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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,2,Mod(11,216)] 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("216.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 59.16
Character \(\chi\) \(=\) 216.59
Dual form 216.2.v.b.11.16

$q$-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+(-0.279321 + 1.38635i) q^{2} +(1.40807 + 1.00863i) q^{3} +(-1.84396 - 0.774477i) q^{4} +(2.25586 - 0.821067i) q^{5} +(-1.79163 + 1.67035i) q^{6} +(1.95474 - 0.344674i) q^{7} +(1.58876 - 2.34005i) q^{8} +(0.965315 + 2.84045i) q^{9} +(0.508180 + 3.35677i) q^{10} +(0.553700 - 1.52128i) q^{11} +(-1.81526 - 2.95040i) q^{12} +(-3.57715 + 4.26309i) q^{13} +(-0.0681610 + 2.80624i) q^{14} +(4.00457 + 1.11922i) q^{15} +(2.80037 + 2.85621i) q^{16} +(-6.40600 - 3.69851i) q^{17} +(-4.20751 + 0.544870i) q^{18} +(-1.39557 - 2.41719i) q^{19} +(-4.79562 - 0.233100i) q^{20} +(3.10006 + 1.48630i) q^{21} +(1.95437 + 1.19255i) q^{22} +(0.465506 - 2.64002i) q^{23} +(4.59734 - 1.69248i) q^{24} +(0.584547 - 0.490493i) q^{25} +(-4.91098 - 6.14998i) q^{26} +(-1.50575 + 4.97320i) q^{27} +(-3.87141 - 0.878338i) q^{28} +(0.138481 - 0.116199i) q^{29} +(-2.67020 + 5.23913i) q^{30} +(-1.76926 - 0.311969i) q^{31} +(-4.74192 + 3.08451i) q^{32} +(2.31406 - 1.58358i) q^{33} +(6.91677 - 7.84792i) q^{34} +(4.12663 - 2.38251i) q^{35} +(0.419863 - 5.98529i) q^{36} +(3.55469 + 2.05230i) q^{37} +(3.74090 - 1.25958i) q^{38} +(-9.33677 + 2.39468i) q^{39} +(1.66268 - 6.58332i) q^{40} +(-2.85334 + 3.40048i) q^{41} +(-2.92645 + 3.88263i) q^{42} +(11.1139 + 4.04511i) q^{43} +(-2.19920 + 2.37635i) q^{44} +(4.50982 + 5.61508i) q^{45} +(3.52998 + 1.38277i) q^{46} +(-2.08588 - 11.8296i) q^{47} +(1.06225 + 6.84629i) q^{48} +(-2.87563 + 1.04664i) q^{49} +(0.516721 + 0.947394i) q^{50} +(-5.28965 - 11.6691i) q^{51} +(9.89779 - 5.09053i) q^{52} +10.1087 q^{53} +(-6.47403 - 3.47662i) q^{54} -3.88642i q^{55} +(2.29906 - 5.12181i) q^{56} +(0.473009 - 4.81119i) q^{57} +(0.122413 + 0.224441i) q^{58} +(-1.96718 - 5.40478i) q^{59} +(-6.51745 - 5.16524i) q^{60} +(-7.60780 + 1.34146i) q^{61} +(0.926693 - 2.36569i) q^{62} +(2.86597 + 5.21963i) q^{63} +(-2.95170 - 7.43555i) q^{64} +(-4.56929 + 12.5540i) q^{65} +(1.54904 + 3.65044i) q^{66} +(1.38817 + 1.16481i) q^{67} +(8.94800 + 11.7812i) q^{68} +(3.31828 - 3.24780i) q^{69} +(2.15035 + 6.38646i) q^{70} +(1.25119 - 2.16713i) q^{71} +(8.18046 + 2.25390i) q^{72} +(-3.79038 - 6.56514i) q^{73} +(-3.83812 + 4.35481i) q^{74} +(1.31781 - 0.101054i) q^{75} +(0.701309 + 5.53804i) q^{76} +(0.557996 - 3.16455i) q^{77} +(-0.711916 - 13.6130i) q^{78} +(-6.77140 - 8.06984i) q^{79} +(8.66239 + 4.14392i) q^{80} +(-7.13633 + 5.48386i) q^{81} +(-3.91727 - 