Properties

Label 900.3.c.q.451.2
Level $900$
Weight $3$
Character 900.451
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(451,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.451");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12239922073600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 32x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 451.2
Root \(1.89639 - 0.635381i\) of defining polynomial
Character \(\chi\) \(=\) 900.451
Dual form 900.3.c.q.451.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.89639 + 0.635381i) q^{2} +(3.19258 - 2.40986i) q^{4} +10.1035i q^{7} +(-4.52320 + 6.59853i) q^{8} +O(q^{10})\) \(q+(-1.89639 + 0.635381i) q^{2} +(3.19258 - 2.40986i) q^{4} +10.1035i q^{7} +(-4.52320 + 6.59853i) q^{8} -10.6555i q^{11} +11.7703 q^{13} +(-6.41959 - 19.1602i) q^{14} +(4.38516 - 15.3873i) q^{16} +16.6320 q^{17} +0.464096i q^{19} +(6.77033 + 20.2071i) q^{22} -42.1327i q^{23} +(-22.3211 + 7.47864i) q^{26} +(24.3481 + 32.2564i) q^{28} -19.5536 q^{29} +9.17534i q^{31} +(1.46084 + 31.9666i) q^{32} +(-31.5407 + 10.5676i) q^{34} +23.5407 q^{37} +(-0.294878 - 0.880107i) q^{38} +44.0525 q^{41} +58.3007i q^{43} +(-25.6784 - 34.0187i) q^{44} +(26.7703 + 79.9000i) q^{46} +41.6433i q^{47} -53.0813 q^{49} +(37.5777 - 28.3648i) q^{52} -80.2381 q^{53} +(-66.6685 - 45.7003i) q^{56} +(37.0813 - 12.4240i) q^{58} -63.9333i q^{59} +80.3923 q^{61} +(-5.82983 - 17.4000i) q^{62} +(-23.0813 - 59.6930i) q^{64} -22.5275i q^{67} +(53.0989 - 40.0807i) q^{68} +61.4860i q^{71} +137.703 q^{73} +(-44.6422 + 14.9573i) q^{74} +(1.11841 + 1.48167i) q^{76} +107.659 q^{77} -138.665i q^{79} +(-83.5407 + 27.9901i) q^{82} +86.2233i q^{83} +(-37.0431 - 110.561i) q^{86} +(70.3110 + 48.1972i) q^{88} +127.212 q^{89} +118.922i q^{91} +(-101.534 - 134.512i) q^{92} +(-26.4593 - 78.9719i) q^{94} +15.0000 q^{97} +(100.663 - 33.7269i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 8 q^{13} - 8 q^{16} - 32 q^{22} + 44 q^{28} - 80 q^{34} + 16 q^{37} + 128 q^{46} - 80 q^{49} + 236 q^{52} - 48 q^{58} + 40 q^{61} + 160 q^{64} + 240 q^{73} - 228 q^{76} - 496 q^{82} + 304 q^{88} - 384 q^{94} + 120 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89639 + 0.635381i −0.948194 + 0.317690i
\(3\) 0 0
\(4\) 3.19258 2.40986i 0.798146 0.602465i
\(5\) 0 0
\(6\) 0 0
\(7\) 10.1035i 1.44336i 0.692226 + 0.721681i \(0.256630\pi\)
−0.692226 + 0.721681i \(0.743370\pi\)
\(8\) −4.52320 + 6.59853i −0.565400 + 0.824817i
\(9\) 0 0
\(10\) 0 0
\(11\) 10.6555i 0.968686i −0.874878 0.484343i \(-0.839059\pi\)
0.874878 0.484343i \(-0.160941\pi\)
\(12\) 0 0
\(13\) 11.7703 0.905410 0.452705 0.891660i \(-0.350459\pi\)
0.452705 + 0.891660i \(0.350459\pi\)
\(14\) −6.41959 19.1602i −0.458542 1.36859i
\(15\) 0 0
\(16\) 4.38516 15.3873i 0.274073 0.961709i
\(17\) 16.6320 0.978350 0.489175 0.872186i \(-0.337298\pi\)
0.489175 + 0.872186i \(0.337298\pi\)
\(18\) 0 0
\(19\) 0.464096i 0.0244261i 0.999925 + 0.0122131i \(0.00388763\pi\)
−0.999925 + 0.0122131i \(0.996112\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.77033 + 20.2071i 0.307742 + 0.918503i
\(23\) 42.1327i 1.83186i −0.401340 0.915929i \(-0.631456\pi\)
0.401340 0.915929i \(-0.368544\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −22.3211 + 7.47864i −0.858505 + 0.287640i
\(27\) 0 0
\(28\) 24.3481 + 32.2564i 0.869574 + 1.15201i
\(29\) −19.5536 −0.674264 −0.337132 0.941457i \(-0.609457\pi\)
−0.337132 + 0.941457i \(0.609457\pi\)
\(30\) 0 0
\(31\) 9.17534i 0.295979i 0.988989 + 0.147989i \(0.0472801\pi\)
−0.988989 + 0.147989i \(0.952720\pi\)
\(32\) 1.46084 + 31.9666i 0.0456514 + 0.998957i
\(33\) 0 0
\(34\) −31.5407 + 10.5676i −0.927666 + 0.310813i
\(35\) 0 0
\(36\) 0 0
\(37\) 23.5407 0.636234 0.318117 0.948051i \(-0.396950\pi\)
0.318117 + 0.948051i \(0.396950\pi\)
\(38\) −0.294878 0.880107i −0.00775994 0.0231607i
\(39\) 0 0
\(40\) 0 0
\(41\) 44.0525 1.07445 0.537226 0.843439i \(-0.319472\pi\)
0.537226 + 0.843439i \(0.319472\pi\)
\(42\) 0 0
\(43\) 58.3007i 1.35583i 0.735140 + 0.677915i \(0.237116\pi\)
−0.735140 + 0.677915i \(0.762884\pi\)
\(44\) −25.6784 34.0187i −0.583599 0.773152i
\(45\) 0 0
\(46\) 26.7703 + 79.9000i 0.581964 + 1.73696i
\(47\) 41.6433i 0.886027i 0.896515 + 0.443014i \(0.146091\pi\)
−0.896515 + 0.443014i \(0.853909\pi\)
\(48\) 0 0
\(49\) −53.0813 −1.08329
\(50\) 0 0
\(51\) 0 0
\(52\) 37.5777 28.3648i 0.722649 0.545477i
\(53\) −80.2381 −1.51393 −0.756963 0.653458i \(-0.773318\pi\)
−0.756963 + 0.653458i \(0.773318\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −66.6685 45.7003i −1.19051 0.816077i
\(57\) 0 0
\(58\) 37.0813 12.4240i 0.639333 0.214207i
\(59\) 63.9333i 1.08361i −0.840503 0.541807i \(-0.817740\pi\)
0.840503 0.541807i \(-0.182260\pi\)
\(60\) 0 0
\(61\) 80.3923 1.31791 0.658953 0.752184i \(-0.270999\pi\)
0.658953 + 0.752184i \(0.270999\pi\)
\(62\) −5.82983 17.4000i −0.0940296 0.280645i
\(63\) 0 0
\(64\) −23.0813 59.6930i −0.360646 0.932703i
\(65\) 0 0
\(66\) 0 0
\(67\) 22.5275i 0.336232i −0.985767 0.168116i \(-0.946232\pi\)
0.985767 0.168116i \(-0.0537683\pi\)
\(68\) 53.0989 40.0807i 0.780866 0.589421i
\(69\) 0 0
\(70\) 0 0
\(71\) 61.4860i 0.866000i 0.901394 + 0.433000i \(0.142545\pi\)
−0.901394 + 0.433000i \(0.857455\pi\)
