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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 2.9
Character \(\chi\) \(=\) 27.2
Dual form 27.5.f.a.14.9

$q$-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+(4.49504 + 0.792597i) q^{2} +(8.84760 + 1.64926i) q^{3} +(4.54211 + 1.65319i) q^{4} +(-3.40574 - 4.05880i) q^{5} +(38.4631 + 14.4261i) q^{6} +(-28.7185 + 10.4527i) q^{7} +(-44.1393 - 25.4838i) q^{8} +(75.5599 + 29.1840i) q^{9} +(-12.0919 - 20.9439i) q^{10} +(13.9368 - 16.6092i) q^{11} +(37.4602 + 22.1179i) q^{12} +(0.515455 + 2.92329i) q^{13} +(-137.376 + 24.2231i) q^{14} +(-23.4386 - 41.5276i) q^{15} +(-237.454 - 199.247i) q^{16} +(-249.786 + 144.214i) q^{17} +(316.514 + 191.072i) q^{18} +(220.850 - 382.524i) q^{19} +(-8.75926 - 24.0659i) q^{20} +(-271.329 + 45.1168i) q^{21} +(75.8110 - 63.6130i) q^{22} +(-301.451 + 828.229i) q^{23} +(-348.497 - 298.268i) q^{24} +(103.655 - 587.858i) q^{25} +13.5489i q^{26} +(620.391 + 382.826i) q^{27} -147.723 q^{28} +(1064.69 + 187.734i) q^{29} +(-72.4427 - 205.246i) q^{30} +(1637.03 + 595.829i) q^{31} +(-385.260 - 459.135i) q^{32} +(150.700 - 123.966i) q^{33} +(-1237.10 + 450.268i) q^{34} +(140.233 + 80.9637i) q^{35} +(294.955 + 257.472i) q^{36} +(293.810 + 508.894i) q^{37} +(1295.92 - 1544.42i) q^{38} +(-0.260731 + 26.7142i) q^{39} +(46.8931 + 265.944i) q^{40} +(-1376.52 + 242.718i) q^{41} +(-1255.40 - 12.2527i) q^{42} +(-1229.63 - 1031.78i) q^{43} +(90.7608 - 52.4008i) q^{44} +(-138.885 - 406.076i) q^{45} +(-2011.49 + 3484.00i) q^{46} +(-754.401 - 2072.70i) q^{47} +(-1772.28 - 2154.48i) q^{48} +(-1123.78 + 942.961i) q^{49} +(931.870 - 2560.29i) q^{50} +(-2447.85 + 863.984i) q^{51} +(-2.49151 + 14.1301i) q^{52} +1536.91i q^{53} +(2485.26 + 2212.54i) q^{54} -114.879 q^{55} +(1533.99 + 270.484i) q^{56} +(2584.88 - 3020.18i) q^{57} +(4637.04 + 1687.74i) q^{58} +(-844.872 - 1006.88i) q^{59} +(-37.8075 - 227.371i) q^{60} +(2815.23 - 1024.66i) q^{61} +(6886.25 + 3975.78i) q^{62} +(-2475.02 - 48.3172i) q^{63} +(1111.94 + 1925.94i) q^{64} +(10.1095 - 12.0481i) q^{65} +(775.659 - 437.790i) q^{66} +(-1421.14 - 8059.70i) q^{67} +(-1372.97 + 242.091i) q^{68} +(-4033.08 + 6830.67i) q^{69} +(566.183 + 475.084i) q^{70} +(-4745.40 + 2739.76i) q^{71} +(-2591.44 - 3213.72i) q^{72} +(-1147.66 + 1987.81i) q^{73} +(917.341 + 2520.37i) q^{74} +(1886.63 - 5030.18i) q^{75} +(1635.51 - 1372.36i) q^{76} +(-226.633 + 622.670i) q^{77} +(-22.3456 + 119.875i) q^{78} +(1006.95 - 5710.70i) q^{79} +1642.36i