Properties

Label 525.4.d.k.274.3
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.3
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.k.274.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.236068i q^{2} -3.00000i q^{3} +7.94427 q^{4} +0.708204 q^{6} -7.00000i q^{7} +3.76393i q^{8} -9.00000 q^{9} -50.4721 q^{11} -23.8328i q^{12} +80.9706i q^{13} +1.65248 q^{14} +62.6656 q^{16} +76.3870i q^{17} -2.12461i q^{18} -4.13777 q^{19} -21.0000 q^{21} -11.9149i q^{22} +204.721i q^{23} +11.2918 q^{24} -19.1146 q^{26} +27.0000i q^{27} -55.6099i q^{28} +91.1672 q^{29} +198.079 q^{31} +44.9048i q^{32} +151.416i q^{33} -18.0325 q^{34} -71.4984 q^{36} +155.666i q^{37} -0.976794i q^{38} +242.912 q^{39} -156.885 q^{41} -4.95743i q^{42} -354.217i q^{43} -400.964 q^{44} -48.3282 q^{46} -175.659i q^{47} -187.997i q^{48} -49.0000 q^{49} +229.161 q^{51} +643.252i q^{52} -200.302i q^{53} -6.37384 q^{54} +26.3475 q^{56} +12.4133i q^{57} +21.5217i q^{58} +312.498 q^{59} -154.170 q^{61} +46.7601i q^{62} +63.0000i q^{63} +490.724 q^{64} -35.7446 q^{66} +734.715i q^{67} +606.839i q^{68} +614.164 q^{69} -678.577 q^{71} -33.8754i q^{72} +60.8003i q^{73} -36.7477 q^{74} -32.8715 q^{76} +353.305i q^{77} +57.3437i q^{78} +1286.26 q^{79} +81.0000 q^{81} -37.0356i q^{82} -116.170i q^{83} -166.830 q^{84} +83.6192 q^{86} -273.502i q^{87} -189.974i q^{88} +916.440 q^{89} +566.794 q^{91} +1626.36i q^{92} -594.237i q^{93} +41.4676 q^{94} +134.714 q^{96} -1416.30i q^{97} -11.5673i q^{98} +454.249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44}+ \cdots + 1656 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.236068i 0.0834626i 0.999129 + 0.0417313i \(0.0132873\pi\)
−0.999129 + 0.0417313i \(0.986713\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 7.94427 0.993034
\(5\) 0 0
\(6\) 0.708204 0.0481872
\(7\) − 7.00000i − 0.377964i
\(8\) 3.76393i 0.166344i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −50.4721 −1.38345 −0.691724 0.722162i \(-0.743148\pi\)
−0.691724 + 0.722162i \(0.743148\pi\)
\(12\) − 23.8328i − 0.573328i
\(13\) 80.9706i 1.72748i 0.503940 + 0.863738i \(0.331883\pi\)
−0.503940 + 0.863738i \(0.668117\pi\)
\(14\) 1.65248 0.0315459
\(15\) 0 0
\(16\) 62.6656 0.979150
\(17\) 76.3870i 1.08980i 0.838502 + 0.544899i \(0.183432\pi\)
−0.838502 + 0.544899i \(0.816568\pi\)
\(18\) − 2.12461i − 0.0278209i
\(19\) −4.13777 −0.0499615 −0.0249808 0.999688i \(-0.507952\pi\)
−0.0249808 + 0.999688i \(0.507952\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) − 11.9149i − 0.115466i
\(23\) 204.721i 1.85597i 0.372615 + 0.927986i \(0.378461\pi\)
−0.372615 + 0.927986i \(0.621539\pi\)
\(24\) 11.2918 0.0960387
\(25\) 0 0
\(26\) −19.1146 −0.144180
\(27\) 27.0000i 0.192450i
\(28\) − 55.6099i − 0.375332i
\(29\) 91.1672 0.583770 0.291885 0.956453i \(-0.405718\pi\)
0.291885 + 0.956453i \(0.405718\pi\)
\(30\) 0 0
\(31\) 198.079 1.14761 0.573807 0.818991i \(-0.305466\pi\)
0.573807 + 0.818991i \(0.305466\pi\)
\(32\) 44.9048i 0.248066i
\(33\) 151.416i 0.798734i
\(34\) −18.0325 −0.0909574
\(35\) 0 0
\(36\) −71.4984 −0.331011
\(37\) 155.666i 0.691656i 0.938298 + 0.345828i \(0.112402\pi\)
−0.938298 + 0.345828i \(0.887598\pi\)
\(38\) − 0.976794i − 0.00416992i
\(39\) 242.912 0.997359
\(40\) 0 0
\(41\) −156.885 −0.597595 −0.298797 0.954317i \(-0.596585\pi\)
−0.298797 + 0.954317i \(0.596585\pi\)
\(42\) − 4.95743i − 0.0182130i
\(43\) − 354.217i − 1.25622i −0.778124 0.628111i \(-0.783828\pi\)
0.778124 0.628111i \(-0.216172\pi\)
\(44\) −400.964 −1.37381
\(45\) 0 0
\(46\) −48.3282 −0.154904
\(47\) − 175.659i − 0.545161i −0.962133 0.272580i \(-0.912123\pi\)
0.962133 0.272580i \(-0.0878771\pi\)
\(48\) − 187.997i − 0.565313i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 229.161 0.629195
\(52\) 643.252i 1.71544i
\(53\) − 200.302i − 0.519124i −0.965726 0.259562i \(-0.916422\pi\)
0.965726 0.259562i \(-0.0835782\pi\)
\(54\) −6.37384 −0.0160624
\(55\) 0 0
\(56\) 26.3475 0.0628721
\(57\) 12.4133i 0.0288453i
\(58\) 21.5217i 0.0487230i
\(59\) 312.498 0.689556 0.344778 0.938684i \(-0.387954\pi\)
0.344778 + 0.938684i \(0.387954\pi\)
\(60\) 0 0
\(61\) −154.170 −0.323598 −0.161799 0.986824i \(-0.551730\pi\)
−0.161799 + 0.986824i \(0.551730\pi\)
\(62\) 46.7601i 0.0957829i
\(63\) 63.0000i 0.125988i
\(64\) 490.724 0.958446
\(65\) 0 0
\(66\) −35.7446 −0.0666644
\(67\) 734.715i 1.33970i 0.742498 + 0.669849i \(0.233641\pi\)
−0.742498 + 0.669849i \(0.766359\pi\)
\(68\) 606.839i 1.08221i
\(69\) 614.164 1.07155
\(70\) 0 0
\(71\) −678.577 −1.13426 −0.567129 0.823629i \(-0.691946\pi\)
−0.567129 + 0.823629i \(0.691946\pi\)
\(72\) − 33.8754i − 0.0554480i
\(73\) 60.8003i 0.0974813i 0.998811 + 0.0487407i \(0.0155208\pi\)
−0.998811 + 0.0487407i \(0.984479\pi\)
\(74\) −36.7477 −0.0577274
\(75\) 0 0
\(76\) −32.8715 −0.0496135
\(77\) 353.305i 0.522894i
\(78\) 57.3437i 0.0832422i
\(79\) 1286.26 1.83184 0.915919 0.401363i \(-0.131463\pi\)
