Properties

Label 1560.4.a.m.1.4
Level $1560$
Weight $4$
Character 1560.1
Self dual yes
Analytic conductor $92.043$
Analytic rank $0$
Dimension $4$
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] = [1560,4,Mod(1,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.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: \(92.0429796090\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 76x^{2} + 50x + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(7.50422\) of defining polynomial
Character \(\chi\) \(=\) 1560.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} +29.9374 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -5.00000 q^{5} +29.9374 q^{7} +9.00000 q^{9} -30.8448 q^{11} +13.0000 q^{13} -15.0000 q^{15} +25.6366 q^{17} +106.958 q^{19} +89.8123 q^{21} +31.6366 q^{23} +25.0000 q^{25} +27.0000 q^{27} -2.43075 q^{29} -46.5272 q^{31} -92.5343 q^{33} -149.687 q^{35} +45.9326 q^{37} +39.0000 q^{39} -281.548 q^{41} +13.2034 q^{43} -45.0000 q^{45} -60.9724 q^{47} +553.250 q^{49} +76.9098 q^{51} +292.614 q^{53} +154.224 q^{55} +320.874 q^{57} +372.847 q^{59} +314.934 q^{61} +269.437 q^{63} -65.0000 q^{65} -25.9113 q^{67} +94.9098 q^{69} +34.0602 q^{71} +1159.87 q^{73} +75.0000 q^{75} -923.413 q^{77} -544.804 q^{79} +81.0000 q^{81} -993.993 q^{83} -128.183 q^{85} -7.29224 q^{87} +528.421 q^{89} +389.187 q^{91} -139.582 q^{93} -534.790 q^{95} +927.730 q^{97} -277.603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9} - 7 q^{11} + 52 q^{13} - 60 q^{15} - 73 q^{17} + 68 q^{19} + 45 q^{21} - 49 q^{23} + 100 q^{25} + 108 q^{27} - 66 q^{29} + 230 q^{31} - 21 q^{33} - 75 q^{35} + 303 q^{37} + 156 q^{39} + 155 q^{41} + 336 q^{43} - 180 q^{45} + 178 q^{47} + 377 q^{49} - 219 q^{51} + 349 q^{53} + 35 q^{55} + 204 q^{57} - 360 q^{59} + 223 q^{61} + 135 q^{63} - 260 q^{65} + 588 q^{67} - 147 q^{69} - 83 q^{71} + 754 q^{73} + 300 q^{75} + 1401 q^{77} + 869 q^{79} + 324 q^{81} - 1044 q^{83} + 365 q^{85} - 198 q^{87} + 1017 q^{89} + 195 q^{91} + 690 q^{93} - 340 q^{95} + 1367 q^{97} - 63 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 29.9374 1.61647 0.808235 0.588860i \(-0.200423\pi\)
0.808235 + 0.588860i \(0.200423\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −30.8448 −0.845458 −0.422729 0.906256i \(-0.638928\pi\)
−0.422729 + 0.906256i \(0.638928\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) 25.6366 0.365752 0.182876 0.983136i \(-0.441459\pi\)
0.182876 + 0.983136i \(0.441459\pi\)
\(18\) 0 0
\(19\) 106.958 1.29146 0.645732 0.763564i \(-0.276552\pi\)
0.645732 + 0.763564i \(0.276552\pi\)
\(20\) 0 0
\(21\) 89.8123 0.933269
\(22\) 0 0
\(23\) 31.6366 0.286813 0.143406 0.989664i \(-0.454194\pi\)
0.143406 + 0.989664i \(0.454194\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −2.43075 −0.0155648 −0.00778239 0.999970i \(-0.502477\pi\)
−0.00778239 + 0.999970i \(0.502477\pi\)
\(30\) 0 0
\(31\) −46.5272 −0.269565 −0.134783 0.990875i \(-0.543034\pi\)
−0.134783 + 0.990875i \(0.543034\pi\)
\(32\) 0 0
\(33\) −92.5343 −0.488126
\(34\) 0 0
\(35\) −149.687 −0.722907
\(36\) 0 0
\(37\) 45.9326 0.204089 0.102044 0.994780i \(-0.467462\pi\)
0.102044 + 0.994780i \(0.467462\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −281.548 −1.07245 −0.536224 0.844076i \(-0.680150\pi\)
−0.536224 + 0.844076i \(0.680150\pi\)
\(42\) 0 0
\(43\) 13.2034 0.0468254 0.0234127 0.999726i \(-0.492547\pi\)
0.0234127 + 0.999726i \(0.492547\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −60.9724 −0.189228 −0.0946142 0.995514i \(-0.530162\pi\)
−0.0946142 + 0.995514i \(0.530162\pi\)
\(48\) 0 0
\(49\) 553.250 1.61298
\(50\) 0 0
\(51\) 76.9098 0.211167
\(52\) 0 0
\(53\) 292.614 0.758370 0.379185 0.925321i \(-0.376204\pi\)
0.379185 + 0.925321i \(0.376204\pi\)
\(54\) 0 0
\(55\) 154.224 0.378101
\(56\) 0 0
\(57\) 320.874 0.745628
\(58\) 0 0
\(59\) 372.847 0.822721 0.411361 0.911473i \(-0.365054\pi\)
0.411361 + 0.911473i \(0.365054\pi\)
\(60\) 0 0
\(61\) 314.934 0.661035 0.330518 0.943800i \(-0.392777\pi\)
0.330518 + 0.943800i \(0.392777\pi\)
\(62\) 0 0
\(63\) 269.437 0.538823
\(64\) 0 0
\(65\) −65.0000 −0.124035
\(66\) 0 0
\(67\) −25.9113 −0.0472473 −0.0236237 0.999721i \(-0.507520\pi\)
