Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(64,171)] 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("171.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 163.1
Root \(0.403374 - 1.68443i\) of defining polynomial
Character \(\chi\) \(=\) 171.163
Dual form 171.2.f.b.64.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.25707 + 2.17731i) q^{2} +(-2.16044 - 3.74200i) q^{4} +(1.66044 - 2.87597i) q^{5} +2.32088 q^{7} +5.83502 q^{8} +(4.17458 + 7.23058i) q^{10} +1.70739 q^{11} +(-2.01414 - 3.48859i) q^{13} +(-2.91751 + 5.05328i) q^{14} +(-3.01414 + 5.22064i) q^{16} +(0.193252 + 4.35461i) q^{19} -14.3492 q^{20} +(-2.14631 + 3.71751i) q^{22} +(-1.17458 - 2.03443i) q^{23} +(-3.01414 - 5.22064i) q^{25} +10.1276 q^{26} +(-5.01414 - 8.68474i) q^{28} +(3.32088 + 5.75194i) q^{29} +6.70739 q^{31} +(-1.74293 - 3.01885i) q^{32} +(3.85369 - 6.67479i) q^{35} -1.00000 q^{37} +(-9.72426 - 5.05328i) q^{38} +(9.68872 - 16.7813i) q^{40} +(-3.32088 + 5.75194i) q^{41} +(-0.353695 + 0.612617i) q^{43} +(-3.68872 - 6.38904i) q^{44} +5.90611 q^{46} +(-3.00000 - 5.19615i) q^{47} -1.61350 q^{49} +15.1559 q^{50} +(-8.70285 + 15.0738i) q^{52} +(-4.98133 - 8.62791i) q^{53} +(2.83502 - 4.91040i) q^{55} +13.5424 q^{56} -16.6983 q^{58} +(0.853695 - 1.47864i) q^{59} +(-1.69325 - 2.93280i) q^{61} +(-8.43165 + 14.6040i) q^{62} -3.29261 q^{64} -13.3774 q^{65} +(4.18872 + 7.25507i) q^{67} +(9.68872 + 16.7813i) q^{70} +(-4.70739 + 8.15344i) q^{71} +(-5.82088 + 10.0821i) q^{73} +(1.25707 - 2.17731i) q^{74} +(15.8774 - 10.1310i) q^{76} +3.96265 q^{77} +(1.67458 - 2.90046i) q^{79} +(10.0096 + 17.3371i) q^{80} +(-8.34916 - 14.4612i) q^{82} +10.0565 q^{83} +(-0.889237 - 1.54020i) q^{86} +9.96265 q^{88} +(-1.33956 - 2.32018i) q^{89} +(-4.67458 - 8.09661i) q^{91} +(-5.07522 + 8.79054i) q^{92} +15.0848 q^{94} +(12.8446 + 6.67479i) q^{95} +(-8.86330 + 15.3517i) q^{97} +(2.02827 - 3.51307i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} - 5 q^{16} + 4 q^{19} - 44 q^{20} - 18 q^{22} + 14 q^{23} - 5 q^{25} + 42 q^{26} - 17 q^{28} + 4 q^{29} + 30 q^{31} - 17 q^{32}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25707 + 2.17731i −0.888882 + 1.53959i −0.0476826 + 0.998863i \(0.515184\pi\)
−0.841199 + 0.540726i \(0.818150\pi\)
\(3\) 0 0
\(4\) −2.16044 − 3.74200i −1.08022 − 1.87100i
\(5\) 1.66044 − 2.87597i 0.742572 − 1.28617i −0.208748 − 0.977969i \(-0.566939\pi\)
0.951320 − 0.308204i \(-0.0997278\pi\)
\(6\) 0 0
