Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [39,4,Mod(4,39)] 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("39.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(39, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.30107449022\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 4.2
Root \(-2.04224i\) of defining polynomial
Character \(\chi\) \(=\) 39.4
Dual form 39.4.j.c.10.2

$q$-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} +(-1.50000 + 2.59808i) q^{3} +(-1.91462 - 3.31622i) q^{4} +12.0825i q^{5} +(5.30590 - 3.06336i) q^{6} +(-25.7533 + 14.8686i) q^{7} +24.1582i q^{8} +(-4.50000 - 7.79423i) q^{9} +(12.3377 - 21.3694i) q^{10} +(24.3038 + 14.0318i) q^{11} +11.4877 q^{12} +(-40.9717 - 22.7667i) q^{13} +60.7308 q^{14} +(-31.3911 - 18.1237i) q^{15} +(9.35146 - 16.1972i) q^{16} +(-25.3278 - 43.8690i) q^{17} +18.3802i q^{18} +(91.0612 - 52.5742i) q^{19} +(40.0681 - 23.1333i) q^{20} -89.2119i q^{21} +(-28.6563 - 49.6342i) q^{22} +(-80.2961 + 139.077i) q^{23} +(-62.7648 - 36.2373i) q^{24} -20.9857 q^{25} +(49.2164 + 82.1030i) q^{26} +27.0000 q^{27} +(98.6155 + 56.9357i) q^{28} +(-70.0525 + 121.334i) q^{29} +(37.0130 + 64.1083i) q^{30} +223.593i q^{31} +(134.294 - 77.5348i) q^{32} +(-72.9113 + 42.0954i) q^{33} +103.451i q^{34} +(-179.650 - 311.163i) q^{35} +(-17.2316 + 29.8460i) q^{36} +(-197.759 - 114.176i) q^{37} -214.739 q^{38} +(120.607 - 72.2975i) q^{39} -291.890 q^{40} +(256.259 + 147.951i) q^{41} +(-91.0962 + 157.783i) q^{42} +(96.0517 + 166.366i) q^{43} -107.462i q^{44} +(94.1734 - 54.3710i) q^{45} +(284.029 - 163.984i) q^{46} -36.9300i q^{47} +(28.0544 + 48.5916i) q^{48} +(270.653 - 468.785i) q^{49} +(37.1160 + 21.4289i) q^{50} +151.967 q^{51} +(2.94589 + 179.461i) q^{52} +149.102 q^{53} +(-47.7531 - 27.5703i) q^{54} +(-169.538 + 293.649i) q^{55} +(-359.200 - 622.152i) q^{56} +315.445i q^{57} +(247.794 - 143.064i) q^{58} +(-380.070 + 219.433i) q^{59} +138.800i q^{60} +(-143.073 - 247.809i) q^{61} +(228.316 - 395.454i) q^{62} +(231.779 + 133.818i) q^{63} -466.313 q^{64} +(275.077 - 495.038i) q^{65} +171.938 q^{66} +(465.166 + 268.564i) q^{67} +(-96.9863 + 167.985i) q^{68} +(-240.888 - 417.231i) q^{69} +733.777i q^{70} +(88.9656 - 51.3643i) q^{71} +(188.294 - 108.712i) q^{72} +75.5209i q^{73} +(233.175 + 403.871i) q^{74} +(31.4786 - 54.5225i) q^{75} +(-348.696 - 201.319i) q^{76} -834.535 q^{77} +(-287.134 + 4.71337i) q^{78} +17.5526 q^{79} +(195.702 + 112.989i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(-302.152 - 523.342i) q^{82} +1463.08i q^{83} +(-295.847 + 170.807i) q^{84} +(530.045 - 306.022i) q^{85} -392.322i q^{86} +(-210.157 - 364.003i) q^{87} +(-338.983 + 587.135i) q^{88} +(-290.036 - 167.453i) q^{89} -222.078 q^{90} +(1393.66 - 22.8773i) q^{91} +614.946 q^{92} +(-580.912 - 335.390i) q^{93} +(-37.7100 + 65.3156i) q^{94} +(635.225 + 1100.24i) q^{95} +465.209i q^{96} +(-648.442 + 374.378i) q^{97} +(-957.374 + 552.740i) q^{98} -252.572i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 15 q^{3} + 30 q^{4} + 30 q^{7} - 45 q^{9} + 40 q^{10} + 60 q^{11} - 180 q^{12} + 25 q^{13} - 60 q^{14} + 45 q^{15} - 250 q^{16} + 105 q^{17} + 180 q^{19} + 510 q^{20} - 290 q^{22} - 60 q^{23} - 960 q^{25}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.76863 − 1.02112i −0.625307 − 0.361021i 0.153626 − 0.988129i \(-0.450905\pi\)
