Properties

Label 6040.2.a.m.1.11
Level $6040$
Weight $2$
Character 6040.1
Self dual yes
Analytic conductor $48.230$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} - 669 x^{3} + 39 x^{2} + 106 x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.45327\) of defining polynomial
Character \(\chi\) \(=\) 6040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.45327 q^{3} -1.00000 q^{5} -3.17952 q^{7} +3.01854 q^{9} +O(q^{10})\) \(q+2.45327 q^{3} -1.00000 q^{5} -3.17952 q^{7} +3.01854 q^{9} -1.84859 q^{11} +0.347557 q^{13} -2.45327 q^{15} +0.562260 q^{17} -3.75513 q^{19} -7.80023 q^{21} +0.184488 q^{23} +1.00000 q^{25} +0.0454727 q^{27} -1.03672 q^{29} +7.93218 q^{31} -4.53509 q^{33} +3.17952 q^{35} +1.09134 q^{37} +0.852651 q^{39} +3.61724 q^{41} +2.30386 q^{43} -3.01854 q^{45} +9.03471 q^{47} +3.10938 q^{49} +1.37938 q^{51} +7.80091 q^{53} +1.84859 q^{55} -9.21235 q^{57} +5.62811 q^{59} +4.83177 q^{61} -9.59751 q^{63} -0.347557 q^{65} +13.2531 q^{67} +0.452599 q^{69} +11.6606 q^{71} +5.05704 q^{73} +2.45327 q^{75} +5.87764 q^{77} +10.3678 q^{79} -8.94405 q^{81} +7.27862 q^{83} -0.562260 q^{85} -2.54335 q^{87} -6.70193 q^{89} -1.10507 q^{91} +19.4598 q^{93} +3.75513 q^{95} -14.3155 q^{97} -5.58003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.45327 1.41640 0.708198 0.706014i \(-0.249508\pi\)
0.708198 + 0.706014i \(0.249508\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.17952 −1.20175 −0.600874 0.799344i \(-0.705181\pi\)
−0.600874 + 0.799344i \(0.705181\pi\)
\(8\) 0 0
\(9\) 3.01854 1.00618
\(10\) 0 0
\(11\) −1.84859 −0.557371 −0.278685 0.960382i \(-0.589899\pi\)
−0.278685 + 0.960382i \(0.589899\pi\)
\(12\) 0 0
\(13\) 0.347557 0.0963950 0.0481975 0.998838i \(-0.484652\pi\)
0.0481975 + 0.998838i \(0.484652\pi\)
\(14\) 0 0
\(15\) −2.45327 −0.633432
\(16\) 0 0
\(17\) 0.562260 0.136368 0.0681841 0.997673i \(-0.478279\pi\)
0.0681841 + 0.997673i \(0.478279\pi\)
\(18\) 0 0
\(19\) −3.75513 −0.861486 −0.430743 0.902475i \(-0.641748\pi\)
−0.430743 + 0.902475i \(0.641748\pi\)
\(20\) 0 0
\(21\) −7.80023 −1.70215
\(22\) 0 0
\(23\) 0.184488 0.0384684 0.0192342 0.999815i \(-0.493877\pi\)
0.0192342 + 0.999815i \(0.493877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.0454727 0.00875123
\(28\) 0 0
\(29\) −1.03672 −0.192514 −0.0962568 0.995357i \(-0.530687\pi\)
−0.0962568 + 0.995357i \(0.530687\pi\)
\(30\) 0 0
\(31\) 7.93218 1.42466 0.712330 0.701844i \(-0.247640\pi\)
0.712330 + 0.701844i \(0.247640\pi\)
\(32\) 0 0
\(33\) −4.53509 −0.789458
\(34\) 0 0
\(35\) 3.17952 0.537438
\(36\) 0 0
\(37\) 1.09134 0.179416 0.0897078 0.995968i \(-0.471407\pi\)
0.0897078 + 0.995968i \(0.471407\pi\)
\(38\) 0 0
\(39\) 0.852651 0.136533
\(40\) 0 0
\(41\) 3.61724 0.564918 0.282459 0.959279i \(-0.408850\pi\)
0.282459 + 0.959279i \(0.408850\pi\)
\(42\) 0 0
\(43\) 2.30386 0.351335 0.175667 0.984450i \(-0.443792\pi\)
0.175667 + 0.984450i \(0.443792\pi\)
\(44\) 0 0
\(45\) −3.01854 −0.449977
\(46\) 0 0
\(47\) 9.03471 1.31785 0.658924 0.752210i \(-0.271012\pi\)
0.658924 + 0.752210i \(0.271012\pi\)
\(48\) 0 0
\(49\) 3.10938 0.444197
\(50\) 0 0
\(51\) 1.37938 0.193151
\(52\) 0 0
\(53\) 7.80091 1.07154 0.535769 0.844365i \(-0.320022\pi\)
0.535769 + 0.844365i \(0.320022\pi\)
\(54\) 0 0
\(55\) 1.84859 0.249264
\(56\) 0 0
\(57\) −9.21235 −1.22021
\(58\) 0 0
\(59\) 5.62811 0.732718 0.366359 0.930474i \(-0.380604\pi\)
0.366359 + 0.930474i \(0.380604\pi\)
\(60\) 0 0
\(61\) 4.83177 0.618645 0.309323 0.950957i \(-0.399898\pi\)
0.309323 + 0.950957i \(0.399898\pi\)
\(62\) 0 0
\(63\) −9.59751 −1.20917
\(64\) 0 0
\(65\) −0.347557 −0.0431091
\(66\) 0 0
\(67\) 13.2531 1.61912 0.809562 0.587035i \(-0.199705\pi\)
0.809562 + 0.587035i \(0.199705\pi\)
\(68\) 0 0
\(69\) 0.452599 0.0544865
\(70\) 0 0
\(71\) 11.6606 1.38386 0.691928 0.721966i \(-0.256761\pi\)
0.691928 + 0.721966i \(0.256761\pi\)
\(72\) 0 0
\(73\) 5.05704 0.591882 0.295941 0.955206i \(-0.404367\pi\)
0.295941 + 0.955206i \(0.404367\pi\)
\(74\) 0 0
\(75\) 2.45327 0.283279
\(76\) 0 0
\(77\) 5.87764 0.669819
\(78\) 0 0
\(79\) 10.3678 1.16647 0.583236 0.812303i \(-0.301786\pi\)
