Properties

Label 9075.2.a.cl.1.2
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.45589 q^{2} -1.00000 q^{3} +4.03138 q^{4} +2.45589 q^{6} -3.28684 q^{7} -4.98884 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.45589 q^{2} -1.00000 q^{3} +4.03138 q^{4} +2.45589 q^{6} -3.28684 q^{7} -4.98884 q^{8} +1.00000 q^{9} -4.03138 q^{12} +0.313133 q^{13} +8.07211 q^{14} +4.18926 q^{16} -5.00000 q^{17} -2.45589 q^{18} -7.45408 q^{19} +3.28684 q^{21} -1.07392 q^{23} +4.98884 q^{24} -0.769020 q^{26} -1.00000 q^{27} -13.2505 q^{28} -5.03647 q^{29} +3.44899 q^{31} -0.310680 q^{32} +12.2794 q^{34} +4.03138 q^{36} -2.63428 q^{37} +18.3064 q^{38} -0.313133 q^{39} +10.8472 q^{41} -8.07211 q^{42} +5.51468 q^{43} +2.63743 q^{46} +11.9982 q^{47} -4.18926 q^{48} +3.80333 q^{49} +5.00000 q^{51} +1.26236 q^{52} -4.93543 q^{53} +2.45589 q^{54} +16.3975 q^{56} +7.45408 q^{57} +12.3690 q^{58} -9.16409 q^{59} +9.18431 q^{61} -8.47033 q^{62} -3.28684 q^{63} -7.61553 q^{64} +15.2739 q^{67} -20.1569 q^{68} +1.07392 q^{69} +3.07211 q^{71} -4.98884 q^{72} +8.65269 q^{73} +6.46950 q^{74} -30.0502 q^{76} +0.769020 q^{78} +5.41446 q^{79} +1.00000 q^{81} -26.6395 q^{82} -16.2454 q^{83} +13.2505 q^{84} -13.5434 q^{86} +5.03647 q^{87} +1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} -3.44899 q^{93} -29.4662 q^{94} +0.310680 q^{96} -0.224082 q^{97} -9.34054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} - 9 q^{12} - 3 q^{13} - 5 q^{14} + 15 q^{16} - 20 q^{17} - 5 q^{18} + 3 q^{19} - 2 q^{21} + 5 q^{23} + 15 q^{24} + 6 q^{26} - 4 q^{27} - 3 q^{28} + 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} + 9 q^{36} + 7 q^{37} - q^{38} + 3 q^{39} + 20 q^{41} + 5 q^{42} + 2 q^{43} + 7 q^{46} + 20 q^{47} - 15 q^{48} + 8 q^{49} + 20 q^{51} + 7 q^{52} - 6 q^{53} + 5 q^{54} + 10 q^{56} - 3 q^{57} + 21 q^{58} - 5 q^{59} - 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 13 q^{67} - 45 q^{68} - 5 q^{69} - 25 q^{71} - 15 q^{72} - 23 q^{73} - 7 q^{74} - 7 q^{76} - 6 q^{78} + 4 q^{81} - 11 q^{82} - 33 q^{83} + 3 q^{84} - 12 q^{86} - 5 q^{87} + 16 q^{89} - 24 q^{91} + q^{93} - 17 q^{94} + 30 q^{96} - 25 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.45589 −1.73657 −0.868287 0.496062i \(-0.834779\pi\)
−0.868287 + 0.496062i \(0.834779\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.03138 2.01569
\(5\) 0 0
\(6\) 2.45589 1.00261
\(7\) −3.28684 −1.24231 −0.621155 0.783688i \(-0.713336\pi\)
−0.621155 + 0.783688i \(0.713336\pi\)
\(8\) −4.98884 −1.76382
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −4.03138 −1.16376
\(13\) 0.313133 0.0868476 0.0434238 0.999057i \(-0.486173\pi\)
0.0434238 + 0.999057i \(0.486173\pi\)
\(14\) 8.07211 2.15736
\(15\) 0 0
\(16\) 4.18926 1.04732
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −2.45589 −0.578858
\(19\) −7.45408 −1.71008 −0.855041 0.518560i \(-0.826468\pi\)
−0.855041 + 0.518560i \(0.826468\pi\)
\(20\) 0 0
\(21\) 3.28684 0.717248
\(22\) 0 0
\(23\) −1.07392 −0.223928 −0.111964 0.993712i \(-0.535714\pi\)
−0.111964 + 0.993712i \(0.535714\pi\)
\(24\) 4.98884 1.01834
\(25\) 0 0
\(26\) −0.769020 −0.150817
\(27\) −1.00000 −0.192450
\(28\) −13.2505 −2.50411
\(29\) −5.03647 −0.935249 −0.467624 0.883927i \(-0.654890\pi\)
−0.467624 + 0.883927i \(0.654890\pi\)
\(30\) 0 0
\(31\) 3.44899 0.619457 0.309728 0.950825i \(-0.399762\pi\)
0.309728 + 0.950825i \(0.399762\pi\)
\(32\) −0.310680 −0.0549210
\(33\) 0 0
\(34\) 12.2794 2.10591
\(35\) 0 0
\(36\) 4.03138 0.671897
\(37\) −2.63428 −0.433073 −0.216537 0.976274i \(-0.569476\pi\)
−0.216537 + 0.976274i \(0.569476\pi\)
\(38\) 18.3064 2.96969
\(39\) −0.313133 −0.0501415
\(40\) 0 0
\(41\) 10.8472 1.69405 0.847024 0.531554i \(-0.178392\pi\)
0.847024 + 0.531554i \(0.178392\pi\)
\(42\) −8.07211 −1.24555
\(43\) 5.51468 0.840980 0.420490 0.907297i \(-0.361858\pi\)
0.420490 + 0.907297i \(0.361858\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.63743 0.388868
\(47\) 11.9982 1.75012 0.875058 0.484018i \(-0.160823\pi\)
0.875058 + 0.484018i \(0.160823\pi\)
\(48\) −4.18926 −0.604668
\(49\) 3.80333 0.543333
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 1.26236 0.175058
\(53\) −4.93543 −0.677934 −0.338967 0.940798i \(-0.610077\pi\)
−0.338967 + 0.940798i \(0.610077\pi\)
\(54\) 2.45589 0.334204
\(55\) 0 0
\(56\) 16.3975 2.19121
\(57\) 7.45408 0.987317
\(58\) 12.3690 1.62413
\(59\) −9.16409 −1.19306 −0.596531 0.802590i \(-0.703455\pi\)
−0.596531 + 0.802590i \(0.703455\pi\)
\(60\) 0 0
\(61\) 9.18431 1.17593 0.587965 0.808886i \(-0.299929\pi\)
0.587965 + 0.808886i \(0.299929\pi\)
\(62\) −8.47033 −1.07573
\(63\) −3.28684 −0.414103
\(64\) −7.61553 −0.951942
\(65\) 0 0
\(66\) 0 0
\(67\) 15.2739 1.86600 0.933000 0.359876i \(-0.117181\pi\)
