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Properties

Label 39.4.j.c.4.5
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 4.5
Root \(5.04537i\) of defining polynomial
Character \(\chi\) \(=\) 39.4
Dual form 39.4.j.c.10.5

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.36942 + 2.52268i 1.54482 + 0.891903i 0.998524 + 0.0543124i \(0.0172967\pi\)
0.546298 + 0.837591i \(0.316037\pi\)
\(3\) −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 \(-0.643802\pi\)
0.436556 − 0.899677i \(-0.356198\pi\)
\(6\) −13.1082 + 7.56805i −0.891903 + 0.514941i
\(7\) −13.3609 + 7.71395i −0.721423 + 0.416514i −0.815276 − 0.579072i \(-0.803415\pi\)
0.0938530 + 0.995586i \(0.470082\pi\)
\(8\) 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 0.784396 − 0.620261i \(-0.212973\pi\)
−0.144964 + 0.989437i \(0.546307\pi\)
\(12\) −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 0.936915 − 0.349556i \(-0.113668\pi\)
−0.771182 + 0.636615i \(0.780334\pi\)
\(18\) − 45.4083i − 0.594602i
\(19\) −39.0399 + 22.5397i −0.471388 + 0.272156i −0.716821 − 0.697258i \(-0.754403\pi\)
0.245433 + 0.969414i \(0.421070\pi\)
\(20\) 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 0.340875 + 0.940109i \(0.389277\pi\)
−0.984595 + 0.174848i \(0.944057\pi\)
\(24\) −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 −0.869669 − 0.493635i \(-0.835668\pi\)
0.862335 + 0.506338i \(0.169001\pi\)
\(30\) 152.249 + 263.704i 0.926561 + 1.60485i
\(31\) − 37.7740i − 0.218852i −0.993995 − 0.109426i \(-0.965099\pi\)
0.993995 − 0.109426i \(-0.0349012\pi\)
\(32\) −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 0.962242 − 0.272195i \(-0.0877497\pi\)
0.245393 + 0.969424i \(0.421083\pi\)
\(38\) −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 0.509648 − 0.860383i \(-0.329776\pi\)
−0.490290 + 0.871559i \(0.663109\pi\)
\(42\) 116.759 − 202.233i 0.428960 − 0.742980i
\(43\) −180.449 − 312.547i −0.639958 − 1.10844i −0.985441 − 0.170015i \(-0.945618\pi\)
0.345483 − 0.938425i \(-0.387715\pi\)
\(44\) 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 \(0.105525\pi\)
−0.945550 + 0.325478i \(0.894475\pi\)
\(48\) −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 0.118744 − 0.992925i \(-0.462113\pi\)
0.919270 + 0.393627i \(0.128780\pi\)
\(60\) 1053.49i 2.26675i
\(61\) −102.894 − 178.218i −0.215971 − 0.374073i 0.737601 − 0.675236i \(-0.235958\pi\)
−0.953573 + 0.301163i \(0.902625\pi\)
\(62\) 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 0.0578203 − 0.998327i \(-0.481585\pi\)
−0.835666 + 0.549237i \(0.814918\pi\)
\(68\) −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 0.237073 − 0.971492i \(-0.423812\pi\)
0.959873 + 0.280435i \(0.0904786\pi\)
\(72\) 371.844 − 214.684i 0.608643 − 0.351400i
\(73\) − 66.1205i − 0.106011i −0.998594 − 0.0530056i \(-0.983120\pi\)
0.998594 − 0.0530056i \(-0.0168801\pi\)
\(74\) 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 \(0.0298155\pi\)
−0.995616 + 0.0935313i \(0.970184\pi\)
\(84\) 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 0.792843 − 0.609426i \(-0.208600\pi\)
−0.131357 + 0.991335i \(0.541933\pi\)
\(90\) −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 −0.911394 − 0.411536i \(-0.864992\pi\)
−0.0992962 + 0.995058i \(0.531659\pi\)
\(98\) −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.4.5 ✓ 10
3.2 odd 2 117.4.q.e.82.1 10
4.3 odd 2 624.4.bv.h.433.1 10
13.4 even 6 507.4.b.i.337.9 10
13.6 odd 12 507.4.a.r.1.2 10
13.7 odd 12 507.4.a.r.1.9 10
13.9 even 3 507.4.b.i.337.2 10
13.10 even 6 inner 39.4.j.c.10.5 yes 10
39.20 even 12 1521.4.a.bk.1.2 10
39.23 odd 6 117.4.q.e.10.1 10
39.32 even 12 1521.4.a.bk.1.9 10
52.23 odd 6 624.4.bv.h.49.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.5 ✓ 10 1.1 even 1 trivial
39.4.j.c.10.5 yes 10 13.10 even 6 inner
117.4.q.e.10.1 10 39.23 odd 6
117.4.q.e.82.1 10 3.2 odd 2
507.4.a.r.1.2 10 13.6 odd 12
507.4.a.r.1.9 10 13.7 odd 12
507.4.b.i.337.2 10 13.9 even 3
507.4.b.i.337.9 10 13.4 even 6
624.4.bv.h.49.5 10 52.23 odd 6
624.4.bv.h.433.1 10 4.3 odd 2
1521.4.a.bk.1.2 10 39.20 even 12
1521.4.a.bk.1.9 10 39.32 even 12