Properties

Label 768.2.o.b.479.3
Level $768$
Weight $2$
Character 768.479
Analytic conductor $6.133$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(95,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.o (of order \(8\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 479.3
Character \(\chi\) \(=\) 768.479
Dual form 768.2.o.b.287.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.44423 + 0.956131i) q^{3} +(-1.56013 - 3.76650i) q^{5} +(-0.838552 - 0.838552i) q^{7} +(1.17163 - 2.76175i) q^{9} +O(q^{10})\) \(q+(-1.44423 + 0.956131i) q^{3} +(-1.56013 - 3.76650i) q^{5} +(-0.838552 - 0.838552i) q^{7} +(1.17163 - 2.76175i) q^{9} +(0.249049 + 0.601256i) q^{11} +(2.05771 + 0.852332i) q^{13} +(5.85446 + 3.94801i) q^{15} -3.23677 q^{17} +(-1.47818 + 3.56865i) q^{19} +(2.01283 + 0.409300i) q^{21} +(-2.58369 - 2.58369i) q^{23} +(-8.21694 + 8.21694i) q^{25} +(0.948494 + 5.10885i) q^{27} +(-3.52027 - 1.45815i) q^{29} +7.63408i q^{31} +(-0.934564 - 0.630232i) q^{33} +(-1.85015 + 4.46665i) q^{35} +(-0.579146 + 0.239890i) q^{37} +(-3.78676 + 0.736474i) q^{39} +(-3.54554 + 3.54554i) q^{41} +(3.19595 - 1.32381i) q^{43} +(-12.2300 - 0.104223i) q^{45} -5.96658i q^{47} -5.59366i q^{49} +(4.67465 - 3.09477i) q^{51} +(0.762825 - 0.315973i) q^{53} +(1.87608 - 1.87608i) q^{55} +(-1.27725 - 6.56731i) q^{57} +(-5.86827 + 2.43072i) q^{59} +(-2.68247 + 6.47607i) q^{61} +(-3.29834 + 1.33340i) q^{63} -9.08011i q^{65} +(-4.78575 - 1.98232i) q^{67} +(6.20180 + 1.26111i) q^{69} +(-10.2094 + 10.2094i) q^{71} +(8.09458 + 8.09458i) q^{73} +(4.01072 - 19.7237i) q^{75} +(0.295345 - 0.713025i) q^{77} -11.5343 q^{79} +(-6.25458 - 6.47149i) q^{81} +(-0.998651 - 0.413655i) q^{83} +(5.04979 + 12.1913i) q^{85} +(6.47828 - 1.25994i) q^{87} +(10.4124 + 10.4124i) q^{89} +(-1.01077 - 2.44022i) q^{91} +(-7.29918 - 11.0254i) q^{93} +15.7475 q^{95} +9.21596 q^{97} +(1.95232 + 0.0166374i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{3} - 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{3} - 8 q^{7} - 4 q^{9} + 8 q^{13} - 8 q^{15} + 8 q^{19} + 4 q^{21} - 8 q^{25} + 28 q^{27} - 8 q^{33} + 8 q^{37} - 28 q^{39} + 8 q^{43} + 4 q^{45} + 16 q^{51} + 24 q^{55} - 4 q^{57} + 40 q^{61} - 56 q^{67} + 4 q^{69} - 8 q^{73} - 16 q^{75} + 16 q^{79} + 48 q^{85} + 52 q^{87} - 40 q^{91} - 8 q^{93} - 16 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{7}{8}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See 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 0
\(3\) −1.44423 + 0.956131i −0.833829 + 0.552022i
\(4\) 0 0
\(5\) −1.56013 3.76650i −0.697713 1.68443i −0.728631 0.684906i \(-0.759843\pi\)
0.0309186 0.999522i \(-0.490157\pi\)
\(6\) 0 0
\(7\) −0.838552 0.838552i −0.316943 0.316943i 0.530649 0.847592i \(-0.321948\pi\)
−0.847592 + 0.530649i \(0.821948\pi\)
\(8\) 0 0
\(9\) 1.17163 2.76175i 0.390542 0.920585i
\(10\) 0 0
\(11\) 0.249049 + 0.601256i 0.0750910 + 0.181286i 0.956968 0.290194i \(-0.0937200\pi\)
−0.881877 + 0.471480i \(0.843720\pi\)
\(12\) 0 0
\(13\) 2.05771 + 0.852332i 0.570706 + 0.236394i 0.649326 0.760510i \(-0.275051\pi\)
−0.0786194 + 0.996905i \(0.525051\pi\)
\(14\) 0 0
\(15\) 5.85446 + 3.94801i 1.51162 + 1.01937i
\(16\) 0 0
\(17\) −3.23677 −0.785032 −0.392516 0.919745i \(-0.628395\pi\)
−0.392516 + 0.919745i \(0.628395\pi\)
\(18\) 0 0
\(19\) −1.47818 + 3.56865i −0.339119 + 0.818705i 0.658682 + 0.752421i \(0.271114\pi\)
−0.997801 + 0.0662838i \(0.978886\pi\)
\(20\) 0 0
\(21\) 2.01283 + 0.409300i 0.439236 + 0.0893166i
\(22\) 0 0
\(23\) −2.58369 2.58369i −0.538737 0.538737i 0.384421 0.923158i \(-0.374401\pi\)
−0.923158 + 0.384421i \(0.874401\pi\)
\(24\) 0 0
\(25\) −8.21694 + 8.21694i −1.64339 + 1.64339i
\(26\) 0 0
\(27\) 0.948494 + 5.10885i 0.182538 + 0.983199i
\(28\) 0 0
\(29\) −3.52027 1.45815i −0.653698 0.270771i 0.0310857 0.999517i \(-0.490104\pi\)
−0.684784 + 0.728746i \(0.740104\pi\)
\(30\) 0 0
\(31\) 7.63408i 1.37112i 0.728015 + 0.685561i \(0.240443\pi\)
−0.728015 + 0.685561i \(0.759557\pi\)
\(32\) 0 0
\(33\) −0.934564 0.630232i −0.162687 0.109709i
\(34\) 0 0
\(35\) −1.85015 + 4.46665i −0.312732 + 0.755002i
\(36\) 0 0
\(37\) −0.579146 + 0.239890i −0.0952110 + 0.0394377i −0.429781 0.902933i \(-0.641409\pi\)
0.334570 + 0.942371i \(0.391409\pi\)
\(38\) 0 0
\(39\) −3.78676 + 0.736474i −0.606367 + 0.117930i
\(40\) 0 0
\(41\) −3.54554 + 3.54554i −0.553720 + 0.553720i −0.927512 0.373792i \(-0.878057\pi\)
0.373792 + 0.927512i \(0.378057\pi\)
\(42\) 0 0
\(43\) 3.19595 1.32381i 0.487378 0.201879i −0.125442 0.992101i \(-0.540035\pi\)
0.612820 + 0.790222i \(0.290035\pi\)
\(44\) 0 0
\(45\) −12.2300 0.104223i −1.82315 0.0155366i
\(46\) 0 0
\(47\) 5.96658i 0.870315i −0.900354 0.435158i \(-0.856693\pi\)
0.900354 0.435158i \(-0.143307\pi\)
\(48\) 0 0
\(49\) 5.59366i 0.799095i
\(50\) 0 0
\(51\) 4.67465 3.09477i 0.654582 0.433355i
\(52\) 0 0
\(53\) 0.762825 0.315973i 0.104782 0.0434022i −0.329677 0.944094i \(-0.606940\pi\)
0.434459 + 0.900692i \(0.356940\pi\)
\(54\) 0 0
\(55\) 1.87608 1.87608i 0.252971 0.252971i
\(56\) 0 0
\(57\) −1.27725 6.56731i −0.169176 0.869862i
\(58\) 0 0
\(59\) −5.86827 + 2.43072i −0.763984 + 0.316452i −0.730433 0.682985i \(-0.760682\pi\)
−0.0335508 + 0.999437i \(0.510682\pi\)
\(60\) 0 0
\(61\) −2.68247 + 6.47607i −0.343456 + 0.829175i 0.653906 + 0.756576i \(0.273129\pi\)
−0.997361 + 0.0725992i \(0.976871\pi\)
\(62\) 0 0
\(63\) −3.29834 + 1.33340i −0.415552 + 0.167993i
\(64\) 0 0
\(65\) 9.08011i 1.12625i
\(66\) 0 0
\(67\) −4.78575 1.98232i −0.584673 0.242179i 0.0706842 0.997499i \(-0.477482\pi\)
