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Properties

Label 171.2.f.b.163.2
Level $171$
Weight $2$
Character 171.163
Analytic conductor $1.365$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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 163.2
Root \(-1.62241 + 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 171.163
Dual form 171.2.f.b.64.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.285997 + 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(-1.33641 + 2.31473i) q^{5} -3.67282 q^{7} -2.10083 q^{8} +(-0.764419 - 1.32401i) q^{10} +3.81681 q^{11} +(-0.0719933 - 0.124696i) q^{13} +(1.05042 - 1.81937i) q^{14} +(-1.07199 + 1.85675i) q^{16} +(4.24482 + 0.990721i) q^{19} -4.47116 q^{20} +(-1.09159 + 1.89070i) q^{22} +(3.76442 + 6.52016i) q^{23} +(-1.07199 - 1.85675i) q^{25} +0.0823593 q^{26} +(-3.07199 - 5.32085i) q^{28} +(-2.67282 - 4.62947i) q^{29} +8.81681 q^{31} +(-2.71400 - 4.70079i) q^{32} +(4.90841 - 8.50161i) q^{35} -1.00000 q^{37} +(-1.70477 + 1.81937i) q^{38} +(2.80757 - 4.86286i) q^{40} +(2.67282 - 4.62947i) q^{41} +(-1.40841 + 2.43943i) q^{43} +(3.19243 + 5.52944i) q^{44} -4.30644 q^{46} +(-3.00000 - 5.19615i) q^{47} +6.48963 q^{49} +1.22635 q^{50} +(0.120432 - 0.208594i) q^{52} +(4.00924 + 6.94420i) q^{53} +(-5.10083 + 8.83490i) q^{55} +7.71598 q^{56} +3.05767 q^{58} +(1.90841 - 3.30545i) q^{59} +(-5.74482 - 9.95031i) q^{61} +(-2.52158 + 4.36750i) q^{62} -1.18319 q^{64} +0.384851 q^{65} +(-2.69243 - 4.66342i) q^{67} +(2.80757 + 4.86286i) q^{70} +(-6.81681 + 11.8071i) q^{71} +(0.172824 - 0.299339i) q^{73} +(0.285997 - 0.495361i) q^{74} +(2.11515 + 6.97815i) q^{76} -14.0185 q^{77} +(-3.26442 + 5.65414i) q^{79} +(-2.86525 - 4.96276i) q^{80} +(1.52884 + 2.64802i) q^{82} +2.28797 q^{83} +(-0.805598 - 1.39534i) q^{86} -8.01847 q^{88} +(-4.33641 - 7.51089i) q^{89} +(0.264419 + 0.457986i) q^{91} +(-6.29721 + 10.9071i) q^{92} +3.43196 q^{94} +(-7.96608 + 8.50161i) q^{95} +(2.95684 - 5.12140i) q^{97} +(-1.85601 + 3.21471i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} - 5 q^{16} + 4 q^{19} - 44 q^{20} - 18 q^{22} + 14 q^{23} - 5 q^{25} + 42 q^{26} - 17 q^{28} + 4 q^{29} + 30 q^{31} - 17 q^{32}+ \cdots - 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.285997 + 0.495361i −0.202230 + 0.350273i −0.949247 − 0.314533i \(-0.898152\pi\)
0.747017 + 0.664805i \(0.231486\pi\)
\(3\) 0 0
\(4\) 0.836412 + 1.44871i 0.418206 + 0.724354i
\(5\) −1.33641 + 2.31473i −0.597662 + 1.03518i 0.395504 + 0.918464i \(0.370570\pi\)
−0.993165 + 0.116716i \(0.962763\pi\)
