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Properties

Label 790.2.j.a.339.1
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 339.1
Character \(\chi\) \(=\) 790.339
Dual form 790.2.j.a.529.1

$q$-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} +(-2.91982 + 1.68576i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-1.75498 - 1.38566i) q^{5} +(1.68576 - 2.91982i) q^{6} +(3.40032 + 1.96318i) q^{7} +1.00000i q^{8} +(4.18358 - 7.24618i) q^{9} +(2.21269 + 0.322524i) q^{10} +(-3.03534 + 5.25736i) q^{11} +3.37152i q^{12} +(-1.99152 + 1.14980i) q^{13} -3.92635 q^{14} +(7.46012 + 1.08740i) q^{15} +(-0.500000 - 0.866025i) q^{16} -1.60022i q^{17} +8.36717i q^{18} +(-2.86492 + 4.96219i) q^{19} +(-2.07750 + 0.827029i) q^{20} -13.2378 q^{21} -6.07068i q^{22} +(-0.462542 - 0.267049i) q^{23} +(-1.68576 - 2.91982i) q^{24} +(1.15991 + 4.86360i) q^{25} +(1.14980 - 1.99152i) q^{26} +18.0955i q^{27} +(3.40032 - 1.96318i) q^{28} +(2.38698 - 4.13437i) q^{29} +(-7.00435 + 2.78835i) q^{30} +(-2.31739 + 4.01384i) q^{31} +(0.866025 + 0.500000i) q^{32} -20.4674i q^{33} +(0.800110 + 1.38583i) q^{34} +(-3.24721 - 8.15701i) q^{35} +(-4.18358 - 7.24618i) q^{36} +(3.06556 - 1.76990i) q^{37} -5.72985i q^{38} +(3.87659 - 6.71444i) q^{39} +(1.38566 - 1.75498i) q^{40} +0.800226 q^{41} +(11.4643 - 6.61889i) q^{42} +(0.198857 - 0.114810i) q^{43} +(3.03534 + 5.25736i) q^{44} +(-17.3828 + 6.91989i) q^{45} +0.534098 q^{46} +(-9.93793 - 5.73767i) q^{47} +(2.91982 + 1.68576i) q^{48} +(4.20812 + 7.28868i) q^{49} +(-3.43631 - 3.63205i) q^{50} +(2.69759 + 4.67236i) q^{51} +2.29960i q^{52} +(-1.08949 - 0.629019i) q^{53} +(-9.04776 - 15.6712i) q^{54} +(12.6119 - 5.02063i) q^{55} +(-1.96318 + 3.40032i) q^{56} -19.3183i q^{57} +4.77396i q^{58} +(-7.14298 - 12.3720i) q^{59} +(4.67178 - 5.91696i) q^{60} +5.78101 q^{61} -4.63478i q^{62} +(28.4511 - 16.4262i) q^{63} -1.00000 q^{64} +(5.08830 + 0.741679i) q^{65} +(10.2337 + 17.7253i) q^{66} -10.0361i q^{67} +(-1.38583 - 0.800110i) q^{68} +1.80072 q^{69} +(6.89067 + 5.44058i) q^{70} +7.98857 q^{71} +(7.24618 + 4.18358i) q^{72} +(2.91883 + 1.68519i) q^{73} +(-1.76990 + 3.06556i) q^{74} +(-11.5856 - 12.2455i) q^{75} +(2.86492 + 4.96219i) q^{76} +(-20.6423 + 11.9178i) q^{77} +7.75317i q^{78} +(-7.11373 - 5.32868i) q^{79} +(-0.322524 + 2.21269i) q^{80} +(-17.9540 - 31.0972i) q^{81} +(-0.693016 + 0.400113i) q^{82} +(3.26814 + 1.88686i) q^{83} +(-6.61889 + 11.4643i) q^{84} +(-2.21736 + 2.80835i) q^{85} +(-0.114810 + 0.198857i) q^{86} +16.0955i q^{87} +(-5.25736 - 3.03534i) q^{88} -11.2394 q^{89} +(11.5940 - 14.6842i) q^{90} -9.02906 q^{91} +(-0.462542 + 0.267049i) q^{92} -15.6263i q^{93} +11.4753 q^{94} +(11.9038 - 4.73875i) q^{95} -3.37152 q^{96} +8.67163i q^{97} +(-7.28868 - 4.20812i) q^{98} +(25.3972 + 43.9892i) 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
\(3\) −2.91982 + 1.68576i −1.68576 + 0.973275i −0.728059 + 0.685514i \(0.759577\pi\)
