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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 19.5
Character \(\chi\) \(=\) 71.19
Dual form 71.2.g.a.15.5

$q$-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+(0.871177 + 2.03822i) q^{2} +(0.685685 - 0.124433i) q^{3} +(-2.01326 + 2.10571i) q^{4} +(-0.768083 - 2.36392i) q^{5} +(0.850976 + 1.28917i) q^{6} +(-1.65970 - 1.45003i) q^{7} +(-1.89530 - 0.711319i) q^{8} +(-2.35402 + 0.883480i) q^{9} +(4.14905 - 3.62491i) q^{10} +(0.236549 + 0.0652833i) q^{11} +(-1.11844 + 1.69437i) q^{12} +(-0.742793 + 0.204998i) q^{13} +(1.50960 - 4.64606i) q^{14} +(-0.820814 - 1.52533i) q^{15} +(0.0600888 + 1.33798i) q^{16} +(6.29398 + 4.57284i) q^{17} +(-3.85150 - 4.02835i) q^{18} +(-0.932339 + 1.73258i) q^{19} +(6.52408 + 3.14183i) q^{20} +(-1.31846 - 0.787745i) q^{21} +(0.0730141 + 0.539011i) q^{22} +(-1.49584 - 6.55371i) q^{23} +(-1.38809 - 0.251902i) q^{24} +(-0.953069 + 0.692445i) q^{25} +(-1.06493 - 1.33539i) q^{26} +(-3.29890 + 1.97100i) q^{27} +(6.39476 - 0.575539i) q^{28} +(1.11272 - 8.21446i) q^{29} +(2.39388 - 3.00183i) q^{30} +(-0.336431 + 7.49122i) q^{31} +(-6.32257 + 3.04479i) q^{32} +(0.170321 + 0.0153292i) q^{33} +(-3.83729 + 16.8123i) q^{34} +(-2.15298 + 5.03714i) q^{35} +(2.87892 - 6.73557i) q^{36} +(-0.840104 + 3.68073i) q^{37} +(-4.34360 - 0.390931i) q^{38} +(-0.483813 + 0.232992i) q^{39} +(-0.225749 + 5.02669i) q^{40} +(2.72472 - 3.41669i) q^{41} +(0.456983 - 3.37358i) q^{42} +(-10.1766 + 0.915913i) q^{43} +(-0.613703 + 0.366670i) q^{44} +(3.89656 + 4.88613i) q^{45} +(12.0548 - 8.75829i) q^{46} +(2.78273 + 0.504992i) q^{47} +(0.207692 + 0.909956i) q^{48} +(-0.287636 - 2.12341i) q^{49} +(-2.24165 - 1.33932i) q^{50} +(4.88470 + 2.35235i) q^{51} +(1.06377 - 1.97682i) q^{52} +(2.93485 + 3.06961i) q^{53} +(-6.89127 - 5.00680i) q^{54} +(-0.0273648 - 0.609325i) q^{55} +(2.11419 + 3.92883i) q^{56} +(-0.423700 + 1.30402i) q^{57} +(17.7123 - 4.88827i) q^{58} +(6.29307 - 9.53360i) q^{59} +(4.86441 + 1.34249i) q^{60} +(8.64549 - 7.55334i) q^{61} +(-15.5618 + 5.84046i) q^{62} +(5.18805 + 1.94711i) q^{63} +(-9.69681 - 8.47185i) q^{64} +(1.05512 + 1.59845i) q^{65} +(0.117136 + 0.360507i) q^{66} +(-1.97431 + 2.06496i) q^{67} +(-22.3005 + 4.04695i) q^{68} +(-1.84118 - 4.30765i) q^{69} -12.1424 q^{70} +(-7.80170 + 3.18332i) q^{71} +5.09003 q^{72} +(0.820375 + 1.91936i) q^{73} +(-8.23402 + 1.49426i) q^{74} +(-0.567342 + 0.593393i) q^{75} +(-1.77126 - 5.45137i) q^{76} +(-0.297936 - 0.451354i) q^{77} +(-0.896376 - 0.783140i) q^{78} +(2.55518 + 0.958974i) q^{79} +(3.11672 - 1.16973i) q^{80} +(3.66371 - 3.20089i) q^{81} +(9.33767 + 2.57703i) q^{82} +(-4.34310 + 6.57952i) q^{83} +(4.31318 - 1.19036i) q^{84} +(5.97552 - 18.3908i) q^{85} +(-10.7325 - 19.9443i) q^{86} +(-0.259176 - 5.77099i) q^{87} +(-0.401894 - 0.291993i) q^{88} +(8.90745 + 9.31646i) q^{89} +(-6.56442 + 12.1987i) q^{90} +(1.53007 + 0.736841i) q^{91} +(16.8117 + 10.0445i) q^{92} +(0.701473 + 5.17848i) q^{93} +(1.39497 + 6.11176i) q^{94} +(4.81178 + 0.873210i) q^{95} +(-3.95642 + 2.87451i) q^{96} +(0.923002 + 1.15741i) q^{97} +(4.07740 - 2.43613i) q^{98} +(-0.614518 + 0.0553076i) 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 \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{8}{35}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.871177 + 2.03822i 0.616015 + 1.44124i 0.879181 + 0.476488i \(0.158091\pi\)
−0.263166 + 0.964751i \(0.584767\pi\)
\(3\) 0.685685 − 0.124433i 0.395880 − 0.0718417i 0.0230338 − 0.999735i \(-0.492667\pi\)
0.372847 + 0.927893i \(0.378382\pi\)
\(4\) −2.01326 + 2.10571i −1.00663 + 1.05285i
\(5\) −0.768083 − 2.36392i −0.343497 − 1.05718i −0.962383 − 0.271695i \(-0.912416\pi\)
0.618886 − 0.785481i \(-0.287584\pi\)
\(6\) 0.850976 + 1.28917i 0.347409 + 0.526303i
\(7\) −1.65970 − 1.45003i −0.627307 − 0.548061i 0.284503 − 0.958675i \(-0.408172\pi\)
−0.911809 + 0.410614i \(0.865314\pi\)
\(8\) −1.89530 − 0.711319i −0.670091 − 0.251489i
\(9\) −2.35402 + 0.883480i −0.784675 + 0.294493i
\(10\) 4.14905 − 3.62491i 1.31204 − 1.14630i
\(11\) 0.236549 + 0.0652833i 0.0713221 + 0.0196837i 0.301516 − 0.953461i \(-0.402507\pi\)
−0.230194 + 0.973145i \(0.573936\pi\)
\(12\) −1.11844 + 1.69437i −0.322867 + 0.489123i
\(13\) −0.742793 + 0.204998i −0.206014 + 0.0568561i −0.367520 − 0.930015i \(-0.619793\pi\)
0.161507 + 0.986872i \(0.448365\pi\)
\(14\) 1.50960 − 4.64606i 0.403457 − 1.24171i
\(15\) −0.820814 − 1.52533i −0.211933 − 0.393838i
\(16\) 0.0600888 + 1.33798i 0.0150222 + 0.334495i
\(17\) 6.29398 + 4.57284i 1.52651 + 1.10908i 0.958137 + 0.286309i \(0.0924282\pi\)
0.568376 + 0.822769i \(0.307572\pi\)
\(18\) −3.85150 − 4.02835i −0.907807 − 0.949491i
\(19\) −0.932339 + 1.73258i −0.213893 + 0.397480i −0.964822 − 0.262903i \(-0.915320\pi\)
0.750929 + 0.660383i \(0.229606\pi\)
\(20\) 6.52408 + 3.14183i 1.45883 + 0.702535i
\(21\) −1.31846 − 0.787745i −0.287712 − 0.171900i
\(22\) 0.0730141 + 0.539011i 0.0155667 + 0.114918i
\(23\) −1.49584 − 6.55371i −0.311905 − 1.36654i −0.851383 − 0.524544i \(-0.824236\pi\)
0.539479 − 0.841999i \(-0.318621\pi\)
\(24\) −1.38809 − 0.251902i −0.283343 − 0.0514192i
\(25\) −0.953069 + 0.692445i −0.190614 + 0.138489i
