Properties

Label 3630.2.a.bs.1.1
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.52625.1
Defining polynomial: \(x^{4} - x^{3} - 24 x^{2} + 19 x + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.15753\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.10899 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.10899 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -1.33343 q^{13} -5.10899 q^{14} -1.00000 q^{15} +1.00000 q^{16} -0.775565 q^{17} +1.00000 q^{18} +0.0785371 q^{19} -1.00000 q^{20} -5.10899 q^{21} +6.64849 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.33343 q^{26} +1.00000 q^{27} -5.10899 q^{28} +2.01882 q^{29} -1.00000 q^{30} +8.49096 q^{31} +1.00000 q^{32} -0.775565 q^{34} +5.10899 q^{35} +1.00000 q^{36} -3.93310 q^{37} +0.0785371 q^{38} -1.33343 q^{39} -1.00000 q^{40} +11.8962 q^{41} -5.10899 q^{42} +6.01163 q^{43} -1.00000 q^{45} +6.64849 q^{46} +6.56950 q^{47} +1.00000 q^{48} +19.1018 q^{49} +1.00000 q^{50} -0.775565 q^{51} -1.33343 q^{52} +10.3451 q^{53} +1.00000 q^{54} -5.10899 q^{56} +0.0785371 q^{57} +2.01882 q^{58} -5.32624 q^{59} -1.00000 q^{60} +3.74585 q^{61} +8.49096 q^{62} -5.10899 q^{63} +1.00000 q^{64} +1.33343 q^{65} +0.588036 q^{67} -0.775565 q^{68} +6.64849 q^{69} +5.10899 q^{70} -2.31506 q^{71} +1.00000 q^{72} -10.9443 q^{73} -3.93310 q^{74} +1.00000 q^{75} +0.0785371 q^{76} -1.33343 q^{78} -12.3267 q^{79} -1.00000 q^{80} +1.00000 q^{81} +11.8962 q^{82} +8.00000 q^{83} -5.10899 q^{84} +0.775565 q^{85} +6.01163 q^{86} +2.01882 q^{87} -9.67821 q^{89} -1.00000 q^{90} +6.81247 q^{91} +6.64849 q^{92} +8.49096 q^{93} +6.56950 q^{94} -0.0785371 q^{95} +1.00000 q^{96} -13.4541 q^{97} +19.1018 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} + 2q^{13} + q^{14} - 4q^{15} + 4q^{16} + 11q^{17} + 4q^{18} + q^{19} - 4q^{20} + q^{21} + 4q^{24} + 4q^{25} + 2q^{26} + 4q^{27} + q^{28} + 9q^{29} - 4q^{30} + 17q^{31} + 4q^{32} + 11q^{34} - q^{35} + 4q^{36} + 8q^{37} + q^{38} + 2q^{39} - 4q^{40} - 11q^{41} + q^{42} + q^{43} - 4q^{45} + 10q^{47} + 4q^{48} + 31q^{49} + 4q^{50} + 11q^{51} + 2q^{52} + 11q^{53} + 4q^{54} + q^{56} + q^{57} + 9q^{58} + 10q^{59} - 4q^{60} - 10q^{61} + 17q^{62} + q^{63} + 4q^{64} - 2q^{65} + 9q^{67} + 11q^{68} - q^{70} + 10q^{71} + 4q^{72} - 8q^{73} + 8q^{74} + 4q^{75} + q^{76} + 2q^{78} - 7q^{79} - 4q^{80} + 4q^{81} - 11q^{82} + 32q^{83} + q^{84} - 11q^{85} + q^{86} + 9q^{87} - 23q^{89} - 4q^{90} + 48q^{91} + 17q^{93} + 10q^{94} - q^{95} + 4q^{96} - 2q^{97} + 31q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −5.10899 −1.93102 −0.965509 0.260370i \(-0.916155\pi\)
−0.965509 + 0.260370i \(0.916155\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −1.33343 −0.369826 −0.184913 0.982755i \(-0.559200\pi\)
−0.184913 + 0.982755i \(0.559200\pi\)
\(14\) −5.10899 −1.36544
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −0.775565 −0.188102 −0.0940511 0.995567i \(-0.529982\pi\)
−0.0940511 + 0.995567i \(0.529982\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.0785371 0.0180176 0.00900882 0.999959i \(-0.497132\pi\)
0.00900882 + 0.999959i \(0.497132\pi\)
\(20\) −1.00000 −0.223607
\(21\) −5.10899 −1.11487
\(22\) 0 0
\(23\) 6.64849 1.38631 0.693153 0.720791i \(-0.256221\pi\)
0.693153 + 0.720791i \(0.256221\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −1.33343 −0.261507
\(27\) 1.00000 0.192450
\(28\) −5.10899 −0.965509
\(29\) 2.01882 0.374886 0.187443 0.982275i \(-0.439980\pi\)
0.187443 + 0.982275i \(0.439980\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.49096 1.52502 0.762511 0.646976i \(-0.223967\pi\)
0.762511 + 0.646976i \(0.223967\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.775565 −0.133008
\(35\) 5.10899 0.863577
\(36\) 1.00000 0.166667
\(37\) −3.93310 −0.646597 −0.323298 0.946297i \(-0.604792\pi\)
−0.323298 + 0.946297i \(0.604792\pi\)
\(38\) 0.0785371 0.0127404
\(39\) −1.33343 −0.213519
\(40\) −1.00000 −0.158114
\(41\) 11.8962 1.85787 0.928936 0.370239i \(-0.120724\pi\)
0.928936 + 0.370239i \(0.120724\pi\)
\(42\) −5.10899 −0.788335
\(43\) 6.01163 0.916765 0.458383 0.888755i \(-0.348429\pi\)
0.458383 + 0.888755i \(0.348429\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 6.64849 0.980266
\(47\) 6.56950 0.958259 0.479130 0.877744i \(-0.340952\pi\)
0.479130 + 0.877744i \(0.340952\pi\)
\(48\) 1.00000 0.144338
\(49\) 19.1018 2.72883
\(50\) 1.00000 0.141421
\(51\) −0.775565 −0.108601
\(52\) −1.33343 −0.184913
\(53\) 10.3451 1.42100 0.710502 0.703696i \(-0.248468\pi\)
0.710502 + 0.703696i \(0.248468\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −5.10899 −0.682718
\(57\) 0.0785371 0.0104025
\(58\) 2.01882 0.265084
\(59\) −5.32624 −0.693417 −0.346709 0.937973i \(-0.612701\pi\)
−0.346709 + 0.937973i \(0.612701\pi\)
\(60\) −1.00000 −0.129099
\(61\) 3.74585 0.479607 0.239803 0.970821i \(-0.422917\pi\)
0.239803 + 0.970821i \(0.422917\pi\)
\(62\) 8.49096 1.07835
\(63\) −5.10899 −0.643673
\(64\) 1.00000 0.125000
\(65\) 1.33343 0.165391
\(66\) 0 0
\(67\) 0.588036 0.0718400 0.0359200 0.999355i \(-0.488564\pi\)
