Properties

Label 575.4.b.i.24.6
Level $575$
Weight $4$
Character 575.24
Analytic conductor $33.926$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (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: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.6
Root \(-0.595043i\) of defining polynomial
Character \(\chi\) \(=\) 575.24
Dual form 575.4.b.i.24.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.404957i q^{2} +7.11323i q^{3} +7.83601 q^{4} -2.88055 q^{6} +13.7888i q^{7} +6.41290i q^{8} -23.5981 q^{9} +O(q^{10})\) \(q+0.404957i q^{2} +7.11323i q^{3} +7.83601 q^{4} -2.88055 q^{6} +13.7888i q^{7} +6.41290i q^{8} -23.5981 q^{9} +24.2317 q^{11} +55.7394i q^{12} -3.05016i q^{13} -5.58389 q^{14} +60.0911 q^{16} +63.1126i q^{17} -9.55621i q^{18} +2.07770 q^{19} -98.0832 q^{21} +9.81282i q^{22} +23.0000i q^{23} -45.6165 q^{24} +1.23518 q^{26} +24.1987i q^{27} +108.049i q^{28} +8.16397 q^{29} -156.989 q^{31} +75.6376i q^{32} +172.366i q^{33} -25.5579 q^{34} -184.915 q^{36} +302.801i q^{37} +0.841380i q^{38} +21.6965 q^{39} -42.7514 q^{41} -39.7195i q^{42} -215.265i q^{43} +189.880 q^{44} -9.31401 q^{46} +247.096i q^{47} +427.442i q^{48} +152.868 q^{49} -448.935 q^{51} -23.9010i q^{52} -600.400i q^{53} -9.79943 q^{54} -88.4265 q^{56} +14.7792i q^{57} +3.30606i q^{58} -92.2014 q^{59} +532.635 q^{61} -63.5740i q^{62} -325.390i q^{63} +450.099 q^{64} -69.8009 q^{66} +30.3010i q^{67} +494.551i q^{68} -163.604 q^{69} -736.349 q^{71} -151.332i q^{72} -349.936i q^{73} -122.622 q^{74} +16.2809 q^{76} +334.127i q^{77} +8.78614i q^{78} -301.545 q^{79} -809.279 q^{81} -17.3125i q^{82} -139.488i q^{83} -768.581 q^{84} +87.1732 q^{86} +58.0722i q^{87} +155.396i q^{88} -859.551 q^{89} +42.0581 q^{91} +180.228i q^{92} -1116.70i q^{93} -100.063 q^{94} -538.028 q^{96} -927.475i q^{97} +61.9051i q^{98} -571.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9} + 46 q^{11} - 186 q^{14} + 564 q^{16} + 322 q^{19} - 120 q^{21} - 210 q^{24} - 514 q^{26} - 802 q^{29} + 64 q^{31} + 1326 q^{34} - 1318 q^{36} - 670 q^{39} - 24 q^{41} - 94 q^{44} - 276 q^{46} + 1476 q^{49} - 1986 q^{51} + 16 q^{54} + 686 q^{56} - 2648 q^{59} - 3346 q^{61} - 4932 q^{64} - 5562 q^{66} + 184 q^{69} - 216 q^{71} - 2916 q^{74} - 6954 q^{76} - 1312 q^{79} - 638 q^{81} + 1436 q^{84} + 224 q^{86} - 1140 q^{89} - 3178 q^{91} + 1896 q^{94} - 11982 q^{96} - 4042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.404957i 0.143174i 0.997434 + 0.0715870i \(0.0228064\pi\)
−0.997434 + 0.0715870i \(0.977194\pi\)
\(3\) 7.11323i 1.36894i 0.729040 + 0.684471i \(0.239967\pi\)
−0.729040 + 0.684471i \(0.760033\pi\)
\(4\) 7.83601 0.979501
\(5\) 0 0
\(6\) −2.88055 −0.195997
\(7\) 13.7888i 0.744527i 0.928127 + 0.372263i \(0.121418\pi\)
−0.928127 + 0.372263i \(0.878582\pi\)
\(8\) 6.41290i 0.283413i
\(9\) −23.5981 −0.874003
\(10\) 0 0
\(11\) 24.2317 0.664195 0.332098 0.943245i \(-0.392244\pi\)
0.332098 + 0.943245i \(0.392244\pi\)
\(12\) 55.7394i 1.34088i
\(13\) − 3.05016i − 0.0650739i −0.999471 0.0325370i \(-0.989641\pi\)
0.999471 0.0325370i \(-0.0103587\pi\)
\(14\) −5.58389 −0.106597
\(15\) 0 0
\(16\) 60.0911 0.938924
\(17\) 63.1126i 0.900415i 0.892924 + 0.450208i \(0.148650\pi\)
−0.892924 + 0.450208i \(0.851350\pi\)
\(18\) − 9.55621i − 0.125134i
\(19\) 2.07770 0.0250872 0.0125436 0.999921i \(-0.496007\pi\)
0.0125436 + 0.999921i \(0.496007\pi\)
\(20\) 0 0
\(21\) −98.0832 −1.01921
\(22\) 9.81282i 0.0950955i
\(23\) 23.0000i 0.208514i
\(24\) −45.6165 −0.387976
\(25\) 0 0
\(26\) 1.23518 0.00931689
\(27\) 24.1987i 0.172483i
\(28\) 108.049i 0.729265i
\(29\) 8.16397 0.0522762 0.0261381 0.999658i \(-0.491679\pi\)
0.0261381 + 0.999658i \(0.491679\pi\)
\(30\) 0 0
\(31\) −156.989 −0.909553 −0.454776 0.890606i \(-0.650281\pi\)
−0.454776 + 0.890606i \(0.650281\pi\)
\(32\) 75.6376i 0.417842i
\(33\) 172.366i 0.909245i
\(34\) −25.5579 −0.128916
\(35\) 0 0
\(36\) −184.915 −0.856087
\(37\) 302.801i 1.34541i 0.739910 + 0.672706i \(0.234868\pi\)
−0.739910 + 0.672706i \(0.765132\pi\)
\(38\) 0.841380i 0.00359184i
\(39\) 21.6965 0.0890824
\(40\) 0 0
\(41\) −42.7514 −0.162845 −0.0814225 0.996680i \(-0.525946\pi\)
−0.0814225 + 0.996680i \(0.525946\pi\)
\(42\) − 39.7195i − 0.145925i
\(43\) − 215.265i − 0.763434i −0.924279 0.381717i \(-0.875333\pi\)
0.924279 0.381717i \(-0.124667\pi\)
\(44\) 189.880 0.650580
\(45\) 0 0
\(46\) −9.31401 −0.0298538
\(47\) 247.096i 0.766866i 0.923569 + 0.383433i \(0.125258\pi\)
−0.923569 + 0.383433i \(0.874742\pi\)
\(48\) 427.442i 1.28533i
\(49\) 152.868 0.445680
\(50\) 0 0
\(51\) −448.935 −1.23262
\(52\) − 23.9010i − 0.0637400i
\(53\) − 600.400i − 1.55606i −0.628225 0.778031i \(-0.716218\pi\)
0.628225 0.778031i \(-0.283782\pi\)
\(54\) −9.79943 −0.0246951
\(55\) 0 0
\(56\) −88.4265 −0.211009
\(57\) 14.7792i 0.0343430i
\(58\) 3.30606i 0.00748460i
\(59\) −92.2014 −0.203451 −0.101725 0.994813i \(-0.532436\pi\)
−0.101725 + 0.994813i \(0.532436\pi\)
\(60\) 0 0
\(61\) 532.635 1.11798 0.558991 0.829174i \(-0.311189\pi\)
