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Properties

Label 171.2.g.c.121.8
Level $171$
Weight $2$
Character 171.121
Analytic conductor $1.365$
Analytic rank $0$
Dimension $32$
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(106,171)] 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("171.106"); 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([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.8
Character \(\chi\) \(=\) 171.121
Dual form 171.2.g.c.106.8

$q$-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+(-0.185445 + 0.321199i) q^{2} +(-0.894876 + 1.48297i) q^{3} +(0.931221 + 1.61292i) q^{4} -3.55521 q^{5} +(-0.310379 - 0.562442i) q^{6} +(-0.124876 - 0.216291i) q^{7} -1.43254 q^{8} +(-1.39839 - 2.65415i) q^{9} +(0.659294 - 1.14193i) q^{10} +(-0.815815 - 1.41303i) q^{11} +(-3.22524 - 0.0623929i) q^{12} +(0.662707 + 1.14784i) q^{13} +0.0926300 q^{14} +(3.18147 - 5.27227i) q^{15} +(-1.59679 + 2.76571i) q^{16} +(3.73000 + 6.46055i) q^{17} +(1.11183 + 0.0430334i) q^{18} +(-4.07660 + 1.54315i) q^{19} +(-3.31069 - 5.73428i) q^{20} +(0.432501 + 0.00836682i) q^{21} +0.605154 q^{22} +(2.24572 + 3.88969i) q^{23} +(1.28194 - 2.12441i) q^{24} +7.63952 q^{25} -0.491582 q^{26} +(5.18741 + 0.301355i) q^{27} +(0.232573 - 0.402829i) q^{28} -4.12725 q^{29} +(1.10346 + 1.99960i) q^{30} +(-4.32871 + 7.49755i) q^{31} +(-2.02477 - 3.50700i) q^{32} +(2.82554 + 0.0546606i) q^{33} -2.76683 q^{34} +(0.443959 + 0.768960i) q^{35} +(2.97872 - 4.72710i) q^{36} +3.10599 q^{37} +(0.260326 - 1.59557i) q^{38} +(-2.29526 - 0.0444022i) q^{39} +5.09297 q^{40} +5.54922 q^{41} +(-0.0828923 + 0.137367i) q^{42} +(5.02032 - 8.69544i) q^{43} +(1.51941 - 2.63169i) q^{44} +(4.97159 + 9.43605i) q^{45} -1.66582 q^{46} +3.36575 q^{47} +(-2.67254 - 4.84295i) q^{48} +(3.46881 - 6.00816i) q^{49} +(-1.41671 + 2.45381i) q^{50} +(-12.9187 - 0.249914i) q^{51} +(-1.23425 + 2.13779i) q^{52} +(0.254182 - 0.440256i) q^{53} +(-1.05877 + 1.61031i) q^{54} +(2.90039 + 5.02363i) q^{55} +(0.178889 + 0.309845i) q^{56} +(1.35962 - 7.42640i) q^{57} +(0.765376 - 1.32567i) q^{58} -10.4624 q^{59} +(11.4664 + 0.221820i) q^{60} +4.14100 q^{61} +(-1.60547 - 2.78076i) q^{62} +(-0.399442 + 0.633898i) q^{63} -4.88521 q^{64} +(-2.35606 - 4.08082i) q^{65} +(-0.541537 + 0.897424i) q^{66} +(0.399675 + 0.692257i) q^{67} +(-6.94690 + 12.0324i) q^{68} +(-7.77793 - 0.150466i) q^{69} -0.329319 q^{70} +(5.60051 + 9.70037i) q^{71} +(2.00325 + 3.80216i) q^{72} +(-1.84754 - 3.20004i) q^{73} +(-0.575989 + 0.997642i) q^{74} +(-6.83642 + 11.3292i) q^{75} +(-6.28519 - 