Properties

Label 84.8.a.c.1.1
Level $84$
Weight $8$
Character 84.1
Self dual yes
Analytic conductor $26.240$
Analytic rank $0$
Dimension $2$
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] = [84,8,Mod(1,84)] 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("84.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 84.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.2403421407\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3649}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(30.7035\) of defining polynomial
Character \(\chi\) \(=\) 84.1

$q$-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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -230.442 q^{5} +343.000 q^{7} +729.000 q^{9} -5027.09 q^{11} -13772.6 q^{13} +6221.93 q^{15} +32474.7 q^{17} +9651.95 q^{19} -9261.00 q^{21} +32034.5 q^{23} -25021.6 q^{25} -19683.0 q^{27} +103672. q^{29} +241936. q^{31} +135731. q^{33} -79041.5 q^{35} -127147. q^{37} +371860. q^{39} +607138. q^{41} +443862. q^{43} -167992. q^{45} +691778. q^{47} +117649. q^{49} -876817. q^{51} +672824. q^{53} +1.15845e6 q^{55} -260603. q^{57} -2.58646e6 q^{59} +1.53593e6 q^{61} +250047. q^{63} +3.17378e6 q^{65} -4.20823e6 q^{67} -864931. q^{69} +1.51772e6 q^{71} +5.47779e6 q^{73} +675584. q^{75} -1.72429e6 q^{77} -6.75855e6 q^{79} +531441. q^{81} +8.36612e6 q^{83} -7.48353e6 q^{85} -2.79914e6 q^{87} -5.17007e6 q^{89} -4.72400e6 q^{91} -6.53227e6 q^{93} -2.22421e6 q^{95} -1.06286e7 q^{97} -3.66475e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} + 264 q^{5} + 686 q^{7} + 1458 q^{9} - 4980 q^{11} - 10148 q^{13} - 7128 q^{15} + 17832 q^{17} + 6256 q^{19} - 18522 q^{21} + 14052 q^{23} + 141326 q^{25} - 39366 q^{27} + 243588 q^{29} + 470824 q^{31}+ \cdots - 3630420 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −230.442 −0.824453 −0.412227 − 0.911081i \(-0.635249\pi\)
−0.412227 + 0.911081i \(0.635249\pi\)
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −5027.09 −1.13879 −0.569393 − 0.822065i \(-0.692822\pi\)
−0.569393 + 0.822065i \(0.692822\pi\)
\(12\) 0 0
\(13\) −13772.6 −1.73866 −0.869329 − 0.494234i \(-0.835449\pi\)
−0.869329 + 0.494234i \(0.835449\pi\)
\(14\) 0 0
\(15\) 6221.93 0.475998
\(16\) 0 0
\(17\) 32474.7 1.60315 0.801575 − 0.597894i \(-0.203996\pi\)
0.801575 + 0.597894i \(0.203996\pi\)
\(18\) 0 0
\(19\) 9651.95 0.322833 0.161416 − 0.986886i \(-0.448394\pi\)
0.161416 + 0.986886i \(0.448394\pi\)
\(20\) 0 0
\(21\) −9261.00 −0.218218
\(22\) 0 0
\(23\) 32034.5 0.548997 0.274499 − 0.961587i \(-0.411488\pi\)
0.274499 + 0.961587i \(0.411488\pi\)
\(24\) 0 0
\(25\) −25021.6 −0.320277
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) 103672. 0.789347 0.394674 − 0.918821i \(-0.370858\pi\)
0.394674 + 0.918821i \(0.370858\pi\)
\(30\) 0 0
\(31\) 241936. 1.45859 0.729297 − 0.684197i \(-0.239847\pi\)
0.729297 + 0.684197i \(0.239847\pi\)
