Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,5,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), 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: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 2.8
Character \(\chi\) \(=\) 27.2
Dual form 27.5.f.a.14.8

$q$-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+(2.89555 + 0.510564i) q^{2} +(-3.30987 + 8.36927i) q^{3} +(-6.91152 - 2.51559i) q^{4} +(31.8237 + 37.9260i) q^{5} +(-13.8570 + 22.5438i) q^{6} +(28.2667 - 10.2882i) q^{7} +(-59.4692 - 34.3346i) q^{8} +(-59.0895 - 55.4025i) q^{9} +(72.7837 + 126.065i) q^{10} +(75.7716 - 90.3011i) q^{11} +(43.9299 - 49.5181i) q^{12} +(3.56465 + 20.2161i) q^{13} +(87.1005 - 15.3582i) q^{14} +(-422.746 + 140.811i) q^{15} +(-64.5172 - 54.1364i) q^{16} +(203.984 - 117.770i) q^{17} +(-142.810 - 190.590i) q^{18} +(47.7809 - 82.7590i) q^{19} +(-124.544 - 342.182i) q^{20} +(-7.45416 + 270.624i) q^{21} +(265.505 - 222.785i) q^{22} +(85.2078 - 234.107i) q^{23} +(484.191 - 384.071i) q^{24} +(-317.105 + 1798.39i) q^{25} +60.3569i q^{26} +(659.257 - 311.160i) q^{27} -221.247 q^{28} +(-1150.78 - 202.914i) q^{29} +(-1295.98 + 191.887i) q^{30} +(562.326 + 204.670i) q^{31} +(547.062 + 651.963i) q^{32} +(504.960 + 933.038i) q^{33} +(650.776 - 236.863i) q^{34} +(1289.74 + 744.633i) q^{35} +(269.028 + 531.560i) q^{36} +(-123.627 - 214.128i) q^{37} +(180.606 - 215.238i) q^{38} +(-180.993 - 37.0794i) q^{39} +(-590.358 - 3348.09i) q^{40} +(-508.880 + 89.7293i) q^{41} +(-159.755 + 779.801i) q^{42} +(-1990.16 - 1669.95i) q^{43} +(-750.857 + 433.508i) q^{44} +(220.750 - 4004.14i) q^{45} +(366.250 - 634.364i) q^{46} +(-176.156 - 483.984i) q^{47} +(666.626 - 360.778i) q^{48} +(-1146.12 + 961.705i) q^{49} +(-1836.39 + 5045.44i) q^{50} +(310.490 + 2097.00i) q^{51} +(26.2183 - 148.692i) q^{52} -3241.10i q^{53} +(2067.78 - 564.389i) q^{54} +5836.10 q^{55} +(-2034.24 - 358.691i) q^{56} +(534.484 + 673.814i) q^{57} +(-3228.55 - 1175.09i) q^{58} +(1210.50 + 1442.62i) q^{59} +(3276.04 + 90.2364i) q^{60} +(-4396.97 + 1600.37i) q^{61} +(1523.75 + 879.736i) q^{62} +(-2240.26 - 958.118i) q^{63} +(1924.95 + 3334.11i) q^{64} +(-653.278 + 778.546i) q^{65} +(985.762 + 2959.48i) q^{66} +(-473.625 - 2686.06i) q^{67} +(-1706.10 + 300.832i) q^{68} +(1677.27 + 1487.99i) q^{69} +(3354.34 + 2814.62i) q^{70} +(5060.73 - 2921.81i) q^{71} +(1611.78 + 5323.56i) q^{72} +(-1626.01 + 2816.33i) q^{73} +(-248.643 - 683.140i) q^{74} +(-14001.7 - 8606.39i) q^{75} +(-538.427 + 451.794i) q^{76} +(1212.77 - 3332.07i) q^{77} +(-505.144 - 