Properties

Label 115.4.a.e.1.3
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $5$
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] = [115,4,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.595043\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.404957 q^{2} -7.11323 q^{3} -7.83601 q^{4} -5.00000 q^{5} -2.88055 q^{6} +13.7888 q^{7} -6.41290 q^{8} +23.5981 q^{9} +O(q^{10})\) \(q+0.404957 q^{2} -7.11323 q^{3} -7.83601 q^{4} -5.00000 q^{5} -2.88055 q^{6} +13.7888 q^{7} -6.41290 q^{8} +23.5981 q^{9} -2.02479 q^{10} +24.2317 q^{11} +55.7394 q^{12} +3.05016 q^{13} +5.58389 q^{14} +35.5662 q^{15} +60.0911 q^{16} +63.1126 q^{17} +9.55621 q^{18} -2.07770 q^{19} +39.1800 q^{20} -98.0832 q^{21} +9.81282 q^{22} -23.0000 q^{23} +45.6165 q^{24} +25.0000 q^{25} +1.23518 q^{26} +24.1987 q^{27} -108.049 q^{28} -8.16397 q^{29} +14.4028 q^{30} -156.989 q^{31} +75.6376 q^{32} -172.366 q^{33} +25.5579 q^{34} -68.9442 q^{35} -184.915 q^{36} +302.801 q^{37} -0.841380 q^{38} -21.6965 q^{39} +32.0645 q^{40} -42.7514 q^{41} -39.7195 q^{42} +215.265 q^{43} -189.880 q^{44} -117.990 q^{45} -9.31401 q^{46} +247.096 q^{47} -427.442 q^{48} -152.868 q^{49} +10.1239 q^{50} -448.935 q^{51} -23.9010 q^{52} +600.400 q^{53} +9.79943 q^{54} -121.159 q^{55} -88.4265 q^{56} +14.7792 q^{57} -3.30606 q^{58} +92.2014 q^{59} -278.697 q^{60} +532.635 q^{61} -63.5740 q^{62} +325.390 q^{63} -450.099 q^{64} -15.2508 q^{65} -69.8009 q^{66} +30.3010 q^{67} -494.551 q^{68} +163.604 q^{69} -27.9194 q^{70} -736.349 q^{71} -151.332 q^{72} +349.936 q^{73} +122.622 q^{74} -177.831 q^{75} +16.2809 q^{76} +334.127 q^{77} -8.78614 q^{78} +301.545 q^{79} -300.456 q^{80} -809.279 q^{81} -17.3125 q^{82} +139.488 q^{83} +768.581 q^{84} -315.563 q^{85} +87.1732 q^{86} +58.0722 q^{87} -155.396 q^{88} +859.551 q^{89} -47.7810 q^{90} +42.0581 q^{91} +180.228 q^{92} +1116.70 q^{93} +100.063 q^{94} +10.3885 q^{95} -538.028 q^{96} -927.475 q^{97} -61.9051 q^{98} +571.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.404957 0.143174 0.0715870 0.997434i \(-0.477194\pi\)
0.0715870 + 0.997434i \(0.477194\pi\)
\(3\) −7.11323 −1.36894 −0.684471 0.729040i \(-0.739967\pi\)
−0.684471 + 0.729040i \(0.739967\pi\)
\(4\) −7.83601 −0.979501
\(5\) −5.00000 −0.447214
\(6\) −2.88055 −0.195997
\(7\) 13.7888 0.744527 0.372263 0.928127i \(-0.378582\pi\)
0.372263 + 0.928127i \(0.378582\pi\)
\(8\) −6.41290 −0.283413
\(9\) 23.5981 0.874003
\(10\) −2.02479 −0.0640293
\(11\) 24.2317 0.664195 0.332098 0.943245i \(-0.392244\pi\)
0.332098 + 0.943245i \(0.392244\pi\)
\(12\) 55.7394 1.34088
\(13\) 3.05016 0.0650739 0.0325370 0.999471i \(-0.489641\pi\)
0.0325370 + 0.999471i \(0.489641\pi\)
\(14\) 5.58389 0.106597
\(15\) 35.5662 0.612210
\(16\) 60.0911 0.938924
\(17\) 63.1126 0.900415 0.450208 0.892924i \(-0.351350\pi\)
0.450208 + 0.892924i \(0.351350\pi\)
\(18\) 9.55621 0.125134
\(19\) −2.07770 −0.0250872 −0.0125436 0.999921i \(-0.503993\pi\)
−0.0125436 + 0.999921i \(0.503993\pi\)
\(20\) 39.1800 0.438046
\(21\) −98.0832 −1.01921
\(22\) 9.81282 0.0950955
\(23\) −23.0000 −0.208514
\(24\) 45.6165 0.387976
\(25\) 25.0000 0.200000
\(26\) 1.23518 0.00931689
\(27\) 24.1987 0.172483
\(28\) −108.049 −0.729265
\(29\) −8.16397 −0.0522762 −0.0261381 0.999658i \(-0.508321\pi\)
−0.0261381 + 0.999658i \(0.508321\pi\)
\(30\) 14.4028 0.0876525
\(31\) −156.989 −0.909553 −0.454776 0.890606i \(-0.650281\pi\)
−0.454776 + 0.890606i \(0.650281\pi\)
\(32\) 75.6376 0.417842
\(33\) −172.366 −0.909245
\(34\) 25.5579 0.128916
\(35\) −68.9442 −0.332963
\(36\) −184.915 −0.856087
\(37\) 302.801 1.34541 0.672706 0.739910i \(-0.265132\pi\)
0.672706 + 0.739910i \(0.265132\pi\)
\(38\) −0.841380 −0.00359184
\(39\) −21.6965 −0.0890824
\(40\) 32.0645 0.126746
\(41\) −42.7514 −0.162845 −0.0814225 0.996680i \(-0.525946\pi\)
−0.0814225 + 0.996680i \(0.525946\pi\)
\(42\) −39.7195 −0.145925
\(43\) 215.265 0.763434 0.381717 0.924279i \(-0.375333\pi\)
0.381717 + 0.924279i \(0.375333\pi\)
\(44\) −189.880 −0.650580
\(45\) −117.990 −0.390866
\(46\) −9.31401 −0.0298538
\(47\) 247.096 0.766866 0.383433 0.923569i \(-0.374742\pi\)
0.383433 + 0.923569i \(0.374742\pi\)
\(48\) −427.442 −1.28533
\(49\) −152.868 −0.445680
\(50\) 10.1239 0.0286348
\(51\) −448.935 −1.23262
\(52\) −23.9010 −0.0637400
\(53\) 600.400 1.55606 0.778031 0.628225i \(-0.216218\pi\)
0.778031 + 0.628225i \(0.216218\pi\)
\(54\) 9.79943 0.0246951
\(55\) −121.159 −0.297037
\(56\) −88.4265 −0.211009
\(57\) 14.7792 0.0343430
\(58\) −3.30606 −0.00748460
\(59\) 92.2014 0.203451 0.101725 0.994813i \(-0.467564\pi\)
0.101725 + 0.994813i \(0.467564\pi\)
\(60\) −278.697 −0.599660
\(61\) 532.635 1.11798 0.558991 0.829174i \(-0.311189\pi\)
0.558991 + 0.829174i \(0.311189\pi\)
\(62\) −63.5740 −0.130224
\(63\) 325.390 0.650719
\(64\) −450.099 −0.879100
\(65\) −15.2508 −0.0291019
\(66\) −69.8009 −0.130180
\(67\) 30.3010 0.0552515 0.0276258 0.999618i \(-0.491205\pi\)
0.0276258 + 0.999618i \(0.491205\pi\)
\(68\) −494.551 −0.881958
\(69\) 163.604 0.285444
\(70\) −27.9194 −0.0476716
\(71\) −736.349 −1.23083 −0.615413 0.788205i \(-0.711011\pi\)
−0.615413 + 0.788205i \(0.711011\pi\)
