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Properties

Label 384.2.v.a.13.7
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 13.7
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.7

$q$-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+(-1.06403 - 0.931580i) q^{2} +(0.290285 + 0.956940i) q^{3} +(0.264317 + 1.98246i) q^{4} +(-0.231940 - 2.35493i) q^{5} +(0.582595 - 1.28864i) q^{6} +(0.430118 - 0.643717i) q^{7} +(1.56558 - 2.35563i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-1.94701 + 2.72179i) q^{10} +(2.50304 - 1.33790i) q^{11} +(-1.82037 + 0.828413i) q^{12} +(-2.62518 - 0.258558i) q^{13} +(-1.05733 + 0.284245i) q^{14} +(2.18620 - 0.905553i) q^{15} +(-3.86027 + 1.04800i) q^{16} +(0.809188 + 0.335177i) q^{17} +(1.40227 + 0.183437i) q^{18} +(0.782804 - 0.642431i) q^{19} +(4.60724 - 1.08226i) q^{20} +(0.740856 + 0.224736i) q^{21} +(-3.90966 - 0.908213i) q^{22} +(1.67434 - 8.41747i) q^{23} +(2.70866 + 0.814361i) q^{24} +(-0.587974 + 0.116955i) q^{25} +(2.55240 + 2.72068i) q^{26} +(-0.773010 - 0.634393i) q^{27} +(1.38983 + 0.682545i) q^{28} +(2.09512 - 3.91970i) q^{29} +(-3.16978 - 1.07308i) q^{30} +(5.38932 - 5.38932i) q^{31} +(5.08374 + 2.48106i) q^{32} +(2.00688 + 2.00688i) q^{33} +(-0.548756 - 1.11046i) q^{34} +(-1.61567 - 0.863594i) q^{35} +(-1.32117 - 1.50151i) q^{36} +(-1.98847 + 2.42296i) q^{37} +(-1.43140 - 0.0456795i) q^{38} +(-0.514625 - 2.58720i) q^{39} +(-5.91045 - 3.14046i) q^{40} +(0.0909338 + 0.0180878i) q^{41} +(-0.578932 - 0.929292i) q^{42} +(1.94008 - 6.39559i) q^{43} +(3.31393 + 4.60853i) q^{44} +(1.50118 + 1.82919i) q^{45} +(-9.62309 + 7.39665i) q^{46} +(-3.55765 + 8.58892i) q^{47} +(-2.12345 - 3.38983i) q^{48} +(2.44941 + 5.91341i) q^{49} +(0.734575 + 0.423301i) q^{50} +(-0.0858492 + 0.871641i) q^{51} +(-0.181301 - 5.27265i) q^{52} +(-3.74120 - 6.99929i) q^{53} +(0.231518 + 1.39513i) q^{54} +(-3.73122 - 5.58416i) q^{55} +(-0.842974 - 2.02099i) q^{56} +(0.842004 + 0.562609i) q^{57} +(-5.88079 + 2.21890i) q^{58} +(1.53129 - 0.150819i) q^{59} +(2.37307 + 4.09469i) q^{60} +(-4.76028 + 1.44402i) q^{61} +(-10.7550 + 0.713812i) q^{62} +0.774192i q^{63} +(-3.09794 - 7.37582i) q^{64} +6.24208i q^{65} +(-0.265811 - 4.00496i) q^{66} +(-5.86212 + 1.77825i) q^{67} +(-0.450591 + 1.69277i) q^{68} +(8.54105 - 0.841220i) q^{69} +(0.914614 + 2.42402i) q^{70} +(-3.79243 - 2.53402i) q^{71} +(0.00698639 + 2.82842i) q^{72} +(6.25875 + 9.36688i) q^{73} +(4.37297 - 0.725680i) q^{74} +(-0.282599 - 0.528706i) q^{75} +(1.48050 + 