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Properties

Label 58.20.a.d.1.12
Level $58$
Weight $20$
Character 58.1
Self dual yes
Analytic conductor $132.714$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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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] = [58,20,Mod(1,58)] 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("58.1"); S:= CuspForms(chi, 20); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 20, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,6144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(132.713684003\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 8734728401 x^{10} + 62781909607608 x^{9} + \cdots - 20\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{32}\cdot 3^{7}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(59090.8\) of defining polynomial
Character \(\chi\) \(=\) 58.1

$q$-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)
 
\(f(q)\) \(=\) \(q+512.000 q^{2} +65393.8 q^{3} +262144. q^{4} -460887. q^{5} +3.34816e7 q^{6} +1.45013e8 q^{7} +1.34218e8 q^{8} +3.11409e9 q^{9} -2.35974e8 q^{10} -6.12067e9 q^{11} +1.71426e10 q^{12} +1.54132e10 q^{13} +7.42464e10 q^{14} -3.01392e10 q^{15} +6.87195e10 q^{16} -1.08544e11 q^{17} +1.59441e12 q^{18} +2.20856e12 q^{19} -1.20819e11 q^{20} +9.48292e12 q^{21} -3.13378e12 q^{22} +1.14560e11 q^{23} +8.77701e12 q^{24} -1.88611e13 q^{25} +7.89158e12 q^{26} +1.27637e14 q^{27} +3.80142e13 q^{28} +1.45071e13 q^{29} -1.54312e13 q^{30} -1.71786e14 q^{31} +3.51844e13 q^{32} -4.00254e14 q^{33} -5.55745e13 q^{34} -6.68344e13 q^{35} +8.16339e14 q^{36} -2.22372e14 q^{37} +1.13078e15 q^{38} +1.00793e15 q^{39} -6.18592e13 q^{40} -2.58387e15 q^{41} +4.85525e15 q^{42} +4.15200e15 q^{43} -1.60450e15 q^{44} -1.43524e15 q^{45} +5.86545e13 q^{46} +4.99087e15 q^{47} +4.49383e15 q^{48} +9.62973e15 q^{49} -9.65687e15 q^{50} -7.09810e15 q^{51} +4.04049e15 q^{52} -1.79138e16 q^{53} +6.53503e16 q^{54} +2.82094e15 q^{55} +1.94632e16 q^{56} +1.44426e17 q^{57} +7.42766e15 q^{58} +5.55625e16 q^{59} -7.90080e15 q^{60} +5.81400e16 q^{61} -8.79546e16 q^{62} +4.51582e17 q^{63} +1.80144e16 q^{64} -7.10376e15 q^{65} -2.04930e17 q^{66} -2.91992e17 q^{67} -2.84541e16 q^{68} +7.49149e15 q^{69} -3.42192e16 q^{70} +4.53562e17 q^{71} +4.17966e17 q^{72} -3.55147e17 q^{73} -1.13855e17 q^{74} -1.23340e18 q^{75} +5.78960e17 q^{76} -8.87574e17 q^{77} +5.16060e17 q^{78} -1.96785e18 q^{79} -3.16719e16 q^{80} +4.72731e18 q^{81} -1.32294e18 q^{82} -5.12097e17 q^{83} +2.48589e18 q^{84} +5.00265e16 q^{85} +2.12582e18 q^{86} +9.48677e17 