Properties

Label 1150.2.a.j.1.1
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,2,Mod(1,1150)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.18279623245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +1.61803 q^{6} +0.618034 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +1.61803 q^{6} +0.618034 q^{7} -1.00000 q^{8} -0.381966 q^{9} -2.85410 q^{11} -1.61803 q^{12} +7.09017 q^{13} -0.618034 q^{14} +1.00000 q^{16} -6.09017 q^{17} +0.381966 q^{18} +1.85410 q^{19} -1.00000 q^{21} +2.85410 q^{22} -1.00000 q^{23} +1.61803 q^{24} -7.09017 q^{26} +5.47214 q^{27} +0.618034 q^{28} -9.23607 q^{29} +9.09017 q^{31} -1.00000 q^{32} +4.61803 q^{33} +6.09017 q^{34} -0.381966 q^{36} -6.47214 q^{37} -1.85410 q^{38} -11.4721 q^{39} +3.32624 q^{41} +1.00000 q^{42} -2.85410 q^{44} +1.00000 q^{46} +3.70820 q^{47} -1.61803 q^{48} -6.61803 q^{49} +9.85410 q^{51} +7.09017 q^{52} -0.472136 q^{53} -5.47214 q^{54} -0.618034 q^{56} -3.00000 q^{57} +9.23607 q^{58} +1.70820 q^{59} -9.32624 q^{61} -9.09017 q^{62} -0.236068 q^{63} +1.00000 q^{64} -4.61803 q^{66} -14.4721 q^{67} -6.09017 q^{68} +1.61803 q^{69} -4.09017 q^{71} +0.381966 q^{72} -3.23607 q^{73} +6.47214 q^{74} +1.85410 q^{76} -1.76393 q^{77} +11.4721 q^{78} +1.52786 q^{79} -7.70820 q^{81} -3.32624 q^{82} +6.94427 q^{83} -1.00000 q^{84} +14.9443 q^{87} +2.85410 q^{88} -10.4721 q^{89} +4.38197 q^{91} -1.00000 q^{92} -14.7082 q^{93} -3.70820 q^{94} +1.61803 q^{96} -12.3820 q^{97} +6.61803 q^{98} +1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{6} - q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{6} - q^{7} - 2 q^{8} - 3 q^{9} + q^{11} - q^{12} + 3 q^{13} + q^{14} + 2 q^{16} - q^{17} + 3 q^{18} - 3 q^{19} - 2 q^{21} - q^{22} - 2 q^{23} + q^{24} - 3 q^{26} + 2 q^{27} - q^{28} - 14 q^{29} + 7 q^{31} - 2 q^{32} + 7 q^{33} + q^{34} - 3 q^{36} - 4 q^{37} + 3 q^{38} - 14 q^{39} - 9 q^{41} + 2 q^{42} + q^{44} + 2 q^{46} - 6 q^{47} - q^{48} - 11 q^{49} + 13 q^{51} + 3 q^{52} + 8 q^{53} - 2 q^{54} + q^{56} - 6 q^{57} + 14 q^{58} - 10 q^{59} - 3 q^{61} - 7 q^{62} + 4 q^{63} + 2 q^{64} - 7 q^{66} - 20 q^{67} - q^{68} + q^{69} + 3 q^{71} + 3 q^{72} - 2 q^{73} + 4 q^{74} - 3 q^{76} - 8 q^{77} + 14 q^{78} + 12 q^{79} - 2 q^{81} + 9 q^{82} - 4 q^{83} - 2 q^{84} + 12 q^{87} - q^{88} - 12 q^{89} + 11 q^{91} - 2 q^{92} - 16 q^{93} + 6 q^{94} + q^{96} - 27 q^{97} + 11 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) 0.618034 0.233595 0.116797 0.993156i \(-0.462737\pi\)
0.116797 + 0.993156i \(0.462737\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) −2.85410 −0.860544 −0.430272 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) −1.61803 −0.467086
\(13\) 7.09017 1.96646 0.983230 0.182372i \(-0.0583774\pi\)
0.983230 + 0.182372i \(0.0583774\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.09017 −1.47708 −0.738542 0.674208i \(-0.764485\pi\)
−0.738542 + 0.674208i \(0.764485\pi\)
\(18\) 0.381966 0.0900303
\(19\) 1.85410 0.425360 0.212680 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.85410 0.608497
\(23\) −1.00000 −0.208514
\(24\) 1.61803 0.330280
\(25\) 0 0
\(26\) −7.09017 −1.39050
\(27\) 5.47214 1.05311
\(28\) 0.618034 0.116797
\(29\) −9.23607 −1.71509 −0.857547 0.514405i \(-0.828013\pi\)
−0.857547 + 0.514405i \(0.828013\pi\)
\(30\) 0 0
\(31\) 9.09017 1.63264 0.816321 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.61803 0.803897
\(34\) 6.09017 1.04446
\(35\) 0 0
\(36\) −0.381966 −0.0636610
\(37\) −6.47214 −1.06401 −0.532006 0.846740i \(-0.678562\pi\)
−0.532006 + 0.846740i \(0.678562\pi\)
\(38\) −1.85410 −0.300775
\(39\) −11.4721 −1.83701
\(40\) 0 0
\(41\) 3.32624 0.519471 0.259736 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) 1.00000 0.154303
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.85410 −0.430272
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.70820 0.540897 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) −1.61803 −0.233543
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) 9.85410 1.37985
\(52\) 7.09017 0.983230
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) −5.47214 −0.744663
\(55\) 0 0
\(56\) −0.618034 −0.0825883
\(57\) −3.00000 −0.397360
\(58\) 9.23607 1.21276
\(59\) 1.70820 0.222389 0.111195 0.993799i \(-0.464532\pi\)
0.111195 + 0.993799i \(0.464532\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) −9.09017 −1.15445
\(63\) −0.236068 −0.0297418
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.61803 −0.568441
\(67\) −14.4721 −1.76805 −0.884026 0.467437i \(-0.845177\pi\)
−0.884026 + 0.467437i \(0.845177\pi\)
\(68\) −6.09017 −0.738542
\(69\) 1.61803 0.194788
\(70\) 0 0
