Properties

Label 475.2.a.i.1.2
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 1.2
Root \(-1.51658\) 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)
 
\(f(q)\) \(=\) \(q-0.816594 q^{2} -1.53844 q^{3} -1.33317 q^{4} +1.25628 q^{6} -5.03316 q^{7} +2.72185 q^{8} -0.633188 q^{9} +O(q^{10})\) \(q-0.816594 q^{2} -1.53844 q^{3} -1.33317 q^{4} +1.25628 q^{6} -5.03316 q^{7} +2.72185 q^{8} -0.633188 q^{9} -3.03316 q^{11} +2.05101 q^{12} +4.57160 q^{13} +4.11005 q^{14} +0.443701 q^{16} +1.07689 q^{17} +0.517058 q^{18} +1.00000 q^{19} +7.74324 q^{21} +2.47686 q^{22} -4.11005 q^{23} -4.18742 q^{24} -3.73315 q^{26} +5.58946 q^{27} +6.71008 q^{28} -1.07689 q^{29} +5.58946 q^{31} -5.80602 q^{32} +4.66635 q^{33} -0.879381 q^{34} +0.844150 q^{36} -0.0947438 q^{37} -0.816594 q^{38} -7.03316 q^{39} +10.6663 q^{41} -6.32308 q^{42} -5.03316 q^{43} +4.04373 q^{44} +3.35624 q^{46} +12.2995 q^{47} -0.682609 q^{48} +18.3327 q^{49} -1.65673 q^{51} -6.09474 q^{52} +4.09474 q^{53} -4.56432 q^{54} -13.6995 q^{56} -1.53844 q^{57} +0.879381 q^{58} -1.39997 q^{59} -5.69951 q^{61} -4.56432 q^{62} +3.18694 q^{63} +3.85376 q^{64} -3.81051 q^{66} -5.28168 q^{67} -1.43568 q^{68} +6.32308 q^{69} -5.67692 q^{71} -1.72344 q^{72} -9.07689 q^{73} +0.0773672 q^{74} -1.33317 q^{76} +15.2664 q^{77} +5.74324 q^{78} -5.39997 q^{79} -6.69951 q^{81} -8.71008 q^{82} -1.95627 q^{83} -10.3231 q^{84} +4.11005 q^{86} +1.65673 q^{87} -8.25581 q^{88} -2.18949 q^{89} -23.0096 q^{91} +5.47941 q^{92} -8.59907 q^{93} -10.0437 q^{94} +8.93225 q^{96} +2.16106 q^{97} -14.9704 q^{98} +1.92056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.816594 −0.577419 −0.288710 0.957417i \(-0.593226\pi\)
−0.288710 + 0.957417i \(0.593226\pi\)
\(3\) −1.53844 −0.888221 −0.444111 0.895972i \(-0.646480\pi\)
−0.444111 + 0.895972i \(0.646480\pi\)
\(4\) −1.33317 −0.666587
\(5\) 0 0
\(6\) 1.25628 0.512876
\(7\) −5.03316 −1.90236 −0.951178 0.308644i \(-0.900125\pi\)
−0.951178 + 0.308644i \(0.900125\pi\)
\(8\) 2.72185 0.962319
\(9\) −0.633188 −0.211063
\(10\) 0 0
\(11\) −3.03316 −0.914532 −0.457266 0.889330i \(-0.651171\pi\)
−0.457266 + 0.889330i \(0.651171\pi\)
\(12\) 2.05101 0.592077
\(13\) 4.57160 1.26793 0.633967 0.773360i \(-0.281425\pi\)
0.633967 + 0.773360i \(0.281425\pi\)
\(14\) 4.11005 1.09846
\(15\) 0 0
\(16\) 0.443701 0.110925
\(17\) 1.07689 0.261184 0.130592 0.991436i \(-0.458312\pi\)
0.130592 + 0.991436i \(0.458312\pi\)
\(18\) 0.517058 0.121872
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 7.74324 1.68971
\(22\) 2.47686 0.528068
\(23\) −4.11005 −0.857004 −0.428502 0.903541i \(-0.640959\pi\)
−0.428502 + 0.903541i \(0.640959\pi\)
\(24\) −4.18742 −0.854753
\(25\) 0 0
\(26\) −3.73315 −0.732130
\(27\) 5.58946 1.07569
\(28\) 6.71008 1.26809
\(29\) −1.07689 −0.199973 −0.0999866 0.994989i \(-0.531880\pi\)
−0.0999866 + 0.994989i \(0.531880\pi\)
\(30\) 0 0
\(31\) 5.58946 1.00390 0.501948 0.864898i \(-0.332617\pi\)
0.501948 + 0.864898i \(0.332617\pi\)
\(32\) −5.80602 −1.02637
\(33\) 4.66635 0.812307
\(34\) −0.879381 −0.150813
\(35\) 0 0
\(36\) 0.844150 0.140692
\(37\) −0.0947438 −0.0155758 −0.00778789 0.999970i \(-0.502479\pi\)
−0.00778789 + 0.999970i \(0.502479\pi\)
\(38\) −0.816594 −0.132469
\(39\) −7.03316 −1.12621
\(40\) 0 0
\(41\) 10.6663 1.66580 0.832902 0.553421i \(-0.186678\pi\)
0.832902 + 0.553421i \(0.186678\pi\)
\(42\) −6.32308 −0.975673
\(43\) −5.03316 −0.767550 −0.383775 0.923427i \(-0.625376\pi\)
−0.383775 + 0.923427i \(0.625376\pi\)
\(44\) 4.04373 0.609615
\(45\) 0 0
\(46\) 3.35624 0.494851
\(47\) 12.2995 1.79407 0.897036 0.441958i \(-0.145716\pi\)
0.897036 + 0.441958i \(0.145716\pi\)
\(48\) −0.682609 −0.0985261
\(49\) 18.3327 2.61896
\(50\) 0 0
\(51\) −1.65673 −0.231989
\(52\) −6.09474 −0.845189
\(53\) 4.09474 0.562456 0.281228 0.959641i \(-0.409258\pi\)
0.281228 + 0.959641i \(0.409258\pi\)
\(54\) −4.56432 −0.621125
\(55\) 0 0
\(56\) −13.6995 −1.83067
\(57\) −1.53844 −0.203772
\(58\) 0.879381 0.115468
\(59\) −1.39997 −0.182261 −0.0911304 0.995839i \(-0.529048\pi\)
−0.0911304 + 0.995839i \(0.529048\pi\)
\(60\) 0 0
\(61\) −5.69951 −0.729747 −0.364874 0.931057i \(-0.618888\pi\)
−0.364874 + 0.931057i \(0.618888\pi\)
\(62\) −4.56432 −0.579669
\(63\) 3.18694 0.401516
\(64\) 3.85376 0.481720
\(65\) 0 0
\(66\) −3.81051 −0.469042
\(67\) −5.28168 −0.645260 −0.322630 0.946525i \(-0.604567\pi\)
−0.322630 + 0.946525i \(0.604567\pi\)
\(68\) −1.43568 −0.174102
\(69\) 6.32308 0.761210
\(70\) 0 0
\(71\) −5.67692 −0.673726 −0.336863 0.941554i \(-0.609366\pi\)
−0.336863 + 0.941554i \(0.609366\pi\)
\(72\) −1.72344 −0.203110
\(73\) −9.07689 −1.06237 −0.531185 0.847256i \(-0.678253\pi\)
−0.531185 + 0.847256i \(0.678253\pi\)
\(74\) 0.0773672 0.00899375
\(75\) 0 0
\(76\) −1.33317 −0.152926
\(77\) 15.2664 1.73977
