Properties

Label 45.3.c.a.26.2
Level $45$
Weight $3$
Character 45.26
Analytic conductor $1.226$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 45.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.22616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.2
Root \(-1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 45.26
Dual form 45.3.c.a.26.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.821854i q^{2} +3.32456 q^{4} -2.23607i q^{5} +0.837722 q^{7} -6.01972i q^{8} +O(q^{10})\) \(q-0.821854i q^{2} +3.32456 q^{4} -2.23607i q^{5} +0.837722 q^{7} -6.01972i q^{8} -1.83772 q^{10} +14.3716i q^{11} -21.8114 q^{13} -0.688486i q^{14} +8.35089 q^{16} +23.5454i q^{17} -6.32456 q^{19} -7.43393i q^{20} +11.8114 q^{22} -38.8723i q^{23} -5.00000 q^{25} +17.9258i q^{26} +2.78505 q^{28} -0.266737i q^{29} +30.2719 q^{31} -30.9421i q^{32} +19.3509 q^{34} -1.87320i q^{35} +9.53950 q^{37} +5.19786i q^{38} -13.4605 q^{40} -19.3028i q^{41} +19.6228 q^{43} +47.7793i q^{44} -31.9473 q^{46} +22.1684i q^{47} -48.2982 q^{49} +4.10927i q^{50} -72.5132 q^{52} +49.0012i q^{53} +32.1359 q^{55} -5.04285i q^{56} -0.219219 q^{58} -73.2351i q^{59} -48.3246 q^{61} -24.8791i q^{62} +7.97367 q^{64} +48.7717i q^{65} +77.2982 q^{67} +78.2780i q^{68} -1.53950 q^{70} -104.044i q^{71} +47.6754 q^{73} -7.84008i q^{74} -21.0263 q^{76} +12.0394i q^{77} +68.2192 q^{79} -18.6732i q^{80} -15.8641 q^{82} +28.2098i q^{83} +52.6491 q^{85} -16.1271i q^{86} +86.5132 q^{88} -53.7774i q^{89} -18.2719 q^{91} -129.233i q^{92} +18.2192 q^{94} +14.1421i q^{95} -114.921 q^{97} +39.6941i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} + 16 q^{7} + O(q^{10}) \) \( 4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55} + 328 q^{58} - 168 q^{61} - 44 q^{64} + 208 q^{67} - 120 q^{70} + 216 q^{73} - 160 q^{76} - 56 q^{79} - 152 q^{82} + 160 q^{85} + 384 q^{88} + 104 q^{91} - 256 q^{94} - 232 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.821854i − 0.410927i −0.978665 0.205464i \(-0.934130\pi\)
0.978665 0.205464i \(-0.0658702\pi\)
\(3\) 0 0
\(4\) 3.32456 0.831139
\(5\) − 2.23607i − 0.447214i
\(6\) 0 0
\(7\) 0.837722 0.119675 0.0598373 0.998208i \(-0.480942\pi\)
0.0598373 + 0.998208i \(0.480942\pi\)
\(8\) − 6.01972i − 0.752465i
\(9\) 0 0
\(10\) −1.83772 −0.183772
\(11\) 14.3716i 1.30651i 0.757137 + 0.653256i \(0.226597\pi\)
−0.757137 + 0.653256i \(0.773403\pi\)
\(12\) 0 0
\(13\) −21.8114 −1.67780 −0.838900 0.544286i \(-0.816801\pi\)
−0.838900 + 0.544286i \(0.816801\pi\)
\(14\) − 0.688486i − 0.0491776i
\(15\) 0 0
\(16\) 8.35089 0.521931
\(17\) 23.5454i 1.38502i 0.721407 + 0.692512i \(0.243496\pi\)
−0.721407 + 0.692512i \(0.756504\pi\)
\(18\) 0 0
\(19\) −6.32456 −0.332871 −0.166436 0.986052i \(-0.553226\pi\)
−0.166436 + 0.986052i \(0.553226\pi\)
\(20\) − 7.43393i − 0.371697i
\(21\) 0 0
\(22\) 11.8114 0.536881
\(23\) − 38.8723i − 1.69010i −0.534689 0.845049i \(-0.679571\pi\)
0.534689 0.845049i \(-0.320429\pi\)
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 17.9258i 0.689453i
\(27\) 0 0
\(28\) 2.78505 0.0994662
\(29\) − 0.266737i − 0.00919784i −0.999989 0.00459892i \(-0.998536\pi\)
0.999989 0.00459892i \(-0.00146389\pi\)
\(30\) 0 0
\(31\) 30.2719 0.976512 0.488256 0.872700i \(-0.337633\pi\)
0.488256 + 0.872700i \(0.337633\pi\)
\(32\) − 30.9421i − 0.966940i
\(33\) 0 0
\(34\) 19.3509 0.569144
\(35\) − 1.87320i − 0.0535201i
\(36\) 0 0
\(37\) 9.53950 0.257824 0.128912 0.991656i \(-0.458851\pi\)
0.128912 + 0.991656i \(0.458851\pi\)
\(38\) 5.19786i 0.136786i
\(39\) 0 0
\(40\) −13.4605 −0.336512
\(41\) − 19.3028i − 0.470799i −0.971899 0.235399i \(-0.924360\pi\)
0.971899 0.235399i \(-0.0756398\pi\)
\(42\) 0 0
\(43\) 19.6228 0.456344 0.228172 0.973621i \(-0.426725\pi\)
0.228172 + 0.973621i \(0.426725\pi\)
\(44\) 47.7793i 1.08589i
\(45\) 0 0
\(46\) −31.9473 −0.694507
\(47\) 22.1684i 0.471669i 0.971793 + 0.235834i \(0.0757823\pi\)
−0.971793 + 0.235834i \(0.924218\pi\)
\(48\) 0 0
\(49\) −48.2982 −0.985678
\(50\) 4.10927i 0.0821854i
\(51\) 0 0
\(52\) −72.5132 −1.39448
\(53\) 49.0012i 0.924552i 0.886736 + 0.462276i \(0.152967\pi\)
−0.886736 + 0.462276i \(0.847033\pi\)
\(54\) 0 0
\(55\) 32.1359 0.584290
\(56\) − 5.04285i − 0.0900509i
\(57\) 0 0
\(58\) −0.219219 −0.00377964
\(59\) − 73.2351i − 1.24127i −0.784098 0.620637i \(-0.786874\pi\)
0.784098 0.620637i \(-0.213126\pi\)
\(60\) 0 0
\(61\) −48.3246 −0.792206 −0.396103 0.918206i \(-0.629638\pi\)
−0.396103 + 0.918206i \(0.629638\pi\)
\(62\) − 24.8791i − 0.401276i
\(63\) 0 0
\(64\) 7.97367 0.124589
\(65\) 48.7717i 0.750335i
\(66\) 0 0
\(67\) 77.2982 1.15370 0.576852 0.816848i \(-0.304281\pi\)
0.576852 + 0.816848i \(0.304281\pi\)
\(68\) 78.2780i 1.15115i
\(69\) 0 0
\(70\) −1.53950 −0.0219929
\(71\) − 104.044i − 1.46541i −0.680548 0.732703i \(-0.738258\pi\)
