Properties

Label 1421.4.a.e.1.4
Level $1421$
Weight $4$
Character 1421.1
Self dual yes
Analytic conductor $83.842$
Analytic rank $1$
Dimension $5$
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] = [1421,4,Mod(1,1421)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1421, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1421.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1421.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: \(83.8417141182\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.328194\) of defining polynomial
Character \(\chi\) \(=\) 1421.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24125 q^{2} -1.84328 q^{3} -2.97681 q^{4} -18.3339 q^{5} -4.13124 q^{6} -24.6017 q^{8} -23.6023 q^{9} +O(q^{10})\) \(q+2.24125 q^{2} -1.84328 q^{3} -2.97681 q^{4} -18.3339 q^{5} -4.13124 q^{6} -24.6017 q^{8} -23.6023 q^{9} -41.0908 q^{10} +52.4385 q^{11} +5.48709 q^{12} +87.5580 q^{13} +33.7945 q^{15} -31.3241 q^{16} -15.4072 q^{17} -52.8987 q^{18} -67.0156 q^{19} +54.5766 q^{20} +117.528 q^{22} +132.679 q^{23} +45.3478 q^{24} +211.133 q^{25} +196.239 q^{26} +93.2741 q^{27} -29.0000 q^{29} +75.7418 q^{30} -90.2221 q^{31} +126.609 q^{32} -96.6587 q^{33} -34.5314 q^{34} +70.2597 q^{36} +11.1247 q^{37} -150.199 q^{38} -161.394 q^{39} +451.047 q^{40} +18.8392 q^{41} -147.756 q^{43} -156.100 q^{44} +432.723 q^{45} +297.366 q^{46} -21.0963 q^{47} +57.7390 q^{48} +473.201 q^{50} +28.3997 q^{51} -260.644 q^{52} -290.454 q^{53} +209.050 q^{54} -961.404 q^{55} +123.528 q^{57} -64.9962 q^{58} +337.343 q^{59} -100.600 q^{60} -84.0147 q^{61} -202.210 q^{62} +534.355 q^{64} -1605.28 q^{65} -216.636 q^{66} +330.821 q^{67} +45.8644 q^{68} -244.564 q^{69} +492.420 q^{71} +580.659 q^{72} +347.053 q^{73} +24.9333 q^{74} -389.176 q^{75} +199.493 q^{76} -361.723 q^{78} -986.297 q^{79} +574.294 q^{80} +465.333 q^{81} +42.2234 q^{82} -594.382 q^{83} +282.475 q^{85} -331.157 q^{86} +53.4550 q^{87} -1290.08 q^{88} -1387.04 q^{89} +969.840 q^{90} -394.960 q^{92} +166.304 q^{93} -47.2820 q^{94} +1228.66 q^{95} -233.375 q^{96} +334.003 q^{97} -1237.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9} + 64 q^{10} + 12 q^{11} + 224 q^{12} - 14 q^{13} - 74 q^{15} + 146 q^{16} - 66 q^{17} - 108 q^{18} - 214 q^{19} - 6 q^{20} - 98 q^{22} + 164 q^{23} - 314 q^{24} + 207 q^{25} - 56 q^{26} - 362 q^{27} - 145 q^{29} - 234 q^{30} - 420 q^{31} - 652 q^{32} + 576 q^{33} - 204 q^{34} - 260 q^{36} + 378 q^{37} + 496 q^{38} - 374 q^{39} + 80 q^{40} + 1158 q^{41} - 204 q^{43} + 784 q^{44} + 1506 q^{45} + 580 q^{46} - 248 q^{47} + 1880 q^{48} + 908 q^{50} + 228 q^{51} - 1482 q^{52} - 554 q^{53} - 918 q^{54} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 636 q^{60} - 618 q^{61} - 1250 q^{62} + 2594 q^{64} - 1656 q^{65} - 2940 q^{66} + 1164 q^{67} - 356 q^{68} + 1968 q^{69} - 692 q^{71} - 2648 q^{72} + 1950 q^{73} - 1832 q^{74} - 3074 q^{75} - 1376 q^{76} - 1302 q^{78} + 272 q^{79} + 890 q^{80} + 1801 q^{81} - 92 q^{82} - 512 q^{83} - 1628 q^{85} + 2446 q^{86} + 232 q^{87} - 6954 q^{88} - 866 q^{89} + 2200 q^{90} + 3468 q^{92} - 40 q^{93} + 5942 q^{94} + 2244 q^{95} - 7386 q^{96} - 1562 q^{97} - 238 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24125 0.792400 0.396200 0.918164i \(-0.370329\pi\)
0.396200 + 0.918164i \(0.370329\pi\)
\(3\) −1.84328 −0.354739 −0.177369 0.984144i \(-0.556759\pi\)
−0.177369 + 0.984144i \(0.556759\pi\)
\(4\) −2.97681 −0.372101
\(5\) −18.3339 −1.63984 −0.819918 0.572481i \(-0.805981\pi\)
−0.819918 + 0.572481i \(0.805981\pi\)
\(6\) −4.13124 −0.281095
\(7\) 0 0
\(8\) −24.6017 −1.08725
\(9\) −23.6023 −0.874160
\(10\) −41.0908 −1.29941
\(11\) 52.4385 1.43735 0.718673 0.695348i \(-0.244750\pi\)
0.718673 + 0.695348i \(0.244750\pi\)
\(12\) 5.48709 0.131999
\(13\) 87.5580 1.86802 0.934008 0.357252i \(-0.116286\pi\)
0.934008 + 0.357252i \(0.116286\pi\)
\(14\) 0 0
\(15\) 33.7945 0.581713
\(16\) −31.3241 −0.489439
\(17\) −15.4072 −0.219812 −0.109906 0.993942i \(-0.535055\pi\)
−0.109906 + 0.993942i \(0.535055\pi\)
\(18\) −52.8987 −0.692685
\(19\) −67.0156 −0.809181 −0.404591 0.914498i \(-0.632586\pi\)
−0.404591 + 0.914498i \(0.632586\pi\)
\(20\) 54.5766 0.610185
\(21\) 0 0
\(22\) 117.528 1.13895
\(23\) 132.679 1.20285 0.601423 0.798931i \(-0.294601\pi\)
0.601423 + 0.798931i \(0.294601\pi\)
\(24\) 45.3478 0.385691
\(25\) 211.133 1.68906
\(26\) 196.239 1.48022
\(27\) 93.2741 0.664837
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 75.7418 0.460950
\(31\) −90.2221 −0.522721 −0.261361 0.965241i \(-0.584171\pi\)
−0.261361 + 0.965241i \(0.584171\pi\)
\(32\) 126.609 0.699422
\(33\) −96.6587 −0.509882
\(34\) −34.5314 −0.174179
\(35\) 0 0
\(36\) 70.2597 0.325276
\(37\) 11.1247 0.0494296 0.0247148 0.999695i \(-0.492132\pi\)
0.0247148 + 0.999695i \(0.492132\pi\)
\(38\) −150.199 −0.641196
\(39\) −161.394 −0.662658
\(40\) 451.047 1.78292
\(41\) 18.8392 0.0717608 0.0358804 0.999356i \(-0.488576\pi\)
0.0358804 + 0.999356i \(0.488576\pi\)
\(42\) 0 0
\(43\) −147.756 −0.524013 −0.262007 0.965066i \(-0.584384\pi\)
