Show commands: Magma / Pari/GP / SageMath

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 2.14
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.14

$q$-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} +(14.6630 + 79.6618i) q^{3} +(-191.930 - 69.8569i) q^{4} +(349.559 + 416.588i) q^{5} +(4.36750 + 582.689i) q^{6} +(22.4234 - 8.16143i) q^{7} +(-2867.39 - 1655.49i) q^{8} +(-6130.99 + 2336.16i) q^{9} +(1956.08 + 3388.03i) q^{10} +(-15469.2 + 18435.5i) q^{11} +(2750.65 - 16313.8i) q^{12} +(-8798.06 - 49896.3i) q^{13} +(169.056 - 29.8091i) q^{14} +(-28060.6 + 33954.9i) q^{15} +(21808.2 + 18299.3i) q^{16} +(-37540.0 + 21673.7i) q^{17} +(-46354.0 + 8891.89i) q^{18} +(-75167.3 + 130194. i) q^{19} +(-37989.4 - 104375. i) q^{20} +(978.948 + 1666.61i) q^{21} +(-132623. + 111284. i) q^{22} +(-108972. + 299398. i) q^{23} +(89834.6 - 252696. i) q^{24} +(16477.1 - 93446.3i) q^{25} -364486. i q^{26} +(-276001. - 454151. i) q^{27} -4873.85 q^{28} +(945594. + 166734. i) q^{29} +(-241215. + 205504. i) q^{30} +(736394. + 268025. i) q^{31} +(676476. + 806193. i) q^{32} +(-1.69543e6 - 961988. i) q^{33} +(-293031. + 106655. i) q^{34} +(11238.2 + 6488.40i) q^{35} +(1.33992e6 - 20087.6i) q^{36} +(1.21793e6 + 2.10952e6i) q^{37} +(-695170. + 828471. i) q^{38} +(3.84582e6 - 1.43250e6i) q^{39} +(-312665. - 1.77321e6i) q^{40} +(2.52414e6 - 445074. i) q^{41} +(4853.51 + 13030.2i) q^{42} +(-2.24939e6 - 1.88746e6i) q^{43} +(4.25687e6 - 2.45770e6i) q^{44} +(-3.11636e6 - 1.73747e6i) q^{45} +(-1.14603e6 + 1.98499e6i) q^{46} +(-569539. - 1.56479e6i) q^{47} +(-1.13798e6 + 2.00561e6i) q^{48} +(-4.41566e6 + 3.70518e6i) q^{49} +(233468. - 641447. i) q^{50} +(-2.27702e6 - 2.67270e6i) q^{51} +(-1.79699e6 + 1.01912e7i) q^{52} +3.88995e6i q^{53} +(-1.38803e6 - 3.56226e6i) q^{54} -1.30874e7 q^{55} +(-77807.6 - 13719.6i) q^{56} +(-1.14736e7 - 4.07893e6i) q^{57} +(6.49088e6 + 2.36249e6i) q^{58} +(6.87370e6 + 8.19175e6i) q^{59} +(7.75765e6 - 4.55675e6i) q^{60} +(-1.49222e7 + 5.43124e6i) q^{61} +(4.88224e6 + 2.81876e6i) q^{62} +(-118411. + 102422. i) q^{63} +(141478. + 245047. i) q^{64} +(1.77108e7 - 2.11069e7i) q^{65} +(-1.08097e7 - 8.93325e6i) q^{66} +(1.45408e6 + 8.24651e6i) q^{67} +(8.71913e6 - 1.53742e6i) q^{68} +(-2.54484e7 - 4.29083e6i) q^{69} +(71513.1 + 60006.6i) q^{70} +(-9.42132e6 + 5.43940e6i) q^{71} +(2.14474e7 + 3.45110e6i) q^{72} +(1.18791e7 - 2.05752e7i) q^{73} +(5.99334e6 + 1.64666e7i) q^{74} +(7.68570e6 - 57607.5i) q^{75} +(2.35218e7 - 1.97371e7i) q^{76} +(-196412. + 539638. i) q^{77} +(2.90356e7 - 5.34446e6i) q^{78} +(-7.07994e6 + 4.01523e7i) q^{79} +1.54817e7i q^{80} +(3.21314e7 - 2.86460e7i) q^{81} +1.84385e7 q^{82} +(-5.20151e7 - 9.17167e6i) q^{83} +(-71465.3 - 388260. i) q^{84} +(-2.21515e7 - 8.06248e6i) q^{85} +(-1.35782e7 - 1.61819e7i) q^{86} +(582937. + 7.77725e7i) q^{87} +(7.48761e7 - 2.72527e7i) q^{88} +(3.60266e7 + 2.07999e7i) q^{89} +(-1.99077e7 - 1.62023e7i) q^{90} +(-604507. - 1.04704e6i) q^{91} +(4.18301e7 - 4.98511e7i) q^{92} +(-1.05536e7 + 6.25925e7i) q^{93} +(-2.08020e6 - 1.17974e7i) q^{94} +(-8.05126e7 + 1.41965e7i) q^{95} +(-5.43036e7 + 6.57105e7i) q^{96} +(-7.44489e7 - 6.24700e7i) q^{97} +(-3.59117e7 + 2.07336e7i) q^{98} +(5.17735e7 - 1.49167e8i) 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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

