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Properties

Label 384.2.v.a.13.12
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 13.12
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.12

$q$-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+(-0.607361 + 1.27715i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(-1.26223 - 1.55138i) q^{4} +(0.269039 + 2.73160i) q^{5} +(1.39846 + 0.210471i) q^{6} +(0.857384 - 1.28317i) q^{7} +(2.74797 - 0.669804i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-3.65206 - 1.31546i) q^{10} +(-5.50570 + 2.94286i) q^{11} +(-1.11818 + 1.65822i) q^{12} +(5.53992 + 0.545635i) q^{13} +(1.11805 + 1.87435i) q^{14} +(2.53588 - 1.05039i) q^{15} +(-0.813572 + 3.91639i) q^{16} +(3.96432 + 1.64207i) q^{17} +(-0.204545 - 1.39934i) q^{18} +(-2.47517 + 2.03132i) q^{19} +(3.89816 - 3.86527i) q^{20} +(-1.47680 - 0.447982i) q^{21} +(-0.414526 - 8.81898i) q^{22} +(-1.24331 + 6.25056i) q^{23} +(-1.43866 - 2.43521i) q^{24} +(-2.48531 + 0.494359i) q^{25} +(-4.06159 + 6.74392i) q^{26} +(0.773010 + 0.634393i) q^{27} +(-3.07289 + 0.289515i) q^{28} +(-1.87108 + 3.50055i) q^{29} +(-0.198680 + 3.87666i) q^{30} +(-1.60293 + 1.60293i) q^{31} +(-4.50769 - 3.41771i) q^{32} +(4.41436 + 4.41436i) q^{33} +(-4.50495 + 4.06570i) q^{34} +(3.73576 + 1.99680i) q^{35} +(1.91140 + 0.588672i) q^{36} +(-3.90148 + 4.75396i) q^{37} +(-1.09098 - 4.39491i) q^{38} +(-1.08601 - 5.45977i) q^{39} +(2.56894 + 7.32615i) q^{40} +(1.19260 + 0.237222i) q^{41} +(1.46909 - 1.61401i) q^{42} +(-1.18922 + 3.92032i) q^{43} +(11.5149 + 4.82689i) q^{44} +(-1.74129 - 2.12177i) q^{45} +(-7.22776 - 5.38424i) q^{46} +(4.43305 - 10.7023i) q^{47} +(3.98392 - 0.358328i) q^{48} +(1.76738 + 4.26683i) q^{49} +(0.878109 - 3.47437i) q^{50} +(0.420586 - 4.27029i) q^{51} +(-6.14615 - 9.28325i) q^{52} +(-3.21836 - 6.02112i) q^{53} +(-1.27971 + 0.601945i) q^{54} +(-9.51994 - 14.2476i) q^{55} +(1.49660 - 4.10038i) q^{56} +(2.66236 + 1.77893i) q^{57} +(-3.33430 - 4.51575i) q^{58} +(12.4050 - 1.22179i) q^{59} +(-4.83041 - 2.60828i) q^{60} +(0.672244 - 0.203923i) q^{61} +(-1.07363 - 3.02074i) q^{62} +1.54325i q^{63} +(7.10273 - 3.68121i) q^{64} +15.2796i q^{65} +(-8.31891 + 2.95669i) q^{66} +(5.38202 - 1.63262i) q^{67} +(-2.45638 - 8.22284i) q^{68} +(6.34233 - 0.624665i) q^{69} +(-4.81917 + 3.55834i) q^{70} +(-4.06093 - 2.71343i) q^{71} +(-1.91273 + 2.08361i) q^{72} +(-3.68754 - 5.51879i) q^{73} +(-3.70192 - 7.87014i) q^{74} +(1.19452 + 2.23479i) q^{75} +(6.27559 + 1.27595i) q^{76} +(-0.944324 + 9.58788i) q^{77} +(7.63254 + 1.92904i) q^{78} +(-1.22482 - 2.95698i) q^{79} +(-10.9169 - 1.16869i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-1.02731 + 1.37905i) q^{82} +(7.62729 + 9.29388i) q^{83} +(1.16906 + 2.85653i) q^{84} +(-3.41893 + 11.2707i) q^{85} +(-4.28456 - 3.89986i) q^{86} +(3.89296 + 0.774358i) q^{87} +(-13.1584 + 11.7746i) q^{88} +(-1.91480 - 9.62637i) q^{89} +(3.76741 - 0.935211i) q^{90} +(5.44998 - 6.64082i) q^{91} +(11.2663 - 5.96076i) q^{92} +(1.99921 + 1.06860i) q^{93} +(10.9760 + 12.1619i) q^{94} +(-6.21467 - 6.21467i) q^{95} +(-1.96204 + 5.30570i) q^{96} +(9.90497 - 9.90497i) q^{97} +(-6.52282 - 0.334296i) q^{98} +(2.94286 - 5.50570i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.607361 + 1.27715i −0.429469 + 0.903082i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) −1.26223 − 1.55138i −0.631113 − 0.775691i
