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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [384,3,Mod(19,384)] 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("384.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([16, 23, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.u (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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(1024\)
Relative dimension: \(64\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 235.50
Character \(\chi\) \(=\) 384.235
Dual form 384.3.u.a.67.50

$q$-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+(1.54308 + 1.27237i) q^{2} +(1.09880 - 1.33889i) q^{3} +(0.762170 + 3.92672i) q^{4} +(0.121452 + 0.400375i) q^{5} +(3.39910 - 0.667937i) q^{6} +(0.354495 - 1.78216i) q^{7} +(-3.82013 + 7.02898i) q^{8} +(-0.585271 - 2.94236i) q^{9} +(-0.322013 + 0.772340i) q^{10} +(9.89699 + 0.974769i) q^{11} +(6.09493 + 3.29422i) q^{12} +(20.9962 + 6.36912i) q^{13} +(2.81458 - 2.29897i) q^{14} +(0.669511 + 0.277320i) q^{15} +(-14.8382 + 5.98565i) q^{16} +(2.44562 + 5.90425i) q^{17} +(2.84064 - 5.28496i) q^{18} +(-18.8365 + 10.0683i) q^{19} +(-1.47959 + 0.782062i) q^{20} +(-1.99661 - 2.43288i) q^{21} +(14.0316 + 14.0967i) q^{22} +(26.4582 + 17.6788i) q^{23} +(5.21349 + 12.8382i) q^{24} +(20.6412 - 13.7920i) q^{25} +(24.2949 + 36.5429i) q^{26} +(-4.58260 - 2.44945i) q^{27} +(7.26824 + 0.0336877i) q^{28} +(-16.6366 + 1.63856i) q^{29} +(0.680253 + 1.27979i) q^{30} +(-5.41658 + 5.41658i) q^{31} +(-30.5124 - 9.64330i) q^{32} +(12.1799 - 12.1799i) q^{33} +(-3.73859 + 12.2224i) q^{34} +(0.756588 - 0.0745174i) q^{35} +(11.1077 - 4.54077i) q^{36} +(-34.0609 - 18.2059i) q^{37} +(-41.8768 - 8.43076i) q^{38} +(31.5982 - 21.1132i) q^{39} +(-3.27819 - 0.675798i) q^{40} +(-8.53618 - 5.70369i) q^{41} +(0.0145872 - 6.29453i) q^{42} +(-14.2836 - 17.4046i) q^{43} +(3.71555 + 39.6056i) q^{44} +(1.10696 - 0.591683i) q^{45} +(18.3331 + 60.9443i) q^{46} +(-33.8168 - 81.6411i) q^{47} +(-8.29009 + 26.4438i) q^{48} +(42.2197 + 17.4880i) q^{49} +(49.3994 + 4.98103i) q^{50} +(10.5924 + 3.21317i) q^{51} +(-9.00708 + 87.3004i) q^{52} +(45.4909 + 4.48046i) q^{53} +(-3.95470 - 9.61043i) q^{54} +(0.811740 + 4.08089i) q^{55} +(11.1726 + 9.29984i) q^{56} +(-7.21718 + 36.2832i) q^{57} +(-27.7564 - 18.6394i) q^{58} +(23.0633 + 76.0296i) q^{59} +(-0.578678 + 2.84034i) q^{60} +(56.3990 - 68.7224i) q^{61} +(-15.2501 + 1.46632i) q^{62} -5.45124 q^{63} +(-34.8132 - 53.7033i) q^{64} +9.17988i q^{65} +(34.2919 - 3.29724i) q^{66} +(-69.4793 - 57.0202i) q^{67} +(-21.3203 + 14.1033i) q^{68} +(52.7424 - 15.9992i) q^{69} +(1.26229 + 