Properties

Label 1323.2.c.d.1322.7
Level $1323$
Weight $2$
Character 1323.1322
Analytic conductor $10.564$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1322.7
Root \(1.65604 + 0.956115i\) of defining polynomial
Character \(\chi\) \(=\) 1323.1322
Dual form 1323.2.c.d.1322.5

$q$-expansion

\(f(q)\) \(=\) \(q+0.656620i q^{2} +1.56885 q^{4} -3.31208 q^{5} +2.34338i q^{8} +O(q^{10})\) \(q+0.656620i q^{2} +1.56885 q^{4} -3.31208 q^{5} +2.34338i q^{8} -2.17478i q^{10} -2.34338i q^{11} -1.58003i q^{13} +1.59899 q^{16} -1.13730 q^{17} +4.44938i q^{19} -5.19615 q^{20} +1.53871 q^{22} +9.39331i q^{23} +5.96986 q^{25} +1.03748 q^{26} +3.65662i q^{29} +7.31873i q^{31} +5.73669i q^{32} -0.746774i q^{34} -5.16784 q^{37} -2.92155 q^{38} -7.76146i q^{40} -9.64553 q^{41} -2.16784 q^{43} -3.67641i q^{44} -6.16784 q^{46} +5.58668 q^{47} +3.91993i q^{50} -2.47883i q^{52} -10.0499i q^{53} +7.76146i q^{55} -2.40101 q^{58} -2.17478 q^{59} +4.00665i q^{61} -4.80563 q^{62} -0.568850 q^{64} +5.23317i q^{65} -10.7668 q^{67} -1.78425 q^{68} +6.39331i q^{71} +10.6830i q^{73} -3.39331i q^{74} +6.98041i q^{76} +1.23317 q^{79} -5.29597 q^{80} -6.33345i q^{82} +1.03748 q^{83} +3.76683 q^{85} -1.42345i q^{86} +5.49143 q^{88} -7.47075 q^{89} +14.7367i q^{92} +3.66833i q^{94} -14.7367i q^{95} -13.5524i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 16q^{4} + O(q^{10}) \) \( 12q - 16q^{4} + 8q^{16} - 40q^{22} + 48q^{25} - 16q^{37} + 20q^{43} - 28q^{46} - 40q^{58} + 28q^{64} - 72q^{67} + 72q^{79} - 12q^{85} + 148q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.656620i 0.464301i 0.972680 + 0.232150i \(0.0745761\pi\)
−0.972680 + 0.232150i \(0.925424\pi\)
\(3\) 0 0
\(4\) 1.56885 0.784425
\(5\) −3.31208 −1.48121 −0.740603 0.671943i \(-0.765460\pi\)
−0.740603 + 0.671943i \(0.765460\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.34338i 0.828510i
\(9\) 0 0
\(10\) − 2.17478i − 0.687725i
\(11\) − 2.34338i − 0.706556i −0.935518 0.353278i \(-0.885067\pi\)
0.935518 0.353278i \(-0.114933\pi\)
\(12\) 0 0
\(13\) − 1.58003i − 0.438221i −0.975700 0.219110i \(-0.929685\pi\)
0.975700 0.219110i \(-0.0703155\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.59899 0.399747
\(17\) −1.13730 −0.275836 −0.137918 0.990444i \(-0.544041\pi\)
−0.137918 + 0.990444i \(0.544041\pi\)
\(18\) 0 0
\(19\) 4.44938i 1.02076i 0.859950 + 0.510379i \(0.170495\pi\)
−0.859950 + 0.510379i \(0.829505\pi\)
\(20\) −5.19615 −1.16190
\(21\) 0 0
\(22\) 1.53871 0.328054
\(23\) 9.39331i 1.95864i 0.202317 + 0.979320i \(0.435153\pi\)
−0.202317 + 0.979320i \(0.564847\pi\)
\(24\) 0 0
\(25\) 5.96986 1.19397
\(26\) 1.03748 0.203466
\(27\) 0 0
\(28\) 0 0
\(29\) 3.65662i 0.679017i 0.940603 + 0.339509i \(0.110261\pi\)
−0.940603 + 0.339509i \(0.889739\pi\)
\(30\) 0 0
\(31\) 7.31873i 1.31448i 0.753680 + 0.657241i \(0.228277\pi\)
−0.753680 + 0.657241i \(0.771723\pi\)
\(32\) 5.73669i 1.01411i
\(33\) 0 0
\(34\) − 0.746774i − 0.128071i
\(35\) 0 0
\(36\) 0 0
\(37\) −5.16784 −0.849587 −0.424794 0.905290i \(-0.639653\pi\)
−0.424794 + 0.905290i \(0.639653\pi\)
\(38\) −2.92155 −0.473938
\(39\) 0 0
\(40\) − 7.76146i − 1.22719i
\(41\) −9.64553 −1.50638 −0.753189 0.657804i \(-0.771486\pi\)
−0.753189 + 0.657804i \(0.771486\pi\)
\(42\) 0 0
\(43\) −2.16784 −0.330592 −0.165296 0.986244i \(-0.552858\pi\)
−0.165296 + 0.986244i \(0.552858\pi\)
\(44\) − 3.67641i − 0.554240i
\(45\) 0 0
\(46\) −6.16784 −0.909398
\(47\) 5.58668 0.814901 0.407450 0.913227i \(-0.366418\pi\)
0.407450 + 0.913227i \(0.366418\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 3.91993i 0.554362i
\(51\) 0 0
\(52\) − 2.47883i − 0.343751i
\(53\) − 10.0499i − 1.38046i −0.723588 0.690232i \(-0.757509\pi\)
0.723588 0.690232i \(-0.242491\pi\)
\(54\) 0 0
\(55\) 7.76146i 1.04655i
\(56\) 0 0
\(57\) 0 0
\(58\) −2.40101 −0.315268
\(59\) −2.17478 −0.283132 −0.141566 0.989929i \(-0.545214\pi\)
−0.141566 + 0.989929i \(0.545214\pi\)
\(60\) 0 0
\(61\) 4.00665i 0.512999i 0.966544 + 0.256500i \(0.0825692\pi\)
−0.966544 + 0.256500i \(0.917431\pi\)
\(62\) −4.80563 −0.610315
\(63\) 0 0
\(64\) −0.568850 −0.0711062
\(65\) 5.23317i 0.649095i
\(66\) 0 0
\(67\) −10.7668 −1.31538 −0.657689 0.753290i \(-0.728466\pi\)
−0.657689 + 0.753290i \(0.728466\pi\)
\(68\) −1.78425 −0.216372
\(69\) 0 0
\(70\) 0 0
\(71\) 6.39331i 0.758746i 0.925244 + 0.379373i \(0.123860\pi\)
−0.925244 + 0.379373i \(0.876140\pi\)
\(72\) 0 0
\(73\) 10.6830i 1.25035i 0.780484 + 0.625176i \(0.214973\pi\)
−0.780484 + 0.625176i \(0.785027\pi\)
\(74\) − 3.39331i − 0.394464i
\(75\) 0 0
