Properties

Label 1150.3.d.b.551.13
Level $1150$
Weight $3$
Character 1150.551
Analytic conductor $31.335$
Analytic rank $0$
Dimension $16$
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] = [1150,3,Mod(551,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, 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("1150.551");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1150.d (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: \(31.3352304014\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 551.13
Root \(-2.98291i\) of defining polynomial
Character \(\chi\) \(=\) 1150.551
Dual form 1150.3.d.b.551.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +0.278523 q^{3} +2.00000 q^{4} +0.393890 q^{6} -8.51262i q^{7} +2.82843 q^{8} -8.92243 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +0.278523 q^{3} +2.00000 q^{4} +0.393890 q^{6} -8.51262i q^{7} +2.82843 q^{8} -8.92243 q^{9} -7.57553i q^{11} +0.557045 q^{12} +2.64076 q^{13} -12.0387i q^{14} +4.00000 q^{16} -7.56057i q^{17} -12.6182 q^{18} +24.2676i q^{19} -2.37096i q^{21} -10.7134i q^{22} +(15.7366 - 16.7738i) q^{23} +0.787781 q^{24} +3.73460 q^{26} -4.99180 q^{27} -17.0252i q^{28} -31.8513 q^{29} -56.5071 q^{31} +5.65685 q^{32} -2.10995i q^{33} -10.6923i q^{34} -17.8449 q^{36} -39.9378i q^{37} +34.3195i q^{38} +0.735511 q^{39} -42.5710 q^{41} -3.35304i q^{42} -20.5721i q^{43} -15.1511i q^{44} +(22.2549 - 23.7217i) q^{46} -84.3049 q^{47} +1.11409 q^{48} -23.4647 q^{49} -2.10579i q^{51} +5.28152 q^{52} -11.9189i q^{53} -7.05947 q^{54} -24.0773i q^{56} +6.75907i q^{57} -45.0446 q^{58} +67.6561 q^{59} -35.1621i q^{61} -79.9131 q^{62} +75.9532i q^{63} +8.00000 q^{64} -2.98393i q^{66} -44.0660i q^{67} -15.1211i q^{68} +(4.38299 - 4.67188i) q^{69} +8.86597 q^{71} -25.2364 q^{72} +87.4150 q^{73} -56.4805i q^{74} +48.5352i q^{76} -64.4876 q^{77} +1.04017 q^{78} -154.217i q^{79} +78.9115 q^{81} -60.2046 q^{82} +141.642i q^{83} -4.74191i q^{84} -29.0933i q^{86} -8.87131 q^{87} -21.4268i q^{88} -63.7252i q^{89} -22.4798i q^{91} +(31.4732 - 33.5476i) q^{92} -15.7385 q^{93} -119.225 q^{94} +1.57556 q^{96} -143.322i q^{97} -33.1841 q^{98} +67.5921i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} - 24 q^{13} + 64 q^{16} + 32 q^{18} - 4 q^{23} - 16 q^{24} + 96 q^{26} + 96 q^{27} - 108 q^{29} - 116 q^{31} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} - 48 q^{52} + 224 q^{54} - 160 q^{58} + 204 q^{59} - 64 q^{62} + 128 q^{64} - 268 q^{69} + 236 q^{71} + 64 q^{72} + 112 q^{73} + 936 q^{77} + 432 q^{78} - 136 q^{81} + 64 q^{82} + 152 q^{87} - 8 q^{92} - 856 q^{93} - 216 q^{94} - 32 q^{96} - 256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 0.278523 0.0928408 0.0464204 0.998922i \(-0.485219\pi\)
0.0464204 + 0.998922i \(0.485219\pi\)
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) 0.393890 0.0656484
\(7\) 8.51262i 1.21609i −0.793903 0.608044i \(-0.791954\pi\)
0.793903 0.608044i \(-0.208046\pi\)
\(8\) 2.82843 0.353553
\(9\) −8.92243 −0.991381
\(10\) 0 0
\(11\) 7.57553i 0.688684i −0.938844 0.344342i \(-0.888102\pi\)
0.938844 0.344342i \(-0.111898\pi\)
\(12\) 0.557045 0.0464204
\(13\) 2.64076 0.203135 0.101568 0.994829i \(-0.467614\pi\)
0.101568 + 0.994829i \(0.467614\pi\)
\(14\) 12.0387i 0.859904i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 7.56057i 0.444739i −0.974962 0.222370i \(-0.928621\pi\)
0.974962 0.222370i \(-0.0713792\pi\)
\(18\) −12.6182 −0.701012
\(19\) 24.2676i 1.27724i 0.769522 + 0.638620i \(0.220495\pi\)
−0.769522 + 0.638620i \(0.779505\pi\)
\(20\) 0 0
\(21\) 2.37096i 0.112903i
\(22\) 10.7134i 0.486973i
\(23\) 15.7366 16.7738i 0.684199 0.729295i
\(24\) 0.787781 0.0328242
\(25\) 0 0
\(26\) 3.73460 0.143638
\(27\) −4.99180 −0.184881
\(28\) 17.0252i 0.608044i
\(29\) −31.8513 −1.09832 −0.549161 0.835717i \(-0.685053\pi\)
−0.549161 + 0.835717i \(0.685053\pi\)
\(30\) 0 0
\(31\) −56.5071 −1.82281 −0.911405 0.411511i \(-0.865001\pi\)
−0.911405 + 0.411511i \(0.865001\pi\)
\(32\) 5.65685 0.176777
\(33\) 2.10995i 0.0639380i
\(34\) 10.6923i 0.314478i
\(35\) 0 0
\(36\) −17.8449 −0.495690
\(37\) 39.9378i 1.07940i −0.841858 0.539700i \(-0.818538\pi\)
0.841858 0.539700i \(-0.181462\pi\)
\(38\) 34.3195i 0.903146i
\(39\) 0.735511 0.0188593
\(40\) 0 0
\(41\) −42.5710 −1.03832 −0.519159 0.854678i \(-0.673755\pi\)
−0.519159 + 0.854678i \(0.673755\pi\)
\(42\) 3.35304i 0.0798342i
\(43\) 20.5721i 0.478420i −0.970968 0.239210i \(-0.923112\pi\)
0.970968 0.239210i \(-0.0768884\pi\)
\(44\) 15.1511i 0.344342i
\(45\) 0 0
\(46\) 22.2549 23.7217i 0.483802 0.515690i
\(47\) −84.3049 −1.79372 −0.896860 0.442314i \(-0.854158\pi\)
−0.896860 + 0.442314i \(0.854158\pi\)
\(48\) 1.11409 0.0232102
\(49\) −23.4647 −0.478871
\(50\) 0 0
\(51\) 2.10579i 0.0412900i
\(52\) 5.28152 0.101568
\(53\) 11.9189i 0.224885i −0.993658 0.112443i \(-0.964133\pi\)
0.993658 0.112443i \(-0.0358674\pi\)
\(54\) −7.05947 −0.130731
\(55\) 0 0
\(56\) 24.0773i 0.429952i
\(57\) 6.75907i 0.118580i
\(58\) −45.0446 −0.776631
\(59\) 67.6561 1.14671 0.573357 0.819306i \(-0.305641\pi\)