−0.0236237 + 0.999721i \(0.507520\pi\)
\(68\) 0 0
\(69\) 94.9098 0.165591
\(70\) 0 0
\(71\) 34.0602 0.0569324 0.0284662 0.999595i \(-0.490938\pi\)
0.0284662 + 0.999595i \(0.490938\pi\)
\(72\) 0 0
\(73\) 1159.87 1.85962 0.929812 0.368034i \(-0.119969\pi\)
0.929812 + 0.368034i \(0.119969\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −923.413 −1.36666
\(78\) 0 0
\(79\) −544.804 −0.775889 −0.387945 0.921683i \(-0.626815\pi\)
−0.387945 + 0.921683i \(0.626815\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −993.993 −1.31452 −0.657259 0.753665i \(-0.728284\pi\)
−0.657259 + 0.753665i \(0.728284\pi\)
\(84\) 0 0
\(85\) −128.183 −0.163569
\(86\) 0 0
\(87\) −7.29224 −0.00898633
\(88\) 0 0
\(89\) 528.421 0.629354 0.314677 0.949199i \(-0.398104\pi\)
0.314677 + 0.949199i \(0.398104\pi\)
\(90\) 0 0
\(91\) 389.187 0.448328
\(92\) 0 0
\(93\) −139.582 −0.155634
\(94\) 0 0
\(95\) −534.790 −0.577561
\(96\) 0 0
\(97\) 927.730 0.971100 0.485550 0.874209i \(-0.338619\pi\)
0.485550 + 0.874209i \(0.338619\pi\)
\(98\) 0 0
\(99\) −277.603 −0.281819
\(100\) 0 0
\(101\) 95.6242 0.0942075 0.0471038 0.998890i \(-0.485001\pi\)
0.0471038 + 0.998890i \(0.485001\pi\)
\(102\) 0 0
\(103\) −621.642 −0.594681 −0.297341 0.954771i \(-0.596100\pi\)
−0.297341 + 0.954771i \(0.596100\pi\)
\(104\) 0 0
\(105\) −449.062 −0.417371
\(106\) 0 0
\(107\) −218.378 −0.197303 −0.0986515 0.995122i \(-0.531453\pi\)
−0.0986515 + 0.995122i \(0.531453\pi\)
\(108\) 0 0
\(109\) 1193.55 1.04882 0.524412 0.851465i \(-0.324285\pi\)
0.524412 + 0.851465i \(0.324285\pi\)
\(110\) 0 0
\(111\) 137.798 0.117831
\(112\) 0 0
\(113\) −1154.34 −0.960983 −0.480492 0.876999i \(-0.659542\pi\)
−0.480492 + 0.876999i \(0.659542\pi\)
\(114\) 0 0
\(115\) −158.183 −0.128266
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 767.494 0.591228
\(120\) 0 0
\(121\) −379.601 −0.285200
\(122\) 0 0
\(123\) −844.643 −0.619178
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −710.316 −0.496302 −0.248151 0.968721i \(-0.579823\pi\)
−0.248151 + 0.968721i \(0.579823\pi\)
\(128\) 0 0
\(129\) 39.6101 0.0270347
\(130\) 0 0
\(131\) 362.881 0.242023 0.121012 0.992651i \(-0.461386\pi\)
0.121012 + 0.992651i \(0.461386\pi\)
\(132\) 0 0
\(133\) 3202.05 2.08761
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) −835.594 −0.521092 −0.260546 0.965461i \(-0.583903\pi\)
−0.260546 + 0.965461i \(0.583903\pi\)
\(138\) 0 0
\(139\) 2539.45 1.54959 0.774797 0.632210i \(-0.217852\pi\)
0.774797 + 0.632210i \(0.217852\pi\)
\(140\) 0 0
\(141\) −182.917 −0.109251
\(142\) 0 0
\(143\) −400.982 −0.234488
\(144\) 0 0
\(145\) 12.1537 0.00696078
\(146\) 0 0
\(147\) 1659.75 0.931252
\(148\) 0 0
\(149\) 2312.03 1.27120 0.635600 0.772019i \(-0.280753\pi\)
0.635600 + 0.772019i \(0.280753\pi\)
\(150\) 0 0
\(151\) −292.005 −0.157371 −0.0786856 0.996899i \(-0.525072\pi\)
−0.0786856 + 0.996899i \(0.525072\pi\)
\(152\) 0 0
\(153\) 230.729 0.121917
\(154\) 0 0
\(155\) 232.636 0.120553
\(156\) 0 0
\(157\) −1604.54 −0.815644 −0.407822 0.913062i \(-0.633712\pi\)
−0.407822 + 0.913062i \(0.633712\pi\)
\(158\) 0 0
\(159\) 877.841 0.437845
\(160\) 0 0
\(161\) 947.119 0.463624
\(162\) 0 0
\(163\) 143.755 0.0690785 0.0345393 0.999403i \(-0.489004\pi\)
0.0345393 + 0.999403i \(0.489004\pi\)
\(164\) 0 0
\(165\) 462.671 0.218296
\(166\) 0 0
\(167\) 486.932 0.225628 0.112814 0.993616i \(-0.464014\pi\)
0.112814 + 0.993616i \(0.464014\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 962.621 0.430488
\(172\) 0 0
\(173\) 2697.35 1.18541 0.592703 0.805421i \(-0.298061\pi\)
0.592703 + 0.805421i \(0.298061\pi\)
\(174\) 0 0
\(175\) 748.436 0.323294
\(176\) 0 0
\(177\) 1118.54 0.474998
\(178\) 0 0
\(179\) 2971.14 1.24063 0.620316 0.784352i \(-0.287004\pi\)
0.620316 + 0.784352i \(0.287004\pi\)
\(180\) 0 0
\(181\) 1247.45 0.512276 0.256138 0.966640i \(-0.417550\pi\)
0.256138 + 0.966640i \(0.417550\pi\)
\(182\) 0 0
\(183\) 944.802 0.381649
\(184\) 0 0
\(185\) −229.663 −0.0912712
\(186\) 0 0
\(187\) −790.755 −0.309228
\(188\) 0 0
\(189\) 808.311 0.311090
\(190\) 0 0
\(191\) 2222.04 0.841785 0.420892 0.907111i \(-0.361717\pi\)
0.420892 + 0.907111i \(0.361717\pi\)
\(192\) 0 0
