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Properties

Label 790.2.k.a.103.5
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 103.5
Character \(\chi\) \(=\) 790.103
Dual form 790.2.k.a.767.5

$q$-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} +(-0.485819 - 1.81310i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(-1.27028 + 1.84022i) q^{5} +(-1.62558 + 0.938529i) q^{6} +(0.662018 - 2.47069i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-0.453235 + 0.261675i) q^{9} +(2.10628 + 0.750711i) q^{10} +(2.90276 + 5.02773i) q^{11} +(1.32728 + 1.32728i) q^{12} +(5.49186 - 1.47154i) q^{13} -2.55784 q^{14} +(3.95362 + 1.40913i) q^{15} +(0.500000 - 0.866025i) q^{16} +(3.59267 + 3.59267i) q^{17} +(0.370065 + 0.370065i) q^{18} +(0.350857 - 0.202568i) q^{19} +(0.179985 - 2.22881i) q^{20} -4.80122 q^{21} +(4.10512 - 4.10512i) q^{22} +(-1.08451 + 4.04744i) q^{23} +(0.938529 - 1.62558i) q^{24} +(-1.77279 - 4.67517i) q^{25} +(-2.84280 - 4.92387i) q^{26} +(-3.28721 - 3.28721i) q^{27} +(0.662018 + 2.47069i) q^{28} +(-2.52800 - 4.37863i) q^{29} +(0.337842 - 4.18361i) q^{30} +(-4.41781 - 7.65187i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(7.70555 - 7.70555i) q^{33} +(2.54040 - 4.40010i) q^{34} +(3.70565 + 4.35671i) q^{35} +(0.261675 - 0.453235i) q^{36} +(3.79929 - 1.01802i) q^{37} +(-0.286474 - 0.286474i) q^{38} +(-5.33610 - 9.24240i) q^{39} +(-2.19945 + 0.403007i) q^{40} +9.19556i q^{41} +(1.24265 + 4.63762i) q^{42} +(1.30464 + 4.86897i) q^{43} +(-5.02773 - 2.90276i) q^{44} +(0.0941950 - 1.16645i) q^{45} +4.19022 q^{46} +(2.74991 - 10.2628i) q^{47} +(-1.81310 - 0.485819i) q^{48} +(0.396155 + 0.228720i) q^{49} +(-4.05704 + 2.92241i) q^{50} +(4.76848 - 8.25925i) q^{51} +(-4.02032 + 4.02032i) q^{52} +(9.21929 + 2.47030i) q^{53} +(-2.32441 + 4.02599i) q^{54} +(-12.9394 - 1.04490i) q^{55} +(2.21516 - 1.27892i) q^{56} +(-0.537728 - 0.537728i) q^{57} +(-3.57514 + 3.57514i) q^{58} +(3.36983 - 5.83672i) q^{59} +(-4.12850 + 0.756469i) q^{60} +1.29350i q^{61} +(-6.24772 + 6.24772i) q^{62} +(0.346468 + 1.29303i) q^{63} +1.00000i q^{64} +(-4.26824 + 11.9755i) q^{65} +(-9.43734 - 5.44865i) q^{66} +(3.91492 - 3.91492i) q^{67} +(-4.90768 - 1.31501i) q^{68} +7.86528 q^{69} +(3.24917 - 4.70698i) q^{70} -8.51116i q^{71} +(-0.505518 - 0.135453i) q^{72} +(1.57973 - 5.89563i) q^{73} +(-1.96666 - 3.40635i) q^{74} +(-7.61530 + 5.48553i) q^{75} +(-0.202568 + 0.350857i) q^{76} +(14.3436 - 3.84336i) q^{77} +(-7.54638 + 7.54638i) q^{78} +(2.06780 + 8.64432i) q^{79} +(0.958535 + 2.02020i) q^{80} +(-5.14808 + 8.91673i) q^{81} +(8.88223 - 2.37999i) q^{82} +(2.42735 - 9.05899i) q^{83} +(4.15798 - 2.40061i) q^{84} +(-11.1750 + 2.04760i) q^{85} +(4.36540 - 2.52037i) q^{86} +(-6.71074 + 6.71074i) q^{87} +(-1.50258 + 5.60770i) q^{88} +7.27543i q^{89} +(-1.15108 + 0.210914i) q^{90} -14.5429i q^{91} +(-1.08451 - 4.04744i) q^{92} +(-11.7273 + 11.7273i) q^{93} -10.6248 q^{94} +(-0.0729181 + 0.902970i) q^{95} +1.87706i q^{96} +(-8.69166 + 8.69166i) q^{97} +(0.118394 - 0.441853i) q^{98} +(-2.63126 - 1.51916i) 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
\(3\) −0.485819 − 1.81310i −0.280488 − 1.04679i −0.952074 − 0.305868i \(-0.901054\pi\)
