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Properties

Label 1728.2.r.a.289.2
Level $1728$
Weight $2$
Character 1728.289
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,0,-12,0,0,0,0,0,0,0,12,0,0,0,12] 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: \(4\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 289.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.289
Dual form 1728.2.r.a.1441.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.00000 + 1.73205i) q^{5} +(2.59808 + 1.50000i) q^{11} +(3.00000 - 1.73205i) q^{13} +3.00000 q^{17} +7.00000i q^{19} +(-1.73205 - 3.00000i) q^{23} +(3.50000 - 6.06218i) q^{25} +(-6.00000 - 3.46410i) q^{29} +(-3.46410 - 6.00000i) q^{31} +10.3923i q^{37} +(1.50000 + 2.59808i) q^{41} +(-4.33013 - 2.50000i) q^{43} +(-1.73205 + 3.00000i) q^{47} +(3.50000 + 6.06218i) q^{49} +13.8564i q^{53} -10.3923 q^{55} +(-7.79423 + 4.50000i) q^{59} +(6.00000 + 3.46410i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(4.33013 - 2.50000i) q^{67} -3.46410 q^{71} -7.00000 q^{73} +(-8.66025 + 15.0000i) q^{79} +(-10.3923 - 6.00000i) q^{83} +(-9.00000 + 5.19615i) q^{85} +6.00000 q^{89} +(-12.1244 - 21.0000i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{5} + 12 q^{13} + 12 q^{17} + 14 q^{25} - 24 q^{29} + 6 q^{41} + 14 q^{49} + 24 q^{61} - 24 q^{65} - 28 q^{73} - 36 q^{85} + 24 q^{89} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −3.00000 + 1.73205i −1.34164 + 0.774597i −0.987048 − 0.160424i \(-0.948714\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 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
\(11\) 2.59808 + 1.50000i 0.783349 + 0.452267i 0.837616 − 0.546259i \(-0.183949\pi\)
−0.0542666 + 0.998526i \(0.517282\pi\)
\(12\) 0 0
\(13\) 3.00000 − 1.73205i 0.832050 − 0.480384i −0.0225039 − 0.999747i \(-0.507164\pi\)
0.854554 + 0.519362i \(0.173830\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 7.00000i 1.60591i 0.596040 + 0.802955i \(0.296740\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.73205 − 3.00000i −0.361158 − 0.625543i 0.626994 − 0.779024i \(-0.284285\pi\)
−0.988152 + 0.153481i \(0.950952\pi\)
\(24\) 0 0
\(25\) 3.50000 − 6.06218i 0.700000 − 1.21244i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 − 3.46410i −1.11417 − 0.643268i −0.174265 − 0.984699i \(-0.555755\pi\)
−0.939907 + 0.341431i \(0.889088\pi\)
\(30\) 0 0
\(31\) −3.46410 − 6.00000i −0.622171 − 1.07763i −0.989081 − 0.147375i \(-0.952918\pi\)
0.366910 − 0.930257i \(-0.380416\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.3923i 1.70848i 0.519875 + 0.854242i \(0.325978\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.50000 + 2.59808i 0.234261 + 0.405751i 0.959058 − 0.283211i \(-0.0913998\pi\)
−0.724797 + 0.688963i \(0.758066\pi\)
\(42\) 0 0
\(43\) −4.33013 − 2.50000i −0.660338 − 0.381246i 0.132068 − 0.991241i \(-0.457838\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.73205 + 3.00000i −0.252646 + 0.437595i −0.964253 − 0.264982i \(-0.914634\pi\)
