Properties

Label 1575.4.a.bi.1.4
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
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] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3030748.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.42371\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.42371 q^{2} +11.5692 q^{4} +7.00000 q^{7} +15.7890 q^{8} +O(q^{10})\) \(q+4.42371 q^{2} +11.5692 q^{4} +7.00000 q^{7} +15.7890 q^{8} -23.3775 q^{11} +3.56918 q^{13} +30.9659 q^{14} -22.7075 q^{16} -57.5082 q^{17} -51.1384 q^{19} -103.415 q^{22} -65.6390 q^{23} +15.7890 q^{26} +80.9843 q^{28} +41.6147 q^{29} -167.538 q^{31} -226.763 q^{32} -254.399 q^{34} -224.538 q^{37} -226.221 q^{38} +196.688 q^{41} +58.9685 q^{43} -270.458 q^{44} -290.368 q^{46} -41.9282 q^{47} +49.0000 q^{49} +41.2925 q^{52} +33.3445 q^{53} +110.523 q^{56} +184.091 q^{58} -229.212 q^{59} -700.613 q^{61} -741.138 q^{62} -821.475 q^{64} -453.890 q^{67} -665.322 q^{68} +930.571 q^{71} +370.182 q^{73} -993.289 q^{74} -591.629 q^{76} -163.642 q^{77} -54.6007 q^{79} +870.091 q^{82} +430.045 q^{83} +260.859 q^{86} -369.107 q^{88} -737.202 q^{89} +24.9843 q^{91} -759.390 q^{92} -185.478 q^{94} +150.371 q^{97} +216.762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 28 q^{7} - 22 q^{13} + 18 q^{16} - 132 q^{19} - 196 q^{22} + 70 q^{28} - 126 q^{31} - 546 q^{34} - 354 q^{37} - 272 q^{43} - 182 q^{46} + 196 q^{49} + 274 q^{52} - 98 q^{58} - 1170 q^{61} - 1726 q^{64} - 38 q^{67} - 188 q^{73} - 988 q^{76} - 690 q^{79} + 2646 q^{82} - 896 q^{88} - 154 q^{91} - 1540 q^{94} + 1980 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42371 1.56402 0.782008 0.623268i \(-0.214195\pi\)
0.782008 + 0.623268i \(0.214195\pi\)
\(3\) 0 0
\(4\) 11.5692 1.44615
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 15.7890 0.697782
\(9\) 0 0
\(10\) 0 0
\(11\) −23.3775 −0.640779 −0.320390 0.947286i \(-0.603814\pi\)
−0.320390 + 0.947286i \(0.603814\pi\)
\(12\) 0 0
\(13\) 3.56918 0.0761471 0.0380735 0.999275i \(-0.487878\pi\)
0.0380735 + 0.999275i \(0.487878\pi\)
\(14\) 30.9659 0.591143
\(15\) 0 0
\(16\) −22.7075 −0.354805
\(17\) −57.5082 −0.820458 −0.410229 0.911983i \(-0.634551\pi\)
−0.410229 + 0.911983i \(0.634551\pi\)
\(18\) 0 0
\(19\) −51.1384 −0.617471 −0.308735 0.951148i \(-0.599906\pi\)
−0.308735 + 0.951148i \(0.599906\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −103.415 −1.00219
\(23\) −65.6390 −0.595073 −0.297537 0.954710i \(-0.596165\pi\)
−0.297537 + 0.954710i \(0.596165\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 15.7890 0.119095
\(27\) 0 0
\(28\) 80.9843 0.546592
\(29\) 41.6147 0.266471 0.133235 0.991084i \(-0.457463\pi\)
0.133235 + 0.991084i \(0.457463\pi\)
\(30\) 0 0
\(31\) −167.538 −0.970666 −0.485333 0.874329i \(-0.661302\pi\)
−0.485333 + 0.874329i \(0.661302\pi\)
\(32\) −226.763 −1.25270
\(33\) 0 0
\(34\) −254.399 −1.28321
\(35\) 0 0
\(36\) 0 0
\(37\) −224.538 −0.997669 −0.498835 0.866697i \(-0.666239\pi\)
−0.498835 + 0.866697i \(0.666239\pi\)
\(38\) −226.221 −0.965734
\(39\) 0 0
\(40\) 0 0
\(41\) 196.688 0.749208 0.374604 0.927185i \(-0.377779\pi\)
0.374604 + 0.927185i \(0.377779\pi\)
\(42\) 0 0
\(43\) 58.9685 0.209131 0.104565 0.994518i \(-0.466655\pi\)
0.104565 + 0.994518i \(0.466655\pi\)
\(44\) −270.458 −0.926661
\(45\) 0 0
\(46\) −290.368 −0.930704
\(47\) −41.9282 −0.130125 −0.0650623 0.997881i \(-0.520725\pi\)
−0.0650623 + 0.997881i \(0.520725\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 41.2925 0.110120
\(53\) 33.3445 0.0864192 0.0432096 0.999066i \(-0.486242\pi\)
0.0432096 + 0.999066i \(0.486242\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 110.523 0.263737
\(57\) 0 0
\(58\) 184.091 0.416765
\(59\) −229.212 −0.505777 −0.252888 0.967495i \(-0.581381\pi\)
−0.252888 + 0.967495i \(0.581381\pi\)
\(60\) 0 0
\(61\) −700.613 −1.47056 −0.735281 0.677762i \(-0.762950\pi\)
−0.735281 + 0.677762i \(0.762950\pi\)
\(62\) −741.138 −1.51814
\(63\) 0 0
\(64\) −821.475 −1.60444
\(65\) 0 0
\(66\) 0 0
\(67\) −453.890 −0.827634 −0.413817 0.910360i \(-0.635805\pi\)
−0.413817 + 0.910360i \(0.635805\pi\)
\(68\) −665.322 −1.18650
\(69\) 0 0
\(70\) 0 0
\(71\) 930.571 1.55547 0.777735 0.628592i \(-0.216368\pi\)
0.777735 + 0.628592i \(0.216368\pi\)
\(72\) 0 0
\(73\) 370.182 0.593514 0.296757 0.954953i \(-0.404095\pi\)