0.0359200 + 0.999355i \(0.488564\pi\)
\(68\) −0.775565 −0.0940511
\(69\) 6.64849 0.800384
\(70\) 5.10899 0.610641
\(71\) −2.31506 −0.274747 −0.137374 0.990519i \(-0.543866\pi\)
−0.137374 + 0.990519i \(0.543866\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.9443 −1.28093 −0.640465 0.767987i \(-0.721258\pi\)
−0.640465 + 0.767987i \(0.721258\pi\)
\(74\) −3.93310 −0.457213
\(75\) 1.00000 0.115470
\(76\) 0.0785371 0.00900882
\(77\) 0 0
\(78\) −1.33343 −0.150981
\(79\) −12.3267 −1.38686 −0.693431 0.720523i \(-0.743902\pi\)
−0.693431 + 0.720523i \(0.743902\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 11.8962 1.31371
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −5.10899 −0.557437
\(85\) 0.775565 0.0841218
\(86\) 6.01163 0.648251
\(87\) 2.01882 0.216440
\(88\) 0 0
\(89\) −9.67821 −1.02589 −0.512944 0.858422i \(-0.671445\pi\)
−0.512944 + 0.858422i \(0.671445\pi\)
\(90\) −1.00000 −0.105409
\(91\) 6.81247 0.714141
\(92\) 6.64849 0.693153
\(93\) 8.49096 0.880471
\(94\) 6.56950 0.677592
\(95\) −0.0785371 −0.00805773
\(96\) 1.00000 0.102062
\(97\) −13.4541 −1.36605 −0.683026 0.730394i \(-0.739336\pi\)
−0.683026 + 0.730394i \(0.739336\pi\)
\(98\) 19.1018 1.92957
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.6408 1.05880 0.529402 0.848371i \(-0.322416\pi\)
0.529402 + 0.848371i \(0.322416\pi\)
\(102\) −0.775565 −0.0767924
\(103\) 16.7087 1.64635 0.823177 0.567785i \(-0.192200\pi\)
0.823177 + 0.567785i \(0.192200\pi\)
\(104\) −1.33343 −0.130753
\(105\) 5.10899 0.498587
\(106\) 10.3451 1.00480
\(107\) −17.5511 −1.69673 −0.848366 0.529410i \(-0.822413\pi\)
−0.848366 + 0.529410i \(0.822413\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.94427 0.856706 0.428353 0.903612i \(-0.359094\pi\)
0.428353 + 0.903612i \(0.359094\pi\)
\(110\) 0 0
\(111\) −3.93310 −0.373313
\(112\) −5.10899 −0.482754
\(113\) −5.72703 −0.538753 −0.269377 0.963035i \(-0.586818\pi\)
−0.269377 + 0.963035i \(0.586818\pi\)
\(114\) 0.0785371 0.00735567
\(115\) −6.64849 −0.619975
\(116\) 2.01882 0.187443
\(117\) −1.33343 −0.123275
\(118\) −5.32624 −0.490320
\(119\) 3.96236 0.363229
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) 3.74585 0.339133
\(123\) 11.8962 1.07264
\(124\) 8.49096 0.762511
\(125\) −1.00000 −0.0894427
\(126\) −5.10899 −0.455145
\(127\) 6.96281 0.617850 0.308925 0.951086i \(-0.400031\pi\)
0.308925 + 0.951086i \(0.400031\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.01163 0.529295
\(130\) 1.33343 0.116949
\(131\) 2.19916 0.192142 0.0960709 0.995374i \(-0.469372\pi\)
0.0960709 + 0.995374i \(0.469372\pi\)
\(132\) 0 0
\(133\) −0.401245 −0.0347924
\(134\) 0.588036 0.0507985
\(135\) −1.00000 −0.0860663
\(136\) −0.775565 −0.0665041
\(137\) −2.17590 −0.185899 −0.0929497 0.995671i \(-0.529630\pi\)
−0.0929497 + 0.995671i \(0.529630\pi\)
\(138\) 6.64849 0.565957
\(139\) −10.3074 −0.874264 −0.437132 0.899397i \(-0.644006\pi\)
−0.437132 + 0.899397i \(0.644006\pi\)
\(140\) 5.10899 0.431789
\(141\) 6.56950 0.553251
\(142\) −2.31506 −0.194276
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.01882 −0.167654
\(146\) −10.9443 −0.905754
\(147\) 19.1018 1.57549
\(148\) −3.93310 −0.323298
\(149\) 15.2477 1.24914 0.624570 0.780969i \(-0.285274\pi\)
0.624570 + 0.780969i \(0.285274\pi\)
\(150\) 1.00000 0.0816497
\(151\) −6.56921 −0.534595 −0.267297 0.963614i \(-0.586131\pi\)
−0.267297 + 0.963614i \(0.586131\pi\)
\(152\) 0.0785371 0.00637020
\(153\) −0.775565 −0.0627007
\(154\) 0 0
\(155\) −8.49096 −0.682010
\(156\) −1.33343 −0.106760
\(157\) −9.30815 −0.742872 −0.371436 0.928459i \(-0.621134\pi\)
−0.371436 + 0.928459i \(0.621134\pi\)
\(158\) −12.3267 −0.980659
\(159\) 10.3451 0.820417
\(160\) −1.00000 −0.0790569
\(161\) −33.9671 −2.67698
\(162\) 1.00000 0.0785674
\(163\) −18.7018 −1.46483 −0.732417 0.680856i \(-0.761608\pi\)
−0.732417 + 0.680856i \(0.761608\pi\)
\(164\) 11.8962 0.928936
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 2.37478 0.183766 0.0918829 0.995770i \(-0.470711\pi\)
0.0918829 + 0.995770i \(0.470711\pi\)
\(168\) −5.10899 −0.394167
\(169\) −11.2220 −0.863229
\(170\) 0.775565 0.0594831
\(171\) 0.0785371 0.00600588
\(172\) 6.01163 0.458383
\(173\) −2.91427 −0.221568 −0.110784 0.993845i \(-0.535336\pi\)
−0.110784 + 0.993845i \(0.535336\pi\)
\(174\) 2.01882 0.153047
\(175\) −5.10899 −0.386204
\(176\) 0 0
\(177\) −5.32624 −0.400345
\(178\) −9.67821 −0.725412
\(179\) 1.56231 0.116772 0.0583861 0.998294i \(-0.481405\pi\)
0.0583861 + 0.998294i \(0.481405\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 18.4721 1.37302 0.686512 0.727119i \(-0.259141\pi\)
0.686512 + 0.727119i \(0.259141\pi\)
\(182\) 6.81247 0.504974
\(183\) 3.74585 0.276901
\(184\) 6.64849 0.490133
\(185\) 3.93310 0.289167
\(186\) 8.49096 0.622587
\(187\) 0 0
\(188\) 6.56950 0.479130
\(189\) −5.10899 −0.371625
\(190\) −0.0785371 −0.00569768
\(191\) 20.7278 1.49981 0.749904 0.661546i \(-0.230100\pi\)
0.749904 + 0.661546i \(0.230100\pi\)
\(192\) 1.00000 0.0721688
\(193\) −1.37079 −0.0986716 −0.0493358 0.998782i \(-0.515710\pi\)
