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Properties

Label 384.2.v.a.13.8
Level $384$
Weight $2$
Character 384.13
Analytic conductor $3.066$
Analytic rank $0$
Dimension $512$
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] = [384,2,Mod(13,384)] 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("384.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 15, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.v (of order \(32\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(32\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 13.8
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.8

$q$-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+(-1.06073 - 0.935332i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(0.250309 + 1.98427i) q^{4} +(-0.385901 - 3.91812i) q^{5} +(-0.587142 + 1.28657i) q^{6} +(-2.38112 + 3.56360i) q^{7} +(1.59044 - 2.33891i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-3.25540 + 4.51703i) q^{10} +(-5.30688 + 2.83658i) q^{11} +(1.82617 - 0.815535i) q^{12} +(2.04975 + 0.201883i) q^{13} +(5.85888 - 1.55289i) q^{14} +(-3.63739 + 1.50666i) q^{15} +(-3.87469 + 0.993364i) q^{16} +(-1.75009 - 0.724912i) q^{17} +(1.40161 + 0.188388i) q^{18} +(3.42826 - 2.81350i) q^{19} +(7.67803 - 1.74648i) q^{20} +(4.10136 + 1.24413i) q^{21} +(8.28232 + 1.95483i) q^{22} +(-0.247363 + 1.24358i) q^{23} +(-2.69988 - 0.843011i) q^{24} +(-10.2988 + 2.04856i) q^{25} +(-1.98541 - 2.13134i) q^{26} +(0.773010 + 0.634393i) q^{27} +(-7.66718 - 3.83280i) q^{28} +(-2.41573 + 4.51951i) q^{29} +(5.26752 + 1.80400i) q^{30} +(-4.26568 + 4.26568i) q^{31} +(5.03914 + 2.57043i) q^{32} +(4.25495 + 4.25495i) q^{33} +(1.17835 + 2.40586i) q^{34} +(14.8815 + 7.95433i) q^{35} +(-1.31053 - 1.51080i) q^{36} +(-1.72240 + 2.09874i) q^{37} +(-6.26802 - 0.222187i) q^{38} +(-0.401822 - 2.02009i) q^{39} +(-9.77788 - 5.32896i) q^{40} +(2.36639 + 0.470705i) q^{41} +(-3.18677 - 5.15582i) q^{42} +(2.56696 - 8.46214i) q^{43} +(-6.95692 - 9.82027i) q^{44} +(2.49766 + 3.04340i) q^{45} +(1.42554 - 1.08774i) q^{46} +(-2.96974 + 7.16958i) q^{47} +(2.07535 + 3.41949i) q^{48} +(-4.35073 - 10.5036i) q^{49} +(12.8404 + 7.45984i) q^{50} +(-0.185673 + 1.88517i) q^{51} +(0.112481 + 4.11780i) q^{52} +(-1.42695 - 2.66964i) q^{53} +(-0.226590 - 1.39594i) q^{54} +(13.1620 + 19.6983i) q^{55} +(4.54789 + 11.2369i) q^{56} +(-3.68752 - 2.46392i) q^{57} +(6.78969 - 2.53449i) q^{58} +(-5.09931 + 0.502238i) q^{59} +(-3.90009 - 6.84044i) q^{60} +(-9.81634 + 2.97775i) q^{61} +(8.51456 - 0.534921i) q^{62} -4.28591i q^{63} +(-2.94098 - 7.43980i) q^{64} -8.10908i q^{65} +(-0.533576 - 8.49315i) q^{66} +(4.02627 - 1.22136i) q^{67} +(1.00036 - 3.65412i) q^{68} +(1.26183 - 0.124280i) q^{69} +(-8.34536 - 22.3566i) q^{70} +(-11.6047 - 7.75400i) q^{71} +(-0.0229781 + 2.82833i) q^{72} +(1.68904 + 2.52783i) q^{73} +(3.79002 - 0.615197i) q^{74} +(4.94995 + 9.26070i) q^{75} +(6.44088 + 6.09836i) q^{76} +(2.52786 - 25.6658i) q^{77} +(-1.46323 + 2.51862i) q^{78} +(-4.52669 - 10.9284i) q^{79} +(5.38737 + 14.7982i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-2.06985 - 2.71266i) q^{82} +(0.813545 + 0.991307i) q^{83} +(-1.44209 + 8.44964i) q^{84} +(-2.16493 + 7.13682i) q^{85} +(-10.6378 + 6.57511i) q^{86} +(5.02615 + 0.999764i) q^{87} +(-1.80578 + 16.9237i) q^{88} +(-0.287119 - 1.44344i) q^{89} +(0.197245 - 5.56438i) q^{90} +(-5.60014 + 6.82379i) q^{91} +(-2.52951 - 0.179557i) q^{92} +(5.32026 + 2.84374i) q^{93} +(9.85604 - 4.82732i) q^{94} +(-12.3466 - 12.3466i) q^{95} +(0.996961 - 5.56831i) q^{96} +(3.16976 - 3.16976i) q^{97} +(-5.20938 + 15.2109i) q^{98} +(2.83658 - 5.30688i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 512 q - 96 q^{50} - 96 q^{52} - 32 q^{54} - 224 q^{56} - 192 q^{60} - 192 q^{62} - 192 q^{64} - 192 q^{66} - 192 q^{68} - 192 q^{70} - 224 q^{74} - 32 q^{76} - 96 q^{78} - 96 q^{80}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(e\left(\frac{15}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06073 − 0.935332i −0.750052 − 0.661379i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) 0.250309 + 1.98427i 0.125155 + 0.992137i
\(5\) −0.385901 − 3.91812i −0.172580 − 1.75224i −0.558317 − 0.829628i \(-0.688553\pi\)
0.385736 − 0.922609i \(-0.373947\pi\)
\(6\) −0.587142 + 1.28657i −0.239700 + 0.525240i
\(7\) −2.38112 + 3.56360i −0.899980 + 1.34691i 0.0376543 + 0.999291i \(0.488011\pi\)
−0.937634 + 0.347624i \(0.886989\pi\)
\(8\) 1.59044 − 2.33891i 0.562307 − 0.826929i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −3.25540 + 4.51703i −1.02945 + 1.42841i
\(11\) −5.30688 + 2.83658i −1.60008 + 0.855262i −0.601682 + 0.798736i \(0.705503\pi\)
−0.998401 + 0.0565264i \(0.981997\pi\)
\(12\) 1.82617 − 0.815535i 0.527170 − 0.235425i
\(13\) 2.04975 + 0.201883i 0.568499 + 0.0559922i 0.378183 − 0.925731i \(-0.376549\pi\)
0.190315 + 0.981723i \(0.439049\pi\)
\(14\) 5.85888 − 1.55289i 1.56585 − 0.415027i
\(15\) −3.63739 + 1.50666i −0.939169 + 0.389017i
\(16\) −3.87469 + 0.993364i −0.968673 + 0.248341i
\(17\) −1.75009 − 0.724912i −0.424460 − 0.175817i 0.160220 − 0.987081i \(-0.448780\pi\)
−0.584680 + 0.811264i \(0.698780\pi\)
\(18\) 1.40161 + 0.188388i 0.330363 + 0.0444035i
\(19\) 3.42826 − 2.81350i 0.786496 − 0.645461i −0.152921 − 0.988238i \(-0.548868\pi\)
0.939416 + 0.342778i \(0.111368\pi\)
\(20\) 7.67803 − 1.74648i 1.71686 − 0.390524i
