Properties

Label 475.2.a.i.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
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] = [475,2,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.78165\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.95594 q^{2} -0.296842 q^{3} +1.82571 q^{4} +0.580605 q^{6} +3.56331 q^{7} +0.340899 q^{8} -2.91188 q^{9} +O(q^{10})\) \(q-1.95594 q^{2} -0.296842 q^{3} +1.82571 q^{4} +0.580605 q^{6} +3.56331 q^{7} +0.340899 q^{8} -2.91188 q^{9} +5.56331 q^{11} -0.541947 q^{12} -5.26647 q^{13} -6.96962 q^{14} -4.31820 q^{16} -1.40632 q^{17} +5.69548 q^{18} +1.00000 q^{19} -1.05774 q^{21} -10.8815 q^{22} +6.96962 q^{23} -0.101193 q^{24} +10.3009 q^{26} +1.75489 q^{27} +6.50557 q^{28} +1.40632 q^{29} +1.75489 q^{31} +7.76435 q^{32} -1.65142 q^{33} +2.75067 q^{34} -5.31626 q^{36} -3.61504 q^{37} -1.95594 q^{38} +1.56331 q^{39} +4.34858 q^{41} +2.06888 q^{42} +3.56331 q^{43} +10.1570 q^{44} -13.6322 q^{46} +8.26046 q^{47} +1.28182 q^{48} +5.69716 q^{49} +0.417453 q^{51} -9.61504 q^{52} +7.61504 q^{53} -3.43247 q^{54} +1.21473 q^{56} -0.296842 q^{57} -2.75067 q^{58} +9.47519 q^{59} +9.21473 q^{61} -3.43247 q^{62} -10.3759 q^{63} -6.55023 q^{64} +3.23009 q^{66} +4.76090 q^{67} -2.56753 q^{68} -2.06888 q^{69} -14.0689 q^{71} -0.992660 q^{72} -6.59368 q^{73} +7.07082 q^{74} +1.82571 q^{76} +19.8238 q^{77} -3.05774 q^{78} +5.47519 q^{79} +8.21473 q^{81} -8.50557 q^{82} +4.15699 q^{83} -1.93112 q^{84} -6.96962 q^{86} -0.417453 q^{87} +1.89653 q^{88} -9.23009 q^{89} -18.7660 q^{91} +12.7245 q^{92} -0.520926 q^{93} -16.1570 q^{94} -2.30478 q^{96} -11.5116 q^{97} -11.1433 q^{98} -16.1997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} + 4 q^{27} + 8 q^{28} + 4 q^{29} + 4 q^{31} + 6 q^{32} - 8 q^{33} - 4 q^{34} + 40 q^{36} + 6 q^{37} + 2 q^{38} - 12 q^{39} + 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} - 38 q^{48} + 20 q^{49} - 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 12 q^{56} - 2 q^{57} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} + 18 q^{67} - 4 q^{68} + 28 q^{69} - 20 q^{71} + 52 q^{72} - 28 q^{73} + 32 q^{74} + 8 q^{76} + 40 q^{77} - 12 q^{78} - 16 q^{79} + 16 q^{81} - 16 q^{82} - 44 q^{84} - 8 q^{86} + 36 q^{87} + 12 q^{88} + 4 q^{89} - 36 q^{91} + 28 q^{92} + 40 q^{93} - 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.95594 −1.38306 −0.691530 0.722348i \(-0.743063\pi\)
−0.691530 + 0.722348i \(0.743063\pi\)
\(3\) −0.296842 −0.171382 −0.0856908 0.996322i \(-0.527310\pi\)
−0.0856908 + 0.996322i \(0.527310\pi\)
\(4\) 1.82571 0.912855
\(5\) 0 0
\(6\) 0.580605 0.237031
\(7\) 3.56331 1.34680 0.673402 0.739277i \(-0.264832\pi\)
0.673402 + 0.739277i \(0.264832\pi\)
\(8\) 0.340899 0.120526
\(9\) −2.91188 −0.970628
\(10\) 0 0
\(11\) 5.56331 1.67740 0.838700 0.544594i \(-0.183316\pi\)
0.838700 + 0.544594i \(0.183316\pi\)
\(12\) −0.541947 −0.156447
\(13\) −5.26647 −1.46065 −0.730327 0.683097i \(-0.760632\pi\)
−0.730327 + 0.683097i \(0.760632\pi\)
\(14\) −6.96962 −1.86271
\(15\) 0 0
\(16\) −4.31820 −1.07955
\(17\) −1.40632 −0.341082 −0.170541 0.985351i \(-0.554552\pi\)
−0.170541 + 0.985351i \(0.554552\pi\)
\(18\) 5.69548 1.34244
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.05774 −0.230817
\(22\) −10.8815 −2.31995
\(23\) 6.96962 1.45327 0.726633 0.687025i \(-0.241084\pi\)
0.726633 + 0.687025i \(0.241084\pi\)
\(24\) −0.101193 −0.0206560
\(25\) 0 0
\(26\) 10.3009 2.02017
\(27\) 1.75489 0.337730
\(28\) 6.50557 1.22944
\(29\) 1.40632 0.261146 0.130573 0.991439i \(-0.458318\pi\)
0.130573 + 0.991439i \(0.458318\pi\)
\(30\) 0 0
\(31\) 1.75489 0.315188 0.157594 0.987504i \(-0.449626\pi\)
0.157594 + 0.987504i \(0.449626\pi\)
\(32\) 7.76435 1.37256
\(33\) −1.65142 −0.287476
\(34\) 2.75067 0.471737
\(35\) 0 0
\(36\) −5.31626 −0.886043
\(37\) −3.61504 −0.594309 −0.297155 0.954829i \(-0.596038\pi\)
−0.297155 + 0.954829i \(0.596038\pi\)
\(38\) −1.95594 −0.317296
\(39\) 1.56331 0.250329
\(40\) 0 0
\(41\) 4.34858 0.679134 0.339567 0.940582i \(-0.389720\pi\)
0.339567 + 0.940582i \(0.389720\pi\)
\(42\) 2.06888 0.319234
\(43\) 3.56331 0.543399 0.271700 0.962382i \(-0.412414\pi\)
0.271700 + 0.962382i \(0.412414\pi\)
\(44\) 10.1570 1.53122
\(45\) 0 0
\(46\) −13.6322 −2.00996
\(47\) 8.26046 1.20491 0.602456 0.798152i \(-0.294189\pi\)
0.602456 + 0.798152i \(0.294189\pi\)
\(48\) 1.28182 0.185015
\(49\) 5.69716 0.813879
\(50\) 0 0
\(51\) 0.417453 0.0584552
\(52\) −9.61504 −1.33337
\(53\) 7.61504 1.04601 0.523003 0.852331i \(-0.324812\pi\)
0.523003 + 0.852331i \(0.324812\pi\)
\(54\) −3.43247 −0.467100
\(55\) 0 0
\(56\) 1.21473 0.162325
\(57\) −0.296842 −0.0393177
\(58\) −2.75067 −0.361181
\(59\) 9.47519 1.23356 0.616782 0.787134i \(-0.288436\pi\)
0.616782 + 0.787134i \(0.288436\pi\)
\(60\) 0 0
\(61\) 9.21473 1.17983 0.589913 0.807467i \(-0.299162\pi\)
0.589913 + 0.807467i \(0.299162\pi\)
