Properties

Label 637.2.a.k.1.2
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.19566\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.195656 q^{2} +0.259788 q^{3} -1.96172 q^{4} -3.93251 q^{5} -0.0508292 q^{6} +0.775135 q^{8} -2.93251 q^{9} +O(q^{10})\) \(q-0.195656 q^{2} +0.259788 q^{3} -1.96172 q^{4} -3.93251 q^{5} -0.0508292 q^{6} +0.775135 q^{8} -2.93251 q^{9} +0.769420 q^{10} +4.50627 q^{11} -0.509632 q^{12} -1.00000 q^{13} -1.02162 q^{15} +3.77178 q^{16} +2.28141 q^{17} +0.573764 q^{18} +1.78768 q^{19} +7.71448 q^{20} -0.881681 q^{22} +1.74021 q^{23} +0.201371 q^{24} +10.4646 q^{25} +0.195656 q^{26} -1.54120 q^{27} +1.65110 q^{29} +0.199886 q^{30} -5.60523 q^{31} -2.28824 q^{32} +1.17068 q^{33} -0.446372 q^{34} +5.75276 q^{36} +7.14407 q^{37} -0.349771 q^{38} -0.259788 q^{39} -3.04823 q^{40} +8.11574 q^{41} +6.81353 q^{43} -8.84004 q^{44} +11.5321 q^{45} -0.340483 q^{46} -3.54543 q^{47} +0.979864 q^{48} -2.04747 q^{50} +0.592684 q^{51} +1.96172 q^{52} +3.28965 q^{53} +0.301545 q^{54} -17.7210 q^{55} +0.464419 q^{57} -0.323048 q^{58} -4.50627 q^{59} +2.00413 q^{60} -7.54467 q^{61} +1.09670 q^{62} -7.09585 q^{64} +3.93251 q^{65} -0.229050 q^{66} -12.6653 q^{67} -4.47548 q^{68} +0.452087 q^{69} +9.54869 q^{71} -2.27309 q^{72} -1.08004 q^{73} -1.39778 q^{74} +2.71859 q^{75} -3.50693 q^{76} +0.0508292 q^{78} +0.791698 q^{79} -14.8326 q^{80} +8.39714 q^{81} -1.58789 q^{82} +7.14643 q^{83} -8.97166 q^{85} -1.33311 q^{86} +0.428937 q^{87} +3.49297 q^{88} +11.2656 q^{89} -2.25633 q^{90} -3.41381 q^{92} -1.45617 q^{93} +0.693685 q^{94} -7.03008 q^{95} -0.594459 q^{96} +8.81353 q^{97} -13.2147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} - 5 q^{13} + 10 q^{16} + 5 q^{17} + 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 4 q^{26} - 3 q^{29} - 13 q^{30} + 6 q^{31} + 22 q^{32} - 8 q^{33} + 22 q^{34} + 7 q^{36} + 4 q^{37} + 10 q^{38} - 28 q^{40} - 14 q^{41} + 2 q^{43} + 32 q^{45} + 3 q^{46} - q^{47} + 23 q^{48} + 9 q^{50} - 8 q^{51} - 8 q^{52} + 17 q^{53} - 23 q^{54} - 16 q^{57} - 27 q^{58} - 11 q^{59} - 29 q^{60} + 11 q^{61} + 23 q^{62} + 9 q^{64} + 2 q^{65} - 21 q^{66} + 13 q^{67} + 32 q^{68} - 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} + 20 q^{75} - 8 q^{76} - 5 q^{78} + 2 q^{79} - 55 q^{80} - 19 q^{81} - 34 q^{82} - 6 q^{83} - 22 q^{85} + 28 q^{86} + 8 q^{87} - 3 q^{88} + 4 q^{89} + 34 q^{90} + 21 q^{92} + 18 q^{93} - 20 q^{94} - 12 q^{95} + 37 q^{96} + 12 q^{97} + 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.195656 −0.138350 −0.0691749 0.997605i \(-0.522037\pi\)
−0.0691749 + 0.997605i \(0.522037\pi\)
\(3\) 0.259788 0.149989 0.0749945 0.997184i \(-0.476106\pi\)
0.0749945 + 0.997184i \(0.476106\pi\)
\(4\) −1.96172 −0.980859
\(5\) −3.93251 −1.75867 −0.879336 0.476202i \(-0.842013\pi\)
−0.879336 + 0.476202i \(0.842013\pi\)
\(6\) −0.0508292 −0.0207509
\(7\) 0 0
\(8\) 0.775135 0.274052
\(9\) −2.93251 −0.977503
\(10\) 0.769420 0.243312
\(11\) 4.50627 1.35869 0.679346 0.733818i \(-0.262263\pi\)
0.679346 + 0.733818i \(0.262263\pi\)
\(12\) −0.509632 −0.147118
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.02162 −0.263781
\(16\) 3.77178 0.942944
\(17\) 2.28141 0.553323 0.276661 0.960967i \(-0.410772\pi\)
0.276661 + 0.960967i \(0.410772\pi\)
\(18\) 0.573764 0.135237
\(19\) 1.78768 0.410123 0.205061 0.978749i \(-0.434261\pi\)
0.205061 + 0.978749i \(0.434261\pi\)
\(20\) 7.71448 1.72501
\(21\) 0 0
\(22\) −0.881681 −0.187975
\(23\) 1.74021 0.362859 0.181430 0.983404i \(-0.441928\pi\)
0.181430 + 0.983404i \(0.441928\pi\)
\(24\) 0.201371 0.0411047
\(25\) 10.4646 2.09293
\(26\) 0.195656 0.0383714
\(27\) −1.54120 −0.296604
\(28\) 0 0
\(29\) 1.65110 0.306602 0.153301 0.988180i \(-0.451010\pi\)
0.153301 + 0.988180i \(0.451010\pi\)
\(30\) 0.199886 0.0364941
\(31\) −5.60523 −1.00673 −0.503365 0.864074i \(-0.667905\pi\)
−0.503365 + 0.864074i \(0.667905\pi\)
\(32\) −2.28824 −0.404508
\(33\) 1.17068 0.203789
\(34\) −0.446372 −0.0765522
\(35\) 0 0
\(36\) 5.75276 0.958793
\(37\) 7.14407 1.17448 0.587239 0.809414i \(-0.300215\pi\)
0.587239 + 0.809414i \(0.300215\pi\)
\(38\) −0.349771 −0.0567404
\(39\) −0.259788 −0.0415994
\(40\) −3.04823 −0.481967
\(41\) 8.11574 1.26746 0.633732 0.773552i \(-0.281522\pi\)
0.633732 + 0.773552i \(0.281522\pi\)
\(42\) 0 0
\(43\) 6.81353 1.03905 0.519527 0.854454i \(-0.326108\pi\)
0.519527 + 0.854454i \(0.326108\pi\)
\(44\) −8.84004 −1.33269
\(45\) 11.5321 1.71911
\(46\) −0.340483 −0.0502015
\(47\) −3.54543 −0.517154 −0.258577 0.965991i \(-0.583254\pi\)
−0.258577 + 0.965991i \(0.583254\pi\)
\(48\) 0.979864 0.141431
\(49\) 0 0
\(50\) −2.04747 −0.289556
\(51\) 0.592684 0.0829923
\(52\) 1.96172 0.272041
\(53\) 3.28965 0.451869 0.225934 0.974143i \(-0.427457\pi\)
0.225934 + 0.974143i \(0.427457\pi\)
\(54\) 0.301545 0.0410351
\(55\) −17.7210 −2.38949
\(56\) 0 0
\(57\) 0.464419 0.0615138
\(58\) −0.323048 −0.0424183
\(59\) −4.50627 −0.586667 −0.293333 0.956010i \(-0.594765\pi\)
−0.293333 + 0.956010i \(0.594765\pi\)
\(60\) 2.00413 0.258732
