Properties

Label 475.3.c.i.151.7
Level $475$
Weight $3$
Character 475.151
Analytic conductor $12.943$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,3,Mod(151,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 151.7
Root \(-0.382406i\) of defining polynomial
Character \(\chi\) \(=\) 475.151
Dual form 475.3.c.i.151.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.382406i q^{2} +3.15259i q^{3} +3.85377 q^{4} +1.20557 q^{6} -4.26742 q^{7} -3.00333i q^{8} -0.938823 q^{9} +O(q^{10})\) \(q-0.382406i q^{2} +3.15259i q^{3} +3.85377 q^{4} +1.20557 q^{6} -4.26742 q^{7} -3.00333i q^{8} -0.938823 q^{9} -9.61517 q^{11} +12.1493i q^{12} +22.7853i q^{13} +1.63189i q^{14} +14.2666 q^{16} -25.9136 q^{17} +0.359012i q^{18} +(8.03195 + 17.2188i) q^{19} -13.4534i q^{21} +3.67690i q^{22} +24.5299 q^{23} +9.46826 q^{24} +8.71325 q^{26} +25.4136i q^{27} -16.4457 q^{28} +21.2643i q^{29} +6.67661i q^{31} -17.4689i q^{32} -30.3127i q^{33} +9.90952i q^{34} -3.61800 q^{36} +35.0237i q^{37} +(6.58458 - 3.07147i) q^{38} -71.8328 q^{39} -72.2427i q^{41} -5.14468 q^{42} -50.2869 q^{43} -37.0546 q^{44} -9.38040i q^{46} +36.1970 q^{47} +44.9766i q^{48} -30.7891 q^{49} -81.6950i q^{51} +87.8094i q^{52} -2.29775i q^{53} +9.71831 q^{54} +12.8165i q^{56} +(-54.2839 + 25.3214i) q^{57} +8.13162 q^{58} +19.0722i q^{59} +50.6664 q^{61} +2.55318 q^{62} +4.00636 q^{63} +50.3861 q^{64} -11.5918 q^{66} -9.20576i q^{67} -99.8650 q^{68} +77.3328i q^{69} +116.977i q^{71} +2.81959i q^{72} +49.2115 q^{73} +13.3933 q^{74} +(30.9532 + 66.3573i) q^{76} +41.0320 q^{77} +27.4693i q^{78} -87.5706i q^{79} -88.5680 q^{81} -27.6260 q^{82} +33.8601 q^{83} -51.8464i q^{84} +19.2300i q^{86} -67.0378 q^{87} +28.8775i q^{88} -7.05272i q^{89} -97.2347i q^{91} +94.5326 q^{92} -21.0486 q^{93} -13.8420i q^{94} +55.0724 q^{96} +16.0742i q^{97} +11.7739i q^{98} +9.02695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.382406i 0.191203i −0.995420 0.0956015i \(-0.969523\pi\)
0.995420 0.0956015i \(-0.0304775\pi\)
\(3\) 3.15259i 1.05086i 0.850836 + 0.525432i \(0.176096\pi\)
−0.850836 + 0.525432i \(0.823904\pi\)
\(4\) 3.85377 0.963441
\(5\) 0 0
\(6\) 1.20557 0.200928
\(7\) −4.26742 −0.609632 −0.304816 0.952411i \(-0.598595\pi\)
−0.304816 + 0.952411i \(0.598595\pi\)
\(8\) 3.00333i 0.375416i
\(9\) −0.938823 −0.104314
\(10\) 0 0
\(11\) −9.61517 −0.874107 −0.437053 0.899436i \(-0.643978\pi\)
−0.437053 + 0.899436i \(0.643978\pi\)
\(12\) 12.1493i 1.01245i
\(13\) 22.7853i 1.75272i 0.481658 + 0.876359i \(0.340035\pi\)
−0.481658 + 0.876359i \(0.659965\pi\)
\(14\) 1.63189i 0.116564i
\(15\) 0 0
\(16\) 14.2666 0.891661
\(17\) −25.9136 −1.52433 −0.762165 0.647383i \(-0.775863\pi\)
−0.762165 + 0.647383i \(0.775863\pi\)
\(18\) 0.359012i 0.0199451i
\(19\) 8.03195 + 17.2188i 0.422734 + 0.906254i
\(20\) 0 0
\(21\) 13.4534i 0.640640i
\(22\) 3.67690i 0.167132i
\(23\) 24.5299 1.06652 0.533260 0.845952i \(-0.320967\pi\)
0.533260 + 0.845952i \(0.320967\pi\)
\(24\) 9.46826 0.394511
\(25\) 0 0
\(26\) 8.71325 0.335125
\(27\) 25.4136i 0.941244i
\(28\) −16.4457 −0.587345
\(29\) 21.2643i 0.733253i 0.930368 + 0.366627i \(0.119487\pi\)
−0.930368 + 0.366627i \(0.880513\pi\)
\(30\) 0 0
\(31\) 6.67661i 0.215375i 0.994185 + 0.107687i \(0.0343445\pi\)
−0.994185 + 0.107687i \(0.965655\pi\)
\(32\) 17.4689i 0.545904i
\(33\) 30.3127i 0.918567i
\(34\) 9.90952i 0.291456i
\(35\) 0 0
\(36\) −3.61800 −0.100500
\(37\) 35.0237i 0.946587i 0.880905 + 0.473293i \(0.156935\pi\)
−0.880905 + 0.473293i \(0.843065\pi\)
\(38\) 6.58458 3.07147i 0.173278 0.0808280i
\(39\) −71.8328 −1.84187
\(40\) 0 0
\(41\) 72.2427i 1.76202i −0.473101 0.881008i \(-0.656865\pi\)
0.473101 0.881008i \(-0.343135\pi\)
\(42\) −5.14468 −0.122492
\(43\) −50.2869 −1.16946 −0.584731 0.811227i \(-0.698800\pi\)
−0.584731 + 0.811227i \(0.698800\pi\)
\(44\) −37.0546 −0.842151
\(45\) 0 0
\(46\) 9.38040i 0.203922i
\(47\) 36.1970 0.770149 0.385075 0.922885i \(-0.374176\pi\)
0.385075 + 0.922885i \(0.374176\pi\)
\(48\) 44.9766i 0.937014i
\(49\) −30.7891 −0.628349
\(50\) 0 0
\(51\) 81.6950i 1.60186i
\(52\) 87.8094i 1.68864i
\(53\) 2.29775i 0.0433537i −0.999765 0.0216768i \(-0.993100\pi\)
0.999765 0.0216768i \(-0.00690050\pi\)
\(54\) 9.71831 0.179969
\(55\) 0 0
\(56\) 12.8165i 0.228866i
\(57\) −54.2839 + 25.3214i −0.952349 + 0.444236i
\(58\) 8.13162 0.140200
\(59\) 19.0722i 0.323257i 0.986852 + 0.161629i \(0.0516747\pi\)
−0.986852 + 0.161629i \(0.948325\pi\)
\(60\) 0 0
\(61\) 50.6664 0.830596 0.415298 0.909685i \(-0.363677\pi\)
0.415298 + 0.909685i \(0.363677\pi\)
\(62\) 2.55318 0.0411803
\(63\) 4.00636 0.0635930
\(64\) 50.3861 0.787282
\(65\) 0 0
\(66\) −11.5918 −0.175633
\(67\) 9.20576i 0.137399i −0.997637 0.0686997i \(-0.978115\pi\)
0.997637 0.0686997i \(-0.0218850\pi\)
\(68\) −99.8650 −1.46860
\(69\) 77.3328i 1.12077i
\(70\) 0 0
\(71\) 116.977i 1.64756i 0.566909 + 0.823780i \(0.308139\pi\)
−0.566909 + 0.823780i \(0.691861\pi\)
\(72\) 2.81959i 0.0391610i
