Properties

Label 1470.3.f.a.391.7
Level $1470$
Weight $3$
Character 1470.391
Analytic conductor $40.055$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,3,Mod(391,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.391");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1470.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.0545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 391.7
Root \(1.72286 + 0.178197i\) of defining polynomial
Character \(\chi\) \(=\) 1470.391
Dual form 1470.3.f.a.391.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} +11.5848 q^{11} +3.46410i q^{12} +7.86371i q^{13} +3.87298 q^{15} +4.00000 q^{16} +27.6066i q^{17} -4.24264 q^{18} -31.4250i q^{19} -4.47214i q^{20} +16.3834 q^{22} +18.1592 q^{23} +4.89898i q^{24} -5.00000 q^{25} +11.1210i q^{26} -5.19615i q^{27} -2.30331 q^{29} +5.47723 q^{30} -5.26382i q^{31} +5.65685 q^{32} +20.0655i q^{33} +39.0416i q^{34} -6.00000 q^{36} +1.98628 q^{37} -44.4417i q^{38} -13.6204 q^{39} -6.32456i q^{40} +22.1905i q^{41} +49.8368 q^{43} +23.1696 q^{44} +6.70820i q^{45} +25.6810 q^{46} +76.6670i q^{47} +6.92820i q^{48} -7.07107 q^{50} -47.8160 q^{51} +15.7274i q^{52} +57.0954 q^{53} -7.34847i q^{54} -25.9044i q^{55} +54.4297 q^{57} -3.25737 q^{58} -70.3495i q^{59} +7.74597 q^{60} +67.6181i q^{61} -7.44416i q^{62} +8.00000 q^{64} +17.5838 q^{65} +28.3768i q^{66} +98.1647 q^{67} +55.2132i q^{68} +31.4526i q^{69} -34.2597 q^{71} -8.48528 q^{72} +19.4915i q^{73} +2.80903 q^{74} -8.66025i q^{75} -62.8501i q^{76} -19.2621 q^{78} +90.4285 q^{79} -8.94427i q^{80} +9.00000 q^{81} +31.3822i q^{82} -133.803i q^{83} +61.7302 q^{85} +70.4799 q^{86} -3.98945i q^{87} +32.7667 q^{88} +11.0647i q^{89} +9.48683i q^{90} +36.3184 q^{92} +9.11720 q^{93} +108.424i q^{94} -70.2685 q^{95} +9.79796i q^{96} +72.3112i q^{97} -34.7544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57} - 208 q^{58} + 64 q^{64} - 40 q^{65} + 240 q^{67} + 8 q^{71} + 32 q^{74} - 192 q^{78} - 24 q^{79} + 72 q^{81} + 120 q^{85} + 80 q^{86} - 96 q^{88} - 48 q^{92} + 264 q^{93} + 80 q^{95} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 1.73205i 0.577350i
\(4\) 2.00000 0.500000
\(5\) − 2.23607i − 0.447214i
\(6\) 2.44949i 0.408248i
\(7\) 0 0
\(8\) 2.82843 0.353553
\(9\) −3.00000 −0.333333
\(10\) − 3.16228i − 0.316228i
\(11\) 11.5848 1.05316 0.526582 0.850125i \(-0.323473\pi\)
0.526582 + 0.850125i \(0.323473\pi\)
\(12\) 3.46410i 0.288675i
\(13\) 7.86371i 0.604901i 0.953165 + 0.302451i \(0.0978047\pi\)
−0.953165 + 0.302451i \(0.902195\pi\)
\(14\) 0 0
\(15\) 3.87298 0.258199
\(16\) 4.00000 0.250000
\(17\) 27.6066i 1.62392i 0.583715 + 0.811959i \(0.301599\pi\)
−0.583715 + 0.811959i \(0.698401\pi\)
\(18\) −4.24264 −0.235702
\(19\) − 31.4250i − 1.65395i −0.562240 0.826974i \(-0.690060\pi\)
0.562240 0.826974i \(-0.309940\pi\)
\(20\) − 4.47214i − 0.223607i
\(21\) 0 0
\(22\) 16.3834 0.744699
\(23\) 18.1592 0.789530 0.394765 0.918782i \(-0.370826\pi\)
0.394765 + 0.918782i \(0.370826\pi\)
\(24\) 4.89898i 0.204124i
\(25\) −5.00000 −0.200000
\(26\) 11.1210i 0.427730i
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) −2.30331 −0.0794244 −0.0397122 0.999211i \(-0.512644\pi\)
−0.0397122 + 0.999211i \(0.512644\pi\)
\(30\) 5.47723 0.182574
\(31\) − 5.26382i − 0.169801i −0.996389 0.0849003i \(-0.972943\pi\)
0.996389 0.0849003i \(-0.0270572\pi\)
\(32\) 5.65685 0.176777
\(33\) 20.0655i 0.608044i
\(34\) 39.0416i 1.14828i
\(35\) 0 0
\(36\) −6.00000 −0.166667
\(37\) 1.98628 0.0536834 0.0268417 0.999640i \(-0.491455\pi\)
0.0268417 + 0.999640i \(0.491455\pi\)
\(38\) − 44.4417i − 1.16952i
\(39\) −13.6204 −0.349240
\(40\) − 6.32456i − 0.158114i
\(41\) 22.1905i 0.541233i 0.962687 + 0.270616i \(0.0872275\pi\)
−0.962687 + 0.270616i \(0.912773\pi\)
\(42\) 0 0
\(43\) 49.8368 1.15900 0.579498 0.814974i \(-0.303249\pi\)
0.579498 + 0.814974i \(0.303249\pi\)
\(44\) 23.1696 0.526582
\(45\) 6.70820i 0.149071i
\(46\) 25.6810 0.558282
\(47\) 76.6670i 1.63121i 0.578607 + 0.815606i \(0.303596\pi\)
−0.578607 + 0.815606i \(0.696404\pi\)
\(48\) 6.92820i 0.144338i
\(49\) 0 0
\(50\) −7.07107 −0.141421
\(51\) −47.8160 −0.937569
\(52\) 15.7274i 0.302451i
\(53\) 57.0954 1.07727 0.538636 0.842539i \(-0.318940\pi\)
0.538636 + 0.842539i \(0.318940\pi\)
\(54\) − 7.34847i − 0.136083i
\(55\) − 25.9044i − 0.470989i
\(56\) 0 0
\(57\) 54.4297 0.954908
\(58\) −3.25737 −0.0561616
\(59\) − 70.3495i − 1.19236i −0.802849 0.596182i \(-0.796684\pi\)
0.802849 0.596182i \(-0.203316\pi\)
\(60\) 7.74597 0.129099
\(61\) 67.6181i 1.10849i 0.832352 + 0.554247i \(0.186994\pi\)
−0.832352 + 0.554247i \(0.813006\pi\)
\(62\) − 7.44416i − 0.120067i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 17.5838 0.270520
\(66\) 28.3768i 0.429952i
\(67\) 98.1647 1.46514 0.732572 0.680689i \(-0.238320\pi\)
0.732572 + 0.680689i \(0.238320\pi\)
\(68\) 55.2132i 0.811959i
\(69\) 31.4526i 0.455835i
\(70\) 0 0
\(71\) −34.2597 −0.482531 −0.241266 0.970459i \(-0.577563\pi\)
−0.241266 + 0.970459i \(0.577563\pi\)
\(72\) −8.48528 −0.117851
\(73\) 19.4915i 0.267006i 0.991048 + 0.133503i \(0.0426226\pi\)