−0.655357 + 0.755319i \(0.727482\pi\)
\(68\) 0 0
\(69\) 6.20180 + 1.26111i 0.746609 + 0.151820i
\(70\) 0 0
\(71\) −10.2094 + 10.2094i −1.21163 + 1.21163i −0.241145 + 0.970489i \(0.577523\pi\)
−0.970489 + 0.241145i \(0.922477\pi\)
\(72\) 0 0
\(73\) 8.09458 + 8.09458i 0.947399 + 0.947399i 0.998684 0.0512851i \(-0.0163317\pi\)
−0.0512851 + 0.998684i \(0.516332\pi\)
\(74\) 0 0
\(75\) 4.01072 19.7237i 0.463118 2.27749i
\(76\) 0 0
\(77\) 0.295345 0.713025i 0.0336576 0.0812567i
\(78\) 0 0
\(79\) −11.5343 −1.29771 −0.648857 0.760910i \(-0.724753\pi\)
−0.648857 + 0.760910i \(0.724753\pi\)
\(80\) 0 0
\(81\) −6.25458 6.47149i −0.694953 0.719055i
\(82\) 0 0
\(83\) −0.998651 0.413655i −0.109616 0.0454045i 0.327201 0.944955i \(-0.393894\pi\)
−0.436818 + 0.899550i \(0.643894\pi\)
\(84\) 0 0
\(85\) 5.04979 + 12.1913i 0.547727 + 1.32233i
\(86\) 0 0
\(87\) 6.47828 1.25994i 0.694544 0.135080i
\(88\) 0 0
\(89\) 10.4124 + 10.4124i 1.10372 + 1.10372i 0.993958 + 0.109759i \(0.0350078\pi\)
0.109759 + 0.993958i \(0.464992\pi\)
\(90\) 0 0
\(91\) −1.01077 2.44022i −0.105958 0.255805i
\(92\) 0 0
\(93\) −7.29918 11.0254i −0.756890 1.14328i
\(94\) 0 0
\(95\) 15.7475 1.61566
\(96\) 0 0
\(97\) 9.21596 0.935739 0.467870 0.883797i \(-0.345022\pi\)
0.467870 + 0.883797i \(0.345022\pi\)
\(98\) 0 0
\(99\) 1.95232 + 0.0166374i 0.196215 + 0.00167212i
\(100\) 0 0
\(101\) −1.61830 3.90691i −0.161026 0.388752i 0.822687 0.568494i \(-0.192474\pi\)
−0.983714 + 0.179742i \(0.942474\pi\)
\(102\) 0 0
\(103\) −4.29846 4.29846i −0.423540 0.423540i 0.462881 0.886421i \(-0.346816\pi\)
−0.886421 + 0.462881i \(0.846816\pi\)
\(104\) 0 0
\(105\) −1.59866 8.21988i −0.156013 0.802178i
\(106\) 0 0
\(107\) 2.82811 + 6.82767i 0.273404 + 0.660055i 0.999624 0.0274065i \(-0.00872486\pi\)
−0.726221 + 0.687462i \(0.758725\pi\)
\(108\) 0 0
\(109\) −12.4025 5.13729i −1.18794 0.492063i −0.300859 0.953669i \(-0.597273\pi\)
−0.887085 + 0.461606i \(0.847273\pi\)
\(110\) 0 0
\(111\) 0.607056 0.900196i 0.0576192 0.0854429i
\(112\) 0 0
\(113\) 3.42949 0.322619 0.161310 0.986904i \(-0.448428\pi\)
0.161310 + 0.986904i \(0.448428\pi\)
\(114\) 0 0
\(115\) −5.70056 + 13.7624i −0.531580 + 1.28335i
\(116\) 0 0
\(117\) 4.76480 4.68428i 0.440506 0.433062i
\(118\) 0 0
\(119\) 2.71420 + 2.71420i 0.248810 + 0.248810i
\(120\) 0 0
\(121\) 7.47869 7.47869i 0.679881 0.679881i
\(122\) 0 0
\(123\) 1.73059 8.51059i 0.156042 0.767374i
\(124\) 0 0
\(125\) 24.9361 + 10.3289i 2.23035 + 0.923842i
\(126\) 0 0
\(127\) 0.724490i 0.0642881i 0.999483 + 0.0321441i \(0.0102335\pi\)
−0.999483 + 0.0321441i \(0.989766\pi\)
\(128\) 0 0
\(129\) −3.34997 + 4.96764i −0.294949 + 0.437376i
\(130\) 0 0
\(131\) 1.44273 3.48306i 0.126052 0.304317i −0.848238 0.529616i \(-0.822336\pi\)
0.974290 + 0.225299i \(0.0723360\pi\)
\(132\) 0 0
\(133\) 4.23204 1.75297i 0.366964 0.152001i
\(134\) 0 0
\(135\) 17.7627 11.5430i 1.52877 0.993462i
\(136\) 0 0
\(137\) 2.20990 2.20990i 0.188805 0.188805i −0.606375 0.795179i \(-0.707377\pi\)
0.795179 + 0.606375i \(0.207377\pi\)
\(138\) 0 0
\(139\) −4.01991 + 1.66510i −0.340964 + 0.141232i −0.546594 0.837398i \(-0.684076\pi\)
0.205629 + 0.978630i \(0.434076\pi\)
\(140\) 0 0
\(141\) 5.70483 + 8.61714i 0.480434 + 0.725694i
\(142\) 0 0
\(143\) 1.44948i 0.121212i
\(144\) 0 0
\(145\) 15.5340i 1.29003i
\(146\) 0 0
\(147\) 5.34827 + 8.07856i 0.441118 + 0.666308i
\(148\) 0 0
\(149\) −5.47285 + 2.26693i −0.448353 + 0.185714i −0.595423 0.803412i \(-0.703016\pi\)
0.147070 + 0.989126i \(0.453016\pi\)
\(150\) 0 0
\(151\) −5.81381 + 5.81381i −0.473121 + 0.473121i −0.902923 0.429802i \(-0.858583\pi\)
0.429802 + 0.902923i \(0.358583\pi\)
\(152\) 0 0
\(153\) −3.79229 + 8.93916i −0.306588 + 0.722688i
\(154\) 0 0
\(155\) 28.7537 11.9102i 2.30956 0.956649i
\(156\) 0 0
\(157\) 3.37305 8.14326i 0.269199 0.649903i −0.730247 0.683183i \(-0.760595\pi\)
0.999446 + 0.0332797i \(0.0105952\pi\)
\(158\) 0 0
\(159\) −0.799588 + 1.18570i −0.0634114 + 0.0940321i
\(160\) 0 0
\(161\) 4.33312i 0.341498i
\(162\) 0 0
\(163\) −13.8401 5.73275i −1.08404 0.449024i −0.232114 0.972688i \(-0.574564\pi\)
−0.851924 + 0.523665i \(0.824564\pi\)
\(164\) 0 0
\(165\) −0.915722 + 4.50328i −0.0712889 + 0.350580i
\(166\) 0 0
\(167\) 6.72756 6.72756i 0.520594 0.520594i −0.397157 0.917751i \(-0.630003\pi\)
0.917751 + 0.397157i \(0.130003\pi\)
\(168\) 0 0
\(169\) −5.68468 5.68468i −0.437283 0.437283i
\(170\) 0 0
\(171\) 8.12386 + 8.26352i 0.621247 + 0.631927i
\(172\) 0 0
\(173\) −0.275804 + 0.665850i −0.0209690 + 0.0506236i −0.934017 0.357228i \(-0.883722\pi\)
0.913048 + 0.407852i \(0.133722\pi\)
\(174\) 0 0
\(175\) 13.7807 1.04172
\(176\) 0 0
\(177\) 6.15107 9.12136i 0.462343 0.685603i
\(178\) 0 0
\(179\) −11.9198 4.93735i −0.890928 0.369035i −0.110203 0.993909i \(-0.535150\pi\)
−0.780725 + 0.624875i \(0.785150\pi\)
\(180\) 0 0
\(181\) 2.66842 + 6.44214i 0.198342 + 0.478841i 0.991489 0.130190i \(-0.0415587\pi\)
−0.793147 + 0.609030i \(0.791559\pi\)
\(182\) 0 0
\(183\) −2.31784 11.9178i −0.171340 0.880986i
\(184\) 0 0
\(185\) 1.80709 + 1.80709i 0.132860 + 0.132860i
\(186\) 0 0
\(187\) −0.806113 1.94613i −0.0589488 0.142315i
\(188\) 0 0
\(189\) 3.48867 5.07940i 0.253764 0.369472i
\(190\) 0 0
\(191\) −1.62308 −0.117442 −0.0587210 0.998274i \(-0.518702\pi\)
−0.0587210 + 0.998274i \(0.518702\pi\)
\(192\) 0 0
\(193\) −10.3575 −0.745548 −0.372774 0.927922i \(-0.621593\pi\)
−0.372774 + 0.927922i \(0.621593\pi\)
\(194\) 0 0
\(195\) 8.68177 + 13.1138i 0.621715 + 0.939099i
