Properties

Label 1155.3.b.a.736.3
Level $1155$
Weight $3$
Character 1155.736
Analytic conductor $31.471$
Analytic rank $0$
Dimension $96$
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] = [1155,3,Mod(736,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1155.736");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1155.b (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: \(31.4714705336\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 736.3
Character \(\chi\) \(=\) 1155.736
Dual form 1155.3.b.a.736.94

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.79404i q^{2} +1.73205 q^{3} -10.3948 q^{4} +2.23607 q^{5} -6.57148i q^{6} -2.64575i q^{7} +24.2621i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q-3.79404i q^{2} +1.73205 q^{3} -10.3948 q^{4} +2.23607 q^{5} -6.57148i q^{6} -2.64575i q^{7} +24.2621i q^{8} +3.00000 q^{9} -8.48374i q^{10} +(10.8209 + 1.97674i) q^{11} -18.0043 q^{12} +11.4229i q^{13} -10.0381 q^{14} +3.87298 q^{15} +50.4723 q^{16} +6.44081i q^{17} -11.3821i q^{18} +15.1396i q^{19} -23.2434 q^{20} -4.58258i q^{21} +(7.49985 - 41.0551i) q^{22} -12.3313 q^{23} +42.0231i q^{24} +5.00000 q^{25} +43.3391 q^{26} +5.19615 q^{27} +27.5020i q^{28} +36.9010i q^{29} -14.6943i q^{30} +37.4956 q^{31} -94.4457i q^{32} +(18.7424 + 3.42382i) q^{33} +24.4367 q^{34} -5.91608i q^{35} -31.1843 q^{36} +71.6335 q^{37} +57.4402 q^{38} +19.7851i q^{39} +54.2516i q^{40} -46.2442i q^{41} -17.3865 q^{42} +0.700871i q^{43} +(-112.481 - 20.5478i) q^{44} +6.70820 q^{45} +46.7855i q^{46} +36.0339 q^{47} +87.4205 q^{48} -7.00000 q^{49} -18.9702i q^{50} +11.1558i q^{51} -118.739i q^{52} -21.6936 q^{53} -19.7144i q^{54} +(24.1963 + 4.42013i) q^{55} +64.1914 q^{56} +26.2225i q^{57} +140.004 q^{58} +80.6121 q^{59} -40.2588 q^{60} +21.0333i q^{61} -142.260i q^{62} -7.93725i q^{63} -156.442 q^{64} +25.5424i q^{65} +(12.9901 - 71.1095i) q^{66} -18.1513 q^{67} -66.9508i q^{68} -21.3584 q^{69} -22.4459 q^{70} -85.1196 q^{71} +72.7862i q^{72} +103.871i q^{73} -271.781i q^{74} +8.66025 q^{75} -157.373i q^{76} +(5.22997 - 28.6295i) q^{77} +75.0655 q^{78} +43.5544i q^{79} +112.859 q^{80} +9.00000 q^{81} -175.452 q^{82} -33.9858i q^{83} +47.6348i q^{84} +14.4021i q^{85} +2.65913 q^{86} +63.9144i q^{87} +(-47.9599 + 262.538i) q^{88} -125.375 q^{89} -25.4512i q^{90} +30.2222 q^{91} +128.181 q^{92} +64.9443 q^{93} -136.714i q^{94} +33.8531i q^{95} -163.585i q^{96} -115.505 q^{97} +26.5583i q^{98} +(32.4628 + 5.93023i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 216 q^{4} + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 96 q - 216 q^{4} + 288 q^{9} + 40 q^{11} - 56 q^{14} + 488 q^{16} + 56 q^{22} + 16 q^{23} + 480 q^{25} - 64 q^{26} + 192 q^{31} + 24 q^{33} + 176 q^{34} - 648 q^{36} - 112 q^{37} - 272 q^{38} - 520 q^{44} + 416 q^{47} - 192 q^{48} - 672 q^{49} + 112 q^{53} - 80 q^{55} + 280 q^{56} - 352 q^{58} + 512 q^{59} - 1112 q^{64} + 288 q^{66} - 304 q^{67} - 480 q^{71} + 224 q^{77} + 240 q^{78} + 864 q^{81} - 720 q^{82} - 432 q^{86} - 376 q^{88} - 32 q^{89} - 384 q^{92} + 384 q^{93} + 272 q^{97} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.79404i 1.89702i −0.316744 0.948511i \(-0.602589\pi\)
0.316744 0.948511i \(-0.397411\pi\)
\(3\) 1.73205 0.577350
\(4\) −10.3948 −2.59869
\(5\) 2.23607 0.447214
\(6\) 6.57148i 1.09525i
\(7\) 2.64575i 0.377964i
\(8\) 24.2621i 3.03276i
\(9\) 3.00000 0.333333
\(10\) 8.48374i 0.848374i
\(11\) 10.8209 + 1.97674i 0.983721 + 0.179704i
\(12\) −18.0043 −1.50036
\(13\) 11.4229i 0.878687i 0.898319 + 0.439343i \(0.144789\pi\)
−0.898319 + 0.439343i \(0.855211\pi\)
\(14\) −10.0381 −0.717007
\(15\) 3.87298 0.258199
\(16\) 50.4723 3.15452
\(17\) 6.44081i 0.378871i 0.981893 + 0.189436i \(0.0606658\pi\)
−0.981893 + 0.189436i \(0.939334\pi\)
\(18\) 11.3821i 0.632341i
\(19\) 15.1396i 0.796820i 0.917207 + 0.398410i \(0.130438\pi\)
−0.917207 + 0.398410i \(0.869562\pi\)
\(20\) −23.2434 −1.16217
\(21\) 4.58258i 0.218218i
\(22\) 7.49985 41.0551i 0.340902 1.86614i
\(23\) −12.3313 −0.536143 −0.268072 0.963399i \(-0.586386\pi\)
−0.268072 + 0.963399i \(0.586386\pi\)
\(24\) 42.0231i 1.75096i
\(25\) 5.00000 0.200000
\(26\) 43.3391 1.66689
\(27\) 5.19615 0.192450
\(28\) 27.5020i 0.982214i
\(29\) 36.9010i 1.27245i 0.771505 + 0.636224i \(0.219504\pi\)
−0.771505 + 0.636224i \(0.780496\pi\)
\(30\) 14.6943i 0.489809i
\(31\) 37.4956 1.20953 0.604767 0.796402i \(-0.293266\pi\)
0.604767 + 0.796402i \(0.293266\pi\)
\(32\) 94.4457i 2.95143i
\(33\) 18.7424 + 3.42382i 0.567951 + 0.103752i
\(34\) 24.4367 0.718727
\(35\) 5.91608i 0.169031i
\(36\) −31.1843 −0.866231
\(37\) 71.6335 1.93604 0.968020 0.250873i \(-0.0807175\pi\)
0.968020 + 0.250873i \(0.0807175\pi\)
\(38\) 57.4402 1.51159
\(39\) 19.7851i 0.507310i
\(40\) 54.2516i 1.35629i
\(41\) 46.2442i 1.12791i −0.825807 0.563953i \(-0.809280\pi\)
0.825807 0.563953i \(-0.190720\pi\)
\(42\) −17.3865 −0.413964
\(43\) 0.700871i 0.0162993i 0.999967 + 0.00814966i \(0.00259415\pi\)
−0.999967 + 0.00814966i \(0.997406\pi\)
\(44\) −112.481 20.5478i −2.55639 0.466995i
\(45\) 6.70820 0.149071
\(46\) 46.7855i 1.01708i
\(47\) 36.0339 0.766679 0.383339 0.923608i \(-0.374774\pi\)
0.383339 + 0.923608i \(0.374774\pi\)
\(48\) 87.4205 1.82126
\(49\) −7.00000 −0.142857
\(50\) 18.9702i 0.379404i
\(51\) 11.1558i 0.218741i
\(52\) 118.739i 2.28344i
\(53\) −21.6936 −0.409313 −0.204656 0.978834i \(-0.565608\pi\)
−0.204656 + 0.978834i \(0.565608\pi\)
\(54\) 19.7144i 0.365082i
\(55\) 24.1963 + 4.42013i 0.439933 + 0.0803660i
