b.863.8 yes 8 8.3 odd 2
1728.2.c.f.1727.1 8 1.1 even 1 trivial
1728.2.c.f.1727.2 8 4.3 odd 2 inner
1728.2.c.f.1727.7 8 3.2 odd 2 inner
1728.2.c.f.1727.8 8 12.11 even 2 inner
2592.2.s.c.863.1 8 72.61 even 6
2592.2.s.c.863.4 8 72.29 odd 6
2592.2.s.c.1727.1 8 72.59 even 6
2592.2.s.c.1727.4 8 72.67 odd 6
2592.2.s.g.863.1 8 72.43 odd 6
2592.2.s.g.863.4 8 72.11 even 6
2592.2.s.g.1727.1 8 72.5 odd 6
2592.2.s.g.1727.4 8 72.13 even 6