\(73\) 49.2115 0.674130 0.337065 0.941481i \(-0.390566\pi\)
0.337065 + 0.941481i \(0.390566\pi\)
\(74\) 13.3933 0.180990
\(75\) 0 0
\(76\) 30.9532 + 66.3573i 0.407279 + 0.873122i
\(77\) 41.0320 0.532883
\(78\) 27.4693i 0.352171i
\(79\) 87.5706i 1.10849i −0.832354 0.554244i \(-0.813007\pi\)
0.832354 0.554244i \(-0.186993\pi\)
\(80\) 0 0
\(81\) −88.5680 −1.09343
\(82\) −27.6260 −0.336903
\(83\) 33.8601 0.407953 0.203976 0.978976i \(-0.434613\pi\)
0.203976 + 0.978976i \(0.434613\pi\)
\(84\) 51.8464i 0.617219i
\(85\) 0 0
\(86\) 19.2300i 0.223605i
\(87\) −67.0378 −0.770549
\(88\) 28.8775i 0.328154i
\(89\) 7.05272i 0.0792440i −0.999215 0.0396220i \(-0.987385\pi\)
0.999215 0.0396220i \(-0.0126154\pi\)
\(90\) 0 0
\(91\) 97.2347i 1.06851i
\(92\) 94.5326 1.02753
\(93\) −21.0486 −0.226329
\(94\) 13.8420i 0.147255i
\(95\) 0 0
\(96\) 55.0724 0.573671
\(97\) 16.0742i 0.165714i 0.996561 + 0.0828568i \(0.0264044\pi\)
−0.996561 + 0.0828568i \(0.973596\pi\)
\(98\) 11.7739i 0.120142i
\(99\) 9.02695 0.0911813
\(100\) 0 0
\(101\) 42.2774 0.418588 0.209294 0.977853i \(-0.432883\pi\)
0.209294 + 0.977853i \(0.432883\pi\)
\(102\) −31.2407 −0.306281
\(103\) 50.6822i 0.492060i −0.969262 0.246030i \(-0.920874\pi\)
0.969262 0.246030i \(-0.0791262\pi\)
\(104\) 68.4318 0.657999
\(105\) 0 0
\(106\) −0.878672 −0.00828936
\(107\) 115.511i 1.07954i −0.841811 0.539772i \(-0.818510\pi\)
0.841811 0.539772i \(-0.181490\pi\)
\(108\) 97.9380i 0.906833i
\(109\) 108.668i 0.996957i 0.866902 + 0.498478i \(0.166108\pi\)
−0.866902 + 0.498478i \(0.833892\pi\)
\(110\) 0 0
\(111\) −110.415 −0.994734
\(112\) −60.8815 −0.543585
\(113\) 40.1073i 0.354932i −0.984127 0.177466i \(-0.943210\pi\)
0.984127 0.177466i \(-0.0567900\pi\)
\(114\) 9.68307 + 20.7585i 0.0849392 + 0.182092i
\(115\) 0 0
\(116\) 81.9478i 0.706447i
\(117\) 21.3914i 0.182833i
\(118\) 7.29332 0.0618078
\(119\) 110.584 0.929280
\(120\) 0 0
\(121\) −28.5484 −0.235938
\(122\) 19.3751i 0.158813i
\(123\) 227.752 1.85164
\(124\) 25.7301i 0.207501i
\(125\) 0 0
\(126\) 1.53206i 0.0121592i
\(127\) 149.933i 1.18057i −0.807194 0.590287i \(-0.799015\pi\)
0.807194 0.590287i \(-0.200985\pi\)
\(128\) 89.1437i 0.696435i
\(129\) 158.534i 1.22894i
\(130\) 0 0
\(131\) 235.661 1.79894 0.899471 0.436980i \(-0.143952\pi\)
0.899471 + 0.436980i \(0.143952\pi\)
\(132\) 116.818i 0.884985i
\(133\) −34.2757 73.4800i −0.257712 0.552481i
\(134\) −3.52034 −0.0262712
\(135\) 0 0
\(136\) 77.8270i 0.572258i
\(137\) 90.1819 0.658262 0.329131 0.944284i \(-0.393244\pi\)
0.329131 + 0.944284i \(0.393244\pi\)
\(138\) 29.5726 0.214294
\(139\) 193.273 1.39045 0.695225 0.718792i \(-0.255305\pi\)
0.695225 + 0.718792i \(0.255305\pi\)
\(140\) 0 0
\(141\) 114.114i 0.809322i
\(142\) 44.7326 0.315019
\(143\) 219.085i 1.53206i
\(144\) −13.3938 −0.0930124
\(145\) 0 0
\(146\) 18.8188i 0.128896i
\(147\) 97.0654i 0.660309i
\(148\) 134.973i 0.911981i
\(149\) −158.141 −1.06135 −0.530673 0.847577i \(-0.678061\pi\)
−0.530673 + 0.847577i \(0.678061\pi\)
\(150\) 0 0
\(151\) 155.289i 1.02841i −0.857669 0.514203i \(-0.828088\pi\)
0.857669 0.514203i \(-0.171912\pi\)
\(152\) 51.7138 24.1226i 0.340222 0.158701i
\(153\) 24.3283 0.159008
\(154\) 15.6909i 0.101889i
\(155\) 0 0
\(156\) −276.827 −1.77453
\(157\) 226.052 1.43982 0.719910 0.694068i \(-0.244183\pi\)
0.719910 + 0.694068i \(0.244183\pi\)
\(158\) −33.4875 −0.211946
\(159\) 7.24385 0.0455588
\(160\) 0 0
\(161\) −104.680 −0.650184
\(162\) 33.8689i 0.209068i
\(163\) −83.8369 −0.514337 −0.257169 0.966367i \(-0.582790\pi\)
−0.257169 + 0.966367i \(0.582790\pi\)
\(164\) 278.406i 1.69760i
\(165\) 0 0
\(166\) 12.9483i 0.0780018i
\(167\) 162.067i 0.970462i −0.874386 0.485231i \(-0.838736\pi\)
0.874386 0.485231i \(-0.161264\pi\)
\(168\) −40.4051 −0.240506
\(169\) −350.172 −2.07202
\(170\) 0 0
\(171\) −7.54058 16.1654i −0.0440969 0.0945347i
\(172\) −193.794 −1.12671
\(173\) 294.504i 1.70234i −0.524894 0.851168i \(-0.675895\pi\)
0.524894 0.851168i \(-0.324105\pi\)
\(174\) 25.6356i 0.147331i
\(175\) 0 0
\(176\) −137.176 −0.779407
\(177\) −60.1267 −0.339699
\(178\) −2.69700 −0.0151517
\(179\) 203.585i 1.13735i 0.822563 + 0.568674i \(0.192543\pi\)
−0.822563 + 0.568674i \(0.807457\pi\)
\(180\) 0 0
\(181\) 298.958i 1.65170i 0.563888 + 0.825851i \(0.309305\pi\)
−0.563888 + 0.825851i \(0.690695\pi\)
\(182\) −37.1832 −0.204303
\(183\) 159.730i 0.872843i
\(184\) 73.6715i 0.400388i
\(185\) 0 0
\(186\) 8.04912i 0.0432748i
\(187\) 249.164 1.33243
\(188\) 139.495 0.741994
\(189\) 108.451i 0.573812i
\(190\) 0 0
\(191\) 123.188 0.644963 0.322481 0.946576i \(-0.395483\pi\)
0.322481 + 0.946576i \(0.395483\pi\)
\(192\) 158.847i 0.827326i
\(193\) 68.6578i 0.355740i −0.984054 0.177870i \(-0.943079\pi\)
0.984054 0.177870i \(-0.0569207\pi\)
\(194\) 6.14688 0.0316849
\(195\) 0 0
\(196\) −118.654 −0.605377
\(197\) 68.1228 0.345801 0.172901 0.984939i \(-0.444686\pi\)
0.172901 + 0.984939i \(0.444686\pi\)
\(198\) 3.45196i 0.0174341i