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

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 0.390597 − 0.920562i \(-0.372269\pi\)
0.0521906 + 0.998637i \(0.483380\pi\)
\(3\) 14.6630 + 79.6618i 0.181025 + 0.983479i
\(4\) −191.930 − 69.8569i −0.749727 − 0.272878i
\(5\) 349.559 + 416.588i 0.559294 + 0.666541i 0.969397 − 0.245498i \(-0.0789516\pi\)
−0.410103 + 0.912039i \(0.634507\pi\)
\(6\) 4.36750 + 582.689i 0.00336998 + 0.449606i
\(7\) 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
\(9\) −6130.99 + 2336.16i −0.934460 + 0.356068i
\(10\) 1956.08 + 3388.03i 0.195608 + 0.338803i
\(11\) −15469.2 + 18435.5i −1.05657 + 1.25917i −0.0918844 + 0.995770i \(0.529289\pi\)
−0.964686 + 0.263402i \(0.915155\pi\)
\(12\) 2750.65 − 16313.8i 0.132651 − 0.786739i
\(13\) −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
\(15\) −28060.6 + 33954.9i −0.554283 + 0.670714i
\(16\) 21808.2 + 18299.3i 0.332767 + 0.279225i
\(17\) −37540.0 + 21673.7i −0.449468 + 0.259501i −0.707606 − 0.706608i \(-0.750225\pi\)
0.258137 + 0.966108i \(0.416891\pi\)
\(18\) −46354.0 + 8891.89i −0.441568 + 0.0847040i
\(19\) −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
\(21\) 978.948 + 1666.61i 0.00503364 + 0.00856954i
\(22\) −132623. + 111284.i −0.566147 + 0.475054i
\(23\) −108972. + 299398.i −0.389407 + 1.06989i 0.577862 + 0.816134i \(0.303887\pi\)
−0.967269 + 0.253753i \(0.918335\pi\)
\(24\) 89834.6 − 252696.i 0.270769 − 0.761645i
\(25\) 16477.1 − 93446.3i 0.0421814 − 0.239223i
\(26\) − 364486.i − 0.797605i
\(27\) −276001. − 454151.i −0.519345 − 0.854565i
\(28\) −4873.85 −0.00792940
\(29\) 945594. + 166734.i 1.33694 + 0.235739i 0.795988 − 0.605312i \(-0.206952\pi\)
0.540955 + 0.841052i \(0.318063\pi\)
\(30\) −241215. + 205504.i −0.297796 + 0.253708i
\(31\) 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
\(33\) −1.69543e6 − 961988.i −1.42963 − 0.811173i
\(34\) −293031. + 106655.i −0.219280 + 0.0798113i
\(35\) 11238.2 + 6488.40i 0.00748904 + 0.00432380i
\(36\) 1.33992e6 − 20087.6i 0.797754 − 0.0119597i
\(37\) 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
\(39\) 3.84582e6 − 1.43250e6i 1.66238 − 0.619207i
\(40\) −312665. − 1.77321e6i −0.122135 − 0.692660i
\(41\) 2.52414e6 − 445074.i 0.893261 − 0.157506i 0.291867 − 0.956459i \(-0.405723\pi\)
0.601394 + 0.798953i \(0.294612\pi\)
\(42\) 4853.51 + 13030.2i 0.00155976 + 0.00418749i
\(43\) −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
\(45\) −3.11636e6 − 1.73747e6i −0.759972 − 0.423709i
\(46\) −1.14603e6 + 1.98499e6i −0.255957 + 0.443330i
\(47\) −569539. − 1.56479e6i −0.116716 − 0.320676i 0.867554 − 0.497342i \(-0.165691\pi\)
−0.984271 + 0.176667i \(0.943468\pi\)
\(48\) −1.13798e6 + 2.00561e6i −0.214373 + 0.377816i
\(49\) −4.41566e6 + 3.70518e6i −0.765969 + 0.642724i
