Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,6,Mod(67,147)] 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("147.67"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3,-18,-65,-33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.2
Root \(-3.69493 + 2.71062i\) of defining polynomial
Character \(\chi\) \(=\) 147.79
Dual form 147.6.e.l.67.2

$q$-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+(4.69493 - 8.13186i) q^{2} +(-4.50000 - 7.79423i) q^{3} +(-28.0848 - 48.6443i) q^{4} +(-35.8645 + 62.1192i) q^{5} -84.5088 q^{6} -226.949 q^{8} +(-40.5000 + 70.1481i) q^{9} +(336.763 + 583.291i) q^{10} +(280.305 + 485.503i) q^{11} +(-252.763 + 437.799i) q^{12} -533.509 q^{13} +645.562 q^{15} +(-166.798 + 288.903i) q^{16} +(502.850 + 870.962i) q^{17} +(380.290 + 658.681i) q^{18} +(684.263 - 1185.18i) q^{19} +4028.99 q^{20} +5264.05 q^{22} +(-1614.04 + 2795.60i) q^{23} +(1021.27 + 1768.90i) q^{24} +(-1010.03 - 1749.42i) q^{25} +(-2504.79 + 4338.42i) q^{26} +729.000 q^{27} -753.456 q^{29} +(3030.87 - 5249.62i) q^{30} +(4103.21 + 7106.97i) q^{31} +(-2064.97 - 3576.64i) q^{32} +(2522.75 - 4369.52i) q^{33} +9443.39 q^{34} +4549.74 q^{36} +(1404.33 - 2432.37i) q^{37} +(-6425.14 - 11128.7i) q^{38} +(2400.79 + 4158.29i) q^{39} +(8139.43 - 14097.9i) q^{40} -245.827 q^{41} -17504.5 q^{43} +(15744.6 - 27270.5i) q^{44} +(-2905.03 - 5031.65i) q^{45} +(15155.6 + 26250.3i) q^{46} +(-8172.74 + 14155.6i) q^{47} +3002.37 q^{48} -18968.1 q^{50} +(4525.65 - 7838.66i) q^{51} +(14983.5 + 25952.2i) q^{52} +(14820.8 + 25670.4i) q^{53} +(3422.61 - 5928.13i) q^{54} -40212.0 q^{55} -12316.7 q^{57} +(-3537.43 + 6127.00i) q^{58} +(-5178.05 - 8968.65i) q^{59} +(-18130.5 - 31402.9i) q^{60} +(477.089 - 826.343i) q^{61} +77057.2 q^{62} -49454.8 q^{64} +(19134.0 - 33141.1i) q^{65} +(-23688.2 - 41029.2i) q^{66} +(9907.60 + 17160.5i) q^{67} +(28244.9 - 48921.6i) q^{68} +29052.7 q^{69} +62125.4 q^{71} +(9191.45 - 15920.1i) q^{72} +(13554.8 + 23477.6i) q^{73} +(-13186.5 - 22839.7i) q^{74} +(-9090.27 + 15744.8i) q^{75} -76869.6 q^{76} +45086.2 q^{78} +(-22343.7 + 38700.4i) q^{79} +(-11964.3 - 20722.8i) q^{80} +(-3280.50 - 5681.99i) q^{81} +(-1154.14 + 1999.03i) q^{82} -15606.6 q^{83} -72137.9 q^{85} +(-82182.5 + 142344. i) q^{86} +(3390.55 + 5872.61i) q^{87} +(-63615.0 - 110184. i) q^{88} +(6817.66 - 11808.5i) q^{89} -54555.6 q^{90} +181320. q^{92} +(36928.9 - 63962.7i) q^{93} +(76740.9 + 132919. i) q^{94} +(49081.6 + 85011.8i) q^{95} +(-18584.8 + 32189.8i) q^{96} +12919.5 q^{97} -45409.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 18 q^{3} - 65 q^{4} - 33 q^{5} - 54 q^{6} - 750 q^{8} - 162 q^{9} + 921 q^{10} + 1137 q^{11} - 585 q^{12} - 1850 q^{13} + 594 q^{15} + 895 q^{16} - 324 q^{17} + 243 q^{18} + 2311 q^{19}+ \cdots - 184194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.69493 − 8.13186i 0.829955 − 1.43752i −0.0681181 − 0.997677i \(-0.521699\pi\)
0.898073 − 0.439847i \(-0.144967\pi\)
