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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 2.4
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.4

$q$-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+(-23.0508 - 4.06447i) q^{2} +(55.5291 - 58.9705i) q^{3} +(274.257 + 99.8213i) q^{4} +(-685.642 - 817.116i) q^{5} +(-1519.67 + 1133.62i) q^{6} +(4384.71 - 1595.90i) q^{7} +(-726.861 - 419.653i) q^{8} +(-394.048 - 6549.16i) q^{9} +(12483.4 + 21621.9i) q^{10} +(7708.06 - 9186.11i) q^{11} +(21115.7 - 10630.1i) q^{12} +(-1038.61 - 5890.22i) q^{13} +(-107557. + 18965.3i) q^{14} +(-86258.8 - 4941.02i) q^{15} +(-42186.5 - 35398.7i) q^{16} +(-55702.5 + 32159.8i) q^{17} +(-17535.8 + 152565. i) q^{18} +(-4680.34 + 8106.59i) q^{19} +(-106476. - 292541. i) q^{20} +(149367. - 347188. i) q^{21} +(-215014. + 180418. i) q^{22} +(22536.6 - 61918.9i) q^{23} +(-65109.1 + 19560.4i) q^{24} +(-129743. + 735808. i) q^{25} +139996. i q^{26} +(-408088. - 340431. i) q^{27} +1.36184e6 q^{28} +(-282095. - 49741.0i) q^{29} +(1.96825e6 + 464491. i) q^{30} +(-179639. - 65383.2i) q^{31} +(966665. + 1.15203e6i) q^{32} +(-113688. - 964645. i) q^{33} +(1.41470e6 - 514908. i) q^{34} +(-4.31038e6 - 2.48860e6i) q^{35} +(545675. - 1.83549e6i) q^{36} +(1.47313e6 + 2.55154e6i) q^{37} +(140834. - 167840. i) q^{38} +(-405022. - 265831. i) q^{39} +(155461. + 881662. i) q^{40} +(2.72382e6 - 480283. i) q^{41} +(-4.85417e6 + 7.39584e6i) q^{42} +(-629226. - 527983. i) q^{43} +(3.03096e6 - 1.74993e6i) q^{44} +(-5.08125e6 + 4.81236e6i) q^{45} +(-771154. + 1.33568e6i) q^{46} +(-865607. - 2.37823e6i) q^{47} +(-4.43005e6 + 522105. i) q^{48} +(1.22627e7 - 1.02896e7i) q^{49} +(5.98134e6 - 1.64336e7i) q^{50} +(-1.19662e6 + 5.07061e6i) q^{51} +(303125. - 1.71911e6i) q^{52} +9.47127e6i q^{53} +(8.02308e6 + 9.50587e6i) q^{54} -1.27911e7 q^{55} +(-3.85680e6 - 680058. i) q^{56} +(218155. + 726153. i) q^{57} +(6.30034e6 + 2.29314e6i) q^{58} +(4.28544e6 + 5.10719e6i) q^{59} +(-2.31639e7 - 9.96558e6i) q^{60} +(-1.64040e7 + 5.97057e6i) q^{61} +(3.87507e6 + 2.23727e6i) q^{62} +(-1.21796e7 - 2.80873e7i) q^{63} +(-1.05510e7 - 1.82748e7i) q^{64} +(-4.10088e6 + 4.88724e6i) q^{65} +(-1.30017e6 + 2.26979e7i) q^{66} +(-3.05843e6 - 1.73452e7i) q^{67} +(-1.84870e7 + 3.25976e6i) q^{68} +(-2.39995e6 - 4.76729e6i) q^{69} +(8.92427e7 + 7.48835e7i) q^{70} +(525382. - 303329. i) q^{71} +(-2.46196e6 + 4.92569e6i) q^{72} +(1.47719e6 - 2.55858e6i) q^{73} +(-2.35862e7 - 6.48025e7i) q^{74} +(3.61865e7 + 4.85097e7i) q^{75} +(-2.09283e6 + 1.75609e6i) q^{76} +(1.91375e7 - 5.25797e7i) q^{77} +(8.25561e6 + 7.77382e6i) q^{78} +(4.44749e6 - 2.52229e7i) q^{79} +5.87421e7i q^{80} +(-4.27362e7 + 5.16136e6i) q^{81} -6.47382e7 q^{82} +(-9.59654e6 - 1.69213e6i) q^{83} +(7.56218e7 - 8.03085e7i) q^{84} +(6.44703e7 + 2.34653e7i) q^{85} +(1.23582e7 + 1.47279e7i) q^{86} +(-1.85977e7 + 1.38732e7i) q^{87} +(-9.45767e6 + 3.44231e6i) q^{88} +(7.49830e7 + 4.32915e7i) q^{89} +(1.36686e8 - 9.02760e7i) q^{90} +(-1.39542e7 - 2.41694e7i) q^{91} +(1.23616e7 - 1.47320e7i) q^{92} +(-1.38309e7 + 6.96273e6i) q^{93} +(1.02866e7 + 5.83384e7i) q^{94} +(9.83306e6 - 1.73383e6i) q^{95} +(1.21614e8 + 6.96619e6i) q^{96} +(-7.07861e7 - 5.93966e7i) q^{97} +(-3.24486e8 + 1.87342e8i) q^{98} +(-6.31986e7 - 4.68615e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.0508 − 4.06447i −1.44067 − 0.254030i −0.601927 − 0.798551i \(-0.705600\pi\)
−0.838747 + 0.544522i \(0.816711\pi\)
\(3\) 55.5291 − 58.9705i 0.685544 − 0.728031i
\(4\) 274.257 + 99.8213i 1.07132 + 0.389927i
\(5\) −685.642 − 817.116i −1.09703 − 1.30739i −0.947900 − 0.318568i \(-0.896798\pi\)
−0.149127 − 0.988818i \(-0.547646\pi\)
\(6\) −1519.67 + 1133.62i −1.17259 + 0.874707i
\(7\) 4384.71 − 1595.90i 1.82620 − 0.664683i 0.832310 − 0.554310i \(-0.187018\pi\)
0.993890 − 0.110372i \(-0.0352044\pi\)
\(8\) −726.861 − 419.653i −0.177456 − 0.102454i
\(9\) −394.048 − 6549.16i −0.0600591 − 0.998195i
\(10\) 12483.4 + 21621.9i 1.24834 + 2.16219i
\(11\) 7708.06 − 9186.11i 0.526471 − 0.627424i −0.435627 − 0.900127i \(-0.643473\pi\)
0.962098 + 0.272703i \(0.0879178\pi\)
\(12\) 21115.7 − 10630.1i 1.01831 − 0.512639i
\(13\) −1038.61 − 5890.22i −0.0363645 − 0.206233i 0.961212 − 0.275810i \(-0.0889461\pi\)
−0.997577 + 0.0695771i \(0.977835\pi\)
\(14\) −107557. + 18965.3i −2.79981 + 0.493682i
\(15\) −86258.8 − 4941.02i −1.70388 − 0.0976004i
\(16\) −42186.5 − 35398.7i −0.643715 − 0.540141i
\(17\) −55702.5 + 32159.8i −0.666928 + 0.385051i −0.794912 − 0.606725i \(-0.792483\pi\)
0.127984 + 0.991776i \(0.459149\pi\)
\(18\) −17535.8 + 152565.i −0.167045 + 1.45333i
\(19\) −4680.34 + 8106.59i −0.0359139 + 0.0622048i −0.883424 − 0.468575i \(-0.844768\pi\)
0.847510 + 0.530780i \(0.178101\pi\)
\(20\) −106476. − 292541.i −0.665477 − 1.82838i
\(21\) 149367. − 347188.i 0.768031 − 1.78520i
\(22\) −215014. + 180418.i −0.917857 + 0.770173i
\(23\) 22536.6 − 61918.9i 0.0805337 − 0.221264i −0.892891 − 0.450274i \(-0.851326\pi\)
0.973424 + 0.229009i \(0.0735486\pi\)
\(24\) −65109.1 + 19560.4i −0.196244 + 0.0589567i
\(25\) −129743. + 735808.i −0.332141 + 1.88367i
\(26\) 139996.i 0.306352i
\(27\) −408088. − 340431.i −0.767890 − 0.640581i
