Properties

Label 1305.4.a.h.1.1
Level $1305$
Weight $4$
Character 1305.1
Self dual yes
Analytic conductor $76.997$
Analytic rank $0$
Dimension $6$
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] = [1305,4,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(76.9974925575\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.42717\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.42717 q^{2} +21.4542 q^{4} +5.00000 q^{5} -10.2400 q^{7} -73.0180 q^{8} -27.1358 q^{10} -12.6318 q^{11} +84.2342 q^{13} +55.5740 q^{14} +224.647 q^{16} -6.89181 q^{17} -79.7045 q^{19} +107.271 q^{20} +68.5548 q^{22} +73.8181 q^{23} +25.0000 q^{25} -457.153 q^{26} -219.690 q^{28} -29.0000 q^{29} +0.254654 q^{31} -635.056 q^{32} +37.4030 q^{34} -51.1998 q^{35} +40.4065 q^{37} +432.570 q^{38} -365.090 q^{40} -208.354 q^{41} -194.427 q^{43} -271.004 q^{44} -400.623 q^{46} +522.168 q^{47} -238.143 q^{49} -135.679 q^{50} +1807.17 q^{52} +294.893 q^{53} -63.1589 q^{55} +747.701 q^{56} +157.388 q^{58} +196.229 q^{59} +189.313 q^{61} -1.38205 q^{62} +1649.38 q^{64} +421.171 q^{65} -570.840 q^{67} -147.858 q^{68} +277.870 q^{70} +1126.87 q^{71} +581.431 q^{73} -219.293 q^{74} -1709.99 q^{76} +129.349 q^{77} -255.631 q^{79} +1123.24 q^{80} +1130.77 q^{82} +983.081 q^{83} -34.4591 q^{85} +1055.19 q^{86} +922.347 q^{88} -678.642 q^{89} -862.555 q^{91} +1583.70 q^{92} -2833.89 q^{94} -398.523 q^{95} -30.8442 q^{97} +1292.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26}+ \cdots + 1433 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.42717 −1.91879 −0.959397 0.282060i \(-0.908982\pi\)
−0.959397 + 0.282060i \(0.908982\pi\)
\(3\) 0 0
\(4\) 21.4542 2.68177
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −10.2400 −0.552906 −0.276453 0.961027i \(-0.589159\pi\)
−0.276453 + 0.961027i \(0.589159\pi\)
\(8\) −73.0180 −3.22697
\(9\) 0 0
\(10\) −27.1358 −0.858111
\(11\) −12.6318 −0.346239 −0.173119 0.984901i \(-0.555385\pi\)
−0.173119 + 0.984901i \(0.555385\pi\)
\(12\) 0 0
\(13\) 84.2342 1.79710 0.898552 0.438866i \(-0.144620\pi\)
0.898552 + 0.438866i \(0.144620\pi\)
\(14\) 55.5740 1.06091
\(15\) 0 0
\(16\) 224.647 3.51012
\(17\) −6.89181 −0.0983241 −0.0491621 0.998791i \(-0.515655\pi\)
−0.0491621 + 0.998791i \(0.515655\pi\)
\(18\) 0 0
\(19\) −79.7045 −0.962393 −0.481196 0.876613i \(-0.659798\pi\)
−0.481196 + 0.876613i \(0.659798\pi\)
\(20\) 107.271 1.19932
\(21\) 0 0
\(22\) 68.5548 0.664361
\(23\) 73.8181 0.669223 0.334612 0.942356i \(-0.391395\pi\)
0.334612 + 0.942356i \(0.391395\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −457.153 −3.44827
\(27\) 0 0
\(28\) −219.690 −1.48277
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 0.254654 0.00147540 0.000737698 1.00000i \(-0.499765\pi\)
0.000737698 1.00000i \(0.499765\pi\)
\(32\) −635.056 −3.50822
\(33\) 0 0
\(34\) 37.4030 0.188664
\(35\) −51.1998 −0.247267
\(36\) 0 0
\(37\) 40.4065 0.179535 0.0897674 0.995963i \(-0.471388\pi\)
0.0897674 + 0.995963i \(0.471388\pi\)
\(38\) 432.570 1.84663
\(39\) 0 0
\(40\) −365.090 −1.44314
\(41\) −208.354 −0.793645 −0.396822 0.917895i \(-0.629887\pi\)
−0.396822 + 0.917895i \(0.629887\pi\)
\(42\) 0 0
\(43\) −194.427 −0.689531 −0.344766 0.938689i \(-0.612042\pi\)
−0.344766 + 0.938689i \(0.612042\pi\)
\(44\) −271.004 −0.928532
\(45\) 0 0
\(46\) −400.623 −1.28410
\(47\) 522.168 1.62055 0.810277 0.586047i \(-0.199317\pi\)
0.810277 + 0.586047i \(0.199317\pi\)
\(48\) 0 0
\(49\) −238.143 −0.694295
\(50\) −135.679 −0.383759
\(51\) 0 0
\(52\) 1807.17 4.81942
\(53\) 294.893 0.764276 0.382138 0.924105i \(-0.375188\pi\)
0.382138 + 0.924105i \(0.375188\pi\)
\(54\) 0 0
\(55\) −63.1589 −0.154843
\(56\) 747.701 1.78421
\(57\) 0 0
\(58\) 157.388 0.356311
\(59\) 196.229 0.432997 0.216498 0.976283i \(-0.430536\pi\)
0.216498 + 0.976283i \(0.430536\pi\)
\(60\) 0 0
\(61\) 189.313 0.397360 0.198680 0.980064i \(-0.436334\pi\)
0.198680 + 0.980064i \(0.436334\pi\)
\(62\) −1.38205 −0.00283098
\(63\) 0 0
\(64\) 1649.38 3.22144
\(65\) 421.171 0.803690
\(66\) 0 0
\(67\) −570.840 −1.04088 −0.520442 0.853897i \(-0.674233\pi\)
−0.520442 + 0.853897i \(0.674233\pi\)
\(68\) −147.858 −0.263683
\(69\) 0 0
\(70\) 277.870 0.474454
\(71\) 1126.87 1.88358 0.941792 0.336196i \(-0.109141\pi\)
0.941792 + 0.336196i \(0.109141\pi\)
\(72\) 0 0
\(73\) 581.431 0.932210 0.466105 0.884729i \(-0.345657\pi\)
0.466105 + 0.884729i \(0.345657\pi\)
\(74\) −219.293 −0.344490
\(75\) 0 0
