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 1729.3
Root \(1.09445 - 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 2592.1729
Dual form 2592.2.i.bh.865.3

$q$-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 \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

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

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
\(5\) 1.32288 − 2.29129i 0.591608 − 1.02470i −0.402408 − 0.915460i \(-0.631827\pi\)
0.994016 − 0.109235i \(-0.0348400\pi\)
\(6\) 0 0
\(7\) −2.18890 − 3.79129i −0.827327 − 1.43297i −0.900128 − 0.435626i \(-0.856527\pi\)
0.0728011 − 0.997346i \(-0.476806\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.79129 − 3.10260i −0.540094 − 0.935470i −0.998898 − 0.0469323i \(-0.985055\pi\)
0.458804 − 0.888537i \(-0.348278\pi\)
\(12\) 0 0
\(13\) 3.29129 − 5.70068i 0.912839 − 1.58108i 0.102804 − 0.994702i \(-0.467218\pi\)
0.810035 − 0.586382i \(-0.199448\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.73205 −0.420084 −0.210042 − 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 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 −0.135096 − 0.990833i \(-0.543134\pi\)
0.925634 − 0.378420i \(-0.123532\pi\)
\(24\) 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 0.996833 + 0.0795232i \(0.0253398\pi\)
−0.429547 + 0.903044i \(0.641327\pi\)
\(30\) 0 0
\(31\) −4.37780 + 7.58258i −0.786276 + 1.36187i 0.141957 + 0.989873i \(0.454661\pi\)
−0.928234 + 0.371998i \(0.878673\pi\)
\(32\) 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 \(-0.878910\pi\)
0.785815 + 0.618462i \(0.212244\pi\)
\(42\) 0 0
\(43\) −1.27520 − 2.20871i −0.194466 − 0.336825i 0.752259 − 0.658867i \(-0.228964\pi\)
−0.946725 + 0.322042i \(0.895631\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 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 \(-0.540056\pi\)
−0.125506 + 0.992093i \(0.540056\pi\)
\(54\) 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 −0.478953 − 0.877841i \(-0.658984\pi\)
0.999709 − 0.0241347i \(-0.00768307\pi\)
\(60\) 0 0
\(61\) 0.708712 + 1.22753i 0.0907413 + 0.157169i 0.907823 − 0.419353i \(-0.137743\pi\)
−0.817082 + 0.576522i \(0.804410\pi\)
\(62\) 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 −0.933347 − 0.358976i \(-0.883126\pi\)
0.777556 + 0.628814i \(0.216459\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.417424 −0.0495392 −0.0247696 − 0.999693i \(-0.507885\pi\)
−0.0247696 + 0.999693i \(0.507885\pi\)
\(72\) 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 −0.999248 − 0.0387857i \(-0.987651\pi\)
0.466034 − 0.884767i \(-0.345682\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.58258 + 13.1334i 0.832296 + 1.44158i 0.896213 + 0.443623i \(0.146307\pi\)
−0.0639175 + 0.997955i \(0.520359\pi\)
\(84\) 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 \(-0.273638\pi\)
0.652697 + 0.757619i \(0.273638\pi\)
\(90\) 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 0.966832 − 0.255414i \(-0.0822118\pi\)
−0.704611 + 0.709594i \(0.748878\pi\)
\(98\) 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.bh.1729.3 8
3.2 odd 2 2592.2.i.bg.1729.1 8
4.3 odd 2 2592.2.i.bg.1729.4 8
9.2 odd 6 2592.2.i.bg.865.1 8
9.4 even 3 2592.2.a.v.1.2 ✓ 4
9.5 odd 6 2592.2.a.w.1.4 yes 4
9.7 even 3 inner 2592.2.i.bh.865.3 8
12.11 even 2 inner 2592.2.i.bh.1729.2 8
36.7 odd 6 2592.2.i.bg.865.4 8
36.11 even 6 inner 2592.2.i.bh.865.2 8
36.23 even 6 2592.2.a.v.1.3 yes 4
36.31 odd 6 2592.2.a.w.1.1 yes 4
72.5 odd 6 5184.2.a.cd.1.2 4
72.13 even 6 5184.2.a.ce.1.4 4
72.59 even 6 5184.2.a.ce.1.1 4
72.67 odd 6 5184.2.a.cd.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.v.1.2 ✓ 4 9.4 even 3
2592.2.a.v.1.3 yes 4 36.23 even 6
2592.2.a.w.1.1 yes 4 36.31 odd 6
2592.2.a.w.1.4 yes 4 9.5 odd 6
2592.2.i.bg.865.1 8 9.2 odd 6
2592.2.i.bg.865.4 8 36.7 odd 6
2592.2.i.bg.1729.1 8 3.2 odd 2
2592.2.i.bg.1729.4 8 4.3 odd 2
2592.2.i.bh.865.2 8 36.11 even 6 inner
2592.2.i.bh.865.3 8 9.7 even 3 inner
2592.2.i.bh.1729.2 8 12.11 even 2 inner
2592.2.i.bh.1729.3 8 1.1 even 1 trivial
5184.2.a.cd.1.2 4 72.5 odd 6
5184.2.a.cd.1.3 4 72.67 odd 6
5184.2.a.ce.1.1 4 72.59 even 6
5184.2.a.ce.1.4 4 72.13 even 6