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Properties

Label 384.2.v.a.13.11
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 13.11
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.11

$q$-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+(-0.612256 - 1.27481i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(-1.25028 + 1.56102i) q^{4} +(0.189500 + 1.92402i) q^{5} +(-1.04219 + 0.955951i) q^{6} +(0.476963 - 0.713825i) q^{7} +(2.75550 + 0.638130i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(2.33674 - 1.41957i) q^{10} +(1.06219 - 0.567753i) q^{11} +(1.85674 + 0.743307i) q^{12} +(2.36283 + 0.232719i) q^{13} +(-1.20202 - 0.170993i) q^{14} +(1.78617 - 0.739855i) q^{15} +(-0.873578 - 3.90344i) q^{16} +(2.31292 + 0.958043i) q^{17} +(1.21732 + 0.719815i) q^{18} +(5.82696 - 4.78206i) q^{19} +(-3.24037 - 2.10976i) q^{20} +(-0.821543 - 0.249212i) q^{21} +(-1.37411 - 1.00648i) q^{22} +(-0.204164 + 1.02640i) q^{23} +(-0.189227 - 2.82209i) q^{24} +(1.23797 - 0.246247i) q^{25} +(-1.14999 - 3.15465i) q^{26} +(0.773010 + 0.634393i) q^{27} +(0.517958 + 1.63703i) q^{28} +(-0.377671 + 0.706573i) q^{29} +(-2.03677 - 1.82404i) q^{30} +(2.13037 - 2.13037i) q^{31} +(-4.44130 + 3.50355i) q^{32} +(-0.851643 - 0.851643i) q^{33} +(-0.194777 - 3.53511i) q^{34} +(1.46380 + 0.782418i) q^{35} +(0.172316 - 1.99256i) q^{36} +(-0.783750 + 0.955002i) q^{37} +(-9.66381 - 4.50042i) q^{38} +(-0.463196 - 2.32864i) q^{39} +(-0.705611 + 5.42258i) q^{40} +(-9.21349 - 1.83268i) q^{41} +(0.185296 + 1.19989i) q^{42} +(2.08700 - 6.87991i) q^{43} +(-0.441766 + 2.36796i) q^{44} +(-1.22649 - 1.49449i) q^{45} +(1.43347 - 0.368150i) q^{46} +(-2.53591 + 6.12224i) q^{47} +(-3.48177 + 1.96907i) q^{48} +(2.39673 + 5.78622i) q^{49} +(-1.07187 - 1.42741i) q^{50} +(0.245385 - 2.49143i) q^{51} +(-3.31749 + 3.39747i) q^{52} +(1.02350 + 1.91484i) q^{53} +(0.335451 - 1.37385i) q^{54} +(1.29366 + 1.93609i) q^{55} +(1.76978 - 1.66258i) q^{56} +(-6.26762 - 4.18789i) q^{57} +(1.13198 + 0.0488553i) q^{58} +(4.98169 - 0.490653i) q^{59} +(-1.07829 + 3.71328i) q^{60} +(4.04828 - 1.22803i) q^{61} +(-4.02015 - 1.41149i) q^{62} +0.858510i q^{63} +(7.18558 + 3.51674i) q^{64} +4.59025i q^{65} +(-0.564260 + 1.60711i) q^{66} +(3.74011 - 1.13455i) q^{67} +(-4.38734 + 2.41269i) q^{68} +(1.04147 - 0.102576i) q^{69} +(0.101213 - 2.34511i) q^{70} +(13.0171 + 8.69775i) q^{71} +(-2.64564 + 1.00029i) q^{72} +(-2.96864 - 4.44289i) q^{73} +(1.69730 + 0.414427i) q^{74} +(-0.595006 - 1.11318i) q^{75} +(0.179547 + 15.0749i) q^{76} +(0.101349 - 1.02902i) q^{77} +(-2.68499 + 2.01621i) q^{78} +(2.08586 + 5.03571i) q^{79} +(7.34478 - 2.42049i) q^{80} +(0.382683 - 0.923880i) q^{81} +(3.30470 + 12.8675i) q^{82} +(-5.32161 - 6.48441i) q^{83} +(1.41619 - 0.970860i) q^{84} +(-1.40500 + 4.63167i) q^{85} +(-10.0484 + 1.55174i) q^{86} +(0.785780 + 0.156301i) q^{87} +(3.28917 - 0.886628i) q^{88} +(-0.166186 - 0.835474i) q^{89} +(-1.15426 + 2.47856i) q^{90} +(1.29310 - 1.57565i) q^{91} +(-1.34697 - 1.60200i) q^{92} +(-2.65705 - 1.42022i) q^{93} +(9.35733 - 0.515569i) q^{94} +(10.3050 + 10.3050i) q^{95} +(4.64193 + 3.23303i) q^{96} +(-11.0074 + 11.0074i) q^{97} +(5.90892 - 6.59803i) q^{98} +(-0.567753 + 1.06219i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.612256 − 1.27481i −0.432931 − 0.901427i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) −1.25028 + 1.56102i −0.625142 + 0.780511i
\(5\) 0.189500 + 1.92402i 0.0847469 + 0.860450i 0.939758 + 0.341840i \(0.111050\pi\)
−0.855011 + 0.518610i \(0.826450\pi\)
\(6\) −1.04219 + 0.955951i −0.425472 + 0.390265i
\(7\) 0.476963 − 0.713825i 0.180275 − 0.269801i −0.730316 − 0.683110i \(-0.760627\pi\)
0.910591 + 0.413309i \(0.135627\pi\)
\(8\) 2.75550 + 0.638130i 0.974217 + 0.225613i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) 2.33674 − 1.41957i 0.738944 − 0.448908i
\(11\) 1.06219 − 0.567753i 0.320263 − 0.171184i −0.303430 − 0.952854i \(-0.598132\pi\)
0.623692 + 0.781670i \(0.285632\pi\)
\(12\) 1.85674 + 0.743307i 0.535996 + 0.214574i
\(13\) 2.36283 + 0.232719i 0.655332 + 0.0645445i 0.420218 − 0.907423i \(-0.361954\pi\)
0.235114 + 0.971968i \(0.424454\pi\)
\(14\) −1.20202 − 0.170993i −0.321252 − 0.0456998i
\(15\) 1.78617 − 0.739855i 0.461187 − 0.191030i
\(16\) −0.873578 − 3.90344i −0.218395 − 0.975861i
\(17\) 2.31292 + 0.958043i 0.560966 + 0.232360i 0.645104 − 0.764094i \(-0.276814\pi\)
−0.0841386 + 0.996454i \(0.526814\pi\)
\(18\) 1.21732 + 0.719815i 0.286925 + 0.169662i
\(19\) 5.82696 − 4.78206i 1.33680 − 1.09708i 0.349714 − 0.936856i \(-0.386279\pi\)
0.987081 − 0.160223i \(-0.0512214\pi\)
\(20\) −3.24037 − 2.10976i −0.724569 − 0.471758i
\(21\) −0.821543 − 0.249212i −0.179275 − 0.0543826i
\(22\) −1.37411 − 1.00648i −0.292961 − 0.214583i
\(23\) −0.204164 + 1.02640i −0.0425711 + 0.214019i −0.996216 − 0.0869159i \(-0.972299\pi\)
0.953645 + 0.300935i \(0.0972989\pi\)
\(24\) −0.189227 − 2.82209i −0.0386259 − 0.576057i
\(25\) 1.23797 − 0.246247i 0.247593 − 0.0492493i
