Properties

Label 525.4.d.j.274.2
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,Mod(274,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("525.274");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not 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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(-2.70156i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.j.274.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} -5.10469 q^{6} +7.00000i q^{7} -22.2984i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} -5.10469 q^{6} +7.00000i q^{7} -22.2984i q^{8} -9.00000 q^{9} +37.4031 q^{11} -15.3141i q^{12} -29.0156i q^{13} +11.9109 q^{14} +2.89531 q^{16} +58.4187i q^{17} +15.3141i q^{18} +54.5969 q^{19} +21.0000 q^{21} -63.6437i q^{22} -161.675i q^{23} -66.8953 q^{24} -49.3719 q^{26} +27.0000i q^{27} +35.7328i q^{28} -137.581 q^{29} +154.659 q^{31} -183.314i q^{32} -112.209i q^{33} +99.4031 q^{34} -45.9422 q^{36} -350.125i q^{37} -92.9000i q^{38} -87.0469 q^{39} +353.769 q^{41} -35.7328i q^{42} +518.156i q^{43} +190.931 q^{44} -275.100 q^{46} -542.219i q^{47} -8.68594i q^{48} -49.0000 q^{49} +175.256 q^{51} -148.116i q^{52} -305.309i q^{53} +45.9422 q^{54} +156.089 q^{56} -163.791i q^{57} +234.103i q^{58} -14.6813 q^{59} -171.069 q^{61} -263.163i q^{62} -63.0000i q^{63} -288.758 q^{64} -190.931 q^{66} +551.956i q^{67} +298.209i q^{68} -485.025 q^{69} -120.334 q^{71} +200.686i q^{72} -284.659i q^{73} -595.759 q^{74} +278.700 q^{76} +261.822i q^{77} +148.116i q^{78} -941.612 q^{79} +81.0000 q^{81} -601.959i q^{82} -377.150i q^{83} +107.198 q^{84} +881.675 q^{86} +412.744i q^{87} -834.031i q^{88} +677.725 q^{89} +203.109 q^{91} -825.300i q^{92} -463.978i q^{93} -922.619 q^{94} -549.942 q^{96} -1225.03i q^{97} +83.3765i q^{98} -336.628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{4} + 18 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{4} + 18 q^{6} - 36 q^{9} + 124 q^{11} - 42 q^{14} + 50 q^{16} + 244 q^{19} + 84 q^{21} - 306 q^{24} - 428 q^{26} - 704 q^{29} + 132 q^{31} + 372 q^{34} + 162 q^{36} + 36 q^{39} + 32 q^{41} - 312 q^{44} - 1920 q^{46} - 196 q^{49} + 240 q^{51} - 162 q^{54} + 714 q^{56} - 1032 q^{59} - 1760 q^{61} + 958 q^{64} + 312 q^{66} - 96 q^{69} + 620 q^{71} - 2716 q^{74} - 1344 q^{76} - 3664 q^{79} + 324 q^{81} - 378 q^{84} + 2912 q^{86} - 1592 q^{89} - 84 q^{91} - 5176 q^{94} - 1854 q^{96} - 1116 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.70156i − 0.601593i −0.953688 0.300797i \(-0.902747\pi\)
0.953688 0.300797i \(-0.0972525\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 5.10469 0.638086
\(5\) 0 0
\(6\) −5.10469 −0.347330
\(7\) 7.00000i 0.377964i
\(8\) − 22.2984i − 0.985461i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 37.4031 1.02522 0.512612 0.858620i \(-0.328678\pi\)
0.512612 + 0.858620i \(0.328678\pi\)
\(12\) − 15.3141i − 0.368399i
\(13\) − 29.0156i − 0.619037i −0.950893 0.309519i \(-0.899832\pi\)
0.950893 0.309519i \(-0.100168\pi\)
\(14\) 11.9109 0.227381
\(15\) 0 0
\(16\) 2.89531 0.0452393
\(17\) 58.4187i 0.833449i 0.909033 + 0.416724i \(0.136822\pi\)
−0.909033 + 0.416724i \(0.863178\pi\)
\(18\) 15.3141i 0.200531i
\(19\) 54.5969 0.659231 0.329615 0.944115i \(-0.393081\pi\)
0.329615 + 0.944115i \(0.393081\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 63.6437i − 0.616768i
\(23\) − 161.675i − 1.46572i −0.680379 0.732860i \(-0.738185\pi\)
0.680379 0.732860i \(-0.261815\pi\)
\(24\) −66.8953 −0.568956
\(25\) 0 0
\(26\) −49.3719 −0.372409
\(27\) 27.0000i 0.192450i
\(28\) 35.7328i 0.241174i
\(29\) −137.581 −0.880972 −0.440486 0.897759i \(-0.645194\pi\)
−0.440486 + 0.897759i \(0.645194\pi\)
\(30\) 0 0
\(31\) 154.659 0.896053 0.448026 0.894020i \(-0.352127\pi\)
0.448026 + 0.894020i \(0.352127\pi\)
\(32\) − 183.314i − 1.01268i
\(33\) − 112.209i − 0.591913i
\(34\) 99.4031 0.501397
\(35\) 0 0
\(36\) −45.9422 −0.212695
\(37\) − 350.125i − 1.55568i −0.628462 0.777840i \(-0.716315\pi\)
0.628462 0.777840i \(-0.283685\pi\)
\(38\) − 92.9000i − 0.396589i
\(39\) −87.0469 −0.357401
\(40\) 0 0
\(41\) 353.769 1.34755 0.673773 0.738938i \(-0.264673\pi\)
0.673773 + 0.738938i \(0.264673\pi\)
\(42\) − 35.7328i − 0.131278i
\(43\) 518.156i 1.83763i 0.394689 + 0.918815i \(0.370852\pi\)
−0.394689 + 0.918815i \(0.629148\pi\)
\(44\) 190.931 0.654181
\(45\) 0 0
\(46\) −275.100 −0.881767
\(47\) − 542.219i − 1.68278i −0.540427 0.841391i \(-0.681737\pi\)
0.540427 0.841391i \(-0.318263\pi\)
\(48\) − 8.68594i − 0.0261189i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 175.256 0.481192
\(52\) − 148.116i − 0.394999i
\(53\) − 305.309i − 0.791273i −0.918407 0.395637i \(-0.870524\pi\)
0.918407 0.395637i \(-0.129476\pi\)
\(54\) 45.9422 0.115777
\(55\) 0 0
\(56\) 156.089 0.372469
\(57\) − 163.791i − 0.380607i
\(58\) 234.103i 0.529987i
\(59\) −14.6813 −0.0323956 −0.0161978 0.999869i \(-0.505156\pi\)
−0.0161978 + 0.999869i \(0.505156\pi\)
\(60\) 0 0
\(61\) −171.069 −0.359067 −0.179534 0.983752i \(-0.557459\pi\)
