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Properties

Label 2.66.a.a.1.1
Level $2$
Weight $66$
Character 2.1
Self dual yes
Analytic conductor $53.514$
Analytic rank $1$
Dimension $2$
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] = [2,66,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 66); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 66, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.5144712945\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1961256803955162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5\cdot 11 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.42861e7\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.29497e9 q^{2} -2.22863e15 q^{3} +1.84467e19 q^{4} +2.44878e22 q^{5} +9.57189e24 q^{6} +8.86741e26 q^{7} -7.92282e28 q^{8} -5.33426e30 q^{9} -1.05174e32 q^{10} -1.95921e33 q^{11} -4.11110e34 q^{12} -3.37675e35 q^{13} -3.80853e36 q^{14} -5.45742e37 q^{15} +3.40282e38 q^{16} -8.16482e38 q^{17} +2.29105e40 q^{18} +4.91157e41 q^{19} +4.51719e41 q^{20} -1.97622e42 q^{21} +8.41475e42 q^{22} +2.14822e43 q^{23} +1.76570e44 q^{24} -2.11086e45 q^{25} +1.45031e45 q^{26} +3.48453e46 q^{27} +1.63575e46 q^{28} +4.84833e47 q^{29} +2.34394e47 q^{30} +2.98599e47 q^{31} -1.46150e48 q^{32} +4.36636e48 q^{33} +3.50676e48 q^{34} +2.17143e49 q^{35} -9.83997e49 q^{36} -3.98371e50 q^{37} -2.10950e51 q^{38} +7.52554e50 q^{39} -1.94012e51 q^{40} +2.36904e52 q^{41} +8.48779e51 q^{42} +5.86037e52 q^{43} -3.61411e52 q^{44} -1.30624e53 q^{45} -9.22653e52 q^{46} -1.98586e54 q^{47} -7.58364e53 q^{48} -7.75201e54 q^{49} +9.06605e54 q^{50} +1.81964e54 q^{51} -6.22901e54 q^{52} -1.09112e56 q^{53} -1.49660e56 q^{54} -4.79767e55 q^{55} -7.02549e55 q^{56} -1.09461e57 q^{57} -2.08234e57 q^{58} -6.12790e57 q^{59} -1.00672e57 q^{60} -1.41421e58 q^{61} -1.28247e57 q^{62} -4.73011e57 q^{63} +6.27710e57 q^{64} -8.26891e57 q^{65} -1.87534e58 q^{66} -1.83150e59 q^{67} -1.50614e58 q^{68} -4.78758e58 q^{69} -9.32622e58 q^{70} +2.55609e60 q^{71} +4.22624e59 q^{72} +4.28287e60 q^{73} +1.71099e60 q^{74} +4.70432e60 q^{75} +9.06024e60 q^{76} -1.73731e60 q^{77} -3.23219e60 q^{78} -2.39333e61 q^{79} +8.33275e60 q^{80} -2.27089e61 q^{81} -1.01749e62 q^{82} -1.47514e62 q^{83} -3.64548e61 q^{84} -1.99938e61 q^{85} -2.51701e62 q^{86} -1.08051e63 q^{87} +1.55225e62 q^{88} +8.60873e61 q^{89} +5.61026e62 q^{90} -2.99431e62 q^{91} +3.96276e62 q^{92} -6.65468e62 q^{93} +8.52919e63 q^{94} +1.20273e64 q^{95} +3.25715e63 q^{96} +6.09845e64 q^{97} +3.32946e64 q^{98} +1.04509e64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8589934592 q^{2} + 11\!\cdots\!92 q^{3} + 36\!\cdots\!32 q^{4} - 74\!\cdots\!00 q^{5} - 49\!\cdots\!32 q^{6} + 27\!\cdots\!24 q^{7} - 15\!\cdots\!72 q^{8} - 42\!\cdots\!54 q^{9} + 32\!\cdots\!00 q^{10}+ \cdots + 21\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.29497e9 −0.707107
