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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 239.1
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 288.239
Dual form 288.4.p.a.47.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.27526 + 5.03723i) q^{3} +(-23.7474 + 12.8475i) q^{9} +(-17.1186 + 9.88344i) q^{11} +24.2022i q^{17} -163.227 q^{19} +(62.5000 + 108.253i) q^{25} +(-95.0000 - 103.238i) q^{27} +(-71.6158 - 73.6266i) q^{33} +(-367.005 - 211.890i) q^{41} +(-281.931 - 488.319i) q^{43} +(-171.500 + 297.047i) q^{49} +(-121.912 + 30.8639i) q^{51} +(-208.156 - 822.213i) q^{57} +(775.346 + 447.646i) q^{59} +(-456.476 + 790.640i) q^{67} -799.089 q^{73} +(-465.593 + 452.878i) q^{75} +(398.883 - 610.191i) q^{81} +(590.327 - 340.825i) q^{83} +1329.36i q^{89} +(455.455 + 788.870i) q^{97} +(279.546 - 454.638i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} - 46 q^{9} + 54 q^{11} - 212 q^{19} + 250 q^{25} - 380 q^{27} + 370 q^{33} - 1566 q^{41} - 290 q^{43} - 686 q^{49} - 1198 q^{51} + 10 q^{57} + 2538 q^{59} + 70 q^{67} + 860 q^{73} - 1250 q^{75}+ \cdots + 2000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) 1.27526 + 5.03723i 0.245423 + 0.969416i
\(4\) 0 0
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) −23.7474 + 12.8475i −0.879535 + 0.475834i
\(10\) 0 0
\(11\) −17.1186 + 9.88344i −0.469224 + 0.270906i −0.715915 − 0.698188i \(-0.753990\pi\)
0.246691 + 0.969094i \(0.420657\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 24.2022i 0.345288i 0.984984 + 0.172644i \(0.0552309\pi\)
−0.984984 + 0.172644i \(0.944769\pi\)
\(18\) 0 0
\(19\) −163.227 −1.97089 −0.985443 − 0.170004i \(-0.945622\pi\)
−0.985443 + 0.170004i \(0.945622\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) −71.6158 − 73.6266i −0.377779 − 0.388386i
\(34\) 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
\(41\) −367.005 − 211.890i −1.39797 − 0.807116i −0.403786 − 0.914854i \(-0.632306\pi\)
−0.994179 + 0.107738i \(0.965639\pi\)
\(42\) 0 0
\(43\) −281.931 − 488.319i −0.999864 − 1.73181i −0.514239 − 0.857647i \(-0.671926\pi\)
−0.485624 − 0.874168i \(-0.661408\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −171.500 + 297.047i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) −121.912 + 30.8639i −0.334727 + 0.0847415i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −208.156 − 822.213i −0.483701 − 1.91061i
\(58\) 0 0
\(59\) 775.346 + 447.646i 1.71087 + 0.987772i 0.933388 + 0.358868i \(0.116837\pi\)
0.777483 + 0.628904i \(0.216496\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −456.476 + 790.640i −0.832350 + 1.44167i 0.0638199 + 0.997961i \(0.479672\pi\)
−0.896170 + 0.443711i \(0.853662\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −799.089 −1.28118 −0.640591 − 0.767882i \(-0.721311\pi\)
−0.640591 + 0.767882i \(0.721311\pi\)
\(74\) 0 0
\(75\) −465.593 + 452.878i −0.716827 + 0.697251i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 398.883 − 610.191i 0.547164 − 0.837025i
\(82\) 0 0
\(83\) 590.327 − 340.825i 0.780684 − 0.450728i −0.0559884 − 0.998431i \(-0.517831\pi\)
0.836673 + 0.547703i \(0.184498\pi\)
\(84\) 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
\(97\) 455.455 + 788.870i 0.476746 + 0.825749i 0.999645 − 0.0266459i \(-0.00848265\pi\)
−0.522898 + 0.852395i \(0.675149\pi\)
\(98\) 0 0
\(99\) 279.546 − 454.638i 0.283792 − 0.461544i
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.239.1 4
3.2 odd 2 864.4.p.a.719.2 4
4.3 odd 2 72.4.l.a.59.2 yes 4
8.3 odd 2 CM 288.4.p.a.239.1 4
8.5 even 2 72.4.l.a.59.2 yes 4
9.2 odd 6 inner 288.4.p.a.47.1 4
9.7 even 3 864.4.p.a.143.2 4
12.11 even 2 216.4.l.a.179.1 4
24.5 odd 2 216.4.l.a.179.1 4
24.11 even 2 864.4.p.a.719.2 4
36.7 odd 6 216.4.l.a.35.1 4
36.11 even 6 72.4.l.a.11.2 ✓ 4
72.11 even 6 inner 288.4.p.a.47.1 4
72.29 odd 6 72.4.l.a.11.2 ✓ 4
72.43 odd 6 864.4.p.a.143.2 4
72.61 even 6 216.4.l.a.35.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.l.a.11.2 ✓ 4 36.11 even 6
72.4.l.a.11.2 ✓ 4 72.29 odd 6
72.4.l.a.59.2 yes 4 4.3 odd 2
72.4.l.a.59.2 yes 4 8.5 even 2
216.4.l.a.35.1 4 36.7 odd 6
216.4.l.a.35.1 4 72.61 even 6
216.4.l.a.179.1 4 12.11 even 2
216.4.l.a.179.1 4 24.5 odd 2
288.4.p.a.47.1 4 9.2 odd 6 inner
288.4.p.a.47.1 4 72.11 even 6 inner
288.4.p.a.239.1 4 1.1 even 1 trivial
288.4.p.a.239.1 4 8.3 odd 2 CM
864.4.p.a.143.2 4 9.7 even 3
864.4.p.a.143.2 4 72.43 odd 6
864.4.p.a.719.2 4 3.2 odd 2
864.4.p.a.719.2 4 24.11 even 2