0.573357 + 0.819306i \(0.305641\pi\)
\(60\) 0 0
\(61\) 35.1621i 0.576428i −0.957566 0.288214i \(-0.906939\pi\)
0.957566 0.288214i \(-0.0930614\pi\)
\(62\) −79.9131 −1.28892
\(63\) 75.9532i 1.20561i
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 2.98393i 0.0452110i
\(67\) 44.0660i 0.657701i −0.944382 0.328850i \(-0.893339\pi\)
0.944382 0.328850i \(-0.106661\pi\)
\(68\) 15.1211i 0.222370i
\(69\) 4.38299 4.67188i 0.0635216 0.0677084i
\(70\) 0 0
\(71\) 8.86597 0.124873 0.0624364 0.998049i \(-0.480113\pi\)
0.0624364 + 0.998049i \(0.480113\pi\)
\(72\) −25.2364 −0.350506
\(73\) 87.4150 1.19747 0.598733 0.800949i \(-0.295671\pi\)
0.598733 + 0.800949i \(0.295671\pi\)
\(74\) 56.4805i 0.763250i
\(75\) 0 0
\(76\) 48.5352i 0.638620i
\(77\) −64.4876 −0.837501
\(78\) 1.04017 0.0133355
\(79\) 154.217i 1.95211i −0.217517 0.976057i \(-0.569796\pi\)
0.217517 0.976057i \(-0.430204\pi\)
\(80\) 0 0
\(81\) 78.9115 0.974216
\(82\) −60.2046 −0.734202
\(83\) 141.642i 1.70653i 0.521476 + 0.853266i \(0.325382\pi\)
−0.521476 + 0.853266i \(0.674618\pi\)
\(84\) 4.74191i 0.0564513i
\(85\) 0 0
\(86\) 29.0933i 0.338294i
\(87\) −8.87131 −0.101969
\(88\) 21.4268i 0.243487i
\(89\) 63.7252i 0.716013i −0.933719 0.358007i \(-0.883457\pi\)
0.933719 0.358007i \(-0.116543\pi\)
\(90\) 0 0
\(91\) 22.4798i 0.247030i
\(92\) 31.4732 33.5476i 0.342100 0.364648i
\(93\) −15.7385 −0.169231
\(94\) −119.225 −1.26835
\(95\) 0 0
\(96\) 1.57556 0.0164121
\(97\) 143.322i 1.47755i −0.673952 0.738775i \(-0.735404\pi\)
0.673952 0.738775i \(-0.264596\pi\)
\(98\) −33.1841 −0.338613
\(99\) 67.5921i 0.682748i
\(100\) 0 0
\(101\) 27.7102 0.274359 0.137179 0.990546i \(-0.456196\pi\)
0.137179 + 0.990546i \(0.456196\pi\)
\(102\) 2.97803i 0.0291964i
\(103\) 133.542i 1.29652i 0.761418 + 0.648261i \(0.224503\pi\)
−0.761418 + 0.648261i \(0.775497\pi\)
\(104\) 7.46919 0.0718192
\(105\) 0 0
\(106\) 16.8559i 0.159018i
\(107\) 50.3091i 0.470179i −0.971974 0.235089i \(-0.924462\pi\)
0.971974 0.235089i \(-0.0755383\pi\)
\(108\) −9.98360 −0.0924407
\(109\) 128.819i 1.18182i 0.806737 + 0.590911i \(0.201232\pi\)
−0.806737 + 0.590911i \(0.798768\pi\)
\(110\) 0 0
\(111\) 11.1236i 0.100212i
\(112\) 34.0505i 0.304022i
\(113\) 86.3028i 0.763742i −0.924216 0.381871i \(-0.875280\pi\)
0.924216 0.381871i \(-0.124720\pi\)
\(114\) 9.55876i 0.0838488i
\(115\) 0 0
\(116\) −63.7027 −0.549161
\(117\) −23.5620 −0.201384
\(118\) 95.6802 0.810849
\(119\) −64.3602 −0.540842
\(120\) 0 0
\(121\) 63.6114 0.525714
\(122\) 49.7267i 0.407596i
\(123\) −11.8570 −0.0963983
\(124\) −113.014 −0.911405
\(125\) 0 0
\(126\) 107.414i 0.852492i
\(127\) 11.0135 0.0867206 0.0433603 0.999059i \(-0.486194\pi\)
0.0433603 + 0.999059i \(0.486194\pi\)
\(128\) 11.3137 0.0883883
\(129\) 5.72978i 0.0444169i
\(130\) 0 0
\(131\) 3.63941 0.0277818 0.0138909 0.999904i \(-0.495578\pi\)
0.0138909 + 0.999904i \(0.495578\pi\)
\(132\) 4.21991i 0.0319690i
\(133\) 206.581 1.55324
\(134\) 62.3187i 0.465065i
\(135\) 0 0
\(136\) 21.3845i 0.157239i
\(137\) 9.69785i 0.0707873i 0.999373 + 0.0353936i \(0.0112685\pi\)
−0.999373 + 0.0353936i \(0.988732\pi\)
\(138\) 6.19849 6.60703i 0.0449166 0.0478771i
\(139\) 8.05485 0.0579486 0.0289743 0.999580i \(-0.490776\pi\)
0.0289743 + 0.999580i \(0.490776\pi\)
\(140\) 0 0
\(141\) −23.4808 −0.166531
\(142\) 12.5384 0.0882984
\(143\) 20.0051i 0.139896i
\(144\) −35.6897 −0.247845
\(145\) 0 0
\(146\) 123.624 0.846737
\(147\) −6.53544 −0.0444588
\(148\) 79.8755i 0.539700i
\(149\) 144.062i 0.966859i −0.875383 0.483430i \(-0.839391\pi\)
0.875383 0.483430i \(-0.160609\pi\)
\(150\) 0 0
\(151\) −109.956 −0.728188 −0.364094 0.931362i \(-0.618621\pi\)
−0.364094 + 0.931362i \(0.618621\pi\)
\(152\) 68.6391i 0.451573i
\(153\) 67.4586i 0.440906i
\(154\) −91.1992 −0.592202
\(155\) 0 0
\(156\) 1.47102 0.00942963
\(157\) 24.8208i 0.158094i −0.996871 0.0790471i \(-0.974812\pi\)
0.996871 0.0790471i \(-0.0251877\pi\)
\(158\) 218.096i 1.38035i
\(159\) 3.31969i 0.0208785i
\(160\) 0 0
\(161\) −142.789 133.960i −0.886887 0.832047i
\(162\) 111.598 0.688875
\(163\) −108.964 −0.668489 −0.334244 0.942486i \(-0.608481\pi\)
−0.334244 + 0.942486i \(0.608481\pi\)
\(164\) −85.1421 −0.519159
\(165\) 0 0
\(166\) 200.312i 1.20670i
\(167\) −72.1383 −0.431966 −0.215983 0.976397i \(-0.569296\pi\)
−0.215983 + 0.976397i \(0.569296\pi\)
\(168\) 6.70608i 0.0399171i
\(169\) −162.026 −0.958736
\(170\) 0 0
\(171\) 216.526i 1.26623i
\(172\) 41.1441i 0.239210i
\(173\) 150.077 0.867497 0.433748 0.901034i \(-0.357191\pi\)
0.433748 + 0.901034i \(0.357191\pi\)
\(174\) −12.5459 −0.0721031
\(175\) 0 0
\(176\) 30.3021i 0.172171i
\(177\) 18.8437 0.106462
\(178\) 90.1210i 0.506298i
\(179\) 207.058 1.15675 0.578375 0.815771i \(-0.303687\pi\)
0.578375 + 0.815771i \(0.303687\pi\)
\(180\) 0 0
\(181\) 331.138i 1.82949i 0.404031 + 0.914746i \(0.367609\pi\)
−0.404031 + 0.914746i \(0.632391\pi\)
\(182\) 31.7912i 0.174677i
\(183\) 9.79343i 0.0535160i
\(184\) 44.5098 47.4434i 0.241901 0.257845i
\(185\) 0 0
\(186\) −22.2576 −0.119665
\(187\) −57.2753 −0.306285
\(188\) −168.610 −0.896860
