Properties

Label 1344.4.b.f.895.4
Level $1344$
Weight $4$
Character 1344.895
Analytic conductor $79.299$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.4
Root \(-3.58293i\) of defining polynomial
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.f.895.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -4.33129i q^{5} +(18.4617 + 1.47188i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -4.33129i q^{5} +(18.4617 + 1.47188i) q^{7} +9.00000 q^{9} -21.1315i q^{11} +36.7607i q^{13} -12.9939i q^{15} -46.5942i q^{17} +89.7520 q^{19} +(55.3850 + 4.41563i) q^{21} +190.308i q^{23} +106.240 q^{25} +27.0000 q^{27} -196.881 q^{29} +87.5824 q^{31} -63.3944i q^{33} +(6.37513 - 79.9629i) q^{35} +17.9215 q^{37} +110.282i q^{39} +374.897i q^{41} -504.442i q^{43} -38.9816i q^{45} +399.286 q^{47} +(338.667 + 54.3467i) q^{49} -139.783i q^{51} +19.8855 q^{53} -91.5265 q^{55} +269.256 q^{57} +74.2866 q^{59} -5.16488i q^{61} +(166.155 + 13.2469i) q^{63} +159.221 q^{65} -276.852i q^{67} +570.924i q^{69} +474.674i q^{71} -653.765i q^{73} +318.720 q^{75} +(31.1029 - 390.122i) q^{77} +586.924i q^{79} +81.0000 q^{81} -441.028 q^{83} -201.813 q^{85} -590.643 q^{87} +862.965i q^{89} +(-54.1072 + 678.664i) q^{91} +262.747 q^{93} -388.742i q^{95} -890.287i q^{97} -190.183i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9} + 56 q^{19} - 12 q^{21} - 656 q^{25} + 216 q^{27} - 240 q^{29} + 320 q^{31} - 600 q^{35} - 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 168 q^{57} + 1824 q^{59} - 36 q^{63} - 816 q^{65} - 1968 q^{75} + 2064 q^{77} + 648 q^{81} - 1680 q^{83} - 2568 q^{85} - 720 q^{87} + 864 q^{91} + 960 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 4.33129i 0.387403i −0.981061 0.193701i \(-0.937951\pi\)
0.981061 0.193701i \(-0.0620493\pi\)
\(6\) 0 0
\(7\) 18.4617 + 1.47188i 0.996837 + 0.0794739i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 21.1315i 0.579216i −0.957145 0.289608i \(-0.906475\pi\)
0.957145 0.289608i \(-0.0935249\pi\)
\(12\) 0 0
\(13\) 36.7607i 0.784275i 0.919907 + 0.392137i \(0.128264\pi\)
−0.919907 + 0.392137i \(0.871736\pi\)
\(14\) 0 0
\(15\) 12.9939i 0.223667i
\(16\) 0 0
\(17\) 46.5942i 0.664750i −0.943147 0.332375i \(-0.892150\pi\)
0.943147 0.332375i \(-0.107850\pi\)
\(18\) 0 0
\(19\) 89.7520 1.08371 0.541856 0.840471i \(-0.317722\pi\)
0.541856 + 0.840471i \(0.317722\pi\)
\(20\) 0 0
\(21\) 55.3850 + 4.41563i 0.575524 + 0.0458843i
\(22\) 0 0
\(23\) 190.308i 1.72530i 0.505800 + 0.862651i \(0.331197\pi\)
−0.505800 + 0.862651i \(0.668803\pi\)
\(24\) 0 0
\(25\) 106.240 0.849919
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −196.881 −1.26069 −0.630343 0.776317i \(-0.717086\pi\)
−0.630343 + 0.776317i \(0.717086\pi\)
\(30\) 0 0
\(31\) 87.5824 0.507428 0.253714 0.967279i \(-0.418348\pi\)
0.253714 + 0.967279i \(0.418348\pi\)
\(32\) 0 0
\(33\) 63.3944i 0.334410i
\(34\) 0 0
\(35\) 6.37513 79.9629i 0.0307884 0.386177i
\(36\) 0 0
\(37\) 17.9215 0.0796289 0.0398145 0.999207i \(-0.487323\pi\)