\(71\) −4.09017 −0.485414 −0.242707 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) 0.381966 0.0450151
\(73\) −3.23607 −0.378753 −0.189377 0.981905i \(-0.560647\pi\)
−0.189377 + 0.981905i \(0.560647\pi\)
\(74\) 6.47214 0.752371
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) −1.76393 −0.201019
\(78\) 11.4721 1.29896
\(79\) 1.52786 0.171898 0.0859491 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) −3.32624 −0.367322
\(83\) 6.94427 0.762233 0.381116 0.924527i \(-0.375540\pi\)
0.381116 + 0.924527i \(0.375540\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 0 0
\(87\) 14.9443 1.60219
\(88\) 2.85410 0.304248
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) 4.38197 0.459355
\(92\) −1.00000 −0.104257
\(93\) −14.7082 −1.52517
\(94\) −3.70820 −0.382472
\(95\) 0 0
\(96\) 1.61803 0.165140
\(97\) −12.3820 −1.25720 −0.628599 0.777730i \(-0.716371\pi\)
−0.628599 + 0.777730i \(0.716371\pi\)
\(98\) 6.61803 0.668522
\(99\) 1.09017 0.109566
\(100\) 0 0
\(101\) −0.291796 −0.0290348 −0.0145174 0.999895i \(-0.504621\pi\)
−0.0145174 + 0.999895i \(0.504621\pi\)
\(102\) −9.85410 −0.975701
\(103\) −16.5623 −1.63193 −0.815966 0.578100i \(-0.803795\pi\)
−0.815966 + 0.578100i \(0.803795\pi\)
\(104\) −7.09017 −0.695248
\(105\) 0 0
\(106\) 0.472136 0.0458579
\(107\) −18.1803 −1.75756 −0.878780 0.477227i \(-0.841642\pi\)
−0.878780 + 0.477227i \(0.841642\pi\)
\(108\) 5.47214 0.526557
\(109\) −11.5623 −1.10747 −0.553734 0.832694i \(-0.686798\pi\)
−0.553734 + 0.832694i \(0.686798\pi\)
\(110\) 0 0
\(111\) 10.4721 0.993971
\(112\) 0.618034 0.0583987
\(113\) −1.05573 −0.0993145 −0.0496573 0.998766i \(-0.515813\pi\)
−0.0496573 + 0.998766i \(0.515813\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) −9.23607 −0.857547
\(117\) −2.70820 −0.250374
\(118\) −1.70820 −0.157253
\(119\) −3.76393 −0.345039
\(120\) 0 0
\(121\) −2.85410 −0.259464
\(122\) 9.32624 0.844358
\(123\) −5.38197 −0.485276
\(124\) 9.09017 0.816321
\(125\) 0 0
\(126\) 0.236068 0.0210306
\(127\) −16.1803 −1.43577 −0.717886 0.696160i \(-0.754890\pi\)
−0.717886 + 0.696160i \(0.754890\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 2.94427 0.257242 0.128621 0.991694i \(-0.458945\pi\)
0.128621 + 0.991694i \(0.458945\pi\)
\(132\) 4.61803 0.401948
\(133\) 1.14590 0.0993620
\(134\) 14.4721 1.25020
\(135\) 0 0
\(136\) 6.09017 0.522228
\(137\) 10.3262 0.882230 0.441115 0.897451i \(-0.354583\pi\)
0.441115 + 0.897451i \(0.354583\pi\)
\(138\) −1.61803 −0.137736
\(139\) 12.7639 1.08262 0.541311 0.840822i \(-0.317928\pi\)
0.541311 + 0.840822i \(0.317928\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 4.09017 0.343239
\(143\) −20.2361 −1.69223
\(144\) −0.381966 −0.0318305
\(145\) 0 0
\(146\) 3.23607 0.267819
\(147\) 10.7082 0.883198
\(148\) −6.47214 −0.532006
\(149\) −7.85410 −0.643433 −0.321717 0.946836i \(-0.604260\pi\)
−0.321717 + 0.946836i \(0.604260\pi\)
\(150\) 0 0
\(151\) −2.56231 −0.208517 −0.104259 0.994550i \(-0.533247\pi\)
−0.104259 + 0.994550i \(0.533247\pi\)
\(152\) −1.85410 −0.150388
\(153\) 2.32624 0.188065
\(154\) 1.76393 0.142142
\(155\) 0 0
\(156\) −11.4721 −0.918506
\(157\) 3.70820 0.295947 0.147973 0.988991i \(-0.452725\pi\)
0.147973 + 0.988991i \(0.452725\pi\)
\(158\) −1.52786 −0.121550
\(159\) 0.763932 0.0605838
\(160\) 0 0
\(161\) −0.618034 −0.0487079
\(162\) 7.70820 0.605614
\(163\) −1.38197 −0.108244 −0.0541220 0.998534i \(-0.517236\pi\)
−0.0541220 + 0.998534i \(0.517236\pi\)
\(164\) 3.32624 0.259736
\(165\) 0 0
\(166\) −6.94427 −0.538980
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 1.00000 0.0771517
\(169\) 37.2705 2.86696
\(170\) 0 0
\(171\) −0.708204 −0.0541577
\(172\) 0 0
\(173\) 1.43769 0.109306 0.0546529 0.998505i \(-0.482595\pi\)
0.0546529 + 0.998505i \(0.482595\pi\)
\(174\) −14.9443 −1.13292
\(175\) 0 0
\(176\) −2.85410 −0.215136
\(177\) −2.76393 −0.207750
\(178\) 10.4721 0.784920
\(179\) 2.18034 0.162966 0.0814831 0.996675i \(-0.474034\pi\)
0.0814831 + 0.996675i \(0.474034\pi\)
\(180\) 0 0
\(181\) −12.1459 −0.902797 −0.451399 0.892322i \(-0.649075\pi\)
−0.451399 + 0.892322i \(0.649075\pi\)
\(182\) −4.38197 −0.324813
\(183\) 15.0902 1.11550
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 14.7082 1.07846
\(187\) 17.3820 1.27110
\(188\) 3.70820 0.270449
\(189\) 3.38197 0.246002
\(190\) 0 0
\(191\) −13.7082 −0.991891 −0.495945 0.868354i \(-0.665178\pi\)
−0.495945 + 0.868354i \(0.665178\pi\)
\(192\) −1.61803 −0.116772
\(193\) −0.763932 −0.0549890 −0.0274945 0.999622i \(-0.508753\pi\)
−0.0274945 + 0.999622i \(0.508753\pi\)
\(194\) 12.3820 0.888973
\(195\) 0 0
\(196\) −6.61803 −0.472717
