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Properties

Label 392.6.a.g.1.2
Level $392$
Weight $6$
Character 392.1
Self dual yes
Analytic conductor $62.870$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [392,6,Mod(1,392)] 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("392.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{86}, \sqrt{134})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 110x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.15111\) of defining polynomial
Character \(\chi\) \(=\) 392.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.30222 q^{3} +81.0956 q^{5} -237.700 q^{9} -732.099 q^{11} +919.678 q^{13} -186.700 q^{15} +1728.85 q^{17} -2443.34 q^{19} +2424.30 q^{23} +3451.50 q^{25} +1106.68 q^{27} +1010.00 q^{29} +6128.04 q^{31} +1685.45 q^{33} -1238.60 q^{37} -2117.30 q^{39} +15388.6 q^{41} +12980.9 q^{43} -19276.4 q^{45} -6557.64 q^{47} -3980.19 q^{51} -31549.8 q^{53} -59370.0 q^{55} +5625.11 q^{57} -21881.4 q^{59} +27003.0 q^{61} +74581.9 q^{65} +8264.07 q^{67} -5581.26 q^{69} +17131.8 q^{71} +9282.54 q^{73} -7946.10 q^{75} +39144.8 q^{79} +55213.2 q^{81} +108829. q^{83} +140202. q^{85} -2325.24 q^{87} -17842.6 q^{89} -14108.1 q^{93} -198144. q^{95} -26643.9 q^{97} +174020. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 92 q^{9} - 352 q^{11} + 112 q^{15} + 1968 q^{23} + 924 q^{25} + 4040 q^{29} + 10504 q^{37} - 9328 q^{39} + 28736 q^{43} + 23584 q^{51} - 28296 q^{53} + 62864 q^{57} + 146320 q^{65} - 72576 q^{67}+ \cdots + 561248 q^{99}+O(q^{100}) \) 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
\(3\) −2.30222 −0.147687 −0.0738437 − 0.997270i \(-0.523527\pi\)
−0.0738437 + 0.997270i \(0.523527\pi\)
\(4\) 0 0
\(5\) 81.0956 1.45068 0.725341 − 0.688390i \(-0.241682\pi\)
0.725341 + 0.688390i \(0.241682\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −237.700 −0.978188
\(10\) 0 0
\(11\) −732.099 −1.82427 −0.912133 − 0.409894i \(-0.865566\pi\)
−0.912133 + 0.409894i \(0.865566\pi\)
\(12\) 0 0
\(13\) 919.678 1.50931 0.754653 − 0.656124i \(-0.227805\pi\)
0.754653 + 0.656124i \(0.227805\pi\)
\(14\) 0 0
\(15\) −186.700 −0.214247
\(16\) 0 0
\(17\) 1728.85 1.45089 0.725446 − 0.688279i \(-0.241634\pi\)
0.725446 + 0.688279i \(0.241634\pi\)
\(18\) 0 0
\(19\) −2443.34 −1.55275 −0.776373 − 0.630274i \(-0.782943\pi\)
−0.776373 + 0.630274i \(0.782943\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2424.30 0.955579 0.477789 − 0.878474i \(-0.341438\pi\)
0.477789 + 0.878474i \(0.341438\pi\)
\(24\) 0 0
\(25\) 3451.50 1.10448
\(26\) 0 0
\(27\) 1106.68 0.292153
\(28\) 0 0
\(29\) 1010.00 0.223011 0.111506 − 0.993764i \(-0.464433\pi\)
0.111506 + 0.993764i \(0.464433\pi\)
\(30\) 0 0
\(31\) 6128.04 1.14530 0.572648 − 0.819802i \(-0.305916\pi\)
0.572648 + 0.819802i \(0.305916\pi\)
\(32\) 0 0
\(33\) 1685.45 0.269421
