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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [209,2,Mod(20,209)] 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("209.20"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([6, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] 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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 58.1
Character \(\chi\) \(=\) 209.58
Dual form 209.2.f.c.191.1

$q$-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+(-2.24608 + 1.63187i) q^{2} +(0.358792 + 1.10425i) q^{3} +(1.76382 - 5.42849i) q^{4} +(-1.10965 - 0.806208i) q^{5} +(-2.60786 - 1.89472i) q^{6} +(0.316566 - 0.974291i) q^{7} +(3.18106 + 9.79028i) q^{8} +(1.33642 - 0.970964i) q^{9} +3.80798 q^{10} +(3.05064 - 1.30137i) q^{11} +6.62725 q^{12} +(0.861031 - 0.625576i) q^{13} +(0.878884 + 2.70493i) q^{14} +(0.492120 - 1.51459i) q^{15} +(-13.8859 - 10.0887i) q^{16} +(3.34920 + 2.43334i) q^{17} +(-1.41721 + 4.36172i) q^{18} +(0.309017 + 0.951057i) q^{19} +(-6.33372 + 4.60171i) q^{20} +1.18944 q^{21} +(-4.72830 + 7.90124i) q^{22} -0.381634 q^{23} +(-9.66957 + 7.02535i) q^{24} +(-0.963733 - 2.96607i) q^{25} +(-0.913083 + 2.81018i) q^{26} +(4.36967 + 3.17475i) q^{27} +(-4.73056 - 3.43695i) q^{28} +(-2.70862 + 8.33627i) q^{29} +(1.36627 + 4.20496i) q^{30} +(3.29151 - 2.39142i) q^{31} +27.0638 q^{32} +(2.53159 + 2.90175i) q^{33} -11.4935 q^{34} +(-1.13676 + 0.825904i) q^{35} +(-2.91366 - 8.96734i) q^{36} +(0.764223 - 2.35204i) q^{37} +(-2.24608 - 1.63187i) q^{38} +(0.999722 + 0.726341i) q^{39} +(4.36314 - 13.4284i) q^{40} +(-3.56878 - 10.9836i) q^{41} +(-2.67157 + 1.94101i) q^{42} +7.89682 q^{43} +(-1.68370 - 18.8558i) q^{44} -2.26575 q^{45} +(0.857180 - 0.622777i) q^{46} +(0.965577 + 2.97174i) q^{47} +(6.15826 - 18.9532i) q^{48} +(4.81409 + 3.49764i) q^{49} +(7.00485 + 5.08932i) q^{50} +(-1.48534 + 4.57142i) q^{51} +(-1.87722 - 5.77750i) q^{52} +(-5.12777 + 3.72554i) q^{53} -14.9954 q^{54} +(-4.43432 - 1.01538i) q^{55} +10.5456 q^{56} +(-0.939330 + 0.682463i) q^{57} +(-7.51994 - 23.1440i) q^{58} +(3.11490 - 9.58667i) q^{59} +(-7.35393 - 5.34294i) q^{60} +(3.76193 + 2.73320i) q^{61} +(-3.49049 + 10.7426i) q^{62} +(-0.522937 - 1.60943i) q^{63} +(-33.0157 + 23.9873i) q^{64} -1.45979 q^{65} +(-10.4214 - 2.38632i) q^{66} +7.87356 q^{67} +(19.1168 - 13.8891i) q^{68} +(-0.136927 - 0.421419i) q^{69} +(1.20548 - 3.71008i) q^{70} +(-4.05936 - 2.94929i) q^{71} +(13.7572 + 9.99521i) q^{72} +(-1.93266 + 5.94812i) q^{73} +(2.12171 + 6.52996i) q^{74} +(2.92949 - 2.12840i) q^{75} +5.70785 q^{76} +(-0.302186 - 3.38419i) q^{77} -3.43074 q^{78} +(-1.60556 + 1.16651i) q^{79} +(7.27488 + 22.3898i) q^{80} +(-0.406514 + 1.25112i) q^{81} +(25.9395 + 18.8462i) q^{82} +(-11.0244 - 8.00971i) q^{83} +(2.09796 - 6.45687i) q^{84} +(-1.75467 - 5.40031i) q^{85} +(-17.7369 + 12.8866i) q^{86} -10.1771 q^{87} +(22.4451 + 25.7269i) q^{88} +3.03261 q^{89} +(5.08906 - 3.69742i) q^{90} +(-0.336919 - 1.03693i) q^{91} +(-0.673135 + 2.07170i) q^{92} +(3.82169 + 2.77662i) q^{93} +(-7.01825 - 5.09906i) q^{94} +(0.423849 - 1.30447i) q^{95} +(9.71029 + 29.8852i) q^{96} +(-11.7735 + 8.55398i) q^{97} -16.5205 q^{98} +(2.81335 - 4.70125i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 3 q^{2} + 3 q^{3} - 13 q^{4} - 9 q^{5} + 10 q^{6} + 4 q^{7} - q^{8} - 21 q^{9} - 22 q^{10} + 8 q^{11} + 20 q^{12} + 13 q^{13} - 20 q^{14} - 9 q^{15} - 41 q^{16} + 21 q^{17} + 27 q^{18} - 10 q^{19}+ \cdots + 110 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24608 + 1.63187i −1.58821 + 1.15391i −0.681783 + 0.731555i \(0.738795\pi\)
−0.906432 + 0.422351i \(0.861205\pi\)
\(3\) 0.358792 + 1.10425i 0.207149 + 0.637538i 0.999618 + 0.0276268i \(0.00879501\pi\)
−0.792470 + 0.609911i \(0.791205\pi\)
\(4\) 1.76382 − 5.42849i 0.881911 − 2.71424i
\(5\) −1.10965 − 0.806208i −0.496251 − 0.360547i 0.311332 − 0.950301i \(-0.399225\pi\)
−0.807583 + 0.589754i \(0.799225\pi\)
\(6\) −2.60786 − 1.89472i −1.06466 − 0.773518i
\(7\) 0.316566 − 0.974291i 0.119651 − 0.368247i −0.873238 − 0.487294i \(-0.837984\pi\)
0.992889 + 0.119047i \(0.0379839\pi\)
\(8\) 3.18106 + 9.79028i 1.12467 + 3.46139i
\(9\) 1.33642 − 0.970964i 0.445473 − 0.323655i
\(10\) 3.80798 1.20419
\(11\) 3.05064 − 1.30137i 0.919804 − 0.392379i
\(12\) 6.62725 1.91312
\(13\) 0.861031 − 0.625576i 0.238807 − 0.173503i −0.461945 − 0.886909i \(-0.652848\pi\)
0.700752 + 0.713405i \(0.252848\pi\)
\(14\) 0.878884 + 2.70493i 0.234892 + 0.722922i
\(15\) 0.492120 − 1.51459i 0.127065 − 0.391066i
\(16\) −13.8859 − 10.0887i −3.47146 − 2.52217i
\(17\) 3.34920 + 2.43334i 0.812301 + 0.590171i 0.914497 − 0.404593i \(-0.132587\pi\)
−0.102196 + 0.994764i \(0.532587\pi\)
\(18\) −1.41721 + 4.36172i −0.334039 + 1.02807i
\(19\) 0.309017 + 0.951057i 0.0708934 + 0.218187i
\(20\) −6.33372 + 4.60171i −1.41626 + 1.02897i
\(21\) 1.18944 0.259557
\(22\) −4.72830 + 7.90124i −1.00808 + 1.68455i
\(23\) −0.381634 −0.0795763 −0.0397881 − 0.999208i \(-0.512668\pi\)