\(193\) 918.417 0.342534 0.171267 0.985225i \(-0.445214\pi\)
0.171267 + 0.985225i \(0.445214\pi\)
\(194\) 0 0
\(195\) −195.000 −0.0716115
\(196\) 0 0
\(197\) −3420.25 −1.23697 −0.618485 0.785797i \(-0.712253\pi\)
−0.618485 + 0.785797i \(0.712253\pi\)
\(198\) 0 0
\(199\) 2900.99 1.03339 0.516697 0.856168i \(-0.327161\pi\)
0.516697 + 0.856168i \(0.327161\pi\)
\(200\) 0 0
\(201\) −77.7340 −0.0272783
\(202\) 0 0
\(203\) −72.7704 −0.0251600
\(204\) 0 0
\(205\) 1407.74 0.479613
\(206\) 0 0
\(207\) 284.729 0.0956042
\(208\) 0 0
\(209\) −3299.09 −1.09188
\(210\) 0 0
\(211\) 4932.98 1.60948 0.804741 0.593627i \(-0.202304\pi\)
0.804741 + 0.593627i \(0.202304\pi\)
\(212\) 0 0
\(213\) 102.180 0.0328699
\(214\) 0 0
\(215\) −66.0168 −0.0209410
\(216\) 0 0
\(217\) −1392.90 −0.435744
\(218\) 0 0
\(219\) 3479.61 1.07365
\(220\) 0 0
\(221\) 333.276 0.101441
\(222\) 0 0
\(223\) −1628.76 −0.489102 −0.244551 0.969636i \(-0.578641\pi\)
−0.244551 + 0.969636i \(0.578641\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 3633.27 1.06233 0.531164 0.847269i \(-0.321755\pi\)
0.531164 + 0.847269i \(0.321755\pi\)
\(228\) 0 0
\(229\) −4430.06 −1.27837 −0.639185 0.769053i \(-0.720728\pi\)
−0.639185 + 0.769053i \(0.720728\pi\)
\(230\) 0 0
\(231\) −2770.24 −0.789040
\(232\) 0 0
\(233\) −1647.65 −0.463266 −0.231633 0.972803i \(-0.574407\pi\)
−0.231633 + 0.972803i \(0.574407\pi\)
\(234\) 0 0
\(235\) 304.862 0.0846255
\(236\) 0 0
\(237\) −1634.41 −0.447960
\(238\) 0 0
\(239\) −1678.88 −0.454385 −0.227193 0.973850i \(-0.572955\pi\)
−0.227193 + 0.973850i \(0.572955\pi\)
\(240\) 0 0
\(241\) 3398.22 0.908292 0.454146 0.890927i \(-0.349944\pi\)
0.454146 + 0.890927i \(0.349944\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −2766.25 −0.721344
\(246\) 0 0
\(247\) 1390.45 0.358188
\(248\) 0 0
\(249\) −2981.98 −0.758937
\(250\) 0 0
\(251\) 4366.48 1.09805 0.549023 0.835807i \(-0.315000\pi\)
0.549023 + 0.835807i \(0.315000\pi\)
\(252\) 0 0
\(253\) −975.823 −0.242488
\(254\) 0 0
\(255\) −384.549 −0.0944368
\(256\) 0 0
\(257\) 1403.98 0.340770 0.170385 0.985378i \(-0.445499\pi\)
0.170385 + 0.985378i \(0.445499\pi\)
\(258\) 0 0
\(259\) 1375.11 0.329903
\(260\) 0 0
\(261\) −21.8767 −0.00518826
\(262\) 0 0
\(263\) 2903.66 0.680788 0.340394 0.940283i \(-0.389440\pi\)
0.340394 + 0.940283i \(0.389440\pi\)
\(264\) 0 0
\(265\) −1463.07 −0.339153
\(266\) 0 0
\(267\) 1585.26 0.363358
\(268\) 0 0
\(269\) 1637.84 0.371229 0.185614 0.982623i \(-0.440572\pi\)
0.185614 + 0.982623i \(0.440572\pi\)
\(270\) 0 0
\(271\) 2030.84 0.455221 0.227610 0.973752i \(-0.426909\pi\)
0.227610 + 0.973752i \(0.426909\pi\)
\(272\) 0 0
\(273\) 1167.56 0.258842
\(274\) 0 0
\(275\) −771.119 −0.169092
\(276\) 0 0
\(277\) −1272.98 −0.276122 −0.138061 0.990424i \(-0.544087\pi\)
−0.138061 + 0.990424i \(0.544087\pi\)
\(278\) 0 0
\(279\) −418.745 −0.0898551
\(280\) 0 0
\(281\) −4926.47 −1.04587 −0.522933 0.852374i \(-0.675162\pi\)
−0.522933 + 0.852374i \(0.675162\pi\)
\(282\) 0 0
\(283\) 3623.18 0.761044 0.380522 0.924772i \(-0.375744\pi\)
0.380522 + 0.924772i \(0.375744\pi\)
\(284\) 0 0
\(285\) −1604.37 −0.333455
\(286\) 0 0
\(287\) −8428.81 −1.73358
\(288\) 0 0
\(289\) −4255.76 −0.866225
\(290\) 0 0
\(291\) 2783.19 0.560665
\(292\) 0 0
\(293\) 7355.41 1.46658 0.733290 0.679916i \(-0.237984\pi\)
0.733290 + 0.679916i \(0.237984\pi\)
\(294\) 0 0
\(295\) −1864.24 −0.367932
\(296\) 0 0
\(297\) −832.808 −0.162709
\(298\) 0 0
\(299\) 411.276 0.0795475
\(300\) 0 0
\(301\) 395.275 0.0756919
\(302\) 0 0
\(303\) 286.872 0.0543907
\(304\) 0 0
\(305\) −1574.67 −0.295624
\(306\) 0 0
\(307\) −3414.14 −0.634707 −0.317353 0.948307i \(-0.602794\pi\)
−0.317353 + 0.948307i \(0.602794\pi\)
\(308\) 0 0
\(309\) −1864.92 −0.343339
\(310\) 0 0
\(311\) −3096.13 −0.564519 −0.282259 0.959338i \(-0.591084\pi\)
−0.282259 + 0.959338i \(0.591084\pi\)
\(312\) 0 0
\(313\) −8912.59 −1.60949 −0.804743 0.593623i \(-0.797697\pi\)
−0.804743 + 0.593623i \(0.797697\pi\)
\(314\) 0 0
\(315\) −1347.18 −0.240969
\(316\) 0 0
\(317\) 8748.65 1.55007 0.775036 0.631917i \(-0.217732\pi\)
0.775036 + 0.631917i \(0.217732\pi\)
\(318\) 0 0
\(319\) 74.9758 0.0131594
\(320\) 0 0