\(3\) −2.22863e15 −0.694380 −0.347190 − 0.937795i \(-0.612864\pi\)
−0.347190 + 0.937795i \(0.612864\pi\)
\(4\) 1.84467e19 0.500000
\(5\) 2.44878e22 0.470353 0.235176 − 0.971953i \(-0.424433\pi\)
0.235176 + 0.971953i \(0.424433\pi\)
\(6\) 9.57189e24 0.491001
\(7\) 8.86741e26 0.303466 0.151733 − 0.988421i \(-0.451515\pi\)
0.151733 + 0.988421i \(0.451515\pi\)
\(8\) −7.92282e28 −0.353553
\(9\) −5.33426e30 −0.517836
\(10\) −1.05174e32 −0.332590
\(11\) −1.95921e33 −0.279782 −0.139891 − 0.990167i \(-0.544675\pi\)
−0.139891 + 0.990167i \(0.544675\pi\)
\(12\) −4.11110e34 −0.347190
\(13\) −3.37675e35 −0.211515 −0.105757 − 0.994392i \(-0.533727\pi\)
−0.105757 + 0.994392i \(0.533727\pi\)
\(14\) −3.80853e36 −0.214583
\(15\) −5.45742e37 −0.326604
\(16\) 3.40282e38 0.250000
\(17\) −8.16482e38 −0.0836290 −0.0418145 − 0.999125i \(-0.513314\pi\)
−0.0418145 + 0.999125i \(0.513314\pi\)
\(18\) 2.29105e40 0.366166
\(19\) 4.91157e41 1.35434 0.677171 − 0.735826i \(-0.263206\pi\)
0.677171 + 0.735826i \(0.263206\pi\)
\(20\) 4.51719e41 0.235176
\(21\) −1.97622e42 −0.210721
\(22\) 8.41475e42 0.197835
\(23\) 2.14822e43 0.119103 0.0595517 − 0.998225i \(-0.481033\pi\)
0.0595517 + 0.998225i \(0.481033\pi\)
\(24\) 1.76570e44 0.245500
\(25\) −2.11086e45 −0.778768
\(26\) 1.45031e45 0.149563
\(27\) 3.48453e46 1.05396
\(28\) 1.63575e46 0.151733
\(29\) 4.84833e47 1.43766 0.718832 − 0.695184i \(-0.244677\pi\)
0.718832 + 0.695184i \(0.244677\pi\)
\(30\) 2.34394e47 0.230944
\(31\) 2.98599e47 0.101352 0.0506762 − 0.998715i \(-0.483862\pi\)
0.0506762 + 0.998715i \(0.483862\pi\)
\(32\) −1.46150e48 −0.176777
\(33\) 4.36636e48 0.194275
\(34\) 3.50676e48 0.0591346
\(35\) 2.17143e49 0.142736
\(36\) −9.83997e49 −0.258918
\(37\) −3.98371e50 −0.430260 −0.215130 − 0.976585i \(-0.569018\pi\)
−0.215130 + 0.976585i \(0.569018\pi\)
\(38\) −2.10950e51 −0.957664
\(39\) 7.52554e50 0.146872
\(40\) −1.94012e51 −0.166295
\(41\) 2.36904e52 0.910115 0.455058 − 0.890462i \(-0.349619\pi\)
0.455058 + 0.890462i \(0.349619\pi\)
\(42\) 8.48779e51 0.149002
\(43\) 5.86037e52 0.478851 0.239425 − 0.970915i \(-0.423041\pi\)
0.239425 + 0.970915i \(0.423041\pi\)
\(44\) −3.61411e52 −0.139891
\(45\) −1.30624e53 −0.243566
