Properties

Label 825.4.a.s.1.3
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 20x + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.26150\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.26150 q^{2} +3.00000 q^{3} +19.6833 q^{4} +15.7845 q^{6} +10.3207 q^{7} +61.4719 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.26150 q^{2} +3.00000 q^{3} +19.6833 q^{4} +15.7845 q^{6} +10.3207 q^{7} +61.4719 q^{8} +9.00000 q^{9} +11.0000 q^{11} +59.0500 q^{12} -63.9817 q^{13} +54.3024 q^{14} +165.967 q^{16} -17.1461 q^{17} +47.3535 q^{18} +90.2104 q^{19} +30.9621 q^{21} +57.8765 q^{22} +212.605 q^{23} +184.416 q^{24} -336.639 q^{26} +27.0000 q^{27} +203.146 q^{28} +57.5461 q^{29} -141.704 q^{31} +381.462 q^{32} +33.0000 q^{33} -90.2140 q^{34} +177.150 q^{36} +257.963 q^{37} +474.642 q^{38} -191.945 q^{39} -225.914 q^{41} +162.907 q^{42} +347.445 q^{43} +216.517 q^{44} +1118.62 q^{46} -404.364 q^{47} +497.902 q^{48} -236.483 q^{49} -51.4382 q^{51} -1259.37 q^{52} -259.568 q^{53} +142.060 q^{54} +634.433 q^{56} +270.631 q^{57} +302.779 q^{58} -853.067 q^{59} -203.699 q^{61} -745.573 q^{62} +92.8864 q^{63} +679.320 q^{64} +173.629 q^{66} -266.890 q^{67} -337.492 q^{68} +637.814 q^{69} +92.4460 q^{71} +553.247 q^{72} +242.026 q^{73} +1357.27 q^{74} +1775.64 q^{76} +113.528 q^{77} -1009.92 q^{78} -1021.60 q^{79} +81.0000 q^{81} -1188.65 q^{82} -706.415 q^{83} +609.438 q^{84} +1828.08 q^{86} +172.638 q^{87} +676.191 q^{88} -440.218 q^{89} -660.336 q^{91} +4184.77 q^{92} -425.111 q^{93} -2127.56 q^{94} +1144.38 q^{96} +197.761 q^{97} -1244.25 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 9 q^{3} + 22 q^{4} + 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{2} + 9 q^{3} + 22 q^{4} + 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} + 33 q^{11} + 66 q^{12} - 56 q^{14} + 50 q^{16} + 218 q^{17} + 36 q^{18} + 146 q^{19} + 12 q^{21} + 44 q^{22} + 200 q^{23} + 144 q^{24} - 508 q^{26} + 81 q^{27} + 340 q^{28} + 68 q^{29} - 68 q^{31} + 688 q^{32} + 99 q^{33} - 176 q^{34} + 198 q^{36} + 390 q^{37} + 316 q^{38} - 196 q^{41} - 168 q^{42} + 524 q^{43} + 242 q^{44} + 1160 q^{46} + 60 q^{47} + 150 q^{48} - 157 q^{49} + 654 q^{51} - 1020 q^{52} + 158 q^{53} + 108 q^{54} + 1368 q^{56} + 438 q^{57} - 1092 q^{58} - 1044 q^{59} + 642 q^{61} - 88 q^{62} + 36 q^{63} + 1166 q^{64} + 132 q^{66} + 236 q^{67} - 144 q^{68} + 600 q^{69} - 544 q^{71} + 432 q^{72} - 900 q^{73} + 1536 q^{74} + 1996 q^{76} + 44 q^{77} - 1524 q^{78} - 1586 q^{79} + 243 q^{81} - 380 q^{82} + 1582 q^{83} + 1020 q^{84} + 3568 q^{86} + 204 q^{87} + 528 q^{88} - 2122 q^{89} - 8 q^{91} + 4128 q^{92} - 204 q^{93} - 2152 q^{94} + 2064 q^{96} - 618 q^{97} - 572 q^{98} + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.26150 1.86022 0.930110 0.367281i \(-0.119711\pi\)
0.930110 + 0.367281i \(0.119711\pi\)
\(3\) 3.00000 0.577350
\(4\) 19.6833 2.46042
\(5\) 0 0
\(6\) 15.7845 1.07400
\(7\) 10.3207 0.557266 0.278633 0.960398i \(-0.410119\pi\)
0.278633 + 0.960398i \(0.410119\pi\)
\(8\) 61.4719 2.71670
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 59.0500 1.42052
\(13\) −63.9817 −1.36502 −0.682512 0.730874i \(-0.739113\pi\)
−0.682512 + 0.730874i \(0.739113\pi\)
\(14\) 54.3024 1.03664
\(15\) 0 0
\(16\) 165.967 2.59324
\(17\) −17.1461 −0.244620 −0.122310 0.992492i \(-0.539030\pi\)
−0.122310 + 0.992492i \(0.539030\pi\)
\(18\) 47.3535 0.620073
\(19\) 90.2104 1.08925 0.544623 0.838681i \(-0.316673\pi\)
0.544623 + 0.838681i \(0.316673\pi\)
\(20\) 0 0
\(21\) 30.9621 0.321738
\(22\) 57.8765 0.560877
\(23\) 212.605 1.92744 0.963721 0.266913i \(-0.0860036\pi\)
0.963721 + 0.266913i \(0.0860036\pi\)
\(24\) 184.416 1.56849
\(25\) 0 0
\(26\) −336.639 −2.53925
\(27\) 27.0000 0.192450
\(28\) 203.146 1.37111
\(29\) 57.5461 0.368484 0.184242 0.982881i \(-0.441017\pi\)
0.184242 + 0.982881i \(0.441017\pi\)
\(30\) 0 0
\(31\) −141.704 −0.820991 −0.410496 0.911863i \(-0.634644\pi\)
−0.410496 + 0.911863i \(0.634644\pi\)
\(32\) 381.462 2.10730
\(33\) 33.0000 0.174078
\(34\) −90.2140 −0.455046
\(35\) 0 0
\(36\) 177.150 0.820139
\(37\) 257.963 1.14619 0.573093 0.819490i \(-0.305743\pi\)
0.573093 + 0.819490i \(0.305743\pi\)
\(38\) 474.642 2.02624
\(39\) −191.945 −0.788097
\(40\) 0 0
\(41\) −225.914 −0.860533 −0.430266 0.902702i \(-0.641580\pi\)
−0.430266 + 0.902702i \(0.641580\pi\)
\(42\) 162.907 0.598503
\(43\) 347.445 1.23221 0.616103 0.787666i \(-0.288711\pi\)
0.616103 + 0.787666i \(0.288711\pi\)
\(44\) 216.517 0.741844
\(45\) 0 0
\(46\) 1118.62 3.58547
\(47\) −404.364 −1.25495 −0.627473 0.778638i \(-0.715911\pi\)
−0.627473 + 0.778638i \(0.715911\pi\)
\(48\) 497.902 1.49721
\(49\) −236.483 −0.689455
\(50\) 0 0
\(51\) −51.4382 −0.141231
\(52\) −1259.37 −3.35853
\(53\) −259.568 −0.672726 −0.336363 0.941732i \(-0.609197\pi\)
−0.336363 + 0.941732i \(0.609197\pi\)
