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Properties

Label 790.2.j.a.339.16
Level $790$
Weight $2$
Character 790.339
Analytic conductor $6.308$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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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(339,790)] 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("790.339"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.j (of order \(6\), degree \(2\), 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: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 339.16
Character \(\chi\) \(=\) 790.339
Dual form 790.2.j.a.529.16

$q$-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+(-0.866025 + 0.500000i) q^{2} +(1.70082 - 0.981971i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.925617 + 2.03549i) q^{5} +(-0.981971 + 1.70082i) q^{6} +(1.62485 + 0.938105i) q^{7} +1.00000i q^{8} +(0.428534 - 0.742242i) q^{9} +(-0.216139 - 2.22560i) q^{10} +(-0.255745 + 0.442964i) q^{11} -1.96394i q^{12} +(-1.65407 + 0.954978i) q^{13} -1.87621 q^{14} +(0.424485 + 4.37094i) q^{15} +(-0.500000 - 0.866025i) q^{16} +6.75763i q^{17} +0.857067i q^{18} +(0.277829 - 0.481214i) q^{19} +(1.29998 + 1.81935i) q^{20} +3.68477 q^{21} -0.511491i q^{22} +(2.27997 + 1.31634i) q^{23} +(0.981971 + 1.70082i) q^{24} +(-3.28647 - 3.76817i) q^{25} +(0.954978 - 1.65407i) q^{26} +4.20860i q^{27} +(1.62485 - 0.938105i) q^{28} +(0.928175 - 1.60765i) q^{29} +(-2.55309 - 3.57311i) q^{30} +(-0.215104 + 0.372572i) q^{31} +(0.866025 + 0.500000i) q^{32} +1.00454i q^{33} +(-3.37882 - 5.85228i) q^{34} +(-3.41349 + 2.43904i) q^{35} +(-0.428534 - 0.742242i) q^{36} +(-5.01675 + 2.89642i) q^{37} +0.555658i q^{38} +(-1.87552 + 3.24850i) q^{39} +(-2.03549 - 0.925617i) q^{40} -2.01483 q^{41} +(-3.19110 + 1.84238i) q^{42} +(2.68810 - 1.55198i) q^{43} +(0.255745 + 0.442964i) q^{44} +(1.11417 + 1.55931i) q^{45} -2.63268 q^{46} +(9.03852 + 5.21839i) q^{47} +(-1.70082 - 0.981971i) q^{48} +(-1.73992 - 3.01362i) q^{49} +(4.73025 + 1.62010i) q^{50} +(6.63580 + 11.4935i) q^{51} +1.90996i q^{52} +(1.92765 + 1.11293i) q^{53} +(-2.10430 - 3.64475i) q^{54} +(-0.664928 - 0.930583i) q^{55} +(-0.938105 + 1.62485i) q^{56} -1.09128i q^{57} +1.85635i q^{58} +(-2.43829 - 4.22324i) q^{59} +(3.99759 + 1.81786i) q^{60} +9.99581 q^{61} -0.430209i q^{62} +(1.39260 - 0.804019i) q^{63} -1.00000 q^{64} +(-0.412816 - 4.25079i) q^{65} +(-0.502269 - 0.869956i) q^{66} -6.76760i q^{67} +(5.85228 + 3.37882i) q^{68} +5.17043 q^{69} +(1.73665 - 3.81901i) q^{70} +11.7190 q^{71} +(0.742242 + 0.428534i) q^{72} +(3.10875 + 1.79484i) q^{73} +(2.89642 - 5.01675i) q^{74} +(-9.28994 - 3.18178i) q^{75} +(-0.277829 - 0.481214i) q^{76} +(-0.831094 + 0.479832i) q^{77} -3.75104i q^{78} +(-6.78517 + 5.74120i) q^{79} +(2.22560 - 0.216139i) q^{80} +(5.41832 + 9.38480i) q^{81} +(1.74490 - 1.00742i) q^{82} +(10.7557 + 6.20980i) q^{83} +(1.84238 - 3.19110i) q^{84} +(-13.7551 - 6.25498i) q^{85} +(-1.55198 + 2.68810i) q^{86} -3.64576i q^{87} +(-0.442964 - 0.255745i) q^{88} -5.43546 q^{89} +(-1.74455 - 0.793315i) q^{90} -3.58348 q^{91} +(2.27997 - 1.31634i) q^{92} +0.844905i q^{93} -10.4368 q^{94} +(0.722345 + 1.01094i) q^{95} +1.96394 q^{96} -1.28632i q^{97} +(3.01362 + 1.73992i) q^{98} +(0.219191 + 0.379650i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 40 q^{4} + 4 q^{6} + 36 q^{9} - 4 q^{10} - 8 q^{11} + 16 q^{14} - 8 q^{15} - 40 q^{16} - 8 q^{19} - 72 q^{21} - 4 q^{24} + 6 q^{25} - 8 q^{29} - 10 q^{30} - 16 q^{31} - 2 q^{35} - 36 q^{36} + 24 q^{39}+ \cdots + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 1.70082 − 0.981971i 0.981971 − 0.566941i 0.0791063 − 0.996866i \(-0.474793\pi\)