\(6\) 0 0
\(7\) −3.67282 −1.38820 −0.694098 − 0.719880i \(-0.744197\pi\)
−0.694098 + 0.719880i \(0.744197\pi\)
\(8\) −2.10083 −0.742756
\(9\) 0 0
\(10\) −0.764419 − 1.32401i −0.241730 − 0.418689i
\(11\) 3.81681 1.15081 0.575406 − 0.817868i \(-0.304844\pi\)
0.575406 + 0.817868i \(0.304844\pi\)
\(12\) 0 0
\(13\) −0.0719933 − 0.124696i −0.0199673 − 0.0345844i 0.855869 − 0.517193i \(-0.173023\pi\)
−0.875836 + 0.482608i \(0.839690\pi\)
\(14\) 1.05042 − 1.81937i 0.280735 − 0.486248i
\(15\) 0 0
\(16\) −1.07199 + 1.85675i −0.267998 + 0.464187i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 4.24482 + 0.990721i 0.973828 + 0.227287i
\(20\) −4.47116 −0.999782
\(21\) 0 0
\(22\) −1.09159 + 1.89070i −0.232729 + 0.403098i
\(23\) 3.76442 + 6.52016i 0.784936 + 1.35955i 0.929038 + 0.369985i \(0.120637\pi\)
−0.144102 + 0.989563i \(0.546029\pi\)
\(24\) 0 0
\(25\) −1.07199 − 1.85675i −0.214399 − 0.371349i
\(26\) 0.0823593 0.0161520
\(27\) 0 0
\(28\) −3.07199 − 5.32085i −0.580552 − 1.00555i
\(29\) −2.67282 − 4.62947i −0.496331 − 0.859670i 0.503660 − 0.863902i \(-0.331986\pi\)
−0.999991 + 0.00423154i \(0.998653\pi\)
\(30\) 0 0
\(31\) 8.81681 1.58355 0.791773 − 0.610816i \(-0.209158\pi\)
0.791773 + 0.610816i \(0.209158\pi\)
\(32\) −2.71400 − 4.70079i −0.479773 − 0.830990i
\(33\) 0 0
\(34\) 0 0
\(35\) 4.90841 − 8.50161i 0.829672 − 1.43703i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −1.70477 + 1.81937i −0.276550 + 0.295141i
\(39\) 0 0
\(40\) 2.80757 − 4.86286i 0.443917 − 0.768886i
\(41\) 2.67282 − 4.62947i 0.417425 − 0.723001i −0.578255 − 0.815856i \(-0.696266\pi\)
0.995680 + 0.0928551i \(0.0295993\pi\)
\(42\) 0 0
\(43\) −1.40841 + 2.43943i −0.214780 + 0.372009i −0.953204 − 0.302327i \(-0.902237\pi\)
0.738425 + 0.674336i \(0.235570\pi\)
\(44\) 3.19243 + 5.52944i 0.481276 + 0.833595i
\(45\) 0 0
\(46\) −4.30644 −0.634951
\(47\) −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
\(49\) 6.48963 0.927091
\(50\) 1.22635 0.173431
\(51\) 0 0
\(52\) 0.120432 − 0.208594i 0.0167009 − 0.0289268i
\(53\) 4.00924 + 6.94420i 0.550711 + 0.953859i 0.998223 + 0.0595815i \(0.0189766\pi\)
−0.447513 + 0.894278i \(0.647690\pi\)
\(54\) 0 0
\(55\) −5.10083 + 8.83490i −0.687796 + 1.19130i
\(56\) 7.71598 1.03109
\(57\) 0 0
\(58\) 3.05767 0.401492
\(59\) 1.90841 − 3.30545i 0.248453 − 0.430334i −0.714644 − 0.699489i \(-0.753411\pi\)
0.963097 + 0.269155i \(0.0867445\pi\)
\(60\) 0 0
\(61\) −5.74482 − 9.95031i −0.735548 − 1.27401i −0.954482 − 0.298268i \(-0.903591\pi\)
0.218934 − 0.975740i \(-0.429742\pi\)
