Properties

Label 289.4.a.f.1.1
Level $289$
Weight $4$
Character 289.1
Self dual yes
Analytic conductor $17.052$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(1,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.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: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 54x^{6} - 20x^{5} + 677x^{4} + 380x^{3} - 2216x^{2} - 1000x + 476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.52230\) of defining polynomial
Character \(\chi\) \(=\) 289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.93651 q^{2} -0.423991 q^{3} +7.49613 q^{4} -1.95080 q^{5} +1.66905 q^{6} -25.4345 q^{7} +1.98349 q^{8} -26.8202 q^{9} +O(q^{10})\) \(q-3.93651 q^{2} -0.423991 q^{3} +7.49613 q^{4} -1.95080 q^{5} +1.66905 q^{6} -25.4345 q^{7} +1.98349 q^{8} -26.8202 q^{9} +7.67934 q^{10} -32.4303 q^{11} -3.17829 q^{12} +54.6855 q^{13} +100.123 q^{14} +0.827121 q^{15} -67.7771 q^{16} +105.578 q^{18} -46.1336 q^{19} -14.6234 q^{20} +10.7840 q^{21} +127.662 q^{22} -75.2978 q^{23} -0.840984 q^{24} -121.194 q^{25} -215.270 q^{26} +22.8193 q^{27} -190.660 q^{28} -157.040 q^{29} -3.25597 q^{30} +252.606 q^{31} +250.937 q^{32} +13.7502 q^{33} +49.6175 q^{35} -201.048 q^{36} -226.214 q^{37} +181.606 q^{38} -23.1862 q^{39} -3.86940 q^{40} +231.163 q^{41} -42.4513 q^{42} +119.642 q^{43} -243.102 q^{44} +52.3209 q^{45} +296.411 q^{46} +188.319 q^{47} +28.7369 q^{48} +303.913 q^{49} +477.083 q^{50} +409.930 q^{52} +468.212 q^{53} -89.8285 q^{54} +63.2650 q^{55} -50.4491 q^{56} +19.5603 q^{57} +618.188 q^{58} +751.217 q^{59} +6.20021 q^{60} -483.210 q^{61} -994.385 q^{62} +682.159 q^{63} -445.601 q^{64} -106.680 q^{65} -54.1277 q^{66} +533.879 q^{67} +31.9256 q^{69} -195.320 q^{70} +78.1521 q^{71} -53.1977 q^{72} -383.013 q^{73} +890.494 q^{74} +51.3854 q^{75} -345.824 q^{76} +824.848 q^{77} +91.2727 q^{78} +1278.85 q^{79} +132.219 q^{80} +714.471 q^{81} -909.975 q^{82} +591.376 q^{83} +80.8382 q^{84} -470.973 q^{86} +66.5834 q^{87} -64.3253 q^{88} +609.729 q^{89} -205.962 q^{90} -1390.90 q^{91} -564.442 q^{92} -107.103 q^{93} -741.321 q^{94} +89.9974 q^{95} -106.395 q^{96} -1592.06 q^{97} -1196.36 q^{98} +869.788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 36 q^{4} + 96 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 36 q^{4} + 96 q^{8} + 8 q^{9} + 88 q^{13} + 252 q^{15} + 420 q^{16} + 428 q^{18} - 52 q^{19} + 260 q^{21} - 140 q^{25} - 268 q^{26} + 120 q^{30} + 2336 q^{32} + 816 q^{33} + 1172 q^{35} - 264 q^{36} + 768 q^{38} - 136 q^{42} - 752 q^{43} + 368 q^{47} + 852 q^{49} + 468 q^{50} + 2564 q^{52} - 1156 q^{53} + 1996 q^{55} + 192 q^{59} - 3160 q^{60} + 3044 q^{64} + 1052 q^{66} + 764 q^{67} + 1812 q^{69} + 544 q^{70} - 1404 q^{72} + 896 q^{76} + 3084 q^{77} - 280 q^{81} + 496 q^{83} - 2952 q^{84} - 4244 q^{86} - 2860 q^{87} + 2156 q^{89} - 4012 q^{93} - 3392 q^{94} - 6728 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.93651 −1.39177 −0.695884 0.718155i \(-0.744987\pi\)
−0.695884 + 0.718155i \(0.744987\pi\)
\(3\) −0.423991 −0.0815972 −0.0407986 0.999167i \(-0.512990\pi\)
−0.0407986 + 0.999167i \(0.512990\pi\)
\(4\) 7.49613 0.937016
\(5\) −1.95080 −0.174485 −0.0872423 0.996187i \(-0.527805\pi\)
−0.0872423 + 0.996187i \(0.527805\pi\)
\(6\) 1.66905 0.113564
\(7\) −25.4345 −1.37333 −0.686667 0.726973i \(-0.740927\pi\)
−0.686667 + 0.726973i \(0.740927\pi\)
\(8\) 1.98349 0.0876588
\(9\) −26.8202 −0.993342
\(10\) 7.67934 0.242842
\(11\) −32.4303 −0.888918 −0.444459 0.895799i \(-0.646604\pi\)
−0.444459 + 0.895799i \(0.646604\pi\)
\(12\) −3.17829 −0.0764579
\(13\) 54.6855 1.16669 0.583347 0.812223i \(-0.301743\pi\)
0.583347 + 0.812223i \(0.301743\pi\)
\(14\) 100.123 1.91136
\(15\) 0.827121 0.0142375
\(16\) −67.7771 −1.05902
\(17\) 0 0
\(18\) 105.578 1.38250
\(19\) −46.1336 −0.557041 −0.278521 0.960430i \(-0.589844\pi\)
−0.278521 + 0.960430i \(0.589844\pi\)
\(20\) −14.6234 −0.163495
\(21\) 10.7840 0.112060
\(22\) 127.662 1.23717
\(23\) −75.2978 −0.682638 −0.341319 0.939948i \(-0.610874\pi\)
−0.341319 + 0.939948i \(0.610874\pi\)
\(24\) −0.840984 −0.00715271
\(25\) −121.194 −0.969555
\(26\) −215.270 −1.62377
\(27\) 22.8193 0.162651
\(28\) −190.660 −1.28684
\(29\) −157.040 −1.00557 −0.502785 0.864412i \(-0.667691\pi\)
−0.502785 + 0.864412i \(0.667691\pi\)
\(30\) −3.25597 −0.0198152
\(31\) 252.606 1.46353 0.731763 0.681559i \(-0.238698\pi\)
0.731763 + 0.681559i \(0.238698\pi\)
\(32\) 250.937 1.38625
\(33\) 13.7502 0.0725332
\(34\) 0 0
\(35\) 49.6175 0.239626
\(36\) −201.048 −0.930777
\(37\) −226.214 −1.00512 −0.502559 0.864543i \(-0.667608\pi\)
−0.502559 + 0.864543i \(0.667608\pi\)
\(38\) 181.606 0.775272
\(39\) −23.1862 −0.0951990
\(40\) −3.86940 −0.0152951
\(41\) 231.163 0.880526 0.440263 0.897869i \(-0.354885\pi\)
0.440263 + 0.897869i \(0.354885\pi\)
\(42\) −42.4513 −0.155962
\(43\) 119.642 0.424309 0.212154 0.977236i \(-0.431952\pi\)
0.212154 + 0.977236i \(0.431952\pi\)
\(44\) −243.102 −0.832931
\(45\) 52.3209 0.173323
\(46\) 296.411 0.950073
\(47\) 188.319 0.584451 0.292225 0.956349i \(-0.405604\pi\)
0.292225 + 0.956349i \(0.405604\pi\)
\(48\) 28.7369 0.0864128
\(49\) 303.913 0.886044
\(50\) 477.083 1.34940