0.583236 + 0.812303i \(0.301786\pi\)
\(80\) 0 0
\(81\) −8.94405 −0.993783
\(82\) 0 0
\(83\) 7.27862 0.798933 0.399466 0.916748i \(-0.369195\pi\)
0.399466 + 0.916748i \(0.369195\pi\)
\(84\) 0 0
\(85\) −0.562260 −0.0609857
\(86\) 0 0
\(87\) −2.54335 −0.272676
\(88\) 0 0
\(89\) −6.70193 −0.710403 −0.355201 0.934790i \(-0.615588\pi\)
−0.355201 + 0.934790i \(0.615588\pi\)
\(90\) 0 0
\(91\) −1.10507 −0.115842
\(92\) 0 0
\(93\) 19.4598 2.01788
\(94\) 0 0
\(95\) 3.75513 0.385268
\(96\) 0 0
\(97\) −14.3155 −1.45352 −0.726761 0.686891i \(-0.758975\pi\)
−0.726761 + 0.686891i \(0.758975\pi\)
\(98\) 0 0
\(99\) −5.58003 −0.560815
\(100\) 0 0
\(101\) 12.9624 1.28980 0.644901 0.764266i \(-0.276898\pi\)
0.644901 + 0.764266i \(0.276898\pi\)
\(102\) 0 0
\(103\) 4.78787 0.471763 0.235881 0.971782i \(-0.424202\pi\)
0.235881 + 0.971782i \(0.424202\pi\)
\(104\) 0 0
\(105\) 7.80023 0.761225
\(106\) 0 0
\(107\) −14.6953 −1.42065 −0.710326 0.703873i \(-0.751453\pi\)
−0.710326 + 0.703873i \(0.751453\pi\)
\(108\) 0 0
\(109\) 14.8774 1.42500 0.712498 0.701674i \(-0.247564\pi\)
0.712498 + 0.701674i \(0.247564\pi\)
\(110\) 0 0
\(111\) 2.67736 0.254124
\(112\) 0 0
\(113\) −0.478687 −0.0450311 −0.0225155 0.999746i \(-0.507168\pi\)
−0.0225155 + 0.999746i \(0.507168\pi\)
\(114\) 0 0
\(115\) −0.184488 −0.0172036
\(116\) 0 0
\(117\) 1.04911 0.0969905
\(118\) 0 0
\(119\) −1.78772 −0.163880
\(120\) 0 0
\(121\) −7.58272 −0.689338
\(122\) 0 0
\(123\) 8.87407 0.800148
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.41528 0.746735 0.373368 0.927683i \(-0.378203\pi\)
0.373368 + 0.927683i \(0.378203\pi\)
\(128\) 0 0
\(129\) 5.65198 0.497629
\(130\) 0 0
\(131\) −11.2435 −0.982345 −0.491172 0.871062i \(-0.663432\pi\)
−0.491172 + 0.871062i \(0.663432\pi\)
\(132\) 0 0
\(133\) 11.9395 1.03529
\(134\) 0 0
\(135\) −0.0454727 −0.00391367
\(136\) 0 0
\(137\) −19.9335 −1.70303 −0.851515 0.524330i \(-0.824316\pi\)
−0.851515 + 0.524330i \(0.824316\pi\)
\(138\) 0 0
\(139\) 5.66713 0.480679 0.240340 0.970689i \(-0.422741\pi\)
0.240340 + 0.970689i \(0.422741\pi\)
\(140\) 0 0
\(141\) 22.1646 1.86659
\(142\) 0 0
\(143\) −0.642490 −0.0537277
\(144\) 0 0
\(145\) 1.03672 0.0860947
\(146\) 0 0
\(147\) 7.62814 0.629159
\(148\) 0 0
\(149\) −7.45349 −0.610613 −0.305307 0.952254i \(-0.598759\pi\)
−0.305307 + 0.952254i \(0.598759\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788
\(152\) 0 0
\(153\) 1.69720 0.137211
\(154\) 0 0
\(155\) −7.93218 −0.637128
\(156\) 0 0
\(157\) −5.31268 −0.423998 −0.211999 0.977270i \(-0.567997\pi\)
−0.211999 + 0.977270i \(0.567997\pi\)
\(158\) 0 0
\(159\) 19.1377 1.51772
\(160\) 0 0
\(161\) −0.586584 −0.0462293
\(162\) 0 0
\(163\) 0.537088 0.0420680 0.0210340 0.999779i \(-0.493304\pi\)
0.0210340 + 0.999779i \(0.493304\pi\)
\(164\) 0 0
\(165\) 4.53509 0.353056
\(166\) 0 0
\(167\) −21.3038 −1.64854 −0.824270 0.566196i \(-0.808414\pi\)
−0.824270 + 0.566196i \(0.808414\pi\)
\(168\) 0 0
\(169\) −12.8792 −0.990708
\(170\) 0 0
\(171\) −11.3350 −0.866809
\(172\) 0 0
\(173\) 12.0816 0.918544 0.459272 0.888296i \(-0.348110\pi\)
0.459272 + 0.888296i \(0.348110\pi\)
\(174\) 0 0
\(175\) −3.17952 −0.240349
\(176\) 0 0
\(177\) 13.8073 1.03782
\(178\) 0 0
\(179\) 10.4481 0.780924 0.390462 0.920619i \(-0.372315\pi\)
0.390462 + 0.920619i \(0.372315\pi\)
\(180\) 0 0
\(181\) −17.0088 −1.26426 −0.632128 0.774864i \(-0.717818\pi\)
−0.632128 + 0.774864i \(0.717818\pi\)
\(182\) 0 0
\(183\) 11.8536 0.876247
\(184\) 0 0
\(185\) −1.09134 −0.0802371
\(186\) 0 0
\(187\) −1.03939 −0.0760076
\(188\) 0 0
\(189\) −0.144582 −0.0105168
\(190\) 0 0
\(191\) −2.10005 −0.151955 −0.0759773 0.997110i \(-0.524208\pi\)
−0.0759773 + 0.997110i \(0.524208\pi\)
\(192\) 0 0
\(193\) 11.6532 0.838818 0.419409 0.907797i \(-0.362237\pi\)
0.419409 + 0.907797i \(0.362237\pi\)
\(194\) 0 0
\(195\) −0.852651 −0.0610596
\(196\) 0 0
\(197\) 7.46171 0.531624 0.265812 0.964025i \(-0.414360\pi\)
0.265812 + 0.964025i \(0.414360\pi\)
\(198\) 0 0
\(199\) −12.3841 −0.877884 −0.438942 0.898515i \(-0.644647\pi\)
−0.438942 + 0.898515i \(0.644647\pi\)
\(200\) 0 0
