Properties

Label 768.3.l.f.319.5
Level $768$
Weight $3$
Character 768.319
Analytic conductor $20.926$
Analytic rank $0$
Dimension $16$
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] = [768,3,Mod(319,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.l (of order \(4\), degree \(2\), 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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} - 155271168 x^{2} + 429981696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 319.5
Root \(2.78422 + 1.60747i\) of defining polynomial
Character \(\chi\) \(=\) 768.319
Dual form 768.3.l.f.703.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 + 1.22474i) q^{3} +(-5.27866 - 5.27866i) q^{5} +12.7431 q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 + 1.22474i) q^{3} +(-5.27866 - 5.27866i) q^{5} +12.7431 q^{7} +3.00000i q^{9} +(-7.46515 + 7.46515i) q^{11} +(-11.9575 + 11.9575i) q^{13} -12.9300i q^{15} -4.55732 q^{17} +(14.6216 + 14.6216i) q^{19} +(15.6070 + 15.6070i) q^{21} +31.6675 q^{23} +30.7285i q^{25} +(-3.67423 + 3.67423i) q^{27} +(38.3491 - 38.3491i) q^{29} +13.8749i q^{31} -18.2858 q^{33} +(-67.2663 - 67.2663i) q^{35} +(28.7278 + 28.7278i) q^{37} -29.2897 q^{39} +3.98569i q^{41} +(-10.7717 + 10.7717i) q^{43} +(15.8360 - 15.8360i) q^{45} -27.1203i q^{47} +113.386 q^{49} +(-5.58155 - 5.58155i) q^{51} +(43.3372 + 43.3372i) q^{53} +78.8120 q^{55} +35.8156i q^{57} +(6.84922 - 6.84922i) q^{59} +(8.18476 - 8.18476i) q^{61} +38.2292i q^{63} +126.239 q^{65} +(26.6785 + 26.6785i) q^{67} +(38.7846 + 38.7846i) q^{69} +16.7372 q^{71} -50.1252i q^{73} +(-37.6346 + 37.6346i) q^{75} +(-95.1289 + 95.1289i) q^{77} +75.2018i q^{79} -9.00000 q^{81} +(-0.382563 - 0.382563i) q^{83} +(24.0565 + 24.0565i) q^{85} +93.9357 q^{87} +165.167i q^{89} +(-152.375 + 152.375i) q^{91} +(-16.9932 + 16.9932i) q^{93} -154.365i q^{95} -63.7481 q^{97} +(-22.3955 - 22.3955i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} - 32 q^{13} + 112 q^{17} + 72 q^{21} - 56 q^{29} - 48 q^{33} + 272 q^{37} - 24 q^{45} + 304 q^{49} + 504 q^{53} - 176 q^{61} + 624 q^{65} + 288 q^{69} - 848 q^{77} - 144 q^{81} + 880 q^{85} + 216 q^{93} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 1.22474 + 1.22474i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) −5.27866 5.27866i −1.05573 1.05573i −0.998352 0.0573795i \(-0.981726\pi\)
−0.0573795 0.998352i \(-0.518274\pi\)
\(6\) 0 0
\(7\) 12.7431 1.82044 0.910219 0.414127i \(-0.135913\pi\)
0.910219 + 0.414127i \(0.135913\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −7.46515 + 7.46515i −0.678650 + 0.678650i −0.959695 0.281045i \(-0.909319\pi\)
0.281045 + 0.959695i \(0.409319\pi\)
\(12\) 0 0
\(13\) −11.9575 + 11.9575i −0.919805 + 0.919805i −0.997015 0.0772099i \(-0.975399\pi\)
0.0772099 + 0.997015i \(0.475399\pi\)
\(14\) 0 0
\(15\) 12.9300i 0.862002i
\(16\) 0 0
\(17\) −4.55732 −0.268078 −0.134039 0.990976i \(-0.542795\pi\)
−0.134039 + 0.990976i \(0.542795\pi\)
\(18\) 0 0
\(19\) 14.6216 + 14.6216i 0.769560 + 0.769560i 0.978029 0.208469i \(-0.0668481\pi\)
−0.208469 + 0.978029i \(0.566848\pi\)
\(20\) 0 0
\(21\) 15.6070 + 15.6070i 0.743191 + 0.743191i
\(22\) 0 0
\(23\) 31.6675 1.37685 0.688424 0.725309i \(-0.258303\pi\)
0.688424 + 0.725309i \(0.258303\pi\)
\(24\) 0 0
\(25\) 30.7285i 1.22914i
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 38.3491 38.3491i 1.32238 1.32238i 0.410539 0.911843i \(-0.365340\pi\)
0.911843 0.410539i \(-0.134660\pi\)
\(30\) 0 0
\(31\) 13.8749i 0.447576i 0.974638 + 0.223788i \(0.0718423\pi\)
−0.974638 + 0.223788i \(0.928158\pi\)
\(32\) 0 0
\(33\) −18.2858 −0.554116
\(34\) 0 0
\(35\) −67.2663 67.2663i −1.92189 1.92189i
\(36\) 0 0
\(37\) 28.7278 + 28.7278i 0.776426 + 0.776426i 0.979221 0.202795i \(-0.0650025\pi\)
−0.202795 + 0.979221i \(0.565003\pi\)
\(38\) 0 0
\(39\) −29.2897 −0.751018
\(40\) 0 0
\(41\) 3.98569i 0.0972120i 0.998818 + 0.0486060i \(0.0154779\pi\)
−0.998818 + 0.0486060i \(0.984522\pi\)
\(42\) 0 0
\(43\) −10.7717 + 10.7717i −0.250504 + 0.250504i −0.821177 0.570673i \(-0.806682\pi\)
0.570673 + 0.821177i \(0.306682\pi\)
\(44\) 0 0
\(45\) 15.8360 15.8360i 0.351911 0.351911i
\(46\) 0 0
\(47\) 27.1203i 0.577027i −0.957476 0.288513i \(-0.906839\pi\)
0.957476 0.288513i \(-0.0931610\pi\)
\(48\) 0 0
\(49\) 113.386 2.31400
\(50\) 0 0
\(51\) −5.58155 5.58155i −0.109442 0.109442i
\(52\) 0 0
\(53\) 43.3372 + 43.3372i 0.817683 + 0.817683i 0.985772 0.168089i \(-0.0537596\pi\)
−0.168089 + 0.985772i \(0.553760\pi\)
\(54\) 0 0
\(55\) 78.8120 1.43295
\(56\) 0 0
\(57\) 35.8156i 0.628343i
\(58\) 0 0
\(59\) 6.84922 6.84922i 0.116089 0.116089i −0.646676 0.762765i \(-0.723841\pi\)
0.762765 + 0.646676i \(0.223841\pi\)
\(60\) 0 0
\(61\) 8.18476 8.18476i 0.134176 0.134176i −0.636829 0.771005i \(-0.719754\pi\)
0.771005 + 0.636829i \(0.219754\pi\)
\(62\) 0 0
\(63\) 38.2292i 0.606813i
\(64\) 0 0
\(65\) 126.239 1.94213
\(66\) 0 0
\(67\) 26.6785 + 26.6785i 0.398187 + 0.398187i 0.877593 0.479406i \(-0.159148\pi\)