0.933000 + 0.359876i \(0.117181\pi\)
\(68\) −20.1569 −2.44438
\(69\) 1.07392 0.129285
\(70\) 0 0
\(71\) 3.07211 0.364593 0.182296 0.983244i \(-0.441647\pi\)
0.182296 + 0.983244i \(0.441647\pi\)
\(72\) −4.98884 −0.587940
\(73\) 8.65269 1.01272 0.506361 0.862322i \(-0.330991\pi\)
0.506361 + 0.862322i \(0.330991\pi\)
\(74\) 6.46950 0.752064
\(75\) 0 0
\(76\) −30.0502 −3.44700
\(77\) 0 0
\(78\) 0.769020 0.0870744
\(79\) 5.41446 0.609175 0.304587 0.952484i \(-0.401481\pi\)
0.304587 + 0.952484i \(0.401481\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −26.6395 −2.94184
\(83\) −16.2454 −1.78317 −0.891583 0.452857i \(-0.850405\pi\)
−0.891583 + 0.452857i \(0.850405\pi\)
\(84\) 13.2505 1.44575
\(85\) 0 0
\(86\) −13.5434 −1.46042
\(87\) 5.03647 0.539966
\(88\) 0 0
\(89\) 1.62118 0.171845 0.0859223 0.996302i \(-0.472616\pi\)
0.0859223 + 0.996302i \(0.472616\pi\)
\(90\) 0 0
\(91\) −1.02922 −0.107892
\(92\) −4.32938 −0.451369
\(93\) −3.44899 −0.357643
\(94\) −29.4662 −3.03921
\(95\) 0 0
\(96\) 0.310680 0.0317086
\(97\) −0.224082 −0.0227521 −0.0113760 0.999935i \(-0.503621\pi\)
−0.0113760 + 0.999935i \(0.503621\pi\)
\(98\) −9.34054 −0.943537
\(99\) 0 0
\(100\) 0 0
\(101\) −0.505326 −0.0502818 −0.0251409 0.999684i \(-0.508003\pi\)
−0.0251409 + 0.999684i \(0.508003\pi\)
\(102\) −12.2794 −1.21585
\(103\) 6.40197 0.630805 0.315402 0.948958i \(-0.397861\pi\)
0.315402 + 0.948958i \(0.397861\pi\)
\(104\) −1.56217 −0.153184
\(105\) 0 0
\(106\) 12.1209 1.17728
\(107\) −2.09249 −0.202289 −0.101144 0.994872i \(-0.532250\pi\)
−0.101144 + 0.994872i \(0.532250\pi\)
\(108\) −4.03138 −0.387920
\(109\) −6.69278 −0.641052 −0.320526 0.947240i \(-0.603860\pi\)
−0.320526 + 0.947240i \(0.603860\pi\)
\(110\) 0 0
\(111\) 2.63428 0.250035
\(112\) −13.7694 −1.30109
\(113\) 10.7941 1.01542 0.507712 0.861527i \(-0.330491\pi\)
0.507712 + 0.861527i \(0.330491\pi\)
\(114\) −18.3064 −1.71455
\(115\) 0 0
\(116\) −20.3039 −1.88517
\(117\) 0.313133 0.0289492
\(118\) 22.5060 2.07184
\(119\) 16.4342 1.50652
\(120\) 0 0
\(121\) 0 0
\(122\) −22.5556 −2.04209
\(123\) −10.8472 −0.978059
\(124\) 13.9042 1.24863
\(125\) 0 0
\(126\) 8.07211 0.719121
\(127\) 17.0033 1.50880 0.754398 0.656417i \(-0.227929\pi\)
0.754398 + 0.656417i \(0.227929\pi\)
\(128\) 19.3242 1.70804
\(129\) −5.51468 −0.485540
\(130\) 0 0
\(131\) −0.0430508 −0.00376136 −0.00188068 0.999998i \(-0.500599\pi\)
−0.00188068 + 0.999998i \(0.500599\pi\)
\(132\) 0 0
\(133\) 24.5004 2.12445
\(134\) −37.5109 −3.24045
\(135\) 0 0
\(136\) 24.9442 2.13895
\(137\) −7.36257 −0.629027 −0.314513 0.949253i \(-0.601841\pi\)
−0.314513 + 0.949253i \(0.601841\pi\)
\(138\) −2.63743 −0.224513
\(139\) 13.2393 1.12295 0.561473 0.827495i \(-0.310235\pi\)
0.561473 + 0.827495i \(0.310235\pi\)
\(140\) 0 0
\(141\) −11.9982 −1.01043
\(142\) −7.54476 −0.633142
\(143\) 0 0
\(144\) 4.18926 0.349105
\(145\) 0 0
\(146\) −21.2500 −1.75867
\(147\) −3.80333 −0.313693
\(148\) −10.6198 −0.872942
\(149\) −5.87858 −0.481592 −0.240796 0.970576i \(-0.577409\pi\)
−0.240796 + 0.970576i \(0.577409\pi\)
\(150\) 0 0
\(151\) 7.62821 0.620775 0.310387 0.950610i \(-0.399541\pi\)
0.310387 + 0.950610i \(0.399541\pi\)
\(152\) 37.1872 3.01628
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 0 0
\(156\) −1.26236 −0.101070
\(157\) 10.1332 0.808719 0.404360 0.914600i \(-0.367494\pi\)
0.404360 + 0.914600i \(0.367494\pi\)
\(158\) −13.2973 −1.05788
\(159\) 4.93543 0.391405
\(160\) 0 0
\(161\) 3.52981 0.278188
\(162\) −2.45589 −0.192953
\(163\) 5.02906 0.393906 0.196953 0.980413i \(-0.436895\pi\)
0.196953 + 0.980413i \(0.436895\pi\)
\(164\) 43.7292 3.41468
\(165\) 0 0
\(166\) 39.8969 3.09660
\(167\) −5.79105 −0.448125 −0.224062 0.974575i \(-0.571932\pi\)
−0.224062 + 0.974575i \(0.571932\pi\)
\(168\) −16.3975 −1.26510
\(169\) −12.9019 −0.992457
\(170\) 0 0
\(171\) −7.45408 −0.570028
\(172\) 22.2318 1.69516
\(173\) −16.0652 −1.22142 −0.610708 0.791856i \(-0.709115\pi\)
−0.610708 + 0.791856i \(0.709115\pi\)
\(174\) −12.3690 −0.937691
\(175\) 0 0
\(176\) 0 0
\(177\) 9.16409 0.688815
\(178\) −3.98143 −0.298421
\(179\) −8.30309 −0.620602 −0.310301 0.950638i \(-0.600430\pi\)
−0.310301 + 0.950638i \(0.600430\pi\)
\(180\) 0 0
\(181\) −6.46425 −0.480484 −0.240242 0.970713i \(-0.577227\pi\)
−0.240242 + 0.970713i \(0.577227\pi\)
\(182\) 2.52765 0.187362
\(183\) −9.18431 −0.678924
\(184\) 5.35762 0.394969
\(185\) 0 0
\(186\) 8.47033 0.621074
\(187\) 0 0
\(188\) 48.3693 3.52769
\(189\) 3.28684 0.239083
\(190\) 0 0
\(191\) 15.3693 1.11208 0.556041 0.831155i \(-0.312320\pi\)
0.556041 + 0.831155i \(0.312320\pi\)
\(192\) 7.61553 0.549604
\(193\) −15.7518 −1.13384 −0.566919 0.823773i \(-0.691865\pi\)
−0.566919 + 0.823773i \(0.691865\pi\)
\(194\) 0.550320 0.0395107
\(195\) 0 0