\(196\) 0 0
\(197\) 0.784033 + 1.89282i 0.0558601 + 0.134858i 0.949346 0.314234i \(-0.101748\pi\)
−0.893486 + 0.449092i \(0.851748\pi\)
\(198\) 0 0
\(199\) −8.78498 8.78498i −0.622751 0.622751i 0.323483 0.946234i \(-0.395146\pi\)
−0.946234 + 0.323483i \(0.895146\pi\)
\(200\) 0 0
\(201\) 8.80711 1.71287i 0.621206 0.120816i
\(202\) 0 0
\(203\) 1.72920 + 4.17466i 0.121366 + 0.293004i
\(204\) 0 0
\(205\) 18.8858 + 7.82274i 1.31904 + 0.546364i
\(206\) 0 0
\(207\) −10.1626 + 4.10840i −0.706353 + 0.285553i
\(208\) 0 0
\(209\) −2.51382 −0.173884
\(210\) 0 0
\(211\) 9.81540 23.6965i 0.675720 1.63133i −0.0960096 0.995380i \(-0.530608\pi\)
0.771729 0.635951i \(-0.219392\pi\)
\(212\) 0 0
\(213\) 4.98325 24.5063i 0.341447 1.67915i
\(214\) 0 0
\(215\) −9.97223 9.97223i −0.680100 0.680100i
\(216\) 0 0
\(217\) 6.40157 6.40157i 0.434567 0.434567i
\(218\) 0 0
\(219\) −19.4300 3.95099i −1.31295 0.266983i
\(220\) 0 0
\(221\) −6.66034 2.75880i −0.448023 0.185577i
\(222\) 0 0
\(223\) 16.3840i 1.09715i −0.836101 0.548576i \(-0.815170\pi\)
0.836101 0.548576i \(-0.184830\pi\)
\(224\) 0 0
\(225\) 13.0660 + 32.3204i 0.871065 + 2.15469i
\(226\) 0 0
\(227\) 9.61652 23.2163i 0.638271 1.54092i −0.190710 0.981646i \(-0.561079\pi\)
0.828981 0.559276i \(-0.188921\pi\)
\(228\) 0 0
\(229\) 11.2556 4.66223i 0.743792 0.308089i 0.0215864 0.999767i \(-0.493128\pi\)
0.722206 + 0.691678i \(0.243128\pi\)
\(230\) 0 0
\(231\) 0.255198 + 1.31216i 0.0167908 + 0.0863340i
\(232\) 0 0
\(233\) −13.5410 + 13.5410i −0.887101 + 0.887101i −0.994244 0.107143i \(-0.965830\pi\)
0.107143 + 0.994244i \(0.465830\pi\)
\(234\) 0 0
\(235\) −22.4731 + 9.30866i −1.46598 + 0.607230i
\(236\) 0 0
\(237\) 16.6583 11.0283i 1.08207 0.716367i
\(238\) 0 0
\(239\) 19.1168i 1.23657i −0.785956 0.618283i \(-0.787829\pi\)
0.785956 0.618283i \(-0.212171\pi\)
\(240\) 0 0
\(241\) 1.71219i 0.110292i −0.998478 0.0551460i \(-0.982438\pi\)
0.998478 0.0551460i \(-0.0175624\pi\)
\(242\) 0 0
\(243\) 15.2207 + 3.36616i 0.976407 + 0.215939i
\(244\) 0 0
\(245\) −21.0685 + 8.72686i −1.34602 + 0.557539i
\(246\) 0 0
\(247\) −6.08335 + 6.08335i −0.387075 + 0.387075i
\(248\) 0 0
\(249\) 1.83780 0.357427i 0.116466 0.0226510i
\(250\) 0 0
\(251\) −25.5253 + 10.5729i −1.61115 + 0.667358i −0.992936 0.118655i \(-0.962142\pi\)
−0.618210 + 0.786013i \(0.712142\pi\)
\(252\) 0 0
\(253\) 0.909996 2.19693i 0.0572110 0.138120i
\(254\) 0 0
\(255\) −18.9495 12.7788i −1.18667 0.800239i
\(256\) 0 0
\(257\) 21.5264i 1.34278i −0.741104 0.671391i \(-0.765697\pi\)
0.741104 0.671391i \(-0.234303\pi\)
\(258\) 0 0
\(259\) 0.686804 + 0.284483i 0.0426759 + 0.0176769i
\(260\) 0 0
\(261\) −8.15149 + 8.01373i −0.504564 + 0.496037i
\(262\) 0 0
\(263\) −10.4924 + 10.4924i −0.646988 + 0.646988i −0.952264 0.305276i \(-0.901251\pi\)
0.305276 + 0.952264i \(0.401251\pi\)
\(264\) 0 0
\(265\) −2.38022 2.38022i −0.146216 0.146216i
\(266\) 0 0
\(267\) −24.9937 5.08235i −1.52959 0.311035i
\(268\) 0 0
\(269\) −4.94856 + 11.9469i −0.301719 + 0.728414i 0.698203 + 0.715900i \(0.253983\pi\)
−0.999922 + 0.0125138i \(0.996017\pi\)
\(270\) 0 0
\(271\) 7.03570 0.427388 0.213694 0.976901i \(-0.431450\pi\)
0.213694 + 0.976901i \(0.431450\pi\)
\(272\) 0 0
\(273\) 3.79296 + 2.55782i 0.229561 + 0.154806i
\(274\) 0 0
\(275\) −6.98690 2.89407i −0.421326 0.174519i
\(276\) 0 0
\(277\) −9.40456 22.7046i −0.565065 1.36419i −0.905671 0.423982i \(-0.860632\pi\)
0.340606 0.940206i \(-0.389368\pi\)
\(278\) 0 0
\(279\) 21.0835 + 8.94430i 1.26223 + 0.535481i
\(280\) 0 0
\(281\) −13.7076 13.7076i −0.817727 0.817727i 0.168051 0.985778i \(-0.446253\pi\)
−0.985778 + 0.168051i \(0.946253\pi\)
\(282\) 0 0
\(283\) 11.3422 + 27.3825i 0.674224 + 1.62772i 0.774358 + 0.632747i \(0.218073\pi\)
−0.100134 + 0.994974i \(0.531927\pi\)
\(284\) 0 0
\(285\) −22.7431 + 15.0567i −1.34718 + 0.891879i
\(286\) 0 0
\(287\) 5.94623 0.350995
\(288\) 0 0
\(289\) −6.52333 −0.383725
\(290\) 0 0
\(291\) −13.3100 + 8.81167i −0.780247 + 0.516549i
\(292\) 0 0
\(293\) 4.07927 + 9.84823i 0.238314 + 0.575340i 0.997109 0.0759896i \(-0.0242116\pi\)
−0.758795 + 0.651329i \(0.774212\pi\)
\(294\) 0 0
\(295\) 18.3106 + 18.3106i 1.06608 + 1.06608i
\(296\) 0 0
\(297\) −2.83551 + 1.84264i −0.164533 + 0.106921i
\(298\) 0 0
\(299\) −3.11433 7.51865i −0.180106 0.434815i
\(300\) 0 0
\(301\) −3.79005 1.56989i −0.218455 0.0904871i
\(302\) 0 0
\(303\) 6.07272 + 4.09520i 0.348869 + 0.235263i
\(304\) 0 0
\(305\) 28.5771 1.63632
\(306\) 0 0
\(307\) 8.04114 19.4130i 0.458932 1.10796i −0.509898 0.860235i \(-0.670317\pi\)
0.968830 0.247725i \(-0.0796830\pi\)
\(308\) 0 0
\(309\) 10.3179 + 2.09809i 0.586963 + 0.119356i
\(310\) 0 0
\(311\) 20.9205 + 20.9205i 1.18629 + 1.18629i 0.978084 + 0.208210i \(0.0667638\pi\)
0.208210 + 0.978084i \(0.433236\pi\)
\(312\) 0 0
\(313\) −8.91198 + 8.91198i −0.503735 + 0.503735i −0.912596 0.408861i \(-0.865926\pi\)
0.408861 + 0.912596i \(0.365926\pi\)
\(314\) 0 0
\(315\) 10.1681 + 10.3429i 0.572909 + 0.582757i
\(316\) 0 0
\(317\) 0.482452 + 0.199838i 0.0270972 + 0.0112240i 0.396191 0.918168i \(-0.370332\pi\)
−0.369094 + 0.929392i \(0.620332\pi\)
\(318\) 0 0
\(319\) 2.47974i 0.138839i
\(320\) 0 0
\(321\) −10.6126 7.15671i −0.592337 0.399448i
\(322\) 0 0
\(323\) 4.78454 11.5509i 0.266219 0.642710i
\(324\) 0 0
\(325\) −23.9116 + 9.90452i −1.32638 + 0.549404i
\(326\) 0 0
\(327\) 22.8240 4.43897i 1.26217 0.245476i
\(328\) 0 0
\(329\) −5.00329 + 5.00329i −0.275840 + 0.275840i
\(330\) 0 0
\(331\) 19.5132 8.08262i 1.07254 0.444261i 0.224654 0.974439i \(-0.427875\pi\)