199.774i) q^{78} +(322.322 - 1827.98i) q^{79} -4169.70i q^{80} +(422.128 + 6547.41i) q^{81} -1519.30 q^{82} +(-1228.73 - 216.659i) q^{83} +(732.299 - 1851.67i) q^{84} +(10958.1 + 3988.42i) q^{85} +(-4910.01 - 5851.52i) q^{86} +(5507.18 - 8959.58i) q^{87} +(-7606.53 + 2768.55i) q^{88} +(6163.24 + 3558.35i) q^{89} +(2683.57 - 11481.5i) q^{90} +(308.749 + 534.769i) q^{91} +(-1177.83 + 1403.68i) q^{92} +(-3574.17 + 4028.83i) q^{93} +(-262.964 - 1491.34i) q^{94} +(4659.29 - 821.558i) q^{95} +(-7267.16 + 2420.60i) q^{96} +(7251.91 + 6085.08i) q^{97} +(-3809.65 + 2199.50i) q^{98} +(-9480.21 + 1137.91i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9} - 3 q^{10} - 492 q^{11} - 339 q^{12} - 6 q^{13} + 1137 q^{14} + 1017 q^{15} - 54 q^{16} - 9 q^{17} + 603 q^{18} - 3 q^{19}+ \cdots - 162405 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.89555 + 0.510564i 0.723889 + 0.127641i 0.523439 − 0.852063i \(-0.324649\pi\)
0.200449 + 0.979704i \(0.435760\pi\)
\(3\) −3.30987 + 8.36927i −0.367764 + 0.929919i
\(4\) −6.91152 − 2.51559i −0.431970 − 0.157224i
\(5\) 31.8237 + 37.9260i 1.27295 + 1.51704i 0.743573 + 0.668655i \(0.233130\pi\)
0.529376 + 0.848387i \(0.322426\pi\)
\(6\) −13.8570 + 22.5438i −0.384916 + 0.626216i
\(7\) 28.2667 − 10.2882i 0.576871 − 0.209964i −0.0370743 − 0.999313i \(-0.511804\pi\)
0.613945 + 0.789349i \(0.289582\pi\)
\(8\) −59.4692 − 34.3346i −0.929207 − 0.536478i
\(9\) −59.0895 − 55.4025i −0.729499 − 0.683981i
\(10\) 72.7837 + 126.065i 0.727837 + 1.26065i
\(11\) 75.7716 − 90.3011i 0.626211 − 0.746290i −0.355914 − 0.934519i \(-0.615830\pi\)
0.982125 + 0.188229i \(0.0602747\pi\)
\(12\) 43.9299 − 49.5181i 0.305069 − 0.343876i
\(13\) 3.56465 + 20.2161i 0.0210926 + 0.119622i 0.993536 − 0.113517i \(-0.0362115\pi\)
−0.972443 + 0.233139i \(0.925100\pi\)
\(14\) 87.1005 − 15.3582i 0.444390 − 0.0783580i
\(15\) −422.746 + 140.811i −1.87887 + 0.625827i
\(16\) −64.5172 − 54.1364i −0.252020 − 0.211470i
\(17\) 203.984 − 117.770i 0.705827 − 0.407509i −0.103687 − 0.994610i \(-0.533064\pi\)
0.809514 + 0.587101i \(0.199731\pi\)
\(18\) −142.810 − 190.590i −0.440772 − 0.588240i
\(19\) 47.7809 − 82.7590i 0.132357 − 0.229249i −0.792228 − 0.610226i \(-0.791079\pi\)
0.924585 + 0.380976i \(0.124412\pi\)
\(20\) −124.544 − 342.182i −0.311360 − 0.855455i
\(21\) −7.45416 + 270.624i −0.0169029 + 0.613660i
\(22\) 265.505 − 222.785i 0.548565 − 0.460300i
\(23\) 85.2078 − 234.107i 0.161073 − 0.442545i −0.832732 − 0.553676i \(-0.813225\pi\)
0.993806 + 0.111130i \(0.0354470\pi\)
\(24\) 484.191 − 384.071i 0.840610 − 0.666790i