0.296757 + 0.954953i \(0.404095\pi\)
\(74\) −993.289 −1.56037
\(75\) 0 0
\(76\) −591.629 −0.892954
\(77\) −163.642 −0.242192
\(78\) 0 0
\(79\) −54.6007 −0.0777602 −0.0388801 0.999244i \(-0.512379\pi\)
−0.0388801 + 0.999244i \(0.512379\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 870.091 1.17177
\(83\) 430.045 0.568718 0.284359 0.958718i \(-0.408219\pi\)
0.284359 + 0.958718i \(0.408219\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 260.859 0.327084
\(87\) 0 0
\(88\) −369.107 −0.447124
\(89\) −737.202 −0.878014 −0.439007 0.898484i \(-0.644670\pi\)
−0.439007 + 0.898484i \(0.644670\pi\)
\(90\) 0 0
\(91\) 24.9843 0.0287809
\(92\) −759.390 −0.860564
\(93\) 0 0
\(94\) −185.478 −0.203517
\(95\) 0 0
\(96\) 0 0
\(97\) 150.371 0.157401 0.0787004 0.996898i \(-0.474923\pi\)
0.0787004 + 0.996898i \(0.474923\pi\)
\(98\) 216.762 0.223431
\(99\) 0 0
\(100\) 0 0
\(101\) 134.423 0.132432 0.0662158 0.997805i \(-0.478907\pi\)
0.0662158 + 0.997805i \(0.478907\pi\)
\(102\) 0 0
\(103\) −232.494 −0.222411 −0.111205 0.993797i \(-0.535471\pi\)
−0.111205 + 0.993797i \(0.535471\pi\)
\(104\) 56.3538 0.0531340
\(105\) 0 0
\(106\) 147.506 0.135161
\(107\) −1539.38 −1.39082 −0.695409 0.718614i \(-0.744777\pi\)
−0.695409 + 0.718614i \(0.744777\pi\)
\(108\) 0 0
\(109\) −503.462 −0.442412 −0.221206 0.975227i \(-0.570999\pi\)
−0.221206 + 0.975227i \(0.570999\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −158.953 −0.134104
\(113\) −267.084 −0.222347 −0.111173 0.993801i \(-0.535461\pi\)
−0.111173 + 0.993801i \(0.535461\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 481.448 0.385356
\(117\) 0 0
\(118\) −1013.97 −0.791043
\(119\) −402.557 −0.310104
\(120\) 0 0
\(121\) −784.494 −0.589402
\(122\) −3099.31 −2.29998
\(123\) 0 0
\(124\) −1938.27 −1.40373
\(125\) 0 0
\(126\) 0 0
\(127\) −296.412 −0.207105 −0.103552 0.994624i \(-0.533021\pi\)
−0.103552 + 0.994624i \(0.533021\pi\)
\(128\) −1819.86 −1.25667
\(129\) 0 0
\(130\) 0 0
\(131\) 2768.20 1.84625 0.923123 0.384504i \(-0.125627\pi\)
0.923123 + 0.384504i \(0.125627\pi\)
\(132\) 0 0
\(133\) −357.969 −0.233382
\(134\) −2007.88 −1.29443
\(135\) 0 0
\(136\) −907.997 −0.572500
\(137\) −813.962 −0.507602 −0.253801 0.967256i \(-0.581681\pi\)
−0.253801 + 0.967256i \(0.581681\pi\)
\(138\) 0 0
\(139\) 99.1631 0.0605101 0.0302550 0.999542i \(-0.490368\pi\)
0.0302550 + 0.999542i \(0.490368\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4116.57 2.43278
\(143\) −83.4384 −0.0487935
\(144\) 0 0
\(145\) 0 0
\(146\) 1637.58 0.928266
\(147\) 0 0
\(148\) −2597.72 −1.44278
\(149\) −1427.06 −0.784624 −0.392312 0.919832i \(-0.628325\pi\)
−0.392312 + 0.919832i \(0.628325\pi\)
\(150\) 0 0
\(151\) 1719.54 0.926718 0.463359 0.886171i \(-0.346644\pi\)
0.463359 + 0.886171i \(0.346644\pi\)
\(152\) −807.423 −0.430860
\(153\) 0 0
\(154\) −723.906 −0.378792
\(155\) 0 0
\(156\) 0 0
\(157\) 3270.08 1.66230 0.831149 0.556049i \(-0.187683\pi\)
0.831149 + 0.556049i \(0.187683\pi\)
\(158\) −241.537 −0.121618
\(159\) 0 0
\(160\) 0 0
\(161\) −459.473 −0.224917
\(162\) 0 0
\(163\) −891.324 −0.428306 −0.214153 0.976800i \(-0.568699\pi\)
−0.214153 + 0.976800i \(0.568699\pi\)
\(164\) 2275.52 1.08347
\(165\) 0 0
\(166\) 1902.39 0.889484
\(167\) −1598.41 −0.740649 −0.370324 0.928902i \(-0.620754\pi\)
−0.370324 + 0.928902i \(0.620754\pi\)
\(168\) 0 0
\(169\) −2184.26 −0.994202
\(170\) 0 0
\(171\) 0 0
\(172\) 682.217 0.302434
\(173\) −2492.49 −1.09538 −0.547690 0.836681i \(-0.684493\pi\)
−0.547690 + 0.836681i \(0.684493\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 530.845 0.227352
\(177\) 0 0
\(178\) −3261.16 −1.37323
\(179\) −1410.34 −0.588904 −0.294452 0.955666i \(-0.595137\pi\)
−0.294452 + 0.955666i \(0.595137\pi\)
\(180\) 0 0
\(181\) 2039.16 0.837399 0.418700 0.908125i \(-0.362486\pi\)
0.418700 + 0.908125i \(0.362486\pi\)
\(182\) 110.523 0.0450138
\(183\) 0 0
\(184\) −1036.37 −0.415231
\(185\) 0 0
\(186\) 0 0
\(187\) 1344.40 0.525732
\(188\) −485.075 −0.188179
\(189\) 0 0
\(190\) 0 0
\(191\) −2576.79 −0.976178 −0.488089 0.872794i \(-0.662306\pi\)
−0.488089 + 0.872794i \(0.662306\pi\)
\(192\) 0 0
\(193\) −2873.61 −1.07175 −0.535874 0.844298i \(-0.680018\pi\)
−0.535874 + 0.844298i \(0.680018\pi\)
\(194\) 665.198 0.246178
\(195\) 0 0
\(196\) 566.890 0.206592