\(76\) −1709.99 −2.58092
\(77\) 129.349 0.191437
\(78\) 0 0
\(79\) −255.631 −0.364060 −0.182030 0.983293i \(-0.558267\pi\)
−0.182030 + 0.983293i \(0.558267\pi\)
\(80\) 1123.24 1.56977
\(81\) 0 0
\(82\) 1130.77 1.52284
\(83\) 983.081 1.30009 0.650043 0.759897i \(-0.274751\pi\)
0.650043 + 0.759897i \(0.274751\pi\)
\(84\) 0 0
\(85\) −34.4591 −0.0439719
\(86\) 1055.19 1.32307
\(87\) 0 0
\(88\) 922.347 1.11730
\(89\) −678.642 −0.808268 −0.404134 0.914700i \(-0.632427\pi\)
−0.404134 + 0.914700i \(0.632427\pi\)
\(90\) 0 0
\(91\) −862.555 −0.993630
\(92\) 1583.70 1.79470
\(93\) 0 0
\(94\) −2833.89 −3.10951
\(95\) −398.523 −0.430395
\(96\) 0 0
\(97\) −30.8442 −0.0322861 −0.0161431 0.999870i \(-0.505139\pi\)
−0.0161431 + 0.999870i \(0.505139\pi\)
\(98\) 1292.44 1.33221
\(99\) 0 0
\(100\) 536.354 0.536354
\(101\) −1749.78 −1.72386 −0.861930 0.507028i \(-0.830744\pi\)
−0.861930 + 0.507028i \(0.830744\pi\)
\(102\) 0 0
\(103\) 1912.26 1.82933 0.914663 0.404217i \(-0.132456\pi\)
0.914663 + 0.404217i \(0.132456\pi\)
\(104\) −6150.61 −5.79920
\(105\) 0 0
\(106\) −1600.43 −1.46649
\(107\) −1719.25 −1.55333 −0.776663 0.629916i \(-0.783089\pi\)
−0.776663 + 0.629916i \(0.783089\pi\)
\(108\) 0 0
\(109\) −779.425 −0.684911 −0.342456 0.939534i \(-0.611259\pi\)
−0.342456 + 0.939534i \(0.611259\pi\)
\(110\) 342.774 0.297111
\(111\) 0 0
\(112\) −2300.38 −1.94076
\(113\) −766.529 −0.638132 −0.319066 0.947732i \(-0.603369\pi\)
−0.319066 + 0.947732i \(0.603369\pi\)
\(114\) 0 0
\(115\) 369.090 0.299286
\(116\) −622.170 −0.497992
\(117\) 0 0
\(118\) −1064.97 −0.830831
\(119\) 70.5719 0.0543640
\(120\) 0 0
\(121\) −1171.44 −0.880119
\(122\) −1027.43 −0.762453
\(123\) 0 0
\(124\) 5.46339 0.00395667
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1363.54 −0.952715 −0.476358 0.879252i \(-0.658043\pi\)
−0.476358 + 0.879252i \(0.658043\pi\)
\(128\) −3870.99 −2.67305
\(129\) 0 0
\(130\) −2285.77 −1.54211
\(131\) 1346.83 0.898269 0.449134 0.893464i \(-0.351732\pi\)
0.449134 + 0.893464i \(0.351732\pi\)
\(132\) 0 0
\(133\) 816.171 0.532113
\(134\) 3098.05 1.99724
\(135\) 0 0
\(136\) 503.226 0.317289
\(137\) 2508.42 1.56430 0.782148 0.623093i \(-0.214124\pi\)
0.782148 + 0.623093i \(0.214124\pi\)
\(138\) 0 0
\(139\) 1363.95 0.832292 0.416146 0.909298i \(-0.363380\pi\)
0.416146 + 0.909298i \(0.363380\pi\)
\(140\) −1098.45 −0.663113
\(141\) 0 0
\(142\) −6115.69 −3.61421
\(143\) −1064.03 −0.622227
\(144\) 0 0
\(145\) −145.000 −0.0830455
\(146\) −3155.52 −1.78872
\(147\) 0 0
\(148\) 866.887 0.481471
\(149\) −2792.25 −1.53523 −0.767617 0.640909i \(-0.778558\pi\)
−0.767617 + 0.640909i \(0.778558\pi\)
\(150\) 0 0
\(151\) 306.493 0.165179 0.0825897 0.996584i \(-0.473681\pi\)
0.0825897 + 0.996584i \(0.473681\pi\)
\(152\) 5819.86 3.10561
\(153\) 0 0
\(154\) −701.999 −0.367329
\(155\) 1.27327 0.000659817 0
\(156\) 0 0
\(157\) −1828.15 −0.929315 −0.464657 0.885491i \(-0.653822\pi\)
−0.464657 + 0.885491i \(0.653822\pi\)
\(158\) 1387.35 0.698556
\(159\) 0 0
\(160\) −3175.28 −1.56892
\(161\) −755.894 −0.370018
\(162\) 0 0
\(163\) 2101.15 1.00966 0.504830 0.863219i \(-0.331555\pi\)
0.504830 + 0.863219i \(0.331555\pi\)
\(164\) −4470.06 −2.12837
\(165\) 0 0
\(166\) −5335.35 −2.49460
\(167\) −1564.53 −0.724950 −0.362475 0.931994i \(-0.618068\pi\)
−0.362475 + 0.931994i \(0.618068\pi\)
\(168\) 0 0
\(169\) 4898.40 2.22959
\(170\) 187.015 0.0843730
\(171\) 0 0
\(172\) −4171.27 −1.84916
\(173\) 2695.23 1.18448 0.592239 0.805762i \(-0.298244\pi\)
0.592239 + 0.805762i \(0.298244\pi\)
\(174\) 0 0
\(175\) −255.999 −0.110581
\(176\) −2837.70 −1.21534
\(177\) 0 0
\(178\) 3683.10 1.55090
\(179\) 3604.16 1.50496 0.752480 0.658615i \(-0.228857\pi\)
0.752480 + 0.658615i \(0.228857\pi\)
\(180\) 0 0
\(181\) −2658.32 −1.09166 −0.545832 0.837894i \(-0.683786\pi\)
−0.545832 + 0.837894i \(0.683786\pi\)
\(182\) 4681.23 1.90657
\(183\) 0 0
\(184\) −5390.05 −2.15956
\(185\) 202.033 0.0802904
\(186\) 0 0
\(187\) 87.0559 0.0340436
\(188\) 11202.7 4.34595
\(189\) 0 0
\(190\) 2162.85 0.825840
\(191\) 145.942 0.0552877 0.0276439 0.999618i \(-0.491200\pi\)
0.0276439 + 0.999618i \(0.491200\pi\)
\(192\) 0 0
\(193\) 1151.46 0.429451 0.214725 0.976674i \(-0.431114\pi\)
0.214725 + 0.976674i \(0.431114\pi\)
\(194\) 167.397 0.0619504
\(195\) 0 0
\(196\) −5109.16 −1.86194
\(197\) −4328.08 −1.56529 −0.782647 0.622466i \(-0.786131\pi\)
−0.782647 + 0.622466i \(0.786131\pi\)
\(198\) 0 0