\(72\) 0 0
\(73\) 137.703 1.88635 0.943173 0.332301i \(-0.107825\pi\)
0.943173 + 0.332301i \(0.107825\pi\)
\(74\) −44.6422 + 14.9573i −0.603274 + 0.202125i
\(75\) 0 0
\(76\) 1.11841 + 1.48167i 0.0147159 + 0.0194956i
\(77\) 107.659 1.39816
\(78\) 0 0
\(79\) 138.665i 1.75525i −0.479347 0.877626i \(-0.659126\pi\)
0.479347 0.877626i \(-0.340874\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −83.5407 + 27.9901i −1.01879 + 0.341343i
\(83\) 86.2233i 1.03883i 0.854521 + 0.519417i \(0.173851\pi\)
−0.854521 + 0.519417i \(0.826149\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −37.0431 110.561i −0.430734 1.28559i
\(87\) 0 0
\(88\) 70.3110 + 48.1972i 0.798989 + 0.547695i
\(89\) 127.212 1.42935 0.714676 0.699456i \(-0.246574\pi\)
0.714676 + 0.699456i \(0.246574\pi\)
\(90\) 0 0
\(91\) 118.922i 1.30683i
\(92\) −101.534 134.512i −1.10363 1.46209i
\(93\) 0 0
\(94\) −26.4593 78.9719i −0.281482 0.840126i
\(95\) 0 0
\(96\) 0 0
\(97\) 15.0000 0.154639 0.0773196 0.997006i \(-0.475364\pi\)
0.0773196 + 0.997006i \(0.475364\pi\)
\(98\) 100.663 33.7269i 1.02717 0.344152i
\(99\) 0 0
\(100\) 0 0
\(101\) 195.764 1.93825 0.969127 0.246563i \(-0.0793012\pi\)
0.969127 + 0.246563i \(0.0793012\pi\)
\(102\) 0 0
\(103\) 15.5661i 0.151127i 0.997141 + 0.0755636i \(0.0240756\pi\)
−0.997141 + 0.0755636i \(0.975924\pi\)
\(104\) −53.2396 + 77.6669i −0.511919 + 0.746797i
\(105\) 0 0
\(106\) 152.163 50.9817i 1.43550 0.480960i
\(107\) 51.3199i 0.479625i 0.970819 + 0.239813i \(0.0770860\pi\)
−0.970819 + 0.239813i \(0.922914\pi\)
\(108\) 0 0
\(109\) −46.8516 −0.429832 −0.214916 0.976633i \(-0.568948\pi\)
−0.214916 + 0.976633i \(0.568948\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 155.466 + 44.3056i 1.38809 + 0.395586i
\(113\) −115.526 −1.02235 −0.511175 0.859477i \(-0.670790\pi\)
−0.511175 + 0.859477i \(0.670790\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −62.4266 + 47.1215i −0.538161 + 0.406220i
\(117\) 0 0
\(118\) 40.6220 + 121.242i 0.344254 + 1.02748i
\(119\) 168.041i 1.41211i
\(120\) 0 0
\(121\) 7.45934 0.0616474
\(122\) −152.455 + 51.0797i −1.24963 + 0.418686i
\(123\) 0 0
\(124\) 22.1113 + 29.2930i 0.178317 + 0.236234i
\(125\) 0 0
\(126\) 0 0
\(127\) 161.656i 1.27289i 0.771324 + 0.636443i \(0.219595\pi\)
−0.771324 + 0.636443i \(0.780405\pi\)
\(128\) 81.6989 + 98.5357i 0.638273 + 0.769810i
\(129\) 0 0
\(130\) 0 0
\(131\) 197.561i 1.50810i 0.656818 + 0.754049i \(0.271902\pi\)
−0.656818 + 0.754049i \(0.728098\pi\)
\(132\) 0 0
\(133\) −4.68901 −0.0352557
\(134\) 14.3136 + 42.7210i 0.106818 + 0.318813i
\(135\) 0 0
\(136\) −75.2297 + 109.747i −0.553159 + 0.806960i
\(137\) 76.4182 0.557797 0.278899 0.960321i \(-0.410031\pi\)
0.278899 + 0.960321i \(0.410031\pi\)
\(138\) 0 0
\(139\) 223.206i 1.60580i 0.596115 + 0.802899i \(0.296710\pi\)
−0.596115 + 0.802899i \(0.703290\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −39.0670 116.601i −0.275120 0.821137i
\(143\) 125.419i 0.877058i
\(144\) 0 0
\(145\) 0 0
\(146\) −261.139 + 87.4940i −1.78862 + 0.599274i
\(147\) 0 0
\(148\) 75.1555 56.7297i 0.507807 0.383308i
\(149\) 166.320 1.11624 0.558119 0.829761i \(-0.311523\pi\)
0.558119 + 0.829761i \(0.311523\pi\)
\(150\) 0 0
\(151\) 112.995i 0.748313i −0.927366 0.374156i \(-0.877932\pi\)
0.927366 0.374156i \(-0.122068\pi\)
\(152\) −3.06236 2.09920i −0.0201471 0.0138105i
\(153\) 0 0
\(154\) −204.163 + 68.4042i −1.32573 + 0.444183i
\(155\) 0 0
\(156\) 0 0
\(157\) 75.9330 0.483649 0.241825 0.970320i \(-0.422254\pi\)
0.241825 + 0.970320i \(0.422254\pi\)
\(158\) 88.1050 + 262.962i 0.557626 + 1.66432i
\(159\) 0 0
\(160\) 0 0
\(161\) 425.689 2.64403
\(162\) 0 0
\(163\) 184.184i 1.12996i 0.825103 + 0.564982i \(0.191117\pi\)
−0.825103 + 0.564982i \(0.808883\pi\)
\(164\) 140.641 106.160i 0.857568 0.647319i
\(165\) 0 0
\(166\) −54.7846 163.513i −0.330028 0.985017i
\(167\) 193.268i 1.15729i −0.815578 0.578647i \(-0.803581\pi\)
0.815578 0.578647i \(-0.196419\pi\)
\(168\) 0 0
\(169\) −30.4593 −0.180233
\(170\) 0 0
\(171\) 0 0
\(172\) 140.496 + 186.130i 0.816840 + 1.08215i
\(173\) −56.8646 −0.328697 −0.164348 0.986402i \(-0.552552\pi\)
−0.164348 + 0.986402i \(0.552552\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −163.961 46.7263i −0.931594 0.265490i
\(177\) 0 0
\(178\) −241.244 + 80.8282i −1.35530 + 0.454091i
\(179\) 272.150i 1.52039i 0.649695 + 0.760195i \(0.274896\pi\)
−0.649695 + 0.760195i \(0.725104\pi\)
\(180\) 0 0
\(181\) −50.2297 −0.277512 −0.138756 0.990327i \(-0.544310\pi\)
−0.138756 + 0.990327i \(0.544310\pi\)
\(182\) −75.5607 225.522i −0.415169 1.23913i
\(183\) 0 0
\(184\) 278.014 + 190.575i 1.51095 + 1.03573i
\(185\) 0 0
\(186\) 0 0
\(187\) 177.223i 0.947714i
\(188\) 100.354 + 132.950i 0.533800 + 0.707179i
\(189\) 0 0
\(190\) 0 0
\(191\) 194.247i 1.01700i −0.861062 0.508500i \(-0.830200\pi\)
0.861062 0.508500i \(-0.169800\pi\)
\(192\) 0 0
\(193\) −22.7033 −0.117634 −0.0588168 0.998269i \(-0.518733\pi\)
−0.0588168 + 0.998269i \(0.518733\pi\)
\(194\) −28.4458 + 9.53071i −0.146628 + 0.0491274i
\(195\) 0 0
\(196\) −169.466 + 127.918i −0.864625 + 0.652645i
\(197\) 154.633 0.784938 0.392469 0.919765i \(-0.371621\pi\)
0.392469 + 0.919765i \(0.371621\pi\)
\(198\) 0 0