\(189\) 42.4933i 0.224832i
\(190\) 0 0
\(191\) 172.749i 0.904447i 0.891905 + 0.452223i \(0.149369\pi\)
−0.891905 + 0.452223i \(0.850631\pi\)
\(192\) 2.22818 0.0116051
\(193\) −136.450 −0.706996 −0.353498 0.935435i \(-0.615008\pi\)
−0.353498 + 0.935435i \(0.615008\pi\)
\(194\) 202.689i 1.04479i
\(195\) 0 0
\(196\) −46.9293 −0.239435
\(197\) 271.090 1.37609 0.688046 0.725667i \(-0.258469\pi\)
0.688046 + 0.725667i \(0.258469\pi\)
\(198\) 95.5896i 0.482776i
\(199\) 257.292i 1.29292i 0.762946 + 0.646462i \(0.223752\pi\)
−0.762946 + 0.646462i \(0.776248\pi\)
\(200\) 0 0
\(201\) 12.2734i 0.0610615i
\(202\) 39.1882 0.194001
\(203\) 271.138i 1.33566i
\(204\) 4.21158i 0.0206450i
\(205\) 0 0
\(206\) 188.857i 0.916780i
\(207\) −140.408 + 149.663i −0.678302 + 0.723009i
\(208\) 10.5630 0.0507838
\(209\) 183.840 0.879615
\(210\) 0 0
\(211\) 54.1944 0.256846 0.128423 0.991720i \(-0.459009\pi\)
0.128423 + 0.991720i \(0.459009\pi\)
\(212\) 23.8378i 0.112443i
\(213\) 2.46937 0.0115933
\(214\) 71.1478i 0.332467i
\(215\) 0 0
\(216\) −14.1189 −0.0653655
\(217\) 481.023i 2.21670i
\(218\) 182.177i 0.835674i
\(219\) 24.3471 0.111174
\(220\) 0 0
\(221\) 19.9656i 0.0903423i
\(222\) 15.7311i 0.0708608i
\(223\) −104.611 −0.469105 −0.234553 0.972103i \(-0.575363\pi\)
−0.234553 + 0.972103i \(0.575363\pi\)
\(224\) 48.1546i 0.214976i
\(225\) 0 0
\(226\) 122.051i 0.540047i
\(227\) 200.484i 0.883190i −0.897215 0.441595i \(-0.854413\pi\)
0.897215 0.441595i \(-0.145587\pi\)
\(228\) 13.5181i 0.0592901i
\(229\) 22.0718i 0.0963834i 0.998838 + 0.0481917i \(0.0153458\pi\)
−0.998838 + 0.0481917i \(0.984654\pi\)
\(230\) 0 0
\(231\) −17.9612 −0.0777543
\(232\) −90.0892 −0.388315
\(233\) 338.632 1.45335 0.726677 0.686979i \(-0.241064\pi\)
0.726677 + 0.686979i \(0.241064\pi\)
\(234\) −33.3217 −0.142400
\(235\) 0 0
\(236\) 135.312 0.573357
\(237\) 42.9529i 0.181236i
\(238\) −91.0191 −0.382433
\(239\) 149.374 0.624997 0.312499 0.949918i \(-0.398834\pi\)
0.312499 + 0.949918i \(0.398834\pi\)
\(240\) 0 0
\(241\) 133.030i 0.551991i 0.961159 + 0.275995i \(0.0890074\pi\)
−0.961159 + 0.275995i \(0.910993\pi\)
\(242\) 89.9601 0.371736
\(243\) 66.9048 0.275328
\(244\) 70.3242i 0.288214i
\(245\) 0 0
\(246\) −16.7683 −0.0681639
\(247\) 64.0848i 0.259453i
\(248\) −159.826 −0.644461
\(249\) 39.4505i 0.158436i
\(250\) 0 0
\(251\) 376.920i 1.50167i −0.660489 0.750836i \(-0.729651\pi\)
0.660489 0.750836i \(-0.270349\pi\)
\(252\) 151.906i 0.602803i
\(253\) −127.070 119.213i −0.502254 0.471197i
\(254\) 15.5755 0.0613208
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −226.085 −0.879708 −0.439854 0.898069i \(-0.644970\pi\)
−0.439854 + 0.898069i \(0.644970\pi\)
\(258\) 8.10314i 0.0314075i
\(259\) −339.975 −1.31264
\(260\) 0 0
\(261\) 284.191 1.08885
\(262\) 5.14690 0.0196447
\(263\) 2.43654i 0.00926441i 0.999989 + 0.00463221i \(0.00147448\pi\)
−0.999989 + 0.00463221i \(0.998526\pi\)
\(264\) 5.96785i 0.0226055i
\(265\) 0 0
\(266\) 292.149 1.09830
\(267\) 17.7489i 0.0664753i
\(268\) 88.1319i 0.328850i
\(269\) −311.337 −1.15739 −0.578694 0.815545i \(-0.696437\pi\)
−0.578694 + 0.815545i \(0.696437\pi\)
\(270\) 0 0
\(271\) 260.751 0.962182 0.481091 0.876671i \(-0.340241\pi\)
0.481091 + 0.876671i \(0.340241\pi\)
\(272\) 30.2423i 0.111185i
\(273\) 6.26112i 0.0229345i
\(274\) 13.7148i 0.0500541i
\(275\) 0 0
\(276\) 8.76599 9.34376i 0.0317608 0.0338542i
\(277\) 182.267 0.658004 0.329002 0.944329i \(-0.393288\pi\)
0.329002 + 0.944329i \(0.393288\pi\)
\(278\) 11.3913 0.0409758
\(279\) 504.180 1.80710
\(280\) 0 0
\(281\) 4.60502i 0.0163880i 0.999966 + 0.00819398i \(0.00260825\pi\)
−0.999966 + 0.00819398i \(0.997392\pi\)
\(282\) −33.2069 −0.117755
\(283\) 332.897i 1.17631i −0.808747 0.588157i \(-0.799854\pi\)
0.808747 0.588157i \(-0.200146\pi\)
\(284\) 17.7319 0.0624364
\(285\) 0 0
\(286\) 28.2915i 0.0989215i
\(287\) 362.391i 1.26269i
\(288\) −50.4729 −0.175253
\(289\) 231.838 0.802207
\(290\) 0 0
\(291\) 39.9185i 0.137177i
\(292\) 174.830 0.598733
\(293\) 336.114i 1.14715i −0.819154 0.573574i \(-0.805557\pi\)
0.819154 0.573574i \(-0.194443\pi\)
\(294\) −9.24251 −0.0314371
\(295\) 0 0
\(296\) 112.961i 0.381625i
\(297\) 37.8155i 0.127325i
\(298\) 203.734i 0.683673i
\(299\) 41.5565 44.2955i 0.138985 0.148146i
\(300\) 0 0
\(301\) −175.122 −0.581801
\(302\) −155.502 −0.514907
\(303\) 7.71792 0.0254717
\(304\) 97.0703i 0.319310i
\(305\) 0 0
\(306\) 95.4009i 0.311768i
\(307\) 457.934 1.49164 0.745821 0.666147i \(-0.232057\pi\)
0.745821 + 0.666147i \(0.232057\pi\)
\(308\) −128.975 −0.418750
\(309\) 37.1944i 0.120370i
\(310\) 0 0
\(311\) −455.620 −1.46502 −0.732508 0.680758i \(-0.761650\pi\)
−0.732508 + 0.680758i \(0.761650\pi\)
\(312\) 2.08034 0.00666775
\(313\) 589.353i 1.88292i 0.337130 + 0.941458i \(0.390544\pi\)
−0.337130 + 0.941458i \(0.609456\pi\)
\(314\) 35.1019i 0.111789i
\(315\) 0 0
\(316\) 308.434i 0.976057i
\(317\) 386.251 1.21846 0.609229 0.792994i \(-0.291479\pi\)
0.609229 + 0.792994i \(0.291479\pi\)
\(318\) 4.69475i 0.0147634i
\(319\) 241.291i 0.756397i
\(320\) 0 0
\(321\) 14.0122i 0.0436518i