\(199\) 1532.72 0.545988 0.272994 0.962016i \(-0.411986\pi\)
0.272994 + 0.962016i \(0.411986\pi\)
\(200\) −1825.45 −0.645394
\(201\) 0 0
\(202\) 9496.36 3.30773
\(203\) 296.959 0.102672
\(204\) 0 0
\(205\) −1041.77 −0.354929
\(206\) −10378.2 −3.51010
\(207\) 0 0
\(208\) 18923.0 6.30805
\(209\) 1006.81 0.333218
\(210\) 0 0
\(211\) 5345.39 1.74404 0.872019 0.489472i \(-0.162810\pi\)
0.872019 + 0.489472i \(0.162810\pi\)
\(212\) 6326.67 2.04961
\(213\) 0 0
\(214\) 9330.65 2.98051
\(215\) −972.135 −0.308368
\(216\) 0 0
\(217\) −2.60765 −0.000815755 0
\(218\) 4230.07 1.31420
\(219\) 0 0
\(220\) −1355.02 −0.415252
\(221\) −580.526 −0.176699
\(222\) 0 0
\(223\) 3783.05 1.13602 0.568008 0.823023i \(-0.307714\pi\)
0.568008 + 0.823023i \(0.307714\pi\)
\(224\) 6502.95 1.93972
\(225\) 0 0
\(226\) 4160.08 1.22444
\(227\) 4348.46 1.27144 0.635721 0.771919i \(-0.280703\pi\)
0.635721 + 0.771919i \(0.280703\pi\)
\(228\) 0 0
\(229\) −3068.67 −0.885517 −0.442759 0.896641i \(-0.646000\pi\)
−0.442759 + 0.896641i \(0.646000\pi\)
\(230\) −2003.12 −0.574268
\(231\) 0 0
\(232\) 2117.52 0.599233
\(233\) 275.696 0.0775170 0.0387585 0.999249i \(-0.487660\pi\)
0.0387585 + 0.999249i \(0.487660\pi\)
\(234\) 0 0
\(235\) 2610.84 0.724734
\(236\) 4209.92 1.16120
\(237\) 0 0
\(238\) −383.006 −0.104313
\(239\) −3132.87 −0.847901 −0.423950 0.905685i \(-0.639357\pi\)
−0.423950 + 0.905685i \(0.639357\pi\)
\(240\) 0 0
\(241\) 5699.61 1.52342 0.761710 0.647918i \(-0.224360\pi\)
0.761710 + 0.647918i \(0.224360\pi\)
\(242\) 6357.59 1.68877
\(243\) 0 0
\(244\) 4061.54 1.06563
\(245\) −1190.72 −0.310498
\(246\) 0 0
\(247\) −6713.85 −1.72952
\(248\) −18.5943 −0.00476105
\(249\) 0 0
\(250\) −678.396 −0.171622
\(251\) −1714.12 −0.431054 −0.215527 0.976498i \(-0.569147\pi\)
−0.215527 + 0.976498i \(0.569147\pi\)
\(252\) 0 0
\(253\) −932.454 −0.231711
\(254\) 7400.17 1.82806
\(255\) 0 0
\(256\) 7813.52 1.90760
\(257\) 803.085 0.194922 0.0974612 0.995239i \(-0.468928\pi\)
0.0974612 + 0.995239i \(0.468928\pi\)
\(258\) 0 0
\(259\) −413.761 −0.0992659
\(260\) 9035.87 2.15531
\(261\) 0 0
\(262\) −7309.48 −1.72359
\(263\) 1694.24 0.397230 0.198615 0.980078i \(-0.436356\pi\)
0.198615 + 0.980078i \(0.436356\pi\)
\(264\) 0 0
\(265\) 1474.46 0.341794
\(266\) −4429.50 −1.02101
\(267\) 0 0
\(268\) −12246.9 −2.79141
\(269\) −3364.95 −0.762693 −0.381347 0.924432i \(-0.624540\pi\)
−0.381347 + 0.924432i \(0.624540\pi\)
\(270\) 0 0
\(271\) 8044.10 1.80312 0.901558 0.432658i \(-0.142424\pi\)
0.901558 + 0.432658i \(0.142424\pi\)
\(272\) −1548.23 −0.345129
\(273\) 0 0
\(274\) −13613.6 −3.00156
\(275\) −315.795 −0.0692477
\(276\) 0 0
\(277\) −6971.54 −1.51220 −0.756100 0.654456i \(-0.772898\pi\)
−0.756100 + 0.654456i \(0.772898\pi\)
\(278\) −7402.38 −1.59700
\(279\) 0 0
\(280\) 3738.51 0.797923
\(281\) 1916.94 0.406957 0.203478 0.979079i \(-0.434775\pi\)
0.203478 + 0.979079i \(0.434775\pi\)
\(282\) 0 0
\(283\) 2036.63 0.427792 0.213896 0.976856i \(-0.431385\pi\)
0.213896 + 0.976856i \(0.431385\pi\)
\(284\) 24176.0 5.05134
\(285\) 0 0
\(286\) 5774.66 1.19393
\(287\) 2133.54 0.438811
\(288\) 0 0
\(289\) −4865.50 −0.990332
\(290\) 786.939 0.159347
\(291\) 0 0
\(292\) 12474.1 2.49997
\(293\) 1754.55 0.349836 0.174918 0.984583i \(-0.444034\pi\)
0.174918 + 0.984583i \(0.444034\pi\)
\(294\) 0 0
\(295\) 981.143 0.193642
\(296\) −2950.40 −0.579353
\(297\) 0 0
\(298\) 15154.0 2.94580
\(299\) 6218.01 1.20266
\(300\) 0 0
\(301\) 1990.92 0.381246
\(302\) −1663.39 −0.316945
\(303\) 0 0
\(304\) −17905.4 −3.37811
\(305\) 946.563 0.177705
\(306\) 0 0
\(307\) 8909.41 1.65631 0.828154 0.560500i \(-0.189391\pi\)
0.828154 + 0.560500i \(0.189391\pi\)
\(308\) 2775.07 0.513391
\(309\) 0 0
\(310\) −6.91026 −0.00126605
\(311\) 7422.88 1.35342 0.676709 0.736251i \(-0.263405\pi\)
0.676709 + 0.736251i \(0.263405\pi\)
\(312\) 0 0
\(313\) −1453.62 −0.262504 −0.131252 0.991349i \(-0.541900\pi\)
−0.131252 + 0.991349i \(0.541900\pi\)
\(314\) 9921.69 1.78316
\(315\) 0 0
\(316\) −5484.35 −0.976326
\(317\) −5203.30 −0.921912 −0.460956 0.887423i \(-0.652493\pi\)
−0.460956 + 0.887423i \(0.652493\pi\)
\(318\) 0 0
\(319\) 366.322 0.0642949
\(320\) 8246.88 1.44067
\(321\) 0 0
\(322\) 4102.37 0.709987
\(323\) 549.309 0.0946264
\(324\) 0 0
\(325\) 2105.86 0.359421
\(326\) −11403.3 −1.93733
\(327\) 0 0
\(328\) 15213.6 2.56107
\(329\) −5346.98 −0.896014
\(330\) 0 0
\(331\) −10985.0 −1.82414 −0.912070 0.410035i \(-0.865516\pi\)
