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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,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, 9); 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, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 14.7
Character \(\chi\) \(=\) 27.14
Dual form 27.9.f.a.2.7

$q$-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+(-18.1654 + 3.20306i) q^{2} +(-0.948605 - 80.9944i) q^{3} +(79.1620 - 28.8126i) q^{4} +(462.775 - 551.514i) q^{5} +(276.662 + 1468.26i) q^{6} +(1642.82 + 597.939i) q^{7} +(2743.73 - 1584.09i) q^{8} +(-6559.20 + 153.664i) q^{9} +(-6639.98 + 11500.8i) q^{10} +(9016.87 + 10745.9i) q^{11} +(-2408.76 - 6384.35i) q^{12} +(9043.29 - 51287.0i) q^{13} +(-31757.8 - 5599.76i) q^{14} +(-45108.6 - 36959.1i) q^{15} +(-61287.6 + 51426.4i) q^{16} +(-49707.0 - 28698.4i) q^{17} +(118659. - 23800.8i) q^{18} +(-116884. - 202450. i) q^{19} +(20743.7 - 56992.7i) q^{20} +(46871.3 - 133627. i) q^{21} +(-198215. - 166322. i) q^{22} +(79587.0 + 218664. i) q^{23} +(-130905. - 220724. i) q^{24} +(-22175.6 - 125764. i) q^{25} +960617. i q^{26} +(18668.0 + 531113. i) q^{27} +147277. q^{28} +(147697. - 26043.0i) q^{29} +(937799. + 526892. i) q^{30} +(-1.38587e6 + 504414. i) q^{31} +(427258. - 509186. i) q^{32} +(861803. - 740510. i) q^{33} +(994872. + 362104. i) q^{34} +(1.09003e6 - 629329. i) q^{35} +(-514812. + 201152. i) q^{36} +(697113. - 1.20744e6i) q^{37} +(2.77171e6 + 3.30320e6i) q^{38} +(-4.16254e6 - 683805. i) q^{39} +(396080. - 2.24628e6i) q^{40} +(-1.41044e6 - 248698. i) q^{41} +(-423424. + 2.57752e6i) q^{42} +(-69449.3 + 58274.9i) q^{43} +(1.02341e6 + 590866. i) q^{44} +(-2.95069e6 + 3.68860e6i) q^{45} +(-2.14612e6 - 3.71720e6i) q^{46} +(2.27848e6 - 6.26006e6i) q^{47} +(4.22339e6 + 4.91517e6i) q^{48} +(-2.07476e6 - 1.74093e6i) q^{49} +(805657. + 2.21352e6i) q^{50} +(-2.27725e6 + 4.05321e6i) q^{51} +(-761828. - 4.32054e6i) q^{52} +396154. i q^{53} +(-2.04030e6 - 9.58810e6i) q^{54} +1.00993e7 q^{55} +(5.45465e6 - 961801. i) q^{56} +(-1.62864e7 + 9.65903e6i) q^{57} +(-2.59957e6 + 946165. i) q^{58} +(-3.41825e6 + 4.07371e6i) q^{59} +(-4.63577e6 - 1.62606e6i) q^{60} +(1.58012e6 + 575117. i) q^{61} +(2.35592e7 - 1.36019e7i) q^{62} +(-1.08675e7 - 3.66956e6i) q^{63} +(4.11030e6 - 7.11925e6i) q^{64} +(-2.41005e7 - 2.87219e7i) q^{65} +(-1.32831e7 + 1.62121e7i) q^{66} +(1.88303e6 - 1.06792e7i) q^{67} +(-4.76178e6 - 839630. i) q^{68} +(1.76350e7 - 6.65353e6i) q^{69} +(-1.77851e7 + 1.49235e7i) q^{70} +(3.79332e7 + 2.19007e7i) q^{71} +(-1.77532e7 + 1.08120e7i) q^{72} +(-9.39455e6 - 1.62718e7i) q^{73} +(-8.79588e6 + 2.41665e7i) q^{74} +(-1.01651e7 + 1.91540e6i) q^{75} +(-1.50859e7 - 1.26586e7i) q^{76} +(8.38774e6 + 2.30451e7i) q^{77} +(7.78046e7 - 911246. i) q^{78} +(171845. + 974580. i) q^{79} +5.75999e7i q^{80} +(4.29995e7 - 2.01582e6i) q^{81} +2.64178e7 q^{82} +(2.50236e7 - 4.41234e6i) q^{83} +(-139708. - 1.19286e7i) q^{84} +(-3.88307e7 + 1.41332e7i) q^{85} +(1.07492e6 - 1.28104e6i) q^{86} +(-2.24944e6 - 1.19379e7i) q^{87} +(4.17623e7 + 1.52002e7i) q^{88} +(-3.33894e7 + 1.92774e7i) q^{89} +(4.17857e7 - 7.64563e7i) q^{90} +(4.55230e7 - 7.88482e7i) q^{91} +(1.26005e7 + 1.50167e7i) q^{92} +(4.21694e7 + 1.11769e8i) q^{93} +(-2.13382e7 + 1.21015e8i) q^{94} +(-1.65745e8 - 2.92253e7i) q^{95} +(-4.16466e7 - 3.41225e7i) q^{96} +(-1.11208e8 + 9.33146e7i) q^{97} +(4.32651e7 + 2.49791e7i) q^{98} +(-6.07947e7 - 6.90988e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −18.1654 + 3.20306i −1.13534 + 0.200191i −0.709566 − 0.704639i \(-0.751109\pi\)
−0.425773 + 0.904830i \(0.639998\pi\)
\(3\) −0.948605 − 80.9944i −0.0117112 − 0.999931i
\(4\) 79.1620 − 28.8126i 0.309227 − 0.112549i
\(5\) 462.775 − 551.514i 0.740441 − 0.882423i −0.256004 − 0.966676i \(-0.582406\pi\)
0.996444 + 0.0842531i \(0.0268504\pi\)
\(6\) 276.662 + 1468.26i 0.213473 + 1.13292i
\(7\) 1642.82 + 597.939i 0.684225 + 0.249037i 0.660660 − 0.750685i \(-0.270276\pi\)
0.0235642 + 0.999722i \(0.492499\pi\)
\(8\) 2743.73 − 1584.09i 0.669855 − 0.386741i
\(9\) −6559.20 + 153.664i −0.999726 + 0.0234207i
\(10\) −6639.98 + 11500.8i −0.663998 + 1.15008i
\(11\) 9016.87 + 10745.9i 0.615864 + 0.733958i 0.980353 − 0.197249i \(-0.0632009\pi\)
−0.364489 + 0.931208i \(0.618756\pi\)
\(12\) −2408.76 − 6384.35i −0.116163 − 0.307887i
\(13\) 9043.29 − 51287.0i 0.316631 − 1.79570i −0.246296 − 0.969195i \(-0.579214\pi\)
0.562927 − 0.826507i \(-0.309675\pi\)
\(14\) −31757.8 − 5599.76i −0.826682 − 0.145766i
\(15\) −45108.6 − 36959.1i −0.891034 − 0.730056i
\(16\) −61287.6 + 51426.4i −0.935175 + 0.784705i
\(17\) −49707.0 − 28698.4i −0.595144 − 0.343606i 0.171985 − 0.985100i \(-0.444982\pi\)
−0.767129 + 0.641493i \(0.778315\pi\)
\(18\) 118659. − 23800.8i 1.13034 − 0.226727i
\(19\) −116884. − 202450.i −0.896896 − 1.55347i −0.831441 − 0.555613i \(-0.812483\pi\)
−0.0654550 − 0.997856i \(-0.520850\pi\)
\(20\) 20743.7 − 56992.7i 0.129648 − 0.356205i
\(21\) 46871.3 − 133627.i 0.241007 − 0.687094i
\(22\) −198215. − 166322.i −0.846147 − 0.710001i
\(23\) 79587.0 + 218664.i 0.284401 + 0.781385i 0.996824 + 0.0796349i \(0.0253754\pi\)
−0.712423 + 0.701750i \(0.752402\pi\)
\(24\) −130905. − 220724.i −0.394559 − 0.665280i
\(25\) −22175.6 − 125764.i −0.0567694 − 0.321955i