−0.179534 + 0.983752i \(0.557459\pi\)
\(62\) − 263.163i − 0.539059i
\(63\) − 63.0000i − 0.125988i
\(64\) −288.758 −0.563980
\(65\) 0 0
\(66\) −190.931 −0.356091
\(67\) 551.956i 1.00645i 0.864155 + 0.503225i \(0.167853\pi\)
−0.864155 + 0.503225i \(0.832147\pi\)
\(68\) 298.209i 0.531812i
\(69\) −485.025 −0.846234
\(70\) 0 0
\(71\) −120.334 −0.201142 −0.100571 0.994930i \(-0.532067\pi\)
−0.100571 + 0.994930i \(0.532067\pi\)
\(72\) 200.686i 0.328487i
\(73\) − 284.659i − 0.456395i −0.973615 0.228198i \(-0.926717\pi\)
0.973615 0.228198i \(-0.0732832\pi\)
\(74\) −595.759 −0.935887
\(75\) 0 0
\(76\) 278.700 0.420646
\(77\) 261.822i 0.387498i
\(78\) 148.116i 0.215010i
\(79\) −941.612 −1.34101 −0.670504 0.741906i \(-0.733922\pi\)
−0.670504 + 0.741906i \(0.733922\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 601.959i − 0.810674i
\(83\) − 377.150i − 0.498766i −0.968405 0.249383i \(-0.919772\pi\)
0.968405 0.249383i \(-0.0802278\pi\)
\(84\) 107.198 0.139242
\(85\) 0 0
\(86\) 881.675 1.10551
\(87\) 412.744i 0.508630i
\(88\) − 834.031i − 1.01032i
\(89\) 677.725 0.807176 0.403588 0.914941i \(-0.367763\pi\)
0.403588 + 0.914941i \(0.367763\pi\)
\(90\) 0 0
\(91\) 203.109 0.233974
\(92\) − 825.300i − 0.935255i
\(93\) − 463.978i − 0.517336i
\(94\) −922.619 −1.01235
\(95\) 0 0
\(96\) −549.942 −0.584669
\(97\) − 1225.03i − 1.28230i −0.767414 0.641151i \(-0.778457\pi\)
0.767414 0.641151i \(-0.221543\pi\)
\(98\) 83.3765i 0.0859419i
\(99\) −336.628 −0.341741
\(100\) 0 0
\(101\) −338.144 −0.333134 −0.166567 0.986030i \(-0.553268\pi\)
−0.166567 + 0.986030i \(0.553268\pi\)
\(102\) − 298.209i − 0.289482i
\(103\) 566.700i 0.542122i 0.962562 + 0.271061i \(0.0873746\pi\)
−0.962562 + 0.271061i \(0.912625\pi\)
\(104\) −647.003 −0.610037
\(105\) 0 0
\(106\) −519.503 −0.476024
\(107\) − 562.531i − 0.508242i −0.967172 0.254121i \(-0.918214\pi\)
0.967172 0.254121i \(-0.0817862\pi\)
\(108\) 137.827i 0.122800i
\(109\) −1830.79 −1.60879 −0.804396 0.594094i \(-0.797511\pi\)
−0.804396 + 0.594094i \(0.797511\pi\)
\(110\) 0 0
\(111\) −1050.37 −0.898173
\(112\) 20.2672i 0.0170988i
\(113\) 31.8032i 0.0264761i 0.999912 + 0.0132380i \(0.00421392\pi\)
−0.999912 + 0.0132380i \(0.995786\pi\)
\(114\) −278.700 −0.228971
\(115\) 0 0
\(116\) −702.309 −0.562136
\(117\) 261.141i 0.206346i
\(118\) 24.9811i 0.0194890i
\(119\) −408.931 −0.315014
\(120\) 0 0
\(121\) 67.9937 0.0510847
\(122\) 291.084i 0.216012i
\(123\) − 1061.31i − 0.778006i
\(124\) 789.488 0.571759
\(125\) 0 0
\(126\) −107.198 −0.0757936
\(127\) 2220.81i 1.55169i 0.630921 + 0.775847i \(0.282677\pi\)
−0.630921 + 0.775847i \(0.717323\pi\)
\(128\) − 975.173i − 0.673390i
\(129\) 1554.47 1.06096
\(130\) 0 0
\(131\) 646.512 0.431191 0.215596 0.976483i \(-0.430831\pi\)
0.215596 + 0.976483i \(0.430831\pi\)
\(132\) − 572.794i − 0.377692i
\(133\) 382.178i 0.249166i
\(134\) 939.188 0.605474
\(135\) 0 0
\(136\) 1302.65 0.821331
\(137\) − 896.009i − 0.558768i −0.960179 0.279384i \(-0.909870\pi\)
0.960179 0.279384i \(-0.0901303\pi\)
\(138\) 825.300i 0.509088i
\(139\) 2313.61 1.41178 0.705891 0.708320i \(-0.250547\pi\)
0.705891 + 0.708320i \(0.250547\pi\)
\(140\) 0 0
\(141\) −1626.66 −0.971554
\(142\) 204.756i 0.121005i
\(143\) − 1085.27i − 0.634652i
\(144\) −26.0578 −0.0150798
\(145\) 0 0
\(146\) −484.366 −0.274564
\(147\) 147.000i 0.0824786i
\(148\) − 1787.28i − 0.992658i
\(149\) −819.337 −0.450488 −0.225244 0.974302i \(-0.572318\pi\)
−0.225244 + 0.974302i \(0.572318\pi\)
\(150\) 0 0
\(151\) 534.744 0.288191 0.144095 0.989564i \(-0.453973\pi\)
0.144095 + 0.989564i \(0.453973\pi\)
\(152\) − 1217.43i − 0.649646i
\(153\) − 525.769i − 0.277816i
\(154\) 445.506 0.233116
\(155\) 0 0
\(156\) −444.347 −0.228053
\(157\) 1564.76i 0.795423i 0.917511 + 0.397711i \(0.130195\pi\)
−0.917511 + 0.397711i \(0.869805\pi\)
\(158\) 1602.21i 0.806741i
\(159\) −915.928 −0.456842
\(160\) 0 0
\(161\) 1131.72 0.553990
\(162\) − 137.827i − 0.0668437i
\(163\) 1114.31i 0.535455i 0.963495 + 0.267728i \(0.0862727\pi\)
−0.963495 + 0.267728i \(0.913727\pi\)
\(164\) 1805.88 0.859850
\(165\) 0 0
\(166\) −641.744 −0.300054
\(167\) − 1774.47i − 0.822231i −0.911583 0.411115i \(-0.865139\pi\)
0.911583 0.411115i \(-0.134861\pi\)
\(168\) − 468.267i − 0.215045i
\(169\) 1355.09 0.616793
\(170\) 0 0
\(171\) −491.372 −0.219744
\(172\) 2645.02i 1.17257i
\(173\) 4215.88i 1.85276i 0.376590 + 0.926380i \(0.377097\pi\)
−0.376590 + 0.926380i \(0.622903\pi\)
\(174\) 702.309 0.305988
\(175\) 0 0
\(176\) 108.294 0.0463804
\(177\) 44.0438i 0.0187036i
\(178\) − 1153.19i − 0.485592i
\(179\) 2430.70 1.01497 0.507483 0.861662i \(-0.330576\pi\)
0.507483 + 0.861662i \(0.330576\pi\)
\(180\) 0 0
\(181\) −2700.91 −1.10916 −0.554578 0.832132i \(-0.687120\pi\)
−0.554578 + 0.832132i \(0.687120\pi\)
\(182\) − 345.603i − 0.140757i
\(183\) 513.206i 0.207308i
\(184\) −3605.10 −1.44441
\(185\) 0 0
\(186\) −789.488 −0.311226
\(187\) 2185.04i 0.854472i
\(188\) − 2767.86i − 1.07376i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 3611.10 1.36801 0.684005 0.729478i \(-0.260237\pi\)
