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Properties

Label 39.4.j.c.10.5
Level $39$
Weight $4$
Character 39.10
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 10.5
Root \(-5.04537i\) of defining polynomial
Character \(\chi\) \(=\) 39.10
Dual form 39.4.j.c.4.5

$q$-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+(4.36942 - 2.52268i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(8.72787 - 15.1171i) q^{4} +20.1174i q^{5} +(-13.1082 - 7.56805i) q^{6} +(-13.3609 - 7.71395i) q^{7} -47.7076i q^{8} +(-4.50000 + 7.79423i) q^{9} +(50.7498 + 87.9013i) q^{10} +(23.3283 - 13.4686i) q^{11} -52.3672 q^{12} +(-3.96071 + 46.7045i) q^{13} -77.8394 q^{14} +(52.2665 - 30.1761i) q^{15} +(-50.5283 - 87.5177i) q^{16} +(11.6167 - 20.1207i) q^{17} +45.4083i q^{18} +(-39.0399 - 22.5397i) q^{19} +(304.117 + 175.582i) q^{20} +46.2837i q^{21} +(67.9540 - 117.700i) q^{22} +(-71.0050 - 122.984i) q^{23} +(-123.948 + 71.5615i) q^{24} -279.710 q^{25} +(100.515 + 214.063i) q^{26} +27.0000 q^{27} +(-233.225 + 134.653i) q^{28} +(-1.14534 - 1.98379i) q^{29} +(152.249 - 263.704i) q^{30} +37.7740i q^{31} +(-111.031 - 64.1035i) q^{32} +(-69.9849 - 40.4058i) q^{33} -117.221i q^{34} +(155.185 - 268.787i) q^{35} +(78.5508 + 136.054i) q^{36} +(271.793 - 156.920i) q^{37} -227.442 q^{38} +(127.283 - 59.7666i) q^{39} +959.753 q^{40} +(5.08201 - 2.93410i) q^{41} +(116.759 + 202.233i) q^{42} +(-180.449 + 312.547i) q^{43} -470.209i q^{44} +(-156.800 - 90.5283i) q^{45} +(-620.501 - 358.246i) q^{46} -209.748i q^{47} +(-151.585 + 262.553i) q^{48} +(-52.4900 - 90.9154i) q^{49} +(-1222.17 + 705.619i) q^{50} -69.7003 q^{51} +(671.469 + 467.505i) q^{52} +276.886 q^{53} +(117.974 - 68.1125i) q^{54} +(270.953 + 469.305i) q^{55} +(-368.014 + 637.419i) q^{56} +135.238i q^{57} +(-10.0089 - 5.77866i) q^{58} +(470.415 + 271.594i) q^{59} -1053.49i q^{60} +(-102.894 + 178.218i) q^{61} +(95.2917 + 165.050i) q^{62} +(120.249 - 69.4255i) q^{63} +161.602 q^{64} +(-939.573 - 79.6791i) q^{65} -407.724 q^{66} +(-426.585 + 246.289i) q^{67} +(-202.778 - 351.222i) q^{68} +(-213.015 + 368.953i) q^{69} -1565.93i q^{70} +(716.081 + 413.430i) q^{71} +(371.844 + 214.684i) q^{72} +66.1205i q^{73} +(791.718 - 1371.30i) q^{74} +(419.564 + 726.707i) q^{75} +(-681.470 + 393.447i) q^{76} -415.584 q^{77} +(405.380 - 582.240i) q^{78} +317.642 q^{79} +(1760.63 - 1016.50i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(14.8036 - 25.6406i) q^{82} -141.450i q^{83} +(699.675 + 403.958i) q^{84} +(404.777 + 233.698i) q^{85} +1820.86i q^{86} +(-3.43602 + 5.95136i) q^{87} +(-642.555 - 1112.94i) q^{88} +(555.399 - 320.660i) q^{89} -913.497 q^{90} +(413.195 - 593.464i) q^{91} -2478.89 q^{92} +(98.1396 - 56.6609i) q^{93} +(-529.129 - 916.478i) q^{94} +(453.440 - 785.381i) q^{95} +384.621i q^{96} +(-965.551 - 557.461i) q^{97} +(-458.702 - 264.832i) q^{98} +242.435i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.36942 − 2.52268i 1.54482 − 0.891903i 0.546298 − 0.837591i \(-0.316037\pi\)
0.998524 − 0.0543124i \(-0.0172967\pi\)
\(3\) −1.50000 − 2.59808i −0.288675 − 0.500000i
\(4\) 8.72787 − 15.1171i 1.09098 − 1.88964i
\(5\) 20.1174i 1.79935i 0.436556 + 0.899677i \(0.356198\pi\)
