Properties

Label 1680.3.bd.a.769.7
Level $1680$
Weight $3$
Character 1680.769
Analytic conductor $45.777$
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] = [1680,3,Mod(769,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.769");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1680.bd (of order \(2\), degree \(1\), 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: \(45.7766844125\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.7
Root \(0.500000 + 0.422343i\) of defining polynomial
Character \(\chi\) \(=\) 1680.769
Dual form 1680.3.bd.a.769.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +(4.91728 - 0.905717i) q^{5} +(1.91369 + 6.73333i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(4.91728 - 0.905717i) q^{5} +(1.91369 + 6.73333i) q^{7} +3.00000 q^{9} -17.5116 q^{11} +4.83531 q^{13} +(-8.51698 + 1.56875i) q^{15} -18.0284 q^{17} -9.13350i q^{19} +(-3.31461 - 11.6625i) q^{21} +3.72515i q^{23} +(23.3594 - 8.90734i) q^{25} -5.19615 q^{27} +1.12582 q^{29} -57.0859i q^{31} +30.3310 q^{33} +(15.5086 + 31.3765i) q^{35} -41.3624i q^{37} -8.37500 q^{39} +11.7156i q^{41} -64.4171i q^{43} +(14.7519 - 2.71715i) q^{45} +77.6614 q^{47} +(-41.6756 + 25.7710i) q^{49} +31.2261 q^{51} -77.5383i q^{53} +(-86.1095 + 15.8606i) q^{55} +15.8197i q^{57} +87.0651i q^{59} +5.36957i q^{61} +(5.74106 + 20.2000i) q^{63} +(23.7766 - 4.37942i) q^{65} +47.1879i q^{67} -6.45216i q^{69} +58.3047 q^{71} +53.4082 q^{73} +(-40.4596 + 15.4280i) q^{75} +(-33.5117 - 117.911i) q^{77} +74.9637 q^{79} +9.00000 q^{81} +28.7890 q^{83} +(-88.6507 + 16.3286i) q^{85} -1.94997 q^{87} +101.499i q^{89} +(9.25327 + 32.5578i) q^{91} +98.8757i q^{93} +(-8.27237 - 44.9120i) q^{95} -107.830 q^{97} -52.5348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

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



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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 4.91728 0.905717i 0.983457 0.181143i
\(6\) 0 0
\(7\) 1.91369 + 6.73333i 0.273384 + 0.961905i
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) −17.5116 −1.59196 −0.795982 0.605321i \(-0.793045\pi\)
−0.795982 + 0.605321i \(0.793045\pi\)
\(12\) 0 0
\(13\) 4.83531 0.371947 0.185973 0.982555i \(-0.440456\pi\)
0.185973 + 0.982555i \(0.440456\pi\)
\(14\) 0 0
\(15\) −8.51698 + 1.56875i −0.567799 + 0.104583i
\(16\) 0 0
\(17\) −18.0284 −1.06049 −0.530247 0.847843i \(-0.677901\pi\)
−0.530247 + 0.847843i \(0.677901\pi\)
\(18\) 0 0
\(19\) 9.13350i 0.480711i −0.970685 0.240355i \(-0.922736\pi\)
0.970685 0.240355i \(-0.0772640\pi\)
\(20\) 0 0
\(21\) −3.31461 11.6625i −0.157838 0.555356i
\(22\) 0 0
\(23\) 3.72515i 0.161963i 0.996716 + 0.0809816i \(0.0258055\pi\)
−0.996716 + 0.0809816i \(0.974194\pi\)
\(24\) 0 0
\(25\) 23.3594 8.90734i 0.934374 0.356294i
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 1.12582 0.0388213 0.0194107 0.999812i \(-0.493821\pi\)
0.0194107 + 0.999812i \(0.493821\pi\)
\(30\) 0 0
\(31\) 57.0859i 1.84148i −0.390175 0.920741i \(-0.627585\pi\)
0.390175 0.920741i \(-0.372415\pi\)
\(32\) 0 0
\(33\) 30.3310 0.919120
\(34\) 0 0
\(35\) 15.5086 + 31.3765i 0.443104 + 0.896470i
\(36\) 0 0
\(37\) 41.3624i 1.11790i −0.829200 0.558952i \(-0.811204\pi\)
0.829200 0.558952i \(-0.188796\pi\)
\(38\) 0 0
\(39\) −8.37500 −0.214744
\(40\) 0 0
\(41\) 11.7156i 0.285745i 0.989741 + 0.142873i \(0.0456340\pi\)
−0.989741 + 0.142873i \(0.954366\pi\)
\(42\) 0 0
\(43\) 64.4171i 1.49807i −0.662529 0.749036i \(-0.730517\pi\)
0.662529 0.749036i \(-0.269483\pi\)
\(44\) 0 0
\(45\) 14.7519 2.71715i 0.327819 0.0603812i
\(46\) 0 0
\(47\) 77.6614 1.65237 0.826185 0.563399i \(-0.190507\pi\)
0.826185 + 0.563399i \(0.190507\pi\)
\(48\) 0 0
\(49\) −41.6756 + 25.7710i −0.850522 + 0.525939i
\(50\) 0 0
\(51\) 31.2261 0.612276
\(52\) 0 0
\(53\) 77.5383i 1.46299i −0.681849 0.731493i \(-0.738824\pi\)
0.681849 0.731493i \(-0.261176\pi\)
\(54\) 0 0
\(55\) −86.1095 + 15.8606i −1.56563 + 0.288374i
\(56\) 0 0
\(57\) 15.8197i 0.277539i
\(58\) 0 0
\(59\) 87.0651i 1.47568i 0.674976 + 0.737839i \(0.264154\pi\)
−0.674976 + 0.737839i \(0.735846\pi\)
\(60\) 0 0
\(61\) 5.36957i 0.0880258i 0.999031 + 0.0440129i \(0.0140143\pi\)
−0.999031 + 0.0440129i \(0.985986\pi\)
\(62\) 0 0
\(63\) 5.74106 + 20.2000i 0.0911280 + 0.320635i
\(64\) 0 0
\(65\) 23.7766 4.37942i 0.365794 0.0673757i
\(66\) 0 0
\(67\) 47.1879i 0.704297i 0.935944 + 0.352149i \(0.114549\pi\)
−0.935944 + 0.352149i \(0.885451\pi\)
\(68\) 0 0
\(69\) 6.45216i 0.0935095i
\(70\) 0 0
\(71\) 58.3047 0.821193 0.410597 0.911817i \(-0.365320\pi\)
0.410597 + 0.911817i \(0.365320\pi\)
\(72\) 0 0
\(73\) 53.4082 0.731619 0.365810 0.930690i \(-0.380792\pi\)
0.365810 + 0.930690i \(0.380792\pi\)
\(74\) 0 0