\(62\) −3.43247 −0.435924
\(63\) −10.3759 −1.30725
\(64\) −6.55023 −0.818779
\(65\) 0 0
\(66\) 3.23009 0.397596
\(67\) 4.76090 0.581636 0.290818 0.956778i \(-0.406073\pi\)
0.290818 + 0.956778i \(0.406073\pi\)
\(68\) −2.56753 −0.311358
\(69\) −2.06888 −0.249063
\(70\) 0 0
\(71\) −14.0689 −1.66967 −0.834834 0.550502i \(-0.814437\pi\)
−0.834834 + 0.550502i \(0.814437\pi\)
\(72\) −0.992660 −0.116986
\(73\) −6.59368 −0.771732 −0.385866 0.922555i \(-0.626097\pi\)
−0.385866 + 0.922555i \(0.626097\pi\)
\(74\) 7.07082 0.821966
\(75\) 0 0
\(76\) 1.82571 0.209423
\(77\) 19.8238 2.25913
\(78\) −3.05774 −0.346221
\(79\) 5.47519 0.616007 0.308004 0.951385i \(-0.400339\pi\)
0.308004 + 0.951385i \(0.400339\pi\)
\(80\) 0 0
\(81\) 8.21473 0.912748
\(82\) −8.50557 −0.939283
\(83\) 4.15699 0.456289 0.228144 0.973627i \(-0.426734\pi\)
0.228144 + 0.973627i \(0.426734\pi\)
\(84\) −1.93112 −0.210703
\(85\) 0 0
\(86\) −6.96962 −0.751554
\(87\) −0.417453 −0.0447557
\(88\) 1.89653 0.202171
\(89\) −9.23009 −0.978387 −0.489194 0.872175i \(-0.662709\pi\)
−0.489194 + 0.872175i \(0.662709\pi\)
\(90\) 0 0
\(91\) −18.7660 −1.96721
\(92\) 12.7245 1.32662
\(93\) −0.520926 −0.0540175
\(94\) −16.1570 −1.66647
\(95\) 0 0
\(96\) −2.30478 −0.235231
\(97\) −11.5116 −1.16882 −0.584411 0.811457i \(-0.698675\pi\)
−0.584411 + 0.811457i \(0.698675\pi\)
\(98\) −11.1433 −1.12564
\(99\) −16.1997 −1.62813
\(100\) 0 0
\(101\) −11.8511 −1.17923 −0.589616 0.807684i \(-0.700721\pi\)
−0.589616 + 0.807684i \(0.700721\pi\)
\(102\) −0.816515 −0.0808470
\(103\) −1.35458 −0.133471 −0.0667354 0.997771i \(-0.521258\pi\)
−0.0667354 + 0.997771i \(0.521258\pi\)
\(104\) −1.79533 −0.176047
\(105\) 0 0
\(106\) −14.8946 −1.44669
\(107\) 7.06287 0.682794 0.341397 0.939919i \(-0.389100\pi\)
0.341397 + 0.939919i \(0.389100\pi\)
\(108\) 3.20393 0.308298
\(109\) 10.1844 0.975484 0.487742 0.872988i \(-0.337821\pi\)
0.487742 + 0.872988i \(0.337821\pi\)
\(110\) 0 0
\(111\) 1.07310 0.101854
\(112\) −15.3871 −1.45394
\(113\) −1.86015 −0.174988 −0.0874940 0.996165i \(-0.527886\pi\)
−0.0874940 + 0.996165i \(0.527886\pi\)
\(114\) 0.580605 0.0543787
\(115\) 0 0
\(116\) 2.56753 0.238389
\(117\) 15.3353 1.41775
\(118\) −18.5329 −1.70609
\(119\) −5.01114 −0.459370
\(120\) 0 0
\(121\) 19.9504 1.81367
\(122\) −18.0235 −1.63177
\(123\) −1.29084 −0.116391
\(124\) 3.20393 0.287721
\(125\) 0 0
\(126\) 20.2947 1.80800
\(127\) 4.89053 0.433964 0.216982 0.976176i \(-0.430379\pi\)
0.216982 + 0.976176i \(0.430379\pi\)
\(128\) −2.71684 −0.240137
\(129\) −1.05774 −0.0931287
\(130\) 0 0
\(131\) 2.81263 0.245741 0.122870 0.992423i \(-0.460790\pi\)
0.122870 + 0.992423i \(0.460790\pi\)
\(132\) −3.01502 −0.262424
\(133\) 3.56331 0.308978
\(134\) −9.31204 −0.804438
\(135\) 0 0
\(136\) −0.479412 −0.0411093
\(137\) −9.23009 −0.788579 −0.394290 0.918986i \(-0.629009\pi\)
−0.394290 + 0.918986i \(0.629009\pi\)
\(138\) 4.04660 0.344470
\(139\) −3.67878 −0.312030 −0.156015 0.987755i \(-0.549865\pi\)
−0.156015 + 0.987755i \(0.549865\pi\)
\(140\) 0 0
\(141\) −2.45205 −0.206500
\(142\) 27.5179 2.30925
\(143\) −29.2990 −2.45010
\(144\) 12.5741 1.04784
\(145\) 0 0
\(146\) 12.8969 1.06735
\(147\) −1.69115 −0.139484
\(148\) −6.60002 −0.542519
\(149\) 7.09925 0.581593 0.290797 0.956785i \(-0.406080\pi\)
0.290797 + 0.956785i \(0.406080\pi\)
\(150\) 0 0
\(151\) −18.3567 −1.49385 −0.746924 0.664910i \(-0.768470\pi\)
−0.746924 + 0.664910i \(0.768470\pi\)
\(152\) 0.340899 0.0276506
\(153\) 4.09503 0.331064
\(154\) −38.7742 −3.12451
\(155\) 0 0
\(156\) 2.85415 0.228515
\(157\) −17.2301 −1.37511 −0.687555 0.726132i \(-0.741316\pi\)
−0.687555 + 0.726132i \(0.741316\pi\)
\(158\) −10.7092 −0.851975
\(159\) −2.26046 −0.179266
\(160\) 0 0
\(161\) 24.8349 1.95726
\(162\) −16.0675 −1.26238
\(163\) −10.8662 −0.851103 −0.425551 0.904934i \(-0.639920\pi\)
−0.425551 + 0.904934i \(0.639920\pi\)
\(164\) 7.93925 0.619951
\(165\) 0 0
\(166\) −8.13083 −0.631075
\(167\) −2.82977 −0.218974 −0.109487 0.993988i \(-0.534921\pi\)
−0.109487 + 0.993988i \(0.534921\pi\)
\(168\) −0.360582 −0.0278195
\(169\) 14.7357 1.13351
\(170\) 0 0
\(171\) −2.91188 −0.222677
\(172\) 6.50557 0.496045
\(173\) 9.26647 0.704516 0.352258 0.935903i \(-0.385414\pi\)
0.352258 + 0.935903i \(0.385414\pi\)
\(174\) 0.816515 0.0618998
\(175\) 0 0
\(176\) −24.0235 −1.81084
\(177\) −2.81263 −0.211410
\(178\) 18.0535 1.35317
\(179\) −3.59067 −0.268379 −0.134190 0.990956i \(-0.542843\pi\)
−0.134190 + 0.990956i \(0.542843\pi\)
\(180\) 0 0
\(181\) −19.7630 −1.46897 −0.734487 0.678623i \(-0.762577\pi\)
−0.734487 + 0.678623i \(0.762577\pi\)
\(182\) 36.7053 2.72078
\(183\) −2.73532 −0.202200
\(184\) 2.37594 0.175157
\(185\) 0 0
\(186\) 1.01890 0.0747095
\(187\) −7.82377 −0.572131
\(188\) 15.0812 1.09991
\(189\) 6.25323 0.454855
\(190\) 0 0
\(191\) 8.31398 0.601579 0.300789 0.953691i \(-0.402750\pi\)