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1238.60 −0.148739 −0.0743696 − 0.997231i \(-0.523694\pi\)
−0.0743696 + 0.997231i \(0.523694\pi\)
\(38\) 0 0
\(39\) −2117.30 −0.222906
\(40\) 0 0
\(41\) 15388.6 1.42968 0.714841 − 0.699287i \(-0.246499\pi\)
0.714841 + 0.699287i \(0.246499\pi\)
\(42\) 0 0
\(43\) 12980.9 1.07062 0.535308 − 0.844657i \(-0.320196\pi\)
0.535308 + 0.844657i \(0.320196\pi\)
\(44\) 0 0
\(45\) −19276.4 −1.41904
\(46\) 0 0
\(47\) −6557.64 −0.433015 −0.216507 − 0.976281i \(-0.569467\pi\)
−0.216507 + 0.976281i \(0.569467\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3980.19 −0.214278
\(52\) 0 0
\(53\) −31549.8 −1.54279 −0.771395 − 0.636357i \(-0.780441\pi\)
−0.771395 + 0.636357i \(0.780441\pi\)
\(54\) 0 0
\(55\) −59370.0 −2.64643
\(56\) 0 0
\(57\) 5625.11 0.229321
\(58\) 0 0
\(59\) −21881.4 −0.818362 −0.409181 − 0.912453i \(-0.634186\pi\)
−0.409181 + 0.912453i \(0.634186\pi\)
\(60\) 0 0
\(61\) 27003.0 0.929153 0.464577 − 0.885533i \(-0.346207\pi\)
0.464577 + 0.885533i \(0.346207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 74581.9 2.18952
\(66\) 0 0
\(67\) 8264.07 0.224909 0.112455 − 0.993657i \(-0.464129\pi\)
0.112455 + 0.993657i \(0.464129\pi\)
\(68\) 0 0
\(69\) −5581.26 −0.141127
\(70\) 0 0
\(71\) 17131.8 0.403327 0.201664 − 0.979455i \(-0.435365\pi\)
0.201664 + 0.979455i \(0.435365\pi\)
\(72\) 0 0
\(73\) 9282.54 0.203873 0.101937 − 0.994791i \(-0.467496\pi\)
0.101937 + 0.994791i \(0.467496\pi\)
\(74\) 0 0
\(75\) −7946.10 −0.163118
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 39144.8 0.705678 0.352839 − 0.935684i \(-0.385216\pi\)
0.352839 + 0.935684i \(0.385216\pi\)
\(80\) 0 0
\(81\) 55213.2 0.935041
\(82\) 0 0
\(83\) 108829. 1.73399 0.866997 − 0.498313i \(-0.166047\pi\)
0.866997 + 0.498313i \(0.166047\pi\)
\(84\) 0 0
\(85\) 140202. 2.10478
\(86\) 0 0
\(87\) −2325.24 −0.0329359
\(88\) 0 0
\(89\) −17842.6 −0.238772 −0.119386 − 0.992848i \(-0.538093\pi\)
−0.119386 + 0.992848i \(0.538093\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −14108.1 −0.169146
\(94\) 0 0
\(95\) −198144. −2.25254
\(96\) 0 0
\(97\) −26643.9 −0.287520 −0.143760 − 0.989613i \(-0.545919\pi\)
−0.143760 + 0.989613i \(0.545919\pi\)
\(98\) 0 0
\(99\) 174020. 1.78448
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 392.6.a.g.1.2 ✓ 4
4.3 odd 2 784.6.a.be.1.3 4
7.2 even 3 392.6.i.n.361.3 8
7.3 odd 6 392.6.i.n.177.2 8
7.4 even 3 392.6.i.n.177.3 8
7.5 odd 6 392.6.i.n.361.2 8
7.6 odd 2 inner 392.6.a.g.1.3 yes 4
28.27 even 2 784.6.a.be.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
392.6.a.g.1.2 ✓ 4 1.1 even 1 trivial
392.6.a.g.1.3 yes 4 7.6 odd 2 inner
392.6.i.n.177.2 8 7.3 odd 6
392.6.i.n.177.3 8 7.4 even 3
392.6.i.n.361.2 8 7.5 odd 6
392.6.i.n.361.3 8 7.2 even 3
784.6.a.be.1.2 4 28.27 even 2
784.6.a.be.1.3 4 4.3 odd 2