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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [28,5,Mod(11,28)] 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("28.11"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 28.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89435896635\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.2
Character \(\chi\) \(=\) 28.11
Dual form 28.5.g.a.23.2

$q$-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+(-3.86347 - 1.03614i) q^{2} +(1.18377 + 0.683453i) q^{3} +(13.8528 + 8.00618i) q^{4} +(-16.2684 - 28.1777i) q^{5} +(-3.86533 - 3.86706i) q^{6} +(-47.8742 - 10.4432i) q^{7} +(-45.2245 - 45.2851i) q^{8} +(-39.5658 - 68.5299i) q^{9} +(33.6565 + 125.720i) q^{10} +(-46.9599 - 27.1123i) q^{11} +(10.9268 + 18.9453i) q^{12} +223.705 q^{13} +(174.140 + 89.9513i) q^{14} -44.4747i q^{15} +(127.802 + 221.817i) q^{16} +(22.6070 - 39.1564i) q^{17} +(81.8548 + 305.759i) q^{18} +(-454.505 + 262.408i) q^{19} +(0.232247 - 520.589i) q^{20} +(-49.5349 - 45.0822i) q^{21} +(153.336 + 153.405i) q^{22} +(640.137 - 369.583i) q^{23} +(-22.5854 - 84.5162i) q^{24} +(-216.822 + 375.547i) q^{25} +(-864.279 - 231.789i) q^{26} -218.885i q^{27} +(-579.583 - 527.957i) q^{28} -392.110 q^{29} +(-46.0820 + 171.827i) q^{30} +(494.434 + 285.461i) q^{31} +(-263.927 - 989.403i) q^{32} +(-37.0600 - 64.1897i) q^{33} +(-127.913 + 127.856i) q^{34} +(484.572 + 1518.88i) q^{35} +(0.564838 - 1266.10i) q^{36} +(162.468 + 281.404i) q^{37} +(2027.86 - 542.878i) q^{38} +(264.817 + 152.892i) q^{39} +(-540.299 + 2011.04i) q^{40} +110.379 q^{41} +(144.665 + 225.499i) q^{42} -2810.88i q^{43} +(-433.462 - 751.552i) q^{44} +(-1287.34 + 2229.75i) q^{45} +(-2856.09 + 764.604i) q^{46} +(-877.553 + 506.656i) q^{47} +(-0.312222 + 349.928i) q^{48} +(2182.88 + 999.920i) q^{49} +(1226.80 - 1226.26i) q^{50} +(53.5231 - 30.9016i) q^{51} +(3098.95 + 1791.02i) q^{52} +(2257.95 - 3910.88i) q^{53} +(-226.795 + 845.655i) q^{54} +1764.30i q^{55} +(1692.17 + 2640.28i) q^{56} -717.375 q^{57} +(1514.91 + 406.281i) q^{58} +(-3889.74 - 2245.74i) q^{59} +(356.073 - 616.101i) q^{60} +(-593.248 - 1027.54i) q^{61} +(-1614.45 - 1615.17i) q^{62} +(1178.51 + 3694.01i) q^{63} +(-5.48196 + 4096.00i) q^{64} +(-3639.33 - 6303.50i) q^{65} +(76.6707 + 286.395i) q^{66} +(-3462.22 - 1998.91i) q^{67} +(626.664 - 361.432i) q^{68} +1010.37 q^{69} +(-298.360 - 6370.23i) q^{70} -2624.77i q^{71} +(-1314.04 + 4890.98i) q^{72} +(-254.442 + 440.707i) q^{73} +(-336.119 - 1255.53i) q^{74} +(-513.337 + 296.375i) q^{75} +(-8397.07 - 3.74612i) q^{76} +(1965.03 + 