Properties

Label 175.4.a.g.1.3
Level $175$
Weight $4$
Character 175.1
Self dual yes
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,4,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, 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("175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(10.3253342510\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.50478\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.504784 q^{2} -4.26379 q^{3} -7.74519 q^{4} -2.15229 q^{6} -7.00000 q^{7} -7.94792 q^{8} -8.82008 q^{9} +O(q^{10})\) \(q+0.504784 q^{2} -4.26379 q^{3} -7.74519 q^{4} -2.15229 q^{6} -7.00000 q^{7} -7.94792 q^{8} -8.82008 q^{9} +54.8800 q^{11} +33.0239 q^{12} -16.0073 q^{13} -3.53349 q^{14} +57.9496 q^{16} +0.422056 q^{17} -4.45223 q^{18} +127.501 q^{19} +29.8465 q^{21} +27.7025 q^{22} -51.1101 q^{23} +33.8883 q^{24} -8.08024 q^{26} +152.729 q^{27} +54.2164 q^{28} +41.4750 q^{29} +192.354 q^{31} +92.8353 q^{32} -233.997 q^{33} +0.213047 q^{34} +68.3132 q^{36} +189.232 q^{37} +64.3605 q^{38} +68.2519 q^{39} -76.3187 q^{41} +15.0660 q^{42} -294.499 q^{43} -425.056 q^{44} -25.7996 q^{46} -540.297 q^{47} -247.085 q^{48} +49.0000 q^{49} -1.79956 q^{51} +123.980 q^{52} +661.316 q^{53} +77.0953 q^{54} +55.6354 q^{56} -543.639 q^{57} +20.9359 q^{58} +410.312 q^{59} +46.0495 q^{61} +97.0974 q^{62} +61.7405 q^{63} -416.735 q^{64} -118.118 q^{66} -10.4074 q^{67} -3.26890 q^{68} +217.923 q^{69} -491.117 q^{71} +70.1012 q^{72} +814.540 q^{73} +95.5215 q^{74} -987.522 q^{76} -384.160 q^{77} +34.4525 q^{78} -858.725 q^{79} -413.064 q^{81} -38.5244 q^{82} +1055.80 q^{83} -231.167 q^{84} -148.658 q^{86} -176.841 q^{87} -436.182 q^{88} +341.567 q^{89} +112.051 q^{91} +395.858 q^{92} -820.159 q^{93} -272.733 q^{94} -395.831 q^{96} +1417.21 q^{97} +24.7344 q^{98} -484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} + 36 q^{4} + q^{6} - 28 q^{7} - 27 q^{8} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 3 q^{3} + 36 q^{4} + q^{6} - 28 q^{7} - 27 q^{8} + 61 q^{9} + 100 q^{11} + 165 q^{12} - 44 q^{13} + 28 q^{14} + 160 q^{16} + 53 q^{17} - 433 q^{18} - 29 q^{19} - 21 q^{21} + 152 q^{22} - 295 q^{23} - 21 q^{24} + 700 q^{26} + 441 q^{27} - 252 q^{28} + 129 q^{29} + 114 q^{31} + 310 q^{32} + 865 q^{33} + 203 q^{34} + 1101 q^{36} - 403 q^{37} - 555 q^{38} + 674 q^{39} + 671 q^{41} - 7 q^{42} + 411 q^{43} + 438 q^{44} - 997 q^{46} + 8 q^{47} + 523 q^{48} + 196 q^{49} - 885 q^{51} - 74 q^{52} - 90 q^{53} - 2777 q^{54} + 189 q^{56} - 233 q^{57} - 673 q^{58} + 1018 q^{59} + 50 q^{61} + 1626 q^{62} - 427 q^{63} - 2421 q^{64} - 3841 q^{66} - 424 q^{67} + 617 q^{68} + 1080 q^{69} + 215 q^{71} - 2940 q^{72} + 1207 q^{73} + 623 q^{74} - 3257 q^{76} - 700 q^{77} - 278 q^{78} - 951 q^{79} + 28 q^{81} - 1695 q^{82} + 3035 q^{83} - 1155 q^{84} - 99 q^{86} - 2210 q^{87} - 163 q^{88} + 2819 q^{89} + 308 q^{91} - 3073 q^{92} + 852 q^{93} - 3056 q^{94} - 1345 q^{96} + 1100 q^{97} - 196 q^{98} + 2383 q^{99}+O(q^{100}) \) 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 \(-0.634570\pi\)
−0.410284 + 0.911958i \(0.634570\pi\)
\(4\) −7.74519 −0.968149
\(5\) 0 0
\(6\) −2.15229 −0.146445
\(7\) −7.00000 −0.377964
\(8\) −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 \(-0.554621\pi\)
−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 \(-0.499042\pi\)
0.00301069 + 0.999995i \(0.499042\pi\)
\(18\) −4.45223 −0.0583000
\(19\) 127.501 1.53952 0.769758 0.638336i \(-0.220377\pi\)
0.769758 + 0.638336i \(0.220377\pi\)
\(20\) 0 0
\(21\) 29.8465 0.310145
\(22\) 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
\(28\) 54.2164 0.365926
