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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,2,Mod(13,216)] 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("216.13"); 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([0, 9, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.t (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: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(34\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 13.14
Character \(\chi\) \(=\) 216.13
Dual form 216.2.t.a.133.14

$q$-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+(-0.565401 + 1.29627i) q^{2} +(-1.67042 + 0.457945i) q^{3} +(-1.36064 - 1.46583i) q^{4} +(1.52309 - 1.81515i) q^{5} +(0.350834 - 2.42424i) q^{6} +(0.767101 + 0.279202i) q^{7} +(2.66942 - 0.934982i) q^{8} +(2.58057 - 1.52992i) q^{9} +(1.49177 + 3.00063i) q^{10} +(3.58256 + 4.26953i) q^{11} +(2.94411 + 1.82544i) q^{12} +(-2.36616 - 0.417219i) q^{13} +(-0.795642 + 0.836511i) q^{14} +(-1.71296 + 3.72954i) q^{15} +(-0.297302 + 3.98894i) q^{16} +(1.52854 - 2.64751i) q^{17} +(0.524128 + 4.21014i) q^{18} +(-0.631691 + 0.364707i) q^{19} +(-4.73308 + 0.237181i) q^{20} +(-1.40924 - 0.115093i) q^{21} +(-7.56006 + 2.22998i) q^{22} +(6.38462 - 2.32381i) q^{23} +(-4.03087 + 2.78426i) q^{24} +(-0.106720 - 0.605238i) q^{25} +(1.87866 - 2.83130i) q^{26} +(-3.61001 + 3.73735i) q^{27} +(-0.634489 - 1.50433i) q^{28} +(9.24326 - 1.62984i) q^{29} +(-3.86600 - 4.32915i) q^{30} +(1.92800 - 0.701734i) q^{31} +(-5.00265 - 2.64073i) q^{32} +(-7.93957 - 5.49127i) q^{33} +(2.56766 + 3.47831i) q^{34} +(1.67516 - 0.967153i) q^{35} +(-5.75383 - 1.70101i) q^{36} +(4.61364 + 2.66369i) q^{37} +(-0.115601 - 1.02505i) q^{38} +(4.14354 - 0.386644i) q^{39} +(2.36864 - 6.26946i) q^{40} +(-1.29167 + 7.32541i) q^{41} +(0.945976 - 1.76168i) q^{42} +(-4.42023 - 5.26782i) q^{43} +(1.38381 - 11.0607i) q^{44} +(1.15342 - 7.01433i) q^{45} +(-0.597580 + 9.59009i) q^{46} +(-10.2350 - 3.72522i) q^{47} +(-1.33009 - 6.79933i) q^{48} +(-4.85182 - 4.07116i) q^{49} +(0.844892 + 0.203864i) q^{50} +(-1.34089 + 5.12243i) q^{51} +(2.60793 + 4.03608i) q^{52} -2.99582i q^{53} +(-2.80352 - 6.79266i) q^{54} +13.2064 q^{55} +(2.30877 + 0.0280815i) q^{56} +(0.888170 - 0.898491i) q^{57} +(-3.11344 + 12.9033i) q^{58} +(0.837520 - 0.998118i) q^{59} +(7.79759 - 2.56368i) q^{60} +(-0.528763 + 1.45277i) q^{61} +(-0.180455 + 2.89597i) q^{62} +(2.40672 - 0.453099i) q^{63} +(6.25162 - 4.99172i) q^{64} +(-4.36120 + 3.65948i) q^{65} +(11.6072 - 7.18708i) q^{66} +(-4.70399 - 0.829441i) q^{67} +(-5.96060 + 1.36174i) q^{68} +(-9.60078 + 6.80553i) q^{69} +(0.306557 + 2.71829i) q^{70} +(-4.37210 + 7.57270i) q^{71} +(5.45819 - 6.49678i) q^{72} +(-2.62609 - 4.54853i) q^{73} +(-6.06142 + 4.47448i) q^{74} +(0.455432 + 0.962127i) q^{75} +(1.39410 + 