\(197\) 22.5623 1.60750 0.803749 0.594969i \(-0.202836\pi\)
0.803749 + 0.594969i \(0.202836\pi\)
\(198\) −1.09017 −0.0774750
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 23.4164 1.65167
\(202\) 0.291796 0.0205307
\(203\) −5.70820 −0.400637
\(204\) 9.85410 0.689925
\(205\) 0 0
\(206\) 16.5623 1.15395
\(207\) 0.381966 0.0265485
\(208\) 7.09017 0.491615
\(209\) −5.29180 −0.366041
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) −0.472136 −0.0324264
\(213\) 6.61803 0.453460
\(214\) 18.1803 1.24278
\(215\) 0 0
\(216\) −5.47214 −0.372332
\(217\) 5.61803 0.381377
\(218\) 11.5623 0.783098
\(219\) 5.23607 0.353821
\(220\) 0 0
\(221\) −43.1803 −2.90462
\(222\) −10.4721 −0.702844
\(223\) 20.9443 1.40253 0.701266 0.712900i \(-0.252618\pi\)
0.701266 + 0.712900i \(0.252618\pi\)
\(224\) −0.618034 −0.0412941
\(225\) 0 0
\(226\) 1.05573 0.0702260
\(227\) 18.7639 1.24541 0.622703 0.782458i \(-0.286035\pi\)
0.622703 + 0.782458i \(0.286035\pi\)
\(228\) −3.00000 −0.198680
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 2.85410 0.187786
\(232\) 9.23607 0.606378
\(233\) −6.29180 −0.412189 −0.206095 0.978532i \(-0.566075\pi\)
−0.206095 + 0.978532i \(0.566075\pi\)
\(234\) 2.70820 0.177041
\(235\) 0 0
\(236\) 1.70820 0.111195
\(237\) −2.47214 −0.160582
\(238\) 3.76393 0.243979
\(239\) −20.3607 −1.31702 −0.658511 0.752571i \(-0.728814\pi\)
−0.658511 + 0.752571i \(0.728814\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 2.85410 0.183469
\(243\) −3.94427 −0.253025
\(244\) −9.32624 −0.597051
\(245\) 0 0
\(246\) 5.38197 0.343142
\(247\) 13.1459 0.836453
\(248\) −9.09017 −0.577226
\(249\) −11.2361 −0.712057
\(250\) 0 0
\(251\) 6.14590 0.387926 0.193963 0.981009i \(-0.437866\pi\)
0.193963 + 0.981009i \(0.437866\pi\)
\(252\) −0.236068 −0.0148709
\(253\) 2.85410 0.179436
\(254\) 16.1803 1.01524
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.81966 0.487777 0.243888 0.969803i \(-0.421577\pi\)
0.243888 + 0.969803i \(0.421577\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 3.52786 0.218369
\(262\) −2.94427 −0.181898
\(263\) −20.7426 −1.27905 −0.639523 0.768772i \(-0.720868\pi\)
−0.639523 + 0.768772i \(0.720868\pi\)
\(264\) −4.61803 −0.284220
\(265\) 0 0
\(266\) −1.14590 −0.0702595
\(267\) 16.9443 1.03697
\(268\) −14.4721 −0.884026
\(269\) −14.1803 −0.864591 −0.432295 0.901732i \(-0.642296\pi\)
−0.432295 + 0.901732i \(0.642296\pi\)
\(270\) 0 0
\(271\) −30.3262 −1.84219 −0.921094 0.389341i \(-0.872703\pi\)
−0.921094 + 0.389341i \(0.872703\pi\)
\(272\) −6.09017 −0.369271
\(273\) −7.09017 −0.429117
\(274\) −10.3262 −0.623831
\(275\) 0 0
\(276\) 1.61803 0.0973942
\(277\) −29.4164 −1.76746 −0.883730 0.467996i \(-0.844976\pi\)
−0.883730 + 0.467996i \(0.844976\pi\)
\(278\) −12.7639 −0.765530
\(279\) −3.47214 −0.207871
\(280\) 0 0
\(281\) 22.7639 1.35798 0.678991 0.734146i \(-0.262417\pi\)
0.678991 + 0.734146i \(0.262417\pi\)
\(282\) 6.00000 0.357295
\(283\) 26.9443 1.60167 0.800835 0.598885i \(-0.204389\pi\)
0.800835 + 0.598885i \(0.204389\pi\)
\(284\) −4.09017 −0.242707
\(285\) 0 0
\(286\) 20.2361 1.19658
\(287\) 2.05573 0.121346
\(288\) 0.381966 0.0225076
\(289\) 20.0902 1.18177
\(290\) 0 0
\(291\) 20.0344 1.17444
\(292\) −3.23607 −0.189377
\(293\) 19.8885 1.16190 0.580951 0.813939i \(-0.302681\pi\)
0.580951 + 0.813939i \(0.302681\pi\)
\(294\) −10.7082 −0.624515
\(295\) 0 0
\(296\) 6.47214 0.376185
\(297\) −15.6180 −0.906250
\(298\) 7.85410 0.454976
\(299\) −7.09017 −0.410035
\(300\) 0 0
\(301\) 0 0
\(302\) 2.56231 0.147444
\(303\) 0.472136 0.0271235
\(304\) 1.85410 0.106340
\(305\) 0 0
\(306\) −2.32624 −0.132982
\(307\) 28.4508 1.62378 0.811888 0.583813i \(-0.198440\pi\)
0.811888 + 0.583813i \(0.198440\pi\)
\(308\) −1.76393 −0.100509
\(309\) 26.7984 1.52451
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 11.4721 0.649482
\(313\) −12.7984 −0.723407 −0.361703 0.932293i \(-0.617805\pi\)
−0.361703 + 0.932293i \(0.617805\pi\)
\(314\) −3.70820 −0.209266
\(315\) 0 0
\(316\) 1.52786 0.0859491
\(317\) 11.0902 0.622886 0.311443 0.950265i \(-0.399188\pi\)
0.311443 + 0.950265i \(0.399188\pi\)
\(318\) −0.763932 −0.0428392
\(319\) 26.3607 1.47591
\(320\) 0 0
\(321\) 29.4164 1.64186
\(322\) 0.618034 0.0344417
\(323\) −11.2918 −0.628292
\(324\) −7.70820 −0.428234
\(325\) 0 0
\(326\) 1.38197 0.0765400
\(327\) 18.7082 1.03457
\(328\) −3.32624 −0.183661
\(329\) 2.29180 0.126351
\(330\) 0 0
\(331\) 19.2361 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(332\) 6.94427 0.381116
\(333\) 2.47214 0.135472
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −13.6738 −0.744857 −0.372429 0.928061i \(-0.621475\pi\)
−0.372429 + 0.928061i \(0.621475\pi\)