0.0398145 + 0.999207i \(0.487323\pi\)
\(38\) 0 0
\(39\) 110.282i 0.452801i
\(40\) 0 0
\(41\) 374.897i 1.42803i 0.700132 + 0.714013i \(0.253124\pi\)
−0.700132 + 0.714013i \(0.746876\pi\)
\(42\) 0 0
\(43\) 504.442i 1.78899i −0.447075 0.894496i \(-0.647534\pi\)
0.447075 0.894496i \(-0.352466\pi\)
\(44\) 0 0
\(45\) 38.9816i 0.129134i
\(46\) 0 0
\(47\) 399.286 1.23919 0.619594 0.784923i \(-0.287297\pi\)
0.619594 + 0.784923i \(0.287297\pi\)
\(48\) 0 0
\(49\) 338.667 + 54.3467i 0.987368 + 0.158445i
\(50\) 0 0
\(51\) 139.783i 0.383794i
\(52\) 0 0
\(53\) 19.8855 0.0515374 0.0257687 0.999668i \(-0.491797\pi\)
0.0257687 + 0.999668i \(0.491797\pi\)
\(54\) 0 0
\(55\) −91.5265 −0.224390
\(56\) 0 0
\(57\) 269.256 0.625681
\(58\) 0 0
\(59\) 74.2866 0.163920 0.0819601 0.996636i \(-0.473882\pi\)
0.0819601 + 0.996636i \(0.473882\pi\)
\(60\) 0 0
\(61\) 5.16488i 0.0108409i −0.999985 0.00542046i \(-0.998275\pi\)
0.999985 0.00542046i \(-0.00172539\pi\)
\(62\) 0 0
\(63\) 166.155 + 13.2469i 0.332279 + 0.0264913i
\(64\) 0 0
\(65\) 159.221 0.303830
\(66\) 0 0
\(67\) 276.852i 0.504818i −0.967621 0.252409i \(-0.918777\pi\)
0.967621 0.252409i \(-0.0812229\pi\)
\(68\) 0 0
\(69\) 570.924i 0.996103i
\(70\) 0 0
\(71\) 474.674i 0.793428i 0.917942 + 0.396714i \(0.129850\pi\)
−0.917942 + 0.396714i \(0.870150\pi\)
\(72\) 0 0
\(73\) 653.765i 1.04818i −0.851662 0.524092i \(-0.824405\pi\)
0.851662 0.524092i \(-0.175595\pi\)
\(74\) 0 0
\(75\) 318.720 0.490701
\(76\) 0 0
\(77\) 31.1029 390.122i 0.0460326 0.577384i
\(78\) 0 0
\(79\) 586.924i 0.835875i 0.908476 + 0.417937i \(0.137247\pi\)
−0.908476 + 0.417937i \(0.862753\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −441.028 −0.583242 −0.291621 0.956534i \(-0.594195\pi\)
−0.291621 + 0.956534i \(0.594195\pi\)
\(84\) 0 0
\(85\) −201.813 −0.257526
\(86\) 0 0
\(87\) −590.643 −0.727857
\(88\) 0 0
\(89\) 862.965i 1.02780i 0.857850 + 0.513899i \(0.171800\pi\)
−0.857850 + 0.513899i \(0.828200\pi\)
\(90\) 0 0
\(91\) −54.1072 + 678.664i −0.0623294 + 0.781794i
\(92\) 0 0
\(93\) 262.747 0.292964
\(94\) 0 0
\(95\) 388.742i 0.419833i
\(96\) 0 0
\(97\) 890.287i 0.931907i −0.884809 0.465953i \(-0.845711\pi\)
0.884809 0.465953i \(-0.154289\pi\)
\(98\) 0 0
\(99\) 190.183i 0.193072i
\(100\) 0 0
\(101\) 684.148i 0.674013i −0.941502 0.337006i \(-0.890586\pi\)
0.941502 0.337006i \(-0.109414\pi\)
\(102\) 0 0
\(103\) −385.671 −0.368944 −0.184472 0.982838i \(-0.559058\pi\)
−0.184472 + 0.982838i \(0.559058\pi\)
\(104\) 0 0
\(105\) 19.1254 239.889i 0.0177757 0.222960i
\(106\) 0 0
\(107\) 341.995i 0.308990i 0.987994 + 0.154495i \(0.0493751\pi\)
−0.987994 + 0.154495i \(0.950625\pi\)
\(108\) 0 0
\(109\) 1807.72 1.58852 0.794259 0.607579i \(-0.207859\pi\)
0.794259 + 0.607579i \(0.207859\pi\)
\(110\) 0 0
\(111\) 53.7644 0.0459738
\(112\) 0 0
\(113\) 707.257 0.588789 0.294394 0.955684i \(-0.404882\pi\)
0.294394 + 0.955684i \(0.404882\pi\)
\(114\) 0 0
\(115\) 824.279 0.668386
\(116\) 0 0
\(117\) 330.846i 0.261425i
