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Properties

Label 27.10.e.a.4.2
Level $27$
Weight $10$
Character 27.4
Analytic conductor $13.906$
Analytic rank $0$
Dimension $156$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,10,Mod(4,27)] 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("27.4"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), 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: \(13.9059675764\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(26\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 4.2
Character \(\chi\) \(=\) 27.4
Dual form 27.10.e.a.7.2

$q$-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+(-39.3878 - 14.3360i) q^{2} +(29.9043 - 137.072i) q^{3} +(953.662 + 800.217i) q^{4} +(27.4469 - 155.659i) q^{5} +(-3142.92 + 4970.25i) q^{6} +(1724.19 - 1446.77i) q^{7} +(-15360.3 - 26604.9i) q^{8} +(-17894.5 - 8198.08i) q^{9} +(-3312.60 + 5737.59i) q^{10} +(-11576.5 - 65653.8i) q^{11} +(138206. - 106790. i) q^{12} +(-88137.2 + 32079.3i) q^{13} +(-88652.9 + 32267.0i) q^{14} +(-20515.7 - 8417.08i) q^{15} +(112919. + 640397. i) q^{16} +(-104455. + 180921. i) q^{17} +(587296. + 579439. i) q^{18} +(149380. + 258734. i) q^{19} +(150736. - 126483. i) q^{20} +(-146751. - 279603. i) q^{21} +(-485238. + 2.75192e6i) q^{22} +(-1.91489e6 - 1.60678e6i) q^{23} +(-4.10612e6 + 1.30987e6i) q^{24} +(1.81186e6 + 659463. i) q^{25} +3.93142e6 q^{26} +(-1.65885e6 + 2.20767e6i) q^{27} +2.80202e6 q^{28} +(6.30782e6 + 2.29586e6i) q^{29} +(687402. + 625643. i) q^{30} +(-3.32722e6 - 2.79187e6i) q^{31} +(2.00177e6 - 1.13526e7i) q^{32} +(-9.34549e6 - 376511. i) q^{33} +(6.70793e6 - 5.62862e6i) q^{34} +(-177879. - 308096. i) q^{35} +(-1.05050e7 - 2.21377e7i) q^{36} +(-7.49482e6 + 1.29814e7i) q^{37} +(-2.17455e6 - 1.23325e7i) q^{38} +(1.76150e6 + 1.30405e7i) q^{39} +(-4.56288e6 + 1.66075e6i) q^{40} +(-7.26229e6 + 2.64326e6i) q^{41} +(1.77180e6 + 1.31168e7i) q^{42} +(-1.61156e6 - 9.13961e6i) q^{43} +(4.14972e7 - 7.18753e7i) q^{44} +(-1.76725e6 + 2.56043e6i) q^{45} +(5.23883e7 + 9.07392e7i) q^{46} +(-4.85098e7 + 4.07046e7i) q^{47} +(9.11573e7 + 3.67254e6i) q^{48} +(-6.12763e6 + 3.47515e7i) q^{49} +(-6.19111e7 - 5.19496e7i) q^{50} +(2.16756e7 + 1.97282e7i) q^{51} +(-1.09724e8 - 3.99361e7i) q^{52} +3.39638e7 q^{53} +(9.69875e7 - 6.31741e7i) q^{54} -1.05374e7 q^{55} +(-6.49752e7 - 2.36490e7i) q^{56} +(3.99324e7 - 1.27386e7i) q^{57} +(-2.15537e8 - 1.80857e8i) q^{58} +(-7.66457e6 + 4.34679e7i) q^{59} +(-1.28296e7 - 2.44441e7i) q^{60} +(3.48913e7 - 2.92773e7i) q^{61} +(9.10276e7 + 1.57664e8i) q^{62} +(-4.27142e7 + 1.17541e7i) q^{63} +(-7.51254e7 + 1.30121e8i) q^{64} +(2.57435e6 + 1.45999e7i) q^{65} +(3.62700e8 + 1.48807e8i) q^{66} +(1.48410e8 - 5.40168e7i) q^{67} +(-2.44391e8 + 