\(197\) 1489.82 0.538810 0.269405 0.963027i \(-0.413173\pi\)
0.269405 + 0.963027i \(0.413173\pi\)
\(198\) 0 0
\(199\) 1311.18 0.467071 0.233536 0.972348i \(-0.424970\pi\)
0.233536 + 0.972348i \(0.424970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 594.648 0.207125
\(203\) 291.303 0.100716
\(204\) 0 0
\(205\) 0 0
\(206\) −1028.48 −0.347854
\(207\) 0 0
\(208\) −81.0472 −0.0270174
\(209\) 1195.49 0.395662
\(210\) 0 0
\(211\) 2671.37 0.871585 0.435793 0.900047i \(-0.356468\pi\)
0.435793 + 0.900047i \(0.356468\pi\)
\(212\) 385.768 0.124975
\(213\) 0 0
\(214\) −6809.77 −2.17526
\(215\) 0 0
\(216\) 0 0
\(217\) −1172.76 −0.366877
\(218\) −2227.17 −0.691940
\(219\) 0 0
\(220\) 0 0
\(221\) −205.257 −0.0624755
\(222\) 0 0
\(223\) −3321.87 −0.997529 −0.498764 0.866738i \(-0.666213\pi\)
−0.498764 + 0.866738i \(0.666213\pi\)
\(224\) −1587.34 −0.473477
\(225\) 0 0
\(226\) −1181.50 −0.347754
\(227\) 1756.95 0.513712 0.256856 0.966450i \(-0.417313\pi\)
0.256856 + 0.966450i \(0.417313\pi\)
\(228\) 0 0
\(229\) 2654.65 0.766045 0.383023 0.923739i \(-0.374883\pi\)
0.383023 + 0.923739i \(0.374883\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 657.054 0.185938
\(233\) 4379.70 1.23143 0.615716 0.787968i \(-0.288867\pi\)
0.615716 + 0.787968i \(0.288867\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2651.79 −0.731427
\(237\) 0 0
\(238\) −1780.80 −0.485008
\(239\) −932.890 −0.252484 −0.126242 0.991999i \(-0.540292\pi\)
−0.126242 + 0.991999i \(0.540292\pi\)
\(240\) 0 0
\(241\) −6451.43 −1.72437 −0.862184 0.506595i \(-0.830904\pi\)
−0.862184 + 0.506595i \(0.830904\pi\)
\(242\) −3470.37 −0.921834
\(243\) 0 0
\(244\) −8105.52 −2.12665
\(245\) 0 0
\(246\) 0 0
\(247\) −182.522 −0.0470186
\(248\) −2645.25 −0.677313
\(249\) 0 0
\(250\) 0 0
\(251\) 5085.23 1.27879 0.639396 0.768878i \(-0.279185\pi\)
0.639396 + 0.768878i \(0.279185\pi\)
\(252\) 0 0
\(253\) 1534.47 0.381311
\(254\) −1311.24 −0.323915
\(255\) 0 0
\(256\) −1478.71 −0.361013
\(257\) 4817.11 1.16919 0.584597 0.811324i \(-0.301253\pi\)
0.584597 + 0.811324i \(0.301253\pi\)
\(258\) 0 0
\(259\) −1571.76 −0.377084
\(260\) 0 0
\(261\) 0 0
\(262\) 12245.7 2.88756
\(263\) 8001.67 1.87606 0.938030 0.346553i \(-0.112648\pi\)
0.938030 + 0.346553i \(0.112648\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1583.55 −0.365013
\(267\) 0 0
\(268\) −5251.13 −1.19688
\(269\) −3477.43 −0.788188 −0.394094 0.919070i \(-0.628942\pi\)
−0.394094 + 0.919070i \(0.628942\pi\)
\(270\) 0 0
\(271\) −4589.09 −1.02866 −0.514331 0.857592i \(-0.671960\pi\)
−0.514331 + 0.857592i \(0.671960\pi\)
\(272\) 1305.87 0.291103
\(273\) 0 0
\(274\) −3600.73 −0.793898
\(275\) 0 0
\(276\) 0 0
\(277\) −1920.87 −0.416657 −0.208329 0.978059i \(-0.566802\pi\)
−0.208329 + 0.978059i \(0.566802\pi\)
\(278\) 438.668 0.0946388
\(279\) 0 0
\(280\) 0 0
\(281\) −392.063 −0.0832331 −0.0416166 0.999134i \(-0.513251\pi\)
−0.0416166 + 0.999134i \(0.513251\pi\)
\(282\) 0 0
\(283\) 7701.96 1.61779 0.808894 0.587955i \(-0.200067\pi\)
0.808894 + 0.587955i \(0.200067\pi\)
\(284\) 10765.9 2.24944
\(285\) 0 0
\(286\) −369.107 −0.0763138
\(287\) 1376.82 0.283174
\(288\) 0 0
\(289\) −1605.81 −0.326849
\(290\) 0 0
\(291\) 0 0
\(292\) 4282.70 0.858309
\(293\) 6657.73 1.32747 0.663735 0.747968i \(-0.268970\pi\)
0.663735 + 0.747968i \(0.268970\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3545.23 −0.696155
\(297\) 0 0
\(298\) −6312.88 −1.22716
\(299\) −234.277 −0.0453131
\(300\) 0 0
\(301\) 412.780 0.0790439
\(302\) 7606.75 1.44940
\(303\) 0 0
\(304\) 1161.23 0.219082
\(305\) 0 0
\(306\) 0 0
\(307\) −5082.53 −0.944870 −0.472435 0.881366i \(-0.656625\pi\)
−0.472435 + 0.881366i \(0.656625\pi\)
\(308\) −1893.21 −0.350245
\(309\) 0 0
\(310\) 0 0
\(311\) −2627.30 −0.479038 −0.239519 0.970892i \(-0.576990\pi\)
−0.239519 + 0.970892i \(0.576990\pi\)
\(312\) 0 0
\(313\) 2378.77 0.429571 0.214786 0.976661i \(-0.431095\pi\)
0.214786 + 0.976661i \(0.431095\pi\)
\(314\) 14465.9 2.59986
\(315\) 0 0
\(316\) −631.685 −0.112453
\(317\) 7116.78 1.26094 0.630470 0.776213i \(-0.282862\pi\)
0.630470 + 0.776213i \(0.282862\pi\)
\(318\) 0 0
\(319\) −972.846 −0.170749
\(320\) 0 0
\(321\) 0 0
\(322\) −2032.57 −0.351773
\(323\) 2940.87 0.506609
\(324\) 0 0
\(325\) 0 0
\(326\) −3942.96 −0.669878
\(327\) 0 0
\(328\) 3105.51 0.522784
\(329\) −293.497 −0.0491825