0.671587 − 0.740926i \(-0.265613\pi\)
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) −1.27028 + 1.84022i −0.568085 + 0.822970i
\(6\) −1.62558 + 0.938529i −0.663641 + 0.383153i
\(7\) 0.662018 − 2.47069i 0.250219 − 0.933832i −0.720468 − 0.693488i \(-0.756073\pi\)
0.970688 − 0.240344i \(-0.0772602\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) −0.453235 + 0.261675i −0.151078 + 0.0872250i
\(10\) 2.10628 + 0.750711i 0.666066 + 0.237396i
\(11\) 2.90276 + 5.02773i 0.875215 + 1.51592i 0.856534 + 0.516091i \(0.172613\pi\)
0.0186810 + 0.999825i \(0.494053\pi\)
\(12\) 1.32728 + 1.32728i 0.383153 + 0.383153i
\(13\) 5.49186 − 1.47154i 1.52317 − 0.408132i 0.602385 − 0.798205i \(-0.294217\pi\)
0.920784 + 0.390074i \(0.127550\pi\)
\(14\) −2.55784 −0.683612
\(15\) 3.95362 + 1.40913i 1.02082 + 0.363835i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) 3.59267 + 3.59267i 0.871350 + 0.871350i 0.992620 − 0.121269i \(-0.0386965\pi\)
−0.121269 + 0.992620i \(0.538696\pi\)
\(18\) 0.370065 + 0.370065i 0.0872250 + 0.0872250i
\(19\) 0.350857 − 0.202568i 0.0804922 − 0.0464722i −0.459214 − 0.888326i \(-0.651869\pi\)
0.539706 + 0.841854i \(0.318535\pi\)
\(20\) 0.179985 − 2.22881i 0.0402458 − 0.498378i
\(21\) −4.80122 −1.04771
\(22\) 4.10512 − 4.10512i 0.875215 − 0.875215i
\(23\) −1.08451 + 4.04744i −0.226135 + 0.843949i 0.755811 + 0.654790i \(0.227243\pi\)
−0.981946 + 0.189159i \(0.939424\pi\)
\(24\) 0.938529 − 1.62558i 0.191577 − 0.331820i
\(25\) −1.77279 − 4.67517i −0.354558 − 0.935034i
\(26\) −2.84280 − 4.92387i −0.557519 − 0.965650i
\(27\) −3.28721 − 3.28721i −0.632624 − 0.632624i
\(28\) 0.662018 + 2.47069i 0.125110 + 0.466916i
\(29\) −2.52800 − 4.37863i −0.469438 − 0.813091i 0.529951 − 0.848028i \(-0.322210\pi\)
−0.999390 + 0.0349371i \(0.988877\pi\)
\(30\) 0.337842 − 4.18361i 0.0616812 − 0.763820i
\(31\) −4.41781 − 7.65187i −0.793462 − 1.37432i −0.923811 − 0.382848i \(-0.874943\pi\)
0.130350 − 0.991468i \(-0.458390\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) 7.70555 − 7.70555i 1.34136 − 1.34136i
\(34\) 2.54040 − 4.40010i 0.435675 − 0.754611i
\(35\) 3.70565 + 4.35671i 0.626369 + 0.736419i
\(36\) 0.261675 − 0.453235i 0.0436125 − 0.0755391i
\(37\) 3.79929 − 1.01802i 0.624599 − 0.167361i 0.0673815 − 0.997727i \(-0.478536\pi\)
0.557218 + 0.830366i \(0.311869\pi\)
\(38\) −0.286474 − 0.286474i −0.0464722 − 0.0464722i
\(39\) −5.33610 − 9.24240i −0.854460 − 1.47997i
\(40\) −2.19945 + 0.403007i −0.347764 + 0.0637211i
\(41\) 9.19556i 1.43610i 0.695989 + 0.718052i \(0.254966\pi\)
−0.695989 + 0.718052i \(0.745034\pi\)
\(42\) 1.24265 + 4.63762i 0.191745 + 0.715601i
\(43\) 1.30464 + 4.86897i 0.198955 + 0.742511i 0.991207 + 0.132317i \(0.0422418\pi\)
−0.792252 + 0.610194i \(0.791092\pi\)
\(44\) −5.02773 − 2.90276i −0.757958 − 0.437607i
\(45\) 0.0941950 − 1.16645i 0.0140418 − 0.173884i
\(46\) 4.19022 0.617814
\(47\) 2.74991 − 10.2628i 0.401116 − 1.49698i −0.409993 − 0.912089i \(-0.634469\pi\)
0.811109 − 0.584896i \(-0.198865\pi\)
\(48\) −1.81310 − 0.485819i −0.261698 − 0.0701219i
\(49\) 0.396155 + 0.228720i 0.0565936 + 0.0326743i
\(50\) −4.05704 + 2.92241i −0.573751 + 0.413291i
\(51\) 4.76848 − 8.25925i 0.667721 − 1.15653i
\(52\) −4.02032 + 4.02032i −0.557519 + 0.557519i
