Properties

Label 560.4.a.r.1.1
Level $560$
Weight $4$
Character 560.1
Self dual yes
Analytic conductor $33.041$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,4,Mod(1,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("560.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.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: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 560.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.65685 q^{3} -5.00000 q^{5} +7.00000 q^{7} +17.3137 q^{9} +O(q^{10})\) \(q-6.65685 q^{3} -5.00000 q^{5} +7.00000 q^{7} +17.3137 q^{9} -38.2548 q^{11} +19.3431 q^{13} +33.2843 q^{15} -87.2254 q^{17} +44.2254 q^{19} -46.5980 q^{21} -218.167 q^{23} +25.0000 q^{25} +64.4802 q^{27} -46.9411 q^{29} -194.558 q^{31} +254.657 q^{33} -35.0000 q^{35} +366.853 q^{37} -128.765 q^{39} -339.362 q^{41} +226.167 q^{43} -86.5685 q^{45} -11.6762 q^{47} +49.0000 q^{49} +580.647 q^{51} -209.019 q^{53} +191.274 q^{55} -294.402 q^{57} +616.000 q^{59} +320.735 q^{61} +121.196 q^{63} -96.7157 q^{65} -14.5097 q^{67} +1452.30 q^{69} +952.000 q^{71} +824.489 q^{73} -166.421 q^{75} -267.784 q^{77} -156.275 q^{79} -896.706 q^{81} +1036.53 q^{83} +436.127 q^{85} +312.480 q^{87} -170.225 q^{89} +135.402 q^{91} +1295.15 q^{93} -221.127 q^{95} +1059.87 q^{97} -662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9} + 14 q^{11} + 50 q^{13} + 10 q^{15} - 50 q^{17} - 36 q^{19} - 14 q^{21} - 244 q^{23} + 50 q^{25} - 86 q^{27} - 26 q^{29} + 120 q^{31} + 498 q^{33} - 70 q^{35} + 564 q^{37} + 14 q^{39} - 328 q^{41} + 260 q^{43} - 60 q^{45} + 350 q^{47} + 98 q^{49} + 754 q^{51} - 56 q^{53} - 70 q^{55} - 668 q^{57} + 1232 q^{59} + 336 q^{61} + 84 q^{63} - 250 q^{65} + 152 q^{67} + 1332 q^{69} + 1904 q^{71} + 676 q^{73} - 50 q^{75} + 98 q^{77} - 1014 q^{79} - 1454 q^{81} + 376 q^{83} + 250 q^{85} + 410 q^{87} - 216 q^{89} + 350 q^{91} + 2760 q^{93} + 180 q^{95} + 2742 q^{97} - 940 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −6.65685 −1.28111 −0.640556 0.767911i \(-0.721296\pi\)
−0.640556 + 0.767911i \(0.721296\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 17.3137 0.641248
\(10\) 0 0
\(11\) −38.2548 −1.04857 −0.524285 0.851543i \(-0.675667\pi\)
−0.524285 + 0.851543i \(0.675667\pi\)
\(12\) 0 0
\(13\) 19.3431 0.412679 0.206339 0.978480i \(-0.433845\pi\)
0.206339 + 0.978480i \(0.433845\pi\)
\(14\) 0 0
\(15\) 33.2843 0.572931
\(16\) 0 0
\(17\) −87.2254 −1.24443 −0.622214 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(18\) 0 0
\(19\) 44.2254 0.534000 0.267000 0.963697i \(-0.413968\pi\)
0.267000 + 0.963697i \(0.413968\pi\)
\(20\) 0 0
\(21\) −46.5980 −0.484215
\(22\) 0 0
\(23\) −218.167 −1.97786 −0.988932 0.148371i \(-0.952597\pi\)
−0.988932 + 0.148371i \(0.952597\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 64.4802 0.459601
\(28\) 0 0
\(29\) −46.9411 −0.300578 −0.150289 0.988642i \(-0.548020\pi\)
−0.150289 + 0.988642i \(0.548020\pi\)
\(30\) 0 0
\(31\) −194.558 −1.12722 −0.563609 0.826042i \(-0.690587\pi\)
−0.563609 + 0.826042i \(0.690587\pi\)
\(32\) 0 0
\(33\) 254.657 1.34334
\(34\) 0 0
\(35\) −35.0000 −0.169031
\(36\) 0 0
\(37\) 366.853 1.63001 0.815003 0.579457i \(-0.196735\pi\)
0.815003 + 0.579457i \(0.196735\pi\)
\(38\) 0 0
\(39\) −128.765 −0.528688
\(40\) 0 0
\(41\) −339.362 −1.29267 −0.646336 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) 0 0
\(43\) 226.167 0.802095 0.401047 0.916057i \(-0.368646\pi\)
0.401047 + 0.916057i \(0.368646\pi\)
\(44\) 0 0
\(45\) −86.5685 −0.286775
\(46\) 0 0
\(47\) −11.6762 −0.0362372 −0.0181186 0.999836i \(-0.505768\pi\)
−0.0181186 + 0.999836i \(0.505768\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 580.647 1.59425
\(52\) 0 0
\(53\) −209.019 −0.541717 −0.270859 0.962619i \(-0.587308\pi\)
−0.270859 + 0.962619i \(0.587308\pi\)
\(54\) 0 0
\(55\) 191.274 0.468935
\(56\) 0 0
\(57\) −294.402 −0.684114
\(58\) 0 0
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 320.735 0.673212 0.336606 0.941646i \(-0.390721\pi\)
0.336606 + 0.941646i \(0.390721\pi\)
\(62\) 0 0
\(63\) 121.196 0.242369
\(64\) 0 0
\(65\) −96.7157 −0.184556
\(66\) 0 0
\(67\) −14.5097 −0.0264573 −0.0132286 0.999912i \(-0.504211\pi\)
−0.0132286 + 0.999912i \(0.504211\pi\)
\(68\) 0 0
\(69\) 1452.30 2.53387