\(54\) 142.060 0.358000
\(55\) 0 0
\(56\) 634.433 1.51392
\(57\) 270.631 0.628877
\(58\) 302.779 0.685462
\(59\) −853.067 −1.88237 −0.941185 0.337891i \(-0.890287\pi\)
−0.941185 + 0.337891i \(0.890287\pi\)
\(60\) 0 0
\(61\) −203.699 −0.427558 −0.213779 0.976882i \(-0.568577\pi\)
−0.213779 + 0.976882i \(0.568577\pi\)
\(62\) −745.573 −1.52722
\(63\) 92.8864 0.185755
\(64\) 679.320 1.32680
\(65\) 0 0
\(66\) 173.629 0.323823
\(67\) −266.890 −0.486653 −0.243327 0.969944i \(-0.578239\pi\)
−0.243327 + 0.969944i \(0.578239\pi\)
\(68\) −337.492 −0.601866
\(69\) 637.814 1.11281
\(70\) 0 0
\(71\) 92.4460 0.154526 0.0772629 0.997011i \(-0.475382\pi\)
0.0772629 + 0.997011i \(0.475382\pi\)
\(72\) 553.247 0.905566
\(73\) 242.026 0.388040 0.194020 0.980998i \(-0.437847\pi\)
0.194020 + 0.980998i \(0.437847\pi\)
\(74\) 1357.27 2.13216
\(75\) 0 0
\(76\) 1775.64 2.68000
\(77\) 113.528 0.168022
\(78\) −1009.92 −1.46603
\(79\) −1021.60 −1.45492 −0.727460 0.686150i \(-0.759299\pi\)
−0.727460 + 0.686150i \(0.759299\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −1188.65 −1.60078
\(83\) −706.415 −0.934206 −0.467103 0.884203i \(-0.654702\pi\)
−0.467103 + 0.884203i \(0.654702\pi\)
\(84\) 609.438 0.791609
\(85\) 0 0
\(86\) 1828.08 2.29217
\(87\) 172.638 0.212745
\(88\) 676.191 0.819116
\(89\) −440.218 −0.524304 −0.262152 0.965027i \(-0.584432\pi\)
−0.262152 + 0.965027i \(0.584432\pi\)
\(90\) 0 0
\(91\) −660.336 −0.760682
\(92\) 4184.77 4.74231
\(93\) −425.111 −0.473999
\(94\) −2127.56 −2.33448
\(95\) 0 0
\(96\) 1144.38 1.21665
\(97\) 197.761 0.207006 0.103503 0.994629i \(-0.466995\pi\)
0.103503 + 0.994629i \(0.466995\pi\)
\(98\) −1244.25 −1.28254
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 1400.62 1.37987 0.689937 0.723870i \(-0.257638\pi\)
0.689937 + 0.723870i \(0.257638\pi\)
\(102\) −270.642 −0.262721
\(103\) 1345.70 1.28734 0.643669 0.765304i \(-0.277411\pi\)
0.643669 + 0.765304i \(0.277411\pi\)
\(104\) −3933.07 −3.70836
\(105\) 0 0
\(106\) −1365.72 −1.25142
\(107\) 889.178 0.803366 0.401683 0.915779i \(-0.368425\pi\)
0.401683 + 0.915779i \(0.368425\pi\)
\(108\) 531.450 0.473508
\(109\) 1256.29 1.10395 0.551974 0.833861i \(-0.313875\pi\)
0.551974 + 0.833861i \(0.313875\pi\)
\(110\) 0 0
\(111\) 773.890 0.661751
\(112\) 1712.90 1.44512
\(113\) 2394.01 1.99301 0.996504 0.0835448i \(-0.0266242\pi\)
0.996504 + 0.0835448i \(0.0266242\pi\)
\(114\) 1423.93 1.16985
\(115\) 0 0
\(116\) 1132.70 0.906626
\(117\) −575.835 −0.455008
\(118\) −4488.41 −3.50162
\(119\) −176.960 −0.136318
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −1071.76 −0.795352
\(123\) −677.742 −0.496829
\(124\) −2789.20 −2.01998
\(125\) 0 0
\(126\) 488.721 0.345546
\(127\) −2065.57 −1.44322 −0.721612 0.692298i \(-0.756598\pi\)
−0.721612 + 0.692298i \(0.756598\pi\)
\(128\) 522.548 0.360837
\(129\) 1042.33 0.711414
\(130\) 0 0
\(131\) 785.526 0.523907 0.261953 0.965081i \(-0.415633\pi\)
0.261953 + 0.965081i \(0.415633\pi\)
\(132\) 649.550 0.428304
\(133\) 931.035 0.607000
\(134\) −1404.24 −0.905282
\(135\) 0 0
\(136\) −1054.00 −0.664558
\(137\) −1276.24 −0.795885 −0.397942 0.917410i \(-0.630276\pi\)
−0.397942 + 0.917410i \(0.630276\pi\)
\(138\) 3355.86 2.07007
\(139\) −2703.21 −1.64952 −0.824760 0.565482i \(-0.808690\pi\)
−0.824760 + 0.565482i \(0.808690\pi\)
\(140\) 0 0
\(141\) −1213.09 −0.724544
\(142\) 486.405 0.287452
\(143\) −703.798 −0.411570
\(144\) 1493.71 0.864413
\(145\) 0 0
\(146\) 1273.42 0.721840
\(147\) −709.449 −0.398057
\(148\) 5077.58 2.82010
\(149\) −2400.99 −1.32011 −0.660056 0.751217i \(-0.729467\pi\)
−0.660056 + 0.751217i \(0.729467\pi\)
\(150\) 0 0
\(151\) −2517.30 −1.35665 −0.678326 0.734761i \(-0.737294\pi\)
−0.678326 + 0.734761i \(0.737294\pi\)
\(152\) 5545.40 2.95916
\(153\) −154.315 −0.0815399
\(154\) 597.326 0.312558
\(155\) 0 0
\(156\) −3778.12 −1.93905
\(157\) −1391.42 −0.707310 −0.353655 0.935376i \(-0.615061\pi\)
−0.353655 + 0.935376i \(0.615061\pi\)
\(158\) −5375.13 −2.70647
\(159\) −778.705 −0.388398
\(160\) 0 0
\(161\) 2194.23 1.07410
\(162\) 426.181 0.206691
\(163\) 2720.53 1.30729 0.653644 0.756802i \(-0.273239\pi\)
0.653644 + 0.756802i \(0.273239\pi\)
\(164\) −4446.74 −2.11727
\(165\) 0 0
\(166\) −3716.80 −1.73783
\(167\) −2950.25 −1.36705 −0.683525 0.729927i \(-0.739554\pi\)
−0.683525 + 0.729927i \(0.739554\pi\)
\(168\) 1903.30 0.874064
\(169\) 1896.65 0.863292
\(170\) 0 0
\(171\) 811.894 0.363082
\(172\) 6838.88 3.03174
\(173\) −537.049 −0.236018 −0.118009 0.993013i \(-0.537651\pi\)
−0.118009 + 0.993013i \(0.537651\pi\)
\(174\) 908.336 0.395752
\(175\) 0 0
\(176\) 1825.64 0.781891
\(177\) −2559.20 −1.08679
\(178\) −2316.21 −0.975320
\(179\) 2891.25 1.20728 0.603638 0.797259i \(-0.293717\pi\)
0.603638 + 0.797259i \(0.293717\pi\)
\(180\) 0 0
\(181\) 435.209 0.178723 0.0893615 0.995999i \(-0.471517\pi\)