\(25\) −317.105 + 1798.39i −0.507368 + 2.87743i
\(26\) 60.3569i 0.0892854i
\(27\) 659.257 − 311.160i 0.904331 − 0.426832i
\(28\) −221.247 −0.282202
\(29\) −1150.78 − 202.914i −1.36835 − 0.241277i −0.559274 − 0.828983i \(-0.688920\pi\)
−0.809074 + 0.587707i \(0.800031\pi\)
\(30\) −1295.98 + 191.887i −1.43997 + 0.213208i
\(31\) 562.326 + 204.670i 0.585147 + 0.212976i 0.617593 − 0.786497i \(-0.288108\pi\)
−0.0324468 + 0.999473i \(0.510330\pi\)
\(32\) 547.062 + 651.963i 0.534240 + 0.636683i
\(33\) 504.960 + 933.038i 0.463691 + 0.856784i
\(34\) 650.776 − 236.863i 0.562955 − 0.204899i
\(35\) 1289.74 + 744.633i 1.05285 + 0.607864i
\(36\) 269.028 + 531.560i 0.207584 + 0.410155i
\(37\) −123.627 − 214.128i −0.0903047 − 0.156412i 0.817335 − 0.576163i \(-0.195451\pi\)
−0.907639 + 0.419751i \(0.862117\pi\)
\(38\) 180.606 − 215.238i 0.125073 − 0.149057i
\(39\) −180.993 − 37.0794i −0.118996 − 0.0243783i
\(40\) −590.358 − 3348.09i −0.368974 − 2.09256i
\(41\) −508.880 + 89.7293i −0.302725 + 0.0533785i −0.322947 − 0.946417i \(-0.604674\pi\)
0.0202226 + 0.999796i \(0.493562\pi\)
\(42\) −159.755 + 779.801i −0.0905641 + 0.442064i
\(43\) −1990.16 − 1669.95i −1.07635 − 0.903161i −0.0807334 − 0.996736i \(-0.525726\pi\)
−0.995612 + 0.0935745i \(0.970171\pi\)
\(44\) −750.857 + 433.508i −0.387840 + 0.223919i
\(45\) 220.750 − 4004.14i 0.109013 − 1.97735i
\(46\) 366.250 − 634.364i 0.173086 − 0.299794i
\(47\) −176.156 − 483.984i −0.0797446 − 0.219097i 0.893413 − 0.449236i \(-0.148304\pi\)
−0.973158 + 0.230140i \(0.926082\pi\)
\(48\) 666.626 − 360.778i 0.289334 − 0.156588i
\(49\) −1146.12 + 961.705i −0.477349 + 0.400544i
\(50\) −1836.39 + 5045.44i −0.734556 + 2.01818i
\(51\) 310.490 + 2097.00i 0.119373 + 0.806229i
\(52\) 26.2183 − 148.692i 0.00969613 − 0.0549895i
\(53\) − 3241.10i − 1.15383i −0.816806 − 0.576913i \(-0.804257\pi\)
0.816806 − 0.576913i \(-0.195743\pi\)
\(54\) 2067.78 − 564.389i 0.709116 − 0.193549i
\(55\) 5836.10 1.92929
\(56\) −2034.24 − 358.691i −0.648673 − 0.114379i
\(57\) 534.484 + 673.814i 0.164507 + 0.207391i
\(58\) −3228.55 − 1175.09i −0.959734 − 0.349315i
\(59\) 1210.50 + 1442.62i 0.347745 + 0.414427i 0.911359 − 0.411611i \(-0.135034\pi\)
−0.563614 + 0.826038i \(0.690589\pi\)
\(60\) 3276.04 + 90.2364i 0.910011 + 0.0250657i
\(61\) −4396.97 + 1600.37i −1.18166 + 0.430090i −0.856789 − 0.515667i \(-0.827544\pi\)
−0.324874 + 0.945757i \(0.605322\pi\)
\(62\) 1523.75 + 879.736i 0.396397 + 0.228860i
\(63\) −2240.26 − 958.118i −0.564438 − 0.241400i
\(64\) 1924.95 + 3334.11i 0.469958 + 0.813991i
