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Properties

Label 117.2.t.c.103.8
Level $117$
Weight $2$
Character 117.103
Analytic conductor $0.934$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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(25,117)] 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("117.25"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,2,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 103.8
Root \(-0.651881 + 1.60470i\) of defining polynomial
Character \(\chi\) \(=\) 117.103
Dual form 117.2.t.c.25.8

$q$-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+(1.41717 - 0.818205i) q^{2} +(0.471101 - 1.66675i) q^{3} +(0.338918 - 0.587023i) q^{4} +(0.950358 + 0.548689i) q^{5} +(-0.696113 - 2.74753i) q^{6} +(-2.77942 + 1.60470i) q^{7} +2.16360i q^{8} +(-2.55613 - 1.57042i) q^{9} +1.79576 q^{10} +(-1.52289 + 0.879239i) q^{11} +(-0.818757 - 0.841439i) q^{12} +(2.19891 - 2.85741i) q^{13} +(-2.62594 + 4.54826i) q^{14} +(1.36224 - 1.32552i) q^{15} +(2.44810 + 4.24024i) q^{16} +1.47360 q^{17} +(-4.90740 - 0.134118i) q^{18} -3.61452i q^{19} +(0.644186 - 0.371921i) q^{20} +(1.36525 + 5.38857i) q^{21} +(-1.43879 + 2.49207i) q^{22} +(-2.34599 + 4.06337i) q^{23} +(3.60619 + 1.01928i) q^{24} +(-1.89788 - 3.28722i) q^{25} +(0.778285 - 5.84860i) q^{26} +(-3.82169 + 3.52060i) q^{27} +2.17544i q^{28} +(0.959085 + 1.66118i) q^{29} +(0.845985 - 2.99309i) q^{30} +(5.68224 + 3.28064i) q^{31} +(3.19130 + 1.84250i) q^{32} +(0.748040 + 2.95249i) q^{33} +(2.08834 - 1.20570i) q^{34} -3.52192 q^{35} +(-1.78819 + 0.968262i) q^{36} -11.6237i q^{37} +(-2.95742 - 5.12239i) q^{38} +(-3.72669 - 5.01117i) q^{39} +(-1.18715 + 2.05620i) q^{40} +(-4.68013 - 2.70208i) q^{41} +(6.34374 + 6.51948i) q^{42} +(0.889142 + 1.54004i) q^{43} +1.19196i q^{44} +(-1.56756 - 2.89498i) q^{45} +7.67798i q^{46} +(-8.90053 + 5.13872i) q^{47} +(8.22074 - 2.08280i) q^{48} +(1.65010 - 2.85806i) q^{49} +(-5.37925 - 3.10571i) q^{50} +(0.694213 - 2.45612i) q^{51} +(-0.932116 - 2.25924i) q^{52} +11.7738 q^{53} +(-2.53542 + 8.11623i) q^{54} -1.92972 q^{55} +(-3.47193 - 6.01355i) q^{56} +(-6.02451 - 1.70280i) q^{57} +(2.71838 + 1.56946i) q^{58} +(4.78585 + 2.76311i) q^{59} +(-0.316423 - 1.24891i) q^{60} +(-0.985148 - 1.70633i) q^{61} +10.7370 q^{62} +(9.62459 + 0.263038i) q^{63} -3.76225 q^{64} +(3.65758 - 1.50905i) q^{65} +(3.47584 + 3.57213i) q^{66} +(-7.15434 - 4.13056i) q^{67} +(0.499428 - 0.865034i) q^{68} +(5.66743 + 5.82443i) q^{69} +(-4.99117 + 2.88165i) q^{70} -5.84860i q^{71} +(3.39776 - 5.53044i) q^{72} -1.24694i q^{73} +(-9.51060 - 16.4728i) q^{74} +(-6.37308 + 1.61468i) q^{75} +(-2.12180 - 1.22502i) q^{76} +(2.82182 - 4.88754i) q^{77} +(-9.38152 - 4.05249i) q^{78} +(-0.242912 - 0.420735i) q^{79} +5.37300i q^{80} +(4.06757 + 8.02838i) q^{81} -8.84340 q^{82} +(13.4351 - 7.75677i) q^{83} +(3.62592 + 1.02485i) q^{84} +(1.40044 + 0.808546i) q^{85} +(2.52014 + 1.45500i) q^{86} +(3.22061 - 0.815971i) q^{87} +(-1.90232 - 3.29492i) q^{88} +13.4245i q^{89} +(-4.59019 - 2.82010i) q^{90} +(-1.52640 + 11.4705i) q^{91} +(1.59019 + 2.75429i) q^{92} +(8.14493 - 7.92538i) q^{93} +(-8.40905 + 14.5649i) q^{94} +(1.98325 - 3.43508i) q^{95} +(4.57442 - 4.45111i) q^{96} +(-5.15756 + 2.97772i) q^{97} -5.40048i q^{98} +(5.27346 + 0.144123i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 12 q^{4} - 2 q^{9} - 16 q^{10} - 2 q^{12} - 4 q^{13} - 18 q^{14} + 4 q^{16} - 12 q^{17} - 10 q^{22} + 24 q^{23} - 12 q^{25} - 12 q^{26} - 22 q^{27} + 12 q^{29} - 54 q^{30} - 12 q^{35} + 50 q^{36}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41717 − 0.818205i 1.00209 − 0.578558i 0.0932254 − 0.995645i \(-0.470282\pi\)
0.908867 + 0.417087i \(0.136949\pi\)
\(3\) 0.471101 − 1.66675i 0.271990 − 0.962300i
\(4\) 0.338918 − 0.587023i 0.169459 − 0.293511i
\(5\) 0.950358 + 0.548689i 0.425013 + 0.245381i 0.697220 − 0.716857i \(-0.254420\pi\)
−0.272207 + 0.962239i \(0.587754\pi\)
\(6\) −0.696113 − 2.74753i −0.284187 − 1.12168i
\(7\) −2.77942 + 1.60470i −1.05052 + 0.606518i −0.922795 − 0.385291i \(-0.874101\pi\)
−0.127725 + 0.991810i \(0.540768\pi\)
\(8\) 2.16360i 0.764949i
\(9\) −2.55613 − 1.57042i −0.852042 − 0.523473i
\(10\) 1.79576 0.567869
\(11\) −1.52289 + 0.879239i −0.459168 + 0.265101i −0.711694 − 0.702489i \(-0.752072\pi\)
0.252527 + 0.967590i \(0.418738\pi\)
\(12\) −0.818757 − 0.841439i −0.236355 − 0.242903i
\(13\) 2.19891 − 2.85741i 0.609868 − 0.792503i
\(14\) −2.62594 + 4.54826i −0.701812 + 1.21557i
\(15\) 1.36224 − 1.32552i 0.351730 − 0.342248i
\(16\) 2.44810 + 4.24024i 0.612026 + 1.06006i
\(17\) 1.47360 0.357399 0.178700 − 0.983904i \(-0.442811\pi\)
0.178700 + 0.983904i \(0.442811\pi\)
\(18\) −4.90740 − 0.134118i −1.15668 − 0.0316120i
\(19\) − 3.61452i − 0.829227i −0.909998 − 0.414614i \(-0.863917\pi\)
0.909998 − 0.414614i \(-0.136083\pi\)
\(20\) 0.644186 − 0.371921i 0.144044 − 0.0831641i
\(21\) 1.36525 + 5.38857i 0.297921 + 1.17588i
\(22\) −1.43879 + 2.49207i −0.306752 + 0.531310i
\(23\) −2.34599 + 4.06337i −0.489172 + 0.847270i −0.999922 − 0.0124586i \(-0.996034\pi\)
0.510751 + 0.859729i \(0.329368\pi\)
\(24\) 3.60619 + 1.01928i 0.736110 + 0.208059i
\(25\) −1.89788 − 3.28722i −0.379576 − 0.657445i
\(26\) 0.778285 − 5.84860i 0.152634 − 1.14700i
\(27\) −3.82169 + 3.52060i −0.735485 + 0.677541i