\(75\) −40.4596 + 15.4280i −0.539461 + 0.205706i
\(76\) 0 0
\(77\) −33.5117 117.911i −0.435217 1.53132i
\(78\) 0 0
\(79\) 74.9637 0.948907 0.474454 0.880280i \(-0.342646\pi\)
0.474454 + 0.880280i \(0.342646\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 28.7890 0.346855 0.173428 0.984847i \(-0.444516\pi\)
0.173428 + 0.984847i \(0.444516\pi\)
\(84\) 0 0
\(85\) −88.6507 + 16.3286i −1.04295 + 0.192101i
\(86\) 0 0
\(87\) −1.94997 −0.0224135
\(88\) 0 0
\(89\) 101.499i 1.14043i 0.821494 + 0.570217i \(0.193141\pi\)
−0.821494 + 0.570217i \(0.806859\pi\)
\(90\) 0 0
\(91\) 9.25327 + 32.5578i 0.101684 + 0.357778i
\(92\) 0 0
\(93\) 98.8757i 1.06318i
\(94\) 0 0
\(95\) −8.27237 44.9120i −0.0870776 0.472758i
\(96\) 0 0
\(97\) −107.830 −1.11165 −0.555827 0.831298i \(-0.687598\pi\)
−0.555827 + 0.831298i \(0.687598\pi\)
\(98\) 0 0
\(99\) −52.5348 −0.530654
\(100\) 0 0
\(101\) 52.6151i 0.520941i −0.965482 0.260471i \(-0.916122\pi\)
0.965482 0.260471i \(-0.0838777\pi\)
\(102\) 0 0
\(103\) 72.4002 0.702915 0.351457 0.936204i \(-0.385686\pi\)
0.351457 + 0.936204i \(0.385686\pi\)
\(104\) 0 0
\(105\) −26.8618 54.3456i −0.255826 0.517577i
\(106\) 0 0
\(107\) 173.562i 1.62207i −0.584994 0.811037i \(-0.698903\pi\)
0.584994 0.811037i \(-0.301097\pi\)
\(108\) 0 0
\(109\) 49.9966 0.458684 0.229342 0.973346i \(-0.426342\pi\)
0.229342 + 0.973346i \(0.426342\pi\)
\(110\) 0 0
\(111\) 71.6418i 0.645422i
\(112\) 0 0
\(113\) 70.1536i 0.620828i 0.950601 + 0.310414i \(0.100468\pi\)
−0.950601 + 0.310414i \(0.899532\pi\)
\(114\) 0 0
\(115\) 3.37394 + 18.3176i 0.0293386 + 0.159284i
\(116\) 0 0
\(117\) 14.5059 0.123982
\(118\) 0 0
\(119\) −34.5007 121.391i −0.289922 1.02009i
\(120\) 0 0
\(121\) 185.656 1.53435
\(122\) 0 0
\(123\) 20.2919i 0.164975i
\(124\) 0 0
\(125\) 106.797 64.9569i 0.854376 0.519655i
\(126\) 0 0
\(127\) 197.945i 1.55863i −0.626635 0.779313i \(-0.715568\pi\)
0.626635 0.779313i \(-0.284432\pi\)
\(128\) 0 0
\(129\) 111.574i 0.864912i
\(130\) 0 0
\(131\) 114.901i 0.877105i 0.898706 + 0.438552i \(0.144509\pi\)
−0.898706 + 0.438552i \(0.855491\pi\)
\(132\) 0 0
\(133\) 61.4989 17.4787i 0.462398 0.131419i
\(134\) 0 0
\(135\) −25.5510 + 4.70625i −0.189266 + 0.0348611i
\(136\) 0 0
\(137\) 95.3358i 0.695882i −0.937516 0.347941i \(-0.886881\pi\)
0.937516 0.347941i \(-0.113119\pi\)
\(138\) 0 0
\(139\) 237.587i 1.70926i −0.519237 0.854630i \(-0.673784\pi\)
0.519237 0.854630i \(-0.326216\pi\)
\(140\) 0 0
\(141\) −134.513 −0.953996
\(142\) 0 0
\(143\) −84.6740 −0.592126
\(144\) 0 0
\(145\) 5.53597 1.01967i 0.0381791 0.00703223i
\(146\) 0 0
\(147\) 72.1842 44.6367i 0.491049 0.303651i
\(148\) 0 0
\(149\) 133.844 0.898285 0.449142 0.893460i \(-0.351730\pi\)
0.449142 + 0.893460i \(0.351730\pi\)
\(150\) 0 0
\(151\) −155.064 −1.02691 −0.513456 0.858116i \(-0.671635\pi\)
−0.513456 + 0.858116i \(0.671635\pi\)
\(152\) 0 0
\(153\) −54.0852 −0.353498
\(154\) 0 0
\(155\) −51.7037 280.708i −0.333572 1.81102i
\(156\) 0 0
\(157\) −104.634 −0.666459 −0.333230 0.942846i \(-0.608138\pi\)
−0.333230 + 0.942846i \(0.608138\pi\)
\(158\) 0 0
\(159\) 134.300i 0.844656i
\(160\) 0 0
\(161\) −25.0827 + 7.12878i −0.155793 + 0.0442782i
\(162\) 0 0
\(163\) 130.909i 0.803120i −0.915833 0.401560i \(-0.868468\pi\)
0.915833 0.401560i \(-0.131532\pi\)
\(164\) 0 0
\(165\) 149.146 27.4713i 0.903915 0.166493i
\(166\) 0 0
\(167\) 102.845 0.615838 0.307919 0.951413i \(-0.400367\pi\)
0.307919 + 0.951413i \(0.400367\pi\)
\(168\) 0 0
\(169\) −145.620 −0.861656
\(170\) 0 0
\(171\) 27.4005i 0.160237i
\(172\) 0 0
\(173\) 197.432 1.14122 0.570612 0.821220i \(-0.306706\pi\)
0.570612 + 0.821220i \(0.306706\pi\)
\(174\) 0 0
\(175\) 104.679 + 140.240i 0.598163 + 0.801374i
\(176\) 0 0
\(177\) 150.801i 0.851984i
\(178\) 0 0
\(179\) 118.822 0.663809 0.331904 0.943313i \(-0.392309\pi\)
0.331904 + 0.943313i \(0.392309\pi\)
\(180\) 0 0
\(181\) 14.4148i 0.0796399i 0.999207 + 0.0398199i \(0.0126784\pi\)
−0.999207 + 0.0398199i \(0.987322\pi\)
\(182\) 0 0
\(183\) 9.30038i 0.0508217i
\(184\) 0 0
\(185\) −37.4627 203.391i −0.202501 1.09941i
\(186\) 0 0
\(187\) 315.706 1.68827
\(188\) 0 0
\(189\) −9.94382 34.9874i −0.0526128 0.185119i
\(190\) 0 0
\(191\) −59.9375 −0.313809 −0.156904 0.987614i \(-0.550151\pi\)
−0.156904 + 0.987614i \(0.550151\pi\)
\(192\) 0 0
\(193\) 245.948i 1.27434i −0.770723 0.637171i \(-0.780105\pi\)
0.770723 0.637171i \(-0.219895\pi\)
\(194\) 0 0
\(195\) −41.1823 + 7.58538i −0.211191 + 0.0388994i
\(196\) 0 0
\(197\) 82.0594i 0.416545i −0.978071 0.208273i \(-0.933216\pi\)
0.978071 0.208273i \(-0.0667842\pi\)
\(198\) 0 0
\(199\) 289.338i 1.45396i −0.686658 0.726981i \(-0.740923\pi\)
0.686658 0.726981i \(-0.259077\pi\)
\(200\) 0 0
\(201\) 81.7319i 0.406626i
\(202\) 0 0
