Properties

Label 768.2.o.b.479.6
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 479.6
Character \(\chi\) \(=\) 768.479
Dual form 768.2.o.b.287.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.345141 - 1.69731i) q^{3} +(1.56013 + 3.76650i) q^{5} +(-0.838552 - 0.838552i) q^{7} +(-2.76175 + 1.17163i) q^{9} +O(q^{10})\) \(q+(-0.345141 - 1.69731i) q^{3} +(1.56013 + 3.76650i) q^{5} +(-0.838552 - 0.838552i) q^{7} +(-2.76175 + 1.17163i) q^{9} +(-0.249049 - 0.601256i) q^{11} +(2.05771 + 0.852332i) q^{13} +(5.85446 - 3.94801i) q^{15} +3.23677 q^{17} +(-1.47818 + 3.56865i) q^{19} +(-1.13387 + 1.71271i) q^{21} +(2.58369 + 2.58369i) q^{23} +(-8.21694 + 8.21694i) q^{25} +(2.94182 + 4.28319i) q^{27} +(3.52027 + 1.45815i) q^{29} +7.63408i q^{31} +(-0.934564 + 0.630232i) q^{33} +(1.85015 - 4.46665i) q^{35} +(-0.579146 + 0.239890i) q^{37} +(0.736474 - 3.78676i) q^{39} +(3.54554 - 3.54554i) q^{41} +(3.19595 - 1.32381i) q^{43} +(-8.72163 - 8.57424i) q^{45} +5.96658i q^{47} -5.59366i q^{49} +(-1.11714 - 5.49382i) q^{51} +(-0.762825 + 0.315973i) q^{53} +(1.87608 - 1.87608i) q^{55} +(6.56731 + 1.27725i) q^{57} +(5.86827 - 2.43072i) q^{59} +(-2.68247 + 6.47607i) q^{61} +(3.29834 + 1.33340i) q^{63} +9.08011i q^{65} +(-4.78575 - 1.98232i) q^{67} +(3.49360 - 5.27708i) q^{69} +(10.2094 - 10.2094i) q^{71} +(8.09458 + 8.09458i) q^{73} +(16.7827 + 11.1107i) q^{75} +(-0.295345 + 0.713025i) q^{77} -11.5343 q^{79} +(6.25458 - 6.47149i) q^{81} +(0.998651 + 0.413655i) q^{83} +(5.04979 + 12.1913i) q^{85} +(1.25994 - 6.47828i) q^{87} +(-10.4124 - 10.4124i) q^{89} +(-1.01077 - 2.44022i) q^{91} +(12.9574 - 2.63484i) q^{93} -15.7475 q^{95} +9.21596 q^{97} +(1.39226 + 1.36873i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{3} - 8 q^{7} - 4 q^{9}+O(q^{10}) \) 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
\(3\) −0.345141 1.69731i −0.199268 0.979945i
\(4\) 0 0
\(5\) 1.56013 + 3.76650i 0.697713 + 1.68443i 0.728631 + 0.684906i \(0.240157\pi\)
−0.0309186 + 0.999522i \(0.509843\pi\)
\(6\) 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
\(9\) −2.76175 + 1.17163i −0.920585 + 0.390542i
\(10\) 0 0
\(11\) −0.249049 0.601256i −0.0750910 0.181286i 0.881877 0.471480i \(-0.156280\pi\)
−0.956968 + 0.290194i \(0.906280\pi\)
\(12\) 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 \(-0.371605\pi\)
0.392516 + 0.919745i \(0.371605\pi\)
\(18\) 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
\(21\) −1.13387 + 1.71271i −0.247430 + 0.373743i
\(22\) 0 0
\(23\) 2.58369 + 2.58369i 0.538737 + 0.538737i 0.923158 0.384421i \(-0.125599\pi\)
−0.384421 + 0.923158i \(0.625599\pi\)
\(24\) 0 0
\(25\) −8.21694 + 8.21694i −1.64339 + 1.64339i
\(26\) 0 0
\(27\) 2.94182 + 4.28319i 0.566153 + 0.824300i
\(28\) 0 0
\(29\) 3.52027 + 1.45815i 0.653698 + 0.270771i 0.684784 0.728746i \(-0.259896\pi\)
−0.0310857 + 0.999517i \(0.509896\pi\)
\(30\) 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
\(39\) 0.736474 3.78676i 0.117930 0.606367i
\(40\) 0 0
\(41\) 3.54554 3.54554i 0.553720 0.553720i −0.373792 0.927512i \(-0.621943\pi\)
0.927512 + 0.373792i \(0.121943\pi\)
\(42\) 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
\(45\) −8.72163 8.57424i −1.30014 1.27817i
\(46\) 0 0
\(47\) 5.96658i 0.870315i 0.900354 + 0.435158i \(0.143307\pi\)
−0.900354 + 0.435158i \(0.856693\pi\)
\(48\) 0 0
\(49\) 5.59366i 0.799095i
\(50\) 0 0
\(51\) −1.11714 5.49382i −0.156431 0.769288i
\(52\) 0 0
\(53\) −0.762825 + 0.315973i −0.104782 + 0.0434022i −0.434459 0.900692i \(-0.643060\pi\)
0.329677 + 0.944094i \(0.393060\pi\)