\(330\) 0 0
\(331\) −9616.07 −1.59682 −0.798410 0.602115i \(-0.794325\pi\)
−0.798410 + 0.602115i \(0.794325\pi\)
\(332\) 4975.27 0.822449
\(333\) 0 0
\(334\) −7070.88 −1.15839
\(335\) 0 0
\(336\) 0 0
\(337\) −11052.0 −1.78648 −0.893238 0.449584i \(-0.851572\pi\)
−0.893238 + 0.449584i \(0.851572\pi\)
\(338\) −9662.53 −1.55495
\(339\) 0 0
\(340\) 0 0
\(341\) 3916.61 0.621983
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 931.054 0.145927
\(345\) 0 0
\(346\) −11026.1 −1.71319
\(347\) 2320.86 0.359050 0.179525 0.983753i \(-0.442544\pi\)
0.179525 + 0.983753i \(0.442544\pi\)
\(348\) 0 0
\(349\) −7462.68 −1.14461 −0.572304 0.820042i \(-0.693950\pi\)
−0.572304 + 0.820042i \(0.693950\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5301.16 0.802706
\(353\) 9430.78 1.42195 0.710977 0.703216i \(-0.248253\pi\)
0.710977 + 0.703216i \(0.248253\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8528.82 −1.26974
\(357\) 0 0
\(358\) −6238.94 −0.921056
\(359\) −7449.39 −1.09516 −0.547582 0.836752i \(-0.684451\pi\)
−0.547582 + 0.836752i \(0.684451\pi\)
\(360\) 0 0
\(361\) −4243.87 −0.618730
\(362\) 9020.63 1.30971
\(363\) 0 0
\(364\) 289.047 0.0416214
\(365\) 0 0
\(366\) 0 0
\(367\) 12464.6 1.77287 0.886436 0.462850i \(-0.153173\pi\)
0.886436 + 0.462850i \(0.153173\pi\)
\(368\) 1490.50 0.211135
\(369\) 0 0
\(370\) 0 0
\(371\) 233.411 0.0326634
\(372\) 0 0
\(373\) 8335.42 1.15708 0.578541 0.815653i \(-0.303622\pi\)
0.578541 + 0.815653i \(0.303622\pi\)
\(374\) 5947.21 0.822254
\(375\) 0 0
\(376\) −662.004 −0.0907986
\(377\) 148.530 0.0202910
\(378\) 0 0
\(379\) −11976.6 −1.62321 −0.811603 0.584209i \(-0.801405\pi\)
−0.811603 + 0.584209i \(0.801405\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −11399.0 −1.52676
\(383\) 1607.28 0.214434 0.107217 0.994236i \(-0.465806\pi\)
0.107217 + 0.994236i \(0.465806\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12712.0 −1.67623
\(387\) 0 0
\(388\) 1739.67 0.227625
\(389\) 10021.6 1.30621 0.653103 0.757269i \(-0.273467\pi\)
0.653103 + 0.757269i \(0.273467\pi\)
\(390\) 0 0
\(391\) 3774.78 0.488233
\(392\) 773.661 0.0996831
\(393\) 0 0
\(394\) 6590.54 0.842708
\(395\) 0 0
\(396\) 0 0
\(397\) 1247.17 0.157666 0.0788331 0.996888i \(-0.474881\pi\)
0.0788331 + 0.996888i \(0.474881\pi\)
\(398\) 5800.28 0.730507
\(399\) 0 0
\(400\) 0 0
\(401\) −5115.73 −0.637075 −0.318538 0.947910i \(-0.603192\pi\)
−0.318538 + 0.947910i \(0.603192\pi\)
\(402\) 0 0
\(403\) −597.972 −0.0739134
\(404\) 1555.16 0.191516
\(405\) 0 0
\(406\) 1288.64 0.157522
\(407\) 5249.12 0.639286
\(408\) 0 0
\(409\) 11412.9 1.37979 0.689894 0.723911i \(-0.257657\pi\)
0.689894 + 0.723911i \(0.257657\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2689.76 −0.321639
\(413\) −1604.48 −0.191166
\(414\) 0 0
\(415\) 0 0
\(416\) −809.359 −0.0953897
\(417\) 0 0
\(418\) 5288.48 0.618823
\(419\) 10144.2 1.18276 0.591381 0.806393i \(-0.298583\pi\)
0.591381 + 0.806393i \(0.298583\pi\)
\(420\) 0 0
\(421\) −195.217 −0.0225993 −0.0112996 0.999936i \(-0.503597\pi\)
−0.0112996 + 0.999936i \(0.503597\pi\)
\(422\) 11817.3 1.36317
\(423\) 0 0
\(424\) 526.476 0.0603017
\(425\) 0 0
\(426\) 0 0
\(427\) −4904.29 −0.555820
\(428\) −17809.4 −2.01133
\(429\) 0 0
\(430\) 0 0
\(431\) 8432.54 0.942415 0.471208 0.882022i \(-0.343818\pi\)
0.471208 + 0.882022i \(0.343818\pi\)
\(432\) 0 0
\(433\) 3039.89 0.337386 0.168693 0.985669i \(-0.446045\pi\)
0.168693 + 0.985669i \(0.446045\pi\)
\(434\) −5187.96 −0.573802
\(435\) 0 0
\(436\) −5824.65 −0.639793
\(437\) 3356.67 0.367440
\(438\) 0 0
\(439\) −8748.73 −0.951148 −0.475574 0.879676i \(-0.657760\pi\)
−0.475574 + 0.879676i \(0.657760\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −907.997 −0.0977127
\(443\) 2184.85 0.234324 0.117162 0.993113i \(-0.462620\pi\)
0.117162 + 0.993113i \(0.462620\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −14695.0 −1.56015
\(447\) 0 0
\(448\) −5750.32 −0.606422
\(449\) −12511.6 −1.31505 −0.657525 0.753432i \(-0.728397\pi\)
−0.657525 + 0.753432i \(0.728397\pi\)
\(450\) 0 0
\(451\) −4598.07 −0.480077
\(452\) −3089.95 −0.321546
\(453\) 0 0
\(454\) 7772.22 0.803455
\(455\) 0 0
\(456\) 0 0
\(457\) 10709.8 1.09625 0.548123 0.836397i \(-0.315342\pi\)
0.548123 + 0.836397i \(0.315342\pi\)
\(458\) 11743.4 1.19811
\(459\) 0 0
\(460\) 0 0