\(203\) 2.15447 + 7.58051i 0.0106131 + 0.0373424i
\(204\) 0 0
\(205\) 10.6110 + 57.6087i 0.0517609 + 0.281018i
\(206\) 0 0
\(207\) 11.1755i 0.0539877i
\(208\) 0 0
\(209\) 159.942i 0.765274i
\(210\) 0 0
\(211\) −33.6995 −0.159713 −0.0798567 0.996806i \(-0.525446\pi\)
−0.0798567 + 0.996806i \(0.525446\pi\)
\(212\) 0 0
\(213\) −100.987 −0.474116
\(214\) 0 0
\(215\) −58.3437 316.757i −0.271366 1.47329i
\(216\) 0 0
\(217\) 384.379 109.245i 1.77133 0.503432i
\(218\) 0 0
\(219\) −92.5057 −0.422401
\(220\) 0 0
\(221\) −87.1728 −0.394447
\(222\) 0 0
\(223\) 10.7556 0.0482313 0.0241156 0.999709i \(-0.492323\pi\)
0.0241156 + 0.999709i \(0.492323\pi\)
\(224\) 0 0
\(225\) 70.0781 26.7220i 0.311458 0.118765i
\(226\) 0 0
\(227\) −416.450 −1.83458 −0.917290 0.398219i \(-0.869628\pi\)
−0.917290 + 0.398219i \(0.869628\pi\)
\(228\) 0 0
\(229\) 388.469i 1.69637i −0.529700 0.848185i \(-0.677695\pi\)
0.529700 0.848185i \(-0.322305\pi\)
\(230\) 0 0
\(231\) 58.0440 + 204.229i 0.251273 + 0.884107i
\(232\) 0 0
\(233\) 70.4036i 0.302162i 0.988521 + 0.151081i \(0.0482754\pi\)
−0.988521 + 0.151081i \(0.951725\pi\)
\(234\) 0 0
\(235\) 381.883 70.3393i 1.62503 0.299316i
\(236\) 0 0
\(237\) −129.841 −0.547852
\(238\) 0 0
\(239\) 313.133 1.31018 0.655091 0.755550i \(-0.272630\pi\)
0.655091 + 0.755550i \(0.272630\pi\)
\(240\) 0 0
\(241\) 198.727i 0.824595i 0.911049 + 0.412297i \(0.135274\pi\)
−0.911049 + 0.412297i \(0.864726\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) −181.589 + 164.470i −0.741181 + 0.671305i
\(246\) 0 0
\(247\) 44.1633i 0.178799i
\(248\) 0 0
\(249\) −49.8640 −0.200257
\(250\) 0 0
\(251\) 300.878i 1.19872i 0.800480 + 0.599359i \(0.204578\pi\)
−0.800480 + 0.599359i \(0.795422\pi\)
\(252\) 0 0
\(253\) 65.2334i 0.257839i
\(254\) 0 0
\(255\) 153.548 28.2820i 0.602147 0.110910i
\(256\) 0 0
\(257\) 306.052 1.19087 0.595433 0.803405i \(-0.296981\pi\)
0.595433 + 0.803405i \(0.296981\pi\)
\(258\) 0 0
\(259\) 278.507 79.1548i 1.07532 0.305617i
\(260\) 0 0
\(261\) 3.37745 0.0129404
\(262\) 0 0
\(263\) 286.135i 1.08797i 0.839096 + 0.543984i \(0.183085\pi\)
−0.839096 + 0.543984i \(0.816915\pi\)
\(264\) 0 0
\(265\) −70.2278 381.278i −0.265010 1.43878i
\(266\) 0 0
\(267\) 175.801i 0.658430i
\(268\) 0 0
\(269\) 188.412i 0.700415i −0.936672 0.350208i \(-0.886111\pi\)
0.936672 0.350208i \(-0.113889\pi\)
\(270\) 0 0
\(271\) 9.59288i 0.0353981i −0.999843 0.0176990i \(-0.994366\pi\)
0.999843 0.0176990i \(-0.00563408\pi\)
\(272\) 0 0
\(273\) −16.0271 56.3917i −0.0587075 0.206563i
\(274\) 0 0
\(275\) −409.060 + 155.982i −1.48749 + 0.567206i
\(276\) 0 0
\(277\) 161.066i 0.581464i 0.956804 + 0.290732i \(0.0938989\pi\)
−0.956804 + 0.290732i \(0.906101\pi\)
\(278\) 0 0
\(279\) 171.258i 0.613827i
\(280\) 0 0
\(281\) −343.107 −1.22102 −0.610510 0.792008i \(-0.709036\pi\)
−0.610510 + 0.792008i \(0.709036\pi\)
\(282\) 0 0
\(283\) 324.748 1.14752 0.573759 0.819024i \(-0.305484\pi\)
0.573759 + 0.819024i \(0.305484\pi\)
\(284\) 0 0
\(285\) 14.3282 + 77.7899i 0.0502743 + 0.272947i
\(286\) 0 0
\(287\) −78.8848 + 22.4199i −0.274860 + 0.0781182i
\(288\) 0 0
\(289\) 36.0229 0.124647
\(290\) 0 0
\(291\) 186.768 0.641813
\(292\) 0 0
\(293\) 100.918 0.344429 0.172215 0.985059i \(-0.444908\pi\)
0.172215 + 0.985059i \(0.444908\pi\)
\(294\) 0 0
\(295\) 78.8563 + 428.124i 0.267310 + 1.45127i
\(296\) 0 0
\(297\) 90.9929 0.306373
\(298\) 0 0
\(299\) 18.0123i 0.0602417i
\(300\) 0 0
\(301\) 433.742 123.274i 1.44100 0.409549i
\(302\) 0 0
\(303\) 91.1320i 0.300766i
\(304\) 0 0
\(305\) 4.86332 + 26.4037i 0.0159453 + 0.0865696i
\(306\) 0 0
\(307\) −41.3057 −0.134546 −0.0672732 0.997735i \(-0.521430\pi\)
−0.0672732 + 0.997735i \(0.521430\pi\)
\(308\) 0 0
\(309\) −125.401 −0.405828
\(310\) 0 0
\(311\) 470.341i 1.51235i 0.654369 + 0.756176i \(0.272934\pi\)
−0.654369 + 0.756176i \(0.727066\pi\)
\(312\) 0 0
\(313\) −160.220 −0.511884 −0.255942 0.966692i \(-0.582386\pi\)
−0.255942 + 0.966692i \(0.582386\pi\)
\(314\) 0 0
\(315\) 46.5259 + 94.1294i 0.147701 + 0.298823i
\(316\) 0 0
\(317\) 268.766i 0.847843i −0.905699 0.423922i \(-0.860653\pi\)
0.905699 0.423922i \(-0.139347\pi\)
\(318\) 0 0
\(319\) −19.7149 −0.0618021
\(320\) 0 0
\(321\) 300.618i 0.936505i
\(322\) 0 0
\(323\) 164.662i 0.509791i
\(324\) 0 0
\(325\) 112.950 43.0697i 0.347537 0.132522i
\(326\) 0 0
\(327\) −86.5967 −0.264822
\(328\) 0 0
\(329\) 148.620 + 522.920i 0.451732 + 1.58942i
\(330\) 0 0
\(331\) −555.018 −1.67679 −0.838395 0.545063i \(-0.816506\pi\)
−0.838395 + 0.545063i \(0.816506\pi\)
\(332\) 0 0
\(333\) 124.087i 0.372634i
\(334\) 0 0
\(335\) 42.7389 + 232.036i 0.127579 + 0.692646i
\(336\) 0 0
\(337\) 47.7972i 0.141831i −0.997482 0.0709157i \(-0.977408\pi\)
0.997482 0.0709157i \(-0.0225921\pi\)
\(338\) 0 0