\(26\) −1.06493 − 1.33539i −0.208851 − 0.261891i
\(27\) −3.29890 + 1.97100i −0.634874 + 0.379320i
\(28\) 6.39476 − 0.575539i 1.20850 − 0.108767i
\(29\) 1.11272 − 8.21446i 0.206628 − 1.52539i −0.529363 − 0.848395i \(-0.677569\pi\)
0.735991 − 0.676992i \(-0.236717\pi\)
\(30\) 2.39388 − 3.00183i 0.437060 − 0.548056i
\(31\) −0.336431 + 7.49122i −0.0604248 + 1.34546i 0.708313 + 0.705899i \(0.249457\pi\)
−0.768738 + 0.639564i \(0.779115\pi\)
\(32\) −6.32257 + 3.04479i −1.11768 + 0.538248i
\(33\) 0.170321 + 0.0153292i 0.0296491 + 0.00266847i
\(34\) −3.83729 + 16.8123i −0.658090 + 2.88328i
\(35\) −2.15298 + 5.03714i −0.363919 + 0.851431i
\(36\) 2.87892 − 6.73557i 0.479820 − 1.12260i
\(37\) −0.840104 + 3.68073i −0.138112 + 0.605109i 0.857737 + 0.514089i \(0.171870\pi\)
−0.995849 + 0.0910201i \(0.970987\pi\)
\(38\) −4.34360 − 0.390931i −0.704625 − 0.0634174i
\(39\) −0.483813 + 0.232992i −0.0774721 + 0.0373086i
\(40\) −0.225749 + 5.02669i −0.0356941 + 0.794790i
\(41\) 2.72472 − 3.41669i 0.425529 − 0.533597i −0.522136 − 0.852862i \(-0.674865\pi\)
0.947665 + 0.319265i \(0.103436\pi\)
\(42\) 0.456983 − 3.37358i 0.0705140 − 0.520555i
\(43\) −10.1766 + 0.915913i −1.55192 + 0.139675i −0.832156 − 0.554542i \(-0.812894\pi\)
−0.719766 + 0.694217i \(0.755751\pi\)
\(44\) −0.613703 + 0.366670i −0.0925192 + 0.0552776i
\(45\) 3.89656 + 4.88613i 0.580865 + 0.728382i
\(46\) 12.0548 − 8.75829i 1.77738 − 1.29134i
\(47\) 2.78273 + 0.504992i 0.405904 + 0.0736606i 0.377669 − 0.925941i \(-0.376726\pi\)
0.0282346 + 0.999601i \(0.491011\pi\)
\(48\) 0.207692 + 0.909956i 0.0299777 + 0.131341i
\(49\) −0.287636 − 2.12341i −0.0410908 − 0.303344i
\(50\) −2.24165 − 1.33932i −0.317017 − 0.189409i
\(51\) 4.88470 + 2.35235i 0.683995 + 0.329395i
\(52\) 1.06377 − 1.97682i 0.147519 − 0.274136i
\(53\) 2.93485 + 3.06961i 0.403133 + 0.421644i 0.892932 − 0.450191i \(-0.148644\pi\)
−0.489799 + 0.871835i \(0.662930\pi\)
\(54\) −6.89127 − 5.00680i −0.937782 − 0.681339i
\(55\) −0.0273648 − 0.609325i −0.00368987 − 0.0821613i
\(56\) 2.11419 + 3.92883i 0.282521 + 0.525012i
\(57\) −0.423700 + 1.30402i −0.0561205 + 0.172721i
\(58\) 17.7123 − 4.88827i 2.32573 − 0.641861i
\(59\) 6.29307 − 9.53360i 0.819288 − 1.24117i −0.148377 − 0.988931i \(-0.547405\pi\)
0.967665 − 0.252238i \(-0.0811666\pi\)
\(60\) 4.86441 + 1.34249i 0.627993 + 0.173315i
\(61\) 8.64549 − 7.55334i 1.10694 − 0.967106i 0.107309 − 0.994226i \(-0.465777\pi\)
0.999632 + 0.0271201i \(0.00863366\pi\)
\(62\) −15.5618 + 5.84046i −1.97636 + 0.741739i
\(63\) 5.18805 + 1.94711i 0.653632 + 0.245312i
\(64\) −9.69681 − 8.47185i −1.21210 − 1.05898i
