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 37.3
Root \(-110.360 - 191.150i\) of defining polynomial
Character \(\chi\) \(=\) 42.37
Dual form 42.12.e.d.25.3

$q$-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+(16.0000 - 27.7128i) q^{2} +(121.500 + 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(-1017.54 + 1762.43i) q^{5} +7776.00 q^{6} +(42078.2 - 14378.9i) q^{7} -32768.0 q^{8} +(-29524.5 + 51137.9i) q^{9} +(32561.3 + 56397.9i) q^{10} +(39381.2 + 68210.2i) q^{11} +(124416. - 215495. i) q^{12} -788558. q^{13} +(274771. - 1.39617e6i) q^{14} -494525. q^{15} +(-524288. + 908093. i) q^{16} +(5.11629e6 + 8.86167e6i) q^{17} +(944784. + 1.63641e6i) q^{18} +(1.03302e7 - 1.78924e7i) q^{19} +2.08392e6 q^{20} +(8.13846e6 + 7.10807e6i) q^{21} +2.52039e6 q^{22} +(-3.67657e6 + 6.36801e6i) q^{23} +(-3.98131e6 - 6.89583e6i) q^{24} +(2.23433e7 + 3.86997e7i) q^{25} +(-1.26169e7 + 2.18532e7i) q^{26} -1.43489e7 q^{27} +(-3.42954e7 - 2.99534e7i) q^{28} +9.93024e7 q^{29} +(-7.91240e6 + 1.37047e7i) q^{30} +(1.00917e8 + 1.74793e8i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} +(-9.56962e6 + 1.65751e7i) q^{33} +3.27443e8 q^{34} +(-1.74744e7 + 8.87911e7i) q^{35} +6.04662e7 q^{36} +(1.95399e8 - 3.38442e8i) q^{37} +(-3.30566e8 - 5.72557e8i) q^{38} +(-9.58098e7 - 1.65947e8i) q^{39} +(3.33428e7 - 5.77514e7i) q^{40} +7.80704e8 q^{41} +(3.27200e8 - 1.11810e8i) q^{42} +2.17607e7 q^{43} +(4.03263e7 - 6.98472e7i) q^{44} +(-6.00848e7 - 1.04070e8i) q^{45} +(1.17650e8 + 2.03776e8i) q^{46} +(1.51556e9 - 2.62504e9i) q^{47} -2.54804e8 q^{48} +(1.56382e9 - 1.21008e9i) q^{49} +1.42997e9 q^{50} +(-1.24326e9 + 2.15339e9i) q^{51} +(4.03742e8 + 6.99301e8i) q^{52} +(1.98519e9 + 3.43845e9i) q^{53} +(-2.29583e8 + 3.97649e8i) q^{54} -1.60288e8 q^{55} +(-1.37882e9 + 4.71168e8i) q^{56} +5.02047e9 q^{57} +(1.58884e9 - 2.75195e9i) q^{58} +(-1.37766e9 - 2.38619e9i) q^{59} +(2.53197e8 + 4.38550e8i) q^{60} +(-5.25879e9 + 9.10849e9i) q^{61} +6.45867e9 q^{62} +(-5.07030e8 + 2.57632e9i) q^{63} +1.07374e9 q^{64} +(8.02391e8 - 1.38978e9i) q^{65} +(3.06228e8 + 5.30402e8i) q^{66} +(1.14527e9 + 1.98367e9i) q^{67} +(5.23908e9 - 9.07435e9i) q^{68} -1.78681e9 q^{69} +(2.18106e9 + 1.90492e9i) q^{70} +4.54366e9 q^{71} +(9.67459e8 - 1.67569e9i) q^{72} +(-6.98203e9 - 1.20932e10i) q^{73} +(-6.25278e9 - 1.08301e10i) q^{74} +(-5.42942e9 + 9.40403e9i) q^{75} -2.11562e10 q^{76} +(2.63788e9 + 2.30390e9i) q^{77} -6.13183e9 q^{78} +(-1.39090e10 + 2.40910e10i) q^{79} +(-1.06697e9 - 1.84805e9i) q^{80} +(-1.74339e9 - 3.01964e9i) q^{81} +(1.24913e10 - 2.16355e10i) q^{82} -6.17383e10 q^{83} +(2.13662e9 - 1.08566e10i) q^{84} -2.08241e10 q^{85} +(3.48172e8 - 6.03052e8i) q^{86} +(1.20652e10 + 2.08976e10i) q^{87} +(-1.29044e9 - 2.23511e9i) q^{88} +(-1.20759e10 + 2.09160e10i) q^{89} -3.84543e9 q^{90} +(-3.31811e10 + 1.13386e10i) q^{91} +7.52962e9 q^{92} +(-2.45228e10 + 4.24747e10i) q^{93} +(-4.84981e10 - 8.40011e10i) q^{94} +(2.10228e10 + 3.64125e10i) q^{95} +(-4.07686e9 + 7.06133e9i) q^{96} -1.03350e11 q^{97} +(-8.51348e9 - 6.26991e10i) q^{98} -4.65084e9 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 \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