−0.0493358 + 0.998782i \(0.515710\pi\)
\(194\) −13.4541 −0.965945
\(195\) 1.33343 0.0954887
\(196\) 19.1018 1.36441
\(197\) 5.01135 0.357044 0.178522 0.983936i \(-0.442868\pi\)
0.178522 + 0.983936i \(0.442868\pi\)
\(198\) 0 0
\(199\) 23.3086 1.65230 0.826152 0.563448i \(-0.190525\pi\)
0.826152 + 0.563448i \(0.190525\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0.588036 0.0414768
\(202\) 10.6408 0.748687
\(203\) −10.3141 −0.723911
\(204\) −0.775565 −0.0543004
\(205\) −11.8962 −0.830866
\(206\) 16.7087 1.16415
\(207\) 6.64849 0.462102
\(208\) −1.33343 −0.0924566
\(209\) 0 0
\(210\) 5.10899 0.352554
\(211\) 20.0842 1.38265 0.691326 0.722543i \(-0.257027\pi\)
0.691326 + 0.722543i \(0.257027\pi\)
\(212\) 10.3451 0.710502
\(213\) −2.31506 −0.158625
\(214\) −17.5511 −1.19977
\(215\) −6.01163 −0.409990
\(216\) 1.00000 0.0680414
\(217\) −43.3802 −2.94484
\(218\) 8.94427 0.605783
\(219\) −10.9443 −0.739545
\(220\) 0 0
\(221\) 1.03416 0.0695651
\(222\) −3.93310 −0.263972
\(223\) 9.10899 0.609983 0.304992 0.952355i \(-0.401346\pi\)
0.304992 + 0.952355i \(0.401346\pi\)
\(224\) −5.10899 −0.341359
\(225\) 1.00000 0.0666667
\(226\) −5.72703 −0.380956
\(227\) −20.6901 −1.37325 −0.686626 0.727011i \(-0.740909\pi\)
−0.686626 + 0.727011i \(0.740909\pi\)
\(228\) 0.0785371 0.00520124
\(229\) 10.9819 0.725705 0.362853 0.931846i \(-0.381803\pi\)
0.362853 + 0.931846i \(0.381803\pi\)
\(230\) −6.64849 −0.438388
\(231\) 0 0
\(232\) 2.01882 0.132542
\(233\) 2.96309 0.194119 0.0970594 0.995279i \(-0.469056\pi\)
0.0970594 + 0.995279i \(0.469056\pi\)
\(234\) −1.33343 −0.0871689
\(235\) −6.56950 −0.428547
\(236\) −5.32624 −0.346709
\(237\) −12.3267 −0.800705
\(238\) 3.96236 0.256841
\(239\) −8.56978 −0.554333 −0.277166 0.960822i \(-0.589395\pi\)
−0.277166 + 0.960822i \(0.589395\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −7.97000 −0.513393 −0.256696 0.966492i \(-0.582634\pi\)
−0.256696 + 0.966492i \(0.582634\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 3.74585 0.239803
\(245\) −19.1018 −1.22037
\(246\) 11.8962 0.758473
\(247\) −0.104723 −0.00666339
\(248\) 8.49096 0.539176
\(249\) 8.00000 0.506979
\(250\) −1.00000 −0.0632456
\(251\) 20.2596 1.27878 0.639388 0.768884i \(-0.279188\pi\)
0.639388 + 0.768884i \(0.279188\pi\)
\(252\) −5.10899 −0.321836
\(253\) 0 0
\(254\) 6.96281 0.436886
\(255\) 0.775565 0.0485678
\(256\) 1.00000 0.0625000
\(257\) −19.9638 −1.24531 −0.622655 0.782497i \(-0.713946\pi\)
−0.622655 + 0.782497i \(0.713946\pi\)
\(258\) 6.01163 0.374268
\(259\) 20.0942 1.24859
\(260\) 1.33343 0.0826957
\(261\) 2.01882 0.124962
\(262\) 2.19916 0.135865
\(263\) 0.696114 0.0429242 0.0214621 0.999770i \(-0.493168\pi\)
0.0214621 + 0.999770i \(0.493168\pi\)
\(264\) 0 0
\(265\) −10.3451 −0.635492
\(266\) −0.401245 −0.0246019
\(267\) −9.67821 −0.592297
\(268\) 0.588036 0.0359200
\(269\) 5.92175 0.361055 0.180528 0.983570i \(-0.442219\pi\)
0.180528 + 0.983570i \(0.442219\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 13.9146 0.845249 0.422625 0.906305i \(-0.361109\pi\)
0.422625 + 0.906305i \(0.361109\pi\)
\(272\) −0.775565 −0.0470255
\(273\) 6.81247 0.412309
\(274\) −2.17590 −0.131451
\(275\) 0 0
\(276\) 6.64849 0.400192
\(277\) 10.1025 0.607003 0.303501 0.952831i \(-0.401844\pi\)
0.303501 + 0.952831i \(0.401844\pi\)
\(278\) −10.3074 −0.618198
\(279\) 8.49096 0.508340
\(280\) 5.10899 0.305321
\(281\) −28.9081 −1.72451 −0.862256 0.506472i \(-0.830949\pi\)
−0.862256 + 0.506472i \(0.830949\pi\)
\(282\) 6.56950 0.391208
\(283\) 13.1211 0.779967 0.389984 0.920822i \(-0.372481\pi\)
0.389984 + 0.920822i \(0.372481\pi\)
\(284\) −2.31506 −0.137374
\(285\) −0.0785371 −0.00465213
\(286\) 0 0
\(287\) −60.7775 −3.58759
\(288\) 1.00000 0.0589256
\(289\) −16.3985 −0.964618
\(290\) −2.01882 −0.118549
\(291\) −13.4541 −0.788691
\(292\) −10.9443 −0.640465
\(293\) −22.6601 −1.32382 −0.661909 0.749584i \(-0.730254\pi\)
−0.661909 + 0.749584i \(0.730254\pi\)
\(294\) 19.1018 1.11404
\(295\) 5.32624 0.310106
\(296\) −3.93310 −0.228607
\(297\) 0 0
\(298\) 15.2477 0.883276
\(299\) −8.86528 −0.512692
\(300\) 1.00000 0.0577350
\(301\) −30.7134 −1.77029
\(302\) −6.56921 −0.378016
\(303\) 10.6408 0.611300
\(304\) 0.0785371 0.00450441
\(305\) −3.74585 −0.214487
\(306\) −0.775565 −0.0443361
\(307\) 29.3158 1.67314 0.836571 0.547859i \(-0.184557\pi\)
0.836571 + 0.547859i \(0.184557\pi\)
\(308\) 0 0
\(309\) 16.7087 0.950522
\(310\) −8.49096 −0.482254
\(311\) 11.3570 0.643995 0.321998 0.946741i \(-0.395646\pi\)
0.321998 + 0.946741i \(0.395646\pi\)
\(312\) −1.33343 −0.0754905
\(313\) −3.17664 −0.179554 −0.0897770 0.995962i \(-0.528615\pi\)
−0.0897770 + 0.995962i \(0.528615\pi\)
\(314\) −9.30815 −0.525290
\(315\) 5.10899 0.287859
\(316\) −12.3267 −0.693431
\(317\) 4.73101 0.265720 0.132860 0.991135i \(-0.457584\pi\)
0.132860 + 0.991135i \(0.457584\pi\)
\(318\) 10.3451 0.580122
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −17.5511 −0.979609
\(322\) −33.9671 −1.89291
\(323\) −0.0609106 −0.00338916
\(324\) 1.00000 0.0555556
