Properties

Label 435.4.a.i.1.1
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $0$
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: \(0\)
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: \( x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 \) 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 1.1
Root \(5.24052\) 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)
 
\(f(q)\) \(=\) \(q-5.24052 q^{2} -3.00000 q^{3} +19.4631 q^{4} +5.00000 q^{5} +15.7216 q^{6} -18.4475 q^{7} -60.0724 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.24052 q^{2} -3.00000 q^{3} +19.4631 q^{4} +5.00000 q^{5} +15.7216 q^{6} -18.4475 q^{7} -60.0724 q^{8} +9.00000 q^{9} -26.2026 q^{10} +11.5384 q^{11} -58.3892 q^{12} -34.4618 q^{13} +96.6746 q^{14} -15.0000 q^{15} +159.106 q^{16} +70.7681 q^{17} -47.1647 q^{18} +3.52127 q^{19} +97.3153 q^{20} +55.3426 q^{21} -60.4671 q^{22} -18.3117 q^{23} +180.217 q^{24} +25.0000 q^{25} +180.598 q^{26} -27.0000 q^{27} -359.045 q^{28} +29.0000 q^{29} +78.6078 q^{30} -120.558 q^{31} -353.221 q^{32} -34.6151 q^{33} -370.862 q^{34} -92.2376 q^{35} +175.168 q^{36} -182.505 q^{37} -18.4533 q^{38} +103.385 q^{39} -300.362 q^{40} +74.4691 q^{41} -290.024 q^{42} -405.899 q^{43} +224.572 q^{44} +45.0000 q^{45} +95.9626 q^{46} -327.143 q^{47} -477.319 q^{48} -2.68899 q^{49} -131.013 q^{50} -212.304 q^{51} -670.732 q^{52} +409.067 q^{53} +141.494 q^{54} +57.6919 q^{55} +1108.19 q^{56} -10.5638 q^{57} -151.975 q^{58} +120.722 q^{59} -291.946 q^{60} +34.6082 q^{61} +631.788 q^{62} -166.028 q^{63} +578.210 q^{64} -172.309 q^{65} +181.401 q^{66} +671.739 q^{67} +1377.36 q^{68} +54.9350 q^{69} +483.373 q^{70} +873.524 q^{71} -540.652 q^{72} +432.426 q^{73} +956.419 q^{74} -75.0000 q^{75} +68.5347 q^{76} -212.854 q^{77} -541.793 q^{78} -548.355 q^{79} +795.532 q^{80} +81.0000 q^{81} -390.257 q^{82} +511.077 q^{83} +1077.14 q^{84} +353.841 q^{85} +2127.12 q^{86} -87.0000 q^{87} -693.138 q^{88} -788.135 q^{89} -235.823 q^{90} +635.734 q^{91} -356.401 q^{92} +361.675 q^{93} +1714.40 q^{94} +17.6063 q^{95} +1059.66 q^{96} +967.070 q^{97} +14.0917 q^{98} +103.845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} - 21 q^{3} + 22 q^{4} + 35 q^{5} + 6 q^{6} - 50 q^{7} - 33 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{2} - 21 q^{3} + 22 q^{4} + 35 q^{5} + 6 q^{6} - 50 q^{7} - 33 q^{8} + 63 q^{9} - 10 q^{10} + 76 q^{11} - 66 q^{12} + 30 q^{13} + 89 q^{14} - 105 q^{15} + 138 q^{16} - 140 q^{17} - 18 q^{18} + 90 q^{19} + 110 q^{20} + 150 q^{21} + 61 q^{22} + 34 q^{23} + 99 q^{24} + 175 q^{25} - 241 q^{26} - 189 q^{27} - 57 q^{28} + 203 q^{29} + 30 q^{30} + 524 q^{31} - 6 q^{32} - 228 q^{33} + 255 q^{34} - 250 q^{35} + 198 q^{36} - 28 q^{37} + 222 q^{38} - 90 q^{39} - 165 q^{40} + 1532 q^{41} - 267 q^{42} - 464 q^{43} + 1475 q^{44} + 315 q^{45} + 72 q^{46} - 360 q^{47} - 414 q^{48} + 569 q^{49} - 50 q^{50} + 420 q^{51} - 205 q^{52} + 282 q^{53} + 54 q^{54} + 380 q^{55} + 1102 q^{56} - 270 q^{57} - 58 q^{58} + 766 q^{59} - 330 q^{60} + 1200 q^{61} + 2856 q^{62} - 450 q^{63} + 701 q^{64} + 150 q^{65} - 183 q^{66} + 1546 q^{67} + 1801 q^{68} - 102 q^{69} + 445 q^{70} + 1802 q^{71} - 297 q^{72} - 220 q^{73} + 1594 q^{74} - 525 q^{75} + 1960 q^{76} + 3222 q^{77} + 723 q^{78} + 1298 q^{79} + 690 q^{80} + 567 q^{81} + 856 q^{82} + 1652 q^{83} + 171 q^{84} - 700 q^{85} + 7628 q^{86} - 609 q^{87} + 550 q^{88} + 2846 q^{89} - 90 q^{90} - 816 q^{91} + 472 q^{92} - 1572 q^{93} + 745 q^{94} + 450 q^{95} + 18 q^{96} + 1110 q^{97} - 761 q^{98} + 684 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.24052 −1.85280 −0.926402 0.376536i \(-0.877115\pi\)
−0.926402 + 0.376536i \(0.877115\pi\)
\(3\) −3.00000 −0.577350
\(4\) 19.4631 2.43288
\(5\) 5.00000 0.447214
\(6\) 15.7216 1.06972
\(7\) −18.4475 −0.996072 −0.498036 0.867156i \(-0.665945\pi\)
−0.498036 + 0.867156i \(0.665945\pi\)
\(8\) −60.0724 −2.65485
\(9\) 9.00000 0.333333
\(10\) −26.2026 −0.828599
\(11\) 11.5384 0.316268 0.158134 0.987418i \(-0.449452\pi\)
0.158134 + 0.987418i \(0.449452\pi\)
\(12\) −58.3892 −1.40463
\(13\) −34.4618 −0.735229 −0.367614 0.929978i \(-0.619825\pi\)
−0.367614 + 0.929978i \(0.619825\pi\)
\(14\) 96.6746 1.84553
\(15\) −15.0000 −0.258199
\(16\) 159.106 2.48604
\(17\) 70.7681 1.00963 0.504817 0.863226i \(-0.331560\pi\)
0.504817 + 0.863226i \(0.331560\pi\)
\(18\) −47.1647 −0.617601
\(19\) 3.52127 0.0425176 0.0212588 0.999774i \(-0.493233\pi\)
0.0212588 + 0.999774i \(0.493233\pi\)
\(20\) 97.3153 1.08802
\(21\) 55.3426 0.575083
\(22\) −60.4671 −0.585983
\(23\) −18.3117 −0.166011 −0.0830053 0.996549i \(-0.526452\pi\)
−0.0830053 + 0.996549i \(0.526452\pi\)
\(24\) 180.217 1.53278
\(25\) 25.0000 0.200000
\(26\) 180.598 1.36224
\(27\) −27.0000 −0.192450
\(28\) −359.045 −2.42333
\(29\) 29.0000 0.185695
\(30\) 78.6078 0.478392
\(31\) −120.558 −0.698481 −0.349240 0.937033i \(-0.613560\pi\)
−0.349240 + 0.937033i \(0.613560\pi\)
\(32\) −353.221 −1.95129
\(33\) −34.6151 −0.182598
\(34\) −370.862 −1.87065
\(35\) −92.2376 −0.445457
\(36\) 175.168 0.810961
\(37\) −182.505 −0.810907 −0.405454 0.914116i \(-0.632886\pi\)
−0.405454 + 0.914116i \(0.632886\pi\)
\(38\) −18.4533 −0.0787768
\(39\) 103.385 0.424485
\(40\) −300.362 −1.18729
