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Properties

Label 790.2.k.a.577.34
Level $790$
Weight $2$
Character 790.577
Analytic conductor $6.308$
Analytic rank $0$
Dimension $160$
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(103,790)] 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("790.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.k (of order \(12\), degree \(4\), 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: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 577.34
Character \(\chi\) \(=\) 790.577
Dual form 790.2.k.a.293.34

$q$-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+(0.965926 - 0.258819i) q^{2} +(1.12333 - 0.300996i) q^{3} +(0.866025 - 0.500000i) q^{4} +(2.15477 + 0.597470i) q^{5} +(1.00715 - 0.581480i) q^{6} +(3.93078 + 1.05325i) q^{7} +(0.707107 - 0.707107i) q^{8} +(-1.42680 + 0.823762i) q^{9} +(2.23598 + 0.0194162i) q^{10} +(-2.50846 - 4.34478i) q^{11} +(0.822337 - 0.822337i) q^{12} +(0.744550 + 2.77870i) q^{13} +4.06944 q^{14} +(2.60036 + 0.0225802i) q^{15} +(0.500000 - 0.866025i) q^{16} +(-1.06947 + 1.06947i) q^{17} +(-1.16498 + 1.16498i) q^{18} +(-3.37239 + 1.94705i) q^{19} +(2.16482 - 0.559961i) q^{20} +4.73260 q^{21} +(-3.54750 - 3.54750i) q^{22} +(-2.61814 - 0.701530i) q^{23} +(0.581480 - 1.00715i) q^{24} +(4.28606 + 2.57482i) q^{25} +(1.43836 + 2.49131i) q^{26} +(-3.82183 + 3.82183i) q^{27} +(3.93078 - 1.05325i) q^{28} +(-3.17698 - 5.50269i) q^{29} +(2.51760 - 0.651212i) q^{30} +(-3.78698 - 6.55924i) q^{31} +(0.258819 - 0.965926i) q^{32} +(-4.12560 - 4.12560i) q^{33} +(-0.756230 + 1.30983i) q^{34} +(7.84064 + 4.61803i) q^{35} +(-0.823762 + 1.42680i) q^{36} +(2.81044 + 10.4887i) q^{37} +(-2.75355 + 2.75355i) q^{38} +(1.67275 + 2.89730i) q^{39} +(1.94613 - 1.10118i) q^{40} -4.33089i q^{41} +(4.57134 - 1.22489i) q^{42} +(-10.1604 + 2.72246i) q^{43} +(-4.34478 - 2.50846i) q^{44} +(-3.56659 + 0.922549i) q^{45} -2.71050 q^{46} +(3.14903 + 0.843780i) q^{47} +(0.300996 - 1.12333i) q^{48} +(8.27952 + 4.78018i) q^{49} +(4.80643 + 1.37777i) q^{50} +(-0.879466 + 1.52328i) q^{51} +(2.03415 + 2.03415i) q^{52} +(2.99496 - 11.1773i) q^{53} +(-2.70244 + 4.68077i) q^{54} +(-2.80928 - 10.8607i) q^{55} +(3.52424 - 2.03472i) q^{56} +(-3.20226 + 3.20226i) q^{57} +(-4.49292 - 4.49292i) q^{58} +(1.99500 - 3.45544i) q^{59} +(2.26327 - 1.28062i) q^{60} -3.82207i q^{61} +(-5.35560 - 5.35560i) q^{62} +(-6.47606 + 1.73525i) q^{63} -1.00000i q^{64} +(-0.0558549 + 6.43230i) q^{65} +(-5.05281 - 2.91724i) q^{66} +(2.71606 + 2.71606i) q^{67} +(-0.391454 + 1.46093i) q^{68} -3.15221 