\(3\) −4.50000 − 7.79423i −0.288675 − 0.500000i
\(4\) −28.0848 − 48.6443i −0.877650 − 1.52013i
\(5\) −35.8645 + 62.1192i −0.641564 + 1.11122i 0.343519 + 0.939146i \(0.388381\pi\)
−0.985084 + 0.172076i \(0.944952\pi\)
\(6\) −84.5088 −0.958349
\(7\) 0 0
\(8\) −226.949 −1.25373
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 336.763 + 583.291i 1.06494 + 1.84453i
\(11\) 280.305 + 485.503i 0.698472 + 1.20979i 0.968996 + 0.247076i \(0.0794698\pi\)
−0.270524 + 0.962713i \(0.587197\pi\)
\(12\) −252.763 + 437.799i −0.506711 + 0.877650i
\(13\) −533.509 −0.875555 −0.437777 − 0.899083i \(-0.644234\pi\)
−0.437777 + 0.899083i \(0.644234\pi\)
\(14\) 0 0
\(15\) 645.562 0.740815
\(16\) −166.798 + 288.903i −0.162889 + 0.282132i
\(17\) 502.850 + 870.962i 0.422004 + 0.730932i 0.996135 − 0.0878311i \(-0.0279936\pi\)
−0.574132 + 0.818763i \(0.694660\pi\)
\(18\) 380.290 + 658.681i 0.276652 + 0.479175i
\(19\) 684.263 − 1185.18i 0.434850 − 0.753182i −0.562434 − 0.826842i \(-0.690135\pi\)
0.997283 + 0.0736606i \(0.0234682\pi\)
\(20\) 4028.99 2.25228
\(21\) 0 0
\(22\) 5264.05 2.31880
\(23\) −1614.04 + 2795.60i −0.636201 + 1.10193i 0.350058 + 0.936728i \(0.386162\pi\)
−0.986259 + 0.165205i \(0.947171\pi\)
\(24\) 1021.27 + 1768.90i 0.361921 + 0.626865i
\(25\) −1010.03 − 1749.42i −0.323209 − 0.559815i
\(26\) −2504.79 + 4338.42i −0.726671 + 1.25863i
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −753.456 −0.166365 −0.0831827 − 0.996534i \(-0.526508\pi\)
−0.0831827 + 0.996534i \(0.526508\pi\)
\(30\) 3030.87 − 5249.62i 0.614843 − 1.06494i
\(31\) 4103.21 + 7106.97i 0.766866 + 1.32825i 0.939254 + 0.343222i \(0.111518\pi\)
−0.172389 + 0.985029i \(0.555148\pi\)
\(32\) −2064.97 − 3576.64i −0.356484 − 0.617448i
\(33\) 2522.75 − 4369.52i 0.403263 − 0.698472i
\(34\) 9443.39 1.40098
\(35\) 0 0
\(36\) 4549.74 0.585100
\(37\) 1404.33 − 2432.37i 0.168642 − 0.292096i −0.769301 − 0.638887i \(-0.779395\pi\)
0.937943 + 0.346791i \(0.112729\pi\)
\(38\) −6425.14 − 11128.7i −0.721811 − 1.25021i
\(39\) 2400.79 + 4158.29i 0.252751 + 0.437777i
\(40\) 8139.43 − 14097.9i 0.804348 − 1.39317i
\(41\) −245.827 −0.0228387 −0.0114193 − 0.999935i \(-0.503635\pi\)
−0.0114193 + 0.999935i \(0.503635\pi\)
\(42\) 0 0
\(43\) −17504.5 −1.44371 −0.721853 − 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(44\) 15744.6 − 27270.5i 1.22603 − 2.12354i
\(45\) −2905.03 − 5031.65i −0.213855 − 0.370407i
\(46\) 15155.6 + 26250.3i 1.05604 + 1.82911i
\(47\) −8172.74 + 14155.6i −0.539663 + 0.934725i 0.459258 + 0.888303i \(0.348115\pi\)
−0.998922 + 0.0464219i \(0.985218\pi\)
\(48\) 3002.37 0.188088
\(49\) 0 0
\(50\) −18968.1 −1.07300
\(51\) 4525.65 − 7838.66i 0.243644 − 0.422004i
\(52\) 14983.5 + 25952.2i 0.768430 + 1.33096i
\(53\) 14820.8 + 25670.4i 0.724741 + 1.25529i 0.959081 + 0.283133i \(0.0913739\pi\)
−0.234340 + 0.972155i \(0.575293\pi\)
\(54\) 3422.61 − 5928.13i 0.159725 − 0.276652i
\(55\) −40212.0 −1.79246
\(56\) 0 0
\(57\) −12316.7 −0.502121