\(321\) −655.135 −0.113913
\(322\) 0 0
\(323\) 2742.04 0.472356
\(324\) 0 0
\(325\) 325.000 0.0554700
\(326\) 0 0
\(327\) 3580.66 0.605539
\(328\) 0 0
\(329\) −1825.36 −0.305882
\(330\) 0 0
\(331\) −2029.88 −0.337077 −0.168538 0.985695i \(-0.553905\pi\)
−0.168538 + 0.985695i \(0.553905\pi\)
\(332\) 0 0
\(333\) 413.394 0.0680296
\(334\) 0 0
\(335\) 129.557 0.0211297
\(336\) 0 0
\(337\) 8400.49 1.35787 0.678937 0.734196i \(-0.262441\pi\)
0.678937 + 0.734196i \(0.262441\pi\)
\(338\) 0 0
\(339\) −3463.02 −0.554824
\(340\) 0 0
\(341\) 1435.12 0.227906
\(342\) 0 0
\(343\) 6294.36 0.990856
\(344\) 0 0
\(345\) −474.549 −0.0740547
\(346\) 0 0
\(347\) 4996.96 0.773057 0.386528 0.922277i \(-0.373674\pi\)
0.386528 + 0.922277i \(0.373674\pi\)
\(348\) 0 0
\(349\) −7433.65 −1.14015 −0.570077 0.821591i \(-0.693087\pi\)
−0.570077 + 0.821591i \(0.693087\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −1136.59 −0.171373 −0.0856863 0.996322i \(-0.527308\pi\)
−0.0856863 + 0.996322i \(0.527308\pi\)
\(354\) 0 0
\(355\) −170.301 −0.0254609
\(356\) 0 0
\(357\) 2302.48 0.341345
\(358\) 0 0
\(359\) 2803.47 0.412149 0.206074 0.978536i \(-0.433931\pi\)
0.206074 + 0.978536i \(0.433931\pi\)
\(360\) 0 0
\(361\) 4581.00 0.667881
\(362\) 0 0
\(363\) −1138.80 −0.164660
\(364\) 0 0
\(365\) −5799.35 −0.831649
\(366\) 0 0
\(367\) −1970.87 −0.280323 −0.140161 0.990129i \(-0.544762\pi\)
−0.140161 + 0.990129i \(0.544762\pi\)
\(368\) 0 0
\(369\) −2533.93 −0.357482
\(370\) 0 0
\(371\) 8760.11 1.22588
\(372\) 0 0
\(373\) 217.387 0.0301766 0.0150883 0.999886i \(-0.495197\pi\)
0.0150883 + 0.999886i \(0.495197\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) −31.5997 −0.00431689
\(378\) 0 0
\(379\) 5166.63 0.700243 0.350121 0.936704i \(-0.386140\pi\)
0.350121 + 0.936704i \(0.386140\pi\)
\(380\) 0 0
\(381\) −2130.95 −0.286540
\(382\) 0 0
\(383\) −11149.3 −1.48747 −0.743736 0.668473i \(-0.766948\pi\)
−0.743736 + 0.668473i \(0.766948\pi\)
\(384\) 0 0
\(385\) 4617.07 0.611188
\(386\) 0 0
\(387\) 118.830 0.0156085
\(388\) 0 0
\(389\) −11092.0 −1.44573 −0.722863 0.690992i \(-0.757174\pi\)
−0.722863 + 0.690992i \(0.757174\pi\)
\(390\) 0 0
\(391\) 811.055 0.104902
\(392\) 0 0
\(393\) 1088.64 0.139732
\(394\) 0 0
\(395\) 2724.02 0.346988
\(396\) 0 0
\(397\) 8055.67 1.01839 0.509197 0.860650i \(-0.329942\pi\)
0.509197 + 0.860650i \(0.329942\pi\)
\(398\) 0 0
\(399\) 9606.14 1.20528
\(400\) 0 0
\(401\) −3318.85 −0.413305 −0.206652 0.978414i \(-0.566257\pi\)
−0.206652 + 0.978414i \(0.566257\pi\)
\(402\) 0 0
\(403\) −604.853 −0.0747640
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) −1416.78 −0.172548
\(408\) 0 0
\(409\) −9684.08 −1.17077 −0.585387 0.810754i \(-0.699058\pi\)
−0.585387 + 0.810754i \(0.699058\pi\)
\(410\) 0 0
\(411\) −2506.78 −0.300853
\(412\) 0 0
\(413\) 11162.1 1.32990
\(414\) 0 0
\(415\) 4969.97 0.587870
\(416\) 0 0
\(417\) 7618.36 0.894658
\(418\) 0 0
\(419\) −641.245 −0.0747658 −0.0373829 0.999301i \(-0.511902\pi\)
−0.0373829 + 0.999301i \(0.511902\pi\)
\(420\) 0 0
\(421\) 12734.8 1.47424 0.737120 0.675762i \(-0.236185\pi\)
0.737120 + 0.675762i \(0.236185\pi\)
\(422\) 0 0
\(423\) −548.751 −0.0630761
\(424\) 0 0
\(425\) 640.915 0.0731505
\(426\) 0 0
\(427\) 9428.31 1.06854
\(428\) 0 0
\(429\) −1202.95 −0.135382
\(430\) 0 0
\(431\) 230.903 0.0258055 0.0129028 0.999917i \(-0.495893\pi\)
0.0129028 + 0.999917i \(0.495893\pi\)
\(432\) 0 0
\(433\) 4530.51 0.502823 0.251411 0.967880i \(-0.419105\pi\)
0.251411 + 0.967880i \(0.419105\pi\)
\(434\) 0 0
\(435\) 36.4612 0.00401881
\(436\) 0 0
\(437\) 3383.79 0.370408
\(438\) 0 0
\(439\) −4862.75 −0.528671 −0.264336 0.964431i \(-0.585153\pi\)
−0.264336 + 0.964431i \(0.585153\pi\)
\(440\) 0 0
\(441\) 4979.25 0.537658
\(442\) 0 0
\(443\) −12774.9 −1.37010 −0.685048 0.728498i \(-0.740219\pi\)
−0.685048 + 0.728498i \(0.740219\pi\)
\(444\) 0 0
\(445\) −2642.10 −0.281456
\(446\) 0 0
\(447\) 6936.08 0.733927
\(448\) 0 0
\(449\) 8513.83 0.894861 0.447431 0.894319i \(-0.352339\pi\)
0.447431 + 0.894319i \(0.352339\pi\)
\(450\) 0 0
\(451\) 8684.27 0.906710
\(452\) 0 0
\(453\) −876.016 −0.0908583
\(454\) 0 0
\(455\) −1945.93 −0.200498
\(456\) 0 0