0.915919 + 0.401363i \(0.131463\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 37.0356i − 0.0498768i
\(83\) − 116.170i − 0.153631i −0.997045 0.0768153i \(-0.975525\pi\)
0.997045 0.0768153i \(-0.0244752\pi\)
\(84\) −166.830 −0.216698
\(85\) 0 0
\(86\) 83.6192 0.104848
\(87\) − 273.502i − 0.337040i
\(88\) − 189.974i − 0.230128i
\(89\) 916.440 1.09149 0.545744 0.837952i \(-0.316247\pi\)
0.545744 + 0.837952i \(0.316247\pi\)
\(90\) 0 0
\(91\) 566.794 0.652925
\(92\) 1626.36i 1.84304i
\(93\) − 594.237i − 0.662575i
\(94\) 41.4676 0.0455006
\(95\) 0 0
\(96\) 134.714 0.143221
\(97\) − 1416.30i − 1.48251i −0.671222 0.741256i \(-0.734230\pi\)
0.671222 0.741256i \(-0.265770\pi\)
\(98\) − 11.5673i − 0.0119232i
\(99\) 454.249 0.461149
\(100\) 0 0
\(101\) −1379.19 −1.35876 −0.679381 0.733785i \(-0.737752\pi\)
−0.679381 + 0.733785i \(0.737752\pi\)
\(102\) 54.0976i 0.0525143i
\(103\) 1308.43i 1.25169i 0.779949 + 0.625844i \(0.215245\pi\)
−0.779949 + 0.625844i \(0.784755\pi\)
\(104\) −304.768 −0.287355
\(105\) 0 0
\(106\) 47.2849 0.0433275
\(107\) 1265.65i 1.14351i 0.820425 + 0.571754i \(0.193737\pi\)
−0.820425 + 0.571754i \(0.806263\pi\)
\(108\) 214.495i 0.191109i
\(109\) −2069.32 −1.81839 −0.909196 0.416368i \(-0.863303\pi\)
−0.909196 + 0.416368i \(0.863303\pi\)
\(110\) 0 0
\(111\) 466.997 0.399328
\(112\) − 438.659i − 0.370084i
\(113\) 1953.89i 1.62661i 0.581840 + 0.813303i \(0.302333\pi\)
−0.581840 + 0.813303i \(0.697667\pi\)
\(114\) −2.93038 −0.00240750
\(115\) 0 0
\(116\) 724.257 0.579703
\(117\) − 728.735i − 0.575826i
\(118\) 73.7709i 0.0575522i
\(119\) 534.709 0.411905
\(120\) 0 0
\(121\) 1216.44 0.913927
\(122\) − 36.3947i − 0.0270083i
\(123\) 470.656i 0.345022i
\(124\) 1573.59 1.13962
\(125\) 0 0
\(126\) −14.8723 −0.0105153
\(127\) 224.251i 0.156685i 0.996926 + 0.0783426i \(0.0249628\pi\)
−0.996926 + 0.0783426i \(0.975037\pi\)
\(128\) 475.083i 0.328061i
\(129\) −1062.65 −0.725280
\(130\) 0 0
\(131\) −490.898 −0.327404 −0.163702 0.986510i \(-0.552344\pi\)
−0.163702 + 0.986510i \(0.552344\pi\)
\(132\) 1202.89i 0.793170i
\(133\) 28.9644i 0.0188837i
\(134\) −173.443 −0.111815
\(135\) 0 0
\(136\) −287.515 −0.181281
\(137\) 1831.12i 1.14192i 0.820977 + 0.570961i \(0.193429\pi\)
−0.820977 + 0.570961i \(0.806571\pi\)
\(138\) 144.984i 0.0894340i
\(139\) 3050.84 1.86165 0.930823 0.365469i \(-0.119091\pi\)
0.930823 + 0.365469i \(0.119091\pi\)
\(140\) 0 0
\(141\) −526.978 −0.314749
\(142\) − 160.190i − 0.0946682i
\(143\) − 4086.76i − 2.38987i
\(144\) −563.991 −0.326383
\(145\) 0 0
\(146\) −14.3530 −0.00813605
\(147\) 147.000i 0.0824786i
\(148\) 1236.65i 0.686838i
\(149\) −2246.55 −1.23520 −0.617599 0.786493i \(-0.711895\pi\)
−0.617599 + 0.786493i \(0.711895\pi\)
\(150\) 0 0
\(151\) −1311.53 −0.706826 −0.353413 0.935467i \(-0.614979\pi\)
−0.353413 + 0.935467i \(0.614979\pi\)
\(152\) − 15.5743i − 0.00831079i
\(153\) − 687.483i − 0.363266i
\(154\) −83.4040 −0.0436421
\(155\) 0 0
\(156\) 1929.76 0.990412
\(157\) 1790.94i 0.910398i 0.890390 + 0.455199i \(0.150432\pi\)
−0.890390 + 0.455199i \(0.849568\pi\)
\(158\) 303.644i 0.152890i
\(159\) −600.906 −0.299716
\(160\) 0 0
\(161\) 1433.05 0.701491
\(162\) 19.1215i 0.00927363i
\(163\) 491.108i 0.235991i 0.993014 + 0.117996i \(0.0376469\pi\)
−0.993014 + 0.117996i \(0.962353\pi\)
\(164\) −1246.34 −0.593432
\(165\) 0 0
\(166\) 27.4241 0.0128224
\(167\) − 826.059i − 0.382769i −0.981515 0.191384i \(-0.938702\pi\)
0.981515 0.191384i \(-0.0612977\pi\)
\(168\) − 79.0426i − 0.0362992i
\(169\) −4359.24 −1.98418
\(170\) 0 0
\(171\) 37.2399 0.0166538
\(172\) − 2813.99i − 1.24747i
\(173\) − 2918.00i − 1.28238i −0.767382 0.641190i \(-0.778441\pi\)
0.767382 0.641190i \(-0.221559\pi\)
\(174\) 64.5650 0.0281302
\(175\) 0 0
\(176\) −3162.87 −1.35460
\(177\) − 937.495i − 0.398116i
\(178\) 216.342i 0.0910984i
\(179\) −955.745 −0.399082 −0.199541 0.979889i \(-0.563945\pi\)
−0.199541 + 0.979889i \(0.563945\pi\)
\(180\) 0 0
\(181\) 206.080 0.0846289 0.0423145 0.999104i \(-0.486527\pi\)
0.0423145 + 0.999104i \(0.486527\pi\)
\(182\) 133.802i 0.0544948i
\(183\) 462.511i 0.186829i
\(184\) −770.557 −0.308730
\(185\) 0 0
\(186\) 140.280 0.0553003
\(187\) − 3855.41i − 1.50768i
\(188\) − 1395.49i − 0.541363i
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −2018.63 −0.764727 −0.382364 0.924012i \(-0.624890\pi\)
−0.382364 + 0.924012i \(0.624890\pi\)
\(192\) − 1472.17i − 0.553359i
\(193\) − 1031.69i − 0.384781i −0.981318 0.192390i \(-0.938376\pi\)
0.981318 0.192390i \(-0.0616240\pi\)
\(194\) 334.344 0.123734
\(195\) 0 0
\(196\) −389.269 −0.141862
\(197\) − 205.955i − 0.0744857i −0.999306 0.0372429i \(-0.988142\pi\)
0.999306 0.0372429i \(-0.0118575\pi\)
\(198\) 107.234i 0.0384887i
\(199\) 1831.38 0.652376 0.326188 0.945305i \(-0.394236\pi\)
0.326188 + 0.945305i \(0.394236\pi\)
\(200\) 0 0
\(201\) 2204.15 0.773475
\(202\) − 325.584i − 0.113406i
\(203\) − 638.170i − 0.220644i
\(204\) 1820.52 0.624812