\(26\) −1.14999 − 3.15465i −0.225531 − 0.618677i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) 0.517958 + 1.63703i 0.0978848 + 0.309370i
\(29\) −0.377671 + 0.706573i −0.0701317 + 0.131207i −0.914544 − 0.404487i \(-0.867450\pi\)
0.844412 + 0.535694i \(0.179950\pi\)
\(30\) −2.03677 − 1.82404i −0.371861 − 0.333024i
\(31\) 2.13037 − 2.13037i 0.382626 − 0.382626i −0.489421 − 0.872047i \(-0.662792\pi\)
0.872047 + 0.489421i \(0.162792\pi\)
\(32\) −4.44130 + 3.50355i −0.785118 + 0.619347i
\(33\) −0.851643 − 0.851643i −0.148252 − 0.148252i
\(34\) −0.194777 − 3.53511i −0.0334040 − 0.606266i
\(35\) 1.46380 + 0.782418i 0.247428 + 0.132253i
\(36\) 0.172316 − 1.99256i 0.0287194 − 0.332094i
\(37\) −0.783750 + 0.955002i −0.128848 + 0.157001i −0.833438 − 0.552613i \(-0.813631\pi\)
0.704590 + 0.709614i \(0.251131\pi\)
\(38\) −9.66381 − 4.50042i −1.56768 − 0.730064i
\(39\) −0.463196 − 2.32864i −0.0741707 − 0.372881i
\(40\) −0.705611 + 5.42258i −0.111567 + 0.857385i
\(41\) −9.21349 − 1.83268i −1.43891 − 0.286216i −0.586862 − 0.809687i \(-0.699637\pi\)
−0.852043 + 0.523471i \(0.824637\pi\)
\(42\) 0.185296 + 1.19989i 0.0285918 + 0.185148i
\(43\) 2.08700 − 6.87991i 0.318264 − 1.04918i −0.641207 − 0.767368i \(-0.721566\pi\)
0.959471 − 0.281808i \(-0.0909341\pi\)
\(44\) −0.441766 + 2.36796i −0.0665987 + 0.356983i
\(45\) −1.22649 − 1.49449i −0.182835 − 0.222785i
\(46\) 1.43347 − 0.368150i 0.211353 − 0.0542808i
\(47\) −2.53591 + 6.12224i −0.369901 + 0.893020i 0.623865 + 0.781532i \(0.285562\pi\)
−0.993766 + 0.111488i \(0.964438\pi\)
\(48\) −3.48177 + 1.96907i −0.502551 + 0.284211i
\(49\) 2.39673 + 5.78622i 0.342390 + 0.826603i
\(50\) −1.07187 − 1.42741i −0.151585 − 0.201866i
\(51\) 0.245385 − 2.49143i 0.0343607 − 0.348870i
\(52\) −3.31749 + 3.39747i −0.460053 + 0.471144i
\(53\) 1.02350 + 1.91484i 0.140589 + 0.263023i 0.942355 − 0.334615i \(-0.108606\pi\)
−0.801766 + 0.597638i \(0.796106\pi\)
\(54\) 0.335451 − 1.37385i 0.0456491 − 0.186958i
\(55\) 1.29366 + 1.93609i 0.174436 + 0.261063i
\(56\) 1.76978 − 1.66258i 0.236497 − 0.222172i
\(57\) −6.26762 − 4.18789i −0.830167 − 0.554700i
\(58\) 1.13198 + 0.0488553i 0.148636 + 0.00641502i
\(59\) 4.98169 − 0.490653i 0.648560 − 0.0638776i 0.231613 − 0.972808i \(-0.425600\pi\)
0.416948 + 0.908930i \(0.363100\pi\)
\(60\) −1.07829 + 3.71328i −0.139206 + 0.479382i
\(61\) 4.04828 − 1.22803i 0.518330 − 0.157234i −0.0202714 − 0.999795i \(-0.506453\pi\)
0.538601 + 0.842561i \(0.318953\pi\)
\(62\) −4.02015 − 1.41149i −0.510560 − 0.179259i
\(63\) 0.858510i 0.108162i
