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Properties

Label 117.2.f.a.61.10
Level $117$
Weight $2$
Character 117.61
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
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] = [117,2,Mod(61,117)] 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("117.61"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 61.10
Character \(\chi\) \(=\) 117.61
Dual form 117.2.f.a.94.10

$q$-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+(1.00395 + 1.73890i) q^{2} +(1.56525 + 0.741609i) q^{3} +(-1.01584 + 1.75949i) q^{4} +(-1.37329 - 2.37860i) q^{5} +(0.281858 + 3.46635i) q^{6} -2.23810 q^{7} -0.0636126 q^{8} +(1.90003 + 2.32161i) q^{9} +(2.75743 - 4.77600i) q^{10} +(-2.31729 - 4.01367i) q^{11} +(-2.89490 + 2.00069i) q^{12} +(-3.57749 + 0.448963i) q^{13} +(-2.24694 - 3.89182i) q^{14} +(-0.385547 - 4.74155i) q^{15} +(1.96782 + 3.40836i) q^{16} +(-0.925543 - 1.60309i) q^{17} +(-2.12950 + 5.63475i) q^{18} +(3.74647 + 6.48908i) q^{19} +5.58016 q^{20} +(-3.50319 - 1.65979i) q^{21} +(4.65290 - 8.05906i) q^{22} +1.89606 q^{23} +(-0.0995698 - 0.0471757i) q^{24} +(-1.27182 + 2.20287i) q^{25} +(-4.37233 - 5.77015i) q^{26} +(1.25230 + 5.04299i) q^{27} +(2.27355 - 3.93791i) q^{28} +(0.859165 + 1.48812i) q^{29} +(7.85799 - 5.43072i) q^{30} +(-0.375036 - 0.649582i) q^{31} +(-4.01480 + 6.95384i) q^{32} +(-0.650575 - 8.00092i) q^{33} +(1.85840 - 3.21885i) q^{34} +(3.07355 + 5.32354i) q^{35} +(-6.01498 + 0.984696i) q^{36} +(1.82852 - 3.16709i) q^{37} +(-7.52256 + 13.0295i) q^{38} +(-5.93263 - 1.95036i) q^{39} +(0.0873583 + 0.151309i) q^{40} -3.68497 q^{41} +(-0.630825 - 7.75804i) q^{42} +4.10049 q^{43} +9.41599 q^{44} +(2.91290 - 7.70765i) q^{45} +(1.90355 + 3.29705i) q^{46} +(3.36320 - 5.82524i) q^{47} +(0.552461 + 6.79429i) q^{48} -1.99091 q^{49} -5.10741 q^{50} +(-0.259845 - 3.19563i) q^{51} +(2.84421 - 6.75062i) q^{52} +2.52368 q^{53} +(-7.51198 + 7.24054i) q^{54} +(-6.36460 + 11.0238i) q^{55} +0.142371 q^{56} +(1.05182 + 12.9355i) q^{57} +(-1.72512 + 2.98800i) q^{58} +(-4.92575 + 8.53165i) q^{59} +(8.73435 + 4.13829i) q^{60} -9.57416 q^{61} +(0.753037 - 1.30430i) q^{62} +(-4.25246 - 5.19599i) q^{63} -8.25141 q^{64} +(5.98082 + 7.89286i) q^{65} +(13.2596 - 9.16383i) q^{66} +12.2572 q^{67} +3.76082 q^{68} +(2.96781 + 1.40614i) q^{69} +(-6.17139 + 10.6892i) q^{70} +(-7.04017 - 12.1939i) q^{71} +(-0.120866 - 0.147684i) q^{72} -11.2098 q^{73} +7.34299 q^{74} +(-3.62439 + 2.50484i) q^{75} -15.2233 q^{76} +(5.18632 + 