5.13823i) q^{76} +(-0.203751 + 0.352907i) q^{77} +(0.439905 - 0.729000i) q^{78} +(-4.92764 + 8.53493i) q^{79} +(5.67691 - 9.83269i) q^{80} +(-5.08898 + 7.42309i) q^{81} +(-1.02907 + 1.78241i) q^{82} +(0.185251 + 0.320865i) q^{83} +(0.389259 + 0.705381i) q^{84} +(-13.2609 - 22.9686i) q^{85} +(1.86198 + 3.22504i) q^{86} +(3.69338 - 6.12059i) q^{87} +(1.16868 + 2.02422i) q^{88} +(4.01034 - 6.94611i) q^{89} +(-3.95281 - 0.152993i) q^{90} +(0.165512 - 0.286675i) q^{91} +(-4.18251 + 7.24433i) q^{92} +(-7.24498 - 13.1287i) q^{93} +(-0.624161 + 1.08108i) q^{94} +(14.4932 - 5.48621i) q^{95} +(7.01269 + 0.135662i) q^{96} +(-3.21577 + 5.56988i) q^{97} +(1.28654 + 2.22836i) q^{98} +(-2.60956 + 4.14127i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9} - 8 q^{10} + 7 q^{11} - 3 q^{12} - 4 q^{13} - 2 q^{14} + q^{15} - 11 q^{16} - 7 q^{17} + 6 q^{18} + 7 q^{19} - 3 q^{20}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.185445 + 0.321199i −0.131129 + 0.227122i −0.924112 − 0.382122i \(-0.875194\pi\)
0.792983 + 0.609244i \(0.208527\pi\)
\(3\) −0.894876 + 1.48297i −0.516657 + 0.856193i
\(4\) 0.931221 + 1.61292i 0.465610 + 0.806461i
\(5\) −3.55521 −1.58994 −0.794969 − 0.606650i \(-0.792513\pi\)
−0.794969 + 0.606650i \(0.792513\pi\)
\(6\) −0.310379 − 0.562442i −0.126712 − 0.229616i
\(7\) −0.124876 − 0.216291i −0.0471985 − 0.0817503i 0.841461 − 0.540318i \(-0.181696\pi\)
−0.888660 + 0.458568i \(0.848363\pi\)
\(8\) −1.43254 −0.506478
\(9\) −1.39839 − 2.65415i −0.466132 − 0.884715i
\(10\) 0.659294 − 1.14193i 0.208487 − 0.361110i
\(11\) −0.815815 − 1.41303i −0.245977 − 0.426045i 0.716429 − 0.697660i \(-0.245776\pi\)
−0.962406 + 0.271615i \(0.912442\pi\)
\(12\) −3.22524 − 0.0623929i −0.931046 − 0.0180113i
\(13\) 0.662707 + 1.14784i 0.183802 + 0.318354i 0.943172 − 0.332305i \(-0.107826\pi\)
−0.759370 + 0.650659i \(0.774493\pi\)
\(14\) 0.0926300 0.0247564
\(15\) 3.18147 − 5.27227i 0.821453 − 1.36129i
\(16\) −1.59679 + 2.76571i −0.399196 + 0.691428i
\(17\) 3.73000 + 6.46055i 0.904658 + 1.56691i 0.821376 + 0.570387i \(0.193207\pi\)
0.0832816 + 0.996526i \(0.473460\pi\)
\(18\) 1.11183 + 0.0430334i 0.262062 + 0.0101431i
\(19\) −4.07660 + 1.54315i −0.935237 + 0.354022i
\(20\) −3.31069 − 5.73428i −0.740292 − 1.28222i
\(21\) 0.432501 + 0.00836682i 0.0943794 + 0.00182579i
\(22\) 0.605154 0.129019
\(23\) 2.24572 + 3.88969i 0.468264 + 0.811057i 0.999342 − 0.0362656i \(-0.0115462\pi\)
−0.531078 + 0.847323i \(0.678213\pi\)
\(24\) 1.28194 − 2.12441i 0.261675 − 0.433643i
\(25\) 7.63952 1.52790
\(26\) −0.491582 −0.0964071