0.0893615 + 0.995999i \(0.471517\pi\)
\(182\) −3474.36 −1.41504
\(183\) −611.098 −0.246851
\(184\) 13069.2 5.23628
\(185\) 0 0
\(186\) −2236.72 −0.881743
\(187\) −188.607 −0.0737556
\(188\) −7959.23 −3.08769
\(189\) 278.659 0.107246
\(190\) 0 0
\(191\) −3779.49 −1.43180 −0.715901 0.698202i \(-0.753984\pi\)
−0.715901 + 0.698202i \(0.753984\pi\)
\(192\) 2037.96 0.766027
\(193\) −3751.91 −1.39932 −0.699660 0.714476i \(-0.746665\pi\)
−0.699660 + 0.714476i \(0.746665\pi\)
\(194\) 1040.52 0.385076
\(195\) 0 0
\(196\) −4654.78 −1.69635
\(197\) 3920.73 1.41797 0.708986 0.705223i \(-0.249153\pi\)
0.708986 + 0.705223i \(0.249153\pi\)
\(198\) 520.888 0.186959
\(199\) −597.084 −0.212694 −0.106347 0.994329i \(-0.533915\pi\)
−0.106347 + 0.994329i \(0.533915\pi\)
\(200\) 0 0
\(201\) −800.669 −0.280969
\(202\) 7369.37 2.56687
\(203\) 593.917 0.205344
\(204\) −1012.48 −0.347488
\(205\) 0 0
\(206\) 7080.40 2.39473
\(207\) 1913.44 0.642480
\(208\) −10618.9 −3.53984
\(209\) 992.314 0.328420
\(210\) 0 0
\(211\) −4384.55 −1.43054 −0.715272 0.698846i \(-0.753697\pi\)
−0.715272 + 0.698846i \(0.753697\pi\)
\(212\) −5109.17 −1.65519
\(213\) 277.338 0.0892155
\(214\) 4678.41 1.49444
\(215\) 0 0
\(216\) 1659.74 0.522829
\(217\) −1462.48 −0.457510
\(218\) 6609.95 2.05359
\(219\) 726.077 0.224035
\(220\) 0 0
\(221\) 1097.03 0.333912
\(222\) 4071.82 1.23100
\(223\) 2333.03 0.700587 0.350294 0.936640i \(-0.386082\pi\)
0.350294 + 0.936640i \(0.386082\pi\)
\(224\) 3936.95 1.17433
\(225\) 0 0
\(226\) 12596.1 3.70743
\(227\) 2120.00 0.619864 0.309932 0.950759i \(-0.399694\pi\)
0.309932 + 0.950759i \(0.399694\pi\)
\(228\) 5326.93 1.54730
\(229\) 2347.12 0.677301 0.338651 0.940912i \(-0.390030\pi\)
0.338651 + 0.940912i \(0.390030\pi\)
\(230\) 0 0
\(231\) 340.583 0.0970075
\(232\) 3537.47 1.00106
\(233\) 375.499 0.105578 0.0527891 0.998606i \(-0.483189\pi\)
0.0527891 + 0.998606i \(0.483189\pi\)
\(234\) −3029.75 −0.846415
\(235\) 0 0
\(236\) −16791.2 −4.63142
\(237\) −3064.79 −0.839998
\(238\) −931.072 −0.253582
\(239\) −1428.15 −0.386524 −0.193262 0.981147i \(-0.561907\pi\)
−0.193262 + 0.981147i \(0.561907\pi\)
\(240\) 0 0
\(241\) 190.819 0.0510032 0.0255016 0.999675i \(-0.491882\pi\)
0.0255016 + 0.999675i \(0.491882\pi\)
\(242\) 636.641 0.169111
\(243\) 243.000 0.0641500
\(244\) −4009.49 −1.05197
\(245\) 0 0
\(246\) −3565.94 −0.924211
\(247\) −5771.81 −1.48685
\(248\) −8710.79 −2.23039
\(249\) −2119.24 −0.539364
\(250\) 0 0
\(251\) −6294.80 −1.58297 −0.791483 0.611191i \(-0.790691\pi\)
−0.791483 + 0.611191i \(0.790691\pi\)
\(252\) 1828.31 0.457036
\(253\) 2338.65 0.581145
\(254\) −10868.0 −2.68471
\(255\) 0 0
\(256\) −2685.18 −0.655561
\(257\) −4459.44 −1.08238 −0.541191 0.840900i \(-0.682026\pi\)
−0.541191 + 0.840900i \(0.682026\pi\)
\(258\) 5484.24 1.32339
\(259\) 2662.36 0.638731
\(260\) 0 0
\(261\) 517.915 0.122828
\(262\) 4133.04 0.974581
\(263\) 4416.65 1.03552 0.517761 0.855525i \(-0.326766\pi\)
0.517761 + 0.855525i \(0.326766\pi\)
\(264\) 2028.57 0.472917
\(265\) 0 0
\(266\) 4898.64 1.12915
\(267\) −1320.65 −0.302707
\(268\) −5253.28 −1.19737
\(269\) 1914.86 0.434020 0.217010 0.976169i \(-0.430370\pi\)
0.217010 + 0.976169i \(0.430370\pi\)
\(270\) 0 0
\(271\) 6088.34 1.36472 0.682362 0.731014i \(-0.260953\pi\)
0.682362 + 0.731014i \(0.260953\pi\)
\(272\) −2845.69 −0.634357
\(273\) −1981.01 −0.439180
\(274\) −6714.91 −1.48052
\(275\) 0 0
\(276\) 12554.3 2.73798
\(277\) 832.321 0.180539 0.0902696 0.995917i \(-0.471227\pi\)
0.0902696 + 0.995917i \(0.471227\pi\)
\(278\) −14222.9 −3.06847
\(279\) −1275.33 −0.273664
\(280\) 0 0
\(281\) −2545.32 −0.540360 −0.270180 0.962810i \(-0.587083\pi\)
−0.270180 + 0.962810i \(0.587083\pi\)
\(282\) −6382.67 −1.34781
\(283\) 5911.71 1.24175 0.620874 0.783911i \(-0.286778\pi\)
0.620874 + 0.783911i \(0.286778\pi\)
\(284\) 1819.65 0.380198
\(285\) 0 0
\(286\) −3703.03 −0.765611
\(287\) −2331.59 −0.479545
\(288\) 3433.15 0.702433
\(289\) −4619.01 −0.940161
\(290\) 0 0
\(291\) 593.282 0.119515
\(292\) 4763.87 0.954742
\(293\) 6871.03 1.37000 0.685000 0.728543i \(-0.259802\pi\)
0.685000 + 0.728543i \(0.259802\pi\)
\(294\) −3732.76 −0.740473
\(295\) 0 0
\(296\) 15857.5 3.11384
\(297\) 297.000 0.0580259
\(298\) −12632.8 −2.45570
\(299\) −13602.8 −2.63100
\(300\) 0 0
\(301\) 3585.88 0.686666
\(302\) −13244.7 −2.52367
\(303\) 4201.87 0.796670
\(304\) 14972.0 2.82468
\(305\) 0 0
\(306\) −811.926 −0.151682
\(307\) −200.179 −0.0372144 −0.0186072 0.999827i \(-0.505923\pi\)
−0.0186072 + 0.999827i \(0.505923\pi\)
\(308\) 2234.61 0.413404
\(309\) 4037.10 0.743245
\(310\) 0 0
\(311\) 5734.93 1.04565 0.522827 0.852439i \(-0.324878\pi\)
0.522827 + 0.852439i \(0.324878\pi\)
\(312\) −11799.2 −2.14102
\(313\) 3077.36 0.555727 0.277864 0.960621i \(-0.410374\pi\)
0.277864 + 0.960621i \(0.410374\pi\)