\(205\) 0 0
\(206\) −308.879 −0.104469
\(207\) − 1842.49i − 0.618657i
\(208\) 5074.07i 1.69146i
\(209\) 208.842 0.0691191
\(210\) 0 0
\(211\) 1030.19 0.336119 0.168060 0.985777i \(-0.446250\pi\)
0.168060 + 0.985777i \(0.446250\pi\)
\(212\) − 1591.25i − 0.515508i
\(213\) 2035.73i 0.654864i
\(214\) −298.780 −0.0954402
\(215\) 0 0
\(216\) −101.626 −0.0320129
\(217\) − 1386.55i − 0.433757i
\(218\) − 488.500i − 0.151768i
\(219\) 182.401 0.0562809
\(220\) 0 0
\(221\) −6185.10 −1.88260
\(222\) 110.243i 0.0333289i
\(223\) − 5368.67i − 1.61217i −0.591803 0.806083i \(-0.701584\pi\)
0.591803 0.806083i \(-0.298416\pi\)
\(224\) 314.334 0.0937603
\(225\) 0 0
\(226\) −461.251 −0.135761
\(227\) − 932.121i − 0.272542i −0.990672 0.136271i \(-0.956488\pi\)
0.990672 0.136271i \(-0.0435118\pi\)
\(228\) 98.6146i 0.0286444i
\(229\) −3163.05 −0.912752 −0.456376 0.889787i \(-0.650853\pi\)
−0.456376 + 0.889787i \(0.650853\pi\)
\(230\) 0 0
\(231\) 1059.91 0.301893
\(232\) 343.147i 0.0971065i
\(233\) − 436.562i − 0.122747i −0.998115 0.0613737i \(-0.980452\pi\)
0.998115 0.0613737i \(-0.0195481\pi\)
\(234\) 172.031 0.0480599
\(235\) 0 0
\(236\) 2482.57 0.684753
\(237\) − 3858.77i − 1.05761i
\(238\) 126.228i 0.0343787i
\(239\) 1980.82 0.536103 0.268051 0.963405i \(-0.413620\pi\)
0.268051 + 0.963405i \(0.413620\pi\)
\(240\) 0 0
\(241\) 5303.55 1.41756 0.708780 0.705430i \(-0.249246\pi\)
0.708780 + 0.705430i \(0.249246\pi\)
\(242\) 287.162i 0.0762787i
\(243\) − 243.000i − 0.0641500i
\(244\) −1224.77 −0.321344
\(245\) 0 0
\(246\) −111.107 −0.0287964
\(247\) − 335.037i − 0.0863074i
\(248\) 745.556i 0.190899i
\(249\) −348.511 −0.0886987
\(250\) 0 0
\(251\) −2996.04 −0.753420 −0.376710 0.926331i \(-0.622945\pi\)
−0.376710 + 0.926331i \(0.622945\pi\)
\(252\) 500.489i 0.125111i
\(253\) − 10332.7i − 2.56764i
\(254\) −52.9384 −0.0130774
\(255\) 0 0
\(256\) 3813.64 0.931065
\(257\) 968.861i 0.235159i 0.993063 + 0.117580i \(0.0375135\pi\)
−0.993063 + 0.117580i \(0.962487\pi\)
\(258\) − 250.858i − 0.0605338i
\(259\) 1089.66 0.261421
\(260\) 0 0
\(261\) −820.505 −0.194590
\(262\) − 115.885i − 0.0273260i
\(263\) 4830.18i 1.13248i 0.824241 + 0.566239i \(0.191602\pi\)
−0.824241 + 0.566239i \(0.808398\pi\)
\(264\) −569.921 −0.132864
\(265\) 0 0
\(266\) −6.83756 −0.00157608
\(267\) − 2749.32i − 0.630171i
\(268\) 5836.78i 1.33037i
\(269\) 4774.97 1.08229 0.541143 0.840930i \(-0.317992\pi\)
0.541143 + 0.840930i \(0.317992\pi\)
\(270\) 0 0
\(271\) 141.909 0.0318094 0.0159047 0.999874i \(-0.494937\pi\)
0.0159047 + 0.999874i \(0.494937\pi\)
\(272\) 4786.84i 1.06708i
\(273\) − 1700.38i − 0.376966i
\(274\) −432.269 −0.0953078
\(275\) 0 0
\(276\) 4879.09 1.06408
\(277\) − 3621.13i − 0.785460i −0.919654 0.392730i \(-0.871531\pi\)
0.919654 0.392730i \(-0.128469\pi\)
\(278\) 720.206i 0.155378i
\(279\) −1782.71 −0.382538
\(280\) 0 0
\(281\) 5790.87 1.22937 0.614687 0.788771i \(-0.289282\pi\)
0.614687 + 0.788771i \(0.289282\pi\)
\(282\) − 124.403i − 0.0262698i
\(283\) − 526.043i − 0.110495i −0.998473 0.0552474i \(-0.982405\pi\)
0.998473 0.0552474i \(-0.0175948\pi\)
\(284\) −5390.80 −1.12636
\(285\) 0 0
\(286\) 964.753 0.199465
\(287\) 1098.20i 0.225870i
\(288\) − 404.143i − 0.0826888i
\(289\) −921.972 −0.187660
\(290\) 0 0
\(291\) −4248.91 −0.855929
\(292\) 483.014i 0.0968023i
\(293\) − 1914.88i − 0.381803i −0.981609 0.190901i \(-0.938859\pi\)
0.981609 0.190901i \(-0.0611411\pi\)
\(294\) −34.7020 −0.00688388
\(295\) 0 0
\(296\) −585.915 −0.115053
\(297\) − 1362.75i − 0.266245i
\(298\) − 530.339i − 0.103093i
\(299\) −16576.4 −3.20615
\(300\) 0 0
\(301\) −2479.52 −0.474807
\(302\) − 309.610i − 0.0589936i
\(303\) 4137.58i 0.784482i
\(304\) −259.296 −0.0489199
\(305\) 0 0
\(306\) 162.293 0.0303191
\(307\) − 6244.17i − 1.16083i −0.814322 0.580413i \(-0.802891\pi\)
0.814322 0.580413i \(-0.197109\pi\)
\(308\) 2806.75i 0.519251i
\(309\) 3925.30 0.722662
\(310\) 0 0
\(311\) −9658.78 −1.76109 −0.880545 0.473962i \(-0.842823\pi\)
−0.880545 + 0.473962i \(0.842823\pi\)
\(312\) 914.303i 0.165905i
\(313\) − 2198.34i − 0.396988i −0.980102 0.198494i \(-0.936395\pi\)
0.980102 0.198494i \(-0.0636051\pi\)
\(314\) −422.783 −0.0759842
\(315\) 0 0
\(316\) 10218.4 1.81908
\(317\) 3030.78i 0.536990i 0.963281 + 0.268495i \(0.0865263\pi\)
−0.963281 + 0.268495i \(0.913474\pi\)
\(318\) − 141.855i − 0.0250151i
\(319\) −4601.40 −0.807615
\(320\) 0 0
\(321\) 3796.96 0.660204
\(322\) 338.297i 0.0585483i
\(323\) − 316.072i − 0.0544480i
\(324\) 643.486 0.110337
\(325\) 0 0
\(326\) −115.935 −0.0196965
\(327\) 6207.96i 1.04985i
\(328\) − 590.506i − 0.0994062i
\(329\) −1229.62 −0.206051
\(330\) 0 0
\(331\) 4753.74 0.789393 0.394696 0.918812i \(-0.370850\pi\)
0.394696 + 0.918812i \(0.370850\pi\)
\(332\) − 922.888i − 0.152560i
\(333\) − 1400.99i − 0.230552i
\(334\) 195.006 0.0319469
\(335\) 0 0
\(336\) −1315.98 −0.213668
\(337\) 8824.40i 1.42640i 0.700962 + 0.713199i \(0.252754\pi\)