\(27\) 5.18741 + 0.301355i 0.998317 + 0.0579958i
\(28\) 0.232573 − 0.402829i 0.0439522 − 0.0761275i
\(29\) −4.12725 −0.766411 −0.383206 − 0.923663i \(-0.625180\pi\)
−0.383206 + 0.923663i \(0.625180\pi\)
\(30\) 1.10346 + 1.99960i 0.201464 + 0.365075i
\(31\) −4.32871 + 7.49755i −0.777460 + 1.34660i 0.155941 + 0.987766i \(0.450159\pi\)
−0.933401 + 0.358834i \(0.883174\pi\)
\(32\) −2.02477 − 3.50700i −0.357932 − 0.619956i
\(33\) 2.82554 + 0.0546606i 0.491863 + 0.00951519i
\(34\) −2.76683 −0.474508
\(35\) 0.443959 + 0.768960i 0.0750428 + 0.129978i
\(36\) 2.97872 − 4.72710i 0.496453 − 0.787849i
\(37\) 3.10599 0.510622 0.255311 − 0.966859i \(-0.417822\pi\)
0.255311 + 0.966859i \(0.417822\pi\)
\(38\) 0.260326 − 1.59557i 0.0422305 − 0.258836i
\(39\) −2.29526 − 0.0444022i −0.367535 − 0.00711004i
\(40\) 5.09297 0.805269
\(41\) 5.54922 0.866643 0.433322 − 0.901239i \(-0.357341\pi\)
0.433322 + 0.901239i \(0.357341\pi\)
\(42\) −0.0828923 + 0.137367i −0.0127906 + 0.0211962i
\(43\) 5.02032 − 8.69544i 0.765591 − 1.32604i −0.174342 − 0.984685i \(-0.555780\pi\)
0.939934 − 0.341358i \(-0.110887\pi\)
\(44\) 1.51941 − 2.63169i 0.229059 − 0.396742i
\(45\) 4.97159 + 9.43605i 0.741121 + 1.40664i
\(46\) −1.66582 −0.245612
\(47\) 3.36575 0.490946 0.245473 − 0.969403i \(-0.421057\pi\)
0.245473 + 0.969403i \(0.421057\pi\)
\(48\) −2.67254 − 4.84295i −0.385748 − 0.699020i
\(49\) 3.46881 − 6.00816i 0.495545 − 0.858308i
\(50\) −1.41671 + 2.45381i −0.200353 + 0.347021i
\(51\) −12.9187 − 0.249914i −1.80898 − 0.0349950i
\(52\) −1.23425 + 2.13779i −0.171160 + 0.296458i
\(53\) 0.254182 − 0.440256i 0.0349146 − 0.0604739i −0.848040 − 0.529932i \(-0.822217\pi\)
0.882955 + 0.469458i \(0.155551\pi\)
\(54\) −1.05877 + 1.61031i −0.144080 + 0.219135i
\(55\) 2.90039 + 5.02363i 0.391089 + 0.677386i
\(56\) 0.178889 + 0.309845i 0.0239050 + 0.0414047i
\(57\) 1.35962 − 7.42640i 0.180085 − 0.983651i
\(58\) 0.765376 − 1.32567i 0.100499 − 0.174069i
\(59\) −10.4624 −1.36209 −0.681045 − 0.732242i \(-0.738474\pi\)
−0.681045 + 0.732242i \(0.738474\pi\)
\(60\) 11.4664 + 0.221820i 1.48031 + 0.0286368i
\(61\) 4.14100 0.530201 0.265100 − 0.964221i \(-0.414595\pi\)
0.265100 + 0.964221i \(0.414595\pi\)
\(62\) −1.60547 − 2.78076i −0.203895 − 0.353157i
\(63\) −0.399442 + 0.633898i −0.0503250 + 0.0798636i
\(64\) −4.88521 −0.610652
\(65\) −2.35606 − 4.08082i −0.292234 − 0.506164i
\(66\) −0.541537 + 0.897424i −0.0666586 + 0.110465i
\(67\) 0.399675 + 0.692257i 0.0488281 + 0.0845727i 0.889406 − 0.457117i \(-0.151118\pi\)
−0.840578 + 0.541690i \(0.817785\pi\)
\(68\) −6.94690 + 12.0324i −0.842436 + 1.45914i