−0.912070 + 0.410035i \(0.865516\pi\)
\(332\) 21091.2 3.48653
\(333\) 0 0
\(334\) 8490.94 1.39103
\(335\) −2854.20 −0.465498
\(336\) 0 0
\(337\) 2295.27 0.371013 0.185507 0.982643i \(-0.440607\pi\)
0.185507 + 0.982643i \(0.440607\pi\)
\(338\) −26584.4 −4.27812
\(339\) 0 0
\(340\) −739.290 −0.117922
\(341\) −3.21674 −0.000510839 0
\(342\) 0 0
\(343\) 5950.88 0.936786
\(344\) 14196.7 2.22510
\(345\) 0 0
\(346\) −14627.5 −2.27277
\(347\) 5313.01 0.821952 0.410976 0.911646i \(-0.365188\pi\)
0.410976 + 0.911646i \(0.365188\pi\)
\(348\) 0 0
\(349\) 2086.08 0.319958 0.159979 0.987120i \(-0.448857\pi\)
0.159979 + 0.987120i \(0.448857\pi\)
\(350\) 1389.35 0.212182
\(351\) 0 0
\(352\) 8021.89 1.21468
\(353\) −5382.37 −0.811543 −0.405772 0.913975i \(-0.632997\pi\)
−0.405772 + 0.913975i \(0.632997\pi\)
\(354\) 0 0
\(355\) 5634.33 0.842364
\(356\) −14559.7 −2.16759
\(357\) 0 0
\(358\) −19560.4 −2.88771
\(359\) 7990.36 1.17469 0.587347 0.809335i \(-0.300173\pi\)
0.587347 + 0.809335i \(0.300173\pi\)
\(360\) 0 0
\(361\) −506.193 −0.0737998
\(362\) 14427.1 2.09468
\(363\) 0 0
\(364\) −18505.4 −2.66469
\(365\) 2907.16 0.416897
\(366\) 0 0
\(367\) −429.201 −0.0610466 −0.0305233 0.999534i \(-0.509717\pi\)
−0.0305233 + 0.999534i \(0.509717\pi\)
\(368\) 16583.0 2.34905
\(369\) 0 0
\(370\) −1096.46 −0.154061
\(371\) −3019.69 −0.422572
\(372\) 0 0
\(373\) 6270.80 0.870482 0.435241 0.900314i \(-0.356663\pi\)
0.435241 + 0.900314i \(0.356663\pi\)
\(374\) −472.467 −0.0653227
\(375\) 0 0
\(376\) −38127.7 −5.22948
\(377\) −2442.79 −0.333714
\(378\) 0 0
\(379\) 10417.0 1.41183 0.705917 0.708294i \(-0.250535\pi\)
0.705917 + 0.708294i \(0.250535\pi\)
\(380\) −8549.96 −1.15422
\(381\) 0 0
\(382\) −792.049 −0.106086
\(383\) −10769.3 −1.43677 −0.718387 0.695644i \(-0.755119\pi\)
−0.718387 + 0.695644i \(0.755119\pi\)
\(384\) 0 0
\(385\) 646.745 0.0856134
\(386\) −6249.18 −0.824028
\(387\) 0 0
\(388\) −661.736 −0.0865839
\(389\) 11121.7 1.44960 0.724798 0.688961i \(-0.241933\pi\)
0.724798 + 0.688961i \(0.241933\pi\)
\(390\) 0 0
\(391\) −508.740 −0.0658008
\(392\) 17388.7 2.24047
\(393\) 0 0
\(394\) 23489.2 3.00348
\(395\) −1278.16 −0.162813
\(396\) 0 0
\(397\) 11286.7 1.42686 0.713432 0.700724i \(-0.247140\pi\)
0.713432 + 0.700724i \(0.247140\pi\)
\(398\) −8318.33 −1.04764
\(399\) 0 0
\(400\) 5616.19 0.702023
\(401\) 15418.0 1.92005 0.960023 0.279922i \(-0.0903086\pi\)
0.960023 + 0.279922i \(0.0903086\pi\)
\(402\) 0 0
\(403\) 21.4506 0.00265144
\(404\) −37540.1 −4.62299
\(405\) 0 0
\(406\) −1611.65 −0.197006
\(407\) −510.406 −0.0621619
\(408\) 0 0
\(409\) 13849.5 1.67437 0.837183 0.546922i \(-0.184201\pi\)
0.837183 + 0.546922i \(0.184201\pi\)
\(410\) 5653.86 0.681035
\(411\) 0 0
\(412\) 41025.9 4.90583
\(413\) −2009.37 −0.239406
\(414\) 0 0
\(415\) 4915.40 0.581416
\(416\) −53493.4 −6.30464
\(417\) 0 0
\(418\) −5464.13 −0.639376
\(419\) 2793.33 0.325688 0.162844 0.986652i \(-0.447933\pi\)
0.162844 + 0.986652i \(0.447933\pi\)
\(420\) 0 0
\(421\) −3643.79 −0.421823 −0.210911 0.977505i \(-0.567643\pi\)
−0.210911 + 0.977505i \(0.567643\pi\)
\(422\) −29010.4 −3.34645
\(423\) 0 0
\(424\) −21532.4 −2.46629
\(425\) −172.295 −0.0196648
\(426\) 0 0
\(427\) −1938.55 −0.219703
\(428\) −36885.0 −4.16566
\(429\) 0 0
\(430\) 5275.94 0.591694
\(431\) −5834.03 −0.652007 −0.326004 0.945369i \(-0.605702\pi\)
−0.326004 + 0.945369i \(0.605702\pi\)
\(432\) 0 0
\(433\) 7919.15 0.878915 0.439457 0.898263i \(-0.355171\pi\)
0.439457 + 0.898263i \(0.355171\pi\)
\(434\) 14.1522 0.00156527
\(435\) 0 0
\(436\) −16721.9 −1.83677
\(437\) −5883.63 −0.644056
\(438\) 0 0
\(439\) 11696.0 1.27158 0.635788 0.771864i \(-0.280675\pi\)
0.635788 + 0.771864i \(0.280675\pi\)
\(440\) 4611.73 0.499672
\(441\) 0 0
\(442\) 3150.61 0.339049
\(443\) 2786.16 0.298814 0.149407 0.988776i \(-0.452264\pi\)
0.149407 + 0.988776i \(0.452264\pi\)
\(444\) 0 0
\(445\) −3393.21 −0.361468
\(446\) −20531.2 −2.17978
\(447\) 0 0
\(448\) −16889.5 −1.78115
\(449\) −11013.8 −1.15762 −0.578811 0.815462i \(-0.696483\pi\)
−0.578811 + 0.815462i \(0.696483\pi\)
\(450\) 0 0
\(451\) 2631.88 0.274790
\(452\) −16445.2 −1.71132
\(453\) 0 0
\(454\) −23599.8 −2.43963
\(455\) −4312.77 −0.444365
\(456\) 0 0
\(457\) 12773.5 1.30748 0.653739 0.756720i \(-0.273199\pi\)
0.653739 + 0.756720i \(0.273199\pi\)
\(458\) 16654.2 1.69912
\(459\) 0 0
\(460\) 7918.52 0.802615
\(461\) 13343.4 1.34808 0.674039 0.738696i \(-0.264558\pi\)
0.674039 + 0.738696i \(0.264558\pi\)
