Properties

Label 725.4.a.c.1.2
Level $725$
Weight $4$
Character 725.1
Self dual yes
Analytic conductor $42.776$
Analytic rank $0$
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] = [725,4,Mod(1,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, 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("725.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(42.7763847542\)
Analytic rank: \(0\)
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 1.2
Root \(0.328194\) of defining polynomial
Character \(\chi\) \(=\) 725.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24125 q^{2} -1.84328 q^{3} -2.97681 q^{4} +4.13124 q^{6} +16.8583 q^{7} +24.6017 q^{8} -23.6023 q^{9} +O(q^{10})\) \(q-2.24125 q^{2} -1.84328 q^{3} -2.97681 q^{4} +4.13124 q^{6} +16.8583 q^{7} +24.6017 q^{8} -23.6023 q^{9} +52.4385 q^{11} +5.48709 q^{12} +87.5580 q^{13} -37.7836 q^{14} -31.3241 q^{16} -15.4072 q^{17} +52.8987 q^{18} +67.0156 q^{19} -31.0745 q^{21} -117.528 q^{22} -132.679 q^{23} -45.3478 q^{24} -196.239 q^{26} +93.2741 q^{27} -50.1839 q^{28} -29.0000 q^{29} +90.2221 q^{31} -126.609 q^{32} -96.6587 q^{33} +34.5314 q^{34} +70.2597 q^{36} -11.1247 q^{37} -150.199 q^{38} -161.394 q^{39} -18.8392 q^{41} +69.6456 q^{42} +147.756 q^{43} -156.100 q^{44} +297.366 q^{46} -21.0963 q^{47} +57.7390 q^{48} -58.7983 q^{49} +28.3997 q^{51} -260.644 q^{52} +290.454 q^{53} -209.050 q^{54} +414.743 q^{56} -123.528 q^{57} +64.9962 q^{58} -337.343 q^{59} +84.0147 q^{61} -202.210 q^{62} -397.895 q^{63} +534.355 q^{64} +216.636 q^{66} -330.821 q^{67} +45.8644 q^{68} +244.564 q^{69} +492.420 q^{71} -580.659 q^{72} +347.053 q^{73} +24.9333 q^{74} -199.493 q^{76} +884.023 q^{77} +361.723 q^{78} -986.297 q^{79} +465.333 q^{81} +42.2234 q^{82} -594.382 q^{83} +92.5029 q^{84} -331.157 q^{86} +53.4550 q^{87} +1290.08 q^{88} +1387.04 q^{89} +1476.08 q^{91} +394.960 q^{92} -166.304 q^{93} +47.2820 q^{94} +233.375 q^{96} +334.003 q^{97} +131.781 q^{98} -1237.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} + 26 q^{4} + 34 q^{6} - 40 q^{7} + 84 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} + 26 q^{4} + 34 q^{6} - 40 q^{7} + 84 q^{8} + 33 q^{9} + 12 q^{11} + 224 q^{12} - 14 q^{13} - 192 q^{14} + 146 q^{16} - 66 q^{17} + 108 q^{18} + 214 q^{19} + 98 q^{22} - 164 q^{23} + 314 q^{24} + 56 q^{26} - 362 q^{27} - 540 q^{28} - 145 q^{29} + 420 q^{31} + 652 q^{32} + 576 q^{33} + 204 q^{34} - 260 q^{36} - 378 q^{37} + 496 q^{38} - 374 q^{39} - 1158 q^{41} - 348 q^{42} + 204 q^{43} + 784 q^{44} + 580 q^{46} - 248 q^{47} + 1880 q^{48} - 283 q^{49} + 228 q^{51} - 1482 q^{52} + 554 q^{53} + 918 q^{54} - 608 q^{56} - 44 q^{57} + 440 q^{59} + 618 q^{61} - 1250 q^{62} - 804 q^{63} + 2594 q^{64} + 2940 q^{66} - 1164 q^{67} - 356 q^{68} - 1968 q^{69} - 692 q^{71} + 2648 q^{72} + 1950 q^{73} - 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 1302 q^{78} + 272 q^{79} + 1801 q^{81} - 92 q^{82} - 512 q^{83} - 3208 q^{84} + 2446 q^{86} + 232 q^{87} + 6954 q^{88} + 866 q^{89} + 2580 q^{91} - 3468 q^{92} + 40 q^{93} - 5942 q^{94} + 7386 q^{96} - 1562 q^{97} + 3408 q^{98} - 238 q^{99}+O(q^{100}) \) 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 \(-0.629671\pi\)
−0.396200 + 0.918164i \(0.629671\pi\)
\(3\) −1.84328 −0.354739 −0.177369 0.984144i \(-0.556759\pi\)
−0.177369 + 0.984144i \(0.556759\pi\)
\(4\) −2.97681 −0.372101
\(5\) 0 0
\(6\) 4.13124 0.281095
\(7\) 16.8583 0.910262 0.455131 0.890425i \(-0.349592\pi\)
0.455131 + 0.890425i \(0.349592\pi\)
\(8\) 24.6017 1.08725
\(9\) −23.6023 −0.874160
\(10\) 0 0
\(11\) 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 \(0.116286\pi\)
\(14\) −37.7836 −0.721292
\(15\) 0 0
\(16\) −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 \(-0.367414\pi\)
0.404591 + 0.914498i \(0.367414\pi\)
\(20\) 0 0
\(21\) −31.0745 −0.322905