−0.479406 + 0.877593i \(0.659148\pi\)
\(68\) 0 0
\(69\) 38.7846 + 38.7846i 0.562096 + 0.562096i
\(70\) 0 0
\(71\) 16.7372 0.235735 0.117868 0.993029i \(-0.462394\pi\)
0.117868 + 0.993029i \(0.462394\pi\)
\(72\) 0 0
\(73\) 50.1252i 0.686646i −0.939217 0.343323i \(-0.888447\pi\)
0.939217 0.343323i \(-0.111553\pi\)
\(74\) 0 0
\(75\) −37.6346 + 37.6346i −0.501794 + 0.501794i
\(76\) 0 0
\(77\) −95.1289 + 95.1289i −1.23544 + 1.23544i
\(78\) 0 0
\(79\) 75.2018i 0.951922i 0.879466 + 0.475961i \(0.157900\pi\)
−0.879466 + 0.475961i \(0.842100\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) −0.382563 0.382563i −0.00460919 0.00460919i 0.704798 0.709408i \(-0.251037\pi\)
−0.709408 + 0.704798i \(0.751037\pi\)
\(84\) 0 0
\(85\) 24.0565 + 24.0565i 0.283018 + 0.283018i
\(86\) 0 0
\(87\) 93.9357 1.07972
\(88\) 0 0
\(89\) 165.167i 1.85581i 0.372817 + 0.927905i \(0.378392\pi\)
−0.372817 + 0.927905i \(0.621608\pi\)
\(90\) 0 0
\(91\) −152.375 + 152.375i −1.67445 + 1.67445i
\(92\) 0 0
\(93\) −16.9932 + 16.9932i −0.182722 + 0.182722i
\(94\) 0 0
\(95\) 154.365i 1.62490i
\(96\) 0 0
\(97\) −63.7481 −0.657197 −0.328599 0.944470i \(-0.606576\pi\)
−0.328599 + 0.944470i \(0.606576\pi\)
\(98\) 0 0
\(99\) −22.3955 22.3955i −0.226217 0.226217i
\(100\) 0 0
\(101\) −46.0210 46.0210i −0.455654 0.455654i 0.441572 0.897226i \(-0.354421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(102\) 0 0
\(103\) 75.8798 0.736697 0.368348 0.929688i \(-0.379923\pi\)
0.368348 + 0.929688i \(0.379923\pi\)
\(104\) 0 0
\(105\) 164.768i 1.56922i
\(106\) 0 0
\(107\) −37.4899 + 37.4899i −0.350373 + 0.350373i −0.860248 0.509875i \(-0.829691\pi\)
0.509875 + 0.860248i \(0.329691\pi\)
\(108\) 0 0
\(109\) −3.75535 + 3.75535i −0.0344527 + 0.0344527i −0.724123 0.689671i \(-0.757755\pi\)
0.689671 + 0.724123i \(0.257755\pi\)
\(110\) 0 0
\(111\) 70.3684i 0.633949i
\(112\) 0 0
\(113\) 27.2277 0.240953 0.120477 0.992716i \(-0.461558\pi\)
0.120477 + 0.992716i \(0.461558\pi\)
\(114\) 0 0
\(115\) −167.162 167.162i −1.45358 1.45358i
\(116\) 0 0
\(117\) −35.8724 35.8724i −0.306602 0.306602i
\(118\) 0 0
\(119\) −58.0742 −0.488019
\(120\) 0 0
\(121\) 9.54301i 0.0788679i
\(122\) 0 0
\(123\) −4.88146 + 4.88146i −0.0396866 + 0.0396866i
\(124\) 0 0
\(125\) 30.2388 30.2388i 0.241910 0.241910i
\(126\) 0 0
\(127\) 154.301i 1.21496i −0.794333 0.607482i \(-0.792180\pi\)
0.794333 0.607482i \(-0.207820\pi\)
\(128\) 0 0
\(129\) −26.3851 −0.204536
\(130\) 0 0
\(131\) −23.9299 23.9299i −0.182671 0.182671i 0.609848 0.792519i \(-0.291231\pi\)
−0.792519 + 0.609848i \(0.791231\pi\)
\(132\) 0 0
\(133\) 186.325 + 186.325i 1.40094 + 1.40094i
\(134\) 0 0
\(135\) 38.7901 0.287334
\(136\) 0 0
\(137\) 239.640i 1.74920i 0.484848 + 0.874598i \(0.338875\pi\)
−0.484848 + 0.874598i \(0.661125\pi\)
\(138\) 0 0
\(139\) 127.817 127.817i 0.919545 0.919545i −0.0774514 0.996996i \(-0.524678\pi\)
0.996996 + 0.0774514i \(0.0246783\pi\)
\(140\) 0 0
\(141\) 33.2154 33.2154i 0.235570 0.235570i
\(142\) 0 0
\(143\) 178.529i 1.24845i
\(144\) 0 0
\(145\) −404.864 −2.79216
\(146\) 0 0
\(147\) 138.869 + 138.869i 0.944685 + 0.944685i
\(148\) 0 0
\(149\) −118.099 118.099i −0.792613 0.792613i 0.189305 0.981918i \(-0.439377\pi\)
−0.981918 + 0.189305i \(0.939377\pi\)
\(150\) 0 0
\(151\) −230.413 −1.52592 −0.762958 0.646448i \(-0.776254\pi\)
−0.762958 + 0.646448i \(0.776254\pi\)
\(152\) 0 0
\(153\) 13.6720i 0.0893592i
\(154\) 0 0
\(155\) 73.2406 73.2406i 0.472520 0.472520i
\(156\) 0 0
\(157\) 159.055 159.055i 1.01309 1.01309i 0.0131763 0.999913i \(-0.495806\pi\)
0.999913 0.0131763i \(-0.00419426\pi\)
\(158\) 0 0
\(159\) 106.154i 0.667635i
\(160\) 0 0
\(161\) 403.541 2.50647
\(162\) 0 0
\(163\) 83.2597 + 83.2597i 0.510796 + 0.510796i 0.914770 0.403974i \(-0.132372\pi\)
−0.403974 + 0.914770i \(0.632372\pi\)
\(164\) 0 0
\(165\) 96.5246 + 96.5246i 0.584997 + 0.584997i
\(166\) 0 0
\(167\) 15.1469 0.0907001 0.0453500 0.998971i \(-0.485560\pi\)
0.0453500 + 0.998971i \(0.485560\pi\)
\(168\) 0 0
\(169\) 116.962i 0.692082i
\(170\) 0 0
\(171\) −43.8649 + 43.8649i −0.256520 + 0.256520i
\(172\) 0 0
\(173\) −36.4769 + 36.4769i −0.210849 + 0.210849i −0.804628 0.593779i \(-0.797635\pi\)
0.593779 + 0.804628i \(0.297635\pi\)
\(174\) 0 0
\(175\) 391.575i 2.23757i
\(176\) 0 0
\(177\) 16.7771 0.0947859
\(178\) 0 0
\(179\) 16.5489 + 16.5489i 0.0924522 + 0.0924522i 0.751820 0.659368i \(-0.229176\pi\)
−0.659368 + 0.751820i \(0.729176\pi\)
\(180\) 0 0
\(181\) −136.141 136.141i −0.752158 0.752158i 0.222724 0.974882i \(-0.428505\pi\)
−0.974882 + 0.222724i \(0.928505\pi\)
\(182\) 0 0
\(183\) 20.0485 0.109555
\(184\) 0 0
\(185\) 303.288i 1.63940i
\(186\) 0 0
\(187\) 34.0211 34.0211i 0.181931 0.181931i
\(188\) 0 0
\(189\) −46.8210 + 46.8210i −0.247730 + 0.247730i
\(190\) 0 0
\(191\) 196.337i 1.02794i 0.857807 + 0.513972i \(0.171827\pi\)
−0.857807 + 0.513972i \(0.828173\pi\)
\(192\) 0 0
\(193\) 166.674 0.863597 0.431798 0.901970i \(-0.357879\pi\)
0.431798 + 0.901970i \(0.357879\pi\)