4.90557i) q^{82} +(0.718131 + 0.855836i) q^{83} +(-4.56529 - 5.14159i) q^{84} +(-17.4878 - 3.08357i) q^{85} +(-8.71230 + 14.2779i) q^{86} +(0.312194 - 0.0239400i) q^{87} +(-2.68018 - 3.71263i) q^{88} +(9.87804 - 5.70309i) q^{89} +(-9.04419 + 4.68380i) q^{90} +(-5.52304 + 9.56619i) q^{91} +(-2.90301 + 4.50756i) q^{92} +(-2.17658 - 2.22381i) q^{93} +(16.9827 + 0.412493i) q^{94} +(-5.13289 - 4.30701i) q^{95} +(-9.78809 - 0.439662i) q^{96} +(-5.25587 - 1.91298i) q^{97} +(-0.647795 - 4.27899i) q^{98} +(4.85561 + 0.104246i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 9 q^{8} - 12 q^{9} - 3 q^{10} - 30 q^{11} - 3 q^{12} + 9 q^{14} - 6 q^{16} - 18 q^{17} + 63 q^{18} - 6 q^{19} - 27 q^{20} - 18 q^{22} - 30 q^{24} - 12 q^{25}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.279321 + 1.38635i −0.197510 + 0.980301i
\(3\) 1.40807 + 1.00863i 0.812949 + 0.582335i
\(4\) −1.84396 − 0.774477i −0.921980 − 0.387238i
\(5\) 2.25586 − 0.821067i 1.00885 − 0.367192i 0.215861 − 0.976424i \(-0.430744\pi\)
0.792992 + 0.609232i \(0.208522\pi\)
\(6\) −1.79163 + 1.67035i −0.731429 + 0.681917i
\(7\) 1.95474 − 0.344674i 0.738823 − 0.130274i 0.208443 − 0.978034i \(-0.433160\pi\)
0.530380 + 0.847760i \(0.322049\pi\)
\(8\) 1.58876 − 2.34005i 0.561710 − 0.827334i
\(9\) 0.965315 + 2.84045i 0.321772 + 0.946817i
\(10\) 0.508180 + 3.35677i 0.160701 + 1.06150i
\(11\) 0.553700 − 1.52128i 0.166947 − 0.458683i −0.827803 − 0.561019i \(-0.810410\pi\)
0.994750 + 0.102336i \(0.0326317\pi\)
\(12\) −1.81526 − 2.95040i −0.524020 − 0.851706i
\(13\) −3.57715 + 4.26309i −0.992124 + 1.18237i −0.00890108 + 0.999960i \(0.502833\pi\)
−0.983223 + 0.182407i \(0.941611\pi\)
\(14\) −0.0681610 + 2.80624i −0.0182168 + 0.750000i
\(15\) 4.00457 + 1.11922i 1.03397 + 0.288982i
\(16\) 2.80037 + 2.85621i 0.700093 + 0.714052i
\(17\) −6.40600 − 3.69851i −1.55368 − 0.897020i −0.997837 − 0.0657345i \(-0.979061\pi\)
−0.555846 − 0.831285i \(-0.687606\pi\)
\(18\) −4.20751 + 0.544870i −0.991719 + 0.128427i
\(19\) −1.39557 − 2.41719i −0.320165 − 0.554543i 0.660357 − 0.750952i \(-0.270405\pi\)
−0.980522 + 0.196410i \(0.937072\pi\)
\(20\) −4.79562 − 0.233100i −1.07233 − 0.0521227i
\(21\) 3.10006 + 1.48630i 0.676489 + 0.324336i
\(22\) 1.95437 + 1.19255i 0.416673 + 0.254253i
\(23\) 0.465506 − 2.64002i 0.0970648 − 0.550482i −0.897030 − 0.441969i \(-0.854280\pi\)
0.994095 − 0.108513i \(-0.0346088\pi\)
\(24\) 4.59734 − 1.69248i 0.938427 − 0.345477i
\(25\) 0.584547 − 0.490493i 0.116909 − 0.0980986i
\(26\) −4.91098 − 6.14998i −0.963122 − 1.20611i
\(27\) −1.50575 + 4.97320i −0.289781 + 0.957093i
\(28\) −3.87141 − 0.878338i −0.731627 − 0.165990i
\(29\) 0.138481 − 0.116199i 0.0257153 − 0.0215777i −0.629839 − 0.776725i \(-0.716879\pi\)
0.655555 + 0.755148i \(0.272435\pi\)
