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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 2.2
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.2

$q$-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+(-29.3275 - 5.17124i) q^{2} +(80.7896 - 5.83408i) q^{3} +(592.802 + 215.762i) q^{4} +(641.275 + 764.242i) q^{5} +(-2399.53 - 246.683i) q^{6} +(548.101 - 199.492i) q^{7} +(-9667.37 - 5581.46i) q^{8} +(6492.93 - 942.667i) q^{9} +(-14855.0 - 25729.5i) q^{10} +(-13801.8 + 16448.3i) q^{11} +(49151.0 + 13972.9i) q^{12} +(1920.53 + 10891.9i) q^{13} +(-17106.1 + 3016.26i) q^{14} +(56267.0 + 58001.6i) q^{15} +(130944. + 109875. i) q^{16} +(-39828.4 + 22994.9i) q^{17} +(-195296. - 5930.36i) q^{18} +(81975.6 - 141986. i) q^{19} +(215255. + 591407. i) q^{20} +(43117.0 - 19314.6i) q^{21} +(489830. - 411016. i) q^{22} +(-95544.6 + 262507. i) q^{23} +(-813586. - 394524. i) q^{24} +(-105001. + 595489. i) q^{25} -329363. i q^{26} +(519062. - 114038. i) q^{27} +367958. q^{28} +(-45440.2 - 8012.33i) q^{29} +(-1.35023e6 - 1.99201e6i) q^{30} +(-821639. - 299052. i) q^{31} +(-1.43517e6 - 1.71036e6i) q^{32} +(-1.01908e6 + 1.40937e6i) q^{33} +(1.28698e6 - 468423. i) q^{34} +(503944. + 290952. i) q^{35} +(4.05241e6 + 842114. i) q^{36} +(1.60612e6 + 2.78188e6i) q^{37} +(-3.13839e6 + 3.74018e6i) q^{38} +(218703. + 868746. i) q^{39} +(-1.93386e6 - 1.09675e7i) q^{40} +(3.61462e6 - 637355. i) q^{41} +(-1.36440e6 + 343481. i) q^{42} +(1.30310e6 + 1.09343e6i) q^{43} +(-1.17306e7 + 6.77269e6i) q^{44} +(4.88418e6 + 4.35766e6i) q^{45} +(4.15957e6 - 7.20459e6i) q^{46} +(-430113. - 1.18173e6i) q^{47} +(1.12199e7 + 8.11280e6i) q^{48} +(-4.15548e6 + 3.48686e6i) q^{49} +(6.15882e6 - 1.69212e7i) q^{50} +(-3.08357e6 + 2.09011e6i) q^{51} +(-1.21156e6 + 6.87110e6i) q^{52} -1.10276e7i q^{53} +(-1.58125e7 + 660264. i) q^{54} -2.14212e7 q^{55} +(-6.41215e6 - 1.13064e6i) q^{56} +(5.79442e6 - 1.19492e7i) q^{57} +(1.29122e6 + 469964. i) q^{58} +(3.23014e6 + 3.84954e6i) q^{59} +(2.08407e7 + 4.65237e7i) q^{60} +(1.28633e7 - 4.68185e6i) q^{61} +(2.25502e7 + 1.30194e7i) q^{62} +(3.37072e6 - 1.81197e6i) q^{63} +(1.13655e7 + 1.96857e7i) q^{64} +(-7.09244e6 + 8.45244e6i) q^{65} +(3.71753e7 - 3.60636e7i) q^{66} +(1.11806e6 + 6.34085e6i) q^{67} +(-2.85718e7 + 5.03798e6i) q^{68} +(-6.18753e6 + 2.17652e7i) q^{69} +(-1.32749e7 - 1.11389e7i) q^{70} +(8.62738e6 - 4.98102e6i) q^{71} +(-6.80310e7 - 2.71269e7i) q^{72} +(-4.76165e6 + 8.24742e6i) q^{73} +(-3.27178e7 - 8.98913e7i) q^{74} +(-5.00884e6 + 4.87219e7i) q^{75} +(7.92305e7 - 6.64823e7i) q^{76} +(-4.28345e6 + 