0.300789 + 0.953691i \(0.402750\pi\)
\(192\) 1.94438 0.140324
\(193\) 22.2514 1.60169 0.800847 0.598869i \(-0.204383\pi\)
0.800847 + 0.598869i \(0.204383\pi\)
\(194\) 22.5160 1.61655
\(195\) 0 0
\(196\) 10.4014 0.742954
\(197\) 8.81263 0.627874 0.313937 0.949444i \(-0.398352\pi\)
0.313937 + 0.949444i \(0.398352\pi\)
\(198\) 31.6857 2.25180
\(199\) 21.0659 1.49332 0.746660 0.665206i \(-0.231656\pi\)
0.746660 + 0.665206i \(0.231656\pi\)
\(200\) 0 0
\(201\) −1.41323 −0.0996818
\(202\) 23.1801 1.63095
\(203\) 5.01114 0.351713
\(204\) 0.762149 0.0533611
\(205\) 0 0
\(206\) 2.64948 0.184598
\(207\) −20.2947 −1.41058
\(208\) 22.7417 1.57685
\(209\) 5.56331 0.384822
\(210\) 0 0
\(211\) 5.34556 0.368004 0.184002 0.982926i \(-0.441095\pi\)
0.184002 + 0.982926i \(0.441095\pi\)
\(212\) 13.9029 0.954853
\(213\) 4.17623 0.286151
\(214\) −13.8146 −0.944345
\(215\) 0 0
\(216\) 0.598242 0.0407052
\(217\) 6.25323 0.424497
\(218\) −19.9200 −1.34915
\(219\) 1.95728 0.132261
\(220\) 0 0
\(221\) 7.40632 0.498203
\(222\) −2.09891 −0.140870
\(223\) 3.10947 0.208226 0.104113 0.994565i \(-0.466800\pi\)
0.104113 + 0.994565i \(0.466800\pi\)
\(224\) 27.6668 1.84856
\(225\) 0 0
\(226\) 3.63834 0.242019
\(227\) −14.4692 −0.960354 −0.480177 0.877172i \(-0.659428\pi\)
−0.480177 + 0.877172i \(0.659428\pi\)
\(228\) −0.541947 −0.0358913
\(229\) −5.21473 −0.344599 −0.172299 0.985045i \(-0.555120\pi\)
−0.172299 + 0.985045i \(0.555120\pi\)
\(230\) 0 0
\(231\) −5.88452 −0.387173
\(232\) 0.479412 0.0314750
\(233\) 3.18737 0.208811 0.104406 0.994535i \(-0.466706\pi\)
0.104406 + 0.994535i \(0.466706\pi\)
\(234\) −29.9950 −1.96084
\(235\) 0 0
\(236\) 17.2990 1.12607
\(237\) −1.62527 −0.105572
\(238\) 9.80150 0.635337
\(239\) 16.5209 1.06865 0.534325 0.845279i \(-0.320566\pi\)
0.534325 + 0.845279i \(0.320566\pi\)
\(240\) 0 0
\(241\) −12.2271 −0.787615 −0.393807 0.919193i \(-0.628842\pi\)
−0.393807 + 0.919193i \(0.628842\pi\)
\(242\) −39.0218 −2.50842
\(243\) −7.70316 −0.494158
\(244\) 16.8234 1.07701
\(245\) 0 0
\(246\) 2.52481 0.160976
\(247\) −5.26647 −0.335097
\(248\) 0.598242 0.0379884
\(249\) −1.23397 −0.0781996
\(250\) 0 0
\(251\) 4.52093 0.285358 0.142679 0.989769i \(-0.454428\pi\)
0.142679 + 0.989769i \(0.454428\pi\)
\(252\) −18.9435 −1.19333
\(253\) 38.7742 2.43771
\(254\) −9.56559 −0.600198
\(255\) 0 0
\(256\) 18.4144 1.15090
\(257\) −15.9290 −0.993625 −0.496813 0.867858i \(-0.665496\pi\)
−0.496813 + 0.867858i \(0.665496\pi\)
\(258\) 2.06888 0.128803
\(259\) −12.8815 −0.800418
\(260\) 0 0
\(261\) −4.09503 −0.253476
\(262\) −5.50135 −0.339874
\(263\) −0.854147 −0.0526689 −0.0263345 0.999653i \(-0.508383\pi\)
−0.0263345 + 0.999653i \(0.508383\pi\)
\(264\) −0.562969 −0.0346483
\(265\) 0 0
\(266\) −6.96962 −0.427335
\(267\) 2.73988 0.167678
\(268\) 8.69202 0.530950
\(269\) 10.3913 0.633569 0.316784 0.948498i \(-0.397397\pi\)
0.316784 + 0.948498i \(0.397397\pi\)
\(270\) 0 0
\(271\) −8.19971 −0.498097 −0.249048 0.968491i \(-0.580118\pi\)
−0.249048 + 0.968491i \(0.580118\pi\)
\(272\) 6.07276 0.368215
\(273\) 5.57054 0.337145
\(274\) 18.0535 1.09065
\(275\) 0 0
\(276\) −3.77717 −0.227359
\(277\) −14.6484 −0.880137 −0.440069 0.897964i \(-0.645046\pi\)
−0.440069 + 0.897964i \(0.645046\pi\)
\(278\) 7.19549 0.431557
\(279\) −5.11005 −0.305931
\(280\) 0 0
\(281\) −31.6129 −1.88587 −0.942935 0.332977i \(-0.891947\pi\)
−0.942935 + 0.332977i \(0.891947\pi\)
\(282\) 4.79607 0.285602
\(283\) −24.7326 −1.47020 −0.735101 0.677957i \(-0.762865\pi\)
−0.735101 + 0.677957i \(0.762865\pi\)
\(284\) −25.6857 −1.52417
\(285\) 0 0
\(286\) 57.3071 3.38864
\(287\) 15.4953 0.914660
\(288\) −22.6089 −1.33224
\(289\) −15.0223 −0.883663
\(290\) 0 0
\(291\) 3.41712 0.200315
\(292\) −12.0382 −0.704480
\(293\) −30.2857 −1.76931 −0.884655 0.466245i \(-0.845606\pi\)
−0.884655 + 0.466245i \(0.845606\pi\)
\(294\) 3.30780 0.192915
\(295\) 0 0
\(296\) −1.23237 −0.0716298
\(297\) 9.76302 0.566508
\(298\) −13.8857 −0.804379
\(299\) −36.7053 −2.12272
\(300\) 0 0
\(301\) 12.6972 0.731852
\(302\) 35.9046 2.06608
\(303\) 3.51791 0.202099
\(304\) −4.31820 −0.247666
\(305\) 0 0
\(306\) −8.00965 −0.457881
\(307\) 23.0629 1.31627 0.658134 0.752901i \(-0.271346\pi\)
0.658134 + 0.752901i \(0.271346\pi\)
\(308\) 36.1925 2.06226
\(309\) 0.402096 0.0228744
\(310\) 0 0
\(311\) −10.3152 −0.584921 −0.292460 0.956278i \(-0.594474\pi\)
−0.292460 + 0.956278i \(0.594474\pi\)
\(312\) 0.532930 0.0301712
\(313\) 4.95038 0.279812 0.139906 0.990165i \(-0.455320\pi\)
0.139906 + 0.990165i \(0.455320\pi\)
\(314\) 33.7011 1.90186
\(315\) 0 0
\(316\) 9.99612 0.562326
\(317\) 14.2433 0.799985 0.399992 0.916518i \(-0.369013\pi\)
0.399992 + 0.916518i \(0.369013\pi\)
\(318\) 4.42134 0.247936
\(319\) 7.82377 0.438047
\(320\) 0 0
\(321\) −2.09656 −0.117018
\(322\) −48.5756 −2.70702
\(323\) −1.40632 −0.0782495
\(324\) 14.9977 0.833207
