Properties

Label 1225.4.a.z.1.3
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
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] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, 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("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(72.2773397570\)
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 1.3
Root \(1.50478\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +O(q^{10})\) \(q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +54.8800 q^{11} -33.0239 q^{12} +16.0073 q^{13} +57.9496 q^{16} -0.422056 q^{17} -4.45223 q^{18} -127.501 q^{19} +27.7025 q^{22} -51.1101 q^{23} -33.8883 q^{24} +8.08024 q^{26} -152.729 q^{27} +41.4750 q^{29} -192.354 q^{31} +92.8353 q^{32} +233.997 q^{33} -0.213047 q^{34} +68.3132 q^{36} +189.232 q^{37} -64.3605 q^{38} +68.2519 q^{39} +76.3187 q^{41} -294.499 q^{43} -425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +247.085 q^{48} -1.79956 q^{51} -123.980 q^{52} +661.316 q^{53} -77.0953 q^{54} -543.639 q^{57} +20.9359 q^{58} -410.312 q^{59} -46.0495 q^{61} -97.0974 q^{62} -416.735 q^{64} +118.118 q^{66} -10.4074 q^{67} +3.26890 q^{68} -217.923 q^{69} -491.117 q^{71} +70.1012 q^{72} -814.540 q^{73} +95.5215 q^{74} +987.522 q^{76} +34.4525 q^{78} -858.725 q^{79} -413.064 q^{81} +38.5244 q^{82} -1055.80 q^{83} -148.658 q^{86} +176.841 q^{87} -436.182 q^{88} -341.567 q^{89} +395.858 q^{92} -820.159 q^{93} +272.733 q^{94} +395.831 q^{96} -1417.21 q^{97} -484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26} - 441 q^{27} + 129 q^{29} - 114 q^{31} + 310 q^{32} - 865 q^{33} - 203 q^{34} + 1101 q^{36} - 403 q^{37} + 555 q^{38} + 674 q^{39} - 671 q^{41} + 411 q^{43} + 438 q^{44} - 997 q^{46} - 8 q^{47} - 523 q^{48} - 885 q^{51} + 74 q^{52} - 90 q^{53} + 2777 q^{54} - 233 q^{57} - 673 q^{58} - 1018 q^{59} - 50 q^{61} - 1626 q^{62} - 2421 q^{64} + 3841 q^{66} - 424 q^{67} - 617 q^{68} - 1080 q^{69} + 215 q^{71} - 2940 q^{72} - 1207 q^{73} + 623 q^{74} + 3257 q^{76} - 278 q^{78} - 951 q^{79} + 28 q^{81} + 1695 q^{82} - 3035 q^{83} - 99 q^{86} + 2210 q^{87} - 163 q^{88} - 2819 q^{89} - 3073 q^{92} + 852 q^{93} + 3056 q^{94} + 1345 q^{96} - 1100 q^{97} + 2383 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.504784 0.178468 0.0892340 0.996011i \(-0.471558\pi\)
0.0892340 + 0.996011i \(0.471558\pi\)
\(3\) 4.26379 0.820567 0.410284 0.911958i \(-0.365430\pi\)
0.410284 + 0.911958i \(0.365430\pi\)
\(4\) −7.74519 −0.968149
\(5\) 0 0
\(6\) 2.15229 0.146445
\(7\) 0 0
\(8\) −7.94792 −0.351252
\(9\) −8.82008 −0.326670
\(10\) 0 0
\(11\) 54.8800 1.50427 0.752134 0.659010i \(-0.229025\pi\)
0.752134 + 0.659010i \(0.229025\pi\)
\(12\) −33.0239 −0.794431
\(13\) 16.0073 0.341510 0.170755 0.985313i \(-0.445379\pi\)
0.170755 + 0.985313i \(0.445379\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 57.9496 0.905462
\(17\) −0.422056 −0.00602139 −0.00301069 0.999995i \(-0.500958\pi\)
−0.00301069 + 0.999995i \(0.500958\pi\)
\(18\) −4.45223 −0.0583000
\(19\) −127.501 −1.53952 −0.769758 0.638336i \(-0.779623\pi\)
−0.769758 + 0.638336i \(0.779623\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 27.7025 0.268464
\(23\) −51.1101 −0.463357 −0.231678 0.972792i \(-0.574422\pi\)
−0.231678 + 0.972792i \(0.574422\pi\)
\(24\) −33.8883 −0.288226
\(25\) 0 0
\(26\) 8.08024 0.0609487
\(27\) −152.729 −1.08862
\(28\) 0 0
\(29\) 41.4750 0.265576 0.132788 0.991144i \(-0.457607\pi\)
0.132788 + 0.991144i \(0.457607\pi\)
\(30\) 0 0
\(31\) −192.354 −1.11445 −0.557224 0.830362i \(-0.688134\pi\)
−0.557224 + 0.830362i \(0.688134\pi\)
\(32\) 92.8353 0.512848
\(33\) 233.997 1.23435
\(34\) −0.213047 −0.00107462
\(35\) 0 0
\(36\) 68.3132 0.316265
\(37\) 189.232 0.840801 0.420400 0.907339i \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(38\) −64.3605 −0.274754
\(39\) 68.2519 0.280232
\(40\) 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 \(0.183487\pi\)
