Properties

Label 490.4.c.g.99.15
Level $490$
Weight $4$
Character 490.99
Analytic conductor $28.911$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,4,Mod(99,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.c (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: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 438 x^{18} + 80439 x^{16} + 8097428 x^{14} + 488971671 x^{12} + 18162509334 x^{10} + 407198911753 x^{8} + 5129420990112 x^{6} + \cdots + 9871083181584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{23}\cdot 5^{2}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.15
Root \(1.53777i\) of defining polynomial
Character \(\chi\) \(=\) 490.99
Dual form 490.4.c.g.99.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -0.537767i q^{3} -4.00000 q^{4} +(8.93458 + 6.72110i) q^{5} +1.07553 q^{6} -8.00000i q^{8} +26.7108 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -0.537767i q^{3} -4.00000 q^{4} +(8.93458 + 6.72110i) q^{5} +1.07553 q^{6} -8.00000i q^{8} +26.7108 q^{9} +(-13.4422 + 17.8692i) q^{10} +14.1793 q^{11} +2.15107i q^{12} -13.9177i q^{13} +(3.61439 - 4.80472i) q^{15} +16.0000 q^{16} -65.7823i q^{17} +53.4216i q^{18} +95.1968 q^{19} +(-35.7383 - 26.8844i) q^{20} +28.3586i q^{22} -69.4781i q^{23} -4.30213 q^{24} +(34.6536 + 120.101i) q^{25} +27.8353 q^{26} -28.8839i q^{27} -127.840 q^{29} +(9.60944 + 7.22877i) q^{30} +98.3654 q^{31} +32.0000i q^{32} -7.62517i q^{33} +131.565 q^{34} -106.843 q^{36} -287.582i q^{37} +190.394i q^{38} -7.48445 q^{39} +(53.7688 - 71.4767i) q^{40} +310.112 q^{41} -197.986i q^{43} -56.7173 q^{44} +(238.650 + 179.526i) q^{45} +138.956 q^{46} +538.883i q^{47} -8.60427i q^{48} +(-240.201 + 69.3071i) q^{50} -35.3755 q^{51} +55.6706i q^{52} +314.867i q^{53} +57.7678 q^{54} +(126.686 + 95.3006i) q^{55} -51.1937i q^{57} -255.680i q^{58} +242.143 q^{59} +(-14.4575 + 19.2189i) q^{60} +440.612 q^{61} +196.731i q^{62} -64.0000 q^{64} +(93.5420 - 124.348i) q^{65} +15.2503 q^{66} +858.375i q^{67} +263.129i q^{68} -37.3630 q^{69} +142.246 q^{71} -213.686i q^{72} +459.680i q^{73} +575.164 q^{74} +(64.5861 - 18.6355i) q^{75} -380.787 q^{76} -14.9689i q^{78} -1193.09 q^{79} +(142.953 + 107.538i) q^{80} +705.659 q^{81} +620.223i q^{82} -403.628i q^{83} +(442.129 - 587.737i) q^{85} +395.973 q^{86} +68.7480i q^{87} -113.435i q^{88} +1084.30 q^{89} +(-359.052 + 477.300i) q^{90} +277.912i q^{92} -52.8977i q^{93} -1077.77 q^{94} +(850.544 + 639.827i) q^{95} +17.2085 q^{96} -166.723i q^{97} +378.741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 80 q^{4} - 316 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 80 q^{4} - 316 q^{9} + 104 q^{11} - 360 q^{15} + 320 q^{16} - 440 q^{25} - 216 q^{29} + 224 q^{30} + 1264 q^{36} - 504 q^{39} - 416 q^{44} + 1600 q^{46} + 952 q^{50} - 296 q^{51} + 1440 q^{60} - 1280 q^{64} + 2732 q^{65} - 1872 q^{71} - 5968 q^{74} - 6424 q^{79} + 2020 q^{81} + 428 q^{85} + 3616 q^{86} + 3568 q^{95} + 624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 0.537767i 0.103493i −0.998660 0.0517466i \(-0.983521\pi\)
0.998660 0.0517466i \(-0.0164788\pi\)
\(4\) −4.00000 −0.500000
\(5\) 8.93458 + 6.72110i 0.799133 + 0.601154i
\(6\) 1.07553 0.0731808
\(7\) 0 0
\(8\) 8.00000i 0.353553i
\(9\) 26.7108 0.989289
\(10\) −13.4422 + 17.8692i −0.425080 + 0.565073i
\(11\) 14.1793 0.388657 0.194328 0.980937i \(-0.437747\pi\)
0.194328 + 0.980937i \(0.437747\pi\)
\(12\) 2.15107i 0.0517466i
\(13\) 13.9177i 0.296928i −0.988918 0.148464i \(-0.952567\pi\)
0.988918 0.148464i \(-0.0474329\pi\)
\(14\) 0 0
\(15\) 3.61439 4.80472i 0.0622154 0.0827049i
\(16\) 16.0000 0.250000
\(17\) 65.7823i 0.938503i −0.883065 0.469251i \(-0.844524\pi\)
0.883065 0.469251i \(-0.155476\pi\)
\(18\) 53.4216i 0.699533i
\(19\) 95.1968 1.14945 0.574727 0.818345i \(-0.305108\pi\)
0.574727 + 0.818345i \(0.305108\pi\)
\(20\) −35.7383 26.8844i −0.399567 0.300577i
\(21\) 0 0
\(22\) 28.3586i 0.274822i
\(23\) 69.4781i 0.629877i −0.949112 0.314939i \(-0.898016\pi\)
0.949112 0.314939i \(-0.101984\pi\)
\(24\) −4.30213 −0.0365904
\(25\) 34.6536 + 120.101i 0.277229 + 0.960804i
\(26\) 27.8353 0.209960
\(27\) 28.8839i 0.205878i
\(28\) 0 0
\(29\) −127.840 −0.818595 −0.409298 0.912401i \(-0.634226\pi\)
−0.409298 + 0.912401i \(0.634226\pi\)
\(30\) 9.60944 + 7.22877i 0.0584812 + 0.0439929i
\(31\) 98.3654 0.569902 0.284951 0.958542i \(-0.408023\pi\)
0.284951 + 0.958542i \(0.408023\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 7.62517i 0.0402234i
\(34\) 131.565 0.663622
\(35\) 0 0
\(36\) −106.843 −0.494645
\(37\) 287.582i 1.27779i −0.769294 0.638895i \(-0.779392\pi\)
0.769294 0.638895i \(-0.220608\pi\)
\(38\) 190.394i 0.812787i
\(39\) −7.48445 −0.0307300
\(40\) 53.7688 71.4767i 0.212540 0.282536i
\(41\) 310.112 1.18125 0.590625 0.806946i \(-0.298881\pi\)
0.590625 + 0.806946i \(0.298881\pi\)
\(42\) 0 0
\(43\) 197.986i 0.702154i −0.936347 0.351077i \(-0.885816\pi\)
0.936347 0.351077i \(-0.114184\pi\)
\(44\) −56.7173 −0.194328
\(45\) 238.650 + 179.526i 0.790574 + 0.594715i
\(46\) 138.956 0.445391
\(47\) 538.883i 1.67243i 0.548402 + 0.836215i \(0.315237\pi\)
−0.548402 + 0.836215i \(0.684763\pi\)
\(48\) 8.60427i 0.0258733i
\(49\) 0 0
\(50\) −240.201 + 69.3071i −0.679391 + 0.196030i
\(51\) −35.3755 −0.0971287
\(52\) 55.6706i 0.148464i
\(53\) 314.867i 0.816043i 0.912972 + 0.408021i \(0.133781\pi\)
−0.912972 + 0.408021i \(0.866219\pi\)
\(54\) 57.7678 0.145578
\(55\) 126.686 + 95.3006i 0.310589 + 0.233642i
\(56\) 0 0
\(57\) 51.1937i 0.118961i
\(58\) 255.680i 0.578834i
