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Properties

Label 392.10.a.f.1.2
Level $392$
Weight $10$
Character 392.1
Self dual yes
Analytic conductor $201.894$
Analytic rank $0$
Dimension $4$
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] = [392,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-70] 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: \(201.894047776\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5576x^{2} - 170673x - 607824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-38.9670\) 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)
 
\(f(q)\) \(=\) \(q-128.448 q^{3} +1333.35 q^{5} -3184.23 q^{9} -22287.0 q^{11} -185165. q^{13} -171266. q^{15} -351739. q^{17} -1.10798e6 q^{19} -1.99049e6 q^{23} -175297. q^{25} +2.93724e6 q^{27} +3.18384e6 q^{29} -7.42986e6 q^{31} +2.86270e6 q^{33} +1.11534e7 q^{37} +2.37840e7 q^{39} -2.22681e7 q^{41} +1.98693e7 q^{43} -4.24571e6 q^{45} +569943. q^{47} +4.51801e7 q^{51} -6.76141e7 q^{53} -2.97164e7 q^{55} +1.42317e8 q^{57} -6.89400e7 q^{59} +9.40859e7 q^{61} -2.46890e8 q^{65} +7.82459e7 q^{67} +2.55673e8 q^{69} +1.61299e8 q^{71} -1.68908e8 q^{73} +2.25164e7 q^{75} -3.52815e8 q^{79} -3.14606e8 q^{81} -1.80933e8 q^{83} -4.68993e8 q^{85} -4.08957e8 q^{87} +8.90321e8 q^{89} +9.54347e8 q^{93} -1.47732e9 q^{95} -5.34984e8 q^{97} +7.09669e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 70 q^{3} - 1022 q^{5} + 12176 q^{9} - 31476 q^{11} - 11466 q^{13} + 98504 q^{15} - 397012 q^{17} - 246610 q^{19} + 1976552 q^{23} + 1086512 q^{25} - 3139108 q^{27} + 3718940 q^{29} - 11118436 q^{31}+ \cdots + 1279453212 q^{99}+O(q^{100}) \) 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
\(3\) −128.448 −0.915546 −0.457773 − 0.889069i \(-0.651353\pi\)
−0.457773 + 0.889069i \(0.651353\pi\)
\(4\) 0 0
\(5\) 1333.35 0.954069 0.477035 − 0.878884i \(-0.341712\pi\)
0.477035 + 0.878884i \(0.341712\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3184.23 −0.161776
\(10\) 0 0
\(11\) −22287.0 −0.458969 −0.229485 − 0.973312i \(-0.573704\pi\)
−0.229485 + 0.973312i \(0.573704\pi\)
\(12\) 0 0
\(13\) −185165. −1.79810 −0.899049 − 0.437849i \(-0.855741\pi\)
−0.899049 + 0.437849i \(0.855741\pi\)
\(14\) 0 0
\(15\) −171266. −0.873494
\(16\) 0 0
\(17\) −351739. −1.02141 −0.510706 − 0.859755i \(-0.670616\pi\)
−0.510706 + 0.859755i \(0.670616\pi\)
\(18\) 0 0
\(19\) −1.10798e6 −1.95047 −0.975235 − 0.221169i \(-0.929013\pi\)
−0.975235 + 0.221169i \(0.929013\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.99049e6 −1.48315 −0.741573 − 0.670872i \(-0.765920\pi\)
−0.741573 + 0.670872i \(0.765920\pi\)
\(24\) 0 0
\(25\) −175297. −0.0897518
\(26\) 0 0
\(27\) 2.93724e6 1.06366
\(28\) 0 0
\(29\) 3.18384e6 0.835912 0.417956 − 0.908467i \(-0.362747\pi\)