\(29\) 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 \(-0.311866\pi\)
0.557224 + 0.830362i \(0.311866\pi\)
\(32\) 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 \(-0.546432\pi\)
−0.145353 + 0.989380i \(0.546432\pi\)
\(42\) 15.0660 0.0553510
\(43\) −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 \(-0.816513\pi\)
−0.838408 + 0.545043i \(0.816513\pi\)
\(48\) −247.085 −0.742992
\(49\) 49.0000 0.142857
\(50\) 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
\(56\) 55.6354 0.132761
\(57\) −543.639 −1.26328
\(58\) 20.9359 0.0473969
\(59\) 410.312 0.905390 0.452695 0.891665i \(-0.350463\pi\)
0.452695 + 0.891665i \(0.350463\pi\)
\(60\) 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
\(63\) 61.7405 0.123469
\(64\) −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 \(-0.273520\pi\)
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
\(78\) 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 \(-0.254016\pi\)
0.698129 + 0.715972i \(0.254016\pi\)
\(84\) −231.167 −0.300267
\(85\) 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 \(-0.434799\pi\)
0.203405 + 0.979095i \(0.434799\pi\)
\(90\) 0 0
\(91\) 112.051 0.129079
\(92\) 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 \(-0.234005\pi\)
0.741731 + 0.670697i \(0.234005\pi\)
\(98\) 24.7344 0.0254954
\(99\) −484.046 −0.491398
\(100\) 0 0
\(101\) −121.051 −0.119258 −0.0596289 0.998221i \(-0.518992\pi\)
−0.0596289 + 0.998221i \(0.518992\pi\)
\(102\) −0.908388 −0.000881802 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 \(-0.884628\pi\)
−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
\(112\) −405.647 −0.342232
\(113\) 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
\(119\) −2.95439 −0.00227587
\(120\) 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
\(126\) 31.1656 0.0220353
\(127\) −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 \(-0.0926954\pi\)
0.957897 + 0.287113i \(0.0926954\pi\)
\(132\) 1812.35 1.19504
\(133\) −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 \(-0.331258\pi\)
0.505636 + 0.862747i \(0.331258\pi\)
\(138\) 110.004 0.0678562
\(139\) 2617.72 1.59735 0.798677 0.601760i \(-0.205534\pi\)
0.798677 + 0.601760i \(0.205534\pi\)
\(140\) 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
\(147\) −208.926 −0.117224
\(148\) −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
\(154\) −193.918 −0.101470
\(155\) 0 0
\(156\) −528.624 −0.271307
\(157\) 3337.59 1.69661 0.848307 0.529505i \(-0.177622\pi\)
0.848307 + 0.529505i \(0.177622\pi\)
\(158\) −433.471 −0.218260
\(159\) −2819.71 −1.40640
\(160\) 0 0
\(161\) 357.771 0.175132
\(162\) −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 \(-0.175489\pi\)
0.851836 + 0.523809i \(0.175489\pi\)
\(168\) −237.218 −0.108939
\(169\) −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 \(-0.571173\pi\)
−0.221737 + 0.975106i \(0.571173\pi\)
\(174\) −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 \(-0.632564\pi\)
−0.404528 + 0.914526i \(0.632564\pi\)
\(182\) 56.5617 0.0230364
\(183\) −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
\(189\) −1069.11 −0.411460
\(190\) 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
\(196\) −379.514 −0.138307
\(197\) −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 \(-0.719068\pi\)
−0.635166 + 0.772376i \(0.719068\pi\)
\(200\) 0 0
\(201\) 44.3749 0.0155720
\(202\) −61.1046 −0.0212837
\(203\) −290.325 −0.100378
\(204\) 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
\(217\) −1346.48 −0.421222
\(218\) 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 \(-0.495123\pi\)
