Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [42,12,Mod(25,42)] 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("42.25"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2704135835\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.2
Root \(-32.2642 + 55.8832i\) of defining polynomial
Character \(\chi\) \(=\) 42.25
Dual form 42.12.e.d.37.2

$q$-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+(16.0000 + 27.7128i) q^{2} +(121.500 - 210.444i) q^{3} +(-512.000 + 886.810i) q^{4} +(-1639.98 - 2840.53i) q^{5} +7776.00 q^{6} +(-17084.7 + 41054.1i) q^{7} -32768.0 q^{8} +(-29524.5 - 51137.9i) q^{9} +(52479.3 - 90896.9i) q^{10} +(101963. - 176604. i) q^{11} +(124416. + 215495. i) q^{12} +1.41901e6 q^{13} +(-1.41108e6 + 183399. i) q^{14} -797030. q^{15} +(-524288. - 908093. i) q^{16} +(-869558. + 1.50612e6i) q^{17} +(944784. - 1.63641e6i) q^{18} +(-9.98892e6 - 1.73013e7i) q^{19} +3.35868e6 q^{20} +(6.56380e6 + 8.58346e6i) q^{21} +6.52561e6 q^{22} +(-1.79055e7 - 3.10132e7i) q^{23} +(-3.98131e6 + 6.89583e6i) q^{24} +(1.90350e7 - 3.29696e7i) q^{25} +(2.27042e7 + 3.93248e7i) q^{26} -1.43489e7 q^{27} +(-2.76598e7 - 3.61706e7i) q^{28} -1.92546e8 q^{29} +(-1.27525e7 - 2.20879e7i) q^{30} +(2.12184e6 - 3.67513e6i) q^{31} +(1.67772e7 - 2.90590e7i) q^{32} +(-2.47769e7 - 4.29149e7i) q^{33} -5.56517e7 q^{34} +(1.44634e8 - 1.87982e7i) q^{35} +6.04662e7 q^{36} +(-2.49099e8 - 4.31452e8i) q^{37} +(3.19646e8 - 5.53642e8i) q^{38} +(1.72410e8 - 2.98623e8i) q^{39} +(5.37388e7 + 9.30784e7i) q^{40} +7.24700e8 q^{41} +(-1.32851e8 + 3.19237e8i) q^{42} +1.43499e9 q^{43} +(1.04410e8 + 1.80843e8i) q^{44} +(-9.68391e7 + 1.67730e8i) q^{45} +(5.72976e8 - 9.92424e8i) q^{46} +(-1.11948e9 - 1.93899e9i) q^{47} -2.54804e8 q^{48} +(-1.39355e9 - 1.40280e9i) q^{49} +1.21824e9 q^{50} +(2.11303e8 + 3.65987e8i) q^{51} +(-7.26534e8 + 1.25839e9i) q^{52} +(1.23956e9 - 2.14698e9i) q^{53} +(-2.29583e8 - 3.97649e8i) q^{54} -6.68866e8 q^{55} +(5.59833e8 - 1.34526e9i) q^{56} -4.85462e9 q^{57} +(-3.08073e9 - 5.33599e9i) q^{58} +(1.11798e7 - 1.93640e7i) q^{59} +(4.08079e8 - 7.06814e8i) q^{60} +(2.07345e8 + 3.59131e8i) q^{61} +1.35797e8 q^{62} +(2.60384e9 - 3.38423e8i) q^{63} +1.07374e9 q^{64} +(-2.32715e9 - 4.03074e9i) q^{65} +(7.92861e8 - 1.37328e9i) q^{66} +(-3.95898e9 + 6.85715e9i) q^{67} +(-8.90427e8 - 1.54227e9i) q^{68} -8.70208e9 q^{69} +(2.83509e9 + 3.70744e9i) q^{70} +1.48562e10 q^{71} +(9.67459e8 + 1.67569e9i) q^{72} +(-1.23247e10 + 2.13471e10i) q^{73} +(7.97117e9 - 