\(339\) 121.510i 0.358435i
\(340\) 0 0
\(341\) 999.666i 2.93157i
\(342\) 0 0
\(343\) −253.279 231.298i −0.738422 0.674338i
\(344\) 0 0
\(345\) −5.84383 31.7271i −0.0169386 0.0919625i
\(346\) 0 0
\(347\) 283.766i 0.817771i −0.912586 0.408885i \(-0.865918\pi\)
0.912586 0.408885i \(-0.134082\pi\)
\(348\) 0 0
\(349\) 20.0321i 0.0573987i 0.999588 + 0.0286993i \(0.00913653\pi\)
−0.999588 + 0.0286993i \(0.990863\pi\)
\(350\) 0 0
\(351\) −25.1250 −0.0715812
\(352\) 0 0
\(353\) −595.015 −1.68559 −0.842797 0.538231i \(-0.819093\pi\)
−0.842797 + 0.538231i \(0.819093\pi\)
\(354\) 0 0
\(355\) 286.701 52.8076i 0.807608 0.148754i
\(356\) 0 0
\(357\) 59.7570 + 210.256i 0.167387 + 0.588952i
\(358\) 0 0
\(359\) −588.390 −1.63897 −0.819485 0.573101i \(-0.805740\pi\)
−0.819485 + 0.573101i \(0.805740\pi\)
\(360\) 0 0
\(361\) 277.579 0.768917
\(362\) 0 0
\(363\) −321.566 −0.885856
\(364\) 0 0
\(365\) 262.623 48.3727i 0.719516 0.132528i
\(366\) 0 0
\(367\) −662.601 −1.80545 −0.902726 0.430215i \(-0.858438\pi\)
−0.902726 + 0.430215i \(0.858438\pi\)
\(368\) 0 0
\(369\) 35.1467i 0.0952485i
\(370\) 0 0
\(371\) 522.091 148.384i 1.40725 0.399957i
\(372\) 0 0
\(373\) 236.349i 0.633644i −0.948485 0.316822i \(-0.897384\pi\)
0.948485 0.316822i \(-0.102616\pi\)
\(374\) 0 0
\(375\) −184.978 + 112.509i −0.493274 + 0.300023i
\(376\) 0 0
\(377\) 5.44368 0.0144395
\(378\) 0 0
\(379\) −344.235 −0.908272 −0.454136 0.890932i \(-0.650052\pi\)
−0.454136 + 0.890932i \(0.650052\pi\)
\(380\) 0 0
\(381\) 342.852i 0.899873i
\(382\) 0 0
\(383\) 65.7097 0.171566 0.0857828 0.996314i \(-0.472661\pi\)
0.0857828 + 0.996314i \(0.472661\pi\)
\(384\) 0 0
\(385\) −271.581 549.452i −0.705406 1.42715i
\(386\) 0 0
\(387\) 193.251i 0.499357i
\(388\) 0 0
\(389\) 161.296 0.414642 0.207321 0.978273i \(-0.433526\pi\)
0.207321 + 0.978273i \(0.433526\pi\)
\(390\) 0 0
\(391\) 67.1585i 0.171761i
\(392\) 0 0
\(393\) 199.014i 0.506397i
\(394\) 0 0
\(395\) 368.618 67.8959i 0.933209 0.171888i
\(396\) 0 0
\(397\) 667.752 1.68200 0.840998 0.541039i \(-0.181969\pi\)
0.840998 + 0.541039i \(0.181969\pi\)
\(398\) 0 0
\(399\) −106.519 + 30.2740i −0.266966 + 0.0758746i
\(400\) 0 0
\(401\) −593.826 −1.48086 −0.740431 0.672132i \(-0.765379\pi\)
−0.740431 + 0.672132i \(0.765379\pi\)
\(402\) 0 0
\(403\) 276.028i 0.684933i
\(404\) 0 0
\(405\) 44.2556 8.15146i 0.109273 0.0201271i
\(406\) 0 0
\(407\) 724.322i 1.77966i
\(408\) 0 0
\(409\) 569.625i 1.39273i −0.717690 0.696363i \(-0.754800\pi\)
0.717690 0.696363i \(-0.245200\pi\)
\(410\) 0 0
\(411\) 165.127i 0.401768i
\(412\) 0 0
\(413\) −586.238 + 166.615i −1.41946 + 0.403427i
\(414\) 0 0
\(415\) 141.564 26.0747i 0.341117 0.0628306i
\(416\) 0 0
\(417\) 411.513i 0.986842i
\(418\) 0 0
\(419\) 209.456i 0.499894i −0.968260 0.249947i \(-0.919587\pi\)
0.968260 0.249947i \(-0.0804132\pi\)
\(420\) 0 0
\(421\) −169.713 −0.403119 −0.201560 0.979476i \(-0.564601\pi\)
−0.201560 + 0.979476i \(0.564601\pi\)
\(422\) 0 0
\(423\) 232.984 0.550790
\(424\) 0 0
\(425\) −421.132 + 160.585i −0.990898 + 0.377847i
\(426\) 0 0
\(427\) −36.1551 + 10.2757i −0.0846725 + 0.0240649i
\(428\) 0 0
\(429\) 146.660 0.341864
\(430\) 0 0
\(431\) −538.427 −1.24925 −0.624625 0.780925i \(-0.714748\pi\)
−0.624625 + 0.780925i \(0.714748\pi\)
\(432\) 0 0
\(433\) 118.314 0.273243 0.136622 0.990623i \(-0.456376\pi\)
0.136622 + 0.990623i \(0.456376\pi\)
\(434\) 0 0
\(435\) −9.58858 + 1.76613i −0.0220427 + 0.00406006i
\(436\) 0 0
\(437\) 34.0237 0.0778575
\(438\) 0 0
\(439\) 434.834i 0.990510i 0.868748 + 0.495255i \(0.164925\pi\)
−0.868748 + 0.495255i \(0.835075\pi\)
\(440\) 0 0
\(441\) −125.027 + 77.3130i −0.283507 + 0.175313i
\(442\) 0 0
\(443\) 555.005i 1.25283i −0.779488 0.626417i \(-0.784521\pi\)
0.779488 0.626417i \(-0.215479\pi\)
\(444\) 0 0
\(445\) 91.9291 + 499.098i 0.206582 + 1.12157i
\(446\) 0 0
\(447\) −231.825 −0.518625
\(448\) 0 0
\(449\) −66.8926 −0.148981 −0.0744906 0.997222i \(-0.523733\pi\)
−0.0744906 + 0.997222i \(0.523733\pi\)
\(450\) 0 0
\(451\) 205.158i 0.454896i
\(452\) 0 0
\(453\) 268.578 0.592888
\(454\) 0 0
\(455\) 74.9891 + 151.715i 0.164811 + 0.333439i
\(456\) 0 0
\(457\) 750.939i 1.64319i 0.570070 + 0.821596i \(0.306916\pi\)
−0.570070 + 0.821596i \(0.693084\pi\)
\(458\) 0 0
\(459\) 93.6783 0.204092
\(460\) 0 0
\(461\) 395.377i 0.857651i −0.903387 0.428826i \(-0.858928\pi\)
0.903387 0.428826i \(-0.141072\pi\)
\(462\) 0 0
\(463\) 423.981i 0.915726i 0.889023 + 0.457863i \(0.151385\pi\)
−0.889023 + 0.457863i \(0.848615\pi\)
\(464\) 0 0
\(465\) 89.5535 + 486.200i 0.192588 + 1.04559i
\(466\) 0 0
\(467\) 584.408 1.25141 0.625705 0.780060i \(-0.284812\pi\)
0.625705 + 0.780060i \(0.284812\pi\)
\(468\) 0 0
\(469\) −317.732 + 90.3030i −0.677467 + 0.192544i
\(470\) 0 0
\(471\) 181.232 0.384780
