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 10.3
Root \(0.917374i\) of defining polynomial
Character \(\chi\) \(=\) 39.10
Dual form 39.4.j.c.4.3

$q$-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+(-0.794469 + 0.458687i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(-3.57921 + 6.19938i) q^{4} +15.4704i q^{5} +(2.38341 + 1.37606i) q^{6} +(17.8257 + 10.2917i) q^{7} -13.9059i q^{8} +(-4.50000 + 7.79423i) q^{9} +(-7.09608 - 12.2908i) q^{10} +(-57.0209 + 32.9210i) q^{11} +21.4753 q^{12} +(19.2429 - 42.7400i) q^{13} -18.8826 q^{14} +(40.1933 - 23.2056i) q^{15} +(-22.2552 - 38.5472i) q^{16} +(22.1478 - 38.3611i) q^{17} -8.25636i q^{18} +(127.352 + 73.5266i) q^{19} +(-95.9069 - 55.3719i) q^{20} -61.7500i q^{21} +(30.2009 - 52.3094i) q^{22} +(26.5793 + 46.0367i) q^{23} +(-36.1287 + 20.8589i) q^{24} -114.334 q^{25} +(4.31639 + 42.7821i) q^{26} +27.0000 q^{27} +(-127.604 + 73.6721i) q^{28} +(19.3128 + 33.4508i) q^{29} +(-21.2882 + 36.8723i) q^{30} -88.3894i q^{31} +(131.705 + 76.0401i) q^{32} +(171.063 + 98.7630i) q^{33} +40.6357i q^{34} +(-159.216 + 275.771i) q^{35} +(-32.2129 - 55.7944i) q^{36} +(68.3803 - 39.4794i) q^{37} -134.903 q^{38} +(-139.906 + 14.1155i) q^{39} +215.131 q^{40} +(307.410 - 177.483i) q^{41} +(28.3239 + 49.0585i) q^{42} +(-203.923 + 353.205i) q^{43} -471.325i q^{44} +(-120.580 - 69.6169i) q^{45} +(-42.2329 - 24.3832i) q^{46} -67.9674i q^{47} +(-66.7657 + 115.642i) q^{48} +(40.3369 + 69.8656i) q^{49} +(90.8345 - 52.4433i) q^{50} -132.887 q^{51} +(196.087 + 272.270i) q^{52} +226.572 q^{53} +(-21.4507 + 12.3845i) q^{54} +(-509.302 - 882.136i) q^{55} +(143.115 - 247.883i) q^{56} -441.160i q^{57} +(-30.6869 - 17.7171i) q^{58} +(-123.002 - 71.0154i) q^{59} +332.231i q^{60} +(-133.416 + 231.083i) q^{61} +(40.5431 + 70.2227i) q^{62} +(-160.431 + 92.6250i) q^{63} +216.569 q^{64} +(661.206 + 297.696i) q^{65} -181.205 q^{66} +(356.098 - 205.593i) q^{67} +(158.543 + 274.605i) q^{68} +(79.7379 - 138.110i) q^{69} -292.122i q^{70} +(-79.2458 - 45.7526i) q^{71} +(108.386 + 62.5767i) q^{72} +63.1328i q^{73} +(-36.2173 + 62.7303i) q^{74} +(171.500 + 297.047i) q^{75} +(-911.638 + 526.335i) q^{76} -1355.25 q^{77} +(104.677 - 75.3875i) q^{78} -287.115 q^{79} +(596.341 - 344.298i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(-162.818 + 282.010i) q^{82} +373.812i q^{83} +(382.812 + 221.016i) q^{84} +(593.463 + 342.636i) q^{85} -374.147i q^{86} +(57.9385 - 100.352i) q^{87} +(457.798 + 792.929i) q^{88} +(103.406 - 59.7013i) q^{89} +127.729 q^{90} +(782.885 - 563.829i) q^{91} -380.532 q^{92} +(-229.643 + 132.584i) q^{93} +(31.1758 + 53.9980i) q^{94} +(-1137.49 + 1970.18i) q^{95} -456.241i q^{96} +(480.341 + 277.325i) q^{97} +(-64.0928 - 37.0040i) q^{98} -592.578i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.794469 + 0.458687i −0.280887 + 0.162170i −0.633825 − 0.773476i \(-0.718516\pi\)
0.352938 + 0.935647i \(0.385183\pi\)
\(3\) −1.50000 − 2.59808i −0.288675 − 0.500000i
\(4\) −3.57921 + 6.19938i −0.447402 + 0.774922i
\(5\) 15.4704i 1.38372i 0.722034 + 0.691858i \(0.243208\pi\)
−0.722034 + 0.691858i \(0.756792\pi\)
\(6\) 2.38341 + 1.37606i 0.162170 + 0.0936291i
