Show commands: Magma / Pari/GP / SageMath

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 8.10
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-7.08460 - 1.24921i) q^{2} +(-191.930 - 69.8569i) q^{4} +(-349.559 - 416.588i) q^{5} +(22.4234 - 8.16143i) q^{7} +(2867.39 + 1655.49i) q^{8} +(1956.08 + 3388.03i) q^{10} +(15469.2 - 18435.5i) q^{11} +(-8798.06 - 49896.3i) q^{13} +(-169.056 + 29.8091i) q^{14} +(21808.2 + 18299.3i) q^{16} +(37540.0 - 21673.7i) q^{17} +(-75167.3 + 130194. i) q^{19} +(37989.4 + 104375. i) q^{20} +(-132623. + 111284. i) q^{22} +(108972. - 299398. i) q^{23} +(16477.1 - 93446.3i) q^{25} +364486. i q^{26} -4873.85 q^{28} +(-945594. - 166734. i) q^{29} +(736394. + 268025. i) q^{31} +(-676476. - 806193. i) q^{32} +(-293031. + 106655. i) q^{34} +(-11238.2 - 6488.40i) q^{35} +(1.21793e6 + 2.10952e6i) q^{37} +(695170. - 828471. i) q^{38} +(-312665. - 1.77321e6i) q^{40} +(-2.52414e6 + 445074. i) q^{41} +(-2.24939e6 - 1.88746e6i) q^{43} +(-4.25687e6 + 2.45770e6i) q^{44} +(-1.14603e6 + 1.98499e6i) q^{46} +(569539. + 1.56479e6i) q^{47} +(-4.41566e6 + 3.70518e6i) q^{49} +(-233468. + 641447. i) q^{50} +(-1.79699e6 + 1.01912e7i) q^{52} -3.88995e6i q^{53} -1.30874e7 q^{55} +(77807.6 + 13719.6i) q^{56} +(6.49088e6 + 2.36249e6i) q^{58} +(-6.87370e6 - 8.19175e6i) q^{59} +(-1.49222e7 + 5.43124e6i) q^{61} +(-4.88224e6 - 2.81876e6i) q^{62} +(141478. + 245047. i) q^{64} +(-1.77108e7 + 2.11069e7i) q^{65} +(1.45408e6 + 8.24651e6i) q^{67} +(-8.71913e6 + 1.53742e6i) q^{68} +(71513.1 + 60006.6i) q^{70} +(9.42132e6 - 5.43940e6i) q^{71} +(1.18791e7 - 2.05752e7i) q^{73} +(-5.99334e6 - 1.64666e7i) q^{74} +(2.35218e7 - 1.97371e7i) q^{76} +(196412. - 539638. i) q^{77} +(-7.07994e6 + 4.01523e7i) q^{79} -1.54817e7i q^{80} +1.84385e7 q^{82} +(5.20151e7 + 9.17167e6i) q^{83} +(-2.21515e7 - 8.06248e6i) q^{85} +(1.35782e7 + 1.61819e7i) q^{86} +(7.48761e7 - 2.72527e7i) q^{88} +(-3.60266e7 - 2.07999e7i) q^{89} +(-604507. - 1.04704e6i) q^{91} +(-4.18301e7 + 4.98511e7i) q^{92} +(-2.08020e6 - 1.17974e7i) q^{94} +(8.05126e7 - 1.41965e7i) q^{95} +(-7.44489e7 - 6.24700e7i) q^{97} +(3.59117e7 - 2.07336e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.08460 − 1.24921i −0.442788 − 0.0780754i −0.0521906 − 0.998637i \(-0.516620\pi\)
−0.390597 + 0.920562i \(0.627731\pi\)
\(3\) 0 0
\(4\) −191.930 − 69.8569i −0.749727 − 0.272878i
\(5\) −349.559 − 416.588i −0.559294 − 0.666541i 0.410103 − 0.912039i \(-0.365493\pi\)
−0.969397 + 0.245498i \(0.921048\pi\)
\(6\) 0 0
\(7\) 22.4234 − 8.16143i 0.00933917 − 0.00339918i −0.337346 − 0.941381i \(-0.609529\pi\)
0.346686 + 0.937981i \(0.387307\pi\)