q^{80} +(4857.59 + 4410.28i) q^{81} -6379.91 q^{82} +(-5853.36 - 1032.10i) q^{83} +(-1306.99 - 243.634i) q^{84} +(1436.04 + 522.676i) q^{85} +(-4709.43 - 5612.48i) q^{86} +(9110.34 + 3416.95i) q^{87} +(-1038.43 + 377.957i) q^{88} +(1843.12 + 1064.13i) q^{89} +(-302.440 - 1935.41i) q^{90} +(-45.3593 - 78.5647i) q^{91} +(-2738.45 + 3263.55i) q^{92} +(13501.1 + 7971.54i) q^{93} +(-1748.25 - 9914.81i) q^{94} +(-2304.75 + 406.389i) q^{95} +(-2651.39 - 4697.63i) q^{96} +(-3155.11 - 2647.45i) q^{97} +(-5798.82 + 3347.95i) q^{98} +(1537.79 - 848.261i) 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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.49504 + 0.792597i 1.12376 + 0.198149i 0.704491 − 0.709713i \(-0.251175\pi\)
0.419269 + 0.907862i \(0.362286\pi\)
\(3\) 8.84760 + 1.64926i 0.983066 + 0.183251i
\(4\) 4.54211 + 1.65319i 0.283882 + 0.103325i
\(5\) −3.40574 − 4.05880i −0.136230 − 0.162352i 0.693617 − 0.720344i \(-0.256016\pi\)
−0.829846 + 0.557992i \(0.811572\pi\)
\(6\) 38.4631 + 14.4261i 1.06842 + 0.400724i
\(7\) −28.7185 + 10.4527i −0.586092 + 0.213320i −0.618010 − 0.786170i \(-0.712061\pi\)
0.0319175 + 0.999491i \(0.489839\pi\)
\(8\) −44.1393 − 25.4838i −0.689677 − 0.398185i
\(9\) 75.5599 + 29.1840i 0.932838 + 0.360296i
\(10\) −12.0919 − 20.9439i −0.120919 − 0.209439i
\(11\) 13.9368 − 16.6092i 0.115180 − 0.137266i −0.705374 − 0.708836i \(-0.749221\pi\)
0.820554 + 0.571569i \(0.193665\pi\)
\(12\) 37.4602 + 22.1179i 0.260140 + 0.153597i
\(13\) 0.515455 + 2.92329i 0.00305003 + 0.0172976i 0.986295 − 0.164993i \(-0.0527601\pi\)
−0.983245 + 0.182290i \(0.941649\pi\)
\(14\) −137.376 + 24.2231i −0.700897 + 0.123587i
\(15\) −23.4386 − 41.5276i −0.104171 − 0.184567i
\(16\) −237.454 − 199.247i −0.927554 − 0.778310i
\(17\) −249.786 + 144.214i −0.864310 + 0.499010i −0.865453 − 0.500989i \(-0.832970\pi\)
0.00114291 + 0.999999i \(0.499636\pi\)
\(18\) 316.514 + 191.072i 0.976894 + 0.589728i
\(19\) 220.850 − 382.524i 0.611774 − 1.05962i −0.379167 − 0.925328i \(-0.623789\pi\)
0.990941 − 0.134296i \(-0.0428772\pi\)
\(20\) −8.75926 − 24.0659i −0.0218982 − 0.0601647i
\(21\) −271.329 + 45.1168i −0.615259 + 0.102306i
\(22\) 75.8110 − 63.6130i 0.156634 − 0.131432i
\(23\) −301.451 + 828.229i −0.569850 + 1.56565i 0.234889 + 0.972022i \(0.424527\pi\)
−0.804739 + 0.593629i \(0.797695\pi\)
\(24\) −348.497 − 298.268i −0.605030 − 0.517826i
\(25\) 103.655 − 587.858i 0.165848 − 0.940573i
\(26\) 13.5489i 0.0200427i
\(27\) 620.391 + 382.826i 0.851017 + 0.525139i