\(51\) 0 0
\(52\) 409.930 1.09321
\(53\) 468.212 1.21347 0.606734 0.794905i \(-0.292479\pi\)
0.606734 + 0.794905i \(0.292479\pi\)
\(54\) −89.8285 −0.226372
\(55\) 63.2650 0.155103
\(56\) −50.4491 −0.120385
\(57\) 19.5603 0.0454530
\(58\) 618.188 1.39952
\(59\) 751.217 1.65763 0.828815 0.559523i \(-0.189016\pi\)
0.828815 + 0.559523i \(0.189016\pi\)
\(60\) 6.20021 0.0133407
\(61\) −483.210 −1.01424 −0.507121 0.861875i \(-0.669290\pi\)
−0.507121 + 0.861875i \(0.669290\pi\)
\(62\) −994.385 −2.03689
\(63\) 682.159 1.36419
\(64\) −445.601 −0.870315
\(65\) −106.680 −0.203570
\(66\) −54.1277 −0.100949
\(67\) 533.879 0.973487 0.486744 0.873545i \(-0.338185\pi\)
0.486744 + 0.873545i \(0.338185\pi\)
\(68\) 0 0
\(69\) 31.9256 0.0557013
\(70\) −195.320 −0.333503
\(71\) 78.1521 0.130633 0.0653165 0.997865i \(-0.479194\pi\)
0.0653165 + 0.997865i \(0.479194\pi\)
\(72\) −53.1977 −0.0870752
\(73\) −383.013 −0.614085 −0.307043 0.951696i \(-0.599339\pi\)
−0.307043 + 0.951696i \(0.599339\pi\)
\(74\) 890.494 1.39889
\(75\) 51.3854 0.0791129
\(76\) −345.824 −0.521957
\(77\) 824.848 1.22078
\(78\) 91.2727 0.132495
\(79\) 1278.85 1.82129 0.910646 0.413187i \(-0.135584\pi\)
0.910646 + 0.413187i \(0.135584\pi\)
\(80\) 132.219 0.184782
\(81\) 714.471 0.980070
\(82\) −909.975 −1.22549
\(83\) 591.376 0.782071 0.391036 0.920376i \(-0.372117\pi\)
0.391036 + 0.920376i \(0.372117\pi\)
\(84\) 80.8382 0.105002
\(85\) 0 0
\(86\) −470.973 −0.590539
\(87\) 66.5834 0.0820516
\(88\) −64.3253 −0.0779215
\(89\) 609.729 0.726192 0.363096 0.931752i \(-0.381720\pi\)
0.363096 + 0.931752i \(0.381720\pi\)
\(90\) −205.962 −0.241225
\(91\) −1390.90 −1.60226
\(92\) −564.442 −0.639643
\(93\) −107.103 −0.119420
\(94\) −741.321 −0.813420
\(95\) 89.9974 0.0971952
\(96\) −106.395 −0.113114
\(97\) −1592.06 −1.66648 −0.833242 0.552908i \(-0.813518\pi\)
−0.833242 + 0.552908i \(0.813518\pi\)
\(98\) −1196.36 −1.23317
\(99\) 869.788 0.883000
\(100\) −908.489 −0.908489
\(101\) −666.897 −0.657017 −0.328508 0.944501i \(-0.606546\pi\)
−0.328508 + 0.944501i \(0.606546\pi\)
\(102\) 0 0
\(103\) −872.373 −0.834538 −0.417269 0.908783i \(-0.637013\pi\)
−0.417269 + 0.908783i \(0.637013\pi\)
\(104\) 108.468 0.102271
\(105\) −21.0374 −0.0195528
\(106\) −1843.12 −1.68887
\(107\) 346.327 0.312904 0.156452 0.987686i \(-0.449994\pi\)
0.156452 + 0.987686i \(0.449994\pi\)
\(108\) 171.056 0.152407
\(109\) 135.821 0.119351 0.0596757 0.998218i \(-0.480993\pi\)
0.0596757 + 0.998218i \(0.480993\pi\)
\(110\) −249.043 −0.215867
\(111\) 95.9128 0.0820147
\(112\) 1723.88 1.45438
\(113\) 885.947 0.737548 0.368774 0.929519i \(-0.379778\pi\)
0.368774 + 0.929519i \(0.379778\pi\)
\(114\) −76.9992 −0.0632600
\(115\) 146.891 0.119110
\(116\) −1177.19 −0.942235
\(117\) −1466.68 −1.15893
\(118\) −2957.18 −2.30703
\(119\) 0 0
\(120\) 1.64059 0.00124804
\(121\) −279.276 −0.209824
\(122\) 1902.16 1.41159
\(123\) −98.0110 −0.0718484
\(124\) 1893.56 1.37135
\(125\) 480.276 0.343657
\(126\) −2685.33 −1.89863
\(127\) −1971.07 −1.37720 −0.688598 0.725144i \(-0.741773\pi\)
−0.688598 + 0.725144i \(0.741773\pi\)
\(128\) −253.384 −0.174970
\(129\) −50.7273 −0.0346224
\(130\) 419.949 0.283323
\(131\) −250.787 −0.167262 −0.0836312 0.996497i \(-0.526652\pi\)
−0.0836312 + 0.996497i \(0.526652\pi\)
\(132\) 103.073 0.0679648
\(133\) 1173.39 0.765003
\(134\) −2101.62 −1.35487
\(135\) −44.5159 −0.0283801
\(136\) 0 0
\(137\) 436.372 0.272130 0.136065 0.990700i \(-0.456554\pi\)
0.136065 + 0.990700i \(0.456554\pi\)
\(138\) −125.676 −0.0775233
\(139\) −83.7228 −0.0510883 −0.0255441 0.999674i \(-0.508132\pi\)
−0.0255441 + 0.999674i \(0.508132\pi\)
\(140\) 371.940 0.224533
\(141\) −79.8457 −0.0476895
\(142\) −307.647 −0.181811
\(143\) −1773.47 −1.03710
\(144\) 1817.80 1.05197
\(145\) 306.353 0.175457
\(146\) 1507.73 0.854664
\(147\) −128.856 −0.0722986
\(148\) −1695.73 −0.941811
\(149\) 2357.45 1.29617 0.648085 0.761568i \(-0.275570\pi\)
0.648085 + 0.761568i \(0.275570\pi\)
\(150\) −202.279 −0.110107
\(151\) 149.746 0.0807031 0.0403515 0.999186i \(-0.487152\pi\)
0.0403515 + 0.999186i \(0.487152\pi\)
\(152\) −91.5058 −0.0488296
\(153\) 0 0
\(154\) −3247.02 −1.69904
\(155\) −492.783 −0.255363
\(156\) −173.807 −0.0892030
\(157\) 119.280 0.0606343 0.0303172 0.999540i \(-0.490348\pi\)
0.0303172 + 0.999540i \(0.490348\pi\)
\(158\) −5034.22 −2.53481
\(159\) −198.518 −0.0990156
\(160\) −489.528 −0.241879
\(161\) 1915.16 0.937489
\(162\) −2812.52 −1.36403
\(163\) −152.626 −0.0733411 −0.0366705 0.999327i \(-0.511675\pi\)
−0.0366705 + 0.999327i \(0.511675\pi\)
\(164\) 1732.83 0.825067
\(165\) −26.8238 −0.0126559
\(166\) −2327.96 −1.08846
\(167\) −2525.07 −1.17003 −0.585017 0.811021i \(-0.698912\pi\)
−0.585017 + 0.811021i \(0.698912\pi\)
\(168\) 21.3900 0.00982306
\(169\) 793.506 0.361177
\(170\) 0 0
\(171\) 1237.31 0.553332
\(172\) 896.854 0.397584
\(173\) 328.274 0.144267 0.0721335 0.997395i \(-0.477019\pi\)
0.0721335 + 0.997395i \(0.477019\pi\)
\(174\) −262.106 −0.114197
\(175\) 3082.52 1.33152
\(176\) 2198.03 0.941379
\(177\) −318.509 −0.135258
\(178\) −2400.20 −1.01069
\(179\) 4155.11 1.73501 0.867506 0.497427i \(-0.165722\pi\)
0.867506 + 0.497427i \(0.165722\pi\)
