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Properties

Label 2.84.a.a.1.3
Level $2$
Weight $84$
Character 2.1
Self dual yes
Analytic conductor $87.254$
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] = [2,84,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: \(0\)
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: \( x^{3} - x^{2} - 287609867501924274375802127400x - 41230865304567060522794640394926417995512500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{7}\cdot 5^{3}\cdot 7\cdot 11\cdot 17 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.97202e14\) 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)
 
\(f(q)\) \(=\) \(q-2.19902e12 q^{2} +2.32406e18 q^{3} +4.83570e24 q^{4} +1.63642e29 q^{5} -5.11067e30 q^{6} -8.77702e33 q^{7} -1.06338e37 q^{8} -3.98544e39 q^{9} -3.59853e41 q^{10} -4.41257e42 q^{11} +1.12385e43 q^{12} -1.03328e45 q^{13} +1.93009e46 q^{14} +3.80315e47 q^{15} +2.33840e49 q^{16} +7.89598e50 q^{17} +8.76407e51 q^{18} +1.66335e53 q^{19} +7.91325e53 q^{20} -2.03984e52 q^{21} +9.70335e54 q^{22} -3.15448e56 q^{23} -2.47137e55 q^{24} +1.64390e58 q^{25} +2.27221e57 q^{26} -1.85374e58 q^{27} -4.24431e58 q^{28} -7.28891e60 q^{29} -8.36321e59 q^{30} +1.29840e62 q^{31} -5.14220e61 q^{32} -1.02551e61 q^{33} -1.73635e63 q^{34} -1.43629e63 q^{35} -1.92724e64 q^{36} +1.79886e65 q^{37} -3.65774e65 q^{38} -2.40141e63 q^{39} -1.74014e66 q^{40} +3.38501e66 q^{41} +4.48565e64 q^{42} -3.48336e67 q^{43} -2.13379e67 q^{44} -6.52186e68 q^{45} +6.93677e68 q^{46} -1.23539e69 q^{47} +5.43460e67 q^{48} -1.38269e70 q^{49} -3.61498e70 q^{50} +1.83508e69 q^{51} -4.99664e69 q^{52} +3.80797e71 q^{53} +4.07641e70 q^{54} -7.22083e71 q^{55} +9.33333e70 q^{56} +3.86573e71 q^{57} +1.60285e73 q^{58} +2.39580e73 q^{59} +1.83909e72 q^{60} -3.26424e73 q^{61} -2.85520e74 q^{62} +3.49803e73 q^{63} +1.13078e74 q^{64} -1.69088e74 q^{65} +2.25512e73 q^{66} -5.45591e75 q^{67} +3.81826e75 q^{68} -7.33121e74 q^{69} +3.15844e75 q^{70} +6.95274e76 q^{71} +4.23804e76 q^{72} +9.48153e76 q^{73} -3.95574e77 q^{74} +3.82053e76 q^{75} +8.04347e77 q^{76} +3.87292e76 q^{77} +5.28076e75 q^{78} -8.48354e78 q^{79} +3.82661e78 q^{80} +1.58622e79 q^{81} -7.44371e78 q^{82} +4.88119e79 q^{83} -9.86404e76 q^{84} +1.29212e80 q^{85} +7.66000e79 q^{86} -1.69399e79 q^{87} +4.69225e79 q^{88} +1.14577e81 q^{89} +1.43417e81 q^{90} +9.06912e78 q^{91} -1.52541e81 q^{92} +3.01755e80 q^{93} +2.71666e81 q^{94} +2.72194e82 q^{95} -1.19508e80 q^{96} +5.10935e82 q^{97} +3.04056e82 q^{98} +1.75860e82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6597069766656 q^{2} - 10\!\cdots\!76 q^{3} + 14\!\cdots\!12 q^{4} - 30\!\cdots\!50 q^{5} + 22\!\cdots\!52 q^{6} - 56\!\cdots\!88 q^{7} - 31\!\cdots\!24 q^{8} - 12\!\cdots\!89 q^{9} + 66\!\cdots\!00 q^{10}+ \cdots + 19\!\cdots\!32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.19902e12 −0.707107
