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Properties

Label 790.2.j.a.339.14
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 339.14
Character \(\chi\) \(=\) 790.339
Dual form 790.2.j.a.529.14

$q$-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} +(0.795047 - 0.459021i) q^{3} +(0.500000 - 0.866025i) q^{4} +(2.06922 + 0.847547i) q^{5} +(-0.459021 + 0.795047i) q^{6} +(-0.0946389 - 0.0546398i) q^{7} +1.00000i q^{8} +(-1.07860 + 1.86819i) q^{9} +(-2.21577 + 0.300611i) q^{10} +(-2.57327 + 4.45703i) q^{11} -0.918041i q^{12} +(-0.970618 + 0.560387i) q^{13} +0.109280 q^{14} +(2.03417 - 0.275974i) q^{15} +(-0.500000 - 0.866025i) q^{16} +1.53320i q^{17} -2.15720i q^{18} +(-3.73350 + 6.46661i) q^{19} +(1.76861 - 1.36822i) q^{20} -0.100323 q^{21} -5.14653i q^{22} +(-7.10813 - 4.10388i) q^{23} +(0.459021 + 0.795047i) q^{24} +(3.56333 + 3.50752i) q^{25} +(0.560387 - 0.970618i) q^{26} +4.73452i q^{27} +(-0.0946389 + 0.0546398i) q^{28} +(-1.54066 + 2.66850i) q^{29} +(-1.62365 + 1.25608i) q^{30} +(2.04667 - 3.54493i) q^{31} +(0.866025 + 0.500000i) q^{32} +4.72473i q^{33} +(-0.766599 - 1.32779i) q^{34} +(-0.149519 - 0.193273i) q^{35} +(1.07860 + 1.86819i) q^{36} +(6.99989 - 4.04139i) q^{37} -7.46700i q^{38} +(-0.514458 + 0.891068i) q^{39} +(-0.847547 + 2.06922i) q^{40} +4.01335 q^{41} +(0.0868824 - 0.0501616i) q^{42} +(10.2060 - 5.89241i) q^{43} +(2.57327 + 4.45703i) q^{44} +(-3.81524 + 2.95153i) q^{45} +8.20776 q^{46} +(-0.354071 - 0.204423i) q^{47} +(-0.795047 - 0.459021i) q^{48} +(-3.49403 - 6.05184i) q^{49} +(-4.83969 - 1.25594i) q^{50} +(0.703770 + 1.21896i) q^{51} +1.12077i q^{52} +(-1.89274 - 1.09277i) q^{53} +(-2.36726 - 4.10022i) q^{54} +(-9.10220 + 7.04160i) q^{55} +(0.0546398 - 0.0946389i) q^{56} +6.85501i q^{57} -3.08132i q^{58} +(6.71377 + 11.6286i) q^{59} +(0.778083 - 1.89963i) q^{60} -0.117047 q^{61} +4.09333i q^{62} +(0.204155 - 0.117869i) q^{63} -1.00000 q^{64} +(-2.48338 + 0.336917i) q^{65} +(-2.36237 - 4.09174i) q^{66} -0.446067i q^{67} +(1.32779 + 0.766599i) q^{68} -7.53506 q^{69} +(0.226123 + 0.0926196i) q^{70} -5.88376 q^{71} +(-1.86819 - 1.07860i) q^{72} +(10.4445 + 6.03013i) q^{73} +(-4.04139 + 6.99989i) q^{74} +(4.44304 + 1.15300i) q^{75} +(3.73350 + 6.46661i) q^{76} +(0.487062 - 0.281206i) q^{77} -1.02892i q^{78} +(-7.97407 - 3.92609i) q^{79} +(-0.300611 - 2.21577i) q^{80} +(-1.06256 - 1.84040i) q^{81} +(-3.47566 + 2.00667i) q^{82} +(9.33528 + 5.38973i) q^{83} +(-0.0501616 + 0.0868824i) q^{84} +(-1.29946 + 3.17252i) q^{85} +(-5.89241 + 10.2060i) q^{86} +2.82878i q^{87} +(-4.45703 - 2.57327i) q^{88} -0.180638 q^{89} +(1.82833 - 4.46372i) q^{90} +0.122478 q^{91} +(-7.10813 + 4.10388i) q^{92} -3.75785i q^{93} +0.408846 q^{94} +(-13.2062 + 10.2165i) q^{95} +0.918041 q^{96} -8.51012i q^{97} +(6.05184 + 3.49403i) q^{98} +(-5.55105 - 9.61471i) 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
\(3\) 0.795047 − 0.459021i 0.459021 − 0.265016i −0.252612 − 0.967568i \(-0.581290\pi\)
0.711632 + 0.702552i \(0.247956\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 2.06922 + 0.847547i 0.925382 + 0.379035i
