↑

Properties

Label 117.2.h.a.16.9
Level $117$
Weight $2$
Character 117.16
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

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(16,117)] 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("117.16"); 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, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.h (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 16.9
Character \(\chi\) \(=\) 117.16
Dual form 117.2.h.a.22.9

$q$-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+1.13584 q^{2} +(1.69295 + 0.365956i) q^{3} -0.709859 q^{4} +(-0.0587384 - 0.101738i) q^{5} +(1.92293 + 0.415669i) q^{6} +(-0.424723 - 0.735641i) q^{7} -3.07798 q^{8} +(2.73215 + 1.23909i) q^{9} +(-0.0667177 - 0.115558i) q^{10} -0.463942 q^{11} +(-1.20175 - 0.259777i) q^{12} +(-3.59350 - 0.294508i) q^{13} +(-0.482419 - 0.835574i) q^{14} +(-0.0622095 - 0.193733i) q^{15} -2.07638 q^{16} +(-1.26296 + 2.18751i) q^{17} +(3.10330 + 1.40741i) q^{18} +(3.09113 - 5.35399i) q^{19} +(0.0416960 + 0.0722196i) q^{20} +(-0.449821 - 1.40083i) q^{21} -0.526966 q^{22} +(-3.32887 + 5.76577i) q^{23} +(-5.21086 - 1.12640i) q^{24} +(2.49310 - 4.31818i) q^{25} +(-4.08166 - 0.334515i) q^{26} +(4.17194 + 3.09756i) q^{27} +(0.301493 + 0.522201i) q^{28} +1.90453 q^{29} +(-0.0706603 - 0.220050i) q^{30} +(0.657577 + 1.13896i) q^{31} +3.79751 q^{32} +(-0.785431 - 0.169783i) q^{33} +(-1.43452 + 2.48467i) q^{34} +(-0.0498951 + 0.0864208i) q^{35} +(-1.93944 - 0.879579i) q^{36} +(-2.01347 - 3.48743i) q^{37} +(3.51104 - 6.08129i) q^{38} +(-5.97584 - 1.81365i) q^{39} +(0.180795 + 0.313147i) q^{40} +(-4.84331 + 8.38887i) q^{41} +(-0.510927 - 1.59113i) q^{42} +(2.10477 + 3.64556i) q^{43} +0.329333 q^{44} +(-0.0344198 - 0.350746i) q^{45} +(-3.78107 + 6.54901i) q^{46} +(1.34586 - 2.33109i) q^{47} +(-3.51521 - 0.759865i) q^{48} +(3.13922 - 5.43729i) q^{49} +(2.83177 - 4.90477i) q^{50} +(-2.93865 + 3.24115i) q^{51} +(2.55088 + 0.209059i) q^{52} -0.389682 q^{53} +(4.73867 + 3.51835i) q^{54} +(0.0272512 + 0.0472005i) q^{55} +(1.30729 + 2.26429i) q^{56} +(7.19244 - 7.93281i) q^{57} +2.16325 q^{58} +11.0732 q^{59} +(0.0441600 + 0.137523i) q^{60} +(3.88380 + 6.72693i) q^{61} +(0.746905 + 1.29368i) q^{62} +(-0.248881 - 2.53615i) q^{63} +8.46614 q^{64} +(0.181114 + 0.382895i) q^{65} +(-0.892126 - 0.192846i) q^{66} +(0.511351 - 0.885686i) q^{67} +(0.896521 - 1.55282i) q^{68} +(-7.74562 + 8.54293i) q^{69} +(-0.0566730 + 0.0981605i) q^{70} +(3.61012 - 6.25291i) q^{71} +(-8.40950 - 3.81389i) q^{72} -3.31321 q^{73} +(-2.28699 - 3.96117i) q^{74} +(5.80095 - 