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Properties

Label 58.22.a.b.1.8
Level $58$
Weight $22$
Character 58.1
Self dual yes
Analytic conductor $162.097$
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] = [58,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12288] 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: \(162.096859686\)
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: \( x^{12} - 2 x^{11} - 85606746065 x^{10} + 168391612240800 x^{9} + \cdots - 43\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{32}\cdot 3^{8}\cdot 5^{3}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(56994.0\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +60416.0 q^{3} +1.04858e6 q^{4} -4.59497e6 q^{5} -6.18660e7 q^{6} +3.08972e7 q^{7} -1.07374e9 q^{8} -6.81026e9 q^{9} +4.70525e9 q^{10} +1.02211e11 q^{11} +6.33507e10 q^{12} +5.47242e11 q^{13} -3.16387e10 q^{14} -2.77610e11 q^{15} +1.09951e12 q^{16} -5.82303e12 q^{17} +6.97371e12 q^{18} +4.16186e13 q^{19} -4.81818e12 q^{20} +1.86668e12 q^{21} -1.04664e14 q^{22} +7.91002e13 q^{23} -6.48712e13 q^{24} -4.55723e14 q^{25} -5.60376e14 q^{26} -1.04342e15 q^{27} +3.23981e13 q^{28} +4.20707e14 q^{29} +2.84272e14 q^{30} -5.33391e14 q^{31} -1.12590e15 q^{32} +6.17519e15 q^{33} +5.96278e15 q^{34} -1.41972e14 q^{35} -7.14108e15 q^{36} +3.20563e16 q^{37} -4.26174e16 q^{38} +3.30621e16 q^{39} +4.93382e15 q^{40} -6.23722e16 q^{41} -1.91148e15 q^{42} -7.65752e16 q^{43} +1.07176e17 q^{44} +3.12930e16 q^{45} -8.09986e16 q^{46} +4.43495e17 q^{47} +6.64281e16 q^{48} -5.57591e17 q^{49} +4.66661e17 q^{50} -3.51804e17 q^{51} +5.73825e17 q^{52} -2.39141e17 q^{53} +1.06846e18 q^{54} -4.69658e17 q^{55} -3.31756e16 q^{56} +2.51443e18 q^{57} -4.30804e17 q^{58} +5.79413e18 q^{59} -2.91095e17 q^{60} +4.17489e18 q^{61} +5.46192e17 q^{62} -2.10418e17 q^{63} +1.15292e18 q^{64} -2.51456e18 q^{65} -6.32340e18 q^{66} -1.52561e19 q^{67} -6.10588e18 q^{68} +4.77892e18 q^{69} +1.45379e17 q^{70} +9.38999e17 q^{71} +7.31247e18 q^{72} +4.95806e19 q^{73} -3.28257e19 q^{74} -2.75330e19 q^{75} +4.36402e19 q^{76} +3.15804e18 q^{77} -3.38556e19 q^{78} -9.35188e19 q^{79} -5.05223e18 q^{80} +8.19847e18 q^{81} +6.38691e19 q^{82} -1.14115e20 q^{83} +1.95736e18 q^{84} +2.67567e19 q^{85} +7.84130e19 q^{86} +2.54174e19 q^{87} -1.09749e20 q^{88} +4.85318e17 q^{89} -3.20440e19 q^{90} +1.69082e19 q^{91} +8.29426e19 q^{92} -3.22253e19 q^{93} -4.54139e20 q^{94} -1.91236e20 q^{95} -6.80223e19 q^{96} -9.83540e20 q^{97} +5.70973e20 q^{98} -6.96086e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12288 q^{2} + 41066 q^{3} + 12582912 q^{4} + 32000498 q^{5} - 42051584 q^{6} + 442380608 q^{7} - 12884901888 q^{8} + 45829788394 q^{9} - 32768509952 q^{10} - 75612722022 q^{11} + 43060822016 q^{12}+ \cdots - 77\!\cdots\!40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) 60416.0 0.590716 0.295358 − 0.955387i \(-0.404561\pi\)
0.295358 + 0.955387i \(0.404561\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −4.59497e6 −0.210425 −0.105213 − 0.994450i \(-0.533552\pi\)
−0.105213 + 0.994450i \(0.533552\pi\)
\(6\) −6.18660e7 −0.417699
\(7\) 3.08972e7 0.0413418 0.0206709 − 0.999786i \(-0.493420\pi\)
0.0206709 + 0.999786i \(0.493420\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −6.81026e9 −0.651055
\(10\) 4.70525e9 0.148793
\(11\) 1.02211e11 1.18816 0.594081 − 0.804405i \(-0.297516\pi\)
0.594081 + 0.804405i \(0.297516\pi\)
\(12\) 6.33507e10 0.295358
\(13\) 5.47242e11 1.10097 0.550483 − 0.834846i \(-0.314444\pi\)
0.550483 + 0.834846i \(0.314444\pi\)
\(14\) −3.16387e10 −0.0292331
\(15\) −2.77610e11 −0.124302
\(16\) 1.09951e12 0.250000
\(17\) −5.82303e12 −0.700543 −0.350272 − 0.936648i \(-0.613911\pi\)
−0.350272 + 0.936648i \(0.613911\pi\)
\(18\) 6.97371e12 0.460365