\(56\) 64.1914 1.14627
\(57\) 26.2225i 0.460044i
\(58\) 140.004 2.41386
\(59\) 80.6121 1.36631 0.683154 0.730275i \(-0.260608\pi\)
0.683154 + 0.730275i \(0.260608\pi\)
\(60\) −40.2588 −0.670980
\(61\) 21.0333i 0.344808i 0.985026 + 0.172404i \(0.0551534\pi\)
−0.985026 + 0.172404i \(0.944847\pi\)
\(62\) 142.260i 2.29451i
\(63\) 7.93725i 0.125988i
\(64\) −156.442 −2.44441
\(65\) 25.5424i 0.392961i
\(66\) 12.9901 71.1095i 0.196820 1.07742i
\(67\) −18.1513 −0.270914 −0.135457 0.990783i \(-0.543250\pi\)
−0.135457 + 0.990783i \(0.543250\pi\)
\(68\) 66.9508i 0.984570i
\(69\) −21.3584 −0.309542
\(70\) −22.4459 −0.320655
\(71\) −85.1196 −1.19887 −0.599434 0.800424i \(-0.704607\pi\)
−0.599434 + 0.800424i \(0.704607\pi\)
\(72\) 72.7862i 1.01092i
\(73\) 103.871i 1.42289i 0.702743 + 0.711444i \(0.251958\pi\)
−0.702743 + 0.711444i \(0.748042\pi\)
\(74\) 271.781i 3.67271i
\(75\) 8.66025 0.115470
\(76\) 157.373i 2.07069i
\(77\) 5.22997 28.6295i 0.0679217 0.371811i
\(78\) 75.0655 0.962378
\(79\) 43.5544i 0.551321i 0.961255 + 0.275661i \(0.0888966\pi\)
−0.961255 + 0.275661i \(0.911103\pi\)
\(80\) 112.859 1.41074
\(81\) 9.00000 0.111111
\(82\) −175.452 −2.13966
\(83\) 33.9858i 0.409467i −0.978818 0.204734i \(-0.934367\pi\)
0.978818 0.204734i \(-0.0656328\pi\)
\(84\) 47.6348i 0.567081i
\(85\) 14.4021i 0.169436i
\(86\) 2.65913 0.0309202
\(87\) 63.9144i 0.734648i
\(88\) −47.9599 + 262.538i −0.544998 + 2.98339i
\(89\) −125.375 −1.40871 −0.704355 0.709847i \(-0.748764\pi\)
−0.704355 + 0.709847i \(0.748764\pi\)
\(90\) 25.4512i 0.282791i
\(91\) 30.2222 0.332112
\(92\) 128.181 1.39327
\(93\) 64.9443 0.698325
\(94\) 136.714i 1.45441i
\(95\) 33.8531i 0.356349i
\(96\) 163.585i 1.70401i
\(97\) −115.505 −1.19077 −0.595386 0.803440i \(-0.703001\pi\)
−0.595386 + 0.803440i \(0.703001\pi\)
\(98\) 26.5583i 0.271003i
\(99\) 32.4628 + 5.93023i 0.327907 + 0.0599013i
\(100\) −51.9739 −0.519739
\(101\) 108.413i 1.07339i −0.843775 0.536696i \(-0.819672\pi\)
0.843775 0.536696i \(-0.180328\pi\)
\(102\) 42.3256 0.414957
\(103\) −6.99136 −0.0678772 −0.0339386 0.999424i \(-0.510805\pi\)
−0.0339386 + 0.999424i \(0.510805\pi\)
\(104\) −277.144 −2.66484
\(105\) 10.2470i 0.0975900i
\(106\) 82.3064i 0.776476i
\(107\) 97.3371i 0.909693i 0.890570 + 0.454846i \(0.150306\pi\)
−0.890570 + 0.454846i \(0.849694\pi\)
\(108\) −54.0128 −0.500119
\(109\) 178.540i 1.63798i −0.573807 0.818990i \(-0.694534\pi\)
0.573807 0.818990i \(-0.305466\pi\)
\(110\) 16.7702 91.8020i 0.152456 0.834563i
\(111\) 124.073 1.11777
\(112\) 133.537i 1.19229i
\(113\) 110.514 0.977999 0.488999 0.872284i \(-0.337362\pi\)
0.488999 + 0.872284i \(0.337362\pi\)
\(114\) 99.4894 0.872714
\(115\) −27.5736 −0.239770
\(116\) 383.577i 3.30670i
\(117\) 34.2688i 0.292896i
\(118\) 305.846i 2.59192i
\(119\) 17.0408 0.143200
\(120\) 93.9666i 0.783055i
\(121\) 113.185 + 42.7804i 0.935413 + 0.353557i
\(122\) 79.8012 0.654108
\(123\) 80.0973i 0.651197i
\(124\) −389.758 −3.14321
\(125\) 11.1803 0.0894427
\(126\) −30.1143 −0.239002
\(127\) 163.597i 1.28817i −0.764955 0.644084i \(-0.777239\pi\)
0.764955 0.644084i \(-0.222761\pi\)
\(128\) 215.766i 1.68567i
\(129\) 1.21394i 0.00941041i
\(130\) 96.9092 0.745455
\(131\) 50.4239i 0.384915i −0.981305 0.192457i \(-0.938354\pi\)
0.981305 0.192457i \(-0.0616458\pi\)
\(132\) −194.823 35.5898i −1.47593 0.269620i
\(133\) 40.0556 0.301170
\(134\) 68.8667i 0.513931i
\(135\) 11.6190 0.0860663
\(136\) −156.267 −1.14902
\(137\) 173.912 1.26943 0.634717 0.772745i \(-0.281117\pi\)
0.634717 + 0.772745i \(0.281117\pi\)
\(138\) 81.0348i 0.587209i
\(139\) 144.788i 1.04164i −0.853667 0.520819i \(-0.825627\pi\)
0.853667 0.520819i \(-0.174373\pi\)
\(140\) 61.4963i 0.439259i
\(141\) 62.4126 0.442642
\(142\) 322.947i 2.27428i
\(143\) −22.5802 + 123.607i −0.157903 + 0.864382i
\(144\) 151.417 1.05151
\(145\) 82.5131i 0.569056i
\(146\) 394.090 2.69925
\(147\) −12.1244 −0.0824786
\(148\) −744.614 −5.03118
\(149\) 33.0981i 0.222135i 0.993813 + 0.111067i \(0.0354269\pi\)
−0.993813 + 0.111067i \(0.964573\pi\)
\(150\) 32.8574i 0.219049i
\(151\) 31.0481i 0.205616i −0.994701 0.102808i \(-0.967217\pi\)
0.994701 0.102808i \(-0.0327828\pi\)
\(152\) −367.317 −2.41656
\(153\) 19.3224i 0.126290i
\(154\) −108.622 19.8427i −0.705335 0.128849i
\(155\) 83.8427 0.540920
\(156\) 205.662i 1.31834i
\(157\) −223.833 −1.42569 −0.712845 0.701322i \(-0.752594\pi\)
−0.712845 + 0.701322i \(0.752594\pi\)
\(158\) 165.247 1.04587
\(159\) −37.5744 −0.236317
\(160\) 211.187i 1.31992i
\(161\) 32.6255i 0.202643i
\(162\) 34.1464i 0.210780i
\(163\) −83.3361 −0.511264 −0.255632 0.966774i \(-0.582284\pi\)
−0.255632 + 0.966774i \(0.582284\pi\)
\(164\) 480.698i 2.93108i
\(165\) 41.9093 + 7.65589i 0.253996 + 0.0463993i
\(166\) −128.944 −0.776768
\(167\) 57.1421i 0.342168i −0.985256 0.171084i \(-0.945273\pi\)
0.985256 0.171084i \(-0.0547270\pi\)
\(168\) 111.183 0.661802
\(169\) 38.5167 0.227910
\(170\) 54.6422 0.321424
\(171\) 45.4187i 0.265607i
\(172\) 7.28539i 0.0423569i
\(173\) 215.222i 1.24406i 0.782993 + 0.622030i \(0.213692\pi\)
−0.782993 + 0.622030i \(0.786308\pi\)
\(174\) 242.494 1.39364
\(175\) 13.2288i 0.0755929i
\(176\) 546.157 + 99.7706i 3.10316 + 0.566879i
\(177\) 139.624 0.788838
\(178\) 475.679i 2.67236i
\(179\) 214.866 1.20037 0.600184 0.799862i \(-0.295094\pi\)
0.600184 + 0.799862i \(0.295094\pi\)
\(180\) −69.7303 −0.387390
\(181\) −235.805 −1.30279 −0.651394 0.758739i \(-0.725816\pi\)
−0.651394 + 0.758739i \(0.725816\pi\)
\(182\) 114.664i 0.630025i