−0.436556 + 0.899677i \(0.643802\pi\)
\(6\) −13.1082 − 7.56805i −0.891903 − 0.514941i
\(7\) −13.3609 − 7.71395i −0.721423 − 0.416514i 0.0938530 − 0.995586i \(-0.470082\pi\)
−0.815276 + 0.579072i \(0.803415\pi\)
\(8\) − 47.7076i − 2.10840i
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 50.7498 + 87.9013i 1.60485 + 2.77968i
\(11\) 23.3283 − 13.4686i 0.639432 − 0.369176i −0.144964 − 0.989437i \(-0.546307\pi\)
0.784396 + 0.620261i \(0.212973\pi\)
\(12\) −52.3672 −1.25976
\(13\) −3.96071 + 46.7045i −0.0845002 + 0.996423i
\(14\) −77.8394 −1.48596
\(15\) 52.2665 − 30.1761i 0.899677 − 0.519429i
\(16\) −50.5283 − 87.5177i −0.789505 − 1.36746i
\(17\) 11.6167 − 20.1207i 0.165733 − 0.287059i −0.771182 − 0.636615i \(-0.780334\pi\)
0.936915 + 0.349556i \(0.113668\pi\)
\(18\) 45.4083i 0.594602i
\(19\) −39.0399 − 22.5397i −0.471388 − 0.272156i 0.245433 − 0.969414i \(-0.421070\pi\)
−0.716821 + 0.697258i \(0.754403\pi\)
\(20\) 304.117 + 175.582i 3.40013 + 1.96307i
\(21\) 46.2837i 0.480949i
\(22\) 67.9540 − 117.700i 0.658539 − 1.14062i
\(23\) −71.0050 − 122.984i −0.643720 − 1.11496i −0.984595 − 0.174848i \(-0.944057\pi\)
0.340875 − 0.940109i \(-0.389277\pi\)
\(24\) −123.948 + 71.5615i −1.05420 + 0.608643i
\(25\) −279.710 −2.23768
\(26\) 100.515 + 214.063i 0.758176 + 1.61466i
\(27\) 27.0000 0.192450
\(28\) −233.225 + 134.653i −1.57412 + 0.908820i
\(29\) −1.14534 − 1.98379i −0.00733394 − 0.0127028i 0.862335 − 0.506338i \(-0.169001\pi\)
−0.869669 + 0.493635i \(0.835668\pi\)
\(30\) 152.249 − 263.704i 0.926561 − 1.60485i
\(31\) 37.7740i 0.218852i 0.993995 + 0.109426i \(0.0349012\pi\)
−0.993995 + 0.109426i \(0.965099\pi\)
\(32\) −111.031 − 64.1035i −0.613363 − 0.354125i
\(33\) −69.9849 − 40.4058i −0.369176 − 0.213144i
\(34\) − 117.221i − 0.591273i
\(35\) 155.185 − 268.787i 0.749456 − 1.29810i
\(36\) 78.5508 + 136.054i 0.363661 + 0.629879i
\(37\) 271.793 − 156.920i 1.20764 − 0.697228i 0.245393 − 0.969424i \(-0.421083\pi\)
0.962242 + 0.272195i \(0.0877497\pi\)
\(38\) −227.442 −0.970947
\(39\) 127.283 − 59.7666i 0.522605 − 0.245393i
\(40\) 959.753 3.79376
\(41\) 5.08201 − 2.93410i 0.0193580 − 0.0111763i −0.490290 − 0.871559i \(-0.663109\pi\)
0.509648 + 0.860383i \(0.329776\pi\)
\(42\) 116.759 + 202.233i 0.428960 + 0.742980i
\(43\) −180.449 + 312.547i −0.639958 + 1.10844i 0.345483 + 0.938425i \(0.387715\pi\)
−0.985441 + 0.170015i \(0.945618\pi\)
\(44\) − 470.209i − 1.61106i
\(45\) −156.800 − 90.5283i −0.519429 − 0.299892i
\(46\) −620.501 − 358.246i −1.98887 − 1.14827i
\(47\) − 209.748i − 0.650956i −0.945550 − 0.325478i \(-0.894475\pi\)
0.945550 − 0.325478i \(-0.105525\pi\)
\(48\) −151.585 + 262.553i −0.455821 + 0.789505i
\(49\) −52.4900 − 90.9154i −0.153032 − 0.265060i
\(50\) −1222.17 + 705.619i −3.45681 + 1.99579i
\(51\) −69.7003 −0.191372
\(52\) 671.469 + 467.505i 1.79069 + 1.24676i
\(53\) 276.886 0.717609 0.358804 − 0.933413i \(-0.383185\pi\)
0.358804 + 0.933413i \(0.383185\pi\)
\(54\) 117.974 − 68.1125i 0.297301 − 0.171647i
\(55\) 270.953 + 469.305i 0.664278 + 1.15056i
\(56\) −368.014 + 637.419i −0.878178 + 1.52105i
\(57\) 135.238i 0.314259i
\(58\) −10.0089 − 5.77866i −0.0226593 − 0.0130823i
