Properties

Label 1856.4.a.y.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $5$
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] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.27399\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.87991 q^{3} +16.8209 q^{5} +5.21997 q^{7} +70.6126 q^{9} +O(q^{10})\) \(q-9.87991 q^{3} +16.8209 q^{5} +5.21997 q^{7} +70.6126 q^{9} +8.55158 q^{11} +11.3429 q^{13} -166.189 q^{15} +68.4740 q^{17} -6.93014 q^{19} -51.5728 q^{21} -132.042 q^{23} +157.944 q^{25} -430.889 q^{27} +29.0000 q^{29} -0.419319 q^{31} -84.4888 q^{33} +87.8048 q^{35} -395.483 q^{37} -112.067 q^{39} -447.209 q^{41} -184.132 q^{43} +1187.77 q^{45} -97.2612 q^{47} -315.752 q^{49} -676.516 q^{51} +209.547 q^{53} +143.845 q^{55} +68.4691 q^{57} -45.9651 q^{59} -427.655 q^{61} +368.596 q^{63} +190.798 q^{65} +405.055 q^{67} +1304.57 q^{69} -557.971 q^{71} -381.988 q^{73} -1560.47 q^{75} +44.6390 q^{77} +577.208 q^{79} +2350.60 q^{81} +353.745 q^{83} +1151.80 q^{85} -286.517 q^{87} -277.871 q^{89} +59.2095 q^{91} +4.14283 q^{93} -116.571 q^{95} +677.917 q^{97} +603.849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 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
\(3\) −9.87991 −1.90139 −0.950695 0.310128i \(-0.899628\pi\)
−0.950695 + 0.310128i \(0.899628\pi\)
\(4\) 0 0
\(5\) 16.8209 1.50451 0.752255 0.658872i \(-0.228966\pi\)
0.752255 + 0.658872i \(0.228966\pi\)
\(6\) 0 0
\(7\) 5.21997 0.281852 0.140926 0.990020i \(-0.454992\pi\)
0.140926 + 0.990020i \(0.454992\pi\)
\(8\) 0 0
\(9\) 70.6126 2.61528
\(10\) 0 0
\(11\) 8.55158 0.234400 0.117200 0.993108i \(-0.462608\pi\)
0.117200 + 0.993108i \(0.462608\pi\)
\(12\) 0 0
\(13\) 11.3429 0.241996 0.120998 0.992653i \(-0.461391\pi\)
0.120998 + 0.992653i \(0.461391\pi\)
\(14\) 0 0
\(15\) −166.189 −2.86066
\(16\) 0 0
\(17\) 68.4740 0.976904 0.488452 0.872591i \(-0.337562\pi\)
0.488452 + 0.872591i \(0.337562\pi\)
\(18\) 0 0
\(19\) −6.93014 −0.0836780 −0.0418390 0.999124i \(-0.513322\pi\)
−0.0418390 + 0.999124i \(0.513322\pi\)
\(20\) 0 0
\(21\) −51.5728 −0.535910
\(22\) 0 0
\(23\) −132.042 −1.19708 −0.598538 0.801095i \(-0.704251\pi\)
−0.598538 + 0.801095i \(0.704251\pi\)
\(24\) 0 0
\(25\) 157.944 1.26355
\(26\) 0 0
\(27\) −430.889 −3.07128
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −0.419319 −0.00242941 −0.00121471 0.999999i \(-0.500387\pi\)
−0.00121471 + 0.999999i \(0.500387\pi\)
\(32\) 0 0
\(33\) −84.4888 −0.445685
\(34\) 0 0
\(35\) 87.8048 0.424049
\(36\) 0 0
\(37\) −395.483 −1.75722 −0.878609 0.477542i \(-0.841528\pi\)
−0.878609 + 0.477542i \(0.841528\pi\)
\(38\) 0 0
\(39\) −112.067 −0.460128
\(40\) 0 0
\(41\) −447.209 −1.70347 −0.851736 0.523971i \(-0.824450\pi\)
−0.851736 + 0.523971i \(0.824450\pi\)
\(42\) 0 0
\(43\) −184.132 −0.653020 −0.326510 0.945194i \(-0.605873\pi\)
−0.326510 + 0.945194i \(0.605873\pi\)
\(44\) 0 0
\(45\) 1187.77 3.93472
\(46\) 0 0
\(47\) −97.2612 −0.301851 −0.150926 0.988545i \(-0.548225\pi\)
−0.150926 + 0.988545i \(0.548225\pi\)
\(48\) 0 0
\(49\) −315.752 −0.920559
\(50\) 0 0
\(51\) −676.516 −1.85748
\(52\) 0 0
\(53\) 209.547 0.543086 0.271543 0.962426i \(-0.412466\pi\)
0.271543 + 0.962426i \(0.412466\pi\)
\(54\) 0 0
\(55\) 143.845 0.352657
\(56\) 0 0
\(57\) 68.4691 0.159105
\(58\) 0 0
\(59\) −45.9651 −0.101426 −0.0507131 0.998713i \(-0.516149\pi\)
−0.0507131 + 0.998713i \(0.516149\pi\)
\(60\) 0 0
\(61\) −427.655 −0.897634 −0.448817 0.893624i \(-0.648154\pi\)
−0.448817 + 0.893624i \(0.648154\pi\)
\(62\) 0 0
\(63\) 368.596 0.737122
\(64\) 0 0
\(65\) 190.798 0.364085
\(66\) 0 0
\(67\) 405.055 0.738588 0.369294 0.929313i \(-0.379600\pi\)
0.369294 + 0.929313i \(0.379600\pi\)
\(68\) 0 0
\(69\) 1304.57 2.27611
\(70\) 0 0
\(71\) −557.971 −0.932662 −0.466331 0.884610i \(-0.654424\pi\)
−0.466331 + 0.884610i \(0.654424\pi\)
\(72\) 0 0
\(73\) −381.988 −0.612443 −0.306222 0.951960i \(-0.599065\pi\)
−0.306222 + 0.951960i \(0.599065\pi\)
\(74\) 0 0
\(75\) −1560.47 −2.40250
\(76\) 0 0
\(77\) 44.6390 0.0660660
\(78\) 0 0
\(79\) 577.208 0.822038 0.411019 0.911627i \(-0.365173\pi\)
0.411019 + 0.911627i \(0.365173\pi\)
\(80\) 0 0