\(78\) 5.74324 0.650294
\(79\) −5.39997 −0.607544 −0.303772 0.952745i \(-0.598246\pi\)
−0.303772 + 0.952745i \(0.598246\pi\)
\(80\) 0 0
\(81\) −6.69951 −0.744390
\(82\) −8.71008 −0.961867
\(83\) −1.95627 −0.214729 −0.107364 0.994220i \(-0.534241\pi\)
−0.107364 + 0.994220i \(0.534241\pi\)
\(84\) −10.3231 −1.12634
\(85\) 0 0
\(86\) 4.11005 0.443198
\(87\) 1.65673 0.177621
\(88\) −8.25581 −0.880072
\(89\) −2.18949 −0.232085 −0.116043 0.993244i \(-0.537021\pi\)
−0.116043 + 0.993244i \(0.537021\pi\)
\(90\) 0 0
\(91\) −23.0096 −2.41206
\(92\) 5.47941 0.571268
\(93\) −8.59907 −0.891682
\(94\) −10.0437 −1.03593
\(95\) 0 0
\(96\) 8.93225 0.911644
\(97\) 2.16106 0.219423 0.109711 0.993963i \(-0.465007\pi\)
0.109711 + 0.993963i \(0.465007\pi\)
\(98\) −14.9704 −1.51224
\(99\) 1.92056 0.193024
\(100\) 0 0
\(101\) 12.5869 1.25244 0.626222 0.779645i \(-0.284600\pi\)
0.626222 + 0.779645i \(0.284600\pi\)
\(102\) 1.35288 0.133955
\(103\) 6.20479 0.611376 0.305688 0.952132i \(-0.401113\pi\)
0.305688 + 0.952132i \(0.401113\pi\)
\(104\) 12.4432 1.22016
\(105\) 0 0
\(106\) −3.34374 −0.324773
\(107\) 12.5481 1.21307 0.606533 0.795058i \(-0.292560\pi\)
0.606533 + 0.795058i \(0.292560\pi\)
\(108\) −7.45172 −0.717042
\(109\) −15.8096 −1.51428 −0.757140 0.653252i \(-0.773404\pi\)
−0.757140 + 0.653252i \(0.773404\pi\)
\(110\) 0 0
\(111\) 0.145758 0.0138347
\(112\) −2.23322 −0.211019
\(113\) 5.49472 0.516899 0.258450 0.966025i \(-0.416788\pi\)
0.258450 + 0.966025i \(0.416788\pi\)
\(114\) 1.25628 0.117662
\(115\) 0 0
\(116\) 1.43568 0.133300
\(117\) −2.89469 −0.267614
\(118\) 1.14321 0.105241
\(119\) −5.42015 −0.496865
\(120\) 0 0
\(121\) −1.79994 −0.163631
\(122\) 4.65418 0.421370
\(123\) −16.4096 −1.47960
\(124\) −7.45172 −0.669184
\(125\) 0 0
\(126\) −2.60243 −0.231843
\(127\) 8.61533 0.764487 0.382244 0.924062i \(-0.375152\pi\)
0.382244 + 0.924062i \(0.375152\pi\)
\(128\) 8.46509 0.748215
\(129\) 7.74324 0.681754
\(130\) 0 0
\(131\) −2.15378 −0.188176 −0.0940882 0.995564i \(-0.529994\pi\)
−0.0940882 + 0.995564i \(0.529994\pi\)
\(132\) −6.22105 −0.541473
\(133\) −5.03316 −0.436430
\(134\) 4.31299 0.372586
\(135\) 0 0
\(136\) 2.93113 0.251342
\(137\) −2.18949 −0.187061 −0.0935303 0.995616i \(-0.529815\pi\)
−0.0935303 + 0.995616i \(0.529815\pi\)
\(138\) −5.16339 −0.439537
\(139\) 22.5196 1.91009 0.955045 0.296460i \(-0.0958062\pi\)
0.955045 + 0.296460i \(0.0958062\pi\)
\(140\) 0 0
\(141\) −18.9222 −1.59353
\(142\) 4.63574 0.389022
\(143\) −13.8664 −1.15957
\(144\) −0.280946 −0.0234122
\(145\) 0 0
\(146\) 7.41213 0.613433
\(147\) −28.2038 −2.32621
\(148\) 0.126310 0.0103826
\(149\) 9.78697 0.801780 0.400890 0.916126i \(-0.368701\pi\)
0.400890 + 0.916126i \(0.368701\pi\)
\(150\) 0 0
\(151\) 5.87683 0.478250 0.239125 0.970989i \(-0.423139\pi\)
0.239125 + 0.970989i \(0.423139\pi\)
\(152\) 2.72185 0.220771
\(153\) −0.681874 −0.0551262
\(154\) −12.4664 −1.00457
\(155\) 0 0
\(156\) 9.37643 0.750715
\(157\) −10.1895 −0.813210 −0.406605 0.913604i \(-0.633287\pi\)
−0.406605 + 0.913604i \(0.633287\pi\)
\(158\) 4.40959 0.350808
\(159\) −6.29954 −0.499586
\(160\) 0 0
\(161\) 20.6865 1.63033
\(162\) 5.47078 0.429825
\(163\) 10.3659 0.811916 0.405958 0.913892i \(-0.366938\pi\)
0.405958 + 0.913892i \(0.366938\pi\)
\(164\) −14.2201 −1.11040
\(165\) 0 0
\(166\) 1.59748 0.123988
\(167\) 15.6048 1.20753 0.603766 0.797161i \(-0.293666\pi\)
0.603766 + 0.797161i \(0.293666\pi\)
\(168\) 21.0759 1.62604
\(169\) 7.89956 0.607659
\(170\) 0 0
\(171\) −0.633188 −0.0484211
\(172\) 6.71008 0.511639
\(173\) −0.571604 −0.0434583 −0.0217291 0.999764i \(-0.506917\pi\)
−0.0217291 + 0.999764i \(0.506917\pi\)
\(174\) −1.35288 −0.102562
\(175\) 0 0
\(176\) −1.34582 −0.101445
\(177\) 2.15378 0.161888
\(178\) 1.78792 0.134010
\(179\) 24.8864 1.86010 0.930050 0.367433i \(-0.119763\pi\)
0.930050 + 0.367433i \(0.119763\pi\)
\(180\) 0 0
\(181\) 6.95372 0.516866 0.258433 0.966029i \(-0.416794\pi\)
0.258433 + 0.966029i \(0.416794\pi\)
\(182\) 18.7895 1.39277
\(183\) 8.76838 0.648177
\(184\) −11.1869 −0.824712
\(185\) 0 0
\(186\) 7.02195 0.514875
\(187\) −3.26638 −0.238861
\(188\) −16.3974 −1.19590
\(189\) −28.1326 −2.04635
\(190\) 0 0
\(191\) −3.91254 −0.283102 −0.141551 0.989931i \(-0.545209\pi\)
−0.141551 + 0.989931i \(0.545209\pi\)
\(192\) −5.92880 −0.427874
\(193\) 9.20734 0.662759 0.331379 0.943498i \(-0.392486\pi\)
0.331379 + 0.943498i \(0.392486\pi\)
\(194\) −1.76471 −0.126699
\(195\) 0 0
\(196\) −24.4407 −1.74576
\(197\) 3.84622 0.274032 0.137016 0.990569i \(-0.456249\pi\)
0.137016 + 0.990569i \(0.456249\pi\)
\(198\) −1.56832 −0.111456
\(199\) −18.2864 −1.29629 −0.648145 0.761517i \(-0.724455\pi\)
−0.648145 + 0.761517i \(0.724455\pi\)
\(200\) 0 0
\(201\) 8.12557 0.573134
\(202\) −10.2784 −0.723185