\(30\) −2.67020 + 5.23913i −0.487509 + 0.956530i
\(31\) −1.76926 − 0.311969i −0.317769 − 0.0560313i 0.0124887 − 0.999922i \(-0.496025\pi\)
−0.330258 + 0.943891i \(0.607136\pi\)
\(32\) −4.74192 + 3.08451i −0.838261 + 0.545269i
\(33\) 2.31406 − 1.58358i 0.402826 − 0.275667i
\(34\) 6.91677 − 7.84792i 1.18622 − 1.34591i
\(35\) 4.12663 − 2.38251i 0.697528 − 0.402718i
\(36\) 0.419863 − 5.98529i 0.0699771 − 0.997549i
\(37\) 3.55469 + 2.05230i 0.584387 + 0.337396i 0.762875 − 0.646546i \(-0.223787\pi\)
−0.178488 + 0.983942i \(0.557120\pi\)
\(38\) 3.74090 − 1.25958i 0.606854 − 0.204331i
\(39\) −9.33677 + 2.39468i −1.49508 + 0.383456i
\(40\) 1.66268 − 6.58332i 0.262892 − 1.04091i
\(41\) −2.85334 + 3.40048i −0.445617 + 0.531066i −0.941360 − 0.337404i \(-0.890451\pi\)
0.495743 + 0.868469i \(0.334896\pi\)
\(42\) −2.92645 + 3.88263i −0.451560 + 0.599103i
\(43\) 11.1139 + 4.04511i 1.69485 + 0.616874i 0.995222 − 0.0976364i \(-0.0311282\pi\)
0.699625 + 0.714510i \(0.253350\pi\)
\(44\) −2.19920 + 2.37635i −0.331541 + 0.358248i
\(45\) 4.50982 + 5.61508i 0.672284 + 0.837047i
\(46\) 3.52998 + 1.38277i 0.520466 + 0.203878i
\(47\) −2.08588 − 11.8296i −0.304257 − 1.72552i −0.626983 − 0.779033i \(-0.715711\pi\)
0.322727 − 0.946492i \(-0.395401\pi\)
\(48\) 1.06225 + 6.84629i 0.153322 + 0.988176i
\(49\) −2.87563 + 1.04664i −0.410804 + 0.149520i
\(50\) 0.516721 + 0.947394i 0.0730754 + 0.133982i
\(51\) −5.28965 − 11.6691i −0.740699 − 1.63400i
\(52\) 9.89779 − 5.09053i 1.37258 − 0.705930i
\(53\) 10.1087 1.38854 0.694271 − 0.719714i \(-0.255727\pi\)
0.694271 + 0.719714i \(0.255727\pi\)
\(54\) −6.47403 − 3.47662i −0.881004 − 0.473108i
\(55\) − 3.88642i − 0.524045i
\(56\) 2.29906 − 5.12181i 0.307224 − 0.684430i
\(57\) 0.473009 − 4.81119i 0.0626516 − 0.637258i
\(58\) 0.122413 + 0.224441i 0.0160736 + 0.0294705i
\(59\) −1.96718 − 5.40478i −0.256105 − 0.703642i −0.999399 − 0.0346769i \(-0.988960\pi\)
0.743294 − 0.668965i \(-0.233262\pi\)
\(60\) −6.51745 − 5.16524i −0.841399 − 0.666830i
\(61\) −7.60780 + 1.34146i −0.974079 + 0.171756i −0.637965 − 0.770065i \(-0.720224\pi\)
−0.336113 + 0.941822i \(0.609113\pi\)
\(62\) 0.926693 − 2.36569i 0.117690 − 0.300443i
\(63\) 2.86597 + 5.21963i 0.361079 + 0.657612i
\(64\) −2.95170 − 7.43555i −0.368963 − 0.929444i
\(65\) −4.56929 + 12.5540i −0.566751 + 1.55714i
\(66\) 1.54904 + 3.65044i 0.190674 + 0.449338i
\(67\) 1.38817 + 1.16481i 0.169592 + 0.142305i 0.723633 − 0.690184i \(-0.242471\pi\)
−0.554041 + 0.832489i \(0.686915\pi\)
\(68\) 8.94800 + 11.7812i 1.08510 + 1.42868i
\(69\) 3.31828 − 3.24780i 0.399474 − 0.390989i
\(70\) 2.15035 + 6.38646i 0.257016 + 0.763328i
\(71\) 1.25119 − 2.16713i 0.148489 − 0.257191i −0.782180 − 0.623052i \(-0.785892\pi\)