\(325\) −1.33343 −0.0739652
\(326\) −18.7018 −1.03579
\(327\) 8.94427 0.494619
\(328\) 11.8962 0.656857
\(329\) −33.5635 −1.85042
\(330\) 0 0
\(331\) 30.0424 1.65128 0.825639 0.564199i \(-0.190815\pi\)
0.825639 + 0.564199i \(0.190815\pi\)
\(332\) 8.00000 0.439057
\(333\) −3.93310 −0.215532
\(334\) 2.37478 0.129942
\(335\) −0.588036 −0.0321278
\(336\) −5.10899 −0.278718
\(337\) 25.3570 1.38128 0.690641 0.723198i \(-0.257328\pi\)
0.690641 + 0.723198i \(0.257328\pi\)
\(338\) −11.2220 −0.610395
\(339\) −5.72703 −0.311049
\(340\) 0.775565 0.0420609
\(341\) 0 0
\(342\) 0.0785371 0.00424680
\(343\) −61.8280 −3.33840
\(344\) 6.01163 0.324126
\(345\) −6.64849 −0.357943
\(346\) −2.91427 −0.156672
\(347\) 3.17664 0.170531 0.0852654 0.996358i \(-0.472826\pi\)
0.0852654 + 0.996358i \(0.472826\pi\)
\(348\) 2.01882 0.108220
\(349\) 6.95865 0.372488 0.186244 0.982504i \(-0.440368\pi\)
0.186244 + 0.982504i \(0.440368\pi\)
\(350\) −5.10899 −0.273087
\(351\) −1.33343 −0.0711731
\(352\) 0 0
\(353\) 8.01163 0.426416 0.213208 0.977007i \(-0.431609\pi\)
0.213208 + 0.977007i \(0.431609\pi\)
\(354\) −5.32624 −0.283086
\(355\) 2.31506 0.122871
\(356\) −9.67821 −0.512944
\(357\) 3.96236 0.209710
\(358\) 1.56231 0.0825704
\(359\) 0.180340 0.00951798 0.00475899 0.999989i \(-0.498485\pi\)
0.00475899 + 0.999989i \(0.498485\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.9938 −0.999675
\(362\) 18.4721 0.970874
\(363\) 0 0
\(364\) 6.81247 0.357070
\(365\) 10.9443 0.572849
\(366\) 3.74585 0.195799
\(367\) 30.8705 1.61142 0.805712 0.592307i \(-0.201783\pi\)
0.805712 + 0.592307i \(0.201783\pi\)
\(368\) 6.64849 0.346576
\(369\) 11.8962 0.619291
\(370\) 3.93310 0.204472
\(371\) −52.8528 −2.74398
\(372\) 8.49096 0.440236
\(373\) −16.1018 −0.833720 −0.416860 0.908971i \(-0.636869\pi\)
−0.416860 + 0.908971i \(0.636869\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 6.56950 0.338796
\(377\) −2.69195 −0.138643
\(378\) −5.10899 −0.262778
\(379\) 32.5487 1.67191 0.835956 0.548796i \(-0.184914\pi\)
0.835956 + 0.548796i \(0.184914\pi\)
\(380\) −0.0785371 −0.00402887
\(381\) 6.96281 0.356716
\(382\) 20.7278 1.06052
\(383\) 2.10825 0.107727 0.0538634 0.998548i \(-0.482846\pi\)
0.0538634 + 0.998548i \(0.482846\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −1.37079 −0.0697714
\(387\) 6.01163 0.305588
\(388\) −13.4541 −0.683026
\(389\) −12.6032 −0.639008 −0.319504 0.947585i \(-0.603516\pi\)
−0.319504 + 0.947585i \(0.603516\pi\)
\(390\) 1.33343 0.0675207
\(391\) −5.15633 −0.260767
\(392\) 19.1018 0.964787
\(393\) 2.19916 0.110933
\(394\) 5.01135 0.252468
\(395\) 12.3267 0.620223
\(396\) 0 0
\(397\) −30.6341 −1.53748 −0.768741 0.639560i \(-0.779116\pi\)
−0.768741 + 0.639560i \(0.779116\pi\)
\(398\) 23.3086 1.16836
\(399\) −0.401245 −0.0200874
\(400\) 1.00000 0.0500000
\(401\) −19.9671 −0.997108 −0.498554 0.866859i \(-0.666136\pi\)
−0.498554 + 0.866859i \(0.666136\pi\)
\(402\) 0.588036 0.0293285
\(403\) −11.3221 −0.563993
\(404\) 10.6408 0.529402
\(405\) −1.00000 −0.0496904
\(406\) −10.3141 −0.511883
\(407\) 0 0
\(408\) −0.775565 −0.0383962
\(409\) 7.13962 0.353032 0.176516 0.984298i \(-0.443517\pi\)
0.176516 + 0.984298i \(0.443517\pi\)
\(410\) −11.8962 −0.587511
\(411\) −2.17590 −0.107329
\(412\) 16.7087 0.823177
\(413\) 27.2117 1.33900
\(414\) 6.64849 0.326755
\(415\) −8.00000 −0.392705
\(416\) −1.33343 −0.0653767
\(417\) −10.3074 −0.504756
\(418\) 0 0
\(419\) −22.3691 −1.09280 −0.546400 0.837524i \(-0.684002\pi\)
−0.546400 + 0.837524i \(0.684002\pi\)
\(420\) 5.10899 0.249293
\(421\) −6.06091 −0.295391 −0.147695 0.989033i \(-0.547186\pi\)
−0.147695 + 0.989033i \(0.547186\pi\)
\(422\) 20.0842 0.977682
\(423\) 6.56950 0.319420
\(424\) 10.3451 0.502401
\(425\) −0.775565 −0.0376204
\(426\) −2.31506 −0.112165
\(427\) −19.1375 −0.926129
\(428\) −17.5511 −0.848366
\(429\) 0 0
\(430\) −6.01163 −0.289907
\(431\) −37.1623 −1.79004 −0.895021 0.446023i \(-0.852840\pi\)
−0.895021 + 0.446023i \(0.852840\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.6540 1.47314 0.736568 0.676364i \(-0.236445\pi\)
0.736568 + 0.676364i \(0.236445\pi\)
\(434\) −43.3802 −2.08232
\(435\) −2.01882 −0.0967951
\(436\) 8.94427 0.428353
\(437\) 0.522153 0.0249780
\(438\) −10.9443 −0.522938
\(439\) −7.06017 −0.336964 −0.168482 0.985705i \(-0.553886\pi\)
−0.168482 + 0.985705i \(0.553886\pi\)
\(440\) 0 0
\(441\) 19.1018 0.909610
\(442\) 1.03416 0.0491900
\(443\) 14.5925 0.693310 0.346655 0.937993i \(-0.387318\pi\)
0.346655 + 0.937993i \(0.387318\pi\)
\(444\) −3.93310 −0.186656
\(445\) 9.67821 0.458791
\(446\) 9.10899 0.431323
\(447\) 15.2477 0.721192
\(448\) −5.10899 −0.241377
\(449\) 12.7687 0.602590 0.301295 0.953531i \(-0.402581\pi\)
0.301295 + 0.953531i \(0.402581\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −5.72703 −0.269377
\(453\) −6.56921 −0.308649
\(454\) −20.6901 −0.971035
\(455\) −6.81247 −0.319374
\(456\) 0.0785371 0.00367783
\(457\) 17.3426 0.811252 0.405626 0.914039i \(-0.367053\pi\)
0.405626 + 0.914039i \(0.367053\pi\)
\(458\) 10.9819 0.513151