0.847887 + 0.530177i \(0.177875\pi\)
\(332\) 0 0
\(333\) −0.0160256 + 1.88052i −0.000878196 + 0.103052i
\(334\) 0 0
\(335\) 21.1182i 1.15381i
\(336\) 0 0
\(337\) 10.2626i 0.559037i 0.960140 + 0.279519i \(0.0901749\pi\)
−0.960140 + 0.279519i \(0.909825\pi\)
\(338\) 0 0
\(339\) −4.95299 + 3.27904i −0.269009 + 0.178093i
\(340\) 0 0
\(341\) −4.59004 + 1.90126i −0.248565 + 0.102959i
\(342\) 0 0
\(343\) −10.5604 + 10.5604i −0.570210 + 0.570210i
\(344\) 0 0
\(345\) −4.92568 25.3266i −0.265190 1.36354i
\(346\) 0 0
\(347\) 4.24383 1.75785i 0.227821 0.0943665i −0.265853 0.964014i \(-0.585654\pi\)
0.493674 + 0.869647i \(0.335654\pi\)
\(348\) 0 0
\(349\) −13.7021 + 33.0798i −0.733457 + 1.77072i −0.102741 + 0.994708i \(0.532761\pi\)
−0.630716 + 0.776014i \(0.717239\pi\)
\(350\) 0 0
\(351\) −2.40271 + 11.3210i −0.128247 + 0.604269i
\(352\) 0 0
\(353\) 4.53983i 0.241631i −0.992675 0.120815i \(-0.961449\pi\)
0.992675 0.120815i \(-0.0385509\pi\)
\(354\) 0 0
\(355\) 54.3817 + 22.5257i 2.88628 + 1.19554i
\(356\) 0 0
\(357\) −6.51507 1.32481i −0.344814 0.0701164i
\(358\) 0 0
\(359\) −2.86633 + 2.86633i −0.151279 + 0.151279i −0.778689 0.627410i \(-0.784115\pi\)
0.627410 + 0.778689i \(0.284115\pi\)
\(360\) 0 0
\(361\) 2.88477 + 2.88477i 0.151830 + 0.151830i
\(362\) 0 0
\(363\) −3.65038 + 17.9516i −0.191595 + 0.942214i
\(364\) 0 0
\(365\) 17.8596 43.1168i 0.934813 2.25684i
\(366\) 0 0
\(367\) −28.9994 −1.51376 −0.756878 0.653557i \(-0.773276\pi\)
−0.756878 + 0.653557i \(0.773276\pi\)
\(368\) 0 0
\(369\) 5.63786 + 13.9460i 0.293495 + 0.725997i
\(370\) 0 0
\(371\) −0.904628 0.374709i −0.0469659 0.0194539i
\(372\) 0 0
\(373\) −7.55879 18.2485i −0.391379 0.944873i −0.989640 0.143572i \(-0.954141\pi\)
0.598261 0.801302i \(-0.295859\pi\)
\(374\) 0 0
\(375\) −45.8893 + 8.92486i −2.36971 + 0.460878i
\(376\) 0 0
\(377\) −6.00088 6.00088i −0.309061 0.309061i
\(378\) 0 0
\(379\) −7.21366 17.4153i −0.370541 0.894564i −0.993659 0.112437i \(-0.964134\pi\)
0.623118 0.782128i \(-0.285866\pi\)
\(380\) 0 0
\(381\) −0.692708 1.04633i −0.0354885 0.0536053i
\(382\) 0 0
\(383\) −18.2276 −0.931388 −0.465694 0.884946i \(-0.654195\pi\)
−0.465694 + 0.884946i \(0.654195\pi\)
\(384\) 0 0
\(385\) −3.14638 −0.160354
\(386\) 0 0
\(387\) 0.0884355 10.3775i 0.00449543 0.527515i
\(388\) 0 0
\(389\) 0.747968 + 1.80575i 0.0379235 + 0.0915554i 0.941706 0.336436i \(-0.109221\pi\)
−0.903783 + 0.427991i \(0.859221\pi\)
\(390\) 0 0
\(391\) 8.36281 + 8.36281i 0.422926 + 0.422926i
\(392\) 0 0
\(393\) 1.24662 + 6.40980i 0.0628837 + 0.323332i
\(394\) 0 0
\(395\) 17.9951 + 43.4440i 0.905432 + 2.18591i
\(396\) 0 0
\(397\) 7.44786 + 3.08500i 0.373797 + 0.154832i 0.561670 0.827361i \(-0.310159\pi\)
−0.187872 + 0.982193i \(0.560159\pi\)
\(398\) 0 0
\(399\) −4.43599 + 6.57807i −0.222077 + 0.329316i
\(400\) 0 0
\(401\) −33.1951 −1.65768 −0.828842 0.559482i \(-0.811000\pi\)
−0.828842 + 0.559482i \(0.811000\pi\)
\(402\) 0 0
\(403\) −6.50677 + 15.7087i −0.324125 + 0.782508i
\(404\) 0 0
\(405\) −14.6169 + 33.6542i −0.726318 + 1.67229i
\(406\) 0 0
\(407\) −0.288471 0.288471i −0.0142990 0.0142990i
\(408\) 0 0
\(409\) 24.3278 24.3278i 1.20293 1.20293i 0.229664 0.973270i \(-0.426237\pi\)
0.973270 0.229664i \(-0.0737629\pi\)
\(410\) 0 0
\(411\) −1.07866 + 5.30457i −0.0532064 + 0.261655i
\(412\) 0 0
\(413\) 6.95913 + 2.88257i 0.342436 + 0.141842i
\(414\) 0 0
\(415\) 4.40677i 0.216320i
\(416\) 0 0
\(417\) 4.21364 6.24836i 0.206343 0.305983i
\(418\) 0 0
\(419\) −8.96150 + 21.6350i −0.437798 + 1.05694i 0.538910 + 0.842363i \(0.318836\pi\)
−0.976708 + 0.214574i \(0.931164\pi\)
\(420\) 0 0
\(421\) −15.8146 + 6.55062i −0.770756 + 0.319258i −0.733179 0.680036i \(-0.761964\pi\)
−0.0375773 + 0.999294i \(0.511964\pi\)
\(422\) 0 0
\(423\) −16.4782 6.99061i −0.801199 0.339895i
\(424\) 0 0
\(425\) 26.5963 26.5963i 1.29011 1.29011i
\(426\) 0 0
\(427\) 7.67991 3.18112i 0.371657 0.153945i
\(428\) 0 0
\(429\) −1.38590 2.09340i −0.0669117 0.101070i
\(430\) 0 0
\(431\) 21.2060i 1.02146i 0.859742 + 0.510729i \(0.170625\pi\)
−0.859742 + 0.510729i \(0.829375\pi\)
\(432\) 0 0
\(433\) 7.73016i 0.371488i −0.982598 0.185744i \(-0.940531\pi\)
0.982598 0.185744i \(-0.0594695\pi\)
\(434\) 0 0
\(435\) −14.8525 22.4347i −0.712124 1.07566i
\(436\) 0 0
\(437\) 13.0395 5.40113i 0.623763 0.258371i
\(438\) 0 0
\(439\) 25.4093 25.4093i 1.21272 1.21272i 0.242594 0.970128i \(-0.422002\pi\)
0.970128 0.242594i \(-0.0779983\pi\)
\(440\) 0 0
\(441\) −15.4483 6.55369i −0.735634 0.312080i
\(442\) 0 0
\(443\) −4.74325 + 1.96472i −0.225359 + 0.0933467i −0.492506 0.870309i \(-0.663919\pi\)
0.267147 + 0.963656i \(0.413919\pi\)
\(444\) 0 0
\(445\) 22.9736 55.4632i 1.08905 2.62921i
\(446\) 0 0
\(447\) 5.73660 8.50674i 0.271332 0.402355i
\(448\) 0 0
\(449\) 13.4003i 0.632399i −0.948693 0.316199i \(-0.897593\pi\)
0.948693 0.316199i \(-0.102407\pi\)
\(450\) 0 0
\(451\) −3.01479 1.24877i −0.141961 0.0588021i
\(452\) 0 0
\(453\) 2.83774 13.9553i 0.133329 0.655676i
\(454\) 0 0
\(455\) −7.61414 + 7.61414i −0.356957 + 0.356957i
\(456\) 0 0
\(457\) −21.8591 21.8591i −1.02253 1.02253i −0.999740 0.0227865i \(-0.992746\pi\)
−0.0227865 0.999740i \(-0.507254\pi\)
\(458\) 0 0
\(459\) −3.07006 16.5362i −0.143298 0.771842i
\(460\) 0 0
\(461\) −8.39897 + 20.2769i −0.391179 + 0.944389i 0.598505 + 0.801119i \(0.295762\pi\)
−0.989684 + 0.143270i \(0.954238\pi\)
\(462\) 0 0
\(463\) 24.1790 1.12369 0.561847 0.827241i \(-0.310091\pi\)
0.561847 + 0.827241i \(0.310091\pi\)
\(464\) 0 0