−0.991048 + 0.133503i \(0.957377\pi\)
\(74\) 2.80903 0.0379599
\(75\) − 8.66025i − 0.115470i
\(76\) − 62.8501i − 0.826974i
\(77\) 0 0
\(78\) −19.2621 −0.246950
\(79\) 90.4285 1.14466 0.572332 0.820022i \(-0.306039\pi\)
0.572332 + 0.820022i \(0.306039\pi\)
\(80\) − 8.94427i − 0.111803i
\(81\) 9.00000 0.111111
\(82\) 31.3822i 0.382709i
\(83\) − 133.803i − 1.61209i −0.591854 0.806045i \(-0.701604\pi\)
0.591854 0.806045i \(-0.298396\pi\)
\(84\) 0 0
\(85\) 61.7302 0.726238
\(86\) 70.4799 0.819534
\(87\) − 3.98945i − 0.0458557i
\(88\) 32.7667 0.372349
\(89\) 11.0647i 0.124323i 0.998066 + 0.0621613i \(0.0197993\pi\)
−0.998066 + 0.0621613i \(0.980201\pi\)
\(90\) 9.48683i 0.105409i
\(91\) 0 0
\(92\) 36.3184 0.394765
\(93\) 9.11720 0.0980344
\(94\) 108.424i 1.15344i
\(95\) −70.2685 −0.739668
\(96\) 9.79796i 0.102062i
\(97\) 72.3112i 0.745476i 0.927937 + 0.372738i \(0.121581\pi\)
−0.927937 + 0.372738i \(0.878419\pi\)
\(98\) 0 0
\(99\) −34.7544 −0.351054
\(100\) −10.0000 −0.100000
\(101\) 50.6533i 0.501518i 0.968050 + 0.250759i \(0.0806801\pi\)
−0.968050 + 0.250759i \(0.919320\pi\)
\(102\) −67.6221 −0.662962
\(103\) 197.965i 1.92199i 0.276572 + 0.960993i \(0.410802\pi\)
−0.276572 + 0.960993i \(0.589198\pi\)
\(104\) 22.2419i 0.213865i
\(105\) 0 0
\(106\) 80.7451 0.761746
\(107\) 147.936 1.38258 0.691289 0.722579i \(-0.257043\pi\)
0.691289 + 0.722579i \(0.257043\pi\)
\(108\) − 10.3923i − 0.0962250i
\(109\) −54.3219 −0.498366 −0.249183 0.968456i \(-0.580162\pi\)
−0.249183 + 0.968456i \(0.580162\pi\)
\(110\) − 36.6343i − 0.333039i
\(111\) 3.44035i 0.0309941i
\(112\) 0 0
\(113\) 47.7883 0.422906 0.211453 0.977388i \(-0.432181\pi\)
0.211453 + 0.977388i \(0.432181\pi\)
\(114\) 76.9753 0.675222
\(115\) − 40.6052i − 0.353089i
\(116\) −4.60662 −0.0397122
\(117\) − 23.5911i − 0.201634i
\(118\) − 99.4892i − 0.843129i
\(119\) 0 0
\(120\) 10.9545 0.0912871
\(121\) 13.2074 0.109152
\(122\) 95.6265i 0.783824i
\(123\) −38.4351 −0.312481
\(124\) − 10.5276i − 0.0849003i
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −101.973 −0.802940 −0.401470 0.915872i \(-0.631501\pi\)
−0.401470 + 0.915872i \(0.631501\pi\)
\(128\) 11.3137 0.0883883
\(129\) 86.3199i 0.669147i
\(130\) 24.8673 0.191287
\(131\) 74.5980i 0.569450i 0.958609 + 0.284725i \(0.0919023\pi\)
−0.958609 + 0.284725i \(0.908098\pi\)
\(132\) 40.1309i 0.304022i
\(133\) 0 0
\(134\) 138.826 1.03601
\(135\) −11.6190 −0.0860663
\(136\) 78.0833i 0.574142i
\(137\) −246.899 −1.80218 −0.901091 0.433630i \(-0.857233\pi\)
−0.901091 + 0.433630i \(0.857233\pi\)
\(138\) 44.4807i 0.322324i
\(139\) 155.917i 1.12170i 0.827916 + 0.560852i \(0.189526\pi\)
−0.827916 + 0.560852i \(0.810474\pi\)
\(140\) 0 0
\(141\) −132.791 −0.941781
\(142\) −48.4506 −0.341201
\(143\) 91.0995i 0.637060i
\(144\) −12.0000 −0.0833333
\(145\) 5.15035i 0.0355197i
\(146\) 27.5651i 0.188802i
\(147\) 0 0
\(148\) 3.97257 0.0268417
\(149\) 163.090 1.09457 0.547283 0.836948i \(-0.315662\pi\)
0.547283 + 0.836948i \(0.315662\pi\)
\(150\) − 12.2474i − 0.0816497i
\(151\) −75.0299 −0.496887 −0.248443 0.968646i \(-0.579919\pi\)
−0.248443 + 0.968646i \(0.579919\pi\)
\(152\) − 88.8834i − 0.584759i
\(153\) − 82.8198i − 0.541306i
\(154\) 0 0
\(155\) −11.7703 −0.0759371
\(156\) −27.2407 −0.174620
\(157\) − 224.731i − 1.43140i −0.698405 0.715702i \(-0.746107\pi\)
0.698405 0.715702i \(-0.253893\pi\)
\(158\) 127.885 0.809400
\(159\) 98.8922i 0.621963i
\(160\) − 12.6491i − 0.0790569i
\(161\) 0 0
\(162\) 12.7279 0.0785674
\(163\) −230.117 −1.41176 −0.705882 0.708330i \(-0.749449\pi\)
−0.705882 + 0.708330i \(0.749449\pi\)
\(164\) 44.3811i 0.270616i
\(165\) 44.8677 0.271926
\(166\) − 189.227i − 1.13992i
\(167\) − 277.788i − 1.66340i −0.555223 0.831702i \(-0.687367\pi\)
0.555223 0.831702i \(-0.312633\pi\)
\(168\) 0 0
\(169\) 107.162 0.634095
\(170\) 87.2997 0.513528
\(171\) 94.2751i 0.551316i
\(172\) 99.6736 0.579498
\(173\) − 126.065i − 0.728702i −0.931262 0.364351i \(-0.881291\pi\)
0.931262 0.364351i \(-0.118709\pi\)
\(174\) − 5.64193i − 0.0324249i
\(175\) 0 0
\(176\) 46.3392 0.263291
\(177\) 121.849 0.688412
\(178\) 15.6479i 0.0879093i
\(179\) 77.1069 0.430765 0.215382 0.976530i \(-0.430900\pi\)
0.215382 + 0.976530i \(0.430900\pi\)
\(180\) 13.4164i 0.0745356i
\(181\) − 212.012i − 1.17134i −0.810551 0.585668i \(-0.800832\pi\)
0.810551 0.585668i \(-0.199168\pi\)
\(182\) 0 0
\(183\) −117.118 −0.639989
\(184\) 51.3619 0.279141
\(185\) − 4.44147i − 0.0240079i
\(186\) 12.8937 0.0693208
\(187\) 319.817i 1.71025i
\(188\) 153.334i 0.815606i
\(189\) 0 0
\(190\) −99.3747 −0.523025
\(191\) 165.696 0.867518 0.433759 0.901029i \(-0.357187\pi\)
0.433759 + 0.901029i \(0.357187\pi\)
\(192\) 13.8564i 0.0721688i
\(193\) −105.299 −0.545593 −0.272796 0.962072i \(-0.587949\pi\)
−0.272796 + 0.962072i \(0.587949\pi\)
\(194\) 102.263i 0.527131i
\(195\) 30.4560i 0.156185i
\(196\) 0 0
\(197\) 244.736 1.24231 0.621156 0.783687i \(-0.286663\pi\)
0.621156 + 0.783687i \(0.286663\pi\)
\(198\) −49.1501 −0.248233
\(199\) 97.9667i 0.492295i 0.969232 + 0.246147i \(0.0791647\pi\)
−0.969232 + 0.246147i \(0.920835\pi\)