0.684005 + 0.729478i \(0.260237\pi\)
\(192\) 866.273i 0.325614i
\(193\) 4468.33i 1.66651i 0.552886 + 0.833257i \(0.313526\pi\)
−0.552886 + 0.833257i \(0.686474\pi\)
\(194\) −2084.47 −0.771425
\(195\) 0 0
\(196\) −250.130 −0.0911551
\(197\) − 434.422i − 0.157113i −0.996910 0.0785566i \(-0.974969\pi\)
0.996910 0.0785566i \(-0.0250311\pi\)
\(198\) 572.794i 0.205589i
\(199\) 468.915 0.167038 0.0835189 0.996506i \(-0.473384\pi\)
0.0835189 + 0.996506i \(0.473384\pi\)
\(200\) 0 0
\(201\) 1655.87 0.581074
\(202\) 575.372i 0.200411i
\(203\) − 963.069i − 0.332976i
\(204\) 894.628 0.307042
\(205\) 0 0
\(206\) 964.275 0.326137
\(207\) 1455.07i 0.488573i
\(208\) − 84.0093i − 0.0280048i
\(209\) 2042.09 0.675859
\(210\) 0 0
\(211\) 3735.51 1.21878 0.609392 0.792869i \(-0.291414\pi\)
0.609392 + 0.792869i \(0.291414\pi\)
\(212\) − 1558.51i − 0.504900i
\(213\) 361.003i 0.116129i
\(214\) −957.182 −0.305755
\(215\) 0 0
\(216\) 602.058 0.189652
\(217\) 1082.62i 0.338676i
\(218\) 3115.21i 0.967838i
\(219\) −853.978 −0.263500
\(220\) 0 0
\(221\) 1695.06 0.515936
\(222\) 1787.28i 0.540334i
\(223\) − 842.806i − 0.253087i −0.991961 0.126544i \(-0.959612\pi\)
0.991961 0.126544i \(-0.0403884\pi\)
\(224\) 1283.20 0.382756
\(225\) 0 0
\(226\) 54.1152 0.0159278
\(227\) 992.150i 0.290094i 0.989425 + 0.145047i \(0.0463333\pi\)
−0.989425 + 0.145047i \(0.953667\pi\)
\(228\) − 836.100i − 0.242860i
\(229\) 6411.39 1.85012 0.925059 0.379825i \(-0.124016\pi\)
0.925059 + 0.379825i \(0.124016\pi\)
\(230\) 0 0
\(231\) 785.466 0.223722
\(232\) 3067.85i 0.868164i
\(233\) 2274.35i 0.639476i 0.947506 + 0.319738i \(0.103595\pi\)
−0.947506 + 0.319738i \(0.896405\pi\)
\(234\) 444.347 0.124136
\(235\) 0 0
\(236\) −74.9433 −0.0206712
\(237\) 2824.84i 0.774232i
\(238\) 695.822i 0.189510i
\(239\) −2863.12 −0.774893 −0.387447 0.921892i \(-0.626643\pi\)
−0.387447 + 0.921892i \(0.626643\pi\)
\(240\) 0 0
\(241\) −5364.23 −1.43378 −0.716889 0.697187i \(-0.754435\pi\)
−0.716889 + 0.697187i \(0.754435\pi\)
\(242\) − 115.696i − 0.0307322i
\(243\) − 243.000i − 0.0641500i
\(244\) −873.252 −0.229116
\(245\) 0 0
\(246\) −1805.88 −0.468043
\(247\) − 1584.16i − 0.408088i
\(248\) − 3448.66i − 0.883025i
\(249\) −1131.45 −0.287963
\(250\) 0 0
\(251\) 5569.81 1.40065 0.700325 0.713824i \(-0.253039\pi\)
0.700325 + 0.713824i \(0.253039\pi\)
\(252\) − 321.595i − 0.0803913i
\(253\) − 6047.15i − 1.50269i
\(254\) 3778.85 0.933489
\(255\) 0 0
\(256\) −3969.38 −0.969087
\(257\) 2095.36i 0.508580i 0.967128 + 0.254290i \(0.0818418\pi\)
−0.967128 + 0.254290i \(0.918158\pi\)
\(258\) − 2645.02i − 0.638264i
\(259\) 2450.87 0.587992
\(260\) 0 0
\(261\) 1238.23 0.293657
\(262\) − 1100.08i − 0.259402i
\(263\) 7465.88i 1.75044i 0.483724 + 0.875220i \(0.339284\pi\)
−0.483724 + 0.875220i \(0.660716\pi\)
\(264\) −2502.09 −0.583308
\(265\) 0 0
\(266\) 650.300 0.149896
\(267\) − 2033.17i − 0.466023i
\(268\) 2817.56i 0.642202i
\(269\) 6521.38 1.47812 0.739062 0.673637i \(-0.235269\pi\)
0.739062 + 0.673637i \(0.235269\pi\)
\(270\) 0 0
\(271\) 2409.70 0.540144 0.270072 0.962840i \(-0.412952\pi\)
0.270072 + 0.962840i \(0.412952\pi\)
\(272\) 169.141i 0.0377046i
\(273\) − 609.328i − 0.135085i
\(274\) −1524.62 −0.336151
\(275\) 0 0
\(276\) −2475.90 −0.539970
\(277\) − 2219.83i − 0.481503i −0.970587 0.240752i \(-0.922606\pi\)
0.970587 0.240752i \(-0.0773939\pi\)
\(278\) − 3936.75i − 0.849319i
\(279\) −1391.93 −0.298684
\(280\) 0 0
\(281\) 5838.56 1.23950 0.619749 0.784800i \(-0.287234\pi\)
0.619749 + 0.784800i \(0.287234\pi\)
\(282\) 2767.86i 0.584480i
\(283\) 3645.04i 0.765636i 0.923824 + 0.382818i \(0.125046\pi\)
−0.923824 + 0.382818i \(0.874954\pi\)
\(284\) −614.269 −0.128346
\(285\) 0 0
\(286\) −1846.66 −0.381802
\(287\) 2476.38i 0.509325i
\(288\) 1649.83i 0.337559i
\(289\) 1500.25 0.305363
\(290\) 0 0
\(291\) −3675.10 −0.740338
\(292\) − 1453.10i − 0.291219i
\(293\) − 3777.91i − 0.753268i −0.926362 0.376634i \(-0.877081\pi\)
0.926362 0.376634i \(-0.122919\pi\)
\(294\) 250.130 0.0496186
\(295\) 0 0
\(296\) −7807.24 −1.53306
\(297\) 1009.88i 0.197304i
\(298\) 1394.15i 0.271011i
\(299\) −4691.10 −0.907336
\(300\) 0 0
\(301\) −3627.09 −0.694559
\(302\) − 909.900i − 0.173374i
\(303\) 1014.43i 0.192335i
\(304\) 158.075 0.0298231
\(305\) 0 0
\(306\) −894.628 −0.167132
\(307\) − 4799.64i − 0.892281i −0.894963 0.446140i \(-0.852798\pi\)
0.894963 0.446140i \(-0.147202\pi\)
\(308\) 1336.52i 0.247257i
\(309\) 1700.10 0.312994
\(310\) 0 0
\(311\) 580.113 0.105772 0.0528861 0.998601i \(-0.483158\pi\)
0.0528861 + 0.998601i \(0.483158\pi\)
\(312\) 1941.01i 0.352205i
\(313\) 6114.78i 1.10424i 0.833764 + 0.552121i \(0.186182\pi\)
−0.833764 + 0.552121i \(0.813818\pi\)
\(314\) 2662.53 0.478521
\(315\) 0 0
\(316\) −4806.64 −0.855679
\(317\) − 4300.63i − 0.761979i −0.924579 0.380989i \(-0.875583\pi\)
0.924579 0.380989i \(-0.124417\pi\)
\(318\) 1558.51i 0.274833i
\(319\) −5145.97 −0.903194
\(320\) 0 0
\(321\) −1687.59 −0.293434