\(194\) 0 0
\(195\) 154.610 + 154.610i 0.792873 + 0.792873i
\(196\) 0 0
\(197\) −233.904 233.904i −1.18733 1.18733i −0.977803 0.209527i \(-0.932808\pi\)
−0.209527 0.977803i \(-0.567192\pi\)
\(198\) 0 0
\(199\) 1.62043 0.00814286 0.00407143 0.999992i \(-0.498704\pi\)
0.00407143 + 0.999992i \(0.498704\pi\)
\(200\) 0 0
\(201\) 65.3487i 0.325118i
\(202\) 0 0
\(203\) 488.685 488.685i 2.40732 2.40732i
\(204\) 0 0
\(205\) 21.0391 21.0391i 0.102630 0.102630i
\(206\) 0 0
\(207\) 95.0025i 0.458949i
\(208\) 0 0
\(209\) −218.305 −1.04452
\(210\) 0 0
\(211\) 194.766 + 194.766i 0.923059 + 0.923059i 0.997244 0.0741852i \(-0.0236356\pi\)
−0.0741852 + 0.997244i \(0.523636\pi\)
\(212\) 0 0
\(213\) 20.4988 + 20.4988i 0.0962385 + 0.0962385i
\(214\) 0 0
\(215\) 113.720 0.528930
\(216\) 0 0
\(217\) 176.808i 0.814784i
\(218\) 0 0
\(219\) 61.3905 61.3905i 0.280322 0.280322i
\(220\) 0 0
\(221\) 54.4940 54.4940i 0.246579 0.246579i
\(222\) 0 0
\(223\) 119.867i 0.537521i −0.963207 0.268760i \(-0.913386\pi\)
0.963207 0.268760i \(-0.0866140\pi\)
\(224\) 0 0
\(225\) −92.1855 −0.409713
\(226\) 0 0
\(227\) −258.655 258.655i −1.13945 1.13945i −0.988549 0.150899i \(-0.951783\pi\)
−0.150899 0.988549i \(-0.548217\pi\)
\(228\) 0 0
\(229\) 143.105 + 143.105i 0.624915 + 0.624915i 0.946784 0.321869i \(-0.104311\pi\)
−0.321869 + 0.946784i \(0.604311\pi\)
\(230\) 0 0
\(231\) −233.017 −1.00873
\(232\) 0 0
\(233\) 215.658i 0.925569i −0.886471 0.462785i \(-0.846850\pi\)
0.886471 0.462785i \(-0.153150\pi\)
\(234\) 0 0
\(235\) −143.159 + 143.159i −0.609185 + 0.609185i
\(236\) 0 0
\(237\) −92.1031 + 92.1031i −0.388620 + 0.388620i
\(238\) 0 0
\(239\) 467.706i 1.95693i −0.206411 0.978465i \(-0.566178\pi\)
0.206411 0.978465i \(-0.433822\pi\)
\(240\) 0 0
\(241\) −243.864 −1.01188 −0.505941 0.862568i \(-0.668855\pi\)
−0.505941 + 0.862568i \(0.668855\pi\)
\(242\) 0 0
\(243\) −11.0227 11.0227i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) −598.525 598.525i −2.44296 2.44296i
\(246\) 0 0
\(247\) −349.675 −1.41569
\(248\) 0 0
\(249\) 0.937084i 0.00376339i
\(250\) 0 0
\(251\) −47.8205 + 47.8205i −0.190520 + 0.190520i −0.795921 0.605401i \(-0.793013\pi\)
0.605401 + 0.795921i \(0.293013\pi\)
\(252\) 0 0
\(253\) −236.403 + 236.403i −0.934398 + 0.934398i
\(254\) 0 0
\(255\) 58.9262i 0.231083i
\(256\) 0 0
\(257\) −198.672 −0.773043 −0.386521 0.922280i \(-0.626323\pi\)
−0.386521 + 0.922280i \(0.626323\pi\)
\(258\) 0 0
\(259\) 366.080 + 366.080i 1.41344 + 1.41344i
\(260\) 0 0
\(261\) 115.047 + 115.047i 0.440794 + 0.440794i
\(262\) 0 0
\(263\) −318.482 −1.21096 −0.605478 0.795862i \(-0.707018\pi\)
−0.605478 + 0.795862i \(0.707018\pi\)
\(264\) 0 0
\(265\) 457.524i 1.72651i
\(266\) 0 0
\(267\) −202.288 + 202.288i −0.757631 + 0.757631i
\(268\) 0 0
\(269\) −80.7988 + 80.7988i −0.300367 + 0.300367i −0.841158 0.540790i \(-0.818125\pi\)
0.540790 + 0.841158i \(0.318125\pi\)
\(270\) 0 0
\(271\) 238.383i 0.879643i −0.898085 0.439822i \(-0.855042\pi\)
0.898085 0.439822i \(-0.144958\pi\)
\(272\) 0 0
\(273\) −373.240 −1.36718
\(274\) 0 0
\(275\) −229.393 229.393i −0.834156 0.834156i
\(276\) 0 0
\(277\) −158.651 158.651i −0.572747 0.572747i 0.360148 0.932895i \(-0.382726\pi\)
−0.932895 + 0.360148i \(0.882726\pi\)
\(278\) 0 0
\(279\) −41.6246 −0.149192
\(280\) 0 0
\(281\) 446.105i 1.58756i 0.608205 + 0.793780i \(0.291890\pi\)
−0.608205 + 0.793780i \(0.708110\pi\)
\(282\) 0 0
\(283\) 191.252 191.252i 0.675801 0.675801i −0.283246 0.959047i \(-0.591411\pi\)
0.959047 + 0.283246i \(0.0914113\pi\)
\(284\) 0 0
\(285\) 189.058 189.058i 0.663362 0.663362i
\(286\) 0 0
\(287\) 50.7900i 0.176968i
\(288\) 0 0
\(289\) −268.231 −0.928134
\(290\) 0 0
\(291\) −78.0752 78.0752i −0.268300 0.268300i
\(292\) 0 0
\(293\) 256.523 + 256.523i 0.875503 + 0.875503i 0.993066 0.117562i \(-0.0375079\pi\)
−0.117562 + 0.993066i \(0.537508\pi\)
\(294\) 0 0
\(295\) −72.3094 −0.245117
\(296\) 0 0
\(297\) 54.8574i 0.184705i
\(298\) 0 0
\(299\) −378.663 + 378.663i −1.26643 + 1.26643i
\(300\) 0 0
\(301\) −137.264 + 137.264i −0.456027 + 0.456027i
\(302\) 0 0
\(303\) 112.728i 0.372040i
\(304\) 0 0
\(305\) −86.4091 −0.283308
\(306\) 0 0
\(307\) 58.8027 + 58.8027i 0.191540 + 0.191540i 0.796361 0.604821i \(-0.206756\pi\)
−0.604821 + 0.796361i \(0.706756\pi\)
\(308\) 0 0
\(309\) 92.9333 + 92.9333i 0.300755 + 0.300755i
\(310\) 0 0
\(311\) −453.650 −1.45868 −0.729341 0.684151i \(-0.760173\pi\)
−0.729341 + 0.684151i \(0.760173\pi\)
\(312\) 0 0
\(313\) 309.389i 0.988464i −0.869330 0.494232i \(-0.835449\pi\)
0.869330 0.494232i \(-0.164551\pi\)
\(314\) 0 0
\(315\) 201.799 201.799i 0.640632 0.640632i
\(316\) 0 0
\(317\) 141.334 141.334i 0.445848 0.445848i −0.448124 0.893972i \(-0.647908\pi\)
0.893972 + 0.448124i \(0.147908\pi\)
\(318\) 0 0
\(319\) 572.563i 1.79487i
\(320\) 0 0
\(321\) −91.8311 −0.286078
\(322\) 0 0
\(323\) −66.6355 66.6355i −0.206302 0.206302i
\(324\) 0 0
\(325\) −367.435 367.435i −1.13057 1.13057i
\(326\) 0 0
