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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2736,2,Mod(577,2736)] 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("2736.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,2,0,2,0,0,0,0,0,1] 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.3
Root \(0.403374 + 1.68443i\) of defining polynomial
Character \(\chi\) \(=\) 2736.577
Dual form 2736.2.s.z.1873.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.66044 + 2.87597i) q^{5} -2.32088 q^{7} -1.70739 q^{11} +(-2.01414 + 3.48859i) q^{13} +(-0.193252 + 4.35461i) q^{19} +(1.17458 - 2.03443i) q^{23} +(-3.01414 + 5.22064i) q^{25} +(3.32088 - 5.75194i) q^{29} -6.70739 q^{31} +(-3.85369 - 6.67479i) q^{35} -1.00000 q^{37} +(-3.32088 - 5.75194i) q^{41} +(0.353695 + 0.612617i) q^{43} +(3.00000 - 5.19615i) q^{47} -1.61350 q^{49} +(-4.98133 + 8.62791i) q^{53} +(-2.83502 - 4.91040i) q^{55} +(-0.853695 - 1.47864i) q^{59} +(-1.69325 + 2.93280i) q^{61} -13.3774 q^{65} +(-4.18872 + 7.25507i) q^{67} +(4.70739 + 8.15344i) q^{71} +(-5.82088 - 10.0821i) q^{73} +3.96265 q^{77} +(-1.67458 - 2.90046i) q^{79} -10.0565 q^{83} +(-1.33956 + 2.32018i) q^{89} +(4.67458 - 8.09661i) q^{91} +(-12.8446 + 6.67479i) q^{95} +(-8.86330 - 15.3517i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(5\) 1.66044 + 2.87597i 0.742572 + 1.28617i 0.951320 + 0.308204i \(0.0997278\pi\)
−0.208748 + 0.977969i \(0.566939\pi\)
\(6\) 0 0
\(7\) −2.32088 −0.877212 −0.438606 − 0.898679i \(-0.644528\pi\)
−0.438606 + 0.898679i \(0.644528\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.70739 −0.514797 −0.257399 − 0.966305i \(-0.582865\pi\)
−0.257399 + 0.966305i \(0.582865\pi\)
\(12\) 0 0
\(13\) −2.01414 + 3.48859i −0.558621 + 0.967560i 0.438991 + 0.898492i \(0.355336\pi\)
−0.997612 + 0.0690685i \(0.977997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −0.193252 + 4.35461i −0.0443351 + 0.999017i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.17458 − 2.03443i 0.244917 − 0.424208i −0.717191 − 0.696876i \(-0.754573\pi\)
0.962108 + 0.272668i \(0.0879061\pi\)
\(24\) 0 0
\(25\) −3.01414 + 5.22064i −0.602827 + 1.04413i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.32088 − 5.75194i 0.616673 − 1.06811i −0.373416 − 0.927664i \(-0.621814\pi\)
0.990089 − 0.140444i \(-0.0448532\pi\)
\(30\) 0 0
\(31\) −6.70739 −1.20468 −0.602341 − 0.798239i \(-0.705765\pi\)
−0.602341 + 0.798239i \(0.705765\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.85369 − 6.67479i −0.651393 − 1.12825i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.32088 − 5.75194i −0.518635 − 0.898302i −0.999766 − 0.0216532i \(-0.993107\pi\)
0.481131 − 0.876649i \(-0.340226\pi\)
\(42\) 0 0
\(43\) 0.353695 + 0.612617i 0.0539379 + 0.0934232i 0.891734 − 0.452561i \(-0.149489\pi\)
−0.837796 + 0.545984i \(0.816156\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 − 5.19615i 0.437595 − 0.757937i −0.559908 − 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) −1.61350 −0.230499
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.98133 + 8.62791i −0.684238 + 1.18513i 0.289438 + 0.957197i \(0.406532\pi\)
−0.973676 + 0.227938i \(0.926802\pi\)
\(54\) 0 0
\(55\) −2.83502 − 4.91040i −0.382274 − 0.662118i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.853695 − 1.47864i −0.111142 − 0.192503i 0.805089 − 0.593154i \(-0.202117\pi\)
−0.916231 + 0.400651i \(0.868784\pi\)
\(60\) 0 0
\(61\) −1.69325 + 2.93280i −0.216799 + 0.375506i −0.953828 − 0.300355i \(-0.902895\pi\)
0.737029 + 0.675861i \(0.236228\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −13.3774 −1.65927
\(66\) 0 0
\(67\) −4.18872 + 7.25507i −0.511733 + 0.886348i 0.488174 + 0.872746i \(0.337663\pi\)
−0.999907 + 0.0136016i \(0.995670\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.70739 + 8.15344i 0.558664 + 0.967635i 0.997608 + 0.0691206i \(0.0220193\pi\)
−0.438944 + 0.898514i \(0.644647\pi\)
\(72\) 0 0
\(73\) −5.82088 − 10.0821i −0.681283 − 1.18002i −0.974590 − 0.223998i \(-0.928089\pi\)
0.293307 − 0.956018i \(-0.405244\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.96265 0.451586
\(78\) 0 0
\(79\) −1.67458 − 2.90046i −0.188405 − 0.326327i 0.756314 − 0.654209i \(-0.226998\pi\)
−0.944719 + 0.327882i \(0.893665\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.0565 −1.10385 −0.551925 − 0.833894i \(-0.686106\pi\)
−0.551925 + 0.833894i \(0.686106\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.33956 + 2.32018i −0.141993 + 0.245939i −0.928247 − 0.371964i \(-0.878684\pi\)
0.786254 + 0.617903i \(0.212018\pi\)
\(90\) 0 0
\(91\) 4.67458 − 8.09661i 0.490029 − 0.848755i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.8446 + 6.67479i −1.31783 + 0.684820i
\(96\) 0 0
\(97\) −8.86330 − 15.3517i −0.899931 − 1.55873i −0.827580 − 0.561347i \(-0.810283\pi\)
−0.0723511 − 0.997379i \(-0.523050\pi\)
\(98\) 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 2736.2.s.z.577.3 6
3.2 odd 2 912.2.q.l.577.1 6
4.3 odd 2 171.2.f.b.64.1 6
12.11 even 2 57.2.e.b.7.3 ✓ 6
19.11 even 3 inner 2736.2.s.z.1873.3 6
57.11 odd 6 912.2.q.l.49.1 6
76.7 odd 6 3249.2.a.y.1.3 3
76.11 odd 6 171.2.f.b.163.1 6
76.31 even 6 3249.2.a.t.1.1 3
228.11 even 6 57.2.e.b.49.3 yes 6
228.83 even 6 1083.2.a.l.1.1 3
228.107 odd 6 1083.2.a.o.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.3 ✓ 6 12.11 even 2
57.2.e.b.49.3 yes 6 228.11 even 6
171.2.f.b.64.1 6 4.3 odd 2
171.2.f.b.163.1 6 76.11 odd 6
912.2.q.l.49.1 6 57.11 odd 6
912.2.q.l.577.1 6 3.2 odd 2
1083.2.a.l.1.1 3 228.83 even 6
1083.2.a.o.1.3 3 228.107 odd 6
2736.2.s.z.577.3 6 1.1 even 1 trivial
2736.2.s.z.1873.3 6 19.11 even 3 inner
3249.2.a.t.1.1 3 76.31 even 6
3249.2.a.y.1.3 3 76.7 odd 6