0.930669 + 0.365862i \(0.119226\pi\)
\(72\) 8.18046 + 2.25390i 0.964077 + 0.265624i
\(73\) −3.79038 − 6.56514i −0.443631 − 0.768391i 0.554325 − 0.832300i \(-0.312976\pi\)
−0.997956 + 0.0639093i \(0.979643\pi\)
\(74\) −3.83812 + 4.35481i −0.446172 + 0.506236i
\(75\) 1.31781 − 0.101054i 0.152168 − 0.0116687i
\(76\) 0.701309 + 5.53804i 0.0804457 + 0.635257i
\(77\) 0.557996 − 3.16455i 0.0635896 − 0.360634i
\(78\) −0.711916 − 13.6130i −0.0806086 − 1.54136i
\(79\) −6.77140 − 8.06984i −0.761842 − 0.907928i 0.236121 − 0.971724i \(-0.424124\pi\)
−0.997963 + 0.0637957i \(0.979679\pi\)
\(80\) 8.66239 + 4.14392i 0.968485 + 0.463304i
\(81\) −7.13633 + 5.48386i −0.792926 + 0.609318i
\(82\) −3.91727 − 4.90557i −0.432590 − 0.541729i
\(83\) 0.718131 + 0.855836i 0.0788251 + 0.0939402i 0.804016 − 0.594608i \(-0.202693\pi\)
−0.725191 + 0.688548i \(0.758248\pi\)
\(84\) −4.56529 − 5.14159i −0.498114 − 0.560994i
\(85\) −17.4878 − 3.08357i −1.89682 − 0.334460i
\(86\) −8.71230 + 14.2779i −0.939471 + 1.53962i
\(87\) 0.312194 − 0.0239400i 0.0334707 − 0.00256664i
\(88\) −2.68018 − 3.71263i −0.285708 − 0.395768i
\(89\) 9.87804 − 5.70309i 1.04707 − 0.604526i 0.125243 − 0.992126i \(-0.460029\pi\)
0.921828 + 0.387600i \(0.126696\pi\)
\(90\) −9.04419 + 4.68380i −0.953341 + 0.493716i
\(91\) −5.52304 + 9.56619i −0.578972 + 1.00281i
\(92\) −2.90301 + 4.50756i −0.302659 + 0.469946i
\(93\) −2.17658 − 2.22381i −0.225701 − 0.230599i
\(94\) 16.9827 + 0.412493i 1.75163 + 0.0425454i
\(95\) −5.13289 − 4.30701i −0.526623 − 0.441890i
\(96\) −9.78809 − 0.439662i −0.998993 − 0.0448728i
\(97\) −5.25587 − 1.91298i −0.533652 − 0.194234i 0.0611161 − 0.998131i \(-0.480534\pi\)
−0.594768 + 0.803897i \(0.702756\pi\)
\(98\) −0.647795 − 4.27899i −0.0654371 − 0.432243i
\(99\) 4.85561 + 0.104246i 0.488007 + 0.0104771i
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 216.2.v.b.59.16 yes 192
3.2 odd 2 648.2.v.b.611.17 192
4.3 odd 2 864.2.bh.b.815.6 192
8.3 odd 2 inner 216.2.v.b.59.32 yes 192
8.5 even 2 864.2.bh.b.815.5 192
24.11 even 2 648.2.v.b.611.1 192
27.11 odd 18 inner 216.2.v.b.11.32 yes 192
27.16 even 9 648.2.v.b.35.1 192
108.11 even 18 864.2.bh.b.335.5 192
216.11 even 18 inner 216.2.v.b.11.16 ✓ 192
216.43 odd 18 648.2.v.b.35.17 192
216.173 odd 18 864.2.bh.b.335.6 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.16 ✓ 192 216.11 even 18 inner
216.2.v.b.11.32 yes 192 27.11 odd 18 inner
216.2.v.b.59.16 yes 192 1.1 even 1 trivial
216.2.v.b.59.32 yes 192 8.3 odd 2 inner
648.2.v.b.35.1 192 27.16 even 9
648.2.v.b.35.17 192 216.43 odd 18
648.2.v.b.611.1 192 24.11 even 2
648.2.v.b.611.17 192 3.2 odd 2
864.2.bh.b.335.5 192 108.11 even 18
864.2.bh.b.335.6 192 216.173 odd 18
864.2.bh.b.815.5 192 8.5 even 2
864.2.bh.b.815.6 192 4.3 odd 2