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 16.1
Root \(4.19493 - 1.84460i\) of defining polynomial
Character \(\chi\) \(=\) 21.16
Dual form 21.6.e.b.4.1

$q$-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+(-3.19493 + 5.53379i) q^{2} +(4.50000 + 7.79423i) q^{3} +(-4.41520 - 7.64735i) q^{4} +(-19.3645 + 33.5404i) q^{5} -57.5088 q^{6} +(-87.5000 - 95.6596i) q^{7} -148.051 q^{8} +(-40.5000 + 70.1481i) q^{9} +(-123.737 - 214.318i) q^{10} +(288.195 + 499.168i) q^{11} +(39.7368 - 68.8262i) q^{12} +391.491 q^{13} +(808.916 - 178.580i) q^{14} -348.562 q^{15} +(614.298 - 1064.00i) q^{16} +(664.850 + 1151.55i) q^{17} +(-258.790 - 448.237i) q^{18} +(-471.237 + 816.206i) q^{19} +341.993 q^{20} +(351.842 - 1112.46i) q^{21} -3683.05 q^{22} +(816.040 - 1413.42i) q^{23} +(-666.228 - 1153.94i) q^{24} +(812.530 + 1407.34i) q^{25} +(-1250.79 + 2166.43i) q^{26} -729.000 q^{27} +(-345.212 + 1091.50i) q^{28} -1463.54 q^{29} +(1113.63 - 1928.87i) q^{30} +(1956.21 + 3388.25i) q^{31} +(1556.47 + 2695.89i) q^{32} +(-2593.75 + 4492.51i) q^{33} -8496.61 q^{34} +(4902.85 - 1082.38i) q^{35} +715.262 q^{36} +(8150.17 - 14116.5i) q^{37} +(-3011.14 - 5215.45i) q^{38} +(1761.71 + 3051.37i) q^{39} +(2866.93 - 4965.67i) q^{40} -13103.8 q^{41} +(5032.02 + 5501.27i) q^{42} +14733.5 q^{43} +(2544.88 - 4407.86i) q^{44} +(-1568.53 - 2716.77i) q^{45} +(5214.38 + 9031.58i) q^{46} +(3407.26 - 5901.55i) q^{47} +11057.4 q^{48} +(-1494.50 + 16740.4i) q^{49} -10383.9 q^{50} +(-5983.65 + 10364.0i) q^{51} +(-1728.51 - 2993.87i) q^{52} +(1005.67 + 1741.87i) q^{53} +(2329.11 - 4034.13i) q^{54} -22323.0 q^{55} +(12954.4 + 14162.5i) q^{56} -8482.26 q^{57} +(4675.93 - 8098.94i) q^{58} +(-25726.6 - 44559.7i) q^{59} +(1538.97 + 2665.57i) q^{60} +(-20548.9 + 35591.8i) q^{61} -24999.8 q^{62} +(10254.1 - 2263.74i) q^{63} +19423.8 q^{64} +(-7581.04 + 13130.8i) q^{65} +(-16573.7 - 28706.6i) q^{66} +(-25289.1 - 43802.0i) q^{67} +(5870.89 - 10168.7i) q^{68} +14688.7 q^{69} +(-9674.63 + 30589.5i) q^{70} +39970.6 q^{71} +(5996.05 - 10385.5i) q^{72} +(27843.3 + 48226.0i) q^{73} +(52078.5 + 90202.6i) q^{74} +(-7312.77 + 12666.1i) q^{75} +8322.42 q^{76} +(22533.2 - 71245.8i) q^{77} -22514.2 q^{78} +(31575.7 - 54690.7i) q^{79} +(23791.2 + 41207.6i) q^{80} +(-3280.50 - 5681.99i) q^{81} +(41865.9 - 72513.8i) q^{82} +45572.4 q^{83} +(-10060.8 + 2221.08i) q^{84} -51498.1 q^{85} +(-47072.5 + 81532.0i) q^{86} +(-6585.95 - 11407.2i) q^{87} +(-42667.5 - 73902.2i) q^{88} +(-7843.34 + 13585.1i) q^{89} +20045.4 q^{90} +(-34255.5 - 37449.9i) q^{91} -14411.9 q^{92} +(-17605.9 + 30494.3i) q^{93} +(21771.9 + 37710.1i) q^{94} +(-18250.6 - 31610.9i) q^{95} +(-14008.3 + 24263.0i) q^{96} +3128.49 q^{97} +(-87863.1 - 61754.8i) q^{98} -46687.6 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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.19493 + 5.53379i −0.564790 + 0.978245i 0.432279 + 0.901740i \(0.357709\pi\)
−0.997069 + 0.0765049i \(0.975624\pi\)
\(3\) 4.50000 + 7.79423i 0.288675 + 0.500000i
\(4\) −4.41520 − 7.64735i −0.137975 − 0.238980i
\(5\) −19.3645 + 33.5404i −0.346403 + 0.599988i −0.985608 − 0.169049i \(-0.945930\pi\)
