Properties

Label 435.4.a.j.1.3
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $1$
Dimension $7$
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] = [435,4,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(25.6658308525\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.26184\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.26184 q^{2} +3.00000 q^{3} -6.40776 q^{4} -5.00000 q^{5} -3.78552 q^{6} -13.4311 q^{7} +18.1803 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.26184 q^{2} +3.00000 q^{3} -6.40776 q^{4} -5.00000 q^{5} -3.78552 q^{6} -13.4311 q^{7} +18.1803 q^{8} +9.00000 q^{9} +6.30920 q^{10} -1.30869 q^{11} -19.2233 q^{12} +85.6641 q^{13} +16.9479 q^{14} -15.0000 q^{15} +28.3214 q^{16} +67.9685 q^{17} -11.3566 q^{18} -63.3429 q^{19} +32.0388 q^{20} -40.2934 q^{21} +1.65136 q^{22} -139.990 q^{23} +54.5409 q^{24} +25.0000 q^{25} -108.094 q^{26} +27.0000 q^{27} +86.0634 q^{28} -29.0000 q^{29} +18.9276 q^{30} -170.385 q^{31} -181.179 q^{32} -3.92607 q^{33} -85.7654 q^{34} +67.1557 q^{35} -57.6698 q^{36} -405.970 q^{37} +79.9286 q^{38} +256.992 q^{39} -90.9015 q^{40} -447.863 q^{41} +50.8438 q^{42} +407.779 q^{43} +8.38576 q^{44} -45.0000 q^{45} +176.645 q^{46} -178.857 q^{47} +84.9643 q^{48} -162.605 q^{49} -31.5460 q^{50} +203.905 q^{51} -548.915 q^{52} +93.3357 q^{53} -34.0697 q^{54} +6.54344 q^{55} -244.182 q^{56} -190.029 q^{57} +36.5934 q^{58} +279.796 q^{59} +96.1164 q^{60} -793.549 q^{61} +214.999 q^{62} -120.880 q^{63} +2.04805 q^{64} -428.321 q^{65} +4.95407 q^{66} +460.471 q^{67} -435.526 q^{68} -419.971 q^{69} -84.7397 q^{70} +803.153 q^{71} +163.623 q^{72} -150.833 q^{73} +512.269 q^{74} +75.0000 q^{75} +405.886 q^{76} +17.5772 q^{77} -324.283 q^{78} -313.814 q^{79} -141.607 q^{80} +81.0000 q^{81} +565.132 q^{82} +250.514 q^{83} +258.190 q^{84} -339.842 q^{85} -514.552 q^{86} -87.0000 q^{87} -23.7923 q^{88} -97.2972 q^{89} +56.7828 q^{90} -1150.57 q^{91} +897.024 q^{92} -511.156 q^{93} +225.689 q^{94} +316.714 q^{95} -543.538 q^{96} -1342.23 q^{97} +205.181 q^{98} -11.7782 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 21 q^{3} + 15 q^{4} - 35 q^{5} - 3 q^{6} - 37 q^{7} - 36 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 21 q^{3} + 15 q^{4} - 35 q^{5} - 3 q^{6} - 37 q^{7} - 36 q^{8} + 63 q^{9} + 5 q^{10} - 11 q^{11} + 45 q^{12} - 133 q^{13} - 75 q^{14} - 105 q^{15} - 53 q^{16} + 21 q^{17} - 9 q^{18} - 170 q^{19} - 75 q^{20} - 111 q^{21} - 369 q^{22} - 68 q^{23} - 108 q^{24} + 175 q^{25} + 181 q^{26} + 189 q^{27} - 637 q^{28} - 203 q^{29} + 15 q^{30} - 480 q^{31} - 779 q^{32} - 33 q^{33} - 897 q^{34} + 185 q^{35} + 135 q^{36} - 1032 q^{37} - 194 q^{38} - 399 q^{39} + 180 q^{40} - 638 q^{41} - 225 q^{42} - 512 q^{43} + 625 q^{44} - 315 q^{45} + 16 q^{46} - 111 q^{47} - 159 q^{48} + 178 q^{49} - 25 q^{50} + 63 q^{51} - 1263 q^{52} + 410 q^{53} - 27 q^{54} + 55 q^{55} + 1174 q^{56} - 510 q^{57} + 29 q^{58} - 426 q^{59} - 225 q^{60} - 1192 q^{61} + 460 q^{62} - 333 q^{63} + 390 q^{64} + 665 q^{65} - 1107 q^{66} - 1671 q^{67} + 1509 q^{68} - 204 q^{69} + 375 q^{70} - 1324 q^{71} - 324 q^{72} - 852 q^{73} + 1780 q^{74} + 525 q^{75} - 564 q^{76} - 2107 q^{77} + 543 q^{78} + 366 q^{79} + 265 q^{80} + 567 q^{81} - 318 q^{82} + 470 q^{83} - 1911 q^{84} - 105 q^{85} - 2196 q^{86} - 609 q^{87} - 2518 q^{88} + 51 q^{89} + 45 q^{90} - 1297 q^{91} - 684 q^{92} - 1440 q^{93} - 1837 q^{94} + 850 q^{95} - 2337 q^{96} - 3322 q^{97} + 1068 q^{98} - 99 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26184 −0.446128 −0.223064 0.974804i \(-0.571606\pi\)
−0.223064 + 0.974804i \(0.571606\pi\)
\(3\) 3.00000 0.577350
\(4\) −6.40776 −0.800970
\(5\) −5.00000 −0.447214
\(6\) −3.78552 −0.257572
\(7\) −13.4311 −0.725213 −0.362606 0.931942i \(-0.618113\pi\)
−0.362606 + 0.931942i \(0.618113\pi\)
\(8\) 18.1803 0.803463
\(9\) 9.00000 0.333333
\(10\) 6.30920 0.199514
\(11\) −1.30869 −0.0358713 −0.0179357 0.999839i \(-0.505709\pi\)
−0.0179357 + 0.999839i \(0.505709\pi\)
\(12\) −19.2233 −0.462440
\(13\) 85.6641 1.82761 0.913806 0.406151i \(-0.133129\pi\)
0.913806 + 0.406151i \(0.133129\pi\)
\(14\) 16.9479 0.323538
\(15\) −15.0000 −0.258199
\(16\) 28.3214 0.442523
\(17\) 67.9685 0.969693 0.484846 0.874599i \(-0.338876\pi\)
0.484846 + 0.874599i \(0.338876\pi\)
\(18\) −11.3566 −0.148709
\(19\) −63.3429 −0.764834 −0.382417 0.923990i \(-0.624908\pi\)
−0.382417 + 0.923990i \(0.624908\pi\)
\(20\) 32.0388 0.358205
\(21\) −40.2934 −0.418702
\(22\) 1.65136 0.0160032
\(23\) −139.990 −1.26913 −0.634565 0.772870i \(-0.718821\pi\)
−0.634565 + 0.772870i \(0.718821\pi\)
\(24\) 54.5409 0.463880
\(25\) 25.0000 0.200000
\(26\) −108.094 −0.815349
\(27\) 27.0000 0.192450
\(28\) 86.0634 0.580874
\(29\) −29.0000 −0.185695
\(30\) 18.9276 0.115190
\(31\) −170.385 −0.987164 −0.493582 0.869699i \(-0.664313\pi\)
−0.493582 + 0.869699i \(0.664313\pi\)
\(32\) −181.179 −1.00088
\(33\) −3.92607 −0.0207103
\(34\) −85.7654 −0.432607
\(35\) 67.1557 0.324325
\(36\) −57.6698 −0.266990
\(37\) −405.970 −1.80381 −0.901905 0.431934i \(-0.857831\pi\)
−0.901905 + 0.431934i \(0.857831\pi\)
\(38\) 79.9286 0.341214
\(39\) 256.992 1.05517
\(40\) −90.9015 −0.359320
\(41\) −447.863 −1.70596 −0.852982 0.521940i \(-0.825208\pi\)
−0.852982 + 0.521940i \(0.825208\pi\)