\(7\) 17.8257 + 10.2917i 0.962497 + 0.555698i 0.896941 − 0.442151i \(-0.145784\pi\)
0.0655563 + 0.997849i \(0.479118\pi\)
\(8\) − 13.9059i − 0.614562i
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) −7.09608 − 12.2908i −0.224398 − 0.388668i
\(11\) −57.0209 + 32.9210i −1.56295 + 0.902369i −0.565992 + 0.824411i \(0.691507\pi\)
−0.996957 + 0.0779583i \(0.975160\pi\)
\(12\) 21.4753 0.516615
\(13\) 19.2429 − 42.7400i 0.410540 − 0.911842i
\(14\) −18.8826 −0.360471
\(15\) 40.1933 − 23.2056i 0.691858 − 0.399444i
\(16\) −22.2552 − 38.5472i −0.347738 − 0.602300i
\(17\) 22.1478 − 38.3611i 0.315979 − 0.547291i −0.663666 − 0.748029i \(-0.731001\pi\)
0.979645 + 0.200738i \(0.0643339\pi\)
\(18\) − 8.25636i − 0.108114i
\(19\) 127.352 + 73.5266i 1.53771 + 0.887798i 0.998972 + 0.0453247i \(0.0144322\pi\)
0.538738 + 0.842473i \(0.318901\pi\)
\(20\) −95.9069 − 55.3719i −1.07227 − 0.619077i
\(21\) − 61.7500i − 0.641665i
\(22\) 30.2009 − 52.3094i 0.292675 − 0.506928i
\(23\) 26.5793 + 46.0367i 0.240964 + 0.417362i 0.960989 − 0.276586i \(-0.0892031\pi\)
−0.720025 + 0.693948i \(0.755870\pi\)
\(24\) −36.1287 + 20.8589i −0.307281 + 0.177409i
\(25\) −114.334 −0.914669
\(26\) 4.31639 + 42.7821i 0.0325582 + 0.322702i
\(27\) 27.0000 0.192450
\(28\) −127.604 + 73.6721i −0.861245 + 0.497240i
\(29\) 19.3128 + 33.4508i 0.123666 + 0.214195i 0.921211 − 0.389064i \(-0.127202\pi\)
−0.797545 + 0.603260i \(0.793868\pi\)
\(30\) −21.2882 + 36.8723i −0.129556 + 0.224398i
\(31\) − 88.3894i − 0.512104i −0.966663 − 0.256052i \(-0.917578\pi\)
0.966663 − 0.256052i \(-0.0824218\pi\)
\(32\) 131.705 + 76.0401i 0.727576 + 0.420066i
\(33\) 171.063 + 98.7630i 0.902369 + 0.520983i
\(34\) 40.6357i 0.204969i
\(35\) −159.216 + 275.771i −0.768928 + 1.33182i
\(36\) −32.2129 − 55.7944i −0.149134 − 0.258307i
\(37\) 68.3803 − 39.4794i 0.303828 − 0.175415i −0.340333 − 0.940305i \(-0.610540\pi\)
0.644161 + 0.764890i \(0.277206\pi\)
\(38\) −134.903 −0.575898
\(39\) −139.906 + 14.1155i −0.574434 + 0.0579561i
\(40\) 215.131 0.850379
\(41\) 307.410 − 177.483i 1.17096 − 0.676054i 0.217053 − 0.976160i \(-0.430355\pi\)
0.953906 + 0.300106i \(0.0970221\pi\)
\(42\) 28.3239 + 49.0585i 0.104059 + 0.180235i
\(43\) −203.923 + 353.205i −0.723208 + 1.25263i 0.236499 + 0.971632i \(0.424000\pi\)
−0.959707 + 0.281002i \(0.909333\pi\)
\(44\) − 471.325i − 1.61489i
\(45\) −120.580 − 69.6169i −0.399444 − 0.230619i
\(46\) −42.2329 − 24.3832i −0.135367 − 0.0781544i
\(47\) − 67.9674i − 0.210938i −0.994423 − 0.105469i \(-0.966366\pi\)
0.994423 − 0.105469i \(-0.0336343\pi\)
\(48\) −66.7657 + 115.642i −0.200767 + 0.347738i
\(49\) 40.3369 + 69.8656i 0.117600 + 0.203690i
\(50\) 90.8345 − 52.4433i 0.256919 − 0.148332i
\(51\) −132.887 −0.364861
\(52\) 196.087 + 272.270i 0.522931 + 0.726097i
\(53\) 226.572 0.587209 0.293604 − 0.955927i \(-0.405145\pi\)
0.293604 + 0.955927i \(0.405145\pi\)
\(54\) −21.4507 + 12.3845i −0.0540568 + 0.0312097i
\(55\) −509.302 − 882.136i −1.24862 − 2.16268i
\(56\) 143.115 − 247.883i 0.341511 − 0.591514i
\(57\) − 441.160i − 1.02514i
\(58\) −30.6869 − 17.7171i −0.0694722 − 0.0401098i
\(59\) −123.002 − 71.0154i −0.271416 − 0.156702i 0.358115 − 0.933677i \(-0.383420\pi\)