0.558991 + 0.829174i \(0.311189\pi\)
\(62\) − 63.5740i − 0.130224i
\(63\) − 325.390i − 0.650719i
\(64\) 450.099 0.879100
\(65\) 0 0
\(66\) −69.8009 −0.130180
\(67\) 30.3010i 0.0552515i 0.999618 + 0.0276258i \(0.00879467\pi\)
−0.999618 + 0.0276258i \(0.991205\pi\)
\(68\) 494.551i 0.881958i
\(69\) −163.604 −0.285444
\(70\) 0 0
\(71\) −736.349 −1.23083 −0.615413 0.788205i \(-0.711011\pi\)
−0.615413 + 0.788205i \(0.711011\pi\)
\(72\) − 151.332i − 0.247704i
\(73\) − 349.936i − 0.561053i −0.959846 0.280527i \(-0.909491\pi\)
0.959846 0.280527i \(-0.0905091\pi\)
\(74\) −122.622 −0.192628
\(75\) 0 0
\(76\) 16.2809 0.0245730
\(77\) 334.127i 0.494511i
\(78\) 8.78614i 0.0127543i
\(79\) −301.545 −0.429449 −0.214725 0.976675i \(-0.568885\pi\)
−0.214725 + 0.976675i \(0.568885\pi\)
\(80\) 0 0
\(81\) −809.279 −1.11012
\(82\) − 17.3125i − 0.0233152i
\(83\) − 139.488i − 0.184468i −0.995737 0.0922340i \(-0.970599\pi\)
0.995737 0.0922340i \(-0.0294008\pi\)
\(84\) −768.581 −0.998322
\(85\) 0 0
\(86\) 87.1732 0.109304
\(87\) 58.0722i 0.0715631i
\(88\) 155.396i 0.188242i
\(89\) −859.551 −1.02373 −0.511866 0.859065i \(-0.671046\pi\)
−0.511866 + 0.859065i \(0.671046\pi\)
\(90\) 0 0
\(91\) 42.0581 0.0484493
\(92\) 180.228i 0.204240i
\(93\) − 1116.70i − 1.24513i
\(94\) −100.063 −0.109795
\(95\) 0 0
\(96\) −538.028 −0.572002
\(97\) − 927.475i − 0.970833i −0.874283 0.485417i \(-0.838668\pi\)
0.874283 0.485417i \(-0.161332\pi\)
\(98\) 61.9051i 0.0638097i
\(99\) −571.823 −0.580508
\(100\) 0 0
\(101\) 1713.34 1.68796 0.843979 0.536376i \(-0.180207\pi\)
0.843979 + 0.536376i \(0.180207\pi\)
\(102\) − 181.799i − 0.176479i
\(103\) 930.516i 0.890160i 0.895491 + 0.445080i \(0.146825\pi\)
−0.895491 + 0.445080i \(0.853175\pi\)
\(104\) 19.5604 0.0184428
\(105\) 0 0
\(106\) 243.136 0.222788
\(107\) 1514.91i 1.36871i 0.729149 + 0.684355i \(0.239916\pi\)
−0.729149 + 0.684355i \(0.760084\pi\)
\(108\) 189.621i 0.168947i
\(109\) −1748.76 −1.53670 −0.768352 0.640027i \(-0.778923\pi\)
−0.768352 + 0.640027i \(0.778923\pi\)
\(110\) 0 0
\(111\) −2153.90 −1.84179
\(112\) 828.586i 0.699054i
\(113\) − 2026.23i − 1.68683i −0.537261 0.843416i \(-0.680541\pi\)
0.537261 0.843416i \(-0.319459\pi\)
\(114\) −5.98493 −0.00491702
\(115\) 0 0
\(116\) 63.9729 0.0512046
\(117\) 71.9778i 0.0568748i
\(118\) − 37.3376i − 0.0291289i
\(119\) −870.249 −0.670383
\(120\) 0 0
\(121\) −743.822 −0.558845
\(122\) 215.694i 0.160066i
\(123\) − 304.100i − 0.222925i
\(124\) −1230.17 −0.890908
\(125\) 0 0
\(126\) 131.769 0.0931660
\(127\) − 2126.49i − 1.48579i −0.669406 0.742897i \(-0.733451\pi\)
0.669406 0.742897i \(-0.266549\pi\)
\(128\) 787.371i 0.543707i
\(129\) 1531.23 1.04510
\(130\) 0 0
\(131\) −1494.23 −0.996576 −0.498288 0.867012i \(-0.666038\pi\)
−0.498288 + 0.867012i \(0.666038\pi\)
\(132\) 1350.66i 0.890606i
\(133\) 28.6491i 0.0186781i
\(134\) −12.2706 −0.00791058
\(135\) 0 0
\(136\) −404.735 −0.255189
\(137\) 2265.31i 1.41269i 0.707867 + 0.706346i \(0.249658\pi\)
−0.707867 + 0.706346i \(0.750342\pi\)
\(138\) − 66.2527i − 0.0408682i
\(139\) 2918.66 1.78099 0.890496 0.454991i \(-0.150358\pi\)
0.890496 + 0.454991i \(0.150358\pi\)
\(140\) 0 0
\(141\) −1757.65 −1.04980
\(142\) − 298.190i − 0.176222i
\(143\) − 73.9106i − 0.0432218i
\(144\) −1418.03 −0.820622
\(145\) 0 0
\(146\) 141.709 0.0803282
\(147\) 1087.39i 0.610110i
\(148\) 2372.75i 1.31783i
\(149\) 549.400 0.302071 0.151036 0.988528i \(-0.451739\pi\)
0.151036 + 0.988528i \(0.451739\pi\)
\(150\) 0 0
\(151\) 335.721 0.180931 0.0904654 0.995900i \(-0.471165\pi\)
0.0904654 + 0.995900i \(0.471165\pi\)
\(152\) 13.3241i 0.00711005i
\(153\) − 1489.34i − 0.786965i
\(154\) −135.307 −0.0708011
\(155\) 0 0
\(156\) 170.014 0.0872563
\(157\) − 1593.69i − 0.810130i −0.914288 0.405065i \(-0.867249\pi\)
0.914288 0.405065i \(-0.132751\pi\)
\(158\) − 122.113i − 0.0614860i
\(159\) 4270.79 2.13016
\(160\) 0 0
\(161\) −317.143 −0.155245
\(162\) − 327.723i − 0.158941i
\(163\) 2767.63i 1.32992i 0.746877 + 0.664962i \(0.231552\pi\)
−0.746877 + 0.664962i \(0.768448\pi\)
\(164\) −335.000 −0.159507
\(165\) 0 0
\(166\) 56.4868 0.0264110
\(167\) 282.867i 0.131071i 0.997850 + 0.0655357i \(0.0208756\pi\)
−0.997850 + 0.0655357i \(0.979124\pi\)
\(168\) − 628.998i − 0.288859i
\(169\) 2187.70 0.995765
\(170\) 0 0
\(171\) −49.0297 −0.0219263
\(172\) − 1686.82i − 0.747784i
\(173\) 2331.63i 1.02468i 0.858782 + 0.512342i \(0.171222\pi\)
−0.858782 + 0.512342i \(0.828778\pi\)
\(174\) −23.5168 −0.0102460
\(175\) 0 0
\(176\) 1456.11 0.623629
\(177\) − 655.850i − 0.278512i
\(178\) − 348.081i − 0.146572i
\(179\) −109.140 −0.0455726 −0.0227863 0.999740i \(-0.507254\pi\)
−0.0227863 + 0.999740i \(0.507254\pi\)
\(180\) 0 0
\(181\) 1476.85 0.606483 0.303242 0.952914i \(-0.401931\pi\)
0.303242 + 0.952914i \(0.401931\pi\)
\(182\) 17.0317i 0.00693667i
\(183\) 3788.76i 1.53045i
\(184\) −147.497 −0.0590957
\(185\) 0 0
\(186\) 452.217 0.178270
\(187\) 1529.33i 0.598051i
\(188\) 1936.25i 0.751147i
\(189\) −333.672 −0.128418