\(76\) 6.98041i 0.800707i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.23317 0.138743 0.0693714 0.997591i \(-0.477901\pi\)
0.0693714 + 0.997591i \(0.477901\pi\)
\(80\) −5.29597 −0.592108
\(81\) 0 0
\(82\) − 6.33345i − 0.699413i
\(83\) 1.03748 0.113878 0.0569390 0.998378i \(-0.481866\pi\)
0.0569390 + 0.998378i \(0.481866\pi\)
\(84\) 0 0
\(85\) 3.76683 0.408570
\(86\) − 1.42345i − 0.153494i
\(87\) 0 0
\(88\) 5.49143 0.585388
\(89\) −7.47075 −0.791898 −0.395949 0.918272i \(-0.629584\pi\)
−0.395949 + 0.918272i \(0.629584\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 14.7367i 1.53641i
\(93\) 0 0
\(94\) 3.66833i 0.378359i
\(95\) − 14.7367i − 1.51195i
\(96\) 0 0
\(97\) − 13.5524i − 1.37603i −0.725695 0.688017i \(-0.758482\pi\)
0.725695 0.688017i \(-0.241518\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 9.36581 0.936581
\(101\) 13.7044 1.36364 0.681819 0.731521i \(-0.261189\pi\)
0.681819 + 0.731521i \(0.261189\pi\)
\(102\) 0 0
\(103\) 4.89211i 0.482033i 0.970521 + 0.241017i \(0.0774809\pi\)
−0.970521 + 0.241017i \(0.922519\pi\)
\(104\) 3.70260 0.363070
\(105\) 0 0
\(106\) 6.59899 0.640950
\(107\) 2.45359i 0.237197i 0.992942 + 0.118599i \(0.0378401\pi\)
−0.992942 + 0.118599i \(0.962160\pi\)
\(108\) 0 0
\(109\) 4.33568 0.415282 0.207641 0.978205i \(-0.433421\pi\)
0.207641 + 0.978205i \(0.433421\pi\)
\(110\) −5.09633 −0.485916
\(111\) 0 0
\(112\) 0 0
\(113\) 7.16013i 0.673569i 0.941582 + 0.336784i \(0.109339\pi\)
−0.941582 + 0.336784i \(0.890661\pi\)
\(114\) 0 0
\(115\) − 31.1114i − 2.90115i
\(116\) 5.73669i 0.532638i
\(117\) 0 0
\(118\) − 1.42800i − 0.131458i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50857 0.500779
\(122\) −2.63085 −0.238186
\(123\) 0 0
\(124\) 11.4820i 1.03111i
\(125\) −3.21226 −0.287313
\(126\) 0 0
\(127\) −17.7065 −1.57120 −0.785601 0.618733i \(-0.787646\pi\)
−0.785601 + 0.618733i \(0.787646\pi\)
\(128\) 11.0999i 0.981098i
\(129\) 0 0
\(130\) −3.43621 −0.301375
\(131\) 13.6046 1.18864 0.594318 0.804230i \(-0.297422\pi\)
0.594318 + 0.804230i \(0.297422\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 7.06972i − 0.610731i
\(135\) 0 0
\(136\) − 2.66513i − 0.228533i
\(137\) 10.1601i 0.868039i 0.900904 + 0.434019i \(0.142905\pi\)
−0.900904 + 0.434019i \(0.857095\pi\)
\(138\) 0 0
\(139\) 3.85463i 0.326945i 0.986548 + 0.163473i \(0.0522695\pi\)
−0.986548 + 0.163473i \(0.947730\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.19798 −0.352286
\(143\) −3.70260 −0.309627
\(144\) 0 0
\(145\) − 12.1110i − 1.00576i
\(146\) −7.01468 −0.580539
\(147\) 0 0
\(148\) −8.10756 −0.666437
\(149\) − 5.08007i − 0.416175i −0.978110 0.208088i \(-0.933276\pi\)
0.978110 0.208088i \(-0.0667239\pi\)
\(150\) 0 0
\(151\) 18.3055 1.48968 0.744842 0.667241i \(-0.232525\pi\)
0.744842 + 0.667241i \(0.232525\pi\)
\(152\) −10.4266 −0.845707
\(153\) 0 0
\(154\) 0 0
\(155\) − 24.2402i − 1.94702i
\(156\) 0 0
\(157\) − 3.31208i − 0.264333i −0.991228 0.132166i \(-0.957807\pi\)
0.991228 0.132166i \(-0.0421933\pi\)
\(158\) 0.809727i 0.0644184i
\(159\) 0 0
\(160\) − 19.0004i − 1.50211i
\(161\) 0 0
\(162\) 0 0
\(163\) −2.29345 −0.179637 −0.0898185 0.995958i \(-0.528629\pi\)
−0.0898185 + 0.995958i \(0.528629\pi\)
\(164\) −15.1324 −1.18164
\(165\) 0 0
\(166\) 0.681229i 0.0528737i
\(167\) −2.66513 −0.206234 −0.103117 0.994669i \(-0.532882\pi\)
−0.103117 + 0.994669i \(0.532882\pi\)
\(168\) 0 0
\(169\) 10.5035 0.807963
\(170\) 2.47338i 0.189699i
\(171\) 0 0
\(172\) −3.40101 −0.259325
\(173\) −2.56530 −0.195036 −0.0975182 0.995234i \(-0.531090\pi\)
−0.0975182 + 0.995234i \(0.531090\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 3.74704i − 0.282444i
\(177\) 0 0
\(178\) − 4.90545i − 0.367679i
\(179\) 13.8365i 1.03419i 0.855927 + 0.517096i \(0.172987\pi\)
−0.855927 + 0.517096i \(0.827013\pi\)
\(180\) 0 0
\(181\) − 4.74008i − 0.352328i −0.984361 0.176164i \(-0.943631\pi\)
0.984361 0.176164i \(-0.0563688\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −22.0121 −1.62275
\(185\) 17.1163 1.25841
\(186\) 0 0
\(187\) 2.66513i 0.194893i
\(188\) 8.76466 0.639228
\(189\) 0 0
\(190\) 9.67641 0.702001
\(191\) − 8.34338i − 0.603706i −0.953355 0.301853i \(-0.902395\pi\)
0.953355 0.301853i \(-0.0976051\pi\)
\(192\) 0 0
\(193\) −17.8442 −1.28446 −0.642229 0.766513i \(-0.721990\pi\)
−0.642229 + 0.766513i \(0.721990\pi\)
\(194\) 8.89876 0.638893
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.1102i − 1.07656i −0.842767 0.538279i \(-0.819075\pi\)
0.842767 0.538279i \(-0.180925\pi\)
\(198\) 0 0
\(199\) 8.66025i 0.613909i 0.951724 + 0.306955i \(0.0993100\pi\)
−0.951724 + 0.306955i \(0.900690\pi\)