1.17687e7i) q^{77} +(-1.92153e6 - 2.66091e7i) q^{78} +(7.11564e6 - 4.03548e7i) q^{79} +1.70532e8i q^{80} +(4.12695e7 - 1.22413e7i) q^{81} -1.09304e8 q^{82} +(-3.59336e7 - 6.33606e6i) q^{83} +(2.97272e7 - 2.14670e6i) q^{84} +(-4.31147e7 - 1.56925e7i) q^{85} +(-3.25623e7 - 3.88062e7i) q^{86} +(-3.71784e6 - 382211. i) q^{87} +(2.25232e8 - 8.19779e7i) q^{88} +(5.60251e7 + 3.23461e7i) q^{89} +(-1.20706e8 - 1.53057e8i) q^{90} +(3.22549e6 + 5.58671e6i) q^{91} +(-1.13278e8 + 1.34999e8i) q^{92} +(-6.81246e7 - 1.93668e7i) q^{93} +(6.50318e6 + 3.68814e7i) q^{94} +(1.61081e8 - 2.84028e7i) q^{95} +(-1.25925e8 - 1.29807e8i) q^{96} +(3.13252e7 + 2.62850e7i) q^{97} +(1.39901e8 - 8.07721e7i) q^{98} +(-7.41086e7 + 1.19808e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.3275 − 5.17124i −1.83297 − 0.323202i −0.852935 − 0.522018i \(-0.825179\pi\)
−0.980037 + 0.198816i \(0.936291\pi\)
\(3\) 80.7896 − 5.83408i 0.997403 − 0.0720257i
\(4\) 592.802 + 215.762i 2.31563 + 0.842821i
\(5\) 641.275 + 764.242i 1.02604 + 1.22279i 0.974563 + 0.224113i \(0.0719485\pi\)
0.0514773 + 0.998674i \(0.483607\pi\)
\(6\) −2399.53 − 246.683i −1.85149 − 0.190342i
\(7\) 548.101 − 199.492i 0.228280 − 0.0830872i −0.225348 − 0.974278i \(-0.572352\pi\)
0.453628 + 0.891191i \(0.350130\pi\)
\(8\) −9667.37 − 5581.46i −2.36020 − 1.36266i
\(9\) 6492.93 − 942.667i 0.989625 − 0.143677i
\(10\) −14855.0 − 25729.5i −1.48550 − 2.57295i
\(11\) −13801.8 + 16448.3i −0.942680 + 1.12344i 0.0495187 + 0.998773i \(0.484231\pi\)
−0.992198 + 0.124669i \(0.960213\pi\)
\(12\) 49151.0 + 13972.9i 2.37032 + 0.673847i
\(13\) 1920.53 + 10891.9i 0.0672431 + 0.381355i 0.999794 + 0.0203151i \(0.00646695\pi\)
−0.932551 + 0.361040i \(0.882422\pi\)
\(14\) −17106.1 + 3016.26i −0.445285 + 0.0785158i
\(15\) 56267.0 + 58001.6i 1.11145 + 1.14571i
\(16\) 130944. + 109875.i 1.99804 + 1.67655i
\(17\) −39828.4 + 22994.9i −0.476867 + 0.275319i −0.719110 − 0.694896i \(-0.755450\pi\)
0.242243 + 0.970216i \(0.422117\pi\)
\(18\) −195296. − 5930.36i −1.86039 − 0.0564925i
\(19\) 81975.6 − 141986.i 0.629028 − 1.08951i −0.358719 − 0.933446i \(-0.616786\pi\)
0.987747 − 0.156063i \(-0.0498804\pi\)
\(20\) 215255. + 591407.i 1.34534 + 3.69629i
\(21\) 43117.0 − 19314.6i 0.221703 − 0.0993135i
\(22\) 489830. − 411016.i 2.09100 − 1.75456i
\(23\) −95544.6 + 262507.i −0.341425 + 0.938056i 0.643557 + 0.765398i \(0.277458\pi\)
−0.984982 + 0.172658i \(0.944764\pi\)
\(24\) −813586. − 394524.i −2.45221 − 1.18913i
\(25\) −105001. + 595489.i −0.268802 + 1.52445i
\(26\) − 329363.i − 0.720745i
\(27\) 519062. − 114038.i 0.976706 − 0.214583i