\(322\) −201.934 189.447i −0.627124 0.588346i
\(323\) 183.477 0.568039
\(324\) 157.823 0.487108
\(325\) 0 0
\(326\) −154.098 −0.472693
\(327\) 35.8789i 0.109721i
\(328\) −120.409 −0.367101
\(329\) 717.655i 2.18132i
\(330\) 0 0
\(331\) 220.712 0.666803 0.333402 0.942785i \(-0.391804\pi\)
0.333402 + 0.942785i \(0.391804\pi\)
\(332\) 283.284i 0.853266i
\(333\) 356.342i 1.07010i
\(334\) −102.019 −0.305446
\(335\) 0 0
\(336\) 9.48382i 0.0282257i
\(337\) 193.563i 0.574371i −0.957875 0.287185i \(-0.907280\pi\)
0.957875 0.287185i \(-0.0927196\pi\)
\(338\) −229.140 −0.677929
\(339\) 24.0373i 0.0709064i
\(340\) 0 0
\(341\) 428.071i 1.25534i
\(342\) 306.213i 0.895361i
\(343\) 217.373i 0.633739i
\(344\) 58.1866i 0.169147i
\(345\) 0 0
\(346\) 212.241 0.613413
\(347\) 262.429 0.756278 0.378139 0.925749i \(-0.376564\pi\)
0.378139 + 0.925749i \(0.376564\pi\)
\(348\) −17.7426 −0.0509846
\(349\) 229.831 0.658543 0.329271 0.944235i \(-0.393197\pi\)
0.329271 + 0.944235i \(0.393197\pi\)
\(350\) 0 0
\(351\) −13.1821 −0.0375560
\(352\) 42.8536i 0.121743i
\(353\) −367.482 −1.04103 −0.520513 0.853854i \(-0.674259\pi\)
−0.520513 + 0.853854i \(0.674259\pi\)
\(354\) 26.6491 0.0752799
\(355\) 0 0
\(356\) 127.450i 0.358007i
\(357\) −17.9258 −0.0502123
\(358\) 292.824 0.817945
\(359\) 239.971i 0.668442i 0.942495 + 0.334221i \(0.108473\pi\)
−0.942495 + 0.334221i \(0.891527\pi\)
\(360\) 0 0
\(361\) −227.915 −0.631344
\(362\) 468.300i 1.29365i
\(363\) 17.7172 0.0488077
\(364\) 44.9595i 0.123515i
\(365\) 0 0
\(366\) 13.8500i 0.0378416i
\(367\) 389.113i 1.06025i −0.847918 0.530127i \(-0.822144\pi\)
0.847918 0.530127i \(-0.177856\pi\)
\(368\) 62.9463 67.0952i 0.171050 0.182324i
\(369\) 379.837 1.02937
\(370\) 0 0
\(371\) −101.461 −0.273480
\(372\) −31.4770 −0.0846156
\(373\) 546.309i 1.46463i 0.680964 + 0.732317i \(0.261561\pi\)
−0.680964 + 0.732317i \(0.738439\pi\)
\(374\) −80.9995 −0.216576
\(375\) 0 0
\(376\) −238.450 −0.634176
\(377\) −84.1117 −0.223108
\(378\) 60.0946i 0.158980i
\(379\) 295.837i 0.780573i −0.920693 0.390287i \(-0.872376\pi\)
0.920693 0.390287i \(-0.127624\pi\)
\(380\) 0 0
\(381\) 3.06751 0.00805122
\(382\) 244.304i 0.639541i
\(383\) 566.033i 1.47789i −0.673765 0.738946i \(-0.735324\pi\)
0.673765 0.738946i \(-0.264676\pi\)
\(384\) 3.15112 0.00820605
\(385\) 0 0
\(386\) −192.970 −0.499921
\(387\) 183.553i 0.474296i
\(388\) 286.645i 0.738775i
\(389\) 196.043i 0.503966i 0.967732 + 0.251983i \(0.0810826\pi\)
−0.967732 + 0.251983i \(0.918917\pi\)
\(390\) 0 0
\(391\) −126.819 118.978i −0.324346 0.304290i
\(392\) −66.3681 −0.169306
\(393\) 1.01366 0.00257928
\(394\) 383.380 0.973045
\(395\) 0 0
\(396\) 135.184i 0.341374i
\(397\) 298.788 0.752616 0.376308 0.926495i \(-0.377194\pi\)
0.376308 + 0.926495i \(0.377194\pi\)
\(398\) 363.866i 0.914236i
\(399\) 57.5374 0.144204
\(400\) 0 0
\(401\) 785.114i 1.95789i −0.204121 0.978946i \(-0.565433\pi\)
0.204121 0.978946i \(-0.434567\pi\)
\(402\) 17.3572i 0.0431770i
\(403\) −149.222 −0.370277
\(404\) 55.4204 0.137179
\(405\) 0 0
\(406\) 383.447i 0.944452i
\(407\) −302.550 −0.743365
\(408\) 5.95607i 0.0145982i
\(409\) 358.186 0.875761 0.437881 0.899033i \(-0.355729\pi\)
0.437881 + 0.899033i \(0.355729\pi\)
\(410\) 0 0
\(411\) 2.70107i 0.00657195i
\(412\) 267.084i 0.648261i
\(413\) 575.930i 1.39450i
\(414\) −198.568 + 211.655i −0.479632 + 0.511245i
\(415\) 0 0
\(416\) 14.9384 0.0359096
\(417\) 2.24346 0.00537999
\(418\) 259.988 0.621982
\(419\) 195.271i 0.466042i 0.972472 + 0.233021i \(0.0748610\pi\)
−0.972472 + 0.233021i \(0.925139\pi\)
\(420\) 0 0
\(421\) 768.042i 1.82433i 0.409826 + 0.912164i \(0.365590\pi\)
−0.409826 + 0.912164i \(0.634410\pi\)
\(422\) 76.6425 0.181617
\(423\) 752.204 1.77826
\(424\) 33.7118i 0.0795089i
\(425\) 0 0
\(426\) 3.49222 0.00819770
\(427\) −299.321 −0.700987
\(428\) 100.618i 0.235089i
\(429\) 5.57188i 0.0129881i
\(430\) 0 0
\(431\) 333.122i 0.772905i −0.922309 0.386453i \(-0.873700\pi\)
0.922309 0.386453i \(-0.126300\pi\)
\(432\) −19.9672 −0.0462204
\(433\) 7.49741i 0.0173150i −0.999963 0.00865752i \(-0.997244\pi\)
0.999963 0.00865752i \(-0.00275581\pi\)
\(434\) 680.270i 1.56744i
\(435\) 0 0
\(436\) 257.637i 0.590911i
\(437\) 407.059 + 381.889i 0.931485 + 0.873887i
\(438\) 34.4319 0.0786117
\(439\) −303.939 −0.692344 −0.346172 0.938171i \(-0.612519\pi\)
−0.346172 + 0.938171i \(0.612519\pi\)
\(440\) 0 0
\(441\) 209.362 0.474743
\(442\) 28.2357i 0.0638816i
\(443\) −637.145 −1.43825 −0.719125 0.694881i \(-0.755457\pi\)
−0.719125 + 0.694881i \(0.755457\pi\)
\(444\) 22.2471i 0.0501062i
\(445\) 0 0
\(446\) −147.942 −0.331708
\(447\) 40.1245i 0.0897640i
\(448\) 68.1009i 0.152011i
\(449\) 88.9331 0.198069 0.0990346 0.995084i \(-0.468425\pi\)
0.0990346 + 0.995084i \(0.468425\pi\)
\(450\) 0 0
\(451\) 322.498i 0.715073i
\(452\) 172.606i 0.381871i
\(453\) −30.6253 −0.0676056
\(454\) 283.527i 0.624509i
\(455\) 0 0
\(456\) 19.1175i 0.0419244i
\(457\) 377.669i 0.826410i 0.910638 + 0.413205i \(0.135591\pi\)
−0.910638 + 0.413205i \(0.864409\pi\)
\(458\) 31.2142i 0.0681534i
\(459\) 37.7408i 0.0822240i
