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Properties

Label 117.4.q.e.10.4
Level $117$
Weight $4$
Character 117.10
Analytic conductor $6.903$
Analytic rank $0$
Dimension $10$
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] = [117,4,Mod(10,117)] 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("117.10"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 10.4
Root \(-2.04224i\) of defining polynomial
Character \(\chi\) \(=\) 117.10
Dual form 117.4.q.e.82.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.76863 - 1.02112i) q^{2} +(-1.91462 + 3.31622i) q^{4} +12.0825i q^{5} +(-25.7533 - 14.8686i) q^{7} +24.1582i q^{8} +(12.3377 + 21.3694i) q^{10} +(-24.3038 + 14.0318i) q^{11} +(-40.9717 + 22.7667i) q^{13} -60.7308 q^{14} +(9.35146 + 16.1972i) q^{16} +(25.3278 - 43.8690i) q^{17} +(91.0612 + 52.5742i) q^{19} +(-40.0681 - 23.1333i) q^{20} +(-28.6563 + 49.6342i) q^{22} +(80.2961 + 139.077i) q^{23} -20.9857 q^{25} +(-49.2164 + 82.1030i) q^{26} +(98.6155 - 56.9357i) q^{28} +(70.0525 + 121.334i) q^{29} -223.593i q^{31} +(-134.294 - 77.5348i) q^{32} -103.451i q^{34} +(179.650 - 311.163i) q^{35} +(-197.759 + 114.176i) q^{37} +214.739 q^{38} -291.890 q^{40} +(-256.259 + 147.951i) q^{41} +(96.0517 - 166.366i) q^{43} -107.462i q^{44} +(284.029 + 163.984i) q^{46} -36.9300i q^{47} +(270.653 + 468.785i) q^{49} +(-37.1160 + 21.4289i) q^{50} +(2.94589 - 179.461i) q^{52} -149.102 q^{53} +(-169.538 - 293.649i) q^{55} +(359.200 - 622.152i) q^{56} +(247.794 + 143.064i) q^{58} +(380.070 + 219.433i) q^{59} +(-143.073 + 247.809i) q^{61} +(-228.316 - 395.454i) q^{62} -466.313 q^{64} +(-275.077 - 495.038i) q^{65} +(465.166 - 268.564i) q^{67} +(96.9863 + 167.985i) q^{68} -733.777i q^{70} +(-88.9656 - 51.3643i) q^{71} -75.5209i q^{73} +(-233.175 + 403.871i) q^{74} +(-348.696 + 201.319i) q^{76} +834.535 q^{77} +17.5526 q^{79} +(-195.702 + 112.989i) q^{80} +(-302.152 + 523.342i) q^{82} +1463.08i q^{83} +(530.045 + 306.022i) q^{85} -392.322i q^{86} +(-338.983 - 587.135i) q^{88} +(290.036 - 167.453i) q^{89} +(1393.66 + 22.8773i) q^{91} -614.946 q^{92} +(-37.7100 - 65.3156i) q^{94} +(-635.225 + 1100.24i) q^{95} +(-648.442 - 374.378i) q^{97} +(957.374 + 552.740i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{4} + 30 q^{7} + 40 q^{10} - 60 q^{11} + 25 q^{13} + 60 q^{14} - 250 q^{16} - 105 q^{17} + 180 q^{19} - 510 q^{20} - 290 q^{22} + 60 q^{23} - 960 q^{25} + 30 q^{26} + 150 q^{28} + 495 q^{29}+ \cdots - 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.76863 − 1.02112i 0.625307 − 0.361021i −0.153626 − 0.988129i \(-0.549095\pi\)
0.778932 + 0.627108i \(0.215762\pi\)
\(3\) 0 0
\(4\) −1.91462 + 3.31622i −0.239328 + 0.414528i
\(5\) 12.0825i 1.08069i 0.841444 + 0.540344i \(0.181706\pi\)