\(54\) 0 0
\(55\) 1.87608 1.87608i 0.252971 0.252971i
\(56\) 0 0
\(57\) 6.56731 + 1.27725i 0.869862 + 0.169176i
\(58\) 0 0
\(59\) 5.86827 2.43072i 0.763984 0.316452i 0.0335508 0.999437i \(-0.489318\pi\)
0.730433 + 0.682985i \(0.239318\pi\)
\(60\) 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
\(69\) 3.49360 5.27708i 0.420580 0.635285i
\(70\) 0 0
\(71\) 10.2094 10.2094i 1.21163 1.21163i 0.241145 0.970489i \(-0.422477\pi\)
0.970489 0.241145i \(-0.0775229\pi\)
\(72\) 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
\(75\) 16.7827 + 11.1107i 1.93790 + 1.28296i
\(76\) 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 0.436818 0.899550i \(-0.356106\pi\)
−0.327201 + 0.944955i \(0.606106\pi\)
\(84\) 0 0
\(85\) 5.04979 + 12.1913i 0.547727 + 1.32233i
\(86\) 0 0
\(87\) 1.25994 6.47828i 0.135080 0.694544i
\(88\) 0 0
\(89\) −10.4124 10.4124i −1.10372 1.10372i −0.993958 0.109759i \(-0.964992\pi\)
−0.109759 0.993958i \(-0.535008\pi\)
\(90\) 0 0
\(91\) −1.01077 2.44022i −0.105958 0.255805i
\(92\) 0 0
\(93\) 12.9574 2.63484i 1.34362 0.273220i
\(94\) 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
\(99\) 1.39226 + 1.36873i 0.139927 + 0.137563i
\(100\) 0 0
\(101\) 1.61830 + 3.90691i 0.161026 + 0.388752i 0.983714 0.179742i \(-0.0575262\pi\)
−0.822687 + 0.568494i \(0.807526\pi\)
\(102\) 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
\(105\) −8.21988 1.59866i −0.802178 0.156013i
\(106\) 0 0
\(107\) −2.82811 6.82767i −0.273404 0.660055i 0.726221 0.687462i \(-0.241275\pi\)
−0.999624 + 0.0274065i \(0.991275\pi\)
\(108\) 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 \(-0.551572\pi\)
−0.161310 + 0.986904i \(0.551572\pi\)
\(114\) 0 0
\(115\) −5.70056 + 13.7624i −0.531580 + 1.28335i
\(116\) 0 0
\(117\) −6.68151 + 0.0569391i −0.617706 + 0.00526402i
\(118\) 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
\(123\) −7.24160 4.79418i −0.652953 0.432277i
\(124\) 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 −0.974290 0.225299i \(-0.927664\pi\)
0.848238 + 0.529616i \(0.177664\pi\)
\(132\) 0 0
\(133\) 4.23204 1.75297i 0.366964 0.152001i
\(134\) 0 0
\(135\) −11.5430 + 17.7627i −0.993462 + 1.52877i
\(136\) 0 0
\(137\) −2.20990 + 2.20990i −0.188805 + 0.188805i −0.795179 0.606375i \(-0.792623\pi\)
0.606375 + 0.795179i \(0.292623\pi\)
\(138\) 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
\(141\) 10.1272 2.05931i 0.852861 0.173426i
\(142\) 0 0
\(143\) 1.44948i 0.121212i
\(144\) 0 0
\(145\) 15.5340i 1.29003i
\(146\) 0 0
\(147\) −9.49420 + 1.93060i −0.783069 + 0.159234i
\(148\) 0 0
\(149\) 5.47285 2.26693i 0.448353 0.185714i −0.147070 0.989126i \(-0.546984\pi\)
0.595423 + 0.803412i \(0.296984\pi\)
\(150\) 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
\(153\) −8.93916 + 3.79229i −0.722688 + 0.306588i
\(154\) 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
\(165\) −3.83181 2.53679i −0.298306 0.197489i
\(166\) 0 0
\(167\) −6.72756 + 6.72756i −0.520594 + 0.520594i −0.917751 0.397157i \(-0.869997\pi\)
0.397157 + 0.917751i \(0.369997\pi\)
\(168\) 0 0
\(169\) −5.68468 5.68468i −0.437283 0.437283i
\(170\) 0 0
\(171\) −0.0987485 11.5876i −0.00755148 0.886128i
\(172\) 0 0
\(173\) 0.275804 0.665850i 0.0209690 0.0506236i −0.913048 0.407852i \(-0.866278\pi\)
0.934017 + 0.357228i \(0.116278\pi\)
\(174\) 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 0.780725 0.624875i \(-0.214850\pi\)
0.110203 + 0.993909i \(0.464850\pi\)
\(180\) 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
\(183\) 11.9178 + 2.31784i 0.880986 + 0.171340i
