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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(116,513)] 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("513.116"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.cd (of order \(18\), degree \(6\), not 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: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 116.8
Character \(\chi\) \(=\) 513.116
Dual form 513.2.cd.a.314.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.388217 - 0.325752i) q^{2} +(-0.302699 - 1.71669i) q^{4} +(2.48025 - 2.95585i) q^{5} -2.21821 q^{7} +(-0.948484 + 1.64282i) q^{8} +(-1.92575 + 0.339561i) q^{10} +(5.52215 - 3.18821i) q^{11} +(0.211324 - 0.580608i) q^{13} +(0.861148 + 0.722589i) q^{14} +(-2.37272 + 0.863601i) q^{16} +(-2.46346 - 0.434375i) q^{17} +(0.375190 + 4.34272i) q^{19} +(-5.82505 - 3.36309i) q^{20} +(-3.18236 - 0.561135i) q^{22} +(-3.50170 + 0.617444i) q^{23} +(-1.71715 - 9.73845i) q^{25} +(-0.271174 + 0.156562i) q^{26} +(0.671451 + 3.80799i) q^{28} +(0.606651 + 0.220803i) q^{29} +(0.0881201 + 0.0508762i) q^{31} +(4.76758 + 1.73526i) q^{32} +(0.814859 + 0.971111i) q^{34} +(-5.50173 + 6.55671i) q^{35} +0.803191i q^{37} +(1.26900 - 1.80814i) q^{38} +(2.50346 + 6.87819i) q^{40} +(0.859419 - 4.87401i) q^{41} +(-0.382351 + 2.16842i) q^{43} +(-7.14472 - 8.51475i) q^{44} +(1.56055 + 0.900985i) q^{46} +(-0.267597 + 0.735218i) q^{47} -2.07952 q^{49} +(-2.50570 + 4.33999i) q^{50} +(-1.06069 - 0.187029i) q^{52} +(-0.994752 - 0.362060i) q^{53} +(4.27244 - 24.2302i) q^{55} +(2.10394 - 3.64413i) q^{56} +(-0.163585 - 0.283338i) q^{58} +(8.73254 - 3.17838i) q^{59} +(2.06000 - 1.72855i) q^{61} +(-0.0176366 - 0.0484563i) q^{62} +(1.23941 + 2.14672i) q^{64} +(-1.19205 - 2.06470i) q^{65} +(-6.42355 - 7.65529i) q^{67} +4.36049i q^{68} +(4.27172 - 0.753220i) q^{70} +(11.9152 - 4.33678i) q^{71} +(-2.46709 - 2.07013i) q^{73} +(0.261641 - 0.311812i) q^{74} +(7.34154 - 1.95862i) q^{76} +(-12.2493 + 7.07214i) q^{77} +(5.23233 + 14.3757i) q^{79} +(-3.33228 + 9.15536i) q^{80} +(-1.92136 + 1.61221i) q^{82} +1.58069i q^{83} +(-7.39396 + 6.20427i) q^{85} +(0.854802 - 0.717264i) q^{86} +12.0959i q^{88} +(1.50252 - 1.26076i) q^{89} +(-0.468762 + 1.28791i) q^{91} +(2.11992 + 5.82444i) q^{92} +(0.343385 - 0.198253i) q^{94} +(13.7670 + 9.66204i) q^{95} +(8.39005 - 9.99888i) q^{97} +(0.807305 + 0.677409i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 9 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} - 12 q^{10} + 9 q^{11} - 6 q^{13} + 9 q^{14} + 9 q^{16} + 27 q^{17} - 15 q^{19} + 18 q^{20} - 6 q^{22} + 9 q^{23} - 3 q^{25} - 72 q^{26} - 36 q^{28} - 18 q^{29}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.388217 − 0.325752i −0.274511 − 0.230342i 0.495130 − 0.868819i \(-0.335120\pi\)
−0.769641 + 0.638477i \(0.779565\pi\)
\(3\) 0 0
