Properties

Label 289.4.a.g.1.11
Level $289$
Weight $4$
Character 289.1
Self dual yes
Analytic conductor $17.052$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(4.91828\) 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)
 
\(f(q)\) \(=\) \(q+3.15292 q^{2} -1.99138 q^{3} +1.94089 q^{4} +5.00761 q^{5} -6.27866 q^{6} -2.78516 q^{7} -19.1039 q^{8} -23.0344 q^{9} +O(q^{10})\) \(q+3.15292 q^{2} -1.99138 q^{3} +1.94089 q^{4} +5.00761 q^{5} -6.27866 q^{6} -2.78516 q^{7} -19.1039 q^{8} -23.0344 q^{9} +15.7886 q^{10} +27.2453 q^{11} -3.86505 q^{12} -59.7352 q^{13} -8.78140 q^{14} -9.97206 q^{15} -75.7601 q^{16} -72.6256 q^{18} +33.2605 q^{19} +9.71922 q^{20} +5.54633 q^{21} +85.9023 q^{22} -210.884 q^{23} +38.0431 q^{24} -99.9239 q^{25} -188.340 q^{26} +99.6376 q^{27} -5.40570 q^{28} +20.0521 q^{29} -31.4411 q^{30} -133.659 q^{31} -86.0343 q^{32} -54.2559 q^{33} -13.9470 q^{35} -44.7072 q^{36} +152.772 q^{37} +104.868 q^{38} +118.956 q^{39} -95.6647 q^{40} +261.840 q^{41} +17.4871 q^{42} -316.820 q^{43} +52.8802 q^{44} -115.347 q^{45} -664.899 q^{46} +329.443 q^{47} +150.867 q^{48} -335.243 q^{49} -315.052 q^{50} -115.939 q^{52} -310.540 q^{53} +314.149 q^{54} +136.434 q^{55} +53.2074 q^{56} -66.2344 q^{57} +63.2227 q^{58} -54.7388 q^{59} -19.3547 q^{60} -818.751 q^{61} -421.417 q^{62} +64.1546 q^{63} +334.822 q^{64} -299.130 q^{65} -171.064 q^{66} +731.181 q^{67} +419.950 q^{69} -43.9738 q^{70} +629.179 q^{71} +440.046 q^{72} +496.481 q^{73} +481.678 q^{74} +198.987 q^{75} +64.5550 q^{76} -75.8828 q^{77} +375.057 q^{78} -90.0262 q^{79} -379.377 q^{80} +423.512 q^{81} +825.559 q^{82} +364.025 q^{83} +10.7648 q^{84} -998.907 q^{86} -39.9315 q^{87} -520.492 q^{88} -192.079 q^{89} -363.680 q^{90} +166.372 q^{91} -409.302 q^{92} +266.167 q^{93} +1038.71 q^{94} +166.556 q^{95} +171.327 q^{96} +1350.20 q^{97} -1056.99 q^{98} -627.580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.15292 1.11472 0.557362 0.830269i \(-0.311813\pi\)
0.557362 + 0.830269i \(0.311813\pi\)
\(3\) −1.99138 −0.383242 −0.191621 0.981469i \(-0.561374\pi\)
−0.191621 + 0.981469i \(0.561374\pi\)
\(4\) 1.94089 0.242611
\(5\) 5.00761 0.447894 0.223947 0.974601i \(-0.428106\pi\)
0.223947 + 0.974601i \(0.428106\pi\)
\(6\) −6.27866 −0.427209
\(7\) −2.78516 −0.150385 −0.0751924 0.997169i \(-0.523957\pi\)
−0.0751924 + 0.997169i \(0.523957\pi\)
\(8\) −19.1039 −0.844280
\(9\) −23.0344 −0.853126
\(10\) 15.7886 0.499279
\(11\) 27.2453 0.746798 0.373399 0.927671i \(-0.378192\pi\)
0.373399 + 0.927671i \(0.378192\pi\)
\(12\) −3.86505 −0.0929787
\(13\) −59.7352 −1.27443 −0.637214 0.770687i \(-0.719913\pi\)
−0.637214 + 0.770687i \(0.719913\pi\)
\(14\) −8.78140 −0.167638
\(15\) −9.97206 −0.171652
\(16\) −75.7601 −1.18375
\(17\) 0 0
\(18\) −72.6256 −0.951000
\(19\) 33.2605 0.401604 0.200802 0.979632i \(-0.435645\pi\)
0.200802 + 0.979632i \(0.435645\pi\)
\(20\) 9.71922 0.108664
\(21\) 5.54633 0.0576337
\(22\) 85.9023 0.832475
\(23\) −210.884 −1.91184 −0.955919 0.293630i \(-0.905137\pi\)
−0.955919 + 0.293630i \(0.905137\pi\)
\(24\) 38.0431 0.323563
\(25\) −99.9239 −0.799391
\(26\) −188.340 −1.42064
\(27\) 99.6376 0.710195
\(28\) −5.40570 −0.0364850
\(29\) 20.0521 0.128400 0.0641998 0.997937i \(-0.479551\pi\)
0.0641998 + 0.997937i \(0.479551\pi\)
\(30\) −31.4411 −0.191344
\(31\) −133.659 −0.774385 −0.387192 0.921999i \(-0.626555\pi\)
−0.387192 + 0.921999i \(0.626555\pi\)
\(32\) −86.0343 −0.475277
\(33\) −54.2559 −0.286204
\(34\) 0 0
\(35\) −13.9470 −0.0673565
\(36\) −44.7072 −0.206978
\(37\) 152.772 0.678800 0.339400 0.940642i \(-0.389776\pi\)
0.339400 + 0.940642i \(0.389776\pi\)
\(38\) 104.868 0.447678
\(39\) 118.956 0.488414
\(40\) −95.6647 −0.378148
\(41\) 261.840 0.997377 0.498689 0.866781i \(-0.333815\pi\)
0.498689 + 0.866781i \(0.333815\pi\)
\(42\) 17.4871 0.0642457
\(43\) −316.820 −1.12359 −0.561797 0.827275i \(-0.689890\pi\)
−0.561797 + 0.827275i \(0.689890\pi\)
\(44\) 52.8802 0.181182
\(45\) −115.347 −0.382110
\(46\) −664.899 −2.13117
\(47\) 329.443 1.02243 0.511215 0.859453i \(-0.329196\pi\)
0.511215 + 0.859453i \(0.329196\pi\)
\(48\) 150.867 0.453663
\(49\) −335.243 −0.977384
\(50\) −315.052 −0.891101
\(51\) 0 0
\(52\) −115.939 −0.309190
\(53\) −310.540 −0.804829 −0.402415 0.915458i \(-0.631829\pi\)
−0.402415 + 0.915458i \(0.631829\pi\)
\(54\) 314.149 0.791672
\(55\) 136.434 0.334487
\(56\) 53.2074 0.126967
\(57\) −66.2344 −0.153911
\(58\) 63.2227 0.143130
\(59\) −54.7388 −0.120786 −0.0603931 0.998175i \(-0.519235\pi\)
−0.0603931 + 0.998175i \(0.519235\pi\)
\(60\) −19.3547 −0.0416446
\(61\) −818.751 −1.71853 −0.859265 0.511531i \(-0.829079\pi\)
−0.859265 + 0.511531i \(0.829079\pi\)
\(62\) −421.417 −0.863226
\(63\) 64.1546 0.128297
\(64\) 334.822 0.653948
\(65\) −299.130 −0.570808
\(66\) −171.064 −0.319039
\(67\) 731.181 1.33325 0.666627 0.745392i \(-0.267738\pi\)
0.666627 + 0.745392i \(0.267738\pi\)
\(68\) 0 0
\(69\) 419.950 0.732696
\(70\) −43.9738 −0.0750839
\(71\) 629.179 1.05169 0.525844 0.850581i \(-0.323750\pi\)
0.525844 + 0.850581i \(0.323750\pi\)