\(200\) −14.1421 −0.0707107
\(201\) 170.026i 0.845902i
\(202\) 71.6346i 0.354627i
\(203\) 0 0
\(204\) −95.6321 −0.468785
\(205\) 49.6196 0.242047
\(206\) 279.964i 1.35905i
\(207\) −54.4776 −0.263177
\(208\) 31.4549i 0.151225i
\(209\) − 364.052i − 1.74188i
\(210\) 0 0
\(211\) −388.914 −1.84319 −0.921597 0.388149i \(-0.873114\pi\)
−0.921597 + 0.388149i \(0.873114\pi\)
\(212\) 114.191 0.538636
\(213\) − 59.3396i − 0.278590i
\(214\) 209.213 0.977630
\(215\) − 111.439i − 0.518319i
\(216\) − 14.6969i − 0.0680414i
\(217\) 0 0
\(218\) −76.8228 −0.352398
\(219\) −33.7602 −0.154156
\(220\) − 51.8088i − 0.235494i
\(221\) −217.090 −0.982310
\(222\) 4.86538i 0.0219161i
\(223\) 251.913i 1.12965i 0.825210 + 0.564827i \(0.191057\pi\)
−0.825210 + 0.564827i \(0.808943\pi\)
\(224\) 0 0
\(225\) 15.0000 0.0666667
\(226\) 67.5829 0.299039
\(227\) 267.822i 1.17983i 0.807465 + 0.589916i \(0.200839\pi\)
−0.807465 + 0.589916i \(0.799161\pi\)
\(228\) 108.859 0.477454
\(229\) − 345.529i − 1.50886i −0.656379 0.754431i \(-0.727913\pi\)
0.656379 0.754431i \(-0.272087\pi\)
\(230\) − 57.4244i − 0.249671i
\(231\) 0 0
\(232\) −6.51474 −0.0280808
\(233\) −288.064 −1.23633 −0.618164 0.786049i \(-0.712123\pi\)
−0.618164 + 0.786049i \(0.712123\pi\)
\(234\) − 33.3629i − 0.142577i
\(235\) 171.433 0.729500
\(236\) − 140.699i − 0.596182i
\(237\) 156.627i 0.660872i
\(238\) 0 0
\(239\) −188.810 −0.789999 −0.395000 0.918681i \(-0.629255\pi\)
−0.395000 + 0.918681i \(0.629255\pi\)
\(240\) 15.4919 0.0645497
\(241\) 97.8480i 0.406008i 0.979178 + 0.203004i \(0.0650705\pi\)
−0.979178 + 0.203004i \(0.934929\pi\)
\(242\) 18.6782 0.0771824
\(243\) 15.5885i 0.0641500i
\(244\) 135.236i 0.554247i
\(245\) 0 0
\(246\) −54.3555 −0.220957
\(247\) 247.117 1.00048
\(248\) − 14.8883i − 0.0600336i
\(249\) 231.754 0.930741
\(250\) 15.8114i 0.0632456i
\(251\) − 241.345i − 0.961533i −0.876849 0.480767i \(-0.840358\pi\)
0.876849 0.480767i \(-0.159642\pi\)
\(252\) 0 0
\(253\) 210.370 0.831504
\(254\) −144.212 −0.567764
\(255\) 106.920i 0.419294i
\(256\) 16.0000 0.0625000
\(257\) 12.5639i 0.0488869i 0.999701 + 0.0244435i \(0.00778137\pi\)
−0.999701 + 0.0244435i \(0.992219\pi\)
\(258\) 122.075i 0.473158i
\(259\) 0 0
\(260\) 35.1676 0.135260
\(261\) 6.90993 0.0264748
\(262\) 105.497i 0.402662i
\(263\) −212.481 −0.807911 −0.403956 0.914779i \(-0.632365\pi\)
−0.403956 + 0.914779i \(0.632365\pi\)
\(264\) 56.7537i 0.214976i
\(265\) − 127.669i − 0.481771i
\(266\) 0 0
\(267\) −19.1646 −0.0717777
\(268\) 196.329 0.732572
\(269\) − 334.597i − 1.24385i −0.783075 0.621927i \(-0.786350\pi\)
0.783075 0.621927i \(-0.213650\pi\)
\(270\) −16.4317 −0.0608581
\(271\) 182.350i 0.672879i 0.941705 + 0.336440i \(0.109223\pi\)
−0.941705 + 0.336440i \(0.890777\pi\)
\(272\) 110.426i 0.405979i
\(273\) 0 0
\(274\) −349.168 −1.27434
\(275\) −57.9240 −0.210633
\(276\) 62.9053i 0.227918i
\(277\) 166.761 0.602026 0.301013 0.953620i \(-0.402675\pi\)
0.301013 + 0.953620i \(0.402675\pi\)
\(278\) 220.500i 0.793164i
\(279\) 15.7915i 0.0566002i
\(280\) 0 0
\(281\) 38.3085 0.136329 0.0681647 0.997674i \(-0.478286\pi\)
0.0681647 + 0.997674i \(0.478286\pi\)
\(282\) −187.795 −0.665940
\(283\) − 385.496i − 1.36218i −0.732201 0.681088i \(-0.761507\pi\)
0.732201 0.681088i \(-0.238493\pi\)
\(284\) −68.5194 −0.241266
\(285\) − 121.709i − 0.427048i
\(286\) 128.834i 0.450469i
\(287\) 0 0
\(288\) −16.9706 −0.0589256
\(289\) −473.125 −1.63711
\(290\) 7.28370i 0.0251162i
\(291\) −125.247 −0.430401
\(292\) 38.9829i 0.133503i
\(293\) 90.9844i 0.310527i 0.987873 + 0.155264i \(0.0496227\pi\)
−0.987873 + 0.155264i \(0.950377\pi\)
\(294\) 0 0
\(295\) −157.306 −0.533242
\(296\) 5.61806 0.0189799
\(297\) − 60.1964i − 0.202681i
\(298\) 230.645 0.773975
\(299\) 142.799i 0.477588i
\(300\) − 17.3205i − 0.0577350i
\(301\) 0 0
\(302\) −106.108 −0.351352
\(303\) −87.7341 −0.289551
\(304\) − 125.700i − 0.413487i
\(305\) 151.199 0.495734
\(306\) − 117.125i − 0.382761i
\(307\) − 508.077i − 1.65497i −0.561485 0.827487i \(-0.689770\pi\)
0.561485 0.827487i \(-0.310230\pi\)
\(308\) 0 0
\(309\) −342.885 −1.10966
\(310\) −16.6457 −0.0536956
\(311\) 104.321i 0.335437i 0.985835 + 0.167719i \(0.0536400\pi\)
−0.985835 + 0.167719i \(0.946360\pi\)
\(312\) −38.5242 −0.123475
\(313\) − 265.869i − 0.849420i −0.905329 0.424710i \(-0.860376\pi\)
0.905329 0.424710i \(-0.139624\pi\)
\(314\) − 317.817i − 1.01216i
\(315\) 0 0
\(316\) 180.857 0.572332
\(317\) −10.7223 −0.0338242 −0.0169121 0.999857i \(-0.505384\pi\)
−0.0169121 + 0.999857i \(0.505384\pi\)
\(318\) 139.855i 0.439795i
\(319\) −26.6834 −0.0836469
\(320\) − 17.8885i − 0.0559017i
\(321\) 256.232i 0.798231i
\(322\) 0 0
\(323\) 867.538 2.68588
\(324\) 18.0000 0.0555556
\(325\) − 39.3186i − 0.120980i
\(326\) −325.435 −0.998267
\(327\) − 94.0883i − 0.287732i
\(328\) 62.7643i 0.191355i
\(329\) 0 0
\(330\) 63.4525 0.192280
\(331\) −352.921 −1.06623 −0.533113 0.846044i \(-0.678978\pi\)
−0.533113 + 0.846044i \(0.678978\pi\)
\(332\) − 267.607i − 0.806045i
\(333\) −5.95885 −0.0178945
\(334\) − 392.852i − 1.17620i
\(335\) − 219.503i − 0.655233i