\(462\) 0 0
\(463\) 17265.4 1.73303 0.866514 0.499153i \(-0.166355\pi\)
0.866514 + 0.499153i \(0.166355\pi\)
\(464\) −6514.78 −0.651812
\(465\) 0 0
\(466\) −1496.25 −0.148739
\(467\) 16016.7 1.58708 0.793538 0.608521i \(-0.208237\pi\)
0.793538 + 0.608521i \(0.208237\pi\)
\(468\) 0 0
\(469\) 5845.38 0.575511
\(470\) −14169.5 −1.39062
\(471\) 0 0
\(472\) −14328.2 −1.39727
\(473\) 2455.96 0.238742
\(474\) 0 0
\(475\) −1992.61 −0.192479
\(476\) 1514.06 0.145792
\(477\) 0 0
\(478\) 17002.6 1.62695
\(479\) 5847.01 0.557738 0.278869 0.960329i \(-0.410040\pi\)
0.278869 + 0.960329i \(0.410040\pi\)
\(480\) 0 0
\(481\) 3403.61 0.322643
\(482\) −30932.7 −2.92313
\(483\) 0 0
\(484\) −25132.2 −2.36028
\(485\) −154.221 −0.0144388
\(486\) 0 0
\(487\) −342.662 −0.0318840 −0.0159420 0.999873i \(-0.505075\pi\)
−0.0159420 + 0.999873i \(0.505075\pi\)
\(488\) −13823.2 −1.28227
\(489\) 0 0
\(490\) 6462.22 0.595782
\(491\) 1673.48 0.153815 0.0769073 0.997038i \(-0.475495\pi\)
0.0769073 + 0.997038i \(0.475495\pi\)
\(492\) 0 0
\(493\) 199.863 0.0182583
\(494\) 36437.2 3.31859
\(495\) 0 0
\(496\) 57.2074 0.00517881
\(497\) −11539.1 −1.04144
\(498\) 0 0
\(499\) 10918.2 0.979487 0.489743 0.871867i \(-0.337090\pi\)
0.489743 + 0.871867i \(0.337090\pi\)
\(500\) 2681.77 0.239865
\(501\) 0 0
\(502\) 9302.84 0.827103
\(503\) −16957.6 −1.50319 −0.751595 0.659625i \(-0.770715\pi\)
−0.751595 + 0.659625i \(0.770715\pi\)
\(504\) 0 0
\(505\) −8748.91 −0.770933
\(506\) 5060.58 0.444606
\(507\) 0 0
\(508\) −29253.6 −2.55496
\(509\) 226.179 0.0196959 0.00984795 0.999952i \(-0.496865\pi\)
0.00984795 + 0.999952i \(0.496865\pi\)
\(510\) 0 0
\(511\) −5953.83 −0.515425
\(512\) −11437.3 −0.987234
\(513\) 0 0
\(514\) −4358.48 −0.374016
\(515\) 9561.30 0.818100
\(516\) 0 0
\(517\) −6595.91 −0.561099
\(518\) 2245.55 0.190471
\(519\) 0 0
\(520\) −30753.0 −2.59348
\(521\) 2051.24 0.172489 0.0862443 0.996274i \(-0.472513\pi\)
0.0862443 + 0.996274i \(0.472513\pi\)
\(522\) 0 0
\(523\) −11922.9 −0.996852 −0.498426 0.866932i \(-0.666088\pi\)
−0.498426 + 0.866932i \(0.666088\pi\)
\(524\) 28895.1 2.40895
\(525\) 0 0
\(526\) −9194.94 −0.762203
\(527\) −1.75503 −0.000145067 0
\(528\) 0 0
\(529\) −6717.89 −0.552140
\(530\) −8002.16 −0.655833
\(531\) 0 0
\(532\) 17510.3 1.42700
\(533\) −17550.5 −1.42626
\(534\) 0 0
\(535\) −8596.24 −0.694669
\(536\) 41681.6 3.35890
\(537\) 0 0
\(538\) 18262.1 1.46345
\(539\) 3008.17 0.240392
\(540\) 0 0
\(541\) 12876.2 1.02327 0.511635 0.859203i \(-0.329040\pi\)
0.511635 + 0.859203i \(0.329040\pi\)
\(542\) −43656.7 −3.45981
\(543\) 0 0
\(544\) 4376.69 0.344943
\(545\) −3897.12 −0.306302
\(546\) 0 0
\(547\) 9065.39 0.708607 0.354304 0.935130i \(-0.384718\pi\)
0.354304 + 0.935130i \(0.384718\pi\)
\(548\) 53816.0 4.19508
\(549\) 0 0
\(550\) 1713.87 0.132872
\(551\) 2311.43 0.178712
\(552\) 0 0
\(553\) 2617.65 0.201291
\(554\) 37835.7 2.90160
\(555\) 0 0
\(556\) 29262.4 2.23202
\(557\) 19520.2 1.48492 0.742458 0.669893i \(-0.233660\pi\)
0.742458 + 0.669893i \(0.233660\pi\)
\(558\) 0 0
\(559\) −16377.4 −1.23916
\(560\) −11501.9 −0.867936
\(561\) 0 0
\(562\) −10403.5 −0.780866
\(563\) −9786.46 −0.732593 −0.366297 0.930498i \(-0.619374\pi\)
−0.366297 + 0.930498i \(0.619374\pi\)
\(564\) 0 0
\(565\) −3832.64 −0.285381
\(566\) −11053.1 −0.820844
\(567\) 0 0
\(568\) −82281.5 −6.07826
\(569\) −13236.9 −0.975257 −0.487628 0.873051i \(-0.662138\pi\)
−0.487628 + 0.873051i \(0.662138\pi\)
\(570\) 0 0
\(571\) −21385.2 −1.56733 −0.783663 0.621186i \(-0.786651\pi\)
−0.783663 + 0.621186i \(0.786651\pi\)
\(572\) −22827.8 −1.66867
\(573\) 0 0
\(574\) −11579.1 −0.841987
\(575\) 1845.45 0.133845
\(576\) 0 0
\(577\) 1531.13 0.110471 0.0552356 0.998473i \(-0.482409\pi\)
0.0552356 + 0.998473i \(0.482409\pi\)
\(578\) 26405.9 1.90024
\(579\) 0 0
\(580\) −3110.85 −0.222709
\(581\) −10066.7 −0.718825
\(582\) 0 0
\(583\) −3725.02 −0.264622
\(584\) −42454.9 −3.00821
\(585\) 0 0
\(586\) −9522.25 −0.671263
\(587\) −2742.12 −0.192810 −0.0964050 0.995342i \(-0.530734\pi\)
−0.0964050 + 0.995342i \(0.530734\pi\)
\(588\) 0 0
\(589\) −20.2971 −0.00141991
\(590\) −5324.83 −0.371559
\(591\) 0 0
\(592\) 9077.22 0.630188
\(593\) 8476.47 0.586993 0.293497 0.955960i \(-0.405181\pi\)
0.293497 + 0.955960i \(0.405181\pi\)
\(594\) 0 0
\(595\) 352.859 0.0243123
\(596\) −59905.3 −4.11714
\(597\) 0 0
\(598\) −33746.2 −2.30767
\(599\) 14839.1 1.01220 0.506100 0.862475i \(-0.331087\pi\)