0.639204 + 0.769037i \(0.279264\pi\)
\(6\) −57.5088 −0.652163
\(7\) −87.5000 − 95.6596i −0.674937 − 0.737876i
\(8\) −148.051 −0.817872
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) −123.737 − 214.318i −0.391290 − 0.677734i
\(11\) 288.195 + 499.168i 0.718133 + 1.24384i 0.961739 + 0.273968i \(0.0883362\pi\)
−0.243606 + 0.969874i \(0.578330\pi\)
\(12\) 39.7368 − 68.8262i 0.0796599 − 0.137975i
\(13\) 391.491 0.642486 0.321243 − 0.946997i \(-0.395899\pi\)
0.321243 + 0.946997i \(0.395899\pi\)
\(14\) 808.916 − 178.580i 1.10302 − 0.243508i
\(15\) −348.562 −0.399992
\(16\) 614.298 − 1064.00i 0.599901 − 1.03906i
\(17\) 664.850 + 1151.55i 0.557958 + 0.966412i 0.997667 + 0.0682711i \(0.0217483\pi\)
−0.439709 + 0.898140i \(0.644918\pi\)
\(18\) −258.790 − 448.237i −0.188263 − 0.326082i
\(19\) −471.237 + 816.206i −0.299471 + 0.518699i −0.976015 − 0.217703i \(-0.930144\pi\)
0.676544 + 0.736402i \(0.263477\pi\)
\(20\) 341.993 0.191180
\(21\) 351.842 − 1112.46i 0.174100 − 0.550475i
\(22\) −3683.05 −1.62238
\(23\) 816.040 − 1413.42i 0.321656 − 0.557124i −0.659174 − 0.751991i \(-0.729094\pi\)
0.980830 + 0.194866i \(0.0624272\pi\)
\(24\) −666.228 − 1153.94i −0.236099 − 0.408936i
\(25\) 812.530 + 1407.34i 0.260009 + 0.450350i
\(26\) −1250.79 + 2166.43i −0.362869 + 0.628508i
\(27\) −729.000 −0.192450
\(28\) −345.212 + 1091.50i −0.0832130 + 0.263105i
\(29\) −1463.54 −0.323155 −0.161577 − 0.986860i \(-0.551658\pi\)
−0.161577 + 0.986860i \(0.551658\pi\)
\(30\) 1113.63 − 1928.87i 0.225911 − 0.391290i
\(31\) 1956.21 + 3388.25i 0.365604 + 0.633245i 0.988873 − 0.148763i \(-0.0475291\pi\)
−0.623269 + 0.782008i \(0.714196\pi\)
\(32\) 1556.47 + 2695.89i 0.268700 + 0.465401i
\(33\) −2593.75 + 4492.51i −0.414614 + 0.718133i
\(34\) −8496.61 −1.26052
\(35\) 4902.85 − 1082.38i 0.676517 − 0.149351i
\(36\) 715.262 0.0919833
\(37\) 8150.17 − 14116.5i 0.978729 − 1.69521i 0.311691 − 0.950184i \(-0.399105\pi\)
0.667038 − 0.745024i \(-0.267562\pi\)
\(38\) −3011.14 − 5215.45i −0.338277 − 0.585912i
\(39\) 1761.71 + 3051.37i 0.185470 + 0.321243i
\(40\) 2866.93 − 4965.67i 0.283314 − 0.490714i
\(41\) −13103.8 −1.21741 −0.608707 − 0.793395i \(-0.708312\pi\)
−0.608707 + 0.793395i \(0.708312\pi\)
\(42\) 5032.02 + 5501.27i 0.440169 + 0.481215i
\(43\) 14733.5 1.21516 0.607582 − 0.794257i \(-0.292140\pi\)
0.607582 + 0.794257i \(0.292140\pi\)
\(44\) 2544.88 − 4407.86i 0.198169 − 0.343238i
\(45\) −1568.53 − 2716.77i −0.115468 − 0.199996i
\(46\) 5214.38 + 9031.58i 0.363336 + 0.629316i
\(47\) 3407.26 − 5901.55i 0.224989 − 0.389692i −0.731327 − 0.682027i \(-0.761099\pi\)
0.956316 + 0.292335i \(0.0944322\pi\)
\(48\) 11057.4 0.692706
\(49\) −1494.50 + 16740.4i −0.0889213 + 0.996039i
\(50\) −10383.9 −0.587403
\(51\) −5983.65 + 10364.0i −0.322137 + 0.557958i
\(52\) −1728.51 − 2993.87i −0.0886470 − 0.153541i
\(53\) 1005.67 + 1741.87i 0.0491775 + 0.0851779i 0.889566 − 0.456806i \(-0.151007\pi\)
−0.840389 + 0.541984i \(0.817673\pi\)
\(54\) 2329.11 − 4034.13i 0.108694 − 0.188263i
\(55\) −22323.0 −0.995054
\(56\) 12954.4 + 14162.5i 0.552012 + 0.603488i
\(57\) −8482.26 −0.345800
\(58\) 4675.93 − 8098.94i 0.182515 − 0.316125i