\(28\) −147.723 −0.188422
\(29\) 1064.69 + 187.734i 1.26598 + 0.223227i 0.766018 − 0.642819i \(-0.222235\pi\)
0.499965 + 0.866046i \(0.333346\pi\)
\(30\) −72.4427 − 205.246i −0.0804919 − 0.228051i
\(31\) 1637.03 + 595.829i 1.70346 + 0.620010i 0.996212 − 0.0869542i \(-0.0277134\pi\)
0.707250 + 0.706964i \(0.249936\pi\)
\(32\) −385.260 − 459.135i −0.376230 − 0.448374i
\(33\) 150.700 − 123.966i 0.138384 − 0.113835i
\(34\) −1237.10 + 450.268i −1.07016 + 0.389505i
\(35\) 140.233 + 80.9637i 0.114476 + 0.0660928i
\(36\) 294.955 + 257.472i 0.227588 + 0.198667i
\(37\) 293.810 + 508.894i 0.214617 + 0.371727i 0.953154 − 0.302486i \(-0.0978165\pi\)
−0.738537 + 0.674213i \(0.764483\pi\)
\(38\) 1295.92 − 1544.42i 0.897451 − 1.06954i
\(39\) −0.260731 + 26.7142i −0.000171421 + 0.0175636i
\(40\) 46.8931 + 265.944i 0.0293082 + 0.166215i
\(41\) −1376.52 + 242.718i −0.818872 + 0.144389i −0.567365 − 0.823466i \(-0.692037\pi\)
−0.251507 + 0.967856i \(0.580926\pi\)
\(42\) −1255.40 − 12.2527i −0.711675 − 0.00694598i
\(43\) −1229.63 − 1031.78i −0.665022 − 0.558020i 0.246566 − 0.969126i \(-0.420698\pi\)
−0.911587 + 0.411107i \(0.865142\pi\)
\(44\) 90.7608 − 52.4008i 0.0468806 − 0.0270665i
\(45\) −138.885 − 406.076i −0.0685853 − 0.200531i
\(46\) −2011.49 + 3484.00i −0.950608 + 1.64650i
\(47\) −754.401 − 2072.70i −0.341513 − 0.938298i −0.984956 − 0.172805i \(-0.944717\pi\)
0.643444 − 0.765494i \(-0.277505\pi\)
\(48\) −1772.28 − 2154.48i −0.769221 − 0.935106i
\(49\) −1123.78 + 942.961i −0.468046 + 0.392737i
\(50\) 931.870 − 2560.29i 0.372748 − 1.02412i
\(51\) −2447.85 + 863.984i −0.941119 + 0.332174i
\(52\) −2.49151 + 14.1301i −0.000921416 + 0.00522561i
\(53\) 1536.91i 0.547138i 0.961852 + 0.273569i \(0.0882042\pi\)
−0.961852 + 0.273569i \(0.911796\pi\)
\(54\) 2485.26 + 2212.54i 0.852283 + 0.758758i
\(55\) −114.879 −0.0379764
\(56\) 1533.99 + 270.484i 0.489155 + 0.0862513i
\(57\) 2584.88 − 3020.18i 0.795592 − 0.929572i
\(58\) 4637.04 + 1687.74i 1.37843 + 0.501707i
\(59\) −844.872 − 1006.88i −0.242710 − 0.289250i 0.630914 − 0.775853i \(-0.282680\pi\)
−0.873623 + 0.486603i \(0.838236\pi\)
\(60\) −37.8075 − 227.371i −0.0105021 − 0.0631587i
\(61\) 2815.23 − 1024.66i 0.756578 − 0.275372i 0.0652071 − 0.997872i \(-0.479229\pi\)
0.691371 + 0.722500i \(0.257007\pi\)
\(62\) 6886.25 + 3975.78i 1.79143 + 1.03428i
\(63\) −2475.02 − 48.3172i −0.623588 − 0.0121736i
\(64\) 1111.94 + 1925.94i 0.271470 + 0.470200i
\(65\) 10.1095 − 12.0481i 0.00239279 − 0.00285162i