\(325\) 0 0
\(326\) 21.2536 1.17713
\(327\) −3.02314 −0.167180
\(328\) 1.48243 0.0818534
\(329\) 29.4346 1.62278
\(330\) 0 0
\(331\) 2.04272 0.112278 0.0561390 0.998423i \(-0.482121\pi\)
0.0561390 + 0.998423i \(0.482121\pi\)
\(332\) 7.58946 0.416526
\(333\) 10.5266 0.576854
\(334\) 5.53487 0.302855
\(335\) 0 0
\(336\) 4.56753 0.249179
\(337\) −34.6951 −1.88996 −0.944980 0.327128i \(-0.893919\pi\)
−0.944980 + 0.327128i \(0.893919\pi\)
\(338\) −28.8221 −1.56772
\(339\) 0.552170 0.0299898
\(340\) 0 0
\(341\) 9.76302 0.528697
\(342\) 5.69548 0.307976
\(343\) −4.64243 −0.250668
\(344\) 1.21473 0.0654938
\(345\) 0 0
\(346\) −18.1247 −0.974388
\(347\) −7.35280 −0.394719 −0.197359 0.980331i \(-0.563237\pi\)
−0.197359 + 0.980331i \(0.563237\pi\)
\(348\) −0.762149 −0.0408555
\(349\) −31.7630 −1.70024 −0.850118 0.526593i \(-0.823469\pi\)
−0.850118 + 0.526593i \(0.823469\pi\)
\(350\) 0 0
\(351\) −9.24209 −0.493306
\(352\) 43.1955 2.30233
\(353\) 7.52179 0.400345 0.200172 0.979761i \(-0.435850\pi\)
0.200172 + 0.979761i \(0.435850\pi\)
\(354\) 5.50135 0.292393
\(355\) 0 0
\(356\) −16.8515 −0.893126
\(357\) 1.48751 0.0787276
\(358\) 7.02314 0.371185
\(359\) −30.3982 −1.60436 −0.802178 0.597085i \(-0.796326\pi\)
−0.802178 + 0.597085i \(0.796326\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 38.6553 2.03168
\(363\) −5.92211 −0.310830
\(364\) −34.2613 −1.79578
\(365\) 0 0
\(366\) 5.35012 0.279655
\(367\) −3.91577 −0.204401 −0.102201 0.994764i \(-0.532588\pi\)
−0.102201 + 0.994764i \(0.532588\pi\)
\(368\) −30.0962 −1.56887
\(369\) −12.6626 −0.659186
\(370\) 0 0
\(371\) 27.1347 1.40877
\(372\) −0.951060 −0.0493102
\(373\) −26.7759 −1.38641 −0.693203 0.720743i \(-0.743801\pi\)
−0.693203 + 0.720743i \(0.743801\pi\)
\(374\) 15.3028 0.791291
\(375\) 0 0
\(376\) 2.81599 0.145223
\(377\) −7.40632 −0.381445
\(378\) −12.2310 −0.629092
\(379\) 14.9504 0.767950 0.383975 0.923344i \(-0.374555\pi\)
0.383975 + 0.923344i \(0.374555\pi\)
\(380\) 0 0
\(381\) −1.45171 −0.0743735
\(382\) −16.2617 −0.832019
\(383\) 27.9910 1.43027 0.715136 0.698985i \(-0.246365\pi\)
0.715136 + 0.698985i \(0.246365\pi\)
\(384\) 0.806471 0.0411551
\(385\) 0 0
\(386\) −43.5225 −2.21524
\(387\) −10.3759 −0.527439
\(388\) −21.0168 −1.06697
\(389\) −35.2036 −1.78489 −0.892447 0.451152i \(-0.851013\pi\)
−0.892447 + 0.451152i \(0.851013\pi\)
\(390\) 0 0
\(391\) −9.80150 −0.495683
\(392\) 1.94216 0.0980937
\(393\) −0.834907 −0.0421155
\(394\) −17.2370 −0.868388
\(395\) 0 0
\(396\) −29.5760 −1.48625
\(397\) 35.9735 1.80546 0.902730 0.430208i \(-0.141560\pi\)
0.902730 + 0.430208i \(0.141560\pi\)
\(398\) −41.2036 −2.06535
\(399\) −1.05774 −0.0529532
\(400\) 0 0
\(401\) 23.2421 1.16065 0.580327 0.814383i \(-0.302925\pi\)
0.580327 + 0.814383i \(0.302925\pi\)
\(402\) 2.76420 0.137866
\(403\) −9.24209 −0.460381
\(404\) −21.6367 −1.07647
\(405\) 0 0
\(406\) −9.80150 −0.486440
\(407\) −20.1116 −0.996895
\(408\) 0.142310 0.00704538
\(409\) −31.8926 −1.57699 −0.788495 0.615041i \(-0.789139\pi\)
−0.788495 + 0.615041i \(0.789139\pi\)
\(410\) 0 0
\(411\) 2.73988 0.135148
\(412\) −2.47307 −0.121840
\(413\) 33.7630 1.66137
\(414\) 39.6953 1.95092
\(415\) 0 0
\(416\) −40.8907 −2.00483
\(417\) 1.09202 0.0534763
\(418\) −10.8815 −0.532232
\(419\) 31.8238 1.55469 0.777346 0.629073i \(-0.216565\pi\)
0.777346 + 0.629073i \(0.216565\pi\)
\(420\) 0 0
\(421\) 0.348578 0.0169887 0.00849433 0.999964i \(-0.497296\pi\)
0.00849433 + 0.999964i \(0.497296\pi\)
\(422\) −10.4556 −0.508971
\(423\) −24.0535 −1.16952
\(424\) 2.59596 0.126071
\(425\) 0 0
\(426\) −8.16847 −0.395763
\(427\) 32.8349 1.58899
\(428\) 12.8948 0.623292
\(429\) 8.69716 0.419903
\(430\) 0 0
\(431\) 29.2764 1.41019 0.705097 0.709111i \(-0.250904\pi\)
0.705097 + 0.709111i \(0.250904\pi\)
\(432\) −7.57799 −0.364596
\(433\) 0.883290 0.0424482 0.0212241 0.999775i \(-0.493244\pi\)
0.0212241 + 0.999775i \(0.493244\pi\)
\(434\) −12.2310 −0.587105
\(435\) 0 0
\(436\) 18.5937 0.890476
\(437\) 6.96962 0.333402
\(438\) −3.82833 −0.182925
\(439\) −13.8584 −0.661424 −0.330712 0.943732i \(-0.607289\pi\)
−0.330712 + 0.943732i \(0.607289\pi\)
\(440\) 0 0
\(441\) −16.5895 −0.789974
\(442\) −14.4863 −0.689044
\(443\) 13.5753 0.644982 0.322491 0.946572i \(-0.395480\pi\)
0.322491 + 0.946572i \(0.395480\pi\)
\(444\) 1.95916 0.0929778
\(445\) 0 0
\(446\) −6.08195 −0.287989
\(447\) −2.10735 −0.0996745
\(448\) −23.3405 −1.10273
\(449\) −15.6334 −0.737785 −0.368893 0.929472i \(-0.620263\pi\)
−0.368893 + 0.929472i \(0.620263\pi\)
\(450\) 0 0
\(451\) 24.1925 1.13918
\(452\) −3.39609 −0.159739
\(453\) 5.44904 0.256018
\(454\) 28.3009 1.32823
\(455\) 0 0
\(456\) −0.101193 −0.00473880
\(457\) −5.30284 −0.248057 −0.124028 0.992279i \(-0.539581\pi\)
−0.124028 + 0.992279i \(0.539581\pi\)
\(458\) 10.1997 0.476601
\(459\) −2.46794 −0.115193
\(460\) 0 0