\(180\) 392.204 0.162406
\(181\) 955.425 0.392355 0.196177 0.980568i \(-0.437147\pi\)
0.196177 + 0.980568i \(0.437147\pi\)
\(182\) 5475.29 2.22997
\(183\) 204.877 0.0827592
\(184\) −149.353 −0.0598393
\(185\) 441.298 0.175378
\(186\) 421.611 0.166204
\(187\) 0 0
\(188\) 1411.67 0.547640
\(189\) −580.397 −0.223374
\(190\) −354.276 −0.135273
\(191\) 889.758 0.337072 0.168536 0.985696i \(-0.446096\pi\)
0.168536 + 0.985696i \(0.446096\pi\)
\(192\) 188.931 0.0710152
\(193\) −1732.07 −0.645997 −0.322999 0.946399i \(-0.604691\pi\)
−0.322999 + 0.946399i \(0.604691\pi\)
\(194\) 6267.16 2.31936
\(195\) 45.2316 0.0166108
\(196\) 2278.17 0.830237
\(197\) −4443.91 −1.60718 −0.803592 0.595181i \(-0.797081\pi\)
−0.803592 + 0.595181i \(0.797081\pi\)
\(198\) −3423.93 −1.22893
\(199\) −3118.38 −1.11083 −0.555417 0.831572i \(-0.687441\pi\)
−0.555417 + 0.831572i \(0.687441\pi\)
\(200\) −240.388 −0.0849901
\(201\) −226.360 −0.0794338
\(202\) 2625.25 0.914415
\(203\) 3994.22 1.38098
\(204\) 0 0
\(205\) −450.952 −0.153638
\(206\) 3434.11 1.16148
\(207\) 2019.50 0.678093
\(208\) −3706.42 −1.23555
\(209\) 1496.13 0.495164
\(210\) 82.8140 0.0272129
\(211\) 4532.53 1.47883 0.739413 0.673252i \(-0.235103\pi\)
0.739413 + 0.673252i \(0.235103\pi\)
\(212\) 3509.78 1.13704
\(213\) −33.1358 −0.0106593
\(214\) −1363.32 −0.435489
\(215\) −233.398 −0.0740354
\(216\) 45.2619 0.0142578
\(217\) −6424.89 −2.00991
\(218\) −534.661 −0.166109
\(219\) 162.394 0.0501076
\(220\) 474.242 0.145334
\(221\) 0 0
\(222\) −377.562 −0.114145
\(223\) 123.574 0.0371082 0.0185541 0.999828i \(-0.494094\pi\)
0.0185541 + 0.999828i \(0.494094\pi\)
\(224\) −6382.46 −1.90378
\(225\) 3250.46 0.963100
\(226\) −3487.54 −1.02649
\(227\) 780.134 0.228103 0.114051 0.993475i \(-0.463617\pi\)
0.114051 + 0.993475i \(0.463617\pi\)
\(228\) 146.626 0.0425902
\(229\) −2776.64 −0.801247 −0.400624 0.916243i \(-0.631206\pi\)
−0.400624 + 0.916243i \(0.631206\pi\)
\(230\) −578.237 −0.165773
\(231\) −349.728 −0.0996122
\(232\) −311.487 −0.0881471
\(233\) 5047.60 1.41922 0.709612 0.704593i \(-0.248870\pi\)
0.709612 + 0.704593i \(0.248870\pi\)
\(234\) 5773.60 1.61296
\(235\) −367.373 −0.101978
\(236\) 5631.22 1.55323
\(237\) −542.222 −0.148612
\(238\) 0 0
\(239\) 2254.20 0.610092 0.305046 0.952338i \(-0.401328\pi\)
0.305046 + 0.952338i \(0.401328\pi\)
\(240\) −56.0599 −0.0150777
\(241\) −1729.12 −0.462168 −0.231084 0.972934i \(-0.574227\pi\)
−0.231084 + 0.972934i \(0.574227\pi\)
\(242\) 1099.37 0.292027
\(243\) −919.051 −0.242622
\(244\) −3622.21 −0.950360
\(245\) −592.873 −0.154601
\(246\) 385.821 0.0999962
\(247\) −2522.84 −0.649897
\(248\) 501.042 0.128291
\(249\) −250.738 −0.0638148
\(250\) −1890.61 −0.478291
\(251\) 1314.25 0.330496 0.165248 0.986252i \(-0.447158\pi\)
0.165248 + 0.986252i \(0.447158\pi\)
\(252\) 5113.55 1.27827
\(253\) 2441.93 0.606809
\(254\) 7759.13 1.91674
\(255\) 0 0
\(256\) 4562.26 1.11383
\(257\) −6039.96 −1.46600 −0.733001 0.680228i \(-0.761881\pi\)
−0.733001 + 0.680228i \(0.761881\pi\)
\(258\) 199.689 0.0481863
\(259\) 5753.64 1.38036
\(260\) −799.690 −0.190749
\(261\) 4211.84 0.998874
\(262\) 987.226 0.232790
\(263\) 990.368 0.232201 0.116100 0.993237i \(-0.462961\pi\)
0.116100 + 0.993237i \(0.462961\pi\)
\(264\) 27.2733 0.00635818
\(265\) −913.386 −0.211732
\(266\) −4619.05 −1.06471
\(267\) −258.520 −0.0592552
\(268\) 4002.02 0.912173
\(269\) 3390.55 0.768497 0.384248 0.923230i \(-0.374461\pi\)
0.384248 + 0.923230i \(0.374461\pi\)
\(270\) 175.237 0.0394985
\(271\) −3706.24 −0.830767 −0.415383 0.909646i \(-0.636353\pi\)
−0.415383 + 0.909646i \(0.636353\pi\)
\(272\) 0 0
\(273\) 589.729 0.130740
\(274\) −1717.79 −0.378742
\(275\) 3930.37 0.861855
\(276\) 239.318 0.0521930
\(277\) 7651.83 1.65976 0.829880 0.557942i \(-0.188409\pi\)
0.829880 + 0.557942i \(0.188409\pi\)
\(278\) 329.576 0.0711030
\(279\) −6774.94 −1.45378
\(280\) 98.4161 0.0210053
\(281\) −3194.87 −0.678256 −0.339128 0.940740i \(-0.610132\pi\)
−0.339128 + 0.940740i \(0.610132\pi\)
\(282\) 314.314 0.0663727
\(283\) −5989.81 −1.25815 −0.629076 0.777344i \(-0.716567\pi\)
−0.629076 + 0.777344i \(0.716567\pi\)
\(284\) 585.838 0.122405
\(285\) −38.1581 −0.00793085
\(286\) 6981.27 1.44340
\(287\) −5879.50 −1.20925
\(288\) −6730.20 −1.37702
\(289\) 0 0
\(290\) −1205.96 −0.244195
\(291\) 675.019 0.135980
\(292\) −2871.11 −0.575408
\(293\) −5372.44 −1.07120 −0.535600 0.844472i \(-0.679914\pi\)
−0.535600 + 0.844472i \(0.679914\pi\)
\(294\) 507.245 0.100623
\(295\) −1465.47 −0.289231
\(296\) −448.694 −0.0881074
\(297\) −740.037 −0.144583
\(298\) −9280.11 −1.80397
\(299\) −4117.70 −0.796430
\(300\) 385.191 0.0741301
\(301\) −3043.04 −0.582717
\(302\) −589.477 −0.112320
\(303\) 282.758 0.0536107
\(304\) 3126.80 0.589916
\(305\) 942.645 0.176970
\(306\) 0 0
\(307\) 7020.91 1.30523 0.652613 0.757692i \(-0.273673\pi\)
0.652613 + 0.757692i \(0.273673\pi\)
\(308\) 6183.16 1.14389
\(309\) 369.878 0.0680960
\(310\) 1939.84 0.355406
\(311\) 1026.01 0.187072 0.0935362 0.995616i \(-0.470183\pi\)
0.0935362 + 0.995616i \(0.470183\pi\)
\(312\) −45.9896 −0.00834503
\(313\) 2490.87 0.449815 0.224907 0.974380i \(-0.427792\pi\)
0.224907 + 0.974380i \(0.427792\pi\)
