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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,5,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, 5); 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, 5, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 20.7
Character \(\chi\) \(=\) 27.20
Dual form 27.5.f.a.23.7

$q$-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+(0.498289 + 1.36904i) q^{2} +(8.88874 + 1.41078i) q^{3} +(10.6307 - 8.92025i) q^{4} +(-15.7451 + 2.77629i) q^{5} +(2.49776 + 12.8720i) q^{6} +(22.4150 + 18.8084i) q^{7} +(37.6967 + 21.7642i) q^{8} +(77.0194 + 25.0800i) q^{9} +(-11.6465 - 20.1723i) q^{10} +(-135.197 - 23.8388i) q^{11} +(107.078 - 64.2922i) q^{12} +(-169.717 - 61.7718i) q^{13} +(-14.5803 + 40.0591i) q^{14} +(-143.871 + 2.46490i) q^{15} +(27.5445 - 156.213i) q^{16} +(-265.471 + 153.270i) q^{17} +(4.04243 + 117.940i) q^{18} +(223.274 - 386.722i) q^{19} +(-142.617 + 169.964i) q^{20} +(172.707 + 198.806i) q^{21} +(-34.7308 - 196.968i) q^{22} +(-125.552 - 149.627i) q^{23} +(304.372 + 246.638i) q^{24} +(-347.107 + 126.337i) q^{25} -263.129i q^{26} +(649.223 + 331.587i) q^{27} +406.064 q^{28} +(320.474 + 880.496i) q^{29} +(-75.0639 - 195.737i) q^{30} +(115.628 - 97.0232i) q^{31} +(913.460 - 161.068i) q^{32} +(-1168.10 - 402.629i) q^{33} +(-342.114 - 287.068i) q^{34} +(-405.145 - 233.911i) q^{35} +(1042.49 - 420.413i) q^{36} +(944.419 + 1635.78i) q^{37} +(640.692 + 112.971i) q^{38} +(-1421.42 - 788.506i) q^{39} +(-653.963 - 238.023i) q^{40} +(796.872 - 2189.39i) q^{41} +(-186.115 + 335.505i) q^{42} +(-432.194 + 2451.10i) q^{43} +(-1649.89 + 952.564i) q^{44} +(-1282.31 - 181.060i) q^{45} +(142.284 - 246.443i) q^{46} +(-441.904 + 526.641i) q^{47} +(465.217 - 1349.68i) q^{48} +(-268.253 - 1521.34i) q^{49} +(-345.919 - 412.250i) q^{50} +(-2575.94 + 987.857i) q^{51} +(-2355.23 + 857.235i) q^{52} -3259.62i q^{53} +(-130.454 + 1054.04i) q^{54} +2194.87 q^{55} +(435.622 + 1196.86i) q^{56} +(2530.20 - 3122.48i) q^{57} +(-1045.74 + 877.483i) q^{58} +(5572.36 - 982.557i) q^{59} +(-1507.47 + 1309.57i) q^{60} +(4765.70 + 3998.90i) q^{61} +(190.444 + 109.953i) q^{62} +(1254.68 + 2010.79i) q^{63} +(-593.306 - 1027.64i) q^{64} +(2843.71 + 501.422i) q^{65} +(-30.8353 - 1799.79i) q^{66} +(-1895.82 - 690.021i) q^{67} +(-1454.95 + 3997.44i) q^{68} +(-904.909 - 1507.12i) q^{69} +(118.353 - 671.214i) q^{70} +(-1215.38 + 701.697i) q^{71} +(2357.53 + 2621.70i) q^{72} +(1186.68 - 2055.39i) q^{73} +(-1768.85 + 2108.04i) q^{74} +(-3263.57 + 633.283i) q^{75} +(-1076.09 - 6102.80i) q^{76} +(-2582.07 - 3077.19i) q^{77} +(371.215 - 2338.88i) q^{78} +(-4163.44 + 1515.37i) q^{79} +2536.06i q^{80} +(5302.98 + 3863.30i) q^{81} +3394.43 q^{82} +(353.761 + 971.950i) q^{83} +(3609.40 + 572.866i) q^{84} +(3754.36 - 3150.28i) q^{85} +(-3571.00 + 629.663i) q^{86} +(1606.43 + 8278.62i) q^{87} +(-4577.64 - 3841.09i) q^{88} +(-11599.9 - 6697.21i) q^{89} +(-391.083 - 1845.75i) q^{90} +(-2642.37 - 4576.73i) q^{91} +(-2669.42 - 470.691i) q^{92} +(1164.66 - 699.289i) q^{93} +(-941.187 - 342.564i) q^{94} +(-2441.82 + 6708.86i) q^{95} +(8346.74 - 143.002i) q^{96} +(1042.57 - 5912.70i) q^{97} +(1949.10 - 1125.32i) q^{98} +(-9814.89 - 5226.79i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9} - 3 q^{10} - 492 q^{11} - 339 q^{12} - 6 q^{13} + 1137 q^{14} + 1017 q^{15} - 54 q^{16} - 9 q^{17} + 603 q^{18} - 3 q^{19}+ \cdots - 162405 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.498289 + 1.36904i 0.124572 + 0.342259i 0.986265 − 0.165171i \(-0.0528176\pi\)
−0.861693 + 0.507430i \(0.830595\pi\)
\(3\) 8.88874 + 1.41078i 0.987638 + 0.156753i
\(4\) 10.6307 − 8.92025i 0.664421 − 0.557516i
\(5\) −15.7451 + 2.77629i −0.629805 + 0.111052i −0.479434 − 0.877578i \(-0.659158\pi\)
−0.150371 + 0.988630i \(0.548047\pi\)
\(6\) 2.49776 + 12.8720i 0.0693821 + 0.357555i
\(7\) 22.4150 + 18.8084i 0.457450 + 0.383846i 0.842192 − 0.539178i \(-0.181265\pi\)
−0.384742 + 0.923024i \(0.625710\pi\)
\(8\) 37.6967 + 21.7642i 0.589011 + 0.340066i
\(9\) 77.0194 + 25.0800i 0.950857 + 0.309630i
\(10\) −11.6465 − 20.1723i −0.116465 − 0.201723i
\(11\) −135.197 − 23.8388i −1.11733 − 0.197015i −0.415661 − 0.909520i \(-0.636450\pi\)
−0.701667 + 0.712505i \(0.747561\pi\)
\(12\) 107.078 − 64.2922i 0.743600 − 0.446474i
\(13\) −169.717 − 61.7718i −1.00424 − 0.365514i −0.213022 − 0.977047i \(-0.568331\pi\)
−0.791219 + 0.611534i \(0.790553\pi\)
\(14\) −14.5803 + 40.0591i −0.0743893 + 0.204383i
\(15\) −143.871 + 2.46490i −0.639427 + 0.0109551i
\(16\) 27.5445 − 156.213i 0.107596 − 0.610206i
\(17\) −265.471 + 153.270i −0.918586 + 0.530346i −0.883184 − 0.469027i \(-0.844605\pi\)
−0.0354023 + 0.999373i \(0.511271\pi\)
\(18\) 4.04243 + 117.940i 0.0124766 + 0.364011i
\(19\) 223.274 − 386.722i 0.618488 − 1.07125i −0.371274 − 0.928523i \(-0.621079\pi\)
0.989762 − 0.142729i \(-0.0455877\pi\)
\(20\) −142.617 + 169.964i −0.356543 + 0.424911i
\(21\) 172.707 + 198.806i 0.391626 + 0.450807i
\(22\) −34.7308 − 196.968i −0.0717578 − 0.406959i
\(23\) −125.552 − 149.627i −0.237338 − 0.282849i 0.634207 − 0.773163i \(-0.281327\pi\)
−0.871546 + 0.490314i \(0.836882\pi\)