0.680548 0.732703i \(-0.261742\pi\)
\(72\) 0 0
\(73\) 47.6754 0.653088 0.326544 0.945182i \(-0.394116\pi\)
0.326544 + 0.945182i \(0.394116\pi\)
\(74\) − 7.84008i − 0.105947i
\(75\) 0 0
\(76\) −21.0263 −0.276662
\(77\) 12.0394i 0.156356i
\(78\) 0 0
\(79\) 68.2192 0.863534 0.431767 0.901985i \(-0.357890\pi\)
0.431767 + 0.901985i \(0.357890\pi\)
\(80\) − 18.6732i − 0.233414i
\(81\) 0 0
\(82\) −15.8641 −0.193464
\(83\) 28.2098i 0.339877i 0.985455 + 0.169938i \(0.0543569\pi\)
−0.985455 + 0.169938i \(0.945643\pi\)
\(84\) 0 0
\(85\) 52.6491 0.619401
\(86\) − 16.1271i − 0.187524i
\(87\) 0 0
\(88\) 86.5132 0.983104
\(89\) − 53.7774i − 0.604240i −0.953270 0.302120i \(-0.902306\pi\)
0.953270 0.302120i \(-0.0976943\pi\)
\(90\) 0 0
\(91\) −18.2719 −0.200790
\(92\) − 129.233i − 1.40471i
\(93\) 0 0
\(94\) 18.2192 0.193821
\(95\) 14.1421i 0.148865i
\(96\) 0 0
\(97\) −114.921 −1.18475 −0.592376 0.805661i \(-0.701810\pi\)
−0.592376 + 0.805661i \(0.701810\pi\)
\(98\) 39.6941i 0.405042i
\(99\) 0 0
\(100\) −16.6228 −0.166228
\(101\) 17.5473i 0.173736i 0.996220 + 0.0868679i \(0.0276858\pi\)
−0.996220 + 0.0868679i \(0.972314\pi\)
\(102\) 0 0
\(103\) −71.5395 −0.694558 −0.347279 0.937762i \(-0.612894\pi\)
−0.347279 + 0.937762i \(0.612894\pi\)
\(104\) 131.298i 1.26248i
\(105\) 0 0
\(106\) 40.2719 0.379923
\(107\) 76.3675i 0.713715i 0.934159 + 0.356858i \(0.116152\pi\)
−0.934159 + 0.356858i \(0.883848\pi\)
\(108\) 0 0
\(109\) −126.921 −1.16441 −0.582206 0.813041i \(-0.697810\pi\)
−0.582206 + 0.813041i \(0.697810\pi\)
\(110\) − 26.4111i − 0.240101i
\(111\) 0 0
\(112\) 6.99573 0.0624618
\(113\) 15.0601i 0.133275i 0.997777 + 0.0666377i \(0.0212272\pi\)
−0.997777 + 0.0666377i \(0.978773\pi\)
\(114\) 0 0
\(115\) −86.9210 −0.755835
\(116\) − 0.886783i − 0.00764468i
\(117\) 0 0
\(118\) −60.1886 −0.510073
\(119\) 19.7245i 0.165752i
\(120\) 0 0
\(121\) −85.5438 −0.706973
\(122\) 39.7157i 0.325539i
\(123\) 0 0
\(124\) 100.641 0.811617
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 158.031 1.24434 0.622168 0.782884i \(-0.286252\pi\)
0.622168 + 0.782884i \(0.286252\pi\)
\(128\) − 130.322i − 1.01814i
\(129\) 0 0
\(130\) 40.0833 0.308333
\(131\) 211.220i 1.61237i 0.591665 + 0.806184i \(0.298471\pi\)
−0.591665 + 0.806184i \(0.701529\pi\)
\(132\) 0 0
\(133\) −5.29822 −0.0398363
\(134\) − 63.5279i − 0.474089i
\(135\) 0 0
\(136\) 141.737 1.04218
\(137\) 69.2592i 0.505542i 0.967526 + 0.252771i \(0.0813419\pi\)
−0.967526 + 0.252771i \(0.918658\pi\)
\(138\) 0 0
\(139\) −159.842 −1.14994 −0.574971 0.818174i \(-0.694987\pi\)
−0.574971 + 0.818174i \(0.694987\pi\)
\(140\) − 6.22757i − 0.0444826i
\(141\) 0 0
\(142\) −85.5089 −0.602175
\(143\) − 313.465i − 2.19206i
\(144\) 0 0
\(145\) −0.596443 −0.00411340
\(146\) − 39.1823i − 0.268372i
\(147\) 0 0
\(148\) 31.7146 0.214288
\(149\) 9.81897i 0.0658991i 0.999457 + 0.0329496i \(0.0104901\pi\)
−0.999457 + 0.0329496i \(0.989510\pi\)
\(150\) 0 0
\(151\) 210.649 1.39503 0.697514 0.716572i \(-0.254290\pi\)
0.697514 + 0.716572i \(0.254290\pi\)
\(152\) 38.0720i 0.250474i
\(153\) 0 0
\(154\) 9.89466 0.0642511
\(155\) − 67.6900i − 0.436710i
\(156\) 0 0
\(157\) 211.276 1.34571 0.672854 0.739775i \(-0.265068\pi\)
0.672854 + 0.739775i \(0.265068\pi\)
\(158\) − 56.0663i − 0.354850i
\(159\) 0 0
\(160\) −69.1886 −0.432429
\(161\) − 32.5642i − 0.202262i
\(162\) 0 0
\(163\) −222.763 −1.36664 −0.683322 0.730117i \(-0.739465\pi\)
−0.683322 + 0.730117i \(0.739465\pi\)
\(164\) − 64.1731i − 0.391299i
\(165\) 0 0
\(166\) 23.1843 0.139665
\(167\) 33.3644i 0.199787i 0.994998 + 0.0998933i \(0.0318501\pi\)
−0.994998 + 0.0998933i \(0.968150\pi\)
\(168\) 0 0
\(169\) 306.737 1.81501
\(170\) − 43.2699i − 0.254529i
\(171\) 0 0
\(172\) 65.2370 0.379285
\(173\) 29.8102i 0.172313i 0.996282 + 0.0861567i \(0.0274586\pi\)
−0.996282 + 0.0861567i \(0.972541\pi\)
\(174\) 0 0
\(175\) −4.18861 −0.0239349
\(176\) 120.016i 0.681909i
\(177\) 0 0
\(178\) −44.1972 −0.248299
\(179\) 111.841i 0.624808i 0.949949 + 0.312404i \(0.101134\pi\)
−0.949949 + 0.312404i \(0.898866\pi\)
\(180\) 0 0
\(181\) −49.0790 −0.271155 −0.135577 0.990767i \(-0.543289\pi\)
−0.135577 + 0.990767i \(0.543289\pi\)
\(182\) 15.0168i 0.0825101i
\(183\) 0 0
\(184\) −234.000 −1.27174
\(185\) − 21.3310i − 0.115303i
\(186\) 0 0
\(187\) −338.386 −1.80955
\(188\) 73.7002i 0.392022i
\(189\) 0 0
\(190\) 11.6228 0.0611725
\(191\) 278.947i 1.46046i 0.683203 + 0.730229i \(0.260586\pi\)
−0.683203 + 0.730229i \(0.739414\pi\)
\(192\) 0 0
\(193\) 89.8947 0.465775 0.232888 0.972504i \(-0.425183\pi\)
0.232888 + 0.972504i \(0.425183\pi\)
\(194\) 94.4483i 0.486847i
\(195\) 0 0
\(196\) −160.570 −0.819235
\(197\) − 212.709i − 1.07974i −0.841748 0.539870i \(-0.818473\pi\)
0.841748 0.539870i \(-0.181527\pi\)
\(198\) 0 0
\(199\) 96.4911 0.484880 0.242440 0.970166i \(-0.422052\pi\)