\(201\) 32.5134 2.29332
\(202\) 0 0
\(203\) 3.29627 0.231353
\(204\) 0 0
\(205\) −3.61724 −0.252639
\(206\) 0 0
\(207\) 0.556884 0.0387061
\(208\) 0 0
\(209\) 6.94170 0.480167
\(210\) 0 0
\(211\) −20.5045 −1.41159 −0.705794 0.708417i \(-0.749410\pi\)
−0.705794 + 0.708417i \(0.749410\pi\)
\(212\) 0 0
\(213\) 28.6066 1.96009
\(214\) 0 0
\(215\) −2.30386 −0.157122
\(216\) 0 0
\(217\) −25.2205 −1.71208
\(218\) 0 0
\(219\) 12.4063 0.838340
\(220\) 0 0
\(221\) 0.195418 0.0131452
\(222\) 0 0
\(223\) −13.1235 −0.878817 −0.439408 0.898287i \(-0.644812\pi\)
−0.439408 + 0.898287i \(0.644812\pi\)
\(224\) 0 0
\(225\) 3.01854 0.201236
\(226\) 0 0
\(227\) 16.9641 1.12595 0.562974 0.826475i \(-0.309657\pi\)
0.562974 + 0.826475i \(0.309657\pi\)
\(228\) 0 0
\(229\) −10.9238 −0.721864 −0.360932 0.932592i \(-0.617541\pi\)
−0.360932 + 0.932592i \(0.617541\pi\)
\(230\) 0 0
\(231\) 14.4194 0.948729
\(232\) 0 0
\(233\) 23.6513 1.54945 0.774724 0.632300i \(-0.217889\pi\)
0.774724 + 0.632300i \(0.217889\pi\)
\(234\) 0 0
\(235\) −9.03471 −0.589359
\(236\) 0 0
\(237\) 25.4351 1.65219
\(238\) 0 0
\(239\) 23.1090 1.49480 0.747399 0.664375i \(-0.231302\pi\)
0.747399 + 0.664375i \(0.231302\pi\)
\(240\) 0 0
\(241\) 8.56228 0.551545 0.275772 0.961223i \(-0.411066\pi\)
0.275772 + 0.961223i \(0.411066\pi\)
\(242\) 0 0
\(243\) −22.0786 −1.41634
\(244\) 0 0
\(245\) −3.10938 −0.198651
\(246\) 0 0
\(247\) −1.30512 −0.0830429
\(248\) 0 0
\(249\) 17.8564 1.13161
\(250\) 0 0
\(251\) 4.57051 0.288488 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(252\) 0 0
\(253\) −0.341043 −0.0214412
\(254\) 0 0
\(255\) −1.37938 −0.0863799
\(256\) 0 0
\(257\) 10.9860 0.685285 0.342643 0.939466i \(-0.388678\pi\)
0.342643 + 0.939466i \(0.388678\pi\)
\(258\) 0 0
\(259\) −3.46995 −0.215612
\(260\) 0 0
\(261\) −3.12937 −0.193703
\(262\) 0 0
\(263\) −10.5717 −0.651880 −0.325940 0.945391i \(-0.605681\pi\)
−0.325940 + 0.945391i \(0.605681\pi\)
\(264\) 0 0
\(265\) −7.80091 −0.479206
\(266\) 0 0
\(267\) −16.4416 −1.00621
\(268\) 0 0
\(269\) 2.38232 0.145252 0.0726262 0.997359i \(-0.476862\pi\)
0.0726262 + 0.997359i \(0.476862\pi\)
\(270\) 0 0
\(271\) 11.9765 0.727519 0.363759 0.931493i \(-0.381493\pi\)
0.363759 + 0.931493i \(0.381493\pi\)
\(272\) 0 0
\(273\) −2.71103 −0.164079
\(274\) 0 0
\(275\) −1.84859 −0.111474
\(276\) 0 0
\(277\) 10.8588 0.652441 0.326220 0.945294i \(-0.394225\pi\)
0.326220 + 0.945294i \(0.394225\pi\)
\(278\) 0 0
\(279\) 23.9436 1.43346
\(280\) 0 0
\(281\) 22.4058 1.33662 0.668310 0.743883i \(-0.267018\pi\)
0.668310 + 0.743883i \(0.267018\pi\)
\(282\) 0 0
\(283\) 15.2818 0.908408 0.454204 0.890898i \(-0.349924\pi\)
0.454204 + 0.890898i \(0.349924\pi\)
\(284\) 0 0
\(285\) 9.21235 0.545692
\(286\) 0 0
\(287\) −11.5011 −0.678889
\(288\) 0 0
\(289\) −16.6839 −0.981404
\(290\) 0 0
\(291\) −35.1198 −2.05876
\(292\) 0 0
\(293\) 11.2760 0.658748 0.329374 0.944200i \(-0.393162\pi\)
0.329374 + 0.944200i \(0.393162\pi\)
\(294\) 0 0
\(295\) −5.62811 −0.327681
\(296\) 0 0
\(297\) −0.0840604 −0.00487768
\(298\) 0 0
\(299\) 0.0641201 0.00370816
\(300\) 0 0
\(301\) −7.32517 −0.422216
\(302\) 0 0
\(303\) 31.8002 1.82687
\(304\) 0 0
\(305\) −4.83177 −0.276667
\(306\) 0 0
\(307\) 6.82435 0.389486 0.194743 0.980854i \(-0.437613\pi\)
0.194743 + 0.980854i \(0.437613\pi\)
\(308\) 0 0
\(309\) 11.7459 0.668203
\(310\) 0 0
\(311\) −14.0236 −0.795207 −0.397604 0.917557i \(-0.630158\pi\)
−0.397604 + 0.917557i \(0.630158\pi\)
\(312\) 0 0
\(313\) 18.9323 1.07011 0.535057 0.844816i \(-0.320290\pi\)
0.535057 + 0.844816i \(0.320290\pi\)
\(314\) 0 0
\(315\) 9.59751 0.540758
\(316\) 0 0
\(317\) −27.1834 −1.52677 −0.763387 0.645941i \(-0.776465\pi\)
−0.763387 + 0.645941i \(0.776465\pi\)
\(318\) 0 0
\(319\) 1.91647 0.107301
\(320\) 0 0
\(321\) −36.0516 −2.01221
\(322\) 0 0
\(323\) −2.11136 −0.117479
\(324\) 0 0
\(325\) 0.347557 0.0192790
\(326\) 0 0
\(327\) 36.4983 2.01836
\(328\) 0 0
\(329\) −28.7261 −1.58372
\(330\) 0 0
\(331\) 31.0143 1.70470 0.852350 0.522972i \(-0.175177\pi\)
0.852350 + 0.522972i \(0.175177\pi\)
\(332\) 0 0
\(333\) 3.29426 0.180524
\(334\) 0 0
