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 2.18
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.18

$q$-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+(14.5831 + 2.57139i) q^{2} +(-73.7172 + 33.5675i) q^{3} +(-34.5070 - 12.5595i) q^{4} +(-51.5257 - 61.4059i) q^{5} +(-1161.34 + 299.961i) q^{6} +(3577.75 - 1302.19i) q^{7} +(-3753.91 - 2167.32i) q^{8} +(4307.45 - 4949.00i) q^{9} +(-593.505 - 1027.98i) q^{10} +(10514.1 - 12530.3i) q^{11} +(2965.35 - 232.459i) q^{12} +(-3509.19 - 19901.6i) q^{13} +(55523.1 - 9790.22i) q^{14} +(5859.57 + 2797.09i) q^{15} +(-41969.1 - 35216.3i) q^{16} +(-41055.8 + 23703.6i) q^{17} +(75541.7 - 61095.5i) q^{18} +(-49120.0 + 85078.3i) q^{19} +(1006.77 + 2766.07i) q^{20} +(-220030. + 216090. i) q^{21} +(185549. - 155694. i) q^{22} +(152612. - 419299. i) q^{23} +(349479. + 33759.6i) q^{24} +(66715.5 - 378363. i) q^{25} -299251. i q^{26} +(-151408. + 509416. i) q^{27} -139812. q^{28} +(51718.6 + 9119.39i) q^{29} +(78258.2 + 55857.4i) q^{30} +(-408929. - 148838. i) q^{31} +(191796. + 228573. i) q^{32} +(-354464. + 1.27663e6i) q^{33} +(-659672. + 240101. i) q^{34} +(-264308. - 152599. i) q^{35} +(-210794. + 116676. i) q^{36} +(199444. + 345448. i) q^{37} +(-935091. + 1.11440e6i) q^{38} +(926735. + 1.34930e6i) q^{39} +(60336.4 + 342185. i) q^{40} +(-3.69299e6 + 651173. i) q^{41} +(-3.76437e6 + 2.58548e6i) q^{42} +(4.65176e6 + 3.90329e6i) q^{43} +(-520185. + 300329. i) q^{44} +(-525842. - 9502.57i) q^{45} +(3.30374e6 - 5.72224e6i) q^{46} +(-2.34037e6 - 6.43011e6i) q^{47} +(4.27597e6 + 1.18725e6i) q^{48} +(6.68849e6 - 5.61231e6i) q^{49} +(1.94584e6 - 5.34614e6i) q^{50} +(2.23085e6 - 3.12550e6i) q^{51} +(-128863. + 730819. i) q^{52} +2.29450e6i q^{53} +(-3.51791e6 + 7.03953e6i) q^{54} -1.31118e6 q^{55} +(-1.62528e7 - 2.86581e6i) q^{56} +(765126. - 7.92057e6i) q^{57} +(730768. + 265978. i) q^{58} +(2.15123e6 + 2.56373e6i) q^{59} +(-167066. - 170112. i) q^{60} +(2.36869e7 - 8.62134e6i) q^{61} +(-5.58072e6 - 3.22203e6i) q^{62} +(8.96643e6 - 2.33154e7i) q^{63} +(9.22193e6 + 1.59729e7i) q^{64} +(-1.04126e6 + 1.24093e6i) q^{65} +(-8.45189e6 + 1.77057e7i) q^{66} +(-3.64008e6 - 2.06439e7i) q^{67} +(1.71442e6 - 302298. i) q^{68} +(2.82464e6 + 3.60323e7i) q^{69} +(-3.46204e6 - 2.90500e6i) q^{70} +(-1.54774e7 + 8.93590e6i) q^{71} +(-2.68958e7 + 9.24245e6i) q^{72} +(-3.39917e6 + 5.88753e6i) q^{73} +(2.02023e6 + 5.55054e6i) q^{74} +(7.78259e6 + 3.01313e7i) q^{75} +(2.76353e6 - 2.31887e6i) q^{76} +(2.13001e7 - 5.85216e7i) q^{77} +(1.00451e7 + 2.20599e7i) q^{78} +(-6.45851e6 + 