\(8\) 2867.39 + 1655.49i 0.700046 + 0.404172i
\(9\) 0 0
\(10\) 1956.08 + 3388.03i 0.195608 + 0.338803i
\(11\) 15469.2 − 18435.5i 1.05657 − 1.25917i 0.0918844 − 0.995770i \(-0.470711\pi\)
0.964686 − 0.263402i \(-0.0848446\pi\)
\(12\) 0 0
\(13\) −8798.06 − 49896.3i −0.308045 − 1.74701i −0.608815 − 0.793312i \(-0.708355\pi\)
0.300771 − 0.953696i \(-0.402756\pi\)
\(14\) −169.056 + 29.8091i −0.00440066 + 0.000775956i
\(15\) 0 0
\(16\) 21808.2 + 18299.3i 0.332767 + 0.279225i
\(17\) 37540.0 − 21673.7i 0.449468 − 0.259501i −0.258137 − 0.966108i \(-0.583109\pi\)
0.707606 + 0.706608i \(0.249775\pi\)
\(18\) 0 0
\(19\) −75167.3 + 130194.i −0.576786 + 0.999023i 0.419059 + 0.907959i \(0.362360\pi\)
−0.995845 + 0.0910639i \(0.970973\pi\)
\(20\) 37989.4 + 104375.i 0.237434 + 0.652343i
\(21\) 0 0
\(22\) −132623. + 111284.i −0.566147 + 0.475054i
\(23\) 108972. − 299398.i 0.389407 − 1.06989i −0.577862 − 0.816134i \(-0.696113\pi\)
0.967269 − 0.253753i \(-0.0816649\pi\)
\(24\) 0 0
\(25\) 16477.1 − 93446.3i 0.0421814 − 0.239223i
\(26\) 364486.i 0.797605i
\(27\) 0 0
\(28\) −4873.85 −0.00792940
\(29\) −945594. − 166734.i −1.33694 − 0.235739i −0.540955 − 0.841052i \(-0.681937\pi\)
−0.795988 + 0.605312i \(0.793048\pi\)
\(30\) 0 0
\(31\) 736394. + 268025.i 0.797376 + 0.290221i 0.708399 − 0.705813i \(-0.249418\pi\)
0.0889773 + 0.996034i \(0.471640\pi\)
\(32\) −676476. − 806193.i −0.645138 − 0.768845i
\(33\) 0 0
\(34\) −293031. + 106655.i −0.219280 + 0.0798113i
\(35\) −11238.2 − 6488.40i −0.00748904 − 0.00432380i
\(36\) 0 0
\(37\) 1.21793e6 + 2.10952e6i 0.649855 + 1.12558i 0.983157 + 0.182761i \(0.0585035\pi\)
−0.333303 + 0.942820i \(0.608163\pi\)
\(38\) 695170. − 828471.i 0.333393 − 0.397322i
\(39\) 0 0
\(40\) −312665. − 1.77321e6i −0.122135 − 0.692660i
\(41\) −2.52414e6 + 445074.i −0.893261 + 0.157506i −0.601394 − 0.798953i \(-0.705388\pi\)
−0.291867 + 0.956459i \(0.594277\pi\)
\(42\) 0 0
\(43\) −2.24939e6 − 1.88746e6i −0.657948 − 0.552084i 0.251523 − 0.967851i \(-0.419068\pi\)
−0.909471 + 0.415768i \(0.863513\pi\)
\(44\) −4.25687e6 + 2.45770e6i −1.13574 + 0.655720i
\(45\) 0 0
\(46\) −1.14603e6 + 1.98499e6i −0.255957 + 0.443330i
\(47\) 569539. + 1.56479e6i 0.116716 + 0.320676i 0.984271 − 0.176667i \(-0.0565315\pi\)
−0.867554 + 0.497342i \(0.834309\pi\)
\(48\) 0 0
\(49\) −4.41566e6 + 3.70518e6i −0.765969 + 0.642724i
\(50\) −233468. + 641447.i −0.0373548 + 0.102631i
\(51\) 0 0
\(52\) −1.79699e6 + 1.01912e7i −0.245771 + 1.39384i
\(53\) − 3.88995e6i − 0.492993i −0.969144 − 0.246496i \(-0.920721\pi\)
0.969144 − 0.246496i \(-0.0792793\pi\)
\(54\) 0 0
\(55\) −1.30874e7 −1.43022
\(56\) 77807.6 + 13719.6i 0.00791170 + 0.00139505i
