Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [78,2,Mod(43,78)] 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("78.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 43.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 78.43
Dual form 78.2.i.a.49.2

$q$-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+(0.866025 + 0.500000i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{4} -0.267949i q^{5} +(-0.866025 + 0.500000i) q^{6} +(0.633975 - 0.366025i) q^{7} +1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.133975 - 0.232051i) q^{10} +(-4.09808 - 2.36603i) q^{11} -1.00000 q^{12} +(2.59808 - 2.50000i) q^{13} +0.732051 q^{14} +(0.232051 + 0.133975i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(-1.13397 - 1.96410i) q^{17} -1.00000i q^{18} +(-1.09808 + 0.633975i) q^{19} +(0.232051 - 0.133975i) q^{20} +0.732051i q^{21} +(-2.36603 - 4.09808i) q^{22} +(-3.09808 + 5.36603i) q^{23} +(-0.866025 - 0.500000i) q^{24} +4.92820 q^{25} +(3.50000 - 0.866025i) q^{26} +1.00000 q^{27} +(0.633975 + 0.366025i) q^{28} +(-1.23205 + 2.13397i) q^{29} +(0.133975 + 0.232051i) q^{30} +5.46410i q^{31} +(-0.866025 + 0.500000i) q^{32} +(4.09808 - 2.36603i) q^{33} -2.26795i q^{34} +(-0.0980762 - 0.169873i) q^{35} +(0.500000 - 0.866025i) q^{36} +(-9.06218 - 5.23205i) q^{37} -1.26795 q^{38} +(0.866025 + 3.50000i) q^{39} +0.267949 q^{40} +(9.86603 + 5.69615i) q^{41} +(-0.366025 + 0.633975i) q^{42} +(3.83013 + 6.63397i) q^{43} -4.73205i q^{44} +(-0.232051 + 0.133975i) q^{45} +(-5.36603 + 3.09808i) q^{46} -8.19615i q^{47} +(-0.500000 - 0.866025i) q^{48} +(-3.23205 + 5.59808i) q^{49} +(4.26795 + 2.46410i) q^{50} +2.26795 q^{51} +(3.46410 + 1.00000i) q^{52} +0.464102 q^{53} +(0.866025 + 0.500000i) q^{54} +(-0.633975 + 1.09808i) q^{55} +(0.366025 + 0.633975i) q^{56} -1.26795i q^{57} +(-2.13397 + 1.23205i) q^{58} +(6.92820 - 4.00000i) q^{59} +0.267949i q^{60} +(-0.598076 - 1.03590i) q^{61} +(-2.73205 + 4.73205i) q^{62} +(-0.633975 - 0.366025i) q^{63} -1.00000 q^{64} +(-0.669873 - 0.696152i) q^{65} +4.73205 q^{66} +(-9.63397 - 5.56218i) q^{67} +(1.13397 - 1.96410i) q^{68} +(-3.09808 - 5.36603i) q^{69} -0.196152i q^{70} +(-1.09808 + 0.633975i) q^{71} +(0.866025 - 0.500000i) q^{72} -9.73205i q^{73} +(-5.23205 - 9.06218i) q^{74} +(-2.46410 + 4.26795i) q^{75} +(-1.09808 - 0.633975i) q^{76} -3.46410 q^{77} +(-1.00000 + 3.46410i) q^{78} -9.46410 q^{79} +(0.232051 + 0.133975i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(5.69615 + 9.86603i) q^{82} -10.1962i q^{83} +(-0.633975 + 0.366025i) q^{84} +(-0.526279 + 0.303848i) q^{85} +7.66025i q^{86} +(-1.23205 - 2.13397i) q^{87} +(2.36603 - 4.09808i) q^{88} +(2.19615 + 1.26795i) q^{89} -0.267949 q^{90} +(0.732051 - 2.53590i) q^{91} -6.19615 