Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(289,864)] 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("864.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.170772624.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.4
Root \(0.335728 + 1.37379i\) of defining polynomial
Character \(\chi\) \(=\) 864.577
Dual form 864.2.i.f.289.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.18614 - 2.05446i) q^{5} +(1.10489 + 1.91373i) q^{7} +(2.96790 + 5.14055i) q^{11} +(-2.18614 + 3.78651i) q^{13} -3.37228 q^{17} +3.72601 q^{19} +(-1.10489 + 1.91373i) q^{23} +(-0.313859 - 0.543620i) q^{25} +(0.186141 + 0.322405i) q^{29} +(4.83090 - 8.36737i) q^{31} +5.24224 q^{35} +4.00000 q^{37} +(0.500000 - 0.866025i) q^{41} +(2.96790 + 5.14055i) q^{43} +(-1.10489 - 1.91373i) q^{47} +(1.05842 - 1.83324i) q^{49} +4.00000 q^{53} +14.0814 q^{55} +(-5.17769 + 8.96801i) q^{59} +(-7.55842 - 13.0916i) q^{61} +(5.18614 + 8.98266i) q^{65} +(-5.17769 + 8.96801i) q^{67} +4.41957 q^{71} +4.62772 q^{73} +(-6.55842 + 11.3595i) q^{77} +(-4.83090 - 8.36737i) q^{79} +(7.04069 + 12.1948i) q^{83} +(-4.00000 + 6.92820i) q^{85} -1.25544 q^{89} -9.66181 q^{91} +(4.41957 - 7.65492i) q^{95} +(-4.50000 - 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 6 q^{13} - 4 q^{17} - 14 q^{25} - 10 q^{29} + 32 q^{37} + 4 q^{41} - 26 q^{49} + 32 q^{53} - 26 q^{61} + 30 q^{65} + 60 q^{73} - 18 q^{77} - 32 q^{85} - 56 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.18614 − 2.05446i 0.530458 − 0.918781i −0.468910 − 0.883246i \(-0.655353\pi\)
0.999368 − 0.0355348i \(-0.0113134\pi\)
\(6\) 0 0
\(7\) 1.10489 + 1.91373i 0.417610 + 0.723322i 0.995699 − 0.0926519i \(-0.0295344\pi\)
−0.578088 + 0.815974i \(0.696201\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.96790 + 5.14055i 0.894855 + 1.54993i 0.833984 + 0.551789i \(0.186055\pi\)
0.0608712 + 0.998146i \(0.480612\pi\)
\(12\) 0 0
\(13\) −2.18614 + 3.78651i −0.606326 + 1.05019i 0.385514 + 0.922702i \(0.374024\pi\)
−0.991840 + 0.127486i \(0.959309\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.37228 −0.817898 −0.408949 − 0.912557i \(-0.634105\pi\)
−0.408949 + 0.912557i \(0.634105\pi\)
\(18\) 0 0
\(19\) 3.72601 0.854805 0.427403 − 0.904061i \(-0.359429\pi\)
0.427403 + 0.904061i \(0.359429\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.10489 + 1.91373i −0.230386 + 0.399041i −0.957922 − 0.287029i \(-0.907332\pi\)
0.727536 + 0.686070i \(0.240666\pi\)
\(24\) 0 0
\(25\) −0.313859 − 0.543620i −0.0627719 − 0.108724i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.186141 + 0.322405i 0.0345655 + 0.0598691i 0.882791 − 0.469767i \(-0.155662\pi\)
−0.848225 + 0.529636i \(0.822329\pi\)
\(30\) 0 0
\(31\) 4.83090 − 8.36737i 0.867656 − 1.50282i 0.00327038 − 0.999995i \(-0.498959\pi\)
0.864386 − 0.502830i \(-0.167708\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.24224 0.886099
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 − 0.866025i 0.0780869 − 0.135250i −0.824338 − 0.566099i \(-0.808452\pi\)
0.902424 + 0.430848i \(0.141786\pi\)
\(42\) 0 0
\(43\) 2.96790 + 5.14055i 0.452600 + 0.783927i 0.998547 − 0.0538934i \(-0.0171631\pi\)
−0.545946 + 0.837820i \(0.683830\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.10489 − 1.91373i −0.161165 − 0.279146i 0.774122 − 0.633037i \(-0.218192\pi\)
−0.935287 + 0.353891i \(0.884859\pi\)