3.66280e7i) q^{79} +4.39169e6i q^{80} +(-5.93843e6 - 4.26351e7i) q^{81} -5.55296e7 q^{82} +(-4.07382e7 - 7.18324e6i) q^{83} +(1.03066e7 - 4.69315e6i) q^{84} +(3.57097e6 + 1.29973e6i) q^{85} +(5.78001e7 + 6.88835e7i) q^{86} +(-4.11867e6 + 1.06381e6i) q^{87} +(-6.66261e7 + 2.42499e7i) q^{88} +(6.06189e7 + 3.49983e7i) q^{89} +(-7.64397e6 - 1.49072e6i) q^{90} +(-3.84708e7 - 6.66334e7i) q^{91} +(-1.05324e7 + 1.25520e7i) q^{92} +(3.51412e7 - 2.75478e6i) q^{93} +(-1.75955e7 - 9.97888e7i) q^{94} +(7.75525e6 - 1.36746e6i) q^{95} +(-2.18113e7 - 1.04117e7i) q^{96} +(1.14315e8 + 9.59217e7i) q^{97} +(1.11970e8 - 6.46461e7i) q^{98} +(-1.67231e7 - 1.06008e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.5831 + 2.57139i 0.911443 + 0.160712i 0.609658 − 0.792664i \(-0.291307\pi\)
0.301784 + 0.953376i \(0.402418\pi\)
\(3\) −73.7172 + 33.5675i −0.910089 + 0.414413i
\(4\) −34.5070 − 12.5595i −0.134793 − 0.0490606i
\(5\) −51.5257 − 61.4059i −0.0824411 − 0.0982495i 0.723247 − 0.690589i \(-0.242649\pi\)
−0.805688 + 0.592340i \(0.798204\pi\)
\(6\) −1161.34 + 299.961i −0.896095 + 0.231452i
\(7\) 3577.75 − 1302.19i 1.49011 − 0.542355i 0.536630 − 0.843817i \(-0.319697\pi\)
0.953478 + 0.301462i \(0.0974748\pi\)
\(8\) −3753.91 − 2167.32i −0.916481 − 0.529130i
\(9\) 4307.45 − 4949.00i 0.656524 − 0.754305i
\(10\) −593.505 − 1027.98i −0.0593505 − 0.102798i
\(11\) 10514.1 − 12530.3i 0.718130 − 0.855834i −0.276318 − 0.961066i \(-0.589114\pi\)
0.994448 + 0.105233i \(0.0335587\pi\)
\(12\) 2965.35 − 232.459i 0.143005 − 0.0112104i
\(13\) −3509.19 − 19901.6i −0.122867 − 0.696811i −0.982552 − 0.185987i \(-0.940452\pi\)
0.859686 − 0.510824i \(-0.170660\pi\)
\(14\) 55523.1 − 9790.22i 1.44531 − 0.254847i
\(15\) 5859.57 + 2797.09i 0.115745 + 0.0552511i
\(16\) −41969.1 − 35216.3i −0.640398 − 0.537358i
\(17\) −41055.8 + 23703.6i −0.491563 + 0.283804i −0.725223 − 0.688514i \(-0.758263\pi\)
0.233660 + 0.972318i \(0.424930\pi\)
\(18\) 75541.7 − 61095.5i 0.719610 − 0.581995i
\(19\) −49120.0 + 85078.3i −0.376916 + 0.652837i −0.990612 − 0.136705i \(-0.956349\pi\)
0.613696 + 0.789542i \(0.289682\pi\)
\(20\) 1006.77 + 2766.07i 0.00629230 + 0.0172879i
\(21\) −220030. + 216090.i −1.13137 + 1.11111i
\(22\) 185549. − 155694.i 0.792077 − 0.664631i
\(23\) 152612. − 419299.i 0.545353 − 1.49835i −0.294564 − 0.955632i \(-0.595175\pi\)
0.839918 − 0.542714i \(-0.182603\pi\)
\(24\) 349479. + 33759.6i 1.05336 + 0.101754i
\(25\) 66715.5 − 378363.i 0.170792 − 0.968608i
\(26\) − 299251.i − 0.654850i
\(27\) −151408. + 509416.i −0.284901 + 0.958557i
\(28\) −139812. −0.227464
