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 106.11
Character \(\chi\) \(=\) 171.106
Dual form 171.2.g.c.121.11

$q$-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+(0.616796 + 1.06832i) q^{2} +(0.506944 + 1.65620i) q^{3} +(0.239126 - 0.414178i) q^{4} -0.551543 q^{5} +(-1.45668 + 1.56312i) q^{6} +(-1.62156 + 2.80862i) q^{7} +3.05715 q^{8} +(-2.48602 + 1.67920i) q^{9} +(-0.340189 - 0.589225i) q^{10} +(2.68677 - 4.65363i) q^{11} +(0.807186 + 0.186076i) q^{12} +(-1.76195 + 3.05178i) q^{13} -4.00069 q^{14} +(-0.279601 - 0.913467i) q^{15} +(1.40739 + 2.43766i) q^{16} +(2.60168 - 4.50624i) q^{17} +(-3.32729 - 1.62014i) q^{18} +(0.164107 - 4.35581i) q^{19} +(-0.131888 + 0.228437i) q^{20} +(-5.47369 - 1.26182i) q^{21} +6.62876 q^{22} +(-1.49233 + 2.58480i) q^{23} +(1.54980 + 5.06326i) q^{24} -4.69580 q^{25} -4.34704 q^{26} +(-4.04137 - 3.26609i) q^{27} +(0.775514 + 1.34323i) q^{28} +5.49841 q^{29} +(0.803420 - 0.862127i) q^{30} +(-2.54567 - 4.40923i) q^{31} +(1.32101 - 2.28806i) q^{32} +(9.06939 + 2.09071i) q^{33} +6.41881 q^{34} +(0.894360 - 1.54908i) q^{35} +(0.101019 + 1.43119i) q^{36} +9.20826 q^{37} +(4.75463 - 2.51133i) q^{38} +(-5.94757 - 1.37106i) q^{39} -1.68615 q^{40} +1.71003 q^{41} +(-2.02812 - 6.62595i) q^{42} +(1.79608 + 3.11091i) q^{43} +(-1.28495 - 2.22561i) q^{44} +(1.37114 - 0.926153i) q^{45} -3.68186 q^{46} -10.6354 q^{47} +(-3.32380 + 3.56667i) q^{48} +(-1.75891 - 3.04653i) q^{49} +(-2.89635 - 5.01663i) q^{50} +(8.78214 + 2.02450i) q^{51} +(0.842653 + 1.45952i) q^{52} +(0.562013 + 0.973435i) q^{53} +(0.996529 - 6.33199i) q^{54} +(-1.48187 + 2.56668i) q^{55} +(-4.95735 + 8.58639i) q^{56} +(7.29730 - 1.93635i) q^{57} +(3.39140 + 5.87407i) q^{58} -7.77857 q^{59} +(-0.445198 - 0.102629i) q^{60} -11.3745 q^{61} +(3.14032 - 5.43919i) q^{62} +(-0.685027 - 9.70521i) q^{63} +8.88872 q^{64} +(0.971789 - 1.68319i) q^{65} +(3.36041 + 10.9786i) q^{66} +(-1.18630 + 2.05473i) q^{67} +(-1.24426 - 2.15512i) q^{68} +(-5.03748 - 1.16126i) q^{69} +2.20655 q^{70} +(-0.507763 + 0.879471i) q^{71} +(-7.60013 + 5.13358i) q^{72} +(-5.98562 + 10.3674i) q^{73} +(5.67962 + 9.83738i) q^{74} +(-2.38051 - 7.77720i) q^{75} +(-1.76484 - 1.10956i) q^{76} +(8.71353 + 15.0923i) q^{77} +(-2.20371 - 7.19958i) q^{78} +(0.568802 + 0.985193i) q^{79} +(-0.776234 - 1.34448i) q^{80} +(3.36055 - 8.34905i) q^{81} +(1.05474 + 1.82686i) q^{82} +(-1.14372 + 1.98099i) q^{83} +(-1.83152 + 