\(72\) 440.046 0.720277
\(73\) 496.481 0.796010 0.398005 0.917383i \(-0.369703\pi\)
0.398005 + 0.917383i \(0.369703\pi\)
\(74\) 481.678 0.756675
\(75\) 198.987 0.306360
\(76\) 64.5550 0.0974337
\(77\) −75.8828 −0.112307
\(78\) 375.057 0.544447
\(79\) −90.0262 −0.128212 −0.0641059 0.997943i \(-0.520420\pi\)
−0.0641059 + 0.997943i \(0.520420\pi\)
\(80\) −379.377 −0.530195
\(81\) 423.512 0.580950
\(82\) 825.559 1.11180
\(83\) 364.025 0.481409 0.240704 0.970599i \(-0.422622\pi\)
0.240704 + 0.970599i \(0.422622\pi\)
\(84\) 10.7648 0.0139826
\(85\) 0 0
\(86\) −998.907 −1.25250
\(87\) −39.9315 −0.0492081
\(88\) −520.492 −0.630507
\(89\) −192.079 −0.228767 −0.114384 0.993437i \(-0.536489\pi\)
−0.114384 + 0.993437i \(0.536489\pi\)
\(90\) −363.680 −0.425948
\(91\) 166.372 0.191654
\(92\) −409.302 −0.463833
\(93\) 266.167 0.296776
\(94\) 1038.71 1.13973
\(95\) 166.556 0.179876
\(96\) 171.327 0.182146
\(97\) 1350.20 1.41332 0.706659 0.707554i \(-0.250202\pi\)
0.706659 + 0.707554i \(0.250202\pi\)
\(98\) −1056.99 −1.08951
\(99\) −627.580 −0.637113
\(100\) −193.941 −0.193941
\(101\) −304.020 −0.299516 −0.149758 0.988723i \(-0.547850\pi\)
−0.149758 + 0.988723i \(0.547850\pi\)
\(102\) 0 0
\(103\) 988.515 0.945643 0.472822 0.881158i \(-0.343236\pi\)
0.472822 + 0.881158i \(0.343236\pi\)
\(104\) 1141.17 1.07597
\(105\) 27.7738 0.0258138
\(106\) −979.107 −0.897163
\(107\) −1175.01 −1.06162 −0.530808 0.847492i \(-0.678111\pi\)
−0.530808 + 0.847492i \(0.678111\pi\)
\(108\) 193.386 0.172301
\(109\) −838.865 −0.737144 −0.368572 0.929599i \(-0.620153\pi\)
−0.368572 + 0.929599i \(0.620153\pi\)
\(110\) 430.165 0.372860
\(111\) −304.228 −0.260144
\(112\) 211.004 0.178018
\(113\) 2026.81 1.68731 0.843655 0.536885i \(-0.180399\pi\)
0.843655 + 0.536885i \(0.180399\pi\)
\(114\) −208.831 −0.171569
\(115\) −1056.02 −0.856301
\(116\) 38.9190 0.0311512
\(117\) 1375.96 1.08725
\(118\) −172.587 −0.134643
\(119\) 0 0
\(120\) 190.505 0.144922
\(121\) −588.691 −0.442292
\(122\) −2581.45 −1.91569
\(123\) −521.423 −0.382237
\(124\) −259.418 −0.187874
\(125\) −1126.33 −0.805937
\(126\) 202.274 0.143016
\(127\) −1600.30 −1.11814 −0.559070 0.829120i \(-0.688842\pi\)
−0.559070 + 0.829120i \(0.688842\pi\)
\(128\) 1743.94 1.20425
\(129\) 630.909 0.430608
\(130\) −943.133 −0.636294
\(131\) 2615.07 1.74412 0.872061 0.489397i \(-0.162783\pi\)
0.872061 + 0.489397i \(0.162783\pi\)
\(132\) −105.305 −0.0694364
\(133\) −92.6360 −0.0603952
\(134\) 2305.35 1.48621
\(135\) 498.946 0.318092
\(136\) 0 0
\(137\) 745.711 0.465039 0.232520 0.972592i \(-0.425303\pi\)
0.232520 + 0.972592i \(0.425303\pi\)
\(138\) 1324.07 0.816754
\(139\) −2532.45 −1.54532 −0.772661 0.634818i \(-0.781075\pi\)
−0.772661 + 0.634818i \(0.781075\pi\)
\(140\) −27.0696 −0.0163414
\(141\) −656.047 −0.391837
\(142\) 1983.75 1.17234
\(143\) −1627.51 −0.951740
\(144\) 1745.09 1.00989
\(145\) 100.413 0.0575094
\(146\) 1565.36 0.887332
\(147\) 667.597 0.374574
\(148\) 296.514 0.164684
\(149\) −1816.70 −0.998858 −0.499429 0.866355i \(-0.666457\pi\)
−0.499429 + 0.866355i \(0.666457\pi\)
\(150\) 627.388 0.341507
\(151\) −2120.50 −1.14280 −0.571402 0.820670i \(-0.693600\pi\)
−0.571402 + 0.820670i \(0.693600\pi\)
\(152\) −635.404 −0.339066
\(153\) 0 0
\(154\) −239.252 −0.125191
\(155\) −669.314 −0.346842
\(156\) 230.880 0.118495
\(157\) −1607.82 −0.817314 −0.408657 0.912688i \(-0.634003\pi\)
−0.408657 + 0.912688i \(0.634003\pi\)
\(158\) −283.845 −0.142921
\(159\) 618.404 0.308444
\(160\) −430.826 −0.212874
\(161\) 587.346 0.287511
\(162\) 1335.30 0.647599
\(163\) −2278.00 −1.09464 −0.547322 0.836922i \(-0.684353\pi\)
−0.547322 + 0.836922i \(0.684353\pi\)
\(164\) 508.202 0.241975
\(165\) −271.692 −0.128189
\(166\) 1147.74 0.536638
\(167\) −1652.12 −0.765538 −0.382769 0.923844i \(-0.625029\pi\)
−0.382769 + 0.923844i \(0.625029\pi\)
\(168\) −105.956 −0.0486590
\(169\) 1371.29 0.624165
\(170\) 0 0
\(171\) −766.135 −0.342619
\(172\) −614.912 −0.272597
\(173\) 2210.82 0.971595 0.485797 0.874071i \(-0.338529\pi\)
0.485797 + 0.874071i \(0.338529\pi\)
\(174\) −125.901 −0.0548534
\(175\) 278.304 0.120216
\(176\) −2064.11 −0.884023
\(177\) 109.006 0.0462903
\(178\) −605.608 −0.255013
\(179\) 20.7548 0.00866640 0.00433320 0.999991i \(-0.498621\pi\)
0.00433320 + 0.999991i \(0.498621\pi\)
\(180\) −223.876 −0.0927042
\(181\) −436.498 −0.179252 −0.0896262 0.995975i \(-0.528567\pi\)
−0.0896262 + 0.995975i \(0.528567\pi\)
\(182\) 524.558 0.213642
\(183\) 1630.45 0.658612
\(184\) 4028.69 1.61413
\(185\) 765.023 0.304030
\(186\) 839.202 0.330824
\(187\) 0 0
\(188\) 639.412 0.248053
\(189\) −277.507 −0.106803
\(190\) 525.136 0.200512
\(191\) 787.808 0.298449 0.149225 0.988803i \(-0.452322\pi\)
0.149225 + 0.988803i \(0.452322\pi\)
\(192\) −666.758 −0.250620
\(193\) 3726.86 1.38998 0.694988 0.719021i \(-0.255410\pi\)
0.694988 + 0.719021i \(0.255410\pi\)
\(194\) 4257.06 1.57546
\(195\) 595.683 0.218758
\(196\) −650.669 −0.237124
\(197\) 451.885 0.163429 0.0817144 0.996656i \(-0.473960\pi\)
0.0817144 + 0.996656i \(0.473960\pi\)