0.847670i) q^{70} +(-88.3282 - 17.5696i) q^{71} +(22.9176 + 7.12633i) q^{72} +(-115.482 + 22.9709i) q^{73} +(-29.3940 - 71.4310i) q^{74} +(4.21455 - 42.7910i) q^{75} +(-53.8922 - 66.2920i) q^{76} +(5.24563 - 17.2925i) q^{77} +(75.6222 + 7.62513i) q^{78} +(6.11467 - 14.7621i) q^{79} +(-4.19863 - 5.21387i) q^{80} +(-8.31492 + 3.44415i) q^{81} +(-5.91479 - 19.6624i) q^{82} +(-3.35907 - 6.28437i) q^{83} +(8.03145 - 9.69438i) q^{84} +(-2.06688 + 1.69625i) q^{85} +(0.104356 - 45.0306i) q^{86} +(-16.0864 + 24.0751i) q^{87} +(-44.6595 + 65.8420i) q^{88} +(70.7889 + 105.943i) q^{89} +(2.46096 + 0.495448i) q^{90} +(18.7939 - 35.1608i) q^{91} +(-49.2541 + 117.368i) q^{92} +(1.30048 + 13.2040i) q^{93} +(51.6953 - 169.006i) q^{94} +(-6.31885 - 6.31885i) q^{95} +(-46.4384 + 30.2568i) q^{96} +(-69.0767 - 69.0767i) q^{97} +(42.8971 + 80.7041i) q^{98} +(-2.92431 - 29.6910i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 1024 q + 1248 q^{50} + 1056 q^{52} + 288 q^{54} + 1568 q^{56} + 576 q^{60} + 192 q^{62} - 192 q^{64} - 576 q^{66} - 960 q^{68} - 1344 q^{70} - 2464 q^{74} - 416 q^{76} - 1440 q^{78} - 1632 q^{80}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.54308 + 1.27237i 0.771538 + 0.636183i
\(3\) 1.09880 − 1.33889i 0.366267 − 0.446298i
\(4\) 0.762170 + 3.92672i 0.190542 + 0.981679i
\(5\) 0.121452 + 0.400375i 0.0242905 + 0.0800749i 0.968248 − 0.249993i \(-0.0804283\pi\)
−0.943957 + 0.330068i \(0.892928\pi\)
\(6\) 3.39910 − 0.667937i 0.566516 − 0.111323i
\(7\) 0.354495 − 1.78216i 0.0506421 − 0.254595i −0.947168 − 0.320738i \(-0.896069\pi\)
0.997810 + 0.0661427i \(0.0210692\pi\)
\(8\) −3.82013 + 7.02898i −0.477517 + 0.878623i
\(9\) −0.585271 − 2.94236i −0.0650301 − 0.326928i
\(10\) −0.322013 + 0.772340i −0.0322013 + 0.0772340i
\(11\) 9.89699 + 0.974769i 0.899726 + 0.0886153i 0.537294 − 0.843395i \(-0.319446\pi\)
0.362432 + 0.932010i \(0.381946\pi\)
\(12\) 6.09493 + 3.29422i 0.507911 + 0.274518i
\(13\) 20.9962 + 6.36912i 1.61509 + 0.489932i 0.963291 − 0.268459i \(-0.0865143\pi\)
0.651800 + 0.758391i \(0.274014\pi\)
\(14\) 2.81458 − 2.29897i 0.201041 − 0.164212i
\(15\) 0.669511 + 0.277320i 0.0446341 + 0.0184880i
\(16\) −14.8382 + 5.98565i −0.927387 + 0.374103i
\(17\) 2.44562 + 5.90425i 0.143860 + 0.347309i 0.979343 − 0.202208i \(-0.0648118\pi\)
−0.835483 + 0.549517i \(0.814812\pi\)
\(18\) 2.84064 − 5.28496i 0.157813 − 0.293609i
\(19\) −18.8365 + 10.0683i −0.991397 + 0.529913i −0.885587 − 0.464474i \(-0.846244\pi\)
−0.105810 + 0.994386i \(0.533744\pi\)
\(20\) −1.47959 + 0.782062i −0.0739795 + 0.0391031i
\(21\) −1.99661 − 2.43288i −0.0950766 − 0.115851i
\(22\) 14.0316 + 14.0967i 0.637798 + 0.640761i
