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 10.2
Root \(2.04224i\) of defining polynomial
Character \(\chi\) \(=\) 39.10
Dual form 39.4.j.c.4.2

$q$-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 \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 −0.778932 − 0.627108i \(-0.784238\pi\)
0.153626 + 0.988129i \(0.450905\pi\)
\(3\) −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 \(-0.818294\pi\)
0.841444 − 0.540344i \(-0.181706\pi\)
\(6\) 5.30590 + 3.06336i 0.361021 + 0.208436i
\(7\) −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
\(9\) −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 −0.128454 − 0.991715i \(-0.541002\pi\)
0.794624 + 0.607102i \(0.207668\pi\)
\(12\) 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 −0.988183 − 0.153280i \(-0.951016\pi\)
0.626836 + 0.779151i \(0.284350\pi\)
\(18\) − 18.3802i − 0.240681i
\(19\) 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
\(21\) 89.2119i 0.927030i
\(22\) −28.6563 + 49.6342i −0.277707 + 0.481002i
\(23\) −80.2961 − 139.077i −0.727951 − 1.26085i −0.957748 − 0.287610i \(-0.907139\pi\)
0.229796 − 0.973239i \(-0.426194\pi\)
\(24\) −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 0.549727 − 0.835344i \(-0.314732\pi\)
−0.998293 + 0.0584051i \(0.981398\pi\)
\(30\) 37.0130 − 64.1083i 0.225254 − 0.390151i
\(31\) − 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
\(33\) −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 −0.870224 − 0.492656i \(-0.836026\pi\)
−0.00845956 + 0.999964i \(0.502693\pi\)
\(38\) −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 0.0750227 − 0.997182i \(-0.476097\pi\)
0.901096 + 0.433619i \(0.142764\pi\)
\(42\) −91.0962 − 157.783i −0.334677 − 0.579678i
\(43\) 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
\(45\) 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 \(0.0182512\pi\)
−0.998357 + 0.0573063i \(0.981749\pi\)
\(48\) 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 0.0181492 − 0.999835i \(-0.494223\pi\)
−0.856808 + 0.515635i \(0.827556\pi\)
\(60\) − 138.800i − 0.298650i
\(61\) −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
\(63\) 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 −0.0118462 − 0.999930i \(-0.503771\pi\)
0.860042 + 0.510224i \(0.170438\pi\)
\(68\) −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 0.572508 − 0.819899i \(-0.305971\pi\)
−0.423800 + 0.905756i \(0.639304\pi\)
\(72\) 188.294 + 108.712i 0.308204 + 0.177942i
\(73\) − 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
\(75\) 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 \(-0.581457\pi\)
0.253122 − 0.967434i \(-0.418543\pi\)
\(84\) −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 −0.662673 − 0.748909i \(-0.730578\pi\)
0.317237 + 0.948346i \(0.397245\pi\)
\(90\) −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 0.120630 − 0.992697i \(-0.461508\pi\)
−0.799386 + 0.600818i \(0.794842\pi\)
\(98\) −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.10.2 yes 10
3.2 odd 2 117.4.q.e.10.4 10
4.3 odd 2 624.4.bv.h.49.2 10
13.2 odd 12 507.4.a.r.1.4 10
13.3 even 3 507.4.b.i.337.4 10
13.4 even 6 inner 39.4.j.c.4.2 ✓ 10
13.10 even 6 507.4.b.i.337.7 10
13.11 odd 12 507.4.a.r.1.7 10
39.2 even 12 1521.4.a.bk.1.7 10
39.11 even 12 1521.4.a.bk.1.4 10
39.17 odd 6 117.4.q.e.82.4 10
52.43 odd 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 13.4 even 6 inner
39.4.j.c.10.2 yes 10 1.1 even 1 trivial
117.4.q.e.10.4 10 3.2 odd 2
117.4.q.e.82.4 10 39.17 odd 6
507.4.a.r.1.4 10 13.2 odd 12
507.4.a.r.1.7 10 13.11 odd 12
507.4.b.i.337.4 10 13.3 even 3
507.4.b.i.337.7 10 13.10 even 6
624.4.bv.h.49.2 10 4.3 odd 2
624.4.bv.h.433.4 10 52.43 odd 6
1521.4.a.bk.1.4 10 39.11 even 12
1521.4.a.bk.1.7 10 39.2 even 12