8.98298i) q^{77} +(-2.56461 - 12.2743i) q^{78} +(-0.753706 + 1.30546i) q^{79} +(5.40475 - 9.36130i) q^{80} +(-1.77976 + 8.82227i) q^{81} +(-3.69953 - 6.40778i) q^{82} +(6.58818 - 11.4111i) q^{83} +(6.47907 - 4.47773i) q^{84} +(-2.54207 + 4.40299i) q^{85} +(4.11669 + 7.13032i) q^{86} +(0.241209 + 2.96644i) q^{87} +(0.147409 + 0.255320i) q^{88} +(-2.97704 + 5.15639i) q^{89} +(16.3272 - 2.67289i) q^{90} +(8.00677 - 1.00482i) q^{91} +(-1.92609 + 3.33609i) q^{92} +(-0.105291 - 1.29489i) q^{93} +13.5060 q^{94} +(10.2900 - 17.8227i) q^{95} +(-11.4412 + 7.90710i) q^{96} +6.13769 q^{97} +(-1.99878 - 3.46199i) q^{98} +(4.91524 - 13.0059i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + q^{2} - q^{3} - 9 q^{4} - 2 q^{5} + 9 q^{6} - 6 q^{7} - 18 q^{8} - 3 q^{9} - 3 q^{11} - 3 q^{12} + 2 q^{14} + 8 q^{15} - 3 q^{16} + 6 q^{17} - 8 q^{18} - 3 q^{19} + 22 q^{20} - 25 q^{21} + 9 q^{22}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00395 + 1.73890i 0.709901 + 1.22959i 0.964893 + 0.262642i \(0.0845939\pi\)
−0.254992 + 0.966943i \(0.582073\pi\)
\(3\) 1.56525 + 0.741609i 0.903699 + 0.428168i
\(4\) −1.01584 + 1.75949i −0.507920 + 0.879744i
\(5\) −1.37329 − 2.37860i −0.614152 − 1.06374i −0.990533 − 0.137277i \(-0.956165\pi\)
0.376381 − 0.926465i \(-0.377168\pi\)
\(6\) 0.281858 + 3.46635i 0.115068 + 1.41513i
\(7\) −2.23810 −0.845922 −0.422961 − 0.906148i \(-0.639009\pi\)
−0.422961 + 0.906148i \(0.639009\pi\)
\(8\) −0.0636126 −0.0224905
\(9\) 1.90003 + 2.32161i 0.633344 + 0.773870i
\(10\) 2.75743 − 4.77600i 0.871975 − 1.51030i
\(11\) −2.31729 − 4.01367i −0.698689 − 1.21017i −0.968921 − 0.247370i \(-0.920434\pi\)
0.270232 − 0.962795i \(-0.412900\pi\)
\(12\) −2.89490 + 2.00069i −0.835685 + 0.577548i
\(13\) −3.57749 + 0.448963i −0.992217 + 0.124520i
\(14\) −2.24694 − 3.89182i −0.600521 − 1.04013i
\(15\) −0.385547 − 4.74155i −0.0995479 − 1.22426i
\(16\) 1.96782 + 3.40836i 0.491954 + 0.852090i
\(17\) −0.925543 − 1.60309i −0.224477 − 0.388806i 0.731685 − 0.681643i \(-0.238734\pi\)
−0.956162 + 0.292837i \(0.905401\pi\)
\(18\) −2.12950 + 5.63475i −0.501928 + 1.32812i
\(19\) 3.74647 + 6.48908i 0.859500 + 1.48870i 0.872407 + 0.488781i \(0.162558\pi\)
−0.0129070 + 0.999917i \(0.504109\pi\)
\(20\) 5.58016 1.24776
\(21\) −3.50319 − 1.65979i −0.764459 − 0.362197i
\(22\) 4.65290 − 8.05906i 0.992001 − 1.71820i
\(23\) 1.89606 0.395356 0.197678 − 0.980267i \(-0.436660\pi\)
0.197678 + 0.980267i \(0.436660\pi\)
\(24\) −0.0995698 − 0.0471757i −0.0203246 − 0.00962970i
\(25\) −1.27182 + 2.20287i −0.254365 + 0.440573i