\(6\) −0.459021 + 0.795047i −0.187394 + 0.324577i
\(7\) −0.0946389 − 0.0546398i −0.0357701 − 0.0206519i 0.482008 − 0.876167i \(-0.339908\pi\)
−0.517778 + 0.855515i \(0.673241\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.07860 + 1.86819i −0.359533 + 0.622730i
\(10\) −2.21577 + 0.300611i −0.700688 + 0.0950617i
\(11\) −2.57327 + 4.45703i −0.775869 + 1.34385i 0.158435 + 0.987369i \(0.449355\pi\)
−0.934305 + 0.356476i \(0.883978\pi\)
\(12\) − 0.918041i − 0.265016i
\(13\) −0.970618 + 0.560387i −0.269201 + 0.155423i −0.628525 − 0.777790i \(-0.716341\pi\)
0.359323 + 0.933213i \(0.383007\pi\)
\(14\) 0.109280 0.0292062
\(15\) 2.03417 − 0.275974i 0.525220 − 0.0712561i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.53320i 0.371855i 0.982563 + 0.185928i \(0.0595290\pi\)
−0.982563 + 0.185928i \(0.940471\pi\)
\(18\) − 2.15720i − 0.508457i
\(19\) −3.73350 + 6.46661i −0.856523 + 1.48354i 0.0187012 + 0.999825i \(0.494047\pi\)
−0.875225 + 0.483717i \(0.839286\pi\)
\(20\) 1.76861 − 1.36822i 0.395472 − 0.305944i
\(21\) −0.100323 −0.0218923
\(22\) − 5.14653i − 1.09724i
\(23\) −7.10813 − 4.10388i −1.48215 − 0.855718i −0.482353 − 0.875977i \(-0.660218\pi\)
−0.999795 + 0.0202591i \(0.993551\pi\)
\(24\) 0.459021 + 0.795047i 0.0936972 + 0.162288i
\(25\) 3.56333 + 3.50752i 0.712665 + 0.701504i
\(26\) 0.560387 − 0.970618i 0.109901 − 0.190354i
\(27\) 4.73452i 0.911159i
\(28\) −0.0946389 + 0.0546398i −0.0178851 + 0.0103259i
\(29\) −1.54066 + 2.66850i −0.286094 + 0.495529i −0.972874 − 0.231336i \(-0.925690\pi\)
0.686780 + 0.726865i \(0.259024\pi\)
\(30\) −1.62365 + 1.25608i −0.296437 + 0.229328i
\(31\) 2.04667 − 3.54493i 0.367592 − 0.636688i −0.621596 − 0.783338i \(-0.713516\pi\)
0.989189 + 0.146649i \(0.0468489\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 4.72473i 0.822470i
\(34\) −0.766599 − 1.32779i −0.131471 − 0.227714i
\(35\) −0.149519 − 0.193273i −0.0252733 − 0.0326690i
\(36\) 1.07860 + 1.86819i 0.179767 + 0.311365i
\(37\) 6.99989 − 4.04139i 1.15078 − 0.664400i 0.201699 − 0.979447i \(-0.435354\pi\)
0.949076 + 0.315047i \(0.102020\pi\)
\(38\) − 7.46700i − 1.21131i
\(39\) −0.514458 + 0.891068i −0.0823792 + 0.142685i
\(40\) −0.847547 + 2.06922i −0.134009 + 0.327172i
\(41\) 4.01335 0.626780 0.313390 − 0.949625i \(-0.398535\pi\)
0.313390 + 0.949625i \(0.398535\pi\)
\(42\) 0.0868824 − 0.0501616i 0.0134062 − 0.00774010i
\(43\) 10.2060 − 5.89241i 1.55639 − 0.898584i 0.558795 − 0.829306i \(-0.311264\pi\)
0.997597 − 0.0692783i \(-0.0220696\pi\)
\(44\) 2.57327 + 4.45703i 0.387935 + 0.671923i
\(45\) −3.81524 + 2.95153i −0.568742 + 0.439988i
\(46\) 8.20776 1.21017
\(47\) −0.354071 − 0.204423i −0.0516466 − 0.0298182i 0.473954 − 0.880549i \(-0.342826\pi\)
−0.525601 + 0.850731i \(0.676159\pi\)
\(48\) −0.795047 − 0.459021i −0.114755 − 0.0662539i
\(49\) −3.49403 − 6.05184i −0.499147 − 0.864548i
\(50\) −4.83969 − 1.25594i −0.684436 − 0.177617i
\(51\) 0.703770 + 1.21896i 0.0985475 + 0.170689i
\(52\) 1.12077i 0.155423i