\(72\) −151.332 −0.247704
\(73\) 349.936 0.561053 0.280527 0.959846i \(-0.409491\pi\)
0.280527 + 0.959846i \(0.409491\pi\)
\(74\) 122.622 0.192628
\(75\) −177.831 −0.273788
\(76\) 16.2809 0.0245730
\(77\) 334.127 0.494511
\(78\) −8.78614 −0.0127543
\(79\) 301.545 0.429449 0.214725 0.976675i \(-0.431115\pi\)
0.214725 + 0.976675i \(0.431115\pi\)
\(80\) −300.456 −0.419900
\(81\) −809.279 −1.11012
\(82\) −17.3125 −0.0233152
\(83\) 139.488 0.184468 0.0922340 0.995737i \(-0.470599\pi\)
0.0922340 + 0.995737i \(0.470599\pi\)
\(84\) 768.581 0.998322
\(85\) −315.563 −0.402678
\(86\) 87.1732 0.109304
\(87\) 58.0722 0.0715631
\(88\) −155.396 −0.188242
\(89\) 859.551 1.02373 0.511866 0.859065i \(-0.328954\pi\)
0.511866 + 0.859065i \(0.328954\pi\)
\(90\) −47.7810 −0.0559618
\(91\) 42.0581 0.0484493
\(92\) 180.228 0.204240
\(93\) 1116.70 1.24513
\(94\) 100.063 0.109795
\(95\) 10.3885 0.0112193
\(96\) −538.028 −0.572002
\(97\) −927.475 −0.970833 −0.485417 0.874283i \(-0.661332\pi\)
−0.485417 + 0.874283i \(0.661332\pi\)
\(98\) −61.9051 −0.0638097
\(99\) 571.823 0.580508
\(100\) −195.900 −0.195900
\(101\) 1713.34 1.68796 0.843979 0.536376i \(-0.180207\pi\)
0.843979 + 0.536376i \(0.180207\pi\)
\(102\) −181.799 −0.176479
\(103\) −930.516 −0.890160 −0.445080 0.895491i \(-0.646825\pi\)
−0.445080 + 0.895491i \(0.646825\pi\)
\(104\) −19.5604 −0.0184428
\(105\) 490.416 0.455806
\(106\) 243.136 0.222788
\(107\) 1514.91 1.36871 0.684355 0.729149i \(-0.260084\pi\)
0.684355 + 0.729149i \(0.260084\pi\)
\(108\) −189.621 −0.168947
\(109\) 1748.76 1.53670 0.768352 0.640027i \(-0.221077\pi\)
0.768352 + 0.640027i \(0.221077\pi\)
\(110\) −49.0641 −0.0425280
\(111\) −2153.90 −1.84179
\(112\) 828.586 0.699054
\(113\) 2026.23 1.68683 0.843416 0.537261i \(-0.180541\pi\)
0.843416 + 0.537261i \(0.180541\pi\)
\(114\) 5.98493 0.00491702
\(115\) 115.000 0.0932505
\(116\) 63.9729 0.0512046
\(117\) 71.9778 0.0568748
\(118\) 37.3376 0.0291289
\(119\) 870.249 0.670383
\(120\) −228.082 −0.173508
\(121\) −743.822 −0.558845
\(122\) 215.694 0.160066
\(123\) 304.100 0.222925
\(124\) 1230.17 0.890908
\(125\) −125.000 −0.0894427
\(126\) 131.769 0.0931660
\(127\) −2126.49 −1.48579 −0.742897 0.669406i \(-0.766549\pi\)
−0.742897 + 0.669406i \(0.766549\pi\)
\(128\) −787.371 −0.543707
\(129\) −1531.23 −1.04510
\(130\) −6.17591 −0.00416664
\(131\) −1494.23 −0.996576 −0.498288 0.867012i \(-0.666038\pi\)
−0.498288 + 0.867012i \(0.666038\pi\)
\(132\) 1350.66 0.890606
\(133\) −28.6491 −0.0186781
\(134\) 12.2706 0.00791058
\(135\) −120.993 −0.0771367
\(136\) −404.735 −0.255189
\(137\) 2265.31 1.41269 0.706346 0.707867i \(-0.250342\pi\)
0.706346 + 0.707867i \(0.250342\pi\)
\(138\) 66.2527 0.0408682
\(139\) −2918.66 −1.78099 −0.890496 0.454991i \(-0.849642\pi\)
−0.890496 + 0.454991i \(0.849642\pi\)
\(140\) 540.247 0.326137
\(141\) −1757.65 −1.04980
\(142\) −298.190 −0.176222
\(143\) 73.9106 0.0432218
\(144\) 1418.03 0.820622
\(145\) 40.8198 0.0233786
\(146\) 141.709 0.0803282
\(147\) 1087.39 0.610110
\(148\) −2372.75 −1.31783
\(149\) −549.400 −0.302071 −0.151036 0.988528i \(-0.548261\pi\)
−0.151036 + 0.988528i \(0.548261\pi\)
\(150\) −72.0139 −0.0391994
\(151\) 335.721 0.180931 0.0904654 0.995900i \(-0.471165\pi\)
0.0904654 + 0.995900i \(0.471165\pi\)
\(152\) 13.3241 0.00711005
\(153\) 1489.34 0.786965
\(154\) 135.307 0.0708011
\(155\) 784.947 0.406764
\(156\) 170.014 0.0872563
\(157\) −1593.69 −0.810130 −0.405065 0.914288i \(-0.632751\pi\)
−0.405065 + 0.914288i \(0.632751\pi\)
\(158\) 122.113 0.0614860
\(159\) −4270.79 −2.13016
\(160\) −378.188 −0.186865
\(161\) −317.143 −0.155245
\(162\) −327.723 −0.158941
\(163\) −2767.63 −1.32992 −0.664962 0.746877i \(-0.731552\pi\)
−0.664962 + 0.746877i \(0.731552\pi\)
\(164\) 335.000 0.159507
\(165\) 861.830 0.406627
\(166\) 56.4868 0.0264110
\(167\) 282.867 0.131071 0.0655357 0.997850i \(-0.479124\pi\)
0.0655357 + 0.997850i \(0.479124\pi\)
\(168\) 628.998 0.288859
\(169\) −2187.70 −0.995765
\(170\) −127.790 −0.0576530
\(171\) −49.0297 −0.0219263
\(172\) −1686.82 −0.747784
\(173\) −2331.63 −1.02468 −0.512342 0.858782i \(-0.671222\pi\)
−0.512342 + 0.858782i \(0.671222\pi\)
\(174\) 23.5168 0.0102460
\(175\) 344.721 0.148905
\(176\) 1456.11 0.623629
\(177\) −655.850 −0.278512
\(178\) 348.081 0.146572
\(179\) 109.140 0.0455726 0.0227863 0.999740i \(-0.492746\pi\)
0.0227863 + 0.999740i \(0.492746\pi\)
\(180\) 924.574 0.382854
\(181\) 1476.85 0.606483 0.303242 0.952914i \(-0.401931\pi\)
0.303242 + 0.952914i \(0.401931\pi\)
\(182\) 17.0317 0.00693667
\(183\) −3788.76 −1.53045
\(184\) 147.497 0.0590957
\(185\) −1514.01 −0.601686
\(186\) 452.217 0.178270
\(187\) 1529.33 0.598051
\(188\) −1936.25 −0.751147
\(189\) 333.672 0.128418
\(190\) 4.20690 0.00160632
\(191\) 2032.16 0.769853 0.384926 0.922947i \(-0.374227\pi\)
0.384926 + 0.922947i \(0.374227\pi\)
\(192\) 3201.66 1.20344
\(193\) 3883.64 1.44845 0.724224 0.689565i \(-0.242198\pi\)
0.724224 + 0.689565i \(0.242198\pi\)
\(194\) −375.588 −0.138998
\(195\) 108.482 0.0398389
\(196\) 1197.88 0.436544
\(197\) 3580.64 1.29498 0.647488 0.762076i \(-0.275820\pi\)
0.647488 + 0.762076i \(0.275820\pi\)