\(41\) 74.4691 0.283661 0.141831 0.989891i \(-0.454701\pi\)
0.141831 + 0.989891i \(0.454701\pi\)
\(42\) −290.024 −1.06552
\(43\) −405.899 −1.43951 −0.719756 0.694227i \(-0.755746\pi\)
−0.719756 + 0.694227i \(0.755746\pi\)
\(44\) 224.572 0.769444
\(45\) 45.0000 0.149071
\(46\) 95.9626 0.307585
\(47\) −327.143 −1.01529 −0.507646 0.861566i \(-0.669484\pi\)
−0.507646 + 0.861566i \(0.669484\pi\)
\(48\) −477.319 −1.43531
\(49\) −2.68899 −0.00783963
\(50\) −131.013 −0.370561
\(51\) −212.304 −0.582913
\(52\) −670.732 −1.78873
\(53\) 409.067 1.06018 0.530092 0.847940i \(-0.322157\pi\)
0.530092 + 0.847940i \(0.322157\pi\)
\(54\) 141.494 0.356572
\(55\) 57.6919 0.141440
\(56\) 1108.19 2.64442
\(57\) −10.5638 −0.0245475
\(58\) −151.975 −0.344057
\(59\) 120.722 0.266384 0.133192 0.991090i \(-0.457477\pi\)
0.133192 + 0.991090i \(0.457477\pi\)
\(60\) −291.946 −0.628168
\(61\) 34.6082 0.0726415 0.0363207 0.999340i \(-0.488436\pi\)
0.0363207 + 0.999340i \(0.488436\pi\)
\(62\) 631.788 1.29415
\(63\) −166.028 −0.332024
\(64\) 578.210 1.12932
\(65\) −172.309 −0.328804
\(66\) 181.401 0.338318
\(67\) 671.739 1.22487 0.612433 0.790523i \(-0.290191\pi\)
0.612433 + 0.790523i \(0.290191\pi\)
\(68\) 1377.36 2.45632
\(69\) 54.9350 0.0958463
\(70\) 483.373 0.825345
\(71\) 873.524 1.46012 0.730058 0.683386i \(-0.239493\pi\)
0.730058 + 0.683386i \(0.239493\pi\)
\(72\) −540.652 −0.884951
\(73\) 432.426 0.693311 0.346655 0.937993i \(-0.387317\pi\)
0.346655 + 0.937993i \(0.387317\pi\)
\(74\) 956.419 1.50245
\(75\) −75.0000 −0.115470
\(76\) 68.5347 0.103440
\(77\) −212.854 −0.315026
\(78\) −541.793 −0.786487
\(79\) −548.355 −0.780946 −0.390473 0.920614i \(-0.627689\pi\)
−0.390473 + 0.920614i \(0.627689\pi\)
\(80\) 795.532 1.11179
\(81\) 81.0000 0.111111
\(82\) −390.257 −0.525569
\(83\) 511.077 0.675879 0.337940 0.941168i \(-0.390270\pi\)
0.337940 + 0.941168i \(0.390270\pi\)
\(84\) 1077.14 1.39911
\(85\) 353.841 0.451522
\(86\) 2127.12 2.66713
\(87\) −87.0000 −0.107211
\(88\) −693.138 −0.839646
\(89\) −788.135 −0.938676 −0.469338 0.883019i \(-0.655507\pi\)
−0.469338 + 0.883019i \(0.655507\pi\)
\(90\) −235.823 −0.276200
\(91\) 635.734 0.732341
\(92\) −356.401 −0.403884
\(93\) 361.675 0.403268
\(94\) 1714.40 1.88114
\(95\) 17.6063 0.0190144
\(96\) 1059.66 1.12658
\(97\) 967.070 1.01228 0.506140 0.862452i \(-0.331072\pi\)
0.506140 + 0.862452i \(0.331072\pi\)
\(98\) 14.0917 0.0145253
\(99\) 103.845 0.105423
\(100\) 486.577 0.486577
\(101\) 510.016 0.502460 0.251230 0.967927i \(-0.419165\pi\)
0.251230 + 0.967927i \(0.419165\pi\)
\(102\) 1112.59 1.08002
\(103\) 1645.72 1.57435 0.787175 0.616730i \(-0.211543\pi\)
0.787175 + 0.616730i \(0.211543\pi\)
\(104\) 2070.20 1.95192
\(105\) 276.713 0.257185
\(106\) −2143.73 −1.96431
\(107\) −1305.35 −1.17937 −0.589687 0.807632i \(-0.700749\pi\)
−0.589687 + 0.807632i \(0.700749\pi\)
\(108\) −525.503 −0.468209
\(109\) 932.125 0.819095 0.409548 0.912289i \(-0.365687\pi\)
0.409548 + 0.912289i \(0.365687\pi\)
\(110\) −302.336 −0.262060
\(111\) 547.514 0.468178
\(112\) −2935.12 −2.47627
\(113\) 1719.89 1.43180 0.715901 0.698202i \(-0.246016\pi\)
0.715901 + 0.698202i \(0.246016\pi\)
\(114\) 55.3598 0.0454818
\(115\) −91.5583 −0.0742422
\(116\) 564.429 0.451775
\(117\) −310.156 −0.245076
\(118\) −632.646 −0.493558
\(119\) −1305.50 −1.00567
\(120\) 901.087 0.685480
\(121\) −1197.87 −0.899974
\(122\) −181.365 −0.134590
\(123\) −223.407 −0.163772
\(124\) −2346.43 −1.69932
\(125\) 125.000 0.0894427
\(126\) 870.072 0.615176
\(127\) −883.792 −0.617511 −0.308756 0.951141i \(-0.599912\pi\)
−0.308756 + 0.951141i \(0.599912\pi\)
\(128\) −204.357 −0.141116
\(129\) 1217.70 0.831102
\(130\) 902.988 0.609210
\(131\) −937.386 −0.625190 −0.312595 0.949887i \(-0.601198\pi\)
−0.312595 + 0.949887i \(0.601198\pi\)
\(132\) −673.717 −0.444239
\(133\) −64.9587 −0.0423506
\(134\) −3520.26 −2.26944
\(135\) −135.000 −0.0860663
\(136\) −4251.21 −2.68043
\(137\) 237.650 0.148203 0.0741014 0.997251i \(-0.476391\pi\)
0.0741014 + 0.997251i \(0.476391\pi\)
\(138\) −287.888 −0.177584
\(139\) 1706.72 1.04146 0.520728 0.853723i \(-0.325661\pi\)
0.520728 + 0.853723i \(0.325661\pi\)
\(140\) −1795.23 −1.08375
\(141\) 981.430 0.586179
\(142\) −4577.72 −2.70531
\(143\) −397.633 −0.232530
\(144\) 1431.96 0.828679
\(145\) 145.000 0.0830455
\(146\) −2266.14 −1.28457
\(147\) 8.06698 0.00452621
\(148\) −3552.10 −1.97284
\(149\) 3546.59 1.94999 0.974994 0.222230i \(-0.0713334\pi\)
0.974994 + 0.222230i \(0.0713334\pi\)
\(150\) 393.039 0.213943
\(151\) 1818.23 0.979901 0.489951 0.871750i \(-0.337015\pi\)
0.489951 + 0.871750i \(0.337015\pi\)
\(152\) −211.531 −0.112878
\(153\) 636.913 0.336545
\(154\) 1115.47 0.583682
\(155\) −602.791 −0.312370
\(156\) 2012.20 1.03272
\(157\) −37.7540 −0.0191917 −0.00959585 0.999954i \(-0.503055\pi\)
−0.00959585 + 0.999954i \(0.503055\pi\)
\(158\) 2873.66 1.44694
\(159\) −1227.20 −0.612097
\(160\) −1766.10 −0.872642
\(161\) 337.805 0.165359
\(162\) −424.482 −0.205867
\(163\) 1541.87 0.740913 0.370457 0.928850i \(-0.379201\pi\)
0.370457 + 0.928850i \(0.379201\pi\)
\(164\) 1449.40 0.690115
\(165\) −173.076 −0.0816601
\(166\) −2678.31 −1.25227
\(167\) 1669.49 0.773586 0.386793 0.922167i \(-0.373583\pi\)
0.386793 + 0.922167i \(0.373583\pi\)
\(168\) −3324.56 −1.52676
