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Properties

Label 2.66.a.b.1.2
Level $2$
Weight $66$
Character 2.1
Self dual yes
Analytic conductor $53.514$
Analytic rank $0$
Dimension $3$
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 = [3] 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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4862367805520722608042x + 130125819203569060903952569933488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11\cdot 13 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.04922e10\) 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)
 
\(f(q)\) \(=\) \(q+4.29497e9 q^{2} -5.99699e13 q^{3} +1.84467e19 q^{4} +6.16250e22 q^{5} -2.57569e23 q^{6} -2.55894e27 q^{7} +7.92282e28 q^{8} -1.02975e31 q^{9} +2.64677e32 q^{10} +8.24813e33 q^{11} -1.10625e33 q^{12} +2.22949e36 q^{13} -1.09906e37 q^{14} -3.69564e36 q^{15} +3.40282e38 q^{16} +8.10668e39 q^{17} -4.42272e40 q^{18} -2.71144e41 q^{19} +1.13678e42 q^{20} +1.53459e41 q^{21} +3.54255e43 q^{22} -1.66211e44 q^{23} -4.75130e42 q^{24} +1.08714e45 q^{25} +9.57560e45 q^{26} +1.23529e45 q^{27} -4.72041e46 q^{28} +5.45865e47 q^{29} -1.58727e46 q^{30} +1.86521e48 q^{31} +1.46150e48 q^{32} -4.94639e47 q^{33} +3.48179e49 q^{34} -1.57695e50 q^{35} -1.89955e50 q^{36} +2.14819e50 q^{37} -1.16455e51 q^{38} -1.33702e50 q^{39} +4.88244e51 q^{40} -7.22402e51 q^{41} +6.59102e50 q^{42} +1.76640e53 q^{43} +1.52151e53 q^{44} -6.34581e53 q^{45} -7.13870e53 q^{46} +2.08814e54 q^{47} -2.04067e52 q^{48} -1.99016e54 q^{49} +4.66921e54 q^{50} -4.86156e53 q^{51} +4.11269e55 q^{52} +1.83129e56 q^{53} +5.30553e54 q^{54} +5.08291e56 q^{55} -2.02740e56 q^{56} +1.62605e55 q^{57} +2.34447e57 q^{58} +3.10277e57 q^{59} -6.81726e55 q^{60} +1.43214e58 q^{61} +8.01103e57 q^{62} +2.63505e58 q^{63} +6.27710e57 q^{64} +1.37393e59 q^{65} -2.12446e57 q^{66} -3.92071e59 q^{67} +1.49542e59 q^{68} +9.96764e57 q^{69} -6.77293e59 q^{70} -1.24089e60 q^{71} -8.15848e59 q^{72} -2.06411e60 q^{73} +9.22642e59 q^{74} -6.51954e58 q^{75} -5.00172e60 q^{76} -2.11065e61 q^{77} -5.74248e59 q^{78} -2.95663e61 q^{79} +2.09699e61 q^{80} +1.06001e62 q^{81} -3.10269e61 q^{82} +2.09374e62 q^{83} +2.83082e60 q^{84} +4.99574e62 q^{85} +7.58665e62 q^{86} -3.27355e61 q^{87} +6.53484e62 q^{88} +1.03182e63 q^{89} -2.72550e63 q^{90} -5.70513e63 q^{91} -3.06605e63 q^{92} -1.11857e62 q^{93} +8.96850e63 q^{94} -1.67092e64 q^{95} -8.76461e61 q^{96} -1.75639e64 q^{97} -8.54768e63 q^{98} -8.49348e64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12884901888 q^{2} + 29\!\cdots\!12 q^{3} + 55\!\cdots\!48 q^{4} + 39\!\cdots\!50 q^{5} + 12\!\cdots\!52 q^{6} + 42\!\cdots\!64 q^{7} + 23\!\cdots\!08 q^{8} + 85\!\cdots\!19 q^{9} + 16\!\cdots\!00 q^{10}+ \cdots - 11\!\cdots\!32 q^{99}+O(q^{100}) \) 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