\(198\) 231.564 0.0831137
\(199\) −2831.17 −1.00853 −0.504263 0.863550i \(-0.668236\pi\)
−0.504263 + 0.863550i \(0.668236\pi\)
\(200\) −160.323 −0.0566826
\(201\) −215.538 −0.0756362
\(202\) 693.830 0.241672
\(203\) −112.572 −0.0389211
\(204\) 3517.86 1.20735
\(205\) 213.757 0.0728265
\(206\) −376.819 −0.127448
\(207\) −542.756 −0.182242
\(208\) 183.287 0.0610995
\(209\) −50.3463 −0.0166628
\(210\) 198.597 0.0652596
\(211\) 2903.80 0.947421 0.473711 0.880681i \(-0.342914\pi\)
0.473711 + 0.880681i \(0.342914\pi\)
\(212\) −4704.74 −1.52417
\(213\) 5237.82 1.68493
\(214\) 613.474 0.195964
\(215\) −1076.33 −0.341418
\(216\) −155.184 −0.0488839
\(217\) −2164.70 −0.677186
\(218\) 708.173 0.220016
\(219\) −2489.17 −0.768049
\(220\) 949.401 0.290948
\(221\) 192.503 0.0585935
\(222\) −872.236 −0.263697
\(223\) 454.760 0.136560 0.0682802 0.997666i \(-0.478249\pi\)
0.0682802 + 0.997666i \(0.478249\pi\)
\(224\) 1042.95 0.311095
\(225\) 589.952 0.174801
\(226\) 820.538 0.241510
\(227\) −2103.24 −0.614966 −0.307483 0.951554i \(-0.599487\pi\)
−0.307483 + 0.951554i \(0.599487\pi\)
\(228\) −115.810 −0.0336390
\(229\) −4647.97 −1.34125 −0.670625 0.741796i \(-0.733974\pi\)
−0.670625 + 0.741796i \(0.733974\pi\)
\(230\) 46.5701 0.0133510
\(231\) −2376.73 −0.676957
\(232\) 52.3548 0.0148158
\(233\) 131.118 0.0368661 0.0184331 0.999830i \(-0.494132\pi\)
0.0184331 + 0.999830i \(0.494132\pi\)
\(234\) 29.1479 0.00814299
\(235\) −1235.48 −0.342953
\(236\) −722.491 −0.199280
\(237\) −2144.96 −0.587891
\(238\) 352.414 0.0959814
\(239\) 3467.77 0.938541 0.469271 0.883054i \(-0.344517\pi\)
0.469271 + 0.883054i \(0.344517\pi\)
\(240\) 2137.21 0.574818
\(241\) 5818.91 1.55531 0.777653 0.628694i \(-0.216410\pi\)
0.777653 + 0.628694i \(0.216410\pi\)
\(242\) −301.216 −0.0800120
\(243\) 5103.22 1.34721
\(244\) −4173.73 −1.09506
\(245\) 764.341 0.199314
\(246\) 123.148 0.0319171
\(247\) −6.33731 −0.00163252
\(248\) 1006.76 0.257779
\(249\) −992.213 −0.252526
\(250\) −50.6196 −0.0128059
\(251\) 4633.30 1.16515 0.582573 0.812779i \(-0.302046\pi\)
0.582573 + 0.812779i \(0.302046\pi\)
\(252\) −2549.76 −0.637380
\(253\) −557.330 −0.138494
\(254\) −861.139 −0.212727
\(255\) 2244.67 0.551243
\(256\) 3281.94 0.801255
\(257\) −5262.78 −1.27737 −0.638683 0.769470i \(-0.720520\pi\)
−0.638683 + 0.769470i \(0.720520\pi\)
\(258\) −620.083 −0.149631
\(259\) 4175.28 1.00170
\(260\) 119.505 0.0285054
\(261\) −192.654 −0.0456896
\(262\) −605.099 −0.142684
\(263\) 1890.26 0.443189 0.221594 0.975139i \(-0.428874\pi\)
0.221594 + 0.975139i \(0.428874\pi\)
\(264\) 1105.37 0.257692
\(265\) −3002.00 −0.695892
\(266\) −11.6016 −0.00267422
\(267\) −6114.18 −1.40143
\(268\) −237.439 −0.0541189
\(269\) 7472.17 1.69363 0.846815 0.531888i \(-0.178517\pi\)
0.846815 + 0.531888i \(0.178517\pi\)
\(270\) −48.9971 −0.0110440
\(271\) 1634.38 0.366352 0.183176 0.983080i \(-0.441362\pi\)
0.183176 + 0.983080i \(0.441362\pi\)
\(272\) 3792.51 0.845421
\(273\) −299.169 −0.0663243
\(274\) 917.355 0.202261
\(275\) 605.794 0.132839
\(276\) −1282.01 −0.279593
\(277\) −590.262 −0.128034 −0.0640170 0.997949i \(-0.520391\pi\)
−0.0640170 + 0.997949i \(0.520391\pi\)
\(278\) −1181.93 −0.254992
\(279\) −3704.65 −0.794952
\(280\) 442.132 0.0943659
\(281\) 2501.14 0.530980 0.265490 0.964114i \(-0.414466\pi\)
0.265490 + 0.964114i \(0.414466\pi\)
\(282\) −711.775 −0.150303
\(283\) −803.901 −0.168859 −0.0844293 0.996429i \(-0.526907\pi\)
−0.0844293 + 0.996429i \(0.526907\pi\)
\(284\) 5770.04 1.20559
\(285\) −73.8959 −0.0153586
\(286\) 29.9306 0.00618823
\(287\) −589.491 −0.121242
\(288\) 1784.90 0.365195
\(289\) −929.797 −0.189252
\(290\) 16.5303 0.00334721
\(291\) 6597.35 1.32901
\(292\) −2742.10 −0.549552
\(293\) −6332.54 −1.26263 −0.631316 0.775526i \(-0.717485\pi\)
−0.631316 + 0.775526i \(0.717485\pi\)
\(294\) 440.345 0.0873518
\(295\) −461.007 −0.0909860
\(296\) −1941.84 −0.381307
\(297\) 586.376 0.114562
\(298\) −222.483 −0.0432487
\(299\) −70.1536 −0.0135688
\(300\) 1393.48 0.268176
\(301\) 2968.26 0.568397
\(302\) 135.952 0.0259046
\(303\) −12187.4 −2.31072
\(304\) −124.851 −0.0235550
\(305\) −2663.17 −0.499977
\(306\) 603.117 0.112673
\(307\) 7317.73 1.36041 0.680203 0.733024i \(-0.261892\pi\)
0.680203 + 0.733024i \(0.261892\pi\)
\(308\) −2618.23 −0.484374
\(309\) 6618.97 1.21858
\(310\) 317.870 0.0582381
\(311\) 2838.86 0.517611 0.258805 0.965930i \(-0.416671\pi\)
0.258805 + 0.965930i \(0.416671\pi\)
\(312\) 139.137 0.0252471
\(313\) 160.132 0.0289175 0.0144588 0.999895i \(-0.495397\pi\)
0.0144588 + 0.999895i \(0.495397\pi\)
\(314\) −645.376 −0.115989
\(315\) −1626.95 −0.291010
\(316\) −2362.91 −0.420646
\(317\) −3330.78 −0.590142 −0.295071 0.955475i \(-0.595343\pi\)
−0.295071 + 0.955475i \(0.595343\pi\)
\(318\) −1729.49 −0.304983
\(319\) −197.827 −0.0347216
\(320\) 2250.50 0.393145
\(321\) −10775.9 −1.87369
\(322\) −128.429 −0.0222270
\(323\) −131.129 −0.0225889
\(324\) 6341.52 1.08737
\(325\) 76.2539 0.0130148
\(326\) −1120.77 −0.190410
\(327\) −12439.3 −2.10366
\(328\) 274.161 0.0461524
\(329\) 3407.17 0.570953
\(330\) 349.004 0.0582184
\(331\) 7337.39 1.21843 0.609214 0.793006i \(-0.291485\pi\)