\(338\) −37.2705 −2.02725
\(339\) 1.70820 0.0927769
\(340\) 0 0
\(341\) −25.9443 −1.40496
\(342\) 0.708204 0.0382953
\(343\) −8.41641 −0.454443
\(344\) 0 0
\(345\) 0 0
\(346\) −1.43769 −0.0772909
\(347\) 6.38197 0.342602 0.171301 0.985219i \(-0.445203\pi\)
0.171301 + 0.985219i \(0.445203\pi\)
\(348\) 14.9443 0.801097
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 38.7984 2.07090
\(352\) 2.85410 0.152124
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 2.76393 0.146901
\(355\) 0 0
\(356\) −10.4721 −0.555022
\(357\) 6.09017 0.322326
\(358\) −2.18034 −0.115235
\(359\) 26.3607 1.39126 0.695632 0.718399i \(-0.255125\pi\)
0.695632 + 0.718399i \(0.255125\pi\)
\(360\) 0 0
\(361\) −15.5623 −0.819069
\(362\) 12.1459 0.638374
\(363\) 4.61803 0.242384
\(364\) 4.38197 0.229677
\(365\) 0 0
\(366\) −15.0902 −0.788776
\(367\) −6.47214 −0.337843 −0.168921 0.985630i \(-0.554028\pi\)
−0.168921 + 0.985630i \(0.554028\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −1.27051 −0.0661401
\(370\) 0 0
\(371\) −0.291796 −0.0151493
\(372\) −14.7082 −0.762585
\(373\) −20.1803 −1.04490 −0.522449 0.852670i \(-0.674982\pi\)
−0.522449 + 0.852670i \(0.674982\pi\)
\(374\) −17.3820 −0.898800
\(375\) 0 0
\(376\) −3.70820 −0.191236
\(377\) −65.4853 −3.37266
\(378\) −3.38197 −0.173950
\(379\) −22.4508 −1.15322 −0.576611 0.817019i \(-0.695625\pi\)
−0.576611 + 0.817019i \(0.695625\pi\)
\(380\) 0 0
\(381\) 26.1803 1.34126
\(382\) 13.7082 0.701373
\(383\) 17.8885 0.914062 0.457031 0.889451i \(-0.348913\pi\)
0.457031 + 0.889451i \(0.348913\pi\)
\(384\) 1.61803 0.0825700
\(385\) 0 0
\(386\) 0.763932 0.0388831
\(387\) 0 0
\(388\) −12.3820 −0.628599
\(389\) 21.3262 1.08128 0.540642 0.841253i \(-0.318182\pi\)
0.540642 + 0.841253i \(0.318182\pi\)
\(390\) 0 0
\(391\) 6.09017 0.307993
\(392\) 6.61803 0.334261
\(393\) −4.76393 −0.240309
\(394\) −22.5623 −1.13667
\(395\) 0 0
\(396\) 1.09017 0.0547831
\(397\) −7.32624 −0.367693 −0.183847 0.982955i \(-0.558855\pi\)
−0.183847 + 0.982955i \(0.558855\pi\)
\(398\) −2.00000 −0.100251
\(399\) −1.85410 −0.0928212
\(400\) 0 0
\(401\) −1.70820 −0.0853036 −0.0426518 0.999090i \(-0.513581\pi\)
−0.0426518 + 0.999090i \(0.513581\pi\)
\(402\) −23.4164 −1.16790
\(403\) 64.4508 3.21053
\(404\) −0.291796 −0.0145174
\(405\) 0 0
\(406\) 5.70820 0.283293
\(407\) 18.4721 0.915630
\(408\) −9.85410 −0.487851
\(409\) −30.2148 −1.49402 −0.747012 0.664810i \(-0.768512\pi\)
−0.747012 + 0.664810i \(0.768512\pi\)
\(410\) 0 0
\(411\) −16.7082 −0.824155
\(412\) −16.5623 −0.815966
\(413\) 1.05573 0.0519490
\(414\) −0.381966 −0.0187726
\(415\) 0 0
\(416\) −7.09017 −0.347624
\(417\) −20.6525 −1.01136
\(418\) 5.29180 0.258830
\(419\) 14.4721 0.707010 0.353505 0.935433i \(-0.384990\pi\)
0.353505 + 0.935433i \(0.384990\pi\)
\(420\) 0 0
\(421\) 13.7426 0.669776 0.334888 0.942258i \(-0.391302\pi\)
0.334888 + 0.942258i \(0.391302\pi\)
\(422\) −14.0000 −0.681509
\(423\) −1.41641 −0.0688681
\(424\) 0.472136 0.0229289
\(425\) 0 0
\(426\) −6.61803 −0.320645
\(427\) −5.76393 −0.278936
\(428\) −18.1803 −0.878780
\(429\) 32.7426 1.58083
\(430\) 0 0
\(431\) 3.34752 0.161245 0.0806223 0.996745i \(-0.474309\pi\)
0.0806223 + 0.996745i \(0.474309\pi\)
\(432\) 5.47214 0.263278
\(433\) −8.50658 −0.408800 −0.204400 0.978887i \(-0.565524\pi\)
−0.204400 + 0.978887i \(0.565524\pi\)
\(434\) −5.61803 −0.269674
\(435\) 0 0
\(436\) −11.5623 −0.553734
\(437\) −1.85410 −0.0886937
\(438\) −5.23607 −0.250189
\(439\) −13.3820 −0.638686 −0.319343 0.947639i \(-0.603462\pi\)
−0.319343 + 0.947639i \(0.603462\pi\)
\(440\) 0 0
\(441\) 2.52786 0.120374
\(442\) 43.1803 2.05388
\(443\) −25.0902 −1.19207 −0.596035 0.802958i \(-0.703258\pi\)
−0.596035 + 0.802958i \(0.703258\pi\)
\(444\) 10.4721 0.496986
\(445\) 0 0
\(446\) −20.9443 −0.991740
\(447\) 12.7082 0.601077
\(448\) 0.618034 0.0291994
\(449\) −1.56231 −0.0737298 −0.0368649 0.999320i \(-0.511737\pi\)
−0.0368649 + 0.999320i \(0.511737\pi\)
\(450\) 0 0
\(451\) −9.49342 −0.447028
\(452\) −1.05573 −0.0496573
\(453\) 4.14590 0.194791
\(454\) −18.7639 −0.880635
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 37.7771 1.76714 0.883569 0.468301i \(-0.155134\pi\)
0.883569 + 0.468301i \(0.155134\pi\)
\(458\) −10.0000 −0.467269
\(459\) −33.3262 −1.55554
\(460\) 0 0
\(461\) −39.2361 −1.82741 −0.913703 0.406383i \(-0.866790\pi\)
−0.913703 + 0.406383i \(0.866790\pi\)
\(462\) −2.85410 −0.132785
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) −9.23607 −0.428774
\(465\) 0 0
\(466\) 6.29180 0.291462
\(467\) 17.1246 0.792433 0.396216 0.918157i \(-0.370323\pi\)
0.396216 + 0.918157i \(0.370323\pi\)