1788.39i) q^{77} +(-864.694 - 865.080i) q^{78} +(8411.13 - 4856.17i) q^{79} +(4171.15 - 7209.77i) q^{80} +(-3055.23 + 5291.82i) q^{81} +(-426.446 - 114.368i) q^{82} +7485.18i q^{83} +(-325.262 - 1021.10i) q^{84} -1471.12 q^{85} +(-2912.46 + 10859.8i) q^{86} +(-464.171 - 267.989i) q^{87} +(895.956 + 3352.73i) q^{88} +(2127.02 + 3684.11i) q^{89} +(7283.94 - 7280.69i) q^{90} +(-10709.7 - 2336.20i) q^{91} +(11826.7 + 5.27614i) q^{92} +(390.199 + 675.844i) q^{93} +(3915.37 - 1048.18i) q^{94} +(14788.1 + 8537.93i) q^{95} +(363.780 - 1351.61i) q^{96} +14601.8 q^{97} +(-7397.44 - 6124.93i) q^{98} +4290.88i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 2 q^{2} - 4 q^{4} - 2 q^{5} - 36 q^{6} - 184 q^{8} + 268 q^{9} + 130 q^{10} - 48 q^{12} + 344 q^{13} - 474 q^{14} - 432 q^{16} - 2 q^{17} - 568 q^{18} + 664 q^{20} + 426 q^{21} + 196 q^{22} - 692 q^{24}+ \cdots - 9682 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.86347 − 1.03614i −0.965868 − 0.259034i
\(3\) 1.18377 + 0.683453i 0.131531 + 0.0759392i 0.564322 − 0.825555i \(-0.309138\pi\)
−0.432791 + 0.901494i \(0.642471\pi\)
\(4\) 13.8528 + 8.00618i 0.865802 + 0.500386i
\(5\) −16.2684 − 28.1777i −0.650736 − 1.12711i −0.982945 − 0.183902i \(-0.941127\pi\)
0.332208 − 0.943206i \(-0.392206\pi\)
\(6\) −3.86533 − 3.86706i −0.107370 − 0.107418i
\(7\) −47.8742 − 10.4432i −0.977025 − 0.213127i
\(8\) −45.2245 − 45.2851i −0.706633 − 0.707580i
\(9\) −39.5658 − 68.5299i −0.488466 − 0.846049i
\(10\) 33.6565 + 125.720i 0.336565 + 1.25720i
\(11\) −46.9599 − 27.1123i −0.388098 − 0.224069i 0.293238 − 0.956040i \(-0.405267\pi\)
−0.681336 + 0.731971i \(0.738601\pi\)
\(12\) 10.9268 + 18.9453i 0.0758805 + 0.131564i
\(13\) 223.705 1.32370 0.661849 − 0.749637i \(-0.269772\pi\)
0.661849 + 0.749637i \(0.269772\pi\)
\(14\) 174.140 + 89.9513i 0.888470 + 0.458935i
\(15\) − 44.4747i − 0.197666i
\(16\) 127.802 + 221.817i 0.499227 + 0.866471i
\(17\) 22.6070 − 39.1564i 0.0782248 − 0.135489i −0.824259 − 0.566213i \(-0.808408\pi\)
0.902484 + 0.430723i \(0.141741\pi\)
\(18\) 81.8548 + 305.759i 0.252638 + 0.943701i
\(19\) −454.505 + 262.408i −1.25902 + 0.726893i −0.972883 − 0.231297i \(-0.925703\pi\)
−0.286133 + 0.958190i \(0.592370\pi\)
\(20\) 0.232247 − 520.589i 0.000580616 − 1.30147i
\(21\) −49.5349 − 45.0822i −0.112324 − 0.102227i
\(22\) 153.336 + 153.405i 0.316810 + 0.316952i
\(23\) 640.137 − 369.583i 1.21009 − 0.698645i 0.247310 − 0.968937i \(-0.420454\pi\)
0.962778 + 0.270292i \(0.0871202\pi\)
\(24\) −22.5854 − 84.5162i −0.0392108 − 0.146730i
\(25\) −216.822 + 375.547i −0.346915 + 0.600874i