\(472\) 0 0
\(473\) 1128.05i 2.38488i
\(474\) 0 0
\(475\) −81.3552 213.353i −0.171274 0.449164i
\(476\) 0 0
\(477\) 232.615i 0.487662i
\(478\) 0 0
\(479\) 734.511i 1.53343i 0.641990 + 0.766713i \(0.278109\pi\)
−0.641990 + 0.766713i \(0.721891\pi\)
\(480\) 0 0
\(481\) 200.000i 0.415801i
\(482\) 0 0
\(483\) 43.4445 12.3474i 0.0899473 0.0255640i
\(484\) 0 0
\(485\) −530.232 + 97.6638i −1.09326 + 0.201369i
\(486\) 0 0
\(487\) 91.6643i 0.188222i 0.995562 + 0.0941111i \(0.0300009\pi\)
−0.995562 + 0.0941111i \(0.969999\pi\)
\(488\) 0 0
\(489\) 226.740i 0.463681i
\(490\) 0 0
\(491\) −460.086 −0.937038 −0.468519 0.883453i \(-0.655212\pi\)
−0.468519 + 0.883453i \(0.655212\pi\)
\(492\) 0 0
\(493\) −20.2967 −0.0411698
\(494\) 0 0
\(495\) −258.328 + 47.5817i −0.521876 + 0.0961246i
\(496\) 0 0
\(497\) 111.577 + 392.585i 0.224501 + 0.789910i
\(498\) 0 0
\(499\) 487.126 0.976204 0.488102 0.872786i \(-0.337689\pi\)
0.488102 + 0.872786i \(0.337689\pi\)
\(500\) 0 0
\(501\) −178.133 −0.355554
\(502\) 0 0
\(503\) −305.145 −0.606650 −0.303325 0.952887i \(-0.598097\pi\)
−0.303325 + 0.952887i \(0.598097\pi\)
\(504\) 0 0
\(505\) −47.6544 258.723i −0.0943651 0.512323i
\(506\) 0 0
\(507\) 252.221 0.497477
\(508\) 0 0
\(509\) 800.434i 1.57256i −0.617869 0.786281i \(-0.712004\pi\)
0.617869 0.786281i \(-0.287996\pi\)
\(510\) 0 0
\(511\) 102.207 + 359.615i 0.200013 + 0.703748i
\(512\) 0 0
\(513\) 47.4591i 0.0925128i
\(514\) 0 0
\(515\) 356.012 65.5741i 0.691286 0.127328i
\(516\) 0 0
\(517\) −1359.97 −2.63051
\(518\) 0 0
\(519\) −341.962 −0.658886
\(520\) 0 0
\(521\) 21.4194i 0.0411121i 0.999789 + 0.0205560i \(0.00654365\pi\)
−0.999789 + 0.0205560i \(0.993456\pi\)
\(522\) 0 0
\(523\) −842.290 −1.61050 −0.805249 0.592937i \(-0.797968\pi\)
−0.805249 + 0.592937i \(0.797968\pi\)
\(524\) 0 0
\(525\) −181.309 242.904i −0.345350 0.462674i
\(526\) 0 0
\(527\) 1029.17i 1.95288i
\(528\) 0 0
\(529\) 515.123 0.973768
\(530\) 0 0
\(531\) 261.195i 0.491893i
\(532\) 0 0
\(533\) 56.6484i 0.106282i
\(534\) 0 0
\(535\) −157.198 853.453i −0.293828 1.59524i
\(536\) 0 0
\(537\) −205.805 −0.383250
\(538\) 0 0
\(539\) 729.806 451.291i 1.35400 0.837275i
\(540\) 0 0
\(541\) −15.6795 −0.0289824 −0.0144912 0.999895i \(-0.504613\pi\)
−0.0144912 + 0.999895i \(0.504613\pi\)
\(542\) 0 0
\(543\) 24.9672i 0.0459801i
\(544\) 0 0
\(545\) 245.847 45.2828i 0.451096 0.0830877i
\(546\) 0 0
\(547\) 315.792i 0.577317i −0.957432 0.288659i \(-0.906791\pi\)
0.957432 0.288659i \(-0.0932092\pi\)
\(548\) 0 0
\(549\) 16.1087i 0.0293419i
\(550\) 0 0
\(551\) 10.2827i 0.0186618i
\(552\) 0 0
\(553\) 143.457 + 504.756i 0.259416 + 0.912759i
\(554\) 0 0
\(555\) 64.8872 + 352.283i 0.116914 + 0.634744i
\(556\) 0 0
\(557\) 358.421i 0.643484i −0.946827 0.321742i \(-0.895732\pi\)
0.946827 0.321742i \(-0.104268\pi\)
\(558\) 0 0
\(559\) 311.477i 0.557203i
\(560\) 0 0
\(561\) −546.819 −0.974721
\(562\) 0 0
\(563\) 622.834 1.10628 0.553138 0.833089i \(-0.313430\pi\)
0.553138 + 0.833089i \(0.313430\pi\)
\(564\) 0 0
\(565\) 63.5393 + 344.965i 0.112459 + 0.610557i
\(566\) 0 0
\(567\) 17.2232 + 60.6000i 0.0303760 + 0.106878i
\(568\) 0 0
\(569\) −185.789 −0.326518 −0.163259 0.986583i \(-0.552201\pi\)
−0.163259 + 0.986583i \(0.552201\pi\)
\(570\) 0 0
\(571\) 486.258 0.851589 0.425795 0.904820i \(-0.359995\pi\)
0.425795 + 0.904820i \(0.359995\pi\)
\(572\) 0 0
\(573\) 103.815 0.181178
\(574\) 0 0
\(575\) 33.1812 + 87.0172i 0.0577064 + 0.151334i
\(576\) 0 0
\(577\) 270.057 0.468037 0.234018 0.972232i \(-0.424812\pi\)
0.234018 + 0.972232i \(0.424812\pi\)
\(578\) 0 0
\(579\) 425.994i 0.735741i
\(580\) 0 0
\(581\) 55.0932 + 193.846i 0.0948247 + 0.333642i
\(582\) 0 0
\(583\) 1357.82i 2.32902i
\(584\) 0 0
\(585\) 71.3298 13.1383i 0.121931 0.0224586i
\(586\) 0 0
\(587\) −400.613 −0.682476 −0.341238 0.939977i \(-0.610846\pi\)
−0.341238 + 0.939977i \(0.610846\pi\)
\(588\) 0 0
\(589\) −521.395 −0.885220
\(590\) 0 0
\(591\) 142.131i 0.240493i
\(592\) 0 0
\(593\) −157.686 −0.265912 −0.132956 0.991122i \(-0.542447\pi\)
−0.132956 + 0.991122i \(0.542447\pi\)
\(594\) 0 0
\(595\) −279.596 565.667i −0.469909 0.950701i
\(596\) 0 0
\(597\) 501.149i 0.839445i
\(598\) 0 0
\(599\) 468.940 0.782872 0.391436 0.920205i \(-0.371978\pi\)
0.391436 + 0.920205i \(0.371978\pi\)
\(600\) 0 0
\(601\) 255.932i 0.425844i −0.977069 0.212922i \(-0.931702\pi\)
0.977069 0.212922i \(-0.0682980\pi\)
\(602\) 0 0
\(603\) 141.564i 0.234766i
\(604\) 0 0
\(605\) 912.923 168.152i 1.50896 0.277937i
\(606\) 0 0
\(607\) 773.553 1.27439 0.637194 0.770704i \(-0.280095\pi\)
0.637194 + 0.770704i \(0.280095\pi\)
\(608\) 0 0
\(609\) −3.73164 13.1298i −0.00612749 0.0215597i
\(610\) 0 0
\(611\) 375.517 0.614594
\(612\) 0 0
\(613\) 443.208i 0.723015i 0.932369 + 0.361507i \(0.117738\pi\)
