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Properties

Label 1728.1.o.a.1279.2
Level $1728$
Weight $1$
Character 1728.1279
Analytic conductor $0.862$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
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] = [1728,1,Mod(127,1728)] 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("1728.127"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.862384341830\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

Embedding invariants

Embedding label 1279.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1279
Dual form 1728.1.o.a.127.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(0.866025 - 0.500000i) q^{7} +(-0.866025 + 0.500000i) q^{11} +(0.500000 - 0.866025i) q^{13} +(0.866025 + 0.500000i) q^{23} +(-0.500000 - 0.866025i) q^{29} +(-0.866025 - 0.500000i) q^{31} -1.00000i q^{35} +(0.500000 - 0.866025i) q^{41} +(-0.866025 + 0.500000i) q^{43} +(-0.866025 + 0.500000i) q^{47} +1.00000i q^{55} +(0.866025 + 0.500000i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-0.500000 - 0.866025i) q^{65} +(0.866025 + 0.500000i) q^{67} -2.00000i q^{71} +(-0.500000 + 0.866025i) q^{77} +(-0.866025 + 0.500000i) q^{79} +(0.866025 - 0.500000i) q^{83} -1.00000i q^{91} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{13} - 2 q^{29} + 2 q^{41} + 2 q^{61} - 2 q^{65} - 2 q^{77} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(6\) 0 0
\(7\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −0.866025 − 0.500000i −0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.00000i − 1.00000i
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(42\) 0 0
\(43\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(66\) 0 0
\(67\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(78\) 0 0
\(79\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) − 1.00000i − 1.00000i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(98\) 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 1728.1.o.a.1279.2 4
3.2 odd 2 576.1.o.a.319.1 4
4.3 odd 2 inner 1728.1.o.a.1279.1 4
8.3 odd 2 864.1.o.a.415.1 4
8.5 even 2 864.1.o.a.415.2 4
9.2 odd 6 576.1.o.a.511.1 4
9.7 even 3 inner 1728.1.o.a.127.1 4
12.11 even 2 576.1.o.a.319.2 4
24.5 odd 2 288.1.o.a.31.2 yes 4
24.11 even 2 288.1.o.a.31.1 ✓ 4
36.7 odd 6 inner 1728.1.o.a.127.2 4
36.11 even 6 576.1.o.a.511.2 4
48.5 odd 4 2304.1.t.a.895.1 4
48.11 even 4 2304.1.t.b.895.1 4
48.29 odd 4 2304.1.t.b.895.2 4
48.35 even 4 2304.1.t.a.895.2 4
72.5 odd 6 2592.1.g.a.2431.2 2
72.11 even 6 288.1.o.a.223.1 yes 4
72.13 even 6 2592.1.g.b.2431.2 2
72.29 odd 6 288.1.o.a.223.2 yes 4
72.43 odd 6 864.1.o.a.127.2 4
72.59 even 6 2592.1.g.a.2431.1 2
72.61 even 6 864.1.o.a.127.1 4
72.67 odd 6 2592.1.g.b.2431.1 2
144.11 even 12 2304.1.t.b.1663.2 4
144.29 odd 12 2304.1.t.b.1663.1 4
144.83 even 12 2304.1.t.a.1663.1 4
144.101 odd 12 2304.1.t.a.1663.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.1.o.a.31.1 ✓ 4 24.11 even 2
288.1.o.a.31.2 yes 4 24.5 odd 2
288.1.o.a.223.1 yes 4 72.11 even 6
288.1.o.a.223.2 yes 4 72.29 odd 6
576.1.o.a.319.1 4 3.2 odd 2
576.1.o.a.319.2 4 12.11 even 2
576.1.o.a.511.1 4 9.2 odd 6
576.1.o.a.511.2 4 36.11 even 6
864.1.o.a.127.1 4 72.61 even 6
864.1.o.a.127.2 4 72.43 odd 6
864.1.o.a.415.1 4 8.3 odd 2
864.1.o.a.415.2 4 8.5 even 2
1728.1.o.a.127.1 4 9.7 even 3 inner
1728.1.o.a.127.2 4 36.7 odd 6 inner
1728.1.o.a.1279.1 4 4.3 odd 2 inner
1728.1.o.a.1279.2 4 1.1 even 1 trivial
2304.1.t.a.895.1 4 48.5 odd 4
2304.1.t.a.895.2 4 48.35 even 4
2304.1.t.a.1663.1 4 144.83 even 12
2304.1.t.a.1663.2 4 144.101 odd 12
2304.1.t.b.895.1 4 48.11 even 4
2304.1.t.b.895.2 4 48.29 odd 4
2304.1.t.b.1663.1 4 144.29 odd 12
2304.1.t.b.1663.2 4 144.11 even 12
2592.1.g.a.2431.1 2 72.59 even 6
2592.1.g.a.2431.2 2 72.5 odd 6
2592.1.g.b.2431.1 2 72.67 odd 6
2592.1.g.b.2431.2 2 72.13 even 6