\(196\) 15.3327 1.09519
\(197\) −16.3940 −1.16802 −0.584010 0.811746i \(-0.698517\pi\)
−0.584010 + 0.811746i \(0.698517\pi\)
\(198\) 0 0
\(199\) 6.96500 0.493736 0.246868 0.969049i \(-0.420599\pi\)
0.246868 + 0.969049i \(0.420599\pi\)
\(200\) 0 0
\(201\) −15.2739 −1.07734
\(202\) 1.24102 0.0873180
\(203\) 16.5541 1.16187
\(204\) 20.1569 1.41127
\(205\) 0 0
\(206\) −15.7225 −1.09544
\(207\) −1.07392 −0.0746427
\(208\) 1.31180 0.0909569
\(209\) 0 0
\(210\) 0 0
\(211\) −19.9531 −1.37363 −0.686814 0.726833i \(-0.740992\pi\)
−0.686814 + 0.726833i \(0.740992\pi\)
\(212\) −19.8966 −1.36650
\(213\) −3.07211 −0.210498
\(214\) 5.13892 0.351289
\(215\) 0 0
\(216\) 4.98884 0.339447
\(217\) −11.3363 −0.769557
\(218\) 16.4367 1.11323
\(219\) −8.65269 −0.584695
\(220\) 0 0
\(221\) −1.56567 −0.105318
\(222\) −6.46950 −0.434204
\(223\) 20.1466 1.34912 0.674559 0.738221i \(-0.264334\pi\)
0.674559 + 0.738221i \(0.264334\pi\)
\(224\) 1.02116 0.0682289
\(225\) 0 0
\(226\) −26.5091 −1.76336
\(227\) −0.533937 −0.0354386 −0.0177193 0.999843i \(-0.505641\pi\)
−0.0177193 + 0.999843i \(0.505641\pi\)
\(228\) 30.0502 1.99012
\(229\) −22.1931 −1.46656 −0.733279 0.679928i \(-0.762011\pi\)
−0.733279 + 0.679928i \(0.762011\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 25.1261 1.64961
\(233\) 4.56567 0.299107 0.149553 0.988754i \(-0.452216\pi\)
0.149553 + 0.988754i \(0.452216\pi\)
\(234\) −0.769020 −0.0502724
\(235\) 0 0
\(236\) −36.9439 −2.40485
\(237\) −5.41446 −0.351707
\(238\) −40.3606 −2.61619
\(239\) −5.86053 −0.379086 −0.189543 0.981872i \(-0.560701\pi\)
−0.189543 + 0.981872i \(0.560701\pi\)
\(240\) 0 0
\(241\) 9.96074 0.641628 0.320814 0.947142i \(-0.396044\pi\)
0.320814 + 0.947142i \(0.396044\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 37.0254 2.37031
\(245\) 0 0
\(246\) 26.6395 1.69847
\(247\) −2.33412 −0.148517
\(248\) −17.2065 −1.09261
\(249\) 16.2454 1.02951
\(250\) 0 0
\(251\) −16.8788 −1.06538 −0.532690 0.846310i \(-0.678819\pi\)
−0.532690 + 0.846310i \(0.678819\pi\)
\(252\) −13.2505 −0.834704
\(253\) 0 0
\(254\) −41.7581 −2.62014
\(255\) 0 0
\(256\) −32.2271 −2.01419
\(257\) −11.1436 −0.695117 −0.347559 0.937658i \(-0.612989\pi\)
−0.347559 + 0.937658i \(0.612989\pi\)
\(258\) 13.5434 0.843177
\(259\) 8.65847 0.538011
\(260\) 0 0
\(261\) −5.03647 −0.311750
\(262\) 0.105728 0.00653189
\(263\) 26.8726 1.65704 0.828519 0.559961i \(-0.189184\pi\)
0.828519 + 0.559961i \(0.189184\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −60.1701 −3.68927
\(267\) −1.62118 −0.0992145
\(268\) 61.5748 3.76128
\(269\) −10.0629 −0.613545 −0.306773 0.951783i \(-0.599249\pi\)
−0.306773 + 0.951783i \(0.599249\pi\)
\(270\) 0 0
\(271\) 10.5441 0.640509 0.320255 0.947331i \(-0.396232\pi\)
0.320255 + 0.947331i \(0.396232\pi\)
\(272\) −20.9463 −1.27006
\(273\) 1.02922 0.0622912
\(274\) 18.0816 1.09235
\(275\) 0 0
\(276\) 4.32938 0.260598
\(277\) 17.9376 1.07777 0.538883 0.842381i \(-0.318847\pi\)
0.538883 + 0.842381i \(0.318847\pi\)
\(278\) −32.5143 −1.95008
\(279\) 3.44899 0.206486
\(280\) 0 0
\(281\) 7.41103 0.442105 0.221052 0.975262i \(-0.429051\pi\)
0.221052 + 0.975262i \(0.429051\pi\)
\(282\) 29.4662 1.75469
\(283\) −5.38684 −0.320214 −0.160107 0.987100i \(-0.551184\pi\)
−0.160107 + 0.987100i \(0.551184\pi\)
\(284\) 12.3848 0.734905
\(285\) 0 0
\(286\) 0 0
\(287\) −35.6530 −2.10453
\(288\) −0.310680 −0.0183070
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0.224082 0.0131359
\(292\) 34.8823 2.04133
\(293\) 1.74006 0.101655 0.0508277 0.998707i \(-0.483814\pi\)
0.0508277 + 0.998707i \(0.483814\pi\)
\(294\) 9.34054 0.544752
\(295\) 0 0
\(296\) 13.1420 0.763864
\(297\) 0 0
\(298\) 14.4371 0.836321
\(299\) −0.336280 −0.0194476
\(300\) 0 0
\(301\) −18.1259 −1.04476
\(302\) −18.7340 −1.07802
\(303\) 0.505326 0.0290302
\(304\) −31.2271 −1.79100
\(305\) 0 0
\(306\) 12.2794 0.701969
\(307\) −21.3566 −1.21889 −0.609444 0.792829i \(-0.708607\pi\)
−0.609444 + 0.792829i \(0.708607\pi\)
\(308\) 0 0
\(309\) −6.40197 −0.364195
\(310\) 0 0
\(311\) 32.8096 1.86046 0.930231 0.366975i \(-0.119607\pi\)
0.930231 + 0.366975i \(0.119607\pi\)
\(312\) 1.56217 0.0884406
\(313\) 3.45852 0.195487 0.0977436 0.995212i \(-0.468837\pi\)
0.0977436 + 0.995212i \(0.468837\pi\)
\(314\) −24.8860 −1.40440
\(315\) 0 0
\(316\) 21.8278 1.22791
\(317\) 2.87566 0.161513 0.0807565 0.996734i \(-0.474266\pi\)
0.0807565 + 0.996734i \(0.474266\pi\)
\(318\) −12.1209 −0.679704
\(319\) 0 0
\(320\) 0 0
\(321\) 2.09249 0.116791
\(322\) −8.66881 −0.483094
\(323\) 37.2704 2.07378
\(324\) 4.03138 0.223966
\(325\) 0 0
\(326\) −12.3508 −0.684048
\(327\) 6.69278 0.370112
\(328\) −54.1150 −2.98800
\(329\) −39.4362 −2.17419
\(330\) 0 0
\(331\) −14.1221 −0.776219 −0.388109 0.921613i \(-0.626872\pi\)