−0.932369 + 0.361507i \(0.882262\pi\)
\(614\) 0 0
\(615\) −18.3788 99.7813i −0.0298842 0.162246i
\(616\) 0 0
\(617\) 363.298i 0.588813i −0.955680 0.294407i \(-0.904878\pi\)
0.955680 0.294407i \(-0.0951220\pi\)
\(618\) 0 0
\(619\) 292.211i 0.472069i −0.971745 0.236035i \(-0.924152\pi\)
0.971745 0.236035i \(-0.0758479\pi\)
\(620\) 0 0
\(621\) 19.3565i 0.0311698i
\(622\) 0 0
\(623\) −683.425 + 194.237i −1.09699 + 0.311777i
\(624\) 0 0
\(625\) 466.319 416.139i 0.746110 0.665823i
\(626\) 0 0
\(627\) 277.028i 0.441831i
\(628\) 0 0
\(629\) 745.698i 1.18553i
\(630\) 0 0
\(631\) −189.221 −0.299874 −0.149937 0.988696i \(-0.547907\pi\)
−0.149937 + 0.988696i \(0.547907\pi\)
\(632\) 0 0
\(633\) 58.3693 0.0922106
\(634\) 0 0
\(635\) −179.283 973.354i −0.282335 1.53284i
\(636\) 0 0
\(637\) −201.514 + 124.611i −0.316349 + 0.195621i
\(638\) 0 0
\(639\) 174.914 0.273731
\(640\) 0 0
\(641\) 1162.94 1.81425 0.907127 0.420857i \(-0.138271\pi\)
0.907127 + 0.420857i \(0.138271\pi\)
\(642\) 0 0
\(643\) 1096.08 1.70464 0.852320 0.523021i \(-0.175195\pi\)
0.852320 + 0.523021i \(0.175195\pi\)
\(644\) 0 0
\(645\) 101.054 + 548.640i 0.156673 + 0.850604i
\(646\) 0 0
\(647\) −262.877 −0.406302 −0.203151 0.979147i \(-0.565118\pi\)
−0.203151 + 0.979147i \(0.565118\pi\)
\(648\) 0 0
\(649\) 1524.65i 2.34923i
\(650\) 0 0
\(651\) −665.763 + 189.217i −1.02268 + 0.290656i
\(652\) 0 0
\(653\) 242.905i 0.371983i 0.982551 + 0.185992i \(0.0595498\pi\)
−0.982551 + 0.185992i \(0.940450\pi\)
\(654\) 0 0
\(655\) 104.068 + 564.999i 0.158882 + 0.862594i
\(656\) 0 0
\(657\) 160.225 0.243873
\(658\) 0 0
\(659\) −1176.49 −1.78527 −0.892634 0.450783i \(-0.851145\pi\)
−0.892634 + 0.450783i \(0.851145\pi\)
\(660\) 0 0
\(661\) 246.444i 0.372835i 0.982471 + 0.186418i \(0.0596877\pi\)
−0.982471 + 0.186418i \(0.940312\pi\)
\(662\) 0 0
\(663\) 150.988 0.227734
\(664\) 0 0
\(665\) 286.577 141.648i 0.430943 0.213005i
\(666\) 0 0
\(667\) 4.19385i 0.00628763i
\(668\) 0 0
\(669\) −18.6292 −0.0278464
\(670\) 0 0
\(671\) 94.0298i 0.140134i
\(672\) 0 0
\(673\) 76.3328i 0.113422i −0.998391 0.0567109i \(-0.981939\pi\)
0.998391 0.0567109i \(-0.0180613\pi\)
\(674\) 0 0
\(675\) −121.379 + 46.2839i −0.179820 + 0.0685687i
\(676\) 0 0
\(677\) −310.277 −0.458312 −0.229156 0.973390i \(-0.573597\pi\)
−0.229156 + 0.973390i \(0.573597\pi\)
\(678\) 0 0
\(679\) −206.354 726.058i −0.303908 1.06930i
\(680\) 0 0
\(681\) 721.312 1.05920
\(682\) 0 0
\(683\) 157.333i 0.230356i −0.993345 0.115178i \(-0.963256\pi\)
0.993345 0.115178i \(-0.0367438\pi\)
\(684\) 0 0
\(685\) −86.3473 468.793i −0.126054 0.684370i
\(686\) 0 0
\(687\) 672.848i 0.979400i
\(688\) 0 0
\(689\) 374.921i 0.544153i
\(690\) 0 0
\(691\) 683.565i 0.989240i −0.869109 0.494620i \(-0.835307\pi\)
0.869109 0.494620i \(-0.164693\pi\)
\(692\) 0 0
\(693\) −100.535 353.734i −0.145072 0.510439i
\(694\) 0 0
\(695\) −215.187 1168.28i −0.309621 1.68098i
\(696\) 0 0
\(697\) 211.213i 0.303031i
\(698\) 0 0
\(699\) 121.943i 0.174453i
\(700\) 0 0
\(701\) −306.153 −0.436738 −0.218369 0.975866i \(-0.570074\pi\)
−0.218369 + 0.975866i \(0.570074\pi\)
\(702\) 0 0
\(703\) −377.784 −0.537388
\(704\) 0 0
\(705\) −661.441 + 121.831i −0.938214 + 0.172810i
\(706\) 0 0
\(707\) 354.275 100.689i 0.501096 0.142417i
\(708\) 0 0
\(709\) 444.082 0.626350 0.313175 0.949695i \(-0.398607\pi\)
0.313175 + 0.949695i \(0.398607\pi\)
\(710\) 0 0
\(711\) 224.891 0.316302
\(712\) 0 0
\(713\) 212.654 0.298252
\(714\) 0 0
\(715\) −416.366 + 76.6907i −0.582330 + 0.107260i
\(716\) 0 0
\(717\) −542.363 −0.756434
\(718\) 0 0
\(719\) 672.926i 0.935919i 0.883750 + 0.467960i \(0.155011\pi\)
−0.883750 + 0.467960i \(0.844989\pi\)
\(720\) 0 0
\(721\) 138.551 + 487.495i 0.192166 + 0.676137i
\(722\) 0 0
\(723\) 344.206i 0.476080i
\(724\) 0 0
\(725\) 26.2984 10.0280i 0.0362736 0.0138318i
\(726\) 0 0
\(727\) 675.927 0.929748 0.464874 0.885377i \(-0.346100\pi\)
0.464874 + 0.885377i \(0.346100\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 1161.34i 1.58870i
\(732\) 0 0
\(733\) 387.908 0.529206 0.264603 0.964357i \(-0.414759\pi\)
0.264603 + 0.964357i \(0.414759\pi\)
\(734\) 0 0
\(735\) 314.522 284.870i 0.427921 0.387578i
\(736\) 0 0
\(737\) 826.336i 1.12122i
\(738\) 0 0
\(739\) 394.643 0.534024 0.267012 0.963693i \(-0.413964\pi\)
0.267012 + 0.963693i \(0.413964\pi\)
\(740\) 0 0
\(741\) 76.4931i 0.103230i
\(742\) 0 0
\(743\) 147.027i 0.197883i −0.995093 0.0989414i \(-0.968454\pi\)
0.995093 0.0989414i \(-0.0315456\pi\)
\(744\) 0 0
\(745\) 658.151 121.225i 0.883424 0.162718i
\(746\) 0 0
\(747\) 86.3670 0.115618
\(748\) 0 0
\(749\) 1168.65 332.144i 1.56028 0.443449i
\(750\) 0 0
\(751\) −103.003 −0.137154 −0.0685771 0.997646i \(-0.521846\pi\)
−0.0685771 + 0.997646i \(0.521846\pi\)