\(199\) 236.094i 1.18640i −0.805054 0.593201i \(-0.797864\pi\)
0.805054 0.593201i \(-0.202136\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −371.244 + 124.384i −1.83784 + 0.615765i
\(203\) 197.561i 0.973206i
\(204\) 0 0
\(205\) 0 0
\(206\) −9.89040 29.5194i −0.0480116 0.143298i
\(207\) 0 0
\(208\) 51.6148 181.114i 0.248148 0.870741i
\(209\) 4.94520 0.0236612
\(210\) 0 0
\(211\) 115.209i 0.546015i −0.962012 0.273007i \(-0.911982\pi\)
0.962012 0.273007i \(-0.0880183\pi\)
\(212\) −256.167 + 193.362i −1.20833 + 0.912087i
\(213\) 0 0
\(214\) −32.6077 97.3225i −0.152372 0.454778i
\(215\) 0 0
\(216\) 0 0
\(217\) −92.7033 −0.427204
\(218\) 88.8489 29.7686i 0.407564 0.136553i
\(219\) 0 0
\(220\) 0 0
\(221\) 195.764 0.885808
\(222\) 0 0
\(223\) 425.170i 1.90659i −0.302039 0.953296i \(-0.597667\pi\)
0.302039 0.953296i \(-0.402333\pi\)
\(224\) −322.976 + 14.7597i −1.44186 + 0.0658915i
\(225\) 0 0
\(226\) 219.081 73.4027i 0.969386 0.324791i
\(227\) 178.208i 0.785055i 0.919740 + 0.392528i \(0.128399\pi\)
−0.919740 + 0.392528i \(0.871601\pi\)
\(228\) 0 0
\(229\) 331.636 1.44819 0.724097 0.689699i \(-0.242257\pi\)
0.724097 + 0.689699i \(0.242257\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 88.4451 129.025i 0.381229 0.556144i
\(233\) −182.952 −0.785200 −0.392600 0.919709i \(-0.628424\pi\)
−0.392600 + 0.919709i \(0.628424\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −154.070 204.112i −0.652840 0.864882i
\(237\) 0 0
\(238\) −106.770 318.672i −0.448615 1.33896i
\(239\) 210.664i 0.881438i −0.897645 0.440719i \(-0.854724\pi\)
0.897645 0.440719i \(-0.145276\pi\)
\(240\) 0 0
\(241\) −3.62198 −0.0150290 −0.00751448 0.999972i \(-0.502392\pi\)
−0.00751448 + 0.999972i \(0.502392\pi\)
\(242\) −14.1458 + 4.73952i −0.0584538 + 0.0195848i
\(243\) 0 0
\(244\) 256.659 193.734i 1.05188 0.793992i
\(245\) 0 0
\(246\) 0 0
\(247\) 5.46257i 0.0221157i
\(248\) −60.5438 41.5019i −0.244128 0.167346i
\(249\) 0 0
\(250\) 0 0
\(251\) 250.839i 0.999357i 0.866211 + 0.499678i \(0.166548\pi\)
−0.866211 + 0.499678i \(0.833452\pi\)
\(252\) 0 0
\(253\) −448.947 −1.77450
\(254\) −102.713 306.564i −0.404384 1.20694i
\(255\) 0 0
\(256\) −217.541 134.952i −0.849768 0.527157i
\(257\) 360.060 1.40101 0.700505 0.713647i \(-0.252958\pi\)
0.700505 + 0.713647i \(0.252958\pi\)
\(258\) 0 0
\(259\) 237.844i 0.918316i
\(260\) 0 0
\(261\) 0 0
\(262\) −125.526 374.652i −0.479108 1.42997i
\(263\) 191.310i 0.727416i 0.931513 + 0.363708i \(0.118489\pi\)
−0.931513 + 0.363708i \(0.881511\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.89219 2.97931i 0.0334293 0.0112004i
\(267\) 0 0
\(268\) −54.2882 71.9210i −0.202568 0.268362i
\(269\) 156.656 0.582365 0.291183 0.956667i \(-0.405951\pi\)
0.291183 + 0.956667i \(0.405951\pi\)
\(270\) 0 0
\(271\) 30.4172i 0.112241i 0.998424 + 0.0561203i \(0.0178730\pi\)
−0.998424 + 0.0561203i \(0.982127\pi\)
\(272\) 72.9339 255.922i 0.268139 0.940888i
\(273\) 0 0
\(274\) −144.919 + 48.5547i −0.528900 + 0.177207i
\(275\) 0 0
\(276\) 0 0
\(277\) −438.421 −1.58275 −0.791373 0.611333i \(-0.790634\pi\)
−0.791373 + 0.611333i \(0.790634\pi\)
\(278\) −141.821 423.285i −0.510146 1.52261i
\(279\) 0 0
\(280\) 0 0
\(281\) −239.589 −0.852630 −0.426315 0.904575i \(-0.640188\pi\)
−0.426315 + 0.904575i \(0.640188\pi\)
\(282\) 0 0
\(283\) 235.523i 0.832238i −0.909310 0.416119i \(-0.863390\pi\)
0.909310 0.416119i \(-0.136610\pi\)
\(284\) 148.173 + 196.299i 0.521734 + 0.691194i
\(285\) 0 0
\(286\) 79.6890 + 237.844i 0.278633 + 0.831622i
\(287\) 445.086i 1.55082i
\(288\) 0 0
\(289\) −12.3780 −0.0428305
\(290\) 0 0
\(291\) 0 0
\(292\) 439.629 331.845i 1.50558 1.13646i
\(293\) −271.955 −0.928173 −0.464086 0.885790i \(-0.653617\pi\)
−0.464086 + 0.885790i \(0.653617\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −106.479 + 155.334i −0.359727 + 0.524777i
\(297\) 0 0
\(298\) −315.407 + 105.676i −1.05841 + 0.354618i
\(299\) 495.916i 1.65858i
\(300\) 0 0
\(301\) −589.043 −1.95695
\(302\) 71.7950 + 214.283i 0.237732 + 0.709546i
\(303\) 0 0
\(304\) 7.14121 + 2.03514i 0.0234908 + 0.00669454i
\(305\) 0 0
\(306\) 0 0
\(307\) 316.352i 1.03046i −0.857052 0.515230i \(-0.827706\pi\)
0.857052 0.515230i \(-0.172294\pi\)
\(308\) 343.709 259.442i 1.11594 0.842344i
\(309\) 0 0
\(310\) 0 0
\(311\) 298.355i 0.959342i 0.877448 + 0.479671i \(0.159244\pi\)
−0.877448 + 0.479671i \(0.840756\pi\)
\(312\) 0 0
\(313\) −191.651 −0.612302 −0.306151 0.951983i \(-0.599041\pi\)
−0.306151 + 0.951983i \(0.599041\pi\)
\(314\) −143.998 + 48.2464i −0.458594 + 0.153651i
\(315\) 0 0
\(316\) −334.163 442.699i −1.05748 1.40095i
\(317\) −232.847 −0.734534 −0.367267 0.930115i \(-0.619707\pi\)
−0.367267 + 0.930115i \(0.619707\pi\)
\(318\) 0 0
\(319\) 208.355i 0.653150i
\(320\) 0 0
\(321\) 0 0
\(322\) −807.273 + 270.475i −2.50706 + 0.839984i
\(323\) 7.71883i 0.0238973i
\(324\) 0 0
\(325\) 0 0
\(326\) −117.027 349.285i −0.358979 1.07143i
\(327\) 0 0
\(328\) −199.258 + 290.682i −0.607495 + 0.886225i
\(329\) −420.744 −1.27886
\(330\) 0 0
\(331\) 200.214i 0.604877i 0.953169 + 0.302438i \(0.0978007\pi\)
−0.953169 + 0.302438i \(0.902199\pi\)
\(332\) 207.786 + 275.275i 0.625861 + 0.829141i
\(333\) 0 0
\(334\) 122.799 + 366.512i 0.367661 + 1.09734i