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 \(-0.816943\pi\)
−0.839144 + 0.543909i \(0.816943\pi\)
\(228\) 4210.59 1.22304
\(229\) −843.125 −0.243298 −0.121649 0.992573i \(-0.538818\pi\)
−0.121649 + 0.992573i \(0.538818\pi\)
\(230\) 0 0
\(231\) 1637.98 0.466541
\(232\) −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
\(238\) −1.49133 −0.000406170 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 \(-0.634131\pi\)
−0.409024 + 0.912524i \(0.634131\pi\)
\(242\) 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 \(-0.811158\pi\)
−0.829120 + 0.559070i \(0.811158\pi\)
\(252\) −478.192 −0.119537
\(253\) −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 \(-0.519816\pi\)
−0.0622139 + 0.998063i \(0.519816\pi\)
\(258\) 633.847 0.152952
\(259\) −1324.63 −0.317793
\(260\) 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
\(266\) −450.524 −0.103847
\(267\) −1456.37 −0.333814
\(268\) 80.6072 0.0183726
\(269\) −549.744 −0.124604 −0.0623020 0.998057i \(-0.519844\pi\)
−0.0623020 + 0.998057i \(0.519844\pi\)
\(270\) 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
\(273\) −477.764 −0.105918
\(274\) 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 \(-0.285275\pi\)
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
\(288\) −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 \(-0.569444\pi\)
−0.216438 + 0.976296i \(0.569444\pi\)
\(294\) −105.462 −0.0209207
\(295\) 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
\(301\) 2061.49 0.394759
\(302\) −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 \(-0.389988\pi\)
0.338775 + 0.940868i \(0.389988\pi\)
\(308\) 2975.39 0.550451
\(309\) −2795.32 −0.514628
\(310\) 0 0
\(311\) 2584.19 0.471177 0.235589 0.971853i \(-0.424298\pi\)
0.235589 + 0.971853i \(0.424298\pi\)
\(312\) −542.461 −0.0984320
\(313\) −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 \(-0.389410\pi\)
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
\(322\) 180.597 0.0312555
\(323\) 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
\(329\) 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
\(333\) −1669.05 −0.274664
\(334\) 1855.95 0.304051
\(335\) 0 0
\(336\) 1729.59 0.280825
\(337\) −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
\(343\) −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 \(-0.435852\pi\)
0.200166 + 0.979762i \(0.435852\pi\)
\(348\) 1369.67 0.210982
\(349\) −8454.24 −1.29669 −0.648345 0.761346i \(-0.724539\pi\)
−0.648345 + 0.761346i \(0.724539\pi\)
\(350\) 0 0
\(351\) −2444.79 −0.371776
\(352\) 5094.80 0.771460
\(353\) 4129.63 0.622658 0.311329 0.950302i \(-0.399226\pi\)
0.311329 + 0.950302i \(0.399226\pi\)
\(354\) −883.111 −0.132590
\(355\) 0 0
\(356\) −2645.50 −0.393852
\(357\) 12.5969 0.00186750
\(358\) 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
\(364\) −867.859 −0.124968
\(365\) 0 0
\(366\) −99.1120 −0.0141548
\(367\) −1643.00 −0.233689 −0.116844 0.993150i \(-0.537278\pi\)
−0.116844 + 0.993150i \(0.537278\pi\)
\(368\) −2961.81 −0.419552
\(369\) 673.137 0.0949650
\(370\) 0 0
\(371\) −4629.21 −0.647808
\(372\) 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
\(378\) −539.667 −0.0734325
\(379\) −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 \(-0.237529\pi\)
0.734261 + 0.678867i \(0.237529\pi\)
\(384\) 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
\(392\) −389.448 −0.0501788
\(393\) −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 \(-0.0938338\pi\)
0.956864 + 0.290537i \(0.0938338\pi\)
\(398\) −1800.12 −0.226714
\(399\) 3805.47 0.477473
\(400\) 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
\(406\) −146.551 −0.0179143
\(407\) 10385.1 1.26479