\(322\) − 1925.70i − 0.333277i
\(323\) 3189.48i 0.549435i
\(324\) 413.480 0.0708984
\(325\) 0 0
\(326\) 1896.06 0.322126
\(327\) 5492.38i 0.928836i
\(328\) − 7888.49i − 1.32795i
\(329\) 3795.53 0.636032
\(330\) 0 0
\(331\) −6687.54 −1.11051 −0.555257 0.831679i \(-0.687380\pi\)
−0.555257 + 0.831679i \(0.687380\pi\)
\(332\) − 1925.23i − 0.318256i
\(333\) 3151.12i 0.518560i
\(334\) −3019.37 −0.494648
\(335\) 0 0
\(336\) 60.8016 0.00987202
\(337\) 5869.28i 0.948723i 0.880330 + 0.474362i \(0.157321\pi\)
−0.880330 + 0.474362i \(0.842679\pi\)
\(338\) − 2305.78i − 0.371058i
\(339\) 95.4097 0.0152860
\(340\) 0 0
\(341\) 5784.74 0.918655
\(342\) 836.100i 0.132196i
\(343\) − 343.000i − 0.0539949i
\(344\) 11554.1 1.81091
\(345\) 0 0
\(346\) 7173.58 1.11461
\(347\) 1937.22i 0.299699i 0.988709 + 0.149850i \(0.0478790\pi\)
−0.988709 + 0.149850i \(0.952121\pi\)
\(348\) 2106.93i 0.324549i
\(349\) 9748.82 1.49525 0.747625 0.664121i \(-0.231194\pi\)
0.747625 + 0.664121i \(0.231194\pi\)
\(350\) 0 0
\(351\) 783.422 0.119134
\(352\) − 6856.52i − 1.03822i
\(353\) 4576.61i 0.690052i 0.938593 + 0.345026i \(0.112130\pi\)
−0.938593 + 0.345026i \(0.887870\pi\)
\(354\) 74.9433 0.0112520
\(355\) 0 0
\(356\) 3459.57 0.515048
\(357\) 1226.79i 0.181873i
\(358\) − 4135.98i − 0.610596i
\(359\) −10849.9 −1.59509 −0.797546 0.603258i \(-0.793869\pi\)
−0.797546 + 0.603258i \(0.793869\pi\)
\(360\) 0 0
\(361\) −3878.18 −0.565415
\(362\) 4595.77i 0.667261i
\(363\) − 203.981i − 0.0294938i
\(364\) 1036.81 0.149296
\(365\) 0 0
\(366\) 873.252 0.124715
\(367\) 11467.7i 1.63108i 0.578699 + 0.815541i \(0.303561\pi\)
−0.578699 + 0.815541i \(0.696439\pi\)
\(368\) − 468.100i − 0.0663081i
\(369\) −3183.92 −0.449182
\(370\) 0 0
\(371\) 2137.17 0.299073
\(372\) − 2368.46i − 0.330105i
\(373\) − 539.982i − 0.0749576i −0.999297 0.0374788i \(-0.988067\pi\)
0.999297 0.0374788i \(-0.0119327\pi\)
\(374\) 3717.99 0.514044
\(375\) 0 0
\(376\) −12090.6 −1.65832
\(377\) 3992.01i 0.545355i
\(378\) 321.595i 0.0437595i
\(379\) −8577.57 −1.16253 −0.581267 0.813713i \(-0.697443\pi\)
−0.581267 + 0.813713i \(0.697443\pi\)
\(380\) 0 0
\(381\) 6662.44 0.895871
\(382\) − 6144.51i − 0.822985i
\(383\) − 8627.96i − 1.15109i −0.817770 0.575546i \(-0.804790\pi\)
0.817770 0.575546i \(-0.195210\pi\)
\(384\) −2925.52 −0.388782
\(385\) 0 0
\(386\) 7603.13 1.00256
\(387\) − 4663.41i − 0.612543i
\(388\) − 6253.42i − 0.818219i
\(389\) −9234.06 −1.20356 −0.601781 0.798661i \(-0.705542\pi\)
−0.601781 + 0.798661i \(0.705542\pi\)
\(390\) 0 0
\(391\) 9444.85 1.22160
\(392\) 1092.62i 0.140780i
\(393\) − 1939.54i − 0.248948i
\(394\) −739.196 −0.0945182
\(395\) 0 0
\(396\) −1718.38 −0.218060
\(397\) 11618.0i 1.46874i 0.678747 + 0.734372i \(0.262523\pi\)
−0.678747 + 0.734372i \(0.737477\pi\)
\(398\) − 797.889i − 0.100489i
\(399\) 1146.53 0.143856
\(400\) 0 0
\(401\) 11157.1 1.38942 0.694711 0.719289i \(-0.255532\pi\)
0.694711 + 0.719289i \(0.255532\pi\)
\(402\) − 2817.56i − 0.349570i
\(403\) − 4487.54i − 0.554690i
\(404\) −1726.12 −0.212568
\(405\) 0 0
\(406\) −1638.72 −0.200316
\(407\) − 13095.8i − 1.59492i
\(408\) − 3907.94i − 0.474196i
\(409\) 7428.08 0.898031 0.449015 0.893524i \(-0.351775\pi\)
0.449015 + 0.893524i \(0.351775\pi\)
\(410\) 0 0
\(411\) −2688.03 −0.322605
\(412\) 2892.83i 0.345921i
\(413\) − 102.769i − 0.0122444i
\(414\) 2475.90 0.293922
\(415\) 0 0
\(416\) −5318.97 −0.626885
\(417\) − 6940.83i − 0.815093i
\(418\) − 3474.75i − 0.406592i
\(419\) −9644.74 −1.12453 −0.562263 0.826959i \(-0.690069\pi\)
−0.562263 + 0.826959i \(0.690069\pi\)
\(420\) 0 0
\(421\) 9918.18 1.14818 0.574088 0.818793i \(-0.305357\pi\)
0.574088 + 0.818793i \(0.305357\pi\)
\(422\) − 6356.21i − 0.733212i
\(423\) 4879.97i 0.560927i
\(424\) −6807.92 −0.779769
\(425\) 0 0
\(426\) 614.269 0.0698625
\(427\) − 1197.48i − 0.135715i
\(428\) − 2871.54i − 0.324302i
\(429\) −3255.82 −0.366417
\(430\) 0 0
\(431\) −16324.1 −1.82437 −0.912185 0.409779i \(-0.865606\pi\)
−0.912185 + 0.409779i \(0.865606\pi\)
\(432\) 78.1735i 0.00870630i
\(433\) 5168.75i 0.573659i 0.957982 + 0.286829i \(0.0926012\pi\)
−0.957982 + 0.286829i \(0.907399\pi\)
\(434\) 1842.14 0.203745
\(435\) 0 0
\(436\) −9345.63 −1.02655
\(437\) − 8826.95i − 0.966248i
\(438\) 1453.10i 0.158520i
\(439\) −18339.0 −1.99378 −0.996892 0.0787782i \(-0.974898\pi\)
−0.996892 + 0.0787782i \(0.974898\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 2884.24i − 0.310383i
\(443\) 1613.28i 0.173023i 0.996251 + 0.0865113i \(0.0275719\pi\)
−0.996251 + 0.0865113i \(0.972428\pi\)
\(444\) −5361.83 −0.573111
\(445\) 0 0
\(446\) −1434.09 −0.152256
\(447\) 2458.01i 0.260089i
\(448\) − 2021.30i − 0.213164i
\(449\) 886.750 0.0932034 0.0466017 0.998914i \(-0.485161\pi\)
0.0466017 + 0.998914i \(0.485161\pi\)
\(450\) 0 0
\(451\) 13232.1 1.38154
\(452\) 162.345i 0.0168940i
\(453\) − 1604.23i − 0.166387i
\(454\) 1688.21 0.174518
\(455\) 0 0
\(456\) −3652.28 −0.375073
\(457\) 7391.22i 0.756557i 0.925692 + 0.378279i \(0.123484\pi\)
