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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [648,2,Mod(35,648)] 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("648.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.v (of order \(18\), degree \(6\), not 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 611.17
Character \(\chi\) \(=\) 648.611
Dual form 648.2.v.b.35.17

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.279321 - 1.38635i) q^{2} +(-1.84396 - 0.774477i) q^{4} +(-2.25586 + 0.821067i) q^{5} +(1.95474 - 0.344674i) q^{7} +(-1.58876 + 2.34005i) q^{8} +(0.508180 + 3.35677i) q^{10} +(-0.553700 + 1.52128i) q^{11} +(-3.57715 + 4.26309i) q^{13} +(0.0681610 - 2.80624i) q^{14} +(2.80037 + 2.85621i) q^{16} +(6.40600 + 3.69851i) q^{17} +(-1.39557 - 2.41719i) q^{19} +(4.79562 + 0.233100i) q^{20} +(1.95437 + 1.19255i) q^{22} +(-0.465506 + 2.64002i) q^{23} +(0.584547 - 0.490493i) q^{25} +(4.91098 + 6.14998i) q^{26} +(-3.87141 - 0.878338i) q^{28} +(-0.138481 + 0.116199i) q^{29} +(-1.76926 - 0.311969i) q^{31} +(4.74192 - 3.08451i) q^{32} +(6.91677 - 7.84792i) q^{34} +(-4.12663 + 2.38251i) q^{35} +(3.55469 + 2.05230i) q^{37} +(-3.74090 + 1.25958i) q^{38} +(1.66268 - 6.58332i) q^{40} +(2.85334 - 3.40048i) q^{41} +(11.1139 + 4.04511i) q^{43} +(2.19920 - 2.37635i) q^{44} +(3.52998 + 1.38277i) q^{46} +(2.08588 + 11.8296i) q^{47} +(-2.87563 + 1.04664i) q^{49} +(-0.516721 - 0.947394i) q^{50} +(9.89779 - 5.09053i) q^{52} -10.1087 q^{53} -3.88642i q^{55} +(-2.29906 + 5.12181i) q^{56} +(0.122413 + 0.224441i) q^{58} +(1.96718 + 5.40478i) q^{59} +(-7.60780 + 1.34146i) q^{61} +(-0.926693 + 2.36569i) q^{62} +(-2.95170 - 7.43555i) q^{64} +(4.56929 - 12.5540i) q^{65} +(1.38817 + 1.16481i) q^{67} +(-8.94800 - 11.7812i) q^{68} +(2.15035 + 6.38646i) q^{70} +(-1.25119 + 2.16713i) q^{71} +(-3.79038 - 6.56514i) q^{73} +(3.83812 - 4.35481i) q^{74} +(0.701309 + 5.53804i) q^{76} +(-0.557996 + 3.16455i) q^{77} +(-6.77140 - 8.06984i) q^{79} +(-8.66239 - 4.14392i) q^{80} +(-3.91727 - 4.90557i) q^{82} +(-0.718131 - 0.855836i) q^{83} +(-17.4878 - 3.08357i) q^{85} +(8.71230 - 14.2779i) q^{86} +(-2.68018 - 3.71263i) q^{88} +(-9.87804 + 5.70309i) q^{89} +(-5.52304 + 9.56619i) q^{91} +(2.90301 - 4.50756i) q^{92} +(16.9827 + 0.412493i) q^{94} +(5.13289 + 4.30701i) q^{95} +(-5.25587 - 1.91298i) q^{97} +(0.647795 + 4.27899i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 6 q^{2} - 6 q^{4} + 9 q^{8} - 3 q^{10} + 30 q^{11} - 9 q^{14} - 6 q^{16} + 18 q^{17} - 6 q^{19} + 27 q^{20} - 18 q^{22} - 12 q^{25} - 12 q^{28} + 36 q^{32} + 12 q^{34} + 18 q^{35} + 102 q^{38} + 9 q^{40}+ \cdots - 162 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{18}\right)\)

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
\(3\) 0 0
\(4\) −1.84396 − 0.774477i −0.921980 − 0.387238i
\(5\) −2.25586 + 0.821067i −1.00885 + 0.367192i −0.792992 − 0.609232i \(-0.791478\pi\)
−0.215861 + 0.976424i \(0.569256\pi\)
\(6\) 0 0
\(7\) 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 0
\(10\) 0.508180 + 3.35677i 0.160701 + 1.06150i
\(11\) −0.553700 + 1.52128i −0.166947 + 0.458683i −0.994750 − 0.102336i \(-0.967368\pi\)
0.827803 + 0.561019i \(0.189590\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) 2.80037 + 2.85621i 0.700093 + 0.714052i
\(17\) 6.40600 + 3.69851i 1.55368 + 0.897020i 0.997837 + 0.0657345i \(0.0209390\pi\)
0.555846 + 0.831285i \(0.312394\pi\)
\(18\) 0 0
\(19\) −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
\(21\) 0 0
\(22\) 1.95437 + 1.19255i 0.416673 + 0.254253i
\(23\) −0.465506 + 2.64002i −0.0970648 + 0.550482i 0.897030 + 0.441969i \(0.145720\pi\)
−0.994095 + 0.108513i \(0.965391\pi\)
\(24\) 0 0
\(25\) 0.584547 − 0.490493i 0.116909 − 0.0980986i