\(64\) 7.18558 + 3.51674i 0.898197 + 0.439592i
\(65\) 4.59025i 0.569350i
\(66\) −0.564260 + 1.60711i −0.0694556 + 0.197821i
\(67\) 3.74011 − 1.13455i 0.456927 − 0.138607i −0.0534268 − 0.998572i \(-0.517014\pi\)
0.510354 + 0.859964i \(0.329514\pi\)
\(68\) −4.38734 + 2.41269i −0.532043 + 0.292582i
\(69\) 1.04147 − 0.102576i 0.125378 − 0.0123487i
\(70\) 0.101213 − 2.34511i 0.0120973 − 0.280294i
\(71\) 13.0171 + 8.69775i 1.54485 + 1.03223i 0.978049 + 0.208375i \(0.0668175\pi\)
0.566797 + 0.823858i \(0.308182\pi\)
\(72\) −2.64564 + 1.00029i −0.311792 + 0.117885i
\(73\) −2.96864 − 4.44289i −0.347453 − 0.520001i 0.616046 − 0.787710i \(-0.288733\pi\)
−0.963500 + 0.267709i \(0.913733\pi\)
\(74\) 1.69730 + 0.414427i 0.197307 + 0.0481761i
\(75\) −0.595006 − 1.11318i −0.0687054 − 0.128539i
\(76\) 0.179547 + 15.0749i 0.0205955 + 1.72921i
\(77\) 0.101349 − 1.02902i 0.0115498 − 0.117267i
\(78\) −2.68499 + 2.01621i −0.304015 + 0.228291i
\(79\) 2.08586 + 5.03571i 0.234678 + 0.566562i 0.996717 − 0.0809692i \(-0.0258015\pi\)
−0.762039 + 0.647531i \(0.775802\pi\)
\(80\) 7.34478 − 2.42049i 0.821171 − 0.270619i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) 3.30470 + 12.8675i 0.364943 + 1.42098i
\(83\) −5.32161 − 6.48441i −0.584123 − 0.711756i 0.394157 − 0.919043i \(-0.371037\pi\)
−0.978280 + 0.207287i \(0.933537\pi\)
\(84\) 1.41619 − 0.970860i 0.154519 − 0.105930i
\(85\) −1.40500 + 4.63167i −0.152394 + 0.502375i
\(86\) −10.0484 + 1.55174i −1.08354 + 0.167329i
\(87\) 0.785780 + 0.156301i 0.0842444 + 0.0167573i
\(88\) 3.28917 − 0.886628i 0.350627 − 0.0945148i
\(89\) −0.166186 − 0.835474i −0.0176157 − 0.0885601i 0.970977 − 0.239171i \(-0.0768758\pi\)
−0.988593 + 0.150611i \(0.951876\pi\)
\(90\) −1.15426 + 2.47856i −0.121670 + 0.261263i
\(91\) 1.29310 − 1.57565i 0.135554 − 0.165173i
\(92\) −1.34697 − 1.60200i −0.140432 − 0.167020i
\(93\) −2.65705 − 1.42022i −0.275524 − 0.147270i
\(94\) 9.35733 − 0.515569i 0.965134 − 0.0531769i
\(95\) 10.3050 + 10.3050i 1.05727 + 1.05727i
\(96\) 4.64193 + 3.23303i 0.473765 + 0.329969i
\(97\) −11.0074 + 11.0074i −1.11763 + 1.11763i −0.125547 + 0.992088i \(0.540069\pi\)
−0.992088 + 0.125547i \(0.959931\pi\)
\(98\) 5.90892 − 6.59803i 0.596891 − 0.666502i
\(99\) −0.567753 + 1.06219i −0.0570613 + 0.106754i
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.13.11 ✓ 512
128.69 even 32 inner 384.2.v.a.325.11 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.11 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.11 yes 512 128.69 even 32 inner