Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,1,Mod(31,288)] 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("288.31"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 288.o (of order \(6\), degree \(2\), 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: \(0.143730723638\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of 16.0.739537035580145664.1

Embedding invariants

Embedding label 31.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 288.31
Dual form 288.1.o.a.223.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(0.500000 - 0.866025i) q^{5} +(0.866025 - 0.500000i) q^{7} -1.00000 q^{9} +(-0.866025 + 0.500000i) q^{11} +(-0.500000 + 0.866025i) q^{13} +(0.866025 + 0.500000i) q^{15} +(0.500000 + 0.866025i) q^{21} +(-0.866025 - 0.500000i) q^{23} -1.00000i q^{27} +(-0.500000 - 0.866025i) q^{29} +(-0.866025 - 0.500000i) q^{31} +(-0.500000 - 0.866025i) q^{33} -1.00000i q^{35} +(-0.866025 - 0.500000i) q^{39} +(-0.500000 + 0.866025i) q^{41} +(0.866025 - 0.500000i) q^{43} +(-0.500000 + 0.866025i) q^{45} +(0.866025 - 0.500000i) q^{47} +1.00000i q^{55} +(0.866025 + 0.500000i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-0.866025 + 0.500000i) q^{63} +(0.500000 + 0.866025i) q^{65} +(-0.866025 - 0.500000i) q^{67} +(0.500000 - 0.866025i) q^{69} +2.00000i q^{71} +(-0.500000 + 0.866025i) q^{77} +(-0.866025 + 0.500000i) q^{79} +1.00000 q^{81} +(0.866025 - 0.500000i) q^{83} +(0.866025 - 0.500000i) q^{87} +1.00000i q^{91} +(0.500000 - 0.866025i) q^{93} +(-0.500000 - 0.866025i) q^{97} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 4 q^{9} - 2 q^{13} + 2 q^{21} - 2 q^{29} - 2 q^{33} - 2 q^{41} - 2 q^{45} - 2 q^{61} + 2 q^{65} + 2 q^{69} - 2 q^{77} + 4 q^{81} + 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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
\(3\) 1.00000i 1.00000i
\(4\) 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
\(9\) −1.00000 −1.00000
\(10\) 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 \(0.333333\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(16\) 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
\(21\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(22\) 0 0
\(23\) −0.866025 − 0.500000i −0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 1.00000i
\(28\) 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
\(33\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(34\) 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
\(39\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(40\) 0 0
\(41\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(46\) 0 0
\(47\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 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 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(64\) 0 0
\(65\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(66\) 0 0
\(67\) −0.866025 − 0.500000i −0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(70\) 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
\(81\) 1.00000 1.00000
\(82\) 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
\(87\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(88\) 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
\(93\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(94\) 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
\(99\) 0.866025 − 0.500000i 0.866025 − 0.500000i
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 288.1.o.a.31.2 yes 4
3.2 odd 2 864.1.o.a.415.2 4
4.3 odd 2 inner 288.1.o.a.31.1 ✓ 4
8.3 odd 2 576.1.o.a.319.2 4
8.5 even 2 576.1.o.a.319.1 4
9.2 odd 6 864.1.o.a.127.1 4
9.4 even 3 2592.1.g.a.2431.2 2
9.5 odd 6 2592.1.g.b.2431.2 2
9.7 even 3 inner 288.1.o.a.223.2 yes 4
12.11 even 2 864.1.o.a.415.1 4
16.3 odd 4 2304.1.t.b.895.1 4
16.5 even 4 2304.1.t.b.895.2 4
16.11 odd 4 2304.1.t.a.895.2 4
16.13 even 4 2304.1.t.a.895.1 4
24.5 odd 2 1728.1.o.a.1279.2 4
24.11 even 2 1728.1.o.a.1279.1 4
36.7 odd 6 inner 288.1.o.a.223.1 yes 4
36.11 even 6 864.1.o.a.127.2 4
36.23 even 6 2592.1.g.b.2431.1 2
36.31 odd 6 2592.1.g.a.2431.1 2
72.11 even 6 1728.1.o.a.127.2 4
72.29 odd 6 1728.1.o.a.127.1 4
72.43 odd 6 576.1.o.a.511.2 4
72.61 even 6 576.1.o.a.511.1 4
144.43 odd 12 2304.1.t.a.1663.1 4
144.61 even 12 2304.1.t.a.1663.2 4
144.115 odd 12 2304.1.t.b.1663.2 4
144.133 even 12 2304.1.t.b.1663.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.1.o.a.31.1 ✓ 4 4.3 odd 2 inner
288.1.o.a.31.2 yes 4 1.1 even 1 trivial
288.1.o.a.223.1 yes 4 36.7 odd 6 inner
288.1.o.a.223.2 yes 4 9.7 even 3 inner
576.1.o.a.319.1 4 8.5 even 2
576.1.o.a.319.2 4 8.3 odd 2
576.1.o.a.511.1 4 72.61 even 6
576.1.o.a.511.2 4 72.43 odd 6
864.1.o.a.127.1 4 9.2 odd 6
864.1.o.a.127.2 4 36.11 even 6
864.1.o.a.415.1 4 12.11 even 2
864.1.o.a.415.2 4 3.2 odd 2
1728.1.o.a.127.1 4 72.29 odd 6
1728.1.o.a.127.2 4 72.11 even 6
1728.1.o.a.1279.1 4 24.11 even 2
1728.1.o.a.1279.2 4 24.5 odd 2
2304.1.t.a.895.1 4 16.13 even 4
2304.1.t.a.895.2 4 16.11 odd 4
2304.1.t.a.1663.1 4 144.43 odd 12
2304.1.t.a.1663.2 4 144.61 even 12
2304.1.t.b.895.1 4 16.3 odd 4
2304.1.t.b.895.2 4 16.5 even 4
2304.1.t.b.1663.1 4 144.133 even 12
2304.1.t.b.1663.2 4 144.115 odd 12
2592.1.g.a.2431.1 2 36.31 odd 6
2592.1.g.a.2431.2 2 9.4 even 3
2592.1.g.b.2431.1 2 36.23 even 6
2592.1.g.b.2431.2 2 9.5 odd 6