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) 824.489 1.32191 0.660953 0.750427i \(-0.270152\pi\)
0.660953 + 0.750427i \(0.270152\pi\)
\(74\) 0 0
\(75\) −166.421 −0.256222
\(76\) 0 0
\(77\) −267.784 −0.396322
\(78\) 0 0
\(79\) −156.275 −0.222561 −0.111280 0.993789i \(-0.535495\pi\)
−0.111280 + 0.993789i \(0.535495\pi\)
\(80\) 0 0
\(81\) −896.706 −1.23005
\(82\) 0 0
\(83\) 1036.53 1.37077 0.685384 0.728182i \(-0.259634\pi\)
0.685384 + 0.728182i \(0.259634\pi\)
\(84\) 0 0
\(85\) 436.127 0.556525
\(86\) 0 0
\(87\) 312.480 0.385074
\(88\) 0 0
\(89\) −170.225 −0.202740 −0.101370 0.994849i \(-0.532323\pi\)
−0.101370 + 0.994849i \(0.532323\pi\)
\(90\) 0 0
\(91\) 135.402 0.155978
\(92\) 0 0
\(93\) 1295.15 1.44409
\(94\) 0 0
\(95\) −221.127 −0.238812
\(96\) 0 0
\(97\) 1059.87 1.10942 0.554710 0.832044i \(-0.312829\pi\)
0.554710 + 0.832044i \(0.312829\pi\)
\(98\) 0 0
\(99\) −662.333 −0.672394
\(100\) 0 0
\(101\) −241.833 −0.238251 −0.119125 0.992879i \(-0.538009\pi\)
−0.119125 + 0.992879i \(0.538009\pi\)
\(102\) 0 0
\(103\) 1679.58 1.60673 0.803367 0.595484i \(-0.203040\pi\)
0.803367 + 0.595484i \(0.203040\pi\)
\(104\) 0 0
\(105\) 232.990 0.216547
\(106\) 0 0
\(107\) −1506.88 −1.36146 −0.680728 0.732537i \(-0.738336\pi\)
−0.680728 + 0.732537i \(0.738336\pi\)
\(108\) 0 0
\(109\) −1252.41 −1.10054 −0.550271 0.834986i \(-0.685476\pi\)
−0.550271 + 0.834986i \(0.685476\pi\)
\(110\) 0 0
\(111\) −2442.09 −2.08822
\(112\) 0 0
\(113\) 1370.20 1.14069 0.570345 0.821405i \(-0.306810\pi\)
0.570345 + 0.821405i \(0.306810\pi\)
\(114\) 0 0
\(115\) 1090.83 0.884528
\(116\) 0 0
\(117\) 334.902 0.264630
\(118\) 0 0
\(119\) −610.578 −0.470349
\(120\) 0 0
\(121\) 132.432 0.0994984
\(122\) 0 0
\(123\) 2259.09 1.65606
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1213.49 −0.847873 −0.423936 0.905692i \(-0.639352\pi\)
−0.423936 + 0.905692i \(0.639352\pi\)
\(128\) 0 0
\(129\) −1505.56 −1.02757
\(130\) 0 0
\(131\) 1982.42 1.32217 0.661087 0.750309i \(-0.270096\pi\)
0.661087 + 0.750309i \(0.270096\pi\)
\(132\) 0 0
\(133\) 309.578 0.201833
\(134\) 0 0
\(135\) −322.401 −0.205540
\(136\) 0 0
\(137\) 2210.95 1.37879 0.689394 0.724386i \(-0.257877\pi\)
0.689394 + 0.724386i \(0.257877\pi\)
\(138\) 0 0
\(139\) −528.039 −0.322213 −0.161107 0.986937i \(-0.551506\pi\)
−0.161107 + 0.986937i \(0.551506\pi\)
\(140\) 0 0
\(141\) 77.7267 0.0464239
\(142\) 0 0
\(143\) −739.969 −0.432722
\(144\) 0 0
\(145\) 234.706 0.134422
\(146\) 0 0
\(147\) −326.186 −0.183016
\(148\) 0 0
\(149\) −328.372 −0.180545 −0.0902727 0.995917i \(-0.528774\pi\)
−0.0902727 + 0.995917i \(0.528774\pi\)
\(150\) 0 0
\(151\) −1029.43 −0.554793 −0.277396 0.960756i \(-0.589472\pi\)
−0.277396 + 0.960756i \(0.589472\pi\)
\(152\) 0 0
\(153\) −1510.20 −0.797987
\(154\) 0 0
\(155\) 972.792 0.504107
\(156\) 0 0
\(157\) 525.098 0.266926 0.133463 0.991054i \(-0.457390\pi\)
0.133463 + 0.991054i \(0.457390\pi\)
\(158\) 0 0
\(159\) 1391.41 0.694001
\(160\) 0 0
\(161\) −1527.17 −0.747562
\(162\) 0 0
\(163\) −1002.63 −0.481790 −0.240895 0.970551i \(-0.577441\pi\)
−0.240895 + 0.970551i \(0.577441\pi\)
\(164\) 0 0
\(165\) −1273.28 −0.600758
\(166\) 0 0
\(167\) 359.422 0.166544 0.0832722 0.996527i \(-0.473463\pi\)
0.0832722 + 0.996527i \(0.473463\pi\)
\(168\) 0 0
\(169\) −1822.84 −0.829696
\(170\) 0 0
\(171\) 765.706 0.342427
\(172\) 0 0
\(173\) 3293.65 1.44747 0.723733 0.690080i \(-0.242425\pi\)
0.723733 + 0.690080i \(0.242425\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) 0 0
\(177\) −4100.62 −1.74137
\(178\) 0 0
\(179\) −2978.82 −1.24384 −0.621921 0.783080i \(-0.713647\pi\)
−0.621921 + 0.783080i \(0.713647\pi\)
\(180\) 0 0
\(181\) 1462.31 0.600514 0.300257 0.953858i \(-0.402928\pi\)
0.300257 + 0.953858i \(0.402928\pi\)
\(182\) 0 0
\(183\) −2135.09 −0.862460
\(184\) 0 0
\(185\) −1834.26 −0.728961
\(186\) 0 0
\(187\) 3336.79 1.30487
\(188\) 0 0
\(189\) 451.362 0.173713
\(190\) 0 0
\(191\) 374.923 0.142034 0.0710169 0.997475i \(-0.477376\pi\)
0.0710169 + 0.997475i \(0.477376\pi\)
\(192\) 0 0
\(193\) 733.028 0.273391 0.136696 0.990613i \(-0.456352\pi\)
0.136696 + 0.990613i \(0.456352\pi\)
\(194\) 0 0
\(195\) 643.823 0.236436
\(196\) 0 0
\(197\) −2093.24 −0.757043 −0.378521 0.925593i \(-0.623567\pi\)
