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Properties

Label 58.20.a.d.1.10
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 1.10
Root \(35629.2\) 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)
 
\(f(q)\) \(=\) \(q+512.000 q^{2} +41932.2 q^{3} +262144. q^{4} +1.58401e6 q^{5} +2.14693e7 q^{6} -1.31195e8 q^{7} +1.34218e8 q^{8} +5.96051e8 q^{9} +8.11014e8 q^{10} +4.80498e9 q^{11} +1.09923e10 q^{12} +7.40899e10 q^{13} -6.71718e10 q^{14} +6.64211e10 q^{15} +6.87195e10 q^{16} +6.74902e10 q^{17} +3.05178e11 q^{18} +7.04202e11 q^{19} +4.15239e11 q^{20} -5.50130e12 q^{21} +2.46015e12 q^{22} -4.08027e12 q^{23} +5.62805e12 q^{24} -1.65644e13 q^{25} +3.79341e13 q^{26} -2.37425e13 q^{27} -3.43920e13 q^{28} +1.45071e13 q^{29} +3.40076e13 q^{30} +2.78960e14 q^{31} +3.51844e13 q^{32} +2.01484e14 q^{33} +3.45550e13 q^{34} -2.07814e14 q^{35} +1.56251e14 q^{36} -1.15290e15 q^{37} +3.60552e14 q^{38} +3.10676e15 q^{39} +2.12602e14 q^{40} +2.11794e15 q^{41} -2.81666e15 q^{42} +5.11418e15 q^{43} +1.25960e15 q^{44} +9.44151e14 q^{45} -2.08910e15 q^{46} -5.31742e15 q^{47} +2.88156e15 q^{48} +5.81321e15 q^{49} -8.48097e15 q^{50} +2.83001e15 q^{51} +1.94222e16 q^{52} -5.47577e15 q^{53} -1.21562e16 q^{54} +7.61115e15 q^{55} -1.76087e16 q^{56} +2.95288e16 q^{57} +7.42766e15 q^{58} +1.27260e17 q^{59} +1.74119e16 q^{60} +5.49821e16 q^{61} +1.42827e17 q^{62} -7.81988e16 q^{63} +1.80144e16 q^{64} +1.17359e17 q^{65} +1.03160e17 q^{66} +1.47906e17 q^{67} +1.76921e16 q^{68} -1.71095e17 q^{69} -1.06401e17 q^{70} +4.25660e17 q^{71} +8.00006e16 q^{72} -9.75824e16 q^{73} -5.90283e17 q^{74} -6.94582e17 q^{75} +1.84602e17 q^{76} -6.30389e17 q^{77} +1.59066e18 q^{78} +1.08218e17 q^{79} +1.08852e17 q^{80} -1.68834e18 q^{81} +1.08438e18 q^{82} +6.83657e16 q^{83} -1.44213e18 q^{84} +1.06905e17 q^{85} +2.61846e18 q^{86} +6.08317e17 q^{87} +6.44914e17 q^{88} +2.95208e18 q^{89} +4.83405e17 q^{90} -9.72022e18 q^{91} -1.06962e18 q^{92} +1.16974e19 q^{93} -2.72252e18 q^{94} +1.11546e18 q^{95} +1.47536e18 q^{96} -6.52783e18 q^{97} +2.97637e18 q^{98} +2.86401e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6144 q^{2} + 75638 q^{3} + 3145728 q^{4} + 7417130 q^{5} + 38726656 q^{6} + 101600432 q^{7} + 1610612736 q^{8} + 3999078122 q^{9} + 3797570560 q^{10} + 2007416390 q^{11} + 19828047872 q^{12} + 57626609050 q^{13}+ \cdots - 28\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) 41932.2 1.22997 0.614987 − 0.788537i \(-0.289161\pi\)
0.614987 + 0.788537i \(0.289161\pi\)
\(4\) 262144. 0.500000
\(5\) 1.58401e6 0.362696 0.181348 − 0.983419i \(-0.441954\pi\)
0.181348 + 0.983419i \(0.441954\pi\)
\(6\) 2.14693e7 0.869723
\(7\) −1.31195e8 −1.22881 −0.614406 − 0.788990i \(-0.710604\pi\)
−0.614406 + 0.788990i \(0.710604\pi\)
\(8\) 1.34218e8 0.353553
\(9\) 5.96051e8 0.512837
\(10\) 8.11014e8 0.256465
\(11\) 4.80498e9 0.614414 0.307207 − 0.951643i \(-0.400606\pi\)
0.307207 + 0.951643i \(0.400606\pi\)
\(12\) 1.09923e10 0.614987
\(13\) 7.40899e10 1.93775 0.968874 − 0.247553i \(-0.0796265\pi\)
0.968874 + 0.247553i \(0.0796265\pi\)
\(14\) −6.71718e10 −0.868902
\(15\) 6.64211e10 0.446107
\(16\) 6.87195e10 0.250000
\(17\) 6.74902e10 0.138031 0.0690153 − 0.997616i \(-0.478014\pi\)
0.0690153 + 0.997616i \(0.478014\pi\)
\(18\) 3.05178e11 0.362631
\(19\) 7.04202e11 0.500655 0.250327 − 0.968161i \(-0.419462\pi\)
0.250327 + 0.968161i \(0.419462\pi\)
\(20\) 4.15239e11 0.181348
\(21\) −5.50130e12 −1.51141
\(22\) 2.46015e12 0.434456