\(198\) −1978.71 −0.710206
\(199\) −3480.16 −1.23971 −0.619855 0.784716i \(-0.712809\pi\)
−0.619855 + 0.784716i \(0.712809\pi\)
\(200\) 1908.93 0.674910
\(201\) −1456.06 −0.510958
\(202\) −958.551 −0.333878
\(203\) −55.8485 −0.0193093
\(204\) 0 0
\(205\) 1311.19 0.446719
\(206\) 3116.71 1.05413
\(207\) 4857.58 1.63104
\(208\) 4525.54 1.50860
\(209\) 906.194 0.299917
\(210\) 87.5686 0.0287753
\(211\) −1818.51 −0.593323 −0.296661 0.954983i \(-0.595873\pi\)
−0.296661 + 0.954983i \(0.595873\pi\)
\(212\) −602.724 −0.195261
\(213\) −1252.94 −0.403051
\(214\) −3704.72 −1.18341
\(215\) −1586.51 −0.503251
\(216\) −1903.46 −0.599603
\(217\) 372.263 0.116456
\(218\) −2644.87 −0.821712
\(219\) −988.684 −0.305064
\(220\) 264.803 0.0811502
\(221\) 0 0
\(222\) −959.205 −0.289989
\(223\) −6507.83 −1.95425 −0.977123 0.212676i \(-0.931782\pi\)
−0.977123 + 0.212676i \(0.931782\pi\)
\(224\) 239.620 0.0714744
\(225\) 2301.69 0.681981
\(226\) 6390.36 1.88089
\(227\) −2427.16 −0.709676 −0.354838 0.934928i \(-0.615464\pi\)
−0.354838 + 0.934928i \(0.615464\pi\)
\(228\) −128.554 −0.0373407
\(229\) 655.706 0.189215 0.0946076 0.995515i \(-0.469840\pi\)
0.0946076 + 0.995515i \(0.469840\pi\)
\(230\) −3329.55 −0.954540
\(231\) 151.112 0.0430408
\(232\) −383.073 −0.108405
\(233\) −1171.70 −0.329445 −0.164723 0.986340i \(-0.552673\pi\)
−0.164723 + 0.986340i \(0.552673\pi\)
\(234\) 4338.30 1.21198
\(235\) 1649.72 0.457940
\(236\) −106.242 −0.0293041
\(237\) 179.277 0.0491361
\(238\) 0 0
\(239\) −5281.06 −1.42930 −0.714651 0.699481i \(-0.753415\pi\)
−0.714651 + 0.699481i \(0.753415\pi\)
\(240\) 755.484 0.203193
\(241\) 1327.84 0.354912 0.177456 0.984129i \(-0.443213\pi\)
0.177456 + 0.984129i \(0.443213\pi\)
\(242\) −1856.09 −0.493034
\(243\) −3533.59 −0.932839
\(244\) −1589.11 −0.416935
\(245\) −1678.77 −0.437765
\(246\) −1644.00 −0.426089
\(247\) −1986.82 −0.511815
\(248\) 2553.41 0.653797
\(249\) −724.912 −0.184496
\(250\) −3551.23 −0.898397
\(251\) −4280.39 −1.07640 −0.538198 0.842818i \(-0.680895\pi\)
−0.538198 + 0.842818i \(0.680895\pi\)
\(252\) 124.517 0.0311263
\(253\) −5745.60 −1.42776
\(254\) −5045.62 −1.24642
\(255\) 0 0
\(256\) 2819.92 0.688458
\(257\) 242.890 0.0589536 0.0294768 0.999565i \(-0.490616\pi\)
0.0294768 + 0.999565i \(0.490616\pi\)
\(258\) 1989.20 0.480010
\(259\) −425.496 −0.102081
\(260\) −580.579 −0.138485
\(261\) −461.889 −0.109541
\(262\) 8245.11 1.94422
\(263\) −3153.49 −0.739364 −0.369682 0.929158i \(-0.620533\pi\)
−0.369682 + 0.929158i \(0.620533\pi\)
\(264\) 1036.50 0.241636
\(265\) −1555.06 −0.360478
\(266\) −292.074 −0.0673240
\(267\) 382.502 0.0876732
\(268\) 1419.14 0.323462
\(269\) 210.898 0.0478018 0.0239009 0.999714i \(-0.492391\pi\)
0.0239009 + 0.999714i \(0.492391\pi\)
\(270\) 1573.14 0.354585
\(271\) −1627.36 −0.364780 −0.182390 0.983226i \(-0.558383\pi\)
−0.182390 + 0.983226i \(0.558383\pi\)
\(272\) 0 0
\(273\) −331.311 −0.0734500
\(274\) 2351.16 0.518391
\(275\) −2722.46 −0.596984
\(276\) 815.076 0.177760
\(277\) 4626.89 1.00362 0.501810 0.864978i \(-0.332668\pi\)
0.501810 + 0.864978i \(0.332668\pi\)
\(278\) −7984.62 −1.72261
\(279\) 3078.76 0.660648
\(280\) 266.442 0.0568677
\(281\) −5076.28 −1.07767 −0.538836 0.842411i \(-0.681136\pi\)
−0.538836 + 0.842411i \(0.681136\pi\)
\(282\) −2068.46 −0.436791
\(283\) 2437.04 0.511898 0.255949 0.966690i \(-0.417612\pi\)
0.255949 + 0.966690i \(0.417612\pi\)
\(284\) 1221.17 0.255151
\(285\) −331.676 −0.0689360
\(286\) −5131.39 −1.06093
\(287\) −729.266 −0.149990
\(288\) 1981.75 0.405471
\(289\) 0 0
\(290\) 316.595 0.0641072
\(291\) −2688.76 −0.541642
\(292\) 963.616 0.193121
\(293\) −3300.30 −0.658041 −0.329020 0.944323i \(-0.606718\pi\)
−0.329020 + 0.944323i \(0.606718\pi\)
\(294\) 2104.88 0.417547
\(295\) −274.111 −0.0540994
\(296\) −2918.54 −0.573097
\(297\) 2714.66 0.530372
\(298\) −5727.91 −1.11345
\(299\) 12597.2 2.43650
\(300\) 386.211 0.0743263
\(301\) 882.395 0.168971
\(302\) −6685.75 −1.27391
\(303\) 605.420 0.114787
\(304\) −2519.82 −0.475399
\(305\) −4099.98 −0.769719
\(306\) 0 0
\(307\) −1186.40 −0.220558 −0.110279 0.993901i \(-0.535174\pi\)
−0.110279 + 0.993901i \(0.535174\pi\)
\(308\) −147.280 −0.0272470
\(309\) −1968.51 −0.362410
\(310\) −2110.29 −0.386634
\(311\) −657.473 −0.119877 −0.0599387 0.998202i \(-0.519091\pi\)
−0.0599387 + 0.998202i \(0.519091\pi\)
\(312\) −2272.51 −0.412358
\(313\) −6073.66 −1.09682 −0.548408 0.836211i \(-0.684766\pi\)
−0.548408 + 0.836211i \(0.684766\pi\)
\(314\) −5069.34 −0.911080
\(315\) 321.261 0.0574635
\(316\) −174.731 −0.0311056
\(317\) 5753.26 1.01935 0.509677 0.860366i \(-0.329765\pi\)
0.509677 + 0.860366i \(0.329765\pi\)
\(318\) 1949.78 0.343830
\(319\) 546.327 0.0958886
\(320\) 1676.66 0.292900
\(321\) 2339.90 0.406855
\(322\) 1851.85 0.320496
\(323\) 0 0
\(324\) 821.991 0.140945
\(325\) 5968.97 1.01877
\(326\) −7182.35 −1.22023
\(327\) 1670.50 0.282504
\(328\) −5002.15 −0.842066
\(329\) −917.553 −0.153758
\(330\) −856.623 −0.142896
\(331\) −89.8427 −0.0149190 −0.00745952 0.999972i \(-0.502374\pi\)