−0.388109 + 0.921613i \(0.626872\pi\)
\(332\) −65.4915 −3.59431
\(333\) −2.63428 −0.144358
\(334\) 14.2222 0.778202
\(335\) 0 0
\(336\) 13.7694 0.751185
\(337\) −15.9490 −0.868796 −0.434398 0.900721i \(-0.643039\pi\)
−0.434398 + 0.900721i \(0.643039\pi\)
\(338\) 31.6857 1.72348
\(339\) −10.7941 −0.586256
\(340\) 0 0
\(341\) 0 0
\(342\) 18.3064 0.989895
\(343\) 10.5070 0.567322
\(344\) −27.5118 −1.48334
\(345\) 0 0
\(346\) 39.4543 2.12108
\(347\) −29.6801 −1.59331 −0.796656 0.604433i \(-0.793400\pi\)
−0.796656 + 0.604433i \(0.793400\pi\)
\(348\) 20.3039 1.08840
\(349\) 31.6937 1.69653 0.848263 0.529574i \(-0.177648\pi\)
0.848263 + 0.529574i \(0.177648\pi\)
\(350\) 0 0
\(351\) −0.313133 −0.0167138
\(352\) 0 0
\(353\) −1.20189 −0.0639703 −0.0319852 0.999488i \(-0.510183\pi\)
−0.0319852 + 0.999488i \(0.510183\pi\)
\(354\) −22.5060 −1.19618
\(355\) 0 0
\(356\) 6.53559 0.346385
\(357\) −16.4342 −0.869791
\(358\) 20.3915 1.07772
\(359\) −11.6591 −0.615343 −0.307671 0.951493i \(-0.599550\pi\)
−0.307671 + 0.951493i \(0.599550\pi\)
\(360\) 0 0
\(361\) 36.5633 1.92438
\(362\) 15.8755 0.834396
\(363\) 0 0
\(364\) −4.14918 −0.217476
\(365\) 0 0
\(366\) 22.5556 1.17900
\(367\) −15.9860 −0.834465 −0.417232 0.908800i \(-0.637000\pi\)
−0.417232 + 0.908800i \(0.637000\pi\)
\(368\) −4.49894 −0.234523
\(369\) 10.8472 0.564683
\(370\) 0 0
\(371\) 16.2220 0.842203
\(372\) −13.9042 −0.720898
\(373\) −0.321975 −0.0166712 −0.00833561 0.999965i \(-0.502653\pi\)
−0.00833561 + 0.999965i \(0.502653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −59.8570 −3.08689
\(377\) −1.57709 −0.0812241
\(378\) −8.07211 −0.415185
\(379\) 11.4174 0.586475 0.293237 0.956040i \(-0.405267\pi\)
0.293237 + 0.956040i \(0.405267\pi\)
\(380\) 0 0
\(381\) −17.0033 −0.871104
\(382\) −37.7452 −1.93121
\(383\) −28.3673 −1.44950 −0.724750 0.689012i \(-0.758045\pi\)
−0.724750 + 0.689012i \(0.758045\pi\)
\(384\) −19.3242 −0.986136
\(385\) 0 0
\(386\) 38.6846 1.96899
\(387\) 5.51468 0.280327
\(388\) −0.903359 −0.0458611
\(389\) 15.1802 0.769666 0.384833 0.922986i \(-0.374259\pi\)
0.384833 + 0.922986i \(0.374259\pi\)
\(390\) 0 0
\(391\) 5.36960 0.271553
\(392\) −18.9742 −0.958341
\(393\) 0.0430508 0.00217163
\(394\) 40.2617 2.02835
\(395\) 0 0
\(396\) 0 0
\(397\) −5.22461 −0.262216 −0.131108 0.991368i \(-0.541853\pi\)
−0.131108 + 0.991368i \(0.541853\pi\)
\(398\) −17.1053 −0.857409
\(399\) −24.5004 −1.22655
\(400\) 0 0
\(401\) 14.0007 0.699160 0.349580 0.936907i \(-0.386324\pi\)
0.349580 + 0.936907i \(0.386324\pi\)
\(402\) 37.5109 1.87087
\(403\) 1.07999 0.0537983
\(404\) −2.03716 −0.101352
\(405\) 0 0
\(406\) −40.6549 −2.01767
\(407\) 0 0
\(408\) −24.9442 −1.23492
\(409\) 33.3112 1.64713 0.823567 0.567218i \(-0.191980\pi\)
0.823567 + 0.567218i \(0.191980\pi\)
\(410\) 0 0
\(411\) 7.36257 0.363169
\(412\) 25.8088 1.27151
\(413\) 30.1209 1.48215
\(414\) 2.63743 0.129623
\(415\) 0 0
\(416\) −0.0972843 −0.00476975
\(417\) −13.2393 −0.648334
\(418\) 0 0
\(419\) −5.28460 −0.258170 −0.129085 0.991634i \(-0.541204\pi\)
−0.129085 + 0.991634i \(0.541204\pi\)
\(420\) 0 0
\(421\) −30.7810 −1.50017 −0.750087 0.661340i \(-0.769988\pi\)
−0.750087 + 0.661340i \(0.769988\pi\)
\(422\) 49.0026 2.38541
\(423\) 11.9982 0.583372
\(424\) 24.6221 1.19575
\(425\) 0 0
\(426\) 7.54476 0.365545
\(427\) −30.1874 −1.46087
\(428\) −8.43562 −0.407751
\(429\) 0 0
\(430\) 0 0
\(431\) 12.3506 0.594910 0.297455 0.954736i \(-0.403862\pi\)
0.297455 + 0.954736i \(0.403862\pi\)
\(432\) −4.18926 −0.201556
\(433\) −1.41287 −0.0678983 −0.0339491 0.999424i \(-0.510808\pi\)
−0.0339491 + 0.999424i \(0.510808\pi\)
\(434\) 27.8406 1.33639
\(435\) 0 0
\(436\) −26.9811 −1.29216
\(437\) 8.00509 0.382935
\(438\) 21.2500 1.01537
\(439\) −7.58532 −0.362028 −0.181014 0.983481i \(-0.557938\pi\)
−0.181014 + 0.983481i \(0.557938\pi\)
\(440\) 0 0
\(441\) 3.80333 0.181111
\(442\) 3.84510 0.182893
\(443\) −11.0662 −0.525771 −0.262885 0.964827i \(-0.584674\pi\)
−0.262885 + 0.964827i \(0.584674\pi\)
\(444\) 10.6198 0.503993
\(445\) 0 0
\(446\) −49.4779 −2.34284
\(447\) 5.87858 0.278047
\(448\) 25.0311 1.18261
\(449\) 6.32856 0.298663 0.149332 0.988787i \(-0.452288\pi\)
0.149332 + 0.988787i \(0.452288\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 43.5152 2.04678
\(453\) −7.62821 −0.358405
\(454\) 1.31129 0.0615418
\(455\) 0 0
\(456\) −37.1872 −1.74145
\(457\) 0.189579 0.00886814 0.00443407 0.999990i \(-0.498589\pi\)
0.00443407 + 0.999990i \(0.498589\pi\)
\(458\) 54.5036 2.54679
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) −26.6198 −1.23981 −0.619904 0.784678i \(-0.712828\pi\)
−0.619904 + 0.784678i \(0.712828\pi\)
\(462\) 0 0
\(463\) −20.9935 −0.975652 −0.487826 0.872941i \(-0.662210\pi\)