\(203\) 5.42015 0.380420
\(204\) 2.20871 0.154641
\(205\) 0 0
\(206\) −5.06680 −0.353020
\(207\) 2.60243 0.180882
\(208\) 2.02842 0.140646
\(209\) −3.03316 −0.209808
\(210\) 0 0
\(211\) −19.2970 −1.32846 −0.664230 0.747529i \(-0.731240\pi\)
−0.664230 + 0.747529i \(0.731240\pi\)
\(212\) −5.45901 −0.374926
\(213\) 8.73362 0.598418
\(214\) −10.2467 −0.700448
\(215\) 0 0
\(216\) 15.2137 1.03516
\(217\) −28.1326 −1.90977
\(218\) 12.9100 0.874375
\(219\) 13.9643 0.943619
\(220\) 0 0
\(221\) 4.92311 0.331164
\(222\) −0.119025 −0.00798844
\(223\) −0.615334 −0.0412058 −0.0206029 0.999788i \(-0.506559\pi\)
−0.0206029 + 0.999788i \(0.506559\pi\)
\(224\) 29.2226 1.95252
\(225\) 0 0
\(226\) −4.48695 −0.298468
\(227\) −17.4712 −1.15960 −0.579801 0.814758i \(-0.696870\pi\)
−0.579801 + 0.814758i \(0.696870\pi\)
\(228\) 2.05101 0.135832
\(229\) 9.69951 0.640962 0.320481 0.947255i \(-0.396156\pi\)
0.320481 + 0.947255i \(0.396156\pi\)
\(230\) 0 0
\(231\) −23.4865 −1.54530
\(232\) −2.93113 −0.192438
\(233\) 8.15378 0.534172 0.267086 0.963673i \(-0.413939\pi\)
0.267086 + 0.963673i \(0.413939\pi\)
\(234\) 2.36378 0.154525
\(235\) 0 0
\(236\) 1.86641 0.121493
\(237\) 8.30756 0.539634
\(238\) 4.42607 0.286899
\(239\) 24.5991 1.59118 0.795591 0.605834i \(-0.207161\pi\)
0.795591 + 0.605834i \(0.207161\pi\)
\(240\) 0 0
\(241\) 25.7738 1.66024 0.830120 0.557585i \(-0.188272\pi\)
0.830120 + 0.557585i \(0.188272\pi\)
\(242\) 1.46982 0.0944838
\(243\) −6.46156 −0.414509
\(244\) 7.59844 0.486440
\(245\) 0 0
\(246\) 13.4000 0.854351
\(247\) 4.57160 0.290884
\(248\) 15.2137 0.966069
\(249\) 3.00961 0.190727
\(250\) 0 0
\(251\) 12.5991 0.795246 0.397623 0.917549i \(-0.369835\pi\)
0.397623 + 0.917549i \(0.369835\pi\)
\(252\) −4.24874 −0.267646
\(253\) 12.4664 0.783758
\(254\) −7.03523 −0.441430
\(255\) 0 0
\(256\) −14.6201 −0.913754
\(257\) −0.182203 −0.0113655 −0.00568275 0.999984i \(-0.501809\pi\)
−0.00568275 + 0.999984i \(0.501809\pi\)
\(258\) −6.32308 −0.393658
\(259\) 0.476860 0.0296307
\(260\) 0 0
\(261\) 0.681874 0.0422069
\(262\) 1.75876 0.108657
\(263\) −7.37643 −0.454850 −0.227425 0.973796i \(-0.573031\pi\)
−0.227425 + 0.973796i \(0.573031\pi\)
\(264\) 12.7011 0.781699
\(265\) 0 0
\(266\) 4.11005 0.252003
\(267\) 3.36841 0.206143
\(268\) 7.04140 0.430122
\(269\) 4.70206 0.286689 0.143345 0.989673i \(-0.454214\pi\)
0.143345 + 0.989673i \(0.454214\pi\)
\(270\) 0 0
\(271\) 9.92056 0.602631 0.301316 0.953524i \(-0.402574\pi\)
0.301316 + 0.953524i \(0.402574\pi\)
\(272\) 0.477817 0.0289719
\(273\) 35.3990 2.14245
\(274\) 1.78792 0.108012
\(275\) 0 0
\(276\) −8.42977 −0.507412
\(277\) 22.6297 1.35969 0.679843 0.733358i \(-0.262048\pi\)
0.679843 + 0.733358i \(0.262048\pi\)
\(278\) −18.3894 −1.10292
\(279\) −3.53918 −0.211885
\(280\) 0 0
\(281\) −3.95386 −0.235868 −0.117934 0.993021i \(-0.537627\pi\)
−0.117934 + 0.993021i \(0.537627\pi\)
\(282\) 15.4517 0.920137
\(283\) 13.0638 0.776561 0.388280 0.921541i \(-0.373069\pi\)
0.388280 + 0.921541i \(0.373069\pi\)
\(284\) 7.56832 0.449097
\(285\) 0 0
\(286\) 11.3232 0.669556
\(287\) −53.6854 −3.16895
\(288\) 3.67631 0.216628
\(289\) −15.8403 −0.931783
\(290\) 0 0
\(291\) −3.32468 −0.194896
\(292\) 12.1011 0.708162
\(293\) 9.69463 0.566366 0.283183 0.959066i \(-0.408610\pi\)
0.283183 + 0.959066i \(0.408610\pi\)
\(294\) 23.0311 1.34320
\(295\) 0 0
\(296\) −0.257878 −0.0149889
\(297\) −16.9537 −0.983755
\(298\) −7.99198 −0.462963
\(299\) −18.7895 −1.08663
\(300\) 0 0
\(301\) 25.3327 1.46015
\(302\) −4.79899 −0.276151
\(303\) −19.3643 −1.11245
\(304\) 0.443701 0.0254480
\(305\) 0 0
\(306\) 0.556814 0.0318309
\(307\) 28.5481 1.62932 0.814662 0.579936i \(-0.196923\pi\)
0.814662 + 0.579936i \(0.196923\pi\)
\(308\) −20.3527 −1.15970
\(309\) −9.54573 −0.543038
\(310\) 0 0
\(311\) 25.4070 1.44070 0.720350 0.693610i \(-0.243981\pi\)
0.720350 + 0.693610i \(0.243981\pi\)
\(312\) −19.1432 −1.08377
\(313\) −16.7999 −0.949589 −0.474794 0.880097i \(-0.657478\pi\)
−0.474794 + 0.880097i \(0.657478\pi\)
\(314\) 8.32068 0.469563
\(315\) 0 0
\(316\) 7.19910 0.404981
\(317\) 31.7505 1.78329 0.891643 0.452738i \(-0.149553\pi\)
0.891643 + 0.452738i \(0.149553\pi\)
\(318\) 5.14416 0.288470
\(319\) 3.26638 0.182882
\(320\) 0 0
\(321\) −19.3045 −1.07747
\(322\) −16.8925 −0.941382
\(323\) 1.07689 0.0599197
\(324\) 8.93161 0.496201
\(325\) 0 0
\(326\) −8.46470 −0.468816
\(327\) 24.3221 1.34502
\(328\) 29.0322 1.60304
\(329\) −61.9055 −3.41296
\(330\) 0 0
\(331\) −9.96429 −0.547687 −0.273843 0.961774i \(-0.588295\pi\)
−0.273843 + 0.961774i \(0.588295\pi\)
\(332\) 2.60805 0.143135
\(333\) 0.0599906 0.00328747
\(334\) −12.7428 −0.697253
\(335\) 0 0
\(336\) 3.43568 0.187432
\(337\) −23.1918 −1.26334 −0.631669 0.775238i \(-0.717630\pi\)
−0.631669 + 0.775238i \(0.717630\pi\)