\(335\) 0 0
\(336\) 0 0
\(337\) 451.029 1.33836 0.669182 0.743099i \(-0.266645\pi\)
0.669182 + 0.743099i \(0.266645\pi\)
\(338\) 57.7628 19.3533i 0.170896 0.0572582i
\(339\) 0 0
\(340\) 0 0
\(341\) 97.7682 0.286710
\(342\) 0 0
\(343\) 41.2357i 0.120221i
\(344\) −384.699 263.706i −1.11831 0.766586i
\(345\) 0 0
\(346\) 107.837 36.1307i 0.311669 0.104424i
\(347\) 128.845i 0.371313i −0.982615 0.185656i \(-0.940559\pi\)
0.982615 0.185656i \(-0.0594411\pi\)
\(348\) 0 0
\(349\) 0.756045 0.00216632 0.00108316 0.999999i \(-0.499655\pi\)
0.00108316 + 0.999999i \(0.499655\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 340.622 15.5661i 0.967676 0.0442219i
\(353\) −55.7392 −0.157902 −0.0789508 0.996879i \(-0.525157\pi\)
−0.0789508 + 0.996879i \(0.525157\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 406.136 306.564i 1.14083 0.861134i
\(357\) 0 0
\(358\) −172.919 516.102i −0.483013 1.44162i
\(359\) 194.247i 0.541078i −0.962709 0.270539i \(-0.912798\pi\)
0.962709 0.270539i \(-0.0872019\pi\)
\(360\) 0 0
\(361\) 360.785 0.999403
\(362\) 95.2550 31.9150i 0.263135 0.0881629i
\(363\) 0 0
\(364\) 286.585 + 379.668i 0.787321 + 1.04304i
\(365\) 0 0
\(366\) 0 0
\(367\) 24.0264i 0.0654671i 0.999464 + 0.0327335i \(0.0104213\pi\)
−0.999464 + 0.0327335i \(0.989579\pi\)
\(368\) −648.311 184.759i −1.76171 0.502062i
\(369\) 0 0
\(370\) 0 0
\(371\) 810.688i 2.18514i
\(372\) 0 0
\(373\) −136.986 −0.367254 −0.183627 0.982996i \(-0.558784\pi\)
−0.183627 + 0.982996i \(0.558784\pi\)
\(374\) 112.604 + 336.083i 0.301080 + 0.898617i
\(375\) 0 0
\(376\) −274.785 188.361i −0.730810 0.500960i
\(377\) −230.153 −0.610485
\(378\) 0 0
\(379\) 103.356i 0.272707i 0.990660 + 0.136353i \(0.0435382\pi\)
−0.990660 + 0.136353i \(0.956462\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 123.421 + 368.368i 0.323091 + 0.964314i
\(383\) 19.3533i 0.0505308i −0.999681 0.0252654i \(-0.991957\pi\)
0.999681 0.0252654i \(-0.00804308\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 43.0543 14.4252i 0.111540 0.0373711i
\(387\) 0 0
\(388\) 47.8887 36.1479i 0.123425 0.0931646i
\(389\) −273.978 −0.704314 −0.352157 0.935941i \(-0.614552\pi\)
−0.352157 + 0.935941i \(0.614552\pi\)
\(390\) 0 0
\(391\) 700.750i 1.79220i
\(392\) 240.097 350.259i 0.612493 0.893518i
\(393\) 0 0
\(394\) −293.244 + 98.2507i −0.744274 + 0.249367i
\(395\) 0 0
\(396\) 0 0
\(397\) −218.230 −0.549697 −0.274848 0.961488i \(-0.588628\pi\)
−0.274848 + 0.961488i \(0.588628\pi\)
\(398\) 150.010 + 447.726i 0.376908 + 1.12494i
\(399\) 0 0
\(400\) 0 0
\(401\) −396.472 −0.988709 −0.494355 0.869260i \(-0.664596\pi\)
−0.494355 + 0.869260i \(0.664596\pi\)
\(402\) 0 0
\(403\) 107.997i 0.267982i
\(404\) 624.991 471.763i 1.54701 1.16773i
\(405\) 0 0
\(406\) 125.526 + 374.652i 0.309178 + 0.922789i
\(407\) 250.839i 0.616311i
\(408\) 0 0
\(409\) 301.976 0.738327 0.369164 0.929364i \(-0.379644\pi\)
0.369164 + 0.929364i \(0.379644\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 37.5121 + 49.6960i 0.0910488 + 0.120621i
\(413\) 645.952 1.56405
\(414\) 0 0
\(415\) 0 0
\(416\) 17.1946 + 376.258i 0.0413332 + 0.904466i
\(417\) 0 0
\(418\) −9.37802 + 3.14209i −0.0224355 + 0.00751695i
\(419\) 420.461i 1.00349i 0.865017 + 0.501743i \(0.167308\pi\)
−0.865017 + 0.501743i \(0.832692\pi\)
\(420\) 0 0
\(421\) −197.378 −0.468831 −0.234416 0.972136i \(-0.575318\pi\)
−0.234416 + 0.972136i \(0.575318\pi\)
\(422\) 73.2017 + 218.481i 0.173464 + 0.517728i
\(423\) 0 0
\(424\) 362.933 529.454i 0.855974 1.24871i
\(425\) 0 0
\(426\) 0 0
\(427\) 812.246i 1.90222i
\(428\) 123.674 + 163.843i 0.288957 + 0.382811i
\(429\) 0 0
\(430\) 0 0
\(431\) 236.869i 0.549581i 0.961504 + 0.274790i \(0.0886084\pi\)
−0.961504 + 0.274790i \(0.911392\pi\)
\(432\) 0 0
\(433\) 364.378 0.841520 0.420760 0.907172i \(-0.361764\pi\)
0.420760 + 0.907172i \(0.361764\pi\)
\(434\) 175.802 58.9019i 0.405073 0.135719i
\(435\) 0 0
\(436\) −149.578 + 112.906i −0.343068 + 0.258958i
\(437\) 19.5536 0.0447452
\(438\) 0 0
\(439\) 72.3679i 0.164847i −0.996597 0.0824236i \(-0.973734\pi\)
0.996597 0.0824236i \(-0.0262660\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −371.244 + 124.384i −0.839918 + 0.281413i
\(443\) 240.295i 0.542428i −0.962519 0.271214i \(-0.912575\pi\)
0.962519 0.271214i \(-0.0874250\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 270.145 + 806.287i 0.605706 + 1.80782i
\(447\) 0 0
\(448\) 603.110 233.203i 1.34623 0.520542i
\(449\) 498.959 1.11127 0.555633 0.831427i \(-0.312476\pi\)
0.555633 + 0.831427i \(0.312476\pi\)
\(450\) 0 0
\(451\) 469.403i 1.04081i
\(452\) −368.825 + 278.400i −0.815984 + 0.615930i
\(453\) 0 0
\(454\) −113.230 337.951i −0.249405 0.744385i
\(455\) 0 0
\(456\) 0 0
\(457\) −171.866 −0.376074 −0.188037 0.982162i \(-0.560213\pi\)
−0.188037 + 0.982162i \(0.560213\pi\)
\(458\) −628.911 + 210.715i −1.37317 + 0.460077i
\(459\) 0 0
\(460\) 0 0
\(461\) −831.825 −1.80439 −0.902196 0.431326i \(-0.858046\pi\)
−0.902196 + 0.431326i \(0.858046\pi\)
\(462\) 0 0
\(463\) 370.011i 0.799160i 0.916698 + 0.399580i \(0.130844\pi\)
−0.916698 + 0.399580i \(0.869156\pi\)
\(464\) −85.7460 + 300.879i −0.184797 + 0.648445i
\(465\) 0 0
\(466\) 346.947 116.244i 0.744522 0.249450i
