Properties

Label 1575.4.a.bl.1.2
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
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] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, 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("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.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: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 32x^{2} - 35x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.50478\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.504784 q^{2} -7.74519 q^{4} -7.00000 q^{7} +7.94792 q^{8} +O(q^{10})\) \(q-0.504784 q^{2} -7.74519 q^{4} -7.00000 q^{7} +7.94792 q^{8} -54.8800 q^{11} -16.0073 q^{13} +3.53349 q^{14} +57.9496 q^{16} -0.422056 q^{17} +127.501 q^{19} +27.7025 q^{22} +51.1101 q^{23} +8.08024 q^{26} +54.2164 q^{28} -41.4750 q^{29} +192.354 q^{31} -92.8353 q^{32} +0.213047 q^{34} +189.232 q^{37} -64.3605 q^{38} +76.3187 q^{41} -294.499 q^{43} +425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +49.0000 q^{49} +123.980 q^{52} -661.316 q^{53} -55.6354 q^{56} +20.9359 q^{58} -410.312 q^{59} +46.0495 q^{61} -97.0974 q^{62} -416.735 q^{64} -10.4074 q^{67} +3.26890 q^{68} +491.117 q^{71} +814.540 q^{73} -95.5215 q^{74} -987.522 q^{76} +384.160 q^{77} -858.725 q^{79} -38.5244 q^{82} -1055.80 q^{83} +148.658 q^{86} -436.182 q^{88} -341.567 q^{89} +112.051 q^{91} -395.858 q^{92} -272.733 q^{94} +1417.21 q^{97} -24.7344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 36 q^{4} - 28 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 36 q^{4} - 28 q^{7} + 27 q^{8} - 100 q^{11} - 44 q^{13} - 28 q^{14} + 160 q^{16} - 53 q^{17} - 29 q^{19} + 152 q^{22} + 295 q^{23} - 700 q^{26} - 252 q^{28} - 129 q^{29} + 114 q^{31} - 310 q^{32} + 203 q^{34} - 403 q^{37} + 555 q^{38} - 671 q^{41} + 411 q^{43} - 438 q^{44} - 997 q^{46} - 8 q^{47} + 196 q^{49} - 74 q^{52} + 90 q^{53} - 189 q^{56} - 673 q^{58} - 1018 q^{59} + 50 q^{61} - 1626 q^{62} - 2421 q^{64} - 424 q^{67} - 617 q^{68} - 215 q^{71} + 1207 q^{73} - 623 q^{74} - 3257 q^{76} + 700 q^{77} - 951 q^{79} - 1695 q^{82} - 3035 q^{83} + 99 q^{86} - 163 q^{88} - 2819 q^{89} + 308 q^{91} + 3073 q^{92} - 3056 q^{94} + 1100 q^{97} + 196 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.504784 −0.178468 −0.0892340 0.996011i \(-0.528442\pi\)
−0.0892340 + 0.996011i \(0.528442\pi\)
\(3\) 0 0
\(4\) −7.74519 −0.968149
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 7.94792 0.351252
\(9\) 0 0
\(10\) 0 0
\(11\) −54.8800 −1.50427 −0.752134 0.659010i \(-0.770975\pi\)
−0.752134 + 0.659010i \(0.770975\pi\)
\(12\) 0 0
\(13\) −16.0073 −0.341510 −0.170755 0.985313i \(-0.554621\pi\)
−0.170755 + 0.985313i \(0.554621\pi\)
\(14\) 3.53349 0.0674546
\(15\) 0 0
\(16\) 57.9496 0.905462
\(17\) −0.422056 −0.00602139 −0.00301069 0.999995i \(-0.500958\pi\)
−0.00301069 + 0.999995i \(0.500958\pi\)
\(18\) 0 0
\(19\) 127.501 1.53952 0.769758 0.638336i \(-0.220377\pi\)
0.769758 + 0.638336i \(0.220377\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 27.7025 0.268464
\(23\) 51.1101 0.463357 0.231678 0.972792i \(-0.425578\pi\)
0.231678 + 0.972792i \(0.425578\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.08024 0.0609487
\(27\) 0 0
\(28\) 54.2164 0.365926
\(29\) −41.4750 −0.265576 −0.132788 0.991144i \(-0.542393\pi\)
−0.132788 + 0.991144i \(0.542393\pi\)
\(30\) 0 0
\(31\) 192.354 1.11445 0.557224 0.830362i \(-0.311866\pi\)
0.557224 + 0.830362i \(0.311866\pi\)
\(32\) −92.8353 −0.512848
\(33\) 0 0
\(34\) 0.213047 0.00107462
\(35\) 0 0
\(36\) 0 0
\(37\) 189.232 0.840801 0.420400 0.907339i \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(38\) −64.3605 −0.274754
\(39\) 0 0
\(40\) 0 0
\(41\) 76.3187 0.290707 0.145353 0.989380i \(-0.453568\pi\)
0.145353 + 0.989380i \(0.453568\pi\)
\(42\) 0 0
\(43\) −294.499 −1.04443 −0.522216 0.852813i \(-0.674895\pi\)
−0.522216 + 0.852813i \(0.674895\pi\)
\(44\) 425.056 1.45636
\(45\) 0 0
\(46\) −25.7996 −0.0826943
\(47\) 540.297 1.67682 0.838408 0.545043i \(-0.183487\pi\)
0.838408 + 0.545043i \(0.183487\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 123.980 0.330633
\(53\) −661.316 −1.71394 −0.856969 0.515368i \(-0.827655\pi\)
−0.856969 + 0.515368i \(0.827655\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −55.6354 −0.132761
\(57\) 0 0
\(58\) 20.9359 0.0473969
\(59\) −410.312 −0.905390 −0.452695 0.891665i \(-0.649537\pi\)