−0.629531 + 0.776976i \(0.716753\pi\)
\(60\) 332.231i 0.714848i
\(61\) −133.416 + 231.083i −0.280035 + 0.485034i −0.971393 − 0.237478i \(-0.923679\pi\)
0.691358 + 0.722512i \(0.257013\pi\)
\(62\) 40.5431 + 70.2227i 0.0830480 + 0.143843i
\(63\) −160.431 + 92.6250i −0.320832 + 0.185233i
\(64\) 216.569 0.422987
\(65\) 661.206 + 297.696i 1.26173 + 0.568071i
\(66\) −181.205 −0.337952
\(67\) 356.098 − 205.593i 0.649318 − 0.374884i −0.138877 − 0.990310i \(-0.544349\pi\)
0.788195 + 0.615426i \(0.211016\pi\)
\(68\) 158.543 + 274.605i 0.282739 + 0.489718i
\(69\) 79.7379 − 138.110i 0.139121 − 0.240964i
\(70\) − 292.122i − 0.498789i
\(71\) −79.2458 − 45.7526i −0.132461 − 0.0764765i 0.432305 − 0.901727i \(-0.357700\pi\)
−0.564766 + 0.825251i \(0.691034\pi\)
\(72\) 108.386 + 62.5767i 0.177409 + 0.102427i
\(73\) 63.1328i 0.101221i 0.998718 + 0.0506105i \(0.0161167\pi\)
−0.998718 + 0.0506105i \(0.983883\pi\)
\(74\) −36.2173 + 62.7303i −0.0568943 + 0.0985439i
\(75\) 171.500 + 297.047i 0.264042 + 0.457335i
\(76\) −911.638 + 526.335i −1.37595 + 0.794404i
\(77\) −1355.25 −2.00578
\(78\) 104.677 − 75.3875i 0.151952 − 0.109435i
\(79\) −287.115 −0.408899 −0.204449 − 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) 596.341 − 344.298i 0.833412 − 0.481170i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) −162.818 + 282.010i −0.219272 + 0.379790i
\(83\) 373.812i 0.494352i 0.968971 + 0.247176i \(0.0795026\pi\)
−0.968971 + 0.247176i \(0.920497\pi\)
\(84\) 382.812 + 221.016i 0.497240 + 0.287082i
\(85\) 593.463 + 342.636i 0.757295 + 0.437224i
\(86\) − 374.147i − 0.469132i
\(87\) 57.9385 − 100.352i 0.0713984 − 0.123666i
\(88\) 457.798 + 792.929i 0.554561 + 0.960528i
\(89\) 103.406 − 59.7013i 0.123157 − 0.0711047i −0.437156 − 0.899386i \(-0.644014\pi\)
0.560313 + 0.828281i \(0.310681\pi\)
\(90\) 127.729 0.149598
\(91\) 782.885 − 563.829i 0.901853 − 0.649509i
\(92\) −380.532 −0.431231
\(93\) −229.643 + 132.584i −0.256052 + 0.147832i
\(94\) 31.1758 + 53.9980i 0.0342078 + 0.0592497i
\(95\) −1137.49 + 1970.18i −1.22846 + 2.12775i
\(96\) − 456.241i − 0.485051i
\(97\) 480.341 + 277.325i 0.502796 + 0.290290i 0.729868 − 0.683589i \(-0.239582\pi\)
−0.227071 + 0.973878i \(0.572915\pi\)
\(98\) −64.0928 − 37.0040i −0.0660648 − 0.0381426i
\(99\) − 592.578i − 0.601579i
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.10.3 yes 10
3.2 odd 2 117.4.q.e.10.3 10
4.3 odd 2 624.4.bv.h.49.4 10
13.2 odd 12 507.4.a.r.1.5 10
13.3 even 3 507.4.b.i.337.5 10
13.4 even 6 inner 39.4.j.c.4.3 ✓ 10
13.10 even 6 507.4.b.i.337.6 10
13.11 odd 12 507.4.a.r.1.6 10
39.2 even 12 1521.4.a.bk.1.6 10
39.11 even 12 1521.4.a.bk.1.5 10
39.17 odd 6 117.4.q.e.82.3 10
52.43 odd 6 624.4.bv.h.433.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 13.4 even 6 inner
39.4.j.c.10.3 yes 10 1.1 even 1 trivial
117.4.q.e.10.3 10 3.2 odd 2
117.4.q.e.82.3 10 39.17 odd 6
507.4.a.r.1.5 10 13.2 odd 12
507.4.a.r.1.6 10 13.11 odd 12
507.4.b.i.337.5 10 13.3 even 3
507.4.b.i.337.6 10 13.10 even 6
624.4.bv.h.49.4 10 4.3 odd 2
624.4.bv.h.433.2 10 52.43 odd 6
1521.4.a.bk.1.5 10 39.11 even 12
1521.4.a.bk.1.6 10 39.2 even 12