\(459\) −0.775565 −0.0362003
\(460\) −6.64849 −0.309987
\(461\) 2.99355 0.139424 0.0697118 0.997567i \(-0.477792\pi\)
0.0697118 + 0.997567i \(0.477792\pi\)
\(462\) 0 0
\(463\) −16.2589 −0.755614 −0.377807 0.925884i \(-0.623322\pi\)
−0.377807 + 0.925884i \(0.623322\pi\)
\(464\) 2.01882 0.0937215
\(465\) −8.49096 −0.393759
\(466\) 2.96309 0.137263
\(467\) −18.5707 −0.859349 −0.429675 0.902984i \(-0.641372\pi\)
−0.429675 + 0.902984i \(0.641372\pi\)
\(468\) −1.33343 −0.0616377
\(469\) −3.00427 −0.138724
\(470\) −6.56950 −0.303028
\(471\) −9.30815 −0.428897
\(472\) −5.32624 −0.245160
\(473\) 0 0
\(474\) −12.3267 −0.566184
\(475\) 0.0785371 0.00360353
\(476\) 3.96236 0.181614
\(477\) 10.3451 0.473668
\(478\) −8.56978 −0.391973
\(479\) 30.5554 1.39611 0.698056 0.716043i \(-0.254049\pi\)
0.698056 + 0.716043i \(0.254049\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 5.24450 0.239129
\(482\) −7.97000 −0.363024
\(483\) −33.9671 −1.54556
\(484\) 0 0
\(485\) 13.4541 0.610917
\(486\) 1.00000 0.0453609
\(487\) −0.508303 −0.0230334 −0.0115167 0.999934i \(-0.503666\pi\)
−0.0115167 + 0.999934i \(0.503666\pi\)
\(488\) 3.74585 0.169567
\(489\) −18.7018 −0.845723
\(490\) −19.1018 −0.862931
\(491\) −39.3943 −1.77784 −0.888921 0.458061i \(-0.848544\pi\)
−0.888921 + 0.458061i \(0.848544\pi\)
\(492\) 11.8962 0.536322
\(493\) −1.56573 −0.0705168
\(494\) −0.104723 −0.00471173
\(495\) 0 0
\(496\) 8.49096 0.381255
\(497\) 11.8276 0.530542
\(498\) 8.00000 0.358489
\(499\) 21.4608 0.960717 0.480358 0.877072i \(-0.340507\pi\)
0.480358 + 0.877072i \(0.340507\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 2.37478 0.106097
\(502\) 20.2596 0.904231
\(503\) 23.5514 1.05011 0.525053 0.851070i \(-0.324046\pi\)
0.525053 + 0.851070i \(0.324046\pi\)
\(504\) −5.10899 −0.227573
\(505\) −10.6408 −0.473511
\(506\) 0 0
\(507\) −11.2220 −0.498385
\(508\) 6.96281 0.308925
\(509\) 38.7178 1.71614 0.858069 0.513535i \(-0.171664\pi\)
0.858069 + 0.513535i \(0.171664\pi\)
\(510\) 0.775565 0.0343426
\(511\) 55.9142 2.47350
\(512\) 1.00000 0.0441942
\(513\) 0.0785371 0.00346750
\(514\) −19.9638 −0.880567
\(515\) −16.7087 −0.736272
\(516\) 6.01163 0.264647
\(517\) 0 0
\(518\) 20.0942 0.882887
\(519\) −2.91427 −0.127922
\(520\) 1.33343 0.0584747
\(521\) 2.53921 0.111245 0.0556225 0.998452i \(-0.482286\pi\)
0.0556225 + 0.998452i \(0.482286\pi\)
\(522\) 2.01882 0.0883615
\(523\) −37.3202 −1.63190 −0.815950 0.578122i \(-0.803786\pi\)
−0.815950 + 0.578122i \(0.803786\pi\)
\(524\) 2.19916 0.0960709
\(525\) −5.10899 −0.222975
\(526\) 0.696114 0.0303520
\(527\) −6.58529 −0.286860
\(528\) 0 0
\(529\) 21.2024 0.921844
\(530\) −10.3451 −0.449361
\(531\) −5.32624 −0.231139
\(532\) −0.401245 −0.0173962
\(533\) −15.8627 −0.687090
\(534\) −9.67821 −0.418817
\(535\) 17.5511 0.758802
\(536\) 0.588036 0.0253993
\(537\) 1.56231 0.0674185
\(538\) 5.92175 0.255305
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 9.23515 0.397050 0.198525 0.980096i \(-0.436385\pi\)
0.198525 + 0.980096i \(0.436385\pi\)
\(542\) 13.9146 0.597681
\(543\) 18.4721 0.792715
\(544\) −0.775565 −0.0332521
\(545\) −8.94427 −0.383131
\(546\) 6.81247 0.291547
\(547\) 25.3758 1.08499 0.542495 0.840059i \(-0.317480\pi\)
0.542495 + 0.840059i \(0.317480\pi\)
\(548\) −2.17590 −0.0929497
\(549\) 3.74585 0.159869
\(550\) 0 0
\(551\) 0.158552 0.00675456
\(552\) 6.64849 0.282978
\(553\) 62.9770 2.67805
\(554\) 10.1025 0.429216
\(555\) 3.93310 0.166951
\(556\) −10.3074 −0.437132
\(557\) 10.6107 0.449589 0.224794 0.974406i \(-0.427829\pi\)
0.224794 + 0.974406i \(0.427829\pi\)
\(558\) 8.49096 0.359451
\(559\) −8.01608 −0.339044
\(560\) 5.10899 0.215894
\(561\) 0 0
\(562\) −28.9081 −1.21941
\(563\) 4.64450 0.195742 0.0978712 0.995199i \(-0.468797\pi\)
0.0978712 + 0.995199i \(0.468797\pi\)
\(564\) 6.56950 0.276626
\(565\) 5.72703 0.240938
\(566\) 13.1211 0.551520
\(567\) −5.10899 −0.214558
\(568\) −2.31506 −0.0971378
\(569\) 35.1105 1.47191 0.735954 0.677032i \(-0.236734\pi\)
0.735954 + 0.677032i \(0.236734\pi\)
\(570\) −0.0785371 −0.00328956
\(571\) −17.7758 −0.743896 −0.371948 0.928254i \(-0.621310\pi\)
−0.371948 + 0.928254i \(0.621310\pi\)
\(572\) 0 0
\(573\) 20.7278 0.865915
\(574\) −60.7775 −2.53681
\(575\) 6.64849 0.277261
\(576\) 1.00000 0.0416667
\(577\) 3.63439 0.151302 0.0756509 0.997134i \(-0.475897\pi\)
0.0756509 + 0.997134i \(0.475897\pi\)
\(578\) −16.3985 −0.682088
\(579\) −1.37079 −0.0569681
\(580\) −2.01882 −0.0838270
\(581\) −40.8719 −1.69565
\(582\) −13.4541 −0.557688
\(583\) 0 0
\(584\) −10.9443 −0.452877
\(585\) 1.33343 0.0551304
\(586\) −22.6601 −0.936081
\(587\) 7.39314 0.305148 0.152574 0.988292i \(-0.451244\pi\)
0.152574 + 0.988292i \(0.451244\pi\)
\(588\) 19.1018 0.787745
\(589\) 0.666855 0.0274773
\(590\) 5.32624 0.219278
\(591\) 5.01135 0.206139
\(592\) −3.93310 −0.161649
\(593\) 35.9315 1.47553 0.737766 0.675057i \(-0.235881\pi\)
0.737766 + 0.675057i \(0.235881\pi\)
\(594\) 0 0
\(595\) −3.96236 −0.162441
\(596\) 15.2477 0.624570
\(597\) 23.3086 0.953958
\(598\) −8.86528 −0.362528