0.711608 + 0.702577i \(0.247967\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.8564i 1.90332i 0.307148 + 0.951662i \(0.400625\pi\)
−0.307148 + 0.951662i \(0.599375\pi\)
\(54\) 0 0
\(55\) −10.3923 −1.40130
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.79423 + 4.50000i −1.01472 + 0.585850i −0.912571 − 0.408919i \(-0.865906\pi\)
−0.102151 + 0.994769i \(0.532573\pi\)
\(60\) 0 0
\(61\) 6.00000 + 3.46410i 0.768221 + 0.443533i 0.832240 − 0.554416i \(-0.187058\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) 4.33013 − 2.50000i 0.529009 − 0.305424i −0.211604 − 0.977356i \(-0.567869\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.66025 + 15.0000i −0.974355 + 1.68763i −0.292306 + 0.956325i \(0.594423\pi\)
−0.682048 + 0.731307i \(0.738911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.3923 − 6.00000i −1.14070 − 0.658586i −0.194099 − 0.980982i \(-0.562178\pi\)
−0.946605 + 0.322396i \(0.895512\pi\)
\(84\) 0 0
\(85\) −9.00000 + 5.19615i −0.976187 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.1244 − 21.0000i −1.24393 − 2.15455i
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.0507673 + 0.0879316i −0.890292 − 0.455389i \(-0.849500\pi\)
0.839525 + 0.543321i \(0.182833\pi\)
\(98\) 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.a.289.2 4
3.2 odd 2 576.2.r.b.97.2 yes 4
4.3 odd 2 inner 1728.2.r.a.289.1 4
8.3 odd 2 1728.2.r.b.289.2 4
8.5 even 2 1728.2.r.b.289.1 4
9.2 odd 6 5184.2.d.c.2593.4 4
9.4 even 3 1728.2.r.b.1441.1 4
9.5 odd 6 576.2.r.a.481.1 yes 4
9.7 even 3 5184.2.d.j.2593.1 4
12.11 even 2 576.2.r.b.97.1 yes 4
24.5 odd 2 576.2.r.a.97.1 ✓ 4
24.11 even 2 576.2.r.a.97.2 yes 4
36.7 odd 6 5184.2.d.j.2593.2 4
36.11 even 6 5184.2.d.c.2593.3 4
36.23 even 6 576.2.r.a.481.2 yes 4
36.31 odd 6 1728.2.r.b.1441.2 4
72.5 odd 6 576.2.r.b.481.2 yes 4
72.11 even 6 5184.2.d.c.2593.2 4
72.13 even 6 inner 1728.2.r.a.1441.2 4
72.29 odd 6 5184.2.d.c.2593.1 4
72.43 odd 6 5184.2.d.j.2593.3 4
72.59 even 6 576.2.r.b.481.1 yes 4
72.61 even 6 5184.2.d.j.2593.4 4
72.67 odd 6 inner 1728.2.r.a.1441.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.a.97.1 ✓ 4 24.5 odd 2
576.2.r.a.97.2 yes 4 24.11 even 2
576.2.r.a.481.1 yes 4 9.5 odd 6
576.2.r.a.481.2 yes 4 36.23 even 6
576.2.r.b.97.1 yes 4 12.11 even 2
576.2.r.b.97.2 yes 4 3.2 odd 2
576.2.r.b.481.1 yes 4 72.59 even 6
576.2.r.b.481.2 yes 4 72.5 odd 6
1728.2.r.a.289.1 4 4.3 odd 2 inner
1728.2.r.a.289.2 4 1.1 even 1 trivial
1728.2.r.a.1441.1 4 72.67 odd 6 inner
1728.2.r.a.1441.2 4 72.13 even 6 inner
1728.2.r.b.289.1 4 8.5 even 2
1728.2.r.b.289.2 4 8.3 odd 2
1728.2.r.b.1441.1 4 9.4 even 3
1728.2.r.b.1441.2 4 36.31 odd 6
5184.2.d.c.2593.1 4 72.29 odd 6
5184.2.d.c.2593.2 4 72.11 even 6
5184.2.d.c.2593.3 4 36.11 even 6
5184.2.d.c.2593.4 4 9.2 odd 6
5184.2.d.j.2593.1 4 9.7 even 3
5184.2.d.j.2593.2 4 36.7 odd 6
5184.2.d.j.2593.3 4 72.43 odd 6
5184.2.d.j.2593.4 4 72.61 even 6