\(59\) 242.143 0.534311 0.267156 0.963653i \(-0.413916\pi\)
0.267156 + 0.963653i \(0.413916\pi\)
\(60\) −14.4575 + 19.2189i −0.0311077 + 0.0413525i
\(61\) 440.612 0.924830 0.462415 0.886664i \(-0.346983\pi\)
0.462415 + 0.886664i \(0.346983\pi\)
\(62\) 196.731i 0.402981i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 93.5420 124.348i 0.178499 0.237285i
\(66\) 15.2503 0.0284422
\(67\) 858.375i 1.56518i 0.622536 + 0.782591i \(0.286102\pi\)
−0.622536 + 0.782591i \(0.713898\pi\)
\(68\) 263.129i 0.469251i
\(69\) −37.3630 −0.0651881
\(70\) 0 0
\(71\) 142.246 0.237768 0.118884 0.992908i \(-0.462068\pi\)
0.118884 + 0.992908i \(0.462068\pi\)
\(72\) 213.686i 0.349767i
\(73\) 459.680i 0.737007i 0.929626 + 0.368503i \(0.120130\pi\)
−0.929626 + 0.368503i \(0.879870\pi\)
\(74\) 575.164 0.903534
\(75\) 64.5861 18.6355i 0.0994367 0.0286913i
\(76\) −380.787 −0.574727
\(77\) 0 0
\(78\) 14.9689i 0.0217294i
\(79\) −1193.09 −1.69915 −0.849574 0.527470i \(-0.823141\pi\)
−0.849574 + 0.527470i \(0.823141\pi\)
\(80\) 142.953 + 107.538i 0.199783 + 0.150288i
\(81\) 705.659 0.967982
\(82\) 620.223i 0.835271i
\(83\) 403.628i 0.533782i −0.963727 0.266891i \(-0.914004\pi\)
0.963727 0.266891i \(-0.0859964\pi\)
\(84\) 0 0
\(85\) 442.129 587.737i 0.564184 0.749989i
\(86\) 395.973 0.496498
\(87\) 68.7480i 0.0847191i
\(88\) 113.435i 0.137411i
\(89\) 1084.30 1.29141 0.645704 0.763587i \(-0.276564\pi\)
0.645704 + 0.763587i \(0.276564\pi\)
\(90\) −359.052 + 477.300i −0.420527 + 0.559020i
\(91\) 0 0
\(92\) 277.912i 0.314939i
\(93\) 52.8977i 0.0589810i
\(94\) −1077.77 −1.18259
\(95\) 850.544 + 639.827i 0.918568 + 0.690999i
\(96\) 17.2085 0.0182952
\(97\) 166.723i 0.174517i −0.996186 0.0872583i \(-0.972189\pi\)
0.996186 0.0872583i \(-0.0278106\pi\)
\(98\) 0 0
\(99\) 378.741 0.384494
\(100\) −138.614 480.402i −0.138614 0.480402i
\(101\) 834.364 0.822003 0.411002 0.911635i \(-0.365179\pi\)
0.411002 + 0.911635i \(0.365179\pi\)
\(102\) 70.7510i 0.0686804i
\(103\) 1415.11i 1.35373i 0.736106 + 0.676867i \(0.236663\pi\)
−0.736106 + 0.676867i \(0.763337\pi\)
\(104\) −111.341 −0.104980
\(105\) 0 0
\(106\) −629.733 −0.577029
\(107\) 1569.05i 1.41763i 0.705396 + 0.708813i \(0.250769\pi\)
−0.705396 + 0.708813i \(0.749231\pi\)
\(108\) 115.536i 0.102939i
\(109\) −750.399 −0.659405 −0.329703 0.944085i \(-0.606948\pi\)
−0.329703 + 0.944085i \(0.606948\pi\)
\(110\) −190.601 + 253.373i −0.165210 + 0.219619i
\(111\) −154.652 −0.132243
\(112\) 0 0
\(113\) 2017.09i 1.67922i −0.543188 0.839611i \(-0.682783\pi\)
0.543188 0.839611i \(-0.317217\pi\)
\(114\) 102.387 0.0841180
\(115\) 466.969 620.758i 0.378653 0.503356i
\(116\) 511.359 0.409298
\(117\) 371.752i 0.293748i
\(118\) 484.287i 0.377815i
\(119\) 0 0
\(120\) −38.4378 28.9151i −0.0292406 0.0219964i
\(121\) −1129.95 −0.848946
\(122\) 881.225i 0.653953i
\(123\) 166.768i 0.122252i
\(124\) −393.462 −0.284951
\(125\) −497.593 + 1305.96i −0.356048 + 0.934468i
\(126\) 0 0
\(127\) 89.7776i 0.0627281i −0.999508 0.0313641i \(-0.990015\pi\)
0.999508 0.0313641i \(-0.00998513\pi\)
\(128\) 128.000i 0.0883883i
\(129\) −106.470 −0.0726682
\(130\) 248.697 + 187.084i 0.167786 + 0.126218i
\(131\) −1406.47 −0.938042 −0.469021 0.883187i \(-0.655393\pi\)
−0.469021 + 0.883187i \(0.655393\pi\)
\(132\) 30.5007i 0.0201117i
\(133\) 0 0
\(134\) −1716.75 −1.10675
\(135\) 194.132 258.066i 0.123764 0.164524i
\(136\) −526.258 −0.331811
\(137\) 646.886i 0.403410i 0.979446 + 0.201705i \(0.0646483\pi\)
−0.979446 + 0.201705i \(0.935352\pi\)
\(138\) 74.7260i 0.0460949i
\(139\) 1309.84 0.799273 0.399636 0.916674i \(-0.369136\pi\)
0.399636 + 0.916674i \(0.369136\pi\)
\(140\) 0 0
\(141\) 289.794 0.173085
\(142\) 284.493i 0.168127i
\(143\) 197.343i 0.115403i
\(144\) 427.373 0.247322
\(145\) −1142.20 859.224i −0.654167 0.492101i
\(146\) −919.360 −0.521142
\(147\) 0 0
\(148\) 1150.33i 0.638895i
\(149\) −1457.54 −0.801385 −0.400692 0.916213i \(-0.631230\pi\)
−0.400692 + 0.916213i \(0.631230\pi\)
\(150\) 37.2711 + 129.172i 0.0202878 + 0.0703124i
\(151\) −1943.86 −1.04761 −0.523804 0.851839i \(-0.675488\pi\)
−0.523804 + 0.851839i \(0.675488\pi\)
\(152\) 761.574i 0.406394i
\(153\) 1757.10i 0.928451i
\(154\) 0 0
\(155\) 878.854 + 661.124i 0.455427 + 0.342598i
\(156\) 29.9378 0.0153650
\(157\) 1322.67i 0.672361i −0.941798 0.336181i \(-0.890865\pi\)
0.941798 0.336181i \(-0.109135\pi\)
\(158\) 2386.17i 1.20148i
\(159\) 169.325 0.0844549
\(160\) −215.075 + 285.907i −0.106270 + 0.141268i
\(161\) 0 0
\(162\) 1411.32i 0.684467i
\(163\) 1133.89i 0.544866i −0.962175 0.272433i \(-0.912172\pi\)
0.962175 0.272433i \(-0.0878283\pi\)
\(164\) −1240.45 −0.590625
\(165\) 51.2495 68.1277i 0.0241804 0.0321438i
\(166\) 807.256 0.377441
\(167\) 483.314i 0.223952i 0.993711 + 0.111976i \(0.0357179\pi\)
−0.993711 + 0.111976i \(0.964282\pi\)
\(168\) 0 0
\(169\) 2003.30 0.911834
\(170\) 1175.47 + 884.259i 0.530322 + 0.398939i
\(171\) 2542.78 1.13714
\(172\) 791.945i 0.351077i
\(173\) 1289.72i 0.566795i −0.959003 0.283398i \(-0.908538\pi\)
0.959003 0.283398i \(-0.0914616\pi\)
\(174\) −137.496 −0.0599054
\(175\) 0 0
\(176\) 226.869 0.0971642
\(177\) 130.217i 0.0552976i
\(178\) 2168.60i 0.913164i
\(179\) −2735.61 −1.14229 −0.571143 0.820851i \(-0.693500\pi\)
−0.571143 + 0.820851i \(0.693500\pi\)
\(180\) −954.600 718.104i −0.395287 0.297357i
\(181\) 1146.35 0.470761 0.235380 0.971903i \(-0.424366\pi\)
0.235380 + 0.971903i \(0.424366\pi\)
\(182\) 0 0
\(183\) 236.947i 0.0957137i
\(184\) −555.825 −0.222695
\(185\) 1932.87 2569.43i 0.768148 1.02112i
\(186\) 105.795 0.0417059