−0.262007 + 0.965066i \(0.584384\pi\)
\(44\) −156.100 −0.534839
\(45\) 432.723 1.43348
\(46\) 297.366 0.953136
\(47\) −21.0963 −0.0654726 −0.0327363 0.999464i \(-0.510422\pi\)
−0.0327363 + 0.999464i \(0.510422\pi\)
\(48\) 57.7390 0.173623
\(49\) 0 0
\(50\) 473.201 1.33841
\(51\) 28.3997 0.0779757
\(52\) −260.644 −0.695092
\(53\) −290.454 −0.752772 −0.376386 0.926463i \(-0.622833\pi\)
−0.376386 + 0.926463i \(0.622833\pi\)
\(54\) 209.050 0.526817
\(55\) −961.404 −2.35701
\(56\) 0 0
\(57\) 123.528 0.287048
\(58\) −64.9962 −0.147145
\(59\) 337.343 0.744379 0.372190 0.928157i \(-0.378607\pi\)
0.372190 + 0.928157i \(0.378607\pi\)
\(60\) −100.600 −0.216456
\(61\) −84.0147 −0.176344 −0.0881720 0.996105i \(-0.528103\pi\)
−0.0881720 + 0.996105i \(0.528103\pi\)
\(62\) −202.210 −0.414205
\(63\) 0 0
\(64\) 534.355 1.04366
\(65\) −1605.28 −3.06324
\(66\) −216.636 −0.404031
\(67\) 330.821 0.603228 0.301614 0.953430i \(-0.402475\pi\)
0.301614 + 0.953430i \(0.402475\pi\)
\(68\) 45.8644 0.0817922
\(69\) −244.564 −0.426696
\(70\) 0 0
\(71\) 492.420 0.823092 0.411546 0.911389i \(-0.364989\pi\)
0.411546 + 0.911389i \(0.364989\pi\)
\(72\) 580.659 0.950434
\(73\) 347.053 0.556431 0.278216 0.960519i \(-0.410257\pi\)
0.278216 + 0.960519i \(0.410257\pi\)
\(74\) 24.9333 0.0391681
\(75\) −389.176 −0.599176
\(76\) 199.493 0.301098
\(77\) 0 0
\(78\) −361.723 −0.525090
\(79\) −986.297 −1.40465 −0.702324 0.711858i \(-0.747854\pi\)
−0.702324 + 0.711858i \(0.747854\pi\)
\(80\) 574.294 0.802600
\(81\) 465.333 0.638317
\(82\) 42.2234 0.0568633
\(83\) −594.382 −0.786048 −0.393024 0.919528i \(-0.628571\pi\)
−0.393024 + 0.919528i \(0.628571\pi\)
\(84\) 0 0
\(85\) 282.475 0.360455
\(86\) −331.157 −0.415228
\(87\) 53.4550 0.0658733
\(88\) −1290.08 −1.56276
\(89\) −1387.04 −1.65197 −0.825987 0.563689i \(-0.809382\pi\)
−0.825987 + 0.563689i \(0.809382\pi\)
\(90\) 969.840 1.13589
\(91\) 0 0
\(92\) −394.960 −0.447581
\(93\) 166.304 0.185430
\(94\) −47.2820 −0.0518805
\(95\) 1228.66 1.32692
\(96\) −233.375 −0.248112
\(97\) 334.003 0.349617 0.174808 0.984602i \(-0.444069\pi\)
0.174808 + 0.984602i \(0.444069\pi\)
\(98\) 0 0
\(99\) −1237.67 −1.25647
\(100\) −628.502 −0.628502
\(101\) 245.919 0.242276 0.121138 0.992636i \(-0.461346\pi\)
0.121138 + 0.992636i \(0.461346\pi\)
\(102\) 63.6508 0.0617880
\(103\) 531.298 0.508255 0.254128 0.967171i \(-0.418212\pi\)
0.254128 + 0.967171i \(0.418212\pi\)
\(104\) −2154.08 −2.03101
\(105\) 0 0
\(106\) −650.979 −0.596497
\(107\) −429.030 −0.387625 −0.193812 0.981039i \(-0.562085\pi\)
−0.193812 + 0.981039i \(0.562085\pi\)
\(108\) −277.659 −0.247387
\(109\) −967.263 −0.849972 −0.424986 0.905200i \(-0.639721\pi\)
−0.424986 + 0.905200i \(0.639721\pi\)
\(110\) −2154.74 −1.86770
\(111\) −20.5060 −0.0175346
\(112\) 0 0
\(113\) −1705.23 −1.41960 −0.709798 0.704405i \(-0.751214\pi\)
−0.709798 + 0.704405i \(0.751214\pi\)
\(114\) 276.858 0.227457
\(115\) −2432.52 −1.97247
\(116\) 86.3275 0.0690975
\(117\) −2066.57 −1.63295
\(118\) 756.070 0.589846
\(119\) 0 0
\(120\) −831.403 −0.632470
\(121\) 1418.80 1.06596
\(122\) −188.298 −0.139735
\(123\) −34.7259 −0.0254563
\(124\) 268.574 0.194505
\(125\) −1579.15 −1.12995
\(126\) 0 0
\(127\) −2670.28 −1.86574 −0.932870 0.360213i \(-0.882704\pi\)
−0.932870 + 0.360213i \(0.882704\pi\)
\(128\) 184.749 0.127576
\(129\) 272.355 0.185888
\(130\) −3597.83 −2.42731
\(131\) −879.993 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(132\) 287.735 0.189728
\(133\) 0 0
\(134\) 741.452 0.477998
\(135\) −1710.08 −1.09022
\(136\) 379.044 0.238991
\(137\) 2064.15 1.28724 0.643620 0.765345i \(-0.277432\pi\)
0.643620 + 0.765345i \(0.277432\pi\)
\(138\) −548.128 −0.338114
\(139\) −605.130 −0.369255 −0.184628 0.982809i \(-0.559108\pi\)
−0.184628 + 0.982809i \(0.559108\pi\)
\(140\) 0 0
\(141\) 38.8863 0.0232257
\(142\) 1103.63 0.652218
\(143\) 4591.41 2.68499
\(144\) 739.322 0.427848
\(145\) 531.684 0.304510
\(146\) 777.832 0.440917
\(147\) 0 0
\(148\) −33.1163 −0.0183928
\(149\) −775.322 −0.426287 −0.213144 0.977021i \(-0.568370\pi\)
−0.213144 + 0.977021i \(0.568370\pi\)
\(150\) −872.240 −0.474787
\(151\) 427.925 0.230623 0.115311 0.993329i \(-0.463213\pi\)
0.115311 + 0.993329i \(0.463213\pi\)
\(152\) 1648.70 0.879785
\(153\) 363.646 0.192151
\(154\) 0 0
\(155\) 1654.12 0.857177
\(156\) 480.438 0.246576
\(157\) 1680.93 0.854474 0.427237 0.904140i \(-0.359487\pi\)
0.427237 + 0.904140i \(0.359487\pi\)
\(158\) −2210.54 −1.11304
\(159\) 535.387 0.267037
\(160\) −2321.24 −1.14694
\(161\) 0 0
\(162\) 1042.93 0.505803
\(163\) 2038.68 0.979645 0.489822 0.871822i \(-0.337062\pi\)
0.489822 + 0.871822i \(0.337062\pi\)
\(164\) −56.0808 −0.0267023
\(165\) 1772.13 0.836123
\(166\) −1332.16 −0.622865
\(167\) −2543.12 −1.17840 −0.589199 0.807988i \(-0.700557\pi\)
−0.589199 + 0.807988i \(0.700557\pi\)
\(168\) 0 0
\(169\) 5469.40 2.48949
\(170\) 633.095 0.285625
\(171\) 1581.73 0.707354
\(172\) 439.842 0.194986
\(173\) 306.031 0.134492 0.0672460 0.997736i \(-0.478579\pi\)