6.39808i) q^{75} +(-2.19426 + 3.80057i) q^{76} +(0.197047 + 0.341295i) q^{77} +(-6.78762 - 2.06002i) q^{78} +(-4.41302 + 7.64357i) q^{79} +(0.121963 + 0.211247i) q^{80} +(5.92931 + 6.77077i) q^{81} +(-5.50125 + 9.52844i) q^{82} +(1.75800 - 3.04495i) q^{83} +(0.319310 + 0.994393i) q^{84} +0.296736 q^{85} +(2.39069 + 4.14079i) q^{86} +(3.22427 + 0.696975i) q^{87} +1.42800 q^{88} +(-6.62760 - 11.4793i) q^{89} +(-0.0390955 - 0.398392i) q^{90} +(1.30959 + 2.76861i) q^{91} +(2.36303 - 4.09288i) q^{92} +(0.696436 + 2.16884i) q^{93} +(1.52868 - 2.64776i) q^{94} -0.726272 q^{95} +(6.42898 + 1.38972i) q^{96} +(-7.87273 - 13.6360i) q^{97} +(3.56567 - 6.17591i) q^{98} +(-1.26756 - 0.574866i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} - q^{3} + 18 q^{4} - 2 q^{5} - 12 q^{6} + 3 q^{7} - 18 q^{8} - 3 q^{9} + 6 q^{11} - 3 q^{12} + 2 q^{14} + 11 q^{15} + 6 q^{16} + 6 q^{17} - 8 q^{18} - 3 q^{19} - 11 q^{20} - 25 q^{21} - 18 q^{22}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.13584 0.803163 0.401581 − 0.915823i \(-0.368461\pi\)
0.401581 + 0.915823i \(0.368461\pi\)
\(3\) 1.69295 + 0.365956i 0.977425 + 0.211285i
\(4\) −0.709859 −0.354929
\(5\) −0.0587384 − 0.101738i −0.0262686 − 0.0454986i 0.852592 − 0.522577i \(-0.175029\pi\)
−0.878861 + 0.477078i \(0.841696\pi\)
\(6\) 1.92293 + 0.415669i 0.785031 + 0.169696i
\(7\) −0.424723 − 0.735641i −0.160530 − 0.278046i 0.774529 − 0.632539i \(-0.217987\pi\)
−0.935059 + 0.354492i \(0.884654\pi\)
\(8\) −3.07798 −1.08823
\(9\) 2.73215 + 1.23909i 0.910717 + 0.413030i
\(10\) −0.0667177 − 0.115558i −0.0210980 − 0.0365428i
\(11\) −0.463942 −0.139884 −0.0699419 − 0.997551i \(-0.522281\pi\)
−0.0699419 + 0.997551i \(0.522281\pi\)
\(12\) −1.20175 − 0.259777i −0.346917 − 0.0749912i
\(13\) −3.59350 − 0.294508i −0.996658 − 0.0816817i
\(14\) −0.482419 − 0.835574i −0.128932 − 0.223316i
\(15\) −0.0622095 − 0.193733i −0.0160624 − 0.0500216i
\(16\) −2.07638 −0.519096
\(17\) −1.26296 + 2.18751i −0.306312 + 0.530548i −0.977553 − 0.210692i \(-0.932428\pi\)
0.671240 + 0.741240i \(0.265762\pi\)
\(18\) 3.10330 + 1.40741i 0.731454 + 0.331730i
\(19\) 3.09113 − 5.35399i 0.709153 − 1.22829i −0.256019 − 0.966672i \(-0.582411\pi\)
0.965172 − 0.261617i \(-0.0842558\pi\)
\(20\) 0.0416960 + 0.0722196i 0.00932350 + 0.0161488i
\(21\) −0.449821 − 1.40083i −0.0981590 − 0.305687i
\(22\) −0.526966 −0.112350
\(23\) −3.32887 + 5.76577i −0.694117 + 1.20225i 0.276361 + 0.961054i \(0.410872\pi\)
−0.970478 + 0.241192i \(0.922462\pi\)
\(24\) −5.21086 − 1.12640i −1.06366 − 0.229926i
\(25\) 2.49310 − 4.31818i 0.498620 − 0.863635i
\(26\) −4.08166 − 0.334515i −0.800479 − 0.0656037i