−0.957702 + 0.287761i \(0.907089\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −1.75498 − 1.38566i −0.784851 − 0.619685i
\(6\) 1.68576 − 2.91982i 0.688209 − 1.19201i
\(7\) 3.40032 + 1.96318i 1.28520 + 0.742011i 0.977794 − 0.209568i \(-0.0672056\pi\)
0.307406 + 0.951578i \(0.400539\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 4.18358 − 7.24618i 1.39453 − 2.41539i
\(10\) 2.21269 + 0.322524i 0.699713 + 0.101991i
\(11\) −3.03534 + 5.25736i −0.915189 + 1.58515i −0.108566 + 0.994089i \(0.534626\pi\)
−0.806624 + 0.591065i \(0.798708\pi\)
\(12\) 3.37152i 0.973275i
\(13\) −1.99152 + 1.14980i −0.552347 + 0.318898i −0.750068 − 0.661361i \(-0.769979\pi\)
0.197721 + 0.980258i \(0.436646\pi\)
\(14\) −3.92635 −1.04936
\(15\) 7.46012 + 1.08740i 1.92619 + 0.280765i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) − 1.60022i − 0.388110i −0.980991 − 0.194055i \(-0.937836\pi\)
0.980991 − 0.194055i \(-0.0621641\pi\)
\(18\) 8.36717i 1.97216i
\(19\) −2.86492 + 4.96219i −0.657259 + 1.13841i 0.324064 + 0.946035i \(0.394951\pi\)
−0.981322 + 0.192370i \(0.938383\pi\)
\(20\) −2.07750 + 0.827029i −0.464544 + 0.184929i
\(21\) −13.2378 −2.88872
\(22\) − 6.07068i − 1.29427i
\(23\) −0.462542 − 0.267049i −0.0964467 − 0.0556835i 0.451001 − 0.892524i \(-0.351067\pi\)
−0.547448 + 0.836840i \(0.684400\pi\)
\(24\) −1.68576 − 2.91982i −0.344105 − 0.596007i
\(25\) 1.15991 + 4.86360i 0.231982 + 0.972720i
\(26\) 1.14980 − 1.99152i 0.225495 − 0.390568i
\(27\) 18.0955i 3.48249i
\(28\) 3.40032 − 1.96318i 0.642600 − 0.371005i
\(29\) 2.38698 − 4.13437i 0.443251 − 0.767733i −0.554678 − 0.832065i \(-0.687158\pi\)
0.997929 + 0.0643322i \(0.0204917\pi\)
\(30\) −7.00435 + 2.78835i −1.27881 + 0.509080i
\(31\) −2.31739 + 4.01384i −0.416216 + 0.720907i −0.995555 − 0.0941795i \(-0.969977\pi\)
0.579339 + 0.815086i \(0.303311\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) − 20.4674i − 3.56292i
\(34\) 0.800110 + 1.38583i 0.137218 + 0.237668i
\(35\) −3.24721 − 8.15701i −0.548878 − 1.37879i
\(36\) −4.18358 − 7.24618i −0.697264 − 1.20770i
\(37\) 3.06556 − 1.76990i 0.503974 − 0.290970i −0.226379 − 0.974039i \(-0.572689\pi\)
0.730353 + 0.683070i \(0.239355\pi\)
\(38\) − 5.72985i − 0.929504i
\(39\) 3.87659 − 6.71444i 0.620750 − 1.07517i
\(40\) 1.38566 − 1.75498i 0.219092 − 0.277487i
\(41\) 0.800226 0.124974 0.0624871 − 0.998046i \(-0.480097\pi\)
0.0624871 + 0.998046i \(0.480097\pi\)
\(42\) 11.4643 − 6.61889i 1.76897 − 1.02132i
\(43\) 0.198857 − 0.114810i 0.0303254 − 0.0175084i −0.484761 − 0.874647i \(-0.661093\pi\)
0.515086 + 0.857138i \(0.327760\pi\)
\(44\) 3.03534 + 5.25736i 0.457595 + 0.792577i
\(45\) −17.3828 + 6.91989i −2.59128 + 1.03156i
\(46\) 0.534098 0.0787484
\(47\) −9.93793 − 5.73767i −1.44960 − 0.836925i −0.451139 − 0.892454i \(-0.648982\pi\)
−0.998457 + 0.0555291i \(0.982315\pi\)
\(48\) 2.91982 + 1.68576i 0.421440 + 0.243319i
\(49\) 4.20812 + 7.28868i 0.601160 + 1.04124i
\(50\) −3.43631 − 3.63205i −0.485968 − 0.513649i
\(51\) 2.69759 + 4.67236i 0.377738 + 0.654261i
\(52\) 2.29960i 0.318898i