\(190\) 0 0
\(191\) 2032.16 0.769853 0.384926 0.922947i \(-0.374227\pi\)
0.384926 + 0.922947i \(0.374227\pi\)
\(192\) 3201.66i 1.20344i
\(193\) − 3883.64i − 1.44845i −0.689565 0.724224i \(-0.742198\pi\)
0.689565 0.724224i \(-0.257802\pi\)
\(194\) 375.588 0.138998
\(195\) 0 0
\(196\) 1197.88 0.436544
\(197\) 3580.64i 1.29498i 0.762076 + 0.647488i \(0.224180\pi\)
−0.762076 + 0.647488i \(0.775820\pi\)
\(198\) − 231.564i − 0.0831137i
\(199\) 2831.17 1.00853 0.504263 0.863550i \(-0.331764\pi\)
0.504263 + 0.863550i \(0.331764\pi\)
\(200\) 0 0
\(201\) −215.538 −0.0756362
\(202\) 693.830i 0.241672i
\(203\) 112.572i 0.0389211i
\(204\) −3517.86 −1.20735
\(205\) 0 0
\(206\) −376.819 −0.127448
\(207\) − 542.756i − 0.182242i
\(208\) − 183.287i − 0.0610995i
\(209\) 50.3463 0.0166628
\(210\) 0 0
\(211\) 2903.80 0.947421 0.473711 0.880681i \(-0.342914\pi\)
0.473711 + 0.880681i \(0.342914\pi\)
\(212\) − 4704.74i − 1.52417i
\(213\) − 5237.82i − 1.68493i
\(214\) −613.474 −0.195964
\(215\) 0 0
\(216\) −155.184 −0.0488839
\(217\) − 2164.70i − 0.677186i
\(218\) − 708.173i − 0.220016i
\(219\) 2489.17 0.768049
\(220\) 0 0
\(221\) 192.503 0.0585935
\(222\) − 872.236i − 0.263697i
\(223\) − 454.760i − 0.136560i −0.997666 0.0682802i \(-0.978249\pi\)
0.997666 0.0682802i \(-0.0217512\pi\)
\(224\) −1042.95 −0.311095
\(225\) 0 0
\(226\) 820.538 0.241510
\(227\) − 2103.24i − 0.614966i −0.951554 0.307483i \(-0.900513\pi\)
0.951554 0.307483i \(-0.0994867\pi\)
\(228\) 115.810i 0.0336390i
\(229\) 4647.97 1.34125 0.670625 0.741796i \(-0.266026\pi\)
0.670625 + 0.741796i \(0.266026\pi\)
\(230\) 0 0
\(231\) −2376.73 −0.676957
\(232\) 52.3548i 0.0148158i
\(233\) − 131.118i − 0.0368661i −0.999830 0.0184331i \(-0.994132\pi\)
0.999830 0.0184331i \(-0.00586776\pi\)
\(234\) −29.1479 −0.00814299
\(235\) 0 0
\(236\) −722.491 −0.199280
\(237\) − 2144.96i − 0.587891i
\(238\) − 352.414i − 0.0959814i
\(239\) −3467.77 −0.938541 −0.469271 0.883054i \(-0.655483\pi\)
−0.469271 + 0.883054i \(0.655483\pi\)
\(240\) 0 0
\(241\) 5818.91 1.55531 0.777653 0.628694i \(-0.216410\pi\)
0.777653 + 0.628694i \(0.216410\pi\)
\(242\) − 301.216i − 0.0800120i
\(243\) − 5103.22i − 1.34721i
\(244\) 4173.73 1.09506
\(245\) 0 0
\(246\) 123.148 0.0319171
\(247\) − 6.33731i − 0.00163252i
\(248\) − 1006.76i − 0.257779i
\(249\) 992.213 0.252526
\(250\) 0 0
\(251\) 4633.30 1.16515 0.582573 0.812779i \(-0.302046\pi\)
0.582573 + 0.812779i \(0.302046\pi\)
\(252\) − 2549.76i − 0.637380i
\(253\) 557.330i 0.138494i
\(254\) 861.139 0.212727
\(255\) 0 0
\(256\) 3281.94 0.801255
\(257\) − 5262.78i − 1.27737i −0.769470 0.638683i \(-0.779480\pi\)
0.769470 0.638683i \(-0.220520\pi\)
\(258\) 620.083i 0.149631i
\(259\) −4175.28 −1.00170
\(260\) 0 0
\(261\) −192.654 −0.0456896
\(262\) − 605.099i − 0.142684i
\(263\) − 1890.26i − 0.443189i −0.975139 0.221594i \(-0.928874\pi\)
0.975139 0.221594i \(-0.0711261\pi\)
\(264\) −1105.37 −0.257692
\(265\) 0 0
\(266\) −11.6016 −0.00267422
\(267\) − 6114.18i − 1.40143i
\(268\) 237.439i 0.0541189i
\(269\) −7472.17 −1.69363 −0.846815 0.531888i \(-0.821483\pi\)
−0.846815 + 0.531888i \(0.821483\pi\)
\(270\) 0 0
\(271\) 1634.38 0.366352 0.183176 0.983080i \(-0.441362\pi\)
0.183176 + 0.983080i \(0.441362\pi\)
\(272\) 3792.51i 0.845421i
\(273\) 299.169i 0.0663243i
\(274\) −917.355 −0.202261
\(275\) 0 0
\(276\) −1282.01 −0.279593
\(277\) − 590.262i − 0.128034i −0.997949 0.0640170i \(-0.979609\pi\)
0.997949 0.0640170i \(-0.0203912\pi\)
\(278\) 1181.93i 0.254992i
\(279\) 3704.65 0.794952
\(280\) 0 0
\(281\) 2501.14 0.530980 0.265490 0.964114i \(-0.414466\pi\)
0.265490 + 0.964114i \(0.414466\pi\)
\(282\) − 711.775i − 0.150303i
\(283\) 803.901i 0.168859i 0.996429 + 0.0844293i \(0.0269067\pi\)
−0.996429 + 0.0844293i \(0.973093\pi\)
\(284\) −5770.04 −1.20559
\(285\) 0 0
\(286\) 29.9306 0.00618823
\(287\) − 589.491i − 0.121242i
\(288\) − 1784.90i − 0.365195i
\(289\) 929.797 0.189252
\(290\) 0 0
\(291\) 6597.35 1.32901
\(292\) − 2742.10i − 0.549552i
\(293\) 6332.54i 1.26263i 0.775526 + 0.631316i \(0.217485\pi\)
−0.775526 + 0.631316i \(0.782515\pi\)
\(294\) −440.345 −0.0873518
\(295\) 0 0
\(296\) −1941.84 −0.381307
\(297\) 586.376i 0.114562i
\(298\) 222.483i 0.0432487i
\(299\) 70.1536 0.0135688
\(300\) 0 0
\(301\) 2968.26 0.568397
\(302\) 135.952i 0.0259046i
\(303\) 12187.4i 2.31072i
\(304\) 124.851 0.0235550
\(305\) 0 0
\(306\) 603.117 0.112673
\(307\) 7317.73i 1.36041i 0.733024 + 0.680203i \(0.238108\pi\)
−0.733024 + 0.680203i \(0.761892\pi\)
\(308\) 2618.23i 0.484374i
\(309\) −6618.97 −1.21858
\(310\) 0 0
\(311\) 2838.86 0.517611 0.258805 0.965930i \(-0.416671\pi\)
0.258805 + 0.965930i \(0.416671\pi\)
\(312\) 139.137i 0.0252471i
\(313\) − 160.132i − 0.0289175i −0.999895 0.0144588i \(-0.995397\pi\)
0.999895 0.0144588i \(-0.00460253\pi\)
\(314\) 645.376 0.115989
\(315\) 0 0
\(316\) −2362.91 −0.420646
\(317\) − 3330.78i − 0.590142i −0.955475 0.295071i \(-0.904657\pi\)
0.955475 0.295071i \(-0.0953433\pi\)