\(461\) 73.0989 0.00738515 0.00369258 0.999993i \(-0.498825\pi\)
0.00369258 + 0.999993i \(0.498825\pi\)
\(462\) 0 0
\(463\) −2307.72 −0.231639 −0.115820 0.993270i \(-0.536949\pi\)
−0.115820 + 0.993270i \(0.536949\pi\)
\(464\) −944.967 −0.0945452
\(465\) 0 0
\(466\) 19374.5 1.92598
\(467\) 977.511 0.0968604 0.0484302 0.998827i \(-0.484578\pi\)
0.0484302 + 0.998827i \(0.484578\pi\)
\(468\) 0 0
\(469\) −3177.23 −0.312816
\(470\) 0 0
\(471\) 0 0
\(472\) −3619.02 −0.352922
\(473\) −1378.53 −0.134007
\(474\) 0 0
\(475\) 0 0
\(476\) −4657.26 −0.448456
\(477\) 0 0
\(478\) −4126.83 −0.394889
\(479\) 8973.08 0.855930 0.427965 0.903795i \(-0.359231\pi\)
0.427965 + 0.903795i \(0.359231\pi\)
\(480\) 0 0
\(481\) −801.415 −0.0759696
\(482\) −28539.2 −2.69694
\(483\) 0 0
\(484\) −9075.95 −0.852362
\(485\) 0 0
\(486\) 0 0
\(487\) −17259.1 −1.60592 −0.802961 0.596031i \(-0.796743\pi\)
−0.802961 + 0.596031i \(0.796743\pi\)
\(488\) −11062.0 −1.02613
\(489\) 0 0
\(490\) 0 0
\(491\) 2612.25 0.240100 0.120050 0.992768i \(-0.461695\pi\)
0.120050 + 0.992768i \(0.461695\pi\)
\(492\) 0 0
\(493\) −2393.18 −0.218628
\(494\) −807.423 −0.0735378
\(495\) 0 0
\(496\) 3804.37 0.344397
\(497\) 6513.99 0.587913
\(498\) 0 0
\(499\) 16056.2 1.44043 0.720215 0.693751i \(-0.244043\pi\)
0.720215 + 0.693751i \(0.244043\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 22495.6 2.00005
\(503\) 20352.4 1.80411 0.902054 0.431623i \(-0.142059\pi\)
0.902054 + 0.431623i \(0.142059\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6788.07 0.596376
\(507\) 0 0
\(508\) −3429.24 −0.299504
\(509\) −19740.3 −1.71900 −0.859502 0.511133i \(-0.829226\pi\)
−0.859502 + 0.511133i \(0.829226\pi\)
\(510\) 0 0
\(511\) 2591.28 0.224327
\(512\) 8017.47 0.692042
\(513\) 0 0
\(514\) 21309.5 1.82864
\(515\) 0 0
\(516\) 0 0
\(517\) 980.175 0.0833812
\(518\) −6953.02 −0.589765
\(519\) 0 0
\(520\) 0 0
\(521\) 19115.6 1.60742 0.803712 0.595019i \(-0.202855\pi\)
0.803712 + 0.595019i \(0.202855\pi\)
\(522\) 0 0
\(523\) 4696.30 0.392648 0.196324 0.980539i \(-0.437100\pi\)
0.196324 + 0.980539i \(0.437100\pi\)
\(524\) 32025.7 2.66994
\(525\) 0 0
\(526\) 35397.0 2.93419
\(527\) 9634.79 0.796391
\(528\) 0 0
\(529\) −7858.52 −0.645888
\(530\) 0 0
\(531\) 0 0
\(532\) −4141.40 −0.337505
\(533\) 702.016 0.0570500
\(534\) 0 0
\(535\) 0 0
\(536\) −7166.46 −0.577508
\(537\) 0 0
\(538\) −15383.1 −1.23274
\(539\) −1145.50 −0.0915399
\(540\) 0 0
\(541\) −1302.32 −0.103496 −0.0517478 0.998660i \(-0.516479\pi\)
−0.0517478 + 0.998660i \(0.516479\pi\)
\(542\) −20300.8 −1.60885
\(543\) 0 0
\(544\) 13040.8 1.02779
\(545\) 0 0
\(546\) 0 0
\(547\) −13145.9 −1.02757 −0.513784 0.857920i \(-0.671757\pi\)
−0.513784 + 0.857920i \(0.671757\pi\)
\(548\) −9416.87 −0.734067
\(549\) 0 0
\(550\) 0 0
\(551\) −2128.11 −0.164538
\(552\) 0 0
\(553\) −382.205 −0.0293906
\(554\) −8497.38 −0.651659
\(555\) 0 0
\(556\) 1147.24 0.0875065
\(557\) 5167.96 0.393130 0.196565 0.980491i \(-0.437021\pi\)
0.196565 + 0.980491i \(0.437021\pi\)
\(558\) 0 0
\(559\) 210.469 0.0159247
\(560\) 0 0
\(561\) 0 0
\(562\) −1734.37 −0.130178
\(563\) 1444.18 0.108108 0.0540542 0.998538i \(-0.482786\pi\)
0.0540542 + 0.998538i \(0.482786\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 34071.2 2.53025
\(567\) 0 0
\(568\) 14692.8 1.08538
\(569\) −22883.1 −1.68596 −0.842980 0.537945i \(-0.819201\pi\)
−0.842980 + 0.537945i \(0.819201\pi\)
\(570\) 0 0
\(571\) 6611.79 0.484580 0.242290 0.970204i \(-0.422102\pi\)
0.242290 + 0.970204i \(0.422102\pi\)
\(572\) −965.313 −0.0705626
\(573\) 0 0
\(574\) 6090.64 0.442889
\(575\) 0 0
\(576\) 0 0
\(577\) 16695.5 1.20458 0.602291 0.798277i \(-0.294255\pi\)
0.602291 + 0.798277i \(0.294255\pi\)
\(578\) −7103.63 −0.511197
\(579\) 0 0
\(580\) 0 0
\(581\) 3010.31 0.214955
\(582\) 0 0
\(583\) −779.510 −0.0553756
\(584\) 5844.81 0.414144
\(585\) 0 0
\(586\) 29451.9 2.07619
\(587\) −16504.8 −1.16052 −0.580261 0.814431i \(-0.697049\pi\)
−0.580261 + 0.814431i \(0.697049\pi\)
\(588\) 0 0
\(589\) 8567.60 0.599358
\(590\) 0 0
\(591\) 0 0
\(592\) 5098.70 0.353978
\(593\) 16410.8 1.13644 0.568222 0.822875i \(-0.307631\pi\)
0.568222 + 0.822875i \(0.307631\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16509.9 −1.13468
\(597\) 0 0
\(598\) −1036.37 −0.0708704
\(599\) 6879.60 0.469270 0.234635 0.972084i \(-0.424610\pi\)
