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 2.3
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.3

$q$-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+(-24.3796 - 4.29877i) q^{2} +(-14.9563 - 79.6072i) q^{3} +(335.322 + 122.047i) q^{4} +(-12.1292 - 14.4550i) q^{5} +(22.4138 + 2005.08i) q^{6} +(-3223.30 + 1173.19i) q^{7} +(-2161.96 - 1248.21i) q^{8} +(-6113.62 + 2381.25i) q^{9} +(233.565 + 404.547i) q^{10} +(1196.41 - 1425.83i) q^{11} +(4700.68 - 28519.4i) q^{12} +(5064.72 + 28723.4i) q^{13} +(83626.0 - 14745.5i) q^{14} +(-969.316 + 1181.76i) q^{15} +(-22637.5 - 18995.2i) q^{16} +(126178. - 72849.1i) q^{17} +(159284. - 31772.9i) q^{18} +(-28539.4 + 49431.6i) q^{19} +(-2302.99 - 6327.42i) q^{20} +(141603. + 239052. i) q^{21} +(-35297.3 + 29618.0i) q^{22} +(34873.5 - 95814.2i) q^{23} +(-67031.5 + 190776. i) q^{24} +(67769.5 - 384340. i) q^{25} -722037. i q^{26} +(281002. + 451074. i) q^{27} -1.22403e6 q^{28} +(-545415. - 96171.4i) q^{29} +(28711.6 - 24644.0i) q^{30} +(171625. + 62466.4i) q^{31} +(881032. + 1.04997e6i) q^{32} +(-131400. - 73918.0i) q^{33} +(-3.38933e6 + 1.23362e6i) q^{34} +(56054.5 + 32363.1i) q^{35} +(-2.34066e6 + 52336.5i) q^{36} +(1.11437e6 + 1.93014e6i) q^{37} +(908272. - 1.08244e6i) q^{38} +(2.21084e6 - 832783. i) q^{39} +(8179.97 + 46390.9i) q^{40} +(4.79560e6 - 845594. i) q^{41} +(-2.42458e6 - 6.43670e6i) q^{42} +(745393. + 625459. i) q^{43} +(575202. - 332093. i) q^{44} +(108574. + 59489.7i) q^{45} +(-1.26208e6 + 2.18599e6i) q^{46} +(-122905. - 337679. i) q^{47} +(-1.17358e6 + 2.08621e6i) q^{48} +(4.59723e6 - 3.85754e6i) q^{49} +(-3.30438e6 + 9.07871e6i) q^{50} +(-7.68647e6 - 8.95516e6i) q^{51} +(-1.80731e6 + 1.02497e7i) q^{52} +7.51606e6i q^{53} +(-4.91164e6 - 1.22049e7i) q^{54} -35121.9 q^{55} +(8.43304e6 + 1.48697e6i) q^{56} +(4.36196e6 + 1.53263e6i) q^{57} +(1.28836e7 + 4.68923e6i) q^{58} +(3.75820e6 + 4.47885e6i) q^{59} +(-469264. + 277969. i) q^{60} +(325842. - 118597. i) q^{61} +(-3.91561e6 - 2.26068e6i) q^{62} +(1.69124e7 - 1.48479e7i) q^{63} +(-1.31830e7 - 2.28336e7i) q^{64} +(353767. - 421603. i) q^{65} +(2.88572e6 + 2.36695e6i) q^{66} +(4.34161e6 + 2.46225e7i) q^{67} +(5.12014e7 - 9.02819e6i) q^{68} +(-8.14908e6 - 1.34316e6i) q^{69} +(-1.22746e6 - 1.02996e6i) q^{70} +(-3.46344e7 + 1.99962e7i) q^{71} +(1.61897e7 + 2.48289e6i) q^{72} +(1.47786e7 - 2.55972e7i) q^{73} +(-1.88705e7 - 5.18464e7i) q^{74} +(-3.16098e7 + 353350. i) q^{75} +(-1.56029e7 + 1.30924e7i) q^{76} +(-2.18364e6 + 5.99949e6i) q^{77} +(-5.74793e7 + 1.07990e7i) q^{78} +(-1.20078e7 + 