0.609214 + 0.793006i \(0.291485\pi\)
\(332\) −1093.03 −0.180687
\(333\) 7145.53 1.17589
\(334\) 114.549 0.0187660
\(335\) −151.505 −0.0247092
\(336\) −5893.93 −0.956965
\(337\) −7160.54 −1.15745 −0.578723 0.815524i \(-0.696449\pi\)
−0.578723 + 0.815524i \(0.696449\pi\)
\(338\) −885.923 −0.142568
\(339\) −14413.1 −2.30918
\(340\) 2472.76 0.394424
\(341\) −3804.13 −0.604121
\(342\) −19.8549 −0.00313928
\(343\) −6837.44 −1.07635
\(344\) −1380.48 −0.216367
\(345\) −818.022 −0.127655
\(346\) −944.209 −0.146708
\(347\) 6740.51 1.04279 0.521397 0.853314i \(-0.325411\pi\)
0.521397 + 0.853314i \(0.325411\pi\)
\(348\) −455.054 −0.0700962
\(349\) −10173.9 −1.56045 −0.780224 0.625501i \(-0.784895\pi\)
−0.780224 + 0.625501i \(0.784895\pi\)
\(350\) 139.597 0.0213194
\(351\) 73.8097 0.0112241
\(352\) 1832.83 0.277529
\(353\) −2552.87 −0.384917 −0.192458 0.981305i \(-0.561646\pi\)
−0.192458 + 0.981305i \(0.561646\pi\)
\(354\) −265.591 −0.0398757
\(355\) 3681.75 0.550442
\(356\) −6735.45 −1.00275
\(357\) −6190.28 −0.917716
\(358\) 44.1970 0.00652481
\(359\) −6677.47 −0.981681 −0.490841 0.871249i \(-0.663310\pi\)
−0.490841 + 0.871249i \(0.663310\pi\)
\(360\) 756.661 0.110776
\(361\) −6854.68 −0.999371
\(362\) 598.062 0.0868326
\(363\) 5290.98 0.765026
\(364\) −329.567 −0.0474561
\(365\) −1749.68 −0.250911
\(366\) −1534.28 −0.219121
\(367\) 11722.8 1.66738 0.833688 0.552236i \(-0.186225\pi\)
0.833688 + 0.552236i \(0.186225\pi\)
\(368\) −1382.10 −0.195779
\(369\) −1008.85 −0.142327
\(370\) −613.108 −0.0861458
\(371\) 8278.82 1.15853
\(372\) −8750.49 −1.21960
\(373\) 8070.71 1.12034 0.560168 0.828379i \(-0.310736\pi\)
0.560168 + 0.828379i \(0.310736\pi\)
\(374\) 619.313 0.0856254
\(375\) 889.154 0.122442
\(376\) −1584.61 −0.217340
\(377\) −24.9014 −0.00340182
\(378\) 135.123 0.0183861
\(379\) −6693.10 −0.907128 −0.453564 0.891224i \(-0.649848\pi\)
−0.453564 + 0.891224i \(0.649848\pi\)
\(380\) −81.4044 −0.0109894
\(381\) 15126.2 2.03397
\(382\) 822.937 0.110223
\(383\) −2502.49 −0.333867 −0.166934 0.985968i \(-0.553387\pi\)
−0.166934 + 0.985968i \(0.553387\pi\)
\(384\) 5600.76 0.744303
\(385\) −1670.64 −0.221152
\(386\) 1572.71 0.207380
\(387\) 5079.85 0.667243
\(388\) 7267.71 0.950933
\(389\) 5487.69 0.715262 0.357631 0.933863i \(-0.383585\pi\)
0.357631 + 0.933863i \(0.383585\pi\)
\(390\) 43.9307 0.00570389
\(391\) −1451.59 −0.187750
\(392\) 980.329 0.126311
\(393\) 10628.8 1.36426
\(394\) 1450.01 0.185407
\(395\) −1507.73 −0.192056
\(396\) −4480.81 −0.568609
\(397\) 7760.91 0.981130 0.490565 0.871405i \(-0.336790\pi\)
0.490565 + 0.871405i \(0.336790\pi\)
\(398\) −1146.50 −0.144395
\(399\) 203.787 0.0255693
\(400\) 1502.28 0.187785
\(401\) 14485.8 1.80395 0.901977 0.431785i \(-0.142116\pi\)
0.901977 + 0.431785i \(0.142116\pi\)
\(402\) −87.2836 −0.0108291
\(403\) −478.842 −0.0591882
\(404\) −13425.8 −1.65336
\(405\) 4046.39 0.496462
\(406\) −45.5867 −0.00557248
\(407\) 7337.41 0.893616
\(408\) 2878.98 0.349340
\(409\) 6664.83 0.805758 0.402879 0.915253i \(-0.368010\pi\)
0.402879 + 0.915253i \(0.368010\pi\)
\(410\) 86.5624 0.0104269
\(411\) −16113.7 −1.93389
\(412\) 7291.53 0.871912
\(413\) 1271.35 0.151475
\(414\) −219.793 −0.0260923
\(415\) −697.442 −0.0824966
\(416\) 230.706 0.0271906
\(417\) 20761.1 2.43807
\(418\) −20.3881 −0.00238568
\(419\) −8437.43 −0.983760 −0.491880 0.870663i \(-0.663690\pi\)
−0.491880 + 0.870663i \(0.663690\pi\)
\(420\) −3842.90 −0.446463
\(421\) 13893.2 1.60834 0.804171 0.594397i \(-0.202609\pi\)
0.804171 + 0.594397i \(0.202609\pi\)
\(422\) 1175.91 0.135646
\(423\) 5831.00 0.670243
\(424\) −3850.31 −0.441008
\(425\) 1577.82 0.180083
\(426\) 2121.09 0.241238
\(427\) 7344.41 0.832368
\(428\) −11870.9 −1.34065
\(429\) −525.743 −0.0591681
\(430\) −435.866 −0.0488822
\(431\) −10611.5 −1.18594 −0.592969 0.805225i \(-0.702044\pi\)
−0.592969 + 0.805225i \(0.702044\pi\)
\(432\) 1454.13 0.161948
\(433\) −7569.77 −0.840139 −0.420069 0.907492i \(-0.637994\pi\)
−0.420069 + 0.907492i \(0.637994\pi\)
\(434\) −876.611 −0.0969555
\(435\) −290.361 −0.0320040
\(436\) −13703.3 −1.50520
\(437\) 47.7871 0.00523105
\(438\) −1008.01 −0.109965
\(439\) −11794.9 −1.28232 −0.641162 0.767406i \(-0.721547\pi\)
−0.641162 + 0.767406i \(0.721547\pi\)
\(440\) 776.980 0.0841842
\(441\) −3607.39 −0.389525
\(442\) 77.9556 0.00838907
\(443\) 11424.0 1.22521 0.612606 0.790388i \(-0.290121\pi\)
0.612606 + 0.790388i \(0.290121\pi\)
\(444\) 16878.0 1.80404
\(445\) −4297.75 −0.457827
\(446\) 184.158 0.0195519
\(447\) 3908.01 0.413518
\(448\) −6206.34 −0.654513
\(449\) 8862.74 0.931533 0.465767 0.884908i \(-0.345779\pi\)
0.465767 + 0.884908i \(0.345779\pi\)
\(450\) 238.905 0.0250269
\(451\) −1035.94 −0.108161
\(452\) −15877.6 −1.65225
\(453\) −2388.06 −0.247684
\(454\) −851.724 −0.0880471
\(455\) −210.290 −0.0216672
\(456\) −94.7774 −0.00973324
\(457\) 5187.22 0.530958 0.265479 0.964117i \(-0.414470\pi\)
0.265479 + 0.964117i \(0.414470\pi\)
\(458\) −1882.23 −0.192032
\(459\) 1527.24 0.155306
\(460\) −901.141 −0.0913390
\(461\) −16816.7 −1.69898 −0.849491 0.527603i \(-0.823091\pi\)
−0.849491 + 0.527603i \(0.823091\pi\)
