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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [71,2,Mod(2,71)] 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("71.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(71, base_ring=CyclotomicField(70)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.g (of order \(35\), degree \(24\), 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: \(0.566937854351\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(5\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label 12.4
Character \(\chi\) \(=\) 71.12
Dual form 71.2.g.a.6.4

$q$-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+(0.0749867 + 0.0784299i) q^{2} +(1.72079 + 0.645822i) q^{3} +(0.0892014 - 1.98622i) q^{4} +(-1.81548 - 1.31902i) q^{5} +(0.0783843 + 0.183389i) q^{6} +(-0.608682 + 4.49347i) q^{7} +(0.325899 - 0.284729i) q^{8} +(0.284802 + 0.248824i) q^{9} +(-0.0326858 - 0.241297i) q^{10} +(-2.41871 + 1.44511i) q^{11} +(1.43624 - 3.36026i) q^{12} +(2.26569 + 1.35368i) q^{13} +(-0.398066 + 0.289212i) q^{14} +(-2.27219 - 3.44222i) q^{15} +(-3.91367 - 0.352237i) q^{16} +(2.02728 - 6.23933i) q^{17} +(0.00184111 + 0.0409955i) q^{18} +(-1.83875 + 2.78559i) q^{19} +(-2.78181 + 3.48828i) q^{20} +(-3.94939 + 7.33920i) q^{21} +(-0.294711 - 0.0813351i) q^{22} +(0.664870 + 0.320185i) q^{23} +(0.744686 - 0.279485i) q^{24} +(0.0110529 + 0.0340172i) q^{25} +(0.0637269 + 0.279206i) q^{26} +(-2.28350 - 4.24346i) q^{27} +(8.87075 + 1.60980i) q^{28} +(1.70914 - 0.471693i) q^{29} +(0.0995892 - 0.436329i) q^{30} +(6.55948 - 0.590364i) q^{31} +(-0.805490 - 1.01005i) q^{32} +(-5.09537 + 0.924673i) q^{33} +(0.641369 - 0.308867i) q^{34} +(7.03203 - 7.35493i) q^{35} +(0.519626 - 0.543486i) q^{36} +(1.96340 - 0.945524i) q^{37} +(-0.356355 + 0.0646690i) q^{38} +(3.02452 + 3.79263i) q^{39} +(-0.967225 + 0.0870518i) q^{40} +(-0.206484 + 0.904666i) q^{41} +(-0.871765 + 0.240592i) q^{42} +(9.88571 + 1.79399i) q^{43} +(2.65456 + 4.93301i) q^{44} +(-0.188847 - 0.827395i) q^{45} +(0.0247444 + 0.0761553i) q^{46} +(-11.8783 + 4.45800i) q^{47} +(-6.50711 - 3.13366i) q^{48} +(-13.0731 - 3.60794i) q^{49} +(-0.00183915 + 0.00341771i) q^{50} +(7.51801 - 9.42728i) q^{51} +(2.89082 - 4.37941i) q^{52} +(0.533306 + 11.8750i) q^{53} +(0.161582 - 0.497298i) q^{54} +(6.29725 + 0.566762i) q^{55} +(1.08105 + 1.63773i) q^{56} +(-4.96309 + 3.60590i) q^{57} +(0.165158 + 0.0986771i) q^{58} +(1.24743 - 2.91850i) q^{59} +(-7.03971 + 4.20603i) q^{60} +(-0.355480 - 2.62426i) q^{61} +(0.538176 + 0.470190i) q^{62} +(-1.29144 + 1.12830i) q^{63} +(-1.03612 + 7.64895i) q^{64} +(-2.32776 - 5.44607i) q^{65} +(-0.454607 - 0.330291i) q^{66} +(0.0241967 - 0.538781i) q^{67} +(-12.2119 - 4.58319i) q^{68} +(0.937317 + 0.980356i) q^{69} +1.10416 q^{70} +(5.65214 - 6.24927i) q^{71} +0.163664 