0.0672460 + 0.997736i \(0.478579\pi\)
\(174\) 119.806 0.0521981
\(175\) 0 0
\(176\) −1642.59 −0.703493
\(177\) −621.817 −0.264060
\(178\) −3108.69 −1.30903
\(179\) −478.797 −0.199927 −0.0999635 0.994991i \(-0.531873\pi\)
−0.0999635 + 0.994991i \(0.531873\pi\)
\(180\) −1288.14 −0.533400
\(181\) 478.433 0.196473 0.0982367 0.995163i \(-0.468680\pi\)
0.0982367 + 0.995163i \(0.468680\pi\)
\(182\) 0 0
\(183\) 154.862 0.0625560
\(184\) −3264.13 −1.30780
\(185\) −203.960 −0.0810565
\(186\) 372.729 0.146934
\(187\) −807.931 −0.315945
\(188\) 62.7998 0.0243625
\(189\) 0 0
\(190\) 2753.73 1.05146
\(191\) 833.106 0.315610 0.157805 0.987470i \(-0.449558\pi\)
0.157805 + 0.987470i \(0.449558\pi\)
\(192\) −984.963 −0.370227
\(193\) −1449.88 −0.540751 −0.270376 0.962755i \(-0.587148\pi\)
−0.270376 + 0.962755i \(0.587148\pi\)
\(194\) 748.582 0.277037
\(195\) 2958.98 1.08665
\(196\) 0 0
\(197\) −1993.27 −0.720886 −0.360443 0.932781i \(-0.617374\pi\)
−0.360443 + 0.932781i \(0.617374\pi\)
\(198\) −2773.93 −0.995628
\(199\) 356.359 0.126943 0.0634714 0.997984i \(-0.479783\pi\)
0.0634714 + 0.997984i \(0.479783\pi\)
\(200\) −5194.23 −1.83644
\(201\) −609.795 −0.213988
\(202\) 551.165 0.191979
\(203\) 0 0
\(204\) −84.5407 −0.0290149
\(205\) −345.397 −0.117676
\(206\) 1190.77 0.402742
\(207\) −3131.53 −1.05148
\(208\) −2742.67 −0.914280
\(209\) −3514.20 −1.16307
\(210\) 0 0
\(211\) 4131.66 1.34803 0.674017 0.738716i \(-0.264567\pi\)
0.674017 + 0.738716i \(0.264567\pi\)
\(212\) 864.626 0.280107
\(213\) −907.666 −0.291982
\(214\) −961.562 −0.307154
\(215\) 2708.95 0.859296
\(216\) −2294.71 −0.722847
\(217\) 0 0
\(218\) −2167.87 −0.673518
\(219\) −639.715 −0.197388
\(220\) 2861.92 0.877048
\(221\) −1349.02 −0.410612
\(222\) −45.9590 −0.0138944
\(223\) −1332.32 −0.400086 −0.200043 0.979787i \(-0.564108\pi\)
−0.200043 + 0.979787i \(0.564108\pi\)
\(224\) 0 0
\(225\) −4983.22 −1.47651
\(226\) −3821.84 −1.12489
\(227\) 1329.33 0.388681 0.194340 0.980934i \(-0.437743\pi\)
0.194340 + 0.980934i \(0.437743\pi\)
\(228\) −367.721 −0.106811
\(229\) −5455.47 −1.57427 −0.787135 0.616780i \(-0.788437\pi\)
−0.787135 + 0.616780i \(0.788437\pi\)
\(230\) −5451.89 −1.56299
\(231\) 0 0
\(232\) 713.451 0.201898
\(233\) −591.158 −0.166215 −0.0831075 0.996541i \(-0.526484\pi\)
−0.0831075 + 0.996541i \(0.526484\pi\)
\(234\) −4631.70 −1.29395
\(235\) 386.778 0.107364
\(236\) −1004.21 −0.276985
\(237\) 1818.02 0.498283
\(238\) 0 0
\(239\) 6946.01 1.87992 0.939959 0.341289i \(-0.110863\pi\)
0.939959 + 0.341289i \(0.110863\pi\)
\(240\) −1058.58 −0.284713
\(241\) −7105.62 −1.89923 −0.949613 0.313426i \(-0.898523\pi\)
−0.949613 + 0.313426i \(0.898523\pi\)
\(242\) 3179.88 0.844670
\(243\) −3376.14 −0.891273
\(244\) 250.096 0.0656179
\(245\) 0 0
\(246\) −77.8293 −0.0201716
\(247\) −5867.75 −1.51156
\(248\) 2219.62 0.568331
\(249\) 1095.61 0.278842
\(250\) −3539.27 −0.895372
\(251\) 4874.53 1.22581 0.612904 0.790158i \(-0.290001\pi\)
0.612904 + 0.790158i \(0.290001\pi\)
\(252\) 0 0
\(253\) 6957.49 1.72891
\(254\) −5984.75 −1.47841
\(255\) −520.679 −0.127867
\(256\) −3860.77 −0.942570
\(257\) −2488.22 −0.603934 −0.301967 0.953318i \(-0.597643\pi\)
−0.301967 + 0.953318i \(0.597643\pi\)
\(258\) 610.415 0.147298
\(259\) 0 0
\(260\) 4778.62 1.13984
\(261\) 684.468 0.162328
\(262\) −1972.28 −0.465068
\(263\) −2812.52 −0.659419 −0.329709 0.944082i \(-0.606951\pi\)
−0.329709 + 0.944082i \(0.606951\pi\)
\(264\) 2377.97 0.554372
\(265\) 5325.16 1.23442
\(266\) 0 0
\(267\) 2556.69 0.586019
\(268\) −984.793 −0.224462
\(269\) 5554.63 1.25900 0.629501 0.777000i \(-0.283259\pi\)
0.629501 + 0.777000i \(0.283259\pi\)
\(270\) −3832.71 −0.863894
\(271\) 3168.41 0.710211 0.355105 0.934826i \(-0.384445\pi\)
0.355105 + 0.934826i \(0.384445\pi\)
\(272\) 482.617 0.107584
\(273\) 0 0
\(274\) 4626.26 1.02001
\(275\) 11071.5 2.42777
\(276\) 728.021 0.158774
\(277\) 3965.64 0.860189 0.430095 0.902784i \(-0.358480\pi\)
0.430095 + 0.902784i \(0.358480\pi\)
\(278\) −1356.25 −0.292598
\(279\) 2129.45 0.456942
\(280\) 0 0
\(281\) 1655.16 0.351383 0.175692 0.984445i \(-0.443784\pi\)
0.175692 + 0.984445i \(0.443784\pi\)
\(282\) 87.1539 0.0184040
\(283\) −7786.09 −1.63546 −0.817730 0.575602i \(-0.804768\pi\)
−0.817730 + 0.575602i \(0.804768\pi\)
\(284\) −1465.84 −0.306274
\(285\) −2264.76 −0.470711
\(286\) 10290.5 2.12758
\(287\) 0 0
\(288\) −2988.27 −0.611407
\(289\) −4675.62 −0.951683
\(290\) 1191.63 0.241294
\(291\) −615.659 −0.124023
\(292\) −1033.11 −0.207049
\(293\) −8090.80 −1.61321 −0.806603 0.591094i \(-0.798696\pi\)
−0.806603 + 0.591094i \(0.798696\pi\)
\(294\) 0 0
\(295\) −6184.83 −1.22066
\(296\) −273.688 −0.0537426
\(297\) 4891.16 0.955601
\(298\) −1737.69 −0.337790
\(299\) 11617.1 2.24694
\(300\) 1158.50 0.222954
\(301\) 0 0
\(302\) 959.086 0.182746
\(303\) −453.297 −0.0859446
\(304\) 2099.20 0.396045
\(305\) 1540.32 0.289175
\(306\) 815.020 0.152260
\(307\) 6129.49 1.13951 0.569753 0.821816i \(-0.307039\pi\)
0.569753 + 0.821816i \(0.307039\pi\)
\(308\) 0 0