\(7\) 2.32088 0.877212 0.438606 − 0.898679i \(-0.355472\pi\)
0.438606 + 0.898679i \(0.355472\pi\)
\(8\) 5.83502 2.06299
\(9\) 0 0
\(10\) 4.17458 + 7.23058i 1.32012 + 2.28651i
\(11\) 1.70739 0.514797 0.257399 − 0.966305i \(-0.417135\pi\)
0.257399 + 0.966305i \(0.417135\pi\)
\(12\) 0 0
\(13\) −2.01414 − 3.48859i −0.558621 − 0.967560i −0.997612 − 0.0690685i \(-0.977997\pi\)
0.438991 − 0.898492i \(-0.355336\pi\)
\(14\) −2.91751 + 5.05328i −0.779738 + 1.35055i
\(15\) 0 0
\(16\) −3.01414 + 5.22064i −0.753534 + 1.30516i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 0.193252 + 4.35461i 0.0443351 + 0.999017i
\(20\) −14.3492 −3.20857
\(21\) 0 0
\(22\) −2.14631 + 3.71751i −0.457594 + 0.792576i
\(23\) −1.17458 − 2.03443i −0.244917 − 0.424208i 0.717191 − 0.696876i \(-0.245427\pi\)
−0.962108 + 0.272668i \(0.912094\pi\)
\(24\) 0 0
\(25\) −3.01414 − 5.22064i −0.602827 − 1.04413i
\(26\) 10.1276 1.98619
\(27\) 0 0
\(28\) −5.01414 − 8.68474i −0.947583 − 1.64126i
\(29\) 3.32088 + 5.75194i 0.616673 + 1.06811i 0.990089 + 0.140444i \(0.0448532\pi\)
−0.373416 + 0.927664i \(0.621814\pi\)
\(30\) 0 0
\(31\) 6.70739 1.20468 0.602341 − 0.798239i \(-0.294235\pi\)
0.602341 + 0.798239i \(0.294235\pi\)
\(32\) −1.74293 − 3.01885i −0.308110 − 0.533662i
\(33\) 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 \(0.526194\pi\)
\(38\) −9.72426 − 5.05328i −1.57748 − 0.819750i
\(39\) 0 0
\(40\) 9.68872 − 16.7813i 1.53192 − 2.65336i
\(41\) −3.32088 + 5.75194i −0.518635 + 0.898302i 0.481131 + 0.876649i \(0.340226\pi\)
−0.999766 + 0.0216532i \(0.993107\pi\)
\(42\) 0 0
\(43\) −0.353695 + 0.612617i −0.0539379 + 0.0934232i −0.891734 − 0.452561i \(-0.850511\pi\)
0.837796 + 0.545984i \(0.183844\pi\)
\(44\) −3.68872 − 6.38904i −0.556095 − 0.963185i
\(45\) 0 0
\(46\) 5.90611 0.870808
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −1.61350 −0.230499
\(50\) 15.1559 2.14337
\(51\) 0 0
\(52\) −8.70285 + 15.0738i −1.20687 + 2.09036i
\(53\) −4.98133 − 8.62791i −0.684238 − 1.18513i −0.973676 − 0.227938i \(-0.926802\pi\)
0.289438 − 0.957197i \(-0.406532\pi\)
\(54\) 0 0
\(55\) 2.83502 − 4.91040i 0.382274 − 0.662118i
\(56\) 13.5424 1.80968
\(57\) 0 0
\(58\) −16.6983 −2.19260
\(59\) 0.853695 − 1.47864i 0.111142 − 0.192503i −0.805089 − 0.593154i \(-0.797883\pi\)
0.916231 + 0.400651i \(0.131216\pi\)
\(60\) 0 0
\(61\) −1.69325 − 2.93280i −0.216799 − 0.375506i 0.737029 − 0.675861i \(-0.236228\pi\)
−0.953828 + 0.300355i \(0.902895\pi\)
\(62\) −8.43165 + 14.6040i −1.07082 + 1.85472i