\(314\) −469.548 −0.0843889
\(315\) −1330.75 −0.238030
\(316\) 9586.44 1.70658
\(317\) −2775.18 −0.491702 −0.245851 0.969308i \(-0.579067\pi\)
−0.245851 + 0.969308i \(0.579067\pi\)
\(318\) 781.467 0.137807
\(319\) 5092.84 0.893869
\(320\) 869.278 0.151857
\(321\) −146.840 −0.0255321
\(322\) −7539.05 −1.30477
\(323\) 0 0
\(324\) 5355.77 0.918341
\(325\) −6627.58 −1.13117
\(326\) 600.815 0.102074
\(327\) −57.5870 −0.00973873
\(328\) 458.510 0.0771859
\(329\) −4789.80 −0.802646
\(330\) 105.592 0.0176141
\(331\) −6017.90 −0.999317 −0.499659 0.866222i \(-0.666541\pi\)
−0.499659 + 0.866222i \(0.666541\pi\)
\(332\) 4433.03 0.732813
\(333\) 6067.11 0.998425
\(334\) 9939.96 1.62841
\(335\) −1041.49 −0.169859
\(336\) −730.908 −0.118674
\(337\) 474.374 0.0766789 0.0383395 0.999265i \(-0.487793\pi\)
0.0383395 + 0.999265i \(0.487793\pi\)
\(338\) −3123.64 −0.502674
\(339\) −375.634 −0.0601818
\(340\) 0 0
\(341\) −8192.07 −1.30095
\(342\) −4870.71 −0.770110
\(343\) 994.158 0.156500
\(344\) 237.310 0.0371944
\(345\) −62.2804 −0.00971903
\(346\) −1292.25 −0.200786
\(347\) −1475.45 −0.228261 −0.114130 0.993466i \(-0.536408\pi\)
−0.114130 + 0.993466i \(0.536408\pi\)
\(348\) 499.118 0.0768837
\(349\) 6855.25 1.05144 0.525721 0.850657i \(-0.323796\pi\)
0.525721 + 0.850657i \(0.323796\pi\)
\(350\) −12134.4 −1.85317
\(351\) 1247.89 0.189764
\(352\) −8137.97 −1.23226
\(353\) −7485.74 −1.12868 −0.564342 0.825541i \(-0.690870\pi\)
−0.564342 + 0.825541i \(0.690870\pi\)
\(354\) 1253.82 0.188247
\(355\) −152.459 −0.0227935
\(356\) 4570.60 0.680454
\(357\) 0 0
\(358\) −16356.6 −2.41473
\(359\) 7169.54 1.05402 0.527011 0.849859i \(-0.323313\pi\)
0.527011 + 0.849859i \(0.323313\pi\)
\(360\) 103.778 0.0151933
\(361\) −4730.69 −0.689705
\(362\) −3761.04 −0.546067
\(363\) 118.411 0.0171211
\(364\) −10426.4 −1.50134
\(365\) 747.180 0.107149
\(366\) −806.500 −0.115182
\(367\) 6608.80 0.939991 0.469995 0.882669i \(-0.344256\pi\)
0.469995 + 0.882669i \(0.344256\pi\)
\(368\) 5103.46 0.722925
\(369\) −6199.84 −0.874663
\(370\) −1737.17 −0.244085
\(371\) −11908.7 −1.66650
\(372\) −802.855 −0.111898
\(373\) −1602.91 −0.222508 −0.111254 0.993792i \(-0.535487\pi\)
−0.111254 + 0.993792i \(0.535487\pi\)
\(374\) 0 0
\(375\) −203.633 −0.0280415
\(376\) 373.530 0.0512323
\(377\) −8587.79 −1.17319
\(378\) 2284.74 0.310885
\(379\) −1743.08 −0.236243 −0.118121 0.992999i \(-0.537687\pi\)
−0.118121 + 0.992999i \(0.537687\pi\)
\(380\) 674.632 0.0910734
\(381\) 835.715 0.112375
\(382\) −3502.55 −0.469125
\(383\) −1155.24 −0.154125 −0.0770627 0.997026i \(-0.524554\pi\)
−0.0770627 + 0.997026i \(0.524554\pi\)
\(384\) 107.432 0.0142771
\(385\) −1609.11 −0.213008
\(386\) 6818.33 0.899078
\(387\) −3208.83 −0.421484
\(388\) −11934.3 −1.56152
\(389\) 3698.95 0.482119 0.241060 0.970510i \(-0.422505\pi\)
0.241060 + 0.970510i \(0.422505\pi\)
\(390\) −178.055 −0.0231183
\(391\) 0 0
\(392\) 602.809 0.0776696
\(393\) 106.332 0.0136481
\(394\) 17493.5 2.23683
\(395\) −2494.78 −0.317788
\(396\) 6520.04 0.827385
\(397\) 13587.9 1.71778 0.858891 0.512159i \(-0.171154\pi\)
0.858891 + 0.512159i \(0.171154\pi\)
\(398\) 12275.5 1.54602
\(399\) −497.505 −0.0624221
\(400\) 8214.20 1.02678
\(401\) 2664.34 0.331798 0.165899 0.986143i \(-0.446947\pi\)
0.165899 + 0.986143i \(0.446947\pi\)
\(402\) 891.068 0.110553
\(403\) 13813.9 1.70749
\(404\) −4999.14 −0.615635
\(405\) −1393.79 −0.171007
\(406\) −15723.3 −1.92201
\(407\) 7336.19 0.893467
\(408\) 0 0
\(409\) 4145.86 0.501222 0.250611 0.968088i \(-0.419368\pi\)
0.250611 + 0.968088i \(0.419368\pi\)
\(410\) 1775.18 0.213829
\(411\) −185.018 −0.0222050
\(412\) −6539.42 −0.781976
\(413\) −19106.8 −2.27648
\(414\) −7949.80 −0.943748
\(415\) −1153.65 −0.136459
\(416\) 13722.6 1.61733
\(417\) 35.4977 0.00416866
\(418\) −5889.52 −0.689153
\(419\) 14247.5 1.66118 0.830592 0.556882i \(-0.188003\pi\)
0.830592 + 0.556882i \(0.188003\pi\)
\(420\) −157.699 −0.0183213
\(421\) 3191.61 0.369477 0.184738 0.982788i \(-0.440856\pi\)
0.184738 + 0.982788i \(0.440856\pi\)
\(422\) −17842.4 −2.05818
\(423\) −5050.77 −0.580560
\(424\) 928.695 0.106371
\(425\) 0 0
\(426\) 130.439 0.0148352
\(427\) 12290.2 1.39289
\(428\) 2596.11 0.293196
\(429\) 751.935 0.0846241
\(430\) 918.774 0.103040
\(431\) −1639.42 −0.183220 −0.0916101 0.995795i \(-0.529201\pi\)
−0.0916101 + 0.995795i \(0.529201\pi\)
\(432\) −1546.63 −0.172250
\(433\) 16579.9 1.84013 0.920066 0.391764i \(-0.128135\pi\)
0.920066 + 0.391764i \(0.128135\pi\)
\(434\) 25291.7 2.79732
\(435\) −129.891 −0.0143168
\(436\) 1018.13 0.111834
\(437\) 3473.76 0.380258
\(438\) −639.266 −0.0697382
\(439\) 7347.09 0.798764 0.399382 0.916785i \(-0.369225\pi\)
0.399382 + 0.916785i \(0.369225\pi\)
\(440\) 125.486 0.0135961
\(441\) −8151.02 −0.880144
\(442\) 0 0
\(443\) −13325.5 −1.42915 −0.714575 0.699559i \(-0.753380\pi\)
−0.714575 + 0.699559i \(0.753380\pi\)
\(444\) 718.974 0.0768491
\(445\) −1189.46 −0.126709
\(446\) −486.451 −0.0516460
\(447\) −999.536 −0.105764
\(448\) 11333.6 1.19523
\(449\) 9686.35 1.01810 0.509050 0.860737i \(-0.329997\pi\)
0.509050 + 0.860737i \(0.329997\pi\)
\(450\) −12795.5 −1.34041
\(451\) −7496.67 −0.782715