\(61\) −7.54467 −0.965996 −0.482998 0.875621i \(-0.660452\pi\)
−0.482998 + 0.875621i \(0.660452\pi\)
\(62\) 1.09670 0.139281
\(63\) 0 0
\(64\) −7.09585 −0.886981
\(65\) 3.93251 0.487768
\(66\) −0.229050 −0.0281942
\(67\) −12.6653 −1.54731 −0.773653 0.633609i \(-0.781573\pi\)
−0.773653 + 0.633609i \(0.781573\pi\)
\(68\) −4.47548 −0.542732
\(69\) 0.452087 0.0544249
\(70\) 0 0
\(71\) 9.54869 1.13322 0.566610 0.823986i \(-0.308254\pi\)
0.566610 + 0.823986i \(0.308254\pi\)
\(72\) −2.27309 −0.267886
\(73\) −1.08004 −0.126409 −0.0632044 0.998001i \(-0.520132\pi\)
−0.0632044 + 0.998001i \(0.520132\pi\)
\(74\) −1.39778 −0.162489
\(75\) 2.71859 0.313916
\(76\) −3.50693 −0.402273
\(77\) 0 0
\(78\) 0.0508292 0.00575528
\(79\) 0.791698 0.0890730 0.0445365 0.999008i \(-0.485819\pi\)
0.0445365 + 0.999008i \(0.485819\pi\)
\(80\) −14.8326 −1.65833
\(81\) 8.39714 0.933016
\(82\) −1.58789 −0.175354
\(83\) 7.14643 0.784422 0.392211 0.919875i \(-0.371710\pi\)
0.392211 + 0.919875i \(0.371710\pi\)
\(84\) 0 0
\(85\) −8.97166 −0.973114
\(86\) −1.33311 −0.143753
\(87\) 0.428937 0.0459869
\(88\) 3.49297 0.372352
\(89\) 11.2656 1.19415 0.597077 0.802184i \(-0.296329\pi\)
0.597077 + 0.802184i \(0.296329\pi\)
\(90\) −2.25633 −0.237838
\(91\) 0 0
\(92\) −3.41381 −0.355914
\(93\) −1.45617 −0.150998
\(94\) 0.693685 0.0715482
\(95\) −7.03008 −0.721271
\(96\) −0.594459 −0.0606717
\(97\) 8.81353 0.894879 0.447439 0.894314i \(-0.352336\pi\)
0.447439 + 0.894314i \(0.352336\pi\)
\(98\) 0 0
\(99\) −13.2147 −1.32813
\(100\) −20.5287 −2.05287
\(101\) 14.3171 1.42461 0.712303 0.701872i \(-0.247652\pi\)
0.712303 + 0.701872i \(0.247652\pi\)
\(102\) −0.115962 −0.0114820
\(103\) 7.49214 0.738223 0.369111 0.929385i \(-0.379662\pi\)
0.369111 + 0.929385i \(0.379662\pi\)
\(104\) −0.775135 −0.0760082
\(105\) 0 0
\(106\) −0.643641 −0.0625160
\(107\) 10.9784 1.06132 0.530660 0.847585i \(-0.321944\pi\)
0.530660 + 0.847585i \(0.321944\pi\)
\(108\) 3.02340 0.290926
\(109\) −12.4463 −1.19214 −0.596068 0.802934i \(-0.703271\pi\)
−0.596068 + 0.802934i \(0.703271\pi\)
\(110\) 3.46722 0.330586
\(111\) 1.85595 0.176159
\(112\) 0 0
\(113\) −1.65110 −0.155323 −0.0776613 0.996980i \(-0.524745\pi\)
−0.0776613 + 0.996980i \(0.524745\pi\)
\(114\) −0.0908666 −0.00851043
\(115\) −6.84340 −0.638150
\(116\) −3.23900 −0.300733
\(117\) 2.93251 0.271111
\(118\) 0.881681 0.0811653
\(119\) 0 0
\(120\) −0.791894 −0.0722897
\(121\) 9.30650 0.846046
\(122\) 1.47616 0.133645
\(123\) 2.10837 0.190106
\(124\) 10.9959 0.987460
\(125\) −21.4897 −1.92210
\(126\) 0 0
\(127\) −4.49297 −0.398687 −0.199343 0.979930i \(-0.563881\pi\)
−0.199343 + 0.979930i \(0.563881\pi\)
\(128\) 5.96483 0.527222
\(129\) 1.77008 0.155847
\(130\) −0.769420 −0.0674826
\(131\) 12.6567 1.10582 0.552911 0.833240i \(-0.313517\pi\)
0.552911 + 0.833240i \(0.313517\pi\)
\(132\) −2.29654 −0.199888
\(133\) 0 0
\(134\) 2.47804 0.214070
\(135\) 6.06077 0.521628
\(136\) 1.76840 0.151639
\(137\) −9.28641 −0.793392 −0.396696 0.917950i \(-0.629843\pi\)
−0.396696 + 0.917950i \(0.629843\pi\)
\(138\) −0.0884536 −0.00752967
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −0.921061 −0.0775673
\(142\) −1.86826 −0.156781
\(143\) −4.50627 −0.376834
\(144\) −11.0608 −0.921731
\(145\) −6.49297 −0.539212
\(146\) 0.211316 0.0174887
\(147\) 0 0
\(148\) −14.0147 −1.15200
\(149\) −15.1649 −1.24235 −0.621177 0.783670i \(-0.713345\pi\)
−0.621177 + 0.783670i \(0.713345\pi\)
\(150\) −0.531909 −0.0434302
\(151\) 5.14159 0.418416 0.209208 0.977871i \(-0.432911\pi\)
0.209208 + 0.977871i \(0.432911\pi\)
\(152\) 1.38570 0.112395
\(153\) −6.69025 −0.540875
\(154\) 0 0
\(155\) 22.0426 1.77051
\(156\) 0.509632 0.0408032
\(157\) 10.7311 0.856438 0.428219 0.903675i \(-0.359141\pi\)
0.428219 + 0.903675i \(0.359141\pi\)
\(158\) −0.154901 −0.0123232
\(159\) 0.854614 0.0677753
\(160\) 8.99853 0.711397
\(161\) 0 0
\(162\) −1.64295 −0.129083
\(163\) 2.37239 0.185820 0.0929101 0.995675i \(-0.470383\pi\)
0.0929101 + 0.995675i \(0.470383\pi\)
\(164\) −15.9208 −1.24320
\(165\) −4.60370 −0.358398
\(166\) −1.39824 −0.108525
\(167\) −12.0784 −0.934653 −0.467327 0.884085i \(-0.654783\pi\)
−0.467327 + 0.884085i \(0.654783\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 1.75536 0.134630
\(171\) −5.24240 −0.400896
\(172\) −13.3662 −1.01917
\(173\) 19.4097 1.47569 0.737846 0.674969i \(-0.235843\pi\)
0.737846 + 0.674969i \(0.235843\pi\)
\(174\) −0.0839242 −0.00636228
\(175\) 0 0
\(176\) 16.9967 1.28117
\(177\) −1.17068 −0.0879935
\(178\) −2.20419 −0.165211
\(179\) 14.6444 1.09457 0.547286 0.836945i \(-0.315661\pi\)
0.547286 + 0.836945i \(0.315661\pi\)
\(180\) −22.6228 −1.68620
\(181\) 9.44627 0.702136 0.351068 0.936350i \(-0.385819\pi\)
0.351068 + 0.936350i \(0.385819\pi\)
\(182\) 0 0
\(183\) −1.96002 −0.144889
\(184\) 1.34890 0.0994422
\(185\) −28.0941 −2.06552
\(186\) 0.284910 0.0208906
\(187\) 10.2807 0.751796
\(188\) 6.95513 0.507255
\(189\) 0 0
\(190\) 1.37548 0.0997878
\(191\) 12.5537 0.908357 0.454179 0.890911i \(-0.349933\pi\)
