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Properties

Label 790.2.k.a.577.24
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 577.24
Character \(\chi\) \(=\) 790.577
Dual form 790.2.k.a.293.24

$q$-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} +(-2.22576 + 0.596391i) q^{3} +(0.866025 - 0.500000i) q^{4} +(-1.86182 + 1.23839i) q^{5} +(-1.99556 + 1.15214i) q^{6} +(-0.394104 - 0.105600i) 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} +(0.0651808 + 0.243258i) 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} +(-0.993188 + 2.00339i) q^{20} +0.940160 q^{21} +(-2.30141 - 2.30141i) q^{22} +(8.08476 + 2.16630i) q^{23} +(-1.15214 + 1.99556i) q^{24} +(1.93276 - 4.61134i) q^{25} +(0.125920 + 0.218099i) q^{26} +(1.12477 - 1.12477i) q^{27} +(-0.394104 + 0.105600i) q^{28} +(4.07268 + 7.05409i) q^{29} +(2.28858 - 4.61637i) q^{30} +(-4.64556 - 8.04635i) q^{31} +(0.258819 - 0.965926i) q^{32} +(5.30308 + 5.30308i) q^{33} +(2.21537 - 3.83714i) q^{34} +(0.864526 - 0.291448i) q^{35} +(1.15485 - 2.00025i) q^{36} +(1.67307 + 6.24398i) q^{37} +(0.786579 - 0.786579i) q^{38} +(-0.290154 - 0.502561i) q^{39} +(-0.440830 + 2.19218i) q^{40} -10.2683i q^{41} +(0.908125 - 0.243331i) q^{42} +(7.98112 - 2.13854i) q^{43} +(-2.81864 - 1.62734i) q^{44} +(-2.29396 + 4.62722i) q^{45} +8.36996 q^{46} +(-10.5459 - 2.82575i) q^{47} +(-0.596391 + 2.22576i) q^{48} +(-5.91801 - 3.41677i) q^{49} +(0.673401 - 4.95445i) q^{50} +(-5.10483 + 8.84183i) q^{51} +(0.178077 + 0.178077i) q^{52} +(1.25827 - 4.69592i) q^{53} +(0.795332 - 1.37755i) q^{54} +(6.52041 + 3.23251i) 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} +(1.01579 - 5.05140i) q^{60} +2.79525i q^{61} +(-6.56981 - 6.56981i) q^{62} +(-0.910259 + 0.243903i) q^{63} -1.00000i q^{64} +(-0.422605 - 0.372184i) q^{65} +(6.49492 + 3.74985i) q^{66} +(7.96370 + 7.96370i) q^{67} +(1.14676 - 4.27977i) q^{68} -19.2867 q^{69} +(0.759636 - 0.505273i) q^{70} -9.08477i q^{71} +(0.597792 - 2.23099i) q^{72} +(8.14686 + 2.18295i) q^{73} +(3.23212 + 5.59820i) q^{74} +(-1.55170 + 11.4164i) q^{75} +(0.556196 - 0.963359i) q^{76} +(0.343694 + 1.28269i) 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} +(-2.65762 - 9.91838i) q^{82} +(-6.75715 - 1.81057i) q^{83} +(0.814203 - 0.470080i) q^{84} +(-1.95321 + 9.71301i) q^{85} +(7.15568 - 4.13133i) q^{86} +(-13.2718 - 13.2718i) q^{87} +(-3.14378 - 0.842374i) q^{88} +5.69350i q^{89} +(-1.01818 + 5.06327i) q^{90} -0.102752i q^{91} +(8.08476 - 2.16630i) q^{92} +(15.1387 + 15.1387i) q^{93} -10.9179 q^{94} +(-1.10481 + 2.22856i) q^{95} +2.30428i q^{96} +(-0.00452422 - 0.00452422i) q^{97} +(-6.60068 - 1.76865i) 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 \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

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

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