\(187\) 932.748i 0.364755i
\(188\) 2155.53i 0.836215i
\(189\) 0 0
\(190\) −1279.65 + 1701.09i −0.488610 + 0.649526i
\(191\) 4247.74 1.60919 0.804597 0.593822i \(-0.202382\pi\)
0.804597 + 0.593822i \(0.202382\pi\)
\(192\) 34.4171i 0.0129367i
\(193\) 5298.59i 1.97617i −0.153903 0.988086i \(-0.549184\pi\)
0.153903 0.988086i \(-0.450816\pi\)
\(194\) 333.445 0.123402
\(195\) −66.8705 50.3038i −0.0245574 0.0184735i
\(196\) 0 0
\(197\) 653.948i 0.236507i 0.992983 + 0.118253i \(0.0377295\pi\)
−0.992983 + 0.118253i \(0.962270\pi\)
\(198\) 757.482i 0.271878i
\(199\) −4484.94 −1.59763 −0.798817 0.601574i \(-0.794540\pi\)
−0.798817 + 0.601574i \(0.794540\pi\)
\(200\) 960.804 277.229i 0.339696 0.0980151i
\(201\) 461.606 0.161986
\(202\) 1668.73i 0.581244i
\(203\) 0 0
\(204\) 141.502 0.0485644
\(205\) 2770.72 + 2084.29i 0.943977 + 0.710113i
\(206\) −2830.21 −0.957234
\(207\) 1855.82i 0.623131i
\(208\) 222.683i 0.0742320i
\(209\) 1349.83 0.446743
\(210\) 0 0
\(211\) 2160.14 0.704787 0.352394 0.935852i \(-0.385368\pi\)
0.352394 + 0.935852i \(0.385368\pi\)
\(212\) 1259.47i 0.408021i
\(213\) 76.4953i 0.0246074i
\(214\) −3138.11 −1.00241
\(215\) 1330.69 1768.93i 0.422103 0.561115i
\(216\) −231.071 −0.0727889
\(217\) 0 0
\(218\) 1500.80i 0.466270i
\(219\) 247.201 0.0762752
\(220\) −506.745 381.203i −0.155294 0.116821i
\(221\) −915.535 −0.278668
\(222\) 309.304i 0.0935096i
\(223\) 6026.45i 1.80969i −0.425742 0.904845i \(-0.639987\pi\)
0.425742 0.904845i \(-0.360013\pi\)
\(224\) 0 0
\(225\) 925.625 + 3207.98i 0.274259 + 0.950513i
\(226\) 4034.19 1.18739
\(227\) 4659.04i 1.36225i −0.732165 0.681127i \(-0.761490\pi\)
0.732165 0.681127i \(-0.238510\pi\)
\(228\) 204.775i 0.0594804i
\(229\) 356.000 0.102730 0.0513649 0.998680i \(-0.483643\pi\)
0.0513649 + 0.998680i \(0.483643\pi\)
\(230\) 1241.52 + 933.938i 0.355927 + 0.267748i
\(231\) 0 0
\(232\) 1022.72i 0.289417i
\(233\) 3308.76i 0.930318i 0.885227 + 0.465159i \(0.154003\pi\)
−0.885227 + 0.465159i \(0.845997\pi\)
\(234\) 743.504 0.207711
\(235\) −3621.89 + 4814.70i −1.00539 + 1.33649i
\(236\) −968.573 −0.267156
\(237\) 641.602i 0.175850i
\(238\) 0 0
\(239\) −985.684 −0.266772 −0.133386 0.991064i \(-0.542585\pi\)
−0.133386 + 0.991064i \(0.542585\pi\)
\(240\) 57.8302 76.8756i 0.0155538 0.0206762i
\(241\) 351.514 0.0939544 0.0469772 0.998896i \(-0.485041\pi\)
0.0469772 + 0.998896i \(0.485041\pi\)
\(242\) 2259.89i 0.600295i
\(243\) 1159.34i 0.306058i
\(244\) −1762.45 −0.462415
\(245\) 0 0
\(246\) 333.535 0.0864449
\(247\) 1324.92i 0.341305i
\(248\) 786.923i 0.201491i
\(249\) −217.058 −0.0552429
\(250\) −2611.92 995.185i −0.660768 0.251764i
\(251\) −16.6514 −0.00418737 −0.00209368 0.999998i \(-0.500666\pi\)
−0.00209368 + 0.999998i \(0.500666\pi\)
\(252\) 0 0
\(253\) 985.152i 0.244806i
\(254\) 179.555 0.0443555
\(255\) −316.066 237.763i −0.0776188 0.0583893i
\(256\) 256.000 0.0625000
\(257\) 7095.34i 1.72216i −0.508469 0.861081i \(-0.669788\pi\)
0.508469 0.861081i \(-0.330212\pi\)
\(258\) 212.941i 0.0513842i
\(259\) 0 0
\(260\) −374.168 + 497.394i −0.0892497 + 0.118643i
\(261\) −3414.70 −0.809827
\(262\) 2812.93i 0.663296i
\(263\) 4943.46i 1.15904i −0.814959 0.579519i \(-0.803240\pi\)
0.814959 0.579519i \(-0.196760\pi\)
\(264\) −61.0013 −0.0142211
\(265\) −2116.25 + 2813.20i −0.490567 + 0.652127i
\(266\) 0 0
\(267\) 583.100i 0.133652i
\(268\) 3433.50i 0.782591i
\(269\) −2145.29 −0.486248 −0.243124 0.969995i \(-0.578172\pi\)
−0.243124 + 0.969995i \(0.578172\pi\)
\(270\) 516.131 + 388.263i 0.116336 + 0.0875146i
\(271\) −8339.99 −1.86944 −0.934720 0.355385i \(-0.884350\pi\)
−0.934720 + 0.355385i \(0.884350\pi\)
\(272\) 1052.52i 0.234626i
\(273\) 0 0
\(274\) −1293.77 −0.285254
\(275\) 491.364 + 1702.94i 0.107747 + 0.373423i
\(276\) 149.452 0.0325940
\(277\) 1094.05i 0.237311i −0.992935 0.118655i \(-0.962142\pi\)
0.992935 0.118655i \(-0.0378584\pi\)
\(278\) 2619.67i 0.565171i
\(279\) 2627.42 0.563798
\(280\) 0 0
\(281\) −5699.48 −1.20997 −0.604987 0.796235i \(-0.706822\pi\)
−0.604987 + 0.796235i \(0.706822\pi\)
\(282\) 579.587i 0.122390i
\(283\) 6101.50i 1.28161i 0.767703 + 0.640806i \(0.221400\pi\)
−0.767703 + 0.640806i \(0.778600\pi\)
\(284\) −568.985 −0.118884
\(285\) 344.078 457.394i 0.0715137 0.0950656i
\(286\) 394.686 0.0816023
\(287\) 0 0
\(288\) 854.746i 0.174883i
\(289\) 585.692 0.119213
\(290\) 1718.45 2284.39i 0.347968 0.462566i
\(291\) −89.6579 −0.0180613
\(292\) 1838.72i 0.368503i
\(293\) 9195.72i 1.83351i −0.399446 0.916757i \(-0.630797\pi\)
0.399446 0.916757i \(-0.369203\pi\)
\(294\) 0 0
\(295\) 2163.45 + 1627.47i 0.426986 + 0.321203i
\(296\) −2300.66 −0.451767
\(297\) 409.554i 0.0800159i
\(298\) 2915.08i 0.566665i
\(299\) −966.972 −0.187028
\(300\) −258.344 + 74.5422i −0.0497184 + 0.0143456i
\(301\) 0 0
\(302\) 3887.72i 0.740771i
\(303\) 448.693i 0.0850718i
\(304\) 1523.15 0.287364
\(305\) 3936.69 + 2961.40i 0.739063 + 0.555965i
\(306\) 3514.20 0.656514
\(307\) 7888.57i 1.46653i 0.679944 + 0.733264i \(0.262004\pi\)
−0.679944 + 0.733264i \(0.737996\pi\)
\(308\) 0 0
\(309\) 760.997 0.140102
\(310\) −1322.25 + 1757.71i −0.242254 + 0.322036i
\(311\) −1745.55 −0.318267 −0.159133 0.987257i \(-0.550870\pi\)
−0.159133 + 0.987257i \(0.550870\pi\)
\(312\) 59.8756i 0.0108647i
\(313\) 7968.41i 1.43898i 0.694502 + 0.719491i \(0.255625\pi\)
−0.694502 + 0.719491i \(0.744375\pi\)
\(314\) 2645.34 0.475431
\(315\) 0 0
\(316\) 4772.34 0.849574
\(317\) 3587.39i 0.635608i 0.948156 + 0.317804i \(0.102945\pi\)
−0.948156 + 0.317804i \(0.897055\pi\)
\(318\) 338.650i 0.0597187i