\(468\) −2.70820 −0.125187
\(469\) −8.94427 −0.413008
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) −1.70820 −0.0786265
\(473\) 0 0
\(474\) 2.47214 0.113549
\(475\) 0 0
\(476\) −3.76393 −0.172520
\(477\) 0.180340 0.00825720
\(478\) 20.3607 0.931276
\(479\) −31.8885 −1.45702 −0.728512 0.685033i \(-0.759788\pi\)
−0.728512 + 0.685033i \(0.759788\pi\)
\(480\) 0 0
\(481\) −45.8885 −2.09234
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) −2.85410 −0.129732
\(485\) 0 0
\(486\) 3.94427 0.178916
\(487\) 19.8197 0.898115 0.449057 0.893503i \(-0.351760\pi\)
0.449057 + 0.893503i \(0.351760\pi\)
\(488\) 9.32624 0.422179
\(489\) 2.23607 0.101118
\(490\) 0 0
\(491\) 6.18034 0.278915 0.139457 0.990228i \(-0.455464\pi\)
0.139457 + 0.990228i \(0.455464\pi\)
\(492\) −5.38197 −0.242638
\(493\) 56.2492 2.53334
\(494\) −13.1459 −0.591462
\(495\) 0 0
\(496\) 9.09017 0.408161
\(497\) −2.52786 −0.113390
\(498\) 11.2361 0.503500
\(499\) 12.3607 0.553340 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(500\) 0 0
\(501\) −12.9443 −0.578307
\(502\) −6.14590 −0.274305
\(503\) −36.3262 −1.61971 −0.809853 0.586632i \(-0.800453\pi\)
−0.809853 + 0.586632i \(0.800453\pi\)
\(504\) 0.236068 0.0105153
\(505\) 0 0
\(506\) −2.85410 −0.126880
\(507\) −60.3050 −2.67824
\(508\) −16.1803 −0.717886
\(509\) 36.6525 1.62459 0.812296 0.583245i \(-0.198217\pi\)
0.812296 + 0.583245i \(0.198217\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 10.1459 0.447952
\(514\) −7.81966 −0.344910
\(515\) 0 0
\(516\) 0 0
\(517\) −10.5836 −0.465466
\(518\) 4.00000 0.175750
\(519\) −2.32624 −0.102111
\(520\) 0 0
\(521\) −15.5279 −0.680288 −0.340144 0.940373i \(-0.610476\pi\)
−0.340144 + 0.940373i \(0.610476\pi\)
\(522\) −3.52786 −0.154410
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 2.94427 0.128621
\(525\) 0 0
\(526\) 20.7426 0.904422
\(527\) −55.3607 −2.41155
\(528\) 4.61803 0.200974
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −0.652476 −0.0283150
\(532\) 1.14590 0.0496810
\(533\) 23.5836 1.02152
\(534\) −16.9443 −0.733250
\(535\) 0 0
\(536\) 14.4721 0.625101
\(537\) −3.52786 −0.152239
\(538\) 14.1803 0.611358
\(539\) 18.8885 0.813587
\(540\) 0 0
\(541\) 22.8328 0.981659 0.490830 0.871256i \(-0.336694\pi\)
0.490830 + 0.871256i \(0.336694\pi\)
\(542\) 30.3262 1.30262
\(543\) 19.6525 0.843368
\(544\) 6.09017 0.261114
\(545\) 0 0
\(546\) 7.09017 0.303431
\(547\) −27.9230 −1.19390 −0.596950 0.802278i \(-0.703621\pi\)
−0.596950 + 0.802278i \(0.703621\pi\)
\(548\) 10.3262 0.441115
\(549\) 3.56231 0.152036
\(550\) 0 0
\(551\) −17.1246 −0.729533
\(552\) −1.61803 −0.0688681
\(553\) 0.944272 0.0401545
\(554\) 29.4164 1.24978
\(555\) 0 0
\(556\) 12.7639 0.541311
\(557\) 22.8328 0.967457 0.483729 0.875218i \(-0.339282\pi\)
0.483729 + 0.875218i \(0.339282\pi\)
\(558\) 3.47214 0.146987
\(559\) 0 0
\(560\) 0 0
\(561\) −28.1246 −1.18742
\(562\) −22.7639 −0.960239
\(563\) 13.8885 0.585332 0.292666 0.956215i \(-0.405458\pi\)
0.292666 + 0.956215i \(0.405458\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −26.9443 −1.13255
\(567\) −4.76393 −0.200066
\(568\) 4.09017 0.171620
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) 15.9787 0.668688 0.334344 0.942451i \(-0.391485\pi\)
0.334344 + 0.942451i \(0.391485\pi\)
\(572\) −20.2361 −0.846113
\(573\) 22.1803 0.926597
\(574\) −2.05573 −0.0858044
\(575\) 0 0
\(576\) −0.381966 −0.0159153
\(577\) 3.52786 0.146867 0.0734335 0.997300i \(-0.476604\pi\)
0.0734335 + 0.997300i \(0.476604\pi\)
\(578\) −20.0902 −0.835641
\(579\) 1.23607 0.0513692
\(580\) 0 0
\(581\) 4.29180 0.178054
\(582\) −20.0344 −0.830454
\(583\) 1.34752 0.0558087
\(584\) 3.23607 0.133909
\(585\) 0 0
\(586\) −19.8885 −0.821588
\(587\) −13.6180 −0.562076 −0.281038 0.959697i \(-0.590679\pi\)
−0.281038 + 0.959697i \(0.590679\pi\)
\(588\) 10.7082 0.441599
\(589\) 16.8541 0.694461
\(590\) 0 0
\(591\) −36.5066 −1.50168
\(592\) −6.47214 −0.266003
\(593\) −39.2361 −1.61123 −0.805616 0.592438i \(-0.798166\pi\)
−0.805616 + 0.592438i \(0.798166\pi\)
\(594\) 15.6180 0.640816
\(595\) 0 0
\(596\) −7.85410 −0.321717
\(597\) −3.23607 −0.132443
\(598\) 7.09017 0.289939
\(599\) −18.3820 −0.751067 −0.375533 0.926809i \(-0.622540\pi\)
−0.375533 + 0.926809i \(0.622540\pi\)
\(600\) 0 0
\(601\) −33.2705 −1.35713 −0.678566 0.734539i \(-0.737398\pi\)
−0.678566 + 0.734539i \(0.737398\pi\)
\(602\) 0 0
\(603\) 5.52786 0.225112
\(604\) −2.56231 −0.104259
\(605\) 0 0
\(606\) −0.472136 −0.0191792
\(607\) −26.4721 −1.07447 −0.537235 0.843432i \(-0.680531\pi\)
−0.537235 + 0.843432i \(0.680531\pi\)
\(608\) −1.85410 −0.0751938
\(609\) 9.23607 0.374264
\(610\) 0 0