−0.452695 + 0.891665i \(0.649537\pi\)
\(60\) 0 0
\(61\) 46.0495 0.0966563 0.0483281 0.998832i \(-0.484611\pi\)
0.0483281 + 0.998832i \(0.484611\pi\)
\(62\) −97.0974 −0.198893
\(63\) 0 0
\(64\) −416.735 −0.813935
\(65\) 0 0
\(66\) 0 0
\(67\) −10.4074 −0.0189771 −0.00948854 0.999955i \(-0.503020\pi\)
−0.00948854 + 0.999955i \(0.503020\pi\)
\(68\) 3.26890 0.00582960
\(69\) 0 0
\(70\) 0 0
\(71\) 491.117 0.820913 0.410456 0.911880i \(-0.365369\pi\)
0.410456 + 0.911880i \(0.365369\pi\)
\(72\) 0 0
\(73\) 814.540 1.30595 0.652977 0.757378i \(-0.273520\pi\)
0.652977 + 0.757378i \(0.273520\pi\)
\(74\) −95.5215 −0.150056
\(75\) 0 0
\(76\) −987.522 −1.49048
\(77\) 384.160 0.568560
\(78\) 0 0
\(79\) −858.725 −1.22296 −0.611482 0.791258i \(-0.709426\pi\)
−0.611482 + 0.791258i \(0.709426\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −38.5244 −0.0518818
\(83\) −1055.80 −1.39626 −0.698129 0.715972i \(-0.745984\pi\)
−0.698129 + 0.715972i \(0.745984\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 148.658 0.186398
\(87\) 0 0
\(88\) −436.182 −0.528376
\(89\) −341.567 −0.406809 −0.203405 0.979095i \(-0.565201\pi\)
−0.203405 + 0.979095i \(0.565201\pi\)
\(90\) 0 0
\(91\) 112.051 0.129079
\(92\) −395.858 −0.448598
\(93\) 0 0
\(94\) −272.733 −0.299258
\(95\) 0 0
\(96\) 0 0
\(97\) 1417.21 1.48346 0.741731 0.670697i \(-0.234005\pi\)
0.741731 + 0.670697i \(0.234005\pi\)
\(98\) −24.7344 −0.0254954
\(99\) 0 0
\(100\) 0 0
\(101\) 121.051 0.119258 0.0596289 0.998221i \(-0.481008\pi\)
0.0596289 + 0.998221i \(0.481008\pi\)
\(102\) 0 0
\(103\) 655.594 0.627161 0.313581 0.949562i \(-0.398471\pi\)
0.313581 + 0.949562i \(0.398471\pi\)
\(104\) −127.225 −0.119956
\(105\) 0 0
\(106\) 333.821 0.305883
\(107\) 2069.81 1.87006 0.935031 0.354567i \(-0.115372\pi\)
0.935031 + 0.354567i \(0.115372\pi\)
\(108\) 0 0
\(109\) 904.218 0.794573 0.397286 0.917695i \(-0.369952\pi\)
0.397286 + 0.917695i \(0.369952\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −405.647 −0.342232
\(113\) −843.350 −0.702086 −0.351043 0.936359i \(-0.614173\pi\)
−0.351043 + 0.936359i \(0.614173\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 321.232 0.257117
\(117\) 0 0
\(118\) 207.119 0.161583
\(119\) 2.95439 0.00227587
\(120\) 0 0
\(121\) 1680.81 1.26282
\(122\) −23.2450 −0.0172501
\(123\) 0 0
\(124\) −1489.82 −1.07895
\(125\) 0 0
\(126\) 0 0
\(127\) −2030.50 −1.41872 −0.709362 0.704845i \(-0.751017\pi\)
−0.709362 + 0.704845i \(0.751017\pi\)
\(128\) 953.043 0.658109
\(129\) 0 0
\(130\) 0 0
\(131\) −2872.47 −1.91579 −0.957897 0.287113i \(-0.907305\pi\)
−0.957897 + 0.287113i \(0.907305\pi\)
\(132\) 0 0
\(133\) −892.509 −0.581882
\(134\) 5.25348 0.00338680
\(135\) 0 0
\(136\) −3.35446 −0.00211502
\(137\) −1621.62 −1.01127 −0.505636 0.862747i \(-0.668742\pi\)
−0.505636 + 0.862747i \(0.668742\pi\)
\(138\) 0 0
\(139\) 2617.72 1.59735 0.798677 0.601760i \(-0.205534\pi\)
0.798677 + 0.601760i \(0.205534\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −247.908 −0.146507
\(143\) 878.482 0.513723
\(144\) 0 0
\(145\) 0 0
\(146\) −411.166 −0.233071
\(147\) 0 0
\(148\) −1465.64 −0.814021
\(149\) −2252.63 −1.23854 −0.619270 0.785178i \(-0.712571\pi\)
−0.619270 + 0.785178i \(0.712571\pi\)
\(150\) 0 0
\(151\) −1849.61 −0.996813 −0.498407 0.866943i \(-0.666081\pi\)
−0.498407 + 0.866943i \(0.666081\pi\)
\(152\) 1013.37 0.540757
\(153\) 0 0
\(154\) −193.918 −0.101470
\(155\) 0 0
\(156\) 0 0
\(157\) 3337.59 1.69661 0.848307 0.529505i \(-0.177622\pi\)
0.848307 + 0.529505i \(0.177622\pi\)
\(158\) 433.471 0.218260
\(159\) 0 0
\(160\) 0 0
\(161\) −357.771 −0.175132
\(162\) 0 0
\(163\) 471.099 0.226376 0.113188 0.993574i \(-0.463894\pi\)
0.113188 + 0.993574i \(0.463894\pi\)
\(164\) −591.103 −0.281447
\(165\) 0 0
\(166\) 532.952 0.249187
\(167\) −3676.72 −1.70367 −0.851836 0.523809i \(-0.824511\pi\)
−0.851836 + 0.523809i \(0.824511\pi\)
\(168\) 0 0
\(169\) −1940.77 −0.883371
\(170\) 0 0
\(171\) 0 0
\(172\) 2280.95 1.01117
\(173\) 1009.11 0.443475 0.221737 0.975106i \(-0.428827\pi\)
0.221737 + 0.975106i \(0.428827\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3180.27 −1.36206
\(177\) 0 0
\(178\) 172.417 0.0726024
\(179\) −3201.27 −1.33673 −0.668364 0.743835i \(-0.733005\pi\)
−0.668364 + 0.743835i \(0.733005\pi\)
\(180\) 0 0
\(181\) −1970.14 −0.809056 −0.404528 0.914526i \(-0.632564\pi\)
−0.404528 + 0.914526i \(0.632564\pi\)
\(182\) −56.5617 −0.0230364
\(183\) 0 0
\(184\) 406.219 0.162755