\(408\) 14.3027 0.00173552
\(409\) 4576.40 0.553273 0.276636 0.960975i \(-0.410780\pi\)
0.276636 + 0.960975i \(0.410780\pi\)
\(410\) 0 0
\(411\) −6914.25 −0.829817
\(412\) −5077.70 −0.607186
\(413\) −2872.18 −0.342205
\(414\) 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 \(-0.629305\pi\)
−0.395142 + 0.918620i \(0.629305\pi\)
\(420\) 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
\(427\) −322.346 −0.0365326
\(428\) 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 \(-0.402055\pi\)
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
\(438\) −1753.13 −0.191250
\(439\) 4188.10 0.455323 0.227662 0.973740i \(-0.426892\pi\)
0.227662 + 0.973740i \(0.426892\pi\)
\(440\) 0 0
\(441\) −432.184 −0.0466671
\(442\) −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
\(448\) 2917.14 0.307639
\(449\) 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 \(-0.318518\pi\)
0.539753 + 0.841823i \(0.318518\pi\)
\(462\) 826.825 0.0832627
\(463\) 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 \(-0.452860\pi\)
0.147555 + 0.989054i \(0.452860\pi\)
\(468\) −1093.51 −0.108008
\(469\) 72.8517 0.00717266
\(470\) 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
\(476\) 22.8823 0.00220338
\(477\) −5832.86 −0.559892
\(478\) 123.008 0.0117704
\(479\) −14214.3 −1.35588 −0.677940 0.735117i \(-0.737127\pi\)
−0.677940 + 0.735117i \(0.737127\pi\)
\(480\) 0 0
\(481\) −3029.11 −0.287142
\(482\) −1544.93 −0.145995
\(483\) −1525.46 −0.143708
\(484\) −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
\(497\) 3437.82 0.310276
\(498\) −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 \(-0.513801\pi\)
−0.0433440 + 0.999060i \(0.513801\pi\)
\(504\) −490.709 −0.0433689
\(505\) 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 \(-0.827186\pi\)
−0.856208 + 0.516632i \(0.827186\pi\)
\(510\) 0 0
\(511\) −5701.78 −0.493604
\(512\) 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
\(518\) −668.650 −0.0567158
\(519\) 4302.63 0.363901
\(520\) 0 0
\(521\) 12522.8 1.05304 0.526519 0.850163i \(-0.323497\pi\)
0.526519 + 0.850163i \(0.323497\pi\)
\(522\) −184.656 −0.0154831
\(523\) −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
\(528\) −13560.0 −1.11766
\(529\) −9554.75 −0.785301
\(530\) 0 0
\(531\) −3618.98 −0.295763
\(532\) 6912.65 0.563349
\(533\) 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
\(539\) 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
\(543\) 8400.25 0.663884
\(544\) 39.1817 0.00308805
\(545\) 0 0
\(546\) −241.167 −0.0189029
\(547\) 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
\(553\) 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 \(-0.235195\pi\)
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 \(-0.517493\pi\)
−0.0549276 + 0.998490i \(0.517493\pi\)
\(564\) −17842.7 −1.33212
\(565\) 0 0
\(566\) 3001.89 0.222931
\(567\) 2891.45 0.214161
\(568\) 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
\(574\) 269.671 0.0196095
\(575\) 0 0
\(576\) 3675.63 0.265888
\(577\) −6165.64 −0.444851 −0.222425 0.974950i \(-0.571397\pi\)
−0.222425 + 0.974950i \(0.571397\pi\)
\(578\) −2479.91 −0.178461
\(579\) 10341.6 0.742283
\(580\) 0 0
\(581\) −7390.62 −0.527736
\(582\) −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 \(-0.801767\pi\)
−0.812268 + 0.583285i \(0.801767\pi\)
\(588\) 1618.17 0.113490
\(589\) 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 \(-0.572761\pi\)
−0.226600 + 0.973988i \(0.572761\pi\)
\(594\) 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 \(-0.502535\pi\)
−0.00796313 + 0.999968i \(0.502535\pi\)
\(602\) 1040.61 0.0704517