\(309\) −979.328 −0.180298
\(310\) 3707.30 0.679228
\(311\) 8167.93 1.48926 0.744632 0.667476i \(-0.232625\pi\)
0.744632 + 0.667476i \(0.232625\pi\)
\(312\) 3970.56 0.720477
\(313\) −1877.25 −0.339005 −0.169502 0.985530i \(-0.554216\pi\)
−0.169502 + 0.985530i \(0.554216\pi\)
\(314\) 3767.37 0.677086
\(315\) 0 0
\(316\) 2936.02 0.522671
\(317\) 1222.93 0.216677 0.108338 0.994114i \(-0.465447\pi\)
0.108338 + 0.994114i \(0.465447\pi\)
\(318\) 1199.93 0.211600
\(319\) −1520.72 −0.266908
\(320\) −9796.82 −1.71143
\(321\) 790.820 0.137506
\(322\) 0 0
\(323\) 1032.52 0.177867
\(324\) −1385.21 −0.237519
\(325\) 18486.4 3.15520
\(326\) 4569.20 0.776271
\(327\) 1782.93 0.301518
\(328\) −463.478 −0.0780222
\(329\) 0 0
\(330\) 3971.79 0.662545
\(331\) 3769.03 0.625876 0.312938 0.949774i \(-0.398687\pi\)
0.312938 + 0.949774i \(0.398687\pi\)
\(332\) 1769.36 0.292489
\(333\) −262.570 −0.0432094
\(334\) −5699.76 −0.933763
\(335\) −6065.25 −0.989195
\(336\) 0 0
\(337\) −10900.2 −1.76193 −0.880967 0.473179i \(-0.843107\pi\)
−0.880967 + 0.473179i \(0.843107\pi\)
\(338\) 12258.3 1.97267
\(339\) 3143.21 0.503586
\(340\) −840.874 −0.134126
\(341\) −4731.11 −0.751332
\(342\) 3545.04 0.560508
\(343\) 0 0
\(344\) 3635.05 0.569735
\(345\) 4483.82 0.699712
\(346\) 685.892 0.106572
\(347\) −8542.30 −1.32154 −0.660771 0.750588i \(-0.729770\pi\)
−0.660771 + 0.750588i \(0.729770\pi\)
\(348\) −159.126 −0.0245116
\(349\) 993.823 0.152430 0.0762151 0.997091i \(-0.475716\pi\)
0.0762151 + 0.997091i \(0.475716\pi\)
\(350\) 0 0
\(351\) 8166.89 1.24193
\(352\) 6639.19 1.00531
\(353\) −8191.10 −1.23504 −0.617519 0.786556i \(-0.711862\pi\)
−0.617519 + 0.786556i \(0.711862\pi\)
\(354\) −1393.65 −0.209241
\(355\) −9027.99 −1.34974
\(356\) 4128.95 0.614702
\(357\) 0 0
\(358\) −1073.10 −0.158422
\(359\) −4703.71 −0.691510 −0.345755 0.938325i \(-0.612377\pi\)
−0.345755 + 0.938325i \(0.612377\pi\)
\(360\) −10645.7 −1.55856
\(361\) −2367.90 −0.345226
\(362\) 1072.29 0.155686
\(363\) −2615.24 −0.378139
\(364\) 0 0
\(365\) −6362.85 −0.912456
\(366\) 347.085 0.0495694
\(367\) −9431.88 −1.34153 −0.670763 0.741672i \(-0.734033\pi\)
−0.670763 + 0.741672i \(0.734033\pi\)
\(368\) −4156.05 −0.588720
\(369\) −444.650 −0.0627305
\(370\) −457.125 −0.0642292
\(371\) 0 0
\(372\) −495.056 −0.0689986
\(373\) −8281.46 −1.14959 −0.574796 0.818297i \(-0.694919\pi\)
−0.574796 + 0.818297i \(0.694919\pi\)
\(374\) −1810.77 −0.250355
\(375\) 2910.81 0.400836
\(376\) 519.006 0.0711854
\(377\) −2539.18 −0.346882
\(378\) 0 0
\(379\) 6875.50 0.931848 0.465924 0.884825i \(-0.345722\pi\)
0.465924 + 0.884825i \(0.345722\pi\)
\(380\) −3657.49 −0.493750
\(381\) 4922.06 0.661850
\(382\) 1867.20 0.250089
\(383\) 4826.61 0.643938 0.321969 0.946750i \(-0.395655\pi\)
0.321969 + 0.946750i \(0.395655\pi\)
\(384\) −340.544 −0.0452560
\(385\) 0 0
\(386\) −3249.55 −0.428492
\(387\) 3487.38 0.458072
\(388\) −994.263 −0.130093
\(389\) 4970.57 0.647861 0.323930 0.946081i \(-0.394996\pi\)
0.323930 + 0.946081i \(0.394996\pi\)
\(390\) 6631.80 0.861062
\(391\) −2044.21 −0.264400
\(392\) 0 0
\(393\) 1622.07 0.208200
\(394\) −4467.41 −0.571230
\(395\) 18082.7 2.30339
\(396\) 3684.31 0.467535
\(397\) 12288.8 1.55355 0.776774 0.629779i \(-0.216855\pi\)
0.776774 + 0.629779i \(0.216855\pi\)
\(398\) 798.689 0.100590
\(399\) 0 0
\(400\) −6613.54 −0.826693
\(401\) −11971.7 −1.49086 −0.745432 0.666581i \(-0.767757\pi\)
−0.745432 + 0.666581i \(0.767757\pi\)
\(402\) −1366.70 −0.169564
\(403\) −7899.66 −0.976452
\(404\) −732.055 −0.0901512
\(405\) −8531.38 −1.04673
\(406\) 0 0
\(407\) 583.365 0.0710475
\(408\) −698.683 −0.0847794
\(409\) −11147.7 −1.34772 −0.673861 0.738858i \(-0.735365\pi\)
−0.673861 + 0.738858i \(0.735365\pi\)
\(410\) −774.120 −0.0932465
\(411\) −3804.79 −0.456634
\(412\) −1581.57 −0.189123
\(413\) 0 0
\(414\) −7018.54 −0.833194
\(415\) 10897.4 1.28899
\(416\) 11085.6 1.30653
\(417\) 1115.42 0.130989
\(418\) −7876.19 −0.921620
\(419\) −11557.3 −1.34752 −0.673762 0.738949i \(-0.735323\pi\)
−0.673762 + 0.738949i \(0.735323\pi\)
\(420\) 0 0
\(421\) −12874.4 −1.49040 −0.745201 0.666840i \(-0.767647\pi\)
−0.745201 + 0.666840i \(0.767647\pi\)
\(422\) 9260.07 1.06818
\(423\) 497.922 0.0572336
\(424\) 7145.67 0.818454
\(425\) −3252.97 −0.371275
\(426\) −2034.30 −0.231367
\(427\) 0 0
\(428\) 1277.14 0.144236
\(429\) −8463.24 −0.952469
\(430\) 6071.42 0.680906
\(431\) −4088.31 −0.456907 −0.228454 0.973555i \(-0.573367\pi\)
−0.228454 + 0.973555i \(0.573367\pi\)
\(432\) −2921.73 −0.325397
\(433\) 3865.90 0.429060 0.214530 0.976717i \(-0.431178\pi\)
0.214530 + 0.976717i \(0.431178\pi\)
\(434\) 0 0
\(435\) −980.040 −0.108021
\(436\) 2879.36 0.316276
\(437\) −8891.56 −0.973320
\(438\) −1433.76 −0.156410
\(439\) −10662.4 −1.15920 −0.579600 0.814901i \(-0.696791\pi\)
−0.579600 + 0.814901i \(0.696791\pi\)
\(440\) 23652.2 2.56267
\(441\) 0 0
\(442\) −3023.50 −0.325369
\(443\) 10288.9 1.10347 0.551736 0.834019i \(-0.313966\pi\)
0.551736 + 0.834019i \(0.313966\pi\)
\(444\) 61.0424 0.00652465