0.242440 + 0.970166i \(0.422052\pi\)
\(200\) 30.0986i 0.150493i
\(201\) 0 0
\(202\) 14.4213 0.0713928
\(203\) − 0.223452i − 0.00110075i
\(204\) 0 0
\(205\) −43.1623 −0.210548
\(206\) 58.7951i 0.285413i
\(207\) 0 0
\(208\) −182.144 −0.875695
\(209\) − 90.8942i − 0.434900i
\(210\) 0 0
\(211\) −65.7893 −0.311798 −0.155899 0.987773i \(-0.549827\pi\)
−0.155899 + 0.987773i \(0.549827\pi\)
\(212\) 162.907i 0.768431i
\(213\) 0 0
\(214\) 62.7630 0.293285
\(215\) − 43.8779i − 0.204083i
\(216\) 0 0
\(217\) 25.3594 0.116864
\(218\) 104.311i 0.478489i
\(219\) 0 0
\(220\) 106.838 0.485626
\(221\) − 513.558i − 2.32379i
\(222\) 0 0
\(223\) 102.302 0.458756 0.229378 0.973337i \(-0.426331\pi\)
0.229378 + 0.973337i \(0.426331\pi\)
\(224\) − 25.9209i − 0.115718i
\(225\) 0 0
\(226\) 12.3772 0.0547665
\(227\) − 12.5296i − 0.0551966i −0.999619 0.0275983i \(-0.991214\pi\)
0.999619 0.0275983i \(-0.00878593\pi\)
\(228\) 0 0
\(229\) 23.2982 0.101739 0.0508695 0.998705i \(-0.483801\pi\)
0.0508695 + 0.998705i \(0.483801\pi\)
\(230\) 71.4364i 0.310593i
\(231\) 0 0
\(232\) −1.60568 −0.00692105
\(233\) 356.382i 1.52954i 0.644306 + 0.764768i \(0.277146\pi\)
−0.644306 + 0.764768i \(0.722854\pi\)
\(234\) 0 0
\(235\) 49.5701 0.210937
\(236\) − 243.474i − 1.03167i
\(237\) 0 0
\(238\) 16.2107 0.0681121
\(239\) − 175.524i − 0.734408i −0.930140 0.367204i \(-0.880315\pi\)
0.930140 0.367204i \(-0.119685\pi\)
\(240\) 0 0
\(241\) 104.438 0.433355 0.216677 0.976243i \(-0.430478\pi\)
0.216677 + 0.976243i \(0.430478\pi\)
\(242\) 70.3045i 0.290515i
\(243\) 0 0
\(244\) −160.658 −0.658433
\(245\) 107.998i 0.440809i
\(246\) 0 0
\(247\) 137.947 0.558491
\(248\) − 182.228i − 0.734791i
\(249\) 0 0
\(250\) 9.18861 0.0367544
\(251\) 130.945i 0.521694i 0.965380 + 0.260847i \(0.0840018\pi\)
−0.965380 + 0.260847i \(0.915998\pi\)
\(252\) 0 0
\(253\) 558.658 2.20813
\(254\) − 129.878i − 0.511331i
\(255\) 0 0
\(256\) −75.2107 −0.293792
\(257\) − 425.641i − 1.65619i −0.560587 0.828095i \(-0.689425\pi\)
0.560587 0.828095i \(-0.310575\pi\)
\(258\) 0 0
\(259\) 7.99145 0.0308550
\(260\) 162.144i 0.623632i
\(261\) 0 0
\(262\) 173.592 0.662566
\(263\) 74.5004i 0.283271i 0.989919 + 0.141636i \(0.0452362\pi\)
−0.989919 + 0.141636i \(0.954764\pi\)
\(264\) 0 0
\(265\) 109.570 0.413472
\(266\) 4.35437i 0.0163698i
\(267\) 0 0
\(268\) 256.982 0.958889
\(269\) 205.067i 0.762331i 0.924507 + 0.381165i \(0.124477\pi\)
−0.924507 + 0.381165i \(0.875523\pi\)
\(270\) 0 0
\(271\) −233.351 −0.861073 −0.430537 0.902573i \(-0.641676\pi\)
−0.430537 + 0.902573i \(0.641676\pi\)
\(272\) 196.625i 0.722886i
\(273\) 0 0
\(274\) 56.9210 0.207741
\(275\) − 71.8582i − 0.261302i
\(276\) 0 0
\(277\) −423.715 −1.52966 −0.764828 0.644235i \(-0.777176\pi\)
−0.764828 + 0.644235i \(0.777176\pi\)
\(278\) 131.367i 0.472543i
\(279\) 0 0
\(280\) −11.2762 −0.0402720
\(281\) − 402.604i − 1.43275i −0.697713 0.716377i \(-0.745799\pi\)
0.697713 0.716377i \(-0.254201\pi\)
\(282\) 0 0
\(283\) −272.333 −0.962308 −0.481154 0.876636i \(-0.659782\pi\)
−0.481154 + 0.876636i \(0.659782\pi\)
\(284\) − 345.900i − 1.21796i
\(285\) 0 0
\(286\) −257.623 −0.900779
\(287\) − 16.1704i − 0.0563427i
\(288\) 0 0
\(289\) −265.386 −0.918290
\(290\) 0.490189i 0.00169031i
\(291\) 0 0
\(292\) 158.500 0.542807
\(293\) 443.188i 1.51259i 0.654232 + 0.756294i \(0.272992\pi\)
−0.654232 + 0.756294i \(0.727008\pi\)
\(294\) 0 0
\(295\) −163.759 −0.555114
\(296\) − 57.4251i − 0.194004i
\(297\) 0 0
\(298\) 8.06976 0.0270797
\(299\) 847.858i 2.83564i
\(300\) 0 0
\(301\) 16.4384 0.0546128
\(302\) − 173.123i − 0.573255i
\(303\) 0 0
\(304\) −52.8157 −0.173736
\(305\) 108.057i 0.354285i
\(306\) 0 0
\(307\) −390.824 −1.27304 −0.636522 0.771259i \(-0.719627\pi\)
−0.636522 + 0.771259i \(0.719627\pi\)
\(308\) 40.0258i 0.129954i
\(309\) 0 0
\(310\) −55.6313 −0.179456
\(311\) 2.97739i 0.00957362i 0.999989 + 0.00478681i \(0.00152369\pi\)
−0.999989 + 0.00478681i \(0.998476\pi\)
\(312\) 0 0
\(313\) 130.105 0.415672 0.207836 0.978164i \(-0.433358\pi\)
0.207836 + 0.978164i \(0.433358\pi\)
\(314\) − 173.638i − 0.552988i
\(315\) 0 0
\(316\) 226.799 0.717717
\(317\) − 131.677i − 0.415384i −0.978194 0.207692i \(-0.933405\pi\)
0.978194 0.207692i \(-0.0665953\pi\)
\(318\) 0 0
\(319\) 3.83345 0.0120171
\(320\) − 17.8297i − 0.0557177i
\(321\) 0 0
\(322\) −26.7630 −0.0831149
\(323\) − 148.914i − 0.461035i
\(324\) 0 0
\(325\) 109.057 0.335560
\(326\) 183.079i 0.561591i
\(327\) 0 0
\(328\) −116.197 −0.354260
\(329\) 18.5710i 0.0564468i
\(330\) 0 0
\(331\) 160.483 0.484842 0.242421 0.970171i \(-0.422059\pi\)
0.242421 + 0.970171i \(0.422059\pi\)
\(332\) 93.7850i 0.282485i
\(333\) 0 0
\(334\) 27.4207 0.0820978
\(335\) − 172.844i − 0.515952i
\(336\) 0 0
\(337\) −128.114 −0.380160 −0.190080 0.981769i \(-0.560875\pi\)
−0.190080 + 0.981769i \(0.560875\pi\)