1.38065e10i) q^{74} +(-4.62550e9 - 8.01161e9i) q^{75} +2.04573e10 q^{76} +(5.50833e9 + 7.20322e9i) q^{77} +1.10342e10 q^{78} +(6.25143e9 + 1.08278e10i) q^{79} +(-1.71964e9 + 2.97851e9i) q^{80} +(-1.74339e9 + 3.01964e9i) q^{81} +(1.15952e10 + 2.00835e10i) q^{82} -5.94072e10 q^{83} +(-1.09726e10 + 1.42611e9i) q^{84} +5.70423e9 q^{85} +(2.29599e10 + 3.97677e10i) q^{86} +(-2.33943e10 + 4.05202e10i) q^{87} +(-3.34111e9 + 5.78697e9i) q^{88} +(2.91964e10 + 5.05696e10i) q^{89} -6.19770e9 q^{90} +(-2.42434e10 + 5.82562e10i) q^{91} +3.66705e10 q^{92} +(-5.15606e8 - 8.93056e8i) q^{93} +(3.58233e10 - 6.20478e10i) q^{94} +(-3.27633e10 + 5.67476e10i) q^{95} +(-4.07686e9 - 7.06133e9i) q^{96} -3.15640e10 q^{97} +(1.65786e10 - 6.10639e10i) q^{98} -1.20416e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} + 972 q^{3} - 4096 q^{4} - 1420 q^{5} + 62208 q^{6} - 66362 q^{7} - 262144 q^{8} - 236196 q^{9} + 45440 q^{10} - 861962 q^{11} + 995328 q^{12} + 836748 q^{13} - 1361216 q^{14} - 690120 q^{15}+ \cdots + 101795988276 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 + 27.7128i 0.353553 + 0.612372i
\(3\) 121.500 − 210.444i 0.288675 − 0.500000i
\(4\) −512.000 + 886.810i −0.250000 + 0.433013i
\(5\) −1639.98 − 2840.53i −0.234695 − 0.406503i 0.724489 − 0.689286i \(-0.242076\pi\)
−0.959184 + 0.282783i \(0.908742\pi\)
\(6\) 7776.00 0.408248
\(7\) −17084.7 + 41054.1i −0.384210 + 0.923246i
\(8\) −32768.0 −0.353553
\(9\) −29524.5 − 51137.9i −0.166667 − 0.288675i
\(10\) 52479.3 − 90896.9i 0.165954 − 0.287441i
\(11\) 101963. − 176604.i 0.190889 − 0.330630i −0.754656 − 0.656121i \(-0.772196\pi\)
0.945545 + 0.325491i \(0.105530\pi\)
\(12\) 124416. + 215495.i 0.144338 + 0.250000i
\(13\) 1.41901e6 1.05998 0.529990 − 0.848004i \(-0.322196\pi\)
0.529990 + 0.848004i \(0.322196\pi\)
\(14\) −1.41108e6 + 183399.i −0.701209 + 0.0911367i
\(15\) −797030. −0.271002
\(16\) −524288. − 908093.i −0.125000 − 0.216506i
\(17\) −869558. + 1.50612e6i −0.148535 + 0.257271i −0.930686 − 0.365818i \(-0.880789\pi\)
0.782151 + 0.623089i \(0.214122\pi\)
\(18\) 944784. − 1.63641e6i 0.117851 − 0.204124i
\(19\) −9.98892e6 − 1.73013e7i −0.925495 − 1.60300i −0.790764 − 0.612121i \(-0.790316\pi\)
−0.134731 − 0.990882i \(-0.543017\pi\)
\(20\) 3.35868e6 0.234695
\(21\) 6.56380e6 + 8.58346e6i 0.350711 + 0.458623i
\(22\) 6.52561e6 0.269958
\(23\) −1.79055e7 − 3.10132e7i −0.580074 − 1.00472i −0.995470 − 0.0950771i \(-0.969690\pi\)
0.415396 − 0.909641i \(-0.363643\pi\)
\(24\) −3.98131e6 + 6.89583e6i −0.102062 + 0.176777i