\(24\) 304.372 + 246.638i 0.528424 + 0.428191i
\(25\) −347.107 + 126.337i −0.555371 + 0.202138i
\(26\) − 263.129i − 0.389244i
\(27\) 649.223 + 331.587i 0.890567 + 0.454852i
\(28\) 406.064 0.517939
\(29\) 320.474 + 880.496i 0.381064 + 1.04696i 0.970909 + 0.239448i \(0.0769663\pi\)
−0.589846 + 0.807516i \(0.700811\pi\)
\(30\) −75.0639 − 195.737i −0.0834043 − 0.217485i
\(31\) 115.628 − 97.0232i 0.120320 − 0.100961i −0.580642 − 0.814159i \(-0.697198\pi\)
0.700962 + 0.713198i \(0.252754\pi\)
\(32\) 913.460 − 161.068i 0.892051 − 0.157293i
\(33\) −1168.10 − 402.629i −1.07263 − 0.369724i
\(34\) −342.114 − 287.068i −0.295946 − 0.248328i
\(35\) −405.145 − 233.911i −0.330731 − 0.190947i
\(36\) 1042.49 − 420.413i 0.804393 − 0.324393i
\(37\) 944.419 + 1635.78i 0.689860 + 1.19487i 0.971883 + 0.235465i \(0.0756614\pi\)
−0.282022 + 0.959408i \(0.591005\pi\)
\(38\) 640.692 + 112.971i 0.443692 + 0.0782349i
\(39\) −1421.42 − 788.506i −0.934531 − 0.518413i
\(40\) −653.963 − 238.023i −0.408727 − 0.148764i
\(41\) 796.872 − 2189.39i 0.474046 − 1.30243i −0.440428 − 0.897788i \(-0.645173\pi\)
0.914474 − 0.404644i \(-0.132604\pi\)
\(42\) −186.115 + 335.505i −0.105507 + 0.190196i
\(43\) −432.194 + 2451.10i −0.233745 + 1.32563i 0.611496 + 0.791247i \(0.290568\pi\)
−0.845241 + 0.534385i \(0.820543\pi\)
\(44\) −1649.89 + 952.564i −0.852215 + 0.492027i
\(45\) −1282.31 − 181.060i −0.633239 − 0.0894123i
\(46\) 142.284 − 246.443i 0.0672419 − 0.116466i
\(47\) −441.904 + 526.641i −0.200047 + 0.238407i −0.856737 − 0.515754i \(-0.827512\pi\)
0.656689 + 0.754161i \(0.271956\pi\)
\(48\) 465.217 − 1349.68i 0.201917 − 0.585797i
\(49\) −268.253 − 1521.34i −0.111726 − 0.633627i
\(50\) −345.919 − 412.250i −0.138368 − 0.164900i
\(51\) −2575.94 + 987.857i −0.990364 + 0.379799i
\(52\) −2355.23 + 857.235i −0.871019 + 0.317025i
\(53\) − 3259.62i − 1.16042i −0.814467 − 0.580210i \(-0.802971\pi\)
0.814467 − 0.580210i \(-0.197029\pi\)
\(54\) −130.454 + 1054.04i −0.0447374 + 0.361467i
\(55\) 2194.87 0.725577
\(56\) 435.622 + 1196.86i 0.138910 + 0.381653i
\(57\) 2530.20 − 3122.48i 0.778764 − 0.961059i
\(58\) −1045.74 + 877.483i −0.310863 + 0.260845i
\(59\) 5572.36 − 982.557i 1.60079 − 0.282263i 0.699225 − 0.714901i \(-0.253528\pi\)
0.901568 + 0.432638i \(0.142417\pi\)
\(60\) −1507.47 + 1309.57i −0.418741 + 0.363769i
\(61\) 4765.70 + 3998.90i 1.28076 + 1.07468i 0.993139 + 0.116939i \(0.0373081\pi\)
0.287619 + 0.957745i \(0.407136\pi\)
\(62\) 190.444 + 109.953i 0.0495433 + 0.0286038i
\(63\) 1254.68 + 2010.79i 0.316119 + 0.506623i
\(64\) −593.306 − 1027.64i −0.144850 − 0.250888i
