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Properties

Label 2.84.a.a.1.2
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 1.2
Root \(-1.56747e14\) 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)
 
\(f(q)\) \(=\) \(q-2.19902e12 q^{2} -7.32556e17 q^{3} +4.83570e24 q^{4} -1.23719e29 q^{5} +1.61091e30 q^{6} +6.61589e34 q^{7} -1.06338e37 q^{8} -3.99030e39 q^{9} +2.72062e41 q^{10} -2.35451e43 q^{11} -3.54242e42 q^{12} +2.17555e46 q^{13} -1.45485e47 q^{14} +9.06313e46 q^{15} +2.33840e49 q^{16} -2.06003e51 q^{17} +8.77477e51 q^{18} -1.54338e53 q^{19} -5.98270e53 q^{20} -4.84651e52 q^{21} +5.17762e55 q^{22} +3.09757e56 q^{23} +7.78987e54 q^{24} +4.96671e57 q^{25} -4.78409e58 q^{26} +5.84663e57 q^{27} +3.19925e59 q^{28} -4.67177e59 q^{29} -1.99300e59 q^{30} -1.70498e61 q^{31} -5.14220e61 q^{32} +1.72481e61 q^{33} +4.53006e63 q^{34} -8.18513e63 q^{35} -1.92959e64 q^{36} -9.03184e64 q^{37} +3.39393e65 q^{38} -1.59371e64 q^{39} +1.31561e66 q^{40} -8.21191e66 q^{41} +1.06576e65 q^{42} -5.19673e67 q^{43} -1.13857e68 q^{44} +4.93677e68 q^{45} -6.81164e68 q^{46} -3.72927e69 q^{47} -1.71301e67 q^{48} -9.52692e69 q^{49} -1.09219e70 q^{50} +1.50909e69 q^{51} +1.05203e71 q^{52} -4.87207e71 q^{53} -1.28569e70 q^{54} +2.91298e72 q^{55} -7.03522e71 q^{56} +1.13061e71 q^{57} +1.02733e72 q^{58} +2.59897e73 q^{59} +4.38266e71 q^{60} -2.95096e73 q^{61} +3.74928e73 q^{62} -2.63994e74 q^{63} +1.13078e74 q^{64} -2.69158e75 q^{65} -3.79289e73 q^{66} +9.82765e75 q^{67} -9.96170e75 q^{68} -2.26915e74 q^{69} +1.79993e76 q^{70} -2.74633e76 q^{71} +4.24322e76 q^{72} +1.97027e77 q^{73} +1.98612e77 q^{74} -3.63839e75 q^{75} -7.46332e77 q^{76} -1.55772e78 q^{77} +3.50461e76 q^{78} +2.44890e78 q^{79} -2.89306e78 q^{80} +1.59204e79 q^{81} +1.80582e79 q^{82} -4.50780e79 q^{83} -2.34363e77 q^{84} +2.54866e80 q^{85} +1.14277e80 q^{86} +3.42233e77 q^{87} +2.50374e80 q^{88} +1.00507e81 q^{89} -1.08561e81 q^{90} +1.43932e81 q^{91} +1.49790e81 q^{92} +1.24899e79 q^{93} +8.20075e81 q^{94} +1.90946e82 q^{95} +3.76695e79 q^{96} +2.37614e82 q^{97} +2.09499e82 q^{98} +9.39519e82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6597069766656 q^{2} - 10\!\cdots\!76 q^{3} + 14\!\cdots\!12 q^{4} - 30\!\cdots\!50 q^{5} + 22\!\cdots\!52 q^{6} - 56\!\cdots\!88 q^{7} - 31\!\cdots\!24 q^{8} - 12\!\cdots\!89 q^{9} + 66\!\cdots\!00 q^{10}+ \cdots + 19\!\cdots\!32 q^{99}+O(q^{100}) \) 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
\(3\) −7.32556e17 −0.0115960 −0.00579801 − 0.999983i \(-0.501846\pi\)
−0.00579801 + 0.999983i \(0.501846\pi\)
\(4\) 4.83570e24 0.500000
\(5\) −1.23719e29 −1.21670 −0.608348 − 0.793670i \(-0.708168\pi\)
−0.608348 + 0.793670i \(0.708168\pi\)
\(6\) 1.61091e30 0.00819962
\(7\) 6.61589e34 0.561073 0.280537 − 0.959843i \(-0.409488\pi\)
0.280537 + 0.959843i \(0.409488\pi\)
\(8\) −1.06338e37 −0.353553
\(9\) −3.99030e39 −0.999866
\(10\) 2.72062e41 0.860334
\(11\) −2.35451e43 −1.42595 −0.712974 − 0.701191i \(-0.752652\pi\)
−0.712974 + 0.701191i \(0.752652\pi\)
\(12\) −3.54242e42 −0.00579801
\(13\) 2.17555e46 1.28505 0.642525 − 0.766265i \(-0.277887\pi\)
0.642525 + 0.766265i \(0.277887\pi\)
\(14\) −1.45485e47 −0.396739
\(15\) 9.06313e46 0.0141088
\(16\) 2.33840e49 0.250000
\(17\) −2.06003e51 −1.77928 −0.889638 − 0.456666i \(-0.849043\pi\)
−0.889638 + 0.456666i \(0.849043\pi\)
\(18\) 8.77477e51 0.707012
\(19\) −1.54338e53 −1.31886 −0.659429 − 0.751767i \(-0.729202\pi\)
−0.659429 + 0.751767i \(0.729202\pi\)
\(20\) −5.98270e53 −0.608348
\(21\) −4.84651e52 −0.00650621
\(22\) 5.17762e55 1.00830