\(27\) 4.17194 + 3.09756i 0.802890 + 0.596127i
\(28\) 0.301493 + 0.522201i 0.0569768 + 0.0986868i
\(29\) 1.90453 0.353662 0.176831 − 0.984241i \(-0.443415\pi\)
0.176831 + 0.984241i \(0.443415\pi\)
\(30\) −0.0706603 − 0.220050i −0.0129007 − 0.0401755i
\(31\) 0.657577 + 1.13896i 0.118104 + 0.204563i 0.919016 − 0.394219i \(-0.128985\pi\)
−0.800912 + 0.598782i \(0.795652\pi\)
\(32\) 3.79751 0.671310
\(33\) −0.785431 − 0.169783i −0.136726 − 0.0295553i
\(34\) −1.43452 + 2.48467i −0.246019 + 0.426117i
\(35\) −0.0498951 + 0.0864208i −0.00843381 + 0.0146078i
\(36\) −1.93944 − 0.879579i −0.323240 − 0.146597i
\(37\) −2.01347 − 3.48743i −0.331012 − 0.573330i 0.651698 − 0.758478i \(-0.274057\pi\)
−0.982711 + 0.185148i \(0.940723\pi\)
\(38\) 3.51104 − 6.08129i 0.569565 − 0.986516i
\(39\) −5.97584 − 1.81365i −0.956900 − 0.290417i
\(40\) 0.180795 + 0.313147i 0.0285863 + 0.0495129i
\(41\) −4.84331 + 8.38887i −0.756399 + 1.31012i 0.188277 + 0.982116i \(0.439710\pi\)
−0.944676 + 0.328005i \(0.893624\pi\)
\(42\) −0.510927 − 1.59113i −0.0788377 − 0.245516i
\(43\) 2.10477 + 3.64556i 0.320974 + 0.555943i 0.980689 − 0.195573i \(-0.0626565\pi\)
−0.659715 + 0.751516i \(0.729323\pi\)
\(44\) 0.329333 0.0496489
\(45\) −0.0344198 − 0.350746i −0.00513100 − 0.0522861i
\(46\) −3.78107 + 6.54901i −0.557489 + 0.965599i
\(47\) 1.34586 − 2.33109i 0.196313 − 0.340025i −0.751017 − 0.660283i \(-0.770436\pi\)
0.947330 + 0.320258i \(0.103770\pi\)
\(48\) −3.51521 − 0.759865i −0.507377 − 0.109677i
\(49\) 3.13922 − 5.43729i 0.448460 − 0.776756i
\(50\) 2.83177 − 4.90477i 0.400473 − 0.693640i
\(51\) −2.93865 + 3.24115i −0.411494 + 0.453852i
\(52\) 2.55088 + 0.209059i 0.353743 + 0.0289912i
\(53\) −0.389682 −0.0535269 −0.0267634 − 0.999642i \(-0.508520\pi\)
−0.0267634 + 0.999642i \(0.508520\pi\)
\(54\) 4.73867 + 3.51835i 0.644852 + 0.478787i
\(55\) 0.0272512 + 0.0472005i 0.00367456 + 0.00636452i
\(56\) 1.30729 + 2.26429i 0.174693 + 0.302578i
\(57\) 7.19244 − 7.93281i 0.952662 − 1.05073i
\(58\) 2.16325 0.284049
\(59\) 11.0732 1.44161 0.720805 − 0.693137i \(-0.243772\pi\)
0.720805 + 0.693137i \(0.243772\pi\)
\(60\) 0.0441600 + 0.137523i 0.00570103 + 0.0177541i
\(61\) 3.88380 + 6.72693i 0.497269 + 0.861295i 0.999995 − 0.00315044i \(-0.00100282\pi\)
−0.502726 + 0.864446i \(0.667669\pi\)
\(62\) 0.746905 + 1.29368i 0.0948570 + 0.164297i
\(63\) −0.248881 − 2.53615i −0.0313560 − 0.319525i
\(64\) 8.46614 1.05827
\(65\) 0.181114 + 0.382895i 0.0224644 + 0.0474922i
\(66\) −0.892126 − 0.192846i −0.109813 − 0.0237378i
\(67\) 0.511351 − 0.885686i 0.0624715 − 0.108204i −0.833098 − 0.553125i \(-0.813435\pi\)