\(338\) −6.45074 −0.350874
\(339\) −8.45331 −0.459121
\(340\) 0 0
\(341\) −16.9537 −0.918095
\(342\) 0.517058 0.0279593
\(343\) −57.0393 −3.07983
\(344\) −13.6995 −0.738628
\(345\) 0 0
\(346\) 0.466769 0.0250936
\(347\) −21.1352 −1.13460 −0.567298 0.823512i \(-0.692011\pi\)
−0.567298 + 0.823512i \(0.692011\pi\)
\(348\) −2.20871 −0.118400
\(349\) −5.04628 −0.270121 −0.135061 0.990837i \(-0.543123\pi\)
−0.135061 + 0.990837i \(0.543123\pi\)
\(350\) 0 0
\(351\) 25.5528 1.36391
\(352\) 17.6106 0.938648
\(353\) −12.5634 −0.668680 −0.334340 0.942452i \(-0.608513\pi\)
−0.334340 + 0.942452i \(0.608513\pi\)
\(354\) −1.75876 −0.0934772
\(355\) 0 0
\(356\) 2.91897 0.154705
\(357\) 8.33861 0.441326
\(358\) −20.3221 −1.07406
\(359\) −17.6534 −0.931709 −0.465855 0.884861i \(-0.654253\pi\)
−0.465855 + 0.884861i \(0.654253\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.67837 −0.298448
\(363\) 2.76911 0.145341
\(364\) 30.6758 1.60785
\(365\) 0 0
\(366\) −7.16021 −0.374270
\(367\) −4.43409 −0.231457 −0.115729 0.993281i \(-0.536920\pi\)
−0.115729 + 0.993281i \(0.536920\pi\)
\(368\) −1.82363 −0.0950634
\(369\) −6.75381 −0.351589
\(370\) 0 0
\(371\) −20.6095 −1.06999
\(372\) 11.4641 0.594384
\(373\) 20.8735 1.08079 0.540396 0.841411i \(-0.318275\pi\)
0.540396 + 0.841411i \(0.318275\pi\)
\(374\) 2.66730 0.137923
\(375\) 0 0
\(376\) 33.4775 1.72647
\(377\) −4.92311 −0.253553
\(378\) 22.9729 1.18160
\(379\) −6.79994 −0.349290 −0.174645 0.984632i \(-0.555878\pi\)
−0.174645 + 0.984632i \(0.555878\pi\)
\(380\) 0 0
\(381\) −13.2542 −0.679034
\(382\) 3.19496 0.163468
\(383\) 10.9078 0.557363 0.278681 0.960384i \(-0.410103\pi\)
0.278681 + 0.960384i \(0.410103\pi\)
\(384\) −13.0231 −0.664581
\(385\) 0 0
\(386\) −7.51866 −0.382690
\(387\) 3.18694 0.162001
\(388\) −2.88107 −0.146264
\(389\) 20.9326 1.06132 0.530662 0.847584i \(-0.321943\pi\)
0.530662 + 0.847584i \(0.321943\pi\)
\(390\) 0 0
\(391\) −4.42607 −0.223836
\(392\) 49.8989 2.52027
\(393\) 3.31347 0.167142
\(394\) −3.14080 −0.158231
\(395\) 0 0
\(396\) −2.56044 −0.128667
\(397\) −13.1221 −0.658578 −0.329289 0.944229i \(-0.606809\pi\)
−0.329289 + 0.944229i \(0.606809\pi\)
\(398\) 14.9326 0.748503
\(399\) 7.74324 0.387647
\(400\) 0 0
\(401\) −11.5528 −0.576919 −0.288459 0.957492i \(-0.593143\pi\)
−0.288459 + 0.957492i \(0.593143\pi\)
\(402\) −6.63530 −0.330939
\(403\) 25.5528 1.27288
\(404\) −16.7805 −0.834863
\(405\) 0 0
\(406\) −4.42607 −0.219662
\(407\) 0.287373 0.0142445
\(408\) −4.50938 −0.223248
\(409\) −18.9433 −0.936686 −0.468343 0.883547i \(-0.655149\pi\)
−0.468343 + 0.883547i \(0.655149\pi\)
\(410\) 0 0
\(411\) 3.36841 0.166151
\(412\) −8.27207 −0.407536
\(413\) 7.04628 0.346725
\(414\) −2.12513 −0.104445
\(415\) 0 0
\(416\) −26.5428 −1.30137
\(417\) −34.6452 −1.69658
\(418\) 2.47686 0.121147
\(419\) 27.2664 1.33205 0.666025 0.745930i \(-0.267994\pi\)
0.666025 + 0.745930i \(0.267994\pi\)
\(420\) 0 0
\(421\) 6.66635 0.324898 0.162449 0.986717i \(-0.448061\pi\)
0.162449 + 0.986717i \(0.448061\pi\)
\(422\) 15.7578 0.767078
\(423\) −7.78792 −0.378662
\(424\) 11.1453 0.541263
\(425\) 0 0
\(426\) −7.13183 −0.345538
\(427\) 28.6865 1.38824
\(428\) −16.7287 −0.808614
\(429\) 21.3327 1.02995
\(430\) 0 0
\(431\) −32.4548 −1.56329 −0.781646 0.623723i \(-0.785619\pi\)
−0.781646 + 0.623723i \(0.785619\pi\)
\(432\) 2.48005 0.119321
\(433\) −33.8168 −1.62513 −0.812567 0.582868i \(-0.801930\pi\)
−0.812567 + 0.582868i \(0.801930\pi\)
\(434\) 22.9729 1.10274
\(435\) 0 0
\(436\) 21.0769 1.00940
\(437\) −4.11005 −0.196610
\(438\) −11.4032 −0.544864
\(439\) −27.8453 −1.32898 −0.664491 0.747296i \(-0.731352\pi\)
−0.664491 + 0.747296i \(0.731352\pi\)
\(440\) 0 0
\(441\) −11.6080 −0.552764
\(442\) −4.02018 −0.191221
\(443\) −22.7754 −1.08209 −0.541047 0.840992i \(-0.681972\pi\)
−0.541047 + 0.840992i \(0.681972\pi\)
\(444\) −0.194321 −0.00922206
\(445\) 0 0
\(446\) 0.502478 0.0237930
\(447\) −15.0567 −0.712158
\(448\) −19.3966 −0.916404
\(449\) 24.8507 1.17278 0.586389 0.810029i \(-0.300549\pi\)
0.586389 + 0.810029i \(0.300549\pi\)
\(450\) 0 0
\(451\) −32.3527 −1.52343
\(452\) −7.32541 −0.344558
\(453\) −9.04118 −0.424792
\(454\) 14.2669 0.669577
\(455\) 0 0
\(456\) −4.18742 −0.196094
\(457\) 7.33270 0.343009 0.171505 0.985183i \(-0.445137\pi\)
0.171505 + 0.985183i \(0.445137\pi\)
\(458\) −7.92056 −0.370104
\(459\) 6.01923 0.280953
\(460\) 0 0
\(461\) 10.3076 0.480071 0.240035 0.970764i \(-0.422841\pi\)
0.240035 + 0.970764i \(0.422841\pi\)
\(462\) 19.1789 0.892284
\(463\) −7.80249 −0.362613 −0.181306 0.983427i \(-0.558033\pi\)
−0.181306 + 0.983427i \(0.558033\pi\)
\(464\) −0.477817 −0.0221821
\(465\) 0 0
\(466\) −6.65833 −0.308441
\(467\) 7.37643 0.341340 0.170670 0.985328i \(-0.445407\pi\)
0.170670 + 0.985328i \(0.445407\pi\)