\(338\) − 252.093i − 0.745837i
\(339\) 0 0
\(340\) 175.035 0.514808
\(341\) 435.056i 1.27583i
\(342\) 0 0
\(343\) −81.5089 −0.237635
\(344\) − 118.124i − 0.343383i
\(345\) 0 0
\(346\) 24.4997 0.0708082
\(347\) 219.637i 0.632959i 0.948599 + 0.316480i \(0.102501\pi\)
−0.948599 + 0.316480i \(0.897499\pi\)
\(348\) 0 0
\(349\) 403.465 1.15606 0.578030 0.816016i \(-0.303822\pi\)
0.578030 + 0.816016i \(0.303822\pi\)
\(350\) 3.44243i 0.00983551i
\(351\) 0 0
\(352\) 444.688 1.26332
\(353\) − 54.8192i − 0.155295i −0.996981 0.0776475i \(-0.975259\pi\)
0.996981 0.0776475i \(-0.0247409\pi\)
\(354\) 0 0
\(355\) −232.649 −0.655350
\(356\) − 178.786i − 0.502207i
\(357\) 0 0
\(358\) 91.9167 0.256751
\(359\) − 480.460i − 1.33833i −0.743114 0.669165i \(-0.766652\pi\)
0.743114 0.669165i \(-0.233348\pi\)
\(360\) 0 0
\(361\) −321.000 −0.889197
\(362\) 40.3358i 0.111425i
\(363\) 0 0
\(364\) −60.7459 −0.166884
\(365\) − 106.606i − 0.292070i
\(366\) 0 0
\(367\) 522.364 1.42333 0.711667 0.702517i \(-0.247940\pi\)
0.711667 + 0.702517i \(0.247940\pi\)
\(368\) − 324.618i − 0.882114i
\(369\) 0 0
\(370\) −17.5310 −0.0473810
\(371\) 41.0494i 0.110645i
\(372\) 0 0
\(373\) −233.285 −0.625428 −0.312714 0.949847i \(-0.601238\pi\)
−0.312714 + 0.949847i \(0.601238\pi\)
\(374\) 278.104i 0.743593i
\(375\) 0 0
\(376\) 133.448 0.354914
\(377\) 5.81791i 0.0154321i
\(378\) 0 0
\(379\) 248.596 0.655927 0.327964 0.944690i \(-0.393638\pi\)
0.327964 + 0.944690i \(0.393638\pi\)
\(380\) 47.0163i 0.123727i
\(381\) 0 0
\(382\) 229.254 0.600142
\(383\) − 468.291i − 1.22269i −0.791364 0.611346i \(-0.790628\pi\)
0.791364 0.611346i \(-0.209372\pi\)
\(384\) 0 0
\(385\) 26.9210 0.0699247
\(386\) − 73.8803i − 0.191400i
\(387\) 0 0
\(388\) −382.061 −0.984694
\(389\) − 484.238i − 1.24483i −0.782688 0.622414i \(-0.786152\pi\)
0.782688 0.622414i \(-0.213848\pi\)
\(390\) 0 0
\(391\) 915.263 2.34083
\(392\) 290.742i 0.741688i
\(393\) 0 0
\(394\) −174.816 −0.443695
\(395\) − 152.543i − 0.386184i
\(396\) 0 0
\(397\) −298.943 −0.753005 −0.376503 0.926416i \(-0.622873\pi\)
−0.376503 + 0.926416i \(0.622873\pi\)
\(398\) − 79.3016i − 0.199250i
\(399\) 0 0
\(400\) −41.7544 −0.104386
\(401\) 467.509i 1.16586i 0.812523 + 0.582929i \(0.198093\pi\)
−0.812523 + 0.582929i \(0.801907\pi\)
\(402\) 0 0
\(403\) −660.272 −1.63839
\(404\) 58.3370i 0.144399i
\(405\) 0 0
\(406\) −0.183645 −0.000452327 0
\(407\) 137.098i 0.336851i
\(408\) 0 0
\(409\) −184.158 −0.450264 −0.225132 0.974328i \(-0.572281\pi\)
−0.225132 + 0.974328i \(0.572281\pi\)
\(410\) 35.4731i 0.0865198i
\(411\) 0 0
\(412\) −237.837 −0.577274
\(413\) − 61.3507i − 0.148549i
\(414\) 0 0
\(415\) 63.0790 0.151998
\(416\) 674.890i 1.62233i
\(417\) 0 0
\(418\) −74.7018 −0.178712
\(419\) 429.840i 1.02587i 0.858427 + 0.512936i \(0.171442\pi\)
−0.858427 + 0.512936i \(0.828558\pi\)
\(420\) 0 0
\(421\) −305.035 −0.724548 −0.362274 0.932072i \(-0.618000\pi\)
−0.362274 + 0.932072i \(0.618000\pi\)
\(422\) 54.0692i 0.128126i
\(423\) 0 0
\(424\) 294.974 0.695693
\(425\) − 117.727i − 0.277005i
\(426\) 0 0
\(427\) −40.4826 −0.0948069
\(428\) 253.888i 0.593196i
\(429\) 0 0
\(430\) −36.0612 −0.0838633
\(431\) 128.880i 0.299025i 0.988760 + 0.149512i \(0.0477704\pi\)
−0.988760 + 0.149512i \(0.952230\pi\)
\(432\) 0 0
\(433\) 243.886 0.563247 0.281624 0.959525i \(-0.409127\pi\)
0.281624 + 0.959525i \(0.409127\pi\)
\(434\) − 20.8418i − 0.0480225i
\(435\) 0 0
\(436\) −421.956 −0.967789
\(437\) 245.850i 0.562585i
\(438\) 0 0
\(439\) −259.614 −0.591376 −0.295688 0.955285i \(-0.595549\pi\)
−0.295688 + 0.955285i \(0.595549\pi\)
\(440\) − 193.449i − 0.439658i
\(441\) 0 0
\(442\) −422.070 −0.954909
\(443\) − 541.011i − 1.22124i −0.791923 0.610622i \(-0.790920\pi\)
0.791923 0.610622i \(-0.209080\pi\)
\(444\) 0 0
\(445\) −120.250 −0.270224
\(446\) − 84.0778i − 0.188515i
\(447\) 0 0
\(448\) 6.67972 0.0149101
\(449\) 791.947i 1.76380i 0.471436 + 0.881900i \(0.343736\pi\)
−0.471436 + 0.881900i \(0.656264\pi\)
\(450\) 0 0
\(451\) 277.412 0.615104
\(452\) 50.0682i 0.110770i
\(453\) 0 0
\(454\) −10.2975 −0.0226818
\(455\) 40.8572i 0.0897960i
\(456\) 0 0
\(457\) 611.359 1.33777 0.668883 0.743367i \(-0.266773\pi\)
0.668883 + 0.743367i \(0.266773\pi\)
\(458\) − 19.1477i − 0.0418073i
\(459\) 0 0
\(460\) −288.974 −0.628204
\(461\) − 586.991i − 1.27330i −0.771153 0.636650i \(-0.780320\pi\)
0.771153 0.636650i \(-0.219680\pi\)
\(462\) 0 0
\(463\) 195.285 0.421781 0.210891 0.977510i \(-0.432364\pi\)
0.210891 + 0.977510i \(0.432364\pi\)
\(464\) − 2.22749i − 0.00480063i
\(465\) 0 0
\(466\) 292.894 0.628528
\(467\) − 753.763i − 1.61405i −0.590515 0.807027i \(-0.701075\pi\)
0.590515 0.807027i \(-0.298925\pi\)
\(468\) 0 0
\(469\) 64.7544 0.138069
\(470\) − 40.7394i − 0.0866796i
\(471\) 0 0
\(472\) −440.855 −0.934014