\(23\) 26.4582 + 17.6788i 1.15036 + 0.768644i 0.976370 − 0.216103i \(-0.0693348\pi\)
0.173987 + 0.984748i \(0.444335\pi\)
\(24\) 5.21349 + 12.8382i 0.217229 + 0.534925i
\(25\) 20.6412 − 13.7920i 0.825648 − 0.551680i
\(26\) 24.2949 + 36.5429i 0.934418 + 1.40550i
\(27\) −4.58260 − 2.44945i −0.169726 − 0.0907203i
\(28\) 7.26824 + 0.0336877i 0.259580 + 0.00120313i
\(29\) −16.6366 + 1.63856i −0.573675 + 0.0565021i −0.380695 − 0.924701i \(-0.624315\pi\)
−0.192980 + 0.981203i \(0.561815\pi\)
\(30\) 0.680253 + 1.27979i 0.0226751 + 0.0426596i
\(31\) −5.41658 + 5.41658i −0.174728 + 0.174728i −0.789053 − 0.614325i \(-0.789428\pi\)
0.614325 + 0.789053i \(0.289428\pi\)
\(32\) −30.5124 − 9.64330i −0.953513 − 0.301353i
\(33\) 12.1799 − 12.1799i 0.369089 − 0.369089i
\(34\) −3.73859 + 12.2224i −0.109958 + 0.359483i
\(35\) 0.756588 − 0.0745174i 0.0216168 − 0.00212907i
\(36\) 11.1077 − 4.54077i 0.308548 − 0.126132i
\(37\) −34.0609 − 18.2059i −0.920564 − 0.492052i −0.0581801 − 0.998306i \(-0.518530\pi\)
−0.862384 + 0.506254i \(0.831030\pi\)
\(38\) −41.8768 − 8.43076i −1.10202 − 0.221862i
\(39\) 31.5982 − 21.1132i 0.810211 − 0.541365i
\(40\) −3.27819 − 0.675798i −0.0819547 − 0.0168950i
\(41\) −8.53618 − 5.70369i −0.208199 − 0.139114i 0.447099 − 0.894484i \(-0.352457\pi\)
−0.655298 + 0.755370i \(0.727457\pi\)
\(42\) 0.0145872 − 6.29453i 0.000347315 − 0.149870i
\(43\) −14.2836 − 17.4046i −0.332177 − 0.404758i 0.579960 − 0.814645i \(-0.303068\pi\)
−0.912136 + 0.409887i \(0.865568\pi\)
\(44\) 3.71555 + 39.6056i 0.0844443 + 0.900128i
\(45\) 1.10696 − 0.591683i 0.0245992 − 0.0131485i
\(46\) 18.3331 + 60.9443i 0.398546 + 1.32488i
\(47\) −33.8168 − 81.6411i −0.719507 − 1.73704i −0.674751 − 0.738046i \(-0.735749\pi\)
−0.0447564 − 0.998998i \(-0.514251\pi\)
\(48\) −8.29009 + 26.4438i −0.172710 + 0.550912i
\(49\) 42.2197 + 17.4880i 0.861626 + 0.356897i
\(50\) 49.3994 + 4.98103i 0.987988 + 0.0996207i
\(51\) 10.5924 + 3.21317i 0.207694 + 0.0630034i
\(52\) −9.00708 + 87.3004i −0.173213 + 1.67885i
\(53\) 45.4909 + 4.48046i 0.858319 + 0.0845370i 0.517594 − 0.855626i \(-0.326828\pi\)
0.340724 + 0.940163i \(0.389328\pi\)
\(54\) −3.95470 − 9.61043i −0.0732352 − 0.177971i
\(55\) 0.811740 + 4.08089i 0.0147589 + 0.0741980i
\(56\) 11.1726 + 9.29984i 0.199510 + 0.166069i
\(57\) −7.21718 + 36.2832i −0.126617 + 0.636548i
\(58\) −27.7564 − 18.6394i −0.478558 − 0.321369i
\(59\) 23.0633 + 76.0296i 0.390904 + 1.28864i 0.902975 + 0.429694i \(0.141378\pi\)
−0.512071 + 0.858943i \(0.671122\pi\)
\(60\) −0.578678 + 2.84034i −0.00964463 + 0.0473391i
\(61\) 56.3990 − 68.7224i 0.924574 − 1.12660i −0.0670902 − 0.997747i \(-0.521372\pi\)