0.506100 + 0.862475i \(0.331087\pi\)
\(600\) 0 0
\(601\) −13198.6 −0.895814 −0.447907 0.894080i \(-0.647830\pi\)
−0.447907 + 0.894080i \(0.647830\pi\)
\(602\) −10805.1 −0.731532
\(603\) 0 0
\(604\) 6575.56 0.442973
\(605\) −5857.19 −0.393601
\(606\) 0 0
\(607\) 2310.67 0.154509 0.0772547 0.997011i \(-0.475385\pi\)
0.0772547 + 0.997011i \(0.475385\pi\)
\(608\) 50616.8 3.37629
\(609\) 0 0
\(610\) −5137.15 −0.340979
\(611\) 43984.4 2.91231
\(612\) 0 0
\(613\) −3944.92 −0.259925 −0.129962 0.991519i \(-0.541486\pi\)
−0.129962 + 0.991519i \(0.541486\pi\)
\(614\) −48352.9 −3.17811
\(615\) 0 0
\(616\) −9444.80 −0.617762
\(617\) 26587.7 1.73482 0.867408 0.497598i \(-0.165784\pi\)
0.867408 + 0.497598i \(0.165784\pi\)
\(618\) 0 0
\(619\) 8691.35 0.564353 0.282177 0.959362i \(-0.408944\pi\)
0.282177 + 0.959362i \(0.408944\pi\)
\(620\) 27.3170 0.00176948
\(621\) 0 0
\(622\) −40285.2 −2.59693
\(623\) 6949.26 0.446896
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 7889.06 0.503690
\(627\) 0 0
\(628\) −39221.5 −2.49221
\(629\) −278.474 −0.0176526
\(630\) 0 0
\(631\) −19396.9 −1.22374 −0.611869 0.790959i \(-0.709582\pi\)
−0.611869 + 0.790959i \(0.709582\pi\)
\(632\) 18665.7 1.17481
\(633\) 0 0
\(634\) 28239.2 1.76896
\(635\) −6817.71 −0.426067
\(636\) 0 0
\(637\) −20059.8 −1.24772
\(638\) −1988.09 −0.123369
\(639\) 0 0
\(640\) −19355.0 −1.19543
\(641\) 11088.7 0.683273 0.341636 0.939832i \(-0.389019\pi\)
0.341636 + 0.939832i \(0.389019\pi\)
\(642\) 0 0
\(643\) −3896.33 −0.238968 −0.119484 0.992836i \(-0.538124\pi\)
−0.119484 + 0.992836i \(0.538124\pi\)
\(644\) −16217.1 −0.992302
\(645\) 0 0
\(646\) −2981.19 −0.181569
\(647\) −14565.2 −0.885034 −0.442517 0.896760i \(-0.645914\pi\)
−0.442517 + 0.896760i \(0.645914\pi\)
\(648\) 0 0
\(649\) −2478.72 −0.149920
\(650\) −11428.8 −0.689655
\(651\) 0 0
\(652\) 45078.3 2.70767
\(653\) −27443.4 −1.64463 −0.822314 0.569034i \(-0.807317\pi\)
−0.822314 + 0.569034i \(0.807317\pi\)
\(654\) 0 0
\(655\) 6734.16 0.401718
\(656\) −46806.2 −2.78579
\(657\) 0 0
\(658\) 29019.0 1.71927
\(659\) −10355.6 −0.612135 −0.306068 0.952010i \(-0.599013\pi\)
−0.306068 + 0.952010i \(0.599013\pi\)
\(660\) 0 0
\(661\) 9704.01 0.571017 0.285508 0.958376i \(-0.407838\pi\)
0.285508 + 0.958376i \(0.407838\pi\)
\(662\) 59617.4 3.50015
\(663\) 0 0
\(664\) −71782.6 −4.19534
\(665\) 4080.85 0.237968
\(666\) 0 0
\(667\) −2140.72 −0.124272
\(668\) −33565.6 −1.94415
\(669\) 0 0
\(670\) 15490.2 0.893194
\(671\) −2391.35 −0.137582
\(672\) 0 0
\(673\) 16876.5 0.966629 0.483315 0.875447i \(-0.339433\pi\)
0.483315 + 0.875447i \(0.339433\pi\)
\(674\) −12456.8 −0.711898
\(675\) 0 0
\(676\) 105091. 5.97924
\(677\) 1095.44 0.0621880 0.0310940 0.999516i \(-0.490101\pi\)
0.0310940 + 0.999516i \(0.490101\pi\)
\(678\) 0 0
\(679\) 315.843 0.0178512
\(680\) 2516.13 0.141896
\(681\) 0 0
\(682\) 17.4578 0.000980195 0
\(683\) 23004.5 1.28879 0.644394 0.764693i \(-0.277110\pi\)
0.644394 + 0.764693i \(0.277110\pi\)
\(684\) 0 0
\(685\) 12542.1 0.699574
\(686\) −32296.4 −1.79750
\(687\) 0 0
\(688\) −43677.5 −2.42034
\(689\) 24840.0 1.37348
\(690\) 0 0
\(691\) 4973.91 0.273830 0.136915 0.990583i \(-0.456281\pi\)
0.136915 + 0.990583i \(0.456281\pi\)
\(692\) 57824.0 3.17650
\(693\) 0 0
\(694\) −28834.6 −1.57716
\(695\) 6819.75 0.372212
\(696\) 0 0
\(697\) 1435.94 0.0780344
\(698\) −11321.5 −0.613933
\(699\) 0 0
\(700\) −5492.24 −0.296553
\(701\) −19281.6 −1.03888 −0.519440 0.854507i \(-0.673859\pi\)
−0.519440 + 0.854507i \(0.673859\pi\)
\(702\) 0 0
\(703\) −3220.58 −0.172783
\(704\) −20834.6 −1.11539
\(705\) 0 0
\(706\) 29211.0 1.55718
\(707\) 17917.7 0.953132
\(708\) 0 0
\(709\) 35391.4 1.87468 0.937342 0.348412i \(-0.113279\pi\)
0.937342 + 0.348412i \(0.113279\pi\)
\(710\) −30578.5 −1.61632
\(711\) 0 0
\(712\) 49553.0 2.60826
\(713\) 18.7981 0.000987369 0
\(714\) 0 0
\(715\) −5320.14 −0.278268
\(716\) 77324.3 4.03596
\(717\) 0 0
\(718\) −43365.0 −2.25399
\(719\) 19325.0 1.00237 0.501183 0.865341i \(-0.332898\pi\)
0.501183 + 0.865341i \(0.332898\pi\)
\(720\) 0 0
\(721\) −19581.5 −1.01145
\(722\) 2747.19 0.141607
\(723\) 0 0
\(724\) −57032.0 −2.92759
\(725\) −725.000 −0.0371391
\(726\) 0 0
\(727\) 16887.9 0.861537 0.430768 0.902462i \(-0.358243\pi\)
0.430768 + 0.902462i \(0.358243\pi\)
\(728\) 62982.0 3.20641
\(729\) 0 0
\(730\) −15777.6 −0.799940
\(731\) 1339.95 0.0677975
\(732\) 0 0
\(733\) 6698.16 0.337520 0.168760 0.985657i \(-0.446024\pi\)