−0.0397881 + 0.999208i \(0.512668\pi\)
\(24\) −9.66957 + 7.02535i −1.97379 + 1.43404i
\(25\) −0.963733 − 2.96607i −0.192747 − 0.593213i
\(26\) −0.913083 + 2.81018i −0.179070 + 0.551122i
\(27\) 4.36967 + 3.17475i 0.840944 + 0.610981i
\(28\) −4.73056 − 3.43695i −0.893992 − 0.649523i
\(29\) −2.70862 + 8.33627i −0.502978 + 1.54801i 0.301166 + 0.953572i \(0.402624\pi\)
−0.804144 + 0.594434i \(0.797376\pi\)
\(30\) 1.36627 + 4.20496i 0.249446 + 0.767717i
\(31\) 3.29151 − 2.39142i 0.591172 − 0.429512i −0.251562 − 0.967841i \(-0.580944\pi\)
0.842734 + 0.538329i \(0.180944\pi\)
\(32\) 27.0638 4.78425
\(33\) 2.53159 + 2.90175i 0.440693 + 0.505129i
\(34\) −11.4935 −1.97111
\(35\) −1.13676 + 0.825904i −0.192147 + 0.139603i
\(36\) −2.91366 − 8.96734i −0.485611 − 1.49456i
\(37\) 0.764223 − 2.35204i 0.125637 − 0.386672i −0.868379 − 0.495901i \(-0.834838\pi\)
0.994017 + 0.109228i \(0.0348380\pi\)
\(38\) −2.24608 − 1.63187i −0.364362 − 0.264724i
\(39\) 0.999722 + 0.726341i 0.160084 + 0.116308i
\(40\) 4.36314 − 13.4284i 0.689874 − 2.12321i
\(41\) −3.56878 − 10.9836i −0.557350 − 1.71535i −0.689654 − 0.724139i \(-0.742238\pi\)
0.132304 − 0.991209i \(-0.457762\pi\)
\(42\) −2.67157 + 1.94101i −0.412233 + 0.299505i
\(43\) 7.89682 1.20425 0.602127 − 0.798400i \(-0.294320\pi\)
0.602127 + 0.798400i \(0.294320\pi\)
\(44\) −1.68370 − 18.8558i −0.253827 − 2.84262i
\(45\) −2.26575 −0.337759
\(46\) 0.857180 − 0.622777i 0.126384 − 0.0918235i
\(47\) 0.965577 + 2.97174i 0.140844 + 0.433473i 0.996453 − 0.0841493i \(-0.0268173\pi\)
−0.855609 + 0.517622i \(0.826817\pi\)
\(48\) 6.15826 − 18.9532i 0.888868 − 2.73565i
\(49\) 4.81409 + 3.49764i 0.687727 + 0.499663i
\(50\) 7.00485 + 5.08932i 0.990635 + 0.719738i
\(51\) −1.48534 + 4.57142i −0.207990 + 0.640126i
\(52\) −1.87722 − 5.77750i −0.260324 − 0.801195i
\(53\) −5.12777 + 3.72554i −0.704353 + 0.511742i −0.881347 − 0.472469i \(-0.843363\pi\)
0.176994 + 0.984212i \(0.443363\pi\)
\(54\) −14.9954 −2.04061
\(55\) −4.43432 − 1.01538i −0.597924 − 0.136914i
\(56\) 10.5456 1.40921
\(57\) −0.939330 + 0.682463i −0.124417 + 0.0903945i
\(58\) −7.51994 − 23.1440i −0.987416 − 3.03896i
\(59\) 3.11490 − 9.58667i 0.405525 − 1.24808i −0.514930 − 0.857232i \(-0.672182\pi\)
0.920456 − 0.390847i \(-0.127818\pi\)
\(60\) −7.35393 − 5.34294i −0.949388 − 0.689770i
\(61\) 3.76193 + 2.73320i 0.481666 + 0.349951i 0.801970 − 0.597364i \(-0.203785\pi\)
−0.320305 + 0.947315i \(0.603785\pi\)
\(62\) −3.49049 + 10.7426i −0.443292 + 1.36431i
\(63\) −0.522937 − 1.60943i −0.0658839 − 0.202770i
