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Properties

Label 192.8.a.o.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 = [1,0,27,0,378] 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 12)
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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +378.000 q^{5} -832.000 q^{7} +729.000 q^{9} +2484.00 q^{11} -14870.0 q^{13} +10206.0 q^{15} -22302.0 q^{17} +16300.0 q^{19} -22464.0 q^{21} -115128. q^{23} +64759.0 q^{25} +19683.0 q^{27} -157086. q^{29} -16456.0 q^{31} +67068.0 q^{33} -314496. q^{35} +149266. q^{37} -401490. q^{39} -241110. q^{41} +443188. q^{43} +275562. q^{45} +922752. q^{47} -131319. q^{49} -602154. q^{51} +697626. q^{53} +938952. q^{55} +440100. q^{57} -870156. q^{59} -2.06706e6 q^{61} -606528. q^{63} -5.62086e6 q^{65} +1.68075e6 q^{67} -3.10846e6 q^{69} -1.07028e6 q^{71} -2.40333e6 q^{73} +1.74849e6 q^{75} -2.06669e6 q^{77} +2.30151e6 q^{79} +531441. q^{81} -4.70869e6 q^{83} -8.43016e6 q^{85} -4.24132e6 q^{87} +4.14369e6 q^{89} +1.23718e7 q^{91} -444312. q^{93} +6.16140e6 q^{95} -1.62297e6 q^{97} +1.81084e6 q^{99} +O(q^{100})\)

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
\(5\) 378.000 1.35237 0.676187 − 0.736730i \(-0.263631\pi\)
0.676187 + 0.736730i \(0.263631\pi\)
\(6\) 0 0
\(7\) −832.000 −0.916812 −0.458406 − 0.888743i \(-0.651579\pi\)
−0.458406 + 0.888743i \(0.651579\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 2484.00 0.562700 0.281350 − 0.959605i \(-0.409218\pi\)
0.281350 + 0.959605i \(0.409218\pi\)
\(12\) 0 0
\(13\) −14870.0 −1.87719 −0.938597 − 0.345015i \(-0.887874\pi\)
−0.938597 + 0.345015i \(0.887874\pi\)
\(14\) 0 0
\(15\) 10206.0 0.780793
\(16\) 0 0
\(17\) −22302.0 −1.10096 −0.550481 − 0.834847i \(-0.685556\pi\)
−0.550481 + 0.834847i \(0.685556\pi\)
\(18\) 0 0
\(19\) 16300.0 0.545193 0.272596 − 0.962128i \(-0.412118\pi\)
0.272596 + 0.962128i \(0.412118\pi\)
\(20\) 0 0
\(21\) −22464.0 −0.529322
\(22\) 0 0
\(23\) −115128. −1.97303 −0.986515 − 0.163673i \(-0.947666\pi\)
−0.986515 + 0.163673i \(0.947666\pi\)
\(24\) 0 0
\(25\) 64759.0 0.828915
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −157086. −1.19604 −0.598018 − 0.801482i \(-0.704045\pi\)
−0.598018 + 0.801482i \(0.704045\pi\)
\(30\) 0 0
\(31\) −16456.0 −0.0992107 −0.0496053 − 0.998769i \(-0.515796\pi\)
−0.0496053 + 0.998769i \(0.515796\pi\)
\(32\) 0 0
\(33\) 67068.0 0.324875
\(34\) 0 0
\(35\) −314496. −1.23987
\(36\) 0 0
\(37\) 149266. 0.484457 0.242228 − 0.970219i \(-0.422122\pi\)
0.242228 + 0.970219i \(0.422122\pi\)
\(38\) 0 0
\(39\) −401490. −1.08380
\(40\) 0 0
\(41\) −241110. −0.546351 −0.273175 − 0.961964i \(-0.588074\pi\)
−0.273175 + 0.961964i \(0.588074\pi\)
\(42\) 0 0
\(43\) 443188. 0.850058 0.425029 − 0.905180i \(-0.360264\pi\)
0.425029 + 0.905180i \(0.360264\pi\)
\(44\) 0 0
\(45\) 275562. 0.450791
\(46\) 0 0
\(47\) 922752. 1.29641 0.648205 − 0.761466i \(-0.275520\pi\)
0.648205 + 0.761466i \(0.275520\pi\)
\(48\) 0 0