−0.700962 + 0.713199i \(0.747246\pi\)
\(338\) − 1029.08i − 0.165605i
\(339\) 5861.67 0.939122
\(340\) 0 0
\(341\) −9997.47 −1.58766
\(342\) 8.79115i 0.00138997i
\(343\) 343.000i 0.0539949i
\(344\) 1333.25 0.208965
\(345\) 0 0
\(346\) 688.847 0.107031
\(347\) − 3413.97i − 0.528160i −0.964501 0.264080i \(-0.914932\pi\)
0.964501 0.264080i \(-0.0850683\pi\)
\(348\) − 2172.77i − 0.334692i
\(349\) 5676.32 0.870621 0.435310 0.900280i \(-0.356639\pi\)
0.435310 + 0.900280i \(0.356639\pi\)
\(350\) 0 0
\(351\) −2186.21 −0.332453
\(352\) − 2266.44i − 0.343187i
\(353\) − 6225.80i − 0.938713i −0.883009 0.469357i \(-0.844486\pi\)
0.883009 0.469357i \(-0.155514\pi\)
\(354\) 221.313 0.0332278
\(355\) 0 0
\(356\) 7280.45 1.08388
\(357\) − 1604.13i − 0.237813i
\(358\) − 225.621i − 0.0333084i
\(359\) 4907.73 0.721505 0.360752 0.932662i \(-0.382520\pi\)
0.360752 + 0.932662i \(0.382520\pi\)
\(360\) 0 0
\(361\) −6841.88 −0.997504
\(362\) 48.6490i 0.00706335i
\(363\) − 3649.31i − 0.527656i
\(364\) 4502.77 0.648377
\(365\) 0 0
\(366\) −109.184 −0.0155933
\(367\) − 3906.48i − 0.555631i −0.960634 0.277816i \(-0.910390\pi\)
0.960634 0.277816i \(-0.0896104\pi\)
\(368\) 12829.0i 1.81728i
\(369\) 1411.97 0.199198
\(370\) 0 0
\(371\) −1402.11 −0.196210
\(372\) − 4720.78i − 0.657960i
\(373\) 2102.52i 0.291862i 0.989295 + 0.145931i \(0.0466178\pi\)
−0.989295 + 0.145931i \(0.953382\pi\)
\(374\) 910.140 0.125835
\(375\) 0 0
\(376\) 661.170 0.0906842
\(377\) 7381.86i 1.00845i
\(378\) 44.6168i 0.00607101i
\(379\) 6612.76 0.896239 0.448120 0.893974i \(-0.352094\pi\)
0.448120 + 0.893974i \(0.352094\pi\)
\(380\) 0 0
\(381\) 672.752 0.0904623
\(382\) − 476.534i − 0.0638262i
\(383\) − 2.37457i 0 0.000316801i −1.00000 0.000158401i \(-0.999950\pi\)
1.00000 0.000158401i \(-5.04204e-5\pi\)
\(384\) 1425.25 0.189406
\(385\) 0 0
\(386\) 243.549 0.0321148
\(387\) 3187.95i 0.418741i
\(388\) − 11251.5i − 1.47218i
\(389\) −7716.98 −1.00583 −0.502913 0.864337i \(-0.667739\pi\)
−0.502913 + 0.864337i \(0.667739\pi\)
\(390\) 0 0
\(391\) −15638.0 −2.02263
\(392\) − 184.433i − 0.0237634i
\(393\) 1472.69i 0.189027i
\(394\) 48.6194 0.00621678
\(395\) 0 0
\(396\) 3608.68 0.457937
\(397\) 6403.95i 0.809584i 0.914409 + 0.404792i \(0.132656\pi\)
−0.914409 + 0.404792i \(0.867344\pi\)
\(398\) 432.329i 0.0544490i
\(399\) 86.8931 0.0109025
\(400\) 0 0
\(401\) −10969.9 −1.36611 −0.683054 0.730368i \(-0.739349\pi\)
−0.683054 + 0.730368i \(0.739349\pi\)
\(402\) 520.328i 0.0645562i
\(403\) 16038.6i 1.98248i
\(404\) −10956.7 −1.34930
\(405\) 0 0
\(406\) 150.652 0.0184155
\(407\) − 7856.78i − 0.956870i
\(408\) 862.546i 0.104663i
\(409\) 400.353 0.0484015 0.0242007 0.999707i \(-0.492296\pi\)
0.0242007 + 0.999707i \(0.492296\pi\)
\(410\) 0 0
\(411\) 5493.37 0.659289
\(412\) 10394.6i 1.24297i
\(413\) − 2187.49i − 0.260628i
\(414\) 434.953 0.0516348
\(415\) 0 0
\(416\) −3635.97 −0.428529
\(417\) − 9152.52i − 1.07482i
\(418\) 49.3009i 0.00576887i
\(419\) 15815.4 1.84399 0.921995 0.387201i \(-0.126558\pi\)
0.921995 + 0.387201i \(0.126558\pi\)
\(420\) 0 0
\(421\) 1936.53 0.224182 0.112091 0.993698i \(-0.464245\pi\)
0.112091 + 0.993698i \(0.464245\pi\)
\(422\) 243.195i 0.0280534i
\(423\) 1580.93i 0.181720i
\(424\) 753.923 0.0863531
\(425\) 0 0
\(426\) −480.571 −0.0546567
\(427\) 1079.19i 0.122309i
\(428\) 10054.7i 1.13554i
\(429\) −12260.3 −1.37979
\(430\) 0 0
\(431\) 2030.91 0.226973 0.113487 0.993540i \(-0.463798\pi\)
0.113487 + 0.993540i \(0.463798\pi\)
\(432\) 1691.97i 0.188438i
\(433\) 10784.1i 1.19689i 0.801165 + 0.598443i \(0.204214\pi\)
−0.801165 + 0.598443i \(0.795786\pi\)
\(434\) 327.321 0.0362025
\(435\) 0 0
\(436\) −16439.2 −1.80573
\(437\) − 847.089i − 0.0927272i
\(438\) 43.0590i 0.00469735i
\(439\) 6304.19 0.685382 0.342691 0.939448i \(-0.388662\pi\)
0.342691 + 0.939448i \(0.388662\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 1460.10i − 0.157127i
\(443\) − 15494.8i − 1.66181i −0.556414 0.830905i \(-0.687823\pi\)
0.556414 0.830905i \(-0.312177\pi\)
\(444\) 3709.95 0.396546
\(445\) 0 0
\(446\) 1267.37 0.134556
\(447\) 6739.65i 0.713142i
\(448\) − 3435.07i − 0.362259i
\(449\) 242.018 0.0254377 0.0127189 0.999919i \(-0.495951\pi\)
0.0127189 + 0.999919i \(0.495951\pi\)
\(450\) 0 0
\(451\) 7918.34 0.826741
\(452\) 15522.2i 1.61528i
\(453\) 3934.59i 0.408086i
\(454\) 220.044 0.0227471
\(455\) 0 0
\(456\) −46.7228 −0.00479824
\(457\) − 11670.4i − 1.19457i −0.802030 0.597283i \(-0.796247\pi\)
0.802030 0.597283i \(-0.203753\pi\)
\(458\) − 746.695i − 0.0761807i
\(459\) −2062.45 −0.209732
\(460\) 0 0
\(461\) 12128.1 1.22530 0.612649 0.790355i \(-0.290104\pi\)
0.612649 + 0.790355i \(0.290104\pi\)
\(462\) 250.212i 0.0251968i
\(463\) − 5161.64i − 0.518103i −0.965864 0.259051i \(-0.916590\pi\)
0.965864 0.259051i \(-0.0834099\pi\)
\(464\) 5713.05 0.571598
\(465\) 0 0
\(466\) 103.058 0.0102448
\(467\) 11680.2i 1.15738i 0.815547 + 0.578691i \(0.196436\pi\)