−0.778932 + 0.627108i \(0.784238\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) −1.91462 − 3.31622i −0.239328 − 0.414528i
\(5\) 12.0825i 1.08069i 0.841444 + 0.540344i \(0.181706\pi\)
−0.841444 + 0.540344i \(0.818294\pi\)
\(6\) 5.30590 − 3.06336i 0.361021 − 0.208436i
\(7\) −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
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) 12.3377 − 21.3694i 0.390151 − 0.675761i
\(11\) 24.3038 + 14.0318i 0.666169 + 0.384613i 0.794624 − 0.607102i \(-0.207668\pi\)
−0.128454 + 0.991715i \(0.541002\pi\)
\(12\) 11.4877 0.276352
\(13\) −40.9717 − 22.7667i −0.874115 − 0.485719i
\(14\) 60.7308 1.15936
\(15\) −31.3911 − 18.1237i −0.540344 − 0.311968i
\(16\) 9.35146 − 16.1972i 0.146117 − 0.253081i
\(17\) −25.3278 − 43.8690i −0.361347 − 0.625871i 0.626836 − 0.779151i \(-0.284350\pi\)
−0.988183 + 0.153280i \(0.951016\pi\)
\(18\) 18.3802i 0.240681i
\(19\) 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
\(21\) − 89.2119i − 0.927030i
\(22\) −28.6563 − 49.6342i −0.277707 − 0.481002i
\(23\) −80.2961 + 139.077i −0.727951 + 1.26085i 0.229796 + 0.973239i \(0.426194\pi\)
−0.957748 + 0.287610i \(0.907139\pi\)
\(24\) −62.7648 − 36.2373i −0.533826 − 0.308204i
\(25\) −20.9857 −0.167886
\(26\) 49.2164 + 82.1030i 0.371235 + 0.619297i
\(27\) 27.0000 0.192450
\(28\) 98.6155 + 56.9357i 0.665592 + 0.384280i
\(29\) −70.0525 + 121.334i −0.448566 + 0.776939i −0.998293 − 0.0584051i \(-0.981398\pi\)
0.549727 + 0.835344i \(0.314732\pi\)
\(30\) 37.0130 + 64.1083i 0.225254 + 0.390151i
\(31\) 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
\(33\) −72.9113 + 42.0954i −0.384613 + 0.222056i
\(34\) 103.451i 0.521815i
\(35\) −179.650 − 311.163i −0.867610 − 1.50274i
\(36\) −17.2316 + 29.8460i −0.0797759 + 0.138176i
\(37\) −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
\(39\) 120.607 − 72.2975i 0.495195 − 0.296843i
\(40\) −291.890 −1.15380
\(41\) 256.259 + 147.951i 0.976119 + 0.563563i 0.901096 − 0.433619i \(-0.142764\pi\)
0.0750227 + 0.997182i \(0.476097\pi\)
\(42\) −91.0962 + 157.783i −0.334677 + 0.579678i
\(43\) 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
\(45\) 94.1734 − 54.3710i 0.311968 − 0.180115i
\(46\) 284.029 − 163.984i 0.910386 − 0.525611i
\(47\) − 36.9300i − 0.114613i −0.998357 − 0.0573063i \(-0.981749\pi\)
0.998357 − 0.0573063i \(-0.0182512\pi\)
\(48\) 28.0544 + 48.5916i 0.0843605 + 0.146117i
\(49\) 270.653 − 468.785i 0.789077 − 1.36672i
\(50\) 37.1160 + 21.4289i 0.104980 + 0.0606102i
\(51\) 151.967 0.417247
\(52\) 2.94589 + 179.461i 0.00785618 + 0.478591i
\(53\) 149.102 0.386429 0.193214 − 0.981157i \(-0.438109\pi\)