1.96535i) q^{84} +(-1.43494 + 2.48538i) q^{85} +(-2.21563 + 3.83759i) q^{86} +(2.78738 + 9.10648i) q^{87} +(8.21387 - 14.2268i) q^{88} +(1.12141 + 1.94234i) q^{89} +(1.83515 + 0.893577i) q^{90} +(-5.71420 - 9.89729i) q^{91} +(0.713711 + 1.23618i) q^{92} +(6.01206 - 6.45137i) q^{93} +(-6.55987 - 11.3620i) q^{94} +(-0.0905122 + 2.40242i) q^{95} +(4.45917 + 1.02795i) q^{96} +(-4.09100 - 7.08582i) q^{97} +(2.16978 - 3.75817i) q^{98} +(1.13502 + 16.0806i) 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 \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.616796 + 1.06832i 0.436140 + 0.755418i 0.997388 − 0.0722305i \(-0.0230117\pi\)
−0.561247 + 0.827648i \(0.689678\pi\)
\(3\) 0.506944 + 1.65620i 0.292684 + 0.956209i
\(4\) 0.239126 − 0.414178i 0.119563 − 0.207089i
\(5\) −0.551543 −0.246658 −0.123329 − 0.992366i \(-0.539357\pi\)
−0.123329 + 0.992366i \(0.539357\pi\)
\(6\) −1.45668 + 1.56312i −0.594686 + 0.638140i
\(7\) −1.62156 + 2.80862i −0.612892 + 1.06156i 0.377858 + 0.925863i \(0.376661\pi\)
−0.990750 + 0.135697i \(0.956673\pi\)
\(8\) 3.05715 1.08087
\(9\) −2.48602 + 1.67920i −0.828672 + 0.559734i
\(10\) −0.340189 − 0.589225i −0.107577 − 0.186329i
\(11\) 2.68677 − 4.65363i 0.810093 − 1.40312i −0.102706 − 0.994712i \(-0.532750\pi\)
0.912799 − 0.408410i \(-0.133917\pi\)
\(12\) 0.807186 + 0.186076i 0.233015 + 0.0537155i
\(13\) −1.76195 + 3.05178i −0.488676 + 0.846411i −0.999915 − 0.0130271i \(-0.995853\pi\)
0.511239 + 0.859438i \(0.329187\pi\)
\(14\) −4.00069 −1.06923
\(15\) −0.279601 − 0.913467i −0.0721927 − 0.235856i
\(16\) 1.40739 + 2.43766i 0.351846 + 0.609416i
\(17\) 2.60168 − 4.50624i 0.630999 − 1.09292i −0.356349 − 0.934353i \(-0.615978\pi\)
0.987348 − 0.158570i \(-0.0506882\pi\)
\(18\) −3.32729 − 1.62014i −0.784251 − 0.381871i
\(19\) 0.164107 − 4.35581i 0.0376488 − 0.999291i
\(20\) −0.131888 + 0.228437i −0.0294911 + 0.0510801i
\(21\) −5.47369 − 1.26182i −1.19446 − 0.275351i
\(22\) 6.62876 1.41326
\(23\) −1.49233 + 2.58480i −0.311173 + 0.538967i −0.978617 − 0.205693i \(-0.934055\pi\)
0.667444 + 0.744660i \(0.267388\pi\)
\(24\) 1.54980 + 5.06326i 0.316352 + 1.03353i
\(25\) −4.69580 −0.939160
\(26\) −4.34704 −0.852525
\(27\) −4.04137 − 3.26609i −0.777762 − 0.628559i
\(28\) 0.775514 + 1.34323i 0.146558 + 0.253847i
\(29\) 5.49841 1.02103 0.510515 − 0.859869i \(-0.329455\pi\)
0.510515 + 0.859869i \(0.329455\pi\)
\(30\) 0.803420 − 0.862127i 0.146684 − 0.157402i
