Properties

Label 289.4.a.b.1.3
Level $289$
Weight $4$
Character 289.1
Self dual yes
Analytic conductor $17.052$
Analytic rank $0$
Dimension $3$
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] = [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: \(3\)
Coefficient field: 3.3.2636.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.287410\) 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)
 
\(f(q)\) \(=\) \(q+4.67129 q^{2} +7.62999 q^{3} +13.8209 q^{4} +11.9174 q^{5} +35.6419 q^{6} -26.1222 q^{7} +27.1912 q^{8} +31.2167 q^{9} +O(q^{10})\) \(q+4.67129 q^{2} +7.62999 q^{3} +13.8209 q^{4} +11.9174 q^{5} +35.6419 q^{6} -26.1222 q^{7} +27.1912 q^{8} +31.2167 q^{9} +55.6696 q^{10} +3.24412 q^{11} +105.453 q^{12} -20.0515 q^{13} -122.024 q^{14} +90.9296 q^{15} +16.4506 q^{16} +145.822 q^{18} +57.3466 q^{19} +164.709 q^{20} -199.312 q^{21} +15.1542 q^{22} -77.0438 q^{23} +207.469 q^{24} +17.0243 q^{25} -93.6662 q^{26} +32.1732 q^{27} -361.033 q^{28} +286.162 q^{29} +424.758 q^{30} +8.54816 q^{31} -140.684 q^{32} +24.7526 q^{33} -311.309 q^{35} +431.443 q^{36} -357.982 q^{37} +267.882 q^{38} -152.992 q^{39} +324.049 q^{40} -194.467 q^{41} -931.044 q^{42} -74.2619 q^{43} +44.8367 q^{44} +372.021 q^{45} -359.894 q^{46} +23.6130 q^{47} +125.518 q^{48} +339.369 q^{49} +79.5255 q^{50} -277.130 q^{52} +104.330 q^{53} +150.290 q^{54} +38.6614 q^{55} -710.295 q^{56} +437.553 q^{57} +1336.75 q^{58} +249.363 q^{59} +1256.73 q^{60} +370.384 q^{61} +39.9309 q^{62} -815.448 q^{63} -788.781 q^{64} -238.961 q^{65} +115.626 q^{66} +939.650 q^{67} -587.843 q^{69} -1454.21 q^{70} +520.197 q^{71} +848.820 q^{72} -348.741 q^{73} -1672.24 q^{74} +129.895 q^{75} +792.583 q^{76} -84.7434 q^{77} -714.672 q^{78} +953.827 q^{79} +196.049 q^{80} -597.369 q^{81} -908.412 q^{82} -1414.28 q^{83} -2754.68 q^{84} -346.899 q^{86} +2183.41 q^{87} +88.2115 q^{88} -486.132 q^{89} +1737.82 q^{90} +523.788 q^{91} -1064.82 q^{92} +65.2223 q^{93} +110.303 q^{94} +683.422 q^{95} -1073.42 q^{96} +685.281 q^{97} +1585.29 q^{98} +101.271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 4 q^{3} + 25 q^{4} + 8 q^{5} + 74 q^{6} - 22 q^{7} - 39 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 4 q^{3} + 25 q^{4} + 8 q^{5} + 74 q^{6} - 22 q^{7} - 39 q^{8} + 59 q^{9} + 56 q^{10} + 28 q^{11} - 22 q^{12} + 30 q^{13} - 92 q^{14} + 108 q^{15} + 137 q^{16} - 103 q^{18} + 80 q^{19} + 168 q^{20} - 192 q^{21} - 286 q^{22} - 142 q^{23} + 666 q^{24} - 223 q^{25} + 26 q^{26} + 20 q^{27} - 476 q^{28} + 456 q^{29} + 400 q^{30} - 230 q^{31} - 71 q^{32} - 332 q^{33} - 332 q^{35} + 1313 q^{36} - 356 q^{37} + 724 q^{38} - 268 q^{39} + 424 q^{40} + 294 q^{41} - 1128 q^{42} + 556 q^{43} + 1122 q^{44} + 384 q^{45} + 704 q^{46} + 640 q^{47} - 774 q^{48} - 269 q^{49} + 547 q^{50} - 774 q^{52} + 302 q^{53} + 1100 q^{54} + 76 q^{55} - 684 q^{56} + 720 q^{57} + 1304 q^{58} + 636 q^{59} + 1328 q^{60} + 84 q^{61} - 508 q^{62} - 1122 q^{63} - 919 q^{64} - 408 q^{65} + 2468 q^{66} + 1008 q^{67} + 576 q^{69} - 1504 q^{70} + 402 q^{71} - 927 q^{72} - 838 q^{73} - 836 q^{74} + 1548 q^{75} - 908 q^{76} - 504 q^{77} - 1308 q^{78} + 594 q^{79} + 40 q^{80} - 505 q^{81} - 358 q^{82} - 2396 q^{83} - 2040 q^{84} - 1264 q^{86} + 1428 q^{87} - 1838 q^{88} - 170 q^{89} + 2008 q^{90} + 1016 q^{91} - 4896 q^{92} + 632 q^{93} - 2016 q^{94} + 472 q^{95} - 678 q^{96} + 270 q^{97} + 2857 q^{98} + 2920 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.67129 1.65155 0.825775 0.564000i \(-0.190738\pi\)
0.825775 + 0.564000i \(0.190738\pi\)
\(3\) 7.62999 1.46839 0.734196 0.678938i \(-0.237559\pi\)
0.734196 + 0.678938i \(0.237559\pi\)
\(4\) 13.8209 1.72762
\(5\) 11.9174 1.06592 0.532962 0.846139i \(-0.321079\pi\)
0.532962 + 0.846139i \(0.321079\pi\)
\(6\) 35.6419 2.42512
\(7\) −26.1222 −1.41047 −0.705233 0.708975i \(-0.749158\pi\)
−0.705233 + 0.708975i \(0.749158\pi\)
\(8\) 27.1912 1.20169
\(9\) 31.2167 1.15617
\(10\) 55.6696 1.76043
\(11\) 3.24412 0.0889216 0.0444608 0.999011i \(-0.485843\pi\)
0.0444608 + 0.999011i \(0.485843\pi\)
\(12\) 105.453 2.53682
\(13\) −20.0515 −0.427790 −0.213895 0.976857i \(-0.568615\pi\)
−0.213895 + 0.976857i \(0.568615\pi\)
\(14\) −122.024 −2.32945
\(15\) 90.9296 1.56519
\(16\) 16.4506 0.257041
\(17\) 0 0
\(18\) 145.822 1.90948
\(19\) 57.3466 0.692432 0.346216 0.938155i \(-0.387466\pi\)
0.346216 + 0.938155i \(0.387466\pi\)
\(20\) 164.709 1.84151
\(21\) −199.312 −2.07112
\(22\) 15.1542 0.146858
\(23\) −77.0438 −0.698467 −0.349233 0.937036i \(-0.613558\pi\)
−0.349233 + 0.937036i \(0.613558\pi\)
\(24\) 207.469 1.76456
\(25\) 17.0243 0.136195
\(26\) −93.6662 −0.706517
\(27\) 32.1732 0.229323
\(28\) −361.033 −2.43674
\(29\) 286.162 1.83238 0.916190 0.400744i \(-0.131248\pi\)
0.916190 + 0.400744i \(0.131248\pi\)
\(30\) 424.758 2.58500
\(31\) 8.54816 0.0495256 0.0247628 0.999693i \(-0.492117\pi\)
0.0247628 + 0.999693i \(0.492117\pi\)
\(32\) −140.684 −0.777178
\(33\) 24.7526 0.130572
\(34\) 0 0
\(35\) −311.309 −1.50345
\(36\) 431.443 1.99742
\(37\) −357.982 −1.59059 −0.795296 0.606221i \(-0.792685\pi\)
−0.795296 + 0.606221i \(0.792685\pi\)
\(38\) 267.882 1.14359
\(39\) −152.992 −0.628164
\(40\) 324.049 1.28091