\(48\) 247.085 0.742992
\(49\) 0 0
\(50\) 0 0
\(51\) −1.79956 −0.00494095
\(52\) −123.980 −0.330633
\(53\) 661.316 1.71394 0.856969 0.515368i \(-0.172345\pi\)
0.856969 + 0.515368i \(0.172345\pi\)
\(54\) −77.0953 −0.194284
\(55\) 0 0
\(56\) 0 0
\(57\) −543.639 −1.26328
\(58\) 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 \(-0.515389\pi\)
−0.0483281 + 0.998832i \(0.515389\pi\)
\(62\) −97.0974 −0.198893
\(63\) 0 0
\(64\) −416.735 −0.813935
\(65\) 0 0
\(66\) 118.118 0.220292
\(67\) −10.4074 −0.0189771 −0.00948854 0.999955i \(-0.503020\pi\)
−0.00948854 + 0.999955i \(0.503020\pi\)
\(68\) 3.26890 0.00582960
\(69\) −217.923 −0.380215
\(70\) 0 0
\(71\) −491.117 −0.820913 −0.410456 0.911880i \(-0.634631\pi\)
−0.410456 + 0.911880i \(0.634631\pi\)
\(72\) 70.1012 0.114743
\(73\) −814.540 −1.30595 −0.652977 0.757378i \(-0.726480\pi\)
−0.652977 + 0.757378i \(0.726480\pi\)
\(74\) 95.5215 0.150056
\(75\) 0 0
\(76\) 987.522 1.49048
\(77\) 0 0
\(78\) 34.4525 0.0500125
\(79\) −858.725 −1.22296 −0.611482 0.791258i \(-0.709426\pi\)
−0.611482 + 0.791258i \(0.709426\pi\)
\(80\) 0 0
\(81\) −413.064 −0.566617
\(82\) 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
\(87\) 176.841 0.217923
\(88\) −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
\(91\) 0 0
\(92\) 395.858 0.448598
\(93\) −820.159 −0.914479
\(94\) 272.733 0.299258
\(95\) 0 0
\(96\) 395.831 0.420826
\(97\) −1417.21 −1.48346 −0.741731 0.670697i \(-0.765995\pi\)
−0.741731 + 0.670697i \(0.765995\pi\)
\(98\) 0 0
\(99\) −484.046 −0.491398
\(100\) 0 0
\(101\) 121.051 0.119258 0.0596289 0.998221i \(-0.481008\pi\)
0.0596289 + 0.998221i \(0.481008\pi\)
\(102\) −0.908388 −0.000881802 0
\(103\) −655.594 −0.627161 −0.313581 0.949562i \(-0.601529\pi\)
−0.313581 + 0.949562i \(0.601529\pi\)
\(104\) −127.225 −0.119956
\(105\) 0 0
\(106\) 333.821 0.305883
\(107\) −2069.81 −1.87006 −0.935031 0.354567i \(-0.884628\pi\)
−0.935031 + 0.354567i \(0.884628\pi\)
\(108\) 1182.92 1.05395
\(109\) 904.218 0.794573 0.397286 0.917695i \(-0.369952\pi\)
0.397286 + 0.917695i \(0.369952\pi\)
\(110\) 0 0
\(111\) 806.848 0.689934
\(112\) 0 0
\(113\) 843.350 0.702086 0.351043 0.936359i \(-0.385827\pi\)
0.351043 + 0.936359i \(0.385827\pi\)
\(114\) −274.420 −0.225454
\(115\) 0 0
\(116\) −321.232 −0.257117
\(117\) −141.186 −0.111561
\(118\) −207.119 −0.161583
\(119\) 0 0
\(120\) 0 0
\(121\) 1680.81 1.26282
\(122\) −23.2450 −0.0172501
\(123\) 325.407 0.238544
\(124\) 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
\(129\) −1255.68 −0.857027
\(130\) 0 0
\(131\) −2872.47 −1.91579 −0.957897 0.287113i \(-0.907305\pi\)
−0.957897 + 0.287113i \(0.907305\pi\)
\(132\) −1812.35 −1.19504
\(133\) 0 0
\(134\) −5.25348 −0.00338680
\(135\) 0 0
\(136\) 3.35446 0.00211502
\(137\) 1621.62 1.01127 0.505636 0.862747i \(-0.331258\pi\)
0.505636 + 0.862747i \(0.331258\pi\)
\(138\) −110.004 −0.0678562
\(139\) −2617.72 −1.59735 −0.798677 0.601760i \(-0.794466\pi\)
−0.798677 + 0.601760i \(0.794466\pi\)
\(140\) 0 0
\(141\) 2303.71 1.37594
\(142\) −247.908 −0.146507
\(143\) 878.482 0.513723
\(144\) −511.120 −0.295787
\(145\) 0 0
\(146\) −411.166 −0.233071
\(147\) 0 0
\(148\) −1465.64 −0.814021
\(149\) 2252.63 1.23854 0.619270 0.785178i \(-0.287429\pi\)
0.619270 + 0.785178i \(0.287429\pi\)
\(150\) 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
\(153\) 3.72257 0.00196700
\(154\) 0 0
\(155\) 0 0
\(156\) −528.624 −0.271307
\(157\) −3337.59 −1.69661 −0.848307 0.529505i \(-0.822378\pi\)
−0.848307 + 0.529505i \(0.822378\pi\)
\(158\) −433.471 −0.218260
\(159\) 2819.71 1.40640
\(160\) 0 0
\(161\) 0 0
\(162\) −208.508 −0.101123
\(163\) 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
\(171\) 1124.57 0.502913
\(172\) 2280.95 1.01117
\(173\) 1009.11 0.443475 0.221737 0.975106i \(-0.428827\pi\)
0.221737 + 0.975106i \(0.428827\pi\)
\(174\) 89.2663 0.0388923
\(175\) 0 0
\(176\) 3180.27 1.36206
\(177\) −1749.48 −0.742934
\(178\) −172.417 −0.0726024