\(49\) −131319. −0.159456
\(50\) 0 0
\(51\) −602154. −0.635641
\(52\) 0 0
\(53\) 697626. 0.643661 0.321830 − 0.946797i \(-0.395702\pi\)
0.321830 + 0.946797i \(0.395702\pi\)
\(54\) 0 0
\(55\) 938952. 0.760981
\(56\) 0 0
\(57\) 440100. 0.314767
\(58\) 0 0
\(59\) −870156. −0.551588 −0.275794 − 0.961217i \(-0.588941\pi\)
−0.275794 + 0.961217i \(0.588941\pi\)
\(60\) 0 0
\(61\) −2.06706e6 −1.16600 −0.583001 − 0.812472i \(-0.698122\pi\)
−0.583001 + 0.812472i \(0.698122\pi\)
\(62\) 0 0
\(63\) −606528. −0.305604
\(64\) 0 0
\(65\) −5.62086e6 −2.53867
\(66\) 0 0
\(67\) 1.68075e6 0.682717 0.341359 − 0.939933i \(-0.389113\pi\)
0.341359 + 0.939933i \(0.389113\pi\)
\(68\) 0 0
\(69\) −3.10846e6 −1.13913
\(70\) 0 0
\(71\) −1.07028e6 −0.354890 −0.177445 − 0.984131i \(-0.556783\pi\)
−0.177445 + 0.984131i \(0.556783\pi\)
\(72\) 0 0
\(73\) −2.40333e6 −0.723076 −0.361538 − 0.932357i \(-0.617748\pi\)
−0.361538 + 0.932357i \(0.617748\pi\)
\(74\) 0 0
\(75\) 1.74849e6 0.478574
\(76\) 0 0
\(77\) −2.06669e6 −0.515890
\(78\) 0 0
\(79\) 2.30151e6 0.525192 0.262596 − 0.964906i \(-0.415421\pi\)
0.262596 + 0.964906i \(0.415421\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −4.70869e6 −0.903914 −0.451957 − 0.892040i \(-0.649274\pi\)
−0.451957 + 0.892040i \(0.649274\pi\)
\(84\) 0 0
\(85\) −8.43016e6 −1.48891
\(86\) 0 0
\(87\) −4.24132e6 −0.690532
\(88\) 0 0
\(89\) 4.14369e6 0.623049 0.311525 − 0.950238i \(-0.399160\pi\)
0.311525 + 0.950238i \(0.399160\pi\)
\(90\) 0 0
\(91\) 1.23718e7 1.72103
\(92\) 0 0
\(93\) −444312. −0.0572793
\(94\) 0 0
\(95\) 6.16140e6 0.737304
\(96\) 0 0
\(97\) −1.62297e6 −0.180555 −0.0902777 − 0.995917i \(-0.528775\pi\)
−0.0902777 + 0.995917i \(0.528775\pi\)
\(98\) 0 0
\(99\) 1.81084e6 0.187567
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.o.1.1 1
3.2 odd 2 576.8.a.d.1.1 1
4.3 odd 2 192.8.a.g.1.1 1
8.3 odd 2 48.8.a.e.1.1 1
8.5 even 2 12.8.a.a.1.1 ✓ 1
12.11 even 2 576.8.a.e.1.1 1
24.5 odd 2 36.8.a.c.1.1 1
24.11 even 2 144.8.a.j.1.1 1
40.13 odd 4 300.8.d.c.49.1 2
40.29 even 2 300.8.a.g.1.1 1
40.37 odd 4 300.8.d.c.49.2 2
56.5 odd 6 588.8.i.a.361.1 2
56.13 odd 2 588.8.a.d.1.1 1
56.37 even 6 588.8.i.h.361.1 2
56.45 odd 6 588.8.i.a.373.1 2
56.53 even 6 588.8.i.h.373.1 2
72.5 odd 6 324.8.e.a.217.1 2
72.13 even 6 324.8.e.f.217.1 2
72.29 odd 6 324.8.e.a.109.1 2
72.61 even 6 324.8.e.f.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.8.a.a.1.1 ✓ 1 8.5 even 2
36.8.a.c.1.1 1 24.5 odd 2
48.8.a.e.1.1 1 8.3 odd 2
144.8.a.j.1.1 1 24.11 even 2
192.8.a.g.1.1 1 4.3 odd 2
192.8.a.o.1.1 1 1.1 even 1 trivial
300.8.a.g.1.1 1 40.29 even 2
300.8.d.c.49.1 2 40.13 odd 4
300.8.d.c.49.2 2 40.37 odd 4
324.8.e.a.109.1 2 72.29 odd 6
324.8.e.a.217.1 2 72.5 odd 6
324.8.e.f.109.1 2 72.61 even 6
324.8.e.f.217.1 2 72.13 even 6
576.8.a.d.1.1 1 3.2 odd 2
576.8.a.e.1.1 1 12.11 even 2
588.8.a.d.1.1 1 56.13 odd 2
588.8.i.a.361.1 2 56.5 odd 6
588.8.i.a.373.1 2 56.45 odd 6
588.8.i.h.361.1 2 56.37 even 6
588.8.i.h.373.1 2 56.53 even 6