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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 2.6
Character \(\chi\) \(=\) 27.2
Dual form 27.5.f.a.14.6

$q$-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+(-0.185922 - 0.0327832i) q^{2} +(3.20767 - 8.40898i) q^{3} +(-15.0016 - 5.46013i) q^{4} +(-10.5929 - 12.6241i) q^{5} +(-0.872050 + 1.45826i) q^{6} +(26.3432 - 9.58813i) q^{7} +(5.22609 + 3.01729i) q^{8} +(-60.4218 - 53.9464i) q^{9} +(1.55560 + 2.69438i) q^{10} +(131.619 - 156.858i) q^{11} +(-94.0342 + 108.634i) q^{12} +(46.4716 + 263.553i) q^{13} +(-5.21212 + 0.919037i) q^{14} +(-140.135 + 48.5815i) q^{15} +(194.798 + 163.455i) q^{16} +(-20.9707 + 12.1075i) q^{17} +(9.46523 + 12.0107i) q^{18} +(-63.5859 + 110.134i) q^{19} +(89.9811 + 247.221i) q^{20} +(3.87375 - 252.275i) q^{21} +(-29.6133 + 24.8485i) q^{22} +(242.681 - 666.761i) q^{23} +(42.1358 - 34.2676i) q^{24} +(61.3709 - 348.052i) q^{25} -50.5240i q^{26} +(-647.447 + 335.043i) q^{27} -447.542 q^{28} +(49.8475 + 8.78945i) q^{29} +(27.6468 - 4.43834i) q^{30} +(710.383 + 258.558i) q^{31} +(-92.9219 - 110.740i) q^{32} +(-896.822 - 1609.93i) q^{33} +(4.29585 - 1.56356i) q^{34} +(-400.093 - 230.994i) q^{35} +(611.868 + 1139.19i) q^{36} +(-409.965 - 710.079i) q^{37} +(15.4326 - 18.3918i) q^{38} +(2365.28 + 454.613i) q^{39} +(-17.2689 - 97.9368i) q^{40} +(466.045 - 82.1763i) q^{41} +(-8.99058 + 46.7766i) q^{42} +(-126.029 - 105.751i) q^{43} +(-2830.96 + 1634.46i) q^{44} +(-40.9844 + 1334.22i) q^{45} +(-66.9784 + 116.010i) q^{46} +(1375.61 + 3779.46i) q^{47} +(1999.33 - 1113.74i) q^{48} +(-1237.24 + 1038.17i) q^{49} +(-22.8205 + 62.6987i) q^{50} +(34.5442 + 215.179i) q^{51} +(741.889 - 4207.46i) q^{52} +959.535i q^{53} +(131.359 - 41.0667i) q^{54} -3374.42 q^{55} +(166.602 + 29.3764i) q^{56} +(722.152 + 887.966i) q^{57} +(-8.97962 - 3.26831i) q^{58} +(-2385.45 - 2842.87i) q^{59} +(2367.50 + 36.3537i) q^{60} +(4410.34 - 1605.23i) q^{61} +(-123.600 - 71.3604i) q^{62} +(-2108.95 - 841.787i) q^{63} +(-2020.68 - 3499.92i) q^{64} +(2834.87 - 3378.46i) q^{65} +(113.961 + 328.723i) q^{66} +(1242.32 + 7045.54i) q^{67} +(380.703 - 67.1281i) q^{68} +(-4828.34 - 4179.45i) q^{69} +(66.8136 + 56.0633i) q^{70} +(-3980.80 + 2298.32i) q^{71} +(-152.998 - 464.238i) q^{72} +(38.8472 - 67.2854i) q^{73} +(52.9430 + 145.460i) q^{74} +(-2729.90 - 1632.50i) q^{75} +(1555.24 - 1305.00i) q^{76} +(1963.30 - 5394.11i) q^{77} +(-424.855 - 162.064i) q^{78} +(-1149.56 + 