\(467\) 527.770i 1.13013i −0.825047 0.565065i \(-0.808851\pi\)
0.825047 0.565065i \(-0.191149\pi\)
\(468\) 0 0
\(469\) 227.608 0.485304
\(470\) 0 0
\(471\) 0 0
\(472\) 421.866 + 289.183i 0.893784 + 0.612676i
\(473\) 621.226 1.31337
\(474\) 0 0
\(475\) 0 0
\(476\) 404.956 + 536.486i 0.850748 + 1.12707i
\(477\) 0 0
\(478\) 133.852 + 399.500i 0.280024 + 0.835775i
\(479\) 532.777i 1.11227i 0.831092 + 0.556135i \(0.187716\pi\)
−0.831092 + 0.556135i \(0.812284\pi\)
\(480\) 0 0
\(481\) 277.081 0.576053
\(482\) 6.86868 2.30134i 0.0142504 0.00477455i
\(483\) 0 0
\(484\) 23.8146 17.9760i 0.0492036 0.0371404i
\(485\) 0 0
\(486\) 0 0
\(487\) 495.073i 1.01658i −0.861187 0.508288i \(-0.830278\pi\)
0.861187 0.508288i \(-0.169722\pi\)
\(488\) −363.631 + 530.471i −0.745144 + 1.08703i
\(489\) 0 0
\(490\) 0 0
\(491\) 518.094i 1.05518i −0.849499 0.527590i \(-0.823096\pi\)
0.849499 0.527590i \(-0.176904\pi\)
\(492\) 0 0
\(493\) −325.215 −0.659666
\(494\) −3.47081 10.3592i −0.00702593 0.0209699i
\(495\) 0 0
\(496\) 141.184 + 40.2354i 0.284645 + 0.0811197i
\(497\) −621.226 −1.24995
\(498\) 0 0
\(499\) 676.579i 1.35587i −0.735122 0.677935i \(-0.762875\pi\)
0.735122 0.677935i \(-0.237125\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −159.378 475.688i −0.317486 0.947585i
\(503\) 35.2804i 0.0701399i 0.999385 + 0.0350700i \(0.0111654\pi\)
−0.999385 + 0.0350700i \(0.988835\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 851.379 285.252i 1.68257 0.563740i
\(507\) 0 0
\(508\) 389.569 + 516.102i 0.766869 + 1.01595i
\(509\) −538.520 −1.05800 −0.528998 0.848623i \(-0.677432\pi\)
−0.528998 + 0.848623i \(0.677432\pi\)
\(510\) 0 0
\(511\) 1391.29i 2.72268i
\(512\) 498.288 + 117.700i 0.973218 + 0.229884i
\(513\) 0 0
\(514\) −682.813 + 228.775i −1.32843 + 0.445088i
\(515\) 0 0
\(516\) 0 0
\(517\) 443.732 0.858282
\(518\) −151.121 451.044i −0.291740 0.870742i
\(519\) 0 0
\(520\) 0 0
\(521\) −73.7237 −0.141504 −0.0707521 0.997494i \(-0.522540\pi\)
−0.0707521 + 0.997494i \(0.522540\pi\)
\(522\) 0 0
\(523\) 465.440i 0.889942i −0.895545 0.444971i \(-0.853214\pi\)
0.895545 0.444971i \(-0.146786\pi\)
\(524\) 476.094 + 630.729i 0.908576 + 1.20368i
\(525\) 0 0
\(526\) −121.555 362.799i −0.231093 0.689732i
\(527\) 152.604i 0.289571i
\(528\) 0 0
\(529\) −1246.17 −2.35570
\(530\) 0 0
\(531\) 0 0
\(532\) −14.9701 + 11.2999i −0.0281392 + 0.0212403i
\(533\) 518.512 0.972819
\(534\) 0 0
\(535\) 0 0
\(536\) 148.649 + 101.897i 0.277330 + 0.190106i
\(537\) 0 0
\(538\) −297.081 + 99.5364i −0.552196 + 0.185012i
\(539\) 565.610i 1.04937i
\(540\) 0 0
\(541\) −48.9857 −0.0905466 −0.0452733 0.998975i \(-0.514416\pi\)
−0.0452733 + 0.998975i \(0.514416\pi\)
\(542\) −19.3265 57.6828i −0.0356577 0.106426i
\(543\) 0 0
\(544\) 24.2967 + 531.668i 0.0446631 + 0.977330i
\(545\) 0 0
\(546\) 0 0
\(547\) 500.536i 0.915056i −0.889195 0.457528i \(-0.848735\pi\)
0.889195 0.457528i \(-0.151265\pi\)
\(548\) 243.971 184.157i 0.445203 0.336053i
\(549\) 0 0
\(550\) 0 0
\(551\) 9.07478i 0.0164696i
\(552\) 0 0
\(553\) 1401.00 2.53346
\(554\) 831.417 278.564i 1.50075 0.502824i
\(555\) 0 0
\(556\) 537.895 + 712.603i 0.967436 + 1.28166i
\(557\) −771.368 −1.38486 −0.692431 0.721484i \(-0.743460\pi\)
−0.692431 + 0.721484i \(0.743460\pi\)
\(558\) 0 0
\(559\) 686.218i 1.22758i
\(560\) 0 0
\(561\) 0 0
\(562\) 454.354 152.230i 0.808459 0.270872i
\(563\) 535.602i 0.951335i 0.879625 + 0.475667i \(0.157793\pi\)
−0.879625 + 0.475667i \(0.842207\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 149.647 + 446.644i 0.264394 + 0.789123i
\(567\) 0 0
\(568\) −405.718 278.114i −0.714292 0.489637i
\(569\) −102.941 −0.180915 −0.0904575 0.995900i \(-0.528833\pi\)
−0.0904575 + 0.995900i \(0.528833\pi\)
\(570\) 0 0
\(571\) 642.662i 1.12550i 0.826626 + 0.562752i \(0.190257\pi\)
−0.826626 + 0.562752i \(0.809743\pi\)
\(572\) −302.243 400.411i −0.528396 0.700020i
\(573\) 0 0
\(574\) −282.799 844.056i −0.492681 1.47048i
\(575\) 0 0
\(576\) 0 0
\(577\) 397.919 0.689634 0.344817 0.938670i \(-0.387941\pi\)
0.344817 + 0.938670i \(0.387941\pi\)
\(578\) 23.4735 7.86476i 0.0406117 0.0136068i
\(579\) 0 0
\(580\) 0 0
\(581\) −871.159 −1.49941
\(582\) 0 0
\(583\) 854.981i 1.46652i
\(584\) −622.860 + 908.640i −1.06654 + 1.55589i
\(585\) 0 0
\(586\) 515.732 172.795i 0.880089 0.294872i
\(587\) 813.625i 1.38607i −0.720903 0.693036i \(-0.756273\pi\)
0.720903 0.693036i \(-0.243727\pi\)
\(588\) 0 0
\(589\) −4.25824 −0.00722961
\(590\) 0 0
\(591\) 0 0
\(592\) 103.230 362.228i 0.174374 0.611872i
\(593\) 731.362 1.23333 0.616663 0.787227i \(-0.288484\pi\)
0.616663 + 0.787227i \(0.288484\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 530.989 400.807i 0.890921 0.672494i
\(597\) 0 0
\(598\) 315.096 + 940.450i 0.526916 + 1.57266i
\(599\) 752.516i 1.25629i −0.778098 0.628143i \(-0.783815\pi\)
0.778098 0.628143i \(-0.216185\pi\)
\(600\) 0 0
\(601\) 1013.81 1.68688 0.843439 0.537226i \(-0.180528\pi\)
0.843439 + 0.537226i \(0.180528\pi\)
\(602\) 1117.05 374.267i 1.85557 0.621705i
\(603\) 0 0
\(604\) −272.302 360.747i −0.450832 0.597262i
\(605\) 0 0
\(606\) 0 0
\(607\) 432.276i 0.712151i 0.934457 + 0.356075i \(0.115885\pi\)
−0.934457 + 0.356075i \(0.884115\pi\)
\(608\) −14.8356 + 0.677973i −0.0244007 + 0.00111509i