\(457\) −14065.5 −1.43973 −0.719865 0.694114i \(-0.755796\pi\)
−0.719865 + 0.694114i \(0.755796\pi\)
\(458\) 0 0
\(459\) 692.188 0.0703891
\(460\) 0 0
\(461\) 2412.53 0.243737 0.121869 0.992546i \(-0.461111\pi\)
0.121869 + 0.992546i \(0.461111\pi\)
\(462\) 0 0
\(463\) 10195.6 1.02339 0.511695 0.859167i \(-0.329018\pi\)
0.511695 + 0.859167i \(0.329018\pi\)
\(464\) 0 0
\(465\) 697.908 0.0696015
\(466\) 0 0
\(467\) −2086.70 −0.206769 −0.103384 0.994641i \(-0.532967\pi\)
−0.103384 + 0.994641i \(0.532967\pi\)
\(468\) 0 0
\(469\) −775.719 −0.0763739
\(470\) 0 0
\(471\) −4813.61 −0.470912
\(472\) 0 0
\(473\) −407.254 −0.0395889
\(474\) 0 0
\(475\) 2673.95 0.258293
\(476\) 0 0
\(477\) 2633.52 0.252790
\(478\) 0 0
\(479\) 10327.2 0.985098 0.492549 0.870285i \(-0.336065\pi\)
0.492549 + 0.870285i \(0.336065\pi\)
\(480\) 0 0
\(481\) 597.124 0.0566040
\(482\) 0 0
\(483\) 2841.36 0.267673
\(484\) 0 0
\(485\) −4638.65 −0.434289
\(486\) 0 0
\(487\) 13131.8 1.22188 0.610942 0.791676i \(-0.290791\pi\)
0.610942 + 0.791676i \(0.290791\pi\)
\(488\) 0 0
\(489\) 431.266 0.0398825
\(490\) 0 0
\(491\) −5201.34 −0.478071 −0.239036 0.971011i \(-0.576831\pi\)
−0.239036 + 0.971011i \(0.576831\pi\)
\(492\) 0 0
\(493\) −62.3161 −0.00569285
\(494\) 0 0
\(495\) 1388.01 0.126034
\(496\) 0 0
\(497\) 1019.67 0.0920295
\(498\) 0 0
\(499\) −11498.6 −1.03156 −0.515781 0.856721i \(-0.672498\pi\)
−0.515781 + 0.856721i \(0.672498\pi\)
\(500\) 0 0
\(501\) 1460.80 0.130267
\(502\) 0 0
\(503\) 4117.25 0.364969 0.182484 0.983209i \(-0.441586\pi\)
0.182484 + 0.983209i \(0.441586\pi\)
\(504\) 0 0
\(505\) −478.121 −0.0421309
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) −3110.48 −0.270864 −0.135432 0.990787i \(-0.543242\pi\)
−0.135432 + 0.990787i \(0.543242\pi\)
\(510\) 0 0
\(511\) 34723.6 3.00603
\(512\) 0 0
\(513\) 2887.86 0.248543
\(514\) 0 0
\(515\) 3108.21 0.265949
\(516\) 0 0
\(517\) 1880.68 0.159985
\(518\) 0 0
\(519\) 8092.04 0.684395
\(520\) 0 0
\(521\) −18391.6 −1.54654 −0.773272 0.634074i \(-0.781381\pi\)
−0.773272 + 0.634074i \(0.781381\pi\)
\(522\) 0 0
\(523\) −2462.46 −0.205881 −0.102941 0.994688i \(-0.532825\pi\)
−0.102941 + 0.994688i \(0.532825\pi\)
\(524\) 0 0
\(525\) 2245.31 0.186654
\(526\) 0 0
\(527\) −1192.80 −0.0985942
\(528\) 0 0
\(529\) −11166.1 −0.917739
\(530\) 0 0
\(531\) 3355.62 0.274240
\(532\) 0 0
\(533\) −3660.12 −0.297443
\(534\) 0 0
\(535\) 1091.89 0.0882366
\(536\) 0 0
\(537\) 8913.41 0.716280
\(538\) 0 0
\(539\) −17064.9 −1.36370
\(540\) 0 0
\(541\) −6060.86 −0.481658 −0.240829 0.970568i \(-0.577419\pi\)
−0.240829 + 0.970568i \(0.577419\pi\)
\(542\) 0 0
\(543\) 3742.34 0.295763
\(544\) 0 0
\(545\) −5967.77 −0.469048
\(546\) 0 0
\(547\) −8474.72 −0.662437 −0.331218 0.943554i \(-0.607460\pi\)
−0.331218 + 0.943554i \(0.607460\pi\)
\(548\) 0 0
\(549\) 2834.40 0.220345
\(550\) 0 0
\(551\) −259.988 −0.0201014
\(552\) 0 0
\(553\) −16310.0 −1.25420
\(554\) 0 0
\(555\) −688.990 −0.0526955
\(556\) 0 0
\(557\) −16771.5 −1.27582 −0.637911 0.770110i \(-0.720201\pi\)
−0.637911 + 0.770110i \(0.720201\pi\)
\(558\) 0 0
\(559\) 171.644 0.0129870
\(560\) 0 0
\(561\) −2372.26 −0.178533
\(562\) 0 0
\(563\) −20461.1 −1.53167 −0.765837 0.643035i \(-0.777675\pi\)
−0.765837 + 0.643035i \(0.777675\pi\)
\(564\) 0 0
\(565\) 5771.70 0.429765
\(566\) 0 0
\(567\) 2424.93 0.179608
\(568\) 0 0
\(569\) −25998.0 −1.91546 −0.957728 0.287675i \(-0.907118\pi\)
−0.957728 + 0.287675i \(0.907118\pi\)
\(570\) 0 0
\(571\) 24951.9 1.82873 0.914366 0.404889i \(-0.132690\pi\)
0.914366 + 0.404889i \(0.132690\pi\)
\(572\) 0 0
\(573\) 6666.11 0.486005
\(574\) 0 0
\(575\) 790.915 0.0573625
\(576\) 0 0
\(577\) −11833.9 −0.853816 −0.426908 0.904295i \(-0.640397\pi\)
−0.426908 + 0.904295i \(0.640397\pi\)
\(578\) 0 0
\(579\) 2755.25 0.197762
\(580\) 0 0
\(581\) −29757.6 −2.12488
\(582\) 0 0
\(583\) −9025.60 −0.641170
\(584\) 0 0
\(585\) −585.000 −0.0413449
\(586\) 0 0
\(587\) −592.026 −0.0416278 −0.0208139 0.999783i \(-0.506626\pi\)
−0.0208139 + 0.999783i \(0.506626\pi\)
\(588\) 0 0
\(589\) −4976.45 −0.348134
\(590\) 0 0
\(591\) −10260.8 −0.714165
\(592\) 0 0
\(593\) 1829.53 0.126694 0.0633472 0.997992i \(-0.479822\pi\)