\(53\) 9.21929 + 2.47030i 1.26637 + 0.339322i 0.828638 − 0.559785i \(-0.189116\pi\)
0.437729 + 0.899107i \(0.355783\pi\)
\(54\) −2.32441 + 4.02599i −0.316312 + 0.547868i
\(55\) −12.9394 − 1.04490i −1.74475 − 0.140895i
\(56\) 2.21516 − 1.27892i 0.296013 − 0.170903i
\(57\) −0.537728 − 0.537728i −0.0712239 − 0.0712239i
\(58\) −3.57514 + 3.57514i −0.469438 + 0.469438i
\(59\) 3.36983 − 5.83672i 0.438715 − 0.759876i −0.558876 − 0.829251i \(-0.688767\pi\)
0.997591 + 0.0693750i \(0.0221005\pi\)
\(60\) −4.12850 + 0.756469i −0.532987 + 0.0976597i
\(61\) 1.29350i 0.165616i 0.996566 + 0.0828081i \(0.0263889\pi\)
−0.996566 + 0.0828081i \(0.973611\pi\)
\(62\) −6.24772 + 6.24772i −0.793462 + 0.793462i
\(63\) 0.346468 + 1.29303i 0.0436508 + 0.162907i
\(64\) 1.00000i 0.125000i
\(65\) −4.26824 + 11.9755i −0.529410 + 1.48538i
\(66\) −9.43734 − 5.44865i −1.16166 − 0.670682i
\(67\) 3.91492 − 3.91492i 0.478283 − 0.478283i −0.426299 − 0.904582i \(-0.640183\pi\)
0.904582 + 0.426299i \(0.140183\pi\)
\(68\) −4.90768 − 1.31501i −0.595143 − 0.159468i
\(69\) 7.86528 0.946869
\(70\) 3.24917 − 4.70698i 0.388350 − 0.562592i
\(71\) − 8.51116i − 1.01009i −0.863093 − 0.505045i \(-0.831476\pi\)
0.863093 − 0.505045i \(-0.168524\pi\)
\(72\) −0.505518 − 0.135453i −0.0595758 − 0.0159633i
\(73\) 1.57973 − 5.89563i 0.184893 − 0.690031i −0.809760 − 0.586761i \(-0.800403\pi\)
0.994653 − 0.103270i \(-0.0329306\pi\)
\(74\) −1.96666 − 3.40635i −0.228619 − 0.395980i
\(75\) −7.61530 + 5.48553i −0.879339 + 0.633414i
\(76\) −0.202568 + 0.350857i −0.0232361 + 0.0402461i
\(77\) 14.3436 − 3.84336i 1.63461 − 0.437992i
\(78\) −7.54638 + 7.54638i −0.854460 + 0.854460i
\(79\) 2.06780 + 8.64432i 0.232645 + 0.972562i
\(80\) 0.958535 + 2.02020i 0.107167 + 0.225865i
\(81\) −5.14808 + 8.91673i −0.572009 + 0.990748i
\(82\) 8.88223 − 2.37999i 0.980878 − 0.262825i
\(83\) 2.42735 − 9.05899i 0.266436 − 0.994353i −0.694929 − 0.719078i \(-0.744564\pi\)
0.961365 − 0.275275i \(-0.0887690\pi\)
\(84\) 4.15798 − 2.40061i 0.453673 − 0.261928i
\(85\) −11.1750 + 2.04760i −1.21210 + 0.222093i
\(86\) 4.36540 − 2.52037i 0.470733 − 0.271778i
\(87\) −6.71074 + 6.71074i −0.719467 + 0.719467i
\(88\) −1.50258 + 5.60770i −0.160175 + 0.597783i
\(89\) 7.27543i 0.771194i 0.922667 + 0.385597i \(0.126005\pi\)
−0.922667 + 0.385597i \(0.873995\pi\)
\(90\) −1.15108 + 0.210914i −0.121335 + 0.0222323i
\(91\) − 14.5429i − 1.52451i
\(92\) −1.08451 − 4.04744i −0.113068 − 0.421974i
\(93\) −11.7273 + 11.7273i −1.21607 + 1.21607i
\(94\) −10.6248 −1.09587
\(95\) −0.0729181 + 0.902970i −0.00748124 + 0.0926428i
\(96\) 1.87706i 0.191577i
\(97\) −8.69166 + 8.69166i −0.882504 + 0.882504i −0.993789 − 0.111284i \(-0.964504\pi\)
0.111284 + 0.993789i \(0.464504\pi\)
\(98\) 0.118394 − 0.441853i 0.0119596 − 0.0446339i
\(99\) −2.63126 − 1.51916i −0.264452 − 0.152681i
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.103.5 ✓ 160
5.2 odd 4 inner 790.2.k.a.577.25 yes 160
79.56 odd 6 inner 790.2.k.a.293.25 yes 160
395.372 even 12 inner 790.2.k.a.767.5 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.5 ✓ 160 1.1 even 1 trivial
790.2.k.a.293.25 yes 160 79.56 odd 6 inner
790.2.k.a.577.25 yes 160 5.2 odd 4 inner
790.2.k.a.767.5 yes 160 395.372 even 12 inner