\(468\) 3.85912 0.178388
\(469\) 26.5835 1.22751
\(470\) 0 0
\(471\) 15.6760 0.722310
\(472\) −3.81051 −0.175393
\(473\) 15.2664 0.701949
\(474\) −6.78390 −0.311595
\(475\) 0 0
\(476\) 7.22601 0.331204
\(477\) −2.59274 −0.118714
\(478\) −20.0875 −0.918779
\(479\) −16.1458 −0.737719 −0.368859 0.929485i \(-0.620252\pi\)
−0.368859 + 0.929485i \(0.620252\pi\)
\(480\) 0 0
\(481\) −0.433131 −0.0197491
\(482\) −21.0468 −0.958654
\(483\) −31.8251 −1.44809
\(484\) 2.39964 0.109074
\(485\) 0 0
\(486\) 5.27647 0.239345
\(487\) 8.40566 0.380897 0.190448 0.981697i \(-0.439006\pi\)
0.190448 + 0.981697i \(0.439006\pi\)
\(488\) −15.5132 −0.702250
\(489\) −15.9473 −0.721162
\(490\) 0 0
\(491\) −36.0855 −1.62852 −0.814259 0.580502i \(-0.802856\pi\)
−0.814259 + 0.580502i \(0.802856\pi\)
\(492\) 21.8768 0.986284
\(493\) −1.15969 −0.0522298
\(494\) −3.73315 −0.167962
\(495\) 0 0
\(496\) 2.48005 0.111357
\(497\) 28.5728 1.28167
\(498\) −2.45763 −0.110129
\(499\) −3.30035 −0.147744 −0.0738720 0.997268i \(-0.523536\pi\)
−0.0738720 + 0.997268i \(0.523536\pi\)
\(500\) 0 0
\(501\) −24.0071 −1.07256
\(502\) −10.2883 −0.459191
\(503\) −3.53530 −0.157631 −0.0788157 0.996889i \(-0.525114\pi\)
−0.0788157 + 0.996889i \(0.525114\pi\)
\(504\) 8.67437 0.386387
\(505\) 0 0
\(506\) −10.1800 −0.452557
\(507\) −12.1530 −0.539736
\(508\) −11.4857 −0.509597
\(509\) 22.4096 0.993287 0.496644 0.867955i \(-0.334566\pi\)
0.496644 + 0.867955i \(0.334566\pi\)
\(510\) 0 0
\(511\) 45.6854 2.02100
\(512\) −4.99151 −0.220596
\(513\) 5.58946 0.246781
\(514\) 0.148786 0.00656266
\(515\) 0 0
\(516\) −10.3231 −0.454448
\(517\) −37.3065 −1.64074
\(518\) −0.389401 −0.0171093
\(519\) 0.879381 0.0386006
\(520\) 0 0
\(521\) 34.4402 1.50885 0.754426 0.656385i \(-0.227915\pi\)
0.754426 + 0.656385i \(0.227915\pi\)
\(522\) −0.556814 −0.0243711
\(523\) 30.4451 1.33127 0.665635 0.746277i \(-0.268161\pi\)
0.665635 + 0.746277i \(0.268161\pi\)
\(524\) 2.87136 0.125436
\(525\) 0 0
\(526\) 6.02355 0.262639
\(527\) 6.01923 0.262202
\(528\) 2.07046 0.0901053
\(529\) −6.10750 −0.265543
\(530\) 0 0
\(531\) 0.886445 0.0384685
\(532\) 6.71008 0.290919
\(533\) 48.7623 2.11213
\(534\) −2.75062 −0.119031
\(535\) 0 0
\(536\) −14.3759 −0.620946
\(537\) −38.2864 −1.65218
\(538\) −3.83967 −0.165540
\(539\) −55.6060 −2.39512
\(540\) 0 0
\(541\) −12.2794 −0.527931 −0.263965 0.964532i \(-0.585030\pi\)
−0.263965 + 0.964532i \(0.585030\pi\)
\(542\) −8.10107 −0.347971
\(543\) −10.6979 −0.459091
\(544\) −6.25244 −0.268071
\(545\) 0 0
\(546\) −28.9066 −1.23709
\(547\) −0.250928 −0.0107289 −0.00536446 0.999986i \(-0.501708\pi\)
−0.00536446 + 0.999986i \(0.501708\pi\)
\(548\) 2.91897 0.124692
\(549\) 3.60886 0.154022
\(550\) 0 0
\(551\) −1.07689 −0.0458770
\(552\) 17.2105 0.732527
\(553\) 27.1789 1.15577
\(554\) −18.4793 −0.785109
\(555\) 0 0
\(556\) −30.0226 −1.27324
\(557\) −1.67596 −0.0710128 −0.0355064 0.999369i \(-0.511304\pi\)
−0.0355064 + 0.999369i \(0.511304\pi\)
\(558\) 2.89007 0.122347
\(559\) −23.0096 −0.973203
\(560\) 0 0
\(561\) 5.02514 0.212162
\(562\) 3.22870 0.136195
\(563\) −20.6605 −0.870737 −0.435368 0.900252i \(-0.643382\pi\)
−0.435368 + 0.900252i \(0.643382\pi\)
\(564\) 25.2265 1.06223
\(565\) 0 0
\(566\) −10.6678 −0.448401
\(567\) 33.7197 1.41609
\(568\) −15.4517 −0.648340
\(569\) 40.9673 1.71744 0.858720 0.512445i \(-0.171260\pi\)
0.858720 + 0.512445i \(0.171260\pi\)
\(570\) 0 0
\(571\) 24.2070 1.01303 0.506515 0.862231i \(-0.330933\pi\)
0.506515 + 0.862231i \(0.330933\pi\)
\(572\) 18.4863 0.772952
\(573\) 6.01923 0.251457
\(574\) 43.8392 1.82981
\(575\) 0 0
\(576\) −2.44016 −0.101673
\(577\) −40.2864 −1.67715 −0.838573 0.544790i \(-0.816609\pi\)
−0.838573 + 0.544790i \(0.816609\pi\)
\(578\) 12.9351 0.538029
\(579\) −14.1650 −0.588677
\(580\) 0 0
\(581\) 9.84622 0.408490
\(582\) 2.71491 0.112537
\(583\) −12.4200 −0.514384
\(584\) −24.7059 −1.02234
\(585\) 0 0
\(586\) −7.91658 −0.327031
\(587\) 34.7754 1.43534 0.717668 0.696385i \(-0.245210\pi\)
0.717668 + 0.696385i \(0.245210\pi\)
\(588\) 37.6006 1.55062
\(589\) 5.58946 0.230310
\(590\) 0 0
\(591\) −5.91720 −0.243401
\(592\) −0.0420379 −0.00172775
\(593\) 35.0864 1.44082 0.720412 0.693546i \(-0.243953\pi\)
0.720412 + 0.693546i \(0.243953\pi\)
\(594\) 13.8443 0.568039
\(595\) 0 0
\(596\) −13.0477 −0.534456
\(597\) 28.1326 1.15139
\(598\) 15.3434 0.627439
\(599\) 30.8201 1.25928 0.629638 0.776889i \(-0.283203\pi\)
0.629638 + 0.776889i \(0.283203\pi\)
\(600\) 0 0
\(601\) −7.10654 −0.289882 −0.144941 0.989440i \(-0.546299\pi\)
−0.144941 + 0.989440i \(0.546299\pi\)
\(602\) −20.6865 −0.843120
\(603\) 3.34430 0.136190
\(604\) −7.83484 −0.318795
\(605\) 0 0
\(606\) 15.8127 0.642349
\(607\) −2.01531 −0.0817987 −0.0408994 0.999163i \(-0.513022\pi\)
