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Properties

Label 2.84.a.b.1.3
Level $2$
Weight $84$
Character 2.1
Self dual yes
Analytic conductor $87.254$
Analytic rank $1$
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] = [2,84,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, 84); 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, 84, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 84 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,6597069766656] 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: \(87.2544256533\)
Analytic rank: \(1\)
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: \( x^{3} - x^{2} - 76392209863211857938006422774x + 4214151671129618412000783695211690286445664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{11}\cdot 5^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.00679e14\) 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)
 
\(f(q)\) \(=\) \(q+2.19902e12 q^{2} +8.79012e19 q^{3} +4.83570e24 q^{4} -8.42086e28 q^{5} +1.93297e32 q^{6} +1.88466e34 q^{7} +1.06338e37 q^{8} +3.73579e39 q^{9} -1.85177e41 q^{10} -2.30063e43 q^{11} +4.25064e44 q^{12} -1.10564e46 q^{13} +4.14441e46 q^{14} -7.40204e48 q^{15} +2.33840e49 q^{16} -6.53898e50 q^{17} +8.21509e51 q^{18} +1.88409e53 q^{19} -4.07208e53 q^{20} +1.65664e54 q^{21} -5.05915e55 q^{22} -3.40745e56 q^{23} +9.34726e56 q^{24} -3.24868e57 q^{25} -2.43132e58 q^{26} -2.24191e58 q^{27} +9.11366e58 q^{28} -3.75005e60 q^{29} -1.62773e61 q^{30} -1.06116e62 q^{31} +5.14220e61 q^{32} -2.02229e63 q^{33} -1.43794e63 q^{34} -1.58705e63 q^{35} +1.80652e64 q^{36} +1.36698e65 q^{37} +4.14317e65 q^{38} -9.71870e65 q^{39} -8.95459e65 q^{40} +2.40641e66 q^{41} +3.64299e66 q^{42} -9.96821e67 q^{43} -1.11252e68 q^{44} -3.14585e68 q^{45} -7.49306e68 q^{46} +2.73088e69 q^{47} +2.05548e69 q^{48} -1.35487e70 q^{49} -7.14391e69 q^{50} -5.74785e70 q^{51} -5.34654e70 q^{52} -2.76408e71 q^{53} -4.93001e70 q^{54} +1.93733e72 q^{55} +2.00412e71 q^{56} +1.65614e73 q^{57} -8.24645e72 q^{58} +9.63861e72 q^{59} -3.57941e73 q^{60} -2.93192e73 q^{61} -2.33352e74 q^{62} +7.04070e73 q^{63} +1.13078e74 q^{64} +9.31042e74 q^{65} -4.44705e75 q^{66} +9.23644e75 q^{67} -3.16206e75 q^{68} -2.99519e76 q^{69} -3.48995e75 q^{70} +9.92557e76 q^{71} +3.97257e76 q^{72} -1.74202e77 q^{73} +3.00603e77 q^{74} -2.85563e77 q^{75} +9.11092e77 q^{76} -4.33592e77 q^{77} -2.13716e78 q^{78} -9.46058e78 q^{79} -1.96914e78 q^{80} -1.68796e79 q^{81} +5.29175e78 q^{82} +6.81845e79 q^{83} +8.01102e78 q^{84} +5.50638e79 q^{85} -2.19203e80 q^{86} -3.29634e80 q^{87} -2.44645e80 q^{88} -1.00922e81 q^{89} -6.91781e80 q^{90} -2.08375e80 q^{91} -1.64774e81 q^{92} -9.32775e81 q^{93} +6.00527e81 q^{94} -1.58657e82 q^{95} +4.52006e81 q^{96} -3.80445e82 q^{97} -2.97940e82 q^{98} -8.59469e82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6597069766656 q^{2} - 16\!