\(465\) −30.1394 + 44.6934i −1.39768 + 2.07261i
\(466\) 0 0
\(467\) −22.1546 9.17674i −1.02519 0.424649i −0.194218 0.980958i \(-0.562217\pi\)
−0.830975 + 0.556309i \(0.812217\pi\)
\(468\) 0 0
\(469\) 2.35082 + 5.67538i 0.108551 + 0.262065i
\(470\) 0 0
\(471\) 2.91455 + 14.9859i 0.134295 + 0.690512i
\(472\) 0 0
\(473\) 1.59190 + 1.59190i 0.0731954 + 0.0731954i
\(474\) 0 0
\(475\) −17.1772 41.4695i −0.788146 1.90275i
\(476\) 0 0
\(477\) 0.0211082 2.47694i 0.000966478 0.113411i
\(478\) 0 0
\(479\) −23.6803 −1.08198 −0.540991 0.841028i \(-0.681951\pi\)
−0.540991 + 0.841028i \(0.681951\pi\)
\(480\) 0 0
\(481\) −1.39618 −0.0636603
\(482\) 0 0
\(483\) −4.14303 6.25804i −0.188514 0.284751i
\(484\) 0 0
\(485\) −14.3781 34.7119i −0.652877 1.57619i
\(486\) 0 0
\(487\) 15.7774 + 15.7774i 0.714942 + 0.714942i 0.967565 0.252623i \(-0.0812931\pi\)
−0.252623 + 0.967565i \(0.581293\pi\)
\(488\) 0 0
\(489\) 25.4696 4.95349i 1.15177 0.224005i
\(490\) 0 0
\(491\) −1.45911 3.52259i −0.0658486 0.158972i 0.887530 0.460751i \(-0.152420\pi\)
−0.953378 + 0.301778i \(0.902420\pi\)
\(492\) 0 0
\(493\) 11.3943 + 4.71968i 0.513174 + 0.212564i
\(494\) 0 0
\(495\) −2.98321 7.37934i −0.134085 0.331677i
\(496\) 0 0
\(497\) 17.1222 0.768037
\(498\) 0 0
\(499\) −12.0066 + 28.9865i −0.537490 + 1.29761i 0.388980 + 0.921246i \(0.372827\pi\)
−0.926470 + 0.376369i \(0.877173\pi\)
\(500\) 0 0
\(501\) −3.28375 + 16.1486i −0.146707 + 0.721466i
\(502\) 0 0
\(503\) −9.02060 9.02060i −0.402208 0.402208i 0.476802 0.879011i \(-0.341796\pi\)
−0.879011 + 0.476802i \(0.841796\pi\)
\(504\) 0 0
\(505\) −12.1906 + 12.1906i −0.542475 + 0.542475i
\(506\) 0 0
\(507\) 13.6453 + 2.77471i 0.606010 + 0.123229i
\(508\) 0 0
\(509\) 13.9019 + 5.75836i 0.616192 + 0.255235i 0.668873 0.743376i \(-0.266777\pi\)
−0.0526815 + 0.998611i \(0.516777\pi\)
\(510\) 0 0
\(511\) 13.5754i 0.600543i
\(512\) 0 0
\(513\) −19.6338 4.16698i −0.866852 0.183977i
\(514\) 0 0
\(515\) −9.48396 + 22.8963i −0.417913 + 1.00893i
\(516\) 0 0
\(517\) 3.58745 1.48597i 0.157776 0.0653528i
\(518\) 0 0
\(519\) −0.238314 1.22535i −0.0104608 0.0537868i
\(520\) 0 0
\(521\) −24.3052 + 24.3052i −1.06483 + 1.06483i −0.0670849 + 0.997747i \(0.521370\pi\)
−0.997747 + 0.0670849i \(0.978630\pi\)
\(522\) 0 0
\(523\) −11.5224 + 4.77273i −0.503839 + 0.208697i −0.620102 0.784522i \(-0.712909\pi\)
0.116262 + 0.993219i \(0.462909\pi\)
\(524\) 0 0
\(525\) −19.9025 + 13.1761i −0.868616 + 0.575053i
\(526\) 0 0
\(527\) 24.7098i 1.07637i
\(528\) 0 0
\(529\) 9.64908i 0.419525i
\(530\) 0 0
\(531\) −0.162381 + 19.0546i −0.00704675 + 0.826900i
\(532\) 0 0
\(533\) −10.3177 + 4.27372i −0.446908 + 0.185115i
\(534\) 0 0
\(535\) 21.3041 21.3041i 0.921058 0.921058i
\(536\) 0 0
\(537\) 21.9357 4.26621i 0.946597 0.184101i
\(538\) 0 0
\(539\) 3.36323 1.39309i 0.144864 0.0600048i
\(540\) 0 0
\(541\) −11.0650 + 26.7133i −0.475722 + 1.14849i 0.485875 + 0.874028i \(0.338501\pi\)
−0.961597 + 0.274466i \(0.911499\pi\)
\(542\) 0 0
\(543\) −10.0134 6.75260i −0.429714 0.289782i
\(544\) 0 0
\(545\) 54.7288i 2.34433i
\(546\) 0 0
\(547\) −5.80786 2.40569i −0.248326 0.102860i 0.255049 0.966928i \(-0.417909\pi\)
−0.503375 + 0.864068i \(0.667909\pi\)
\(548\) 0 0
\(549\) 14.7424 + 14.9959i 0.629192 + 0.640008i
\(550\) 0 0
\(551\) 10.4072 10.4072i 0.443363 0.443363i
\(552\) 0 0
\(553\) 9.67214 + 9.67214i 0.411301 + 0.411301i
\(554\) 0 0
\(555\) −4.33767 0.882047i −0.184124 0.0374408i
\(556\) 0 0
\(557\) 7.71096 18.6159i 0.326724 0.788781i −0.672108 0.740453i \(-0.734611\pi\)
0.998832 0.0483277i \(-0.0153892\pi\)
\(558\) 0 0
\(559\) 7.70467 0.325873
\(560\) 0 0
\(561\) 3.02497 + 2.03992i 0.127714 + 0.0861253i
\(562\) 0 0
\(563\) 11.4466 + 4.74136i 0.482419 + 0.199824i 0.610620 0.791924i \(-0.290920\pi\)
−0.128201 + 0.991748i \(0.540920\pi\)
\(564\) 0 0
\(565\) −5.35046 12.9172i −0.225096 0.543429i
\(566\) 0 0
\(567\) −0.181895 + 10.6715i −0.00763889 + 0.448160i
\(568\) 0 0
\(569\) 27.9002 + 27.9002i 1.16964 + 1.16964i 0.982295 + 0.187343i \(0.0599874\pi\)
0.187343 + 0.982295i \(0.440013\pi\)
\(570\) 0 0
\(571\) 9.45224 + 22.8197i 0.395564 + 0.954976i 0.988705 + 0.149877i \(0.0478878\pi\)
−0.593141 + 0.805099i \(0.702112\pi\)
\(572\) 0 0
\(573\) 2.34411 1.55188i 0.0979266 0.0648307i
\(574\) 0 0
\(575\) 42.4601 1.77071
\(576\) 0 0
\(577\) −16.6076 −0.691383 −0.345691 0.938348i \(-0.612356\pi\)
−0.345691 + 0.938348i \(0.612356\pi\)
\(578\) 0 0
\(579\) 14.9586 9.90310i 0.621659 0.411559i
\(580\) 0 0
\(581\) 0.490550 + 1.18429i 0.0203514 + 0.0491327i
\(582\) 0 0
\(583\) 0.379961 + 0.379961i 0.0157364 + 0.0157364i
\(584\) 0 0
\(585\) −25.0770 10.6385i −1.03681 0.439848i
\(586\) 0 0
\(587\) −5.82661 14.0667i −0.240490 0.580594i 0.756842 0.653598i \(-0.226741\pi\)
−0.997332 + 0.0730041i \(0.976741\pi\)
\(588\) 0 0
\(589\) −27.2434 11.2846i −1.12254 0.464973i
\(590\) 0 0
\(591\) −2.94212 1.98404i −0.121022 0.0816126i
\(592\) 0 0
\(593\) 0.760076 0.0312126 0.0156063 0.999878i \(-0.495032\pi\)
0.0156063 + 0.999878i \(0.495032\pi\)
\(594\) 0 0
\(595\) 5.98850 14.4575i 0.245505 0.592701i
\(596\) 0 0
\(597\) 21.0872 + 4.28798i 0.863041 + 0.175495i
\(598\) 0 0
\(599\) −18.9662 18.9662i −0.774937 0.774937i 0.204028 0.978965i \(-0.434597\pi\)
−0.978965 + 0.204028i \(0.934597\pi\)
\(600\) 0 0
\(601\) −15.3756 + 15.3756i −0.627183 + 0.627183i −0.947358 0.320175i \(-0.896258\pi\)
0.320175 + 0.947358i \(0.396258\pi\)
\(602\) 0 0
\(603\) −11.0818 + 10.8945i −0.451286 + 0.443660i
\(604\) 0 0
\(605\) −39.8362 16.5007i −1.61957 0.670849i