\(473\) 282.011i 0.596218i
\(474\) 0 0
\(475\) 31.6228 0.0665743
\(476\) 65.5752i 0.137763i
\(477\) 0 0
\(478\) −144.255 −0.301788
\(479\) − 614.848i − 1.28361i −0.766869 0.641803i \(-0.778186\pi\)
0.766869 0.641803i \(-0.221814\pi\)
\(480\) 0 0
\(481\) −208.070 −0.432577
\(482\) − 85.8332i − 0.178077i
\(483\) 0 0
\(484\) −284.395 −0.587593
\(485\) 256.971i 0.529837i
\(486\) 0 0
\(487\) 478.197 0.981924 0.490962 0.871181i \(-0.336645\pi\)
0.490962 + 0.871181i \(0.336645\pi\)
\(488\) 290.900i 0.596107i
\(489\) 0 0
\(490\) 88.7587 0.181140
\(491\) 617.223i 1.25707i 0.777780 + 0.628537i \(0.216346\pi\)
−0.777780 + 0.628537i \(0.783654\pi\)
\(492\) 0 0
\(493\) 6.28043 0.0127392
\(494\) − 113.373i − 0.229499i
\(495\) 0 0
\(496\) 252.797 0.509672
\(497\) − 87.1599i − 0.175372i
\(498\) 0 0
\(499\) 783.096 1.56933 0.784665 0.619919i \(-0.212835\pi\)
0.784665 + 0.619919i \(0.212835\pi\)
\(500\) 37.1697i 0.0743393i
\(501\) 0 0
\(502\) 107.618 0.214378
\(503\) − 369.395i − 0.734383i −0.930145 0.367191i \(-0.880319\pi\)
0.930145 0.367191i \(-0.119681\pi\)
\(504\) 0 0
\(505\) 39.2370 0.0776970
\(506\) − 459.135i − 0.907382i
\(507\) 0 0
\(508\) 525.381 1.03422
\(509\) − 225.635i − 0.443291i −0.975127 0.221645i \(-0.928857\pi\)
0.975127 0.221645i \(-0.0711427\pi\)
\(510\) 0 0
\(511\) 39.9388 0.0781581
\(512\) − 459.474i − 0.897410i
\(513\) 0 0
\(514\) −349.815 −0.680574
\(515\) 159.967i 0.310616i
\(516\) 0 0
\(517\) −318.596 −0.616241
\(518\) − 6.56781i − 0.0126792i
\(519\) 0 0
\(520\) 293.592 0.564600
\(521\) − 1007.73i − 1.93422i −0.254367 0.967108i \(-0.581867\pi\)
0.254367 0.967108i \(-0.418133\pi\)
\(522\) 0 0
\(523\) 935.517 1.78875 0.894376 0.447316i \(-0.147620\pi\)
0.894376 + 0.447316i \(0.147620\pi\)
\(524\) 702.213i 1.34010i
\(525\) 0 0
\(526\) 61.2285 0.116404
\(527\) 712.764i 1.35249i
\(528\) 0 0
\(529\) −982.052 −1.85643
\(530\) − 90.0507i − 0.169907i
\(531\) 0 0
\(532\) −17.6142 −0.0331095
\(533\) 421.020i 0.789906i
\(534\) 0 0
\(535\) 170.763 0.319183
\(536\) − 465.314i − 0.868122i
\(537\) 0 0
\(538\) 168.535 0.313263
\(539\) − 694.124i − 1.28780i
\(540\) 0 0
\(541\) −399.149 −0.737798 −0.368899 0.929469i \(-0.620265\pi\)
−0.368899 + 0.929469i \(0.620265\pi\)
\(542\) 191.780i 0.353838i
\(543\) 0 0
\(544\) 728.544 1.33923
\(545\) 283.804i 0.520741i
\(546\) 0 0
\(547\) 480.833 0.879036 0.439518 0.898234i \(-0.355149\pi\)
0.439518 + 0.898234i \(0.355149\pi\)
\(548\) 230.256i 0.420175i
\(549\) 0 0
\(550\) −59.0569 −0.107376
\(551\) 1.68699i 0.00306170i
\(552\) 0 0
\(553\) 57.1488 0.103343
\(554\) 348.232i 0.628577i
\(555\) 0 0
\(556\) −531.404 −0.955762
\(557\) 751.542i 1.34927i 0.738152 + 0.674634i \(0.235699\pi\)
−0.738152 + 0.674634i \(0.764301\pi\)
\(558\) 0 0
\(559\) −428.000 −0.765653
\(560\) − 15.6429i − 0.0279338i
\(561\) 0 0
\(562\) −330.882 −0.588758
\(563\) 670.820i 1.19151i 0.803166 + 0.595755i \(0.203147\pi\)
−0.803166 + 0.595755i \(0.796853\pi\)
\(564\) 0 0
\(565\) 33.6754 0.0596026
\(566\) 223.818i 0.395438i
\(567\) 0 0
\(568\) −626.315 −1.10267
\(569\) − 275.325i − 0.483875i −0.970292 0.241937i \(-0.922217\pi\)
0.970292 0.241937i \(-0.0777829\pi\)
\(570\) 0 0
\(571\) −900.289 −1.57669 −0.788344 0.615235i \(-0.789061\pi\)
−0.788344 + 0.615235i \(0.789061\pi\)
\(572\) − 1042.13i − 1.82191i
\(573\) 0 0
\(574\) −13.2897 −0.0231527
\(575\) 194.361i 0.338020i
\(576\) 0 0
\(577\) −596.236 −1.03334 −0.516669 0.856185i \(-0.672828\pi\)
−0.516669 + 0.856185i \(0.672828\pi\)
\(578\) 218.108i 0.377350i
\(579\) 0 0
\(580\) −1.98291 −0.00341880
\(581\) 23.6320i 0.0406746i
\(582\) 0 0
\(583\) −704.228 −1.20794
\(584\) − 286.993i − 0.491426i
\(585\) 0 0
\(586\) 364.236 0.621564
\(587\) − 497.431i − 0.847412i −0.905800 0.423706i \(-0.860729\pi\)
0.905800 0.423706i \(-0.139271\pi\)
\(588\) 0 0
\(589\) −191.456 −0.325053
\(590\) 134.586i 0.228112i
\(591\) 0 0
\(592\) 79.6633 0.134566
\(593\) 898.856i 1.51578i 0.652384 + 0.757889i \(0.273769\pi\)
−0.652384 + 0.757889i \(0.726231\pi\)
\(594\) 0 0
\(595\) 44.1053 0.0741266
\(596\) 32.6437i 0.0547713i
\(597\) 0 0
\(598\) 696.816 1.16524
\(599\) − 28.5701i − 0.0476964i −0.999716 0.0238482i \(-0.992408\pi\)
0.999716 0.0238482i \(-0.00759183\pi\)
\(600\) 0 0
\(601\) −20.2107 −0.0336284 −0.0168142 0.999859i \(-0.505352\pi\)
−0.0168142 + 0.999859i \(0.505352\pi\)
\(602\) − 13.5100i − 0.0224419i
\(603\) 0 0
\(604\) 700.315 1.15946
\(605\) 191.282i 0.316168i
\(606\) 0 0
\(607\) −713.626 −1.17566 −0.587831 0.808984i \(-0.700018\pi\)
−0.587831 + 0.808984i \(0.700018\pi\)
\(608\) 195.695i 0.321867i
\(609\) 0 0
\(610\) 88.8071 0.145585
\(611\) − 483.524i − 0.791365i
\(612\) 0 0
\(613\) −76.1530 −0.124230 −0.0621150 0.998069i \(-0.519785\pi\)
−0.0621150 + 0.998069i \(0.519785\pi\)
\(614\) 321.201i 0.523128i
\(615\) 0 0
\(616\) 72.4740 0.117653