\(26\) −864.279 − 231.789i −1.27852 − 0.342884i
\(27\) − 218.885i − 0.300253i
\(28\) −579.583 − 527.957i −0.739265 − 0.673415i
\(29\) −392.110 −0.466243 −0.233122 − 0.972448i \(-0.574894\pi\)
−0.233122 + 0.972448i \(0.574894\pi\)
\(30\) −46.0820 + 171.827i −0.0512022 + 0.190919i
\(31\) 494.434 + 285.461i 0.514499 + 0.297046i 0.734681 − 0.678413i \(-0.237332\pi\)
−0.220182 + 0.975459i \(0.570665\pi\)
\(32\) −263.927 − 989.403i −0.257742 − 0.966214i
\(33\) −37.0600 − 64.1897i −0.0340312 − 0.0589438i
\(34\) −127.913 + 127.856i −0.110651 + 0.110602i
\(35\) 484.572 + 1518.88i 0.395569 + 1.23990i
\(36\) 0.564838 − 1266.10i 0.000435832 − 0.976933i
\(37\) 162.468 + 281.404i 0.118677 + 0.205554i 0.919244 − 0.393689i \(-0.128801\pi\)
−0.800567 + 0.599244i \(0.795468\pi\)
\(38\) 2027.86 − 542.878i 1.40433 − 0.375954i
\(39\) 264.817 + 152.892i 0.174107 + 0.100521i
\(40\) −540.299 + 2011.04i −0.337687 + 1.25690i
\(41\) 110.379 0.0656627 0.0328314 − 0.999461i \(-0.489548\pi\)
0.0328314 + 0.999461i \(0.489548\pi\)
\(42\) 144.665 + 225.499i 0.0820098 + 0.127834i
\(43\) − 2810.88i − 1.52022i −0.649796 − 0.760109i \(-0.725146\pi\)
0.649796 − 0.760109i \(-0.274854\pi\)
\(44\) −433.462 − 751.552i −0.223896 − 0.388198i
\(45\) −1287.34 + 2229.75i −0.635726 + 1.10111i
\(46\) −2856.09 + 764.604i −1.34976 + 0.361344i
\(47\) −877.553 + 506.656i −0.397263 + 0.229360i −0.685302 − 0.728259i \(-0.740330\pi\)
0.288040 + 0.957619i \(0.406997\pi\)
\(48\) −0.312222 + 349.928i −0.000135513 + 0.151878i
\(49\) 2182.88 + 999.920i 0.909154 + 0.416460i
\(50\) 1226.80 − 1226.26i 0.490721 − 0.490503i
\(51\) 53.5231 − 30.9016i 0.0205779 − 0.0118807i
\(52\) 3098.95 + 1791.02i 1.14606 + 0.662361i
\(53\) 2257.95 − 3910.88i 0.803827 − 1.39227i −0.113253 − 0.993566i \(-0.536127\pi\)
0.917080 − 0.398703i \(-0.130539\pi\)
\(54\) −226.795 + 845.655i −0.0777760 + 0.290005i
\(55\) 1764.30i 0.583238i
\(56\) 1692.17 + 2640.28i 0.539594 + 0.841925i
\(57\) −717.375 −0.220799
\(58\) 1514.91 + 406.281i 0.450329 + 0.120773i
\(59\) −3889.74 − 2245.74i −1.11742 − 0.645143i −0.176679 − 0.984269i \(-0.556535\pi\)
−0.940741 + 0.339126i \(0.889869\pi\)
\(60\) 356.073 − 616.101i 0.0989091 − 0.171139i
\(61\) −593.248 − 1027.54i −0.159432 − 0.276145i 0.775232 − 0.631677i \(-0.217633\pi\)
−0.934664 + 0.355532i \(0.884300\pi\)
\(62\) −1614.45 − 1615.17i −0.419993 − 0.420180i
\(63\) 1178.51 + 3694.01i 0.296928 + 0.930716i
\(64\) −5.48196 + 4096.00i −0.00133837 + 0.999999i
\(65\) −3639.33 − 6303.50i −0.861379 − 1.49195i
\(66\) 76.6707 + 286.395i 0.0176012 + 0.0657471i
\(67\) −3462.22 − 1998.91i −0.771268 − 0.445292i 0.0620589 − 0.998072i \(-0.480233\pi\)