\(32\) 0 0
\(33\) 135731. 0.657479
\(34\) 0 0
\(35\) −79041.5 −0.311614
\(36\) 0 0
\(37\) −127147. −0.412666 −0.206333 − 0.978482i \(-0.566153\pi\)
−0.206333 + 0.978482i \(0.566153\pi\)
\(38\) 0 0
\(39\) 371860. 1.00381
\(40\) 0 0
\(41\) 607138. 1.37576 0.687882 − 0.725823i \(-0.258541\pi\)
0.687882 + 0.725823i \(0.258541\pi\)
\(42\) 0 0
\(43\) 443862. 0.851350 0.425675 − 0.904876i \(-0.360037\pi\)
0.425675 + 0.904876i \(0.360037\pi\)
\(44\) 0 0
\(45\) −167992. −0.274818
\(46\) 0 0
\(47\) 691778. 0.971905 0.485953 − 0.873985i \(-0.338473\pi\)
0.485953 + 0.873985i \(0.338473\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −876817. −0.925579
\(52\) 0 0
\(53\) 672824. 0.620778 0.310389 − 0.950610i \(-0.399541\pi\)
0.310389 + 0.950610i \(0.399541\pi\)
\(54\) 0 0
\(55\) 1.15845e6 0.938877
\(56\) 0 0
\(57\) −260603. −0.186388
\(58\) 0 0
\(59\) −2.58646e6 −1.63955 −0.819775 − 0.572686i \(-0.805901\pi\)
−0.819775 + 0.572686i \(0.805901\pi\)
\(60\) 0 0
\(61\) 1.53593e6 0.866397 0.433198 − 0.901299i \(-0.357385\pi\)
0.433198 + 0.901299i \(0.357385\pi\)
\(62\) 0 0
\(63\) 250047. 0.125988
\(64\) 0 0
\(65\) 3.17378e6 1.43344
\(66\) 0 0
\(67\) −4.20823e6 −1.70937 −0.854687 − 0.519143i \(-0.826251\pi\)
−0.854687 + 0.519143i \(0.826251\pi\)
\(68\) 0 0
\(69\) −864931. −0.316964
\(70\) 0 0
\(71\) 1.51772e6 0.503254 0.251627 − 0.967824i \(-0.419034\pi\)
0.251627 + 0.967824i \(0.419034\pi\)
\(72\) 0 0
\(73\) 5.47779e6 1.64807 0.824034 − 0.566540i \(-0.191718\pi\)
0.824034 + 0.566540i \(0.191718\pi\)
\(74\) 0 0
\(75\) 675584. 0.184912
\(76\) 0 0
\(77\) −1.72429e6 −0.430421
\(78\) 0 0
\(79\) −6.75855e6 −1.54226 −0.771132 − 0.636675i \(-0.780309\pi\)
−0.771132 + 0.636675i \(0.780309\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 8.36612e6 1.60602 0.803009 − 0.595966i \(-0.203231\pi\)
0.803009 + 0.595966i \(0.203231\pi\)
\(84\) 0 0
\(85\) −7.48353e6 −1.32172
\(86\) 0 0
\(87\) −2.79914e6 −0.455730
\(88\) 0 0
\(89\) −5.17007e6 −0.777377 −0.388688 − 0.921369i \(-0.627072\pi\)
−0.388688 + 0.921369i \(0.627072\pi\)
\(90\) 0 0
\(91\) −4.72400e6 −0.657151
\(92\) 0 0
\(93\) −6.53227e6 −0.842120
\(94\) 0 0
\(95\) −2.22421e6 −0.266160
\(96\) 0 0
\(97\) −1.06286e7 −1.18242 −0.591212 − 0.806516i \(-0.701350\pi\)
−0.591212 + 0.806516i \(0.701350\pi\)
\(98\) 0 0
\(99\) −3.66475e6 −0.379596
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 84.8.a.c.1.1 ✓ 2
3.2 odd 2 252.8.a.c.1.2 2
4.3 odd 2 336.8.a.q.1.1 2
7.2 even 3 588.8.i.k.361.2 4
7.3 odd 6 588.8.i.j.373.1 4
7.4 even 3 588.8.i.k.373.2 4
7.5 odd 6 588.8.i.j.361.1 4
7.6 odd 2 588.8.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.c.1.1 ✓ 2 1.1 even 1 trivial
252.8.a.c.1.2 2 3.2 odd 2
336.8.a.q.1.1 2 4.3 odd 2
588.8.a.f.1.2 2 7.6 odd 2
588.8.i.j.361.1 4 7.5 odd 6
588.8.i.j.373.1 4 7.3 odd 6
588.8.i.k.361.2 4 7.2 even 3
588.8.i.k.373.2 4 7.4 even 3