\(66\) 775.659 − 437.790i 0.178067 − 0.100503i
\(67\) −1421.14 − 8059.70i −0.316583 − 1.79543i −0.563200 − 0.826320i \(-0.690430\pi\)
0.246617 − 0.969113i \(-0.420681\pi\)
\(68\) −1372.97 + 242.091i −0.296922 + 0.0523554i
\(69\) −4033.08 + 6830.67i −0.847108 + 1.43471i
\(70\) 566.183 + 475.084i 0.115547 + 0.0969558i
\(71\) −4745.40 + 2739.76i −0.941360 + 0.543495i −0.890386 − 0.455205i \(-0.849566\pi\)
−0.0509738 + 0.998700i \(0.516233\pi\)
\(72\) −2591.44 − 3213.72i −0.499892 − 0.619930i
\(73\) −1147.66 + 1987.81i −0.215362 + 0.373017i −0.953384 − 0.301759i \(-0.902426\pi\)
0.738023 + 0.674776i \(0.235760\pi\)
\(74\) 917.341 + 2520.37i 0.167520 + 0.460258i
\(75\) 1886.63 − 5030.18i 0.335401 − 0.894254i
\(76\) 1635.51 − 1372.36i 0.283157 − 0.237597i
\(77\) −226.633 + 622.670i −0.0382246 + 0.105021i
\(78\) −22.3456 + 119.875i −0.00367285 + 0.0197033i
\(79\) 1006.95 − 5710.70i 0.161344 − 0.915029i −0.791410 − 0.611286i \(-0.790652\pi\)
0.952754 − 0.303743i \(-0.0982365\pi\)
\(80\) 1642.36i 0.256619i
\(81\) 4857.59 + 4410.28i 0.740373 + 0.672196i
\(82\) −6379.91 −0.948826
\(83\) −5853.36 − 1032.10i −0.849667 − 0.149819i −0.268174 − 0.963371i \(-0.586420\pi\)
−0.581493 + 0.813551i \(0.697531\pi\)
\(84\) −1306.99 − 243.634i −0.185232 − 0.0345286i
\(85\) 1436.04 + 522.676i 0.198760 + 0.0723427i
\(86\) −4709.43 − 5612.48i −0.636754 − 0.758854i
\(87\) 9110.34 + 3416.95i 1.20364 + 0.451440i
\(88\) −1038.43 + 377.957i −0.134095 + 0.0488064i
\(89\) 1843.12 + 1064.13i 0.232688 + 0.134343i 0.611812 − 0.791003i \(-0.290441\pi\)
−0.379123 + 0.925346i \(0.623774\pi\)
\(90\) −302.440 − 1935.41i −0.0373383 − 0.238939i
\(91\) −45.3593 − 78.5647i −0.00547752 − 0.00948734i
\(92\) −2738.45 + 3263.55i −0.323540 + 0.385581i
\(93\) 13501.1 + 7971.54i 1.56100 + 0.921672i
\(94\) −1748.25 − 9914.81i −0.197855 − 1.12209i
\(95\) −2304.75 + 406.389i −0.255374 + 0.0450293i
\(96\) −2651.39 − 4697.63i −0.287694 − 0.509726i
\(97\) −3155.11 − 2647.45i −0.335329 − 0.281374i 0.459538 − 0.888158i \(-0.348015\pi\)
−0.794867 + 0.606784i \(0.792459\pi\)
\(98\) −5798.82 + 3347.95i −0.603792 + 0.348599i
\(99\) 1537.79 − 848.261i 0.156901 − 0.0865484i
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.2.9 ✓ 66
3.2 odd 2 81.5.f.a.8.3 66
27.13 even 9 81.5.f.a.71.3 66
27.14 odd 18 inner 27.5.f.a.14.9 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.f.a.2.9 ✓ 66 1.1 even 1 trivial
27.5.f.a.14.9 yes 66 27.14 odd 18 inner
81.5.f.a.8.3 66 3.2 odd 2
81.5.f.a.71.3 66 27.13 even 9