q^{92} +(-4.73205 - 2.73205i) q^{93} +(4.09808 - 7.09808i) q^{94} +(0.169873 + 0.294229i) q^{95} -1.00000i q^{96} +(5.19615 - 3.00000i) q^{97} +(-5.59808 + 3.23205i) q^{98} +4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} + 6 q^{7} - 2 q^{9} + 4 q^{10} - 6 q^{11} - 4 q^{12} - 4 q^{14} - 6 q^{15} - 2 q^{16} - 8 q^{17} + 6 q^{19} - 6 q^{20} - 6 q^{22} - 2 q^{23} - 8 q^{25} + 14 q^{26} + 4 q^{27} + 6 q^{28}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.612372 + 0.353553i
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) − 0.267949i − 0.119831i −0.998203 − 0.0599153i \(-0.980917\pi\)
0.998203 − 0.0599153i \(-0.0190830\pi\)
\(6\) −0.866025 + 0.500000i −0.353553 + 0.204124i
\(7\) 0.633975 − 0.366025i 0.239620 − 0.138345i −0.375382 − 0.926870i \(-0.622489\pi\)
0.615002 + 0.788526i \(0.289155\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0.133975 − 0.232051i 0.0423665 − 0.0733809i
\(11\) −4.09808 − 2.36603i −1.23562 − 0.713384i −0.267421 − 0.963580i \(-0.586172\pi\)
−0.968195 + 0.250196i \(0.919505\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.59808 − 2.50000i 0.720577 − 0.693375i
\(14\) 0.732051 0.195649
\(15\) 0.232051 + 0.133975i 0.0599153 + 0.0345921i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.13397 − 1.96410i −0.275029 − 0.476365i 0.695113 − 0.718900i \(-0.255354\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.09808 + 0.633975i −0.251916 + 0.145444i −0.620641 − 0.784095i \(-0.713128\pi\)
0.368725 + 0.929538i \(0.379794\pi\)
\(20\) 0.232051 − 0.133975i 0.0518881 − 0.0299576i
\(21\) 0.732051i 0.159747i
\(22\) −2.36603 − 4.09808i −0.504438 − 0.873713i
\(23\) −3.09808 + 5.36603i −0.645994 + 1.11889i 0.338078 + 0.941118i \(0.390223\pi\)
−0.984071 + 0.177775i \(0.943110\pi\)
\(24\) −0.866025 − 0.500000i −0.176777 − 0.102062i
\(25\) 4.92820 0.985641
\(26\) 3.50000 − 0.866025i 0.686406 − 0.169842i
\(27\) 1.00000 0.192450
\(28\) 0.633975 + 0.366025i 0.119810 + 0.0691723i
\(29\) −1.23205 + 2.13397i −0.228786 + 0.396269i −0.957449 − 0.288604i \(-0.906809\pi\)
0.728663 + 0.684873i \(0.240142\pi\)
\(30\) 0.133975 + 0.232051i 0.0244603 + 0.0423665i
\(31\) 5.46410i 0.981382i 0.871334 + 0.490691i \(0.163256\pi\)
−0.871334 + 0.490691i \(0.836744\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 4.09808 − 2.36603i 0.713384 − 0.411872i
\(34\) − 2.26795i − 0.388950i
\(35\) −0.0980762 − 0.169873i −0.0165779 − 0.0287138i
\(36\) 0.500000 − 0.866025i 0.0833333 − 0.144338i
\(37\) −9.06218 − 5.23205i −1.48981 − 0.860144i −0.489881 − 0.871789i \(-0.662960\pi\)
−0.999932 + 0.0116456i \(0.996293\pi\)
\(38\) −1.26795 −0.205689
\(39\) 0.866025 + 3.50000i 0.138675 + 0.560449i
\(40\) 0.267949 0.0423665