q^{87} -8.21503e17 q^{88} -5.40416e17 q^{89} -7.34844e17 q^{90} +2.23511e18 q^{91} +3.00311e16 q^{92} -1.12338e19 q^{93} +2.55533e18 q^{94} -1.01790e18 q^{95} +2.30084e18 q^{96} +2.57826e18 q^{97} +4.93042e18 q^{98} -1.90603e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6144 q^{2} + 75638 q^{3} + 3145728 q^{4} + 7417130 q^{5} + 38726656 q^{6} + 101600432 q^{7} + 1610612736 q^{8} + 3999078122 q^{9} + 3797570560 q^{10} + 2007416390 q^{11} + 19828047872 q^{12} + 57626609050 q^{13}+ \cdots - 28\!\cdots\!60 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 512.000 0.707107
\(3\) 65393.8 1.91816 0.959080 − 0.283137i \(-0.0913750\pi\)
0.959080 + 0.283137i \(0.0913750\pi\)
\(4\) 262144. 0.500000
\(5\) −460887. −0.105531 −0.0527654 − 0.998607i \(-0.516804\pi\)
−0.0527654 + 0.998607i \(0.516804\pi\)
\(6\) 3.34816e7 1.35634
\(7\) 1.45013e8 1.35823 0.679116 − 0.734031i \(-0.262363\pi\)
0.679116 + 0.734031i \(0.262363\pi\)
\(8\) 1.34218e8 0.353553
\(9\) 3.11409e9 2.67933
\(10\) −2.35974e8 −0.0746216
\(11\) −6.12067e9 −0.782652 −0.391326 − 0.920252i \(-0.627983\pi\)
−0.391326 + 0.920252i \(0.627983\pi\)
\(12\) 1.71426e10 0.959080
\(13\) 1.54132e10 0.403118 0.201559 − 0.979476i \(-0.435399\pi\)
0.201559 + 0.979476i \(0.435399\pi\)
\(14\) 7.42464e10 0.960415
\(15\) −3.01392e10 −0.202425
\(16\) 6.87195e10 0.250000
\(17\) −1.08544e11 −0.221994 −0.110997 − 0.993821i \(-0.535404\pi\)
−0.110997 + 0.993821i \(0.535404\pi\)
\(18\) 1.59441e12 1.89458
\(19\) 2.20856e12 1.57018 0.785090 − 0.619382i \(-0.212617\pi\)
0.785090 + 0.619382i \(0.212617\pi\)
\(20\) −1.20819e11 −0.0527654
\(21\) 9.48292e12 2.60531
\(22\) −3.13378e12 −0.553418
\(23\) 1.14560e11 0.0132622 0.00663112 − 0.999978i \(-0.497889\pi\)
0.00663112 + 0.999978i \(0.497889\pi\)
\(24\) 8.77701e12 0.678172
\(25\) −1.88611e13 −0.988863
\(26\) 7.89158e12 0.285047
\(27\) 1.27637e14 3.22123
\(28\) 3.80142e13 0.679116
\(29\) 1.45071e13 0.185695
\(30\) −1.54312e13 −0.143136
\(31\) −1.71786e14 −1.16695 −0.583476 − 0.812131i \(-0.698308\pi\)
−0.583476 + 0.812131i \(0.698308\pi\)
\(32\) 3.51844e13 0.176777
\(33\) −4.00254e14 −1.50125
\(34\) −5.55745e13 −0.156973
\(35\) −6.68344e13 −0.143335
\(36\) 8.16339e14 1.33967
\(37\) −2.22372e14 −0.281296 −0.140648 − 0.990060i \(-0.544919\pi\)
−0.140648 + 0.990060i \(0.544919\pi\)
\(38\) 1.13078e15 1.11028
\(39\) 1.00793e15 0.773244
\(40\) −6.18592e13 −0.0373108
\(41\) −2.58387e15 −1.23260 −0.616302 − 0.787510i \(-0.711370\pi\)
−0.616302 + 0.787510i \(0.711370\pi\)
\(42\) 4.85525e15 1.84223
\(43\) 4.15200e15 1.25982 0.629908 − 0.776670i \(-0.283093\pi\)