\(335\) −13.2531 −0.724094
\(336\) 0 0
\(337\) 9.11446 0.496496 0.248248 0.968696i \(-0.420145\pi\)
0.248248 + 0.968696i \(0.420145\pi\)
\(338\) 0 0
\(339\) −1.17435 −0.0637818
\(340\) 0 0
\(341\) −14.6633 −0.794064
\(342\) 0 0
\(343\) 12.3703 0.667935
\(344\) 0 0
\(345\) −0.452599 −0.0243671
\(346\) 0 0
\(347\) 0.683614 0.0366983 0.0183492 0.999832i \(-0.494159\pi\)
0.0183492 + 0.999832i \(0.494159\pi\)
\(348\) 0 0
\(349\) 1.92961 0.103290 0.0516448 0.998666i \(-0.483554\pi\)
0.0516448 + 0.998666i \(0.483554\pi\)
\(350\) 0 0
\(351\) 0.0158044 0.000843574 0
\(352\) 0 0
\(353\) −4.76551 −0.253642 −0.126821 0.991926i \(-0.540477\pi\)
−0.126821 + 0.991926i \(0.540477\pi\)
\(354\) 0 0
\(355\) −11.6606 −0.618879
\(356\) 0 0
\(357\) −4.38576 −0.232119
\(358\) 0 0
\(359\) 0.0594328 0.00313674 0.00156837 0.999999i \(-0.499501\pi\)
0.00156837 + 0.999999i \(0.499501\pi\)
\(360\) 0 0
\(361\) −4.89900 −0.257842
\(362\) 0 0
\(363\) −18.6025 −0.976375
\(364\) 0 0
\(365\) −5.05704 −0.264698
\(366\) 0 0
\(367\) 25.5393 1.33314 0.666570 0.745443i \(-0.267762\pi\)
0.666570 + 0.745443i \(0.267762\pi\)
\(368\) 0 0
\(369\) 10.9188 0.568409
\(370\) 0 0
\(371\) −24.8032 −1.28772
\(372\) 0 0
\(373\) 18.4093 0.953198 0.476599 0.879121i \(-0.341869\pi\)
0.476599 + 0.879121i \(0.341869\pi\)
\(374\) 0 0
\(375\) −2.45327 −0.126686
\(376\) 0 0
\(377\) −0.360318 −0.0185573
\(378\) 0 0
\(379\) −7.54292 −0.387454 −0.193727 0.981056i \(-0.562058\pi\)
−0.193727 + 0.981056i \(0.562058\pi\)
\(380\) 0 0
\(381\) 20.6450 1.05767
\(382\) 0 0
\(383\) 3.60748 0.184333 0.0921667 0.995744i \(-0.470621\pi\)
0.0921667 + 0.995744i \(0.470621\pi\)
\(384\) 0 0
\(385\) −5.87764 −0.299552
\(386\) 0 0
\(387\) 6.95427 0.353505
\(388\) 0 0
\(389\) −3.80568 −0.192956 −0.0964778 0.995335i \(-0.530758\pi\)
−0.0964778 + 0.995335i \(0.530758\pi\)
\(390\) 0 0
\(391\) 0.103730 0.00524587
\(392\) 0 0
\(393\) −27.5832 −1.39139
\(394\) 0 0
\(395\) −10.3678 −0.521662
\(396\) 0 0
\(397\) −2.33774 −0.117328 −0.0586639 0.998278i \(-0.518684\pi\)
−0.0586639 + 0.998278i \(0.518684\pi\)
\(398\) 0 0
\(399\) 29.2909 1.46638
\(400\) 0 0
\(401\) −18.8370 −0.940675 −0.470338 0.882487i \(-0.655868\pi\)
−0.470338 + 0.882487i \(0.655868\pi\)
\(402\) 0 0
\(403\) 2.75688 0.137330
\(404\) 0 0
\(405\) 8.94405 0.444433
\(406\) 0 0
\(407\) −2.01745 −0.100001
\(408\) 0 0
\(409\) 1.85637 0.0917916 0.0458958 0.998946i \(-0.485386\pi\)
0.0458958 + 0.998946i \(0.485386\pi\)
\(410\) 0 0
\(411\) −48.9022 −2.41217
\(412\) 0 0
\(413\) −17.8947 −0.880542
\(414\) 0 0
\(415\) −7.27862 −0.357293
\(416\) 0 0
\(417\) 13.9030 0.680832
\(418\) 0 0
\(419\) 15.1243 0.738868 0.369434 0.929257i \(-0.379552\pi\)
0.369434 + 0.929257i \(0.379552\pi\)
\(420\) 0 0
\(421\) −10.0385 −0.489246 −0.244623 0.969618i \(-0.578664\pi\)
−0.244623 + 0.969618i \(0.578664\pi\)
\(422\) 0 0
\(423\) 27.2716 1.32599
\(424\) 0 0
\(425\) 0.562260 0.0272736
\(426\) 0 0
\(427\) −15.3627 −0.743455
\(428\) 0 0
\(429\) −1.57620 −0.0760998
\(430\) 0 0
\(431\) 16.0366 0.772454 0.386227 0.922404i \(-0.373778\pi\)
0.386227 + 0.922404i \(0.373778\pi\)
\(432\) 0 0
\(433\) −24.2093 −1.16342 −0.581711 0.813395i \(-0.697617\pi\)
−0.581711 + 0.813395i \(0.697617\pi\)
\(434\) 0 0
\(435\) 2.54335 0.121944
\(436\) 0 0
\(437\) −0.692776 −0.0331400
\(438\) 0 0
\(439\) 15.0185 0.716796 0.358398 0.933569i \(-0.383323\pi\)
0.358398 + 0.933569i \(0.383323\pi\)
\(440\) 0 0
\(441\) 9.38577 0.446941
\(442\) 0 0
\(443\) 13.0636 0.620669 0.310334 0.950627i \(-0.399559\pi\)
0.310334 + 0.950627i \(0.399559\pi\)
\(444\) 0 0
\(445\) 6.70193 0.317702
\(446\) 0 0
\(447\) −18.2854 −0.864871
\(448\) 0 0
\(449\) 27.0002 1.27422 0.637109 0.770774i \(-0.280130\pi\)
0.637109 + 0.770774i \(0.280130\pi\)
\(450\) 0 0
\(451\) −6.68680 −0.314869
\(452\) 0 0
\(453\) 2.45327 0.115265
\(454\) 0 0
\(455\) 1.10507 0.0518063
\(456\) 0 0
\(457\) −23.5842 −1.10322 −0.551612 0.834101i \(-0.685987\pi\)
−0.551612 + 0.834101i \(0.685987\pi\)
\(458\) 0 0
\(459\) 0.0255675 0.00119339
\(460\) 0 0
\(461\) −28.3311 −1.31951 −0.659755 0.751480i \(-0.729340\pi\)
−0.659755 + 0.751480i \(0.729340\pi\)
