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Properties

Label 384.2.v.a.13.9
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 13.9
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.9

$q$-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+(-0.922626 + 1.07180i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(-0.297523 - 1.97775i) q^{4} +(-0.0692954 - 0.703568i) q^{5} +(1.29348 + 0.571770i) q^{6} +(-2.09840 + 3.14048i) q^{7} +(2.39426 + 1.50583i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(0.818020 + 0.574859i) q^{10} +(2.74124 - 1.46523i) q^{11} +(-1.80622 + 0.858821i) q^{12} +(0.988056 + 0.0973150i) q^{13} +(-1.42993 - 5.14656i) q^{14} +(-0.653157 + 0.270546i) q^{15} +(-3.82296 + 1.17685i) q^{16} +(5.69460 + 2.35878i) q^{17} +(0.171674 - 1.40376i) q^{18} +(4.13124 - 3.39042i) q^{19} +(-1.37086 + 0.346376i) q^{20} +(3.61438 + 1.09641i) q^{21} +(-0.958710 + 4.28993i) q^{22} +(1.75976 - 8.84689i) q^{23} +(0.745978 - 2.72828i) q^{24} +(4.41372 - 0.877944i) q^{25} +(-1.01591 + 0.969216i) q^{26} +(0.773010 + 0.634393i) q^{27} +(6.83539 + 3.21574i) q^{28} +(4.38728 - 8.20803i) q^{29} +(0.312647 - 0.949669i) q^{30} +(-6.32341 + 6.32341i) q^{31} +(2.26581 - 5.18325i) q^{32} +(-2.19787 - 2.19787i) q^{33} +(-7.78214 + 3.92722i) q^{34} +(2.35495 + 1.25875i) q^{35} +(1.34616 + 1.47914i) q^{36} +(0.539535 - 0.657425i) q^{37} +(-0.177726 + 7.55596i) q^{38} +(-0.193693 - 0.973759i) q^{39} +(0.893545 - 1.78887i) q^{40} +(4.53327 + 0.901723i) q^{41} +(-4.50986 + 2.86233i) q^{42} +(-3.21185 + 10.5881i) q^{43} +(-3.71343 - 4.98555i) q^{44} +(0.448498 + 0.546497i) q^{45} +(7.85853 + 10.0485i) q^{46} +(-3.20109 + 7.72811i) q^{47} +(2.23592 + 3.31672i) q^{48} +(-2.78054 - 6.71281i) q^{49} +(-3.13123 + 5.54065i) q^{50} +(0.604157 - 6.13411i) q^{51} +(-0.101505 - 1.98308i) q^{52} +(3.54717 + 6.63629i) q^{53} +(-1.39314 + 0.243207i) q^{54} +(-1.22084 - 1.82712i) q^{55} +(-9.75315 + 4.35927i) q^{56} +(-4.44366 - 2.96916i) q^{57} +(4.74957 + 12.2752i) q^{58} +(-4.92635 + 0.485203i) q^{59} +(0.729401 + 1.21128i) q^{60} +(5.12187 - 1.55370i) q^{61} +(-0.943307 - 12.6116i) q^{62} -3.77702i q^{63} +(3.46493 + 7.21071i) q^{64} -0.701908i q^{65} +(4.38350 - 0.327872i) q^{66} +(3.72518 - 1.13002i) q^{67} +(2.97080 - 11.9643i) q^{68} +(-8.97678 + 0.884136i) q^{69} +(-3.52186 + 1.36269i) q^{70} +(-5.87698 - 3.92687i) q^{71} +(-2.82735 + 0.0781222i) q^{72} +(-3.51161 - 5.25550i) q^{73} +(0.206841 + 1.18483i) q^{74} +(-2.12138 - 3.96881i) q^{75} +(-7.93453 - 7.16181i) q^{76} +(-1.15072 + 11.6834i) q^{77} +(1.22238 + 0.690815i) q^{78} +(-2.98662 - 7.21034i) q^{79} +(1.09291 + 2.60816i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-5.14898 + 4.02682i) q^{82} +(0.745612 + 0.908531i) q^{83} +(1.09306 - 7.47454i) q^{84} +(1.26495 - 4.16999i) q^{85} +(-8.38497 - 13.2113i) q^{86} +(-9.12815 - 1.81570i) q^{87} +(8.76963 + 0.619734i) q^{88} +(-0.652421 - 3.27994i) q^{89} +(-0.999533 - 0.0235103i) q^{90} +(-2.37895 + 2.89876i) q^{91} +(-18.0205 - 0.848199i) q^{92} +(7.88672 + 4.21554i) q^{93} +(-5.32961 - 10.5611i) q^{94} +(-2.67167 - 2.67167i) q^{95} +(-5.61779 - 0.663628i) q^{96} +(3.71723 - 3.71723i) q^{97} +(9.76020 + 3.21322i) q^{98} +(-1.46523 + 2.74124i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.922626 + 1.07180i −0.652395 + 0.757879i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) −0.297523 − 1.97775i −0.148761 − 0.988873i
\(5\) −0.0692954 − 0.703568i −0.0309898 − 0.314645i −0.998166 − 0.0605379i \(-0.980718\pi\)
0.967176 − 0.254107i \(-0.0817816\pi\)
\(6\) 1.29348 + 0.571770i 0.528059 + 0.233424i
\(7\) −2.09840 + 3.14048i −0.793121 + 1.18699i 0.185768 + 0.982594i \(0.440523\pi\)
−0.978888 + 0.204395i \(0.934477\pi\)
\(8\) 2.39426 + 1.50583i 0.846497 + 0.532393i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) 0.818020 + 0.574859i 0.258680 + 0.181786i
\(11\) 2.74124 − 1.46523i 0.826516 − 0.441782i −0.00318153 − 0.999995i \(-0.501013\pi\)
0.829698 + 0.558213i \(0.188513\pi\)
\(12\) −1.80622 + 0.858821i −0.521410 + 0.247920i
\(13\) 0.988056 + 0.0973150i 0.274037 + 0.0269903i 0.234102 − 0.972212i \(-0.424785\pi\)
0.0399350 + 0.999202i \(0.487285\pi\)
\(14\) −1.42993 − 5.14656i −0.382166 − 1.37548i
\(15\) −0.653157 + 0.270546i −0.168644 + 0.0698548i
\(16\) −3.82296 + 1.17685i −0.955740 + 0.294212i
\(17\) 5.69460 + 2.35878i 1.38114 + 0.572089i 0.944787 − 0.327685i \(-0.106269\pi\)
0.436357 + 0.899774i \(0.356269\pi\)
\(18\) 0.171674 − 1.40376i 0.0404639 − 0.330868i
\(19\) 4.13124 − 3.39042i 0.947771 − 0.777816i −0.0273186 − 0.999627i \(-0.508697\pi\)
0.975090 + 0.221811i \(0.0711969\pi\)
\(20\) −1.37086 + 0.346376i −0.306534 + 0.0774520i
\(21\) 3.61438 + 1.09641i 0.788723 + 0.239257i
\(22\) −0.958710 + 4.28993i −0.204398 + 0.914616i
\(23\) 1.75976 − 8.84689i 0.366935 − 1.84470i −0.150008 − 0.988685i \(-0.547930\pi\)
0.516943 − 0.856020i \(-0.327070\pi\)
\(24\) 0.745978 − 2.72828i 0.152272 − 0.556908i
\(25\) 4.41372 − 0.877944i 0.882744 − 0.175589i