\(169\) −1009.39 −0.459438
\(170\) −1854.31 −0.836582
\(171\) 31.6914 0.0141725
\(172\) −7900.04 −3.50216
\(173\) −728.623 −0.320209 −0.160105 0.987100i \(-0.551183\pi\)
−0.160105 + 0.987100i \(0.551183\pi\)
\(174\) 455.925 0.198641
\(175\) −461.188 −0.199214
\(176\) 1835.83 0.786255
\(177\) −362.166 −0.153797
\(178\) 4130.24 1.73918
\(179\) 1926.46 0.804416 0.402208 0.915548i \(-0.368243\pi\)
0.402208 + 0.915548i \(0.368243\pi\)
\(180\) 875.838 0.362673
\(181\) 3695.62 1.51764 0.758821 0.651299i \(-0.225776\pi\)
0.758821 + 0.651299i \(0.225776\pi\)
\(182\) −3331.58 −1.35689
\(183\) −103.825 −0.0419396
\(184\) 1100.03 0.440733
\(185\) −912.523 −0.362649
\(186\) −1895.36 −0.747177
\(187\) 816.549 0.319315
\(188\) −6367.21 −2.47009
\(189\) 498.083 0.191694
\(190\) −92.2664 −0.0352300
\(191\) 2490.19 0.943370 0.471685 0.881767i \(-0.343646\pi\)
0.471685 + 0.881767i \(0.343646\pi\)
\(192\) −1734.63 −0.652012
\(193\) 4493.27 1.67582 0.837909 0.545809i \(-0.183778\pi\)
0.837909 + 0.545809i \(0.183778\pi\)
\(194\) −5067.95 −1.87555
\(195\) 516.927 0.189835
\(196\) −52.3361 −0.0190729
\(197\) 1990.23 0.719787 0.359894 0.932993i \(-0.382813\pi\)
0.359894 + 0.932993i \(0.382813\pi\)
\(198\) −544.204 −0.195328
\(199\) −83.8021 −0.0298521 −0.0149261 0.999889i \(-0.504751\pi\)
−0.0149261 + 0.999889i \(0.504751\pi\)
\(200\) −1501.81 −0.530970
\(201\) −2015.22 −0.707176
\(202\) −2672.75 −0.930960
\(203\) −534.978 −0.184966
\(204\) −4132.09 −1.41816
\(205\) 372.345 0.126857
\(206\) −8624.45 −2.91696
\(207\) −164.805 −0.0553369
\(208\) −5483.09 −1.82781
\(209\) 40.6297 0.0134470
\(210\) −1450.12 −0.476513
\(211\) −4369.95 −1.42578 −0.712891 0.701275i \(-0.752615\pi\)
−0.712891 + 0.701275i \(0.752615\pi\)
\(212\) 7961.71 2.57930
\(213\) −2620.57 −0.842998
\(214\) 6840.72 2.18515
\(215\) −2029.49 −0.643769
\(216\) 1621.96 0.510926
\(217\) 2224.00 0.695738
\(218\) −4884.82 −1.51762
\(219\) −1297.28 −0.400283
\(220\) 1122.86 0.344106
\(221\) −2438.79 −0.742313
\(222\) −2869.26 −0.867441
\(223\) 3633.76 1.09119 0.545594 0.838050i \(-0.316304\pi\)
0.545594 + 0.838050i \(0.316304\pi\)
\(224\) 6516.05 1.94362
\(225\) 225.000 0.0666667
\(226\) −9013.12 −2.65285
\(227\) −3692.30 −1.07959 −0.539794 0.841797i \(-0.681498\pi\)
−0.539794 + 0.841797i \(0.681498\pi\)
\(228\) −205.604 −0.0597213
\(229\) 4833.73 1.39486 0.697428 0.716655i \(-0.254328\pi\)
0.697428 + 0.716655i \(0.254328\pi\)
\(230\) 479.813 0.137556
\(231\) 638.563 0.181880
\(232\) −1742.10 −0.492994
\(233\) −4834.45 −1.35929 −0.679647 0.733539i \(-0.737867\pi\)
−0.679647 + 0.733539i \(0.737867\pi\)
\(234\) 1625.38 0.454078
\(235\) −1635.72 −0.454053
\(236\) 2349.62 0.648081
\(237\) 1645.06 0.450879
\(238\) 6841.48 1.86331
\(239\) 4019.80 1.08795 0.543974 0.839102i \(-0.316919\pi\)
0.543974 + 0.839102i \(0.316919\pi\)
\(240\) −2386.60 −0.641892
\(241\) 1076.81 0.287816 0.143908 0.989591i \(-0.454033\pi\)
0.143908 + 0.989591i \(0.454033\pi\)
\(242\) 6277.44 1.66748
\(243\) −243.000 −0.0641500
\(244\) 673.582 0.176728
\(245\) −13.4450 −0.00350599
\(246\) 1170.77 0.303437
\(247\) −121.349 −0.0312602
\(248\) 7242.23 1.85436
\(249\) −1533.23 −0.390219
\(250\) −655.065 −0.165720
\(251\) −3179.99 −0.799677 −0.399839 0.916586i \(-0.630934\pi\)
−0.399839 + 0.916586i \(0.630934\pi\)
\(252\) −3231.41 −0.807776
\(253\) −211.287 −0.0525039
\(254\) 4631.53 1.14413
\(255\) −1061.52 −0.260687
\(256\) −3554.74 −0.867858
\(257\) 3125.29 0.758561 0.379280 0.925282i \(-0.376172\pi\)
0.379280 + 0.925282i \(0.376172\pi\)
\(258\) −6381.37 −1.53987
\(259\) 3366.76 0.807722
\(260\) −3353.66 −0.799943
\(261\) 261.000 0.0618984
\(262\) 4912.39 1.15835
\(263\) −4474.30 −1.04904 −0.524519 0.851399i \(-0.675755\pi\)
−0.524519 + 0.851399i \(0.675755\pi\)
\(264\) 2079.42 0.484770
\(265\) 2045.34 0.474129
\(266\) 340.417 0.0784674
\(267\) 2364.41 0.541945
\(268\) 13074.1 2.97995
\(269\) 4053.04 0.918655 0.459327 0.888267i \(-0.348091\pi\)
0.459327 + 0.888267i \(0.348091\pi\)
\(270\) 707.470 0.159464
\(271\) −8045.96 −1.80353 −0.901766 0.432225i \(-0.857729\pi\)
−0.901766 + 0.432225i \(0.857729\pi\)
\(272\) 11259.7 2.50999
\(273\) −1907.20 −0.422817
\(274\) −1245.41 −0.274591
\(275\) 288.459 0.0632537
\(276\) 1069.20 0.233183
\(277\) 4625.74 1.00337 0.501685 0.865050i \(-0.332714\pi\)
0.501685 + 0.865050i \(0.332714\pi\)
\(278\) −8944.11 −1.92961
\(279\) −1085.02 −0.232827
\(280\) 5540.94 1.18262
\(281\) −69.7964 −0.0148174 −0.00740872 0.999973i \(-0.502358\pi\)
−0.00740872 + 0.999973i \(0.502358\pi\)
\(282\) −5143.20 −1.08608
\(283\) 1903.58 0.399845 0.199922 0.979812i \(-0.435931\pi\)
0.199922 + 0.979812i \(0.435931\pi\)
\(284\) 17001.4 3.55229
\(285\) −52.8190 −0.0109780
\(286\) 2083.80 0.430832
\(287\) −1373.77 −0.282547
\(288\) −3178.99 −0.650429
\(289\) 95.1246 0.0193618
\(290\) −759.876 −0.153867
\(291\) −2901.21 −0.584440
\(292\) 8416.34 1.68674
\(293\) 623.198 0.124258 0.0621290 0.998068i \(-0.480211\pi\)
0.0621290 + 0.998068i \(0.480211\pi\)
\(294\) −42.2752 −0.00838619
\(295\) 603.610 0.119131
\(296\) 10963.5 2.15284
\(297\) −311.536 −0.0608659
\(298\) −18586.0 −3.61295
\(299\) 631.052 0.122056
\(300\) −1459.73 −0.280925
\(301\) 7487.83 1.43386
\(302\) −9528.45 −1.81557
\(303\) −1530.05 −0.290095
