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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 14.22
Character \(\chi\) \(=\) 27.14
Dual form 27.9.f.a.2.22

$q$-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+(26.8116 - 4.72761i) q^{2} +(-20.8997 - 78.2573i) q^{3} +(455.951 - 165.953i) q^{4} +(232.968 - 277.641i) q^{5} +(-930.324 - 1999.40i) q^{6} +(45.0827 + 16.4088i) q^{7} +(5404.32 - 3120.19i) q^{8} +(-5687.41 + 3271.10i) q^{9} +(4933.68 - 8545.38i) q^{10} +(-735.313 - 876.312i) q^{11} +(-22516.2 - 32213.1i) q^{12} +(6621.57 - 37552.8i) q^{13} +(1286.31 + 226.812i) q^{14} +(-26596.4 - 12428.9i) q^{15} +(34993.7 - 29363.2i) q^{16} +(67597.9 + 39027.7i) q^{17} +(-137024. + 114591. i) q^{18} +(113236. + 196131. i) q^{19} +(60146.9 - 165252. i) q^{20} +(341.892 - 3870.99i) q^{21} +(-23857.8 - 20019.1i) q^{22} +(-82558.0 - 226826. i) q^{23} +(-357126. - 357717. i) q^{24} +(45021.1 + 255328. i) q^{25} -1.03816e6i q^{26} +(374853. + 376716. i) q^{27} +23278.6 q^{28} +(572834. - 101006. i) q^{29} +(-771850. - 207501. i) q^{30} +(-24300.0 + 8844.49i) q^{31} +(-227456. + 271071. i) q^{32} +(-53210.0 + 75858.3i) q^{33} +(1.99692e6 + 726818. i) q^{34} +(15058.6 - 8694.08i) q^{35} +(-2.05033e6 + 2.43530e6i) q^{36} +(-1.45625e6 + 2.52229e6i) q^{37} +(3.96328e6 + 4.72325e6i) q^{38} +(-3.07717e6 + 266655. i) q^{39} +(392744. - 2.22736e6i) q^{40} +(535074. + 94347.9i) q^{41} +(-9133.85 - 105404. i) q^{42} +(4.54620e6 - 3.81472e6i) q^{43} +(-480693. - 277528. i) q^{44} +(-416794. + 2.34112e6i) q^{45} +(-3.28586e6 - 5.69128e6i) q^{46} +(-3.01327e6 + 8.27890e6i) q^{47} +(-3.02924e6 - 2.12483e6i) q^{48} +(-4.41433e6 - 3.70406e6i) q^{49} +(2.41418e6 + 6.63290e6i) q^{50} +(1.64143e6 - 6.10569e6i) q^{51} +(-3.21287e6 - 1.82211e7i) q^{52} -3.89255e6i q^{53} +(1.18314e7 + 8.32821e6i) q^{54} -414605. q^{55} +(294840. - 51988.2i) q^{56} +(1.29821e7 - 1.29606e7i) q^{57} +(1.48811e7 - 5.41627e6i) q^{58} +(2.64580e6 - 3.15314e6i) q^{59} +(-1.41892e7 - 1.25322e6i) q^{60} +(-1.75097e7 - 6.37302e6i) q^{61} +(-609710. + 352016. i) q^{62} +(-310078. + 54146.8i) q^{63} +(-1.06641e7 + 1.84708e7i) q^{64} +(-8.88358e6 - 1.05870e7i) q^{65} +(-1.06802e6 + 2.28544e6i) q^{66} +(-493265. + 2.79744e6i) q^{67} +(3.72981e7 + 6.57666e6i) q^{68} +(-1.60254e7 + 1.12014e7i) q^{69} +(362643. - 304293. i) q^{70} +(893484. + 515853. i) q^{71} +(-2.05301e7 + 3.54239e7i) q^{72} +(-1.26668e7 - 2.19395e7i) q^{73} +(-2.71199e7 + 7.45114e7i) q^{74} +(1.90403e7 - 8.85949e6i) q^{75} +(8.41786e7 + 7.06343e7i) q^{76} +(-18770.7 - 51572.1i) q^{77} +(-8.12433e7 + 