\(199\) 263.929 1.32628 0.663139 0.748496i \(-0.269224\pi\)
0.663139 + 0.748496i \(0.269224\pi\)
\(200\) 0 0
\(201\) 29.0220 0.144388
\(202\) 16.1671i 0.0800354i
\(203\) 90.7440i 0.447015i
\(204\) 314.833i 1.54330i
\(205\) 0 0
\(206\) −19.3812 −0.0940834
\(207\) −23.0293 −0.111253
\(208\) 325.069i 1.56283i
\(209\) −77.2286 165.562i −0.369515 0.792162i
\(210\) 0 0
\(211\) 411.603i 1.95072i 0.220613 + 0.975362i \(0.429194\pi\)
−0.220613 + 0.975362i \(0.570806\pi\)
\(212\) 8.85497i 0.0417687i
\(213\) −368.780 −1.73136
\(214\) −44.1722 −0.206412
\(215\) 0 0
\(216\) 76.3253 0.353358
\(217\) 28.4919i 0.131299i
\(218\) 41.5554 0.190621
\(219\) 155.144i 0.708418i
\(220\) 0 0
\(221\) 590.450i 2.67172i
\(222\) 42.2235i 0.190196i
\(223\) 123.041i 0.551751i 0.961193 + 0.275876i \(0.0889678\pi\)
−0.961193 + 0.275876i \(0.911032\pi\)
\(224\) 74.5474i 0.332801i
\(225\) 0 0
\(226\) −15.3373 −0.0678641
\(227\) 278.989i 1.22903i −0.788907 0.614513i \(-0.789352\pi\)
0.788907 0.614513i \(-0.210648\pi\)
\(228\) −209.197 + 97.5829i −0.917532 + 0.427995i
\(229\) −11.9531 −0.0521969 −0.0260984 0.999659i \(-0.508308\pi\)
−0.0260984 + 0.999659i \(0.508308\pi\)
\(230\) 0 0
\(231\) 129.357i 0.559988i
\(232\) 63.8638 0.275275
\(233\) 215.911 0.926658 0.463329 0.886186i \(-0.346655\pi\)
0.463329 + 0.886186i \(0.346655\pi\)
\(234\) −8.18020 −0.0349581
\(235\) 0 0
\(236\) 73.4997i 0.311439i
\(237\) 276.074 1.16487
\(238\) 42.2881i 0.177681i
\(239\) −3.19514 −0.0133688 −0.00668438 0.999978i \(-0.502128\pi\)
−0.00668438 + 0.999978i \(0.502128\pi\)
\(240\) 0 0
\(241\) 68.7508i 0.285273i −0.989775 0.142637i \(-0.954442\pi\)
0.989775 0.142637i \(-0.0455580\pi\)
\(242\) 10.9171i 0.0451120i
\(243\) 50.4964i 0.207804i
\(244\) 195.256 0.800231
\(245\) 0 0
\(246\) 87.0936i 0.354039i
\(247\) −392.337 + 183.011i −1.58841 + 0.740934i
\(248\) 20.0521 0.0808550
\(249\) 106.747i 0.428703i
\(250\) 0 0
\(251\) 26.9054 0.107193 0.0535964 0.998563i \(-0.482932\pi\)
0.0535964 + 0.998563i \(0.482932\pi\)
\(252\) 15.4396 0.0612681
\(253\) −235.860 −0.932251
\(254\) −57.3352 −0.225729
\(255\) 0 0
\(256\) 167.455 0.654122
\(257\) 413.078i 1.60731i 0.595097 + 0.803654i \(0.297114\pi\)
−0.595097 + 0.803654i \(0.702886\pi\)
\(258\) −60.6243 −0.234978
\(259\) 149.461i 0.577070i
\(260\) 0 0
\(261\) 19.9635i 0.0764883i
\(262\) 90.1184i 0.343963i
\(263\) 227.053 0.863319 0.431660 0.902037i \(-0.357928\pi\)
0.431660 + 0.902037i \(0.357928\pi\)
\(264\) −91.0390 −0.344845
\(265\) 0 0
\(266\) −28.0992 + 13.1072i −0.105636 + 0.0492754i
\(267\) 22.2343 0.0832746
\(268\) 35.4768i 0.132376i
\(269\) 404.662i 1.50432i 0.658981 + 0.752160i \(0.270988\pi\)
−0.658981 + 0.752160i \(0.729012\pi\)
\(270\) 0 0
\(271\) 162.057 0.597995 0.298997 0.954254i \(-0.403348\pi\)
0.298997 + 0.954254i \(0.403348\pi\)
\(272\) −369.698 −1.35918
\(273\) 306.541 1.12286
\(274\) 34.4861i 0.125862i
\(275\) 0 0
\(276\) 298.023i 1.07979i
\(277\) −433.591 −1.56531 −0.782654 0.622456i \(-0.786135\pi\)
−0.782654 + 0.622456i \(0.786135\pi\)
\(278\) 73.9086i 0.265858i
\(279\) 6.26816i 0.0224665i
\(280\) 0 0
\(281\) 521.733i 1.85670i −0.371704 0.928351i \(-0.621226\pi\)
0.371704 0.928351i \(-0.378774\pi\)
\(282\) 43.6380 0.154745
\(283\) 375.661 1.32742 0.663712 0.747988i \(-0.268980\pi\)
0.663712 + 0.747988i \(0.268980\pi\)
\(284\) 450.801i 1.58733i
\(285\) 0 0
\(286\) −83.7794 −0.292935
\(287\) 308.290i 1.07418i
\(288\) 16.4002i 0.0569453i
\(289\) 382.515 1.32358
\(290\) 0 0
\(291\) −50.6754 −0.174142
\(292\) 189.649 0.649484
\(293\) 145.752i 0.497448i 0.968574 + 0.248724i \(0.0800113\pi\)
−0.968574 + 0.248724i \(0.919989\pi\)
\(294\) −37.1184 −0.126253
\(295\) 0 0
\(296\) 105.188 0.355364
\(297\) 244.356i 0.822748i
\(298\) 60.4739i 0.202933i
\(299\) 558.923i 1.86931i
\(300\) 0 0
\(301\) 214.595 0.712941
\(302\) −59.3835 −0.196634
\(303\) 133.283i 0.439879i
\(304\) 114.588 + 245.654i 0.376935 + 0.808071i
\(305\) 0 0
\(306\) 9.30329i 0.0304029i
\(307\) 199.397i 0.649502i 0.945800 + 0.324751i \(0.105281\pi\)
−0.945800 + 0.324751i \(0.894719\pi\)
\(308\) 158.128 0.513402
\(309\) 159.780 0.517088
\(310\) 0 0
\(311\) 309.298 0.994528 0.497264 0.867599i \(-0.334338\pi\)
0.497264 + 0.867599i \(0.334338\pi\)
\(312\) 215.738i 0.691467i
\(313\) −315.997 −1.00957 −0.504787 0.863244i \(-0.668429\pi\)
−0.504787 + 0.863244i \(0.668429\pi\)
\(314\) 86.4435i 0.275298i
\(315\) 0 0
\(316\) 337.477i 1.06796i
\(317\) 59.9579i 0.189142i 0.995518 + 0.0945708i \(0.0301479\pi\)
−0.995518 + 0.0945708i \(0.969852\pi\)
\(318\) 2.77009i 0.00871098i
\(319\) 204.460i 0.640942i
\(320\) 0 0
\(321\) 364.159 1.13445
\(322\) 40.0301i 0.124317i
\(323\) −208.137 446.202i −0.644386 1.38143i
\(324\) −341.320 −1.05346
\(325\) 0 0
\(326\) 32.0598i 0.0983428i
\(327\) −342.587 −1.04767
\(328\) −216.968 −0.661489
\(329\) −154.468 −0.469508
\(330\) 0 0
\(331\) 125.718i 0.379814i −0.981802 0.189907i \(-0.939181\pi\)
0.981802 0.189907i \(-0.0608186\pi\)
\(332\) 130.489 0.393039