\(19\) 4.16186e13 1.55731 0.778654 − 0.627454i \(-0.215903\pi\)
0.778654 + 0.627454i \(0.215903\pi\)
\(20\) −4.81818e12 −0.105213
\(21\) 1.86668e12 0.0244213
\(22\) −1.04664e14 −0.840157
\(23\) 7.91002e13 0.398140 0.199070 − 0.979985i \(-0.436208\pi\)
0.199070 + 0.979985i \(0.436208\pi\)
\(24\) −6.48712e13 −0.208850
\(25\) −4.55723e14 −0.955721
\(26\) −5.60376e14 −0.778501
\(27\) −1.04342e15 −0.975304
\(28\) 3.23981e13 0.0206709
\(29\) 4.20707e14 0.185695
\(30\) 2.84272e14 0.0878945
\(31\) −5.33391e14 −0.116882 −0.0584410 − 0.998291i \(-0.518613\pi\)
−0.0584410 + 0.998291i \(0.518613\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 6.17519e15 0.701866
\(34\) 5.96278e15 0.495359
\(35\) −1.41972e14 −0.00869937
\(36\) −7.14108e15 −0.325527
\(37\) 3.20563e16 1.09596 0.547981 − 0.836491i \(-0.315396\pi\)
0.547981 + 0.836491i \(0.315396\pi\)
\(38\) −4.26174e16 −1.10118
\(39\) 3.30621e16 0.650358
\(40\) 4.93382e15 0.0743966
\(41\) −6.23722e16 −0.725706 −0.362853 − 0.931846i \(-0.618197\pi\)
−0.362853 + 0.931846i \(0.618197\pi\)
\(42\) −1.91148e15 −0.0172684
\(43\) −7.65752e16 −0.540343 −0.270171 − 0.962812i \(-0.587080\pi\)
−0.270171 + 0.962812i \(0.587080\pi\)
\(44\) 1.07176e17 0.594081
\(45\) 3.12930e16 0.136998
\(46\) −8.09986e16 −0.281527
\(47\) 4.43495e17 1.22988 0.614938 − 0.788576i \(-0.289181\pi\)
0.614938 + 0.788576i \(0.289181\pi\)
\(48\) 6.64281e16 0.147679
\(49\) −5.57591e17 −0.998291
\(50\) 4.66661e17 0.675797
\(51\) −3.51804e17 −0.413822
\(52\) 5.73825e17 0.550483
\(53\) −2.39141e17 −0.187826 −0.0939132 − 0.995580i \(-0.529938\pi\)
−0.0939132 + 0.995580i \(0.529938\pi\)
\(54\) 1.06846e18 0.689644
\(55\) −4.69658e17 −0.250019
\(56\) −3.31756e16 −0.0146165
\(57\) 2.51443e18 0.919926
\(58\) −4.30804e17 −0.131306
\(59\) 5.79413e18 1.47585 0.737924 − 0.674884i \(-0.235806\pi\)
0.737924 + 0.674884i \(0.235806\pi\)
\(60\) −2.91095e17 −0.0621508
\(61\) 4.17489e18 0.749345 0.374672 − 0.927157i \(-0.377755\pi\)
0.374672 + 0.927157i \(0.377755\pi\)
\(62\) 5.46192e17 0.0826481
\(63\) −2.10418e17 −0.0269158
\(64\) 1.15292e18 0.125000
\(65\) −2.51456e18 −0.231671
\(66\) −6.32340e18 −0.496294
\(67\) −1.52561e19 −1.02249 −0.511243 − 0.859436i \(-0.670815\pi\)
−0.511243 + 0.859436i \(0.670815\pi\)
\(68\) −6.10588e18 −0.350272
\(69\) 4.77892e18 0.235187
\(70\) 1.45379e17 0.00615138
\(71\) 9.38999e17 0.0342336 0.0171168 − 0.999853i \(-0.494551\pi\)
0.0171168 + 0.999853i \(0.494551\pi\)
\(72\) 7.31247e18 0.230183
\(73\) 4.95806e19 1.35027 0.675137 − 0.737692i \(-0.264084\pi\)
0.675137 + 0.737692i \(0.264084\pi\)
\(74\) −3.28257e19 −0.774963
\(75\) −2.75330e19 −0.564560
\(76\) 4.36402e19 0.778654
\(77\) 3.15804e18 0.0491208
\(78\) −3.38556e19 −0.459873
\(79\) −9.35188e19 −1.11126 −0.555629 − 0.831431i \(-0.687522\pi\)
−0.555629 + 0.831431i \(0.687522\pi\)
\(80\) −5.05223e18 −0.0526063
\(81\) 8.19847e18 0.0749273
\(82\) 6.38691e19 0.513151
\(83\) −1.14115e20 −0.807277 −0.403639 − 0.914918i \(-0.632255\pi\)
−0.403639 + 0.914918i \(0.632255\pi\)
\(84\) 1.95736e18 0.0122106
\(85\) 2.67567e19 0.147412
\(86\) 7.84130e19 0.382080
\(87\) 2.54174e19 0.109693
\(88\) −1.09749e20 −0.420079
\(89\) 4.85318e17 0.00164980 0.000824901 − 1.00000i \(-0.499737\pi\)
0.000824901 1.00000i \(0.499737\pi\)
\(90\) −3.20440e19 −0.0968725
\(91\) 1.69082e19 0.0455159
\(92\) 8.29426e19 0.199070
\(93\) −3.22253e19 −0.0690441
\(94\) −4.54139e20 −0.869653
\(95\) −1.91236e20 −0.327697
\(96\) −6.80223e19 −0.104425
\(97\) −9.83540e20 −1.35422 −0.677110 − 0.735882i \(-0.736768\pi\)
−0.677110 + 0.735882i \(0.736768\pi\)
\(98\) 5.70973e20 0.705898
\(99\) −6.96086e20 −0.773559
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 58.22.a.b.1.8 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.22.a.b.1.8 ✓ 12 1.1 even 1 trivial