\(3\) −2.22576 + 0.596391i −1.28504 + 0.344326i −0.835776 − 0.549071i \(-0.814982\pi\)
−0.449268 + 0.893397i \(0.648315\pi\)
\(4\) 0.866025 − 0.500000i 0.433013 − 0.250000i
\(5\) −1.86182 + 1.23839i −0.832632 + 0.553827i
\(6\) −1.99556 + 1.15214i −0.814685 + 0.470358i
\(7\) −0.394104 − 0.105600i −0.148957 − 0.0399130i 0.183570 − 0.983007i \(-0.441235\pi\)
−0.332527 + 0.943094i \(0.607901\pi\)
\(8\) 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
\(13\) 0.0651808 + 0.243258i 0.0180779 + 0.0674677i 0.974376 − 0.224927i \(-0.0722145\pi\)
−0.956298 + 0.292395i \(0.905548\pi\)
\(14\) −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 \(-0.569442\pi\)
0.976298 + 0.216431i \(0.0694417\pi\)
\(18\) 1.63320 − 1.63320i 0.384948 − 0.384948i
\(19\) 0.963359 − 0.556196i 0.221010 − 0.127600i −0.385408 − 0.922746i \(-0.625939\pi\)
0.606418 + 0.795146i \(0.292606\pi\)
\(20\) −0.993188 + 2.00339i −0.222084 + 0.447972i
\(21\) 0.940160 0.205160
\(22\) −2.30141 − 2.30141i −0.490662 − 0.490662i
\(23\) 8.08476 + 2.16630i 1.68579 + 0.451706i 0.969297 − 0.245891i \(-0.0790805\pi\)
0.716491 + 0.697597i \(0.245747\pi\)
\(24\) −1.15214 + 1.99556i −0.235179 + 0.407342i
\(25\) 1.93276 − 4.61134i 0.386552 − 0.922268i
\(26\) 0.125920 + 0.218099i 0.0246949 + 0.0427728i
\(27\) 1.12477 − 1.12477i 0.216462 − 0.216462i
\(28\) −0.394104 + 0.105600i −0.0744787 + 0.0199565i
\(29\) 4.07268 + 7.05409i 0.756278 + 1.30991i 0.944736 + 0.327831i \(0.106318\pi\)
−0.188458 + 0.982081i \(0.560349\pi\)
\(30\) 2.28858 − 4.61637i 0.417836 − 0.842830i
\(31\) −4.64556 − 8.04635i −0.834367 − 1.44517i −0.894545 − 0.446979i \(-0.852500\pi\)
0.0601773 − 0.998188i \(-0.480833\pi\)
\(32\) 0.258819 − 0.965926i 0.0457532 − 0.170753i
\(33\) 5.30308 + 5.30308i 0.923148 + 0.923148i
\(34\) 2.21537 − 3.83714i 0.379933 − 0.658064i
\(35\) 0.864526 − 0.291448i 0.146132 − 0.0492637i
\(36\) 1.15485 − 2.00025i 0.192474 − 0.333375i
\(37\) 1.67307 + 6.24398i 0.275051 + 1.02650i 0.955807 + 0.293994i \(0.0949847\pi\)
−0.680756 + 0.732510i \(0.738349\pi\)
\(38\) 0.786579 − 0.786579i 0.127600 − 0.127600i
\(39\) −0.290154 − 0.502561i −0.0464618 − 0.0804742i
\(40\) −0.440830 + 2.19218i −0.0697013 + 0.346615i
\(41\) − 10.2683i − 1.60363i −0.597571 − 0.801816i \(-0.703867\pi\)
0.597571 − 0.801816i \(-0.296133\pi\)
\(42\) 0.908125 − 0.243331i 0.140127 − 0.0375469i
\(43\) 7.98112 − 2.13854i 1.21711 − 0.326124i 0.407562 − 0.913178i \(-0.366379\pi\)
0.809548 + 0.587054i \(0.199712\pi\)
\(44\) −2.81864 − 1.62734i −0.424926 − 0.245331i
\(45\) −2.29396 + 4.62722i −0.341963 + 0.689784i
\(46\) 8.36996 1.23408
\(47\) −10.5459 − 2.82575i −1.53827 − 0.412179i −0.612565 − 0.790420i \(-0.709862\pi\)
−0.925707 + 0.378242i \(0.876529\pi\)
\(48\) −0.596391 + 2.22576i −0.0860816 + 0.321261i
\(49\) −5.91801 − 3.41677i −0.845430 − 0.488109i