−0.487826 + 0.872941i \(0.662210\pi\)
\(464\) −21.0991 −0.979501
\(465\) 0 0
\(466\) −11.2128 −0.519421
\(467\) −7.89989 −0.365563 −0.182782 0.983154i \(-0.558510\pi\)
−0.182782 + 0.983154i \(0.558510\pi\)
\(468\) 1.26236 0.0583526
\(469\) −50.2028 −2.31815
\(470\) 0 0
\(471\) −10.1332 −0.466914
\(472\) 45.7182 2.10435
\(473\) 0 0
\(474\) 13.2973 0.610766
\(475\) 0 0
\(476\) 66.2525 3.03668
\(477\) −4.93543 −0.225978
\(478\) 14.3928 0.658311
\(479\) 39.9728 1.82640 0.913201 0.407509i \(-0.133602\pi\)
0.913201 + 0.407509i \(0.133602\pi\)
\(480\) 0 0
\(481\) −0.824882 −0.0376114
\(482\) −24.4624 −1.11423
\(483\) −3.52981 −0.160612
\(484\) 0 0
\(485\) 0 0
\(486\) 2.45589 0.111401
\(487\) 9.93556 0.450223 0.225112 0.974333i \(-0.427725\pi\)
0.225112 + 0.974333i \(0.427725\pi\)
\(488\) −45.8190 −2.07413
\(489\) −5.02906 −0.227422
\(490\) 0 0
\(491\) 4.97349 0.224451 0.112225 0.993683i \(-0.464202\pi\)
0.112225 + 0.993683i \(0.464202\pi\)
\(492\) −43.7292 −1.97146
\(493\) 25.1823 1.13416
\(494\) 5.73234 0.257910
\(495\) 0 0
\(496\) 14.4487 0.648767
\(497\) −10.0975 −0.452937
\(498\) −39.8969 −1.78782
\(499\) 43.7757 1.95967 0.979834 0.199812i \(-0.0640332\pi\)
0.979834 + 0.199812i \(0.0640332\pi\)
\(500\) 0 0
\(501\) 5.79105 0.258725
\(502\) 41.4524 1.85011
\(503\) −6.28236 −0.280117 −0.140058 0.990143i \(-0.544729\pi\)
−0.140058 + 0.990143i \(0.544729\pi\)
\(504\) 16.3975 0.730404
\(505\) 0 0
\(506\) 0 0
\(507\) 12.9019 0.572996
\(508\) 68.5467 3.04127
\(509\) 24.8381 1.10093 0.550465 0.834858i \(-0.314450\pi\)
0.550465 + 0.834858i \(0.314450\pi\)
\(510\) 0 0
\(511\) −28.4400 −1.25811
\(512\) 40.4976 1.78976
\(513\) 7.45408 0.329106
\(514\) 27.3674 1.20712
\(515\) 0 0
\(516\) −22.2318 −0.978699
\(517\) 0 0
\(518\) −21.2642 −0.934296
\(519\) 16.0652 0.705185
\(520\) 0 0
\(521\) −6.94869 −0.304428 −0.152214 0.988348i \(-0.548640\pi\)
−0.152214 + 0.988348i \(0.548640\pi\)
\(522\) 12.3690 0.541376
\(523\) 26.7510 1.16974 0.584869 0.811128i \(-0.301146\pi\)
0.584869 + 0.811128i \(0.301146\pi\)
\(524\) −0.173554 −0.00758175
\(525\) 0 0
\(526\) −65.9962 −2.87757
\(527\) −17.2449 −0.751202
\(528\) 0 0
\(529\) −21.8467 −0.949856
\(530\) 0 0
\(531\) −9.16409 −0.397688
\(532\) 98.7703 4.28224
\(533\) 3.39662 0.147124
\(534\) 3.98143 0.172293
\(535\) 0 0
\(536\) −76.1989 −3.29129
\(537\) 8.30309 0.358305
\(538\) 24.7133 1.06547
\(539\) 0 0
\(540\) 0 0
\(541\) 14.5084 0.623767 0.311883 0.950120i \(-0.399040\pi\)
0.311883 + 0.950120i \(0.399040\pi\)
\(542\) −25.8951 −1.11229
\(543\) 6.46425 0.277408
\(544\) 1.55340 0.0666015
\(545\) 0 0
\(546\) −2.52765 −0.108173
\(547\) 26.7346 1.14309 0.571543 0.820572i \(-0.306345\pi\)
0.571543 + 0.820572i \(0.306345\pi\)
\(548\) −29.6813 −1.26792
\(549\) 9.18431 0.391977
\(550\) 0 0
\(551\) 37.5422 1.59935
\(552\) −5.35762 −0.228035
\(553\) −17.7965 −0.756784
\(554\) −44.0527 −1.87162
\(555\) 0 0
\(556\) 53.3728 2.26351
\(557\) −17.2444 −0.730670 −0.365335 0.930876i \(-0.619046\pi\)
−0.365335 + 0.930876i \(0.619046\pi\)
\(558\) −8.47033 −0.358578
\(559\) 1.72683 0.0730371
\(560\) 0 0
\(561\) 0 0
\(562\) −18.2006 −0.767748
\(563\) 0.831914 0.0350610 0.0175305 0.999846i \(-0.494420\pi\)
0.0175305 + 0.999846i \(0.494420\pi\)
\(564\) −48.3693 −2.03671
\(565\) 0 0
\(566\) 13.2295 0.556076
\(567\) −3.28684 −0.138034
\(568\) −15.3263 −0.643076
\(569\) 11.5961 0.486132 0.243066 0.970010i \(-0.421847\pi\)
0.243066 + 0.970010i \(0.421847\pi\)
\(570\) 0 0
\(571\) 21.8414 0.914034 0.457017 0.889458i \(-0.348918\pi\)
0.457017 + 0.889458i \(0.348918\pi\)
\(572\) 0 0
\(573\) −15.3693 −0.642061
\(574\) 87.5598 3.65468
\(575\) 0 0
\(576\) −7.61553 −0.317314
\(577\) −9.74587 −0.405726 −0.202863 0.979207i \(-0.565025\pi\)
−0.202863 + 0.979207i \(0.565025\pi\)
\(578\) −19.6471 −0.817211
\(579\) 15.7518 0.654622
\(580\) 0 0
\(581\) 53.3961 2.21524
\(582\) −0.550320 −0.0228115
\(583\) 0 0
\(584\) −43.1669 −1.78626
\(585\) 0 0
\(586\) −4.27339 −0.176532
\(587\) −22.9441 −0.947005 −0.473502 0.880793i \(-0.657010\pi\)
−0.473502 + 0.880793i \(0.657010\pi\)
\(588\) −15.3327 −0.632308
\(589\) −25.7090 −1.05932
\(590\) 0 0
\(591\) 16.3940 0.674357
\(592\) −11.0357 −0.453565
\(593\) −28.7819 −1.18193 −0.590965 0.806697i \(-0.701253\pi\)
−0.590965 + 0.806697i \(0.701253\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −23.6988 −0.970741
\(597\) −6.96500 −0.285059
\(598\) 0.825867 0.0337722
\(599\) −29.1951 −1.19288 −0.596440 0.802657i \(-0.703419\pi\)
−0.596440 + 0.802657i \(0.703419\pi\)
\(600\) 0 0
\(601\) −6.68087 −0.272518 −0.136259 0.990673i \(-0.543508\pi\)
−0.136259 + 0.990673i \(0.543508\pi\)
\(602\) 44.5151 1.81430
\(603\) 15.2739 0.622000
\(604\) 30.7522 1.25129
\(605\) 0 0
\(606\) −1.24102 −0.0504131