\(183\) 36.4307i 0.199075i
\(184\) 299.183i 1.62599i
\(185\) 160.177 0.865824
\(186\) 246.401i 1.32474i
\(187\) −12.7318 + 69.6955i −0.0680846 + 0.372703i
\(188\) −374.564 −1.99236
\(189\) 13.7477i 0.0727393i
\(190\) 128.440 0.676002
\(191\) 186.716 0.977568 0.488784 0.872405i \(-0.337441\pi\)
0.488784 + 0.872405i \(0.337441\pi\)
\(192\) −270.966 −1.41128
\(193\) 303.729i 1.57372i 0.617130 + 0.786862i \(0.288295\pi\)
−0.617130 + 0.786862i \(0.711705\pi\)
\(194\) 438.231i 2.25892i
\(195\) 44.2408i 0.226876i
\(196\) 72.7634 0.371242
\(197\) 220.622i 1.11991i 0.828523 + 0.559955i \(0.189182\pi\)
−0.828523 + 0.559955i \(0.810818\pi\)
\(198\) 22.4995 123.165i 0.113634 0.622047i
\(199\) 122.270 0.614425 0.307212 0.951641i \(-0.400604\pi\)
0.307212 + 0.951641i \(0.400604\pi\)
\(200\) 121.310i 0.606552i
\(201\) −31.4389 −0.156413
\(202\) −411.322 −2.03625
\(203\) 97.6308 0.480940
\(204\) 115.962i 0.568442i
\(205\) 103.405i 0.504415i
\(206\) 26.5255i 0.128765i
\(207\) −36.9939 −0.178714
\(208\) 576.541i 2.77183i
\(209\) −29.9270 + 163.824i −0.143192 + 0.783848i
\(210\) −38.8774 −0.185130
\(211\) 226.174i 1.07192i 0.844245 + 0.535958i \(0.180049\pi\)
−0.844245 + 0.535958i \(0.819951\pi\)
\(212\) 225.500 1.06368
\(213\) −147.431 −0.692166
\(214\) 369.301 1.72571
\(215\) 1.56719i 0.00728928i
\(216\) 126.069i 0.583655i
\(217\) 99.2040i 0.457161i
\(218\) −677.388 −3.10729
\(219\) 179.909i 0.821504i
\(220\) −251.515 45.9463i −1.14325 0.208847i
\(221\) −73.5729 −0.332909
\(222\) 470.738i 2.12044i
\(223\) 343.952 1.54239 0.771193 0.636601i \(-0.219660\pi\)
0.771193 + 0.636601i \(0.219660\pi\)
\(224\) −249.880 −1.11554
\(225\) 15.0000 0.0666667
\(226\) 419.294i 1.85529i
\(227\) 40.9631i 0.180454i −0.995921 0.0902271i \(-0.971241\pi\)
0.995921 0.0902271i \(-0.0287593\pi\)
\(228\) 272.577i 1.19551i
\(229\) 34.1390 0.149079 0.0745394 0.997218i \(-0.476251\pi\)
0.0745394 + 0.997218i \(0.476251\pi\)
\(230\) 104.615i 0.454850i
\(231\) 9.05857 49.5877i 0.0392146 0.214665i
\(232\) −895.294 −3.85902
\(233\) 356.584i 1.53041i −0.643790 0.765203i \(-0.722639\pi\)
0.643790 0.765203i \(-0.277361\pi\)
\(234\) 130.017 0.555629
\(235\) 80.5743 0.342869
\(236\) −837.945 −3.55061
\(237\) 75.4384i 0.318305i
\(238\) 64.6535i 0.271653i
\(239\) 85.5789i 0.358071i −0.983843 0.179035i \(-0.942702\pi\)
0.983843 0.179035i \(-0.0572977\pi\)
\(240\) 195.478 0.814493
\(241\) 57.4930i 0.238560i 0.992861 + 0.119280i \(0.0380586\pi\)
−0.992861 + 0.119280i \(0.961941\pi\)
\(242\) 162.311 429.429i 0.670705 1.77450i
\(243\) 15.5885 0.0641500
\(244\) 218.636i 0.896050i
\(245\) −15.6525 −0.0638877
\(246\) −303.893 −1.23534
\(247\) −172.938 −0.700155
\(248\) 909.720i 3.66823i
\(249\) 58.8651i 0.236406i
\(250\) 42.4187i 0.169675i
\(251\) −285.926 −1.13915 −0.569574 0.821940i \(-0.692892\pi\)
−0.569574 + 0.821940i \(0.692892\pi\)
\(252\) 82.5060i 0.327405i
\(253\) −133.436 24.3758i −0.527415 0.0963470i
\(254\) −620.695 −2.44368
\(255\) 24.9451i 0.0978241i
\(256\) 192.858 0.753350
\(257\) 226.034 0.879510 0.439755 0.898118i \(-0.355065\pi\)
0.439755 + 0.898118i \(0.355065\pi\)
\(258\) 4.60576 0.0178518
\(259\) 189.524i 0.731754i
\(260\) 265.508i 1.02118i
\(261\) 110.703i 0.424149i
\(262\) −191.310 −0.730192
\(263\) 339.512i 1.29092i 0.763794 + 0.645461i \(0.223335\pi\)
−0.763794 + 0.645461i \(0.776665\pi\)
\(264\) −83.0689 + 454.729i −0.314655 + 1.72246i
\(265\) −48.5083 −0.183050
\(266\) 151.973i 0.571326i
\(267\) −217.156 −0.813320
\(268\) 188.678 0.704024
\(269\) 75.4600 0.280521 0.140260 0.990115i \(-0.455206\pi\)
0.140260 + 0.990115i \(0.455206\pi\)
\(270\) 44.0828i 0.163270i
\(271\) 46.9854i 0.173378i 0.996235 + 0.0866889i \(0.0276286\pi\)
−0.996235 + 0.0866889i \(0.972371\pi\)
\(272\) 325.082i 1.19515i
\(273\) 52.3464 0.191745
\(274\) 659.831i 2.40814i
\(275\) 54.1046 + 9.88371i 0.196744 + 0.0359408i
\(276\) 222.016 0.804406
\(277\) 482.230i 1.74090i 0.492255 + 0.870451i \(0.336173\pi\)
−0.492255 + 0.870451i \(0.663827\pi\)
\(278\) −549.331 −1.97601
\(279\) 112.487 0.403178
\(280\) 143.536 0.512630
\(281\) 533.222i 1.89759i −0.315894 0.948794i \(-0.602304\pi\)
0.315894 0.948794i \(-0.397696\pi\)
\(282\) 236.796i 0.839702i
\(283\) 104.404i 0.368920i 0.982840 + 0.184460i \(0.0590535\pi\)
−0.982840 + 0.184460i \(0.940946\pi\)
\(284\) 884.799 3.11549
\(285\) 58.6353i 0.205738i
\(286\) 468.969 + 85.6702i 1.63975 + 0.299546i
\(287\) −122.351 −0.426309
\(288\) 283.337i 0.983810i
\(289\) 247.516 0.856457
\(290\) 313.058 1.07951
\(291\) −200.060 −0.687492
\(292\) 1079.71i 3.69765i
\(293\) 125.564i 0.428545i −0.976774 0.214272i \(-0.931262\pi\)
0.976774 0.214272i \(-0.0687380\pi\)
\(294\) 46.0003i 0.156464i
\(295\) 180.254 0.611031
\(296\) 1737.98i 5.87154i
\(297\) 56.2272 + 10.2715i 0.189317 + 0.0345840i
\(298\) 125.576 0.421395
\(299\) 140.859i 0.471102i
\(300\) −90.0214 −0.300071
\(301\) 1.85433 0.00616056
\(302\) −117.798 −0.390059
\(303\) 187.776i 0.619724i
\(304\) 764.129i 2.51358i
\(305\) 47.0318i 0.154203i
\(306\) 73.3101 0.239576
\(307\) 357.902i 1.16580i 0.812543 + 0.582902i \(0.198083\pi\)
−0.812543 + 0.582902i \(0.801917\pi\)
\(308\) −54.3644 + 297.597i −0.176508 + 0.966224i
\(309\) −12.1094 −0.0391889
\(310\) 318.103i 1.02614i
\(311\) 497.178 1.59864 0.799321 0.600905i \(-0.205193\pi\)
0.799321 + 0.600905i \(0.205193\pi\)
\(312\) −480.027 −1.53855
\(313\) 168.118 0.537118 0.268559 0.963263i \(-0.413453\pi\)
0.268559 + 0.963263i \(0.413453\pi\)
\(314\) 849.233i 2.70456i