\(50\) 233468. − 641447.i 0.0373548 − 0.102631i
\(51\) −2.27702e6 − 2.67270e6i −0.336578 − 0.395066i
\(52\) −1.79699e6 + 1.01912e7i −0.245771 + 1.39384i
\(53\) 3.88995e6i 0.492993i 0.969144 + 0.246496i \(0.0792793\pi\)
−0.969144 + 0.246496i \(0.920721\pi\)
\(54\) −1.38803e6 − 3.56226e6i −0.163239 − 0.418939i
\(55\) −1.30874e7 −1.43022
\(56\) −77807.6 − 13719.6i −0.00791170 − 0.00139505i
\(57\) −1.14736e7 − 4.07893e6i −1.08693 − 0.386409i
\(58\) 6.49088e6 + 2.36249e6i 0.573577 + 0.208765i
\(59\) 6.87370e6 + 8.19175e6i 0.567260 + 0.676034i 0.971066 − 0.238810i \(-0.0767574\pi\)
−0.403806 + 0.914845i \(0.632313\pi\)
\(60\) 7.75765e6 − 4.55675e6i 0.598584 − 0.351601i
\(61\) −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
\(63\) −118411. + 102422.i −0.00751675 + 0.00650178i
\(64\) 141478. + 245047.i 0.00843275 + 0.0146060i
\(65\) 1.77108e7 − 2.11069e7i 0.992165 − 1.18242i
\(66\) −1.08097e7 − 8.93325e6i −0.569692 − 0.470797i
\(67\) 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
\(69\) −2.54484e7 − 4.29083e6i −1.12270 − 0.189298i
\(70\) 71513.1 + 60006.6i 0.00297847 + 0.00249924i
\(71\) −9.42132e6 + 5.43940e6i −0.370747 + 0.214051i −0.673785 − 0.738927i \(-0.735333\pi\)
0.303037 + 0.952979i \(0.401999\pi\)
\(72\) 2.14474e7 + 3.45110e6i 0.798078 + 0.128419i
\(73\) 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
\(75\) 7.68570e6 − 57607.5i 0.242906 − 0.00182068i
\(76\) 2.35218e7 − 1.97371e7i 0.705044 − 0.591602i
\(77\) −196412. + 539638.i −0.00558734 + 0.0153511i
\(78\) 2.90356e7 − 5.34446e6i 0.784427 − 0.144386i
\(79\) −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
\(81\) 3.21314e7 − 2.86460e7i 0.746432 − 0.665462i
\(82\) 1.84385e7 0.407822
\(83\) −5.20151e7 − 9.17167e6i −1.09602 − 0.193257i −0.403729 − 0.914879i \(-0.632286\pi\)
−0.692288 + 0.721621i \(0.743397\pi\)
\(84\) −71465.3 − 388260.i −0.00143542 − 0.00779839i
\(85\) −2.21515e7 − 8.06248e6i −0.424353 − 0.154452i
\(86\) −1.35782e7 − 1.61819e7i −0.248227 − 0.295825i
\(87\) 582937. + 7.77725e7i 0.0101752 + 1.35753i
\(88\) 7.48761e7 − 2.72527e7i 1.24857 − 0.454442i
\(89\) 3.60266e7 + 2.07999e7i 0.574199 + 0.331514i 0.758825 − 0.651295i \(-0.225774\pi\)
−0.184625 + 0.982809i \(0.559107\pi\)
\(90\) −1.99077e7 − 1.62023e7i −0.303425 − 0.246948i
\(91\) −604507. − 1.04704e6i −0.00881528 − 0.0152685i
\(92\) 4.18301e7 − 4.98511e7i 0.583898 − 0.695863i
\(93\) −1.05536e7 + 6.25925e7i −0.141082 + 0.836739i
\(94\) −2.08020e6 − 1.17974e7i −0.0266437 − 0.151104i
\(95\) −8.05126e7 + 1.41965e7i −0.988483 + 0.174296i
\(96\) −5.43036e7 + 6.57105e7i −0.639357 + 0.773659i
\(97\) −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
\(99\) 5.17735e7 − 1.49167e8i 0.538973 − 1.55286i
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.2.14 ✓ 138
3.2 odd 2 81.9.f.a.8.10 138
27.13 even 9 81.9.f.a.71.10 138
27.14 odd 18 inner 27.9.f.a.14.14 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.14 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.14 yes 138 27.14 odd 18 inner
81.9.f.a.8.10 138 3.2 odd 2
81.9.f.a.71.10 138 27.13 even 9