\(118\) 0 0
\(119\) 68.5810 860.207i 0.0528303 0.662648i
\(120\) 0 0
\(121\) 884.462 0.664509
\(122\) 0 0
\(123\) 1124.69i 0.824471i
\(124\) 0 0
\(125\) 1001.57i 0.716664i
\(126\) 0 0
\(127\) 1282.50i 0.896092i −0.894010 0.448046i \(-0.852120\pi\)
0.894010 0.448046i \(-0.147880\pi\)
\(128\) 0 0
\(129\) 1513.33i 1.03288i
\(130\) 0 0
\(131\) −2295.10 −1.53071 −0.765357 0.643606i \(-0.777438\pi\)
−0.765357 + 0.643606i \(0.777438\pi\)
\(132\) 0 0
\(133\) 1656.97 + 132.104i 1.08028 + 0.0861268i
\(134\) 0 0
\(135\) 116.945i 0.0745557i
\(136\) 0 0
\(137\) 375.692 0.234288 0.117144 0.993115i \(-0.462626\pi\)
0.117144 + 0.993115i \(0.462626\pi\)
\(138\) 0 0
\(139\) 1346.64 0.821728 0.410864 0.911697i \(-0.365227\pi\)
0.410864 + 0.911697i \(0.365227\pi\)
\(140\) 0 0
\(141\) 1197.86 0.715445
\(142\) 0 0
\(143\) 776.806 0.454264
\(144\) 0 0
\(145\) 852.749i 0.488393i
\(146\) 0 0
\(147\) 1016.00 + 163.040i 0.570057 + 0.0914783i
\(148\) 0 0
\(149\) 2440.27 1.34171 0.670855 0.741588i \(-0.265927\pi\)
0.670855 + 0.741588i \(0.265927\pi\)
\(150\) 0 0
\(151\) 1896.11i 1.02188i −0.859618 0.510938i \(-0.829298\pi\)
0.859618 0.510938i \(-0.170702\pi\)
\(152\) 0 0
\(153\) 419.348i 0.221583i
\(154\) 0 0
\(155\) 379.345i 0.196579i
\(156\) 0 0
\(157\) 3198.03i 1.62567i 0.582493 + 0.812836i \(0.302077\pi\)
−0.582493 + 0.812836i \(0.697923\pi\)
\(158\) 0 0
\(159\) 59.6565 0.0297551
\(160\) 0 0
\(161\) −280.110 + 3513.40i −0.137116 + 1.71984i
\(162\) 0 0
\(163\) 2106.63i 1.01229i 0.862447 + 0.506147i \(0.168931\pi\)
−0.862447 + 0.506147i \(0.831069\pi\)
\(164\) 0 0
\(165\) −274.580 −0.129551
\(166\) 0 0
\(167\) 82.8763 0.0384022 0.0192011 0.999816i \(-0.493888\pi\)
0.0192011 + 0.999816i \(0.493888\pi\)
\(168\) 0 0
\(169\) 845.654 0.384913
\(170\) 0 0
\(171\) 807.768 0.361237
\(172\) 0 0
\(173\) 4383.57i 1.92646i −0.268687 0.963228i \(-0.586590\pi\)
0.268687 0.963228i \(-0.413410\pi\)
\(174\) 0 0
\(175\) 1961.37 + 156.372i 0.847231 + 0.0675464i
\(176\) 0 0
\(177\) 222.860 0.0946394
\(178\) 0 0
\(179\) 2994.66i 1.25045i 0.780443 + 0.625227i \(0.214994\pi\)
−0.780443 + 0.625227i \(0.785006\pi\)
\(180\) 0 0
\(181\) 518.569i 0.212956i −0.994315 0.106478i \(-0.966043\pi\)
0.994315 0.106478i \(-0.0339573\pi\)
\(182\) 0 0
\(183\) 15.4947i 0.00625900i
\(184\) 0 0
\(185\) 77.6231i 0.0308485i
\(186\) 0 0
\(187\) −984.603 −0.385034
\(188\) 0 0
\(189\) 498.465 + 39.7407i 0.191841 + 0.0152948i
\(190\) 0 0
\(191\) 2580.23i 0.977482i −0.872429 0.488741i \(-0.837456\pi\)
0.872429 0.488741i \(-0.162544\pi\)
\(192\) 0 0
\(193\) 861.276 0.321223 0.160611 0.987018i \(-0.448653\pi\)
0.160611 + 0.987018i \(0.448653\pi\)
\(194\) 0 0
\(195\) 477.664 0.175416
\(196\) 0 0
\(197\) 2206.28 0.797922 0.398961 0.916968i \(-0.369371\pi\)
0.398961 + 0.916968i \(0.369371\pi\)
\(198\) 0 0
\(199\) −2855.66 −1.01725 −0.508624 0.860989i \(-0.669846\pi\)
−0.508624 + 0.860989i \(0.669846\pi\)
\(200\) 0 0
\(201\) 830.555i 0.291457i
\(202\) 0 0