\(611\) 26.2918 1.06365
\(612\) 2.32624 0.0940326
\(613\) −19.3050 −0.779720 −0.389860 0.920874i \(-0.627477\pi\)
−0.389860 + 0.920874i \(0.627477\pi\)
\(614\) −28.4508 −1.14818
\(615\) 0 0
\(616\) 1.76393 0.0710708
\(617\) −34.0902 −1.37242 −0.686209 0.727404i \(-0.740727\pi\)
−0.686209 + 0.727404i \(0.740727\pi\)
\(618\) −26.7984 −1.07799
\(619\) −2.79837 −0.112476 −0.0562381 0.998417i \(-0.517911\pi\)
−0.0562381 + 0.998417i \(0.517911\pi\)
\(620\) 0 0
\(621\) −5.47214 −0.219589
\(622\) −4.00000 −0.160385
\(623\) −6.47214 −0.259301
\(624\) −11.4721 −0.459253
\(625\) 0 0
\(626\) 12.7984 0.511526
\(627\) 8.56231 0.341946
\(628\) 3.70820 0.147973
\(629\) 39.4164 1.57164
\(630\) 0 0
\(631\) 42.0689 1.67474 0.837368 0.546640i \(-0.184093\pi\)
0.837368 + 0.546640i \(0.184093\pi\)
\(632\) −1.52786 −0.0607752
\(633\) −22.6525 −0.900355
\(634\) −11.0902 −0.440447
\(635\) 0 0
\(636\) 0.763932 0.0302919
\(637\) −46.9230 −1.85916
\(638\) −26.3607 −1.04363
\(639\) 1.56231 0.0618039
\(640\) 0 0
\(641\) 0.360680 0.0142460 0.00712300 0.999975i \(-0.497733\pi\)
0.00712300 + 0.999975i \(0.497733\pi\)
\(642\) −29.4164 −1.16097
\(643\) 8.29180 0.326997 0.163498 0.986544i \(-0.447722\pi\)
0.163498 + 0.986544i \(0.447722\pi\)
\(644\) −0.618034 −0.0243540
\(645\) 0 0
\(646\) 11.2918 0.444270
\(647\) 36.2492 1.42510 0.712552 0.701619i \(-0.247539\pi\)
0.712552 + 0.701619i \(0.247539\pi\)
\(648\) 7.70820 0.302807
\(649\) −4.87539 −0.191376
\(650\) 0 0
\(651\) −9.09017 −0.356272
\(652\) −1.38197 −0.0541220
\(653\) 8.03444 0.314412 0.157206 0.987566i \(-0.449751\pi\)
0.157206 + 0.987566i \(0.449751\pi\)
\(654\) −18.7082 −0.731549
\(655\) 0 0
\(656\) 3.32624 0.129868
\(657\) 1.23607 0.0482236
\(658\) −2.29180 −0.0893435
\(659\) −46.2492 −1.80161 −0.900807 0.434220i \(-0.857024\pi\)
−0.900807 + 0.434220i \(0.857024\pi\)
\(660\) 0 0
\(661\) 18.6738 0.726325 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(662\) −19.2361 −0.747631
\(663\) 69.8673 2.71342
\(664\) −6.94427 −0.269490
\(665\) 0 0
\(666\) −2.47214 −0.0957933
\(667\) 9.23607 0.357622
\(668\) 8.00000 0.309529
\(669\) −33.8885 −1.31021
\(670\) 0 0
\(671\) 26.6180 1.02758
\(672\) 1.00000 0.0385758
\(673\) 10.9443 0.421871 0.210935 0.977500i \(-0.432349\pi\)
0.210935 + 0.977500i \(0.432349\pi\)
\(674\) 13.6738 0.526694
\(675\) 0 0
\(676\) 37.2705 1.43348
\(677\) 50.9443 1.95795 0.978974 0.203986i \(-0.0653899\pi\)
0.978974 + 0.203986i \(0.0653899\pi\)
\(678\) −1.70820 −0.0656032
\(679\) −7.65248 −0.293675
\(680\) 0 0
\(681\) −30.3607 −1.16342
\(682\) 25.9443 0.993458
\(683\) 31.5623 1.20770 0.603849 0.797099i \(-0.293633\pi\)
0.603849 + 0.797099i \(0.293633\pi\)
\(684\) −0.708204 −0.0270789
\(685\) 0 0
\(686\) 8.41641 0.321340
\(687\) −16.1803 −0.617318
\(688\) 0 0
\(689\) −3.34752 −0.127531
\(690\) 0 0
\(691\) −29.2361 −1.11219 −0.556096 0.831118i \(-0.687701\pi\)
−0.556096 + 0.831118i \(0.687701\pi\)
\(692\) 1.43769 0.0546529
\(693\) 0.673762 0.0255941
\(694\) −6.38197 −0.242256
\(695\) 0 0
\(696\) −14.9443 −0.566461
\(697\) −20.2574 −0.767302
\(698\) 2.00000 0.0757011
\(699\) 10.1803 0.385056
\(700\) 0 0
\(701\) 43.3394 1.63691 0.818453 0.574573i \(-0.194832\pi\)
0.818453 + 0.574573i \(0.194832\pi\)
\(702\) −38.7984 −1.46435
\(703\) −12.0000 −0.452589
\(704\) −2.85410 −0.107568
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) −0.180340 −0.00678238
\(708\) −2.76393 −0.103875
\(709\) −26.0902 −0.979837 −0.489918 0.871768i \(-0.662973\pi\)
−0.489918 + 0.871768i \(0.662973\pi\)
\(710\) 0 0
\(711\) −0.583592 −0.0218864
\(712\) 10.4721 0.392460
\(713\) −9.09017 −0.340430
\(714\) −6.09017 −0.227919
\(715\) 0 0
\(716\) 2.18034 0.0814831
\(717\) 32.9443 1.23033
\(718\) −26.3607 −0.983772
\(719\) 35.2705 1.31537 0.657684 0.753294i \(-0.271536\pi\)
0.657684 + 0.753294i \(0.271536\pi\)
\(720\) 0 0
\(721\) −10.2361 −0.381211
\(722\) 15.5623 0.579169
\(723\) 0 0
\(724\) −12.1459 −0.451399
\(725\) 0 0
\(726\) −4.61803 −0.171391
\(727\) −28.2016 −1.04594 −0.522970 0.852351i \(-0.675176\pi\)
−0.522970 + 0.852351i \(0.675176\pi\)
\(728\) −4.38197 −0.162406
\(729\) 29.5066 1.09284
\(730\) 0 0
\(731\) 0 0
\(732\) 15.0902 0.557749
\(733\) 29.4164 1.08652 0.543260 0.839565i \(-0.317190\pi\)
0.543260 + 0.839565i \(0.317190\pi\)
\(734\) 6.47214 0.238891
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 41.3050 1.52149
\(738\) 1.27051 0.0467681
\(739\) −13.8885 −0.510898 −0.255449 0.966822i \(-0.582223\pi\)
−0.255449 + 0.966822i \(0.582223\pi\)
\(740\) 0 0
\(741\) −21.2705 −0.781392
\(742\) 0.291796 0.0107122
\(743\) −33.6312 −1.23381 −0.616904 0.787038i \(-0.711613\pi\)