0.454179 + 0.890911i \(0.349933\pi\)
\(192\) −1.84342 −0.133037
\(193\) −9.36859 −0.674366 −0.337183 0.941439i \(-0.609474\pi\)
−0.337183 + 0.941439i \(0.609474\pi\)
\(194\) −1.72442 −0.123806
\(195\) 1.02162 0.0731598
\(196\) 0 0
\(197\) −7.62276 −0.543099 −0.271550 0.962424i \(-0.587536\pi\)
−0.271550 + 0.962424i \(0.587536\pi\)
\(198\) 2.58554 0.183746
\(199\) −13.5289 −0.959036 −0.479518 0.877532i \(-0.659188\pi\)
−0.479518 + 0.877532i \(0.659188\pi\)
\(200\) 8.11151 0.573570
\(201\) −3.29029 −0.232079
\(202\) −2.80123 −0.197094
\(203\) 0 0
\(204\) −1.16268 −0.0814038
\(205\) −31.9152 −2.22906
\(206\) −1.46588 −0.102133
\(207\) −5.10319 −0.354696
\(208\) −3.77178 −0.261526
\(209\) 8.05579 0.557231
\(210\) 0 0
\(211\) −15.7995 −1.08768 −0.543840 0.839189i \(-0.683030\pi\)
−0.543840 + 0.839189i \(0.683030\pi\)
\(212\) −6.45338 −0.443220
\(213\) 2.48064 0.169971
\(214\) −2.14799 −0.146833
\(215\) −26.7943 −1.82736
\(216\) −1.19464 −0.0812847
\(217\) 0 0
\(218\) 2.43519 0.164932
\(219\) −0.280581 −0.0189599
\(220\) 34.7636 2.34376
\(221\) −2.28141 −0.153464
\(222\) −0.363128 −0.0243715
\(223\) −22.4737 −1.50495 −0.752474 0.658622i \(-0.771139\pi\)
−0.752474 + 0.658622i \(0.771139\pi\)
\(224\) 0 0
\(225\) −30.6876 −2.04584
\(226\) 0.323048 0.0214889
\(227\) 9.20249 0.610791 0.305395 0.952226i \(-0.401211\pi\)
0.305395 + 0.952226i \(0.401211\pi\)
\(228\) −0.911060 −0.0603364
\(229\) −15.2922 −1.01054 −0.505269 0.862962i \(-0.668607\pi\)
−0.505269 + 0.862962i \(0.668607\pi\)
\(230\) 1.33895 0.0882880
\(231\) 0 0
\(232\) 1.27983 0.0840247
\(233\) −8.05579 −0.527752 −0.263876 0.964557i \(-0.585001\pi\)
−0.263876 + 0.964557i \(0.585001\pi\)
\(234\) −0.573764 −0.0375081
\(235\) 13.9424 0.909504
\(236\) 8.84004 0.575438
\(237\) 0.205674 0.0133600
\(238\) 0 0
\(239\) 21.7258 1.40533 0.702663 0.711523i \(-0.251994\pi\)
0.702663 + 0.711523i \(0.251994\pi\)
\(240\) −3.85333 −0.248731
\(241\) 20.4980 1.32039 0.660195 0.751094i \(-0.270473\pi\)
0.660195 + 0.751094i \(0.270473\pi\)
\(242\) −1.82088 −0.117050
\(243\) 6.80507 0.436546
\(244\) 14.8005 0.947507
\(245\) 0 0
\(246\) −0.412517 −0.0263011
\(247\) −1.78768 −0.113748
\(248\) −4.34481 −0.275896
\(249\) 1.85656 0.117655
\(250\) 4.20460 0.265922
\(251\) −2.60871 −0.164660 −0.0823301 0.996605i \(-0.526236\pi\)
−0.0823301 + 0.996605i \(0.526236\pi\)
\(252\) 0 0
\(253\) 7.84187 0.493014
\(254\) 0.879078 0.0551583
\(255\) −2.33073 −0.145956
\(256\) 13.0246 0.814040
\(257\) 8.99676 0.561202 0.280601 0.959824i \(-0.409466\pi\)
0.280601 + 0.959824i \(0.409466\pi\)
\(258\) −0.346327 −0.0215614
\(259\) 0 0
\(260\) −7.71448 −0.478432
\(261\) −4.84187 −0.299704
\(262\) −2.47637 −0.152990
\(263\) 1.43392 0.0884194 0.0442097 0.999022i \(-0.485923\pi\)
0.0442097 + 0.999022i \(0.485923\pi\)
\(264\) 0.907433 0.0558487
\(265\) −12.9366 −0.794689
\(266\) 0 0
\(267\) 2.92668 0.179110
\(268\) 24.8457 1.51769
\(269\) 8.16832 0.498031 0.249016 0.968499i \(-0.419893\pi\)
0.249016 + 0.968499i \(0.419893\pi\)
\(270\) −1.18583 −0.0721672
\(271\) 0.212317 0.0128973 0.00644867 0.999979i \(-0.497947\pi\)
0.00644867 + 0.999979i \(0.497947\pi\)
\(272\) 8.60497 0.521753
\(273\) 0 0
\(274\) 1.81695 0.109766
\(275\) 47.1565 2.84364
\(276\) −0.886867 −0.0533831
\(277\) 22.9749 1.38043 0.690215 0.723604i \(-0.257516\pi\)
0.690215 + 0.723604i \(0.257516\pi\)
\(278\) −0.782625 −0.0469387
\(279\) 16.4374 0.984081
\(280\) 0 0
\(281\) −0.345228 −0.0205946 −0.0102973 0.999947i \(-0.503278\pi\)
−0.0102973 + 0.999947i \(0.503278\pi\)
\(282\) 0.180211 0.0107314
\(283\) 28.9715 1.72217 0.861087 0.508457i \(-0.169784\pi\)
0.861087 + 0.508457i \(0.169784\pi\)
\(284\) −18.7318 −1.11153
\(285\) −1.82633 −0.108183
\(286\) 0.881681 0.0521349
\(287\) 0 0
\(288\) 6.71029 0.395408
\(289\) −11.7952 −0.693834
\(290\) 1.27039 0.0745999
\(291\) 2.28965 0.134222
\(292\) 2.11873 0.123989
\(293\) −31.5427 −1.84274 −0.921372 0.388682i \(-0.872930\pi\)
−0.921372 + 0.388682i \(0.872930\pi\)
\(294\) 0 0
\(295\) 17.7210 1.03175
\(296\) 5.53762 0.321868
\(297\) −6.94506 −0.402993
\(298\) 2.96710 0.171880
\(299\) −1.74021 −0.100639
\(300\) −5.33311 −0.307907
\(301\) 0 0
\(302\) −1.00598 −0.0578879
\(303\) 3.71942 0.213675
\(304\) 6.74274 0.386723
\(305\) 29.6695 1.69887
\(306\) 1.30899 0.0748300
\(307\) 18.1941 1.03839 0.519197 0.854655i \(-0.326231\pi\)
0.519197 + 0.854655i \(0.326231\pi\)
\(308\) 0 0
\(309\) 1.94637 0.110725
\(310\) −4.31278 −0.244949
\(311\) 0.376623 0.0213563 0.0106782 0.999943i \(-0.496601\pi\)
0.0106782 + 0.999943i \(0.496601\pi\)
\(312\) −0.201371 −0.0114004
\(313\) −10.9883 −0.621095 −0.310548 0.950558i \(-0.600512\pi\)
−0.310548 + 0.950558i \(0.600512\pi\)
\(314\) −2.09961 −0.118488
\(315\) 0 0
\(316\) −1.55309 −0.0873680
\(317\) 26.1806 1.47045 0.735225 0.677823i \(-0.237077\pi\)
0.735225 + 0.677823i \(0.237077\pi\)
\(318\) −0.167211 −0.00937670
\(319\) 7.44031 0.416578
\(320\) 27.9045 1.55991
\(321\) 2.85206 0.159186
\(322\) 0 0
\(323\) 4.07844 0.226930
\(324\) −16.4728 −0.915158