\(203\) −3634.75 289.785i −1.25670 0.100192i
\(204\) 0 0
\(205\) 1623.79 0.553221
\(206\) 0 0
\(207\) 1712.77i 0.575100i
\(208\) 0 0
\(209\) 1896.59i 0.627703i
\(210\) 0 0
\(211\) 2314.89i 0.755279i −0.925953 0.377640i \(-0.876736\pi\)
0.925953 0.377640i \(-0.123264\pi\)
\(212\) 0 0
\(213\) 1424.02i 0.458086i
\(214\) 0 0
\(215\) −2184.89 −0.693060
\(216\) 0 0
\(217\) 1616.92 + 128.911i 0.505823 + 0.0403273i
\(218\) 0 0
\(219\) 1961.29i 0.605169i
\(220\) 0 0
\(221\) 1712.83 0.521347
\(222\) 0 0
\(223\) −2776.63 −0.833799 −0.416899 0.908953i \(-0.636883\pi\)
−0.416899 + 0.908953i \(0.636883\pi\)
\(224\) 0 0
\(225\) 956.159 0.283306
\(226\) 0 0
\(227\) 2451.68 0.716846 0.358423 0.933559i \(-0.383315\pi\)
0.358423 + 0.933559i \(0.383315\pi\)
\(228\) 0 0
\(229\) 362.328i 0.104556i 0.998633 + 0.0522780i \(0.0166482\pi\)
−0.998633 + 0.0522780i \(0.983352\pi\)
\(230\) 0 0
\(231\) 93.3088 1170.37i 0.0265769 0.333353i
\(232\) 0 0
\(233\) 6493.96 1.82589 0.912947 0.408078i \(-0.133801\pi\)
0.912947 + 0.408078i \(0.133801\pi\)
\(234\) 0 0
\(235\) 1729.42i 0.480065i
\(236\) 0 0
\(237\) 1760.77i 0.482593i
\(238\) 0 0
\(239\) 272.296i 0.0736962i 0.999321 + 0.0368481i \(0.0117318\pi\)
−0.999321 + 0.0368481i \(0.988268\pi\)
\(240\) 0 0
\(241\) 5941.82i 1.58816i −0.607814 0.794079i \(-0.707953\pi\)
0.607814 0.794079i \(-0.292047\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 235.391 1466.87i 0.0613820 0.382509i
\(246\) 0 0
\(247\) 3299.34i 0.849928i
\(248\) 0 0
\(249\) −1323.08 −0.336735
\(250\) 0 0
\(251\) −3810.08 −0.958129 −0.479065 0.877780i \(-0.659024\pi\)
−0.479065 + 0.877780i \(0.659024\pi\)
\(252\) 0 0
\(253\) 4021.48 0.999322
\(254\) 0 0
\(255\) −605.439 −0.148683
\(256\) 0 0
\(257\) 3955.09i 0.959967i 0.877278 + 0.479983i \(0.159357\pi\)
−0.877278 + 0.479983i \(0.840643\pi\)
\(258\) 0 0
\(259\) 330.860 + 26.3782i 0.0793771 + 0.00632842i
\(260\) 0 0
\(261\) −1771.93 −0.420228
\(262\) 0 0
\(263\) 5506.53i 1.29105i 0.763737 + 0.645527i \(0.223362\pi\)
−0.763737 + 0.645527i \(0.776638\pi\)
\(264\) 0 0
\(265\) 86.1299i 0.0199657i
\(266\) 0 0
\(267\) 2588.90i 0.593400i
\(268\) 0 0
\(269\) 7132.82i 1.61671i 0.588694 + 0.808356i \(0.299642\pi\)
−0.588694 + 0.808356i \(0.700358\pi\)
\(270\) 0 0
\(271\) 5017.04 1.12459 0.562294 0.826937i \(-0.309919\pi\)
0.562294 + 0.826937i \(0.309919\pi\)
\(272\) 0 0
\(273\) −162.322 + 2035.99i −0.0359859 + 0.451369i
\(274\) 0 0
\(275\) 2245.00i 0.492287i
\(276\) 0 0
\(277\) −6117.95 −1.32705 −0.663524 0.748155i \(-0.730940\pi\)
−0.663524 + 0.748155i \(0.730940\pi\)
\(278\) 0 0
\(279\) 788.242 0.169143
\(280\) 0 0
\(281\) 5293.44 1.12377 0.561886 0.827214i \(-0.310076\pi\)
0.561886 + 0.827214i \(0.310076\pi\)
\(282\) 0 0
\(283\) 402.940 0.0846371 0.0423185 0.999104i \(-0.486526\pi\)
0.0423185 + 0.999104i \(0.486526\pi\)
\(284\) 0 0
\(285\) 1166.23i 0.242391i
\(286\) 0 0
\(287\) −551.803 + 6921.23i −0.113491 + 1.42351i
\(288\) 0 0
\(289\) 2741.98 0.558107
\(290\) 0 0
\(291\) 2670.86i 0.538037i