−0.378521 + 0.925593i \(0.623567\pi\)
\(198\) 0 0
\(199\) −2865.04 −1.02059 −0.510295 0.860000i \(-0.670464\pi\)
−0.510295 + 0.860000i \(0.670464\pi\)
\(200\) 0 0
\(201\) 96.5887 0.0338948
\(202\) 0 0
\(203\) −328.588 −0.113608
\(204\) 0 0
\(205\) 1696.81 0.578100
\(206\) 0 0
\(207\) −3777.27 −1.26830
\(208\) 0 0
\(209\) −1691.84 −0.559936
\(210\) 0 0
\(211\) −5643.65 −1.84135 −0.920674 0.390331i \(-0.872360\pi\)
−0.920674 + 0.390331i \(0.872360\pi\)
\(212\) 0 0
\(213\) −6337.33 −2.03862
\(214\) 0 0
\(215\) −1130.83 −0.358708
\(216\) 0 0
\(217\) −1361.91 −0.426048
\(218\) 0 0
\(219\) −5488.51 −1.69351
\(220\) 0 0
\(221\) −1687.21 −0.513549
\(222\) 0 0
\(223\) 6369.16 1.91260 0.956302 0.292381i \(-0.0944477\pi\)
0.956302 + 0.292381i \(0.0944477\pi\)
\(224\) 0 0
\(225\) 432.843 0.128250
\(226\) 0 0
\(227\) 1015.67 0.296972 0.148486 0.988914i \(-0.452560\pi\)
0.148486 + 0.988914i \(0.452560\pi\)
\(228\) 0 0
\(229\) 4108.35 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(230\) 0 0
\(231\) 1782.60 0.507733
\(232\) 0 0
\(233\) 608.431 0.171071 0.0855357 0.996335i \(-0.472740\pi\)
0.0855357 + 0.996335i \(0.472740\pi\)
\(234\) 0 0
\(235\) 58.3810 0.0162058
\(236\) 0 0
\(237\) 1040.30 0.285126
\(238\) 0 0
\(239\) 5054.44 1.36797 0.683985 0.729496i \(-0.260245\pi\)
0.683985 + 0.729496i \(0.260245\pi\)
\(240\) 0 0
\(241\) 4.86782 0.00130109 0.000650547 1.00000i \(-0.499793\pi\)
0.000650547 1.00000i \(0.499793\pi\)
\(242\) 0 0
\(243\) 4228.27 1.11623
\(244\) 0 0
\(245\) −245.000 −0.0638877
\(246\) 0 0
\(247\) 855.458 0.220370
\(248\) 0 0
\(249\) −6900.02 −1.75611
\(250\) 0 0
\(251\) 547.921 0.137787 0.0688934 0.997624i \(-0.478053\pi\)
0.0688934 + 0.997624i \(0.478053\pi\)
\(252\) 0 0
\(253\) 8345.92 2.07393
\(254\) 0 0
\(255\) −2903.23 −0.712971
\(256\) 0 0
\(257\) −1774.61 −0.430729 −0.215364 0.976534i \(-0.569094\pi\)
−0.215364 + 0.976534i \(0.569094\pi\)
\(258\) 0 0
\(259\) 2567.97 0.616084
\(260\) 0 0
\(261\) −812.725 −0.192745
\(262\) 0 0
\(263\) 1199.09 0.281138 0.140569 0.990071i \(-0.455107\pi\)
0.140569 + 0.990071i \(0.455107\pi\)
\(264\) 0 0
\(265\) 1045.10 0.242263
\(266\) 0 0
\(267\) 1133.17 0.259733
\(268\) 0 0
\(269\) 3250.29 0.736706 0.368353 0.929686i \(-0.379922\pi\)
0.368353 + 0.929686i \(0.379922\pi\)
\(270\) 0 0
\(271\) 896.143 0.200874 0.100437 0.994943i \(-0.467976\pi\)
0.100437 + 0.994943i \(0.467976\pi\)
\(272\) 0 0
\(273\) −901.352 −0.199825
\(274\) 0 0
\(275\) −956.371 −0.209714
\(276\) 0 0
\(277\) −386.562 −0.0838492 −0.0419246 0.999121i \(-0.513349\pi\)
−0.0419246 + 0.999121i \(0.513349\pi\)
\(278\) 0 0
\(279\) −3368.53 −0.722826
\(280\) 0 0
\(281\) −3335.10 −0.708025 −0.354013 0.935241i \(-0.615183\pi\)
−0.354013 + 0.935241i \(0.615183\pi\)
\(282\) 0 0
\(283\) −5412.26 −1.13684 −0.568419 0.822739i \(-0.692445\pi\)
−0.568419 + 0.822739i \(0.692445\pi\)
\(284\) 0 0
\(285\) 1472.01 0.305945
\(286\) 0 0
\(287\) −2375.54 −0.488584
\(288\) 0 0
\(289\) 2695.27 0.548600
\(290\) 0 0
\(291\) −7055.42 −1.42129
\(292\) 0 0
\(293\) 282.211 0.0562695 0.0281347 0.999604i \(-0.491043\pi\)
0.0281347 + 0.999604i \(0.491043\pi\)
\(294\) 0 0
\(295\) −3080.00 −0.607880
\(296\) 0 0
\(297\) −2466.68 −0.481924
\(298\) 0 0
\(299\) −4220.03 −0.816222
\(300\) 0 0
\(301\) 1583.17 0.303163
\(302\) 0 0
\(303\) 1609.85 0.305226
\(304\) 0 0
\(305\) −1603.68 −0.301069
\(306\) 0 0
\(307\) −1919.67 −0.356878 −0.178439 0.983951i \(-0.557105\pi\)
−0.178439 + 0.983951i \(0.557105\pi\)
\(308\) 0 0
\(309\) −11180.7 −2.05841
\(310\) 0 0
\(311\) −1213.31 −0.221223 −0.110612 0.993864i \(-0.535281\pi\)
−0.110612 + 0.993864i \(0.535281\pi\)
\(312\) 0 0
\(313\) −1434.00 −0.258960 −0.129480 0.991582i \(-0.541331\pi\)
−0.129480 + 0.991582i \(0.541331\pi\)
\(314\) 0 0
\(315\) −605.980 −0.108391
\(316\) 0 0
\(317\) 6496.95 1.15112 0.575560 0.817760i \(-0.304784\pi\)
0.575560 + 0.817760i \(0.304784\pi\)
\(318\) 0 0
\(319\) 1795.72 0.315176
\(320\) 0 0
\(321\) 10031.1 1.74418
\(322\) 0 0
\(323\) −3857.58 −0.664524
\(324\) 0 0
\(325\) 483.579 0.0825357
\(326\) 0 0
\(327\) 8337.11 1.40992
\(328\) 0 0
\(329\) −81.7333 −0.0136964
\(330\) 0 0
\(331\) 9683.88 1.60808 0.804039 0.594576i \(-0.202680\pi\)
0.804039 + 0.594576i \(0.202680\pi\)
\(332\) 0 0
\(333\) 6351.58 1.04524
\(334\) 0 0