\(22\) −117.528 −1.13895
\(23\) −132.679 −1.20285 −0.601423 0.798931i \(-0.705399\pi\)
−0.601423 + 0.798931i \(0.705399\pi\)
\(24\) −45.3478 −0.385691
\(25\) 0 0
\(26\) −196.239 −1.48022
\(27\) 93.2741 0.664837
\(28\) −50.1839 −0.338710
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 90.2221 0.522721 0.261361 0.965241i \(-0.415829\pi\)
0.261361 + 0.965241i \(0.415829\pi\)
\(32\) −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 \(-0.507868\pi\)
−0.0247148 + 0.999695i \(0.507868\pi\)
\(38\) −150.199 −0.641196
\(39\) −161.394 −0.662658
\(40\) 0 0
\(41\) −18.8392 −0.0717608 −0.0358804 0.999356i \(-0.511424\pi\)
−0.0358804 + 0.999356i \(0.511424\pi\)
\(42\) 69.6456 0.255870
\(43\) 147.756 0.524013 0.262007 0.965066i \(-0.415616\pi\)
0.262007 + 0.965066i \(0.415616\pi\)
\(44\) −156.100 −0.534839
\(45\) 0 0
\(46\) 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
\(49\) −58.7983 −0.171424
\(50\) 0 0
\(51\) 28.3997 0.0779757
\(52\) −260.644 −0.695092
\(53\) 290.454 0.752772 0.376386 0.926463i \(-0.377167\pi\)
0.376386 + 0.926463i \(0.377167\pi\)
\(54\) −209.050 −0.526817
\(55\) 0 0
\(56\) 414.743 0.989686
\(57\) −123.528 −0.287048
\(58\) 64.9962 0.147145
\(59\) −337.343 −0.744379 −0.372190 0.928157i \(-0.621393\pi\)
−0.372190 + 0.928157i \(0.621393\pi\)
\(60\) 0 0
\(61\) 84.0147 0.176344 0.0881720 0.996105i \(-0.471897\pi\)
0.0881720 + 0.996105i \(0.471897\pi\)
\(62\) −202.210 −0.414205
\(63\) −397.895 −0.795715
\(64\) 534.355 1.04366
\(65\) 0 0
\(66\) 216.636 0.404031
\(67\) −330.821 −0.603228 −0.301614 0.953430i \(-0.597525\pi\)
−0.301614 + 0.953430i \(0.597525\pi\)
\(68\) 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
\(75\) 0 0
\(76\) −199.493 −0.301098
\(77\) 884.023 1.30836
\(78\) 361.723 0.525090
\(79\) −986.297 −1.40465 −0.702324 0.711858i \(-0.747854\pi\)
−0.702324 + 0.711858i \(0.747854\pi\)
\(80\) 0 0
\(81\) 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 \(0.628571\pi\)
\(84\) 92.5029 0.120153
\(85\) 0 0
\(86\) −331.157 −0.415228
\(87\) 53.4550 0.0658733
\(88\) 1290.08 1.56276
\(89\) 1387.04 1.65197 0.825987 0.563689i \(-0.190618\pi\)
0.825987 + 0.563689i \(0.190618\pi\)
\(90\) 0 0
\(91\) 1476.08 1.70038
\(92\) 394.960 0.447581
\(93\) −166.304 −0.185430
\(94\) 47.2820 0.0518805
\(95\) 0 0
\(96\) 233.375 0.248112
\(97\) 334.003 0.349617 0.174808 0.984602i \(-0.444069\pi\)
0.174808 + 0.984602i \(0.444069\pi\)
\(98\) 131.781 0.135836
\(99\) −1237.67 −1.25647
\(100\) 0 0
\(101\) −245.919 −0.242276 −0.121138 0.992636i \(-0.538654\pi\)
−0.121138 + 0.992636i \(0.538654\pi\)
\(102\) −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 \(-0.437915\pi\)
0.193812 + 0.981039i \(0.437915\pi\)
\(108\) −277.659 −0.247387
\(109\) −967.263 −0.849972 −0.424986 0.905200i \(-0.639721\pi\)
−0.424986 + 0.905200i \(0.639721\pi\)
\(110\) 0 0
\(111\) 20.5060 0.0175346
\(112\) −528.070 −0.445518
\(113\) 1705.23 1.41960 0.709798 0.704405i \(-0.248786\pi\)
0.709798 + 0.704405i \(0.248786\pi\)
\(114\) 276.858 0.227457
\(115\) 0 0
\(116\) 86.3275 0.0690975
\(117\) −2066.57 −1.63295
\(118\) 756.070 0.589846
\(119\) −259.739 −0.200086
\(120\) 0 0
\(121\) 1418.80 1.06596
\(122\) −188.298 −0.139735
\(123\) 34.7259 0.0254563
\(124\) −268.574 −0.194505
\(125\) 0 0
\(126\) 891.781 0.630525
\(127\) 2670.28 1.86574 0.932870 0.360213i \(-0.117296\pi\)
0.932870 + 0.360213i \(0.117296\pi\)
\(128\) −184.749 −0.127576
\(129\) −272.355 −0.185888
\(130\) 0 0
\(131\) 879.993 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(132\) 287.735 0.189728
\(133\) 1129.77 0.736567
\(134\) 741.452 0.477998
\(135\) 0 0
\(136\) −379.044 −0.238991
\(137\) −2064.15 −1.28724 −0.643620 0.765345i \(-0.722568\pi\)
−0.643620 + 0.765345i \(0.722568\pi\)
\(138\) −548.128 −0.338114
\(139\) 605.130 0.369255 0.184628 0.982809i \(-0.440892\pi\)