\(4\) −0.302699 − 1.71669i −0.151349 − 0.858345i
\(5\) 2.48025 − 2.95585i 1.10920 − 1.32190i 0.167339 − 0.985899i \(-0.446483\pi\)
0.941863 − 0.335996i \(-0.109073\pi\)
\(6\) 0 0
\(7\) −2.21821 −0.838406 −0.419203 − 0.907892i \(-0.637691\pi\)
−0.419203 + 0.907892i \(0.637691\pi\)
\(8\) −0.948484 + 1.64282i −0.335340 + 0.580825i
\(9\) 0 0
\(10\) −1.92575 + 0.339561i −0.608975 + 0.107379i
\(11\) 5.52215 − 3.18821i 1.66499 − 0.961282i 0.694712 − 0.719288i \(-0.255532\pi\)
0.970278 − 0.241994i \(-0.0778015\pi\)
\(12\) 0 0
\(13\) 0.211324 − 0.580608i 0.0586108 − 0.161032i −0.906931 − 0.421278i \(-0.861582\pi\)
0.965542 + 0.260246i \(0.0838038\pi\)
\(14\) 0.861148 + 0.722589i 0.230151 + 0.193120i
\(15\) 0 0
\(16\) −2.37272 + 0.863601i −0.593181 + 0.215900i
\(17\) −2.46346 − 0.434375i −0.597478 − 0.105351i −0.133273 − 0.991079i \(-0.542549\pi\)
−0.464205 + 0.885728i \(0.653660\pi\)
\(18\) 0 0
\(19\) 0.375190 + 4.34272i 0.0860744 + 0.996289i
\(20\) −5.82505 − 3.36309i −1.30252 − 0.752010i
\(21\) 0 0
\(22\) −3.18236 − 0.561135i −0.678481 − 0.119634i
\(23\) −3.50170 + 0.617444i −0.730155 + 0.128746i −0.526353 − 0.850266i \(-0.676441\pi\)
−0.203802 + 0.979012i \(0.565330\pi\)
\(24\) 0 0
\(25\) −1.71715 − 9.73845i −0.343430 − 1.94769i
\(26\) −0.271174 + 0.156562i −0.0531816 + 0.0307044i
\(27\) 0 0
\(28\) 0.671451 + 3.80799i 0.126892 + 0.719642i
\(29\) 0.606651 + 0.220803i 0.112652 + 0.0410021i 0.397731 − 0.917502i \(-0.369798\pi\)
−0.285079 + 0.958504i \(0.592020\pi\)
\(30\) 0 0
\(31\) 0.0881201 + 0.0508762i 0.0158268 + 0.00913763i 0.507893 − 0.861420i \(-0.330425\pi\)
−0.492066 + 0.870558i \(0.663758\pi\)
\(32\) 4.76758 + 1.73526i 0.842798 + 0.306753i
\(33\) 0 0
\(34\) 0.814859 + 0.971111i 0.139747 + 0.166544i
\(35\) −5.50173 + 6.55671i −0.929962 + 1.10829i
\(36\) 0 0
\(37\) 0.803191i 0.132044i 0.997818 + 0.0660219i \(0.0210307\pi\)
−0.997818 + 0.0660219i \(0.978969\pi\)
\(38\) 1.26900 − 1.80814i 0.205858 − 0.293318i
\(39\) 0 0
\(40\) 2.50346 + 6.87819i 0.395831 + 1.08754i
\(41\) 0.859419 − 4.87401i 0.134219 − 0.761192i −0.841182 − 0.540752i \(-0.818140\pi\)
0.975401 − 0.220440i \(-0.0707493\pi\)
\(42\) 0 0
\(43\) −0.382351 + 2.16842i −0.0583079 + 0.330681i −0.999983 − 0.00584302i \(-0.998140\pi\)
0.941675 + 0.336524i \(0.109251\pi\)
\(44\) −7.14472 − 8.51475i −1.07711 − 1.28365i
\(45\) 0 0
\(46\) 1.56055 + 0.900985i 0.230091 + 0.132843i
\(47\) −0.267597 + 0.735218i −0.0390331 + 0.107243i −0.957678 − 0.287842i \(-0.907062\pi\)
0.918645 + 0.395084i \(0.129285\pi\)
\(48\) 0 0
\(49\) −2.07952 −0.297075
\(50\) −2.50570 + 4.33999i −0.354359 + 0.613768i
\(51\) 0 0
\(52\) −1.06069 − 0.187029i −0.147092 − 0.0259362i
\(53\) −0.994752 − 0.362060i −0.136640 − 0.0497328i 0.272795 − 0.962072i \(-0.412052\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(54\) 0 0
\(55\) 4.27244 − 24.2302i 0.576095 − 3.26720i