\(336\) 0 0
\(337\) 220.634 0.654699 0.327349 0.944903i \(-0.393845\pi\)
0.327349 + 0.944903i \(0.393845\pi\)
\(338\) 151.550 0.448373
\(339\) 82.7718i 0.244165i
\(340\) 123.460 0.363119
\(341\) − 60.9802i − 0.178828i
\(342\) 133.325i 0.389839i
\(343\) 0 0
\(344\) 140.960 0.409767
\(345\) 70.3302 0.203856
\(346\) − 178.283i − 0.515270i
\(347\) −256.009 −0.737777 −0.368888 0.929474i \(-0.620262\pi\)
−0.368888 + 0.929474i \(0.620262\pi\)
\(348\) − 7.97889i − 0.0229279i
\(349\) − 407.250i − 1.16691i −0.812147 0.583453i \(-0.801701\pi\)
0.812147 0.583453i \(-0.198299\pi\)
\(350\) 0 0
\(351\) 40.8611 0.116413
\(352\) 65.5335 0.186175
\(353\) − 583.238i − 1.65223i −0.563501 0.826116i \(-0.690546\pi\)
0.563501 0.826116i \(-0.309454\pi\)
\(354\) 172.320 0.486781
\(355\) 76.6071i 0.215795i
\(356\) 22.1294i 0.0621613i
\(357\) 0 0
\(358\) 109.046 0.304597
\(359\) 691.528 1.92626 0.963131 0.269032i \(-0.0867038\pi\)
0.963131 + 0.269032i \(0.0867038\pi\)
\(360\) 18.9737i 0.0527046i
\(361\) −626.532 −1.73555
\(362\) − 299.830i − 0.828260i
\(363\) 22.8760i 0.0630192i
\(364\) 0 0
\(365\) 43.5842 0.119409
\(366\) −165.630 −0.452541
\(367\) 229.727i 0.625959i 0.949760 + 0.312980i \(0.101327\pi\)
−0.949760 + 0.312980i \(0.898673\pi\)
\(368\) 72.6368 0.197382
\(369\) − 66.5716i − 0.180411i
\(370\) − 6.28118i − 0.0169762i
\(371\) 0 0
\(372\) 18.2344 0.0490172
\(373\) −404.609 −1.08474 −0.542371 0.840139i \(-0.682473\pi\)
−0.542371 + 0.840139i \(0.682473\pi\)
\(374\) 452.289i 1.20933i
\(375\) −19.3649 −0.0516398
\(376\) 216.847i 0.576721i
\(377\) − 18.1126i − 0.0480439i
\(378\) 0 0
\(379\) 265.866 0.701493 0.350746 0.936471i \(-0.385928\pi\)
0.350746 + 0.936471i \(0.385928\pi\)
\(380\) −140.537 −0.369834
\(381\) − 176.623i − 0.463578i
\(382\) 234.329 0.613428
\(383\) − 306.404i − 0.800010i −0.916513 0.400005i \(-0.869008\pi\)
0.916513 0.400005i \(-0.130992\pi\)
\(384\) 19.5959i 0.0510310i
\(385\) 0 0
\(386\) −148.916 −0.385792
\(387\) −149.510 −0.386332
\(388\) 144.622i 0.372738i
\(389\) 106.354 0.273405 0.136702 0.990612i \(-0.456350\pi\)
0.136702 + 0.990612i \(0.456350\pi\)
\(390\) 43.0713i 0.110439i
\(391\) 501.314i 1.28213i
\(392\) 0 0
\(393\) −129.207 −0.328772
\(394\) 346.108 0.878448
\(395\) − 202.204i − 0.511909i
\(396\) −69.5088 −0.175527
\(397\) 217.827i 0.548682i 0.961633 + 0.274341i \(0.0884597\pi\)
−0.961633 + 0.274341i \(0.911540\pi\)
\(398\) 138.546i 0.348105i
\(399\) 0 0
\(400\) −20.0000 −0.0500000
\(401\) 61.9814 0.154567 0.0772836 0.997009i \(-0.475375\pi\)
0.0772836 + 0.997009i \(0.475375\pi\)
\(402\) 240.453i 0.598143i
\(403\) 41.3932 0.102713
\(404\) 101.307i 0.250759i
\(405\) − 20.1246i − 0.0496904i
\(406\) 0 0
\(407\) 23.0107 0.0565373
\(408\) −135.244 −0.331481
\(409\) − 434.360i − 1.06201i −0.847370 0.531003i \(-0.821815\pi\)
0.847370 0.531003i \(-0.178185\pi\)
\(410\) 70.1726 0.171153
\(411\) − 427.642i − 1.04049i
\(412\) 395.929i 0.960993i
\(413\) 0 0
\(414\) −77.0429 −0.186094
\(415\) −299.194 −0.720949
\(416\) 44.4839i 0.106932i
\(417\) −270.056 −0.647616
\(418\) − 514.848i − 1.23169i
\(419\) 457.221i 1.09122i 0.838040 + 0.545609i \(0.183702\pi\)
−0.838040 + 0.545609i \(0.816298\pi\)
\(420\) 0 0
\(421\) −653.149 −1.55142 −0.775712 0.631088i \(-0.782609\pi\)
−0.775712 + 0.631088i \(0.782609\pi\)
\(422\) −550.007 −1.30333
\(423\) − 230.001i − 0.543738i
\(424\) 161.490 0.380873
\(425\) − 138.033i − 0.324784i
\(426\) − 83.9188i − 0.196993i
\(427\) 0 0
\(428\) 295.872 0.691289
\(429\) −157.789 −0.367807
\(430\) − 157.598i − 0.366507i
\(431\) −470.256 −1.09108 −0.545540 0.838085i \(-0.683676\pi\)
−0.545540 + 0.838085i \(0.683676\pi\)
\(432\) − 20.7846i − 0.0481125i
\(433\) 32.0299i 0.0739719i 0.999316 + 0.0369860i \(0.0117757\pi\)
−0.999316 + 0.0369860i \(0.988224\pi\)
\(434\) 0 0
\(435\) −8.92068 −0.0205073
\(436\) −108.644 −0.249183
\(437\) − 570.653i − 1.30584i
\(438\) −47.7441 −0.109005
\(439\) − 382.239i − 0.870703i −0.900260 0.435352i \(-0.856624\pi\)
0.900260 0.435352i \(-0.143376\pi\)
\(440\) − 73.2687i − 0.166520i
\(441\) 0 0
\(442\) −307.012 −0.694598
\(443\) 123.742 0.279326 0.139663 0.990199i \(-0.455398\pi\)
0.139663 + 0.990199i \(0.455398\pi\)
\(444\) 6.88069i 0.0154971i
\(445\) 24.7414 0.0555987
\(446\) 356.258i 0.798785i
\(447\) 282.481i 0.631948i
\(448\) 0 0
\(449\) −266.985 −0.594622 −0.297311 0.954781i \(-0.596090\pi\)
−0.297311 + 0.954781i \(0.596090\pi\)
\(450\) 21.2132 0.0471405
\(451\) 257.073i 0.570006i
\(452\) 95.5767 0.211453
\(453\) − 129.956i − 0.286878i
\(454\) 378.757i 0.834267i
\(455\) 0 0
\(456\) 153.951 0.337611
\(457\) −449.340 −0.983238 −0.491619 0.870810i \(-0.663595\pi\)
−0.491619 + 0.870810i \(0.663595\pi\)
\(458\) − 488.652i − 1.06693i
\(459\) 143.448 0.312523
\(460\) − 81.2104i − 0.176544i
\(461\) − 105.528i − 0.228912i −0.993428 0.114456i \(-0.963487\pi\)
0.993428 0.114456i \(-0.0365125\pi\)
\(462\) 0 0
\(463\) 588.555 1.27118 0.635588 0.772028i \(-0.280758\pi\)
0.635588 + 0.772028i \(0.280758\pi\)
\(464\) −9.21323 −0.0198561
\(465\) − 20.3867i − 0.0438423i
\(466\) −407.385 −0.874216