\(292\) 0 0
\(293\) 900.733i 0.179595i −0.995960 0.0897976i \(-0.971378\pi\)
0.995960 0.0897976i \(-0.0286220\pi\)
\(294\) 0 0
\(295\) 321.757i 0.0635031i
\(296\) 0 0
\(297\) 570.549i 0.111470i
\(298\) 0 0
\(299\) −6995.84 −1.35311
\(300\) 0 0
\(301\) 742.477 9312.85i 0.142178 1.78333i
\(302\) 0 0
\(303\) 2052.44i 0.389141i
\(304\) 0 0
\(305\) −22.3706 −0.00419980
\(306\) 0 0
\(307\) 2809.28 0.522262 0.261131 0.965303i \(-0.415905\pi\)
0.261131 + 0.965303i \(0.415905\pi\)
\(308\) 0 0
\(309\) −1157.01 −0.213010
\(310\) 0 0
\(311\) −8837.72 −1.61139 −0.805693 0.592333i \(-0.798207\pi\)
−0.805693 + 0.592333i \(0.798207\pi\)
\(312\) 0 0
\(313\) 7844.03i 1.41652i 0.705952 + 0.708260i \(0.250519\pi\)
−0.705952 + 0.708260i \(0.749481\pi\)
\(314\) 0 0
\(315\) 57.3762 719.666i 0.0102628 0.128726i
\(316\) 0 0
\(317\) 5733.24 1.01581 0.507904 0.861414i \(-0.330421\pi\)
0.507904 + 0.861414i \(0.330421\pi\)
\(318\) 0 0
\(319\) 4160.38i 0.730209i
\(320\) 0 0
\(321\) 1025.99i 0.178395i
\(322\) 0 0
\(323\) 4181.92i 0.720398i
\(324\) 0 0
\(325\) 3905.45i 0.666570i
\(326\) 0 0
\(327\) 5423.17 0.917132
\(328\) 0 0
\(329\) 7371.49 + 587.700i 1.23527 + 0.0984831i
\(330\) 0 0
\(331\) 6075.19i 1.00883i −0.863461 0.504415i \(-0.831708\pi\)
0.863461 0.504415i \(-0.168292\pi\)
\(332\) 0 0
\(333\) 161.293 0.0265430
\(334\) 0 0
\(335\) −1199.13 −0.195568
\(336\) 0 0
\(337\) −1154.68 −0.186646 −0.0933230 0.995636i \(-0.529749\pi\)
−0.0933230 + 0.995636i \(0.529749\pi\)
\(338\) 0 0
\(339\) 2121.77 0.339937
\(340\) 0 0
\(341\) 1850.74i 0.293910i
\(342\) 0 0
\(343\) 6172.37 + 1501.81i 0.971652 + 0.236414i
\(344\) 0 0
\(345\) 2472.84 0.385893
\(346\) 0 0
\(347\) 3063.66i 0.473965i −0.971514 0.236982i \(-0.923842\pi\)
0.971514 0.236982i \(-0.0761583\pi\)
\(348\) 0 0
\(349\) 7903.85i 1.21227i 0.795361 + 0.606136i \(0.207281\pi\)
−0.795361 + 0.606136i \(0.792719\pi\)
\(350\) 0 0
\(351\) 992.538i 0.150934i
\(352\) 0 0
\(353\) 10440.0i 1.57411i −0.616880 0.787057i \(-0.711603\pi\)
0.616880 0.787057i \(-0.288397\pi\)
\(354\) 0 0
\(355\) 2055.95 0.307376
\(356\) 0 0
\(357\) 205.743 2580.62i 0.0305016 0.382580i
\(358\) 0 0
\(359\) 7891.79i 1.16020i −0.814544 0.580101i \(-0.803013\pi\)
0.814544 0.580101i \(-0.196987\pi\)
\(360\) 0 0
\(361\) 1196.42 0.174431
\(362\) 0 0
\(363\) 2653.38 0.383655
\(364\) 0 0
\(365\) −2831.65 −0.406069
\(366\) 0 0
\(367\) 69.3175 0.00985924 0.00492962 0.999988i \(-0.498431\pi\)
0.00492962 + 0.999988i \(0.498431\pi\)
\(368\) 0 0
\(369\) 3374.07i 0.476009i
\(370\) 0 0
\(371\) 367.120 + 29.2690i 0.0513744 + 0.00409588i
\(372\) 0 0
\(373\) 1221.71 0.169591 0.0847956 0.996398i \(-0.472976\pi\)
0.0847956 + 0.996398i \(0.472976\pi\)
\(374\) 0 0
\(375\) 3004.70i 0.413766i
\(376\) 0 0
\(377\) 7237.47i 0.988724i
\(378\) 0 0
\(379\) 9565.80i 1.29647i 0.761440 + 0.648235i \(0.224493\pi\)
−0.761440 + 0.648235i \(0.775507\pi\)
\(380\) 0 0
\(381\) 3847.51i 0.517359i
\(382\) 0 0
\(383\) −5413.33 −0.722215 −0.361107 0.932524i \(-0.617601\pi\)