\(21\) 4.10136 + 1.24413i 0.894990 + 0.271492i
\(22\) 8.28232 + 1.95483i 1.76580 + 0.416771i
\(23\) −0.247363 + 1.24358i −0.0515787 + 0.259304i −0.997967 − 0.0637264i \(-0.979701\pi\)
0.946389 + 0.323030i \(0.104701\pi\)
\(24\) −2.69988 − 0.843011i −0.551110 − 0.172079i
\(25\) −10.2988 + 2.04856i −2.05977 + 0.409713i
\(26\) −1.98541 − 2.13134i −0.389371 − 0.417990i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) −7.66718 − 3.83280i −1.44896 − 0.724331i
\(29\) −2.41573 + 4.51951i −0.448590 + 0.839252i 0.551398 + 0.834242i \(0.314095\pi\)
−0.999987 + 0.00500979i \(0.998405\pi\)
\(30\) 5.26752 + 1.80400i 0.961713 + 0.329365i
\(31\) −4.26568 + 4.26568i −0.766138 + 0.766138i −0.977424 − 0.211286i \(-0.932235\pi\)
0.211286 + 0.977424i \(0.432235\pi\)
\(32\) 5.03914 + 2.57043i 0.890802 + 0.454392i
\(33\) 4.25495 + 4.25495i 0.740691 + 0.740691i
\(34\) 1.17835 + 2.40586i 0.202085 + 0.412601i
\(35\) 14.8815 + 7.95433i 2.51543 + 1.34453i
\(36\) −1.31053 − 1.51080i −0.218421 − 0.251800i
\(37\) −1.72240 + 2.09874i −0.283160 + 0.345031i −0.895094 − 0.445877i \(-0.852892\pi\)
0.611934 + 0.790909i \(0.290392\pi\)
\(38\) −6.26802 − 0.222187i −1.01681 − 0.0360435i
\(39\) −0.401822 − 2.02009i −0.0643429 − 0.323474i
\(40\) −9.77788 − 5.32896i −1.54602 − 0.842583i
\(41\) 2.36639 + 0.470705i 0.369569 + 0.0735118i 0.376381 − 0.926465i \(-0.377168\pi\)
−0.00681215 + 0.999977i \(0.502168\pi\)
\(42\) −3.18677 − 5.15582i −0.491729 − 0.795561i
\(43\) 2.56696 − 8.46214i 0.391458 − 1.29046i −0.510960 − 0.859604i \(-0.670710\pi\)
0.902419 − 0.430860i \(-0.141790\pi\)
\(44\) −6.95692 − 9.82027i −1.04880 − 1.48046i
\(45\) 2.49766 + 3.04340i 0.372329 + 0.453684i
\(46\) 1.42554 − 1.08774i 0.210185 − 0.160378i
\(47\) −2.96974 + 7.16958i −0.433181 + 1.04579i 0.545075 + 0.838388i \(0.316501\pi\)
−0.978255 + 0.207404i \(0.933499\pi\)
\(48\) 2.07535 + 3.41949i 0.299551 + 0.493561i
\(49\) −4.35073 − 10.5036i −0.621533 − 1.50051i
\(50\) 12.8404 + 7.45984i 1.81591 + 1.05498i
\(51\) −0.185673 + 1.88517i −0.0259994 + 0.263976i
\(52\) 0.112481 + 4.11780i 0.0155982 + 0.571036i
\(53\) −1.42695 − 2.66964i −0.196007 − 0.366704i 0.764648 − 0.644448i \(-0.222913\pi\)
−0.960655 + 0.277745i \(0.910413\pi\)
\(54\) −0.226590 − 1.39594i −0.0308349 − 0.189964i
\(55\) 13.1620 + 19.6983i 1.77476 + 2.65612i
\(56\) 4.54789 + 11.2369i 0.607738 + 1.50160i
\(57\) −3.68752 − 2.46392i −0.488424 − 0.326354i
\(58\) 6.78969 − 2.53449i 0.891530 − 0.332794i
\(59\) −5.09931 + 0.502238i −0.663874 + 0.0653858i −0.424341 − 0.905502i \(-0.639494\pi\)
−0.239533 + 0.970888i \(0.576994\pi\)
\(60\) −3.90009 − 6.84044i −0.503499 − 0.883098i
\(61\) −9.81634 + 2.97775i −1.25685 + 0.381262i −0.847344 − 0.531044i \(-0.821800\pi\)