6.80994e7i) q^{79} +557622. i q^{80} +(3.17060e7 - 2.91162e7i) q^{81} -1.20550e8 q^{82} +(8.38708e7 + 1.47887e7i) q^{83} +(1.83069e7 + 9.74416e7i) q^{84} +(-2.58348e6 - 940308. i) q^{85} +(-1.54837e7 - 1.84527e7i) q^{86} +(501437. + 4.48574e7i) q^{87} +(-4.36632e6 + 1.58921e6i) q^{88} +(-6.63371e7 - 3.82997e7i) q^{89} +(-2.39126e6 - 1.91707e6i) q^{90} +(-5.00231e7 - 8.66425e7i) q^{91} +(2.33877e7 - 2.78724e7i) q^{92} +(2.40591e6 - 1.45969e7i) q^{93} +(1.54476e6 + 8.76080e6i) q^{94} +(1.06069e6 - 187029. i) q^{95} +(7.04085e7 - 8.58402e7i) q^{96} +(8.83737e7 + 7.41543e7i) q^{97} +(-1.28661e8 + 7.42825e7i) q^{98} +(-3.91915e6 + 1.15659e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.3796 − 4.29877i −1.52372 − 0.268673i −0.651828 − 0.758367i \(-0.725998\pi\)
−0.871895 + 0.489694i \(0.837109\pi\)
\(3\) −14.9563 − 79.6072i −0.184645 − 0.982805i
\(4\) 335.322 + 122.047i 1.30985 + 0.476747i
\(5\) −12.1292 − 14.4550i −0.0194067 − 0.0231280i 0.756254 − 0.654278i \(-0.227028\pi\)
−0.775661 + 0.631150i \(0.782583\pi\)
\(6\) 22.4138 + 2005.08i 0.0172946 + 1.54713i
\(7\) −3223.30 + 1173.19i −1.34248 + 0.488624i −0.910594 − 0.413302i \(-0.864375\pi\)
−0.431890 + 0.901926i \(0.642153\pi\)
\(8\) −2161.96 − 1248.21i −0.527822 − 0.304738i
\(9\) −6113.62 + 2381.25i −0.931812 + 0.362941i
\(10\) 233.565 + 404.547i 0.0233565 + 0.0404547i
\(11\) 1196.41 − 1425.83i 0.0817165 − 0.0973860i −0.723637 − 0.690181i \(-0.757531\pi\)
0.805354 + 0.592795i \(0.201975\pi\)
\(12\) 4700.68 − 28519.4i 0.226692 − 1.37536i
\(13\) 5064.72 + 28723.4i 0.177330 + 1.00569i 0.935420 + 0.353538i \(0.115021\pi\)
−0.758090 + 0.652149i \(0.773867\pi\)
\(14\) 83626.0 − 14745.5i 2.17685 − 0.383838i
\(15\) −969.316 + 1181.76i −0.0191470 + 0.0233435i
\(16\) −22637.5 − 18995.2i −0.345422 − 0.289843i
\(17\) 126178. − 72849.1i 1.51074 − 0.872225i 0.510816 − 0.859690i \(-0.329343\pi\)
0.999921 − 0.0125351i \(-0.00399015\pi\)
\(18\) 159284. − 31772.9i 1.51734 − 0.302668i
\(19\) −28539.4 + 49431.6i −0.218993 + 0.379307i −0.954500 − 0.298210i \(-0.903610\pi\)
0.735508 + 0.677517i \(0.236944\pi\)
\(20\) −2302.99 − 6327.42i −0.0143937 − 0.0395464i
\(21\) 141603. + 239052.i 0.728106 + 1.22918i
\(22\) −35297.3 + 29618.0i −0.150678 + 0.126434i
\(23\) 34873.5 − 95814.2i 0.124619 − 0.342388i −0.861657 − 0.507490i \(-0.830573\pi\)
0.986276 + 0.165103i \(0.0527955\pi\)
\(24\) −67031.5 + 190776.i −0.202039 + 0.575015i
\(25\) 67769.5 − 384340.i 0.173490 − 0.983910i
\(26\) − 722037.i − 1.58003i
\(27\) 281002. + 451074.i 0.528755 + 0.848775i
\(28\) −1.22403e6 −1.99141