\(333\) 32.8811i 0.0987420i
\(334\) −61.9755 −0.185555
\(335\) 0 0
\(336\) 191.934i 0.571233i
\(337\) 124.397i 0.369130i 0.982820 + 0.184565i \(0.0590877\pi\)
−0.982820 + 0.184565i \(0.940912\pi\)
\(338\) 133.908i 0.396177i
\(339\) 126.442 0.372985
\(340\) 0 0
\(341\) 64.1968i 0.188260i
\(342\) −6.18176 + 2.88356i −0.0180753 + 0.00843147i
\(343\) 340.494 0.992694
\(344\) 151.028i 0.439035i
\(345\) 0 0
\(346\) −112.620 −0.325492
\(347\) 155.887 0.449241 0.224621 0.974446i \(-0.427886\pi\)
0.224621 + 0.974446i \(0.427886\pi\)
\(348\) −258.348 −0.742379
\(349\) −546.288 −1.56530 −0.782648 0.622465i \(-0.786131\pi\)
−0.782648 + 0.622465i \(0.786131\pi\)
\(350\) 0 0
\(351\) −579.057 −1.64974
\(352\) 167.967i 0.477179i
\(353\) −491.756 −1.39308 −0.696538 0.717520i \(-0.745277\pi\)
−0.696538 + 0.717520i \(0.745277\pi\)
\(354\) 22.9928i 0.0649515i
\(355\) 0 0
\(356\) 27.1795i 0.0763469i
\(357\) 348.627i 0.976546i
\(358\) 77.8522 0.217464
\(359\) −175.785 −0.489651 −0.244825 0.969567i \(-0.578731\pi\)
−0.244825 + 0.969567i \(0.578731\pi\)
\(360\) 0 0
\(361\) −231.976 + 276.601i −0.642592 + 0.766209i
\(362\) 114.323 0.315811
\(363\) 90.0015i 0.247938i
\(364\) 374.720i 1.02945i
\(365\) 0 0
\(366\) 61.0818 0.166890
\(367\) 450.517 1.22757 0.613783 0.789475i \(-0.289647\pi\)
0.613783 + 0.789475i \(0.289647\pi\)
\(368\) 349.958 0.950973
\(369\) 67.8231i 0.183802i
\(370\) 0 0
\(371\) 9.80546i 0.0264298i
\(372\) −81.1164 −0.218055
\(373\) 566.983i 1.52006i 0.649887 + 0.760031i \(0.274816\pi\)
−0.649887 + 0.760031i \(0.725184\pi\)
\(374\) 95.2817i 0.254764i
\(375\) 0 0
\(376\) 108.712i 0.289126i
\(377\) −484.515 −1.28519
\(378\) −41.4722 −0.109715
\(379\) 333.683i 0.880431i −0.897892 0.440216i \(-0.854902\pi\)
0.897892 0.440216i \(-0.145098\pi\)
\(380\) 0 0
\(381\) 472.677 1.24062
\(382\) 47.1078i 0.123319i
\(383\) 14.7262i 0.0384496i −0.999815 0.0192248i \(-0.993880\pi\)
0.999815 0.0192248i \(-0.00611982\pi\)
\(384\) 281.033 0.731858
\(385\) 0 0
\(386\) −26.2552 −0.0680186
\(387\) 47.2105 0.121991
\(388\) 61.9463i 0.159655i
\(389\) −522.712 −1.34373 −0.671866 0.740673i \(-0.734507\pi\)
−0.671866 + 0.740673i \(0.734507\pi\)
\(390\) 0 0
\(391\) −635.659 −1.62573
\(392\) 92.4697i 0.235892i
\(393\) 742.944i 1.89044i
\(394\) 26.0506i 0.0661182i
\(395\) 0 0
\(396\) 34.7877 0.0878478
\(397\) −207.249 −0.522037 −0.261019 0.965334i \(-0.584058\pi\)
−0.261019 + 0.965334i \(0.584058\pi\)
\(398\) 100.928i 0.253588i
\(399\) 231.652 108.057i 0.580582 0.270820i
\(400\) 0 0
\(401\) 279.133i 0.696093i −0.937477 0.348047i \(-0.886845\pi\)
0.937477 0.348047i \(-0.113155\pi\)
\(402\) 11.0982i 0.0276074i
\(403\) −152.129 −0.377491
\(404\) 162.927 0.403285
\(405\) 0 0
\(406\) −34.7011 −0.0854706
\(407\) 336.759i 0.827418i
\(408\) −245.357 −0.601365
\(409\) 511.459i 1.25051i 0.780420 + 0.625256i \(0.215005\pi\)
−0.780420 + 0.625256i \(0.784995\pi\)
\(410\) 0 0
\(411\) 284.307i 0.691744i
\(412\) 195.317i 0.474071i
\(413\) 81.3891i 0.197068i
\(414\) 8.80653i 0.0212718i
\(415\) 0 0
\(416\) 398.036 0.956816
\(417\) 609.309i 1.46117i
\(418\) −63.3119 + 29.5327i −0.151464 + 0.0706523i
\(419\) −486.210 −1.16041 −0.580203 0.814472i \(-0.697027\pi\)
−0.580203 + 0.814472i \(0.697027\pi\)
\(420\) 0 0
\(421\) 499.876i 1.18735i −0.804704 0.593676i \(-0.797676\pi\)
0.804704 0.593676i \(-0.202324\pi\)
\(422\) 157.399 0.372984
\(423\) −33.9826 −0.0803371
\(424\) −6.90088 −0.0162757
\(425\) 0 0
\(426\) 141.024i 0.331041i
\(427\) −216.215 −0.506358
\(428\) 445.153i 1.04008i
\(429\) 690.685 1.60999
\(430\) 0 0
\(431\) 666.010i 1.54527i −0.634852 0.772634i \(-0.718939\pi\)
0.634852 0.772634i \(-0.281061\pi\)
\(432\) 362.565i 0.839270i
\(433\) 36.6550i 0.0846536i 0.999104 + 0.0423268i \(0.0134771\pi\)
−0.999104 + 0.0423268i \(0.986523\pi\)
\(434\) −10.8955 −0.0251048
\(435\) 0 0
\(436\) 418.782i 0.960509i
\(437\) 197.023 + 422.377i 0.450854 + 0.966537i
\(438\) 59.3278 0.135452
\(439\) 206.265i 0.469852i −0.972013 0.234926i \(-0.924515\pi\)
0.972013 0.234926i \(-0.0754848\pi\)
\(440\) 0 0
\(441\) 28.9055 0.0655454
\(442\) −225.792 −0.510841
\(443\) −611.151 −1.37957 −0.689787 0.724012i \(-0.742296\pi\)
−0.689787 + 0.724012i \(0.742296\pi\)
\(444\) −425.515 −0.958367
\(445\) 0 0
\(446\) 47.0515 0.105497
\(447\) 498.552i 1.11533i
\(448\) −215.019 −0.479952
\(449\) 0.285011i 0.000634768i 1.00000 0.000317384i \(0.000101026\pi\)
−1.00000 0.000317384i \(0.999899\pi\)
\(450\) 0 0
\(451\) 694.626i 1.54019i
\(452\) 154.564i 0.341956i
\(453\) 489.563 1.08071
\(454\) −106.687 −0.234994
\(455\) 0 0
\(456\) 76.0486 + 163.032i 0.166773 + 0.357527i
\(457\) 333.707 0.730213 0.365107 0.930966i \(-0.381033\pi\)
0.365107 + 0.930966i \(0.381033\pi\)
\(458\) 4.57093i 0.00998020i
\(459\) 658.558i 1.43477i
\(460\) 0 0
\(461\) −888.555 −1.92745 −0.963725 0.266895i \(-0.914002\pi\)
−0.963725 + 0.266895i \(0.914002\pi\)
\(462\) 49.4670 0.107071
\(463\) 245.546 0.530337 0.265169 0.964202i \(-0.414572\pi\)