−0.925692 + 0.378279i \(0.876516\pi\)
\(458\) − 10909.4i − 1.11302i
\(459\) −1577.31 −0.160397
\(460\) 0 0
\(461\) −7133.35 −0.720679 −0.360340 0.932821i \(-0.617339\pi\)
−0.360340 + 0.932821i \(0.617339\pi\)
\(462\) − 1336.52i − 0.134590i
\(463\) − 14461.8i − 1.45162i −0.687897 0.725808i \(-0.741466\pi\)
0.687897 0.725808i \(-0.258534\pi\)
\(464\) −398.341 −0.0398546
\(465\) 0 0
\(466\) 3869.95 0.384704
\(467\) − 16393.5i − 1.62441i −0.583370 0.812206i \(-0.698266\pi\)
0.583370 0.812206i \(-0.301734\pi\)
\(468\) 1333.04i 0.131666i
\(469\) −3863.69 −0.380403
\(470\) 0 0
\(471\) 4694.28 0.459238
\(472\) 327.370i 0.0319246i
\(473\) 19380.7i 1.88398i
\(474\) 4806.64 0.465772
\(475\) 0 0
\(476\) −2087.47 −0.201006
\(477\) 2747.78i 0.263758i
\(478\) 4871.77i 0.466171i
\(479\) 12991.4 1.23923 0.619617 0.784904i \(-0.287288\pi\)
0.619617 + 0.784904i \(0.287288\pi\)
\(480\) 0 0
\(481\) −10159.1 −0.963025
\(482\) 9127.57i 0.862551i
\(483\) − 3395.17i − 0.319846i
\(484\) 347.087 0.0325964
\(485\) 0 0
\(486\) −413.480 −0.0385922
\(487\) − 12863.5i − 1.19692i −0.801152 0.598461i \(-0.795779\pi\)
0.801152 0.598461i \(-0.204221\pi\)
\(488\) 3814.57i 0.353847i
\(489\) 3342.92 0.309145
\(490\) 0 0
\(491\) −4898.10 −0.450200 −0.225100 0.974336i \(-0.572271\pi\)
−0.225100 + 0.974336i \(0.572271\pi\)
\(492\) − 5417.63i − 0.496435i
\(493\) − 8037.32i − 0.734245i
\(494\) −2695.55 −0.245503
\(495\) 0 0
\(496\) 447.787 0.0405368
\(497\) − 842.340i − 0.0760244i
\(498\) 1925.23i 0.173236i
\(499\) −10308.0 −0.924746 −0.462373 0.886686i \(-0.653002\pi\)
−0.462373 + 0.886686i \(0.653002\pi\)
\(500\) 0 0
\(501\) −5323.41 −0.474715
\(502\) − 9477.37i − 0.842621i
\(503\) 15119.6i 1.34026i 0.742244 + 0.670130i \(0.233762\pi\)
−0.742244 + 0.670130i \(0.766238\pi\)
\(504\) −1404.80 −0.124156
\(505\) 0 0
\(506\) −10289.6 −0.904009
\(507\) − 4065.28i − 0.356105i
\(508\) 11336.6i 0.990114i
\(509\) −14183.8 −1.23514 −0.617571 0.786515i \(-0.711883\pi\)
−0.617571 + 0.786515i \(0.711883\pi\)
\(510\) 0 0
\(511\) 1992.62 0.172501
\(512\) − 1047.24i − 0.0903943i
\(513\) 1474.12i 0.126869i
\(514\) 3565.39 0.305958
\(515\) 0 0
\(516\) 7935.07 0.676981
\(517\) − 20280.7i − 1.72523i
\(518\) − 4170.32i − 0.353732i
\(519\) 12647.6 1.06969
\(520\) 0 0
\(521\) −7464.08 −0.627653 −0.313827 0.949480i \(-0.601611\pi\)
−0.313827 + 0.949480i \(0.601611\pi\)
\(522\) − 2106.93i − 0.176662i
\(523\) 16642.9i 1.39148i 0.718295 + 0.695739i \(0.244923\pi\)
−0.718295 + 0.695739i \(0.755077\pi\)
\(524\) 3300.24 0.275137
\(525\) 0 0
\(526\) 12703.7 1.05305
\(527\) 9035.01i 0.746814i
\(528\) − 324.881i − 0.0267777i
\(529\) −13971.8 −1.14834
\(530\) 0 0
\(531\) 132.132 0.0107985
\(532\) 1950.90i 0.158989i
\(533\) − 10264.8i − 0.834181i
\(534\) −3459.57 −0.280356
\(535\) 0 0
\(536\) 12307.8 0.991818
\(537\) − 7292.09i − 0.585991i
\(538\) − 11096.5i − 0.889230i
\(539\) −1832.75 −0.146461
\(540\) 0 0
\(541\) 67.8755 0.00539408 0.00269704 0.999996i \(-0.499142\pi\)
0.00269704 + 0.999996i \(0.499142\pi\)
\(542\) − 4100.26i − 0.324947i
\(543\) 8102.74i 0.640372i
\(544\) 10709.0 0.844014
\(545\) 0 0
\(546\) −1036.81 −0.0812662
\(547\) − 9212.91i − 0.720138i −0.932926 0.360069i \(-0.882753\pi\)
0.932926 0.360069i \(-0.117247\pi\)
\(548\) − 4573.85i − 0.356542i
\(549\) 1539.62 0.119689
\(550\) 0 0
\(551\) −7511.51 −0.580764
\(552\) 10815.3i 0.833931i
\(553\) − 6591.29i − 0.506854i
\(554\) −3777.17 −0.289669
\(555\) 0 0
\(556\) 11810.2 0.900838
\(557\) 16699.6i 1.27035i 0.772370 + 0.635173i \(0.219071\pi\)
−0.772370 + 0.635173i \(0.780929\pi\)
\(558\) 2368.46i 0.179686i
\(559\) 15034.6 1.13756
\(560\) 0 0
\(561\) 6555.13 0.493329
\(562\) − 9934.67i − 0.745674i
\(563\) − 14772.8i − 1.10586i −0.833227 0.552931i \(-0.813509\pi\)
0.833227 0.552931i \(-0.186491\pi\)
\(564\) −8303.57 −0.619935
\(565\) 0 0
\(566\) 6202.26 0.460601
\(567\) 567.000i 0.0419961i
\(568\) 2683.27i 0.198217i
\(569\) −5663.76 −0.417289 −0.208644 0.977992i \(-0.566905\pi\)
−0.208644 + 0.977992i \(0.566905\pi\)
\(570\) 0 0
\(571\) 5579.58 0.408929 0.204464 0.978874i \(-0.434455\pi\)
0.204464 + 0.978874i \(0.434455\pi\)
\(572\) − 5539.99i − 0.404962i
\(573\) − 10833.3i − 0.789821i
\(574\) 4213.72 0.306406
\(575\) 0 0
\(576\) 2598.82 0.187993
\(577\) − 2301.23i − 0.166034i −0.996548 0.0830170i \(-0.973544\pi\)
0.996548 0.0830170i \(-0.0264556\pi\)
\(578\) − 2552.77i − 0.183704i
\(579\) 13405.0 0.962162
\(580\) 0 0
\(581\) 2640.05 0.188516
\(582\) 6253.42i 0.445382i
\(583\) − 11419.5i − 0.811232i
\(584\) −6347.46 −0.449760
\(585\) 0 0
\(586\) −6428.34 −0.453161
\(587\) − 16470.4i − 1.15810i −0.815291 0.579052i \(-0.803423\pi\)
0.815291 0.579052i \(-0.196577\pi\)
\(588\) 750.389i 0.0526284i
\(589\) 8443.92 0.590706
\(590\) 0 0
\(591\) −1303.27 −0.0907093
\(592\) − 1013.72i − 0.0703779i
\(593\) 13570.0i 0.939715i 0.882742 + 0.469858i \(0.155695\pi\)
−0.882742 + 0.469858i \(0.844305\pi\)
\(594\) 1718.38 0.118697
\(595\) 0 0
\(596\) −4182.46 −0.287450
\(597\) − 1406.75i − 0.0964393i