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

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
\(5\) −1017.54 + 1762.43i −0.145619 + 0.252219i −0.929604 − 0.368561i \(-0.879851\pi\)
0.783985 + 0.620780i \(0.213184\pi\)
\(6\) 7776.00 0.408248
\(7\) 42078.2 − 14378.9i 0.946276 − 0.323360i
\(8\) −32768.0 −0.353553
\(9\) −29524.5 + 51137.9i −0.166667 + 0.288675i
\(10\) 32561.3 + 56397.9i 0.102968 + 0.178346i
\(11\) 39381.2 + 68210.2i 0.0737274 + 0.127700i 0.900532 − 0.434789i \(-0.143177\pi\)
−0.826805 + 0.562489i \(0.809844\pi\)
\(12\) 124416. − 215495.i 0.144338 − 0.250000i
\(13\) −788558. −0.589041 −0.294520 − 0.955645i \(-0.595160\pi\)
−0.294520 + 0.955645i \(0.595160\pi\)
\(14\) 274771. − 1.39617e6i 0.136542 − 0.693798i
\(15\) −494525. −0.168146
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) 5.11629e6 + 8.86167e6i 0.873949 + 1.51372i 0.857878 + 0.513853i \(0.171782\pi\)
0.0160707 + 0.999871i \(0.494884\pi\)
\(18\) 944784. + 1.63641e6i 0.117851 + 0.204124i
\(19\) 1.03302e7 − 1.78924e7i 0.957113 − 1.65777i 0.227656 − 0.973742i \(-0.426894\pi\)
0.729457 − 0.684027i \(-0.239773\pi\)
\(20\) 2.08392e6 0.145619
\(21\) 8.13846e6 + 7.10807e6i 0.434846 + 0.379792i
\(22\) 2.52039e6 0.104266
\(23\) −3.67657e6 + 6.36801e6i −0.119108 + 0.206301i −0.919414 − 0.393290i \(-0.871337\pi\)
0.800307 + 0.599591i \(0.204670\pi\)
\(24\) −3.98131e6 − 6.89583e6i −0.102062 − 0.176777i
\(25\) 2.23433e7 + 3.86997e7i 0.457590 + 0.792570i
\(26\) −1.26169e7 + 2.18532e7i −0.208257 + 0.360712i
\(27\) −1.43489e7 −0.192450
\(28\) −3.42954e7 − 2.99534e7i −0.376588 − 0.328909i
\(29\) 9.93024e7 0.899023 0.449511 − 0.893275i \(-0.351598\pi\)
0.449511 + 0.893275i \(0.351598\pi\)
\(30\) −7.91240e6 + 1.37047e7i −0.0594486 + 0.102968i
\(31\) 1.00917e8 + 1.74793e8i 0.633102 + 1.09657i 0.986914 + 0.161249i \(0.0515521\pi\)
−0.353811 + 0.935317i \(0.615115\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) −9.56962e6 + 1.65751e7i −0.0425665 + 0.0737274i
\(34\) 3.27443e8 1.23595
\(35\) −1.74744e7 + 8.87911e7i −0.0562379 + 0.285756i
\(36\) 6.04662e7 0.166667
\(37\) 1.95399e8 − 3.38442e8i 0.463248 − 0.802370i −0.535872 − 0.844299i \(-0.680017\pi\)
0.999121 + 0.0419295i \(0.0133505\pi\)
\(38\) −3.30566e8 − 5.72557e8i −0.676781 − 1.17222i
\(39\) −9.58098e7 − 1.65947e8i −0.170041 − 0.294520i
\(40\) 3.33428e7 − 5.77514e7i 0.0514840 − 0.0891729i
\(41\) 7.80704e8 1.05239 0.526193 − 0.850365i \(-0.323619\pi\)
0.526193 + 0.850365i \(0.323619\pi\)
\(42\) 3.27200e8 − 1.11810e8i 0.386316 − 0.132011i
\(43\) 2.17607e7 0.0225734 0.0112867 − 0.999936i \(-0.496407\pi\)
0.0112867 + 0.999936i \(0.496407\pi\)
\(44\) 4.03263e7 − 6.98472e7i 0.0368637 − 0.0638498i
\(45\) −6.00848e7 − 1.04070e8i −0.0485396 − 0.0840730i
\(46\) 1.17650e8 + 2.03776e8i 0.0842219 + 0.145877i
\(47\) 1.51556e9 − 2.62504e9i 0.963910 − 1.66954i 0.251393 − 0.967885i \(-0.419111\pi\)
0.712517 − 0.701655i \(-0.247555\pi\)
\(48\) −2.54804e8 −0.144338
\(49\) 1.56382e9 − 1.21008e9i 0.790876 − 0.611976i
\(50\) 1.42997e9 0.647131
\(51\) −1.24326e9 + 2.15339e9i −0.504575 + 0.873949i
\(52\) 4.03742e8 + 6.99301e8i 0.147260 + 0.255062i
\(53\) 1.98519e9 + 3.43845e9i 0.652057 + 1.12940i 0.982623 + 0.185612i \(0.0594269\pi\)
