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Properties

Label 117.2.h.a.16.1
Level $117$
Weight $2$
Character 117.16
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(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 16.1
Character \(\chi\) \(=\) 117.16
Dual form 117.2.h.a.22.1

$q$-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-2.65628 q^{2} +(1.38934 + 1.03428i) q^{3} +5.05585 q^{4} +(0.324360 + 0.561808i) q^{5} +(-3.69049 - 2.74733i) q^{6} +(-0.773958 - 1.34053i) q^{7} -8.11720 q^{8} +(0.860545 + 2.87393i) q^{9} +(-0.861592 - 1.49232i) q^{10} +5.07008 q^{11} +(7.02430 + 5.22914i) q^{12} +(-0.445184 + 3.57796i) q^{13} +(2.05585 + 3.56084i) q^{14} +(-0.130417 + 1.11602i) q^{15} +11.4499 q^{16} +(-0.103828 + 0.179835i) q^{17} +(-2.28585 - 7.63397i) q^{18} +(-1.79488 + 3.10883i) q^{19} +(1.63991 + 2.84041i) q^{20} +(0.311190 - 2.66295i) q^{21} -13.4676 q^{22} +(1.60137 - 2.77365i) q^{23} +(-11.2776 - 8.39543i) q^{24} +(2.28958 - 3.96567i) q^{25} +(1.18254 - 9.50408i) q^{26} +(-1.77684 + 4.88291i) q^{27} +(-3.91301 - 6.77754i) q^{28} -6.83026 q^{29} +(0.346426 - 2.96447i) q^{30} +(1.58024 + 2.73705i) q^{31} -14.1798 q^{32} +(7.04407 + 5.24386i) q^{33} +(0.275796 - 0.477692i) q^{34} +(0.502082 - 0.869631i) q^{35} +(4.35079 + 14.5301i) q^{36} +(-4.71300 - 8.16316i) q^{37} +(4.76772 - 8.25793i) q^{38} +(-4.31911 + 4.51057i) q^{39} +(-2.63289 - 4.56031i) q^{40} +(4.30114 - 7.44979i) q^{41} +(-0.826610 + 7.07355i) q^{42} +(-2.99929 - 5.19492i) q^{43} +25.6335 q^{44} +(-1.33547 + 1.41565i) q^{45} +(-4.25368 + 7.36759i) q^{46} +(-1.42859 + 2.47438i) q^{47} +(15.9078 + 11.8424i) q^{48} +(2.30198 - 3.98714i) q^{49} +(-6.08178 + 10.5340i) q^{50} +(-0.330251 + 0.142466i) q^{51} +(-2.25078 + 18.0896i) q^{52} -2.48667 q^{53} +(4.71980 - 12.9704i) q^{54} +(1.64453 + 2.84841i) q^{55} +(6.28237 + 10.8814i) q^{56} +(-5.70909 + 2.46282i) q^{57} +18.1431 q^{58} -2.98403 q^{59} +(-0.659371 + 5.64243i) q^{60} +(-4.02238 - 6.96697i) q^{61} +(-4.19756 - 7.27038i) q^{62} +(3.18657 - 3.37789i) q^{63} +14.7657 q^{64} +(-2.15453 + 0.910439i) q^{65} +(-18.7111 - 13.9292i) q^{66} +(-2.47432 + 4.28565i) q^{67} +(-0.524937 + 0.909217i) q^{68} +(5.09356 - 2.19729i) q^{69} +(-1.33367 + 2.30999i) q^{70} +(-0.787066 + 1.36324i) q^{71} +(-6.98522 - 23.3282i) q^{72} +3.03817 q^{73} +(12.5191 + 21.6837i) q^{74} +(7.28261 - 3.14162i) q^{75} +(-9.07465 + 15.7178i) q^{76} +(-3.92403 - 6.79661i) q^{77} +(11.4728 - 11.9814i) q^{78} +(3.23418 - 5.60177i) q^{79} +(3.71389 + 6.43264i) q^{80} +(-7.51892 + 4.94629i) q^{81} +(-11.4251 + 19.7888i) q^{82} +(1.24623 - 2.15854i) q^{83} +(1.57333 - 13.4635i) q^{84} -0.134710 q^{85} +(7.96696 + 13.7992i) q^{86} +(-9.48957 - 7.06437i) q^{87} -41.1548 q^{88} +(-1.76275 - 3.05317i) q^{89} +(3.54738 - 3.76036i) q^{90} +(5.14094 - 2.17241i) q^{91} +(8.09626 - 14.0231i) q^{92} +(-0.635376 + 5.43710i) q^{93} +(3.79473 - 6.57267i) q^{94} -2.32875 q^{95} +(-19.7006 - 14.6658i) q^{96} +(4.69325 + 8.12894i) q^{97} +(-6.11471 + 10.5910i) q^{98} +(4.36303 + 14.5710i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.65628 −1.87828 −0.939138 − 0.343539i \(-0.888374\pi\)
−0.939138 + 0.343539i \(0.888374\pi\)
\(3\) 1.38934 + 1.03428i 0.802137 + 0.597140i
\(4\) 5.05585 2.52792
\(5\) 0.324360 + 0.561808i 0.145058 + 0.251248i 0.929395 − 0.369088i \(-0.120330\pi\)
−0.784336 + 0.620336i \(0.786996\pi\)
\(6\) −3.69049 − 2.74733i −1.50664 − 1.12159i
\(7\) −0.773958 − 1.34053i −0.292529 − 0.506674i 0.681878 − 0.731466i \(-0.261163\pi\)
−0.974407 + 0.224791i \(0.927830\pi\)
\(8\) −8.11720 −2.86986
\(9\) 0.860545 + 2.87393i 0.286848 + 0.957976i
\(10\) −0.861592 − 1.49232i −0.272459 − 0.471913i
\(11\) 5.07008 1.52869 0.764343 − 0.644810i \(-0.223063\pi\)
0.764343 + 0.644810i \(0.223063\pi\)
\(12\) 7.02430 + 5.22914i 2.02774 + 1.50952i
\(13\) −0.445184 + 3.57796i −0.123472 + 0.992348i
\(14\) 2.05585 + 3.56084i 0.549450 + 0.951675i
\(15\) −0.130417 + 1.11602i −0.0336736 + 0.288155i
\(16\) 11.4499 2.86247
\(17\) −0.103828 + 0.179835i −0.0251819 + 0.0436163i −0.878342 − 0.478033i \(-0.841350\pi\)
0.853160 + 0.521650i \(0.174683\pi\)
\(18\) −2.28585 − 7.63397i −0.538781 − 1.79934i
\(19\) −1.79488 + 3.10883i −0.411774 + 0.713214i −0.995084 − 0.0990360i \(-0.968424\pi\)
0.583310 + 0.812250i \(0.301757\pi\)
\(20\) 1.63991 + 2.84041i 0.366696 + 0.635136i
\(21\) 0.311190 − 2.66295i 0.0679073 − 0.581103i
\(22\) −13.4676 −2.87130
\(23\) 1.60137 − 2.77365i 0.333908 − 0.578345i −0.649367 − 0.760475i \(-0.724966\pi\)
0.983274 + 0.182130i \(0.0582993\pi\)
\(24\) −11.2776 − 8.39543i −2.30202 − 1.71371i
\(25\) 2.28958 − 3.96567i 0.457916 − 0.793134i
\(26\) 1.18254 − 9.50408i 0.231914 − 1.86390i
\(27\) −1.77684 + 4.88291i −0.341954 + 0.939717i
\(28\) −3.91301 − 6.77754i −0.739490 − 1.28083i
\(29\) −6.83026 −1.26835 −0.634173 − 0.773191i \(-0.718659\pi\)
−0.634173 + 0.773191i \(0.718659\pi\)