\(603\) 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 \(-0.338546\pi\)
0.485750 + 0.874098i \(0.338546\pi\)
\(608\) 11836.6 0.789537
\(609\) 1237.89 0.0823672
\(610\) 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
\(616\) 3053.27 0.199708
\(617\) 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 \(-0.627249\pi\)
−0.389201 + 0.921153i \(0.627249\pi\)
\(620\) 0 0
\(621\) −7806.02 −0.504420
\(622\) 1304.46 0.0840900
\(623\) −2390.97 −0.153759
\(624\) 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
\(637\) −784.359 −0.0487872
\(638\) 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 \(-0.647932\pi\)
−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 \(-0.337896\pi\)
0.487536 + 0.873103i \(0.337896\pi\)
\(648\) 3283.00 0.199025
\(649\) 22517.9 1.36195
\(650\) 0 0
\(651\) 5741.12 0.345641
\(652\) −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
\(658\) 1909.13 0.113109
\(659\) 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 \(-0.248006\pi\)
0.711522 + 0.702664i \(0.248006\pi\)
\(662\) −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
\(672\) 2770.81 0.159057
\(673\) −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 \(-0.314405\pi\)
0.550584 + 0.834780i \(0.314405\pi\)
\(678\) −1815.14 −0.102817
\(679\) −9920.47 −0.560696
\(680\) 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
\(686\) −173.141 −0.00963636
\(687\) 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 \(-0.665807\pi\)
−0.497660 + 0.867372i \(0.665807\pi\)
\(692\) 7815.75 0.429350
\(693\) 3388.32 0.185731
\(694\) 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
\(707\) 847.358 0.0450752
\(708\) 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
\(714\) 6.35871 0.000333290 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 \(-0.560631\pi\)
−0.189330 + 0.981914i \(0.560631\pi\)
\(720\) 0 0
\(721\) −4589.16 −0.237045
\(722\) 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 \(-0.629082\pi\)
−0.394499 + 0.918896i \(0.629082\pi\)
\(728\) −890.574 −0.0453391
\(729\) 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 \(-0.386705\pi\)
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
\(738\) 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
\(742\) −2336.75 −0.115613
\(743\) −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
\(749\) 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
\(753\) 28116.0 1.36070
\(754\) −335.128 −0.0161865
\(755\) 0 0
\(756\) 8280.43 0.398355
\(757\) 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 \(-0.392162\pi\)
0.332340 + 0.943160i \(0.392162\pi\)
\(762\) 4370.23 0.207765
\(763\) −6329.53 −0.300320
\(764\) −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 \(-0.202858\pi\)
0.803707 + 0.595025i \(0.202858\pi\)
\(770\) 0 0
\(771\) 2185.82 0.102101
\(772\) 18785.5 0.875786
\(773\) −8769.31 −0.408034 −0.204017 0.978967i \(-0.565400\pi\)
−0.204017 + 0.978967i \(0.565400\pi\)
\(774\) 1311.18 0.0608905
\(775\) 0 0
\(776\) −11263.9 −0.521069
\(777\) 5647.94 0.260770
\(778\) 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
\(784\) 2839.53 0.129352
\(785\) 0 0
\(786\) −6182.40 −0.280558
\(787\) 3546.15 0.160618 0.0803092 0.996770i \(-0.474409\pi\)
0.0803092 + 0.996770i \(0.474409\pi\)
\(788\) 24215.4 1.09472
\(789\) −15179.3 −0.684912
\(790\) 0 0
\(791\) −5903.45 −0.265363
\(792\) 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 \(-0.367703\pi\)
0.403760 + 0.914865i \(0.367703\pi\)
\(798\) 1920.94 0.0852137
\(799\) −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 \(-0.583090\pi\)
−0.258080 + 0.966123i \(0.583090\pi\)
\(812\) 2248.62 0.0971813