\(200\) 13.9897i 0.989218i
\(201\) 0 0
\(202\) 8.99858i 0.633138i
\(203\) 0 0
\(204\) 0 0
\(205\) 31.9468 2.23126
\(206\) −3.21226 −0.223809
\(207\) 0 0
\(208\) − 2.52645i − 0.175177i
\(209\) 10.4266 0.721222
\(210\) 0 0
\(211\) −14.5809 −1.00379 −0.501897 0.864928i \(-0.667364\pi\)
−0.501897 + 0.864928i \(0.667364\pi\)
\(212\) − 15.7668i − 1.08287i
\(213\) 0 0
\(214\) −1.61107 −0.110131
\(215\) 7.18005 0.489675
\(216\) 0 0
\(217\) 0 0
\(218\) 2.84689i 0.192816i
\(219\) 0 0
\(220\) 12.1766i 0.820943i
\(221\) 1.79696i 0.120877i
\(222\) 0 0
\(223\) 10.5832i 0.708703i 0.935112 + 0.354351i \(0.115298\pi\)
−0.935112 + 0.354351i \(0.884702\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4.70149 −0.312738
\(227\) 6.72398 0.446286 0.223143 0.974786i \(-0.428368\pi\)
0.223143 + 0.974786i \(0.428368\pi\)
\(228\) 0 0
\(229\) − 12.2630i − 0.810364i −0.914236 0.405182i \(-0.867208\pi\)
0.914236 0.405182i \(-0.132792\pi\)
\(230\) 20.4284 1.34701
\(231\) 0 0
\(232\) −8.56885 −0.562573
\(233\) 5.62648i 0.368603i 0.982870 + 0.184302i \(0.0590023\pi\)
−0.982870 + 0.184302i \(0.940998\pi\)
\(234\) 0 0
\(235\) −18.5035 −1.20704
\(236\) −3.41190 −0.222096
\(237\) 0 0
\(238\) 0 0
\(239\) − 15.9595i − 1.03234i −0.856488 0.516168i \(-0.827358\pi\)
0.856488 0.516168i \(-0.172642\pi\)
\(240\) 0 0
\(241\) 4.58806i 0.295543i 0.989022 + 0.147771i \(0.0472100\pi\)
−0.989022 + 0.147771i \(0.952790\pi\)
\(242\) 3.61704i 0.232512i
\(243\) 0 0
\(244\) 6.28583i 0.402409i
\(245\) 0 0
\(246\) 0 0
\(247\) 7.03014 0.447317
\(248\) −17.1506 −1.08906
\(249\) 0 0
\(250\) − 2.10923i − 0.133400i
\(251\) 13.8042 0.871314 0.435657 0.900113i \(-0.356516\pi\)
0.435657 + 0.900113i \(0.356516\pi\)
\(252\) 0 0
\(253\) 22.0121 1.38389
\(254\) − 11.6265i − 0.729510i
\(255\) 0 0
\(256\) −8.42609 −0.526631
\(257\) −8.96430 −0.559178 −0.279589 0.960120i \(-0.590198\pi\)
−0.279589 + 0.960120i \(0.590198\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 8.21006i 0.509166i
\(261\) 0 0
\(262\) 8.93303i 0.551885i
\(263\) − 28.5534i − 1.76068i −0.474343 0.880340i \(-0.657314\pi\)
0.474343 0.880340i \(-0.342686\pi\)
\(264\) 0 0
\(265\) 33.2861i 2.04475i
\(266\) 0 0
\(267\) 0 0
\(268\) −16.8915 −1.03181
\(269\) 23.9313 1.45912 0.729559 0.683918i \(-0.239725\pi\)
0.729559 + 0.683918i \(0.239725\pi\)
\(270\) 0 0
\(271\) 14.6032i 0.887080i 0.896255 + 0.443540i \(0.146278\pi\)
−0.896255 + 0.443540i \(0.853722\pi\)
\(272\) −1.81853 −0.110265
\(273\) 0 0
\(274\) −6.67135 −0.403031
\(275\) − 13.9897i − 0.843608i
\(276\) 0 0
\(277\) 22.3830 1.34486 0.672431 0.740160i \(-0.265250\pi\)
0.672431 + 0.740160i \(0.265250\pi\)
\(278\) −2.53103 −0.151801
\(279\) 0 0
\(280\) 0 0
\(281\) − 11.6067i − 0.692397i −0.938161 0.346199i \(-0.887472\pi\)
0.938161 0.346199i \(-0.112528\pi\)
\(282\) 0 0
\(283\) 14.9937i 0.891283i 0.895211 + 0.445642i \(0.147024\pi\)
−0.895211 + 0.445642i \(0.852976\pi\)
\(284\) 10.0301i 0.595179i
\(285\) 0 0
\(286\) − 2.43121i − 0.143760i
\(287\) 0 0
\(288\) 0 0
\(289\) −15.7065 −0.923915
\(290\) 7.95234 0.466977
\(291\) 0 0
\(292\) 16.7600i 0.980807i
\(293\) 16.3352 0.954314 0.477157 0.878818i \(-0.341667\pi\)
0.477157 + 0.878818i \(0.341667\pi\)
\(294\) 0 0
\(295\) 7.20304 0.419377
\(296\) − 12.1102i − 0.703891i
\(297\) 0 0
\(298\) 3.33568 0.193231
\(299\) 14.8417 0.858317
\(300\) 0 0
\(301\) 0 0
\(302\) 12.0198i 0.691661i
\(303\) 0 0
\(304\) 7.11450i 0.408045i
\(305\) − 13.2703i − 0.759857i
\(306\) 0 0
\(307\) − 3.17340i − 0.181115i −0.995891 0.0905577i \(-0.971135\pi\)
0.995891 0.0905577i \(-0.0288650\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 15.9166 0.904003
\(311\) 25.1342 1.42523 0.712614 0.701556i \(-0.247511\pi\)
0.712614 + 0.701556i \(0.247511\pi\)
\(312\) 0 0
\(313\) − 18.6441i − 1.05383i −0.849919 0.526914i \(-0.823349\pi\)
0.849919 0.526914i \(-0.176651\pi\)
\(314\) 2.17478 0.122730
\(315\) 0 0
\(316\) 1.93466 0.108833
\(317\) − 7.98965i − 0.448744i −0.974504 0.224372i \(-0.927967\pi\)
0.974504 0.224372i \(-0.0720330\pi\)
\(318\) 0 0
\(319\) 8.56885 0.479763
\(320\) 1.88407 0.105323
\(321\) 0 0
\(322\) 0 0
\(323\) − 5.06028i − 0.281561i
\(324\) 0 0
\(325\) − 9.43254i − 0.523223i
\(326\) − 1.50593i − 0.0834056i
\(327\) 0 0
\(328\) − 22.6031i − 1.24805i
\(329\) 0 0
\(330\) 0 0
\(331\) −8.93466 −0.491094 −0.245547 0.969385i \(-0.578967\pi\)
−0.245547 + 0.969385i \(0.578967\pi\)
\(332\) 1.62765 0.0893287
\(333\) 0 0
\(334\) − 1.74998i − 0.0957544i
\(335\) 35.6606 1.94835
\(336\) 0 0
\(337\) 4.24526 0.231254 0.115627 0.993293i \(-0.463112\pi\)
0.115627 + 0.993293i \(0.463112\pi\)