\(327\) −9.19869 −0.0281305
\(328\) 0 0
\(329\) 345.595i 1.05044i
\(330\) 0 0
\(331\) −119.102 + 119.102i −0.359826 + 0.359826i −0.863749 0.503923i \(-0.831890\pi\)
0.503923 + 0.863749i \(0.331890\pi\)
\(332\) 0 0
\(333\) −86.1833 + 86.1833i −0.258809 + 0.258809i
\(334\) 0 0
\(335\) 281.654i 0.840757i
\(336\) 0 0
\(337\) 270.363 0.802265 0.401133 0.916020i \(-0.368617\pi\)
0.401133 + 0.916020i \(0.368617\pi\)
\(338\) 0 0
\(339\) 33.3470 + 33.3470i 0.0983687 + 0.0983687i
\(340\) 0 0
\(341\) −103.578 103.578i −0.303747 0.303747i
\(342\) 0 0
\(343\) 820.473 2.39205
\(344\) 0 0
\(345\) 409.462i 1.18684i
\(346\) 0 0
\(347\) 362.402 362.402i 1.04439 1.04439i 0.0454174 0.998968i \(-0.485538\pi\)
0.998968 0.0454174i \(-0.0144618\pi\)
\(348\) 0 0
\(349\) 14.3277 14.3277i 0.0410537 0.0410537i −0.686282 0.727336i \(-0.740758\pi\)
0.727336 + 0.686282i \(0.240758\pi\)
\(350\) 0 0
\(351\) 87.8691i 0.250339i
\(352\) 0 0
\(353\) 370.167 1.04863 0.524316 0.851524i \(-0.324321\pi\)
0.524316 + 0.851524i \(0.324321\pi\)
\(354\) 0 0
\(355\) −88.3500 88.3500i −0.248873 0.248873i
\(356\) 0 0
\(357\) −71.1261 71.1261i −0.199233 0.199233i
\(358\) 0 0
\(359\) −129.009 −0.359356 −0.179678 0.983725i \(-0.557506\pi\)
−0.179678 + 0.983725i \(0.557506\pi\)
\(360\) 0 0
\(361\) 66.5846i 0.184445i
\(362\) 0 0
\(363\) −11.6878 + 11.6878i −0.0321977 + 0.0321977i
\(364\) 0 0
\(365\) −264.594 + 264.594i −0.724914 + 0.724914i
\(366\) 0 0
\(367\) 608.337i 1.65759i 0.559550 + 0.828797i \(0.310974\pi\)
−0.559550 + 0.828797i \(0.689026\pi\)
\(368\) 0 0
\(369\) −11.9571 −0.0324040
\(370\) 0 0
\(371\) 552.249 + 552.249i 1.48854 + 1.48854i
\(372\) 0 0
\(373\) 193.032 + 193.032i 0.517513 + 0.517513i 0.916818 0.399305i \(-0.130748\pi\)
−0.399305 + 0.916818i \(0.630748\pi\)
\(374\) 0 0
\(375\) 74.0696 0.197519
\(376\) 0 0
\(377\) 917.116i 2.43267i
\(378\) 0 0
\(379\) 192.727 192.727i 0.508515 0.508515i −0.405555 0.914071i \(-0.632922\pi\)
0.914071 + 0.405555i \(0.132922\pi\)
\(380\) 0 0
\(381\) 188.979 188.979i 0.496007 0.496007i
\(382\) 0 0
\(383\) 144.237i 0.376598i −0.982112 0.188299i \(-0.939703\pi\)
0.982112 0.188299i \(-0.0602974\pi\)
\(384\) 0 0
\(385\) 1004.31 2.60859
\(386\) 0 0
\(387\) −32.3150 32.3150i −0.0835014 0.0835014i
\(388\) 0 0
\(389\) 186.426 + 186.426i 0.479244 + 0.479244i 0.904890 0.425646i \(-0.139953\pi\)
−0.425646 + 0.904890i \(0.639953\pi\)
\(390\) 0 0
\(391\) −144.319 −0.369102
\(392\) 0 0
\(393\) 58.6161i 0.149150i
\(394\) 0 0
\(395\) 396.965 396.965i 1.00497 1.00497i
\(396\) 0 0
\(397\) −299.949 + 299.949i −0.755540 + 0.755540i −0.975507 0.219968i \(-0.929405\pi\)
0.219968 + 0.975507i \(0.429405\pi\)
\(398\) 0 0
\(399\) 456.400i 1.14386i
\(400\) 0 0
\(401\) 532.680 1.32838 0.664189 0.747565i \(-0.268777\pi\)
0.664189 + 0.747565i \(0.268777\pi\)
\(402\) 0 0
\(403\) −165.908 165.908i −0.411682 0.411682i
\(404\) 0 0
\(405\) 47.5079 + 47.5079i 0.117304 + 0.117304i
\(406\) 0 0
\(407\) −428.914 −1.05384
\(408\) 0 0
\(409\) 373.085i 0.912187i 0.889932 + 0.456094i \(0.150752\pi\)
−0.889932 + 0.456094i \(0.849248\pi\)
\(410\) 0 0
\(411\) −293.498 + 293.498i −0.714106 + 0.714106i
\(412\) 0 0
\(413\) 87.2801 87.2801i 0.211332 0.211332i
\(414\) 0 0
\(415\) 4.03884i 0.00973214i
\(416\) 0 0
\(417\) 313.086 0.750805
\(418\) 0 0
\(419\) 91.9363 + 91.9363i 0.219418 + 0.219418i 0.808253 0.588835i \(-0.200413\pi\)
−0.588835 + 0.808253i \(0.700413\pi\)
\(420\) 0 0
\(421\) 110.915 + 110.915i 0.263456 + 0.263456i 0.826457 0.563001i \(-0.190353\pi\)
−0.563001 + 0.826457i \(0.690353\pi\)
\(422\) 0 0
\(423\) 81.3608 0.192342
\(424\) 0 0
\(425\) 140.040i 0.329505i
\(426\) 0 0
\(427\) 104.299 104.299i 0.244260 0.244260i
\(428\) 0 0
\(429\) 218.652 218.652i 0.509678 0.509678i
\(430\) 0 0
\(431\) 19.9392i 0.0462626i −0.999732 0.0231313i \(-0.992636\pi\)
0.999732 0.0231313i \(-0.00736358\pi\)
\(432\) 0 0
\(433\) −707.556 −1.63408 −0.817039 0.576582i \(-0.804386\pi\)
−0.817039 + 0.576582i \(0.804386\pi\)
\(434\) 0 0
\(435\) −495.854 495.854i −1.13990 1.13990i
\(436\) 0 0
\(437\) 463.031 + 463.031i 1.05957 + 1.05957i
\(438\) 0 0
\(439\) −25.5403 −0.0581784 −0.0290892 0.999577i \(-0.509261\pi\)
−0.0290892 + 0.999577i \(0.509261\pi\)
\(440\) 0 0
\(441\) 340.157i 0.771332i
\(442\) 0 0
\(443\) −468.137 + 468.137i −1.05674 + 1.05674i −0.0584533 + 0.998290i \(0.518617\pi\)
−0.998290 + 0.0584533i \(0.981383\pi\)
\(444\) 0 0
\(445\) 871.861 871.861i 1.95924 1.95924i
\(446\) 0 0
\(447\) 289.283i 0.647166i
\(448\) 0 0
\(449\) −724.735 −1.61411 −0.807055 0.590476i \(-0.798940\pi\)
−0.807055 + 0.590476i \(0.798940\pi\)
\(450\) 0 0
\(451\) −29.7538 29.7538i −0.0659730 0.0659730i
\(452\) 0 0
\(453\) −282.198 282.198i −0.622953 0.622953i
\(454\) 0 0
\(455\) 1608.67 3.53554
\(456\) 0 0
\(457\) 100.661i 0.220266i −0.993917 0.110133i \(-0.964872\pi\)
0.993917 0.110133i \(-0.0351276\pi\)
\(458\) 0 0
\(459\) 16.7447 16.7447i 0.0364807 0.0364807i
\(460\) 0 0
\(461\) 376.240 376.240i 0.816139 0.816139i −0.169407 0.985546i \(-0.554185\pi\)