\(28\) 2.17544i 0.411120i
\(29\) 0.959085 + 1.66118i 0.178098 + 0.308474i 0.941229 − 0.337769i \(-0.109672\pi\)
−0.763131 + 0.646244i \(0.776339\pi\)
\(30\) 0.845985 − 2.99309i 0.154455 − 0.546461i
\(31\) 5.68224 + 3.28064i 1.02056 + 0.589221i 0.914266 − 0.405114i \(-0.132768\pi\)
0.106294 + 0.994335i \(0.466101\pi\)
\(32\) 3.19130 + 1.84250i 0.564148 + 0.325711i
\(33\) 0.748040 + 2.95249i 0.130217 + 0.513962i
\(34\) 2.08834 − 1.20570i 0.358147 − 0.206776i
\(35\) −3.52192 −0.595313
\(36\) −1.78819 + 0.968262i −0.298031 + 0.161377i
\(37\) − 11.6237i − 1.91093i −0.295103 − 0.955465i \(-0.595354\pi\)
0.295103 − 0.955465i \(-0.404646\pi\)
\(38\) −2.95742 − 5.12239i −0.479756 − 0.830962i
\(39\) −3.72669 − 5.01117i −0.596748 − 0.802429i
\(40\) −1.18715 + 2.05620i −0.187704 + 0.325113i
\(41\) −4.68013 − 2.70208i −0.730914 − 0.421993i 0.0878426 − 0.996134i \(-0.472003\pi\)
−0.818756 + 0.574141i \(0.805336\pi\)
\(42\) 6.34374 + 6.51948i 0.978861 + 1.00598i
\(43\) 0.889142 + 1.54004i 0.135593 + 0.234854i 0.925824 − 0.377955i \(-0.123373\pi\)
−0.790231 + 0.612809i \(0.790039\pi\)
\(44\) 1.19196i 0.179695i
\(45\) −1.56756 − 2.89498i −0.233679 − 0.431558i
\(46\) 7.67798i 1.13206i
\(47\) −8.90053 + 5.13872i −1.29828 + 0.749559i −0.980106 − 0.198475i \(-0.936401\pi\)
−0.318169 + 0.948034i \(0.603068\pi\)
\(48\) 8.22074 − 2.08280i 1.18656 − 0.300626i
\(49\) 1.65010 − 2.85806i 0.235729 − 0.408294i
\(50\) −5.37925 − 3.10571i −0.760740 − 0.439214i
\(51\) 0.694213 − 2.45612i 0.0972093 − 0.343925i
\(52\) −0.932116 − 2.25924i −0.129261 − 0.313300i
\(53\) 11.7738 1.61726 0.808628 − 0.588320i \(-0.200211\pi\)
0.808628 + 0.588320i \(0.200211\pi\)
\(54\) −2.53542 + 8.11623i −0.345027 + 1.10448i
\(55\) −1.92972 −0.260203
\(56\) −3.47193 − 6.01355i −0.463956 − 0.803595i
\(57\) −6.02451 − 1.70280i −0.797965 − 0.225542i
\(58\) 2.71838 + 1.56946i 0.356940 + 0.206080i
\(59\) 4.78585 + 2.76311i 0.623064 + 0.359726i 0.778061 − 0.628189i \(-0.216203\pi\)
−0.154997 + 0.987915i \(0.549537\pi\)
\(60\) −0.316423 − 1.24891i −0.0408501 − 0.161234i
\(61\) −0.985148 − 1.70633i −0.126135 − 0.218473i 0.796041 − 0.605243i \(-0.206924\pi\)
−0.922176 + 0.386770i \(0.873591\pi\)
\(62\) 10.7370 1.36359
\(63\) 9.62459 + 0.263038i 1.21258 + 0.0331397i
\(64\) −3.76225 −0.470282
\(65\) 3.65758 − 1.50905i 0.453667 − 0.187174i
\(66\) 3.47584 + 3.57213i 0.427846 + 0.439699i
\(67\) −7.15434 − 4.13056i −0.874042 − 0.504628i −0.00535270 − 0.999986i \(-0.501704\pi\)
−0.868689 + 0.495357i \(0.835037\pi\)
\(68\) 0.499428 − 0.865034i 0.0605645 − 0.104901i
\(69\) 5.66743 + 5.82443i 0.682278 + 0.701179i
\(70\) −4.99117 + 2.88165i −0.596558 + 0.344423i
