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Properties

Label 171.2.f.b.64.2
Level $171$
Weight $2$
Character 171.64
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 64.2
Root \(-1.62241 - 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 171.64
Dual form 171.2.f.b.163.2

$q$-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 \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

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

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 0.747017 − 0.664805i \(-0.231486\pi\)
−0.949247 + 0.314533i \(0.898152\pi\)
\(3\) 0 0
\(4\) 0.836412 − 1.44871i 0.418206 − 0.724354i
\(5\) −1.33641 − 2.31473i −0.597662 − 1.03518i −0.993165 − 0.116716i \(-0.962763\pi\)
0.395504 − 0.918464i \(-0.370570\pi\)
\(6\) 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 −0.875836 − 0.482608i \(-0.839690\pi\)
0.855869 + 0.517193i \(0.173023\pi\)
\(14\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 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 −0.144102 − 0.989563i \(-0.546029\pi\)
0.929038 − 0.369985i \(-0.120637\pi\)
\(24\) 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 −0.999991 − 0.00423154i \(-0.998653\pi\)
0.503660 + 0.863902i \(0.331986\pi\)
\(30\) 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 0.995680 − 0.0928551i \(-0.0295993\pi\)
−0.578255 + 0.815856i \(0.696266\pi\)
\(42\) 0 0
\(43\) −1.40841 − 2.43943i −0.214780 − 0.372009i 0.738425 − 0.674336i \(-0.235570\pi\)
−0.953204 + 0.302327i \(0.902237\pi\)
\(44\) 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 −0.997503 − 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 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 −0.447513 − 0.894278i \(-0.647690\pi\)
0.998223 − 0.0595815i \(-0.0189766\pi\)
\(54\) 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 0.963097 − 0.269155i \(-0.0867445\pi\)
−0.714644 + 0.699489i \(0.753411\pi\)
\(60\) 0 0
\(61\) −5.74482 + 9.95031i −0.735548 + 1.27401i 0.218934 + 0.975740i \(0.429742\pi\)
−0.954482 + 0.298268i \(0.903591\pi\)
\(62\) −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 −0.982300 − 0.187313i \(-0.940022\pi\)
0.653368 + 0.757040i \(0.273355\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 2.80757 − 4.86286i 0.335569 − 0.581223i
\(71\) −6.81681 − 11.8071i −0.809007 − 1.40124i −0.913553 − 0.406720i \(-0.866672\pi\)
0.104546 − 0.994520i \(-0.466661\pi\)
\(72\) 0 0
\(73\) 0.172824 + 0.299339i 0.0202275 + 0.0350350i 0.875962 − 0.482380i \(-0.160228\pi\)
−0.855734 + 0.517415i \(0.826894\pi\)
\(74\) 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 0.621863 − 0.783126i \(-0.286376\pi\)
−0.989139 + 0.146986i \(0.953043\pi\)
\(80\) −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 −0.998943 − 0.0459717i \(-0.985362\pi\)
0.539284 + 0.842124i \(0.318695\pi\)
\(90\) 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 0.976186 − 0.216935i \(-0.0696059\pi\)
−0.675964 + 0.736935i \(0.736273\pi\)
\(98\) −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.64.2 6
3.2 odd 2 57.2.e.b.7.2 ✓ 6
4.3 odd 2 2736.2.s.z.577.1 6
12.11 even 2 912.2.q.l.577.3 6
19.7 even 3 3249.2.a.y.1.2 3
19.11 even 3 inner 171.2.f.b.163.2 6
19.12 odd 6 3249.2.a.t.1.2 3
57.11 odd 6 57.2.e.b.49.2 yes 6
57.26 odd 6 1083.2.a.l.1.2 3
57.50 even 6 1083.2.a.o.1.2 3
76.11 odd 6 2736.2.s.z.1873.1 6
228.11 even 6 912.2.q.l.49.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.2 ✓ 6 3.2 odd 2
57.2.e.b.49.2 yes 6 57.11 odd 6
171.2.f.b.64.2 6 1.1 even 1 trivial
171.2.f.b.163.2 6 19.11 even 3 inner
912.2.q.l.49.3 6 228.11 even 6
912.2.q.l.577.3 6 12.11 even 2
1083.2.a.l.1.2 3 57.26 odd 6
1083.2.a.o.1.2 3 57.50 even 6
2736.2.s.z.577.1 6 4.3 odd 2
2736.2.s.z.1873.1 6 76.11 odd 6
3249.2.a.t.1.2 3 19.12 odd 6
3249.2.a.y.1.2 3 19.7 even 3