2.16971e7i) q^{78} +(-1.00312e7 - 5.68895e7i) q^{79} -1.65564e7i q^{80} +(2.16465e7 - 3.72082e7i) q^{81} +1.47922e7 q^{82} +(1.09503e7 - 1.93083e6i) q^{83} +(-486515. - 1.82172e6i) q^{84} +(2.65838e7 - 9.67572e6i) q^{85} +(1.03856e8 - 1.23771e8i) q^{86} +(-1.98765e7 - 4.27175e7i) q^{87} +(-6.70813e6 - 2.44156e6i) q^{88} +(-2.45007e7 + 1.41455e7i) q^{89} +(-107023. + 6.47396e7i) q^{90} +(914714. - 1.58433e6i) q^{91} +(-7.52848e7 - 8.97209e7i) q^{92} +(1.20001e6 + 1.71681e6i) q^{93} +(-4.16513e7 + 2.36216e8i) q^{94} +(8.08344e7 + 1.42533e7i) q^{95} +(2.59670e7 + 1.21348e7i) q^{96} +(-4.04609e7 + 3.39507e7i) q^{97} +(-1.35867e8 - 7.84427e7i) q^{98} +(7.04853e6 + 2.57866e6i) 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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 26.8116 − 4.72761i 1.67573 − 0.295476i 0.746609 − 0.665263i \(-0.231680\pi\)
0.929117 + 0.369787i \(0.120569\pi\)
\(3\) −20.8997 − 78.2573i −0.258021 − 0.966139i
\(4\) 455.951 − 165.953i 1.78106 − 0.648252i
\(5\) 232.968 − 277.641i 0.372749 − 0.444225i −0.546763 − 0.837288i \(-0.684140\pi\)
0.919512 + 0.393063i \(0.128584\pi\)
\(6\) −930.324 − 1999.40i −0.717842 − 1.54275i
\(7\) 45.0827 + 16.4088i 0.0187766 + 0.00683414i 0.351391 − 0.936229i \(-0.385709\pi\)
−0.332615 + 0.943063i \(0.607931\pi\)
\(8\) 5404.32 − 3120.19i 1.31941 − 0.761764i
\(9\) −5687.41 + 3271.10i −0.866851 + 0.498568i
\(10\) 4933.68 − 8545.38i 0.493368 − 0.854538i
\(11\) −735.313 − 876.312i −0.0502229 − 0.0598533i 0.740348 − 0.672224i \(-0.234661\pi\)
−0.790571 + 0.612370i \(0.790216\pi\)
\(12\) −22516.2 − 32213.1i −1.08585 − 1.55349i
\(13\) 6621.57 − 37552.8i 0.231840 − 1.31483i −0.617328 − 0.786706i \(-0.711785\pi\)
0.849168 − 0.528123i \(-0.177104\pi\)
\(14\) 1286.31 + 226.812i 0.0334838 + 0.00590410i
\(15\) −26596.4 − 12428.9i −0.525360 − 0.245508i
\(16\) 34993.7 − 29363.2i 0.533962 − 0.448047i
\(17\) 67597.9 + 39027.7i 0.809352 + 0.467280i 0.846731 − 0.532022i \(-0.178567\pi\)
−0.0373788 + 0.999301i \(0.511901\pi\)
\(18\) −137024. + 114591.i −1.30529 + 1.09160i
\(19\) 113236. + 196131.i 0.868903 + 1.50498i 0.863119 + 0.505000i \(0.168508\pi\)
0.00578360 + 0.999983i \(0.498159\pi\)
\(20\) 60146.9 − 165252.i 0.375918 − 1.03283i
\(21\) 341.892 − 3870.99i 0.00175797 − 0.0199042i
\(22\) −23857.8 − 20019.1i −0.101845 − 0.0854581i
\(23\) −82558.0 − 226826.i −0.295018 − 0.810554i −0.995313 − 0.0967014i \(-0.969171\pi\)
0.700296 − 0.713853i \(-0.253051\pi\)
\(24\) −357126. − 357717.i −1.07641 − 1.07819i
\(25\) 45021.1 + 255328.i 0.115254 + 0.653639i
\(26\) − 1.03816e6i − 2.27179i
\(27\) 374853. + 376716.i 0.705351 + 0.708858i
