Properties

Label 1680.3.bd.a.769.5
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 769.5
Root \(0.500000 + 4.96598i\) of defining polynomial
Character \(\chi\) \(=\) 1680.769
Dual form 1680.3.bd.a.769.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +(-1.38028 - 4.80571i) q^{5} +(-5.24961 + 4.63050i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(-1.38028 - 4.80571i) q^{5} +(-5.24961 + 4.63050i) q^{7} +3.00000 q^{9} -11.7671 q^{11} -24.8280 q^{13} +(2.39072 + 8.32373i) q^{15} -7.26100 q^{17} +23.0050i q^{19} +(9.09260 - 8.02027i) q^{21} +26.4223i q^{23} +(-21.1896 + 13.2665i) q^{25} -5.19615 q^{27} +57.0861 q^{29} -10.2222i q^{31} +20.3812 q^{33} +(29.4988 + 18.8367i) q^{35} -14.2196i q^{37} +43.0034 q^{39} +16.2271i q^{41} -82.3114i q^{43} +(-4.14085 - 14.4171i) q^{45} -51.6631 q^{47} +(6.11686 - 48.6167i) q^{49} +12.5764 q^{51} +64.8273i q^{53} +(16.2419 + 56.5491i) q^{55} -39.8459i q^{57} -81.7685i q^{59} -13.1240i q^{61} +(-15.7488 + 13.8915i) q^{63} +(34.2697 + 119.316i) q^{65} +22.4035i q^{67} -45.7648i q^{69} -91.5022 q^{71} +71.9256 q^{73} +(36.7015 - 22.9782i) q^{75} +(61.7726 - 54.4875i) q^{77} -45.2802 q^{79} +9.00000 q^{81} +17.7328 q^{83} +(10.0222 + 34.8943i) q^{85} -98.8761 q^{87} +77.5800i q^{89} +(130.338 - 114.966i) q^{91} +17.7054i q^{93} +(110.555 - 31.7534i) q^{95} +6.15741 q^{97} -35.3012 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9}+O(q^{10}) \) 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
\(5\) −1.38028 4.80571i −0.276057 0.961141i
\(6\) 0 0
\(7\) −5.24961 + 4.63050i −0.749945 + 0.661501i
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) −11.7671 −1.06973 −0.534867 0.844936i \(-0.679638\pi\)
−0.534867 + 0.844936i \(0.679638\pi\)
\(12\) 0 0
\(13\) −24.8280 −1.90985 −0.954925 0.296848i \(-0.904065\pi\)
−0.954925 + 0.296848i \(0.904065\pi\)
\(14\) 0 0
\(15\) 2.39072 + 8.32373i 0.159381 + 0.554915i
\(16\) 0 0
\(17\) −7.26100 −0.427118 −0.213559 0.976930i \(-0.568506\pi\)
−0.213559 + 0.976930i \(0.568506\pi\)
\(18\) 0 0
\(19\) 23.0050i 1.21079i 0.795925 + 0.605395i \(0.206985\pi\)
−0.795925 + 0.605395i \(0.793015\pi\)
\(20\) 0 0
\(21\) 9.09260 8.02027i 0.432981 0.381918i
\(22\) 0 0
\(23\) 26.4223i 1.14880i 0.818576 + 0.574398i \(0.194764\pi\)
−0.818576 + 0.574398i \(0.805236\pi\)
\(24\) 0 0
\(25\) −21.1896 + 13.2665i −0.847586 + 0.530659i
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 57.0861 1.96849 0.984243 0.176819i \(-0.0565806\pi\)
0.984243 + 0.176819i \(0.0565806\pi\)
\(30\) 0 0
\(31\) 10.2222i 0.329749i −0.986315 0.164874i \(-0.947278\pi\)
0.986315 0.164874i \(-0.0527219\pi\)
\(32\) 0 0
\(33\) 20.3812 0.617612
\(34\) 0 0
\(35\) 29.4988 + 18.8367i 0.842823 + 0.538191i
\(36\) 0 0
\(37\) 14.2196i 0.384314i −0.981364 0.192157i \(-0.938452\pi\)
0.981364 0.192157i \(-0.0615482\pi\)
\(38\) 0 0
\(39\) 43.0034 1.10265
\(40\) 0 0
\(41\) 16.2271i 0.395782i 0.980224 + 0.197891i \(0.0634093\pi\)
−0.980224 + 0.197891i \(0.936591\pi\)
\(42\) 0 0
\(43\) 82.3114i 1.91422i −0.289725 0.957110i \(-0.593564\pi\)
0.289725 0.957110i \(-0.406436\pi\)
\(44\) 0 0
\(45\) −4.14085 14.4171i −0.0920188 0.320380i
\(46\) 0 0
\(47\) −51.6631 −1.09922 −0.549608 0.835423i \(-0.685223\pi\)
−0.549608 + 0.835423i \(0.685223\pi\)
\(48\) 0 0
\(49\) 6.11686 48.6167i 0.124834 0.992178i
\(50\) 0 0
\(51\) 12.5764 0.246597
\(52\) 0 0
\(53\) 64.8273i 1.22316i 0.791184 + 0.611579i \(0.209465\pi\)
−0.791184 + 0.611579i \(0.790535\pi\)
\(54\) 0 0
\(55\) 16.2419 + 56.5491i 0.295307 + 1.02817i
\(56\) 0 0
\(57\) 39.8459i 0.699050i
\(58\) 0 0
\(59\) 81.7685i 1.38591i −0.720982 0.692953i \(-0.756309\pi\)
0.720982 0.692953i \(-0.243691\pi\)
\(60\) 0 0
\(61\) 13.1240i 0.215148i −0.994197 0.107574i \(-0.965692\pi\)
0.994197 0.107574i \(-0.0343083\pi\)
\(62\) 0 0
\(63\) −15.7488 + 13.8915i −0.249982 + 0.220500i
\(64\) 0 0
\(65\) 34.2697 + 119.316i 0.527226 + 1.83564i
\(66\) 0 0
\(67\) 22.4035i 0.334381i 0.985925 + 0.167190i \(0.0534695\pi\)
−0.985925 + 0.167190i \(0.946531\pi\)
\(68\) 0 0
\(69\) 45.7648i 0.663257i
\(70\) 0 0
\(71\) −91.5022 −1.28876 −0.644382 0.764704i \(-0.722885\pi\)
−0.644382 + 0.764704i \(0.722885\pi\)
\(72\) 0 0
\(73\) 71.9256 0.985282 0.492641 0.870233i \(-0.336032\pi\)
0.492641 + 0.870233i \(0.336032\pi\)
\(74\) 0 0
\(75\) 36.7015 22.9782i 0.489354 0.306376i
\(76\) 0 0
\(77\) 61.7726 54.4875i 0.802242 0.707630i
\(78\) 0 0
\(79\) −45.2802 −0.573167 −0.286584 0.958055i \(-0.592520\pi\)
−0.286584 + 0.958055i \(0.592520\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 17.7328 0.213648 0.106824 0.994278i \(-0.465932\pi\)
0.106824 + 0.994278i \(0.465932\pi\)
\(84\) 0 0
\(85\) 10.0222 + 34.8943i 0.117909 + 0.410521i