\(460\) 0 0
\(461\) −585.070 −1.26913 −0.634566 0.772869i \(-0.718821\pi\)
−0.634566 + 0.772869i \(0.718821\pi\)
\(462\) −25.4010 −0.0549806
\(463\) 225.877 0.487854 0.243927 0.969794i \(-0.421564\pi\)
0.243927 + 0.969794i \(0.421564\pi\)
\(464\) −127.405 −0.274580
\(465\) 0 0
\(466\) 478.897 1.02768
\(467\) 702.984i 1.50532i −0.658410 0.752659i \(-0.728771\pi\)
0.658410 0.752659i \(-0.271229\pi\)
\(468\) −47.1240 −0.100692
\(469\) −375.117 −0.799822
\(470\) 0 0
\(471\) 6.91315i 0.0146776i
\(472\) 191.360 0.405424
\(473\) −155.844 −0.329480
\(474\) 60.7446i 0.128153i
\(475\) 0 0
\(476\) −128.720 −0.270421
\(477\) 106.346i 0.222947i
\(478\) 211.247 0.441940
\(479\) 291.706i 0.608989i 0.952514 + 0.304494i \(0.0984875\pi\)
−0.952514 + 0.304494i \(0.901512\pi\)
\(480\) 0 0
\(481\) 105.466i 0.219264i
\(482\) 188.132i 0.390316i
\(483\) −39.7699 37.3107i −0.0823394 0.0772479i
\(484\) 127.223 0.262857
\(485\) 0 0
\(486\) 94.6177 0.194687
\(487\) −205.957 −0.422909 −0.211454 0.977388i \(-0.567820\pi\)
−0.211454 + 0.977388i \(0.567820\pi\)
\(488\) 99.4534i 0.203798i
\(489\) −30.3488 −0.0620630
\(490\) 0 0
\(491\) −192.478 −0.392012 −0.196006 0.980603i \(-0.562797\pi\)
−0.196006 + 0.980603i \(0.562797\pi\)
\(492\) −23.7140 −0.0481992
\(493\) 240.814i 0.488467i
\(494\) 90.6296i 0.183461i
\(495\) 0 0
\(496\) −226.028 −0.455703
\(497\) 75.4726i 0.151856i
\(498\) 55.7915i 0.112031i
\(499\) 887.386 1.77833 0.889164 0.457589i \(-0.151287\pi\)
0.889164 + 0.457589i \(0.151287\pi\)
\(500\) 0 0
\(501\) −20.0921 −0.0401041
\(502\) 533.045i 1.06184i
\(503\) 330.810i 0.657673i −0.944387 0.328837i \(-0.893343\pi\)
0.944387 0.328837i \(-0.106657\pi\)
\(504\) 214.828i 0.426246i
\(505\) 0 0
\(506\) −179.704 168.592i −0.355147 0.333187i
\(507\) −45.1280 −0.0890099
\(508\) 22.0270 0.0433603
\(509\) −407.928 −0.801430 −0.400715 0.916203i \(-0.631238\pi\)
−0.400715 + 0.916203i \(0.631238\pi\)
\(510\) 0 0
\(511\) 744.131i 1.45622i
\(512\) 22.6274 0.0441942
\(513\) 121.139i 0.236138i
\(514\) −319.733 −0.622048
\(515\) 0 0
\(516\) 11.4596i 0.0222085i
\(517\) 638.654i 1.23531i
\(518\) −480.797 −0.928180
\(519\) 41.7998 0.0805391
\(520\) 0 0
\(521\) 458.780i 0.880576i 0.897857 + 0.440288i \(0.145124\pi\)
−0.897857 + 0.440288i \(0.854876\pi\)
\(522\) 401.907 0.769937
\(523\) 404.987i 0.774354i −0.922005 0.387177i \(-0.873450\pi\)
0.922005 0.387177i \(-0.126550\pi\)
\(524\) 7.27882 0.0138909
\(525\) 0 0
\(526\) 3.44579i 0.00655093i
\(527\) 427.226i 0.810675i
\(528\) 8.43982i 0.0159845i
\(529\) −33.7199 527.924i −0.0637427 0.997966i
\(530\) 0 0
\(531\) −603.656 −1.13683
\(532\) 413.161 0.776619
\(533\) −112.420 −0.210919
\(534\) 25.1007i 0.0470051i
\(535\) 0 0
\(536\) 124.637i 0.232532i
\(537\) 57.6703 0.107394
\(538\) −440.298 −0.818397
\(539\) 177.757i 0.329791i
\(540\) 0 0
\(541\) 1047.51 1.93625 0.968123 0.250477i \(-0.0805875\pi\)
0.968123 + 0.250477i \(0.0805875\pi\)
\(542\) 368.758 0.680366
\(543\) 92.2294i 0.169851i
\(544\) 42.7690i 0.0786195i
\(545\) 0 0
\(546\) 8.85457i 0.0162172i
\(547\) −123.784 −0.226296 −0.113148 0.993578i \(-0.536093\pi\)
−0.113148 + 0.993578i \(0.536093\pi\)
\(548\) 19.3957i 0.0353936i
\(549\) 313.731i 0.571459i
\(550\) 0 0
\(551\) 772.955i 1.40282i
\(552\) 12.3970 13.2141i 0.0224583 0.0239385i
\(553\) −1312.79 −2.37394
\(554\) 257.765 0.465279
\(555\) 0 0
\(556\) 16.1097 0.0289743
\(557\) 246.292i 0.442176i 0.975254 + 0.221088i \(0.0709608\pi\)
−0.975254 + 0.221088i \(0.929039\pi\)
\(558\) 713.019 1.27781
\(559\) 54.3259i 0.0971840i
\(560\) 0 0
\(561\) −15.9525 −0.0284357
\(562\) 6.51248i 0.0115880i
\(563\) 574.776i 1.02092i 0.859902 + 0.510458i \(0.170524\pi\)
−0.859902 + 0.510458i \(0.829476\pi\)
\(564\) −46.9616 −0.0832653
\(565\) 0 0
\(566\) 470.787i 0.831780i
\(567\) 671.743i 1.18473i
\(568\) 25.0768 0.0441492
\(569\) 794.332i 1.39601i −0.716091 0.698007i \(-0.754070\pi\)
0.716091 0.698007i \(-0.245930\pi\)
\(570\) 0 0
\(571\) 416.688i 0.729751i 0.931056 + 0.364876i \(0.118889\pi\)
−0.931056 + 0.364876i \(0.881111\pi\)
\(572\) 40.0103i 0.0699480i
\(573\) 48.1146i 0.0839696i
\(574\) 512.498i 0.892854i
\(575\) 0 0
\(576\) −71.3794 −0.123923
\(577\) 282.647 0.489856 0.244928 0.969541i \(-0.421236\pi\)
0.244928 + 0.969541i \(0.421236\pi\)
\(578\) 327.868 0.567246
\(579\) −38.0044 −0.0656381
\(580\) 0 0
\(581\) 1205.75 2.07529
\(582\) 56.4533i 0.0969988i
\(583\) −90.2920 −0.154875
\(584\) 247.247 0.423368
\(585\) 0 0
\(586\) 475.337i 0.811156i
\(587\) 168.123 0.286410 0.143205 0.989693i \(-0.454259\pi\)
0.143205 + 0.989693i \(0.454259\pi\)
\(588\) −13.0709 −0.0222294
\(589\) 1371.29i 2.32817i
\(590\) 0 0
\(591\) 75.5048 0.127758
\(592\) 159.751i 0.269850i
\(593\) 419.364 0.707191 0.353595 0.935398i \(-0.384959\pi\)
0.353595 + 0.935398i \(0.384959\pi\)
\(594\) 53.4792i 0.0900323i
\(595\) 0 0
\(596\) 288.124i 0.483430i
\(597\) 71.6616i 0.120036i
\(598\) 58.7698 62.6433i 0.0982773 0.104755i
\(599\) −519.774 −0.867736 −0.433868 0.900977i \(-0.642852\pi\)
−0.433868 + 0.900977i \(0.642852\pi\)
\(600\) 0 0