\(29\) −545415. − 96171.4i −0.771144 − 0.135973i −0.225786 − 0.974177i \(-0.572495\pi\)
−0.545358 + 0.838203i \(0.683606\pi\)
\(30\) 28711.6 − 24644.0i 0.0354464 − 0.0304247i
\(31\) 171625. + 62466.4i 0.185838 + 0.0676394i 0.433263 − 0.901268i \(-0.357362\pi\)
−0.247425 + 0.968907i \(0.579584\pi\)
\(32\) 881032. + 1.04997e6i 0.840217 + 1.00133i
\(33\) −131400. − 73918.0i −0.110800 − 0.0623296i
\(34\) −3.38933e6 + 1.23362e6i −2.53629 + 0.923134i
\(35\) 56054.5 + 32363.1i 0.0373541 + 0.0215664i
\(36\) −2.34066e6 + 52336.5i −1.39357 + 0.0311598i
\(37\) 1.11437e6 + 1.93014e6i 0.594595 + 1.02987i 0.993604 + 0.112922i \(0.0360210\pi\)
−0.399009 + 0.916947i \(0.630646\pi\)
\(38\) 908272. − 1.08244e6i 0.435594 − 0.519120i
\(39\) 2.21084e6 − 832783.i 0.955652 − 0.359976i
\(40\) 8179.97 + 46390.9i 0.00319530 + 0.0181214i
\(41\) 4.79560e6 − 845594.i 1.69710 − 0.299245i 0.760421 − 0.649430i \(-0.224992\pi\)
0.936680 + 0.350185i \(0.113881\pi\)
\(42\) −2.42458e6 − 6.43670e6i −0.779184 − 2.06855i
\(43\) 745393. + 625459.i 0.218028 + 0.182947i 0.745259 − 0.666775i \(-0.232326\pi\)
−0.527232 + 0.849722i \(0.676770\pi\)
\(44\) 575202. − 332093.i 0.153465 − 0.0886031i
\(45\) 108574. + 59489.7i 0.0264775 + 0.0145075i
\(46\) −1.26208e6 + 2.18599e6i −0.281875 + 0.488222i
\(47\) −122905. − 337679.i −0.0251871 − 0.0692010i 0.926462 − 0.376389i \(-0.122834\pi\)
−0.951649 + 0.307188i \(0.900612\pi\)
\(48\) −1.17358e6 + 2.08621e6i −0.221079 + 0.393000i
\(49\) 4.59723e6 − 3.85754e6i 0.797466 − 0.669153i
\(50\) −3.30438e6 + 9.07871e6i −0.528701 + 1.45259i
\(51\) −7.68647e6 − 8.95516e6i −1.13618 − 1.32371i
\(52\) −1.80731e6 + 1.02497e7i −0.247183 + 1.40184i
\(53\) 7.51606e6i 0.952548i 0.879297 + 0.476274i \(0.158013\pi\)
−0.879297 + 0.476274i \(0.841987\pi\)
\(54\) −4.91164e6 − 1.22049e7i −0.577632 − 1.43536i
\(55\) −35121.9 −0.00383819
\(56\) 8.43304e6 + 1.48697e6i 0.857496 + 0.151200i
\(57\) 4.36196e6 + 1.53263e6i 0.413221 + 0.145190i
\(58\) 1.28836e7 + 4.68923e6i 1.13848 + 0.414372i
\(59\) 3.75820e6 + 4.47885e6i 0.310150 + 0.369622i 0.898492 − 0.438990i \(-0.144664\pi\)
−0.588342 + 0.808612i \(0.700219\pi\)
\(60\) −469264. + 277969.i −0.0362086 + 0.0214482i
\(61\) 325842. − 118597.i 0.0235336 − 0.00856552i −0.330227 − 0.943902i \(-0.607125\pi\)
0.353760 + 0.935336i \(0.384903\pi\)
\(62\) −3.91561e6 − 2.26068e6i −0.264992 − 0.152993i
\(63\) 1.69124e7 − 1.48479e7i 1.07360 − 0.942548i
\(64\) −1.31830e7 − 2.28336e7i −0.785769 − 1.36099i
\(65\) 353767. − 421603.i 0.0198182 − 0.0236184i
\(66\) 2.88572e6 + 2.36695e6i 0.152082 + 0.124742i