\(338\) 6.89682i 0.375138i
\(339\) 0 0
\(340\) 5.90958 0.320492
\(341\) 17.1506 0.928755
\(342\) 0 0
\(343\) 0 0
\(344\) − 5.08007i − 0.273899i
\(345\) 0 0
\(346\) − 1.68443i − 0.0905556i
\(347\) 2.62648i 0.140997i 0.997512 + 0.0704985i \(0.0224590\pi\)
−0.997512 + 0.0704985i \(0.977541\pi\)
\(348\) 0 0
\(349\) − 32.4918i − 1.73924i −0.493718 0.869622i \(-0.664363\pi\)
0.493718 0.869622i \(-0.335637\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 13.4432 0.716527
\(353\) −23.3842 −1.24461 −0.622307 0.782773i \(-0.713805\pi\)
−0.622307 + 0.782773i \(0.713805\pi\)
\(354\) 0 0
\(355\) − 21.1751i − 1.12386i
\(356\) −11.7205 −0.621185
\(357\) 0 0
\(358\) −9.08536 −0.480176
\(359\) − 21.3504i − 1.12683i −0.826174 0.563416i \(-0.809487\pi\)
0.826174 0.563416i \(-0.190513\pi\)
\(360\) 0 0
\(361\) −0.796965 −0.0419455
\(362\) 3.11243 0.163586
\(363\) 0 0
\(364\) 0 0
\(365\) − 35.3830i − 1.85203i
\(366\) 0 0
\(367\) − 15.5229i − 0.810289i −0.914253 0.405145i \(-0.867221\pi\)
0.914253 0.405145i \(-0.132779\pi\)
\(368\) 15.0198i 0.782961i
\(369\) 0 0
\(370\) 11.2389i 0.584283i
\(371\) 0 0
\(372\) 0 0
\(373\) −23.3528 −1.20916 −0.604582 0.796543i \(-0.706660\pi\)
−0.604582 + 0.796543i \(0.706660\pi\)
\(374\) −1.74998 −0.0904891
\(375\) 0 0
\(376\) 13.0917i 0.675153i
\(377\) 5.77756 0.297559
\(378\) 0 0
\(379\) 2.53871 0.130405 0.0652024 0.997872i \(-0.479231\pi\)
0.0652024 + 0.997872i \(0.479231\pi\)
\(380\) − 23.1196i − 1.18601i
\(381\) 0 0
\(382\) 5.47843 0.280301
\(383\) −16.2041 −0.827993 −0.413996 0.910278i \(-0.635867\pi\)
−0.413996 + 0.910278i \(0.635867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 11.7169i − 0.596374i
\(387\) 0 0
\(388\) − 21.2616i − 1.07939i
\(389\) 22.3330i 1.13233i 0.824292 + 0.566165i \(0.191573\pi\)
−0.824292 + 0.566165i \(0.808427\pi\)
\(390\) 0 0
\(391\) − 10.6830i − 0.540263i
\(392\) 0 0
\(393\) 0 0
\(394\) 9.92167 0.499847
\(395\) −4.08437 −0.205507
\(396\) 0 0
\(397\) 32.6914i 1.64073i 0.571837 + 0.820367i \(0.306231\pi\)
−0.571837 + 0.820367i \(0.693769\pi\)
\(398\) −5.68650 −0.285038
\(399\) 0 0
\(400\) 9.54574 0.477287
\(401\) 28.9270i 1.44454i 0.691609 + 0.722272i \(0.256902\pi\)
−0.691609 + 0.722272i \(0.743098\pi\)
\(402\) 0 0
\(403\) 11.5638 0.576033
\(404\) 21.5001 1.06967
\(405\) 0 0
\(406\) 0 0
\(407\) 12.1102i 0.600281i
\(408\) 0 0
\(409\) − 1.05082i − 0.0519598i −0.999662 0.0259799i \(-0.991729\pi\)
0.999662 0.0259799i \(-0.00827059\pi\)
\(410\) 20.9769i 1.03597i
\(411\) 0 0
\(412\) 7.67498i 0.378119i
\(413\) 0 0
\(414\) 0 0
\(415\) −3.43621 −0.168677
\(416\) 9.06412 0.444405
\(417\) 0 0
\(418\) 6.84631i 0.334864i
\(419\) −3.12120 −0.152480 −0.0762402 0.997089i \(-0.524292\pi\)
−0.0762402 + 0.997089i \(0.524292\pi\)
\(420\) 0 0
\(421\) −8.70655 −0.424331 −0.212166 0.977234i \(-0.568052\pi\)
−0.212166 + 0.977234i \(0.568052\pi\)
\(422\) − 9.57414i − 0.466062i
\(423\) 0 0
\(424\) 23.5508 1.14373
\(425\) −6.78952 −0.329340
\(426\) 0 0
\(427\) 0 0
\(428\) 3.84931i 0.186063i
\(429\) 0 0
\(430\) 4.71457i 0.227357i
\(431\) − 27.2831i − 1.31418i −0.753812 0.657090i \(-0.771787\pi\)
0.753812 0.657090i \(-0.228213\pi\)
\(432\) 0 0
\(433\) − 12.8711i − 0.618547i −0.950973 0.309274i \(-0.899914\pi\)
0.950973 0.309274i \(-0.100086\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.80202 0.325758
\(437\) −41.7944 −1.99930
\(438\) 0 0
\(439\) 18.4445i 0.880306i 0.897923 + 0.440153i \(0.145076\pi\)
−0.897923 + 0.440153i \(0.854924\pi\)
\(440\) −18.1880 −0.867081
\(441\) 0 0
\(442\) −1.17992 −0.0561233
\(443\) − 2.87000i − 0.136358i −0.997673 0.0681790i \(-0.978281\pi\)
0.997673 0.0681790i \(-0.0217189\pi\)
\(444\) 0 0
\(445\) 24.7437 1.17296
\(446\) −6.94914 −0.329051
\(447\) 0 0
\(448\) 0 0
\(449\) − 1.81675i − 0.0857380i −0.999081 0.0428690i \(-0.986350\pi\)
0.999081 0.0428690i \(-0.0136498\pi\)
\(450\) 0 0
\(451\) 22.6031i 1.06434i
\(452\) 11.2332i 0.528364i
\(453\) 0 0
\(454\) 4.41510i 0.207211i
\(455\) 0 0
\(456\) 0 0
\(457\) 6.15575 0.287954 0.143977 0.989581i \(-0.454011\pi\)
0.143977 + 0.989581i \(0.454011\pi\)
\(458\) 8.05216 0.376253
\(459\) 0 0
\(460\) − 48.8091i − 2.27573i
\(461\) −17.0507 −0.794132 −0.397066 0.917790i \(-0.629972\pi\)
−0.397066 + 0.917790i \(0.629972\pi\)
\(462\) 0 0
\(463\) 29.4131 1.36694 0.683471 0.729977i \(-0.260469\pi\)
0.683471 + 0.729977i \(0.260469\pi\)
\(464\) 5.84689i 0.271435i
\(465\) 0 0
\(466\) −3.69446 −0.171143
\(467\) 27.1181 1.25487 0.627437 0.778667i \(-0.284104\pi\)
0.627437 + 0.778667i \(0.284104\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 12.1498i − 0.560428i