\(65\) 1.05512 + 1.59845i 0.130872 + 0.198263i
\(66\) 0.117136 + 0.360507i 0.0144184 + 0.0443753i
\(67\) −1.97431 + 2.06496i −0.241200 + 0.252275i −0.832303 − 0.554320i \(-0.812978\pi\)
0.591103 + 0.806596i \(0.298692\pi\)
\(68\) −22.3005 + 4.04695i −2.70434 + 0.490765i
\(69\) −1.84118 − 4.30765i −0.221652 − 0.518580i
\(70\) −12.1424 −1.45130
\(71\) −7.80170 + 3.18332i −0.925891 + 0.377791i
\(72\) 5.09003 0.599865
\(73\) 0.820375 + 1.91936i 0.0960177 + 0.224644i 0.960651 − 0.277760i \(-0.0895919\pi\)
−0.864633 + 0.502404i \(0.832449\pi\)
\(74\) −8.23402 + 1.49426i −0.957186 + 0.173704i
\(75\) −0.567342 + 0.593393i −0.0655110 + 0.0685191i
\(76\) −1.77126 − 5.45137i −0.203177 − 0.625315i
\(77\) −0.297936 − 0.451354i −0.0339530 − 0.0514366i
\(78\) −0.896376 − 0.783140i −0.101495 − 0.0886732i
\(79\) 2.55518 + 0.958974i 0.287480 + 0.107893i 0.490949 − 0.871188i \(-0.336650\pi\)
−0.203469 + 0.979081i \(0.565222\pi\)
\(80\) 3.11672 − 1.16973i 0.348460 − 0.130779i
\(81\) 3.66371 − 3.20089i 0.407079 − 0.355654i
\(82\) 9.33767 + 2.57703i 1.03117 + 0.284586i
\(83\) −4.34310 + 6.57952i −0.476718 + 0.722196i −0.990769 − 0.135563i \(-0.956716\pi\)
0.514051 + 0.857760i \(0.328144\pi\)
\(84\) 4.31318 − 1.19036i 0.470606 − 0.129879i
\(85\) 5.97552 − 18.3908i 0.648136 − 1.99476i
\(86\) −10.7325 − 19.9443i −1.15731 − 2.15065i
\(87\) −0.259176 − 5.77099i −0.0277865 − 0.618715i
\(88\) −0.401894 − 0.291993i −0.0428421 − 0.0311266i
\(89\) 8.90745 + 9.31646i 0.944188 + 0.987543i 0.999941 − 0.0109018i \(-0.00347020\pi\)
−0.0557529 + 0.998445i \(0.517756\pi\)
\(90\) −6.56442 + 12.1987i −0.691950 + 1.28586i
\(91\) 1.53007 + 0.736841i 0.160394 + 0.0772419i
\(92\) 16.8117 + 10.0445i 1.75274 + 1.04722i
\(93\) 0.701473 + 5.17848i 0.0727394 + 0.536984i
\(94\) 1.39497 + 6.11176i 0.143880 + 0.630380i
\(95\) 4.81178 + 0.873210i 0.493678 + 0.0895894i
\(96\) −3.95642 + 2.87451i −0.403800 + 0.293378i
\(97\) 0.923002 + 1.15741i 0.0937166 + 0.117517i 0.826479 − 0.562968i \(-0.190341\pi\)
−0.732762 + 0.680485i \(0.761769\pi\)
\(98\) 4.07740 − 2.43613i 0.411879 − 0.246086i
\(99\) −0.614518 + 0.0553076i −0.0617614 + 0.00555862i
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.19.5 yes 120
3.2 odd 2 639.2.v.a.19.1 120
71.15 even 35 inner 71.2.g.a.15.5 ✓ 120
71.21 odd 70 5041.2.a.t.1.5 60
71.50 even 35 5041.2.a.s.1.5 60
213.86 odd 70 639.2.v.a.370.1 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.15.5 ✓ 120 71.15 even 35 inner
71.2.g.a.19.5 yes 120 1.1 even 1 trivial
639.2.v.a.19.1 120 3.2 odd 2
639.2.v.a.370.1 120 213.86 odd 70
5041.2.a.s.1.5 60 71.50 even 35
5041.2.a.t.1.5 60 71.21 odd 70