0.234635 + 0.972084i \(0.424610\pi\)
\(600\) 0 0
\(601\) −979.314 −0.0664676 −0.0332338 0.999448i \(-0.510581\pi\)
−0.0332338 + 0.999448i \(0.510581\pi\)
\(602\) 1826.02 0.123626
\(603\) 0 0
\(604\) 19893.7 1.34017
\(605\) 0 0
\(606\) 0 0
\(607\) −5654.83 −0.378126 −0.189063 0.981965i \(-0.560545\pi\)
−0.189063 + 0.981965i \(0.560545\pi\)
\(608\) 11596.3 0.773507
\(609\) 0 0
\(610\) 0 0
\(611\) −149.649 −0.00990861
\(612\) 0 0
\(613\) −8619.72 −0.567940 −0.283970 0.958833i \(-0.591652\pi\)
−0.283970 + 0.958833i \(0.591652\pi\)
\(614\) −22483.6 −1.47779
\(615\) 0 0
\(616\) −2583.75 −0.168997
\(617\) −7061.35 −0.460744 −0.230372 0.973103i \(-0.573994\pi\)
−0.230372 + 0.973103i \(0.573994\pi\)
\(618\) 0 0
\(619\) 5725.08 0.371745 0.185873 0.982574i \(-0.440489\pi\)
0.185873 + 0.982574i \(0.440489\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −11622.4 −0.749223
\(623\) −5160.41 −0.331858
\(624\) 0 0
\(625\) 0 0
\(626\) 10523.0 0.671857
\(627\) 0 0
\(628\) 37832.2 2.40393
\(629\) 12912.8 0.818546
\(630\) 0 0
\(631\) −3621.56 −0.228482 −0.114241 0.993453i \(-0.536444\pi\)
−0.114241 + 0.993453i \(0.536444\pi\)
\(632\) −862.090 −0.0542597
\(633\) 0 0
\(634\) 31482.5 1.97213
\(635\) 0 0
\(636\) 0 0
\(637\) 174.890 0.0108782
\(638\) −4303.58 −0.267054
\(639\) 0 0
\(640\) 0 0
\(641\) 20656.0 1.27280 0.636398 0.771361i \(-0.280424\pi\)
0.636398 + 0.771361i \(0.280424\pi\)
\(642\) 0 0
\(643\) 15856.5 0.972504 0.486252 0.873819i \(-0.338364\pi\)
0.486252 + 0.873819i \(0.338364\pi\)
\(644\) −5315.73 −0.325262
\(645\) 0 0
\(646\) 13009.6 0.792344
\(647\) −5476.66 −0.332782 −0.166391 0.986060i \(-0.553211\pi\)
−0.166391 + 0.986060i \(0.553211\pi\)
\(648\) 0 0
\(649\) 5358.39 0.324091
\(650\) 0 0
\(651\) 0 0
\(652\) −10311.9 −0.619394
\(653\) −18293.1 −1.09627 −0.548135 0.836390i \(-0.684662\pi\)
−0.548135 + 0.836390i \(0.684662\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4466.31 −0.265823
\(657\) 0 0
\(658\) −1298.35 −0.0769222
\(659\) −32480.3 −1.91996 −0.959979 0.280073i \(-0.909641\pi\)
−0.959979 + 0.280073i \(0.909641\pi\)
\(660\) 0 0
\(661\) −19403.4 −1.14176 −0.570880 0.821034i \(-0.693398\pi\)
−0.570880 + 0.821034i \(0.693398\pi\)
\(662\) −42538.7 −2.49745
\(663\) 0 0
\(664\) 6789.98 0.396841
\(665\) 0 0
\(666\) 0 0
\(667\) −2731.55 −0.158570
\(668\) −18492.2 −1.07109
\(669\) 0 0
\(670\) 0 0
\(671\) 16378.6 0.942306
\(672\) 0 0
\(673\) −3449.20 −0.197559 −0.0987793 0.995109i \(-0.531494\pi\)
−0.0987793 + 0.995109i \(0.531494\pi\)
\(674\) −48890.9 −2.79408
\(675\) 0 0
\(676\) −25270.1 −1.43776
\(677\) −23683.0 −1.34448 −0.672239 0.740334i \(-0.734667\pi\)
−0.672239 + 0.740334i \(0.734667\pi\)
\(678\) 0 0
\(679\) 1052.60 0.0594919
\(680\) 0 0
\(681\) 0 0
\(682\) 17325.9 0.972792
\(683\) 14101.5 0.790012 0.395006 0.918679i \(-0.370742\pi\)
0.395006 + 0.918679i \(0.370742\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1517.33 0.0844489
\(687\) 0 0
\(688\) −1339.03 −0.0742006
\(689\) 119.012 0.00658057
\(690\) 0 0
\(691\) 1327.04 0.0730577 0.0365289 0.999333i \(-0.488370\pi\)
0.0365289 + 0.999333i \(0.488370\pi\)
\(692\) −28836.1 −1.58408
\(693\) 0 0
\(694\) 10266.8 0.561560
\(695\) 0 0
\(696\) 0 0
\(697\) −11311.2 −0.614694
\(698\) −33012.7 −1.79019
\(699\) 0 0
\(700\) 0 0
\(701\) 9073.19 0.488858 0.244429 0.969667i \(-0.421399\pi\)
0.244429 + 0.969667i \(0.421399\pi\)
\(702\) 0 0
\(703\) 11482.5 0.616032
\(704\) 19204.0 1.02809
\(705\) 0 0
\(706\) 41719.0 2.22396
\(707\) 940.961 0.0500544
\(708\) 0 0
\(709\) −13479.5 −0.714010 −0.357005 0.934102i \(-0.616202\pi\)
−0.357005 + 0.934102i \(0.616202\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11639.7 −0.612662
\(713\) 10997.0 0.577618
\(714\) 0 0
\(715\) 0 0
\(716\) −16316.5 −0.851642
\(717\) 0 0
\(718\) −32953.9 −1.71285
\(719\) −26389.8 −1.36881 −0.684404 0.729103i \(-0.739938\pi\)
−0.684404 + 0.729103i \(0.739938\pi\)
\(720\) 0 0
\(721\) −1627.46 −0.0840633
\(722\) −18773.6 −0.967704
\(723\) 0 0
\(724\) 23591.4 1.21100
\(725\) 0 0
\(726\) 0 0
\(727\) −7983.43 −0.407275 −0.203638 0.979046i \(-0.565276\pi\)
−0.203638 + 0.979046i \(0.565276\pi\)
\(728\) 394.476 0.0200828
\(729\) 0 0
\(730\) 0 0
\(731\) −3391.17 −0.171583
\(732\) 0 0
\(733\) 24350.2 1.22700 0.613502 0.789693i \(-0.289760\pi\)