\(30\) 0.346426 − 2.96447i 0.0632484 − 0.541236i
\(31\) 1.58024 + 2.73705i 0.283819 + 0.491589i 0.972322 − 0.233645i \(-0.0750653\pi\)
−0.688503 + 0.725233i \(0.741732\pi\)
\(32\) −14.1798 −2.50665
\(33\) 7.04407 + 5.24386i 1.22622 + 0.912839i
\(34\) 0.275796 − 0.477692i 0.0472986 − 0.0819235i
\(35\) 0.502082 − 0.869631i 0.0848673 − 0.146995i
\(36\) 4.35079 + 14.5301i 0.725131 + 2.42169i
\(37\) −4.71300 − 8.16316i −0.774813 − 1.34202i −0.934900 − 0.354912i \(-0.884511\pi\)
0.160087 − 0.987103i \(-0.448823\pi\)
\(38\) 4.76772 − 8.25793i 0.773426 − 1.33961i
\(39\) −4.31911 + 4.51057i −0.691612 + 0.722269i
\(40\) −2.63289 − 4.56031i −0.416297 − 0.721048i
\(41\) 4.30114 − 7.44979i 0.671725 − 1.16346i −0.305689 − 0.952131i \(-0.598887\pi\)
0.977415 − 0.211331i \(-0.0677798\pi\)
\(42\) −0.826610 + 7.07355i −0.127549 + 1.09147i
\(43\) −2.99929 − 5.19492i −0.457387 − 0.792217i 0.541435 − 0.840743i \(-0.317881\pi\)
−0.998822 + 0.0485254i \(0.984548\pi\)
\(44\) 25.6335 3.86440
\(45\) −1.33547 + 1.41565i −0.199080 + 0.211032i
\(46\) −4.25368 + 7.36759i −0.627171 + 1.08629i
\(47\) −1.42859 + 2.47438i −0.208381 + 0.360926i −0.951205 − 0.308561i \(-0.900153\pi\)
0.742824 + 0.669487i \(0.233486\pi\)
\(48\) 15.9078 + 11.8424i 2.29610 + 1.70930i
\(49\) 2.30198 − 3.98714i 0.328854 − 0.569592i
\(50\) −6.08178 + 10.5340i −0.860093 + 1.48973i
\(51\) −0.330251 + 0.142466i −0.0462444 + 0.0199492i
\(52\) −2.25078 + 18.0896i −0.312128 + 2.50858i
\(53\) −2.48667 −0.341571 −0.170786 − 0.985308i \(-0.554631\pi\)
−0.170786 + 0.985308i \(0.554631\pi\)
\(54\) 4.71980 − 12.9704i 0.642283 − 1.76505i
\(55\) 1.64453 + 2.84841i 0.221748 + 0.384079i
\(56\) 6.28237 + 10.8814i 0.839517 + 1.45409i
\(57\) −5.70909 + 2.46282i −0.756188 + 0.326209i
\(58\) 18.1431 2.38231
\(59\) −2.98403 −0.388487 −0.194244 − 0.980953i \(-0.562225\pi\)
−0.194244 + 0.980953i \(0.562225\pi\)
\(60\) −0.659371 + 5.64243i −0.0851244 + 0.728435i
\(61\) −4.02238 − 6.96697i −0.515013 − 0.892029i −0.999848 − 0.0174233i \(-0.994454\pi\)
0.484835 − 0.874606i \(-0.338880\pi\)
\(62\) −4.19756 − 7.27038i −0.533090 − 0.923339i
\(63\) 3.18657 − 3.37789i 0.401471 − 0.425574i
\(64\) 14.7657 1.84572
\(65\) −2.15453 + 0.910439i −0.267236 + 0.112926i
\(66\) −18.7111 − 13.9292i −2.30317 − 1.71456i
\(67\) −2.47432 + 4.28565i −0.302286 + 0.523575i −0.976653 − 0.214821i \(-0.931083\pi\)
0.674367 + 0.738396i \(0.264417\pi\)
\(68\) −0.524937 + 0.909217i −0.0636579 + 0.110259i
