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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 20.4
Character \(\chi\) \(=\) 27.20
Dual form 27.5.f.a.23.4

$q$-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+(-1.40615 - 3.86336i) q^{2} +(4.52187 - 7.78156i) q^{3} +(-0.691617 + 0.580335i) q^{4} +(-11.8424 + 2.08814i) q^{5} +(-36.4214 - 6.52758i) q^{6} +(13.5416 + 11.3628i) q^{7} +(-53.7534 - 31.0345i) q^{8} +(-40.1054 - 70.3744i) q^{9} +(24.7194 + 42.8153i) q^{10} +(10.9334 + 1.92785i) q^{11} +(1.38852 + 8.00606i) q^{12} +(272.944 + 99.3436i) q^{13} +(24.8570 - 68.2939i) q^{14} +(-37.3008 + 101.595i) q^{15} +(-46.8208 + 265.534i) q^{16} +(329.418 - 190.190i) q^{17} +(-215.488 + 253.899i) q^{18} +(72.9105 - 126.285i) q^{19} +(6.97859 - 8.31676i) q^{20} +(149.653 - 53.9940i) q^{21} +(-7.92598 - 44.9505i) q^{22} +(-57.0807 - 68.0262i) q^{23} +(-484.563 + 277.951i) q^{24} +(-451.426 + 164.305i) q^{25} -1194.17i q^{26} +(-728.974 - 6.14076i) q^{27} -15.9598 q^{28} +(310.157 + 852.151i) q^{29} +(444.948 + 1.24936i) q^{30} +(663.014 - 556.335i) q^{31} +(113.675 - 20.0439i) q^{32} +(64.4410 - 76.3613i) q^{33} +(-1197.98 - 1005.23i) q^{34} +(-184.092 - 106.286i) q^{35} +(68.5783 + 25.3975i) q^{36} +(398.049 + 689.441i) q^{37} +(-590.407 - 104.105i) q^{38} +(2007.27 - 1674.71i) q^{39} +(701.374 + 255.279i) q^{40} +(-612.946 + 1684.06i) q^{41} +(-419.034 - 502.242i) q^{42} +(-253.959 + 1440.27i) q^{43} +(-8.68051 + 5.01170i) q^{44} +(621.896 + 749.656i) q^{45} +(-182.546 + 316.179i) q^{46} +(-2547.10 + 3035.52i) q^{47} +(1854.55 + 1565.05i) q^{48} +(-362.666 - 2056.78i) q^{49} +(1269.54 + 1512.98i) q^{50} +(9.61253 - 3423.40i) q^{51} +(-246.425 + 89.6915i) q^{52} +526.213i q^{53} +(1001.32 + 2824.93i) q^{54} -133.503 q^{55} +(-375.270 - 1031.04i) q^{56} +(-653.001 - 1138.40i) q^{57} +(2856.04 - 2396.50i) q^{58} +(6213.52 - 1095.61i) q^{59} +(-33.1611 - 91.9116i) q^{60} +(-5105.31 - 4283.86i) q^{61} +(-3081.62 - 1779.17i) q^{62} +(256.555 - 1408.69i) q^{63} +(1919.76 + 3325.13i) q^{64} +(-3439.76 - 606.522i) q^{65} +(-385.625 - 141.584i) q^{66} +(-5993.11 - 2181.31i) q^{67} +(-117.457 + 322.711i) q^{68} +(-787.461 + 136.572i) q^{69} +(-151.759 + 860.669i) q^{70} +(1965.51 - 1134.79i) q^{71} +(-28.2336 + 5027.51i) q^{72} +(-3190.17 + 5525.54i) q^{73} +(2103.84 - 2507.26i) q^{74} +(-762.733 + 4255.76i) q^{75} +(22.8613 + 129.653i) q^{76} +(126.150 + 150.340i) q^{77} +(-9292.55 - 5399.90i) q^{78} +(8933.61 - 