−0.409509 + 0.912306i \(0.634300\pi\)
\(62\) 8.51456 − 0.534921i 1.08135 − 0.0679351i
\(63\) − 4.28591i − 0.539974i
\(64\) −2.94098 − 7.43980i −0.367622 − 0.929975i
\(65\) − 8.10908i − 1.00581i
\(66\) −0.533576 − 8.49315i −0.0656787 − 1.04543i
\(67\) 4.02627 − 1.22136i 0.491887 − 0.149212i −0.0345899 − 0.999402i \(-0.511013\pi\)
0.526477 + 0.850189i \(0.323513\pi\)
\(68\) 1.00036 − 3.65412i 0.121312 − 0.443127i
\(69\) 1.26183 − 0.124280i 0.151907 − 0.0149615i
\(70\) −8.34536 − 22.3566i −0.997462 − 2.67212i
\(71\) −11.6047 − 7.75400i −1.37722 − 0.920231i −0.377240 − 0.926116i \(-0.623127\pi\)
−0.999983 + 0.00588476i \(0.998127\pi\)
\(72\) −0.0229781 + 2.82833i −0.00270799 + 0.333322i
\(73\) 1.68904 + 2.52783i 0.197687 + 0.295860i 0.917048 − 0.398777i \(-0.130565\pi\)
−0.719361 + 0.694637i \(0.755565\pi\)
\(74\) 3.79002 − 0.615197i 0.440581 − 0.0715152i
\(75\) 4.94995 + 9.26070i 0.571570 + 1.06933i
\(76\) 6.44088 + 6.09836i 0.738819 + 0.699530i
\(77\) 2.52786 − 25.6658i 0.288077 − 2.92489i
\(78\) −1.46323 + 2.51862i −0.165678 + 0.285177i
\(79\) −4.52669 − 10.9284i −0.509292 − 1.22954i −0.944292 − 0.329109i \(-0.893252\pi\)
0.435000 − 0.900431i \(-0.356748\pi\)
\(80\) 5.38737 + 14.7982i 0.602326 + 1.65449i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −2.06985 − 2.71266i −0.228576 − 0.299563i
\(83\) 0.813545 + 0.991307i 0.0892981 + 0.108810i 0.815743 − 0.578415i \(-0.196329\pi\)
−0.726444 + 0.687225i \(0.758829\pi\)
\(84\) −1.44209 + 8.44964i −0.157345 + 0.921931i
\(85\) −2.16493 + 7.13682i −0.234820 + 0.774097i
\(86\) −10.6378 + 6.57511i −1.14710 + 0.709013i
\(87\) 5.02615 + 0.999764i 0.538860 + 0.107186i
\(88\) −1.80578 + 16.9237i −0.192497 + 1.80407i
\(89\) −0.287119 − 1.44344i −0.0304345 − 0.153005i 0.962580 − 0.270998i \(-0.0873538\pi\)
−0.993014 + 0.117994i \(0.962354\pi\)
\(90\) 0.197245 − 5.56438i 0.0207914 − 0.586537i
\(91\) −5.60014 + 6.82379i −0.587054 + 0.715327i
\(92\) −2.52951 − 0.179557i −0.263720 − 0.0187201i
\(93\) 5.32026 + 2.84374i 0.551685 + 0.294882i
\(94\) 9.85604 − 4.82732i 1.01657 − 0.497900i
\(95\) −12.3466 − 12.3466i −1.26673 − 1.26673i
\(96\) 0.996961 − 5.56831i 0.101752 − 0.568313i
\(97\) 3.16976 − 3.16976i 0.321840 − 0.321840i −0.527632 − 0.849473i \(-0.676920\pi\)
0.849473 + 0.527632i \(0.176920\pi\)
\(98\) −5.20938 + 15.2109i −0.526227 + 1.53653i
\(99\) 2.83658 − 5.30688i 0.285087 − 0.533361i
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 384.2.v.a.13.8 ✓ 512
128.69 even 32 inner 384.2.v.a.325.8 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.8 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.8 yes 512 128.69 even 32 inner