−0.361107 + 0.932524i \(0.617601\pi\)
\(384\) 0 0
\(385\) −1689.73 134.716i −0.223680 0.0178331i
\(386\) 0 0
\(387\) 4539.98i 0.596331i
\(388\) 0 0
\(389\) −5222.11 −0.680647 −0.340323 0.940308i \(-0.610537\pi\)
−0.340323 + 0.940308i \(0.610537\pi\)
\(390\) 0 0
\(391\) 8867.24 1.14689
\(392\) 0 0
\(393\) −6885.29 −0.883758
\(394\) 0 0
\(395\) 2542.14 0.323820
\(396\) 0 0
\(397\) 13550.8i 1.71308i −0.516079 0.856541i \(-0.672609\pi\)
0.516079 0.856541i \(-0.327391\pi\)
\(398\) 0 0
\(399\) 4970.92 + 396.312i 0.623702 + 0.0497254i
\(400\) 0 0
\(401\) −14908.0 −1.85654 −0.928268 0.371912i \(-0.878702\pi\)
−0.928268 + 0.371912i \(0.878702\pi\)
\(402\) 0 0
\(403\) 3219.59i 0.397963i
\(404\) 0 0
\(405\) 350.835i 0.0430447i
\(406\) 0 0
\(407\) 378.707i 0.0461223i
\(408\) 0 0
\(409\) 5137.85i 0.621150i −0.950549 0.310575i \(-0.899478\pi\)
0.950549 0.310575i \(-0.100522\pi\)
\(410\) 0 0
\(411\) 1127.07 0.135266
\(412\) 0 0
\(413\) 1371.46 + 109.341i 0.163402 + 0.0130274i
\(414\) 0 0
\(415\) 1910.22i 0.225949i
\(416\) 0 0
\(417\) 4039.91 0.474425
\(418\) 0 0
\(419\) 11861.5 1.38299 0.691494 0.722382i \(-0.256953\pi\)
0.691494 + 0.722382i \(0.256953\pi\)
\(420\) 0 0
\(421\) −5965.71 −0.690619 −0.345310 0.938489i \(-0.612226\pi\)
−0.345310 + 0.938489i \(0.612226\pi\)
\(422\) 0 0
\(423\) 3593.57 0.413063
\(424\) 0 0
\(425\) 4950.16i 0.564984i
\(426\) 0 0
\(427\) 7.60208 95.3524i 0.000861570 0.0108066i
\(428\) 0 0
\(429\) 2330.42 0.262270
\(430\) 0 0
\(431\) 9360.31i 1.04610i −0.852301 0.523051i \(-0.824794\pi\)
0.852301 0.523051i \(-0.175206\pi\)
\(432\) 0 0
\(433\) 9022.58i 1.00138i 0.865627 + 0.500690i \(0.166920\pi\)
−0.865627 + 0.500690i \(0.833080\pi\)
\(434\) 0 0
\(435\) 2558.25i 0.281974i
\(436\) 0 0
\(437\) 17080.5i 1.86973i
\(438\) 0 0
\(439\) 2551.97 0.277446 0.138723 0.990331i \(-0.455700\pi\)
0.138723 + 0.990331i \(0.455700\pi\)
\(440\) 0 0
\(441\) 3048.00 + 489.120i 0.329123 + 0.0528150i
\(442\) 0 0
\(443\) 14077.7i 1.50983i 0.655824 + 0.754914i \(0.272321\pi\)
−0.655824 + 0.754914i \(0.727679\pi\)
\(444\) 0 0
\(445\) 3737.75 0.398172
\(446\) 0 0
\(447\) 7320.82 0.774637
\(448\) 0 0
\(449\) −15244.7 −1.60232 −0.801162 0.598448i \(-0.795784\pi\)
−0.801162 + 0.598448i \(0.795784\pi\)
\(450\) 0 0
\(451\) 7922.12 0.827135
\(452\) 0 0
\(453\) 5688.33i 0.589980i
\(454\) 0 0
\(455\) 2939.49 + 234.354i 0.302869 + 0.0241466i
\(456\) 0 0
\(457\) 2533.83 0.259360 0.129680 0.991556i \(-0.458605\pi\)
0.129680 + 0.991556i \(0.458605\pi\)
\(458\) 0 0
\(459\) 1258.04i 0.127931i
\(460\) 0 0
\(461\) 9315.46i 0.941137i 0.882363 + 0.470569i \(0.155951\pi\)
−0.882363 + 0.470569i \(0.844049\pi\)
\(462\) 0 0
\(463\) 15998.0i 1.60581i −0.596104 0.802907i \(-0.703286\pi\)
0.596104 0.802907i \(-0.296714\pi\)
\(464\) 0 0
\(465\) 1138.04i 0.113495i
\(466\) 0 0
\(467\) −13869.3 −1.37430 −0.687148 0.726518i \(-0.741138\pi\)
−0.687148 + 0.726518i \(0.741138\pi\)
\(468\) 0 0
\(469\) 407.492 5111.15i 0.0401199 0.503221i
