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Properties

Label 790.2.m.c.141.3
Level $790$
Weight $2$
Character 790.141
Analytic conductor $6.308$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [790,2,Mod(21,790)] 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("790.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([0, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,4,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label 141.3
Character \(\chi\) \(=\) 790.141
Dual form 790.2.m.c.381.3

$q$-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+(0.970942 + 0.239316i) q^{2} +(0.589522 - 0.522271i) q^{3} +(0.885456 + 0.464723i) q^{4} +(0.568065 - 0.822984i) q^{5} +(0.697380 - 0.366013i) q^{6} +(-3.19730 + 2.83256i) q^{7} +(0.748511 + 0.663123i) q^{8} +(-0.286841 + 2.36235i) q^{9} +(0.748511 - 0.663123i) q^{10} +(2.15469 + 3.12161i) q^{11} +(0.764708 - 0.188484i) q^{12} +(1.92120 + 1.00832i) q^{13} +(-3.78226 + 1.98509i) q^{14} +(-0.0949339 - 0.781851i) q^{15} +(0.568065 + 0.822984i) q^{16} +(1.81937 + 0.954881i) q^{17} +(-0.843852 + 2.22505i) q^{18} +(0.390392 - 1.02938i) q^{19} +(0.885456 - 0.464723i) q^{20} +(-0.405514 + 3.33971i) q^{21} +(1.34503 + 3.54655i) q^{22} -2.67399 q^{23} +0.787594 q^{24} +(-0.354605 - 0.935016i) q^{25} +(1.62406 + 1.43879i) q^{26} +(2.40690 + 3.48699i) q^{27} +(-4.14742 + 1.02225i) q^{28} +(-0.852628 - 7.02202i) q^{29} +(0.0949339 - 0.781851i) q^{30} +(8.38854 + 2.06759i) q^{31} +(0.354605 + 0.935016i) q^{32} +(2.90057 + 0.714925i) q^{33} +(1.53799 + 1.36254i) q^{34} +(0.514878 + 4.24040i) q^{35} +(-1.35182 + 1.95845i) q^{36} +(1.36671 - 3.60372i) q^{37} +(0.625395 - 0.906041i) q^{38} +(1.65921 - 0.408958i) q^{39} +(0.970942 - 0.239316i) q^{40} +(0.129883 - 0.188168i) q^{41} +(-1.19298 + 3.14562i) q^{42} +(-1.65605 + 2.39920i) q^{43} +(0.457200 + 3.76538i) q^{44} +(1.78123 + 1.57803i) q^{45} +(-2.59629 - 0.639928i) q^{46} +(-3.78325 - 9.97561i) q^{47} +(0.764708 + 0.188484i) q^{48} +(1.35557 - 11.1641i) q^{49} +(-0.120537 - 0.992709i) q^{50} +(1.57127 - 0.387283i) q^{51} +(1.23255 + 1.78565i) q^{52} +(-0.467698 - 0.414344i) q^{53} +(1.50247 + 3.96168i) q^{54} +3.79304 q^{55} -4.27154 q^{56} +(-0.307471 - 0.810733i) q^{57} +(0.852628 - 7.02202i) q^{58} +(-3.84479 + 2.01790i) q^{59} +(0.279285 - 0.736413i) q^{60} +(2.42985 - 6.40700i) q^{61} +(7.64997 + 4.01502i) q^{62} +(-5.77437 - 8.36561i) q^{63} +(0.120537 + 0.992709i) q^{64} +(1.92120 - 1.00832i) q^{65} +(2.64519 + 1.38830i) q^{66} +(-12.2085 + 3.00912i) q^{67} +(1.16722 + 1.69101i) q^{68} +(-1.57638 + 1.39655i) q^{69} +(-0.514878 + 