1.38207i) q^{76} +(0.215371 - 2.18670i) q^{77} +(-1.86260 + 3.23227i) q^{78} +(4.49830 + 10.8599i) q^{79} +(3.36331 + 8.84760i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-0.0799059 - 0.103958i) q^{82} +(0.226883 + 0.276458i) q^{83} +(-0.249709 + 1.52812i) q^{84} +(0.601634 - 1.98332i) q^{85} +(-8.02231 + 4.99776i) q^{86} +(4.35910 + 0.867080i) q^{87} +(0.767101 - 7.99080i) q^{88} +(1.91512 + 9.62797i) q^{89} +(0.106740 - 3.34479i) q^{90} +(-1.29558 + 1.57866i) q^{91} +(17.1298 + 1.09442i) q^{92} +(6.72169 + 3.59282i) q^{93} +(11.7867 - 5.82463i) q^{94} +(-1.69444 - 1.69444i) q^{95} +(-0.898491 + 5.58504i) q^{96} +(-7.99181 + 7.99181i) q^{97} +(2.90256 - 8.57387i) q^{98} +(-1.33790 + 2.50304i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06403 − 0.931580i −0.752382 − 0.658727i
\(3\) 0.290285 + 0.956940i 0.167596 + 0.552490i
\(4\) 0.264317 + 1.98246i 0.132159 + 0.991229i
\(5\) −0.231940 − 2.35493i −0.103727 − 1.05316i −0.896321 − 0.443407i \(-0.853770\pi\)
0.792594 − 0.609750i \(-0.208730\pi\)
\(6\) 0.582595 − 1.28864i 0.237843 − 0.526083i
\(7\) 0.430118 − 0.643717i 0.162569 − 0.243302i −0.741238 − 0.671243i \(-0.765761\pi\)
0.903807 + 0.427941i \(0.140761\pi\)
\(8\) 1.56558 − 2.35563i 0.553515 − 0.832839i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −1.94701 + 2.72179i −0.615700 + 0.860704i
\(11\) 2.50304 − 1.33790i 0.754694 − 0.403392i −0.0487208 − 0.998812i \(-0.515514\pi\)
0.803414 + 0.595420i \(0.203014\pi\)
\(12\) −1.82037 + 0.828413i −0.525494 + 0.239142i
\(13\) −2.62518 − 0.258558i −0.728094 − 0.0717110i −0.272823 − 0.962064i \(-0.587957\pi\)
−0.455270 + 0.890353i \(0.650457\pi\)
\(14\) −1.05733 + 0.284245i −0.282584 + 0.0759676i
\(15\) 2.18620 − 0.905553i 0.564474 − 0.233813i
\(16\) −3.86027 + 1.04800i −0.965068 + 0.261999i
\(17\) 0.809188 + 0.335177i 0.196257 + 0.0812923i 0.478647 − 0.878007i \(-0.341127\pi\)
−0.282390 + 0.959300i \(0.591127\pi\)
\(18\) 1.40227 + 0.183437i 0.330517 + 0.0432366i
\(19\) 0.782804 − 0.642431i 0.179588 − 0.147384i −0.540305 − 0.841469i \(-0.681691\pi\)
0.719892 + 0.694086i \(0.244191\pi\)
\(20\) 4.60724 − 1.08226i 1.03021 − 0.242001i
\(21\) 0.740856 + 0.224736i 0.161668 + 0.0490414i
\(22\) −3.90966 − 0.908213i −0.833543 − 0.193632i
\(23\) 1.67434 − 8.41747i 0.349124 − 1.75516i −0.263377 − 0.964693i \(-0.584836\pi\)
0.612501 − 0.790470i \(-0.290164\pi\)
\(24\) 2.70866 + 0.814361i 0.552902 + 0.166231i
\(25\) −0.587974 + 0.116955i −0.117595 + 0.0233911i
\(26\) 2.55240 + 2.72068i 0.500567 + 0.533569i