\(470\) 0 0
\(471\) 9594.08i 0.938582i
\(472\) 0 0
\(473\) −10659.6 −1.03621
\(474\) 0 0
\(475\) 9535.25 0.921068
\(476\) 0 0
\(477\) 178.970 0.0171791
\(478\) 0 0
\(479\) −16338.3 −1.55849 −0.779245 0.626719i \(-0.784397\pi\)
−0.779245 + 0.626719i \(0.784397\pi\)
\(480\) 0 0
\(481\) 658.805i 0.0624510i
\(482\) 0 0
\(483\) −840.330 + 10540.2i −0.0791642 + 0.992952i
\(484\) 0 0
\(485\) −3856.09 −0.361023
\(486\) 0 0
\(487\) 5149.77i 0.479175i 0.970875 + 0.239588i \(0.0770122\pi\)
−0.970875 + 0.239588i \(0.922988\pi\)
\(488\) 0 0
\(489\) 6319.89i 0.584448i
\(490\) 0 0
\(491\) 1425.88i 0.131057i 0.997851 + 0.0655287i \(0.0208734\pi\)
−0.997851 + 0.0655287i \(0.979127\pi\)
\(492\) 0 0
\(493\) 9173.51i 0.838041i
\(494\) 0 0
\(495\) −823.739 −0.0747966
\(496\) 0 0
\(497\) −698.662 + 8763.28i −0.0630569 + 0.790919i
\(498\) 0 0
\(499\) 9498.21i 0.852101i 0.904699 + 0.426050i \(0.140095\pi\)
−0.904699 + 0.426050i \(0.859905\pi\)
\(500\) 0 0
\(501\) 248.629 0.0221715
\(502\) 0 0
\(503\) −16057.1 −1.42336 −0.711680 0.702503i \(-0.752066\pi\)
−0.711680 + 0.702503i \(0.752066\pi\)
\(504\) 0 0
\(505\) −2963.25 −0.261114
\(506\) 0 0
\(507\) 2536.96 0.222230
\(508\) 0 0
\(509\) 3313.42i 0.288536i 0.989539 + 0.144268i \(0.0460828\pi\)
−0.989539 + 0.144268i \(0.953917\pi\)
\(510\) 0 0
\(511\) 962.262 12069.6i 0.0833032 1.04487i
\(512\) 0 0
\(513\) 2423.30 0.208560
\(514\) 0 0
\(515\) 1670.45i 0.142930i
\(516\) 0 0
\(517\) 8437.49i 0.717757i
\(518\) 0 0
\(519\) 13150.7i 1.11224i
\(520\) 0 0
\(521\) 7862.04i 0.661118i −0.943785 0.330559i \(-0.892763\pi\)
0.943785 0.330559i \(-0.107237\pi\)
\(522\) 0 0
\(523\) −17648.7 −1.47557 −0.737784 0.675037i \(-0.764127\pi\)
−0.737784 + 0.675037i \(0.764127\pi\)
\(524\) 0 0
\(525\) 5884.10 + 469.116i 0.489149 + 0.0389979i
\(526\) 0 0
\(527\) 4080.83i 0.337313i
\(528\) 0 0
\(529\) −24050.1 −1.97666
\(530\) 0 0
\(531\) 668.580 0.0546401
\(532\) 0 0
\(533\) −13781.5 −1.11997
\(534\) 0 0
\(535\) 1481.28 0.119704
\(536\) 0 0
\(537\) 8983.97i 0.721950i
\(538\) 0 0
\(539\) 1148.42 7156.53i 0.0917739 0.571899i
\(540\) 0 0
\(541\) −744.076 −0.0591319 −0.0295659 0.999563i \(-0.509413\pi\)
−0.0295659 + 0.999563i \(0.509413\pi\)
\(542\) 0 0
\(543\) 1555.71i 0.122950i
\(544\) 0 0
\(545\) 7829.78i 0.615396i
\(546\) 0 0
\(547\) 9139.33i 0.714387i −0.934031 0.357193i \(-0.883734\pi\)
0.934031 0.357193i \(-0.116266\pi\)
\(548\) 0 0
\(549\) 46.4840i 0.00361364i
\(550\) 0 0
\(551\) −17670.5 −1.36622
\(552\) 0 0
\(553\) −863.880 + 10835.6i −0.0664303 + 0.833231i
\(554\) 0 0
\(555\) 232.869i 0.0178104i
\(556\) 0 0
\(557\) −6925.09 −0.526796 −0.263398 0.964687i \(-0.584843\pi\)
−0.263398 + 0.964687i \(0.584843\pi\)
\(558\) 0 0
\(559\) 18543.6 1.40306
\(560\) 0 0
\(561\) −2953.81 −0.222299
\(562\) 0 0
\(563\) −22776.2 −1.70497 −0.852487 0.522748i \(-0.824907\pi\)
−0.852487 + 0.522748i \(0.824907\pi\)
\(564\) 0 0
\(565\) 3063.34i 0.228098i
\(566\) 0 0
\(567\) 1495.40 + 119.222i 0.110760 + 0.00883044i