\(752\) 0 0
\(753\) 521.136i 0.692080i
\(754\) 0 0
\(755\) −762.492 + 140.444i −1.00992 + 0.186018i
\(756\) 0 0
\(757\) 547.747i 0.723575i 0.932261 + 0.361788i \(0.117833\pi\)
−0.932261 + 0.361788i \(0.882167\pi\)
\(758\) 0 0
\(759\) 112.988i 0.148864i
\(760\) 0 0
\(761\) 1223.78i 1.60813i 0.594544 + 0.804063i \(0.297332\pi\)
−0.594544 + 0.804063i \(0.702668\pi\)
\(762\) 0 0
\(763\) 95.6779 + 336.644i 0.125397 + 0.441211i
\(764\) 0 0
\(765\) −265.952 + 48.9859i −0.347650 + 0.0640338i
\(766\) 0 0
\(767\) 420.986i 0.548874i
\(768\) 0 0
\(769\) 194.874i 0.253413i −0.991940 0.126706i \(-0.959559\pi\)
0.991940 0.126706i \(-0.0404406\pi\)
\(770\) 0 0
\(771\) −530.098 −0.687547
\(772\) 0 0
\(773\) 1017.59 1.31642 0.658209 0.752835i \(-0.271314\pi\)
0.658209 + 0.752835i \(0.271314\pi\)
\(774\) 0 0
\(775\) −508.484 1333.49i −0.656108 1.72063i
\(776\) 0 0
\(777\) −482.388 + 137.100i −0.620834 + 0.176448i
\(778\) 0 0
\(779\) 107.004 0.137361
\(780\) 0 0
\(781\) −1021.01 −1.30731
\(782\) 0 0
\(783\) −5.84992 −0.00747117
\(784\) 0 0
\(785\) −514.515 + 94.7689i −0.655434 + 0.120725i
\(786\) 0 0
\(787\) −222.094 −0.282203 −0.141101 0.989995i \(-0.545064\pi\)
−0.141101 + 0.989995i \(0.545064\pi\)
\(788\) 0 0
\(789\) 495.601i 0.628138i
\(790\) 0 0
\(791\) −472.367 + 134.252i −0.597178 + 0.169724i
\(792\) 0 0
\(793\) 25.9636i 0.0327409i
\(794\) 0 0
\(795\) 121.638 + 660.392i 0.153004 + 0.830682i
\(796\) 0 0
\(797\) 340.349 0.427038 0.213519 0.976939i \(-0.431507\pi\)
0.213519 + 0.976939i \(0.431507\pi\)
\(798\) 0 0
\(799\) −1400.11 −1.75233
\(800\) 0 0
\(801\) 304.496i 0.380145i
\(802\) 0 0
\(803\) −935.263 −1.16471
\(804\) 0 0
\(805\) −116.882 + 57.7721i −0.145195 + 0.0717666i
\(806\) 0 0
\(807\) 326.339i 0.404385i
\(808\) 0 0
\(809\) −462.403 −0.571573 −0.285787 0.958293i \(-0.592255\pi\)
−0.285787 + 0.958293i \(0.592255\pi\)
\(810\) 0 0
\(811\) 1420.09i 1.75103i 0.483190 + 0.875516i \(0.339478\pi\)
−0.483190 + 0.875516i \(0.660522\pi\)
\(812\) 0 0
\(813\) 16.6154i 0.0204371i
\(814\) 0 0
\(815\) −118.566 643.714i −0.145480 0.789833i
\(816\) 0 0
\(817\) −588.354 −0.720140
\(818\) 0 0
\(819\) 27.7598 + 97.6733i 0.0338948 + 0.119259i
\(820\) 0 0
\(821\) −651.695 −0.793782 −0.396891 0.917866i \(-0.629911\pi\)
−0.396891 + 0.917866i \(0.629911\pi\)
\(822\) 0 0
\(823\) 604.705i 0.734756i 0.930072 + 0.367378i \(0.119745\pi\)
−0.930072 + 0.367378i \(0.880255\pi\)
\(824\) 0 0
\(825\) 708.512 270.168i 0.858802 0.327477i
\(826\) 0 0
\(827\) 504.280i 0.609771i −0.952389 0.304885i \(-0.901382\pi\)
0.952389 0.304885i \(-0.0986181\pi\)
\(828\) 0 0
\(829\) 814.575i 0.982599i 0.870991 + 0.491300i \(0.163478\pi\)
−0.870991 + 0.491300i \(0.836522\pi\)
\(830\) 0 0
\(831\) 278.974i 0.335709i
\(832\) 0 0
\(833\) 751.344 464.610i 0.901973 0.557755i
\(834\) 0 0
\(835\) 505.718 93.1484i 0.605650 0.111555i
\(836\) 0 0
\(837\) 296.627i 0.354393i
\(838\) 0 0
\(839\) 128.459i 0.153109i 0.997065 + 0.0765547i \(0.0243920\pi\)
−0.997065 + 0.0765547i \(0.975608\pi\)
\(840\) 0 0
\(841\) −839.733 −0.998493
\(842\) 0 0
\(843\) 594.278 0.704956
\(844\) 0 0
\(845\) −716.054 + 131.890i −0.847401 + 0.156083i
\(846\) 0 0
\(847\) 355.288 + 1250.08i 0.419466 + 1.47590i
\(848\) 0 0
\(849\) −562.480 −0.662520
\(850\) 0 0
\(851\) 154.081 0.181059
\(852\) 0 0
\(853\) −636.175 −0.745808 −0.372904 0.927870i \(-0.621638\pi\)
−0.372904 + 0.927870i \(0.621638\pi\)
\(854\) 0 0
\(855\) −24.8171 134.736i −0.0290259 0.157586i
\(856\) 0 0
\(857\) −494.679 −0.577221 −0.288611 0.957447i \(-0.593193\pi\)
−0.288611 + 0.957447i \(0.593193\pi\)
\(858\) 0 0
\(859\) 745.414i 0.867770i 0.900968 + 0.433885i \(0.142858\pi\)
−0.900968 + 0.433885i \(0.857142\pi\)
\(860\) 0 0
\(861\) 136.632 38.8325i 0.158690 0.0451016i
\(862\) 0 0
\(863\) 593.451i 0.687661i 0.939032 + 0.343830i \(0.111725\pi\)
−0.939032 + 0.343830i \(0.888275\pi\)
\(864\) 0 0
\(865\) 970.828 178.817i 1.12234 0.206725i
\(866\) 0 0
\(867\) −62.3935 −0.0719648
\(868\) 0 0
\(869\) −1312.73 −1.51063
\(870\) 0 0
\(871\) 228.168i 0.261961i
\(872\) 0 0
\(873\) −323.491 −0.370551
\(874\) 0 0
\(875\) 641.753 + 594.793i 0.733432 + 0.679763i
\(876\) 0 0
\(877\) 291.879i 0.332815i 0.986057 + 0.166407i \(0.0532167\pi\)
−0.986057 + 0.166407i \(0.946783\pi\)
\(878\) 0 0
\(879\) −174.795 −0.198856
\(880\) 0 0
\(881\) 1226.42i 1.39208i 0.718004 + 0.696039i \(0.245056\pi\)
−0.718004 + 0.696039i \(0.754944\pi\)
\(882\) 0 0
\(883\) 1132.03i 1.28202i −0.767531 0.641012i \(-0.778515\pi\)
0.767531 0.641012i \(-0.221485\pi\)
\(884\) 0 0
\(885\) −136.583 741.532i −0.154331 0.837889i
\(886\) 0 0
\(887\) −959.088 −1.08127 −0.540636 0.841257i \(-0.681816\pi\)
−0.540636 + 0.841257i \(0.681816\pi\)
\(888\) 0 0
\(889\) 1332.83 378.806i 1.49925 0.426103i