0.985546 + 0.169407i \(0.0541852\pi\)
\(462\) 0 0
\(463\) 845.968i 1.82714i −0.406677 0.913572i \(-0.633312\pi\)
0.406677 0.913572i \(-0.366688\pi\)
\(464\) 0 0
\(465\) 179.402 0.385811
\(466\) 0 0
\(467\) −234.037 234.037i −0.501150 0.501150i 0.410646 0.911795i \(-0.365303\pi\)
−0.911795 + 0.410646i \(0.865303\pi\)
\(468\) 0 0
\(469\) 339.966 + 339.966i 0.724874 + 0.724874i
\(470\) 0 0
\(471\) 389.604 0.827184
\(472\) 0 0
\(473\) 160.824i 0.340009i
\(474\) 0 0
\(475\) −449.301 + 449.301i −0.945897 + 0.945897i
\(476\) 0 0
\(477\) −130.012 + 130.012i −0.272561 + 0.272561i
\(478\) 0 0
\(479\) 161.916i 0.338029i 0.985614 + 0.169015i \(0.0540585\pi\)
−0.985614 + 0.169015i \(0.945941\pi\)
\(480\) 0 0
\(481\) −687.023 −1.42832
\(482\) 0 0
\(483\) 494.235 + 494.235i 1.02326 + 1.02326i
\(484\) 0 0
\(485\) 336.505 + 336.505i 0.693824 + 0.693824i
\(486\) 0 0
\(487\) 137.720 0.282793 0.141396 0.989953i \(-0.454841\pi\)
0.141396 + 0.989953i \(0.454841\pi\)
\(488\) 0 0
\(489\) 203.944i 0.417063i
\(490\) 0 0
\(491\) −43.5877 + 43.5877i −0.0887734 + 0.0887734i −0.750099 0.661326i \(-0.769994\pi\)
0.661326 + 0.750099i \(0.269994\pi\)
\(492\) 0 0
\(493\) −174.769 + 174.769i −0.354501 + 0.354501i
\(494\) 0 0
\(495\) 236.436i 0.477648i
\(496\) 0 0
\(497\) 213.283 0.429141
\(498\) 0 0
\(499\) −529.133 529.133i −1.06039 1.06039i −0.998056 0.0623308i \(-0.980147\pi\)
−0.0623308 0.998056i \(-0.519853\pi\)
\(500\) 0 0
\(501\) 18.5511 + 18.5511i 0.0370282 + 0.0370282i
\(502\) 0 0
\(503\) −391.263 −0.777859 −0.388929 0.921268i \(-0.627155\pi\)
−0.388929 + 0.921268i \(0.627155\pi\)
\(504\) 0 0
\(505\) 485.859i 0.962097i
\(506\) 0 0
\(507\) 143.249 143.249i 0.282541 0.282541i
\(508\) 0 0
\(509\) −605.614 + 605.614i −1.18981 + 1.18981i −0.212692 + 0.977119i \(0.568223\pi\)
−0.977119 + 0.212692i \(0.931777\pi\)
\(510\) 0 0
\(511\) 638.748i 1.25000i
\(512\) 0 0
\(513\) −107.447 −0.209448
\(514\) 0 0
\(515\) −400.543 400.543i −0.777754 0.777754i
\(516\) 0 0
\(517\) 202.457 + 202.457i 0.391599 + 0.391599i
\(518\) 0 0
\(519\) −89.3498 −0.172158
\(520\) 0 0
\(521\) 288.250i 0.553262i −0.960976 0.276631i \(-0.910782\pi\)
0.960976 0.276631i \(-0.0892180\pi\)
\(522\) 0 0
\(523\) −641.538 + 641.538i −1.22665 + 1.22665i −0.261427 + 0.965223i \(0.584193\pi\)
−0.965223 + 0.261427i \(0.915807\pi\)
\(524\) 0 0
\(525\) −479.580 + 479.580i −0.913485 + 0.913485i
\(526\) 0 0
\(527\) 63.2321i 0.119985i
\(528\) 0 0
\(529\) 473.831 0.895710
\(530\) 0 0
\(531\) 20.5477 + 20.5477i 0.0386962 + 0.0386962i
\(532\) 0 0
\(533\) −47.6588 47.6588i −0.0894161 0.0894161i
\(534\) 0 0
\(535\) 395.793 0.739799
\(536\) 0 0
\(537\) 40.5365i 0.0754869i
\(538\) 0 0
\(539\) −846.442 + 846.442i −1.57039 + 1.57039i
\(540\) 0 0
\(541\) 539.789 539.789i 0.997762 0.997762i −0.00223575 0.999998i \(-0.500712\pi\)
0.999998 + 0.00223575i \(0.000711663\pi\)
\(542\) 0 0
\(543\) 333.475i 0.614134i
\(544\) 0 0
\(545\) 39.6464 0.0727457
\(546\) 0 0
\(547\) −404.757 404.757i −0.739958 0.739958i 0.232612 0.972570i \(-0.425273\pi\)
−0.972570 + 0.232612i \(0.925273\pi\)
\(548\) 0 0
\(549\) 24.5543 + 24.5543i 0.0447254 + 0.0447254i
\(550\) 0 0
\(551\) 1121.45 2.03530
\(552\) 0 0
\(553\) 958.302i 1.73292i
\(554\) 0 0
\(555\) 371.451 371.451i 0.669280 0.669280i
\(556\) 0 0
\(557\) 127.187 127.187i 0.228343 0.228343i −0.583657 0.812000i \(-0.698379\pi\)
0.812000 + 0.583657i \(0.198379\pi\)
\(558\) 0 0
\(559\) 257.604i 0.460830i
\(560\) 0 0
\(561\) 83.3343 0.148546
\(562\) 0 0
\(563\) 292.853 + 292.853i 0.520164 + 0.520164i 0.917621 0.397457i \(-0.130107\pi\)
−0.397457 + 0.917621i \(0.630107\pi\)
\(564\) 0 0
\(565\) −143.726 143.726i −0.254382 0.254382i
\(566\) 0 0
\(567\) −114.688 −0.202271
\(568\) 0 0
\(569\) 414.222i 0.727982i 0.931403 + 0.363991i \(0.118586\pi\)
−0.931403 + 0.363991i \(0.881414\pi\)
\(570\) 0 0
\(571\) 203.517 203.517i 0.356422 0.356422i −0.506070 0.862492i \(-0.668902\pi\)
0.862492 + 0.506070i \(0.168902\pi\)
\(572\) 0 0
\(573\) −240.463 + 240.463i −0.419657 + 0.419657i
\(574\) 0 0
\(575\) 973.095i 1.69234i
\(576\) 0 0
\(577\) 474.909 0.823065 0.411533 0.911395i \(-0.364994\pi\)
0.411533 + 0.911395i \(0.364994\pi\)
\(578\) 0 0
\(579\) 204.133 + 204.133i 0.352562 + 0.352562i
\(580\) 0 0
\(581\) −4.87503 4.87503i −0.00839075 0.00839075i
\(582\) 0 0
\(583\) −647.037 −1.10984
\(584\) 0 0
\(585\) 378.716i 0.647378i
\(586\) 0 0
\(587\) 9.33802 9.33802i 0.0159080 0.0159080i −0.699108 0.715016i \(-0.746419\pi\)
0.715016 + 0.699108i \(0.246419\pi\)
\(588\) 0 0
\(589\) −202.873 + 202.873i −0.344436 + 0.344436i
\(590\) 0 0
\(591\) 572.945i 0.969451i
\(592\) 0 0
\(593\) 332.523 0.560748 0.280374 0.959891i \(-0.409542\pi\)
0.280374 + 0.959891i \(0.409542\pi\)
\(594\) 0 0
\(595\) 306.554 + 306.554i 0.515217 + 0.515217i
\(596\) 0 0
\(597\) 1.98461 + 1.98461i 0.00332431 + 0.00332431i
\(598\) 0 0
\(599\) 1044.75 1.74415 0.872076 0.489371i \(-0.162773\pi\)
0.872076 + 0.489371i \(0.162773\pi\)