\(607\) 42.6108 1.72952 0.864759 0.502188i \(-0.167471\pi\)
0.864759 + 0.502188i \(0.167471\pi\)
\(608\) 2.31583 0.0939194
\(609\) −16.5541 −0.670805
\(610\) 0 0
\(611\) 3.75703 0.151993
\(612\) −20.1569 −0.814794
\(613\) 5.83156 0.235535 0.117767 0.993041i \(-0.462426\pi\)
0.117767 + 0.993041i \(0.462426\pi\)
\(614\) 52.4495 2.11669
\(615\) 0 0
\(616\) 0 0
\(617\) 33.6386 1.35424 0.677119 0.735874i \(-0.263228\pi\)
0.677119 + 0.735874i \(0.263228\pi\)
\(618\) 15.7225 0.632452
\(619\) 42.5616 1.71070 0.855349 0.518053i \(-0.173343\pi\)
0.855349 + 0.518053i \(0.173343\pi\)
\(620\) 0 0
\(621\) 1.07392 0.0430950
\(622\) −80.5766 −3.23083
\(623\) −5.32856 −0.213484
\(624\) −1.31180 −0.0525140
\(625\) 0 0
\(626\) −8.49374 −0.339478
\(627\) 0 0
\(628\) 40.8509 1.63013
\(629\) 13.1714 0.525179
\(630\) 0 0
\(631\) −8.89989 −0.354299 −0.177149 0.984184i \(-0.556688\pi\)
−0.177149 + 0.984184i \(0.556688\pi\)
\(632\) −27.0119 −1.07448
\(633\) 19.9531 0.793065
\(634\) −7.06228 −0.280479
\(635\) 0 0
\(636\) 19.8966 0.788951
\(637\) 1.19095 0.0471871
\(638\) 0 0
\(639\) 3.07211 0.121531
\(640\) 0 0
\(641\) −6.16806 −0.243624 −0.121812 0.992553i \(-0.538870\pi\)
−0.121812 + 0.992553i \(0.538870\pi\)
\(642\) −5.13892 −0.202817
\(643\) −4.35335 −0.171680 −0.0858398 0.996309i \(-0.527357\pi\)
−0.0858398 + 0.996309i \(0.527357\pi\)
\(644\) 14.2300 0.560740
\(645\) 0 0
\(646\) −91.5318 −3.60127
\(647\) −13.4933 −0.530478 −0.265239 0.964183i \(-0.585451\pi\)
−0.265239 + 0.964183i \(0.585451\pi\)
\(648\) −4.98884 −0.195980
\(649\) 0 0
\(650\) 0 0
\(651\) 11.3363 0.444304
\(652\) 20.2741 0.793993
\(653\) −27.4481 −1.07413 −0.537064 0.843541i \(-0.680467\pi\)
−0.537064 + 0.843541i \(0.680467\pi\)
\(654\) −16.4367 −0.642726
\(655\) 0 0
\(656\) 45.4418 1.77420
\(657\) 8.65269 0.337574
\(658\) 96.8507 3.77563
\(659\) −18.7768 −0.731441 −0.365721 0.930725i \(-0.619177\pi\)
−0.365721 + 0.930725i \(0.619177\pi\)
\(660\) 0 0
\(661\) −21.6525 −0.842184 −0.421092 0.907018i \(-0.638353\pi\)
−0.421092 + 0.907018i \(0.638353\pi\)
\(662\) 34.6822 1.34796
\(663\) 1.56567 0.0608055
\(664\) 81.0458 3.14519
\(665\) 0 0
\(666\) 6.46950 0.250688
\(667\) 5.40877 0.209428
\(668\) −23.3459 −0.903281
\(669\) −20.1466 −0.778914
\(670\) 0 0
\(671\) 0 0
\(672\) −1.02116 −0.0393919
\(673\) −23.3021 −0.898232 −0.449116 0.893474i \(-0.648261\pi\)
−0.449116 + 0.893474i \(0.648261\pi\)
\(674\) 39.1689 1.50873
\(675\) 0 0
\(676\) −52.0127 −2.00049
\(677\) −33.2808 −1.27909 −0.639543 0.768756i \(-0.720876\pi\)
−0.639543 + 0.768756i \(0.720876\pi\)
\(678\) 26.5091 1.01808
\(679\) 0.736522 0.0282651
\(680\) 0 0
\(681\) 0.533937 0.0204605
\(682\) 0 0
\(683\) −16.9244 −0.647593 −0.323796 0.946127i \(-0.604959\pi\)
−0.323796 + 0.946127i \(0.604959\pi\)
\(684\) −30.0502 −1.14900
\(685\) 0 0
\(686\) −25.8039 −0.985197
\(687\) 22.1931 0.846718
\(688\) 23.1024 0.880772
\(689\) −1.54545 −0.0588769
\(690\) 0 0
\(691\) −48.4335 −1.84250 −0.921249 0.388973i \(-0.872830\pi\)
−0.921249 + 0.388973i \(0.872830\pi\)
\(692\) −64.7650 −2.46200
\(693\) 0 0
\(694\) 72.8910 2.76690
\(695\) 0 0
\(696\) −25.1261 −0.952403
\(697\) −54.2360 −2.05434
\(698\) −77.8362 −2.94614
\(699\) −4.56567 −0.172689
\(700\) 0 0
\(701\) −45.4161 −1.71534 −0.857672 0.514197i \(-0.828090\pi\)
−0.857672 + 0.514197i \(0.828090\pi\)
\(702\) 0.769020 0.0290248
\(703\) 19.6361 0.740591
\(704\) 0 0
\(705\) 0 0
\(706\) 2.95171 0.111089
\(707\) 1.66093 0.0624655
\(708\) 36.9439 1.38844
\(709\) −1.76497 −0.0662848 −0.0331424 0.999451i \(-0.510551\pi\)
−0.0331424 + 0.999451i \(0.510551\pi\)
\(710\) 0 0
\(711\) 5.41446 0.203058
\(712\) −8.08780 −0.303103
\(713\) −3.70394 −0.138714
\(714\) 40.3606 1.51046
\(715\) 0 0
\(716\) −33.4729 −1.25094
\(717\) 5.86053 0.218865
\(718\) 28.6334 1.06859
\(719\) −3.29998 −0.123069 −0.0615343 0.998105i \(-0.519599\pi\)
−0.0615343 + 0.998105i \(0.519599\pi\)
\(720\) 0 0
\(721\) −21.0423 −0.783655
\(722\) −89.7952 −3.34183
\(723\) −9.96074 −0.370444
\(724\) −26.0599 −0.968507
\(725\) 0 0
\(726\) 0 0
\(727\) −11.7838 −0.437037 −0.218519 0.975833i \(-0.570122\pi\)
−0.218519 + 0.975833i \(0.570122\pi\)
\(728\) 5.13461 0.190301
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −27.5734 −1.01984
\(732\) −37.0254 −1.36850
\(733\) −5.73108 −0.211682 −0.105841 0.994383i \(-0.533754\pi\)
−0.105841 + 0.994383i \(0.533754\pi\)
\(734\) 39.2599 1.44911
\(735\) 0 0
\(736\) 0.333646 0.0122983
\(737\) 0 0
\(738\) −26.6395 −0.980614
\(739\) −21.0551 −0.774524 −0.387262 0.921970i \(-0.626579\pi\)
−0.387262 + 0.921970i \(0.626579\pi\)
\(740\) 0 0
\(741\) 2.33412 0.0857461
\(742\) −39.8393 −1.46255
\(743\) −13.1283 −0.481630 −0.240815 0.970571i \(-0.577415\pi\)
−0.240815 + 0.970571i \(0.577415\pi\)
\(744\) 17.2065 0.630819