\(42\) 50.8438 0.186795
\(43\) 407.779 1.44618 0.723090 0.690754i \(-0.242721\pi\)
0.723090 + 0.690754i \(0.242721\pi\)
\(44\) 8.38576 0.0287318
\(45\) −45.0000 −0.149071
\(46\) 176.645 0.566194
\(47\) −178.857 −0.555086 −0.277543 0.960713i \(-0.589520\pi\)
−0.277543 + 0.960713i \(0.589520\pi\)
\(48\) 84.9643 0.255491
\(49\) −162.605 −0.474066
\(50\) −31.5460 −0.0892256
\(51\) 203.905 0.559852
\(52\) −548.915 −1.46386
\(53\) 93.3357 0.241899 0.120949 0.992659i \(-0.461406\pi\)
0.120949 + 0.992659i \(0.461406\pi\)
\(54\) −34.0697 −0.0858574
\(55\) 6.54344 0.0160421
\(56\) −244.182 −0.582682
\(57\) −190.029 −0.441577
\(58\) 36.5934 0.0828439
\(59\) 279.796 0.617396 0.308698 0.951160i \(-0.400107\pi\)
0.308698 + 0.951160i \(0.400107\pi\)
\(60\) 96.1164 0.206810
\(61\) −793.549 −1.66563 −0.832816 0.553550i \(-0.813273\pi\)
−0.832816 + 0.553550i \(0.813273\pi\)
\(62\) 214.999 0.440402
\(63\) −120.880 −0.241738
\(64\) 2.04805 0.00400010
\(65\) −428.321 −0.817333
\(66\) 4.95407 0.00923945
\(67\) 460.471 0.839635 0.419818 0.907609i \(-0.362094\pi\)
0.419818 + 0.907609i \(0.362094\pi\)
\(68\) −435.526 −0.776695
\(69\) −419.971 −0.732732
\(70\) −84.7397 −0.144690
\(71\) 803.153 1.34249 0.671244 0.741236i \(-0.265760\pi\)
0.671244 + 0.741236i \(0.265760\pi\)
\(72\) 163.623 0.267821
\(73\) −150.833 −0.241832 −0.120916 0.992663i \(-0.538583\pi\)
−0.120916 + 0.992663i \(0.538583\pi\)
\(74\) 512.269 0.804730
\(75\) 75.0000 0.115470
\(76\) 405.886 0.612609
\(77\) 17.5772 0.0260143
\(78\) −324.283 −0.470742
\(79\) −313.814 −0.446922 −0.223461 0.974713i \(-0.571735\pi\)
−0.223461 + 0.974713i \(0.571735\pi\)
\(80\) −141.607 −0.197902
\(81\) 81.0000 0.111111
\(82\) 565.132 0.761078
\(83\) 250.514 0.331295 0.165648 0.986185i \(-0.447029\pi\)
0.165648 + 0.986185i \(0.447029\pi\)
\(84\) 258.190 0.335368
\(85\) −339.842 −0.433660
\(86\) −514.552 −0.645181
\(87\) −87.0000 −0.107211
\(88\) −23.7923 −0.0288213
\(89\) −97.2972 −0.115882 −0.0579409 0.998320i \(-0.518453\pi\)
−0.0579409 + 0.998320i \(0.518453\pi\)
\(90\) 56.7828 0.0665048
\(91\) −1150.57 −1.32541
\(92\) 897.024 1.01653
\(93\) −511.156 −0.569940
\(94\) 225.689 0.247639
\(95\) 316.714 0.342044
\(96\) −543.538 −0.577861
\(97\) −1342.23 −1.40497 −0.702487 0.711697i \(-0.747927\pi\)
−0.702487 + 0.711697i \(0.747927\pi\)
\(98\) 205.181 0.211494
\(99\) −11.7782 −0.0119571
\(100\) −160.194 −0.160194
\(101\) 73.7087 0.0726167 0.0363084 0.999341i \(-0.488440\pi\)
0.0363084 + 0.999341i \(0.488440\pi\)
\(102\) −257.296 −0.249766
\(103\) −1307.09 −1.25040 −0.625199 0.780465i \(-0.714982\pi\)
−0.625199 + 0.780465i \(0.714982\pi\)
\(104\) 1557.40 1.46842
\(105\) 201.467 0.187249
\(106\) −117.775 −0.107918
\(107\) 845.454 0.763861 0.381930 0.924191i \(-0.375259\pi\)
0.381930 + 0.924191i \(0.375259\pi\)
\(108\) −173.009 −0.154147
\(109\) −1502.99 −1.32074 −0.660370 0.750941i \(-0.729600\pi\)
−0.660370 + 0.750941i \(0.729600\pi\)
\(110\) −8.25678 −0.00715685
\(111\) −1217.91 −1.04143
\(112\) −380.389 −0.320923
\(113\) −317.218 −0.264083 −0.132042 0.991244i \(-0.542153\pi\)
−0.132042 + 0.991244i \(0.542153\pi\)
\(114\) 239.786 0.197000
\(115\) 699.951 0.567572
\(116\) 185.825 0.148736
\(117\) 770.977 0.609204
\(118\) −353.058 −0.275438
\(119\) −912.893 −0.703234
\(120\) −272.704 −0.207453
\(121\) −1329.29 −0.998713
\(122\) 1001.33 0.743085
\(123\) −1343.59 −0.984939
\(124\) 1091.79 0.790689
\(125\) −125.000 −0.0894427
\(126\) 152.531 0.107846
\(127\) −1936.01 −1.35270 −0.676349 0.736581i \(-0.736439\pi\)
−0.676349 + 0.736581i \(0.736439\pi\)
\(128\) 1446.85 0.999100
\(129\) 1223.34 0.834952
\(130\) 540.472 0.364635
\(131\) −1234.54 −0.823379 −0.411689 0.911324i \(-0.635061\pi\)
−0.411689 + 0.911324i \(0.635061\pi\)
\(132\) 25.1573 0.0165883
\(133\) 850.767 0.554668
\(134\) −581.042 −0.374585
\(135\) −135.000 −0.0860663
\(136\) 1235.69 0.779112
\(137\) −886.190 −0.552645 −0.276322 0.961065i \(-0.589116\pi\)
−0.276322 + 0.961065i \(0.589116\pi\)
\(138\) 529.936 0.326892
\(139\) 1582.12 0.965421 0.482711 0.875780i \(-0.339652\pi\)
0.482711 + 0.875780i \(0.339652\pi\)
\(140\) −430.317 −0.259775
\(141\) −536.572 −0.320479
\(142\) −1013.45 −0.598922
\(143\) −112.108 −0.0655588
\(144\) 254.893 0.147508
\(145\) 145.000 0.0830455
\(146\) 190.328 0.107888
\(147\) −487.814 −0.273702
\(148\) 2601.36 1.44480
\(149\) −1649.57 −0.906969 −0.453485 0.891264i \(-0.649819\pi\)
−0.453485 + 0.891264i \(0.649819\pi\)
\(150\) −94.6380 −0.0515144
\(151\) 343.852 0.185313 0.0926565 0.995698i \(-0.470464\pi\)
0.0926565 + 0.995698i \(0.470464\pi\)
\(152\) −1151.59 −0.614516
\(153\) 611.716 0.323231
\(154\) −22.1796 −0.0116057
\(155\) 851.926 0.441473
\(156\) −1646.75 −0.845161
\(157\) 2493.83 1.26770 0.633851 0.773455i \(-0.281473\pi\)
0.633851 + 0.773455i \(0.281473\pi\)
\(158\) 395.983 0.199384
\(159\) 280.007 0.139660
\(160\) 905.897 0.447609
\(161\) 1880.23 0.920389
\(162\) −102.209 −0.0495698
\(163\) −1915.12 −0.920270 −0.460135 0.887849i \(-0.652199\pi\)
−0.460135 + 0.887849i \(0.652199\pi\)
\(164\) 2869.80 1.36643
\(165\) 19.6303 0.00926194
\(166\) −316.109 −0.147800
\(167\) 99.5806 0.0461424 0.0230712 0.999734i \(-0.492656\pi\)
0.0230712 + 0.999734i \(0.492656\pi\)
\(168\) −732.546 −0.336411
\(169\) 5141.34 2.34016
\(170\) 428.827 0.193468