\(461\) 0.374734 0.0174531 0.00872656 0.999962i \(-0.497222\pi\)
0.00872656 + 0.999962i \(0.497222\pi\)
\(462\) 11.5098 0.535484
\(463\) −6.65564 −0.309314 −0.154657 0.987968i \(-0.549427\pi\)
−0.154657 + 0.987968i \(0.549427\pi\)
\(464\) −6.07276 −0.281921
\(465\) 0 0
\(466\) −6.23431 −0.288799
\(467\) 0.854147 0.0395252 0.0197626 0.999805i \(-0.493709\pi\)
0.0197626 + 0.999805i \(0.493709\pi\)
\(468\) 27.9979 1.29420
\(469\) 16.9645 0.783349
\(470\) 0 0
\(471\) 5.11461 0.235669
\(472\) 3.23009 0.148677
\(473\) 19.8238 0.911498
\(474\) 3.17893 0.146013
\(475\) 0 0
\(476\) −9.14889 −0.419339
\(477\) −22.1741 −1.01528
\(478\) −32.3140 −1.47801
\(479\) −17.0731 −0.780090 −0.390045 0.920796i \(-0.627540\pi\)
−0.390045 + 0.920796i \(0.627540\pi\)
\(480\) 0 0
\(481\) 19.0385 0.868081
\(482\) 23.9154 1.08932
\(483\) −7.37204 −0.335439
\(484\) 36.4236 1.65562
\(485\) 0 0
\(486\) 15.0669 0.683450
\(487\) −12.8259 −0.581197 −0.290598 0.956845i \(-0.593854\pi\)
−0.290598 + 0.956845i \(0.593854\pi\)
\(488\) 3.14129 0.142200
\(489\) 3.22553 0.145863
\(490\) 0 0
\(491\) −10.4054 −0.469591 −0.234796 0.972045i \(-0.575442\pi\)
−0.234796 + 0.972045i \(0.575442\pi\)
\(492\) −2.35670 −0.106248
\(493\) −1.97773 −0.0890723
\(494\) 10.3009 0.463460
\(495\) 0 0
\(496\) −7.57799 −0.340262
\(497\) −50.1317 −2.24872
\(498\) 2.41357 0.108155
\(499\) 36.1612 1.61880 0.809399 0.587258i \(-0.199793\pi\)
0.809399 + 0.587258i \(0.199793\pi\)
\(500\) 0 0
\(501\) 0.839995 0.0375282
\(502\) −8.84267 −0.394668
\(503\) −33.2536 −1.48270 −0.741352 0.671117i \(-0.765815\pi\)
−0.741352 + 0.671117i \(0.765815\pi\)
\(504\) −3.53715 −0.157557
\(505\) 0 0
\(506\) −75.8400 −3.37150
\(507\) −4.37416 −0.194263
\(508\) 8.92869 0.396146
\(509\) 7.29084 0.323161 0.161580 0.986860i \(-0.448341\pi\)
0.161580 + 0.986860i \(0.448341\pi\)
\(510\) 0 0
\(511\) −23.4953 −1.03937
\(512\) −30.5839 −1.35163
\(513\) 1.75489 0.0774805
\(514\) 31.1563 1.37424
\(515\) 0 0
\(516\) −1.93112 −0.0850130
\(517\) 45.9555 2.02112
\(518\) 25.1955 1.10703
\(519\) −2.75067 −0.120741
\(520\) 0 0
\(521\) −9.87849 −0.432785 −0.216392 0.976306i \(-0.569429\pi\)
−0.216392 + 0.976306i \(0.569429\pi\)
\(522\) 8.00965 0.350573
\(523\) 11.1925 0.489414 0.244707 0.969597i \(-0.421308\pi\)
0.244707 + 0.969597i \(0.421308\pi\)
\(524\) 5.13505 0.224326
\(525\) 0 0
\(526\) 1.67066 0.0728443
\(527\) −2.46794 −0.107505
\(528\) 7.13117 0.310344
\(529\) 25.5756 1.11198
\(530\) 0 0
\(531\) −27.5907 −1.19733
\(532\) 6.50557 0.282052
\(533\) −22.9016 −0.991980
\(534\) −5.35904 −0.231908
\(535\) 0 0
\(536\) 1.62299 0.0701023
\(537\) 1.06586 0.0459953
\(538\) −20.3248 −0.876263
\(539\) 31.6950 1.36520
\(540\) 0 0
\(541\) 2.22587 0.0956974 0.0478487 0.998855i \(-0.484763\pi\)
0.0478487 + 0.998855i \(0.484763\pi\)
\(542\) 16.0382 0.688898
\(543\) 5.86649 0.251755
\(544\) −10.9191 −0.468154
\(545\) 0 0
\(546\) −10.8957 −0.466291
\(547\) 34.9675 1.49510 0.747552 0.664204i \(-0.231229\pi\)
0.747552 + 0.664204i \(0.231229\pi\)
\(548\) −16.8515 −0.719859
\(549\) −26.8322 −1.14517
\(550\) 0 0
\(551\) 1.40632 0.0599111
\(552\) −0.705278 −0.0300186
\(553\) 19.5098 0.829641
\(554\) 28.6514 1.21728
\(555\) 0 0
\(556\) −6.71640 −0.284839
\(557\) 8.88539 0.376486 0.188243 0.982122i \(-0.439721\pi\)
0.188243 + 0.982122i \(0.439721\pi\)
\(558\) 9.99497 0.423121
\(559\) −18.7660 −0.793719
\(560\) 0 0
\(561\) 2.32242 0.0980527
\(562\) 61.8331 2.60827
\(563\) 29.6767 1.25072 0.625362 0.780335i \(-0.284952\pi\)
0.625362 + 0.780335i \(0.284952\pi\)
\(564\) −4.47674 −0.188505
\(565\) 0 0
\(566\) 48.3756 2.03338
\(567\) 29.2716 1.22929
\(568\) −4.79607 −0.201239
\(569\) −22.1152 −0.927116 −0.463558 0.886067i \(-0.653427\pi\)
−0.463558 + 0.886067i \(0.653427\pi\)
\(570\) 0 0
\(571\) −33.2656 −1.39212 −0.696060 0.717983i \(-0.745065\pi\)
−0.696060 + 0.717983i \(0.745065\pi\)
\(572\) −53.4914 −2.23659
\(573\) −2.46794 −0.103100
\(574\) −30.3080 −1.26503
\(575\) 0 0
\(576\) 19.0735 0.794730
\(577\) −0.934140 −0.0388887 −0.0194444 0.999811i \(-0.506190\pi\)
−0.0194444 + 0.999811i \(0.506190\pi\)
\(578\) 29.3827 1.22216
\(579\) −6.60516 −0.274501
\(580\) 0 0
\(581\) 14.8126 0.614532
\(582\) −6.68368 −0.277047
\(583\) 42.3648 1.75457
\(584\) −2.24778 −0.0930139
\(585\) 0 0
\(586\) 59.2371 2.44706
\(587\) −1.57531 −0.0650200 −0.0325100 0.999471i \(-0.510350\pi\)
−0.0325100 + 0.999471i \(0.510350\pi\)
\(588\) −3.08756 −0.127329
\(589\) 1.75489 0.0723092
\(590\) 0 0
\(591\) −2.61596 −0.107606
\(592\) 15.6105 0.641587
\(593\) −26.0162 −1.06836 −0.534180 0.845371i \(-0.679379\pi\)
−0.534180 + 0.845371i \(0.679379\pi\)
\(594\) −19.0959 −0.783514
\(595\) 0 0
\(596\) 12.9612 0.530911
\(597\) −6.25323 −0.255928
\(598\) 71.7934 2.93585
\(599\) 19.5359 0.798217 0.399109 0.916904i \(-0.369320\pi\)
0.399109 + 0.916904i \(0.369320\pi\)
\(600\) 0 0
\(601\) 43.5299 1.77562 0.887811 0.460208i \(-0.152225\pi\)
