Properties

Label 1575.4.a.bh.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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 1.1
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 \(-0.785805\pi\)
−0.782008 + 0.623268i \(0.785805\pi\)
\(3\) 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 \(-0.512122\pi\)
−0.0380735 + 0.999275i \(0.512122\pi\)
\(14\) 30.9659 0.591143
\(15\) 0 0
\(16\) −22.7075 −0.354805
\(17\) 57.5082 0.820458 0.410229 0.911983i \(-0.365449\pi\)
0.410229 + 0.911983i \(0.365449\pi\)
\(18\) 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 \(-0.403835\pi\)
0.297537 + 0.954710i \(0.403835\pi\)
\(24\) 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 \(-0.333761\pi\)
0.498835 + 0.866697i \(0.333761\pi\)
\(38\) 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 \(-0.533345\pi\)
−0.104565 + 0.994518i \(0.533345\pi\)
\(44\) −270.458 −0.926661
\(45\) 0 0
\(46\) −290.368 −0.930704
\(47\) 41.9282 0.130125 0.0650623 0.997881i \(-0.479275\pi\)
0.0650623 + 0.997881i \(0.479275\pi\)
\(48\) 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 \(-0.513758\pi\)
−0.0432096 + 0.999066i \(0.513758\pi\)
\(54\) 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 \(-0.364195\pi\)
0.413817 + 0.910360i \(0.364195\pi\)
\(68\) 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 \(-0.595905\pi\)
−0.296757 + 0.954953i \(0.595905\pi\)
\(74\) −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 \(-0.591781\pi\)
−0.284359 + 0.958718i \(0.591781\pi\)
\(84\) 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 \(-0.525077\pi\)
−0.0787004 + 0.996898i \(0.525077\pi\)
\(98\) −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 \(-0.464529\pi\)
0.111205 + 0.993797i \(0.464529\pi\)
\(104\) 56.3538 0.0531340
\(105\) 0 0
\(106\) 147.506 0.135161
\(107\) 1539.38 1.39082 0.695409 0.718614i \(-0.255223\pi\)
0.695409 + 0.718614i \(0.255223\pi\)
\(108\) 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 \(-0.464539\pi\)
0.111173 + 0.993801i \(0.464539\pi\)
\(114\) 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 \(-0.466979\pi\)
0.103552 + 0.994624i \(0.466979\pi\)
\(128\) 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 \(-0.418319\pi\)
0.253801 + 0.967256i \(0.418319\pi\)
\(138\) 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 \(-0.812317\pi\)
−0.831149 + 0.556049i \(0.812317\pi\)
\(158\) 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 \(-0.431301\pi\)
0.214153 + 0.976800i \(0.431301\pi\)
\(164\) 2275.52 1.08347
\(165\) 0 0
\(166\) 1902.39 0.889484
\(167\) 1598.41 0.740649 0.370324 0.928902i \(-0.379246\pi\)
0.370324 + 0.928902i \(0.379246\pi\)
\(168\) 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 \(-0.315507\pi\)
0.547690 + 0.836681i \(0.315507\pi\)
\(174\) 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 \(-0.319982\pi\)
0.535874 + 0.844298i \(0.319982\pi\)
\(194\) 665.198 0.246178
\(195\) 0 0
\(196\) 566.890 0.206592
\(197\) −1489.82 −0.538810 −0.269405 0.963027i \(-0.586827\pi\)
−0.269405 + 0.963027i \(0.586827\pi\)
\(198\) 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 \(-0.333787\pi\)
0.498764 + 0.866738i \(0.333787\pi\)
\(224\) −1587.34 −0.473477
\(225\) 0 0
\(226\) −1181.50 −0.347754
\(227\) −1756.95 −0.513712 −0.256856 0.966450i \(-0.582687\pi\)
−0.256856 + 0.966450i \(0.582687\pi\)
\(228\) 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 \(-0.711133\pi\)
−0.615716 + 0.787968i \(0.711133\pi\)
\(234\) 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 \(-0.698747\pi\)
−0.584597 + 0.811324i \(0.698747\pi\)
\(258\) 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 \(-0.887352\pi\)
−0.938030 + 0.346553i \(0.887352\pi\)
\(264\) 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 \(-0.433198\pi\)
0.208329 + 0.978059i \(0.433198\pi\)