0.184628 + 0.982809i \(0.440892\pi\)
\(140\) 0 0
\(141\) 38.8863 0.0232257
\(142\) −1103.63 −0.652218
\(143\) 4591.41 2.68499
\(144\) 739.322 0.427848
\(145\) 0 0
\(146\) −777.832 −0.440917
\(147\) 108.381 0.0608106
\(148\) 33.1163 0.0183928
\(149\) −775.322 −0.426287 −0.213144 0.977021i \(-0.568370\pi\)
−0.213144 + 0.977021i \(0.568370\pi\)
\(150\) 0 0
\(151\) 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
\(154\) −1981.31 −1.03675
\(155\) 0 0
\(156\) 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
\(160\) 0 0
\(161\) −2236.74 −1.09490
\(162\) −1042.93 −0.505803
\(163\) −2038.68 −0.979645 −0.489822 0.871822i \(-0.662938\pi\)
−0.489822 + 0.871822i \(0.662938\pi\)
\(164\) 56.0808 0.0267023
\(165\) 0 0
\(166\) 1332.16 0.622865
\(167\) −2543.12 −1.17840 −0.589199 0.807988i \(-0.700557\pi\)
−0.589199 + 0.807988i \(0.700557\pi\)
\(168\) −764.486 −0.351080
\(169\) 5469.40 2.48949
\(170\) 0 0
\(171\) −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 \(0.531873\pi\)
\(180\) 0 0
\(181\) −478.433 −0.196473 −0.0982367 0.995163i \(-0.531320\pi\)
−0.0982367 + 0.995163i \(0.531320\pi\)
\(182\) −3308.25 −1.34739
\(183\) −154.862 −0.0625560
\(184\) −3264.13 −1.30780
\(185\) 0 0
\(186\) 372.729 0.146934
\(187\) −807.931 −0.315945
\(188\) 62.7998 0.0243625
\(189\) 1572.44 0.605176
\(190\) 0 0
\(191\) 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 \(-0.412852\pi\)
0.270376 + 0.962755i \(0.412852\pi\)
\(194\) −748.582 −0.277037
\(195\) 0 0
\(196\) 175.031 0.0637870
\(197\) 1993.27 0.720886 0.360443 0.932781i \(-0.382626\pi\)
0.360443 + 0.932781i \(0.382626\pi\)
\(198\) 2773.93 0.995628
\(199\) −356.359 −0.126943 −0.0634714 0.997984i \(-0.520217\pi\)
−0.0634714 + 0.997984i \(0.520217\pi\)
\(200\) 0 0
\(201\) 609.795 0.213988
\(202\) 551.165 0.191979
\(203\) −488.890 −0.169031
\(204\) −84.5407 −0.0290149
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) 2294.71 0.722847
\(217\) 1520.99 0.475813
\(218\) 2167.87 0.673518
\(219\) −639.715 −0.197388
\(220\) 0 0
\(221\) −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 \(0.564108\pi\)
\(224\) −2134.41 −0.636657
\(225\) 0 0
\(226\) −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 \(-0.211563\pi\)
0.787135 + 0.616780i \(0.211563\pi\)
\(230\) 0 0
\(231\) −1629.50 −0.464126
\(232\) −713.451 −0.201898
\(233\) 591.158 0.166215 0.0831075 0.996541i \(-0.473516\pi\)
0.0831075 + 0.996541i \(0.473516\pi\)
\(234\) 4631.70 1.29395
\(235\) 0 0
\(236\) 1004.21 0.276985
\(237\) 1818.02 0.498283
\(238\) 582.139 0.158548
\(239\) 6946.01 1.87992 0.939959 0.341289i \(-0.110863\pi\)
0.939959 + 0.341289i \(0.110863\pi\)
\(240\) 0 0
\(241\) 7105.62 1.89923 0.949613 0.313426i \(-0.101477\pi\)
0.949613 + 0.313426i \(0.101477\pi\)
\(242\) −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
\(250\) 0 0
\(251\) −4874.53 −1.22581 −0.612904 0.790158i \(-0.709999\pi\)
−0.612904 + 0.790158i \(0.709999\pi\)
\(252\) 1184.46 0.296087
\(253\) −6957.49 −1.72891
\(254\) −5984.75 −1.47841
\(255\) 0 0
\(256\) −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
\(259\) −187.544 −0.0449939
\(260\) 0 0
\(261\) 684.468 0.162328
\(262\) −1972.28 −0.465068
\(263\) 2812.52 0.659419 0.329709 0.944082i \(-0.393049\pi\)
0.329709 + 0.944082i \(0.393049\pi\)
\(264\) −2377.97 −0.554372
\(265\) 0 0
\(266\) −2532.09 −0.583656
\(267\) −2556.69 −0.586019
\(268\) 984.793 0.224462
\(269\) −5554.63 −1.25900 −0.629501 0.777000i \(-0.716741\pi\)
−0.629501 + 0.777000i \(0.716741\pi\)
\(270\) 0 0
\(271\) −3168.41 −0.710211 −0.355105 0.934826i \(-0.615555\pi\)
−0.355105 + 0.934826i \(0.615555\pi\)
\(272\) 482.617 0.107584
\(273\) −2720.82 −0.603192
\(274\) 4626.26 1.02001
\(275\) 0 0
\(276\) −728.021 −0.158774
\(277\) −3965.64 −0.860189 −0.430095 0.902784i \(-0.641520\pi\)
−0.430095 + 0.902784i \(0.641520\pi\)