\(325\) −10.4646 −0.580473
\(326\) −0.464174 −0.0257082
\(327\) −3.23340 −0.178807
\(328\) 6.29079 0.347351
\(329\) 0 0
\(330\) 0.900743 0.0495843
\(331\) −34.0932 −1.87393 −0.936967 0.349419i \(-0.886379\pi\)
−0.936967 + 0.349419i \(0.886379\pi\)
\(332\) −14.0193 −0.769408
\(333\) −20.9501 −1.14806
\(334\) 2.36321 0.129309
\(335\) 49.8062 2.72120
\(336\) 0 0
\(337\) 14.7532 0.803657 0.401829 0.915715i \(-0.368375\pi\)
0.401829 + 0.915715i \(0.368375\pi\)
\(338\) −0.195656 −0.0106423
\(339\) −0.428937 −0.0232967
\(340\) 17.5999 0.954487
\(341\) −25.2587 −1.36784
\(342\) 1.02571 0.0554639
\(343\) 0 0
\(344\) 5.28141 0.284754
\(345\) −1.77784 −0.0957155
\(346\) −3.79763 −0.204162
\(347\) 29.9466 1.60762 0.803809 0.594888i \(-0.202804\pi\)
0.803809 + 0.594888i \(0.202804\pi\)
\(348\) −0.841454 −0.0451066
\(349\) 13.4793 0.721532 0.360766 0.932656i \(-0.382515\pi\)
0.360766 + 0.932656i \(0.382515\pi\)
\(350\) 0 0
\(351\) 1.54120 0.0822630
\(352\) −10.3114 −0.549602
\(353\) −0.163532 −0.00870392 −0.00435196 0.999991i \(-0.501385\pi\)
−0.00435196 + 0.999991i \(0.501385\pi\)
\(354\) 0.229050 0.0121739
\(355\) −37.5503 −1.99296
\(356\) −22.1000 −1.17130
\(357\) 0 0
\(358\) −2.86527 −0.151434
\(359\) 8.92130 0.470848 0.235424 0.971893i \(-0.424352\pi\)
0.235424 + 0.971893i \(0.424352\pi\)
\(360\) 8.93895 0.471124
\(361\) −15.8042 −0.831799
\(362\) −1.84822 −0.0971404
\(363\) 2.41772 0.126898
\(364\) 0 0
\(365\) 4.24726 0.222312
\(366\) 0.383490 0.0200453
\(367\) −36.6552 −1.91339 −0.956693 0.291098i \(-0.905979\pi\)
−0.956693 + 0.291098i \(0.905979\pi\)
\(368\) 6.56369 0.342156
\(369\) −23.7995 −1.23895
\(370\) 5.49679 0.285765
\(371\) 0 0
\(372\) 2.85660 0.148108
\(373\) 27.1274 1.40460 0.702302 0.711879i \(-0.252156\pi\)
0.702302 + 0.711879i \(0.252156\pi\)
\(374\) −2.01147 −0.104011
\(375\) −5.58278 −0.288294
\(376\) −2.74819 −0.141727
\(377\) −1.65110 −0.0850360
\(378\) 0 0
\(379\) −15.8943 −0.816434 −0.408217 0.912885i \(-0.633849\pi\)
−0.408217 + 0.912885i \(0.633849\pi\)
\(380\) 13.7910 0.707465
\(381\) −1.16722 −0.0597986
\(382\) −2.45622 −0.125671
\(383\) −1.15079 −0.0588025 −0.0294013 0.999568i \(-0.509360\pi\)
−0.0294013 + 0.999568i \(0.509360\pi\)
\(384\) 1.54959 0.0790774
\(385\) 0 0
\(386\) 1.83302 0.0932985
\(387\) −19.9808 −1.01568
\(388\) −17.2897 −0.877750
\(389\) −14.3130 −0.725699 −0.362850 0.931848i \(-0.618196\pi\)
−0.362850 + 0.931848i \(0.618196\pi\)
\(390\) −0.199886 −0.0101216
\(391\) 3.97013 0.200778
\(392\) 0 0
\(393\) 3.28807 0.165861
\(394\) 1.49144 0.0751377
\(395\) −3.11336 −0.156650
\(396\) 25.9235 1.30271
\(397\) 25.9176 1.30077 0.650383 0.759607i \(-0.274609\pi\)
0.650383 + 0.759607i \(0.274609\pi\)
\(398\) 2.64701 0.132682
\(399\) 0 0
\(400\) 39.4703 1.97351
\(401\) −4.29631 −0.214548 −0.107274 0.994230i \(-0.534212\pi\)
−0.107274 + 0.994230i \(0.534212\pi\)
\(402\) 0.643765 0.0321081
\(403\) 5.60523 0.279216
\(404\) −28.0861 −1.39734
\(405\) −33.0219 −1.64087
\(406\) 0 0
\(407\) 32.1931 1.59575
\(408\) 0.459410 0.0227442
\(409\) 24.7071 1.22169 0.610844 0.791751i \(-0.290830\pi\)
0.610844 + 0.791751i \(0.290830\pi\)
\(410\) 6.24441 0.308389
\(411\) −2.41250 −0.119000
\(412\) −14.6975 −0.724093
\(413\) 0 0
\(414\) 0.998471 0.0490722
\(415\) −28.1034 −1.37954
\(416\) 2.28824 0.112190
\(417\) 1.03915 0.0508876
\(418\) −1.57617 −0.0770928
\(419\) 24.9293 1.21787 0.608937 0.793218i \(-0.291596\pi\)
0.608937 + 0.793218i \(0.291596\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 3.09127 0.150480
\(423\) 10.3970 0.505520
\(424\) 2.54993 0.123835
\(425\) 23.8741 1.15806
\(426\) −0.485352 −0.0235154
\(427\) 0 0
\(428\) −21.5365 −1.04101
\(429\) −1.17068 −0.0565209
\(430\) 5.24247 0.252814
\(431\) −5.68851 −0.274006 −0.137003 0.990571i \(-0.543747\pi\)
−0.137003 + 0.990571i \(0.543747\pi\)
\(432\) −5.81305 −0.279681
\(433\) −12.2598 −0.589169 −0.294584 0.955625i \(-0.595181\pi\)
−0.294584 + 0.955625i \(0.595181\pi\)
\(434\) 0 0
\(435\) −1.68680 −0.0808758
\(436\) 24.4161 1.16932
\(437\) 3.11095 0.148817
\(438\) 0.0548975 0.00262310
\(439\) −5.02317 −0.239743 −0.119871 0.992789i \(-0.538248\pi\)
−0.119871 + 0.992789i \(0.538248\pi\)
\(440\) −13.7361 −0.654845
\(441\) 0 0
\(442\) 0.446372 0.0212317
\(443\) 0.578803 0.0274997 0.0137499 0.999905i \(-0.495623\pi\)
0.0137499 + 0.999905i \(0.495623\pi\)
\(444\) −3.64085 −0.172787
\(445\) −44.3022 −2.10012
\(446\) 4.39711 0.208209
\(447\) −3.93966 −0.186339
\(448\) 0 0
\(449\) −7.36359 −0.347509 −0.173755 0.984789i \(-0.555590\pi\)
−0.173755 + 0.984789i \(0.555590\pi\)
\(450\) 6.00423 0.283042
\(451\) 36.5717 1.72210
\(452\) 3.23900 0.152350
\(453\) 1.33572 0.0627578
\(454\) −1.80052 −0.0845028
\(455\) 0 0
\(456\) 0.359988 0.0168580
\(457\) 7.91824 0.370399 0.185200 0.982701i \(-0.440707\pi\)
0.185200 + 0.982701i \(0.440707\pi\)
\(458\) 2.99202 0.139808
\(459\) −3.51610 −0.164118
\(460\) 13.4248 0.625936
\(461\) 9.53600 0.444136 0.222068 0.975031i \(-0.428719\pi\)
0.222068 + 0.975031i \(0.428719\pi\)
\(462\) 0 0