\(29\) 51718.6 + 9119.39i 0.0731232 + 0.0128936i 0.210090 − 0.977682i \(-0.432624\pi\)
−0.136967 + 0.990576i \(0.543735\pi\)
\(30\) 78258.2 + 55857.4i 0.0966150 + 0.0689597i
\(31\) −408929. − 148838.i −0.442793 − 0.161163i 0.110995 − 0.993821i \(-0.464596\pi\)
−0.553788 + 0.832657i \(0.686818\pi\)
\(32\) 191796. + 228573.i 0.182911 + 0.217984i
\(33\) −354464. + 1.27663e6i −0.298893 + 1.07649i
\(34\) −659672. + 240101.i −0.493642 + 0.179671i
\(35\) −264308. − 152599.i −0.176132 − 0.101690i
\(36\) −210794. + 116676.i −0.125501 + 0.0694656i
\(37\) 199444. + 345448.i 0.106418 + 0.184321i 0.914317 − 0.405000i \(-0.132729\pi\)
−0.807899 + 0.589321i \(0.799395\pi\)
\(38\) −935091. + 1.11440e6i −0.448456 + 0.534449i
\(39\) 926735. + 1.34930e6i 0.400587 + 0.583243i
\(40\) 60336.4 + 342185.i 0.0235689 + 0.133666i
\(41\) −3.69299e6 + 651173.i −1.30690 + 0.230442i −0.783365 − 0.621562i \(-0.786498\pi\)
−0.523535 + 0.852004i \(0.675387\pi\)
\(42\) −3.76437e6 + 2.58548e6i −1.20975 + 0.830890i
\(43\) 4.65176e6 + 3.90329e6i 1.36064 + 1.14171i 0.975784 + 0.218736i \(0.0701934\pi\)
0.384856 + 0.922977i \(0.374251\pi\)
\(44\) −520185. + 300329.i −0.138787 + 0.0801285i
\(45\) −525842. − 9502.57i −0.128235 − 0.00231735i
\(46\) 3.30374e6 − 5.72224e6i 0.737860 − 1.27801i
\(47\) −2.34037e6 − 6.43011e6i −0.479615 − 1.31773i −0.909821 − 0.415001i \(-0.863781\pi\)
0.430206 − 0.902731i \(-0.358441\pi\)
\(48\) 4.27597e6 + 1.18725e6i 0.805507 + 0.223654i
\(49\) 6.68849e6 − 5.61231e6i 1.16023 − 0.973548i
\(50\) 1.94584e6 − 5.34614e6i 0.311334 − 0.855383i
\(51\) 2.23085e6 − 3.12550e6i 0.329754 − 0.461997i
\(52\) −128863. + 730819.i −0.0176244 + 0.0999532i
\(53\) 2.29450e6i 0.290793i 0.989373 + 0.145397i \(0.0464458\pi\)
−0.989373 + 0.145397i \(0.953554\pi\)
\(54\) −3.51791e6 + 7.03953e6i −0.413723 + 0.827883i
\(55\) −1.31118e6 −0.143289
\(56\) −1.62528e7 − 2.86581e6i −1.65263 − 0.291404i
\(57\) 765126. − 7.92057e6i 0.0724826 − 0.750338i
\(58\) 730768. + 265978.i 0.0645754 + 0.0235035i
\(59\) 2.15123e6 + 2.56373e6i 0.177533 + 0.211575i 0.847471 − 0.530841i \(-0.178124\pi\)
−0.669938 + 0.742417i \(0.733680\pi\)
\(60\) −167066. − 170112.i −0.0128909 − 0.0131260i
\(61\) 2.36869e7 − 8.62134e6i 1.71076 − 0.622666i 0.713785 − 0.700365i \(-0.246979\pi\)
0.996977 + 0.0776989i \(0.0247573\pi\)
\(62\) −5.58072e6 − 3.22203e6i −0.377679 − 0.218053i
\(63\) 8.96643e6 − 2.33154e7i 0.569190 − 1.48007i
\(64\) 9.22193e6 + 1.59729e7i 0.549670 + 0.952056i
\(65\) −1.04126e6 + 1.24093e6i −0.0583321 + 0.0695174i
\(66\) −8.45189e6 + 1.77057e7i −0.445428 + 0.933121i