3251.57i) q^{79} -3242.33i q^{80} +(-3344.11 + 5644.79i) q^{81} +7368.01 q^{82} +(-905.297 - 2487.28i) q^{83} +(-72.1682 + 124.192i) q^{84} +(-3503.96 + 2940.17i) q^{85} +(5921.41 - 1044.10i) q^{86} +(8033.55 + 1439.80i) q^{87} +(-527.876 - 442.941i) q^{88} +(4547.60 + 2625.56i) q^{89} +(2021.72 - 3456.74i) q^{90} +(2567.29 + 4446.67i) q^{91} +(78.9560 + 13.9221i) q^{92} +(-1331.09 - 7674.95i) q^{93} +(15308.9 + 5571.99i) q^{94} +(-599.736 + 1647.76i) q^{95} +(358.049 - 975.202i) q^{96} +(-1949.57 + 11056.5i) q^{97} +(-7436.14 + 4293.26i) q^{98} +(-302.817 - 846.747i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40615 − 3.86336i −0.351537 − 0.965841i −0.981877 − 0.189521i \(-0.939306\pi\)
0.630339 − 0.776320i \(-0.282916\pi\)
\(3\) 4.52187 − 7.78156i 0.502430 − 0.864618i
\(4\) −0.691617 + 0.580335i −0.0432260 + 0.0362710i
\(5\) −11.8424 + 2.08814i −0.473696 + 0.0835254i −0.405399 − 0.914140i \(-0.632868\pi\)
−0.0682967 + 0.997665i \(0.521756\pi\)
\(6\) −36.4214 − 6.52758i −1.01171 − 0.181322i
\(7\) 13.5416 + 11.3628i 0.276360 + 0.231893i 0.770424 − 0.637532i \(-0.220045\pi\)
−0.494064 + 0.869425i \(0.664489\pi\)
\(8\) −53.7534 − 31.0345i −0.839896 − 0.484914i
\(9\) −40.1054 − 70.3744i −0.495129 − 0.868820i
\(10\) 24.7194 + 42.8153i 0.247194 + 0.428153i
\(11\) 10.9334 + 1.92785i 0.0903585 + 0.0159326i 0.218645 − 0.975805i \(-0.429836\pi\)
−0.128286 + 0.991737i \(0.540948\pi\)
\(12\) 1.38852 + 8.00606i 0.00964248 + 0.0555976i
\(13\) 272.944 + 99.3436i 1.61505 + 0.587832i 0.982431 − 0.186628i \(-0.0597558\pi\)
0.632624 + 0.774459i \(0.281978\pi\)
\(14\) 24.8570 − 68.2939i 0.126821 − 0.348438i
\(15\) −37.3008 + 101.595i −0.165781 + 0.451532i
\(16\) −46.8208 + 265.534i −0.182894 + 1.03724i
\(17\) 329.418 − 190.190i 1.13986 − 0.658096i 0.193460 − 0.981108i \(-0.438029\pi\)
0.946395 + 0.323012i \(0.104696\pi\)
\(18\) −215.488 + 253.899i −0.665085 + 0.783638i
\(19\) 72.9105 − 126.285i 0.201968 − 0.349819i −0.747194 − 0.664606i \(-0.768600\pi\)
0.949163 + 0.314787i \(0.101933\pi\)
\(20\) 6.97859 − 8.31676i 0.0174465 − 0.0207919i
\(21\) 149.653 − 53.9940i 0.339350 − 0.122435i
\(22\) −7.92598 − 44.9505i −0.0163760 − 0.0928729i
\(23\) −57.0807 − 68.0262i −0.107903 − 0.128594i 0.709390 − 0.704816i \(-0.248971\pi\)
−0.817293 + 0.576222i \(0.804526\pi\)
\(24\) −484.563 + 277.951i −0.841255 + 0.482554i
\(25\) −451.426 + 164.305i −0.722281 + 0.262889i
\(26\) − 1194.17i − 1.76653i
\(27\) −728.974 − 6.14076i −0.999965 − 0.00842355i