\(179\) 3201.27 1.33673 0.668364 0.743835i \(-0.266995\pi\)
0.668364 + 0.743835i \(0.266995\pi\)
\(180\) 0 0
\(181\) 1970.14 0.809056 0.404528 0.914526i \(-0.367436\pi\)
0.404528 + 0.914526i \(0.367436\pi\)
\(182\) 0 0
\(183\) −196.345 −0.0793130
\(184\) 406.219 0.162755
\(185\) 0 0
\(186\) −414.003 −0.163205
\(187\) −23.1624 −0.00905778
\(188\) −4184.70 −1.62341
\(189\) 0 0
\(190\) 0 0
\(191\) 3953.39 1.49768 0.748842 0.662749i \(-0.230610\pi\)
0.748842 + 0.662749i \(0.230610\pi\)
\(192\) −1776.87 −0.667888
\(193\) −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
\(196\) 0 0
\(197\) −3126.50 −1.13073 −0.565366 0.824840i \(-0.691265\pi\)
−0.565366 + 0.824840i \(0.691265\pi\)
\(198\) −244.338 −0.0876989
\(199\) 3566.13 1.27033 0.635166 0.772376i \(-0.280932\pi\)
0.635166 + 0.772376i \(0.280932\pi\)
\(200\) 0 0
\(201\) −44.3749 −0.0155720
\(202\) 61.1046 0.0212837
\(203\) 0 0
\(204\) 13.9379 0.00478358
\(205\) 0 0
\(206\) −330.933 −0.111928
\(207\) 450.795 0.151364
\(208\) 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
\(213\) −2094.02 −0.673614
\(214\) −1044.81 −0.333746
\(215\) 0 0
\(216\) 1213.88 0.382380
\(217\) 0 0
\(218\) 456.435 0.141806
\(219\) −3473.03 −1.07162
\(220\) 0 0
\(221\) −6.75599 −0.00205637
\(222\) 407.284 0.123131
\(223\) −102.041 −0.0306419 −0.0153210 0.999883i \(-0.504877\pi\)
−0.0153210 + 0.999883i \(0.504877\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 425.709 0.125300
\(227\) 5739.91 1.67829 0.839144 0.543909i \(-0.183057\pi\)
0.839144 + 0.543909i \(0.183057\pi\)
\(228\) 4210.59 1.22304
\(229\) 843.125 0.243298 0.121649 0.992573i \(-0.461182\pi\)
0.121649 + 0.992573i \(0.461182\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −329.640 −0.0932841
\(233\) 2017.65 0.567300 0.283650 0.958928i \(-0.408455\pi\)
0.283650 + 0.958928i \(0.408455\pi\)
\(234\) −71.2683 −0.0199101
\(235\) 0 0
\(236\) 3177.94 0.876553
\(237\) −3661.43 −1.00352
\(238\) 0 0
\(239\) 243.685 0.0659526 0.0329763 0.999456i \(-0.489501\pi\)
0.0329763 + 0.999456i \(0.489501\pi\)
\(240\) 0 0
\(241\) 3060.58 0.818048 0.409024 0.912524i \(-0.365869\pi\)
0.409024 + 0.912524i \(0.365869\pi\)
\(242\) 848.448 0.225373
\(243\) 2362.47 0.623674
\(244\) 356.662 0.0935777
\(245\) 0 0
\(246\) 164.260 0.0425725
\(247\) −2040.95 −0.525760
\(248\) 1528.82 0.391452
\(249\) −4501.72 −1.14572
\(250\) 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 \(0.480184\pi\)
\(258\) −633.847 −0.152952
\(259\) 0 0
\(260\) 0 0
\(261\) −365.813 −0.0867557
\(262\) −1449.98 −0.341908
\(263\) 3560.04 0.834681 0.417341 0.908750i \(-0.362962\pi\)
0.417341 + 0.908750i \(0.362962\pi\)
\(264\) −1859.79 −0.433568
\(265\) 0 0
\(266\) 0 0
\(267\) −1456.37 −0.333814
\(268\) 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 \(-0.354222\pi\)
0.442134 + 0.896949i \(0.354222\pi\)
\(272\) −24.4580 −0.00545214
\(273\) 0 0
\(274\) 818.567 0.180480
\(275\) 0 0
\(276\) 1687.86 0.368105
\(277\) 1254.67 0.272150 0.136075 0.990699i \(-0.456551\pi\)
0.136075 + 0.990699i \(0.456551\pi\)
\(278\) −1321.38 −0.285077
\(279\) 1696.58 0.364056
\(280\) 0 0
\(281\) −681.496 −0.144679 −0.0723393 0.997380i \(-0.523046\pi\)
−0.0723393 + 0.997380i \(0.523046\pi\)
\(282\) 1162.88 0.245561
\(283\) −5946.88 −1.24914 −0.624568 0.780970i \(-0.714725\pi\)
−0.624568 + 0.780970i \(0.714725\pi\)
\(284\) 3803.79 0.794766
\(285\) 0 0
\(286\) 443.444 0.0916831
\(287\) 0 0
\(288\) −818.815 −0.167532
\(289\) −4912.82 −0.999964
\(290\) 0 0
\(291\) −6042.69 −1.21728
\(292\) 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
\(297\) −8381.79 −1.63758
\(298\) 1137.09 0.221040
\(299\) −818.137 −0.158241
\(300\) 0 0
\(301\) 0 0
\(302\) −933.651 −0.177899
\(303\) 516.137 0.0978590
\(304\) −7388.64 −1.39397
\(305\) 0 0
\(306\) 1.87909 0.000351047 0
\(307\) −3644.59 −0.677549 −0.338775 0.940868i \(-0.610012\pi\)
−0.338775 + 0.940868i \(0.610012\pi\)
\(308\) 0 0
\(309\) −2795.32 −0.514628
\(310\) 0 0
