Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,2,Mod(47,288)] 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("288.47"); S:= CuspForms(chi, 2); 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, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 47.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 288.47
Dual form 288.2.p.a.239.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.724745 + 1.57313i) q^{3} +(-1.94949 + 2.28024i) q^{9} +(5.72474 + 3.30518i) q^{11} +2.36773i q^{17} -6.34847 q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 - 1.41421i) q^{27} +(-1.05051 + 11.4012i) q^{33} +(9.39898 - 5.42650i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-3.50000 - 6.06218i) q^{49} +(-3.72474 + 1.71600i) q^{51} +(-4.60102 - 9.98698i) q^{57} +(1.62372 - 0.937458i) q^{59} +(0.174235 + 0.301783i) q^{67} -15.6969 q^{73} +(8.62372 + 0.794593i) q^{75} +(-1.39898 - 8.89060i) q^{81} +(-2.44949 - 1.41421i) q^{83} +5.65685i q^{89} +(4.84847 - 8.39780i) q^{97} +(-18.6969 + 6.61037i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{9} + 18 q^{11} + 4 q^{19} + 10 q^{25} - 20 q^{27} - 14 q^{33} + 18 q^{41} + 10 q^{43} - 14 q^{49} - 10 q^{51} - 38 q^{57} - 18 q^{59} - 14 q^{67} - 4 q^{73} + 10 q^{75} + 14 q^{81}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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
\(3\) 0.724745 + 1.57313i 0.418432 + 0.908248i
\(4\) 0 0
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −1.94949 + 2.28024i −0.649830 + 0.760080i
\(10\) 0 0
\(11\) 5.72474 + 3.30518i 1.72608 + 0.996550i 0.904534 + 0.426401i \(0.140219\pi\)
0.821541 + 0.570149i \(0.193114\pi\)
\(12\) 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
\(17\) 2.36773i 0.574258i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) −6.34847 −1.45644 −0.728219 − 0.685344i \(-0.759652\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) 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
\(27\) −5.00000 − 1.41421i −0.962250 − 0.272166i
\(28\) 0 0
\(29\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) −1.05051 + 11.4012i −0.182870 + 1.98469i
\(34\) 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
\(41\) 9.39898 − 5.42650i 1.46787 − 0.847477i 0.468521 − 0.883452i \(-0.344787\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) 6.17423 − 10.6941i 0.941562 − 1.63083i 0.179069 − 0.983836i \(-0.442691\pi\)
0.762493 − 0.646997i \(-0.223975\pi\)
\(44\) 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
\(51\) −3.72474 + 1.71600i −0.521569 + 0.240288i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.60102 − 9.98698i −0.609420 − 1.32281i
\(58\) 0 0
\(59\) 1.62372 − 0.937458i 0.211391 − 0.122047i −0.390567 − 0.920575i \(-0.627721\pi\)
0.601958 + 0.798528i \(0.294388\pi\)
\(60\) 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
\(67\) 0.174235 + 0.301783i 0.0212861 + 0.0368687i 0.876472 − 0.481452i \(-0.159891\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −15.6969 −1.83719 −0.918594 − 0.395203i \(-0.870674\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 8.62372 + 0.794593i 0.995782 + 0.0917517i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −1.39898 − 8.89060i −0.155442 − 0.987845i
\(82\) 0 0
\(83\) −2.44949 − 1.41421i −0.268866 − 0.155230i 0.359506 − 0.933143i \(-0.382945\pi\)
−0.628372 + 0.777913i \(0.716279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−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
\(97\) 4.84847 − 8.39780i 0.492287 − 0.852667i −0.507673 − 0.861550i \(-0.669494\pi\)
0.999961 + 0.00888289i \(0.00282755\pi\)
\(98\) 0 0
\(99\) −18.6969 + 6.61037i −1.87911 + 0.664367i
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 288.2.p.a.47.2 4
3.2 odd 2 864.2.p.a.143.1 4
4.3 odd 2 72.2.l.a.11.2 ✓ 4
8.3 odd 2 CM 288.2.p.a.47.2 4
8.5 even 2 72.2.l.a.11.2 ✓ 4
9.2 odd 6 2592.2.f.a.1295.4 4
9.4 even 3 864.2.p.a.719.1 4
9.5 odd 6 inner 288.2.p.a.239.2 4
9.7 even 3 2592.2.f.a.1295.1 4
12.11 even 2 216.2.l.a.35.1 4
24.5 odd 2 216.2.l.a.35.1 4
24.11 even 2 864.2.p.a.143.1 4
36.7 odd 6 648.2.f.a.323.2 4
36.11 even 6 648.2.f.a.323.3 4
36.23 even 6 72.2.l.a.59.2 yes 4
36.31 odd 6 216.2.l.a.179.1 4
72.5 odd 6 72.2.l.a.59.2 yes 4
72.11 even 6 2592.2.f.a.1295.4 4
72.13 even 6 216.2.l.a.179.1 4
72.29 odd 6 648.2.f.a.323.3 4
72.43 odd 6 2592.2.f.a.1295.1 4
72.59 even 6 inner 288.2.p.a.239.2 4
72.61 even 6 648.2.f.a.323.2 4
72.67 odd 6 864.2.p.a.719.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.l.a.11.2 ✓ 4 4.3 odd 2
72.2.l.a.11.2 ✓ 4 8.5 even 2
72.2.l.a.59.2 yes 4 36.23 even 6
72.2.l.a.59.2 yes 4 72.5 odd 6
216.2.l.a.35.1 4 12.11 even 2
216.2.l.a.35.1 4 24.5 odd 2
216.2.l.a.179.1 4 36.31 odd 6
216.2.l.a.179.1 4 72.13 even 6
288.2.p.a.47.2 4 1.1 even 1 trivial
288.2.p.a.47.2 4 8.3 odd 2 CM
288.2.p.a.239.2 4 9.5 odd 6 inner
288.2.p.a.239.2 4 72.59 even 6 inner
648.2.f.a.323.2 4 36.7 odd 6
648.2.f.a.323.2 4 72.61 even 6
648.2.f.a.323.3 4 36.11 even 6
648.2.f.a.323.3 4 72.29 odd 6
864.2.p.a.143.1 4 3.2 odd 2
864.2.p.a.143.1 4 24.11 even 2
864.2.p.a.719.1 4 9.4 even 3
864.2.p.a.719.1 4 72.67 odd 6
2592.2.f.a.1295.1 4 9.7 even 3
2592.2.f.a.1295.1 4 72.43 odd 6
2592.2.f.a.1295.4 4 9.2 odd 6
2592.2.f.a.1295.4 4 72.11 even 6