\(3\) −5.99699e13 −0.0186850 −0.00934248 − 0.999956i \(-0.502974\pi\)
−0.00934248 + 0.999956i \(0.502974\pi\)
\(4\) 1.84467e19 0.500000
\(5\) 6.16250e22 1.18367 0.591837 − 0.806058i \(-0.298403\pi\)
0.591837 + 0.806058i \(0.298403\pi\)
\(6\) −2.57569e23 −0.0132123
\(7\) −2.55894e27 −0.875736 −0.437868 − 0.899039i \(-0.644266\pi\)
−0.437868 + 0.899039i \(0.644266\pi\)
\(8\) 7.92282e28 0.353553
\(9\) −1.02975e31 −0.999651
\(10\) 2.64677e32 0.836983
\(11\) 8.24813e33 1.17786 0.588930 − 0.808184i \(-0.299550\pi\)
0.588930 + 0.808184i \(0.299550\pi\)
\(12\) −1.10625e33 −0.00934248
\(13\) 2.22949e36 1.39652 0.698260 − 0.715844i \(-0.253958\pi\)
0.698260 + 0.715844i \(0.253958\pi\)
\(14\) −1.09906e37 −0.619239
\(15\) −3.69564e36 −0.0221169
\(16\) 3.40282e38 0.250000
\(17\) 8.10668e39 0.830334 0.415167 − 0.909745i \(-0.363723\pi\)
0.415167 + 0.909745i \(0.363723\pi\)
\(18\) −4.42272e40 −0.706860
\(19\) −2.71144e41 −0.747666 −0.373833 − 0.927496i \(-0.621957\pi\)
−0.373833 + 0.927496i \(0.621957\pi\)
\(20\) 1.13678e42 0.591837
\(21\) 1.53459e41 0.0163631
\(22\) 3.54255e43 0.832872
\(23\) −1.66211e44 −0.921519 −0.460760 − 0.887525i \(-0.652423\pi\)
−0.460760 + 0.887525i \(0.652423\pi\)
\(24\) −4.75130e42 −0.00660613
\(25\) 1.08714e45 0.401082
\(26\) 9.57560e45 0.987489
\(27\) 1.23529e45 0.0373634
\(28\) −4.72041e46 −0.437868
\(29\) 5.45865e47 1.61864 0.809320 − 0.587368i \(-0.199836\pi\)
0.809320 + 0.587368i \(0.199836\pi\)
\(30\) −1.58727e46 −0.0156390
\(31\) 1.86521e48 0.633101 0.316551 − 0.948576i \(-0.397475\pi\)
0.316551 + 0.948576i \(0.397475\pi\)
\(32\) 1.46150e48 0.176777
\(33\) −4.94639e47 −0.0220083
\(34\) 3.48179e49 0.587135
\(35\) −1.57695e50 −1.03659
\(36\) −1.89955e50 −0.499825
\(37\) 2.14819e50 0.232016 0.116008 − 0.993248i \(-0.462990\pi\)
0.116008 + 0.993248i \(0.462990\pi\)
\(38\) −1.16455e51 −0.528680
\(39\) −1.33702e50 −0.0260939
\(40\) 4.88244e51 0.418492
\(41\) −7.22402e51 −0.277526 −0.138763 − 0.990326i \(-0.544313\pi\)
−0.138763 + 0.990326i \(0.544313\pi\)
\(42\) 6.59102e50 0.0115705
\(43\) 1.76640e53 1.44333 0.721664 − 0.692243i \(-0.243377\pi\)
0.721664 + 0.692243i \(0.243377\pi\)
\(44\) 1.52151e53 0.588930
\(45\) −6.34581e53 −1.18326