\(890\) 0 0
\(891\) −157.604 −0.176885
\(892\) 0 0
\(893\) 709.321i 0.794312i
\(894\) 0 0
\(895\) 584.280 107.619i 0.652827 0.120245i
\(896\) 0 0
\(897\) 31.1982i 0.0347806i
\(898\) 0 0
\(899\) 64.2684i 0.0714887i
\(900\) 0 0
\(901\) 1397.89i 1.55149i
\(902\) 0 0
\(903\) −751.263 + 213.517i −0.831964 + 0.236453i
\(904\) 0 0
\(905\) 13.0558 + 70.8817i 0.0144262 + 0.0783224i
\(906\) 0 0
\(907\) 989.799i 1.09129i 0.838017 + 0.545644i \(0.183715\pi\)
−0.838017 + 0.545644i \(0.816285\pi\)
\(908\) 0 0
\(909\) 157.845i 0.173647i
\(910\) 0 0
\(911\) −994.781 −1.09197 −0.545983 0.837796i \(-0.683844\pi\)
−0.545983 + 0.837796i \(0.683844\pi\)
\(912\) 0 0
\(913\) −504.141 −0.552181
\(914\) 0 0
\(915\) −8.42351 45.7326i −0.00920602 0.0499810i
\(916\) 0 0
\(917\) −773.665 + 219.884i −0.843691 + 0.239786i
\(918\) 0 0
\(919\) 887.212 0.965410 0.482705 0.875783i \(-0.339654\pi\)
0.482705 + 0.875783i \(0.339654\pi\)
\(920\) 0 0
\(921\) 71.5436 0.0776804
\(922\) 0 0
\(923\) 281.921 0.305440
\(924\) 0 0
\(925\) −368.429 966.199i −0.398302 1.04454i
\(926\) 0 0
\(927\) 217.201 0.234305
\(928\) 0 0
\(929\) 1224.49i 1.31808i −0.752109 0.659039i \(-0.770963\pi\)
0.752109 0.659039i \(-0.229037\pi\)
\(930\) 0 0
\(931\) 235.380 + 380.644i 0.252825 + 0.408855i
\(932\) 0 0
\(933\) 814.655i 0.873156i
\(934\) 0 0
\(935\) 1552.42 285.940i 1.66034 0.305819i
\(936\) 0 0
\(937\) 807.001 0.861260 0.430630 0.902529i \(-0.358291\pi\)
0.430630 + 0.902529i \(0.358291\pi\)
\(938\) 0 0
\(939\) 277.508 0.295536
\(940\) 0 0
\(941\) 1246.48i 1.32464i 0.749223 + 0.662318i \(0.230427\pi\)
−0.749223 + 0.662318i \(0.769573\pi\)
\(942\) 0 0
\(943\) −43.6423 −0.0462802
\(944\) 0 0
\(945\) −80.5853 163.037i −0.0852754 0.172526i
\(946\) 0 0
\(947\) 675.978i 0.713810i −0.934141 0.356905i \(-0.883832\pi\)
0.934141 0.356905i \(-0.116168\pi\)
\(948\) 0 0
\(949\) 258.245 0.272123
\(950\) 0 0
\(951\) 465.517i 0.489502i
\(952\) 0 0
\(953\) 1098.24i 1.15240i 0.817308 + 0.576201i \(0.195466\pi\)
−0.817308 + 0.576201i \(0.804534\pi\)
\(954\) 0 0
\(955\) −294.730 + 54.2864i −0.308618 + 0.0568444i
\(956\) 0 0
\(957\) 34.1472 0.0356815
\(958\) 0 0
\(959\) 641.928 182.443i 0.669372 0.190243i
\(960\) 0 0
\(961\) −2297.80 −2.39105
\(962\) 0 0
\(963\) 520.686i 0.540692i
\(964\) 0 0
\(965\) −222.759 1209.40i −0.230839 1.25326i
\(966\) 0 0
\(967\) 1345.41i 1.39132i −0.718370 0.695661i \(-0.755111\pi\)
0.718370 0.695661i \(-0.244889\pi\)
\(968\) 0 0
\(969\) 285.204i 0.294328i
\(970\) 0 0
\(971\) 1230.36i 1.26710i 0.773701 + 0.633551i \(0.218403\pi\)
−0.773701 + 0.633551i \(0.781597\pi\)
\(972\) 0 0
\(973\) 1599.75 454.668i 1.64415 0.467284i
\(974\) 0 0
\(975\) −195.635 + 74.5990i −0.200651 + 0.0765118i
\(976\) 0 0
\(977\) 228.116i 0.233487i −0.993162 0.116743i \(-0.962755\pi\)
0.993162 0.116743i \(-0.0372455\pi\)
\(978\) 0 0
\(979\) 1777.40i 1.81553i
\(980\) 0 0
\(981\) 149.990 0.152895
\(982\) 0 0
\(983\) 1721.09 1.75085 0.875425 0.483354i \(-0.160582\pi\)
0.875425 + 0.483354i \(0.160582\pi\)
\(984\) 0 0
\(985\) −74.3227 403.510i −0.0754545 0.409654i
\(986\) 0 0
\(987\) −257.417 905.724i −0.260807 0.917654i
\(988\) 0 0
\(989\) 239.964 0.242633
\(990\) 0 0
\(991\) 1911.05 1.92841 0.964205 0.265159i \(-0.0854244\pi\)
0.964205 + 0.265159i \(0.0854244\pi\)
\(992\) 0 0
\(993\) 961.319 0.968095
\(994\) 0 0
\(995\) −262.059 1422.76i −0.263376 1.42991i
\(996\) 0 0
\(997\) −31.3627 −0.0314571 −0.0157285 0.999876i \(-0.505007\pi\)
−0.0157285 + 0.999876i \(0.505007\pi\)
\(998\) 0 0
\(999\) 214.925i 0.215141i
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 1680.3.bd.a.769.7 16
4.3 odd 2 210.3.h.a.139.16 yes 16
5.4 even 2 inner 1680.3.bd.a.769.9 16
7.6 odd 2 inner 1680.3.bd.a.769.10 16
12.11 even 2 630.3.h.e.559.1 16
20.3 even 4 1050.3.f.e.601.13 16
20.7 even 4 1050.3.f.e.601.4 16
20.19 odd 2 210.3.h.a.139.1 16
28.27 even 2 210.3.h.a.139.9 yes 16
35.34 odd 2 inner 1680.3.bd.a.769.8 16
60.59 even 2 630.3.h.e.559.16 16
84.83 odd 2 630.3.h.e.559.8 16
140.27 odd 4 1050.3.f.e.601.8 16
140.83 odd 4 1050.3.f.e.601.9 16
140.139 even 2 210.3.h.a.139.8 yes 16
420.419 odd 2 630.3.h.e.559.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.h.a.139.1 16 20.19 odd 2
210.3.h.a.139.8 yes 16 140.139 even 2
210.3.h.a.139.9 yes 16 28.27 even 2
210.3.h.a.139.16 yes 16 4.3 odd 2
630.3.h.e.559.1 16 12.11 even 2
630.3.h.e.559.8 16 84.83 odd 2
630.3.h.e.559.9 16 420.419 odd 2
630.3.h.e.559.16 16 60.59 even 2
1050.3.f.e.601.4 16 20.7 even 4
1050.3.f.e.601.8 16 140.27 odd 4
1050.3.f.e.601.9 16 140.83 odd 4
1050.3.f.e.601.13 16 20.3 even 4
1680.3.bd.a.769.7 16 1.1 even 1 trivial
1680.3.bd.a.769.8 16 35.34 odd 2 inner
1680.3.bd.a.769.9 16 5.4 even 2 inner
1680.3.bd.a.769.10 16 7.6 odd 2 inner