\(609\) 0 0
\(610\) 0 0
\(611\) 490.155i 0.802218i
\(612\) 0 0
\(613\) −213.971 −0.349056 −0.174528 0.984652i \(-0.555840\pi\)
−0.174528 + 0.984652i \(0.555840\pi\)
\(614\) 201.004 + 599.926i 0.327368 + 0.977077i
\(615\) 0 0
\(616\) −486.962 + 710.389i −0.790522 + 1.15323i
\(617\) −814.966 −1.32085 −0.660426 0.750891i \(-0.729624\pi\)
−0.660426 + 0.750891i \(0.729624\pi\)
\(618\) 0 0
\(619\) 1035.45i 1.67278i −0.548133 0.836391i \(-0.684661\pi\)
0.548133 0.836391i \(-0.315339\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −189.569 565.798i −0.304774 0.909643i
\(623\) 1285.29i 2.06307i
\(624\) 0 0
\(625\) 0 0
\(626\) 363.444 121.771i 0.580581 0.194522i
\(627\) 0 0
\(628\) 242.422 182.988i 0.386023 0.291382i
\(629\) 391.527 0.622460
\(630\) 0 0
\(631\) 67.7270i 0.107333i 0.998559 + 0.0536664i \(0.0170907\pi\)
−0.998559 + 0.0536664i \(0.982909\pi\)
\(632\) 914.985 + 627.209i 1.44776 + 0.992419i
\(633\) 0 0
\(634\) 441.569 147.947i 0.696481 0.233355i
\(635\) 0 0
\(636\) 0 0
\(637\) −624.785 −0.980824
\(638\) −132.385 395.122i −0.207499 0.619313i
\(639\) 0 0
\(640\) 0 0
\(641\) −851.379 −1.32820 −0.664102 0.747642i \(-0.731186\pi\)
−0.664102 + 0.747642i \(0.731186\pi\)
\(642\) 0 0
\(643\) 562.800i 0.875272i 0.899152 + 0.437636i \(0.144184\pi\)
−0.899152 + 0.437636i \(0.855816\pi\)
\(644\) 1359.05 1025.85i 2.11032 1.59294i
\(645\) 0 0
\(646\) −4.90440 14.6379i −0.00759195 0.0226593i
\(647\) 331.188i 0.511883i −0.966692 0.255942i \(-0.917615\pi\)
0.966692 0.255942i \(-0.0823855\pi\)
\(648\) 0 0
\(649\) −681.244 −1.04968
\(650\) 0 0
\(651\) 0 0
\(652\) 443.857 + 588.023i 0.680763 + 0.901875i
\(653\) −804.177 −1.23151 −0.615756 0.787937i \(-0.711149\pi\)
−0.615756 + 0.787937i \(0.711149\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 193.177 677.851i 0.294478 1.03331i
\(657\) 0 0
\(658\) 797.895 267.333i 1.21261 0.406281i
\(659\) 1061.53i 1.61081i −0.592722 0.805407i \(-0.701947\pi\)
0.592722 0.805407i \(-0.298053\pi\)
\(660\) 0 0
\(661\) −117.646 −0.177982 −0.0889910 0.996032i \(-0.528364\pi\)
−0.0889910 + 0.996032i \(0.528364\pi\)
\(662\) −127.212 379.684i −0.192164 0.573541i
\(663\) 0 0
\(664\) −568.947 390.005i −0.856848 0.587357i
\(665\) 0 0
\(666\) 0 0
\(667\) 823.848i 1.23516i
\(668\) −465.749 617.025i −0.697229 0.923690i
\(669\) 0 0
\(670\) 0 0
\(671\) 856.624i 1.27664i
\(672\) 0 0
\(673\) 984.785 1.46328 0.731638 0.681693i \(-0.238756\pi\)
0.731638 + 0.681693i \(0.238756\pi\)
\(674\) −855.326 + 286.575i −1.26903 + 0.425185i
\(675\) 0 0
\(676\) −97.2440 + 73.4027i −0.143852 + 0.108584i
\(677\) 99.5646 0.147067 0.0735337 0.997293i \(-0.476572\pi\)
0.0735337 + 0.997293i \(0.476572\pi\)
\(678\) 0 0
\(679\) 151.553i 0.223200i
\(680\) 0 0
\(681\) 0 0
\(682\) −185.407 + 62.1201i −0.271857 + 0.0910851i
\(683\) 434.807i 0.636614i 0.947988 + 0.318307i \(0.103114\pi\)
−0.947988 + 0.318307i \(0.896886\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.2004 + 78.1990i 0.0381930 + 0.113993i
\(687\) 0 0
\(688\) 897.093 + 255.658i 1.30391 + 0.371596i
\(689\) −944.429 −1.37072
\(690\) 0 0
\(691\) 1153.87i 1.66986i −0.550358 0.834929i \(-0.685509\pi\)
0.550358 0.834929i \(-0.314491\pi\)
\(692\) −181.545 + 137.036i −0.262348 + 0.198028i
\(693\) 0 0
\(694\) 81.8659 + 244.341i 0.117962 + 0.352077i
\(695\) 0 0
\(696\) 0 0
\(697\) 732.679 1.05119
\(698\) −1.43375 + 0.480376i −0.00205409 + 0.000688218i
\(699\) 0 0
\(700\) 0 0
\(701\) −479.405 −0.683887 −0.341944 0.939720i \(-0.611085\pi\)
−0.341944 + 0.939720i \(0.611085\pi\)
\(702\) 0 0
\(703\) 10.9251i 0.0155407i
\(704\) −636.061 + 245.944i −0.903496 + 0.349352i
\(705\) 0 0
\(706\) 105.703 35.4156i 0.149721 0.0501638i
\(707\) 1977.90i 2.79760i
\(708\) 0 0
\(709\) 932.880 1.31577 0.657884 0.753119i \(-0.271451\pi\)
0.657884 + 0.753119i \(0.271451\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −575.407 + 839.415i −0.808155 + 1.17895i
\(713\) 386.582 0.542191
\(714\) 0 0
\(715\) 0 0
\(716\) 655.842 + 868.860i 0.915981 + 1.21349i
\(717\) 0 0
\(718\) 123.421 + 368.368i 0.171895 + 0.513047i
\(719\) 141.836i 0.197268i 0.995124 + 0.0986341i \(0.0314473\pi\)
−0.995124 + 0.0986341i \(0.968553\pi\)
\(720\) 0 0
\(721\) −157.273 −0.218131
\(722\) −684.188 + 229.236i −0.947629 + 0.317501i
\(723\) 0 0
\(724\) −160.362 + 121.046i −0.221495 + 0.167191i
\(725\) 0 0
\(726\) 0 0
\(727\) 891.575i 1.22638i −0.789937 0.613188i \(-0.789887\pi\)
0.789937 0.613188i \(-0.210113\pi\)
\(728\) −784.710 537.907i −1.07790 0.738884i
\(729\) 0 0
\(730\) 0 0
\(731\) 969.655i 1.32648i
\(732\) 0 0
\(733\) −614.593 −0.838463 −0.419232 0.907879i \(-0.637700\pi\)
−0.419232 + 0.907879i \(0.637700\pi\)
\(734\) −15.2659 45.5634i −0.0207983 0.0620755i
\(735\) 0 0
\(736\) 1346.84 61.5494i 1.82995 0.0836269i
\(737\) −240.043 −0.325703
\(738\) 0 0
\(739\) 817.138i 1.10573i −0.833269 0.552867i \(-0.813534\pi\)
0.833269 0.552867i \(-0.186466\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 515.096 + 1537.38i 0.694199 + 2.07194i
\(743\) 868.860i 1.16939i −0.811252 0.584697i \(-0.801213\pi\)
0.811252 0.584697i \(-0.198787\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 259.778 87.0381i 0.348228 0.116673i
\(747\) 0 0