\(599\) −20.5554 −0.839871 −0.419935 0.907554i \(-0.637947\pi\)
−0.419935 + 0.907554i \(0.637947\pi\)
\(600\) 1.00000 0.0408248
\(601\) 5.12136 0.208905 0.104452 0.994530i \(-0.466691\pi\)
0.104452 + 0.994530i \(0.466691\pi\)
\(602\) −30.7134 −1.25178
\(603\) 0.588036 0.0239467
\(604\) −6.56921 −0.267297
\(605\) 0 0
\(606\) 10.6408 0.432255
\(607\) 37.3946 1.51780 0.758900 0.651207i \(-0.225737\pi\)
0.758900 + 0.651207i \(0.225737\pi\)
\(608\) 0.0785371 0.00318510
\(609\) −10.3141 −0.417950
\(610\) −3.74585 −0.151665
\(611\) −8.75995 −0.354389
\(612\) −0.775565 −0.0313504
\(613\) 8.39685 0.339145 0.169573 0.985518i \(-0.445761\pi\)
0.169573 + 0.985518i \(0.445761\pi\)
\(614\) 29.3158 1.18309
\(615\) −11.8962 −0.479701
\(616\) 0 0
\(617\) −25.9029 −1.04281 −0.521406 0.853309i \(-0.674592\pi\)
−0.521406 + 0.853309i \(0.674592\pi\)
\(618\) 16.7087 0.672121
\(619\) 25.1441 1.01063 0.505313 0.862936i \(-0.331377\pi\)
0.505313 + 0.862936i \(0.331377\pi\)
\(620\) −8.49096 −0.341005
\(621\) 6.64849 0.266795
\(622\) 11.3570 0.455373
\(623\) 49.4459 1.98101
\(624\) −1.33343 −0.0533798
\(625\) 1.00000 0.0400000
\(626\) −3.17664 −0.126964
\(627\) 0 0
\(628\) −9.30815 −0.371436
\(629\) 3.05037 0.121626
\(630\) 5.10899 0.203547
\(631\) −42.0436 −1.67373 −0.836864 0.547411i \(-0.815613\pi\)
−0.836864 + 0.547411i \(0.815613\pi\)
\(632\) −12.3267 −0.490330
\(633\) 20.0842 0.798274
\(634\) 4.73101 0.187893
\(635\) −6.96281 −0.276311
\(636\) 10.3451 0.410208
\(637\) −25.4709 −1.00919
\(638\) 0 0
\(639\) −2.31506 −0.0915824
\(640\) −1.00000 −0.0395285
\(641\) −2.11562 −0.0835619 −0.0417809 0.999127i \(-0.513303\pi\)
−0.0417809 + 0.999127i \(0.513303\pi\)
\(642\) −17.5511 −0.692688
\(643\) −39.4827 −1.55704 −0.778522 0.627617i \(-0.784030\pi\)
−0.778522 + 0.627617i \(0.784030\pi\)
\(644\) −33.9671 −1.33849
\(645\) −6.01163 −0.236708
\(646\) −0.0609106 −0.00239649
\(647\) −16.1454 −0.634743 −0.317371 0.948301i \(-0.602800\pi\)
−0.317371 + 0.948301i \(0.602800\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −1.33343 −0.0523013
\(651\) −43.3802 −1.70021
\(652\) −18.7018 −0.732417
\(653\) −1.29134 −0.0505340 −0.0252670 0.999681i \(-0.508044\pi\)
−0.0252670 + 0.999681i \(0.508044\pi\)
\(654\) 8.94427 0.349749
\(655\) −2.19916 −0.0859284
\(656\) 11.8962 0.464468
\(657\) −10.9443 −0.426977
\(658\) −33.5635 −1.30844
\(659\) 23.7015 0.923278 0.461639 0.887068i \(-0.347262\pi\)
0.461639 + 0.887068i \(0.347262\pi\)
\(660\) 0 0
\(661\) −17.0796 −0.664318 −0.332159 0.943223i \(-0.607777\pi\)
−0.332159 + 0.943223i \(0.607777\pi\)
\(662\) 30.0424 1.16763
\(663\) 1.03416 0.0401634
\(664\) 8.00000 0.310460
\(665\) 0.401245 0.0155596
\(666\) −3.93310 −0.152404
\(667\) 13.4221 0.519707
\(668\) 2.37478 0.0918829
\(669\) 9.10899 0.352174
\(670\) −0.588036 −0.0227178
\(671\) 0 0
\(672\) −5.10899 −0.197084
\(673\) 8.51069 0.328063 0.164032 0.986455i \(-0.447550\pi\)
0.164032 + 0.986455i \(0.447550\pi\)
\(674\) 25.3570 0.976714
\(675\) 1.00000 0.0384900
\(676\) −11.2220 −0.431614
\(677\) −22.7125 −0.872911 −0.436456 0.899726i \(-0.643766\pi\)
−0.436456 + 0.899726i \(0.643766\pi\)
\(678\) −5.72703 −0.219945
\(679\) 68.7367 2.63787
\(680\) 0.775565 0.0297416
\(681\) −20.6901 −0.792847
\(682\) 0 0
\(683\) −11.0419 −0.422507 −0.211254 0.977431i \(-0.567755\pi\)
−0.211254 + 0.977431i \(0.567755\pi\)
\(684\) 0.0785371 0.00300294
\(685\) 2.17590 0.0831367
\(686\) −61.8280 −2.36060
\(687\) 10.9819 0.418986
\(688\) 6.01163 0.229191
\(689\) −13.7944 −0.525524
\(690\) −6.64849 −0.253104
\(691\) 9.49934 0.361372 0.180686 0.983541i \(-0.442168\pi\)
0.180686 + 0.983541i \(0.442168\pi\)
\(692\) −2.91427 −0.110784
\(693\) 0 0
\(694\) 3.17664 0.120583
\(695\) 10.3074 0.390983
\(696\) 2.01882 0.0765233
\(697\) −9.22627 −0.349470
\(698\) 6.95865 0.263389
\(699\) 2.96309 0.112075
\(700\) −5.10899 −0.193102
\(701\) −25.7924 −0.974165 −0.487082 0.873356i \(-0.661939\pi\)
−0.487082 + 0.873356i \(0.661939\pi\)
\(702\) −1.33343 −0.0503270
\(703\) −0.308894 −0.0116501
\(704\) 0 0
\(705\) −6.56950 −0.247422
\(706\) 8.01163 0.301522
\(707\) −54.3640 −2.04457
\(708\) −5.32624 −0.200172
\(709\) −13.7924 −0.517984 −0.258992 0.965880i \(-0.583390\pi\)
−0.258992 + 0.965880i \(0.583390\pi\)
\(710\) 2.31506 0.0868827
\(711\) −12.3267 −0.462287
\(712\) −9.67821 −0.362706
\(713\) 56.4520 2.11415
\(714\) 3.96236 0.148287
\(715\) 0 0
\(716\) 1.56231 0.0583861
\(717\) −8.56978 −0.320044
\(718\) 0.180340 0.00673022
\(719\) −19.4035 −0.723629 −0.361814 0.932250i \(-0.617843\pi\)
−0.361814 + 0.932250i \(0.617843\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −85.3644 −3.17914
\(722\) −18.9938 −0.706877
\(723\) −7.97000 −0.296408
\(724\) 18.4721 0.686512
\(725\) 2.01882 0.0749772
\(726\) 0 0
\(727\) −1.37752 −0.0510895 −0.0255447 0.999674i \(-0.508132\pi\)
−0.0255447 + 0.999674i \(0.508132\pi\)
\(728\) 6.81247 0.252487
\(729\) 1.00000 0.0370370
\(730\) 10.9443 0.405066
\(731\) −4.66241 −0.172446
\(732\) 3.74585 0.138451
\(733\) −21.8778 −0.808076 −0.404038 0.914742i \(-0.632394\pi\)