−0.00745952 + 0.999972i \(0.502374\pi\)
\(332\) 706.532 0.116795
\(333\) −3519.01 −0.579101
\(334\) −5209.00 −0.853364
\(335\) 3661.47 0.597156
\(336\) −420.190 −0.0682240
\(337\) −847.308 −0.136961 −0.0684804 0.997652i \(-0.521815\pi\)
−0.0684804 + 0.997652i \(0.521815\pi\)
\(338\) 4323.56 0.695772
\(339\) −4036.15 −0.646648
\(340\) 0 0
\(341\) −3641.60 −0.578309
\(342\) −2415.56 −0.381926
\(343\) 1889.02 0.297369
\(344\) 6052.48 0.948628
\(345\) 2102.94 0.328170
\(346\) 6970.55 1.08306
\(347\) 590.322 0.0913260 0.0456630 0.998957i \(-0.485460\pi\)
0.0456630 + 0.998957i \(0.485460\pi\)
\(348\) −77.5026 −0.0119384
\(349\) 9387.68 1.43986 0.719930 0.694047i \(-0.244174\pi\)
0.719930 + 0.694047i \(0.244174\pi\)
\(350\) 877.471 0.134008
\(351\) −5951.87 −0.905092
\(352\) −2344.03 −0.354936
\(353\) −6176.09 −0.931218 −0.465609 0.884990i \(-0.654165\pi\)
−0.465609 + 0.884990i \(0.654165\pi\)
\(354\) 343.687 0.0516009
\(355\) 3150.68 0.471045
\(356\) −372.804 −0.0555015
\(357\) 0 0
\(358\) 65.4382 0.00966066
\(359\) 7151.14 1.05132 0.525658 0.850696i \(-0.323819\pi\)
0.525658 + 0.850696i \(0.323819\pi\)
\(360\) 2203.58 0.322608
\(361\) −5752.74 −0.838714
\(362\) −1376.24 −0.199817
\(363\) 1172.31 0.169505
\(364\) 322.910 0.0464975
\(365\) 2486.18 0.356528
\(366\) 5140.66 0.734171
\(367\) −3358.05 −0.477626 −0.238813 0.971066i \(-0.576758\pi\)
−0.238813 + 0.971066i \(0.576758\pi\)
\(368\) 15976.6 2.26314
\(369\) −6031.32 −0.850888
\(370\) 2412.05 0.338910
\(371\) 864.905 0.121034
\(372\) 516.600 0.0720013
\(373\) 8379.14 1.16315 0.581576 0.813492i \(-0.302436\pi\)
0.581576 + 0.813492i \(0.302436\pi\)
\(374\) 0 0
\(375\) 2242.95 0.308868
\(376\) −6293.64 −0.863217
\(377\) −1197.82 −0.163636
\(378\) −874.957 −0.119055
\(379\) −6395.71 −0.866821 −0.433411 0.901196i \(-0.642690\pi\)
−0.433411 + 0.901196i \(0.642690\pi\)
\(380\) 323.266 0.0436400
\(381\) 3186.81 0.428518
\(382\) 2483.89 0.332689
\(383\) −204.687 −0.0273081 −0.0136541 0.999907i \(-0.504346\pi\)
−0.0136541 + 0.999907i \(0.504346\pi\)
\(384\) −3472.85 −0.461518
\(385\) −379.991 −0.0503017
\(386\) 11750.5 1.54944
\(387\) 7297.75 0.958567
\(388\) 2620.59 0.342887
\(389\) 5770.56 0.752131 0.376066 0.926593i \(-0.377277\pi\)
0.376066 + 0.926593i \(0.377277\pi\)
\(390\) 1878.14 0.243854
\(391\) 0 0
\(392\) 6404.44 0.825186
\(393\) −5207.61 −0.668420
\(394\) 1424.76 0.182178
\(395\) −450.816 −0.0574253
\(396\) −1218.06 −0.154571
\(397\) 7144.37 0.903188 0.451594 0.892223i \(-0.350855\pi\)
0.451594 + 0.892223i \(0.350855\pi\)
\(398\) −10972.7 −1.38194
\(399\) 184.474 0.0231459
\(400\) 7570.24 0.946280
\(401\) −6364.52 −0.792591 −0.396296 0.918123i \(-0.629704\pi\)
−0.396296 + 0.918123i \(0.629704\pi\)
\(402\) −4590.84 −0.569578
\(403\) 7984.16 0.986897
\(404\) −590.070 −0.0726660
\(405\) 2120.78 0.260204
\(406\) −176.086 −0.0215246
\(407\) 4162.33 0.506926
\(408\) 0 0
\(409\) 2997.87 0.362433 0.181217 0.983443i \(-0.441997\pi\)
0.181217 + 0.983443i \(0.441997\pi\)
\(410\) 4134.07 0.497969
\(411\) −1485.00 −0.178222
\(412\) 1918.60 0.229424
\(413\) 152.457 0.0181644
\(414\) 15315.5 1.81816
\(415\) 1822.89 0.215620
\(416\) 5139.27 0.605705
\(417\) 5043.08 0.592232
\(418\) 2857.15 0.334325
\(419\) −11747.3 −1.36967 −0.684835 0.728698i \(-0.740126\pi\)
−0.684835 + 0.728698i \(0.740126\pi\)
\(420\) 53.9060 0.00626272
\(421\) 8842.15 1.02361 0.511805 0.859102i \(-0.328977\pi\)
0.511805 + 0.859102i \(0.328977\pi\)
\(422\) −5733.60 −0.661391
\(423\) −7588.52 −0.872261
\(424\) 5932.52 0.679501
\(425\) 0 0
\(426\) −3950.41 −0.449291
\(427\) 2280.36 0.258441
\(428\) −2280.57 −0.257560
\(429\) 3240.98 0.364746
\(430\) −5002.13 −0.560987
\(431\) 3099.38 0.346385 0.173192 0.984888i \(-0.444592\pi\)
0.173192 + 0.984888i \(0.444592\pi\)
\(432\) −7548.55 −0.840694
\(433\) 10072.8 1.11794 0.558970 0.829188i \(-0.311197\pi\)
0.558970 + 0.829188i \(0.311197\pi\)
\(434\) 1173.72 0.129816
\(435\) −199.961 −0.0220400
\(436\) −1628.14 −0.178839
\(437\) −7014.09 −0.767802
\(438\) −3117.24 −0.340063
\(439\) 17357.9 1.88712 0.943561 0.331199i \(-0.107453\pi\)
0.943561 + 0.331199i \(0.107453\pi\)
\(440\) −2606.42 −0.282400
\(441\) 7722.12 0.833832
\(442\) 0 0
\(443\) 6979.90 0.748589 0.374295 0.927310i \(-0.377885\pi\)
0.374295 + 0.927310i \(0.377885\pi\)
\(444\) −590.472 −0.0631139
\(445\) −961.855 −0.102464
\(446\) −20518.7 −2.17845
\(447\) 3617.74 0.382804
\(448\) −932.533 −0.0983439
\(449\) 9230.50 0.970188 0.485094 0.874462i \(-0.338785\pi\)
0.485094 + 0.874462i \(0.338785\pi\)
\(450\) 7257.03 0.760221
\(451\) 7133.91 0.744840
\(452\) 3933.81 0.409361
\(453\) 4222.72 0.437970
\(454\) −7652.64 −0.791093
\(455\) 833.127 0.0858409
\(456\) 1265.33 0.129944
\(457\) 12250.6 1.25396 0.626979 0.779036i \(-0.284291\pi\)
0.626979 + 0.779036i \(0.284291\pi\)
\(458\) 2067.39 0.210923
\(459\) 0 0
\(460\) −2049.62 −0.207748
\(461\) 6259.50 0.632395 0.316198 0.948693i \(-0.397594\pi\)
0.316198 + 0.948693i \(0.397594\pi\)
\(462\) 476.442 0.0479786
\(463\) 10195.0 1.02333 0.511665 0.859185i \(-0.329029\pi\)