8.89510e7i) q^{68} +(-2.77508e8 + 2.14428e8i) q^{69} +(2.58941e6 + 1.46853e7i) q^{70} +(-6.24349e7 + 1.08140e8i) q^{71} +(5.67561e7 + 6.02005e8i) q^{72} +(-8.15654e7 - 1.41275e8i) q^{73} +(4.81305e8 - 4.03863e8i) q^{74} +(1.44576e8 - 2.28635e8i) q^{75} +(-6.45854e7 + 3.66282e8i) q^{76} +(-1.14946e8 - 9.64512e7i) q^{77} +(1.17566e8 - 5.38887e8i) q^{78} +(-3.10563e8 - 1.13036e8i) q^{79} +1.02783e8 q^{80} +(2.53004e8 + 2.93401e8i) q^{81} +3.23939e8 q^{82} +(-4.43080e8 - 1.61268e8i) q^{83} +(8.37925e7 - 3.84079e8i) q^{84} +(2.52951e7 + 2.12251e7i) q^{85} +(-6.75495e7 + 3.83092e8i) q^{86} +(5.03328e8 - 7.95969e8i) q^{87} +(-1.56889e9 + 1.31646e9i) q^{88} +(5.75741e7 + 9.97212e7i) q^{89} +(1.06314e8 - 7.55142e7i) q^{90} +(-1.05554e8 + 1.82825e8i) q^{91} +(-5.40380e8 - 3.06465e9i) q^{92} +(-4.82185e8 + 3.72580e8i) q^{93} +(2.49423e9 - 9.07827e8i) q^{94} +(4.43744e7 - 1.61510e7i) q^{95} +(-1.49626e9 - 6.13879e8i) q^{96} +(986733. + 5.59604e6i) q^{97} +(7.39551e8 - 1.28094e9i) q^{98} +(-3.31079e8 + 1.26975e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9} - 3 q^{10} - 121767 q^{11} + 319935 q^{12} - 6 q^{13} - 720417 q^{14} - 706356 q^{15} + 1530 q^{16} + 1002249 q^{17}+ \cdots + 5248107342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −39.3878 − 14.3360i −1.74071 − 0.633567i −0.741413 − 0.671049i \(-0.765844\pi\)
−0.999297 + 0.0374827i \(0.988066\pi\)
\(3\) 29.9043 − 137.072i 0.213151 − 0.977019i
\(4\) 953.662 + 800.217i 1.86262 + 1.56292i
\(5\) 27.4469 − 155.659i 0.0196394 − 0.111381i −0.973412 − 0.229061i \(-0.926435\pi\)
0.993052 + 0.117680i \(0.0375457\pi\)
\(6\) −3142.92 + 4970.25i −0.990041 + 1.56566i
\(7\) 1724.19 − 1446.77i 0.271422 − 0.227750i −0.496909 − 0.867802i \(-0.665532\pi\)
0.768331 + 0.640053i \(0.221087\pi\)
\(8\) −15360.3 − 26604.9i −1.32585 − 2.29644i
\(9\) −17894.5 − 8198.08i −0.909133 − 0.416506i
\(10\) −3312.60 + 5737.59i −0.104754 + 0.181439i
\(11\) −11576.5 − 65653.8i −0.238403 − 1.35205i −0.835327 − 0.549753i \(-0.814722\pi\)
0.596924 − 0.802298i \(-0.296389\pi\)
\(12\) 138206. − 106790.i 1.92403 − 1.48668i
\(13\) −88137.2 + 32079.3i −0.855883 + 0.311516i −0.732436 − 0.680835i \(-0.761617\pi\)
−0.123446 + 0.992351i \(0.539395\pi\)
\(14\) −88652.9 + 32267.0i −0.616761 + 0.224483i
\(15\) −20515.7 − 8417.08i −0.104635 − 0.0429290i
\(16\) 112919. + 640397.i 0.430753 + 2.44292i
\(17\) −104455. + 180921.i −0.303325 + 0.525375i −0.976887 − 0.213756i \(-0.931430\pi\)
0.673562 + 0.739131i \(0.264764\pi\)
\(18\) 587296. + 579439.i 1.31865 + 1.30101i
\(19\) 149380. + 258734.i 0.262968 + 0.455473i 0.967029 − 0.254666i \(-0.0819654\pi\)
−0.704061 + 0.710139i \(0.748632\pi\)
\(20\) 150736. − 126483.i 0.210660 − 0.176765i
\(21\) −146751. − 279603.i −0.164662 − 0.313729i
\(22\) −485238. + 2.75192e6i −0.441624 + 2.50457i