q^{69} +(8.76871 + 2.43137i) q^{70} -5.73101i q^{71} +(-0.426411 + 1.59139i) q^{72} +(14.6894 + 3.93601i) q^{73} +(5.42936 + 9.40393i) q^{74} +(5.58968 + 1.60229i) q^{75} +(-1.94705 + 3.37239i) q^{76} +(-5.28407 - 19.7204i) q^{77} +(2.36563 + 2.36563i) q^{78} +(-0.612394 - 8.86707i) q^{79} +(1.59481 - 1.56735i) q^{80} +(-0.671545 + 1.16315i) q^{81} +(-1.12092 - 4.18332i) q^{82} +(13.8016 + 3.69812i) q^{83} +(4.09855 - 2.36630i) q^{84} +(-2.94344 + 1.66549i) q^{85} +(-9.10952 + 5.25938i) q^{86} +(-5.22509 - 5.22509i) q^{87} +(-4.84597 - 1.29848i) q^{88} +8.66863i q^{89} +(-3.20629 + 1.81422i) q^{90} +11.7066i q^{91} +(-2.61814 + 0.701530i) q^{92} +(-6.22834 - 6.22834i) q^{93} +3.26012 q^{94} +(-8.43003 + 2.18054i) q^{95} -1.16296i q^{96} +(-13.5739 - 13.5739i) q^{97} +(9.23460 + 2.47440i) q^{98} +(7.15813 + 4.13275i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q - 12 q^{7} - 8 q^{10} - 16 q^{11} + 80 q^{16} - 32 q^{18} + 48 q^{21} - 8 q^{22} - 12 q^{28} - 24 q^{31} - 36 q^{35} + 72 q^{36} - 36 q^{37} + 8 q^{38} + 20 q^{42} - 48 q^{43} - 4 q^{45} + 16 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.965926 − 0.258819i 0.683013 − 0.183013i
\(3\) 1.12333 − 0.300996i 0.648557 − 0.173780i 0.0804804 − 0.996756i \(-0.474355\pi\)
0.568076 + 0.822976i \(0.307688\pi\)
\(4\) 0.866025 − 0.500000i 0.433013 − 0.250000i
\(5\) 2.15477 + 0.597470i 0.963642 + 0.267197i
\(6\) 1.00715 − 0.581480i 0.411168 − 0.237388i
\(7\) 3.93078 + 1.05325i 1.48570 + 0.398091i 0.908281 − 0.418361i \(-0.137395\pi\)
0.577414 + 0.816451i \(0.304062\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) −1.42680 + 0.823762i −0.475599 + 0.274587i
\(10\) 2.23598 + 0.0194162i 0.707080 + 0.00613993i
\(11\) −2.50846 − 4.34478i −0.756329 − 1.31000i −0.944711 − 0.327905i \(-0.893657\pi\)
0.188381 − 0.982096i \(-0.439676\pi\)
\(12\) 0.822337 − 0.822337i 0.237388 − 0.237388i
\(13\) 0.744550 + 2.77870i 0.206501 + 0.770672i 0.988987 + 0.148004i \(0.0472847\pi\)
−0.782486 + 0.622668i \(0.786049\pi\)
\(14\) 4.06944 1.08760
\(15\) 2.60036 + 0.0225802i 0.671410 + 0.00583019i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −1.06947 + 1.06947i −0.259385 + 0.259385i −0.824804 − 0.565419i \(-0.808715\pi\)
0.565419 + 0.824804i \(0.308715\pi\)
\(18\) −1.16498 + 1.16498i −0.274587 + 0.274587i
\(19\) −3.37239 + 1.94705i −0.773680 + 0.446684i −0.834186 − 0.551484i \(-0.814062\pi\)
0.0605060 + 0.998168i \(0.480729\pi\)
\(20\) 2.16482 − 0.559961i 0.484068 − 0.125211i
\(21\) 4.73260 1.03274
\(22\) −3.54750 − 3.54750i −0.756329 − 0.756329i