0.887811 + 0.460208i \(0.152225\pi\)
\(602\) −24.8349 −1.01220
\(603\) −13.8632 −0.564552
\(604\) −33.5140 −1.36367
\(605\) 0 0
\(606\) −6.88083 −0.279515
\(607\) 12.5847 0.510796 0.255398 0.966836i \(-0.417794\pi\)
0.255398 + 0.966836i \(0.417794\pi\)
\(608\) 7.76435 0.314886
\(609\) −1.48751 −0.0602771
\(610\) 0 0
\(611\) −43.5034 −1.75996
\(612\) 7.47634 0.302213
\(613\) 18.2412 0.736756 0.368378 0.929676i \(-0.379913\pi\)
0.368378 + 0.929676i \(0.379913\pi\)
\(614\) −45.1097 −1.82048
\(615\) 0 0
\(616\) 6.75791 0.272284
\(617\) −26.7314 −1.07617 −0.538084 0.842892i \(-0.680852\pi\)
−0.538084 + 0.842892i \(0.680852\pi\)
\(618\) −0.786477 −0.0316367
\(619\) 2.92690 0.117642 0.0588211 0.998269i \(-0.481266\pi\)
0.0588211 + 0.998269i \(0.481266\pi\)
\(620\) 0 0
\(621\) 12.2310 0.490811
\(622\) 20.1759 0.808980
\(623\) −32.8896 −1.31770
\(624\) −6.75067 −0.270243
\(625\) 0 0
\(626\) −9.68267 −0.386997
\(627\) −1.65142 −0.0659514
\(628\) −31.4572 −1.25528
\(629\) 5.08389 0.202708
\(630\) 0 0
\(631\) −13.2493 −0.527447 −0.263724 0.964598i \(-0.584951\pi\)
−0.263724 + 0.964598i \(0.584951\pi\)
\(632\) 1.86649 0.0742450
\(633\) −1.58679 −0.0630691
\(634\) −27.8591 −1.10643
\(635\) 0 0
\(636\) −4.12695 −0.163644
\(637\) −30.0039 −1.18880
\(638\) −15.3028 −0.605845
\(639\) 40.9669 1.62063
\(640\) 0 0
\(641\) 6.93026 0.273729 0.136864 0.990590i \(-0.456298\pi\)
0.136864 + 0.990590i \(0.456298\pi\)
\(642\) 4.10074 0.161843
\(643\) −14.4794 −0.571012 −0.285506 0.958377i \(-0.592162\pi\)
−0.285506 + 0.958377i \(0.592162\pi\)
\(644\) 45.3414 1.78670
\(645\) 0 0
\(646\) 2.75067 0.108224
\(647\) −6.35549 −0.249860 −0.124930 0.992166i \(-0.539871\pi\)
−0.124930 + 0.992166i \(0.539871\pi\)
\(648\) 2.80040 0.110010
\(649\) 52.7134 2.06918
\(650\) 0 0
\(651\) −1.85622 −0.0727510
\(652\) −19.8385 −0.776934
\(653\) 34.4030 1.34629 0.673146 0.739509i \(-0.264942\pi\)
0.673146 + 0.739509i \(0.264942\pi\)
\(654\) 5.91309 0.231220
\(655\) 0 0
\(656\) −18.7780 −0.733159
\(657\) 19.2000 0.749065
\(658\) −57.5723 −2.24440
\(659\) −19.6214 −0.764341 −0.382170 0.924092i \(-0.624823\pi\)
−0.382170 + 0.924092i \(0.624823\pi\)
\(660\) 0 0
\(661\) 39.8054 1.54825 0.774126 0.633032i \(-0.218190\pi\)
0.774126 + 0.633032i \(0.218190\pi\)
\(662\) −3.99544 −0.155287
\(663\) −2.19850 −0.0853828
\(664\) 1.41712 0.0549947
\(665\) 0 0
\(666\) −20.5894 −0.797823
\(667\) 9.80150 0.379515
\(668\) −5.16635 −0.199892
\(669\) −0.923022 −0.0356861
\(670\) 0 0
\(671\) 51.2644 1.97904
\(672\) −8.21266 −0.316810
\(673\) 8.90374 0.343214 0.171607 0.985166i \(-0.445104\pi\)
0.171607 + 0.985166i \(0.445104\pi\)
\(674\) 67.8615 2.61393
\(675\) 0 0
\(676\) 26.9030 1.03473
\(677\) −22.2695 −0.855886 −0.427943 0.903806i \(-0.640762\pi\)
−0.427943 + 0.903806i \(0.640762\pi\)
\(678\) −1.08001 −0.0414776
\(679\) −41.0193 −1.57417
\(680\) 0 0
\(681\) 4.29506 0.164587
\(682\) −19.0959 −0.731220
\(683\) −15.4054 −0.589472 −0.294736 0.955579i \(-0.595232\pi\)
−0.294736 + 0.955579i \(0.595232\pi\)
\(684\) −5.31626 −0.203272
\(685\) 0 0
\(686\) 9.08033 0.346689
\(687\) 1.54795 0.0590580
\(688\) −15.3871 −0.586627
\(689\) −40.1044 −1.52785
\(690\) 0 0
\(691\) −17.8773 −0.680084 −0.340042 0.940410i \(-0.610441\pi\)
−0.340042 + 0.940410i \(0.610441\pi\)
\(692\) 16.9179 0.643122
\(693\) −57.7245 −2.19277
\(694\) 14.3817 0.545920
\(695\) 0 0
\(696\) −0.142310 −0.00539423
\(697\) −6.11548 −0.231640
\(698\) 62.1266 2.35153
\(699\) −0.946144 −0.0357864
\(700\) 0 0
\(701\) 35.3609 1.33556 0.667782 0.744357i \(-0.267244\pi\)
0.667782 + 0.744357i \(0.267244\pi\)
\(702\) 18.0770 0.682272
\(703\) −3.61504 −0.136344
\(704\) −36.4409 −1.37342
\(705\) 0 0
\(706\) −14.7122 −0.553701
\(707\) −42.2292 −1.58819
\(708\) −5.13505 −0.192987
\(709\) 6.90410 0.259289 0.129644 0.991561i \(-0.458616\pi\)
0.129644 + 0.991561i \(0.458616\pi\)
\(710\) 0 0
\(711\) −15.9431 −0.597914
\(712\) −3.14653 −0.117921
\(713\) 12.2310 0.458053
\(714\) −2.90949 −0.108885
\(715\) 0 0
\(716\) −6.55552 −0.244991
\(717\) −4.90410 −0.183147
\(718\) 59.4572 2.21892
\(719\) −38.9431 −1.45233 −0.726167 0.687518i \(-0.758700\pi\)
−0.726167 + 0.687518i \(0.758700\pi\)
\(720\) 0 0
\(721\) −4.82678 −0.179759
\(722\) −1.95594 −0.0727926
\(723\) 3.62951 0.134983
\(724\) −36.0816 −1.34096
\(725\) 0 0
\(726\) 11.5833 0.429897
\(727\) 6.18347 0.229332 0.114666 0.993404i \(-0.463420\pi\)
0.114666 + 0.993404i \(0.463420\pi\)
\(728\) −6.39733 −0.237101
\(729\) −22.3576 −0.828058
\(730\) 0 0
\(731\) −5.01114 −0.185344
\(732\) −4.99390 −0.184580
\(733\) 0.688715 0.0254383 0.0127191 0.999919i \(-0.495951\pi\)
0.0127191 + 0.999919i \(0.495951\pi\)
\(734\) 7.65902 0.282699
\(735\) 0 0
\(736\) 54.1146 1.99469
\(737\) 26.4863 0.975636
\(738\) 24.7672 0.911695
\(739\) −26.2532 −0.965741 −0.482870 0.875692i \(-0.660406\pi\)