−0.815547 + 0.578691i \(0.803564\pi\)
\(468\) − 5789.27i − 0.571814i
\(469\) 5143.01 0.506358
\(470\) 0 0
\(471\) 5372.82 0.525619
\(472\) 1176.22i 0.114703i
\(473\) 17878.1i 1.73792i
\(474\) 910.932 0.0882711
\(475\) 0 0
\(476\) 4247.87 0.409036
\(477\) 1802.72i 0.173041i
\(478\) 467.608i 0.0447445i
\(479\) 18458.7 1.76075 0.880373 0.474282i \(-0.157292\pi\)
0.880373 + 0.474282i \(0.157292\pi\)
\(480\) 0 0
\(481\) −12604.3 −1.19482
\(482\) 1252.00i 0.118313i
\(483\) − 4299.15i − 0.405006i
\(484\) 9663.70 0.907560
\(485\) 0 0
\(486\) 57.3645 0.00535413
\(487\) 8630.11i 0.803014i 0.915856 + 0.401507i \(0.131513\pi\)
−0.915856 + 0.401507i \(0.868487\pi\)
\(488\) − 580.286i − 0.0538286i
\(489\) 1473.33 0.136250
\(490\) 0 0
\(491\) 17801.6 1.63621 0.818103 0.575072i \(-0.195026\pi\)
0.818103 + 0.575072i \(0.195026\pi\)
\(492\) 3739.02i 0.342618i
\(493\) 6963.99i 0.636191i
\(494\) 79.0916 0.00720344
\(495\) 0 0
\(496\) 12412.7 1.12369
\(497\) 4750.04i 0.428709i
\(498\) − 82.2723i − 0.00740303i
\(499\) −200.167 −0.0179574 −0.00897868 0.999960i \(-0.502858\pi\)
−0.00897868 + 0.999960i \(0.502858\pi\)
\(500\) 0 0
\(501\) −2478.18 −0.220992
\(502\) − 707.269i − 0.0628824i
\(503\) − 16400.4i − 1.45379i −0.686747 0.726896i \(-0.740962\pi\)
0.686747 0.726896i \(-0.259038\pi\)
\(504\) −237.128 −0.0209574
\(505\) 0 0
\(506\) 2439.23 0.214302
\(507\) 13077.7i 1.14556i
\(508\) 1781.51i 0.155594i
\(509\) 15006.6 1.30679 0.653394 0.757018i \(-0.273344\pi\)
0.653394 + 0.757018i \(0.273344\pi\)
\(510\) 0 0
\(511\) 425.602 0.0368445
\(512\) 4700.94i 0.405770i
\(513\) − 111.720i − 0.00961510i
\(514\) −228.717 −0.0196270
\(515\) 0 0
\(516\) −8441.98 −0.720228
\(517\) 8865.91i 0.754201i
\(518\) 257.234i 0.0218189i
\(519\) −8754.01 −0.740382
\(520\) 0 0
\(521\) 7113.18 0.598146 0.299073 0.954230i \(-0.403323\pi\)
0.299073 + 0.954230i \(0.403323\pi\)
\(522\) − 193.695i − 0.0162410i
\(523\) 8888.46i 0.743146i 0.928404 + 0.371573i \(0.121181\pi\)
−0.928404 + 0.371573i \(0.878819\pi\)
\(524\) −3899.83 −0.325123
\(525\) 0 0
\(526\) −1140.25 −0.0945196
\(527\) 15130.7i 1.25067i
\(528\) 9488.60i 0.782081i
\(529\) −29743.8 −2.44463
\(530\) 0 0
\(531\) −2812.49 −0.229852
\(532\) 230.101i 0.0187521i
\(533\) − 12703.1i − 1.03233i
\(534\) 649.026 0.0525957
\(535\) 0 0
\(536\) −2765.42 −0.222850
\(537\) 2867.23i 0.230410i
\(538\) 1127.22i 0.0903305i
\(539\) 2473.13 0.197635
\(540\) 0 0
\(541\) −653.827 −0.0519597 −0.0259799 0.999662i \(-0.508271\pi\)
−0.0259799 + 0.999662i \(0.508271\pi\)
\(542\) 33.5001i 0.00265489i
\(543\) − 618.241i − 0.0488605i
\(544\) −3430.14 −0.270342
\(545\) 0 0
\(546\) 401.406 0.0314626
\(547\) 1138.52i 0.0889940i 0.999010 + 0.0444970i \(0.0141685\pi\)
−0.999010 + 0.0444970i \(0.985831\pi\)
\(548\) 14546.9i 1.13397i
\(549\) 1387.53 0.107866
\(550\) 0 0
\(551\) −377.229 −0.0291660
\(552\) 2311.67i 0.178245i
\(553\) − 9003.80i − 0.692370i
\(554\) 854.832 0.0655566
\(555\) 0 0
\(556\) 24236.7 1.84868
\(557\) − 19804.8i − 1.50657i −0.657696 0.753283i \(-0.728469\pi\)
0.657696 0.753283i \(-0.271531\pi\)
\(558\) − 420.841i − 0.0319276i
\(559\) 28681.1 2.17009
\(560\) 0 0
\(561\) −11566.2 −0.870458
\(562\) 1367.04i 0.102607i
\(563\) − 10276.3i − 0.769265i −0.923070 0.384632i \(-0.874328\pi\)
0.923070 0.384632i \(-0.125672\pi\)
\(564\) −4186.46 −0.312556
\(565\) 0 0
\(566\) 124.182 0.00922219
\(567\) − 567.000i − 0.0419961i
\(568\) − 2554.12i − 0.188677i
\(569\) −4139.03 −0.304951 −0.152475 0.988307i \(-0.548724\pi\)
−0.152475 + 0.988307i \(0.548724\pi\)
\(570\) 0 0
\(571\) −4486.81 −0.328839 −0.164420 0.986390i \(-0.552575\pi\)
−0.164420 + 0.986390i \(0.552575\pi\)
\(572\) − 32466.3i − 2.37323i
\(573\) 6055.89i 0.441516i
\(574\) −259.249 −0.0188517
\(575\) 0 0
\(576\) −4416.52 −0.319482
\(577\) − 1104.77i − 0.0797093i −0.999205 0.0398547i \(-0.987311\pi\)
0.999205 0.0398547i \(-0.0126895\pi\)
\(578\) − 217.648i − 0.0156626i
\(579\) −3095.07 −0.222153
\(580\) 0 0
\(581\) −813.192 −0.0580669
\(582\) − 1003.03i − 0.0714381i
\(583\) 10109.7i 0.718181i
\(584\) −228.848 −0.0162154
\(585\) 0 0
\(586\) 452.041 0.0318663
\(587\) − 10413.2i − 0.732199i −0.930576 0.366100i \(-0.880693\pi\)
0.930576 0.366100i \(-0.119307\pi\)
\(588\) 1167.81i 0.0819041i
\(589\) −819.605 −0.0573365
\(590\) 0 0
\(591\) −617.865 −0.0430044
\(592\) 9754.89i 0.677235i
\(593\) 3235.16i 0.224034i 0.993706 + 0.112017i \(0.0357311\pi\)
−0.993706 + 0.112017i \(0.964269\pi\)
\(594\) 321.701 0.0222215
\(595\) 0 0
\(596\) −17847.2 −1.22659
\(597\) − 5494.13i − 0.376649i
\(598\) − 3913.16i − 0.267594i
\(599\) −569.048 −0.0388158 −0.0194079 0.999812i \(-0.506178\pi\)
−0.0194079 + 0.999812i \(0.506178\pi\)
\(600\) 0 0
\(601\) 3760.89 0.255258 0.127629 0.991822i \(-0.459263\pi\)
0.127629 + 0.991822i \(0.459263\pi\)
\(602\) − 585.335i − 0.0396287i
\(603\) − 6612.44i − 0.446566i
\(604\) −10419.1 −0.701902
\(605\) 0 0
\(606\) −976.751 −0.0654749