q^{72} +(-3.71196 - 3.88240i) q^{73} +(0.221386 + 0.0830876i) q^{74} +(-0.00294945 + 0.0656745i) q^{75} +(5.36878 + 3.90065i) q^{76} +(-5.02135 - 11.7480i) q^{77} +(-0.0706568 + 0.521610i) q^{78} +(-6.98479 + 6.10243i) q^{79} +(6.64057 + 5.80169i) q^{80} +(-1.34120 - 9.90111i) q^{81} +(-0.0864365 + 0.0516434i) q^{82} +(-0.514996 + 1.20489i) q^{83} +(14.2250 + 8.49905i) q^{84} +(-11.9103 + 8.65333i) q^{85} +(0.600594 + 0.909861i) q^{86} +(3.24569 + 0.292118i) q^{87} +(-0.376789 + 1.15964i) q^{88} +(-0.0743855 - 1.65632i) q^{89} +(0.0507315 - 0.0768549i) q^{90} +(-7.46183 + 9.35683i) q^{91} +(0.695266 - 1.29202i) q^{92} +(11.6687 + 3.22036i) q^{93} +(-1.24035 - 0.597323i) q^{94} +(7.01246 - 2.63182i) q^{95} +(-0.733761 - 2.25829i) q^{96} +(-2.13103 - 9.33664i) q^{97} +(-0.697336 - 1.29587i) q^{98} +(-1.04843 - 0.190263i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 22 q^{2} - 20 q^{3} - 18 q^{4} - 20 q^{5} - 20 q^{6} - 27 q^{7} - 27 q^{8} - 11 q^{9} - 8 q^{10} - 27 q^{11} + 3 q^{12} - 31 q^{13} + 2 q^{14} + 12 q^{15} + 30 q^{16} + 9 q^{17} + 27 q^{18} - 31 q^{19}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{19}{35}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0749867 + 0.0784299i 0.0530236 + 0.0554583i 0.748775 − 0.662825i \(-0.230643\pi\)
−0.695751 + 0.718283i \(0.744928\pi\)
\(3\) 1.72079 + 0.645822i 0.993496 + 0.372865i 0.794585 − 0.607152i \(-0.207688\pi\)
0.198910 + 0.980018i \(0.436260\pi\)
\(4\) 0.0892014 − 1.98622i 0.0446007 − 0.993112i
\(5\) −1.81548 − 1.31902i −0.811906 − 0.589884i 0.102477 − 0.994735i \(-0.467323\pi\)
−0.914382 + 0.404851i \(0.867323\pi\)
\(6\) 0.0783843 + 0.183389i 0.0320002 + 0.0748683i
\(7\) −0.608682 + 4.49347i −0.230060 + 1.69837i 0.395718 + 0.918372i \(0.370496\pi\)
−0.625778 + 0.780001i \(0.715218\pi\)
\(8\) 0.325899 − 0.284729i 0.115223 − 0.100667i
\(9\) 0.284802 + 0.248824i 0.0949341 + 0.0829414i
\(10\) −0.0326858 − 0.241297i −0.0103362 − 0.0763047i
\(11\) −2.41871 + 1.44511i −0.729269 + 0.435718i −0.828965 − 0.559300i \(-0.811070\pi\)
0.0996964 + 0.995018i \(0.468213\pi\)
\(12\) 1.43624 − 3.36026i 0.414608 − 0.970022i
\(13\) 2.26569 + 1.35368i 0.628388 + 0.375444i 0.791505 − 0.611162i \(-0.209298\pi\)
−0.163117 + 0.986607i \(0.552155\pi\)
\(14\) −0.398066 + 0.289212i −0.106388 + 0.0772951i
\(15\) −2.27219 − 3.44222i −0.586678 − 0.888779i
\(16\) −3.91367 − 0.352237i −0.978419 − 0.0880593i
\(17\) 2.02728 − 6.23933i 0.491688 − 1.51326i −0.330368 − 0.943852i \(-0.607173\pi\)
0.822056 − 0.569407i \(-0.192827\pi\)
\(18\) 0.00184111 + 0.0409955i 0.000433954 + 0.00966274i
\(19\) −1.83875 + 2.78559i −0.421838 + 0.639058i −0.981902 − 0.189392i \(-0.939348\pi\)
0.560063 + 0.828450i \(0.310777\pi\)
\(20\) −2.78181 + 3.48828i −0.622032 + 0.780004i
\(21\) −3.94939 + 7.33920i −0.861828 + 1.60155i