\(23\) −4.08027e12 −0.472361 −0.236181 − 0.971709i \(-0.575896\pi\)
−0.236181 + 0.971709i \(0.575896\pi\)
\(24\) 5.62805e12 0.434862
\(25\) −1.65644e13 −0.868451
\(26\) 3.79341e13 1.37020
\(27\) −2.37425e13 −0.599198
\(28\) −3.43920e13 −0.614406
\(29\) 1.45071e13 0.185695
\(30\) 3.40076e13 0.315446
\(31\) 2.78960e14 1.89498 0.947492 − 0.319778i \(-0.103608\pi\)
0.947492 + 0.319778i \(0.103608\pi\)
\(32\) 3.51844e13 0.176777
\(33\) 2.01484e14 0.755714
\(34\) 3.45550e13 0.0976024
\(35\) −2.07814e14 −0.445686
\(36\) 1.56251e14 0.256419
\(37\) −1.15290e15 −1.45839 −0.729195 − 0.684306i \(-0.760105\pi\)
−0.729195 + 0.684306i \(0.760105\pi\)
\(38\) 3.60552e14 0.354016
\(39\) 3.10676e15 2.38338
\(40\) 2.12602e14 0.128233
\(41\) 2.11794e15 1.01034 0.505168 − 0.863021i \(-0.331430\pi\)
0.505168 + 0.863021i \(0.331430\pi\)
\(42\) −2.81666e15 −1.06873
\(43\) 5.11418e15 1.55176 0.775882 − 0.630878i \(-0.217305\pi\)
0.775882 + 0.630878i \(0.217305\pi\)
\(44\) 1.25960e15 0.307207
\(45\) 9.44151e14 0.186004
\(46\) −2.08910e15 −0.334010
\(47\) −5.31742e15 −0.693061 −0.346531 − 0.938039i \(-0.612640\pi\)
−0.346531 + 0.938039i \(0.612640\pi\)
\(48\) 2.88156e15 0.307494
\(49\) 5.81321e15 0.509980
\(50\) −8.48097e15 −0.614088
\(51\) 2.83001e15 0.169774
\(52\) 1.94222e16 0.968874
\(53\) −5.47577e15 −0.227942 −0.113971 − 0.993484i \(-0.536357\pi\)
−0.113971 + 0.993484i \(0.536357\pi\)
\(54\) −1.21562e16 −0.423697
\(55\) 7.61115e15 0.222846
\(56\) −1.76087e16 −0.434451
\(57\) 2.95288e16 0.615792
\(58\) 7.42766e15 0.131306
\(59\) 1.27260e17 1.91248 0.956239 − 0.292588i \(-0.0945163\pi\)
0.956239 + 0.292588i \(0.0945163\pi\)
\(60\) 1.74119e16 0.223054
\(61\) 5.49821e16 0.601988 0.300994 − 0.953626i \(-0.402682\pi\)
0.300994 + 0.953626i \(0.402682\pi\)
\(62\) 1.42827e17 1.33996
\(63\) −7.81988e16 −0.630181
\(64\) 1.80144e16 0.125000
\(65\) 1.17359e17 0.702814
\(66\) 1.03160e17 0.534370
\(67\) 1.47906e17 0.664166 0.332083 − 0.943250i \(-0.392249\pi\)
0.332083 + 0.943250i \(0.392249\pi\)
\(68\) 1.76921e16 0.0690153
\(69\) −1.71095e17 −0.580992
\(70\) −1.06401e17 −0.315148
\(71\) 4.25660e17 1.10181 0.550907 − 0.834566i \(-0.314282\pi\)
0.550907 + 0.834566i \(0.314282\pi\)
\(72\) 8.00006e16 0.181315
\(73\) −9.75824e16 −0.194001 −0.0970006 − 0.995284i \(-0.530925\pi\)
−0.0970006 + 0.995284i \(0.530925\pi\)
\(74\) −5.90283e17 −1.03124
\(75\) −6.94582e17 −1.06817
\(76\) 1.84602e17 0.250327
\(77\) −6.30389e17 −0.755000
\(78\) 1.59066e18 1.68530
\(79\) 1.08218e17 0.101588 0.0507940 − 0.998709i \(-0.483825\pi\)
0.0507940 + 0.998709i \(0.483825\pi\)
\(80\) 1.08852e17 0.0906741
\(81\) −1.68834e18 −1.24984
\(82\) 1.08438e18 0.714416
\(83\) 6.83657e16 0.0401417 0.0200709 − 0.999799i \(-0.493611\pi\)
0.0200709 + 0.999799i \(0.493611\pi\)
\(84\) −1.44213e18 −0.755704
\(85\) 1.06905e17 0.0500632
\(86\) 2.61846e18 1.09726
\(87\) 6.08317e17 0.228401
\(88\) 6.44914e17 0.217228
\(89\) 2.95208e18 0.893147 0.446574 − 0.894747i \(-0.352644\pi\)
0.446574 + 0.894747i \(0.352644\pi\)
\(90\) 4.83405e17 0.131525
\(91\) −9.72022e18 −2.38113
\(92\) −1.06962e18 −0.236181
\(93\) 1.16974e19 2.33078
\(94\) −2.72252e18 −0.490068
\(95\) 1.11546e18 0.181586
\(96\) 1.47536e18 0.217431
\(97\) −6.52783e18 −0.871840 −0.435920 − 0.899985i \(-0.643577\pi\)
−0.435920 + 0.899985i \(0.643577\pi\)
\(98\) 2.97637e18 0.360611
\(99\) 2.86401e18 0.315094
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.1.10 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.20.a.d.1.10 ✓ 12 1.1 even 1 trivial