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Properties

Label 790.2.k.a.103.4
Level $790$
Weight $2$
Character 790.103
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 103.4
Character \(\chi\) \(=\) 790.103
Dual form 790.2.k.a.767.4

$q$-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+(-0.258819 - 0.965926i) q^{2} +(-0.596391 - 2.22576i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(2.00339 - 0.993188i) q^{5} +(-1.99556 + 1.15214i) q^{6} +(-0.105600 + 0.394104i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.00025 + 1.15485i) q^{9} +(-1.47786 - 1.67807i) q^{10} +(-1.62734 - 2.81864i) q^{11} +(1.62937 + 1.62937i) q^{12} +(-0.243258 + 0.0651808i) q^{13} +0.408007 q^{14} +(-3.40540 - 3.86674i) q^{15} +(0.500000 - 0.866025i) q^{16} +(-3.13301 - 3.13301i) q^{17} +(1.63320 + 1.63320i) q^{18} +(-0.963359 + 0.556196i) q^{19} +(-1.23839 + 1.86182i) q^{20} +0.940160 q^{21} +(-2.30141 + 2.30141i) q^{22} +(-2.16630 + 8.08476i) q^{23} +(1.15214 - 1.99556i) q^{24} +(3.02716 - 3.97949i) q^{25} +(0.125920 + 0.218099i) q^{26} +(-1.12477 - 1.12477i) q^{27} +(-0.105600 - 0.394104i) q^{28} +(-4.07268 - 7.05409i) q^{29} +(-2.85360 + 4.29015i) q^{30} +(-4.64556 - 8.04635i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(-5.30308 + 5.30308i) q^{33} +(-2.21537 + 3.83714i) q^{34} +(0.179862 + 0.894426i) q^{35} +(1.15485 - 2.00025i) q^{36} +(6.24398 - 1.67307i) q^{37} +(0.786579 + 0.786579i) q^{38} +(0.290154 + 0.502561i) q^{39} +(2.11890 + 0.714322i) q^{40} -10.2683i q^{41} +(-0.243331 - 0.908125i) q^{42} +(2.13854 + 7.98112i) q^{43} +(2.81864 + 1.62734i) q^{44} +(-2.86031 + 4.30023i) q^{45} +8.36996 q^{46} +(-2.82575 + 10.5459i) q^{47} +(-2.22576 - 0.596391i) q^{48} +(5.91801 + 3.41677i) q^{49} +(-4.62738 - 1.89404i) q^{50} +(-5.10483 + 8.84183i) q^{51} +(0.178077 - 0.178077i) q^{52} +(4.69592 + 1.25827i) q^{53} +(-0.795332 + 1.37755i) q^{54} +(-6.05964 - 4.03058i) q^{55} +(-0.353344 + 0.204003i) q^{56} +(1.81250 + 1.81250i) q^{57} +(-5.75964 + 5.75964i) q^{58} +(-7.55853 + 13.0918i) q^{59} +(4.88254 + 1.64600i) q^{60} +2.79525i q^{61} +(-6.56981 + 6.56981i) q^{62} +(-0.243903 - 0.910259i) q^{63} +1.00000i q^{64} +(-0.422605 + 0.372184i) q^{65} +(6.49492 + 3.74985i) q^{66} +(7.96370 - 7.96370i) q^{67} +(4.27977 + 1.14676i) q^{68} +19.2867 q^{69} +(0.817397 - 0.405227i) q^{70} -9.08477i q^{71} +(-2.23099 - 0.597792i) q^{72} +(-2.18295 + 8.14686i) q^{73} +(-3.23212 - 5.59820i) q^{74} +(-10.6628 - 4.36439i) q^{75} +(0.556196 - 0.963359i) q^{76} +(1.28269 - 0.343694i) q^{77} +(0.410339 - 0.410339i) q^{78} +(8.76978 - 1.44604i) q^{79} +(0.141570 - 2.23158i) q^{80} +(-5.29720 + 9.17502i) q^{81} +(-9.91838 + 2.65762i) q^{82} +(1.81057 - 6.75715i) q^{83} +(-0.814203 + 0.470080i) q^{84} +(-9.38832 - 3.16498i) q^{85} +(7.15568 - 4.13133i) q^{86} +(-13.2718 + 13.2718i) q^{87} +(0.842374 - 3.14378i) q^{88} -5.69350i q^{89} +(4.89401 + 1.64986i) q^{90} -0.102752i q^{91} +(-2.16630 - 8.08476i) q^{92} +(-15.1387 + 15.1387i) q^{93} +10.9179 q^{94} +(-1.37758 + 2.07107i) q^{95} +2.30428i q^{96} +(-0.00452422 + 0.00452422i) q^{97} +(1.76865 - 6.60068i) q^{98} +(6.51019 + 3.75866i) 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 \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.258819 − 0.965926i −0.183013 − 0.683013i