\(53\) −1.08949 − 0.629019i −0.149653 − 0.0864024i 0.423303 − 0.905988i \(-0.360870\pi\)
−0.572957 + 0.819586i \(0.694204\pi\)
\(54\) −9.04776 − 15.6712i −1.23124 − 2.13258i
\(55\) 12.6119 − 5.02063i 1.70058 − 0.676981i
\(56\) −1.96318 + 3.40032i −0.262340 + 0.454387i
\(57\) − 19.3183i − 2.55877i
\(58\) 4.77396i 0.626852i
\(59\) −7.14298 − 12.3720i −0.929937 − 1.61070i −0.783422 − 0.621490i \(-0.786528\pi\)
−0.146515 − 0.989208i \(-0.546806\pi\)
\(60\) 4.67178 − 5.91696i 0.603124 − 0.763876i
\(61\) 5.78101 0.740183 0.370091 − 0.928995i \(-0.379326\pi\)
0.370091 + 0.928995i \(0.379326\pi\)
\(62\) − 4.63478i − 0.588618i
\(63\) 28.4511 − 16.4262i 3.58450 − 2.06951i
\(64\) −1.00000 −0.125000
\(65\) 5.08830 + 0.741679i 0.631126 + 0.0919939i
\(66\) 10.2337 + 17.7253i 1.25968 + 2.18184i
\(67\) − 10.0361i − 1.22611i −0.790040 − 0.613055i \(-0.789940\pi\)
0.790040 − 0.613055i \(-0.210060\pi\)
\(68\) −1.38583 − 0.800110i −0.168057 − 0.0970275i
\(69\) 1.80072 0.216782
\(70\) 6.89067 + 5.44058i 0.823592 + 0.650273i
\(71\) 7.98857 0.948069 0.474035 − 0.880506i \(-0.342797\pi\)
0.474035 + 0.880506i \(0.342797\pi\)
\(72\) 7.24618 + 4.18358i 0.853970 + 0.493040i
\(73\) 2.91883 + 1.68519i 0.341623 + 0.197236i 0.660990 − 0.750395i \(-0.270137\pi\)
−0.319366 + 0.947631i \(0.603470\pi\)
\(74\) −1.76990 + 3.06556i −0.205747 + 0.356364i
\(75\) −11.5856 − 12.2455i −1.33779 − 1.41399i
\(76\) 2.86492 + 4.96219i 0.328629 + 0.569203i
\(77\) −20.6423 + 11.9178i −2.35240 + 1.35816i
\(78\) 7.75317i 0.877874i
\(79\) −7.11373 − 5.32868i −0.800357 − 0.599523i
\(80\) −0.322524 + 2.21269i −0.0360593 + 0.247386i
\(81\) −17.9540 − 31.0972i −1.99489 − 3.45525i
\(82\) −0.693016 + 0.400113i −0.0765308 + 0.0441851i
\(83\) 3.26814 + 1.88686i 0.358725 + 0.207110i 0.668521 − 0.743693i \(-0.266927\pi\)
−0.309796 + 0.950803i \(0.600261\pi\)
\(84\) −6.61889 + 11.4643i −0.722180 + 1.25085i
\(85\) −2.21736 + 2.80835i −0.240506 + 0.304609i
\(86\) −0.114810 + 0.198857i −0.0123803 + 0.0214433i
\(87\) 16.0955i 1.72562i
\(88\) −5.25736 − 3.03534i −0.560437 − 0.323568i
\(89\) −11.2394 −1.19137 −0.595687 − 0.803217i \(-0.703120\pi\)
−0.595687 + 0.803217i \(0.703120\pi\)
\(90\) 11.5940 − 14.6842i 1.22212 − 1.54785i
\(91\) −9.02906 −0.946502
\(92\) −0.462542 + 0.267049i −0.0482234 + 0.0278418i
\(93\) − 15.6263i − 1.62037i
\(94\) 11.4753 1.18359
\(95\) 11.9038 − 4.73875i 1.22130 − 0.486185i
\(96\) −3.37152 −0.344105
\(97\) 8.67163i 0.880471i 0.897882 + 0.440235i \(0.145105\pi\)
−0.897882 + 0.440235i \(0.854895\pi\)
\(98\) −7.28868 − 4.20812i −0.736268 − 0.425084i
\(99\) 25.3972 + 43.9892i 2.55251 + 4.42108i
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.339.1 ✓ 80
5.4 even 2 inner 790.2.j.a.339.40 yes 80
79.55 even 3 inner 790.2.j.a.529.40 yes 80
395.134 even 6 inner 790.2.j.a.529.1 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.j.a.339.1 ✓ 80 1.1 even 1 trivial
790.2.j.a.339.40 yes 80 5.4 even 2 inner
790.2.j.a.529.1 yes 80 395.134 even 6 inner
790.2.j.a.529.40 yes 80 79.55 even 3 inner