\(86\) 0 0
\(87\) −98.8761 −1.13651
\(88\) 0 0
\(89\) 77.5800i 0.871686i 0.900023 + 0.435843i \(0.143550\pi\)
−0.900023 + 0.435843i \(0.856450\pi\)
\(90\) 0 0
\(91\) 130.338 114.966i 1.43228 1.26337i
\(92\) 0 0
\(93\) 17.7054i 0.190381i
\(94\) 0 0
\(95\) 110.555 31.7534i 1.16374 0.334247i
\(96\) 0 0
\(97\) 6.15741 0.0634785 0.0317392 0.999496i \(-0.489895\pi\)
0.0317392 + 0.999496i \(0.489895\pi\)
\(98\) 0 0
\(99\) −35.3012 −0.356578
\(100\) 0 0
\(101\) 142.983i 1.41568i −0.706374 0.707839i \(-0.749670\pi\)
0.706374 0.707839i \(-0.250330\pi\)
\(102\) 0 0
\(103\) 14.5292 0.141060 0.0705302 0.997510i \(-0.477531\pi\)
0.0705302 + 0.997510i \(0.477531\pi\)
\(104\) 0 0
\(105\) −51.0934 32.6261i −0.486604 0.310725i
\(106\) 0 0
\(107\) 78.4228i 0.732923i 0.930433 + 0.366462i \(0.119431\pi\)
−0.930433 + 0.366462i \(0.880569\pi\)
\(108\) 0 0
\(109\) −74.7424 −0.685710 −0.342855 0.939388i \(-0.611394\pi\)
−0.342855 + 0.939388i \(0.611394\pi\)
\(110\) 0 0
\(111\) 24.6291i 0.221884i
\(112\) 0 0
\(113\) 85.2417i 0.754351i −0.926142 0.377175i \(-0.876895\pi\)
0.926142 0.377175i \(-0.123105\pi\)
\(114\) 0 0
\(115\) 126.978 36.4702i 1.10415 0.317132i
\(116\) 0 0
\(117\) −74.4841 −0.636617
\(118\) 0 0
\(119\) 38.1175 33.6221i 0.320315 0.282539i
\(120\) 0 0
\(121\) 17.4642 0.144332
\(122\) 0 0
\(123\) 28.1061i 0.228505i
\(124\) 0 0
\(125\) 93.0025 + 83.5197i 0.744020 + 0.668158i
\(126\) 0 0
\(127\) 66.7734i 0.525775i 0.964827 + 0.262887i \(0.0846747\pi\)
−0.964827 + 0.262887i \(0.915325\pi\)
\(128\) 0 0
\(129\) 142.568i 1.10518i
\(130\) 0 0
\(131\) 28.8330i 0.220099i −0.993926 0.110049i \(-0.964899\pi\)
0.993926 0.110049i \(-0.0351009\pi\)
\(132\) 0 0
\(133\) −106.525 120.767i −0.800939 0.908026i
\(134\) 0 0
\(135\) 7.17216 + 24.9712i 0.0531271 + 0.184972i
\(136\) 0 0
\(137\) 119.450i 0.871897i −0.899972 0.435948i \(-0.856413\pi\)
0.899972 0.435948i \(-0.143587\pi\)
\(138\) 0 0
\(139\) 200.948i 1.44567i −0.691021 0.722834i \(-0.742839\pi\)
0.691021 0.722834i \(-0.257161\pi\)
\(140\) 0 0
\(141\) 89.4832 0.634633
\(142\) 0 0
\(143\) 292.154 2.04303
\(144\) 0 0
\(145\) −78.7950 274.339i −0.543414 1.89199i
\(146\) 0 0
\(147\) −10.5947 + 84.2066i −0.0720729 + 0.572834i
\(148\) 0 0
\(149\) −5.70484 −0.0382875 −0.0191438 0.999817i \(-0.506094\pi\)
−0.0191438 + 0.999817i \(0.506094\pi\)
\(150\) 0 0
\(151\) 216.490 1.43371 0.716854 0.697223i \(-0.245581\pi\)
0.716854 + 0.697223i \(0.245581\pi\)
\(152\) 0 0
\(153\) −21.7830 −0.142373
\(154\) 0 0
\(155\) −49.1250 + 14.1095i −0.316935 + 0.0910293i
\(156\) 0 0
\(157\) 108.287 0.689726 0.344863 0.938653i \(-0.387925\pi\)
0.344863 + 0.938653i \(0.387925\pi\)
\(158\) 0 0
\(159\) 112.284i 0.706190i
\(160\) 0 0
\(161\) −122.349 138.707i −0.759929 0.861533i
\(162\) 0 0
\(163\) 234.395i 1.43800i −0.695008 0.719002i \(-0.744599\pi\)
0.695008 0.719002i \(-0.255401\pi\)
\(164\) 0 0
\(165\) −28.1318 97.9460i −0.170496 0.593612i
\(166\) 0 0
\(167\) 41.8830 0.250796 0.125398 0.992106i \(-0.459979\pi\)
0.125398 + 0.992106i \(0.459979\pi\)
\(168\) 0 0
\(169\) 447.432 2.64753
\(170\) 0 0
\(171\) 69.0151i 0.403597i
\(172\) 0 0
\(173\) −13.0301 −0.0753183 −0.0376592 0.999291i \(-0.511990\pi\)
−0.0376592 + 0.999291i \(0.511990\pi\)
\(174\) 0 0
\(175\) 49.8070 167.763i 0.284611 0.958643i
\(176\) 0 0
\(177\) 141.627i 0.800154i
\(178\) 0 0
\(179\) 24.1573 0.134957 0.0674786 0.997721i \(-0.478505\pi\)
0.0674786 + 0.997721i \(0.478505\pi\)
\(180\) 0 0
\(181\) 43.5107i 0.240390i 0.992750 + 0.120195i \(0.0383521\pi\)
−0.992750 + 0.120195i \(0.961648\pi\)
\(182\) 0 0
\(183\) 22.7315i 0.124216i
\(184\) 0 0
\(185\) −68.3353 + 19.6271i −0.369380 + 0.106092i
\(186\) 0 0
\(187\) 85.4408 0.456903
\(188\) 0 0
\(189\) 27.2778 24.0608i 0.144327 0.127306i
\(190\) 0 0
\(191\) 246.572 1.29095 0.645477 0.763780i \(-0.276659\pi\)
0.645477 + 0.763780i \(0.276659\pi\)
\(192\) 0 0
\(193\) 161.875i 0.838733i 0.907817 + 0.419366i \(0.137748\pi\)
−0.907817 + 0.419366i \(0.862252\pi\)
\(194\) 0 0
\(195\) −59.3569 206.662i −0.304394 1.05980i
\(196\) 0 0
\(197\) 44.8380i 0.227604i 0.993503 + 0.113802i \(0.0363029\pi\)
−0.993503 + 0.113802i \(0.963697\pi\)
\(198\) 0 0
\(199\) 114.767i 0.576718i 0.957522 + 0.288359i \(0.0931097\pi\)
−0.957522 + 0.288359i \(0.906890\pi\)
\(200\) 0 0
\(201\) 38.8040i 0.193055i
\(202\) 0 0
\(203\) −299.680 + 264.338i −1.47626 + 1.30216i
\(204\) 0 0
\(205\) 77.9826 22.3980i 0.380403 0.109258i
\(206\) 0 0
\(207\) 79.2669i 0.382932i
\(208\) 0 0
\(209\) 270.702i 1.29522i
\(210\) 0 0
\(211\) 219.602 1.04077 0.520384 0.853932i \(-0.325789\pi\)
0.520384 + 0.853932i \(0.325789\pi\)
\(212\) 0 0
\(213\) 158.486 0.744068
\(214\) 0 0