\(25\) 1.90350e7 − 3.29696e7i 0.389837 − 0.675217i
\(26\) 2.27042e7 + 3.93248e7i 0.374759 + 0.649102i
\(27\) −1.43489e7 −0.192450
\(28\) −2.76598e7 − 3.61706e7i −0.303724 − 0.397179i
\(29\) −1.92546e8 −1.74319 −0.871596 − 0.490225i \(-0.836915\pi\)
−0.871596 + 0.490225i \(0.836915\pi\)
\(30\) −1.27525e7 − 2.20879e7i −0.0958137 − 0.165954i
\(31\) 2.12184e6 − 3.67513e6i 0.0133114 − 0.0230559i −0.859293 − 0.511484i \(-0.829096\pi\)
0.872604 + 0.488428i \(0.162429\pi\)
\(32\) 1.67772e7 − 2.90590e7i 0.0883883 − 0.153093i
\(33\) −2.47769e7 − 4.29149e7i −0.110210 − 0.190889i
\(34\) −5.56517e7 −0.210061
\(35\) 1.44634e8 − 1.87982e7i 0.465474 − 0.0604981i
\(36\) 6.04662e7 0.166667
\(37\) −2.49099e8 − 4.31452e8i −0.590558 − 1.02288i −0.994157 − 0.107941i \(-0.965574\pi\)
0.403599 − 0.914936i \(-0.367759\pi\)
\(38\) 3.19646e8 − 5.53642e8i 0.654423 − 1.13349i
\(39\) 1.72410e8 − 2.98623e8i 0.305990 − 0.529990i
\(40\) 5.37388e7 + 9.30784e7i 0.0829771 + 0.143721i
\(41\) 7.24700e8 0.976893 0.488446 − 0.872594i \(-0.337564\pi\)
0.488446 + 0.872594i \(0.337564\pi\)
\(42\) −1.32851e8 + 3.19237e8i −0.156853 + 0.376913i
\(43\) 1.43499e9 1.48858 0.744292 − 0.667855i \(-0.232787\pi\)
0.744292 + 0.667855i \(0.232787\pi\)
\(44\) 1.04410e8 + 1.80843e8i 0.0954446 + 0.165315i
\(45\) −9.68391e7 + 1.67730e8i −0.0782316 + 0.135501i
\(46\) 5.72976e8 − 9.92424e8i 0.410174 − 0.710443i
\(47\) −1.11948e9 − 1.93899e9i −0.711996 − 1.23321i −0.964107 − 0.265515i \(-0.914458\pi\)
0.252111 − 0.967698i \(-0.418875\pi\)
\(48\) −2.54804e8 −0.144338
\(49\) −1.39355e9 − 1.40280e9i −0.704765 − 0.709441i
\(50\) 1.21824e9 0.551312
\(51\) 2.11303e8 + 3.65987e8i 0.0857568 + 0.148535i
\(52\) −7.26534e8 + 1.25839e9i −0.264995 + 0.458985i
\(53\) 1.23956e9 − 2.14698e9i 0.407146 − 0.705197i −0.587423 − 0.809280i \(-0.699857\pi\)
0.994569 + 0.104083i \(0.0331907\pi\)
\(54\) −2.29583e8 − 3.97649e8i −0.0680414 − 0.117851i
\(55\) −6.68866e8 −0.179203
\(56\) 5.59833e8 − 1.34526e9i 0.135839 − 0.326417i
\(57\) −4.85462e9 −1.06867
\(58\) −3.08073e9 − 5.33599e9i −0.616311 − 1.06748i
\(59\) 1.11798e7 − 1.93640e7i 0.00203586 − 0.00352622i −0.865006 − 0.501762i \(-0.832685\pi\)
0.867042 + 0.498236i \(0.166019\pi\)
\(60\) 4.08079e8 − 7.06814e8i 0.0677505 − 0.117347i
\(61\) 2.07345e8 + 3.59131e8i 0.0314325 + 0.0544426i 0.881314 − 0.472532i \(-0.156660\pi\)
−0.849881 + 0.526974i \(0.823326\pi\)
\(62\) 1.35797e8 0.0188251
\(63\) 2.60384e9 − 3.38423e8i 0.330553 − 0.0429623i
\(64\) 1.07374e9 0.125000