\(184\) 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
\(189\) 1.12481 6.05854i 0.0818180 0.440694i
\(190\) 0 0
\(191\) 1.62308 0.117442 0.0587210 0.998274i \(-0.481298\pi\)
0.0587210 + 0.998274i \(0.481298\pi\)
\(192\) 0 0
\(193\) −10.3575 −0.745548 −0.372774 0.927922i \(-0.621593\pi\)
−0.372774 + 0.927922i \(0.621593\pi\)
\(194\) 0 0
\(195\) 15.4118 3.13392i 1.10366 0.224425i
\(196\) 0 0
\(197\) −0.784033 1.89282i −0.0558601 0.134858i 0.893486 0.449092i \(-0.148252\pi\)
−0.949346 + 0.314234i \(0.898252\pi\)
\(198\) 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
\(201\) −1.71287 + 8.80711i −0.120816 + 0.621206i
\(202\) 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
\(213\) −20.8523 13.8049i −1.42877 0.945896i
\(214\) 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
\(219\) 10.9453 16.5328i 0.739613 1.11718i
\(220\) 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 \(0.438921\pi\)
−0.828981 + 0.559276i \(0.811079\pi\)
\(228\) 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
\(231\) 1.31216 + 0.255198i 0.0863340 + 0.0167908i
\(232\) 0 0
\(233\) 13.5410 13.5410i 0.887101 0.887101i −0.107143 0.994244i \(-0.534170\pi\)
0.994244 + 0.107143i \(0.0341701\pi\)
\(234\) 0 0
\(235\) −22.4731 + 9.30866i −1.46598 + 0.607230i
\(236\) 0 0
\(237\) 3.98098 + 19.5774i 0.258592 + 1.27169i
\(238\) 0 0
\(239\) 19.1168i 1.23657i 0.785956 + 0.618283i \(0.212171\pi\)
−0.785956 + 0.618283i \(0.787829\pi\)
\(240\) 0 0
\(241\) 1.71219i 0.110292i −0.998478 0.0551460i \(-0.982438\pi\)
0.998478 0.0551460i \(-0.0175624\pi\)
\(242\) 0 0
\(243\) −13.1429 8.38241i −0.843116 0.537732i
\(244\) 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
\(249\) 0.357427 1.83780i 0.0226510 0.116466i
\(250\) 0 0
\(251\) 25.5253 10.5729i 1.61115 0.667358i 0.618210 0.786013i \(-0.287858\pi\)
0.992936 + 0.118655i \(0.0378582\pi\)
\(252\) 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 \(0.234303\pi\)
−0.741104 + 0.671391i \(0.765697\pi\)
\(258\) 0 0
\(259\) 0.686804 + 0.284483i 0.0426759 + 0.0176769i
\(260\) 0 0
\(261\) −11.4305 + 0.0974097i −0.707532 + 0.00602951i
\(262\) 0 0
\(263\) 10.4924 10.4924i 0.646988 0.646988i −0.305276 0.952264i \(-0.598749\pi\)
0.952264 + 0.305276i \(0.0987488\pi\)
\(264\) 0 0
\(265\) −2.38022 2.38022i −0.146216 0.146216i
\(266\) 0 0
\(267\) −14.0794 + 21.2670i −0.861647 + 1.30152i
\(268\) 0 0
\(269\) 4.94856 11.9469i 0.301719 0.728414i −0.698203 0.715900i \(-0.746017\pi\)
0.999922 0.0125138i \(-0.00398337\pi\)
\(270\) 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
\(279\) −8.94430 21.0835i −0.535481 1.26223i
\(280\) 0 0
\(281\) 13.7076 + 13.7076i 0.817727 + 0.817727i 0.985778 0.168051i \(-0.0537473\pi\)
−0.168051 + 0.985778i \(0.553747\pi\)
\(282\) 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
\(285\) 5.43511 + 26.7284i 0.321948 + 1.58326i
\(286\) 0 0
\(287\) −5.94623 −0.350995
\(288\) 0 0
\(289\) −6.52333 −0.383725
\(290\) 0 0
\(291\) −3.18081 15.6424i −0.186462 0.916973i
\(292\) 0 0
\(293\) −4.07927 9.84823i −0.238314 0.575340i 0.758795 0.651329i \(-0.225788\pi\)
−0.997109 + 0.0759896i \(0.975788\pi\)
\(294\) 0 0
\(295\) 18.3106 + 18.3106i 1.06608 + 1.06608i
\(296\) 0 0
\(297\) 1.84264 2.83551i 0.106921 0.164533i
\(298\) 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
\(309\) −5.81226 + 8.77941i −0.330648 + 0.499443i
\(310\) 0 0
\(311\) −20.9205 20.9205i −1.18629 1.18629i −0.978084 0.208210i \(-0.933236\pi\)
−0.208210 0.978084i \(-0.566764\pi\)
\(312\) 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