\(59\) 470.415 + 271.594i 1.03801 + 0.599298i 0.919270 − 0.393627i \(-0.128780\pi\)
0.118744 + 0.992925i \(0.462113\pi\)
\(60\) − 1053.49i − 2.26675i
\(61\) −102.894 + 178.218i −0.215971 + 0.374073i −0.953573 − 0.301163i \(-0.902625\pi\)
0.737601 + 0.675236i \(0.235958\pi\)
\(62\) 95.2917 + 165.050i 0.195195 + 0.338087i
\(63\) 120.249 − 69.4255i 0.240474 − 0.138838i
\(64\) 161.602 0.315629
\(65\) −939.573 − 79.6791i −1.79292 − 0.152046i
\(66\) −407.724 −0.760415
\(67\) −426.585 + 246.289i −0.777846 + 0.449090i −0.835666 − 0.549237i \(-0.814918\pi\)
0.0578203 + 0.998327i \(0.481585\pi\)
\(68\) −202.778 − 351.222i −0.361625 − 0.626352i
\(69\) −213.015 + 368.953i −0.371652 + 0.643720i
\(70\) − 1565.93i − 2.67377i
\(71\) 716.081 + 413.430i 1.19695 + 0.691057i 0.959873 − 0.280435i \(-0.0904786\pi\)
0.237073 + 0.971492i \(0.423812\pi\)
\(72\) 371.844 + 214.684i 0.608643 + 0.351400i
\(73\) 66.1205i 0.106011i 0.998594 + 0.0530056i \(0.0168801\pi\)
−0.998594 + 0.0530056i \(0.983120\pi\)
\(74\) 791.718 − 1371.30i 1.24372 − 2.15419i
\(75\) 419.564 + 726.707i 0.645962 + 1.11884i
\(76\) −681.470 + 393.447i −1.02855 + 0.593835i
\(77\) −415.584 −0.615068
\(78\) 405.380 − 582.240i 0.588465 − 0.845201i
\(79\) 317.642 0.452374 0.226187 − 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(80\) 1760.63 − 1016.50i 2.46055 − 1.42060i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) 14.8036 − 25.6406i 0.0199364 − 0.0345309i
\(83\) − 141.450i − 0.187063i −0.995616 − 0.0935313i \(-0.970184\pi\)
0.995616 − 0.0935313i \(-0.0298155\pi\)
\(84\) 699.675 + 403.958i 0.908820 + 0.524707i
\(85\) 404.777 + 233.698i 0.516520 + 0.298213i
\(86\) 1820.86i 2.28312i
\(87\) −3.43602 + 5.95136i −0.00423425 + 0.00733394i
\(88\) −642.555 − 1112.94i −0.778371 − 1.34818i
\(89\) 555.399 − 320.660i 0.661486 − 0.381909i −0.131357 − 0.991335i \(-0.541933\pi\)
0.792843 + 0.609426i \(0.208600\pi\)
\(90\) −913.497 −1.06990
\(91\) 413.195 − 593.464i 0.475985 − 0.683648i
\(92\) −2478.89 −2.80915
\(93\) 98.1396 − 56.6609i 0.109426 − 0.0631771i
\(94\) −529.129 − 916.478i −0.580590 − 1.00561i
\(95\) 453.440 − 785.381i 0.489705 − 0.848194i
\(96\) 384.621i 0.408909i
\(97\) −965.551 − 557.461i −1.01069 − 0.583522i −0.0992962 − 0.995058i \(-0.531659\pi\)
−0.911394 + 0.411536i \(0.864992\pi\)
\(98\) −458.702 − 264.832i −0.472815 − 0.272980i
\(99\) 242.435i 0.246117i
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.10.5 yes 10
3.2 odd 2 117.4.q.e.10.1 10
4.3 odd 2 624.4.bv.h.49.5 10
13.2 odd 12 507.4.a.r.1.9 10
13.3 even 3 507.4.b.i.337.9 10
13.4 even 6 inner 39.4.j.c.4.5 ✓ 10
13.10 even 6 507.4.b.i.337.2 10
13.11 odd 12 507.4.a.r.1.2 10
39.2 even 12 1521.4.a.bk.1.2 10
39.11 even 12 1521.4.a.bk.1.9 10
39.17 odd 6 117.4.q.e.82.1 10
52.43 odd 6 624.4.bv.h.433.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.5 ✓ 10 13.4 even 6 inner
39.4.j.c.10.5 yes 10 1.1 even 1 trivial
117.4.q.e.10.1 10 3.2 odd 2
117.4.q.e.82.1 10 39.17 odd 6
507.4.a.r.1.2 10 13.11 odd 12
507.4.a.r.1.9 10 13.2 odd 12
507.4.b.i.337.2 10 13.10 even 6
507.4.b.i.337.9 10 13.3 even 3
624.4.bv.h.49.5 10 4.3 odd 2
624.4.bv.h.433.1 10 52.43 odd 6
1521.4.a.bk.1.2 10 39.2 even 12
1521.4.a.bk.1.9 10 39.11 even 12