\(56\) 2.10394 − 3.64413i 0.281151 − 0.486968i
\(57\) 0 0
\(58\) −0.163585 − 0.283338i −0.0214798 − 0.0372040i
\(59\) 8.73254 − 3.17838i 1.13688 − 0.413790i 0.296094 − 0.955159i \(-0.404316\pi\)
0.840785 + 0.541368i \(0.182094\pi\)
\(60\) 0 0
\(61\) 2.06000 − 1.72855i 0.263756 − 0.221318i −0.501313 − 0.865266i \(-0.667149\pi\)
0.765069 + 0.643948i \(0.222705\pi\)
\(62\) −0.0176366 − 0.0484563i −0.00223986 − 0.00615396i
\(63\) 0 0
\(64\) 1.23941 + 2.14672i 0.154926 + 0.268340i
\(65\) −1.19205 − 2.06470i −0.147856 − 0.256094i
\(66\) 0 0
\(67\) −6.42355 − 7.65529i −0.784761 − 0.935242i 0.214377 − 0.976751i \(-0.431228\pi\)
−0.999138 + 0.0415090i \(0.986783\pi\)
\(68\) 4.36049i 0.528787i
\(69\) 0 0
\(70\) 4.27172 − 0.753220i 0.510569 − 0.0900271i
\(71\) 11.9152 − 4.33678i 1.41408 − 0.514681i 0.481752 − 0.876308i \(-0.340001\pi\)
0.932323 + 0.361626i \(0.117778\pi\)
\(72\) 0 0
\(73\) −2.46709 − 2.07013i −0.288751 − 0.242291i 0.486893 − 0.873462i \(-0.338130\pi\)
−0.775644 + 0.631171i \(0.782575\pi\)
\(74\) 0.261641 − 0.311812i 0.0304152 − 0.0362474i
\(75\) 0 0
\(76\) 7.34154 − 1.95862i 0.842132 − 0.224669i
\(77\) −12.2493 + 7.07214i −1.39594 + 0.805945i
\(78\) 0 0
\(79\) 5.23233 + 14.3757i 0.588683 + 1.61739i 0.772915 + 0.634510i \(0.218798\pi\)
−0.184232 + 0.982883i \(0.558980\pi\)
\(80\) −3.33228 + 9.15536i −0.372560 + 1.02360i
\(81\) 0 0
\(82\) −1.92136 + 1.61221i −0.212179 + 0.178039i
\(83\) 1.58069i 0.173504i 0.996230 + 0.0867518i \(0.0276487\pi\)
−0.996230 + 0.0867518i \(0.972351\pi\)
\(84\) 0 0
\(85\) −7.39396 + 6.20427i −0.801987 + 0.672947i
\(86\) 0.854802 − 0.717264i 0.0921757 − 0.0773446i
\(87\) 0 0
\(88\) 12.0959i 1.28942i
\(89\) 1.50252 − 1.26076i 0.159267 − 0.133641i −0.559672 − 0.828714i \(-0.689073\pi\)
0.718938 + 0.695074i \(0.244628\pi\)
\(90\) 0 0
\(91\) −0.468762 + 1.28791i −0.0491396 + 0.135010i
\(92\) 2.11992 + 5.82444i 0.221017 + 0.607240i
\(93\) 0 0
\(94\) 0.343385 − 0.198253i 0.0354174 − 0.0204483i
\(95\) 13.7670 + 9.66204i 1.41246 + 0.991304i
\(96\) 0 0
\(97\) 8.39005 − 9.99888i 0.851881 − 1.01523i −0.147775 − 0.989021i \(-0.547211\pi\)
0.999656 − 0.0262112i \(-0.00834425\pi\)
\(98\) 0.807305 + 0.677409i 0.0815501 + 0.0684287i
\(99\) 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 513.2.cd.a.116.8 108
3.2 odd 2 171.2.bd.a.2.11 yes 108
9.4 even 3 171.2.x.a.59.11 yes 108
9.5 odd 6 513.2.bo.a.287.8 108
19.10 odd 18 513.2.bo.a.143.8 108
57.29 even 18 171.2.x.a.29.11 ✓ 108
171.67 odd 18 171.2.bd.a.86.11 yes 108
171.86 even 18 inner 513.2.cd.a.314.8 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.29.11 ✓ 108 57.29 even 18
171.2.x.a.59.11 yes 108 9.4 even 3
171.2.bd.a.2.11 yes 108 3.2 odd 2
171.2.bd.a.86.11 yes 108 171.67 odd 18
513.2.bo.a.143.8 108 19.10 odd 18
513.2.bo.a.287.8 108 9.5 odd 6
513.2.cd.a.116.8 108 1.1 even 1 trivial
513.2.cd.a.314.8 108 171.86 even 18 inner