\(278\) −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 \(-0.799933\pi\)
−0.808894 + 0.587955i \(0.799933\pi\)
\(284\) 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 \(-0.731030\pi\)
−0.663735 + 0.747968i \(0.731030\pi\)
\(294\) 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 \(-0.343375\pi\)
0.472435 + 0.881366i \(0.343375\pi\)
\(308\) 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 \(-0.568905\pi\)
−0.214786 + 0.976661i \(0.568905\pi\)
\(314\) 14465.9 2.59986
\(315\) 0 0
\(316\) −631.685 −0.112453
\(317\) −7116.78 −1.26094 −0.630470 0.776213i \(-0.717138\pi\)
−0.630470 + 0.776213i \(0.717138\pi\)
\(318\) 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 \(-0.148428\pi\)
0.893238 + 0.449584i \(0.148428\pi\)
\(338\) 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 \(-0.557456\pi\)
−0.179525 + 0.983753i \(0.557456\pi\)
\(348\) 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 \(-0.751747\pi\)
−0.710977 + 0.703216i \(0.751747\pi\)
\(354\) 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 \(-0.846827\pi\)
−0.886436 + 0.462850i \(0.846827\pi\)
\(368\) −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 \(-0.696378\pi\)
−0.578541 + 0.815653i \(0.696378\pi\)
\(374\) 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 \(-0.534194\pi\)
−0.107217 + 0.994236i \(0.534194\pi\)
\(384\) 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 \(-0.525119\pi\)
−0.0788331 + 0.996888i \(0.525119\pi\)
\(398\) −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 \(-0.553955\pi\)
−0.168693 + 0.985669i \(0.553955\pi\)
\(434\) −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 \(-0.537380\pi\)
−0.117162 + 0.993113i \(0.537380\pi\)
\(444\) 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 \(-0.684658\pi\)
−0.548123 + 0.836397i \(0.684658\pi\)
\(458\) −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 \(-0.463051\pi\)
0.115820 + 0.993270i \(0.463051\pi\)
\(464\) −944.967 −0.0945452
\(465\) 0 0
\(466\) 19374.5 1.92598
\(467\) −977.511 −0.0968604 −0.0484302 0.998827i \(-0.515422\pi\)
−0.0484302 + 0.998827i \(0.515422\pi\)
\(468\) 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 \(-0.203257\pi\)
0.802961 + 0.596031i \(0.203257\pi\)
\(488\) 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 \(-0.857941\pi\)
−0.902054 + 0.431623i \(0.857941\pi\)
\(504\) 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 \(-0.562900\pi\)
−0.196324 + 0.980539i \(0.562900\pi\)
\(524\) 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 \(-0.328243\pi\)
0.513784 + 0.857920i \(0.328243\pi\)
\(548\) 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 \(-0.562979\pi\)
−0.196565 + 0.980491i \(0.562979\pi\)
\(558\) 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 \(-0.517214\pi\)
−0.0540542 + 0.998538i \(0.517214\pi\)
\(564\) 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 \(-0.705745\pi\)
−0.602291 + 0.798277i \(0.705745\pi\)
\(578\) 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 \(-0.302951\pi\)
0.580261 + 0.814431i \(0.302951\pi\)
\(588\) 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 \(-0.692369\pi\)
−0.568222 + 0.822875i \(0.692369\pi\)
\(594\) 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 \(-0.439455\pi\)
0.189063 + 0.981965i \(0.439455\pi\)
\(608\) −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 \(-0.408348\pi\)
0.283970 + 0.958833i \(0.408348\pi\)
\(614\) −22483.6 −1.47779
\(615\) 0 0
\(616\) −2583.75 −0.168997
\(617\) 7061.35 0.460744 0.230372 0.973103i \(-0.426006\pi\)
0.230372 + 0.973103i \(0.426006\pi\)
\(618\) 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 \(-0.661636\pi\)
−0.486252 + 0.873819i \(0.661636\pi\)
\(644\) −5315.73 −0.325262
\(645\) 0 0
\(646\) 13009.6 0.792344