\(28\) −15.9598 −0.0203569
\(29\) 310.157 + 852.151i 0.368796 + 1.01326i 0.975820 + 0.218575i \(0.0701409\pi\)
−0.607024 + 0.794683i \(0.707637\pi\)
\(30\) 444.948 + 1.24936i 0.494387 + 0.00138818i
\(31\) 663.014 − 556.335i 0.689921 − 0.578912i −0.228966 − 0.973434i \(-0.573534\pi\)
0.918886 + 0.394522i \(0.129090\pi\)
\(32\) 113.675 − 20.0439i 0.111010 − 0.0195741i
\(33\) 64.4410 − 76.3613i 0.0591745 − 0.0701206i
\(34\) −1197.98 − 1005.23i −1.03632 − 0.869574i
\(35\) −184.092 − 106.286i −0.150279 − 0.0867639i
\(36\) 68.5783 + 25.3975i 0.0529154 + 0.0195968i
\(37\) 398.049 + 689.441i 0.290759 + 0.503609i 0.973989 − 0.226594i \(-0.0727589\pi\)
−0.683231 + 0.730203i \(0.739426\pi\)
\(38\) −590.407 − 104.105i −0.408869 − 0.0720946i
\(39\) 2007.27 − 1674.71i 1.31970 − 1.10106i
\(40\) 701.374 + 255.279i 0.438358 + 0.159549i
\(41\) −612.946 + 1684.06i −0.364632 + 1.00182i 0.612739 + 0.790285i \(0.290068\pi\)
−0.977371 + 0.211532i \(0.932155\pi\)
\(42\) −419.034 − 502.242i −0.237547 − 0.284718i
\(43\) −253.959 + 1440.27i −0.137350 + 0.778948i 0.835845 + 0.548965i \(0.184978\pi\)
−0.973195 + 0.229983i \(0.926133\pi\)
\(44\) −8.68051 + 5.01170i −0.00448373 + 0.00258869i
\(45\) 621.896 + 749.656i 0.307109 + 0.370201i
\(46\) −182.546 + 316.179i −0.0862693 + 0.149423i
\(47\) −2547.10 + 3035.52i −1.15306 + 1.37416i −0.237787 + 0.971317i \(0.576422\pi\)
−0.915270 + 0.402842i \(0.868022\pi\)
\(48\) 1854.55 + 1565.05i 0.804927 + 0.679275i
\(49\) −362.666 − 2056.78i −0.151048 − 0.856636i
\(50\) 1269.54 + 1512.98i 0.507818 + 0.605193i
\(51\) 9.61253 − 3423.40i 0.00369570 − 1.31619i
\(52\) −246.425 + 89.6915i −0.0911336 + 0.0331699i
\(53\) 526.213i 0.187331i 0.995604 + 0.0936655i \(0.0298584\pi\)
−0.995604 + 0.0936655i \(0.970142\pi\)
\(54\) 1001.32 + 2824.93i 0.343389 + 0.968768i
\(55\) −133.503 −0.0441333
\(56\) −375.270 − 1031.04i −0.119665 − 0.328777i
\(57\) −653.001 − 1138.40i −0.200985 − 0.350385i
\(58\) 2856.04 − 2396.50i 0.849001 − 0.712397i
\(59\) 6213.52 − 1095.61i 1.78498 − 0.314740i 0.819085 − 0.573672i \(-0.194482\pi\)
0.965895 + 0.258932i \(0.0833706\pi\)
\(60\) −33.1611 − 91.9116i −0.00921142 − 0.0255310i
\(61\) −5105.31 − 4283.86i −1.37203 − 1.15127i −0.972059 − 0.234736i \(-0.924577\pi\)
−0.399966 − 0.916530i \(-0.630978\pi\)
\(62\) −3081.62 − 1779.17i −0.801670 − 0.462844i
\(63\) 256.555 − 1408.69i 0.0646398 − 0.354924i
\(64\) 1919.76 + 3325.13i 0.468692 + 0.811798i
\(65\) −3439.76 − 606.522i −0.814144 − 0.143556i
