Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,6,Mod(4,21)] 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("21.4"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(3.36806021607\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.2
Root \(-3.69493 - 2.71062i\) of defining polynomial
Character \(\chi\) \(=\) 21.4
Dual form 21.6.e.b.16.2

$q$-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} +(-87.5000 - 95.6596i) q^{7} -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} +(367.084 - 1160.65i) q^{14} +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} +(-1139.34 + 251.527i) q^{21} +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} +(7110.71 - 1569.80i) q^{28} -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} +(2804.15 - 8866.21i) q^{35} +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} +(-7394.52 - 8084.07i) q^{42} -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} +(-1494.50 + 16740.4i) q^{49} -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} +(19858.1 + 21709.9i) q^{56} -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} +(-3166.58 + 10012.2i) q^{63} -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} +(85264.1 - 18823.3i) q^{70} +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} +(-70969.7 + 15667.6i) q^{77} +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} +(19762.8 - 62486.6i) q^{84} -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} +(-46682.0 - 51035.2i) q^{91} +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} +(-143147. + 66442.1i) q^{98} -45409.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 18 q^{3} - 65 q^{4} + 33 q^{5} + 54 q^{6} - 350 q^{7} - 750 q^{8} - 162 q^{9} - 921 q^{10} + 1137 q^{11} + 585 q^{12} + 1850 q^{13} + 2352 q^{14} + 594 q^{15} + 895 q^{16} + 324 q^{17} + 243 q^{18}+ \cdots - 184194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.69493 + 8.13186i 0.829955 + 1.43752i 0.898073 + 0.439847i \(0.144967\pi\)
−0.0681181 + 0.997677i \(0.521699\pi\)
\(3\) 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 0.985084 + 0.172076i \(0.0550476\pi\)
−0.343519 + 0.939146i \(0.611619\pi\)
\(6\) 84.5088 0.958349
\(7\) −87.5000 − 95.6596i −0.674937 − 0.737876i
\(8\) −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 −0.270524 − 0.962713i \(-0.587197\pi\)
0.968996 − 0.247076i \(-0.0794698\pi\)
\(12\) 252.763 + 437.799i 0.506711 + 0.877650i
\(13\) 533.509 0.875555 0.437777 − 0.899083i \(-0.355766\pi\)
0.437777 + 0.899083i \(0.355766\pi\)
\(14\) 367.084 − 1160.65i 0.500547 − 1.58264i
\(15\) 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 \(-0.972006\pi\)
0.574132 + 0.818763i \(0.305340\pi\)
\(18\) 380.290 − 658.681i 0.276652 − 0.479175i
\(19\) −684.263 − 1185.18i −0.434850 − 0.753182i 0.562434 − 0.826842i \(-0.309865\pi\)
−0.997283 + 0.0736606i \(0.976532\pi\)
\(20\) −4028.99 −2.25228
\(21\) −1139.34 + 251.527i −0.563775 + 0.124462i
\(22\) 5264.05 2.31880
\(23\) −1614.04 − 2795.60i −0.636201 − 1.10193i −0.986259 − 0.165205i \(-0.947171\pi\)
0.350058 − 0.936728i \(-0.386162\pi\)
\(24\) −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
\(28\) 7110.71 − 1569.80i 1.71403 − 0.378398i
\(29\) −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 0.172389 + 0.985029i \(0.444852\pi\)
−0.939254 + 0.343222i \(0.888482\pi\)