\(462\) 0 0
\(463\) 16.0551 0.746145 0.373073 0.927802i \(-0.378304\pi\)
0.373073 + 0.927802i \(0.378304\pi\)
\(464\) 0 0
\(465\) −19.4598 −0.902425
\(466\) 0 0
\(467\) −39.6316 −1.83393 −0.916967 0.398963i \(-0.869370\pi\)
−0.916967 + 0.398963i \(0.869370\pi\)
\(468\) 0 0
\(469\) −42.1385 −1.94578
\(470\) 0 0
\(471\) −13.0334 −0.600550
\(472\) 0 0
\(473\) −4.25888 −0.195824
\(474\) 0 0
\(475\) −3.75513 −0.172297
\(476\) 0 0
\(477\) 23.5473 1.07816
\(478\) 0 0
\(479\) 31.8513 1.45532 0.727662 0.685936i \(-0.240607\pi\)
0.727662 + 0.685936i \(0.240607\pi\)
\(480\) 0 0
\(481\) 0.379304 0.0172948
\(482\) 0 0
\(483\) −1.43905 −0.0654790
\(484\) 0 0
\(485\) 14.3155 0.650034
\(486\) 0 0
\(487\) −25.7700 −1.16775 −0.583875 0.811844i \(-0.698464\pi\)
−0.583875 + 0.811844i \(0.698464\pi\)
\(488\) 0 0
\(489\) 1.31762 0.0595850
\(490\) 0 0
\(491\) −1.01715 −0.0459032 −0.0229516 0.999737i \(-0.507306\pi\)
−0.0229516 + 0.999737i \(0.507306\pi\)
\(492\) 0 0
\(493\) −0.582905 −0.0262527
\(494\) 0 0
\(495\) 5.58003 0.250804
\(496\) 0 0
\(497\) −37.0751 −1.66305
\(498\) 0 0
\(499\) −8.88659 −0.397818 −0.198909 0.980018i \(-0.563740\pi\)
−0.198909 + 0.980018i \(0.563740\pi\)
\(500\) 0 0
\(501\) −52.2641 −2.33499
\(502\) 0 0
\(503\) −26.3462 −1.17472 −0.587359 0.809326i \(-0.699832\pi\)
−0.587359 + 0.809326i \(0.699832\pi\)
\(504\) 0 0
\(505\) −12.9624 −0.576817
\(506\) 0 0
\(507\) −31.5962 −1.40324
\(508\) 0 0
\(509\) 24.8057 1.09949 0.549746 0.835332i \(-0.314724\pi\)
0.549746 + 0.835332i \(0.314724\pi\)
\(510\) 0 0
\(511\) −16.0790 −0.711293
\(512\) 0 0
\(513\) −0.170756 −0.00753906
\(514\) 0 0
\(515\) −4.78787 −0.210979
\(516\) 0 0
\(517\) −16.7015 −0.734530
\(518\) 0 0
\(519\) 29.6393 1.30102
\(520\) 0 0
\(521\) 6.22150 0.272569 0.136284 0.990670i \(-0.456484\pi\)
0.136284 + 0.990670i \(0.456484\pi\)
\(522\) 0 0
\(523\) 1.99802 0.0873672 0.0436836 0.999045i \(-0.486091\pi\)
0.0436836 + 0.999045i \(0.486091\pi\)
\(524\) 0 0
\(525\) −7.80023 −0.340430
\(526\) 0 0
\(527\) 4.45995 0.194278
\(528\) 0 0
\(529\) −22.9660 −0.998520
\(530\) 0 0
\(531\) 16.9887 0.737245
\(532\) 0 0
\(533\) 1.25720 0.0544553
\(534\) 0 0
\(535\) 14.6953 0.635335
\(536\) 0 0
\(537\) 25.6319 1.10610
\(538\) 0 0
\(539\) −5.74796 −0.247582
\(540\) 0 0
\(541\) −11.2836 −0.485118 −0.242559 0.970137i \(-0.577987\pi\)
−0.242559 + 0.970137i \(0.577987\pi\)
\(542\) 0 0
\(543\) −41.7272 −1.79069
\(544\) 0 0
\(545\) −14.8774 −0.637277
\(546\) 0 0
\(547\) 5.89315 0.251973 0.125987 0.992032i \(-0.459790\pi\)
0.125987 + 0.992032i \(0.459790\pi\)
\(548\) 0 0
\(549\) 14.5849 0.622468
\(550\) 0 0
\(551\) 3.89301 0.165848
\(552\) 0 0
\(553\) −32.9648 −1.40181
\(554\) 0 0
\(555\) −2.67736 −0.113648
\(556\) 0 0
\(557\) 34.3228 1.45431 0.727153 0.686476i \(-0.240843\pi\)
0.727153 + 0.686476i \(0.240843\pi\)
\(558\) 0 0
\(559\) 0.800721 0.0338669
\(560\) 0 0
\(561\) −2.54990 −0.107657
\(562\) 0 0
\(563\) 28.8583 1.21623 0.608117 0.793847i \(-0.291925\pi\)
0.608117 + 0.793847i \(0.291925\pi\)
\(564\) 0 0
\(565\) 0.478687 0.0201385
\(566\) 0 0
\(567\) 28.4378 1.19428
\(568\) 0 0
\(569\) −17.9690 −0.753300 −0.376650 0.926356i \(-0.622924\pi\)
−0.376650 + 0.926356i \(0.622924\pi\)
\(570\) 0 0
\(571\) 13.1563 0.550573 0.275286 0.961362i \(-0.411227\pi\)
0.275286 + 0.961362i \(0.411227\pi\)
\(572\) 0 0
\(573\) −5.15200 −0.215228
\(574\) 0 0
\(575\) 0.184488 0.00769368
\(576\) 0 0
\(577\) 17.0147 0.708329 0.354165 0.935183i \(-0.384765\pi\)
0.354165 + 0.935183i \(0.384765\pi\)
\(578\) 0 0
\(579\) 28.5885 1.18810
\(580\) 0 0
\(581\) −23.1426 −0.960115
\(582\) 0 0
\(583\) −14.4207 −0.597244
\(584\) 0 0
\(585\) −1.04911 −0.0433755
\(586\) 0 0
\(587\) 31.7045 1.30859 0.654293 0.756241i \(-0.272966\pi\)
0.654293 + 0.756241i \(0.272966\pi\)
\(588\) 0 0
\(589\) −29.7863 −1.22733
\(590\) 0 0
\(591\) 18.3056 0.752991
\(592\) 0 0
\(593\) −15.3419 −0.630018 −0.315009 0.949089i \(-0.602008\pi\)
−0.315009 + 0.949089i \(0.602008\pi\)
\(594\) 0 0
\(595\) 1.78772 0.0732894
\(596\) 0 0
\(597\) −30.3815 −1.24343
\(598\) 0 0
\(599\) 7.89873 0.322733 0.161367 0.986895i \(-0.448410\pi\)