\(598\) 7982.20i 0.545847i
\(599\) −27814.1 −1.89725 −0.948625 0.316403i \(-0.897525\pi\)
−0.948625 + 0.316403i \(0.897525\pi\)
\(600\) 0 0
\(601\) 20646.1 1.40128 0.700641 0.713514i \(-0.252898\pi\)
0.700641 + 0.713514i \(0.252898\pi\)
\(602\) 6171.72i 0.417842i
\(603\) − 4967.61i − 0.335483i
\(604\) 2729.70 0.183891
\(605\) 0 0
\(606\) 1726.12 0.115707
\(607\) 3315.28i 0.221686i 0.993838 + 0.110843i \(0.0353550\pi\)
−0.993838 + 0.110843i \(0.964645\pi\)
\(608\) − 10008.4i − 0.667588i
\(609\) −2889.21 −0.192244
\(610\) 0 0
\(611\) −15732.8 −1.04170
\(612\) − 2683.88i − 0.177271i
\(613\) 11113.9i 0.732278i 0.930560 + 0.366139i \(0.119320\pi\)
−0.930560 + 0.366139i \(0.880680\pi\)
\(614\) −8166.89 −0.536790
\(615\) 0 0
\(616\) 5838.22 0.381865
\(617\) 7871.34i 0.513595i 0.966465 + 0.256797i \(0.0826673\pi\)
−0.966465 + 0.256797i \(0.917333\pi\)
\(618\) − 2892.83i − 0.188295i
\(619\) 19107.1 1.24068 0.620339 0.784334i \(-0.286995\pi\)
0.620339 + 0.784334i \(0.286995\pi\)
\(620\) 0 0
\(621\) 4365.22 0.282078
\(622\) − 987.098i − 0.0636319i
\(623\) 4744.07i 0.305084i
\(624\) −252.028 −0.0161686
\(625\) 0 0
\(626\) 10404.7 0.664305
\(627\) − 6126.28i − 0.390208i
\(628\) 7987.60i 0.507548i
\(629\) 20453.9 1.29658
\(630\) 0 0
\(631\) −25769.9 −1.62580 −0.812902 0.582401i \(-0.802113\pi\)
−0.812902 + 0.582401i \(0.802113\pi\)
\(632\) 20996.5i 1.32151i
\(633\) − 11206.5i − 0.703665i
\(634\) −7317.79 −0.458401
\(635\) 0 0
\(636\) −4675.53 −0.291504
\(637\) 1421.77i 0.0884339i
\(638\) 8756.19i 0.543355i
\(639\) 1083.01 0.0670472
\(640\) 0 0
\(641\) −1954.61 −0.120440 −0.0602202 0.998185i \(-0.519180\pi\)
−0.0602202 + 0.998185i \(0.519180\pi\)
\(642\) 2871.54i 0.176528i
\(643\) 19396.5i 1.18961i 0.803868 + 0.594807i \(0.202772\pi\)
−0.803868 + 0.594807i \(0.797228\pi\)
\(644\) 5777.10 0.353493
\(645\) 0 0
\(646\) 5427.10 0.330536
\(647\) 31264.3i 1.89973i 0.312661 + 0.949865i \(0.398780\pi\)
−0.312661 + 0.949865i \(0.601220\pi\)
\(648\) − 1806.17i − 0.109496i
\(649\) −549.126 −0.0332127
\(650\) 0 0
\(651\) 3247.85 0.195535
\(652\) 5688.18i 0.341666i
\(653\) − 6442.75i − 0.386102i −0.981189 0.193051i \(-0.938162\pi\)
0.981189 0.193051i \(-0.0618382\pi\)
\(654\) 9345.63 0.558781
\(655\) 0 0
\(656\) 1024.27 0.0609620
\(657\) 2561.93i 0.152132i
\(658\) − 6458.33i − 0.382632i
\(659\) 3584.70 0.211897 0.105949 0.994372i \(-0.466212\pi\)
0.105949 + 0.994372i \(0.466212\pi\)
\(660\) 0 0
\(661\) 6294.43 0.370386 0.185193 0.982702i \(-0.440709\pi\)
0.185193 + 0.982702i \(0.440709\pi\)
\(662\) 11379.3i 0.668078i
\(663\) − 5085.17i − 0.297876i
\(664\) −8409.85 −0.491514
\(665\) 0 0
\(666\) 5361.83 0.311962
\(667\) 22243.4i 1.29126i
\(668\) − 9058.11i − 0.524654i
\(669\) −2528.42 −0.146120
\(670\) 0 0
\(671\) −6398.51 −0.368125
\(672\) − 3849.60i − 0.220984i
\(673\) 10233.9i 0.586162i 0.956088 + 0.293081i \(0.0946806\pi\)
−0.956088 + 0.293081i \(0.905319\pi\)
\(674\) 9986.94 0.570745
\(675\) 0 0
\(676\) 6917.33 0.393567
\(677\) − 7100.75i − 0.403108i −0.979477 0.201554i \(-0.935401\pi\)
0.979477 0.201554i \(-0.0645992\pi\)
\(678\) − 162.345i − 0.00919593i
\(679\) 8575.24 0.484665
\(680\) 0 0
\(681\) 2976.45 0.167486
\(682\) − 9843.10i − 0.552657i
\(683\) 35274.6i 1.97620i 0.153813 + 0.988100i \(0.450845\pi\)
−0.153813 + 0.988100i \(0.549155\pi\)
\(684\) −2508.30 −0.140215
\(685\) 0 0
\(686\) −583.636 −0.0324830
\(687\) − 19234.2i − 1.06817i
\(688\) 1500.22i 0.0831330i
\(689\) −8858.74 −0.489828
\(690\) 0 0
\(691\) 4945.12 0.272245 0.136122 0.990692i \(-0.456536\pi\)
0.136122 + 0.990692i \(0.456536\pi\)
\(692\) 21520.8i 1.18222i
\(693\) − 2356.40i − 0.129166i
\(694\) 3296.31 0.180297
\(695\) 0 0
\(696\) 9203.54 0.501235
\(697\) 20666.7i 1.12311i
\(698\) − 16588.2i − 0.899532i
\(699\) 6823.06 0.369201
\(700\) 0 0
\(701\) −15300.4 −0.824379 −0.412190 0.911098i \(-0.635236\pi\)
−0.412190 + 0.911098i \(0.635236\pi\)
\(702\) − 1333.04i − 0.0716701i
\(703\) − 19115.7i − 1.02555i
\(704\) −10800.4 −0.578206
\(705\) 0 0
\(706\) 7787.38 0.415130
\(707\) − 2367.01i − 0.125913i
\(708\) 224.830i 0.0119345i
\(709\) 28297.4 1.49892 0.749458 0.662052i \(-0.230314\pi\)
0.749458 + 0.662052i \(0.230314\pi\)
\(710\) 0 0
\(711\) 8474.51 0.447003
\(712\) − 15112.2i − 0.795441i
\(713\) − 25004.5i − 1.31336i
\(714\) 2087.47 0.109414
\(715\) 0 0
\(716\) 12407.9 0.647635
\(717\) 8589.35i 0.447385i
\(718\) 18461.9i 0.959597i
\(719\) −8548.96 −0.443425 −0.221712 0.975112i \(-0.571165\pi\)
−0.221712 + 0.975112i \(0.571165\pi\)
\(720\) 0 0
\(721\) −3966.90 −0.204903
\(722\) 6598.97i 0.340150i
\(723\) 16092.7i 0.827792i
\(724\) −13787.3 −0.707737
\(725\) 0 0
\(726\) −347.087 −0.0177432
\(727\) 14345.3i 0.731827i 0.930649 + 0.365913i \(0.119243\pi\)
−0.930649 + 0.365913i \(0.880757\pi\)
\(728\) − 4529.02i − 0.230572i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −30270.0 −1.53157
\(732\) 2619.76i 0.132280i
\(733\) 22624.0i 1.14002i 0.821637 + 0.570012i \(0.193061\pi\)
−0.821637 + 0.570012i \(0.806939\pi\)