\(50\) 0.673401 − 4.95445i 0.0952333 − 0.700664i
\(51\) −5.10483 + 8.84183i −0.714820 + 1.23810i
\(52\) 0.178077 + 0.178077i 0.0246949 + 0.0246949i
\(53\) 1.25827 − 4.69592i 0.172836 − 0.645034i −0.824074 − 0.566482i \(-0.808304\pi\)
0.996910 − 0.0785513i \(-0.0250294\pi\)
\(54\) 0.795332 − 1.37755i 0.108231 − 0.187461i
\(55\) 6.52041 + 3.23251i 0.879212 + 0.435872i
\(56\) −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 \(-0.390285\pi\)
0.646141 − 0.763218i \(-0.276382\pi\)
\(60\) 1.01579 − 5.05140i 0.131138 − 0.652133i
\(61\) 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
\(63\) −0.910259 + 0.243903i −0.114682 + 0.0307289i
\(64\) − 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 0.999643 − 0.0267226i \(-0.00850709\pi\)
−0.0267226 + 0.999643i \(0.508507\pi\)
\(68\) 1.14676 − 4.27977i 0.139065 − 0.518999i
\(69\) −19.2867 −2.32184
\(70\) 0.759636 − 0.505273i 0.0907939 − 0.0603917i
\(71\) − 9.08477i − 1.07816i −0.842254 − 0.539082i \(-0.818771\pi\)
0.842254 − 0.539082i \(-0.181229\pi\)
\(72\) 0.597792 − 2.23099i 0.0704505 − 0.262925i
\(73\) 8.14686 + 2.18295i 0.953518 + 0.255494i 0.701855 − 0.712320i \(-0.252356\pi\)
0.251664 + 0.967815i \(0.419022\pi\)
\(74\) 3.23212 + 5.59820i 0.375727 + 0.650778i
\(75\) −1.55170 + 11.4164i −0.179175 + 1.31825i
\(76\) 0.556196 − 0.963359i 0.0638000 − 0.110505i
\(77\) 0.343694 + 1.28269i 0.0391676 + 0.146176i
\(78\) −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
\(82\) −2.65762 − 9.91838i −0.293485 − 1.09530i
\(83\) −6.75715 − 1.81057i −0.741693 − 0.198736i −0.131863 − 0.991268i \(-0.542096\pi\)
−0.609830 + 0.792532i \(0.708762\pi\)
\(84\) 0.814203 − 0.470080i 0.0888368 − 0.0512900i
\(85\) −1.95321 + 9.71301i −0.211855 + 1.05352i
\(86\) 7.15568 − 4.13133i 0.771617 − 0.445493i
\(87\) −13.2718 − 13.2718i −1.42289 − 1.42289i
\(88\) −3.14378 − 0.842374i −0.335128 − 0.0897974i
\(89\) 5.69350i 0.603510i 0.953385 + 0.301755i \(0.0975725\pi\)
−0.953385 + 0.301755i \(0.902428\pi\)
\(90\) −1.01818 + 5.06327i −0.107326 + 0.533715i
\(91\) − 0.102752i − 0.0107714i
\(92\) 8.08476 − 2.16630i 0.842894 − 0.225853i
\(93\) 15.1387 + 15.1387i 1.56981 + 1.56981i
\(94\) −10.9179 −1.12609
\(95\) −1.10481 + 2.22856i −0.113351 + 0.228645i
\(96\) 2.30428i 0.235179i
\(97\) −0.00452422 − 0.00452422i −0.000459364 − 0.000459364i 0.706877 − 0.707336i \(-0.250103\pi\)
−0.707336 + 0.706877i \(0.750103\pi\)
\(98\) −6.60068 − 1.76865i −0.666770 − 0.178660i
\(99\) −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.577.24 yes 160
5.3 odd 4 inner 790.2.k.a.103.4 ✓ 160
79.56 odd 6 inner 790.2.k.a.767.4 yes 160
395.293 even 12 inner 790.2.k.a.293.24 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.4 ✓ 160 5.3 odd 4 inner
790.2.k.a.293.24 yes 160 395.293 even 12 inner
790.2.k.a.577.24 yes 160 1.1 even 1 trivial
790.2.k.a.767.4 yes 160 79.56 odd 6 inner