0.613502 + 0.789693i \(0.289760\pi\)
\(734\) 55139.5 2.77280
\(735\) 0 0
\(736\) 14884.5 0.745450
\(737\) 10610.8 0.530331
\(738\) 0 0
\(739\) 1936.37 0.0963880 0.0481940 0.998838i \(-0.484653\pi\)
0.0481940 + 0.998838i \(0.484653\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1032.54 0.0510861
\(743\) −27212.0 −1.34362 −0.671811 0.740722i \(-0.734483\pi\)
−0.671811 + 0.740722i \(0.734483\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 36873.5 1.80970
\(747\) 0 0
\(748\) 15553.6 0.760287
\(749\) −10775.7 −0.525680
\(750\) 0 0
\(751\) −13542.6 −0.658026 −0.329013 0.944325i \(-0.606716\pi\)
−0.329013 + 0.944325i \(0.606716\pi\)
\(752\) 952.086 0.0461689
\(753\) 0 0
\(754\) 657.054 0.0317354
\(755\) 0 0
\(756\) 0 0
\(757\) −6328.67 −0.303857 −0.151928 0.988392i \(-0.548548\pi\)
−0.151928 + 0.988392i \(0.548548\pi\)
\(758\) −52980.8 −2.53872
\(759\) 0 0
\(760\) 0 0
\(761\) −26036.0 −1.24021 −0.620107 0.784517i \(-0.712911\pi\)
−0.620107 + 0.784517i \(0.712911\pi\)
\(762\) 0 0
\(763\) −3524.24 −0.167216
\(764\) −29811.4 −1.41170
\(765\) 0 0
\(766\) 7110.15 0.335379
\(767\) −818.097 −0.0385134
\(768\) 0 0
\(769\) −6092.16 −0.285681 −0.142841 0.989746i \(-0.545624\pi\)
−0.142841 + 0.989746i \(0.545624\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −33245.3 −1.54990
\(773\) 16076.6 0.748040 0.374020 0.927421i \(-0.377979\pi\)
0.374020 + 0.927421i \(0.377979\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2374.21 0.109831
\(777\) 0 0
\(778\) 44332.5 2.04293
\(779\) −10058.3 −0.462614
\(780\) 0 0
\(781\) −21754.4 −0.996713
\(782\) 16698.5 0.763604
\(783\) 0 0
\(784\) −1112.67 −0.0506865
\(785\) 0 0
\(786\) 0 0
\(787\) −41683.3 −1.88799 −0.943995 0.329961i \(-0.892964\pi\)
−0.943995 + 0.329961i \(0.892964\pi\)
\(788\) 17236.0 0.779199
\(789\) 0 0
\(790\) 0 0
\(791\) −1869.59 −0.0840392
\(792\) 0 0
\(793\) −2500.61 −0.111979
\(794\) 5517.10 0.246593
\(795\) 0 0
\(796\) 15169.3 0.675454
\(797\) −37247.7 −1.65543 −0.827717 0.561145i \(-0.810361\pi\)
−0.827717 + 0.561145i \(0.810361\pi\)
\(798\) 0 0
\(799\) 2411.21 0.106762
\(800\) 0 0
\(801\) 0 0
\(802\) −22630.5 −0.996396
\(803\) −8653.92 −0.380312
\(804\) 0 0
\(805\) 0 0
\(806\) −2645.25 −0.115602
\(807\) 0 0
\(808\) 2122.40 0.0924083
\(809\) 12084.0 0.525156 0.262578 0.964911i \(-0.415427\pi\)
0.262578 + 0.964911i \(0.415427\pi\)
\(810\) 0 0
\(811\) −43963.8 −1.90355 −0.951775 0.306796i \(-0.900743\pi\)
−0.951775 + 0.306796i \(0.900743\pi\)
\(812\) 3370.13 0.145651
\(813\) 0 0
\(814\) 23220.6 0.999854
\(815\) 0 0
\(816\) 0 0
\(817\) −3015.55 −0.129132
\(818\) 50487.4 2.15801
\(819\) 0 0
\(820\) 0 0
\(821\) 6761.26 0.287417 0.143709 0.989620i \(-0.454097\pi\)
0.143709 + 0.989620i \(0.454097\pi\)
\(822\) 0 0
\(823\) 21729.5 0.920343 0.460172 0.887830i \(-0.347788\pi\)
0.460172 + 0.887830i \(0.347788\pi\)
\(824\) −3670.84 −0.155194
\(825\) 0 0
\(826\) −7097.76 −0.298986
\(827\) 27297.7 1.14780 0.573902 0.818924i \(-0.305429\pi\)
0.573902 + 0.818924i \(0.305429\pi\)
\(828\) 0 0
\(829\) −14194.9 −0.594703 −0.297352 0.954768i \(-0.596103\pi\)
−0.297352 + 0.954768i \(0.596103\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2931.99 −0.122174
\(833\) −2817.90 −0.117208
\(834\) 0 0
\(835\) 0 0
\(836\) 13830.8 0.572186
\(837\) 0 0
\(838\) 44875.0 1.84986
\(839\) 30955.4 1.27378 0.636889 0.770955i \(-0.280221\pi\)
0.636889 + 0.770955i \(0.280221\pi\)
\(840\) 0 0
\(841\) −22657.2 −0.928993
\(842\) −863.583 −0.0353457
\(843\) 0 0
\(844\) 30905.5 1.26044
\(845\) 0 0
\(846\) 0 0
\(847\) −5491.46 −0.222773
\(848\) −757.171 −0.0306620
\(849\) 0 0
\(850\) 0 0
\(851\) 14738.4 0.593686
\(852\) 0 0
\(853\) 21925.6 0.880094 0.440047 0.897975i \(-0.354962\pi\)
0.440047 + 0.897975i \(0.354962\pi\)
\(854\) −21695.1 −0.869312
\(855\) 0 0
\(856\) −24305.3 −0.970487
\(857\) −28288.5 −1.12756 −0.563780 0.825925i \(-0.690653\pi\)
−0.563780 + 0.825925i \(0.690653\pi\)
\(858\) 0 0
\(859\) −12483.8 −0.495858 −0.247929 0.968778i \(-0.579750\pi\)
−0.247929 + 0.968778i \(0.579750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 37303.1 1.47395
\(863\) 13344.0 0.526346 0.263173 0.964749i \(-0.415231\pi\)
0.263173 + 0.964749i \(0.415231\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13447.6 0.527677
\(867\) 0 0
\(868\) −13567.9 −0.530559
\(869\) 1276.43 0.0498271