\(315\) 17.7482i 0.0563436i
\(316\) 452.738i 1.43271i
\(317\) −81.4840 −0.257047 −0.128524 0.991706i \(-0.541024\pi\)
−0.128524 + 0.991706i \(0.541024\pi\)
\(318\) 142.559i 0.448299i
\(319\) −72.9437 + 399.303i −0.228664 + 1.25173i
\(320\) −349.816 −1.09317
\(321\) 168.593i 0.525211i
\(322\) 123.783 0.384418
\(323\) −97.5111 −0.301892
\(324\) −93.5530 −0.288744
\(325\) 57.1146i 0.175737i
\(326\) 316.181i 0.969880i
\(327\) 309.240i 0.945689i
\(328\) 1121.98 3.42067
\(329\) 95.3368i 0.289777i
\(330\) 29.0468 159.006i 0.0880206 0.481835i
\(331\) −103.610 −0.313020 −0.156510 0.987676i \(-0.550024\pi\)
−0.156510 + 0.987676i \(0.550024\pi\)
\(332\) 353.274i 1.06408i
\(333\) 214.900 0.645347
\(334\) −216.800 −0.649101
\(335\) −40.5875 −0.121157
\(336\) 231.293i 0.688372i
\(337\) 17.0836i 0.0506932i 0.999679 + 0.0253466i \(0.00806894\pi\)
−0.999679 + 0.0253466i \(0.991931\pi\)
\(338\) 146.134i 0.432350i
\(339\) 191.416 0.564648
\(340\) 149.706i 0.440313i
\(341\) 405.737 + 74.1191i 1.18984 + 0.217358i
\(342\) 172.321 0.503862
\(343\) 18.5203i 0.0539949i
\(344\) −17.0046 −0.0494319
\(345\) −47.7589 −0.138432
\(346\) 816.563 2.36001
\(347\) 649.237i 1.87100i −0.353327 0.935500i \(-0.614950\pi\)
0.353327 0.935500i \(-0.385050\pi\)
\(348\) 664.375i 1.90912i
\(349\) 276.848i 0.793259i −0.917979 0.396630i \(-0.870180\pi\)
0.917979 0.396630i \(-0.129820\pi\)
\(350\) −50.1905 −0.143401
\(351\) 59.3553i 0.169103i
\(352\) 186.695 1021.99i 0.530383 2.90338i
\(353\) 174.078 0.493138 0.246569 0.969125i \(-0.420697\pi\)
0.246569 + 0.969125i \(0.420697\pi\)
\(354\) 529.741i 1.49644i
\(355\) −190.333 −0.536150
\(356\) 1303.25 3.66081
\(357\) 29.5155 0.0826764
\(358\) 815.211i 2.27712i
\(359\) 54.5892i 0.152059i −0.997106 0.0760295i \(-0.975776\pi\)
0.997106 0.0760295i \(-0.0242243\pi\)
\(360\) 162.755i 0.452097i
\(361\) 131.793 0.365078
\(362\) 894.654i 2.47142i
\(363\) 196.042 + 74.0978i 0.540061 + 0.204126i
\(364\) −314.153 −0.863058
\(365\) 232.262i 0.636335i
\(366\) 138.220 0.377650
\(367\) −720.700 −1.96376 −0.981880 0.189502i \(-0.939313\pi\)
−0.981880 + 0.189502i \(0.939313\pi\)
\(368\) −622.388 −1.69127
\(369\) 138.733i 0.375969i
\(370\) 607.720i 1.64249i
\(371\) 57.3958i 0.154706i
\(372\) −675.081 −1.81473
\(373\) 482.361i 1.29319i −0.762832 0.646596i \(-0.776192\pi\)
0.762832 0.646596i \(-0.223808\pi\)
\(374\) 264.428 + 48.3051i 0.707027 + 0.129158i
\(375\) 19.3649 0.0516398
\(376\) 874.257i 2.32515i
\(377\) −421.517 −1.11808
\(378\) −52.1595 −0.137988
\(379\) −78.1531 −0.206209 −0.103104 0.994671i \(-0.532878\pi\)
−0.103104 + 0.994671i \(0.532878\pi\)
\(380\) 351.896i 0.926041i
\(381\) 283.359i 0.743724i
\(382\) 708.407i 1.85447i
\(383\) −39.3013 −0.102614 −0.0513071 0.998683i \(-0.516339\pi\)
−0.0513071 + 0.998683i \(0.516339\pi\)
\(384\) 373.718i 0.973224i
\(385\) 11.6946 64.0175i 0.0303755 0.166279i
\(386\) 1152.36 2.98539
\(387\) 2.10261i 0.00543311i
\(388\) 1200.65 3.09445
\(389\) −185.063 −0.475741 −0.237870 0.971297i \(-0.576449\pi\)
−0.237870 + 0.971297i \(0.576449\pi\)
\(390\) 167.852 0.430389
\(391\) 79.4235i 0.203129i
\(392\) 169.834i 0.433251i
\(393\) 87.3367i 0.222231i
\(394\) 837.051 2.12449
\(395\) 97.3905i 0.246558i
\(396\) −337.443 61.6434i −0.852130 0.155665i
\(397\) 647.463 1.63089 0.815445 0.578835i \(-0.196492\pi\)
0.815445 + 0.578835i \(0.196492\pi\)
\(398\) 463.900i 1.16558i
\(399\) 69.3783 0.173880
\(400\) 252.361 0.630903
\(401\) 81.9779 0.204434 0.102217 0.994762i \(-0.467406\pi\)
0.102217 + 0.994762i \(0.467406\pi\)
\(402\) 119.281i 0.296718i
\(403\) 428.309i 1.06280i
\(404\) 1126.93i 2.78942i
\(405\) 20.1246 0.0496904
\(406\) 370.416i 0.912354i
\(407\) 775.141 + 141.601i 1.90452 + 0.347914i
\(408\) −270.663 −0.663389
\(409\) 618.678i 1.51266i −0.654190 0.756331i \(-0.726990\pi\)
0.654190 0.756331i \(-0.273010\pi\)
\(410\) −392.324 −0.956887
\(411\) 301.225 0.732908
\(412\) 72.6736 0.176392
\(413\) 213.280i 0.516416i
\(414\) 140.356i 0.339025i
\(415\) 75.9945i 0.183119i
\(416\) 1078.85 2.59338
\(417\) 250.780i 0.601390i
\(418\) 621.557 + 113.545i 1.48698 + 0.271638i
\(419\) 360.854 0.861228 0.430614 0.902536i \(-0.358297\pi\)
0.430614 + 0.902536i \(0.358297\pi\)
\(420\) 106.515i 0.253607i
\(421\) −779.460 −1.85145 −0.925725 0.378197i \(-0.876544\pi\)
−0.925725 + 0.378197i \(0.876544\pi\)
\(422\) 858.115 2.03345
\(423\) 108.102 0.255560
\(424\) 526.331i 1.24135i
\(425\) 32.2040i 0.0757742i
\(426\) 559.361i 1.31305i
\(427\) 55.6488 0.130325
\(428\) 1011.80i 2.36401i
\(429\) −39.1100 + 214.093i −0.0911656 + 0.499051i
\(430\) 5.94600 0.0138279
\(431\) 282.147i 0.654634i 0.944915 + 0.327317i \(0.106144\pi\)
−0.944915 + 0.327317i \(0.893856\pi\)
\(432\) 262.262 0.607087
\(433\) −670.676 −1.54891 −0.774453 0.632631i \(-0.781975\pi\)
−0.774453 + 0.632631i \(0.781975\pi\)
\(434\) −376.384 −0.867245
\(435\) 142.917i 0.328544i
\(436\) 1855.88i 4.25661i
\(437\) 186.691i 0.427210i
\(438\) 682.584 1.55841
\(439\) 426.904i 0.972447i 0.873835 + 0.486223i \(0.161626\pi\)
−0.873835 + 0.486223i \(0.838374\pi\)
\(440\) −107.241 + 587.053i −0.243731 + 1.33421i
\(441\) −21.0000 −0.0476190
\(442\) 279.139i 0.631536i
\(443\) 11.0112 0.0248560 0.0124280 0.999923i \(-0.496044\pi\)
0.0124280 + 0.999923i \(0.496044\pi\)
\(444\) −1289.71 −2.90475
\(445\) −280.348 −0.629995
\(446\) 1304.97i 2.92594i
\(447\) 57.3275i 0.128250i
\(448\) 413.908i 0.923901i
\(449\) 428.106 0.953464 0.476732 0.879049i \(-0.341821\pi\)
0.476732 + 0.879049i \(0.341821\pi\)
\(450\) 56.9107i 0.126468i