\(314\) −7320.97 −1.31575
\(315\) 0 0
\(316\) −20108.5 −3.57971
\(317\) −2142.38 −0.379584 −0.189792 0.981824i \(-0.560781\pi\)
−0.189792 + 0.981824i \(0.560781\pi\)
\(318\) −4097.15 −0.722506
\(319\) 633.007 0.111102
\(320\) 0 0
\(321\) 2667.54 0.463823
\(322\) 11544.9 1.99806
\(323\) −1546.75 −0.266451
\(324\) 1594.35 0.273380
\(325\) 0 0
\(326\) 14314.0 2.43184
\(327\) 3768.86 0.637365
\(328\) −13887.4 −2.33781
\(329\) −4173.32 −0.699339
\(330\) 0 0
\(331\) −1618.23 −0.268719 −0.134359 0.990933i \(-0.542898\pi\)
−0.134359 + 0.990933i \(0.542898\pi\)
\(332\) −13904.6 −2.29854
\(333\) 2321.67 0.382062
\(334\) −15522.7 −2.54301
\(335\) 0 0
\(336\) 5138.70 0.834343
\(337\) −2406.47 −0.388988 −0.194494 0.980904i \(-0.562306\pi\)
−0.194494 + 0.980904i \(0.562306\pi\)
\(338\) 9979.23 1.60591
\(339\) 7182.04 1.15066
\(340\) 0 0
\(341\) −1558.74 −0.247538
\(342\) 4271.78 0.675413
\(343\) −5980.67 −0.941475
\(344\) 21358.1 3.34753
\(345\) 0 0
\(346\) −2825.68 −0.439045
\(347\) 6612.89 1.02305 0.511525 0.859268i \(-0.329081\pi\)
0.511525 + 0.859268i \(0.329081\pi\)
\(348\) 3398.10 0.523441
\(349\) 349.871 0.0536623 0.0268311 0.999640i \(-0.491458\pi\)
0.0268311 + 0.999640i \(0.491458\pi\)
\(350\) 0 0
\(351\) −1727.50 −0.262699
\(352\) 4196.08 0.635374
\(353\) −1723.29 −0.259835 −0.129917 0.991525i \(-0.541471\pi\)
−0.129917 + 0.991525i \(0.541471\pi\)
\(354\) −13465.2 −2.02166
\(355\) 0 0
\(356\) −8664.97 −1.29001
\(357\) −530.879 −0.0787033
\(358\) 15212.3 2.24580
\(359\) 5875.74 0.863816 0.431908 0.901918i \(-0.357841\pi\)
0.431908 + 0.901918i \(0.357841\pi\)
\(360\) 0 0
\(361\) 1278.92 0.186458
\(362\) 2289.85 0.332464
\(363\) 363.000 0.0524864
\(364\) −12997.6 −1.87159
\(365\) 0 0
\(366\) −3215.29 −0.459197
\(367\) 5368.28 0.763548 0.381774 0.924256i \(-0.375313\pi\)
0.381774 + 0.924256i \(0.375313\pi\)
\(368\) 35285.5 4.99832
\(369\) −2033.23 −0.286844
\(370\) 0 0
\(371\) −2678.93 −0.374887
\(372\) −8367.60 −1.16624
\(373\) −10393.9 −1.44282 −0.721412 0.692506i \(-0.756507\pi\)
−0.721412 + 0.692506i \(0.756507\pi\)
\(374\) −992.354 −0.137202
\(375\) 0 0
\(376\) −24857.0 −3.40931
\(377\) −3681.90 −0.502990
\(378\) 1466.16 0.199501
\(379\) 10918.9 1.47986 0.739928 0.672686i \(-0.234860\pi\)
0.739928 + 0.672686i \(0.234860\pi\)
\(380\) 0 0
\(381\) −6196.70 −0.833246
\(382\) −19885.8 −2.66347
\(383\) 11663.6 1.55609 0.778044 0.628210i \(-0.216212\pi\)
0.778044 + 0.628210i \(0.216212\pi\)
\(384\) 1567.64 0.208329
\(385\) 0 0
\(386\) −19740.7 −2.60304
\(387\) 3127.00 0.410735
\(388\) 3892.59 0.509321
\(389\) −5827.00 −0.759487 −0.379744 0.925092i \(-0.623988\pi\)
−0.379744 + 0.925092i \(0.623988\pi\)
\(390\) 0 0
\(391\) −3645.34 −0.471490
\(392\) −14537.1 −1.87304
\(393\) 2356.58 0.302478
\(394\) 20628.9 2.63774
\(395\) 0 0
\(396\) 1948.65 0.247281
\(397\) −7366.99 −0.931332 −0.465666 0.884961i \(-0.654185\pi\)
−0.465666 + 0.884961i \(0.654185\pi\)
\(398\) −3141.55 −0.395658
\(399\) 2793.11 0.350452
\(400\) 0 0
\(401\) 14604.2 1.81870 0.909349 0.416035i \(-0.136581\pi\)
0.909349 + 0.416035i \(0.136581\pi\)
\(402\) −4212.72 −0.522665
\(403\) 9066.43 1.12067
\(404\) 27569.0 3.39507
\(405\) 0 0
\(406\) 3124.89 0.381985
\(407\) 2837.60 0.345588
\(408\) −3162.00 −0.383683
\(409\) 12581.2 1.52103 0.760515 0.649320i \(-0.224946\pi\)
0.760515 + 0.649320i \(0.224946\pi\)
\(410\) 0 0
\(411\) −3828.71 −0.459504
\(412\) 26487.9 3.16739
\(413\) −8804.26 −1.04898
\(414\) 10067.6 1.19516
\(415\) 0 0
\(416\) −24406.6 −2.87651
\(417\) −8109.63 −0.952351
\(418\) 5221.06 0.610934
\(419\) −3776.01 −0.440263 −0.220131 0.975470i \(-0.570649\pi\)
−0.220131 + 0.975470i \(0.570649\pi\)
\(420\) 0 0
\(421\) 12683.4 1.46830 0.734148 0.678989i \(-0.237582\pi\)
0.734148 + 0.678989i \(0.237582\pi\)
\(422\) −23069.3 −2.66113
\(423\) −3639.27 −0.418316
\(424\) −15956.2 −1.82759
\(425\) 0 0
\(426\) 1459.21 0.165960
\(427\) −2102.32 −0.238263
\(428\) 17502.0 1.97662
\(429\) −2111.39 −0.237620
\(430\) 0 0
\(431\) 14152.6 1.58168 0.790841 0.612022i \(-0.209644\pi\)
0.790841 + 0.612022i \(0.209644\pi\)
\(432\) 4481.12 0.499069
\(433\) 10950.2 1.21532 0.607661 0.794197i \(-0.292108\pi\)
0.607661 + 0.794197i \(0.292108\pi\)
\(434\) −7694.84 −0.851070
\(435\) 0 0
\(436\) 24727.9 2.71618
\(437\) 19179.2 2.09946
\(438\) 3820.25 0.416755
\(439\) −11221.0 −1.21993 −0.609964 0.792429i \(-0.708816\pi\)
−0.609964 + 0.792429i \(0.708816\pi\)
\(440\) 0 0
\(441\) −2128.35 −0.229818
\(442\) 5772.04 0.621149
\(443\) −9647.11 −1.03465 −0.517323 0.855790i \(-0.673071\pi\)
−0.517323 + 0.855790i \(0.673071\pi\)
\(444\) 15232.7 1.62818
\(445\) 0 0
\(446\) 12275.2 1.30325
\(447\) −7202.96 −0.762167
\(448\) 7011.07 0.739379
\(449\) 6482.03 0.681305 0.340652 0.940189i \(-0.389352\pi\)
0.340652 + 0.940189i \(0.389352\pi\)
\(450\) 0 0