\(26\) −1.01591 + 0.969216i −0.199236 + 0.190079i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) 6.83539 + 3.21574i 1.29177 + 0.607718i
\(29\) 4.38728 − 8.20803i 0.814698 − 1.52419i −0.0364915 − 0.999334i \(-0.511618\pi\)
0.851189 − 0.524859i \(-0.175882\pi\)
\(30\) 0.312647 − 0.949669i 0.0570813 − 0.173385i
\(31\) −6.32341 + 6.32341i −1.13572 + 1.13572i −0.146509 + 0.989209i \(0.546804\pi\)
−0.989209 + 0.146509i \(0.953196\pi\)
\(32\) 2.26581 − 5.18325i 0.400543 − 0.916278i
\(33\) −2.19787 − 2.19787i −0.382601 − 0.382601i
\(34\) −7.78214 + 3.92722i −1.33463 + 0.673512i
\(35\) 2.35495 + 1.25875i 0.398059 + 0.212767i
\(36\) 1.34616 + 1.47914i 0.224360 + 0.246524i
\(37\) 0.539535 − 0.657425i 0.0886989 − 0.108080i −0.726770 − 0.686881i \(-0.758979\pi\)
0.815469 + 0.578801i \(0.196479\pi\)
\(38\) −0.177726 + 7.55596i −0.0288310 + 1.22574i
\(39\) −0.193693 − 0.973759i −0.0310157 − 0.155926i
\(40\) 0.893545 − 1.78887i 0.141282 − 0.282845i
\(41\) 4.53327 + 0.901723i 0.707977 + 0.140825i 0.535929 − 0.844263i \(-0.319961\pi\)
0.172048 + 0.985089i \(0.444961\pi\)
\(42\) −4.50986 + 2.86233i −0.695887 + 0.441667i
\(43\) −3.21185 + 10.5881i −0.489803 + 1.61466i 0.268859 + 0.963180i \(0.413353\pi\)
−0.758662 + 0.651485i \(0.774147\pi\)
\(44\) −3.71343 − 4.98555i −0.559820 − 0.751600i
\(45\) 0.448498 + 0.546497i 0.0668582 + 0.0814669i
\(46\) 7.85853 + 10.0485i 1.15868 + 1.48157i
\(47\) −3.20109 + 7.72811i −0.466927 + 1.12726i 0.498571 + 0.866849i \(0.333858\pi\)
−0.965498 + 0.260412i \(0.916142\pi\)
\(48\) 2.23592 + 3.31672i 0.322727 + 0.478728i
\(49\) −2.78054 − 6.71281i −0.397219 − 0.958972i
\(50\) −3.13123 + 5.54065i −0.442823 + 0.783567i
\(51\) 0.604157 − 6.13411i 0.0845990 − 0.858948i
\(52\) −0.101505 − 1.98308i −0.0140762 − 0.275003i
\(53\) 3.54717 + 6.63629i 0.487241 + 0.911565i 0.998636 + 0.0522184i \(0.0166292\pi\)
−0.511394 + 0.859346i \(0.670871\pi\)
\(54\) −1.39314 + 0.243207i −0.189583 + 0.0330963i
\(55\) −1.22084 − 1.82712i −0.164618 − 0.246368i
\(56\) −9.75315 + 4.35927i −1.30332 + 0.582532i
\(57\) −4.44366 − 2.96916i −0.588578 − 0.393275i
\(58\) 4.74957 + 12.2752i 0.623649 + 1.61182i
\(59\) −4.92635 + 0.485203i −0.641357 + 0.0631681i −0.413467 − 0.910519i \(-0.635682\pi\)
−0.227889 + 0.973687i \(0.573182\pi\)
\(60\) 0.729401 + 1.21128i 0.0941653 + 0.156376i
\(61\) 5.12187 − 1.55370i 0.655789 − 0.198931i 0.0551825 − 0.998476i \(-0.482426\pi\)
0.600606 + 0.799545i \(0.294926\pi\)
\(62\) −0.943307 − 12.6116i −0.119800 − 1.60167i
\(63\) − 3.77702i − 0.475860i