\(26\) 960617.i 2.10212i
\(27\) 18668.0 + 531113.i 0.0351271 + 0.999383i
\(28\) 147277. 0.239609
\(29\) 147697. − 26043.0i 0.208824 − 0.0368213i −0.0682575 − 0.997668i \(-0.521744\pi\)
0.277081 + 0.960846i \(0.410633\pi\)
\(30\) 937799. + 526892.i 1.15778 + 0.650484i
\(31\) −1.38587e6 + 504414.i −1.50063 + 0.546186i −0.956225 − 0.292633i \(-0.905468\pi\)
−0.544409 + 0.838820i \(0.683246\pi\)
\(32\) 427258. − 509186.i 0.407465 − 0.485598i
\(33\) 861803. − 740510.i 0.726695 − 0.624417i
\(34\) 994872. + 362104.i 0.744477 + 0.270968i
\(35\) 1.09003e6 − 629329.i 0.726384 − 0.419378i
\(36\) −514812. + 201152.i −0.306506 + 0.119761i
\(37\) 697113. − 1.20744e6i 0.371960 − 0.644254i −0.617907 − 0.786251i \(-0.712019\pi\)
0.989867 + 0.141998i \(0.0453525\pi\)
\(38\) 2.77171e6 + 3.30320e6i 1.32927 + 1.58416i
\(39\) −4.16254e6 − 683805.i −1.79929 − 0.295579i
\(40\) 396080. − 2.24628e6i 0.154719 − 0.877454i
\(41\) −1.41044e6 − 248698.i −0.499136 − 0.0880111i −0.0815882 − 0.996666i \(-0.525999\pi\)
−0.417548 + 0.908655i \(0.637110\pi\)
\(42\) −423424. + 2.57752e6i −0.136075 + 0.828333i
\(43\) −69449.3 + 58274.9i −0.0203139 + 0.0170454i −0.652888 − 0.757454i \(-0.726443\pi\)
0.632574 + 0.774500i \(0.281998\pi\)
\(44\) 1.02341e6 + 590866.i 0.273048 + 0.157644i
\(45\) −2.95069e6 + 3.68860e6i −0.719570 + 0.899522i
\(46\) −2.14612e6 − 3.71720e6i −0.479318 − 0.830203i
\(47\) 2.27848e6 − 6.26006e6i 0.466931 − 1.28288i −0.453248 − 0.891385i \(-0.649735\pi\)
0.920179 − 0.391498i \(-0.128043\pi\)
\(48\) 4.22339e6 + 4.91517e6i 0.795603 + 0.925921i
\(49\) −2.07476e6 − 1.74093e6i −0.359901 − 0.301993i
\(50\) 805657. + 2.21352e6i 0.128905 + 0.354164i
\(51\) −2.27725e6 + 4.05321e6i −0.336613 + 0.599127i
\(52\) −761828. − 4.32054e6i −0.104194 − 0.590915i
\(53\) 396154.i 0.0502065i 0.999685 + 0.0251033i \(0.00799146\pi\)
−0.999685 + 0.0251033i \(0.992009\pi\)
\(54\) −2.04030e6 − 9.58810e6i −0.239949 − 1.12761i
\(55\) 1.00993e7 1.10367
\(56\) 5.45465e6 − 961801.i 0.554644 − 0.0977987i
\(57\) −1.62864e7 + 9.65903e6i −1.54286 + 0.915027i
\(58\) −2.59957e6 + 946165.i −0.229715 + 0.0836093i
\(59\) −3.41825e6 + 4.07371e6i −0.282095 + 0.336188i −0.888422 − 0.459028i \(-0.848198\pi\)
0.606327 + 0.795216i \(0.292642\pi\)
\(60\) −4.63577e6 − 1.62606e6i −0.357698 − 0.125467i
\(61\) 1.58012e6 + 575117.i 0.114123 + 0.0415372i 0.398450 − 0.917190i \(-0.369548\pi\)
−0.284328 + 0.958727i \(0.591770\pi\)
\(62\) 2.35592e7 − 1.36019e7i 1.59439 − 0.920520i
\(63\) −1.08675e7 − 3.66956e6i −0.689869 − 0.232944i
\(64\) 4.11030e6 − 7.11925e6i 0.244993 − 0.424340i
\(65\) −2.41005e7 − 2.87219e7i −1.35012 − 1.60901i