\(311\) −2584.19 −0.471177 −0.235589 0.971853i \(-0.575702\pi\)
−0.235589 + 0.971853i \(0.575702\pi\)
\(312\) −542.461 −0.0984320
\(313\) 6693.63 1.20878 0.604388 0.796690i \(-0.293418\pi\)
0.604388 + 0.796690i \(0.293418\pi\)
\(314\) −1684.76 −0.302791
\(315\) 0 0
\(316\) 6650.99 1.18401
\(317\) 3843.36 0.680962 0.340481 0.940251i \(-0.389410\pi\)
0.340481 + 0.940251i \(0.389410\pi\)
\(318\) 1423.35 0.250998
\(319\) 2276.15 0.399498
\(320\) 0 0
\(321\) −8825.26 −1.53451
\(322\) 0 0
\(323\) 53.8126 0.00927002
\(324\) 3199.26 0.548570
\(325\) 0 0
\(326\) 237.803 0.0404009
\(327\) 3855.40 0.652000
\(328\) −606.575 −0.102111
\(329\) 0 0
\(330\) 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
\(333\) −1669.05 −0.274664
\(334\) −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
\(339\) 3595.87 0.576108
\(340\) 0 0
\(341\) −10556.4 −1.67643
\(342\) 567.665 0.0897538
\(343\) 0 0
\(344\) 2340.65 0.366859
\(345\) 0 0
\(346\) 509.382 0.0791461
\(347\) 2587.70 0.400332 0.200166 0.979762i \(-0.435852\pi\)
0.200166 + 0.979762i \(0.435852\pi\)
\(348\) −1369.67 −0.210982
\(349\) 8454.24 1.29669 0.648345 0.761346i \(-0.275461\pi\)
0.648345 + 0.761346i \(0.275461\pi\)
\(350\) 0 0
\(351\) −2444.79 −0.371776
\(352\) 5094.80 0.771460
\(353\) −4129.63 −0.622658 −0.311329 0.950302i \(-0.600774\pi\)
−0.311329 + 0.950302i \(0.600774\pi\)
\(354\) −883.111 −0.132590
\(355\) 0 0
\(356\) 2645.50 0.393852
\(357\) 0 0
\(358\) 1615.95 0.238563
\(359\) −4536.69 −0.666957 −0.333478 0.942758i \(-0.608222\pi\)
−0.333478 + 0.942758i \(0.608222\pi\)
\(360\) 0 0
\(361\) 9397.56 1.37011
\(362\) 994.492 0.144390
\(363\) 7166.65 1.03623
\(364\) 0 0
\(365\) 0 0
\(366\) −99.1120 −0.0141548
\(367\) 1643.00 0.233689 0.116844 0.993150i \(-0.462722\pi\)
0.116844 + 0.993150i \(0.462722\pi\)
\(368\) −2961.81 −0.419552
\(369\) −673.137 −0.0949650
\(370\) 0 0
\(371\) 0 0
\(372\) 6352.29 0.885352
\(373\) −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
\(381\) −8657.63 −1.16416
\(382\) 1995.61 0.267289
\(383\) −11007.2 −1.46852 −0.734261 0.678867i \(-0.762471\pi\)
−0.734261 + 0.678867i \(0.762471\pi\)
\(384\) −4063.58 −0.540023
\(385\) 0 0
\(386\) −1224.33 −0.161442
\(387\) 2597.50 0.341184
\(388\) 10976.6 1.43621
\(389\) 6162.79 0.803254 0.401627 0.915803i \(-0.368445\pi\)
0.401627 + 0.915803i \(0.368445\pi\)
\(390\) 0 0
\(391\) 21.5713 0.00279005
\(392\) 0 0
\(393\) −12247.6 −1.57204
\(394\) −1578.21 −0.201799
\(395\) 0 0
\(396\) 3749.03 0.475747
\(397\) −15137.9 −1.91373 −0.956864 0.290537i \(-0.906166\pi\)
−0.956864 + 0.290537i \(0.906166\pi\)
\(398\) 1800.12 0.226714
\(399\) 0 0
\(400\) 0 0
\(401\) 3302.29 0.411243 0.205622 0.978632i \(-0.434078\pi\)
0.205622 + 0.978632i \(0.434078\pi\)
\(402\) −22.3997 −0.00277910
\(403\) −3079.08 −0.380596
\(404\) −937.564 −0.115459
\(405\) 0 0
\(406\) 0 0
\(407\) 10385.1 1.26479
\(408\) 14.3027 0.00173552
\(409\) −4576.40 −0.553273 −0.276636 0.960975i \(-0.589220\pi\)
−0.276636 + 0.960975i \(0.589220\pi\)
\(410\) 0 0
\(411\) 6914.25 0.829817
\(412\) 5077.70 0.607186
\(413\) 0 0
\(414\) 227.554 0.0270137
\(415\) 0 0
\(416\) 1486.05 0.175143
\(417\) −11161.4 −1.31074
\(418\) −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
\(423\) −4765.46 −0.547765
\(424\) −5256.08 −0.602024
\(425\) 0 0
\(426\) −1057.03 −0.120219
\(427\) 0 0
\(428\) 16031.1 1.81050
\(429\) 3745.67 0.421544
\(430\) 0 0
\(431\) −3435.10 −0.383905 −0.191952 0.981404i \(-0.561482\pi\)
−0.191952 + 0.981404i \(0.561482\pi\)
\(432\) −8850.60 −0.985705
\(433\) −5457.82 −0.605742 −0.302871 0.953032i \(-0.597945\pi\)
−0.302871 + 0.953032i \(0.597945\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7003.35 −0.769265
\(437\) 6516.61 0.713344
\(438\) −1753.13 −0.191250
\(439\) −4188.10 −0.455323 −0.227662 0.973740i \(-0.573108\pi\)
−0.227662 + 0.973740i \(0.573108\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.41031 −0.000366996 0
\(443\) −2097.50 −0.224955 −0.112478 0.993654i \(-0.535879\pi\)