\(745\) 0 0
\(746\) 0.790734 0.0289508
\(747\) −16.2454 −0.594389
\(748\) 0 0
\(749\) 6.87768 0.251305
\(750\) 0 0
\(751\) 25.6251 0.935073 0.467537 0.883974i \(-0.345142\pi\)
0.467537 + 0.883974i \(0.345142\pi\)
\(752\) 50.2636 1.83292
\(753\) 16.8788 0.615098
\(754\) 3.87315 0.141052
\(755\) 0 0
\(756\) 13.2505 0.481916
\(757\) −31.1970 −1.13387 −0.566936 0.823762i \(-0.691871\pi\)
−0.566936 + 0.823762i \(0.691871\pi\)
\(758\) −28.0400 −1.01846
\(759\) 0 0
\(760\) 0 0
\(761\) 11.3761 0.412382 0.206191 0.978512i \(-0.433893\pi\)
0.206191 + 0.978512i \(0.433893\pi\)
\(762\) 41.7581 1.51274
\(763\) 21.9981 0.796385
\(764\) 61.9594 2.24161
\(765\) 0 0
\(766\) 69.6668 2.51717
\(767\) −2.86958 −0.103615
\(768\) 32.2271 1.16290
\(769\) 10.3938 0.374811 0.187405 0.982283i \(-0.439992\pi\)
0.187405 + 0.982283i \(0.439992\pi\)
\(770\) 0 0
\(771\) 11.1436 0.401326
\(772\) −63.5014 −2.28547
\(773\) −14.0348 −0.504796 −0.252398 0.967623i \(-0.581219\pi\)
−0.252398 + 0.967623i \(0.581219\pi\)
\(774\) −13.5434 −0.486808
\(775\) 0 0
\(776\) 1.11791 0.0401306
\(777\) −8.65847 −0.310621
\(778\) −37.2808 −1.33658
\(779\) −80.8559 −2.89696
\(780\) 0 0
\(781\) 0 0
\(782\) −13.1871 −0.471571
\(783\) 5.03647 0.179989
\(784\) 15.9331 0.569041
\(785\) 0 0
\(786\) −0.105728 −0.00377119
\(787\) 13.8176 0.492545 0.246273 0.969201i \(-0.420794\pi\)
0.246273 + 0.969201i \(0.420794\pi\)
\(788\) −66.0902 −2.35437
\(789\) −26.8726 −0.956691
\(790\) 0 0
\(791\) −35.4785 −1.26147
\(792\) 0 0
\(793\) 2.87591 0.102127
\(794\) 12.8311 0.455357
\(795\) 0 0
\(796\) 28.0786 0.995218
\(797\) 5.38594 0.190780 0.0953898 0.995440i \(-0.469590\pi\)
0.0953898 + 0.995440i \(0.469590\pi\)
\(798\) 60.1701 2.13000
\(799\) −59.9910 −2.12233
\(800\) 0 0
\(801\) 1.62118 0.0572815
\(802\) −34.3840 −1.21414
\(803\) 0 0
\(804\) −61.5748 −2.17157
\(805\) 0 0
\(806\) −2.65234 −0.0934248
\(807\) 10.0629 0.354231
\(808\) 2.52099 0.0886880
\(809\) 23.7748 0.835876 0.417938 0.908476i \(-0.362753\pi\)
0.417938 + 0.908476i \(0.362753\pi\)
\(810\) 0 0
\(811\) 9.46335 0.332303 0.166152 0.986100i \(-0.446866\pi\)
0.166152 + 0.986100i \(0.446866\pi\)
\(812\) 66.7358 2.34197
\(813\) −10.5441 −0.369798
\(814\) 0 0
\(815\) 0 0
\(816\) 20.9463 0.733268
\(817\) −41.1068 −1.43815
\(818\) −81.8086 −2.86037
\(819\) −1.02922 −0.0359639
\(820\) 0 0
\(821\) −10.4189 −0.363622 −0.181811 0.983334i \(-0.558196\pi\)
−0.181811 + 0.983334i \(0.558196\pi\)
\(822\) −18.0816 −0.630670
\(823\) −24.3540 −0.848928 −0.424464 0.905445i \(-0.639537\pi\)
−0.424464 + 0.905445i \(0.639537\pi\)
\(824\) −31.9384 −1.11263
\(825\) 0 0
\(826\) −73.9736 −2.57387
\(827\) 22.0006 0.765037 0.382519 0.923948i \(-0.375057\pi\)
0.382519 + 0.923948i \(0.375057\pi\)
\(828\) −4.32938 −0.150456
\(829\) 3.03289 0.105337 0.0526683 0.998612i \(-0.483227\pi\)
0.0526683 + 0.998612i \(0.483227\pi\)
\(830\) 0 0
\(831\) −17.9376 −0.622248
\(832\) −2.38468 −0.0826738
\(833\) −19.0166 −0.658888
\(834\) 32.5143 1.12588
\(835\) 0 0
\(836\) 0 0
\(837\) −3.44899 −0.119214
\(838\) 12.9784 0.448331
\(839\) −48.5383 −1.67573 −0.837864 0.545879i \(-0.816196\pi\)
−0.837864 + 0.545879i \(0.816196\pi\)
\(840\) 0 0
\(841\) −3.63399 −0.125310
\(842\) 75.5946 2.60516
\(843\) −7.41103 −0.255249
\(844\) −80.4386 −2.76881
\(845\) 0 0
\(846\) −29.4662 −1.01307
\(847\) 0 0
\(848\) −20.6758 −0.710011
\(849\) 5.38684 0.184876
\(850\) 0 0
\(851\) 2.82901 0.0969773
\(852\) −12.3848 −0.424298
\(853\) 11.7632 0.402766 0.201383 0.979513i \(-0.435456\pi\)
0.201383 + 0.979513i \(0.435456\pi\)
\(854\) 74.1368 2.53691
\(855\) 0 0
\(856\) 10.4391 0.356801
\(857\) −1.61311 −0.0551026 −0.0275513 0.999620i \(-0.508771\pi\)
−0.0275513 + 0.999620i \(0.508771\pi\)
\(858\) 0 0
\(859\) 47.3263 1.61475 0.807376 0.590038i \(-0.200887\pi\)
0.807376 + 0.590038i \(0.200887\pi\)
\(860\) 0 0
\(861\) 35.6530 1.21505
\(862\) −30.3318 −1.03310
\(863\) 9.97233 0.339462 0.169731 0.985490i \(-0.445710\pi\)
0.169731 + 0.985490i \(0.445710\pi\)
\(864\) 0.310680 0.0105695
\(865\) 0 0
\(866\) 3.46985 0.117910
\(867\) −8.00000 −0.271694
\(868\) −45.7009 −1.55119
\(869\) 0 0
\(870\) 0 0
\(871\) 4.78276 0.162058
\(872\) 33.3892 1.13070
\(873\) −0.224082 −0.00758402
\(874\) −19.6596 −0.664996
\(875\) 0 0
\(876\) −34.8823 −1.17856
\(877\) −26.0057 −0.878151 −0.439076 0.898450i \(-0.644694\pi\)
−0.439076 + 0.898450i \(0.644694\pi\)
\(878\) 18.6287 0.628688
\(879\) −1.74006 −0.0586907
\(880\) 0 0
\(881\) 10.4081 0.350657 0.175329 0.984510i \(-0.443901\pi\)
0.175329 + 0.984510i \(0.443901\pi\)
\(882\) −9.34054 −0.314512
\(883\) 53.7283 1.80810 0.904051 0.427424i \(-0.140579\pi\)
0.904051 + 0.427424i \(0.140579\pi\)
\(884\) −6.31180 −0.212289
\(885\) 0 0
\(886\) 27.1773 0.913040