0.265169 + 0.964202i \(0.414572\pi\)
\(464\) 303.369i 0.653813i
\(465\) 0 0
\(466\) 82.5658i 0.177180i
\(467\) 32.4279 0.0694389 0.0347194 0.999397i \(-0.488946\pi\)
0.0347194 + 0.999397i \(0.488946\pi\)
\(468\) 82.4375i 0.176148i
\(469\) 39.2849i 0.0837631i
\(470\) 0 0
\(471\) 712.648i 1.51305i
\(472\) 57.2800 0.121356
\(473\) 483.517 1.02223
\(474\) 105.572i 0.222727i
\(475\) 0 0
\(476\) 426.166 0.895307
\(477\) 2.15718i 0.00452238i
\(478\) 1.22184i 0.00255615i
\(479\) −345.842 −0.722008 −0.361004 0.932564i \(-0.617566\pi\)
−0.361004 + 0.932564i \(0.617566\pi\)
\(480\) 0 0
\(481\) −798.027 −1.65910
\(482\) −26.2907 −0.0545451
\(483\) 330.012i 0.683255i
\(484\) −110.019 −0.227312
\(485\) 0 0
\(486\) −19.3101 −0.0397328
\(487\) 345.826i 0.710114i −0.934845 0.355057i \(-0.884461\pi\)
0.934845 0.355057i \(-0.115539\pi\)
\(488\) 152.168i 0.311819i
\(489\) 264.303i 0.540498i
\(490\) 0 0
\(491\) 122.626 0.249748 0.124874 0.992173i \(-0.460147\pi\)
0.124874 + 0.992173i \(0.460147\pi\)
\(492\) 877.701 1.78395
\(493\) 551.036i 1.11772i
\(494\) 69.9844 + 150.032i 0.141669 + 0.303708i
\(495\) 0 0
\(496\) 95.2523i 0.192041i
\(497\) 499.190i 1.00441i
\(498\) 40.8207 0.0819693
\(499\) 4.16222 0.00834111 0.00417056 0.999991i \(-0.498672\pi\)
0.00417056 + 0.999991i \(0.498672\pi\)
\(500\) 0 0
\(501\) 510.931 1.01982
\(502\) 10.2888i 0.0204956i
\(503\) 444.782 0.884258 0.442129 0.896951i \(-0.354223\pi\)
0.442129 + 0.896951i \(0.354223\pi\)
\(504\) 12.0324i 0.0238738i
\(505\) 0 0
\(506\) 90.1942i 0.178249i
\(507\) 1103.95i 2.17741i
\(508\) 577.806i 1.13741i
\(509\) 39.1447i 0.0769051i −0.999260 0.0384526i \(-0.987757\pi\)
0.999260 0.0384526i \(-0.0122429\pi\)
\(510\) 0 0
\(511\) −210.006 −0.410971
\(512\) 420.611i 0.821505i
\(513\) −437.592 + 204.121i −0.853006 + 0.397896i
\(514\) 157.964 0.307322
\(515\) 0 0
\(516\) 610.952i 1.18402i
\(517\) −348.041 −0.673193
\(518\) −57.1548 −0.110337
\(519\) 928.450 1.78892
\(520\) 0 0
\(521\) 93.5181i 0.179497i −0.995964 0.0897487i \(-0.971394\pi\)
0.995964 0.0897487i \(-0.0286064\pi\)
\(522\) −7.63415 −0.0146248
\(523\) 831.457i 1.58978i 0.606751 + 0.794892i \(0.292473\pi\)
−0.606751 + 0.794892i \(0.707527\pi\)
\(524\) 908.184 1.73318
\(525\) 0 0
\(526\) 86.8264i 0.165069i
\(527\) 173.015i 0.328302i
\(528\) 432.458i 0.819050i
\(529\) 72.7180 0.137463
\(530\) 0 0
\(531\) 17.9054i 0.0337201i
\(532\) −132.091 283.175i −0.248291 0.532283i
\(533\) 1646.07 3.08832
\(534\) 8.50254i 0.0159224i
\(535\) 0 0
\(536\) −27.6479 −0.0515819
\(537\) −641.821 −1.19520
\(538\) 154.745 0.287630
\(539\) 296.042 0.549244
\(540\) 0 0
\(541\) 663.633 1.22668 0.613339 0.789820i \(-0.289826\pi\)
0.613339 + 0.789820i \(0.289826\pi\)
\(542\) 61.9714i 0.114338i
\(543\) −942.492 −1.73571
\(544\) 452.683i 0.832138i
\(545\) 0 0
\(546\) 117.223i 0.214695i
\(547\) 110.471i 0.201957i −0.994889 0.100979i \(-0.967803\pi\)
0.994889 0.100979i \(-0.0321974\pi\)
\(548\) 347.540 0.634197
\(549\) −47.5668 −0.0866425
\(550\) 0 0
\(551\) −366.147 + 170.794i −0.664514 + 0.309971i
\(552\) 232.256 0.420753
\(553\) 373.701i 0.675770i
\(554\) 165.808i 0.299292i
\(555\) 0 0
\(556\) 744.827 1.33962
\(557\) −697.017 −1.25138 −0.625688 0.780073i \(-0.715182\pi\)
−0.625688 + 0.780073i \(0.715182\pi\)
\(558\) −2.39698 −0.00429567
\(559\) 1145.80i 2.04974i
\(560\) 0 0
\(561\) 785.511i 1.40020i
\(562\) −199.514 −0.355007
\(563\) 576.361i 1.02373i −0.859065 0.511866i \(-0.828954\pi\)
0.859065 0.511866i \(-0.171046\pi\)
\(564\) 439.770i 0.779734i
\(565\) 0 0
\(566\) 143.655i 0.253808i
\(567\) 377.957 0.666591
\(568\) 351.320 0.618520
\(569\) 616.679i 1.08379i 0.840445 + 0.541897i \(0.182294\pi\)
−0.840445 + 0.541897i \(0.817706\pi\)
\(570\) 0 0
\(571\) 251.262 0.440039 0.220019 0.975496i \(-0.429388\pi\)
0.220019 + 0.975496i \(0.429388\pi\)
\(572\) 844.302i 1.47605i
\(573\) 388.361i 0.677768i
\(574\) 117.892 0.205387
\(575\) 0 0
\(576\) −47.3036 −0.0821243
\(577\) 814.466 1.41155 0.705777 0.708434i \(-0.250598\pi\)
0.705777 + 0.708434i \(0.250598\pi\)
\(578\) 146.276i 0.253073i
\(579\) 216.450 0.373834
\(580\) 0 0
\(581\) −144.495 −0.248701
\(582\) 19.3786i 0.0332965i
\(583\) 22.0932i 0.0378958i
\(584\) 147.798i 0.253079i
\(585\) 0 0
\(586\) 55.7366 0.0951136
\(587\) 954.562 1.62617 0.813085 0.582145i \(-0.197786\pi\)
0.813085 + 0.582145i \(0.197786\pi\)
\(588\) 374.067i 0.636169i
\(589\) −114.963 + 53.6262i −0.195184 + 0.0910462i
\(590\) 0 0
\(591\) 214.763i 0.363390i
\(592\) 499.668i 0.844034i
\(593\) −730.770 −1.23233 −0.616163 0.787618i \(-0.711314\pi\)
−0.616163 + 0.787618i \(0.711314\pi\)
\(594\) −93.4432 −0.157312
\(595\) 0 0
\(596\) −609.437 −1.02255
\(597\) 832.061i 1.39374i
\(598\) 213.736 0.357417
\(599\) 195.522i 0.326414i −0.986592 0.163207i \(-0.947816\pi\)
0.986592 0.163207i \(-0.0521838\pi\)
\(600\) 0 0
\(601\) 385.902i 0.642100i 0.947062 + 0.321050i \(0.104036\pi\)
−0.947062 + 0.321050i \(0.895964\pi\)
