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Properties

Label 39.4.j.c.4.1
Level $39$
Weight $4$
Character 39.4
Analytic conductor $2.301$
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] = [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 4.1
Root \(-5.36472i\) of defining polynomial
Character \(\chi\) \(=\) 39.4
Dual form 39.4.j.c.10.1

$q$-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+(-4.64599 - 2.68236i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(10.3901 + 17.9962i) q^{4} -2.69631i q^{5} +(13.9380 - 8.04709i) q^{6} +(13.1657 - 7.60123i) q^{7} -68.5626i q^{8} +(-4.50000 - 7.79423i) q^{9} +(-7.23249 + 12.5270i) q^{10} +(57.9240 + 33.4424i) q^{11} -62.3408 q^{12} +(46.8650 - 0.818689i) q^{13} -81.5570 q^{14} +(7.00522 + 4.04447i) q^{15} +(-100.789 + 174.571i) q^{16} +(2.08177 + 3.60573i) q^{17} +48.2825i q^{18} +(-22.5903 + 13.0425i) q^{19} +(48.5235 - 28.0150i) q^{20} +45.6074i q^{21} +(-179.409 - 310.746i) q^{22} +(23.6621 - 40.9839i) q^{23} +(178.131 + 102.844i) q^{24} +117.730 q^{25} +(-219.930 - 121.905i) q^{26} +27.0000 q^{27} +(273.587 + 157.956i) q^{28} +(-128.503 + 222.575i) q^{29} +(-21.6975 - 37.5811i) q^{30} -206.242i q^{31} +(461.511 - 266.453i) q^{32} +(-173.772 + 100.327i) q^{33} -22.3362i q^{34} +(-20.4953 - 35.4989i) q^{35} +(93.5112 - 161.966i) q^{36} +(-152.149 - 87.8430i) q^{37} +139.939 q^{38} +(-68.1705 + 122.987i) q^{39} -184.866 q^{40} +(135.501 + 78.2313i) q^{41} +(122.336 - 211.891i) q^{42} +(25.9922 + 45.0199i) q^{43} +1389.88i q^{44} +(-21.0157 + 12.1334i) q^{45} +(-219.868 + 126.941i) q^{46} -354.222i q^{47} +(-302.366 - 523.714i) q^{48} +(-55.9425 + 96.8953i) q^{49} +(-546.972 - 315.794i) q^{50} -12.4906 q^{51} +(501.667 + 834.888i) q^{52} -10.4723 q^{53} +(-125.442 - 72.4238i) q^{54} +(90.1712 - 156.181i) q^{55} +(-521.160 - 902.676i) q^{56} -78.2550i q^{57} +(1194.05 - 689.386i) q^{58} +(385.480 - 222.557i) q^{59} +168.090i q^{60} +(-59.8481 - 103.660i) q^{61} +(-553.216 + 958.199i) q^{62} +(-118.491 - 68.4111i) q^{63} -1246.28 q^{64} +(-2.20744 - 126.363i) q^{65} +1076.46 q^{66} +(-19.4057 - 11.2039i) q^{67} +(-43.2597 + 74.9281i) q^{68} +(70.9863 + 122.952i) q^{69} +219.903i q^{70} +(-246.997 + 142.604i) q^{71} +(-534.393 + 308.532i) q^{72} +740.989i q^{73} +(471.253 + 816.235i) q^{74} +(-176.595 + 305.871i) q^{75} +(-469.432 - 271.027i) q^{76} +1016.81 q^{77} +(646.615 - 388.538i) q^{78} -547.679 q^{79} +(470.698 + 271.758i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(-419.689 - 726.923i) q^{82} -603.056i q^{83} +(-820.762 + 473.867i) q^{84} +(9.72218 - 5.61310i) q^{85} -278.882i q^{86} +(-385.510 - 667.724i) q^{87} +(2292.90 - 3971.42i) q^{88} +(-186.774 - 107.834i) q^{89} +130.185 q^{90} +(610.789 - 367.011i) q^{91} +983.409 q^{92} +(535.833 + 309.363i) q^{93} +(-950.153 + 1645.71i) q^{94} +(35.1666 + 60.9104i) q^{95} +1598.72i q^{96} +(-1253.58 + 723.752i) q^{97} +(519.817 - 300.116i) q^{98} -601.964i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.64599 − 2.68236i −1.64260 − 0.948358i −0.979903 − 0.199475i \(-0.936076\pi\)
−0.662702 − 0.748883i \(-0.730590\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) 10.3901 + 17.9962i 1.29877 + 2.24953i