−0.112478 + 0.993654i \(0.535879\pi\)
\(444\) −6249.19 −0.667959
\(445\) 0 0
\(446\) −51.5085 −0.00546860
\(447\) 9604.74 1.01631
\(448\) 0 0
\(449\) 11318.2 1.18962 0.594809 0.803867i \(-0.297228\pi\)
0.594809 + 0.803867i \(0.297228\pi\)
\(450\) 0 0
\(451\) 4188.37 0.437301
\(452\) −6531.91 −0.679724
\(453\) −7886.34 −0.817952
\(454\) 2897.41 0.299521
\(455\) 0 0
\(456\) 4320.79 0.443728
\(457\) 11696.4 1.19723 0.598617 0.801036i \(-0.295717\pi\)
0.598617 + 0.801036i \(0.295717\pi\)
\(458\) 425.596 0.0434209
\(459\) 64.4603 0.00655501
\(460\) 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 \(0.547140\pi\)
\(468\) 1093.51 0.108008
\(469\) 0 0
\(470\) 0 0
\(471\) −14230.8 −1.39219
\(472\) 3261.12 0.318020
\(473\) −16162.1 −1.57111
\(474\) −1848.23 −0.179097
\(475\) 0 0
\(476\) 0 0
\(477\) −5832.86 −0.559892
\(478\) 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
\(486\) 1192.54 0.111306
\(487\) −18097.0 −1.68389 −0.841946 0.539562i \(-0.818590\pi\)
−0.841946 + 0.539562i \(0.818590\pi\)
\(488\) 365.998 0.0339507
\(489\) 2008.67 0.185757
\(490\) 0 0
\(491\) −13750.8 −1.26388 −0.631940 0.775017i \(-0.717741\pi\)
−0.631940 + 0.775017i \(0.717741\pi\)
\(492\) −2520.34 −0.230947
\(493\) −17.5048 −0.00159914
\(494\) −1030.24 −0.0938314
\(495\) 0 0
\(496\) −11146.9 −1.00909
\(497\) 0 0
\(498\) −2272.40 −0.204475
\(499\) 15684.2 1.40706 0.703528 0.710667i \(-0.251607\pi\)
0.703528 + 0.710667i \(0.251607\pi\)
\(500\) 0 0
\(501\) −15676.8 −1.39798
\(502\) 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
\(507\) −8275.02 −0.724865
\(508\) 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
\(511\) 0 0
\(512\) 9064.39 0.782409
\(513\) 19473.2 1.67595
\(514\) 258.775 0.0222064
\(515\) 0 0
\(516\) 9725.49 0.829730
\(517\) 29651.5 2.52238
\(518\) 0 0
\(519\) 4302.63 0.363901
\(520\) 0 0
\(521\) −12522.8 −1.05304 −0.526519 0.850163i \(-0.676503\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(522\) −184.656 −0.0154831
\(523\) 1104.42 0.0923382 0.0461691 0.998934i \(-0.485299\pi\)
0.0461691 + 0.998934i \(0.485299\pi\)
\(524\) 22247.8 1.85477
\(525\) 0 0
\(526\) 1797.05 0.148964
\(527\) 81.1843 0.00671052
\(528\) 13560.0 1.11766
\(529\) −9554.75 −0.785301
\(530\) 0 0
\(531\) 3618.98 0.295763
\(532\) 0 0
\(533\) 1221.66 0.0992794
\(534\) −735.152 −0.0595751
\(535\) 0 0
\(536\) 82.7170 0.00666573
\(537\) 13649.5 1.09687
\(538\) 277.502 0.0222378
\(539\) 0 0
\(540\) 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
\(543\) 8400.25 0.663884
\(544\) −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
\(549\) 406.160 0.0315747
\(550\) 0 0
\(551\) −5288.11 −0.408859
\(552\) 1732.03 0.133551
\(553\) 0 0
\(554\) 633.335 0.0485701
\(555\) 0 0
\(556\) 20274.8 1.54648
\(557\) 19435.1 1.47844 0.739219 0.673465i \(-0.235195\pi\)
0.739219 + 0.673465i \(0.235195\pi\)
\(558\) 856.407 0.0649724
\(559\) −4714.14 −0.356685
\(560\) 0 0
\(561\) −98.7598 −0.00743252
\(562\) −344.008 −0.0258205
\(563\) 1467.52 0.109855 0.0549276 0.998490i \(-0.482507\pi\)
0.0549276 + 0.998490i \(0.482507\pi\)
\(564\) −17842.7 −1.33212
\(565\) 0 0
\(566\) −3001.89 −0.222931
\(567\) 0 0
\(568\) 3903.35 0.288347
\(569\) 15080.3 1.11107 0.555536 0.831493i \(-0.312513\pi\)
0.555536 + 0.831493i \(0.312513\pi\)
\(570\) 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
\(573\) 16856.5 1.22895
\(574\) 0 0
\(575\) 0 0
\(576\) 3675.63 0.265888
\(577\) 6165.64 0.444851 0.222425 0.974950i \(-0.428603\pi\)
0.222425 + 0.974950i \(0.428603\pi\)
\(578\) −2479.91 −0.178461
\(579\) −10341.6 −0.742283
\(580\) 0 0
\(581\) 0 0
\(582\) −3050.25 −0.217246
\(583\) 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
\(591\) −13330.8 −0.927841
\(592\) 10965.9 0.761313
\(593\) 6544.43 0.453200 0.226600 0.973988i \(-0.427239\pi\)
0.226600 + 0.973988i \(0.427239\pi\)
\(594\) −4230.99 −0.292255
\(595\) 0 0
\(596\) −17447.0 −1.19909
\(597\) 15205.2 1.04239
\(598\) −412.982 −0.0282410