\(318\) 1729.49i 0.304983i
\(319\) 197.827 0.0347216
\(320\) 0 0
\(321\) −10775.9 −1.87369
\(322\) − 128.429i − 0.0222270i
\(323\) 131.129i 0.0225889i
\(324\) −6341.52 −1.08737
\(325\) 0 0
\(326\) −1120.77 −0.190410
\(327\) − 12439.3i − 2.10366i
\(328\) − 274.161i − 0.0461524i
\(329\) −3407.17 −0.570953
\(330\) 0 0
\(331\) 7337.39 1.21843 0.609214 0.793006i \(-0.291485\pi\)
0.609214 + 0.793006i \(0.291485\pi\)
\(332\) − 1093.03i − 0.180687i
\(333\) − 7145.53i − 1.17589i
\(334\) −114.549 −0.0187660
\(335\) 0 0
\(336\) −5893.93 −0.956965
\(337\) − 7160.54i − 1.15745i −0.815524 0.578723i \(-0.803551\pi\)
0.815524 0.578723i \(-0.196449\pi\)
\(338\) 885.923i 0.142568i
\(339\) 14413.1 2.30918
\(340\) 0 0
\(341\) −3804.13 −0.604121
\(342\) − 19.8549i − 0.00313928i
\(343\) 6837.44i 1.07635i
\(344\) 1380.48 0.216367
\(345\) 0 0
\(346\) −944.209 −0.146708
\(347\) 6740.51i 1.04279i 0.853314 + 0.521397i \(0.174589\pi\)
−0.853314 + 0.521397i \(0.825411\pi\)
\(348\) 455.054i 0.0700962i
\(349\) 10173.9 1.56045 0.780224 0.625501i \(-0.215105\pi\)
0.780224 + 0.625501i \(0.215105\pi\)
\(350\) 0 0
\(351\) 73.8097 0.0112241
\(352\) 1832.83i 0.277529i
\(353\) 2552.87i 0.384917i 0.981305 + 0.192458i \(0.0616461\pi\)
−0.981305 + 0.192458i \(0.938354\pi\)
\(354\) 265.591 0.0398757
\(355\) 0 0
\(356\) −6735.45 −1.00275
\(357\) − 6190.28i − 0.917716i
\(358\) − 44.1970i − 0.00652481i
\(359\) 6677.47 0.981681 0.490841 0.871249i \(-0.336690\pi\)
0.490841 + 0.871249i \(0.336690\pi\)
\(360\) 0 0
\(361\) −6854.68 −0.999371
\(362\) 598.062i 0.0868326i
\(363\) − 5290.98i − 0.765026i
\(364\) 329.567 0.0474561
\(365\) 0 0
\(366\) −1534.28 −0.219121
\(367\) 11722.8i 1.66738i 0.552236 + 0.833688i \(0.313775\pi\)
−0.552236 + 0.833688i \(0.686225\pi\)
\(368\) 1382.10i 0.195779i
\(369\) 1008.85 0.142327
\(370\) 0 0
\(371\) 8278.82 1.15853
\(372\) − 8750.49i − 1.21960i
\(373\) − 8070.71i − 1.12034i −0.828379 0.560168i \(-0.810736\pi\)
0.828379 0.560168i \(-0.189264\pi\)
\(374\) −619.313 −0.0856254
\(375\) 0 0
\(376\) −1584.61 −0.217340
\(377\) − 24.9014i − 0.00340182i
\(378\) − 135.123i − 0.0183861i
\(379\) 6693.10 0.907128 0.453564 0.891224i \(-0.350152\pi\)
0.453564 + 0.891224i \(0.350152\pi\)
\(380\) 0 0
\(381\) 15126.2 2.03397
\(382\) 822.937i 0.110223i
\(383\) 2502.49i 0.333867i 0.985968 + 0.166934i \(0.0533866\pi\)
−0.985968 + 0.166934i \(0.946613\pi\)
\(384\) −5600.76 −0.744303
\(385\) 0 0
\(386\) 1572.71 0.207380
\(387\) 5079.85i 0.667243i
\(388\) − 7267.71i − 0.950933i
\(389\) −5487.69 −0.715262 −0.357631 0.933863i \(-0.616415\pi\)
−0.357631 + 0.933863i \(0.616415\pi\)
\(390\) 0 0
\(391\) −1451.59 −0.187750
\(392\) 980.329i 0.126311i
\(393\) − 10628.8i − 1.36426i
\(394\) −1450.01 −0.185407
\(395\) 0 0
\(396\) −4480.81 −0.568609
\(397\) 7760.91i 0.981130i 0.871405 + 0.490565i \(0.163210\pi\)
−0.871405 + 0.490565i \(0.836790\pi\)
\(398\) 1146.50i 0.144395i
\(399\) −203.787 −0.0255693
\(400\) 0 0
\(401\) 14485.8 1.80395 0.901977 0.431785i \(-0.142116\pi\)
0.901977 + 0.431785i \(0.142116\pi\)
\(402\) − 87.2836i − 0.0108291i
\(403\) 478.842i 0.0591882i
\(404\) 13425.8 1.65336
\(405\) 0 0
\(406\) −45.5867 −0.00557248
\(407\) 7337.41i 0.893616i
\(408\) − 2878.98i − 0.349340i
\(409\) −6664.83 −0.805758 −0.402879 0.915253i \(-0.631990\pi\)
−0.402879 + 0.915253i \(0.631990\pi\)
\(410\) 0 0
\(411\) −16113.7 −1.93389
\(412\) 7291.53i 0.871912i
\(413\) − 1271.35i − 0.151475i
\(414\) 219.793 0.0260923
\(415\) 0 0
\(416\) 230.706 0.0271906
\(417\) 20761.1i 2.43807i
\(418\) 20.3881i 0.00238568i
\(419\) 8437.43 0.983760 0.491880 0.870663i \(-0.336310\pi\)
0.491880 + 0.870663i \(0.336310\pi\)
\(420\) 0 0
\(421\) 13893.2 1.60834 0.804171 0.594397i \(-0.202609\pi\)
0.804171 + 0.594397i \(0.202609\pi\)
\(422\) 1175.91i 0.135646i
\(423\) − 5831.00i − 0.670243i
\(424\) 3850.31 0.441008
\(425\) 0 0
\(426\) 2121.09 0.241238
\(427\) 7344.41i 0.832368i
\(428\) 11870.9i 1.34065i
\(429\) 525.743 0.0591681
\(430\) 0 0
\(431\) −10611.5 −1.18594 −0.592969 0.805225i \(-0.702044\pi\)
−0.592969 + 0.805225i \(0.702044\pi\)
\(432\) 1454.13i 0.161948i
\(433\) 7569.77i 0.840139i 0.907492 + 0.420069i \(0.137994\pi\)
−0.907492 + 0.420069i \(0.862006\pi\)
\(434\) 876.611 0.0969555
\(435\) 0 0
\(436\) −13703.3 −1.50520
\(437\) 47.7871i 0.00523105i
\(438\) 1008.01i 0.109965i
\(439\) 11794.9 1.28232 0.641162 0.767406i \(-0.278453\pi\)
0.641162 + 0.767406i \(0.278453\pi\)
\(440\) 0 0
\(441\) −3607.39 −0.389525
\(442\) 77.9556i 0.00838907i
\(443\) − 11424.0i − 1.22521i −0.790388 0.612606i \(-0.790121\pi\)
0.790388 0.612606i \(-0.209879\pi\)
\(444\) −16878.0 −1.80404
\(445\) 0 0
\(446\) 184.158 0.0195519
\(447\) 3908.01i 0.413518i
\(448\) 6206.34i 0.654513i
\(449\) −8862.74 −0.931533 −0.465767 0.884908i \(-0.654221\pi\)
−0.465767 + 0.884908i \(0.654221\pi\)
\(450\) 0 0
\(451\) −1035.94 −0.108161
\(452\) − 15877.6i − 1.65225i
\(453\) 2388.06i 0.247684i
\(454\) 851.724 0.0880471
\(455\) 0 0
\(456\) −94.7774 −0.00973324