\(319\) −1812.68 −0.318153
\(320\) −571.813 430.151i −0.0998917 0.0751442i
\(321\) 843.784 0.146715
\(322\) 0 0
\(323\) 6262.26i 1.07877i
\(324\) −2822.64 −0.483991
\(325\) 1671.52 482.297i 0.285290 0.0823169i
\(326\) 2267.78 0.385278
\(327\) 403.540i 0.0682440i
\(328\) 2480.89i 0.417635i
\(329\) 0 0
\(330\) 136.255 + 102.499i 0.0227291 + 0.0170981i
\(331\) −1442.47 −0.239532 −0.119766 0.992802i \(-0.538214\pi\)
−0.119766 + 0.992802i \(0.538214\pi\)
\(332\) 1614.51i 0.266891i
\(333\) 7681.55i 1.26410i
\(334\) −966.627 −0.158358
\(335\) −5769.23 + 7669.22i −0.940915 + 1.25079i
\(336\) 0 0
\(337\) 5613.50i 0.907380i 0.891160 + 0.453690i \(0.149893\pi\)
−0.891160 + 0.453690i \(0.850107\pi\)
\(338\) 4006.60i 0.644764i
\(339\) −1084.73 −0.173788
\(340\) −1768.52 + 2350.95i −0.282092 + 0.374994i
\(341\) 1394.75 0.221496
\(342\) 5085.57i 0.804082i
\(343\) 0 0
\(344\) −1583.89 −0.248249
\(345\) −333.823 251.121i −0.0520940 0.0391880i
\(346\) 2579.44 0.400785
\(347\) 11217.9i 1.73547i 0.497029 + 0.867734i \(0.334424\pi\)
−0.497029 + 0.867734i \(0.665576\pi\)
\(348\) 274.992i 0.0423595i
\(349\) −5462.91 −0.837888 −0.418944 0.908012i \(-0.637600\pi\)
−0.418944 + 0.908012i \(0.637600\pi\)
\(350\) 0 0
\(351\) −401.996 −0.0611309
\(352\) 453.738i 0.0687055i
\(353\) 1434.34i 0.216268i 0.994136 + 0.108134i \(0.0344875\pi\)
−0.994136 + 0.108134i \(0.965513\pi\)
\(354\) 260.433 0.0391013
\(355\) 1270.91 + 956.052i 0.190008 + 0.142935i
\(356\) −4337.19 −0.645704
\(357\) 0 0
\(358\) 5471.22i 0.807718i
\(359\) −174.484 −0.0256515 −0.0128258 0.999918i \(-0.504083\pi\)
−0.0128258 + 0.999918i \(0.504083\pi\)
\(360\) 1436.21 1909.20i 0.210263 0.279510i
\(361\) 2203.43 0.321246
\(362\) 2292.70i 0.332878i
\(363\) 607.648i 0.0878602i
\(364\) 0 0
\(365\) −3089.56 + 4107.05i −0.443054 + 0.588967i
\(366\) 473.893 0.0676798
\(367\) 12258.5i 1.74356i 0.489893 + 0.871782i \(0.337036\pi\)
−0.489893 + 0.871782i \(0.662964\pi\)
\(368\) 1111.65i 0.157469i
\(369\) 8283.33 1.16860
\(370\) 5138.85 + 3865.74i 0.722044 + 0.543163i
\(371\) 0 0
\(372\) 211.591i 0.0294905i
\(373\) 5921.30i 0.821966i −0.911643 0.410983i \(-0.865186\pi\)
0.911643 0.410983i \(-0.134814\pi\)
\(374\) 1865.50 0.257921
\(375\) 702.301 + 267.589i 0.0967111 + 0.0368486i
\(376\) 4311.07 0.591293
\(377\) 1779.23i 0.243064i
\(378\) 0 0
\(379\) 6322.18 0.856857 0.428428 0.903576i \(-0.359067\pi\)
0.428428 + 0.903576i \(0.359067\pi\)
\(380\) −3402.17 2559.31i −0.459284 0.345499i
\(381\) −48.2794 −0.00649194
\(382\) 8495.49i 1.13787i
\(383\) 11997.0i 1.60057i 0.599619 + 0.800285i \(0.295319\pi\)
−0.599619 + 0.800285i \(0.704681\pi\)
\(384\) −68.8341 −0.00914760
\(385\) 0 0
\(386\) 10597.2 1.39736
\(387\) 5288.37i 0.694633i
\(388\) 666.891i 0.0872583i
\(389\) 14636.9 1.90776 0.953879 0.300191i \(-0.0970505\pi\)
0.953879 + 0.300191i \(0.0970505\pi\)
\(390\) 100.608 133.741i 0.0130627 0.0173647i
\(391\) −4570.43 −0.591142
\(392\) 0 0
\(393\) 756.351i 0.0970810i
\(394\) −1307.90 −0.167236
\(395\) −10659.7 8018.85i −1.35785 1.02145i
\(396\) −1514.96 −0.192247
\(397\) 6298.50i 0.796254i 0.917330 + 0.398127i \(0.130340\pi\)
−0.917330 + 0.398127i \(0.869660\pi\)
\(398\) 8969.88i 1.12970i
\(399\) 0 0
\(400\) 554.457 + 1921.61i 0.0693071 + 0.240201i
\(401\) −8289.93 −1.03237 −0.516183 0.856478i \(-0.672648\pi\)
−0.516183 + 0.856478i \(0.672648\pi\)
\(402\) 923.211i 0.114541i
\(403\) 1369.02i 0.169220i
\(404\) −3337.46 −0.411002
\(405\) 6304.77 + 4742.81i 0.773547 + 0.581906i
\(406\) 0 0
\(407\) 4077.72i 0.496622i
\(408\) 283.004i 0.0343402i
\(409\) 9522.06 1.15119 0.575593 0.817736i \(-0.304771\pi\)
0.575593 + 0.817736i \(0.304771\pi\)
\(410\) −4168.58 + 5541.44i −0.502126 + 0.667493i
\(411\) 347.874 0.0417503
\(412\) 5660.42i 0.676867i
\(413\) 0 0
\(414\) 3711.63 0.440620
\(415\) 2712.82 3606.25i 0.320885 0.426563i
\(416\) 445.365 0.0524899
\(417\) 704.387i 0.0827194i
\(418\) 2699.65i 0.315895i
\(419\) −8936.51 −1.04195 −0.520975 0.853572i \(-0.674432\pi\)
−0.520975 + 0.853572i \(0.674432\pi\)
\(420\) 0 0
\(421\) −10741.8 −1.24352 −0.621760 0.783207i \(-0.713582\pi\)
−0.621760 + 0.783207i \(0.713582\pi\)
\(422\) 4320.28i 0.498360i
\(423\) 14394.0i 1.65452i
\(424\) 2518.93 0.288515
\(425\) 7900.48 2279.59i 0.901717 0.260180i
\(426\) 152.991 0.0174001
\(427\) 0 0
\(428\) 6276.21i 0.708813i
\(429\) −106.124 −0.0119434
\(430\) 3537.85 + 2661.37i 0.396768 + 0.298472i
\(431\) 7886.50 0.881391 0.440695 0.897657i \(-0.354732\pi\)
0.440695 + 0.897657i \(0.354732\pi\)
\(432\) 462.142i 0.0514695i
\(433\) 3756.12i 0.416877i −0.978035 0.208438i \(-0.933162\pi\)
0.978035 0.208438i \(-0.0668381\pi\)
\(434\) 0 0
\(435\) −462.062 + 614.235i −0.0509292 + 0.0677018i
\(436\) 3001.60 0.329703
\(437\) 6614.09i 0.724016i
\(438\) 494.401i 0.0539347i
\(439\) −6920.65 −0.752402 −0.376201 0.926538i \(-0.622770\pi\)
−0.376201 + 0.926538i \(0.622770\pi\)
\(440\) 762.405 1013.49i 0.0826051 0.109810i
\(441\) 0 0
\(442\) 1831.07i 0.197048i
\(443\) 11618.9i 1.24611i −0.782176 0.623057i \(-0.785890\pi\)
0.782176 0.623057i \(-0.214110\pi\)
\(444\) 618.608 0.0661213
\(445\) 9687.75 + 7287.68i 1.03201 + 0.776335i
\(446\) 12052.9 1.27964
\(447\) 783.817i 0.0829379i
\(448\) 0 0
\(449\) −17715.7 −1.86204 −0.931018 0.364973i \(-0.881078\pi\)
−0.931018 + 0.364973i \(0.881078\pi\)
\(450\) −6415.96 + 1851.25i −0.672114 + 0.193931i
\(451\) 4397.17 0.459101
\(452\) 8068.37i 0.839611i
\(453\) 1045.34i 0.108420i
\(454\) 9318.09 0.963259
\(455\) 0 0
\(456\) −409.549 −0.0420590