\(41\) −194.467 −0.740748 −0.370374 0.928883i \(-0.620770\pi\)
−0.370374 + 0.928883i \(0.620770\pi\)
\(42\) −931.044 −3.42055
\(43\) −74.2619 −0.263368 −0.131684 0.991292i \(-0.542038\pi\)
−0.131684 + 0.991292i \(0.542038\pi\)
\(44\) 44.8367 0.153622
\(45\) 372.021 1.23239
\(46\) −359.894 −1.15355
\(47\) 23.6130 0.0732831 0.0366416 0.999328i \(-0.488334\pi\)
0.0366416 + 0.999328i \(0.488334\pi\)
\(48\) 125.518 0.377437
\(49\) 339.369 0.989415
\(50\) 79.5255 0.224932
\(51\) 0 0
\(52\) −277.130 −0.739058
\(53\) 104.330 0.270393 0.135197 0.990819i \(-0.456833\pi\)
0.135197 + 0.990819i \(0.456833\pi\)
\(54\) 150.290 0.378739
\(55\) 38.6614 0.0947837
\(56\) −710.295 −1.69495
\(57\) 437.553 1.01676
\(58\) 1336.75 3.02627
\(59\) 249.363 0.550243 0.275122 0.961409i \(-0.411282\pi\)
0.275122 + 0.961409i \(0.411282\pi\)
\(60\) 1256.73 2.70405
\(61\) 370.384 0.777424 0.388712 0.921359i \(-0.372920\pi\)
0.388712 + 0.921359i \(0.372920\pi\)
\(62\) 39.9309 0.0817940
\(63\) −815.448 −1.63074
\(64\) −788.781 −1.54059
\(65\) −238.961 −0.455992
\(66\) 115.626 0.215646
\(67\) 939.650 1.71338 0.856691 0.515830i \(-0.172517\pi\)
0.856691 + 0.515830i \(0.172517\pi\)
\(68\) 0 0
\(69\) −587.843 −1.02562
\(70\) −1454.21 −2.48302
\(71\) 520.197 0.869522 0.434761 0.900546i \(-0.356833\pi\)
0.434761 + 0.900546i \(0.356833\pi\)
\(72\) 848.820 1.38937
\(73\) −348.741 −0.559137 −0.279568 0.960126i \(-0.590191\pi\)
−0.279568 + 0.960126i \(0.590191\pi\)
\(74\) −1672.24 −2.62694
\(75\) 129.895 0.199987
\(76\) 792.583 1.19626
\(77\) −84.7434 −0.125421
\(78\) −714.672 −1.03744
\(79\) 953.827 1.35840 0.679202 0.733951i \(-0.262326\pi\)
0.679202 + 0.733951i \(0.262326\pi\)
\(80\) 196.049 0.273986
\(81\) −597.369 −0.819437
\(82\) −908.412 −1.22338
\(83\) −1414.28 −1.87033 −0.935166 0.354211i \(-0.884750\pi\)
−0.935166 + 0.354211i \(0.884750\pi\)
\(84\) −2754.68 −3.57809
\(85\) 0 0
\(86\) −346.899 −0.434966
\(87\) 2183.41 2.69065
\(88\) 88.2115 0.106857
\(89\) −486.132 −0.578987 −0.289493 0.957180i \(-0.593487\pi\)
−0.289493 + 0.957180i \(0.593487\pi\)
\(90\) 1737.82 2.03536
\(91\) 523.788 0.603384
\(92\) −1064.82 −1.20668
\(93\) 65.2223 0.0727230
\(94\) 110.303 0.121031
\(95\) 683.422 0.738080
\(96\) −1073.42 −1.14120
\(97\) 685.281 0.717317 0.358659 0.933469i \(-0.383234\pi\)
0.358659 + 0.933469i \(0.383234\pi\)
\(98\) 1585.29 1.63407
\(99\) 101.271 0.102809
\(100\) 235.292 0.235292
\(101\) 864.755 0.851944 0.425972 0.904736i \(-0.359932\pi\)
0.425972 + 0.904736i \(0.359932\pi\)
\(102\) 0 0
\(103\) 1880.91 1.79933 0.899665 0.436580i \(-0.143810\pi\)
0.899665 + 0.436580i \(0.143810\pi\)
\(104\) −545.224 −0.514073
\(105\) −2375.28 −2.20765
\(106\) 487.355 0.446567
\(107\) 32.8149 0.0296480 0.0148240 0.999890i \(-0.495281\pi\)
0.0148240 + 0.999890i \(0.495281\pi\)
\(108\) 444.663 0.396183
\(109\) −528.727 −0.464613 −0.232307 0.972643i \(-0.574627\pi\)
−0.232307 + 0.972643i \(0.574627\pi\)
\(110\) 180.599 0.156540
\(111\) −2731.40 −2.33561
\(112\) −429.727 −0.362548
\(113\) −414.691 −0.345229 −0.172614 0.984989i \(-0.555221\pi\)
−0.172614 + 0.984989i \(0.555221\pi\)
\(114\) 2043.94 1.67923
\(115\) −918.161 −0.744513
\(116\) 3955.03 3.16565
\(117\) −625.940 −0.494600
\(118\) 1164.85 0.908754
\(119\) 0 0
\(120\) 2472.49 1.88088
\(121\) −1320.48 −0.992093
\(122\) 1730.17 1.28395
\(123\) −1483.78 −1.08771
\(124\) 118.143 0.0855613
\(125\) −1286.79 −0.920751
\(126\) −3809.19 −2.69325
\(127\) 596.093 0.416494 0.208247 0.978076i \(-0.433224\pi\)
0.208247 + 0.978076i \(0.433224\pi\)
\(128\) −2559.15 −1.76718
\(129\) −566.617 −0.386728
\(130\) −1116.26 −0.753094
\(131\) −121.819 −0.0812472 −0.0406236 0.999175i \(-0.512934\pi\)
−0.0406236 + 0.999175i \(0.512934\pi\)
\(132\) 342.103 0.225578
\(133\) −1498.02 −0.976652
\(134\) 4389.38 2.82973
\(135\) 383.420 0.244441
\(136\) 0 0
\(137\) −897.365 −0.559614 −0.279807 0.960056i \(-0.590270\pi\)
−0.279807 + 0.960056i \(0.590270\pi\)
\(138\) −2745.98 −1.69387
\(139\) 2113.61 1.28974 0.644871 0.764292i \(-0.276911\pi\)
0.644871 + 0.764292i \(0.276911\pi\)
\(140\) −4302.57 −2.59738
\(141\) 180.167 0.107608
\(142\) 2429.99 1.43606
\(143\) −65.0493 −0.0380398
\(144\) 513.534 0.297184
\(145\) 3410.31 1.95318
\(146\) −1629.07 −0.923442
\(147\) 2589.38 1.45285
\(148\) −4947.65 −2.74793
\(149\) 2580.76 1.41895 0.709476 0.704729i \(-0.248932\pi\)
0.709476 + 0.704729i \(0.248932\pi\)
\(150\) 606.778 0.330288
\(151\) 1342.77 0.723662 0.361831 0.932244i \(-0.382152\pi\)
0.361831 + 0.932244i \(0.382152\pi\)
\(152\) 1559.32 0.832091
\(153\) 0 0
\(154\) −395.861 −0.207139
\(155\) 101.872 0.0527906
\(156\) −2114.50 −1.08523
\(157\) −2495.82 −1.26871 −0.634357 0.773041i \(-0.718735\pi\)
−0.634357 + 0.773041i \(0.718735\pi\)
\(158\) 4455.60 2.24347
\(159\) 796.036 0.397043
\(160\) −1676.59 −0.828413
\(161\) 2012.55 0.985164
\(162\) −2790.48 −1.35334
\(163\) −1961.58 −0.942595 −0.471297 0.881974i \(-0.656214\pi\)
−0.471297 + 0.881974i \(0.656214\pi\)
\(164\) −2687.72 −1.27973
\(165\) 294.986 0.139180
\(166\) −6606.51 −3.08894
\(167\) −2179.24 −1.00979 −0.504894 0.863182i \(-0.668468\pi\)
−0.504894 + 0.863182i \(0.668468\pi\)
\(168\) −5419.54 −2.48885
\(169\) −1794.94 −0.816995
\(170\) 0 0
\(171\) 1790.17 0.800571
\(172\) −1026.37 −0.454999