\(606\) 0 0
\(607\) 8.77741i 0.356264i 0.984007 + 0.178132i \(0.0570054\pi\)
−0.984007 + 0.178132i \(0.942995\pi\)
\(608\) 0 0
\(609\) −6.48890 4.37585i −0.262943 0.177318i
\(610\) 0 0
\(611\) 5.08551 12.2775i 0.205738 0.496695i
\(612\) 0 0
\(613\) 4.08951 1.69393i 0.165174 0.0684172i −0.298564 0.954389i \(-0.596508\pi\)
0.463738 + 0.885972i \(0.346508\pi\)
\(614\) 0 0
\(615\) −34.7550 + 6.75939i −1.40146 + 0.272565i
\(616\) 0 0
\(617\) −10.3859 + 10.3859i −0.418123 + 0.418123i −0.884556 0.466434i \(-0.845539\pi\)
0.466434 + 0.884556i \(0.345539\pi\)
\(618\) 0 0
\(619\) 20.6486 8.55293i 0.829937 0.343771i 0.0730593 0.997328i \(-0.476724\pi\)
0.756878 + 0.653556i \(0.226724\pi\)
\(620\) 0 0
\(621\) 10.7491 15.6503i 0.431346 0.628025i
\(622\) 0 0
\(623\) 17.4628i 0.699630i
\(624\) 0 0
\(625\) 51.9336i 2.07734i
\(626\) 0 0
\(627\) 3.63054 2.40354i 0.144990 0.0959880i
\(628\) 0 0
\(629\) 1.87456 0.776468i 0.0747436 0.0309598i
\(630\) 0 0
\(631\) 13.2657 13.2657i 0.528098 0.528098i −0.391907 0.920005i \(-0.628185\pi\)
0.920005 + 0.391907i \(0.128185\pi\)
\(632\) 0 0
\(633\) 8.48118 + 43.6081i 0.337097 + 1.73326i
\(634\) 0 0
\(635\) 2.72879 1.13030i 0.108289 0.0448546i
\(636\) 0 0
\(637\) 4.76766 11.5101i 0.188901 0.456048i
\(638\) 0 0
\(639\) 16.2343 + 40.1575i 0.642218 + 1.58861i
\(640\) 0 0
\(641\) 3.60149i 0.142250i −0.997467 0.0711252i \(-0.977341\pi\)
0.997467 0.0711252i \(-0.0226590\pi\)
\(642\) 0 0
\(643\) −16.5202 6.84287i −0.651492 0.269857i 0.0323617 0.999476i \(-0.489697\pi\)
−0.683853 + 0.729620i \(0.739697\pi\)
\(644\) 0 0
\(645\) 23.9370 + 4.86748i 0.942518 + 0.191657i
\(646\) 0 0
\(647\) 23.6599 23.6599i 0.930167 0.930167i −0.0675491 0.997716i \(-0.521518\pi\)
0.997716 + 0.0675491i \(0.0215179\pi\)
\(648\) 0 0
\(649\) −2.92297 2.92297i −0.114737 0.114737i
\(650\) 0 0
\(651\) −3.12463 + 15.3661i −0.122464 + 0.602246i
\(652\) 0 0
\(653\) −12.1760 + 29.3955i −0.476484 + 1.15033i 0.484763 + 0.874646i \(0.338906\pi\)
−0.961247 + 0.275689i \(0.911094\pi\)
\(654\) 0 0
\(655\) −15.3698 −0.600547
\(656\) 0 0
\(657\) 31.8391 12.8714i 1.24216 0.502162i
\(658\) 0 0
\(659\) 41.6508 + 17.2523i 1.62248 + 0.672055i 0.994360 0.106059i \(-0.0338233\pi\)
0.628123 + 0.778114i \(0.283823\pi\)
\(660\) 0 0
\(661\) −2.34448 5.66007i −0.0911896 0.220151i 0.871704 0.490033i \(-0.163015\pi\)
−0.962893 + 0.269882i \(0.913015\pi\)
\(662\) 0 0
\(663\) 12.2569 2.38380i 0.476017 0.0925790i
\(664\) 0 0
\(665\) −13.2051 13.2051i −0.512071 0.512071i
\(666\) 0 0
\(667\) 5.32790 + 12.8627i 0.206297 + 0.498046i
\(668\) 0 0
\(669\) 15.6652 + 23.6623i 0.605652 + 0.914837i
\(670\) 0 0
\(671\) −4.56184 −0.176108
\(672\) 0 0
\(673\) 45.2206 1.74313 0.871563 0.490283i \(-0.163107\pi\)
0.871563 + 0.490283i \(0.163107\pi\)
\(674\) 0 0
\(675\) −49.7728 34.1854i −1.91576 1.31580i
\(676\) 0 0
\(677\) 19.1300 + 46.1838i 0.735224 + 1.77499i 0.624338 + 0.781155i \(0.285369\pi\)
0.110886 + 0.993833i \(0.464631\pi\)
\(678\) 0 0
\(679\) −7.72806 7.72806i −0.296576 0.296576i
\(680\) 0 0
\(681\) 8.30935 + 42.7245i 0.318415 + 1.63721i
\(682\) 0 0
\(683\) −14.1282 34.1085i −0.540600 1.30512i −0.924300 0.381666i \(-0.875350\pi\)
0.383700 0.923458i \(-0.374650\pi\)
\(684\) 0 0
\(685\) −11.7713 4.87584i −0.449759 0.186296i
\(686\) 0 0
\(687\) −11.7981 + 17.4952i −0.450124 + 0.667484i
\(688\) 0 0
\(689\) 1.83899 0.0700599
\(690\) 0 0
\(691\) 1.84991 4.46608i 0.0703740 0.169898i −0.884779 0.466011i \(-0.845691\pi\)
0.955153 + 0.296113i \(0.0956906\pi\)
\(692\) 0 0
\(693\) −1.62317 1.65107i −0.0616590 0.0627189i
\(694\) 0 0
\(695\) 12.5432 + 12.5432i 0.475791 + 0.475791i
\(696\) 0 0
\(697\) 11.4761 11.4761i 0.434688 0.434688i
\(698\) 0 0
\(699\) 6.60942 32.5034i 0.249991 1.22939i
\(700\) 0 0
\(701\) −9.68643 4.01225i −0.365851 0.151541i 0.192182 0.981359i \(-0.438444\pi\)
−0.558033 + 0.829819i \(0.688444\pi\)
\(702\) 0 0
\(703\) 2.42137i 0.0913238i
\(704\) 0 0
\(705\) 23.5561 34.9311i 0.887175 1.31558i
\(706\) 0 0
\(707\) −1.91912 + 4.63317i −0.0721761 + 0.174248i
\(708\) 0 0
\(709\) 38.1564 15.8049i 1.43299 0.593565i 0.474904 0.880038i \(-0.342483\pi\)
0.958088 + 0.286473i \(0.0924827\pi\)
\(710\) 0 0
\(711\) −13.5139 + 31.8550i −0.506812 + 1.19466i
\(712\) 0 0
\(713\) 19.7241 19.7241i 0.738674 0.738674i
\(714\) 0 0
\(715\) 5.45948 2.26139i 0.204173 0.0845711i
\(716\) 0 0
\(717\) 18.2782 + 27.6092i 0.682612 + 1.03108i
\(718\) 0 0
\(719\) 46.6233i 1.73876i −0.494147 0.869378i \(-0.664519\pi\)
0.494147 0.869378i \(-0.335481\pi\)
\(720\) 0 0
\(721\) 7.20896i 0.268476i
\(722\) 0 0
\(723\) 1.63708 + 2.47281i 0.0608837 + 0.0919647i
\(724\) 0 0
\(725\) 40.9074 16.9444i 1.51926 0.629298i
\(726\) 0 0
\(727\) −25.2103 + 25.2103i −0.934997 + 0.934997i −0.998013 0.0630153i \(-0.979928\pi\)
0.0630153 + 0.998013i \(0.479928\pi\)
\(728\) 0 0
\(729\) −25.2007 + 9.69143i −0.933360 + 0.358942i
\(730\) 0 0
\(731\) −10.3446 + 4.28486i −0.382608 + 0.158481i
\(732\) 0 0
\(733\) 2.25020 5.43246i 0.0831129 0.200652i −0.876860 0.480747i \(-0.840366\pi\)
0.959973 + 0.280094i \(0.0903657\pi\)
\(734\) 0 0
\(735\) 22.0838 32.7479i 0.814575 1.20792i
\(736\) 0 0
\(737\) 3.37116i 0.124178i
\(738\) 0 0
\(739\) −10.5170 4.35630i −0.386876 0.160249i 0.180763 0.983527i \(-0.442143\pi\)
−0.567639 + 0.823277i \(0.692143\pi\)
\(740\) 0 0
\(741\) 2.96931 14.6023i 0.109080 0.536428i
\(742\) 0 0
\(743\) 15.4609 15.4609i 0.567205 0.567205i −0.364139 0.931345i \(-0.618637\pi\)
0.931345 + 0.364139i \(0.118637\pi\)
\(744\) 0 0