\(601\) 1089.19 1.81230 0.906151 0.422955i \(-0.139007\pi\)
0.906151 + 0.422955i \(0.139007\pi\)
\(602\) −247.660 −0.411395
\(603\) 393.175i 0.652032i
\(604\) −219.913 −0.364094
\(605\) 0 0
\(606\) 10.9148 0.0180112
\(607\) −890.673 −1.46734 −0.733668 0.679508i \(-0.762193\pi\)
−0.733668 + 0.679508i \(0.762193\pi\)
\(608\) 137.278i 0.225786i
\(609\) 75.5181i 0.124003i
\(610\) 0 0
\(611\) −222.629 −0.364368
\(612\) 134.917i 0.220453i
\(613\) 629.354i 1.02668i −0.858186 0.513339i \(-0.828408\pi\)
0.858186 0.513339i \(-0.171592\pi\)
\(614\) 647.616 1.05475
\(615\) 0 0
\(616\) −182.398 −0.296101
\(617\) 157.187i 0.254760i 0.991854 + 0.127380i \(0.0406568\pi\)
−0.991854 + 0.127380i \(0.959343\pi\)
\(618\) 52.6008i 0.0851146i
\(619\) 462.293i 0.746838i −0.927663 0.373419i \(-0.878185\pi\)
0.927663 0.373419i \(-0.121815\pi\)
\(620\) 0 0
\(621\) −78.5539 + 83.7314i −0.126496 + 0.134833i
\(622\) −644.344 −1.03592
\(623\) −542.468 −0.870735
\(624\) 2.94204 0.00471481
\(625\) 0 0
\(626\) 833.471i 1.33142i
\(627\) 51.2035 0.0816642
\(628\) 49.6416i 0.0790471i
\(629\) −301.952 −0.480051
\(630\) 0 0
\(631\) 411.630i 0.652345i −0.945310 0.326173i \(-0.894241\pi\)
0.945310 0.326173i \(-0.105759\pi\)
\(632\) 436.191i 0.690176i
\(633\) 15.0944 0.0238458
\(634\) 546.242 0.861581
\(635\) 0 0
\(636\) 6.63937i 0.0104393i
\(637\) −61.9645 −0.0972756
\(638\) 341.236i 0.534853i
\(639\) −79.1060 −0.123797
\(640\) 0 0
\(641\) 251.089i 0.391715i 0.980632 + 0.195857i \(0.0627490\pi\)
−0.980632 + 0.195857i \(0.937251\pi\)
\(642\) 19.8163i 0.0308665i
\(643\) 102.907i 0.160042i 0.996793 + 0.0800211i \(0.0254988\pi\)
−0.996793 + 0.0800211i \(0.974501\pi\)
\(644\) −285.578 267.919i −0.443444 0.416023i
\(645\) 0 0
\(646\) 259.475 0.401664
\(647\) 437.368 0.675993 0.337997 0.941147i \(-0.390251\pi\)
0.337997 + 0.941147i \(0.390251\pi\)
\(648\) 223.195 0.344437
\(649\) 512.530i 0.789723i
\(650\) 0 0
\(651\) 133.976i 0.205800i
\(652\) −217.927 −0.334244
\(653\) 94.4001 0.144564 0.0722819 0.997384i \(-0.476972\pi\)
0.0722819 + 0.997384i \(0.476972\pi\)
\(654\) 50.7404i 0.0775847i
\(655\) 0 0
\(656\) −170.284 −0.259580
\(657\) −779.954 −1.18714
\(658\) 1014.92i 1.54243i
\(659\) 759.727i 1.15285i 0.817151 + 0.576424i \(0.195552\pi\)
−0.817151 + 0.576424i \(0.804448\pi\)
\(660\) 0 0
\(661\) 581.754i 0.880112i 0.897970 + 0.440056i \(0.145041\pi\)
−0.897970 + 0.440056i \(0.854959\pi\)
\(662\) 312.134 0.471501
\(663\) 5.56088i 0.00838745i
\(664\) 400.625i 0.603350i
\(665\) 0 0
\(666\) 503.943i 0.756672i
\(667\) −501.231 + 534.267i −0.751471 + 0.801001i
\(668\) −144.277 −0.215983
\(669\) −29.1364 −0.0435521
\(670\) 0 0
\(671\) −266.371 −0.396977
\(672\) 13.4122i 0.0199586i
\(673\) 831.173 1.23503 0.617513 0.786560i \(-0.288140\pi\)
0.617513 + 0.786560i \(0.288140\pi\)
\(674\) 273.739i 0.406142i
\(675\) 0 0
\(676\) −324.053 −0.479368
\(677\) 106.706i 0.157616i −0.996890 0.0788079i \(-0.974889\pi\)
0.996890 0.0788079i \(-0.0251114\pi\)
\(678\) 33.9938i 0.0501384i
\(679\) −1220.05 −1.79683
\(680\) 0 0
\(681\) 55.8393i 0.0819961i
\(682\) 605.384i 0.887660i
\(683\) −380.675 −0.557357 −0.278678 0.960385i \(-0.589896\pi\)
−0.278678 + 0.960385i \(0.589896\pi\)
\(684\) 433.051i 0.633116i
\(685\) 0 0
\(686\) 307.411i 0.448121i
\(687\) 6.14749i 0.00894832i
\(688\) 82.2883i 0.119605i
\(689\) 31.4750i 0.0456821i
\(690\) 0 0
\(691\) −318.845 −0.461425 −0.230713 0.973022i \(-0.574106\pi\)
−0.230713 + 0.973022i \(0.574106\pi\)
\(692\) 300.154 0.433748
\(693\) 575.385 0.830282
\(694\) 371.130 0.534769
\(695\) 0 0
\(696\) −25.0919 −0.0360515
\(697\) 321.861i 0.461781i
\(698\) 325.031 0.465660
\(699\) 94.3165 0.134931
\(700\) 0 0
\(701\) 112.754i 0.160848i 0.996761 + 0.0804239i \(0.0256274\pi\)
−0.996761 + 0.0804239i \(0.974373\pi\)
\(702\) −18.6424 −0.0265561
\(703\) 969.193 1.37865
\(704\) 60.6042i 0.0860855i
\(705\) 0 0
\(706\) −519.698 −0.736116
\(707\) 235.887i 0.333644i
\(708\) 37.6875 0.0532309
\(709\) 465.481i 0.656531i −0.944585 0.328266i \(-0.893536\pi\)
0.944585 0.328266i \(-0.106464\pi\)
\(710\) 0 0
\(711\) 1375.99i 1.93529i
\(712\) 180.242i 0.253149i
\(713\) −889.229 + 947.838i −1.24717 + 1.32937i
\(714\) −25.3509 −0.0355054
\(715\) 0 0
\(716\) 414.116 0.578375
\(717\) 41.6041 0.0580253
\(718\) 339.370i 0.472660i
\(719\) −347.913 −0.483885 −0.241943 0.970291i \(-0.577785\pi\)
−0.241943 + 0.970291i \(0.577785\pi\)
\(720\) 0 0
\(721\) 1136.79 1.57669
\(722\) −322.321 −0.446428
\(723\) 37.0518i 0.0512473i
\(724\) 662.276i 0.914746i
\(725\) 0 0
\(726\) 25.0559 0.0345123
\(727\) 88.3077i 0.121469i −0.998154 0.0607343i \(-0.980656\pi\)
0.998154 0.0607343i \(-0.0193442\pi\)
\(728\) 63.5824i 0.0873385i
\(729\) −691.569 −0.948654
\(730\) 0 0
\(731\) −155.536 −0.212772
\(732\) 19.5869i 0.0267580i
\(733\) 510.848i 0.696928i 0.937322 + 0.348464i \(0.113297\pi\)
−0.937322 + 0.348464i \(0.886703\pi\)
\(734\) 550.289i 0.749712i
\(735\) 0 0
\(736\) 89.0196 94.8869i 0.120950 0.128922i
\(737\) −333.823 −0.452948
\(738\) 537.171 0.727873
\(739\) −416.763 −0.563955 −0.281978 0.959421i \(-0.590990\pi\)