\(335\) 72.5483 0.0118321
\(336\) 0 0
\(337\) 29.1319 0.00470895 0.00235447 0.999997i \(-0.499251\pi\)
0.00235447 + 0.999997i \(0.499251\pi\)
\(338\) 0 0
\(339\) −9121.24 −1.46135
\(340\) 0 0
\(341\) 7442.80 1.18197
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −7261.51 −1.13318
\(346\) 0 0
\(347\) 7848.58 1.21422 0.607110 0.794618i \(-0.292329\pi\)
0.607110 + 0.794618i \(0.292329\pi\)
\(348\) 0 0
\(349\) −10269.6 −1.57513 −0.787567 0.616229i \(-0.788659\pi\)
−0.787567 + 0.616229i \(0.788659\pi\)
\(350\) 0 0
\(351\) 1247.25 0.189668
\(352\) 0 0
\(353\) 2799.93 0.422168 0.211084 0.977468i \(-0.432301\pi\)
0.211084 + 0.977468i \(0.432301\pi\)
\(354\) 0 0
\(355\) −4760.00 −0.711647
\(356\) 0 0
\(357\) 4064.53 0.602570
\(358\) 0 0
\(359\) 3163.29 0.465048 0.232524 0.972591i \(-0.425302\pi\)
0.232524 + 0.972591i \(0.425302\pi\)
\(360\) 0 0
\(361\) −4903.11 −0.714844
\(362\) 0 0
\(363\) −881.583 −0.127469
\(364\) 0 0
\(365\) −4122.45 −0.591175
\(366\) 0 0
\(367\) −3182.85 −0.452706 −0.226353 0.974045i \(-0.572680\pi\)
−0.226353 + 0.974045i \(0.572680\pi\)
\(368\) 0 0
\(369\) −5875.62 −0.828923
\(370\) 0 0
\(371\) −1463.14 −0.204750
\(372\) 0 0
\(373\) −2615.14 −0.363021 −0.181510 0.983389i \(-0.558099\pi\)
−0.181510 + 0.983389i \(0.558099\pi\)
\(374\) 0 0
\(375\) 832.107 0.114586
\(376\) 0 0
\(377\) −907.989 −0.124042
\(378\) 0 0
\(379\) 672.434 0.0911362 0.0455681 0.998961i \(-0.485490\pi\)
0.0455681 + 0.998961i \(0.485490\pi\)
\(380\) 0 0
\(381\) 8078.03 1.08622
\(382\) 0 0
\(383\) −1169.86 −0.156075 −0.0780377 0.996950i \(-0.524865\pi\)
−0.0780377 + 0.996950i \(0.524865\pi\)
\(384\) 0 0
\(385\) 1338.92 0.177241
\(386\) 0 0
\(387\) 3915.78 0.514342
\(388\) 0 0
\(389\) −1122.22 −0.146269 −0.0731347 0.997322i \(-0.523300\pi\)
−0.0731347 + 0.997322i \(0.523300\pi\)
\(390\) 0 0
\(391\) 19029.7 2.46131
\(392\) 0 0
\(393\) −13196.7 −1.69385
\(394\) 0 0
\(395\) 781.375 0.0995323
\(396\) 0 0
\(397\) −1985.93 −0.251060 −0.125530 0.992090i \(-0.540063\pi\)
−0.125530 + 0.992090i \(0.540063\pi\)
\(398\) 0 0
\(399\) −2060.81 −0.258571
\(400\) 0 0
\(401\) −4172.38 −0.519597 −0.259799 0.965663i \(-0.583656\pi\)
−0.259799 + 0.965663i \(0.583656\pi\)
\(402\) 0 0
\(403\) −3763.37 −0.465178
\(404\) 0 0
\(405\) 4483.53 0.550095
\(406\) 0 0
\(407\) −14033.9 −1.70918
\(408\) 0 0
\(409\) −11700.8 −1.41459 −0.707295 0.706919i \(-0.750085\pi\)
−0.707295 + 0.706919i \(0.750085\pi\)
\(410\) 0 0
\(411\) −14718.0 −1.76638
\(412\) 0 0
\(413\) 4312.00 0.513752
\(414\) 0 0
\(415\) −5182.64 −0.613026
\(416\) 0 0
\(417\) 3515.08 0.412791
\(418\) 0 0
\(419\) −2733.20 −0.318677 −0.159339 0.987224i \(-0.550936\pi\)
−0.159339 + 0.987224i \(0.550936\pi\)
\(420\) 0 0
\(421\) 13549.4 1.56854 0.784272 0.620417i \(-0.213037\pi\)
0.784272 + 0.620417i \(0.213037\pi\)
\(422\) 0 0
\(423\) −202.158 −0.0232370
\(424\) 0 0
\(425\) −2180.63 −0.248885
\(426\) 0 0
\(427\) 2245.15 0.254450
\(428\) 0 0
\(429\) 4925.86 0.554366
\(430\) 0 0
\(431\) 6429.25 0.718530 0.359265 0.933236i \(-0.383027\pi\)
0.359265 + 0.933236i \(0.383027\pi\)
\(432\) 0 0
\(433\) 8022.03 0.890333 0.445166 0.895448i \(-0.353145\pi\)
0.445166 + 0.895448i \(0.353145\pi\)
\(434\) 0 0
\(435\) −1562.40 −0.172210
\(436\) 0 0
\(437\) −9648.50 −1.05618
\(438\) 0 0
\(439\) 5569.88 0.605549 0.302774 0.953062i \(-0.402087\pi\)
0.302774 + 0.953062i \(0.402087\pi\)
\(440\) 0 0
\(441\) 848.372 0.0916069
\(442\) 0 0
\(443\) 5486.21 0.588392 0.294196 0.955745i \(-0.404948\pi\)
0.294196 + 0.955745i \(0.404948\pi\)
\(444\) 0 0
\(445\) 851.127 0.0906681
\(446\) 0 0
\(447\) 2185.92 0.231299
\(448\) 0 0
\(449\) −7232.67 −0.760203 −0.380101 0.924945i \(-0.624111\pi\)
−0.380101 + 0.924945i \(0.624111\pi\)
\(450\) 0 0
\(451\) 12982.3 1.35546
\(452\) 0 0
\(453\) 6852.76 0.710752
\(454\) 0 0
\(455\) −677.010 −0.0697554
\(456\) 0 0
\(457\) −2900.51 −0.296893 −0.148446 0.988920i \(-0.547427\pi\)
−0.148446 + 0.988920i \(0.547427\pi\)
\(458\) 0 0
\(459\) −5624.31 −0.571940
\(460\) 0 0
\(461\) 6073.57 0.613611 0.306805 0.951772i \(-0.400740\pi\)
0.306805 + 0.951772i \(0.400740\pi\)
\(462\) 0 0
\(463\) 18922.8 1.89939 0.949693 0.313183i \(-0.101395\pi\)
0.949693 + 0.313183i \(0.101395\pi\)
\(464\) 0 0
\(465\) −6475.74 −0.645817
\(466\) 0 0