\(69\) −7.77793 − 0.150466i −0.936353 − 0.0181139i
\(70\) −0.329319 −0.0393612
\(71\) 5.60051 + 9.70037i 0.664658 + 1.15122i 0.979378 + 0.202037i \(0.0647563\pi\)
−0.314719 + 0.949185i \(0.601910\pi\)
\(72\) 2.00325 + 3.80216i 0.236086 + 0.448089i
\(73\) −1.84754 − 3.20004i −0.216239 − 0.374537i 0.737416 − 0.675439i \(-0.236046\pi\)
−0.953655 + 0.300902i \(0.902712\pi\)
\(74\) −0.575989 + 0.997642i −0.0669573 + 0.115974i
\(75\) −6.83642 + 11.3292i −0.789402 + 1.30818i
\(76\) −6.28519 − 5.13823i −0.720961 − 0.589396i
\(77\) −0.203751 + 0.352907i −0.0232195 + 0.0402174i
\(78\) 0.439905 − 0.729000i 0.0498094 − 0.0825430i
\(79\) −4.92764 + 8.53493i −0.554403 + 0.960255i 0.443546 + 0.896251i \(0.353720\pi\)
−0.997950 + 0.0640032i \(0.979613\pi\)
\(80\) 5.67691 − 9.83269i 0.634698 − 1.09933i
\(81\) −5.08898 + 7.42309i −0.565443 + 0.824788i
\(82\) −1.02907 + 1.78241i −0.113642 + 0.196834i
\(83\) 0.185251 + 0.320865i 0.0203340 + 0.0352195i 0.876013 − 0.482287i \(-0.160194\pi\)
−0.855679 + 0.517506i \(0.826860\pi\)
\(84\) 0.389259 + 0.705381i 0.0424716 + 0.0769634i
\(85\) −13.2609 − 22.9686i −1.43835 − 2.49130i
\(86\) 1.86198 + 3.22504i 0.200782 + 0.347765i
\(87\) 3.69338 − 6.12059i 0.395972 − 0.656196i
\(88\) 1.16868 + 2.02422i 0.124582 + 0.215783i
\(89\) 4.01034 − 6.94611i 0.425095 − 0.736286i −0.571334 − 0.820717i \(-0.693574\pi\)
0.996429 + 0.0844315i \(0.0269074\pi\)
\(90\) −3.95281 − 0.152993i −0.416662 − 0.0161269i
\(91\) 0.165512 − 0.286675i 0.0173504 − 0.0300517i
\(92\) −4.18251 + 7.24433i −0.436057 + 0.755273i
\(93\) −7.24498 − 13.1287i −0.751269 − 1.36139i
\(94\) −0.624161 + 1.08108i −0.0643772 + 0.111505i
\(95\) 14.4932 − 5.48621i 1.48697 − 0.562873i
\(96\) 7.01269 + 0.135662i 0.715729 + 0.0138459i
\(97\) −3.21577 + 5.56988i −0.326512 + 0.565536i −0.981817 − 0.189829i \(-0.939207\pi\)
0.655305 + 0.755364i \(0.272540\pi\)
\(98\) 1.28654 + 2.22836i 0.129961 + 0.225098i
\(99\) −2.60956 + 4.14127i −0.262271 + 0.416213i
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.g.c.121.8 yes 32
3.2 odd 2 513.2.g.c.64.9 32
9.2 odd 6 513.2.h.c.235.8 32
9.7 even 3 171.2.h.c.7.9 yes 32
19.11 even 3 171.2.h.c.49.9 yes 32
57.11 odd 6 513.2.h.c.334.8 32
171.11 odd 6 513.2.g.c.505.9 32
171.106 even 3 inner 171.2.g.c.106.8 ✓ 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.8 ✓ 32 171.106 even 3 inner
171.2.g.c.121.8 yes 32 1.1 even 1 trivial
171.2.h.c.7.9 yes 32 9.7 even 3
171.2.h.c.49.9 yes 32 19.11 even 3
513.2.g.c.64.9 32 3.2 odd 2
513.2.g.c.505.9 32 171.11 odd 6
513.2.h.c.235.8 32 9.2 odd 6
513.2.h.c.334.8 32 57.11 odd 6