\(185\) 0 0
\(186\) 0 0
\(187\) 23.1624 0.00905778
\(188\) −4184.70 −1.62341
\(189\) 0 0
\(190\) 0 0
\(191\) −3953.39 −1.49768 −0.748842 0.662749i \(-0.769390\pi\)
−0.748842 + 0.662749i \(0.769390\pi\)
\(192\) 0 0
\(193\) −2425.45 −0.904598 −0.452299 0.891866i \(-0.649396\pi\)
−0.452299 + 0.891866i \(0.649396\pi\)
\(194\) −715.385 −0.264751
\(195\) 0 0
\(196\) −379.514 −0.138307
\(197\) 3126.50 1.13073 0.565366 0.824840i \(-0.308735\pi\)
0.565366 + 0.824840i \(0.308735\pi\)
\(198\) 0 0
\(199\) −3566.13 −1.27033 −0.635166 0.772376i \(-0.719068\pi\)
−0.635166 + 0.772376i \(0.719068\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −61.1046 −0.0212837
\(203\) 290.325 0.100378
\(204\) 0 0
\(205\) 0 0
\(206\) −330.933 −0.111928
\(207\) 0 0
\(208\) −927.618 −0.309225
\(209\) −6997.27 −2.31584
\(210\) 0 0
\(211\) −3699.18 −1.20693 −0.603465 0.797389i \(-0.706214\pi\)
−0.603465 + 0.797389i \(0.706214\pi\)
\(212\) 5122.02 1.65935
\(213\) 0 0
\(214\) −1044.81 −0.333746
\(215\) 0 0
\(216\) 0 0
\(217\) −1346.48 −0.421222
\(218\) −456.435 −0.141806
\(219\) 0 0
\(220\) 0 0
\(221\) 6.75599 0.00205637
\(222\) 0 0
\(223\) 102.041 0.0306419 0.0153210 0.999883i \(-0.495123\pi\)
0.0153210 + 0.999883i \(0.495123\pi\)
\(224\) 649.847 0.193838
\(225\) 0 0
\(226\) 425.709 0.125300
\(227\) 5739.91 1.67829 0.839144 0.543909i \(-0.183057\pi\)
0.839144 + 0.543909i \(0.183057\pi\)
\(228\) 0 0
\(229\) −843.125 −0.243298 −0.121649 0.992573i \(-0.538818\pi\)
−0.121649 + 0.992573i \(0.538818\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −329.640 −0.0932841
\(233\) −2017.65 −0.567300 −0.283650 0.958928i \(-0.591545\pi\)
−0.283650 + 0.958928i \(0.591545\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3177.94 0.876553
\(237\) 0 0
\(238\) −1.49133 −0.000406170 0
\(239\) −243.685 −0.0659526 −0.0329763 0.999456i \(-0.510499\pi\)
−0.0329763 + 0.999456i \(0.510499\pi\)
\(240\) 0 0
\(241\) −3060.58 −0.818048 −0.409024 0.912524i \(-0.634131\pi\)
−0.409024 + 0.912524i \(0.634131\pi\)
\(242\) −848.448 −0.225373
\(243\) 0 0
\(244\) −356.662 −0.0935777
\(245\) 0 0
\(246\) 0 0
\(247\) −2040.95 −0.525760
\(248\) 1528.82 0.391452
\(249\) 0 0
\(250\) 0 0
\(251\) 6594.14 1.65824 0.829120 0.559070i \(-0.188842\pi\)
0.829120 + 0.559070i \(0.188842\pi\)
\(252\) 0 0
\(253\) −2804.92 −0.697012
\(254\) 1024.96 0.253197
\(255\) 0 0
\(256\) 2852.80 0.696484
\(257\) 512.646 0.124428 0.0622139 0.998063i \(-0.480184\pi\)
0.0622139 + 0.998063i \(0.480184\pi\)
\(258\) 0 0
\(259\) −1324.63 −0.317793
\(260\) 0 0
\(261\) 0 0
\(262\) 1449.98 0.341908
\(263\) −3560.04 −0.834681 −0.417341 0.908750i \(-0.637038\pi\)
−0.417341 + 0.908750i \(0.637038\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 450.524 0.103847
\(267\) 0 0
\(268\) 80.6072 0.0183726
\(269\) 549.744 0.124604 0.0623020 0.998057i \(-0.480156\pi\)
0.0623020 + 0.998057i \(0.480156\pi\)
\(270\) 0 0
\(271\) −3944.91 −0.884267 −0.442134 0.896949i \(-0.645778\pi\)
−0.442134 + 0.896949i \(0.645778\pi\)
\(272\) −24.4580 −0.00545214
\(273\) 0 0
\(274\) 818.567 0.180480
\(275\) 0 0
\(276\) 0 0
\(277\) 1254.67 0.272150 0.136075 0.990699i \(-0.456551\pi\)
0.136075 + 0.990699i \(0.456551\pi\)
\(278\) −1321.38 −0.285077
\(279\) 0 0
\(280\) 0 0
\(281\) 681.496 0.144679 0.0723393 0.997380i \(-0.476954\pi\)
0.0723393 + 0.997380i \(0.476954\pi\)
\(282\) 0 0
\(283\) 5946.88 1.24914 0.624568 0.780970i \(-0.285275\pi\)
0.624568 + 0.780970i \(0.285275\pi\)
\(284\) −3803.79 −0.794766
\(285\) 0 0
\(286\) −443.444 −0.0916831
\(287\) −534.231 −0.109877
\(288\) 0 0
\(289\) −4912.82 −0.999964
\(290\) 0 0
\(291\) 0 0
\(292\) −6308.77 −1.26436
\(293\) 2171.03 0.432877 0.216438 0.976296i \(-0.430556\pi\)
0.216438 + 0.976296i \(0.430556\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1504.00 0.295333
\(297\) 0 0
\(298\) 1137.09 0.221040
\(299\) −818.137 −0.158241
\(300\) 0 0
\(301\) 2061.49 0.394759
\(302\) 933.651 0.177899
\(303\) 0 0
\(304\) 7388.64 1.39397
\(305\) 0 0
\(306\) 0 0
\(307\) 3644.59 0.677549 0.338775 0.940868i \(-0.389988\pi\)
0.338775 + 0.940868i \(0.389988\pi\)
\(308\) −2975.39 −0.550451
\(309\) 0 0
\(310\) 0 0
\(311\) −2584.19 −0.471177 −0.235589 0.971853i \(-0.575702\pi\)
−0.235589 + 0.971853i \(0.575702\pi\)
\(312\) 0 0
\(313\) −6693.63 −1.20878 −0.604388 0.796690i \(-0.706582\pi\)
−0.604388 + 0.796690i \(0.706582\pi\)
\(314\) −1684.76 −0.302791
\(315\) 0 0
\(316\) 6650.99 1.18401