\(462\) −962.472 −0.0969226
\(463\) 6351.26 0.637512 0.318756 0.947837i \(-0.396735\pi\)
0.318756 + 0.947837i \(0.396735\pi\)
\(464\) −490.582 −0.0490834
\(465\) −5583.51 −0.556837
\(466\) 53.0970 0.00527827
\(467\) 1057.02 0.104739 0.0523693 0.998628i \(-0.483323\pi\)
0.0523693 + 0.998628i \(0.483323\pi\)
\(468\) −564.019 −0.0557089
\(469\) 417.815 0.0411363
\(470\) −500.317 −0.0491020
\(471\) 11336.3 1.10902
\(472\) −591.279 −0.0576606
\(473\) 5216.25 0.507069
\(474\) −868.618 −0.0841707
\(475\) −51.9425 −0.00501745
\(476\) −6819.28 −0.656641
\(477\) 14168.3 1.36000
\(478\) 1404.30 0.134375
\(479\) −10138.2 −0.967073 −0.483537 0.875324i \(-0.660648\pi\)
−0.483537 + 0.875324i \(0.660648\pi\)
\(480\) 2690.14 0.255807
\(481\) 923.591 0.0875512
\(482\) 2356.41 0.222679
\(483\) 2255.91 0.212521
\(484\) 5828.60 0.547389
\(485\) 4637.38 0.434170
\(486\) 2066.59 0.192885
\(487\) −8874.74 −0.825776 −0.412888 0.910782i \(-0.635480\pi\)
−0.412888 + 0.910782i \(0.635480\pi\)
\(488\) −3415.74 −0.316851
\(489\) 19686.8 1.82059
\(490\) 309.525 0.0285366
\(491\) −21452.2 −1.97174 −0.985869 0.167521i \(-0.946424\pi\)
−0.985869 + 0.167521i \(0.946424\pi\)
\(492\) −2382.93 −0.218356
\(493\) −515.249 −0.0470703
\(494\) −2.56634 −0.000233735 0
\(495\) −2859.11 −0.259611
\(496\) −9433.67 −0.854001
\(497\) −10153.4 −0.916382
\(498\) −401.804 −0.0361551
\(499\) 9696.91 0.869926 0.434963 0.900448i \(-0.356761\pi\)
0.434963 + 0.900448i \(0.356761\pi\)
\(500\) 979.501 0.0876093
\(501\) −2012.10 −0.179429
\(502\) 1876.29 0.166818
\(503\) −8892.32 −0.788249 −0.394124 0.919057i \(-0.628952\pi\)
−0.394124 + 0.919057i \(0.628952\pi\)
\(504\) −2086.69 −0.184422
\(505\) −8566.71 −0.754878
\(506\) −225.695 −0.0198288
\(507\) 15561.6 1.36315
\(508\) 16663.2 1.45534
\(509\) −1084.94 −0.0944778 −0.0472389 0.998884i \(-0.515042\pi\)
−0.0472389 + 0.998884i \(0.515042\pi\)
\(510\) 908.997 0.0789236
\(511\) 4825.20 0.417719
\(512\) 7628.02 0.658426
\(513\) −50.2776 −0.00432712
\(514\) −2131.20 −0.182885
\(515\) 4652.58 0.398091
\(516\) 11998.7 1.02367
\(517\) 5987.58 0.509349
\(518\) 1690.81 0.143417
\(519\) 16585.4 1.40273
\(520\) 97.8018 0.00824787
\(521\) 4578.35 0.384993 0.192496 0.981298i \(-0.438342\pi\)
0.192496 + 0.981298i \(0.438342\pi\)
\(522\) −78.0166 −0.00654156
\(523\) 7298.28 0.610194 0.305097 0.952321i \(-0.401311\pi\)
0.305097 + 0.952321i \(0.401311\pi\)
\(524\) 11708.8 0.976148
\(525\) −2452.08 −0.203843
\(526\) 765.476 0.0634531
\(527\) −9908.02 −0.818975
\(528\) −10357.7 −0.853712
\(529\) 529.000 0.0434783
\(530\) −1215.68 −0.0996337
\(531\) 2175.77 0.177817
\(532\) 224.494 0.0182952
\(533\) −130.398 −0.0105970
\(534\) −2475.98 −0.200648
\(535\) −7574.56 −0.612106
\(536\) −194.317 −0.0156590
\(537\) −776.338 −0.0623863
\(538\) 3025.91 0.242484
\(539\) −3704.26 −0.296018
\(540\) 948.106 0.0755555
\(541\) −14971.5 −1.18979 −0.594895 0.803803i \(-0.702806\pi\)
−0.594895 + 0.803803i \(0.702806\pi\)
\(542\) 661.854 0.0524521
\(543\) −10505.2 −0.830241
\(544\) 4773.69 0.376232
\(545\) −8743.80 −0.687235
\(546\) −121.151 −0.00949591
\(547\) −19510.9 −1.52509 −0.762547 0.646932i \(-0.776052\pi\)
−0.762547 + 0.646932i \(0.776052\pi\)
\(548\) −17751.0 −1.38373
\(549\) 12569.2 0.977119
\(550\) 245.320 0.0190191
\(551\) 16.9623 0.00131147
\(552\) −1049.18 −0.0808986
\(553\) 4157.96 0.319737
\(554\) −239.031 −0.0183311
\(555\) 10769.5 0.823674
\(556\) 22870.7 1.74448
\(557\) −13098.1 −0.996378 −0.498189 0.867068i \(-0.666001\pi\)
−0.498189 + 0.867068i \(0.666001\pi\)
\(558\) −1500.22 −0.113816
\(559\) 656.592 0.0496796
\(560\) −4142.93 −0.312626
\(561\) −10878.5 −0.818698
\(562\) 1012.85 0.0760226
\(563\) −4086.19 −0.305883 −0.152942 0.988235i \(-0.548875\pi\)
−0.152942 + 0.988235i \(0.548875\pi\)
\(564\) 13773.0 1.02828
\(565\) −10131.2 −0.754374
\(566\) −325.546 −0.0241762
\(567\) −11159.0 −0.826516
\(568\) 4722.14 0.348832
\(569\) −5021.20 −0.369946 −0.184973 0.982744i \(-0.559220\pi\)
−0.184973 + 0.982744i \(0.559220\pi\)
\(570\) −29.9247 −0.00219896
\(571\) −8277.56 −0.606664 −0.303332 0.952885i \(-0.598099\pi\)
−0.303332 + 0.952885i \(0.598099\pi\)
\(572\) −579.164 −0.0423358
\(573\) −14455.2 −1.05388
\(574\) −238.719 −0.0173588
\(575\) −575.000 −0.0417029
\(576\) −10621.5 −0.768336
\(577\) −19548.4 −1.41042 −0.705210 0.708998i \(-0.749147\pi\)
−0.705210 + 0.708998i \(0.749147\pi\)
\(578\) −376.528 −0.0270960
\(579\) −27625.2 −1.98284
\(580\) −319.865 −0.0228994
\(581\) 1923.38 0.137341
\(582\) 2671.64 0.190280
\(583\) 14548.7 1.03353
\(584\) −2244.10 −0.159010
\(585\) −359.889 −0.0254352
\(586\) −2564.41 −0.180776
\(587\) −17479.7 −1.22907 −0.614536 0.788889i \(-0.710657\pi\)
−0.614536 + 0.788889i \(0.710657\pi\)
\(588\) −8520.77 −0.597603
\(589\) 326.177 0.0228182
\(590\) −186.688 −0.0130268
\(591\) −25469.9 −1.77275
\(592\) 18195.7 1.26324
\(593\) 513.405 0.0355531 0.0177766 0.999842i \(-0.494341\pi\)
0.0177766 + 0.999842i \(0.494341\pi\)
\(594\) 237.457 0.0164023
\(595\) −4351.25 −0.299805
\(596\) 4305.10 0.295879
\(597\) 20138.8 1.38061
\(598\) −28.4092 −0.00194271
\(599\) 13706.8 0.934964 0.467482 0.884003i \(-0.345161\pi\)