0.193214 + 0.981157i \(0.438109\pi\)
\(54\) −47.7531 − 27.5703i −0.120340 − 0.0694785i
\(55\) −169.538 + 293.649i −0.415647 + 0.719921i
\(56\) −359.200 − 622.152i −0.857144 − 1.48462i
\(57\) 315.445i 0.733013i
\(58\) 247.794 − 143.064i 0.560983 − 0.323884i
\(59\) −380.070 + 219.433i −0.838659 + 0.484200i −0.856808 − 0.515635i \(-0.827556\pi\)
0.0181492 + 0.999835i \(0.494223\pi\)
\(60\) 138.800i 0.298650i
\(61\) −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
\(63\) 231.779 + 133.818i 0.463515 + 0.267611i
\(64\) −466.313 −0.910768
\(65\) 275.077 − 495.038i 0.524910 − 0.944645i
\(66\) 171.938 0.320668
\(67\) 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
\(69\) −240.888 − 417.231i −0.420283 − 0.727951i
\(70\) 733.777i 1.25290i
\(71\) 88.9656 − 51.3643i 0.148708 − 0.0858567i −0.423800 − 0.905756i \(-0.639304\pi\)
0.572508 + 0.819899i \(0.305971\pi\)
\(72\) 188.294 − 108.712i 0.308204 − 0.177942i
\(73\) 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
\(75\) 31.4786 − 54.5225i 0.0484644 − 0.0839428i
\(76\) −348.696 − 201.319i −0.526291 − 0.303854i
\(77\) −834.535 −1.23512
\(78\) −287.134 + 4.71337i −0.416815 + 0.00684211i
\(79\) 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
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) −302.152 − 523.342i −0.406916 − 0.704799i
\(83\) 1463.08i 1.93487i 0.253122 + 0.967434i \(0.418543\pi\)
−0.253122 + 0.967434i \(0.581457\pi\)
\(84\) −295.847 + 170.807i −0.384280 + 0.221864i
\(85\) 530.045 − 306.022i 0.676371 − 0.390503i
\(86\) − 392.322i − 0.491920i
\(87\) −210.157 − 364.003i −0.258980 − 0.448566i
\(88\) −338.983 + 587.135i −0.410633 + 0.711236i
\(89\) −290.036 − 167.453i −0.345436 − 0.199438i 0.317237 − 0.948346i \(-0.397245\pi\)
−0.662673 + 0.748909i \(0.730578\pi\)
\(90\) −222.078 −0.260101
\(91\) 1393.66 − 22.8773i 1.60545 − 0.0263538i
\(92\) 614.946 0.696876
\(93\) −580.912 − 335.390i −0.647718 − 0.373960i
\(94\) −37.7100 + 65.3156i −0.0413776 + 0.0716680i
\(95\) 635.225 + 1100.24i 0.686029 + 1.18824i
\(96\) 465.209i 0.494585i
\(97\) −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
\(99\) − 252.572i − 0.256409i
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 39.4.j.c.4.2 ✓ 10
3.2 odd 2 117.4.q.e.82.4 10
4.3 odd 2 624.4.bv.h.433.4 10
13.4 even 6 507.4.b.i.337.4 10
13.6 odd 12 507.4.a.r.1.7 10
13.7 odd 12 507.4.a.r.1.4 10
13.9 even 3 507.4.b.i.337.7 10
13.10 even 6 inner 39.4.j.c.10.2 yes 10
39.20 even 12 1521.4.a.bk.1.7 10
39.23 odd 6 117.4.q.e.10.4 10
39.32 even 12 1521.4.a.bk.1.4 10
52.23 odd 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 1.1 even 1 trivial
39.4.j.c.10.2 yes 10 13.10 even 6 inner
117.4.q.e.10.4 10 39.23 odd 6
117.4.q.e.82.4 10 3.2 odd 2
507.4.a.r.1.4 10 13.7 odd 12
507.4.a.r.1.7 10 13.6 odd 12
507.4.b.i.337.4 10 13.4 even 6
507.4.b.i.337.7 10 13.9 even 3
624.4.bv.h.49.2 10 52.23 odd 6
624.4.bv.h.433.4 10 4.3 odd 2
1521.4.a.bk.1.4 10 39.32 even 12
1521.4.a.bk.1.7 10 39.20 even 12