\(457\) 2724.63i 0.278890i 0.990230 + 0.139445i \(0.0445319\pi\)
−0.990230 + 0.139445i \(0.955468\pi\)
\(458\) 712.000i 0.0726410i
\(459\) −1900.05 −0.193217
\(460\) −1867.88 + 2483.03i −0.189327 + 0.251678i
\(461\) 11108.3 1.12227 0.561135 0.827724i \(-0.310365\pi\)
0.561135 + 0.827724i \(0.310365\pi\)
\(462\) 0 0
\(463\) 14640.7i 1.46957i −0.678301 0.734784i \(-0.737284\pi\)
0.678301 0.734784i \(-0.262716\pi\)
\(464\) −2045.44 −0.204649
\(465\) 355.531 472.619i 0.0354566 0.0471337i
\(466\) −6617.52 −0.657834
\(467\) 1242.76i 0.123143i 0.998103 + 0.0615716i \(0.0196113\pi\)
−0.998103 + 0.0615716i \(0.980389\pi\)
\(468\) 1487.01i 0.146874i
\(469\) 0 0
\(470\) −9629.40 7243.78i −0.945045 0.710916i
\(471\) −711.289 −0.0695849
\(472\) 1937.15i 0.188908i
\(473\) 2807.31i 0.272897i
\(474\) −1283.20 −0.124345
\(475\) 3298.91 + 11433.2i 0.318662 + 1.10440i
\(476\) 0 0
\(477\) 8410.34i 0.807302i
\(478\) 1971.37i 0.188637i
\(479\) −3197.61 −0.305015 −0.152508 0.988302i \(-0.548735\pi\)
−0.152508 + 0.988302i \(0.548735\pi\)
\(480\) 153.751 + 115.660i 0.0146203 + 0.0109982i
\(481\) −4002.47 −0.379411
\(482\) 703.028i 0.0664358i
\(483\) 0 0
\(484\) 4519.79 0.424473
\(485\) 1120.56 1489.60i 0.104911 0.139462i
\(486\) 2318.69 0.216415
\(487\) 7448.42i 0.693060i 0.938039 + 0.346530i \(0.112640\pi\)
−0.938039 + 0.346530i \(0.887360\pi\)
\(488\) 3524.90i 0.326977i
\(489\) −609.769 −0.0563900
\(490\) 0 0
\(491\) −20139.2 −1.85106 −0.925528 0.378679i \(-0.876378\pi\)
−0.925528 + 0.378679i \(0.876378\pi\)
\(492\) 667.071i 0.0611258i
\(493\) 8409.59i 0.768254i
\(494\) 2649.83 0.241339
\(495\) 3383.89 + 2545.56i 0.307262 + 0.231140i
\(496\) 1573.85 0.142475
\(497\) 0 0
\(498\) 434.115i 0.0390626i
\(499\) 17794.2 1.59635 0.798174 0.602427i \(-0.205799\pi\)
0.798174 + 0.602427i \(0.205799\pi\)
\(500\) 1990.37 5223.83i 0.178024 0.467234i
\(501\) 259.910 0.0231775
\(502\) 33.3029i 0.00296092i
\(503\) 16316.2i 1.44633i −0.690674 0.723166i \(-0.742686\pi\)
0.690674 0.723166i \(-0.257314\pi\)
\(504\) 0 0
\(505\) 7454.70 + 5607.85i 0.656890 + 0.494150i
\(506\) 1970.30 0.173104
\(507\) 1077.31i 0.0943687i
\(508\) 359.110i 0.0313641i
\(509\) 8586.36 0.747709 0.373854 0.927487i \(-0.378036\pi\)
0.373854 + 0.927487i \(0.378036\pi\)
\(510\) 475.525 632.131i 0.0412875 0.0548848i
\(511\) 0 0
\(512\) 512.000i 0.0441942i
\(513\) 2749.65i 0.236647i
\(514\) 14190.7 1.21775
\(515\) −9511.07 + 12643.4i −0.813802 + 1.08181i
\(516\) 425.882 0.0363341
\(517\) 7641.00i 0.650001i
\(518\) 0 0
\(519\) −693.568 −0.0586595
\(520\) −994.788 748.336i −0.0838929 0.0631091i
\(521\) −8784.84 −0.738715 −0.369358 0.929287i \(-0.620422\pi\)
−0.369358 + 0.929287i \(0.620422\pi\)
\(522\) 6829.41i 0.572634i
\(523\) 6156.83i 0.514760i 0.966310 + 0.257380i \(0.0828592\pi\)
−0.966310 + 0.257380i \(0.917141\pi\)
\(524\) 5625.86 0.469021
\(525\) 0 0
\(526\) 9886.93 0.819564
\(527\) 6470.70i 0.534854i
\(528\) 122.003i 0.0100558i
\(529\) 7339.80 0.603254
\(530\) −5626.41 4232.50i −0.461124 0.346883i
\(531\) 6467.84 0.528588
\(532\) 0 0
\(533\) 4316.03i 0.350746i
\(534\) 1166.20 0.0945063
\(535\) −10545.8 + 14018.8i −0.852211 + 1.13287i
\(536\) 6867.00 0.553375
\(537\) 1471.12i 0.118219i
\(538\) 4290.58i 0.343829i
\(539\) 0 0
\(540\) −776.526 + 1032.26i −0.0618822 + 0.0822620i
\(541\) −9921.09 −0.788431 −0.394215 0.919018i \(-0.628984\pi\)
−0.394215 + 0.919018i \(0.628984\pi\)
\(542\) 16680.0i 1.32189i
\(543\) 616.470i 0.0487206i
\(544\) 2105.03 0.165905
\(545\) −6704.50 5043.51i −0.526953 0.396404i
\(546\) 0 0
\(547\) 17559.1i 1.37253i 0.727353 + 0.686264i \(0.240751\pi\)
−0.727353 + 0.686264i \(0.759249\pi\)
\(548\) 2587.54i 0.201705i
\(549\) 11769.1 0.914924
\(550\) −3405.89 + 982.728i −0.264050 + 0.0761885i
\(551\) −12169.9 −0.940938
\(552\) 298.904i 0.0230475i
\(553\) 0 0
\(554\) 2188.10 0.167804
\(555\) −1381.75 1039.43i −0.105679 0.0794981i
\(556\) −5239.35 −0.399636
\(557\) 642.121i 0.0488466i −0.999702 0.0244233i \(-0.992225\pi\)
0.999702 0.0244233i \(-0.00777495\pi\)
\(558\) 5254.84i 0.398665i
\(559\) −2755.51 −0.208489
\(560\) 0 0
\(561\) −501.601 −0.0377497
\(562\) 11399.0i 0.855581i
\(563\) 3933.27i 0.294436i 0.989104 + 0.147218i \(0.0470320\pi\)
−0.989104 + 0.147218i \(0.952968\pi\)
\(564\) −1159.17 −0.0865426
\(565\) 13557.1 18021.9i 1.00947 1.34192i
\(566\) −12203.0 −0.906237
\(567\) 0 0
\(568\) 1137.97i 0.0840637i
\(569\) 15117.4 1.11380 0.556902 0.830578i \(-0.311990\pi\)
0.556902 + 0.830578i \(0.311990\pi\)
\(570\) 914.788 + 688.156i 0.0672215 + 0.0505678i
\(571\) −20883.6 −1.53056 −0.765282 0.643695i \(-0.777400\pi\)
−0.765282 + 0.643695i \(0.777400\pi\)
\(572\) 789.372i 0.0577015i
\(573\) 2284.30i 0.166541i
\(574\) 0 0
\(575\) 8344.35 2407.66i 0.605189 0.174620i
\(576\) −1709.49 −0.123661
\(577\) 738.564i 0.0532874i −0.999645 0.0266437i \(-0.991518\pi\)
0.999645 0.0266437i \(-0.00848196\pi\)
\(578\) 1171.38i 0.0842961i
\(579\) −2849.41 −0.204520
\(580\) 4568.78 + 3436.90i 0.327083 + 0.246051i
\(581\) 0 0
\(582\) 179.316i 0.0127713i
\(583\) 4464.60i 0.317161i
\(584\) 3677.44 0.260571
\(585\) 2498.58 3321.45i 0.176587 0.234744i
\(586\) 18391.4 1.29649
\(587\) 9348.74i 0.657349i −0.944443 0.328674i \(-0.893398\pi\)
0.944443 0.328674i \(-0.106602\pi\)
\(588\) 0 0
\(589\) 9364.07 0.655076
\(590\) −3254.94 + 4326.90i −0.227125 + 0.301925i
\(591\) 351.671 0.0244769
\(592\) 4601.31i 0.319447i
\(593\) 20470.1i 1.41755i −0.705434 0.708775i \(-0.749248\pi\)
0.705434 0.708775i \(-0.250752\pi\)
\(594\) 819.108 0.0565798
\(595\) 0 0
\(596\) 5830.16 0.400692