−0.482870 + 0.875692i \(0.660406\pi\)
\(740\) 0 0
\(741\) 1.56331 0.0574295
\(742\) −53.0740 −1.94841
\(743\) 32.5688 1.19483 0.597416 0.801931i \(-0.296194\pi\)
0.597416 + 0.801931i \(0.296194\pi\)
\(744\) −0.177583 −0.00651052
\(745\) 0 0
\(746\) 52.3722 1.91748
\(747\) −12.1047 −0.442887
\(748\) −14.2839 −0.522273
\(749\) 25.1672 0.919589
\(750\) 0 0
\(751\) 45.4833 1.65971 0.829855 0.557979i \(-0.188423\pi\)
0.829855 + 0.557979i \(0.188423\pi\)
\(752\) −35.6703 −1.30076
\(753\) −1.34200 −0.0489052
\(754\) 14.4863 0.527561
\(755\) 0 0
\(756\) 11.4166 0.415217
\(757\) −2.74947 −0.0999311 −0.0499656 0.998751i \(-0.515911\pi\)
−0.0499656 + 0.998751i \(0.515911\pi\)
\(758\) −29.2421 −1.06212
\(759\) −11.5098 −0.417779
\(760\) 0 0
\(761\) −33.2978 −1.20704 −0.603521 0.797347i \(-0.706236\pi\)
−0.603521 + 0.797347i \(0.706236\pi\)
\(762\) 2.83947 0.102863
\(763\) 36.2900 1.31379
\(764\) 15.1789 0.549154
\(765\) 0 0
\(766\) −54.7488 −1.97815
\(767\) −49.9008 −1.80181
\(768\) −5.46617 −0.197244
\(769\) −19.1540 −0.690710 −0.345355 0.938472i \(-0.612241\pi\)
−0.345355 + 0.938472i \(0.612241\pi\)
\(770\) 0 0
\(771\) 4.72840 0.170289
\(772\) 40.6247 1.46212
\(773\) −0.569309 −0.0204766 −0.0102383 0.999948i \(-0.503259\pi\)
−0.0102383 + 0.999948i \(0.503259\pi\)
\(774\) 20.2947 0.729479
\(775\) 0 0
\(776\) −3.92429 −0.140874
\(777\) 3.82377 0.137177
\(778\) 68.8562 2.46862
\(779\) 4.34858 0.155804
\(780\) 0 0
\(781\) −78.2695 −2.80070
\(782\) 19.1712 0.685559
\(783\) 2.46794 0.0881969
\(784\) −24.6015 −0.878624
\(785\) 0 0
\(786\) 1.63303 0.0582483
\(787\) −15.9991 −0.570307 −0.285153 0.958482i \(-0.592044\pi\)
−0.285153 + 0.958482i \(0.592044\pi\)
\(788\) 16.0893 0.573158
\(789\) 0.253546 0.00902649
\(790\) 0 0
\(791\) −6.62828 −0.235675
\(792\) −5.52247 −0.196232
\(793\) −48.5290 −1.72332
\(794\) −70.3621 −2.49706
\(795\) 0 0
\(796\) 38.4602 1.36318
\(797\) 35.7528 1.26643 0.633214 0.773976i \(-0.281735\pi\)
0.633214 + 0.773976i \(0.281735\pi\)
\(798\) 2.06888 0.0732374
\(799\) −11.6168 −0.410974
\(800\) 0 0
\(801\) 26.8769 0.949650
\(802\) −45.4602 −1.60526
\(803\) −36.6827 −1.29450
\(804\) −2.58016 −0.0909951
\(805\) 0 0
\(806\) 18.0770 0.636735
\(807\) −3.08457 −0.108582
\(808\) −4.04004 −0.142128
\(809\) 23.2036 0.815796 0.407898 0.913028i \(-0.366262\pi\)
0.407898 + 0.913028i \(0.366262\pi\)
\(810\) 0 0
\(811\) −21.7549 −0.763918 −0.381959 0.924179i \(-0.624750\pi\)
−0.381959 + 0.924179i \(0.624750\pi\)
\(812\) 9.14889 0.321063
\(813\) 2.43402 0.0853647
\(814\) 39.3371 1.37877
\(815\) 0 0
\(816\) −1.80265 −0.0631053
\(817\) 3.56331 0.124664
\(818\) 62.3802 2.18107
\(819\) 54.6445 1.90943
\(820\) 0 0
\(821\) 52.2532 1.82365 0.911825 0.410579i \(-0.134673\pi\)
0.911825 + 0.410579i \(0.134673\pi\)
\(822\) −5.35904 −0.186918
\(823\) 9.44783 0.329331 0.164665 0.986349i \(-0.447346\pi\)
0.164665 + 0.986349i \(0.447346\pi\)
\(824\) −0.461775 −0.0160867
\(825\) 0 0
\(826\) −66.0385 −2.29777
\(827\) 50.2216 1.74638 0.873188 0.487383i \(-0.162048\pi\)
0.873188 + 0.487383i \(0.162048\pi\)
\(828\) −37.0523 −1.28766
\(829\) 44.2066 1.53536 0.767680 0.640834i \(-0.221411\pi\)
0.767680 + 0.640834i \(0.221411\pi\)
\(830\) 0 0
\(831\) 4.34826 0.150839
\(832\) 34.4966 1.19595
\(833\) −8.01200 −0.277599
\(834\) −2.13592 −0.0739609
\(835\) 0 0
\(836\) 10.1570 0.351287
\(837\) 3.07965 0.106448
\(838\) −62.2455 −2.15023
\(839\) −30.4033 −1.04964 −0.524819 0.851214i \(-0.675867\pi\)
−0.524819 + 0.851214i \(0.675867\pi\)
\(840\) 0 0
\(841\) −27.0223 −0.931803
\(842\) −0.681799 −0.0234963
\(843\) 9.38404 0.323204
\(844\) 9.75945 0.335934
\(845\) 0 0
\(846\) 47.0473 1.61752
\(847\) 71.0893 2.44266
\(848\) −32.8833 −1.12922
\(849\) 7.34168 0.251966
\(850\) 0 0
\(851\) −25.1955 −0.863690
\(852\) 7.62459 0.261214
\(853\) −3.93925 −0.134877 −0.0674386 0.997723i \(-0.521483\pi\)
−0.0674386 + 0.997723i \(0.521483\pi\)
\(854\) −64.2232 −2.19767
\(855\) 0 0
\(856\) 2.40773 0.0822945
\(857\) −27.4388 −0.937292 −0.468646 0.883386i \(-0.655258\pi\)
−0.468646 + 0.883386i \(0.655258\pi\)
\(858\) −17.0111 −0.580751
\(859\) 32.4517 1.10724 0.553619 0.832770i \(-0.313246\pi\)
0.553619 + 0.832770i \(0.313246\pi\)
\(860\) 0 0
\(861\) −4.59966 −0.156756
\(862\) −57.2629 −1.95038
\(863\) 2.10861 0.0717778 0.0358889 0.999356i \(-0.488574\pi\)
0.0358889 + 0.999356i \(0.488574\pi\)
\(864\) 13.6256 0.463553
\(865\) 0 0
\(866\) −1.72766 −0.0587084
\(867\) 4.45924 0.151444
\(868\) 11.4166 0.387504
\(869\) 30.4602 1.03329
\(870\) 0 0
\(871\) −25.0731 −0.849569
\(872\) 3.47184 0.117571
\(873\) 33.5204 1.13449
\(874\) −13.6322 −0.461115
\(875\) 0 0
\(876\) 3.57343 0.120735
\(877\) 37.6613 1.27173 0.635866 0.771799i \(-0.280643\pi\)
0.635866 + 0.771799i \(0.280643\pi\)
\(878\) 27.1062 0.914789
\(879\) 8.99007 0.303227
\(880\) 0 0
\(881\) −39.8818 −1.34365 −0.671827 0.740708i \(-0.734490\pi\)