\(23\) −2.61814 − 0.701530i −0.545921 − 0.146279i −0.0246906 − 0.999695i \(-0.507860\pi\)
−0.521230 + 0.853416i \(0.674527\pi\)
\(24\) 0.581480 − 1.00715i 0.118694 − 0.205584i
\(25\) 4.28606 + 2.57482i 0.857212 + 0.514964i
\(26\) 1.43836 + 2.49131i 0.282085 + 0.488586i
\(27\) −3.82183 + 3.82183i −0.735512 + 0.735512i
\(28\) 3.93078 − 1.05325i 0.742848 − 0.199045i
\(29\) −3.17698 − 5.50269i −0.589950 − 1.02182i −0.994238 − 0.107192i \(-0.965814\pi\)
0.404289 − 0.914632i \(-0.367519\pi\)
\(30\) 2.51760 − 0.651212i 0.459648 − 0.118894i
\(31\) −3.78698 − 6.55924i −0.680161 − 1.17807i −0.974931 − 0.222506i \(-0.928576\pi\)
0.294770 − 0.955568i \(-0.404757\pi\)
\(32\) 0.258819 − 0.965926i 0.0457532 − 0.170753i
\(33\) −4.12560 − 4.12560i −0.718175 − 0.718175i
\(34\) −0.756230 + 1.30983i −0.129692 + 0.224634i
\(35\) 7.84064 + 4.61803i 1.32531 + 0.780590i
\(36\) −0.823762 + 1.42680i −0.137294 + 0.237800i
\(37\) 2.81044 + 10.4887i 0.462034 + 1.72433i 0.666540 + 0.745469i \(0.267775\pi\)
−0.204506 + 0.978865i \(0.565559\pi\)
\(38\) −2.75355 + 2.75355i −0.446684 + 0.446684i
\(39\) 1.67275 + 2.89730i 0.267855 + 0.463939i
\(40\) 1.94613 − 1.10118i 0.307710 − 0.174111i
\(41\) − 4.33089i − 0.676371i −0.941079 − 0.338185i \(-0.890187\pi\)
0.941079 − 0.338185i \(-0.109813\pi\)
\(42\) 4.57134 − 1.22489i 0.705373 − 0.189004i
\(43\) −10.1604 + 2.72246i −1.54944 + 0.415171i −0.929301 − 0.369322i \(-0.879590\pi\)
−0.620138 + 0.784493i \(0.712923\pi\)
\(44\) −4.34478 − 2.50846i −0.655001 − 0.378165i
\(45\) −3.56659 + 0.922549i −0.531676 + 0.137525i
\(46\) −2.71050 −0.399642
\(47\) 3.14903 + 0.843780i 0.459333 + 0.123078i 0.481063 − 0.876686i \(-0.340251\pi\)
−0.0217299 + 0.999764i \(0.506917\pi\)
\(48\) 0.300996 − 1.12333i 0.0434451 − 0.162139i
\(49\) 8.27952 + 4.78018i 1.18279 + 0.682883i
\(50\) 4.80643 + 1.37777i 0.679732 + 0.194846i
\(51\) −0.879466 + 1.52328i −0.123150 + 0.213302i
\(52\) 2.03415 + 2.03415i 0.282085 + 0.282085i
\(53\) 2.99496 − 11.1773i 0.411390 − 1.53533i −0.380569 − 0.924752i \(-0.624272\pi\)
0.791959 − 0.610574i \(-0.209061\pi\)
\(54\) −2.70244 + 4.68077i −0.367756 + 0.636972i
\(55\) −2.80928 − 10.8607i −0.378803 − 1.46446i
\(56\) 3.52424 − 2.03472i 0.470947 − 0.271901i
\(57\) −3.20226 + 3.20226i −0.424150 + 0.424150i
\(58\) −4.49292 − 4.49292i −0.589950 − 0.589950i
\(59\) 1.99500 − 3.45544i 0.259727 − 0.449860i −0.706442 − 0.707771i \(-0.749701\pi\)
0.966169 + 0.257911i \(0.0830342\pi\)
\(60\) 2.26327 − 1.28062i 0.292187 − 0.165328i
\(61\) − 3.82207i − 0.489365i −0.969603 − 0.244683i \(-0.921316\pi\)
0.969603 − 0.244683i \(-0.0786838\pi\)
