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Properties

Label 192.8.a.i.1.1
Level $192$
Weight $8$
Character 192.1
Self dual yes
Analytic conductor $59.978$
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] = [192,8,Mod(1,192)] 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("192.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,27,0,-390] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.9779248930\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 192.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} -390.000 q^{5} -64.0000 q^{7} +729.000 q^{9} +948.000 q^{11} +5098.00 q^{13} -10530.0 q^{15} +28386.0 q^{17} +8620.00 q^{19} -1728.00 q^{21} -15288.0 q^{23} +73975.0 q^{25} +19683.0 q^{27} -36510.0 q^{29} -276808. q^{31} +25596.0 q^{33} +24960.0 q^{35} -268526. q^{37} +137646. q^{39} -629718. q^{41} -685772. q^{43} -284310. q^{45} +583296. q^{47} -819447. q^{49} +766422. q^{51} +428058. q^{53} -369720. q^{55} +232740. q^{57} -1.30638e6 q^{59} -300662. q^{61} -46656.0 q^{63} -1.98822e6 q^{65} +507244. q^{67} -412776. q^{69} +5.56063e6 q^{71} +1.36908e6 q^{73} +1.99733e6 q^{75} -60672.0 q^{77} -6.91372e6 q^{79} +531441. q^{81} +4.37675e6 q^{83} -1.10705e7 q^{85} -985770. q^{87} -8.52831e6 q^{89} -326272. q^{91} -7.47382e6 q^{93} -3.36180e6 q^{95} -8.82681e6 q^{97} +691092. q^{99} +O(q^{100})\)

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
\(5\) −390.000 −1.39531 −0.697653 − 0.716436i \(-0.745772\pi\)
−0.697653 + 0.716436i \(0.745772\pi\)
\(6\) 0 0
\(7\) −64.0000 −0.0705240 −0.0352620 − 0.999378i \(-0.511227\pi\)
−0.0352620 + 0.999378i \(0.511227\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 948.000 0.214750 0.107375 − 0.994219i \(-0.465755\pi\)
0.107375 + 0.994219i \(0.465755\pi\)
\(12\) 0 0
\(13\) 5098.00 0.643573 0.321787 − 0.946812i \(-0.395717\pi\)
0.321787 + 0.946812i \(0.395717\pi\)
\(14\) 0 0
\(15\) −10530.0 −0.805581
\(16\) 0 0
\(17\) 28386.0 1.40131 0.700653 − 0.713502i \(-0.252892\pi\)
0.700653 + 0.713502i \(0.252892\pi\)
\(18\) 0 0
\(19\) 8620.00 0.288317 0.144158 − 0.989555i \(-0.453953\pi\)
0.144158 + 0.989555i \(0.453953\pi\)
\(20\) 0 0
\(21\) −1728.00 −0.0407170
\(22\) 0 0
\(23\) −15288.0 −0.262001 −0.131001 − 0.991382i \(-0.541819\pi\)
−0.131001 + 0.991382i \(0.541819\pi\)
\(24\) 0 0
\(25\) 73975.0 0.946880
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −36510.0 −0.277983 −0.138992 − 0.990294i \(-0.544386\pi\)
−0.138992 + 0.990294i \(0.544386\pi\)
\(30\) 0 0
\(31\) −276808. −1.66883 −0.834416 − 0.551135i \(-0.814195\pi\)
−0.834416 + 0.551135i \(0.814195\pi\)
\(32\) 0 0
\(33\) 25596.0 0.123986
\(34\) 0 0
\(35\) 24960.0 0.0984026
\(36\) 0 0
\(37\) −268526. −0.871526 −0.435763 − 0.900061i \(-0.643521\pi\)
−0.435763 + 0.900061i \(0.643521\pi\)
\(38\) 0 0
\(39\) 137646. 0.371567
\(40\) 0 0
\(41\) −629718. −1.42693 −0.713465 − 0.700691i \(-0.752875\pi\)
−0.713465 + 0.700691i \(0.752875\pi\)
\(42\) 0 0
\(43\) −685772. −1.31535 −0.657673 − 0.753303i \(-0.728459\pi\)
−0.657673 + 0.753303i \(0.728459\pi\)
\(44\) 0 0
\(45\) −284310. −0.465102
\(46\) 0 0
\(47\) 583296. 0.819495 0.409748 − 0.912199i \(-0.365617\pi\)
0.409748 + 0.912199i \(0.365617\pi\)
\(48\) 0 0
\(49\) −819447. −0.995026
\(50\) 0 0
\(51\) 766422. 0.809044