\(31\) −2.54567 − 4.40923i −0.457216 − 0.791921i 0.541597 − 0.840638i \(-0.317820\pi\)
−0.998813 + 0.0487176i \(0.984487\pi\)
\(32\) 1.32101 − 2.28806i 0.233524 − 0.404475i
\(33\) 9.06939 + 2.09071i 1.57878 + 0.363947i
\(34\) 6.41881 1.10082
\(35\) 0.894360 − 1.54908i 0.151174 − 0.261842i
\(36\) 0.101019 + 1.43119i 0.0168364 + 0.238532i
\(37\) 9.20826 1.51383 0.756914 − 0.653514i \(-0.226706\pi\)
0.756914 + 0.653514i \(0.226706\pi\)
\(38\) 4.75463 − 2.51133i 0.771302 − 0.407391i
\(39\) −5.94757 − 1.37106i −0.952374 − 0.219545i
\(40\) −1.68615 −0.266604
\(41\) 1.71003 0.267061 0.133531 − 0.991045i \(-0.457369\pi\)
0.133531 + 0.991045i \(0.457369\pi\)
\(42\) −2.02812 − 6.62595i −0.312946 − 1.02241i
\(43\) 1.79608 + 3.11091i 0.273900 + 0.474409i 0.969857 − 0.243675i \(-0.0783529\pi\)
−0.695957 + 0.718083i \(0.745020\pi\)
\(44\) −1.28495 − 2.22561i −0.193714 − 0.335523i
\(45\) 1.37114 − 0.926153i 0.204398 − 0.138063i
\(46\) −3.68186 −0.542860
\(47\) −10.6354 −1.55133 −0.775666 − 0.631143i \(-0.782586\pi\)
−0.775666 + 0.631143i \(0.782586\pi\)
\(48\) −3.32380 + 3.56667i −0.479749 + 0.514805i
\(49\) −1.75891 − 3.04653i −0.251273 − 0.435218i
\(50\) −2.89635 − 5.01663i −0.409606 − 0.709458i
\(51\) 8.78214 + 2.02450i 1.22975 + 0.283486i
\(52\) 0.842653 + 1.45952i 0.116855 + 0.202399i
\(53\) 0.562013 + 0.973435i 0.0771985 + 0.133712i 0.902040 − 0.431652i \(-0.142069\pi\)
−0.824842 + 0.565364i \(0.808736\pi\)
\(54\) 0.996529 − 6.33199i 0.135610 − 0.861675i
\(55\) −1.48187 + 2.56668i −0.199815 + 0.346091i
\(56\) −4.95735 + 8.58639i −0.662454 + 1.14740i
\(57\) 7.29730 − 1.93635i 0.966550 − 0.256476i
\(58\) 3.39140 + 5.87407i 0.445312 + 0.771303i
\(59\) −7.77857 −1.01268 −0.506342 − 0.862333i \(-0.669003\pi\)
−0.506342 + 0.862333i \(0.669003\pi\)
\(60\) −0.445198 − 0.102629i −0.0574748 − 0.0132493i
\(61\) −11.3745 −1.45635 −0.728176 − 0.685391i \(-0.759631\pi\)
−0.728176 + 0.685391i \(0.759631\pi\)
\(62\) 3.14032 − 5.43919i 0.398820 − 0.690777i
\(63\) −0.685027 − 9.70521i −0.0863052 − 1.22274i
\(64\) 8.88872 1.11109
\(65\) 0.971789 − 1.68319i 0.120536 − 0.208774i
\(66\) 3.36041 + 10.9786i 0.413638 + 1.35137i
\(67\) −1.18630 + 2.05473i −0.144929 + 0.251025i −0.929347 − 0.369209i \(-0.879629\pi\)
0.784417 + 0.620233i \(0.212962\pi\)
\(68\) −1.24426 − 2.15512i −0.150888 − 0.261346i
\(69\) −5.03748 − 1.16126i −0.606441 − 0.139799i
\(70\) 2.20655 0.263733
\(71\) −0.507763 + 0.879471i −0.0602604 + 0.104374i −0.894582 − 0.446904i \(-0.852526\pi\)