\(304\) 560.256 0.105700
\(305\) 173.041 0.0324863
\(306\) −3337.76 −0.623552
\(307\) −6899.90 −1.28273 −0.641365 0.767236i \(-0.721632\pi\)
−0.641365 + 0.767236i \(0.721632\pi\)
\(308\) −4142.80 −0.766422
\(309\) −4937.17 −0.908951
\(310\) 3158.94 0.578761
\(311\) −181.369 −0.0330692 −0.0165346 0.999863i \(-0.505263\pi\)
−0.0165346 + 0.999863i \(0.505263\pi\)
\(312\) −6210.61 −1.12694
\(313\) −3839.22 −0.693308 −0.346654 0.937993i \(-0.612682\pi\)
−0.346654 + 0.937993i \(0.612682\pi\)
\(314\) 197.851 0.0355585
\(315\) −830.138 −0.148486
\(316\) −10672.7 −1.89995
\(317\) 7000.08 1.24026 0.620132 0.784497i \(-0.287079\pi\)
0.620132 + 0.784497i \(0.287079\pi\)
\(318\) 6431.18 1.13410
\(319\) 334.613 0.0587296
\(320\) 2891.05 0.505046
\(321\) 3916.05 0.680912
\(322\) −1770.27 −0.306377
\(323\) 249.193 0.0429272
\(324\) 1576.51 0.270320
\(325\) −861.544 −0.147046
\(326\) −8080.22 −1.37277
\(327\) −2796.38 −0.472905
\(328\) −4473.54 −0.753079
\(329\) 6034.98 1.01130
\(330\) 907.007 0.151300
\(331\) −1176.84 −0.195423 −0.0977115 0.995215i \(-0.531152\pi\)
−0.0977115 + 0.995215i \(0.531152\pi\)
\(332\) 9947.12 1.64433
\(333\) −1642.54 −0.270302
\(334\) −8748.99 −1.43330
\(335\) 3358.69 0.547776
\(336\) 8805.35 1.42968
\(337\) −174.049 −0.0281336 −0.0140668 0.999901i \(-0.504478\pi\)
−0.0140668 + 0.999901i \(0.504478\pi\)
\(338\) 5289.71 0.851249
\(339\) −5159.67 −0.826652
\(340\) 6886.82 1.09850
\(341\) −1391.05 −0.220907
\(342\) −166.080 −0.0262589
\(343\) 6377.10 1.00388
\(344\) 24383.3 3.82169
\(345\) 274.675 0.0428637
\(346\) 3818.37 0.593285
\(347\) −10895.5 −1.68560 −0.842799 0.538228i \(-0.819094\pi\)
−0.842799 + 0.538228i \(0.819094\pi\)
\(348\) −1693.29 −0.260832
\(349\) −11820.3 −1.81298 −0.906488 0.422232i \(-0.861247\pi\)
−0.906488 + 0.422232i \(0.861247\pi\)
\(350\) 2416.87 0.369105
\(351\) 930.468 0.141495
\(352\) −4075.60 −0.617131
\(353\) 9795.13 1.47689 0.738445 0.674314i \(-0.235561\pi\)
0.738445 + 0.674314i \(0.235561\pi\)
\(354\) 1897.94 0.284956
\(355\) 4367.62 0.652983
\(356\) −15339.5 −2.28369
\(357\) 3916.49 0.580623
\(358\) −10095.7 −1.49042
\(359\) 8894.06 1.30755 0.653775 0.756689i \(-0.273184\pi\)
0.653775 + 0.756689i \(0.273184\pi\)
\(360\) −2703.26 −0.395762
\(361\) −6846.60 −0.998192
\(362\) −19367.0 −2.81189
\(363\) 3593.60 0.519600
\(364\) 12373.3 1.78170
\(365\) 2162.13 0.310058
\(366\) 544.096 0.0777058
\(367\) 11113.1 1.58066 0.790329 0.612683i \(-0.209910\pi\)
0.790329 + 0.612683i \(0.209910\pi\)
\(368\) −2913.50 −0.412708
\(369\) 670.222 0.0945538
\(370\) 4782.10 0.671917
\(371\) −7546.28 −1.05602
\(372\) 7039.30 0.981104
\(373\) 13469.8 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(374\) −4279.14 −0.591629
\(375\) −375.000 −0.0516398
\(376\) 19652.3 2.69545
\(377\) −999.391 −0.136529
\(378\) −2610.21 −0.355172
\(379\) −3750.71 −0.508341 −0.254170 0.967159i \(-0.581802\pi\)
−0.254170 + 0.967159i \(0.581802\pi\)
\(380\) 342.673 0.0462599
\(381\) 2651.38 0.356520
\(382\) −13049.9 −1.74788
\(383\) 12404.3 1.65490 0.827451 0.561537i \(-0.189790\pi\)
0.827451 + 0.561537i \(0.189790\pi\)
\(384\) 613.072 0.0814732
\(385\) −1064.27 −0.140884
\(386\) −23547.1 −3.10496
\(387\) −3653.09 −0.479837
\(388\) 18822.1 2.46276
\(389\) −7718.67 −1.00605 −0.503023 0.864273i \(-0.667779\pi\)
−0.503023 + 0.864273i \(0.667779\pi\)
\(390\) −2708.97 −0.351728
\(391\) −1295.88 −0.167610
\(392\) 161.534 0.0208131
\(393\) 2812.16 0.360953
\(394\) −10429.8 −1.33362
\(395\) −2741.77 −0.349250
\(396\) 2021.15 0.256481
\(397\) −6747.40 −0.853004 −0.426502 0.904487i \(-0.640254\pi\)
−0.426502 + 0.904487i \(0.640254\pi\)
\(398\) 439.167 0.0553101
\(399\) 194.876 0.0244511
\(400\) 3977.66 0.497207
\(401\) −5047.40 −0.628567 −0.314283 0.949329i \(-0.601764\pi\)
−0.314283 + 0.949329i \(0.601764\pi\)
\(402\) 10560.8 1.31026
\(403\) 4154.65 0.513543
\(404\) 9926.47 1.22243
\(405\) 405.000 0.0496904
\(406\) 2803.56 0.342706
\(407\) −2105.81 −0.256464
\(408\) 12753.6 1.54755
\(409\) 5317.14 0.642826 0.321413 0.946939i \(-0.395842\pi\)
0.321413 + 0.946939i \(0.395842\pi\)
\(410\) −1951.28 −0.235042
\(411\) −712.949 −0.0855649
\(412\) 32030.8 3.83021
\(413\) −2227.02 −0.265338
\(414\) 863.663 0.102528
\(415\) 2555.38 0.302262
\(416\) 12172.6 1.43464
\(417\) −5120.17 −0.601284
\(418\) −212.921 −0.0249146
\(419\) 2953.43 0.344355 0.172177 0.985066i \(-0.444920\pi\)
0.172177 + 0.985066i \(0.444920\pi\)
\(420\) 5385.68 0.625701
\(421\) 7427.88 0.859888 0.429944 0.902855i \(-0.358533\pi\)
0.429944 + 0.902855i \(0.358533\pi\)
\(422\) 22900.8 2.64170
\(423\) −2944.29 −0.338431
\(424\) −24573.7 −2.81463
\(425\) 1769.20 0.201927
\(426\) 13733.2 1.56191
\(427\) −638.436 −0.0723562
\(428\) −25406.1 −2.86928
\(429\) 1192.90 0.134251
\(430\) 10635.6 1.19278
\(431\) 3302.08 0.369038 0.184519 0.982829i \(-0.440927\pi\)
0.184519 + 0.982829i \(0.440927\pi\)
\(432\) −4295.87 −0.478438
\(433\) 9352.89 1.03804 0.519020 0.854762i \(-0.326297\pi\)
0.519020 + 0.854762i \(0.326297\pi\)
\(434\) −11654.9 −1.28907
\(435\) −435.000 −0.0479463
\(436\) 18142.0 1.99276
\(437\) −64.4802 −0.00705837
\(438\) 6798.42 0.741646
\(439\) −6109.74 −0.664241 −0.332120 0.943237i \(-0.607764\pi\)
−0.332120 + 0.943237i \(0.607764\pi\)
\(440\) −3465.69 −0.375501
\(441\) −24.2009 −0.00261321