\(171\) −570.086 −0.254945
\(172\) −2612.95 −1.15835
\(173\) 2139.30 0.940160 0.470080 0.882624i \(-0.344225\pi\)
0.470080 + 0.882624i \(0.344225\pi\)
\(174\) 109.780 0.0478299
\(175\) −335.778 −0.145043
\(176\) −37.0640 −0.0158739
\(177\) 839.389 0.356454
\(178\) 122.773 0.0516981
\(179\) 1863.54 0.778142 0.389071 0.921208i \(-0.372796\pi\)
0.389071 + 0.921208i \(0.372796\pi\)
\(180\) 288.349 0.119402
\(181\) −3008.76 −1.23558 −0.617788 0.786345i \(-0.711971\pi\)
−0.617788 + 0.786345i \(0.711971\pi\)
\(182\) 1451.83 0.591301
\(183\) −2380.65 −0.961653
\(184\) −2545.06 −1.01970
\(185\) 2029.85 0.806689
\(186\) 644.997 0.254266
\(187\) −88.9496 −0.0347842
\(188\) 1146.07 0.444607
\(189\) −362.641 −0.139567
\(190\) −399.643 −0.152596
\(191\) 5175.24 1.96056 0.980280 0.197611i \(-0.0633184\pi\)
0.980280 + 0.197611i \(0.0633184\pi\)
\(192\) 6.14415 0.00230946
\(193\) −3407.89 −1.27101 −0.635505 0.772097i \(-0.719208\pi\)
−0.635505 + 0.772097i \(0.719208\pi\)
\(194\) 1693.68 0.626798
\(195\) −1284.96 −0.471887
\(196\) 1041.93 0.379713
\(197\) −2662.23 −0.962823 −0.481412 0.876495i \(-0.659876\pi\)
−0.481412 + 0.876495i \(0.659876\pi\)
\(198\) 14.8622 0.00533440
\(199\) 1124.12 0.400437 0.200218 0.979751i \(-0.435835\pi\)
0.200218 + 0.979751i \(0.435835\pi\)
\(200\) 454.507 0.160693
\(201\) 1381.41 0.484764
\(202\) −93.0086 −0.0323964
\(203\) 389.503 0.134669
\(204\) −1306.58 −0.448425
\(205\) 2239.32 0.762930
\(206\) 1649.33 0.557837
\(207\) −1259.91 −0.423043
\(208\) 2426.13 0.808759
\(209\) 82.8961 0.0274356
\(210\) −254.219 −0.0835371
\(211\) 109.248 0.0356442 0.0178221 0.999841i \(-0.494327\pi\)
0.0178221 + 0.999841i \(0.494327\pi\)
\(212\) −598.072 −0.193754
\(213\) 2409.46 0.775086
\(214\) −1066.83 −0.340780
\(215\) −2038.89 −0.646751
\(216\) 490.868 0.154627
\(217\) 2288.47 0.715904
\(218\) 1896.54 0.589219
\(219\) −452.500 −0.139622
\(220\) −41.9288 −0.0128493
\(221\) 5822.46 1.77222
\(222\) 1536.81 0.464611
\(223\) −4655.65 −1.39805 −0.699026 0.715097i \(-0.746383\pi\)
−0.699026 + 0.715097i \(0.746383\pi\)
\(224\) 2433.45 0.725854
\(225\) 225.000 0.0666667
\(226\) 400.279 0.117815
\(227\) 3715.56 1.08639 0.543195 0.839606i \(-0.317214\pi\)
0.543195 + 0.839606i \(0.317214\pi\)
\(228\) 1217.66 0.353690
\(229\) 2437.74 0.703451 0.351726 0.936103i \(-0.385595\pi\)
0.351726 + 0.936103i \(0.385595\pi\)
\(230\) −883.227 −0.253210
\(231\) 52.7315 0.0150194
\(232\) −527.228 −0.149199
\(233\) −2704.29 −0.760360 −0.380180 0.924912i \(-0.624138\pi\)
−0.380180 + 0.924912i \(0.624138\pi\)
\(234\) −972.850 −0.271783
\(235\) 894.287 0.248242
\(236\) −1792.87 −0.494516
\(237\) −941.441 −0.258030
\(238\) 1151.93 0.313732
\(239\) 6018.67 1.62893 0.814467 0.580210i \(-0.197030\pi\)
0.814467 + 0.580210i \(0.197030\pi\)
\(240\) −424.822 −0.114259
\(241\) −1244.90 −0.332743 −0.166372 0.986063i \(-0.553205\pi\)
−0.166372 + 0.986063i \(0.553205\pi\)
\(242\) 1677.35 0.445554
\(243\) 243.000 0.0641500
\(244\) 5084.87 1.33412
\(245\) 813.024 0.212009
\(246\) 1695.40 0.439409
\(247\) −5426.21 −1.39782
\(248\) −3097.65 −0.793150
\(249\) 751.543 0.191273
\(250\) 157.730 0.0399029
\(251\) 2477.48 0.623016 0.311508 0.950244i \(-0.399166\pi\)
0.311508 + 0.950244i \(0.399166\pi\)
\(252\) 774.571 0.193625
\(253\) 183.204 0.0455254
\(254\) 2442.93 0.603477
\(255\) −1019.53 −0.250374
\(256\) −1842.08 −0.449727
\(257\) −4778.08 −1.15972 −0.579860 0.814716i \(-0.696893\pi\)
−0.579860 + 0.814716i \(0.696893\pi\)
\(258\) −1543.66 −0.372495
\(259\) 5452.63 1.30815
\(260\) 2744.58 0.654659
\(261\) −261.000 −0.0618984
\(262\) 1557.80 0.367332
\(263\) −1571.56 −0.368467 −0.184233 0.982883i \(-0.558980\pi\)
−0.184233 + 0.982883i \(0.558980\pi\)
\(264\) −71.3770 −0.0166400
\(265\) −466.678 −0.108180
\(266\) −1073.53 −0.247453
\(267\) −291.892 −0.0669044
\(268\) −2950.59 −0.672522
\(269\) 4946.58 1.12118 0.560591 0.828093i \(-0.310574\pi\)
0.560591 + 0.828093i \(0.310574\pi\)
\(270\) 170.348 0.0383966
\(271\) −914.942 −0.205088 −0.102544 0.994728i \(-0.532698\pi\)
−0.102544 + 0.994728i \(0.532698\pi\)
\(272\) 1924.97 0.429111
\(273\) −3451.70 −0.765224
\(274\) 1118.23 0.246550
\(275\) −32.7172 −0.00717426
\(276\) 2691.07 0.586896
\(277\) −2441.56 −0.529599 −0.264799 0.964304i \(-0.585306\pi\)
−0.264799 + 0.964304i \(0.585306\pi\)
\(278\) −1996.38 −0.430701
\(279\) −1533.47 −0.329055
\(280\) 1220.91 0.260583
\(281\) −5311.04 −1.12751 −0.563755 0.825942i \(-0.690644\pi\)
−0.563755 + 0.825942i \(0.690644\pi\)
\(282\) 677.068 0.142975
\(283\) 669.167 0.140558 0.0702789 0.997527i \(-0.477611\pi\)
0.0702789 + 0.997527i \(0.477611\pi\)
\(284\) −5146.41 −1.07529
\(285\) 950.143 0.197479
\(286\) 141.462 0.0292476
\(287\) 6015.31 1.23719
\(288\) −1630.62 −0.333628
\(289\) −293.286 −0.0596960
\(290\) −182.967 −0.0370489
\(291\) −4026.68 −0.811162
\(292\) 966.504 0.193700
\(293\) −7468.02 −1.48903 −0.744516 0.667604i \(-0.767320\pi\)
−0.744516 + 0.667604i \(0.767320\pi\)
\(294\) 615.544 0.122106
\(295\) −1398.98 −0.276108
\(296\) −7380.65 −1.44930
\(297\) −35.3346 −0.00690344
\(298\) 2081.50 0.404624
\(299\) −11992.1 −2.31948
\(300\) −480.582 −0.0924880
\(301\) −5476.93 −1.04879
\(302\) −433.886 −0.0826733
\(303\) 221.126 0.0419253
\(304\) −1793.96 −0.338456
\(305\) 3967.74 0.744893
\(306\) −771.888 −0.144202