6519.49i) q^{79} -4190.62i q^{80} +(740.576 + 6519.07i) q^{81} -89.3423 q^{82} +(6993.89 + 1233.21i) q^{83} +(-1435.57 + 3763.37i) q^{84} +(374.987 + 136.484i) q^{85} +(19.9649 + 23.7932i) q^{86} +(233.804 - 390.973i) q^{87} +(1161.14 - 422.620i) q^{88} +(8595.86 + 4962.82i) q^{89} +(51.3599 - 246.718i) q^{90} +(3751.19 + 6497.26i) q^{91} +(-7281.21 + 8677.40i) q^{92} +(4452.88 - 5144.22i) q^{93} +(-131.855 - 747.784i) q^{94} +(2063.91 - 363.923i) q^{95} +(-1229.27 + 426.161i) q^{96} +(391.006 + 328.093i) q^{97} +(264.066 - 152.458i) q^{98} +(-16414.6 + 2377.23i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.185922 − 0.0327832i −0.0464806 − 0.00819579i 0.150360 − 0.988631i \(-0.451957\pi\)
−0.196840 + 0.980436i \(0.563068\pi\)
\(3\) 3.20767 − 8.40898i 0.356407 − 0.934331i
\(4\) −15.0016 − 5.46013i −0.937599 − 0.341258i
\(5\) −10.5929 − 12.6241i −0.423717 − 0.504966i 0.511382 − 0.859354i \(-0.329134\pi\)
−0.935099 + 0.354388i \(0.884689\pi\)
\(6\) −0.872050 + 1.45826i −0.0242236 + 0.0405072i
\(7\) 26.3432 − 9.58813i 0.537616 − 0.195676i −0.0589198 − 0.998263i \(-0.518766\pi\)
0.596536 + 0.802587i \(0.296543\pi\)
\(8\) 5.22609 + 3.01729i 0.0816577 + 0.0471451i
\(9\) −60.4218 − 53.9464i −0.745948 − 0.666005i
\(10\) 1.55560 + 2.69438i 0.0155560 + 0.0269438i
\(11\) 131.619 − 156.858i 1.08776 − 1.29634i 0.135595 − 0.990764i \(-0.456705\pi\)
0.952166 − 0.305580i \(-0.0988503\pi\)
\(12\) −94.0342 + 108.634i −0.653015 + 0.754401i
\(13\) 46.4716 + 263.553i 0.274980 + 1.55949i 0.739026 + 0.673676i \(0.235286\pi\)
−0.464047 + 0.885811i \(0.653603\pi\)
\(14\) −5.21212 + 0.919037i −0.0265924 + 0.00468897i
\(15\) −140.135 + 48.5815i −0.622821 + 0.215918i
\(16\) 194.798 + 163.455i 0.760929 + 0.638495i
\(17\) −20.9707 + 12.1075i −0.0725631 + 0.0418943i −0.535843 − 0.844318i \(-0.680006\pi\)
0.463279 + 0.886212i \(0.346673\pi\)
\(18\) 9.46523 + 12.0107i 0.0292137 + 0.0370699i
\(19\) −63.5859 + 110.134i −0.176138 + 0.305080i −0.940555 − 0.339642i \(-0.889694\pi\)
0.764416 + 0.644723i \(0.223027\pi\)
\(20\) 89.9811 + 247.221i 0.224953 + 0.618052i
\(21\) 3.87375 − 252.275i 0.00878402 − 0.572052i
\(22\) −29.6133 + 24.8485i −0.0611844 + 0.0513398i
\(23\) 242.681 − 666.761i 0.458755 − 1.26042i −0.467659 − 0.883909i \(-0.654902\pi\)
0.926413 − 0.376509i \(-0.122875\pi\)
\(24\) 42.1358 − 34.2676i 0.0731525 − 0.0594924i
\(25\) 61.3709 − 348.052i 0.0981935 − 0.556883i
\(26\) − 50.5240i − 0.0747396i
\(27\) −647.447 + 335.043i −0.888130 + 0.459593i