0.429714i) q^{76} +(1.55613 + 4.27542i) q^{77} +(-1.84157 + 5.58977i) q^{78} +(-0.739506 - 4.19395i) q^{79} +(6.78770 + 6.61516i) q^{80} +(4.31872 - 7.89612i) q^{81} +(-8.76541 - 5.81615i) q^{82} +(-14.1745 + 2.49935i) q^{83} +(1.74876 + 2.22230i) q^{84} +(-2.47752 - 6.80694i) q^{85} +(9.32773 - 2.75138i) q^{86} +(-14.6937 + 6.95541i) q^{87} +(13.5553 + 8.04754i) q^{88} +(4.96560 + 8.60067i) q^{89} +(8.44033 + 5.46106i) q^{90} +(-1.69860 - 0.980687i) q^{91} +(-12.0935 - 6.19687i) q^{92} +(-2.89920 + 2.05510i) q^{93} +(10.6158 - 11.1611i) q^{94} +(-0.300125 + 1.70209i) q^{95} +(9.56582 + 2.12018i) q^{96} +(-11.8735 + 9.96305i) q^{97} +(8.02056 - 3.98744i) q^{98} +(15.7771 + 5.53682i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 204 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9} - 3 q^{10} + 3 q^{12} - 21 q^{14} - 12 q^{15} - 6 q^{16} - 6 q^{17} - 27 q^{18} + 15 q^{20} - 6 q^{22} - 12 q^{23} - 12 q^{25} - 30 q^{26}+ \cdots + 54 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.565401 + 1.29627i −0.399799 + 0.916603i
\(3\) −1.67042 + 0.457945i −0.964415 + 0.264395i
\(4\) −1.36064 − 1.46583i −0.680321 − 0.732914i
\(5\) 1.52309 − 1.81515i 0.681147 − 0.811760i −0.309108 − 0.951027i \(-0.600030\pi\)
0.990255 + 0.139268i \(0.0444748\pi\)
\(6\) 0.350834 − 2.42424i 0.143227 − 0.989690i
\(7\) 0.767101 + 0.279202i 0.289937 + 0.105528i 0.482894 − 0.875679i \(-0.339586\pi\)
−0.192957 + 0.981207i \(0.561808\pi\)
\(8\) 2.66942 − 0.934982i 0.943783 − 0.330566i
\(9\) 2.58057 − 1.52992i 0.860191 − 0.509972i
\(10\) 1.49177 + 3.00063i 0.471739 + 0.948882i
\(11\) 3.58256 + 4.26953i 1.08018 + 1.28731i 0.955460 + 0.295120i \(0.0953598\pi\)
0.124723 + 0.992192i \(0.460196\pi\)
\(12\) 2.94411 + 1.82544i 0.849890 + 0.526960i
\(13\) −2.36616 − 0.417219i −0.656256 − 0.115716i −0.164402 − 0.986393i \(-0.552569\pi\)
−0.491854 + 0.870678i \(0.663681\pi\)
\(14\) −0.795642 + 0.836511i −0.212644 + 0.223567i
\(15\) −1.71296 + 3.72954i −0.442283 + 0.962964i
\(16\) −0.297302 + 3.98894i −0.0743256 + 0.997234i
\(17\) 1.52854 − 2.64751i 0.370726 − 0.642116i −0.618951 − 0.785429i \(-0.712442\pi\)
0.989677 + 0.143313i \(0.0457755\pi\)
\(18\) 0.524128 + 4.21014i 0.123538 + 0.992340i
\(19\) −0.631691 + 0.364707i −0.144920 + 0.0836695i −0.570707 − 0.821154i \(-0.693331\pi\)
0.425787 + 0.904823i \(0.359997\pi\)
\(20\) −4.73308 + 0.237181i −1.05835 + 0.0530352i
\(21\) −1.40924 − 0.115093i −0.307521 − 0.0251154i
\(22\) −7.56006 + 2.22998i −1.61181 + 0.475433i
\(23\) 6.38462 − 2.32381i 1.33128 − 0.484548i 0.424228 − 0.905556i \(-0.360546\pi\)
0.907057 + 0.421008i \(0.138323\pi\)
\(24\) −4.03087 + 2.78426i −0.822798 + 0.568334i
\(25\) −0.106720 − 0.605238i −0.0213440 − 0.121048i
\(26\) 1.87866 − 2.83130i 0.368436 − 0.555263i
\(27\) −3.61001 + 3.73735i −0.694747 + 0.719254i