\(452\) 6641.17 0.691094
\(453\) −63.4910 −0.00658514
\(454\) −3071.01 −0.317466
\(455\) 2713.36 0.279570
\(456\) 38.7976 0.00398436
\(457\) 698.065 0.0714532 0.0357266 0.999362i \(-0.488625\pi\)
0.0357266 + 0.999362i \(0.488625\pi\)
\(458\) 10930.3 1.11515
\(459\) 0 0
\(460\) 1101.11 0.111608
\(461\) 6505.58 0.657256 0.328628 0.944459i \(-0.393414\pi\)
0.328628 + 0.944459i \(0.393414\pi\)
\(462\) 1376.71 0.138637
\(463\) −1181.74 −0.118618 −0.0593089 0.998240i \(-0.518890\pi\)
−0.0593089 + 0.998240i \(0.518890\pi\)
\(464\) 10643.7 1.06492
\(465\) 208.935 0.0208369
\(466\) −19869.9 −1.97523
\(467\) 2950.96 0.292407 0.146204 0.989255i \(-0.453295\pi\)
0.146204 + 0.989255i \(0.453295\pi\)
\(468\) −10994.4 −1.08593
\(469\) −13578.9 −1.33692
\(470\) 1446.17 0.141929
\(471\) −50.5737 −0.00494759
\(472\) 1490.03 0.145306
\(473\) −3880.03 −0.377176
\(474\) 2134.46 0.206834
\(475\) 5591.14 0.540082
\(476\) 0 0
\(477\) −12557.5 −1.20539
\(478\) −8873.69 −0.849107
\(479\) 8890.21 0.848025 0.424013 0.905656i \(-0.360621\pi\)
0.424013 + 0.905656i \(0.360621\pi\)
\(480\) 207.556 0.0197366
\(481\) −12370.6 −1.17267
\(482\) 6806.70 0.643230
\(483\) −812.011 −0.0764965
\(484\) −2093.49 −0.196609
\(485\) 3105.78 0.290776
\(486\) 3617.85 0.337673
\(487\) −7104.47 −0.661056 −0.330528 0.943796i \(-0.607227\pi\)
−0.330528 + 0.943796i \(0.607227\pi\)
\(488\) −958.444 −0.0889072
\(489\) 64.7121 0.00598442
\(490\) 2333.85 0.215169
\(491\) 7638.33 0.702063 0.351031 0.936364i \(-0.385831\pi\)
0.351031 + 0.936364i \(0.385831\pi\)
\(492\) −734.703 −0.0673231
\(493\) 0 0
\(494\) 9931.20 0.904505
\(495\) −1696.78 −0.154070
\(496\) −17120.9 −1.54990
\(497\) −1987.76 −0.179403
\(498\) 987.033 0.0888153
\(499\) −20202.5 −1.81241 −0.906203 0.422843i \(-0.861032\pi\)
−0.906203 + 0.422843i \(0.861032\pi\)
\(500\) 3600.21 0.322012
\(501\) 1070.61 0.0954714
\(502\) −5173.54 −0.459973
\(503\) 20469.5 1.81449 0.907244 0.420604i \(-0.138182\pi\)
0.907244 + 0.420604i \(0.138182\pi\)
\(504\) 1353.06 0.119583
\(505\) 1300.98 0.114639
\(506\) −9612.68 −0.844537
\(507\) −336.439 −0.0294710
\(508\) −14775.4 −1.29045
\(509\) −19503.1 −1.69835 −0.849173 0.528116i \(-0.822899\pi\)
−0.849173 + 0.528116i \(0.822899\pi\)
\(510\) 0 0
\(511\) 9741.73 0.843344
\(512\) −15932.3 −1.37523
\(513\) −1052.74 −0.0906033
\(514\) 23776.4 2.04033
\(515\) 1701.82 0.145614
\(516\) −380.258 −0.0324417
\(517\) −6107.25 −0.519529
\(518\) −22649.3 −1.92114
\(519\) −139.185 −0.0117718
\(520\) −211.600 −0.0178447
\(521\) 6963.51 0.585561 0.292780 0.956180i \(-0.405420\pi\)
0.292780 + 0.956180i \(0.405420\pi\)
\(522\) −16580.0 −1.39020
\(523\) 2507.22 0.209623 0.104812 0.994492i \(-0.466576\pi\)
0.104812 + 0.994492i \(0.466576\pi\)
\(524\) −1879.93 −0.156728
\(525\) −1306.96 −0.108648
\(526\) −3898.60 −0.323169
\(527\) 0 0
\(528\) −931.946 −0.0768139
\(529\) −6497.24 −0.534005
\(530\) 3595.56 0.294681
\(531\) −20147.8 −1.64659
\(532\) 8795.85 0.716820
\(533\) 12641.3 1.02730
\(534\) 1017.67 0.0824695
\(535\) −675.614 −0.0545969
\(536\) 1058.94 0.0853348
\(537\) −1761.73 −0.141572
\(538\) −13346.9 −1.06957
\(539\) −9855.99 −0.787620
\(540\) −333.697 −0.0265926
\(541\) 19376.1 1.53982 0.769912 0.638149i \(-0.220300\pi\)
0.769912 + 0.638149i \(0.220300\pi\)
\(542\) 14589.6 1.15623
\(543\) −405.092 −0.0320150
\(544\) 0 0
\(545\) −264.960 −0.0208250
\(546\) −2321.47 −0.181960
\(547\) −2956.31 −0.231083 −0.115542 0.993303i \(-0.536860\pi\)
−0.115542 + 0.993303i \(0.536860\pi\)
\(548\) 3271.10 0.254990
\(549\) 12959.8 1.00749
\(550\) −15471.9 −1.19950
\(551\) 7244.81 0.560144
\(552\) 63.3242 0.00488271
\(553\) −32526.9 −2.50124
\(554\) −30121.5 −2.31000
\(555\) −187.106 −0.0143103
\(556\) −627.597 −0.0478706
\(557\) 3920.00 0.298197 0.149099 0.988822i \(-0.452363\pi\)
0.149099 + 0.988822i \(0.452363\pi\)
\(558\) 26669.6 2.02333
\(559\) 6542.70 0.495039
\(560\) −3362.93 −0.253768
\(561\) 0 0
\(562\) 12576.6 0.943975
\(563\) 20821.5 1.55865 0.779326 0.626618i \(-0.215561\pi\)
0.779326 + 0.626618i \(0.215561\pi\)
\(564\) −598.534 −0.0446859
\(565\) −1728.30 −0.128691
\(566\) 23579.0 1.75106
\(567\) −18172.2 −1.34596
\(568\) 155.014 0.0114511
\(569\) −7849.66 −0.578339 −0.289170 0.957278i \(-0.593379\pi\)
−0.289170 + 0.957278i \(0.593379\pi\)
\(570\) 150.210 0.0110379
\(571\) 24535.7 1.79822 0.899112 0.437719i \(-0.144214\pi\)
0.899112 + 0.437719i \(0.144214\pi\)
\(572\) −13294.1 −0.971776
\(573\) −377.250 −0.0275041
\(574\) 23144.7 1.68300
\(575\) 9125.67 0.661855
\(576\) 11951.1 0.864521
\(577\) −7549.37 −0.544687 −0.272343 0.962200i \(-0.587799\pi\)
−0.272343 + 0.962200i \(0.587799\pi\)
\(578\) 0 0
\(579\) 734.384 0.0527115
\(580\) 2296.46 0.164406
\(581\) −15041.3 −1.07404
\(582\) −2657.22 −0.189253
\(583\) −15184.2 −1.07867
\(584\) −759.703 −0.0538300
\(585\) 2861.19 0.202215
\(586\) 21148.7 1.49086
\(587\) 6451.77 0.453651 0.226826 0.973935i \(-0.427165\pi\)
0.226826 + 0.973935i \(0.427165\pi\)
\(588\) −965.925 −0.0677450
\(589\) −11653.6 −0.815244
\(590\) 5768.85 0.402542
\(591\) 1884.18 0.131142
\(592\) 15332.1 1.06444
\(593\) 176.420 0.0122170 0.00610850 0.999981i \(-0.498056\pi\)
0.00610850 + 0.999981i \(0.498056\pi\)