0.511665 + 0.859185i \(0.329029\pi\)
\(464\) −1519.15 −0.151993
\(465\) 1332.86 0.132924
\(466\) −3694.28 −0.367241
\(467\) −14783.0 −1.46483 −0.732417 0.680856i \(-0.761608\pi\)
−0.732417 + 0.680856i \(0.761608\pi\)
\(468\) 2670.59 0.263778
\(469\) −2036.46 −0.200501
\(470\) 5201.43 0.510477
\(471\) 3201.79 0.313229
\(472\) 1045.72 0.101977
\(473\) −8631.86 −0.839098
\(474\) 565.244 0.0547732
\(475\) −3323.52 −0.321039
\(476\) 0 0
\(477\) 7153.10 0.686620
\(478\) −16650.7 −1.59328
\(479\) −16370.0 −1.56152 −0.780758 0.624834i \(-0.785167\pi\)
−0.780758 + 0.624834i \(0.785167\pi\)
\(480\) 857.939 0.0815820
\(481\) −9125.87 −0.865081
\(482\) 4186.57 0.395629
\(483\) −1169.63 −0.110186
\(484\) −1142.58 −0.107305
\(485\) 6761.26 0.633017
\(486\) −11141.1 −1.03986
\(487\) −4106.31 −0.382084 −0.191042 0.981582i \(-0.561187\pi\)
−0.191042 + 0.981582i \(0.561187\pi\)
\(488\) 15641.3 1.45092
\(489\) 4536.37 0.419513
\(490\) −5293.01 −0.487987
\(491\) −878.001 −0.0806999 −0.0403499 0.999186i \(-0.512847\pi\)
−0.0403499 + 0.999186i \(0.512847\pi\)
\(492\) −1012.02 −0.0927349
\(493\) 0 0
\(494\) −6264.28 −0.570533
\(495\) −3142.68 −0.285359
\(496\) 10126.0 0.916679
\(497\) −1752.37 −0.158158
\(498\) −2285.59 −0.205662
\(499\) −13819.6 −1.23978 −0.619892 0.784687i \(-0.712824\pi\)
−0.619892 + 0.784687i \(0.712824\pi\)
\(500\) −2186.08 −0.195529
\(501\) 3290.00 0.293386
\(502\) −13495.7 −1.19989
\(503\) 4721.43 0.418525 0.209263 0.977859i \(-0.432894\pi\)
0.209263 + 0.977859i \(0.432894\pi\)
\(504\) −1225.60 −0.108319
\(505\) −1522.41 −0.134152
\(506\) −18115.4 −1.59156
\(507\) −2730.76 −0.239206
\(508\) −3106.01 −0.271273
\(509\) −16554.3 −1.44156 −0.720782 0.693161i \(-0.756217\pi\)
−0.720782 + 0.693161i \(0.756217\pi\)
\(510\) 0 0
\(511\) −1382.78 −0.119708
\(512\) −5060.53 −0.436808
\(513\) 3314.00 0.285217
\(514\) 765.813 0.0657171
\(515\) 4950.09 0.423548
\(516\) 1224.53 0.104470
\(517\) 8975.79 0.763548
\(518\) −1341.55 −0.113792
\(519\) −4402.60 −0.372356
\(520\) 5714.55 0.481922
\(521\) 14755.5 1.24079 0.620394 0.784290i \(-0.286973\pi\)
0.620394 + 0.784290i \(0.286973\pi\)
\(522\) −1456.30 −0.122108
\(523\) 7800.86 0.652214 0.326107 0.945333i \(-0.394263\pi\)
0.326107 + 0.945333i \(0.394263\pi\)
\(524\) 5075.57 0.423144
\(525\) −554.210 −0.0460719
\(526\) −9942.71 −0.824188
\(527\) 0 0
\(528\) 4110.43 0.338795
\(529\) 32304.9 2.65512
\(530\) −4902.98 −0.401834
\(531\) 1260.88 0.103046
\(532\) −179.796 −0.0146525
\(533\) −15641.0 −1.27108
\(534\) 1206.00 0.0977315
\(535\) −5884.01 −0.475491
\(536\) −13968.4 −1.12564
\(537\) −41.3307 −0.00332133
\(538\) 664.944 0.0532859
\(539\) −9133.81 −0.729909
\(540\) 968.399 0.0771727
\(541\) −23028.5 −1.83007 −0.915037 0.403369i \(-0.867839\pi\)
−0.915037 + 0.403369i \(0.867839\pi\)
\(542\) −5130.95 −0.406629
\(543\) 869.235 0.0686970
\(544\) 0 0
\(545\) −4200.71 −0.330162
\(546\) −1044.60 −0.0818765
\(547\) −2641.77 −0.206497 −0.103249 0.994656i \(-0.532924\pi\)
−0.103249 + 0.994656i \(0.532924\pi\)
\(548\) 1447.34 0.112824
\(549\) 18859.4 1.46612
\(550\) −8583.69 −0.665473
\(551\) 666.944 0.0515658
\(552\) −8022.67 −0.618601
\(553\) 250.738 0.0192811
\(554\) 14588.2 1.11876
\(555\) −1523.45 −0.116517
\(556\) −4915.21 −0.374913
\(557\) −19800.8 −1.50626 −0.753129 0.657873i \(-0.771456\pi\)
−0.753129 + 0.657873i \(0.771456\pi\)
\(558\) 9707.08 0.736440
\(559\) 18925.3 1.43194
\(560\) 1056.63 0.0797333
\(561\) 0 0
\(562\) −16005.1 −1.20131
\(563\) −9733.93 −0.728661 −0.364331 0.931270i \(-0.618702\pi\)
−0.364331 + 0.931270i \(0.618702\pi\)
\(564\) −1273.31 −0.0950642
\(565\) 10149.5 0.755736
\(566\) 7683.80 0.570626
\(567\) −1179.55 −0.0873660
\(568\) −12019.8 −0.887919
\(569\) 6340.67 0.467161 0.233581 0.972337i \(-0.424956\pi\)
0.233581 + 0.972337i \(0.424956\pi\)
\(570\) −1045.75 −0.0768447
\(571\) 12377.7 0.907166 0.453583 0.891214i \(-0.350146\pi\)
0.453583 + 0.891214i \(0.350146\pi\)
\(572\) −3158.81 −0.230903
\(573\) −1568.83 −0.114378
\(574\) −2299.32 −0.167198
\(575\) 21072.3 1.52831
\(576\) −7712.41 −0.557900
\(577\) −36.6040 −0.00264098 −0.00132049 0.999999i \(-0.500420\pi\)
−0.00132049 + 0.999999i \(0.500420\pi\)
\(578\) 0 0
\(579\) −7421.61 −0.532697
\(580\) 194.891 0.0139524
\(581\) −1013.87 −0.0723965
\(582\) −8477.44 −0.603782
\(583\) −8460.77 −0.601045
\(584\) −9484.72 −0.672055
\(585\) 6890.29 0.486971
\(586\) −10405.6 −0.733534
\(587\) −20350.4 −1.43092 −0.715462 0.698651i \(-0.753784\pi\)
−0.715462 + 0.698651i \(0.753784\pi\)
\(588\) 1295.73 0.0908760
\(589\) −4445.58 −0.310996
\(590\) −864.248 −0.0603060
\(591\) −899.876 −0.0626328
\(592\) −11574.0 −0.803530
\(593\) −20387.4 −1.41182 −0.705910 0.708301i \(-0.749462\pi\)
−0.705910 + 0.708301i \(0.749462\pi\)
\(594\) 8559.10 0.591219
\(595\) 0 0
\(596\) −3526.02 −0.242334
\(597\) 6930.33 0.475108
\(598\) 39717.8 2.71603
\(599\) 316.417 0.0215834 0.0107917 0.999942i \(-0.496565\pi\)
0.0107917 + 0.999942i \(0.496565\pi\)
\(600\) −3801.41 −0.258653
\(601\) 1331.85 0.0903950 0.0451975 0.998978i \(-0.485608\pi\)
