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Properties

Label 384.2.v.a.13.18
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 13.18
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.18

$q$-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+(0.147414 - 1.40651i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(-1.95654 - 0.414679i) q^{4} +(0.138503 + 1.40624i) q^{5} +(-1.38874 + 0.267222i) q^{6} +(-1.77662 + 2.65890i) q^{7} +(-0.871671 + 2.69076i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(1.99831 + 0.0124945i) q^{10} +(-3.73022 + 1.99385i) q^{11} +(0.171130 + 1.99267i) q^{12} +(-1.62654 - 0.160201i) q^{13} +(3.47786 + 2.89079i) q^{14} +(1.30549 - 0.540750i) q^{15} +(3.65608 + 1.62267i) q^{16} +(-2.98818 - 1.23775i) q^{17} +(0.658844 + 1.25137i) q^{18} +(-4.49007 + 3.68491i) q^{19} +(0.312153 - 2.80880i) q^{20} +(3.06013 + 0.928280i) q^{21} +(2.25448 + 5.54052i) q^{22} +(0.342886 - 1.72381i) q^{23} +(2.82793 + 0.0530505i) q^{24} +(2.94559 - 0.585914i) q^{25} +(-0.465099 + 2.26413i) q^{26} +(0.773010 + 0.634393i) q^{27} +(4.57861 - 4.46551i) q^{28} +(3.62350 - 6.77909i) q^{29} +(-0.568123 - 1.91589i) q^{30} +(-0.217873 + 0.217873i) q^{31} +(2.82126 - 4.90311i) q^{32} +(2.99082 + 2.99082i) q^{33} +(-2.18140 + 4.02044i) q^{34} +(-3.98512 - 2.13009i) q^{35} +(1.85719 - 0.742202i) q^{36} +(0.411530 - 0.501451i) q^{37} +(4.52096 + 6.85854i) q^{38} +(0.318858 + 1.60301i) q^{39} +(-3.90459 - 0.853103i) q^{40} +(-6.27044 - 1.24727i) q^{41} +(1.75674 - 4.16726i) q^{42} +(-2.72454 + 8.98162i) q^{43} +(8.12513 - 2.35419i) q^{44} +(-0.896428 - 1.09230i) q^{45} +(-2.37400 - 0.736386i) q^{46} +(2.42875 - 5.86353i) q^{47} +(0.491493 - 3.96969i) q^{48} +(-1.23457 - 2.98052i) q^{49} +(-0.389872 - 4.22937i) q^{50} +(-0.317025 + 3.21881i) q^{51} +(3.11596 + 0.987931i) q^{52} +(3.57542 + 6.68915i) q^{53} +(1.00623 - 0.993728i) q^{54} +(-3.32048 - 4.96945i) q^{55} +(-5.60583 - 7.09813i) q^{56} +(4.82963 + 3.22706i) q^{57} +(-9.00069 - 6.09582i) q^{58} +(2.79553 - 0.275336i) q^{59} +(-2.77847 + 0.516641i) q^{60} +(-7.76083 + 2.35422i) q^{61} +(0.274323 + 0.338559i) q^{62} -3.19783i q^{63} +(-6.48038 - 4.69091i) q^{64} -2.30950i q^{65} +(4.64750 - 3.76573i) q^{66} +(-11.5975 + 3.51805i) q^{67} +(5.33322 + 3.66083i) q^{68} +(-1.74911 + 0.172273i) q^{69} +(-3.58346 + 5.29111i) q^{70} +(-2.71896 - 1.81675i) q^{71} +(-0.770138 - 2.72156i) q^{72} +(5.84796 + 8.75209i) q^{73} +(-0.644630 - 0.652742i) q^{74} +(-1.41574 - 2.64867i) q^{75} +(10.3130 - 5.34772i) q^{76} +(1.32575 - 13.4606i) q^{77} +(2.30165 - 0.212171i) q^{78} +(5.24635 + 12.6658i) q^{79} +(-1.77549 + 5.36609i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-2.67864 + 8.63556i) q^{82} +(-9.14474 - 11.1429i) q^{83} +(-5.60232 - 3.08519i) q^{84} +(1.32670 - 4.37354i) q^{85} +(12.2311 + 5.15611i) q^{86} +(-7.53903 - 1.49961i) q^{87} +(-2.11343 - 11.7751i) q^{88} +(2.83142 + 14.2345i) q^{89} +(-1.66848 + 1.09981i) q^{90} +(3.31570 - 4.04019i) q^{91} +(-1.38570 + 3.23051i) q^{92} +(0.271737 + 0.145247i) q^{93} +(-7.88908 - 4.28043i) q^{94} +(-5.80376 - 5.80376i) q^{95} +(-5.51095 - 1.27648i) q^{96} +(7.48649 - 7.48649i) q^{97} +(-4.37413 + 1.29707i) q^{98} +(1.99385 - 3.73022i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.147414 − 1.40651i 0.104238 − 0.994552i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) −1.95654 − 0.414679i −0.978269 − 0.207339i
\(5\) 0.138503 + 1.40624i 0.0619404 + 0.628891i 0.975172 + 0.221450i \(0.0710789\pi\)
−0.913231 + 0.407441i \(0.866421\pi\)
\(6\) −1.38874 + 0.267222i −0.566950 + 0.109093i
\(7\) −1.77662 + 2.65890i −0.671498 + 1.00497i 0.326710 + 0.945125i \(0.394060\pi\)
−0.998208 + 0.0598433i \(0.980940\pi\)
\(8\) −0.871671 + 2.69076i −0.308182 + 0.951327i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) 1.99831 + 0.0124945i 0.631922 + 0.00395110i
\(11\) −3.73022 + 1.99385i −1.12470 + 0.601167i −0.925484 − 0.378787i \(-0.876341\pi\)
−0.199221 + 0.979955i \(0.563841\pi\)
\(12\) 0.171130 + 1.99267i 0.0494011 + 0.575233i
\(13\) −1.62654 − 0.160201i −0.451122 − 0.0444316i −0.130095 − 0.991502i \(-0.541528\pi\)
−0.321027 + 0.947070i \(0.604028\pi\)
\(14\) 3.47786 + 2.89079i 0.929498 + 0.772596i
\(15\) 1.30549 − 0.540750i 0.337075 − 0.139621i
\(16\) 3.65608 + 1.62267i 0.914021 + 0.405667i
\(17\) −2.98818 − 1.23775i −0.724740 − 0.300197i −0.0103517 − 0.999946i \(-0.503295\pi\)
−0.714389 + 0.699749i \(0.753295\pi\)
\(18\) 0.658844 + 1.25137i 0.155291 + 0.294950i
\(19\) −4.49007 + 3.68491i −1.03009 + 0.845376i −0.988111 − 0.153743i \(-0.950867\pi\)
−0.0419822 + 0.999118i \(0.513367\pi\)
\(20\) 0.312153 − 2.80880i 0.0697995 − 0.628068i
\(21\) 3.06013 + 0.928280i 0.667775 + 0.202567i
\(22\) 2.25448 + 5.54052i 0.480656 + 1.18124i
\(23\) 0.342886 − 1.72381i 0.0714968 − 0.359439i −0.928430 − 0.371506i \(-0.878841\pi\)
0.999927 + 0.0120678i \(0.00384140\pi\)
\(24\) 2.82793 + 0.0530505i 0.577249 + 0.0108289i
\(25\) 2.94559 − 0.585914i 0.589118 − 0.117183i