\(28\) 1.36184e6 2.21562
\(29\) −282095. − 49741.0i −0.398845 − 0.0703271i −0.0293717 − 0.999569i \(-0.509351\pi\)
−0.369473 + 0.929241i \(0.620462\pi\)
\(30\) 1.96825e6 + 464491.i 2.42994 + 0.573446i
\(31\) −179639. − 65383.2i −0.194515 − 0.0707978i 0.242926 − 0.970045i \(-0.421893\pi\)
−0.437441 + 0.899247i \(0.644115\pi\)
\(32\) 966665. + 1.15203e6i 0.921884 + 1.09866i
\(33\) −113688. − 964645.i −0.0958651 − 0.813414i
\(34\) 1.41470e6 − 514908.i 1.05864 − 0.385313i
\(35\) −4.31038e6 − 2.48860e6i −2.87239 − 1.65837i
\(36\) 545675. − 1.83549e6i 0.324881 − 1.09280i
\(37\) 1.47313e6 + 2.55154e6i 0.786023 + 1.36143i 0.928386 + 0.371617i \(0.121196\pi\)
−0.142363 + 0.989814i \(0.545470\pi\)
\(38\) 140834. − 167840.i 0.0675421 − 0.0804935i
\(39\) −405022. − 265831.i −0.175074 − 0.114907i
\(40\) 155461. + 881662.i 0.0607269 + 0.344399i
\(41\) 2.72382e6 − 480283.i 0.963924 − 0.169966i 0.330530 − 0.943795i \(-0.392772\pi\)
0.633394 + 0.773830i \(0.281661\pi\)
\(42\) −4.85417e6 + 7.39584e6i −1.55998 + 2.37679i
\(43\) −629226. − 527983.i −0.184049 − 0.154435i 0.546108 − 0.837715i \(-0.316109\pi\)
−0.730157 + 0.683279i \(0.760553\pi\)
\(44\) 3.03096e6 − 1.74993e6i 0.808666 − 0.466884i
\(45\) −5.08125e6 + 4.81236e6i −1.23914 + 1.17357i
\(46\) −771154. + 1.33568e6i −0.172230 + 0.298312i
\(47\) −865607. − 2.37823e6i −0.177390 − 0.487375i 0.818850 − 0.574007i \(-0.194612\pi\)
−0.996240 + 0.0866320i \(0.972390\pi\)
\(48\) −4.43005e6 + 522105.i −0.834534 + 0.0983542i
\(49\) 1.22627e7 − 1.02896e7i 2.12716 − 1.78490i
\(50\) 5.98134e6 − 1.64336e7i 0.957014 − 2.62938i
\(51\) −1.19662e6 + 5.07061e6i −0.176879 + 0.749514i
\(52\) 303125. − 1.71911e6i 0.0414580 − 0.235120i
\(53\) 9.47127e6i 1.20034i 0.799872 + 0.600171i \(0.204901\pi\)
−0.799872 + 0.600171i \(0.795099\pi\)
\(54\) 8.02308e6 + 9.50587e6i 0.943552 + 1.11794i
\(55\) −1.27911e7 −1.39784
\(56\) −3.85680e6 − 680058.i −0.392170 − 0.0691502i
\(57\) 218155. + 726153.i 0.0206664 + 0.0687906i
\(58\) 6.30034e6 + 2.29314e6i 0.556740 + 0.202637i
\(59\) 4.28544e6 + 5.10719e6i 0.353661 + 0.421477i 0.913318 − 0.407247i \(-0.133511\pi\)
−0.559656 + 0.828725i \(0.689067\pi\)
\(60\) −2.31639e7 − 9.96558e6i −1.78733 − 0.768949i
\(61\) −1.64040e7 + 5.97057e6i −1.18476 + 0.431218i −0.857881 − 0.513848i \(-0.828220\pi\)
−0.326880 + 0.945066i \(0.605997\pi\)
\(62\) 3.87507e6 + 2.23727e6i 0.262248 + 0.151409i
\(63\) −1.21796e7 − 2.80873e7i −0.773163 − 1.78298i
\(64\) −1.05510e7 − 1.82748e7i −0.628887 − 1.08926i
\(65\) −4.10088e6 + 4.88724e6i −0.229733 + 0.273786i
\(66\) −1.30017e6 + 2.26979e7i −0.0685209 + 1.19622i