\(451\) 91.4128 500.405i 0.202689 1.10955i
\(452\) −1148.77 −2.54152
\(453\) 53.7768i 0.118713i
\(454\) −155.416 −0.342326
\(455\) 67.5789 0.148525
\(456\) −636.212 −1.39520
\(457\) 419.530i 0.918009i 0.888434 + 0.459004i \(0.151794\pi\)
−0.888434 + 0.459004i \(0.848206\pi\)
\(458\) 129.525i 0.282806i
\(459\) 33.4674i 0.0729138i
\(460\) 286.621 0.623090
\(461\) 50.8690i 0.110345i 0.998477 + 0.0551724i \(0.0175708\pi\)
−0.998477 + 0.0551724i \(0.982429\pi\)
\(462\) −188.138 34.3686i −0.407225 0.0743910i
\(463\) 213.799 0.461769 0.230884 0.972981i \(-0.425838\pi\)
0.230884 + 0.972981i \(0.425838\pi\)
\(464\) 1862.48i 4.01396i
\(465\) 145.220 0.312301
\(466\) −1352.90 −2.90321
\(467\) −421.138 −0.901794 −0.450897 0.892576i \(-0.648896\pi\)
−0.450897 + 0.892576i \(0.648896\pi\)
\(468\) 356.216i 0.761146i
\(469\) 48.0237i 0.102396i
\(470\) 305.702i 0.650431i
\(471\) −387.691 −0.823122
\(472\) 1955.82i 4.14368i
\(473\) −1.38544 + 7.58407i −0.00292905 + 0.0160340i
\(474\) 286.217 0.603832
\(475\) 75.6979i 0.159364i
\(476\) −177.135 −0.372132
\(477\) −65.0808 −0.136438
\(478\) −324.690 −0.679268
\(479\) 465.160i 0.971107i 0.874207 + 0.485553i \(0.161382\pi\)
−0.874207 + 0.485553i \(0.838618\pi\)
\(480\) 365.787i 0.762056i
\(481\) 818.264i 1.70117i
\(482\) 218.131 0.452554
\(483\) 56.5091i 0.116996i
\(484\) −1176.53 444.692i −2.43085 0.918786i
\(485\) −258.277 −0.532529
\(486\) 59.1433i 0.121694i
\(487\) 596.434 1.22471 0.612355 0.790583i \(-0.290222\pi\)
0.612355 + 0.790583i \(0.290222\pi\)
\(488\) −510.311 −1.04572
\(489\) −144.342 −0.295179
\(490\) 59.3862i 0.121196i
\(491\) 459.925i 0.936710i 0.883540 + 0.468355i \(0.155153\pi\)
−0.883540 + 0.468355i \(0.844847\pi\)
\(492\) 832.593i 1.69226i
\(493\) −237.672 −0.482093
\(494\) 656.136i 1.32821i
\(495\) 72.5890 + 13.2604i 0.146644 + 0.0267887i
\(496\) 1892.49 3.81550
\(497\) 225.205i 0.453129i
\(498\) −223.337 −0.448467
\(499\) −536.311 −1.07477 −0.537386 0.843336i \(-0.680588\pi\)
−0.537386 + 0.843336i \(0.680588\pi\)
\(500\) −116.217 −0.232434
\(501\) 98.9730i 0.197551i
\(502\) 1084.82i 2.16099i
\(503\) 706.964i 1.40549i −0.711439 0.702747i \(-0.751956\pi\)
0.711439 0.702747i \(-0.248044\pi\)
\(504\) 192.574 0.382092
\(505\) 242.418i 0.480036i
\(506\) −92.4828 + 506.262i −0.182772 + 1.00052i
\(507\) 66.7130 0.131584
\(508\) 1700.56i 3.34755i
\(509\) −716.152 −1.40698 −0.703490 0.710706i \(-0.748376\pi\)
−0.703490 + 0.710706i \(0.748376\pi\)
\(510\) 94.6430 0.185574
\(511\) 274.816 0.537801
\(512\) 131.355i 0.256553i
\(513\) 78.6676i 0.153348i
\(514\) 857.583i 1.66845i
\(515\) −15.6331 −0.0303556
\(516\) 12.6187i 0.0244548i
\(517\) 389.920 + 71.2298i 0.754198 + 0.137775i
\(518\) −719.064 −1.38815
\(519\) 372.776i 0.718259i
\(520\) −619.712 −1.19175
\(521\) −540.716 −1.03784 −0.518921 0.854822i \(-0.673666\pi\)
−0.518921 + 0.854822i \(0.673666\pi\)
\(522\) 420.012 0.804620
\(523\) 830.514i 1.58798i 0.607930 + 0.793991i \(0.292000\pi\)
−0.607930 + 0.793991i \(0.708000\pi\)
\(524\) 524.145i 1.00028i
\(525\) 22.9129i 0.0436436i
\(526\) 1288.12 2.44891
\(527\) 241.502i 0.458258i
\(528\) 945.971 + 172.808i 1.79161 + 0.327288i
\(529\) −376.939 −0.712551
\(530\) 184.043i 0.347251i
\(531\) 241.836 0.455436
\(532\) −416.369 −0.782648
\(533\) 528.244 0.991077
\(534\) 823.901i 1.54289i
\(535\) 217.652i 0.406827i
\(536\) 440.387i 0.821618i
\(537\) 372.159 0.693033
\(538\) 286.299i 0.532154i
\(539\) −75.7465 13.8372i −0.140532 0.0256720i
\(540\) −120.776 −0.223660
\(541\) 563.661i 1.04189i 0.853591 + 0.520944i \(0.174420\pi\)
−0.853591 + 0.520944i \(0.825580\pi\)
\(542\) 178.265 0.328902
\(543\) −408.426 −0.752166
\(544\) 608.307 1.11821
\(545\) 399.227i 0.732527i
\(546\) 198.605i 0.363745i
\(547\) 313.843i 0.573753i −0.957968 0.286877i \(-0.907383\pi\)
0.957968 0.286877i \(-0.0926170\pi\)
\(548\) −1807.78 −3.29887
\(549\) 63.0998i 0.114936i
\(550\) 37.4992 205.275i 0.0681804 0.373228i
\(551\) −558.665 −1.01391
\(552\) 518.199i 0.938767i
\(553\) 115.234 0.208380
\(554\) 1829.60 3.30253
\(555\) 277.435 0.499883
\(556\) 1505.03i 2.70690i
\(557\) 326.419i 0.586031i −0.956108 0.293016i \(-0.905341\pi\)
0.956108 0.293016i \(-0.0946588\pi\)
\(558\) 426.780i 0.764838i
\(559\) −8.00599 −0.0143220
\(560\) 298.598i 0.533211i
\(561\) −22.0522 + 120.716i −0.0393087 + 0.215180i
\(562\) −2023.07 −3.59977
\(563\) 861.940i 1.53098i −0.643450 0.765488i \(-0.722497\pi\)
0.643450 0.765488i \(-0.277503\pi\)
\(564\) −648.765 −1.15029
\(565\) 247.116 0.437374
\(566\) 396.114 0.699849
\(567\) 23.8118i 0.0419961i
\(568\) 2065.18i 3.63587i
\(569\) 955.752i 1.67971i −0.542814 0.839853i \(-0.682641\pi\)
0.542814 0.839853i \(-0.317359\pi\)
\(570\) 222.465 0.390290
\(571\) 601.731i 1.05382i −0.849921 0.526910i \(-0.823351\pi\)
0.849921 0.526910i \(-0.176649\pi\)
\(572\) 234.716 1284.86i 0.410343 2.24627i
\(573\) 323.401 0.564399
\(574\) 464.204i 0.808717i
\(575\) −61.6565 −0.107229
\(576\) −469.327 −0.814804
\(577\) −861.319 −1.49275 −0.746377 0.665524i \(-0.768208\pi\)
−0.746377 + 0.665524i \(0.768208\pi\)
\(578\) 939.087i 1.62472i
\(579\) 526.073i 0.908590i
\(580\) 857.705i 1.47880i
\(581\) −89.9179 −0.154764
\(582\) 759.038i 1.30419i
\(583\) −234.745 42.8826i −0.402650 0.0735551i
\(584\) −2520.12 −4.31527
\(585\) 76.6273i 0.130987i
\(586\) −476.394 −0.812959
\(587\) −354.226 −0.603451 −0.301726 0.953395i \(-0.597563\pi\)
−0.301726 + 0.953395i \(0.597563\pi\)
\(588\) 126.030 0.214337
\(589\) 567.667i 0.963782i