\(5\) 0.269039 + 2.73160i 0.120318 + 1.22161i 0.845851 + 0.533420i \(0.179093\pi\)
−0.725533 + 0.688187i \(0.758407\pi\)
\(6\) 1.39846 + 0.210471i 0.570921 + 0.0859244i
\(7\) 0.857384 − 1.28317i 0.324061 − 0.484991i −0.633292 − 0.773913i \(-0.718297\pi\)
0.957353 + 0.288922i \(0.0932968\pi\)
\(8\) 2.74797 − 0.669804i 0.971556 − 0.236811i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −3.65206 − 1.31546i −1.15488 − 0.415985i
\(11\) −5.50570 + 2.94286i −1.66003 + 0.887305i −0.670681 + 0.741746i \(0.733998\pi\)
−0.989350 + 0.145559i \(0.953502\pi\)
\(12\) −1.11818 + 1.65822i −0.322789 + 0.478686i
\(13\) 5.53992 + 0.545635i 1.53650 + 0.151332i 0.830555 − 0.556937i \(-0.188023\pi\)
0.705943 + 0.708268i \(0.250523\pi\)
\(14\) 1.11805 + 1.87435i 0.298812 + 0.500942i
\(15\) 2.53588 − 1.05039i 0.654761 − 0.271211i
\(16\) −0.813572 + 3.91639i −0.203393 + 0.979097i
\(17\) 3.96432 + 1.64207i 0.961489 + 0.398262i 0.807537 − 0.589817i \(-0.200800\pi\)
0.153952 + 0.988078i \(0.450800\pi\)
\(18\) −0.204545 − 1.39934i −0.0482116 − 0.329828i
\(19\) −2.47517 + 2.03132i −0.567844 + 0.466017i −0.873974 − 0.485972i \(-0.838466\pi\)
0.306130 + 0.951990i \(0.400966\pi\)
\(20\) 3.89816 − 3.86527i 0.871655 − 0.864301i
\(21\) −1.47680 − 0.447982i −0.322264 − 0.0977576i
\(22\) −0.414526 − 8.81898i −0.0883772 − 1.88021i
\(23\) −1.24331 + 6.25056i −0.259249 + 1.30333i 0.603364 + 0.797466i \(0.293827\pi\)
−0.862612 + 0.505865i \(0.831173\pi\)
\(24\) −1.43866 − 2.43521i −0.293665 − 0.497086i
\(25\) −2.48531 + 0.494359i −0.497062 + 0.0988718i
\(26\) −4.06159 + 6.74392i −0.796543 + 1.32259i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) −3.07289 + 0.289515i −0.580722 + 0.0547132i
\(29\) −1.87108 + 3.50055i −0.347451 + 0.650035i −0.993572 − 0.113200i \(-0.963890\pi\)
0.646121 + 0.763235i \(0.276390\pi\)
\(30\) −0.198680 + 3.87666i −0.0362739 + 0.707779i
\(31\) −1.60293 + 1.60293i −0.287895 + 0.287895i −0.836247 − 0.548353i \(-0.815255\pi\)
0.548353 + 0.836247i \(0.315255\pi\)
\(32\) −4.50769 − 3.41771i −0.796854 − 0.604172i
\(33\) 4.41436 + 4.41436i 0.768441 + 0.768441i
\(34\) −4.50495 + 4.06570i −0.772592 + 0.697262i
\(35\) 3.73576 + 1.99680i 0.631458 + 0.337522i
\(36\) 1.91140 + 0.588672i 0.318567 + 0.0981120i
\(37\) −3.90148 + 4.75396i −0.641399 + 0.781547i −0.987664 − 0.156590i \(-0.949950\pi\)
0.346265 + 0.938137i \(0.387450\pi\)
\(38\) −1.09098 − 4.39491i −0.176980 − 0.712949i
\(39\) −1.08601 − 5.45977i −0.173902 − 0.874262i
\(40\) 2.56894 + 7.32615i 0.406186 + 1.15837i
\(41\) 1.19260 + 0.237222i 0.186252 + 0.0370479i 0.287335 − 0.957830i \(-0.407231\pi\)
−0.101083 + 0.994878i \(0.532231\pi\)
\(42\) 1.46909 − 1.61401i 0.226685 − 0.249047i
\(43\) −1.18922 + 3.92032i −0.181354 + 0.597844i 0.818378 + 0.574681i \(0.194874\pi\)
−0.999732 + 0.0231631i \(0.992626\pi\)
\(44\) 11.5149 + 4.82689i 1.73594 + 0.727681i
\(45\) −1.74129 − 2.12177i −0.259576 − 0.316295i
\(46\) −7.22776 − 5.38424i −1.06568 − 0.793863i
\(47\) 4.43305 − 10.7023i 0.646627 − 1.56110i −0.170951 − 0.985279i \(-0.554684\pi\)
0.817579 − 0.575817i \(-0.195316\pi\)
\(48\) 3.98392 − 0.358328i 0.575029 − 0.0517202i
\(49\) 1.76738 + 4.26683i 0.252483 + 0.609547i
\(50\) 0.878109 − 3.47437i 0.124183 − 0.491350i