\(28\) −447.542 −0.570844
\(29\) 49.8475 + 8.78945i 0.0592717 + 0.0104512i 0.203205 − 0.979136i \(-0.434864\pi\)
−0.143934 + 0.989587i \(0.545975\pi\)
\(30\) 27.6468 − 4.43834i 0.0307187 − 0.00493149i
\(31\) 710.383 + 258.558i 0.739212 + 0.269051i 0.684059 − 0.729427i \(-0.260213\pi\)
0.0551531 + 0.998478i \(0.482435\pi\)
\(32\) −92.9219 − 110.740i −0.0907440 − 0.108145i
\(33\) −896.822 − 1609.93i −0.823528 − 1.47836i
\(34\) 4.29585 − 1.56356i 0.00371613 − 0.00135256i
\(35\) −400.093 − 230.994i −0.326607 − 0.188566i
\(36\) 611.868 + 1139.19i 0.472120 + 0.879006i
\(37\) −409.965 − 710.079i −0.299463 − 0.518685i 0.676550 − 0.736396i \(-0.263474\pi\)
−0.976013 + 0.217712i \(0.930141\pi\)
\(38\) 15.4326 − 18.3918i 0.0106874 − 0.0127367i
\(39\) 2365.28 + 454.613i 1.55508 + 0.298891i
\(40\) −17.2689 − 97.9368i −0.0107931 − 0.0612105i
\(41\) 466.045 − 82.1763i 0.277243 − 0.0488854i −0.0332979 − 0.999445i \(-0.510601\pi\)
0.310541 + 0.950560i \(0.399490\pi\)
\(42\) −8.99058 + 46.7766i −0.00509670 + 0.0265173i
\(43\) −126.029 − 105.751i −0.0681609 − 0.0571938i 0.608071 − 0.793883i \(-0.291944\pi\)
−0.676232 + 0.736689i \(0.736388\pi\)
\(44\) −2830.96 + 1634.46i −1.46227 + 0.844244i
\(45\) −40.9844 + 1334.22i −0.0202392 + 0.658875i
\(46\) −66.9784 + 116.010i −0.0316533 + 0.0548251i
\(47\) 1375.61 + 3779.46i 0.622731 + 1.71094i 0.700202 + 0.713945i \(0.253093\pi\)
−0.0774712 + 0.996995i \(0.524685\pi\)
\(48\) 1999.33 − 1113.74i 0.867766 − 0.483395i
\(49\) −1237.24 + 1038.17i −0.515303 + 0.432390i
\(50\) −22.8205 + 62.6987i −0.00912819 + 0.0250795i
\(51\) 34.5442 + 215.179i 0.0132811 + 0.0827294i
\(52\) 741.889 − 4207.46i 0.274367 − 1.55601i
\(53\) 959.535i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(54\) 131.359 − 41.0667i 0.0450476 − 0.0140832i
\(55\) −3374.42 −1.11551
\(56\) 166.602 + 29.3764i 0.0531257 + 0.00936749i
\(57\) 722.152 + 887.966i 0.222269 + 0.273304i
\(58\) −8.97962 − 3.26831i −0.00266933 − 0.000971556i
\(59\) −2385.45 − 2842.87i −0.685278 − 0.816683i 0.305498 − 0.952193i \(-0.401177\pi\)
−0.990776 + 0.135510i \(0.956733\pi\)
\(60\) 2367.50 + 36.3537i 0.657640 + 0.0100983i
\(61\) 4410.34 − 1605.23i 1.18526 − 0.431398i 0.327201 − 0.944955i \(-0.393895\pi\)
0.858056 + 0.513557i \(0.171672\pi\)
\(62\) −123.600 − 71.3604i −0.0321539 − 0.0185641i
\(63\) −2108.95 − 841.787i −0.531355 − 0.212091i
\(64\) −2020.68 − 3499.92i −0.493330 − 0.854472i
\(65\) 2834.87 − 3378.46i 0.670974 − 0.799636i