\(307\) −9927.63 −1.84560 −0.922801 0.385278i \(-0.874106\pi\)
−0.922801 + 0.385278i \(0.874106\pi\)
\(308\) −112.630 −0.0208367
\(309\) −3921.26 −0.721918
\(310\) −1075.00 −0.196954
\(311\) 3131.42 0.570953 0.285477 0.958386i \(-0.407848\pi\)
0.285477 + 0.958386i \(0.407848\pi\)
\(312\) 4672.20 0.847792
\(313\) 8434.75 1.52320 0.761598 0.648050i \(-0.224415\pi\)
0.761598 + 0.648050i \(0.224415\pi\)
\(314\) −3146.82 −0.565558
\(315\) 604.401 0.108108
\(316\) 2010.84 0.357971
\(317\) −9162.80 −1.62345 −0.811725 0.584039i \(-0.801471\pi\)
−0.811725 + 0.584039i \(0.801471\pi\)
\(318\) −353.324 −0.0623064
\(319\) 37.9520 0.00666114
\(320\) −10.2403 −0.00178890
\(321\) 2536.36 0.441015
\(322\) −2372.55 −0.410611
\(323\) −4305.32 −0.741654
\(324\) −519.028 −0.0889966
\(325\) 2141.60 0.365522
\(326\) 2416.58 0.410558
\(327\) −4508.98 −0.762529
\(328\) −8142.29 −1.37068
\(329\) 2402.26 0.402555
\(330\) −24.7703 −0.00413201
\(331\) 5194.87 0.862647 0.431323 0.902197i \(-0.358047\pi\)
0.431323 + 0.902197i \(0.358047\pi\)
\(332\) −1605.23 −0.265357
\(333\) −3653.73 −0.601270
\(334\) −125.655 −0.0205854
\(335\) −2302.36 −0.375496
\(336\) −1141.17 −0.185285
\(337\) −3217.15 −0.520027 −0.260014 0.965605i \(-0.583727\pi\)
−0.260014 + 0.965605i \(0.583727\pi\)
\(338\) −6487.55 −1.04401
\(339\) −951.655 −0.152468
\(340\) 2177.63 0.347348
\(341\) 222.981 0.0354109
\(342\) 719.358 0.113738
\(343\) 6790.84 1.06901
\(344\) 7413.54 1.16195
\(345\) 2099.85 0.327688
\(346\) −2699.45 −0.419432
\(347\) −609.761 −0.0943334 −0.0471667 0.998887i \(-0.515019\pi\)
−0.0471667 + 0.998887i \(0.515019\pi\)
\(348\) 557.475 0.0858730
\(349\) 4406.40 0.675843 0.337921 0.941174i \(-0.390276\pi\)
0.337921 + 0.941174i \(0.390276\pi\)
\(350\) 423.699 0.0647075
\(351\) 2312.93 0.351724
\(352\) 237.108 0.0359031
\(353\) 2480.09 0.373943 0.186971 0.982365i \(-0.440133\pi\)
0.186971 + 0.982365i \(0.440133\pi\)
\(354\) −1059.18 −0.159024
\(355\) −4015.76 −0.600379
\(356\) 623.457 0.0928178
\(357\) −2738.68 −0.406012
\(358\) −2351.49 −0.347151
\(359\) 6578.82 0.967177 0.483589 0.875295i \(-0.339333\pi\)
0.483589 + 0.875295i \(0.339333\pi\)
\(360\) −818.113 −0.119773
\(361\) −2846.68 −0.415028
\(362\) 3796.57 0.551225
\(363\) −3987.86 −0.576607
\(364\) 7372.55 1.06161
\(365\) 754.167 0.108150
\(366\) 3004.00 0.429020
\(367\) −731.114 −0.103989 −0.0519943 0.998647i \(-0.516558\pi\)
−0.0519943 + 0.998647i \(0.516558\pi\)
\(368\) −3964.72 −0.561618
\(369\) −4030.77 −0.568655
\(370\) −2561.34 −0.359886
\(371\) −1253.60 −0.175428
\(372\) 3275.36 0.456504
\(373\) −1673.03 −0.232242 −0.116121 0.993235i \(-0.537046\pi\)
−0.116121 + 0.993235i \(0.537046\pi\)
\(374\) 112.240 0.0155182
\(375\) −375.000 −0.0516398
\(376\) −3251.68 −0.445991
\(377\) −2484.26 −0.339379
\(378\) 457.594 0.0622649
\(379\) 3760.44 0.509659 0.254830 0.966986i \(-0.417981\pi\)
0.254830 + 0.966986i \(0.417981\pi\)
\(380\) −2029.43 −0.273967
\(381\) −5808.02 −0.780981
\(382\) −6530.33 −0.874661
\(383\) −12540.8 −1.67312 −0.836562 0.547872i \(-0.815438\pi\)
−0.836562 + 0.547872i \(0.815438\pi\)
\(384\) 4340.55 0.576831
\(385\) −87.8859 −0.0116340
\(386\) 4300.21 0.567033
\(387\) 3670.01 0.482060
\(388\) 8600.66 1.12534
\(389\) 5779.21 0.753259 0.376629 0.926364i \(-0.377083\pi\)
0.376629 + 0.926364i \(0.377083\pi\)
\(390\) 1621.42 0.210522
\(391\) −9514.92 −1.23067
\(392\) −2956.20 −0.380895
\(393\) −3703.63 −0.475378
\(394\) 3359.31 0.429542
\(395\) 1569.07 0.199869
\(396\) 75.4719 0.00957728
\(397\) 14394.6 1.81976 0.909881 0.414869i \(-0.136173\pi\)
0.909881 + 0.414869i \(0.136173\pi\)
\(398\) −1418.46 −0.178646
\(399\) 2552.30 0.320238
\(400\) 708.036 0.0885045
\(401\) 8620.31 1.07351 0.536756 0.843738i \(-0.319650\pi\)
0.536756 + 0.843738i \(0.319650\pi\)
\(402\) −1743.12 −0.216267
\(403\) −14595.9 −1.80415
\(404\) −472.308 −0.0581638
\(405\) −405.000 −0.0496904
\(406\) −491.490 −0.0600795
\(407\) 531.288 0.0647051
\(408\) 3707.06 0.449821
\(409\) −14058.0 −1.69957 −0.849783 0.527133i \(-0.823267\pi\)
−0.849783 + 0.527133i \(0.823267\pi\)
\(410\) −2825.66 −0.340365
\(411\) −2658.57 −0.319070
\(412\) 8375.49 1.00153
\(413\) −3757.98 −0.447744
\(414\) 1589.81 0.188731
\(415\) −1252.57 −0.148160
\(416\) −15520.6 −1.82923
\(417\) 4746.36 0.557386
\(418\) −104.602 −0.0122398
\(419\) −308.722 −0.0359954 −0.0179977 0.999838i \(-0.505729\pi\)
−0.0179977 + 0.999838i \(0.505729\pi\)
\(420\) −1290.95 −0.149981
\(421\) 7341.48 0.849886 0.424943 0.905220i \(-0.360294\pi\)
0.424943 + 0.905220i \(0.360294\pi\)
\(422\) −137.853 −0.0159019
\(423\) −1609.72 −0.185029
\(424\) 1696.87 0.194357
\(425\) 1699.21 0.193939
\(426\) −3040.35 −0.345788
\(427\) 10658.3 1.20794
\(428\) −5417.47 −0.611830
\(429\) −336.323 −0.0378504
\(430\) 2572.76 0.288534
\(431\) −16660.8 −1.86201 −0.931003 0.365013i \(-0.881065\pi\)
−0.931003 + 0.365013i \(0.881065\pi\)
\(432\) 764.679 0.0851635
\(433\) 10091.5 1.12001 0.560006 0.828488i \(-0.310799\pi\)
0.560006 + 0.828488i \(0.310799\pi\)
\(434\) −2887.68 −0.319385
\(435\) 435.000 0.0479463
\(436\) 9630.82 1.05787
\(437\) 8867.38 0.970674
\(438\) 570.983 0.0622891
\(439\) 5149.74 0.559871 0.279936 0.960019i \(-0.409687\pi\)
0.279936 + 0.960019i \(0.409687\pi\)
\(440\) 118.962 0.0128893
\(441\) −1463.44 −0.158022