\(278\) −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
\(285\) 0 0
\(286\) −10290.5 −2.12758
\(287\) −317.597 −0.0653211
\(288\) 2988.27 0.611407
\(289\) −4675.62 −0.951683
\(290\) 0 0
\(291\) −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 \(0.798696\pi\)
\(294\) −242.910 −0.0481863
\(295\) 0 0
\(296\) −273.688 −0.0537426
\(297\) 4891.16 0.955601
\(298\) 1737.69 0.337790
\(299\) −11617.1 −2.24694
\(300\) 0 0
\(301\) 2490.91 0.476989
\(302\) −959.086 −0.182746
\(303\) 453.297 0.0859446
\(304\) −2099.20 −0.396045
\(305\) 0 0
\(306\) −815.020 −0.152260
\(307\) 6129.49 1.13951 0.569753 0.821816i \(-0.307039\pi\)
0.569753 + 0.821816i \(0.307039\pi\)
\(308\) −2631.57 −0.486843
\(309\) −979.328 −0.180298
\(310\) 0 0
\(311\) −8167.93 −1.48926 −0.744632 0.667476i \(-0.767375\pi\)
−0.744632 + 0.667476i \(0.767375\pi\)
\(312\) −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 \(-0.534553\pi\)
−0.108338 + 0.994114i \(0.534553\pi\)
\(318\) 1199.93 0.211600
\(319\) −1520.72 −0.266908
\(320\) 0 0
\(321\) −790.820 −0.137506
\(322\) 5013.08 0.867603
\(323\) −1032.52 −0.177867
\(324\) −1385.21 −0.237519
\(325\) 0 0
\(326\) 4569.20 0.776271
\(327\) 1782.93 0.301518
\(328\) −463.478 −0.0780222
\(329\) −355.648 −0.0595972
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) 973.380 0.158042
\(337\) 10900.2 1.76193 0.880967 0.473179i \(-0.156893\pi\)
0.880967 + 0.473179i \(0.156893\pi\)
\(338\) −12258.3 −1.97267
\(339\) −3143.21 −0.503586
\(340\) 0 0
\(341\) 4731.11 0.751332
\(342\) 3545.04 0.560508
\(343\) −6773.63 −1.06630
\(344\) 3635.05 0.569735
\(345\) 0 0
\(346\) −685.892 −0.106572
\(347\) 8542.30 1.32154 0.660771 0.750588i \(-0.270230\pi\)
0.660771 + 0.750588i \(0.270230\pi\)
\(348\) −159.126 −0.0245116
\(349\) −993.823 −0.152430 −0.0762151 0.997091i \(-0.524284\pi\)
−0.0762151 + 0.997091i \(0.524284\pi\)
\(350\) 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
\(355\) 0 0
\(356\) −4128.95 −0.614702
\(357\) 478.771 0.0709783
\(358\) 1073.10 0.158422
\(359\) −4703.71 −0.691510 −0.345755 0.938325i \(-0.612377\pi\)
−0.345755 + 0.938325i \(0.612377\pi\)
\(360\) 0 0
\(361\) −2367.90 −0.345226
\(362\) 1072.29 0.155686
\(363\) −2615.24 −0.378139
\(364\) −4394.00 −0.632715
\(365\) 0 0
\(366\) 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
\(370\) 0 0
\(371\) 4896.55 0.685219
\(372\) 495.056 0.0689986
\(373\) 8281.46 1.14959 0.574796 0.818297i \(-0.305081\pi\)
0.574796 + 0.818297i \(0.305081\pi\)
\(374\) 1810.77 0.250355
\(375\) 0 0
\(376\) −519.006 −0.0711854
\(377\) −2539.18 −0.346882
\(378\) −3524.23 −0.479542
\(379\) 6875.50 0.931848 0.465924 0.884825i \(-0.345722\pi\)
0.465924 + 0.884825i \(0.345722\pi\)
\(380\) 0 0
\(381\) −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 \(0.394996\pi\)
\(390\) 0 0
\(391\) 2044.21 0.264400
\(392\) −1446.54 −0.186381
\(393\) −1622.07 −0.208200
\(394\) −4467.41 −0.571230
\(395\) 0 0
\(396\) 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
\(399\) −2082.48 −0.261289
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 1095.72 0.133941
\(407\) −583.365 −0.0710475
\(408\) 698.683 0.0847794
\(409\) 11147.7 1.34772 0.673861 0.738858i \(-0.264635\pi\)
0.673861 + 0.738858i \(0.264635\pi\)
\(410\) 0 0
\(411\) 3804.79 0.456634
\(412\) −1581.57 −0.189123
\(413\) −5687.03 −0.677580
\(414\) −7018.54 −0.833194
\(415\) 0 0
\(416\) −11085.6 −1.30653
\(417\) −1115.42 −0.130989
\(418\) −7876.19 −0.921620
\(419\) 11557.3 1.34752 0.673762 0.738949i \(-0.264677\pi\)
0.673762 + 0.738949i \(0.264677\pi\)
\(420\) 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
\(425\) 0 0
\(426\) 2034.30 0.231367
\(427\) 1416.34 0.160519
\(428\) −1277.14 −0.144236
\(429\) −8463.24 −0.952469
\(430\) 0 0
\(431\) −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 \(0.431178\pi\)