\(600\) 0 0
\(601\) 635.025i 1.05661i −0.849053 0.528307i \(-0.822827\pi\)
0.849053 0.528307i \(-0.177173\pi\)
\(602\) 0 0
\(603\) −80.0355 + 80.0355i −0.132729 + 0.132729i
\(604\) 0 0
\(605\) 50.3743 50.3743i 0.0832633 0.0832633i
\(606\) 0 0
\(607\) 813.850i 1.34078i −0.742011 0.670388i \(-0.766128\pi\)
0.742011 0.670388i \(-0.233872\pi\)
\(608\) 0 0
\(609\) 1197.03 1.96556
\(610\) 0 0
\(611\) 324.289 + 324.289i 0.530752 + 0.530752i
\(612\) 0 0
\(613\) 429.836 + 429.836i 0.701201 + 0.701201i 0.964668 0.263468i \(-0.0848662\pi\)
−0.263468 + 0.964668i \(0.584866\pi\)
\(614\) 0 0
\(615\) 51.5351 0.0837969
\(616\) 0 0
\(617\) 333.971i 0.541282i −0.962680 0.270641i \(-0.912764\pi\)
0.962680 0.270641i \(-0.0872357\pi\)
\(618\) 0 0
\(619\) 114.066 114.066i 0.184275 0.184275i −0.608941 0.793216i \(-0.708405\pi\)
0.793216 + 0.608941i \(0.208405\pi\)
\(620\) 0 0
\(621\) −116.354 + 116.354i −0.187365 + 0.187365i
\(622\) 0 0
\(623\) 2104.74i 3.37839i
\(624\) 0 0
\(625\) 448.972 0.718355
\(626\) 0 0
\(627\) −267.369 267.369i −0.426425 0.426425i
\(628\) 0 0
\(629\) −130.922 130.922i −0.208142 0.208142i
\(630\) 0 0
\(631\) 160.396 0.254193 0.127096 0.991890i \(-0.459434\pi\)
0.127096 + 0.991890i \(0.459434\pi\)
\(632\) 0 0
\(633\) 477.076i 0.753675i
\(634\) 0 0
\(635\) −814.500 + 814.500i −1.28268 + 1.28268i
\(636\) 0 0
\(637\) −1355.81 + 1355.81i −2.12843 + 2.12843i
\(638\) 0 0
\(639\) 50.2116i 0.0785784i
\(640\) 0 0
\(641\) 309.175 0.482333 0.241166 0.970484i \(-0.422470\pi\)
0.241166 + 0.970484i \(0.422470\pi\)
\(642\) 0 0
\(643\) −392.517 392.517i −0.610446 0.610446i 0.332616 0.943062i \(-0.392069\pi\)
−0.943062 + 0.332616i \(0.892069\pi\)
\(644\) 0 0
\(645\) 139.278 + 139.278i 0.215935 + 0.215935i
\(646\) 0 0
\(647\) −199.770 −0.308763 −0.154381 0.988011i \(-0.549338\pi\)
−0.154381 + 0.988011i \(0.549338\pi\)
\(648\) 0 0
\(649\) 102.261i 0.157567i
\(650\) 0 0
\(651\) −216.545 + 216.545i −0.332634 + 0.332634i
\(652\) 0 0
\(653\) 632.149 632.149i 0.968068 0.968068i −0.0314373 0.999506i \(-0.510008\pi\)
0.999506 + 0.0314373i \(0.0100084\pi\)
\(654\) 0 0
\(655\) 252.636i 0.385704i
\(656\) 0 0
\(657\) 150.375 0.228882
\(658\) 0 0
\(659\) 62.1301 + 62.1301i 0.0942793 + 0.0942793i 0.752673 0.658394i \(-0.228764\pi\)
−0.658394 + 0.752673i \(0.728764\pi\)
\(660\) 0 0
\(661\) 48.1550 + 48.1550i 0.0728517 + 0.0728517i 0.742594 0.669742i \(-0.233595\pi\)
−0.669742 + 0.742594i \(0.733595\pi\)
\(662\) 0 0
\(663\) 133.482 0.201331
\(664\) 0 0
\(665\) 1967.09i 2.95803i
\(666\) 0 0
\(667\) 1214.42 1214.42i 1.82072 1.82072i
\(668\) 0 0
\(669\) 146.807 146.807i 0.219442 0.219442i
\(670\) 0 0
\(671\) 122.201i 0.182118i
\(672\) 0 0
\(673\) 88.3501 0.131278 0.0656390 0.997843i \(-0.479091\pi\)
0.0656390 + 0.997843i \(0.479091\pi\)
\(674\) 0 0
\(675\) −112.904 112.904i −0.167265 0.167265i
\(676\) 0 0
\(677\) 405.589 + 405.589i 0.599097 + 0.599097i 0.940072 0.340975i \(-0.110757\pi\)
−0.340975 + 0.940072i \(0.610757\pi\)
\(678\) 0 0
\(679\) −812.347 −1.19639
\(680\) 0 0
\(681\) 633.572i 0.930356i
\(682\) 0 0
\(683\) 572.903 572.903i 0.838803 0.838803i −0.149898 0.988701i \(-0.547895\pi\)
0.988701 + 0.149898i \(0.0478946\pi\)
\(684\) 0 0
\(685\) 1264.98 1264.98i 1.84668 1.84668i
\(686\) 0 0
\(687\) 350.535i 0.510241i
\(688\) 0 0
\(689\) −1036.41 −1.50422
\(690\) 0 0
\(691\) −210.640 210.640i −0.304833 0.304833i 0.538068 0.842901i \(-0.319154\pi\)
−0.842901 + 0.538068i \(0.819154\pi\)
\(692\) 0 0
\(693\) −285.387 285.387i −0.411814 0.411814i
\(694\) 0 0
\(695\) −1349.40 −1.94159
\(696\) 0 0
\(697\) 18.1641i 0.0260604i
\(698\) 0 0
\(699\) 264.126 264.126i 0.377862 0.377862i
\(700\) 0 0
\(701\) 419.735 419.735i 0.598766 0.598766i −0.341218 0.939984i \(-0.610840\pi\)
0.939984 + 0.341218i \(0.110840\pi\)
\(702\) 0 0
\(703\) 840.094i 1.19501i
\(704\) 0 0
\(705\) −350.665 −0.497398
\(706\) 0 0
\(707\) −586.449 586.449i −0.829490 0.829490i
\(708\) 0 0
\(709\) 228.651 + 228.651i 0.322497 + 0.322497i 0.849724 0.527227i \(-0.176768\pi\)
−0.527227 + 0.849724i \(0.676768\pi\)
\(710\) 0 0
\(711\) −225.605 −0.317307
\(712\) 0 0
\(713\) 439.382i 0.616244i
\(714\) 0 0
\(715\) −942.392 + 942.392i −1.31803 + 1.31803i
\(716\) 0 0
\(717\) 572.821 572.821i 0.798914 0.798914i
\(718\) 0 0
\(719\) 1144.39i 1.59165i −0.605528 0.795824i \(-0.707038\pi\)
0.605528 0.795824i \(-0.292962\pi\)
\(720\) 0 0
\(721\) 966.941 1.34111
\(722\) 0 0
\(723\) −298.671 298.671i −0.413099 0.413099i
\(724\) 0 0
\(725\) 1178.41 + 1178.41i 1.62539 + 1.62539i
\(726\) 0 0
\(727\) 646.638 0.889461 0.444731 0.895664i \(-0.353299\pi\)
0.444731 + 0.895664i \(0.353299\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 49.0900 49.0900i 0.0671545 0.0671545i
\(732\) 0 0
\(733\) −485.081 + 485.081i −0.661775 + 0.661775i −0.955798 0.294023i \(-0.905006\pi\)
0.294023 + 0.955798i \(0.405006\pi\)
\(734\) 0 0
\(735\) 1466.08i 1.99467i
\(736\) 0 0
\(737\) −398.318 −0.540459
\(738\) 0 0