\(22\) −0.294711 − 0.0813351i −0.0628326 − 0.0173407i
\(23\) 0.664870 + 0.320185i 0.138635 + 0.0667631i 0.501915 − 0.864917i \(-0.332629\pi\)
−0.363280 + 0.931680i \(0.618343\pi\)
\(24\) 0.744686 − 0.279485i 0.152008 − 0.0570497i
\(25\) 0.0110529 + 0.0340172i 0.00221057 + 0.00680345i
\(26\) 0.0637269 + 0.279206i 0.0124979 + 0.0547568i
\(27\) −2.28350 − 4.24346i −0.439460 − 0.816654i
\(28\) 8.87075 + 1.60980i 1.67641 + 0.304224i
\(29\) 1.70914 − 0.471693i 0.317379 − 0.0875911i −0.103718 − 0.994607i \(-0.533074\pi\)
0.421097 + 0.907016i \(0.361645\pi\)
\(30\) 0.0995892 − 0.436329i 0.0181824 − 0.0796624i
\(31\) 6.55948 − 0.590364i 1.17812 − 0.106032i 0.516778 − 0.856120i \(-0.327131\pi\)
0.661339 + 0.750087i \(0.269988\pi\)
\(32\) −0.805490 − 1.01005i −0.142392 − 0.178554i
\(33\) −5.09537 + 0.924673i −0.886990 + 0.160965i
\(34\) 0.641369 − 0.308867i 0.109994 − 0.0529703i
\(35\) 7.03203 − 7.35493i 1.18863 − 1.24321i
\(36\) 0.519626 − 0.543486i 0.0866043 − 0.0905810i
\(37\) 1.96340 − 0.945524i 0.322781 − 0.155443i −0.265475 − 0.964118i \(-0.585529\pi\)
0.588256 + 0.808675i \(0.299815\pi\)
\(38\) −0.356355 + 0.0646690i −0.0578085 + 0.0104907i
\(39\) 3.02452 + 3.79263i 0.484311 + 0.607307i
\(40\) −0.967225 + 0.0870518i −0.152932 + 0.0137641i
\(41\) −0.206484 + 0.904666i −0.0322474 + 0.141285i −0.988489 − 0.151293i \(-0.951656\pi\)
0.956241 + 0.292579i \(0.0945133\pi\)
\(42\) −0.871765 + 0.240592i −0.134516 + 0.0371241i
\(43\) 9.88571 + 1.79399i 1.50756 + 0.273581i 0.868543 − 0.495614i \(-0.165057\pi\)
0.639014 + 0.769195i \(0.279343\pi\)
\(44\) 2.65456 + 4.93301i 0.400191 + 0.743679i
\(45\) −0.188847 − 0.827395i −0.0281517 − 0.123341i
\(46\) 0.0247444 + 0.0761553i 0.00364836 + 0.0112285i
\(47\) −11.8783 + 4.45800i −1.73263 + 0.650266i −0.999553 − 0.0298840i \(-0.990486\pi\)
−0.733073 + 0.680150i \(0.761915\pi\)
\(48\) −6.50711 − 3.13366i −0.939221 − 0.452305i
\(49\) −13.0731 − 3.60794i −1.86758 − 0.515420i
\(50\) −0.00183915 + 0.00341771i −0.000260095 + 0.000483338i
\(51\) 7.51801 − 9.42728i 1.05273 − 1.32008i
\(52\) 2.89082 − 4.37941i 0.400885 − 0.607315i
\(53\) 0.533306 + 11.8750i 0.0732552 + 1.63115i 0.613534 + 0.789668i \(0.289747\pi\)
−0.540279 + 0.841486i \(0.681681\pi\)
\(54\) 0.161582 − 0.497298i 0.0219885 − 0.0676737i
\(55\) 6.29725 + 0.566762i 0.849120 + 0.0764222i
\(56\) 1.08105 + 1.63773i 0.144462 + 0.218850i
\(57\) −4.96309 + 3.60590i −0.657377 + 0.477613i
\(58\) 0.165158 + 0.0986771i 0.0216863 + 0.0129569i
\(59\) 1.24743 − 2.91850i 0.162401 − 0.379956i −0.818488 − 0.574524i \(-0.805187\pi\)
0.980889 + 0.194567i \(0.0623302\pi\)
\(60\) −7.03971 + 4.20603i −0.908823 + 0.542996i
\(61\) −0.355480 − 2.62426i −0.0455146 − 0.336002i −0.999362 − 0.0357109i \(-0.988630\pi\)
0.953848 − 0.300291i \(-0.0970838\pi\)