\(434\) −3408.91 −0.377035
\(435\) 0 0
\(436\) 2879.36 0.316276
\(437\) −8891.56 −0.973320
\(438\) 1433.76 0.156410
\(439\) 10662.4 1.15920 0.579600 0.814901i \(-0.303209\pi\)
0.579600 + 0.814901i \(0.303209\pi\)
\(440\) 0 0
\(441\) 1387.78 0.149852
\(442\) 3023.50 0.325369
\(443\) −10288.9 −1.10347 −0.551736 0.834019i \(-0.686034\pi\)
−0.551736 + 0.834019i \(0.686034\pi\)
\(444\) −61.0424 −0.00652465
\(445\) 0 0
\(446\) 2986.07 0.317028
\(447\) 1429.13 0.151221
\(448\) 9008.30 0.950005
\(449\) 12426.2 1.30608 0.653041 0.757323i \(-0.273493\pi\)
0.653041 + 0.757323i \(0.273493\pi\)
\(450\) 0 0
\(451\) −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 \(-0.316354\pi\)
0.545462 + 0.838136i \(0.316354\pi\)
\(458\) −12227.1 −1.24745
\(459\) −1437.09 −0.146139
\(460\) 0 0
\(461\) 9819.21 0.992031 0.496016 0.868314i \(-0.334796\pi\)
0.496016 + 0.868314i \(0.334796\pi\)
\(462\) 3652.11 0.367774
\(463\) 19210.5 1.92827 0.964135 0.265412i \(-0.0855081\pi\)
0.964135 + 0.265412i \(0.0855081\pi\)
\(464\) 908.399 0.0908865
\(465\) 0 0
\(466\) −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
\(469\) −5577.08 −0.549095
\(470\) 0 0
\(471\) −3098.41 −0.303115
\(472\) −8299.24 −0.809329
\(473\) 7748.10 0.753188
\(474\) −4074.63 −0.394839
\(475\) 0 0
\(476\) 773.194 0.0744523
\(477\) −6855.39 −0.658043
\(478\) −15567.7 −1.48965
\(479\) −253.709 −0.0242009 −0.0121005 0.999927i \(-0.503852\pi\)
−0.0121005 + 0.999927i \(0.503852\pi\)
\(480\) 0 0
\(481\) −974.060 −0.0923354
\(482\) −15925.5 −1.50495
\(483\) 4122.93 0.388405
\(484\) −4223.50 −0.396647
\(485\) 0 0
\(486\) 7566.76 0.706245
\(487\) 13255.1 1.23336 0.616680 0.787214i \(-0.288477\pi\)
0.616680 + 0.787214i \(0.288477\pi\)
\(488\) 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
\(495\) 0 0
\(496\) −2826.12 −0.255840
\(497\) 8301.36 0.749229
\(498\) −2455.54 −0.220954
\(499\) 8090.41 0.725805 0.362902 0.931827i \(-0.381786\pi\)
0.362902 + 0.931827i \(0.381786\pi\)
\(500\) 0 0
\(501\) 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 \(0.183977\pi\)
\(504\) −9788.91 −0.865144
\(505\) 0 0
\(506\) 15593.4 1.36999
\(507\) −10081.6 −0.883117
\(508\) −7948.92 −0.694245
\(509\) 4265.15 0.371413 0.185707 0.982605i \(-0.440543\pi\)
0.185707 + 0.982605i \(0.440543\pi\)
\(510\) 0 0
\(511\) 5850.72 0.506498
\(512\) 10130.9 0.874469
\(513\) 6250.82 0.537974
\(514\) 5576.72 0.478558
\(515\) 0 0
\(516\) 810.750 0.0691691
\(517\) −1106.26 −0.0941068
\(518\) 420.333 0.0356532
\(519\) −564.100 −0.0477096
\(520\) 0 0
\(521\) −3324.96 −0.279595 −0.139798 0.990180i \(-0.544645\pi\)
−0.139798 + 0.990180i \(0.544645\pi\)
\(522\) −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
\(530\) 0 0
\(531\) 7962.09 0.650707
\(532\) −3363.11 −0.274078
\(533\) −1649.52 −0.134050
\(534\) 5730.18 0.464362
\(535\) 0 0
\(536\) −8138.78 −0.655862
\(537\) 882.555 0.0709219
\(538\) 12449.3 0.997634
\(539\) −3083.29 −0.246395
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) 6098.03 0.477970
\(547\) −1612.94 −0.126078 −0.0630389 0.998011i \(-0.520079\pi\)
−0.0630389 + 0.998011i \(0.520079\pi\)
\(548\) 6144.58 0.478984
\(549\) −1982.94 −0.154153
\(550\) 0 0
\(551\) −1943.45 −0.150261
\(552\) 6016.70 0.463927
\(553\) −16627.3 −1.27860
\(554\) 8887.99 0.681614
\(555\) 0 0
\(556\) −1801.36 −0.137400
\(557\) 7803.94 0.593651 0.296826 0.954932i \(-0.404072\pi\)
0.296826 + 0.954932i \(0.404072\pi\)
\(558\) 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
\(565\) 0 0
\(566\) 17450.6 1.29594
\(567\) 7844.72 0.581035
\(568\) 12114.4 0.894910
\(569\) 1554.46 0.114528 0.0572640 0.998359i \(-0.481762\pi\)
0.0572640 + 0.998359i \(0.481762\pi\)
\(570\) 0 0
\(571\) 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
\(574\) 711.813 0.0517605
\(575\) 0 0
\(576\) −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