\(48\) 0 0
\(49\) 1.05842 − 1.83324i 0.151203 − 0.261892i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 14.0814 1.89873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.17769 + 8.96801i −0.674077 + 1.16754i 0.302661 + 0.953098i \(0.402125\pi\)
−0.976738 + 0.214437i \(0.931208\pi\)
\(60\) 0 0
\(61\) −7.55842 − 13.0916i −0.967757 − 1.67620i −0.702019 − 0.712159i \(-0.747718\pi\)
−0.265738 − 0.964045i \(-0.585616\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.18614 + 8.98266i 0.643262 + 1.11416i
\(66\) 0 0
\(67\) −5.17769 + 8.96801i −0.632555 + 1.09562i 0.354473 + 0.935066i \(0.384660\pi\)
−0.987028 + 0.160551i \(0.948673\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.41957 0.524507 0.262253 − 0.964999i \(-0.415534\pi\)
0.262253 + 0.964999i \(0.415534\pi\)
\(72\) 0 0
\(73\) 4.62772 0.541634 0.270817 − 0.962631i \(-0.412706\pi\)
0.270817 + 0.962631i \(0.412706\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.55842 + 11.3595i −0.747402 + 1.29454i
\(78\) 0 0
\(79\) −4.83090 − 8.36737i −0.543519 − 0.941403i −0.998698 − 0.0510030i \(-0.983758\pi\)
0.455179 − 0.890400i \(-0.349575\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.04069 + 12.1948i 0.772816 + 1.33856i 0.936014 + 0.351963i \(0.114486\pi\)
−0.163198 + 0.986593i \(0.552181\pi\)
\(84\) 0 0
\(85\) −4.00000 + 6.92820i −0.433861 + 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.25544 −0.133076 −0.0665380 − 0.997784i \(-0.521195\pi\)
−0.0665380 + 0.997784i \(0.521195\pi\)
\(90\) 0 0
\(91\) −9.66181 −1.01283
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.41957 − 7.65492i 0.453439 − 0.785379i
\(96\) 0 0
\(97\) −4.50000 − 7.79423i −0.456906 − 0.791384i 0.541890 − 0.840450i \(-0.317709\pi\)
−0.998796 + 0.0490655i \(0.984376\pi\)
\(98\) 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 864.2.i.f.577.4 8
3.2 odd 2 288.2.i.f.193.4 yes 8
4.3 odd 2 inner 864.2.i.f.577.3 8
8.3 odd 2 1728.2.i.n.577.1 8
8.5 even 2 1728.2.i.n.577.2 8
9.2 odd 6 288.2.i.f.97.4 yes 8
9.4 even 3 2592.2.a.x.1.1 4
9.5 odd 6 2592.2.a.u.1.3 4
9.7 even 3 inner 864.2.i.f.289.4 8
12.11 even 2 288.2.i.f.193.1 yes 8
24.5 odd 2 576.2.i.n.193.1 8
24.11 even 2 576.2.i.n.193.4 8
36.7 odd 6 inner 864.2.i.f.289.3 8
36.11 even 6 288.2.i.f.97.1 ✓ 8
36.23 even 6 2592.2.a.u.1.4 4
36.31 odd 6 2592.2.a.x.1.2 4
72.5 odd 6 5184.2.a.cf.1.1 4
72.11 even 6 576.2.i.n.385.4 8
72.13 even 6 5184.2.a.cc.1.3 4
72.29 odd 6 576.2.i.n.385.1 8
72.43 odd 6 1728.2.i.n.1153.1 8
72.59 even 6 5184.2.a.cf.1.2 4
72.61 even 6 1728.2.i.n.1153.2 8
72.67 odd 6 5184.2.a.cc.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.f.97.1 ✓ 8 36.11 even 6
288.2.i.f.97.4 yes 8 9.2 odd 6
288.2.i.f.193.1 yes 8 12.11 even 2
288.2.i.f.193.4 yes 8 3.2 odd 2
576.2.i.n.193.1 8 24.5 odd 2
576.2.i.n.193.4 8 24.11 even 2
576.2.i.n.385.1 8 72.29 odd 6
576.2.i.n.385.4 8 72.11 even 6
864.2.i.f.289.3 8 36.7 odd 6 inner
864.2.i.f.289.4 8 9.7 even 3 inner
864.2.i.f.577.3 8 4.3 odd 2 inner
864.2.i.f.577.4 8 1.1 even 1 trivial
1728.2.i.n.577.1 8 8.3 odd 2
1728.2.i.n.577.2 8 8.5 even 2
1728.2.i.n.1153.1 8 72.43 odd 6
1728.2.i.n.1153.2 8 72.61 even 6
2592.2.a.u.1.3 4 9.5 odd 6
2592.2.a.u.1.4 4 36.23 even 6
2592.2.a.x.1.1 4 9.4 even 3
2592.2.a.x.1.2 4 36.31 odd 6
5184.2.a.cc.1.3 4 72.13 even 6
5184.2.a.cc.1.4 4 72.67 odd 6
5184.2.a.cf.1.1 4 72.5 odd 6
5184.2.a.cf.1.2 4 72.59 even 6