4.24040i) q^{70} +(2.32743 + 2.06193i) q^{71} +(-1.78123 + 1.57803i) q^{72} +(0.920622 - 0.483180i) q^{73} +(2.18942 - 3.17192i) q^{74} +(-0.697380 - 0.366013i) q^{75} +(0.824052 - 0.730047i) q^{76} +(-15.7313 - 3.87742i) q^{77} +1.70886 q^{78} +(4.89888 + 7.41626i) q^{79} +1.00000 q^{80} +(-3.69156 - 0.909888i) q^{81} +(0.171141 - 0.151617i) q^{82} +(-1.65714 - 0.869736i) q^{83} +(-1.91111 + 2.76872i) q^{84} +(1.81937 - 0.954881i) q^{85} +(-2.18209 + 1.93317i) q^{86} +(-4.17004 - 3.69434i) q^{87} +(-0.457200 + 3.76538i) q^{88} +(0.668674 - 0.592393i) q^{89} +(1.35182 + 1.95845i) q^{90} +(-8.99877 + 2.21800i) q^{91} +(-2.36770 - 1.24266i) q^{92} +(6.02507 - 3.16220i) q^{93} +(-1.28600 - 10.5911i) q^{94} +(-0.625395 - 0.906041i) q^{95} +(0.697380 + 0.366013i) q^{96} +(-4.17471 + 11.0078i) q^{97} +(3.98792 - 10.5153i) q^{98} +(-7.99237 + 4.19472i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{2} - q^{3} - 4 q^{4} - 4 q^{5} + q^{6} - 9 q^{7} + 4 q^{8} - 21 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 17 q^{13} - 4 q^{14} - 14 q^{15} - 4 q^{16} - 6 q^{17} + 21 q^{18} - 34 q^{19} - 4 q^{20}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{10}{13}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.970942 + 0.239316i 0.686560 + 0.169222i
\(3\) 0.589522 − 0.522271i 0.340361 − 0.301533i −0.475576 − 0.879675i \(-0.657760\pi\)
0.815937 + 0.578141i \(0.196222\pi\)
\(4\) 0.885456 + 0.464723i 0.442728 + 0.232362i
\(5\) 0.568065 − 0.822984i 0.254046 − 0.368050i
\(6\) 0.697380 − 0.366013i 0.284704 − 0.149424i
\(7\) −3.19730 + 2.83256i −1.20846 + 1.07061i −0.212847 + 0.977085i \(0.568274\pi\)
−0.995617 + 0.0935208i \(0.970188\pi\)
\(8\) 0.748511 + 0.663123i 0.264639 + 0.234449i
\(9\) −0.286841 + 2.36235i −0.0956136 + 0.787449i
\(10\) 0.748511 − 0.663123i 0.236700 − 0.209698i
\(11\) 2.15469 + 3.12161i 0.649664 + 0.941201i 0.999990 + 0.00457281i \(0.00145558\pi\)
−0.350326 + 0.936628i \(0.613929\pi\)
\(12\) 0.764708 − 0.188484i 0.220752 − 0.0544105i
\(13\) 1.92120 + 1.00832i 0.532844 + 0.279658i 0.709610 − 0.704595i \(-0.248871\pi\)
−0.176766 + 0.984253i \(0.556563\pi\)
\(14\) −3.78226 + 1.98509i −1.01085 + 0.530536i
\(15\) −0.0949339 − 0.781851i −0.0245118 − 0.201873i
\(16\) 0.568065 + 0.822984i 0.142016 + 0.205746i
\(17\) 1.81937 + 0.954881i 0.441263 + 0.231593i 0.670697 − 0.741732i \(-0.265995\pi\)
−0.229434 + 0.973324i \(0.573687\pi\)
\(18\) −0.843852 + 2.22505i −0.198898 + 0.524450i
\(19\) 0.390392 − 1.02938i 0.0895622 − 0.236156i −0.882749 − 0.469844i \(-0.844310\pi\)
0.972312 + 0.233688i \(0.0750794\pi\)
\(20\) 0.885456 − 0.464723i 0.197994 − 0.103915i