\(51\) 0.420586 − 4.27029i 0.0588939 − 0.597960i
\(52\) −6.14615 − 9.28325i −0.852317 − 1.28736i
\(53\) −3.21836 − 6.02112i −0.442075 − 0.827065i 0.557921 − 0.829894i \(-0.311599\pi\)
−0.999996 + 0.00282966i \(0.999099\pi\)
\(54\) −1.27971 + 0.601945i −0.174147 + 0.0819143i
\(55\) −9.51994 − 14.2476i −1.28367 − 1.92115i
\(56\) 1.49660 − 4.10038i 0.199992 − 0.547937i
\(57\) 2.66236 + 1.77893i 0.352638 + 0.235625i
\(58\) −3.33430 − 4.51575i −0.437816 − 0.592947i
\(59\) 12.4050 − 1.22179i 1.61500 − 0.159063i 0.750209 − 0.661201i \(-0.229953\pi\)
0.864788 + 0.502137i \(0.167453\pi\)
\(60\) −4.83041 − 2.60828i −0.623604 − 0.336727i
\(61\) 0.672244 − 0.203923i 0.0860720 − 0.0261097i −0.246955 − 0.969027i \(-0.579430\pi\)
0.333027 + 0.942917i \(0.391930\pi\)
\(62\) −1.07363 − 3.02074i −0.136351 − 0.383634i
\(63\) 1.54325i 0.194431i
\(64\) 7.10273 − 3.68121i 0.887841 − 0.460151i
\(65\) 15.2796i 1.89520i
\(66\) −8.31891 + 2.95669i −1.02399 + 0.363944i
\(67\) 5.38202 − 1.63262i 0.657519 − 0.199456i 0.0561434 − 0.998423i \(-0.482120\pi\)
0.601375 + 0.798967i \(0.294620\pi\)
\(68\) −2.45638 − 8.22284i −0.297880 − 0.997166i
\(69\) 6.34233 − 0.624665i 0.763526 − 0.0752008i
\(70\) −4.81917 + 3.55834i −0.576001 + 0.425304i
\(71\) −4.06093 − 2.71343i −0.481944 − 0.322025i 0.290755 − 0.956798i \(-0.406094\pi\)
−0.772699 + 0.634773i \(0.781094\pi\)
\(72\) −1.91273 + 2.08361i −0.225418 + 0.245556i
\(73\) −3.68754 − 5.51879i −0.431593 − 0.645925i 0.550387 − 0.834910i \(-0.314480\pi\)
−0.981980 + 0.188985i \(0.939480\pi\)
\(74\) −3.70192 − 7.87014i −0.430340 − 0.914885i
\(75\) 1.19452 + 2.23479i 0.137931 + 0.258051i
\(76\) 6.27559 + 1.27595i 0.719859 + 0.146362i
\(77\) −0.944324 + 9.58788i −0.107616 + 1.09264i
\(78\) 7.63254 + 1.92904i 0.864215 + 0.218421i
\(79\) −1.22482 − 2.95698i −0.137803 − 0.332687i 0.839880 − 0.542773i \(-0.182626\pi\)
−0.977683 + 0.210086i \(0.932626\pi\)
\(80\) −10.9169 − 1.16869i −1.22054 − 0.130663i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −1.02731 + 1.37905i −0.113447 + 0.152290i
\(83\) 7.62729 + 9.29388i 0.837204 + 1.02014i 0.999443 + 0.0333841i \(0.0106285\pi\)
−0.162239 + 0.986751i \(0.551872\pi\)
\(84\) 1.16906 + 2.85653i 0.127555 + 0.311673i
\(85\) −3.41893 + 11.2707i −0.370835 + 1.22248i
\(86\) −4.28456 − 3.89986i −0.462016 − 0.420533i
\(87\) 3.89296 + 0.774358i 0.417369 + 0.0830199i
\(88\) −13.1584 + 11.7746i −1.40269 + 1.25518i
\(89\) −1.91480 − 9.62637i −0.202969 − 1.02039i −0.939123 − 0.343580i \(-0.888360\pi\)
0.736154 − 0.676814i \(-0.236640\pi\)
\(90\) 3.76741 − 0.935211i 0.397120 − 0.0985799i
\(91\) 5.44998 − 6.64082i 0.571313 − 0.696147i
\(92\) 11.2663 − 5.96076i 1.17460 − 0.621452i
\(93\) 1.99921 + 1.06860i 0.207309 + 0.110809i
\(94\) 10.9760 + 12.1619i 1.13209 + 1.25440i
\(95\) −6.21467 − 6.21467i −0.637612 − 0.637612i
\(96\) −1.96204 + 5.30570i −0.200250 + 0.541510i
\(97\) 9.90497 − 9.90497i 1.00570 − 1.00570i 0.00571375 − 0.999984i \(-0.498181\pi\)
0.999984 − 0.00571375i \(-0.00181875\pi\)
\(98\) −6.52282 − 0.334296i −0.658904 − 0.0337690i
\(99\) 2.94286 − 5.50570i 0.295768 − 0.553343i
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.13.12 ✓ 512
128.69 even 32 inner 384.2.v.a.325.12 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.12 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.12 yes 512 128.69 even 32 inner