\(46\) −7.13870e53 −0.651613
\(47\) 2.08814e54 0.947501 0.473751 − 0.880659i \(-0.342900\pi\)
0.473751 + 0.880659i \(0.342900\pi\)
\(48\) −2.04067e52 −0.00467124
\(49\) −1.99016e54 −0.233086
\(50\) 4.66921e54 0.283608
\(51\) −4.86156e53 −0.0155148
\(52\) 4.11269e55 0.698260
\(53\) 1.83129e56 1.67414 0.837068 − 0.547099i \(-0.184268\pi\)
0.837068 + 0.547099i \(0.184268\pi\)
\(54\) 5.30553e54 0.0264199
\(55\) 5.08291e56 1.39420
\(56\) −2.02740e56 −0.309620
\(57\) 1.62605e55 0.0139701
\(58\) 2.34447e57 1.14455
\(59\) 3.10277e57 0.869078 0.434539 − 0.900653i \(-0.356911\pi\)
0.434539 + 0.900653i \(0.356911\pi\)
\(60\) −6.81726e55 −0.0110584
\(61\) 1.43214e58 1.35758 0.678792 − 0.734331i \(-0.262504\pi\)
0.678792 + 0.734331i \(0.262504\pi\)
\(62\) 8.01103e57 0.447670
\(63\) 2.63505e58 0.875431
\(64\) 6.27710e57 0.125000
\(65\) 1.37393e59 1.65302
\(66\) −2.12446e57 −0.0155622
\(67\) −3.92071e59 −1.76171 −0.880854 − 0.473387i \(-0.843031\pi\)
−0.880854 + 0.473387i \(0.843031\pi\)
\(68\) 1.49542e59 0.415167
\(69\) 9.96764e57 0.0172186
\(70\) −6.77293e59 −0.732977
\(71\) −1.24089e60 −0.846911 −0.423456 − 0.905917i \(-0.639183\pi\)
−0.423456 + 0.905917i \(0.639183\pi\)
\(72\) −8.15848e59 −0.353430
\(73\) −2.06411e60 −0.571136 −0.285568 − 0.958358i \(-0.592182\pi\)
−0.285568 + 0.958358i \(0.592182\pi\)
\(74\) 9.22642e59 0.164060
\(75\) −6.51954e58 −0.00749421
\(76\) −5.00172e60 −0.373833
\(77\) −2.11065e61 −1.03149
\(78\) −5.74248e59 −0.0184512
\(79\) −2.95663e61 −0.627938 −0.313969 − 0.949433i \(-0.601659\pi\)
−0.313969 + 0.949433i \(0.601659\pi\)
\(80\) 2.09699e61 0.295918
\(81\) 1.06001e62 0.998953
\(82\) −3.10269e61 −0.196240
\(83\) 2.09374e62 0.893068 0.446534 − 0.894767i \(-0.352658\pi\)
0.446534 + 0.894767i \(0.352658\pi\)
\(84\) 2.83082e60 0.00818155
\(85\) 4.99574e62 0.982844
\(86\) 7.58665e62 1.02059
\(87\) −3.27355e61 −0.0302442
\(88\) 6.53484e62 0.416436
\(89\) 1.03182e63 0.455437 0.227718 − 0.973727i \(-0.426873\pi\)
0.227718 + 0.973727i \(0.426873\pi\)
\(90\) −2.72550e63 −0.836691
\(91\) −5.70513e63 −1.22298
\(92\) −3.06605e63 −0.460760
\(93\) −1.11857e62 −0.0118295
\(94\) 8.96850e63 0.669985
\(95\) −1.67092e64 −0.884993
\(96\) −8.76461e61 −0.00330307
\(97\) −1.75639e64 −0.472650 −0.236325 − 0.971674i \(-0.575943\pi\)
−0.236325 + 0.971674i \(0.575943\pi\)
\(98\) −8.54768e63 −0.164817
\(99\) −8.49348e64 −1.17745
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.b.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.66.a.b.1.2 ✓ 3 1.1 even 1 trivial