\(66\) −385.625 − 141.584i −0.0885274 − 0.0325031i
\(67\) −5993.11 − 2181.31i −1.33507 − 0.485924i −0.426811 − 0.904341i \(-0.640363\pi\)
−0.908255 + 0.418417i \(0.862585\pi\)
\(68\) −117.457 + 322.711i −0.0254017 + 0.0697905i
\(69\) −787.461 + 136.572i −0.165398 + 0.0286856i
\(70\) −151.759 + 860.669i −0.0309713 + 0.175647i
\(71\) 1965.51 − 1134.79i 0.389904 − 0.225111i −0.292214 − 0.956353i \(-0.594392\pi\)
0.682119 + 0.731242i \(0.261059\pi\)
\(72\) −28.2336 + 5027.51i −0.00544630 + 0.969814i
\(73\) −3190.17 + 5525.54i −0.598643 + 1.03688i 0.394378 + 0.918948i \(0.370960\pi\)
−0.993022 + 0.117933i \(0.962373\pi\)
\(74\) 2103.84 − 2507.26i 0.384194 − 0.457864i
\(75\) −762.733 + 4255.76i −0.135597 + 0.756580i
\(76\) 22.8613 + 129.653i 0.00395799 + 0.0224469i
\(77\) 126.150 + 150.340i 0.0212768 + 0.0253567i
\(78\) −9292.55 − 5399.90i −1.52737 − 0.887558i
\(79\) 8933.61 − 3251.57i 1.43144 − 0.521001i 0.494096 − 0.869408i \(-0.335499\pi\)
0.937344 + 0.348406i \(0.113277\pi\)
\(80\) − 3242.33i − 0.506614i
\(81\) −3344.11 + 5644.79i −0.509695 + 0.860355i
\(82\) 7368.01 1.09578
\(83\) −905.297 − 2487.28i −0.131412 − 0.361051i 0.856483 − 0.516175i \(-0.172645\pi\)
−0.987895 + 0.155124i \(0.950422\pi\)
\(84\) −72.1682 + 124.192i −0.0102279 + 0.0176010i
\(85\) −3503.96 + 2940.17i −0.484977 + 0.406944i
\(86\) 5921.41 − 1044.10i 0.800623 − 0.141171i
\(87\) 8033.55 + 1439.80i 1.06138 + 0.190224i
\(88\) −527.876 − 442.941i −0.0681658 − 0.0571979i
\(89\) 4547.60 + 2625.56i 0.574119 + 0.331468i 0.758793 − 0.651332i \(-0.225790\pi\)
−0.184674 + 0.982800i \(0.559123\pi\)
\(90\) 2021.72 − 3456.74i 0.249595 − 0.426758i
\(91\) 2567.29 + 4446.67i 0.310021 + 0.536973i
\(92\) 78.9560 + 13.9221i 0.00932845 + 0.00164486i
\(93\) −1331.09 − 7674.95i −0.153901 − 0.887380i
\(94\) 15308.9 + 5571.99i 1.73256 + 0.630601i
\(95\) −599.736 + 1647.76i −0.0664528 + 0.182577i
\(96\) 358.049 − 975.202i 0.0388508 − 0.105816i
\(97\) −1949.57 + 11056.5i −0.207202 + 1.17510i 0.686734 + 0.726909i \(0.259044\pi\)
−0.893936 + 0.448194i \(0.852067\pi\)
\(98\) −7436.14 + 4293.26i −0.774275 + 0.447028i
\(99\) −302.817 − 846.747i −0.0308965 − 0.0863940i
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.20.4 ✓ 66
3.2 odd 2 81.5.f.a.62.8 66
27.4 even 9 81.5.f.a.17.8 66
27.23 odd 18 inner 27.5.f.a.23.4 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.f.a.20.4 ✓ 66 1.1 even 1 trivial
27.5.f.a.23.4 yes 66 27.23 odd 18 inner
81.5.f.a.17.8 66 27.4 even 9
81.5.f.a.62.8 66 3.2 odd 2