\(748\) −427.081 565.798i −0.570964 0.756414i
\(749\) −518.512 −0.692273
\(750\) 0 0
\(751\) 100.822i 0.134250i −0.997745 0.0671252i \(-0.978617\pi\)
0.997745 0.0671252i \(-0.0213827\pi\)
\(752\) 640.779 + 182.613i 0.852100 + 0.242836i
\(753\) 0 0
\(754\) 436.459 146.235i 0.578859 0.193945i
\(755\) 0 0
\(756\) 0 0
\(757\) 33.6363 0.0444336 0.0222168 0.999753i \(-0.492928\pi\)
0.0222168 + 0.999753i \(0.492928\pi\)
\(758\) −65.6703 196.003i −0.0866363 0.258579i
\(759\) 0 0
\(760\) 0 0
\(761\) −685.059 −0.900209 −0.450105 0.892976i \(-0.648613\pi\)
−0.450105 + 0.892976i \(0.648613\pi\)
\(762\) 0 0
\(763\) 473.367i 0.620402i
\(764\) −468.108 620.150i −0.612707 0.811714i
\(765\) 0 0
\(766\) 12.2967 + 36.7013i 0.0160531 + 0.0479130i
\(767\) 752.516i 0.981116i
\(768\) 0 0
\(769\) 23.6505 0.0307549 0.0153775 0.999882i \(-0.495105\pi\)
0.0153775 + 0.999882i \(0.495105\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −72.4821 + 54.7117i −0.0938888 + 0.0708701i
\(773\) 947.350 1.22555 0.612775 0.790257i \(-0.290053\pi\)
0.612775 + 0.790257i \(0.290053\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −67.8480 + 98.9780i −0.0874330 + 0.127549i
\(777\) 0 0
\(778\) 519.569 174.080i 0.667827 0.223754i
\(779\) 20.4446i 0.0262447i
\(780\) 0 0
\(781\) 655.167 0.838882
\(782\) 445.243 + 1328.89i 0.569364 + 1.69935i
\(783\) 0 0
\(784\) −232.770 + 816.780i −0.296901 + 1.04181i
\(785\) 0 0
\(786\) 0 0
\(787\) 1054.73i 1.34019i 0.742275 + 0.670096i \(0.233747\pi\)
−0.742275 + 0.670096i \(0.766253\pi\)
\(788\) 493.678 372.643i 0.626495 0.472897i
\(789\) 0 0
\(790\) 0 0
\(791\) 1167.22i 1.47562i
\(792\) 0 0
\(793\) 946.244 1.19325
\(794\) 413.848 138.659i 0.521220 0.174633i
\(795\) 0 0
\(796\) −568.953 753.749i −0.714765 0.946921i
\(797\) 810.475 1.01691 0.508454 0.861089i \(-0.330217\pi\)
0.508454 + 0.861089i \(0.330217\pi\)
\(798\) 0 0
\(799\) 692.609i 0.866845i
\(800\) 0 0
\(801\) 0 0
\(802\) 751.866 251.911i 0.937489 0.314103i
\(803\) 1467.30i 1.82728i
\(804\) 0 0
\(805\) 0 0
\(806\) −68.6191 204.804i −0.0851353 0.254099i
\(807\) 0 0
\(808\) −885.478 + 1291.75i −1.09589 + 1.59870i
\(809\) 768.673 0.950152 0.475076 0.879945i \(-0.342421\pi\)
0.475076 + 0.879945i \(0.342421\pi\)
\(810\) 0 0
\(811\) 744.199i 0.917632i 0.888531 + 0.458816i \(0.151726\pi\)
−0.888531 + 0.458816i \(0.848274\pi\)
\(812\) −476.094 630.729i −0.586322 0.776760i
\(813\) 0 0
\(814\) 159.378 + 475.688i 0.195796 + 0.584383i
\(815\) 0 0
\(816\) 0 0
\(817\) −27.0571 −0.0331177
\(818\) −572.664 + 191.870i −0.700078 + 0.234559i
\(819\) 0 0
\(820\) 0 0
\(821\) 205.427 0.250215 0.125108 0.992143i \(-0.460072\pi\)
0.125108 + 0.992143i \(0.460072\pi\)
\(822\) 0 0
\(823\) 1287.79i 1.56475i 0.622808 + 0.782375i \(0.285992\pi\)
−0.622808 + 0.782375i \(0.714008\pi\)
\(824\) −102.713 70.4086i −0.124652 0.0854473i
\(825\) 0 0
\(826\) −1224.98 + 410.425i −1.48302 + 0.496883i
\(827\) 255.621i 0.309094i −0.987985 0.154547i \(-0.950608\pi\)
0.987985 0.154547i \(-0.0493918\pi\)
\(828\) 0 0
\(829\) −14.0286 −0.0169223 −0.00846114 0.999964i \(-0.502693\pi\)
−0.00846114 + 0.999964i \(0.502693\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −271.675 702.606i −0.326532 0.844479i
\(833\) −882.846 −1.05984
\(834\) 0 0
\(835\) 0 0
\(836\) 15.7880 11.9172i 0.0188851 0.0142551i
\(837\) 0 0
\(838\) −267.153 797.357i −0.318798 0.951500i
\(839\) 333.636i 0.397659i −0.980034 0.198829i \(-0.936286\pi\)
0.980034 0.198829i \(-0.0637140\pi\)
\(840\) 0 0
\(841\) −458.655 −0.545369
\(842\) 374.306 125.410i 0.444543 0.148943i
\(843\) 0 0
\(844\) −277.638 367.815i −0.328955 0.435799i
\(845\) 0 0
\(846\) 0 0
\(847\) 75.3657i 0.0889795i
\(848\) −351.857 + 1234.65i −0.414926 + 1.45596i
\(849\) 0 0
\(850\) 0 0
\(851\) 991.832i 1.16549i
\(852\) 0 0
\(853\) −1563.83 −1.83333 −0.916663 0.399661i \(-0.869128\pi\)
−0.916663 + 0.399661i \(0.869128\pi\)
\(854\) −516.086 1540.33i −0.604316 1.80367i
\(855\) 0 0
\(856\) −338.636 232.130i −0.395603 0.271180i
\(857\) 1283.36 1.49750 0.748749 0.662854i \(-0.230655\pi\)
0.748749 + 0.662854i \(0.230655\pi\)
\(858\) 0 0
\(859\) 1448.91i 1.68674i 0.537330 + 0.843372i \(0.319433\pi\)
−0.537330 + 0.843372i \(0.680567\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −150.502 449.196i −0.174597 0.521109i
\(863\) 755.677i 0.875640i −0.899063 0.437820i \(-0.855751\pi\)
0.899063 0.437820i \(-0.144249\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −691.002 + 231.519i −0.797924 + 0.267343i
\(867\) 0 0
\(868\) −295.963 + 223.402i −0.340971 + 0.257375i
\(869\) −1477.55 −1.70029
\(870\) 0 0
\(871\) 265.157i 0.304428i
\(872\) 211.919 309.152i 0.243027 0.354532i
\(873\) 0 0
\(874\) −37.0813 + 12.4240i −0.0424271 + 0.0142151i
\(875\) 0 0
\(876\) 0 0
\(877\) 248.612 0.283480 0.141740 0.989904i \(-0.454730\pi\)
0.141740 + 0.989904i \(0.454730\pi\)
\(878\) 45.9812 + 137.238i 0.0523704 + 0.156307i
\(879\) 0 0
\(880\) 0 0
\(881\) 460.306 0.522481 0.261240 0.965274i \(-0.415868\pi\)
0.261240 + 0.965274i \(0.415868\pi\)
\(882\) 0 0
\(883\) 554.195i 0.627628i −0.949485 0.313814i \(-0.898393\pi\)
0.949485 0.313814i \(-0.101607\pi\)
\(884\) 624.991 471.763i 0.707004 0.533668i
\(885\) 0 0
\(886\) 152.679 + 455.694i 0.172324 + 0.514327i