\(317\) −3843.36 −0.680962 −0.340481 0.940251i \(-0.610590\pi\)
−0.340481 + 0.940251i \(0.610590\pi\)
\(318\) 0 0
\(319\) 2276.15 0.399498
\(320\) 0 0
\(321\) 0 0
\(322\) 180.597 0.0312555
\(323\) −53.8126 −0.00927002
\(324\) 0 0
\(325\) 0 0
\(326\) −237.803 −0.0404009
\(327\) 0 0
\(328\) 606.575 0.102111
\(329\) −3782.08 −0.633777
\(330\) 0 0
\(331\) −821.922 −0.136486 −0.0682431 0.997669i \(-0.521739\pi\)
−0.0682431 + 0.997669i \(0.521739\pi\)
\(332\) 8177.39 1.35179
\(333\) 0 0
\(334\) 1855.95 0.304051
\(335\) 0 0
\(336\) 0 0
\(337\) −1032.55 −0.166904 −0.0834520 0.996512i \(-0.526595\pi\)
−0.0834520 + 0.996512i \(0.526595\pi\)
\(338\) 979.667 0.157653
\(339\) 0 0
\(340\) 0 0
\(341\) −10556.4 −1.67643
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −2340.65 −0.366859
\(345\) 0 0
\(346\) −509.382 −0.0791461
\(347\) −2587.70 −0.400332 −0.200166 0.979762i \(-0.564148\pi\)
−0.200166 + 0.979762i \(0.564148\pi\)
\(348\) 0 0
\(349\) −8454.24 −1.29669 −0.648345 0.761346i \(-0.724539\pi\)
−0.648345 + 0.761346i \(0.724539\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5094.80 0.771460
\(353\) −4129.63 −0.622658 −0.311329 0.950302i \(-0.600774\pi\)
−0.311329 + 0.950302i \(0.600774\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2645.50 0.393852
\(357\) 0 0
\(358\) 1615.95 0.238563
\(359\) 4536.69 0.666957 0.333478 0.942758i \(-0.391778\pi\)
0.333478 + 0.942758i \(0.391778\pi\)
\(360\) 0 0
\(361\) 9397.56 1.37011
\(362\) 994.492 0.144390
\(363\) 0 0
\(364\) −867.859 −0.124968
\(365\) 0 0
\(366\) 0 0
\(367\) −1643.00 −0.233689 −0.116844 0.993150i \(-0.537278\pi\)
−0.116844 + 0.993150i \(0.537278\pi\)
\(368\) 2961.81 0.419552
\(369\) 0 0
\(370\) 0 0
\(371\) 4629.21 0.647808
\(372\) 0 0
\(373\) −9019.53 −1.25205 −0.626024 0.779804i \(-0.715319\pi\)
−0.626024 + 0.779804i \(0.715319\pi\)
\(374\) −11.6920 −0.00161652
\(375\) 0 0
\(376\) 4294.23 0.588984
\(377\) 663.904 0.0906971
\(378\) 0 0
\(379\) −2463.63 −0.333900 −0.166950 0.985965i \(-0.553392\pi\)
−0.166950 + 0.985965i \(0.553392\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1995.61 0.267289
\(383\) −11007.2 −1.46852 −0.734261 0.678867i \(-0.762471\pi\)
−0.734261 + 0.678867i \(0.762471\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1224.33 0.161442
\(387\) 0 0
\(388\) −10976.6 −1.43621
\(389\) −6162.79 −0.803254 −0.401627 0.915803i \(-0.631555\pi\)
−0.401627 + 0.915803i \(0.631555\pi\)
\(390\) 0 0
\(391\) −21.5713 −0.00279005
\(392\) 389.448 0.0501788
\(393\) 0 0
\(394\) −1578.21 −0.201799
\(395\) 0 0
\(396\) 0 0
\(397\) 15137.9 1.91373 0.956864 0.290537i \(-0.0938338\pi\)
0.956864 + 0.290537i \(0.0938338\pi\)
\(398\) 1800.12 0.226714
\(399\) 0 0
\(400\) 0 0
\(401\) −3302.29 −0.411243 −0.205622 0.978632i \(-0.565922\pi\)
−0.205622 + 0.978632i \(0.565922\pi\)
\(402\) 0 0
\(403\) −3079.08 −0.380596
\(404\) −937.564 −0.115459
\(405\) 0 0
\(406\) −146.551 −0.0179143
\(407\) −10385.1 −1.26479
\(408\) 0 0
\(409\) 4576.40 0.553273 0.276636 0.960975i \(-0.410780\pi\)
0.276636 + 0.960975i \(0.410780\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −5077.70 −0.607186
\(413\) 2872.18 0.342205
\(414\) 0 0
\(415\) 0 0
\(416\) 1486.05 0.175143
\(417\) 0 0
\(418\) 3532.11 0.413304
\(419\) 6778.05 0.790285 0.395142 0.918620i \(-0.370695\pi\)
0.395142 + 0.918620i \(0.370695\pi\)
\(420\) 0 0
\(421\) −3952.68 −0.457582 −0.228791 0.973476i \(-0.573477\pi\)
−0.228791 + 0.973476i \(0.573477\pi\)
\(422\) 1867.29 0.215398
\(423\) 0 0
\(424\) −5256.08 −0.602024
\(425\) 0 0
\(426\) 0 0
\(427\) −322.346 −0.0365326
\(428\) −16031.1 −1.81050
\(429\) 0 0
\(430\) 0 0
\(431\) 3435.10 0.383905 0.191952 0.981404i \(-0.438518\pi\)
0.191952 + 0.981404i \(0.438518\pi\)
\(432\) 0 0
\(433\) 5457.82 0.605742 0.302871 0.953032i \(-0.402055\pi\)
0.302871 + 0.953032i \(0.402055\pi\)
\(434\) 679.682 0.0751746
\(435\) 0 0
\(436\) −7003.35 −0.769265
\(437\) 6516.61 0.713344
\(438\) 0 0
\(439\) 4188.10 0.455323 0.227662 0.973740i \(-0.426892\pi\)
0.227662 + 0.973740i \(0.426892\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.41031 −0.000366996 0
\(443\) 2097.50 0.224955 0.112478 0.993654i \(-0.464121\pi\)
0.112478 + 0.993654i \(0.464121\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −51.5085 −0.00546860
\(447\) 0 0
\(448\) 2917.14 0.307639
\(449\) −11318.2 −1.18962 −0.594809 0.803867i \(-0.702772\pi\)
−0.594809 + 0.803867i \(0.702772\pi\)
\(450\) 0 0
\(451\) −4188.37 −0.437301
