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Properties

Label 84.8.a.a.1.1
Level $84$
Weight $8$
Character 84.1
Self dual yes
Analytic conductor $26.240$
Analytic rank $1$
Dimension $1$
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 = [1,0,-27] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} +100.000 q^{5} -343.000 q^{7} +729.000 q^{9} +2774.00 q^{11} -3294.00 q^{13} -2700.00 q^{15} +5900.00 q^{17} +6644.00 q^{19} +9261.00 q^{21} +1982.00 q^{23} -68125.0 q^{25} -19683.0 q^{27} -208106. q^{29} -117792. q^{31} -74898.0 q^{33} -34300.0 q^{35} -335686. q^{37} +88938.0 q^{39} -265488. q^{41} -93292.0 q^{43} +72900.0 q^{45} -657516. q^{47} +117649. q^{49} -159300. q^{51} -608718. q^{53} +277400. q^{55} -179388. q^{57} -536120. q^{59} -1.79709e6 q^{61} -250047. q^{63} -329400. q^{65} +2.12318e6 q^{67} -53514.0 q^{69} -1.19121e6 q^{71} +1.05643e6 q^{73} +1.83938e6 q^{75} -951482. q^{77} +998484. q^{79} +531441. q^{81} +3.89800e6 q^{83} +590000. q^{85} +5.61886e6 q^{87} -4.62235e6 q^{89} +1.12984e6 q^{91} +3.18038e6 q^{93} +664400. q^{95} +1.52877e7 q^{97} +2.02225e6 q^{99} +O(q^{100})\)

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
\(5\) 100.000 0.357771 0.178885 − 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 2774.00 0.628394 0.314197 − 0.949358i \(-0.398265\pi\)
0.314197 + 0.949358i \(0.398265\pi\)
\(12\) 0 0
\(13\) −3294.00 −0.415836 −0.207918 − 0.978146i \(-0.566669\pi\)
−0.207918 + 0.978146i \(0.566669\pi\)
\(14\) 0 0
\(15\) −2700.00 −0.206559
\(16\) 0 0
\(17\) 5900.00 0.291260 0.145630 − 0.989339i \(-0.453479\pi\)
0.145630 + 0.989339i \(0.453479\pi\)
\(18\) 0 0
\(19\) 6644.00 0.222225 0.111112 − 0.993808i \(-0.464559\pi\)
0.111112 + 0.993808i \(0.464559\pi\)
\(20\) 0 0
\(21\) 9261.00 0.218218
\(22\) 0 0
\(23\) 1982.00 0.0339669 0.0169835 − 0.999856i \(-0.494594\pi\)
0.0169835 + 0.999856i \(0.494594\pi\)
\(24\) 0 0
\(25\) −68125.0 −0.872000
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −208106. −1.58450 −0.792249 − 0.610198i \(-0.791090\pi\)
−0.792249 + 0.610198i \(0.791090\pi\)
\(30\) 0 0
\(31\) −117792. −0.710150 −0.355075 − 0.934838i \(-0.615545\pi\)
−0.355075 + 0.934838i \(0.615545\pi\)
\(32\) 0 0
\(33\) −74898.0 −0.362803
\(34\) 0 0
\(35\) −34300.0 −0.135225
\(36\) 0 0
\(37\) −335686. −1.08950 −0.544750 − 0.838599i \(-0.683375\pi\)
−0.544750 + 0.838599i \(0.683375\pi\)
\(38\) 0 0
\(39\) 88938.0 0.240083
\(40\) 0 0
\(41\) −265488. −0.601591 −0.300796 − 0.953689i \(-0.597252\pi\)
−0.300796 + 0.953689i \(0.597252\pi\)
\(42\) 0 0
\(43\) −93292.0 −0.178939 −0.0894695 − 0.995990i \(-0.528517\pi\)
−0.0894695 + 0.995990i \(0.528517\pi\)
\(44\) 0 0
\(45\) 72900.0 0.119257
\(46\) 0 0
\(47\) −657516. −0.923770 −0.461885 − 0.886940i \(-0.652827\pi\)
−0.461885 + 0.886940i \(0.652827\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −159300. −0.168159
\(52\) 0 0
\(53\) −608718. −0.561630 −0.280815 − 0.959762i \(-0.590605\pi\)
−0.280815 + 0.959762i \(0.590605\pi\)
\(54\) 0 0
\(55\) 277400. 0.224821
\(56\) 0 0
\(57\) −179388. −0.128301
\(58\) 0 0
\(59\) −536120. −0.339844 −0.169922 − 0.985457i \(-0.554352\pi\)
−0.169922 + 0.985457i \(0.554352\pi\)
\(60\) 0 0
\(61\) −1.79709e6 −1.01371 −0.506857 − 0.862030i \(-0.669193\pi\)
−0.506857 + 0.862030i \(0.669193\pi\)
\(62\) 0 0
\(63\) −250047. −0.125988
\(64\) 0 0
\(65\) −329400. −0.148774
\(66\) 0 0
\(67\) 2.12318e6 0.862431 0.431215 − 0.902249i \(-0.358085\pi\)
0.431215 + 0.902249i \(0.358085\pi\)
\(68\) 0 0
\(69\) −53514.0 −0.0196108
\(70\) 0 0
\(71\) −1.19121e6 −0.394990 −0.197495 − 0.980304i \(-0.563281\pi\)
−0.197495 + 0.980304i \(0.563281\pi\)
\(72\) 0 0
\(73\) 1.05643e6 0.317842 0.158921 − 0.987291i \(-0.449199\pi\)
0.158921 + 0.987291i \(0.449199\pi\)
\(74\) 0 0
\(75\) 1.83938e6 0.503449
\(76\) 0 0
\(77\) −951482. −0.237511
\(78\) 0 0
\(79\) 998484. 0.227849 0.113924 − 0.993489i \(-0.463658\pi\)
0.113924 + 0.993489i \(0.463658\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 3.89800e6 0.748288 0.374144 − 0.927371i \(-0.377937\pi\)
0.374144 + 0.927371i \(0.377937\pi\)
\(84\) 0 0
\(85\) 590000. 0.104204
\(86\) 0 0
\(87\) 5.61886e6 0.914810
\(88\) 0 0
\(89\) −4.62235e6 −0.695021 −0.347511 − 0.937676i \(-0.612973\pi\)
−0.347511 + 0.937676i \(0.612973\pi\)
\(90\) 0 0
\(91\) 1.12984e6 0.157171
\(92\) 0 0
\(93\) 3.18038e6 0.410005
\(94\) 0 0
\(95\) 664400. 0.0795055
\(96\) 0 0
\(97\) 1.52877e7 1.70075 0.850377 − 0.526174i \(-0.176374\pi\)
0.850377 + 0.526174i \(0.176374\pi\)
\(98\) 0 0
\(99\) 2.02225e6 0.209465
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.a.1.1 ✓ 1
3.2 odd 2 252.8.a.a.1.1 1
4.3 odd 2 336.8.a.j.1.1 1
7.2 even 3 588.8.i.f.361.1 2
7.3 odd 6 588.8.i.c.373.1 2
7.4 even 3 588.8.i.f.373.1 2
7.5 odd 6 588.8.i.c.361.1 2
7.6 odd 2 588.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
252.8.a.a.1.1 1 3.2 odd 2
336.8.a.j.1.1 1 4.3 odd 2
588.8.a.c.1.1 1 7.6 odd 2
588.8.i.c.361.1 2 7.5 odd 6
588.8.i.c.373.1 2 7.3 odd 6
588.8.i.f.361.1 2 7.2 even 3
588.8.i.f.373.1 2 7.4 even 3