\(315\) 0.123597 + 14.5035i 0.00696391 + 0.817179i
\(316\) 0 0
\(317\) −0.482452 0.199838i −0.0270972 0.0112240i 0.369094 0.929392i \(-0.379668\pi\)
−0.396191 + 0.918168i \(0.629668\pi\)
\(318\) 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
\(327\) −4.43897 + 22.8240i −0.245476 + 1.26217i
\(328\) 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
\(333\) 1.31840 1.34106i 0.0722477 0.0734896i
\(334\) 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
\(339\) 1.18366 + 5.82092i 0.0642875 + 0.316149i
\(340\) 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
\(345\) 25.3266 + 4.92568i 1.36354 + 0.265190i
\(346\) 0 0
\(347\) −4.24383 + 1.75785i −0.227821 + 0.0943665i −0.493674 0.869647i \(-0.664346\pi\)
0.265853 + 0.964014i \(0.414346\pi\)
\(348\) 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 \(0.0385509\pi\)
−0.992675 + 0.120815i \(0.961449\pi\)
\(354\) 0 0
\(355\) 54.3817 + 22.5257i 2.88628 + 1.19554i
\(356\) 0 0
\(357\) −3.67007 + 5.54363i −0.194241 + 0.293400i
\(358\) 0 0
\(359\) 2.86633 2.86633i 0.151279 0.151279i −0.627410 0.778689i \(-0.715885\pi\)
0.778689 + 0.627410i \(0.215885\pi\)
\(360\) 0 0
\(361\) 2.88477 + 2.88477i 0.151830 + 0.151830i
\(362\) 0 0
\(363\) −15.2749 10.1125i −0.801724 0.530768i
\(364\) 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
\(375\) −8.92486 + 45.8893i −0.460878 + 2.36971i
\(376\) 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
\(381\) 1.22969 0.250052i 0.0629988 0.0128105i
\(382\) 0 0
\(383\) 18.2276 0.931388 0.465694 0.884946i \(-0.345805\pi\)
0.465694 + 0.884946i \(0.345805\pi\)
\(384\) 0 0
\(385\) −3.14638 −0.160354
\(386\) 0 0
\(387\) −7.27543 + 7.40050i −0.369831 + 0.376189i
\(388\) 0 0
\(389\) −0.747968 1.80575i −0.0379235 0.0915554i 0.903783 0.427991i \(-0.140779\pi\)
−0.941706 + 0.336436i \(0.890779\pi\)
\(390\) 0 0
\(391\) 8.36281 + 8.36281i 0.422926 + 0.422926i
\(392\) 0 0
\(393\) 6.40980 + 1.24662i 0.323332 + 0.0628837i
\(394\) 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 \(-0.189000\pi\)
0.828842 + 0.559482i \(0.189000\pi\)
\(402\) 0 0
\(403\) −6.50677 + 15.7087i −0.324125 + 0.782508i
\(404\) 0 0
\(405\) 34.1328 + 13.4614i 1.69607 + 0.668905i
\(406\) 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
\(411\) 4.51363 + 2.98817i 0.222641 + 0.147396i
\(412\) 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 \(-0.681164\pi\)
0.976708 0.214574i \(-0.0688363\pi\)
\(420\) 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
\(423\) −6.99061 16.4782i −0.339895 0.801199i
\(424\) 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
\(429\) −2.46023 + 0.500277i −0.118781 + 0.0241536i
\(430\) 0 0
\(431\) 21.2060i 1.02146i −0.859742 0.510729i \(-0.829375\pi\)
0.859742 0.510729i \(-0.170625\pi\)
\(432\) 0 0
\(433\) 7.73016i 0.371488i −0.982598 0.185744i \(-0.940531\pi\)
0.982598 0.185744i \(-0.0594695\pi\)
\(434\) 0 0
\(435\) 26.3661 5.36143i 1.26416 0.257061i
\(436\) 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
\(441\) 6.55369 + 15.4483i 0.312080 + 0.735634i
\(442\) 0 0
\(443\) 4.74325 1.96472i 0.225359 0.0933467i −0.267147 0.963656i \(-0.586081\pi\)
0.492506 + 0.870309i \(0.336081\pi\)
\(444\) 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 \(0.102407\pi\)
−0.948693 + 0.316199i \(0.897593\pi\)
\(450\) 0 0
\(451\) −3.01479 1.24877i −0.141961 0.0588021i
\(452\) 0 0
\(453\) 11.8745 + 7.86128i 0.557911 + 0.369355i
\(454\) 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
\(459\) 9.52198 + 13.8637i 0.444448 + 0.647102i
\(460\) 0 0