\(452\) 6531.91 0.679724
\(453\) 0 0
\(454\) −2897.41 −0.299521
\(455\) 0 0
\(456\) 0 0
\(457\) 11696.4 1.19723 0.598617 0.801036i \(-0.295717\pi\)
0.598617 + 0.801036i \(0.295717\pi\)
\(458\) 425.596 0.0434209
\(459\) 0 0
\(460\) 0 0
\(461\) −10685.0 −1.07951 −0.539753 0.841823i \(-0.681482\pi\)
−0.539753 + 0.841823i \(0.681482\pi\)
\(462\) 0 0
\(463\) 10407.2 1.04463 0.522315 0.852752i \(-0.325068\pi\)
0.522315 + 0.852752i \(0.325068\pi\)
\(464\) −2403.46 −0.240469
\(465\) 0 0
\(466\) 1018.48 0.101245
\(467\) −2978.24 −0.295111 −0.147555 0.989054i \(-0.547140\pi\)
−0.147555 + 0.989054i \(0.547140\pi\)
\(468\) 0 0
\(469\) 72.8517 0.00717266
\(470\) 0 0
\(471\) 0 0
\(472\) −3261.12 −0.318020
\(473\) 16162.1 1.57111
\(474\) 0 0
\(475\) 0 0
\(476\) −22.8823 −0.00220338
\(477\) 0 0
\(478\) 123.008 0.0117704
\(479\) 14214.3 1.35588 0.677940 0.735117i \(-0.262873\pi\)
0.677940 + 0.735117i \(0.262873\pi\)
\(480\) 0 0
\(481\) −3029.11 −0.287142
\(482\) 1544.93 0.145995
\(483\) 0 0
\(484\) −13018.2 −1.22260
\(485\) 0 0
\(486\) 0 0
\(487\) −18097.0 −1.68389 −0.841946 0.539562i \(-0.818590\pi\)
−0.841946 + 0.539562i \(0.818590\pi\)
\(488\) 365.998 0.0339507
\(489\) 0 0
\(490\) 0 0
\(491\) 13750.8 1.26388 0.631940 0.775017i \(-0.282259\pi\)
0.631940 + 0.775017i \(0.282259\pi\)
\(492\) 0 0
\(493\) 17.5048 0.00159914
\(494\) 1030.24 0.0938314
\(495\) 0 0
\(496\) 11146.9 1.00909
\(497\) −3437.82 −0.310276
\(498\) 0 0
\(499\) 15684.2 1.40706 0.703528 0.710667i \(-0.251607\pi\)
0.703528 + 0.710667i \(0.251607\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3328.61 −0.295943
\(503\) 977.937 0.0866880 0.0433440 0.999060i \(-0.486199\pi\)
0.0433440 + 0.999060i \(0.486199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1415.88 0.124394
\(507\) 0 0
\(508\) 15726.6 1.37354
\(509\) 19664.6 1.71242 0.856208 0.516632i \(-0.172814\pi\)
0.856208 + 0.516632i \(0.172814\pi\)
\(510\) 0 0
\(511\) −5701.78 −0.493604
\(512\) −9064.39 −0.782409
\(513\) 0 0
\(514\) −258.775 −0.0222064
\(515\) 0 0
\(516\) 0 0
\(517\) −29651.5 −2.52238
\(518\) 668.650 0.0567158
\(519\) 0 0
\(520\) 0 0
\(521\) −12522.8 −1.05304 −0.526519 0.850163i \(-0.676503\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(522\) 0 0
\(523\) −1104.42 −0.0923382 −0.0461691 0.998934i \(-0.514701\pi\)
−0.0461691 + 0.998934i \(0.514701\pi\)
\(524\) 22247.8 1.85477
\(525\) 0 0
\(526\) 1797.05 0.148964
\(527\) −81.1843 −0.00671052
\(528\) 0 0
\(529\) −9554.75 −0.785301
\(530\) 0 0
\(531\) 0 0
\(532\) 6912.65 0.563349
\(533\) −1221.66 −0.0992794
\(534\) 0 0
\(535\) 0 0
\(536\) −82.7170 −0.00666573
\(537\) 0 0
\(538\) −277.502 −0.0222378
\(539\) −2689.12 −0.214895
\(540\) 0 0
\(541\) −2126.45 −0.168989 −0.0844945 0.996424i \(-0.526928\pi\)
−0.0844945 + 0.996424i \(0.526928\pi\)
\(542\) 1991.33 0.157813
\(543\) 0 0
\(544\) 39.1817 0.00308805
\(545\) 0 0
\(546\) 0 0
\(547\) 2137.04 0.167044 0.0835220 0.996506i \(-0.473383\pi\)
0.0835220 + 0.996506i \(0.473383\pi\)
\(548\) 12559.8 0.979063
\(549\) 0 0
\(550\) 0 0
\(551\) −5288.11 −0.408859
\(552\) 0 0
\(553\) 6011.08 0.462237
\(554\) −633.335 −0.0485701
\(555\) 0 0
\(556\) −20274.8 −1.54648
\(557\) −19435.1 −1.47844 −0.739219 0.673465i \(-0.764805\pi\)
−0.739219 + 0.673465i \(0.764805\pi\)
\(558\) 0 0
\(559\) 4714.14 0.356685
\(560\) 0 0
\(561\) 0 0
\(562\) −344.008 −0.0258205
\(563\) 1467.52 0.109855 0.0549276 0.998490i \(-0.482507\pi\)
0.0549276 + 0.998490i \(0.482507\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3001.89 −0.222931
\(567\) 0 0
\(568\) 3903.35 0.288347
\(569\) −15080.3 −1.11107 −0.555536 0.831493i \(-0.687487\pi\)
−0.555536 + 0.831493i \(0.687487\pi\)
\(570\) 0 0
\(571\) 15800.8 1.15804 0.579022 0.815312i \(-0.303435\pi\)
0.579022 + 0.815312i \(0.303435\pi\)
\(572\) −6804.02 −0.497361
\(573\) 0 0
\(574\) 269.671 0.0196095
\(575\) 0 0
\(576\) 0 0
\(577\) −6165.64 −0.444851 −0.222425 0.974950i \(-0.571397\pi\)
−0.222425 + 0.974950i \(0.571397\pi\)
\(578\) 2479.91 0.178461
\(579\) 0 0
\(580\) 0 0
\(581\) 7390.62 0.527736
\(582\) 0 0
\(583\) 36293.0 2.57822
\(584\) 6473.89 0.458719
\(585\) 0 0
\(586\) −1095.90 −0.0772547
\(587\) 23104.0 1.62454 0.812268 0.583285i \(-0.198233\pi\)
0.812268 + 0.583285i \(0.198233\pi\)
\(588\) 0 0
\(589\) 24525.4 1.71571
\(590\) 0 0
\(591\) 0 0
\(592\) 10965.9 0.761313
\(593\) 6544.43 0.453200 0.226600 0.973988i \(-0.427239\pi\)
0.226600 + 0.973988i \(0.427239\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17447.0 1.19909