\(580\) 0 0
\(581\) −10020.3 −0.715509
\(582\) 1379.84 0.0982756
\(583\) 15231.0 1.08199
\(584\) 8538.11 0.604982
\(585\) 0 0
\(586\) 18133.5 1.27831
\(587\) −6437.47 −0.452645 −0.226323 0.974052i \(-0.572670\pi\)
−0.226323 + 0.974052i \(0.572670\pi\)
\(588\) −322.631 −0.0226277
\(589\) 6046.29 0.422976
\(590\) 0 0
\(591\) −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 \(0.317514\pi\)
\(600\) 0 0
\(601\) 117.190 0.00795385 0.00397692 0.999992i \(-0.498734\pi\)
0.00397692 + 0.999992i \(0.498734\pi\)
\(602\) −5582.75 −0.377966
\(603\) 7808.16 0.527318
\(604\) −1273.85 −0.0858151
\(605\) 0 0
\(606\) −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
\(609\) 901.160 0.0599620
\(610\) 0 0
\(611\) −1847.15 −0.122304
\(612\) −1082.51 −0.0714995
\(613\) 12719.2 0.838050 0.419025 0.907975i \(-0.362372\pi\)
0.419025 + 0.907975i \(0.362372\pi\)
\(614\) −13737.7 −0.902945
\(615\) 0 0
\(616\) 21748.5 1.42252
\(617\) −12736.9 −0.831064 −0.415532 0.909578i \(-0.636405\pi\)
−0.415532 + 0.909578i \(0.636405\pi\)
\(618\) 2194.92 0.142868
\(619\) 28083.4 1.82353 0.911767 0.410709i \(-0.134719\pi\)
0.911767 + 0.410709i \(0.134719\pi\)
\(620\) 0 0
\(621\) −12375.5 −0.799697
\(622\) 18306.4 1.18009
\(623\) 23383.1 1.50373
\(624\) 5055.51 0.324331
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 1593.75 0.0993650
\(637\) −5148.26 −0.320222
\(638\) 3408.30 0.211498
\(639\) −11622.3 −0.719514
\(640\) 0 0
\(641\) −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 \(0.488542\pi\)
\(644\) 6658.35 0.407416
\(645\) 0 0
\(646\) 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
\(650\) 0 0
\(651\) −2803.60 −0.168789
\(652\) 6068.78 0.364527
\(653\) 5282.40 0.316564 0.158282 0.987394i \(-0.449404\pi\)
0.158282 + 0.987394i \(0.449404\pi\)
\(654\) −3995.99 −0.238923
\(655\) 0 0
\(656\) 590.122 0.0351225
\(657\) −8191.26 −0.486410
\(658\) 797.094 0.0472249
\(659\) −19243.7 −1.13752 −0.568761 0.822503i \(-0.692577\pi\)
−0.568761 + 0.822503i \(0.692577\pi\)
\(660\) 0 0
\(661\) 29196.9 1.71804 0.859021 0.511940i \(-0.171073\pi\)
0.859021 + 0.511940i \(0.171073\pi\)
\(662\) −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
\(670\) 0 0
\(671\) 4405.61 0.253467
\(672\) 3934.31 0.225847
\(673\) −19924.5 −1.14121 −0.570605 0.821224i \(-0.693291\pi\)
−0.570605 + 0.821224i \(0.693291\pi\)
\(674\) −24430.0 −1.39616
\(675\) 0 0
\(676\) −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
\(679\) 5630.71 0.318243
\(680\) 0 0
\(681\) −2450.32 −0.137880
\(682\) −10603.6 −0.595355
\(683\) −29295.8 −1.64125 −0.820624 0.571468i \(-0.806374\pi\)
−0.820624 + 0.571468i \(0.806374\pi\)
\(684\) 4708.50 0.263208
\(685\) 0 0
\(686\) 15181.4 0.844938
\(687\) −10055.9 −0.558455
\(688\) −4628.32 −0.256472
\(689\) 25431.5 1.40619
\(690\) 0 0
\(691\) −32759.8 −1.80353 −0.901766 0.432225i \(-0.857729\pi\)
−0.901766 + 0.432225i \(0.857729\pi\)
\(692\) −910.998 −0.0500447
\(693\) −20865.0 −1.14372
\(694\) −19145.4 −1.04719
\(695\) 0 0
\(696\) 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
\(705\) 0 0
\(706\) 18358.3 0.978644
\(707\) −4145.77 −0.220534
\(708\) −1851.03 −0.0982572
\(709\) −31863.5 −1.68781 −0.843906 0.536492i \(-0.819749\pi\)
−0.843906 + 0.536492i \(0.819749\pi\)
\(710\) 0 0
\(711\) 23278.9 1.22789
\(712\) 34123.6 1.79612
\(713\) −11970.6 −0.628753
\(714\) −1073.04 −0.0562432
\(715\) 0 0
\(716\) 1425.29 0.0743931
\(717\) −12803.4 −0.666879
\(718\) 10542.2 0.547953
\(719\) 7944.76 0.412085 0.206043 0.978543i \(-0.433941\pi\)
0.206043 + 0.978543i \(0.433941\pi\)
\(720\) 0 0
\(721\) 8956.76 0.462645
\(722\) 5307.06 0.273557
\(723\) −13097.6 −0.673729
\(724\) 1424.21 0.0731080
\(725\) 0 0
\(726\) 5861.39 0.299637
\(727\) −28640.3 −1.46109 −0.730543 0.682866i \(-0.760733\pi\)
−0.730543 + 0.682866i \(0.760733\pi\)