\(607\) − 2224.05i − 0.148717i −0.997232 0.0743585i \(-0.976309\pi\)
0.997232 0.0743585i \(-0.0236909\pi\)
\(608\) − 185.806i − 0.0123938i
\(609\) −1914.51 −0.127389
\(610\) 0 0
\(611\) 14223.2 0.941753
\(612\) − 5461.55i − 0.360735i
\(613\) − 5914.50i − 0.389697i −0.980833 0.194849i \(-0.937578\pi\)
0.980833 0.194849i \(-0.0624216\pi\)
\(614\) 1474.05 0.0968856
\(615\) 0 0
\(616\) −1329.82 −0.0869802
\(617\) − 18591.2i − 1.21306i −0.795062 0.606528i \(-0.792562\pi\)
0.795062 0.606528i \(-0.207438\pi\)
\(618\) 926.638i 0.0603153i
\(619\) −5125.97 −0.332844 −0.166422 0.986055i \(-0.553221\pi\)
−0.166422 + 0.986055i \(0.553221\pi\)
\(620\) 0 0
\(621\) −5527.48 −0.357182
\(622\) − 2280.13i − 0.146985i
\(623\) − 6415.08i − 0.412544i
\(624\) 15222.2 0.976565
\(625\) 0 0
\(626\) 518.957 0.0331337
\(627\) − 626.526i − 0.0399060i
\(628\) 14227.7i 0.904056i
\(629\) −11890.8 −0.753765
\(630\) 0 0
\(631\) −10649.0 −0.671839 −0.335919 0.941891i \(-0.609047\pi\)
−0.335919 + 0.941891i \(0.609047\pi\)
\(632\) 4841.38i 0.304715i
\(633\) − 3090.57i − 0.194058i
\(634\) −715.471 −0.0448186
\(635\) 0 0
\(636\) −4773.76 −0.297629
\(637\) − 3967.56i − 0.246782i
\(638\) − 1086.24i − 0.0674056i
\(639\) 6107.20 0.378086
\(640\) 0 0
\(641\) 24025.7 1.48044 0.740218 0.672367i \(-0.234722\pi\)
0.740218 + 0.672367i \(0.234722\pi\)
\(642\) 896.341i 0.0551024i
\(643\) − 14929.3i − 0.915638i −0.889045 0.457819i \(-0.848631\pi\)
0.889045 0.457819i \(-0.151369\pi\)
\(644\) 11384.5 0.696605
\(645\) 0 0
\(646\) 74.6144 0.00454437
\(647\) 14479.1i 0.879801i 0.898046 + 0.439901i \(0.144986\pi\)
−0.898046 + 0.439901i \(0.855014\pi\)
\(648\) 304.878i 0.0184827i
\(649\) −15772.5 −0.953965
\(650\) 0 0
\(651\) −4159.66 −0.250430
\(652\) 3901.50i 0.234347i
\(653\) − 898.168i − 0.0538255i −0.999638 0.0269127i \(-0.991432\pi\)
0.999638 0.0269127i \(-0.00856762\pi\)
\(654\) −1465.50 −0.0876232
\(655\) 0 0
\(656\) −9831.33 −0.585135
\(657\) − 547.203i − 0.0324938i
\(658\) − 290.273i − 0.0171976i
\(659\) 30198.0 1.78505 0.892526 0.450997i \(-0.148931\pi\)
0.892526 + 0.450997i \(0.148931\pi\)
\(660\) 0 0
\(661\) −19337.8 −1.13790 −0.568952 0.822371i \(-0.692651\pi\)
−0.568952 + 0.822371i \(0.692651\pi\)
\(662\) 1122.20i 0.0658848i
\(663\) 18555.3i 1.08692i
\(664\) 437.257 0.0255555
\(665\) 0 0
\(666\) 330.729 0.0192425
\(667\) 18663.9i 1.08346i
\(668\) − 6562.44i − 0.380102i
\(669\) −16106.0 −0.930784
\(670\) 0 0
\(671\) 7781.30 0.447681
\(672\) − 943.001i − 0.0541325i
\(673\) − 10132.2i − 0.580336i −0.956976 0.290168i \(-0.906289\pi\)
0.956976 0.290168i \(-0.0937113\pi\)
\(674\) −2083.16 −0.119051
\(675\) 0 0
\(676\) −34631.0 −1.97035
\(677\) − 33177.3i − 1.88347i −0.336361 0.941733i \(-0.609196\pi\)
0.336361 0.941733i \(-0.390804\pi\)
\(678\) 1383.75i 0.0783816i
\(679\) −9914.11 −0.560337
\(680\) 0 0
\(681\) −2796.36 −0.157352
\(682\) − 2360.08i − 0.132511i
\(683\) 11423.6i 0.639987i 0.947420 + 0.319994i \(0.103681\pi\)
−0.947420 + 0.319994i \(0.896319\pi\)
\(684\) 295.844 0.0165378
\(685\) 0 0
\(686\) −80.9713 −0.00450656
\(687\) 9489.15i 0.526978i
\(688\) − 22197.2i − 1.23003i
\(689\) 16218.6 0.896775
\(690\) 0 0
\(691\) −19737.4 −1.08661 −0.543304 0.839536i \(-0.682827\pi\)
−0.543304 + 0.839536i \(0.682827\pi\)
\(692\) − 23181.4i − 1.27345i
\(693\) − 3179.74i − 0.174298i
\(694\) 805.929 0.0440816
\(695\) 0 0
\(696\) 1029.44 0.0560645
\(697\) − 11984.0i − 0.651258i
\(698\) 1340.00i 0.0726643i
\(699\) −1309.69 −0.0708682
\(700\) 0 0
\(701\) 11307.8 0.609258 0.304629 0.952471i \(-0.401468\pi\)
0.304629 + 0.952471i \(0.401468\pi\)
\(702\) − 516.093i − 0.0277474i
\(703\) − 644.108i − 0.0345562i
\(704\) −24767.9 −1.32596
\(705\) 0 0
\(706\) 1469.71 0.0783475
\(707\) 9654.36i 0.513564i
\(708\) − 7447.72i − 0.395342i
\(709\) −30859.2 −1.63461 −0.817307 0.576202i \(-0.804534\pi\)
−0.817307 + 0.576202i \(0.804534\pi\)
\(710\) 0 0
\(711\) −11576.3 −0.610613
\(712\) 3449.42i 0.181562i
\(713\) 40551.0i 2.12994i
\(714\) 378.683 0.0198485
\(715\) 0 0
\(716\) −7592.69 −0.396302
\(717\) − 5942.46i − 0.309519i
\(718\) 1158.56i 0.0602187i
\(719\) −33152.4 −1.71958 −0.859789 0.510650i \(-0.829405\pi\)
−0.859789 + 0.510650i \(0.829405\pi\)
\(720\) 0 0
\(721\) 9159.03 0.473093
\(722\) − 1615.15i − 0.0832543i
\(723\) − 15910.7i − 0.818428i
\(724\) 1637.16 0.0840394
\(725\) 0 0
\(726\) 861.485 0.0440395
\(727\) − 16743.0i − 0.854146i −0.904217 0.427073i \(-0.859545\pi\)
0.904217 0.427073i \(-0.140455\pi\)
\(728\) 2133.37i 0.108610i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 27057.5 1.36903
\(732\) 3674.31i 0.185528i
\(733\) 8827.55i 0.444820i 0.974953 + 0.222410i \(0.0713924\pi\)
−0.974953 + 0.222410i \(0.928608\pi\)
\(734\) 922.195 0.0463744
\(735\) 0 0
\(736\) −9192.97 −0.460404
\(737\) − 37082.6i − 1.85340i
\(738\) 333.321i 0.0166256i
\(739\) −36154.0 −1.79966 −0.899829 0.436243i \(-0.856309\pi\)
−0.899829 + 0.436243i \(0.856309\pi\)