\(597\) 0 0
\(598\) 412.982 0.0282410
\(599\) −3215.71 −0.219349 −0.109675 0.993968i \(-0.534981\pi\)
−0.109675 + 0.993968i \(0.534981\pi\)
\(600\) 0 0
\(601\) −234.653 −0.0159263 −0.00796313 0.999968i \(-0.502535\pi\)
−0.00796313 + 0.999968i \(0.502535\pi\)
\(602\) −1040.61 −0.0704517
\(603\) 0 0
\(604\) 14325.6 0.965064
\(605\) 0 0
\(606\) 0 0
\(607\) 14528.7 0.971501 0.485750 0.874098i \(-0.338546\pi\)
0.485750 + 0.874098i \(0.338546\pi\)
\(608\) −11836.6 −0.789537
\(609\) 0 0
\(610\) 0 0
\(611\) −8648.71 −0.572650
\(612\) 0 0
\(613\) −11332.7 −0.746696 −0.373348 0.927691i \(-0.621790\pi\)
−0.373348 + 0.927691i \(0.621790\pi\)
\(614\) −1839.73 −0.120921
\(615\) 0 0
\(616\) 3053.27 0.199708
\(617\) −17400.2 −1.13534 −0.567672 0.823255i \(-0.692156\pi\)
−0.567672 + 0.823255i \(0.692156\pi\)
\(618\) 0 0
\(619\) −11987.8 −0.778402 −0.389201 0.921153i \(-0.627249\pi\)
−0.389201 + 0.921153i \(0.627249\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1304.46 0.0840900
\(623\) 2390.97 0.153759
\(624\) 0 0
\(625\) 0 0
\(626\) 3378.84 0.215728
\(627\) 0 0
\(628\) −25850.3 −1.64258
\(629\) −79.8667 −0.00506279
\(630\) 0 0
\(631\) −3800.38 −0.239764 −0.119882 0.992788i \(-0.538252\pi\)
−0.119882 + 0.992788i \(0.538252\pi\)
\(632\) −6825.08 −0.429568
\(633\) 0 0
\(634\) 1940.07 0.121530
\(635\) 0 0
\(636\) 0 0
\(637\) −784.359 −0.0487872
\(638\) −1148.96 −0.0712976
\(639\) 0 0
\(640\) 0 0
\(641\) −12294.2 −0.757552 −0.378776 0.925488i \(-0.623655\pi\)
−0.378776 + 0.925488i \(0.623655\pi\)
\(642\) 0 0
\(643\) −14615.4 −0.896382 −0.448191 0.893938i \(-0.647932\pi\)
−0.448191 + 0.893938i \(0.647932\pi\)
\(644\) 2771.01 0.169554
\(645\) 0 0
\(646\) 27.1637 0.00165440
\(647\) −16047.0 −0.975071 −0.487536 0.873103i \(-0.662104\pi\)
−0.487536 + 0.873103i \(0.662104\pi\)
\(648\) 0 0
\(649\) 22517.9 1.36195
\(650\) 0 0
\(651\) 0 0
\(652\) −3648.75 −0.219166
\(653\) 623.425 0.0373607 0.0186803 0.999826i \(-0.494054\pi\)
0.0186803 + 0.999826i \(0.494054\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4422.63 0.263224
\(657\) 0 0
\(658\) 1909.13 0.113109
\(659\) −963.459 −0.0569515 −0.0284758 0.999594i \(-0.509065\pi\)
−0.0284758 + 0.999594i \(0.509065\pi\)
\(660\) 0 0
\(661\) 24183.6 1.42304 0.711522 0.702664i \(-0.248006\pi\)
0.711522 + 0.702664i \(0.248006\pi\)
\(662\) 414.893 0.0243584
\(663\) 0 0
\(664\) −8391.43 −0.490438
\(665\) 0 0
\(666\) 0 0
\(667\) −2119.79 −0.123057
\(668\) 28476.9 1.64941
\(669\) 0 0
\(670\) 0 0
\(671\) −2527.20 −0.145397
\(672\) 0 0
\(673\) −14167.0 −0.811437 −0.405718 0.913998i \(-0.632979\pi\)
−0.405718 + 0.913998i \(0.632979\pi\)
\(674\) 521.215 0.0297870
\(675\) 0 0
\(676\) 15031.6 0.855235
\(677\) −19397.1 −1.10117 −0.550584 0.834780i \(-0.685595\pi\)
−0.550584 + 0.834780i \(0.685595\pi\)
\(678\) 0 0
\(679\) −9920.47 −0.560696
\(680\) 0 0
\(681\) 0 0
\(682\) 5328.71 0.299189
\(683\) 20984.8 1.17564 0.587820 0.808992i \(-0.299986\pi\)
0.587820 + 0.808992i \(0.299986\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 173.141 0.00963636
\(687\) 0 0
\(688\) −17066.1 −0.945694
\(689\) 10585.9 0.585328
\(690\) 0 0
\(691\) −18079.2 −0.995320 −0.497660 0.867372i \(-0.665807\pi\)
−0.497660 + 0.867372i \(0.665807\pi\)
\(692\) −7815.75 −0.429350
\(693\) 0 0
\(694\) 1306.23 0.0714465
\(695\) 0 0
\(696\) 0 0
\(697\) −32.2107 −0.00175046
\(698\) 4267.56 0.231418
\(699\) 0 0
\(700\) 0 0
\(701\) −18660.6 −1.00542 −0.502711 0.864455i \(-0.667664\pi\)
−0.502711 + 0.864455i \(0.667664\pi\)
\(702\) 0 0
\(703\) 24127.4 1.29443
\(704\) 22870.4 1.22438
\(705\) 0 0
\(706\) 2084.57 0.111125
\(707\) −847.358 −0.0450752
\(708\) 0 0
\(709\) −24545.4 −1.30017 −0.650085 0.759862i \(-0.725267\pi\)
−0.650085 + 0.759862i \(0.725267\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2714.74 −0.142892
\(713\) 9831.26 0.516387
\(714\) 0 0
\(715\) 0 0
\(716\) 24794.5 1.29415
\(717\) 0 0
\(718\) −2290.05 −0.119030
\(719\) 7300.32 0.378659 0.189330 0.981914i \(-0.439369\pi\)
0.189330 + 0.981914i \(0.439369\pi\)
\(720\) 0 0
\(721\) −4589.16 −0.237045
\(722\) −4743.74 −0.244520
\(723\) 0 0
\(724\) 15259.1 0.783286
\(725\) 0 0
\(726\) 0 0
\(727\) −15466.0 −0.788998 −0.394499 0.918896i \(-0.629082\pi\)
−0.394499 + 0.918896i \(0.629082\pi\)
\(728\) 890.574 0.0453391
\(729\) 0 0
\(730\) 0 0
\(731\) 124.295 0.00628893
\(732\) 0 0
\(733\) 13830.5 0.696917 0.348459 0.937324i \(-0.386705\pi\)