\(173\) −3111.45 −1.36739 −0.683697 0.729766i \(-0.739629\pi\)
−0.683697 + 0.729766i \(0.739629\pi\)
\(174\) 10199.4 4.44374
\(175\) −444.713 −0.192098
\(176\) 53.3677 0.0228565
\(177\) 1902.64 0.807972
\(178\) −2270.86 −0.956226
\(179\) 810.106 0.338269 0.169135 0.985593i \(-0.445903\pi\)
0.169135 + 0.985593i \(0.445903\pi\)
\(180\) 5141.68 2.12910
\(181\) 3356.23 1.37827 0.689134 0.724634i \(-0.257991\pi\)
0.689134 + 0.724634i \(0.257991\pi\)
\(182\) 2446.77 0.996519
\(183\) 2826.03 1.14156
\(184\) −2094.92 −0.839343
\(185\) −4266.22 −1.69545
\(186\) 304.672 0.120106
\(187\) 0 0
\(188\) 326.353 0.126605
\(189\) −840.434 −0.323453
\(190\) 3192.46 1.21898
\(191\) 1338.41 0.507038 0.253519 0.967330i \(-0.418412\pi\)
0.253519 + 0.967330i \(0.418412\pi\)
\(192\) −6018.39 −2.26219
\(193\) 227.465 0.0848358 0.0424179 0.999100i \(-0.486494\pi\)
0.0424179 + 0.999100i \(0.486494\pi\)
\(194\) 3201.15 1.18468
\(195\) −1823.27 −0.669575
\(196\) 4690.40 1.70933
\(197\) −815.549 −0.294952 −0.147476 0.989066i \(-0.547115\pi\)
−0.147476 + 0.989066i \(0.547115\pi\)
\(198\) 473.064 0.169794
\(199\) −1866.90 −0.665030 −0.332515 0.943098i \(-0.607897\pi\)
−0.332515 + 0.943098i \(0.607897\pi\)
\(200\) 462.912 0.163664
\(201\) 7169.52 2.51591
\(202\) 4039.52 1.40703
\(203\) −7475.19 −2.58451
\(204\) 0 0
\(205\) −2317.54 −0.789581
\(206\) 8786.25 2.97168
\(207\) −2405.05 −0.807549
\(208\) −329.859 −0.109960
\(209\) 186.039 0.0615721
\(210\) −11095.6 −3.64605
\(211\) 1102.88 0.359836 0.179918 0.983682i \(-0.442417\pi\)
0.179918 + 0.983682i \(0.442417\pi\)
\(212\) 1441.94 0.467135
\(213\) 3969.10 1.27680
\(214\) 153.288 0.0489651
\(215\) −885.008 −0.280731
\(216\) 874.828 0.275576
\(217\) −223.297 −0.0698542
\(218\) −2469.84 −0.767332
\(219\) −2660.89 −0.821032
\(220\) 534.337 0.163750
\(221\) 0 0
\(222\) −12759.2 −3.85738
\(223\) −568.848 −0.170820 −0.0854100 0.996346i \(-0.527220\pi\)
−0.0854100 + 0.996346i \(0.527220\pi\)
\(224\) 3674.98 1.09618
\(225\) 531.442 0.157464
\(226\) −1937.14 −0.570163
\(227\) −2106.99 −0.616061 −0.308030 0.951377i \(-0.599670\pi\)
−0.308030 + 0.951377i \(0.599670\pi\)
\(228\) 6047.39 1.75657
\(229\) 4336.30 1.25131 0.625656 0.780099i \(-0.284831\pi\)
0.625656 + 0.780099i \(0.284831\pi\)
\(230\) −4289.00 −1.22960
\(231\) −646.591 −0.184167
\(232\) 7781.11 2.20196
\(233\) 4517.39 1.27014 0.635072 0.772453i \(-0.280970\pi\)
0.635072 + 0.772453i \(0.280970\pi\)
\(234\) −2923.95 −0.816856
\(235\) 281.405 0.0781143
\(236\) 3446.43 0.950609
\(237\) 7277.69 1.99467
\(238\) 0 0
\(239\) 5300.88 1.43467 0.717333 0.696731i \(-0.245363\pi\)
0.717333 + 0.696731i \(0.245363\pi\)
\(240\) 1495.85 0.402319
\(241\) 1368.82 0.365864 0.182932 0.983126i \(-0.441441\pi\)
0.182932 + 0.983126i \(0.441441\pi\)
\(242\) −6168.32 −1.63849
\(243\) −5426.60 −1.43258
\(244\) 5119.05 1.34309
\(245\) 4044.40 1.05464
\(246\) −6931.17 −1.79640
\(247\) −1149.88 −0.296216
\(248\) 232.435 0.0595146
\(249\) −10790.9 −2.74638
\(250\) −6010.96 −1.52067
\(251\) −5547.63 −1.39507 −0.697536 0.716549i \(-0.745720\pi\)
−0.697536 + 0.716549i \(0.745720\pi\)
\(252\) −11270.3 −2.81730
\(253\) −249.939 −0.0621088
\(254\) 2784.52 0.687860
\(255\) 0 0
\(256\) −5644.28 −1.37800
\(257\) −193.949 −0.0470748 −0.0235374 0.999723i \(-0.507493\pi\)
−0.0235374 + 0.999723i \(0.507493\pi\)
\(258\) −2646.83 −0.638700
\(259\) 9351.29 2.24348
\(260\) −3302.67 −0.787780
\(261\) 8933.04 2.11855
\(262\) −569.052 −0.134184
\(263\) −1345.63 −0.315494 −0.157747 0.987480i \(-0.550423\pi\)
−0.157747 + 0.987480i \(0.550423\pi\)
\(264\) 673.052 0.156907
\(265\) 1243.34 0.288218
\(266\) −6997.67 −1.61299
\(267\) −3709.18 −0.850180
\(268\) 12986.8 2.96007
\(269\) 3083.04 0.698797 0.349398 0.936974i \(-0.386386\pi\)
0.349398 + 0.936974i \(0.386386\pi\)
\(270\) 1791.07 0.403707
\(271\) −422.163 −0.0946294 −0.0473147 0.998880i \(-0.515066\pi\)
−0.0473147 + 0.998880i \(0.515066\pi\)
\(272\) 0 0
\(273\) 3996.50 0.886004
\(274\) −4191.85 −0.924230
\(275\) 55.2288 0.0121106
\(276\) −8124.54 −1.77188
\(277\) 8260.00 1.79168 0.895840 0.444377i \(-0.146575\pi\)
0.895840 + 0.444377i \(0.146575\pi\)
\(278\) 9873.28 2.13007
\(279\) 266.845 0.0572602
\(280\) −8464.86 −1.80669
\(281\) 3321.91 0.705226 0.352613 0.935769i \(-0.385293\pi\)
0.352613 + 0.935769i \(0.385293\pi\)
\(282\) 841.611 0.177721
\(283\) 7954.43 1.67082 0.835409 0.549629i \(-0.185231\pi\)
0.835409 + 0.549629i \(0.185231\pi\)
\(284\) 7189.61 1.50220
\(285\) 5214.50 1.08379
\(286\) −303.864 −0.0628246
\(287\) 5079.91 1.04480
\(288\) −4391.69 −0.898552
\(289\) 0 0
\(290\) 15930.5 3.22577
\(291\) 5228.69 1.05330
\(292\) −4819.92 −0.965974
\(293\) −1171.99 −0.233681 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 12095.8 2.39945
\(295\) 2971.76 0.586517
\(296\) −9733.98 −1.91141
\(297\) 104.374 0.0203918
\(298\) 12055.5 2.34347
\(299\) 1544.84 0.298798
\(300\) 1795.27 0.345500
\(301\) 1939.88 0.371472
\(302\) 6272.46 1.19516
\(303\) 6598.07 1.25099
\(304\) 943.387 0.177983
\(305\) 4414.01 0.828675
\(306\) 0 0
\(307\) 865.763 0.160950 0.0804751 0.996757i \(-0.474356\pi\)
0.0804751 + 0.996757i \(0.474356\pi\)
\(308\) −1171.23 −0.216679
\(309\) 14351.3 2.64212
\(310\) 475.872 0.0871862