−0.330566 + 0.943783i \(0.607240\pi\)
\(54\) −2.29583e8 + 3.97649e8i −0.0680414 + 0.117851i
\(55\) −1.60288e8 −0.0429443
\(56\) −1.37882e9 + 4.71168e8i −0.334559 + 0.114325i
\(57\) 5.02047e9 1.10518
\(58\) 1.58884e9 − 2.75195e9i 0.317853 − 0.550537i
\(59\) −1.37766e9 − 2.38619e9i −0.250875 − 0.434528i 0.712892 − 0.701274i \(-0.247385\pi\)
−0.963767 + 0.266746i \(0.914052\pi\)
\(60\) 2.53197e8 + 4.38550e8i 0.0420365 + 0.0728093i
\(61\) −5.25879e9 + 9.10849e9i −0.797208 + 1.38080i 0.124220 + 0.992255i \(0.460357\pi\)
−0.921428 + 0.388550i \(0.872976\pi\)
\(62\) 6.45867e9 0.895342
\(63\) −5.07030e8 + 2.57632e9i −0.0643666 + 0.327060i
\(64\) 1.07374e9 0.125000
\(65\) 8.02391e8 − 1.38978e9i 0.0857753 − 0.148567i
\(66\) 3.06228e8 + 5.30402e8i 0.0300991 + 0.0521331i
\(67\) 1.14527e9 + 1.98367e9i 0.103633 + 0.179497i 0.913179 − 0.407559i \(-0.133620\pi\)
−0.809546 + 0.587056i \(0.800287\pi\)
\(68\) 5.23908e9 − 9.07435e9i 0.436974 − 0.756862i
\(69\) −1.78681e9 −0.137534
\(70\) 2.18106e9 + 1.90492e9i 0.155106 + 0.135469i
\(71\) 4.54366e9 0.298872 0.149436 − 0.988771i \(-0.452254\pi\)
0.149436 + 0.988771i \(0.452254\pi\)
\(72\) 9.67459e8 − 1.67569e9i 0.0589256 − 0.102062i
\(73\) −6.98203e9 − 1.20932e10i −0.394191 − 0.682758i 0.598807 − 0.800893i \(-0.295642\pi\)
−0.992997 + 0.118135i \(0.962308\pi\)
\(74\) −6.25278e9 − 1.08301e10i −0.327566 − 0.567361i
\(75\) −5.42942e9 + 9.40403e9i −0.264190 + 0.457590i
\(76\) −2.11562e10 −0.957113
\(77\) 2.63788e9 + 2.30390e9i 0.111059 + 0.0969985i
\(78\) −6.13183e9 −0.240475
\(79\) −1.39090e10 + 2.40910e10i −0.508564 + 0.880859i 0.491387 + 0.870941i \(0.336490\pi\)
−0.999951 + 0.00991713i \(0.996843\pi\)
\(80\) −1.06697e9 − 1.84805e9i −0.0364047 − 0.0630547i
\(81\) −1.74339e9 − 3.01964e9i −0.0555556 − 0.0962250i
\(82\) 1.24913e10 − 2.16355e10i 0.372075 − 0.644452i
\(83\) −6.17383e10 −1.72038 −0.860192 − 0.509971i \(-0.829656\pi\)
−0.860192 + 0.509971i \(0.829656\pi\)
\(84\) 2.13662e9 − 1.08566e10i 0.0557431 − 0.283242i
\(85\) −2.08241e10 −0.509053
\(86\) 3.48172e8 − 6.03052e8i 0.00798091 − 0.0138233i
\(87\) 1.20652e10 + 2.08976e10i 0.259526 + 0.449511i
\(88\) −1.29044e9 − 2.23511e9i −0.0260666 − 0.0451486i
\(89\) −1.20759e10 + 2.09160e10i −0.229231 + 0.397040i −0.957580 − 0.288166i \(-0.906954\pi\)
0.728350 + 0.685206i \(0.240288\pi\)
\(90\) −3.84543e9 −0.0686453
\(91\) −3.31811e10 + 1.13386e10i −0.557395 + 0.190472i
\(92\) 7.52962e9 0.119108
\(93\) −2.45228e10 + 4.24747e10i −0.365522 + 0.633102i
\(94\) −4.84981e10 − 8.40011e10i −0.681587 − 1.18054i
\(95\) 2.10228e10 + 3.64125e10i 0.278747 + 0.482804i
\(96\) −4.07686e9 + 7.06133e9i −0.0510310 + 0.0883883i
\(97\) −1.03350e11 −1.22198 −0.610991 − 0.791637i \(-0.709229\pi\)
−0.610991 + 0.791637i \(0.709229\pi\)
\(98\) −8.51348e9 − 6.26991e10i −0.0951401 − 0.700677i
\(99\) −4.65084e9 −0.0491516
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.37.3 yes 8
3.2 odd 2 126.12.g.d.37.2 8
7.4 even 3 inner 42.12.e.d.25.3 ✓ 8
21.11 odd 6 126.12.g.d.109.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.d.25.3 ✓ 8 7.4 even 3 inner
42.12.e.d.37.3 yes 8 1.1 even 1 trivial
126.12.g.d.37.2 8 3.2 odd 2
126.12.g.d.109.2 8 21.11 odd 6