\(64\) −33.0157 + 23.9873i −4.12696 + 2.99841i
\(65\) −1.45979 −0.181064
\(66\) −10.4214 − 2.38632i −1.28279 − 0.293736i
\(67\) 7.87356 0.961909 0.480954 − 0.876746i \(-0.340290\pi\)
0.480954 + 0.876746i \(0.340290\pi\)
\(68\) 19.1168 − 13.8891i 2.31825 − 1.68431i
\(69\) −0.136927 − 0.421419i −0.0164841 − 0.0507329i
\(70\) 1.20548 − 3.71008i 0.144082 − 0.443440i
\(71\) −4.05936 − 2.94929i −0.481757 − 0.350017i 0.320249 − 0.947334i \(-0.396234\pi\)
−0.802005 + 0.597317i \(0.796234\pi\)
\(72\) 13.7572 + 9.99521i 1.62131 + 1.17795i
\(73\) −1.93266 + 5.94812i −0.226201 + 0.696174i 0.771967 + 0.635663i \(0.219273\pi\)
−0.998168 + 0.0605114i \(0.980727\pi\)
\(74\) 2.12171 + 6.52996i 0.246644 + 0.759092i
\(75\) 2.92949 − 2.12840i 0.338269 − 0.245767i
\(76\) 5.70785 0.654735
\(77\) −0.302186 − 3.38419i −0.0344373 − 0.385664i
\(78\) −3.43074 −0.388455
\(79\) −1.60556 + 1.16651i −0.180640 + 0.131242i −0.674431 − 0.738338i \(-0.735611\pi\)
0.493791 + 0.869581i \(0.335611\pi\)
\(80\) 7.27488 + 22.3898i 0.813356 + 2.50325i
\(81\) −0.406514 + 1.25112i −0.0451682 + 0.139014i
\(82\) 25.9395 + 18.8462i 2.86454 + 2.08121i
\(83\) −11.0244 − 8.00971i −1.21009 − 0.879180i −0.214848 − 0.976647i \(-0.568926\pi\)
−0.995239 + 0.0974678i \(0.968926\pi\)
\(84\) 2.09796 − 6.45687i 0.228907 − 0.704502i
\(85\) −1.75467 − 5.40031i −0.190320 − 0.585746i
\(86\) −17.7369 + 12.8866i −1.91261 + 1.38960i
\(87\) −10.1771 −1.09110
\(88\) 22.4451 + 25.7269i 2.39265 + 2.74250i
\(89\) 3.03261 0.321456 0.160728 − 0.986999i \(-0.448616\pi\)
0.160728 + 0.986999i \(0.448616\pi\)
\(90\) 5.08906 − 3.69742i 0.536434 − 0.389742i
\(91\) −0.336919 − 1.03693i −0.0353187 − 0.108700i
\(92\) −0.673135 + 2.07170i −0.0701792 + 0.215989i
\(93\) 3.82169 + 2.77662i 0.396291 + 0.287922i
\(94\) −7.01825 − 5.09906i −0.723877 − 0.525928i
\(95\) 0.423849 − 1.30447i 0.0434859 − 0.133836i
\(96\) 9.71029 + 29.8852i 0.991052 + 3.05014i
\(97\) −11.7735 + 8.55398i −1.19542 + 0.868525i −0.993827 − 0.110944i \(-0.964613\pi\)
−0.201596 + 0.979469i \(0.564613\pi\)
\(98\) −16.5205 −1.66882
\(99\) 2.81335 − 4.70125i 0.282752 − 0.472493i
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 209.2.f.c.58.1 ✓ 40
11.2 odd 10 2299.2.a.y.1.1 20
11.4 even 5 inner 209.2.f.c.191.1 yes 40
11.9 even 5 2299.2.a.x.1.20 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.f.c.58.1 ✓ 40 1.1 even 1 trivial
209.2.f.c.191.1 yes 40 11.4 even 5 inner
2299.2.a.x.1.20 20 11.9 even 5
2299.2.a.y.1.1 20 11.2 odd 10