−0.404038 + 0.914742i \(0.632394\pi\)
\(734\) 30.8705 1.13945
\(735\) −19.1018 −0.704581
\(736\) 6.64849 0.245067
\(737\) 0 0
\(738\) 11.8962 0.437905
\(739\) −0.525400 −0.0193272 −0.00966358 0.999953i \(-0.503076\pi\)
−0.00966358 + 0.999953i \(0.503076\pi\)
\(740\) 3.93310 0.144583
\(741\) −0.104723 −0.00384711
\(742\) −52.8528 −1.94029
\(743\) −47.8868 −1.75680 −0.878399 0.477929i \(-0.841388\pi\)
−0.878399 + 0.477929i \(0.841388\pi\)
\(744\) 8.49096 0.311294
\(745\) −15.2477 −0.558633
\(746\) −16.1018 −0.589529
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 89.6686 3.27642
\(750\) −1.00000 −0.0365148
\(751\) 36.1543 1.31929 0.659645 0.751577i \(-0.270707\pi\)
0.659645 + 0.751577i \(0.270707\pi\)
\(752\) 6.56950 0.239565
\(753\) 20.2596 0.738301
\(754\) −2.69195 −0.0980352
\(755\) 6.56921 0.239078
\(756\) −5.10899 −0.185812
\(757\) −43.6114 −1.58508 −0.792542 0.609818i \(-0.791243\pi\)
−0.792542 + 0.609818i \(0.791243\pi\)
\(758\) 32.5487 1.18222
\(759\) 0 0
\(760\) −0.0785371 −0.00284884
\(761\) 1.10226 0.0399569 0.0199784 0.999800i \(-0.493640\pi\)
0.0199784 + 0.999800i \(0.493640\pi\)
\(762\) 6.96281 0.252236
\(763\) −45.6962 −1.65431
\(764\) 20.7278 0.749904
\(765\) 0.775565 0.0280406
\(766\) 2.10825 0.0761743
\(767\) 7.10215 0.256444
\(768\) 1.00000 0.0360844
\(769\) 26.9983 0.973583 0.486791 0.873518i \(-0.338167\pi\)
0.486791 + 0.873518i \(0.338167\pi\)
\(770\) 0 0
\(771\) −19.9638 −0.718980
\(772\) −1.37079 −0.0493358
\(773\) −23.5077 −0.845512 −0.422756 0.906244i \(-0.638937\pi\)
−0.422756 + 0.906244i \(0.638937\pi\)
\(774\) 6.01163 0.216084
\(775\) 8.49096 0.305004
\(776\) −13.4541 −0.482972
\(777\) 20.0942 0.720874
\(778\) −12.6032 −0.451847
\(779\) 0.934292 0.0334745
\(780\) 1.33343 0.0477444
\(781\) 0 0
\(782\) −5.15633 −0.184390
\(783\) 2.01882 0.0721468
\(784\) 19.1018 0.682207
\(785\) 9.30815 0.332222
\(786\) 2.19916 0.0784415
\(787\) 32.3634 1.15363 0.576816 0.816874i \(-0.304295\pi\)
0.576816 + 0.816874i \(0.304295\pi\)
\(788\) 5.01135 0.178522
\(789\) 0.696114 0.0247823
\(790\) 12.3267 0.438564
\(791\) 29.2593 1.04034
\(792\) 0 0
\(793\) −4.99482 −0.177371
\(794\) −30.6341 −1.08716
\(795\) −10.3451 −0.366901
\(796\) 23.3086 0.826152
\(797\) −40.4031 −1.43115 −0.715575 0.698536i \(-0.753835\pi\)
−0.715575 + 0.698536i \(0.753835\pi\)
\(798\) −0.401245 −0.0142039
\(799\) −5.09507 −0.180251
\(800\) 1.00000 0.0353553
\(801\) −9.67821 −0.341963
\(802\) −19.9671 −0.705062
\(803\) 0 0
\(804\) 0.588036 0.0207384
\(805\) 33.9671 1.19718
\(806\) −11.3221 −0.398803
\(807\) 5.92175 0.208455
\(808\) 10.6408 0.374344
\(809\) −12.2365 −0.430213 −0.215107 0.976591i \(-0.569010\pi\)
−0.215107 + 0.976591i \(0.569010\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 15.1684 0.532635 0.266318 0.963885i \(-0.414193\pi\)
0.266318 + 0.963885i \(0.414193\pi\)
\(812\) −10.3141 −0.361956
\(813\) 13.9146 0.488005
\(814\) 0 0
\(815\) 18.7018 0.655094
\(816\) −0.775565 −0.0271502
\(817\) 0.472136 0.0165179
\(818\) 7.13962 0.249631
\(819\) 6.81247 0.238047
\(820\) −11.8962 −0.415433
\(821\) 14.4730 0.505113 0.252556 0.967582i \(-0.418729\pi\)
0.252556 + 0.967582i \(0.418729\pi\)
\(822\) −2.17590 −0.0758931
\(823\) 3.54640 0.123620 0.0618099 0.998088i \(-0.480313\pi\)
0.0618099 + 0.998088i \(0.480313\pi\)
\(824\) 16.7087 0.582074
\(825\) 0 0
\(826\) 27.2117 0.946816
\(827\) 37.2217 1.29432 0.647162 0.762352i \(-0.275956\pi\)
0.647162 + 0.762352i \(0.275956\pi\)
\(828\) 6.64849 0.231051
\(829\) −40.8705 −1.41949 −0.709745 0.704459i \(-0.751190\pi\)
−0.709745 + 0.704459i \(0.751190\pi\)
\(830\) −8.00000 −0.277684
\(831\) 10.1025 0.350453
\(832\) −1.33343 −0.0462283
\(833\) −14.8147 −0.513299
\(834\) −10.3074 −0.356917
\(835\) −2.37478 −0.0821825
\(836\) 0 0
\(837\) 8.49096 0.293490
\(838\) −22.3691 −0.772727
\(839\) −16.4136 −0.566661 −0.283330 0.959022i \(-0.591439\pi\)
−0.283330 + 0.959022i \(0.591439\pi\)
\(840\) 5.10899 0.176277
\(841\) −24.9244 −0.859461
\(842\) −6.06091 −0.208873
\(843\) −28.9081 −0.995648
\(844\) 20.0842 0.691326
\(845\) 11.2220 0.386048
\(846\) 6.56950 0.225864
\(847\) 0 0
\(848\) 10.3451 0.355251
\(849\) 13.1211 0.450314
\(850\) −0.775565 −0.0266017
\(851\) −26.1491 −0.896381
\(852\) −2.31506 −0.0793127
\(853\) −18.1880 −0.622745 −0.311372 0.950288i \(-0.600789\pi\)
−0.311372 + 0.950288i \(0.600789\pi\)
\(854\) −19.1375 −0.654872
\(855\) −0.0785371 −0.00268591
\(856\) −17.5511 −0.599885
\(857\) −48.5679 −1.65905 −0.829525 0.558470i \(-0.811389\pi\)
−0.829525 + 0.558470i \(0.811389\pi\)
\(858\) 0 0
\(859\) −29.1564 −0.994805 −0.497402 0.867520i \(-0.665713\pi\)
−0.497402 + 0.867520i \(0.665713\pi\)
\(860\) −6.01163 −0.204995
\(861\) −60.7775 −2.07129
\(862\) −37.1623 −1.26575
\(863\) −29.7157 −1.01153 −0.505767 0.862670i \(-0.668790\pi\)
−0.505767 + 0.862670i \(0.668790\pi\)
\(864\) 1.00000 0.0340207
\(865\) 2.91427 0.0990883
\(866\) 30.6540 1.04166
\(867\) −16.3985 −0.556922
\(868\) −43.3802 −1.47242
\(869\) 0 0
\(870\) −2.01882 −0.0684445
\(871\) −0.784103 −0.0265683