\(461\) 8.39897 20.2769i 0.391179 0.944389i −0.598505 0.801119i \(-0.704238\pi\)
0.989684 0.143270i \(-0.0457618\pi\)
\(462\) 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 0.830975 0.556309i \(-0.187783\pi\)
0.194218 + 0.980958i \(0.437783\pi\)
\(468\) 0 0
\(469\) 2.35082 + 5.67538i 0.108551 + 0.262065i
\(470\) 0 0
\(471\) −14.9859 2.91455i −0.690512 0.134295i
\(472\) 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
\(477\) 1.73653 1.76639i 0.0795105 0.0808773i
\(478\) 0 0
\(479\) 23.6803 1.08198 0.540991 0.841028i \(-0.318049\pi\)
0.540991 + 0.841028i \(0.318049\pi\)
\(480\) 0 0
\(481\) −1.39618 −0.0636603
\(482\) 0 0
\(483\) −7.35467 + 1.49554i −0.334649 + 0.0680494i
\(484\) 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
\(489\) −4.95349 + 25.4696i −0.224005 + 1.15177i
\(490\) 0 0
\(491\) 1.45911 + 3.52259i 0.0658486 + 0.158972i 0.953378 0.301778i \(-0.0975802\pi\)
−0.887530 + 0.460751i \(0.847580\pi\)
\(492\) 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
\(501\) 13.7407 + 9.09682i 0.613891 + 0.406416i
\(502\) 0 0
\(503\) 9.02060 + 9.02060i 0.402208 + 0.402208i 0.879011 0.476802i \(-0.158204\pi\)
−0.476802 + 0.879011i \(0.658204\pi\)
\(504\) 0 0
\(505\) −12.1906 + 12.1906i −0.542475 + 0.542475i
\(506\) 0 0
\(507\) −7.68668 + 11.6107i −0.341377 + 0.515650i
\(508\) 0 0
\(509\) −13.9019 5.75836i −0.616192 0.255235i 0.0526815 0.998611i \(-0.483223\pi\)
−0.668873 + 0.743376i \(0.733223\pi\)
\(510\) 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
\(519\) −1.22535 0.238314i −0.0537868 0.0104608i
\(520\) 0 0
\(521\) 24.3052 24.3052i 1.06483 1.06483i 0.0670849 0.997747i \(-0.478630\pi\)
0.997747 0.0670849i \(-0.0213698\pi\)
\(522\) 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
\(525\) −4.75627 23.3901i −0.207581 1.02083i
\(526\) 0 0
\(527\) 24.7098i 1.07637i
\(528\) 0 0
\(529\) 9.64908i 0.419525i
\(530\) 0 0
\(531\) −13.3588 + 13.5885i −0.579724 + 0.589689i
\(532\) 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
\(537\) 4.26621 21.9357i 0.184101 0.946597i
\(538\) 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
\(549\) −0.179200 21.0282i −0.00764806 0.897460i
\(550\) 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
\(555\) −2.44350 + 3.69090i −0.103721 + 0.156670i
\(556\) 0 0
\(557\) −7.71096 + 18.6159i −0.326724 + 0.788781i 0.672108 + 0.740453i \(0.265389\pi\)
−0.998832 + 0.0483277i \(0.984611\pi\)
\(558\) 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 0.128201 0.991748i \(-0.459080\pi\)
−0.610620 + 0.791924i \(0.709080\pi\)
\(564\) 0 0
\(565\) −5.35046 12.9172i −0.225096 0.543429i
\(566\) 0 0
\(567\) −10.6715 + 0.181895i −0.448160 + 0.00763889i
\(568\) 0 0
\(569\) −27.9002 27.9002i −1.16964 1.16964i −0.982295 0.187343i \(-0.940013\pi\)
−0.187343 0.982295i \(-0.559987\pi\)
\(570\) 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
\(573\) −0.560193 2.75488i −0.0234024 0.115087i
\(574\) 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
\(579\) 3.57479 + 17.5799i 0.148563 + 0.730596i
\(580\) 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
\(585\) −10.6385 25.0770i −0.439848 1.03681i
\(586\) 0 0
\(587\) 5.82661 + 14.0667i 0.240490 + 0.580594i 0.997332 0.0730041i \(-0.0232586\pi\)
−0.756842 + 0.653598i \(0.773259\pi\)
\(588\) 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 \(-0.504968\pi\)
−0.0156063 + 0.999878i \(0.504968\pi\)
\(594\) 0 0
\(595\) 5.98850 14.4575i 0.245505 0.592701i
\(596\) 0 0
\(597\) −11.8788 + 17.9429i −0.486168 + 0.734356i
\(598\) 0 0
\(599\) 18.9662 + 18.9662i 0.774937 + 0.774937i 0.978965 0.204028i \(-0.0654033\pi\)