\(81\) 2350.60 3.22442
\(82\) 0 0
\(83\) 353.745 0.467813 0.233907 0.972259i \(-0.424849\pi\)
0.233907 + 0.972259i \(0.424849\pi\)
\(84\) 0 0
\(85\) 1151.80 1.46976
\(86\) 0 0
\(87\) −286.517 −0.353079
\(88\) 0 0
\(89\) −277.871 −0.330947 −0.165473 0.986214i \(-0.552915\pi\)
−0.165473 + 0.986214i \(0.552915\pi\)
\(90\) 0 0
\(91\) 59.2095 0.0682070
\(92\) 0 0
\(93\) 4.14283 0.00461926
\(94\) 0 0
\(95\) −116.571 −0.125894
\(96\) 0 0
\(97\) 677.917 0.709609 0.354804 0.934941i \(-0.384547\pi\)
0.354804 + 0.934941i \(0.384547\pi\)
\(98\) 0 0
\(99\) 603.849 0.613022
\(100\) 0 0
\(101\) −567.816 −0.559404 −0.279702 0.960087i \(-0.590236\pi\)
−0.279702 + 0.960087i \(0.590236\pi\)
\(102\) 0 0
\(103\) 319.205 0.305362 0.152681 0.988276i \(-0.451209\pi\)
0.152681 + 0.988276i \(0.451209\pi\)
\(104\) 0 0
\(105\) −867.503 −0.806282
\(106\) 0 0
\(107\) 79.6547 0.0719674 0.0359837 0.999352i \(-0.488544\pi\)
0.0359837 + 0.999352i \(0.488544\pi\)
\(108\) 0 0
\(109\) 1708.43 1.50126 0.750632 0.660721i \(-0.229749\pi\)
0.750632 + 0.660721i \(0.229749\pi\)
\(110\) 0 0
\(111\) 3907.34 3.34116
\(112\) 0 0
\(113\) 1050.31 0.874383 0.437191 0.899369i \(-0.355973\pi\)
0.437191 + 0.899369i \(0.355973\pi\)
\(114\) 0 0
\(115\) −2221.08 −1.80101
\(116\) 0 0
\(117\) 800.950 0.632887
\(118\) 0 0
\(119\) 357.432 0.275342
\(120\) 0 0
\(121\) −1257.87 −0.945057
\(122\) 0 0
\(123\) 4418.39 3.23896
\(124\) 0 0
\(125\) 554.143 0.396513
\(126\) 0 0
\(127\) 366.926 0.256373 0.128187 0.991750i \(-0.459084\pi\)
0.128187 + 0.991750i \(0.459084\pi\)
\(128\) 0 0
\(129\) 1819.21 1.24165
\(130\) 0 0
\(131\) −2310.86 −1.54123 −0.770614 0.637302i \(-0.780050\pi\)
−0.770614 + 0.637302i \(0.780050\pi\)
\(132\) 0 0
\(133\) −36.1751 −0.0235848
\(134\) 0 0
\(135\) −7247.95 −4.62077
\(136\) 0 0
\(137\) 1899.65 1.18466 0.592329 0.805696i \(-0.298209\pi\)
0.592329 + 0.805696i \(0.298209\pi\)
\(138\) 0 0
\(139\) −1309.49 −0.799061 −0.399531 0.916720i \(-0.630827\pi\)
−0.399531 + 0.916720i \(0.630827\pi\)
\(140\) 0 0
\(141\) 960.932 0.573937
\(142\) 0 0
\(143\) 96.9994 0.0567238
\(144\) 0 0
\(145\) 487.807 0.279380
\(146\) 0 0
\(147\) 3119.60 1.75034
\(148\) 0 0
\(149\) −1782.98 −0.980320 −0.490160 0.871632i \(-0.663062\pi\)
−0.490160 + 0.871632i \(0.663062\pi\)
\(150\) 0 0
\(151\) −1631.96 −0.879515 −0.439758 0.898116i \(-0.644936\pi\)
−0.439758 + 0.898116i \(0.644936\pi\)
\(152\) 0 0
\(153\) 4835.12 2.55488
\(154\) 0 0
\(155\) −7.05333 −0.00365508
\(156\) 0 0
\(157\) −852.817 −0.433517 −0.216759 0.976225i \(-0.569548\pi\)
−0.216759 + 0.976225i \(0.569548\pi\)
\(158\) 0 0
\(159\) −2070.31 −1.03262
\(160\) 0 0
\(161\) −689.257 −0.337398
\(162\) 0 0
\(163\) −3280.24 −1.57625 −0.788123 0.615518i \(-0.788947\pi\)
−0.788123 + 0.615518i \(0.788947\pi\)
\(164\) 0 0
\(165\) −1421.18 −0.670538
\(166\) 0 0
\(167\) −1682.26 −0.779504 −0.389752 0.920920i \(-0.627439\pi\)
−0.389752 + 0.920920i \(0.627439\pi\)
\(168\) 0 0
\(169\) −2068.34 −0.941438
\(170\) 0 0
\(171\) −489.355 −0.218842
\(172\) 0 0
\(173\) 1590.22 0.698854 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(174\) 0 0
\(175\) 824.462 0.356134
\(176\) 0 0
\(177\) 454.131 0.192851
\(178\) 0 0
\(179\) 1794.60 0.749356 0.374678 0.927155i \(-0.377753\pi\)
0.374678 + 0.927155i \(0.377753\pi\)
\(180\) 0 0
\(181\) −2353.41 −0.966450 −0.483225 0.875496i \(-0.660535\pi\)
−0.483225 + 0.875496i \(0.660535\pi\)
\(182\) 0 0
\(183\) 4225.20 1.70675
\(184\) 0 0
\(185\) −6652.40 −2.64375
\(186\) 0 0
\(187\) 585.560 0.228986
\(188\) 0 0
\(189\) −2249.23 −0.865646
\(190\) 0 0
\(191\) −2184.35 −0.827508 −0.413754 0.910389i \(-0.635783\pi\)
−0.413754 + 0.910389i \(0.635783\pi\)
\(192\) 0 0
\(193\) −3109.71 −1.15980 −0.579901 0.814687i \(-0.696909\pi\)
−0.579901 + 0.814687i \(0.696909\pi\)
\(194\) 0 0
\(195\) −1885.06 −0.692267
\(196\) 0 0
\(197\) −923.756 −0.334086 −0.167043 0.985950i \(-0.553422\pi\)
−0.167043 + 0.985950i \(0.553422\pi\)
\(198\) 0 0
\(199\) 4548.71 1.62035 0.810174 0.586189i \(-0.199372\pi\)
0.810174 + 0.586189i \(0.199372\pi\)
\(200\) 0 0
\(201\) −4001.91 −1.40434
\(202\) 0 0
\(203\) 151.379 0.0523386
\(204\) 0 0
\(205\) −7522.48 −2.56289
