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 82.4
Root \(2.04224i\) of defining polynomial
Character \(\chi\) \(=\) 117.82
Dual form 117.4.q.e.10.4

$q$-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 \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

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

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 0.778932 − 0.627108i \(-0.215762\pi\)
−0.153626 + 0.988129i \(0.549095\pi\)
\(3\) 0 0
\(4\) −1.91462 − 3.31622i −0.239328 − 0.414528i
\(5\) − 12.0825i − 1.08069i −0.841444 − 0.540344i \(-0.818294\pi\)
0.841444 − 0.540344i \(-0.181706\pi\)
\(6\) 0 0
\(7\) −25.7533 + 14.8686i −1.39055 + 0.802832i −0.993375 − 0.114914i \(-0.963341\pi\)
−0.397170 + 0.917745i \(0.630008\pi\)
\(8\) − 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 0.128454 − 0.991715i \(-0.458998\pi\)
−0.794624 + 0.607102i \(0.792332\pi\)
\(12\) 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 0.988183 − 0.153280i \(-0.0489838\pi\)
−0.626836 + 0.779151i \(0.715650\pi\)
\(18\) 0 0
\(19\) 91.0612 − 52.5742i 1.09952 − 0.634808i 0.163425 − 0.986556i \(-0.447746\pi\)
0.936095 + 0.351748i \(0.114413\pi\)
\(20\) −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 −0.229796 − 0.973239i \(-0.573806\pi\)
0.957748 − 0.287610i \(-0.0928607\pi\)
\(24\) 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 −0.549727 − 0.835344i \(-0.685268\pi\)
0.998293 + 0.0584051i \(0.0186015\pi\)
\(30\) 0 0
\(31\) 223.593i 1.29544i 0.761880 + 0.647718i \(0.224276\pi\)
−0.761880 + 0.647718i \(0.775724\pi\)
\(32\) −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 −0.00845956 − 0.999964i \(-0.502693\pi\)
−0.870224 + 0.492656i \(0.836026\pi\)
\(38\) 214.739 0.916716
\(39\) 0 0
\(40\) −291.890 −1.15380
\(41\) −256.259 − 147.951i −0.976119 − 0.563563i −0.0750227 − 0.997182i \(-0.523903\pi\)
−0.901096 + 0.433619i \(0.857236\pi\)
\(42\) 0 0
\(43\) 96.0517 + 166.366i 0.340645 + 0.590015i 0.984553 − 0.175088i \(-0.0560211\pi\)
−0.643907 + 0.765103i \(0.722688\pi\)
\(44\) 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 \(0.0182512\pi\)
−0.998357 + 0.0573063i \(0.981749\pi\)
\(48\) 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 −0.0181492 − 0.999835i \(-0.505777\pi\)
0.856808 + 0.515635i \(0.172444\pi\)
\(60\) 0 0
\(61\) −143.073 − 247.809i −0.300305 − 0.520143i 0.675900 − 0.736993i \(-0.263755\pi\)
−0.976205 + 0.216850i \(0.930422\pi\)
\(62\) −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 0.860042 − 0.510224i \(-0.170438\pi\)
−0.0118462 + 0.999930i \(0.503771\pi\)
\(68\) 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 −0.572508 − 0.819899i \(-0.694029\pi\)
0.423800 + 0.905756i \(0.360696\pi\)
\(72\) 0 0
\(73\) 75.5209i 0.121083i 0.998166 + 0.0605414i \(0.0192827\pi\)
−0.998166 + 0.0605414i \(0.980717\pi\)
\(74\) −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 \(-0.581457\pi\)
0.253122 − 0.967434i \(-0.418543\pi\)
\(84\) 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 0.662673 − 0.748909i \(-0.269422\pi\)
−0.317237 + 0.948346i \(0.602755\pi\)
\(90\) 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 −0.799386 − 0.600818i \(-0.794842\pi\)
0.120630 + 0.992697i \(0.461508\pi\)
\(98\) 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.82.4 10
3.2 odd 2 39.4.j.c.4.2 ✓ 10
12.11 even 2 624.4.bv.h.433.4 10
13.6 odd 12 1521.4.a.bk.1.4 10
13.7 odd 12 1521.4.a.bk.1.7 10
13.10 even 6 inner 117.4.q.e.10.4 10
39.17 odd 6 507.4.b.i.337.4 10
39.20 even 12 507.4.a.r.1.4 10
39.23 odd 6 39.4.j.c.10.2 yes 10
39.32 even 12 507.4.a.r.1.7 10
39.35 odd 6 507.4.b.i.337.7 10
156.23 even 6 624.4.bv.h.49.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.2 ✓ 10 3.2 odd 2
39.4.j.c.10.2 yes 10 39.23 odd 6
117.4.q.e.10.4 10 13.10 even 6 inner
117.4.q.e.82.4 10 1.1 even 1 trivial
507.4.a.r.1.4 10 39.20 even 12
507.4.a.r.1.7 10 39.32 even 12
507.4.b.i.337.4 10 39.17 odd 6
507.4.b.i.337.7 10 39.35 odd 6
624.4.bv.h.49.2 10 156.23 even 6
624.4.bv.h.433.4 10 12.11 even 2
1521.4.a.bk.1.4 10 13.6 odd 12
1521.4.a.bk.1.7 10 13.7 odd 12