\(740\) 0 0
\(741\) −1005.11 −0.0498296
\(742\) − 330.994i − 0.0163762i
\(743\) 1820.69i 0.0898987i 0.998989 + 0.0449494i \(0.0143126\pi\)
−0.998989 + 0.0449494i \(0.985687\pi\)
\(744\) 2236.67 0.110215
\(745\) 0 0
\(746\) −496.338 −0.0243596
\(747\) 1045.53i 0.0512102i
\(748\) − 30628.5i − 1.49718i
\(749\) 8859.57 0.432205
\(750\) 0 0
\(751\) 27764.4 1.34905 0.674526 0.738251i \(-0.264348\pi\)
0.674526 + 0.738251i \(0.264348\pi\)
\(752\) − 11007.8i − 0.533795i
\(753\) 8988.12i 0.434987i
\(754\) −1742.62 −0.0841678
\(755\) 0 0
\(756\) 1501.47 0.0722326
\(757\) − 13518.3i − 0.649050i −0.945877 0.324525i \(-0.894795\pi\)
0.945877 0.324525i \(-0.105205\pi\)
\(758\) 1561.06i 0.0748025i
\(759\) −30998.2 −1.48243
\(760\) 0 0
\(761\) 30695.2 1.46216 0.731078 0.682294i \(-0.239017\pi\)
0.731078 + 0.682294i \(0.239017\pi\)
\(762\) 158.815i 0.00755022i
\(763\) 14485.2i 0.687288i
\(764\) −16036.5 −0.759400
\(765\) 0 0
\(766\) 0.560560 2.64410e−5 0
\(767\) 25303.2i 1.19119i
\(768\) − 11440.9i − 0.537551i
\(769\) −4536.39 −0.212726 −0.106363 0.994327i \(-0.533921\pi\)
−0.106363 + 0.994327i \(0.533921\pi\)
\(770\) 0 0
\(771\) 2906.58 0.135769
\(772\) − 8196.03i − 0.382100i
\(773\) − 31238.9i − 1.45354i −0.686883 0.726768i \(-0.741022\pi\)
0.686883 0.726768i \(-0.258978\pi\)
\(774\) −752.573 −0.0349492
\(775\) 0 0
\(776\) 5330.86 0.246607
\(777\) − 3268.98i − 0.150932i
\(778\) − 1821.73i − 0.0839490i
\(779\) 649.155 0.0298567
\(780\) 0 0
\(781\) 34249.2 1.56919
\(782\) − 3691.64i − 0.168814i
\(783\) 2461.51i 0.112347i
\(784\) −3070.62 −0.139879
\(785\) 0 0
\(786\) −347.656 −0.0157767
\(787\) 39597.2i 1.79350i 0.442534 + 0.896752i \(0.354079\pi\)
−0.442534 + 0.896752i \(0.645921\pi\)
\(788\) − 1636.16i − 0.0739669i
\(789\) 14490.5 0.653836
\(790\) 0 0
\(791\) 13677.2 0.614799
\(792\) 1709.76i 0.0767093i
\(793\) − 12483.3i − 0.559008i
\(794\) −1511.77 −0.0675700
\(795\) 0 0
\(796\) 14549.0 0.647832
\(797\) − 16567.0i − 0.736302i −0.929766 0.368151i \(-0.879991\pi\)
0.929766 0.368151i \(-0.120009\pi\)
\(798\) 20.5127i 0 0.000909951i
\(799\) 13418.1 0.594115
\(800\) 0 0
\(801\) −8247.96 −0.363829
\(802\) − 2589.63i − 0.114019i
\(803\) − 3068.72i − 0.134860i
\(804\) 17510.3 0.768087
\(805\) 0 0
\(806\) −3786.19 −0.165463
\(807\) − 14324.9i − 0.624859i
\(808\) − 5191.20i − 0.226022i
\(809\) 12141.6 0.527657 0.263828 0.964570i \(-0.415015\pi\)
0.263828 + 0.964570i \(0.415015\pi\)
\(810\) 0 0
\(811\) 30295.6 1.31174 0.655870 0.754873i \(-0.272302\pi\)
0.655870 + 0.754873i \(0.272302\pi\)
\(812\) − 5069.80i − 0.219107i
\(813\) − 425.726i − 0.0183651i
\(814\) 1854.73 0.0798629
\(815\) 0 0
\(816\) 14360.5 0.616077
\(817\) 1465.67i 0.0627628i
\(818\) 94.5106i 0.00403971i
\(819\) −5101.15 −0.217642
\(820\) 0 0
\(821\) −14914.8 −0.634018 −0.317009 0.948423i \(-0.602678\pi\)
−0.317009 + 0.948423i \(0.602678\pi\)
\(822\) 1296.81i 0.0550260i
\(823\) 31077.6i 1.31628i 0.752897 + 0.658138i \(0.228656\pi\)
−0.752897 + 0.658138i \(0.771344\pi\)
\(824\) −4924.85 −0.208210
\(825\) 0 0
\(826\) 516.396 0.0217527
\(827\) 15527.8i 0.652908i 0.945213 + 0.326454i \(0.105854\pi\)
−0.945213 + 0.326454i \(0.894146\pi\)
\(828\) − 14637.3i − 0.614348i
\(829\) 40221.5 1.68510 0.842551 0.538617i \(-0.181053\pi\)
0.842551 + 0.538617i \(0.181053\pi\)
\(830\) 0 0
\(831\) −10863.4 −0.453486
\(832\) 39734.2i 1.65569i
\(833\) − 3742.96i − 0.155685i
\(834\) 2160.62 0.0897075
\(835\) 0 0
\(836\) 1659.10 0.0686377
\(837\) 5348.13i 0.220858i
\(838\) 3733.51i 0.153904i
\(839\) 21153.7 0.870448 0.435224 0.900322i \(-0.356669\pi\)
0.435224 + 0.900322i \(0.356669\pi\)
\(840\) 0 0
\(841\) −16077.5 −0.659213
\(842\) 457.152i 0.0187108i
\(843\) − 17372.6i − 0.709780i
\(844\) 8184.10 0.333778
\(845\) 0 0
\(846\) −373.208 −0.0151669
\(847\) − 8515.06i − 0.345432i
\(848\) − 12552.0i − 0.508301i
\(849\) −1578.13 −0.0637942
\(850\) 0 0
\(851\) −31868.1 −1.28369
\(852\) 16172.4i 0.650302i
\(853\) − 636.075i − 0.0255320i −0.999919 0.0127660i \(-0.995936\pi\)
0.999919 0.0127660i \(-0.00406366\pi\)
\(854\) −254.763 −0.0102082
\(855\) 0 0
\(856\) −4763.83 −0.190215
\(857\) − 3941.46i − 0.157104i −0.996910 0.0785518i \(-0.974970\pi\)
0.996910 0.0785518i \(-0.0250296\pi\)
\(858\) − 2894.26i − 0.115161i
\(859\) 21781.6 0.865165 0.432583 0.901594i \(-0.357602\pi\)
0.432583 + 0.901594i \(0.357602\pi\)
\(860\) 0 0
\(861\) 3294.59 0.130406
\(862\) 479.432i 0.0189438i
\(863\) 44697.9i 1.76308i 0.472112 + 0.881538i \(0.343492\pi\)
−0.472112 + 0.881538i \(0.656508\pi\)
\(864\) −1212.43 −0.0477404
\(865\) 0 0
\(866\) −2545.79 −0.0998953
\(867\) 2765.92i 0.108345i
\(868\) − 11015.2i − 0.430736i
\(869\) −64920.1 −2.53425
\(870\) 0 0
\(871\) −59490.3 −2.31430
\(872\) − 7788.78i − 0.302478i
\(873\) 12746.7i 0.494171i
\(874\) 199.971 0.00773926
\(875\) 0 0
\(876\) 1449.04 0.0558888
\(877\) − 20171.7i − 0.776682i −0.921516 0.388341i \(-0.873048\pi\)
0.921516 0.388341i \(-0.126952\pi\)