\(26\) −4.37233 − 5.77015i −0.857484 − 1.13162i
\(27\) 1.25230 + 5.04299i 0.241006 + 0.970524i
\(28\) 2.27355 − 3.93791i 0.429661 − 0.744194i
\(29\) 0.859165 + 1.48812i 0.159543 + 0.276336i 0.934704 − 0.355427i \(-0.115665\pi\)
−0.775161 + 0.631764i \(0.782331\pi\)
\(30\) 7.85799 − 5.43072i 1.43467 − 0.991509i
\(31\) −0.375036 − 0.649582i −0.0673585 − 0.116668i 0.830379 − 0.557199i \(-0.188124\pi\)
−0.897738 + 0.440530i \(0.854790\pi\)
\(32\) −4.01480 + 6.95384i −0.709723 + 1.22928i
\(33\) −0.650575 − 8.00092i −0.113251 − 1.39278i
\(34\) 1.85840 − 3.21885i 0.318713 − 0.552028i
\(35\) 3.07355 + 5.32354i 0.519524 + 0.899843i
\(36\) −6.01498 + 0.984696i −1.00250 + 0.164116i
\(37\) 1.82852 − 3.16709i 0.300607 − 0.520666i −0.675667 − 0.737207i \(-0.736144\pi\)
0.976274 + 0.216541i \(0.0694774\pi\)
\(38\) −7.52256 + 13.0295i −1.22032 + 2.11366i
\(39\) −5.93263 − 1.95036i −0.949981 − 0.312307i
\(40\) 0.0873583 + 0.151309i 0.0138126 + 0.0239240i
\(41\) −3.68497 −0.575495 −0.287748 − 0.957706i \(-0.592906\pi\)
−0.287748 + 0.957706i \(0.592906\pi\)
\(42\) −0.630825 − 7.75804i −0.0973385 − 1.19709i
\(43\) 4.10049 0.625318 0.312659 − 0.949865i \(-0.398780\pi\)
0.312659 + 0.949865i \(0.398780\pi\)
\(44\) 9.41599 1.41951
\(45\) 2.91290 − 7.70765i 0.434229 − 1.14899i
\(46\) 1.90355 + 3.29705i 0.280664 + 0.486124i
\(47\) 3.36320 − 5.82524i 0.490573 − 0.849698i −0.509368 − 0.860549i \(-0.670121\pi\)
0.999941 + 0.0108509i \(0.00345403\pi\)
\(48\) 0.552461 + 6.79429i 0.0797409 + 0.980672i
\(49\) −1.99091 −0.284416
\(50\) −5.10741 −0.722296
\(51\) −0.259845 − 3.19563i −0.0363855 − 0.447478i
\(52\) 2.84421 − 6.75062i 0.394421 − 0.936143i
\(53\) 2.52368 0.346654 0.173327 − 0.984864i \(-0.444548\pi\)
0.173327 + 0.984864i \(0.444548\pi\)
\(54\) −7.51198 + 7.24054i −1.02225 + 0.985313i
\(55\) −6.36460 + 11.0238i −0.858203 + 1.48645i
\(56\) 0.142371 0.0190252
\(57\) 1.05182 + 12.9355i 0.139316 + 1.71334i
\(58\) −1.72512 + 2.98800i −0.226519 + 0.392343i
\(59\) −4.92575 + 8.53165i −0.641278 + 1.11073i 0.343870 + 0.939017i \(0.388262\pi\)
−0.985148 + 0.171709i \(0.945071\pi\)
\(60\) 8.73435 + 4.13829i 1.12760 + 0.534251i
\(61\) −9.57416 −1.22585 −0.612923 − 0.790143i \(-0.710006\pi\)
−0.612923 + 0.790143i \(0.710006\pi\)
\(62\) 0.753037 − 1.30430i 0.0956358 − 0.165646i
\(63\) −4.25246 − 5.19599i −0.535759 − 0.654634i
\(64\) −8.25141 −1.03143
\(65\) 5.98082 + 7.89286i 0.741829 + 0.978989i
\(66\) 13.2596 − 9.16383i 1.63215 − 1.12799i
\(67\) 12.2572 1.49746 0.748728 − 0.662877i \(-0.230665\pi\)