\(745\) 17.0768 + 17.0768i 0.625644 + 0.625644i
\(746\) 0 0
\(747\) −2.31246 + 2.27338i −0.0846085 + 0.0831787i
\(748\) 0 0
\(749\) 3.35383 8.09687i 0.122546 0.295853i
\(750\) 0 0
\(751\) 34.5867 1.26209 0.631043 0.775748i \(-0.282627\pi\)
0.631043 + 0.775748i \(0.282627\pi\)
\(752\) 0 0
\(753\) 26.7555 39.6754i 0.975023 1.44585i
\(754\) 0 0
\(755\) 30.9680 + 12.8274i 1.12704 + 0.466836i
\(756\) 0 0
\(757\) 6.82795 + 16.4841i 0.248166 + 0.599126i 0.998048 0.0624455i \(-0.0198900\pi\)
−0.749882 + 0.661571i \(0.769890\pi\)
\(758\) 0 0
\(759\) 0.786300 + 4.04295i 0.0285409 + 0.146750i
\(760\) 0 0
\(761\) −10.6555 10.6555i −0.386261 0.386261i 0.487090 0.873352i \(-0.338058\pi\)
−0.873352 + 0.487090i \(0.838058\pi\)
\(762\) 0 0
\(763\) 6.09226 + 14.7080i 0.220555 + 0.532466i
\(764\) 0 0
\(765\) 39.5858 + 0.337346i 1.43123 + 0.0121968i
\(766\) 0 0
\(767\) −14.1470 −0.510818
\(768\) 0 0
\(769\) 4.53950 0.163699 0.0818493 0.996645i \(-0.473917\pi\)
0.0818493 + 0.996645i \(0.473917\pi\)
\(770\) 0 0
\(771\) 20.5821 + 31.0892i 0.741245 + 1.11965i
\(772\) 0 0
\(773\) −10.7578 25.9715i −0.386929 0.934130i −0.990587 0.136886i \(-0.956290\pi\)
0.603657 0.797244i \(-0.293710\pi\)
\(774\) 0 0
\(775\) −62.7288 62.7288i −2.25328 2.25328i
\(776\) 0 0
\(777\) −1.26391 + 0.245814i −0.0453425 + 0.00881851i
\(778\) 0 0
\(779\) −7.41183 17.8938i −0.265557 0.641110i
\(780\) 0 0
\(781\) −8.68111 3.59583i −0.310635 0.128669i
\(782\) 0 0
\(783\) 4.11049 19.3676i 0.146897 0.692141i
\(784\) 0 0
\(785\) −35.9340 −1.28254
\(786\) 0 0
\(787\) 18.2001 43.9390i 0.648764 1.56626i −0.165786 0.986162i \(-0.553016\pi\)
0.814550 0.580093i \(-0.196984\pi\)
\(788\) 0 0
\(789\) 5.12137 25.1855i 0.182325 0.896629i
\(790\) 0 0
\(791\) −2.87581 2.87581i −0.102252 0.102252i
\(792\) 0 0
\(793\) −11.0395 + 11.0395i −0.392025 + 0.392025i
\(794\) 0 0
\(795\) 5.71340 + 1.16179i 0.202633 + 0.0412046i
\(796\) 0 0
\(797\) −0.989560 0.409889i −0.0350520 0.0145190i 0.365089 0.930973i \(-0.381039\pi\)
−0.400141 + 0.916454i \(0.631039\pi\)
\(798\) 0 0
\(799\) 19.3124i 0.683225i
\(800\) 0 0
\(801\) 40.9561 16.5571i 1.44711 0.585017i
\(802\) 0 0
\(803\) −2.85097 + 6.88286i −0.100609 + 0.242891i
\(804\) 0 0
\(805\) 16.3207 6.76024i 0.575228 0.238267i
\(806\) 0 0
\(807\) −4.27590 21.9856i −0.150519 0.773928i
\(808\) 0 0
\(809\) 16.4456 16.4456i 0.578196 0.578196i −0.356210 0.934406i \(-0.615931\pi\)
0.934406 + 0.356210i \(0.115931\pi\)
\(810\) 0 0
\(811\) 30.7926 12.7547i 1.08127 0.447878i 0.230317 0.973116i \(-0.426024\pi\)
0.850956 + 0.525238i \(0.176024\pi\)
\(812\) 0 0
\(813\) −10.1612 + 6.72705i −0.356369 + 0.235928i
\(814\) 0 0
\(815\) 61.0725i 2.13927i
\(816\) 0 0
\(817\) 13.3621i 0.467480i
\(818\) 0 0
\(819\) −7.92354 0.0675235i −0.276871 0.00235946i
\(820\) 0 0
\(821\) 30.3992 12.5918i 1.06094 0.439455i 0.217155 0.976137i \(-0.430322\pi\)
0.843785 + 0.536682i \(0.180322\pi\)
\(822\) 0 0
\(823\) −7.58486 + 7.58486i −0.264392 + 0.264392i −0.826835 0.562444i \(-0.809861\pi\)
0.562444 + 0.826835i \(0.309861\pi\)
\(824\) 0 0
\(825\) 12.8578 2.50068i 0.447652 0.0870624i
\(826\) 0 0
\(827\) 34.9305 14.4687i 1.21465 0.503126i 0.318947 0.947773i \(-0.396671\pi\)
0.895706 + 0.444647i \(0.146671\pi\)
\(828\) 0 0
\(829\) 1.61267 3.89332i 0.0560102 0.135221i −0.893397 0.449268i \(-0.851685\pi\)
0.949407 + 0.314047i \(0.101685\pi\)
\(830\) 0 0
\(831\) 35.2910 + 23.7988i 1.22423 + 0.825571i
\(832\) 0 0
\(833\) 18.1054i 0.627315i
\(834\) 0 0
\(835\) −35.8352 14.8434i −1.24013 0.513678i
\(836\) 0 0
\(837\) −39.0014 + 7.24088i −1.34809 + 0.250282i
\(838\) 0 0
\(839\) −13.7057 + 13.7057i −0.473173 + 0.473173i −0.902940 0.429767i \(-0.858596\pi\)
0.429767 + 0.902940i \(0.358596\pi\)
\(840\) 0 0
\(841\) −10.2400 10.2400i −0.353102 0.353102i
\(842\) 0 0
\(843\) 32.9033 + 6.69074i 1.13325 + 0.230441i
\(844\) 0 0
\(845\) −12.5425 + 30.2802i −0.431474 + 1.04167i
\(846\) 0 0
\(847\) −12.5425 −0.430967
\(848\) 0 0
\(849\) −42.5621 28.7021i −1.46073 0.985055i
\(850\) 0 0
\(851\) 2.11614 + 0.876532i 0.0725402 + 0.0300471i
\(852\) 0 0
\(853\) 13.8120 + 33.3450i 0.472913 + 1.14171i 0.962870 + 0.269966i \(0.0870124\pi\)
−0.489957 + 0.871746i \(0.662988\pi\)
\(854\) 0 0
\(855\) 18.4502 43.4907i 0.630983 1.48735i
\(856\) 0 0
\(857\) 24.4023 + 24.4023i 0.833565 + 0.833565i 0.988003 0.154438i \(-0.0493565\pi\)
−0.154438 + 0.988003i \(0.549357\pi\)
\(858\) 0 0
\(859\) −1.51555 3.65886i −0.0517099 0.124839i 0.895914 0.444228i \(-0.146522\pi\)
−0.947623 + 0.319390i \(0.896522\pi\)
\(860\) 0 0
\(861\) −8.58776 + 5.68538i −0.292670 + 0.193757i
\(862\) 0 0
\(863\) 37.8790 1.28942 0.644708 0.764429i \(-0.276979\pi\)
0.644708 + 0.764429i \(0.276979\pi\)
\(864\) 0 0
\(865\) 2.93821 0.0999021
\(866\) 0 0
\(867\) 9.42122 6.23716i 0.319961 0.211825i
\(868\) 0 0
\(869\) −2.87261 6.93509i −0.0974466 0.235257i
\(870\) 0 0
\(871\) −8.15810 8.15810i −0.276427 0.276427i
\(872\) 0 0
\(873\) 10.7977 25.4522i 0.365446 0.861427i
\(874\) 0 0
\(875\) −12.2489 29.5715i −0.414089 0.999699i
\(876\) 0 0
\(877\) −8.89587 3.68479i −0.300392 0.124426i 0.227397 0.973802i \(-0.426979\pi\)
−0.527789 + 0.849376i \(0.676979\pi\)
\(878\) 0 0
\(879\) −15.3076 10.3228i −0.516313 0.348181i
\(880\) 0 0
\(881\) −8.66239 −0.291843 −0.145922 0.989296i \(-0.546615\pi\)
−0.145922 + 0.989296i \(0.546615\pi\)
\(882\) 0 0
\(883\) −10.6679 + 25.7547i −0.359005 + 0.866715i 0.636435 + 0.771330i \(0.280408\pi\)
−0.995440 + 0.0953849i \(0.969592\pi\)
\(884\) 0 0
\(885\) −43.9521 8.93746i −1.47743 0.300429i
\(886\) 0 0