\(3\) −0.596391 − 2.22576i −0.344326 − 1.28504i −0.893397 − 0.449268i \(-0.851685\pi\)
0.549071 − 0.835776i \(-0.314982\pi\)
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) 2.00339 − 0.993188i 0.895944 − 0.444167i
\(6\) −1.99556 + 1.15214i −0.814685 + 0.470358i
\(7\) −0.105600 + 0.394104i −0.0399130 + 0.148957i −0.983007 − 0.183570i \(-0.941235\pi\)
0.943094 + 0.332527i \(0.107901\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) −2.00025 + 1.15485i −0.666750 + 0.384948i
\(10\) −1.47786 − 1.67807i −0.467341 − 0.530653i
\(11\) −1.62734 − 2.81864i −0.490662 − 0.849852i 0.509280 − 0.860601i \(-0.329912\pi\)
−0.999942 + 0.0107491i \(0.996578\pi\)
\(12\) 1.62937 + 1.62937i 0.470358 + 0.470358i
\(13\) −0.243258 + 0.0651808i −0.0674677 + 0.0180779i −0.292395 − 0.956298i \(-0.594452\pi\)
0.224927 + 0.974376i \(0.427786\pi\)
\(14\) 0.408007 0.109044
\(15\) −3.40540 − 3.86674i −0.879271 − 0.998388i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −3.13301 − 3.13301i −0.759867 − 0.759867i 0.216431 − 0.976298i \(-0.430558\pi\)
−0.976298 + 0.216431i \(0.930558\pi\)
\(18\) 1.63320 + 1.63320i 0.384948 + 0.384948i
\(19\) −0.963359 + 0.556196i −0.221010 + 0.127600i −0.606418 − 0.795146i \(-0.707394\pi\)
0.385408 + 0.922746i \(0.374061\pi\)
\(20\) −1.23839 + 1.86182i −0.276913 + 0.416316i
\(21\) 0.940160 0.205160
\(22\) −2.30141 + 2.30141i −0.490662 + 0.490662i
\(23\) −2.16630 + 8.08476i −0.451706 + 1.68579i 0.245891 + 0.969297i \(0.420919\pi\)
−0.697597 + 0.716491i \(0.745747\pi\)
\(24\) 1.15214 − 1.99556i 0.235179 − 0.407342i
\(25\) 3.02716 − 3.97949i 0.605431 − 0.795898i
\(26\) 0.125920 + 0.218099i 0.0246949 + 0.0427728i
\(27\) −1.12477 − 1.12477i −0.216462 − 0.216462i
\(28\) −0.105600 − 0.394104i −0.0199565 − 0.0744787i
\(29\) −4.07268 − 7.05409i −0.756278 − 1.30991i −0.944736 − 0.327831i \(-0.893682\pi\)
0.188458 − 0.982081i \(-0.439651\pi\)
\(30\) −2.85360 + 4.29015i −0.520994 + 0.783271i
\(31\) −4.64556 − 8.04635i −0.834367 − 1.44517i −0.894545 − 0.446979i \(-0.852500\pi\)
0.0601773 − 0.998188i \(-0.480833\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) −5.30308 + 5.30308i −0.923148 + 0.923148i
\(34\) −2.21537 + 3.83714i −0.379933 + 0.658064i
\(35\) 0.179862 + 0.894426i 0.0304022 + 0.151186i
\(36\) 1.15485 − 2.00025i 0.192474 − 0.333375i
\(37\) 6.24398 − 1.67307i 1.02650 − 0.275051i 0.293994 − 0.955807i \(-0.405015\pi\)
0.732510 + 0.680756i \(0.238349\pi\)
\(38\) 0.786579 + 0.786579i 0.127600 + 0.127600i
\(39\) 0.290154 + 0.502561i 0.0464618 + 0.0804742i
\(40\) 2.11890 + 0.714322i 0.335028 + 0.112944i
\(41\) − 10.2683i − 1.60363i −0.597571 − 0.801816i \(-0.703867\pi\)
0.597571 − 0.801816i \(-0.296133\pi\)
\(42\) −0.243331 − 0.908125i −0.0375469 − 0.140127i
\(43\) 2.13854 + 7.98112i 0.326124 + 1.21711i 0.913178 + 0.407562i \(0.133621\pi\)
−0.587054 + 0.809548i \(0.699712\pi\)
\(44\) 2.81864 + 1.62734i 0.424926 + 0.245331i
\(45\) −2.86031 + 4.30023i −0.426389 + 0.641041i
\(46\) 8.36996 1.23408
\(47\) −2.82575 + 10.5459i −0.412179 + 1.53827i 0.378242 + 0.925707i \(0.376529\pi\)
−0.790420 + 0.612565i \(0.790138\pi\)
\(48\) −2.22576 − 0.596391i −0.321261 − 0.0860816i
\(49\) 5.91801 + 3.41677i 0.845430 + 0.488109i
\(50\) −4.62738 − 1.89404i −0.654410 − 0.267858i
\(51\) −5.10483 + 8.84183i −0.714820 + 1.23810i