0.348459 + 0.937324i \(0.386705\pi\)
\(734\) 829.358 0.0417060
\(735\) 0 0
\(736\) −4744.83 −0.237631
\(737\) 571.157 0.0285466
\(738\) 0 0
\(739\) 11714.5 0.583118 0.291559 0.956553i \(-0.405826\pi\)
0.291559 + 0.956553i \(0.405826\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2336.75 −0.115613
\(743\) 28963.4 1.43010 0.715050 0.699074i \(-0.246404\pi\)
0.715050 + 0.699074i \(0.246404\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4552.91 0.223450
\(747\) 0 0
\(748\) −179.397 −0.00876928
\(749\) −14488.7 −0.706817
\(750\) 0 0
\(751\) 6331.99 0.307667 0.153833 0.988097i \(-0.450838\pi\)
0.153833 + 0.988097i \(0.450838\pi\)
\(752\) 31310.0 1.51829
\(753\) 0 0
\(754\) −335.128 −0.0161865
\(755\) 0 0
\(756\) 0 0
\(757\) 6256.80 0.300406 0.150203 0.988655i \(-0.452007\pi\)
0.150203 + 0.988655i \(0.452007\pi\)
\(758\) 1243.60 0.0595905
\(759\) 0 0
\(760\) 0 0
\(761\) −13953.7 −0.664679 −0.332340 0.943160i \(-0.607838\pi\)
−0.332340 + 0.943160i \(0.607838\pi\)
\(762\) 0 0
\(763\) −6329.53 −0.300320
\(764\) 30619.8 1.44998
\(765\) 0 0
\(766\) 5556.28 0.262084
\(767\) 6568.00 0.309200
\(768\) 0 0
\(769\) 34278.1 1.60741 0.803707 0.595025i \(-0.202858\pi\)
0.803707 + 0.595025i \(0.202858\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18785.5 0.875786
\(773\) 8769.31 0.408034 0.204017 0.978967i \(-0.434600\pi\)
0.204017 + 0.978967i \(0.434600\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 11263.9 0.521069
\(777\) 0 0
\(778\) 3110.88 0.143355
\(779\) 9730.73 0.447547
\(780\) 0 0
\(781\) −26952.5 −1.23487
\(782\) 10.8889 0.000497934 0
\(783\) 0 0
\(784\) 2839.53 0.129352
\(785\) 0 0
\(786\) 0 0
\(787\) 3546.15 0.160618 0.0803092 0.996770i \(-0.474409\pi\)
0.0803092 + 0.996770i \(0.474409\pi\)
\(788\) −24215.4 −1.09472
\(789\) 0 0
\(790\) 0 0
\(791\) 5903.45 0.265363
\(792\) 0 0
\(793\) −737.130 −0.0330091
\(794\) −7641.37 −0.341539
\(795\) 0 0
\(796\) 27620.3 1.22987
\(797\) −18169.4 −0.807520 −0.403760 0.914865i \(-0.632297\pi\)
−0.403760 + 0.914865i \(0.632297\pi\)
\(798\) 0 0
\(799\) −228.035 −0.0100968
\(800\) 0 0
\(801\) 0 0
\(802\) 1666.94 0.0733938
\(803\) −44702.0 −1.96451
\(804\) 0 0
\(805\) 0 0
\(806\) 1554.27 0.0679241
\(807\) 0 0
\(808\) 962.104 0.0418895
\(809\) −11853.0 −0.515115 −0.257557 0.966263i \(-0.582918\pi\)
−0.257557 + 0.966263i \(0.582918\pi\)
\(810\) 0 0
\(811\) −11921.1 −0.516161 −0.258080 0.966123i \(-0.583090\pi\)
−0.258080 + 0.966123i \(0.583090\pi\)
\(812\) −2248.62 −0.0971813
\(813\) 0 0
\(814\) 5242.22 0.225724
\(815\) 0 0
\(816\) 0 0
\(817\) −37548.9 −1.60792
\(818\) −2310.09 −0.0987415
\(819\) 0 0
\(820\) 0 0
\(821\) 42094.1 1.78940 0.894698 0.446671i \(-0.147390\pi\)
0.894698 + 0.446671i \(0.147390\pi\)
\(822\) 0 0
\(823\) −1622.17 −0.0687064 −0.0343532 0.999410i \(-0.510937\pi\)
−0.0343532 + 0.999410i \(0.510937\pi\)
\(824\) 5210.61 0.220291
\(825\) 0 0
\(826\) −1449.83 −0.0610727
\(827\) 28171.7 1.18455 0.592277 0.805735i \(-0.298229\pi\)
0.592277 + 0.805735i \(0.298229\pi\)
\(828\) 0 0
\(829\) −9244.23 −0.387293 −0.193646 0.981071i \(-0.562031\pi\)
−0.193646 + 0.981071i \(0.562031\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6670.81 0.277967
\(833\) −20.6807 −0.000860198 0
\(834\) 0 0
\(835\) 0 0
\(836\) 54195.2 2.24208
\(837\) 0 0
\(838\) −3421.45 −0.141041
\(839\) 43012.5 1.76991 0.884956 0.465674i \(-0.154188\pi\)
0.884956 + 0.465674i \(0.154188\pi\)
\(840\) 0 0
\(841\) −22668.8 −0.929469
\(842\) 1995.25 0.0816637
\(843\) 0 0
\(844\) 28650.9 1.16849
\(845\) 0 0
\(846\) 0 0
\(847\) −11765.7 −0.477302
\(848\) −38323.0 −1.55191
\(849\) 0 0
\(850\) 0 0
\(851\) 9671.70 0.389591
\(852\) 0 0
\(853\) 19756.9 0.793041 0.396521 0.918026i \(-0.370217\pi\)
0.396521 + 0.918026i \(0.370217\pi\)
\(854\) 162.715 0.00651991
\(855\) 0 0
\(856\) 16450.7 0.656862
\(857\) 325.658 0.0129805 0.00649024 0.999979i \(-0.497934\pi\)
0.00649024 + 0.999979i \(0.497934\pi\)
\(858\) 0 0
\(859\) 3275.22 0.130092 0.0650461 0.997882i \(-0.479281\pi\)
0.0650461 + 0.997882i \(0.479281\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1733.98 −0.0685147
\(863\) 44585.0 1.75862 0.879311 0.476247i \(-0.158003\pi\)
0.879311 + 0.476247i \(0.158003\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2755.02 −0.108106
\(867\) 0 0
\(868\) 10428.8 0.407805
\(869\) 47126.9 1.83967
\(870\) 0 0
\(871\) 166.594 0.00648087
\(872\) 7186.65 0.279095
\(873\) 0 0
\(874\) −3289.48 −0.127309