0.467482 + 0.884003i \(0.345161\pi\)
\(600\) 1140.41 0.0775952
\(601\) −24403.7 −1.65632 −0.828159 0.560493i \(-0.810612\pi\)
−0.828159 + 0.560493i \(0.810612\pi\)
\(602\) 1202.02 0.0813796
\(603\) 715.045 0.0482900
\(604\) −2630.71 −0.177222
\(605\) 3719.11 0.249923
\(606\) −4935.37 −0.330835
\(607\) 16304.7 1.09026 0.545129 0.838352i \(-0.316481\pi\)
0.545129 + 0.838352i \(0.316481\pi\)
\(608\) −157.152 −0.0104825
\(609\) 800.748 0.0532807
\(610\) −1078.47 −0.0715837
\(611\) 753.683 0.0499030
\(612\) −11670.5 −0.770834
\(613\) 7754.74 0.510948 0.255474 0.966816i \(-0.417769\pi\)
0.255474 + 0.966816i \(0.417769\pi\)
\(614\) 2963.36 0.194775
\(615\) −1520.50 −0.0996952
\(616\) −2142.73 −0.140151
\(617\) −1574.90 −0.102760 −0.0513801 0.998679i \(-0.516362\pi\)
−0.0513801 + 0.998679i \(0.516362\pi\)
\(618\) 2680.40 0.174468
\(619\) −9160.25 −0.594800 −0.297400 0.954753i \(-0.596120\pi\)
−0.297400 + 0.954753i \(0.596120\pi\)
\(620\) −6150.85 −0.398426
\(621\) −556.570 −0.0359652
\(622\) 1149.62 0.0741083
\(623\) 11852.2 0.762196
\(624\) −1303.76 −0.0836416
\(625\) 625.000 0.0400000
\(626\) 64.8465 0.00414024
\(627\) 358.125 0.0228104
\(628\) 12488.2 0.793523
\(629\) 19110.6 1.21143
\(630\) −658.845 −0.0416651
\(631\) 11663.9 0.735871 0.367935 0.929851i \(-0.380065\pi\)
0.367935 + 0.929851i \(0.380065\pi\)
\(632\) −1933.78 −0.121712
\(633\) −20655.4 −1.29697
\(634\) −1348.82 −0.0844930
\(635\) 10632.5 0.664467
\(636\) 33465.9 2.08649
\(637\) −466.272 −0.0290021
\(638\) −80.1115 −0.00497123
\(639\) −17376.4 −1.07574
\(640\) 3936.86 0.243153
\(641\) −27074.5 −1.66830 −0.834148 0.551541i \(-0.814040\pi\)
−0.834148 + 0.551541i \(0.814040\pi\)
\(642\) −4363.78 −0.268263
\(643\) −4463.82 −0.273773 −0.136886 0.990587i \(-0.543710\pi\)
−0.136886 + 0.990587i \(0.543710\pi\)
\(644\) 2485.14 0.152062
\(645\) 7656.16 0.467381
\(646\) −53.1017 −0.00323415
\(647\) −11755.0 −0.714277 −0.357139 0.934051i \(-0.616248\pi\)
−0.357139 + 0.934051i \(0.616248\pi\)
\(648\) 5189.83 0.314623
\(649\) 2234.20 0.135131
\(650\) 30.8796 0.00186338
\(651\) 15398.0 0.927029
\(652\) 21687.2 1.30266
\(653\) −1236.64 −0.0741094 −0.0370547 0.999313i \(-0.511798\pi\)
−0.0370547 + 0.999313i \(0.511798\pi\)
\(654\) −5037.40 −0.301189
\(655\) 7471.15 0.445682
\(656\) −2568.98 −0.152899
\(657\) 8257.81 0.490362
\(658\) 1379.76 0.0817456
\(659\) 23646.9 1.39780 0.698901 0.715218i \(-0.253673\pi\)
0.698901 + 0.715218i \(0.253673\pi\)
\(660\) −6753.31 −0.398291
\(661\) 10150.5 0.597290 0.298645 0.954364i \(-0.403465\pi\)
0.298645 + 0.954364i \(0.403465\pi\)
\(662\) 2971.33 0.174447
\(663\) −1369.32 −0.0802112
\(664\) −894.526 −0.0522806
\(665\) 143.245 0.00835311
\(666\) 2893.63 0.168357
\(667\) 187.771 0.0109003
\(668\) −2216.55 −0.128385
\(669\) −3234.81 −0.186943
\(670\) −61.3530 −0.00353772
\(671\) 12906.7 0.742558
\(672\) −7418.77 −0.425871
\(673\) 13941.8 0.798540 0.399270 0.916833i \(-0.369264\pi\)
0.399270 + 0.916833i \(0.369264\pi\)
\(674\) −2899.71 −0.165716
\(675\) 604.967 0.0344966
\(676\) 17142.8 0.975353
\(677\) −13370.4 −0.759035 −0.379518 0.925185i \(-0.623910\pi\)
−0.379518 + 0.925185i \(0.623910\pi\)
\(678\) −5836.68 −0.330614
\(679\) −12788.8 −0.722812
\(680\) 2023.68 0.114124
\(681\) 14960.9 0.841852
\(682\) −1540.51 −0.0864943
\(683\) 15659.3 0.877288 0.438644 0.898661i \(-0.355459\pi\)
0.438644 + 0.898661i \(0.355459\pi\)
\(684\) 384.198 0.0214768
\(685\) −11326.6 −0.631775
\(686\) −2768.87 −0.154105
\(687\) 33062.1 1.83609
\(688\) 12935.5 0.716806
\(689\) 1831.31 0.101259
\(690\) −331.264 −0.0182768
\(691\) −9631.82 −0.530263 −0.265131 0.964212i \(-0.585415\pi\)
−0.265131 + 0.964212i \(0.585415\pi\)
\(692\) 18270.7 1.00368
\(693\) 7884.77 0.432204
\(694\) 2729.62 0.149301
\(695\) 14593.3 0.796484
\(696\) −372.412 −0.0202819
\(697\) −2698.15 −0.146628
\(698\) −4119.99 −0.223415
\(699\) −932.671 −0.0504676
\(700\) −2701.24 −0.145853
\(701\) 21140.9 1.13906 0.569530 0.821971i \(-0.307125\pi\)
0.569530 + 0.821971i \(0.307125\pi\)
\(702\) 29.8898 0.00160700
\(703\) −629.131 −0.0337527
\(704\) −10906.7 −0.583894
\(705\) 8788.27 0.469483
\(706\) −1033.80 −0.0551101
\(707\) 23625.0 1.25673
\(708\) 5139.25 0.272803
\(709\) −15917.4 −0.843144 −0.421572 0.906795i \(-0.638522\pi\)
−0.421572 + 0.906795i \(0.638522\pi\)
\(710\) 1490.95 0.0788089
\(711\) 7115.89 0.375340
\(712\) −5512.22 −0.290139
\(713\) 3610.76 0.189655
\(714\) −2506.80 −0.131393
\(715\) −369.553 −0.0193294
\(716\) −855.221 −0.0446384
\(717\) −24667.1 −1.28481
\(718\) −2704.09 −0.140551
\(719\) 3323.34 0.172378 0.0861889 0.996279i \(-0.472531\pi\)
0.0861889 + 0.996279i \(0.472531\pi\)
\(720\) −7090.17 −0.366993
\(721\) −12830.7 −0.662748
\(722\) −2775.85 −0.143084
\(723\) −41391.2 −2.12912
\(724\) −11572.6 −0.594051
\(725\) −204.099 −0.0104552
\(726\) 2142.62 0.109532
\(727\) −9877.52 −0.503902 −0.251951 0.967740i \(-0.581072\pi\)
−0.251951 + 0.967740i \(0.581072\pi\)
\(728\) −269.714 −0.0137312
\(729\) −14449.9 −0.734130
\(730\) −708.545 −0.0359239
\(731\) 13586.0 0.687407
\(732\) 29688.7 1.49908
\(733\) 28951.5 1.45886 0.729432 0.684053i \(-0.239784\pi\)
0.729432 + 0.684053i \(0.239784\pi\)