\(463\) 2.16049 0.100406 0.0502032 0.998739i \(-0.484013\pi\)
0.0502032 + 0.998739i \(0.484013\pi\)
\(464\) 6.22758 0.289108
\(465\) 5.72642 0.265556
\(466\) 1.57617 0.0730145
\(467\) 8.11900 0.375702 0.187851 0.982198i \(-0.439848\pi\)
0.187851 + 0.982198i \(0.439848\pi\)
\(468\) −5.75276 −0.265921
\(469\) 0 0
\(470\) −2.72792 −0.125830
\(471\) 2.78783 0.128456
\(472\) −3.49297 −0.160777
\(473\) 30.7036 1.41176
\(474\) −0.0402414 −0.00184835
\(475\) 18.7074 0.858357
\(476\) 0 0
\(477\) −9.64694 −0.441703
\(478\) −4.25079 −0.194427
\(479\) −14.5533 −0.664955 −0.332478 0.943111i \(-0.607885\pi\)
−0.332478 + 0.943111i \(0.607885\pi\)
\(480\) 2.33772 0.106702
\(481\) −7.14407 −0.325742
\(482\) −4.01056 −0.182676
\(483\) 0 0
\(484\) −18.2567 −0.829852
\(485\) −34.6593 −1.57380
\(486\) −1.33146 −0.0603960
\(487\) −33.2590 −1.50711 −0.753554 0.657386i \(-0.771662\pi\)
−0.753554 + 0.657386i \(0.771662\pi\)
\(488\) −5.84814 −0.264733
\(489\) 0.616320 0.0278710
\(490\) 0 0
\(491\) −22.5563 −1.01795 −0.508977 0.860780i \(-0.669976\pi\)
−0.508977 + 0.860780i \(0.669976\pi\)
\(492\) −4.13604 −0.186467
\(493\) 3.76684 0.169650
\(494\) 0.349771 0.0157370
\(495\) 51.9669 2.33574
\(496\) −21.1417 −0.949290
\(497\) 0 0
\(498\) −0.363248 −0.0162775
\(499\) −11.3854 −0.509682 −0.254841 0.966983i \(-0.582023\pi\)
−0.254841 + 0.966983i \(0.582023\pi\)
\(500\) 42.1568 1.88531
\(501\) −3.13782 −0.140188
\(502\) 0.510410 0.0227807
\(503\) 8.81825 0.393186 0.196593 0.980485i \(-0.437012\pi\)
0.196593 + 0.980485i \(0.437012\pi\)
\(504\) 0 0
\(505\) −56.3022 −2.50541
\(506\) −1.53431 −0.0682084
\(507\) 0.259788 0.0115376
\(508\) 8.81394 0.391056
\(509\) −19.2838 −0.854738 −0.427369 0.904077i \(-0.640559\pi\)
−0.427369 + 0.904077i \(0.640559\pi\)
\(510\) 0.456023 0.0201930
\(511\) 0 0
\(512\) −14.4780 −0.639844
\(513\) −2.75517 −0.121644
\(514\) −1.76027 −0.0776423
\(515\) −29.4629 −1.29829
\(516\) −3.47239 −0.152864
\(517\) −15.9767 −0.702653
\(518\) 0 0
\(519\) 5.04241 0.221337
\(520\) 3.04823 0.133674
\(521\) −25.1168 −1.10039 −0.550193 0.835037i \(-0.685446\pi\)
−0.550193 + 0.835037i \(0.685446\pi\)
\(522\) 0.947342 0.0414640
\(523\) 29.9648 1.31027 0.655134 0.755513i \(-0.272612\pi\)
0.655134 + 0.755513i \(0.272612\pi\)
\(524\) −24.8289 −1.08466
\(525\) 0 0
\(526\) −0.280556 −0.0122328
\(527\) −12.7878 −0.557046
\(528\) 4.41554 0.192162
\(529\) −19.9717 −0.868333
\(530\) 2.53113 0.109945
\(531\) 13.2147 0.573469
\(532\) 0 0
\(533\) −8.11574 −0.351532
\(534\) −0.572623 −0.0247798
\(535\) −43.1726 −1.86651
\(536\) −9.81728 −0.424042
\(537\) 3.80444 0.164174
\(538\) −1.59818 −0.0689026
\(539\) 0 0
\(540\) −11.8895 −0.511644
\(541\) 5.09973 0.219255 0.109627 0.993973i \(-0.465034\pi\)
0.109627 + 0.993973i \(0.465034\pi\)
\(542\) −0.0415412 −0.00178435
\(543\) 2.45403 0.105313
\(544\) −5.22042 −0.223823
\(545\) 48.9451 2.09658
\(546\) 0 0
\(547\) 2.92025 0.124861 0.0624305 0.998049i \(-0.480115\pi\)
0.0624305 + 0.998049i \(0.480115\pi\)
\(548\) 18.2173 0.778206
\(549\) 22.1248 0.944265
\(550\) −9.22647 −0.393418
\(551\) 2.95165 0.125744
\(552\) 0.350428 0.0149152
\(553\) 0 0
\(554\) −4.49519 −0.190982
\(555\) −7.29853 −0.309805
\(556\) −7.84687 −0.332782
\(557\) 25.9874 1.10112 0.550561 0.834795i \(-0.314414\pi\)
0.550561 + 0.834795i \(0.314414\pi\)
\(558\) −3.21608 −0.136148
\(559\) −6.81353 −0.288182
\(560\) 0 0
\(561\) 2.67079 0.112761
\(562\) 0.0675460 0.00284925
\(563\) −3.65069 −0.153858 −0.0769291 0.997037i \(-0.524512\pi\)
−0.0769291 + 0.997037i \(0.524512\pi\)
\(564\) 1.80686 0.0760826
\(565\) 6.49297 0.273161
\(566\) −5.66845 −0.238263
\(567\) 0 0
\(568\) 7.40152 0.310561
\(569\) −25.3533 −1.06286 −0.531432 0.847101i \(-0.678346\pi\)
−0.531432 + 0.847101i \(0.678346\pi\)
\(570\) 0.357334 0.0149671
\(571\) 27.7253 1.16027 0.580133 0.814522i \(-0.303000\pi\)
0.580133 + 0.814522i \(0.303000\pi\)
\(572\) 8.84004 0.369621
\(573\) 3.26132 0.136244
\(574\) 0 0
\(575\) 18.2107 0.759438
\(576\) 20.8086 0.867027
\(577\) −39.5754 −1.64755 −0.823773 0.566920i \(-0.808135\pi\)
−0.823773 + 0.566920i \(0.808135\pi\)
\(578\) 2.30780 0.0959918
\(579\) −2.43385 −0.101147
\(580\) 12.7374 0.528891
\(581\) 0 0
\(582\) −0.447985 −0.0185696
\(583\) 14.8241 0.613951
\(584\) −0.837175 −0.0346426
\(585\) −11.5321 −0.476795
\(586\) 6.17153 0.254943
\(587\) 8.24177 0.340174 0.170087 0.985429i \(-0.445595\pi\)
0.170087 + 0.985429i \(0.445595\pi\)
\(588\) 0 0
\(589\) −10.0204 −0.412882
\(590\) −3.46722 −0.142743
\(591\) −1.98031 −0.0814589
\(592\) 26.9458 1.10747
\(593\) 11.9230 0.489618 0.244809 0.969571i \(-0.421275\pi\)
0.244809 + 0.969571i \(0.421275\pi\)
\(594\) 1.35884 0.0557540
\(595\) 0 0
\(596\) 29.7492 1.21857
\(597\) −3.51464 −0.143845
\(598\) 0.340483 0.0139234
\(599\) −35.6158 −1.45522 −0.727611 0.685990i \(-0.759369\pi\)
−0.727611 + 0.685990i \(0.759369\pi\)
\(600\) 2.10728 0.0860292
\(601\) −38.9252 −1.58779 −0.793896 0.608054i \(-0.791950\pi\)
−0.793896 + 0.608054i \(0.791950\pi\)
\(602\) 0 0
\(603\) 37.1410 1.51250