−0.0408994 + 0.999163i \(0.513022\pi\)
\(608\) −5.80602 −0.235465
\(609\) −8.33861 −0.337897
\(610\) 0 0
\(611\) 56.2286 2.27477
\(612\) 0.909056 0.0367464
\(613\) 11.6096 0.468909 0.234455 0.972127i \(-0.424670\pi\)
0.234455 + 0.972127i \(0.424670\pi\)
\(614\) −23.3122 −0.940803
\(615\) 0 0
\(616\) 41.5528 1.67421
\(617\) −12.4307 −0.500442 −0.250221 0.968189i \(-0.580503\pi\)
−0.250221 + 0.968189i \(0.580503\pi\)
\(618\) 7.79499 0.313560
\(619\) 3.85424 0.154915 0.0774575 0.996996i \(-0.475320\pi\)
0.0774575 + 0.996996i \(0.475320\pi\)
\(620\) 0 0
\(621\) −22.9729 −0.921873
\(622\) −20.7472 −0.831888
\(623\) 11.0200 0.441509
\(624\) −3.12062 −0.124925
\(625\) 0 0
\(626\) 13.7187 0.548311
\(627\) 4.66635 0.186356
\(628\) 13.5844 0.542075
\(629\) −0.102029 −0.00406814
\(630\) 0 0
\(631\) −16.8794 −0.671958 −0.335979 0.941870i \(-0.609067\pi\)
−0.335979 + 0.941870i \(0.609067\pi\)
\(632\) −14.6979 −0.584652
\(633\) 29.6873 1.17997
\(634\) −25.9273 −1.02970
\(635\) 0 0
\(636\) 8.39838 0.333017
\(637\) 83.8098 3.32067
\(638\) −2.66730 −0.105600
\(639\) 3.59456 0.142199
\(640\) 0 0
\(641\) 43.4855 1.71757 0.858787 0.512332i \(-0.171218\pi\)
0.858787 + 0.512332i \(0.171218\pi\)
\(642\) 15.7639 0.622153
\(643\) −11.0689 −0.436514 −0.218257 0.975891i \(-0.570037\pi\)
−0.218257 + 0.975891i \(0.570037\pi\)
\(644\) −27.5787 −1.08675
\(645\) 0 0
\(646\) −0.879381 −0.0345988
\(647\) −5.61766 −0.220853 −0.110427 0.993884i \(-0.535222\pi\)
−0.110427 + 0.993884i \(0.535222\pi\)
\(648\) −18.2351 −0.716341
\(649\) 4.24634 0.166683
\(650\) 0 0
\(651\) 43.2805 1.69630
\(652\) −13.8195 −0.541213
\(653\) −44.5211 −1.74224 −0.871122 0.491066i \(-0.836607\pi\)
−0.871122 + 0.491066i \(0.836607\pi\)
\(654\) −19.8613 −0.776639
\(655\) 0 0
\(656\) 4.73267 0.184780
\(657\) 5.74738 0.224227
\(658\) 50.5517 1.97071
\(659\) −6.89154 −0.268456 −0.134228 0.990950i \(-0.542856\pi\)
−0.134228 + 0.990950i \(0.542856\pi\)
\(660\) 0 0
\(661\) −44.3989 −1.72692 −0.863458 0.504421i \(-0.831706\pi\)
−0.863458 + 0.504421i \(0.831706\pi\)
\(662\) 8.13678 0.316245
\(663\) −7.57393 −0.294147
\(664\) −5.32468 −0.206637
\(665\) 0 0
\(666\) −0.0489880 −0.00189825
\(667\) 4.42607 0.171378
\(668\) −20.8039 −0.804926
\(669\) 0.946657 0.0365999
\(670\) 0 0
\(671\) 17.2875 0.667377
\(672\) −44.9574 −1.73427
\(673\) −38.6214 −1.48875 −0.744374 0.667763i \(-0.767252\pi\)
−0.744374 + 0.667763i \(0.767252\pi\)
\(674\) 18.9383 0.729476
\(675\) 0 0
\(676\) −10.5315 −0.405057
\(677\) −43.3917 −1.66768 −0.833840 0.552006i \(-0.813862\pi\)
−0.833840 + 0.552006i \(0.813862\pi\)
\(678\) 6.90293 0.265105
\(679\) −10.8770 −0.417420
\(680\) 0 0
\(681\) 26.8784 1.02998
\(682\) 13.8443 0.530126
\(683\) 34.7123 1.32823 0.664114 0.747631i \(-0.268809\pi\)
0.664114 + 0.747631i \(0.268809\pi\)
\(684\) 0.844150 0.0322769
\(685\) 0 0
\(686\) 46.5779 1.77835
\(687\) −14.9222 −0.569316
\(688\) −2.23322 −0.0851406
\(689\) 18.7195 0.713158
\(690\) 0 0
\(691\) 2.94570 0.112060 0.0560299 0.998429i \(-0.482156\pi\)
0.0560299 + 0.998429i \(0.482156\pi\)
\(692\) 0.762048 0.0289687
\(693\) −9.66649 −0.367200
\(694\) 17.2589 0.655138
\(695\) 0 0
\(696\) 4.50938 0.170928
\(697\) 11.4865 0.435081
\(698\) 4.12076 0.155973
\(699\) −12.5441 −0.474463
\(700\) 0 0
\(701\) 18.5920 0.702210 0.351105 0.936336i \(-0.385806\pi\)
0.351105 + 0.936336i \(0.385806\pi\)
\(702\) −20.8663 −0.787546
\(703\) −0.0947438 −0.00357333
\(704\) −11.6891 −0.440549
\(705\) 0 0
\(706\) 10.2592 0.386109
\(707\) −63.3519 −2.38259
\(708\) −2.87136 −0.107912
\(709\) 39.8443 1.49638 0.748192 0.663482i \(-0.230922\pi\)
0.748192 + 0.663482i \(0.230922\pi\)
\(710\) 0 0
\(711\) 3.41920 0.128230
\(712\) −5.95946 −0.223340
\(713\) −22.9729 −0.860344
\(714\) −6.80926 −0.254830
\(715\) 0 0
\(716\) −33.1780 −1.23992
\(717\) −37.8443 −1.41332
\(718\) 14.4156 0.537987
\(719\) 21.2321 0.791824 0.395912 0.918288i \(-0.370428\pi\)
0.395912 + 0.918288i \(0.370428\pi\)
\(720\) 0 0
\(721\) −31.2297 −1.16306
\(722\) −0.816594 −0.0303905
\(723\) −39.6516 −1.47466
\(724\) −9.27052 −0.344536
\(725\) 0 0
\(726\) −2.26124 −0.0839225
\(727\) 49.1658 1.82346 0.911729 0.410792i \(-0.134748\pi\)
0.911729 + 0.410792i \(0.134748\pi\)
\(728\) −62.6287 −2.32118
\(729\) 30.0393 1.11257
\(730\) 0 0
\(731\) −5.42015 −0.200472
\(732\) −11.6898 −0.432066
\(733\) −1.60498 −0.0592815 −0.0296407 0.999561i \(-0.509436\pi\)
−0.0296407 + 0.999561i \(0.509436\pi\)
\(734\) 3.62085 0.133648
\(735\) 0 0
\(736\) 23.8630 0.879603
\(737\) 16.0202 0.590111
\(738\) 5.51512 0.203014
\(739\) 8.13264 0.299164 0.149582 0.988749i \(-0.452207\pi\)
0.149582 + 0.988749i \(0.452207\pi\)
\(740\) 0 0
\(741\) −7.03316 −0.258370
\(742\) 16.8296 0.617834
\(743\) 42.9261 1.57481 0.787403 0.616439i \(-0.211425\pi\)