\(457\) 5187.22i 0.530958i 0.964117 + 0.265479i \(0.0855301\pi\)
−0.964117 + 0.265479i \(0.914470\pi\)
\(458\) 1882.23i 0.192032i
\(459\) −1527.24 −0.155306
\(460\) 0 0
\(461\) −16816.7 −1.69898 −0.849491 0.527603i \(-0.823091\pi\)
−0.849491 + 0.527603i \(0.823091\pi\)
\(462\) − 962.472i − 0.0969226i
\(463\) − 6351.26i − 0.637512i −0.947837 0.318756i \(-0.896735\pi\)
0.947837 0.318756i \(-0.103265\pi\)
\(464\) 490.582 0.0490834
\(465\) 0 0
\(466\) 53.0970 0.00527827
\(467\) 1057.02i 0.104739i 0.998628 + 0.0523693i \(0.0166773\pi\)
−0.998628 + 0.0523693i \(0.983323\pi\)
\(468\) 564.019i 0.0557089i
\(469\) −417.815 −0.0411363
\(470\) 0 0
\(471\) 11336.3 1.10902
\(472\) − 591.279i − 0.0576606i
\(473\) − 5216.25i − 0.507069i
\(474\) 868.618 0.0841707
\(475\) 0 0
\(476\) −6819.28 −0.656641
\(477\) 14168.3i 1.36000i
\(478\) − 1404.30i − 0.134375i
\(479\) 10138.2 0.967073 0.483537 0.875324i \(-0.339352\pi\)
0.483537 + 0.875324i \(0.339352\pi\)
\(480\) 0 0
\(481\) 923.591 0.0875512
\(482\) 2356.41i 0.222679i
\(483\) − 2255.91i − 0.212521i
\(484\) −5828.60 −0.547389
\(485\) 0 0
\(486\) 2066.59 0.192885
\(487\) − 8874.74i − 0.825776i −0.910782 0.412888i \(-0.864520\pi\)
0.910782 0.412888i \(-0.135480\pi\)
\(488\) 3415.74i 0.316851i
\(489\) −19686.8 −1.82059
\(490\) 0 0
\(491\) −21452.2 −1.97174 −0.985869 0.167521i \(-0.946424\pi\)
−0.985869 + 0.167521i \(0.946424\pi\)
\(492\) − 2382.93i − 0.218356i
\(493\) 515.249i 0.0470703i
\(494\) 2.56634 0.000233735 0
\(495\) 0 0
\(496\) −9433.67 −0.854001
\(497\) − 10153.4i − 0.916382i
\(498\) 401.804i 0.0361551i
\(499\) −9696.91 −0.869926 −0.434963 0.900448i \(-0.643239\pi\)
−0.434963 + 0.900448i \(0.643239\pi\)
\(500\) 0 0
\(501\) −2012.10 −0.179429
\(502\) 1876.29i 0.166818i
\(503\) 8892.32i 0.788249i 0.919057 + 0.394124i \(0.128952\pi\)
−0.919057 + 0.394124i \(0.871048\pi\)
\(504\) 2086.69 0.184422
\(505\) 0 0
\(506\) −225.695 −0.0198288
\(507\) 15561.6i 1.36315i
\(508\) − 16663.2i − 1.45534i
\(509\) 1084.94 0.0944778 0.0472389 0.998884i \(-0.484958\pi\)
0.0472389 + 0.998884i \(0.484958\pi\)
\(510\) 0 0
\(511\) 4825.20 0.417719
\(512\) 7628.02i 0.658426i
\(513\) 50.2776i 0.00432712i
\(514\) 2131.20 0.182885
\(515\) 0 0
\(516\) 11998.7 1.02367
\(517\) 5987.58i 0.509349i
\(518\) − 1690.81i − 0.143417i
\(519\) −16585.4 −1.40273
\(520\) 0 0
\(521\) 4578.35 0.384993 0.192496 0.981298i \(-0.438342\pi\)
0.192496 + 0.981298i \(0.438342\pi\)
\(522\) − 78.0166i − 0.00654156i
\(523\) − 7298.28i − 0.610194i −0.952321 0.305097i \(-0.901311\pi\)
0.952321 0.305097i \(-0.0986888\pi\)
\(524\) −11708.8 −0.976148
\(525\) 0 0
\(526\) 765.476 0.0634531
\(527\) − 9908.02i − 0.818975i
\(528\) 10357.7i 0.853712i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 2175.77 0.177817
\(532\) 224.494i 0.0182952i
\(533\) 130.398i 0.0105970i
\(534\) 2475.98 0.200648
\(535\) 0 0
\(536\) −194.317 −0.0156590
\(537\) − 776.338i − 0.0623863i
\(538\) − 3025.91i − 0.242484i
\(539\) 3704.26 0.296018
\(540\) 0 0
\(541\) −14971.5 −1.18979 −0.594895 0.803803i \(-0.702806\pi\)
−0.594895 + 0.803803i \(0.702806\pi\)
\(542\) 661.854i 0.0524521i
\(543\) 10505.2i 0.830241i
\(544\) −4773.69 −0.376232
\(545\) 0 0
\(546\) −121.151 −0.00949591
\(547\) − 19510.9i − 1.52509i −0.646932 0.762547i \(-0.723948\pi\)
0.646932 0.762547i \(-0.276052\pi\)
\(548\) 17751.0i 1.38373i
\(549\) −12569.2 −0.977119
\(550\) 0 0
\(551\) 16.9623 0.00131147
\(552\) − 1049.18i − 0.0808986i
\(553\) − 4157.96i − 0.319737i
\(554\) 239.031 0.0183311
\(555\) 0 0
\(556\) 22870.7 1.74448
\(557\) − 13098.1i − 0.996378i −0.867068 0.498189i \(-0.833999\pi\)
0.867068 0.498189i \(-0.166001\pi\)
\(558\) 1500.22i 0.113816i
\(559\) −656.592 −0.0496796
\(560\) 0 0
\(561\) −10878.5 −0.818698
\(562\) 1012.85i 0.0760226i
\(563\) 4086.19i 0.305883i 0.988235 + 0.152942i \(0.0488746\pi\)
−0.988235 + 0.152942i \(0.951125\pi\)
\(564\) −13773.0 −1.02828
\(565\) 0 0
\(566\) −325.546 −0.0241762
\(567\) − 11159.0i − 0.826516i
\(568\) − 4722.14i − 0.348832i
\(569\) 5021.20 0.369946 0.184973 0.982744i \(-0.440780\pi\)
0.184973 + 0.982744i \(0.440780\pi\)
\(570\) 0 0
\(571\) −8277.56 −0.606664 −0.303332 0.952885i \(-0.598099\pi\)
−0.303332 + 0.952885i \(0.598099\pi\)
\(572\) − 579.164i − 0.0423358i
\(573\) 14455.2i 1.05388i
\(574\) 238.719 0.0173588
\(575\) 0 0
\(576\) −10621.5 −0.768336
\(577\) − 19548.4i − 1.41042i −0.708998 0.705210i \(-0.750853\pi\)
0.708998 0.705210i \(-0.249147\pi\)
\(578\) 376.528i 0.0270960i
\(579\) 27625.2 1.98284
\(580\) 0 0
\(581\) 1923.38 0.137341
\(582\) 2671.64i 0.190280i
\(583\) − 14548.7i − 1.03353i
\(584\) 2244.10 0.159010
\(585\) 0 0
\(586\) −2564.41 −0.180776
\(587\) − 17479.7i − 1.22907i −0.788889 0.614536i \(-0.789343\pi\)
0.788889 0.614536i \(-0.210657\pi\)
\(588\) 8520.77i 0.597603i
\(589\) −326.177 −0.0228182
\(590\) 0 0
\(591\) −25469.9 −1.77275
\(592\) 18195.7i 1.26324i
\(593\) − 513.405i − 0.0355531i −0.999842 0.0177766i \(-0.994341\pi\)
0.999842 0.0177766i \(-0.00565875\pi\)
\(594\) −237.457 −0.0164023