\(590\) 683.893i 1.15914i
\(591\) 382.129i 0.646580i
\(592\) 3615.50 6.10727
\(593\) 297.046i 0.500921i 0.968127 + 0.250460i \(0.0805820\pi\)
−0.968127 + 0.250460i \(0.919418\pi\)
\(594\) 38.9704 213.328i 0.0656067 0.359139i
\(595\) 38.1043 0.0640409
\(596\) 344.047i 0.577260i
\(597\) 211.779 0.354738
\(598\) −534.427 −0.893691
\(599\) 1009.59 1.68547 0.842733 0.538332i \(-0.180945\pi\)
0.842733 + 0.538332i \(0.180945\pi\)
\(600\) 210.116i 0.350193i
\(601\) 619.434i 1.03067i 0.856988 + 0.515336i \(0.172333\pi\)
−0.856988 + 0.515336i \(0.827667\pi\)
\(602\) 7.03541i 0.0116867i
\(603\) −54.4538 −0.0903048
\(604\) 322.738i 0.534334i
\(605\) 253.089 + 95.6598i 0.418329 + 0.158115i
\(606\) −712.431 −1.17563
\(607\) 65.3491i 0.107659i 0.998550 + 0.0538296i \(0.0171428\pi\)
−0.998550 + 0.0538296i \(0.982857\pi\)
\(608\) 1429.87 2.35176
\(609\) 169.101 0.277671
\(610\) 178.441 0.292526
\(611\) 411.613i 0.673671i
\(612\) 200.852i 0.328190i
\(613\) 930.121i 1.51733i −0.651483 0.758663i \(-0.725853\pi\)
0.651483 0.758663i \(-0.274147\pi\)
\(614\) 1357.89 2.21156
\(615\) 179.103i 0.291224i
\(616\) 694.610 + 126.890i 1.12761 + 0.205990i
\(617\) −365.168 −0.591844 −0.295922 0.955212i \(-0.595627\pi\)
−0.295922 + 0.955212i \(0.595627\pi\)
\(618\) 45.9435i 0.0743423i
\(619\) −544.199 −0.879159 −0.439579 0.898204i \(-0.644872\pi\)
−0.439579 + 0.898204i \(0.644872\pi\)
\(620\) −871.526 −1.40569
\(621\) −64.0753 −0.103181
\(622\) 1886.31i 3.03266i
\(623\) 331.712i 0.532443i
\(624\) 998.598i 1.60032i
\(625\) 25.0000 0.0400000
\(626\) 637.847i 1.01893i
\(627\) −51.8352 + 283.752i −0.0826717 + 0.452555i
\(628\) 2326.70 3.70493
\(629\) 461.378i 0.733510i
\(630\) −67.3376 −0.106885
\(631\) −548.112 −0.868640 −0.434320 0.900759i \(-0.643011\pi\)
−0.434320 + 0.900759i \(0.643011\pi\)
\(632\) −1056.72 −1.67202
\(633\) 391.745i 0.618871i
\(634\) 309.154i 0.487624i
\(635\) 365.815i 0.576086i
\(636\) 390.577 0.614115
\(637\) 79.9605i 0.125527i
\(638\) 1514.97 + 276.752i 2.37457 + 0.433780i
\(639\) −255.359 −0.399622
\(640\) 482.468i 0.753856i
\(641\) −169.697 −0.264738 −0.132369 0.991200i \(-0.542258\pi\)
−0.132369 + 0.991200i \(0.542258\pi\)
\(642\) 639.649 0.996338
\(643\) −1243.73 −1.93426 −0.967132 0.254274i \(-0.918164\pi\)
−0.967132 + 0.254274i \(0.918164\pi\)
\(644\) 339.135i 0.526607i
\(645\) 2.71446i 0.00420847i
\(646\) 369.962i 0.572696i
\(647\) −477.079 −0.737371 −0.368685 0.929554i \(-0.620192\pi\)
−0.368685 + 0.929554i \(0.620192\pi\)
\(648\) 218.359i 0.336973i
\(649\) 872.298 + 159.349i 1.34406 + 0.245531i
\(650\) 216.695 0.333378
\(651\) 171.826i 0.263942i
\(652\) 866.260 1.32862
\(653\) −80.0744 −0.122625 −0.0613127 0.998119i \(-0.519529\pi\)
−0.0613127 + 0.998119i \(0.519529\pi\)
\(654\) −1173.27 −1.79399
\(655\) 112.751i 0.172139i
\(656\) 2334.05i 3.55800i
\(657\) 311.612i 0.474296i
\(658\) −361.712 −0.549714
\(659\) 1258.14i 1.90917i −0.297941 0.954584i \(-0.596300\pi\)
0.297941 0.954584i \(-0.403700\pi\)
\(660\) −435.638 79.5813i −0.660057 0.120578i
\(661\) 284.579 0.430528 0.215264 0.976556i \(-0.430939\pi\)
0.215264 + 0.976556i \(0.430939\pi\)
\(662\) 393.100i 0.593806i
\(663\) −127.432 −0.192205
\(664\) 824.565 1.24181
\(665\) 89.5670 0.134687
\(666\) 815.342i 1.22424i
\(667\) 455.037i 0.682214i
\(668\) 593.979i 0.889191i
\(669\) 595.743 0.890497
\(670\) 153.991i 0.229837i
\(671\) −41.5774 + 227.600i −0.0619633 + 0.339195i
\(672\) −432.805 −0.644055
\(673\) 577.479i 0.858067i 0.903289 + 0.429033i \(0.141146\pi\)
−0.903289 + 0.429033i \(0.858854\pi\)
\(674\) 64.8160 0.0961662
\(675\) 25.9808 0.0384900
\(676\) −400.373 −0.592268
\(677\) 115.713i 0.170921i 0.996342 + 0.0854604i \(0.0272361\pi\)
−0.996342 + 0.0854604i \(0.972764\pi\)
\(678\) 726.239i 1.07115i
\(679\) 305.597i 0.450069i
\(680\) −349.424 −0.513859
\(681\) 70.9502i 0.104185i
\(682\) 281.211 1539.38i 0.412333 2.25716i
\(683\) −64.0207 −0.0937346 −0.0468673 0.998901i \(-0.514924\pi\)
−0.0468673 + 0.998901i \(0.514924\pi\)
\(684\) 472.118i 0.690230i
\(685\) 388.880 0.567708
\(686\) 70.2667 0.102430
\(687\) 59.1305 0.0860706
\(688\) 35.3745i 0.0514164i
\(689\) 247.804i 0.359658i
\(690\) 181.199i 0.262608i
\(691\) −1183.60 −1.71288 −0.856440 0.516247i \(-0.827329\pi\)
−0.856440 + 0.516247i \(0.827329\pi\)
\(692\) 2237.19i 3.23293i
\(693\) 15.6899 85.8885i 0.0226406 0.123937i
\(694\) −2463.23 −3.54933
\(695\) 323.755i 0.465834i
\(696\) −1550.69 −2.22801
\(697\) 297.850 0.427331
\(698\) −1050.37 −1.50483
\(699\) 617.622i 0.883580i
\(700\) 137.510i 0.196443i
\(701\) 235.724i 0.336268i −0.985764 0.168134i \(-0.946226\pi\)
0.985764 0.168134i \(-0.0537741\pi\)
\(702\) 225.197 0.320793
\(703\) 1084.50i 1.54268i
\(704\) −1692.85 309.246i −2.40462 0.439270i
\(705\) 139.559 0.197956
\(706\) 660.458i 0.935493i
\(707\) −286.833 −0.405704
\(708\) −1451.36 −2.04995
\(709\) 480.239 0.677346 0.338673 0.940904i \(-0.390022\pi\)
0.338673 + 0.940904i \(0.390022\pi\)
\(710\) 722.132i 1.01709i
\(711\) 130.663i 0.183774i
\(712\) 3041.86i 4.27228i
\(713\) −462.369 −0.648484
\(714\) 111.983i 0.156839i
\(715\) −50.4908 + 276.393i −0.0706165 + 0.386564i
\(716\) −2233.48 −3.11939
\(717\) 148.227i 0.206732i
\(718\) −207.114 −0.288459
\(719\) 1258.52 1.75038 0.875191 0.483778i \(-0.160736\pi\)
0.875191 + 0.483778i \(0.160736\pi\)
\(720\) 338.578 0.470247
\(721\) 18.4974i 0.0256552i
\(722\) 500.029i 0.692561i
\(723\) 99.5807i 0.137733i
\(724\) 2451.14 3.38555
\(725\) 184.505i 0.254489i
\(726\) 281.130 743.793i 0.387232 1.02451i