\(734\) 4747.24 0.238725
\(735\) −5436.93 −0.272849
\(736\) −1739.66 −0.0871262
\(737\) 734.246 0.0366978
\(738\) −408.541 −0.0203775
\(739\) −31009.5 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(740\) 11863.8 0.589353
\(741\) 45.0788 0.00223483
\(742\) 3352.57 0.165871
\(743\) −13761.0 −0.679465 −0.339733 0.940522i \(-0.610337\pi\)
−0.339733 + 0.940522i \(0.610337\pi\)
\(744\) −7161.31 −0.352885
\(745\) 2747.00 0.135090
\(746\) 3268.29 0.160403
\(747\) 3291.66 0.161226
\(748\) −11983.8 −0.585792
\(749\) 20888.9 1.01904
\(750\) 360.069 0.0175305
\(751\) 32197.4 1.56445 0.782223 0.622998i \(-0.214086\pi\)
0.782223 + 0.622998i \(0.214086\pi\)
\(752\) 14848.3 0.720029
\(753\) −32957.8 −1.59502
\(754\) −10.0840 −0.000487052 0
\(755\) −1678.60 −0.0809148
\(756\) −2614.65 −0.125786
\(757\) 26139.9 1.25505 0.627524 0.778597i \(-0.284068\pi\)
0.627524 + 0.778597i \(0.284068\pi\)
\(758\) −2710.42 −0.129877
\(759\) 3964.42 0.189591
\(760\) −66.6205 −0.00317971
\(761\) −24004.0 −1.14342 −0.571712 0.820454i \(-0.693721\pi\)
−0.571712 + 0.820454i \(0.693721\pi\)
\(762\) 6125.48 0.291211
\(763\) 24113.3 1.14412
\(764\) −15924.0 −0.754072
\(765\) −7446.68 −0.351942
\(766\) −1013.40 −0.0478011
\(767\) 281.229 0.0132393
\(768\) −23345.2 −1.09687
\(769\) −33733.4 −1.58187 −0.790934 0.611902i \(-0.790405\pi\)
−0.790934 + 0.611902i \(0.790405\pi\)
\(770\) −676.537 −0.0316632
\(771\) 37435.3 1.74864
\(772\) −30432.2 −1.41876
\(773\) 40247.1 1.87269 0.936344 0.351083i \(-0.114186\pi\)
0.936344 + 0.351083i \(0.114186\pi\)
\(774\) 2057.12 0.0955318
\(775\) −3924.74 −0.181911
\(776\) 5947.81 0.275147
\(777\) −29699.7 −1.37126
\(778\) 2222.28 0.102407
\(779\) 88.8246 0.00408533
\(780\) −850.069 −0.0390222
\(781\) −17843.0 −0.817508
\(782\) −587.832 −0.0268808
\(783\) −197.557 −0.00901676
\(784\) −9186.02 −0.418459
\(785\) 7968.45 0.362301
\(786\) 4304.21 0.195326
\(787\) −16327.3 −0.739522 −0.369761 0.929127i \(-0.620560\pi\)
−0.369761 + 0.929127i \(0.620560\pi\)
\(788\) −28057.9 −1.26843
\(789\) −13445.9 −0.606700
\(790\) −610.565 −0.0274974
\(791\) 27939.4 1.25589
\(792\) −3667.04 −0.164524
\(793\) 1624.62 0.0727515
\(794\) 3142.83 0.140472
\(795\) 21353.9 0.952636
\(796\) 22185.1 0.987852
\(797\) −29358.8 −1.30482 −0.652410 0.757866i \(-0.726242\pi\)
−0.652410 + 0.757866i \(0.726242\pi\)
\(798\) 82.5252 0.00366085
\(799\) 15594.9 0.690498
\(800\) 1890.94 0.0835685
\(801\) 20283.7 0.894745
\(802\) 5866.12 0.258279
\(803\) 8479.56 0.372649
\(804\) 1688.96 0.0740857
\(805\) 1585.72 0.0694275
\(806\) −193.911 −0.00847420
\(807\) −53151.3 −2.31848
\(808\) −10987.5 −0.478389
\(809\) −2426.36 −0.105447 −0.0527234 0.998609i \(-0.516790\pi\)
−0.0527234 + 0.998609i \(0.516790\pi\)
\(810\) 1638.62 0.0710804
\(811\) −31170.9 −1.34964 −0.674819 0.737983i \(-0.735778\pi\)
−0.674819 + 0.737983i \(0.735778\pi\)
\(812\) 882.112 0.0381232
\(813\) −11625.7 −0.501515
\(814\) 2971.34 0.127943
\(815\) 13838.2 0.594760
\(816\) −26977.0 −1.15733
\(817\) −447.257 −0.0191524
\(818\) 2698.97 0.115363
\(819\) 992.490 0.0423448
\(820\) −1675.00 −0.0713336
\(821\) −6679.93 −0.283960 −0.141980 0.989870i \(-0.545347\pi\)
−0.141980 + 0.989870i \(0.545347\pi\)
\(822\) −6525.36 −0.276883
\(823\) 29299.3 1.24096 0.620480 0.784222i \(-0.286938\pi\)
0.620480 + 0.784222i \(0.286938\pi\)
\(824\) 5967.31 0.252283
\(825\) −4309.15 −0.181849
\(826\) 514.842 0.0216872
\(827\) 39806.7 1.67378 0.836889 0.547372i \(-0.184372\pi\)
0.836889 + 0.547372i \(0.184372\pi\)
\(828\) 4253.04 0.178506
\(829\) −17854.1 −0.748007 −0.374004 0.927427i \(-0.622015\pi\)
−0.374004 + 0.927427i \(0.622015\pi\)
\(830\) −282.434 −0.0118114
\(831\) 4198.67 0.175271
\(832\) −1372.87 −0.0572065
\(833\) −9647.91 −0.401297
\(834\) 8407.37 0.349069
\(835\) −1414.34 −0.0586169
\(836\) 394.514 0.0163212
\(837\) −3798.94 −0.156882
\(838\) −3416.80 −0.140849
\(839\) −6656.92 −0.273924 −0.136962 0.990576i \(-0.543734\pi\)
−0.136962 + 0.990576i \(0.543734\pi\)
\(840\) −3144.99 −0.129181
\(841\) −24322.3 −0.997267
\(842\) 5626.14 0.230273
\(843\) −17791.2 −0.726881
\(844\) −22754.2 −0.928000
\(845\) 10938.5 0.445320
\(846\) 2361.31 0.0959614
\(847\) −10256.4 −0.416075
\(848\) 36078.7 1.46102
\(849\) 5718.34 0.231158
\(850\) 638.948 0.0257832
\(851\) −6964.43 −0.280538
\(852\) −41043.6 −1.65039
\(853\) −30008.7 −1.20455 −0.602273 0.798290i \(-0.705738\pi\)
−0.602273 + 0.798290i \(0.705738\pi\)
\(854\) 2974.17 0.119173
\(855\) 245.149 0.00980574
\(856\) −9714.98 −0.387910
\(857\) 24281.5 0.967843 0.483922 0.875111i \(-0.339212\pi\)
0.483922 + 0.875111i \(0.339212\pi\)
\(858\) −212.903 −0.00847133
\(859\) 30635.5 1.21685 0.608423 0.793613i \(-0.291802\pi\)
0.608423 + 0.793613i \(0.291802\pi\)
\(860\) 8434.10 0.334419
\(861\) 4193.19 0.165974
\(862\) −4297.22 −0.169796
\(863\) −26572.1 −1.04812 −0.524058 0.851683i \(-0.675582\pi\)
−0.524058 + 0.851683i \(0.675582\pi\)
\(864\) 1830.33 0.0720707
\(865\) 11658.1 0.458253
\(866\) −3065.43 −0.120286
\(867\) 6613.87 0.259076
\(868\) 16962.6 0.663305
\(869\) 7306.97 0.285238
\(870\) −117.584 −0.00458214
\(871\) 92.4227 0.00359543
\(872\) −11214.6 −0.435522