\(32\) −2064.97 + 3576.64i −0.356484 + 0.617448i
\(33\) −2522.75 − 4369.52i −0.403263 − 0.698472i
\(34\) −9443.39 −1.40098
\(35\) 2804.15 − 8866.21i 0.386929 − 1.22340i
\(36\) 4549.74 0.585100
\(37\) 1404.33 + 2432.37i 0.168642 + 0.292096i 0.937943 − 0.346791i \(-0.112729\pi\)
−0.769301 + 0.638887i \(0.779395\pi\)
\(38\) 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 \(-0.496365\pi\)
0.0114193 + 0.999935i \(0.496365\pi\)
\(42\) −7394.52 − 8084.07i −0.646825 − 0.707143i
\(43\) −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 0.998922 + 0.0464219i \(0.0147819\pi\)
−0.459258 + 0.888303i \(0.651885\pi\)
\(48\) −3002.37 −0.188088
\(49\) −1494.50 + 16740.4i −0.0889213 + 0.996039i
\(50\) −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 −0.234340 − 0.972155i \(-0.575293\pi\)
0.959081 − 0.283133i \(-0.0913739\pi\)
\(54\) −3422.61 − 5928.13i −0.159725 − 0.276652i
\(55\) 40212.0 1.79246
\(56\) 19858.1 + 21709.9i 0.846188 + 0.925097i
\(57\) −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 \(-0.771298\pi\)
0.946460 + 0.322821i \(0.104631\pi\)
\(60\) −18130.5 + 31402.9i −0.650176 + 1.12614i
\(61\) −477.089 − 826.343i −0.0164163 − 0.0284339i 0.857701 − 0.514150i \(-0.171892\pi\)
−0.874117 + 0.485716i \(0.838559\pi\)
\(62\) −77057.2 −2.54586
\(63\) −3166.58 + 10012.2i −0.100517 + 0.317817i
\(64\) −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 −0.699130 − 0.714994i \(-0.746429\pi\)
0.968768 + 0.247967i \(0.0797626\pi\)
\(68\) −28244.9 − 48921.6i −0.740743 − 1.28300i
\(69\) −29052.7 −0.734622
\(70\) 85264.1 − 18823.3i 2.07980 − 0.459147i
\(71\) 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 \(-0.929555\pi\)
0.677905 + 0.735149i \(0.262888\pi\)
\(74\) −13186.5 + 22839.7i −0.279930 + 0.484853i
\(75\) 9090.27 + 15744.8i 0.186605 + 0.323209i
\(76\) 76869.6 1.52658
\(77\) −70969.7 + 15667.6i −1.36410 + 0.301145i
\(78\) 45086.2 0.839087
\(79\) −22343.7 − 38700.4i −0.402798 − 0.697666i 0.591265 − 0.806477i \(-0.298629\pi\)
−0.994062 + 0.108812i \(0.965295\pi\)
\(80\) 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 \(-0.460321\pi\)
0.124332 + 0.992241i \(0.460321\pi\)
\(84\) 19762.8 − 62486.6i 0.305599 − 0.966248i
\(85\) −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 \(-0.195748\pi\)
−0.908031 + 0.418903i \(0.862415\pi\)
\(90\) 54555.6 0.709959
\(91\) −46682.0 − 51035.2i −0.590944 − 0.646050i
\(92\) 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 \(-0.522207\pi\)
−0.0697086 + 0.997567i \(0.522207\pi\)
\(98\) −143147. + 66442.1i −1.50563 + 0.698841i
\(99\) −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 21.6.e.b.4.2 ✓ 4
3.2 odd 2 63.6.e.c.46.1 4
4.3 odd 2 336.6.q.e.193.2 4
7.2 even 3 inner 21.6.e.b.16.2 yes 4
7.3 odd 6 147.6.a.k.1.1 2
7.4 even 3 147.6.a.i.1.1 2
7.5 odd 6 147.6.e.l.79.2 4
7.6 odd 2 147.6.e.l.67.2 4
21.2 odd 6 63.6.e.c.37.1 4
21.11 odd 6 441.6.a.t.1.2 2
21.17 even 6 441.6.a.s.1.2 2
28.23 odd 6 336.6.q.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 ✓ 4 1.1 even 1 trivial
21.6.e.b.16.2 yes 4 7.2 even 3 inner
63.6.e.c.37.1 4 21.2 odd 6
63.6.e.c.46.1 4 3.2 odd 2
147.6.a.i.1.1 2 7.4 even 3
147.6.a.k.1.1 2 7.3 odd 6
147.6.e.l.67.2 4 7.6 odd 2
147.6.e.l.79.2 4 7.5 odd 6
336.6.q.e.193.2 4 4.3 odd 2
336.6.q.e.289.2 4 28.23 odd 6
441.6.a.s.1.2 2 21.17 even 6
441.6.a.t.1.2 2 21.11 odd 6