\(647\) 5476.66 0.332782 0.166391 0.986060i \(-0.446789\pi\)
0.166391 + 0.986060i \(0.446789\pi\)
\(648\) 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 \(-0.315338\pi\)
0.548135 + 0.836390i \(0.315338\pi\)
\(654\) 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 \(-0.468506\pi\)
0.0987793 + 0.995109i \(0.468506\pi\)
\(674\) −48890.9 −2.79408
\(675\) 0 0
\(676\) −25270.1 −1.43776
\(677\) 23683.0 1.34448 0.672239 0.740334i \(-0.265333\pi\)
0.672239 + 0.740334i \(0.265333\pi\)
\(678\) 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 \(-0.629258\pi\)
−0.395006 + 0.918679i \(0.629258\pi\)
\(684\) 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 \(-0.434724\pi\)
0.203638 + 0.979046i \(0.434724\pi\)
\(728\) −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 \(-0.710240\pi\)
−0.613502 + 0.789693i \(0.710240\pi\)
\(734\) 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 \(-0.265517\pi\)
0.671811 + 0.740722i \(0.265517\pi\)
\(744\) 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 \(-0.451452\pi\)
0.151928 + 0.988392i \(0.451452\pi\)
\(758\) 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 \(-0.622021\pi\)
−0.374020 + 0.927421i \(0.622021\pi\)
\(774\) 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 \(-0.107036\pi\)
0.943995 + 0.329961i \(0.107036\pi\)
\(788\) −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 \(-0.189639\pi\)
0.827717 + 0.561145i \(0.189639\pi\)
\(798\) 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 \(-0.652212\pi\)
−0.460172 + 0.887830i \(0.652212\pi\)
\(824\) −3670.84 −0.155194
\(825\) 0 0
\(826\) −7097.76 −0.298986
\(827\) −27297.7 −1.14780 −0.573902 0.818924i \(-0.694571\pi\)
−0.573902 + 0.818924i \(0.694571\pi\)
\(828\) 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 \(-0.645038\pi\)
−0.440047 + 0.897975i \(0.645038\pi\)
\(854\) −21695.1 −0.869312
\(855\) 0 0
\(856\) −24305.3 −0.970487
\(857\) 28288.5 1.12756 0.563780 0.825925i \(-0.309347\pi\)
0.563780 + 0.825925i \(0.309347\pi\)
\(858\) 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 \(-0.584769\pi\)
−0.263173 + 0.964749i \(0.584769\pi\)
\(864\) 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 \(-0.379492\pi\)
0.369609 + 0.929187i \(0.379492\pi\)
\(878\) 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 \(-0.749945\pi\)
−0.706985 + 0.707229i \(0.749945\pi\)
\(884\) −2374.65 −0.0903487
\(885\) 0 0
\(886\) 9665.15 0.366487
\(887\) −4568.24 −0.172927 −0.0864636 0.996255i \(-0.527557\pi\)
−0.0864636 + 0.996255i \(0.527557\pi\)
\(888\) 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 \(-0.495476\pi\)
0.0142132 + 0.999899i \(0.495476\pi\)
\(908\) −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 \(-0.413824\pi\)
0.267435 + 0.963576i \(0.413824\pi\)
\(938\) 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 \(-0.555847\pi\)
−0.174550 + 0.984648i \(0.555847\pi\)
\(948\) 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 \(-0.583395\pi\)
−0.259007 + 0.965875i \(0.583395\pi\)
\(954\) 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 \(-0.702229\pi\)
−0.593437 + 0.804881i \(0.702229\pi\)
\(968\) 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 \(-0.423771\pi\)
0.237197 + 0.971462i \(0.423771\pi\)
\(978\) 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 \(-0.694045\pi\)
−0.572547 + 0.819872i \(0.694045\pi\)
\(984\) 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 \(-0.240209\pi\)
0.728520 + 0.685025i \(0.240209\pi\)
\(998\) −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.bh.1.1 4
3.2 odd 2 inner 1575.4.a.bh.1.4 yes 4
5.4 even 2 1575.4.a.bi.1.4 yes 4
15.14 odd 2 1575.4.a.bi.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.1 4 1.1 even 1 trivial
1575.4.a.bh.1.4 yes 4 3.2 odd 2 inner
1575.4.a.bi.1.1 yes 4 15.14 odd 2
1575.4.a.bi.1.4 yes 4 5.4 even 2