\(67\) −3.64008e6 − 2.06439e7i −0.180639 − 1.02446i −0.931431 − 0.363917i \(-0.881439\pi\)
0.750792 − 0.660539i \(-0.229672\pi\)
\(68\) 1.71442e6 − 302298.i 0.0801829 − 0.0141384i
\(69\) 2.82464e6 + 3.60323e7i 0.124614 + 1.58963i
\(70\) −3.46204e6 − 2.90500e6i −0.144192 − 0.120991i
\(71\) −1.54774e7 + 8.93590e6i −0.609068 + 0.351646i −0.772601 − 0.634892i \(-0.781044\pi\)
0.163533 + 0.986538i \(0.447711\pi\)
\(72\) −2.68958e7 + 9.24245e6i −1.00082 + 0.343920i
\(73\) −3.39917e6 + 5.88753e6i −0.119696 + 0.207320i −0.919647 − 0.392745i \(-0.871525\pi\)
0.799951 + 0.600065i \(0.204859\pi\)
\(74\) 2.02023e6 + 5.55054e6i 0.0673712 + 0.185101i
\(75\) 7.78259e6 + 3.01313e7i 0.245968 + 0.952298i
\(76\) 2.76353e6 − 2.31887e6i 0.0828341 − 0.0695061i
\(77\) 2.13001e7 − 5.85216e7i 0.605925 − 1.66477i
\(78\) 1.00451e7 + 2.20599e7i 0.271378 + 0.595971i
\(79\) −6.45851e6 + 3.66280e7i −0.165815 + 0.940384i 0.782405 + 0.622770i \(0.213993\pi\)
−0.948220 + 0.317614i \(0.897118\pi\)
\(80\) 4.39169e6i 0.107219i
\(81\) −5.93843e6 − 4.26351e7i −0.137953 − 0.990439i
\(82\) −5.55296e7 −1.22820
\(83\) −4.07382e7 − 7.18324e6i −0.858399 − 0.151359i −0.272910 − 0.962040i \(-0.587986\pi\)
−0.585489 + 0.810681i \(0.699097\pi\)
\(84\) 1.03066e7 − 4.69315e6i 0.207013 − 0.0942642i
\(85\) 3.57097e6 + 1.29973e6i 0.0684086 + 0.0248987i
\(86\) 5.78001e7 + 6.88835e7i 1.05666 + 1.25928i
\(87\) −4.11867e6 + 1.06381e6i −0.0718919 + 0.0185689i
\(88\) −6.66261e7 + 2.42499e7i −1.11100 + 0.404371i
\(89\) 6.06189e7 + 3.49983e7i 0.966158 + 0.557812i 0.898063 − 0.439867i \(-0.144975\pi\)
0.0680951 + 0.997679i \(0.478308\pi\)
\(90\) −7.64397e6 − 1.49072e6i −0.116506 − 0.0227210i
\(91\) −3.84708e7 − 6.66334e7i −0.561004 − 0.971687i
\(92\) −1.05324e7 + 1.25520e7i −0.147020 + 0.175211i
\(93\) 3.51412e7 − 2.75478e6i 0.469769 − 0.0368261i
\(94\) −1.75955e7 − 9.97888e7i −0.225366 − 1.27812i
\(95\) 7.75525e6 − 1.36746e6i 0.0952142 − 0.0167888i
\(96\) −2.18113e7 − 1.04117e7i −0.256801 − 0.122585i
\(97\) 1.14315e8 + 9.59217e7i 1.29127 + 1.08350i 0.991584 + 0.129463i \(0.0413255\pi\)
0.299683 + 0.954039i \(0.403119\pi\)
\(98\) 1.11970e8 − 6.46461e7i 1.21394 − 0.700871i
\(99\) −1.67231e7 − 1.06008e8i −0.174091 − 1.10356i
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.2.18 ✓ 138
3.2 odd 2 81.9.f.a.8.6 138
27.13 even 9 81.9.f.a.71.6 138
27.14 odd 18 inner 27.9.f.a.14.18 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.18 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.18 yes 138 27.14 odd 18 inner
81.9.f.a.8.6 138 3.2 odd 2
81.9.f.a.71.6 138 27.13 even 9