\(215\) −395.565 + 113.613i −1.83984 + 0.528433i
\(216\) 0 0
\(217\) 47.3340 + 53.6626i 0.218129 + 0.247293i
\(218\) 0 0
\(219\) −124.579 −0.568853
\(220\) 0 0
\(221\) 180.277 0.815731
\(222\) 0 0
\(223\) −3.57102 −0.0160136 −0.00800678 0.999968i \(-0.502549\pi\)
−0.00800678 + 0.999968i \(0.502549\pi\)
\(224\) 0 0
\(225\) −63.5689 + 39.7994i −0.282529 + 0.176886i
\(226\) 0 0
\(227\) 68.0647 0.299844 0.149922 0.988698i \(-0.452098\pi\)
0.149922 + 0.988698i \(0.452098\pi\)
\(228\) 0 0
\(229\) 180.019i 0.786109i 0.919515 + 0.393054i \(0.128582\pi\)
−0.919515 + 0.393054i \(0.871418\pi\)
\(230\) 0 0
\(231\) −106.993 + 94.3752i −0.463175 + 0.408550i
\(232\) 0 0
\(233\) 115.207i 0.494449i 0.968958 + 0.247225i \(0.0795185\pi\)
−0.968958 + 0.247225i \(0.920481\pi\)
\(234\) 0 0
\(235\) 71.3097 + 248.278i 0.303446 + 1.05650i
\(236\) 0 0
\(237\) 78.4276 0.330918
\(238\) 0 0
\(239\) −15.6873 −0.0656374 −0.0328187 0.999461i \(-0.510448\pi\)
−0.0328187 + 0.999461i \(0.510448\pi\)
\(240\) 0 0
\(241\) 303.442i 1.25909i 0.776962 + 0.629547i \(0.216760\pi\)
−0.776962 + 0.629547i \(0.783240\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) −242.081 + 37.7089i −0.988084 + 0.153914i
\(246\) 0 0
\(247\) 571.170i 2.31243i
\(248\) 0 0
\(249\) −30.7141 −0.123350
\(250\) 0 0
\(251\) 80.5241i 0.320813i 0.987051 + 0.160407i \(0.0512805\pi\)
−0.987051 + 0.160407i \(0.948719\pi\)
\(252\) 0 0
\(253\) 310.913i 1.22891i
\(254\) 0 0
\(255\) −17.3590 60.4386i −0.0680746 0.237014i
\(256\) 0 0
\(257\) −39.1012 −0.152145 −0.0760723 0.997102i \(-0.524238\pi\)
−0.0760723 + 0.997102i \(0.524238\pi\)
\(258\) 0 0
\(259\) 65.8439 + 74.6474i 0.254224 + 0.288214i
\(260\) 0 0
\(261\) 171.258 0.656162
\(262\) 0 0
\(263\) 79.1644i 0.301005i 0.988610 + 0.150503i \(0.0480892\pi\)
−0.988610 + 0.150503i \(0.951911\pi\)
\(264\) 0 0
\(265\) 311.541 89.4801i 1.17563 0.337661i
\(266\) 0 0
\(267\) 134.373i 0.503268i
\(268\) 0 0
\(269\) 53.2775i 0.198058i −0.995085 0.0990288i \(-0.968426\pi\)
0.995085 0.0990288i \(-0.0315736\pi\)
\(270\) 0 0
\(271\) 237.638i 0.876892i 0.898757 + 0.438446i \(0.144471\pi\)
−0.898757 + 0.438446i \(0.855529\pi\)
\(272\) 0 0
\(273\) −225.751 + 199.128i −0.826928 + 0.729405i
\(274\) 0 0
\(275\) 249.340 156.108i 0.906692 0.567664i
\(276\) 0 0
\(277\) 168.185i 0.607167i 0.952805 + 0.303583i \(0.0981831\pi\)
−0.952805 + 0.303583i \(0.901817\pi\)
\(278\) 0 0
\(279\) 30.6666i 0.109916i
\(280\) 0 0
\(281\) 128.175 0.456140 0.228070 0.973645i \(-0.426758\pi\)
0.228070 + 0.973645i \(0.426758\pi\)
\(282\) 0 0
\(283\) 72.7293 0.256994 0.128497 0.991710i \(-0.458985\pi\)
0.128497 + 0.991710i \(0.458985\pi\)
\(284\) 0 0
\(285\) −191.488 + 54.9986i −0.671886 + 0.192977i
\(286\) 0 0
\(287\) −75.1396 85.1859i −0.261810 0.296815i
\(288\) 0 0
\(289\) −236.278 −0.817570
\(290\) 0 0
\(291\) −10.6649 −0.0366493
\(292\) 0 0
\(293\) 273.216 0.932477 0.466238 0.884659i \(-0.345609\pi\)
0.466238 + 0.884659i \(0.345609\pi\)
\(294\) 0 0
\(295\) −392.955 + 112.864i −1.33205 + 0.382589i
\(296\) 0 0
\(297\) 61.1436 0.205871
\(298\) 0 0
\(299\) 656.014i 2.19403i
\(300\) 0 0
\(301\) 381.143 + 432.103i 1.26626 + 1.43556i
\(302\) 0 0
\(303\) 247.655i 0.817342i
\(304\) 0 0
\(305\) −63.0703 + 18.1149i −0.206788 + 0.0593931i
\(306\) 0 0
\(307\) 336.086 1.09474 0.547372 0.836889i \(-0.315628\pi\)
0.547372 + 0.836889i \(0.315628\pi\)
\(308\) 0 0
\(309\) −25.1653 −0.0814412
\(310\) 0 0
\(311\) 497.276i 1.59896i 0.600693 + 0.799480i \(0.294891\pi\)
−0.600693 + 0.799480i \(0.705109\pi\)
\(312\) 0 0
\(313\) 252.024 0.805188 0.402594 0.915379i \(-0.368109\pi\)
0.402594 + 0.915379i \(0.368109\pi\)
\(314\) 0 0
\(315\) 88.4964 + 56.5101i 0.280941 + 0.179397i
\(316\) 0 0
\(317\) 270.203i 0.852377i 0.904634 + 0.426188i \(0.140144\pi\)
−0.904634 + 0.426188i \(0.859856\pi\)
\(318\) 0 0
\(319\) −671.737 −2.10576
\(320\) 0 0
\(321\) 135.832i 0.423153i
\(322\) 0 0
\(323\) 167.040i 0.517150i
\(324\) 0 0
\(325\) 526.097 329.380i 1.61876 1.01348i
\(326\) 0 0
\(327\) 129.458 0.395895
\(328\) 0 0
\(329\) 271.211 239.226i 0.824351 0.727132i
\(330\) 0 0
\(331\) 306.813 0.926926 0.463463 0.886116i \(-0.346607\pi\)
0.463463 + 0.886116i \(0.346607\pi\)
\(332\) 0 0
\(333\) 42.6588i 0.128105i
\(334\) 0 0
\(335\) 107.665 30.9232i 0.321387 0.0923080i
\(336\) 0 0
\(337\) 485.903i 1.44185i −0.693014 0.720924i \(-0.743718\pi\)
0.693014 0.720924i \(-0.256282\pi\)
\(338\) 0 0
\(339\) 147.643i 0.435525i
\(340\) 0 0
\(341\) 120.286i 0.352744i
\(342\) 0 0
\(343\) 193.009 + 283.543i 0.562708 + 0.826656i
\(344\) 0 0
\(345\) −219.932 + 63.1683i −0.637484 + 0.183097i
\(346\) 0 0
\(347\) 8.02712i 0.0231329i −0.999933 0.0115665i \(-0.996318\pi\)