\(595\) 0 0
\(596\) 4305.10 0.295879
\(597\) 20138.8i 1.38061i
\(598\) 28.4092i 0.00194271i
\(599\) −13706.8 −0.934964 −0.467482 0.884003i \(-0.654839\pi\)
−0.467482 + 0.884003i \(0.654839\pi\)
\(600\) 0 0
\(601\) −24403.7 −1.65632 −0.828159 0.560493i \(-0.810612\pi\)
−0.828159 + 0.560493i \(0.810612\pi\)
\(602\) 1202.02i 0.0813796i
\(603\) − 715.045i − 0.0482900i
\(604\) 2630.71 0.177222
\(605\) 0 0
\(606\) −4935.37 −0.330835
\(607\) 16304.7i 1.09026i 0.838352 + 0.545129i \(0.183519\pi\)
−0.838352 + 0.545129i \(0.816481\pi\)
\(608\) 157.152i 0.0104825i
\(609\) −800.748 −0.0532807
\(610\) 0 0
\(611\) 753.683 0.0499030
\(612\) − 11670.5i − 0.770834i
\(613\) − 7754.74i − 0.510948i −0.966816 0.255474i \(-0.917769\pi\)
0.966816 0.255474i \(-0.0822315\pi\)
\(614\) −2963.36 −0.194775
\(615\) 0 0
\(616\) −2142.73 −0.140151
\(617\) − 1574.90i − 0.102760i −0.998679 0.0513801i \(-0.983638\pi\)
0.998679 0.0513801i \(-0.0163620\pi\)
\(618\) − 2680.40i − 0.174468i
\(619\) 9160.25 0.594800 0.297400 0.954753i \(-0.403880\pi\)
0.297400 + 0.954753i \(0.403880\pi\)
\(620\) 0 0
\(621\) −556.570 −0.0359652
\(622\) 1149.62i 0.0741083i
\(623\) − 11852.2i − 0.762196i
\(624\) 1303.76 0.0836416
\(625\) 0 0
\(626\) 64.8465 0.00414024
\(627\) 358.125i 0.0228104i
\(628\) − 12488.2i − 0.793523i
\(629\) −19110.6 −1.21143
\(630\) 0 0
\(631\) 11663.9 0.735871 0.367935 0.929851i \(-0.380065\pi\)
0.367935 + 0.929851i \(0.380065\pi\)
\(632\) − 1933.78i − 0.121712i
\(633\) 20655.4i 1.29697i
\(634\) 1348.82 0.0844930
\(635\) 0 0
\(636\) 33465.9 2.08649
\(637\) − 466.272i − 0.0290021i
\(638\) 80.1115i 0.00497123i
\(639\) 17376.4 1.07574
\(640\) 0 0
\(641\) −27074.5 −1.66830 −0.834148 0.551541i \(-0.814040\pi\)
−0.834148 + 0.551541i \(0.814040\pi\)
\(642\) − 4363.78i − 0.268263i
\(643\) 4463.82i 0.273773i 0.990587 + 0.136886i \(0.0437096\pi\)
−0.990587 + 0.136886i \(0.956290\pi\)
\(644\) −2485.14 −0.152062
\(645\) 0 0
\(646\) −53.1017 −0.00323415
\(647\) − 11755.0i − 0.714277i −0.934051 0.357139i \(-0.883752\pi\)
0.934051 0.357139i \(-0.116248\pi\)
\(648\) − 5189.83i − 0.314623i
\(649\) −2234.20 −0.135131
\(650\) 0 0
\(651\) 15398.0 0.927029
\(652\) 21687.2i 1.30266i
\(653\) 1236.64i 0.0741094i 0.999313 + 0.0370547i \(0.0117976\pi\)
−0.999313 + 0.0370547i \(0.988202\pi\)
\(654\) 5037.40 0.301189
\(655\) 0 0
\(656\) −2568.98 −0.152899
\(657\) 8257.81i 0.490362i
\(658\) − 1379.76i − 0.0817456i
\(659\) −23646.9 −1.39780 −0.698901 0.715218i \(-0.746327\pi\)
−0.698901 + 0.715218i \(0.746327\pi\)
\(660\) 0 0
\(661\) 10150.5 0.597290 0.298645 0.954364i \(-0.403465\pi\)
0.298645 + 0.954364i \(0.403465\pi\)
\(662\) 2971.33i 0.174447i
\(663\) 1369.32i 0.0802112i
\(664\) 894.526 0.0522806
\(665\) 0 0
\(666\) 2893.63 0.168357
\(667\) 187.771i 0.0109003i
\(668\) 2216.55i 0.128385i
\(669\) 3234.81 0.186943
\(670\) 0 0
\(671\) 12906.7 0.742558
\(672\) − 7418.77i − 0.425871i
\(673\) − 13941.8i − 0.798540i −0.916833 0.399270i \(-0.869264\pi\)
0.916833 0.399270i \(-0.130736\pi\)
\(674\) 2899.71 0.165716
\(675\) 0 0
\(676\) 17142.8 0.975353
\(677\) − 13370.4i − 0.759035i −0.925185 0.379518i \(-0.876090\pi\)
0.925185 0.379518i \(-0.123910\pi\)
\(678\) 5836.68i 0.330614i
\(679\) 12788.8 0.722812
\(680\) 0 0
\(681\) 14960.9 0.841852
\(682\) − 1540.51i − 0.0864943i
\(683\) − 15659.3i − 0.877288i −0.898661 0.438644i \(-0.855459\pi\)
0.898661 0.438644i \(-0.144541\pi\)
\(684\) −384.198 −0.0214768
\(685\) 0 0
\(686\) −2768.87 −0.154105
\(687\) 33062.1i 1.83609i
\(688\) − 12935.5i − 0.716806i
\(689\) −1831.31 −0.101259
\(690\) 0 0
\(691\) −9631.82 −0.530263 −0.265131 0.964212i \(-0.585415\pi\)
−0.265131 + 0.964212i \(0.585415\pi\)
\(692\) 18270.7i 1.00368i
\(693\) − 7884.77i − 0.432204i
\(694\) −2729.62 −0.149301
\(695\) 0 0
\(696\) −372.412 −0.0202819
\(697\) − 2698.15i − 0.146628i
\(698\) 4119.99i 0.223415i
\(699\) 932.671 0.0504676
\(700\) 0 0
\(701\) 21140.9 1.13906 0.569530 0.821971i \(-0.307125\pi\)
0.569530 + 0.821971i \(0.307125\pi\)
\(702\) 29.8898i 0.00160700i
\(703\) 629.131i 0.0337527i
\(704\) 10906.7 0.583894
\(705\) 0 0
\(706\) −1033.80 −0.0551101
\(707\) 23625.0i 1.25673i
\(708\) − 5139.25i − 0.272803i
\(709\) 15917.4 0.843144 0.421572 0.906795i \(-0.361478\pi\)
0.421572 + 0.906795i \(0.361478\pi\)
\(710\) 0 0
\(711\) 7115.89 0.375340
\(712\) − 5512.22i − 0.290139i
\(713\) − 3610.76i − 0.189655i
\(714\) 2506.80 0.131393
\(715\) 0 0
\(716\) −855.221 −0.0446384
\(717\) − 24667.1i − 1.28481i
\(718\) 2704.09i 0.140551i
\(719\) −3323.34 −0.172378 −0.0861889 0.996279i \(-0.527469\pi\)
−0.0861889 + 0.996279i \(0.527469\pi\)
\(720\) 0 0
\(721\) −12830.7 −0.662748
\(722\) − 2775.85i − 0.143084i
\(723\) 41391.2i 2.12912i
\(724\) 11572.6 0.594051
\(725\) 0 0
\(726\) 2142.62 0.109532
\(727\) − 9877.52i − 0.503902i −0.967740 0.251951i \(-0.918928\pi\)
0.967740 0.251951i \(-0.0810722\pi\)
\(728\) 269.714i 0.0137312i
\(729\) 14449.9 0.734130
\(730\) 0 0
\(731\) 13586.0 0.687407
\(732\) 29688.7i 1.49908i
