Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2592,2,Mod(865,2592)] 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("2592.865"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,-4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(-1.09445 - 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 2592.865
Dual form 2592.2.i.bg.1729.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.32288 - 2.29129i) q^{5} +(-2.18890 + 3.79129i) q^{7} +(1.79129 - 3.10260i) q^{11} +(3.29129 + 5.70068i) q^{13} +1.73205 q^{17} -2.55040 q^{19} +(-3.79129 - 6.56670i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(-3.05493 + 5.29129i) q^{29} +(-4.37780 - 7.58258i) q^{31} +11.5826 q^{35} +2.58258 q^{37} +(0.913701 + 1.58258i) q^{41} +(-1.27520 + 2.20871i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-6.08258 - 10.5353i) q^{49} +1.82740 q^{53} -9.47860 q^{55} +(-4.00000 - 6.92820i) q^{59} +(0.708712 - 1.22753i) q^{61} +(8.70793 - 15.0826i) q^{65} +(-1.27520 - 2.20871i) q^{67} +0.417424 q^{71} -6.16515 q^{73} +(7.84190 + 13.5826i) q^{77} +(-4.73930 + 8.20871i) q^{79} +(-7.58258 + 13.1334i) q^{83} +(-2.29129 - 3.96863i) q^{85} -12.3151 q^{89} -28.8172 q^{91} +(3.37386 + 5.84370i) q^{95} +(2.58258 - 4.47315i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{11} + 8 q^{13} - 12 q^{23} - 8 q^{25} + 56 q^{35} - 16 q^{37} - 32 q^{47} - 12 q^{49} - 32 q^{59} + 24 q^{61} + 40 q^{71} + 24 q^{73} - 24 q^{83} - 28 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) −1.32288 − 2.29129i −0.591608 − 1.02470i −0.994016 − 0.109235i \(-0.965160\pi\)
0.402408 − 0.915460i \(-0.368173\pi\)
\(6\) 0 0
\(7\) −2.18890 + 3.79129i −0.827327 + 1.43297i 0.0728011 + 0.997346i \(0.476806\pi\)
−0.900128 + 0.435626i \(0.856527\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.79129 − 3.10260i 0.540094 − 0.935470i −0.458804 − 0.888537i \(-0.651722\pi\)
0.998898 − 0.0469323i \(-0.0149445\pi\)
\(12\) 0 0
\(13\) 3.29129 + 5.70068i 0.912839 + 1.58108i 0.810035 + 0.586382i \(0.199448\pi\)
0.102804 + 0.994702i \(0.467218\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205 0.420084 0.210042 − 0.977692i \(-0.432640\pi\)
0.210042 + 0.977692i \(0.432640\pi\)
\(18\) 0 0
\(19\) −2.55040 −0.585102 −0.292551 − 0.956250i \(-0.594504\pi\)
−0.292551 + 0.956250i \(0.594504\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.79129 − 6.56670i −0.790538 − 1.36925i −0.925634 − 0.378420i \(-0.876468\pi\)
0.135096 − 0.990833i \(-0.456866\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.05493 + 5.29129i −0.567286 + 0.982567i 0.429547 + 0.903044i \(0.358673\pi\)
−0.996833 + 0.0795232i \(0.974660\pi\)
\(30\) 0 0
\(31\) −4.37780 − 7.58258i −0.786276 − 1.36187i −0.928234 − 0.371998i \(-0.878673\pi\)
0.141957 − 0.989873i \(-0.454661\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.5826 1.95781
\(36\) 0 0
\(37\) 2.58258 0.424573 0.212286 − 0.977207i \(-0.431909\pi\)
0.212286 + 0.977207i \(0.431909\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.913701 + 1.58258i 0.142696 + 0.247157i 0.928511 − 0.371305i \(-0.121090\pi\)
−0.785815 + 0.618462i \(0.787756\pi\)
\(42\) 0 0
\(43\) −1.27520 + 2.20871i −0.194466 + 0.336825i −0.946725 − 0.322042i \(-0.895631\pi\)