\(21\) −0.405514 + 3.33971i −0.0884905 + 0.728785i
\(22\) 1.34503 + 3.54655i 0.286761 + 0.756128i
\(23\) −2.67399 −0.557565 −0.278783 − 0.960354i \(-0.589931\pi\)
−0.278783 + 0.960354i \(0.589931\pi\)
\(24\) 0.787594 0.160767
\(25\) −0.354605 − 0.935016i −0.0709210 − 0.187003i
\(26\) 1.62406 + 1.43879i 0.318505 + 0.282171i
\(27\) 2.40690 + 3.48699i 0.463208 + 0.671072i
\(28\) −4.14742 + 1.02225i −0.783789 + 0.193187i
\(29\) −0.852628 − 7.02202i −0.158329 − 1.30396i −0.827760 − 0.561082i \(-0.810385\pi\)
0.669431 − 0.742874i \(-0.266538\pi\)
\(30\) 0.0949339 − 0.781851i 0.0173325 − 0.142746i
\(31\) 8.38854 + 2.06759i 1.50663 + 0.371350i 0.904299 − 0.426900i \(-0.140394\pi\)
0.602327 + 0.798250i \(0.294240\pi\)
\(32\) 0.354605 + 0.935016i 0.0626859 + 0.165289i
\(33\) 2.90057 + 0.714925i 0.504924 + 0.124453i
\(34\) 1.53799 + 1.36254i 0.263763 + 0.233673i
\(35\) 0.514878 + 4.24040i 0.0870302 + 0.716758i
\(36\) −1.35182 + 1.95845i −0.225304 + 0.326409i
\(37\) 1.36671 − 3.60372i 0.224686 − 0.592447i −0.774509 − 0.632562i \(-0.782003\pi\)
0.999195 + 0.0401151i \(0.0127725\pi\)
\(38\) 0.625395 − 0.906041i 0.101452 − 0.146979i
\(39\) 1.65921 − 0.408958i 0.265686 − 0.0654856i
\(40\) 0.970942 − 0.239316i 0.153519 − 0.0378391i
\(41\) 0.129883 − 0.188168i 0.0202843 − 0.0293869i −0.812711 − 0.582668i \(-0.802009\pi\)
0.832995 + 0.553281i \(0.186624\pi\)
\(42\) −1.19298 + 3.14562i −0.184080 + 0.485380i
\(43\) −1.65605 + 2.39920i −0.252545 + 0.365875i −0.928667 − 0.370915i \(-0.879044\pi\)
0.676122 + 0.736790i \(0.263659\pi\)
\(44\) 0.457200 + 3.76538i 0.0689255 + 0.567653i
\(45\) 1.78123 + 1.57803i 0.265530 + 0.235239i
\(46\) −2.59629 − 0.639928i −0.382802 − 0.0943522i
\(47\) −3.78325 − 9.97561i −0.551844 − 1.45509i −0.863231 − 0.504809i \(-0.831563\pi\)
0.311387 − 0.950283i \(-0.399206\pi\)
\(48\) 0.764708 + 0.188484i 0.110376 + 0.0272053i
\(49\) 1.35557 − 11.1641i 0.193652 − 1.59487i
\(50\) −0.120537 − 0.992709i −0.0170465 − 0.140390i
\(51\) 1.57127 − 0.387283i 0.220022 − 0.0542304i
\(52\) 1.23255 + 1.78565i 0.170923 + 0.247625i
\(53\) −0.467698 − 0.414344i −0.0642433 − 0.0569146i 0.630390 − 0.776278i \(-0.282895\pi\)
−0.694633 + 0.719364i \(0.744433\pi\)
\(54\) 1.50247 + 3.96168i 0.204460 + 0.539116i
\(55\) 3.79304 0.511453
\(56\) −4.27154 −0.570809
\(57\) −0.307471 − 0.810733i −0.0407255 − 0.107384i
\(58\) 0.852628 − 7.02202i 0.111955 − 0.922036i
\(59\) −3.84479 + 2.01790i −0.500549 + 0.262708i −0.696048 − 0.717996i \(-0.745060\pi\)
0.195499 + 0.980704i \(0.437367\pi\)
\(60\) 0.279285 − 0.736413i 0.0360555 − 0.0950705i