\(873\) −21886.6 −0.848511
\(874\) 19.3517 0.000748950 0
\(875\) −1723.60 −0.0665925
\(876\) 19505.2 0.752305
\(877\) −37158.4 −1.43073 −0.715366 0.698750i \(-0.753740\pi\)
−0.715366 + 0.698750i \(0.753740\pi\)
\(878\) −4776.43 −0.183595
\(879\) 45044.9 1.72847
\(880\) −7280.57 −0.278895
\(881\) −3353.12 −0.128229 −0.0641143 0.997943i \(-0.520422\pi\)
−0.0641143 + 0.997943i \(0.520422\pi\)
\(882\) −1460.84 −0.0557699
\(883\) 16292.5 0.620936 0.310468 0.950584i \(-0.399514\pi\)
0.310468 + 0.950584i \(0.399514\pi\)
\(884\) −1508.46 −0.0573924
\(885\) 3279.25 0.124555
\(886\) 4626.22 0.175418
\(887\) −5949.82 −0.225226 −0.112613 0.993639i \(-0.535922\pi\)
−0.112613 + 0.993639i \(0.535922\pi\)
\(888\) 13812.7 0.521988
\(889\) −29321.9 −1.10621
\(890\) −1740.41 −0.0655489
\(891\) −19610.2 −0.737338
\(892\) −3563.50 −0.133761
\(893\) −513.393 −0.0192386
\(894\) 1582.58 0.0592050
\(895\) −545.700 −0.0203807
\(896\) −10856.9 −0.404804
\(897\) 499.019 0.0185750
\(898\) 3589.03 0.133371
\(899\) 1281.66 0.0475480
\(900\) −4622.87 −0.171217
\(901\) 37892.8 1.40110
\(902\) −419.512 −0.0154858
\(903\) −21113.9 −0.778102
\(904\) −12994.0 −0.478070
\(905\) −7384.26 −0.271228
\(906\) −967.062 −0.0354619
\(907\) −44105.2 −1.61465 −0.807326 0.590106i \(-0.799086\pi\)
−0.807326 + 0.590106i \(0.799086\pi\)
\(908\) 16481.0 0.602360
\(909\) 40431.6 1.47528
\(910\) −85.1586 −0.00310218
\(911\) 4385.18 0.159481 0.0797407 0.996816i \(-0.474591\pi\)
0.0797407 + 0.996816i \(0.474591\pi\)
\(912\) 888.097 0.0322454
\(913\) 3380.05 0.122523
\(914\) 2100.60 0.0760194
\(915\) 18943.8 0.684439
\(916\) 36421.5 1.31376
\(917\) −20603.7 −0.741978
\(918\) 618.468 0.0222358
\(919\) 30027.0 1.07780 0.538901 0.842369i \(-0.318840\pi\)
0.538901 + 0.842369i \(0.318840\pi\)
\(920\) −737.484 −0.0264284
\(921\) −52052.7 −1.86232
\(922\) −6810.03 −0.243250
\(923\) −2245.98 −0.0800946
\(924\) 18624.1 0.663080
\(925\) 7570.03 0.269082
\(926\) 2571.99 0.0912751
\(927\) −21958.4 −0.778002
\(928\) −617.503 −0.0218432
\(929\) −5457.52 −0.192740 −0.0963700 0.995346i \(-0.530723\pi\)
−0.0963700 + 0.995346i \(0.530723\pi\)
\(930\) −2261.08 −0.0797246
\(931\) 317.614 0.0111809
\(932\) −1027.44 −0.0361104
\(933\) −20193.5 −0.708579
\(934\) 428.046 0.0149958
\(935\) −7646.65 −0.267457
\(936\) −461.587 −0.0161191
\(937\) 3039.15 0.105960 0.0529802 0.998596i \(-0.483128\pi\)
0.0529802 + 0.998596i \(0.483128\pi\)
\(938\) 169.197 0.00588964
\(939\) −1139.05 −0.0395864
\(940\) 9681.25 0.335923
\(941\) 25791.0 0.893479 0.446740 0.894664i \(-0.352585\pi\)
0.446740 + 0.894664i \(0.352585\pi\)
\(942\) 4590.71 0.158783
\(943\) 983.282 0.0339555
\(944\) 5540.48 0.191025
\(945\) −1668.36 −0.0574304
\(946\) 2112.36 0.0725991
\(947\) 18339.3 0.629299 0.314650 0.949208i \(-0.398113\pi\)
0.314650 + 0.949208i \(0.398113\pi\)
\(948\) 16807.9 0.575840
\(949\) 1067.36 0.0365099
\(950\) −21.0345 −0.000718368 0
\(951\) 23692.6 0.807871
\(952\) −5580.83 −0.189995
\(953\) 12658.7 0.430278 0.215139 0.976583i \(-0.430979\pi\)
0.215139 + 0.976583i \(0.430979\pi\)
\(954\) 5737.55 0.194717
\(955\) −10160.8 −0.344289
\(956\) −27173.5 −0.919302
\(957\) 1407.19 0.0475319
\(958\) −4105.55 −0.138460
\(959\) 31236.0 1.05179
\(960\) −16008.3 −0.538193
\(961\) −5145.31 −0.172714
\(962\) 374.015 0.0125351
\(963\) 35749.0 1.19626
\(964\) −45597.0 −1.52342
\(965\) −19418.2 −0.647765
\(966\) 913.548 0.0304275
\(967\) 50470.3 1.67840 0.839201 0.543822i \(-0.183023\pi\)
0.839201 + 0.543822i \(0.183023\pi\)
\(968\) 4770.06 0.158384
\(969\) 932.752 0.0309229
\(970\) 1877.94 0.0621618
\(971\) 30778.2 1.01722 0.508610 0.860997i \(-0.330160\pi\)
0.508610 + 0.860997i \(0.330160\pi\)
\(972\) −39988.9 −1.31959
\(973\) −40245.0 −1.32600
\(974\) −3593.89 −0.118230
\(975\) −542.412 −0.0178165
\(976\) 32006.6 1.04970
\(977\) −23575.5 −0.772004 −0.386002 0.922498i \(-0.626144\pi\)
−0.386002 + 0.922498i \(0.626144\pi\)
\(978\) 7972.31 0.260661
\(979\) 20828.4 0.679958
\(980\) −5989.38 −0.195228
\(981\) 41267.4 1.34308
\(982\) −8687.21 −0.282301
\(983\) −44798.3 −1.45355 −0.726777 0.686874i \(-0.758982\pi\)
−0.726777 + 0.686874i \(0.758982\pi\)
\(984\) −1950.17 −0.0631799
\(985\) −17903.2 −0.579131
\(986\) −208.654 −0.00673924
\(987\) −24236.0 −0.781601
\(988\) 49.6592 0.00159906
\(989\) −4951.10 −0.159187
\(990\) −1157.82 −0.0371696
\(991\) −12153.5 −0.389574 −0.194787 0.980846i \(-0.562402\pi\)
−0.194787 + 0.980846i \(0.562402\pi\)
\(992\) −11874.3 −0.380050
\(993\) −52192.6 −1.66796
\(994\) −4111.69 −0.131202
\(995\) 14155.9 0.451026
\(996\) 7774.99 0.247350
\(997\) 36538.1 1.16065 0.580327 0.814383i \(-0.302925\pi\)
0.580327 + 0.814383i \(0.302925\pi\)
\(998\) 3926.83 0.124551
\(999\) 7327.40 0.232061
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 115.4.a.e.1.3 5
3.2 odd 2 1035.4.a.k.1.3 5
4.3 odd 2 1840.4.a.n.1.5 5
5.2 odd 4 575.4.b.i.24.6 10
5.3 odd 4 575.4.b.i.24.5 10
5.4 even 2 575.4.a.j.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.3 5 1.1 even 1 trivial
575.4.a.j.1.3 5 5.4 even 2
575.4.b.i.24.5 10 5.3 odd 4
575.4.b.i.24.6 10 5.2 odd 4
1035.4.a.k.1.3 5 3.2 odd 2
1840.4.a.n.1.5 5 4.3 odd 2