0.168760 + 0.985657i \(0.446024\pi\)
\(734\) 2329.35 0.117136
\(735\) 0 0
\(736\) −46878.6 −2.34778
\(737\) 7210.73 0.360394
\(738\) 0 0
\(739\) 4934.50 0.245627 0.122814 0.992430i \(-0.460808\pi\)
0.122814 + 0.992430i \(0.460808\pi\)
\(740\) 4334.44 0.215320
\(741\) 0 0
\(742\) 16388.4 0.810829
\(743\) 5140.65 0.253825 0.126913 0.991914i \(-0.459493\pi\)
0.126913 + 0.991914i \(0.459493\pi\)
\(744\) 0 0
\(745\) −13961.2 −0.686578
\(746\) −34032.7 −1.67027
\(747\) 0 0
\(748\) 1867.71 0.0912971
\(749\) 17605.0 0.858843
\(750\) 0 0
\(751\) 9625.37 0.467690 0.233845 0.972274i \(-0.424869\pi\)
0.233845 + 0.972274i \(0.424869\pi\)
\(752\) 117304. 5.68834
\(753\) 0 0
\(754\) 13257.4 0.640328
\(755\) 1532.47 0.0738704
\(756\) 0 0
\(757\) −5128.83 −0.246249 −0.123125 0.992391i \(-0.539291\pi\)
−0.123125 + 0.992391i \(0.539291\pi\)
\(758\) −56534.8 −2.70902
\(759\) 0 0
\(760\) 29099.3 1.38887
\(761\) 25024.7 1.19204 0.596022 0.802968i \(-0.296747\pi\)
0.596022 + 0.802968i \(0.296747\pi\)
\(762\) 0 0
\(763\) 7981.28 0.378692
\(764\) 3131.05 0.148269
\(765\) 0 0
\(766\) 58446.7 2.75687
\(767\) 16529.2 0.778140
\(768\) 0 0
\(769\) −26474.8 −1.24149 −0.620746 0.784012i \(-0.713170\pi\)
−0.620746 + 0.784012i \(0.713170\pi\)
\(770\) −3509.99 −0.164274
\(771\) 0 0
\(772\) 24703.6 1.15169
\(773\) 24685.1 1.14859 0.574295 0.818648i \(-0.305276\pi\)
0.574295 + 0.818648i \(0.305276\pi\)
\(774\) 0 0
\(775\) 6.36636 0.000295079 0
\(776\) 2252.18 0.104186
\(777\) 0 0
\(778\) −60359.4 −2.78148
\(779\) 16606.8 0.763798
\(780\) 0 0
\(781\) −14234.3 −0.652170
\(782\) 2761.02 0.126258
\(783\) 0 0
\(784\) −53498.3 −2.43706
\(785\) −9140.76 −0.415602
\(786\) 0 0
\(787\) −35383.4 −1.60265 −0.801323 0.598232i \(-0.795870\pi\)
−0.801323 + 0.598232i \(0.795870\pi\)
\(788\) −92855.3 −4.19776
\(789\) 0 0
\(790\) 6936.77 0.312404
\(791\) 7849.22 0.352827
\(792\) 0 0
\(793\) 15946.6 0.714098
\(794\) −61255.0 −2.73786
\(795\) 0 0
\(796\) 32883.2 1.46421
\(797\) 24869.4 1.10529 0.552647 0.833415i \(-0.313618\pi\)
0.552647 + 0.833415i \(0.313618\pi\)
\(798\) 0 0
\(799\) −3598.69 −0.159340
\(800\) −15876.4 −0.701644
\(801\) 0 0
\(802\) −83676.1 −3.68417
\(803\) −7344.51 −0.322767
\(804\) 0 0
\(805\) −3779.47 −0.165477
\(806\) −116.416 −0.00508757
\(807\) 0 0
\(808\) 127765. 5.56284
\(809\) −834.568 −0.0362693 −0.0181346 0.999836i \(-0.505773\pi\)
−0.0181346 + 0.999836i \(0.505773\pi\)
\(810\) 0 0
\(811\) −17419.5 −0.754232 −0.377116 0.926166i \(-0.623084\pi\)
−0.377116 + 0.926166i \(0.623084\pi\)
\(812\) 6371.00 0.275343
\(813\) 0 0
\(814\) 2770.06 0.119276
\(815\) 10505.7 0.451533
\(816\) 0 0
\(817\) 15496.7 0.663600
\(818\) −75163.8 −3.21276
\(819\) 0 0
\(820\) −22350.3 −0.951837
\(821\) −20163.7 −0.857148 −0.428574 0.903507i \(-0.640984\pi\)
−0.428574 + 0.903507i \(0.640984\pi\)
\(822\) 0 0
\(823\) 10456.9 0.442898 0.221449 0.975172i \(-0.428921\pi\)
0.221449 + 0.975172i \(0.428921\pi\)
\(824\) −139629. −5.90318
\(825\) 0 0
\(826\) 10905.2 0.459371
\(827\) 10943.6 0.460154 0.230077 0.973172i \(-0.426102\pi\)
0.230077 + 0.973172i \(0.426102\pi\)
\(828\) 0 0
\(829\) −33775.8 −1.41506 −0.707528 0.706685i \(-0.750190\pi\)
−0.707528 + 0.706685i \(0.750190\pi\)
\(830\) −26676.7 −1.11562
\(831\) 0 0
\(832\) 138934. 5.78926
\(833\) 1641.24 0.0682660
\(834\) 0 0
\(835\) −7822.63 −0.324207
\(836\) 21600.3 0.893613
\(837\) 0 0
\(838\) −15159.9 −0.624927
\(839\) −39087.3 −1.60840 −0.804198 0.594361i \(-0.797405\pi\)
−0.804198 + 0.594361i \(0.797405\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 19775.4 0.809391
\(843\) 0 0
\(844\) 114681. 4.67711
\(845\) 24492.0 0.997101
\(846\) 0 0
\(847\) 11995.5 0.486623
\(848\) 66246.9 2.68270
\(849\) 0 0
\(850\) 935.076 0.0377327
\(851\) 2982.73 0.120149
\(852\) 0 0
\(853\) 47274.9 1.89761 0.948806 0.315859i \(-0.102293\pi\)
0.948806 + 0.315859i \(0.102293\pi\)
\(854\) 10520.9 0.421565
\(855\) 0 0
\(856\) 125536. 5.01254
\(857\) −15482.0 −0.617099 −0.308550 0.951208i \(-0.599844\pi\)
−0.308550 + 0.951208i \(0.599844\pi\)
\(858\) 0 0
\(859\) −32719.1 −1.29960 −0.649802 0.760103i \(-0.725148\pi\)
−0.649802 + 0.760103i \(0.725148\pi\)
\(860\) −20856.3 −0.826971
\(861\) 0 0
\(862\) 31662.2 1.25107
\(863\) 1741.34 0.0686858 0.0343429 0.999410i \(-0.489066\pi\)
0.0343429 + 0.999410i \(0.489066\pi\)
\(864\) 0 0
\(865\) 13476.2 0.529715
\(866\) −42978.6 −1.68646
\(867\) 0 0
\(868\) −55.9449 −0.00218767