\(875\) 0 0
\(876\) 0 0
\(877\) −16433.8 −0.632761 −0.316380 0.948632i \(-0.602468\pi\)
−0.316380 + 0.948632i \(0.602468\pi\)
\(878\) −2114.08 −0.0812607
\(879\) 0 0
\(880\) 0 0
\(881\) 39080.2 1.49449 0.747245 0.664549i \(-0.231376\pi\)
0.747245 + 0.664549i \(0.231376\pi\)
\(882\) 0 0
\(883\) −48548.2 −1.85026 −0.925129 0.379652i \(-0.876044\pi\)
−0.925129 + 0.379652i \(0.876044\pi\)
\(884\) −52.3264 −0.00199087
\(885\) 0 0
\(886\) −1058.78 −0.0401473
\(887\) −13447.1 −0.509029 −0.254514 0.967069i \(-0.581916\pi\)
−0.254514 + 0.967069i \(0.581916\pi\)
\(888\) 0 0
\(889\) 14213.5 0.536227
\(890\) 0 0
\(891\) 0 0
\(892\) −790.325 −0.0296660
\(893\) 68888.5 2.58148
\(894\) 0 0
\(895\) 0 0
\(896\) −6671.30 −0.248742
\(897\) 0 0
\(898\) 5713.24 0.212309
\(899\) −7977.90 −0.295971
\(900\) 0 0
\(901\) 279.112 0.0103203
\(902\) 2114.22 0.0780442
\(903\) 0 0
\(904\) −6702.87 −0.246609
\(905\) 0 0
\(906\) 0 0
\(907\) −15750.3 −0.576604 −0.288302 0.957539i \(-0.593091\pi\)
−0.288302 + 0.957539i \(0.593091\pi\)
\(908\) −44456.7 −1.62483
\(909\) 0 0
\(910\) 0 0
\(911\) −31449.7 −1.14377 −0.571886 0.820333i \(-0.693788\pi\)
−0.571886 + 0.820333i \(0.693788\pi\)
\(912\) 0 0
\(913\) 57942.4 2.10034
\(914\) −5904.16 −0.213668
\(915\) 0 0
\(916\) 6530.17 0.235549
\(917\) 20107.3 0.724102
\(918\) 0 0
\(919\) −30355.6 −1.08960 −0.544798 0.838567i \(-0.683394\pi\)
−0.544798 + 0.838567i \(0.683394\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 5393.64 0.192657
\(923\) −7861.46 −0.280350
\(924\) 0 0
\(925\) 0 0
\(926\) −5253.39 −0.186433
\(927\) 0 0
\(928\) 3850.34 0.136200
\(929\) 28533.8 1.00771 0.503856 0.863788i \(-0.331914\pi\)
0.503856 + 0.863788i \(0.331914\pi\)
\(930\) 0 0
\(931\) 6247.56 0.219931
\(932\) 15627.1 0.549231
\(933\) 0 0
\(934\) 1503.37 0.0526678
\(935\) 0 0
\(936\) 0 0
\(937\) 19269.1 0.671817 0.335908 0.941895i \(-0.390957\pi\)
0.335908 + 0.941895i \(0.390957\pi\)
\(938\) −36.7743 −0.00128009
\(939\) 0 0
\(940\) 0 0
\(941\) 2097.84 0.0726756 0.0363378 0.999340i \(-0.488431\pi\)
0.0363378 + 0.999340i \(0.488431\pi\)
\(942\) 0 0
\(943\) 3900.66 0.134701
\(944\) −23777.4 −0.819797
\(945\) 0 0
\(946\) −8158.36 −0.280392
\(947\) 5072.14 0.174047 0.0870235 0.996206i \(-0.472264\pi\)
0.0870235 + 0.996206i \(0.472264\pi\)
\(948\) 0 0
\(949\) −13038.6 −0.445997
\(950\) 0 0
\(951\) 0 0
\(952\) 23.4813 0.000799403 0
\(953\) −23464.1 −0.797564 −0.398782 0.917046i \(-0.630567\pi\)
−0.398782 + 0.917046i \(0.630567\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1887.39 0.0638519
\(957\) 0 0
\(958\) −7175.13 −0.241981
\(959\) 11351.3 0.382225
\(960\) 0 0
\(961\) 7209.24 0.241994
\(962\) 1529.04 0.0512457
\(963\) 0 0
\(964\) 23704.8 0.791992
\(965\) 0 0
\(966\) 0 0
\(967\) 6994.27 0.232596 0.116298 0.993214i \(-0.462897\pi\)
0.116298 + 0.993214i \(0.462897\pi\)
\(968\) 13359.0 0.443568
\(969\) 0 0
\(970\) 0 0
\(971\) −37752.5 −1.24772 −0.623860 0.781536i \(-0.714436\pi\)
−0.623860 + 0.781536i \(0.714436\pi\)
\(972\) 0 0
\(973\) −18324.0 −0.603743
\(974\) 9135.09 0.300521
\(975\) 0 0
\(976\) 2668.55 0.0875186
\(977\) −6380.21 −0.208926 −0.104463 0.994529i \(-0.533312\pi\)
−0.104463 + 0.994529i \(0.533312\pi\)
\(978\) 0 0
\(979\) 18745.2 0.611950
\(980\) 0 0
\(981\) 0 0
\(982\) −6941.18 −0.225562
\(983\) 17330.0 0.562299 0.281150 0.959664i \(-0.409284\pi\)
0.281150 + 0.959664i \(0.409284\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.83612 −0.000285395 0
\(987\) 0 0
\(988\) 15807.6 0.509014
\(989\) −15051.9 −0.483945
\(990\) 0 0
\(991\) 32191.8 1.03189 0.515947 0.856621i \(-0.327440\pi\)
0.515947 + 0.856621i \(0.327440\pi\)
\(992\) −17857.3 −0.571542
\(993\) 0 0
\(994\) 1735.35 0.0553743
\(995\) 0 0
\(996\) 0 0
\(997\) 2678.20 0.0850748 0.0425374 0.999095i \(-0.486456\pi\)
0.0425374 + 0.999095i \(0.486456\pi\)
\(998\) −7917.13 −0.251114
\(999\) 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 1575.4.a.bl.1.2 4
3.2 odd 2 175.4.a.g.1.3 4
5.4 even 2 1575.4.a.bg.1.3 4
15.2 even 4 175.4.b.f.99.5 8
15.8 even 4 175.4.b.f.99.4 8
15.14 odd 2 175.4.a.h.1.2 yes 4
21.20 even 2 1225.4.a.z.1.3 4
105.104 even 2 1225.4.a.bd.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.3 4 3.2 odd 2
175.4.a.h.1.2 yes 4 15.14 odd 2
175.4.b.f.99.4 8 15.8 even 4
175.4.b.f.99.5 8 15.2 even 4
1225.4.a.z.1.3 4 21.20 even 2
1225.4.a.bd.1.2 4 105.104 even 2
1575.4.a.bg.1.3 4 5.4 even 2
1575.4.a.bl.1.2 4 1.1 even 1 trivial