0.895570 + 0.444922i \(0.146768\pi\)
\(68\) 0.896521 − 1.55282i 0.108719 − 0.188307i
\(69\) −7.74562 + 8.54293i −0.932463 + 1.02845i
\(70\) −0.0566730 + 0.0981605i −0.00677372 + 0.0117324i
\(71\) 3.61012 − 6.25291i 0.428442 − 0.742083i −0.568293 − 0.822826i \(-0.692396\pi\)
0.996735 + 0.0807430i \(0.0257293\pi\)
\(72\) −8.40950 − 3.81389i −0.991069 − 0.449471i
\(73\) −3.31321 −0.387782 −0.193891 − 0.981023i \(-0.562111\pi\)
−0.193891 + 0.981023i \(0.562111\pi\)
\(74\) −2.28699 − 3.96117i −0.265857 − 0.460477i
\(75\) 5.80095 − 6.39808i 0.669836 − 0.738787i
\(76\) −2.19426 + 3.80057i −0.251699 + 0.435956i
\(77\) 0.197047 + 0.341295i 0.0224556 + 0.0388942i
\(78\) −6.78762 − 2.06002i −0.768547 − 0.233252i
\(79\) −4.41302 + 7.64357i −0.496503 + 0.859969i −0.999992 − 0.00403289i \(-0.998716\pi\)
0.503489 + 0.864002i \(0.332050\pi\)
\(80\) 0.121963 + 0.211247i 0.0136359 + 0.0236181i
\(81\) 5.92931 + 6.77077i 0.658812 + 0.752307i
\(82\) −5.50125 + 9.52844i −0.607511 + 1.05224i
\(83\) 1.75800 − 3.04495i 0.192966 − 0.334227i −0.753266 − 0.657716i \(-0.771523\pi\)
0.946232 + 0.323489i \(0.104856\pi\)
\(84\) 0.319310 + 0.994393i 0.0348395 + 0.108497i
\(85\) 0.296736 0.0321856
\(86\) 2.39069 + 4.14079i 0.257794 + 0.446513i
\(87\) 3.22427 + 0.696975i 0.345678 + 0.0747236i
\(88\) 1.42800 0.152226
\(89\) −6.62760 − 11.4793i −0.702525 − 1.21681i −0.967577 − 0.252574i \(-0.918723\pi\)
0.265053 − 0.964234i \(-0.414611\pi\)
\(90\) −0.0390955 − 0.398392i −0.00412103 − 0.0419942i
\(91\) 1.30959 + 2.76861i 0.137282 + 0.290230i
\(92\) 2.36303 − 4.09288i 0.246362 − 0.426712i
\(93\) 0.696436 + 2.16884i 0.0722170 + 0.224898i
\(94\) 1.52868 − 2.64776i 0.157672 − 0.273095i
\(95\) −0.726272 −0.0745139
\(96\) 6.42898 + 1.38972i 0.656155 + 0.141838i
\(97\) −7.87273 − 13.6360i −0.799354 − 1.38452i −0.920037 − 0.391831i \(-0.871842\pi\)
0.120683 − 0.992691i \(-0.461492\pi\)
\(98\) 3.56567 − 6.17591i 0.360187 − 0.623861i
\(99\) −1.26756 − 0.574866i −0.127395 − 0.0577762i
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.h.a.16.9 yes 24
3.2 odd 2 351.2.h.a.289.4 24
9.4 even 3 117.2.f.a.94.4 yes 24
9.5 odd 6 351.2.f.a.172.9 24
13.9 even 3 117.2.f.a.61.4 ✓ 24
39.35 odd 6 351.2.f.a.100.9 24
117.22 even 3 inner 117.2.h.a.22.9 yes 24
117.113 odd 6 351.2.h.a.334.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.4 ✓ 24 13.9 even 3
117.2.f.a.94.4 yes 24 9.4 even 3
117.2.h.a.16.9 yes 24 1.1 even 1 trivial
117.2.h.a.22.9 yes 24 117.22 even 3 inner
351.2.f.a.100.9 24 39.35 odd 6
351.2.f.a.172.9 24 9.5 odd 6
351.2.h.a.289.4 24 3.2 odd 2
351.2.h.a.334.4 24 117.113 odd 6