0.161367 + 0.986895i \(0.448410\pi\)
\(600\) 0 0
\(601\) 6.83231 0.278696 0.139348 0.990243i \(-0.455499\pi\)
0.139348 + 0.990243i \(0.455499\pi\)
\(602\) 0 0
\(603\) 40.0049 1.62913
\(604\) 0 0
\(605\) 7.58272 0.308281
\(606\) 0 0
\(607\) 14.6498 0.594618 0.297309 0.954781i \(-0.403911\pi\)
0.297309 + 0.954781i \(0.403911\pi\)
\(608\) 0 0
\(609\) 8.08664 0.327687
\(610\) 0 0
\(611\) 3.14008 0.127034
\(612\) 0 0
\(613\) 3.41504 0.137932 0.0689660 0.997619i \(-0.478030\pi\)
0.0689660 + 0.997619i \(0.478030\pi\)
\(614\) 0 0
\(615\) −8.87407 −0.357837
\(616\) 0 0
\(617\) 24.8508 1.00046 0.500228 0.865894i \(-0.333250\pi\)
0.500228 + 0.865894i \(0.333250\pi\)
\(618\) 0 0
\(619\) −29.0389 −1.16717 −0.583586 0.812051i \(-0.698351\pi\)
−0.583586 + 0.812051i \(0.698351\pi\)
\(620\) 0 0
\(621\) 0.00838917 0.000336646 0
\(622\) 0 0
\(623\) 21.3089 0.853725
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 17.0299 0.680107
\(628\) 0 0
\(629\) 0.613619 0.0244666
\(630\) 0 0
\(631\) 2.76540 0.110089 0.0550445 0.998484i \(-0.482470\pi\)
0.0550445 + 0.998484i \(0.482470\pi\)
\(632\) 0 0
\(633\) −50.3031 −1.99937
\(634\) 0 0
\(635\) −8.41528 −0.333950
\(636\) 0 0
\(637\) 1.08069 0.0428183
\(638\) 0 0
\(639\) 35.1979 1.39241
\(640\) 0 0
\(641\) 0.291889 0.0115289 0.00576446 0.999983i \(-0.498165\pi\)
0.00576446 + 0.999983i \(0.498165\pi\)
\(642\) 0 0
\(643\) −8.09175 −0.319108 −0.159554 0.987189i \(-0.551006\pi\)
−0.159554 + 0.987189i \(0.551006\pi\)
\(644\) 0 0
\(645\) −5.65198 −0.222547
\(646\) 0 0
\(647\) −12.4961 −0.491273 −0.245636 0.969362i \(-0.578997\pi\)
−0.245636 + 0.969362i \(0.578997\pi\)
\(648\) 0 0
\(649\) −10.4041 −0.408395
\(650\) 0 0
\(651\) −61.8728 −2.42499
\(652\) 0 0
\(653\) −8.17861 −0.320054 −0.160027 0.987113i \(-0.551158\pi\)
−0.160027 + 0.987113i \(0.551158\pi\)
\(654\) 0 0
\(655\) 11.2435 0.439318
\(656\) 0 0
\(657\) 15.2649 0.595539
\(658\) 0 0
\(659\) 45.2257 1.76174 0.880871 0.473357i \(-0.156958\pi\)
0.880871 + 0.473357i \(0.156958\pi\)
\(660\) 0 0
\(661\) 22.4453 0.873021 0.436511 0.899699i \(-0.356214\pi\)
0.436511 + 0.899699i \(0.356214\pi\)
\(662\) 0 0
\(663\) 0.479412 0.0186188
\(664\) 0 0
\(665\) −11.9395 −0.462995
\(666\) 0 0
\(667\) −0.191262 −0.00740569
\(668\) 0 0
\(669\) −32.1956 −1.24475
\(670\) 0 0
\(671\) −8.93197 −0.344815
\(672\) 0 0
\(673\) −6.59172 −0.254092 −0.127046 0.991897i \(-0.540550\pi\)
−0.127046 + 0.991897i \(0.540550\pi\)
\(674\) 0 0
\(675\) 0.0454727 0.00175025
\(676\) 0 0
\(677\) 6.28244 0.241454 0.120727 0.992686i \(-0.461477\pi\)
0.120727 + 0.992686i \(0.461477\pi\)
\(678\) 0 0
\(679\) 45.5166 1.74677
\(680\) 0 0
\(681\) 41.6176 1.59479
\(682\) 0 0
\(683\) −46.6355 −1.78446 −0.892230 0.451582i \(-0.850860\pi\)
−0.892230 + 0.451582i \(0.850860\pi\)
\(684\) 0 0
\(685\) 19.9335 0.761618
\(686\) 0 0
\(687\) −26.7990 −1.02245
\(688\) 0 0
\(689\) 2.71126 0.103291
\(690\) 0 0
\(691\) 19.9584 0.759254 0.379627 0.925140i \(-0.376052\pi\)
0.379627 + 0.925140i \(0.376052\pi\)
\(692\) 0 0
\(693\) 17.7419 0.673957
\(694\) 0 0
\(695\) −5.66713 −0.214966
\(696\) 0 0
\(697\) 2.03383 0.0770369
\(698\) 0 0
\(699\) 58.0230 2.19463
\(700\) 0 0
\(701\) −35.9484 −1.35775 −0.678876 0.734253i \(-0.737533\pi\)
−0.678876 + 0.734253i \(0.737533\pi\)
\(702\) 0 0
\(703\) −4.09813 −0.154564
\(704\) 0 0
\(705\) −22.1646 −0.834766
\(706\) 0 0
\(707\) −41.2141 −1.55002
\(708\) 0 0
\(709\) −49.6979 −1.86644 −0.933221 0.359303i \(-0.883015\pi\)
−0.933221 + 0.359303i \(0.883015\pi\)
\(710\) 0 0
\(711\) 31.2957 1.17368
\(712\) 0 0
\(713\) 1.46339 0.0548044
\(714\) 0 0
\(715\) 0.642490 0.0240278
\(716\) 0 0
\(717\) 56.6927 2.11723
\(718\) 0 0
\(719\) 13.6081 0.507496 0.253748 0.967270i \(-0.418337\pi\)
0.253748 + 0.967270i \(0.418337\pi\)
\(720\) 0 0
\(721\) −15.2231 −0.566940
\(722\) 0 0
\(723\) 21.0056 0.781206
\(724\) 0 0
\(725\) −1.03672 −0.0385027
\(726\) 0 0
\(727\) −41.0886 −1.52389 −0.761945 0.647642i \(-0.775755\pi\)
−0.761945 + 0.647642i \(0.775755\pi\)
\(728\) 0 0
\(729\) −27.3326 −1.01232
\(730\) 0 0
\(731\) 1.29537 0.0479109
\(732\) 0 0