\(813\) 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
\(819\) −988.301 −0.0421661
\(820\) 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
\(826\) −1449.83 −0.0610727
\(827\) −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 \(-0.562031\pi\)
−0.193646 + 0.981071i \(0.562031\pi\)
\(830\) 0 0
\(831\) −5349.63 −0.223317
\(832\) 6670.81 0.277967
\(833\) 20.6807 0.000860198 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 \(-0.845812\pi\)
−0.884956 + 0.465674i \(0.845812\pi\)
\(840\) 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
\(847\) −11765.7 −0.477302
\(848\) 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 \(-0.370217\pi\)
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 \(-0.502066\pi\)
−0.00649024 + 0.999979i \(0.502066\pi\)
\(858\) 1890.75 0.0752321
\(859\) 3275.22 0.130092 0.0650461 0.997882i \(-0.479281\pi\)
0.0650461 + 0.997882i \(0.479281\pi\)
\(860\) 0 0
\(861\) −2277.85 −0.0901613
\(862\) −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
\(868\) 10428.8 0.407805
\(869\) −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 \(-0.768624\pi\)
−0.747245 + 0.664549i \(0.768624\pi\)
\(882\) −218.159 −0.00832858
\(883\) −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 \(-0.418084\pi\)
0.254514 + 0.967069i \(0.418084\pi\)
\(888\) 6412.76 0.242340
\(889\) 14213.5 0.536227
\(890\) 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
\(896\) 6671.30 0.248742
\(897\) −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
\(903\) −8789.76 −0.323926
\(904\) −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
\(917\) −20107.3 −0.724102
\(918\) 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
\(924\) −12686.5 −0.451682
\(925\) 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 \(-0.668086\pi\)
−0.503856 + 0.863788i \(0.668086\pi\)
\(930\) 0 0
\(931\) 6247.56 0.219931
\(932\) −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 \(-0.390957\pi\)
0.335908 + 0.941895i \(0.390957\pi\)
\(938\) 36.7743 0.00128009
\(939\) 28540.3 0.991881
\(940\) 0 0
\(941\) −2097.84 −0.0726756 −0.0363378 0.999340i \(-0.511569\pi\)
−0.0363378 + 0.999340i \(0.511569\pi\)
\(942\) −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
\(952\) 23.4813 0.000799403 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
\(959\) −11351.3 −0.382225
\(960\) 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
\(966\) −770.028 −0.0256472
\(967\) 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 \(-0.285564\pi\)
0.623860 + 0.781536i \(0.285564\pi\)
\(972\) 18297.8 0.603809
\(973\) −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 \(-0.466688\pi\)
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 \(-0.590716\pi\)
−0.281150 + 0.959664i \(0.590716\pi\)
\(984\) −2586.31 −0.0837891
\(985\) 0 0
\(986\) 8.83612 0.000285395 0
\(987\) −16126.0 −0.520057
\(988\) 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
\(994\) 1735.35 0.0553743
\(995\) 0 0
\(996\) 34866.7 1.10923
\(997\) 2678.20 0.0850748 0.0425374 0.999095i \(-0.486456\pi\)
0.0425374 + 0.999095i \(0.486456\pi\)
\(998\) 7917.13 0.251114
\(999\) 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 175.4.a.g.1.3 4
3.2 odd 2 1575.4.a.bl.1.2 4
5.2 odd 4 175.4.b.f.99.5 8
5.3 odd 4 175.4.b.f.99.4 8
5.4 even 2 175.4.a.h.1.2 yes 4
7.6 odd 2 1225.4.a.z.1.3 4
15.14 odd 2 1575.4.a.bg.1.3 4
35.34 odd 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 1.1 even 1 trivial
175.4.a.h.1.2 yes 4 5.4 even 2
175.4.b.f.99.4 8 5.3 odd 4
175.4.b.f.99.5 8 5.2 odd 4
1225.4.a.z.1.3 4 7.6 odd 2
1225.4.a.bd.1.2 4 35.34 odd 2
1575.4.a.bg.1.3 4 15.14 odd 2
1575.4.a.bl.1.2 4 3.2 odd 2