−0.841444 + 0.540344i \(0.818294\pi\)
\(6\) 0 0
\(7\) −25.7533 − 14.8686i −1.39055 − 0.802832i −0.397170 − 0.917745i \(-0.630008\pi\)
−0.993375 + 0.114914i \(0.963341\pi\)
\(8\) 24.1582i 1.06765i
\(9\) 0 0
\(10\) 12.3377 + 21.3694i 0.390151 + 0.675761i
\(11\) −24.3038 + 14.0318i −0.666169 + 0.384613i −0.794624 − 0.607102i \(-0.792332\pi\)
0.128454 + 0.991715i \(0.458998\pi\)
\(12\) 0 0
\(13\) −40.9717 + 22.7667i −0.874115 + 0.485719i
\(14\) −60.7308 −1.15936
\(15\) 0 0
\(16\) 9.35146 + 16.1972i 0.146117 + 0.253081i
\(17\) 25.3278 − 43.8690i 0.361347 − 0.625871i −0.626836 − 0.779151i \(-0.715650\pi\)
0.988183 + 0.153280i \(0.0489838\pi\)
\(18\) 0 0
\(19\) 91.0612 + 52.5742i 1.09952 + 0.634808i 0.936095 − 0.351748i \(-0.114413\pi\)
0.163425 + 0.986556i \(0.447746\pi\)
\(20\) −40.0681 − 23.1333i −0.447975 − 0.258639i
\(21\) 0 0
\(22\) −28.6563 + 49.6342i −0.277707 + 0.481002i
\(23\) 80.2961 + 139.077i 0.727951 + 1.26085i 0.957748 + 0.287610i \(0.0928607\pi\)
−0.229796 + 0.973239i \(0.573806\pi\)
\(24\) 0 0
\(25\) −20.9857 −0.167886
\(26\) −49.2164 + 82.1030i −0.371235 + 0.619297i
\(27\) 0 0
\(28\) 98.6155 − 56.9357i 0.665592 − 0.384280i
\(29\) 70.0525 + 121.334i 0.448566 + 0.776939i 0.998293 − 0.0584051i \(-0.0186015\pi\)
−0.549727 + 0.835344i \(0.685268\pi\)
\(30\) 0 0
\(31\) − 223.593i − 1.29544i −0.761880 − 0.647718i \(-0.775724\pi\)
0.761880 − 0.647718i \(-0.224276\pi\)
\(32\) −134.294 − 77.5348i −0.741878 − 0.428323i
\(33\) 0 0
\(34\) − 103.451i − 0.521815i
\(35\) 179.650 − 311.163i 0.867610 − 1.50274i
\(36\) 0 0
\(37\) −197.759 + 114.176i −0.878684 + 0.507308i −0.870224 − 0.492656i \(-0.836026\pi\)
−0.00845956 + 0.999964i \(0.502693\pi\)
\(38\) 214.739 0.916716
\(39\) 0 0
\(40\) −291.890 −1.15380
\(41\) −256.259 + 147.951i −0.976119 + 0.563563i −0.901096 − 0.433619i \(-0.857236\pi\)
−0.0750227 + 0.997182i \(0.523903\pi\)
\(42\) 0 0
\(43\) 96.0517 − 166.366i 0.340645 − 0.590015i −0.643907 − 0.765103i \(-0.722688\pi\)
0.984553 + 0.175088i \(0.0560211\pi\)
\(44\) − 107.462i − 0.368194i
\(45\) 0 0
\(46\) 284.029 + 163.984i 0.910386 + 0.525611i
\(47\) − 36.9300i − 0.114613i −0.998357 − 0.0573063i \(-0.981749\pi\)
0.998357 − 0.0573063i \(-0.0182512\pi\)
\(48\) 0 0
\(49\) 270.653 + 468.785i 0.789077 + 1.36672i
\(50\) −37.1160 + 21.4289i −0.104980 + 0.0606102i
\(51\) 0 0
\(52\) 2.94589 − 179.461i 0.00785618 − 0.478591i
\(53\) −149.102 −0.386429 −0.193214 − 0.981157i \(-0.561891\pi\)
−0.193214 + 0.981157i \(0.561891\pi\)
\(54\) 0 0
\(55\) −169.538 − 293.649i −0.415647 − 0.719921i
\(56\) 359.200 − 622.152i 0.857144 − 1.48462i
\(57\) 0 0
\(58\) 247.794 + 143.064i 0.560983 + 0.323884i
\(59\) 380.070 + 219.433i 0.838659 + 0.484200i 0.856808 − 0.515635i \(-0.172444\pi\)