\(28\) −0.634489 − 1.50433i −0.119907 − 0.284292i
\(29\) 9.24326 − 1.62984i 1.71643 − 0.302653i 0.773045 − 0.634351i \(-0.218733\pi\)
0.943385 + 0.331698i \(0.107622\pi\)
\(30\) −3.86600 − 4.32915i −0.705831 − 0.790390i
\(31\) 1.92800 − 0.701734i 0.346279 − 0.126035i −0.163025 − 0.986622i \(-0.552125\pi\)
0.509304 + 0.860587i \(0.329903\pi\)
\(32\) −5.00265 − 2.64073i −0.884352 − 0.466820i
\(33\) −7.93957 − 5.49127i −1.38210 − 0.955908i
\(34\) 2.56766 + 3.47831i 0.440350 + 0.596526i
\(35\) 1.67516 − 0.967153i 0.283154 − 0.163479i
\(36\) −5.75383 − 1.70101i −0.958972 − 0.283501i
\(37\) 4.61364 + 2.66369i 0.758478 + 0.437907i 0.828749 − 0.559621i \(-0.189053\pi\)
−0.0702711 + 0.997528i \(0.522386\pi\)
\(38\) −0.115601 − 1.02505i −0.0187529 − 0.166285i
\(39\) 4.14354 − 0.386644i 0.663497 − 0.0619126i
\(40\) 2.36864 − 6.26946i 0.374515 − 0.991289i
\(41\) −1.29167 + 7.32541i −0.201725 + 1.14404i 0.700787 + 0.713371i \(0.252832\pi\)
−0.902511 + 0.430666i \(0.858279\pi\)
\(42\) 0.945976 − 1.76168i 0.145967 − 0.271833i
\(43\) −4.42023 − 5.26782i −0.674078 − 0.803335i 0.315255 − 0.949007i \(-0.397910\pi\)
−0.989333 + 0.145672i \(0.953466\pi\)
\(44\) 1.38381 − 11.0607i 0.208617 − 1.66747i
\(45\) 1.15342 − 7.01433i 0.171942 − 1.04563i
\(46\) −0.597580 + 9.59009i −0.0881083 + 1.41398i
\(47\) −10.2350 − 3.72522i −1.49292 − 0.543380i −0.538706 − 0.842494i \(-0.681087\pi\)
−0.954217 + 0.299114i \(0.903309\pi\)
\(48\) −1.33009 − 6.79933i −0.191983 − 0.981398i
\(49\) −4.85182 − 4.07116i −0.693117 − 0.581594i
\(50\) 0.844892 + 0.203864i 0.119486 + 0.0288308i
\(51\) −1.34089 + 5.12243i −0.187762 + 0.717284i
\(52\) 2.60793 + 4.03608i 0.361655 + 0.559703i
\(53\) − 2.99582i − 0.411507i −0.978604 − 0.205754i \(-0.934035\pi\)
0.978604 − 0.205754i \(-0.0659645\pi\)
\(54\) −2.80352 − 6.79266i −0.381511 − 0.924364i
\(55\) 13.2064 1.78075
\(56\) 2.30877 + 0.0280815i 0.308522 + 0.00375255i
\(57\) 0.888170 − 0.898491i 0.117641 − 0.119008i
\(58\) −3.11344 + 12.9033i −0.408815 + 1.69429i
\(59\) 0.837520 − 0.998118i 0.109036 − 0.129944i −0.708766 − 0.705443i \(-0.750748\pi\)
0.817802 + 0.575500i \(0.195192\pi\)
\(60\) 7.79759 − 2.56368i 1.00666 − 0.330970i
\(61\) −0.528763 + 1.45277i −0.0677012 + 0.186008i −0.968929 − 0.247337i \(-0.920444\pi\)
0.901228 + 0.433345i \(0.142667\pi\)
\(62\) −0.180455 + 2.89597i −0.0229177 + 0.367789i
\(63\) 2.40672 − 0.453099i 0.303218 − 0.0570851i
\(64\) 6.25162 − 4.99172i 0.781452 − 0.623965i
\(65\) −4.36120 + 3.65948i −0.540940 + 0.453903i
\(66\) 11.6072 − 7.18708i 1.42875 − 0.884668i
\(67\) −4.70399 − 0.829441i −0.574684 − 0.101332i −0.121250 − 0.992622i \(-0.538690\pi\)
−0.453434 + 0.891290i \(0.649801\pi\)
\(68\) −5.96060 + 1.36174i −0.722829 + 0.165135i
\(69\) −9.60078 + 6.80553i −1.15580 + 0.819289i
\(70\) 0.306557 + 2.71829i 0.0366406 + 0.324898i
