Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,10,Mod(1,288)] 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("288.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4288,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(148.330320815\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 555x^{2} + 14000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{5}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-22.9894\) of defining polynomial
Character \(\chi\) \(=\) 288.1

$q$-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)
 
\(f(q)\) \(=\) \(q+132.850 q^{5} +4801.33 q^{7} +60245.9 q^{11} +128542. q^{13} -518306. q^{17} -1.08877e6 q^{19} -2.31357e6 q^{23} -1.93548e6 q^{25} +3.12467e6 q^{29} +3.39227e6 q^{31} +637858. q^{35} +7.76108e6 q^{37} +1.81304e6 q^{41} +8.20315e6 q^{43} -2.97857e7 q^{47} -1.73008e7 q^{49} -4.81930e7 q^{53} +8.00368e6 q^{55} +1.01052e8 q^{59} -6.26936e7 q^{61} +1.70768e7 q^{65} -2.74029e8 q^{67} +2.45525e8 q^{71} +2.60358e8 q^{73} +2.89260e8 q^{77} +4.88935e8 q^{79} -7.25779e8 q^{83} -6.88570e7 q^{85} -6.96452e8 q^{89} +6.17173e8 q^{91} -1.44643e8 q^{95} -9.39522e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4288 q^{5} + 128616 q^{13} - 386432 q^{17} + 2590892 q^{25} + 1245376 q^{29} + 53020792 q^{37} + 16360832 q^{41} + 141308132 q^{49} - 143107904 q^{53} + 71932760 q^{61} + 326656128 q^{65} - 254795496 q^{73}+ \cdots - 2467261512 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 132.850 0.0950599 0.0475299 − 0.998870i \(-0.484865\pi\)
0.0475299 + 0.998870i \(0.484865\pi\)
\(6\) 0 0
\(7\) 4801.33 0.755824 0.377912 − 0.925842i \(-0.376642\pi\)
0.377912 + 0.925842i \(0.376642\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 60245.9 1.24068 0.620341 − 0.784332i \(-0.286994\pi\)
0.620341 + 0.784332i \(0.286994\pi\)
\(12\) 0 0
\(13\) 128542. 1.24825 0.624123 − 0.781326i \(-0.285457\pi\)
0.624123 + 0.781326i \(0.285457\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −518306. −1.50510 −0.752551 − 0.658534i \(-0.771177\pi\)
−0.752551 + 0.658534i \(0.771177\pi\)
\(18\) 0 0
\(19\) −1.08877e6 −1.91666 −0.958329 − 0.285668i \(-0.907785\pi\)
−0.958329 + 0.285668i \(0.907785\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.31357e6 −1.72388 −0.861940 − 0.507011i \(-0.830750\pi\)
−0.861940 + 0.507011i \(0.830750\pi\)
\(24\) 0 0
\(25\) −1.93548e6 −0.990964
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.12467e6 0.820376 0.410188 − 0.912001i \(-0.365463\pi\)
0.410188 + 0.912001i \(0.365463\pi\)
\(30\) 0 0
\(31\) 3.39227e6 0.659726 0.329863 − 0.944029i \(-0.392998\pi\)
0.329863 + 0.944029i \(0.392998\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 637858. 0.0718485
\(36\) 0 0
\(37\) 7.76108e6 0.680792 0.340396 − 0.940282i \(-0.389439\pi\)
0.340396 + 0.940282i \(0.389439\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.81304e6 0.100203 0.0501015 − 0.998744i \(-0.484046\pi\)
0.0501015 + 0.998744i \(0.484046\pi\)
\(42\) 0 0
\(43\) 8.20315e6 0.365908 0.182954 − 0.983121i \(-0.441434\pi\)
0.182954 + 0.983121i \(0.441434\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.97857e7 −0.890363 −0.445182 − 0.895440i \(-0.646861\pi\)
−0.445182 + 0.895440i \(0.646861\pi\)
\(48\) 0 0
\(49\) −1.73008e7 −0.428731
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.81930e7 −0.838962 −0.419481 − 0.907764i \(-0.637788\pi\)
−0.419481 + 0.907764i \(0.637788\pi\)
\(54\) 0 0
\(55\) 8.00368e6 0.117939
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.01052e8 1.08570 0.542849 − 0.839830i \(-0.317345\pi\)
0.542849 + 0.839830i \(0.317345\pi\)
\(60\) 0 0
\(61\) −6.26936e7 −0.579747 −0.289874 − 0.957065i \(-0.593613\pi\)
−0.289874 + 0.957065i \(0.593613\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.70768e7 0.118658
\(66\) 0 0
\(67\) −2.74029e8 −1.66135 −0.830674 − 0.556759i \(-0.812045\pi\)
−0.830674 + 0.556759i \(0.812045\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.45525e8 1.14666 0.573328 − 0.819326i \(-0.305652\pi\)
0.573328 + 0.819326i \(0.305652\pi\)
\(72\) 0 0
\(73\) 2.60358e8 1.07304 0.536522 − 0.843886i \(-0.319738\pi\)
0.536522 + 0.843886i \(0.319738\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.89260e8 0.937736
\(78\) 0 0
\(79\) 4.88935e8 1.41231 0.706154 − 0.708058i \(-0.250429\pi\)
0.706154 + 0.708058i \(0.250429\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.25779e8 −1.67862 −0.839311 − 0.543652i \(-0.817041\pi\)
−0.839311 + 0.543652i \(0.817041\pi\)
\(84\) 0 0
\(85\) −6.88570e7 −0.143075
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.96452e8 −1.17662 −0.588310 − 0.808636i \(-0.700206\pi\)
−0.588310 + 0.808636i \(0.700206\pi\)
\(90\) 0 0
\(91\) 6.17173e8 0.943454
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.44643e8 −0.182197
\(96\) 0 0
\(97\) −9.39522e8 −1.07754 −0.538771 − 0.842452i \(-0.681111\pi\)
−0.538771 + 0.842452i \(0.681111\pi\)
\(98\) 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 288.10.a.p.1.4 yes 4
3.2 odd 2 288.10.a.s.1.2 yes 4
4.3 odd 2 inner 288.10.a.p.1.3 ✓ 4
12.11 even 2 288.10.a.s.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.10.a.p.1.3 ✓ 4 4.3 odd 2 inner
288.10.a.p.1.4 yes 4 1.1 even 1 trivial
288.10.a.s.1.1 yes 4 12.11 even 2
288.10.a.s.1.2 yes 4 3.2 odd 2