\(471\) 0 0
\(472\) − 5.09633i − 0.234578i
\(473\) 5.08007i 0.233582i
\(474\) 0 0
\(475\) 26.5622i 1.21876i
\(476\) 0 0
\(477\) 0 0
\(478\) 10.4793 0.479314
\(479\) 21.5569 0.984960 0.492480 0.870324i \(-0.336090\pi\)
0.492480 + 0.870324i \(0.336090\pi\)
\(480\) 0 0
\(481\) 8.16532i 0.372307i
\(482\) −3.01261 −0.137221
\(483\) 0 0
\(484\) 8.64212 0.392824
\(485\) 44.8865i 2.03819i
\(486\) 0 0
\(487\) −14.5809 −0.660725 −0.330363 0.943854i \(-0.607171\pi\)
−0.330363 + 0.943854i \(0.607171\pi\)
\(488\) −9.38910 −0.425025
\(489\) 0 0
\(490\) 0 0
\(491\) − 0.283102i − 0.0127762i −0.999980 0.00638811i \(-0.997967\pi\)
0.999980 0.00638811i \(-0.00203341\pi\)
\(492\) 0 0
\(493\) − 4.15867i − 0.187297i
\(494\) 4.61613i 0.207690i
\(495\) 0 0
\(496\) 11.7026i 0.525461i
\(497\) 0 0
\(498\) 0 0
\(499\) 12.3055 0.550871 0.275436 0.961319i \(-0.411178\pi\)
0.275436 + 0.961319i \(0.411178\pi\)
\(500\) −5.03955 −0.225375
\(501\) 0 0
\(502\) 9.06412i 0.404552i
\(503\) −1.78425 −0.0795559 −0.0397779 0.999209i \(-0.512665\pi\)
−0.0397779 + 0.999209i \(0.512665\pi\)
\(504\) 0 0
\(505\) −45.3900 −2.01983
\(506\) 14.4536i 0.642540i
\(507\) 0 0
\(508\) −27.7789 −1.23249
\(509\) −39.0725 −1.73186 −0.865928 0.500168i \(-0.833272\pi\)
−0.865928 + 0.500168i \(0.833272\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 16.6670i 0.736583i
\(513\) 0 0
\(514\) − 5.88614i − 0.259627i
\(515\) − 16.2030i − 0.713991i
\(516\) 0 0
\(517\) − 13.0917i − 0.575773i
\(518\) 0 0
\(519\) 0 0
\(520\) −12.2633 −0.537782
\(521\) 20.3628 0.892111 0.446056 0.895005i \(-0.352828\pi\)
0.446056 + 0.895005i \(0.352828\pi\)
\(522\) 0 0
\(523\) 1.43259i 0.0626426i 0.999509 + 0.0313213i \(0.00997151\pi\)
−0.999509 + 0.0313213i \(0.990028\pi\)
\(524\) 21.3435 0.932396
\(525\) 0 0
\(526\) 18.7488 0.817485
\(527\) − 8.32359i − 0.362581i
\(528\) 0 0
\(529\) −65.2342 −2.83627
\(530\) −21.8564 −0.949380
\(531\) 0 0
\(532\) 0 0
\(533\) 15.2402i 0.660126i
\(534\) 0 0
\(535\) − 8.12647i − 0.351338i
\(536\) − 25.2308i − 1.08980i
\(537\) 0 0
\(538\) 15.7138i 0.677470i
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0904 −0.433821 −0.216910 0.976192i \(-0.569598\pi\)
−0.216910 + 0.976192i \(0.569598\pi\)
\(542\) −9.58875 −0.411872
\(543\) 0 0
\(544\) − 6.52433i − 0.279729i
\(545\) −14.3601 −0.615119
\(546\) 0 0
\(547\) 7.30048 0.312146 0.156073 0.987746i \(-0.450117\pi\)
0.156073 + 0.987746i \(0.450117\pi\)
\(548\) 15.9397i 0.680911i
\(549\) 0 0
\(550\) 9.18589 0.391688
\(551\) −16.2697 −0.693112
\(552\) 0 0
\(553\) 0 0
\(554\) 14.6971i 0.624420i
\(555\) 0 0
\(556\) 6.04733i 0.256464i
\(557\) 22.8597i 0.968595i 0.874903 + 0.484297i \(0.160925\pi\)
−0.874903 + 0.484297i \(0.839075\pi\)
\(558\) 0 0
\(559\) 3.42524i 0.144872i
\(560\) 0 0
\(561\) 0 0
\(562\) 7.62119 0.321481
\(563\) 21.1096 0.889663 0.444832 0.895614i \(-0.353264\pi\)
0.444832 + 0.895614i \(0.353264\pi\)
\(564\) 0 0
\(565\) − 23.7149i − 0.997694i
\(566\) −9.84517 −0.413824
\(567\) 0 0
\(568\) −14.9819 −0.628629
\(569\) 26.4536i 1.10899i 0.832186 + 0.554496i \(0.187089\pi\)
−0.832186 + 0.554496i \(0.812911\pi\)
\(570\) 0 0
\(571\) 12.0301 0.503446 0.251723 0.967799i \(-0.419003\pi\)
0.251723 + 0.967799i \(0.419003\pi\)
\(572\) −5.80883 −0.242879
\(573\) 0 0
\(574\) 0 0
\(575\) 56.0767i 2.33856i
\(576\) 0 0
\(577\) 36.8413i 1.53372i 0.641812 + 0.766862i \(0.278183\pi\)
−0.641812 + 0.766862i \(0.721817\pi\)
\(578\) − 10.3132i − 0.428974i
\(579\) 0 0
\(580\) − 19.0004i − 0.788947i
\(581\) 0 0
\(582\) 0 0
\(583\) −23.5508 −0.975374
\(584\) −25.0343 −1.03593
\(585\) 0 0
\(586\) 10.7260i 0.443089i
\(587\) −28.7712 −1.18752 −0.593758 0.804644i \(-0.702356\pi\)
−0.593758 + 0.804644i \(0.702356\pi\)
\(588\) 0 0
\(589\) −32.5638 −1.34177
\(590\) 4.72966i 0.194717i
\(591\) 0 0
\(592\) −8.26331 −0.339620
\(593\) −10.5832 −0.434599 −0.217300 0.976105i \(-0.569725\pi\)
−0.217300 + 0.976105i \(0.569725\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 7.96986i − 0.326458i
\(597\) 0 0
\(598\) 9.74535i 0.398517i
\(599\) − 18.1927i − 0.743333i −0.928366 0.371666i \(-0.878786\pi\)
0.928366 0.371666i \(-0.121214\pi\)
\(600\) 0 0
\(601\) 26.4101i 1.07729i 0.842532 + 0.538646i \(0.181064\pi\)
−0.842532 + 0.538646i \(0.818936\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 28.7186 1.16854
\(605\) −18.2448 −0.741757
\(606\) 0 0
\(607\) − 24.9776i − 1.01381i −0.862003 0.506904i \(-0.830790\pi\)
0.862003 0.506904i \(-0.169210\pi\)
\(608\) −25.5247 −1.03516
\(609\) 0 0
\(610\) 8.71358 0.352802
\(611\) − 8.82710i − 0.357106i
\(612\) 0 0