\(733\) 1.94145 0.0717090 0.0358545 0.999357i \(-0.488585\pi\)
0.0358545 + 0.999357i \(0.488585\pi\)
\(734\) 0 0
\(735\) −7.62814 −0.281368
\(736\) 0 0
\(737\) −24.4995 −0.902452
\(738\) 0 0
\(739\) 10.3848 0.382011 0.191005 0.981589i \(-0.438825\pi\)
0.191005 + 0.981589i \(0.438825\pi\)
\(740\) 0 0
\(741\) −3.20182 −0.117622
\(742\) 0 0
\(743\) −11.8130 −0.433378 −0.216689 0.976241i \(-0.569526\pi\)
−0.216689 + 0.976241i \(0.569526\pi\)
\(744\) 0 0
\(745\) 7.45349 0.273075
\(746\) 0 0
\(747\) 21.9708 0.803869
\(748\) 0 0
\(749\) 46.7242 1.70727
\(750\) 0 0
\(751\) 10.7633 0.392760 0.196380 0.980528i \(-0.437081\pi\)
0.196380 + 0.980528i \(0.437081\pi\)
\(752\) 0 0
\(753\) 11.2127 0.408613
\(754\) 0 0
\(755\) −1.00000 −0.0363937
\(756\) 0 0
\(757\) −50.8259 −1.84730 −0.923649 0.383240i \(-0.874808\pi\)
−0.923649 + 0.383240i \(0.874808\pi\)
\(758\) 0 0
\(759\) −0.836670 −0.0303692
\(760\) 0 0
\(761\) −22.4407 −0.813475 −0.406737 0.913545i \(-0.633334\pi\)
−0.406737 + 0.913545i \(0.633334\pi\)
\(762\) 0 0
\(763\) −47.3030 −1.71248
\(764\) 0 0
\(765\) −1.69720 −0.0613625
\(766\) 0 0
\(767\) 1.95609 0.0706303
\(768\) 0 0
\(769\) −10.5434 −0.380205 −0.190102 0.981764i \(-0.560882\pi\)
−0.190102 + 0.981764i \(0.560882\pi\)
\(770\) 0 0
\(771\) 26.9515 0.970635
\(772\) 0 0
\(773\) −23.0253 −0.828162 −0.414081 0.910240i \(-0.635897\pi\)
−0.414081 + 0.910240i \(0.635897\pi\)
\(774\) 0 0
\(775\) 7.93218 0.284932
\(776\) 0 0
\(777\) −8.51273 −0.305392
\(778\) 0 0
\(779\) −13.5832 −0.486669
\(780\) 0 0
\(781\) −21.5556 −0.771321
\(782\) 0 0
\(783\) −0.0471424 −0.00168473
\(784\) 0 0
\(785\) 5.31268 0.189618
\(786\) 0 0
\(787\) 20.4695 0.729659 0.364830 0.931074i \(-0.381127\pi\)
0.364830 + 0.931074i \(0.381127\pi\)
\(788\) 0 0
\(789\) −25.9353 −0.923320
\(790\) 0 0
\(791\) 1.52200 0.0541160
\(792\) 0 0
\(793\) 1.67932 0.0596343
\(794\) 0 0
\(795\) −19.1377 −0.678746
\(796\) 0 0
\(797\) 17.9438 0.635602 0.317801 0.948157i \(-0.397056\pi\)
0.317801 + 0.948157i \(0.397056\pi\)
\(798\) 0 0
\(799\) 5.07986 0.179712
\(800\) 0 0
\(801\) −20.2300 −0.714792
\(802\) 0 0
\(803\) −9.34840 −0.329898
\(804\) 0 0
\(805\) 0.586584 0.0206744
\(806\) 0 0
\(807\) 5.84447 0.205735
\(808\) 0 0
\(809\) 24.7656 0.870713 0.435356 0.900258i \(-0.356622\pi\)
0.435356 + 0.900258i \(0.356622\pi\)
\(810\) 0 0
\(811\) 7.01277 0.246252 0.123126 0.992391i \(-0.460708\pi\)
0.123126 + 0.992391i \(0.460708\pi\)
\(812\) 0 0
\(813\) 29.3815 1.03045
\(814\) 0 0
\(815\) −0.537088 −0.0188134
\(816\) 0 0
\(817\) −8.65128 −0.302670
\(818\) 0 0
\(819\) −3.33568 −0.116558
\(820\) 0 0
\(821\) 31.6819 1.10571 0.552853 0.833279i \(-0.313539\pi\)
0.552853 + 0.833279i \(0.313539\pi\)
\(822\) 0 0
\(823\) −9.56321 −0.333352 −0.166676 0.986012i \(-0.553303\pi\)
−0.166676 + 0.986012i \(0.553303\pi\)
\(824\) 0 0
\(825\) −4.53509 −0.157892
\(826\) 0 0
\(827\) 11.1021 0.386058 0.193029 0.981193i \(-0.438169\pi\)
0.193029 + 0.981193i \(0.438169\pi\)
\(828\) 0 0
\(829\) −11.3884 −0.395534 −0.197767 0.980249i \(-0.563369\pi\)
−0.197767 + 0.980249i \(0.563369\pi\)
\(830\) 0 0
\(831\) 26.6395 0.924115
\(832\) 0 0
\(833\) 1.74828 0.0605743
\(834\) 0 0
\(835\) 21.3038 0.737250
\(836\) 0 0
\(837\) 0.360698 0.0124675
\(838\) 0 0
\(839\) 11.6880 0.403515 0.201757 0.979436i \(-0.435335\pi\)
0.201757 + 0.979436i \(0.435335\pi\)
\(840\) 0 0
\(841\) −27.9252 −0.962939
\(842\) 0 0
\(843\) 54.9675 1.89318
\(844\) 0 0
\(845\) 12.8792 0.443058
\(846\) 0 0
\(847\) 24.1094 0.828410
\(848\) 0 0
\(849\) 37.4904 1.28667
\(850\) 0 0
\(851\) 0.201340 0.00690183
\(852\) 0 0
\(853\) 35.2687 1.20758 0.603789 0.797144i \(-0.293657\pi\)
0.603789 + 0.797144i \(0.293657\pi\)
\(854\) 0 0
\(855\) 11.3350 0.387649
\(856\) 0 0
\(857\) −16.4026 −0.560303 −0.280152 0.959956i \(-0.590385\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(858\) 0 0
\(859\) 21.4579 0.732135 0.366067 0.930588i \(-0.380704\pi\)
0.366067 + 0.930588i \(0.380704\pi\)
\(860\) 0 0
\(861\) −28.2153 −0.961576
\(862\) 0 0
\(863\) 13.5031 0.459652 0.229826 0.973232i \(-0.426184\pi\)
0.229826 + 0.973232i \(0.426184\pi\)