0.0451975 + 0.998978i \(0.485608\pi\)
\(602\) 2782.12 0.188357
\(603\) −16842.3 −1.13743
\(604\) −4115.65 −0.277257
\(605\) −2947.93 −0.198100
\(606\) 1908.84 0.127956
\(607\) 13873.2 0.927668 0.463834 0.885922i \(-0.346473\pi\)
0.463834 + 0.885922i \(0.346473\pi\)
\(608\) −2861.54 −0.190873
\(609\) 111.216 0.00740014
\(610\) −12926.9 −0.858025
\(611\) −19679.3 −1.30301
\(612\) 0 0
\(613\) 15297.0 1.00790 0.503948 0.863734i \(-0.331880\pi\)
0.503948 + 0.863734i \(0.331880\pi\)
\(614\) −3740.61 −0.245861
\(615\) −2611.08 −0.171201
\(616\) 1449.66 0.0948186
\(617\) 15116.8 0.986354 0.493177 0.869929i \(-0.335835\pi\)
0.493177 + 0.869929i \(0.335835\pi\)
\(618\) −6206.55 −0.403987
\(619\) 22412.1 1.45528 0.727639 0.685960i \(-0.240618\pi\)
0.727639 + 0.685960i \(0.240618\pi\)
\(620\) −1299.06 −0.0841479
\(621\) −21011.9 −1.35778
\(622\) −2072.96 −0.133630
\(623\) 534.971 0.0344031
\(624\) −9012.08 −0.578160
\(625\) 6850.26 0.438417
\(626\) −19149.7 −1.22265
\(627\) −1804.58 −0.114941
\(628\) −3120.61 −0.198290
\(629\) 0 0
\(630\) 1012.91 0.0640560
\(631\) −4830.94 −0.304781 −0.152390 0.988320i \(-0.548697\pi\)
−0.152390 + 0.988320i \(0.548697\pi\)
\(632\) 1719.85 0.108247
\(633\) 3621.34 0.227386
\(634\) 18139.6 1.13630
\(635\) −8013.68 −0.500808
\(636\) 1200.25 0.0748320
\(637\) 20025.8 1.24561
\(638\) 1722.53 0.106889
\(639\) −14492.8 −0.897222
\(640\) 8732.96 0.539376
\(641\) −23007.3 −1.41768 −0.708839 0.705370i \(-0.750781\pi\)
−0.708839 + 0.705370i \(0.750781\pi\)
\(642\) 7377.52 0.453532
\(643\) 5689.90 0.348970 0.174485 0.984660i \(-0.444174\pi\)
0.174485 + 0.984660i \(0.444174\pi\)
\(644\) 1139.97 0.0697535
\(645\) 3159.35 0.192867
\(646\) 0 0
\(647\) −15949.0 −0.969122 −0.484561 0.874758i \(-0.661021\pi\)
−0.484561 + 0.874758i \(0.661021\pi\)
\(648\) −8090.72 −0.490484
\(649\) −1491.38 −0.0902029
\(650\) 18819.7 1.13564
\(651\) −741.318 −0.0446307
\(652\) −4421.35 −0.265573
\(653\) −10764.6 −0.645101 −0.322550 0.946552i \(-0.604540\pi\)
−0.322550 + 0.946552i \(0.604540\pi\)
\(654\) 5266.95 0.314914
\(655\) 13095.3 0.781182
\(656\) −19837.0 −1.18065
\(657\) −11436.1 −0.679097
\(658\) −2892.97 −0.171398
\(659\) 25208.2 1.49010 0.745048 0.667011i \(-0.232427\pi\)
0.745048 + 0.667011i \(0.232427\pi\)
\(660\) −527.325 −0.0311001
\(661\) 14419.5 0.848492 0.424246 0.905547i \(-0.360539\pi\)
0.424246 + 0.905547i \(0.360539\pi\)
\(662\) −283.267 −0.0166306
\(663\) 0 0
\(664\) −6954.28 −0.406444
\(665\) −463.885 −0.0270506
\(666\) −11095.2 −0.645539
\(667\) −4228.67 −0.245479
\(668\) −3206.58 −0.185728
\(669\) 12959.6 0.748948
\(670\) 11544.3 0.665665
\(671\) −22307.2 −1.28339
\(672\) −477.174 −0.0273920
\(673\) −2110.64 −0.120890 −0.0604451 0.998172i \(-0.519252\pi\)
−0.0604451 + 0.998172i \(0.519252\pi\)
\(674\) −2671.49 −0.152674
\(675\) −9956.17 −0.567723
\(676\) 2661.52 0.151429
\(677\) 11944.5 0.678087 0.339043 0.940771i \(-0.389897\pi\)
0.339043 + 0.940771i \(0.389897\pi\)
\(678\) −12725.6 −0.720834
\(679\) −3760.52 −0.212541
\(680\) 0 0
\(681\) 4833.41 0.271977
\(682\) −11481.6 −0.644656
\(683\) 19085.8 1.06925 0.534626 0.845089i \(-0.320453\pi\)
0.534626 + 0.845089i \(0.320453\pi\)
\(684\) −1486.98 −0.0831232
\(685\) 3734.23 0.208288
\(686\) 5955.92 0.331484
\(687\) −1305.76 −0.0725152
\(688\) 24002.3 1.33006
\(689\) 18550.2 1.02570
\(690\) 6630.41 0.365819
\(691\) −28316.8 −1.55893 −0.779465 0.626445i \(-0.784509\pi\)
−0.779465 + 0.626445i \(0.784509\pi\)
\(692\) 4290.97 0.235720
\(693\) 1747.91 0.0958121
\(694\) 1861.24 0.101803
\(695\) −12681.5 −0.692141
\(696\) 762.846 0.0415454
\(697\) 0 0
\(698\) 29598.6 1.60505
\(699\) 2333.31 0.126257
\(700\) 540.158 0.0291658
\(701\) −5916.75 −0.318791 −0.159396 0.987215i \(-0.550955\pi\)
−0.159396 + 0.987215i \(0.550955\pi\)
\(702\) −18765.8 −1.00893
\(703\) 5081.28 0.272609
\(704\) 9122.33 0.488368
\(705\) −3285.22 −0.175502
\(706\) −19472.7 −1.03805
\(707\) 846.747 0.0450427
\(708\) 211.568 0.0112305
\(709\) −18499.8 −0.979936 −0.489968 0.871740i \(-0.662992\pi\)
−0.489968 + 0.871740i \(0.662992\pi\)
\(710\) 9933.85 0.525085
\(711\) 2073.70 0.109381
\(712\) 3669.45 0.193144
\(713\) 28186.6 1.48050
\(714\) 0 0
\(715\) −8149.91 −0.426279
\(716\) 40.2828 0.00210257
\(717\) 10516.6 0.547768
\(718\) 22546.9 1.17193
\(719\) −22206.5 −1.15182 −0.575912 0.817511i \(-0.695353\pi\)
−0.575912 + 0.817511i \(0.695353\pi\)
\(720\) 8738.72 0.452323
\(721\) −2753.18 −0.142210
\(722\) −18137.9 −0.934935
\(723\) −2644.24 −0.136017
\(724\) −847.195 −0.0434886
\(725\) −2003.69 −0.102641
\(726\) 3696.19 0.188951
\(727\) 3777.02 0.192685 0.0963424 0.995348i \(-0.469286\pi\)
0.0963424 + 0.995348i \(0.469286\pi\)
\(728\) −3178.36 −0.161810
\(729\) −4398.10 −0.223447
\(730\) 7838.73 0.397431
\(731\) 0 0
\(732\) 3164.52 0.159787
\(733\) −19956.4 −1.00560 −0.502801 0.864402i \(-0.667697\pi\)
−0.502801 + 0.864402i \(0.667697\pi\)
\(734\) −10587.7 −0.532422
\(735\) 3343.06 0.167770
\(736\) 18143.2 0.908652
\(737\) 19921.3 0.995671
\(738\) −19016.2 −0.948506