0.787403 + 0.616439i \(0.211425\pi\)
\(744\) −23.4054 −0.858083
\(745\) 0 0
\(746\) −17.0452 −0.624070
\(747\) 1.23869 0.0453212
\(748\) 4.35465 0.159222
\(749\) −63.1564 −2.30768
\(750\) 0 0
\(751\) 4.05685 0.148037 0.0740183 0.997257i \(-0.476418\pi\)
0.0740183 + 0.997257i \(0.476418\pi\)
\(752\) 5.45731 0.199008
\(753\) −19.3830 −0.706355
\(754\) 4.02018 0.146406
\(755\) 0 0
\(756\) 37.5057 1.36407
\(757\) −22.6151 −0.821960 −0.410980 0.911644i \(-0.634813\pi\)
−0.410980 + 0.911644i \(0.634813\pi\)
\(758\) 5.55279 0.201687
\(759\) −19.1789 −0.696151
\(760\) 0 0
\(761\) −41.3609 −1.49933 −0.749666 0.661817i \(-0.769786\pi\)
−0.749666 + 0.661817i \(0.769786\pi\)
\(762\) 10.8233 0.392087
\(763\) 79.5720 2.88070
\(764\) 5.21610 0.188712
\(765\) 0 0
\(766\) −8.90725 −0.321832
\(767\) −6.40011 −0.231095
\(768\) 22.4922 0.811616
\(769\) 17.9196 0.646197 0.323099 0.946365i \(-0.395275\pi\)
0.323099 + 0.946365i \(0.395275\pi\)
\(770\) 0 0
\(771\) 0.280309 0.0100951
\(772\) −12.2750 −0.441787
\(773\) 21.9043 0.787843 0.393921 0.919144i \(-0.371118\pi\)
0.393921 + 0.919144i \(0.371118\pi\)
\(774\) −2.60243 −0.0935426
\(775\) 0 0
\(776\) 5.88209 0.211155
\(777\) −0.733623 −0.0263186
\(778\) −17.0934 −0.612829
\(779\) 10.6663 0.382162
\(780\) 0 0
\(781\) 17.2190 0.616144
\(782\) 3.61430 0.129247
\(783\) −6.01923 −0.215110
\(784\) 8.13423 0.290508
\(785\) 0 0
\(786\) −2.70576 −0.0965112
\(787\) 31.6354 1.12768 0.563840 0.825884i \(-0.309324\pi\)
0.563840 + 0.825884i \(0.309324\pi\)
\(788\) −5.12768 −0.182666
\(789\) 11.3482 0.404007
\(790\) 0 0
\(791\) −27.6558 −0.983326
\(792\) 5.22748 0.185750
\(793\) −26.0559 −0.925272
\(794\) 10.7154 0.380275
\(795\) 0 0
\(796\) 24.3790 0.864090
\(797\) 15.4486 0.547217 0.273608 0.961841i \(-0.411783\pi\)
0.273608 + 0.961841i \(0.411783\pi\)
\(798\) −6.32308 −0.223835
\(799\) 13.2452 0.468583
\(800\) 0 0
\(801\) 1.38636 0.0489845
\(802\) 9.43394 0.333124
\(803\) 27.5317 0.971571
\(804\) −10.8328 −0.382044
\(805\) 0 0
\(806\) −20.8663 −0.734983
\(807\) −7.23385 −0.254644
\(808\) 34.2597 1.20525
\(809\) −32.9326 −1.15785 −0.578924 0.815382i \(-0.696527\pi\)
−0.578924 + 0.815382i \(0.696527\pi\)
\(810\) 0 0
\(811\) −25.5895 −0.898567 −0.449284 0.893389i \(-0.648321\pi\)
−0.449284 + 0.893389i \(0.648321\pi\)
\(812\) −7.22601 −0.253583
\(813\) −15.2622 −0.535270
\(814\) −0.234667 −0.00822508
\(815\) 0 0
\(816\) −0.735094 −0.0257334
\(817\) −5.03316 −0.176088
\(818\) 15.4690 0.540860
\(819\) 14.5694 0.509097
\(820\) 0 0
\(821\) 17.8674 0.623575 0.311788 0.950152i \(-0.399072\pi\)
0.311788 + 0.950152i \(0.399072\pi\)
\(822\) −2.75062 −0.0959389
\(823\) 18.4533 0.643242 0.321621 0.946868i \(-0.395772\pi\)
0.321621 + 0.946868i \(0.395772\pi\)
\(824\) 16.8885 0.588339
\(825\) 0 0
\(826\) −5.75395 −0.200206
\(827\) −47.5460 −1.65334 −0.826668 0.562690i \(-0.809767\pi\)
−0.826668 + 0.562690i \(0.809767\pi\)
\(828\) −3.46950 −0.120573
\(829\) 19.0308 0.660965 0.330483 0.943812i \(-0.392788\pi\)
0.330483 + 0.943812i \(0.392788\pi\)
\(830\) 0 0
\(831\) −34.8145 −1.20770
\(832\) 17.6179 0.610790
\(833\) 19.7423 0.684029
\(834\) 28.2911 0.979640
\(835\) 0 0
\(836\) 4.04373 0.139855
\(837\) 31.2420 1.07988
\(838\) −22.2656 −0.769151
\(839\) 3.04022 0.104960 0.0524801 0.998622i \(-0.483287\pi\)
0.0524801 + 0.998622i \(0.483287\pi\)
\(840\) 0 0
\(841\) −27.8403 −0.960011
\(842\) −5.44370 −0.187602
\(843\) 6.08280 0.209503
\(844\) 25.7262 0.885534
\(845\) 0 0
\(846\) 6.35957 0.218647
\(847\) 9.05940 0.311285
\(848\) 1.81684 0.0623906
\(849\) −20.0979 −0.689758
\(850\) 0 0
\(851\) 0.389401 0.0133485
\(852\) −11.6434 −0.398898
\(853\) 18.2201 0.623844 0.311922 0.950108i \(-0.399027\pi\)
0.311922 + 0.950108i \(0.399027\pi\)
\(854\) −23.4253 −0.801596
\(855\) 0 0
\(856\) 34.1539 1.16736
\(857\) −19.3611 −0.661363 −0.330682 0.943742i \(-0.607279\pi\)
−0.330682 + 0.943742i \(0.607279\pi\)
\(858\) −17.4202 −0.594714
\(859\) 3.44129 0.117415 0.0587077 0.998275i \(-0.481302\pi\)
0.0587077 + 0.998275i \(0.481302\pi\)
\(860\) 0 0
\(861\) 82.5921 2.81473
\(862\) 26.5024 0.902674
\(863\) 26.5471 0.903674 0.451837 0.892101i \(-0.350769\pi\)
0.451837 + 0.892101i \(0.350769\pi\)
\(864\) −32.4525 −1.10406
\(865\) 0 0
\(866\) 27.6146 0.938383
\(867\) 24.3694 0.827630
\(868\) 37.5057 1.27303
\(869\) 16.3790 0.555619
\(870\) 0 0
\(871\) −24.1458 −0.818148
\(872\) −43.0312 −1.45722
\(873\) −1.36836 −0.0463120
\(874\) 3.35624 0.113527
\(875\) 0 0
\(876\) −18.6168 −0.629004
\(877\) −20.5495 −0.693908 −0.346954 0.937882i \(-0.612784\pi\)
−0.346954 + 0.937882i \(0.612784\pi\)
\(878\) 22.7383 0.767380
\(879\) −14.9146 −0.503059
\(880\) 0 0
\(881\) −31.1911 −1.05085 −0.525427 0.850839i \(-0.676094\pi\)
−0.525427 + 0.850839i \(0.676094\pi\)
\(882\) 9.47906 0.319177