\(597\) 2411.85i 0.165344i
\(598\) 1933.94i 0.132249i
\(599\) −2394.09 −0.163306 −0.0816528 0.996661i \(-0.526020\pi\)
−0.0816528 + 0.996661i \(0.526020\pi\)
\(600\) −149.084 516.688i −0.0101439 0.0351562i
\(601\) 5007.10 0.339840 0.169920 0.985458i \(-0.445649\pi\)
0.169920 + 0.985458i \(0.445649\pi\)
\(602\) 0 0
\(603\) 22927.9i 1.54842i
\(604\) 7775.43 0.523804
\(605\) −10095.6 7594.49i −0.678421 0.510347i
\(606\) 897.387 0.0601549
\(607\) 15037.7i 1.00554i 0.864422 + 0.502768i \(0.167685\pi\)
−0.864422 + 0.502768i \(0.832315\pi\)
\(608\) 3046.30i 0.203197i
\(609\) 0 0
\(610\) −5922.80 + 7873.37i −0.393127 + 0.522596i
\(611\) 7500.00 0.496591
\(612\) 7028.39i 0.464225i
\(613\) 29396.1i 1.93687i −0.249276 0.968433i \(-0.580193\pi\)
0.249276 0.968433i \(-0.419807\pi\)
\(614\) −15777.1 −1.03699
\(615\) 1120.86 1490.00i 0.0734919 0.0976953i
\(616\) 0 0
\(617\) 14648.4i 0.955788i −0.878417 0.477894i \(-0.841400\pi\)
0.878417 0.477894i \(-0.158600\pi\)
\(618\) 1521.99i 0.0990673i
\(619\) −8336.14 −0.541289 −0.270644 0.962679i \(-0.587237\pi\)
−0.270644 + 0.962679i \(0.587237\pi\)
\(620\) −3515.42 2644.50i −0.227714 0.171299i
\(621\) −2006.80 −0.129678
\(622\) 3491.10i 0.225049i
\(623\) 0 0
\(624\) −119.751 −0.00768251
\(625\) −13223.3 + 8323.82i −0.846289 + 0.532725i
\(626\) −15936.8 −1.01751
\(627\) 725.891i 0.0462349i
\(628\) 5290.69i 0.336181i
\(629\) −18917.8 −1.19921
\(630\) 0 0
\(631\) 5599.86 0.353292 0.176646 0.984274i \(-0.443475\pi\)
0.176646 + 0.984274i \(0.443475\pi\)
\(632\) 9544.69i 0.600739i
\(633\) 1161.65i 0.0729407i
\(634\) −7174.77 −0.449442
\(635\) 603.404 802.125i 0.0377092 0.0501282i
\(636\) −677.299 −0.0422275
\(637\) 0 0
\(638\) 3625.36i 0.224968i
\(639\) 3799.51 0.235221
\(640\) 860.301 1143.63i 0.0531350 0.0706341i
\(641\) 18625.8 1.14770 0.573850 0.818960i \(-0.305449\pi\)
0.573850 + 0.818960i \(0.305449\pi\)
\(642\) 1687.57i 0.103743i
\(643\) 9741.95i 0.597489i 0.954333 + 0.298744i \(0.0965678\pi\)
−0.954333 + 0.298744i \(0.903432\pi\)
\(644\) 0 0
\(645\) −951.269 715.599i −0.0580716 0.0436848i
\(646\) 12524.5 0.762803
\(647\) 15071.8i 0.915820i −0.888999 0.457910i \(-0.848598\pi\)
0.888999 0.457910i \(-0.151402\pi\)
\(648\) 5645.27i 0.342233i
\(649\) 3433.43 0.207664
\(650\) 964.593 + 3343.04i 0.0582068 + 0.201730i
\(651\) 0 0
\(652\) 4535.56i 0.272433i
\(653\) 22142.9i 1.32698i 0.748186 + 0.663489i \(0.230925\pi\)
−0.748186 + 0.663489i \(0.769075\pi\)
\(654\) −807.079 −0.0482558
\(655\) −12566.2 9453.00i −0.749621 0.563907i
\(656\) 4961.79 0.295313
\(657\) 12278.4i 0.729113i
\(658\) 0 0
\(659\) 12874.3 0.761022 0.380511 0.924776i \(-0.375748\pi\)
0.380511 + 0.924776i \(0.375748\pi\)
\(660\) −204.998 + 272.511i −0.0120902 + 0.0160719i
\(661\) 13184.1 0.775799 0.387899 0.921702i \(-0.373201\pi\)
0.387899 + 0.921702i \(0.373201\pi\)
\(662\) 2884.94i 0.169375i
\(663\) 492.344i 0.0288402i
\(664\) −3229.02 −0.188720
\(665\) 0 0
\(666\) 15363.1 0.893856
\(667\) 8882.06i 0.515615i
\(668\) 1933.25i 0.111976i
\(669\) −3240.82 −0.187291
\(670\) −15338.4 11538.5i −0.884442 0.665327i
\(671\) 6247.58 0.359441
\(672\) 0 0
\(673\) 5699.50i 0.326448i 0.986589 + 0.163224i \(0.0521893\pi\)
−0.986589 + 0.163224i \(0.947811\pi\)
\(674\) −11227.0 −0.641615
\(675\) 3468.97 1000.93i 0.197808 0.0570753i
\(676\) −8013.20 −0.455917
\(677\) 23186.5i 1.31629i −0.752889 0.658147i \(-0.771340\pi\)
0.752889 0.658147i \(-0.228660\pi\)
\(678\) 2169.45i 0.122887i
\(679\) 0 0
\(680\) −4701.90 3537.04i −0.265161 0.199469i
\(681\) −2505.48 −0.140984
\(682\) 2789.51i 0.156621i
\(683\) 9258.04i 0.518666i 0.965788 + 0.259333i \(0.0835027\pi\)
−0.965788 + 0.259333i \(0.916497\pi\)
\(684\) −10171.1 −0.568572
\(685\) −4347.79 + 5779.66i −0.242512 + 0.322379i
\(686\) 0 0
\(687\) 191.445i 0.0106318i
\(688\) 3167.78i 0.175539i
\(689\) 4382.21 0.242306
\(690\) 502.241 667.646i 0.0277101 0.0368360i
\(691\) −30149.5 −1.65983 −0.829913 0.557892i \(-0.811610\pi\)
−0.829913 + 0.557892i \(0.811610\pi\)
\(692\) 5158.88i 0.283398i
\(693\) 0 0
\(694\) −22435.8 −1.22716
\(695\) 11702.8 + 8803.55i 0.638726 + 0.480486i
\(696\) 549.984 0.0299527
\(697\) 20399.8i 1.10861i
\(698\) 10925.8i 0.592476i
\(699\) 1779.34 0.0962817
\(700\) 0 0
\(701\) −9299.00 −0.501025 −0.250512 0.968113i \(-0.580599\pi\)
−0.250512 + 0.968113i \(0.580599\pi\)
\(702\) 803.992i 0.0432261i
\(703\) 27376.9i 1.46876i
\(704\) −907.476 −0.0485821
\(705\) 2589.19 + 1947.73i 0.138318 + 0.104051i
\(706\) −2868.69 −0.152924
\(707\) 0 0
\(708\) 520.867i 0.0276488i
\(709\) −26855.2 −1.42252 −0.711262 0.702927i \(-0.751876\pi\)
−0.711262 + 0.702927i \(0.751876\pi\)
\(710\) −1912.10 + 2541.82i −0.101070 + 0.134356i
\(711\) −31868.3 −1.68095
\(712\) 8674.39i 0.456582i
\(713\) 6834.24i 0.358968i
\(714\) 0 0
\(715\) 1326.36 1763.18i 0.0693750 0.0922225i
\(716\) 10942.4 0.571143
\(717\) 530.068i 0.0276091i
\(718\) 348.967i 0.0181384i
\(719\) −29252.7 −1.51730 −0.758651 0.651498i \(-0.774141\pi\)
−0.758651 + 0.651498i \(0.774141\pi\)
\(720\) 3818.40 + 2872.42i 0.197644 + 0.148679i
\(721\) 0 0
\(722\) 4406.86i 0.227155i
\(723\) 189.033i 0.00972365i
\(724\) −4585.41 −0.235380
\(725\) −4430.11 15353.6i −0.226938 0.786509i
\(726\) −1215.30 −0.0621265
\(727\) 2271.26i 0.115869i −0.998320 0.0579343i \(-0.981549\pi\)
0.998320 0.0579343i \(-0.0184514\pi\)
\(728\) 0 0
\(729\) 18429.3 0.936307
\(730\) −8214.10 6179.11i −0.416462 0.313287i
\(731\) −13024.0 −0.658974
\(732\) 947.787i 0.0478568i
\(733\) 7665.14i 0.386246i −0.981175 0.193123i \(-0.938138\pi\)