\(63\) 0 0
\(64\) −3.29261 −0.411576
\(65\) −13.3774 −1.65927
\(66\) 0 0
\(67\) 4.18872 + 7.25507i 0.511733 + 0.886348i 0.999907 + 0.0136016i \(0.00432967\pi\)
−0.488174 + 0.872746i \(0.662337\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 9.68872 + 16.7813i 1.15802 + 2.00575i
\(71\) −4.70739 + 8.15344i −0.558664 + 0.967635i 0.438944 + 0.898514i \(0.355353\pi\)
−0.997608 + 0.0691206i \(0.977981\pi\)
\(72\) 0 0
\(73\) −5.82088 + 10.0821i −0.681283 + 1.18002i 0.293307 + 0.956018i \(0.405244\pi\)
−0.974590 + 0.223998i \(0.928089\pi\)
\(74\) 1.25707 − 2.17731i 0.146131 − 0.253107i
\(75\) 0 0
\(76\) 15.8774 − 10.1310i 1.82127 − 1.16211i
\(77\) 3.96265 0.451586
\(78\) 0 0
\(79\) 1.67458 − 2.90046i 0.188405 − 0.326327i −0.756314 − 0.654209i \(-0.773002\pi\)
0.944719 + 0.327882i \(0.106335\pi\)
\(80\) 10.0096 + 17.3371i 1.11911 + 1.93835i
\(81\) 0 0
\(82\) −8.34916 − 14.4612i −0.922010 − 1.59697i
\(83\) 10.0565 1.10385 0.551925 − 0.833894i \(-0.313894\pi\)
0.551925 + 0.833894i \(0.313894\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.889237 − 1.54020i −0.0958889 − 0.166084i
\(87\) 0 0
\(88\) 9.96265 1.06202
\(89\) −1.33956 − 2.32018i −0.141993 − 0.245939i 0.786254 − 0.617903i \(-0.212018\pi\)
−0.928247 + 0.371964i \(0.878684\pi\)
\(90\) 0 0
\(91\) −4.67458 − 8.09661i −0.490029 − 0.848755i
\(92\) −5.07522 + 8.79054i −0.529128 + 0.916477i
\(93\) 0 0
\(94\) 15.0848 1.55588
\(95\) 12.8446 + 6.67479i 1.31783 + 0.684820i
\(96\) 0 0
\(97\) −8.86330 + 15.3517i −0.899931 + 1.55873i −0.0723511 + 0.997379i \(0.523050\pi\)
−0.827580 + 0.561347i \(0.810283\pi\)
\(98\) 2.02827 − 3.51307i 0.204887 − 0.354874i
\(99\) 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 171.2.f.b.163.1 6
3.2 odd 2 57.2.e.b.49.3 yes 6
4.3 odd 2 2736.2.s.z.1873.3 6
12.11 even 2 912.2.q.l.49.1 6
19.7 even 3 inner 171.2.f.b.64.1 6
19.8 odd 6 3249.2.a.t.1.1 3
19.11 even 3 3249.2.a.y.1.3 3
57.8 even 6 1083.2.a.o.1.3 3
57.11 odd 6 1083.2.a.l.1.1 3
57.26 odd 6 57.2.e.b.7.3 ✓ 6
76.7 odd 6 2736.2.s.z.577.3 6
228.83 even 6 912.2.q.l.577.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.3 ✓ 6 57.26 odd 6
57.2.e.b.49.3 yes 6 3.2 odd 2
171.2.f.b.64.1 6 19.7 even 3 inner
171.2.f.b.163.1 6 1.1 even 1 trivial
912.2.q.l.49.1 6 12.11 even 2
912.2.q.l.577.1 6 228.83 even 6
1083.2.a.l.1.1 3 57.11 odd 6
1083.2.a.o.1.3 3 57.8 even 6
2736.2.s.z.577.3 6 76.7 odd 6
2736.2.s.z.1873.3 6 4.3 odd 2
3249.2.a.t.1.1 3 19.8 odd 6
3249.2.a.y.1.3 3 19.11 even 3