\(445\) 25429.8 2.70897
\(446\) −2986.07 −0.317028
\(447\) 1429.13 0.151221
\(448\) 0 0
\(449\) 12426.2 1.30608 0.653041 0.757323i \(-0.273493\pi\)
0.653041 + 0.757323i \(0.273493\pi\)
\(450\) −11168.6 −1.16999
\(451\) 987.901 0.103145
\(452\) 5076.14 0.528234
\(453\) −788.784 −0.0818108
\(454\) 2979.35 0.307991
\(455\) 0 0
\(456\) −3039.01 −0.312094
\(457\) −10657.8 −1.09092 −0.545462 0.838136i \(-0.683646\pi\)
−0.545462 + 0.838136i \(0.683646\pi\)
\(458\) −12227.1 −1.24745
\(459\) −1437.09 −0.146139
\(460\) 7241.17 0.733959
\(461\) −9819.21 −0.992031 −0.496016 0.868314i \(-0.665204\pi\)
−0.496016 + 0.868314i \(0.665204\pi\)
\(462\) 0 0
\(463\) −19210.5 −1.92827 −0.964135 0.265412i \(-0.914492\pi\)
−0.964135 + 0.265412i \(0.914492\pi\)
\(464\) 908.399 0.0908865
\(465\) −3049.01 −0.304074
\(466\) −1324.93 −0.131709
\(467\) 345.566 0.0342417 0.0171208 0.999853i \(-0.494550\pi\)
0.0171208 + 0.999853i \(0.494550\pi\)
\(468\) 6151.80 0.607622
\(469\) 0 0
\(470\) 866.865 0.0850756
\(471\) −3098.41 −0.303115
\(472\) −8299.24 −0.809329
\(473\) −7748.10 −0.753188
\(474\) 4074.63 0.394839
\(475\) −14149.2 −1.36676
\(476\) 0 0
\(477\) 6855.39 0.658043
\(478\) 15567.7 1.48965
\(479\) 253.709 0.0242009 0.0121005 0.999927i \(-0.496148\pi\)
0.0121005 + 0.999927i \(0.496148\pi\)
\(480\) 4278.68 0.406863
\(481\) 974.060 0.0923354
\(482\) −15925.5 −1.50495
\(483\) 0 0
\(484\) −4223.50 −0.396647
\(485\) −6123.58 −0.573314
\(486\) −7566.76 −0.706245
\(487\) −13255.1 −1.23336 −0.616680 0.787214i \(-0.711523\pi\)
−0.616680 + 0.787214i \(0.711523\pi\)
\(488\) 2066.91 0.191731
\(489\) −3757.86 −0.347518
\(490\) 0 0
\(491\) −6454.57 −0.593260 −0.296630 0.954993i \(-0.595863\pi\)
−0.296630 + 0.954993i \(0.595863\pi\)
\(492\) 103.372 0.00947234
\(493\) 446.809 0.0408180
\(494\) −13151.1 −1.19776
\(495\) 22691.4 2.06041
\(496\) 2826.12 0.255840
\(497\) 0 0
\(498\) 2455.54 0.220954
\(499\) 8090.41 0.725805 0.362902 0.931827i \(-0.381786\pi\)
0.362902 + 0.931827i \(0.381786\pi\)
\(500\) 4700.84 0.420455
\(501\) 4687.67 0.418023
\(502\) 10925.0 0.971330
\(503\) 18897.4 1.67513 0.837567 0.546334i \(-0.183977\pi\)
0.837567 + 0.546334i \(0.183977\pi\)
\(504\) 0 0
\(505\) −4508.66 −0.397293
\(506\) 15593.4 1.36999
\(507\) −10081.6 −0.883117
\(508\) 7948.92 0.694245
\(509\) −4265.15 −0.371413 −0.185707 0.982605i \(-0.559457\pi\)
−0.185707 + 0.982605i \(0.559457\pi\)
\(510\) −1166.97 −0.101322
\(511\) 0 0
\(512\) −10130.9 −0.874469
\(513\) −6250.82 −0.537974
\(514\) −5576.72 −0.478558
\(515\) −9740.77 −0.833455
\(516\) −810.750 −0.0691691
\(517\) −1106.26 −0.0941068
\(518\) 0 0
\(519\) −564.100 −0.0477096
\(520\) 39492.7 3.33052
\(521\) 3324.96 0.279595 0.139798 0.990180i \(-0.455355\pi\)
0.139798 + 0.990180i \(0.455355\pi\)
\(522\) 1534.06 0.128628
\(523\) −13017.6 −1.08838 −0.544188 0.838964i \(-0.683162\pi\)
−0.544188 + 0.838964i \(0.683162\pi\)
\(524\) 2619.57 0.218390
\(525\) 0 0
\(526\) −6303.54 −0.522524
\(527\) 1390.07 0.114900
\(528\) 3027.75 0.249556
\(529\) 5436.69 0.446839
\(530\) 11935.0 0.978156
\(531\) −7962.09 −0.650707
\(532\) 0 0
\(533\) 1649.52 0.134050
\(534\) 5730.18 0.464362
\(535\) 7865.80 0.635641
\(536\) −8138.78 −0.655862
\(537\) 882.555 0.0709219
\(538\) 12449.3 0.997634
\(539\) 0 0
\(540\) 5090.59 0.405674
\(541\) −17906.8 −1.42305 −0.711527 0.702658i \(-0.751996\pi\)
−0.711527 + 0.702658i \(0.751996\pi\)
\(542\) 7101.19 0.562771
\(543\) −881.885 −0.0696967
\(544\) −1950.69 −0.153741
\(545\) 17733.7 1.39382
\(546\) 0 0
\(547\) 1612.94 0.126078 0.0630389 0.998011i \(-0.479921\pi\)
0.0630389 + 0.998011i \(0.479921\pi\)
\(548\) −6144.58 −0.478984
\(549\) 1982.94 0.154153
\(550\) 24813.9 1.92376
\(551\) 1943.45 0.150261
\(552\) 6016.70 0.463927
\(553\) 0 0
\(554\) 8887.99 0.681614
\(555\) 375.955 0.0287539
\(556\) 1801.36 0.137400
\(557\) −7803.94 −0.593651 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(558\) 4772.63 0.362081
\(559\) −12937.2 −0.978865
\(560\) 0 0
\(561\) 1489.24 0.112078
\(562\) 3709.63 0.278436
\(563\) 12329.5 0.922958 0.461479 0.887151i \(-0.347319\pi\)
0.461479 + 0.887151i \(0.347319\pi\)
\(564\) −115.757 −0.00864231
\(565\) 31263.5 2.32790
\(566\) −17450.6 −1.29594
\(567\) 0 0
\(568\) −12114.4 −0.894910
\(569\) 1554.46 0.114528 0.0572640 0.998359i \(-0.481762\pi\)
0.0572640 + 0.998359i \(0.481762\pi\)
\(570\) −5075.88 −0.372992
\(571\) 15951.8 1.16911 0.584556 0.811353i \(-0.301269\pi\)
0.584556 + 0.811353i \(0.301269\pi\)
\(572\) −13667.8 −0.999087
\(573\) −1535.64 −0.111959
\(574\) 0 0
\(575\) 28012.9 2.03168
\(576\) −12612.0 −0.912328
\(577\) −10491.7 −0.756975 −0.378488 0.925606i \(-0.623556\pi\)
−0.378488 + 0.925606i \(0.623556\pi\)
\(578\) −10479.2 −0.754114
\(579\) 2672.54 0.191825
\(580\) −1582.72 −0.113309
\(581\) 0 0
\(582\) −1379.84 −0.0982756
\(583\) −15231.0 −1.08199
\(584\) −8538.11 −0.604982
\(585\) 37888.4 2.67776
\(586\) −18133.5 −1.27831
\(587\) −6437.47 −0.452645 −0.226323 0.974052i \(-0.572670\pi\)
−0.226323 + 0.974052i \(0.572670\pi\)
\(588\) 0 0