\(46\) −9.22653e52 −0.0842188
\(47\) −1.98586e54 −0.901089 −0.450544 − 0.892754i \(-0.648770\pi\)
−0.450544 + 0.892754i \(0.648770\pi\)
\(48\) −7.58364e53 −0.173595
\(49\) −7.75201e54 −0.907908
\(50\) 9.06605e54 0.550672
\(51\) 1.81964e54 0.0580703
\(52\) −6.22901e54 −0.105757
\(53\) −1.09112e56 −0.997488 −0.498744 − 0.866749i \(-0.666205\pi\)
−0.498744 + 0.866749i \(0.666205\pi\)
\(54\) −1.49660e56 −0.745259
\(55\) −4.79767e55 −0.131596
\(56\) −7.02549e55 −0.107292
\(57\) −1.09461e57 −0.940428
\(58\) −2.08234e57 −1.01658
\(59\) −6.12790e57 −1.71641 −0.858206 − 0.513306i \(-0.828421\pi\)
−0.858206 + 0.513306i \(0.828421\pi\)
\(60\) −1.00672e57 −0.163302
\(61\) −1.41421e58 −1.34059 −0.670294 − 0.742096i \(-0.733832\pi\)
−0.670294 + 0.742096i \(0.733832\pi\)
\(62\) −1.28247e57 −0.0716670
\(63\) −4.73011e57 −0.157146
\(64\) 6.27710e57 0.125000
\(65\) −8.26891e57 −0.0994866
\(66\) −1.87534e58 −0.137373
\(67\) −1.83150e59 −0.822953 −0.411476 − 0.911420i \(-0.634987\pi\)
−0.411476 + 0.911420i \(0.634987\pi\)
\(68\) −1.50614e58 −0.0418145
\(69\) −4.78758e58 −0.0827030
\(70\) −9.32622e58 −0.100930
\(71\) 2.55609e60 1.74454 0.872269 − 0.489026i \(-0.162648\pi\)
0.872269 + 0.489026i \(0.162648\pi\)
\(72\) 4.22624e59 0.183083
\(73\) 4.28287e60 1.18506 0.592532 − 0.805547i \(-0.298129\pi\)
0.592532 + 0.805547i \(0.298129\pi\)
\(74\) 1.71099e60 0.304240
\(75\) 4.70432e60 0.540761
\(76\) 9.06024e60 0.677171
\(77\) −1.73731e60 −0.0849043
\(78\) −3.23219e60 −0.103854
\(79\) −2.39333e61 −0.508303 −0.254151 − 0.967164i \(-0.581796\pi\)
−0.254151 + 0.967164i \(0.581796\pi\)
\(80\) 8.33275e60 0.117588
\(81\) −2.27089e61 −0.214009
\(82\) −1.01749e62 −0.643549
\(83\) −1.47514e62 −0.629211 −0.314605 − 0.949223i \(-0.601872\pi\)
−0.314605 + 0.949223i \(0.601872\pi\)
\(84\) −3.64548e61 −0.105361
\(85\) −1.99938e61 −0.0393351
\(86\) −2.51701e62 −0.338598
\(87\) −1.08051e63 −0.998285
\(88\) 1.55225e62 0.0989177
\(89\) 8.60873e61 0.0379983 0.0189991 − 0.999819i \(-0.493952\pi\)
0.0189991 + 0.999819i \(0.493952\pi\)
\(90\) 5.61026e62 0.172227
\(91\) −2.99431e62 −0.0641876
\(92\) 3.96276e62 0.0595517
\(93\) −6.65468e62 −0.0703771
\(94\) 8.52919e63 0.637166
\(95\) 1.20273e64 0.637018
\(96\) 3.25715e63 0.122750
\(97\) 6.09845e64 1.64111 0.820556 − 0.571566i \(-0.193664\pi\)
0.820556 + 0.571566i \(0.193664\pi\)
\(98\) 3.32946e64 0.641988
\(99\) 1.04509e64 0.144881
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 2.66.a.a.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.66.a.a.1.1 ✓ 2 1.1 even 1 trivial