\(864\) 0 0
\(865\) −12.0816 −0.410785
\(866\) 0 0
\(867\) −40.9300 −1.39006
\(868\) 0 0
\(869\) −19.1659 −0.650158
\(870\) 0 0
\(871\) 4.60621 0.156075
\(872\) 0 0
\(873\) −43.2119 −1.46250
\(874\) 0 0
\(875\) 3.17952 0.107488
\(876\) 0 0
\(877\) −15.8614 −0.535600 −0.267800 0.963475i \(-0.586297\pi\)
−0.267800 + 0.963475i \(0.586297\pi\)
\(878\) 0 0
\(879\) 27.6630 0.933048
\(880\) 0 0
\(881\) 12.3293 0.415386 0.207693 0.978194i \(-0.433404\pi\)
0.207693 + 0.978194i \(0.433404\pi\)
\(882\) 0 0
\(883\) 18.0825 0.608523 0.304262 0.952588i \(-0.401590\pi\)
0.304262 + 0.952588i \(0.401590\pi\)
\(884\) 0 0
\(885\) −13.8073 −0.464127
\(886\) 0 0
\(887\) 5.49410 0.184474 0.0922368 0.995737i \(-0.470598\pi\)
0.0922368 + 0.995737i \(0.470598\pi\)
\(888\) 0 0
\(889\) −26.7566 −0.897387
\(890\) 0 0
\(891\) 16.5339 0.553906
\(892\) 0 0
\(893\) −33.9265 −1.13531
\(894\) 0 0
\(895\) −10.4481 −0.349240
\(896\) 0 0
\(897\) 0.157304 0.00525222
\(898\) 0 0
\(899\) −8.22343 −0.274267
\(900\) 0 0
\(901\) 4.38614 0.146124
\(902\) 0 0
\(903\) −17.9706 −0.598025
\(904\) 0 0
\(905\) 17.0088 0.565392
\(906\) 0 0
\(907\) 26.0553 0.865152 0.432576 0.901598i \(-0.357605\pi\)
0.432576 + 0.901598i \(0.357605\pi\)
\(908\) 0 0
\(909\) 39.1273 1.29777
\(910\) 0 0
\(911\) −16.3194 −0.540686 −0.270343 0.962764i \(-0.587137\pi\)
−0.270343 + 0.962764i \(0.587137\pi\)
\(912\) 0 0
\(913\) −13.4552 −0.445302
\(914\) 0 0
\(915\) −11.8536 −0.391869
\(916\) 0 0
\(917\) 35.7488 1.18053
\(918\) 0 0
\(919\) 26.2762 0.866773 0.433387 0.901208i \(-0.357318\pi\)
0.433387 + 0.901208i \(0.357318\pi\)
\(920\) 0 0
\(921\) 16.7420 0.551667
\(922\) 0 0
\(923\) 4.05272 0.133397
\(924\) 0 0
\(925\) 1.09134 0.0358831
\(926\) 0 0
\(927\) 14.4524 0.474678
\(928\) 0 0
\(929\) −46.7450 −1.53365 −0.766826 0.641855i \(-0.778165\pi\)
−0.766826 + 0.641855i \(0.778165\pi\)
\(930\) 0 0
\(931\) −11.6761 −0.382669
\(932\) 0 0
\(933\) −34.4038 −1.12633
\(934\) 0 0
\(935\) 1.03939 0.0339917
\(936\) 0 0
\(937\) −27.4327 −0.896188 −0.448094 0.893986i \(-0.647897\pi\)
−0.448094 + 0.893986i \(0.647897\pi\)
\(938\) 0 0
\(939\) 46.4460 1.51571
\(940\) 0 0
\(941\) 6.51900 0.212513 0.106257 0.994339i \(-0.466113\pi\)
0.106257 + 0.994339i \(0.466113\pi\)
\(942\) 0 0
\(943\) 0.667338 0.0217315
\(944\) 0 0
\(945\) 0.144582 0.00470324
\(946\) 0 0
\(947\) 43.7858 1.42285 0.711424 0.702763i \(-0.248051\pi\)
0.711424 + 0.702763i \(0.248051\pi\)
\(948\) 0 0
\(949\) 1.75761 0.0570545
\(950\) 0 0
\(951\) −66.6883 −2.16252
\(952\) 0 0
\(953\) 59.0036 1.91131 0.955657 0.294483i \(-0.0951473\pi\)
0.955657 + 0.294483i \(0.0951473\pi\)
\(954\) 0 0
\(955\) 2.10005 0.0679561
\(956\) 0 0
\(957\) 4.70161 0.151981
\(958\) 0 0
\(959\) 63.3789 2.04661
\(960\) 0 0
\(961\) 31.9194 1.02966
\(962\) 0 0
\(963\) −44.3584 −1.42943
\(964\) 0 0
\(965\) −11.6532 −0.375131
\(966\) 0 0
\(967\) 39.4813 1.26963 0.634817 0.772663i \(-0.281076\pi\)
0.634817 + 0.772663i \(0.281076\pi\)
\(968\) 0 0
\(969\) −5.17974 −0.166397
\(970\) 0 0
\(971\) −30.8018 −0.988475 −0.494238 0.869327i \(-0.664553\pi\)
−0.494238 + 0.869327i \(0.664553\pi\)
\(972\) 0 0
\(973\) −18.0188 −0.577655
\(974\) 0 0
\(975\) 0.852651 0.0273067
\(976\) 0 0
\(977\) −34.0177 −1.08832 −0.544161 0.838981i \(-0.683152\pi\)
−0.544161 + 0.838981i \(0.683152\pi\)
\(978\) 0 0
\(979\) 12.3891 0.395958
\(980\) 0 0
\(981\) 44.9079 1.43380
\(982\) 0 0
\(983\) 28.6619 0.914172 0.457086 0.889423i \(-0.348893\pi\)
0.457086 + 0.889423i \(0.348893\pi\)
\(984\) 0 0
\(985\) −7.46171 −0.237750
\(986\) 0 0
\(987\) −70.4728 −2.24317
\(988\) 0 0
\(989\) 0.425034 0.0135153
\(990\) 0 0
\(991\) −13.9754 −0.443942 −0.221971 0.975053i \(-0.571249\pi\)
−0.221971 + 0.975053i \(0.571249\pi\)
\(992\) 0 0
\(993\) 76.0865 2.41453
\(994\) 0 0
\(995\) 12.3841 0.392602
\(996\) 0 0
\(997\) −1.75327 −0.0555266 −0.0277633 0.999615i \(-0.508838\pi\)
−0.0277633 + 0.999615i \(0.508838\pi\)
\(998\) 0 0
\(999\) 0.0496263 0.00157011
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 6040.2.a.m.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6040.2.a.m.1.11 12 1.1 even 1 trivial