\(52\) 0.178077 − 0.178077i 0.0246949 − 0.0246949i
\(53\) 4.69592 + 1.25827i 0.645034 + 0.172836i 0.566482 − 0.824074i \(-0.308304\pi\)
0.0785513 + 0.996910i \(0.474971\pi\)
\(54\) −0.795332 + 1.37755i −0.108231 + 0.187461i
\(55\) −6.05964 − 4.03058i −0.817082 − 0.543484i
\(56\) −0.353344 + 0.204003i −0.0472176 + 0.0272611i
\(57\) 1.81250 + 1.81250i 0.240071 + 0.240071i
\(58\) −5.75964 + 5.75964i −0.756278 + 0.756278i
\(59\) −7.55853 + 13.0918i −0.984037 + 1.70440i −0.337895 + 0.941184i \(0.609715\pi\)
−0.646141 + 0.763218i \(0.723618\pi\)
\(60\) 4.88254 + 1.64600i 0.630333 + 0.212497i
\(61\) 2.79525i 0.357896i 0.983859 + 0.178948i \(0.0572693\pi\)
−0.983859 + 0.178948i \(0.942731\pi\)
\(62\) −6.56981 + 6.56981i −0.834367 + 0.834367i
\(63\) −0.243903 − 0.910259i −0.0307289 − 0.114682i
\(64\) 1.00000i 0.125000i
\(65\) −0.422605 + 0.372184i −0.0524176 + 0.0461637i
\(66\) 6.49492 + 3.74985i 0.799470 + 0.461574i
\(67\) 7.96370 − 7.96370i 0.972920 − 0.972920i −0.0267226 − 0.999643i \(-0.508507\pi\)
0.999643 + 0.0267226i \(0.00850709\pi\)
\(68\) 4.27977 + 1.14676i 0.518999 + 0.139065i
\(69\) 19.2867 2.32184
\(70\) 0.817397 − 0.405227i 0.0976977 − 0.0484339i
\(71\) − 9.08477i − 1.07816i −0.842254 − 0.539082i \(-0.818771\pi\)
0.842254 − 0.539082i \(-0.181229\pi\)
\(72\) −2.23099 − 0.597792i −0.262925 − 0.0704505i
\(73\) −2.18295 + 8.14686i −0.255494 + 0.953518i 0.712320 + 0.701855i \(0.247644\pi\)
−0.967815 + 0.251664i \(0.919022\pi\)
\(74\) −3.23212 − 5.59820i −0.375727 − 0.650778i
\(75\) −10.6628 − 4.36439i −1.23123 − 0.503957i
\(76\) 0.556196 − 0.963359i 0.0638000 − 0.110505i
\(77\) 1.28269 − 0.343694i 0.146176 − 0.0391676i
\(78\) 0.410339 − 0.410339i 0.0464618 − 0.0464618i
\(79\) 8.76978 − 1.44604i 0.986677 − 0.162692i
\(80\) 0.141570 − 2.23158i 0.0158280 − 0.249498i
\(81\) −5.29720 + 9.17502i −0.588578 + 1.01945i
\(82\) −9.91838 + 2.65762i −1.09530 + 0.293485i
\(83\) 1.81057 − 6.75715i 0.198736 − 0.741693i −0.792532 − 0.609830i \(-0.791238\pi\)
0.991268 − 0.131863i \(-0.0420958\pi\)
\(84\) −0.814203 + 0.470080i −0.0888368 + 0.0512900i
\(85\) −9.38832 − 3.16498i −1.01831 − 0.343290i
\(86\) 7.15568 − 4.13133i 0.771617 − 0.445493i
\(87\) −13.2718 + 13.2718i −1.42289 + 1.42289i
\(88\) 0.842374 − 3.14378i 0.0897974 − 0.335128i
\(89\) − 5.69350i − 0.603510i −0.953385 − 0.301755i \(-0.902428\pi\)
0.953385 − 0.301755i \(-0.0975725\pi\)
\(90\) 4.89401 + 1.64986i 0.515874 + 0.173911i
\(91\) − 0.102752i − 0.0107714i
\(92\) −2.16630 − 8.08476i −0.225853 − 0.842894i
\(93\) −15.1387 + 15.1387i −1.56981 + 1.56981i
\(94\) 10.9179 1.12609
\(95\) −1.37758 + 2.07107i −0.141337 + 0.212488i
\(96\) 2.30428i 0.235179i
\(97\) −0.00452422 + 0.00452422i −0.000459364 + 0.000459364i −0.707336 − 0.706877i \(-0.750103\pi\)
0.706877 + 0.707336i \(0.250103\pi\)
\(98\) 1.76865 − 6.60068i 0.178660 − 0.666770i
\(99\) 6.51019 + 3.75866i 0.654298 + 0.377759i
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.103.4 ✓ 160
5.2 odd 4 inner 790.2.k.a.577.24 yes 160
79.56 odd 6 inner 790.2.k.a.293.24 yes 160
395.372 even 12 inner 790.2.k.a.767.4 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.4 ✓ 160 1.1 even 1 trivial
790.2.k.a.293.24 yes 160 79.56 odd 6 inner
790.2.k.a.577.24 yes 160 5.2 odd 4 inner
790.2.k.a.767.4 yes 160 395.372 even 12 inner