\(734\) 19512.9 0.981248
\(735\) 0 0
\(736\) −29637.3 −1.48430
\(737\) 20644.9i 1.03184i
\(738\) 5417.63i 0.270225i
\(739\) −14837.3 −0.738566 −0.369283 0.929317i \(-0.620397\pi\)
−0.369283 + 0.929317i \(0.620397\pi\)
\(740\) 0 0
\(741\) −4752.49 −0.235610
\(742\) − 3636.52i − 0.179920i
\(743\) − 13073.1i − 0.645497i −0.946485 0.322749i \(-0.895393\pi\)
0.946485 0.322749i \(-0.104607\pi\)
\(744\) −10346.0 −0.509815
\(745\) 0 0
\(746\) −918.813 −0.0450940
\(747\) 3394.35i 0.166255i
\(748\) 11154.0i 0.545226i
\(749\) 3937.72 0.192098
\(750\) 0 0
\(751\) 16213.3 0.787791 0.393895 0.919155i \(-0.371127\pi\)
0.393895 + 0.919155i \(0.371127\pi\)
\(752\) − 1569.89i − 0.0761278i
\(753\) − 16709.4i − 0.808665i
\(754\) 6792.65 0.328082
\(755\) 0 0
\(756\) −964.786 −0.0464139
\(757\) 19903.9i 0.955642i 0.878457 + 0.477821i \(0.158573\pi\)
−0.878457 + 0.477821i \(0.841427\pi\)
\(758\) 14595.3i 0.699372i
\(759\) −18141.4 −0.867580
\(760\) 0 0
\(761\) −30125.5 −1.43502 −0.717509 0.696549i \(-0.754718\pi\)
−0.717509 + 0.696549i \(0.754718\pi\)
\(762\) − 11336.6i − 0.538950i
\(763\) − 12815.6i − 0.608066i
\(764\) 18433.5 0.872907
\(765\) 0 0
\(766\) −14681.0 −0.692489
\(767\) 425.986i 0.0200541i
\(768\) 11908.1i 0.559503i
\(769\) 36049.1 1.69046 0.845230 0.534402i \(-0.179463\pi\)
0.845230 + 0.534402i \(0.179463\pi\)
\(770\) 0 0
\(771\) 6286.09 0.293629
\(772\) 22809.4i 1.06338i
\(773\) 9644.77i 0.448769i 0.974501 + 0.224384i \(0.0720371\pi\)
−0.974501 + 0.224384i \(0.927963\pi\)
\(774\) −7935.07 −0.368502
\(775\) 0 0
\(776\) −27316.4 −1.26366
\(777\) − 7352.62i − 0.339477i
\(778\) 15712.3i 0.724054i
\(779\) 19314.7 0.888344
\(780\) 0 0
\(781\) −4500.88 −0.206215
\(782\) − 16071.0i − 0.734908i
\(783\) − 3714.69i − 0.169543i
\(784\) −141.870 −0.00646275
\(785\) 0 0
\(786\) −3300.24 −0.149766
\(787\) − 25218.6i − 1.14224i −0.820865 0.571122i \(-0.806508\pi\)
0.820865 0.571122i \(-0.193492\pi\)
\(788\) − 2217.59i − 0.100252i
\(789\) 22397.6 1.01062
\(790\) 0 0
\(791\) −222.623 −0.0100070
\(792\) 7506.28i 0.336773i
\(793\) 4963.67i 0.222276i
\(794\) 19768.8 0.883586
\(795\) 0 0
\(796\) 2393.67 0.106584
\(797\) 32042.3i 1.42409i 0.702135 + 0.712044i \(0.252230\pi\)
−0.702135 + 0.712044i \(0.747770\pi\)
\(798\) − 1950.90i − 0.0865427i
\(799\) 31675.7 1.40251
\(800\) 0 0
\(801\) −6099.52 −0.269059
\(802\) − 18984.5i − 0.835867i
\(803\) − 10647.1i − 0.467908i
\(804\) 8452.69 0.370775
\(805\) 0 0
\(806\) −7635.82 −0.333698
\(807\) − 19564.1i − 0.853396i
\(808\) 7540.07i 0.328291i
\(809\) −3427.90 −0.148972 −0.0744860 0.997222i \(-0.523732\pi\)
−0.0744860 + 0.997222i \(0.523732\pi\)
\(810\) 0 0
\(811\) 23094.4 0.999943 0.499972 0.866042i \(-0.333344\pi\)
0.499972 + 0.866042i \(0.333344\pi\)
\(812\) − 4916.16i − 0.212467i
\(813\) − 7229.11i − 0.311852i
\(814\) −22283.3 −0.959494
\(815\) 0 0
\(816\) 507.422 0.0217688
\(817\) 28289.7i 1.21142i
\(818\) − 12639.3i − 0.540249i
\(819\) −1827.98 −0.0779914
\(820\) 0 0
\(821\) 474.741 0.0201810 0.0100905 0.999949i \(-0.496788\pi\)
0.0100905 + 0.999949i \(0.496788\pi\)
\(822\) 4573.85i 0.194077i
\(823\) − 24159.8i − 1.02328i −0.859201 0.511638i \(-0.829039\pi\)
0.859201 0.511638i \(-0.170961\pi\)
\(824\) 12636.5 0.534240
\(825\) 0 0
\(826\) −174.868 −0.00736613
\(827\) − 7566.35i − 0.318147i −0.987267 0.159074i \(-0.949149\pi\)
0.987267 0.159074i \(-0.0508507\pi\)
\(828\) 7427.70i 0.311752i
\(829\) 10580.9 0.443295 0.221648 0.975127i \(-0.428857\pi\)
0.221648 + 0.975127i \(0.428857\pi\)
\(830\) 0 0
\(831\) −6659.48 −0.277996
\(832\) 8378.49i 0.349125i
\(833\) − 2862.52i − 0.119064i
\(834\) −11810.2 −0.490354
\(835\) 0 0
\(836\) 10424.2 0.431256
\(837\) 4175.80i 0.172445i
\(838\) 16411.1i 0.676507i
\(839\) −15315.6 −0.630218 −0.315109 0.949055i \(-0.602041\pi\)
−0.315109 + 0.949055i \(0.602041\pi\)
\(840\) 0 0
\(841\) −5460.40 −0.223888
\(842\) − 16876.4i − 0.690735i
\(843\) − 17515.7i − 0.715625i
\(844\) 19068.6 0.777688
\(845\) 0 0
\(846\) 8303.57 0.337450
\(847\) 475.956i 0.0193082i
\(848\) − 883.966i − 0.0357966i
\(849\) 10935.1 0.442040
\(850\) 0 0
\(851\) −56606.4 −2.28019
\(852\) 1842.81i 0.0741004i
\(853\) − 18598.2i − 0.746528i −0.927725 0.373264i \(-0.878239\pi\)
0.927725 0.373264i \(-0.121761\pi\)
\(854\) −2037.59 −0.0816450
\(855\) 0 0
\(856\) −12543.6 −0.500853
\(857\) 41775.3i 1.66513i 0.553926 + 0.832566i \(0.313129\pi\)
−0.553926 + 0.832566i \(0.686871\pi\)
\(858\) 5539.99i 0.220434i
\(859\) −32414.7 −1.28752 −0.643758 0.765229i \(-0.722626\pi\)
−0.643758 + 0.765229i \(0.722626\pi\)
\(860\) 0 0
\(861\) 7429.14 0.294059
\(862\) 27776.4i 1.09753i
\(863\) 31299.7i 1.23459i 0.786730 + 0.617297i \(0.211772\pi\)
−0.786730 + 0.617297i \(0.788228\pi\)
\(864\) 4949.48 0.194890
\(865\) 0 0
\(866\) 8794.94 0.345109
\(867\) − 4500.75i − 0.176302i
\(868\) 5526.41i 0.216104i
\(869\) −35219.2 −1.37483
\(870\) 0 0
\(871\) 16015.4 0.623030
\(872\) 40823.8i 1.58540i
\(873\) 11025.3i 0.427434i
\(874\) −15019.6 −0.581288
\(875\) 0 0
\(876\) −4359.29 −0.168136