\(887\) 251.593i 0.283644i 0.989892 + 0.141822i \(0.0452961\pi\)
−0.989892 + 0.141822i \(0.954704\pi\)
\(888\) 0 0
\(889\) −1633.30 −1.83723
\(890\) 0 0
\(891\) 0 0
\(892\) −1024.60 1357.39i −1.14865 1.52174i
\(893\) −19.3265 −0.0216422
\(894\) 0 0
\(895\) 0 0
\(896\) −995.558 + 825.448i −1.11111 + 0.921259i
\(897\) 0 0
\(898\) −946.220 + 317.029i −1.05370 + 0.353039i
\(899\) 179.411i 0.199568i
\(900\) 0 0
\(901\) −1334.52 −1.48115
\(902\) 298.250 + 890.171i 0.330654 + 0.986886i
\(903\) 0 0
\(904\) 522.545 762.299i 0.578037 0.843251i
\(905\) 0 0
\(906\) 0 0
\(907\) 447.842i 0.493762i −0.969046 0.246881i \(-0.920594\pi\)
0.969046 0.246881i \(-0.0794056\pi\)
\(908\) 429.455 + 568.942i 0.472968 + 0.626588i
\(909\) 0 0
\(910\) 0 0
\(911\) 96.7664i 0.106220i 0.998589 + 0.0531100i \(0.0169134\pi\)
−0.998589 + 0.0531100i \(0.983087\pi\)
\(912\) 0 0
\(913\) 918.756 1.00630
\(914\) 325.925 109.200i 0.356592 0.119475i
\(915\) 0 0
\(916\) 1058.78 799.196i 1.15587 0.872485i
\(917\) −1996.06 −2.17673
\(918\) 0 0
\(919\) 234.739i 0.255429i 0.991811 + 0.127715i \(0.0407641\pi\)
−0.991811 + 0.127715i \(0.959236\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1577.46 528.526i 1.71092 0.573238i
\(923\) 723.711i 0.784085i
\(924\) 0 0
\(925\) 0 0
\(926\) −235.098 701.685i −0.253886 0.757759i
\(927\) 0 0
\(928\) −28.5648 625.064i −0.0307811 0.673561i
\(929\) −552.902 −0.595158 −0.297579 0.954697i \(-0.596179\pi\)
−0.297579 + 0.954697i \(0.596179\pi\)
\(930\) 0 0
\(931\) 24.6348i 0.0264606i
\(932\) −584.088 + 440.887i −0.626704 + 0.473055i
\(933\) 0 0
\(934\) 335.335 + 1000.86i 0.359031 + 1.07158i
\(935\) 0 0
\(936\) 0 0
\(937\) −957.870 −1.02227 −0.511137 0.859499i \(-0.670775\pi\)
−0.511137 + 0.859499i \(0.670775\pi\)
\(938\) −431.633 + 144.618i −0.460163 + 0.154177i
\(939\) 0 0
\(940\) 0 0
\(941\) 1125.13 1.19567 0.597837 0.801618i \(-0.296027\pi\)
0.597837 + 0.801618i \(0.296027\pi\)
\(942\) 0 0
\(943\) 1856.05i 1.96824i
\(944\) −983.763 280.358i −1.04212 0.296989i
\(945\) 0 0
\(946\) −1178.09 + 394.715i −1.24533 + 0.417246i
\(947\) 256.824i 0.271198i −0.990764 0.135599i \(-0.956704\pi\)
0.990764 0.135599i \(-0.0432959\pi\)
\(948\) 0 0
\(949\) 1620.81 1.70792
\(950\) 0 0
\(951\) 0 0
\(952\) −1108.83 760.085i −1.16473 0.798409i
\(953\) −575.377 −0.603753 −0.301877 0.953347i \(-0.597613\pi\)
−0.301877 + 0.953347i \(0.597613\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −507.670 672.561i −0.531035 0.703516i
\(957\) 0 0
\(958\) −338.516 1010.35i −0.353357 1.05465i
\(959\) 772.094i 0.805103i
\(960\) 0 0
\(961\) 876.813 0.912397
\(962\) −525.454 + 176.052i −0.546210 + 0.183006i
\(963\) 0 0
\(964\) −11.5635 + 8.72845i −0.0119953 + 0.00905441i
\(965\) 0 0
\(966\) 0 0
\(967\) 860.976i 0.890358i −0.895442 0.445179i \(-0.853140\pi\)
0.895442 0.445179i \(-0.146860\pi\)
\(968\) −33.7401 + 49.2207i −0.0348555 + 0.0508478i
\(969\) 0 0
\(970\) 0 0
\(971\) 177.116i 0.182406i 0.995832 + 0.0912030i \(0.0290712\pi\)
−0.995832 + 0.0912030i \(0.970929\pi\)
\(972\) 0 0
\(973\) −2255.17 −2.31775
\(974\) 314.560 + 938.851i 0.322957 + 0.963913i
\(975\) 0 0
\(976\) 352.534 1237.02i 0.361202 1.26744i
\(977\) 755.634 0.773423 0.386711 0.922201i \(-0.373611\pi\)
0.386711 + 0.922201i \(0.373611\pi\)
\(978\) 0 0
\(979\) 1355.52i 1.38459i
\(980\) 0 0
\(981\) 0 0
\(982\) 329.187 + 982.507i 0.335221 + 1.00052i
\(983\) 1108.93i 1.12811i −0.825738 0.564054i \(-0.809241\pi\)
0.825738 0.564054i \(-0.190759\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 616.735 206.636i 0.625492 0.209570i
\(987\) 0 0
\(988\) 13.1640 + 17.4397i 0.0133239 + 0.0176515i
\(989\) 2456.37 2.48369
\(990\) 0 0
\(991\) 516.421i 0.521111i 0.965459 + 0.260556i \(0.0839058\pi\)
−0.965459 + 0.260556i \(0.916094\pi\)
\(992\) −293.305 + 13.4037i −0.295670 + 0.0135118i
\(993\) 0 0
\(994\) 1178.09 394.715i 1.18520 0.397098i
\(995\) 0 0
\(996\) 0 0
\(997\) −1127.03 −1.13042 −0.565212 0.824946i \(-0.691206\pi\)
−0.565212 + 0.824946i \(0.691206\pi\)
\(998\) 429.885 + 1283.06i 0.430747 + 1.28563i
\(999\) 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 900.3.c.q.451.2 yes 8
3.2 odd 2 inner 900.3.c.q.451.7 yes 8
4.3 odd 2 inner 900.3.c.q.451.1 yes 8
5.2 odd 4 900.3.f.g.199.5 16
5.3 odd 4 900.3.f.g.199.12 16
5.4 even 2 900.3.c.p.451.7 yes 8
12.11 even 2 inner 900.3.c.q.451.8 yes 8
15.2 even 4 900.3.f.g.199.11 16
15.8 even 4 900.3.f.g.199.6 16
15.14 odd 2 900.3.c.p.451.2 yes 8
20.3 even 4 900.3.f.g.199.7 16
20.7 even 4 900.3.f.g.199.10 16
20.19 odd 2 900.3.c.p.451.8 yes 8
60.23 odd 4 900.3.f.g.199.9 16
60.47 odd 4 900.3.f.g.199.8 16
60.59 even 2 900.3.c.p.451.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.3.c.p.451.1 8 60.59 even 2
900.3.c.p.451.2 yes 8 15.14 odd 2
900.3.c.p.451.7 yes 8 5.4 even 2
900.3.c.p.451.8 yes 8 20.19 odd 2
900.3.c.q.451.1 yes 8 4.3 odd 2 inner
900.3.c.q.451.2 yes 8 1.1 even 1 trivial
900.3.c.q.451.7 yes 8 3.2 odd 2 inner
900.3.c.q.451.8 yes 8 12.11 even 2 inner
900.3.f.g.199.5 16 5.2 odd 4
900.3.f.g.199.6 16 15.8 even 4
900.3.f.g.199.7 16 20.3 even 4
900.3.f.g.199.8 16 60.47 odd 4
900.3.f.g.199.9 16 60.23 odd 4
900.3.f.g.199.10 16 20.7 even 4
900.3.f.g.199.11 16 15.2 even 4
900.3.f.g.199.12 16 5.3 odd 4