\(206\) 0 0
\(207\) −9323.85 −3.13069
\(208\) 0 0
\(209\) −59.2636 −0.0196141
\(210\) 0 0
\(211\) 318.664 0.103970 0.0519851 0.998648i \(-0.483445\pi\)
0.0519851 + 0.998648i \(0.483445\pi\)
\(212\) 0 0
\(213\) 5512.70 1.77335
\(214\) 0 0
\(215\) −3097.27 −0.982475
\(216\) 0 0
\(217\) −2.18883 −0.000684735 0
\(218\) 0 0
\(219\) 3774.01 1.16449
\(220\) 0 0
\(221\) 776.691 0.236407
\(222\) 0 0
\(223\) 1706.70 0.512509 0.256254 0.966609i \(-0.417512\pi\)
0.256254 + 0.966609i \(0.417512\pi\)
\(224\) 0 0
\(225\) 11152.8 3.30454
\(226\) 0 0
\(227\) −3043.63 −0.889925 −0.444962 0.895549i \(-0.646783\pi\)
−0.444962 + 0.895549i \(0.646783\pi\)
\(228\) 0 0
\(229\) 2621.57 0.756500 0.378250 0.925704i \(-0.376526\pi\)
0.378250 + 0.925704i \(0.376526\pi\)
\(230\) 0 0
\(231\) −441.029 −0.125617
\(232\) 0 0
\(233\) −2778.73 −0.781291 −0.390646 0.920541i \(-0.627748\pi\)
−0.390646 + 0.920541i \(0.627748\pi\)
\(234\) 0 0
\(235\) −1636.02 −0.454138
\(236\) 0 0
\(237\) −5702.76 −1.56301
\(238\) 0 0
\(239\) 4722.67 1.27818 0.639088 0.769133i \(-0.279312\pi\)
0.639088 + 0.769133i \(0.279312\pi\)
\(240\) 0 0
\(241\) −4704.13 −1.25734 −0.628672 0.777671i \(-0.716401\pi\)
−0.628672 + 0.777671i \(0.716401\pi\)
\(242\) 0 0
\(243\) −11589.7 −3.05959
\(244\) 0 0
\(245\) −5311.24 −1.38499
\(246\) 0 0
\(247\) −78.6076 −0.0202497
\(248\) 0 0
\(249\) −3494.96 −0.889496
\(250\) 0 0
\(251\) −4449.07 −1.11882 −0.559408 0.828893i \(-0.688971\pi\)
−0.559408 + 0.828893i \(0.688971\pi\)
\(252\) 0 0
\(253\) −1129.17 −0.280594
\(254\) 0 0
\(255\) −11379.6 −2.79459
\(256\) 0 0
\(257\) 7226.68 1.75404 0.877019 0.480456i \(-0.159529\pi\)
0.877019 + 0.480456i \(0.159529\pi\)
\(258\) 0 0
\(259\) −2064.41 −0.495275
\(260\) 0 0
\(261\) 2047.77 0.485646
\(262\) 0 0
\(263\) −4789.59 −1.12296 −0.561481 0.827490i \(-0.689768\pi\)
−0.561481 + 0.827490i \(0.689768\pi\)
\(264\) 0 0
\(265\) 3524.78 0.817078
\(266\) 0 0
\(267\) 2745.34 0.629259
\(268\) 0 0
\(269\) 47.0448 0.0106631 0.00533154 0.999986i \(-0.498303\pi\)
0.00533154 + 0.999986i \(0.498303\pi\)
\(270\) 0 0
\(271\) 7721.78 1.73087 0.865433 0.501025i \(-0.167044\pi\)
0.865433 + 0.501025i \(0.167044\pi\)
\(272\) 0 0
\(273\) −584.984 −0.129688
\(274\) 0 0
\(275\) 1350.67 0.296176
\(276\) 0 0
\(277\) 7554.66 1.63868 0.819342 0.573305i \(-0.194339\pi\)
0.819342 + 0.573305i \(0.194339\pi\)
\(278\) 0 0
\(279\) −29.6092 −0.00635360
\(280\) 0 0
\(281\) −2094.83 −0.444723 −0.222361 0.974964i \(-0.571377\pi\)
−0.222361 + 0.974964i \(0.571377\pi\)
\(282\) 0 0
\(283\) 8200.83 1.72257 0.861287 0.508119i \(-0.169659\pi\)
0.861287 + 0.508119i \(0.169659\pi\)
\(284\) 0 0
\(285\) 1151.71 0.239374
\(286\) 0 0
\(287\) −2334.42 −0.480127
\(288\) 0 0
\(289\) −224.318 −0.0456580
\(290\) 0 0
\(291\) −6697.76 −1.34924
\(292\) 0 0
\(293\) −3518.87 −0.701620 −0.350810 0.936447i \(-0.614094\pi\)
−0.350810 + 0.936447i \(0.614094\pi\)
\(294\) 0 0
\(295\) −773.175 −0.152597
\(296\) 0 0
\(297\) −3684.78 −0.719907
\(298\) 0 0
\(299\) −1497.74 −0.289687
\(300\) 0 0
\(301\) −961.163 −0.184055
\(302\) 0 0
\(303\) 5609.97 1.06364
\(304\) 0 0
\(305\) −7193.56 −1.35050
\(306\) 0 0
\(307\) 8725.36 1.62209 0.811046 0.584982i \(-0.198898\pi\)
0.811046 + 0.584982i \(0.198898\pi\)
\(308\) 0 0
\(309\) −3153.72 −0.580611
\(310\) 0 0
\(311\) 2640.09 0.481370 0.240685 0.970603i \(-0.422628\pi\)
0.240685 + 0.970603i \(0.422628\pi\)
\(312\) 0 0
\(313\) −1938.00 −0.349976 −0.174988 0.984571i \(-0.555989\pi\)
−0.174988 + 0.984571i \(0.555989\pi\)
\(314\) 0 0
\(315\) 6200.12 1.10901
\(316\) 0 0
\(317\) 5546.11 0.982651 0.491326 0.870976i \(-0.336512\pi\)
0.491326 + 0.870976i \(0.336512\pi\)
\(318\) 0 0
\(319\) 247.996 0.0435270
\(320\) 0 0
\(321\) −786.981 −0.136838
\(322\) 0 0
\(323\) −474.534 −0.0817454
\(324\) 0 0
\(325\) 1791.53 0.305774
\(326\) 0 0
\(327\) −16879.1 −2.85449
\(328\) 0 0
\(329\) −507.701 −0.0850773
\(330\) 0 0
\(331\) −185.492 −0.0308023 −0.0154012 0.999881i \(-0.504903\pi\)
−0.0154012 + 0.999881i \(0.504903\pi\)
\(332\) 0 0
\(333\) −27926.1 −4.59562
\(334\) 0 0
\(335\) 6813.41 1.11121
\(336\) 0 0
\(337\) −8508.32 −1.37530 −0.687652 0.726040i \(-0.741359\pi\)