\(41\) 9.86603 + 5.69615i 1.54081 + 0.889590i 0.998788 + 0.0492283i \(0.0156762\pi\)
0.542027 + 0.840361i \(0.317657\pi\)
\(42\) −0.366025 + 0.633975i −0.0564789 + 0.0978244i
\(43\) 3.83013 + 6.63397i 0.584089 + 1.01167i 0.994988 + 0.0999910i \(0.0318814\pi\)
−0.410899 + 0.911681i \(0.634785\pi\)
\(44\) − 4.73205i − 0.713384i
\(45\) −0.232051 + 0.133975i −0.0345921 + 0.0199718i
\(46\) −5.36603 + 3.09808i −0.791177 + 0.456786i
\(47\) − 8.19615i − 1.19553i −0.801671 − 0.597766i \(-0.796055\pi\)
0.801671 − 0.597766i \(-0.203945\pi\)
\(48\) −0.500000 − 0.866025i −0.0721688 − 0.125000i
\(49\) −3.23205 + 5.59808i −0.461722 + 0.799725i
\(50\) 4.26795 + 2.46410i 0.603579 + 0.348477i
\(51\) 2.26795 0.317576
\(52\) 3.46410 + 1.00000i 0.480384 + 0.138675i
\(53\) 0.464102 0.0637493 0.0318746 − 0.999492i \(-0.489852\pi\)
0.0318746 + 0.999492i \(0.489852\pi\)
\(54\) 0.866025 + 0.500000i 0.117851 + 0.0680414i
\(55\) −0.633975 + 1.09808i −0.0854851 + 0.148065i
\(56\) 0.366025 + 0.633975i 0.0489122 + 0.0847184i
\(57\) − 1.26795i − 0.167944i
\(58\) −2.13397 + 1.23205i −0.280205 + 0.161776i
\(59\) 6.92820 − 4.00000i 0.901975 − 0.520756i 0.0241347 − 0.999709i \(-0.492317\pi\)
0.877841 + 0.478953i \(0.158984\pi\)
\(60\) 0.267949i 0.0345921i
\(61\) −0.598076 − 1.03590i −0.0765758 − 0.132633i 0.825195 − 0.564848i \(-0.191065\pi\)
−0.901770 + 0.432215i \(0.857732\pi\)
\(62\) −2.73205 + 4.73205i −0.346971 + 0.600971i
\(63\) −0.633975 − 0.366025i −0.0798733 − 0.0461149i
\(64\) −1.00000 −0.125000
\(65\) −0.669873 − 0.696152i −0.0830875 − 0.0863471i
\(66\) 4.73205 0.582475
\(67\) −9.63397 − 5.56218i −1.17698 − 0.679528i −0.221664 − 0.975123i \(-0.571149\pi\)
−0.955313 + 0.295595i \(0.904482\pi\)
\(68\) 1.13397 − 1.96410i 0.137515 − 0.238182i
\(69\) −3.09808 − 5.36603i −0.372965 − 0.645994i
\(70\) − 0.196152i − 0.0234447i
\(71\) −1.09808 + 0.633975i −0.130318 + 0.0752389i −0.563742 − 0.825951i \(-0.690639\pi\)
0.433424 + 0.901190i \(0.357305\pi\)
\(72\) 0.866025 − 0.500000i 0.102062 − 0.0589256i
\(73\) − 9.73205i − 1.13905i −0.821974 − 0.569525i \(-0.807127\pi\)
0.821974 − 0.569525i \(-0.192873\pi\)
\(74\) −5.23205 − 9.06218i −0.608214 − 1.05346i
\(75\) −2.46410 + 4.26795i −0.284530 + 0.492820i
\(76\) −1.09808 − 0.633975i −0.125958 − 0.0727219i
\(77\) −3.46410 −0.394771
\(78\) −1.00000 + 3.46410i −0.113228 + 0.392232i
\(79\) −9.46410 −1.06479 −0.532397 − 0.846495i \(-0.678709\pi\)
−0.532397 + 0.846495i \(0.678709\pi\)
\(80\) 0.232051 + 0.133975i 0.0259441 + 0.0149788i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 5.69615 + 9.86603i 0.629035 + 1.08952i
\(83\) − 10.1962i − 1.11917i −0.828772 − 0.559587i \(-0.810960\pi\)
0.828772 − 0.559587i \(-0.189040\pi\)
\(84\) −0.633975 + 0.366025i −0.0691723 + 0.0399366i