\cdots\!24 q^{3} + 14\!\cdots\!12 q^{4} - 89\!\cdots\!70 q^{5} - 37\!\cdots\!48 q^{6} + 72\!\cdots\!88 q^{7} + 31\!\cdots\!24 q^{8} + 29\!\cdots\!11 q^{9} - 19\!\cdots\!40 q^{10}+ \cdots - 90\!\cdots\!68 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.19902e12 0.707107
\(3\) 8.79012e19 1.39144 0.695718 − 0.718315i \(-0.255087\pi\)
0.695718 + 0.718315i \(0.255087\pi\)
\(4\) 4.83570e24 0.500000
\(5\) −8.42086e28 −0.828135 −0.414067 − 0.910246i \(-0.635892\pi\)
−0.414067 + 0.910246i \(0.635892\pi\)
\(6\) 1.93297e32 0.983893
\(7\) 1.88466e34 0.159832 0.0799162 − 0.996802i \(-0.474535\pi\)
0.0799162 + 0.996802i \(0.474535\pi\)
\(8\) 1.06338e37 0.353553
\(9\) 3.73579e39 0.936091
\(10\) −1.85177e41 −0.585580
\(11\) −2.30063e43 −1.39332 −0.696660 − 0.717401i \(-0.745332\pi\)
−0.696660 + 0.717401i \(0.745332\pi\)
\(12\) 4.25064e44 0.695718
\(13\) −1.10564e46 −0.653076 −0.326538 − 0.945184i \(-0.605882\pi\)
−0.326538 + 0.945184i \(0.605882\pi\)
\(14\) 4.14441e46 0.113019
\(15\) −7.40204e48 −1.15230
\(16\) 2.33840e49 0.250000
\(17\) −6.53898e50 −0.564781 −0.282390 − 0.959300i \(-0.591127\pi\)
−0.282390 + 0.959300i \(0.591127\pi\)
\(18\) 8.21509e51 0.661917
\(19\) 1.88409e53 1.61001 0.805004 − 0.593269i \(-0.202163\pi\)
0.805004 + 0.593269i \(0.202163\pi\)
\(20\) −4.07208e53 −0.414067
\(21\) 1.65664e54 0.222396
\(22\) −5.05915e55 −0.985227
\(23\) −3.40745e56 −1.04888 −0.524438 − 0.851448i \(-0.675725\pi\)
−0.524438 + 0.851448i \(0.675725\pi\)
\(24\) 9.34726e56 0.491947
\(25\) −3.24868e57 −0.314193
\(26\) −2.43132e58 −0.461795
\(27\) −2.24191e58 −0.0889246
\(28\) 9.11366e58 0.0799162
\(29\) −3.75005e60 −0.766514 −0.383257 − 0.923642i \(-0.625198\pi\)
−0.383257 + 0.923642i \(0.625198\pi\)
\(30\) −1.62773e61 −0.814796
\(31\) −1.06116e62 −1.36230 −0.681148 − 0.732146i \(-0.738519\pi\)
−0.681148 + 0.732146i \(0.738519\pi\)
\(32\) 5.14220e61 0.176777
\(33\) −2.02229e63 −1.93872
\(34\) −1.43794e63 −0.399360
\(35\) −1.58705e63 −0.132363
\(36\) 1.80652e64 0.468046
\(37\) 1.36698e65 1.13603 0.568017 − 0.823017i \(-0.307711\pi\)
0.568017 + 0.823017i \(0.307711\pi\)
\(38\) 4.14317e65 1.13845
\(39\) −9.71870e65 −0.908713
\(40\) −8.95459e65 −0.292790
\(41\) 2.40641e66 0.282384 0.141192 − 0.989982i \(-0.454907\pi\)
0.141192 + 0.989982i \(0.454907\pi\)
\(42\) 3.64299e66 0.157258
\(43\) −9.96821e67 −1.62060 −0.810301 − 0.586015i \(-0.800696\pi\)