\(870\) 0 0
\(871\) −1620.01 −0.0630219
\(872\) −7949.17 −0.308707
\(873\) 0 0
\(874\) 14848.9 0.574683
\(875\) 0 0
\(876\) 0 0
\(877\) −19198.7 −0.739219 −0.369609 0.929187i \(-0.620508\pi\)
−0.369609 + 0.929187i \(0.620508\pi\)
\(878\) −38701.8 −1.48761
\(879\) 0 0
\(880\) 0 0
\(881\) 31063.7 1.18793 0.593963 0.804492i \(-0.297562\pi\)
0.593963 + 0.804492i \(0.297562\pi\)
\(882\) 0 0
\(883\) 37100.6 1.41397 0.706985 0.707229i \(-0.250055\pi\)
0.706985 + 0.707229i \(0.250055\pi\)
\(884\) −2374.65 −0.0903487
\(885\) 0 0
\(886\) 9665.15 0.366487
\(887\) 4568.24 0.172927 0.0864636 0.996255i \(-0.472443\pi\)
0.0864636 + 0.996255i \(0.472443\pi\)
\(888\) 0 0
\(889\) −2074.88 −0.0782782
\(890\) 0 0
\(891\) 0 0
\(892\) −38431.3 −1.44257
\(893\) 2144.14 0.0803481
\(894\) 0 0
\(895\) 0 0
\(896\) −12739.0 −0.474977
\(897\) 0 0
\(898\) −55347.5 −2.05676
\(899\) −6972.03 −0.258654
\(900\) 0 0
\(901\) −1917.58 −0.0709033
\(902\) −20340.5 −0.750849
\(903\) 0 0
\(904\) −4216.99 −0.155150
\(905\) 0 0
\(906\) 0 0
\(907\) −776.485 −0.0284264 −0.0142132 0.999899i \(-0.504524\pi\)
−0.0142132 + 0.999899i \(0.504524\pi\)
\(908\) 20326.4 0.742904
\(909\) 0 0
\(910\) 0 0
\(911\) −24605.9 −0.894872 −0.447436 0.894316i \(-0.647663\pi\)
−0.447436 + 0.894316i \(0.647663\pi\)
\(912\) 0 0
\(913\) −10053.4 −0.364423
\(914\) 47377.2 1.71455
\(915\) 0 0
\(916\) 30712.1 1.10781
\(917\) 19377.4 0.697816
\(918\) 0 0
\(919\) −156.897 −0.00563173 −0.00281587 0.999996i \(-0.500896\pi\)
−0.00281587 + 0.999996i \(0.500896\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 323.368 0.0115505
\(923\) 3321.37 0.118445
\(924\) 0 0
\(925\) 0 0
\(926\) −10208.7 −0.362287
\(927\) 0 0
\(928\) −9436.69 −0.333809
\(929\) 2740.94 0.0968002 0.0484001 0.998828i \(-0.484588\pi\)
0.0484001 + 0.998828i \(0.484588\pi\)
\(930\) 0 0
\(931\) −2505.78 −0.0882101
\(932\) 50669.5 1.78083
\(933\) 0 0
\(934\) 4324.22 0.151491
\(935\) 0 0
\(936\) 0 0
\(937\) −15341.1 −0.534870 −0.267435 0.963576i \(-0.586176\pi\)
−0.267435 + 0.963576i \(0.586176\pi\)
\(938\) −14055.1 −0.489250
\(939\) 0 0
\(940\) 0 0
\(941\) −40269.2 −1.39505 −0.697523 0.716563i \(-0.745714\pi\)
−0.697523 + 0.716563i \(0.745714\pi\)
\(942\) 0 0
\(943\) −12910.4 −0.445834
\(944\) 5204.83 0.179452
\(945\) 0 0
\(946\) −6098.23 −0.209588
\(947\) 10173.6 0.349100 0.174550 0.984648i \(-0.444153\pi\)
0.174550 + 0.984648i \(0.444153\pi\)
\(948\) 0 0
\(949\) 1321.25 0.0451944
\(950\) 0 0
\(951\) 0 0
\(952\) −6355.98 −0.216385
\(953\) 15239.9 0.518015 0.259007 0.965875i \(-0.416605\pi\)
0.259007 + 0.965875i \(0.416605\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −10792.8 −0.365129
\(957\) 0 0
\(958\) 39694.3 1.33869
\(959\) −5697.73 −0.191855
\(960\) 0 0
\(961\) −1722.13 −0.0578069
\(962\) −3545.23 −0.118818
\(963\) 0 0
\(964\) −74637.7 −2.49369
\(965\) 0 0
\(966\) 0 0
\(967\) 35689.8 1.18687 0.593437 0.804881i \(-0.297771\pi\)
0.593437 + 0.804881i \(0.297771\pi\)
\(968\) −12386.4 −0.411274
\(969\) 0 0
\(970\) 0 0
\(971\) 22841.5 0.754913 0.377456 0.926027i \(-0.376799\pi\)
0.377456 + 0.926027i \(0.376799\pi\)
\(972\) 0 0
\(973\) 694.142 0.0228707
\(974\) −76349.2 −2.51169
\(975\) 0 0
\(976\) 15909.2 0.521763
\(977\) −14487.1 −0.474394 −0.237197 0.971462i \(-0.576229\pi\)
−0.237197 + 0.971462i \(0.576229\pi\)
\(978\) 0 0
\(979\) 17233.9 0.562613
\(980\) 0 0
\(981\) 0 0
\(982\) 11555.8 0.375520
\(983\) 35291.6 1.14509 0.572547 0.819872i \(-0.305955\pi\)
0.572547 + 0.819872i \(0.305955\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −10586.7 −0.341938
\(987\) 0 0
\(988\) −2111.63 −0.0679958
\(989\) −3870.64 −0.124448
\(990\) 0 0
\(991\) −29615.5 −0.949310 −0.474655 0.880172i \(-0.657427\pi\)
−0.474655 + 0.880172i \(0.657427\pi\)
\(992\) 37991.4 1.21596
\(993\) 0 0
\(994\) 28816.0 0.919505
\(995\) 0 0
\(996\) 0 0
\(997\) −45868.5 −1.45704 −0.728520 0.685025i \(-0.759791\pi\)
−0.728520 + 0.685025i \(0.759791\pi\)
\(998\) 71027.9 2.25286
\(999\) 0 0
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 1575.4.a.bi.1.4 yes 4
3.2 odd 2 inner 1575.4.a.bi.1.1 yes 4
5.4 even 2 1575.4.a.bh.1.1 4
15.14 odd 2 1575.4.a.bh.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.1 4 5.4 even 2
1575.4.a.bh.1.4 yes 4 15.14 odd 2
1575.4.a.bi.1.1 yes 4 3.2 odd 2 inner
1575.4.a.bi.1.4 yes 4 1.1 even 1 trivial