\(61\) 2.42985 − 6.40700i 0.311111 − 0.820332i −0.684694 − 0.728831i \(-0.740064\pi\)
0.995805 − 0.0915016i \(-0.0291667\pi\)
\(62\) 7.64997 + 4.01502i 0.971548 + 0.509908i
\(63\) −5.77437 − 8.36561i −0.727502 − 1.05397i
\(64\) 0.120537 + 0.992709i 0.0150671 + 0.124089i
\(65\) 1.92120 − 1.00832i 0.238295 − 0.125067i
\(66\) 2.64519 + 1.38830i 0.325600 + 0.170888i
\(67\) −12.2085 + 3.00912i −1.49150 + 0.367623i −0.899001 − 0.437946i \(-0.855706\pi\)
−0.592502 + 0.805569i \(0.701860\pi\)
\(68\) 1.16722 + 1.69101i 0.141546 + 0.205065i
\(69\) −1.57638 + 1.39655i −0.189773 + 0.168125i
\(70\) −0.514878 + 4.24040i −0.0615397 + 0.506825i
\(71\) 2.32743 + 2.06193i 0.276215 + 0.244706i 0.789845 − 0.613307i \(-0.210161\pi\)
−0.513630 + 0.858012i \(0.671699\pi\)
\(72\) −1.78123 + 1.57803i −0.209920 + 0.185973i
\(73\) 0.920622 − 0.483180i 0.107751 − 0.0565519i −0.409987 − 0.912091i \(-0.634467\pi\)
0.517737 + 0.855540i \(0.326774\pi\)
\(74\) 2.18942 − 3.17192i 0.254515 − 0.368729i
\(75\) −0.697380 − 0.366013i −0.0805265 − 0.0422635i
\(76\) 0.824052 − 0.730047i 0.0945253 − 0.0837421i
\(77\) −15.7313 − 3.87742i −1.79275 − 0.441874i
\(78\) 1.70886 0.193491
\(79\) 4.89888 + 7.41626i 0.551167 + 0.834395i
\(80\) 1.00000 0.111803
\(81\) −3.69156 − 0.909888i −0.410174 − 0.101099i
\(82\) 0.171141 − 0.151617i 0.0188993 − 0.0167433i
\(83\) −1.65714 − 0.869736i −0.181895 − 0.0954660i 0.371318 − 0.928506i \(-0.378906\pi\)
−0.553213 + 0.833040i \(0.686598\pi\)
\(84\) −1.91111 + 2.76872i −0.208519 + 0.302092i
\(85\) 1.81937 − 0.954881i 0.197339 − 0.103571i
\(86\) −2.18209 + 1.93317i −0.235301 + 0.208459i
\(87\) −4.17004 − 3.69434i −0.447075 − 0.396074i
\(88\) −0.457200 + 3.76538i −0.0487377 + 0.401391i
\(89\) 0.668674 − 0.592393i 0.0708793 − 0.0627936i −0.626939 − 0.779068i \(-0.715693\pi\)
0.697819 + 0.716275i \(0.254154\pi\)
\(90\) 1.35182 + 1.95845i 0.142495 + 0.206439i
\(91\) −8.99877 + 2.21800i −0.943327 + 0.232509i
\(92\) −2.36770 − 1.24266i −0.246850 − 0.129557i
\(93\) 6.02507 − 3.16220i 0.624771 − 0.327905i
\(94\) −1.28600 − 10.5911i −0.132640 − 1.09239i
\(95\) −0.625395 − 0.906041i −0.0641642 − 0.0929579i
\(96\) 0.697380 + 0.366013i 0.0711760 + 0.0373560i
\(97\) −4.17471 + 11.0078i −0.423878 + 1.11767i 0.537699 + 0.843137i \(0.319294\pi\)
−0.961577 + 0.274536i \(0.911476\pi\)
\(98\) 3.98792 − 10.5153i 0.402841 − 1.06220i
\(99\) −7.99237 + 4.19472i −0.803264 + 0.421585i
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 790.2.m.c.141.3 ✓ 48
79.65 even 13 inner 790.2.m.c.381.3 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.m.c.141.3 ✓ 48 1.1 even 1 trivial
790.2.m.c.381.3 yes 48 79.65 even 13 inner