\(442\) 12780.6 1.37536
\(443\) 14282.0 1.53173 0.765865 0.643002i \(-0.222311\pi\)
0.765865 + 0.643002i \(0.222311\pi\)
\(444\) 10656.3 1.13902
\(445\) −3940.68 −0.419789
\(446\) −19042.8 −2.02176
\(447\) −10639.8 −1.12583
\(448\) −10666.5 −1.12488
\(449\) −16406.0 −1.72439 −0.862193 0.506581i \(-0.830909\pi\)
−0.862193 + 0.506581i \(0.830909\pi\)
\(450\) −1179.12 −0.123520
\(451\) 859.252 0.0897131
\(452\) 33474.3 3.48341
\(453\) −5454.68 −0.565746
\(454\) 19349.6 2.00026
\(455\) 3178.67 0.327513
\(456\) 634.593 0.0651701
\(457\) −15726.2 −1.60971 −0.804856 0.593470i \(-0.797757\pi\)
−0.804856 + 0.593470i \(0.797757\pi\)
\(458\) −25331.3 −2.58440
\(459\) −1910.74 −0.194304
\(460\) −1782.00 −0.180623
\(461\) −14091.2 −1.42363 −0.711816 0.702366i \(-0.752127\pi\)
−0.711816 + 0.702366i \(0.752127\pi\)
\(462\) −3346.41 −0.336989
\(463\) −8794.40 −0.882743 −0.441372 0.897324i \(-0.645508\pi\)
−0.441372 + 0.897324i \(0.645508\pi\)
\(464\) 4614.08 0.461645
\(465\) 1808.37 0.180347
\(466\) 25335.0 2.51850
\(467\) −8746.05 −0.866636 −0.433318 0.901241i \(-0.642657\pi\)
−0.433318 + 0.901241i \(0.642657\pi\)
\(468\) −6036.59 −0.596242
\(469\) −12391.9 −1.22005
\(470\) 8572.00 0.841270
\(471\) 113.262 0.0110803
\(472\) −7252.06 −0.707210
\(473\) −4683.41 −0.455272
\(474\) −8620.99 −0.835391
\(475\) 88.0317 0.00850352
\(476\) −25409.0 −2.44668
\(477\) 3681.61 0.353395
\(478\) −21065.9 −2.01575
\(479\) −11452.6 −1.09245 −0.546225 0.837638i \(-0.683936\pi\)
−0.546225 + 0.837638i \(0.683936\pi\)
\(480\) 5298.31 0.503820
\(481\) 6289.43 0.596203
\(482\) −5643.06 −0.533266
\(483\) −1013.41 −0.0954698
\(484\) −23314.1 −2.18953
\(485\) 4835.35 0.452705
\(486\) 1273.45 0.118857
\(487\) 1635.31 0.152163 0.0760813 0.997102i \(-0.475759\pi\)
0.0760813 + 0.997102i \(0.475759\pi\)
\(488\) −2079.00 −0.192852
\(489\) −4625.62 −0.427766
\(490\) 70.4586 0.00649591
\(491\) 2106.92 0.193654 0.0968270 0.995301i \(-0.469131\pi\)
0.0968270 + 0.995301i \(0.469131\pi\)
\(492\) −4348.19 −0.398438
\(493\) 2052.28 0.187484
\(494\) 635.933 0.0579190
\(495\) 519.227 0.0471465
\(496\) −19181.6 −1.73645
\(497\) −16114.3 −1.45438
\(498\) 8034.93 0.722999
\(499\) −8865.74 −0.795361 −0.397681 0.917524i \(-0.630185\pi\)
−0.397681 + 0.917524i \(0.630185\pi\)
\(500\) 2432.88 0.217604
\(501\) −5008.46 −0.446630
\(502\) 16664.8 1.48165
\(503\) −14591.9 −1.29348 −0.646738 0.762712i \(-0.723867\pi\)
−0.646738 + 0.762712i \(0.723867\pi\)
\(504\) 9973.69 0.881475
\(505\) 2550.08 0.224707
\(506\) 1107.25 0.0972794
\(507\) 3028.16 0.265257
\(508\) −17201.3 −1.50233
\(509\) 13600.3 1.18432 0.592162 0.805819i \(-0.298274\pi\)
0.592162 + 0.805819i \(0.298274\pi\)
\(510\) 5562.93 0.483001
\(511\) −7977.19 −0.690588
\(512\) 20263.6 1.74909
\(513\) −95.0742 −0.00818251
\(514\) −16378.1 −1.40546
\(515\) 8228.62 0.704070
\(516\) 23700.1 2.02197
\(517\) −3774.70 −0.321105
\(518\) −17643.6 −1.49655
\(519\) 2185.87 0.184873
\(520\) 10351.0 0.872927
\(521\) 8355.92 0.702647 0.351324 0.936254i \(-0.385732\pi\)
0.351324 + 0.936254i \(0.385732\pi\)
\(522\) −1367.78 −0.114686
\(523\) 926.372 0.0774520 0.0387260 0.999250i \(-0.487670\pi\)
0.0387260 + 0.999250i \(0.487670\pi\)
\(524\) −18244.4 −1.52101
\(525\) 1383.56 0.115017
\(526\) 23447.7 1.94366
\(527\) −8531.68 −0.705210
\(528\) −5507.49 −0.453944
\(529\) −11831.7 −0.972440
\(530\) −10718.6 −0.878467
\(531\) 1086.50 0.0887947
\(532\) −1264.29 −0.103034
\(533\) −2566.34 −0.208556
\(534\) −12390.7 −1.00412
\(535\) −6526.75 −0.527432
\(536\) −40353.0 −3.25184
\(537\) −5779.38 −0.464430
\(538\) −21240.0 −1.70209
\(539\) −31.0266 −0.00247943
\(540\) −2627.51 −0.209389
\(541\) 13503.6 1.07313 0.536565 0.843859i \(-0.319722\pi\)
0.536565 + 0.843859i \(0.319722\pi\)
\(542\) 42165.0 3.34159
\(543\) −11086.9 −0.876211
\(544\) −24996.8 −1.97009
\(545\) 4660.63 0.366311
\(546\) 9994.74 0.783398
\(547\) −20425.4 −1.59658 −0.798288 0.602275i \(-0.794261\pi\)
−0.798288 + 0.602275i \(0.794261\pi\)
\(548\) 4625.39 0.360560
\(549\) 311.474 0.0242138
\(550\) −1511.68 −0.117197
\(551\) 102.117 0.00789532
\(552\) −3300.08 −0.254458
\(553\) 10115.8 0.777879
\(554\) −24241.3 −1.85905
\(555\) 2737.57 0.209375
\(556\) 33218.0 2.53374
\(557\) −3929.12 −0.298891 −0.149446 0.988770i \(-0.547749\pi\)
−0.149446 + 0.988770i \(0.547749\pi\)
\(558\) 5686.09 0.431383
\(559\) 13988.0 1.05837
\(560\) −14675.6 −1.10742
\(561\) −2449.65 −0.184357
\(562\) 365.769 0.0274538
\(563\) 4452.67 0.333318 0.166659 0.986015i \(-0.446702\pi\)
0.166659 + 0.986015i \(0.446702\pi\)
\(564\) 19101.6 1.42611
\(565\) 8599.45 0.640322
\(566\) −9975.75 −0.740834
\(567\) −1494.25 −0.110675
\(568\) −52474.7 −3.87639
\(569\) −5462.20 −0.402438 −0.201219 0.979546i \(-0.564490\pi\)
−0.201219 + 0.979546i \(0.564490\pi\)
\(570\) 276.799 0.0203401
\(571\) −7228.87 −0.529805 −0.264902 0.964275i \(-0.585340\pi\)
−0.264902 + 0.964275i \(0.585340\pi\)
\(572\) −7739.16 −0.565717
\(573\) −7470.57 −0.544655
\(574\) 7199.27 0.523505
\(575\) −457.791 −0.0332021
\(576\) 5203.89 0.376439
\(577\) 12771.6 0.921468 0.460734 0.887538i \(-0.347586\pi\)
0.460734 + 0.887538i \(0.347586\pi\)
\(578\) −498.502 −0.0358736
\(579\) −13479.8 −0.967534
\(580\) 2822.14 0.202040
\(581\) −9428.10 −0.673224