0.834321 + 0.551278i \(0.185860\pi\)
\(72\) −7.60013 + 5.13358i −0.895683 + 0.604998i
\(73\) −5.98562 + 10.3674i −0.700564 + 1.21341i 0.267704 + 0.963501i \(0.413735\pi\)
−0.968269 + 0.249912i \(0.919598\pi\)
\(74\) 5.67962 + 9.83738i 0.660242 + 1.14357i
\(75\) −2.38051 − 7.77720i −0.274877 − 0.898033i
\(76\) −1.76484 − 1.10956i −0.202441 − 0.127275i
\(77\) 8.71353 + 15.0923i 0.992999 + 1.71992i
\(78\) −2.20371 − 7.19958i −0.249521 − 0.815192i
\(79\) 0.568802 + 0.985193i 0.0639952 + 0.110843i 0.896248 − 0.443554i \(-0.146282\pi\)
−0.832253 + 0.554397i \(0.812949\pi\)
\(80\) −0.776234 − 1.34448i −0.0867856 − 0.150317i
\(81\) 3.36055 − 8.34905i 0.373395 − 0.927673i
\(82\) 1.05474 + 1.82686i 0.116476 + 0.201743i
\(83\) −1.14372 + 1.98099i −0.125540 + 0.217441i −0.921944 − 0.387324i \(-0.873400\pi\)
0.796404 + 0.604765i \(0.206733\pi\)
\(84\) −1.83152 + 1.96535i −0.199835 + 0.214437i
\(85\) −1.43494 + 2.48538i −0.155641 + 0.269578i
\(86\) −2.21563 + 3.83759i −0.238918 + 0.413818i
\(87\) 2.78738 + 9.10648i 0.298839 + 0.976318i
\(88\) 8.21387 − 14.2268i 0.875601 − 1.51659i
\(89\) 1.12141 + 1.94234i 0.118870 + 0.205888i 0.919320 − 0.393511i \(-0.128740\pi\)
−0.800450 + 0.599399i \(0.795406\pi\)
\(90\) 1.83515 + 0.893577i 0.193441 + 0.0941913i
\(91\) −5.71420 − 9.89729i −0.599011 − 1.03752i
\(92\) 0.713711 + 1.23618i 0.0744095 + 0.128881i
\(93\) 6.01206 − 6.45137i 0.623422 − 0.668976i
\(94\) −6.55987 − 11.3620i −0.676599 − 1.17190i
\(95\) −0.0905122 + 2.40242i −0.00928635 + 0.246483i
\(96\) 4.45917 + 1.02795i 0.455112 + 0.104914i
\(97\) −4.09100 − 7.08582i −0.415378 − 0.719456i 0.580090 − 0.814552i \(-0.303017\pi\)
−0.995468 + 0.0950967i \(0.969684\pi\)
\(98\) 2.16978 − 3.75817i 0.219181 − 0.379633i
\(99\) 1.13502 + 16.0806i 0.114074 + 1.61616i
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.106.11 ✓ 32
3.2 odd 2 513.2.g.c.505.6 32
9.4 even 3 171.2.h.c.49.6 yes 32
9.5 odd 6 513.2.h.c.334.11 32
19.7 even 3 171.2.h.c.7.6 yes 32
57.26 odd 6 513.2.h.c.235.11 32
171.121 even 3 inner 171.2.g.c.121.11 yes 32
171.140 odd 6 513.2.g.c.64.6 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.11 ✓ 32 1.1 even 1 trivial
171.2.g.c.121.11 yes 32 171.121 even 3 inner
171.2.h.c.7.6 yes 32 19.7 even 3
171.2.h.c.49.6 yes 32 9.4 even 3
513.2.g.c.64.6 32 171.140 odd 6
513.2.g.c.505.6 32 3.2 odd 2
513.2.h.c.235.11 32 57.26 odd 6
513.2.h.c.334.11 32 9.5 odd 6