0.991664 − 0.128850i \(-0.0411285\pi\)
\(62\) −15.2501 + 1.46632i −0.245969 + 0.0236504i
\(63\) −5.45124 −0.0865276
\(64\) −34.8132 − 53.7033i −0.543956 − 0.839114i
\(65\) 9.17988i 0.141229i
\(66\) 34.2919 − 3.29724i 0.519574 − 0.0499581i
\(67\) −69.4793 − 57.0202i −1.03700 − 0.851047i −0.0479871 − 0.998848i \(-0.515281\pi\)
−0.989017 + 0.147800i \(0.952781\pi\)
\(68\) −21.3203 + 14.1033i −0.313534 + 0.207401i
\(69\) 52.7424 − 15.9992i 0.764383 − 0.231873i
\(70\) 1.26229 + 0.847670i 0.0180327 + 0.0121096i
\(71\) −88.3282 − 17.5696i −1.24406 − 0.247459i −0.471194 − 0.882030i \(-0.656177\pi\)
−0.772865 + 0.634571i \(0.781177\pi\)
\(72\) 22.9176 + 7.12633i 0.318300 + 0.0989769i
\(73\) −115.482 + 22.9709i −1.58195 + 0.314670i −0.906327 − 0.422577i \(-0.861126\pi\)
−0.675624 + 0.737246i \(0.736126\pi\)
\(74\) −29.3940 − 71.4310i −0.397216 − 0.965284i
\(75\) 4.21455 − 42.7910i 0.0561940 − 0.570547i
\(76\) −53.8922 − 66.2920i −0.709107 − 0.872263i
\(77\) 5.24563 − 17.2925i 0.0681250 − 0.224578i
\(78\) 75.6222 + 7.62513i 0.969516 + 0.0977581i
\(79\) 6.11467 − 14.7621i 0.0774009 − 0.186862i −0.880443 − 0.474153i \(-0.842754\pi\)
0.957844 + 0.287290i \(0.0927545\pi\)
\(80\) −4.19863 − 5.21387i −0.0524829 − 0.0651733i
\(81\) −8.31492 + 3.44415i −0.102653 + 0.0425204i
\(82\) −5.91479 − 19.6624i −0.0721316 − 0.239785i
\(83\) −3.35907 − 6.28437i −0.0404707 − 0.0757153i 0.860873 − 0.508820i \(-0.169918\pi\)
−0.901344 + 0.433105i \(0.857418\pi\)
\(84\) 8.03145 − 9.69438i 0.0956126 − 0.115409i
\(85\) −2.06688 + 1.69625i −0.0243163 + 0.0199559i
\(86\) 0.104356 − 45.0306i 0.00121344 − 0.523612i
\(87\) −16.0864 + 24.0751i −0.184902 + 0.276725i
\(88\) −44.6595 + 65.8420i −0.507494 + 0.748205i
\(89\) 70.7889 + 105.943i 0.795381 + 1.19037i 0.978290 + 0.207242i \(0.0664488\pi\)
−0.182908 + 0.983130i \(0.558551\pi\)
\(90\) 2.46096 + 0.495448i 0.0273441 + 0.00550498i
\(91\) 18.7939 − 35.1608i 0.206526 − 0.386383i
\(92\) −49.2541 + 117.368i −0.535370 + 1.27574i
\(93\) 1.30048 + 13.2040i 0.0139836 + 0.141978i
\(94\) 51.6953 − 169.006i 0.549950 − 1.79793i
\(95\) −6.31885 − 6.31885i −0.0665142 − 0.0665142i
\(96\) −46.4384 + 30.2568i −0.483734 + 0.315175i
\(97\) −69.0767 − 69.0767i −0.712131 − 0.712131i 0.254850 − 0.966981i \(-0.417974\pi\)
−0.966981 + 0.254850i \(0.917974\pi\)
\(98\) 42.8971 + 80.7041i 0.437725 + 0.823511i
\(99\) −2.92431 − 29.6910i −0.0295384 − 0.299909i
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 384.3.u.a.235.50 yes 1024
128.67 odd 32 inner 384.3.u.a.67.50 ✓ 1024
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.u.a.67.50 ✓ 1024 128.67 odd 32 inner
384.3.u.a.235.50 yes 1024 1.1 even 1 trivial