0.0633472 + 0.997992i \(0.479822\pi\)
\(594\) 0 0
\(595\) −3837.47 −0.264405
\(596\) 0 0
\(597\) 8702.96 0.596631
\(598\) 0 0
\(599\) −28333.7 −1.93269 −0.966346 0.257245i \(-0.917185\pi\)
−0.966346 + 0.257245i \(0.917185\pi\)
\(600\) 0 0
\(601\) −24753.2 −1.68004 −0.840022 0.542553i \(-0.817458\pi\)
−0.840022 + 0.542553i \(0.817458\pi\)
\(602\) 0 0
\(603\) −233.202 −0.0157491
\(604\) 0 0
\(605\) 1898.01 0.127545
\(606\) 0 0
\(607\) 7302.18 0.488280 0.244140 0.969740i \(-0.421494\pi\)
0.244140 + 0.969740i \(0.421494\pi\)
\(608\) 0 0
\(609\) −218.311 −0.0145261
\(610\) 0 0
\(611\) −792.641 −0.0524825
\(612\) 0 0
\(613\) −26820.0 −1.76713 −0.883563 0.468312i \(-0.844862\pi\)
−0.883563 + 0.468312i \(0.844862\pi\)
\(614\) 0 0
\(615\) 4223.21 0.276905
\(616\) 0 0
\(617\) −5701.67 −0.372027 −0.186013 0.982547i \(-0.559557\pi\)
−0.186013 + 0.982547i \(0.559557\pi\)
\(618\) 0 0
\(619\) −9385.86 −0.609450 −0.304725 0.952440i \(-0.598565\pi\)
−0.304725 + 0.952440i \(0.598565\pi\)
\(620\) 0 0
\(621\) 854.188 0.0551971
\(622\) 0 0
\(623\) 15819.6 1.01733
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −9897.27 −0.630397
\(628\) 0 0
\(629\) 1177.56 0.0746459
\(630\) 0 0
\(631\) 1551.34 0.0978730 0.0489365 0.998802i \(-0.484417\pi\)
0.0489365 + 0.998802i \(0.484417\pi\)
\(632\) 0 0
\(633\) 14798.9 0.929234
\(634\) 0 0
\(635\) 3551.58 0.221953
\(636\) 0 0
\(637\) 7192.26 0.447359
\(638\) 0 0
\(639\) 306.541 0.0189775
\(640\) 0 0
\(641\) 3746.46 0.230852 0.115426 0.993316i \(-0.463177\pi\)
0.115426 + 0.993316i \(0.463177\pi\)
\(642\) 0 0
\(643\) −9748.79 −0.597908 −0.298954 0.954268i \(-0.596638\pi\)
−0.298954 + 0.954268i \(0.596638\pi\)
\(644\) 0 0
\(645\) −198.050 −0.0120903
\(646\) 0 0
\(647\) 22052.5 1.33999 0.669995 0.742366i \(-0.266296\pi\)
0.669995 + 0.742366i \(0.266296\pi\)
\(648\) 0 0
\(649\) −11500.4 −0.695577
\(650\) 0 0
\(651\) −4178.71 −0.251577
\(652\) 0 0
\(653\) −14505.6 −0.869293 −0.434646 0.900601i \(-0.643127\pi\)
−0.434646 + 0.900601i \(0.643127\pi\)
\(654\) 0 0
\(655\) −1814.40 −0.108236
\(656\) 0 0
\(657\) 10438.8 0.619875
\(658\) 0 0
\(659\) −24976.9 −1.47642 −0.738210 0.674571i \(-0.764329\pi\)
−0.738210 + 0.674571i \(0.764329\pi\)
\(660\) 0 0
\(661\) 29608.3 1.74225 0.871126 0.491059i \(-0.163390\pi\)
0.871126 + 0.491059i \(0.163390\pi\)
\(662\) 0 0
\(663\) 999.828 0.0585672
\(664\) 0 0
\(665\) −16010.2 −0.933609
\(666\) 0 0
\(667\) −76.9006 −0.00446417
\(668\) 0 0
\(669\) −4886.28 −0.282383
\(670\) 0 0
\(671\) −9714.06 −0.558878
\(672\) 0 0
\(673\) −17420.1 −0.997762 −0.498881 0.866671i \(-0.666256\pi\)
−0.498881 + 0.866671i \(0.666256\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) −18655.3 −1.05906 −0.529529 0.848292i \(-0.677631\pi\)
−0.529529 + 0.848292i \(0.677631\pi\)
\(678\) 0 0
\(679\) 27773.9 1.56975
\(680\) 0 0
\(681\) 10899.8 0.613336
\(682\) 0 0
\(683\) −29518.2 −1.65371 −0.826855 0.562415i \(-0.809872\pi\)
−0.826855 + 0.562415i \(0.809872\pi\)
\(684\) 0 0
\(685\) 4177.97 0.233039
\(686\) 0 0
\(687\) −13290.2 −0.738067
\(688\) 0 0
\(689\) 3803.98 0.210334
\(690\) 0 0
\(691\) 6814.27 0.375148 0.187574 0.982250i \(-0.439938\pi\)
0.187574 + 0.982250i \(0.439938\pi\)
\(692\) 0 0
\(693\) −8310.72 −0.455553
\(694\) 0 0
\(695\) −12697.3 −0.692999
\(696\) 0 0
\(697\) −7217.92 −0.392250
\(698\) 0 0
\(699\) −4942.94 −0.267467
\(700\) 0 0
\(701\) −13833.0 −0.745314 −0.372657 0.927969i \(-0.621553\pi\)
−0.372657 + 0.927969i \(0.621553\pi\)
\(702\) 0 0
\(703\) 4912.86 0.263573
\(704\) 0 0
\(705\) 914.586 0.0488586
\(706\) 0 0
\(707\) 2862.74 0.152284
\(708\) 0 0
\(709\) −14817.0 −0.784860 −0.392430 0.919782i \(-0.628365\pi\)
−0.392430 + 0.919782i \(0.628365\pi\)
\(710\) 0 0
\(711\) −4903.24 −0.258630
\(712\) 0 0
\(713\) −1471.96 −0.0773147
\(714\) 0 0
\(715\) 2004.91 0.104866
\(716\) 0 0
\(717\) −5036.65 −0.262339
\(718\) 0 0
\(719\) −5475.18 −0.283992 −0.141996 0.989867i \(-0.545352\pi\)
−0.141996 + 0.989867i \(0.545352\pi\)
\(720\) 0 0
\(721\) −18610.4 −0.961284
\(722\) 0 0
\(723\) 10194.7 0.524403
\(724\) 0 0
\(725\) −60.7687 −0.00311296
\(726\) 0 0
\(727\) 1245.89 0.0635593 0.0317796 0.999495i \(-0.489883\pi\)