\(467\) 6776.71 0.671496 0.335748 0.941952i \(-0.391011\pi\)
0.335748 + 0.941952i \(0.391011\pi\)
\(468\) 0 0
\(469\) −101.568 −0.00999991
\(470\) 0 0
\(471\) −3495.50 −0.341962
\(472\) 0 0
\(473\) −8651.96 −0.841052
\(474\) 0 0
\(475\) 1105.63 0.106800
\(476\) 0 0
\(477\) −3618.90 −0.347375
\(478\) 0 0
\(479\) −2397.32 −0.228677 −0.114338 0.993442i \(-0.536475\pi\)
−0.114338 + 0.993442i \(0.536475\pi\)
\(480\) 0 0
\(481\) 7096.09 0.672669
\(482\) 0 0
\(483\) 10166.1 0.957711
\(484\) 0 0
\(485\) −5299.37 −0.496148
\(486\) 0 0
\(487\) −5586.17 −0.519781 −0.259890 0.965638i \(-0.583686\pi\)
−0.259890 + 0.965638i \(0.583686\pi\)
\(488\) 0 0
\(489\) 6674.33 0.617227
\(490\) 0 0
\(491\) −537.392 −0.0493934 −0.0246967 0.999695i \(-0.507862\pi\)
−0.0246967 + 0.999695i \(0.507862\pi\)
\(492\) 0 0
\(493\) 4094.46 0.374047
\(494\) 0 0
\(495\) 3311.67 0.300704
\(496\) 0 0
\(497\) 6664.00 0.601451
\(498\) 0 0
\(499\) −598.965 −0.0537342 −0.0268671 0.999639i \(-0.508553\pi\)
−0.0268671 + 0.999639i \(0.508553\pi\)
\(500\) 0 0
\(501\) −2392.62 −0.213362
\(502\) 0 0
\(503\) −4426.76 −0.392405 −0.196202 0.980563i \(-0.562861\pi\)
−0.196202 + 0.980563i \(0.562861\pi\)
\(504\) 0 0
\(505\) 1209.17 0.106549
\(506\) 0 0
\(507\) 12134.4 1.06293
\(508\) 0 0
\(509\) 17727.7 1.54374 0.771872 0.635779i \(-0.219321\pi\)
0.771872 + 0.635779i \(0.219321\pi\)
\(510\) 0 0
\(511\) 5771.43 0.499634
\(512\) 0 0
\(513\) 2851.66 0.245427
\(514\) 0 0
\(515\) −8397.88 −0.718553
\(516\) 0 0
\(517\) 446.671 0.0379972
\(518\) 0 0
\(519\) −21925.4 −1.85437
\(520\) 0 0
\(521\) 8662.79 0.728453 0.364226 0.931310i \(-0.381333\pi\)
0.364226 + 0.931310i \(0.381333\pi\)
\(522\) 0 0
\(523\) 7770.40 0.649667 0.324833 0.945771i \(-0.394692\pi\)
0.324833 + 0.945771i \(0.394692\pi\)
\(524\) 0 0
\(525\) −1164.95 −0.0968430
\(526\) 0 0
\(527\) 16970.4 1.40274
\(528\) 0 0
\(529\) 35429.6 2.91194
\(530\) 0 0
\(531\) 10665.2 0.871624
\(532\) 0 0
\(533\) −6564.34 −0.533458
\(534\) 0 0
\(535\) 7534.41 0.608861
\(536\) 0 0
\(537\) 19829.6 1.59350
\(538\) 0 0
\(539\) −1874.49 −0.149796
\(540\) 0 0
\(541\) 21641.0 1.71981 0.859906 0.510453i \(-0.170522\pi\)
0.859906 + 0.510453i \(0.170522\pi\)
\(542\) 0 0
\(543\) −9734.41 −0.769325
\(544\) 0 0
\(545\) 6262.05 0.492177
\(546\) 0 0
\(547\) −7489.29 −0.585409 −0.292705 0.956203i \(-0.594555\pi\)
−0.292705 + 0.956203i \(0.594555\pi\)
\(548\) 0 0
\(549\) 5553.11 0.431696
\(550\) 0 0
\(551\) −2075.99 −0.160508
\(552\) 0 0
\(553\) −1093.93 −0.0841201
\(554\) 0 0
\(555\) 12210.4 0.933881
\(556\) 0 0
\(557\) 25297.9 1.92443 0.962214 0.272295i \(-0.0877826\pi\)
0.962214 + 0.272295i \(0.0877826\pi\)
\(558\) 0 0
\(559\) 4374.77 0.331007
\(560\) 0 0
\(561\) −22212.5 −1.67168
\(562\) 0 0
\(563\) −15661.3 −1.17237 −0.586186 0.810177i \(-0.699371\pi\)
−0.586186 + 0.810177i \(0.699371\pi\)
\(564\) 0 0
\(565\) −6851.02 −0.510132
\(566\) 0 0
\(567\) −6276.94 −0.464915
\(568\) 0 0
\(569\) −9982.75 −0.735498 −0.367749 0.929925i \(-0.619871\pi\)
−0.367749 + 0.929925i \(0.619871\pi\)
\(570\) 0 0
\(571\) 11583.6 0.848966 0.424483 0.905436i \(-0.360456\pi\)
0.424483 + 0.905436i \(0.360456\pi\)
\(572\) 0 0
\(573\) −2495.81 −0.181961
\(574\) 0 0
\(575\) −5454.16 −0.395573
\(576\) 0 0
\(577\) −595.378 −0.0429565 −0.0214783 0.999769i \(-0.506837\pi\)
−0.0214783 + 0.999769i \(0.506837\pi\)
\(578\) 0 0
\(579\) −4879.66 −0.350245
\(580\) 0 0
\(581\) 7255.70 0.518102
\(582\) 0 0
\(583\) 7996.00 0.568028
\(584\) 0 0
\(585\) −1674.51 −0.118346
\(586\) 0 0
\(587\) 15750.3 1.10747 0.553736 0.832693i \(-0.313202\pi\)
0.553736 + 0.832693i \(0.313202\pi\)
\(588\) 0 0
\(589\) −8604.42 −0.601934
\(590\) 0 0
\(591\) 13934.4 0.969856
\(592\) 0 0
\(593\) −417.878 −0.0289379 −0.0144690 0.999895i \(-0.504606\pi\)
−0.0144690 + 0.999895i \(0.504606\pi\)
\(594\) 0 0
\(595\) 3052.89 0.210347
\(596\) 0 0
\(597\) 19072.2 1.30749
\(598\) 0 0
\(599\) 19997.3 1.36406 0.682028 0.731326i \(-0.261098\pi\)
0.682028 + 0.731326i \(0.261098\pi\)
\(600\) 0 0
\(601\) −15992.6 −1.08545 −0.542723 0.839912i \(-0.682607\pi\)
−0.542723 + 0.839912i \(0.682607\pi\)
\(602\) 0 0
\(603\) −251.216 −0.0169657
\(604\) 0 0
\(605\) −662.162 −0.0444970
\(606\) 0 0
\(607\) −14159.2 −0.946793 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(608\) 0 0