\(26\) 4.91098 + 6.14998i 0.963122 + 1.20611i
\(27\) 0 0
\(28\) −3.87141 − 0.878338i −0.731627 − 0.165990i
\(29\) −0.138481 + 0.116199i −0.0257153 + 0.0215777i −0.655555 − 0.755148i \(-0.727565\pi\)
0.629839 + 0.776725i \(0.283121\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) 6.91677 − 7.84792i 1.18622 − 1.34591i
\(35\) −4.12663 + 2.38251i −0.697528 + 0.402718i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) 1.66268 − 6.58332i 0.262892 − 1.04091i
\(41\) 2.85334 − 3.40048i 0.445617 − 0.531066i −0.495743 − 0.868469i \(-0.665104\pi\)
0.941360 + 0.337404i \(0.109549\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) 3.52998 + 1.38277i 0.520466 + 0.203878i
\(47\) 2.08588 + 11.8296i 0.304257 + 1.72552i 0.626983 + 0.779033i \(0.284289\pi\)
−0.322727 + 0.946492i \(0.604599\pi\)
\(48\) 0 0
\(49\) −2.87563 + 1.04664i −0.410804 + 0.149520i
\(50\) −0.516721 − 0.947394i −0.0730754 − 0.133982i
\(51\) 0 0
\(52\) 9.89779 − 5.09053i 1.37258 − 0.705930i
\(53\) −10.1087 −1.38854 −0.694271 − 0.719714i \(-0.744273\pi\)
−0.694271 + 0.719714i \(0.744273\pi\)
\(54\) 0 0
\(55\) − 3.88642i − 0.524045i
\(56\) −2.29906 + 5.12181i −0.307224 + 0.684430i
\(57\) 0 0
\(58\) 0.122413 + 0.224441i 0.0160736 + 0.0294705i
\(59\) 1.96718 + 5.40478i 0.256105 + 0.703642i 0.999399 + 0.0346769i \(0.0110402\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(60\) 0 0
\(61\) −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
\(63\) 0 0
\(64\) −2.95170 − 7.43555i −0.368963 − 0.929444i
\(65\) 4.56929 − 12.5540i 0.566751 − 1.55714i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) 2.15035 + 6.38646i 0.257016 + 0.763328i
\(71\) −1.25119 + 2.16713i −0.148489 + 0.257191i −0.930669 − 0.365862i \(-0.880774\pi\)
0.782180 + 0.623052i \(0.214108\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 0 0
\(76\) 0.701309 + 5.53804i 0.0804457 + 0.635257i
\(77\) −0.557996 + 3.16455i −0.0635896 + 0.360634i
\(78\) 0 0
\(79\) −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
\(81\) 0 0
\(82\) −3.91727 − 4.90557i −0.432590 − 0.541729i
\(83\) −0.718131 − 0.855836i −0.0788251 − 0.0939402i 0.725191 − 0.688548i \(-0.241752\pi\)
−0.804016 + 0.594608i \(0.797307\pi\)
\(84\) 0 0
\(85\) −17.4878 − 3.08357i −1.89682 − 0.334460i
\(86\) 8.71230 − 14.2779i 0.939471 − 1.53962i
\(87\) 0 0
\(88\) −2.68018 − 3.71263i −0.285708 − 0.395768i
\(89\) −9.87804 + 5.70309i −1.04707 + 0.604526i −0.921828 − 0.387600i \(-0.873304\pi\)
−0.125243 + 0.992126i \(0.539971\pi\)
\(90\) 0 0
\(91\) −5.52304 + 9.56619i −0.578972 + 1.00281i
\(92\) 2.90301 − 4.50756i 0.302659 − 0.469946i
\(93\) 0 0
\(94\) 16.9827 + 0.412493i 1.75163 + 0.0425454i
\(95\) 5.13289 + 4.30701i 0.526623 + 0.441890i
\(96\) 0 0
\(97\) −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
\(99\) 0 0
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 648.2.v.b.611.17 192
3.2 odd 2 216.2.v.b.59.16 yes 192
8.3 odd 2 inner 648.2.v.b.611.1 192
12.11 even 2 864.2.bh.b.815.6 192
24.5 odd 2 864.2.bh.b.815.5 192
24.11 even 2 216.2.v.b.59.32 yes 192
27.11 odd 18 inner 648.2.v.b.35.1 192
27.16 even 9 216.2.v.b.11.32 yes 192
108.43 odd 18 864.2.bh.b.335.5 192
216.11 even 18 inner 648.2.v.b.35.17 192
216.43 odd 18 216.2.v.b.11.16 ✓ 192
216.205 even 18 864.2.bh.b.335.6 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.16 ✓ 192 216.43 odd 18
216.2.v.b.11.32 yes 192 27.16 even 9
216.2.v.b.59.16 yes 192 3.2 odd 2
216.2.v.b.59.32 yes 192 24.11 even 2
648.2.v.b.35.1 192 27.11 odd 18 inner
648.2.v.b.35.17 192 216.11 even 18 inner
648.2.v.b.611.1 192 8.3 odd 2 inner
648.2.v.b.611.17 192 1.1 even 1 trivial
864.2.bh.b.335.5 192 108.43 odd 18
864.2.bh.b.335.6 192 216.205 even 18
864.2.bh.b.815.5 192 24.5 odd 2
864.2.bh.b.815.6 192 12.11 even 2