\(23\) 3.09757e56 0.953491 0.476745 − 0.879041i \(-0.341816\pi\)
0.476745 + 0.879041i \(0.341816\pi\)
\(24\) 7.78987e54 0.00409981
\(25\) 4.96671e57 0.480350
\(26\) −4.78409e58 −0.908668
\(27\) 5.84663e57 0.0231905
\(28\) 3.19925e59 0.280537
\(29\) −4.67177e59 −0.0954914 −0.0477457 − 0.998860i \(-0.515204\pi\)
−0.0477457 + 0.998860i \(0.515204\pi\)
\(30\) −1.99300e59 −0.00997645
\(31\) −1.70498e61 −0.218881 −0.109441 − 0.993993i \(-0.534906\pi\)
−0.109441 + 0.993993i \(0.534906\pi\)
\(32\) −5.14220e61 −0.176777
\(33\) 1.72481e61 0.0165353
\(34\) 4.53006e63 1.25814
\(35\) −8.18513e63 −0.682656
\(36\) −1.92959e64 −0.499933
\(37\) −9.03184e64 −0.750593 −0.375297 − 0.926905i \(-0.622459\pi\)
−0.375297 + 0.926905i \(0.622459\pi\)
\(38\) 3.39393e65 0.932574
\(39\) −1.59371e64 −0.0149015
\(40\) 1.31561e66 0.430167
\(41\) −8.21191e66 −0.963638 −0.481819 − 0.876271i \(-0.660024\pi\)
−0.481819 + 0.876271i \(0.660024\pi\)
\(42\) 1.06576e65 0.00460059
\(43\) −5.19673e67 −0.844869 −0.422435 − 0.906393i \(-0.638824\pi\)
−0.422435 + 0.906393i \(0.638824\pi\)
\(44\) −1.13857e68 −0.712974
\(45\) 4.93677e68 1.21653
\(46\) −6.81164e68 −0.674220
\(47\) −3.72927e69 −1.51204 −0.756019 − 0.654550i \(-0.772858\pi\)
−0.756019 + 0.654550i \(0.772858\pi\)
\(48\) −1.71301e67 −0.00289900
\(49\) −9.52692e69 −0.685197
\(50\) −1.09219e70 −0.339659
\(51\) 1.50909e69 0.0206325
\(52\) 1.05203e71 0.642525
\(53\) −4.87207e71 −1.34979 −0.674893 − 0.737916i \(-0.735810\pi\)
−0.674893 + 0.737916i \(0.735810\pi\)
\(54\) −1.28569e70 −0.0163981
\(55\) 2.91298e72 1.73495
\(56\) −7.03522e71 −0.198369
\(57\) 1.13061e71 0.0152935
\(58\) 1.02733e72 0.0675226
\(59\) 2.59897e73 0.840316 0.420158 − 0.907451i \(-0.361975\pi\)
0.420158 + 0.907451i \(0.361975\pi\)
\(60\) 4.38266e71 0.00705441
\(61\) −2.95096e73 −0.239208 −0.119604 − 0.992822i \(-0.538162\pi\)
−0.119604 + 0.992822i \(0.538162\pi\)
\(62\) 3.74928e73 0.154772
\(63\) −2.63994e74 −0.560998
\(64\) 1.13078e74 0.125000
\(65\) −2.69158e75 −1.56352
\(66\) −3.79289e73 −0.0116922
\(67\) 9.82765e75 1.62310 0.811550 − 0.584283i \(-0.198624\pi\)
0.811550 + 0.584283i \(0.198624\pi\)
\(68\) −9.96170e75 −0.889638
\(69\) −2.26915e74 −0.0110567
\(70\) 1.79993e76 0.482711
\(71\) −2.74633e76 −0.408819 −0.204410 − 0.978885i \(-0.565527\pi\)
−0.204410 + 0.978885i \(0.565527\pi\)
\(72\) 4.24322e76 0.353506
\(73\) 1.97027e77 0.926035 0.463017 − 0.886349i \(-0.346767\pi\)
0.463017 + 0.886349i \(0.346767\pi\)
\(74\) 1.98612e77 0.530750
\(75\) −3.63839e75 −0.00557015
\(76\) −7.46332e77 −0.659429
\(77\) −1.55772e78 −0.800061
\(78\) 3.50461e76 0.0105369
\(79\) 2.44890e78 0.433957 0.216978 − 0.976176i \(-0.430380\pi\)
0.216978 + 0.976176i \(0.430380\pi\)
\(80\) −2.89306e78 −0.304174
\(81\) 1.59204e79 0.999597
\(82\) 1.80582e79 0.681395
\(83\) −4.50780e79 −1.02855 −0.514273 − 0.857627i \(-0.671938\pi\)
−0.514273 + 0.857627i \(0.671938\pi\)
\(84\) −2.34363e77 −0.00325311
\(85\) 2.54866e80 2.16484
\(86\) 1.14277e80 0.597413
\(87\) 3.42233e77 0.00110732
\(88\) 2.50374e80 0.504149
\(89\) 1.00507e81 1.26623 0.633115 − 0.774058i \(-0.281776\pi\)
0.633115 + 0.774058i \(0.281776\pi\)
\(90\) −1.08561e81 −0.860219
\(91\) 1.43932e81 0.721007
\(92\) 1.49790e81 0.476745
\(93\) 1.24899e79 0.00253815
\(94\) 8.20075e81 1.06917
\(95\) 1.90946e82 1.60465
\(96\) 3.76695e79 0.00204990
\(97\) 2.37614e82 0.841096 0.420548 − 0.907270i \(-0.361838\pi\)
0.420548 + 0.907270i \(0.361838\pi\)
\(98\) 2.09499e82 0.484507
\(99\) 9.39519e82 1.42576
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.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.84.a.a.1.2 ✓ 3 1.1 even 1 trivial