0.981175 0.193123i \(-0.0618617\pi\)
\(734\) −24517.0 −1.23289
\(735\) 0 0
\(736\) 2223.30 0.111348
\(737\) 12171.2i 0.608319i
\(738\) 16566.7i 0.826324i
\(739\) 4464.70 0.222242 0.111121 0.993807i \(-0.464556\pi\)
0.111121 + 0.993807i \(0.464556\pi\)
\(740\) −7731.48 + 10277.7i −0.384074 + 0.510562i
\(741\) −712.496 −0.0353228
\(742\) 0 0
\(743\) 26438.3i 1.30542i −0.757608 0.652710i \(-0.773632\pi\)
0.757608 0.652710i \(-0.226368\pi\)
\(744\) −423.181 −0.0208529
\(745\) −13022.5 9796.28i −0.640413 0.481755i
\(746\) 11842.6 0.581217
\(747\) 10781.2i 0.528065i
\(748\) 3730.99i 0.182378i
\(749\) 0 0
\(750\) −535.177 + 1404.60i −0.0260559 + 0.0683851i
\(751\) −30714.1 −1.49238 −0.746188 0.665735i \(-0.768118\pi\)
−0.746188 + 0.665735i \(0.768118\pi\)
\(752\) 8622.13i 0.418108i
\(753\) 8.95459i 0.000433364i
\(754\) −3558.46 −0.171872
\(755\) −17367.6 13064.9i −0.837179 0.629774i
\(756\) 0 0
\(757\) 7199.44i 0.345665i 0.984951 + 0.172832i \(0.0552919\pi\)
−0.984951 + 0.172832i \(0.944708\pi\)
\(758\) 12644.4i 0.605889i
\(759\) −529.782 −0.0253358
\(760\) 5118.62 6804.35i 0.244305 0.324763i
\(761\) −16957.6 −0.807768 −0.403884 0.914810i \(-0.632340\pi\)
−0.403884 + 0.914810i \(0.632340\pi\)
\(762\) 96.5588i 0.00459049i
\(763\) 0 0
\(764\) −16991.0 −0.804597
\(765\) 11809.6 15698.9i 0.558141 0.741956i
\(766\) −23994.0 −1.13177
\(767\) 3370.07i 0.158652i
\(768\) 137.668i 0.00646833i
\(769\) 9879.39 0.463277 0.231638 0.972802i \(-0.425591\pi\)
0.231638 + 0.972802i \(0.425591\pi\)
\(770\) 0 0
\(771\) −3815.64 −0.178232
\(772\) 21194.4i 0.988086i
\(773\) 21797.7i 1.01424i −0.861875 0.507121i \(-0.830710\pi\)
0.861875 0.507121i \(-0.169290\pi\)
\(774\) 10576.7 0.491180
\(775\) 3408.71 + 11813.7i 0.157993 + 0.547564i
\(776\) −1333.78 −0.0617010
\(777\) 0 0
\(778\) 29273.7i 1.34899i
\(779\) 29521.6 1.35779
\(780\) 267.482 + 201.215i 0.0122787 + 0.00923674i
\(781\) 2016.96 0.0924102
\(782\) 9140.85i 0.418000i
\(783\) 3692.51i 0.168531i
\(784\) 0 0
\(785\) 8889.82 11817.5i 0.404193 0.537306i
\(786\) −1512.70 −0.0686467
\(787\) 5954.44i 0.269699i 0.990866 + 0.134849i \(0.0430550\pi\)
−0.990866 + 0.134849i \(0.956945\pi\)
\(788\) 2615.79i 0.118253i
\(789\) −2658.43 −0.119953
\(790\) 16037.7 21319.5i 0.722273 0.960142i
\(791\) 0 0
\(792\) 3029.93i 0.135939i
\(793\) 6132.29i 0.274608i
\(794\) −12597.0 −0.563037
\(795\) 1512.85 + 1138.05i 0.0674908 + 0.0507704i
\(796\) 17939.8 0.798817
\(797\) 30426.4i 1.35227i −0.736779 0.676134i \(-0.763654\pi\)
0.736779 0.676134i \(-0.236346\pi\)
\(798\) 0 0
\(799\) 35449.0 1.56958
\(800\) −3843.22 + 1108.91i −0.169848 + 0.0490075i
\(801\) 28962.5 1.27758
\(802\) 16579.9i 0.729994i
\(803\) 6517.95i 0.286443i
\(804\) −1846.42 −0.0809929
\(805\) 0 0
\(806\) 2738.03 0.119656
\(807\) 1153.67i 0.0503234i
\(808\) 6674.91i 0.290622i
\(809\) −12450.3 −0.541074 −0.270537 0.962710i \(-0.587201\pi\)
−0.270537 + 0.962710i \(0.587201\pi\)
\(810\) −9485.61 + 12609.5i −0.411470 + 0.546980i
\(811\) 3330.98 0.144225 0.0721126 0.997397i \(-0.477026\pi\)
0.0721126 + 0.997397i \(0.477026\pi\)
\(812\) 0 0
\(813\) 4484.97i 0.193474i
\(814\) 8155.44 0.351165
\(815\) 7620.99 10130.8i 0.327548 0.435421i
\(816\) −566.008 −0.0242822
\(817\) 18847.7i 0.807094i
\(818\) 19044.1i 0.814012i
\(819\) 0 0
\(820\) −11082.9 8337.17i −0.471989 0.355057i
\(821\) 4257.43 0.180981 0.0904905 0.995897i \(-0.471157\pi\)
0.0904905 + 0.995897i \(0.471157\pi\)
\(822\) 695.748i 0.0295219i
\(823\) 9845.13i 0.416986i 0.978024 + 0.208493i \(0.0668559\pi\)
−0.978024 + 0.208493i \(0.933144\pi\)
\(824\) 11320.8 0.478617
\(825\) 915.786 264.239i 0.0386468 0.0111511i
\(826\) 0 0
\(827\) 31609.5i 1.32911i −0.747242 0.664553i \(-0.768622\pi\)
0.747242 0.664553i \(-0.231378\pi\)
\(828\) 7423.26i 0.311565i
\(829\) 448.283 0.0187811 0.00939054 0.999956i \(-0.497011\pi\)
0.00939054 + 0.999956i \(0.497011\pi\)
\(830\) 7212.49 + 5425.65i 0.301626 + 0.226900i
\(831\) −588.344 −0.0245601
\(832\) 890.730i 0.0371160i
\(833\) 0 0
\(834\) 1408.77 0.0584914
\(835\) −3248.40 + 4318.21i −0.134629 + 0.178967i
\(836\) −5399.30 −0.223372
\(837\) 2841.18i 0.117330i
\(838\) 17873.0i 0.736770i
\(839\) 14956.1 0.615424 0.307712 0.951480i \(-0.400437\pi\)
0.307712 + 0.951480i \(0.400437\pi\)
\(840\) 0 0
\(841\) −8045.98 −0.329902
\(842\) 21483.6i 0.879302i
\(843\) 3064.99i 0.125224i
\(844\) −8640.55 −0.352394
\(845\) 17898.6 + 13464.4i 0.728677 + 0.548152i
\(846\) −28788.0 −1.16992
\(847\) 0 0
\(848\) 5037.87i 0.204011i
\(849\) 3281.18 0.132638
\(850\) 4559.18 + 15801.0i 0.183975 + 0.637610i
\(851\) −19980.7 −0.804851
\(852\) 305.981i 0.0123037i
\(853\) 45363.9i 1.82090i 0.413615 + 0.910452i \(0.364266\pi\)
−0.413615 + 0.910452i \(0.635734\pi\)
\(854\) 0 0
\(855\) 22718.7 + 17090.3i 0.908729 + 0.683598i
\(856\) 12552.4 0.501207
\(857\) 1714.18i 0.0683258i −0.999416 0.0341629i \(-0.989123\pi\)
0.999416 0.0341629i \(-0.0108765\pi\)
\(858\) 212.249i 0.00844529i
\(859\) 34820.1 1.38306 0.691529 0.722348i \(-0.256937\pi\)
0.691529 + 0.722348i \(0.256937\pi\)
\(860\) −5322.74 + 7075.70i −0.211051 + 0.280557i
\(861\) 0 0
\(862\) 15773.0i 0.623237i
\(863\) 7309.71i 0.288326i 0.989554 + 0.144163i \(0.0460490\pi\)
−0.989554 + 0.144163i \(0.953951\pi\)
\(864\) 924.284 0.0363944
\(865\) 8668.34 11523.1i 0.340731 0.452945i
\(866\) 7512.24 0.294776
\(867\) 314.966i 0.0123377i
\(868\) 0 0
\(869\) −16917.1 −0.660385
\(870\) −1228.47 924.125i −0.0478724 0.0360124i
\(871\) 11946.6 0.464746
\(872\) 6003.19i 0.233135i
\(873\) 4453.30i 0.172647i
\(874\) 13228.2 0.511956