\(739\) 23268.4 1.15824 0.579121 0.815241i \(-0.303396\pi\)
0.579121 + 0.815241i \(0.303396\pi\)
\(740\) 1484.83 0.0737612
\(741\) 3956.52 0.196149
\(742\) 2726.97 0.134920
\(743\) 12587.1 0.621502 0.310751 0.950491i \(-0.399420\pi\)
0.310751 + 0.950491i \(0.399420\pi\)
\(744\) −5084.82 −0.250562
\(745\) −9097.32 −0.447383
\(746\) 26418.7 1.29659
\(747\) −8385.09 −0.410702
\(748\) 0 0
\(749\) 3272.61 0.159651
\(750\) 7071.85 0.344303
\(751\) 2662.79 0.129383 0.0646914 0.997905i \(-0.479394\pi\)
0.0646914 + 0.997905i \(0.479394\pi\)
\(752\) −24958.6 −1.21030
\(753\) 8523.89 0.412520
\(754\) −3776.62 −0.182409
\(755\) −10618.6 −0.511855
\(756\) −538.611 −0.0259115
\(757\) 26430.2 1.26899 0.634493 0.772929i \(-0.281209\pi\)
0.634493 + 0.772929i \(0.281209\pi\)
\(758\) −20165.1 −0.966267
\(759\) 11441.7 0.547176
\(760\) −3181.86 −0.151866
\(761\) −8469.75 −0.403454 −0.201727 0.979442i \(-0.564655\pi\)
−0.201727 + 0.979442i \(0.564655\pi\)
\(762\) 10047.8 0.477680
\(763\) 2336.38 0.110855
\(764\) 1529.05 0.0724071
\(765\) 0 0
\(766\) −645.361 −0.0304410
\(767\) 3269.83 0.153933
\(768\) −5615.54 −0.263846
\(769\) −8452.54 −0.396367 −0.198184 0.980165i \(-0.563504\pi\)
−0.198184 + 0.980165i \(0.563504\pi\)
\(770\) −1198.08 −0.0560725
\(771\) −483.687 −0.0225935
\(772\) 7233.43 0.337224
\(773\) 33936.2 1.57904 0.789521 0.613724i \(-0.210329\pi\)
0.789521 + 0.613724i \(0.210329\pi\)
\(774\) 23009.2 1.06854
\(775\) 13355.8 0.619036
\(776\) −25794.0 −1.19324
\(777\) 847.324 0.0391217
\(778\) 18194.1 0.838419
\(779\) 8708.91 0.400551
\(780\) 1156.15 0.0530730
\(781\) 17142.2 0.785399
\(782\) 0 0
\(783\) 1997.95 0.0911887
\(784\) 25398.0 1.15698
\(785\) −8051.35 −0.366070
\(786\) −16419.2 −0.745105
\(787\) −34379.1 −1.55716 −0.778578 0.627547i \(-0.784059\pi\)
−0.778578 + 0.627547i \(0.784059\pi\)
\(788\) 877.059 0.0396497
\(789\) 6279.81 0.283355
\(790\) −1421.39 −0.0640134
\(791\) −5645.00 −0.253746
\(792\) 11989.2 0.537902
\(793\) 48908.2 2.19014
\(794\) 22525.6 1.00681
\(795\) 3096.72 0.138150
\(796\) −6754.61 −0.300768
\(797\) −3329.26 −0.147965 −0.0739826 0.997260i \(-0.523571\pi\)
−0.0739826 + 0.997260i \(0.523571\pi\)
\(798\) 581.630 0.0258014
\(799\) 0 0
\(800\) 8596.87 0.379932
\(801\) 4424.42 0.195167
\(802\) −20066.8 −0.883521
\(803\) 13526.8 0.594459
\(804\) −2826.05 −0.123964
\(805\) 2941.20 0.128775
\(806\) 25173.4 1.10012
\(807\) −419.979 −0.0183196
\(808\) 5807.97 0.252876
\(809\) −7217.64 −0.313670 −0.156835 0.987625i \(-0.550129\pi\)
−0.156835 + 0.987625i \(0.550129\pi\)
\(810\) 6686.66 0.290056
\(811\) 36564.1 1.58316 0.791579 0.611067i \(-0.209259\pi\)
0.791579 + 0.611067i \(0.209259\pi\)
\(812\) −108.396 −0.00468466
\(813\) 3240.70 0.139799
\(814\) 13123.5 0.565083
\(815\) −11407.3 −0.490284
\(816\) 0 0
\(817\) −10537.6 −0.451240
\(818\) 9452.04 0.404013
\(819\) −3832.29 −0.163505
\(820\) 2544.88 0.108379
\(821\) 23140.5 0.983691 0.491845 0.870683i \(-0.336323\pi\)
0.491845 + 0.870683i \(0.336323\pi\)
\(822\) −4682.07 −0.198669
\(823\) −41287.9 −1.74873 −0.874365 0.485269i \(-0.838722\pi\)
−0.874365 + 0.485269i \(0.838722\pi\)
\(824\) −18884.5 −0.798388
\(825\) 5421.46 0.228789
\(826\) 480.683 0.0202483
\(827\) −10008.0 −0.420811 −0.210405 0.977614i \(-0.567478\pi\)
−0.210405 + 0.977614i \(0.567478\pi\)
\(828\) 9428.02 0.395708
\(829\) −44643.6 −1.87037 −0.935185 0.354159i \(-0.884767\pi\)
−0.935185 + 0.354159i \(0.884767\pi\)
\(830\) 5747.43 0.240357
\(831\) −9213.90 −0.384629
\(832\) −20000.6 −0.833410
\(833\) 0 0
\(834\) 15900.4 0.660176
\(835\) −8273.17 −0.342880
\(836\) 1758.82 0.0727633
\(837\) −13317.5 −0.549964
\(838\) −37038.2 −1.52681
\(839\) 14235.3 0.585765 0.292882 0.956148i \(-0.405386\pi\)
0.292882 + 0.956148i \(0.405386\pi\)
\(840\) −530.588 −0.0217941
\(841\) −23986.9 −0.983514
\(842\) 27878.6 1.14104
\(843\) 10108.8 0.413008
\(844\) −3529.52 −0.143947
\(845\) 6866.88 0.279560
\(846\) −23926.0 −0.972331
\(847\) 1639.60 0.0665140
\(848\) 23526.5 0.952717
\(849\) −4853.08 −0.196181
\(850\) 0 0
\(851\) −32217.1 −1.29775
\(852\) −2431.81 −0.0977846
\(853\) −27729.0 −1.11304 −0.556520 0.830834i \(-0.687864\pi\)
−0.556520 + 0.830834i \(0.687864\pi\)
\(854\) 7189.78 0.288090
\(855\) −3836.51 −0.153457
\(856\) 22447.3 0.896301
\(857\) −31280.7 −1.24682 −0.623411 0.781894i \(-0.714254\pi\)
−0.623411 + 0.781894i \(0.714254\pi\)
\(858\) 10218.6 0.406592
\(859\) −55.2890 −0.00219608 −0.00109804 0.999999i \(-0.500350\pi\)
−0.00109804 + 0.999999i \(0.500350\pi\)
\(860\) −3079.24 −0.122094
\(861\) 1452.25 0.0574826
\(862\) 9772.09 0.386124
\(863\) −22900.4 −0.903291 −0.451646 0.892197i \(-0.649163\pi\)
−0.451646 + 0.892197i \(0.649163\pi\)
\(864\) −8572.25 −0.337539
\(865\) 11070.9 0.435172
\(866\) 31758.7 1.24620
\(867\) 0 0
\(868\) 722.522 0.0282535
\(869\) −2452.79 −0.0957484
\(870\) −630.461 −0.0245685
\(871\) −43677.2 −1.69913
\(872\) 16025.6 0.622356
\(873\) −31101.0 −1.20574
\(874\) −22114.9 −0.855888
\(875\) 3137.02 0.121201
\(876\) −1918.93 −0.0740120
\(877\) −1782.50 −0.0686326 −0.0343163 0.999411i \(-0.510925\pi\)