\(66\) −1.32831e7 + 1.62121e7i −0.700043 + 0.854404i
\(67\) 1.88303e6 − 1.06792e7i 0.0934456 − 0.529956i −0.901767 − 0.432222i \(-0.857730\pi\)
0.995213 − 0.0977338i \(-0.0311594\pi\)
\(68\) −4.76178e6 − 839630.i −0.222707 − 0.0392692i
\(69\) 1.76350e7 − 6.65353e6i 0.778001 − 0.293532i
\(70\) −1.77851e7 + 1.49235e7i −0.740737 + 0.621552i
\(71\) 3.79332e7 + 2.19007e7i 1.49275 + 0.861837i 0.999965 − 0.00831631i \(-0.00264719\pi\)
0.492781 + 0.870154i \(0.335981\pi\)
\(72\) −1.77532e7 + 1.08120e7i −0.660613 + 0.402323i
\(73\) −9.39455e6 − 1.62718e7i −0.330815 − 0.572988i 0.651857 − 0.758342i \(-0.273990\pi\)
−0.982672 + 0.185354i \(0.940657\pi\)
\(74\) −8.79588e6 + 2.41665e7i −0.293327 + 0.805910i
\(75\) −1.01651e7 + 1.91540e6i −0.321268 + 0.0605360i
\(76\) −1.50859e7 − 1.26586e7i −0.452186 − 0.379429i
\(77\) 8.38774e6 + 2.30451e7i 0.238606 + 0.655565i
\(78\) 7.78046e7 − 911246.i 2.10197 − 0.0246183i
\(79\) 171845. + 974580.i 0.00441192 + 0.0250213i 0.986934 − 0.161124i \(-0.0515118\pi\)
−0.982522 + 0.186145i \(0.940401\pi\)
\(80\) 5.75999e7i 1.40625i
\(81\) 4.29995e7 − 2.01582e6i 0.998903 − 0.0468286i
\(82\) 2.64178e7 0.584308
\(83\) 2.50236e7 − 4.41234e6i 0.527276 − 0.0929730i 0.0963279 − 0.995350i \(-0.469290\pi\)
0.430948 + 0.902377i \(0.358179\pi\)
\(84\) −139708. − 1.19286e7i −0.00280611 − 0.239593i
\(85\) −3.88307e7 + 1.41332e7i −0.743875 + 0.270748i
\(86\) 1.07492e6 − 1.28104e6i 0.0196509 − 0.0234190i
\(87\) −2.24944e6 − 1.19379e7i −0.0392643 − 0.208378i
\(88\) 4.17623e7 + 1.52002e7i 0.696391 + 0.253466i
\(89\) −3.33894e7 + 1.92774e7i −0.532168 + 0.307247i −0.741899 − 0.670512i \(-0.766075\pi\)
0.209731 + 0.977759i \(0.432741\pi\)
\(90\) 4.17857e7 − 7.64563e7i 0.636880 − 1.16531i
\(91\) 4.55230e7 − 7.88482e7i 0.663843 − 1.14981i
\(92\) 1.26005e7 + 1.50167e7i 0.175889 + 0.209616i
\(93\) 4.21694e7 + 1.11769e8i 0.563723 + 1.49413i
\(94\) −2.13382e7 + 1.21015e8i −0.273304 + 1.54998i
\(95\) −1.65745e8 − 2.92253e7i −2.03491 − 0.358810i
\(96\) −4.16466e7 − 3.41225e7i −0.490337 − 0.401750i
\(97\) −1.11208e8 + 9.33146e7i −1.25617 + 1.05405i −0.260093 + 0.965584i \(0.583753\pi\)
−0.996079 + 0.0884697i \(0.971802\pi\)
\(98\) 4.32651e7 + 2.49791e7i 0.469066 + 0.270815i
\(99\) −6.07947e7 − 6.90988e7i −0.632885 − 0.719333i
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 27.9.f.a.14.7 yes 138
3.2 odd 2 81.9.f.a.71.17 138
27.2 odd 18 inner 27.9.f.a.2.7 ✓ 138
27.25 even 9 81.9.f.a.8.17 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.7 ✓ 138 27.2 odd 18 inner
27.9.f.a.14.7 yes 138 1.1 even 1 trivial
81.9.f.a.8.17 138 27.25 even 9
81.9.f.a.71.17 138 3.2 odd 2