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Properties

Label 1728.2.r.c.289.4
Level $1728$
Weight $2$
Character 1728.289
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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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 = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 289.4
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1728.289
Dual form 1728.2.r.c.1441.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.71940 + 2.72474i) q^{11} +1.89898 q^{17} -8.34847i q^{19} +(-2.50000 + 4.33013i) q^{25} +(6.39898 + 11.0834i) q^{41} +(-2.03383 - 1.17423i) q^{43} +(3.50000 + 6.06218i) q^{49} +(8.00853 - 4.62372i) q^{59} +(12.4261 - 7.17423i) q^{67} +13.6969 q^{73} +(15.5885 + 9.00000i) q^{83} -18.0000 q^{89} +(9.84847 - 17.0580i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{17} - 20 q^{25} + 12 q^{41} + 28 q^{49} - 8 q^{73} - 144 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.71940 + 2.72474i 1.42295 + 0.821541i 0.996550 − 0.0829925i \(-0.0264478\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.89898 0.460570 0.230285 − 0.973123i \(-0.426034\pi\)
0.230285 + 0.973123i \(0.426034\pi\)
\(18\) 0 0
\(19\) − 8.34847i − 1.91527i −0.287984 − 0.957635i \(-0.592985\pi\)
0.287984 − 0.957635i \(-0.407015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.39898 + 11.0834i 0.999353 + 1.73093i 0.530831 + 0.847477i \(0.321880\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −2.03383 − 1.17423i −0.310157 − 0.179069i 0.336840 − 0.941562i \(-0.390642\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00853 − 4.62372i 1.04262 − 0.601958i 0.122047 − 0.992524i \(-0.461054\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.4261 − 7.17423i 1.51809 − 0.876472i 0.518321 − 0.855186i \(-0.326557\pi\)
0.999773 − 0.0212861i \(-0.00677610\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.6969 1.60311 0.801553 − 0.597924i \(-0.204008\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.5885 + 9.00000i 1.71106 + 0.987878i 0.933143 + 0.359506i \(0.117055\pi\)
0.777913 + 0.628372i \(0.216279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.84847 − 17.0580i 0.999961 − 1.73198i 0.492287 − 0.870433i \(-0.336161\pi\)
0.507673 − 0.861550i \(-0.330506\pi\)
\(98\) 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.c.289.4 8
3.2 odd 2 576.2.r.c.97.2 ✓ 8
4.3 odd 2 inner 1728.2.r.c.289.1 8
8.3 odd 2 CM 1728.2.r.c.289.4 8
8.5 even 2 inner 1728.2.r.c.289.1 8
9.2 odd 6 5184.2.d.l.2593.4 4
9.4 even 3 inner 1728.2.r.c.1441.1 8
9.5 odd 6 576.2.r.c.481.3 yes 8
9.7 even 3 5184.2.d.e.2593.1 4
12.11 even 2 576.2.r.c.97.3 yes 8
24.5 odd 2 576.2.r.c.97.3 yes 8
24.11 even 2 576.2.r.c.97.2 ✓ 8
36.7 odd 6 5184.2.d.e.2593.4 4
36.11 even 6 5184.2.d.l.2593.1 4
36.23 even 6 576.2.r.c.481.2 yes 8
36.31 odd 6 inner 1728.2.r.c.1441.4 8
72.5 odd 6 576.2.r.c.481.2 yes 8
72.11 even 6 5184.2.d.l.2593.4 4
72.13 even 6 inner 1728.2.r.c.1441.4 8
72.29 odd 6 5184.2.d.l.2593.1 4
72.43 odd 6 5184.2.d.e.2593.1 4
72.59 even 6 576.2.r.c.481.3 yes 8
72.61 even 6 5184.2.d.e.2593.4 4
72.67 odd 6 inner 1728.2.r.c.1441.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.c.97.2 ✓ 8 3.2 odd 2
576.2.r.c.97.2 ✓ 8 24.11 even 2
576.2.r.c.97.3 yes 8 12.11 even 2
576.2.r.c.97.3 yes 8 24.5 odd 2
576.2.r.c.481.2 yes 8 36.23 even 6
576.2.r.c.481.2 yes 8 72.5 odd 6
576.2.r.c.481.3 yes 8 9.5 odd 6
576.2.r.c.481.3 yes 8 72.59 even 6
1728.2.r.c.289.1 8 4.3 odd 2 inner
1728.2.r.c.289.1 8 8.5 even 2 inner
1728.2.r.c.289.4 8 1.1 even 1 trivial
1728.2.r.c.289.4 8 8.3 odd 2 CM
1728.2.r.c.1441.1 8 9.4 even 3 inner
1728.2.r.c.1441.1 8 72.67 odd 6 inner
1728.2.r.c.1441.4 8 36.31 odd 6 inner
1728.2.r.c.1441.4 8 72.13 even 6 inner
5184.2.d.e.2593.1 4 9.7 even 3
5184.2.d.e.2593.1 4 72.43 odd 6
5184.2.d.e.2593.4 4 36.7 odd 6
5184.2.d.e.2593.4 4 72.61 even 6
5184.2.d.l.2593.1 4 36.11 even 6
5184.2.d.l.2593.1 4 72.29 odd 6
5184.2.d.l.2593.4 4 9.2 odd 6
5184.2.d.l.2593.4 4 72.11 even 6