\(582\) 15203.9 1.08285
\(583\) 4719.97 0.335303
\(584\) −25976.9 −1.84064
\(585\) −1550.78 −0.109601
\(586\) −3265.88 −0.230226
\(587\) 6986.93 0.491280 0.245640 0.969361i \(-0.421002\pi\)
0.245640 + 0.969361i \(0.421002\pi\)
\(588\) 157.008 0.0110117
\(589\) −424.518 −0.0296977
\(590\) −3163.23 −0.220726
\(591\) −5970.69 −0.415569
\(592\) −29037.6 −2.01595
\(593\) 4323.42 0.299396 0.149698 0.988732i \(-0.452170\pi\)
0.149698 + 0.988732i \(0.452170\pi\)
\(594\) 1632.61 0.112773
\(595\) −6527.48 −0.449749
\(596\) 69027.6 4.74409
\(597\) 251.406 0.0172351
\(598\) −3307.04 −0.226145
\(599\) 13443.7 0.917019 0.458510 0.888689i \(-0.348383\pi\)
0.458510 + 0.888689i \(0.348383\pi\)
\(600\) 4505.43 0.306556
\(601\) −7179.72 −0.487299 −0.243650 0.969863i \(-0.578345\pi\)
−0.243650 + 0.969863i \(0.578345\pi\)
\(602\) −39240.1 −2.65666
\(603\) 6045.65 0.408288
\(604\) 35388.2 2.38399
\(605\) −5989.33 −0.402481
\(606\) 8018.24 0.537490
\(607\) −8560.81 −0.572442 −0.286221 0.958164i \(-0.592399\pi\)
−0.286221 + 0.958164i \(0.592399\pi\)
\(608\) −1243.79 −0.0829641
\(609\) 1604.93 0.106790
\(610\) −906.826 −0.0601907
\(611\) 11273.9 0.746472
\(612\) 12396.3 0.818774
\(613\) 19634.3 1.29367 0.646837 0.762628i \(-0.276091\pi\)
0.646837 + 0.762628i \(0.276091\pi\)
\(614\) 36159.1 2.37665
\(615\) −1117.04 −0.0732410
\(616\) 12786.7 0.836348
\(617\) 14268.5 0.931003 0.465502 0.885047i \(-0.345874\pi\)
0.465502 + 0.885047i \(0.345874\pi\)
\(618\) 25873.3 1.68411
\(619\) 1663.48 0.108014 0.0540071 0.998541i \(-0.482801\pi\)
0.0540071 + 0.998541i \(0.482801\pi\)
\(620\) −11732.2 −0.759960
\(621\) 494.415 0.0319488
\(622\) 950.470 0.0612707
\(623\) 14539.1 0.934990
\(624\) 16449.3 1.05528
\(625\) 625.000 0.0400000
\(626\) 20119.5 1.28456
\(627\) −121.889 −0.00776361
\(628\) −734.808 −0.0466912
\(629\) −12915.5 −0.818720
\(630\) 4350.36 0.275115
\(631\) 13448.3 0.848444 0.424222 0.905558i \(-0.360548\pi\)
0.424222 + 0.905558i \(0.360548\pi\)
\(632\) 32941.0 2.07330
\(633\) 13109.9 0.823176
\(634\) −36684.1 −2.29797
\(635\) −4418.96 −0.276159
\(636\) −23885.1 −1.48916
\(637\) 92.6675 0.00576392
\(638\) −1753.55 −0.108814
\(639\) 7861.71 0.486705
\(640\) −1021.79 −0.0631088
\(641\) 7644.92 0.471070 0.235535 0.971866i \(-0.424316\pi\)
0.235535 + 0.971866i \(0.424316\pi\)
\(642\) −20522.1 −1.26160
\(643\) 1814.60 0.111292 0.0556462 0.998451i \(-0.482278\pi\)
0.0556462 + 0.998451i \(0.482278\pi\)
\(644\) 6574.71 0.402298
\(645\) 6088.48 0.371680
\(646\) −1305.90 −0.0795357
\(647\) 18746.6 1.13911 0.569556 0.821952i \(-0.307115\pi\)
0.569556 + 0.821952i \(0.307115\pi\)
\(648\) −4865.87 −0.294984
\(649\) 1392.94 0.0842489
\(650\) 4514.94 0.272447
\(651\) −6672.00 −0.401684
\(652\) 30009.6 1.80256
\(653\) 8299.18 0.497354 0.248677 0.968586i \(-0.420004\pi\)
0.248677 + 0.968586i \(0.420004\pi\)
\(654\) 14654.5 0.876200
\(655\) −4686.93 −0.279593
\(656\) 11848.5 0.705192
\(657\) 3891.84 0.231104
\(658\) −31626.4 −1.87375
\(659\) 12624.7 0.746262 0.373131 0.927779i \(-0.378284\pi\)
0.373131 + 0.927779i \(0.378284\pi\)
\(660\) −3368.58 −0.198670
\(661\) −10912.8 −0.642144 −0.321072 0.947055i \(-0.604043\pi\)
−0.321072 + 0.947055i \(0.604043\pi\)
\(662\) 6167.26 0.362081
\(663\) 7316.38 0.428574
\(664\) −30701.6 −1.79436
\(665\) −324.793 −0.0189398
\(666\) 8607.77 0.500817
\(667\) −531.038 −0.0308274
\(668\) 32493.4 1.88204
\(669\) −10901.3 −0.629998
\(670\) −17601.3 −1.01492
\(671\) 399.323 0.0229742
\(672\) −19548.1 −1.12215
\(673\) 6292.15 0.360393 0.180197 0.983631i \(-0.442327\pi\)
0.180197 + 0.983631i \(0.442327\pi\)
\(674\) 912.105 0.0521261
\(675\) −675.000 −0.0384900
\(676\) −19645.7 −1.11776
\(677\) 16630.8 0.944125 0.472062 0.881565i \(-0.343510\pi\)
0.472062 + 0.881565i \(0.343510\pi\)
\(678\) 27039.4 1.53162
\(679\) −17840.0 −1.00830
\(680\) −21256.1 −1.19872
\(681\) 11076.9 0.623300
\(682\) 7289.81 0.409298
\(683\) 11843.7 0.663521 0.331761 0.943364i \(-0.392357\pi\)
0.331761 + 0.943364i \(0.392357\pi\)
\(684\) 616.812 0.0344801
\(685\) 1188.25 0.0662783
\(686\) −33419.4 −1.86000
\(687\) −14501.2 −0.805321
\(688\) −64581.1 −3.57868
\(689\) −14097.2 −0.779478
\(690\) −1439.44 −0.0794181
\(691\) 20458.3 1.12630 0.563149 0.826356i \(-0.309590\pi\)
0.563149 + 0.826356i \(0.309590\pi\)
\(692\) −14181.2 −0.779032
\(693\) −1915.69 −0.105009
\(694\) 57098.2 3.12308
\(695\) 8533.61 0.465753
\(696\) 5226.30 0.284630
\(697\) 5270.03 0.286394
\(698\) 61944.8 3.35909
\(699\) 14503.4 0.784789
\(700\) −8976.13 −0.484666
\(701\) 23987.6 1.29244 0.646220 0.763151i \(-0.276349\pi\)
0.646220 + 0.763151i \(0.276349\pi\)
\(702\) −4876.14 −0.262162
\(703\) −642.648 −0.0344778
\(704\) 6671.61 0.357167
\(705\) 4907.15 0.262147
\(706\) −51331.6 −2.73639
\(707\) −9408.52 −0.500487
\(708\) −7048.86 −0.374170
\(709\) 6275.84 0.332432 0.166216 0.986089i \(-0.446845\pi\)
0.166216 + 0.986089i \(0.446845\pi\)
\(710\) −22888.6 −1.20985
\(711\) −4935.19 −0.260315
\(712\) 47345.2 2.49205
\(713\) 2207.62 0.115955
\(714\) −20524.4 −1.07578
\(715\) −1988.16 −0.103990
\(716\) 37494.8 1.95705
\(717\) −12059.4 −0.628127
\(718\) −46609.5 −2.42263
\(719\) −15544.4 −0.806273 −0.403136 0.915140i \(-0.632080\pi\)
−0.403136 + 0.915140i \(0.632080\pi\)
\(720\) 7159.79 0.370597
\(721\) −30359.5 −1.56817