\(875\) 0 0
\(876\) −988.803 −0.0381376
\(877\) 2992.90i 0.115237i 0.998339 + 0.0576186i \(0.0183507\pi\)
−0.998339 + 0.0576186i \(0.981649\pi\)
\(878\) 13841.3i 0.532029i
\(879\) −4945.15 −0.189756
\(880\) 2026.98 + 1524.81i 0.0776472 + 0.0584106i
\(881\) 28667.6 1.09629 0.548147 0.836382i \(-0.315333\pi\)
0.548147 + 0.836382i \(0.315333\pi\)
\(882\) 0 0
\(883\) 23488.0i 0.895169i 0.894242 + 0.447585i \(0.147716\pi\)
−0.894242 + 0.447585i \(0.852284\pi\)
\(884\) 3662.14 0.139334
\(885\) 875.199 1163.43i 0.0332424 0.0441902i
\(886\) 23237.7 0.881136
\(887\) 22944.5i 0.868546i 0.900781 + 0.434273i \(0.142995\pi\)
−0.900781 + 0.434273i \(0.857005\pi\)
\(888\) 1237.22i 0.0467548i
\(889\) 0 0
\(890\) −14575.4 + 19375.5i −0.548952 + 0.729740i
\(891\) 10005.8 0.376213
\(892\) 24105.8i 0.904845i
\(893\) 51300.0i 1.92238i
\(894\) −1567.63 −0.0586460
\(895\) −24441.5 18386.3i −0.912839 0.686689i
\(896\) 0 0
\(897\) 520.005i 0.0193562i
\(898\) 35431.3i 1.31666i
\(899\) −12575.0 −0.466519
\(900\) −3702.50 12831.9i −0.137130 0.475256i
\(901\) 20712.7 0.765858
\(902\) 8794.34i 0.324634i
\(903\) 0 0
\(904\) −16136.7 −0.593695
\(905\) 10242.2 + 7704.75i 0.376201 + 0.283000i
\(906\) −2090.68 −0.0766648
\(907\) 14063.6i 0.514855i 0.966298 + 0.257428i \(0.0828749\pi\)
−0.966298 + 0.257428i \(0.917125\pi\)
\(908\) 18636.2i 0.681127i
\(909\) 22286.5 0.813199
\(910\) 0 0
\(911\) −1591.22 −0.0578700 −0.0289350 0.999581i \(-0.509212\pi\)
−0.0289350 + 0.999581i \(0.509212\pi\)
\(912\) 819.099i 0.0297402i
\(913\) 5723.17i 0.207458i
\(914\) −5449.26 −0.197205
\(915\) 1592.54 2117.02i 0.0575386 0.0764880i
\(916\) −1424.00 −0.0513649
\(917\) 0 0
\(918\) 3800.10i 0.136625i
\(919\) −33319.1 −1.19597 −0.597984 0.801508i \(-0.704031\pi\)
−0.597984 + 0.801508i \(0.704031\pi\)
\(920\) −4966.06 3735.75i −0.177963 0.133874i
\(921\) 4242.21 0.151776
\(922\) 22216.7i 0.793565i
\(923\) 1979.74i 0.0706000i
\(924\) 0 0
\(925\) 34538.8 9965.75i 1.22771 0.354240i
\(926\) 29281.4 1.03914
\(927\) 37798.6i 1.33923i
\(928\) 4090.87i 0.144709i
\(929\) 44409.8 1.56839 0.784197 0.620512i \(-0.213075\pi\)
0.784197 + 0.620512i \(0.213075\pi\)
\(930\) 945.237 + 711.061i 0.0333285 + 0.0250716i
\(931\) 0 0
\(932\) 13235.0i 0.465159i
\(933\) 938.698i 0.0329385i
\(934\) −2485.51 −0.0870754
\(935\) 6269.09 8333.71i 0.219274 0.291488i
\(936\) −2974.02 −0.103855
\(937\) 25069.7i 0.874055i −0.899448 0.437028i \(-0.856031\pi\)
0.899448 0.437028i \(-0.143969\pi\)
\(938\) 0 0
\(939\) 4285.15 0.148925
\(940\) 14487.6 19258.8i 0.502694 0.668247i
\(941\) 27867.4 0.965409 0.482704 0.875783i \(-0.339655\pi\)
0.482704 + 0.875783i \(0.339655\pi\)
\(942\) 1422.58i 0.0492039i
\(943\) 21546.0i 0.744043i
\(944\) 3874.29 0.133578
\(945\) 0 0
\(946\) 5614.62 0.192967
\(947\) 31947.8i 1.09627i −0.836391 0.548133i \(-0.815339\pi\)
0.836391 0.548133i \(-0.184661\pi\)
\(948\) 2566.41i 0.0879252i
\(949\) 6397.67 0.218838
\(950\) −22866.4 + 6597.82i −0.780929 + 0.225328i
\(951\) 1929.18 0.0657811
\(952\) 0 0
\(953\) 13026.8i 0.442792i 0.975184 + 0.221396i \(0.0710613\pi\)
−0.975184 + 0.221396i \(0.928939\pi\)
\(954\) −16820.7 −0.570849
\(955\) 37951.8 + 28549.5i 1.28596 + 0.967372i
\(956\) 3942.74 0.133386
\(957\) 974.800i 0.0329266i
\(958\) 6395.21i 0.215678i
\(959\) 0 0
\(960\) −231.321 + 307.502i −0.00777692 + 0.0103381i
\(961\) −20115.2 −0.675212
\(962\) 8004.94i 0.268284i
\(963\) 41910.7i 1.40244i
\(964\) −1406.06 −0.0469772
\(965\) 35612.4 47340.7i 1.18798 1.57923i
\(966\) 0 0
\(967\) 25825.8i 0.858844i −0.903104 0.429422i \(-0.858717\pi\)
0.903104 0.429422i \(-0.141283\pi\)
\(968\) 9039.58i 0.300148i
\(969\) −3367.64 −0.111645
\(970\) 2979.19 + 2241.12i 0.0986146 + 0.0741835i
\(971\) −27715.9 −0.916011 −0.458005 0.888949i \(-0.651436\pi\)
−0.458005 + 0.888949i \(0.651436\pi\)
\(972\) 4637.38i 0.153029i
\(973\) 0 0
\(974\) −14896.8 −0.490067
\(975\) −259.363 898.887i −0.00851925 0.0295256i
\(976\) 7049.80 0.231207
\(977\) 16522.4i 0.541043i 0.962714 + 0.270521i \(0.0871961\pi\)
−0.962714 + 0.270521i \(0.912804\pi\)
\(978\) 1219.54i 0.0398737i
\(979\) 15374.6 0.501915
\(980\) 0 0
\(981\) −20043.8 −0.652342
\(982\) 40278.4i 1.30889i
\(983\) 16046.6i 0.520659i −0.965520 0.260330i \(-0.916169\pi\)
0.965520 0.260330i \(-0.0838312\pi\)
\(984\) −1334.14 −0.0432224
\(985\) −4395.25 + 5842.75i −0.142177 + 0.189001i
\(986\) −16819.2 −0.543237
\(987\) 0 0
\(988\) 5299.67i 0.170653i
\(989\) −13755.7 −0.442271
\(990\) −5091.11 + 6767.79i −0.163441 + 0.217267i
\(991\) 7725.20 0.247628 0.123814 0.992305i \(-0.460487\pi\)
0.123814 + 0.992305i \(0.460487\pi\)
\(992\) 3147.69i 0.100745i
\(993\) 775.711i 0.0247900i
\(994\) 0 0
\(995\) −40071.1 30143.8i −1.27672 0.960423i
\(996\) 868.231 0.0276214
\(997\) 11167.4i 0.354738i −0.984144 0.177369i \(-0.943241\pi\)
0.984144 0.177369i \(-0.0567586\pi\)
\(998\) 35588.4i 1.12879i
\(999\) −8306.49 −0.263069
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 490.4.c.g.99.15 yes 20
5.2 odd 4 2450.4.a.db.1.5 10
5.3 odd 4 2450.4.a.dc.1.6 10
5.4 even 2 inner 490.4.c.g.99.6 yes 20
7.6 odd 2 inner 490.4.c.g.99.16 yes 20
35.13 even 4 2450.4.a.dc.1.5 10
35.27 even 4 2450.4.a.db.1.6 10
35.34 odd 2 inner 490.4.c.g.99.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.4.c.g.99.5 20 35.34 odd 2 inner
490.4.c.g.99.6 yes 20 5.4 even 2 inner
490.4.c.g.99.15 yes 20 1.1 even 1 trivial
490.4.c.g.99.16 yes 20 7.6 odd 2 inner
2450.4.a.db.1.5 10 5.2 odd 4
2450.4.a.db.1.6 10 35.27 even 4
2450.4.a.dc.1.5 10 35.13 even 4
2450.4.a.dc.1.6 10 5.3 odd 4