0.999933 0.0115665i \(-0.00368180\pi\)
\(348\) 0 0
\(349\) 649.899i 1.86217i −0.364798 0.931087i \(-0.618862\pi\)
0.364798 0.931087i \(-0.381138\pi\)
\(350\) 0 0
\(351\) 129.010 0.367551
\(352\) 0 0
\(353\) −396.496 −1.12322 −0.561608 0.827403i \(-0.689817\pi\)
−0.561608 + 0.827403i \(0.689817\pi\)
\(354\) 0 0
\(355\) 126.299 + 439.733i 0.355772 + 1.23868i
\(356\) 0 0
\(357\) −66.0214 + 58.2352i −0.184934 + 0.163124i
\(358\) 0 0
\(359\) −398.981 −1.11137 −0.555684 0.831394i \(-0.687543\pi\)
−0.555684 + 0.831394i \(0.687543\pi\)
\(360\) 0 0
\(361\) −168.231 −0.466014
\(362\) 0 0
\(363\) −30.2489 −0.0833303
\(364\) 0 0
\(365\) −99.2777 345.653i −0.271994 0.946996i
\(366\) 0 0
\(367\) −506.200 −1.37929 −0.689646 0.724147i \(-0.742234\pi\)
−0.689646 + 0.724147i \(0.742234\pi\)
\(368\) 0 0
\(369\) 48.6812i 0.131927i
\(370\) 0 0
\(371\) −300.183 340.318i −0.809119 0.917300i
\(372\) 0 0
\(373\) 278.751i 0.747322i −0.927565 0.373661i \(-0.878102\pi\)
0.927565 0.373661i \(-0.121898\pi\)
\(374\) 0 0
\(375\) −161.085 144.660i −0.429560 0.385761i
\(376\) 0 0
\(377\) −1417.34 −3.75951
\(378\) 0 0
\(379\) 515.959 1.36137 0.680685 0.732576i \(-0.261682\pi\)
0.680685 + 0.732576i \(0.261682\pi\)
\(380\) 0 0
\(381\) 115.655i 0.303556i
\(382\) 0 0
\(383\) −238.715 −0.623276 −0.311638 0.950201i \(-0.600878\pi\)
−0.311638 + 0.950201i \(0.600878\pi\)
\(384\) 0 0
\(385\) −347.115 221.653i −0.901597 0.575722i
\(386\) 0 0
\(387\) 246.934i 0.638073i
\(388\) 0 0
\(389\) −170.748 −0.438941 −0.219471 0.975619i \(-0.570433\pi\)
−0.219471 + 0.975619i \(0.570433\pi\)
\(390\) 0 0
\(391\) 191.852i 0.490671i
\(392\) 0 0
\(393\) 49.9402i 0.127074i
\(394\) 0 0
\(395\) 62.4995 + 217.603i 0.158227 + 0.550895i
\(396\) 0 0
\(397\) −80.7184 −0.203321 −0.101660 0.994819i \(-0.532416\pi\)
−0.101660 + 0.994819i \(0.532416\pi\)
\(398\) 0 0
\(399\) 184.506 + 209.175i 0.462422 + 0.524249i
\(400\) 0 0
\(401\) 469.375 1.17051 0.585255 0.810849i \(-0.300994\pi\)
0.585255 + 0.810849i \(0.300994\pi\)
\(402\) 0 0
\(403\) 253.798i 0.629770i
\(404\) 0 0
\(405\) −12.4225 43.2514i −0.0306729 0.106793i
\(406\) 0 0
\(407\) 167.323i 0.411114i
\(408\) 0 0
\(409\) 671.238i 1.64117i 0.571525 + 0.820585i \(0.306352\pi\)
−0.571525 + 0.820585i \(0.693648\pi\)
\(410\) 0 0
\(411\) 206.893i 0.503390i
\(412\) 0 0
\(413\) 378.629 + 429.253i 0.916778 + 1.03935i
\(414\) 0 0
\(415\) −24.4763 85.2185i −0.0589789 0.205346i
\(416\) 0 0
\(417\) 348.052i 0.834657i
\(418\) 0 0
\(419\) 16.1628i 0.0385747i −0.999814 0.0192874i \(-0.993860\pi\)
0.999814 0.0192874i \(-0.00613974\pi\)
\(420\) 0 0
\(421\) −463.356 −1.10061 −0.550304 0.834964i \(-0.685488\pi\)
−0.550304 + 0.834964i \(0.685488\pi\)
\(422\) 0 0
\(423\) −154.989 −0.366405
\(424\) 0 0
\(425\) 153.858 96.3279i 0.362019 0.226654i
\(426\) 0 0
\(427\) 60.7709 + 68.8961i 0.142321 + 0.161349i
\(428\) 0 0
\(429\) −506.025 −1.17955
\(430\) 0 0
\(431\) −425.457 −0.987140 −0.493570 0.869706i \(-0.664308\pi\)
−0.493570 + 0.869706i \(0.664308\pi\)
\(432\) 0 0
\(433\) 480.947 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(434\) 0 0
\(435\) 136.477 + 475.169i 0.313740 + 1.09234i
\(436\) 0 0
\(437\) −607.845 −1.39095
\(438\) 0 0
\(439\) 73.3796i 0.167152i −0.996501 0.0835759i \(-0.973366\pi\)
0.996501 0.0835759i \(-0.0266341\pi\)
\(440\) 0 0
\(441\) 18.3506 145.850i 0.0416113 0.330726i
\(442\) 0 0
\(443\) 72.4361i 0.163513i 0.996652 + 0.0817563i \(0.0260529\pi\)
−0.996652 + 0.0817563i \(0.973947\pi\)
\(444\) 0 0
\(445\) 372.827 107.082i 0.837813 0.240635i
\(446\) 0 0
\(447\) 9.88107 0.0221053
\(448\) 0 0
\(449\) −721.760 −1.60748 −0.803742 0.594978i \(-0.797161\pi\)
−0.803742 + 0.594978i \(0.797161\pi\)
\(450\) 0 0
\(451\) 190.945i 0.423382i
\(452\) 0 0
\(453\) −374.972 −0.827752
\(454\) 0 0
\(455\) −732.397 467.678i −1.60966 1.02786i
\(456\) 0 0
\(457\) 622.396i 1.36192i −0.732322 0.680958i \(-0.761564\pi\)
0.732322 0.680958i \(-0.238436\pi\)
\(458\) 0 0
\(459\) 37.7293 0.0821989
\(460\) 0 0
\(461\) 600.196i 1.30194i 0.759102 + 0.650971i \(0.225638\pi\)
−0.759102 + 0.650971i \(0.774362\pi\)
\(462\) 0 0
\(463\) 127.513i 0.275406i −0.990474 0.137703i \(-0.956028\pi\)
0.990474 0.137703i \(-0.0439720\pi\)
\(464\) 0 0
\(465\) 85.0869 24.4384i 0.182983 0.0525558i
\(466\) 0 0
\(467\) 11.4438 0.0245048 0.0122524 0.999925i \(-0.496100\pi\)
0.0122524 + 0.999925i \(0.496100\pi\)
\(468\) 0 0
\(469\) −103.740 117.610i −0.221193 0.250767i
\(470\) 0 0
\(471\) −187.559 −0.398214
\(472\) 0 0
\(473\) 968.566i 2.04771i
\(474\) 0 0
\(475\) −305.195 487.468i −0.642517 1.02625i
\(476\) 0 0
\(477\) 194.482i 0.407719i
\(478\) 0 0
\(479\) 205.867i 0.429784i −0.976638 0.214892i \(-0.931060\pi\)