\(27\) −0.773010 − 0.634393i −0.148766 − 0.122089i
\(28\) 1.38983 + 0.682545i 0.262653 + 0.128989i
\(29\) 2.09512 − 3.91970i 0.389055 − 0.727870i −0.608801 − 0.793323i \(-0.708349\pi\)
0.997856 + 0.0654527i \(0.0208491\pi\)
\(30\) −3.16978 − 1.07308i −0.578719 − 0.195917i
\(31\) 5.38932 − 5.38932i 0.967950 − 0.967950i −0.0315517 − 0.999502i \(-0.510045\pi\)
0.999502 + 0.0315517i \(0.0100449\pi\)
\(32\) 5.08374 + 2.48106i 0.898686 + 0.438593i
\(33\) 2.00688 + 2.00688i 0.349354 + 0.349354i
\(34\) −0.548756 − 1.11046i −0.0941109 − 0.190442i
\(35\) −1.61567 − 0.863594i −0.273098 − 0.145974i
\(36\) −1.32117 − 1.50151i −0.220194 − 0.250251i
\(37\) −1.98847 + 2.42296i −0.326903 + 0.398332i −0.910364 − 0.413809i \(-0.864198\pi\)
0.583461 + 0.812141i \(0.301698\pi\)
\(38\) −1.43140 − 0.0456795i −0.232204 − 0.00741019i
\(39\) −0.514625 − 2.58720i −0.0824060 − 0.414283i
\(40\) −5.91045 − 3.14046i −0.934525 − 0.496550i
\(41\) 0.0909338 + 0.0180878i 0.0142015 + 0.00282485i 0.202186 − 0.979347i \(-0.435195\pi\)
−0.187984 + 0.982172i \(0.560195\pi\)
\(42\) −0.578932 − 0.929292i −0.0893312 − 0.143393i
\(43\) 1.94008 − 6.39559i 0.295860 − 0.975319i −0.675431 − 0.737423i \(-0.736042\pi\)
0.971291 − 0.237896i \(-0.0764576\pi\)
\(44\) 3.31393 + 4.60853i 0.499593 + 0.694762i
\(45\) 1.50118 + 1.82919i 0.223783 + 0.272680i
\(46\) −9.62309 + 7.39665i −1.41885 + 1.09058i
\(47\) −3.55765 + 8.58892i −0.518936 + 1.25282i 0.419621 + 0.907699i \(0.362163\pi\)
−0.938557 + 0.345123i \(0.887837\pi\)
\(48\) −2.12345 − 3.38983i −0.306493 − 0.489280i
\(49\) 2.44941 + 5.91341i 0.349916 + 0.844773i
\(50\) 0.734575 + 0.423301i 0.103885 + 0.0598638i
\(51\) −0.0858492 + 0.871641i −0.0120213 + 0.122054i
\(52\) −0.181301 − 5.27265i −0.0251419 − 0.731185i
\(53\) −3.74120 − 6.99929i −0.513893 − 0.961427i −0.996338 − 0.0855003i \(-0.972751\pi\)
0.482445 − 0.875926i \(-0.339749\pi\)
\(54\) 0.231518 + 1.39513i 0.0315056 + 0.189854i
\(55\) −3.73122 − 5.58416i −0.503117 − 0.752968i
\(56\) −0.842974 − 2.02099i −0.112647 − 0.270066i
\(57\) 0.842004 + 0.562609i 0.111526 + 0.0745194i
\(58\) −5.88079 + 2.21890i −0.772186 + 0.291356i
\(59\) 1.53129 − 0.150819i 0.199357 − 0.0196350i 0.00215431 − 0.999998i \(-0.499314\pi\)
0.197203 + 0.980363i \(0.436814\pi\)
\(60\) 2.37307 + 4.09469i 0.306362 + 0.528623i
\(61\) −4.76028 + 1.44402i −0.609492 + 0.184887i −0.579917 − 0.814675i \(-0.696915\pi\)
−0.0295743 + 0.999563i \(0.509415\pi\)
\(62\) −10.7550 + 0.713812i −1.36588 + 0.0906542i
\(63\) 0.774192i 0.0975390i