\(739\) 7.07050 + 7.07050i 0.00956766 + 0.00956766i 0.711874 0.702307i \(-0.247847\pi\)
−0.702307 + 0.711874i \(0.747847\pi\)
\(740\) 0 0
\(741\) −428.263 428.263i −0.577953 0.577953i
\(742\) 0 0
\(743\) −1272.93 −1.71322 −0.856612 0.515961i \(-0.827435\pi\)
−0.856612 + 0.515961i \(0.827435\pi\)
\(744\) 0 0
\(745\) 1246.81i 1.67357i
\(746\) 0 0
\(747\) 1.14769 1.14769i 0.00153640 0.00153640i
\(748\) 0 0
\(749\) −477.736 + 477.736i −0.637832 + 0.637832i
\(750\) 0 0
\(751\) 176.056i 0.234428i −0.993107 0.117214i \(-0.962604\pi\)
0.993107 0.117214i \(-0.0373964\pi\)
\(752\) 0 0
\(753\) −117.136 −0.155559
\(754\) 0 0
\(755\) 1216.27 + 1216.27i 1.61096 + 1.61096i
\(756\) 0 0
\(757\) −840.383 840.383i −1.11015 1.11015i −0.993130 0.117019i \(-0.962666\pi\)
−0.117019 0.993130i \(-0.537334\pi\)
\(758\) 0 0
\(759\) −579.066 −0.762933
\(760\) 0 0
\(761\) 525.952i 0.691132i −0.938394 0.345566i \(-0.887687\pi\)
0.938394 0.345566i \(-0.112313\pi\)
\(762\) 0 0
\(763\) −47.8547 + 47.8547i −0.0627191 + 0.0627191i
\(764\) 0 0
\(765\) −72.1696 + 72.1696i −0.0943394 + 0.0943394i
\(766\) 0 0
\(767\) 163.799i 0.213558i
\(768\) 0 0
\(769\) −805.196 −1.04707 −0.523534 0.852005i \(-0.675387\pi\)
−0.523534 + 0.852005i \(0.675387\pi\)
\(770\) 0 0
\(771\) −243.322 243.322i −0.315593 0.315593i
\(772\) 0 0
\(773\) 91.8885 + 91.8885i 0.118873 + 0.118873i 0.764041 0.645168i \(-0.223213\pi\)
−0.645168 + 0.764041i \(0.723213\pi\)
\(774\) 0 0
\(775\) −426.353 −0.550133
\(776\) 0 0
\(777\) 896.709i 1.15407i
\(778\) 0 0
\(779\) −58.2774 + 58.2774i −0.0748105 + 0.0748105i
\(780\) 0 0
\(781\) −124.946 + 124.946i −0.159982 + 0.159982i
\(782\) 0 0
\(783\) 281.807i 0.359907i
\(784\) 0 0
\(785\) −1679.19 −2.13910
\(786\) 0 0
\(787\) 250.015 + 250.015i 0.317681 + 0.317681i 0.847876 0.530195i \(-0.177881\pi\)
−0.530195 + 0.847876i \(0.677881\pi\)
\(788\) 0 0
\(789\) −390.059 390.059i −0.494371 0.494371i
\(790\) 0 0
\(791\) 346.965 0.438640
\(792\) 0 0
\(793\) 195.738i 0.246832i
\(794\) 0 0
\(795\) 560.351 560.351i 0.704844 0.704844i
\(796\) 0 0
\(797\) −428.762 + 428.762i −0.537969 + 0.537969i −0.922932 0.384963i \(-0.874214\pi\)
0.384963 + 0.922932i \(0.374214\pi\)
\(798\) 0 0
\(799\) 123.596i 0.154688i
\(800\) 0 0
\(801\) −495.501 −0.618603
\(802\) 0 0
\(803\) 374.192 + 374.192i 0.465992 + 0.465992i
\(804\) 0 0
\(805\) −2130.16 2130.16i −2.64616 2.64616i
\(806\) 0 0
\(807\) −197.916 −0.245249
\(808\) 0 0
\(809\) 960.576i 1.18736i 0.804700 + 0.593681i \(0.202326\pi\)
−0.804700 + 0.593681i \(0.797674\pi\)
\(810\) 0 0
\(811\) 1095.66 1095.66i 1.35100 1.35100i 0.466451 0.884547i \(-0.345532\pi\)
0.884547 0.466451i \(-0.154468\pi\)
\(812\) 0 0
\(813\) 291.959 291.959i 0.359113 0.359113i
\(814\) 0 0
\(815\) 878.999i 1.07853i
\(816\) 0 0
\(817\) −314.999 −0.385556
\(818\) 0 0
\(819\) −457.124 457.124i −0.558149 0.558149i
\(820\) 0 0
\(821\) 987.644 + 987.644i 1.20298 + 1.20298i 0.973257 + 0.229721i \(0.0737813\pi\)
0.229721 + 0.973257i \(0.426219\pi\)
\(822\) 0 0
\(823\) 1265.91 1.53817 0.769083 0.639149i \(-0.220713\pi\)
0.769083 + 0.639149i \(0.220713\pi\)
\(824\) 0 0
\(825\) 561.896i 0.681085i
\(826\) 0 0
\(827\) −246.989 + 246.989i −0.298657 + 0.298657i −0.840488 0.541831i \(-0.817731\pi\)
0.541831 + 0.840488i \(0.317731\pi\)
\(828\) 0 0
\(829\) 70.0908 70.0908i 0.0845486 0.0845486i −0.663568 0.748116i \(-0.730959\pi\)
0.748116 + 0.663568i \(0.230959\pi\)
\(830\) 0 0
\(831\) 388.614i 0.467646i
\(832\) 0 0
\(833\) −516.735 −0.620330
\(834\) 0 0
\(835\) −79.9554 79.9554i −0.0957550 0.0957550i
\(836\) 0 0
\(837\) −50.9795 50.9795i −0.0609074 0.0609074i
\(838\) 0 0
\(839\) 850.570 1.01379 0.506895 0.862008i \(-0.330793\pi\)
0.506895 + 0.862008i \(0.330793\pi\)
\(840\) 0 0
\(841\) 2100.30i 2.49739i
\(842\) 0 0
\(843\) −546.364 + 546.364i −0.648119 + 0.648119i
\(844\) 0 0
\(845\) −617.402 + 617.402i −0.730653 + 0.730653i
\(846\) 0 0
\(847\) 121.607i 0.143574i
\(848\) 0 0
\(849\) 468.469 0.551790
\(850\) 0 0
\(851\) 909.737 + 909.737i 1.06902 + 1.06902i
\(852\) 0 0
\(853\) −958.146 958.146i −1.12327 1.12327i −0.991247 0.132019i \(-0.957854\pi\)
−0.132019 0.991247i \(-0.542146\pi\)
\(854\) 0 0
\(855\) 463.096 0.541633
\(856\) 0 0
\(857\) 574.307i 0.670137i −0.942194 0.335068i \(-0.891241\pi\)
0.942194 0.335068i \(-0.108759\pi\)
\(858\) 0 0
\(859\) 167.633 167.633i 0.195149 0.195149i −0.602768 0.797917i \(-0.705935\pi\)
0.797917 + 0.602768i \(0.205935\pi\)
\(860\) 0 0
\(861\) −62.2047 + 62.2047i −0.0722471 + 0.0722471i
\(862\) 0 0
\(863\) 1451.78i 1.68225i 0.540843 + 0.841124i \(0.318105\pi\)
−0.540843 + 0.841124i \(0.681895\pi\)
\(864\) 0 0
\(865\) 385.098 0.445200
\(866\) 0 0
\(867\) −328.514 328.514i −0.378909 0.378909i
\(868\) 0 0
\(869\) −561.393 561.393i −0.646022 0.646022i
\(870\) 0 0
\(871\) −638.015 −0.732508
\(872\) 0 0
\(873\) 191.244i 0.219066i
\(874\) 0 0
\(875\) 385.335 385.335i 0.440383 0.440383i
\(876\) 0 0
\(877\) −696.542 + 696.542i −0.794233 + 0.794233i −0.982179 0.187946i \(-0.939817\pi\)