0.976638 0.214892i \(-0.0689400\pi\)
\(480\) 0 0
\(481\) 353.045i 0.733981i
\(482\) 0 0
\(483\) 211.914 + 240.247i 0.438745 + 0.497406i
\(484\) 0 0
\(485\) −8.49897 29.5907i −0.0175236 0.0610118i
\(486\) 0 0
\(487\) 603.451i 1.23912i −0.784950 0.619559i \(-0.787311\pi\)
0.784950 0.619559i \(-0.212689\pi\)
\(488\) 0 0
\(489\) 405.983i 0.830232i
\(490\) 0 0
\(491\) 522.576 1.06431 0.532154 0.846647i \(-0.321383\pi\)
0.532154 + 0.846647i \(0.321383\pi\)
\(492\) 0 0
\(493\) −414.503 −0.840776
\(494\) 0 0
\(495\) 48.7257 + 169.647i 0.0984358 + 0.342722i
\(496\) 0 0
\(497\) 480.351 423.701i 0.966501 0.852518i
\(498\) 0 0
\(499\) −124.569 −0.249636 −0.124818 0.992180i \(-0.539835\pi\)
−0.124818 + 0.992180i \(0.539835\pi\)
\(500\) 0 0
\(501\) −72.5435 −0.144797
\(502\) 0 0
\(503\) −33.8355 −0.0672673 −0.0336337 0.999434i \(-0.510708\pi\)
−0.0336337 + 0.999434i \(0.510708\pi\)
\(504\) 0 0
\(505\) −687.137 + 197.358i −1.36067 + 0.390807i
\(506\) 0 0
\(507\) −774.975 −1.52855
\(508\) 0 0
\(509\) 630.938i 1.23956i 0.784774 + 0.619782i \(0.212779\pi\)
−0.784774 + 0.619782i \(0.787221\pi\)
\(510\) 0 0
\(511\) −377.582 + 333.052i −0.738907 + 0.651765i
\(512\) 0 0
\(513\) 119.538i 0.233017i
\(514\) 0 0
\(515\) −20.0544 69.8232i −0.0389406 0.135579i
\(516\) 0 0
\(517\) 607.924 1.17587
\(518\) 0 0
\(519\) 22.5687 0.0434850
\(520\) 0 0
\(521\) 492.821i 0.945914i 0.881086 + 0.472957i \(0.156813\pi\)
−0.881086 + 0.472957i \(0.843187\pi\)
\(522\) 0 0
\(523\) −331.944 −0.634692 −0.317346 0.948310i \(-0.602792\pi\)
−0.317346 + 0.948310i \(0.602792\pi\)
\(524\) 0 0
\(525\) −86.2682 + 290.573i −0.164320 + 0.553473i
\(526\) 0 0
\(527\) 74.2235i 0.140842i
\(528\) 0 0
\(529\) −169.138 −0.319731
\(530\) 0 0
\(531\) 245.306i 0.461969i
\(532\) 0 0
\(533\) 402.887i 0.755885i
\(534\) 0 0
\(535\) 376.877 108.246i 0.704443 0.202328i
\(536\) 0 0
\(537\) −41.8417 −0.0779175
\(538\) 0 0
\(539\) −71.9776 + 572.077i −0.133539 + 1.06137i
\(540\) 0 0
\(541\) −96.8104 −0.178947 −0.0894735 0.995989i \(-0.528518\pi\)
−0.0894735 + 0.995989i \(0.528518\pi\)
\(542\) 0 0
\(543\) 75.3627i 0.138789i
\(544\) 0 0
\(545\) 103.166 + 359.190i 0.189295 + 0.659065i
\(546\) 0 0
\(547\) 772.034i 1.41140i −0.708512 0.705699i \(-0.750633\pi\)
0.708512 0.705699i \(-0.249367\pi\)
\(548\) 0 0
\(549\) 39.3721i 0.0717161i
\(550\) 0 0
\(551\) 1313.27i 2.38343i
\(552\) 0 0
\(553\) 237.704 209.670i 0.429844 0.379150i
\(554\) 0 0
\(555\) 118.360 33.9951i 0.213261 0.0612524i
\(556\) 0 0
\(557\) 42.3682i 0.0760650i 0.999277 + 0.0380325i \(0.0121090\pi\)
−0.999277 + 0.0380325i \(0.987891\pi\)
\(558\) 0 0
\(559\) 2043.63i 3.65587i
\(560\) 0 0
\(561\) −147.988 −0.263793
\(562\) 0 0
\(563\) 370.393 0.657891 0.328946 0.944349i \(-0.393307\pi\)
0.328946 + 0.944349i \(0.393307\pi\)
\(564\) 0 0
\(565\) −409.646 + 117.658i −0.725038 + 0.208244i
\(566\) 0 0
\(567\) −47.2465 + 41.6745i −0.0833272 + 0.0735001i
\(568\) 0 0
\(569\) −419.012 −0.736401 −0.368201 0.929746i \(-0.620026\pi\)
−0.368201 + 0.929746i \(0.620026\pi\)
\(570\) 0 0
\(571\) 113.210 0.198266 0.0991329 0.995074i \(-0.468393\pi\)
0.0991329 + 0.995074i \(0.468393\pi\)
\(572\) 0 0
\(573\) −427.075 −0.745332
\(574\) 0 0
\(575\) −350.531 559.879i −0.609618 0.973702i
\(576\) 0 0
\(577\) 604.749 1.04809 0.524046 0.851690i \(-0.324422\pi\)
0.524046 + 0.851690i \(0.324422\pi\)
\(578\) 0 0
\(579\) 280.376i 0.484242i
\(580\) 0 0
\(581\) −93.0902 + 82.1117i −0.160224 + 0.141328i
\(582\) 0 0
\(583\) 762.829i 1.30845i
\(584\) 0 0
\(585\) 102.809 + 357.949i 0.175742 + 0.611878i
\(586\) 0 0
\(587\) −458.151 −0.780496 −0.390248 0.920710i \(-0.627611\pi\)
−0.390248 + 0.920710i \(0.627611\pi\)
\(588\) 0 0
\(589\) 235.162 0.399257
\(590\) 0 0
\(591\) 77.6616i 0.131407i
\(592\) 0 0
\(593\) 780.457 1.31612 0.658059 0.752967i \(-0.271378\pi\)
0.658059 + 0.752967i \(0.271378\pi\)
\(594\) 0 0
\(595\) −214.191 136.773i −0.359985 0.229871i
\(596\) 0 0
\(597\) 198.782i 0.332968i
\(598\) 0 0
\(599\) 11.7391 0.0195977 0.00979887 0.999952i \(-0.496881\pi\)
0.00979887 + 0.999952i \(0.496881\pi\)
\(600\) 0 0
\(601\) 182.878i 0.304290i 0.988358 + 0.152145i \(0.0486181\pi\)
−0.988358 + 0.152145i \(0.951382\pi\)
\(602\) 0 0
\(603\) 67.2106i 0.111460i
\(604\) 0 0
\(605\) −24.1056 83.9279i −0.0398439 0.138724i
\(606\) 0 0
\(607\) 777.171 1.28035 0.640174 0.768230i \(-0.278862\pi\)
0.640174 + 0.768230i \(0.278862\pi\)
\(608\) 0 0
\(609\) 519.061 457.846i 0.852317 0.751800i
\(610\) 0 0
\(611\) 1282.69 2.09934
\(612\) 0 0
\(613\) 106.169i 0.173196i −0.996243 0.0865978i \(-0.972400\pi\)
0.996243 0.0865978i \(-0.0275995\pi\)
\(614\) 0 0
\(615\) −135.070 + 38.7944i −0.219626 + 0.0630803i
\(616\) 0 0