\(609\) 2187.36 0.145544
\(610\) 0 0
\(611\) −225.854 −0.0149543
\(612\) 0 0
\(613\) −4629.41 −0.305025 −0.152512 0.988302i \(-0.548736\pi\)
−0.152512 + 0.988302i \(0.548736\pi\)
\(614\) 0 0
\(615\) −11295.4 −0.740611
\(616\) 0 0
\(617\) −23165.3 −1.51151 −0.755753 0.654857i \(-0.772729\pi\)
−0.755753 + 0.654857i \(0.772729\pi\)
\(618\) 0 0
\(619\) −12370.6 −0.803258 −0.401629 0.915803i \(-0.631556\pi\)
−0.401629 + 0.915803i \(0.631556\pi\)
\(620\) 0 0
\(621\) −14067.4 −0.909028
\(622\) 0 0
\(623\) −1191.58 −0.0766285
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 11262.3 0.717341
\(628\) 0 0
\(629\) −31998.9 −2.02842
\(630\) 0 0
\(631\) −13980.2 −0.882002 −0.441001 0.897507i \(-0.645377\pi\)
−0.441001 + 0.897507i \(0.645377\pi\)
\(632\) 0 0
\(633\) 37568.9 2.35897
\(634\) 0 0
\(635\) 6067.45 0.379180
\(636\) 0 0
\(637\) 947.814 0.0589541
\(638\) 0 0
\(639\) 16482.7 1.02041
\(640\) 0 0
\(641\) −16060.9 −0.989655 −0.494828 0.868991i \(-0.664769\pi\)
−0.494828 + 0.868991i \(0.664769\pi\)
\(642\) 0 0
\(643\) 4502.17 0.276125 0.138063 0.990424i \(-0.455913\pi\)
0.138063 + 0.990424i \(0.455913\pi\)
\(644\) 0 0
\(645\) 7527.79 0.459545
\(646\) 0 0
\(647\) −29414.8 −1.78735 −0.893675 0.448715i \(-0.851882\pi\)
−0.893675 + 0.448715i \(0.851882\pi\)
\(648\) 0 0
\(649\) −23565.0 −1.42528
\(650\) 0 0
\(651\) 9066.03 0.545815
\(652\) 0 0
\(653\) 13013.6 0.779882 0.389941 0.920840i \(-0.372495\pi\)
0.389941 + 0.920840i \(0.372495\pi\)
\(654\) 0 0
\(655\) −9912.09 −0.591294
\(656\) 0 0
\(657\) 14275.0 0.847671
\(658\) 0 0
\(659\) −23474.2 −1.38759 −0.693797 0.720171i \(-0.744063\pi\)
−0.693797 + 0.720171i \(0.744063\pi\)
\(660\) 0 0
\(661\) −9266.36 −0.545264 −0.272632 0.962118i \(-0.587894\pi\)
−0.272632 + 0.962118i \(0.587894\pi\)
\(662\) 0 0
\(663\) 11231.5 0.657914
\(664\) 0 0
\(665\) −1547.89 −0.0902625
\(666\) 0 0
\(667\) 10241.0 0.594501
\(668\) 0 0
\(669\) −42398.6 −2.45026
\(670\) 0 0
\(671\) −12269.7 −0.705909
\(672\) 0 0
\(673\) −25067.2 −1.43576 −0.717882 0.696164i \(-0.754888\pi\)
−0.717882 + 0.696164i \(0.754888\pi\)
\(674\) 0 0
\(675\) 1612.01 0.0919202
\(676\) 0 0
\(677\) −22409.6 −1.27219 −0.636093 0.771613i \(-0.719450\pi\)
−0.636093 + 0.771613i \(0.719450\pi\)
\(678\) 0 0
\(679\) 7419.11 0.419322
\(680\) 0 0
\(681\) −6761.20 −0.380455
\(682\) 0 0
\(683\) 8757.53 0.490626 0.245313 0.969444i \(-0.421109\pi\)
0.245313 + 0.969444i \(0.421109\pi\)
\(684\) 0 0
\(685\) −11054.7 −0.616613
\(686\) 0 0
\(687\) −27348.7 −1.51880
\(688\) 0 0
\(689\) −4043.09 −0.223555
\(690\) 0 0
\(691\) −8468.42 −0.466214 −0.233107 0.972451i \(-0.574889\pi\)
−0.233107 + 0.972451i \(0.574889\pi\)
\(692\) 0 0
\(693\) −4636.33 −0.254141
\(694\) 0 0
\(695\) 2640.19 0.144098
\(696\) 0 0
\(697\) 29601.0 1.60864
\(698\) 0 0
\(699\) −4050.23 −0.219162
\(700\) 0 0
\(701\) 15996.9 0.861906 0.430953 0.902374i \(-0.358177\pi\)
0.430953 + 0.902374i \(0.358177\pi\)
\(702\) 0 0
\(703\) 16224.2 0.870423
\(704\) 0 0
\(705\) −388.633 −0.0207614
\(706\) 0 0
\(707\) −1692.83 −0.0900503
\(708\) 0 0
\(709\) 19903.0 1.05426 0.527131 0.849784i \(-0.323268\pi\)
0.527131 + 0.849784i \(0.323268\pi\)
\(710\) 0 0
\(711\) −2705.70 −0.142717
\(712\) 0 0
\(713\) 42446.1 2.22948
\(714\) 0 0
\(715\) 3699.84 0.193519
\(716\) 0 0
\(717\) −33646.7 −1.75252
\(718\) 0 0
\(719\) 11073.1 0.574347 0.287174 0.957879i \(-0.407284\pi\)
0.287174 + 0.957879i \(0.407284\pi\)
\(720\) 0 0
\(721\) 11757.0 0.607288
\(722\) 0 0
\(723\) −32.4043 −0.00166685
\(724\) 0 0
\(725\) −1173.53 −0.0601155
\(726\) 0 0
\(727\) −31652.7 −1.61476 −0.807382 0.590029i \(-0.799116\pi\)
−0.807382 + 0.590029i \(0.799116\pi\)
\(728\) 0 0
\(729\) −3935.94 −0.199967
\(730\) 0 0
\(731\) −19727.5 −0.998149
\(732\) 0 0
\(733\) 16958.3 0.854528 0.427264 0.904127i \(-0.359477\pi\)
0.427264 + 0.904127i \(0.359477\pi\)
\(734\) 0 0
\(735\) 1630.93 0.0818473
\(736\) 0 0
\(737\) 555.065 0.0277423
\(738\) 0 0
\(739\) 11616.6 0.578245 0.289123 0.957292i \(-0.406636\pi\)
0.289123 + 0.957292i \(0.406636\pi\)
\(740\) 0 0
\(741\) −5694.66 −0.282319
\(742\) 0 0
\(743\) −15928.0 −0.786464 −0.393232 0.919439i \(-0.628643\pi\)
−0.393232 + 0.919439i \(0.628643\pi\)
\(744\) 0 0
\(745\) 1641.86 0.0807423
\(746\) 0 0
\(747\) 17946.1 0.879003