\(67\) 4.34161e6 + 2.46225e7i 0.215453 + 1.22189i 0.880119 + 0.474753i \(0.157462\pi\)
−0.664667 + 0.747140i \(0.731426\pi\)
\(68\) 5.12014e7 − 9.02819e6i 2.39467 − 0.422246i
\(69\) −8.14908e6 − 1.34316e6i −0.359511 − 0.0592559i
\(70\) −1.22746e6 − 1.02996e6i −0.0511230 − 0.0428973i
\(71\) −3.46344e7 + 1.99962e7i −1.36293 + 0.786888i −0.990013 − 0.140976i \(-0.954976\pi\)
−0.372918 + 0.927864i \(0.621643\pi\)
\(72\) 1.61897e7 + 2.48289e6i 0.602433 + 0.0923907i
\(73\) 1.47786e7 − 2.55972e7i 0.520405 − 0.901367i −0.479314 − 0.877644i \(-0.659114\pi\)
0.999719 − 0.0237238i \(-0.00755222\pi\)
\(74\) −1.88705e7 − 5.18464e7i −0.629300 − 1.72899i
\(75\) −3.16098e7 + 353350.i −0.999026 + 0.0111676i
\(76\) −1.56029e7 + 1.30924e7i −0.467681 + 0.392431i
\(77\) −2.18364e6 + 5.99949e6i −0.0621180 + 0.170668i
\(78\) −5.74793e7 + 1.07990e7i −1.55286 + 0.291745i
\(79\) −1.20078e7 + 6.80994e7i −0.308286 + 1.74838i 0.299334 + 0.954148i \(0.403235\pi\)
−0.607620 + 0.794228i \(0.707876\pi\)
\(80\) 557622.i 0.0136138i
\(81\) 3.17060e7 − 2.91162e7i 0.736548 − 0.676385i
\(82\) −1.20550e8 −2.66631
\(83\) 8.38708e7 + 1.47887e7i 1.76725 + 0.311614i 0.960293 − 0.278993i \(-0.0900007\pi\)
0.806959 + 0.590607i \(0.201112\pi\)
\(84\) 1.83069e7 + 9.74416e7i 0.367704 + 1.95716i
\(85\) −2.58348e6 − 940308.i −0.0494913 − 0.0180133i
\(86\) −1.54837e7 − 1.84527e7i −0.283061 − 0.337339i
\(87\) 501437. + 4.48574e7i 0.00875265 + 0.782991i
\(88\) −4.36632e6 + 1.58921e6i −0.0728090 + 0.0265003i
\(89\) −6.63371e7 − 3.82997e7i −1.05730 − 0.610430i −0.132612 − 0.991168i \(-0.542337\pi\)
−0.924683 + 0.380738i \(0.875670\pi\)
\(90\) −2.39126e6 − 1.91707e6i −0.0364466 − 0.0292192i
\(91\) −5.00231e7 − 8.66425e7i −0.729466 − 1.26347i
\(92\) 2.33877e7 − 2.78724e7i 0.326465 − 0.389066i
\(93\) 2.40591e6 − 1.45969e7i 0.0321623 − 0.195131i
\(94\) 1.54476e6 + 8.76080e6i 0.0197857 + 0.112210i
\(95\) 1.06069e6 − 187029.i 0.0130225 − 0.00229622i
\(96\) 7.04085e7 − 8.58402e7i 0.828972 − 1.01066i
\(97\) 8.83737e7 + 7.41543e7i 0.998243 + 0.837625i 0.986740 − 0.162309i \(-0.0518940\pi\)
0.0115027 + 0.999934i \(0.496338\pi\)
\(98\) −1.28661e8 + 7.42825e7i −1.39490 + 0.805346i
\(99\) −3.91915e6 + 1.15659e7i −0.0407991 + 0.120404i
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.2.3 ✓ 138
3.2 odd 2 81.9.f.a.8.21 138
27.13 even 9 81.9.f.a.71.21 138
27.14 odd 18 inner 27.9.f.a.14.3 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.3 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.3 yes 138 27.14 odd 18 inner
81.9.f.a.8.21 138 3.2 odd 2
81.9.f.a.71.21 138 27.13 even 9