\(28\) 23278.6 0.0378725
\(29\) 572834. − 101006.i 0.809910 − 0.142809i 0.246667 − 0.969100i \(-0.420665\pi\)
0.563243 + 0.826291i \(0.309553\pi\)
\(30\) −771850. − 207501.i −0.952902 − 0.256174i
\(31\) −24300.0 + 8844.49i −0.0263124 + 0.00957693i −0.355143 − 0.934812i \(-0.615568\pi\)
0.328830 + 0.944389i \(0.393346\pi\)
\(32\) −227456. + 271071.i −0.216919 + 0.258514i
\(33\) −53210.0 + 75858.3i −0.0448681 + 0.0639657i
\(34\) 1.99692e6 + 726818.i 1.49432 + 0.543889i
\(35\) 15058.6 − 8694.08i 0.0100349 − 0.00579364i
\(36\) −2.05033e6 + 2.43530e6i −1.22071 + 1.44992i
\(37\) −1.45625e6 + 2.52229e6i −0.777013 + 1.34583i 0.156643 + 0.987655i \(0.449933\pi\)
−0.933656 + 0.358171i \(0.883401\pi\)
\(38\) 3.96328e6 + 4.72325e6i 1.90073 + 2.26520i
\(39\) −3.07717e6 + 266655.i −1.33013 + 0.115263i
\(40\) 392744. − 2.22736e6i 0.153416 − 0.870064i
\(41\) 535074. + 94347.9i 0.189356 + 0.0333885i 0.267522 − 0.963552i \(-0.413795\pi\)
−0.0781660 + 0.996940i \(0.524906\pi\)
\(42\) −9133.85 − 105404.i −0.00293533 − 0.0338734i
\(43\) 4.54620e6 − 3.81472e6i 1.32976 − 1.11581i 0.345632 − 0.938370i \(-0.387665\pi\)
0.984133 − 0.177435i \(-0.0567799\pi\)
\(44\) −480693. − 277528.i −0.128250 − 0.0740451i
\(45\) −416794. + 2.34112e6i −0.101642 + 0.570918i
\(46\) −3.28586e6 − 5.69128e6i −0.733868 − 1.27110i
\(47\) −3.01327e6 + 8.27890e6i −0.617515 + 1.69661i 0.0954743 + 0.995432i \(0.469563\pi\)
−0.712989 + 0.701175i \(0.752659\pi\)
\(48\) −3.02924e6 − 2.12483e6i −0.570649 − 0.400276i
\(49\) −4.41433e6 − 3.70406e6i −0.765739 − 0.642531i
\(50\) 2.41418e6 + 6.63290e6i 0.386269 + 1.06126i
\(51\) 1.64143e6 − 6.10569e6i 0.242628 − 0.902515i
\(52\) −3.21287e6 − 1.82211e7i −0.439420 − 2.49208i
\(53\) − 3.89255e6i − 0.493322i −0.969102 − 0.246661i \(-0.920667\pi\)
0.969102 − 0.246661i \(-0.0793335\pi\)
\(54\) 1.18314e7 + 8.32821e6i 1.39143 + 0.979438i
\(55\) −414605. −0.0453089
\(56\) 294840. − 51988.2i 0.0299802 − 0.00528631i
\(57\) 1.29821e7 − 1.29606e7i 1.22983 − 1.22780i
\(58\) 1.48811e7 − 5.41627e6i 1.31499 − 0.478618i
\(59\) 2.64580e6 − 3.15314e6i 0.218348 − 0.260217i −0.645741 − 0.763557i \(-0.723451\pi\)
0.864088 + 0.503340i \(0.167896\pi\)
\(60\) −1.41892e7 − 1.25322e6i −1.09485 − 0.0966988i
\(61\) −1.75097e7 − 6.37302e6i −1.26462 − 0.460284i −0.379303 − 0.925273i \(-0.623836\pi\)
−0.885317 + 0.464989i \(0.846058\pi\)
\(62\) −609710. + 352016.i −0.0412626 + 0.0238230i
\(63\) −310078. + 54146.8i −0.0196838 + 0.00343725i
\(64\) −1.06641e7 + 1.84708e7i −0.635631 + 1.10094i
\(65\) −8.88358e6 − 1.05870e7i −0.497662 − 0.593090i