\(617\) 680.569i 1.10303i −0.834165 0.551515i \(-0.814050\pi\)
0.834165 0.551515i \(-0.185950\pi\)
\(618\) 0 0
\(619\) 861.054i 1.39104i −0.718507 0.695520i \(-0.755174\pi\)
0.718507 0.695520i \(-0.244826\pi\)
\(620\) 0 0
\(621\) 137.294i 0.221086i
\(622\) 0 0
\(623\) −359.235 407.265i −0.576621 0.653716i
\(624\) 0 0
\(625\) 273.002 562.223i 0.436803 0.899557i
\(626\) 0 0
\(627\) 468.870i 0.747798i
\(628\) 0 0
\(629\) 103.249i 0.164147i
\(630\) 0 0
\(631\) 489.247 0.775352 0.387676 0.921796i \(-0.373278\pi\)
0.387676 + 0.921796i \(0.373278\pi\)
\(632\) 0 0
\(633\) −380.362 −0.600888
\(634\) 0 0
\(635\) 320.893 92.1661i 0.505344 0.145144i
\(636\) 0 0
\(637\) −151.870 + 1207.06i −0.238414 + 1.89491i
\(638\) 0 0
\(639\) −274.507 −0.429588
\(640\) 0 0
\(641\) 962.813 1.50205 0.751024 0.660275i \(-0.229560\pi\)
0.751024 + 0.660275i \(0.229560\pi\)
\(642\) 0 0
\(643\) 581.069 0.903684 0.451842 0.892098i \(-0.350767\pi\)
0.451842 + 0.892098i \(0.350767\pi\)
\(644\) 0 0
\(645\) 685.138 196.784i 1.06223 0.305091i
\(646\) 0 0
\(647\) 339.036 0.524012 0.262006 0.965066i \(-0.415616\pi\)
0.262006 + 0.965066i \(0.415616\pi\)
\(648\) 0 0
\(649\) 962.177i 1.48255i
\(650\) 0 0
\(651\) −81.9849 92.9464i −0.125937 0.142775i
\(652\) 0 0
\(653\) 289.383i 0.443159i 0.975142 + 0.221580i \(0.0711213\pi\)
−0.975142 + 0.221580i \(0.928879\pi\)
\(654\) 0 0
\(655\) −138.563 + 39.7976i −0.211546 + 0.0607598i
\(656\) 0 0
\(657\) 215.777 0.328427
\(658\) 0 0
\(659\) −62.6627 −0.0950875 −0.0475438 0.998869i \(-0.515139\pi\)
−0.0475438 + 0.998869i \(0.515139\pi\)
\(660\) 0 0
\(661\) 148.826i 0.225153i 0.993643 + 0.112577i \(0.0359104\pi\)
−0.993643 + 0.112577i \(0.964090\pi\)
\(662\) 0 0
\(663\) −312.248 −0.470962
\(664\) 0 0
\(665\) −433.339 + 678.620i −0.651637 + 1.02048i
\(666\) 0 0
\(667\) 1508.35i 2.26139i
\(668\) 0 0
\(669\) 6.18519 0.00924543
\(670\) 0 0
\(671\) 154.432i 0.230152i
\(672\) 0 0
\(673\) 833.049i 1.23781i −0.785464 0.618907i \(-0.787576\pi\)
0.785464 0.618907i \(-0.212424\pi\)
\(674\) 0 0
\(675\) 110.105 68.9346i 0.163118 0.102125i
\(676\) 0 0
\(677\) 391.042 0.577611 0.288805 0.957388i \(-0.406742\pi\)
0.288805 + 0.957388i \(0.406742\pi\)
\(678\) 0 0
\(679\) −32.3240 + 28.5119i −0.0476053 + 0.0419910i
\(680\) 0 0
\(681\) −117.891 −0.173115
\(682\) 0 0
\(683\) 992.775i 1.45355i 0.686875 + 0.726775i \(0.258982\pi\)
−0.686875 + 0.726775i \(0.741018\pi\)
\(684\) 0 0
\(685\) −574.041 + 164.875i −0.838016 + 0.240693i
\(686\) 0 0
\(687\) 311.802i 0.453860i
\(688\) 0 0
\(689\) 1609.54i 2.33605i
\(690\) 0 0
\(691\) 697.871i 1.00994i 0.863136 + 0.504972i \(0.168497\pi\)
−0.863136 + 0.504972i \(0.831503\pi\)
\(692\) 0 0
\(693\) 185.318 163.463i 0.267414 0.235877i
\(694\) 0 0
\(695\) −965.697 + 277.365i −1.38949 + 0.399086i
\(696\) 0 0
\(697\) 117.825i 0.169046i
\(698\) 0 0
\(699\) 199.544i 0.285470i
\(700\) 0 0
\(701\) −222.920 −0.318003 −0.159002 0.987278i \(-0.550828\pi\)
−0.159002 + 0.987278i \(0.550828\pi\)
\(702\) 0 0
\(703\) 327.122 0.465323
\(704\) 0 0
\(705\) −123.512 430.030i −0.175194 0.609972i
\(706\) 0 0
\(707\) 662.085 + 750.608i 0.936472 + 1.06168i
\(708\) 0 0
\(709\) −267.679 −0.377544 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(710\) 0 0
\(711\) −135.841 −0.191056
\(712\) 0 0
\(713\) 270.094 0.378814
\(714\) 0 0
\(715\) −403.255 1404.00i −0.563992 1.96364i
\(716\) 0 0
\(717\) 27.1713 0.0378958
\(718\) 0 0
\(719\) 217.258i 0.302167i 0.988521 + 0.151083i \(0.0482762\pi\)
−0.988521 + 0.151083i \(0.951724\pi\)
\(720\) 0 0
\(721\) −76.2728 + 67.2776i −0.105787 + 0.0933115i
\(722\) 0 0
\(723\) 525.577i 0.726939i
\(724\) 0 0
\(725\) −1209.63 + 757.331i −1.66846 + 1.04459i
\(726\) 0 0
\(727\) −771.678 −1.06146 −0.530728 0.847543i \(-0.678081\pi\)
−0.530728 + 0.847543i \(0.678081\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 597.664i 0.817598i
\(732\) 0 0
\(733\) −1103.98 −1.50611 −0.753053 0.657960i \(-0.771420\pi\)
−0.753053 + 0.657960i \(0.771420\pi\)
\(734\) 0 0
\(735\) 419.296 65.3138i 0.570471 0.0888623i
\(736\) 0 0
\(737\) 263.624i 0.357699i
\(738\) 0 0
\(739\) −264.136 −0.357423 −0.178712 0.983901i \(-0.557193\pi\)
−0.178712 + 0.983901i \(0.557193\pi\)
\(740\) 0 0
\(741\) 989.295i 1.33508i
\(742\) 0 0
\(743\) 1069.03i 1.43880i −0.694595 0.719401i \(-0.744417\pi\)
0.694595 0.719401i \(-0.255583\pi\)
\(744\) 0 0
\(745\) 7.87429 + 27.4158i 0.0105695 + 0.0367997i
\(746\) 0 0
\(747\) 53.1983 0.0712160
\(748\) 0 0
\(749\) −363.137 411.689i −0.484829 0.549652i
\(750\) 0 0
\(751\) −987.190 −1.31450 −0.657250 0.753673i \(-0.728280\pi\)
−0.657250 + 0.753673i \(0.728280\pi\)
\(752\) 0 0
\(753\) 139.472i 0.185222i
\(754\) 0 0
\(755\) −298.817 1040.39i −0.395785 1.37800i