\(604\) −10.0863 −0.410408
\(605\) −36.5979 −1.48792
\(606\) −0.727728 −0.0295619
\(607\) 13.6966 0.555926 0.277963 0.960592i \(-0.410341\pi\)
0.277963 + 0.960592i \(0.410341\pi\)
\(608\) −4.09065 −0.165898
\(609\) 0 0
\(610\) −5.80502 −0.235039
\(611\) 3.54543 0.143433
\(612\) 13.1244 0.530522
\(613\) 3.16112 0.127676 0.0638382 0.997960i \(-0.479666\pi\)
0.0638382 + 0.997960i \(0.479666\pi\)
\(614\) −3.55979 −0.143662
\(615\) −8.29120 −0.334334
\(616\) 0 0
\(617\) 20.9297 0.842597 0.421299 0.906922i \(-0.361574\pi\)
0.421299 + 0.906922i \(0.361574\pi\)
\(618\) −0.380820 −0.0153188
\(619\) −30.9544 −1.24416 −0.622082 0.782952i \(-0.713713\pi\)
−0.622082 + 0.782952i \(0.713713\pi\)
\(620\) −43.2414 −1.73662
\(621\) −2.68201 −0.107625
\(622\) −0.0736887 −0.00295465
\(623\) 0 0
\(624\) −0.979864 −0.0392260
\(625\) 32.1854 1.28742
\(626\) 2.14993 0.0859285
\(627\) 2.09280 0.0835784
\(628\) −21.0515 −0.840045
\(629\) 16.2986 0.649866
\(630\) 0 0
\(631\) −15.1218 −0.601988 −0.300994 0.953626i \(-0.597318\pi\)
−0.300994 + 0.953626i \(0.597318\pi\)
\(632\) 0.613673 0.0244106
\(633\) −4.10452 −0.163140
\(634\) −5.12240 −0.203437
\(635\) 17.6687 0.701159
\(636\) −1.67651 −0.0664780
\(637\) 0 0
\(638\) −1.45574 −0.0576335
\(639\) −28.0016 −1.10773
\(640\) −23.4568 −0.927210
\(641\) −47.2414 −1.86592 −0.932962 0.359976i \(-0.882785\pi\)
−0.932962 + 0.359976i \(0.882785\pi\)
\(642\) −0.558023 −0.0220234
\(643\) −39.9249 −1.57448 −0.787241 0.616645i \(-0.788491\pi\)
−0.787241 + 0.616645i \(0.788491\pi\)
\(644\) 0 0
\(645\) −6.96085 −0.274083
\(646\) −0.797972 −0.0313958
\(647\) 29.8278 1.17265 0.586327 0.810075i \(-0.300574\pi\)
0.586327 + 0.810075i \(0.300574\pi\)
\(648\) 6.50892 0.255695
\(649\) −20.3065 −0.797100
\(650\) 2.04747 0.0803084
\(651\) 0 0
\(652\) −4.65397 −0.182263
\(653\) 25.1549 0.984387 0.492194 0.870486i \(-0.336195\pi\)
0.492194 + 0.870486i \(0.336195\pi\)
\(654\) 0.632635 0.0247380
\(655\) −49.7727 −1.94478
\(656\) 30.6107 1.19515
\(657\) 3.16722 0.123565
\(658\) 0 0
\(659\) 17.3155 0.674517 0.337258 0.941412i \(-0.390500\pi\)
0.337258 + 0.941412i \(0.390500\pi\)
\(660\) 9.03117 0.351538
\(661\) 9.20074 0.357867 0.178934 0.983861i \(-0.442735\pi\)
0.178934 + 0.983861i \(0.442735\pi\)
\(662\) 6.67055 0.259258
\(663\) −0.592684 −0.0230179
\(664\) 5.53945 0.214972
\(665\) 0 0
\(666\) 4.09901 0.158833
\(667\) 2.87327 0.111253
\(668\) 23.6944 0.916763
\(669\) −5.83840 −0.225725
\(670\) −9.74490 −0.376478
\(671\) −33.9984 −1.31249
\(672\) 0 0
\(673\) 17.3609 0.669212 0.334606 0.942358i \(-0.391397\pi\)
0.334606 + 0.942358i \(0.391397\pi\)
\(674\) −2.88655 −0.111186
\(675\) −16.1281 −0.620770
\(676\) −1.96172 −0.0754507
\(677\) −49.9825 −1.92098 −0.960492 0.278307i \(-0.910227\pi\)
−0.960492 + 0.278307i \(0.910227\pi\)
\(678\) 0.0839242 0.00322309
\(679\) 0 0
\(680\) −6.95425 −0.266683
\(681\) 2.39070 0.0916118
\(682\) 4.94202 0.189240
\(683\) 33.6153 1.28626 0.643128 0.765759i \(-0.277636\pi\)
0.643128 + 0.765759i \(0.277636\pi\)
\(684\) 10.2841 0.393223
\(685\) 36.5189 1.39532
\(686\) 0 0
\(687\) −3.97274 −0.151570
\(688\) 25.6991 0.979770
\(689\) −3.28965 −0.125326
\(690\) 0.347845 0.0132422
\(691\) 15.1309 0.575607 0.287803 0.957689i \(-0.407075\pi\)
0.287803 + 0.957689i \(0.407075\pi\)
\(692\) −38.0764 −1.44745
\(693\) 0 0
\(694\) −5.85924 −0.222414
\(695\) −15.7300 −0.596674
\(696\) 0.332484 0.0126028
\(697\) 18.5153 0.701317
\(698\) −2.63731 −0.0998238
\(699\) −2.09280 −0.0791570
\(700\) 0 0
\(701\) 2.02467 0.0764705 0.0382353 0.999269i \(-0.487826\pi\)
0.0382353 + 0.999269i \(0.487826\pi\)
\(702\) −0.301545 −0.0113811
\(703\) 12.7713 0.481680
\(704\) −31.9758 −1.20513
\(705\) 3.62208 0.136415
\(706\) 0.0319960 0.00120419
\(707\) 0 0
\(708\) 2.29654 0.0863093
\(709\) −30.4553 −1.14377 −0.571886 0.820333i \(-0.693788\pi\)
−0.571886 + 0.820333i \(0.693788\pi\)
\(710\) 7.34695 0.275726
\(711\) −2.32166 −0.0870691
\(712\) 8.73238 0.327260
\(713\) −9.75429 −0.365301
\(714\) 0 0
\(715\) 17.7210 0.662727
\(716\) −28.7282 −1.07362
\(717\) 5.64411 0.210783
\(718\) −1.74551 −0.0651418
\(719\) 24.4246 0.910883 0.455442 0.890266i \(-0.349481\pi\)
0.455442 + 0.890266i \(0.349481\pi\)
\(720\) 43.4966 1.62102
\(721\) 0 0
\(722\) 3.09219 0.115079
\(723\) 5.32514 0.198044
\(724\) −18.5309 −0.688697
\(725\) 17.2782 0.641695
\(726\) −0.473043 −0.0175563
\(727\) 3.09307 0.114716 0.0573578 0.998354i \(-0.481732\pi\)
0.0573578 + 0.998354i \(0.481732\pi\)
\(728\) 0 0
\(729\) −23.4236 −0.867539
\(730\) −0.831003 −0.0307568
\(731\) 15.5445 0.574933
\(732\) 3.84501 0.142115
\(733\) 8.41427 0.310788 0.155394 0.987853i \(-0.450335\pi\)
0.155394 + 0.987853i \(0.450335\pi\)
\(734\) 7.17182 0.264717
\(735\) 0 0
\(736\) −3.98203 −0.146779
\(737\) −57.0731 −2.10231
\(738\) 4.65652 0.171409
\(739\) −7.22758 −0.265871 −0.132936 0.991125i \(-0.542440\pi\)
−0.132936 + 0.991125i \(0.542440\pi\)
\(740\) 55.1128 2.02599
\(741\) −0.464419 −0.0170609
\(742\) 0 0
\(743\) −53.9092 −1.97774 −0.988869 0.148791i \(-0.952462\pi\)