\(64\) −3.09794 − 7.37582i −0.387243 − 0.921978i
\(65\) 6.24208i 0.774235i
\(66\) −0.265811 − 4.00496i −0.0327190 − 0.492976i
\(67\) −5.86212 + 1.77825i −0.716172 + 0.217248i −0.627281 − 0.778793i \(-0.715832\pi\)
−0.0888909 + 0.996041i \(0.528332\pi\)
\(68\) −0.450591 + 1.69277i −0.0546422 + 0.205279i
\(69\) 8.54105 − 0.841220i 1.02822 − 0.101271i
\(70\) 0.914614 + 2.42402i 0.109317 + 0.289725i
\(71\) −3.79243 − 2.53402i −0.450079 − 0.300733i 0.309795 − 0.950803i \(-0.399739\pi\)
−0.759874 + 0.650070i \(0.774739\pi\)
\(72\) 0.00698639 + 2.82842i 0.000823354 + 0.333332i
\(73\) 6.25875 + 9.36688i 0.732531 + 1.09631i 0.991460 + 0.130414i \(0.0416306\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(74\) 4.37297 − 0.725680i 0.508348 − 0.0843586i
\(75\) −0.282599 − 0.528706i −0.0326317 − 0.0610497i
\(76\) 1.48050 + 1.38207i 0.169825 + 0.158534i
\(77\) 0.215371 − 2.18670i 0.0245438 − 0.249198i
\(78\) −1.86260 + 3.23227i −0.210898 + 0.365982i
\(79\) 4.49830 + 10.8599i 0.506098 + 1.22183i 0.946112 + 0.323839i \(0.104974\pi\)
−0.440014 + 0.897991i \(0.645026\pi\)
\(80\) 3.36331 + 8.84760i 0.376029 + 0.989192i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −0.0799059 − 0.103958i −0.00882413 − 0.0114802i
\(83\) 0.226883 + 0.276458i 0.0249037 + 0.0303452i 0.785313 − 0.619098i \(-0.212502\pi\)
−0.760410 + 0.649444i \(0.775002\pi\)
\(84\) −0.249709 + 1.52812i −0.0272455 + 0.166731i
\(85\) 0.601634 − 1.98332i 0.0652564 − 0.215121i
\(86\) −8.02231 + 4.99776i −0.865068 + 0.538922i
\(87\) 4.35910 + 0.867080i 0.467345 + 0.0929607i
\(88\) 0.767101 − 7.99080i 0.0817732 − 0.851822i
\(89\) 1.91512 + 9.62797i 0.203003 + 1.02056i 0.939089 + 0.343675i \(0.111672\pi\)
−0.736086 + 0.676888i \(0.763328\pi\)
\(90\) 0.106740 − 3.34479i 0.0112514 − 0.352571i
\(91\) −1.29558 + 1.57866i −0.135813 + 0.165489i
\(92\) 17.1298 + 1.09442i 1.78591 + 0.114101i
\(93\) 6.72169 + 3.59282i 0.697007 + 0.372558i
\(94\) 11.7867 − 5.82463i 1.21571 − 0.600765i
\(95\) −1.69444 − 1.69444i −0.173846 − 0.173846i
\(96\) −0.898491 + 5.58504i −0.0917019 + 0.570021i
\(97\) −7.99181 + 7.99181i −0.811446 + 0.811446i −0.984851 − 0.173405i \(-0.944523\pi\)
0.173405 + 0.984851i \(0.444523\pi\)
\(98\) 2.90256 − 8.57387i 0.293203 − 0.866091i
\(99\) −1.33790 + 2.50304i −0.134464 + 0.251565i
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.13.7 ✓ 512
128.69 even 32 inner 384.2.v.a.325.7 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.7 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.7 yes 512 128.69 even 32 inner