0.0317796 + 0.999495i \(0.489883\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 338.489 0.0171265
\(732\) 0 0
\(733\) −5575.21 −0.280935 −0.140467 0.990085i \(-0.544861\pi\)
−0.140467 + 0.990085i \(0.544861\pi\)
\(734\) 0 0
\(735\) −8298.76 −0.416468
\(736\) 0 0
\(737\) 799.228 0.0399457
\(738\) 0 0
\(739\) −14543.3 −0.723930 −0.361965 0.932192i \(-0.617894\pi\)
−0.361965 + 0.932192i \(0.617894\pi\)
\(740\) 0 0
\(741\) 4171.36 0.206800
\(742\) 0 0
\(743\) 27422.3 1.35400 0.677002 0.735981i \(-0.263279\pi\)
0.677002 + 0.735981i \(0.263279\pi\)
\(744\) 0 0
\(745\) −11560.1 −0.568498
\(746\) 0 0
\(747\) −8945.94 −0.438172
\(748\) 0 0
\(749\) −6537.69 −0.318934
\(750\) 0 0
\(751\) 22289.4 1.08302 0.541512 0.840693i \(-0.317852\pi\)
0.541512 + 0.840693i \(0.317852\pi\)
\(752\) 0 0
\(753\) 13099.4 0.633957
\(754\) 0 0
\(755\) 1460.03 0.0703785
\(756\) 0 0
\(757\) −23198.2 −1.11381 −0.556904 0.830577i \(-0.688011\pi\)
−0.556904 + 0.830577i \(0.688011\pi\)
\(758\) 0 0
\(759\) −2927.47 −0.140001
\(760\) 0 0
\(761\) −7255.28 −0.345603 −0.172801 0.984957i \(-0.555282\pi\)
−0.172801 + 0.984957i \(0.555282\pi\)
\(762\) 0 0
\(763\) 35732.0 1.69539
\(764\) 0 0
\(765\) −1153.65 −0.0545231
\(766\) 0 0
\(767\) 4847.01 0.228182
\(768\) 0 0
\(769\) −3230.75 −0.151500 −0.0757502 0.997127i \(-0.524135\pi\)
−0.0757502 + 0.997127i \(0.524135\pi\)
\(770\) 0 0
\(771\) 4211.94 0.196743
\(772\) 0 0
\(773\) 23646.1 1.10025 0.550123 0.835084i \(-0.314581\pi\)
0.550123 + 0.835084i \(0.314581\pi\)
\(774\) 0 0
\(775\) −1163.18 −0.0539131
\(776\) 0 0
\(777\) 4125.32 0.190470
\(778\) 0 0
\(779\) −30113.7 −1.38503
\(780\) 0 0
\(781\) −1050.58 −0.0481339
\(782\) 0 0
\(783\) −65.6302 −0.00299544
\(784\) 0 0
\(785\) 8022.69 0.364767
\(786\) 0 0
\(787\) 4303.98 0.194943 0.0974717 0.995238i \(-0.468924\pi\)
0.0974717 + 0.995238i \(0.468924\pi\)
\(788\) 0 0
\(789\) 8710.97 0.393053
\(790\) 0 0
\(791\) −34558.0 −1.55340
\(792\) 0 0
\(793\) 4094.14 0.183338
\(794\) 0 0
\(795\) −4389.21 −0.195810
\(796\) 0 0
\(797\) −14133.4 −0.628142 −0.314071 0.949399i \(-0.601693\pi\)
−0.314071 + 0.949399i \(0.601693\pi\)
\(798\) 0 0
\(799\) −1563.12 −0.0692107
\(800\) 0 0
\(801\) 4755.79 0.209785
\(802\) 0 0
\(803\) −35775.9 −1.57224
\(804\) 0 0
\(805\) −4735.60 −0.207339
\(806\) 0 0
\(807\) 4913.51 0.214329
\(808\) 0 0
\(809\) −8149.34 −0.354160 −0.177080 0.984196i \(-0.556665\pi\)
−0.177080 + 0.984196i \(0.556665\pi\)
\(810\) 0 0
\(811\) −762.204 −0.0330020 −0.0165010 0.999864i \(-0.505253\pi\)
−0.0165010 + 0.999864i \(0.505253\pi\)
\(812\) 0 0
\(813\) 6092.53 0.262822
\(814\) 0 0
\(815\) −718.777 −0.0308929
\(816\) 0 0
\(817\) 1412.20 0.0604734
\(818\) 0 0
\(819\) 3502.68 0.149443
\(820\) 0 0
\(821\) −15118.9 −0.642694 −0.321347 0.946961i \(-0.604136\pi\)
−0.321347 + 0.946961i \(0.604136\pi\)
\(822\) 0 0
\(823\) 46721.1 1.97885 0.989426 0.145038i \(-0.0463303\pi\)
0.989426 + 0.145038i \(0.0463303\pi\)
\(824\) 0 0
\(825\) −2313.36 −0.0976251
\(826\) 0 0
\(827\) −667.106 −0.0280502 −0.0140251 0.999902i \(-0.504464\pi\)
−0.0140251 + 0.999902i \(0.504464\pi\)
\(828\) 0 0
\(829\) 14816.1 0.620728 0.310364 0.950618i \(-0.399549\pi\)
0.310364 + 0.950618i \(0.399549\pi\)
\(830\) 0 0
\(831\) −3818.93 −0.159419
\(832\) 0 0
\(833\) 14183.5 0.589949
\(834\) 0 0
\(835\) −2434.66 −0.100904
\(836\) 0 0
\(837\) −1256.23 −0.0518779
\(838\) 0 0
\(839\) −14951.3 −0.615226 −0.307613 0.951511i \(-0.599530\pi\)
−0.307613 + 0.951511i \(0.599530\pi\)
\(840\) 0 0
\(841\) −24383.1 −0.999758
\(842\) 0 0
\(843\) −14779.4 −0.603831
\(844\) 0 0
\(845\) −845.000 −0.0344010
\(846\) 0 0
\(847\) −11364.3 −0.461017
\(848\) 0 0
\(849\) 10869.5 0.439389
\(850\) 0 0
\(851\) 1453.15 0.0585352
\(852\) 0 0
\(853\) 38357.3 1.53966 0.769829 0.638251i \(-0.220342\pi\)
0.769829 + 0.638251i \(0.220342\pi\)
\(854\) 0 0
\(855\) −4813.11 −0.192520
\(856\) 0 0
\(857\) −30786.7 −1.22713 −0.613566 0.789643i \(-0.710266\pi\)
−0.613566 + 0.789643i \(0.710266\pi\)
\(858\) 0 0
\(859\) −34100.2 −1.35446 −0.677231 0.735770i \(-0.736820\pi\)
−0.677231 + 0.735770i \(0.736820\pi\)
\(860\) 0 0
\(861\) −25286.4 −1.00088
\(862\) 0 0
\(863\) 17823.0 0.703014 0.351507 0.936185i \(-0.385669\pi\)