\(442\) −7347.02 −0.790638
\(443\) 15820.5 1.69674 0.848371 0.529402i \(-0.177584\pi\)
0.848371 + 0.529402i \(0.177584\pi\)
\(444\) 7804.07 0.834154
\(445\) 486.486 0.0518239
\(446\) 5874.69 0.623710
\(447\) −4948.72 −0.523639
\(448\) −27.5076 −0.00290092
\(449\) −13903.0 −1.46130 −0.730651 0.682751i \(-0.760783\pi\)
−0.730651 + 0.682751i \(0.760783\pi\)
\(450\) −283.914 −0.0297419
\(451\) 586.114 0.0611952
\(452\) 2032.66 0.211523
\(453\) 1031.55 0.106990
\(454\) −4688.45 −0.484669
\(455\) 5752.83 0.592740
\(456\) −3454.78 −0.354791
\(457\) 4315.24 0.441704 0.220852 0.975307i \(-0.429116\pi\)
0.220852 + 0.975307i \(0.429116\pi\)
\(458\) −3076.04 −0.313829
\(459\) 1835.15 0.186617
\(460\) −4485.12 −0.454608
\(461\) 5936.08 0.599720 0.299860 0.953983i \(-0.403060\pi\)
0.299860 + 0.953983i \(0.403060\pi\)
\(462\) −66.5388 −0.00670057
\(463\) −12894.4 −1.29428 −0.647141 0.762370i \(-0.724036\pi\)
−0.647141 + 0.762370i \(0.724036\pi\)
\(464\) −821.322 −0.0821744
\(465\) 2555.78 0.254885
\(466\) 3412.38 0.339218
\(467\) 12653.8 1.25385 0.626924 0.779081i \(-0.284314\pi\)
0.626924 + 0.779081i \(0.284314\pi\)
\(468\) −4940.24 −0.487954
\(469\) −6184.65 −0.608914
\(470\) −1128.45 −0.110748
\(471\) 7481.49 0.731909
\(472\) 5086.78 0.496055
\(473\) −533.656 −0.0518764
\(474\) 1187.95 0.115115
\(475\) −1583.57 −0.152967
\(476\) 5849.60 0.563269
\(477\) 840.021 0.0806330
\(478\) −7594.60 −0.726713
\(479\) 8991.57 0.857694 0.428847 0.903377i \(-0.358920\pi\)
0.428847 + 0.903377i \(0.358920\pi\)
\(480\) 2717.69 0.258427
\(481\) −34777.0 −3.29667
\(482\) 1570.87 0.148446
\(483\) 5640.68 0.531387
\(484\) 8517.75 0.799939
\(485\) 6711.13 0.628323
\(486\) −306.627 −0.0286191
\(487\) 7671.39 0.713806 0.356903 0.934141i \(-0.383833\pi\)
0.356903 + 0.934141i \(0.383833\pi\)
\(488\) −14426.9 −1.33827
\(489\) −5745.37 −0.531318
\(490\) −1025.91 −0.0945831
\(491\) 10322.9 0.948809 0.474405 0.880307i \(-0.342663\pi\)
0.474405 + 0.880307i \(0.342663\pi\)
\(492\) 8609.40 0.788906
\(493\) −1971.09 −0.180067
\(494\) 6847.01 0.623607
\(495\) 58.8910 0.00534738
\(496\) −4825.56 −0.436843
\(497\) −10787.3 −0.973590
\(498\) −948.327 −0.0853324
\(499\) −4385.18 −0.393402 −0.196701 0.980464i \(-0.563023\pi\)
−0.196701 + 0.980464i \(0.563023\pi\)
\(500\) 800.970 0.0716409
\(501\) 298.742 0.0266403
\(502\) −3126.18 −0.277945
\(503\) 8327.07 0.738143 0.369071 0.929401i \(-0.379676\pi\)
0.369071 + 0.929401i \(0.379676\pi\)
\(504\) −2197.64 −0.194227
\(505\) −368.544 −0.0324752
\(506\) −231.174 −0.0203101
\(507\) 15424.0 1.35109
\(508\) 12405.5 1.08347
\(509\) −20580.8 −1.79220 −0.896098 0.443856i \(-0.853610\pi\)
−0.896098 + 0.443856i \(0.853610\pi\)
\(510\) 1286.48 0.111699
\(511\) 2025.86 0.175379
\(512\) −9250.40 −0.798465
\(513\) −1710.26 −0.147192
\(514\) 6029.17 0.517384
\(515\) 6535.43 0.559195
\(516\) −7838.85 −0.668771
\(517\) 234.069 0.0199117
\(518\) −6880.35 −0.583601
\(519\) 6417.89 0.542802
\(520\) −7786.99 −0.656697
\(521\) −15109.3 −1.27054 −0.635270 0.772290i \(-0.719111\pi\)
−0.635270 + 0.772290i \(0.719111\pi\)
\(522\) 329.340 0.0276146
\(523\) −507.100 −0.0423976 −0.0211988 0.999775i \(-0.506748\pi\)
−0.0211988 + 0.999775i \(0.506748\pi\)
\(524\) 7910.66 0.659502
\(525\) −1007.33 −0.0837404
\(526\) 1983.06 0.164383
\(527\) −11580.8 −0.957246
\(528\) −111.192 −0.00916478
\(529\) 7430.26 0.610690
\(530\) 588.874 0.0482623
\(531\) 2518.17 0.205799
\(532\) −5451.51 −0.444272
\(533\) −38365.8 −3.11784
\(534\) 368.320 0.0298479
\(535\) −4227.27 −0.341609
\(536\) 8371.51 0.674616
\(537\) 5590.61 0.449260
\(538\) −6241.79 −0.500191
\(539\) 212.799 0.0170054
\(540\) 865.047 0.0689365
\(541\) 8249.02 0.655551 0.327776 0.944756i \(-0.393701\pi\)
0.327776 + 0.944756i \(0.393701\pi\)
\(542\) 1154.51 0.0914954
\(543\) −9026.27 −0.713360
\(544\) −12314.5 −0.970551
\(545\) 7514.96 0.590653
\(546\) 4355.49 0.341388
\(547\) −19432.6 −1.51897 −0.759486 0.650524i \(-0.774550\pi\)
−0.759486 + 0.650524i \(0.774550\pi\)
\(548\) 5678.49 0.442652
\(549\) −7141.94 −0.555210
\(550\) 41.2839 0.00320064
\(551\) 1836.94 0.142026
\(552\) −7635.19 −0.588723
\(553\) 4214.87 0.324113
\(554\) 3080.85 0.236269
\(555\) 6089.54 0.465742
\(556\) −10137.8 −0.773273
\(557\) 18807.6 1.43071 0.715354 0.698762i \(-0.246265\pi\)
0.715354 + 0.698762i \(0.246265\pi\)
\(558\) 1934.99 0.146801
\(559\) 34932.0 2.64305
\(560\) 1901.94 0.143521
\(561\) −266.849 −0.0200826
\(562\) 6701.69 0.503013
\(563\) 7337.15 0.549243 0.274622 0.961552i \(-0.411447\pi\)
0.274622 + 0.961552i \(0.411447\pi\)
\(564\) 3438.22 0.256694
\(565\) 1586.09 0.118102
\(566\) −844.382 −0.0627068
\(567\) −1087.92 −0.0805792
\(568\) 14601.6 1.07864
\(569\) 5459.12 0.402211 0.201106 0.979570i \(-0.435547\pi\)
0.201106 + 0.979570i \(0.435547\pi\)
\(570\) −1198.93 −0.0881011
\(571\) 5306.93 0.388946 0.194473 0.980908i \(-0.437700\pi\)
0.194473 + 0.980908i \(0.437700\pi\)
\(572\) 718.359 0.0525107
\(573\) 15525.7 1.13193
\(574\) −7590.36 −0.551944
\(575\) −3499.76 −0.253826
\(576\) 18.4325 0.00133337
\(577\) 16127.1 1.16357 0.581785 0.813343i \(-0.302355\pi\)
0.581785 + 0.813343i \(0.302355\pi\)
\(578\) 370.081 0.0266320
\(579\) −10223.7 −0.733818
\(580\) −929.125 −0.0665169
\(581\) −3364.69 −0.240260
\(582\) 5081.03 0.361882
\(583\) −122.147 −0.00867723