\(722\) 35879.8 1.84945
\(723\) −3230.44 −0.166170
\(724\) 71928.1 3.69225
\(725\) 725.000 0.0371391
\(726\) −18832.3 −0.962718
\(727\) −25002.9 −1.27552 −0.637762 0.770233i \(-0.720140\pi\)
−0.637762 + 0.770233i \(0.720140\pi\)
\(728\) −38190.1 −1.94426
\(729\) 729.000 0.0370370
\(730\) −11330.7 −0.574477
\(731\) −28724.7 −1.45338
\(732\) −2020.75 −0.102034
\(733\) −16981.1 −0.855678 −0.427839 0.903855i \(-0.640725\pi\)
−0.427839 + 0.903855i \(0.640725\pi\)
\(734\) −58238.7 −2.92865
\(735\) 40.3349 0.00202418
\(736\) 6468.06 0.323934
\(737\) 7750.78 0.387386
\(738\) −3512.31 −0.175190
\(739\) 920.528 0.0458216 0.0229108 0.999738i \(-0.492707\pi\)
0.0229108 + 0.999738i \(0.492707\pi\)
\(740\) −17760.5 −0.882282
\(741\) 364.047 0.0180481
\(742\) 39546.4 1.95660
\(743\) 6708.21 0.331225 0.165613 0.986191i \(-0.447040\pi\)
0.165613 + 0.986191i \(0.447040\pi\)
\(744\) −21726.7 −1.07062
\(745\) 17733.0 0.872061
\(746\) −70588.9 −3.46440
\(747\) 4599.69 0.225293
\(748\) 15892.5 0.776857
\(749\) 24080.5 1.17474
\(750\) 1965.20 0.0956784
\(751\) 34311.4 1.66717 0.833583 0.552394i \(-0.186286\pi\)
0.833583 + 0.552394i \(0.186286\pi\)
\(752\) −52050.6 −2.52405
\(753\) 9539.96 0.461694
\(754\) 5237.33 0.252961
\(755\) 9091.13 0.438225
\(756\) 9694.22 0.466370
\(757\) −12405.6 −0.595628 −0.297814 0.954624i \(-0.596257\pi\)
−0.297814 + 0.954624i \(0.596257\pi\)
\(758\) 19655.7 0.941856
\(759\) 633.860 0.0303131
\(760\) −1057.66 −0.0504805
\(761\) −18869.8 −0.898857 −0.449428 0.893316i \(-0.648372\pi\)
−0.449428 + 0.893316i \(0.648372\pi\)
\(762\) −13894.6 −0.660562
\(763\) −17195.4 −0.815878
\(764\) 48466.7 2.29511
\(765\) 3184.56 0.150507
\(766\) −65004.8 −3.06621
\(767\) −4160.29 −0.195853
\(768\) 10664.2 0.501058
\(769\) 38269.7 1.79459 0.897295 0.441432i \(-0.145529\pi\)
0.897295 + 0.441432i \(0.145529\pi\)
\(770\) 5577.34 0.261030
\(771\) −9375.86 −0.437955
\(772\) 87452.9 4.07707
\(773\) 37398.4 1.74014 0.870069 0.492930i \(-0.164074\pi\)
0.870069 + 0.492930i \(0.164074\pi\)
\(774\) 19144.1 0.889044
\(775\) −3013.96 −0.139696
\(776\) −58094.2 −2.68745
\(777\) −10100.3 −0.466339
\(778\) 40449.9 1.86401
\(779\) 262.226 0.0120606
\(780\) 10061.0 0.461847
\(781\) 10079.0 0.461788
\(782\) 6791.09 0.310549
\(783\) −783.000 −0.0357371
\(784\) −427.836 −0.0194896
\(785\) −188.770 −0.00858279
\(786\) −14737.2 −0.668776
\(787\) −22352.2 −1.01241 −0.506206 0.862413i \(-0.668952\pi\)
−0.506206 + 0.862413i \(0.668952\pi\)
\(788\) 38736.0 1.75116
\(789\) 13422.9 0.605663
\(790\) 14368.3 0.647091
\(791\) −31727.7 −1.42618
\(792\) −6238.25 −0.279882
\(793\) −1192.66 −0.0534081
\(794\) 35359.9 1.58045
\(795\) −6136.01 −0.273738
\(796\) −1631.05 −0.0726267
\(797\) −29993.8 −1.33304 −0.666521 0.745487i \(-0.732217\pi\)
−0.666521 + 0.745487i \(0.732217\pi\)
\(798\) −1021.25 −0.0453032
\(799\) −23151.3 −1.02507
\(800\) −8830.52 −0.390258
\(801\) −7093.22 −0.312892
\(802\) 26451.0 1.16461
\(803\) 4989.50 0.219272
\(804\) −39222.3 −1.72048
\(805\) 1689.02 0.0739506
\(806\) −21772.5 −0.951495
\(807\) −12159.1 −0.530385
\(808\) −30637.9 −1.33396
\(809\) 28328.9 1.23114 0.615569 0.788083i \(-0.288926\pi\)
0.615569 + 0.788083i \(0.288926\pi\)
\(810\) −2122.41 −0.0920666
\(811\) 4087.14 0.176965 0.0884826 0.996078i \(-0.471798\pi\)
0.0884826 + 0.996078i \(0.471798\pi\)
\(812\) −10412.3 −0.450001
\(813\) 24137.9 1.04127
\(814\) 11035.5 0.475178
\(815\) 7709.37 0.331347
\(816\) −33779.0 −1.44914
\(817\) −1429.28 −0.0612046
\(818\) −27864.6 −1.19103
\(819\) 5721.61 0.244114
\(820\) 7246.98 0.308629
\(821\) −9552.90 −0.406088 −0.203044 0.979170i \(-0.565084\pi\)
−0.203044 + 0.979170i \(0.565084\pi\)
\(822\) 3736.23 0.158535
\(823\) 26863.5 1.13779 0.568895 0.822410i \(-0.307371\pi\)
0.568895 + 0.822410i \(0.307371\pi\)
\(824\) −98862.6 −4.17966
\(825\) −865.378 −0.0365195
\(826\) 11670.8 0.491619
\(827\) −43901.7 −1.84596 −0.922981 0.384846i \(-0.874254\pi\)
−0.922981 + 0.384846i \(0.874254\pi\)
\(828\) −3207.61 −0.134628
\(829\) −9622.68 −0.403148 −0.201574 0.979473i \(-0.564606\pi\)
−0.201574 + 0.979473i \(0.564606\pi\)
\(830\) −13391.5 −0.560033
\(831\) −13877.2 −0.579296
\(832\) −19926.2 −0.830307
\(833\) −190.295 −0.00791516
\(834\) 26832.3 1.11406
\(835\) 8347.44 0.345958
\(836\) 790.779 0.0327149
\(837\) 3255.07 0.134423
\(838\) −15477.5 −0.638022
\(839\) 26742.6 1.10043 0.550214 0.835024i \(-0.314546\pi\)
0.550214 + 0.835024i \(0.314546\pi\)
\(840\) −16622.8 −0.682788
\(841\) 841.000 0.0344828
\(842\) −38926.0 −1.59320
\(843\) 209.389 0.00855486
\(844\) −85052.7 −3.46876
\(845\) −5046.93 −0.205467
\(846\) 15429.6 0.627046
\(847\) 22097.7 0.896440
\(848\) 65085.2 2.63566
\(849\) −5710.74 −0.230851
\(850\) −9271.54 −0.374131
\(851\) 3341.96 0.134619
\(852\) −51004.3 −2.05092
\(853\) −16600.5 −0.666341 −0.333171 0.942867i \(-0.608119\pi\)
−0.333171 + 0.942867i \(0.608119\pi\)
\(854\) 3345.74 0.134062
\(855\) 158.457 0.00633815
\(856\) 78415.6 3.13106
\(857\) 15752.2 0.627869 0.313935 0.949445i \(-0.398353\pi\)
0.313935 + 0.949445i \(0.398353\pi\)
\(858\) −6251.41 −0.248741
\(859\) −35249.8 −1.40012 −0.700062 0.714082i \(-0.746844\pi\)
−0.700062 + 0.714082i \(0.746844\pi\)
\(860\) −39500.2 −1.56622
\(861\) 4121.31 0.163129
\(862\) −17304.6 −0.683755