\(311\) −6994.83 −1.27537 −0.637685 0.770297i \(-0.720108\pi\)
−0.637685 + 0.770297i \(0.720108\pi\)
\(312\) −4160.05 −0.754861
\(313\) −3442.33 −0.621635 −0.310818 0.950470i \(-0.600603\pi\)
−0.310818 + 0.950470i \(0.600603\pi\)
\(314\) −11658.7 −2.09534
\(315\) −9718.02 −1.73825
\(316\) 13182.8 2.34680
\(317\) −2066.15 −0.366078 −0.183039 0.983106i \(-0.558593\pi\)
−0.183039 + 0.983106i \(0.558593\pi\)
\(318\) 3718.52 0.655736
\(319\) 928.344 0.162938
\(320\) −9400.22 −1.64215
\(321\) 250.377 0.0435349
\(322\) 9401.21 1.62705
\(323\) 0 0
\(324\) −8256.20 −1.41567
\(325\) −341.362 −0.0582627
\(326\) −9163.11 −1.55674
\(327\) −4034.18 −0.682234
\(328\) −5287.80 −0.890152
\(329\) −616.823 −0.103363
\(330\) 1377.96 0.229862
\(331\) 9027.44 1.49907 0.749536 0.661964i \(-0.230277\pi\)
0.749536 + 0.661964i \(0.230277\pi\)
\(332\) −19546.7 −3.23121
\(333\) −11175.0 −1.83900
\(334\) −10179.8 −1.66771
\(335\) 11198.2 1.82633
\(336\) −3278.81 −0.532362
\(337\) −204.309 −0.0330250 −0.0165125 0.999864i \(-0.505256\pi\)
−0.0165125 + 0.999864i \(0.505256\pi\)
\(338\) −8384.67 −1.34931
\(339\) −3164.09 −0.506931
\(340\) 0 0
\(341\) 27.7312 0.00440390
\(342\) 8362.39 1.32218
\(343\) 94.8397 0.0149296
\(344\) −2019.27 −0.316488
\(345\) −7005.56 −1.09324
\(346\) −14534.5 −2.25832
\(347\) −143.063 −0.0221326 −0.0110663 0.999939i \(-0.503523\pi\)
−0.0110663 + 0.999939i \(0.503523\pi\)
\(348\) 30176.8 4.64841
\(349\) −3998.42 −0.613268 −0.306634 0.951828i \(-0.599203\pi\)
−0.306634 + 0.951828i \(0.599203\pi\)
\(350\) −2077.38 −0.317259
\(351\) −645.119 −0.0981024
\(352\) −456.396 −0.0691079
\(353\) −5809.57 −0.875956 −0.437978 0.898986i \(-0.644305\pi\)
−0.437978 + 0.898986i \(0.644305\pi\)
\(354\) 8887.77 1.33441
\(355\) 6199.39 0.926844
\(356\) −6718.79 −1.00027
\(357\) 0 0
\(358\) 3784.24 0.558668
\(359\) −4895.37 −0.719687 −0.359844 0.933013i \(-0.617170\pi\)
−0.359844 + 0.933013i \(0.617170\pi\)
\(360\) 10115.7 1.48096
\(361\) −3570.37 −0.520538
\(362\) 15677.9 2.27628
\(363\) −10075.2 −1.45678
\(364\) 7239.24 1.04242
\(365\) −4156.08 −0.595998
\(366\) 13201.2 1.88535
\(367\) −528.151 −0.0751206 −0.0375603 0.999294i \(-0.511959\pi\)
−0.0375603 + 0.999294i \(0.511959\pi\)
\(368\) −1267.42 −0.179535
\(369\) −6070.62 −0.856433
\(370\) −19928.7 −2.80012
\(371\) −2725.33 −0.381380
\(372\) 901.433 0.125637
\(373\) −10113.5 −1.40390 −0.701950 0.712226i \(-0.747687\pi\)
−0.701950 + 0.712226i \(0.747687\pi\)
\(374\) 0 0
\(375\) −9818.18 −1.35202
\(376\) 642.066 0.0880639
\(377\) −5737.98 −0.783875
\(378\) −3925.91 −0.534198
\(379\) −729.385 −0.0988548 −0.0494274 0.998778i \(-0.515740\pi\)
−0.0494274 + 0.998778i \(0.515740\pi\)
\(380\) 9445.52 1.27512
\(381\) 4548.18 0.611576
\(382\) 6252.12 0.837399
\(383\) −1608.08 −0.214540 −0.107270 0.994230i \(-0.534211\pi\)
−0.107270 + 0.994230i \(0.534211\pi\)
\(384\) −19526.3 −2.59491
\(385\) −1009.92 −0.133689
\(386\) 1062.56 0.140111
\(387\) −2318.21 −0.304499
\(388\) 9471.22 1.23925
\(389\) 9824.09 1.28047 0.640233 0.768181i \(-0.278838\pi\)
0.640233 + 0.768181i \(0.278838\pi\)
\(390\) −8517.02 −1.10584
\(391\) 0 0
\(392\) 9227.87 1.18897
\(393\) −929.478 −0.119303
\(394\) −3809.66 −0.487127
\(395\) 11367.1 1.44796
\(396\) 1399.65 0.177614
\(397\) −2876.88 −0.363694 −0.181847 0.983327i \(-0.558208\pi\)
−0.181847 + 0.983327i \(0.558208\pi\)
\(398\) −8720.82 −1.09833
\(399\) −11429.9 −1.43411
\(400\) 280.061 0.0350076
\(401\) 6515.91 0.811444 0.405722 0.913996i \(-0.367020\pi\)
0.405722 + 0.913996i \(0.367020\pi\)
\(402\) 33490.9 4.15516
\(403\) −171.403 −0.0211866
\(404\) 11951.7 1.47183
\(405\) −7119.09 −0.873457
\(406\) −34918.8 −4.26845
\(407\) −1161.34 −0.141438
\(408\) 0 0
\(409\) −8870.10 −1.07237 −0.536183 0.844101i \(-0.680134\pi\)
−0.536183 + 0.844101i \(0.680134\pi\)
\(410\) −10825.9 −1.30403
\(411\) −6846.89 −0.821732
\(412\) 25995.9 3.10855
\(413\) −6513.92 −0.776099
\(414\) −11234.7 −1.33371
\(415\) −16854.5 −1.99363
\(416\) 2820.93 0.332469
\(417\) 16126.8 1.89385
\(418\) 869.041 0.101689
\(419\) 1009.53 0.117706 0.0588531 0.998267i \(-0.481256\pi\)
0.0588531 + 0.998267i \(0.481256\pi\)
\(420\) −32828.6 −3.81398
\(421\) −3253.60 −0.376652 −0.188326 0.982107i \(-0.560306\pi\)
−0.188326 + 0.982107i \(0.560306\pi\)
\(422\) 5151.87 0.594287
\(423\) 737.119 0.0847280
\(424\) 2836.86 0.324930
\(425\) 0 0
\(426\) 18540.8 2.10870
\(427\) −9675.25 −1.09653
\(428\) 453.532 0.0512203
\(429\) −496.325 −0.0558573
\(430\) −4134.13 −0.463640
\(431\) 2352.51 0.262915 0.131457 0.991322i \(-0.458034\pi\)
0.131457 + 0.991322i \(0.458034\pi\)
\(432\) 529.269 0.0589455
\(433\) −5860.51 −0.650434 −0.325217 0.945639i \(-0.605437\pi\)
−0.325217 + 0.945639i \(0.605437\pi\)
\(434\) −1043.08 −0.115368
\(435\) 26020.6 2.86803
\(436\) −7307.50 −0.802674
\(437\) −4418.20 −0.483641
\(438\) −12429.8 −1.35597
\(439\) −2894.17 −0.314650 −0.157325 0.987547i \(-0.550287\pi\)
−0.157325 + 0.987547i \(0.550287\pi\)
\(440\) 1051.25 0.113901
\(441\) 10594.0 1.14394
\(442\) 0 0
\(443\) −8256.85 −0.885541 −0.442771 0.896635i \(-0.646004\pi\)
−0.442771 + 0.896635i \(0.646004\pi\)
\(444\) −37750.5 −4.03504
\(445\) −5793.42 −0.617156
\(446\) −2657.25 −0.282118
\(447\) 19691.1 2.08358
\(448\) 20604.7 2.17295