\(733\) − 28951.5i − 1.45886i −0.684053 0.729432i \(-0.739784\pi\)
0.684053 0.729432i \(-0.260216\pi\)
\(734\) −4747.24 −0.238725
\(735\) 0 0
\(736\) −1739.66 −0.0871262
\(737\) 734.246i 0.0366978i
\(738\) 408.541i 0.0203775i
\(739\) 31009.5 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(740\) 0 0
\(741\) 45.0788 0.00223483
\(742\) 3352.57i 0.165871i
\(743\) 13761.0i 0.679465i 0.940522 + 0.339733i \(0.110337\pi\)
−0.940522 + 0.339733i \(0.889663\pi\)
\(744\) 7161.31 0.352885
\(745\) 0 0
\(746\) 3268.29 0.160403
\(747\) 3291.66i 0.161226i
\(748\) 11983.8i 0.585792i
\(749\) −20888.9 −1.01904
\(750\) 0 0
\(751\) 32197.4 1.56445 0.782223 0.622998i \(-0.214086\pi\)
0.782223 + 0.622998i \(0.214086\pi\)
\(752\) 14848.3i 0.720029i
\(753\) 32957.8i 1.59502i
\(754\) 10.0840 0.000487052 0
\(755\) 0 0
\(756\) −2614.65 −0.125786
\(757\) 26139.9i 1.25505i 0.778597 + 0.627524i \(0.215932\pi\)
−0.778597 + 0.627524i \(0.784068\pi\)
\(758\) 2710.42i 0.129877i
\(759\) −3964.42 −0.189591
\(760\) 0 0
\(761\) −24004.0 −1.14342 −0.571712 0.820454i \(-0.693721\pi\)
−0.571712 + 0.820454i \(0.693721\pi\)
\(762\) 6125.48i 0.291211i
\(763\) − 24113.3i − 1.14412i
\(764\) 15924.0 0.754072
\(765\) 0 0
\(766\) −1013.40 −0.0478011
\(767\) 281.229i 0.0132393i
\(768\) 23345.2i 1.09687i
\(769\) 33733.4 1.58187 0.790934 0.611902i \(-0.209595\pi\)
0.790934 + 0.611902i \(0.209595\pi\)
\(770\) 0 0
\(771\) 37435.3 1.74864
\(772\) − 30432.2i − 1.41876i
\(773\) − 40247.1i − 1.87269i −0.351083 0.936344i \(-0.614186\pi\)
0.351083 0.936344i \(-0.385814\pi\)
\(774\) −2057.12 −0.0955318
\(775\) 0 0
\(776\) 5947.81 0.275147
\(777\) − 29699.7i − 1.37126i
\(778\) − 2222.28i − 0.102407i
\(779\) −88.8246 −0.00408533
\(780\) 0 0
\(781\) −17843.0 −0.817508
\(782\) − 587.832i − 0.0268808i
\(783\) 197.557i 0.00901676i
\(784\) 9186.02 0.418459
\(785\) 0 0
\(786\) 4304.21 0.195326
\(787\) − 16327.3i − 0.739522i −0.929127 0.369761i \(-0.879440\pi\)
0.929127 0.369761i \(-0.120560\pi\)
\(788\) 28057.9i 1.26843i
\(789\) 13445.9 0.606700
\(790\) 0 0
\(791\) 27939.4 1.25589
\(792\) − 3667.04i − 0.164524i
\(793\) − 1624.62i − 0.0727515i
\(794\) −3142.83 −0.140472
\(795\) 0 0
\(796\) 22185.1 0.987852
\(797\) − 29358.8i − 1.30482i −0.757866 0.652410i \(-0.773758\pi\)
0.757866 0.652410i \(-0.226242\pi\)
\(798\) − 82.5252i − 0.00366085i
\(799\) −15594.9 −0.690498
\(800\) 0 0
\(801\) 20283.7 0.894745
\(802\) 5866.12i 0.258279i
\(803\) − 8479.56i − 0.372649i
\(804\) −1688.96 −0.0740857
\(805\) 0 0
\(806\) −193.911 −0.00847420
\(807\) − 53151.3i − 2.31848i
\(808\) 10987.5i 0.478389i
\(809\) 2426.36 0.105447 0.0527234 0.998609i \(-0.483210\pi\)
0.0527234 + 0.998609i \(0.483210\pi\)
\(810\) 0 0
\(811\) −31170.9 −1.34964 −0.674819 0.737983i \(-0.735778\pi\)
−0.674819 + 0.737983i \(0.735778\pi\)
\(812\) 882.112i 0.0381232i
\(813\) 11625.7i 0.501515i
\(814\) −2971.34 −0.127943
\(815\) 0 0
\(816\) −26977.0 −1.15733
\(817\) − 447.257i − 0.0191524i
\(818\) − 2698.97i − 0.115363i
\(819\) −992.490 −0.0423448
\(820\) 0 0
\(821\) −6679.93 −0.283960 −0.141980 0.989870i \(-0.545347\pi\)
−0.141980 + 0.989870i \(0.545347\pi\)
\(822\) − 6525.36i − 0.276883i
\(823\) − 29299.3i − 1.24096i −0.784222 0.620480i \(-0.786938\pi\)
0.784222 0.620480i \(-0.213062\pi\)
\(824\) −5967.31 −0.252283
\(825\) 0 0
\(826\) 514.842 0.0216872
\(827\) 39806.7i 1.67378i 0.547372 + 0.836889i \(0.315628\pi\)
−0.547372 + 0.836889i \(0.684372\pi\)
\(828\) − 4253.04i − 0.178506i
\(829\) 17854.1 0.748007 0.374004 0.927427i \(-0.377985\pi\)
0.374004 + 0.927427i \(0.377985\pi\)
\(830\) 0 0
\(831\) 4198.67 0.175271
\(832\) − 1372.87i − 0.0572065i
\(833\) 9647.91i 0.401297i
\(834\) −8407.37 −0.349069
\(835\) 0 0
\(836\) 394.514 0.0163212
\(837\) − 3798.94i − 0.156882i
\(838\) 3416.80i 0.140849i
\(839\) 6656.92 0.273924 0.136962 0.990576i \(-0.456266\pi\)
0.136962 + 0.990576i \(0.456266\pi\)
\(840\) 0 0
\(841\) −24322.3 −0.997267
\(842\) 5626.14i 0.230273i
\(843\) 17791.2i 0.726881i
\(844\) 22754.2 0.928000
\(845\) 0 0
\(846\) 2361.31 0.0959614
\(847\) − 10256.4i − 0.416075i
\(848\) − 36078.7i − 1.46102i
\(849\) −5718.34 −0.231158
\(850\) 0 0
\(851\) −6964.43 −0.280538
\(852\) − 41043.6i − 1.65039i
\(853\) 30008.7i 1.20455i 0.798290 + 0.602273i \(0.205738\pi\)
−0.798290 + 0.602273i \(0.794262\pi\)
\(854\) −2974.17 −0.119173
\(855\) 0 0
\(856\) −9714.98 −0.387910
\(857\) 24281.5i 0.967843i 0.875111 + 0.483922i \(0.160788\pi\)
−0.875111 + 0.483922i \(0.839212\pi\)
\(858\) 212.903i 0.00847133i
\(859\) −30635.5 −1.21685 −0.608423 0.793613i \(-0.708198\pi\)
−0.608423 + 0.793613i \(0.708198\pi\)
\(860\) 0 0
\(861\) 4193.19 0.165974
\(862\) − 4297.22i − 0.169796i
\(863\) 26572.1i 1.04812i 0.851683 + 0.524058i \(0.175582\pi\)
−0.851683 + 0.524058i \(0.824418\pi\)
\(864\) −1830.33 −0.0720707
\(865\) 0 0
\(866\) −3065.43 −0.120286
\(867\) 6613.87i 0.259076i
\(868\) − 16962.6i − 0.663305i
\(869\) −7306.97 −0.285238
\(870\) 0 0
\(871\) 92.4227 0.00359543