\(584\) −2742.19 −0.194303
\(585\) −3854.89 −0.272444
\(586\) 9423.45 0.664299
\(587\) 9109.13 0.640501 0.320250 0.947333i \(-0.396233\pi\)
0.320250 + 0.947333i \(0.396233\pi\)
\(588\) 3125.80 0.219227
\(589\) 10792.7 0.755017
\(590\) 1765.29 0.123180
\(591\) −7986.69 −0.555886
\(592\) −11497.6 −0.798227
\(593\) 28352.1 1.96337 0.981687 0.190500i \(-0.0610110\pi\)
0.981687 + 0.190500i \(0.0610110\pi\)
\(594\) 44.5866 0.00307982
\(595\) 4564.47 0.314496
\(596\) 10570.1 0.726455
\(597\) 3372.37 0.231192
\(598\) 15132.2 1.03478
\(599\) −15496.1 −1.05702 −0.528508 0.848928i \(-0.677248\pi\)
−0.528508 + 0.848928i \(0.677248\pi\)
\(600\) 1363.52 0.0927759
\(601\) 7637.92 0.518398 0.259199 0.965824i \(-0.416541\pi\)
0.259199 + 0.965824i \(0.416541\pi\)
\(602\) 6911.01 0.467894
\(603\) 4144.24 0.279878
\(604\) −2203.32 −0.148430
\(605\) 6646.44 0.446638
\(606\) −279.026 −0.0187040
\(607\) −12012.8 −0.803268 −0.401634 0.915800i \(-0.631558\pi\)
−0.401634 + 0.915800i \(0.631558\pi\)
\(608\) 11476.4 0.765511
\(609\) 1168.51 0.0777510
\(610\) −5006.66 −0.332318
\(611\) −15321.7 −1.01448
\(612\) −3919.73 −0.258898
\(613\) −6113.47 −0.402807 −0.201404 0.979508i \(-0.564550\pi\)
−0.201404 + 0.979508i \(0.564550\pi\)
\(614\) 12527.1 0.823374
\(615\) 6717.95 0.440478
\(616\) 319.558 0.0209016
\(617\) −28775.9 −1.87759 −0.938797 0.344472i \(-0.888058\pi\)
−0.938797 + 0.344472i \(0.888058\pi\)
\(618\) 4948.00 0.322068
\(619\) −7092.81 −0.460556 −0.230278 0.973125i \(-0.573964\pi\)
−0.230278 + 0.973125i \(0.573964\pi\)
\(620\) −5458.94 −0.353607
\(621\) −3779.74 −0.244244
\(622\) −3951.35 −0.254718
\(623\) 1306.81 0.0840390
\(624\) 7278.39 0.466937
\(625\) 625.000 0.0400000
\(626\) −10643.3 −0.679540
\(627\) 248.688 0.0158400
\(628\) −15979.9 −1.01539
\(629\) −27593.1 −1.74914
\(630\) −762.657 −0.0482302
\(631\) −2283.67 −0.144075 −0.0720377 0.997402i \(-0.522950\pi\)
−0.0720377 + 0.997402i \(0.522950\pi\)
\(632\) −5705.23 −0.359085
\(633\) 327.743 0.0205792
\(634\) 11562.0 0.724267
\(635\) 9680.03 0.604945
\(636\) −1794.22 −0.111864
\(637\) −13929.4 −0.866409
\(638\) −47.8893 −0.00297172
\(639\) 7228.38 0.447496
\(640\) −7234.26 −0.446811
\(641\) −17599.2 −1.08444 −0.542220 0.840236i \(-0.682416\pi\)
−0.542220 + 0.840236i \(0.682416\pi\)
\(642\) −3200.48 −0.196749
\(643\) 9248.80 0.567242 0.283621 0.958936i \(-0.408464\pi\)
0.283621 + 0.958936i \(0.408464\pi\)
\(644\) −12048.0 −0.737204
\(645\) −6116.68 −0.373402
\(646\) 5432.63 0.330873
\(647\) −12201.5 −0.741409 −0.370704 0.928751i \(-0.620884\pi\)
−0.370704 + 0.928751i \(0.620884\pi\)
\(648\) 1472.60 0.0892737
\(649\) −366.166 −0.0221468
\(650\) −2702.36 −0.163070
\(651\) 6865.40 0.413328
\(652\) 12271.6 0.737108
\(653\) 9769.75 0.585482 0.292741 0.956192i \(-0.405433\pi\)
0.292741 + 0.956192i \(0.405433\pi\)
\(654\) 5689.61 0.340186
\(655\) 6172.72 0.368226
\(656\) −12684.1 −0.754927
\(657\) −1357.50 −0.0806105
\(658\) −3031.26 −0.179591
\(659\) −15386.3 −0.909506 −0.454753 0.890618i \(-0.650272\pi\)
−0.454753 + 0.890618i \(0.650272\pi\)
\(660\) −125.786 −0.00741853
\(661\) 16886.7 0.993671 0.496835 0.867845i \(-0.334495\pi\)
0.496835 + 0.867845i \(0.334495\pi\)
\(662\) −6555.10 −0.384851
\(663\) 17467.4 1.02319
\(664\) 4554.42 0.266183
\(665\) −4253.83 −0.248055
\(666\) 4610.42 0.268243
\(667\) 4059.72 0.235671
\(668\) −638.089 −0.0369587
\(669\) −13967.0 −0.807165
\(670\) 2905.21 0.167519
\(671\) 1038.51 0.0597484
\(672\) 7300.34 0.419072
\(673\) 26462.8 1.51570 0.757851 0.652428i \(-0.226249\pi\)
0.757851 + 0.652428i \(0.226249\pi\)
\(674\) 4059.53 0.231999
\(675\) 675.000 0.0384900
\(676\) −32944.5 −1.87440
\(677\) 28533.1 1.61982 0.809908 0.586556i \(-0.199517\pi\)
0.809908 + 0.586556i \(0.199517\pi\)
\(678\) 1200.84 0.0680204
\(679\) 18027.6 1.01890
\(680\) −6178.43 −0.348430
\(681\) 11146.7 0.627228
\(682\) −281.367 −0.0157978
\(683\) 27990.7 1.56813 0.784066 0.620677i \(-0.213142\pi\)
0.784066 + 0.620677i \(0.213142\pi\)
\(684\) 3652.97 0.204203
\(685\) 4430.95 0.247150
\(686\) −8568.96 −0.476916
\(687\) 7313.22 0.406138
\(688\) 11548.9 0.639967
\(689\) 7995.52 0.442097
\(690\) −2649.68 −0.146191
\(691\) 9248.47 0.509158 0.254579 0.967052i \(-0.418063\pi\)
0.254579 + 0.967052i \(0.418063\pi\)
\(692\) −13708.1 −0.753040
\(693\) 158.195 0.00867145
\(694\) 769.421 0.0420848
\(695\) −7910.59 −0.431749
\(696\) −1581.69 −0.0861403
\(697\) −30440.6 −1.65426
\(698\) −5560.17 −0.301512
\(699\) −8112.87 −0.438994
\(700\) 2151.59 0.116175
\(701\) 16805.6 0.905474 0.452737 0.891644i \(-0.350448\pi\)
0.452737 + 0.891644i \(0.350448\pi\)
\(702\) −2918.55 −0.156914
\(703\) 25715.3 1.37962
\(704\) −2.68026 −0.000143489 0
\(705\) 2682.86 0.143322
\(706\) −3129.48 −0.166826
\(707\) −989.991 −0.0526626
\(708\) −5378.60 −0.285509
\(709\) 28586.0 1.51421 0.757103 0.653296i \(-0.226614\pi\)
0.757103 + 0.653296i \(0.226614\pi\)
\(710\) 5067.25 0.267846
\(711\) −2824.32 −0.148974
\(712\) −1768.89 −0.0931067
\(713\) 23852.3 1.25284
\(714\) 3455.78 0.181133
\(715\) 560.538 0.0293188
\(716\) −11941.1 −0.623268
\(717\) 18056.0 0.940465
\(718\) −8301.42 −0.431485
\(719\) −37753.6 −1.95824 −0.979119 0.203289i \(-0.934837\pi\)
−0.979119 + 0.203289i \(0.934837\pi\)
\(720\) −1274.46 −0.0659674
\(721\) 17555.6 0.906805
\(722\) 3592.05 0.185156