\(26\) −0.465099 + 2.26413i −0.0912134 + 0.444033i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) 4.57861 − 4.46551i 0.865275 − 0.843901i
\(29\) 3.62350 − 6.77909i 0.672866 − 1.25884i −0.281315 − 0.959616i \(-0.590771\pi\)
0.954181 − 0.299229i \(-0.0967295\pi\)
\(30\) −0.568123 − 1.91589i −0.103725 − 0.349793i
\(31\) −0.217873 + 0.217873i −0.0391312 + 0.0391312i −0.726402 − 0.687270i \(-0.758809\pi\)
0.687270 + 0.726402i \(0.258809\pi\)
\(32\) 2.82126 − 4.90311i 0.498733 − 0.866756i
\(33\) 2.99082 + 2.99082i 0.520635 + 0.520635i
\(34\) −2.18140 + 4.02044i −0.374107 + 0.689501i
\(35\) −3.98512 − 2.13009i −0.673609 − 0.360051i
\(36\) 1.85719 − 0.742202i 0.309531 − 0.123700i
\(37\) 0.411530 − 0.501451i 0.0676551 − 0.0824380i −0.738083 − 0.674710i \(-0.764269\pi\)
0.805738 + 0.592272i \(0.201769\pi\)
\(38\) 4.52096 + 6.85854i 0.733396 + 1.11260i
\(39\) 0.318858 + 1.60301i 0.0510582 + 0.256687i
\(40\) −3.90459 − 0.853103i −0.617370 − 0.134887i
\(41\) −6.27044 − 1.24727i −0.979278 − 0.194790i −0.320599 − 0.947215i \(-0.603884\pi\)
−0.658678 + 0.752425i \(0.728884\pi\)
\(42\) 1.75674 − 4.16726i 0.271071 − 0.643022i
\(43\) −2.72454 + 8.98162i −0.415489 + 1.36968i 0.460631 + 0.887592i \(0.347623\pi\)
−0.876120 + 0.482092i \(0.839877\pi\)
\(44\) 8.12513 − 2.35419i 1.22491 − 0.354908i
\(45\) −0.896428 − 1.09230i −0.133632 − 0.162831i
\(46\) −2.37400 − 0.736386i −0.350028 − 0.108574i
\(47\) 2.42875 − 5.86353i 0.354270 − 0.855284i −0.641813 − 0.766861i \(-0.721817\pi\)
0.996083 − 0.0884224i \(-0.0281825\pi\)
\(48\) 0.491493 − 3.96969i 0.0709409 − 0.572975i
\(49\) −1.23457 − 2.98052i −0.176368 − 0.425789i
\(50\) −0.389872 − 4.22937i −0.0551363 − 0.598123i
\(51\) −0.317025 + 3.21881i −0.0443924 + 0.450723i
\(52\) 3.11596 + 0.987931i 0.432106 + 0.137001i
\(53\) 3.57542 + 6.68915i 0.491122 + 0.918825i 0.998373 + 0.0570266i \(0.0181620\pi\)
−0.507250 + 0.861799i \(0.669338\pi\)
\(54\) 1.00623 − 0.993728i 0.136931 − 0.135229i
\(55\) −3.32048 − 4.96945i −0.447733 − 0.670080i
\(56\) −5.60583 − 7.09813i −0.749110 − 0.948528i
\(57\) 4.82963 + 3.22706i 0.639701 + 0.427434i
\(58\) −9.00069 − 6.09582i −1.18185 − 0.800420i
\(59\) 2.79553 − 0.275336i 0.363947 − 0.0358456i 0.0856089 − 0.996329i \(-0.472716\pi\)
0.278338 + 0.960483i \(0.410216\pi\)
\(60\) −2.77847 + 0.516641i −0.358699 + 0.0666981i
\(61\) −7.76083 + 2.35422i −0.993673 + 0.301427i −0.744930 − 0.667142i \(-0.767517\pi\)
−0.248743 + 0.968570i \(0.580017\pi\)
\(62\) 0.274323 + 0.338559i 0.0348391 + 0.0429970i
\(63\) − 3.19783i − 0.402888i