\(602\) 82.0626i 0.136317i
\(603\) 8.64258i 0.0143326i
\(604\) 598.448i 0.990808i
\(605\) 0 0
\(606\) 50.9684 0.0841062
\(607\) 368.570i 0.607200i −0.952800 0.303600i \(-0.901811\pi\)
0.952800 0.303600i \(-0.0981886\pi\)
\(608\) 300.794 140.310i 0.494728 0.230772i
\(609\) 286.079 0.469751
\(610\) 0 0
\(611\) 824.761i 1.34986i
\(612\) 93.7555 0.153195
\(613\) −675.526 −1.10200 −0.551000 0.834505i \(-0.685754\pi\)
−0.551000 + 0.834505i \(0.685754\pi\)
\(614\) 76.2507 0.124187
\(615\) 0 0
\(616\) 123.233i 0.200053i
\(617\) −433.371 −0.702384 −0.351192 0.936303i \(-0.614224\pi\)
−0.351192 + 0.936303i \(0.614224\pi\)
\(618\) 61.1009i 0.0988688i
\(619\) 316.106 0.510672 0.255336 0.966852i \(-0.417814\pi\)
0.255336 + 0.966852i \(0.417814\pi\)
\(620\) 0 0
\(621\) 623.394i 1.00385i
\(622\) 118.277i 0.190157i
\(623\) 30.0969i 0.0483097i
\(624\) −1024.81 −1.64232
\(625\) 0 0
\(626\) 120.839i 0.193034i
\(627\) 521.949 243.470i 0.832454 0.388309i
\(628\) 871.150 1.38718
\(629\) 907.591i 1.44291i
\(630\) 0 0
\(631\) −614.891 −0.974471 −0.487235 0.873271i \(-0.661995\pi\)
−0.487235 + 0.873271i \(0.661995\pi\)
\(632\) −263.003 −0.416144
\(633\) −1297.61 −2.04994
\(634\) 22.9283 0.0361645
\(635\) 0 0
\(636\) 27.9161 0.0438932
\(637\) 701.540i 1.10132i
\(638\) −78.1869 −0.122550
\(639\) 109.821i 0.171863i
\(640\) 0 0
\(641\) 3.80543i 0.00593670i 0.999996 + 0.00296835i \(0.000944857\pi\)
−0.999996 + 0.00296835i \(0.999055\pi\)
\(642\) 139.257i 0.216911i
\(643\) −786.552 −1.22325 −0.611627 0.791146i \(-0.709485\pi\)
−0.611627 + 0.791146i \(0.709485\pi\)
\(644\) −403.411 −0.626414
\(645\) 0 0
\(646\) −170.630 + 79.5927i −0.264134 + 0.123209i
\(647\) 766.493 1.18469 0.592344 0.805685i \(-0.298203\pi\)
0.592344 + 0.805685i \(0.298203\pi\)
\(648\) 265.999i 0.410492i
\(649\) 183.382i 0.282561i
\(650\) 0 0
\(651\) 89.8234 0.137978
\(652\) −323.088 −0.495534
\(653\) −360.741 −0.552436 −0.276218 0.961095i \(-0.589081\pi\)
−0.276218 + 0.961095i \(0.589081\pi\)
\(654\) 131.007i 0.200317i
\(655\) 0 0
\(656\) 1030.66i 1.57112i
\(657\) −46.2008 −0.0703209
\(658\) 59.0695i 0.0897713i
\(659\) 594.422i 0.902006i −0.892522 0.451003i \(-0.851066\pi\)
0.892522 0.451003i \(-0.148934\pi\)
\(660\) 0 0
\(661\) 143.641i 0.217309i −0.994080 0.108655i \(-0.965346\pi\)
0.994080 0.108655i \(-0.0346542\pi\)
\(662\) −48.0755 −0.0726215
\(663\) 1861.45 2.80761
\(664\) 101.693i 0.153152i
\(665\) 0 0
\(666\) −12.5739 −0.0188798
\(667\) 521.613i 0.782029i
\(668\) 624.569i 0.934983i
\(669\) −387.896 −0.579815
\(670\) 0 0
\(671\) −487.166 −0.726030
\(672\) −235.017 −0.349728
\(673\) 80.3740i 0.119426i −0.998216 0.0597132i \(-0.980981\pi\)
0.998216 0.0597132i \(-0.0190186\pi\)
\(674\) 47.5701 0.0705788
\(675\) 0 0
\(676\) −1349.48 −1.99627
\(677\) 643.046i 0.949846i 0.880027 + 0.474923i \(0.157524\pi\)
−0.880027 + 0.474923i \(0.842476\pi\)
\(678\) 48.3522i 0.0713159i
\(679\) 68.5955i 0.101024i
\(680\) 0 0
\(681\) 879.538 1.29154
\(682\) −24.5492 −0.0359959
\(683\) 1189.25i 1.74121i 0.491982 + 0.870606i \(0.336273\pi\)
−0.491982 + 0.870606i \(0.663727\pi\)
\(684\) −29.0596 62.2978i −0.0424848 0.0910786i
\(685\) 0 0
\(686\) 130.207i 0.189806i
\(687\) 37.6832i 0.0548518i
\(688\) −717.421 −1.04276
\(689\) 52.3549 0.0759868
\(690\) 0 0
\(691\) 454.252 0.657383 0.328692 0.944437i \(-0.393392\pi\)
0.328692 + 0.944437i \(0.393392\pi\)
\(692\) 1134.95i 1.64010i
\(693\) −38.5218 −0.0555870
\(694\) 59.6120i 0.0858963i
\(695\) 0 0
\(696\) 201.336i 0.289276i
\(697\) 1872.07i 2.68589i
\(698\) 208.904i 0.299289i
\(699\) 680.680i 0.973791i
\(700\) 0 0
\(701\) −183.307 −0.261494 −0.130747 0.991416i \(-0.541738\pi\)
−0.130747 + 0.991416i \(0.541738\pi\)
\(702\) 221.435i 0.315434i
\(703\) −603.067 + 281.309i −0.857848 + 0.400155i
\(704\) −484.471 −0.688169
\(705\) 0 0
\(706\) 188.050i 0.266360i
\(707\) −180.416 −0.255185
\(708\) −231.714 −0.327280
\(709\) 1209.49 1.70591 0.852953 0.521987i \(-0.174809\pi\)
0.852953 + 0.521987i \(0.174809\pi\)
\(710\) 0 0
\(711\) 82.2133i 0.115631i
\(712\) −21.1816 −0.0297495
\(713\) 163.777i 0.229701i
\(714\) 133.317 0.186719
\(715\) 0 0
\(716\) 784.570i 1.09577i
\(717\) 10.0730i 0.0140487i
\(718\) 67.2211i 0.0936227i
\(719\) −721.858 −1.00397 −0.501987 0.864875i \(-0.667398\pi\)
−0.501987 + 0.864875i \(0.667398\pi\)
\(720\) 0 0
\(721\) 216.283i 0.299976i
\(722\) 105.774 + 88.7089i 0.146501 + 0.122866i
\(723\) 216.743 0.299783
\(724\) 1152.11i 1.59132i
\(725\) 0 0
\(726\) −34.4171 −0.0474065
\(727\) −141.344 −0.194421 −0.0972103 0.995264i \(-0.530992\pi\)
−0.0972103 + 0.995264i \(0.530992\pi\)
\(728\) −292.028 −0.401137
\(729\) −637.918 −0.875059
\(730\) 0 0
\(731\) 1303.11 1.78265
\(732\) 615.563i 0.840933i
\(733\) −270.218 −0.368647 −0.184324 0.982866i \(-0.559009\pi\)
−0.184324 + 0.982866i \(0.559009\pi\)
\(734\) 172.280i 0.234714i
\(735\) 0 0
\(736\) 428.512i 0.582217i
\(737\) 88.5150i 0.120102i
\(738\) 25.9360 0.0351436
\(739\) −955.610 −1.29311 −0.646556 0.762866i \(-0.723791\pi\)