\(451\) −2485.05 −0.259460
\(452\) 47122.2 4.90363
\(453\) −7551.89 −0.783264
\(454\) 11154.4 1.15308
\(455\) 0 0
\(456\) 16636.2 1.70847
\(457\) −11319.8 −1.15868 −0.579342 0.815085i \(-0.696690\pi\)
−0.579342 + 0.815085i \(0.696690\pi\)
\(458\) 12349.4 1.25993
\(459\) −462.944 −0.0470771
\(460\) 0 0
\(461\) −8406.73 −0.849329 −0.424664 0.905351i \(-0.639608\pi\)
−0.424664 + 0.905351i \(0.639608\pi\)
\(462\) 1791.98 0.180455
\(463\) −9758.56 −0.979523 −0.489761 0.871857i \(-0.662916\pi\)
−0.489761 + 0.871857i \(0.662916\pi\)
\(464\) 9550.78 0.955568
\(465\) 0 0
\(466\) 1975.68 0.196399
\(467\) 16388.7 1.62394 0.811969 0.583701i \(-0.198396\pi\)
0.811969 + 0.583701i \(0.198396\pi\)
\(468\) −11334.4 −1.11951
\(469\) −2754.49 −0.271195
\(470\) 0 0
\(471\) −4174.27 −0.408365
\(472\) −52439.6 −5.11384
\(473\) 3821.89 0.371524
\(474\) −16125.4 −1.56258
\(475\) 0 0
\(476\) −3483.16 −0.335400
\(477\) −2336.11 −0.224242
\(478\) −7514.20 −0.719020
\(479\) 13829.0 1.31913 0.659567 0.751646i \(-0.270740\pi\)
0.659567 + 0.751646i \(0.270740\pi\)
\(480\) 0 0
\(481\) −16504.9 −1.56457
\(482\) 1004.00 0.0948771
\(483\) 6582.69 0.620130
\(484\) 2381.68 0.223674
\(485\) 0 0
\(486\) 1278.54 0.119333
\(487\) −13264.4 −1.23423 −0.617113 0.786875i \(-0.711698\pi\)
−0.617113 + 0.786875i \(0.711698\pi\)
\(488\) −12521.8 −1.16155
\(489\) 8161.58 0.754763
\(490\) 0 0
\(491\) −7468.22 −0.686428 −0.343214 0.939257i \(-0.611516\pi\)
−0.343214 + 0.939257i \(0.611516\pi\)
\(492\) −13340.2 −1.22241
\(493\) −986.690 −0.0901385
\(494\) −30368.4 −2.76586
\(495\) 0 0
\(496\) −23518.2 −2.12903
\(497\) 954.109 0.0861119
\(498\) −11150.4 −1.00334
\(499\) −5276.64 −0.473377 −0.236688 0.971586i \(-0.576062\pi\)
−0.236688 + 0.971586i \(0.576062\pi\)
\(500\) 0 0
\(501\) −8850.76 −0.789267
\(502\) −33120.1 −2.94466
\(503\) 10956.8 0.971253 0.485626 0.874166i \(-0.338592\pi\)
0.485626 + 0.874166i \(0.338592\pi\)
\(504\) 5709.90 0.504641
\(505\) 0 0
\(506\) 12304.8 1.08106
\(507\) 5689.96 0.498422
\(508\) −40657.3 −3.55093
\(509\) 12734.4 1.10892 0.554462 0.832209i \(-0.312924\pi\)
0.554462 + 0.832209i \(0.312924\pi\)
\(510\) 0 0
\(511\) 2497.87 0.216242
\(512\) −18308.4 −1.58032
\(513\) 2435.68 0.209626
\(514\) −23463.3 −2.01347
\(515\) 0 0
\(516\) 20516.6 1.75038
\(517\) −4448.00 −0.378381
\(518\) 14008.0 1.18818
\(519\) −1611.15 −0.136265
\(520\) 0 0
\(521\) −5650.70 −0.475166 −0.237583 0.971367i \(-0.576355\pi\)
−0.237583 + 0.971367i \(0.576355\pi\)
\(522\) 2725.01 0.228487
\(523\) −14103.2 −1.17914 −0.589572 0.807716i \(-0.700703\pi\)
−0.589572 + 0.807716i \(0.700703\pi\)
\(524\) 15461.8 1.28903
\(525\) 0 0
\(526\) 23238.2 1.92630
\(527\) 2429.66 0.200830
\(528\) 5476.92 0.451425
\(529\) 33033.8 2.71503
\(530\) 0 0
\(531\) −7677.60 −0.627457
\(532\) 18325.9 1.49347
\(533\) 14454.4 1.17465
\(534\) −6948.62 −0.563102
\(535\) 0 0
\(536\) −16406.2 −1.32209
\(537\) 8673.75 0.697021
\(538\) 10075.0 0.807372
\(539\) −2601.31 −0.207878
\(540\) 0 0
\(541\) −6391.90 −0.507965 −0.253983 0.967209i \(-0.581741\pi\)
−0.253983 + 0.967209i \(0.581741\pi\)
\(542\) 32033.8 2.53869
\(543\) 1305.63 0.103186
\(544\) −6540.57 −0.515486
\(545\) 0 0
\(546\) −10423.1 −0.816971
\(547\) −20786.7 −1.62482 −0.812409 0.583088i \(-0.801844\pi\)
−0.812409 + 0.583088i \(0.801844\pi\)
\(548\) −25120.6 −1.95821
\(549\) −1833.29 −0.142519
\(550\) 0 0
\(551\) 5191.26 0.401370
\(552\) 39207.6 3.02317
\(553\) −10543.6 −0.810777
\(554\) 4379.26 0.335843
\(555\) 0 0
\(556\) −53208.2 −4.05851
\(557\) 15125.9 1.15064 0.575320 0.817928i \(-0.304877\pi\)
0.575320 + 0.817928i \(0.304877\pi\)
\(558\) −6710.16 −0.509075
\(559\) −22230.1 −1.68199
\(560\) 0 0
\(561\) −565.820 −0.0425828
\(562\) −13392.2 −1.00519
\(563\) −8706.42 −0.651744 −0.325872 0.945414i \(-0.605658\pi\)
−0.325872 + 0.945414i \(0.605658\pi\)
\(564\) −23877.7 −1.78268
\(565\) 0 0
\(566\) 31104.4 2.30992
\(567\) 835.977 0.0619184
\(568\) 5682.83 0.419800
\(569\) 7067.76 0.520731 0.260366 0.965510i \(-0.416157\pi\)
0.260366 + 0.965510i \(0.416157\pi\)
\(570\) 0 0
\(571\) −3326.42 −0.243794 −0.121897 0.992543i \(-0.538898\pi\)
−0.121897 + 0.992543i \(0.538898\pi\)
\(572\) −13853.1 −1.01264
\(573\) −11338.5 −0.826651
\(574\) −12267.7 −0.892060
\(575\) 0 0
\(576\) 6113.88 0.442266
\(577\) 3308.06 0.238676 0.119338 0.992854i \(-0.461923\pi\)
0.119338 + 0.992854i \(0.461923\pi\)
\(578\) −24302.9 −1.74891
\(579\) −11255.7 −0.807897
\(580\) 0 0
\(581\) −7290.70 −0.520601
\(582\) 3121.55 0.222324
\(583\) −2855.25 −0.202834
\(584\) 14877.8 1.05419
\(585\) 0 0
\(586\) 36151.9 2.54850
\(587\) 5694.88 0.400431 0.200215 0.979752i \(-0.435836\pi\)
0.200215 + 0.979752i \(0.435836\pi\)
\(588\) −13964.3 −0.979387
\(589\) −12783.1 −0.894262
\(590\) 0 0
\(591\) 11762.2 0.818666
\(592\) 42813.5 2.97234