−0.988869 + 0.148791i \(0.952462\pi\)
\(744\) −1.12873 −0.0413813
\(745\) 59.6360 2.18489
\(746\) −5.30765 −0.194327
\(747\) −20.9570 −0.766776
\(748\) −20.1678 −0.737406
\(749\) 0 0
\(750\) 1.09231 0.0398854
\(751\) 29.2442 1.06714 0.533568 0.845757i \(-0.320851\pi\)
0.533568 + 0.845757i \(0.320851\pi\)
\(752\) −13.3726 −0.487647
\(753\) −0.677712 −0.0246972
\(754\) 0.323048 0.0117647
\(755\) −20.2193 −0.735857
\(756\) 0 0
\(757\) −22.0597 −0.801773 −0.400887 0.916128i \(-0.631298\pi\)
−0.400887 + 0.916128i \(0.631298\pi\)
\(758\) 3.10981 0.112954
\(759\) 2.03723 0.0739467
\(760\) −5.44926 −0.197666
\(761\) −17.8161 −0.645833 −0.322917 0.946427i \(-0.604663\pi\)
−0.322917 + 0.946427i \(0.604663\pi\)
\(762\) 0.228374 0.00827313
\(763\) 0 0
\(764\) −24.6269 −0.890971
\(765\) 26.3095 0.951222
\(766\) 0.225159 0.00813532
\(767\) 4.50627 0.162712
\(768\) 3.38365 0.122097
\(769\) 11.3069 0.407738 0.203869 0.978998i \(-0.434648\pi\)
0.203869 + 0.978998i \(0.434648\pi\)
\(770\) 0 0
\(771\) 2.33725 0.0841741
\(772\) 18.3785 0.661458
\(773\) 1.92821 0.0693528 0.0346764 0.999399i \(-0.488960\pi\)
0.0346764 + 0.999399i \(0.488960\pi\)
\(774\) 3.90936 0.140519
\(775\) −58.6567 −2.10701
\(776\) 6.83168 0.245243
\(777\) 0 0
\(778\) 2.80043 0.100400
\(779\) 14.5084 0.519816
\(780\) −2.00413 −0.0717594
\(781\) 43.0290 1.53970
\(782\) −0.776782 −0.0277777
\(783\) −2.54467 −0.0909392
\(784\) 0 0
\(785\) −42.2003 −1.50619
\(786\) −0.643331 −0.0229469
\(787\) −5.53155 −0.197178 −0.0985892 0.995128i \(-0.531433\pi\)
−0.0985892 + 0.995128i \(0.531433\pi\)
\(788\) 14.9537 0.532704
\(789\) 0.372516 0.0132619
\(790\) 0.609148 0.0216725
\(791\) 0 0
\(792\) −10.2432 −0.363975
\(793\) 7.54467 0.267919
\(794\) −5.07093 −0.179961
\(795\) −3.36078 −0.119195
\(796\) 26.5398 0.940679
\(797\) 13.8038 0.488955 0.244477 0.969655i \(-0.421384\pi\)
0.244477 + 0.969655i \(0.421384\pi\)
\(798\) 0 0
\(799\) −8.08857 −0.286153
\(800\) −23.9456 −0.846605
\(801\) −33.0365 −1.16729
\(802\) 0.840601 0.0296826
\(803\) −4.86695 −0.171751
\(804\) 6.45461 0.227637
\(805\) 0 0
\(806\) −1.09670 −0.0386296
\(807\) 2.12204 0.0746992
\(808\) 11.0977 0.390415
\(809\) 28.2996 0.994961 0.497480 0.867475i \(-0.334259\pi\)
0.497480 + 0.867475i \(0.334259\pi\)
\(810\) 6.46093 0.227014
\(811\) 12.2124 0.428837 0.214418 0.976742i \(-0.431214\pi\)
0.214418 + 0.976742i \(0.431214\pi\)
\(812\) 0 0
\(813\) 0.0551575 0.00193446
\(814\) −6.29879 −0.220772
\(815\) −9.32946 −0.326797
\(816\) 2.23547 0.0782571
\(817\) 12.1804 0.426140
\(818\) −4.83410 −0.169020
\(819\) 0 0
\(820\) 62.6087 2.18639
\(821\) −41.4011 −1.44491 −0.722453 0.691420i \(-0.756986\pi\)
−0.722453 + 0.691420i \(0.756986\pi\)
\(822\) 0.472021 0.0164636
\(823\) 47.1752 1.64443 0.822213 0.569180i \(-0.192739\pi\)
0.822213 + 0.569180i \(0.192739\pi\)
\(824\) 5.80742 0.202311
\(825\) 12.2507 0.426515
\(826\) 0 0
\(827\) 21.1124 0.734150 0.367075 0.930191i \(-0.380359\pi\)
0.367075 + 0.930191i \(0.380359\pi\)
\(828\) 10.0110 0.347907
\(829\) 0.636752 0.0221153 0.0110577 0.999939i \(-0.496480\pi\)
0.0110577 + 0.999939i \(0.496480\pi\)
\(830\) 5.49861 0.190859
\(831\) 5.96862 0.207049
\(832\) 7.09585 0.246004
\(833\) 0 0
\(834\) −0.203317 −0.00704029
\(835\) 47.4983 1.64375
\(836\) −15.8032 −0.546565
\(837\) 8.63877 0.298600
\(838\) −4.87757 −0.168493
\(839\) 26.9432 0.930183 0.465092 0.885263i \(-0.346021\pi\)
0.465092 + 0.885263i \(0.346021\pi\)
\(840\) 0 0
\(841\) −26.2739 −0.905995
\(842\) 1.95656 0.0674276
\(843\) −0.0896862 −0.00308896
\(844\) 30.9941 1.06686
\(845\) −3.93251 −0.135282
\(846\) −2.03424 −0.0699386
\(847\) 0 0
\(848\) 12.4078 0.426087
\(849\) 7.52645 0.258307
\(850\) −4.67112 −0.160218
\(851\) 12.4322 0.426170
\(852\) −4.86631 −0.166717
\(853\) −6.74784 −0.231042 −0.115521 0.993305i \(-0.536854\pi\)
−0.115521 + 0.993305i \(0.536854\pi\)
\(854\) 0 0
\(855\) 20.6158 0.705045
\(856\) 8.50973 0.290856
\(857\) 45.0268 1.53809 0.769043 0.639197i \(-0.220733\pi\)
0.769043 + 0.639197i \(0.220733\pi\)
\(858\) 0.229050 0.00781965
\(859\) 36.7270 1.25311 0.626554 0.779378i \(-0.284465\pi\)
0.626554 + 0.779378i \(0.284465\pi\)
\(860\) 52.5629 1.79238
\(861\) 0 0
\(862\) 1.11299 0.0379087
\(863\) −43.4275 −1.47829 −0.739144 0.673547i \(-0.764770\pi\)
−0.739144 + 0.673547i \(0.764770\pi\)
\(864\) 3.52663 0.119978
\(865\) −76.3288 −2.59526
\(866\) 2.39871 0.0815114
\(867\) −3.06425 −0.104067
\(868\) 0 0
\(869\) 3.56761 0.121023
\(870\) 0.330033 0.0111892
\(871\) 12.6653 0.429146
\(872\) −9.64754 −0.326707
\(873\) −25.8458 −0.874747
\(874\) −0.608676 −0.0205888
\(875\) 0 0
\(876\) 0.550422 0.0185970
\(877\) 40.0081 1.35098 0.675488 0.737371i \(-0.263933\pi\)
0.675488 + 0.737371i \(0.263933\pi\)
\(878\) 0.982815 0.0331684
\(879\) −8.19443 −0.276391
\(880\) −66.8395 −2.25316
\(881\) −35.4308 −1.19370 −0.596848 0.802355i \(-0.703580\pi\)
−0.596848 + 0.802355i \(0.703580\pi\)
\(882\) 0 0
\(883\) 22.6654 0.762751 0.381375 0.924420i \(-0.375451\pi\)
0.381375 + 0.924420i \(0.375451\pi\)