\(65\) −653.278 + 778.546i −0.154622 + 0.184271i
\(66\) 985.762 + 2959.48i 0.226300 + 0.679403i
\(67\) −473.625 − 2686.06i −0.105508 − 0.598365i −0.991016 − 0.133742i \(-0.957301\pi\)
0.885508 − 0.464624i \(-0.153810\pi\)
\(68\) −1706.10 + 300.832i −0.368967 + 0.0650588i
\(69\) 1677.27 + 1487.99i 0.352295 + 0.312537i
\(70\) 3354.34 + 2814.62i 0.684559 + 0.574413i
\(71\) 5060.73 − 2921.81i 1.00391 − 0.579610i 0.0945101 − 0.995524i \(-0.469872\pi\)
0.909404 + 0.415914i \(0.136538\pi\)
\(72\) 1611.78 + 5323.56i 0.310915 + 1.02692i
\(73\) −1626.01 + 2816.33i −0.305124 + 0.528491i −0.977289 − 0.211911i \(-0.932031\pi\)
0.672165 + 0.740402i \(0.265365\pi\)
\(74\) −248.643 − 683.140i −0.0454059 − 0.124752i
\(75\) −14001.7 − 8606.39i −2.48918 − 1.53003i
\(76\) −538.427 + 451.794i −0.0932179 + 0.0782191i
\(77\) 1212.77 − 3332.07i 0.204549 − 0.561995i
\(78\) −505.144 − 199.774i −0.0830282 − 0.0328359i
\(79\) 322.322 − 1827.98i 0.0516459 − 0.292899i −0.948035 − 0.318167i \(-0.896933\pi\)
0.999681 + 0.0252681i \(0.00804395\pi\)
\(80\) − 4169.70i − 0.651516i
\(81\) 422.128 + 6547.41i 0.0643390 + 0.997928i
\(82\) −1519.30 −0.225952
\(83\) −1228.73 − 216.659i −0.178362 − 0.0314500i 0.0837539 − 0.996486i \(-0.473309\pi\)
−0.262116 + 0.965037i \(0.584420\pi\)
\(84\) 732.299 − 1851.67i 0.103784 − 0.262425i
\(85\) 10958.1 + 3988.42i 1.51669 + 0.552030i
\(86\) −4910.01 − 5851.52i −0.663874 − 0.791174i
\(87\) 5507.18 − 8959.58i 0.727597 − 1.18372i
\(88\) −7606.53 + 2768.55i −0.982248 + 0.357509i
\(89\) 6163.24 + 3558.35i 0.778089 + 0.449230i 0.835753 − 0.549106i \(-0.185032\pi\)
−0.0576637 + 0.998336i \(0.518365\pi\)
\(90\) 2683.57 − 11481.5i 0.331305 − 1.41747i
\(91\) 308.749 + 534.769i 0.0372840 + 0.0645779i
\(92\) −1177.83 + 1403.68i −0.139158 + 0.165842i
\(93\) −3574.17 + 4028.83i −0.413246 + 0.465814i
\(94\) −262.964 − 1491.34i −0.0297605 − 0.168780i
\(95\) 4659.29 − 821.558i 0.516265 − 0.0910314i
\(96\) −7267.16 + 2420.60i −0.788538 + 0.262651i
\(97\) 7251.91 + 6085.08i 0.770742 + 0.646730i 0.940899 − 0.338687i \(-0.109983\pi\)
−0.170157 + 0.985417i \(0.554427\pi\)
\(98\) −3809.65 + 2199.50i −0.396673 + 0.229020i
\(99\) −9480.21 + 1137.91i −0.967269 + 0.116101i
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 27.5.f.a.2.8 ✓ 66
3.2 odd 2 81.5.f.a.8.4 66
27.13 even 9 81.5.f.a.71.4 66
27.14 odd 18 inner 27.5.f.a.14.8 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.f.a.2.8 ✓ 66 1.1 even 1 trivial
27.5.f.a.14.8 yes 66 27.14 odd 18 inner
81.5.f.a.8.4 66 3.2 odd 2
81.5.f.a.71.4 66 27.13 even 9