\(593\) 3907.69 0.270606 0.135303 0.990804i \(-0.456799\pi\)
0.135303 + 0.990804i \(0.456799\pi\)
\(594\) 1562.66 0.107941
\(595\) 0 0
\(596\) −47259.5 −3.24803
\(597\) −1791.25 −0.122799
\(598\) −71571.1 −4.89425
\(599\) 10727.2 0.731720 0.365860 0.930670i \(-0.380775\pi\)
0.365860 + 0.930670i \(0.380775\pi\)
\(600\) 0 0
\(601\) 3348.98 0.227301 0.113650 0.993521i \(-0.463746\pi\)
0.113650 + 0.993521i \(0.463746\pi\)
\(602\) 18867.1 1.27735
\(603\) −2402.01 −0.162218
\(604\) −49548.8 −3.33793
\(605\) 0 0
\(606\) 22108.1 1.48198
\(607\) 21539.9 1.44032 0.720162 0.693806i \(-0.244068\pi\)
0.720162 + 0.693806i \(0.244068\pi\)
\(608\) 34411.8 2.29537
\(609\) 1781.75 0.118555
\(610\) 0 0
\(611\) 25871.9 1.71303
\(612\) −3037.43 −0.200622
\(613\) 9284.33 0.611730 0.305865 0.952075i \(-0.401054\pi\)
0.305865 + 0.952075i \(0.401054\pi\)
\(614\) −1053.24 −0.0692270
\(615\) 0 0
\(616\) 6978.77 0.456465
\(617\) 20711.3 1.35139 0.675695 0.737181i \(-0.263844\pi\)
0.675695 + 0.737181i \(0.263844\pi\)
\(618\) 21241.2 1.38260
\(619\) −13282.9 −0.862496 −0.431248 0.902233i \(-0.641927\pi\)
−0.431248 + 0.902233i \(0.641927\pi\)
\(620\) 0 0
\(621\) 5740.33 0.370936
\(622\) 30174.3 1.94515
\(623\) −4543.36 −0.292177
\(624\) −31856.6 −2.04373
\(625\) 0 0
\(626\) 16191.5 1.03378
\(627\) 2976.94 0.189613
\(628\) −27387.9 −1.74028
\(629\) −4423.06 −0.280380
\(630\) 0 0
\(631\) −14789.9 −0.933086 −0.466543 0.884498i \(-0.654501\pi\)
−0.466543 + 0.884498i \(0.654501\pi\)
\(632\) −62799.5 −3.95258
\(633\) −13153.6 −0.825925
\(634\) −11272.1 −0.706109
\(635\) 0 0
\(636\) −15327.5 −0.955622
\(637\) 15130.6 0.941123
\(638\) 3330.57 0.206675
\(639\) 832.014 0.0515086
\(640\) 0 0
\(641\) −5808.52 −0.357914 −0.178957 0.983857i \(-0.557272\pi\)
−0.178957 + 0.983857i \(0.557272\pi\)
\(642\) 14035.2 0.862813
\(643\) 18891.2 1.15862 0.579311 0.815106i \(-0.303322\pi\)
0.579311 + 0.815106i \(0.303322\pi\)
\(644\) 43189.8 2.64273
\(645\) 0 0
\(646\) −8138.24 −0.495657
\(647\) 243.046 0.0147684 0.00738418 0.999973i \(-0.497650\pi\)
0.00738418 + 0.999973i \(0.497650\pi\)
\(648\) 4979.22 0.301855
\(649\) −9383.74 −0.567556
\(650\) 0 0
\(651\) −4387.44 −0.264144
\(652\) 53549.0 3.21648
\(653\) 15920.4 0.954081 0.477041 0.878881i \(-0.341709\pi\)
0.477041 + 0.878881i \(0.341709\pi\)
\(654\) 19829.8 1.18564
\(655\) 0 0
\(656\) −37494.4 −2.23157
\(657\) 2178.23 0.129347
\(658\) −21957.9 −1.30092
\(659\) −7476.38 −0.441940 −0.220970 0.975281i \(-0.570922\pi\)
−0.220970 + 0.975281i \(0.570922\pi\)
\(660\) 0 0
\(661\) −14920.1 −0.877948 −0.438974 0.898500i \(-0.644658\pi\)
−0.438974 + 0.898500i \(0.644658\pi\)
\(662\) −8514.30 −0.499876
\(663\) 3291.10 0.192784
\(664\) −43424.7 −2.53796
\(665\) 0 0
\(666\) 12215.5 0.710720
\(667\) 12234.6 0.710232
\(668\) −58070.8 −3.36351
\(669\) 6999.08 0.404484
\(670\) 0 0
\(671\) −2240.69 −0.128914
\(672\) 11810.9 0.677997
\(673\) −563.692 −0.0322864 −0.0161432 0.999870i \(-0.505139\pi\)
−0.0161432 + 0.999870i \(0.505139\pi\)
\(674\) −12661.6 −0.723602
\(675\) 0 0
\(676\) 37332.5 2.12406
\(677\) −13280.2 −0.753914 −0.376957 0.926231i \(-0.623030\pi\)
−0.376957 + 0.926231i \(0.623030\pi\)
\(678\) 37788.3 2.14049
\(679\) 2041.03 0.115357
\(680\) 0 0
\(681\) 6359.99 0.357879
\(682\) −8201.30 −0.460475
\(683\) −6856.80 −0.384141 −0.192070 0.981381i \(-0.561520\pi\)
−0.192070 + 0.981381i \(0.561520\pi\)
\(684\) 15980.8 0.893334
\(685\) 0 0
\(686\) −31467.3 −1.75135
\(687\) 7041.36 0.391040
\(688\) 57664.5 3.19541
\(689\) 16607.6 0.918287
\(690\) 0 0
\(691\) −28374.6 −1.56211 −0.781057 0.624460i \(-0.785319\pi\)
−0.781057 + 0.624460i \(0.785319\pi\)
\(692\) −10570.9 −0.580702
\(693\) 1021.75 0.0560073
\(694\) 34793.7 1.90310
\(695\) 0 0
\(696\) 10612.4 0.577963
\(697\) 3873.54 0.210503
\(698\) 1840.84 0.0998237
\(699\) 1126.50 0.0609556
\(700\) 0 0
\(701\) 667.753 0.0359781 0.0179891 0.999838i \(-0.494274\pi\)
0.0179891 + 0.999838i \(0.494274\pi\)
\(702\) −9089.26 −0.488678
\(703\) 23271.0 1.24848
\(704\) 7472.52 0.400044
\(705\) 0 0
\(706\) −9067.10 −0.483349
\(707\) 14455.4 0.768956
\(708\) −50373.6 −2.67395
\(709\) 23667.8 1.25368 0.626842 0.779147i \(-0.284347\pi\)
0.626842 + 0.779147i \(0.284347\pi\)
\(710\) 0 0
\(711\) −9194.37 −0.484973
\(712\) −27061.0 −1.42438
\(713\) −30126.9 −1.58241
\(714\) −2793.22 −0.146405
\(715\) 0 0
\(716\) 56909.5 2.97040
\(717\) −4284.45 −0.223160
\(718\) 30915.2 1.60689
\(719\) 11835.5 0.613896 0.306948 0.951726i \(-0.400692\pi\)
0.306948 + 0.951726i \(0.400692\pi\)
\(720\) 0 0
\(721\) 13888.6 0.717390
\(722\) 6729.01 0.346853
\(723\) 572.458 0.0294467
\(724\) 8566.38 0.439733
\(725\) 0 0
\(726\) 1909.92 0.0976362
\(727\) −15633.2 −0.797530 −0.398765 0.917053i \(-0.630561\pi\)
−0.398765 + 0.917053i \(0.630561\pi\)
\(728\) −40592.1 −2.06654
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −5957.31 −0.301422