\(613\) −14.1126 −0.570003 −0.285002 0.958527i \(-0.591994\pi\)
−0.285002 + 0.958527i \(0.591994\pi\)
\(614\) 2.08372 0.0840920
\(615\) 0 0
\(616\) 0 0
\(617\) 14.6265i 0.588840i 0.955676 + 0.294420i \(0.0951264\pi\)
−0.955676 + 0.294420i \(0.904874\pi\)
\(618\) 0 0
\(619\) 26.5442i 1.06690i 0.845831 + 0.533452i \(0.179105\pi\)
−0.845831 + 0.533452i \(0.820895\pi\)
\(620\) − 38.0292i − 1.52729i
\(621\) 0 0
\(622\) 16.5036i 0.661734i
\(623\) 0 0
\(624\) 0 0
\(625\) −19.2101 −0.768403
\(626\) 12.2421 0.489293
\(627\) 0 0
\(628\) − 5.19615i − 0.207349i
\(629\) 5.87738 0.234347
\(630\) 0 0
\(631\) −8.20304 −0.326558 −0.163279 0.986580i \(-0.552207\pi\)
−0.163279 + 0.986580i \(0.552207\pi\)
\(632\) 2.88979i 0.114950i
\(633\) 0 0
\(634\) 5.24617 0.208352
\(635\) 58.6455 2.32727
\(636\) 0 0
\(637\) 0 0
\(638\) 5.62648i 0.222755i
\(639\) 0 0
\(640\) − 36.7636i − 1.45321i
\(641\) − 16.0499i − 0.633934i −0.948437 0.316967i \(-0.897336\pi\)
0.948437 0.316967i \(-0.102664\pi\)
\(642\) 0 0
\(643\) − 26.4101i − 1.04151i −0.853705 0.520757i \(-0.825650\pi\)
0.853705 0.520757i \(-0.174350\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.32268 0.130729
\(647\) −31.7015 −1.24632 −0.623158 0.782096i \(-0.714151\pi\)
−0.623158 + 0.782096i \(0.714151\pi\)
\(648\) 0 0
\(649\) 5.09633i 0.200048i
\(650\) 6.19360 0.242933
\(651\) 0 0
\(652\) −3.59808 −0.140912
\(653\) − 13.2901i − 0.520083i −0.965597 0.260041i \(-0.916264\pi\)
0.965597 0.260041i \(-0.0837362\pi\)
\(654\) 0 0
\(655\) −45.0594 −1.76062
\(656\) −15.4231 −0.602171
\(657\) 0 0
\(658\) 0 0
\(659\) 36.2125i 1.41064i 0.708890 + 0.705319i \(0.249196\pi\)
−0.708890 + 0.705319i \(0.750804\pi\)
\(660\) 0 0
\(661\) − 50.7930i − 1.97562i −0.155674 0.987809i \(-0.549755\pi\)
0.155674 0.987809i \(-0.450245\pi\)
\(662\) − 5.86668i − 0.228015i
\(663\) 0 0
\(664\) 2.43121i 0.0943491i
\(665\) 0 0
\(666\) 0 0
\(667\) −34.3478 −1.32995
\(668\) −4.18118 −0.161775
\(669\) 0 0
\(670\) 23.4155i 0.904618i
\(671\) 9.38910 0.362462
\(672\) 0 0
\(673\) 3.32865 0.128310 0.0641550 0.997940i \(-0.479565\pi\)
0.0641550 + 0.997940i \(0.479565\pi\)
\(674\) 2.78752i 0.107371i
\(675\) 0 0
\(676\) 16.4784 0.633786
\(677\) −19.9380 −0.766280 −0.383140 0.923690i \(-0.625157\pi\)
−0.383140 + 0.923690i \(0.625157\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 8.82710i 0.338504i
\(681\) 0 0
\(682\) 11.2614i 0.431222i
\(683\) 28.6636i 1.09678i 0.836221 + 0.548392i \(0.184760\pi\)
−0.836221 + 0.548392i \(0.815240\pi\)
\(684\) 0 0
\(685\) − 33.6512i − 1.28574i
\(686\) 0 0
\(687\) 0 0
\(688\) −3.46635 −0.132153
\(689\) −15.8792 −0.604948
\(690\) 0 0
\(691\) − 20.3674i − 0.774812i −0.921909 0.387406i \(-0.873371\pi\)
0.921909 0.387406i \(-0.126629\pi\)
\(692\) −4.02458 −0.152991
\(693\) 0 0
\(694\) −1.72460 −0.0654650
\(695\) − 12.7668i − 0.484273i
\(696\) 0 0
\(697\) 10.9699 0.415513
\(698\) 21.3347 0.807532
\(699\) 0 0
\(700\) 0 0
\(701\) − 49.1172i − 1.85513i −0.373659 0.927566i \(-0.621897\pi\)
0.373659 0.927566i \(-0.378103\pi\)
\(702\) 0 0
\(703\) − 22.9937i − 0.867222i
\(704\) 1.33303i 0.0502405i
\(705\) 0 0
\(706\) − 15.3545i − 0.577876i
\(707\) 0 0
\(708\) 0 0
\(709\) 23.3658 0.877522 0.438761 0.898604i \(-0.355418\pi\)
0.438761 + 0.898604i \(0.355418\pi\)
\(710\) 13.9040 0.521809
\(711\) 0 0
\(712\) − 17.5068i − 0.656095i
\(713\) −68.7471 −2.57460
\(714\) 0 0
\(715\) 12.2633 0.458622
\(716\) 21.7075i 0.811246i
\(717\) 0 0
\(718\) 14.0191 0.523189
\(719\) −12.6669 −0.472396 −0.236198 0.971705i \(-0.575901\pi\)
−0.236198 + 0.971705i \(0.575901\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 0.523303i − 0.0194753i
\(723\) 0 0
\(724\) − 7.43648i − 0.276374i
\(725\) 21.8295i 0.810728i
\(726\) 0 0
\(727\) − 2.70398i − 0.100285i −0.998742 0.0501426i \(-0.984032\pi\)
0.998742 0.0501426i \(-0.0159676\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 23.2332 0.859898
\(731\) 2.46548 0.0911891
\(732\) 0 0
\(733\) 21.8087i 0.805524i 0.915305 + 0.402762i \(0.131950\pi\)
−0.915305 + 0.402762i \(0.868050\pi\)
\(734\) 10.1927 0.376218
\(735\) 0 0
\(736\) −53.8865 −1.98628
\(737\) 25.2308i 0.929387i
\(738\) 0 0
\(739\) 32.1267 1.18180 0.590899 0.806745i \(-0.298773\pi\)
0.590899 + 0.806745i \(0.298773\pi\)
\(740\) 26.8529 0.987131
\(741\) 0 0
\(742\) 0 0
\(743\) 41.2728i 1.51415i 0.653328 + 0.757075i \(0.273372\pi\)
−0.653328 + 0.757075i \(0.726628\pi\)
\(744\) 0 0
\(745\) 16.8256i 0.616442i
\(746\) − 15.3339i − 0.561415i
\(747\) 0 0
\(748\) 4.18118i 0.152879i
\(749\) 0 0
\(750\) 0 0
\(751\) 45.8091 1.67160 0.835798 0.549037i \(-0.185005\pi\)