\(449\) −15487.1 −1.62779 −0.813897 0.581009i \(-0.802658\pi\)
−0.813897 + 0.581009i \(0.802658\pi\)
\(450\) 2482.52 0.260060
\(451\) −630.874 −0.0658685
\(452\) −5731.42 −0.596423
\(453\) 10245.3 1.06262
\(454\) −9842.35 −1.01745
\(455\) 6242.19 0.643162
\(456\) 11897.6 1.22184
\(457\) −16055.6 −1.64343 −0.821716 0.569897i \(-0.806983\pi\)
−0.821716 + 0.569897i \(0.806983\pi\)
\(458\) 20256.1 2.06661
\(459\) 0 0
\(460\) −12689.8 −1.28623
\(461\) 14064.0 1.42088 0.710440 0.703758i \(-0.248496\pi\)
0.710440 + 0.703758i \(0.248496\pi\)
\(462\) −3020.41 −0.304161
\(463\) −8071.30 −0.810162 −0.405081 0.914281i \(-0.632757\pi\)
−0.405081 + 0.914281i \(0.632757\pi\)
\(464\) 4707.55 0.470997
\(465\) 777.280 0.0775172
\(466\) 21102.0 2.09771
\(467\) 8582.41 0.850421 0.425211 0.905094i \(-0.360200\pi\)
0.425211 + 0.905094i \(0.360200\pi\)
\(468\) −8651.07 −0.854479
\(469\) −24545.7 −2.41667
\(470\) 1314.53 0.129010
\(471\) −19043.1 −1.86297
\(472\) 6780.49 0.661224
\(473\) −240.914 −0.0234191
\(474\) 33996.2 3.29429
\(475\) 976.286 0.0943054
\(476\) 0 0
\(477\) 3256.84 0.312621
\(478\) 24761.9 2.36942
\(479\) 6320.96 0.602948 0.301474 0.953474i \(-0.402521\pi\)
0.301474 + 0.953474i \(0.402521\pi\)
\(480\) −12792.4 −1.21643
\(481\) 7178.07 0.680441
\(482\) 6394.13 0.604242
\(483\) 15355.8 1.44661
\(484\) −18250.2 −1.71396
\(485\) 8166.77 0.764606
\(486\) −25349.2 −2.36597
\(487\) −7336.47 −0.682643 −0.341321 0.939947i \(-0.610874\pi\)
−0.341321 + 0.939947i \(0.610874\pi\)
\(488\) 10071.2 0.934225
\(489\) −14966.8 −1.38410
\(490\) 18892.6 1.74179
\(491\) 6672.53 0.613294 0.306647 0.951823i \(-0.400793\pi\)
0.306647 + 0.951823i \(0.400793\pi\)
\(492\) −20507.2 −1.87914
\(493\) 0 0
\(494\) −5371.43 −0.489215
\(495\) 1206.88 0.109586
\(496\) 140.623 0.0127301
\(497\) −13588.7 −1.22643
\(498\) −50407.6 −4.53578
\(499\) −17920.9 −1.60772 −0.803858 0.594821i \(-0.797223\pi\)
−0.803858 + 0.594821i \(0.797223\pi\)
\(500\) −17784.6 −1.59070
\(501\) −16627.6 −1.48276
\(502\) −25914.6 −2.30403
\(503\) 11325.3 1.00392 0.501959 0.864891i \(-0.332613\pi\)
0.501959 + 0.864891i \(0.332613\pi\)
\(504\) −22173.0 −1.95965
\(505\) 10305.6 0.908108
\(506\) −1167.54 −0.102576
\(507\) −13695.4 −1.19967
\(508\) 8238.56 0.719541
\(509\) −8313.78 −0.723972 −0.361986 0.932184i \(-0.617901\pi\)
−0.361986 + 0.932184i \(0.617901\pi\)
\(510\) 0 0
\(511\) 9109.87 0.788644
\(512\) −5892.85 −0.508651
\(513\) 1845.02 0.158791
\(514\) −905.993 −0.0777464
\(515\) 22415.5 1.91795
\(516\) −7831.17 −0.668117
\(517\) 76.6033 0.00651646
\(518\) 43682.6 3.70521
\(519\) −23740.3 −2.00787
\(520\) −6497.65 −0.547963
\(521\) −5121.64 −0.430677 −0.215339 0.976539i \(-0.569086\pi\)
−0.215339 + 0.976539i \(0.569086\pi\)
\(522\) 41728.8 3.49889
\(523\) 13378.5 1.11855 0.559275 0.828982i \(-0.311080\pi\)
0.559275 + 0.828982i \(0.311080\pi\)
\(524\) −1683.65 −0.140364
\(525\) −3393.15 −0.282075
\(526\) −6285.81 −0.521054
\(527\) 0 0
\(528\) 407.195 0.0335623
\(529\) −6231.26 −0.512144
\(530\) 5808.01 0.476007
\(531\) 7784.29 0.636176
\(532\) −20704.0 −1.68728
\(533\) 3899.35 0.316885
\(534\) −17326.6 −1.40411
\(535\) 391.068 0.0316025
\(536\) 25550.2 2.05896
\(537\) 6181.10 0.496712
\(538\) 14401.8 1.15410
\(539\) 1100.95 0.0879804
\(540\) 5299.23 0.422301
\(541\) −9906.81 −0.787296 −0.393648 0.919261i \(-0.628787\pi\)
−0.393648 + 0.919261i \(0.628787\pi\)
\(542\) −1972.04 −0.156285
\(543\) 25608.0 2.02384
\(544\) 0 0
\(545\) −6301.05 −0.495243
\(546\) 18668.8 1.46328
\(547\) −16399.6 −1.28189 −0.640947 0.767585i \(-0.721458\pi\)
−0.640947 + 0.767585i \(0.721458\pi\)
\(548\) −12402.4 −0.966798
\(549\) 11562.2 0.898836
\(550\) 257.990 0.0200013
\(551\) 16410.4 1.26880
\(552\) −15984.2 −1.23248
\(553\) −24916.1 −1.91598
\(554\) 38584.8 2.95905
\(555\) −32551.2 −2.48959
\(556\) 29212.0 2.22818
\(557\) 22044.3 1.67692 0.838461 0.544962i \(-0.183456\pi\)
0.838461 + 0.544962i \(0.183456\pi\)
\(558\) 1246.51 0.0945681
\(559\) 1489.06 0.112666
\(560\) −5121.22 −0.386448
\(561\) 0 0
\(562\) 15517.6 1.16472
\(563\) 12048.8 0.901947 0.450973 0.892537i \(-0.351077\pi\)
0.450973 + 0.892537i \(0.351077\pi\)
\(564\) 2490.07 0.185906
\(565\) −4942.04 −0.367988
\(566\) 37157.4 2.75944
\(567\) 15604.6 1.15579
\(568\) 14144.8 1.04490
\(569\) −23785.4 −1.75243 −0.876217 0.481916i \(-0.839941\pi\)
−0.876217 + 0.481916i \(0.839941\pi\)
\(570\) 24358.4 1.78993
\(571\) 10878.3 0.797271 0.398635 0.917110i \(-0.369484\pi\)
0.398635 + 0.917110i \(0.369484\pi\)
\(572\) −899.041 −0.0657182
\(573\) 10212.1 0.744531
\(574\) 23729.7 1.72554
\(575\) −1311.62 −0.0951274
\(576\) −24623.1 −1.78119
\(577\) 6315.86 0.455689 0.227845 0.973698i \(-0.426832\pi\)
0.227845 + 0.973698i \(0.426832\pi\)
\(578\) 0 0
\(579\) 1735.56 0.124572
\(580\) 47133.7 3.37434
\(581\) 36944.1 2.63804
\(582\) 24424.7 1.73958
\(583\) 338.459 0.0240438
\(584\) −9482.69 −0.671912
\(585\) −7459.58 −0.527206
\(586\) −5474.72 −0.385936
\(587\) 18192.1 1.27916 0.639581 0.768724i \(-0.279108\pi\)
0.639581 + 0.768724i \(0.279108\pi\)
\(588\) 35787.7 2.50996
\(589\) 490.207 0.0342931
\(590\) 13882.0 0.968663
\(591\) −6222.63 −0.433104
\(592\) −5889.03 −0.408848
\(593\) −9828.72 −0.680636 −0.340318 0.940310i \(-0.610535\pi\)