\(617\) − 201.599i − 0.326741i −0.986565 0.163371i \(-0.947763\pi\)
0.986565 0.163371i \(-0.0522367\pi\)
\(618\) 0 0
\(619\) −204.263 −0.329989 −0.164995 0.986294i \(-0.552761\pi\)
−0.164995 + 0.986294i \(0.552761\pi\)
\(620\) − 225.039i − 0.362966i
\(621\) 0 0
\(622\) 2.44699 0.00393406
\(623\) − 45.0505i − 0.0723122i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) − 106.928i − 0.170811i
\(627\) 0 0
\(628\) 702.399 1.11847
\(629\) 224.611i 0.357093i
\(630\) 0 0
\(631\) −639.903 −1.01411 −0.507055 0.861914i \(-0.669266\pi\)
−0.507055 + 0.861914i \(0.669266\pi\)
\(632\) − 410.660i − 0.649779i
\(633\) 0 0
\(634\) −108.219 −0.170693
\(635\) − 353.367i − 0.556484i
\(636\) 0 0
\(637\) 1053.45 1.65377
\(638\) − 3.15054i − 0.00493815i
\(639\) 0 0
\(640\) −291.408 −0.455325
\(641\) 570.709i 0.890342i 0.895446 + 0.445171i \(0.146857\pi\)
−0.895446 + 0.445171i \(0.853143\pi\)
\(642\) 0 0
\(643\) 453.693 0.705587 0.352794 0.935701i \(-0.385232\pi\)
0.352794 + 0.935701i \(0.385232\pi\)
\(644\) − 108.261i − 0.168108i
\(645\) 0 0
\(646\) −122.386 −0.189452
\(647\) 983.536i 1.52015i 0.649837 + 0.760074i \(0.274837\pi\)
−0.649837 + 0.760074i \(0.725163\pi\)
\(648\) 0 0
\(649\) 1052.51 1.62174
\(650\) − 89.6289i − 0.137891i
\(651\) 0 0
\(652\) −740.588 −1.13587
\(653\) − 544.478i − 0.833811i −0.908950 0.416905i \(-0.863115\pi\)
0.908950 0.416905i \(-0.136885\pi\)
\(654\) 0 0
\(655\) 472.302 0.721073
\(656\) − 161.195i − 0.245724i
\(657\) 0 0
\(658\) 15.2626 0.0231955
\(659\) 107.933i 0.163783i 0.996641 + 0.0818916i \(0.0260961\pi\)
−0.996641 + 0.0818916i \(0.973904\pi\)
\(660\) 0 0
\(661\) 345.728 0.523038 0.261519 0.965198i \(-0.415777\pi\)
0.261519 + 0.965198i \(0.415777\pi\)
\(662\) − 131.893i − 0.199235i
\(663\) 0 0
\(664\) 169.815 0.255745
\(665\) 11.8472i 0.0178153i
\(666\) 0 0
\(667\) −10.3687 −0.0155452
\(668\) 110.922i 0.166050i
\(669\) 0 0
\(670\) −142.053 −0.212019
\(671\) − 694.503i − 1.03503i
\(672\) 0 0
\(673\) −1204.78 −1.79016 −0.895082 0.445902i \(-0.852883\pi\)
−0.895082 + 0.445902i \(0.852883\pi\)
\(674\) 105.291i 0.156218i
\(675\) 0 0
\(676\) 1019.76 1.50853
\(677\) − 574.598i − 0.848742i −0.905488 0.424371i \(-0.860495\pi\)
0.905488 0.424371i \(-0.139505\pi\)
\(678\) 0 0
\(679\) −96.2719 −0.141785
\(680\) − 316.933i − 0.466078i
\(681\) 0 0
\(682\) 357.553 0.524271
\(683\) − 334.387i − 0.489585i −0.969575 0.244792i \(-0.921280\pi\)
0.969575 0.244792i \(-0.0787198\pi\)
\(684\) 0 0
\(685\) 154.868 0.226085
\(686\) 66.9884i 0.0976508i
\(687\) 0 0
\(688\) 163.868 0.238180
\(689\) − 1068.79i − 1.55121i
\(690\) 0 0
\(691\) 901.149 1.30412 0.652061 0.758166i \(-0.273904\pi\)
0.652061 + 0.758166i \(0.273904\pi\)
\(692\) 99.1057i 0.143216i
\(693\) 0 0
\(694\) 180.510 0.260100
\(695\) 357.418i 0.514270i
\(696\) 0 0
\(697\) 454.491 0.652068
\(698\) − 331.589i − 0.475056i
\(699\) 0 0
\(700\) −13.9253 −0.0198932
\(701\) 887.934i 1.26667i 0.773879 + 0.633334i \(0.218314\pi\)
−0.773879 + 0.633334i \(0.781686\pi\)
\(702\) 0 0
\(703\) −60.3331 −0.0858223
\(704\) 114.595i 0.162776i
\(705\) 0 0
\(706\) −45.0534 −0.0638150
\(707\) 14.6998i 0.0207918i
\(708\) 0 0
\(709\) −1026.35 −1.44760 −0.723801 0.690008i \(-0.757607\pi\)
−0.723801 + 0.690008i \(0.757607\pi\)
\(710\) 191.204i 0.269301i
\(711\) 0 0
\(712\) −323.725 −0.454669
\(713\) − 1176.74i − 1.65040i
\(714\) 0 0
\(715\) −700.930 −0.980321
\(716\) 371.820i 0.519302i
\(717\) 0 0
\(718\) −394.868 −0.549956
\(719\) 740.080i 1.02932i 0.857395 + 0.514659i \(0.172081\pi\)
−0.857395 + 0.514659i \(0.827919\pi\)
\(720\) 0 0
\(721\) −59.9302 −0.0831210
\(722\) 263.815i 0.365395i
\(723\) 0 0
\(724\) −163.166 −0.225367
\(725\) 1.33369i 0.00183957i
\(726\) 0 0
\(727\) 1063.75 1.46321 0.731603 0.681731i \(-0.238773\pi\)
0.731603 + 0.681731i \(0.238773\pi\)
\(728\) 109.992i 0.151087i
\(729\) 0 0
\(730\) −87.6142 −0.120019
\(731\) 462.026i 0.632047i
\(732\) 0 0
\(733\) 134.749 0.183833 0.0919164 0.995767i \(-0.470701\pi\)
0.0919164 + 0.995767i \(0.470701\pi\)
\(734\) − 429.307i − 0.584887i
\(735\) 0 0
\(736\) −1202.79 −1.63422
\(737\) 1110.90i 1.50733i
\(738\) 0 0
\(739\) −711.429 −0.962692 −0.481346 0.876531i \(-0.659852\pi\)
−0.481346 + 0.876531i \(0.659852\pi\)
\(740\) − 70.9160i − 0.0958324i
\(741\) 0 0
\(742\) 33.7367 0.0454672
\(743\) − 466.330i − 0.627631i −0.949484 0.313816i \(-0.898393\pi\)
0.949484 0.313816i \(-0.101607\pi\)
\(744\) 0 0
\(745\) 21.9559 0.0294710
\(746\) 191.726i 0.257005i
\(747\) 0 0
\(748\) −1124.98 −1.50399
\(749\) 63.9748i 0.0854136i
\(750\) 0 0
\(751\) 227.359 0.302742 0.151371 0.988477i \(-0.451631\pi\)
0.151371 + 0.988477i \(0.451631\pi\)
\(752\) 185.126i 0.246178i
\(753\) 0 0
\(754\) 4.78147 0.00634148
\(755\) − 471.026i − 0.623875i
\(756\) 0 0
\(757\) −552.258 −0.729535 −0.364768 0.931099i \(-0.618852\pi\)