−0.646556 + 0.762866i \(0.723791\pi\)
\(740\) 0 0
\(741\) −576.958 1236.88i −0.778620 1.66920i
\(742\) 3.74967 0.00505346
\(743\) 1090.15i 1.46722i −0.679569 0.733612i \(-0.737833\pi\)
0.679569 0.733612i \(-0.262167\pi\)
\(744\) 63.2159i 0.0849676i
\(745\) 0 0
\(746\) 216.818 0.290640
\(747\) −31.7886 −0.0425551
\(748\) 960.219 1.28372
\(749\) 492.935i 0.658125i
\(750\) 0 0
\(751\) 729.113i 0.970856i −0.874277 0.485428i \(-0.838664\pi\)
0.874277 0.485428i \(-0.161336\pi\)
\(752\) 516.407 0.686712
\(753\) 84.8216i 0.112645i
\(754\) 185.282i 0.245732i
\(755\) 0 0
\(756\) 417.943i 0.552835i
\(757\) 200.136 0.264381 0.132191 0.991224i \(-0.457799\pi\)
0.132191 + 0.991224i \(0.457799\pi\)
\(758\) −127.603 −0.168341
\(759\) 743.569i 0.979669i
\(760\) 0 0
\(761\) 649.201 0.853090 0.426545 0.904466i \(-0.359731\pi\)
0.426545 + 0.904466i \(0.359731\pi\)
\(762\) 180.754i 0.237211i
\(763\) 463.734i 0.607777i
\(764\) 474.737 0.621384
\(765\) 0 0
\(766\) −5.63138 −0.00735168
\(767\) −434.566 −0.566579
\(768\) 527.917i 0.687392i
\(769\) −975.651 −1.26873 −0.634363 0.773035i \(-0.718738\pi\)
−0.634363 + 0.773035i \(0.718738\pi\)
\(770\) 0 0
\(771\) −1302.27 −1.68906
\(772\) 264.591i 0.342735i
\(773\) 1411.33i 1.82579i −0.408196 0.912894i \(-0.633842\pi\)
0.408196 0.912894i \(-0.366158\pi\)
\(774\) 18.0536i 0.0233250i
\(775\) 0 0
\(776\) 48.2761 0.0622115
\(777\) 471.189 0.606421
\(778\) 199.888i 0.256926i
\(779\) 1243.93 580.249i 1.59683 0.744864i
\(780\) 0 0
\(781\) 1124.75i 1.44014i
\(782\) 243.080i 0.310844i
\(783\) −540.403 −0.690170
\(784\) −439.255 −0.560274
\(785\) 0 0
\(786\) 284.106 0.361458
\(787\) 318.102i 0.404196i 0.979365 + 0.202098i \(0.0647759\pi\)
−0.979365 + 0.202098i \(0.935224\pi\)
\(788\) 262.529 0.333159
\(789\) 715.805i 0.907230i
\(790\) 0 0
\(791\) 171.155i 0.216378i
\(792\) 27.1109i 0.0342309i
\(793\) 1154.45i 1.45580i
\(794\) 79.2532i 0.0998152i
\(795\) 0 0
\(796\) 1017.12 1.27779
\(797\) 27.1975i 0.0341249i −0.999854 0.0170624i \(-0.994569\pi\)
0.999854 0.0170624i \(-0.00543141\pi\)
\(798\) −41.3218 88.5853i −0.0517817 0.111009i
\(799\) −937.995 −1.17396
\(800\) 0 0
\(801\) 6.62125i 0.00826623i
\(802\) −106.742 −0.133095
\(803\) −473.177 −0.589261
\(804\) 111.844 0.139109
\(805\) 0 0
\(806\) 58.1750i 0.0721774i
\(807\) −1275.73 −1.58083
\(808\) 126.973i 0.157145i
\(809\) 1023.22 1.26480 0.632398 0.774643i \(-0.282071\pi\)
0.632398 + 0.774643i \(0.282071\pi\)
\(810\) 0 0
\(811\) 1153.64i 1.42249i 0.702944 + 0.711245i \(0.251869\pi\)
−0.702944 + 0.711245i \(0.748131\pi\)
\(812\) 349.706i 0.430673i
\(813\) 510.898i 0.628411i
\(814\) −128.779 −0.158205
\(815\) 0 0
\(816\) 1165.51i 1.42832i
\(817\) −403.901 865.880i −0.494371 1.05983i
\(818\) 195.585 0.239102
\(819\) 91.2862i 0.111461i
\(820\) 0 0
\(821\) 201.567 0.245515 0.122757 0.992437i \(-0.460826\pi\)
0.122757 + 0.992437i \(0.460826\pi\)
\(822\) 108.721 0.132263
\(823\) −194.061 −0.235797 −0.117898 0.993026i \(-0.537616\pi\)
−0.117898 + 0.993026i \(0.537616\pi\)
\(824\) −152.215 −0.184727
\(825\) 0 0
\(826\) −31.1237 −0.0376800
\(827\) 252.357i 0.305147i −0.988292 0.152574i \(-0.951244\pi\)
0.988292 0.152574i \(-0.0487561\pi\)
\(828\) −88.7494 −0.107185
\(829\) 80.0611i 0.0965756i −0.998833 0.0482878i \(-0.984624\pi\)
0.998833 0.0482878i \(-0.0153765\pi\)
\(830\) 0 0
\(831\) 1366.93i 1.64493i
\(832\) 1148.06i 1.37988i
\(833\) 797.856 0.957811
\(834\) 233.004 0.279381
\(835\) 0 0
\(836\) −297.621 638.037i −0.356006 0.763202i
\(837\) −169.677 −0.202720
\(838\) 185.930i 0.221873i
\(839\) 1347.70i 1.60632i 0.595765 + 0.803159i \(0.296849\pi\)
−0.595765 + 0.803159i \(0.703151\pi\)
\(840\) 0 0
\(841\) 388.828 0.462340
\(842\) −191.155 −0.227025
\(843\) 1644.81 1.95114
\(844\) 1586.22i 1.87941i
\(845\) 0 0
\(846\) 12.9952i 0.0153607i
\(847\) 121.828 0.143835
\(848\) 32.7810i 0.0386568i
\(849\) 1184.31i 1.39494i
\(850\) 0 0
\(851\) 859.130i 1.00955i
\(852\) −1421.19 −1.66806
\(853\) 1254.19 1.47033 0.735165 0.677888i \(-0.237105\pi\)
0.735165 + 0.677888i \(0.237105\pi\)
\(854\) 82.6819i 0.0968172i
\(855\) 0 0
\(856\) −346.918 −0.405278
\(857\) 1044.59i 1.21889i 0.792828 + 0.609445i \(0.208608\pi\)
−0.792828 + 0.609445i \(0.791392\pi\)
\(858\) 264.122i 0.307835i
\(859\) 803.394 0.935267 0.467634 0.883922i \(-0.345107\pi\)
0.467634 + 0.883922i \(0.345107\pi\)
\(860\) 0 0
\(861\) −971.912 −1.12882
\(862\) −254.686 −0.295460
\(863\) 852.534i 0.987873i 0.869498 + 0.493936i \(0.164442\pi\)
−0.869498 + 0.493936i \(0.835558\pi\)
\(864\) 443.948 0.513829
\(865\) 0 0
\(866\) 14.0171 0.0161860
\(867\) 1205.91i 1.39090i
\(868\) 109.801i 0.126499i
\(869\) 842.007i 0.968937i
\(870\) 0 0
\(871\) 209.756 0.240822
\(872\) 326.366 0.374274
\(873\) 15.0908i 0.0172862i
\(874\) 161.519 75.3429i 0.184805 0.0862046i
\(875\) 0 0
\(876\) 597.887i 0.682519i
\(877\) 134.051i 0.152852i 0.997075 + 0.0764260i \(0.0243509\pi\)
−0.997075 + 0.0764260i \(0.975649\pi\)