\(62\) 0.538176 + 0.470190i 0.0683484 + 0.0597142i
\(63\) −1.29144 + 1.12830i −0.162706 + 0.142152i
\(64\) −1.03612 + 7.64895i −0.129515 + 0.956118i
\(65\) −2.32776 − 5.44607i −0.288723 − 0.675501i
\(66\) −0.454607 − 0.330291i −0.0559582 − 0.0406560i
\(67\) 0.0241967 − 0.538781i 0.00295610 − 0.0658226i −0.996973 − 0.0777518i \(-0.975226\pi\)
0.999929 + 0.0119292i \(0.00379726\pi\)
\(68\) −12.2119 − 4.58319i −1.48091 − 0.555793i
\(69\) 0.937317 + 0.980356i 0.112840 + 0.118021i
\(70\) 1.10416 0.131972
\(71\) 5.65214 − 6.24927i 0.670785 − 0.741652i
\(72\) 0.163664 0.0192880
\(73\) −3.71196 − 3.88240i −0.434452 − 0.454401i 0.468932 − 0.883234i \(-0.344639\pi\)
−0.903383 + 0.428834i \(0.858925\pi\)
\(74\) 0.221386 + 0.0830876i 0.0257356 + 0.00965874i
\(75\) −0.00294945 + 0.0656745i −0.000340573 + 0.00758344i
\(76\) 5.36878 + 3.90065i 0.615842 + 0.447435i
\(77\) −5.02135 − 11.7480i −0.572236 − 1.33881i
\(78\) −0.0706568 + 0.521610i −0.00800031 + 0.0590607i
\(79\) −6.98479 + 6.10243i −0.785850 + 0.686577i −0.954206 − 0.299150i \(-0.903297\pi\)
0.168356 + 0.985726i \(0.446154\pi\)
\(80\) 6.64057 + 5.80169i 0.742439 + 0.648649i
\(81\) −1.34120 − 9.90111i −0.149022 − 1.10012i
\(82\) −0.0864365 + 0.0516434i −0.00954531 + 0.00570306i
\(83\) −0.514996 + 1.20489i −0.0565281 + 0.132254i −0.945411 − 0.325880i \(-0.894339\pi\)
0.888883 + 0.458134i \(0.151482\pi\)
\(84\) 14.2250 + 8.49905i 1.55208 + 0.927322i
\(85\) −11.9103 + 8.65333i −1.29185 + 0.938585i
\(86\) 0.600594 + 0.909861i 0.0647637 + 0.0981128i
\(87\) 3.24569 + 0.292118i 0.347975 + 0.0313183i
\(88\) −0.376789 + 1.15964i −0.0401659 + 0.123618i
\(89\) −0.0743855 − 1.65632i −0.00788485 − 0.175570i −0.999071 − 0.0430936i \(-0.986279\pi\)
0.991186 − 0.132476i \(-0.0422928\pi\)
\(90\) 0.0507315 − 0.0768549i 0.00534757 − 0.00810122i
\(91\) −7.46183 + 9.35683i −0.782212 + 0.980863i
\(92\) 0.695266 − 1.29202i 0.0724865 − 0.134702i
\(93\) 11.6687 + 3.22036i 1.20999 + 0.333936i
\(94\) −1.24035 − 0.597323i −0.127933 − 0.0616092i
\(95\) 7.01246 − 2.63182i 0.719463 − 0.270019i
\(96\) −0.733761 − 2.25829i −0.0748892 − 0.230485i
\(97\) −2.13103 − 9.33664i −0.216373 − 0.947992i −0.960133 − 0.279545i \(-0.909816\pi\)
0.743760 − 0.668447i \(-0.233041\pi\)
\(98\) −0.697336 − 1.29587i −0.0704415 − 0.130902i
\(99\) −1.04843 − 0.190263i −0.105372 − 0.0191221i
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 71.2.g.a.12.4 yes 120
3.2 odd 2 639.2.v.a.154.2 120
71.6 even 35 inner 71.2.g.a.6.4 ✓ 120
71.19 even 35 5041.2.a.s.1.32 60
71.52 odd 70 5041.2.a.t.1.32 60
213.77 odd 70 639.2.v.a.361.2 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.6.4 ✓ 120 71.6 even 35 inner
71.2.g.a.12.4 yes 120 1.1 even 1 trivial
639.2.v.a.154.2 120 3.2 odd 2
639.2.v.a.361.2 120 213.77 odd 70
5041.2.a.s.1.32 60 71.19 even 35
5041.2.a.t.1.32 60 71.52 odd 70