−0.204028 + 0.978965i \(0.565403\pi\)
\(600\) 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
\(603\) 15.5396 0.132427i 0.632822 0.00539284i
\(604\) 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
\(615\) 6.75939 34.7550i 0.272565 1.40146i
\(616\) 0 0
\(617\) 10.3859 10.3859i 0.418123 0.418123i −0.466434 0.884556i \(-0.654461\pi\)
0.884556 + 0.466434i \(0.154461\pi\)
\(618\) 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
\(621\) −3.46570 + 18.6672i −0.139074 + 0.749088i
\(622\) 0 0
\(623\) 17.4628i 0.699630i
\(624\) 0 0
\(625\) 51.9336i 2.07734i
\(626\) 0 0
\(627\) −0.867622 4.26674i −0.0346495 0.170397i
\(628\) 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
\(633\) −43.6081 8.48118i −1.73326 0.337097i
\(634\) 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 \(0.0226590\pi\)
−0.997467 + 0.0711252i \(0.977341\pi\)
\(642\) 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
\(645\) 13.4842 20.3678i 0.530939 0.801983i
\(646\) 0 0
\(647\) −23.6599 + 23.6599i −0.930167 + 0.930167i −0.997716 0.0675491i \(-0.978482\pi\)
0.0675491 + 0.997716i \(0.478482\pi\)
\(648\) 0 0
\(649\) −2.92297 2.92297i −0.114737 0.114737i
\(650\) 0 0
\(651\) −13.0749 8.65604i −0.512447 0.339257i
\(652\) 0 0
\(653\) 12.1760 29.3955i 0.476484 1.15033i −0.484763 0.874646i \(-0.661094\pi\)
0.961247 0.275689i \(-0.0889058\pi\)
\(654\) 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 −0.628123 0.778114i \(-0.716177\pi\)
−0.994360 + 0.106059i \(0.966177\pi\)
\(660\) 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
\(663\) 2.38380 12.2569i 0.0925790 0.476017i
\(664\) 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
\(669\) −27.8087 + 5.65478i −1.07515 + 0.218627i
\(670\) 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
\(675\) −59.3674 11.0220i −2.28505 0.424236i
\(676\) 0 0
\(677\) −19.1300 46.1838i −0.735224 1.77499i −0.624338 0.781155i \(-0.714631\pi\)
−0.110886 0.993833i \(-0.535369\pi\)
\(678\) 0 0
\(679\) −7.72806 7.72806i −0.296576 0.296576i
\(680\) 0 0
\(681\) 42.7245 + 8.30935i 1.63721 + 0.318415i
\(682\) 0 0
\(683\) 14.1282 + 34.1085i 0.540600 + 1.30512i 0.924300 + 0.381666i \(0.124650\pi\)
−0.383700 + 0.923458i \(0.625350\pi\)
\(684\) 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
\(693\) −0.0197302 2.31523i −0.000749487 0.0879485i
\(694\) 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
\(699\) −27.6569 18.3098i −1.04608 0.692540i
\(700\) 0 0
\(701\) 9.68643 + 4.01225i 0.365851 + 0.151541i 0.558033 0.829819i \(-0.311556\pi\)
−0.192182 + 0.981359i \(0.561556\pi\)
\(702\) 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
\(711\) 31.8550 13.5139i 1.19466 0.506812i
\(712\) 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
\(717\) 32.4473 6.59802i 1.21177 0.246407i
\(718\) 0 0
\(719\) 46.6233i 1.73876i 0.494147 + 0.869378i \(0.335481\pi\)
−0.494147 + 0.869378i \(0.664519\pi\)
\(720\) 0 0
\(721\) 7.20896i 0.268476i
\(722\) 0 0
\(723\) −2.90613 + 0.590948i −0.108080 + 0.0219776i
\(724\) 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
\(729\) −9.69143 + 25.2007i −0.358942 + 0.933360i
\(730\) 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
\(741\) 12.4250 + 8.22575i 0.456443 + 0.302180i
\(742\) 0 0
\(743\) −15.4609 + 15.4609i −0.567205 + 0.567205i −0.931345 0.364139i \(-0.881363\pi\)
0.364139 + 0.931345i \(0.381363\pi\)
\(744\) 0 0
\(745\) 17.0768 + 17.0768i 0.625644 + 0.625644i
\(746\) 0 0
\(747\) −3.24268 + 0.0276338i −0.118643 + 0.00101107i
\(748\) 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