−0.616904 + 0.787038i \(0.711613\pi\)
\(744\) 14.7082 0.539229
\(745\) 0 0
\(746\) 20.1803 0.738855
\(747\) −2.65248 −0.0970490
\(748\) 17.3820 0.635548
\(749\) −11.2361 −0.410557
\(750\) 0 0
\(751\) −47.0132 −1.71553 −0.857767 0.514038i \(-0.828149\pi\)
−0.857767 + 0.514038i \(0.828149\pi\)
\(752\) 3.70820 0.135224
\(753\) −9.94427 −0.362389
\(754\) 65.4853 2.38483
\(755\) 0 0
\(756\) 3.38197 0.123001
\(757\) 17.8885 0.650170 0.325085 0.945685i \(-0.394607\pi\)
0.325085 + 0.945685i \(0.394607\pi\)
\(758\) 22.4508 0.815452
\(759\) −4.61803 −0.167624
\(760\) 0 0
\(761\) 46.8673 1.69894 0.849468 0.527640i \(-0.176923\pi\)
0.849468 + 0.527640i \(0.176923\pi\)
\(762\) −26.1803 −0.948414
\(763\) −7.14590 −0.258699
\(764\) −13.7082 −0.495945
\(765\) 0 0
\(766\) −17.8885 −0.646339
\(767\) 12.1115 0.437319
\(768\) −1.61803 −0.0583858
\(769\) −6.58359 −0.237410 −0.118705 0.992930i \(-0.537874\pi\)
−0.118705 + 0.992930i \(0.537874\pi\)
\(770\) 0 0
\(771\) −12.6525 −0.455668
\(772\) −0.763932 −0.0274945
\(773\) −28.9443 −1.04105 −0.520527 0.853845i \(-0.674264\pi\)
−0.520527 + 0.853845i \(0.674264\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.3820 0.444487
\(777\) 6.47214 0.232187
\(778\) −21.3262 −0.764583
\(779\) 6.16718 0.220962
\(780\) 0 0
\(781\) 11.6738 0.417720
\(782\) −6.09017 −0.217784
\(783\) −50.5410 −1.80619
\(784\) −6.61803 −0.236358
\(785\) 0 0
\(786\) 4.76393 0.169924
\(787\) 2.87539 0.102497 0.0512483 0.998686i \(-0.483680\pi\)
0.0512483 + 0.998686i \(0.483680\pi\)
\(788\) 22.5623 0.803749
\(789\) 33.5623 1.19485
\(790\) 0 0
\(791\) −0.652476 −0.0231994
\(792\) −1.09017 −0.0387375
\(793\) −66.1246 −2.34815
\(794\) 7.32624 0.259998
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) −13.7082 −0.485569 −0.242785 0.970080i \(-0.578061\pi\)
−0.242785 + 0.970080i \(0.578061\pi\)
\(798\) 1.85410 0.0656345
\(799\) −22.5836 −0.798950
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 1.70820 0.0603188
\(803\) 9.23607 0.325934
\(804\) 23.4164 0.825833
\(805\) 0 0
\(806\) −64.4508 −2.27018
\(807\) 22.9443 0.807677
\(808\) 0.291796 0.0102653
\(809\) −46.7426 −1.64338 −0.821692 0.569932i \(-0.806970\pi\)
−0.821692 + 0.569932i \(0.806970\pi\)
\(810\) 0 0
\(811\) 21.8197 0.766192 0.383096 0.923709i \(-0.374858\pi\)
0.383096 + 0.923709i \(0.374858\pi\)
\(812\) −5.70820 −0.200319
\(813\) 49.0689 1.72092
\(814\) −18.4721 −0.647448
\(815\) 0 0
\(816\) 9.85410 0.344963
\(817\) 0 0
\(818\) 30.2148 1.05644
\(819\) −1.67376 −0.0584860
\(820\) 0 0
\(821\) −33.0557 −1.15365 −0.576826 0.816867i \(-0.695709\pi\)
−0.576826 + 0.816867i \(0.695709\pi\)
\(822\) 16.7082 0.582766
\(823\) −25.4164 −0.885960 −0.442980 0.896531i \(-0.646079\pi\)
−0.442980 + 0.896531i \(0.646079\pi\)
\(824\) 16.5623 0.576975
\(825\) 0 0
\(826\) −1.05573 −0.0367335
\(827\) −21.7082 −0.754868 −0.377434 0.926036i \(-0.623194\pi\)
−0.377434 + 0.926036i \(0.623194\pi\)
\(828\) 0.381966 0.0132742
\(829\) 18.9443 0.657962 0.328981 0.944337i \(-0.393295\pi\)
0.328981 + 0.944337i \(0.393295\pi\)
\(830\) 0 0
\(831\) 47.5967 1.65111
\(832\) 7.09017 0.245807
\(833\) 40.3050 1.39648
\(834\) 20.6525 0.715137
\(835\) 0 0
\(836\) −5.29180 −0.183021
\(837\) 49.7426 1.71936
\(838\) −14.4721 −0.499932
\(839\) 33.0132 1.13974 0.569870 0.821735i \(-0.306993\pi\)
0.569870 + 0.821735i \(0.306993\pi\)
\(840\) 0 0
\(841\) 56.3050 1.94155
\(842\) −13.7426 −0.473603
\(843\) −36.8328 −1.26859
\(844\) 14.0000 0.481900
\(845\) 0 0
\(846\) 1.41641 0.0486971
\(847\) −1.76393 −0.0606094
\(848\) −0.472136 −0.0162132
\(849\) −43.5967 −1.49624
\(850\) 0 0
\(851\) 6.47214 0.221862
\(852\) 6.61803 0.226730
\(853\) 10.7984 0.369729 0.184865 0.982764i \(-0.440815\pi\)
0.184865 + 0.982764i \(0.440815\pi\)
\(854\) 5.76393 0.197238
\(855\) 0 0
\(856\) 18.1803 0.621391
\(857\) 6.58359 0.224891 0.112446 0.993658i \(-0.464132\pi\)
0.112446 + 0.993658i \(0.464132\pi\)
\(858\) −32.7426 −1.11782
\(859\) −24.0689 −0.821220 −0.410610 0.911811i \(-0.634684\pi\)
−0.410610 + 0.911811i \(0.634684\pi\)
\(860\) 0 0
\(861\) −3.32624 −0.113358
\(862\) −3.34752 −0.114017
\(863\) −32.7639 −1.11530 −0.557649 0.830077i \(-0.688296\pi\)
−0.557649 + 0.830077i \(0.688296\pi\)
\(864\) −5.47214 −0.186166
\(865\) 0 0
\(866\) 8.50658 0.289065
\(867\) −32.5066 −1.10398
\(868\) 5.61803 0.190688
\(869\) −4.36068 −0.147926
\(870\) 0 0
\(871\) −102.610 −3.47680
\(872\) 11.5623 0.391549
\(873\) 4.72949 0.160069
\(874\) 1.85410 0.0627159
\(875\) 0 0
\(876\) 5.23607 0.176910
\(877\) −18.7426 −0.632894 −0.316447 0.948610i \(-0.602490\pi\)
−0.316447 + 0.948610i \(0.602490\pi\)
\(878\) 13.3820 0.451619
\(879\) −32.1803 −1.08542