\(723\) −3734.70 −0.192109
\(724\) 19279.4 0.989659
\(725\) −725.000 −0.0371391
\(726\) 5032.05 0.257241
\(727\) −10769.6 −0.549412 −0.274706 0.961528i \(-0.588581\pi\)
−0.274706 + 0.961528i \(0.588581\pi\)
\(728\) −20917.6 −1.06492
\(729\) 729.000 0.0370370
\(730\) −951.638 −0.0482489
\(731\) 27716.1 1.40235
\(732\) 15254.6 0.770255
\(733\) −29083.9 −1.46554 −0.732768 0.680479i \(-0.761772\pi\)
−0.732768 + 0.680479i \(0.761772\pi\)
\(734\) 922.549 0.0463923
\(735\) 2439.07 0.122403
\(736\) 25363.4 1.27025
\(737\) −602.614 −0.0301188
\(738\) 5086.19 0.253693
\(739\) 5594.45 0.278478 0.139239 0.990259i \(-0.455534\pi\)
0.139239 + 0.990259i \(0.455534\pi\)
\(740\) −13006.8 −0.646133
\(741\) −16278.6 −0.807032
\(742\) 1581.85 0.0782634
\(743\) 8984.59 0.443624 0.221812 0.975089i \(-0.428803\pi\)
0.221812 + 0.975089i \(0.428803\pi\)
\(744\) −9292.96 −0.457925
\(745\) 8247.87 0.405609
\(746\) 2111.10 0.103610
\(747\) 2254.63 0.110432
\(748\) 569.967 0.0278611
\(749\) −11355.4 −0.553962
\(750\) 473.190 0.0230379
\(751\) 4621.64 0.224562 0.112281 0.993676i \(-0.464184\pi\)
0.112281 + 0.993676i \(0.464184\pi\)
\(752\) −5065.50 −0.245638
\(753\) 7432.44 0.359699
\(754\) 3134.74 0.151406
\(755\) −1719.26 −0.0828745
\(756\) 2323.71 0.111789
\(757\) −5554.76 −0.266699 −0.133349 0.991069i \(-0.542573\pi\)
−0.133349 + 0.991069i \(0.542573\pi\)
\(758\) −4745.07 −0.227373
\(759\) 549.611 0.0262841
\(760\) 5757.96 0.274820
\(761\) 28686.9 1.36649 0.683245 0.730189i \(-0.260568\pi\)
0.683245 + 0.730189i \(0.260568\pi\)
\(762\) 7328.79 0.348417
\(763\) 20186.9 0.957817
\(764\) −33161.7 −1.57035
\(765\) −3058.58 −0.144553
\(766\) 15824.5 0.746427
\(767\) 23968.5 1.12836
\(768\) −5526.24 −0.259650
\(769\) 26311.7 1.23384 0.616921 0.787025i \(-0.288380\pi\)
0.616921 + 0.787025i \(0.288380\pi\)
\(770\) 110.898 0.00519024
\(771\) −14334.2 −0.669565
\(772\) 21836.9 1.01804
\(773\) −25176.4 −1.17145 −0.585727 0.810508i \(-0.699191\pi\)
−0.585727 + 0.810508i \(0.699191\pi\)
\(774\) −4630.97 −0.215060
\(775\) −4259.63 −0.197433
\(776\) −24402.1 −1.12884
\(777\) 16357.9 0.755259
\(778\) −7292.44 −0.336050
\(779\) 28369.0 1.30478
\(780\) 8233.73 0.377968
\(781\) −1051.08 −0.0481568
\(782\) 12006.3 0.549034
\(783\) −783.000 −0.0357371
\(784\) −4605.20 −0.209785
\(785\) −12469.2 −0.566934
\(786\) 4673.39 0.212079
\(787\) −27651.8 −1.25245 −0.626226 0.779641i \(-0.715401\pi\)
−0.626226 + 0.779641i \(0.715401\pi\)
\(788\) 17058.9 0.771192
\(789\) −4714.69 −0.212734
\(790\) −1979.91 −0.0891673
\(791\) 4260.60 0.191516
\(792\) −214.131 −0.00960709
\(793\) −67978.7 −3.04413
\(794\) −18163.7 −0.811847
\(795\) −1400.03 −0.0624580
\(796\) −7203.11 −0.320738
\(797\) −35053.5 −1.55792 −0.778958 0.627076i \(-0.784252\pi\)
−0.778958 + 0.627076i \(0.784252\pi\)
\(798\) −3220.59 −0.142867
\(799\) −12156.7 −0.538262
\(800\) −4529.49 −0.200177
\(801\) −875.675 −0.0386273
\(802\) −10877.5 −0.478923
\(803\) 197.394 0.00867482
\(804\) −8851.77 −0.388281
\(805\) −9401.13 −0.411610
\(806\) 18417.7 0.804883
\(807\) 14839.7 0.647315
\(808\) 1340.05 0.0583449
\(809\) −23824.8 −1.03540 −0.517699 0.855563i \(-0.673211\pi\)
−0.517699 + 0.855563i \(0.673211\pi\)
\(810\) 511.045 0.0221683
\(811\) 19361.6 0.838322 0.419161 0.907912i \(-0.362324\pi\)
0.419161 + 0.907912i \(0.362324\pi\)
\(812\) −2495.84 −0.107866
\(813\) −2744.83 −0.118407
\(814\) −670.401 −0.0288667
\(815\) 9575.61 0.411557
\(816\) 5774.90 0.247747
\(817\) −25829.9 −1.10609
\(818\) 17738.9 0.758224
\(819\) −10355.1 −0.441803
\(820\) −14349.0 −0.611084
\(821\) −14914.6 −0.634012 −0.317006 0.948424i \(-0.602678\pi\)
−0.317006 + 0.948424i \(0.602678\pi\)
\(822\) 3354.69 0.142346
\(823\) 7639.98 0.323588 0.161794 0.986825i \(-0.448272\pi\)
0.161794 + 0.986825i \(0.448272\pi\)
\(824\) −23763.2 −1.00465
\(825\) −98.1517 −0.00414206
\(826\) 4741.97 0.199751
\(827\) 12462.1 0.524003 0.262001 0.965068i \(-0.415618\pi\)
0.262001 + 0.965068i \(0.415618\pi\)
\(828\) 8073.21 0.338845
\(829\) −26986.5 −1.13062 −0.565309 0.824880i \(-0.691243\pi\)
−0.565309 + 0.824880i \(0.691243\pi\)
\(830\) 1580.54 0.0660982
\(831\) −7324.67 −0.305764
\(832\) 175.444 0.00731063
\(833\) −11052.0 −0.459699
\(834\) −5989.14 −0.248666
\(835\) −497.903 −0.0206355
\(836\) −531.178 −0.0219751
\(837\) −4600.40 −0.189980
\(838\) 389.558 0.0160585
\(839\) 23448.1 0.964860 0.482430 0.875935i \(-0.339754\pi\)
0.482430 + 0.875935i \(0.339754\pi\)
\(840\) 3662.73 0.150448
\(841\) 841.000 0.0344828
\(842\) −9263.78 −0.379158
\(843\) −15933.1 −0.650968
\(844\) −700.034 −0.0285499
\(845\) −25706.7 −1.04655
\(846\) 2031.20 0.0825464
\(847\) 17853.8 0.724280
\(848\) 2643.40 0.107046
\(849\) 2007.50 0.0811511
\(850\) −2144.13 −0.0865214
\(851\) 56831.8 2.28927
\(852\) −15439.2 −0.620821
\(853\) −30230.4 −1.21344 −0.606722 0.794914i \(-0.707516\pi\)
−0.606722 + 0.794914i \(0.707516\pi\)
\(854\) −13449.0 −0.538895
\(855\) 2850.43 0.114015
\(856\) 15370.6 0.613734
\(857\) 32088.1 1.27900 0.639502 0.768789i \(-0.279140\pi\)
0.639502 + 0.768789i \(0.279140\pi\)
\(858\) 424.386 0.0168861
\(859\) −43201.3 −1.71596 −0.857980 0.513683i \(-0.828281\pi\)
−0.857980 + 0.513683i \(0.828281\pi\)
\(860\) 13064.7 0.518028
\(861\) 18045.9 0.714290
\(862\) 21023.3 0.830692