\(727\) 194.359 0.267344 0.133672 0.991026i \(-0.457323\pi\)
0.133672 + 0.991026i \(0.457323\pi\)
\(728\) 733.254i 1.00722i
\(729\) 27.0000 0.0370370
\(730\) 881.213 1.20714
\(731\) −4.51417 −0.00617534
\(732\) 378.689i 0.517335i
\(733\) 351.523i 0.479567i −0.970826 0.239783i \(-0.922924\pi\)
0.970826 0.239783i \(-0.0770764\pi\)
\(734\) 2734.37i 3.72530i
\(735\) −27.1109 −0.0368856
\(736\) 1164.64i 1.58239i
\(737\) −196.414 35.8804i −0.266504 0.0486844i
\(738\) −526.357 −0.713221
\(739\) 774.198i 1.04763i −0.851832 0.523815i \(-0.824508\pi\)
0.851832 0.523815i \(-0.175492\pi\)
\(740\) −1665.01 −2.25001
\(741\) −299.538 −0.404235
\(742\) 217.762 0.293480
\(743\) 1409.74i 1.89737i 0.316226 + 0.948684i \(0.397584\pi\)
−0.316226 + 0.948684i \(0.602416\pi\)
\(744\) 1575.68i 2.11785i
\(745\) 74.0095i 0.0993417i
\(746\) −1830.10 −2.45322
\(747\) 101.957i 0.136489i
\(748\) 132.344 724.469i 0.176931 0.968542i
\(749\) 257.530 0.343831
\(750\) 73.4714i 0.0979618i
\(751\) −274.755 −0.365853 −0.182926 0.983127i \(-0.558557\pi\)
−0.182926 + 0.983127i \(0.558557\pi\)
\(752\) 1818.71 2.41850
\(753\) −495.238 −0.657687
\(754\) 1599.25i 2.12103i
\(755\) 69.4256i 0.0919544i
\(756\) 142.905i 0.189027i
\(757\) 84.0067 0.110973 0.0554866 0.998459i \(-0.482329\pi\)
0.0554866 + 0.998459i \(0.482329\pi\)
\(758\) 296.516i 0.391182i
\(759\) −231.118 42.2201i −0.304503 0.0556260i
\(760\) −821.347 −1.08072
\(761\) 197.644i 0.259717i −0.991533 0.129858i \(-0.958548\pi\)
0.991533 0.129858i \(-0.0414522\pi\)
\(762\) −1075.08 −1.41086
\(763\) −472.372 −0.619099
\(764\) −1940.87 −2.54040
\(765\) 43.2063i 0.0564788i
\(766\) 149.111i 0.194662i
\(767\) 920.826i 1.20056i
\(768\) 334.039 0.434947
\(769\) 735.394i 0.956299i −0.878279 0.478149i \(-0.841308\pi\)
0.878279 0.478149i \(-0.158692\pi\)
\(770\) −242.885 44.3697i −0.315435 0.0576230i
\(771\) 391.503 0.507785
\(772\) 3157.19i 4.08962i
\(773\) −580.926 −0.751521 −0.375761 0.926717i \(-0.622619\pi\)
−0.375761 + 0.926717i \(0.622619\pi\)
\(774\) 7.97740 0.0103067
\(775\) 187.478 0.241907
\(776\) 2802.39i 3.61132i
\(777\) 328.266i 0.422479i
\(778\) 702.138i 0.902491i
\(779\) 700.117 0.898739
\(780\) 459.873i 0.589581i
\(781\) −921.073 168.259i −1.17935 0.215441i
\(782\) −301.336 −0.385340
\(783\) 191.743i 0.244883i
\(784\) −353.306 −0.450645
\(785\) −500.506 −0.637588
\(786\) −331.359 −0.421577
\(787\) 863.761i 1.09754i 0.835975 + 0.548768i \(0.184903\pi\)
−0.835975 + 0.548768i \(0.815097\pi\)
\(788\) 2293.32i 2.91030i
\(789\) 588.053i 0.745314i
\(790\) 369.504 0.467727
\(791\) 292.392i 0.369649i
\(792\) −143.880 + 787.614i −0.181666 + 0.994462i
\(793\) −240.262 −0.302978
\(794\) 2456.50i 3.09383i
\(795\) −84.0189 −0.105684
\(796\) −1270.97 −1.59670
\(797\) −557.124 −0.699027 −0.349513 0.936931i \(-0.613653\pi\)
−0.349513 + 0.936931i \(0.613653\pi\)
\(798\) 263.224i 0.329855i
\(799\) 232.088i 0.290472i
\(800\) 472.229i 0.590286i
\(801\) −376.126 −0.469570
\(802\) 311.028i 0.387815i
\(803\) −205.326 + 1123.98i −0.255698 + 1.39972i
\(804\) 326.801 0.406468
\(805\) 72.9529i 0.0906247i
\(806\) 1625.02 2.01616
\(807\) 130.701 0.161959
\(808\) 2630.31 3.25534
\(809\) 1364.17i 1.68624i −0.537722 0.843122i \(-0.680715\pi\)
0.537722 0.843122i \(-0.319285\pi\)
\(810\) 76.3537i 0.0942638i
\(811\) 872.097i 1.07534i 0.843157 + 0.537668i \(0.180695\pi\)
−0.843157 + 0.537668i \(0.819305\pi\)
\(812\) −1014.85 −1.24982
\(813\) 81.3811i 0.100100i
\(814\) 537.240 2940.92i 0.660000 3.61292i
\(815\) −186.345 −0.228644
\(816\) 563.059i 0.690023i
\(817\) −10.6109 −0.0129876
\(818\) −2347.29 −2.86955
\(819\) 90.6667 0.110704
\(820\) 1074.87i 1.31082i
\(821\) 433.155i 0.527595i 0.964578 + 0.263797i \(0.0849750\pi\)
−0.964578 + 0.263797i \(0.915025\pi\)
\(822\) 1142.86i 1.39034i
\(823\) −1317.04 −1.60029 −0.800146 0.599806i \(-0.795245\pi\)
−0.800146 + 0.599806i \(0.795245\pi\)
\(824\) 169.625i 0.205855i
\(825\) 93.7120 + 17.1191i 0.113590 + 0.0207504i
\(826\) −809.193 −0.979652
\(827\) 167.202i 0.202179i 0.994877 + 0.101089i \(0.0322328\pi\)
−0.994877 + 0.101089i \(0.967767\pi\)
\(828\) 384.543 0.464424
\(829\) 1.28049 0.00154461 0.000772307 1.00000i \(-0.499754\pi\)
0.000772307 1.00000i \(0.499754\pi\)
\(830\) −288.326 −0.347381
\(831\) 835.246i 1.00511i
\(832\) 1787.03i 2.14787i
\(833\) 45.0857i 0.0541244i
\(834\) −951.469 −1.14085
\(835\) 127.774i 0.153022i
\(836\) 311.085 1702.92i 0.372111 2.03698i
\(837\) 194.833 0.232775
\(838\) 1369.10i 1.63377i
\(839\) 985.680 1.17483 0.587413 0.809287i \(-0.300146\pi\)
0.587413 + 0.809287i \(0.300146\pi\)
\(840\) 248.612 0.295967
\(841\) −520.682 −0.619122
\(842\) 2957.31i 3.51224i
\(843\) 923.568i 1.09557i
\(844\) 2351.03i 2.78558i
\(845\) 86.1261 0.101924
\(846\) 410.143i 0.484802i
\(847\) 113.186 299.459i 0.133632 0.353553i
\(848\) −1094.92 −1.29118
\(849\) 180.833i 0.212996i
\(850\) 122.184 0.143745
\(851\) −883.333 −1.03799
\(852\) 1532.52 1.79873
\(853\) 824.902i 0.967060i 0.875328 + 0.483530i \(0.160646\pi\)
−0.875328 + 0.483530i \(0.839354\pi\)
\(854\) 211.134i 0.247230i
\(855\) 101.559i 0.118783i
\(856\) −2361.60 −2.75888
\(857\) 1101.72i 1.28555i −0.766054 0.642776i \(-0.777783\pi\)
0.766054 0.642776i \(-0.222217\pi\)
\(858\) 812.279 + 148.385i 0.946712 + 0.172943i
\(859\) 1300.88 1.51441 0.757207 0.653174i \(-0.226563\pi\)
0.757207 + 0.653174i \(0.226563\pi\)
\(860\) 16.2906i 0.0189426i
\(861\) −211.917 −0.246129
\(862\) 1070.48 1.24185
\(863\) −1353.77 −1.56868 −0.784338 0.620334i \(-0.786997\pi\)