\(28\) 367958. 0.598641
\(29\) −45440.2 − 8012.33i −0.0642463 − 0.0113284i 0.141433 − 0.989948i \(-0.454829\pi\)
−0.205679 + 0.978620i \(0.565940\pi\)
\(30\) −1.35023e6 − 1.99201e6i −1.66696 − 2.45928i
\(31\) −821639. − 299052.i −0.889681 − 0.323817i −0.143571 − 0.989640i \(-0.545859\pi\)
−0.746110 + 0.665823i \(0.768081\pi\)
\(32\) −1.43517e6 − 1.71036e6i −1.36868 − 1.63113i
\(33\) −1.01908e6 + 1.40937e6i −0.859315 + 1.18842i
\(34\) 1.28698e6 − 468423.i 0.963067 − 0.350528i
\(35\) 503944. + 290952.i 0.335823 + 0.193887i
\(36\) 4.05241e6 + 842114.i 2.41270 + 0.501373i
\(37\) 1.60612e6 + 2.78188e6i 0.856980 + 1.48433i 0.874796 + 0.484492i \(0.160996\pi\)
−0.0178154 + 0.999841i \(0.505671\pi\)
\(38\) −3.13839e6 + 3.74018e6i −1.50512 + 1.79374i
\(39\) 218703. + 868746.i 0.0945358 + 0.375521i
\(40\) −1.93386e6 − 1.09675e7i −0.755414 − 4.28417i
\(41\) 3.61462e6 − 637355.i 1.27917 − 0.225552i 0.507541 − 0.861628i \(-0.330555\pi\)
0.771626 + 0.636076i \(0.219444\pi\)
\(42\) −1.36440e6 + 343481.i −0.438473 + 0.110384i
\(43\) 1.30310e6 + 1.09343e6i 0.381156 + 0.319828i 0.813156 − 0.582046i \(-0.197747\pi\)
−0.432000 + 0.901874i \(0.642192\pi\)
\(44\) −1.17306e7 + 6.77269e6i −3.12976 + 1.80697i
\(45\) 4.88418e6 + 4.35766e6i 1.19108 + 1.06268i
\(46\) 4.15957e6 − 7.20459e6i 0.929004 − 1.60908i
\(47\) −430113. − 1.18173e6i −0.0881437 − 0.242173i 0.887787 − 0.460255i \(-0.152242\pi\)
−0.975930 + 0.218082i \(0.930020\pi\)
\(48\) 1.12199e7 + 8.11280e6i 2.11361 + 1.52829i
\(49\) −4.15548e6 + 3.48686e6i −0.720836 + 0.604853i
\(50\) 6.15882e6 − 1.69212e7i 0.985412 − 2.70740i
\(51\) −3.08357e6 + 2.09011e6i −0.455798 + 0.308951i
\(52\) −1.21156e6 + 6.87110e6i −0.165703 + 0.939751i
\(53\) − 1.10276e7i − 1.39758i −0.715328 − 0.698788i \(-0.753723\pi\)
0.715328 − 0.698788i \(-0.246277\pi\)
\(54\) −1.58125e7 + 660264.i −1.85963 + 0.0776502i
\(55\) −2.14212e7 −2.34096
\(56\) −6.41215e6 − 1.13064e6i −0.652006 − 0.114966i
\(57\) 5.79442e6 − 1.19492e7i 0.548922 − 1.13199i
\(58\) 1.29122e6 + 469964.i 0.114100 + 0.0415291i
\(59\) 3.23014e6 + 3.84954e6i 0.266572 + 0.317688i 0.882681 − 0.469973i \(-0.155736\pi\)
−0.616109 + 0.787661i \(0.711292\pi\)
\(60\) 2.08407e7 + 4.65237e7i 1.60808 + 3.58979i
\(61\) 1.28633e7 − 4.68185e6i 0.929036 − 0.338141i 0.167208 − 0.985922i \(-0.446525\pi\)
0.761827 + 0.647780i \(0.224303\pi\)
\(62\) 2.25502e7 + 1.30194e7i 1.52610 + 0.881095i
\(63\) 3.37072e6 − 1.81197e6i 0.213974 − 0.115024i
\(64\) 1.13655e7 + 1.96857e7i 0.677440 + 1.17336i
\(65\) −7.09244e6 + 8.45244e6i −0.397322 + 0.473509i
\(66\) 3.71753e7 − 3.60636e7i 1.95920 − 1.90061i