\(887\) 40.6246 1.36404 0.682021 0.731333i \(-0.261101\pi\)
0.682021 + 0.731333i \(0.261101\pi\)
\(888\) −13.1420 −0.441017
\(889\) −55.8871 −1.87439
\(890\) 0 0
\(891\) 0 0
\(892\) 81.2188 2.71941
\(893\) −89.4354 −2.99284
\(894\) −14.4371 −0.482850
\(895\) 0 0
\(896\) −63.5157 −2.12191
\(897\) 0.336280 0.0112281
\(898\) −15.5422 −0.518651
\(899\) −17.3707 −0.579346
\(900\) 0 0
\(901\) 24.6772 0.822115
\(902\) 0 0
\(903\) 18.1259 0.603191
\(904\) −53.8501 −1.79103
\(905\) 0 0
\(906\) 18.7340 0.622396
\(907\) 19.4070 0.644398 0.322199 0.946672i \(-0.395578\pi\)
0.322199 + 0.946672i \(0.395578\pi\)
\(908\) −2.15250 −0.0714333
\(909\) −0.505326 −0.0167606
\(910\) 0 0
\(911\) 10.7208 0.355195 0.177597 0.984103i \(-0.443168\pi\)
0.177597 + 0.984103i \(0.443168\pi\)
\(912\) 31.2271 1.03403
\(913\) 0 0
\(914\) −0.465585 −0.0154002
\(915\) 0 0
\(916\) −89.4686 −2.95613
\(917\) 0.141501 0.00467278
\(918\) −12.2794 −0.405282
\(919\) −23.1310 −0.763021 −0.381511 0.924364i \(-0.624596\pi\)
−0.381511 + 0.924364i \(0.624596\pi\)
\(920\) 0 0
\(921\) 21.3566 0.703725
\(922\) 65.3752 2.15302
\(923\) 0.961981 0.0316640
\(924\) 0 0
\(925\) 0 0
\(926\) 51.5577 1.69429
\(927\) 6.40197 0.210268
\(928\) 1.56473 0.0513648
\(929\) 26.2273 0.860489 0.430245 0.902712i \(-0.358427\pi\)
0.430245 + 0.902712i \(0.358427\pi\)
\(930\) 0 0
\(931\) −28.3503 −0.929144
\(932\) 18.4059 0.602907
\(933\) −32.8096 −1.07414
\(934\) 19.4012 0.634828
\(935\) 0 0
\(936\) −1.56217 −0.0510612
\(937\) −23.4011 −0.764480 −0.382240 0.924063i \(-0.624847\pi\)
−0.382240 + 0.924063i \(0.624847\pi\)
\(938\) 123.292 4.02564
\(939\) −3.45852 −0.112865
\(940\) 0 0
\(941\) −10.9687 −0.357570 −0.178785 0.983888i \(-0.557217\pi\)
−0.178785 + 0.983888i \(0.557217\pi\)
\(942\) 24.8860 0.810831
\(943\) −11.6490 −0.379345
\(944\) −38.3908 −1.24951
\(945\) 0 0
\(946\) 0 0
\(947\) 13.3652 0.434310 0.217155 0.976137i \(-0.430322\pi\)
0.217155 + 0.976137i \(0.430322\pi\)
\(948\) −21.8278 −0.708933
\(949\) 2.70945 0.0879524
\(950\) 0 0
\(951\) −2.87566 −0.0932495
\(952\) −81.9876 −2.65723
\(953\) 21.8242 0.706956 0.353478 0.935443i \(-0.384999\pi\)
0.353478 + 0.935443i \(0.384999\pi\)
\(954\) 12.1209 0.392427
\(955\) 0 0
\(956\) −23.6260 −0.764120
\(957\) 0 0
\(958\) −98.1686 −3.17168
\(959\) 24.1996 0.781446
\(960\) 0 0
\(961\) −19.1045 −0.616273
\(962\) 2.02582 0.0653150
\(963\) −2.09249 −0.0674295
\(964\) 40.1555 1.29332
\(965\) 0 0
\(966\) 8.66881 0.278914
\(967\) −16.6600 −0.535750 −0.267875 0.963454i \(-0.586321\pi\)
−0.267875 + 0.963454i \(0.586321\pi\)
\(968\) 0 0
\(969\) −37.2704 −1.19730
\(970\) 0 0
\(971\) 11.1032 0.356320 0.178160 0.984002i \(-0.442986\pi\)
0.178160 + 0.984002i \(0.442986\pi\)
\(972\) −4.03138 −0.129307
\(973\) −43.5156 −1.39505
\(974\) −24.4006 −0.781846
\(975\) 0 0
\(976\) 38.4755 1.23157
\(977\) −18.8144 −0.601926 −0.300963 0.953636i \(-0.597308\pi\)
−0.300963 + 0.953636i \(0.597308\pi\)
\(978\) 12.3508 0.394935
\(979\) 0 0
\(980\) 0 0
\(981\) −6.69278 −0.213684
\(982\) −12.2143 −0.389775
\(983\) 1.37848 0.0439667 0.0219833 0.999758i \(-0.493002\pi\)
0.0219833 + 0.999758i \(0.493002\pi\)
\(984\) 54.1150 1.72512
\(985\) 0 0
\(986\) −61.8450 −1.96955
\(987\) 39.4362 1.25527
\(988\) −9.40973 −0.299363
\(989\) −5.92233 −0.188319
\(990\) 0 0
\(991\) −46.3186 −1.47136 −0.735680 0.677329i \(-0.763137\pi\)
−0.735680 + 0.677329i \(0.763137\pi\)
\(992\) −1.07153 −0.0340212
\(993\) 14.1221 0.448150
\(994\) 24.7984 0.786558
\(995\) 0 0
\(996\) 65.4915 2.07518
\(997\) −14.5470 −0.460709 −0.230355 0.973107i \(-0.573989\pi\)
−0.230355 + 0.973107i \(0.573989\pi\)
\(998\) −107.508 −3.40311
\(999\) 2.63428 0.0833450
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 9075.2.a.cl.1.2 4
5.4 even 2 1815.2.a.x.1.3 4
11.3 even 5 825.2.n.k.526.1 8
11.4 even 5 825.2.n.k.676.1 8
11.10 odd 2 9075.2.a.dj.1.3 4
15.14 odd 2 5445.2.a.be.1.2 4
55.3 odd 20 825.2.bx.h.724.4 16
55.4 even 10 165.2.m.a.16.2 8
55.14 even 10 165.2.m.a.31.2 yes 8
55.37 odd 20 825.2.bx.h.49.4 16
55.47 odd 20 825.2.bx.h.724.1 16
55.48 odd 20 825.2.bx.h.49.1 16
55.54 odd 2 1815.2.a.o.1.2 4
165.14 odd 10 495.2.n.d.361.1 8
165.59 odd 10 495.2.n.d.181.1 8
165.164 even 2 5445.2.a.bv.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.16.2 8 55.4 even 10
165.2.m.a.31.2 yes 8 55.14 even 10
495.2.n.d.181.1 8 165.59 odd 10
495.2.n.d.361.1 8 165.14 odd 10
825.2.n.k.526.1 8 11.3 even 5
825.2.n.k.676.1 8 11.4 even 5
825.2.bx.h.49.1 16 55.48 odd 20
825.2.bx.h.49.4 16 55.37 odd 20
825.2.bx.h.724.1 16 55.47 odd 20
825.2.bx.h.724.4 16 55.3 odd 20
1815.2.a.o.1.2 4 55.54 odd 2
1815.2.a.x.1.3 4 5.4 even 2
5445.2.a.be.1.2 4 15.14 odd 2
5445.2.a.bv.1.3 4 165.164 even 2
9075.2.a.cl.1.2 4 1.1 even 1 trivial
9075.2.a.dj.1.3 4 11.10 odd 2