\(756\) 0 0
\(757\) 1173.09i 1.54966i −0.632169 0.774830i \(-0.717835\pi\)
0.632169 0.774830i \(-0.282165\pi\)
\(758\) 0 0
\(759\) 538.518i 0.709509i
\(760\) 0 0
\(761\) 271.712i 0.357045i 0.983936 + 0.178523i \(0.0571318\pi\)
−0.983936 + 0.178523i \(0.942868\pi\)
\(762\) 0 0
\(763\) 392.369 346.095i 0.514245 0.453598i
\(764\) 0 0
\(765\) 30.0667 + 104.683i 0.0393029 + 0.136840i
\(766\) 0 0
\(767\) 2030.15i 2.64687i
\(768\) 0 0
\(769\) 1031.82i 1.34177i −0.741563 0.670883i \(-0.765915\pi\)
0.741563 0.670883i \(-0.234085\pi\)
\(770\) 0 0
\(771\) 67.7252 0.0878407
\(772\) 0 0
\(773\) −877.988 −1.13582 −0.567910 0.823091i \(-0.692248\pi\)
−0.567910 + 0.823091i \(0.692248\pi\)
\(774\) 0 0
\(775\) 135.613 + 216.605i 0.174984 + 0.279490i
\(776\) 0 0
\(777\) −114.045 129.293i −0.146776 0.166400i
\(778\) 0 0
\(779\) −373.304 −0.479210
\(780\) 0 0
\(781\) 1076.71 1.37863
\(782\) 0 0
\(783\) −296.628 −0.378835
\(784\) 0 0
\(785\) −149.467 520.396i −0.190403 0.662924i
\(786\) 0 0
\(787\) −142.727 −0.181356 −0.0906780 0.995880i \(-0.528903\pi\)
−0.0906780 + 0.995880i \(0.528903\pi\)
\(788\) 0 0
\(789\) 137.117i 0.173786i
\(790\) 0 0
\(791\) 394.712 + 447.486i 0.499004 + 0.565721i
\(792\) 0 0
\(793\) 325.844i 0.410901i
\(794\) 0 0
\(795\) −539.605 + 154.984i −0.678749 + 0.194948i
\(796\) 0 0
\(797\) 317.415 0.398262 0.199131 0.979973i \(-0.436188\pi\)
0.199131 + 0.979973i \(0.436188\pi\)
\(798\) 0 0
\(799\) 375.126 0.469495
\(800\) 0 0
\(801\) 232.740i 0.290562i
\(802\) 0 0
\(803\) −846.354 −1.05399
\(804\) 0 0
\(805\) −497.709 + 779.426i −0.618272 + 0.968231i
\(806\) 0 0
\(807\) 92.2793i 0.114349i
\(808\) 0 0
\(809\) −1392.39 −1.72112 −0.860560 0.509350i \(-0.829886\pi\)
−0.860560 + 0.509350i \(0.829886\pi\)
\(810\) 0 0
\(811\) 1087.46i 1.34089i 0.741958 + 0.670447i \(0.233898\pi\)
−0.741958 + 0.670447i \(0.766102\pi\)
\(812\) 0 0
\(813\) 411.601i 0.506274i
\(814\) 0 0
\(815\) −1126.43 + 323.531i −1.38212 + 0.396970i
\(816\) 0 0
\(817\) 1893.58 2.31772
\(818\) 0 0
\(819\) 391.013 344.899i 0.477427 0.421122i
\(820\) 0 0
\(821\) 172.333 0.209906 0.104953 0.994477i \(-0.466531\pi\)
0.104953 + 0.994477i \(0.466531\pi\)
\(822\) 0 0
\(823\) 215.439i 0.261773i −0.991397 0.130886i \(-0.958218\pi\)
0.991397 0.130886i \(-0.0417823\pi\)
\(824\) 0 0
\(825\) −431.870 + 270.386i −0.523479 + 0.327741i
\(826\) 0 0
\(827\) 797.435i 0.964250i −0.876102 0.482125i \(-0.839865\pi\)
0.876102 0.482125i \(-0.160135\pi\)
\(828\) 0 0
\(829\) 389.170i 0.469445i −0.972062 0.234723i \(-0.924582\pi\)
0.972062 0.234723i \(-0.0754182\pi\)
\(830\) 0 0
\(831\) 291.305i 0.350548i
\(832\) 0 0
\(833\) −44.4146 + 353.006i −0.0533188 + 0.423777i
\(834\) 0 0
\(835\) −57.8104 201.277i −0.0692340 0.241051i
\(836\) 0 0
\(837\) 53.1162i 0.0634602i
\(838\) 0 0
\(839\) 280.262i 0.334043i 0.985953 + 0.167021i \(0.0534149\pi\)
−0.985953 + 0.167021i \(0.946585\pi\)
\(840\) 0 0
\(841\) 2417.83 2.87494
\(842\) 0 0
\(843\) −222.006 −0.263353
\(844\) 0 0
\(845\) −617.582 2150.23i −0.730867 2.54465i
\(846\) 0 0
\(847\) −91.6804 + 80.8681i −0.108241 + 0.0954760i
\(848\) 0 0
\(849\) −125.971 −0.148376
\(850\) 0 0
\(851\) 375.715 0.441498
\(852\) 0 0
\(853\) −245.874 −0.288246 −0.144123 0.989560i \(-0.546036\pi\)
−0.144123 + 0.989560i \(0.546036\pi\)
\(854\) 0 0
\(855\) 331.666 95.2603i 0.387914 0.111416i
\(856\) 0 0
\(857\) 798.850 0.932147 0.466074 0.884746i \(-0.345668\pi\)
0.466074 + 0.884746i \(0.345668\pi\)
\(858\) 0 0
\(859\) 1274.93i 1.48420i −0.670287 0.742102i \(-0.733829\pi\)
0.670287 0.742102i \(-0.266171\pi\)
\(860\) 0 0
\(861\) 130.146 + 147.546i 0.151156 + 0.171366i
\(862\) 0 0
\(863\) 736.114i 0.852971i 0.904494 + 0.426486i \(0.140249\pi\)
−0.904494 + 0.426486i \(0.859751\pi\)
\(864\) 0 0
\(865\) 17.9852 + 62.6187i 0.0207921 + 0.0723915i
\(866\) 0 0
\(867\) 409.245 0.472024
\(868\) 0 0
\(869\) 532.816 0.613137
\(870\) 0 0
\(871\) 556.236i 0.638617i
\(872\) 0 0
\(873\) 18.4722 0.0211595
\(874\) 0 0
\(875\) −874.965 7.79790i −0.999960 0.00891189i
\(876\) 0 0
\(877\) 1103.15i 1.25787i 0.777457 + 0.628936i \(0.216509\pi\)
−0.777457 + 0.628936i \(0.783491\pi\)
\(878\) 0 0
\(879\) −473.223 −0.538366
\(880\) 0 0
\(881\) 269.425i 0.305817i 0.988240 + 0.152909i \(0.0488640\pi\)
−0.988240 + 0.152909i \(0.951136\pi\)
\(882\) 0 0
\(883\) 1077.49i 1.22026i −0.792302 0.610129i \(-0.791118\pi\)
0.792302 0.610129i \(-0.208882\pi\)
\(884\) 0 0
\(885\) 680.619 195.486i 0.769061 0.220888i
\(886\) 0 0
\(887\) 500.908 0.564721 0.282361 0.959308i \(-0.408883\pi\)
0.282361 + 0.959308i \(0.408883\pi\)
\(888\) 0 0
\(889\) −309.194 350.534i −0.347800 0.394302i
\(890\) 0 0
\(891\) −105.904 −0.118859
\(892\) 0 0