\(53\) −1.89274 − 1.09277i −0.259987 − 0.150104i 0.364341 − 0.931265i \(-0.381294\pi\)
−0.624329 + 0.781162i \(0.714627\pi\)
\(54\) −2.36726 − 4.10022i −0.322143 − 0.557969i
\(55\) −9.10220 + 7.04160i −1.22734 + 0.949489i
\(56\) 0.0546398 − 0.0946389i 0.00730155 − 0.0126467i
\(57\) 6.85501i 0.907968i
\(58\) − 3.08132i − 0.404598i
\(59\) 6.71377 + 11.6286i 0.874059 + 1.51391i 0.857762 + 0.514047i \(0.171854\pi\)
0.0162970 + 0.999867i \(0.494812\pi\)
\(60\) 0.778083 − 1.89963i 0.100450 − 0.245241i
\(61\) −0.117047 −0.0149863 −0.00749317 − 0.999972i \(-0.502385\pi\)
−0.00749317 + 0.999972i \(0.502385\pi\)
\(62\) 4.09333i 0.519854i
\(63\) 0.204155 − 0.117869i 0.0257211 − 0.0148501i
\(64\) −1.00000 −0.125000
\(65\) −2.48338 + 0.336917i −0.308025 + 0.0417894i
\(66\) −2.36237 − 4.09174i −0.290787 − 0.503658i
\(67\) − 0.446067i − 0.0544957i −0.999629 − 0.0272479i \(-0.991326\pi\)
0.999629 − 0.0272479i \(-0.00867434\pi\)
\(68\) 1.32779 + 0.766599i 0.161018 + 0.0929638i
\(69\) −7.53506 −0.907115
\(70\) 0.226123 + 0.0926196i 0.0270269 + 0.0110702i
\(71\) −5.88376 −0.698274 −0.349137 − 0.937072i \(-0.613525\pi\)
−0.349137 + 0.937072i \(0.613525\pi\)
\(72\) −1.86819 − 1.07860i −0.220168 − 0.127114i
\(73\) 10.4445 + 6.03013i 1.22244 + 0.705774i 0.965437 − 0.260638i \(-0.0839328\pi\)
0.256999 + 0.966412i \(0.417266\pi\)
\(74\) −4.04139 + 6.99989i −0.469802 + 0.813721i
\(75\) 4.44304 + 1.15300i 0.513038 + 0.133137i
\(76\) 3.73350 + 6.46661i 0.428262 + 0.741771i
\(77\) 0.487062 − 0.281206i 0.0555059 − 0.0320463i
\(78\) − 1.02892i − 0.116502i
\(79\) −7.97407 − 3.92609i −0.897153 − 0.441720i
\(80\) −0.300611 − 2.21577i −0.0336094 − 0.247731i
\(81\) −1.06256 − 1.84040i −0.118062 − 0.204489i
\(82\) −3.47566 + 2.00667i −0.383822 + 0.221600i
\(83\) 9.33528 + 5.38973i 1.02468 + 0.591599i 0.915456 − 0.402418i \(-0.131830\pi\)
0.109224 + 0.994017i \(0.465163\pi\)
\(84\) −0.0501616 + 0.0868824i −0.00547308 + 0.00947964i
\(85\) −1.29946 + 3.17252i −0.140946 + 0.344108i
\(86\) −5.89241 + 10.2060i −0.635395 + 1.10054i
\(87\) 2.82878i 0.303277i
\(88\) −4.45703 − 2.57327i −0.475121 − 0.274311i
\(89\) −0.180638 −0.0191476 −0.00957381 − 0.999954i \(-0.503047\pi\)
−0.00957381 + 0.999954i \(0.503047\pi\)
\(90\) 1.82833 − 4.46372i 0.192723 − 0.470517i
\(91\) 0.122478 0.0128391
\(92\) −7.10813 + 4.10388i −0.741074 + 0.427859i
\(93\) − 3.75785i − 0.389671i
\(94\) 0.408846 0.0421692
\(95\) −13.2062 + 10.2165i −1.35493 + 1.04819i
\(96\) 0.918041 0.0936972
\(97\) − 8.51012i − 0.864072i −0.901856 − 0.432036i \(-0.857795\pi\)
0.901856 − 0.432036i \(-0.142205\pi\)
\(98\) 6.05184 + 3.49403i 0.611328 + 0.352950i
\(99\) −5.55105 − 9.61471i −0.557902 − 0.966314i
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.339.14 ✓ 80
5.4 even 2 inner 790.2.j.a.339.27 yes 80
79.55 even 3 inner 790.2.j.a.529.27 yes 80
395.134 even 6 inner 790.2.j.a.529.14 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.j.a.339.14 ✓ 80 1.1 even 1 trivial
790.2.j.a.339.27 yes 80 5.4 even 2 inner
790.2.j.a.529.14 yes 80 395.134 even 6 inner
790.2.j.a.529.27 yes 80 79.55 even 3 inner