\(59\) −25726.6 − 44559.7i −0.962170 − 1.66653i −0.717035 − 0.697037i \(-0.754501\pi\)
−0.245135 − 0.969489i \(-0.578832\pi\)
\(60\) 1538.97 + 2665.57i 0.0551889 + 0.0955900i
\(61\) −20548.9 + 35591.8i −0.707073 + 1.22469i 0.258865 + 0.965913i \(0.416651\pi\)
−0.965938 + 0.258773i \(0.916682\pi\)
\(62\) −24999.8 −0.825958
\(63\) 10254.1 − 2263.74i 0.325496 − 0.0718581i
\(64\) 19423.8 0.592766
\(65\) −7581.04 + 13130.8i −0.222559 + 0.385484i
\(66\) −16573.7 − 28706.6i −0.468340 − 0.811188i
\(67\) −25289.1 − 43802.0i −0.688250 − 1.19208i −0.972404 − 0.233305i \(-0.925046\pi\)
0.284153 − 0.958779i \(-0.408288\pi\)
\(68\) 5870.89 − 10168.7i 0.153968 − 0.266681i
\(69\) 14688.7 0.371416
\(70\) −9674.63 + 30589.5i −0.235988 + 0.746151i
\(71\) 39970.6 0.941012 0.470506 − 0.882397i \(-0.344071\pi\)
0.470506 + 0.882397i \(0.344071\pi\)
\(72\) 5996.05 − 10385.5i 0.136312 − 0.236099i
\(73\) 27843.3 + 48226.0i 0.611524 + 1.05919i 0.990984 + 0.133983i \(0.0427767\pi\)
−0.379459 + 0.925208i \(0.623890\pi\)
\(74\) 52078.5 + 90202.6i 1.10555 + 1.91487i
\(75\) −7312.77 + 12666.1i −0.150117 + 0.260009i
\(76\) 8322.42 0.165278
\(77\) 22533.2 − 71245.8i 0.433107 − 1.36941i
\(78\) −22514.2 −0.419006
\(79\) 31575.7 − 54690.7i 0.569226 − 0.985929i −0.427416 − 0.904055i \(-0.640576\pi\)
0.996643 − 0.0818739i \(-0.0260905\pi\)
\(80\) 23791.2 + 41207.6i 0.415615 + 0.719867i
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 41865.9 − 72513.8i 0.687583 − 1.19093i
\(83\) 45572.4 0.726116 0.363058 − 0.931766i \(-0.381733\pi\)
0.363058 + 0.931766i \(0.381733\pi\)
\(84\) −10060.8 + 2221.08i −0.155574 + 0.0343453i
\(85\) −51498.1 −0.773114
\(86\) −47072.5 + 81532.0i −0.686312 + 1.18873i
\(87\) −6585.95 − 11407.2i −0.0932868 − 0.161577i
\(88\) −42667.5 − 73902.2i −0.587341 − 1.01730i
\(89\) −7843.34 + 13585.1i −0.104961 + 0.181797i −0.913722 − 0.406340i \(-0.866805\pi\)
0.808762 + 0.588137i \(0.200138\pi\)
\(90\) 20045.4 0.260860
\(91\) −34255.5 − 37449.9i −0.433637 − 0.474075i
\(92\) −14411.9 −0.177522
\(93\) −17605.9 + 30494.3i −0.211082 + 0.365604i
\(94\) 21771.9 + 37710.1i 0.254143 + 0.440188i
\(95\) −18250.6 − 31610.9i −0.207476 − 0.359358i
\(96\) −14008.3 + 24263.0i −0.155134 + 0.268700i
\(97\) 3128.49 0.0337603 0.0168801 − 0.999858i \(-0.494627\pi\)
0.0168801 + 0.999858i \(0.494627\pi\)
\(98\) −87863.1 − 61754.8i −0.924148 − 0.649539i
\(99\) −46687.6 −0.478755
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.16.1 yes 4
3.2 odd 2 63.6.e.c.37.2 4
4.3 odd 2 336.6.q.e.289.1 4
7.2 even 3 147.6.a.i.1.2 2
7.3 odd 6 147.6.e.l.67.1 4
7.4 even 3 inner 21.6.e.b.4.1 ✓ 4
7.5 odd 6 147.6.a.k.1.2 2
7.6 odd 2 147.6.e.l.79.1 4
21.2 odd 6 441.6.a.t.1.1 2
21.5 even 6 441.6.a.s.1.1 2
21.11 odd 6 63.6.e.c.46.2 4
28.11 odd 6 336.6.q.e.193.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 ✓ 4 7.4 even 3 inner
21.6.e.b.16.1 yes 4 1.1 even 1 trivial
63.6.e.c.37.2 4 3.2 odd 2
63.6.e.c.46.2 4 21.11 odd 6
147.6.a.i.1.2 2 7.2 even 3
147.6.a.k.1.2 2 7.5 odd 6
147.6.e.l.67.1 4 7.3 odd 6
147.6.e.l.79.1 4 7.6 odd 2
336.6.q.e.193.1 4 28.11 odd 6
336.6.q.e.289.1 4 4.3 odd 2
441.6.a.s.1.1 2 21.5 even 6
441.6.a.t.1.1 2 21.2 odd 6