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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(8,81)] 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("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), not 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: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 71.17
Character \(\chi\) \(=\) 81.71
Dual form 81.9.f.a.8.17

$q$-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} +(79.1620 - 28.8126i) q^{4} +(-462.775 + 551.514i) q^{5} +(1642.82 + 597.939i) q^{7} +(-2743.73 + 1584.09i) q^{8} +(-6639.98 + 11500.8i) q^{10} +(-9016.87 - 10745.9i) q^{11} +(9043.29 - 51287.0i) q^{13} +(31757.8 + 5599.76i) q^{14} +(-61287.6 + 51426.4i) q^{16} +(49707.0 + 28698.4i) q^{17} +(-116884. - 202450. i) q^{19} +(-20743.7 + 56992.7i) q^{20} +(-198215. - 166322. i) q^{22} +(-79587.0 - 218664. i) q^{23} +(-22175.6 - 125764. i) q^{25} -960617. i q^{26} +147277. q^{28} +(-147697. + 26043.0i) q^{29} +(-1.38587e6 + 504414. i) q^{31} +(-427258. + 509186. i) q^{32} +(994872. + 362104. i) q^{34} +(-1.09003e6 + 629329. i) q^{35} +(697113. - 1.20744e6i) q^{37} +(-2.77171e6 - 3.30320e6i) q^{38} +(396080. - 2.24628e6i) q^{40} +(1.41044e6 + 248698. i) q^{41} +(-69449.3 + 58274.9i) q^{43} +(-1.02341e6 - 590866. i) q^{44} +(-2.14612e6 - 3.71720e6i) q^{46} +(-2.27848e6 + 6.26006e6i) q^{47} +(-2.07476e6 - 1.74093e6i) q^{49} +(-805657. - 2.21352e6i) q^{50} +(-761828. - 4.32054e6i) q^{52} -396154. i q^{53} +1.00993e7 q^{55} +(-5.45465e6 + 961801. i) q^{56} +(-2.59957e6 + 946165. i) q^{58} +(3.41825e6 - 4.07371e6i) q^{59} +(1.58012e6 + 575117. i) q^{61} +(-2.35592e7 + 1.36019e7i) q^{62} +(4.11030e6 - 7.11925e6i) q^{64} +(2.41005e7 + 2.87219e7i) q^{65} +(1.88303e6 - 1.06792e7i) q^{67} +(4.76178e6 + 839630. i) q^{68} +(-1.77851e7 + 1.49235e7i) q^{70} +(-3.79332e7 - 2.19007e7i) q^{71} +(-9.39455e6 - 1.62718e7i) q^{73} +(8.79588e6 - 2.41665e7i) q^{74} +(-1.50859e7 - 1.26586e7i) q^{76} +(-8.38774e6 - 2.30451e7i) q^{77} +(171845. + 974580. i) q^{79} -5.75999e7i q^{80} +2.64178e7 q^{82} +(-2.50236e7 + 4.41234e6i) q^{83} +(-3.88307e7 + 1.41332e7i) q^{85} +(-1.07492e6 + 1.28104e6i) q^{86} +(4.17623e7 + 1.52002e7i) q^{88} +(3.33894e7 - 1.92774e7i) q^{89} +(4.55230e7 - 7.88482e7i) q^{91} +(-1.26005e7 - 1.50167e7i) q^{92} +(-2.13382e7 + 1.21015e8i) q^{94} +(1.65745e8 + 2.92253e7i) q^{95} +(-1.11208e8 + 9.33146e7i) q^{97} +(-4.32651e7 - 2.49791e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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 0.425773 − 0.904830i \(-0.360002\pi\)
0.709566 + 0.704639i \(0.248891\pi\)
\(3\) 0 0
\(4\) 79.1620 − 28.8126i 0.309227 − 0.112549i
\(5\) −462.775 + 551.514i −0.740441 + 0.882423i −0.996444 − 0.0842531i \(-0.973150\pi\)
0.256004 + 0.966676i \(0.417594\pi\)
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) −6639.98 + 11500.8i −0.663998 + 1.15008i
\(11\) −9016.87 − 10745.9i −0.615864 − 0.733958i 0.364489 − 0.931208i \(-0.381244\pi\)
−0.980353 + 0.197249i \(0.936799\pi\)
\(12\) 0 0
\(13\) 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
\(15\) 0 0
\(16\) −61287.6 + 51426.4i −0.935175 + 0.784705i
\(17\) 49707.0 + 28698.4i 0.595144 + 0.343606i 0.767129 − 0.641493i \(-0.221685\pi\)
−0.171985 + 0.985100i \(0.555018\pi\)
\(18\) 0 0
\(19\) −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