−0.687652 + 0.726040i \(0.741359\pi\)
\(338\) 0 0
\(339\) −10377.0 −1.66254
\(340\) 0 0
\(341\) −3.58584 −0.000569454 0
\(342\) 0 0
\(343\) −3438.67 −0.541313
\(344\) 0 0
\(345\) 21944.0 3.42442
\(346\) 0 0
\(347\) −7853.53 −1.21498 −0.607492 0.794326i \(-0.707824\pi\)
−0.607492 + 0.794326i \(0.707824\pi\)
\(348\) 0 0
\(349\) −6328.33 −0.970624 −0.485312 0.874341i \(-0.661294\pi\)
−0.485312 + 0.874341i \(0.661294\pi\)
\(350\) 0 0
\(351\) −4887.51 −0.743237
\(352\) 0 0
\(353\) −7973.45 −1.20222 −0.601111 0.799166i \(-0.705275\pi\)
−0.601111 + 0.799166i \(0.705275\pi\)
\(354\) 0 0
\(355\) −9385.59 −1.40320
\(356\) 0 0
\(357\) −3531.40 −0.523533
\(358\) 0 0
\(359\) −10059.4 −1.47887 −0.739435 0.673228i \(-0.764907\pi\)
−0.739435 + 0.673228i \(0.764907\pi\)
\(360\) 0 0
\(361\) −6810.97 −0.992998
\(362\) 0 0
\(363\) 12427.6 1.79692
\(364\) 0 0
\(365\) −6425.40 −0.921426
\(366\) 0 0
\(367\) −3032.67 −0.431346 −0.215673 0.976466i \(-0.569195\pi\)
−0.215673 + 0.976466i \(0.569195\pi\)
\(368\) 0 0
\(369\) −31578.6 −4.45506
\(370\) 0 0
\(371\) 1093.83 0.153070
\(372\) 0 0
\(373\) 11109.4 1.54215 0.771077 0.636742i \(-0.219718\pi\)
0.771077 + 0.636742i \(0.219718\pi\)
\(374\) 0 0
\(375\) −5474.89 −0.753925
\(376\) 0 0
\(377\) 328.943 0.0449375
\(378\) 0 0
\(379\) −4510.20 −0.611275 −0.305638 0.952148i \(-0.598870\pi\)
−0.305638 + 0.952148i \(0.598870\pi\)
\(380\) 0 0
\(381\) −3625.20 −0.487466
\(382\) 0 0
\(383\) 7810.96 1.04209 0.521047 0.853528i \(-0.325542\pi\)
0.521047 + 0.853528i \(0.325542\pi\)
\(384\) 0 0
\(385\) 750.869 0.0993970
\(386\) 0 0
\(387\) −13002.0 −1.70783
\(388\) 0 0
\(389\) −9221.90 −1.20198 −0.600989 0.799258i \(-0.705226\pi\)
−0.600989 + 0.799258i \(0.705226\pi\)
\(390\) 0 0
\(391\) −9041.46 −1.16943
\(392\) 0 0
\(393\) 22831.1 2.93047
\(394\) 0 0
\(395\) 9709.17 1.23676
\(396\) 0 0
\(397\) 10034.3 1.26854 0.634268 0.773113i \(-0.281301\pi\)
0.634268 + 0.773113i \(0.281301\pi\)
\(398\) 0 0
\(399\) 357.407 0.0448439
\(400\) 0 0
\(401\) −8193.64 −1.02038 −0.510188 0.860063i \(-0.670424\pi\)
−0.510188 + 0.860063i \(0.670424\pi\)
\(402\) 0 0
\(403\) −4.75628 −0.000587908 0
\(404\) 0 0
\(405\) 39539.3 4.85117
\(406\) 0 0
\(407\) −3382.01 −0.411892
\(408\) 0 0
\(409\) 3921.69 0.474120 0.237060 0.971495i \(-0.423816\pi\)
0.237060 + 0.971495i \(0.423816\pi\)
\(410\) 0 0
\(411\) −18768.4 −2.25250
\(412\) 0 0
\(413\) −239.936 −0.0285872
\(414\) 0 0
\(415\) 5950.31 0.703830
\(416\) 0 0
\(417\) 12937.6 1.51933
\(418\) 0 0
\(419\) −3026.66 −0.352892 −0.176446 0.984310i \(-0.556460\pi\)
−0.176446 + 0.984310i \(0.556460\pi\)
\(420\) 0 0
\(421\) −4823.43 −0.558384 −0.279192 0.960235i \(-0.590067\pi\)
−0.279192 + 0.960235i \(0.590067\pi\)
\(422\) 0 0
\(423\) −6867.87 −0.789426
\(424\) 0 0
\(425\) 10815.0 1.23437
\(426\) 0 0
\(427\) −2232.35 −0.253000
\(428\) 0 0
\(429\) −958.346 −0.107854
\(430\) 0 0
\(431\) −2029.51 −0.226816 −0.113408 0.993548i \(-0.536177\pi\)
−0.113408 + 0.993548i \(0.536177\pi\)
\(432\) 0 0
\(433\) −535.808 −0.0594672 −0.0297336 0.999558i \(-0.509466\pi\)
−0.0297336 + 0.999558i \(0.509466\pi\)
\(434\) 0 0
\(435\) −4819.49 −0.531211
\(436\) 0 0
\(437\) 915.072 0.100169
\(438\) 0 0
\(439\) 8791.74 0.955824 0.477912 0.878408i \(-0.341394\pi\)
0.477912 + 0.878408i \(0.341394\pi\)
\(440\) 0 0
\(441\) −22296.1 −2.40752
\(442\) 0 0
\(443\) −13916.9 −1.49258 −0.746291 0.665619i \(-0.768167\pi\)
−0.746291 + 0.665619i \(0.768167\pi\)
\(444\) 0 0
\(445\) −4674.05 −0.497912
\(446\) 0 0
\(447\) 17615.7 1.86397
\(448\) 0 0
\(449\) 2129.54 0.223829 0.111915 0.993718i \(-0.464302\pi\)
0.111915 + 0.993718i \(0.464302\pi\)
\(450\) 0 0
\(451\) −3824.35 −0.399294
\(452\) 0 0
\(453\) 16123.6 1.67230
\(454\) 0 0
\(455\) 995.958 0.102618
\(456\) 0 0
\(457\) 1932.25 0.197783 0.0988915 0.995098i \(-0.468470\pi\)
0.0988915 + 0.995098i \(0.468470\pi\)
\(458\) 0 0
\(459\) −29504.6 −3.00035
\(460\) 0 0
\(461\) 16518.3 1.66884 0.834418 0.551132i \(-0.185804\pi\)
0.834418 + 0.551132i \(0.185804\pi\)
\(462\) 0 0
\(463\) 11535.2 1.15785 0.578926 0.815380i \(-0.303472\pi\)
0.578926 + 0.815380i \(0.303472\pi\)
\(464\) 0 0