\(57\) 0 0
\(58\) 6.49088e6 + 2.36249e6i 0.573577 + 0.208765i
\(59\) −6.87370e6 − 8.19175e6i −0.567260 − 0.676034i 0.403806 − 0.914845i \(-0.367687\pi\)
−0.971066 + 0.238810i \(0.923243\pi\)
\(60\) 0 0
\(61\) −1.49222e7 + 5.43124e6i −1.07774 + 0.392265i −0.819065 − 0.573700i \(-0.805508\pi\)
−0.258673 + 0.965965i \(0.583285\pi\)
\(62\) −4.88224e6 − 2.81876e6i −0.330409 − 0.190762i
\(63\) 0 0
\(64\) 141478. + 245047.i 0.00843275 + 0.0146060i
\(65\) −1.77108e7 + 2.11069e7i −0.992165 + 1.18242i
\(66\) 0 0
\(67\) 1.45408e6 + 8.24651e6i 0.0721588 + 0.409233i 0.999396 + 0.0347579i \(0.0110660\pi\)
−0.927237 + 0.374475i \(0.877823\pi\)
\(68\) −8.71913e6 + 1.53742e6i −0.407791 + 0.0719045i
\(69\) 0 0
\(70\) 71513.1 + 60006.6i 0.00297847 + 0.00249924i
\(71\) 9.42132e6 − 5.43940e6i 0.370747 − 0.214051i −0.303037 − 0.952979i \(-0.598001\pi\)
0.673785 + 0.738927i \(0.264667\pi\)
\(72\) 0 0
\(73\) 1.18791e7 − 2.05752e7i 0.418305 − 0.724525i −0.577464 − 0.816416i \(-0.695958\pi\)
0.995769 + 0.0918908i \(0.0292911\pi\)
\(74\) −5.99334e6 − 1.64666e7i −0.199867 − 0.549131i
\(75\) 0 0
\(76\) 2.35218e7 − 1.97371e7i 0.705044 − 0.591602i
\(77\) 196412. − 539638.i 0.00558734 − 0.0153511i
\(78\) 0 0
\(79\) −7.07994e6 + 4.01523e7i −0.181770 + 1.03087i 0.748266 + 0.663398i \(0.230886\pi\)
−0.930036 + 0.367468i \(0.880225\pi\)
\(80\) − 1.54817e7i − 0.377972i
\(81\) 0 0
\(82\) 1.84385e7 0.407822
\(83\) 5.20151e7 + 9.17167e6i 1.09602 + 0.193257i 0.692288 − 0.721621i \(-0.256603\pi\)
0.403729 + 0.914879i \(0.367714\pi\)
\(84\) 0 0
\(85\) −2.21515e7 − 8.06248e6i −0.424353 − 0.154452i
\(86\) 1.35782e7 + 1.61819e7i 0.248227 + 0.295825i
\(87\) 0 0
\(88\) 7.48761e7 − 2.72527e7i 1.24857 − 0.454442i
\(89\) −3.60266e7 − 2.07999e7i −0.574199 − 0.331514i 0.184625 − 0.982809i \(-0.440893\pi\)
−0.758825 + 0.651295i \(0.774226\pi\)
\(90\) 0 0
\(91\) −604507. − 1.04704e6i −0.00881528 − 0.0152685i
\(92\) −4.18301e7 + 4.98511e7i −0.583898 + 0.695863i
\(93\) 0 0
\(94\) −2.08020e6 − 1.17974e7i −0.0266437 − 0.151104i
\(95\) 8.05126e7 − 1.41965e7i 0.988483 − 0.174296i
\(96\) 0 0
\(97\) −7.44489e7 − 6.24700e7i −0.840952 − 0.705642i 0.116826 − 0.993152i \(-0.462728\pi\)
−0.957778 + 0.287510i \(0.907172\pi\)
\(98\) 3.59117e7 − 2.07336e7i 0.389343 − 0.224787i
\(99\) 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.8.10 138
3.2 odd 2 27.9.f.a.2.14 ✓ 138
27.13 even 9 27.9.f.a.14.14 yes 138
27.14 odd 18 inner 81.9.f.a.71.10 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.14 ✓ 138 3.2 odd 2
27.9.f.a.14.14 yes 138 27.13 even 9
81.9.f.a.8.10 138 1.1 even 1 trivial
81.9.f.a.71.10 138 27.14 odd 18 inner