\(872\) − 11214.6i − 0.435522i
\(873\) 21886.6i 0.848511i
\(874\) −19.3517 −0.000748950 0
\(875\) 0 0
\(876\) 19505.2 0.752305
\(877\) − 37158.4i − 1.43073i −0.698750 0.715366i \(-0.746260\pi\)
0.698750 0.715366i \(-0.253740\pi\)
\(878\) 4776.43i 0.183595i
\(879\) −45044.9 −1.72847
\(880\) 0 0
\(881\) −3353.12 −0.128229 −0.0641143 0.997943i \(-0.520422\pi\)
−0.0641143 + 0.997943i \(0.520422\pi\)
\(882\) − 1460.84i − 0.0557699i
\(883\) − 16292.5i − 0.620936i −0.950584 0.310468i \(-0.899514\pi\)
0.950584 0.310468i \(-0.100486\pi\)
\(884\) 1508.46 0.0573924
\(885\) 0 0
\(886\) 4626.22 0.175418
\(887\) − 5949.82i − 0.225226i −0.993639 0.112613i \(-0.964078\pi\)
0.993639 0.112613i \(-0.0359220\pi\)
\(888\) − 13812.7i − 0.521988i
\(889\) 29321.9 1.10621
\(890\) 0 0
\(891\) −19610.2 −0.737338
\(892\) − 3563.50i − 0.133761i
\(893\) 513.393i 0.0192386i
\(894\) −1582.58 −0.0592050
\(895\) 0 0
\(896\) −10856.9 −0.404804
\(897\) 499.019i 0.0185750i
\(898\) − 3589.03i − 0.133371i
\(899\) −1281.66 −0.0475480
\(900\) 0 0
\(901\) 37892.8 1.40110
\(902\) − 419.512i − 0.0154858i
\(903\) 21113.9i 0.778102i
\(904\) 12994.0 0.478070
\(905\) 0 0
\(906\) −967.062 −0.0354619
\(907\) − 44105.2i − 1.61465i −0.590106 0.807326i \(-0.700914\pi\)
0.590106 0.807326i \(-0.299086\pi\)
\(908\) − 16481.0i − 0.602360i
\(909\) −40431.6 −1.47528
\(910\) 0 0
\(911\) 4385.18 0.159481 0.0797407 0.996816i \(-0.474591\pi\)
0.0797407 + 0.996816i \(0.474591\pi\)
\(912\) 888.097i 0.0322454i
\(913\) − 3380.05i − 0.122523i
\(914\) −2100.60 −0.0760194
\(915\) 0 0
\(916\) 36421.5 1.31376
\(917\) − 20603.7i − 0.741978i
\(918\) − 618.468i − 0.0222358i
\(919\) −30027.0 −1.07780 −0.538901 0.842369i \(-0.681160\pi\)
−0.538901 + 0.842369i \(0.681160\pi\)
\(920\) 0 0
\(921\) −52052.7 −1.86232
\(922\) − 6810.03i − 0.243250i
\(923\) 2245.98i 0.0800946i
\(924\) −18624.1 −0.663080
\(925\) 0 0
\(926\) 2571.99 0.0912751
\(927\) − 21958.4i − 0.778002i
\(928\) 617.503i 0.0218432i
\(929\) 5457.52 0.192740 0.0963700 0.995346i \(-0.469277\pi\)
0.0963700 + 0.995346i \(0.469277\pi\)
\(930\) 0 0
\(931\) 317.614 0.0111809
\(932\) − 1027.44i − 0.0361104i
\(933\) 20193.5i 0.708579i
\(934\) −428.046 −0.0149958
\(935\) 0 0
\(936\) −461.587 −0.0161191
\(937\) 3039.15i 0.105960i 0.998596 + 0.0529802i \(0.0168720\pi\)
−0.998596 + 0.0529802i \(0.983128\pi\)
\(938\) − 169.197i − 0.00588964i
\(939\) 1139.05 0.0395864
\(940\) 0 0
\(941\) 25791.0 0.893479 0.446740 0.894664i \(-0.352585\pi\)
0.446740 + 0.894664i \(0.352585\pi\)
\(942\) 4590.71i 0.158783i
\(943\) − 983.282i − 0.0339555i
\(944\) −5540.48 −0.191025
\(945\) 0 0
\(946\) 2112.36 0.0725991
\(947\) 18339.3i 0.629299i 0.949208 + 0.314650i \(0.101887\pi\)
−0.949208 + 0.314650i \(0.898113\pi\)
\(948\) − 16807.9i − 0.575840i
\(949\) −1067.36 −0.0365099
\(950\) 0 0
\(951\) 23692.6 0.807871
\(952\) − 5580.83i − 0.189995i
\(953\) − 12658.7i − 0.430278i −0.976583 0.215139i \(-0.930979\pi\)
0.976583 0.215139i \(-0.0690205\pi\)
\(954\) −5737.55 −0.194717
\(955\) 0 0
\(956\) −27173.5 −0.919302
\(957\) 1407.19i 0.0475319i
\(958\) 4105.55i 0.138460i
\(959\) −31236.0 −1.05179
\(960\) 0 0
\(961\) −5145.31 −0.172714
\(962\) 374.015i 0.0125351i
\(963\) − 35749.0i − 1.19626i
\(964\) 45597.0 1.52342
\(965\) 0 0
\(966\) 913.548 0.0304275
\(967\) 50470.3i 1.67840i 0.543822 + 0.839201i \(0.316977\pi\)
−0.543822 + 0.839201i \(0.683023\pi\)
\(968\) − 4770.06i − 0.158384i
\(969\) −932.752 −0.0309229
\(970\) 0 0
\(971\) 30778.2 1.01722 0.508610 0.860997i \(-0.330160\pi\)
0.508610 + 0.860997i \(0.330160\pi\)
\(972\) − 39988.9i − 1.31959i
\(973\) 40245.0i 1.32600i
\(974\) 3593.89 0.118230
\(975\) 0 0
\(976\) 32006.6 1.04970
\(977\) − 23575.5i − 0.772004i −0.922498 0.386002i \(-0.873856\pi\)
0.922498 0.386002i \(-0.126144\pi\)
\(978\) − 7972.31i − 0.260661i
\(979\) −20828.4 −0.679958
\(980\) 0 0
\(981\) 41267.4 1.34308
\(982\) − 8687.21i − 0.282301i
\(983\) 44798.3i 1.45355i 0.686874 + 0.726777i \(0.258982\pi\)
−0.686874 + 0.726777i \(0.741018\pi\)
\(984\) 1950.17 0.0631799
\(985\) 0 0
\(986\) −208.654 −0.00673924
\(987\) − 24236.0i − 0.781601i
\(988\) − 49.6592i − 0.00159906i
\(989\) 4951.10 0.159187
\(990\) 0 0
\(991\) −12153.5 −0.389574 −0.194787 0.980846i \(-0.562402\pi\)
−0.194787 + 0.980846i \(0.562402\pi\)
\(992\) − 11874.3i − 0.380050i
\(993\) 52192.6i 1.66796i
\(994\) 4111.69 0.131202
\(995\) 0 0
\(996\) 7774.99 0.247350
\(997\) 36538.1i 1.16065i 0.814383 + 0.580327i \(0.197075\pi\)
−0.814383 + 0.580327i \(0.802925\pi\)
\(998\) − 3926.83i − 0.124551i
\(999\) −7327.40 −0.232061
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 575.4.b.i.24.6 10
5.2 odd 4 575.4.a.j.1.3 5
5.3 odd 4 115.4.a.e.1.3 5
5.4 even 2 inner 575.4.b.i.24.5 10
15.8 even 4 1035.4.a.k.1.3 5
20.3 even 4 1840.4.a.n.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.3 5 5.3 odd 4
575.4.a.j.1.3 5 5.2 odd 4
575.4.b.i.24.5 10 5.4 even 2 inner
575.4.b.i.24.6 10 1.1 even 1 trivial
1035.4.a.k.1.3 5 15.8 even 4
1840.4.a.n.1.5 5 20.3 even 4