\(594\) 2913.16 0.201227
\(595\) 0 0
\(596\) 17671.7 1.21453
\(597\) 1322.16 0.0906408
\(598\) 16209.4 1.10845
\(599\) 11248.7 0.767292 0.383646 0.923480i \(-0.374668\pi\)
0.383646 + 0.923480i \(0.374668\pi\)
\(600\) 101.923 0.00693495
\(601\) 25418.5 1.72519 0.862596 0.505893i \(-0.168837\pi\)
0.862596 + 0.505893i \(0.168837\pi\)
\(602\) 11979.0 0.811007
\(603\) −14318.7 −0.967006
\(604\) 1122.52 0.0756201
\(605\) 544.812 0.0366112
\(606\) −1113.08 −0.0746136
\(607\) 19680.4 1.31599 0.657993 0.753024i \(-0.271406\pi\)
0.657993 + 0.753024i \(0.271406\pi\)
\(608\) −11576.7 −0.772196
\(609\) −1693.51 −0.112684
\(610\) −3710.74 −0.246300
\(611\) 10298.3 0.681876
\(612\) 0 0
\(613\) 3619.34 0.238473 0.119236 0.992866i \(-0.461955\pi\)
0.119236 + 0.992866i \(0.461955\pi\)
\(614\) −27637.9 −1.81657
\(615\) 191.200 0.0125364
\(616\) 1636.08 0.107012
\(617\) −19744.9 −1.28833 −0.644164 0.764887i \(-0.722795\pi\)
−0.644164 + 0.764887i \(0.722795\pi\)
\(618\) −1456.03 −0.0947737
\(619\) −18695.9 −1.21397 −0.606987 0.794712i \(-0.707622\pi\)
−0.606987 + 0.794712i \(0.707622\pi\)
\(620\) −3693.96 −0.239279
\(621\) −1718.24 −0.111032
\(622\) −4038.89 −0.260361
\(623\) −15508.1 −0.997304
\(624\) 1571.49 0.100817
\(625\) 14212.4 0.909592
\(626\) −9805.33 −0.626038
\(627\) −634.345 −0.0404040
\(628\) 894.139 0.0568153
\(629\) 0 0
\(630\) 5238.53 0.331283
\(631\) −10087.2 −0.636394 −0.318197 0.948025i \(-0.603077\pi\)
−0.318197 + 0.948025i \(0.603077\pi\)
\(632\) 2536.59 0.159652
\(633\) −1921.75 −0.120668
\(634\) 10924.5 0.684335
\(635\) 3845.15 0.240300
\(636\) −1488.11 −0.0927792
\(637\) 16619.6 1.03374
\(638\) −20048.0 −1.24406
\(639\) −2096.06 −0.129763
\(640\) 494.300 0.0305296
\(641\) −10265.1 −0.632523 −0.316261 0.948672i \(-0.602428\pi\)
−0.316261 + 0.948672i \(0.602428\pi\)
\(642\) 578.036 0.0355347
\(643\) −7778.26 −0.477052 −0.238526 0.971136i \(-0.576664\pi\)
−0.238526 + 0.971136i \(0.576664\pi\)
\(644\) 14356.3 0.878443
\(645\) 98.9587 0.00604108
\(646\) 0 0
\(647\) 5015.28 0.304746 0.152373 0.988323i \(-0.451308\pi\)
0.152373 + 0.988323i \(0.451308\pi\)
\(648\) 1417.15 0.0859118
\(649\) −24362.2 −1.47350
\(650\) 26089.5 1.57433
\(651\) 2724.10 0.164003
\(652\) −1144.10 −0.0687218
\(653\) 14563.9 0.872784 0.436392 0.899757i \(-0.356256\pi\)
0.436392 + 0.899757i \(0.356256\pi\)
\(654\) 226.692 0.0135541
\(655\) 489.235 0.0291847
\(656\) −15667.5 −0.932491
\(657\) 10272.5 0.609997
\(658\) 18855.1 1.11710
\(659\) −6199.21 −0.366445 −0.183222 0.983072i \(-0.558653\pi\)
−0.183222 + 0.983072i \(0.558653\pi\)
\(660\) −201.075 −0.0118588
\(661\) 15131.8 0.890406 0.445203 0.895430i \(-0.353132\pi\)
0.445203 + 0.895430i \(0.353132\pi\)
\(662\) 23689.6 1.39082
\(663\) 0 0
\(664\) 1172.99 0.0685554
\(665\) −2289.04 −0.133481
\(666\) −23883.3 −1.38958
\(667\) 11824.7 0.686440
\(668\) −18928.2 −1.09634
\(669\) −52.3943 −0.00302792
\(670\) 4099.84 0.236404
\(671\) 15670.6 0.901577
\(672\) 2706.11 0.155343
\(673\) −27043.8 −1.54898 −0.774488 0.632589i \(-0.781992\pi\)
−0.774488 + 0.632589i \(0.781992\pi\)
\(674\) −1867.38 −0.106719
\(675\) −2765.57 −0.157699
\(676\) 5948.22 0.338429
\(677\) 2671.45 0.151658 0.0758288 0.997121i \(-0.475840\pi\)
0.0758288 + 0.997121i \(0.475840\pi\)
\(678\) 1478.69 0.0837591
\(679\) 40493.2 2.28864
\(680\) 0 0
\(681\) −330.770 −0.0186125
\(682\) 32248.2 1.81063
\(683\) 14786.6 0.828393 0.414197 0.910187i \(-0.364063\pi\)
0.414197 + 0.910187i \(0.364063\pi\)
\(684\) 9275.07 0.518481
\(685\) −851.274 −0.0474825
\(686\) −3913.52 −0.217812
\(687\) 1177.27 0.0653795
\(688\) −8109.01 −0.449350
\(689\) 25604.4 1.41575
\(690\) 245.168 0.0135266
\(691\) −18670.1 −1.02785 −0.513924 0.857836i \(-0.671809\pi\)
−0.513924 + 0.857836i \(0.671809\pi\)
\(692\) 2460.78 0.135181
\(693\) −22122.6 −1.21265
\(694\) 5808.14 0.317686
\(695\) 163.326 0.00891413
\(696\) 132.068 0.00719255
\(697\) 0 0
\(698\) −26985.8 −1.46336
\(699\) −2140.14 −0.115805
\(700\) 23106.9 1.24766
\(701\) −27668.5 −1.49076 −0.745381 0.666638i \(-0.767733\pi\)
−0.745381 + 0.666638i \(0.767733\pi\)
\(702\) −4912.32 −0.264107
\(703\) 10436.1 0.559892
\(704\) 14451.0 0.773639
\(705\) 155.763 0.00832109
\(706\) 29467.7 1.57087
\(707\) 16962.2 0.902303
\(708\) −2387.59 −0.126739
\(709\) 10439.9 0.553002 0.276501 0.961014i \(-0.410825\pi\)
0.276501 + 0.961014i \(0.410825\pi\)
\(710\) 600.157 0.0317232
\(711\) −34299.1 −1.80917
\(712\) 1209.39 0.0636572
\(713\) −19020.6 −0.999059
\(714\) 0 0
\(715\) 3459.68 0.180957
\(716\) 31147.2 1.62573
\(717\) −955.761 −0.0497818
\(718\) −28223.0 −1.46695
\(719\) 17815.1 0.924050 0.462025 0.886867i \(-0.347123\pi\)
0.462025 + 0.886867i \(0.347123\pi\)
\(720\) −3546.16 −0.183552
\(721\) 22188.4 1.14610
\(722\) 18622.4 0.959909
\(723\) 733.132 0.0377116
\(724\) 7161.99 0.367643
\(725\) 19032.3 0.974955
\(726\) −466.125 −0.0238286
\(727\) 4296.42 0.219182 0.109591 0.993977i \(-0.465046\pi\)
0.109591 + 0.993977i \(0.465046\pi\)
\(728\) −2758.84 −0.140452
\(729\) −18901.0 −0.960273
\(730\) −2941.28 −0.149126
\(731\) 0 0
\(732\) 1535.78 0.0775467
\(733\) −28559.1 −1.43909 −0.719546 0.694445i \(-0.755650\pi\)
−0.719546 + 0.694445i \(0.755650\pi\)