\(599\) 3215.71 0.219349 0.109675 0.993968i \(-0.465019\pi\)
0.109675 + 0.993968i \(0.465019\pi\)
\(600\) 0 0
\(601\) 234.653 0.0159263 0.00796313 0.999968i \(-0.497465\pi\)
0.00796313 + 0.999968i \(0.497465\pi\)
\(602\) 0 0
\(603\) 91.7939 0.00619923
\(604\) 14325.6 0.965064
\(605\) 0 0
\(606\) 260.537 0.0174647
\(607\) −14528.7 −0.971501 −0.485750 0.874098i \(-0.661454\pi\)
−0.485750 + 0.874098i \(0.661454\pi\)
\(608\) −11836.6 −0.789537
\(609\) 0 0
\(610\) 0 0
\(611\) 8648.71 0.572650
\(612\) −28.8320 −0.00190435
\(613\) −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
\(616\) 0 0
\(617\) 17400.2 1.13534 0.567672 0.823255i \(-0.307844\pi\)
0.567672 + 0.823255i \(0.307844\pi\)
\(618\) −1411.03 −0.0918446
\(619\) 11987.8 0.778402 0.389201 0.921153i \(-0.372751\pi\)
0.389201 + 0.921153i \(0.372751\pi\)
\(620\) 0 0
\(621\) 7806.02 0.504420
\(622\) −1304.46 −0.0840900
\(623\) 0 0
\(624\) 3955.17 0.253740
\(625\) 0 0
\(626\) 3378.84 0.215728
\(627\) −29834.9 −1.90030
\(628\) 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
\(633\) −15772.6 −0.990368
\(634\) 1940.07 0.121530
\(635\) 0 0
\(636\) −21839.2 −1.36161
\(637\) 0 0
\(638\) 1148.96 0.0712976
\(639\) 4331.69 0.268167
\(640\) 0 0
\(641\) 12294.2 0.757552 0.378776 0.925488i \(-0.376345\pi\)
0.378776 + 0.925488i \(0.376345\pi\)
\(642\) −4454.85 −0.273861
\(643\) 14615.4 0.896382 0.448191 0.893938i \(-0.352068\pi\)
0.448191 + 0.893938i \(0.352068\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 27.1637 0.00165440
\(647\) −16047.0 −0.975071 −0.487536 0.873103i \(-0.662104\pi\)
−0.487536 + 0.873103i \(0.662104\pi\)
\(648\) 3283.00 0.199025
\(649\) −22517.9 −1.36195
\(650\) 0 0
\(651\) 0 0
\(652\) −3648.75 −0.219166
\(653\) −623.425 −0.0373607 −0.0186803 0.999826i \(-0.505946\pi\)
−0.0186803 + 0.999826i \(0.505946\pi\)
\(654\) 1946.14 0.116361
\(655\) 0 0
\(656\) 4422.63 0.263224
\(657\) 7184.30 0.426616
\(658\) 0 0
\(659\) 963.459 0.0569515 0.0284758 0.999594i \(-0.490935\pi\)
0.0284758 + 0.999594i \(0.490935\pi\)
\(660\) 0 0
\(661\) −24183.6 −1.42304 −0.711522 0.702664i \(-0.751994\pi\)
−0.711522 + 0.702664i \(0.751994\pi\)
\(662\) −414.893 −0.0243584
\(663\) −28.8061 −0.00168739
\(664\) 8391.43 0.490438
\(665\) 0 0
\(666\) −842.507 −0.0490187
\(667\) −2119.79 −0.123057
\(668\) 28476.9 1.64941
\(669\) −435.080 −0.0251438
\(670\) 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 \(0.685595\pi\)
\(678\) 1815.14 0.102817
\(679\) 0 0
\(680\) 0 0
\(681\) 24473.8 1.37715
\(682\) −5328.71 −0.299189
\(683\) −20984.8 −1.17564 −0.587820 0.808992i \(-0.700014\pi\)
−0.587820 + 0.808992i \(0.700014\pi\)
\(684\) −8710.02 −0.486895
\(685\) 0 0
\(686\) 0 0
\(687\) 3594.91 0.199643
\(688\) −17066.1 −0.945694
\(689\) 10585.9 0.585328
\(690\) 0 0
\(691\) 18079.2 0.995320 0.497660 0.867372i \(-0.334193\pi\)
0.497660 + 0.867372i \(0.334193\pi\)
\(692\) −7815.75 −0.429350
\(693\) 0 0
\(694\) 1306.23 0.0714465
\(695\) 0 0
\(696\) −1405.52 −0.0765459
\(697\) −32.2107 −0.00175046
\(698\) 4267.56 0.231418
\(699\) 8602.86 0.465508
\(700\) 0 0
\(701\) 18660.6 1.00542 0.502711 0.864455i \(-0.332336\pi\)
0.502711 + 0.864455i \(0.332336\pi\)
\(702\) −1234.09 −0.0663500
\(703\) −24127.4 −1.29443
\(704\) −22870.4 −1.22438
\(705\) 0 0
\(706\) −2084.57 −0.111125
\(707\) 0 0
\(708\) 13550.1 0.719271
\(709\) −24545.4 −1.30017 −0.650085 0.759862i \(-0.725267\pi\)
−0.650085 + 0.759862i \(0.725267\pi\)
\(710\) 0 0
\(711\) 7574.02 0.399505
\(712\) 2714.74 0.142892
\(713\) 9831.26 0.516387
\(714\) 0 0
\(715\) 0 0
\(716\) −24794.5 −1.29415
\(717\) 1039.02 0.0541185
\(718\) −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
\(721\) 0 0
\(722\) 4743.74 0.244520
\(723\) 13049.7 0.671263
\(724\) −15259.1 −0.783286
\(725\) 0 0
\(726\) 3617.61 0.184934
\(727\) 15466.0 0.788998 0.394499 0.918896i \(-0.370918\pi\)
0.394499 + 0.918896i \(0.370918\pi\)
\(728\) 0 0
\(729\) 21225.8 1.07838
\(730\) 0 0
