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Properties

Label 2.84.a.b.1.1
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 1.1
Root \(2.43002e14\) 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)
 
\(f(q)\) \(=\) \(q+2.19902e12 q^{2} -8.12053e19 q^{3} +4.83570e24 q^{4} -9.96318e28 q^{5} -1.78572e32 q^{6} -5.66053e34 q^{7} +1.06338e37 q^{8} +2.60347e39 q^{9} -2.19093e41 q^{10} -1.84354e42 q^{11} -3.92685e44 q^{12} +2.57649e46 q^{13} -1.24476e47 q^{14} +8.09064e48 q^{15} +2.33840e49 q^{16} +6.69632e50 q^{17} +5.72509e51 q^{18} +7.59111e52 q^{19} -4.81790e53 q^{20} +4.59665e54 q^{21} -4.05399e54 q^{22} -4.69781e56 q^{23} -8.63523e56 q^{24} -4.13261e56 q^{25} +5.66575e58 q^{26} +1.12662e59 q^{27} -2.73726e59 q^{28} +2.67824e60 q^{29} +1.77915e61 q^{30} +1.41131e62 q^{31} +5.14220e61 q^{32} +1.49705e62 q^{33} +1.47254e63 q^{34} +5.63969e63 q^{35} +1.25896e64 q^{36} -1.78238e65 q^{37} +1.66930e65 q^{38} -2.09224e66 q^{39} -1.05947e66 q^{40} +4.67079e66 q^{41} +1.01081e67 q^{42} +3.40824e67 q^{43} -8.91482e66 q^{44} -2.59388e68 q^{45} -1.03306e69 q^{46} -2.61516e69 q^{47} -1.89891e69 q^{48} -1.06998e70 q^{49} -9.08771e68 q^{50} -5.43777e70 q^{51} +1.24591e71 q^{52} +1.87235e71 q^{53} +2.47746e71 q^{54} +1.83675e71 q^{55} -6.01931e71 q^{56} -6.16439e72 q^{57} +5.88951e72 q^{58} +4.41947e73 q^{59} +3.91239e73 q^{60} -2.05477e74 q^{61} +3.10351e74 q^{62} -1.47370e74 q^{63} +1.13078e74 q^{64} -2.56700e75 q^{65} +3.29206e74 q^{66} -1.07076e76 q^{67} +3.23814e75 q^{68} +3.81487e76 q^{69} +1.24018e76 q^{70} +4.83952e76 q^{71} +2.76848e76 q^{72} -1.47922e77 q^{73} -3.91949e77 q^{74} +3.35590e76 q^{75} +3.67084e77 q^{76} +1.04354e77 q^{77} -4.60089e78 q^{78} +6.65039e78 q^{79} -2.32979e78 q^{80} -1.95388e79 q^{81} +1.02712e79 q^{82} -3.99360e79 q^{83} +2.22281e79 q^{84} -6.67166e79 q^{85} +7.49481e79 q^{86} -2.17487e80 q^{87} -1.96039e79 q^{88} -5.21870e80 q^{89} -5.70401e80 q^{90} -1.45843e81 q^{91} -2.27172e81 q^{92} -1.14606e82 q^{93} -5.75080e81 q^{94} -7.56316e81 q^{95} -4.17574e81 q^{96} +1.44659e82 q^{97} -2.35290e82 q^{98} -4.79960e81 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6597069766656 q^{2} - 16\!\cdots\!24 q^{3} + 14\!\cdots\!12 q^{4} - 89\!\cdots\!70 q^{5} - 37\!\cdots\!48 q^{6} + 72\!\cdots\!88 q^{7} + 31\!\cdots\!24 q^{8} + 29\!\cdots\!11 q^{9} - 19\!\cdots\!40 q^{10}+ \cdots - 90\!\cdots\!68 q^{99}+O(q^{100}) \) 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
\(3\) −8.12053e19 −1.28544 −0.642721 − 0.766100i \(-0.722195\pi\)
−0.642721 + 0.766100i \(0.722195\pi\)
\(4\) 4.83570e24 0.500000
\(5\) −9.96318e28 −0.979812 −0.489906 − 0.871775i \(-0.662969\pi\)
−0.489906 + 0.871775i \(0.662969\pi\)
\(6\) −1.78572e32 −0.908945
\(7\) −5.66053e34 −0.480052 −0.240026 − 0.970766i \(-0.577156\pi\)
−0.240026 + 0.970766i \(0.577156\pi\)
\(8\) 1.06338e37 0.353553
\(9\) 2.60347e39 0.652362
\(10\) −2.19093e41 −0.692832
\(11\) −1.84354e42 −0.111649 −0.0558247 − 0.998441i \(-0.517779\pi\)
−0.0558247 + 0.998441i \(0.517779\pi\)
\(12\) −3.92685e44 −0.642721
\(13\) 2.57649e46 1.52187 0.760937 − 0.648826i \(-0.224740\pi\)
0.760937 + 0.648826i \(0.224740\pi\)
\(14\) −1.24476e47 −0.339448
\(15\) 8.09064e48 1.25949
\(16\) 2.33840e49 0.250000
\(17\) 6.69632e50 0.578370 0.289185 − 0.957273i \(-0.406616\pi\)
0.289185 + 0.957273i \(0.406616\pi\)
\(18\) 5.72509e51 0.461289
\(19\) 7.59111e52 0.648681 0.324340 − 0.945940i \(-0.394858\pi\)
0.324340 + 0.945940i \(0.394858\pi\)
\(20\) −4.81790e53 −0.489906
\(21\) 4.59665e54 0.617079
\(22\) −4.05399e54 −0.0789480