\(732\) −12028.5 −0.607356
\(733\) 14870.7 0.749335 0.374668 0.927159i \(-0.377757\pi\)
0.374668 + 0.927159i \(0.377757\pi\)
\(734\) 28245.2 1.42037
\(735\) 0 0
\(736\) 81100.6 4.06169
\(737\) −2935.79 −0.146731
\(738\) −10697.8 −0.533593
\(739\) 20850.3 1.03788 0.518939 0.854812i \(-0.326327\pi\)
0.518939 + 0.854812i \(0.326327\pi\)
\(740\) 0 0
\(741\) −17315.4 −0.858432
\(742\) −14095.2 −0.697372
\(743\) 29254.7 1.44448 0.722242 0.691641i \(-0.243112\pi\)
0.722242 + 0.691641i \(0.243112\pi\)
\(744\) −26132.4 −1.28771
\(745\) 0 0
\(746\) −54687.2 −2.68397
\(747\) −6357.73 −0.311402
\(748\) −3712.41 −0.181470
\(749\) 9176.95 0.447688
\(750\) 0 0
\(751\) 10936.8 0.531411 0.265705 0.964054i \(-0.414395\pi\)
0.265705 + 0.964054i \(0.414395\pi\)
\(752\) −67111.2 −3.25438
\(753\) −18884.4 −0.913926
\(754\) −19372.3 −0.935672
\(755\) 0 0
\(756\) 5484.94 0.263870
\(757\) 8476.34 0.406972 0.203486 0.979078i \(-0.434773\pi\)
0.203486 + 0.979078i \(0.434773\pi\)
\(758\) 57449.6 2.75286
\(759\) 7015.96 0.335524
\(760\) 0 0
\(761\) 913.964 0.0435364 0.0217682 0.999763i \(-0.493070\pi\)
0.0217682 + 0.999763i \(0.493070\pi\)
\(762\) −32603.9 −1.55002
\(763\) 12965.8 0.615193
\(764\) −74393.0 −3.52283
\(765\) 0 0
\(766\) 61367.9 2.89466
\(767\) 54580.6 2.56948
\(768\) −8055.53 −0.378488
\(769\) 32215.2 1.51067 0.755337 0.655337i \(-0.227473\pi\)
0.755337 + 0.655337i \(0.227473\pi\)
\(770\) 0 0
\(771\) −13378.3 −0.624914
\(772\) −73850.2 −3.44291
\(773\) −72.6900 −0.00338225 −0.00169112 0.999999i \(-0.500538\pi\)
−0.00169112 + 0.999999i \(0.500538\pi\)
\(774\) 16452.7 0.764058
\(775\) 0 0
\(776\) 12156.7 0.562373
\(777\) 7987.09 0.368771
\(778\) −30658.7 −1.41281
\(779\) −20379.8 −0.937332
\(780\) 0 0
\(781\) 1016.91 0.0465913
\(782\) −19179.9 −0.877075
\(783\) 1553.75 0.0709148
\(784\) −39248.5 −1.78792
\(785\) 0 0
\(786\) 12399.1 0.562675
\(787\) −487.318 −0.0220724 −0.0110362 0.999939i \(-0.503513\pi\)
−0.0110362 + 0.999939i \(0.503513\pi\)
\(788\) 77173.1 3.48880
\(789\) 13249.9 0.597859
\(790\) 0 0
\(791\) 24707.9 1.11064
\(792\) 6085.72 0.273039
\(793\) 13033.0 0.583627
\(794\) −38761.4 −1.73248
\(795\) 0 0
\(796\) −11752.6 −0.523317
\(797\) −31379.9 −1.39465 −0.697324 0.716756i \(-0.745626\pi\)
−0.697324 + 0.716756i \(0.745626\pi\)
\(798\) 14695.9 0.651917
\(799\) 6933.25 0.306985
\(800\) 0 0
\(801\) −3961.96 −0.174768
\(802\) 76839.8 3.38318
\(803\) 2662.28 0.116999
\(804\) −15759.8 −0.691302
\(805\) 0 0
\(806\) 47703.0 2.08470
\(807\) 5744.59 0.250581
\(808\) 86099.0 3.74870
\(809\) 1824.26 0.0792802 0.0396401 0.999214i \(-0.487379\pi\)
0.0396401 + 0.999214i \(0.487379\pi\)
\(810\) 0 0
\(811\) −4364.52 −0.188976 −0.0944878 0.995526i \(-0.530121\pi\)
−0.0944878 + 0.995526i \(0.530121\pi\)
\(812\) 11690.3 0.505232
\(813\) 18265.0 0.787924
\(814\) 14930.0 0.642870
\(815\) 0 0
\(816\) −8537.06 −0.366246
\(817\) 31343.1 1.34218
\(818\) 66196.1 2.82945
\(819\) −5943.02 −0.253561
\(820\) 0 0
\(821\) 3306.16 0.140543 0.0702714 0.997528i \(-0.477613\pi\)
0.0702714 + 0.997528i \(0.477613\pi\)
\(822\) −20144.7 −0.854779
\(823\) 19183.7 0.812519 0.406259 0.913758i \(-0.366833\pi\)
0.406259 + 0.913758i \(0.366833\pi\)
\(824\) 82722.8 3.49731
\(825\) 0 0
\(826\) −46323.6 −1.95134
\(827\) −20646.4 −0.868131 −0.434066 0.900881i \(-0.642921\pi\)
−0.434066 + 0.900881i \(0.642921\pi\)
\(828\) 37663.0 1.58077
\(829\) 5345.44 0.223950 0.111975 0.993711i \(-0.464282\pi\)
0.111975 + 0.993711i \(0.464282\pi\)
\(830\) 0 0
\(831\) 2496.96 0.104234
\(832\) −43464.0 −1.81111
\(833\) 4054.75 0.168654
\(834\) −42668.8 −1.77158
\(835\) 0 0
\(836\) 19532.1 0.808051
\(837\) −3826.00 −0.158000
\(838\) −19867.5 −0.818985
\(839\) 29284.9 1.20504 0.602519 0.798105i \(-0.294164\pi\)
0.602519 + 0.798105i \(0.294164\pi\)
\(840\) 0 0
\(841\) −21077.4 −0.864219
\(842\) 66733.8 2.73135
\(843\) −7635.97 −0.311977
\(844\) −86302.6 −3.51974
\(845\) 0 0
\(846\) −19148.0 −0.778159
\(847\) 1248.81 0.0506605
\(848\) −43079.9 −1.74454
\(849\) 17735.1 0.716923
\(850\) 0 0
\(851\) 54844.2 2.20921
\(852\) 5458.94 0.219507
\(853\) 8070.62 0.323954 0.161977 0.986795i \(-0.448213\pi\)
0.161977 + 0.986795i \(0.448213\pi\)
\(854\) −11061.4 −0.443222
\(855\) 0 0
\(856\) 54659.5 2.18250
\(857\) 11344.2 0.452169 0.226085 0.974108i \(-0.427407\pi\)
0.226085 + 0.974108i \(0.427407\pi\)
\(858\) −11109.1 −0.442026
\(859\) 25470.6 1.01170 0.505848 0.862623i \(-0.331180\pi\)
0.505848 + 0.862623i \(0.331180\pi\)
\(860\) 0 0
\(861\) −6994.78 −0.276866
\(862\) 74463.6 2.94228
\(863\) 14558.4 0.574243 0.287122 0.957894i \(-0.407302\pi\)
0.287122 + 0.957894i \(0.407302\pi\)
\(864\) 10299.5 0.405550
\(865\) 0 0
\(866\) 57614.6 2.26077
\(867\) −13857.0 −0.542802
\(868\) −28786.5 −1.12567
\(869\) −11237.6 −0.438675
\(870\) 0 0
\(871\) 17076.0 0.664294
