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Properties

Label 117.2.f.a.61.4
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 61.4
Character \(\chi\) \(=\) 117.61
Dual form 117.2.f.a.94.4

$q$-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+(-0.567922 - 0.983670i) q^{2} +(-0.529547 - 1.64911i) q^{3} +(0.354929 - 0.614756i) q^{4} +(-0.0587384 - 0.101738i) q^{5} +(-1.32144 + 1.45747i) q^{6} +0.849445 q^{7} -3.07798 q^{8} +(-2.43916 + 1.74657i) q^{9} +(-0.0667177 + 0.115558i) q^{10} +(0.231971 + 0.401786i) q^{11} +(-1.20175 - 0.259777i) q^{12} +(1.54170 - 3.25932i) q^{13} +(-0.482419 - 0.835574i) q^{14} +(-0.136673 + 0.150741i) q^{15} +(1.03819 + 1.79820i) q^{16} +(-1.26296 - 2.18751i) q^{17} +(3.10330 + 1.40741i) q^{18} +(3.09113 + 5.35399i) q^{19} -0.0833920 q^{20} +(-0.449821 - 1.40083i) q^{21} +(0.263483 - 0.456366i) q^{22} +6.65773 q^{23} +(1.62993 + 5.07594i) 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} +(-0.952265 - 1.64937i) q^{29} +(0.225899 + 0.0488315i) q^{30} +(0.657577 + 1.13896i) q^{31} +(-1.89875 + 3.28874i) q^{32} +(0.539751 - 0.595312i) q^{33} +(-1.43452 + 2.48467i) q^{34} +(-0.0498951 - 0.0864208i) q^{35} +(0.207983 + 2.11940i) q^{36} +(-2.01347 + 3.48743i) q^{37} +(3.51104 - 6.08129i) q^{38} +(-6.19139 - 0.816478i) q^{39} +(0.180795 + 0.313147i) q^{40} +9.68663 q^{41} +(-1.12249 + 1.23804i) q^{42} -4.20953 q^{43} +0.329333 q^{44} +(0.320965 + 0.145564i) q^{45} +(-3.78107 - 6.54901i) q^{46} +(1.34586 - 2.33109i) q^{47} +(2.41567 - 2.66433i) q^{48} -6.27844 q^{49} -5.66354 q^{50} +(-2.93865 + 3.24115i) q^{51} +(-1.45649 - 2.10460i) q^{52} -0.389682 q^{53} +(0.677644 - 5.86299i) q^{54} +(0.0272512 - 0.0472005i) q^{55} -2.61457 q^{56} +(7.19244 - 7.93281i) q^{57} +(-1.08162 + 1.87343i) q^{58} +(-5.53661 + 9.58969i) q^{59} +(0.0441600 + 0.137523i) q^{60} -7.76759 q^{61} +(0.746905 - 1.29368i) q^{62} +(-2.07193 + 1.48361i) q^{63} +8.46614 q^{64} +(-0.422153 + 0.0345978i) q^{65} +(-0.892126 - 0.192846i) q^{66} -1.02270 q^{67} -1.79304 q^{68} +(-3.52558 - 10.9794i) q^{69} +(-0.0566730 + 0.0981605i) q^{70} +(3.61012 + 6.25291i) q^{71} +(7.50768 - 5.37589i) q^{72} -3.31321 q^{73} +4.57397 q^{74} +(-8.44138 - 1.82473i) q^{75} +4.38853 q^{76} +(0.197047 + 0.341295i) q^{77} +(2.71308 + 6.55398i) q^{78} +(-4.41302 + 7.64357i) q^{79} +(0.121963 - 0.211247i) q^{80} +(2.89900 - 8.52032i) q^{81} +(-5.50125 - 9.52844i) q^{82} +(1.75800 - 3.04495i) q^{83} +(-1.02082 - 0.220667i) q^{84} +(-0.148368 + 0.256981i) q^{85} +(2.39069 + 4.14079i) q^{86} +(-2.21573 + 2.44382i) q^{87} +(-0.714002 - 1.23669i) 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} +(1.53005 - 1.68755i) q^{93} -3.05736 q^{94} +(0.363136 - 0.628970i) q^{95} +(6.42898 + 1.38972i) q^{96} +15.7455 q^{97} +(3.56567 + 6.17591i) q^{98} +(-1.26756 - 0.574866i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.567922 − 0.983670i −0.401581 − 0.695559i 0.592336 − 0.805691i \(-0.298206\pi\)
−0.993917 + 0.110132i \(0.964873\pi\)