\(52\) 0 0
\(53\) 428058. 0.394945 0.197473 − 0.980308i \(-0.436727\pi\)
0.197473 + 0.980308i \(0.436727\pi\)
\(54\) 0 0
\(55\) −369720. −0.299643
\(56\) 0 0
\(57\) 232740. 0.166460
\(58\) 0 0
\(59\) −1.30638e6 −0.828109 −0.414054 − 0.910252i \(-0.635888\pi\)
−0.414054 + 0.910252i \(0.635888\pi\)
\(60\) 0 0
\(61\) −300662. −0.169599 −0.0847997 − 0.996398i \(-0.527025\pi\)
−0.0847997 + 0.996398i \(0.527025\pi\)
\(62\) 0 0
\(63\) −46656.0 −0.0235080
\(64\) 0 0
\(65\) −1.98822e6 −0.897982
\(66\) 0 0
\(67\) 507244. 0.206042 0.103021 − 0.994679i \(-0.467149\pi\)
0.103021 + 0.994679i \(0.467149\pi\)
\(68\) 0 0
\(69\) −412776. −0.151266
\(70\) 0 0
\(71\) 5.56063e6 1.84383 0.921913 − 0.387397i \(-0.126626\pi\)
0.921913 + 0.387397i \(0.126626\pi\)
\(72\) 0 0
\(73\) 1.36908e6 0.411907 0.205954 − 0.978562i \(-0.433970\pi\)
0.205954 + 0.978562i \(0.433970\pi\)
\(74\) 0 0
\(75\) 1.99733e6 0.546681
\(76\) 0 0
\(77\) −60672.0 −0.0151451
\(78\) 0 0
\(79\) −6.91372e6 −1.57767 −0.788836 − 0.614603i \(-0.789316\pi\)
−0.788836 + 0.614603i \(0.789316\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 4.37675e6 0.840191 0.420096 − 0.907480i \(-0.361997\pi\)
0.420096 + 0.907480i \(0.361997\pi\)
\(84\) 0 0
\(85\) −1.10705e7 −1.95525
\(86\) 0 0
\(87\) −985770. −0.160494
\(88\) 0 0
\(89\) −8.52831e6 −1.28232 −0.641162 − 0.767405i \(-0.721547\pi\)
−0.641162 + 0.767405i \(0.721547\pi\)
\(90\) 0 0
\(91\) −326272. −0.0453874
\(92\) 0 0
\(93\) −7.47382e6 −0.963501
\(94\) 0 0
\(95\) −3.36180e6 −0.402290
\(96\) 0 0
\(97\) −8.82681e6 −0.981981 −0.490990 − 0.871165i \(-0.663365\pi\)
−0.490990 + 0.871165i \(0.663365\pi\)
\(98\) 0 0
\(99\) 691092. 0.0715835
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 192.8.a.i.1.1 1
3.2 odd 2 576.8.a.w.1.1 1
4.3 odd 2 192.8.a.a.1.1 1
8.3 odd 2 48.8.a.g.1.1 1
8.5 even 2 3.8.a.a.1.1 ✓ 1
12.11 even 2 576.8.a.x.1.1 1
24.5 odd 2 9.8.a.a.1.1 1
24.11 even 2 144.8.a.b.1.1 1
40.13 odd 4 75.8.b.c.49.1 2
40.29 even 2 75.8.a.a.1.1 1
40.37 odd 4 75.8.b.c.49.2 2
56.5 odd 6 147.8.e.a.67.1 2
56.13 odd 2 147.8.a.b.1.1 1
56.37 even 6 147.8.e.b.67.1 2
56.45 odd 6 147.8.e.a.79.1 2
56.53 even 6 147.8.e.b.79.1 2
72.5 odd 6 81.8.c.c.55.1 2
72.13 even 6 81.8.c.a.55.1 2
72.29 odd 6 81.8.c.c.28.1 2
72.61 even 6 81.8.c.a.28.1 2
88.21 odd 2 363.8.a.b.1.1 1
104.77 even 2 507.8.a.a.1.1 1
120.29 odd 2 225.8.a.i.1.1 1
120.53 even 4 225.8.b.f.199.2 2
120.77 even 4 225.8.b.f.199.1 2
168.125 even 2 441.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.8.a.a.1.1 ✓ 1 8.5 even 2
9.8.a.a.1.1 1 24.5 odd 2
48.8.a.g.1.1 1 8.3 odd 2
75.8.a.a.1.1 1 40.29 even 2
75.8.b.c.49.1 2 40.13 odd 4
75.8.b.c.49.2 2 40.37 odd 4
81.8.c.a.28.1 2 72.61 even 6
81.8.c.a.55.1 2 72.13 even 6
81.8.c.c.28.1 2 72.29 odd 6
81.8.c.c.55.1 2 72.5 odd 6
144.8.a.b.1.1 1 24.11 even 2
147.8.a.b.1.1 1 56.13 odd 2
147.8.e.a.67.1 2 56.5 odd 6
147.8.e.a.79.1 2 56.45 odd 6
147.8.e.b.67.1 2 56.37 even 6
147.8.e.b.79.1 2 56.53 even 6
192.8.a.a.1.1 1 4.3 odd 2
192.8.a.i.1.1 1 1.1 even 1 trivial
225.8.a.i.1.1 1 120.29 odd 2
225.8.b.f.199.1 2 120.77 even 4
225.8.b.f.199.2 2 120.53 even 4
363.8.a.b.1.1 1 88.21 odd 2
441.8.a.a.1.1 1 168.125 even 2
507.8.a.a.1.1 1 104.77 even 2
576.8.a.w.1.1 1 3.2 odd 2
576.8.a.x.1.1 1 12.11 even 2