\(748\) 0 0
\(749\) −10548.2 −0.514582
\(750\) 0 0
\(751\) −25571.9 −1.24252 −0.621260 0.783604i \(-0.713379\pi\)
−0.621260 + 0.783604i \(0.713379\pi\)
\(752\) 0 0
\(753\) −3647.43 −0.176520
\(754\) 0 0
\(755\) 5147.14 0.248111
\(756\) 0 0
\(757\) 6202.41 0.297794 0.148897 0.988853i \(-0.452428\pi\)
0.148897 + 0.988853i \(0.452428\pi\)
\(758\) 0 0
\(759\) −55557.6 −2.65693
\(760\) 0 0
\(761\) −29199.1 −1.39089 −0.695444 0.718580i \(-0.744792\pi\)
−0.695444 + 0.718580i \(0.744792\pi\)
\(762\) 0 0
\(763\) −8766.87 −0.415966
\(764\) 0 0
\(765\) 7550.98 0.356871
\(766\) 0 0
\(767\) 11915.4 0.560938
\(768\) 0 0
\(769\) 21838.2 1.02407 0.512033 0.858966i \(-0.328892\pi\)
0.512033 + 0.858966i \(0.328892\pi\)
\(770\) 0 0
\(771\) 11813.3 0.551812
\(772\) 0 0
\(773\) −25544.8 −1.18859 −0.594296 0.804246i \(-0.702569\pi\)
−0.594296 + 0.804246i \(0.702569\pi\)
\(774\) 0 0
\(775\) −4863.96 −0.225443
\(776\) 0 0
\(777\) −17094.6 −0.789273
\(778\) 0 0
\(779\) −15008.4 −0.690286
\(780\) 0 0
\(781\) −36418.6 −1.66858
\(782\) 0 0
\(783\) −3026.77 −0.138146
\(784\) 0 0
\(785\) −2625.49 −0.119373
\(786\) 0 0
\(787\) 37223.2 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(788\) 0 0
\(789\) −7982.19 −0.360169
\(790\) 0 0
\(791\) 9591.42 0.431140
\(792\) 0 0
\(793\) 6204.03 0.277820
\(794\) 0 0
\(795\) −6957.06 −0.310366
\(796\) 0 0
\(797\) −40384.6 −1.79485 −0.897425 0.441168i \(-0.854564\pi\)
−0.897425 + 0.441168i \(0.854564\pi\)
\(798\) 0 0
\(799\) 1018.46 0.0450945
\(800\) 0 0
\(801\) −2947.23 −0.130007
\(802\) 0 0
\(803\) −31540.7 −1.38611
\(804\) 0 0
\(805\) 7635.83 0.334320
\(806\) 0 0
\(807\) −21636.7 −0.943803
\(808\) 0 0
\(809\) −1955.76 −0.0849948 −0.0424974 0.999097i \(-0.513531\pi\)
−0.0424974 + 0.999097i \(0.513531\pi\)
\(810\) 0 0
\(811\) 34301.8 1.48520 0.742600 0.669735i \(-0.233592\pi\)
0.742600 + 0.669735i \(0.233592\pi\)
\(812\) 0 0
\(813\) −5965.49 −0.257342
\(814\) 0 0
\(815\) 5013.13 0.215463
\(816\) 0 0
\(817\) 10002.3 0.428319
\(818\) 0 0
\(819\) 2344.31 0.100021
\(820\) 0 0
\(821\) −13665.6 −0.580918 −0.290459 0.956887i \(-0.593808\pi\)
−0.290459 + 0.956887i \(0.593808\pi\)
\(822\) 0 0
\(823\) 21519.5 0.911449 0.455724 0.890121i \(-0.349380\pi\)
0.455724 + 0.890121i \(0.349380\pi\)
\(824\) 0 0
\(825\) 6366.42 0.268667
\(826\) 0 0
\(827\) −35220.6 −1.48094 −0.740471 0.672088i \(-0.765398\pi\)
−0.740471 + 0.672088i \(0.765398\pi\)
\(828\) 0 0
\(829\) 31365.5 1.31408 0.657039 0.753857i \(-0.271809\pi\)
0.657039 + 0.753857i \(0.271809\pi\)
\(830\) 0 0
\(831\) 2573.28 0.107420
\(832\) 0 0
\(833\) −4274.04 −0.177775
\(834\) 0 0
\(835\) −1797.11 −0.0744810
\(836\) 0 0
\(837\) −12545.2 −0.518070
\(838\) 0 0
\(839\) 28287.1 1.16398 0.581990 0.813196i \(-0.302274\pi\)
0.581990 + 0.813196i \(0.302274\pi\)
\(840\) 0 0
\(841\) −22185.5 −0.909653
\(842\) 0 0
\(843\) 22201.2 0.907060
\(844\) 0 0
\(845\) 9114.21 0.371051
\(846\) 0 0
\(847\) 927.026 0.0376068
\(848\) 0 0
\(849\) 36028.6 1.45642
\(850\) 0 0
\(851\) −80035.0 −3.22393
\(852\) 0 0
\(853\) 9405.41 0.377533 0.188766 0.982022i \(-0.439551\pi\)
0.188766 + 0.982022i \(0.439551\pi\)
\(854\) 0 0
\(855\) −3828.53 −0.153138
\(856\) 0 0
\(857\) −27966.9 −1.11474 −0.557369 0.830265i \(-0.688189\pi\)
−0.557369 + 0.830265i \(0.688189\pi\)
\(858\) 0 0
\(859\) −6281.11 −0.249486 −0.124743 0.992189i \(-0.539811\pi\)
−0.124743 + 0.992189i \(0.539811\pi\)
\(860\) 0 0
\(861\) 15813.6 0.625931
\(862\) 0 0
\(863\) 4757.13 0.187642 0.0938208 0.995589i \(-0.470092\pi\)
0.0938208 + 0.995589i \(0.470092\pi\)
\(864\) 0 0
\(865\) −16468.3 −0.647327
\(866\) 0 0
\(867\) −17942.0 −0.702818
\(868\) 0 0
\(869\) 5978.28 0.233371
\(870\) 0 0
\(871\) −280.663 −0.0109184
\(872\) 0 0
\(873\) 18350.3 0.711414
\(874\) 0 0
\(875\) −875.000 −0.0338062
\(876\) 0 0
\(877\) −30240.5 −1.16437 −0.582184 0.813057i \(-0.697802\pi\)
−0.582184 + 0.813057i \(0.697802\pi\)
\(878\) 0 0
\(879\) −1878.64 −0.0720875
\(880\) 0 0
\(881\) 44875.5 1.71611 0.858056 0.513556i \(-0.171672\pi\)
0.858056 + 0.513556i \(0.171672\pi\)
\(882\) 0 0
\(883\) −4892.13 −0.186448 −0.0932238 0.995645i \(-0.529717\pi\)
−0.0932238 + 0.995645i \(0.529717\pi\)
\(884\) 0 0
\(885\) 20503.1 0.778762
\(886\) 0 0
\(887\) −1761.40 −0.0666765 −0.0333382 0.999444i \(-0.510614\pi\)