\(5\) − 2.69631i − 0.241165i −0.992703 − 0.120583i \(-0.961524\pi\)
0.992703 − 0.120583i \(-0.0384763\pi\)
\(6\) 13.9380 − 8.04709i 0.948358 − 0.547535i
\(7\) 13.1657 − 7.60123i 0.710882 − 0.410428i −0.100505 − 0.994937i \(-0.532046\pi\)
0.811388 + 0.584509i \(0.198713\pi\)
\(8\) − 68.5626i − 3.03007i
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −7.23249 + 12.5270i −0.228711 + 0.396140i
\(11\) 57.9240 + 33.4424i 1.58770 + 0.916661i 0.993685 + 0.112209i \(0.0357927\pi\)
0.594018 + 0.804451i \(0.297541\pi\)
\(12\) −62.3408 −1.49969
\(13\) 46.8650 − 0.818689i 0.999847 − 0.0174664i
\(14\) −81.5570 −1.55693
\(15\) 7.00522 + 4.04447i 0.120583 + 0.0696185i
\(16\) −100.789 + 174.571i −1.57482 + 2.72767i
\(17\) 2.08177 + 3.60573i 0.0297002 + 0.0514422i 0.880493 − 0.474058i \(-0.157211\pi\)
−0.850793 + 0.525501i \(0.823878\pi\)
\(18\) 48.2825i 0.632239i
\(19\) −22.5903 + 13.0425i −0.272766 + 0.157482i −0.630144 − 0.776478i \(-0.717004\pi\)
0.357378 + 0.933960i \(0.383671\pi\)
\(20\) 48.5235 − 28.0150i 0.542509 − 0.313218i
\(21\) 45.6074i 0.473921i
\(22\) −179.409 − 310.746i −1.73865 − 3.01142i
\(23\) 23.6621 − 40.9839i 0.214517 − 0.371554i −0.738606 − 0.674137i \(-0.764516\pi\)
0.953123 + 0.302583i \(0.0978490\pi\)
\(24\) 178.131 + 102.844i 1.51503 + 0.874705i
\(25\) 117.730 0.941839
\(26\) −219.930 − 121.905i −1.65892 − 0.919523i
\(27\) 27.0000 0.192450
\(28\) 273.587 + 157.956i 1.84654 + 1.06610i
\(29\) −128.503 + 222.575i −0.822845 + 1.42521i 0.0807106 + 0.996738i \(0.474281\pi\)
−0.903555 + 0.428471i \(0.859052\pi\)
\(30\) −21.6975 − 37.5811i −0.132047 − 0.228711i
\(31\) − 206.242i − 1.19491i −0.801903 − 0.597455i \(-0.796179\pi\)
0.801903 − 0.597455i \(-0.203821\pi\)
\(32\) 461.511 − 266.453i 2.54951 − 1.47196i
\(33\) −173.772 + 100.327i −0.916661 + 0.529234i
\(34\) − 22.3362i − 0.112666i
\(35\) −20.4953 − 35.4989i −0.0989811 − 0.171440i
\(36\) 93.5112 − 161.966i 0.432922 − 0.749843i
\(37\) −152.149 − 87.8430i −0.676029 − 0.390305i 0.122328 − 0.992490i \(-0.460964\pi\)
−0.798357 + 0.602184i \(0.794297\pi\)
\(38\) 139.939 0.597396
\(39\) −68.1705 + 122.987i −0.279898 + 0.504966i
\(40\) −184.866 −0.730748
\(41\) 135.501 + 78.2313i 0.516137 + 0.297992i 0.735353 − 0.677684i \(-0.237016\pi\)
−0.219216 + 0.975676i \(0.570350\pi\)
\(42\) 122.336 − 211.891i 0.449447 − 0.778466i
\(43\) 25.9922 + 45.0199i 0.0921809 + 0.159662i 0.908429 − 0.418040i \(-0.137283\pi\)
−0.816248 + 0.577702i \(0.803950\pi\)
\(44\) 1389.88i 4.76211i
\(45\) −21.0157 + 12.1334i −0.0696185 + 0.0401942i
\(46\) −219.868 + 126.941i −0.704733 + 0.406878i
\(47\) − 354.222i − 1.09933i −0.835384 − 0.549666i \(-0.814755\pi\)
0.835384 − 0.549666i \(-0.185245\pi\)
\(48\) −302.366 − 523.714i −0.909225 − 1.57482i
\(49\) −55.9425 + 96.8953i −0.163098 + 0.282494i
\(50\) −546.972 − 315.794i −1.54707 − 0.893201i
\(51\) −12.4906 −0.0342948
\(52\) 501.667 + 834.888i 1.33786 + 2.22650i
\(53\) −10.4723 −0.0271412 −0.0135706 − 0.999908i \(-0.504320\pi\)
−0.0135706 + 0.999908i \(0.504320\pi\)
\(54\) −125.442 − 72.4238i −0.316119 − 0.182512i
\(55\) 90.1712 − 156.181i 0.221067 − 0.382899i
\(56\) −521.160 − 902.676i −1.24362 − 2.15402i
\(57\) − 78.2550i − 0.181844i
\(58\) 1194.05 − 689.386i 2.70322 − 1.56070i