\(71\) −4.37210 + 7.57270i −0.518873 + 0.898714i 0.480887 + 0.876783i \(0.340315\pi\)
−0.999759 + 0.0219314i \(0.993018\pi\)
\(72\) 5.45819 − 6.49678i 0.643254 − 0.765653i
\(73\) −2.62609 − 4.54853i −0.307361 − 0.532365i 0.670423 − 0.741979i \(-0.266113\pi\)
−0.977784 + 0.209614i \(0.932779\pi\)
\(74\) −6.06142 + 4.47448i −0.704626 + 0.520148i
\(75\) 0.455432 + 0.962127i 0.0525887 + 0.111097i
\(76\) 1.39410 + 0.429714i 0.159915 + 0.0492916i
\(77\) 1.55613 + 4.27542i 0.177337 + 0.487230i
\(78\) −1.84157 + 5.58977i −0.208516 + 0.632916i
\(79\) −0.739506 − 4.19395i −0.0832010 − 0.471856i −0.997730 − 0.0673365i \(-0.978550\pi\)
0.914529 − 0.404520i \(-0.132561\pi\)
\(80\) 6.78770 + 6.61516i 0.758888 + 0.739598i
\(81\) 4.31872 − 7.89612i 0.479857 − 0.877347i
\(82\) −8.76541 − 5.81615i −0.967978 − 0.642286i
\(83\) −14.1745 + 2.49935i −1.55585 + 0.274339i −0.884408 − 0.466716i \(-0.845437\pi\)
−0.671445 + 0.741054i \(0.734326\pi\)
\(84\) 1.74876 + 2.22230i 0.190805 + 0.242473i
\(85\) −2.47752 − 6.80694i −0.268725 − 0.738316i
\(86\) 9.32773 − 2.75138i 1.00584 − 0.296689i
\(87\) −14.6937 + 6.95541i −1.57533 + 0.745698i
\(88\) 13.5553 + 8.04754i 1.44500 + 0.857871i
\(89\) 4.96560 + 8.60067i 0.526352 + 0.911669i 0.999529 + 0.0307010i \(0.00977397\pi\)
−0.473176 + 0.880968i \(0.656893\pi\)
\(90\) 8.44033 + 5.46106i 0.889689 + 0.575646i
\(91\) −1.69860 − 0.980687i −0.178062 − 0.102804i
\(92\) −12.0935 − 6.19687i −1.26083 − 0.646069i
\(93\) −2.89920 + 2.05510i −0.300633 + 0.213104i
\(94\) 10.6158 − 11.1611i 1.09493 − 1.15118i
\(95\) −0.300125 + 1.70209i −0.0307922 + 0.174631i
\(96\) 9.56582 + 2.12018i 0.976307 + 0.216390i
\(97\) −11.8735 + 9.96305i −1.20557 + 1.01159i −0.206119 + 0.978527i \(0.566083\pi\)
−0.999453 + 0.0330678i \(0.989472\pi\)
\(98\) 8.02056 − 3.98744i 0.810199 − 0.402792i
\(99\) 15.7771 + 5.53682i 1.58566 + 0.556471i
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.t.a.13.14 ✓ 204
3.2 odd 2 648.2.t.a.253.21 204
4.3 odd 2 864.2.bf.a.337.32 204
8.3 odd 2 864.2.bf.a.337.3 204
8.5 even 2 inner 216.2.t.a.13.25 yes 204
24.5 odd 2 648.2.t.a.253.10 204
27.2 odd 18 648.2.t.a.397.10 204
27.25 even 9 inner 216.2.t.a.133.25 yes 204
108.79 odd 18 864.2.bf.a.241.3 204
216.29 odd 18 648.2.t.a.397.21 204
216.133 even 18 inner 216.2.t.a.133.14 yes 204
216.187 odd 18 864.2.bf.a.241.32 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.t.a.13.14 ✓ 204 1.1 even 1 trivial
216.2.t.a.13.25 yes 204 8.5 even 2 inner
216.2.t.a.133.14 yes 204 216.133 even 18 inner
216.2.t.a.133.25 yes 204 27.25 even 9 inner
648.2.t.a.253.10 204 24.5 odd 2
648.2.t.a.253.21 204 3.2 odd 2
648.2.t.a.397.10 204 27.2 odd 18
648.2.t.a.397.21 204 216.29 odd 18
864.2.bf.a.241.3 204 108.79 odd 18
864.2.bf.a.241.32 204 216.187 odd 18
864.2.bf.a.337.3 204 8.3 odd 2
864.2.bf.a.337.32 204 4.3 odd 2