\(21\) 0 0
\(22\) −198215. − 166322.i −0.846147 − 0.710001i
\(23\) −79587.0 − 218664.i −0.284401 − 0.781385i −0.996824 − 0.0796349i \(-0.974625\pi\)
0.712423 − 0.701750i \(-0.247598\pi\)
\(24\) 0 0
\(25\) −22175.6 − 125764.i −0.0567694 − 0.321955i
\(26\) − 960617.i − 2.10212i
\(27\) 0 0
\(28\) 147277. 0.239609
\(29\) −147697. + 26043.0i −0.208824 + 0.0368213i −0.277081 − 0.960846i \(-0.589367\pi\)
0.0682575 + 0.997668i \(0.478256\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) 994872. + 362104.i 0.744477 + 0.270968i
\(35\) −1.09003e6 + 629329.i −0.726384 + 0.419378i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) 396080. − 2.24628e6i 0.154719 − 0.877454i
\(41\) 1.41044e6 + 248698.i 0.499136 + 0.0880111i 0.417548 − 0.908655i \(-0.362890\pi\)
0.0815882 + 0.996666i \(0.474001\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −2.14612e6 − 3.71720e6i −0.479318 − 0.830203i
\(47\) −2.27848e6 + 6.26006e6i −0.466931 + 1.28288i 0.453248 + 0.891385i \(0.350265\pi\)
−0.920179 + 0.391498i \(0.871957\pi\)
\(48\) 0 0
\(49\) −2.07476e6 − 1.74093e6i −0.359901 − 0.301993i
\(50\) −805657. − 2.21352e6i −0.128905 − 0.354164i
\(51\) 0 0
\(52\) −761828. − 4.32054e6i −0.104194 − 0.590915i
\(53\) − 396154.i − 0.0502065i −0.999685 − 0.0251033i \(-0.992009\pi\)
0.999685 − 0.0251033i \(-0.00799146\pi\)
\(54\) 0 0
\(55\) 1.00993e7 1.10367
\(56\) −5.45465e6 + 961801.i −0.554644 + 0.0977987i
\(57\) 0 0
\(58\) −2.59957e6 + 946165.i −0.229715 + 0.0836093i
\(59\) 3.41825e6 − 4.07371e6i 0.282095 − 0.336188i −0.606327 − 0.795216i \(-0.707358\pi\)
0.888422 + 0.459028i \(0.151802\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) 4.11030e6 − 7.11925e6i 0.244993 − 0.424340i
\(65\) 2.41005e7 + 2.87219e7i 1.35012 + 1.60901i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) −1.77851e7 + 1.49235e7i −0.740737 + 0.621552i
\(71\) −3.79332e7 − 2.19007e7i −1.49275 − 0.861837i −0.492781 − 0.870154i \(-0.664019\pi\)
−0.999965 + 0.00831631i \(0.997353\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 0 0
\(76\) −1.50859e7 − 1.26586e7i −0.452186 − 0.379429i
\(77\) −8.38774e6 − 2.30451e7i −0.238606 − 0.655565i
\(78\) 0 0
\(79\) 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
\(81\) 0 0
\(82\) 2.64178e7 0.584308
\(83\) −2.50236e7 + 4.41234e6i −0.527276 + 0.0929730i −0.430948 − 0.902377i \(-0.641821\pi\)
−0.0963279 + 0.995350i \(0.530710\pi\)
\(84\) 0 0
\(85\) −3.88307e7 + 1.41332e7i −0.743875 + 0.270748i
\(86\) −1.07492e6 + 1.28104e6i −0.0196509 + 0.0234190i
\(87\) 0 0
\(88\) 4.17623e7 + 1.52002e7i 0.696391 + 0.253466i
\(89\) 3.33894e7 − 1.92774e7i 0.532168 − 0.307247i −0.209731 − 0.977759i \(-0.567259\pi\)
0.741899 + 0.670512i \(0.233925\pi\)
\(90\) 0 0
\(91\) 4.55230e7 − 7.88482e7i 0.663843 − 1.14981i
\(92\) −1.26005e7 − 1.50167e7i −0.175889 − 0.209616i
\(93\) 0 0
\(94\) −2.13382e7 + 1.21015e8i −0.273304 + 1.54998i
\(95\) 1.65745e8 + 2.92253e7i 2.03491 + 0.358810i
\(96\) 0 0
\(97\) −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
\(99\) 0 0
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 81.9.f.a.71.17 138
3.2 odd 2 27.9.f.a.14.7 yes 138
27.2 odd 18 inner 81.9.f.a.8.17 138
27.25 even 9 27.9.f.a.2.7 ✓ 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.7 ✓ 138 27.25 even 9
27.9.f.a.14.7 yes 138 3.2 odd 2
81.9.f.a.8.17 138 27.2 odd 18 inner
81.9.f.a.71.17 138 1.1 even 1 trivial