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 4.1
Root \(4.19493 + 1.84460i\) of defining polynomial
Character \(\chi\) \(=\) 21.4
Dual form 21.6.e.b.16.1

$q$-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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.19493 − 5.53379i −0.564790 − 0.978245i −0.997069 − 0.0765049i \(-0.975624\pi\)
0.432279 − 0.901740i \(-0.357709\pi\)
\(3\) 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 0.639204 − 0.769037i \(-0.279264\pi\)
−0.985608 + 0.169049i \(0.945930\pi\)
\(6\) −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 −0.243606 − 0.969874i \(-0.578330\pi\)
0.961739 − 0.273968i \(-0.0883362\pi\)
\(12\) 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 −0.439709 − 0.898140i \(-0.644918\pi\)
0.997667 − 0.0682711i \(-0.0217483\pi\)
\(18\) −258.790 + 448.237i −0.188263 + 0.326082i
\(19\) −471.237 − 816.206i −0.299471 − 0.518699i 0.676544 − 0.736402i \(-0.263477\pi\)
−0.976015 + 0.217703i \(0.930144\pi\)
\(20\) 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 0.980830 − 0.194866i \(-0.0624272\pi\)
−0.659174 + 0.751991i \(0.729094\pi\)
\(24\) −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 −0.623269 − 0.782008i \(-0.714196\pi\)
0.988873 + 0.148763i \(0.0475291\pi\)
\(32\) 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 0.667038 + 0.745024i \(0.267562\pi\)
0.311691 + 0.950184i \(0.399105\pi\)
\(38\) −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 0.956316 − 0.292335i \(-0.0944322\pi\)
−0.731327 + 0.682027i \(0.761099\pi\)
\(48\) 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 −0.840389 − 0.541984i \(-0.817673\pi\)
0.889566 + 0.456806i \(0.151007\pi\)
\(54\) 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 −0.245135 + 0.969489i \(0.578832\pi\)
−0.717035 + 0.697037i \(0.754501\pi\)
\(60\) 1538.97 − 2665.57i 0.0551889 − 0.0955900i
\(61\) −20548.9 − 35591.8i −0.707073 − 1.22469i −0.965938 − 0.258773i \(-0.916682\pi\)
0.258865 − 0.965913i \(-0.416651\pi\)
\(62\) −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 0.284153 + 0.958779i \(0.408288\pi\)
−0.972404 + 0.233305i \(0.925046\pi\)
\(68\) 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 −0.379459 − 0.925208i \(-0.623890\pi\)
0.990984 − 0.133983i \(-0.0427767\pi\)
\(74\) 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 0.996643 + 0.0818739i \(0.0260905\pi\)
−0.427416 + 0.904055i \(0.640576\pi\)
\(80\) 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 0.808762 − 0.588137i \(-0.200138\pi\)
−0.913722 + 0.406340i \(0.866805\pi\)
\(90\) 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.4.1 ✓ 4
3.2 odd 2 63.6.e.c.46.2 4
4.3 odd 2 336.6.q.e.193.1 4
7.2 even 3 inner 21.6.e.b.16.1 yes 4
7.3 odd 6 147.6.a.k.1.2 2
7.4 even 3 147.6.a.i.1.2 2
7.5 odd 6 147.6.e.l.79.1 4
7.6 odd 2 147.6.e.l.67.1 4
21.2 odd 6 63.6.e.c.37.2 4
21.11 odd 6 441.6.a.t.1.1 2
21.17 even 6 441.6.a.s.1.1 2
28.23 odd 6 336.6.q.e.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 ✓ 4 1.1 even 1 trivial
21.6.e.b.16.1 yes 4 7.2 even 3 inner
63.6.e.c.37.2 4 21.2 odd 6
63.6.e.c.46.2 4 3.2 odd 2
147.6.a.i.1.2 2 7.4 even 3
147.6.a.k.1.2 2 7.3 odd 6
147.6.e.l.67.1 4 7.6 odd 2
147.6.e.l.79.1 4 7.5 odd 6
336.6.q.e.193.1 4 4.3 odd 2
336.6.q.e.289.1 4 28.23 odd 6
441.6.a.s.1.1 2 21.17 even 6
441.6.a.t.1.1 2 21.11 odd 6