\(869\) 3229.08 0.126052
\(870\) 0 0
\(871\) −48084.3 −1.87058
\(872\) 56912.0 2.21019
\(873\) 0 0
\(874\) 31931.5 1.23581
\(875\) −1280.00 −0.0494534
\(876\) 0 0
\(877\) 11848.6 0.456213 0.228106 0.973636i \(-0.426747\pi\)
0.228106 + 0.973636i \(0.426747\pi\)
\(878\) −63476.4 −2.43989
\(879\) 0 0
\(880\) −14188.5 −0.543516
\(881\) 29332.3 1.12171 0.560857 0.827913i \(-0.310472\pi\)
0.560857 + 0.827913i \(0.310472\pi\)
\(882\) 0 0
\(883\) −25723.3 −0.980360 −0.490180 0.871621i \(-0.663069\pi\)
−0.490180 + 0.871621i \(0.663069\pi\)
\(884\) −12454.7 −0.473865
\(885\) 0 0
\(886\) −15121.0 −0.573362
\(887\) −10959.8 −0.414875 −0.207437 0.978248i \(-0.566512\pi\)
−0.207437 + 0.978248i \(0.566512\pi\)
\(888\) 0 0
\(889\) 13962.6 0.526762
\(890\) 18415.5 0.693583
\(891\) 0 0
\(892\) 81162.1 3.04653
\(893\) −41619.2 −1.55961
\(894\) 0 0
\(895\) 18020.8 0.673039
\(896\) 39638.8 1.47795
\(897\) 0 0
\(898\) 59773.6 2.22124
\(899\) −7.38497 −0.000273974 0
\(900\) 0 0
\(901\) −2032.34 −0.0751467
\(902\) −14283.7 −0.527266
\(903\) 0 0
\(904\) 55970.3 2.05923
\(905\) −13291.6 −0.488207
\(906\) 0 0
\(907\) −3053.42 −0.111783 −0.0558915 0.998437i \(-0.517800\pi\)
−0.0558915 + 0.998437i \(0.517800\pi\)
\(908\) 93292.4 3.40971
\(909\) 0 0
\(910\) 23406.2 0.852644
\(911\) 32880.5 1.19581 0.597903 0.801568i \(-0.296001\pi\)
0.597903 + 0.801568i \(0.296001\pi\)
\(912\) 0 0
\(913\) −12418.1 −0.450140
\(914\) −69323.8 −2.50878
\(915\) 0 0
\(916\) −65835.7 −2.37475
\(917\) −13791.5 −0.496658
\(918\) 0 0
\(919\) −49962.9 −1.79339 −0.896694 0.442650i \(-0.854038\pi\)
−0.896694 + 0.442650i \(0.854038\pi\)
\(920\) −26950.2 −0.965786
\(921\) 0 0
\(922\) −72416.8 −2.58668
\(923\) 94920.7 3.38500
\(924\) 0 0
\(925\) 1010.16 0.0359070
\(926\) −93702.3 −3.32532
\(927\) 0 0
\(928\) 18416.6 0.651460
\(929\) 48163.6 1.70096 0.850482 0.526004i \(-0.176310\pi\)
0.850482 + 0.526004i \(0.176310\pi\)
\(930\) 0 0
\(931\) 18981.1 0.668185
\(932\) 5914.83 0.207883
\(933\) 0 0
\(934\) −86925.3 −3.04527
\(935\) 435.279 0.0152248
\(936\) 0 0
\(937\) 32244.1 1.12419 0.562097 0.827071i \(-0.309995\pi\)
0.562097 + 0.827071i \(0.309995\pi\)
\(938\) −31723.9 −1.10429
\(939\) 0 0
\(940\) 56013.4 1.94357
\(941\) −37391.5 −1.29535 −0.647677 0.761915i \(-0.724260\pi\)
−0.647677 + 0.761915i \(0.724260\pi\)
\(942\) 0 0
\(943\) −15380.3 −0.531125
\(944\) 44082.3 1.51987
\(945\) 0 0
\(946\) −13328.9 −0.458097
\(947\) −17887.8 −0.613807 −0.306903 0.951741i \(-0.599293\pi\)
−0.306903 + 0.951741i \(0.599293\pi\)
\(948\) 0 0
\(949\) 48976.4 1.67528
\(950\) 10814.2 0.369327
\(951\) 0 0
\(952\) −5153.02 −0.175431
\(953\) −20302.3 −0.690090 −0.345045 0.938586i \(-0.612136\pi\)
−0.345045 + 0.938586i \(0.612136\pi\)
\(954\) 0 0
\(955\) 729.708 0.0247254
\(956\) −67213.0 −2.27387
\(957\) 0 0
\(958\) −31732.7 −1.07018
\(959\) −25686.1 −0.864908
\(960\) 0 0
\(961\) −29790.9 −0.999998
\(962\) −18472.0 −0.619085
\(963\) 0 0
\(964\) 122280. 4.08546
\(965\) 5757.31 0.192056
\(966\) 0 0
\(967\) −19313.4 −0.642273 −0.321137 0.947033i \(-0.604065\pi\)
−0.321137 + 0.947033i \(0.604065\pi\)
\(968\) 85536.0 2.84012
\(969\) 0 0
\(970\) 836.983 0.0277050
\(971\) 25297.3 0.836077 0.418038 0.908429i \(-0.362718\pi\)
0.418038 + 0.908429i \(0.362718\pi\)
\(972\) 0 0
\(973\) −13966.8 −0.460179
\(974\) 1859.68 0.0611787
\(975\) 0 0
\(976\) 42528.6 1.39478
\(977\) −21461.4 −0.702776 −0.351388 0.936230i \(-0.614290\pi\)
−0.351388 + 0.936230i \(0.614290\pi\)
\(978\) 0 0
\(979\) 8572.45 0.279854
\(980\) −25545.8 −0.832684
\(981\) 0 0
\(982\) −9082.24 −0.295138
\(983\) −14059.8 −0.456193 −0.228096 0.973639i \(-0.573250\pi\)
−0.228096 + 0.973639i \(0.573250\pi\)
\(984\) 0 0
\(985\) −21640.4 −0.700021
\(986\) −1084.69 −0.0350340
\(987\) 0 0
\(988\) −144040. −4.63818
\(989\) −14352.2 −0.461450
\(990\) 0 0
\(991\) −60932.3 −1.95316 −0.976578 0.215162i \(-0.930972\pi\)
−0.976578 + 0.215162i \(0.930972\pi\)
\(992\) −161.720 −0.00517601
\(993\) 0 0
\(994\) 62624.5 1.99832
\(995\) 7663.60 0.244173
\(996\) 0 0
\(997\) −4183.32 −0.132886 −0.0664428 0.997790i \(-0.521165\pi\)
−0.0664428 + 0.997790i \(0.521165\pi\)
\(998\) −59254.7 −1.87943
\(999\) 0 0
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 1305.4.a.h.1.1 6
3.2 odd 2 435.4.a.h.1.6 6
15.14 odd 2 2175.4.a.k.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.h.1.6 6 3.2 odd 2
1305.4.a.h.1.1 6 1.1 even 1 trivial
2175.4.a.k.1.1 6 15.14 odd 2