\(465\) 69.6863 0.00694973
\(466\) 0 0
\(467\) −15667.8 −1.55250 −0.776250 0.630425i \(-0.782880\pi\)
−0.776250 + 0.630425i \(0.782880\pi\)
\(468\) 0 0
\(469\) 2114.38 0.208172
\(470\) 0 0
\(471\) 8425.75 0.824285
\(472\) 0 0
\(473\) −1574.62 −0.153068
\(474\) 0 0
\(475\) −1094.57 −0.105731
\(476\) 0 0
\(477\) 14796.7 1.42032
\(478\) 0 0
\(479\) 4672.28 0.445682 0.222841 0.974855i \(-0.428467\pi\)
0.222841 + 0.974855i \(0.428467\pi\)
\(480\) 0 0
\(481\) −4485.92 −0.425239
\(482\) 0 0
\(483\) 6809.80 0.641525
\(484\) 0 0
\(485\) 11403.2 1.06761
\(486\) 0 0
\(487\) −8465.76 −0.787722 −0.393861 0.919170i \(-0.628861\pi\)
−0.393861 + 0.919170i \(0.628861\pi\)
\(488\) 0 0
\(489\) 32408.4 2.99706
\(490\) 0 0
\(491\) 12302.6 1.13077 0.565385 0.824827i \(-0.308728\pi\)
0.565385 + 0.824827i \(0.308728\pi\)
\(492\) 0 0
\(493\) 1985.74 0.181407
\(494\) 0 0
\(495\) 10157.3 0.922297
\(496\) 0 0
\(497\) −2912.59 −0.262873
\(498\) 0 0
\(499\) 2781.31 0.249516 0.124758 0.992187i \(-0.460185\pi\)
0.124758 + 0.992187i \(0.460185\pi\)
\(500\) 0 0
\(501\) 16620.6 1.48214
\(502\) 0 0
\(503\) −13673.9 −1.21211 −0.606054 0.795424i \(-0.707248\pi\)
−0.606054 + 0.795424i \(0.707248\pi\)
\(504\) 0 0
\(505\) −9551.18 −0.841628
\(506\) 0 0
\(507\) 20435.0 1.79004
\(508\) 0 0
\(509\) −7431.68 −0.647158 −0.323579 0.946201i \(-0.604886\pi\)
−0.323579 + 0.946201i \(0.604886\pi\)
\(510\) 0 0
\(511\) −1993.97 −0.172618
\(512\) 0 0
\(513\) 2986.12 0.256999
\(514\) 0 0
\(515\) 5369.33 0.459419
\(516\) 0 0
\(517\) −831.737 −0.0707538
\(518\) 0 0
\(519\) −15711.2 −1.32879
\(520\) 0 0
\(521\) 7797.86 0.655721 0.327860 0.944726i \(-0.393672\pi\)
0.327860 + 0.944726i \(0.393672\pi\)
\(522\) 0 0
\(523\) 16670.9 1.39382 0.696908 0.717160i \(-0.254558\pi\)
0.696908 + 0.717160i \(0.254558\pi\)
\(524\) 0 0
\(525\) −8145.61 −0.677149
\(526\) 0 0
\(527\) −28.7124 −0.00237331
\(528\) 0 0
\(529\) 5268.18 0.432990
\(530\) 0 0
\(531\) −3245.71 −0.265258
\(532\) 0 0
\(533\) −5072.64 −0.412233
\(534\) 0 0
\(535\) 1339.87 0.108276
\(536\) 0 0
\(537\) −17730.5 −1.42482
\(538\) 0 0
\(539\) −2700.18 −0.215779
\(540\) 0 0
\(541\) −13400.1 −1.06491 −0.532455 0.846458i \(-0.678730\pi\)
−0.532455 + 0.846458i \(0.678730\pi\)
\(542\) 0 0
\(543\) 23251.5 1.83760
\(544\) 0 0
\(545\) 28737.4 2.25867
\(546\) 0 0
\(547\) 17339.4 1.35535 0.677677 0.735360i \(-0.262987\pi\)
0.677677 + 0.735360i \(0.262987\pi\)
\(548\) 0 0
\(549\) −30197.9 −2.34757
\(550\) 0 0
\(551\) −200.974 −0.0155386
\(552\) 0 0
\(553\) 3013.01 0.231693
\(554\) 0 0
\(555\) 65725.1 5.02680
\(556\) 0 0
\(557\) 7790.90 0.592659 0.296330 0.955086i \(-0.404237\pi\)
0.296330 + 0.955086i \(0.404237\pi\)
\(558\) 0 0
\(559\) −2088.58 −0.158028
\(560\) 0 0
\(561\) −5785.28 −0.435392
\(562\) 0 0
\(563\) 21514.2 1.61051 0.805253 0.592932i \(-0.202030\pi\)
0.805253 + 0.592932i \(0.202030\pi\)
\(564\) 0 0
\(565\) 17667.3 1.31552
\(566\) 0 0
\(567\) 12270.1 0.908808
\(568\) 0 0
\(569\) 16993.5 1.25203 0.626015 0.779811i \(-0.284685\pi\)
0.626015 + 0.779811i \(0.284685\pi\)
\(570\) 0 0
\(571\) −16791.4 −1.23064 −0.615321 0.788277i \(-0.710973\pi\)
−0.615321 + 0.788277i \(0.710973\pi\)
\(572\) 0 0
\(573\) 21581.2 1.57342
\(574\) 0 0
\(575\) −20855.3 −1.51256
\(576\) 0 0
\(577\) 8108.87 0.585055 0.292527 0.956257i \(-0.405504\pi\)
0.292527 + 0.956257i \(0.405504\pi\)
\(578\) 0 0
\(579\) 30723.7 2.20524
\(580\) 0 0
\(581\) 1846.54 0.131854
\(582\) 0 0
\(583\) 1791.96 0.127299
\(584\) 0 0
\(585\) 13472.7 0.952185
\(586\) 0 0
\(587\) −16076.3 −1.13039 −0.565197 0.824956i \(-0.691200\pi\)
−0.565197 + 0.824956i \(0.691200\pi\)
\(588\) 0 0
\(589\) 2.90594 0.000203289 0
\(590\) 0 0
\(591\) 9126.63 0.635227
\(592\) 0 0
\(593\) −4341.83 −0.300671 −0.150335 0.988635i \(-0.548035\pi\)
−0.150335 + 0.988635i \(0.548035\pi\)
\(594\) 0 0
\(595\) 6012.34 0.414255
\(596\) 0 0
\(597\) −44940.8 −3.08091
\(598\) 0 0
\(599\) −10540.1 −0.718960 −0.359480 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(600\) 0 0
\(601\) 16485.6 1.11890 0.559451 0.828863i \(-0.311012\pi\)
0.559451 + 0.828863i \(0.311012\pi\)
\(602\) 0 0
\(603\) 28602.0 1.93162
\(604\) 0 0
\(605\) −21158.6 −1.42185