\(69\) 5.09356 − 2.19729i 0.613193 − 0.264523i
\(70\) −1.33367 + 2.30999i −0.159404 + 0.276096i
\(71\) −0.787066 + 1.36324i −0.0934076 + 0.161787i −0.908943 − 0.416921i \(-0.863109\pi\)
0.815535 + 0.578707i \(0.196443\pi\)
\(72\) −6.98522 − 23.3282i −0.823216 − 2.74926i
\(73\) 3.03817 0.355591 0.177796 − 0.984067i \(-0.443103\pi\)
0.177796 + 0.984067i \(0.443103\pi\)
\(74\) 12.5191 + 21.6837i 1.45531 + 2.52068i
\(75\) 7.28261 − 3.14162i 0.840924 − 0.362763i
\(76\) −9.07465 + 15.7178i −1.04093 + 1.80295i
\(77\) −3.92403 − 6.79661i −0.447184 − 0.774546i
\(78\) 11.4728 − 11.9814i 1.29904 − 1.35662i
\(79\) 3.23418 − 5.60177i 0.363874 − 0.630248i −0.624721 − 0.780848i \(-0.714787\pi\)
0.988595 + 0.150600i \(0.0481205\pi\)
\(80\) 3.71389 + 6.43264i 0.415225 + 0.719191i
\(81\) −7.51892 + 4.94629i −0.835436 + 0.549588i
\(82\) −11.4251 + 19.7888i −1.26169 + 2.18530i
\(83\) 1.24623 − 2.15854i 0.136792 − 0.236931i −0.789489 − 0.613765i \(-0.789654\pi\)
0.926281 + 0.376835i \(0.122987\pi\)
\(84\) 1.57333 − 13.4635i 0.171664 − 1.46898i
\(85\) −0.134710 −0.0146114
\(86\) 7.96696 + 13.7992i 0.859099 + 1.48800i
\(87\) −9.48957 − 7.06437i −1.01739 − 0.757380i
\(88\) −41.1548 −4.38712
\(89\) −1.76275 − 3.05317i −0.186851 − 0.323635i 0.757348 − 0.653012i \(-0.226495\pi\)
−0.944199 + 0.329376i \(0.893161\pi\)
\(90\) 3.54738 − 3.76036i 0.373927 − 0.396377i
\(91\) 5.14094 − 2.17241i 0.538916 − 0.227730i
\(92\) 8.09626 − 14.0231i 0.844093 − 1.46201i
\(93\) −0.635376 + 5.43710i −0.0658854 + 0.563801i
\(94\) 3.79473 − 6.57267i 0.391397 − 0.677919i
\(95\) −2.32875 −0.238925
\(96\) −19.7006 − 14.6658i −2.01068 − 1.49682i
\(97\) 4.69325 + 8.12894i 0.476527 + 0.825369i 0.999638 − 0.0268952i \(-0.00856206\pi\)
−0.523111 + 0.852265i \(0.675229\pi\)
\(98\) −6.11471 + 10.5910i −0.617679 + 1.06985i
\(99\) 4.36303 + 14.5710i 0.438501 + 1.46444i
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.16.1 yes 24
3.2 odd 2 351.2.h.a.289.12 24
9.4 even 3 117.2.f.a.94.12 yes 24
9.5 odd 6 351.2.f.a.172.1 24
13.9 even 3 117.2.f.a.61.12 ✓ 24
39.35 odd 6 351.2.f.a.100.1 24
117.22 even 3 inner 117.2.h.a.22.1 yes 24
117.113 odd 6 351.2.h.a.334.12 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.12 ✓ 24 13.9 even 3
117.2.f.a.94.12 yes 24 9.4 even 3
117.2.h.a.16.1 yes 24 1.1 even 1 trivial
117.2.h.a.22.1 yes 24 117.22 even 3 inner
351.2.f.a.100.1 24 39.35 odd 6
351.2.f.a.172.1 24 9.5 odd 6
351.2.h.a.289.12 24 3.2 odd 2
351.2.h.a.334.12 24 117.113 odd 6