\(863\) −36020.7 −1.42081 −0.710404 0.703794i \(-0.751488\pi\)
−0.710404 + 0.703794i \(0.751488\pi\)
\(864\) 9536.96 0.375526
\(865\) −3643.12 −0.143202
\(866\) −49014.0 −1.92328
\(867\) −285.374 −0.0111785
\(868\) 43285.9 1.69265
\(869\) −6327.12 −0.246988
\(870\) 2279.63 0.0888352
\(871\) −23149.3 −0.900556
\(872\) −55995.0 −2.17458
\(873\) 8703.63 0.337426
\(874\) 337.910 0.0130778
\(875\) −2305.94 −0.0890914
\(876\) −25249.0 −0.973842
\(877\) 1353.74 0.0521239 0.0260620 0.999660i \(-0.491703\pi\)
0.0260620 + 0.999660i \(0.491703\pi\)
\(878\) 32018.2 1.23071
\(879\) −1869.59 −0.0717404
\(880\) 9179.15 0.351624
\(881\) −1774.07 −0.0678435 −0.0339217 0.999424i \(-0.510800\pi\)
−0.0339217 + 0.999424i \(0.510800\pi\)
\(882\) 126.826 0.00484177
\(883\) 1483.09 0.0565233 0.0282616 0.999601i \(-0.491003\pi\)
0.0282616 + 0.999601i \(0.491003\pi\)
\(884\) −47466.4 −1.80596
\(885\) −1810.83 −0.0687801
\(886\) −74844.9 −2.83799
\(887\) 18524.8 0.701244 0.350622 0.936517i \(-0.385970\pi\)
0.350622 + 0.936517i \(0.385970\pi\)
\(888\) −32890.5 −1.24294
\(889\) 16303.8 0.615086
\(890\) 20651.2 0.777786
\(891\) 934.609 0.0351409
\(892\) 70724.2 2.65473
\(893\) −1151.96 −0.0431678
\(894\) 55758.0 2.08594
\(895\) 9632.30 0.359746
\(896\) 3769.89 0.140561
\(897\) −1893.16 −0.0704689
\(898\) 85976.2 3.19495
\(899\) −3496.19 −0.129705
\(900\) 4379.19 0.162192
\(901\) 28948.9 1.07040
\(902\) −4502.93 −0.166221
\(903\) −22463.5 −0.827838
\(904\) −103318. −3.80122
\(905\) 18478.1 0.678710
\(906\) 28585.4 1.04822
\(907\) −26687.1 −0.976991 −0.488495 0.872567i \(-0.662454\pi\)
−0.488495 + 0.872567i \(0.662454\pi\)
\(908\) −71863.4 −2.62651
\(909\) 4590.14 0.167487
\(910\) −16657.9 −0.606817
\(911\) −4443.10 −0.161588 −0.0807939 0.996731i \(-0.525746\pi\)
−0.0807939 + 0.996731i \(0.525746\pi\)
\(912\) −1680.77 −0.0610261
\(913\) 5897.00 0.213759
\(914\) 82413.2 2.98248
\(915\) −519.124 −0.0187559
\(916\) 94079.3 3.39352
\(917\) 17292.5 0.622734
\(918\) 10013.3 0.360008
\(919\) 17674.0 0.634398 0.317199 0.948359i \(-0.397258\pi\)
0.317199 + 0.948359i \(0.397258\pi\)
\(920\) 5500.13 0.197102
\(921\) 20699.7 0.740585
\(922\) 73845.4 2.63771
\(923\) −30103.2 −1.07352
\(924\) 12428.4 0.442494
\(925\) −4562.61 −0.162181
\(926\) 46087.2 1.63555
\(927\) 14811.5 0.524783
\(928\) −10243.4 −0.362345
\(929\) 10760.5 0.380021 0.190010 0.981782i \(-0.439148\pi\)
0.190010 + 0.981782i \(0.439148\pi\)
\(930\) −9476.82 −0.334148
\(931\) −9.46867 −0.000333322 0
\(932\) −94093.2 −3.30700
\(933\) 544.108 0.0190925
\(934\) 45833.9 1.60571
\(935\) 4082.75 0.142802
\(936\) 18631.8 0.650641
\(937\) 22210.3 0.774365 0.387182 0.922003i \(-0.373448\pi\)
0.387182 + 0.922003i \(0.373448\pi\)
\(938\) 64940.1 2.26052
\(939\) 11517.6 0.400281
\(940\) −31836.0 −1.10466
\(941\) 34977.1 1.21171 0.605855 0.795575i \(-0.292831\pi\)
0.605855 + 0.795575i \(0.292831\pi\)
\(942\) −593.552 −0.0205297
\(943\) −1363.65 −0.0470908
\(944\) 19207.6 0.662241
\(945\) 2490.42 0.0857283
\(946\) 24543.5 0.843530
\(947\) 42798.4 1.46860 0.734298 0.678827i \(-0.237511\pi\)
0.734298 + 0.678827i \(0.237511\pi\)
\(948\) 32018.0 1.09694
\(949\) −14902.2 −0.509742
\(950\) −461.332 −0.0157554
\(951\) −21000.2 −0.716067
\(952\) 78424.3 2.66990
\(953\) −15363.5 −0.522217 −0.261108 0.965309i \(-0.584088\pi\)
−0.261108 + 0.965309i \(0.584088\pi\)
\(954\) −19293.5 −0.654771
\(955\) 12450.9 0.421888
\(956\) 78237.7 2.64685
\(957\) −1003.84 −0.0339075
\(958\) 60017.7 2.02410
\(959\) −4384.05 −0.147621
\(960\) −8673.16 −0.291588
\(961\) −15256.7 −0.512125
\(962\) −32959.9 −1.10465
\(963\) −11748.2 −0.393125
\(964\) 20958.1 0.700222
\(965\) 22466.4 0.749449
\(966\) 5310.82 0.176887
\(967\) −44310.2 −1.47355 −0.736774 0.676139i \(-0.763652\pi\)
−0.736774 + 0.676139i \(0.763652\pi\)
\(968\) 71958.7 2.38930
\(969\) −747.580 −0.0247840
\(970\) −25339.8 −0.838774
\(971\) 50163.2 1.65789 0.828946 0.559328i \(-0.188941\pi\)
0.828946 + 0.559328i \(0.188941\pi\)
\(972\) −4729.52 −0.156070
\(973\) −31484.8 −1.03736
\(974\) −8569.90 −0.281927
\(975\) 2584.63 0.0848969
\(976\) 5506.39 0.180589
\(977\) −2549.65 −0.0834908 −0.0417454 0.999128i \(-0.513292\pi\)
−0.0417454 + 0.999128i \(0.513292\pi\)
\(978\) 24240.7 0.792568
\(979\) −9093.80 −0.296874
\(980\) −261.680 −0.00852966
\(981\) 8389.13 0.273032
\(982\) −11041.4 −0.358803
\(983\) 25382.0 0.823561 0.411781 0.911283i \(-0.364907\pi\)
0.411781 + 0.911283i \(0.364907\pi\)
\(984\) 13420.6 0.434790
\(985\) 9951.15 0.321899
\(986\) −10755.0 −0.347372
\(987\) −18104.9 −0.583877
\(988\) −2361.83 −0.0760523
\(989\) 7432.68 0.238974
\(990\) −2721.02 −0.0873532
\(991\) −44583.2 −1.42909 −0.714547 0.699588i \(-0.753367\pi\)
−0.714547 + 0.699588i \(0.753367\pi\)
\(992\) 42583.7 1.36294
\(993\) 3530.52 0.112828
\(994\) 84447.6 2.69468
\(995\) −419.010 −0.0133503
\(996\) −29841.4 −0.949357
\(997\) −39668.0 −1.26008 −0.630039 0.776563i \(-0.716961\pi\)
−0.630039 + 0.776563i \(0.716961\pi\)
\(998\) 46461.1 1.47365
\(999\) 4927.62 0.156059
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.i.1.1 7
3.2 odd 2 1305.4.a.n.1.7 7
5.4 even 2 2175.4.a.n.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.i.1.1 7 1.1 even 1 trivial
1305.4.a.n.1.7 7 3.2 odd 2
2175.4.a.n.1.7 7 5.4 even 2