−0.364768 + 0.931099i \(0.618852\pi\)
\(758\) − 204.310i − 0.269538i
\(759\) 0 0
\(760\) 85.1317 0.112015
\(761\) 303.051i 0.398228i 0.979976 + 0.199114i \(0.0638064\pi\)
−0.979976 + 0.199114i \(0.936194\pi\)
\(762\) 0 0
\(763\) −106.325 −0.139351
\(764\) 927.376i 1.21384i
\(765\) 0 0
\(766\) −384.867 −0.502437
\(767\) 1597.36i 2.08261i
\(768\) 0 0
\(769\) 739.684 0.961878 0.480939 0.876754i \(-0.340296\pi\)
0.480939 + 0.876754i \(0.340296\pi\)
\(770\) − 22.1251i − 0.0287339i
\(771\) 0 0
\(772\) 298.860 0.387124
\(773\) − 603.479i − 0.780697i −0.920667 0.390348i \(-0.872355\pi\)
0.920667 0.390348i \(-0.127645\pi\)
\(774\) 0 0
\(775\) −151.359 −0.195302
\(776\) 691.792i 0.891485i
\(777\) 0 0
\(778\) −397.973 −0.511533
\(779\) 122.081i 0.156715i
\(780\) 0 0
\(781\) 1495.28 1.91457
\(782\) − 752.213i − 0.961909i
\(783\) 0 0
\(784\) −403.333 −0.514455
\(785\) − 472.428i − 0.601819i
\(786\) 0 0
\(787\) −660.483 −0.839241 −0.419620 0.907700i \(-0.637837\pi\)
−0.419620 + 0.907700i \(0.637837\pi\)
\(788\) − 707.162i − 0.897414i
\(789\) 0 0
\(790\) −125.368 −0.158694
\(791\) 12.6162i 0.0159497i
\(792\) 0 0
\(793\) 1054.03 1.32916
\(794\) 245.688i 0.309430i
\(795\) 0 0
\(796\) 320.790 0.403003
\(797\) − 634.969i − 0.796699i −0.917234 0.398349i \(-0.869583\pi\)
0.917234 0.398349i \(-0.130417\pi\)
\(798\) 0 0
\(799\) −521.964 −0.653272
\(800\) 154.710i 0.193388i
\(801\) 0 0
\(802\) 384.224 0.479083
\(803\) 685.174i 0.853268i
\(804\) 0 0
\(805\) −72.8157 −0.0904542
\(806\) 542.647i 0.673260i
\(807\) 0 0
\(808\) 105.630 0.130730
\(809\) − 405.048i − 0.500677i −0.968158 0.250339i \(-0.919458\pi\)
0.968158 0.250339i \(-0.0805419\pi\)
\(810\) 0 0
\(811\) 406.034 0.500659 0.250329 0.968161i \(-0.419461\pi\)
0.250329 + 0.968161i \(0.419461\pi\)
\(812\) − 0.742878i 0 0.000914874i
\(813\) 0 0
\(814\) 112.675 0.138421
\(815\) 498.113i 0.611182i
\(816\) 0 0
\(817\) −124.105 −0.151904
\(818\) 151.351i 0.185026i
\(819\) 0 0
\(820\) −143.495 −0.174994
\(821\) − 550.073i − 0.670003i −0.942218 0.335002i \(-0.891263\pi\)
0.942218 0.335002i \(-0.108737\pi\)
\(822\) 0 0
\(823\) −1472.51 −1.78920 −0.894600 0.446868i \(-0.852540\pi\)
−0.894600 + 0.446868i \(0.852540\pi\)
\(824\) 430.648i 0.522631i
\(825\) 0 0
\(826\) −50.4213 −0.0610428
\(827\) − 1510.47i − 1.82644i −0.407466 0.913220i \(-0.633587\pi\)
0.407466 0.913220i \(-0.366413\pi\)
\(828\) 0 0
\(829\) 712.692 0.859701 0.429850 0.902900i \(-0.358566\pi\)
0.429850 + 0.902900i \(0.358566\pi\)
\(830\) − 51.8418i − 0.0624599i
\(831\) 0 0
\(832\) −173.917 −0.209035
\(833\) − 1137.20i − 1.36519i
\(834\) 0 0
\(835\) 74.6050 0.0893473
\(836\) − 302.183i − 0.361463i
\(837\) 0 0
\(838\) 353.266 0.421559
\(839\) − 530.579i − 0.632394i −0.948694 0.316197i \(-0.897594\pi\)
0.948694 0.316197i \(-0.102406\pi\)
\(840\) 0 0
\(841\) 840.929 0.999915
\(842\) 250.694i 0.297737i
\(843\) 0 0
\(844\) −218.720 −0.259147
\(845\) − 685.884i − 0.811697i
\(846\) 0 0
\(847\) −71.6619 −0.0846068
\(848\) 409.204i 0.482552i
\(849\) 0 0
\(850\) −96.7544 −0.113829
\(851\) − 370.822i − 0.435748i
\(852\) 0 0
\(853\) 215.232 0.252324 0.126162 0.992010i \(-0.459734\pi\)
0.126162 + 0.992010i \(0.459734\pi\)
\(854\) 33.2708i 0.0389587i
\(855\) 0 0
\(856\) 459.711 0.537046
\(857\) 4.97441i 0.00580445i 0.999996 + 0.00290222i \(0.000923808\pi\)
−0.999996 + 0.00290222i \(0.999076\pi\)
\(858\) 0 0
\(859\) 1652.28 1.92349 0.961746 0.273941i \(-0.0883273\pi\)
0.961746 + 0.273941i \(0.0883273\pi\)
\(860\) − 145.874i − 0.169621i
\(861\) 0 0
\(862\) 105.920 0.122877
\(863\) 379.077i 0.439255i 0.975584 + 0.219627i \(0.0704841\pi\)
−0.975584 + 0.219627i \(0.929516\pi\)
\(864\) 0 0
\(865\) 66.6577 0.0770609
\(866\) − 200.439i − 0.231454i
\(867\) 0 0
\(868\) 84.3088 0.0971300
\(869\) 980.421i 1.12822i
\(870\) 0 0
\(871\) −1685.98 −1.93568
\(872\) 764.029i 0.876180i
\(873\) 0 0
\(874\) 202.053 0.231182
\(875\) 9.36602i 0.0107040i
\(876\) 0 0
\(877\) 1101.60 1.25610 0.628051 0.778173i \(-0.283853\pi\)
0.628051 + 0.778173i \(0.283853\pi\)
\(878\) 213.365i 0.243013i
\(879\) 0 0
\(880\) 268.364 0.304959
\(881\) 184.877i 0.209850i 0.994480 + 0.104925i \(0.0334602\pi\)
−0.994480 + 0.104925i \(0.966540\pi\)
\(882\) 0 0
\(883\) 978.236 1.10786 0.553928 0.832565i \(-0.313128\pi\)
0.553928 + 0.832565i \(0.313128\pi\)
\(884\) − 1707.35i − 1.93139i
\(885\) 0 0
\(886\) −444.632 −0.501842
\(887\) − 667.937i − 0.753029i −0.926411 0.376514i \(-0.877123\pi\)
0.926411 0.376514i \(-0.122877\pi\)
\(888\) 0 0
\(889\) 132.386 0.148915
\(890\) 98.8279i 0.111043i
\(891\) 0 0
\(892\) 340.110 0.381290
\(893\) − 140.205i − 0.157005i
\(894\) 0 0
\(895\) 250.083 0.279423
\(896\) − 109.173i − 0.121845i
\(897\) 0 0
\(898\) 650.865 0.724794
\(899\) − 8.07464i − 0.00898180i
\(900\) 0 0
\(901\) −1153.75 −1.28053
\(902\) − 227.992i − 0.252763i