\(863\) −15650.5 −0.617323 −0.308662 0.951172i \(-0.599881\pi\)
−0.308662 + 0.951172i \(0.599881\pi\)
\(864\) −4891.85 −0.192620
\(865\) −10696.5 −0.420452
\(866\) −12733.8 −0.499669
\(867\) −879.859 −0.0344655
\(868\) −14663.9 −0.573418
\(869\) 410.685 0.0160317
\(870\) −548.901 −0.0213902
\(871\) 39445.9 1.53453
\(872\) −27324.9 −1.06117
\(873\) −12080.0 −0.468324
\(874\) −11189.2 −0.433045
\(875\) 1678.89 0.0648650
\(876\) 2899.51 0.111833
\(877\) −37921.1 −1.46010 −0.730049 0.683395i \(-0.760503\pi\)
−0.730049 + 0.683395i \(0.760503\pi\)
\(878\) −6498.14 −0.249774
\(879\) −22404.1 −0.859693
\(880\) 185.320 0.00709901
\(881\) 19844.4 0.758881 0.379441 0.925216i \(-0.376116\pi\)
0.379441 + 0.925216i \(0.376116\pi\)
\(882\) 1846.63 0.0704981
\(883\) 33386.5 1.27242 0.636210 0.771516i \(-0.280501\pi\)
0.636210 + 0.771516i \(0.280501\pi\)
\(884\) −37308.9 −1.41950
\(885\) −4196.95 −0.159411
\(886\) −19963.0 −0.756964
\(887\) −25988.8 −0.983788 −0.491894 0.870655i \(-0.663695\pi\)
−0.491894 + 0.870655i \(0.663695\pi\)
\(888\) −22141.9 −0.836751
\(889\) 26002.7 0.980994
\(890\) −613.867 −0.0231201
\(891\) −106.004 −0.00398570
\(892\) 29832.3 1.11980
\(893\) 11329.3 0.424549
\(894\) 6244.50 0.233610
\(895\) −9317.69 −0.347996
\(896\) −19432.9 −0.724560
\(897\) −35976.4 −1.33915
\(898\) 17543.4 0.651928
\(899\) 4941.17 0.183312
\(900\) −1441.75 −0.0533980
\(901\) 6343.88 0.234568
\(902\) −739.582 −0.0273009
\(903\) −16430.8 −0.605518
\(904\) −5767.12 −0.212181
\(905\) 15043.8 0.552566
\(906\) −1301.66 −0.0477314
\(907\) −26383.8 −0.965887 −0.482943 0.875652i \(-0.660432\pi\)
−0.482943 + 0.875652i \(0.660432\pi\)
\(908\) −23808.4 −0.870166
\(909\) 663.378 0.0242056
\(910\) −7259.15 −0.264438
\(911\) −23448.5 −0.852779 −0.426390 0.904540i \(-0.640215\pi\)
−0.426390 + 0.904540i \(0.640215\pi\)
\(912\) −5381.89 −0.195408
\(913\) −327.845 −0.0118840
\(914\) −5445.15 −0.197056
\(915\) 11903.2 0.430064
\(916\) −15620.4 −0.563443
\(917\) 16581.3 0.597125
\(918\) −2315.66 −0.0832553
\(919\) −1649.45 −0.0592060 −0.0296030 0.999562i \(-0.509424\pi\)
−0.0296030 + 0.999562i \(0.509424\pi\)
\(920\) 12725.3 0.456023
\(921\) −29782.9 −1.06556
\(922\) −7490.39 −0.267552
\(923\) 68801.4 2.45355
\(924\) −337.891 −0.0120301
\(925\) −10149.2 −0.360762
\(926\) 16270.7 0.577415
\(927\) −11763.8 −0.416799
\(928\) 5254.20 0.185860
\(929\) 48039.5 1.69658 0.848291 0.529531i \(-0.177632\pi\)
0.848291 + 0.529531i \(0.177632\pi\)
\(930\) −3224.99 −0.113711
\(931\) 10299.9 0.362582
\(932\) 17328.4 0.609026
\(933\) 9394.26 0.329640
\(934\) −15967.0 −0.559376
\(935\) 444.748 0.0155559
\(936\) 14016.6 0.489473
\(937\) −28647.5 −0.998796 −0.499398 0.866373i \(-0.666445\pi\)
−0.499398 + 0.866373i \(0.666445\pi\)
\(938\) 7804.04 0.271654
\(939\) 25304.2 0.879417
\(940\) −5730.37 −0.198834
\(941\) 9541.12 0.330533 0.165267 0.986249i \(-0.447152\pi\)
0.165267 + 0.986249i \(0.447152\pi\)
\(942\) −9440.45 −0.326525
\(943\) 62696.5 2.16509
\(944\) 7924.24 0.273212
\(945\) 1813.20 0.0624164
\(946\) 673.388 0.0231435
\(947\) 44596.6 1.53030 0.765151 0.643851i \(-0.222664\pi\)
0.765151 + 0.643851i \(0.222664\pi\)
\(948\) 6032.53 0.206674
\(949\) −12921.0 −0.441974
\(950\) 1998.22 0.0682428
\(951\) −27488.4 −0.937300
\(952\) −16596.7 −0.565022
\(953\) 437.931 0.0148856 0.00744279 0.999972i \(-0.497631\pi\)
0.00744279 + 0.999972i \(0.497631\pi\)
\(954\) −1059.97 −0.0359726
\(955\) −25876.2 −0.876790
\(956\) −38566.2 −1.30473
\(957\) 113.856 0.00384581
\(958\) −11345.9 −0.382641
\(959\) 11902.5 0.400785
\(960\) −30.7208 −0.00103282
\(961\) −759.861 −0.0255064
\(962\) 43883.1 1.47073
\(963\) 7609.09 0.254620
\(964\) 7977.02 0.266517
\(965\) 17039.4 0.568413
\(966\) −7117.64 −0.237067
\(967\) −37762.6 −1.25580 −0.627902 0.778292i \(-0.716086\pi\)
−0.627902 + 0.778292i \(0.716086\pi\)
\(968\) −24166.8 −0.802429
\(969\) −12916.0 −0.428194
\(970\) −8468.38 −0.280313
\(971\) 16153.0 0.533858 0.266929 0.963716i \(-0.413991\pi\)
0.266929 + 0.963716i \(0.413991\pi\)
\(972\) −1557.09 −0.0513822
\(973\) −21249.6 −0.700136
\(974\) −9680.07 −0.318449
\(975\) 6424.81 0.211034
\(976\) −22474.4 −0.737079
\(977\) 33700.3 1.10355 0.551775 0.833993i \(-0.313951\pi\)
0.551775 + 0.833993i \(0.313951\pi\)
\(978\) 7249.74 0.237036
\(979\) 127.332 0.00415683
\(980\) −5209.66 −0.169813
\(981\) −13526.9 −0.440247
\(982\) −13025.8 −0.423290
\(983\) −7637.13 −0.247799 −0.123900 0.992295i \(-0.539540\pi\)
−0.123900 + 0.992295i \(0.539540\pi\)
\(984\) −24426.9 −0.791362
\(985\) 13311.2 0.430588
\(986\) 2487.20 0.0803331
\(987\) 7206.77 0.232415
\(988\) 34769.9 1.11961
\(989\) −57085.1 −1.83539
\(990\) −74.3110 −0.00238562
\(991\) 9339.90 0.299386 0.149693 0.988733i \(-0.452171\pi\)
0.149693 + 0.988733i \(0.452171\pi\)
\(992\) 30870.3 0.988038
\(993\) 15584.6 0.498049
\(994\) 13611.8 0.434346
\(995\) −5620.61 −0.179081
\(996\) −4815.70 −0.153204
\(997\) −11144.7 −0.354017 −0.177009 0.984209i \(-0.556642\pi\)
−0.177009 + 0.984209i \(0.556642\pi\)
\(998\) 5533.39 0.175508
\(999\) −10961.2 −0.347144
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 435.4.a.j.1.3 7
3.2 odd 2 1305.4.a.m.1.5 7
5.4 even 2 2175.4.a.m.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.j.1.3 7 1.1 even 1 trivial
1305.4.a.m.1.5 7 3.2 odd 2
2175.4.a.m.1.5 7 5.4 even 2