\(880\) 0 0
\(881\) 8.58359 0.289189 0.144594 0.989491i \(-0.453812\pi\)
0.144594 + 0.989491i \(0.453812\pi\)
\(882\) −2.52786 −0.0851176
\(883\) −15.5623 −0.523713 −0.261857 0.965107i \(-0.584335\pi\)
−0.261857 + 0.965107i \(0.584335\pi\)
\(884\) −43.1803 −1.45231
\(885\) 0 0
\(886\) 25.0902 0.842921
\(887\) 5.16718 0.173497 0.0867485 0.996230i \(-0.472352\pi\)
0.0867485 + 0.996230i \(0.472352\pi\)
\(888\) −10.4721 −0.351422
\(889\) −10.0000 −0.335389
\(890\) 0 0
\(891\) 22.0000 0.737028
\(892\) 20.9443 0.701266
\(893\) 6.87539 0.230076
\(894\) −12.7082 −0.425026
\(895\) 0 0
\(896\) −0.618034 −0.0206471
\(897\) 11.4721 0.383043
\(898\) 1.56231 0.0521348
\(899\) −83.9574 −2.80014
\(900\) 0 0
\(901\) 2.87539 0.0957931
\(902\) 9.49342 0.316096
\(903\) 0 0
\(904\) 1.05573 0.0351130
\(905\) 0 0
\(906\) −4.14590 −0.137738
\(907\) −7.12461 −0.236569 −0.118284 0.992980i \(-0.537739\pi\)
−0.118284 + 0.992980i \(0.537739\pi\)
\(908\) 18.7639 0.622703
\(909\) 0.111456 0.00369677
\(910\) 0 0
\(911\) 36.0689 1.19502 0.597508 0.801863i \(-0.296158\pi\)
0.597508 + 0.801863i \(0.296158\pi\)
\(912\) −3.00000 −0.0993399
\(913\) −19.8197 −0.655935
\(914\) −37.7771 −1.24955
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 1.81966 0.0600905
\(918\) 33.3262 1.09993
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −46.0344 −1.51689
\(922\) 39.2361 1.29217
\(923\) −29.0000 −0.954547
\(924\) 2.85410 0.0938931
\(925\) 0 0
\(926\) 2.00000 0.0657241
\(927\) 6.32624 0.207781
\(928\) 9.23607 0.303189
\(929\) −3.52786 −0.115745 −0.0578727 0.998324i \(-0.518432\pi\)
−0.0578727 + 0.998324i \(0.518432\pi\)
\(930\) 0 0
\(931\) −12.2705 −0.402150
\(932\) −6.29180 −0.206095
\(933\) −6.47214 −0.211888
\(934\) −17.1246 −0.560334
\(935\) 0 0
\(936\) 2.70820 0.0885204
\(937\) 36.7984 1.20215 0.601075 0.799192i \(-0.294739\pi\)
0.601075 + 0.799192i \(0.294739\pi\)
\(938\) 8.94427 0.292041
\(939\) 20.7082 0.675787
\(940\) 0 0
\(941\) −22.4934 −0.733265 −0.366632 0.930366i \(-0.619489\pi\)
−0.366632 + 0.930366i \(0.619489\pi\)
\(942\) 6.00000 0.195491
\(943\) −3.32624 −0.108317
\(944\) 1.70820 0.0555973
\(945\) 0 0
\(946\) 0 0
\(947\) 54.6869 1.77709 0.888543 0.458793i \(-0.151718\pi\)
0.888543 + 0.458793i \(0.151718\pi\)
\(948\) −2.47214 −0.0802912
\(949\) −22.9443 −0.744803
\(950\) 0 0
\(951\) −17.9443 −0.581883
\(952\) 3.76393 0.121990
\(953\) −3.79837 −0.123041 −0.0615207 0.998106i \(-0.519595\pi\)
−0.0615207 + 0.998106i \(0.519595\pi\)
\(954\) −0.180340 −0.00583872
\(955\) 0 0
\(956\) −20.3607 −0.658511
\(957\) −42.6525 −1.37876
\(958\) 31.8885 1.03027
\(959\) 6.38197 0.206084
\(960\) 0 0
\(961\) 51.6312 1.66552
\(962\) 45.8885 1.47951
\(963\) 6.94427 0.223776
\(964\) 0 0
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) 16.5410 0.531923 0.265962 0.963984i \(-0.414311\pi\)
0.265962 + 0.963984i \(0.414311\pi\)
\(968\) 2.85410 0.0917343
\(969\) 18.2705 0.586933
\(970\) 0 0
\(971\) 34.2705 1.09979 0.549896 0.835233i \(-0.314667\pi\)
0.549896 + 0.835233i \(0.314667\pi\)
\(972\) −3.94427 −0.126513
\(973\) 7.88854 0.252895
\(974\) −19.8197 −0.635063
\(975\) 0 0
\(976\) −9.32624 −0.298526
\(977\) 23.5623 0.753825 0.376912 0.926249i \(-0.376986\pi\)
0.376912 + 0.926249i \(0.376986\pi\)
\(978\) −2.23607 −0.0715016
\(979\) 29.8885 0.955242
\(980\) 0 0
\(981\) 4.41641 0.141005
\(982\) −6.18034 −0.197223
\(983\) 14.2705 0.455159 0.227579 0.973760i \(-0.426919\pi\)
0.227579 + 0.973760i \(0.426919\pi\)
\(984\) 5.38197 0.171571
\(985\) 0 0
\(986\) −56.2492 −1.79134
\(987\) −3.70820 −0.118033
\(988\) 13.1459 0.418227
\(989\) 0 0
\(990\) 0 0
\(991\) 27.5066 0.873775 0.436888 0.899516i \(-0.356081\pi\)
0.436888 + 0.899516i \(0.356081\pi\)
\(992\) −9.09017 −0.288613
\(993\) −31.1246 −0.987710
\(994\) 2.52786 0.0801790
\(995\) 0 0
\(996\) −11.2361 −0.356028
\(997\) −57.1935 −1.81134 −0.905668 0.423987i \(-0.860630\pi\)
−0.905668 + 0.423987i \(0.860630\pi\)
\(998\) −12.3607 −0.391270
\(999\) −35.4164 −1.12053
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 1150.2.a.j.1.1 2
4.3 odd 2 9200.2.a.bu.1.2 2
5.2 odd 4 1150.2.b.i.599.2 4
5.3 odd 4 1150.2.b.i.599.3 4
5.4 even 2 230.2.a.c.1.2 2
15.14 odd 2 2070.2.a.u.1.1 2
20.19 odd 2 1840.2.a.l.1.1 2
40.19 odd 2 7360.2.a.bn.1.2 2
40.29 even 2 7360.2.a.bh.1.1 2
115.114 odd 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 5.4 even 2
1150.2.a.j.1.1 2 1.1 even 1 trivial
1150.2.b.i.599.2 4 5.2 odd 4
1150.2.b.i.599.3 4 5.3 odd 4
1840.2.a.l.1.1 2 20.19 odd 2
2070.2.a.u.1.1 2 15.14 odd 2
5290.2.a.o.1.2 2 115.114 odd 2
7360.2.a.bh.1.1 2 40.29 even 2
7360.2.a.bn.1.2 2 40.19 odd 2
9200.2.a.bu.1.2 2 4.3 odd 2