\(589\) 6046.29 0.422976
\(590\) −13861.7 −0.967251
\(591\) 3674.15 0.255726
\(592\) −348.473 −0.0241928
\(593\) −11240.6 −0.778411 −0.389205 0.921151i \(-0.627250\pi\)
−0.389205 + 0.921151i \(0.627250\pi\)
\(594\) 10962.3 0.757219
\(595\) 0 0
\(596\) 2307.99 0.158622
\(597\) −656.869 −0.0450316
\(598\) 26036.8 1.78047
\(599\) 15903.5 1.08481 0.542405 0.840117i \(-0.317514\pi\)
0.542405 + 0.840117i \(0.317514\pi\)
\(600\) 9574.41 0.651456
\(601\) −117.190 −0.00795385 −0.00397692 0.999992i \(-0.501266\pi\)
−0.00397692 + 0.999992i \(0.501266\pi\)
\(602\) 0 0
\(603\) −7808.16 −0.527318
\(604\) −1273.85 −0.0858151
\(605\) −26012.1 −1.74801
\(606\) −1015.95 −0.0681025
\(607\) 22047.8 1.47429 0.737143 0.675737i \(-0.236175\pi\)
0.737143 + 0.675737i \(0.236175\pi\)
\(608\) −8484.78 −0.565959
\(609\) 0 0
\(610\) 3452.24 0.229143
\(611\) −1847.15 −0.122304
\(612\) −1082.51 −0.0714995
\(613\) −12719.2 −0.838050 −0.419025 0.907975i \(-0.637628\pi\)
−0.419025 + 0.907975i \(0.637628\pi\)
\(614\) 13737.7 0.902945
\(615\) 636.662 0.0417442
\(616\) 0 0
\(617\) 12736.9 0.831064 0.415532 0.909578i \(-0.363595\pi\)
0.415532 + 0.909578i \(0.363595\pi\)
\(618\) −2194.92 −0.142868
\(619\) −28083.4 −1.82353 −0.911767 0.410709i \(-0.865281\pi\)
−0.911767 + 0.410709i \(0.865281\pi\)
\(620\) −4924.02 −0.318957
\(621\) 12375.5 0.799697
\(622\) 18306.4 1.18009
\(623\) 0 0
\(624\) 5055.51 0.324331
\(625\) 2560.44 0.163868
\(626\) −4207.38 −0.268628
\(627\) 6477.64 0.412587
\(628\) −5003.80 −0.317951
\(629\) −171.401 −0.0108652
\(630\) 0 0
\(631\) 281.496 0.0177594 0.00887969 0.999961i \(-0.497173\pi\)
0.00887969 + 0.999961i \(0.497173\pi\)
\(632\) 24264.6 1.52721
\(633\) −7615.79 −0.478200
\(634\) 2740.89 0.171695
\(635\) 48956.7 3.05951
\(636\) −1593.75 −0.0993650
\(637\) 0 0
\(638\) −3408.30 −0.211498
\(639\) −11622.3 −0.719514
\(640\) −3387.18 −0.209203
\(641\) −8440.98 −0.520123 −0.260061 0.965592i \(-0.583743\pi\)
−0.260061 + 0.965592i \(0.583743\pi\)
\(642\) 1772.42 0.108959
\(643\) 1173.61 0.0719792 0.0359896 0.999352i \(-0.488542\pi\)
0.0359896 + 0.999352i \(0.488542\pi\)
\(644\) 0 0
\(645\) −4993.34 −0.304825
\(646\) 2314.14 0.140942
\(647\) 10845.7 0.659025 0.329513 0.944151i \(-0.393116\pi\)
0.329513 + 0.944151i \(0.393116\pi\)
\(648\) −11448.0 −0.694012
\(649\) 17689.8 1.06993
\(650\) 41432.5 2.50018
\(651\) 0 0
\(652\) −6068.78 −0.364527
\(653\) −5282.40 −0.316564 −0.158282 0.987394i \(-0.550596\pi\)
−0.158282 + 0.987394i \(0.550596\pi\)
\(654\) 3995.99 0.238923
\(655\) 16133.7 0.962437
\(656\) −590.122 −0.0351225
\(657\) −8191.26 −0.486410
\(658\) 0 0
\(659\) −19243.7 −1.13752 −0.568761 0.822503i \(-0.692577\pi\)
−0.568761 + 0.822503i \(0.692577\pi\)
\(660\) −5275.31 −0.311123
\(661\) −29196.9 −1.71804 −0.859021 0.511940i \(-0.828927\pi\)
−0.859021 + 0.511940i \(0.828927\pi\)
\(662\) 8447.34 0.495944
\(663\) 2486.62 0.145660
\(664\) 14622.8 0.854633
\(665\) 0 0
\(666\) −588.484 −0.0342392
\(667\) −3847.69 −0.223363
\(668\) 7570.39 0.438484
\(669\) 2455.84 0.141926
\(670\) −13593.7 −0.783838
\(671\) −4405.61 −0.253467
\(672\) 0 0
\(673\) 19924.5 1.14121 0.570605 0.821224i \(-0.306709\pi\)
0.570605 + 0.821224i \(0.306709\pi\)
\(674\) −24430.0 −1.39616
\(675\) 19693.2 1.12295
\(676\) −16281.4 −0.926341
\(677\) 4980.43 0.282738 0.141369 0.989957i \(-0.454850\pi\)
0.141369 + 0.989957i \(0.454850\pi\)
\(678\) 7044.70 0.399041
\(679\) 0 0
\(680\) −6949.37 −0.391906
\(681\) −2450.32 −0.137880
\(682\) −10603.6 −0.595355
\(683\) 29295.8 1.64125 0.820624 0.571468i \(-0.193626\pi\)
0.820624 + 0.571468i \(0.193626\pi\)
\(684\) −4708.50 −0.263208
\(685\) −37843.9 −2.11086
\(686\) 0 0
\(687\) 10055.9 0.558455
\(688\) 4628.32 0.256472
\(689\) −25431.5 −1.40619
\(690\) 10049.3 0.554452
\(691\) 32759.8 1.80353 0.901766 0.432225i \(-0.142271\pi\)
0.901766 + 0.432225i \(0.142271\pi\)
\(692\) −910.998 −0.0500447
\(693\) 0 0
\(694\) −19145.4 −1.04719
\(695\) 11094.4 0.605518
\(696\) −1315.09 −0.0716210
\(697\) −290.260 −0.0157739
\(698\) 2227.40 0.120786
\(699\) 1089.67 0.0589629
\(700\) 0 0
\(701\) 27958.5 1.50639 0.753195 0.657797i \(-0.228512\pi\)
0.753195 + 0.657797i \(0.228512\pi\)
\(702\) 18304.0 0.984104
\(703\) −745.532 −0.0399975
\(704\) 28020.8 1.50010
\(705\) −712.939 −0.0380863
\(706\) −18358.3 −0.978644
\(707\) 0 0
\(708\) 1851.03 0.0982572
\(709\) −31863.5 −1.68781 −0.843906 0.536492i \(-0.819749\pi\)
−0.843906 + 0.536492i \(0.819749\pi\)
\(710\) −20234.0 −1.06953
\(711\) 23278.9 1.22789
\(712\) 34123.6 1.79612
\(713\) −11970.6 −0.628753
\(714\) 0 0
\(715\) −84178.6 −4.40294
\(716\) 1425.29 0.0743931
\(717\) −12803.4 −0.666879
\(718\) −10542.2 −0.547953
\(719\) −7944.76 −0.412085 −0.206043 0.978543i \(-0.566059\pi\)
−0.206043 + 0.978543i \(0.566059\pi\)
\(720\) −13554.7 −0.701601
\(721\) 0 0
\(722\) −5307.06 −0.273557
\(723\) 13097.6 0.673729
\(724\) −1424.21 −0.0731080
\(725\) −6122.85 −0.313651
\(726\) −5861.39 −0.299637
\(727\) −28640.3 −1.46109 −0.730543 0.682866i \(-0.760733\pi\)