−0.281978 + 0.959421i \(0.590990\pi\)
\(740\) 0 0
\(741\) 17.8491i 0.0240878i
\(742\) −143.488 −0.193380
\(743\) 633.469i 0.852583i −0.904586 0.426291i \(-0.859820\pi\)
0.904586 0.426291i \(-0.140180\pi\)
\(744\) −44.5152 −0.0598323
\(745\) 0 0
\(746\) 772.597i 1.03565i
\(747\) 1263.79i 1.69182i
\(748\) −114.551 −0.153142
\(749\) −428.262 −0.571779
\(750\) 0 0
\(751\) 1282.52i 1.70775i 0.520475 + 0.853877i \(0.325755\pi\)
−0.520475 + 0.853877i \(0.674245\pi\)
\(752\) −337.219 −0.448430
\(753\) 104.981i 0.139416i
\(754\) −118.952 −0.157761
\(755\) 0 0
\(756\) 84.9866i 0.112416i
\(757\) 1233.64i 1.62964i −0.579716 0.814819i \(-0.696836\pi\)
0.579716 0.814819i \(-0.303164\pi\)
\(758\) 418.377i 0.551949i
\(759\) −35.3919 33.2035i −0.0466297 0.0437463i
\(760\) 0 0
\(761\) −262.559 −0.345018 −0.172509 0.985008i \(-0.555187\pi\)
−0.172509 + 0.985008i \(0.555187\pi\)
\(762\) 4.33812 0.00569307
\(763\) 1096.58 1.43720
\(764\) 345.499i 0.452223i
\(765\) 0 0
\(766\) 800.491i 1.04503i
\(767\) 178.663 0.232938
\(768\) 4.45636 0.00580255
\(769\) 801.510i 1.04228i −0.853473 0.521138i \(-0.825508\pi\)
0.853473 0.521138i \(-0.174492\pi\)
\(770\) 0 0
\(771\) −62.9698 −0.0816729
\(772\) −272.900 −0.353498
\(773\) 1143.79i 1.47967i −0.672786 0.739837i \(-0.734903\pi\)
0.672786 0.739837i \(-0.265097\pi\)
\(774\) 259.583i 0.335378i
\(775\) 0 0
\(776\) 405.377i 0.522393i
\(777\) −94.6907 −0.121867
\(778\) 277.246i 0.356357i
\(779\) 1033.10i 1.32618i
\(780\) 0 0
\(781\) 67.1644i 0.0859979i
\(782\) −179.350 168.260i −0.229347 0.215166i
\(783\) 158.995 0.203059
\(784\) −93.8587 −0.119718
\(785\) 0 0
\(786\) 1.43353 0.00182383
\(787\) 111.467i 0.141635i −0.997489 0.0708175i \(-0.977439\pi\)
0.997489 0.0708175i \(-0.0225608\pi\)
\(788\) 542.181 0.688046
\(789\) 0.678632i 0.000860116i
\(790\) 0 0
\(791\) −734.663 −0.928777
\(792\) 191.179i 0.241388i
\(793\) 92.8546i 0.117093i
\(794\) 422.551 0.532180
\(795\) 0 0
\(796\) 514.584i 0.646462i
\(797\) 231.014i 0.289855i −0.989442 0.144927i \(-0.953705\pi\)
0.989442 0.144927i \(-0.0462949\pi\)
\(798\) 81.3701 0.101968
\(799\) 637.393i 0.797738i
\(800\) 0 0
\(801\) 568.583i 0.709842i
\(802\) 1110.32i 1.38444i
\(803\) 662.215i 0.824676i
\(804\) 24.5467i 0.0305307i
\(805\) 0 0
\(806\) −211.031 −0.261825
\(807\) −86.7145 −0.107453
\(808\) 78.3763 0.0970004
\(809\) −165.201 −0.204204 −0.102102 0.994774i \(-0.532557\pi\)
−0.102102 + 0.994774i \(0.532557\pi\)
\(810\) 0 0
\(811\) 317.322 0.391273 0.195636 0.980677i \(-0.437323\pi\)
0.195636 + 0.980677i \(0.437323\pi\)
\(812\) 542.276i 0.667828i
\(813\) 72.6251 0.0893298
\(814\) −427.870 −0.525638
\(815\) 0 0
\(816\) 8.42315i 0.0103225i
\(817\) 499.234 0.611058
\(818\) 506.552 0.619257
\(819\) 200.574i 0.244901i
\(820\) 0 0
\(821\) −1389.82 −1.69284 −0.846421 0.532514i \(-0.821247\pi\)
−0.846421 + 0.532514i \(0.821247\pi\)
\(822\) 3.81989i 0.00464707i
\(823\) −948.001 −1.15188 −0.575942 0.817490i \(-0.695365\pi\)
−0.575942 + 0.817490i \(0.695365\pi\)
\(824\) 377.713i 0.458390i
\(825\) 0 0
\(826\) 814.489i 0.986064i
\(827\) 1177.54i 1.42387i −0.702245 0.711935i \(-0.747819\pi\)
0.702245 0.711935i \(-0.252181\pi\)
\(828\) −280.817 + 299.326i −0.339151 + 0.361505i
\(829\) −1278.78 −1.54256 −0.771279 0.636497i \(-0.780383\pi\)
−0.771279 + 0.636497i \(0.780383\pi\)
\(830\) 0 0
\(831\) 50.7655 0.0610897
\(832\) 21.1261 0.0253919
\(833\) 177.406i 0.212973i
\(834\) 3.17273 0.00380423
\(835\) 0 0
\(836\) 367.679 0.439808
\(837\) 282.072 0.337004
\(838\) 276.156i 0.329541i
\(839\) 1435.77i 1.71128i −0.517570 0.855641i \(-0.673163\pi\)
0.517570 0.855641i \(-0.326837\pi\)
\(840\) 0 0
\(841\) 173.507 0.206311
\(842\) 1086.18i 1.28999i
\(843\) 1.28260i 0.00152147i
\(844\) 108.389 0.128423
\(845\) 0 0
\(846\) 1063.78 1.25742
\(847\) 541.500i 0.639315i
\(848\) 47.6757i 0.0562213i
\(849\) 92.7193i 0.109210i
\(850\) 0 0
\(851\) −669.908 628.484i −0.787201 0.738524i
\(852\) 4.93875 0.00579665
\(853\) −812.996 −0.953102 −0.476551 0.879147i \(-0.658113\pi\)
−0.476551 + 0.879147i \(0.658113\pi\)
\(854\) −423.304 −0.495673
\(855\) 0 0
\(856\) 142.296i 0.166233i
\(857\) 958.841 1.11883 0.559417 0.828886i \(-0.311025\pi\)
0.559417 + 0.828886i \(0.311025\pi\)
\(858\) 7.87983i 0.00918395i
\(859\) −383.654 −0.446628 −0.223314 0.974747i \(-0.571688\pi\)
−0.223314 + 0.974747i \(0.571688\pi\)
\(860\) 0 0
\(861\) 100.934i 0.117229i
\(862\) 471.106i 0.546526i
\(863\) −1017.48 −1.17900 −0.589502 0.807767i \(-0.700676\pi\)
−0.589502 + 0.807767i \(0.700676\pi\)
\(864\) −28.2379 −0.0326827
\(865\) 0 0
\(866\) 10.6029i 0.0122436i
\(867\) 64.5721 0.0744776
\(868\) 962.047i 1.10835i
\(869\) −1168.27 −1.34439
\(870\) 0 0
\(871\) 116.368i 0.133602i
\(872\) 364.354i 0.417837i
\(873\) 1278.78i 1.46482i
\(874\) 575.669 + 540.072i 0.658660 + 0.617932i
\(875\) 0 0
\(876\) 48.6941 0.0555869
\(877\) −1443.83 −1.64633 −0.823163 0.567806i \(-0.807793\pi\)
−0.823163 + 0.567806i \(0.807793\pi\)
\(878\) −429.835 −0.489561
\(879\) 93.6154i 0.106502i
\(880\) 0 0
\(881\) 894.988i 1.01588i −0.861393 0.507939i \(-0.830408\pi\)