0.351507 + 0.936185i \(0.385669\pi\)
\(864\) 0 0
\(865\) −13486.7 −0.530130
\(866\) 0 0
\(867\) −12767.3 −0.500115
\(868\) 0 0
\(869\) 16804.3 0.655982
\(870\) 0 0
\(871\) −336.847 −0.0131041
\(872\) 0 0
\(873\) 8349.57 0.323700
\(874\) 0 0
\(875\) −3742.18 −0.144581
\(876\) 0 0
\(877\) 43937.2 1.69174 0.845870 0.533390i \(-0.179082\pi\)
0.845870 + 0.533390i \(0.179082\pi\)
\(878\) 0 0
\(879\) 22066.2 0.846730
\(880\) 0 0
\(881\) 21509.0 0.822540 0.411270 0.911514i \(-0.365085\pi\)
0.411270 + 0.911514i \(0.365085\pi\)
\(882\) 0 0
\(883\) 22046.4 0.840228 0.420114 0.907471i \(-0.361990\pi\)
0.420114 + 0.907471i \(0.361990\pi\)
\(884\) 0 0
\(885\) −5592.71 −0.212426
\(886\) 0 0
\(887\) −9604.36 −0.363566 −0.181783 0.983339i \(-0.558187\pi\)
−0.181783 + 0.983339i \(0.558187\pi\)
\(888\) 0 0
\(889\) −21265.0 −0.802257
\(890\) 0 0
\(891\) −2498.43 −0.0939398
\(892\) 0 0
\(893\) −6521.48 −0.244382
\(894\) 0 0
\(895\) −14855.7 −0.554828
\(896\) 0 0
\(897\) 1233.83 0.0459268
\(898\) 0 0
\(899\) 113.096 0.00419572
\(900\) 0 0
\(901\) 7501.62 0.277375
\(902\) 0 0
\(903\) 1185.82 0.0437007
\(904\) 0 0
\(905\) −6237.23 −0.229097
\(906\) 0 0
\(907\) −17819.4 −0.652353 −0.326177 0.945309i \(-0.605760\pi\)
−0.326177 + 0.945309i \(0.605760\pi\)
\(908\) 0 0
\(909\) 860.617 0.0314025
\(910\) 0 0
\(911\) −10034.7 −0.364944 −0.182472 0.983211i \(-0.558410\pi\)
−0.182472 + 0.983211i \(0.558410\pi\)
\(912\) 0 0
\(913\) 30659.5 1.11137
\(914\) 0 0
\(915\) −4724.01 −0.170679
\(916\) 0 0
\(917\) 10863.7 0.391223
\(918\) 0 0
\(919\) −54693.5 −1.96319 −0.981595 0.190976i \(-0.938835\pi\)
−0.981595 + 0.190976i \(0.938835\pi\)
\(920\) 0 0
\(921\) −10242.4 −0.366448
\(922\) 0 0
\(923\) 442.782 0.0157902
\(924\) 0 0
\(925\) 1148.32 0.0408177
\(926\) 0 0
\(927\) −5594.77 −0.198227
\(928\) 0 0
\(929\) 37434.8 1.32206 0.661031 0.750359i \(-0.270119\pi\)
0.661031 + 0.750359i \(0.270119\pi\)
\(930\) 0 0
\(931\) 59174.5 2.08310
\(932\) 0 0
\(933\) −9288.38 −0.325925
\(934\) 0 0
\(935\) 3953.77 0.138291
\(936\) 0 0
\(937\) 22416.0 0.781534 0.390767 0.920490i \(-0.372210\pi\)
0.390767 + 0.920490i \(0.372210\pi\)
\(938\) 0 0
\(939\) −26737.8 −0.929237
\(940\) 0 0
\(941\) 42374.1 1.46797 0.733983 0.679168i \(-0.237659\pi\)
0.733983 + 0.679168i \(0.237659\pi\)
\(942\) 0 0
\(943\) −8907.21 −0.307591
\(944\) 0 0
\(945\) −4041.55 −0.139124
\(946\) 0 0
\(947\) 23229.9 0.797117 0.398559 0.917143i \(-0.369511\pi\)
0.398559 + 0.917143i \(0.369511\pi\)
\(948\) 0 0
\(949\) 15078.3 0.515767
\(950\) 0 0
\(951\) 26245.9 0.894935
\(952\) 0 0
\(953\) −24063.4 −0.817933 −0.408967 0.912549i \(-0.634111\pi\)
−0.408967 + 0.912549i \(0.634111\pi\)
\(954\) 0 0
\(955\) −11110.2 −0.376458
\(956\) 0 0
\(957\) 224.927 0.00759757
\(958\) 0 0
\(959\) −25015.5 −0.842330
\(960\) 0 0
\(961\) −27626.2 −0.927335
\(962\) 0 0
\(963\) −1965.40 −0.0657677
\(964\) 0 0
\(965\) −4592.08 −0.153186
\(966\) 0 0
\(967\) −21899.6 −0.728276 −0.364138 0.931345i \(-0.618636\pi\)
−0.364138 + 0.931345i \(0.618636\pi\)
\(968\) 0 0
\(969\) 8226.11 0.272715
\(970\) 0 0
\(971\) 5125.99 0.169414 0.0847070 0.996406i \(-0.473005\pi\)
0.0847070 + 0.996406i \(0.473005\pi\)
\(972\) 0 0
\(973\) 76024.7 2.50487
\(974\) 0 0
\(975\) 975.000 0.0320256
\(976\) 0 0
\(977\) −40536.5 −1.32741 −0.663705 0.747995i \(-0.731017\pi\)
−0.663705 + 0.747995i \(0.731017\pi\)
\(978\) 0 0
\(979\) −16299.0 −0.532093
\(980\) 0 0
\(981\) 10742.0 0.349608
\(982\) 0 0
\(983\) −1634.18 −0.0530235 −0.0265118 0.999649i \(-0.508440\pi\)
−0.0265118 + 0.999649i \(0.508440\pi\)
\(984\) 0 0
\(985\) 17101.3 0.553190
\(986\) 0 0
\(987\) −5476.07 −0.176601
\(988\) 0 0
\(989\) 417.709 0.0134301
\(990\) 0 0
\(991\) −21750.2 −0.697191 −0.348596 0.937273i \(-0.613341\pi\)
−0.348596 + 0.937273i \(0.613341\pi\)
\(992\) 0 0
\(993\) −6089.65 −0.194611
\(994\) 0 0
\(995\) −14504.9 −0.462148
\(996\) 0 0
\(997\) −2618.14 −0.0831667 −0.0415834 0.999135i \(-0.513240\pi\)
−0.0415834 + 0.999135i \(0.513240\pi\)
\(998\) 0 0
\(999\) 1240.18 0.0392769
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 1560.4.a.m.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.4.a.m.1.4 4 1.1 even 1 trivial