−0.730543 + 0.682866i \(0.760733\pi\)
\(728\) 0 0
\(729\) −6340.84 −0.322148
\(730\) −14260.7 −0.723031
\(731\) 2276.51 0.115184
\(732\) −460.996 −0.0232772
\(733\) −11852.7 −0.597258 −0.298629 0.954369i \(-0.596529\pi\)
−0.298629 + 0.954369i \(0.596529\pi\)
\(734\) −21139.2 −1.06303
\(735\) 0 0
\(736\) 16798.3 0.841297
\(737\) 17347.8 0.867047
\(738\) −996.570 −0.0497076
\(739\) −24052.5 −1.19727 −0.598636 0.801021i \(-0.704290\pi\)
−0.598636 + 0.801021i \(0.704290\pi\)
\(740\) 607.151 0.0301612
\(741\) 10815.9 0.536210
\(742\) 0 0
\(743\) −12530.4 −0.618704 −0.309352 0.950948i \(-0.600112\pi\)
−0.309352 + 0.950948i \(0.600112\pi\)
\(744\) −4091.37 −0.201609
\(745\) 14214.7 0.699042
\(746\) −18560.8 −0.910937
\(747\) 14028.8 0.687132
\(748\) 2405.06 0.117564
\(749\) 0 0
\(750\) 6523.85 0.317623
\(751\) 30921.6 1.50246 0.751229 0.660042i \(-0.229461\pi\)
0.751229 + 0.660042i \(0.229461\pi\)
\(752\) 660.823 0.0320449
\(753\) −8985.11 −0.434841
\(754\) −5690.93 −0.274869
\(755\) −7845.54 −0.378184
\(756\) 0 0
\(757\) −6257.40 −0.300435 −0.150217 0.988653i \(-0.547997\pi\)
−0.150217 + 0.988653i \(0.547997\pi\)
\(758\) 15409.7 0.738397
\(759\) −12824.6 −0.613310
\(760\) −30227.2 −1.44270
\(761\) 20094.8 0.957211 0.478605 0.878030i \(-0.341142\pi\)
0.478605 + 0.878030i \(0.341142\pi\)
\(762\) 11031.6 0.524450
\(763\) 0 0
\(764\) −2480.00 −0.117439
\(765\) −6667.06 −0.315095
\(766\) 10817.6 0.510257
\(767\) 29537.1 1.39051
\(768\) 7116.46 0.334366
\(769\) 26647.8 1.24960 0.624801 0.780784i \(-0.285180\pi\)
0.624801 + 0.780784i \(0.285180\pi\)
\(770\) 0 0
\(771\) 4586.48 0.214239
\(772\) 4316.03 0.201214
\(773\) 18513.2 0.861414 0.430707 0.902492i \(-0.358264\pi\)
0.430707 + 0.902492i \(0.358264\pi\)
\(774\) 7816.09 0.362976
\(775\) −19048.8 −0.882909
\(776\) −8217.05 −0.380122
\(777\) 0 0
\(778\) 11140.3 0.513365
\(779\) −1262.52 −0.0580675
\(780\) −8808.32 −0.404344
\(781\) 25821.8 1.18307
\(782\) −4581.58 −0.209510
\(783\) −2704.95 −0.123457
\(784\) 0 0
\(785\) −30818.0 −1.40120
\(786\) 3635.46 0.164978
\(787\) 29524.5 1.33727 0.668636 0.743590i \(-0.266878\pi\)
0.668636 + 0.743590i \(0.266878\pi\)
\(788\) 5933.59 0.268243
\(789\) 5184.25 0.233921
\(790\) 40527.8 1.82521
\(791\) 0 0
\(792\) 30448.9 1.36610
\(793\) −7356.16 −0.329413
\(794\) 27542.3 1.23103
\(795\) −9815.74 −0.437897
\(796\) −1060.81 −0.0472356
\(797\) −38789.4 −1.72395 −0.861976 0.506949i \(-0.830773\pi\)
−0.861976 + 0.506949i \(0.830773\pi\)
\(798\) 0 0
\(799\) 325.035 0.0143916
\(800\) 26731.3 1.18137
\(801\) 32737.3 1.44409
\(802\) −26831.5 −1.18136
\(803\) 18199.0 0.799785
\(804\) 1815.25 0.0796254
\(805\) 0 0
\(806\) −17705.1 −0.773741
\(807\) −10238.7 −0.446617
\(808\) −6050.04 −0.263415
\(809\) −11552.0 −0.502037 −0.251018 0.967982i \(-0.580765\pi\)
−0.251018 + 0.967982i \(0.580765\pi\)
\(810\) −19120.9 −0.829433
\(811\) −26939.1 −1.16641 −0.583205 0.812325i \(-0.698202\pi\)
−0.583205 + 0.812325i \(0.698202\pi\)
\(812\) 0 0
\(813\) −5840.25 −0.251939
\(814\) 1307.47 0.0562981
\(815\) −37377.1 −1.60646
\(816\) −889.596 −0.0381643
\(817\) 9901.96 0.424022
\(818\) −24984.8 −1.06794
\(819\) 0 0
\(820\) 1028.18 0.0437874
\(821\) −7558.67 −0.321315 −0.160657 0.987010i \(-0.551361\pi\)
−0.160657 + 0.987010i \(0.551361\pi\)
\(822\) −8527.48 −0.361837
\(823\) −3201.90 −0.135615 −0.0678076 0.997698i \(-0.521600\pi\)
−0.0678076 + 0.997698i \(0.521600\pi\)
\(824\) −13070.8 −0.552603
\(825\) −20407.8 −0.861223
\(826\) 0 0
\(827\) −11479.1 −0.482670 −0.241335 0.970442i \(-0.577585\pi\)
−0.241335 + 0.970442i \(0.577585\pi\)
\(828\) 9321.98 0.391257
\(829\) 1667.94 0.0698794 0.0349397 0.999389i \(-0.488876\pi\)
0.0349397 + 0.999389i \(0.488876\pi\)
\(830\) 24423.7 1.02140
\(831\) −7309.78 −0.305142
\(832\) 46787.0 1.94958
\(833\) 0 0
\(834\) 2499.94 0.103796
\(835\) 46625.3 1.93238
\(836\) 10461.1 0.432781
\(837\) −8415.38 −0.347525
\(838\) −25902.8 −1.06778
\(839\) −15210.3 −0.625884 −0.312942 0.949772i \(-0.601315\pi\)
−0.312942 + 0.949772i \(0.601315\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) −28854.7 −1.18100
\(843\) −3050.92 −0.124649
\(844\) −12299.2 −0.501606
\(845\) −100276. −4.08235
\(846\) 1115.97 0.0453519
\(847\) 0 0
\(848\) 9098.20 0.368436
\(849\) 14351.9 0.580161
\(850\) −7290.70 −0.294199
\(851\) 1476.02 0.0594563
\(852\) 2701.95 0.108647
\(853\) 18281.2 0.733805 0.366902 0.930259i \(-0.380418\pi\)
0.366902 + 0.930259i \(0.380418\pi\)
\(854\) 0 0
\(855\) −28999.2 −1.15994
\(856\) 10554.9 0.421447
\(857\) −585.415 −0.0233342 −0.0116671 0.999932i \(-0.503714\pi\)
−0.0116671 + 0.999932i \(0.503714\pi\)
\(858\) −18968.2 −0.754737
\(859\) 935.611 0.0371626 0.0185813 0.999827i \(-0.494085\pi\)
0.0185813 + 0.999827i \(0.494085\pi\)
\(860\) −8064.02 −0.319745
\(861\) 0 0
\(862\) −9162.92 −0.362054
\(863\) −12110.7 −0.477696 −0.238848 0.971057i \(-0.576770\pi\)
−0.238848 + 0.971057i \(0.576770\pi\)
\(864\) 11809.3 0.465002
\(865\) −5610.75 −0.220545
\(866\) 8664.43 0.339988
\(867\) 8618.46 0.337599
\(868\) 0 0
\(869\) −51720.0 −2.01896