\(872\) 8.94427 0.302891
\(873\) −13.4541 −0.455351
\(874\) 0.522153 0.0176621
\(875\) 5.10899 0.172715
\(876\) −10.9443 −0.369773
\(877\) −26.7194 −0.902249 −0.451125 0.892461i \(-0.648977\pi\)
−0.451125 + 0.892461i \(0.648977\pi\)
\(878\) −7.06017 −0.238269
\(879\) −22.6601 −0.764307
\(880\) 0 0
\(881\) 2.72012 0.0916431 0.0458216 0.998950i \(-0.485409\pi\)
0.0458216 + 0.998950i \(0.485409\pi\)
\(882\) 19.1018 0.643191
\(883\) 34.7250 1.16859 0.584295 0.811541i \(-0.301371\pi\)
0.584295 + 0.811541i \(0.301371\pi\)
\(884\) 1.03416 0.0347825
\(885\) 5.32624 0.179040
\(886\) 14.5925 0.490244
\(887\) 37.2657 1.25126 0.625630 0.780120i \(-0.284842\pi\)
0.625630 + 0.780120i \(0.284842\pi\)
\(888\) −3.93310 −0.131986
\(889\) −35.5730 −1.19308
\(890\) 9.67821 0.324414
\(891\) 0 0
\(892\) 9.10899 0.304992
\(893\) 0.515949 0.0172656
\(894\) 15.2477 0.509959
\(895\) −1.56231 −0.0522221
\(896\) −5.10899 −0.170679
\(897\) −8.86528 −0.296003
\(898\) 12.7687 0.426096
\(899\) 17.1417 0.571709
\(900\) 1.00000 0.0333333
\(901\) −8.02327 −0.267294
\(902\) 0 0
\(903\) −30.7134 −1.02208
\(904\) −5.72703 −0.190478
\(905\) −18.4721 −0.614035
\(906\) −6.56921 −0.218247
\(907\) −39.7936 −1.32133 −0.660663 0.750682i \(-0.729725\pi\)
−0.660663 + 0.750682i \(0.729725\pi\)
\(908\) −20.6901 −0.686626
\(909\) 10.6408 0.352934
\(910\) −6.81247 −0.225831
\(911\) −12.8234 −0.424857 −0.212429 0.977177i \(-0.568137\pi\)
−0.212429 + 0.977177i \(0.568137\pi\)
\(912\) 0.0785371 0.00260062
\(913\) 0 0
\(914\) 17.3426 0.573642
\(915\) −3.74585 −0.123834
\(916\) 10.9819 0.362853
\(917\) −11.2355 −0.371029
\(918\) −0.775565 −0.0255975
\(919\) 37.1139 1.22427 0.612137 0.790752i \(-0.290310\pi\)
0.612137 + 0.790752i \(0.290310\pi\)
\(920\) −6.64849 −0.219194
\(921\) 29.3158 0.965988
\(922\) 2.99355 0.0985873
\(923\) 3.08697 0.101609
\(924\) 0 0
\(925\) −3.93310 −0.129319
\(926\) −16.2589 −0.534300
\(927\) 16.7087 0.548784
\(928\) 2.01882 0.0662711
\(929\) 9.97000 0.327105 0.163553 0.986535i \(-0.447705\pi\)
0.163553 + 0.986535i \(0.447705\pi\)
\(930\) −8.49096 −0.278429
\(931\) 1.50020 0.0491670
\(932\) 2.96309 0.0970594
\(933\) 11.3570 0.371811
\(934\) −18.5707 −0.607652
\(935\) 0 0
\(936\) −1.33343 −0.0435844
\(937\) −11.9767 −0.391263 −0.195631 0.980677i \(-0.562676\pi\)
−0.195631 + 0.980677i \(0.562676\pi\)
\(938\) −3.00427 −0.0980929
\(939\) −3.17664 −0.103666
\(940\) −6.56950 −0.214273
\(941\) −5.50996 −0.179619 −0.0898097 0.995959i \(-0.528626\pi\)
−0.0898097 + 0.995959i \(0.528626\pi\)
\(942\) −9.30815 −0.303276
\(943\) 79.0917 2.57558
\(944\) −5.32624 −0.173354
\(945\) 5.10899 0.166196
\(946\) 0 0
\(947\) −27.3946 −0.890206 −0.445103 0.895479i \(-0.646833\pi\)
−0.445103 + 0.895479i \(0.646833\pi\)
\(948\) −12.3267 −0.400352
\(949\) 14.5934 0.473722
\(950\) 0.0785371 0.00254808
\(951\) 4.73101 0.153414
\(952\) 3.96236 0.128421
\(953\) −7.98985 −0.258816 −0.129408 0.991591i \(-0.541308\pi\)
−0.129408 + 0.991591i \(0.541308\pi\)
\(954\) 10.3451 0.334934
\(955\) −20.7278 −0.670735
\(956\) −8.56978 −0.277166
\(957\) 0 0
\(958\) 30.5554 0.987200
\(959\) 11.1166 0.358975
\(960\) −1.00000 −0.0322749
\(961\) 41.0964 1.32569
\(962\) 5.24450 0.169089
\(963\) −17.5511 −0.565577
\(964\) −7.97000 −0.256696
\(965\) 1.37079 0.0441273
\(966\) −33.9671 −1.09287
\(967\) 19.0966 0.614106 0.307053 0.951692i \(-0.400657\pi\)
0.307053 + 0.951692i \(0.400657\pi\)
\(968\) 0 0
\(969\) −0.0609106 −0.00195673
\(970\) 13.4541 0.431984
\(971\) 0.149777 0.00480656 0.00240328 0.999997i \(-0.499235\pi\)
0.00240328 + 0.999997i \(0.499235\pi\)
\(972\) 1.00000 0.0320750
\(973\) 52.6605 1.68822
\(974\) −0.508303 −0.0162871
\(975\) −1.33343 −0.0427039
\(976\) 3.74585 0.119902
\(977\) −19.1384 −0.612292 −0.306146 0.951984i \(-0.599040\pi\)
−0.306146 + 0.951984i \(0.599040\pi\)
\(978\) −18.7018 −0.598016
\(979\) 0 0
\(980\) −19.1018 −0.610185
\(981\) 8.94427 0.285569
\(982\) −39.3943 −1.25712
\(983\) 8.58575 0.273843 0.136921 0.990582i \(-0.456279\pi\)
0.136921 + 0.990582i \(0.456279\pi\)
\(984\) 11.8962 0.379237
\(985\) −5.01135 −0.159675
\(986\) −1.56573 −0.0498629
\(987\) −33.5635 −1.06834
\(988\) −0.104723 −0.00333170
\(989\) 39.9683 1.27092
\(990\) 0 0
\(991\) −11.3086 −0.359230 −0.179615 0.983737i \(-0.557485\pi\)
−0.179615 + 0.983737i \(0.557485\pi\)
\(992\) 8.49096 0.269588
\(993\) 30.0424 0.953366
\(994\) 11.8276 0.375150
\(995\) −23.3086 −0.738933
\(996\) 8.00000 0.253490
\(997\) 37.0321 1.17282 0.586409 0.810015i \(-0.300541\pi\)
0.586409 + 0.810015i \(0.300541\pi\)
\(998\) 21.4608 0.679329
\(999\) −3.93310 −0.124438
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 3630.2.a.bs.1.1 4
11.2 odd 10 330.2.m.f.301.2 yes 8
11.6 odd 10 330.2.m.f.91.2 8
11.10 odd 2 3630.2.a.bq.1.4 4
33.2 even 10 990.2.n.i.631.2 8
33.17 even 10 990.2.n.i.91.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.f.91.2 8 11.6 odd 10
330.2.m.f.301.2 yes 8 11.2 odd 10
990.2.n.i.91.2 8 33.17 even 10
990.2.n.i.631.2 8 33.2 even 10
3630.2.a.bq.1.4 4 11.10 odd 2
3630.2.a.bs.1.1 4 1.1 even 1 trivial