\(606\) 0 0
\(607\) 13326.5 0.891112 0.445556 0.895254i \(-0.353006\pi\)
0.445556 + 0.895254i \(0.353006\pi\)
\(608\) 0 0
\(609\) −1495.61 −0.0995161
\(610\) 0 0
\(611\) −1103.22 −0.0730467
\(612\) 0 0
\(613\) −20459.4 −1.34804 −0.674019 0.738714i \(-0.735433\pi\)
−0.674019 + 0.738714i \(0.735433\pi\)
\(614\) 0 0
\(615\) 74321.4 4.87305
\(616\) 0 0
\(617\) −3108.50 −0.202826 −0.101413 0.994844i \(-0.532336\pi\)
−0.101413 + 0.994844i \(0.532336\pi\)
\(618\) 0 0
\(619\) −10426.3 −0.677012 −0.338506 0.940964i \(-0.609922\pi\)
−0.338506 + 0.940964i \(0.609922\pi\)
\(620\) 0 0
\(621\) 56895.5 3.67655
\(622\) 0 0
\(623\) −1450.48 −0.0932780
\(624\) 0 0
\(625\) −10421.8 −0.666992
\(626\) 0 0
\(627\) 585.519 0.0372941
\(628\) 0 0
\(629\) −27080.3 −1.71663
\(630\) 0 0
\(631\) 18018.2 1.13675 0.568377 0.822768i \(-0.307572\pi\)
0.568377 + 0.822768i \(0.307572\pi\)
\(632\) 0 0
\(633\) −3148.37 −0.197688
\(634\) 0 0
\(635\) 6172.04 0.385716
\(636\) 0 0
\(637\) −3581.53 −0.222772
\(638\) 0 0
\(639\) −39399.8 −2.43917
\(640\) 0 0
\(641\) 15178.7 0.935295 0.467648 0.883915i \(-0.345102\pi\)
0.467648 + 0.883915i \(0.345102\pi\)
\(642\) 0 0
\(643\) −23957.1 −1.46932 −0.734662 0.678433i \(-0.762659\pi\)
−0.734662 + 0.678433i \(0.762659\pi\)
\(644\) 0 0
\(645\) 30600.7 1.86807
\(646\) 0 0
\(647\) −12835.2 −0.779915 −0.389958 0.920833i \(-0.627510\pi\)
−0.389958 + 0.920833i \(0.627510\pi\)
\(648\) 0 0
\(649\) −393.074 −0.0237743
\(650\) 0 0
\(651\) 21.6255 0.00130195
\(652\) 0 0
\(653\) 4355.97 0.261045 0.130523 0.991445i \(-0.458335\pi\)
0.130523 + 0.991445i \(0.458335\pi\)
\(654\) 0 0
\(655\) −38870.8 −2.31879
\(656\) 0 0
\(657\) −26973.2 −1.60171
\(658\) 0 0
\(659\) 32704.1 1.93319 0.966594 0.256313i \(-0.0825079\pi\)
0.966594 + 0.256313i \(0.0825079\pi\)
\(660\) 0 0
\(661\) 2839.66 0.167095 0.0835475 0.996504i \(-0.473375\pi\)
0.0835475 + 0.996504i \(0.473375\pi\)
\(662\) 0 0
\(663\) −7673.64 −0.449501
\(664\) 0 0
\(665\) −608.499 −0.0354836
\(666\) 0 0
\(667\) −3829.23 −0.222291
\(668\) 0 0
\(669\) −16862.1 −0.974478
\(670\) 0 0
\(671\) −3657.13 −0.210405
\(672\) 0 0
\(673\) −18569.9 −1.06362 −0.531810 0.846864i \(-0.678488\pi\)
−0.531810 + 0.846864i \(0.678488\pi\)
\(674\) 0 0
\(675\) −68056.1 −3.88071
\(676\) 0 0
\(677\) −18106.5 −1.02790 −0.513952 0.857819i \(-0.671819\pi\)
−0.513952 + 0.857819i \(0.671819\pi\)
\(678\) 0 0
\(679\) 3538.71 0.200005
\(680\) 0 0
\(681\) 30070.8 1.69209
\(682\) 0 0
\(683\) 5510.97 0.308743 0.154371 0.988013i \(-0.450665\pi\)
0.154371 + 0.988013i \(0.450665\pi\)
\(684\) 0 0
\(685\) 31953.9 1.78233
\(686\) 0 0
\(687\) −25900.9 −1.43840
\(688\) 0 0
\(689\) 2376.87 0.131425
\(690\) 0 0
\(691\) −24652.0 −1.35717 −0.678587 0.734520i \(-0.737408\pi\)
−0.678587 + 0.734520i \(0.737408\pi\)
\(692\) 0 0
\(693\) 3152.08 0.172781
\(694\) 0 0
\(695\) −22026.9 −1.20220
\(696\) 0 0
\(697\) −30622.2 −1.66413
\(698\) 0 0
\(699\) 27453.6 1.48554
\(700\) 0 0
\(701\) 3399.10 0.183142 0.0915708 0.995799i \(-0.470811\pi\)
0.0915708 + 0.995799i \(0.470811\pi\)
\(702\) 0 0
\(703\) 2740.75 0.147041
\(704\) 0 0
\(705\) 16163.8 0.863493
\(706\) 0 0
\(707\) −2963.98 −0.157669
\(708\) 0 0
\(709\) −26329.1 −1.39465 −0.697327 0.716753i \(-0.745627\pi\)
−0.697327 + 0.716753i \(0.745627\pi\)
\(710\) 0 0
\(711\) 40758.2 2.14986
\(712\) 0 0
\(713\) 55.3678 0.00290819
\(714\) 0 0
\(715\) 1631.62 0.0853415
\(716\) 0 0
\(717\) −46659.6 −2.43031
\(718\) 0 0
\(719\) 22345.6 1.15904 0.579521 0.814957i \(-0.303240\pi\)
0.579521 + 0.814957i \(0.303240\pi\)
\(720\) 0 0
\(721\) 1666.24 0.0860668
\(722\) 0 0
\(723\) 46476.4 2.39070
\(724\) 0 0
\(725\) 4580.37 0.234635
\(726\) 0 0
\(727\) −29862.8 −1.52345 −0.761727 0.647898i \(-0.775648\pi\)
−0.761727 + 0.647898i \(0.775648\pi\)
\(728\) 0 0
\(729\) 51039.2 2.59306
\(730\) 0 0
\(731\) −12608.2 −0.637938
\(732\) 0 0
\(733\) 987.919 0.0497812 0.0248906 0.999690i \(-0.492076\pi\)
0.0248906 + 0.999690i \(0.492076\pi\)
\(734\) 0 0
\(735\) 52474.6 2.63341
\(736\) 0 0
\(737\) 3463.86 0.173125
\(738\) 0 0
\(739\) 35870.9 1.78556 0.892782 0.450489i \(-0.148750\pi\)
0.892782 + 0.450489i \(0.148750\pi\)