\(65\) 2843.71 + 501.422i 0.673067 + 0.118680i
\(66\) −30.8353 − 1799.79i −0.00707881 − 0.413176i
\(67\) −1895.82 − 690.021i −0.422325 − 0.153714i 0.122110 − 0.992517i \(-0.461034\pi\)
−0.544435 + 0.838803i \(0.683256\pi\)
\(68\) −1454.95 + 3997.44i −0.314652 + 0.864499i
\(69\) −904.909 − 1507.12i −0.190067 − 0.316556i
\(70\) 118.353 − 671.214i 0.0241537 − 0.136982i
\(71\) −1215.38 + 701.697i −0.241098 + 0.139198i −0.615681 − 0.787995i \(-0.711119\pi\)
0.374583 + 0.927193i \(0.377786\pi\)
\(72\) 2357.53 + 2621.70i 0.454771 + 0.505730i
\(73\) 1186.68 − 2055.39i 0.222684 − 0.385699i −0.732938 − 0.680295i \(-0.761852\pi\)
0.955622 + 0.294596i \(0.0951850\pi\)
\(74\) −1768.85 + 2108.04i −0.323019 + 0.384959i
\(75\) −3263.57 + 633.283i −0.580191 + 0.112584i
\(76\) −1076.09 − 6102.80i −0.186303 − 1.05658i
\(77\) −2582.07 − 3077.19i −0.435498 − 0.519006i
\(78\) 371.215 − 2338.88i 0.0610150 − 0.384432i
\(79\) −4163.44 + 1515.37i −0.667112 + 0.242809i −0.653304 − 0.757096i \(-0.726618\pi\)
−0.0138079 + 0.999905i \(0.504395\pi\)
\(80\) 2536.06i 0.396259i
\(81\) 5302.98 + 3863.30i 0.808259 + 0.588828i
\(82\) 3394.43 0.504822
\(83\) 353.761 + 971.950i 0.0513515 + 0.141087i 0.962717 − 0.270511i \(-0.0871928\pi\)
−0.911365 + 0.411599i \(0.864971\pi\)
\(84\) 3609.40 + 572.866i 0.511536 + 0.0811884i
\(85\) 3754.36 − 3150.28i 0.519634 − 0.436025i
\(86\) −3571.00 + 629.663i −0.482828 + 0.0851357i
\(87\) 1606.43 + 8278.62i 0.212238 + 1.09375i
\(88\) −4577.64 − 3841.09i −0.591121 − 0.496009i
\(89\) −11599.9 − 6697.21i −1.46445 − 0.845500i −0.465237 − 0.885186i \(-0.654031\pi\)
−0.999212 + 0.0396855i \(0.987364\pi\)
\(90\) −391.083 − 1845.75i −0.0482819 − 0.227870i
\(91\) −2642.37 − 4576.73i −0.319089 − 0.552678i
\(92\) −2669.42 − 470.691i −0.315385 − 0.0556109i
\(93\) 1164.66 − 699.289i 0.134659 − 0.0808520i
\(94\) −941.187 − 342.564i −0.106517 − 0.0387691i
\(95\) −2441.82 + 6708.86i −0.270562 + 0.743364i
\(96\) 8346.74 − 143.002i 0.905680 − 0.0155167i
\(97\) 1042.57 − 5912.70i 0.110806 − 0.628409i −0.877936 − 0.478777i \(-0.841080\pi\)
0.988742 − 0.149632i \(-0.0478088\pi\)
\(98\) 1949.10 − 1125.32i 0.202947 − 0.117172i
\(99\) −9814.89 − 5226.79i −1.00142 − 0.533291i
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 27.5.f.a.20.7 ✓ 66
3.2 odd 2 81.5.f.a.62.5 66
27.4 even 9 81.5.f.a.17.5 66
27.23 odd 18 inner 27.5.f.a.23.7 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.f.a.20.7 ✓ 66 1.1 even 1 trivial
27.5.f.a.23.7 yes 66 27.23 odd 18 inner
81.5.f.a.17.5 66 27.4 even 9
81.5.f.a.62.5 66 3.2 odd 2