\(467\) − 365.657i − 0.782992i −0.920180 0.391496i \(-0.871958\pi\)
0.920180 0.391496i \(-0.128042\pi\)
\(468\) − 47.1823i − 0.100817i
\(469\) 0 0
\(470\) 242.442 0.515835
\(471\) 389.245 0.826422
\(472\) − 198.978i − 0.421565i
\(473\) 577.349 1.22061
\(474\) 221.504i 0.467307i
\(475\) 157.125i 0.330790i
\(476\) 0 0
\(477\) −171.286 −0.359091
\(478\) −267.017 −0.558614
\(479\) − 487.176i − 1.01707i −0.861042 0.508535i \(-0.830187\pi\)
0.861042 0.508535i \(-0.169813\pi\)
\(480\) 21.9089 0.0456435
\(481\) 15.6196i 0.0324731i
\(482\) 138.378i 0.287091i
\(483\) 0 0
\(484\) 26.4149 0.0545762
\(485\) 161.693 0.333387
\(486\) 22.0454i 0.0453609i
\(487\) −195.774 −0.402000 −0.201000 0.979591i \(-0.564419\pi\)
−0.201000 + 0.979591i \(0.564419\pi\)
\(488\) 191.253i 0.391912i
\(489\) − 398.575i − 0.815082i
\(490\) 0 0
\(491\) −349.221 −0.711244 −0.355622 0.934630i \(-0.615731\pi\)
−0.355622 + 0.934630i \(0.615731\pi\)
\(492\) −76.8703 −0.156240
\(493\) − 63.5865i − 0.128979i
\(494\) 349.477 0.707443
\(495\) 77.7132i 0.156996i
\(496\) − 21.0553i − 0.0424501i
\(497\) 0 0
\(498\) 327.750 0.658133
\(499\) −335.439 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(500\) 22.3607i 0.0447214i
\(501\) 481.144 0.960366
\(502\) − 341.313i − 0.679907i
\(503\) − 523.663i − 1.04108i −0.853837 0.520540i \(-0.825731\pi\)
0.853837 0.520540i \(-0.174269\pi\)
\(504\) 0 0
\(505\) 113.264 0.224286
\(506\) 297.509 0.587962
\(507\) 185.610i 0.366095i
\(508\) −203.947 −0.401470
\(509\) 482.354i 0.947651i 0.880619 + 0.473825i \(0.157127\pi\)
−0.880619 + 0.473825i \(0.842873\pi\)
\(510\) 151.208i 0.296485i
\(511\) 0 0
\(512\) 22.6274 0.0441942
\(513\) −163.289 −0.318303
\(514\) 17.7681i 0.0345683i
\(515\) 442.662 0.859538
\(516\) 172.640i 0.334573i
\(517\) 888.171i 1.71793i
\(518\) 0 0
\(519\) 218.352 0.420716
\(520\) 49.7345 0.0956433
\(521\) − 71.3039i − 0.136860i −0.997656 0.0684298i \(-0.978201\pi\)
0.997656 0.0684298i \(-0.0217989\pi\)
\(522\) 9.77211 0.0187205
\(523\) 602.128i 1.15130i 0.817697 + 0.575648i \(0.195250\pi\)
−0.817697 + 0.575648i \(0.804750\pi\)
\(524\) 149.196i 0.284725i
\(525\) 0 0
\(526\) −300.493 −0.571279
\(527\) 145.316 0.275742
\(528\) 80.2618i 0.152011i
\(529\) −199.244 −0.376643
\(530\) − 180.552i − 0.340663i
\(531\) 211.049i 0.397455i
\(532\) 0 0
\(533\) −174.500 −0.327392
\(534\) −27.1029 −0.0507545
\(535\) − 330.794i − 0.618307i
\(536\) 277.652 0.518007
\(537\) 133.553i 0.248702i
\(538\) − 473.191i − 0.879537i
\(539\) 0 0
\(540\) −23.2379 −0.0430331
\(541\) −603.314 −1.11518 −0.557591 0.830116i \(-0.688274\pi\)
−0.557591 + 0.830116i \(0.688274\pi\)
\(542\) 257.882i 0.475798i
\(543\) 367.215 0.676271
\(544\) 156.167i 0.287071i
\(545\) 121.468i 0.222876i
\(546\) 0 0
\(547\) −879.935 −1.60866 −0.804328 0.594185i \(-0.797475\pi\)
−0.804328 + 0.594185i \(0.797475\pi\)
\(548\) −493.798 −0.901091
\(549\) − 202.854i − 0.369498i
\(550\) −81.9169 −0.148940
\(551\) 72.3815i 0.131364i
\(552\) 88.9615i 0.161162i
\(553\) 0 0
\(554\) 235.836 0.425697
\(555\) 7.69285 0.0138610
\(556\) 311.834i 0.560852i
\(557\) −882.807 −1.58493 −0.792466 0.609916i \(-0.791203\pi\)
−0.792466 + 0.609916i \(0.791203\pi\)
\(558\) 22.3325i 0.0400224i
\(559\) 391.903i 0.701078i
\(560\) 0 0
\(561\) −553.939 −0.987414
\(562\) 54.1765 0.0963994
\(563\) − 521.156i − 0.925676i −0.886443 0.462838i \(-0.846831\pi\)
0.886443 0.462838i \(-0.153169\pi\)
\(564\) −265.582 −0.470891
\(565\) − 106.858i − 0.189129i
\(566\) − 545.174i − 0.963204i
\(567\) 0 0
\(568\) −96.9011 −0.170601
\(569\) 244.801 0.430230 0.215115 0.976589i \(-0.430987\pi\)
0.215115 + 0.976589i \(0.430987\pi\)
\(570\) − 172.122i − 0.301968i
\(571\) −416.253 −0.728989 −0.364495 0.931205i \(-0.618758\pi\)
−0.364495 + 0.931205i \(0.618758\pi\)
\(572\) 182.199i 0.318530i
\(573\) 286.994i 0.500862i
\(574\) 0 0
\(575\) −90.7959 −0.157906
\(576\) −24.0000 −0.0416667
\(577\) 773.568i 1.34067i 0.742058 + 0.670336i \(0.233850\pi\)
−0.742058 + 0.670336i \(0.766150\pi\)
\(578\) −669.099 −1.15761
\(579\) − 182.384i − 0.314998i
\(580\) 10.3007i 0.0177598i
\(581\) 0 0
\(582\) −177.125 −0.304339
\(583\) 661.439 1.13454
\(584\) 55.1302i 0.0944010i
\(585\) −52.7514 −0.0901733
\(586\) 128.671i 0.219576i
\(587\) − 633.860i − 1.07983i −0.841720 0.539915i \(-0.818456\pi\)
0.841720 0.539915i \(-0.181544\pi\)
\(588\) 0 0
\(589\) −165.416 −0.280841
\(590\) −222.465 −0.377059
\(591\) 423.894i 0.717250i
\(592\) 7.94514 0.0134208
\(593\) − 92.8412i − 0.156562i −0.996931 0.0782809i \(-0.975057\pi\)
0.996931 0.0782809i \(-0.0249431\pi\)
\(594\) − 85.1305i − 0.143317i
\(595\) 0 0
\(596\) 326.181 0.547283
\(597\) −169.683 −0.284227
\(598\) 201.948i 0.337705i
\(599\) −615.361 −1.02731 −0.513657 0.857996i \(-0.671709\pi\)
−0.513657 + 0.857996i \(0.671709\pi\)
\(600\) − 24.4949i − 0.0408248i
\(601\) 821.399i 1.36672i 0.730081 + 0.683360i \(0.239482\pi\)
−0.730081 + 0.683360i \(0.760518\pi\)
\(602\) 0 0
\(603\) −294.494 −0.488381
\(604\) −150.060 −0.248443
\(605\) − 29.5328i − 0.0488145i
\(606\) −124.075 −0.204744
\(607\) 1004.56i 1.65497i 0.561490 + 0.827483i \(0.310228\pi\)