\(731\) 124.295 0.00628893
\(732\) 1520.73 0.0767868
\(733\) −13830.5 −0.696917 −0.348459 0.937324i \(-0.613295\pi\)
−0.348459 + 0.937324i \(0.613295\pi\)
\(734\) 829.358 0.0417060
\(735\) 0 0
\(736\) −4744.83 −0.237631
\(737\) −571.157 −0.0285466
\(738\) −339.788 −0.0169482
\(739\) 11714.5 0.583118 0.291559 0.956553i \(-0.405826\pi\)
0.291559 + 0.956553i \(0.405826\pi\)
\(740\) 0 0
\(741\) −8702.20 −0.431422
\(742\) 0 0
\(743\) −28963.4 −1.43010 −0.715050 0.699074i \(-0.753596\pi\)
−0.715050 + 0.699074i \(0.753596\pi\)
\(744\) 6518.56 0.321212
\(745\) 0 0
\(746\) −4552.91 −0.223450
\(747\) 9312.26 0.456115
\(748\) 179.397 0.00876928
\(749\) 0 0
\(750\) 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
\(753\) 28116.0 1.36070
\(754\) 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
\(759\) −11959.6 −0.571945
\(760\) 0 0
\(761\) −13953.7 −0.664679 −0.332340 0.943160i \(-0.607838\pi\)
−0.332340 + 0.943160i \(0.607838\pi\)
\(762\) −4370.23 −0.207765
\(763\) 0 0
\(764\) −30619.8 −1.44998
\(765\) 0 0
\(766\) −5556.28 −0.262084
\(767\) −6568.00 −0.309200
\(768\) 12163.7 0.571512
\(769\) −34278.1 −1.60741 −0.803707 0.595025i \(-0.797142\pi\)
−0.803707 + 0.595025i \(0.797142\pi\)
\(770\) 0 0
\(771\) 2185.82 0.102101
\(772\) 18785.5 0.875786
\(773\) 8769.31 0.408034 0.204017 0.978967i \(-0.434600\pi\)
0.204017 + 0.978967i \(0.434600\pi\)
\(774\) 1311.18 0.0608905
\(775\) 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
\(783\) −6334.45 −0.289112
\(784\) 0 0
\(785\) 0 0
\(786\) −6182.40 −0.280558
\(787\) −3546.15 −0.160618 −0.0803092 0.996770i \(-0.525591\pi\)
−0.0803092 + 0.996770i \(0.525591\pi\)
\(788\) 24215.4 1.09472
\(789\) 15179.3 0.684912
\(790\) 0 0
\(791\) 0 0
\(792\) 3847.16 0.172604
\(793\) −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
\(801\) 3012.65 0.132892
\(802\) 1666.94 0.0733938
\(803\) −44702.0 −1.96451
\(804\) 343.692 0.0150760
\(805\) 0 0
\(806\) −1554.27 −0.0679241
\(807\) 2343.99 0.102246
\(808\) −962.104 −0.0418895
\(809\) 11853.0 0.515115 0.257557 0.966263i \(-0.417082\pi\)
0.257557 + 0.966263i \(0.417082\pi\)
\(810\) 0 0
\(811\) 11921.1 0.516161 0.258080 0.966123i \(-0.416910\pi\)
0.258080 + 0.966123i \(0.416910\pi\)
\(812\) 0 0
\(813\) 16820.3 0.725601
\(814\) 5242.22 0.225724
\(815\) 0 0
\(816\) −104.284 −0.00447384
\(817\) 37548.9 1.60792
\(818\) −2310.09 −0.0987415
\(819\) 0 0
\(820\) 0 0
\(821\) −42094.1 −1.78940 −0.894698 0.446671i \(-0.852610\pi\)
−0.894698 + 0.446671i \(0.852610\pi\)
\(822\) 3490.20 0.148096
\(823\) −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
\(826\) 0 0
\(827\) −28171.7 −1.18455 −0.592277 0.805735i \(-0.701771\pi\)
−0.592277 + 0.805735i \(0.701771\pi\)
\(828\) −3491.50 −0.146543
\(829\) 9244.23 0.387293 0.193646 0.981071i \(-0.437969\pi\)
0.193646 + 0.981071i \(0.437969\pi\)
\(830\) 0 0
\(831\) 5349.63 0.223317
\(832\) −6670.81 −0.277967
\(833\) 0 0
\(834\) −5634.10 −0.233924
\(835\) 0 0
\(836\) 54195.2 2.24208
\(837\) 29378.2 1.21321
\(838\) 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
\(843\) −2905.76 −0.118718
\(844\) 28650.9 1.16849
\(845\) 0 0
\(846\) −2405.53 −0.0977585
\(847\) 0 0
\(848\) 38323.0 1.55191
\(849\) −25356.3 −1.02500
\(850\) 0 0
\(851\) −9671.70 −0.389591
\(852\) 16218.6 0.652159
\(853\) −19756.9 −0.793041 −0.396521 0.918026i \(-0.629783\pi\)
−0.396521 + 0.918026i \(0.629783\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16450.7 0.656862
\(857\) 325.658 0.0129805 0.00649024 0.999979i \(-0.497934\pi\)
0.00649024 + 0.999979i \(0.497934\pi\)
\(858\) 1890.75 0.0752321
\(859\) −3275.22 −0.130092 −0.0650461 0.997882i \(-0.520719\pi\)
−0.0650461 + 0.997882i \(0.520719\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1733.98 −0.0685147
\(863\) −44585.0 −1.75862 −0.879311 0.476247i \(-0.841997\pi\)
−0.879311 + 0.476247i \(0.841997\pi\)
\(864\) −14178.7 −0.558297
\(865\) 0 0
\(866\) −2755.02 −0.108106
\(867\) −20947.3 −0.820537
\(868\) 0 0