\(870\) −2196.51 −0.0855963
\(871\) 28966.1 1.12684
\(872\) 23796.4 0.924136
\(873\) −7883.24 −0.305621
\(874\) −19928.2 −0.771260
\(875\) 0 0
\(876\) 1904.31 0.0734483
\(877\) 5841.41 0.224915 0.112458 0.993657i \(-0.464128\pi\)
0.112458 + 0.993657i \(0.464128\pi\)
\(878\) −23897.1 −0.918551
\(879\) 14913.6 0.572267
\(880\) 30115.1 1.15361
\(881\) 47826.2 1.82895 0.914476 0.404641i \(-0.132603\pi\)
0.914476 + 0.404641i \(0.132603\pi\)
\(882\) 0 0
\(883\) 17337.5 0.660761 0.330381 0.943848i \(-0.392823\pi\)
0.330381 + 0.943848i \(0.392823\pi\)
\(884\) 4015.79 0.152789
\(885\) 11400.4 0.433015
\(886\) 23059.9 0.874392
\(887\) −35181.5 −1.33177 −0.665885 0.746055i \(-0.731946\pi\)
−0.665885 + 0.746055i \(0.731946\pi\)
\(888\) 504.483 0.0190646
\(889\) 0 0
\(890\) 56994.6 2.14659
\(891\) 24401.4 0.917482
\(892\) 3966.08 0.148872
\(893\) 1413.78 0.0529792
\(894\) 3203.04 0.119827
\(895\) 8778.22 0.327848
\(896\) 0 0
\(897\) −21413.5 −0.797075
\(898\) 27850.3 1.03494
\(899\) 2616.44 0.0970669
\(900\) 14834.1 0.549412
\(901\) 4475.08 0.165468
\(902\) 2214.13 0.0817322
\(903\) 0 0
\(904\) 41951.6 1.54346
\(905\) −8771.56 −0.322184
\(906\) −1767.86 −0.0648270
\(907\) −28184.4 −1.03180 −0.515902 0.856648i \(-0.672543\pi\)
−0.515902 + 0.856648i \(0.672543\pi\)
\(908\) −3957.15 −0.144629
\(909\) −5804.26 −0.211788
\(910\) 0 0
\(911\) −34136.6 −1.24149 −0.620745 0.784013i \(-0.713170\pi\)
−0.620745 + 0.784013i \(0.713170\pi\)
\(912\) −3869.41 −0.140492
\(913\) −31168.5 −1.12982
\(914\) −23886.8 −0.864448
\(915\) −2839.23 −0.102582
\(916\) 16239.9 0.585788
\(917\) 0 0
\(918\) −3220.88 −0.115801
\(919\) 15512.0 0.556794 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(920\) 59844.4 2.14458
\(921\) −11298.3 −0.404227
\(922\) −22007.3 −0.786086
\(923\) 43115.3 1.53755
\(924\) 0 0
\(925\) 2348.80 0.0834897
\(926\) −43055.5 −1.52796
\(927\) −12539.9 −0.444297
\(928\) −3671.66 −0.129879
\(929\) 3100.72 0.109506 0.0547531 0.998500i \(-0.482563\pi\)
0.0547531 + 0.998500i \(0.482563\pi\)
\(930\) −6833.58 −0.240948
\(931\) 0 0
\(932\) 1759.77 0.0618488
\(933\) −15055.8 −0.528299
\(934\) 774.498 0.0271331
\(935\) 14812.5 0.518098
\(936\) 50841.3 1.77543
\(937\) 19638.8 0.684708 0.342354 0.939571i \(-0.388776\pi\)
0.342354 + 0.939571i \(0.388776\pi\)
\(938\) 0 0
\(939\) 3460.29 0.120258
\(940\) −1151.37 −0.0399504
\(941\) 50033.6 1.73332 0.866658 0.498903i \(-0.166264\pi\)
0.866658 + 0.498903i \(0.166264\pi\)
\(942\) −6944.30 −0.240189
\(943\) 2499.57 0.0863172
\(944\) −10567.0 −0.364328
\(945\) 0 0
\(946\) −17365.4 −0.596827
\(947\) 19758.4 0.677994 0.338997 0.940787i \(-0.389912\pi\)
0.338997 + 0.940787i \(0.389912\pi\)
\(948\) −5411.90 −0.185412
\(949\) 30387.3 1.03942
\(950\) −31711.8 −1.08302
\(951\) −2254.20 −0.0768636
\(952\) 0 0
\(953\) 33843.4 1.15036 0.575180 0.818027i \(-0.304932\pi\)
0.575180 + 0.818027i \(0.304932\pi\)
\(954\) 15364.6 0.521434
\(955\) −15274.1 −0.517548
\(956\) −20677.0 −0.699520
\(957\) 2803.10 0.0946828
\(958\) 568.624 0.0191768
\(959\) 0 0
\(960\) 18058.2 0.607112
\(961\) −21651.0 −0.726762
\(962\) 2183.11 0.0731666
\(963\) 10126.1 0.338846
\(964\) 21152.1 0.706705
\(965\) 26582.1 0.886743
\(966\) 0 0
\(967\) 35236.9 1.17181 0.585906 0.810379i \(-0.300739\pi\)
0.585906 + 0.810379i \(0.300739\pi\)
\(968\) −34904.9 −1.15897
\(969\) −1903.23 −0.0630964
\(970\) −13724.5 −0.454294
\(971\) 4505.40 0.148903 0.0744517 0.997225i \(-0.476279\pi\)
0.0744517 + 0.997225i \(0.476279\pi\)
\(972\) 10050.1 0.331644
\(973\) 0 0
\(974\) −29708.0 −0.977314
\(975\) −34075.5 −1.11927
\(976\) 2631.69 0.0863096
\(977\) −15167.4 −0.496671 −0.248336 0.968674i \(-0.579884\pi\)
−0.248336 + 0.968674i \(0.579884\pi\)
\(978\) −8422.29 −0.275373
\(979\) −72734.2 −2.37446
\(980\) 0 0
\(981\) 22829.7 0.743012
\(982\) −14466.3 −0.470099
\(983\) −25342.5 −0.822278 −0.411139 0.911573i \(-0.634869\pi\)
−0.411139 + 0.911573i \(0.634869\pi\)
\(984\) 854.318 0.0276775
\(985\) 36544.4 1.18213
\(986\) 1001.41 0.0323442
\(987\) 0 0
\(988\) 17467.2 0.562455
\(989\) −19604.1 −0.630307
\(990\) 50857.0 1.63267
\(991\) 40576.1 1.30065 0.650324 0.759657i \(-0.274633\pi\)
0.650324 + 0.759657i \(0.274633\pi\)
\(992\) −11422.9 −0.365603
\(993\) −6947.37 −0.222022
\(994\) 0 0
\(995\) −6533.46 −0.208166
\(996\) −3261.43 −0.103757
\(997\) −7442.49 −0.236415 −0.118208 0.992989i \(-0.537715\pi\)
−0.118208 + 0.992989i \(0.537715\pi\)
\(998\) 18132.6 0.575128
\(999\) 1037.65 0.0328627
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 1421.4.a.e.1.4 5
7.6 odd 2 29.4.a.b.1.4 5
21.20 even 2 261.4.a.f.1.2 5
28.27 even 2 464.4.a.l.1.3 5
35.34 odd 2 725.4.a.c.1.2 5
56.13 odd 2 1856.4.a.y.1.3 5
56.27 even 2 1856.4.a.bb.1.3 5
203.202 odd 2 841.4.a.b.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.4 5 7.6 odd 2
261.4.a.f.1.2 5 21.20 even 2
464.4.a.l.1.3 5 28.27 even 2
725.4.a.c.1.2 5 35.34 odd 2
841.4.a.b.1.2 5 203.202 odd 2
1421.4.a.e.1.4 5 1.1 even 1 trivial
1856.4.a.y.1.3 5 56.13 odd 2
1856.4.a.bb.1.3 5 56.27 even 2