\(23\) −4.69781e56 −1.44607 −0.723036 − 0.690810i \(-0.757254\pi\)
−0.723036 + 0.690810i \(0.757254\pi\)
\(24\) −8.63523e56 −0.454472
\(25\) −4.13261e56 −0.0399682
\(26\) 5.66575e58 1.07613
\(27\) 1.12662e59 0.446869
\(28\) −2.73726e59 −0.240026
\(29\) 2.67824e60 0.547435 0.273718 − 0.961810i \(-0.411747\pi\)
0.273718 + 0.961810i \(0.411747\pi\)
\(30\) 1.77915e61 0.890595
\(31\) 1.41131e62 1.81181 0.905907 − 0.423477i \(-0.139191\pi\)
0.905907 + 0.423477i \(0.139191\pi\)
\(32\) 5.14220e61 0.176777
\(33\) 1.49705e62 0.143519
\(34\) 1.47254e63 0.408969
\(35\) 5.63969e63 0.470361
\(36\) 1.25896e64 0.326181
\(37\) −1.78238e65 −1.48125 −0.740625 − 0.671918i \(-0.765471\pi\)
−0.740625 + 0.671918i \(0.765471\pi\)
\(38\) 1.66930e65 0.458687
\(39\) −2.09224e66 −1.95628
\(40\) −1.05947e66 −0.346416
\(41\) 4.67079e66 0.548101 0.274051 − 0.961715i \(-0.411636\pi\)
0.274051 + 0.961715i \(0.411636\pi\)
\(42\) 1.01081e67 0.436341
\(43\) 3.40824e67 0.554102 0.277051 − 0.960855i \(-0.410643\pi\)
0.277051 + 0.960855i \(0.410643\pi\)
\(44\) −8.91482e66 −0.0558247
\(45\) −2.59388e68 −0.639192
\(46\) −1.03306e69 −1.02253
\(47\) −2.61516e69 −1.06032 −0.530160 − 0.847897i \(-0.677868\pi\)
−0.530160 + 0.847897i \(0.677868\pi\)
\(48\) −1.89891e69 −0.321361
\(49\) −1.06998e70 −0.769550
\(50\) −9.08771e68 −0.0282618
\(51\) −5.43777e70 −0.743461
\(52\) 1.24591e71 0.760937
\(53\) 1.87235e71 0.518726 0.259363 − 0.965780i \(-0.416487\pi\)
0.259363 + 0.965780i \(0.416487\pi\)
\(54\) 2.47746e71 0.315984
\(55\) 1.83675e71 0.109395
\(56\) −6.01931e71 −0.169724
\(57\) −6.16439e72 −0.833842
\(58\) 5.88951e72 0.387095
\(59\) 4.41947e73 1.42893 0.714466 − 0.699670i \(-0.246670\pi\)
0.714466 + 0.699670i \(0.246670\pi\)
\(60\) 3.91239e73 0.629746
\(61\) −2.05477e74 −1.66561 −0.832807 − 0.553563i \(-0.813268\pi\)
−0.832807 + 0.553563i \(0.813268\pi\)
\(62\) 3.10351e74 1.28115
\(63\) −1.47370e74 −0.313168
\(64\) 1.13078e74 0.125000
\(65\) −2.56700e75 −1.49115
\(66\) 3.29206e74 0.101483
\(67\) −1.07076e76 −1.76842 −0.884212 − 0.467086i \(-0.845304\pi\)
−0.884212 + 0.467086i \(0.845304\pi\)
\(68\) 3.23814e75 0.289185
\(69\) 3.81487e76 1.85884
\(70\) 1.24018e76 0.332595
\(71\) 4.83952e76 0.720412 0.360206 − 0.932873i \(-0.382706\pi\)
0.360206 + 0.932873i \(0.382706\pi\)
\(72\) 2.76848e76 0.230645
\(73\) −1.47922e77 −0.695237 −0.347618 − 0.937636i \(-0.613010\pi\)
−0.347618 + 0.937636i \(0.613010\pi\)
\(74\) −3.91949e77 −1.04740
\(75\) 3.35590e76 0.0513768
\(76\) 3.67084e77 0.324340
\(77\) 1.04354e77 0.0535975
\(78\) −4.60089e78 −1.38330
\(79\) 6.65039e78 1.17848 0.589241 − 0.807957i \(-0.299427\pi\)
0.589241 + 0.807957i \(0.299427\pi\)
\(80\) −2.32979e78 −0.244953
\(81\) −1.95388e79 −1.22679
\(82\) 1.02712e79 0.387566
\(83\) −3.99360e79 −0.911220 −0.455610 − 0.890180i \(-0.650579\pi\)
−0.455610 + 0.890180i \(0.650579\pi\)
\(84\) 2.22281e79 0.308540
\(85\) −6.67166e79 −0.566694
\(86\) 7.49481e79 0.391809
\(87\) −2.17487e80 −0.703696
\(88\) −1.96039e79 −0.0394740
\(89\) −5.21870e80 −0.657472 −0.328736 − 0.944422i \(-0.606623\pi\)
−0.328736 + 0.944422i \(0.606623\pi\)
\(90\) −5.70401e80 −0.451977
\(91\) −1.45843e81 −0.730579
\(92\) −2.27172e81 −0.723036
\(93\) −1.14606e82 −2.32898
\(94\) −5.75080e81 −0.749760
\(95\) −7.56316e81 −0.635585
\(96\) −4.17574e81 −0.227236
\(97\) 1.44659e82 0.512059 0.256029 − 0.966669i \(-0.417586\pi\)
0.256029 + 0.966669i \(0.417586\pi\)
\(98\) −2.35290e82 −0.544154
\(99\) −4.79960e81 −0.0728358
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.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.84.a.b.1.1 ✓ 3 1.1 even 1 trivial