−0.561490 + 0.827483i \(0.689772\pi\)
\(608\) − 177.767i − 0.292380i
\(609\) 0 0
\(610\) 213.827 0.350537
\(611\) −602.887 −0.986722
\(612\) − 165.640i − 0.270653i
\(613\) 1113.75 1.81688 0.908438 0.418019i \(-0.137276\pi\)
0.908438 + 0.418019i \(0.137276\pi\)
\(614\) − 718.529i − 1.17024i
\(615\) 85.9436i 0.139746i
\(616\) 0 0
\(617\) −463.256 −0.750820 −0.375410 0.926859i \(-0.622498\pi\)
−0.375410 + 0.926859i \(0.622498\pi\)
\(618\) −484.912 −0.784648
\(619\) 561.311i 0.906802i 0.891307 + 0.453401i \(0.149790\pi\)
−0.891307 + 0.453401i \(0.850210\pi\)
\(620\) −23.5405 −0.0379686
\(621\) − 94.3579i − 0.151945i
\(622\) 147.532i 0.237190i
\(623\) 0 0
\(624\) −54.4814 −0.0873100
\(625\) 25.0000 0.0400000
\(626\) − 375.995i − 0.600631i
\(627\) 630.557 1.00567
\(628\) − 449.461i − 0.715702i
\(629\) 54.8346i 0.0871774i
\(630\) 0 0
\(631\) 244.533 0.387533 0.193767 0.981048i \(-0.437930\pi\)
0.193767 + 0.981048i \(0.437930\pi\)
\(632\) 255.770 0.404700
\(633\) − 673.618i − 1.06417i
\(634\) −15.1636 −0.0239173
\(635\) 228.019i 0.359086i
\(636\) 197.784i 0.310982i
\(637\) 0 0
\(638\) −37.7360 −0.0591473
\(639\) 102.779 0.160844
\(640\) − 25.2982i − 0.0395285i
\(641\) 651.769 1.01680 0.508400 0.861121i \(-0.330237\pi\)
0.508400 + 0.861121i \(0.330237\pi\)
\(642\) 362.367i 0.564435i
\(643\) − 76.4894i − 0.118957i −0.998230 0.0594786i \(-0.981056\pi\)
0.998230 0.0594786i \(-0.0189438\pi\)
\(644\) 0 0
\(645\) 193.017 0.299251
\(646\) 1226.88 1.89920
\(647\) 163.891i 0.253309i 0.991947 + 0.126654i \(0.0404239\pi\)
−0.991947 + 0.126654i \(0.959576\pi\)
\(648\) 25.4558 0.0392837
\(649\) − 814.985i − 1.25575i
\(650\) − 55.6049i − 0.0855459i
\(651\) 0 0
\(652\) −460.235 −0.705882
\(653\) −1114.86 −1.70729 −0.853647 0.520852i \(-0.825614\pi\)
−0.853647 + 0.520852i \(0.825614\pi\)
\(654\) − 133.061i − 0.203457i
\(655\) 166.806 0.254666
\(656\) 88.7622i 0.135308i
\(657\) − 58.4744i − 0.0890021i
\(658\) 0 0
\(659\) 1164.66 1.76732 0.883660 0.468130i \(-0.155072\pi\)
0.883660 + 0.468130i \(0.155072\pi\)
\(660\) 89.7354 0.135963
\(661\) 625.948i 0.946971i 0.880802 + 0.473485i \(0.157004\pi\)
−0.880802 + 0.473485i \(0.842996\pi\)
\(662\) −499.105 −0.753936
\(663\) − 376.012i − 0.567137i
\(664\) − 378.453i − 0.569960i
\(665\) 0 0
\(666\) −8.42709 −0.0126533
\(667\) −41.8262 −0.0627080
\(668\) − 555.577i − 0.831702i
\(669\) −436.326 −0.652206
\(670\) − 310.424i − 0.463319i
\(671\) 783.342i 1.16743i
\(672\) 0 0
\(673\) −38.0207 −0.0564943 −0.0282471 0.999601i \(-0.508993\pi\)
−0.0282471 + 0.999601i \(0.508993\pi\)
\(674\) 312.023 0.462942
\(675\) 25.9808i 0.0384900i
\(676\) 214.324 0.317047
\(677\) − 781.713i − 1.15467i −0.816506 0.577336i \(-0.804092\pi\)
0.816506 0.577336i \(-0.195908\pi\)
\(678\) 117.057i 0.172651i
\(679\) 0 0
\(680\) 174.599 0.256764
\(681\) −463.881 −0.681176
\(682\) − 86.2391i − 0.126450i
\(683\) −134.438 −0.196834 −0.0984172 0.995145i \(-0.531378\pi\)
−0.0984172 + 0.995145i \(0.531378\pi\)
\(684\) 188.550i 0.275658i
\(685\) 552.083i 0.805960i
\(686\) 0 0
\(687\) 598.474 0.871142
\(688\) 199.347 0.289749
\(689\) 448.982i 0.651643i
\(690\) 99.4620 0.144148
\(691\) 454.089i 0.657148i 0.944478 + 0.328574i \(0.106568\pi\)
−0.944478 + 0.328574i \(0.893432\pi\)
\(692\) − 252.131i − 0.364351i
\(693\) 0 0
\(694\) −362.051 −0.521687
\(695\) 348.641 0.501641
\(696\) − 11.2839i − 0.0162124i
\(697\) −612.605 −0.878917
\(698\) − 575.939i − 0.825127i
\(699\) − 498.942i − 0.713794i
\(700\) 0 0
\(701\) 982.015 1.40088 0.700439 0.713713i \(-0.252988\pi\)
0.700439 + 0.713713i \(0.252988\pi\)
\(702\) 57.7863 0.0823166
\(703\) − 62.4191i − 0.0887896i
\(704\) 92.6784 0.131645
\(705\) 296.930i 0.421177i
\(706\) − 824.822i − 1.16830i
\(707\) 0 0
\(708\) 243.698 0.344206
\(709\) −333.072 −0.469778 −0.234889 0.972022i \(-0.575473\pi\)
−0.234889 + 0.972022i \(0.575473\pi\)
\(710\) 108.339i 0.152590i
\(711\) −271.285 −0.381555
\(712\) 31.2957i 0.0439547i
\(713\) − 95.5866i − 0.134063i
\(714\) 0 0
\(715\) 203.705 0.284902
\(716\) 154.214 0.215382
\(717\) − 327.028i − 0.456106i
\(718\) 977.969 1.36207
\(719\) − 1219.05i − 1.69547i −0.530416 0.847737i \(-0.677964\pi\)
0.530416 0.847737i \(-0.322036\pi\)
\(720\) 26.8328i 0.0372678i
\(721\) 0 0
\(722\) −886.051 −1.22722
\(723\) −169.478 −0.234409
\(724\) − 424.024i − 0.585668i
\(725\) 11.5165 0.0158849
\(726\) 32.3515i 0.0445613i
\(727\) 215.108i 0.295885i 0.988996 + 0.147942i \(0.0472650\pi\)
−0.988996 + 0.147942i \(0.952735\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 61.6374 0.0844348
\(731\) 1375.83i 1.88211i
\(732\) −234.236 −0.319995
\(733\) 749.455i 1.02245i 0.859447 + 0.511225i \(0.170808\pi\)
−0.859447 + 0.511225i \(0.829192\pi\)
\(734\) 324.883i 0.442620i
\(735\) 0 0
\(736\) 102.724 0.139570
\(737\) 1137.22 1.54304
\(738\) − 94.1465i − 0.127570i
\(739\) −934.204 −1.26415 −0.632073 0.774909i \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(740\) − 8.88294i − 0.0120040i
\(741\) 428.020i 0.577625i
\(742\) 0 0
\(743\) 1232.47 1.65878 0.829389 0.558671i \(-0.188689\pi\)