\(66\) −1.06802e6 + 2.28544e6i −0.0562863 + 0.120446i
\(67\) −493265. + 2.79744e6i −0.0244783 + 0.138823i −0.994598 − 0.103805i \(-0.966898\pi\)
0.970119 + 0.242628i \(0.0780094\pi\)
\(68\) 3.72981e7 + 6.57666e6i 1.74442 + 0.307588i
\(69\) −1.60254e7 + 1.12014e7i −0.706988 + 0.494168i
\(70\) 362643. − 304293.i 0.0151038 − 0.0126736i
\(71\) 893484. + 515853.i 0.0351603 + 0.0202998i 0.517477 − 0.855697i \(-0.326871\pi\)
−0.482317 + 0.875997i \(0.660205\pi\)
\(72\) −2.05301e7 + 3.54239e7i −0.763944 + 1.31815i
\(73\) −1.26668e7 − 2.19395e7i −0.446042 − 0.772567i 0.552082 − 0.833790i \(-0.313833\pi\)
−0.998124 + 0.0612227i \(0.980500\pi\)
\(74\) −2.71199e7 + 7.45114e7i −0.904402 + 2.48482i
\(75\) 1.90403e7 − 8.85949e6i 0.601768 − 0.280004i
\(76\) 8.41786e7 + 7.06343e7i 2.52318 + 2.11720i
\(77\) −18770.7 − 51572.1i −0.000533971 − 0.00146707i
\(78\) −8.12433e7 + 2.16971e7i −2.19487 + 0.586170i
\(79\) −1.00312e7 − 5.68895e7i −0.257539 − 1.46057i −0.789471 − 0.613788i \(-0.789645\pi\)
0.531932 − 0.846787i \(-0.321466\pi\)
\(80\) − 1.65564e7i − 0.404209i
\(81\) 2.16465e7 − 3.72082e7i 0.502861 − 0.864368i
\(82\) 1.47922e7 0.327174
\(83\) 1.09503e7 − 1.93083e6i 0.230735 − 0.0406848i −0.0570850 − 0.998369i \(-0.518181\pi\)
0.287820 + 0.957684i \(0.407069\pi\)
\(84\) −486515. − 1.82172e6i −0.00977189 − 0.0365901i
\(85\) 2.65838e7 − 9.67572e6i 0.509263 − 0.185356i
\(86\) 1.03856e8 − 1.23771e8i 1.89863 − 2.26270i
\(87\) −1.98765e7 − 4.27175e7i −0.346947 − 0.745639i
\(88\) −6.70813e6 − 2.44156e6i −0.111859 − 0.0407133i
\(89\) −2.45007e7 + 1.41455e7i −0.390497 + 0.225454i −0.682376 − 0.731002i \(-0.739053\pi\)
0.291878 + 0.956455i \(0.405720\pi\)
\(90\) −107023. + 6.47396e7i −0.00163120 + 0.986734i
\(91\) 914714. − 1.58433e6i 0.0133389 − 0.0231036i
\(92\) −7.52848e7 − 8.97209e7i −1.05089 − 1.25240i
\(93\) 1.20001e6 + 1.71681e6i 0.0160418 + 0.0229504i
\(94\) −4.16513e7 + 2.36216e8i −0.533479 + 3.02551i
\(95\) 8.08344e7 + 1.42533e7i 0.992434 + 0.174993i
\(96\) 2.59670e7 + 1.21348e7i 0.305730 + 0.142872i
\(97\) −4.04609e7 + 3.39507e7i −0.457034 + 0.383497i −0.842038 − 0.539418i \(-0.818644\pi\)
0.385005 + 0.922915i \(0.374200\pi\)
\(98\) −1.35867e8 − 7.84427e7i −1.47302 − 0.850449i
\(99\) 7.04853e6 + 2.57866e6i 0.0733767 + 0.0268444i
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.14.22 yes 138
3.2 odd 2 81.9.f.a.71.2 138
27.2 odd 18 inner 27.9.f.a.2.22 ✓ 138
27.25 even 9 81.9.f.a.8.2 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.22 ✓ 138 27.2 odd 18 inner
27.9.f.a.14.22 yes 138 1.1 even 1 trivial
81.9.f.a.8.2 138 27.25 even 9
81.9.f.a.71.2 138 3.2 odd 2