\(759\) −4.04295 0.786300i −0.146750 0.0285409i
\(760\) 0 0
\(761\) 10.6555 + 10.6555i 0.386261 + 0.386261i 0.873352 0.487090i \(-0.161942\pi\)
−0.487090 + 0.873352i \(0.661942\pi\)
\(762\) 0 0
\(763\) 6.09226 + 14.7080i 0.220555 + 0.532466i
\(764\) 0 0
\(765\) −28.2299 27.7528i −1.02065 1.00341i
\(766\) 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
\(771\) 36.5371 7.42966i 1.31585 0.267573i
\(772\) 0 0
\(773\) 10.7578 + 25.9715i 0.386929 + 0.934130i 0.990587 + 0.136886i \(0.0437095\pi\)
−0.603657 + 0.797244i \(0.706290\pi\)
\(774\) 0 0
\(775\) −62.7288 62.7288i −2.25328 2.25328i
\(776\) 0 0
\(777\) 0.245814 1.26391i 0.00881851 0.0453425i
\(778\) 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
\(789\) −21.4302 14.1875i −0.762936 0.505089i
\(790\) 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
\(795\) −3.21847 + 4.86149i −0.114147 + 0.172419i
\(796\) 0 0
\(797\) 0.989560 + 0.409889i 0.0350520 + 0.0145190i 0.400141 0.916454i \(-0.368961\pi\)
−0.365089 + 0.930973i \(0.618961\pi\)
\(798\) 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
\(807\) −21.9856 4.27590i −0.773928 0.150519i
\(808\) 0 0
\(809\) −16.4456 + 16.4456i −0.578196 + 0.578196i −0.934406 0.356210i \(-0.884069\pi\)
0.356210 + 0.934406i \(0.384069\pi\)
\(810\) 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
\(813\) −2.42831 11.9418i −0.0851646 0.418817i
\(814\) 0 0
\(815\) 61.0725i 2.13927i
\(816\) 0 0
\(817\) 13.3621i 0.467480i
\(818\) 0 0
\(819\) 5.65054 + 5.55504i 0.197446 + 0.194109i
\(820\) 0 0
\(821\) −30.3992 + 12.5918i −1.06094 + 0.439455i −0.843785 0.536682i \(-0.819678\pi\)
−0.217155 + 0.976137i \(0.569678\pi\)
\(822\) 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
\(825\) 2.50068 12.8578i 0.0870624 0.447652i
\(826\) 0 0
\(827\) −34.9305 + 14.4687i −1.21465 + 0.503126i −0.895706 0.444647i \(-0.853329\pi\)
−0.318947 + 0.947773i \(0.603329\pi\)
\(828\) 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
\(837\) −32.6982 + 22.4581i −1.13022 + 0.776264i
\(838\) 0 0
\(839\) 13.7057 13.7057i 0.473173 0.473173i −0.429767 0.902940i \(-0.641404\pi\)
0.902940 + 0.429767i \(0.141404\pi\)
\(840\) 0 0
\(841\) −10.2400 10.2400i −0.353102 0.353102i
\(842\) 0 0
\(843\) 18.5351 27.9972i 0.638381 0.964274i
\(844\) 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
\(855\) 43.4907 18.4502i 1.48735 0.630983i
\(856\) 0 0
\(857\) −24.4023 24.4023i −0.833565 0.833565i 0.154438 0.988003i \(-0.450643\pi\)
−0.988003 + 0.154438i \(0.950643\pi\)
\(858\) 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
\(861\) 2.05229 + 10.0926i 0.0699419 + 0.343956i
\(862\) 0 0
\(863\) −37.8790 −1.28942 −0.644708 0.764429i \(-0.723021\pi\)
−0.644708 + 0.764429i \(0.723021\pi\)
\(864\) 0 0
\(865\) 2.93821 0.0999021
\(866\) 0 0
\(867\) 2.25147 + 11.0721i 0.0764640 + 0.376030i
\(868\) 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
\(873\) −25.4522 + 10.7977i −0.861427 + 0.365446i
\(874\) 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 \(-0.453385\pi\)
0.145922 + 0.989296i \(0.453385\pi\)
\(882\) 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
\(885\) 24.7591 37.3985i 0.832267 1.25714i
\(886\) 0 0
\(887\) 13.1549 + 13.1549i 0.441698 + 0.441698i 0.892582 0.450885i \(-0.148891\pi\)
−0.450885 + 0.892582i \(0.648891\pi\)
\(888\) 0 0
\(889\) 0.607523 0.607523i 0.0203757 0.0203757i
\(890\) 0 0
\(891\) −5.44872 2.14889i −0.182539 0.0719905i
\(892\) 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
\(903\) −1.35650 + 6.97475i −0.0451414 + 0.232105i