0.829389 + 0.558671i \(0.188689\pi\)
\(744\) 25.7873 0.0346604
\(745\) − 364.681i − 0.489505i
\(746\) −572.203 −0.767028
\(747\) 401.410i 0.537363i
\(748\) 639.634i 0.855125i
\(749\) 0 0
\(750\) −27.3861 −0.0365148
\(751\) 786.706 1.04754 0.523772 0.851858i \(-0.324524\pi\)
0.523772 + 0.851858i \(0.324524\pi\)
\(752\) 306.668i 0.407803i
\(753\) 418.022 0.555142
\(754\) − 25.6150i − 0.0339722i
\(755\) 167.772i 0.222214i
\(756\) 0 0
\(757\) 1303.18 1.72151 0.860753 0.509023i \(-0.169993\pi\)
0.860753 + 0.509023i \(0.169993\pi\)
\(758\) 375.991 0.496030
\(759\) 364.372i 0.480069i
\(760\) −198.749 −0.261512
\(761\) − 694.880i − 0.913114i −0.889694 0.456557i \(-0.849082\pi\)
0.889694 0.456557i \(-0.150918\pi\)
\(762\) − 249.783i − 0.327799i
\(763\) 0 0
\(764\) 331.392 0.433759
\(765\) −185.191 −0.242079
\(766\) − 433.320i − 0.565692i
\(767\) 553.209 0.721263
\(768\) 27.7128i 0.0360844i
\(769\) 263.988i 0.343287i 0.985159 + 0.171644i \(0.0549077\pi\)
−0.985159 + 0.171644i \(0.945092\pi\)
\(770\) 0 0
\(771\) −21.7614 −0.0282249
\(772\) −210.599 −0.272796
\(773\) − 157.360i − 0.203571i −0.994806 0.101786i \(-0.967544\pi\)
0.994806 0.101786i \(-0.0324556\pi\)
\(774\) −211.440 −0.273178
\(775\) 26.3191i 0.0339601i
\(776\) 204.527i 0.263566i
\(777\) 0 0
\(778\) 150.408 0.193326
\(779\) 697.338 0.895171
\(780\) 60.9121i 0.0780924i
\(781\) −396.892 −0.508184
\(782\) 708.964i 0.906604i
\(783\) 11.9683i 0.0152852i
\(784\) 0 0
\(785\) −502.513 −0.640144
\(786\) −182.727 −0.232477
\(787\) 982.258i 1.24810i 0.781383 + 0.624052i \(0.214515\pi\)
−0.781383 + 0.624052i \(0.785485\pi\)
\(788\) 489.471 0.621156
\(789\) − 368.027i − 0.466448i
\(790\) − 285.960i − 0.361975i
\(791\) 0 0
\(792\) −98.3002 −0.124116
\(793\) −531.730 −0.670529
\(794\) 308.053i 0.387977i
\(795\) 221.130 0.278150
\(796\) 195.933i 0.246147i
\(797\) − 946.927i − 1.18811i −0.804423 0.594057i \(-0.797525\pi\)
0.804423 0.594057i \(-0.202475\pi\)
\(798\) 0 0
\(799\) −2116.52 −2.64896
\(800\) −28.2843 −0.0353553
\(801\) − 33.1941i − 0.0414408i
\(802\) 87.6550 0.109295
\(803\) 225.805i 0.281201i
\(804\) 340.052i 0.422951i
\(805\) 0 0
\(806\) 58.5388 0.0726287
\(807\) 579.538 0.718139
\(808\) 143.269i 0.177313i
\(809\) 429.136 0.530452 0.265226 0.964186i \(-0.414553\pi\)
0.265226 + 0.964186i \(0.414553\pi\)
\(810\) − 28.4605i − 0.0351364i
\(811\) − 948.404i − 1.16943i −0.811241 0.584713i \(-0.801207\pi\)
0.811241 0.584713i \(-0.198793\pi\)
\(812\) 0 0
\(813\) −315.840 −0.388487
\(814\) 32.5420 0.0399779
\(815\) 514.558i 0.631360i
\(816\) −191.264 −0.234392
\(817\) − 1566.12i − 1.91692i
\(818\) − 614.278i − 0.750952i
\(819\) 0 0
\(820\) 99.2391 0.121023
\(821\) −403.768 −0.491800 −0.245900 0.969295i \(-0.579083\pi\)
−0.245900 + 0.969295i \(0.579083\pi\)
\(822\) − 604.776i − 0.735738i
\(823\) 866.028 1.05228 0.526141 0.850397i \(-0.323638\pi\)
0.526141 + 0.850397i \(0.323638\pi\)
\(824\) 559.928i 0.679525i
\(825\) − 100.327i − 0.121609i
\(826\) 0 0
\(827\) −363.528 −0.439574 −0.219787 0.975548i \(-0.570536\pi\)
−0.219787 + 0.975548i \(0.570536\pi\)
\(828\) −108.955 −0.131588
\(829\) 240.890i 0.290579i 0.989389 + 0.145290i \(0.0464114\pi\)
−0.989389 + 0.145290i \(0.953589\pi\)
\(830\) −423.124 −0.509788
\(831\) 288.839i 0.347580i
\(832\) 62.9097i 0.0756126i
\(833\) 0 0
\(834\) −381.917 −0.457934
\(835\) −621.154 −0.743897
\(836\) − 728.105i − 0.870939i
\(837\) −27.3516 −0.0326781
\(838\) 646.607i 0.771608i
\(839\) − 214.638i − 0.255826i −0.991785 0.127913i \(-0.959172\pi\)
0.991785 0.127913i \(-0.0408278\pi\)
\(840\) 0 0
\(841\) −835.695 −0.993692
\(842\) −923.692 −1.09702
\(843\) 66.3523i 0.0787098i
\(844\) −777.827 −0.921597
\(845\) − 239.621i − 0.283576i
\(846\) − 325.271i − 0.384481i
\(847\) 0 0
\(848\) 228.382 0.269318
\(849\) 667.699 0.786453
\(850\) − 195.208i − 0.229657i
\(851\) 36.0693 0.0423846
\(852\) − 118.679i − 0.139295i
\(853\) − 1388.39i − 1.62765i −0.581110 0.813825i \(-0.697382\pi\)
0.581110 0.813825i \(-0.302618\pi\)
\(854\) 0 0
\(855\) 210.805 0.246556
\(856\) 418.426 0.488815
\(857\) 1096.98i 1.28002i 0.768366 + 0.640011i \(0.221070\pi\)
−0.768366 + 0.640011i \(0.778930\pi\)
\(858\) −223.147 −0.260078
\(859\) 270.552i 0.314961i 0.987522 + 0.157481i \(0.0503372\pi\)
−0.987522 + 0.157481i \(0.949663\pi\)
\(860\) − 222.877i − 0.259159i
\(861\) 0 0
\(862\) −665.042 −0.771511
\(863\) 268.214 0.310792 0.155396 0.987852i \(-0.450335\pi\)
0.155396 + 0.987852i \(0.450335\pi\)
\(864\) − 29.3939i − 0.0340207i
\(865\) −281.891 −0.325885
\(866\) 45.2970i 0.0523061i
\(867\) − 819.476i − 0.945186i
\(868\) 0 0
\(869\) 1047.60 1.20552
\(870\) −12.6157 −0.0145009
\(871\) 771.939i 0.886268i
\(872\) −153.646 −0.176199
\(873\) − 216.934i − 0.248492i
\(874\) − 807.025i − 0.923370i
\(875\) 0 0
\(876\) −67.5204 −0.0770781
\(877\) 507.411 0.578575 0.289288 0.957242i \(-0.406582\pi\)
0.289288 + 0.957242i \(0.406582\pi\)
\(878\) − 540.567i − 0.615680i
\(879\) −157.590 −0.179283
\(880\) − 103.618i − 0.117747i
\(881\) 833.545i 0.946135i 0.881026 + 0.473067i \(0.156853\pi\)
−0.881026 + 0.473067i \(0.843147\pi\)
\(882\) 0 0
