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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5041,2,Mod(1,5041)] 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("5041.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5041, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5041 = 71^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5041.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,-2,-2,46,10,4,-25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.2525876589\)
Analytic rank: \(1\)
Dimension: \(60\)
Twist minimal: no (minimal twist has level 71)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 5041.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.21659 q^{2} +0.696884 q^{3} +2.91329 q^{4} -2.48557 q^{5} -1.54471 q^{6} -2.20390 q^{7} -2.02439 q^{8} -2.51435 q^{9} +5.50950 q^{10} -0.245392 q^{11} +2.03022 q^{12} +0.770562 q^{13} +4.88516 q^{14} -1.73215 q^{15} -1.33933 q^{16} -7.77978 q^{17} +5.57330 q^{18} -1.96751 q^{19} -7.24118 q^{20} -1.53587 q^{21} +0.543934 q^{22} +6.72225 q^{23} -1.41076 q^{24} +1.17806 q^{25} -1.70802 q^{26} -3.84287 q^{27} -6.42061 q^{28} +8.28948 q^{29} +3.83948 q^{30} +7.49877 q^{31} +7.01753 q^{32} -0.171010 q^{33} +17.2446 q^{34} +5.47796 q^{35} -7.32503 q^{36} +3.77539 q^{37} +4.36116 q^{38} +0.536992 q^{39} +5.03176 q^{40} +4.37011 q^{41} +3.40439 q^{42} +10.2178 q^{43} -0.714897 q^{44} +6.24960 q^{45} -14.9005 q^{46} +2.82818 q^{47} -0.933357 q^{48} -2.14280 q^{49} -2.61128 q^{50} -5.42161 q^{51} +2.24487 q^{52} -4.24687 q^{53} +8.51807 q^{54} +0.609939 q^{55} +4.46156 q^{56} -1.37112 q^{57} -18.3744 q^{58} -11.4233 q^{59} -5.04627 q^{60} +11.4803 q^{61} -16.6217 q^{62} +5.54139 q^{63} -12.8763 q^{64} -1.91528 q^{65} +0.379059 q^{66} +2.85691 q^{67} -22.6648 q^{68} +4.68463 q^{69} -12.1424 q^{70} +5.09003 q^{72} -2.08734 q^{73} -8.36851 q^{74} +0.820970 q^{75} -5.73191 q^{76} +0.540820 q^{77} -1.19029 q^{78} +2.72920 q^{79} +3.32900 q^{80} +4.86502 q^{81} -9.68675 q^{82} +7.88369 q^{83} -4.47442 q^{84} +19.3372 q^{85} -22.6486 q^{86} +5.77681 q^{87} +0.496769 q^{88} -12.8895 q^{89} -13.8528 q^{90} -1.69824 q^{91} +19.5839 q^{92} +5.22578 q^{93} -6.26894 q^{94} +4.89037 q^{95} +4.89040 q^{96} +1.48038 q^{97} +4.74973 q^{98} +0.617002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 2 q^{2} - 2 q^{3} + 46 q^{4} + 10 q^{5} + 4 q^{6} - 25 q^{7} + 9 q^{8} + 32 q^{9} + 18 q^{10} - 53 q^{11} + 44 q^{12} - 13 q^{13} - 49 q^{14} - 20 q^{15} + 26 q^{16} - 33 q^{17} - 48 q^{18} + 3 q^{19}+ \cdots - 135 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.21659 −1.56737 −0.783684 − 0.621159i \(-0.786662\pi\)
−0.783684 + 0.621159i \(0.786662\pi\)
\(3\) 0.696884 0.402346 0.201173 − 0.979556i \(-0.435525\pi\)
0.201173 + 0.979556i \(0.435525\pi\)
\(4\) 2.91329 1.45664
\(5\) −2.48557 −1.11158 −0.555790 − 0.831322i \(-0.687584\pi\)
−0.555790 + 0.831322i \(0.687584\pi\)
\(6\) −1.54471 −0.630625
\(7\) −2.20390 −0.832998 −0.416499 − 0.909136i \(-0.636743\pi\)
−0.416499 + 0.909136i \(0.636743\pi\)
\(8\) −2.02439 −0.715729
\(9\) −2.51435 −0.838117
\(10\) 5.50950 1.74226
\(11\) −0.245392 −0.0739884 −0.0369942 − 0.999315i \(-0.511778\pi\)
−0.0369942 + 0.999315i \(0.511778\pi\)
\(12\) 2.03022 0.586075
\(13\) 0.770562 0.213715 0.106858 − 0.994274i \(-0.465921\pi\)
0.106858 + 0.994274i \(0.465921\pi\)
\(14\) 4.88516 1.30561
\(15\) −1.73215 −0.447240
\(16\) −1.33933 −0.334832
\(17\) −7.77978 −1.88687 −0.943437 − 0.331551i \(-0.892428\pi\)
−0.943437 + 0.331551i \(0.892428\pi\)
\(18\) 5.57330 1.31364
\(19\) −1.96751 −0.451377 −0.225688 − 0.974200i \(-0.572463\pi\)
−0.225688 + 0.974200i \(0.572463\pi\)
\(20\) −7.24118 −1.61918