\(62\) −2.52158 + 4.36750i −0.320241 + 0.554673i
\(63\) 0 0
\(64\) −1.18319 −0.147899
\(65\) 0.384851 0.0477348
\(66\) 0 0
\(67\) −2.69243 − 4.66342i −0.328932 − 0.569727i 0.653368 − 0.757040i \(-0.273355\pi\)
−0.982300 + 0.187313i \(0.940022\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 2.80757 + 4.86286i 0.335569 + 0.581223i
\(71\) −6.81681 + 11.8071i −0.809007 + 1.40124i 0.104546 + 0.994520i \(0.466661\pi\)
−0.913553 + 0.406720i \(0.866672\pi\)
\(72\) 0 0
\(73\) 0.172824 − 0.299339i 0.0202275 − 0.0350350i −0.855734 − 0.517415i \(-0.826894\pi\)
0.875962 + 0.482380i \(0.160228\pi\)
\(74\) 0.285997 − 0.495361i 0.0332464 − 0.0575845i
\(75\) 0 0
\(76\) 2.11515 + 6.97815i 0.242624 + 0.800449i
\(77\) −14.0185 −1.59755
\(78\) 0 0
\(79\) −3.26442 + 5.65414i −0.367276 + 0.636140i −0.989139 − 0.146986i \(-0.953043\pi\)
0.621863 + 0.783126i \(0.286376\pi\)
\(80\) −2.86525 − 4.96276i −0.320345 − 0.554853i
\(81\) 0 0
\(82\) 1.52884 + 2.64802i 0.168832 + 0.292425i
\(83\) 2.28797 0.251138 0.125569 − 0.992085i \(-0.459924\pi\)
0.125569 + 0.992085i \(0.459924\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.805598 − 1.39534i −0.0868699 − 0.150463i
\(87\) 0 0
\(88\) −8.01847 −0.854772
\(89\) −4.33641 − 7.51089i −0.459659 − 0.796152i 0.539284 − 0.842124i \(-0.318695\pi\)
−0.998943 + 0.0459717i \(0.985362\pi\)
\(90\) 0 0
\(91\) 0.264419 + 0.457986i 0.0277186 + 0.0480100i
\(92\) −6.29721 + 10.9071i −0.656529 + 1.13714i
\(93\) 0 0
\(94\) 3.43196 0.353980
\(95\) −7.96608 + 8.50161i −0.817303 + 0.872246i
\(96\) 0 0
\(97\) 2.95684 − 5.12140i 0.300222 − 0.520000i −0.675964 − 0.736935i \(-0.736273\pi\)
0.976186 + 0.216935i \(0.0696059\pi\)
\(98\) −1.85601 + 3.21471i −0.187486 + 0.324735i
\(99\) 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.163.2 6
3.2 odd 2 57.2.e.b.49.2 yes 6
4.3 odd 2 2736.2.s.z.1873.1 6
12.11 even 2 912.2.q.l.49.3 6
19.7 even 3 inner 171.2.f.b.64.2 6
19.8 odd 6 3249.2.a.t.1.2 3
19.11 even 3 3249.2.a.y.1.2 3
57.8 even 6 1083.2.a.o.1.2 3
57.11 odd 6 1083.2.a.l.1.2 3
57.26 odd 6 57.2.e.b.7.2 ✓ 6
76.7 odd 6 2736.2.s.z.577.1 6
228.83 even 6 912.2.q.l.577.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.2 ✓ 6 57.26 odd 6
57.2.e.b.49.2 yes 6 3.2 odd 2
171.2.f.b.64.2 6 19.7 even 3 inner
171.2.f.b.163.2 6 1.1 even 1 trivial
912.2.q.l.49.3 6 12.11 even 2
912.2.q.l.577.3 6 228.83 even 6
1083.2.a.l.1.2 3 57.11 odd 6
1083.2.a.o.1.2 3 57.8 even 6
2736.2.s.z.577.1 6 76.7 odd 6
2736.2.s.z.1873.1 6 4.3 odd 2
3249.2.a.t.1.2 3 19.8 odd 6
3249.2.a.y.1.2 3 19.11 even 3