−0.340318 + 0.940310i \(0.610535\pi\)
\(594\) 487.559 0.0336781
\(595\) 0 0
\(596\) 35668.5 2.45140
\(597\) −14244.4 −0.976524
\(598\) 7216.40 0.493479
\(599\) 4662.57 0.318043 0.159021 0.987275i \(-0.449166\pi\)
0.159021 + 0.987275i \(0.449166\pi\)
\(600\) 3532.01 0.240323
\(601\) −21658.6 −1.47000 −0.735001 0.678066i \(-0.762818\pi\)
−0.735001 + 0.678066i \(0.762818\pi\)
\(602\) 9061.76 0.613504
\(603\) 29332.8 1.98097
\(604\) 18558.3 1.25021
\(605\) −15736.6 −1.05750
\(606\) 30821.5 2.06607
\(607\) −25764.7 −1.72283 −0.861415 0.507902i \(-0.830421\pi\)
−0.861415 + 0.507902i \(0.830421\pi\)
\(608\) −8067.76 −0.538143
\(609\) −57035.6 −3.79507
\(610\) 20619.1 1.36860
\(611\) −473.475 −0.0313498
\(612\) 0 0
\(613\) 16018.1 1.05541 0.527705 0.849428i \(-0.323053\pi\)
0.527705 + 0.849428i \(0.323053\pi\)
\(614\) 4044.23 0.265817
\(615\) −17682.8 −1.15941
\(616\) −2304.28 −0.150718
\(617\) −22250.3 −1.45180 −0.725902 0.687798i \(-0.758578\pi\)
−0.725902 + 0.687798i \(0.758578\pi\)
\(618\) 67038.9 4.36360
\(619\) 3765.95 0.244534 0.122267 0.992497i \(-0.460984\pi\)
0.122267 + 0.992497i \(0.460984\pi\)
\(620\) 1407.96 0.0912018
\(621\) −2478.74 −0.160175
\(622\) −32674.9 −2.10634
\(623\) 12698.8 0.816642
\(624\) −2516.82 −0.161464
\(625\) −17463.2 −1.11765
\(626\) −16080.1 −1.02666
\(627\) 1419.47 0.0904120
\(628\) −34494.5 −2.19185
\(629\) 0 0
\(630\) −45395.7 −2.87080
\(631\) −20806.5 −1.31267 −0.656334 0.754470i \(-0.727894\pi\)
−0.656334 + 0.754470i \(0.727894\pi\)
\(632\) 25935.7 1.63239
\(633\) 8414.96 0.528380
\(634\) −9651.60 −0.604596
\(635\) 7103.88 0.443951
\(636\) 11002.0 0.685937
\(637\) −6804.85 −0.423262
\(638\) 4336.56 0.269100
\(639\) 16238.8 1.00532
\(640\) −30498.4 −1.88368
\(641\) −2439.58 −0.150324 −0.0751620 0.997171i \(-0.523947\pi\)
−0.0751620 + 0.997171i \(0.523947\pi\)
\(642\) 1169.58 0.0719000
\(643\) 19320.1 1.18493 0.592466 0.805595i \(-0.298154\pi\)
0.592466 + 0.805595i \(0.298154\pi\)
\(644\) 27815.4 1.70199
\(645\) −6752.60 −0.412222
\(646\) 0 0
\(647\) −14067.1 −0.854766 −0.427383 0.904071i \(-0.640564\pi\)
−0.427383 + 0.904071i \(0.640564\pi\)
\(648\) −16243.2 −0.984712
\(649\) 808.963 0.0489285
\(650\) −1594.60 −0.0962238
\(651\) −1703.75 −0.102573
\(652\) −27110.9 −1.62844
\(653\) −15893.7 −0.952478 −0.476239 0.879316i \(-0.658000\pi\)
−0.476239 + 0.879316i \(0.658000\pi\)
\(654\) −18844.8 −1.12674
\(655\) −1451.77 −0.0866033
\(656\) −3199.11 −0.190403
\(657\) −10886.5 −0.646459
\(658\) −2881.36 −0.170710
\(659\) −9653.54 −0.570635 −0.285318 0.958433i \(-0.592099\pi\)
−0.285318 + 0.958433i \(0.592099\pi\)
\(660\) 4076.98 0.240449
\(661\) 5389.06 0.317111 0.158555 0.987350i \(-0.449316\pi\)
0.158555 + 0.987350i \(0.449316\pi\)
\(662\) 42169.7 2.47579
\(663\) 0 0
\(664\) −38456.0 −2.24757
\(665\) −17852.5 −1.04104
\(666\) −52201.7 −3.03720
\(667\) −22047.0 −1.27986
\(668\) −30119.1 −1.74453
\(669\) −4340.30 −0.250831
\(670\) 52309.9 3.01628
\(671\) 1201.57 0.0691297
\(672\) 28040.1 1.60963
\(673\) 3032.18 0.173673 0.0868366 0.996223i \(-0.472324\pi\)
0.0868366 + 0.996223i \(0.472324\pi\)
\(674\) −954.386 −0.0545424
\(675\) 547.726 0.0312326
\(676\) −24807.7 −1.41145
\(677\) 22029.2 1.25059 0.625295 0.780388i \(-0.284979\pi\)
0.625295 + 0.780388i \(0.284979\pi\)
\(678\) −14780.4 −0.837222
\(679\) −17901.1 −1.01175
\(680\) 0 0
\(681\) −16076.3 −0.904618
\(682\) 129.540 0.00727326
\(683\) 9040.72 0.506491 0.253246 0.967402i \(-0.418502\pi\)
0.253246 + 0.967402i \(0.418502\pi\)
\(684\) 24741.8 1.38308
\(685\) −10694.3 −0.596506
\(686\) 443.023 0.0246570
\(687\) 33085.9 1.83742
\(688\) −1221.65 −0.0676964
\(689\) −2091.97 −0.115672
\(690\) −32725.0 −1.80553
\(691\) 22863.5 1.25871 0.629355 0.777118i \(-0.283319\pi\)
0.629355 + 0.777118i \(0.283319\pi\)
\(692\) −43003.1 −2.36233
\(693\) −2645.41 −0.145008
\(694\) −668.288 −0.0365531
\(695\) 25188.7 1.37477
\(696\) 59369.7 3.23334
\(697\) 0 0
\(698\) −18677.8 −1.01284
\(699\) 34467.6 1.86507
\(700\) −6146.34 −0.331871
\(701\) 1753.00 0.0944507 0.0472253 0.998884i \(-0.484962\pi\)
0.0472253 + 0.998884i \(0.484962\pi\)
\(702\) −3013.54 −0.162021
\(703\) −20529.1 −1.10138
\(704\) −2558.90 −0.136992
\(705\) 2147.12 0.114702
\(706\) −27138.2 −1.44668
\(707\) −22589.3 −1.20164
\(708\) 26296.2 1.39587
\(709\) −11547.0 −0.611645 −0.305823 0.952089i \(-0.598931\pi\)
−0.305823 + 0.952089i \(0.598931\pi\)
\(710\) 28959.2 1.53073
\(711\) 29775.3 1.57055
\(712\) −13218.5 −0.695765
\(713\) −658.582 −0.0345920
\(714\) 0 0
\(715\) −775.218 −0.0405476
\(716\) 11196.4 0.584399
\(717\) 40445.6 2.10665
\(718\) −22867.7 −1.18860
\(719\) 10289.8 0.533720 0.266860 0.963735i \(-0.414014\pi\)
0.266860 + 0.963735i \(0.414014\pi\)
\(720\) 6119.99 0.316776
\(721\) −49133.4 −2.53790
\(722\) −16678.2 −0.859695
\(723\) 10444.0 0.537231
\(724\) 46386.2 2.38112
\(725\) 4871.72 0.249560
\(726\) −47064.2 −2.40595
\(727\) 2950.10 0.150499 0.0752497 0.997165i \(-0.476025\pi\)
0.0752497 + 0.997165i \(0.476025\pi\)
\(728\) 14242.5 0.725083
\(729\) −25275.9 −1.28415
\(730\) −19414.2 −0.984320
\(731\) 0 0
\(732\) 39058.3 1.97218
\(733\) 24348.2 1.22691 0.613453 0.789731i \(-0.289780\pi\)
0.613453 + 0.789731i \(0.289780\pi\)