\(740\) 0 0
\(741\) 776.636 0.0385026
\(742\) 0 0
\(743\) 8322.20 0.410918 0.205459 0.978666i \(-0.434131\pi\)
0.205459 + 0.978666i \(0.434131\pi\)
\(744\) 0 0
\(745\) −29991.4 −1.47490
\(746\) 0 0
\(747\) 24978.8 1.22346
\(748\) 0 0
\(749\) 415.795 0.0202841
\(750\) 0 0
\(751\) 20781.2 1.00974 0.504872 0.863194i \(-0.331540\pi\)
0.504872 + 0.863194i \(0.331540\pi\)
\(752\) 0 0
\(753\) 43956.4 2.12730
\(754\) 0 0
\(755\) −27451.0 −1.32324
\(756\) 0 0
\(757\) 16424.7 0.788596 0.394298 0.918983i \(-0.370988\pi\)
0.394298 + 0.918983i \(0.370988\pi\)
\(758\) 0 0
\(759\) 11156.1 0.533519
\(760\) 0 0
\(761\) 416.312 0.0198309 0.00991543 0.999951i \(-0.496844\pi\)
0.00991543 + 0.999951i \(0.496844\pi\)
\(762\) 0 0
\(763\) 8917.95 0.423134
\(764\) 0 0
\(765\) 81331.3 3.84384
\(766\) 0 0
\(767\) −521.376 −0.0245447
\(768\) 0 0
\(769\) −37129.2 −1.74111 −0.870554 0.492073i \(-0.836239\pi\)
−0.870554 + 0.492073i \(0.836239\pi\)
\(770\) 0 0
\(771\) −71398.9 −3.33511
\(772\) 0 0
\(773\) 2182.83 0.101567 0.0507833 0.998710i \(-0.483828\pi\)
0.0507833 + 0.998710i \(0.483828\pi\)
\(774\) 0 0
\(775\) −66.2287 −0.00306969
\(776\) 0 0
\(777\) 20396.2 0.941711
\(778\) 0 0
\(779\) 3099.22 0.142543
\(780\) 0 0
\(781\) −4771.53 −0.218616
\(782\) 0 0
\(783\) −12495.8 −0.570322
\(784\) 0 0
\(785\) −14345.2 −0.652231
\(786\) 0 0
\(787\) −7068.15 −0.320143 −0.160071 0.987105i \(-0.551172\pi\)
−0.160071 + 0.987105i \(0.551172\pi\)
\(788\) 0 0
\(789\) 47320.7 2.13519
\(790\) 0 0
\(791\) 5482.61 0.246446
\(792\) 0 0
\(793\) −4850.84 −0.217224
\(794\) 0 0
\(795\) −34824.5 −1.55358
\(796\) 0 0
\(797\) −15086.0 −0.670482 −0.335241 0.942132i \(-0.608818\pi\)
−0.335241 + 0.942132i \(0.608818\pi\)
\(798\) 0 0
\(799\) −6659.86 −0.294880
\(800\) 0 0
\(801\) −19621.2 −0.865519
\(802\) 0 0
\(803\) −3266.60 −0.143557
\(804\) 0 0
\(805\) −11593.9 −0.507619
\(806\) 0 0
\(807\) −464.798 −0.0202747
\(808\) 0 0
\(809\) −21798.3 −0.947326 −0.473663 0.880706i \(-0.657069\pi\)
−0.473663 + 0.880706i \(0.657069\pi\)
\(810\) 0 0
\(811\) −5569.61 −0.241153 −0.120577 0.992704i \(-0.538474\pi\)
−0.120577 + 0.992704i \(0.538474\pi\)
\(812\) 0 0
\(813\) −76290.4 −3.29105
\(814\) 0 0
\(815\) −55176.6 −2.37148
\(816\) 0 0
\(817\) 1276.06 0.0546434
\(818\) 0 0
\(819\) 4180.93 0.178381
\(820\) 0 0
\(821\) −21281.8 −0.904676 −0.452338 0.891847i \(-0.649410\pi\)
−0.452338 + 0.891847i \(0.649410\pi\)
\(822\) 0 0
\(823\) 16799.0 0.711514 0.355757 0.934578i \(-0.384223\pi\)
0.355757 + 0.934578i \(0.384223\pi\)
\(824\) 0 0
\(825\) −13344.5 −0.563145
\(826\) 0 0
\(827\) −7360.62 −0.309497 −0.154748 0.987954i \(-0.549457\pi\)
−0.154748 + 0.987954i \(0.549457\pi\)
\(828\) 0 0
\(829\) 11634.6 0.487438 0.243719 0.969846i \(-0.421633\pi\)
0.243719 + 0.969846i \(0.421633\pi\)
\(830\) 0 0
\(831\) −74639.4 −3.11578
\(832\) 0 0
\(833\) −21620.8 −0.899299
\(834\) 0 0
\(835\) −28297.2 −1.17277
\(836\) 0 0
\(837\) 180.680 0.00746141
\(838\) 0 0
\(839\) −32151.0 −1.32297 −0.661486 0.749957i \(-0.730074\pi\)
−0.661486 + 0.749957i \(0.730074\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 20696.7 0.845592
\(844\) 0 0
\(845\) −34791.4 −1.41640
\(846\) 0 0
\(847\) −6566.05 −0.266366
\(848\) 0 0
\(849\) −81023.4 −3.27528
\(850\) 0 0
\(851\) 52220.6 2.10352
\(852\) 0 0
\(853\) −35044.9 −1.40670 −0.703349 0.710844i \(-0.748313\pi\)
−0.703349 + 0.710844i \(0.748313\pi\)
\(854\) 0 0
\(855\) −8231.41 −0.329249
\(856\) 0 0
\(857\) 7143.05 0.284716 0.142358 0.989815i \(-0.454532\pi\)
0.142358 + 0.989815i \(0.454532\pi\)
\(858\) 0 0
\(859\) −33960.3 −1.34891 −0.674454 0.738317i \(-0.735621\pi\)
−0.674454 + 0.738317i \(0.735621\pi\)
\(860\) 0 0
\(861\) 23063.9 0.912909
\(862\) 0 0
\(863\) 45823.4 1.80747 0.903736 0.428090i \(-0.140814\pi\)
0.903736 + 0.428090i \(0.140814\pi\)
\(864\) 0 0
\(865\) 26748.9 1.05143
\(866\) 0 0
\(867\) 2216.24 0.0868136
\(868\) 0 0
\(869\) 4936.04 0.192685
\(870\) 0 0
\(871\) 4594.49 0.178735
\(872\) 0 0
\(873\) 47869.5 1.85583
\(874\) 0 0
\(875\) 2892.61 0.111758
\(876\) 0 0
\(877\) 21281.3 0.819406 0.409703 0.912219i \(-0.365632\pi\)
0.409703 + 0.912219i \(0.365632\pi\)
\(878\) 0 0