\(869\) −47126.9 −1.83967
\(870\) 0 0
\(871\) −166.594 −0.00648087
\(872\) −7186.65 −0.279095
\(873\) 12499.9 0.484602
\(874\) 3289.48 0.127309
\(875\) 0 0
\(876\) 26899.3 1.03749
\(877\) −16433.8 −0.632761 −0.316380 0.948632i \(-0.602468\pi\)
−0.316380 + 0.948632i \(0.602468\pi\)
\(878\) −2114.08 −0.0812607
\(879\) 9256.82 0.355205
\(880\) 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 \(0.581916\pi\)
\(888\) −6412.76 −0.242340
\(889\) 0 0
\(890\) 0 0
\(891\) −22669.0 −0.852344
\(892\) 790.325 0.0296660
\(893\) −68888.5 −2.58148
\(894\) 4848.32 0.181378
\(895\) 0 0
\(896\) 0 0
\(897\) −3488.37 −0.129847
\(898\) 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
\(906\) −3980.89 −0.145978
\(907\) −15750.3 −0.576604 −0.288302 0.957539i \(-0.593091\pi\)
−0.288302 + 0.957539i \(0.593091\pi\)
\(908\) −44456.7 −1.62483
\(909\) −1067.68 −0.0389579
\(910\) 0 0
\(911\) 31449.7 1.14377 0.571886 0.820333i \(-0.306212\pi\)
0.571886 + 0.820333i \(0.306212\pi\)
\(912\) −31503.6 −1.14385
\(913\) −57942.4 −2.10034
\(914\) 5904.16 0.213668
\(915\) 0 0
\(916\) −6530.17 −0.235549
\(917\) 0 0
\(918\) 32.5385 0.00116986
\(919\) −30355.6 −1.08960 −0.544798 0.838567i \(-0.683394\pi\)
−0.544798 + 0.838567i \(0.683394\pi\)
\(920\) 0 0
\(921\) −15539.8 −0.555975
\(922\) −5393.64 −0.192657
\(923\) −7861.46 −0.280350
\(924\) 0 0
\(925\) 0 0
\(926\) 5253.39 0.186433
\(927\) 5782.39 0.204874
\(928\) 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
\(931\) 0 0
\(932\) −15627.1 −0.549231
\(933\) −11018.5 −0.386632
\(934\) −1503.37 −0.0526678
\(935\) 0 0
\(936\) 1122.13 0.0391860
\(937\) −19269.1 −0.671817 −0.335908 0.941895i \(-0.609043\pi\)
−0.335908 + 0.941895i \(0.609043\pi\)
\(938\) 0 0
\(939\) 28540.3 0.991881
\(940\) 0 0
\(941\) 2097.84 0.0726756 0.0363378 0.999340i \(-0.488431\pi\)
0.0363378 + 0.999340i \(0.488431\pi\)
\(942\) −7183.46 −0.248461
\(943\) −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 \(-0.527736\pi\)
−0.0870235 + 0.996206i \(0.527736\pi\)
\(948\) 28358.5 0.971561
\(949\) −13038.6 −0.445997
\(950\) 0 0
\(951\) 16387.3 0.558775
\(952\) 0 0
\(953\) 23464.1 0.797564 0.398782 0.917046i \(-0.369433\pi\)
0.398782 + 0.917046i \(0.369433\pi\)
\(954\) −2944.33 −0.0999227
\(955\) 0 0
\(956\) −1887.39 −0.0638519
\(957\) 9705.02 0.327815
\(958\) 7175.13 0.241981
\(959\) 0 0
\(960\) 0 0
\(961\) 7209.24 0.241994
\(962\) 1529.04 0.0512457
\(963\) 18255.9 0.610892
\(964\) −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
\(969\) 229.446 0.00760667
\(970\) 0 0
\(971\) −37752.5 −1.24772 −0.623860 0.781536i \(-0.714436\pi\)
−0.623860 + 0.781536i \(0.714436\pi\)
\(972\) −18297.8 −0.603809
\(973\) 0 0
\(974\) −9135.09 −0.300521
\(975\) 0 0
\(976\) −2668.55 −0.0875186
\(977\) 6380.21 0.208926 0.104463 0.994529i \(-0.466688\pi\)
0.104463 + 0.994529i \(0.466688\pi\)
\(978\) 1013.94 0.0331517
\(979\) −18745.2 −0.611950
\(980\) 0 0
\(981\) −7975.28 −0.259563
\(982\) −6941.18 −0.225562
\(983\) 17330.0 0.562299 0.281150 0.959664i \(-0.409284\pi\)
0.281150 + 0.959664i \(0.409284\pi\)
\(984\) −2586.31 −0.0837891
\(985\) 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
\(993\) −3504.50 −0.111996
\(994\) 0 0
\(995\) 0 0
\(996\) 34866.7 1.10923
\(997\) −2678.20 −0.0850748 −0.0425374 0.999095i \(-0.513544\pi\)
−0.0425374 + 0.999095i \(0.513544\pi\)
\(998\) 7917.13 0.251114
\(999\) −28901.4 −0.915314
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 1225.4.a.z.1.3 4
5.4 even 2 1225.4.a.bd.1.2 4
7.6 odd 2 175.4.a.g.1.3 4
21.20 even 2 1575.4.a.bl.1.2 4
35.13 even 4 175.4.b.f.99.4 8
35.27 even 4 175.4.b.f.99.5 8
35.34 odd 2 175.4.a.h.1.2 yes 4
105.104 even 2 1575.4.a.bg.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.3 4 7.6 odd 2
175.4.a.h.1.2 yes 4 35.34 odd 2
175.4.b.f.99.4 8 35.13 even 4
175.4.b.f.99.5 8 35.27 even 4
1225.4.a.z.1.3 4 1.1 even 1 trivial
1225.4.a.bd.1.2 4 5.4 even 2
1575.4.a.bg.1.3 4 105.104 even 2
1575.4.a.bl.1.2 4 21.20 even 2