\(734\) −2467.15 −0.124065
\(735\) 30858.7 1.54863
\(736\) 10838.8 0.542833
\(737\) 3048.33 0.152357
\(738\) −28357.6 −1.41444
\(739\) −29233.5 −1.45517 −0.727585 0.686017i \(-0.759358\pi\)
−0.727585 + 0.686017i \(0.759358\pi\)
\(740\) −58963.1 −2.92909
\(741\) −8773.59 −0.434961
\(742\) −12730.8 −0.629868
\(743\) −15340.6 −0.757457 −0.378729 0.925508i \(-0.623639\pi\)
−0.378729 + 0.925508i \(0.623639\pi\)
\(744\) 1773.47 0.0873908
\(745\) 30755.9 1.51250
\(746\) −47242.9 −2.31861
\(747\) −44149.2 −2.16243
\(748\) 0 0
\(749\) −857.197 −0.0418175
\(750\) −45863.5 −2.23293
\(751\) −39862.6 −1.93689 −0.968446 0.249223i \(-0.919825\pi\)
−0.968446 + 0.249223i \(0.919825\pi\)
\(752\) 388.448 0.0188368
\(753\) −42328.3 −2.04851
\(754\) −26803.7 −1.29461
\(755\) 16002.3 0.771369
\(756\) −11615.6 −0.558802
\(757\) 26375.1 1.26634 0.633169 0.774013i \(-0.281754\pi\)
0.633169 + 0.774013i \(0.281754\pi\)
\(758\) −3407.17 −0.163264
\(759\) −1907.03 −0.0912000
\(760\) 18583.1 0.886946
\(761\) 7848.63 0.373867 0.186933 0.982373i \(-0.440145\pi\)
0.186933 + 0.982373i \(0.440145\pi\)
\(762\) 21245.9 1.01005
\(763\) 13811.5 0.655322
\(764\) 18498.1 0.875967
\(765\) 0 0
\(766\) −7511.78 −0.354323
\(767\) −5000.10 −0.235389
\(768\) −43065.8 −2.02344
\(769\) 31818.9 1.49209 0.746046 0.665895i \(-0.231950\pi\)
0.746046 + 0.665895i \(0.231950\pi\)
\(770\) −4717.63 −0.220794
\(771\) −1479.83 −0.0691242
\(772\) 3143.78 0.146564
\(773\) −29559.8 −1.37541 −0.687706 0.725990i \(-0.741382\pi\)
−0.687706 + 0.725990i \(0.741382\pi\)
\(774\) −10829.0 −0.502896
\(775\) 145.527 0.00674512
\(776\) 18633.6 0.861996
\(777\) 71350.2 3.29430
\(778\) 45891.2 2.11475
\(779\) −11152.0 −0.512917
\(780\) −25199.3 −1.15677
\(781\) 1687.58 0.0773193
\(782\) 0 0
\(783\) 9206.75 0.420208
\(784\) 5582.84 0.254320
\(785\) −29743.6 −1.35235
\(786\) −4341.86 −0.197034
\(787\) 28038.7 1.26998 0.634989 0.772521i \(-0.281005\pi\)
0.634989 + 0.772521i \(0.281005\pi\)
\(788\) −11271.6 −0.509563
\(789\) −10267.1 −0.463269
\(790\) 53099.2 2.39137
\(791\) 10832.6 0.486934
\(792\) 2753.67 0.123545
\(793\) −7426.75 −0.332574
\(794\) −13438.7 −0.600659
\(795\) 9486.68 0.423217
\(796\) −25802.3 −1.14892
\(797\) 5320.45 0.236462 0.118231 0.992986i \(-0.462278\pi\)
0.118231 + 0.992986i \(0.462278\pi\)
\(798\) −53392.2 −2.36850
\(799\) 0 0
\(800\) −2395.05 −0.105847
\(801\) −15175.4 −0.669409
\(802\) 30437.7 1.34014
\(803\) −1131.35 −0.0497194
\(804\) 99089.4 4.34653
\(805\) 23984.4 1.05011
\(806\) −800.673 −0.0349907
\(807\) 23523.6 1.02611
\(808\) 23513.8 1.02378
\(809\) −30934.0 −1.34435 −0.672177 0.740390i \(-0.734641\pi\)
−0.672177 + 0.740390i \(0.734641\pi\)
\(810\) −33255.3 −1.44256
\(811\) 40364.5 1.74771 0.873854 0.486189i \(-0.161613\pi\)
0.873854 + 0.486189i \(0.161613\pi\)
\(812\) −103314. −4.46504
\(813\) −3221.10 −0.138953
\(814\) −5424.94 −0.233592
\(815\) −23376.9 −1.00473
\(816\) 0 0
\(817\) −4258.66 −0.182364
\(818\) −41434.8 −1.77107
\(819\) 16350.9 0.697616
\(820\) −32030.6 −1.36409
\(821\) −19799.7 −0.841672 −0.420836 0.907137i \(-0.638263\pi\)
−0.420836 + 0.907137i \(0.638263\pi\)
\(822\) −31983.8 −1.35713
\(823\) −18756.4 −0.794419 −0.397210 0.917728i \(-0.630021\pi\)
−0.397210 + 0.917728i \(0.630021\pi\)
\(824\) 51144.1 2.16225
\(825\) 421.395 0.0177832
\(826\) −30428.4 −1.28177
\(827\) −20958.0 −0.881234 −0.440617 0.897695i \(-0.645240\pi\)
−0.440617 + 0.897695i \(0.645240\pi\)
\(828\) −33240.0 −1.39513
\(829\) −31320.3 −1.31218 −0.656091 0.754682i \(-0.727791\pi\)
−0.656091 + 0.754682i \(0.727791\pi\)
\(830\) −78732.4 −3.29258
\(831\) 63023.7 2.63089
\(832\) 15816.2 0.659049
\(833\) 0 0
\(834\) 75333.0 3.12778
\(835\) −25970.8 −1.07636
\(836\) 2571.23 0.106373
\(837\) 275.021 0.0113574
\(838\) 4715.82 0.194398
\(839\) 30290.6 1.24642 0.623212 0.782053i \(-0.285827\pi\)
0.623212 + 0.782053i \(0.285827\pi\)
\(840\) −64586.8 −2.65292
\(841\) 57499.9 2.35762
\(842\) −15198.5 −0.622060
\(843\) 25346.1 1.03555
\(844\) 15242.8 0.621659
\(845\) −21391.0 −0.870855
\(846\) 3443.29 0.139933
\(847\) 34493.7 1.39931
\(848\) 1716.29 0.0695021
\(849\) 60692.2 2.45342
\(850\) 0 0
\(851\) 27580.3 1.11098
\(852\) 54856.6 2.20582
\(853\) 21111.8 0.847425 0.423712 0.905797i \(-0.360727\pi\)
0.423712 + 0.905797i \(0.360727\pi\)
\(854\) −45195.9 −1.81097
\(855\) 21334.2 0.853348
\(856\) 892.277 0.0356278
\(857\) 39983.0 1.59369 0.796845 0.604184i \(-0.206501\pi\)
0.796845 + 0.604184i \(0.206501\pi\)
\(858\) −2318.48 −0.0922512
\(859\) −39503.3 −1.56907 −0.784537 0.620082i \(-0.787099\pi\)
−0.784537 + 0.620082i \(0.787099\pi\)
\(860\) −12231.6 −0.484995
\(861\) 38759.6 1.53418
\(862\) 10989.2 0.434217
\(863\) 26019.9 1.02634 0.513168 0.858288i \(-0.328472\pi\)
0.513168 + 0.858288i \(0.328472\pi\)
\(864\) −4526.26 −0.178225
\(865\) −37080.4 −1.45754
\(866\) −27376.1 −1.07422
\(867\) 0 0
\(868\) −3086.17 −0.120681
\(869\) 3094.33 0.120791
\(870\) 121550. 4.73669
\(871\) −18841.4 −0.732968
\(872\) −14376.7 −0.558323
\(873\) 21392.2 0.829343
\(874\) −20638.7 −0.798757
\(875\) 33613.8 1.29869
\(876\) −36775.9 −1.41843
\(877\) −15038.3 −0.579027 −0.289514 0.957174i \(-0.593494\pi\)
−0.289514 + 0.957174i \(0.593494\pi\)
\(878\) −13519.5 −0.519660
