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Properties

Label 58.20.a.d.1.11
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 1.11
Root \(42042.8\) 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)
 
\(f(q)\) \(=\) \(q+512.000 q^{2} +48345.8 q^{3} +262144. q^{4} +3.03871e6 q^{5} +2.47531e7 q^{6} -610691. q^{7} +1.34218e8 q^{8} +1.17506e9 q^{9} +1.55582e9 q^{10} +1.12099e10 q^{11} +1.26736e10 q^{12} -5.22340e10 q^{13} -3.12674e8 q^{14} +1.46909e11 q^{15} +6.87195e10 q^{16} +7.03061e11 q^{17} +6.01630e11 q^{18} +4.13587e10 q^{19} +7.96580e11 q^{20} -2.95244e10 q^{21} +5.73946e12 q^{22} +6.41372e12 q^{23} +6.48887e12 q^{24} -9.83971e12 q^{25} -2.67438e13 q^{26} +6.18701e11 q^{27} -1.60089e11 q^{28} +1.45071e13 q^{29} +7.52175e13 q^{30} -9.17854e12 q^{31} +3.51844e13 q^{32} +5.41951e14 q^{33} +3.59967e14 q^{34} -1.85571e12 q^{35} +3.08035e14 q^{36} +1.03967e15 q^{37} +2.11757e13 q^{38} -2.52530e15 q^{39} +4.07849e14 q^{40} +1.35021e15 q^{41} -1.51165e13 q^{42} -2.68236e15 q^{43} +2.93860e15 q^{44} +3.57067e15 q^{45} +3.28383e15 q^{46} +6.46670e15 q^{47} +3.32230e15 q^{48} -1.13985e16 q^{49} -5.03793e15 q^{50} +3.39901e16 q^{51} -1.36928e16 q^{52} +8.82547e15 q^{53} +3.16775e14 q^{54} +3.40636e16 q^{55} -8.19656e13 q^{56} +1.99952e15 q^{57} +7.42766e15 q^{58} +5.33433e16 q^{59} +3.85113e16 q^{60} +2.49873e16 q^{61} -4.69941e15 q^{62} -7.17598e14 q^{63} +1.80144e16 q^{64} -1.58724e17 q^{65} +2.77479e17 q^{66} +3.93018e17 q^{67} +1.84303e17 q^{68} +3.10077e17 q^{69} -9.50126e14 q^{70} -7.04603e17 q^{71} +1.57714e17 q^{72} +5.85730e16 q^{73} +5.32312e17 q^{74} -4.75709e17 q^{75} +1.08419e16 q^{76} -6.84578e15 q^{77} -1.29295e18 q^{78} -2.53949e17 q^{79} +2.08819e17 q^{80} -1.33581e18 q^{81} +6.91306e17 q^{82} +1.20424e18 q^{83} -7.73964e15 q^{84} +2.13640e18 q^{85} -1.37337e18 q^{86} +7.01360e17 q^{87} +1.50457e18 q^{88} -1.43556e18 q^{89} +1.82818e18 q^{90} +3.18988e16 q^{91} +1.68132e18 q^{92} -4.43744e17 q^{93} +3.31095e18 q^{94} +1.25677e17 q^{95} +1.70102e18 q^{96} -7.32634e18 q^{97} -5.83604e18 q^{98} +1.31723e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6144 q^{2} + 75638 q^{3} + 3145728 q^{4} + 7417130 q^{5} + 38726656 q^{6} + 101600432 q^{7} + 1610612736 q^{8} + 3999078122 q^{9} + 3797570560 q^{10} + 2007416390 q^{11} + 19828047872 q^{12} + 57626609050 q^{13}+ \cdots - 28\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) 48345.8 1.41810 0.709051 − 0.705158i \(-0.249124\pi\)
0.709051 + 0.705158i \(0.249124\pi\)
\(4\) 262144. 0.500000
\(5\) 3.03871e6 0.695784 0.347892 − 0.937535i \(-0.386898\pi\)
0.347892 + 0.937535i \(0.386898\pi\)
\(6\) 2.47531e7 1.00275
\(7\) −610691. −0.00571992 −0.00285996 − 0.999996i \(-0.500910\pi\)
−0.00285996 + 0.999996i \(0.500910\pi\)
\(8\) 1.34218e8 0.353553
\(9\) 1.17506e9 1.01101
\(10\) 1.55582e9 0.491994
\(11\) 1.12099e10 1.43341 0.716705 − 0.697376i \(-0.245649\pi\)
0.716705 + 0.697376i \(0.245649\pi\)
\(12\) 1.26736e10 0.709051
\(13\) −5.22340e10 −1.36613 −0.683064 − 0.730358i \(-0.739353\pi\)
−0.683064 + 0.730358i \(0.739353\pi\)
\(14\) −3.12674e8 −0.00404460
\(15\) 1.46909e11 0.986692
\(16\) 6.87195e10 0.250000
\(17\) 7.03061e11 1.43790 0.718949 − 0.695063i \(-0.244624\pi\)
0.718949 + 0.695063i \(0.244624\pi\)
\(18\) 6.01630e11 0.714893
\(19\) 4.13587e10 0.0294041 0.0147020 − 0.999892i \(-0.495320\pi\)
0.0147020 + 0.999892i \(0.495320\pi\)
\(20\) 7.96580e11 0.347892
\(21\) −2.95244e10 −0.00811143
\(22\) 5.73946e12 1.01357