\(65\) −2.32715e9 − 4.03074e9i −0.248772 − 0.430885i
\(66\) 7.92861e8 − 1.37328e9i 0.0779302 − 0.134979i
\(67\) −3.95898e9 + 6.85715e9i −0.358238 + 0.620486i −0.987667 − 0.156572i \(-0.949956\pi\)
0.629429 + 0.777058i \(0.283289\pi\)
\(68\) −8.90427e8 − 1.54227e9i −0.0742676 − 0.128635i
\(69\) −8.70208e9 −0.669812
\(70\) 2.83509e9 + 3.70744e9i 0.201617 + 0.263654i
\(71\) 1.48562e10 0.977208 0.488604 − 0.872506i \(-0.337506\pi\)
0.488604 + 0.872506i \(0.337506\pi\)
\(72\) 9.67459e8 + 1.67569e9i 0.0589256 + 0.102062i
\(73\) −1.23247e10 + 2.13471e10i −0.695828 + 1.20521i 0.274073 + 0.961709i \(0.411629\pi\)
−0.969901 + 0.243500i \(0.921704\pi\)
\(74\) 7.97117e9 − 1.38065e10i 0.417588 − 0.723283i
\(75\) −4.62550e9 − 8.01161e9i −0.225072 − 0.389837i
\(76\) 2.04573e10 0.925495
\(77\) 5.50833e9 + 7.20322e9i 0.231911 + 0.303269i
\(78\) 1.10342e10 0.432735
\(79\) 6.25143e9 + 1.08278e10i 0.228576 + 0.395905i 0.957386 − 0.288810i \(-0.0932597\pi\)
−0.728810 + 0.684716i \(0.759926\pi\)
\(80\) −1.71964e9 + 2.97851e9i −0.0586737 + 0.101626i
\(81\) −1.74339e9 + 3.01964e9i −0.0555556 + 0.0962250i
\(82\) 1.15952e10 + 2.00835e10i 0.345384 + 0.598222i
\(83\) −5.94072e10 −1.65542 −0.827712 − 0.561153i \(-0.810358\pi\)
−0.827712 + 0.561153i \(0.810358\pi\)
\(84\) −1.09726e10 + 1.42611e9i −0.286267 + 0.0372064i
\(85\) 5.70423e9 0.139442
\(86\) 2.29599e10 + 3.97677e10i 0.526294 + 0.911567i
\(87\) −2.33943e10 + 4.05202e10i −0.503216 + 0.871596i
\(88\) −3.34111e9 + 5.78697e9i −0.0674895 + 0.116895i
\(89\) 2.91964e10 + 5.05696e10i 0.554222 + 0.959941i 0.997964 + 0.0637858i \(0.0203174\pi\)
−0.443742 + 0.896155i \(0.646349\pi\)
\(90\) −6.19770e9 −0.110636
\(91\) −2.42434e10 + 5.82562e10i −0.407255 + 0.978622i
\(92\) 3.66705e10 0.580074
\(93\) −5.15606e8 − 8.93056e8i −0.00768532 − 0.0133114i
\(94\) 3.58233e10 − 6.20478e10i 0.503457 − 0.872013i
\(95\) −3.27633e10 + 5.67476e10i −0.434417 + 0.752433i
\(96\) −4.07686e9 − 7.06133e9i −0.0510310 − 0.0883883i
\(97\) −3.15640e10 −0.373206 −0.186603 − 0.982435i \(-0.559748\pi\)
−0.186603 + 0.982435i \(0.559748\pi\)
\(98\) 1.65786e10 − 6.10639e10i 0.185270 − 0.682404i
\(99\) −1.20416e10 −0.127259
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 42.12.e.d.25.2 ✓ 8
3.2 odd 2 126.12.g.d.109.3 8
7.2 even 3 inner 42.12.e.d.37.2 yes 8
21.2 odd 6 126.12.g.d.37.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.d.25.2 ✓ 8 1.1 even 1 trivial
42.12.e.d.37.2 yes 8 7.2 even 3 inner
126.12.g.d.37.3 8 21.2 odd 6
126.12.g.d.109.3 8 3.2 odd 2