\(734\) −26015.6 −1.30825
\(735\) 251.373 0.0126150
\(736\) −18895.0 −0.946304
\(737\) −17313.8 −0.865351
\(738\) 24405.7 1.21733
\(739\) −10646.9 −0.529974 −0.264987 0.964252i \(-0.585368\pi\)
−0.264987 + 0.964252i \(0.585368\pi\)
\(740\) 3308.03 0.164332
\(741\) 1069.66 0.0530298
\(742\) 46878.8 2.31937
\(743\) 13964.7 0.689521 0.344760 0.938691i \(-0.387960\pi\)
0.344760 + 0.938691i \(0.387960\pi\)
\(744\) −212.437 −0.0104682
\(745\) −4598.90 −0.226162
\(746\) 6309.87 0.309679
\(747\) −15860.8 −0.776864
\(748\) 0 0
\(749\) −8808.65 −0.429721
\(750\) 801.602 0.0390272
\(751\) 26761.0 1.30030 0.650149 0.759807i \(-0.274707\pi\)
0.650149 + 0.759807i \(0.274707\pi\)
\(752\) −12763.7 −0.618943
\(753\) −557.229 −0.0269675
\(754\) 33805.9 1.63281
\(755\) −292.124 −0.0140815
\(756\) −4350.73 −0.209305
\(757\) −1399.40 −0.0671889 −0.0335945 0.999436i \(-0.510695\pi\)
−0.0335945 + 0.999436i \(0.510695\pi\)
\(758\) 6861.66 0.328795
\(759\) −1035.36 −0.0495139
\(760\) 178.509 0.00852002
\(761\) −29282.9 −1.39488 −0.697440 0.716644i \(-0.745677\pi\)
−0.697440 + 0.716644i \(0.745677\pi\)
\(762\) −3289.80 −0.156400
\(763\) −3454.54 −0.163909
\(764\) 6669.74 0.315841
\(765\) 0 0
\(766\) 4547.62 0.214507
\(767\) 41080.7 1.93395
\(768\) −1934.36 −0.0908856
\(769\) 11804.9 0.553571 0.276786 0.960932i \(-0.410731\pi\)
0.276786 + 0.960932i \(0.410731\pi\)
\(770\) 6334.29 0.296457
\(771\) 2560.89 0.119622
\(772\) −12983.9 −0.605310
\(773\) −18421.7 −0.857155 −0.428578 0.903505i \(-0.640985\pi\)
−0.428578 + 0.903505i \(0.640985\pi\)
\(774\) 12631.6 0.586607
\(775\) −30614.4 −1.41897
\(776\) −3157.84 −0.146082
\(777\) −2439.49 −0.112634
\(778\) −14561.0 −0.670998
\(779\) −10664.4 −0.490489
\(780\) 339.062 0.0155646
\(781\) −2534.49 −0.116122
\(782\) 0 0
\(783\) −3583.53 −0.163557
\(784\) −20598.3 −0.938335
\(785\) −232.691 −0.0105798
\(786\) −418.575 −0.0189950
\(787\) 22391.5 1.01419 0.507096 0.861889i \(-0.330719\pi\)
0.507096 + 0.861889i \(0.330719\pi\)
\(788\) −33312.1 −1.50596
\(789\) −419.908 −0.0189469
\(790\) 9820.74 0.442286
\(791\) −22533.6 −1.01290
\(792\) 1725.22 0.0774027
\(793\) −26424.6 −1.18331
\(794\) −53489.1 −2.39075
\(795\) 387.268 0.0172767
\(796\) −23375.7 −1.04087
\(797\) −13676.1 −0.607818 −0.303909 0.952701i \(-0.598292\pi\)
−0.303909 + 0.952701i \(0.598292\pi\)
\(798\) 1958.43 0.0868770
\(799\) 0 0
\(800\) −30412.2 −1.34404
\(801\) −16353.1 −0.721357
\(802\) −10488.2 −0.461786
\(803\) 12421.2 0.545872
\(804\) −1696.82 −0.0744308
\(805\) −3736.09 −0.163578
\(806\) −54378.5 −2.37643
\(807\) −1437.56 −0.0627071
\(808\) −1322.79 −0.0575933
\(809\) 100.880 0.00438410 0.00219205 0.999998i \(-0.499302\pi\)
0.00219205 + 0.999998i \(0.499302\pi\)
\(810\) 5486.67 0.238002
\(811\) −16719.6 −0.723926 −0.361963 0.932192i \(-0.617893\pi\)
−0.361963 + 0.932192i \(0.617893\pi\)
\(812\) 29941.2 1.29400
\(813\) 1571.41 0.0677882
\(814\) −28879.0 −1.24350
\(815\) 297.743 0.0127969
\(816\) 0 0
\(817\) −5519.54 −0.236358
\(818\) −16320.2 −0.697584
\(819\) 37304.2 1.59159
\(820\) −3380.39 −0.143962
\(821\) 4932.52 0.209679 0.104839 0.994489i \(-0.466567\pi\)
0.104839 + 0.994489i \(0.466567\pi\)
\(822\) 728.326 0.0309042
\(823\) −26576.3 −1.12563 −0.562814 0.826583i \(-0.690281\pi\)
−0.562814 + 0.826583i \(0.690281\pi\)
\(824\) −1730.35 −0.0731547
\(825\) −1666.44 −0.0703249
\(826\) 75214.2 3.16833
\(827\) 31136.7 1.30923 0.654613 0.755965i \(-0.272832\pi\)
0.654613 + 0.755965i \(0.272832\pi\)
\(828\) 15138.5 0.635384
\(829\) −17274.9 −0.723742 −0.361871 0.932228i \(-0.617862\pi\)
−0.361871 + 0.932228i \(0.617862\pi\)
\(830\) 4541.37 0.189920
\(831\) −3244.31 −0.135432
\(832\) −24367.9 −1.01539
\(833\) 0 0
\(834\) −139.737 −0.00580180
\(835\) 4925.90 0.204153
\(836\) 11215.2 0.463977
\(837\) 5764.28 0.238044
\(838\) −56085.4 −2.31198
\(839\) 5614.44 0.231027 0.115514 0.993306i \(-0.463149\pi\)
0.115514 + 0.993306i \(0.463149\pi\)
\(840\) −41.7276 −0.00171397
\(841\) 272.434 0.0111704
\(842\) −12563.8 −0.514226
\(843\) 1354.60 0.0553438
\(844\) 33976.4 1.38568
\(845\) −1547.97 −0.0630198
\(846\) 19882.4 0.808004
\(847\) 7103.25 0.288159
\(848\) −31734.0 −1.28508
\(849\) 2539.63 0.102662
\(850\) 0 0
\(851\) 17033.4 0.686132
\(852\) −248.390 −0.00998792
\(853\) 3799.83 0.152525 0.0762625 0.997088i \(-0.475701\pi\)
0.0762625 + 0.997088i \(0.475701\pi\)
\(854\) −48380.5 −1.93858
\(855\) −2413.75 −0.0965480
\(856\) 686.938 0.0274288
\(857\) −42210.3 −1.68247 −0.841235 0.540669i \(-0.818171\pi\)
−0.841235 + 0.540669i \(0.818171\pi\)
\(858\) −2960.00 −0.117777
\(859\) 3132.98 0.124442 0.0622212 0.998062i \(-0.480182\pi\)
0.0622212 + 0.998062i \(0.480182\pi\)
\(860\) −1749.58 −0.0693724
\(861\) 2492.86 0.0986718
\(862\) 6453.58 0.255000
\(863\) 31934.9 1.25965 0.629824 0.776738i \(-0.283127\pi\)
0.629824 + 0.776738i \(0.283127\pi\)
\(864\) 5726.22 0.225474
\(865\) −640.396 −0.0251724
\(866\) −65266.8 −2.56104
\(867\) 0 0
\(868\) −48161.8 −1.88332
\(869\) −41473.5 −1.61898
\(870\) 511.317 0.0199256
\(871\) 29195.4 1.13576
\(872\) 269.400 0.0104622
\(873\) 42699.4 1.65539
\(874\) −13674.5 −0.529230
\(875\) −12215.6 −0.471956
\(876\) 1217.33 0.0469517