0.752259 + 0.658867i \(0.228964\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 + 6.92820i −0.583460 + 1.01058i 0.411606 + 0.911362i \(0.364968\pi\)
−0.995066 + 0.0992202i \(0.968365\pi\)
\(48\) 0 0
\(49\) −6.08258 − 10.5353i −0.868939 − 1.50505i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.82740 0.251013 0.125506 − 0.992093i \(-0.459944\pi\)
0.125506 + 0.992093i \(0.459944\pi\)
\(54\) 0 0
\(55\) −9.47860 −1.27809
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 − 6.92820i −0.520756 − 0.901975i −0.999709 − 0.0241347i \(-0.992317\pi\)
0.478953 − 0.877841i \(-0.341016\pi\)
\(60\) 0 0
\(61\) 0.708712 − 1.22753i 0.0907413 − 0.157169i −0.817082 − 0.576522i \(-0.804410\pi\)
0.907823 + 0.419353i \(0.137743\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.70793 − 15.0826i 1.08009 − 1.87076i
\(66\) 0 0
\(67\) −1.27520 − 2.20871i −0.155791 − 0.269837i 0.777556 − 0.628814i \(-0.216459\pi\)
−0.933347 + 0.358976i \(0.883126\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.417424 0.0495392 0.0247696 − 0.999693i \(-0.492115\pi\)
0.0247696 + 0.999693i \(0.492115\pi\)
\(72\) 0 0
\(73\) −6.16515 −0.721576 −0.360788 − 0.932648i \(-0.617492\pi\)
−0.360788 + 0.932648i \(0.617492\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.84190 + 13.5826i 0.893668 + 1.54788i
\(78\) 0 0
\(79\) −4.73930 + 8.20871i −0.533213 + 0.923552i 0.466034 + 0.884767i \(0.345682\pi\)
−0.999248 + 0.0387857i \(0.987651\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.58258 + 13.1334i −0.832296 + 1.44158i 0.0639175 + 0.997955i \(0.479641\pi\)
−0.896213 + 0.443623i \(0.853693\pi\)
\(84\) 0 0
\(85\) −2.29129 − 3.96863i −0.248525 − 0.430458i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.3151 −1.30539 −0.652697 − 0.757619i \(-0.726362\pi\)
−0.652697 + 0.757619i \(0.726362\pi\)
\(90\) 0 0
\(91\) −28.8172 −3.02086
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.37386 + 5.84370i 0.346151 + 0.599551i
\(96\) 0 0
\(97\) 2.58258 − 4.47315i 0.262221 − 0.454180i −0.704611 − 0.709594i \(-0.748878\pi\)
0.966832 + 0.255414i \(0.0822118\pi\)
\(98\) 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 2592.2.i.bg.865.1 8
3.2 odd 2 2592.2.i.bh.865.3 8
4.3 odd 2 2592.2.i.bh.865.2 8
9.2 odd 6 2592.2.a.v.1.2 ✓ 4
9.4 even 3 inner 2592.2.i.bg.1729.1 8
9.5 odd 6 2592.2.i.bh.1729.3 8
9.7 even 3 2592.2.a.w.1.4 yes 4
12.11 even 2 inner 2592.2.i.bg.865.4 8
36.7 odd 6 2592.2.a.v.1.3 yes 4
36.11 even 6 2592.2.a.w.1.1 yes 4
36.23 even 6 inner 2592.2.i.bg.1729.4 8
36.31 odd 6 2592.2.i.bh.1729.2 8
72.11 even 6 5184.2.a.cd.1.3 4
72.29 odd 6 5184.2.a.ce.1.4 4
72.43 odd 6 5184.2.a.ce.1.1 4
72.61 even 6 5184.2.a.cd.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.v.1.2 ✓ 4 9.2 odd 6
2592.2.a.v.1.3 yes 4 36.7 odd 6
2592.2.a.w.1.1 yes 4 36.11 even 6
2592.2.a.w.1.4 yes 4 9.7 even 3
2592.2.i.bg.865.1 8 1.1 even 1 trivial
2592.2.i.bg.865.4 8 12.11 even 2 inner
2592.2.i.bg.1729.1 8 9.4 even 3 inner
2592.2.i.bg.1729.4 8 36.23 even 6 inner
2592.2.i.bh.865.2 8 4.3 odd 2
2592.2.i.bh.865.3 8 3.2 odd 2
2592.2.i.bh.1729.2 8 36.31 odd 6
2592.2.i.bh.1729.3 8 9.5 odd 6
5184.2.a.cd.1.2 4 72.61 even 6
5184.2.a.cd.1.3 4 72.11 even 6
5184.2.a.ce.1.1 4 72.43 odd 6
5184.2.a.ce.1.4 4 72.29 odd 6