\(878\) −78.8771 −0.0898372
\(879\) −459.497 −0.522750
\(880\) 0 0
\(881\) 1395.05 1.58348 0.791742 0.610855i \(-0.209174\pi\)
0.791742 + 0.610855i \(0.209174\pi\)
\(882\) 11.0536i 0.0125325i
\(883\) −879.025 −0.995498 −0.497749 0.867321i \(-0.665840\pi\)
−0.497749 + 0.867321i \(0.665840\pi\)
\(884\) 2275.46i 2.57405i
\(885\) 0 0
\(886\) 233.708i 0.263779i
\(887\) 1198.89i 1.35163i 0.737072 + 0.675814i \(0.236208\pi\)
−0.737072 + 0.675814i \(0.763792\pi\)
\(888\) 331.614i 0.373439i
\(889\) 639.827i 0.719715i
\(890\) 0 0
\(891\) 851.597 0.955776
\(892\) 474.170i 0.531580i
\(893\) 290.733 + 623.270i 0.325568 + 0.697951i
\(894\) −190.650 −0.213254
\(895\) 0 0
\(896\) 380.414i 0.424569i
\(897\) −1762.06 −1.96439
\(898\) 0.108990 0.000121370
\(899\) −141.974 −0.157924
\(900\) 0 0
\(901\) 59.5429i 0.0660853i
\(902\) 265.629 0.294489
\(903\) 676.531i 0.749204i
\(904\) −120.455 −0.133247
\(905\) 0 0
\(906\) 187.212i 0.206636i
\(907\) 793.689i 0.875071i −0.899201 0.437535i \(-0.855851\pi\)
0.899201 0.437535i \(-0.144149\pi\)
\(908\) 1075.16i 1.18409i
\(909\) −39.6910 −0.0436645
\(910\) 0 0
\(911\) 136.125i 0.149423i 0.997205 + 0.0747116i \(0.0238036\pi\)
−0.997205 + 0.0747116i \(0.976196\pi\)
\(912\) −774.445 + 361.250i −0.849172 + 0.396108i
\(913\) −325.571 −0.356594
\(914\) 127.612i 0.139619i
\(915\) 0 0
\(916\) −46.0644 −0.0502886
\(917\) −1005.67 −1.09669
\(918\) −251.836 −0.274332
\(919\) 124.377 0.135339 0.0676695 0.997708i \(-0.478444\pi\)
0.0676695 + 0.997708i \(0.478444\pi\)
\(920\) 0 0
\(921\) −628.617 −0.682538
\(922\) 339.789i 0.368534i
\(923\) −2665.36 −2.88771
\(924\) 498.512i 0.539515i
\(925\) 0 0
\(926\) 93.8984i 0.101402i
\(927\) 47.5816i 0.0513286i
\(928\) 371.465 0.400286
\(929\) 412.151 0.443650 0.221825 0.975086i \(-0.428799\pi\)
0.221825 + 0.975086i \(0.428799\pi\)
\(930\) 0 0
\(931\) −247.296 530.152i −0.265624 0.569443i
\(932\) 832.071 0.892780
\(933\) 975.090i 1.04511i
\(934\) 12.4006i 0.0132769i
\(935\) 0 0
\(936\) −64.2454 −0.0686382
\(937\) 703.752 0.751069 0.375535 0.926808i \(-0.377459\pi\)
0.375535 + 0.926808i \(0.377459\pi\)
\(938\) 15.0228 0.0160158
\(939\) 996.209i 1.06092i
\(940\) 0 0
\(941\) 556.128i 0.590997i −0.955343 0.295498i \(-0.904514\pi\)
0.955343 0.295498i \(-0.0954857\pi\)
\(942\) 272.521 0.289300
\(943\) 1772.11i 1.87922i
\(944\) 272.095i 0.288236i
\(945\) 0 0
\(946\) 184.900i 0.195454i
\(947\) 729.330 0.770148 0.385074 0.922886i \(-0.374176\pi\)
0.385074 + 0.922886i \(0.374176\pi\)
\(948\) 1063.93 1.12228
\(949\) 1121.30i 1.18156i
\(950\) 0 0
\(951\) −189.023 −0.198762
\(952\) 332.121i 0.348867i
\(953\) 1856.42i 1.94798i 0.226592 + 0.973990i \(0.427242\pi\)
−0.226592 + 0.973990i \(0.572758\pi\)
\(954\) 0.824918 0.000864693
\(955\) 0 0
\(956\) −12.3133 −0.0128800
\(957\) 644.580 0.673542
\(958\) 132.252i 0.138050i
\(959\) −384.845 −0.401298
\(960\) 0 0
\(961\) 916.423 0.953614
\(962\) 305.171i 0.317225i
\(963\) 108.445i 0.112611i
\(964\) 264.949i 0.274844i
\(965\) 0 0
\(966\) −126.199 −0.130640
\(967\) 1029.19 1.06432 0.532158 0.846645i \(-0.321381\pi\)
0.532158 + 0.846645i \(0.321381\pi\)
\(968\) 85.7403i 0.0885747i
\(969\) 1406.69 656.170i 1.45169 0.677162i
\(970\) 0 0
\(971\) 231.062i 0.237963i −0.992896 0.118981i \(-0.962037\pi\)
0.992896 0.118981i \(-0.0379629\pi\)
\(972\) 194.601i 0.200207i
\(973\) −824.776 −0.847663
\(974\) −132.246 −0.135776
\(975\) 0 0
\(976\) 722.835 0.740610
\(977\) 491.283i 0.502849i 0.967877 + 0.251424i \(0.0808990\pi\)
−0.967877 + 0.251424i \(0.919101\pi\)
\(978\) −101.071 −0.103345
\(979\) 67.8131i 0.0692677i
\(980\) 0 0
\(981\) 102.020i 0.103996i
\(982\) 46.8930i 0.0477525i
\(983\) 363.273i 0.369555i 0.982780 + 0.184778i \(0.0591565\pi\)
−0.982780 + 0.184778i \(0.940844\pi\)
\(984\) 684.012i 0.695135i
\(985\) 0 0
\(986\) −210.719 −0.213711
\(987\) 486.974i 0.493388i
\(988\) −1511.97 + 705.280i −1.53034 + 0.713846i
\(989\) −1233.53 −1.24725
\(990\) 0 0
\(991\) 548.658i 0.553640i 0.960922 + 0.276820i \(0.0892806\pi\)
−0.960922 + 0.276820i \(0.910719\pi\)
\(992\) 116.633 0.117574
\(993\) 396.338 0.399132
\(994\) −190.893 −0.192045
\(995\) 0 0
\(996\) 411.378i 0.413030i
\(997\) −863.800 −0.866400 −0.433200 0.901298i \(-0.642616\pi\)
−0.433200 + 0.901298i \(0.642616\pi\)
\(998\) 1.59166i 0.00159485i
\(999\) −890.078 −0.890969
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 475.3.c.i.151.7 yes 14
5.2 odd 4 475.3.d.d.474.16 28
5.3 odd 4 475.3.d.d.474.13 28
5.4 even 2 475.3.c.h.151.8 yes 14
19.18 odd 2 inner 475.3.c.i.151.8 yes 14
95.18 even 4 475.3.d.d.474.15 28
95.37 even 4 475.3.d.d.474.14 28
95.94 odd 2 475.3.c.h.151.7 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.3.c.h.151.7 14 95.94 odd 2
475.3.c.h.151.8 yes 14 5.4 even 2
475.3.c.i.151.7 yes 14 1.1 even 1 trivial
475.3.c.i.151.8 yes 14 19.18 odd 2 inner
475.3.d.d.474.13 28 5.3 odd 4
475.3.d.d.474.14 28 95.37 even 4
475.3.d.d.474.15 28 95.18 even 4
475.3.d.d.474.16 28 5.2 odd 4