\(878\) 1488.22i 0.0572038i
\(879\) −5744.63 −0.220434
\(880\) 0 0
\(881\) 11577.6 0.442744 0.221372 0.975189i \(-0.428946\pi\)
0.221372 + 0.975189i \(0.428946\pi\)
\(882\) 104.106i 0.00397441i
\(883\) 35388.3i 1.34871i 0.738407 + 0.674355i \(0.235578\pi\)
−0.738407 + 0.674355i \(0.764422\pi\)
\(884\) −49136.1 −1.86949
\(885\) 0 0
\(886\) 3657.83 0.138699
\(887\) − 41705.0i − 1.57871i −0.613936 0.789356i \(-0.710415\pi\)
0.613936 0.789356i \(-0.289585\pi\)
\(888\) 1757.74i 0.0664257i
\(889\) 1569.75 0.0592215
\(890\) 0 0
\(891\) −4088.24 −0.153716
\(892\) − 42650.2i − 1.60093i
\(893\) 726.838i 0.0272371i
\(894\) −1591.02 −0.0595207
\(895\) 0 0
\(896\) 3325.58 0.123995
\(897\) 49729.2i 1.85107i
\(898\) 57.1328i 0.00212310i
\(899\) 18058.3 0.669942
\(900\) 0 0
\(901\) 15300.5 0.565740
\(902\) 1869.27i 0.0690020i
\(903\) 7438.55i 0.274130i
\(904\) −7354.31 −0.270576
\(905\) 0 0
\(906\) −928.830 −0.0340600
\(907\) − 28645.1i − 1.04867i −0.851511 0.524336i \(-0.824313\pi\)
0.851511 0.524336i \(-0.175687\pi\)
\(908\) − 7405.02i − 0.270643i
\(909\) 12412.8 0.452921
\(910\) 0 0
\(911\) 13337.8 0.485074 0.242537 0.970142i \(-0.422020\pi\)
0.242537 + 0.970142i \(0.422020\pi\)
\(912\) 777.887i 0.0282439i
\(913\) 5863.36i 0.212540i
\(914\) 2755.00 0.0997016
\(915\) 0 0
\(916\) −25128.1 −0.906394
\(917\) 3436.29i 0.123747i
\(918\) − 486.878i − 0.0175048i
\(919\) 28911.0 1.03774 0.518871 0.854853i \(-0.326353\pi\)
0.518871 + 0.854853i \(0.326353\pi\)
\(920\) 0 0
\(921\) −18732.5 −0.670203
\(922\) 2863.06i 0.102267i
\(923\) − 54944.8i − 1.95940i
\(924\) 8420.25 0.299790
\(925\) 0 0
\(926\) 1218.50 0.0432422
\(927\) − 11775.9i − 0.417229i
\(928\) 4093.84i 0.144814i
\(929\) −7093.88 −0.250530 −0.125265 0.992123i \(-0.539978\pi\)
−0.125265 + 0.992123i \(0.539978\pi\)
\(930\) 0 0
\(931\) 202.751 0.00713736
\(932\) − 3468.17i − 0.121892i
\(933\) 28976.3i 1.01677i
\(934\) −2757.33 −0.0965981
\(935\) 0 0
\(936\) 2742.91 0.0957851
\(937\) − 19271.1i − 0.671888i −0.941882 0.335944i \(-0.890945\pi\)
0.941882 0.335944i \(-0.109055\pi\)
\(938\) 1214.10i 0.0422620i
\(939\) −6595.01 −0.229201
\(940\) 0 0
\(941\) −18115.2 −0.627563 −0.313782 0.949495i \(-0.601596\pi\)
−0.313782 + 0.949495i \(0.601596\pi\)
\(942\) 1268.35i 0.0438695i
\(943\) − 32117.8i − 1.10912i
\(944\) 19582.9 0.675180
\(945\) 0 0
\(946\) −4220.44 −0.145051
\(947\) − 2475.55i − 0.0849468i −0.999098 0.0424734i \(-0.986476\pi\)
0.999098 0.0424734i \(-0.0135238\pi\)
\(948\) − 30655.1i − 1.05024i
\(949\) −4923.04 −0.168397
\(950\) 0 0
\(951\) 9092.35 0.310031
\(952\) 2012.61i 0.0685179i
\(953\) − 12866.6i − 0.437344i −0.975798 0.218672i \(-0.929827\pi\)
0.975798 0.218672i \(-0.0701725\pi\)
\(954\) −425.564 −0.0144425
\(955\) 0 0
\(956\) 15736.2 0.532368
\(957\) 13804.2i 0.466277i
\(958\) 4357.50i 0.146957i
\(959\) 12817.9 0.431606
\(960\) 0 0
\(961\) 9444.26 0.317017
\(962\) − 2975.48i − 0.0997228i
\(963\) − 11390.9i − 0.381169i
\(964\) 42132.9 1.40768
\(965\) 0 0
\(966\) 1014.89 0.0338029
\(967\) − 2142.86i − 0.0712614i −0.999365 0.0356307i \(-0.988656\pi\)
0.999365 0.0356307i \(-0.0113440\pi\)
\(968\) 4578.58i 0.152026i
\(969\) −948.215 −0.0314356
\(970\) 0 0
\(971\) 2879.06 0.0951529 0.0475765 0.998868i \(-0.484850\pi\)
0.0475765 + 0.998868i \(0.484850\pi\)
\(972\) − 1930.46i − 0.0637032i
\(973\) − 21355.9i − 0.703636i
\(974\) −2037.29 −0.0670217
\(975\) 0 0
\(976\) −9661.18 −0.316851
\(977\) 48741.1i 1.59607i 0.602608 + 0.798037i \(0.294128\pi\)
−0.602608 + 0.798037i \(0.705872\pi\)
\(978\) 347.805i 0.0113718i
\(979\) −46254.7 −1.51002
\(980\) 0 0
\(981\) 18623.9 0.606131
\(982\) 4202.40i 0.136562i
\(983\) 45756.8i 1.48466i 0.670037 + 0.742328i \(0.266278\pi\)
−0.670037 + 0.742328i \(0.733722\pi\)
\(984\) −1771.52 −0.0573922
\(985\) 0 0
\(986\) −1643.97 −0.0530982
\(987\) 3688.85i 0.118964i
\(988\) − 2661.63i − 0.0857062i
\(989\) 72515.7 2.33151
\(990\) 0 0
\(991\) −51552.1 −1.65248 −0.826240 0.563319i \(-0.809524\pi\)
−0.826240 + 0.563319i \(0.809524\pi\)
\(992\) 8894.70i 0.284684i
\(993\) − 14261.2i − 0.455756i
\(994\) −1121.33 −0.0357812
\(995\) 0 0
\(996\) −2768.67 −0.0880808
\(997\) 25565.3i 0.812097i 0.913852 + 0.406048i \(0.133094\pi\)
−0.913852 + 0.406048i \(0.866906\pi\)
\(998\) − 47.2531i − 0.00149877i
\(999\) −4202.97 −0.133109
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.k.274.3 4
5.2 odd 4 525.4.a.o.1.1 2
5.3 odd 4 105.4.a.d.1.2 2
5.4 even 2 inner 525.4.d.k.274.2 4
15.2 even 4 1575.4.a.n.1.2 2
15.8 even 4 315.4.a.l.1.1 2
20.3 even 4 1680.4.a.bd.1.2 2
35.13 even 4 735.4.a.m.1.2 2
105.83 odd 4 2205.4.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.d.1.2 2 5.3 odd 4
315.4.a.l.1.1 2 15.8 even 4
525.4.a.o.1.1 2 5.2 odd 4
525.4.d.k.274.2 4 5.4 even 2 inner
525.4.d.k.274.3 4 1.1 even 1 trivial
735.4.a.m.1.2 2 35.13 even 4
1575.4.a.n.1.2 2 15.2 even 4
1680.4.a.bd.1.2 2 20.3 even 4
2205.4.a.be.1.1 2 105.83 odd 4