−0.810301 + 0.586015i \(0.800696\pi\)
\(44\) −1.11252e68 −0.696660
\(45\) −3.14585e68 −0.775210
\(46\) −7.49306e68 −0.741668
\(47\) 2.73088e69 1.10724 0.553619 − 0.832770i \(-0.313246\pi\)
0.553619 + 0.832770i \(0.313246\pi\)
\(48\) 2.05548e69 0.347859
\(49\) −1.35487e70 −0.974454
\(50\) −7.14391e69 −0.222168
\(51\) −5.74785e70 −0.785856
\(52\) −5.34654e70 −0.326538
\(53\) −2.76408e71 −0.765778 −0.382889 − 0.923794i \(-0.625071\pi\)
−0.382889 + 0.923794i \(0.625071\pi\)
\(54\) −4.93001e70 −0.0628792
\(55\) 1.93733e72 1.15386
\(56\) 2.00412e71 0.0565093
\(57\) 1.65614e73 2.24022
\(58\) −8.24645e72 −0.542007
\(59\) 9.63861e72 0.311642 0.155821 − 0.987785i \(-0.450198\pi\)
0.155821 + 0.987785i \(0.450198\pi\)
\(60\) −3.57941e73 −0.576148
\(61\) −2.93192e73 −0.237664 −0.118832 − 0.992914i \(-0.537915\pi\)
−0.118832 + 0.992914i \(0.537915\pi\)
\(62\) −2.33352e74 −0.963289
\(63\) 7.04070e73 0.149618
\(64\) 1.13078e74 0.125000
\(65\) 9.31042e74 0.540835
\(66\) −4.44705e75 −1.37088
\(67\) 9.23644e75 1.52546 0.762729 − 0.646718i \(-0.223859\pi\)
0.762729 + 0.646718i \(0.223859\pi\)
\(68\) −3.16206e75 −0.282390
\(69\) −2.99519e76 −1.45944
\(70\) −3.48995e75 −0.0935946
\(71\) 9.92557e76 1.47752 0.738761 − 0.673967i \(-0.235411\pi\)
0.738761 + 0.673967i \(0.235411\pi\)
\(72\) 3.97257e76 0.330958
\(73\) −1.74202e77 −0.818757 −0.409378 − 0.912365i \(-0.634254\pi\)
−0.409378 + 0.912365i \(0.634254\pi\)
\(74\) 3.00603e77 0.803298
\(75\) −2.85563e77 −0.437179
\(76\) 9.11092e77 0.805004
\(77\) −4.33592e77 −0.222698
\(78\) −2.13716e78 −0.642557
\(79\) −9.46058e78 −1.67646 −0.838230 − 0.545317i \(-0.816409\pi\)
−0.838230 + 0.545317i \(0.816409\pi\)
\(80\) −1.96914e78 −0.207034
\(81\) −1.68796e79 −1.05982
\(82\) 5.29175e78 0.199675
\(83\) 6.81845e79 1.55577 0.777885 − 0.628407i \(-0.216293\pi\)
0.777885 + 0.628407i \(0.216293\pi\)
\(84\) 8.01102e78 0.111198
\(85\) 5.50638e79 0.467715
\(86\) −2.19203e80 −1.14594
\(87\) −3.29634e80 −1.06655
\(88\) −2.44645e80 −0.492613
\(89\) −1.00922e81 −1.27146 −0.635728 − 0.771913i \(-0.719300\pi\)
−0.635728 + 0.771913i \(0.719300\pi\)
\(90\) −6.91781e80 −0.548156
\(91\) −2.08375e80 −0.104383
\(92\) −1.64774e81 −0.524438
\(93\) −9.32775e81 −1.89555
\(94\) 6.00527e81 0.782936
\(95\) −1.58657e82 −1.33330
\(96\) 4.52006e81 0.245973
\(97\) −3.80445e82 −1.34668 −0.673340 − 0.739333i \(-0.735141\pi\)
−0.673340 + 0.739333i \(0.735141\pi\)
\(98\) −2.97940e82 −0.689043
\(99\) −8.59469e82 −1.30428
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 2.84.a.b.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.84.a.b.1.3 ✓ 3 1.1 even 1 trivial