0.861393 0.507939i \(-0.169592\pi\)
\(882\) 296.082 0.335694
\(883\) 870.115 0.985407 0.492704 0.870197i \(-0.336009\pi\)
0.492704 + 0.870197i \(0.336009\pi\)
\(884\) 39.9313i 0.0451711i
\(885\) 0 0
\(886\) −901.058 −1.01700
\(887\) 529.330 0.596764 0.298382 0.954446i \(-0.403553\pi\)
0.298382 + 0.954446i \(0.403553\pi\)
\(888\) 31.4622i 0.0354304i
\(889\) 93.7539i 0.105460i
\(890\) 0 0
\(891\) 597.796i 0.670927i
\(892\) −209.221 −0.234553
\(893\) 2045.87i 2.29101i
\(894\) 56.7446i 0.0634728i
\(895\) 0 0
\(896\) 96.3093i 0.107488i
\(897\) 11.5744 12.3373i 0.0129035 0.0137540i
\(898\) 125.770 0.140056
\(899\) 1799.83 2.00203
\(900\) 0 0
\(901\) −90.1138 −0.100015
\(902\) 456.081i 0.505633i
\(903\) −48.7755 −0.0540149
\(904\) 244.101i 0.270023i
\(905\) 0 0
\(906\) −43.3107 −0.0478044
\(907\) 691.482i 0.762384i −0.924496 0.381192i \(-0.875514\pi\)
0.924496 0.381192i \(-0.124486\pi\)
\(908\) 400.968i 0.441595i
\(909\) −247.242 −0.271994
\(910\) 0 0
\(911\) 1794.49i 1.96980i −0.173115 0.984902i \(-0.555383\pi\)
0.173115 0.984902i \(-0.444617\pi\)
\(912\) 27.0363i 0.0296450i
\(913\) 1073.01 1.17526
\(914\) 534.105i 0.584360i
\(915\) 0 0
\(916\) 44.1436i 0.0481917i
\(917\) 30.9809i 0.0337851i
\(918\) 53.3736i 0.0581412i
\(919\) 428.705i 0.466491i 0.972418 + 0.233245i \(0.0749345\pi\)
−0.972418 + 0.233245i \(0.925065\pi\)
\(920\) 0 0
\(921\) 127.545 0.138485
\(922\) −827.413 −0.897411
\(923\) 23.4129 0.0253661
\(924\) −35.9225 −0.0388771
\(925\) 0 0
\(926\) 319.438 0.344965
\(927\) 1191.52i 1.28535i
\(928\) −180.178 −0.194158
\(929\) 1577.01 1.69754 0.848768 0.528766i \(-0.177345\pi\)
0.848768 + 0.528766i \(0.177345\pi\)
\(930\) 0 0
\(931\) 569.431i 0.611633i
\(932\) 677.263 0.726677
\(933\) −126.900 −0.136013
\(934\) 994.169i 1.06442i
\(935\) 0 0
\(936\) −66.6433 −0.0712001
\(937\) 141.317i 0.150818i −0.997153 0.0754091i \(-0.975974\pi\)
0.997153 0.0754091i \(-0.0240263\pi\)
\(938\) −530.495 −0.565560
\(939\) 164.148i 0.174812i
\(940\) 0 0
\(941\) 1506.21i 1.60065i 0.599566 + 0.800325i \(0.295340\pi\)
−0.599566 + 0.800325i \(0.704660\pi\)
\(942\) 9.77667i 0.0103786i
\(943\) −669.923 + 714.078i −0.710417 + 0.757240i
\(944\) 270.624 0.286678
\(945\) 0 0
\(946\) −220.397 −0.232978
\(947\) −1689.84 −1.78441 −0.892205 0.451632i \(-0.850842\pi\)
−0.892205 + 0.451632i \(0.850842\pi\)
\(948\) 85.9058i 0.0906179i
\(949\) 230.842 0.243248
\(950\) 0 0
\(951\) 107.580 0.113123
\(952\) −182.038 −0.191217
\(953\) 990.482i 1.03933i −0.854370 0.519665i \(-0.826057\pi\)
0.854370 0.519665i \(-0.173943\pi\)
\(954\) 150.395i 0.157647i
\(955\) 0 0
\(956\) 298.749 0.312499
\(957\) 67.2049i 0.0702245i
\(958\) 412.534i 0.430620i
\(959\) 82.5541 0.0860836
\(960\) 0 0
\(961\) 2232.05 2.32264
\(962\) 149.151i 0.155043i
\(963\) 448.879i 0.466126i
\(964\) 266.059i 0.275995i
\(965\) 0 0
\(966\) −56.2432 52.7654i −0.0582227 0.0546225i
\(967\) −1528.09 −1.58024 −0.790118 0.612954i \(-0.789981\pi\)
−0.790118 + 0.612954i \(0.789981\pi\)
\(968\) 179.920 0.185868
\(969\) 51.1024 0.0527372
\(970\) 0 0
\(971\) 616.239i 0.634643i −0.948318 0.317322i \(-0.897217\pi\)
0.948318 0.317322i \(-0.102783\pi\)
\(972\) 133.810 0.137664
\(973\) 68.5679i 0.0704706i
\(974\) −291.267 −0.299042
\(975\) 0 0
\(976\) 140.648i 0.144107i
\(977\) 1393.79i 1.42660i 0.700859 + 0.713299i \(0.252800\pi\)
−0.700859 + 0.713299i \(0.747200\pi\)
\(978\) −42.9197 −0.0438852
\(979\) −482.752 −0.493107
\(980\) 0 0
\(981\) 1149.37i 1.17164i
\(982\) −272.205 −0.277195
\(983\) 1033.22i 1.05109i 0.850767 + 0.525543i \(0.176138\pi\)
−0.850767 + 0.525543i \(0.823862\pi\)
\(984\) −33.5367 −0.0340820
\(985\) 0 0
\(986\) 340.563i 0.345398i
\(987\) 199.883i 0.202516i
\(988\) 128.170i 0.129726i
\(989\) −345.071 323.734i −0.348909 0.327335i
\(990\) 0 0
\(991\) −1122.80 −1.13300 −0.566501 0.824061i \(-0.691703\pi\)
−0.566501 + 0.824061i \(0.691703\pi\)
\(992\) −319.652 −0.322230
\(993\) 61.4732 0.0619066
\(994\) 106.734i 0.107379i
\(995\) 0 0
\(996\) 78.9011i 0.0792179i
\(997\) 718.650 0.720812 0.360406 0.932795i \(-0.382638\pi\)
0.360406 + 0.932795i \(0.382638\pi\)
\(998\) 1254.95 1.25747
\(999\) 199.361i 0.199561i
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 1150.3.d.b.551.13 16
5.2 odd 4 1150.3.c.c.1149.8 32
5.3 odd 4 1150.3.c.c.1149.25 32
5.4 even 2 230.3.d.a.91.3 16
15.14 odd 2 2070.3.c.a.91.16 16
20.19 odd 2 1840.3.k.d.321.9 16
23.22 odd 2 inner 1150.3.d.b.551.14 16
115.22 even 4 1150.3.c.c.1149.26 32
115.68 even 4 1150.3.c.c.1149.7 32
115.114 odd 2 230.3.d.a.91.4 yes 16
345.344 even 2 2070.3.c.a.91.9 16
460.459 even 2 1840.3.k.d.321.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.d.a.91.3 16 5.4 even 2
230.3.d.a.91.4 yes 16 115.114 odd 2
1150.3.c.c.1149.7 32 115.68 even 4
1150.3.c.c.1149.8 32 5.2 odd 4
1150.3.c.c.1149.25 32 5.3 odd 4
1150.3.c.c.1149.26 32 115.22 even 4
1150.3.d.b.551.13 16 1.1 even 1 trivial
1150.3.d.b.551.14 16 23.22 odd 2 inner
1840.3.k.d.321.9 16 20.19 odd 2
1840.3.k.d.321.10 16 460.459 even 2
2070.3.c.a.91.9 16 345.344 even 2
2070.3.c.a.91.16 16 15.14 odd 2