\(903\) 0 0
\(904\) 90.6577 0.100285
\(905\) 109.744i 0.121264i
\(906\) 0 0
\(907\) −832.622 −0.917996 −0.458998 0.888437i \(-0.651791\pi\)
−0.458998 + 0.888437i \(0.651791\pi\)
\(908\) − 41.6554i − 0.0458760i
\(909\) 0 0
\(910\) 33.5787 0.0368996
\(911\) 565.263i 0.620486i 0.950657 + 0.310243i \(0.100410\pi\)
−0.950657 + 0.310243i \(0.899590\pi\)
\(912\) 0 0
\(913\) −405.421 −0.444053
\(914\) − 502.448i − 0.549725i
\(915\) 0 0
\(916\) 77.4562 0.0845592
\(917\) 176.944i 0.192959i
\(918\) 0 0
\(919\) −1185.75 −1.29026 −0.645128 0.764075i \(-0.723196\pi\)
−0.645128 + 0.764075i \(0.723196\pi\)
\(920\) 523.240i 0.568739i
\(921\) 0 0
\(922\) −482.421 −0.523234
\(923\) 2269.34i 2.45866i
\(924\) 0 0
\(925\) −47.6975 −0.0515649
\(926\) − 160.496i − 0.173321i
\(927\) 0 0
\(928\) −8.25341 −0.00889376
\(929\) 942.995i 1.01506i 0.861633 + 0.507532i \(0.169442\pi\)
−0.861633 + 0.507532i \(0.830558\pi\)
\(930\) 0 0
\(931\) 305.465 0.328104
\(932\) 1184.81i 1.27126i
\(933\) 0 0
\(934\) −619.483 −0.663258
\(935\) 756.654i 0.809255i
\(936\) 0 0
\(937\) −531.281 −0.567002 −0.283501 0.958972i \(-0.591496\pi\)
−0.283501 + 0.958972i \(0.591496\pi\)
\(938\) − 53.2187i − 0.0567364i
\(939\) 0 0
\(940\) 164.799 0.175318
\(941\) − 595.650i − 0.632996i −0.948593 0.316498i \(-0.897493\pi\)
0.948593 0.316498i \(-0.102507\pi\)
\(942\) 0 0
\(943\) −750.342 −0.795696
\(944\) − 611.578i − 0.647859i
\(945\) 0 0
\(946\) 231.772 0.245002
\(947\) − 782.246i − 0.826026i −0.910725 0.413013i \(-0.864476\pi\)
0.910725 0.413013i \(-0.135524\pi\)
\(948\) 0 0
\(949\) −1039.87 −1.09575
\(950\) − 25.9893i − 0.0273572i
\(951\) 0 0
\(952\) 118.736 0.124723
\(953\) 426.708i 0.447752i 0.974618 + 0.223876i \(0.0718711\pi\)
−0.974618 + 0.223876i \(0.928129\pi\)
\(954\) 0 0
\(955\) 623.745 0.653136
\(956\) − 583.538i − 0.610395i
\(957\) 0 0
\(958\) −505.315 −0.527469
\(959\) 58.0200i 0.0605005i
\(960\) 0 0
\(961\) −44.6128 −0.0464234
\(962\) 171.003i 0.177758i
\(963\) 0 0
\(964\) 347.211 0.360178
\(965\) − 201.011i − 0.208301i
\(966\) 0 0
\(967\) −1210.91 −1.25223 −0.626116 0.779730i \(-0.715356\pi\)
−0.626116 + 0.779730i \(0.715356\pi\)
\(968\) 514.949i 0.531973i
\(969\) 0 0
\(970\) 211.193 0.217725
\(971\) − 1193.69i − 1.22934i −0.788784 0.614670i \(-0.789289\pi\)
0.788784 0.614670i \(-0.210711\pi\)
\(972\) 0 0
\(973\) −133.903 −0.137619
\(974\) − 393.008i − 0.403499i
\(975\) 0 0
\(976\) −403.553 −0.413476
\(977\) 909.475i 0.930886i 0.885078 + 0.465443i \(0.154105\pi\)
−0.885078 + 0.465443i \(0.845895\pi\)
\(978\) 0 0
\(979\) 772.868 0.789447
\(980\) 359.046i 0.366373i
\(981\) 0 0
\(982\) 507.268 0.516566
\(983\) 1560.88i 1.58788i 0.607999 + 0.793938i \(0.291972\pi\)
−0.607999 + 0.793938i \(0.708028\pi\)
\(984\) 0 0
\(985\) −475.631 −0.482874
\(986\) − 5.16160i − 0.00523489i
\(987\) 0 0
\(988\) 458.614 0.464184
\(989\) − 762.782i − 0.771265i
\(990\) 0 0
\(991\) −1692.63 −1.70800 −0.854001 0.520271i \(-0.825831\pi\)
−0.854001 + 0.520271i \(0.825831\pi\)
\(992\) − 936.675i − 0.944229i
\(993\) 0 0
\(994\) −71.6327 −0.0720651
\(995\) − 215.761i − 0.216845i
\(996\) 0 0
\(997\) 16.5744 0.0166243 0.00831213 0.999965i \(-0.497354\pi\)
0.00831213 + 0.999965i \(0.497354\pi\)
\(998\) − 643.591i − 0.644881i
\(999\) 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 45.3.c.a.26.2 4
3.2 odd 2 inner 45.3.c.a.26.3 yes 4
4.3 odd 2 720.3.l.a.161.2 4
5.2 odd 4 225.3.d.b.224.6 8
5.3 odd 4 225.3.d.b.224.3 8
5.4 even 2 225.3.c.c.26.3 4
8.3 odd 2 2880.3.l.c.1601.4 4
8.5 even 2 2880.3.l.g.1601.3 4
9.2 odd 6 405.3.i.d.296.3 8
9.4 even 3 405.3.i.d.26.3 8
9.5 odd 6 405.3.i.d.26.2 8
9.7 even 3 405.3.i.d.296.2 8
12.11 even 2 720.3.l.a.161.4 4
15.2 even 4 225.3.d.b.224.4 8
15.8 even 4 225.3.d.b.224.5 8
15.14 odd 2 225.3.c.c.26.2 4
20.3 even 4 3600.3.c.i.449.5 8
20.7 even 4 3600.3.c.i.449.3 8
20.19 odd 2 3600.3.l.v.1601.1 4
24.5 odd 2 2880.3.l.g.1601.1 4
24.11 even 2 2880.3.l.c.1601.2 4
60.23 odd 4 3600.3.c.i.449.6 8
60.47 odd 4 3600.3.c.i.449.4 8
60.59 even 2 3600.3.l.v.1601.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.3.c.a.26.2 4 1.1 even 1 trivial
45.3.c.a.26.3 yes 4 3.2 odd 2 inner
225.3.c.c.26.2 4 15.14 odd 2
225.3.c.c.26.3 4 5.4 even 2
225.3.d.b.224.3 8 5.3 odd 4
225.3.d.b.224.4 8 15.2 even 4
225.3.d.b.224.5 8 15.8 even 4
225.3.d.b.224.6 8 5.2 odd 4
405.3.i.d.26.2 8 9.5 odd 6
405.3.i.d.26.3 8 9.4 even 3
405.3.i.d.296.2 8 9.7 even 3
405.3.i.d.296.3 8 9.2 odd 6
720.3.l.a.161.2 4 4.3 odd 2
720.3.l.a.161.4 4 12.11 even 2
2880.3.l.c.1601.2 4 24.11 even 2
2880.3.l.c.1601.4 4 8.3 odd 2
2880.3.l.g.1601.1 4 24.5 odd 2
2880.3.l.g.1601.3 4 8.5 even 2
3600.3.c.i.449.3 8 20.7 even 4
3600.3.c.i.449.4 8 60.47 odd 4
3600.3.c.i.449.5 8 20.3 even 4
3600.3.c.i.449.6 8 60.23 odd 4
3600.3.l.v.1601.1 4 20.19 odd 2
3600.3.l.v.1601.2 4 60.59 even 2