\(904\) 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
\(909\) −9.04678 8.89389i −0.300063 0.294992i
\(910\) 0 0
\(911\) 52.5495i 1.74104i −0.492132 0.870520i \(-0.663782\pi\)
0.492132 0.870520i \(-0.336218\pi\)
\(912\) 0 0
\(913\) 0.703466i 0.0232813i
\(914\) 0 0
\(915\) 9.86314 + 48.5043i 0.326065 + 1.60350i
\(916\) 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
\(921\) −35.7253 6.94810i −1.17719 0.228948i
\(922\) 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 \(-0.727915\pi\)
0.656385 0.754426i \(-0.272085\pi\)
\(930\) 0 0
\(931\) 19.9618 + 8.26846i 0.654223 + 0.270988i
\(932\) 0 0
\(933\) −28.2882 + 42.7293i −0.926113 + 1.39889i
\(934\) 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
\(939\) 18.2023 + 12.0505i 0.594011 + 0.393255i
\(940\) 0 0
\(941\) 15.8263 38.2081i 0.515924 1.24555i −0.424464 0.905445i \(-0.639537\pi\)
0.940387 0.340105i \(-0.110463\pi\)
\(942\) 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 0.809674 0.586880i \(-0.199644\pi\)
0.157539 + 0.987513i \(0.449644\pi\)
\(948\) 0 0
\(949\) 9.75704 + 23.5556i 0.316727 + 0.764646i
\(950\) 0 0
\(951\) −0.172674 + 0.887846i −0.00559934 + 0.0287904i
\(952\) 0 0
\(953\) 12.6032 + 12.6032i 0.408259 + 0.408259i 0.881131 0.472872i \(-0.156783\pi\)
−0.472872 + 0.881131i \(0.656783\pi\)
\(954\) 0 0
\(955\) 2.53222 + 6.11333i 0.0819408 + 0.197823i
\(956\) 0 0
\(957\) −4.20889 + 0.855860i −0.136054 + 0.0276660i
\(958\) 0 0
\(959\) 3.70623 0.119681
\(960\) 0 0
\(961\) −27.2792 −0.879975
\(962\) 0 0
\(963\) 15.8100 + 15.5428i 0.509471 + 0.500861i
\(964\) 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
\(969\) 21.2569 + 4.13418i 0.682869 + 0.132809i
\(970\) 0 0
\(971\) −2.40697 5.81095i −0.0772435 0.186482i 0.880541 0.473971i \(-0.157180\pi\)
−0.957784 + 0.287488i \(0.907180\pi\)
\(972\) 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 \(-0.625195\pi\)
−0.383249 + 0.923645i \(0.625195\pi\)
\(978\) 0 0
\(979\) −3.66735 + 8.85376i −0.117209 + 0.282967i
\(980\) 0 0
\(981\) 40.2717 0.343191i 1.28578 0.0109572i
\(982\) 0 0
\(983\) −21.9489 21.9489i −0.700063 0.700063i 0.264361 0.964424i \(-0.414839\pi\)
−0.964424 + 0.264361i \(0.914839\pi\)
\(984\) 0 0
\(985\) 5.90611 5.90611i 0.188184 0.188184i
\(986\) 0 0
\(987\) −10.2190 6.76531i −0.325274 0.215342i
\(988\) 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
\(999\) −2.73123 1.77488i −0.0864124 0.0561547i
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.479.6 56
3.2 odd 2 inner 768.2.o.b.479.3 56
4.3 odd 2 768.2.o.a.479.9 56
8.3 odd 2 384.2.o.a.239.6 56
8.5 even 2 96.2.o.a.83.13 yes 56
12.11 even 2 768.2.o.a.479.12 56
24.5 odd 2 96.2.o.a.83.2 yes 56
24.11 even 2 384.2.o.a.239.3 56
32.5 even 8 768.2.o.a.287.12 56
32.11 odd 8 96.2.o.a.59.2 56
32.21 even 8 384.2.o.a.143.3 56
32.27 odd 8 inner 768.2.o.b.287.3 56
96.5 odd 8 768.2.o.a.287.9 56
96.11 even 8 96.2.o.a.59.13 yes 56
96.53 odd 8 384.2.o.a.143.6 56
96.59 even 8 inner 768.2.o.b.287.6 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.o.a.59.2 56 32.11 odd 8
96.2.o.a.59.13 yes 56 96.11 even 8
96.2.o.a.83.2 yes 56 24.5 odd 2
96.2.o.a.83.13 yes 56 8.5 even 2
384.2.o.a.143.3 56 32.21 even 8
384.2.o.a.143.6 56 96.53 odd 8
384.2.o.a.239.3 56 24.11 even 2
384.2.o.a.239.6 56 8.3 odd 2
768.2.o.a.287.9 56 96.5 odd 8
768.2.o.a.287.12 56 32.5 even 8
768.2.o.a.479.9 56 4.3 odd 2
768.2.o.a.479.12 56 12.11 even 2
768.2.o.b.287.3 56 32.27 odd 8 inner
768.2.o.b.287.6 56 96.59 even 8 inner
768.2.o.b.479.3 56 3.2 odd 2 inner
768.2.o.b.479.6 56 1.1 even 1 trivial