0.902865 + 0.429925i \(0.141460\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −0.925617 + 2.03549i −0.413948 + 0.910300i
\(6\) −0.981971 + 1.70082i −0.400888 + 0.694358i
\(7\) 1.62485 + 0.938105i 0.614134 + 0.354570i 0.774582 − 0.632474i \(-0.217960\pi\)
−0.160448 + 0.987044i \(0.551294\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.428534 − 0.742242i 0.142845 − 0.247414i
\(10\) −0.216139 − 2.22560i −0.0683493 − 0.703796i
\(11\) −0.255745 + 0.442964i −0.0771101 + 0.133559i −0.902002 − 0.431732i \(-0.857903\pi\)
0.824892 + 0.565291i \(0.191236\pi\)
\(12\) − 1.96394i − 0.566941i
\(13\) −1.65407 + 0.954978i −0.458756 + 0.264863i −0.711521 − 0.702665i \(-0.751993\pi\)
0.252765 + 0.967528i \(0.418660\pi\)
\(14\) −1.87621 −0.501438
\(15\) 0.424485 + 4.37094i 0.109602 + 1.12857i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 6.75763i 1.63897i 0.573103 + 0.819483i \(0.305739\pi\)
−0.573103 + 0.819483i \(0.694261\pi\)
\(18\) 0.857067i 0.202013i
\(19\) 0.277829 − 0.481214i 0.0637384 − 0.110398i −0.832395 − 0.554182i \(-0.813031\pi\)
0.896134 + 0.443784i \(0.146364\pi\)
\(20\) 1.29998 + 1.81935i 0.290685 + 0.406820i
\(21\) 3.68477 0.804082
\(22\) − 0.511491i − 0.109050i
\(23\) 2.27997 + 1.31634i 0.475407 + 0.274476i 0.718500 − 0.695527i \(-0.244829\pi\)
−0.243094 + 0.970003i \(0.578162\pi\)
\(24\) 0.981971 + 1.70082i 0.200444 + 0.347179i
\(25\) −3.28647 − 3.76817i −0.657294 − 0.753635i
\(26\) 0.954978 − 1.65407i 0.187287 − 0.324390i
\(27\) 4.20860i 0.809945i
\(28\) 1.62485 − 0.938105i 0.307067 − 0.177285i
\(29\) 0.928175 − 1.60765i 0.172358 − 0.298532i −0.766886 − 0.641783i \(-0.778195\pi\)
0.939244 + 0.343251i \(0.111528\pi\)
\(30\) −2.55309 − 3.57311i −0.466128 − 0.652357i
\(31\) −0.215104 + 0.372572i −0.0386339 + 0.0669158i −0.884696 − 0.466169i \(-0.845634\pi\)
0.846062 + 0.533085i \(0.178967\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 1.00454i 0.174868i
\(34\) −3.37882 − 5.85228i −0.579462 − 1.00366i
\(35\) −3.41349 + 2.43904i −0.576985 + 0.412273i
\(36\) −0.428534 − 0.742242i −0.0714223 − 0.123707i
\(37\) −5.01675 + 2.89642i −0.824749 + 0.476169i −0.852051 − 0.523458i \(-0.824642\pi\)
0.0273024 + 0.999627i \(0.491308\pi\)
\(38\) 0.555658i 0.0901397i
\(39\) −1.87552 + 3.24850i −0.300324 + 0.520176i
\(40\) −2.03549 − 0.925617i −0.321840 − 0.146353i
\(41\) −2.01483 −0.314664 −0.157332 − 0.987546i \(-0.550289\pi\)
−0.157332 + 0.987546i \(0.550289\pi\)
\(42\) −3.19110 + 1.84238i −0.492398 + 0.284286i
\(43\) 2.68810 − 1.55198i 0.409932 − 0.236674i −0.280829 − 0.959758i \(-0.590609\pi\)
0.690760 + 0.723084i \(0.257276\pi\)
\(44\) 0.255745 + 0.442964i 0.0385551 + 0.0667793i
\(45\) 1.11417 + 1.55931i 0.166091 + 0.232448i
\(46\) −2.63268 −0.388168
\(47\) 9.03852 + 5.21839i 1.31840 + 0.761181i 0.983472 − 0.181062i \(-0.0579533\pi\)
0.334932 + 0.942242i \(0.391287\pi\)