−0.0343163 + 0.999411i \(0.510925\pi\)
\(878\) 54728.0 2.10362
\(879\) 6572.17 0.252189
\(880\) −10336.3 −0.395949
\(881\) −17232.6 −0.659001 −0.329500 0.944155i \(-0.606880\pi\)
−0.329500 + 0.944155i \(0.606880\pi\)
\(882\) 24347.2 0.929493
\(883\) 10188.2 0.388291 0.194145 0.980973i \(-0.437807\pi\)
0.194145 + 0.980973i \(0.437807\pi\)
\(884\) 0 0
\(885\) 545.859 0.0207332
\(886\) 22007.1 0.834471
\(887\) −15671.6 −0.593238 −0.296619 0.954996i \(-0.595859\pi\)
−0.296619 + 0.954996i \(0.595859\pi\)
\(888\) 5811.93 0.219635
\(889\) 4457.10 0.168151
\(890\) −3032.65 −0.114219
\(891\) 11538.7 0.433852
\(892\) −12631.0 −0.474122
\(893\) 10957.4 0.410612
\(894\) 11406.4 0.426721
\(895\) 103.932 0.00388163
\(896\) −4857.16 −0.181101
\(897\) −25085.8 −0.933768
\(898\) 29103.0 1.08149
\(899\) −2680.15 −0.0994307
\(900\) 4467.32 0.165456
\(901\) 0 0
\(902\) 22492.6 0.830291
\(903\) −1757.19 −0.0647569
\(904\) −38719.9 −1.42456
\(905\) −2185.81 −0.0802861
\(906\) 13313.9 0.488216
\(907\) 16908.3 0.618999 0.309500 0.950900i \(-0.399838\pi\)
0.309500 + 0.950900i \(0.399838\pi\)
\(908\) −4710.86 −0.172175
\(909\) 7002.92 0.255525
\(910\) 2626.78 0.0956890
\(911\) 36100.6 1.31292 0.656458 0.754362i \(-0.272054\pi\)
0.656458 + 0.754362i \(0.272054\pi\)
\(912\) 5017.92 0.182193
\(913\) 9917.98 0.359515
\(914\) 38625.1 1.39782
\(915\) 8164.63 0.294988
\(916\) 1272.65 0.0459057
\(917\) −7283.41 −0.262290
\(918\) 0 0
\(919\) 20016.2 0.718470 0.359235 0.933247i \(-0.383038\pi\)
0.359235 + 0.933247i \(0.383038\pi\)
\(920\) 20174.1 0.722958
\(921\) 2362.57 0.0845269
\(922\) 19735.7 0.704947
\(923\) −37584.1 −1.34030
\(924\) 293.291 0.0104422
\(925\) −15265.6 −0.542626
\(926\) 32144.0 1.14073
\(927\) −22769.8 −0.806753
\(928\) −1725.17 −0.0610253
\(929\) −30630.3 −1.08175 −0.540875 0.841103i \(-0.681907\pi\)
−0.540875 + 0.841103i \(0.681907\pi\)
\(930\) 4202.40 0.148174
\(931\) −11150.3 −0.392522
\(932\) −2274.15 −0.0799272
\(933\) 1309.28 0.0459420
\(934\) −46609.7 −1.63289
\(935\) 0 0
\(936\) −26286.2 −0.917941
\(937\) 3883.68 0.135405 0.0677025 0.997706i \(-0.478433\pi\)
0.0677025 + 0.997706i \(0.478433\pi\)
\(938\) −6420.79 −0.223503
\(939\) 12095.0 0.420346
\(940\) 3201.93 0.111101
\(941\) 21696.0 0.751613 0.375807 0.926698i \(-0.377366\pi\)
0.375807 + 0.926698i \(0.377366\pi\)
\(942\) 10095.0 0.349164
\(943\) −55217.7 −1.90682
\(944\) 4147.02 0.142981
\(945\) −1389.65 −0.0478362
\(946\) −27215.6 −0.935364
\(947\) −15904.8 −0.545763 −0.272882 0.962048i \(-0.587977\pi\)
−0.272882 + 0.962048i \(0.587977\pi\)
\(948\) 347.956 0.0119210
\(949\) −29657.4 −1.01446
\(950\) −10478.8 −0.357870
\(951\) −11456.9 −0.390659
\(952\) 0 0
\(953\) −81.8493 −0.00278212 −0.00139106 0.999999i \(-0.500443\pi\)
−0.00139106 + 0.999999i \(0.500443\pi\)
\(954\) 22553.1 0.765393
\(955\) 3945.03 0.133674
\(956\) −10250.0 −0.346765
\(957\) −1087.95 −0.0367485
\(958\) −51613.4 −1.74066
\(959\) −2076.93 −0.0699348
\(960\) −3338.86 −0.112251
\(961\) −11926.2 −0.400328
\(962\) −28773.1 −0.964327
\(963\) 27065.7 0.905692
\(964\) 2577.19 0.0861056
\(965\) 18662.7 0.622562
\(966\) −3687.75 −0.122827
\(967\) −19279.7 −0.641151 −0.320576 0.947223i \(-0.603876\pi\)
−0.320576 + 0.947223i \(0.603876\pi\)
\(968\) 11246.3 0.373419
\(969\) 0 0
\(970\) 21317.7 0.705639
\(971\) −53776.3 −1.77731 −0.888653 0.458581i \(-0.848358\pi\)
−0.888653 + 0.458581i \(0.848358\pi\)
\(972\) −6858.31 −0.226317
\(973\) 7053.30 0.232393
\(974\) −12946.9 −0.425918
\(975\) −11886.5 −0.390433
\(976\) 62028.6 2.03431
\(977\) 54738.2 1.79246 0.896228 0.443594i \(-0.146297\pi\)
0.896228 + 0.443594i \(0.146297\pi\)
\(978\) 14302.8 0.467641
\(979\) −5233.25 −0.170843
\(980\) −3258.30 −0.106207
\(981\) 19322.7 0.628876
\(982\) −2768.27 −0.0899581
\(983\) 3405.32 0.110491 0.0552456 0.998473i \(-0.482406\pi\)
0.0552456 + 0.998473i \(0.482406\pi\)
\(984\) 9961.19 0.322715
\(985\) 2262.86 0.0731988
\(986\) 0 0
\(987\) 1827.20 0.0589264
\(988\) −3856.20 −0.124172
\(989\) 66812.1 2.14813
\(990\) −9908.60 −0.318097
\(991\) 29925.7 0.959256 0.479628 0.877472i \(-0.340772\pi\)
0.479628 + 0.877472i \(0.340772\pi\)
\(992\) 11499.3 0.368047
\(993\) 178.911 0.00571759
\(994\) −5525.07 −0.176302
\(995\) −17427.3 −0.555259
\(996\) −1406.98 −0.0447608
\(997\) 32714.5 1.03920 0.519598 0.854411i \(-0.326082\pi\)
0.519598 + 0.854411i \(0.326082\pi\)
\(998\) −43572.2 −1.38202
\(999\) 15221.8 0.482080
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.g.1.11 12
17.3 odd 16 17.4.d.a.9.3 yes 12
17.4 even 4 289.4.b.e.288.2 12
17.6 odd 16 17.4.d.a.2.3 12
17.13 even 4 289.4.b.e.288.1 12
17.16 even 2 inner 289.4.a.g.1.12 12
51.20 even 16 153.4.l.a.145.1 12
51.23 even 16 153.4.l.a.19.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.2.3 12 17.6 odd 16
17.4.d.a.9.3 yes 12 17.3 odd 16
153.4.l.a.19.1 12 51.23 even 16
153.4.l.a.145.1 12 51.20 even 16
289.4.a.g.1.11 12 1.1 even 1 trivial
289.4.a.g.1.12 12 17.16 even 2 inner
289.4.b.e.288.1 12 17.13 even 4
289.4.b.e.288.2 12 17.4 even 4