\(728\) 36314.1 1.84875
\(729\) −6340.84 −0.322148
\(730\) 0 0
\(731\) −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 \(0.704290\pi\)
\(740\) 0 0
\(741\) −10815.9 −0.536210
\(742\) −10974.4 −0.542968
\(743\) 12530.4 0.618704 0.309352 0.950948i \(-0.399888\pi\)
0.309352 + 0.950948i \(0.399888\pi\)
\(744\) −4091.37 −0.201609
\(745\) 0 0
\(746\) −18560.8 −0.910937
\(747\) 14028.8 0.687132
\(748\) 2405.06 0.117564
\(749\) 7232.71 0.352840
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) −4680.86 −0.225187
\(757\) 6257.40 0.300435 0.150217 0.988653i \(-0.452003\pi\)
0.150217 + 0.988653i \(0.452003\pi\)
\(758\) −15409.7 −0.738397
\(759\) 12824.6 0.613310
\(760\) 0 0
\(761\) −20094.8 −0.957211 −0.478605 0.878030i \(-0.658858\pi\)
−0.478605 + 0.878030i \(0.658858\pi\)
\(762\) 11031.6 0.524450
\(763\) −16306.4 −0.773697
\(764\) −2480.00 −0.117439
\(765\) 0 0
\(766\) −10817.6 −0.510257
\(767\) −29537.1 −1.39051
\(768\) 7116.46 0.334366
\(769\) −26647.8 −1.24960 −0.624801 0.780784i \(-0.714820\pi\)
−0.624801 + 0.780784i \(0.714820\pi\)
\(770\) 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
\(775\) 0 0
\(776\) 8217.05 0.380122
\(777\) 345.696 0.0159611
\(778\) −11140.3 −0.513365
\(779\) −1262.52 −0.0580675
\(780\) 0 0
\(781\) 25821.8 1.18307
\(782\) −4581.58 −0.209510
\(783\) −2704.95 −0.123457
\(784\) 1841.80 0.0839014
\(785\) 0 0
\(786\) 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
\(790\) 0 0
\(791\) 28747.2 1.29220
\(792\) −30448.9 −1.36610
\(793\) 7356.16 0.329413
\(794\) −27542.3 −1.23103
\(795\) 0 0
\(796\) 1060.81 0.0472356
\(797\) −38789.4 −1.72395 −0.861976 0.506949i \(-0.830773\pi\)
−0.861976 + 0.506949i \(0.830773\pi\)
\(798\) 4667.34 0.207045
\(799\) 325.035 0.0143916
\(800\) 0 0
\(801\) −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 \(0.580765\pi\)
\(810\) 0 0
\(811\) 26939.1 1.16641 0.583205 0.812325i \(-0.301798\pi\)
0.583205 + 0.812325i \(0.301798\pi\)
\(812\) 1455.33 0.0628968
\(813\) 5840.25 0.251939
\(814\) 1307.47 0.0562981
\(815\) 0 0
\(816\) −889.596 −0.0381643
\(817\) 9901.96 0.424022
\(818\) −24984.8 −1.06794
\(819\) −34838.9 −1.48641
\(820\) 0 0
\(821\) −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 \(-0.478400\pi\)
0.0678076 + 0.997698i \(0.478400\pi\)
\(824\) 13070.8 0.552603
\(825\) 0 0
\(826\) 12746.0 0.536915
\(827\) 11479.1 0.482670 0.241335 0.970442i \(-0.422415\pi\)
0.241335 + 0.970442i \(0.422415\pi\)
\(828\) −9321.98 −0.391257
\(829\) −1667.94 −0.0698794 −0.0349397 0.999389i \(-0.511124\pi\)
−0.0349397 + 0.999389i \(0.511124\pi\)
\(830\) 0 0
\(831\) 7309.78 0.305142
\(832\) 46787.0 1.94958
\(833\) 905.917 0.0376809
\(834\) 2499.94 0.103796
\(835\) 0 0
\(836\) −10461.1 −0.432781
\(837\) 8415.38 0.347525
\(838\) −25902.8 −1.06778
\(839\) 15210.3 0.625884 0.312942 0.949772i \(-0.398685\pi\)
0.312942 + 0.949772i \(0.398685\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 28854.7 1.18100
\(843\) −3050.92 −0.124649
\(844\) −12299.2 −0.501606
\(845\) 0 0
\(846\) −1115.97 −0.0453519
\(847\) 23918.5 0.970306
\(848\) −9098.20 −0.368436
\(849\) 14351.9 0.580161
\(850\) 0 0
\(851\) 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 \(0.380418\pi\)
\(854\) −3174.38 −0.127195
\(855\) 0 0
\(856\) 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 \(-0.505915\pi\)
−0.0185813 + 0.999827i \(0.505915\pi\)
\(860\) 0 0
\(861\) 585.419 0.0231719
\(862\) 9162.92 0.362054
\(863\) 12110.7 0.477696 0.238848 0.971057i \(-0.423230\pi\)
0.238848 + 0.971057i \(0.423230\pi\)
\(864\) −11809.3 −0.465002
\(865\) 0 0
\(866\) −8664.43 −0.339988
\(867\) 8618.46 0.337599
\(868\) −4527.70 −0.177051
\(869\) −51720.0 −2.01896
\(870\) 0 0
\(871\) −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 \(-0.535872\pi\)
−0.112458 + 0.993657i \(0.535872\pi\)
\(878\) −23897.1 −0.918551