0.417956 + 0.908467i \(0.362747\pi\)
\(30\) 0 0
\(31\) −7.42986e6 −1.44495 −0.722476 − 0.691396i \(-0.756996\pi\)
−0.722476 + 0.691396i \(0.756996\pi\)
\(32\) 0 0
\(33\) 2.86270e6 0.420207
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.11534e7 0.978360 0.489180 − 0.872183i \(-0.337296\pi\)
0.489180 + 0.872183i \(0.337296\pi\)
\(38\) 0 0
\(39\) 2.37840e7 1.64624
\(40\) 0 0
\(41\) −2.22681e7 −1.23071 −0.615354 − 0.788251i \(-0.710987\pi\)
−0.615354 + 0.788251i \(0.710987\pi\)
\(42\) 0 0
\(43\) 1.98693e7 0.886287 0.443144 − 0.896451i \(-0.353863\pi\)
0.443144 + 0.896451i \(0.353863\pi\)
\(44\) 0 0
\(45\) −4.24571e6 −0.154345
\(46\) 0 0
\(47\) 569943. 0.0170369 0.00851847 − 0.999964i \(-0.497288\pi\)
0.00851847 + 0.999964i \(0.497288\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.51801e7 0.935149
\(52\) 0 0
\(53\) −6.76141e7 −1.17705 −0.588527 − 0.808478i \(-0.700292\pi\)
−0.588527 + 0.808478i \(0.700292\pi\)
\(54\) 0 0
\(55\) −2.97164e7 −0.437889
\(56\) 0 0
\(57\) 1.42317e8 1.78575
\(58\) 0 0
\(59\) −6.89400e7 −0.740691 −0.370345 − 0.928894i \(-0.620761\pi\)
−0.370345 + 0.928894i \(0.620761\pi\)
\(60\) 0 0
\(61\) 9.40859e7 0.870042 0.435021 − 0.900420i \(-0.356741\pi\)
0.435021 + 0.900420i \(0.356741\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.46890e8 −1.71551
\(66\) 0 0
\(67\) 7.82459e7 0.474378 0.237189 − 0.971463i \(-0.423774\pi\)
0.237189 + 0.971463i \(0.423774\pi\)
\(68\) 0 0
\(69\) 2.55673e8 1.35789
\(70\) 0 0
\(71\) 1.61299e8 0.753302 0.376651 − 0.926355i \(-0.377076\pi\)
0.376651 + 0.926355i \(0.377076\pi\)
\(72\) 0 0
\(73\) −1.68908e8 −0.696142 −0.348071 − 0.937468i \(-0.613163\pi\)
−0.348071 + 0.937468i \(0.613163\pi\)
\(74\) 0 0
\(75\) 2.25164e7 0.0821719
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.52815e8 −1.01912 −0.509559 − 0.860436i \(-0.670192\pi\)
−0.509559 + 0.860436i \(0.670192\pi\)
\(80\) 0 0
\(81\) −3.14606e8 −0.812053
\(82\) 0 0
\(83\) −1.80933e8 −0.418471 −0.209235 − 0.977865i \(-0.567098\pi\)
−0.209235 + 0.977865i \(0.567098\pi\)
\(84\) 0 0
\(85\) −4.68993e8 −0.974498
\(86\) 0 0
\(87\) −4.08957e8 −0.765316
\(88\) 0 0
\(89\) 8.90321e8 1.50415 0.752076 − 0.659076i \(-0.229053\pi\)
0.752076 + 0.659076i \(0.229053\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.54347e8 1.32292
\(94\) 0 0
\(95\) −1.47732e9 −1.86088
\(96\) 0 0
\(97\) −5.34984e8 −0.613576 −0.306788 − 0.951778i \(-0.599254\pi\)
−0.306788 + 0.951778i \(0.599254\pi\)
\(98\) 0 0
\(99\) 7.09669e7 0.0742501
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 392.10.a.f.1.2 4
7.6 odd 2 56.10.a.c.1.3 ✓ 4
28.27 even 2 112.10.a.k.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.c.1.3 ✓ 4 7.6 odd 2
112.10.a.k.1.2 4 28.27 even 2
392.10.a.f.1.2 4 1.1 even 1 trivial