\(3\) 2.32406e18 0.0367888 0.0183944 − 0.999831i \(-0.494145\pi\)
0.0183944 + 0.999831i \(0.494145\pi\)
\(4\) 4.83570e24 0.500000
\(5\) 1.63642e29 1.60931 0.804656 − 0.593742i \(-0.202350\pi\)
0.804656 + 0.593742i \(0.202350\pi\)
\(6\) −5.11067e30 −0.0260136
\(7\) −8.77702e33 −0.0744352 −0.0372176 − 0.999307i \(-0.511849\pi\)
−0.0372176 + 0.999307i \(0.511849\pi\)
\(8\) −1.06338e37 −0.353553
\(9\) −3.98544e39 −0.998647
\(10\) −3.59853e41 −1.13796
\(11\) −4.41257e42 −0.267236 −0.133618 − 0.991033i \(-0.542660\pi\)
−0.133618 + 0.991033i \(0.542660\pi\)
\(12\) 1.12385e43 0.0183944
\(13\) −1.03328e45 −0.0610336 −0.0305168 − 0.999534i \(-0.509715\pi\)
−0.0305168 + 0.999534i \(0.509715\pi\)
\(14\) 1.93009e46 0.0526336
\(15\) 3.80315e47 0.0592047
\(16\) 2.33840e49 0.250000
\(17\) 7.89598e50 0.681987 0.340993 − 0.940066i \(-0.389237\pi\)
0.340993 + 0.940066i \(0.389237\pi\)
\(18\) 8.76407e51 0.706150
\(19\) 1.66335e53 1.42138 0.710688 − 0.703507i \(-0.248384\pi\)
0.710688 + 0.703507i \(0.248384\pi\)
\(20\) 7.91325e53 0.804656
\(21\) −2.03984e52 −0.00273838
\(22\) 9.70335e54 0.188965
\(23\) −3.15448e56 −0.971007 −0.485504 − 0.874235i \(-0.661364\pi\)
−0.485504 + 0.874235i \(0.661364\pi\)
\(24\) −2.47137e55 −0.0130068
\(25\) 1.64390e58 1.58988
\(26\) 2.27221e57 0.0431573
\(27\) −1.85374e58 −0.0735279
\(28\) −4.24431e58 −0.0372176
\(29\) −7.28891e60 −1.48986 −0.744930 − 0.667142i \(-0.767517\pi\)
−0.744930 + 0.667142i \(0.767517\pi\)
\(30\) −8.36321e59 −0.0418640
\(31\) 1.29840e62 1.66685 0.833425 − 0.552632i \(-0.186377\pi\)
0.833425 + 0.552632i \(0.186377\pi\)
\(32\) −5.14220e61 −0.176777
\(33\) −1.02551e61 −0.00983131
\(34\) −1.73635e63 −0.482238
\(35\) −1.43629e63 −0.119789
\(36\) −1.92724e64 −0.499323
\(37\) 1.79886e65 1.49495 0.747475 − 0.664290i \(-0.231266\pi\)
0.747475 + 0.664290i \(0.231266\pi\)
\(38\) −3.65774e65 −1.00506
\(39\) −2.40141e63 −0.00224536
\(40\) −1.74014e66 −0.568978
\(41\) 3.38501e66 0.397219 0.198609 − 0.980079i \(-0.436358\pi\)
0.198609 + 0.980079i \(0.436358\pi\)
\(42\) 4.48565e64 0.00193633
\(43\) −3.48336e67 −0.566315 −0.283157 − 0.959073i \(-0.591382\pi\)
−0.283157 + 0.959073i \(0.591382\pi\)
\(44\) −2.13379e67 −0.133618