\(879\) 14913.6 0.572267
\(880\) 0 0
\(881\) −47826.2 −1.82895 −0.914476 0.404641i \(-0.867397\pi\)
−0.914476 + 0.404641i \(0.867397\pi\)
\(882\) −3110.35 −0.118743
\(883\) −17337.5 −0.660761 −0.330381 0.943848i \(-0.607177\pi\)
−0.330381 + 0.943848i \(0.607177\pi\)
\(884\) 4015.79 0.152789
\(885\) 0 0
\(886\) 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
\(889\) 45016.3 1.69831
\(890\) 0 0
\(891\) 24401.4 0.917482
\(892\) 3966.08 0.148872
\(893\) −1413.78 −0.0529792
\(894\) −3203.04 −0.119827
\(895\) 0 0
\(896\) −3114.56 −0.116127
\(897\) 21413.5 0.797075
\(898\) −27850.3 −1.03494
\(899\) −2616.44 −0.0970669
\(900\) 0 0
\(901\) −4475.08 −0.165468
\(902\) 2214.13 0.0817322
\(903\) −4591.44 −0.169207
\(904\) 41951.6 1.54346
\(905\) 0 0
\(906\) 1767.86 0.0648270
\(907\) 28184.4 1.03180 0.515902 0.856648i \(-0.327457\pi\)
0.515902 + 0.856648i \(0.327457\pi\)
\(908\) −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
\(915\) 0 0
\(916\) −16239.9 −0.585788
\(917\) 14835.2 0.534242
\(918\) 3220.88 0.115801
\(919\) 15512.0 0.556794 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(920\) 0 0
\(921\) −11298.3 −0.404227
\(922\) −22007.3 −0.786086
\(923\) 43115.3 1.53755
\(924\) 4850.71 0.172702
\(925\) 0 0
\(926\) −43055.5 −1.52796
\(927\) −12539.9 −0.444297
\(928\) 3671.66 0.129879
\(929\) −3100.72 −0.109506 −0.0547531 0.998500i \(-0.517437\pi\)
−0.0547531 + 0.998500i \(0.517437\pi\)
\(930\) 0 0
\(931\) −3940.40 −0.138713
\(932\) −1759.77 −0.0618488
\(933\) 15055.8 0.528299
\(934\) −774.498 −0.0271331
\(935\) 0 0
\(936\) −50841.3 −1.77543
\(937\) 19638.8 0.684708 0.342354 0.939571i \(-0.388776\pi\)
0.342354 + 0.939571i \(0.388776\pi\)
\(938\) 12499.6 0.435103
\(939\) 3460.29 0.120258
\(940\) 0 0
\(941\) −50033.6 −1.73332 −0.866658 0.498903i \(-0.833736\pi\)
−0.866658 + 0.498903i \(0.833736\pi\)
\(942\) 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 \(-0.610088\pi\)
−0.338997 + 0.940787i \(0.610088\pi\)
\(948\) −5411.90 −0.185412
\(949\) 30387.3 1.03942
\(950\) 0 0
\(951\) 2254.20 0.0768636
\(952\) −6390.03 −0.217544
\(953\) −33843.4 −1.15036 −0.575180 0.818027i \(-0.695068\pi\)
−0.575180 + 0.818027i \(0.695068\pi\)
\(954\) 15364.6 0.521434
\(955\) 0 0
\(956\) −20677.0 −0.699520
\(957\) 2803.10 0.0946828
\(958\) 568.624 0.0191768
\(959\) −34798.0 −1.17173
\(960\) 0 0
\(961\) −21651.0 −0.726762
\(962\) 2183.11 0.0731666
\(963\) −10126.1 −0.338846
\(964\) −21152.1 −0.706705
\(965\) 0 0
\(966\) −9240.50 −0.307772
\(967\) −35236.9 −1.17181 −0.585906 0.810379i \(-0.699261\pi\)
−0.585906 + 0.810379i \(0.699261\pi\)
\(968\) 34904.9 1.15897
\(969\) 1903.23 0.0630964
\(970\) 0 0
\(971\) −4505.40 −0.148903 −0.0744517 0.997225i \(-0.523721\pi\)
−0.0744517 + 0.997225i \(0.523721\pi\)
\(972\) 10050.1 0.331644
\(973\) 10201.5 0.336119
\(974\) −29708.0 −0.977314
\(975\) 0 0
\(976\) −2631.69 −0.0863096
\(977\) 15167.4 0.496671 0.248336 0.968674i \(-0.420116\pi\)
0.248336 + 0.968674i \(0.420116\pi\)
\(978\) −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
\(985\) 0 0
\(986\) −1001.41 −0.0323442
\(987\) 655.557 0.0211414
\(988\) −17467.2 −0.562455
\(989\) −19604.1 −0.630307
\(990\) 0 0
\(991\) 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
\(994\) −18605.4 −0.593689
\(995\) 0 0
\(996\) −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 725.4.a.c.1.2 5
5.4 even 2 29.4.a.b.1.4 5
15.14 odd 2 261.4.a.f.1.2 5
20.19 odd 2 464.4.a.l.1.3 5
35.34 odd 2 1421.4.a.e.1.4 5
40.19 odd 2 1856.4.a.bb.1.3 5
40.29 even 2 1856.4.a.y.1.3 5
145.144 even 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 5.4 even 2
261.4.a.f.1.2 5 15.14 odd 2
464.4.a.l.1.3 5 20.19 odd 2
725.4.a.c.1.2 5 1.1 even 1 trivial
841.4.a.b.1.2 5 145.144 even 2
1421.4.a.e.1.4 5 35.34 odd 2
1856.4.a.y.1.3 5 40.29 even 2
1856.4.a.bb.1.3 5 40.19 odd 2