−0.0181492 + 0.999835i \(0.505777\pi\)
\(60\) 0 0
\(61\) −143.073 + 247.809i −0.300305 + 0.520143i −0.976205 − 0.216850i \(-0.930422\pi\)
0.675900 + 0.736993i \(0.263755\pi\)
\(62\) −228.316 − 395.454i −0.467679 − 0.810044i
\(63\) 0 0
\(64\) −466.313 −0.910768
\(65\) −275.077 − 495.038i −0.524910 − 0.944645i
\(66\) 0 0
\(67\) 465.166 − 268.564i 0.848195 − 0.489706i −0.0118462 − 0.999930i \(-0.503771\pi\)
0.860042 + 0.510224i \(0.170438\pi\)
\(68\) 96.9863 + 167.985i 0.172961 + 0.299576i
\(69\) 0 0
\(70\) − 733.777i − 1.25290i
\(71\) −88.9656 − 51.3643i −0.148708 − 0.0858567i 0.423800 − 0.905756i \(-0.360696\pi\)
−0.572508 + 0.819899i \(0.694029\pi\)
\(72\) 0 0
\(73\) − 75.5209i − 0.121083i −0.998166 − 0.0605414i \(-0.980717\pi\)
0.998166 − 0.0605414i \(-0.0192827\pi\)
\(74\) −233.175 + 403.871i −0.366298 + 0.634446i
\(75\) 0 0
\(76\) −348.696 + 201.319i −0.526291 + 0.303854i
\(77\) 834.535 1.23512
\(78\) 0 0
\(79\) 17.5526 0.0249978 0.0124989 − 0.999922i \(-0.496021\pi\)
0.0124989 + 0.999922i \(0.496021\pi\)
\(80\) −195.702 + 112.989i −0.273502 + 0.157906i
\(81\) 0 0
\(82\) −302.152 + 523.342i −0.406916 + 0.704799i
\(83\) 1463.08i 1.93487i 0.253122 + 0.967434i \(0.418543\pi\)
−0.253122 + 0.967434i \(0.581457\pi\)
\(84\) 0 0
\(85\) 530.045 + 306.022i 0.676371 + 0.390503i
\(86\) − 392.322i − 0.491920i
\(87\) 0 0
\(88\) −338.983 − 587.135i −0.410633 − 0.711236i
\(89\) 290.036 − 167.453i 0.345436 − 0.199438i −0.317237 − 0.948346i \(-0.602755\pi\)
0.662673 + 0.748909i \(0.269422\pi\)
\(90\) 0 0
\(91\) 1393.66 + 22.8773i 1.60545 + 0.0263538i
\(92\) −614.946 −0.696876
\(93\) 0 0
\(94\) −37.7100 − 65.3156i −0.0413776 − 0.0716680i
\(95\) −635.225 + 1100.24i −0.686029 + 1.18824i
\(96\) 0 0
\(97\) −648.442 − 374.378i −0.678756 − 0.391880i 0.120630 − 0.992697i \(-0.461508\pi\)
−0.799386 + 0.600818i \(0.794842\pi\)
\(98\) 957.374 + 552.740i 0.986830 + 0.569747i
\(99\) 0 0
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 117.4.q.e.10.4 10
3.2 odd 2 39.4.j.c.10.2 yes 10
12.11 even 2 624.4.bv.h.49.2 10
13.2 odd 12 1521.4.a.bk.1.7 10
13.4 even 6 inner 117.4.q.e.82.4 10
13.11 odd 12 1521.4.a.bk.1.4 10
39.2 even 12 507.4.a.r.1.4 10
39.11 even 12 507.4.a.r.1.7 10
39.17 odd 6 39.4.j.c.4.2 ✓ 10
39.23 odd 6 507.4.b.i.337.7 10
39.29 odd 6 507.4.b.i.337.4 10
156.95 even 6 624.4.bv.h.433.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.2 ✓ 10 39.17 odd 6
39.4.j.c.10.2 yes 10 3.2 odd 2
117.4.q.e.10.4 10 1.1 even 1 trivial
117.4.q.e.82.4 10 13.4 even 6 inner
507.4.a.r.1.4 10 39.2 even 12
507.4.a.r.1.7 10 39.11 even 12
507.4.b.i.337.4 10 39.29 odd 6
507.4.b.i.337.7 10 39.23 odd 6
624.4.bv.h.49.2 10 12.11 even 2
624.4.bv.h.433.4 10 156.95 even 6
1521.4.a.bk.1.4 10 13.11 odd 12
1521.4.a.bk.1.7 10 13.2 odd 12