\(877\) 15340.7 0.590670 0.295335 0.955394i \(-0.404569\pi\)
0.295335 + 0.955394i \(0.404569\pi\)
\(878\) −28921.9 −1.11169
\(879\) 2277.87 0.0874068
\(880\) −4287.91 −0.164256
\(881\) 46287.9 1.77012 0.885062 0.465472i \(-0.154116\pi\)
0.885062 + 0.465472i \(0.154116\pi\)
\(882\) 32086.6 1.22496
\(883\) −26455.9 −1.00828 −0.504140 0.863622i \(-0.668190\pi\)
−0.504140 + 0.863622i \(0.668190\pi\)
\(884\) 0 0
\(885\) 621.348 0.0236004
\(886\) 52456.0 1.98904
\(887\) −18030.9 −0.682546 −0.341273 0.939964i \(-0.610858\pi\)
−0.341273 + 0.939964i \(0.610858\pi\)
\(888\) 190.242 0.00718932
\(889\) 50133.0 1.89135
\(890\) 4682.31 0.176350
\(891\) −23170.5 −0.871202
\(892\) 926.327 0.0347710
\(893\) −8687.85 −0.325563
\(894\) 3934.69 0.147199
\(895\) −8105.77 −0.302733
\(896\) 6444.68 0.240292
\(897\) 1745.87 0.0649864
\(898\) −38130.4 −1.41696
\(899\) −39669.1 −1.47168
\(900\) 24365.9 0.902440
\(901\) 0 0
\(902\) 29510.7 1.08936
\(903\) 1290.22 0.0475481
\(904\) 1757.27 0.0646526
\(905\) −1863.84 −0.0684599
\(906\) 249.933 0.00916498
\(907\) −45743.1 −1.67461 −0.837307 0.546733i \(-0.815871\pi\)
−0.837307 + 0.546733i \(0.815871\pi\)
\(908\) 5847.99 0.213736
\(909\) 17886.3 0.652642
\(910\) −10681.2 −0.389096
\(911\) 1701.39 0.0618766 0.0309383 0.999521i \(-0.490150\pi\)
0.0309383 + 0.999521i \(0.490150\pi\)
\(912\) −1325.74 −0.0481355
\(913\) −19178.5 −0.695197
\(914\) −2747.94 −0.0994462
\(915\) −399.673 −0.0144402
\(916\) −20814.1 −0.750782
\(917\) 6378.64 0.229707
\(918\) 0 0
\(919\) 30814.5 1.10607 0.553035 0.833158i \(-0.313470\pi\)
0.553035 + 0.833158i \(0.313470\pi\)
\(920\) 291.357 0.0104410
\(921\) −2976.80 −0.106503
\(922\) −25609.3 −0.914747
\(923\) 4273.79 0.152409
\(924\) −2621.61 −0.0933383
\(925\) 27415.9 0.974517
\(926\) 4651.93 0.165088
\(927\) 23397.2 0.828982
\(928\) −39407.1 −1.39397
\(929\) 4093.21 0.144558 0.0722788 0.997384i \(-0.476973\pi\)
0.0722788 + 0.997384i \(0.476973\pi\)
\(930\) −822.477 −0.0290001
\(931\) −14020.6 −0.493563
\(932\) 37837.4 1.32984
\(933\) −435.018 −0.0152646
\(934\) −11616.5 −0.406963
\(935\) 0 0
\(936\) −2909.15 −0.101590
\(937\) −16517.3 −0.575877 −0.287939 0.957649i \(-0.592970\pi\)
−0.287939 + 0.957649i \(0.592970\pi\)
\(938\) 53453.6 1.86068
\(939\) −1056.11 −0.0367036
\(940\) −2753.88 −0.0955548
\(941\) −6860.74 −0.237677 −0.118838 0.992914i \(-0.537917\pi\)
−0.118838 + 0.992914i \(0.537917\pi\)
\(942\) 199.084 0.00688589
\(943\) −17406.0 −0.601080
\(944\) −50915.3 −1.75546
\(945\) 1132.24 0.0389754
\(946\) 15273.8 0.524941
\(947\) −33298.8 −1.14262 −0.571312 0.820733i \(-0.693565\pi\)
−0.571312 + 0.820733i \(0.693565\pi\)
\(948\) −4064.57 −0.139252
\(949\) −20945.2 −0.716450
\(950\) −22009.6 −0.751669
\(951\) 1176.65 0.0401215
\(952\) 0 0
\(953\) −24857.5 −0.844925 −0.422462 0.906380i \(-0.638834\pi\)
−0.422462 + 0.906380i \(0.638834\pi\)
\(954\) 49432.9 1.67762
\(955\) −1735.74 −0.0588138
\(956\) 16897.8 0.571666
\(957\) −2159.32 −0.0729372
\(958\) −34996.4 −1.18025
\(959\) −11098.9 −0.373725
\(960\) −368.566 −0.0123911
\(961\) 34018.6 1.14191
\(962\) 48697.1 1.63208
\(963\) −9288.57 −0.310820
\(964\) −12961.7 −0.433059
\(965\) 3378.93 0.112717
\(966\) 3196.49 0.106465
\(967\) 15832.2 0.526504 0.263252 0.964727i \(-0.415205\pi\)
0.263252 + 0.964727i \(0.415205\pi\)
\(968\) −553.943 −0.0183930
\(969\) 0 0
\(970\) −12226.0 −0.404693
\(971\) −39521.9 −1.30620 −0.653098 0.757273i \(-0.726531\pi\)
−0.653098 + 0.757273i \(0.726531\pi\)
\(972\) −6889.32 −0.227341
\(973\) 2129.45 0.0701612
\(974\) 27966.8 0.920036
\(975\) 2810.03 0.0923007
\(976\) 32750.6 1.07410
\(977\) −11587.3 −0.379438 −0.189719 0.981838i \(-0.560758\pi\)
−0.189719 + 0.981838i \(0.560758\pi\)
\(978\) −254.740 −0.00832893
\(979\) −19773.7 −0.645525
\(980\) −4444.25 −0.144864
\(981\) −3642.75 −0.118557
\(982\) −30068.4 −0.977108
\(983\) −32119.5 −1.04217 −0.521085 0.853505i \(-0.674472\pi\)
−0.521085 + 0.853505i \(0.674472\pi\)
\(984\) −194.404 −0.00629815
\(985\) 8669.17 0.280429
\(986\) 0 0
\(987\) 2030.83 0.0654936
\(988\) −18911.5 −0.608964
\(989\) −9008.80 −0.289649
\(990\) 6679.40 0.214429
\(991\) 18717.3 0.599976 0.299988 0.953943i \(-0.403017\pi\)
0.299988 + 0.953943i \(0.403017\pi\)
\(992\) 63388.2 2.02881
\(993\) 2551.54 0.0815414
\(994\) 7824.83 0.249687
\(995\) 6083.32 0.193823
\(996\) −1879.56 −0.0597955
\(997\) −11611.3 −0.368840 −0.184420 0.982848i \(-0.559041\pi\)
−0.184420 + 0.982848i \(0.559041\pi\)
\(998\) 79527.6 2.52245
\(999\) −5162.05 −0.163483
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 289.4.a.f.1.1 8
17.2 even 8 17.4.c.a.4.1 8
17.4 even 4 289.4.b.c.288.8 8
17.9 even 8 17.4.c.a.13.4 yes 8
17.13 even 4 289.4.b.c.288.7 8
17.16 even 2 inner 289.4.a.f.1.2 8
51.2 odd 8 153.4.f.a.55.4 8
51.26 odd 8 153.4.f.a.64.1 8
68.19 odd 8 272.4.o.e.225.3 8
68.43 odd 8 272.4.o.e.81.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.c.a.4.1 8 17.2 even 8
17.4.c.a.13.4 yes 8 17.9 even 8
153.4.f.a.55.4 8 51.2 odd 8
153.4.f.a.64.1 8 51.26 odd 8
272.4.o.e.81.3 8 68.43 odd 8
272.4.o.e.225.3 8 68.19 odd 8
289.4.a.f.1.1 8 1.1 even 1 trivial
289.4.a.f.1.2 8 17.16 even 2 inner
289.4.b.c.288.7 8 17.13 even 4
289.4.b.c.288.8 8 17.4 even 4