\(48\) −1.70082 − 0.981971i −0.245493 − 0.141735i
\(49\) −1.73992 − 3.01362i −0.248560 − 0.430518i
\(50\) 4.73025 + 1.62010i 0.668959 + 0.229117i
\(51\) 6.63580 + 11.4935i 0.929197 + 1.60942i
\(52\) 1.90996i 0.264863i
\(53\) 1.92765 + 1.11293i 0.264783 + 0.152872i 0.626514 − 0.779410i \(-0.284481\pi\)
−0.361732 + 0.932282i \(0.617814\pi\)
\(54\) −2.10430 − 3.64475i −0.286359 − 0.495988i
\(55\) −0.664928 − 0.930583i −0.0896589 − 0.125480i
\(56\) −0.938105 + 1.62485i −0.125360 + 0.217129i
\(57\) − 1.09128i − 0.144544i
\(58\) 1.85635i 0.243751i
\(59\) −2.43829 − 4.22324i −0.317438 − 0.549819i 0.662514 − 0.749049i \(-0.269489\pi\)
−0.979953 + 0.199230i \(0.936156\pi\)
\(60\) 3.99759 + 1.81786i 0.516087 + 0.234684i
\(61\) 9.99581 1.27983 0.639916 − 0.768445i \(-0.278969\pi\)
0.639916 + 0.768445i \(0.278969\pi\)
\(62\) − 0.430209i − 0.0546366i
\(63\) 1.39260 − 0.804019i 0.175451 − 0.101297i
\(64\) −1.00000 −0.125000
\(65\) −0.412816 − 4.25079i −0.0512036 − 0.527246i
\(66\) −0.502269 − 0.869956i −0.0618251 − 0.107084i
\(67\) − 6.76760i − 0.826794i −0.910551 − 0.413397i \(-0.864342\pi\)
0.910551 − 0.413397i \(-0.135658\pi\)
\(68\) 5.85228 + 3.37882i 0.709693 + 0.409742i
\(69\) 5.17043 0.622447
\(70\) 1.73665 − 3.81901i 0.207570 − 0.456460i
\(71\) 11.7190 1.39079 0.695395 − 0.718628i \(-0.255229\pi\)
0.695395 + 0.718628i \(0.255229\pi\)
\(72\) 0.742242 + 0.428534i 0.0874740 + 0.0505032i
\(73\) 3.10875 + 1.79484i 0.363852 + 0.210070i 0.670769 − 0.741666i \(-0.265964\pi\)
−0.306917 + 0.951736i \(0.599297\pi\)
\(74\) 2.89642 − 5.01675i 0.336702 − 0.583186i
\(75\) −9.28994 − 3.18178i −1.07271 − 0.367400i
\(76\) −0.277829 − 0.481214i −0.0318692 − 0.0551991i
\(77\) −0.831094 + 0.479832i −0.0947119 + 0.0546820i
\(78\) − 3.75104i − 0.424722i
\(79\) −6.78517 + 5.74120i −0.763392 + 0.645936i
\(80\) 2.22560 − 0.216139i 0.248829 − 0.0241651i
\(81\) 5.41832 + 9.38480i 0.602035 + 1.04276i
\(82\) 1.74490 − 1.00742i 0.192692 − 0.111251i
\(83\) 10.7557 + 6.20980i 1.18059 + 0.681615i 0.956151 − 0.292873i \(-0.0946114\pi\)
0.224440 + 0.974488i \(0.427945\pi\)
\(84\) 1.84238 − 3.19110i 0.201021 − 0.348178i
\(85\) −13.7551 − 6.25498i −1.49195 − 0.678447i
\(86\) −1.55198 + 2.68810i −0.167354 + 0.289866i
\(87\) − 3.64576i − 0.390867i
\(88\) −0.442964 − 0.255745i −0.0472201 − 0.0272626i
\(89\) −5.43546 −0.576158 −0.288079 − 0.957607i \(-0.593017\pi\)
−0.288079 + 0.957607i \(0.593017\pi\)
\(90\) −1.74455 − 0.793315i −0.183892 − 0.0836228i
\(91\) −3.58348 −0.375651
\(92\) 2.27997 − 1.31634i 0.237703 − 0.137238i
\(93\) 0.844905i 0.0876125i
\(94\) −10.4368 −1.07647
\(95\) 0.722345 + 1.01094i 0.0741111 + 0.103720i
\(96\) 1.96394 0.200444
\(97\) − 1.28632i − 0.130606i −0.997865 − 0.0653032i \(-0.979199\pi\)
0.997865 − 0.0653032i \(-0.0208014\pi\)
\(98\) 3.01362 + 1.73992i 0.304422 + 0.175758i
\(99\) 0.219191 + 0.379650i 0.0220295 + 0.0381562i
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.j.a.339.16 ✓ 80
5.4 even 2 inner 790.2.j.a.339.25 yes 80
79.55 even 3 inner 790.2.j.a.529.25 yes 80
395.134 even 6 inner 790.2.j.a.529.16 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.j.a.339.16 ✓ 80 1.1 even 1 trivial
790.2.j.a.339.25 yes 80 5.4 even 2 inner
790.2.j.a.529.16 yes 80 395.134 even 6 inner
790.2.j.a.529.25 yes 80 79.55 even 3 inner