\(67\) 1.11806e6 + 6.34085e6i 0.0554839 + 0.314665i 0.999901 − 0.0140853i \(-0.00448365\pi\)
−0.944417 + 0.328750i \(0.893373\pi\)
\(68\) −2.85718e7 + 5.03798e6i −1.33629 + 0.235625i
\(69\) −6.18753e6 + 2.17652e7i −0.272974 + 0.960211i
\(70\) −1.32749e7 − 1.11389e7i −0.552889 − 0.463929i
\(71\) 8.62738e6 − 4.98102e6i 0.339505 − 0.196013i −0.320548 − 0.947232i \(-0.603867\pi\)
0.660053 + 0.751219i \(0.270534\pi\)
\(72\) −6.80310e7 − 2.71269e7i −2.53149 − 1.00942i
\(73\) −4.76165e6 + 8.24742e6i −0.167674 + 0.290420i −0.937602 − 0.347711i \(-0.886959\pi\)
0.769928 + 0.638131i \(0.220292\pi\)
\(74\) −3.27178e7 − 8.98913e7i −1.09108 − 2.99772i
\(75\) −5.00884e6 + 4.87219e7i −0.158304 + 1.53985i
\(76\) 7.92305e7 − 6.64823e7i 2.37486 − 1.99274i
\(77\) −4.28345e6 + 1.17687e7i −0.121851 + 0.334784i
\(78\) −1.92153e6 − 2.66091e7i −0.0519122 − 0.718873i
\(79\) 7.11564e6 − 4.03548e7i 0.182686 − 1.03606i −0.746206 − 0.665715i \(-0.768127\pi\)
0.928893 − 0.370349i \(-0.120762\pi\)
\(80\) 1.70532e8i 4.16339i
\(81\) 4.12695e7 − 1.22413e7i 0.958714 − 0.284373i
\(82\) −1.09304e8 −2.41758
\(83\) −3.59336e7 − 6.33606e6i −0.757160 − 0.133508i −0.218276 − 0.975887i \(-0.570043\pi\)
−0.538885 + 0.842379i \(0.681154\pi\)
\(84\) 2.97272e7 − 2.14670e6i 0.597086 − 0.0431175i
\(85\) −4.31147e7 − 1.56925e7i −0.825941 − 0.300618i
\(86\) −3.25623e7 − 3.88062e7i −0.595280 − 0.709427i
\(87\) −3.71784e6 − 382211.i −0.0648954 − 0.00667155i
\(88\) 2.25232e8 − 8.19779e7i 3.75578 − 1.36699i
\(89\) 5.60251e7 + 3.23461e7i 0.892941 + 0.515540i 0.874903 − 0.484298i \(-0.160925\pi\)
0.0180375 + 0.999837i \(0.494258\pi\)
\(90\) −1.20706e8 − 1.53057e8i −1.83976 − 2.33283i
\(91\) 3.22549e6 + 5.58671e6i 0.0470360 + 0.0814687i
\(92\) −1.13278e8 + 1.34999e8i −1.58123 + 1.88443i
\(93\) −6.81246e7 − 1.93668e7i −0.910694 − 0.258897i
\(94\) 6.50318e6 + 3.68814e7i 0.0832941 + 0.472384i
\(95\) 1.61081e8 − 2.84028e7i 1.97765 − 0.348712i
\(96\) −1.25925e8 − 1.29807e8i −1.48261 − 1.52831i
\(97\) 3.13252e7 + 2.62850e7i 0.353840 + 0.296907i 0.802330 − 0.596881i \(-0.203594\pi\)
−0.448490 + 0.893788i \(0.648038\pi\)
\(98\) 1.39901e8 − 8.07721e7i 1.51676 − 0.875703i
\(99\) −7.41086e7 + 1.19808e8i −0.771486 + 1.24723i
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.2.2 ✓ 138
3.2 odd 2 81.9.f.a.8.22 138
27.13 even 9 81.9.f.a.71.22 138
27.14 odd 18 inner 27.9.f.a.14.2 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.2 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.2 yes 138 27.14 odd 18 inner
81.9.f.a.8.22 138 3.2 odd 2
81.9.f.a.71.22 138 27.13 even 9