0.835798 + 0.549037i \(0.185005\pi\)
\(752\) 8.93303 0.325754
\(753\) 0 0
\(754\) 3.79366i 0.138157i
\(755\) −60.6294 −2.20653
\(756\) 0 0
\(757\) 50.3427 1.82974 0.914868 0.403752i \(-0.132294\pi\)
0.914868 + 0.403752i \(0.132294\pi\)
\(758\) 1.66697i 0.0605471i
\(759\) 0 0
\(760\) 34.5337 1.25267
\(761\) 30.3646 1.10072 0.550358 0.834929i \(-0.314491\pi\)
0.550358 + 0.834929i \(0.314491\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 13.0895i − 0.473562i
\(765\) 0 0
\(766\) − 10.6400i − 0.384438i
\(767\) 3.43621i 0.124074i
\(768\) 0 0
\(769\) 41.3383i 1.49070i 0.666675 + 0.745349i \(0.267717\pi\)
−0.666675 + 0.745349i \(0.732283\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −27.9949 −1.00756
\(773\) 13.9951 0.503368 0.251684 0.967809i \(-0.419016\pi\)
0.251684 + 0.967809i \(0.419016\pi\)
\(774\) 0 0
\(775\) 43.6918i 1.56946i
\(776\) 31.7583 1.14006
\(777\) 0 0
\(778\) −14.6643 −0.525741
\(779\) − 42.9166i − 1.53765i
\(780\) 0 0
\(781\) 14.9819 0.536096
\(782\) 7.01468 0.250845
\(783\) 0 0
\(784\) 0 0
\(785\) 10.9699i 0.391531i
\(786\) 0 0
\(787\) − 13.4868i − 0.480753i −0.970680 0.240377i \(-0.922729\pi\)
0.970680 0.240377i \(-0.0772709\pi\)
\(788\) − 23.7056i − 0.844478i
\(789\) 0 0
\(790\) − 2.68188i − 0.0954170i
\(791\) 0 0
\(792\) 0 0
\(793\) 6.33062 0.224807
\(794\) −21.4658 −0.761794
\(795\) 0 0
\(796\) 13.5866i 0.481566i
\(797\) −8.31735 −0.294616 −0.147308 0.989091i \(-0.547061\pi\)
−0.147308 + 0.989091i \(0.547061\pi\)
\(798\) 0 0
\(799\) −6.35373 −0.224779
\(800\) 34.2472i 1.21082i
\(801\) 0 0
\(802\) −18.9940 −0.670703
\(803\) 25.0343 0.883443
\(804\) 0 0
\(805\) 0 0
\(806\) 7.59302i 0.267453i
\(807\) 0 0
\(808\) 32.1146i 1.12979i
\(809\) 14.5440i 0.511340i 0.966764 + 0.255670i \(0.0822960\pi\)
−0.966764 + 0.255670i \(0.917704\pi\)
\(810\) 0 0
\(811\) − 41.8287i − 1.46880i −0.678715 0.734401i \(-0.737463\pi\)
0.678715 0.734401i \(-0.262537\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −7.95181 −0.278711
\(815\) 7.59609 0.266079
\(816\) 0 0
\(817\) − 9.64553i − 0.337454i
\(818\) 0.689991 0.0241250
\(819\) 0 0
\(820\) 50.1196 1.75025
\(821\) 46.4828i 1.62226i 0.584865 + 0.811131i \(0.301148\pi\)
−0.584865 + 0.811131i \(0.698852\pi\)
\(822\) 0 0
\(823\) 7.73669 0.269684 0.134842 0.990867i \(-0.456947\pi\)
0.134842 + 0.990867i \(0.456947\pi\)
\(824\) −11.4641 −0.399369
\(825\) 0 0
\(826\) 0 0
\(827\) 20.8898i 0.726409i 0.931709 + 0.363205i \(0.118317\pi\)
−0.931709 + 0.363205i \(0.881683\pi\)
\(828\) 0 0
\(829\) − 47.6062i − 1.65343i −0.562619 0.826716i \(-0.690206\pi\)
0.562619 0.826716i \(-0.309794\pi\)
\(830\) − 2.25628i − 0.0783168i
\(831\) 0 0
\(832\) 0.898798i 0.0311602i
\(833\) 0 0
\(834\) 0 0
\(835\) 8.82710 0.305475
\(836\) 16.3577 0.565744
\(837\) 0 0
\(838\) − 2.04944i − 0.0707968i
\(839\) −22.4035 −0.773455 −0.386727 0.922194i \(-0.626395\pi\)
−0.386727 + 0.922194i \(0.626395\pi\)
\(840\) 0 0
\(841\) 15.6291 0.538935
\(842\) − 5.71690i − 0.197017i
\(843\) 0 0
\(844\) −22.8753 −0.787400
\(845\) −34.7885 −1.19676
\(846\) 0 0
\(847\) 0 0
\(848\) − 16.0697i − 0.551836i
\(849\) 0 0
\(850\) − 4.45814i − 0.152913i
\(851\) − 48.5431i − 1.66404i
\(852\) 0 0
\(853\) 48.4273i 1.65812i 0.559160 + 0.829060i \(0.311124\pi\)
−0.559160 + 0.829060i \(0.688876\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5.74968 −0.196520
\(857\) −49.0655 −1.67605 −0.838023 0.545636i \(-0.816288\pi\)
−0.838023 + 0.545636i \(0.816288\pi\)
\(858\) 0 0
\(859\) − 12.0245i − 0.410272i −0.978733 0.205136i \(-0.934236\pi\)
0.978733 0.205136i \(-0.0657636\pi\)
\(860\) 11.2644 0.384113
\(861\) 0 0
\(862\) 17.9146 0.610175
\(863\) − 46.0499i − 1.56756i −0.621040 0.783779i \(-0.713290\pi\)
0.621040 0.783779i \(-0.286710\pi\)
\(864\) 0 0
\(865\) 8.49649 0.288889
\(866\) 8.45145 0.287192
\(867\) 0 0
\(868\) 0 0
\(869\) − 2.88979i − 0.0980296i
\(870\) 0 0
\(871\) 17.0119i 0.576426i
\(872\) 10.1601i 0.344066i
\(873\) 0 0
\(874\) − 27.4430i − 0.928275i
\(875\) 0 0
\(876\) 0 0
\(877\) 13.4734 0.454964 0.227482 0.973782i \(-0.426951\pi\)
0.227482 + 0.973782i \(0.426951\pi\)
\(878\) −12.1110 −0.408727
\(879\) 0 0
\(880\) 12.4105i 0.418357i
\(881\) 25.5247 0.859949 0.429974 0.902841i \(-0.358523\pi\)
0.429974 + 0.902841i \(0.358523\pi\)
\(882\) 0 0
\(883\) 6.45532 0.217239 0.108619 0.994083i \(-0.465357\pi\)
0.108619 + 0.994083i \(0.465357\pi\)
\(884\) 2.81917i 0.0948189i
\(885\) 0 0
\(886\) 1.88450 0.0633111
\(887\) −33.1208 −1.11209 −0.556043 0.831153i \(-0.687681\pi\)
−0.556043 + 0.831153i \(0.687681\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 16.2472i 0.544608i