\(3\) −0.529547 − 1.64911i −0.305734 − 0.952117i
\(4\) 0.354929 − 0.614756i 0.177465 − 0.307378i
\(5\) −0.0587384 − 0.101738i −0.0262686 − 0.0454986i 0.852592 − 0.522577i \(-0.175029\pi\)
−0.878861 + 0.477078i \(0.841696\pi\)
\(6\) −1.32144 + 1.45747i −0.539477 + 0.595009i
\(7\) 0.849445 0.321060 0.160530 − 0.987031i \(-0.448680\pi\)
0.160530 + 0.987031i \(0.448680\pi\)
\(8\) −3.07798 −1.08823
\(9\) −2.43916 + 1.74657i −0.813053 + 0.582189i
\(10\) −0.0667177 + 0.115558i −0.0210980 + 0.0365428i
\(11\) 0.231971 + 0.401786i 0.0699419 + 0.121143i 0.898876 − 0.438204i \(-0.144385\pi\)
−0.828934 + 0.559347i \(0.811052\pi\)
\(12\) −1.20175 − 0.259777i −0.346917 − 0.0749912i
\(13\) 1.54170 − 3.25932i 0.427591 − 0.903972i
\(14\) −0.482419 − 0.835574i −0.128932 − 0.223316i
\(15\) −0.136673 + 0.150741i −0.0352888 + 0.0389213i
\(16\) 1.03819 + 1.79820i 0.259548 + 0.449550i
\(17\) −1.26296 − 2.18751i −0.306312 − 0.530548i 0.671240 − 0.741240i \(-0.265762\pi\)
−0.977553 + 0.210692i \(0.932428\pi\)
\(18\) 3.10330 + 1.40741i 0.731454 + 0.331730i
\(19\) 3.09113 + 5.35399i 0.709153 + 1.22829i 0.965172 + 0.261617i \(0.0842558\pi\)
−0.256019 + 0.966672i \(0.582411\pi\)
\(20\) −0.0833920 −0.0186470
\(21\) −0.449821 − 1.40083i −0.0981590 − 0.305687i
\(22\) 0.263483 − 0.456366i 0.0561748 − 0.0972975i
\(23\) 6.65773 1.38823 0.694117 − 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(24\) 1.62993 + 5.07594i 0.332709 + 1.03612i
\(25\) 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
\(29\) −0.952265 − 1.64937i −0.176831 − 0.306281i 0.763962 − 0.645261i \(-0.223251\pi\)
−0.940793 + 0.338980i \(0.889918\pi\)
\(30\) 0.225899 + 0.0488315i 0.0412434 + 0.00891537i
\(31\) 0.657577 + 1.13896i 0.118104 + 0.204563i 0.919016 − 0.394219i \(-0.128985\pi\)
−0.800912 + 0.598782i \(0.795652\pi\)
\(32\) −1.89875 + 3.28874i −0.335655 + 0.581372i
\(33\) 0.539751 − 0.595312i 0.0939586 − 0.103630i
\(34\) −1.43452 + 2.48467i −0.246019 + 0.426117i
\(35\) −0.0498951 − 0.0864208i −0.00843381 − 0.0146078i
\(36\) 0.207983 + 2.11940i 0.0346639 + 0.353233i
\(37\) −2.01347 + 3.48743i −0.331012 + 0.573330i −0.982711 − 0.185148i \(-0.940723\pi\)
0.651698 + 0.758478i \(0.274057\pi\)
\(38\) 3.51104 − 6.08129i 0.569565 − 0.986516i
\(39\) −6.19139 − 0.816478i −0.991417 − 0.130741i
\(40\) 0.180795 + 0.313147i 0.0285863 + 0.0495129i
\(41\) 9.68663 1.51280 0.756399 − 0.654111i \(-0.226957\pi\)
0.756399 + 0.654111i \(0.226957\pi\)
\(42\) −1.12249 + 1.23804i −0.173204 + 0.191034i
\(43\) −4.20953 −0.641948 −0.320974 − 0.947088i \(-0.604010\pi\)
−0.320974 + 0.947088i \(0.604010\pi\)
\(44\) 0.329333 0.0496489
\(45\) 0.320965 + 0.145564i 0.0478466 + 0.0216995i
\(46\) −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 \(0.103770\pi\)
\(48\) 2.41567 − 2.66433i 0.348672 − 0.384563i
\(49\) −6.27844 −0.896920
\(50\) −5.66354 −0.800946
\(51\) −2.93865 + 3.24115i −0.411494 + 0.453852i
\(52\) −1.45649 − 2.10460i −0.201979 − 0.291855i
\(53\) −0.389682 −0.0535269 −0.0267634 − 0.999642i \(-0.508520\pi\)