\(59\) 385.480 − 222.557i 0.850597 − 0.491092i −0.0102552 − 0.999947i \(-0.503264\pi\)
0.860852 + 0.508855i \(0.169931\pi\)
\(60\) 168.090i 0.361673i
\(61\) −59.8481 − 103.660i −0.125619 − 0.217579i 0.796356 − 0.604829i \(-0.206758\pi\)
−0.921975 + 0.387250i \(0.873425\pi\)
\(62\) −553.216 + 958.199i −1.13320 + 1.96276i
\(63\) −118.491 − 68.4111i −0.236961 − 0.136809i
\(64\) −1246.28 −2.43413
\(65\) −2.20744 − 126.363i −0.00421230 − 0.241129i
\(66\) 1076.46 2.00761
\(67\) −19.4057 − 11.2039i −0.0353848 − 0.0204294i 0.482203 − 0.876059i \(-0.339837\pi\)
−0.517588 + 0.855630i \(0.673170\pi\)
\(68\) −43.2597 + 74.9281i −0.0771473 + 0.133623i
\(69\) 70.9863 + 122.952i 0.123851 + 0.214517i
\(70\) 219.903i 0.375478i
\(71\) −246.997 + 142.604i −0.412861 + 0.238365i −0.692018 − 0.721880i \(-0.743278\pi\)
0.279157 + 0.960245i \(0.409945\pi\)
\(72\) −534.393 + 308.532i −0.874705 + 0.505011i
\(73\) 740.989i 1.18803i 0.804454 + 0.594015i \(0.202458\pi\)
−0.804454 + 0.594015i \(0.797542\pi\)
\(74\) 471.253 + 816.235i 0.740299 + 1.28223i
\(75\) −176.595 + 305.871i −0.271886 + 0.470920i
\(76\) −469.432 − 271.027i −0.708520 − 0.409064i
\(77\) 1016.81 1.50489
\(78\) 646.615 − 388.538i 0.938650 − 0.564016i
\(79\) −547.679 −0.779983 −0.389992 − 0.920818i \(-0.627522\pi\)
−0.389992 + 0.920818i \(0.627522\pi\)
\(80\) 470.698 + 271.758i 0.657821 + 0.379793i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) −419.689 − 726.923i −0.565206 − 0.978966i
\(83\) − 603.056i − 0.797518i −0.917056 − 0.398759i \(-0.869441\pi\)
0.917056 − 0.398759i \(-0.130559\pi\)
\(84\) −820.762 + 473.867i −1.06610 + 0.615513i
\(85\) 9.72218 − 5.61310i 0.0124061 − 0.00716266i
\(86\) − 278.882i − 0.349682i
\(87\) −385.510 − 667.724i −0.475070 − 0.822845i
\(88\) 2292.90 − 3971.42i 2.77754 − 4.81085i
\(89\) −186.774 − 107.834i −0.222450 − 0.128431i 0.384634 − 0.923069i \(-0.374328\pi\)
−0.607084 + 0.794638i \(0.707661\pi\)
\(90\) 130.185 0.152474
\(91\) 610.789 − 367.011i 0.703605 − 0.422782i
\(92\) 983.409 1.11443
\(93\) 535.833 + 309.363i 0.597455 + 0.344941i
\(94\) −950.153 + 1645.71i −1.04256 + 1.80577i
\(95\) 35.1666 + 60.9104i 0.0379792 + 0.0657818i
\(96\) 1598.72i 1.69967i
\(97\) −1253.58 + 723.752i −1.31218 + 0.757586i −0.982457 − 0.186492i \(-0.940288\pi\)
−0.329722 + 0.944078i \(0.606955\pi\)
\(98\) 519.817 − 300.116i 0.535810 − 0.309350i
\(99\) − 601.964i − 0.611107i
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.4.1 ✓ 10
3.2 odd 2 117.4.q.e.82.5 10
4.3 odd 2 624.4.bv.h.433.3 10
13.4 even 6 507.4.b.i.337.1 10
13.6 odd 12 507.4.a.r.1.10 10
13.7 odd 12 507.4.a.r.1.1 10
13.9 even 3 507.4.b.i.337.10 10
13.10 even 6 inner 39.4.j.c.10.1 yes 10
39.20 even 12 1521.4.a.bk.1.10 10
39.23 odd 6 117.4.q.e.10.5 10
39.32 even 12 1521.4.a.bk.1.1 10
52.23 odd 6 624.4.bv.h.49.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.1 ✓ 10 1.1 even 1 trivial
39.4.j.c.10.1 yes 10 13.10 even 6 inner
117.4.q.e.10.5 10 39.23 odd 6
117.4.q.e.82.5 10 3.2 odd 2
507.4.a.r.1.1 10 13.7 odd 12
507.4.a.r.1.10 10 13.6 odd 12
507.4.b.i.337.1 10 13.4 even 6
507.4.b.i.337.10 10 13.9 even 3
624.4.bv.h.49.3 10 52.23 odd 6
624.4.bv.h.433.3 10 4.3 odd 2
1521.4.a.bk.1.1 10 39.32 even 12
1521.4.a.bk.1.10 10 39.20 even 12