−0.0333382 + 0.999444i \(0.510614\pi\)
\(888\) 0 0
\(889\) −8494.43 −0.320466
\(890\) 0 0
\(891\) 34303.3 1.28979
\(892\) 0 0
\(893\) −516.384 −0.0193507
\(894\) 0 0
\(895\) 14894.1 0.556263
\(896\) 0 0
\(897\) 28092.1 1.04567
\(898\) 0 0
\(899\) 9132.79 0.338816
\(900\) 0 0
\(901\) 18231.8 0.674128
\(902\) 0 0
\(903\) −10538.9 −0.388386
\(904\) 0 0
\(905\) −7311.57 −0.268558
\(906\) 0 0
\(907\) −23689.1 −0.867238 −0.433619 0.901096i \(-0.642764\pi\)
−0.433619 + 0.901096i \(0.642764\pi\)
\(908\) 0 0
\(909\) −4187.03 −0.152778
\(910\) 0 0
\(911\) 13877.3 0.504692 0.252346 0.967637i \(-0.418798\pi\)
0.252346 + 0.967637i \(0.418798\pi\)
\(912\) 0 0
\(913\) −39652.2 −1.43735
\(914\) 0 0
\(915\) 10675.4 0.385704
\(916\) 0 0
\(917\) 13876.9 0.499735
\(918\) 0 0
\(919\) −14331.6 −0.514426 −0.257213 0.966355i \(-0.582804\pi\)
−0.257213 + 0.966355i \(0.582804\pi\)
\(920\) 0 0
\(921\) 12779.0 0.457201
\(922\) 0 0
\(923\) 18414.7 0.656692
\(924\) 0 0
\(925\) 9171.32 0.326001
\(926\) 0 0
\(927\) 29079.7 1.03032
\(928\) 0 0
\(929\) 16668.4 0.588668 0.294334 0.955703i \(-0.404902\pi\)
0.294334 + 0.955703i \(0.404902\pi\)
\(930\) 0 0
\(931\) 2167.04 0.0762857
\(932\) 0 0
\(933\) 8076.82 0.283412
\(934\) 0 0
\(935\) −16684.0 −0.583555
\(936\) 0 0
\(937\) 30384.9 1.05937 0.529685 0.848194i \(-0.322310\pi\)
0.529685 + 0.848194i \(0.322310\pi\)
\(938\) 0 0
\(939\) 9545.94 0.331757
\(940\) 0 0
\(941\) −1196.35 −0.0414452 −0.0207226 0.999785i \(-0.506597\pi\)
−0.0207226 + 0.999785i \(0.506597\pi\)
\(942\) 0 0
\(943\) 74037.5 2.55673
\(944\) 0 0
\(945\) −2256.81 −0.0776867
\(946\) 0 0
\(947\) −1788.41 −0.0613681 −0.0306840 0.999529i \(-0.509769\pi\)
−0.0306840 + 0.999529i \(0.509769\pi\)
\(948\) 0 0
\(949\) 15948.2 0.545523
\(950\) 0 0
\(951\) −43249.2 −1.47471
\(952\) 0 0
\(953\) 8578.60 0.291593 0.145796 0.989315i \(-0.453426\pi\)
0.145796 + 0.989315i \(0.453426\pi\)
\(954\) 0 0
\(955\) −1874.61 −0.0635194
\(956\) 0 0
\(957\) −11953.9 −0.403776
\(958\) 0 0
\(959\) 15476.6 0.521133
\(960\) 0 0
\(961\) 8061.99 0.270618
\(962\) 0 0
\(963\) −26089.7 −0.873031
\(964\) 0 0
\(965\) −3665.14 −0.122264
\(966\) 0 0
\(967\) 55459.3 1.84431 0.922156 0.386818i \(-0.126426\pi\)
0.922156 + 0.386818i \(0.126426\pi\)
\(968\) 0 0
\(969\) 25679.3 0.851330
\(970\) 0 0
\(971\) −22047.3 −0.728662 −0.364331 0.931270i \(-0.618702\pi\)
−0.364331 + 0.931270i \(0.618702\pi\)
\(972\) 0 0
\(973\) −3696.27 −0.121785
\(974\) 0 0
\(975\) −3219.11 −0.105738
\(976\) 0 0
\(977\) 14402.3 0.471617 0.235809 0.971800i \(-0.424226\pi\)
0.235809 + 0.971800i \(0.424226\pi\)
\(978\) 0 0
\(979\) 6511.94 0.212587
\(980\) 0 0
\(981\) −21683.9 −0.705721
\(982\) 0 0
\(983\) −7817.11 −0.253639 −0.126819 0.991926i \(-0.540477\pi\)
−0.126819 + 0.991926i \(0.540477\pi\)
\(984\) 0 0
\(985\) 10466.2 0.338560
\(986\) 0 0
\(987\) 544.087 0.0175466
\(988\) 0 0
\(989\) −49342.0 −1.58643
\(990\) 0 0
\(991\) −24501.6 −0.785386 −0.392693 0.919670i \(-0.628457\pi\)
−0.392693 + 0.919670i \(0.628457\pi\)
\(992\) 0 0
\(993\) −64464.2 −2.06013
\(994\) 0 0
\(995\) 14325.2 0.456421
\(996\) 0 0
\(997\) −50696.0 −1.61039 −0.805195 0.593010i \(-0.797940\pi\)
−0.805195 + 0.593010i \(0.797940\pi\)
\(998\) 0 0
\(999\) 23654.8 0.749152
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 560.4.a.r.1.1 2
4.3 odd 2 35.4.a.b.1.1 2
8.3 odd 2 2240.4.a.bn.1.1 2
8.5 even 2 2240.4.a.bo.1.2 2
12.11 even 2 315.4.a.f.1.2 2
20.3 even 4 175.4.b.c.99.2 4
20.7 even 4 175.4.b.c.99.3 4
20.19 odd 2 175.4.a.c.1.2 2
28.3 even 6 245.4.e.i.226.2 4
28.11 odd 6 245.4.e.h.226.2 4
28.19 even 6 245.4.e.i.116.2 4
28.23 odd 6 245.4.e.h.116.2 4
28.27 even 2 245.4.a.k.1.1 2
60.59 even 2 1575.4.a.z.1.1 2
84.83 odd 2 2205.4.a.u.1.2 2
140.139 even 2 1225.4.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 4.3 odd 2
175.4.a.c.1.2 2 20.19 odd 2
175.4.b.c.99.2 4 20.3 even 4
175.4.b.c.99.3 4 20.7 even 4
245.4.a.k.1.1 2 28.27 even 2
245.4.e.h.116.2 4 28.23 odd 6
245.4.e.h.226.2 4 28.11 odd 6
245.4.e.i.116.2 4 28.19 even 6
245.4.e.i.226.2 4 28.3 even 6
315.4.a.f.1.2 2 12.11 even 2
560.4.a.r.1.1 2 1.1 even 1 trivial
1225.4.a.m.1.2 2 140.139 even 2
1575.4.a.z.1.1 2 60.59 even 2
2205.4.a.u.1.2 2 84.83 odd 2
2240.4.a.bn.1.1 2 8.3 odd 2
2240.4.a.bo.1.2 2 8.5 even 2