\(883\) −1350.99 −1.53000 −0.765002 0.644028i \(-0.777262\pi\)
−0.765002 + 0.644028i \(0.777262\pi\)
\(884\) −434.181 −0.491155
\(885\) − 272.463i − 0.307867i
\(886\) 174.997 0.197514
\(887\) − 374.129i − 0.421792i −0.977509 0.210896i \(-0.932362\pi\)
0.977509 0.210896i \(-0.0676381\pi\)
\(888\) 9.73077i 0.0109581i
\(889\) 0 0
\(890\) 34.9897 0.0393142
\(891\) 104.263 0.117018
\(892\) 503.825i 0.564827i
\(893\) 2409.26 2.69794
\(894\) 399.488i 0.446855i
\(895\) − 172.416i − 0.192644i
\(896\) 0 0
\(897\) −247.335 −0.275735
\(898\) −377.574 −0.420461
\(899\) 12.1242i 0.0134863i
\(900\) 30.0000 0.0333333
\(901\) 1576.21i 1.74940i
\(902\) 363.556i 0.403055i
\(903\) 0 0
\(904\) 135.166 0.149520
\(905\) −474.073 −0.523837
\(906\) − 183.785i − 0.202853i
\(907\) −357.516 −0.394174 −0.197087 0.980386i \(-0.563148\pi\)
−0.197087 + 0.980386i \(0.563148\pi\)
\(908\) 535.644i 0.589916i
\(909\) − 151.960i − 0.167173i
\(910\) 0 0
\(911\) −1503.55 −1.65044 −0.825219 0.564813i \(-0.808948\pi\)
−0.825219 + 0.564813i \(0.808948\pi\)
\(912\) 217.719 0.238727
\(913\) − 1550.09i − 1.69779i
\(914\) −635.462 −0.695254
\(915\) 261.884i 0.286212i
\(916\) − 691.059i − 0.754431i
\(917\) 0 0
\(918\) 202.866 0.220987
\(919\) −624.998 −0.680085 −0.340043 0.940410i \(-0.610442\pi\)
−0.340043 + 0.940410i \(0.610442\pi\)
\(920\) − 114.849i − 0.124836i
\(921\) 880.015 0.955499
\(922\) − 149.240i − 0.161865i
\(923\) − 269.409i − 0.291884i
\(924\) 0 0
\(925\) −9.93142 −0.0107367
\(926\) 832.342 0.898858
\(927\) − 593.894i − 0.640662i
\(928\) −13.0295 −0.0140404
\(929\) 1088.75i 1.17196i 0.810327 + 0.585978i \(0.199290\pi\)
−0.810327 + 0.585978i \(0.800710\pi\)
\(930\) − 28.8311i − 0.0310012i
\(931\) 0 0
\(932\) −576.129 −0.618164
\(933\) −180.689 −0.193665
\(934\) − 517.117i − 0.553659i
\(935\) 715.132 0.764847
\(936\) − 66.7258i − 0.0712883i
\(937\) 252.836i 0.269835i 0.990857 + 0.134918i \(0.0430770\pi\)
−0.990857 + 0.134918i \(0.956923\pi\)
\(938\) 0 0
\(939\) 460.498 0.490413
\(940\) 342.865 0.364750
\(941\) − 383.664i − 0.407719i −0.979000 0.203859i \(-0.934651\pi\)
0.979000 0.203859i \(-0.0653486\pi\)
\(942\) 550.475 0.584369
\(943\) 402.962i 0.427319i
\(944\) − 281.398i − 0.298091i
\(945\) 0 0
\(946\) 816.495 0.863103
\(947\) −281.784 −0.297554 −0.148777 0.988871i \(-0.547534\pi\)
−0.148777 + 0.988871i \(0.547534\pi\)
\(948\) 313.253i 0.330436i
\(949\) −153.275 −0.161512
\(950\) 222.208i 0.233904i
\(951\) − 18.5715i − 0.0195284i
\(952\) 0 0
\(953\) −332.322 −0.348711 −0.174356 0.984683i \(-0.555784\pi\)
−0.174356 + 0.984683i \(0.555784\pi\)
\(954\) −242.235 −0.253915
\(955\) − 370.507i − 0.387966i
\(956\) −377.620 −0.395000
\(957\) − 46.2169i − 0.0482935i
\(958\) − 688.971i − 0.719177i
\(959\) 0 0
\(960\) 30.9839 0.0322749
\(961\) 933.292 0.971168
\(962\) 22.0894i 0.0229620i
\(963\) −443.807 −0.460859
\(964\) 195.696i 0.203004i
\(965\) 235.457i 0.243997i
\(966\) 0 0
\(967\) 1155.53 1.19496 0.597482 0.801882i \(-0.296168\pi\)
0.597482 + 0.801882i \(0.296168\pi\)
\(968\) 37.3563 0.0385912
\(969\) 1502.62i 1.55069i
\(970\) 228.668 0.235740
\(971\) 1510.28i 1.55539i 0.628645 + 0.777693i \(0.283610\pi\)
−0.628645 + 0.777693i \(0.716390\pi\)
\(972\) 31.1769i 0.0320750i
\(973\) 0 0
\(974\) −276.866 −0.284257
\(975\) 68.1018 0.0698480
\(976\) 270.473i 0.277124i
\(977\) 801.616 0.820487 0.410244 0.911976i \(-0.365444\pi\)
0.410244 + 0.911976i \(0.365444\pi\)
\(978\) − 563.670i − 0.576350i
\(979\) 128.182i 0.130932i
\(980\) 0 0
\(981\) 162.966 0.166122
\(982\) −493.873 −0.502926
\(983\) − 1094.79i − 1.11372i −0.830605 0.556862i \(-0.812005\pi\)
0.830605 0.556862i \(-0.187995\pi\)
\(984\) −108.711 −0.110479
\(985\) − 547.245i − 0.555579i
\(986\) − 89.9249i − 0.0912018i
\(987\) 0 0
\(988\) 494.235 0.500238
\(989\) 904.996 0.915062
\(990\) 109.903i 0.111013i
\(991\) 43.9925 0.0443920 0.0221960 0.999754i \(-0.492934\pi\)
0.0221960 + 0.999754i \(0.492934\pi\)
\(992\) − 29.7766i − 0.0300168i
\(993\) − 611.277i − 0.615586i
\(994\) 0 0
\(995\) 219.060 0.220161
\(996\) 463.509 0.465370
\(997\) 12.2561i 0.0122929i 0.999981 + 0.00614647i \(0.00195649\pi\)
−0.999981 + 0.00614647i \(0.998044\pi\)
\(998\) −474.382 −0.475333
\(999\) − 10.3210i − 0.0103314i
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 1470.3.f.a.391.7 8
7.4 even 3 210.3.o.a.61.2 yes 8
7.5 odd 6 210.3.o.a.31.2 8
7.6 odd 2 inner 1470.3.f.a.391.6 8
21.5 even 6 630.3.v.b.451.3 8
21.11 odd 6 630.3.v.b.271.3 8
35.4 even 6 1050.3.p.b.901.3 8
35.12 even 12 1050.3.q.c.199.7 16
35.18 odd 12 1050.3.q.c.649.7 16
35.19 odd 6 1050.3.p.b.451.3 8
35.32 odd 12 1050.3.q.c.649.2 16
35.33 even 12 1050.3.q.c.199.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.o.a.31.2 8 7.5 odd 6
210.3.o.a.61.2 yes 8 7.4 even 3
630.3.v.b.271.3 8 21.11 odd 6
630.3.v.b.451.3 8 21.5 even 6
1050.3.p.b.451.3 8 35.19 odd 6
1050.3.p.b.901.3 8 35.4 even 6
1050.3.q.c.199.2 16 35.33 even 12
1050.3.q.c.199.7 16 35.12 even 12
1050.3.q.c.649.2 16 35.32 odd 12
1050.3.q.c.649.7 16 35.18 odd 12
1470.3.f.a.391.6 8 7.6 odd 2 inner
1470.3.f.a.391.7 8 1.1 even 1 trivial