\(71\) − 5.84860i − 0.694101i −0.937846 − 0.347051i \(-0.887183\pi\)
0.937846 − 0.347051i \(-0.112817\pi\)
\(72\) 3.39776 − 5.53044i 0.400430 − 0.651769i
\(73\) − 1.24694i − 0.145943i −0.997334 − 0.0729714i \(-0.976752\pi\)
0.997334 − 0.0729714i \(-0.0232482\pi\)
\(74\) −9.51060 − 16.4728i −1.10558 − 1.91493i
\(75\) −6.37308 + 1.61468i −0.735900 + 0.186447i
\(76\) −2.12180 − 1.22502i −0.243388 − 0.140520i
\(77\) 2.82182 − 4.88754i 0.321577 − 0.556987i
\(78\) −9.38152 − 4.05249i −1.06225 − 0.458854i
\(79\) −0.242912 − 0.420735i −0.0273297 − 0.0473364i 0.852037 − 0.523481i \(-0.175367\pi\)
−0.879367 + 0.476145i \(0.842034\pi\)
\(80\) 5.37300i 0.600719i
\(81\) 4.06757 + 8.02838i 0.451952 + 0.892042i
\(82\) −8.84340 −0.976590
\(83\) 13.4351 − 7.75677i 1.47470 − 0.851417i 0.475103 − 0.879930i \(-0.342411\pi\)
0.999593 + 0.0285134i \(0.00907732\pi\)
\(84\) 3.62592 + 1.02485i 0.395620 + 0.111821i
\(85\) 1.40044 + 0.808546i 0.151899 + 0.0876991i
\(86\) 2.52014 + 1.45500i 0.271753 + 0.156897i
\(87\) 3.22061 − 0.815971i 0.345286 − 0.0874813i
\(88\) −1.90232 − 3.29492i −0.202788 − 0.351240i
\(89\) 13.4245i 1.42300i 0.702688 + 0.711498i \(0.251983\pi\)
−0.702688 + 0.711498i \(0.748017\pi\)
\(90\) −4.59019 − 2.82010i −0.483849 − 0.297264i
\(91\) −1.52640 + 11.4705i −0.160011 + 1.20244i
\(92\) 1.59019 + 2.75429i 0.165789 + 0.287155i
\(93\) 8.14493 − 7.92538i 0.844590 − 0.821823i
\(94\) −8.40905 + 14.5649i −0.867327 + 1.50225i
\(95\) 1.98325 − 3.43508i 0.203477 − 0.352432i
\(96\) 4.57442 − 4.45111i 0.466874 − 0.454289i
\(97\) −5.15756 + 2.97772i −0.523671 + 0.302342i −0.738435 − 0.674324i \(-0.764435\pi\)
0.214764 + 0.976666i \(0.431102\pi\)
\(98\) − 5.40048i − 0.545531i
\(99\) 5.27346 + 0.144123i 0.530003 + 0.0144849i
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.t.c.103.8 yes 20
3.2 odd 2 351.2.t.c.64.3 20
9.2 odd 6 351.2.t.c.181.8 20
9.4 even 3 1053.2.b.j.649.8 10
9.5 odd 6 1053.2.b.i.649.3 10
9.7 even 3 inner 117.2.t.c.25.3 ✓ 20
13.12 even 2 inner 117.2.t.c.103.3 yes 20
39.38 odd 2 351.2.t.c.64.8 20
117.25 even 6 inner 117.2.t.c.25.8 yes 20
117.38 odd 6 351.2.t.c.181.3 20
117.77 odd 6 1053.2.b.i.649.8 10
117.103 even 6 1053.2.b.j.649.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.3 ✓ 20 9.7 even 3 inner
117.2.t.c.25.8 yes 20 117.25 even 6 inner
117.2.t.c.103.3 yes 20 13.12 even 2 inner
117.2.t.c.103.8 yes 20 1.1 even 1 trivial
351.2.t.c.64.3 20 3.2 odd 2
351.2.t.c.64.8 20 39.38 odd 2
351.2.t.c.181.3 20 117.38 odd 6
351.2.t.c.181.8 20 9.2 odd 6
1053.2.b.i.649.3 10 9.5 odd 6
1053.2.b.i.649.8 10 117.77 odd 6
1053.2.b.j.649.3 10 117.103 even 6
1053.2.b.j.649.8 10 9.4 even 3