\(879\) −8942.30 −0.343136
\(880\) 636.004 0.0243633
\(881\) 18334.3 0.701133 0.350567 0.936538i \(-0.385989\pi\)
0.350567 + 0.936538i \(0.385989\pi\)
\(882\) 49487.5 1.88927
\(883\) 26659.5 1.01604 0.508020 0.861345i \(-0.330378\pi\)
0.508020 + 0.861345i \(0.330378\pi\)
\(884\) 0 0
\(885\) 22674.5 0.861237
\(886\) −38570.1 −1.46251
\(887\) −11473.7 −0.434327 −0.217163 0.976135i \(-0.569680\pi\)
−0.217163 + 0.976135i \(0.569680\pi\)
\(888\) −74270.1 −2.80669
\(889\) −15571.3 −0.587450
\(890\) −27062.7 −1.01926
\(891\) −1937.94 −0.0728656
\(892\) −7862.01 −0.295111
\(893\) 1354.12 0.0507436
\(894\) 91983.0 3.44113
\(895\) 9654.36 0.360569
\(896\) 66850.7 2.49255
\(897\) 11787.1 0.438752
\(898\) −72344.6 −2.68838
\(899\) 2446.16 0.0907498
\(900\) 7345.03 0.272038
\(901\) 0 0
\(902\) −2946.99 −0.108785
\(903\) 14801.3 0.545466
\(904\) −11276.0 −0.414860
\(905\) 39997.5 1.46913
\(906\) 47858.8 1.75497
\(907\) 20361.6 0.745421 0.372710 0.927948i \(-0.378429\pi\)
0.372710 + 0.927948i \(0.378429\pi\)
\(908\) −29120.5 −1.06432
\(909\) 26994.8 0.984995
\(910\) 29159.1 1.06221
\(911\) 19261.9 0.700523 0.350262 0.936652i \(-0.386093\pi\)
0.350262 + 0.936652i \(0.386093\pi\)
\(912\) 7198.03 0.261349
\(913\) −4588.09 −0.166313
\(914\) −75000.3 −2.71421
\(915\) 33678.9 1.21682
\(916\) 59931.7 2.16179
\(917\) 3182.18 0.114596
\(918\) 0 0
\(919\) −21191.8 −0.760666 −0.380333 0.924850i \(-0.624191\pi\)
−0.380333 + 0.924850i \(0.624191\pi\)
\(920\) −24965.9 −0.894677
\(921\) 6605.76 0.236338
\(922\) 65697.0 2.34665
\(923\) −10430.7 −0.371973
\(924\) −8936.49 −0.318170
\(925\) −6094.40 −0.216630
\(926\) −37703.4 −1.33802
\(927\) 58715.6 2.08034
\(928\) −40258.5 −1.42409
\(929\) −40815.0 −1.44144 −0.720719 0.693227i \(-0.756188\pi\)
−0.720719 + 0.693227i \(0.756188\pi\)
\(930\) 3630.90 0.128024
\(931\) 19461.7 0.685102
\(932\) 62434.5 2.19432
\(933\) −53370.4 −1.87274
\(934\) 40090.9 1.40451
\(935\) 0 0
\(936\) −17020.1 −0.594358
\(937\) −38439.1 −1.34018 −0.670092 0.742278i \(-0.733745\pi\)
−0.670092 + 0.742278i \(0.733745\pi\)
\(938\) −114660. −3.99124
\(939\) −26264.9 −0.912804
\(940\) 3889.28 0.134951
\(941\) 2244.08 0.0777415 0.0388708 0.999244i \(-0.487624\pi\)
0.0388708 + 0.999244i \(0.487624\pi\)
\(942\) −88955.6 −3.07678
\(943\) 14982.5 0.517388
\(944\) 4102.18 0.141435
\(945\) −10015.8 −0.344776
\(946\) −1125.38 −0.0386778
\(947\) 42289.0 1.45112 0.725559 0.688160i \(-0.241581\pi\)
0.725559 + 0.688160i \(0.241581\pi\)
\(948\) 100584. 3.44602
\(949\) 6992.76 0.239193
\(950\) 4560.51 0.155750
\(951\) −15764.7 −0.537546
\(952\) 0 0
\(953\) 37426.2 1.27214 0.636072 0.771629i \(-0.280558\pi\)
0.636072 + 0.771629i \(0.280558\pi\)
\(954\) 15213.6 0.516309
\(955\) 15950.4 0.540464
\(956\) 73263.0 2.47855
\(957\) 7083.25 0.239257
\(958\) 29527.0 0.995799
\(959\) 23441.2 0.789317
\(960\) −71723.5 −2.41132
\(961\) −29717.9 −0.997547
\(962\) 33530.8 1.12378
\(963\) 1024.37 0.0342782
\(964\) 18918.3 0.632072
\(965\) 2710.80 0.0904286
\(966\) 71731.1 2.38914
\(967\) 1088.56 0.0362003 0.0181001 0.999836i \(-0.494238\pi\)
0.0181001 + 0.999836i \(0.494238\pi\)
\(968\) −35905.4 −1.19219
\(969\) 0 0
\(970\) 38149.3 1.26278
\(971\) 39506.5 1.30569 0.652845 0.757491i \(-0.273575\pi\)
0.652845 + 0.757491i \(0.273575\pi\)
\(972\) −75000.6 −2.47494
\(973\) −55212.1 −1.81914
\(974\) −34270.8 −1.12742
\(975\) −2604.59 −0.0855525
\(976\) 6093.05 0.199830
\(977\) −43326.8 −1.41878 −0.709389 0.704817i \(-0.751029\pi\)
−0.709389 + 0.704817i \(0.751029\pi\)
\(978\) −69914.4 −2.28591
\(979\) −1577.07 −0.0514845
\(980\) 55897.3 1.82202
\(981\) −16505.1 −0.537174
\(982\) 31169.3 1.01288
\(983\) −10664.1 −0.346014 −0.173007 0.984921i \(-0.555348\pi\)
−0.173007 + 0.984921i \(0.555348\pi\)
\(984\) −40345.9 −1.30709
\(985\) −9719.22 −0.314396
\(986\) 0 0
\(987\) −4706.35 −0.151778
\(988\) −15892.4 −0.511747
\(989\) 5721.42 0.183954
\(990\) 5637.69 0.180987
\(991\) −15461.4 −0.495609 −0.247804 0.968810i \(-0.579709\pi\)
−0.247804 + 0.968810i \(0.579709\pi\)
\(992\) −1202.59 −0.0384902
\(993\) 68879.2 2.20122
\(994\) −63476.7 −2.02551
\(995\) −22248.6 −0.708871
\(996\) −149141. −4.74469
\(997\) −46061.1 −1.46316 −0.731579 0.681756i \(-0.761216\pi\)
−0.731579 + 0.681756i \(0.761216\pi\)
\(998\) −83713.7 −2.65522
\(999\) −11517.4 −0.364760
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.b.1.3 3
17.4 even 4 289.4.b.b.288.1 6
17.13 even 4 289.4.b.b.288.2 6
17.16 even 2 17.4.a.b.1.3 3
51.50 odd 2 153.4.a.g.1.1 3
68.67 odd 2 272.4.a.h.1.3 3
85.33 odd 4 425.4.b.f.324.2 6
85.67 odd 4 425.4.b.f.324.5 6
85.84 even 2 425.4.a.g.1.1 3
119.118 odd 2 833.4.a.d.1.3 3
136.67 odd 2 1088.4.a.x.1.1 3
136.101 even 2 1088.4.a.v.1.3 3
187.186 odd 2 2057.4.a.e.1.1 3
204.203 even 2 2448.4.a.bi.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.3 3 17.16 even 2
153.4.a.g.1.1 3 51.50 odd 2
272.4.a.h.1.3 3 68.67 odd 2
289.4.a.b.1.3 3 1.1 even 1 trivial
289.4.b.b.288.1 6 17.4 even 4
289.4.b.b.288.2 6 17.13 even 4
425.4.a.g.1.1 3 85.84 even 2
425.4.b.f.324.2 6 85.33 odd 4
425.4.b.f.324.5 6 85.67 odd 4
833.4.a.d.1.3 3 119.118 odd 2
1088.4.a.v.1.3 3 136.101 even 2
1088.4.a.x.1.1 3 136.67 odd 2
2057.4.a.e.1.1 3 187.186 odd 2
2448.4.a.bi.1.3 3 204.203 even 2