\(879\) 34766.1 1.33405
\(880\) 0 0
\(881\) 1359.32 0.0519826 0.0259913 0.999662i \(-0.491726\pi\)
0.0259913 + 0.999662i \(0.491726\pi\)
\(882\) 0 0
\(883\) 47928.2 1.82663 0.913313 0.407257i \(-0.133515\pi\)
0.913313 + 0.407257i \(0.133515\pi\)
\(884\) 0 0
\(885\) 7638.90 0.290146
\(886\) 0 0
\(887\) 3100.58 0.117370 0.0586851 0.998277i \(-0.481309\pi\)
0.0586851 + 0.998277i \(0.481309\pi\)
\(888\) 0 0
\(889\) 1915.34 0.0722593
\(890\) 0 0
\(891\) 20101.3 0.755803
\(892\) 0 0
\(893\) 674.033 0.0252583
\(894\) 0 0
\(895\) 30186.8 1.12741
\(896\) 0 0
\(897\) 14797.5 0.550808
\(898\) 0 0
\(899\) −12.1602 −0.000451131 0
\(900\) 0 0
\(901\) 14348.5 0.530543
\(902\) 0 0
\(903\) 9496.21 0.349960
\(904\) 0 0
\(905\) −39586.5 −1.45403
\(906\) 0 0
\(907\) −23428.1 −0.857680 −0.428840 0.903380i \(-0.641078\pi\)
−0.428840 + 0.903380i \(0.641078\pi\)
\(908\) 0 0
\(909\) −40094.9 −1.46300
\(910\) 0 0
\(911\) −12868.3 −0.467999 −0.234000 0.972237i \(-0.575181\pi\)
−0.234000 + 0.972237i \(0.575181\pi\)
\(912\) 0 0
\(913\) 3025.07 0.109655
\(914\) 0 0
\(915\) 71071.7 2.56782
\(916\) 0 0
\(917\) −12062.6 −0.434398
\(918\) 0 0
\(919\) 3938.41 0.141367 0.0706834 0.997499i \(-0.477482\pi\)
0.0706834 + 0.997499i \(0.477482\pi\)
\(920\) 0 0
\(921\) −86205.7 −3.08423
\(922\) 0 0
\(923\) −6328.99 −0.225700
\(924\) 0 0
\(925\) −62464.1 −2.22033
\(926\) 0 0
\(927\) 22539.9 0.798607
\(928\) 0 0
\(929\) −15856.7 −0.560001 −0.280001 0.960000i \(-0.590335\pi\)
−0.280001 + 0.960000i \(0.590335\pi\)
\(930\) 0 0
\(931\) 2188.20 0.0770306
\(932\) 0 0
\(933\) −26083.9 −0.915271
\(934\) 0 0
\(935\) 9849.67 0.344512
\(936\) 0 0
\(937\) −18559.7 −0.647085 −0.323542 0.946214i \(-0.604874\pi\)
−0.323542 + 0.946214i \(0.604874\pi\)
\(938\) 0 0
\(939\) 19147.3 0.665441
\(940\) 0 0
\(941\) −24125.6 −0.835785 −0.417892 0.908497i \(-0.637231\pi\)
−0.417892 + 0.908497i \(0.637231\pi\)
\(942\) 0 0
\(943\) 59050.6 2.03919
\(944\) 0 0
\(945\) −37834.1 −1.30237
\(946\) 0 0
\(947\) 6883.31 0.236196 0.118098 0.993002i \(-0.462320\pi\)
0.118098 + 0.993002i \(0.462320\pi\)
\(948\) 0 0
\(949\) −4332.84 −0.148209
\(950\) 0 0
\(951\) −54795.0 −1.86840
\(952\) 0 0
\(953\) 37573.8 1.27716 0.638580 0.769555i \(-0.279522\pi\)
0.638580 + 0.769555i \(0.279522\pi\)
\(954\) 0 0
\(955\) −36742.8 −1.24499
\(956\) 0 0
\(957\) −2450.18 −0.0827617
\(958\) 0 0
\(959\) 9916.13 0.333898
\(960\) 0 0
\(961\) −29790.8 −0.999994
\(962\) 0 0
\(963\) 5624.63 0.188215
\(964\) 0 0
\(965\) −52308.2 −1.74493
\(966\) 0 0
\(967\) −1584.28 −0.0526856 −0.0263428 0.999653i \(-0.508386\pi\)
−0.0263428 + 0.999653i \(0.508386\pi\)
\(968\) 0 0
\(969\) 4688.35 0.155430
\(970\) 0 0
\(971\) −36569.1 −1.20861 −0.604304 0.796754i \(-0.706549\pi\)
−0.604304 + 0.796754i \(0.706549\pi\)
\(972\) 0 0
\(973\) −6835.50 −0.225217
\(974\) 0 0
\(975\) −17700.2 −0.581395
\(976\) 0 0
\(977\) −33583.0 −1.09971 −0.549854 0.835261i \(-0.685317\pi\)
−0.549854 + 0.835261i \(0.685317\pi\)
\(978\) 0 0
\(979\) −2376.23 −0.0775738
\(980\) 0 0
\(981\) 120637. 3.92623
\(982\) 0 0
\(983\) −25900.6 −0.840387 −0.420193 0.907435i \(-0.638038\pi\)
−0.420193 + 0.907435i \(0.638038\pi\)
\(984\) 0 0
\(985\) −15538.4 −0.502635
\(986\) 0 0
\(987\) 5016.04 0.161765
\(988\) 0 0
\(989\) 24313.2 0.781714
\(990\) 0 0
\(991\) 29604.3 0.948951 0.474475 0.880269i \(-0.342638\pi\)
0.474475 + 0.880269i \(0.342638\pi\)
\(992\) 0 0
\(993\) 1832.65 0.0585672
\(994\) 0 0
\(995\) 76513.5 2.43783
\(996\) 0 0
\(997\) −39267.2 −1.24735 −0.623673 0.781685i \(-0.714360\pi\)
−0.623673 + 0.781685i \(0.714360\pi\)
\(998\) 0 0
\(999\) 170409. 5.39691
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 1856.4.a.y.1.1 5
4.3 odd 2 1856.4.a.bb.1.5 5
8.3 odd 2 464.4.a.l.1.1 5
8.5 even 2 29.4.a.b.1.3 5
24.5 odd 2 261.4.a.f.1.3 5
40.29 even 2 725.4.a.c.1.3 5
56.13 odd 2 1421.4.a.e.1.3 5
232.173 even 2 841.4.a.b.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.3 5 8.5 even 2
261.4.a.f.1.3 5 24.5 odd 2
464.4.a.l.1.1 5 8.3 odd 2
725.4.a.c.1.3 5 40.29 even 2
841.4.a.b.1.3 5 232.173 even 2
1421.4.a.e.1.3 5 56.13 odd 2
1856.4.a.y.1.1 5 1.1 even 1 trivial
1856.4.a.bb.1.5 5 4.3 odd 2