−0.784338 + 0.620334i \(0.786997\pi\)
\(864\) 490.754i 0.568003i
\(865\) 481.252i 0.556361i
\(866\) 2544.58i 2.93831i
\(867\) 428.710 0.494476
\(868\) 1031.20i 1.18802i
\(869\) −86.0957 + 471.299i −0.0990745 + 0.542346i
\(870\) 542.233 0.623256
\(871\) 207.341i 0.238049i
\(872\) 4331.75 4.96760
\(873\) −346.515 −0.396924
\(874\) −708.312 −0.810426
\(875\) 29.5804i 0.0338062i
\(876\) 1870.12i 2.13484i
\(877\) 341.336i 0.389209i −0.980882 0.194604i \(-0.937658\pi\)
0.980882 0.194604i \(-0.0623423\pi\)
\(878\) 1619.69 1.84475
\(879\) 217.483i 0.247420i
\(880\) 1221.24 + 223.094i 1.38778 + 0.253516i
\(881\) 1281.15 1.45420 0.727101 0.686531i \(-0.240867\pi\)
0.727101 + 0.686531i \(0.240867\pi\)
\(882\) 79.6749i 0.0903344i
\(883\) −1227.03 −1.38961 −0.694806 0.719197i \(-0.744510\pi\)
−0.694806 + 0.719197i \(0.744510\pi\)
\(884\) 764.774 0.865128
\(885\) 312.209 0.352779
\(886\) 41.7770i 0.0471523i
\(887\) 902.044i 1.01696i 0.861074 + 0.508480i \(0.169793\pi\)
−0.861074 + 0.508480i \(0.830207\pi\)
\(888\) 3010.26i 3.38994i
\(889\) −432.838 −0.486882
\(890\) 1063.65i 1.19511i
\(891\) 97.3884 + 17.7907i 0.109302 + 0.0199671i
\(892\) −3575.31 −4.00819
\(893\) 545.538i 0.610905i
\(894\) 217.503 0.243292
\(895\) 480.455 0.536821
\(896\) 570.864 0.637125
\(897\) 243.976i 0.271991i
\(898\) 1624.25i 1.80874i
\(899\) 1383.62i 1.53907i
\(900\) −155.922 −0.173246
\(901\) 139.724i 0.155077i
\(902\) −1898.56 346.824i −2.10483 0.384506i
\(903\) 3.21179 0.00355680
\(904\) 2681.29i 2.96603i
\(905\) −527.276 −0.582625
\(906\) −204.032 −0.225201
\(907\) 202.474 0.223235 0.111617 0.993751i \(-0.464397\pi\)
0.111617 + 0.993751i \(0.464397\pi\)
\(908\) 425.802i 0.468945i
\(909\) 325.238i 0.357798i
\(910\) 256.398i 0.281756i
\(911\) 18.2211 0.0200012 0.0100006 0.999950i \(-0.496817\pi\)
0.0100006 + 0.999950i \(0.496817\pi\)
\(912\) 1323.51i 1.45122i
\(913\) 67.1811 367.758i 0.0735828 0.402801i
\(914\) 1591.72 1.74148
\(915\) 81.4615i 0.0890290i
\(916\) −354.868 −0.387410
\(917\) −133.409 −0.145484
\(918\) 126.977 0.138319
\(919\) 366.701i 0.399021i −0.979896 0.199511i \(-0.936065\pi\)
0.979896 0.199511i \(-0.0639353\pi\)
\(920\) 668.993i 0.727166i
\(921\) 619.904i 0.673077i
\(922\) 192.999 0.209327
\(923\) 972.315i 1.05343i
\(924\) −94.1618 + 515.453i −0.101907 + 0.557850i
\(925\) 358.167 0.387208
\(926\) 811.162i 0.875985i
\(927\) −20.9741 −0.0226257
\(928\) 3485.14 3.75554
\(929\) 822.900 0.885791 0.442895 0.896573i \(-0.353951\pi\)
0.442895 + 0.896573i \(0.353951\pi\)
\(930\) 550.970i 0.592441i
\(931\) 105.977i 0.113831i
\(932\) 3706.61i 3.97705i
\(933\) 861.137 0.922976
\(934\) 1597.82i 1.71072i
\(935\) −28.4692 + 155.844i −0.0304484 + 0.166678i
\(936\) −831.431 −0.888281
\(937\) 333.255i 0.355662i −0.984061 0.177831i \(-0.943092\pi\)
0.984061 0.177831i \(-0.0569080\pi\)
\(938\) 182.204 0.194248
\(939\) 291.189 0.310105
\(940\) −837.552 −0.891012
\(941\) 1394.00i 1.48140i −0.671834 0.740702i \(-0.734493\pi\)
0.671834 0.740702i \(-0.265507\pi\)
\(942\) 1470.92i 1.56148i
\(943\) 570.250i 0.604719i
\(944\) 4068.68 4.31004
\(945\) 30.7409i 0.0325300i
\(946\) 28.7743 + 5.25642i 0.0304168 + 0.00555647i
\(947\) −29.1583 −0.0307902 −0.0153951 0.999881i \(-0.504901\pi\)
−0.0153951 + 0.999881i \(0.504901\pi\)
\(948\) 784.165i 0.827178i
\(949\) −1186.51 −1.25027
\(950\) 287.201 0.302317
\(951\) −141.134 −0.148406
\(952\) 413.444i 0.434290i
\(953\) 1138.36i 1.19450i −0.802055 0.597250i \(-0.796260\pi\)
0.802055 0.597250i \(-0.203740\pi\)
\(954\) 246.919i 0.258825i
\(955\) 417.509 0.437182
\(956\) 889.574i 0.930516i
\(957\) −126.342 + 691.613i −0.132019 + 0.722688i
\(958\) 1764.84 1.84221
\(959\) 460.129i 0.479801i
\(960\) −605.899 −0.631144
\(961\) 444.919 0.462975
\(962\) 3104.53 3.22716
\(963\) 292.011i 0.303231i
\(964\) 597.626i 0.619944i
\(965\) 679.158i 0.703790i
\(966\) 214.398 0.221944
\(967\) 169.986i 0.175787i 0.996130 + 0.0878937i \(0.0280136\pi\)
−0.996130 + 0.0878937i \(0.971986\pi\)
\(968\) −1037.94 + 2746.10i −1.07225 + 2.83688i
\(969\) −168.894 −0.174297
\(970\) 979.913i 1.01022i
\(971\) 1098.16 1.13096 0.565479 0.824763i \(-0.308691\pi\)
0.565479 + 0.824763i \(0.308691\pi\)
\(972\) −162.039 −0.166706
\(973\) −383.072 −0.393702
\(974\) 2262.90i 2.32330i
\(975\) 98.9254i 0.101462i
\(976\) 1061.60i 1.08770i
\(977\) −921.219 −0.942906 −0.471453 0.881891i \(-0.656270\pi\)
−0.471453 + 0.881891i \(0.656270\pi\)
\(978\) 547.641i 0.559960i
\(979\) −1356.68 247.835i −1.38578 0.253151i
\(980\) 162.704 0.166024
\(981\) 535.620i 0.545994i
\(982\) 1744.97 1.77696
\(983\) −1261.35 −1.28317 −0.641583 0.767053i \(-0.721722\pi\)
−0.641583 + 0.767053i \(0.721722\pi\)
\(984\) 1943.32 1.97492
\(985\) 493.326i 0.500839i
\(986\) 901.739i 0.914542i
\(987\) 165.128i 0.167303i
\(988\) 1797.65 1.81949
\(989\) 8.64264i 0.00873876i
\(990\) 50.3105 275.406i 0.0508187 0.278188i
\(991\) 446.382 0.450436 0.225218 0.974308i \(-0.427691\pi\)
0.225218 + 0.974308i \(0.427691\pi\)
\(992\) 3541.30i 3.56986i
\(993\) −179.457 −0.180722
\(994\) 854.439 0.859596
\(995\) 273.405 0.274779
\(996\) 611.889i 0.614347i
\(997\) 584.900i 0.586660i 0.956011 + 0.293330i \(0.0947634\pi\)
−0.956011 + 0.293330i \(0.905237\pi\)
\(998\) 2034.79i 2.03887i
\(999\) 372.219 0.372591
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 1155.3.b.a.736.3 96
11.10 odd 2 inner 1155.3.b.a.736.94 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.3.b.a.736.3 96 1.1 even 1 trivial
1155.3.b.a.736.94 yes 96 11.10 odd 2 inner