\(893\) 1188.51i 1.33092i
\(894\) 0 0
\(895\) −33.3439 116.093i −0.0372558 0.129713i
\(896\) 0 0
\(897\) 1136.25i 1.26672i
\(898\) 0 0
\(899\) 583.546i 0.649106i
\(900\) 0 0
\(901\) 470.712i 0.522433i
\(902\) 0 0
\(903\) −660.160 748.425i −0.731074 0.828820i
\(904\) 0 0
\(905\) 209.100 60.0570i 0.231049 0.0663614i
\(906\) 0 0
\(907\) 789.727i 0.870702i −0.900261 0.435351i \(-0.856624\pi\)
0.900261 0.435351i \(-0.143376\pi\)
\(908\) 0 0
\(909\) 428.950i 0.471893i
\(910\) 0 0
\(911\) −885.912 −0.972461 −0.486231 0.873830i \(-0.661629\pi\)
−0.486231 + 0.873830i \(0.661629\pi\)
\(912\) 0 0
\(913\) −208.663 −0.228547
\(914\) 0 0
\(915\) 109.241 31.3759i 0.119389 0.0342906i
\(916\) 0 0
\(917\) 133.511 + 151.362i 0.145596 + 0.165062i
\(918\) 0 0
\(919\) 1411.37 1.53577 0.767883 0.640590i \(-0.221310\pi\)
0.767883 + 0.640590i \(0.221310\pi\)
\(920\) 0 0
\(921\) −582.119 −0.632051
\(922\) 0 0
\(923\) 2271.82 2.46134
\(924\) 0 0
\(925\) 188.644 + 301.308i 0.203939 + 0.325739i
\(926\) 0 0
\(927\) 43.5877 0.0470201
\(928\) 0 0
\(929\) 1394.66i 1.50125i 0.660727 + 0.750627i \(0.270248\pi\)
−0.660727 + 0.750627i \(0.729752\pi\)
\(930\) 0 0
\(931\) 1118.43 + 140.719i 1.20132 + 0.151148i
\(932\) 0 0
\(933\) 861.308i 0.923160i
\(934\) 0 0
\(935\) −117.933 410.604i −0.126131 0.439148i
\(936\) 0 0
\(937\) −448.660 −0.478826 −0.239413 0.970918i \(-0.576955\pi\)
−0.239413 + 0.970918i \(0.576955\pi\)
\(938\) 0 0
\(939\) −436.518 −0.464876
\(940\) 0 0
\(941\) 1061.98i 1.12856i 0.825583 + 0.564281i \(0.190847\pi\)
−0.825583 + 0.564281i \(0.809153\pi\)
\(942\) 0 0
\(943\) −428.757 −0.454673
\(944\) 0 0
\(945\) −153.280 97.8783i −0.162201 0.103575i
\(946\) 0 0
\(947\) 470.708i 0.497051i −0.968625 0.248526i \(-0.920054\pi\)
0.968625 0.248526i \(-0.0799460\pi\)
\(948\) 0 0
\(949\) −1785.77 −1.88174
\(950\) 0 0
\(951\) 468.006i 0.492120i
\(952\) 0 0
\(953\) 625.282i 0.656119i 0.944657 + 0.328060i \(0.106395\pi\)
−0.944657 + 0.328060i \(0.893605\pi\)
\(954\) 0 0
\(955\) −340.339 1184.95i −0.356376 1.24079i
\(956\) 0 0
\(957\) 1163.48 1.21576
\(958\) 0 0
\(959\) 553.113 + 627.065i 0.576760 + 0.653874i
\(960\) 0 0
\(961\) 856.506 0.891266
\(962\) 0 0
\(963\) 235.268i 0.244308i
\(964\) 0 0
\(965\) 777.926 223.434i 0.806141 0.231538i
\(966\) 0 0
\(967\) 593.257i 0.613503i 0.951790 + 0.306751i \(0.0992420\pi\)
−0.951790 + 0.306751i \(0.900758\pi\)
\(968\) 0 0
\(969\) 289.321i 0.298577i
\(970\) 0 0
\(971\) 1633.10i 1.68187i −0.541133 0.840937i \(-0.682004\pi\)
0.541133 0.840937i \(-0.317996\pi\)
\(972\) 0 0
\(973\) 930.490 + 1054.90i 0.956311 + 1.08417i
\(974\) 0 0
\(975\) −911.227 + 570.504i −0.934592 + 0.585132i
\(976\) 0 0
\(977\) 1137.81i 1.16459i 0.812977 + 0.582296i \(0.197846\pi\)
−0.812977 + 0.582296i \(0.802154\pi\)
\(978\) 0 0
\(979\) 912.891i 0.932472i
\(980\) 0 0
\(981\) −224.227 −0.228570
\(982\) 0 0
\(983\) −746.446 −0.759355 −0.379677 0.925119i \(-0.623965\pi\)
−0.379677 + 0.925119i \(0.623965\pi\)
\(984\) 0 0
\(985\) 215.478 61.8891i 0.218760 0.0628315i
\(986\) 0 0
\(987\) −469.752 + 414.352i −0.475939 + 0.419810i
\(988\) 0 0
\(989\) 2174.86 2.19905
\(990\) 0 0
\(991\) 765.218 0.772168 0.386084 0.922464i \(-0.373828\pi\)
0.386084 + 0.922464i \(0.373828\pi\)
\(992\) 0 0
\(993\) −531.415 −0.535161
\(994\) 0 0
\(995\) 551.536 158.411i 0.554308 0.159207i
\(996\) 0 0
\(997\) −1799.05 −1.80446 −0.902230 0.431255i \(-0.858071\pi\)
−0.902230 + 0.431255i \(0.858071\pi\)
\(998\) 0 0
\(999\) 73.8872i 0.0739612i
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.769.5 16
4.3 odd 2 210.3.h.a.139.7 yes 16
5.4 even 2 inner 1680.3.bd.a.769.11 16
7.6 odd 2 inner 1680.3.bd.a.769.12 16
12.11 even 2 630.3.h.e.559.13 16
20.3 even 4 1050.3.f.e.601.5 16
20.7 even 4 1050.3.f.e.601.12 16
20.19 odd 2 210.3.h.a.139.10 yes 16
28.27 even 2 210.3.h.a.139.2 16
35.34 odd 2 inner 1680.3.bd.a.769.6 16
60.59 even 2 630.3.h.e.559.4 16
84.83 odd 2 630.3.h.e.559.12 16
140.27 odd 4 1050.3.f.e.601.16 16
140.83 odd 4 1050.3.f.e.601.1 16
140.139 even 2 210.3.h.a.139.15 yes 16
420.419 odd 2 630.3.h.e.559.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.h.a.139.2 16 28.27 even 2
210.3.h.a.139.7 yes 16 4.3 odd 2
210.3.h.a.139.10 yes 16 20.19 odd 2
210.3.h.a.139.15 yes 16 140.139 even 2
630.3.h.e.559.4 16 60.59 even 2
630.3.h.e.559.5 16 420.419 odd 2
630.3.h.e.559.12 16 84.83 odd 2
630.3.h.e.559.13 16 12.11 even 2
1050.3.f.e.601.1 16 140.83 odd 4
1050.3.f.e.601.5 16 20.3 even 4
1050.3.f.e.601.12 16 20.7 even 4
1050.3.f.e.601.16 16 140.27 odd 4
1680.3.bd.a.769.5 16 1.1 even 1 trivial
1680.3.bd.a.769.6 16 35.34 odd 2 inner
1680.3.bd.a.769.11 16 5.4 even 2 inner
1680.3.bd.a.769.12 16 7.6 odd 2 inner