\(85\) −0.526279 + 0.303848i −0.0570830 + 0.0329569i
\(86\) 7.66025i 0.826026i
\(87\) −1.23205 − 2.13397i −0.132090 − 0.228786i
\(88\) 2.36603 − 4.09808i 0.252219 − 0.436856i
\(89\) 2.19615 + 1.26795i 0.232792 + 0.134402i 0.611859 − 0.790967i \(-0.290422\pi\)
−0.379068 + 0.925369i \(0.623755\pi\)
\(90\) −0.267949 −0.0282443
\(91\) 0.732051 − 2.53590i 0.0767398 − 0.265834i
\(92\) −6.19615 −0.645994
\(93\) −4.73205 − 2.73205i −0.490691 − 0.283300i
\(94\) 4.09808 − 7.09808i 0.422684 − 0.732111i
\(95\) 0.169873 + 0.294229i 0.0174286 + 0.0301872i
\(96\) − 1.00000i − 0.102062i
\(97\) 5.19615 − 3.00000i 0.527589 − 0.304604i −0.212445 − 0.977173i \(-0.568143\pi\)
0.740034 + 0.672569i \(0.234809\pi\)
\(98\) −5.59808 + 3.23205i −0.565491 + 0.326486i
\(99\) 4.73205i 0.475589i
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 78.2.i.a.43.2 ✓ 4
3.2 odd 2 234.2.l.c.199.1 4
4.3 odd 2 624.2.bv.e.433.1 4
5.2 odd 4 1950.2.y.b.199.2 4
5.3 odd 4 1950.2.y.g.199.1 4
5.4 even 2 1950.2.bc.d.901.1 4
12.11 even 2 1872.2.by.h.433.2 4
13.2 odd 12 1014.2.e.i.991.2 4
13.3 even 3 1014.2.i.a.361.1 4
13.4 even 6 1014.2.b.e.337.3 4
13.5 odd 4 1014.2.e.i.529.2 4
13.6 odd 12 1014.2.a.i.1.2 2
13.7 odd 12 1014.2.a.k.1.1 2
13.8 odd 4 1014.2.e.g.529.1 4
13.9 even 3 1014.2.b.e.337.2 4
13.10 even 6 inner 78.2.i.a.49.2 yes 4
13.11 odd 12 1014.2.e.g.991.1 4
13.12 even 2 1014.2.i.a.823.1 4
39.17 odd 6 3042.2.b.i.1351.2 4
39.20 even 12 3042.2.a.p.1.2 2
39.23 odd 6 234.2.l.c.127.1 4
39.32 even 12 3042.2.a.y.1.1 2
39.35 odd 6 3042.2.b.i.1351.3 4
52.7 even 12 8112.2.a.bp.1.1 2
52.19 even 12 8112.2.a.bj.1.2 2
52.23 odd 6 624.2.bv.e.49.2 4
65.23 odd 12 1950.2.y.b.49.2 4
65.49 even 6 1950.2.bc.d.751.1 4
65.62 odd 12 1950.2.y.g.49.1 4
156.23 even 6 1872.2.by.h.1297.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.2 ✓ 4 1.1 even 1 trivial
78.2.i.a.49.2 yes 4 13.10 even 6 inner
234.2.l.c.127.1 4 39.23 odd 6
234.2.l.c.199.1 4 3.2 odd 2
624.2.bv.e.49.2 4 52.23 odd 6
624.2.bv.e.433.1 4 4.3 odd 2
1014.2.a.i.1.2 2 13.6 odd 12
1014.2.a.k.1.1 2 13.7 odd 12
1014.2.b.e.337.2 4 13.9 even 3
1014.2.b.e.337.3 4 13.4 even 6
1014.2.e.g.529.1 4 13.8 odd 4
1014.2.e.g.991.1 4 13.11 odd 12
1014.2.e.i.529.2 4 13.5 odd 4
1014.2.e.i.991.2 4 13.2 odd 12
1014.2.i.a.361.1 4 13.3 even 3
1014.2.i.a.823.1 4 13.12 even 2
1872.2.by.h.433.2 4 12.11 even 2
1872.2.by.h.1297.1 4 156.23 even 6
1950.2.y.b.49.2 4 65.23 odd 12
1950.2.y.b.199.2 4 5.2 odd 4
1950.2.y.g.49.1 4 65.62 odd 12
1950.2.y.g.199.1 4 5.3 odd 4
1950.2.bc.d.751.1 4 65.49 even 6
1950.2.bc.d.901.1 4 5.4 even 2
3042.2.a.p.1.2 2 39.20 even 12
3042.2.a.y.1.1 2 39.32 even 12
3042.2.b.i.1351.2 4 39.17 odd 6
3042.2.b.i.1351.3 4 39.35 odd 6
8112.2.a.bj.1.2 2 52.19 even 12
8112.2.a.bp.1.1 2 52.7 even 12