\(887\) −13.1549 13.1549i −0.441698 0.441698i 0.450885 0.892582i \(-0.351109\pi\)
−0.892582 + 0.450885i \(0.851109\pi\)
\(888\) 0 0
\(889\) 0.607523 0.607523i 0.0203757 0.0203757i
\(890\) 0 0
\(891\) 2.33333 5.37232i 0.0781696 0.179980i
\(892\) 0 0
\(893\) 21.2927 + 8.81971i 0.712532 + 0.295140i
\(894\) 0 0
\(895\) 52.5988i 1.75818i
\(896\) 0 0
\(897\) 11.6866 + 7.88099i 0.390205 + 0.263139i
\(898\) 0 0
\(899\) 11.1316 26.8741i 0.371260 0.896300i
\(900\) 0 0
\(901\) −2.46909 + 1.02273i −0.0822573 + 0.0340721i
\(902\) 0 0
\(903\) 6.97475 1.35650i 0.232105 0.0451414i
\(904\) 0 0
\(905\) 20.1012 20.1012i 0.668187 0.668187i
\(906\) 0 0
\(907\) −24.0939 + 9.98004i −0.800026 + 0.331382i −0.744967 0.667101i \(-0.767535\pi\)
−0.0550594 + 0.998483i \(0.517535\pi\)
\(908\) 0 0
\(909\) −12.6860 0.108108i −0.420767 0.00358573i
\(910\) 0 0
\(911\) 52.5495i 1.74104i 0.492132 + 0.870520i \(0.336218\pi\)
−0.492132 + 0.870520i \(0.663782\pi\)
\(912\) 0 0
\(913\) 0.703466i 0.0232813i
\(914\) 0 0
\(915\) −41.2720 + 27.3234i −1.36441 + 0.903285i
\(916\) 0 0
\(917\) −4.13053 + 1.71092i −0.136402 + 0.0564997i
\(918\) 0 0
\(919\) −32.3967 + 32.3967i −1.06867 + 1.06867i −0.0712063 + 0.997462i \(0.522685\pi\)
−0.997462 + 0.0712063i \(0.977315\pi\)
\(920\) 0 0
\(921\) 6.94810 + 35.7253i 0.228948 + 1.17719i
\(922\) 0 0
\(923\) −29.7098 + 12.3062i −0.977911 + 0.405064i
\(924\) 0 0
\(925\) 2.78764 6.72996i 0.0916571 0.221280i
\(926\) 0 0
\(927\) −16.9075 + 6.83510i −0.555315 + 0.224494i
\(928\) 0 0
\(929\) 45.9891i 1.50885i 0.656385 + 0.754426i \(0.272085\pi\)
−0.656385 + 0.754426i \(0.727915\pi\)
\(930\) 0 0
\(931\) 19.9618 + 8.26846i 0.654223 + 0.270988i
\(932\) 0 0
\(933\) −50.2169 10.2114i −1.64403 0.334306i
\(934\) 0 0
\(935\) −6.07244 + 6.07244i −0.198590 + 0.198590i
\(936\) 0 0
\(937\) −19.7725 19.7725i −0.645940 0.645940i 0.306069 0.952009i \(-0.400986\pi\)
−0.952009 + 0.306069i \(0.900986\pi\)
\(938\) 0 0
\(939\) 4.34997 21.3920i 0.141956 0.698102i
\(940\) 0 0
\(941\) −15.8263 + 38.2081i −0.515924 + 1.24555i 0.424464 + 0.905445i \(0.360463\pi\)
−0.940387 + 0.340105i \(0.889537\pi\)
\(942\) 0 0
\(943\) 18.3211 0.596619
\(944\) 0 0
\(945\) −24.5743 5.21554i −0.799403 0.169661i
\(946\) 0 0
\(947\) −29.7644 12.3288i −0.967213 0.400633i −0.157539 0.987513i \(-0.550356\pi\)
−0.809674 + 0.586880i \(0.800356\pi\)
\(948\) 0 0
\(949\) 9.75704 + 23.5556i 0.316727 + 0.764646i
\(950\) 0 0
\(951\) −0.887846 + 0.172674i −0.0287904 + 0.00559934i
\(952\) 0 0
\(953\) −12.6032 12.6032i −0.408259 0.408259i 0.472872 0.881131i \(-0.343217\pi\)
−0.881131 + 0.472872i \(0.843217\pi\)
\(954\) 0 0
\(955\) 2.53222 + 6.11333i 0.0819408 + 0.197823i
\(956\) 0 0
\(957\) 2.37095 + 3.58132i 0.0766420 + 0.115768i
\(958\) 0 0
\(959\) −3.70623 −0.119681
\(960\) 0 0
\(961\) −27.2792 −0.879975
\(962\) 0 0
\(963\) 22.1698 + 0.188929i 0.714413 + 0.00608815i
\(964\) 0 0
\(965\) 16.1590 + 39.0114i 0.520178 + 1.25582i
\(966\) 0 0
\(967\) 25.3843 + 25.3843i 0.816305 + 0.816305i 0.985571 0.169265i \(-0.0541395\pi\)
−0.169265 + 0.985571i \(0.554139\pi\)
\(968\) 0 0
\(969\) 4.13418 + 21.2569i 0.132809 + 0.682869i
\(970\) 0 0
\(971\) 2.40697 + 5.81095i 0.0772435 + 0.186482i 0.957784 0.287488i \(-0.0928203\pi\)
−0.880541 + 0.473971i \(0.842820\pi\)
\(972\) 0 0
\(973\) 4.76718 + 1.97463i 0.152829 + 0.0633037i
\(974\) 0 0
\(975\) 25.0640 37.1671i 0.802690 1.19030i
\(976\) 0 0
\(977\) 23.9584 0.766497 0.383249 0.923645i \(-0.374805\pi\)
0.383249 + 0.923645i \(0.374805\pi\)
\(978\) 0 0
\(979\) −3.66735 + 8.85376i −0.117209 + 0.282967i
\(980\) 0 0
\(981\) −28.7190 + 28.2337i −0.916928 + 0.901432i
\(982\) 0 0
\(983\) 21.9489 + 21.9489i 0.700063 + 0.700063i 0.964424 0.264361i \(-0.0851611\pi\)
−0.264361 + 0.964424i \(0.585161\pi\)
\(984\) 0 0
\(985\) 5.90611 5.90611i 0.188184 0.188184i
\(986\) 0 0
\(987\) 2.44212 12.0097i 0.0777336 0.382274i
\(988\) 0 0
\(989\) −11.6777 4.83705i −0.371328 0.153809i
\(990\) 0 0
\(991\) 31.9976i 1.01644i −0.861229 0.508218i \(-0.830304\pi\)
0.861229 0.508218i \(-0.169696\pi\)
\(992\) 0 0
\(993\) −20.4536 + 30.3304i −0.649074 + 0.962505i
\(994\) 0 0
\(995\) −19.3829 + 46.7944i −0.614478 + 1.48348i
\(996\) 0 0
\(997\) −29.5281 + 12.2309i −0.935164 + 0.387358i −0.797635 0.603140i \(-0.793916\pi\)
−0.137529 + 0.990498i \(0.543916\pi\)
\(998\) 0 0
\(999\) −1.77488 2.73123i −0.0561547 0.0864124i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 768.2.o.b.479.3 56
3.2 odd 2 inner 768.2.o.b.479.6 56
4.3 odd 2 768.2.o.a.479.12 56
8.3 odd 2 384.2.o.a.239.3 56
8.5 even 2 96.2.o.a.83.2 yes 56
12.11 even 2 768.2.o.a.479.9 56
24.5 odd 2 96.2.o.a.83.13 yes 56
24.11 even 2 384.2.o.a.239.6 56
32.5 even 8 768.2.o.a.287.9 56
32.11 odd 8 96.2.o.a.59.13 yes 56
32.21 even 8 384.2.o.a.143.6 56
32.27 odd 8 inner 768.2.o.b.287.6 56
96.5 odd 8 768.2.o.a.287.12 56
96.11 even 8 96.2.o.a.59.2 56
96.53 odd 8 384.2.o.a.143.3 56
96.59 even 8 inner 768.2.o.b.287.3 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.o.a.59.2 56 96.11 even 8
96.2.o.a.59.13 yes 56 32.11 odd 8
96.2.o.a.83.2 yes 56 8.5 even 2
96.2.o.a.83.13 yes 56 24.5 odd 2
384.2.o.a.143.3 56 96.53 odd 8
384.2.o.a.143.6 56 32.21 even 8
384.2.o.a.239.3 56 8.3 odd 2
384.2.o.a.239.6 56 24.11 even 2
768.2.o.a.287.9 56 32.5 even 8
768.2.o.a.287.12 56 96.5 odd 8
768.2.o.a.479.9 56 12.11 even 2
768.2.o.a.479.12 56 4.3 odd 2
768.2.o.b.287.3 56 96.59 even 8 inner
768.2.o.b.287.6 56 32.27 odd 8 inner
768.2.o.b.479.3 56 1.1 even 1 trivial
768.2.o.b.479.6 56 3.2 odd 2 inner