−0.833327 + 0.552781i \(0.813567\pi\)
\(68\) 626.664 − 361.432i 0.135524 − 0.0781643i
\(69\) 1010.37 0.212218
\(70\) −298.360 − 6370.23i −0.0608899 − 1.30005i
\(71\) − 2624.77i − 0.520684i −0.965516 − 0.260342i \(-0.916165\pi\)
0.965516 − 0.260342i \(-0.0838354\pi\)
\(72\) −1314.04 + 4890.98i −0.253480 + 0.943475i
\(73\) −254.442 + 440.707i −0.0477467 + 0.0826998i −0.888911 − 0.458080i \(-0.848537\pi\)
0.841164 + 0.540780i \(0.181871\pi\)
\(74\) −336.119 − 1255.53i −0.0613805 − 0.229280i
\(75\) −513.337 + 296.375i −0.0912599 + 0.0526889i
\(76\) −8397.07 − 3.74612i −1.45379 − 0.000648567i
\(77\) 1965.03 + 1788.39i 0.331427 + 0.301635i
\(78\) −864.694 − 865.080i −0.142126 − 0.142189i
\(79\) 8411.13 − 4856.17i 1.34772 − 0.778107i 0.359795 − 0.933031i \(-0.382847\pi\)
0.987926 + 0.154924i \(0.0495134\pi\)
\(80\) 4171.15 − 7209.77i 0.651742 − 1.12653i
\(81\) −3055.23 + 5291.82i −0.465665 + 0.806556i
\(82\) −426.446 − 114.368i −0.0634215 − 0.0170089i
\(83\) 7485.18i 1.08654i 0.839558 + 0.543270i \(0.182814\pi\)
−0.839558 + 0.543270i \(0.817186\pi\)
\(84\) −325.262 − 1021.10i −0.0460973 − 0.144714i
\(85\) −1471.12 −0.203615
\(86\) −2912.46 + 10859.8i −0.393789 + 1.46833i
\(87\) −464.171 − 267.989i −0.0613252 − 0.0354061i
\(88\) 895.956 + 3352.73i 0.115697 + 0.432945i
\(89\) 2127.02 + 3684.11i 0.268530 + 0.465107i 0.968482 − 0.249082i \(-0.0801289\pi\)
−0.699953 + 0.714189i \(0.746796\pi\)
\(90\) 7283.94 − 7280.69i 0.899252 − 0.898851i
\(91\) −10709.7 − 2336.20i −1.29329 − 0.282115i
\(92\) 11826.7 + 5.27614i 1.39729 + 0.000623363i
\(93\) 390.199 + 675.844i 0.0451149 + 0.0781413i
\(94\) 3915.37 − 1048.18i 0.443115 − 0.118626i
\(95\) 14788.1 + 8537.93i 1.63857 + 0.946031i
\(96\) 363.780 − 1351.61i 0.0394726 − 0.146659i
\(97\) 14601.8 1.55190 0.775949 − 0.630796i \(-0.217272\pi\)
0.775949 + 0.630796i \(0.217272\pi\)
\(98\) −7397.44 − 6124.93i −0.770246 − 0.637747i
\(99\) 4290.88i 0.437800i
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 28.5.g.a.11.2 ✓ 28
4.3 odd 2 inner 28.5.g.a.11.8 yes 28
7.2 even 3 inner 28.5.g.a.23.8 yes 28
7.3 odd 6 196.5.c.g.99.11 14
7.4 even 3 196.5.c.h.99.11 14
28.3 even 6 196.5.c.g.99.12 14
28.11 odd 6 196.5.c.h.99.12 14
28.23 odd 6 inner 28.5.g.a.23.2 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.g.a.11.2 ✓ 28 1.1 even 1 trivial
28.5.g.a.11.8 yes 28 4.3 odd 2 inner
28.5.g.a.23.2 yes 28 28.23 odd 6 inner
28.5.g.a.23.8 yes 28 7.2 even 3 inner
196.5.c.g.99.11 14 7.3 odd 6
196.5.c.g.99.12 14 28.3 even 6
196.5.c.h.99.11 14 7.4 even 3
196.5.c.h.99.12 14 28.11 odd 6