\(21\) −1.53587 −0.335154
\(22\) 0.543934 0.115967
\(23\) 6.72225 1.40169 0.700843 − 0.713315i \(-0.252807\pi\)
0.700843 + 0.713315i \(0.252807\pi\)
\(24\) −1.41076 −0.287971
\(25\) 1.17806 0.235612
\(26\) −1.70802 −0.334971
\(27\) −3.84287 −0.739560
\(28\) −6.42061 −1.21338
\(29\) 8.28948 1.53932 0.769659 − 0.638455i \(-0.220426\pi\)
0.769659 + 0.638455i \(0.220426\pi\)
\(30\) 3.83948 0.700991
\(31\) 7.49877 1.34682 0.673410 − 0.739270i \(-0.264829\pi\)
0.673410 + 0.739270i \(0.264829\pi\)
\(32\) 7.01753 1.24053
\(33\) −0.171010 −0.0297690
\(34\) 17.2446 2.95743
\(35\) 5.47796 0.925944
\(36\) −7.32503 −1.22084
\(37\) 3.77539 0.620671 0.310335 − 0.950627i \(-0.399559\pi\)
0.310335 + 0.950627i \(0.399559\pi\)
\(38\) 4.36116 0.707474
\(39\) 0.536992 0.0859876
\(40\) 5.03176 0.795591
\(41\) 4.37011 0.682496 0.341248 − 0.939973i \(-0.389150\pi\)
0.341248 + 0.939973i \(0.389150\pi\)
\(42\) 3.40439 0.525309
\(43\) 10.2178 1.55819 0.779097 − 0.626903i \(-0.215678\pi\)
0.779097 + 0.626903i \(0.215678\pi\)
\(44\) −0.714897 −0.107775
\(45\) 6.24960 0.931635
\(46\) −14.9005 −2.19696
\(47\) 2.82818 0.412533 0.206267 − 0.978496i \(-0.433869\pi\)
0.206267 + 0.978496i \(0.433869\pi\)
\(48\) −0.933357 −0.134719
\(49\) −2.14280 −0.306115
\(50\) −2.61128 −0.369290
\(51\) −5.42161 −0.759177
\(52\) 2.24487 0.311307
\(53\) −4.24687 −0.583352 −0.291676 − 0.956517i \(-0.594213\pi\)
−0.291676 + 0.956517i \(0.594213\pi\)
\(54\) 8.51807 1.15916
\(55\) 0.609939 0.0822441
\(56\) 4.46156 0.596201
\(57\) −1.37112 −0.181610
\(58\) −18.3744 −2.41268
\(59\) −11.4233 −1.48719 −0.743595 − 0.668630i \(-0.766881\pi\)
−0.743595 + 0.668630i \(0.766881\pi\)
\(60\) −5.04627 −0.651470
\(61\) 11.4803 1.46990 0.734951 − 0.678120i \(-0.237205\pi\)
0.734951 + 0.678120i \(0.237205\pi\)
\(62\) −16.6217 −2.11096
\(63\) 5.54139 0.698150
\(64\) −12.8763 −1.60954
\(65\) −1.91528 −0.237562
\(66\) 0.379059 0.0466590
\(67\) 2.85691 0.349028 0.174514 − 0.984655i \(-0.444165\pi\)
0.174514 + 0.984655i \(0.444165\pi\)
\(68\) −22.6648 −2.74851
\(69\) 4.68463 0.563963
\(70\) −12.1424 −1.45130
\(71\) 0 0
\(72\) 5.09003 0.599865
\(73\) −2.08734 −0.244304 −0.122152 − 0.992511i \(-0.538980\pi\)
−0.122152 + 0.992511i \(0.538980\pi\)
\(74\) −8.36851 −0.972819
\(75\) 0.820970 0.0947975
\(76\) −5.73191 −0.657495
\(77\) 0.540820 0.0616322
\(78\) −1.19029 −0.134774
\(79\) 2.72920 0.307060 0.153530 − 0.988144i \(-0.450936\pi\)
0.153530 + 0.988144i \(0.450936\pi\)
\(80\) 3.32900 0.372193
\(81\) 4.86502 0.540558
\(82\) −9.68675 −1.06972
\(83\) 7.88369 0.865348 0.432674 − 0.901550i \(-0.357570\pi\)
0.432674 + 0.901550i \(0.357570\pi\)
\(84\) −4.47442 −0.488199
\(85\) 19.3372 2.09741
\(86\) −22.6486 −2.44226
\(87\) 5.77681 0.619339
\(88\) 0.496769 0.0529557
\(89\) −12.8895 −1.36628 −0.683142 − 0.730286i \(-0.739387\pi\)
−0.683142 + 0.730286i \(0.739387\pi\)
\(90\) −13.8528 −1.46022
\(91\) −1.69824 −0.178024
\(92\) 19.5839 2.04176
\(93\) 5.22578 0.541888
\(94\) −6.26894 −0.646591
\(95\) 4.89037 0.501742
\(96\) 4.89040 0.499125
\(97\) 1.48038 0.150310 0.0751549 − 0.997172i \(-0.476055\pi\)
0.0751549 + 0.997172i \(0.476055\pi\)
\(98\) 4.74973 0.479795
\(99\) 0.617002 0.0620110
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 5041.2.a.s.1.5 60
71.27 even 35 71.2.g.a.19.5 yes 120
71.50 even 35 71.2.g.a.15.5 ✓ 120
71.70 odd 2 5041.2.a.t.1.5 60
213.50 odd 70 639.2.v.a.370.1 120
213.98 odd 70 639.2.v.a.19.1 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.15.5 ✓ 120 71.50 even 35
71.2.g.a.19.5 yes 120 71.27 even 35
639.2.v.a.19.1 120 213.98 odd 70
639.2.v.a.370.1 120 213.50 odd 70
5041.2.a.s.1.5 60 1.1 even 1 trivial
5041.2.a.t.1.5 60 71.70 odd 2