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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,5,Mod(127,288)] 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("288.127"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.7705493681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 288.127
Dual form 288.5.g.d.127.2

$q$-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)
 
\(f(q)\) \(=\) \(q-34.8328 q^{5} -73.6656i q^{7} +171.331i q^{11} -288.663 q^{13} +197.666 q^{17} -83.3313i q^{19} -515.988i q^{23} +588.325 q^{25} +1220.83 q^{29} +426.322i q^{31} +2565.98i q^{35} +1403.31 q^{37} +1014.35 q^{41} +2708.64i q^{43} +4273.31i q^{47} -3025.63 q^{49} +854.870 q^{53} -5967.95i q^{55} -705.263i q^{59} -537.988 q^{61} +10054.9 q^{65} -2041.24i q^{67} +8207.95i q^{71} +1179.33 q^{73} +12621.2 q^{77} +124.898i q^{79} +1860.72i q^{83} -6885.25 q^{85} -9863.14 q^{89} +21264.5i q^{91} +2902.66i q^{95} +15661.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{5} - 296 q^{13} + 576 q^{17} + 636 q^{25} + 3488 q^{29} + 1320 q^{37} + 6848 q^{41} - 3516 q^{49} + 11040 q^{53} + 424 q^{61} + 25408 q^{65} + 3000 q^{73} + 28160 q^{77} - 10368 q^{85} - 384 q^{89}+ \cdots + 11128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) −34.8328 −1.39331 −0.696656 − 0.717405i \(-0.745330\pi\)
−0.696656 + 0.717405i \(0.745330\pi\)
\(6\) 0 0
\(7\) − 73.6656i − 1.50338i −0.659516 − 0.751690i \(-0.729239\pi\)
0.659516 − 0.751690i \(-0.270761\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 171.331i 1.41596i 0.706232 + 0.707980i \(0.250394\pi\)
−0.706232 + 0.707980i \(0.749606\pi\)
\(12\) 0 0
\(13\) −288.663 −1.70806 −0.854031 − 0.520222i \(-0.825849\pi\)
−0.854031 + 0.520222i \(0.825849\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 197.666 0.683964 0.341982 − 0.939706i \(-0.388902\pi\)
0.341982 + 0.939706i \(0.388902\pi\)
\(18\) 0 0
\(19\) − 83.3313i − 0.230835i −0.993317 − 0.115417i \(-0.963179\pi\)
0.993317 − 0.115417i \(-0.0368205\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 515.988i − 0.975402i −0.873011 − 0.487701i \(-0.837836\pi\)
0.873011 − 0.487701i \(-0.162164\pi\)
\(24\) 0 0
\(25\) 588.325 0.941320
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1220.83 1.45164 0.725818 − 0.687886i \(-0.241461\pi\)
0.725818 + 0.687886i \(0.241461\pi\)
\(30\) 0 0
\(31\) 426.322i 0.443623i 0.975090 + 0.221812i \(0.0711970\pi\)
−0.975090 + 0.221812i \(0.928803\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2565.98i 2.09468i
\(36\) 0 0
\(37\) 1403.31 1.02506 0.512532 − 0.858668i \(-0.328708\pi\)
0.512532 + 0.858668i \(0.328708\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1014.35 0.603419 0.301709 − 0.953400i \(-0.402443\pi\)
0.301709 + 0.953400i \(0.402443\pi\)
\(42\) 0 0
\(43\) 2708.64i 1.46492i 0.680808 + 0.732462i \(0.261629\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4273.31i 1.93450i 0.253824 + 0.967250i \(0.418312\pi\)
−0.253824 + 0.967250i \(0.581688\pi\)
\(48\) 0 0
\(49\) −3025.63 −1.26015
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 854.870 0.304333 0.152166 − 0.988355i \(-0.451375\pi\)
0.152166 + 0.988355i \(0.451375\pi\)
\(54\) 0 0
\(55\) − 5967.95i − 1.97288i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 705.263i − 0.202604i −0.994856 − 0.101302i \(-0.967699\pi\)
0.994856 − 0.101302i \(-0.0323008\pi\)
\(60\) 0 0
\(61\) −537.988 −0.144581 −0.0722907 − 0.997384i \(-0.523031\pi\)
−0.0722907 + 0.997384i \(0.523031\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10054.9 2.37986
\(66\) 0 0
\(67\) − 2041.24i − 0.454720i −0.973811 − 0.227360i \(-0.926991\pi\)
0.973811 − 0.227360i \(-0.0730094\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8207.95i 1.62824i 0.580698 + 0.814119i \(0.302780\pi\)
−0.580698 + 0.814119i \(0.697220\pi\)
\(72\) 0 0
\(73\) 1179.33 0.221303 0.110652 − 0.993859i \(-0.464706\pi\)
0.110652 + 0.993859i \(0.464706\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12621.2 2.12873
\(78\) 0 0
\(79\) 124.898i 0.0200124i 0.999950 + 0.0100062i \(0.00318513\pi\)
−0.999950 + 0.0100062i \(0.996815\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1860.72i 0.270100i 0.990839 + 0.135050i \(0.0431195\pi\)
−0.990839 + 0.135050i \(0.956881\pi\)
\(84\) 0 0
\(85\) −6885.25 −0.952976
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9863.14 −1.24519 −0.622595 − 0.782544i \(-0.713921\pi\)
−0.622595 + 0.782544i \(0.713921\pi\)
\(90\) 0 0
\(91\) 21264.5i 2.56787i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2902.66i 0.321625i
\(96\) 0 0
\(97\) 15661.8 1.66455 0.832275 − 0.554363i \(-0.187038\pi\)
0.832275 + 0.554363i \(0.187038\pi\)
\(98\) 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 288.5.g.d.127.1 ✓ 4
3.2 odd 2 288.5.g.e.127.3 yes 4
4.3 odd 2 inner 288.5.g.d.127.2 yes 4
8.3 odd 2 576.5.g.n.127.4 4
8.5 even 2 576.5.g.n.127.3 4
12.11 even 2 288.5.g.e.127.4 yes 4
24.5 odd 2 576.5.g.k.127.1 4
24.11 even 2 576.5.g.k.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.5.g.d.127.1 ✓ 4 1.1 even 1 trivial
288.5.g.d.127.2 yes 4 4.3 odd 2 inner
288.5.g.e.127.3 yes 4 3.2 odd 2
288.5.g.e.127.4 yes 4 12.11 even 2
576.5.g.k.127.1 4 24.5 odd 2
576.5.g.k.127.2 4 24.11 even 2
576.5.g.n.127.3 4 8.5 even 2
576.5.g.n.127.4 4 8.3 odd 2