\(23\) −1.91489e6 − 1.60678e6i −1.42681 − 1.19724i −0.947566 − 0.319561i \(-0.896465\pi\)
−0.479249 − 0.877679i \(-0.659091\pi\)
\(24\) −4.10612e6 + 1.30987e6i −2.52628 + 0.805894i
\(25\) 1.81186e6 + 659463.i 0.927673 + 0.337645i
\(26\) 3.93142e6 1.68721
\(27\) −1.65885e6 + 2.20767e6i −0.600717 + 0.799462i
\(28\) 2.80202e6 0.861511
\(29\) 6.30782e6 + 2.29586e6i 1.65611 + 0.602773i 0.989744 − 0.142855i \(-0.0456283\pi\)
0.666362 + 0.745628i \(0.267851\pi\)
\(30\) 687402. + 625643.i 0.154941 + 0.141020i
\(31\) −3.32722e6 − 2.79187e6i −0.647074 − 0.542959i 0.259108 − 0.965848i \(-0.416572\pi\)
−0.906181 + 0.422889i \(0.861016\pi\)
\(32\) 2.00177e6 − 1.13526e7i 0.337474 − 1.91391i
\(33\) −9.34549e6 − 376511.i −1.37180 − 0.0552668i
\(34\) 6.70793e6 − 5.62862e6i 0.860862 − 0.722349i
\(35\) −177879. − 308096.i −0.0200364 − 0.0347040i
\(36\) −1.05050e7 − 2.21377e7i −1.04240 − 2.19670i
\(37\) −7.49482e6 + 1.29814e7i −0.657436 + 1.13871i 0.323841 + 0.946111i \(0.395026\pi\)
−0.981277 + 0.192601i \(0.938308\pi\)
\(38\) −2.17455e6 − 1.23325e7i −0.169178 − 0.959455i
\(39\) 1.76150e6 + 1.30405e7i 0.121925 + 0.902614i
\(40\) −4.56288e6 + 1.66075e6i −0.281818 + 0.102574i
\(41\) −7.26229e6 + 2.64326e6i −0.401371 + 0.146087i −0.534815 − 0.844969i \(-0.679619\pi\)
0.133444 + 0.991056i \(0.457396\pi\)
\(42\) 1.77180e6 + 1.31168e7i 0.0878605 + 0.650436i
\(43\) −1.61156e6 − 9.13961e6i −0.0718850 − 0.407680i −0.999423 − 0.0339524i \(-0.989191\pi\)
0.927538 − 0.373728i \(-0.121921\pi\)
\(44\) 4.14972e7 − 7.18753e7i 1.66910 − 2.89096i
\(45\) −1.76725e6 + 2.56043e6i −0.0642455 + 0.0930799i
\(46\) 5.23883e7 + 9.07392e7i 1.72514 + 2.98803i
\(47\) −4.85098e7 + 4.07046e7i −1.45007 + 1.21675i −0.517542 + 0.855658i \(0.673153\pi\)
−0.932529 + 0.361096i \(0.882403\pi\)
\(48\) 9.11573e7 + 3.67254e6i 2.47860 + 0.0998575i
\(49\) −6.12763e6 + 3.47515e7i −0.151848 + 0.861175i
\(50\) −6.19111e7 − 5.19496e7i −1.40089 − 1.17548i
\(51\) 2.16756e7 + 1.97282e7i 0.448647 + 0.408339i
\(52\) −1.09724e8 − 3.99361e7i −2.08106 − 0.757444i
\(53\) 3.39638e7 0.591256 0.295628 − 0.955303i \(-0.404471\pi\)
0.295628 + 0.955303i \(0.404471\pi\)
\(54\) 9.69875e7 − 6.31741e7i 1.55219 − 1.01104i
\(55\) −1.05374e7 −0.155274
\(56\) −6.49752e7 − 2.36490e7i −0.882880 − 0.321342i
\(57\) 3.99324e7 − 1.27386e7i 0.501058 − 0.159840i
\(58\) −2.15537e8 − 1.80857e8i −2.50090 − 2.09851i
\(59\) −7.66457e6 + 4.34679e7i −0.0823481 + 0.467019i 0.915549 + 0.402206i \(0.131756\pi\)
−0.997897 + 0.0648136i \(0.979355\pi\)
\(60\) −1.28296e7 − 2.44441e7i −0.127800 − 0.243497i
\(61\) 3.48913e7 − 2.92773e7i 0.322651 − 0.270737i −0.467046 − 0.884233i \(-0.654682\pi\)
0.789698 + 0.613496i \(0.210237\pi\)
\(62\) 9.10276e7 + 1.57664e8i 0.782367 + 1.35510i