−0.671827 + 0.740708i \(0.734490\pi\)
\(882\) 32.4480 1.09258
\(883\) −36.0458 −1.21304 −0.606518 0.795070i \(-0.707434\pi\)
−0.606518 + 0.795070i \(0.707434\pi\)
\(884\) 13.5218 0.454787
\(885\) 0 0
\(886\) −26.5525 −0.892050
\(887\) 26.2057 0.879902 0.439951 0.898022i \(-0.354996\pi\)
0.439951 + 0.898022i \(0.354996\pi\)
\(888\) 0.365818 0.0122760
\(889\) 17.4264 0.584464
\(890\) 0 0
\(891\) 45.7011 1.53104
\(892\) 5.67700 0.190080
\(893\) 8.26046 0.276426
\(894\) 4.12186 0.137856
\(895\) 0 0
\(896\) −9.68093 −0.323417
\(897\) 10.8957 0.363796
\(898\) 30.5780 1.02040
\(899\) 2.46794 0.0823103
\(900\) 0 0
\(901\) −10.7092 −0.356774
\(902\) −47.3191 −1.57555
\(903\) −3.76905 −0.125426
\(904\) −0.634123 −0.0210906
\(905\) 0 0
\(906\) −10.6580 −0.354088
\(907\) 22.8036 0.757182 0.378591 0.925564i \(-0.376409\pi\)
0.378591 + 0.925564i \(0.376409\pi\)
\(908\) −26.4166 −0.876664
\(909\) 34.5091 1.14460
\(910\) 0 0
\(911\) 4.44146 0.147152 0.0735761 0.997290i \(-0.476559\pi\)
0.0735761 + 0.997290i \(0.476559\pi\)
\(912\) 1.28182 0.0424454
\(913\) 23.1266 0.765379
\(914\) 10.3721 0.343077
\(915\) 0 0
\(916\) −9.52059 −0.314569
\(917\) 10.0223 0.330965
\(918\) 4.82714 0.159319
\(919\) 37.0111 1.22088 0.610442 0.792061i \(-0.290992\pi\)
0.610442 + 0.792061i \(0.290992\pi\)
\(920\) 0 0
\(921\) −6.84602 −0.225584
\(922\) −0.732959 −0.0241387
\(923\) 74.0932 2.43881
\(924\) −10.7434 −0.353433
\(925\) 0 0
\(926\) 13.0181 0.427800
\(927\) 3.94438 0.129550
\(928\) 10.9191 0.358438
\(929\) −3.04185 −0.0997999 −0.0499000 0.998754i \(-0.515890\pi\)
−0.0499000 + 0.998754i \(0.515890\pi\)
\(930\) 0 0
\(931\) 5.69716 0.186717
\(932\) 5.81921 0.190615
\(933\) 3.06198 0.100245
\(934\) −1.67066 −0.0546657
\(935\) 0 0
\(936\) 5.22781 0.170876
\(937\) −38.6099 −1.26133 −0.630666 0.776055i \(-0.717218\pi\)
−0.630666 + 0.776055i \(0.717218\pi\)
\(938\) −33.1817 −1.08342
\(939\) −1.46948 −0.0479547
\(940\) 0 0
\(941\) 2.69716 0.0879248 0.0439624 0.999033i \(-0.486002\pi\)
0.0439624 + 0.999033i \(0.486002\pi\)
\(942\) −10.0039 −0.325944
\(943\) 30.3080 0.986963
\(944\) −40.9158 −1.33170
\(945\) 0 0
\(946\) −38.7742 −1.26066
\(947\) 20.1877 0.656012 0.328006 0.944676i \(-0.393623\pi\)
0.328006 + 0.944676i \(0.393623\pi\)
\(948\) −2.96727 −0.0963723
\(949\) 34.7254 1.12723
\(950\) 0 0
\(951\) −4.22801 −0.137103
\(952\) −1.70829 −0.0553661
\(953\) −5.18559 −0.167978 −0.0839888 0.996467i \(-0.526766\pi\)
−0.0839888 + 0.996467i \(0.526766\pi\)
\(954\) 43.3713 1.40420
\(955\) 0 0
\(956\) 30.1624 0.975523
\(957\) −2.32242 −0.0750732
\(958\) 33.3940 1.07891
\(959\) −32.8896 −1.06206
\(960\) 0 0
\(961\) −27.9203 −0.900656
\(962\) −37.2382 −1.20061
\(963\) −20.5663 −0.662739
\(964\) −22.3231 −0.718979
\(965\) 0 0
\(966\) 14.4193 0.463933
\(967\) −58.0054 −1.86533 −0.932665 0.360744i \(-0.882523\pi\)
−0.932665 + 0.360744i \(0.882523\pi\)
\(968\) 6.80107 0.218595
\(969\) 0.417453 0.0134105
\(970\) 0 0
\(971\) −24.3221 −0.780533 −0.390267 0.920702i \(-0.627617\pi\)
−0.390267 + 0.920702i \(0.627617\pi\)
\(972\) −14.0637 −0.451095
\(973\) −13.1086 −0.420244
\(974\) 25.0867 0.803830
\(975\) 0 0
\(976\) −39.7911 −1.27368
\(977\) 36.1134 1.15537 0.577685 0.816260i \(-0.303956\pi\)
0.577685 + 0.816260i \(0.303956\pi\)
\(978\) −6.30895 −0.201738
\(979\) −51.3498 −1.64115
\(980\) 0 0
\(981\) −29.6557 −0.946832
\(982\) 20.3525 0.649473
\(983\) −21.1603 −0.674909 −0.337455 0.941342i \(-0.609566\pi\)
−0.337455 + 0.941342i \(0.609566\pi\)
\(984\) −0.440046 −0.0140282
\(985\) 0 0
\(986\) 3.86832 0.123192
\(987\) −8.73741 −0.278115
\(988\) −9.61504 −0.305895
\(989\) 24.8349 0.789704
\(990\) 0 0
\(991\) 20.2652 0.643746 0.321873 0.946783i \(-0.395688\pi\)
0.321873 + 0.946783i \(0.395688\pi\)
\(992\) 13.6256 0.432614
\(993\) −0.606364 −0.0192424
\(994\) 98.0548 3.11011
\(995\) 0 0
\(996\) −2.25287 −0.0713849
\(997\) −24.2875 −0.769193 −0.384597 0.923085i \(-0.625659\pi\)
−0.384597 + 0.923085i \(0.625659\pi\)
\(998\) −70.7293 −2.23890
\(999\) −6.34402 −0.200716
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 475.2.a.i.1.1 4
3.2 odd 2 4275.2.a.bo.1.4 4
4.3 odd 2 7600.2.a.cf.1.2 4
5.2 odd 4 475.2.b.e.324.3 8
5.3 odd 4 475.2.b.e.324.6 8
5.4 even 2 95.2.a.b.1.4 4
15.14 odd 2 855.2.a.m.1.1 4
19.18 odd 2 9025.2.a.bf.1.4 4
20.19 odd 2 1520.2.a.t.1.3 4
35.34 odd 2 4655.2.a.y.1.4 4
40.19 odd 2 6080.2.a.ch.1.2 4
40.29 even 2 6080.2.a.cc.1.3 4
95.94 odd 2 1805.2.a.p.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.4 4 5.4 even 2
475.2.a.i.1.1 4 1.1 even 1 trivial
475.2.b.e.324.3 8 5.2 odd 4
475.2.b.e.324.6 8 5.3 odd 4
855.2.a.m.1.1 4 15.14 odd 2
1520.2.a.t.1.3 4 20.19 odd 2
1805.2.a.p.1.1 4 95.94 odd 2
4275.2.a.bo.1.4 4 3.2 odd 2
4655.2.a.y.1.4 4 35.34 odd 2
6080.2.a.cc.1.3 4 40.29 even 2
6080.2.a.ch.1.2 4 40.19 odd 2
7600.2.a.cf.1.2 4 4.3 odd 2
9025.2.a.bf.1.4 4 19.18 odd 2