0.629908 + 0.776670i \(0.283093\pi\)
\(44\) −1.60450e15 −0.391326
\(45\) −1.43524e15 −0.282753
\(46\) 5.86545e13 0.00937782
\(47\) 4.99087e15 0.650499 0.325250 − 0.945628i \(-0.394552\pi\)
0.325250 + 0.945628i \(0.394552\pi\)
\(48\) 4.49383e15 0.479540
\(49\) 9.62973e15 0.844795
\(50\) −9.65687e15 −0.699232
\(51\) −7.09810e15 −0.425819
\(52\) 4.04049e15 0.201559
\(53\) −1.79138e16 −0.745706 −0.372853 − 0.927890i \(-0.621620\pi\)
−0.372853 + 0.927890i \(0.621620\pi\)
\(54\) 6.53503e16 2.27775
\(55\) 2.82094e15 0.0825939
\(56\) 1.94632e16 0.480208
\(57\) 1.44426e17 3.01185
\(58\) 7.42766e15 0.131306
\(59\) 5.55625e16 0.835002 0.417501 − 0.908676i \(-0.362906\pi\)
0.417501 + 0.908676i \(0.362906\pi\)
\(60\) −7.90080e15 −0.101213
\(61\) 5.81400e16 0.636564 0.318282 − 0.947996i \(-0.396894\pi\)
0.318282 + 0.947996i \(0.396894\pi\)
\(62\) −8.79546e16 −0.825159
\(63\) 4.51582e17 3.63916
\(64\) 1.80144e16 0.125000
\(65\) −7.10376e15 −0.0425414
\(66\) −2.04930e17 −1.06154
\(67\) −2.91992e17 −1.31117 −0.655586 − 0.755120i \(-0.727578\pi\)
−0.655586 + 0.755120i \(0.727578\pi\)
\(68\) −2.84541e16 −0.110997
\(69\) 7.49149e15 0.0254391
\(70\) −3.42192e16 −0.101353
\(71\) 4.53562e17 1.17404 0.587019 − 0.809573i \(-0.300301\pi\)
0.587019 + 0.809573i \(0.300301\pi\)
\(72\) 4.17966e17 0.947288
\(73\) −3.55147e17 −0.706059 −0.353030 − 0.935612i \(-0.614849\pi\)
−0.353030 + 0.935612i \(0.614849\pi\)
\(74\) −1.13855e17 −0.198906
\(75\) −1.23340e18 −1.89680
\(76\) 5.78960e17 0.785090
\(77\) −8.87574e17 −1.06302
\(78\) 5.16060e17 0.546766
\(79\) −1.96785e18 −1.84729 −0.923646 − 0.383247i \(-0.874806\pi\)
−0.923646 + 0.383247i \(0.874806\pi\)
\(80\) −3.16719e16 −0.0263827
\(81\) 4.72731e18 3.49950
\(82\) −1.32294e18 −0.871583
\(83\) −5.12097e17 −0.300684 −0.150342 − 0.988634i \(-0.548038\pi\)
−0.150342 + 0.988634i \(0.548038\pi\)
\(84\) 2.48589e18 1.30265
\(85\) 5.00265e16 0.0234272
\(86\) 2.12582e18 0.890824
\(87\) 9.48677e17 0.356193
\(88\) −8.21503e17 −0.276709
\(89\) −5.40416e17 −0.163502 −0.0817510 − 0.996653i \(-0.526051\pi\)
−0.0817510 + 0.996653i \(0.526051\pi\)
\(90\) −7.34844e17 −0.199936
\(91\) 2.23511e18 0.547528
\(92\) 3.00311e16 0.00663112
\(93\) −1.12338e19 −2.23840
\(94\) 2.55533e18 0.459973
\(95\) −1.01790e18 −0.165702
\(96\) 2.30084e18 0.339086
\(97\) 2.57826e18 0.344346 0.172173 − 0.985067i \(-0.444921\pi\)
0.172173 + 0.985067i \(0.444921\pi\)
\(98\) 4.93042e18 0.597360
\(99\) −1.90603e19 −2.09699
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 58.20.a.d.1.12 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.20.a.d.1.12 ✓ 12 1.1 even 1 trivial