\(58\) −3537.43 + 6127.00i −0.138076 + 0.239154i
\(59\) −5178.05 − 8968.65i −0.193659 − 0.335426i 0.752801 − 0.658248i \(-0.228702\pi\)
−0.946460 + 0.322821i \(0.895369\pi\)
\(60\) −18130.5 − 31402.9i −0.650176 − 1.12614i
\(61\) 477.089 − 826.343i 0.0164163 − 0.0284339i −0.857701 − 0.514150i \(-0.828108\pi\)
0.874117 + 0.485716i \(0.161441\pi\)
\(62\) 77057.2 2.54586
\(63\) 0 0
\(64\) −49454.8 −1.50924
\(65\) 19134.0 − 33141.1i 0.561725 − 0.972935i
\(66\) −23688.2 − 41029.2i −0.669380 − 1.15940i
\(67\) 9907.60 + 17160.5i 0.269638 + 0.467027i 0.968768 − 0.247967i \(-0.0797626\pi\)
−0.699130 + 0.714994i \(0.746429\pi\)
\(68\) 28244.9 − 48921.6i 0.740743 − 1.28300i
\(69\) 29052.7 0.734622
\(70\) 0 0
\(71\) 62125.4 1.46259 0.731296 − 0.682060i \(-0.238916\pi\)
0.731296 + 0.682060i \(0.238916\pi\)
\(72\) 9191.45 − 15920.1i 0.208955 − 0.361921i
\(73\) 13554.8 + 23477.6i 0.297705 + 0.515641i 0.975611 − 0.219509i \(-0.0704454\pi\)
−0.677905 + 0.735149i \(0.737112\pi\)
\(74\) −13186.5 − 22839.7i −0.279930 − 0.484853i
\(75\) −9090.27 + 15744.8i −0.186605 + 0.323209i
\(76\) −76869.6 −1.52658
\(77\) 0 0
\(78\) 45086.2 0.839087
\(79\) −22343.7 + 38700.4i −0.402798 + 0.697666i −0.994062 − 0.108812i \(-0.965295\pi\)
0.591265 + 0.806477i \(0.298629\pi\)
\(80\) −11964.3 − 20722.8i −0.209008 − 0.362012i
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) −1154.14 + 1999.03i −0.0189551 + 0.0328311i
\(83\) −15606.6 −0.248665 −0.124332 − 0.992241i \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) −82182.5 + 142344.i −1.19821 + 2.07536i
\(87\) 3390.55 + 5872.61i 0.0480255 + 0.0831827i
\(88\) −63615.0 − 110184.i −0.875696 − 1.51675i
\(89\) 6817.66 − 11808.5i 0.0912347 − 0.158023i −0.816796 − 0.576926i \(-0.804252\pi\)
0.908031 + 0.418903i \(0.137585\pi\)
\(90\) −54555.6 −0.709959
\(91\) 0 0
\(92\) 181320. 2.23345
\(93\) 36928.9 − 63962.7i 0.442750 − 0.766866i
\(94\) 76740.9 + 132919.i 0.895793 + 1.55156i
\(95\) 49081.6 + 85011.8i 0.557968 + 0.966429i
\(96\) −18584.8 + 32189.8i −0.205816 + 0.356484i
\(97\) 12919.5 0.139417 0.0697086 − 0.997567i \(-0.477793\pi\)
0.0697086 + 0.997567i \(0.477793\pi\)
\(98\) 0 0
\(99\) −45409.4 −0.465648
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 147.6.e.l.79.2 4
7.2 even 3 147.6.a.k.1.1 2
7.3 odd 6 21.6.e.b.4.2 ✓ 4
7.4 even 3 inner 147.6.e.l.67.2 4
7.5 odd 6 147.6.a.i.1.1 2
7.6 odd 2 21.6.e.b.16.2 yes 4
21.2 odd 6 441.6.a.s.1.2 2
21.5 even 6 441.6.a.t.1.2 2
21.17 even 6 63.6.e.c.46.1 4
21.20 even 2 63.6.e.c.37.1 4
28.3 even 6 336.6.q.e.193.2 4
28.27 even 2 336.6.q.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 ✓ 4 7.3 odd 6
21.6.e.b.16.2 yes 4 7.6 odd 2
63.6.e.c.37.1 4 21.20 even 2
63.6.e.c.46.1 4 21.17 even 6
147.6.a.i.1.1 2 7.5 odd 6
147.6.a.k.1.1 2 7.2 even 3
147.6.e.l.67.2 4 7.4 even 3 inner
147.6.e.l.79.2 4 1.1 even 1 trivial
336.6.q.e.193.2 4 28.3 even 6
336.6.q.e.289.2 4 28.27 even 2
441.6.a.s.1.2 2 21.2 odd 6
441.6.a.t.1.2 2 21.5 even 6