\(66\) 113.961 + 328.723i 0.0261618 + 0.0754644i
\(67\) 1242.32 + 7045.54i 0.276747 + 1.56951i 0.733356 + 0.679845i \(0.237953\pi\)
−0.456608 + 0.889668i \(0.650936\pi\)
\(68\) 380.703 − 67.1281i 0.0823319 − 0.0145173i
\(69\) −4828.34 − 4179.45i −1.01414 − 0.877851i
\(70\) 66.8136 + 56.0633i 0.0136354 + 0.0114415i
\(71\) −3980.80 + 2298.32i −0.789685 + 0.455925i −0.839852 − 0.542816i \(-0.817358\pi\)
0.0501666 + 0.998741i \(0.484025\pi\)
\(72\) −152.998 − 464.238i −0.0295135 − 0.0895522i
\(73\) 38.8472 − 67.2854i 0.00728978 − 0.0126263i −0.862358 − 0.506300i \(-0.831013\pi\)
0.869647 + 0.493674i \(0.164346\pi\)
\(74\) 52.9430 + 145.460i 0.00966819 + 0.0265631i
\(75\) −2729.90 − 1632.50i −0.485316 − 0.290222i
\(76\) 1555.24 − 1305.00i 0.269258 − 0.225935i
\(77\) 1963.30 − 5394.11i 0.331134 − 0.909784i
\(78\) −424.855 − 162.064i −0.0698315 − 0.0266377i
\(79\) −1149.56 + 6519.49i −0.184195 + 1.04462i 0.742790 + 0.669525i \(0.233502\pi\)
−0.926985 + 0.375099i \(0.877609\pi\)
\(80\) − 4190.62i − 0.654784i
\(81\) 740.576 + 6519.07i 0.112876 + 0.993609i
\(82\) −89.3423 −0.0132871
\(83\) 6993.89 + 1233.21i 1.01523 + 0.179012i 0.656416 − 0.754399i \(-0.272072\pi\)
0.358810 + 0.933411i \(0.383183\pi\)
\(84\) −1435.57 + 3763.37i −0.203453 + 0.533358i
\(85\) 374.987 + 136.484i 0.0519014 + 0.0188906i
\(86\) 19.9649 + 23.7932i 0.00269941 + 0.00321703i
\(87\) 233.804 − 390.973i 0.0308897 − 0.0516544i
\(88\) 1161.14 − 422.620i 0.149940 − 0.0545738i
\(89\) 8595.86 + 4962.82i 1.08520 + 0.626540i 0.932294 − 0.361701i \(-0.117804\pi\)
0.152905 + 0.988241i \(0.451137\pi\)
\(90\) 51.3599 − 246.718i 0.00634073 − 0.0304591i
\(91\) 3751.19 + 6497.26i 0.452988 + 0.784598i
\(92\) −7281.21 + 8677.40i −0.860256 + 1.02521i
\(93\) 4452.88 − 5144.22i 0.514843 − 0.594777i
\(94\) −131.855 − 747.784i −0.0149224 − 0.0846293i
\(95\) 2063.91 − 363.923i 0.228688 − 0.0403238i
\(96\) −1229.27 + 426.161i −0.133385 + 0.0462414i
\(97\) 391.006 + 328.093i 0.0415566 + 0.0348701i 0.663329 − 0.748328i \(-0.269143\pi\)
−0.621773 + 0.783198i \(0.713587\pi\)
\(98\) 264.066 − 152.458i 0.0274954 − 0.0158745i
\(99\) −16414.6 + 2377.23i −1.67478 + 0.242550i
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.2.6 ✓ 66
3.2 odd 2 81.5.f.a.8.6 66
27.13 even 9 81.5.f.a.71.6 66
27.14 odd 18 inner 27.5.f.a.14.6 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.f.a.2.6 ✓ 66 1.1 even 1 trivial
27.5.f.a.14.6 yes 66 27.14 odd 18 inner
81.5.f.a.8.6 66 3.2 odd 2
81.5.f.a.71.6 66 27.13 even 9