0.187946 + 0.982179i \(0.439817\pi\)
\(878\) 0 0
\(879\) 628.349i 0.714846i
\(880\) 0 0
\(881\) −483.595 −0.548916 −0.274458 0.961599i \(-0.588498\pi\)
−0.274458 + 0.961599i \(0.588498\pi\)
\(882\) 0 0
\(883\) 1013.54 + 1013.54i 1.14784 + 1.14784i 0.986977 + 0.160863i \(0.0514277\pi\)
0.160863 + 0.986977i \(0.448572\pi\)
\(884\) 0 0
\(885\) −88.5606 88.5606i −0.100068 0.100068i
\(886\) 0 0
\(887\) −1132.45 −1.27672 −0.638359 0.769739i \(-0.720386\pi\)
−0.638359 + 0.769739i \(0.720386\pi\)
\(888\) 0 0
\(889\) 1966.26i 2.21177i
\(890\) 0 0
\(891\) 67.1864 67.1864i 0.0754056 0.0754056i
\(892\) 0 0
\(893\) 396.543 396.543i 0.444057 0.444057i
\(894\) 0 0
\(895\) 174.712i 0.195209i
\(896\) 0 0
\(897\) −927.531 −1.03404
\(898\) 0 0
\(899\) 532.088 + 532.088i 0.591866 + 0.591866i
\(900\) 0 0
\(901\) −197.501 197.501i −0.219202 0.219202i
\(902\) 0 0
\(903\) −336.227 −0.372345
\(904\) 0 0
\(905\) 1437.28i 1.58815i
\(906\) 0 0
\(907\) −213.492 + 213.492i −0.235383 + 0.235383i −0.814935 0.579552i \(-0.803228\pi\)
0.579552 + 0.814935i \(0.303228\pi\)
\(908\) 0 0
\(909\) 138.063 138.063i 0.151885 0.151885i
\(910\) 0 0
\(911\) 1383.69i 1.51887i −0.650584 0.759435i \(-0.725476\pi\)
0.650584 0.759435i \(-0.274524\pi\)
\(912\) 0 0
\(913\) 5.71178 0.00625606
\(914\) 0 0
\(915\) −105.829 105.829i −0.115660 0.115660i
\(916\) 0 0
\(917\) −304.941 304.941i −0.332542 0.332542i
\(918\) 0 0
\(919\) 841.973 0.916184 0.458092 0.888905i \(-0.348533\pi\)
0.458092 + 0.888905i \(0.348533\pi\)
\(920\) 0 0
\(921\) 144.037i 0.156391i
\(922\) 0 0
\(923\) −200.134 + 200.134i −0.216830 + 0.216830i
\(924\) 0 0
\(925\) −882.761 + 882.761i −0.954336 + 0.954336i
\(926\) 0 0
\(927\) 227.639i 0.245566i
\(928\) 0 0
\(929\) −1467.41 −1.57956 −0.789780 0.613390i \(-0.789805\pi\)
−0.789780 + 0.613390i \(0.789805\pi\)
\(930\) 0 0
\(931\) 1657.89 + 1657.89i 1.78076 + 1.78076i
\(932\) 0 0
\(933\) −555.605 555.605i −0.595504 0.595504i
\(934\) 0 0
\(935\) −359.171 −0.384141
\(936\) 0 0
\(937\) 868.448i 0.926839i 0.886139 + 0.463419i \(0.153378\pi\)
−0.886139 + 0.463419i \(0.846622\pi\)
\(938\) 0 0
\(939\) 378.923 378.923i 0.403539 0.403539i
\(940\) 0 0
\(941\) 877.521 877.521i 0.932541 0.932541i −0.0653233 0.997864i \(-0.520808\pi\)
0.997864 + 0.0653233i \(0.0208079\pi\)
\(942\) 0 0
\(943\) 126.217i 0.133846i
\(944\) 0 0
\(945\) 494.305 0.523074
\(946\) 0 0
\(947\) 588.434 + 588.434i 0.621366 + 0.621366i 0.945881 0.324514i \(-0.105201\pi\)
−0.324514 + 0.945881i \(0.605201\pi\)
\(948\) 0 0
\(949\) 599.370 + 599.370i 0.631580 + 0.631580i
\(950\) 0 0
\(951\) 346.196 0.364033
\(952\) 0 0
\(953\) 622.721i 0.653432i −0.945122 0.326716i \(-0.894058\pi\)
0.945122 0.326716i \(-0.105942\pi\)
\(954\) 0 0
\(955\) 1036.40 1036.40i 1.08523 1.08523i
\(956\) 0 0
\(957\) −701.244 + 701.244i −0.732753 + 0.732753i
\(958\) 0 0
\(959\) 3053.75i 3.18430i
\(960\) 0 0
\(961\) 768.489 0.799676
\(962\) 0 0
\(963\) −112.470 112.470i −0.116791 0.116791i
\(964\) 0 0
\(965\) −879.816 879.816i −0.911727 0.911727i
\(966\) 0 0
\(967\) 1405.67 1.45364 0.726820 0.686828i \(-0.240998\pi\)
0.726820 + 0.686828i \(0.240998\pi\)
\(968\) 0 0
\(969\) 163.223i 0.168445i
\(970\) 0 0
\(971\) 400.839 400.839i 0.412810 0.412810i −0.469906 0.882716i \(-0.655712\pi\)
0.882716 + 0.469906i \(0.155712\pi\)
\(972\) 0 0
\(973\) 1628.78 1628.78i 1.67397 1.67397i
\(974\) 0 0
\(975\) 900.028i 0.923106i
\(976\) 0 0
\(977\) 63.0363 0.0645203 0.0322601 0.999480i \(-0.489729\pi\)
0.0322601 + 0.999480i \(0.489729\pi\)
\(978\) 0 0
\(979\) −1233.00 1233.00i −1.25945 1.25945i
\(980\) 0 0
\(981\) −11.2660 11.2660i −0.0114842 0.0114842i
\(982\) 0 0
\(983\) −1179.73 −1.20013 −0.600064 0.799952i \(-0.704858\pi\)
−0.600064 + 0.799952i \(0.704858\pi\)
\(984\) 0 0
\(985\) 2469.40i 2.50700i
\(986\) 0 0
\(987\) 423.266 423.266i 0.428841 0.428841i
\(988\) 0 0
\(989\) −341.112 + 341.112i −0.344906 + 0.344906i
\(990\) 0 0
\(991\) 1827.60i 1.84420i −0.386952 0.922100i \(-0.626472\pi\)
0.386952 0.922100i \(-0.373528\pi\)
\(992\) 0 0
\(993\) −291.740 −0.293797
\(994\) 0 0
\(995\) −8.55369 8.55369i −0.00859668 0.00859668i
\(996\) 0 0
\(997\) 53.3327 + 53.3327i 0.0534932 + 0.0534932i 0.733347 0.679854i \(-0.237957\pi\)
−0.679854 + 0.733347i \(0.737957\pi\)
\(998\) 0 0
\(999\) −211.105 −0.211316
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 768.3.l.f.319.5 yes 16
4.3 odd 2 inner 768.3.l.f.319.1 yes 16
8.3 odd 2 768.3.l.e.319.8 yes 16
8.5 even 2 768.3.l.e.319.4 16
16.3 odd 4 inner 768.3.l.f.703.5 yes 16
16.5 even 4 768.3.l.e.703.8 yes 16
16.11 odd 4 768.3.l.e.703.4 yes 16
16.13 even 4 inner 768.3.l.f.703.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.3.l.e.319.4 16 8.5 even 2
768.3.l.e.319.8 yes 16 8.3 odd 2
768.3.l.e.703.4 yes 16 16.11 odd 4
768.3.l.e.703.8 yes 16 16.5 even 4
768.3.l.f.319.1 yes 16 4.3 odd 2 inner
768.3.l.f.319.5 yes 16 1.1 even 1 trivial
768.3.l.f.703.1 yes 16 16.13 even 4 inner
768.3.l.f.703.5 yes 16 16.3 odd 4 inner