−0.0267634 + 0.999642i \(0.508520\pi\)
\(54\) 0.677644 − 5.86299i 0.0922156 − 0.797851i
\(55\) 0.0272512 − 0.0472005i 0.00367456 − 0.00636452i
\(56\) −2.61457 −0.349387
\(57\) 7.19244 − 7.93281i 0.952662 − 1.05073i
\(58\) −1.08162 + 1.87343i −0.142024 + 0.245993i
\(59\) −5.53661 + 9.58969i −0.720805 + 1.24847i 0.239872 + 0.970805i \(0.422895\pi\)
−0.960677 + 0.277667i \(0.910439\pi\)
\(60\) 0.0441600 + 0.137523i 0.00570103 + 0.0177541i
\(61\) −7.76759 −0.994538 −0.497269 − 0.867596i \(-0.665664\pi\)
−0.497269 + 0.867596i \(0.665664\pi\)
\(62\) 0.746905 − 1.29368i 0.0948570 − 0.164297i
\(63\) −2.07193 + 1.48361i −0.261039 + 0.186918i
\(64\) 8.46614 1.05827
\(65\) −0.422153 + 0.0345978i −0.0523617 + 0.00429133i
\(66\) −0.892126 − 0.192846i −0.109813 − 0.0237378i
\(67\) −1.02270 −0.124943 −0.0624715 − 0.998047i \(-0.519898\pi\)
−0.0624715 + 0.998047i \(0.519898\pi\)
\(68\) −1.79304 −0.217438
\(69\) −3.52558 − 10.9794i −0.424430 − 1.32176i
\(70\) −0.0566730 + 0.0981605i −0.00677372 + 0.0117324i
\(71\) 3.61012 + 6.25291i 0.428442 + 0.742083i 0.996735 − 0.0807430i \(-0.0257293\pi\)
−0.568293 + 0.822826i \(0.692396\pi\)
\(72\) 7.50768 − 5.37589i 0.884788 − 0.633555i
\(73\) −3.31321 −0.387782 −0.193891 − 0.981023i \(-0.562111\pi\)
−0.193891 + 0.981023i \(0.562111\pi\)
\(74\) 4.57397 0.531713
\(75\) −8.44138 − 1.82473i −0.974727 − 0.210702i
\(76\) 4.38853 0.503398
\(77\) 0.197047 + 0.341295i 0.0224556 + 0.0388942i
\(78\) 2.71308 + 6.55398i 0.307196 + 0.742092i
\(79\) −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
\(81\) 2.89900 − 8.52032i 0.322111 − 0.946702i
\(82\) −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 \(0.104856\pi\)
\(84\) −1.02082 − 0.220667i −0.111381 − 0.0240767i
\(85\) −0.148368 + 0.256981i −0.0160928 + 0.0278735i
\(86\) 2.39069 + 4.14079i 0.257794 + 0.446513i
\(87\) −2.21573 + 2.44382i −0.237552 + 0.262004i
\(88\) −0.714002 − 1.23669i −0.0761128 − 0.131831i
\(89\) −6.62760 + 11.4793i −0.702525 + 1.21681i 0.265053 + 0.964234i \(0.414611\pi\)
−0.967577 + 0.252574i \(0.918723\pi\)
\(90\) −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
\(93\) 1.53005 − 1.68755i 0.158659 − 0.174991i
\(94\) −3.05736 −0.315343
\(95\) 0.363136 − 0.628970i 0.0372569 − 0.0645309i
\(96\) 6.42898 + 1.38972i 0.656155 + 0.141838i
\(97\) 15.7455 1.59871 0.799354 − 0.600860i \(-0.205175\pi\)
0.799354 + 0.600860i \(0.205175\pi\)
\(98\) 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.f.a.61.4 ✓ 24
3.2 odd 2 351.2.f.a.100.9 24
9.4 even 3 117.2.h.a.22.9 yes 24
9.5 odd 6 351.2.h.a.334.4 24
13.3 even 3 117.2.h.a.16.9 yes 24
39.29 odd 6 351.2.h.a.289.4 24
117.68 odd 6 351.2.f.a.172.9 24
117.94 even 3 inner 117.2.f.a.94.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.4 ✓ 24 1.1 even 1 trivial
117.2.f.a.94.4 yes 24 117.94 even 3 inner
117.2.h.a.16.9 yes 24 13.3 even 3
117.2.h.a.22.9 yes 24 9.4 even 3
351.2.f.a.100.9 24 3.2 odd 2
351.2.f.a.172.9 24 117.68 odd 6
351.2.h.a.289.4 24 39.29 odd 6
351.2.h.a.334.4 24 9.5 odd 6