\(883\) −14.1861 −0.477401 −0.238701 0.971093i \(-0.576721\pi\)
−0.238701 + 0.971093i \(0.576721\pi\)
\(884\) −6.56336 −0.220750
\(885\) 0 0
\(886\) 18.5983 0.624822
\(887\) −46.6046 −1.56483 −0.782415 0.622757i \(-0.786012\pi\)
−0.782415 + 0.622757i \(0.786012\pi\)
\(888\) 0.396732 0.0133134
\(889\) −43.3623 −1.45433
\(890\) 0 0
\(891\) 20.3207 0.680768
\(892\) 0.820347 0.0274672
\(893\) 12.2995 0.411588
\(894\) 12.2952 0.411214
\(895\) 0 0
\(896\) −42.6061 −1.42337
\(897\) 28.9066 0.965164
\(898\) −20.2930 −0.677185
\(899\) −6.01923 −0.200752
\(900\) 0 0
\(901\) 4.40959 0.146905
\(902\) 26.4191 0.879658
\(903\) −38.9729 −1.29694
\(904\) 14.9558 0.497422
\(905\) 0 0
\(906\) 7.38297 0.245283
\(907\) 0.754028 0.0250371 0.0125185 0.999922i \(-0.496015\pi\)
0.0125185 + 0.999922i \(0.496015\pi\)
\(908\) 23.2921 0.772976
\(909\) −7.96988 −0.264344
\(910\) 0 0
\(911\) −53.1413 −1.76065 −0.880325 0.474371i \(-0.842675\pi\)
−0.880325 + 0.474371i \(0.842675\pi\)
\(912\) −0.682609 −0.0226034
\(913\) 5.93368 0.196376
\(914\) −5.98784 −0.198060
\(915\) 0 0
\(916\) −12.9311 −0.427257
\(917\) 10.8403 0.357979
\(918\) −4.91527 −0.162228
\(919\) 37.4202 1.23438 0.617189 0.786815i \(-0.288272\pi\)
0.617189 + 0.786815i \(0.288272\pi\)
\(920\) 0 0
\(921\) −43.9196 −1.44720
\(922\) −8.41709 −0.277202
\(923\) −25.9526 −0.854241
\(924\) 31.3116 1.03007
\(925\) 0 0
\(926\) 6.37147 0.209379
\(927\) −3.92880 −0.129039
\(928\) 6.25244 0.205247
\(929\) −19.1981 −0.629871 −0.314935 0.949113i \(-0.601983\pi\)
−0.314935 + 0.949113i \(0.601983\pi\)
\(930\) 0 0
\(931\) 18.3327 0.600830
\(932\) −10.8704 −0.356072
\(933\) −39.0873 −1.27966
\(934\) −6.02355 −0.197096
\(935\) 0 0
\(936\) −7.87890 −0.257530
\(937\) 20.0095 0.653681 0.326840 0.945080i \(-0.394016\pi\)
0.326840 + 0.945080i \(0.394016\pi\)
\(938\) −21.7080 −0.708790
\(939\) 25.8458 0.843445
\(940\) 0 0
\(941\) 15.3327 0.499832 0.249916 0.968268i \(-0.419597\pi\)
0.249916 + 0.968268i \(0.419597\pi\)
\(942\) −12.8009 −0.417076
\(943\) −43.8392 −1.42760
\(944\) −0.621168 −0.0202173
\(945\) 0 0
\(946\) −12.4664 −0.405319
\(947\) 29.8217 0.969076 0.484538 0.874770i \(-0.338988\pi\)
0.484538 + 0.874770i \(0.338988\pi\)
\(948\) −11.0754 −0.359713
\(949\) −41.4959 −1.34702
\(950\) 0 0
\(951\) −48.8464 −1.58395
\(952\) −14.7529 −0.478143
\(953\) −31.4938 −1.02018 −0.510091 0.860120i \(-0.670388\pi\)
−0.510091 + 0.860120i \(0.670388\pi\)
\(954\) 2.11722 0.0685475
\(955\) 0 0
\(956\) −32.7948 −1.06066
\(957\) −5.02514 −0.162440
\(958\) 13.1845 0.425973
\(959\) 11.0200 0.355856
\(960\) 0 0
\(961\) 0.242050 0.00780806
\(962\) 0.353692 0.0114035
\(963\) −7.94528 −0.256033
\(964\) −34.3610 −1.10669
\(965\) 0 0
\(966\) 25.9882 0.836156
\(967\) −1.16143 −0.0373490 −0.0186745 0.999826i \(-0.505945\pi\)
−0.0186745 + 0.999826i \(0.505945\pi\)
\(968\) −4.89917 −0.157465
\(969\) −1.65673 −0.0532220
\(970\) 0 0
\(971\) 18.4557 0.592272 0.296136 0.955146i \(-0.404302\pi\)
0.296136 + 0.955146i \(0.404302\pi\)
\(972\) 8.61438 0.276306
\(973\) −113.345 −3.63367
\(974\) −6.86401 −0.219937
\(975\) 0 0
\(976\) −2.52888 −0.0809474
\(977\) −5.62735 −0.180035 −0.0900175 0.995940i \(-0.528692\pi\)
−0.0900175 + 0.995940i \(0.528692\pi\)
\(978\) 13.0225 0.416413
\(979\) 6.64107 0.212249
\(980\) 0 0
\(981\) 10.0104 0.319608
\(982\) 29.4672 0.940338
\(983\) 25.1228 0.801293 0.400647 0.916233i \(-0.368786\pi\)
0.400647 + 0.916233i \(0.368786\pi\)
\(984\) −44.6644 −1.42385
\(985\) 0 0
\(986\) 0.946996 0.0301585
\(987\) 95.2382 3.03147
\(988\) −6.09474 −0.193900
\(989\) 20.6865 0.657793
\(990\) 0 0
\(991\) −41.8749 −1.33020 −0.665100 0.746754i \(-0.731611\pi\)
−0.665100 + 0.746754i \(0.731611\pi\)
\(992\) −32.4525 −1.03037
\(993\) 15.3295 0.486467
\(994\) −23.3324 −0.740059
\(995\) 0 0
\(996\) −4.01234 −0.127136
\(997\) 37.0346 1.17290 0.586449 0.809986i \(-0.300525\pi\)
0.586449 + 0.809986i \(0.300525\pi\)
\(998\) 2.69505 0.0853102
\(999\) −0.529566 −0.0167547
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.1.2 4
3.2 odd 2 4275.2.a.bo.1.3 4
4.3 odd 2 7600.2.a.cf.1.3 4
5.2 odd 4 475.2.b.e.324.4 8
5.3 odd 4 475.2.b.e.324.5 8
5.4 even 2 95.2.a.b.1.3 4
15.14 odd 2 855.2.a.m.1.2 4
19.18 odd 2 9025.2.a.bf.1.3 4
20.19 odd 2 1520.2.a.t.1.2 4
35.34 odd 2 4655.2.a.y.1.3 4
40.19 odd 2 6080.2.a.ch.1.3 4
40.29 even 2 6080.2.a.cc.1.2 4
95.94 odd 2 1805.2.a.p.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.3 4 5.4 even 2
475.2.a.i.1.2 4 1.1 even 1 trivial
475.2.b.e.324.4 8 5.2 odd 4
475.2.b.e.324.5 8 5.3 odd 4
855.2.a.m.1.2 4 15.14 odd 2
1520.2.a.t.1.2 4 20.19 odd 2
1805.2.a.p.1.2 4 95.94 odd 2
4275.2.a.bo.1.3 4 3.2 odd 2
4655.2.a.y.1.3 4 35.34 odd 2
6080.2.a.cc.1.2 4 40.29 even 2
6080.2.a.ch.1.3 4 40.19 odd 2
7600.2.a.cf.1.3 4 4.3 odd 2
9025.2.a.bf.1.3 4 19.18 odd 2