\(62\) −5.35560 − 5.35560i −0.680161 − 0.680161i
\(63\) −6.47606 + 1.73525i −0.815906 + 0.218621i
\(64\) − 1.00000i − 0.125000i
\(65\) −0.0558549 + 6.43230i −0.00692795 + 0.797828i
\(66\) −5.05281 − 2.91724i −0.621958 − 0.359087i
\(67\) 2.71606 + 2.71606i 0.331820 + 0.331820i 0.853277 − 0.521457i \(-0.174611\pi\)
−0.521457 + 0.853277i \(0.674611\pi\)
\(68\) −0.391454 + 1.46093i −0.0474707 + 0.177163i
\(69\) −3.15221 −0.379481
\(70\) 8.76871 + 2.43137i 1.04806 + 0.290604i
\(71\) − 5.73101i − 0.680146i −0.940399 − 0.340073i \(-0.889548\pi\)
0.940399 − 0.340073i \(-0.110452\pi\)
\(72\) −0.426411 + 1.59139i −0.0502530 + 0.187547i
\(73\) 14.6894 + 3.93601i 1.71926 + 0.460675i 0.977665 − 0.210169i \(-0.0674015\pi\)
0.741598 + 0.670844i \(0.234068\pi\)
\(74\) 5.42936 + 9.40393i 0.631150 + 1.09318i
\(75\) 5.58968 + 1.60229i 0.645441 + 0.185017i
\(76\) −1.94705 + 3.37239i −0.223342 + 0.386840i
\(77\) −5.28407 − 19.7204i −0.602176 − 2.24735i
\(78\) 2.36563 + 2.36563i 0.267855 + 0.267855i
\(79\) −0.612394 − 8.86707i −0.0688997 − 0.997624i
\(80\) 1.59481 − 1.56735i 0.178305 − 0.175235i
\(81\) −0.671545 + 1.16315i −0.0746161 + 0.129239i
\(82\) −1.12092 − 4.18332i −0.123784 − 0.461970i
\(83\) 13.8016 + 3.69812i 1.51492 + 0.405921i 0.918065 − 0.396430i \(-0.129751\pi\)
0.596853 + 0.802351i \(0.296418\pi\)
\(84\) 4.09855 − 2.36630i 0.447189 − 0.258184i
\(85\) −2.94344 + 1.66549i −0.319261 + 0.180647i
\(86\) −9.10952 + 5.25938i −0.982305 + 0.567134i
\(87\) −5.22509 − 5.22509i −0.560189 − 0.560189i
\(88\) −4.84597 − 1.29848i −0.516583 − 0.138418i
\(89\) 8.66863i 0.918873i 0.888211 + 0.459436i \(0.151949\pi\)
−0.888211 + 0.459436i \(0.848051\pi\)
\(90\) −3.20629 + 1.81422i −0.337973 + 0.191235i
\(91\) 11.7066i 1.22719i
\(92\) −2.61814 + 0.701530i −0.272960 + 0.0731395i
\(93\) −6.22834 − 6.22834i −0.645849 − 0.645849i
\(94\) 3.26012 0.336255
\(95\) −8.43003 + 2.18054i −0.864903 + 0.223719i
\(96\) − 1.16296i − 0.118694i
\(97\) −13.5739 − 13.5739i −1.37822 − 1.37822i −0.847631 − 0.530586i \(-0.821972\pi\)
−0.530586 − 0.847631i \(-0.678028\pi\)
\(98\) 9.23460 + 2.47440i 0.932836 + 0.249953i
\(99\) 7.15813 + 4.13275i 0.719420 + 0.415357i
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.k.a.577.34 yes 160
5.3 odd 4 inner 790.2.k.a.103.14 ✓ 160
79.56 odd 6 inner 790.2.k.a.767.14 yes 160
395.293 even 12 inner 790.2.k.a.293.34 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.14 ✓ 160 5.3 odd 4 inner
790.2.k.a.293.34 yes 160 395.293 even 12 inner
790.2.k.a.577.34 yes 160 1.1 even 1 trivial
790.2.k.a.767.14 yes 160 79.56 odd 6 inner