\(872\) 77226.3 2.99910
\(873\) 1779.85 0.0690019
\(874\) 100911. 3.90546
\(875\) 0 0
\(876\) 14291.6 0.551220
\(877\) −185.528 −0.00714350 −0.00357175 0.999994i \(-0.501137\pi\)
−0.00357175 + 0.999994i \(0.501137\pi\)
\(878\) −59039.1 −2.26933
\(879\) 20613.1 0.790969
\(880\) 0 0
\(881\) 18950.1 0.724681 0.362340 0.932046i \(-0.381978\pi\)
0.362340 + 0.932046i \(0.381978\pi\)
\(882\) −11198.3 −0.427513
\(883\) −21258.3 −0.810189 −0.405095 0.914275i \(-0.632761\pi\)
−0.405095 + 0.914275i \(0.632761\pi\)
\(884\) 21593.3 0.821563
\(885\) 0 0
\(886\) −50758.2 −1.92467
\(887\) −35707.4 −1.35168 −0.675839 0.737049i \(-0.736219\pi\)
−0.675839 + 0.737049i \(0.736219\pi\)
\(888\) 47572.5 1.79778
\(889\) −21318.1 −0.804259
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 45921.8 1.72374
\(893\) −36477.8 −1.36695
\(894\) −37898.4 −1.41780
\(895\) 0 0
\(896\) 5393.06 0.201082
\(897\) −40808.4 −1.51901
\(898\) 34105.2 1.26738
\(899\) −8154.49 −0.302522
\(900\) 0 0
\(901\) 4450.58 0.164562
\(902\) −13075.1 −0.482653
\(903\) 10757.6 0.396447
\(904\) 147165. 5.41440
\(905\) 0 0
\(906\) −39734.2 −1.45704
\(907\) 6542.52 0.239516 0.119758 0.992803i \(-0.461788\pi\)
0.119758 + 0.992803i \(0.461788\pi\)
\(908\) 41728.6 1.52512
\(909\) 12605.6 0.459958
\(910\) 0 0
\(911\) 31171.2 1.13364 0.566821 0.823841i \(-0.308173\pi\)
0.566821 + 0.823841i \(0.308173\pi\)
\(912\) 44915.9 1.63083
\(913\) −7770.56 −0.281674
\(914\) −59559.2 −2.15541
\(915\) 0 0
\(916\) 46199.2 1.66644
\(917\) 8107.19 0.291955
\(918\) −2435.78 −0.0875737
\(919\) 12031.9 0.431877 0.215938 0.976407i \(-0.430719\pi\)
0.215938 + 0.976407i \(0.430719\pi\)
\(920\) 0 0
\(921\) −600.537 −0.0214857
\(922\) −44232.0 −1.57994
\(923\) −5914.85 −0.210931
\(924\) 6703.82 0.238679
\(925\) 0 0
\(926\) −51344.7 −1.82213
\(927\) 12111.3 0.429113
\(928\) 21951.6 0.776506
\(929\) −12546.2 −0.443085 −0.221542 0.975151i \(-0.571109\pi\)
−0.221542 + 0.975151i \(0.571109\pi\)
\(930\) 0 0
\(931\) −21333.2 −0.750986
\(932\) 7391.07 0.259767
\(933\) 17204.8 0.603708
\(934\) 86229.1 3.02088
\(935\) 0 0
\(936\) −35397.7 −1.23612
\(937\) −17909.8 −0.624427 −0.312214 0.950012i \(-0.601070\pi\)
−0.312214 + 0.950012i \(0.601070\pi\)
\(938\) −14492.7 −0.504483
\(939\) 9232.08 0.320849
\(940\) 0 0
\(941\) 829.893 0.0287500 0.0143750 0.999897i \(-0.495424\pi\)
0.0143750 + 0.999897i \(0.495424\pi\)
\(942\) −21962.9 −0.759650
\(943\) −48030.4 −1.65863
\(944\) −141581. −4.88144
\(945\) 0 0
\(946\) 20108.9 0.691116
\(947\) 17654.9 0.605814 0.302907 0.953020i \(-0.402043\pi\)
0.302907 + 0.953020i \(0.402043\pi\)
\(948\) −60325.4 −2.06675
\(949\) −15485.2 −0.529685
\(950\) 0 0
\(951\) −6427.14 −0.219153
\(952\) −10878.0 −0.370335
\(953\) −30736.6 −1.04476 −0.522380 0.852713i \(-0.674956\pi\)
−0.522380 + 0.852713i \(0.674956\pi\)
\(954\) −12291.5 −0.417139
\(955\) 0 0
\(956\) −28110.8 −0.951012
\(957\) 1899.02 0.0641449
\(958\) 72761.5 2.45388
\(959\) −13171.7 −0.443519
\(960\) 0 0
\(961\) −9711.08 −0.325974
\(962\) −86840.6 −2.91045
\(963\) 8002.61 0.267789
\(964\) 3755.97 0.125489
\(965\) 0 0
\(966\) 34634.8 1.15358
\(967\) −23645.2 −0.786327 −0.393163 0.919469i \(-0.628619\pi\)
−0.393163 + 0.919469i \(0.628619\pi\)
\(968\) 7438.10 0.246973
\(969\) −4640.26 −0.153836
\(970\) 0 0
\(971\) 27402.0 0.905635 0.452818 0.891603i \(-0.350419\pi\)
0.452818 + 0.891603i \(0.350419\pi\)
\(972\) 4783.05 0.157836
\(973\) −27899.1 −0.919222
\(974\) −69790.6 −2.29593
\(975\) 0 0
\(976\) −33807.5 −1.10876
\(977\) 49118.7 1.60844 0.804220 0.594332i \(-0.202584\pi\)
0.804220 + 0.594332i \(0.202584\pi\)
\(978\) 42942.1 1.40403
\(979\) −4842.40 −0.158084
\(980\) 0 0
\(981\) 11306.6 0.367983
\(982\) −39294.0 −1.27691
\(983\) −52630.5 −1.70768 −0.853842 0.520533i \(-0.825733\pi\)
−0.853842 + 0.520533i \(0.825733\pi\)
\(984\) −41662.1 −1.34973
\(985\) 0 0
\(986\) −5191.47 −0.167677
\(987\) −12520.0 −0.403764
\(988\) −113609. −3.65827
\(989\) 73868.4 2.37500
\(990\) 0 0
\(991\) 45472.7 1.45761 0.728803 0.684724i \(-0.240077\pi\)
0.728803 + 0.684724i \(0.240077\pi\)
\(992\) −54054.5 −1.73007
\(993\) −4854.68 −0.155145
\(994\) 5020.04 0.160187
\(995\) 0 0
\(996\) −41713.8 −1.32706
\(997\) −55068.1 −1.74927 −0.874637 0.484779i \(-0.838900\pi\)
−0.874637 + 0.484779i \(0.838900\pi\)
\(998\) −27763.0 −0.880585
\(999\) 6965.01 0.220584
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 825.4.a.s.1.3 3
3.2 odd 2 2475.4.a.s.1.1 3
5.2 odd 4 825.4.c.l.199.6 6
5.3 odd 4 825.4.c.l.199.1 6
5.4 even 2 165.4.a.d.1.1 3
15.14 odd 2 495.4.a.l.1.3 3
55.54 odd 2 1815.4.a.s.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.1 3 5.4 even 2
495.4.a.l.1.3 3 15.14 odd 2
825.4.a.s.1.3 3 1.1 even 1 trivial
825.4.c.l.199.1 6 5.3 odd 4
825.4.c.l.199.6 6 5.2 odd 4
1815.4.a.s.1.3 3 55.54 odd 2
2475.4.a.s.1.1 3 3.2 odd 2