\(891\) 0 0
\(892\) 16.6034i 0.555924i
\(893\) 24.8572i 0.831816i
\(894\) 0 0
\(895\) − 45.8277i − 1.53185i
\(896\) 0 0
\(897\) 0 0
\(898\) 1.19292 0.0398082
\(899\) −26.7618 −0.892556
\(900\) 0 0
\(901\) 11.4298i 0.380781i
\(902\) −14.8417 −0.494174
\(903\) 0 0
\(904\) −16.7789 −0.558058
\(905\) 15.6995i 0.521870i
\(906\) 0 0
\(907\) 6.21512 0.206370 0.103185 0.994662i \(-0.467097\pi\)
0.103185 + 0.994662i \(0.467097\pi\)
\(908\) 10.5489 0.350078
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0475i 0.597941i 0.954262 + 0.298970i \(0.0966432\pi\)
−0.954262 + 0.298970i \(0.903357\pi\)
\(912\) 0 0
\(913\) − 2.43121i − 0.0804611i
\(914\) 4.04199i 0.133697i
\(915\) 0 0
\(916\) − 19.2389i − 0.635670i
\(917\) 0 0
\(918\) 0 0
\(919\) −8.25825 −0.272415 −0.136207 0.990680i \(-0.543491\pi\)
−0.136207 + 0.990680i \(0.543491\pi\)
\(920\) 72.9057 2.40363
\(921\) 0 0
\(922\) − 11.1959i − 0.368716i
\(923\) 10.1016 0.332498
\(924\) 0 0
\(925\) −30.8513 −1.01438
\(926\) 19.3132i 0.634672i
\(927\) 0 0
\(928\) −20.9769 −0.688600
\(929\) −34.7709 −1.14080 −0.570399 0.821368i \(-0.693211\pi\)
−0.570399 + 0.821368i \(0.693211\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.82710i 0.289141i
\(933\) 0 0
\(934\) 17.8063i 0.582639i
\(935\) − 8.82710i − 0.288677i
\(936\) 0 0
\(937\) 51.0703i 1.66839i 0.551466 + 0.834197i \(0.314069\pi\)
−0.551466 + 0.834197i \(0.685931\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −29.0292 −0.946829
\(941\) 3.47744 0.113361 0.0566807 0.998392i \(-0.481948\pi\)
0.0566807 + 0.998392i \(0.481948\pi\)
\(942\) 0 0
\(943\) − 90.6034i − 2.95045i
\(944\) −3.47744 −0.113181
\(945\) 0 0
\(946\) −3.33568 −0.108452
\(947\) 14.3401i 0.465989i 0.972478 + 0.232995i \(0.0748525\pi\)
−0.972478 + 0.232995i \(0.925148\pi\)
\(948\) 0 0
\(949\) 16.8794 0.547930
\(950\) −17.4413 −0.565869
\(951\) 0 0
\(952\) 0 0
\(953\) 7.53697i 0.244147i 0.992521 + 0.122073i \(0.0389543\pi\)
−0.992521 + 0.122073i \(0.961046\pi\)
\(954\) 0 0
\(955\) 27.6339i 0.894213i
\(956\) − 25.0381i − 0.809789i
\(957\) 0 0
\(958\) 14.1547i 0.457318i
\(959\) 0 0
\(960\) 0 0
\(961\) −22.5638 −0.727864
\(962\) −5.36152 −0.172862
\(963\) 0 0
\(964\) 7.19797i 0.231831i
\(965\) 59.1015 1.90255
\(966\) 0 0
\(967\) −11.6161 −0.373550 −0.186775 0.982403i \(-0.559803\pi\)
−0.186775 + 0.982403i \(0.559803\pi\)
\(968\) 12.9087i 0.414901i
\(969\) 0 0
\(970\) −29.4734 −0.946333
\(971\) −35.4952 −1.13910 −0.569548 0.821958i \(-0.692882\pi\)
−0.569548 + 0.821958i \(0.692882\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 9.57414i − 0.306775i
\(975\) 0 0
\(976\) 6.40659i 0.205070i
\(977\) 32.0094i 1.02407i 0.858964 + 0.512036i \(0.171109\pi\)
−0.858964 + 0.512036i \(0.828891\pi\)
\(978\) 0 0
\(979\) 17.5068i 0.559520i
\(980\) 0 0
\(981\) 0 0
\(982\) 0.185891 0.00593201
\(983\) 49.4648 1.57768 0.788841 0.614598i \(-0.210682\pi\)
0.788841 + 0.614598i \(0.210682\pi\)
\(984\) 0 0
\(985\) 50.0462i 1.59460i
\(986\) 2.73067 0.0869623
\(987\) 0 0
\(988\) 11.0292 0.350887
\(989\) − 20.3632i − 0.647511i
\(990\) 0 0
\(991\) 17.9518 0.570258 0.285129 0.958489i \(-0.407964\pi\)
0.285129 + 0.958489i \(0.407964\pi\)
\(992\) −41.9853 −1.33303
\(993\) 0 0
\(994\) 0 0
\(995\) − 28.6834i − 0.909326i
\(996\) 0 0
\(997\) 33.5037i 1.06107i 0.847662 + 0.530537i \(0.178010\pi\)
−0.847662 + 0.530537i \(0.821990\pi\)
\(998\) 8.08007i 0.255770i
\(999\) 0 0
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 1323.2.c.d.1322.7 12
3.2 odd 2 inner 1323.2.c.d.1322.6 12
7.2 even 3 189.2.p.d.80.4 yes 12
7.3 odd 6 189.2.p.d.26.3 12
7.6 odd 2 inner 1323.2.c.d.1322.8 12
21.2 odd 6 189.2.p.d.80.3 yes 12
21.17 even 6 189.2.p.d.26.4 yes 12
21.20 even 2 inner 1323.2.c.d.1322.5 12
63.2 odd 6 567.2.s.f.458.4 12
63.16 even 3 567.2.s.f.458.3 12
63.23 odd 6 567.2.i.f.269.3 12
63.31 odd 6 567.2.s.f.26.4 12
63.38 even 6 567.2.i.f.215.3 12
63.52 odd 6 567.2.i.f.215.4 12
63.58 even 3 567.2.i.f.269.4 12
63.59 even 6 567.2.s.f.26.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.d.26.3 12 7.3 odd 6
189.2.p.d.26.4 yes 12 21.17 even 6
189.2.p.d.80.3 yes 12 21.2 odd 6
189.2.p.d.80.4 yes 12 7.2 even 3
567.2.i.f.215.3 12 63.38 even 6
567.2.i.f.215.4 12 63.52 odd 6
567.2.i.f.269.3 12 63.23 odd 6
567.2.i.f.269.4 12 63.58 even 3
567.2.s.f.26.3 12 63.59 even 6
567.2.s.f.26.4 12 63.31 odd 6
567.2.s.f.458.3 12 63.16 even 3
567.2.s.f.458.4 12 63.2 odd 6
1323.2.c.d.1322.5 12 21.20 even 2 inner
1323.2.c.d.1322.6 12 3.2 odd 2 inner
1323.2.c.d.1322.7 12 1.1 even 1 trivial
1323.2.c.d.1322.8 12 7.6 odd 2 inner