\(64\) 3.46493 + 7.21071i 0.433116 + 0.901338i
\(65\) − 0.701908i − 0.0870609i
\(66\) 4.38350 − 0.327872i 0.539572 − 0.0403583i
\(67\) 3.72518 − 1.13002i 0.455103 − 0.138054i −0.0544070 − 0.998519i \(-0.517327\pi\)
0.509510 + 0.860465i \(0.329827\pi\)
\(68\) 2.97080 − 11.9643i 0.360262 − 1.45088i
\(69\) −8.97678 + 0.884136i −1.08068 + 0.106437i
\(70\) −3.52186 + 1.36269i −0.420943 + 0.162872i
\(71\) −5.87698 − 3.92687i −0.697469 − 0.466034i 0.155608 − 0.987819i \(-0.450266\pi\)
−0.853077 + 0.521785i \(0.825266\pi\)
\(72\) −2.82735 + 0.0781222i −0.333206 + 0.00920679i
\(73\) −3.51161 − 5.25550i −0.411003 − 0.615110i 0.566995 − 0.823721i \(-0.308106\pi\)
−0.977999 + 0.208611i \(0.933106\pi\)
\(74\) 0.206841 + 1.18483i 0.0240448 + 0.137734i
\(75\) −2.12138 − 3.96881i −0.244955 − 0.458279i
\(76\) −7.93453 − 7.16181i −0.910153 − 0.821516i
\(77\) −1.15072 + 11.6834i −0.131137 + 1.33145i
\(78\) 1.22238 + 0.690815i 0.138408 + 0.0782194i
\(79\) −2.98662 − 7.21034i −0.336021 − 0.811227i −0.998090 − 0.0617836i \(-0.980321\pi\)
0.662068 − 0.749443i \(-0.269679\pi\)
\(80\) 1.09291 + 2.60816i 0.122191 + 0.291601i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −5.14898 + 4.02682i −0.568610 + 0.444687i
\(83\) 0.745612 + 0.908531i 0.0818415 + 0.0997242i 0.812321 − 0.583210i \(-0.198204\pi\)
−0.730480 + 0.682934i \(0.760704\pi\)
\(84\) 1.09306 − 7.47454i 0.119263 − 0.815539i
\(85\) 1.26495 − 4.16999i 0.137203 − 0.452299i
\(86\) −8.38497 − 13.2113i −0.904175 − 1.42461i
\(87\) −9.12815 − 1.81570i −0.978641 − 0.194664i
\(88\) 8.76963 + 0.619734i 0.934846 + 0.0660638i
\(89\) −0.652421 − 3.27994i −0.0691564 − 0.347673i 0.930679 − 0.365838i \(-0.119218\pi\)
−0.999835 + 0.0181650i \(0.994218\pi\)
\(90\) −0.999533 − 0.0235103i −0.105360 − 0.00247820i
\(91\) −2.37895 + 2.89876i −0.249382 + 0.303873i
\(92\) −18.0205 − 0.848199i −1.87876 − 0.0884309i
\(93\) 7.88672 + 4.21554i 0.817814 + 0.437131i
\(94\) −5.32961 − 10.5611i −0.549707 − 1.08929i
\(95\) −2.67167 − 2.67167i −0.274107 − 0.274107i
\(96\) −5.61779 − 0.663628i −0.573364 − 0.0677313i
\(97\) 3.71723 − 3.71723i 0.377427 − 0.377427i −0.492746 − 0.870173i \(-0.664007\pi\)
0.870173 + 0.492746i \(0.164007\pi\)
\(98\) 9.76020 + 3.21322i 0.985929 + 0.324585i
\(99\) −1.46523 + 2.74124i −0.147261 + 0.275505i
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.13.9 ✓ 512
128.69 even 32 inner 384.2.v.a.325.9 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.9 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.9 yes 512 128.69 even 32 inner