\(63\) −4.27142e7 + 1.17541e7i −0.341617 + 0.0940063i
\(64\) −7.51254e7 + 1.30121e8i −0.559728 + 0.969478i
\(65\) 2.57435e6 + 1.45999e7i 0.0178878 + 0.101447i
\(66\) 3.62700e8 + 1.48807e8i 2.35288 + 0.965327i
\(67\) 1.48410e8 − 5.40168e7i 0.899760 − 0.327486i 0.149603 − 0.988746i \(-0.452200\pi\)
0.750157 + 0.661260i \(0.229978\pi\)
\(68\) −2.44391e8 + 8.89510e7i −1.38610 + 0.504500i
\(69\) −2.77508e8 + 2.14428e8i −1.47385 + 1.13883i
\(70\) 2.58941e6 + 1.46853e7i 0.0128902 + 0.0731040i
\(71\) −6.24349e7 + 1.08140e8i −0.291585 + 0.505039i −0.974185 − 0.225753i \(-0.927516\pi\)
0.682600 + 0.730792i \(0.260849\pi\)
\(72\) 5.67561e7 + 6.02005e8i 0.248895 + 2.64000i
\(73\) −8.15654e7 − 1.41275e8i −0.336166 − 0.582256i 0.647542 − 0.762029i \(-0.275797\pi\)
−0.983708 + 0.179773i \(0.942464\pi\)
\(74\) 4.81305e8 − 4.03863e8i 1.86586 − 1.56564i
\(75\) 1.44576e8 − 2.28635e8i 0.527620 − 0.834385i
\(76\) −6.45854e7 + 3.66282e8i −0.222061 + 1.25937i
\(77\) −1.14946e8 − 9.64512e7i −0.372637 − 0.312680i
\(78\) 1.17566e8 − 5.38887e8i 0.359631 − 1.64844i
\(79\) −3.10563e8 − 1.13036e8i −0.897072 − 0.326508i −0.147993 − 0.988988i \(-0.547281\pi\)
−0.749079 + 0.662481i \(0.769504\pi\)
\(80\) 1.02783e8 0.280554
\(81\) 2.53004e8 + 2.93401e8i 0.653046 + 0.757318i
\(82\) 3.23939e8 0.791227
\(83\) −4.43080e8 − 1.61268e8i −1.02478 − 0.372989i −0.225689 − 0.974199i \(-0.572463\pi\)
−0.799091 + 0.601210i \(0.794685\pi\)
\(84\) 8.37925e7 − 3.84079e8i 0.183632 − 0.841713i
\(85\) 2.52951e7 + 2.12251e7i 0.0525595 + 0.0441026i
\(86\) −6.75495e7 + 3.83092e8i −0.133162 + 0.755197i
\(87\) 5.03328e8 − 7.95969e8i 0.941922 − 1.48957i
\(88\) −1.56889e9 + 1.31646e9i −2.78882 + 2.34010i
\(89\) 5.75741e7 + 9.97212e7i 0.0972684 + 0.168474i 0.910553 − 0.413392i \(-0.135656\pi\)
−0.813285 + 0.581866i \(0.802323\pi\)
\(90\) 1.06314e8 − 7.55142e7i 0.170805 − 0.121321i
\(91\) −1.05554e8 + 1.82825e8i −0.161357 + 0.279479i
\(92\) −5.40380e8 − 3.06465e9i −0.786419 − 4.46001i
\(93\) −4.82185e8 + 3.72580e8i −0.668406 + 0.516471i
\(94\) 2.49423e9 − 9.07827e8i 3.29505 − 1.19930i
\(95\) 4.43744e7 − 1.61510e7i 0.0558955 − 0.0203443i
\(96\) −1.49626e9 − 6.13879e8i −1.79799 − 0.737670i
\(97\) 986733. + 5.59604e6i 0.00113169 + 0.00641812i 0.985368 − 0.170438i \(-0.0545183\pi\)
−0.984237 + 0.176856i \(0.943407\pi\)
\(98\) 7.39551e8 − 1.28094e9i 0.809936 − 1.40285i
\(99\) −3.31079e8 + 1.26975e9i −0.346397 + 1.32849i
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 27.10.e.a.4.2 ✓ 156
3.2 odd 2 81.10.e.a.64.25 156
27.7 even 9 inner 27.10.e.a.7.2 yes 156
27.20 odd 18 81.10.e.a.19.25 156
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.10.e.a.4.2 ✓ 156 1.1 even 1 trivial
27.10.e.a.7.2 yes 156 27.7 even 9 inner
81.10.e.a.19.25 156 27.20 odd 18
81.10.e.a.64.25 156 3.2 odd 2