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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(16.9925500817\)
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 288.4.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)
 
\(f(q)\) \(=\) \(q+(3.72474 - 3.62302i) q^{3} +(0.747449 - 26.9897i) q^{9} +(44.1186 + 25.4719i) q^{11} -131.682i q^{17} +57.2270 q^{19} +(62.5000 - 108.253i) q^{25} +(-95.0000 - 103.238i) q^{27} +(256.616 - 64.9663i) q^{33} +(-415.995 + 240.175i) q^{41} +(136.931 - 237.172i) q^{43} +(-171.500 - 297.047i) q^{49} +(-477.088 - 490.483i) q^{51} +(213.156 - 207.335i) q^{57} +(493.654 - 285.011i) q^{59} +(491.476 + 851.262i) q^{67} +1229.09 q^{73} +(-159.407 - 629.654i) q^{75} +(-727.883 - 40.3468i) q^{81} +(-590.327 - 340.825i) q^{83} +1329.36i q^{89} +(499.545 - 865.238i) q^{97} +(720.454 - 1171.71i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} - 46 q^{9} + 54 q^{11} - 212 q^{19} + 250 q^{25} - 380 q^{27} + 370 q^{33} - 1566 q^{41} - 290 q^{43} - 686 q^{49} - 1198 q^{51} + 10 q^{57} + 2538 q^{59} + 70 q^{67} + 860 q^{73} - 1250 q^{75}+ \cdots + 2000 q^{99}+O(q^{100}) \) 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
\(3\) 3.72474 − 3.62302i 0.716827 − 0.697251i
\(4\) 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
\(9\) 0.747449 − 26.9897i 0.0276833 − 0.999617i
\(10\) 0 0
\(11\) 44.1186 + 25.4719i 1.20930 + 0.698188i 0.962606 − 0.270906i \(-0.0873234\pi\)
0.246691 + 0.969094i \(0.420657\pi\)
\(12\) 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
\(17\) − 131.682i − 1.87869i −0.342978 − 0.939343i \(-0.611436\pi\)
0.342978 − 0.939343i \(-0.388564\pi\)
\(18\) 0 0
\(19\) 57.2270 0.690989 0.345494 − 0.938421i \(-0.387711\pi\)
0.345494 + 0.938421i \(0.387711\pi\)
\(20\) 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
\(25\) 62.5000 − 108.253i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) −95.0000 − 103.238i −0.677139 − 0.735855i
\(28\) 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
\(33\) 256.616 − 64.9663i 1.35367 − 0.342703i
\(34\) 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
\(41\) −415.995 + 240.175i −1.58457 + 0.914854i −0.590394 + 0.807116i \(0.701027\pi\)
−0.994179 + 0.107738i \(0.965639\pi\)
\(42\) 0 0
\(43\) 136.931 − 237.172i 0.485624 − 0.841126i −0.514239 − 0.857647i \(-0.671926\pi\)
0.999864 + 0.0165210i \(0.00525904\pi\)
\(44\) 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
\(49\) −171.500 − 297.047i −0.500000 − 0.866025i
\(50\) 0 0
\(51\) −477.088 − 490.483i −1.30992 − 1.34669i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 213.156 − 207.335i 0.495320 − 0.481792i
\(58\) 0 0
\(59\) 493.654 − 285.011i 1.08929 − 0.628904i 0.155905 − 0.987772i \(-0.450171\pi\)
0.933388 + 0.358868i \(0.116837\pi\)
\(60\) 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
\(67\) 491.476 + 851.262i 0.896170 + 1.55221i 0.832350 + 0.554250i \(0.186995\pi\)
0.0638199 + 0.997961i \(0.479672\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1229.09 1.97060 0.985301 − 0.170827i \(-0.0546438\pi\)
0.985301 + 0.170827i \(0.0546438\pi\)
\(74\) 0 0
\(75\) −159.407 − 629.654i −0.245423 − 0.969416i
\(76\) 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
\(81\) −727.883 − 40.3468i −0.998467 − 0.0553454i
\(82\) 0 0
\(83\) −590.327 − 340.825i −0.780684 − 0.450728i 0.0559884 − 0.998431i \(-0.482169\pi\)
−0.836673 + 0.547703i \(0.815502\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1329.36i 1.58328i 0.610988 + 0.791640i \(0.290773\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 499.545 − 865.238i 0.522898 − 0.905687i −0.476746 − 0.879041i \(-0.658184\pi\)
0.999645 − 0.0266459i \(-0.00848265\pi\)
\(98\) 0 0
\(99\) 720.454 − 1171.71i 0.731398 − 1.18951i
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 288.4.p.a.47.2 4
3.2 odd 2 864.4.p.a.143.1 4
4.3 odd 2 72.4.l.a.11.1 ✓ 4
8.3 odd 2 CM 288.4.p.a.47.2 4
8.5 even 2 72.4.l.a.11.1 ✓ 4
9.4 even 3 864.4.p.a.719.1 4
9.5 odd 6 inner 288.4.p.a.239.2 4
12.11 even 2 216.4.l.a.35.2 4
24.5 odd 2 216.4.l.a.35.2 4
24.11 even 2 864.4.p.a.143.1 4
36.23 even 6 72.4.l.a.59.1 yes 4
36.31 odd 6 216.4.l.a.179.2 4
72.5 odd 6 72.4.l.a.59.1 yes 4
72.13 even 6 216.4.l.a.179.2 4
72.59 even 6 inner 288.4.p.a.239.2 4
72.67 odd 6 864.4.p.a.719.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.l.a.11.1 ✓ 4 4.3 odd 2
72.4.l.a.11.1 ✓ 4 8.5 even 2
72.4.l.a.59.1 yes 4 36.23 even 6
72.4.l.a.59.1 yes 4 72.5 odd 6
216.4.l.a.35.2 4 12.11 even 2
216.4.l.a.35.2 4 24.5 odd 2
216.4.l.a.179.2 4 36.31 odd 6
216.4.l.a.179.2 4 72.13 even 6
288.4.p.a.47.2 4 1.1 even 1 trivial
288.4.p.a.47.2 4 8.3 odd 2 CM
288.4.p.a.239.2 4 9.5 odd 6 inner
288.4.p.a.239.2 4 72.59 even 6 inner
864.4.p.a.143.1 4 3.2 odd 2
864.4.p.a.143.1 4 24.11 even 2
864.4.p.a.719.1 4 9.4 even 3
864.4.p.a.719.1 4 72.67 odd 6