\(64\) −6.48038 − 4.69091i −0.810048 − 0.586364i
\(65\) − 2.30950i − 0.286459i
\(66\) 4.64750 − 3.76573i 0.572068 − 0.463529i
\(67\) −11.5975 + 3.51805i −1.41686 + 0.429798i −0.903578 − 0.428424i \(-0.859069\pi\)
−0.513278 + 0.858223i \(0.671569\pi\)
\(68\) 5.33322 + 3.66083i 0.646748 + 0.443941i
\(69\) −1.74911 + 0.172273i −0.210569 + 0.0207392i
\(70\) −3.58346 + 5.29111i −0.428305 + 0.632408i
\(71\) −2.71896 − 1.81675i −0.322681 − 0.215609i 0.383670 − 0.923470i \(-0.374660\pi\)
−0.706351 + 0.707861i \(0.749660\pi\)
\(72\) −0.770138 − 2.72156i −0.0907617 − 0.320739i
\(73\) 5.84796 + 8.75209i 0.684452 + 1.02435i 0.997219 + 0.0745245i \(0.0237439\pi\)
−0.312768 + 0.949830i \(0.601256\pi\)
\(74\) −0.644630 − 0.652742i −0.0749367 − 0.0758797i
\(75\) −1.41574 − 2.64867i −0.163476 − 0.305842i
\(76\) 10.3130 − 5.34772i 1.18299 − 0.613426i
\(77\) 1.32575 − 13.4606i 0.151083 − 1.53398i
\(78\) 2.30165 − 0.212171i 0.260611 − 0.0240236i
\(79\) 5.24635 + 12.6658i 0.590261 + 1.42502i 0.883251 + 0.468900i \(0.155350\pi\)
−0.292990 + 0.956115i \(0.594650\pi\)
\(80\) −1.77549 + 5.36609i −0.198506 + 0.599947i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −2.67864 + 8.63556i −0.295807 + 0.953638i
\(83\) −9.14474 − 11.1429i −1.00377 − 1.22309i −0.974700 − 0.223515i \(-0.928247\pi\)
−0.0290657 − 0.999578i \(-0.509253\pi\)
\(84\) −5.60232 − 3.08519i −0.611263 − 0.336621i
\(85\) 1.32670 − 4.37354i 0.143901 − 0.474377i
\(86\) 12.2311 + 5.15611i 1.31891 + 0.555998i
\(87\) −7.53903 − 1.49961i −0.808268 − 0.160775i
\(88\) −2.11343 − 11.7751i −0.225293 − 1.25523i
\(89\) 2.83142 + 14.2345i 0.300130 + 1.50885i 0.776786 + 0.629765i \(0.216849\pi\)
−0.476656 + 0.879090i \(0.658151\pi\)
\(90\) −1.66848 + 1.09981i −0.175873 + 0.115931i
\(91\) 3.31570 − 4.04019i 0.347580 − 0.423527i
\(92\) −1.38570 + 3.23051i −0.144469 + 0.336804i
\(93\) 0.271737 + 0.145247i 0.0281778 + 0.0150614i
\(94\) −7.88908 − 4.28043i −0.813696 − 0.441493i
\(95\) −5.80376 − 5.80376i −0.595454 − 0.595454i
\(96\) −5.51095 − 1.27648i −0.562459 − 0.130280i
\(97\) 7.48649 − 7.48649i 0.760138 − 0.760138i −0.216209 − 0.976347i \(-0.569369\pi\)
0.976347 + 0.216209i \(0.0693693\pi\)
\(98\) −4.37413 + 1.29707i −0.441854 + 0.131024i
\(99\) 1.99385 − 3.73022i 0.200389 − 0.374902i
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.13.18 ✓ 512
128.69 even 32 inner 384.2.v.a.325.18 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.18 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.18 yes 512 128.69 even 32 inner