\(877\) − 19973.0i − 0.769031i −0.923119 0.384515i \(-0.874369\pi\)
0.923119 0.384515i \(-0.125631\pi\)
\(878\) 31204.9i 1.19945i
\(879\) −11333.7 −0.434900
\(880\) 0 0
\(881\) 17367.9 0.664176 0.332088 0.943248i \(-0.392247\pi\)
0.332088 + 0.943248i \(0.392247\pi\)
\(882\) − 750.389i − 0.0286473i
\(883\) 14364.2i 0.547446i 0.961809 + 0.273723i \(0.0882551\pi\)
−0.961809 + 0.273723i \(0.911745\pi\)
\(884\) 8652.73 0.329211
\(885\) 0 0
\(886\) 2745.09 0.104089
\(887\) − 33738.1i − 1.27713i −0.769568 0.638564i \(-0.779529\pi\)
0.769568 0.638564i \(-0.220471\pi\)
\(888\) 23421.7i 0.885114i
\(889\) −15545.7 −0.586485
\(890\) 0 0
\(891\) 3029.65 0.113914
\(892\) − 4302.26i − 0.161491i
\(893\) − 29603.4i − 1.10934i
\(894\) 4182.46 0.156468
\(895\) 0 0
\(896\) 6826.21 0.254518
\(897\) 14073.3i 0.523850i
\(898\) − 1508.86i − 0.0560705i
\(899\) −21278.2 −0.789398
\(900\) 0 0
\(901\) 17835.8 0.659485
\(902\) − 22515.2i − 0.831123i
\(903\) 10881.3i 0.401004i
\(904\) 709.162 0.0260911
\(905\) 0 0
\(906\) −2729.70 −0.100097
\(907\) − 32998.6i − 1.20805i −0.796966 0.604024i \(-0.793563\pi\)
0.796966 0.604024i \(-0.206437\pi\)
\(908\) 5064.62i 0.185105i
\(909\) 3043.29 0.111045
\(910\) 0 0
\(911\) 33446.3 1.21638 0.608192 0.793790i \(-0.291895\pi\)
0.608192 + 0.793790i \(0.291895\pi\)
\(912\) − 474.225i − 0.0172184i
\(913\) − 14106.6i − 0.511347i
\(914\) 12576.6 0.455139
\(915\) 0 0
\(916\) 32728.2 1.18053
\(917\) 4525.59i 0.162975i
\(918\) 2683.88i 0.0964939i
\(919\) −41708.7 −1.49711 −0.748554 0.663074i \(-0.769252\pi\)
−0.748554 + 0.663074i \(0.769252\pi\)
\(920\) 0 0
\(921\) −14398.9 −0.515158
\(922\) 12137.8i 0.433556i
\(923\) 3491.58i 0.124514i
\(924\) 4009.56 0.142754
\(925\) 0 0
\(926\) −24607.7 −0.873283
\(927\) − 5100.30i − 0.180707i
\(928\) 25220.6i 0.892140i
\(929\) −49024.6 −1.73137 −0.865686 0.500587i \(-0.833117\pi\)
−0.865686 + 0.500587i \(0.833117\pi\)
\(930\) 0 0
\(931\) −2675.25 −0.0941758
\(932\) 11609.9i 0.408040i
\(933\) − 1740.34i − 0.0610677i
\(934\) −27894.6 −0.977235
\(935\) 0 0
\(936\) 5823.03 0.203346
\(937\) − 5447.58i − 0.189930i −0.995481 0.0949651i \(-0.969726\pi\)
0.995481 0.0949651i \(-0.0302739\pi\)
\(938\) 6574.31i 0.228848i
\(939\) 18344.4 0.637535
\(940\) 0 0
\(941\) 4125.75 0.142928 0.0714642 0.997443i \(-0.477233\pi\)
0.0714642 + 0.997443i \(0.477233\pi\)
\(942\) − 7987.60i − 0.276274i
\(943\) − 57195.5i − 1.97513i
\(944\) −42.5069 −0.00146555
\(945\) 0 0
\(946\) 32977.4 1.13339
\(947\) 17332.3i 0.594747i 0.954761 + 0.297374i \(0.0961107\pi\)
−0.954761 + 0.297374i \(0.903889\pi\)
\(948\) 14419.9i 0.494026i
\(949\) −8259.57 −0.282526
\(950\) 0 0
\(951\) −12901.9 −0.439929
\(952\) 9118.53i 0.310434i
\(953\) − 56839.4i − 1.93201i −0.258517 0.966007i \(-0.583234\pi\)
0.258517 0.966007i \(-0.416766\pi\)
\(954\) 4675.53 0.158675
\(955\) 0 0
\(956\) −14615.3 −0.494449
\(957\) 15437.9i 0.521459i
\(958\) − 22105.7i − 0.745515i
\(959\) 6272.07 0.211195
\(960\) 0 0
\(961\) −5871.48 −0.197089
\(962\) 17286.3i 0.579349i
\(963\) 5062.78i 0.169414i
\(964\) −27382.7 −0.914873
\(965\) 0 0
\(966\) −5777.10 −0.192417
\(967\) − 13284.2i − 0.441769i −0.975300 0.220884i \(-0.929106\pi\)
0.975300 0.220884i \(-0.0708943\pi\)
\(968\) − 1516.15i − 0.0503420i
\(969\) 9568.44 0.317216
\(970\) 0 0
\(971\) 12153.0 0.401658 0.200829 0.979626i \(-0.435636\pi\)
0.200829 + 0.979626i \(0.435636\pi\)
\(972\) − 1240.44i − 0.0409332i
\(973\) 16195.3i 0.533604i
\(974\) −21888.1 −0.720060
\(975\) 0 0
\(976\) −495.298 −0.0162440
\(977\) − 37999.3i − 1.24433i −0.782888 0.622163i \(-0.786254\pi\)
0.782888 0.622163i \(-0.213746\pi\)
\(978\) − 5688.18i − 0.185980i
\(979\) 25349.0 0.827537
\(980\) 0 0
\(981\) 16477.1 0.536264
\(982\) 8334.42i 0.270837i
\(983\) − 22375.0i − 0.725993i −0.931791 0.362996i \(-0.881754\pi\)
0.931791 0.362996i \(-0.118246\pi\)
\(984\) −23665.5 −0.766695
\(985\) 0 0
\(986\) −13676.0 −0.441717
\(987\) − 11386.6i − 0.367213i
\(988\) − 8086.65i − 0.260395i
\(989\) 83772.9 2.69345
\(990\) 0 0
\(991\) 18985.3 0.608564 0.304282 0.952582i \(-0.401583\pi\)
0.304282 + 0.952582i \(0.401583\pi\)
\(992\) − 28351.2i − 0.907412i
\(993\) 20062.6i 0.641156i
\(994\) −1433.29 −0.0457358
\(995\) 0 0
\(996\) −5775.70 −0.183745
\(997\) − 56476.5i − 1.79401i −0.442019 0.897006i \(-0.645738\pi\)
0.442019 0.897006i \(-0.354262\pi\)
\(998\) 17539.6i 0.556321i
\(999\) 9453.37 0.299391
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 525.4.d.j.274.2 4
5.2 odd 4 525.4.a.i.1.2 2
5.3 odd 4 105.4.a.g.1.1 2
5.4 even 2 inner 525.4.d.j.274.3 4
15.2 even 4 1575.4.a.y.1.1 2
15.8 even 4 315.4.a.g.1.2 2
20.3 even 4 1680.4.a.y.1.1 2
35.13 even 4 735.4.a.q.1.1 2
105.83 odd 4 2205.4.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.1 2 5.3 odd 4
315.4.a.g.1.2 2 15.8 even 4
525.4.a.i.1.2 2 5.2 odd 4
525.4.d.j.274.2 4 1.1 even 1 trivial
525.4.d.j.274.3 4 5.4 even 2 inner
735.4.a.q.1.1 2 35.13 even 4
1575.4.a.y.1.1 2 15.2 even 4
1680.4.a.y.1.1 2 20.3 even 4
2205.4.a.v.1.2 2 105.83 odd 4