\(45\) −6.52186e68 −1.60713
\(46\) 6.93677e68 0.686606
\(47\) −1.23539e69 −0.500892 −0.250446 − 0.968131i \(-0.580577\pi\)
−0.250446 + 0.968131i \(0.580577\pi\)
\(48\) 5.43460e67 0.00919721
\(49\) −1.38269e70 −0.994459
\(50\) −3.61498e70 −1.12422
\(51\) 1.83508e69 0.0250895
\(52\) −4.99664e69 −0.0305168
\(53\) 3.80797e71 1.05498 0.527491 − 0.849560i \(-0.323133\pi\)
0.527491 + 0.849560i \(0.323133\pi\)
\(54\) 4.07641e70 0.0519921
\(55\) −7.22083e71 −0.430067
\(56\) 9.33333e70 0.0263168
\(57\) 3.86573e71 0.0522908
\(58\) 1.60285e73 1.05349
\(59\) 2.39580e73 0.774627 0.387313 − 0.921948i \(-0.373403\pi\)
0.387313 + 0.921948i \(0.373403\pi\)
\(60\) 1.83909e72 0.0296023
\(61\) −3.26424e73 −0.264602 −0.132301 − 0.991210i \(-0.542237\pi\)
−0.132301 + 0.991210i \(0.542237\pi\)
\(62\) −2.85520e74 −1.17864
\(63\) 3.49803e73 0.0743345
\(64\) 1.13078e74 0.125000
\(65\) −1.69088e74 −0.0982221
\(66\) 2.25512e73 0.00695179
\(67\) −5.45591e75 −0.901079 −0.450539 − 0.892757i \(-0.648768\pi\)
−0.450539 + 0.892757i \(0.648768\pi\)
\(68\) 3.81826e75 0.340993
\(69\) −7.33121e74 −0.0357222
\(70\) 3.15844e75 0.0847039
\(71\) 6.95274e76 1.03499 0.517493 − 0.855688i \(-0.326865\pi\)
0.517493 + 0.855688i \(0.326865\pi\)
\(72\) 4.23804e76 0.353075
\(73\) 9.48153e76 0.445636 0.222818 − 0.974860i \(-0.428474\pi\)
0.222818 + 0.974860i \(0.428474\pi\)
\(74\) −3.95574e77 −1.05709
\(75\) 3.82053e76 0.0584900
\(76\) 8.04347e77 0.710688
\(77\) 3.87292e76 0.0198918
\(78\) 5.28076e75 0.00158771
\(79\) −8.48354e78 −1.50333 −0.751663 − 0.659548i \(-0.770748\pi\)
−0.751663 + 0.659548i \(0.770748\pi\)
\(80\) 3.82661e78 0.402328
\(81\) 1.58622e79 0.995942
\(82\) −7.44371e78 −0.280876
\(83\) 4.88119e79 1.11374 0.556871 − 0.830599i \(-0.312002\pi\)
0.556871 + 0.830599i \(0.312002\pi\)
\(84\) −9.86404e76 −0.00136919
\(85\) 1.29212e80 1.09753
\(86\) 7.66000e79 0.400445
\(87\) −1.69399e79 −0.0548102
\(88\) 4.69225e79 0.0944823
\(89\) 1.14577e81 1.44348 0.721742 − 0.692162i \(-0.243342\pi\)
0.721742 + 0.692162i \(0.243342\pi\)
\(90\) 1.43417e81 1.13642
\(91\) 9.06912e78 0.00454305
\(92\) −1.52541e81 −0.485504
\(93\) 3.01755e80 0.0613215
\(94\) 2.71666e81 0.354184
\(95\) 2.72194e82 2.28744
\(96\) −1.19508e80 −0.00650341
\(97\) 5.10935e82 1.80858 0.904292 − 0.426915i \(-0.140400\pi\)
0.904292 + 0.426915i \(0.140400\pi\)
\(98\) 3.04056e82 0.703189
\(99\) 1.75860e82 0.266875
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.a.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.84.a.a.1.3 ✓ 3 1.1 even 1 trivial