\(23\) 6.41372e12 0.742499 0.371249 − 0.928533i \(-0.378930\pi\)
0.371249 + 0.928533i \(0.378930\pi\)
\(24\) 6.48887e12 0.501374
\(25\) −9.83971e12 −0.515884
\(26\) −2.67438e13 −0.965999
\(27\) 6.18701e11 0.0156144
\(28\) −1.60089e11 −0.00285996
\(29\) 1.45071e13 0.185695
\(30\) 7.52175e13 0.697697
\(31\) −9.17854e12 −0.0623502 −0.0311751 − 0.999514i \(-0.509925\pi\)
−0.0311751 + 0.999514i \(0.509925\pi\)
\(32\) 3.51844e13 0.176777
\(33\) 5.41951e14 2.03272
\(34\) 3.59967e14 1.01675
\(35\) −1.85571e12 −0.00397983
\(36\) 3.08035e14 0.505505
\(37\) 1.03967e15 1.31516 0.657582 − 0.753383i \(-0.271579\pi\)
0.657582 + 0.753383i \(0.271579\pi\)
\(38\) 2.11757e13 0.0207918
\(39\) −2.52530e15 −1.93731
\(40\) 4.07849e14 0.245997
\(41\) 1.35021e15 0.644101 0.322050 − 0.946723i \(-0.395628\pi\)
0.322050 + 0.946723i \(0.395628\pi\)
\(42\) −1.51165e13 −0.00573565
\(43\) −2.68236e15 −0.813893 −0.406947 − 0.913452i \(-0.633406\pi\)
−0.406947 + 0.913452i \(0.633406\pi\)
\(44\) 2.93860e15 0.716705
\(45\) 3.57067e15 0.703445
\(46\) 3.28383e15 0.525026
\(47\) 6.46670e15 0.842856 0.421428 − 0.906862i \(-0.361529\pi\)
0.421428 + 0.906862i \(0.361529\pi\)
\(48\) 3.32230e15 0.354525
\(49\) −1.13985e16 −0.999967
\(50\) −5.03793e15 −0.364785
\(51\) 3.39901e16 2.03908
\(52\) −1.36928e16 −0.683064
\(53\) 8.82547e15 0.367381 0.183691 − 0.982984i \(-0.441196\pi\)
0.183691 + 0.982984i \(0.441196\pi\)
\(54\) 3.16775e14 0.0110410
\(55\) 3.40636e16 0.997345
\(56\) −8.19656e13 −0.00202230
\(57\) 1.99952e15 0.0416980
\(58\) 7.42766e15 0.131306
\(59\) 5.33433e16 0.801653 0.400826 − 0.916154i \(-0.368723\pi\)
0.400826 + 0.916154i \(0.368723\pi\)
\(60\) 3.85113e16 0.493346
\(61\) 2.49873e16 0.273581 0.136790 − 0.990600i \(-0.456321\pi\)
0.136790 + 0.990600i \(0.456321\pi\)
\(62\) −4.69941e15 −0.0440882
\(63\) −7.17598e14 −0.00578290
\(64\) 1.80144e16 0.125000
\(65\) −1.58724e17 −0.950531
\(66\) 2.77479e17 1.43735
\(67\) 3.93018e17 1.76483 0.882414 − 0.470475i \(-0.155917\pi\)
0.882414 + 0.470475i \(0.155917\pi\)
\(68\) 1.84303e17 0.718949
\(69\) 3.10077e17 1.05294
\(70\) −9.50126e14 −0.00281417
\(71\) −7.04603e17 −1.82385 −0.911927 − 0.410353i \(-0.865406\pi\)
−0.911927 + 0.410353i \(0.865406\pi\)
\(72\) 1.57714e17 0.357446
\(73\) 5.85730e16 0.116448 0.0582238 − 0.998304i \(-0.481456\pi\)
0.0582238 + 0.998304i \(0.481456\pi\)
\(74\) 5.32312e17 0.929961
\(75\) −4.75709e17 −0.731576
\(76\) 1.08419e16 0.0147020
\(77\) −6.84578e15 −0.00819900
\(78\) −1.29295e18 −1.36988
\(79\) −2.53949e17 −0.238390 −0.119195 − 0.992871i \(-0.538031\pi\)
−0.119195 + 0.992871i \(0.538031\pi\)
\(80\) 2.08819e17 0.173946
\(81\) −1.33581e18 −0.988868
\(82\) 6.91306e17 0.455448
\(83\) 1.20424e18 0.707083 0.353541 − 0.935419i \(-0.384977\pi\)
0.353541 + 0.935419i \(0.384977\pi\)
\(84\) −7.73964e15 −0.00405571
\(85\) 2.13640e18 1.00047
\(86\) −1.37337e18 −0.575509
\(87\) 7.01360e17 0.263335
\(88\) 1.50457e18 0.506787
\(89\) −1.43556e18 −0.434326 −0.217163 − 0.976135i \(-0.569680\pi\)
−0.217163 + 0.976135i \(0.569680\pi\)
\(90\) 1.82818e18 0.497411
\(91\) 3.18988e16 0.00781415
\(92\) 1.68132e18 0.371249
\(93\) −4.43744e17 −0.0884189
\(94\) 3.31095e18 0.595989
\(95\) 1.25677e17 0.0204589
\(96\) 1.70102e18 0.250687
\(97\) −7.32634e18 −0.978488 −0.489244 − 0.872147i \(-0.662727\pi\)
−0.489244 + 0.872147i \(0.662727\pi\)
\(98\) −5.83604e18 −0.707084
\(99\) 1.31723e19 1.44919
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.1.11 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.20.a.d.1.11 ✓ 12 1.1 even 1 trivial