0.748728 + 0.662877i \(0.230665\pi\)
\(68\) 3.76082 0.456066
\(69\) 2.96781 + 1.40614i 0.357283 + 0.169279i
\(70\) −6.17139 + 10.6892i −0.737622 + 1.27760i
\(71\) −7.04017 − 12.1939i −0.835514 − 1.44715i −0.893611 − 0.448842i \(-0.851837\pi\)
0.0580970 − 0.998311i \(-0.481497\pi\)
\(72\) −0.120866 − 0.147684i −0.0142442 − 0.0174047i
\(73\) −11.2098 −1.31200 −0.656001 − 0.754760i \(-0.727753\pi\)
−0.656001 + 0.754760i \(0.727753\pi\)
\(74\) 7.34299 0.853605
\(75\) −3.62439 + 2.50484i −0.418509 + 0.289234i
\(76\) −15.2233 −1.74623
\(77\) 5.18632 + 8.98298i 0.591037 + 1.02371i
\(78\) −2.56461 − 12.2743i −0.290385 − 1.38979i
\(79\) −0.753706 + 1.30546i −0.0847985 + 0.146875i −0.905305 − 0.424761i \(-0.860358\pi\)
0.820507 + 0.571637i \(0.193691\pi\)
\(80\) 5.40475 − 9.36130i 0.604269 − 1.04663i
\(81\) −1.77976 + 8.82227i −0.197751 + 0.980252i
\(82\) −3.69953 − 6.40778i −0.408545 − 0.707621i
\(83\) 6.58818 − 11.4111i 0.723146 − 1.25253i −0.236586 − 0.971611i \(-0.576028\pi\)
0.959732 − 0.280916i \(-0.0906382\pi\)
\(84\) 6.47907 − 4.47773i 0.706924 − 0.488561i
\(85\) −2.54207 + 4.40299i −0.275726 + 0.477572i
\(86\) 4.11669 + 7.13032i 0.443914 + 0.768882i
\(87\) 0.241209 + 2.96644i 0.0258603 + 0.318036i
\(88\) 0.147409 + 0.255320i 0.0157138 + 0.0272172i
\(89\) −2.97704 + 5.15639i −0.315566 + 0.546576i −0.979558 − 0.201164i \(-0.935528\pi\)
0.663992 + 0.747740i \(0.268861\pi\)
\(90\) 16.3272 − 2.67289i 1.72104 − 0.281747i
\(91\) 8.00677 − 1.00482i 0.839338 − 0.105334i
\(92\) −1.92609 + 3.33609i −0.200809 + 0.347812i
\(93\) −0.105291 − 1.29489i −0.0109181 − 0.134274i
\(94\) 13.5060 1.39304
\(95\) 10.2900 − 17.8227i 1.05573 − 1.82857i
\(96\) −11.4412 + 7.90710i −1.16771 + 0.807015i
\(97\) 6.13769 0.623188 0.311594 − 0.950215i \(-0.399137\pi\)
0.311594 + 0.950215i \(0.399137\pi\)
\(98\) −1.99878 − 3.46199i −0.201908 − 0.349714i
\(99\) 4.91524 − 13.0059i 0.494001 − 1.30715i
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 117.2.f.a.61.10 ✓ 24
3.2 odd 2 351.2.f.a.100.3 24
9.4 even 3 117.2.h.a.22.3 yes 24
9.5 odd 6 351.2.h.a.334.10 24
13.3 even 3 117.2.h.a.16.3 yes 24
39.29 odd 6 351.2.h.a.289.10 24
117.68 odd 6 351.2.f.a.172.3 24
117.94 even 3 inner 117.2.f.a.94.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.10 ✓ 24 1.1 even 1 trivial
117.2.f.a.94.10 yes 24 117.94 even 3 inner
117.2.h.a.16.3 yes 24 13.3 even 3
117.2.h.a.22.3 yes 24 9.4 even 3
351.2.f.a.100.3 24 3.2 odd 2
351.2.f.a.172.3 24 117.68 odd 6
351.2.h.a.289.10 24 39.29 odd 6
351.2.h.a.334.10 24 9.5 odd 6