\(884\) 4.47548 0.150527
\(885\) 4.60370 0.154752
\(886\) −0.113246 −0.00380459
\(887\) −44.6881 −1.50048 −0.750240 0.661166i \(-0.770062\pi\)
−0.750240 + 0.661166i \(0.770062\pi\)
\(888\) 1.43861 0.0482766
\(889\) 0 0
\(890\) 8.66800 0.290552
\(891\) 37.8398 1.26768
\(892\) 44.0870 1.47614
\(893\) −6.33810 −0.212096
\(894\) 0.770818 0.0257800
\(895\) −57.5892 −1.92499
\(896\) 0 0
\(897\) −0.452087 −0.0150947
\(898\) 1.44073 0.0480779
\(899\) −9.25480 −0.308665
\(900\) 60.2005 2.00668
\(901\) 7.50505 0.250029
\(902\) −7.15549 −0.238252
\(903\) 0 0
\(904\) −1.27983 −0.0425664
\(905\) −37.1476 −1.23483
\(906\) −0.261343 −0.00868254
\(907\) 54.4748 1.80881 0.904403 0.426680i \(-0.140317\pi\)
0.904403 + 0.426680i \(0.140317\pi\)
\(908\) −18.0527 −0.599100
\(909\) −41.9851 −1.39256
\(910\) 0 0
\(911\) −27.4793 −0.910431 −0.455215 0.890381i \(-0.650438\pi\)
−0.455215 + 0.890381i \(0.650438\pi\)
\(912\) 1.75169 0.0580041
\(913\) 32.2038 1.06579
\(914\) −1.54925 −0.0512447
\(915\) 7.70779 0.254812
\(916\) 29.9990 0.991196
\(917\) 0 0
\(918\) 0.687947 0.0227056
\(919\) −48.2880 −1.59287 −0.796437 0.604722i \(-0.793284\pi\)
−0.796437 + 0.604722i \(0.793284\pi\)
\(920\) −5.30456 −0.174886
\(921\) 4.72662 0.155747
\(922\) −1.86578 −0.0614461
\(923\) −9.54869 −0.314299
\(924\) 0 0
\(925\) 74.7601 2.45810
\(926\) −0.422713 −0.0138912
\(927\) −21.9708 −0.721615
\(928\) −3.77812 −0.124023
\(929\) −43.3154 −1.42113 −0.710566 0.703631i \(-0.751561\pi\)
−0.710566 + 0.703631i \(0.751561\pi\)
\(930\) −1.12041 −0.0367397
\(931\) 0 0
\(932\) 15.8032 0.517651
\(933\) 0.0978423 0.00320321
\(934\) −1.58853 −0.0519784
\(935\) −40.4288 −1.32216
\(936\) 2.27309 0.0742983
\(937\) 37.2211 1.21596 0.607980 0.793952i \(-0.291980\pi\)
0.607980 + 0.793952i \(0.291980\pi\)
\(938\) 0 0
\(939\) −2.85463 −0.0931574
\(940\) −27.3511 −0.892095
\(941\) −15.9751 −0.520773 −0.260386 0.965504i \(-0.583850\pi\)
−0.260386 + 0.965504i \(0.583850\pi\)
\(942\) −0.545456 −0.0177719
\(943\) 14.1231 0.459911
\(944\) −16.9967 −0.553194
\(945\) 0 0
\(946\) −6.00736 −0.195316
\(947\) 27.7572 0.901988 0.450994 0.892527i \(-0.351070\pi\)
0.450994 + 0.892527i \(0.351070\pi\)
\(948\) −0.403474 −0.0131042
\(949\) 1.08004 0.0350595
\(950\) −3.66023 −0.118754
\(951\) 6.80142 0.220551
\(952\) 0 0
\(953\) 12.0303 0.389700 0.194850 0.980833i \(-0.437578\pi\)
0.194850 + 0.980833i \(0.437578\pi\)
\(954\) 1.88748 0.0611096
\(955\) −49.3677 −1.59750
\(956\) −42.6199 −1.37843
\(957\) 1.93291 0.0624820
\(958\) 2.84744 0.0919965
\(959\) 0 0
\(960\) 7.24926 0.233969
\(961\) 0.418620 0.0135039
\(962\) 1.39778 0.0450663
\(963\) −32.1942 −1.03744
\(964\) −40.2113 −1.29512
\(965\) 36.8421 1.18599
\(966\) 0 0
\(967\) 5.40788 0.173906 0.0869528 0.996212i \(-0.472287\pi\)
0.0869528 + 0.996212i \(0.472287\pi\)
\(968\) 7.21380 0.231860
\(969\) 1.05953 0.0340370
\(970\) 6.78131 0.217735
\(971\) −42.2752 −1.35668 −0.678338 0.734750i \(-0.737300\pi\)
−0.678338 + 0.734750i \(0.737300\pi\)
\(972\) −13.3496 −0.428190
\(973\) 0 0
\(974\) 6.50733 0.208508
\(975\) −2.71859 −0.0870646
\(976\) −28.4568 −0.910881
\(977\) −10.8302 −0.346488 −0.173244 0.984879i \(-0.555425\pi\)
−0.173244 + 0.984879i \(0.555425\pi\)
\(978\) −0.120587 −0.00385594
\(979\) 50.7660 1.62249
\(980\) 0 0
\(981\) 36.4988 1.16532
\(982\) 4.41329 0.140834
\(983\) −21.5610 −0.687688 −0.343844 0.939027i \(-0.611729\pi\)
−0.343844 + 0.939027i \(0.611729\pi\)
\(984\) 1.63427 0.0520988
\(985\) 29.9766 0.955134
\(986\) −0.737005 −0.0234710
\(987\) 0 0
\(988\) 3.50693 0.111570
\(989\) 11.8570 0.377030
\(990\) −10.1677 −0.323149
\(991\) 8.62624 0.274022 0.137011 0.990570i \(-0.456251\pi\)
0.137011 + 0.990570i \(0.456251\pi\)
\(992\) 12.8261 0.407230
\(993\) −8.85703 −0.281069
\(994\) 0 0
\(995\) 53.2024 1.68663
\(996\) −3.64205 −0.115403
\(997\) −21.6967 −0.687142 −0.343571 0.939127i \(-0.611637\pi\)
−0.343571 + 0.939127i \(0.611637\pi\)
\(998\) 2.22763 0.0705144
\(999\) −11.0104 −0.348354
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 637.2.a.k.1.2 5
3.2 odd 2 5733.2.a.bm.1.4 5
7.2 even 3 637.2.e.m.508.4 10
7.3 odd 6 91.2.e.c.79.4 yes 10
7.4 even 3 637.2.e.m.79.4 10
7.5 odd 6 91.2.e.c.53.4 10
7.6 odd 2 637.2.a.l.1.2 5
13.12 even 2 8281.2.a.bx.1.4 5
21.5 even 6 819.2.j.h.235.2 10
21.17 even 6 819.2.j.h.352.2 10
21.20 even 2 5733.2.a.bl.1.4 5
28.3 even 6 1456.2.r.p.625.3 10
28.19 even 6 1456.2.r.p.417.3 10
91.12 odd 6 1183.2.e.f.508.2 10
91.38 odd 6 1183.2.e.f.170.2 10
91.90 odd 2 8281.2.a.bw.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.4 10 7.5 odd 6
91.2.e.c.79.4 yes 10 7.3 odd 6
637.2.a.k.1.2 5 1.1 even 1 trivial
637.2.a.l.1.2 5 7.6 odd 2
637.2.e.m.79.4 10 7.4 even 3
637.2.e.m.508.4 10 7.2 even 3
819.2.j.h.235.2 10 21.5 even 6
819.2.j.h.352.2 10 21.17 even 6
1183.2.e.f.170.2 10 91.38 odd 6
1183.2.e.f.508.2 10 91.12 odd 6
1456.2.r.p.417.3 10 28.19 even 6
1456.2.r.p.625.3 10 28.3 even 6
5733.2.a.bl.1.4 5 21.20 even 2
5733.2.a.bm.1.4 5 3.2 odd 2
8281.2.a.bw.1.4 5 91.90 odd 2
8281.2.a.bx.1.4 5 13.12 even 2