\(67\) −3.05843e6 − 1.73452e7i −0.151775 − 0.860756i −0.961676 − 0.274190i \(-0.911590\pi\)
0.809901 − 0.586567i \(-0.199521\pi\)
\(68\) −1.84870e7 + 3.25976e6i −0.864632 + 0.152458i
\(69\) −2.39995e6 − 4.76729e6i −0.105878 − 0.210318i
\(70\) 8.92427e7 + 7.48835e7i 3.71690 + 3.11885i
\(71\) 525382. − 303329.i 0.0206748 − 0.0119366i −0.489627 − 0.871932i \(-0.662867\pi\)
0.510302 + 0.859995i \(0.329534\pi\)
\(72\) −2.46196e6 + 4.92569e6i −0.0916116 + 0.183289i
\(73\) 1.47719e6 − 2.55858e6i 0.0520171 − 0.0900963i −0.838844 − 0.544371i \(-0.816768\pi\)
0.890862 + 0.454275i \(0.150102\pi\)
\(74\) −2.35862e7 − 6.48025e7i −0.786558 − 2.16105i
\(75\) 3.61865e7 + 4.85097e7i 1.14367 + 1.53315i
\(76\) −2.09283e6 + 1.75609e6i −0.0627305 + 0.0526371i
\(77\) 1.91375e7 − 5.25797e7i 0.544404 − 1.49574i
\(78\) 8.25561e6 + 7.77382e6i 0.223034 + 0.210018i
\(79\) 4.44749e6 − 2.52229e7i 0.114184 − 0.647571i −0.872967 − 0.487780i \(-0.837807\pi\)
0.987151 − 0.159791i \(-0.0510821\pi\)
\(80\) 5.87421e7i 1.43413i
\(81\) −4.27362e7 + 5.16136e6i −0.992786 + 0.119901i
\(82\) −6.47382e7 −1.43188
\(83\) −9.59654e6 − 1.69213e6i −0.202210 − 0.0356551i 0.0716257 − 0.997432i \(-0.477181\pi\)
−0.273836 + 0.961777i \(0.588292\pi\)
\(84\) 7.56218e7 − 8.03085e7i 1.51890 − 1.61304i
\(85\) 6.44703e7 + 2.34653e7i 1.23505 + 0.449521i
\(86\) 1.23582e7 + 1.47279e7i 0.225923 + 0.269245i
\(87\) −1.85977e7 + 1.38732e7i −0.324626 + 0.242159i
\(88\) −9.45767e6 + 3.44231e6i −0.157708 + 0.0574010i
\(89\) 7.49830e7 + 4.32915e7i 1.19510 + 0.689989i 0.959458 − 0.281851i \(-0.0909485\pi\)
0.235639 + 0.971841i \(0.424282\pi\)
\(90\) 1.36686e8 − 9.02760e7i 2.08332 − 1.37595i
\(91\) −1.39542e7 − 2.41694e7i −0.203488 − 0.352452i
\(92\) 1.23616e7 − 1.47320e7i 0.172554 − 0.205642i
\(93\) −1.38309e7 + 6.96273e6i −0.184892 + 0.0930782i
\(94\) 1.02866e7 + 5.83384e7i 0.131753 + 0.747210i
\(95\) 9.83306e6 − 1.73383e6i 0.120724 − 0.0212869i
\(96\) 1.21614e8 + 6.96619e6i 1.43185 + 0.0820183i
\(97\) −7.07861e7 − 5.93966e7i −0.799578 − 0.670926i 0.148518 − 0.988910i \(-0.452550\pi\)
−0.948096 + 0.317984i \(0.896994\pi\)
\(98\) −3.24486e8 + 1.87342e8i −3.51796 + 2.03110i
\(99\) −6.31986e7 − 4.68615e7i −0.657911 − 0.487838i
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.2.4 ✓ 138
3.2 odd 2 81.9.f.a.8.20 138
27.13 even 9 81.9.f.a.71.20 138
27.14 odd 18 inner 27.9.f.a.14.4 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.4 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.4 yes 138 27.14 odd 18 inner
81.9.f.a.8.20 138 3.2 odd 2
81.9.f.a.71.20 138 27.13 even 9