\(568\) 0 0
\(569\) −9189.15 −0.677028 −0.338514 0.940961i \(-0.609924\pi\)
−0.338514 + 0.940961i \(0.609924\pi\)
\(570\) 0 0
\(571\) 7124.04i 0.522122i 0.965322 + 0.261061i \(0.0840724\pi\)
−0.965322 + 0.261061i \(0.915928\pi\)
\(572\) 0 0
\(573\) 7740.70i 0.564349i
\(574\) 0 0
\(575\) 20218.3i 1.46637i
\(576\) 0 0
\(577\) 20110.0i 1.45094i 0.688255 + 0.725469i \(0.258377\pi\)
−0.688255 + 0.725469i \(0.741623\pi\)
\(578\) 0 0
\(579\) 2583.83 0.185458
\(580\) 0 0
\(581\) −8142.11 649.139i −0.581397 0.0463525i
\(582\) 0 0
\(583\) 420.210i 0.0298513i
\(584\) 0 0
\(585\) 1432.99 0.101277
\(586\) 0 0
\(587\) −11857.5 −0.833753 −0.416876 0.908963i \(-0.636875\pi\)
−0.416876 + 0.908963i \(0.636875\pi\)
\(588\) 0 0
\(589\) 7860.70 0.549906
\(590\) 0 0
\(591\) 6618.83 0.460681
\(592\) 0 0
\(593\) 195.777i 0.0135575i −0.999977 0.00677876i \(-0.997842\pi\)
0.999977 0.00677876i \(-0.00215776\pi\)
\(594\) 0 0
\(595\) −3725.81 297.044i −0.256711 0.0204666i
\(596\) 0 0
\(597\) −8566.98 −0.587308
\(598\) 0 0
\(599\) 12375.6i 0.844161i −0.906558 0.422081i \(-0.861300\pi\)
0.906558 0.422081i \(-0.138700\pi\)
\(600\) 0 0
\(601\) 20034.3i 1.35976i −0.733322 0.679882i \(-0.762031\pi\)
0.733322 0.679882i \(-0.237969\pi\)
\(602\) 0 0
\(603\) 2491.67i 0.168273i
\(604\) 0 0
\(605\) 3830.86i 0.257433i
\(606\) 0 0
\(607\) −14743.8 −0.985883 −0.492941 0.870062i \(-0.664078\pi\)
−0.492941 + 0.870062i \(0.664078\pi\)
\(608\) 0 0
\(609\) −10904.3 869.354i −0.725555 0.0578457i
\(610\) 0 0
\(611\) 14678.0i 0.971864i
\(612\) 0 0
\(613\) −25967.4 −1.71095 −0.855476 0.517843i \(-0.826735\pi\)
−0.855476 + 0.517843i \(0.826735\pi\)
\(614\) 0 0
\(615\) 4871.37 0.319402
\(616\) 0 0
\(617\) 23573.3 1.53813 0.769064 0.639172i \(-0.220723\pi\)
0.769064 + 0.639172i \(0.220723\pi\)
\(618\) 0 0
\(619\) 19905.1 1.29250 0.646248 0.763127i \(-0.276337\pi\)
0.646248 + 0.763127i \(0.276337\pi\)
\(620\) 0 0
\(621\) 5138.31i 0.332034i
\(622\) 0 0
\(623\) −1270.18 + 15931.8i −0.0816832 + 1.02455i
\(624\) 0 0
\(625\) 8941.90 0.572282
\(626\) 0 0
\(627\) 5689.77i 0.362404i
\(628\) 0 0
\(629\) 835.036i 0.0529333i
\(630\) 0 0
\(631\) 17257.1i 1.08874i 0.838845 + 0.544370i \(0.183231\pi\)
−0.838845 + 0.544370i \(0.816769\pi\)
\(632\) 0 0
\(633\) 6944.68i 0.436061i
\(634\) 0 0
\(635\) −5554.89 −0.347148
\(636\) 0 0
\(637\) −1997.82 + 12449.6i −0.124265 + 0.774368i
\(638\) 0 0
\(639\) 4272.06i 0.264476i
\(640\) 0 0
\(641\) −22540.0 −1.38889 −0.694443 0.719547i \(-0.744349\pi\)
−0.694443 + 0.719547i \(0.744349\pi\)
\(642\) 0 0
\(643\) −2084.87 −0.127868 −0.0639341 0.997954i \(-0.520365\pi\)
−0.0639341 + 0.997954i \(0.520365\pi\)
\(644\) 0 0
\(645\) −6554.66 −0.400139
\(646\) 0 0
\(647\) −19323.6 −1.17417 −0.587087 0.809524i \(-0.699725\pi\)
−0.587087 + 0.809524i \(0.699725\pi\)
\(648\) 0 0
\(649\) 1569.78i 0.0949452i
\(650\) 0 0
\(651\) 4850.76 + 386.732i 0.292037 + 0.0232830i
\(652\) 0 0
\(653\) 10267.1 0.615287 0.307643 0.951502i \(-0.400460\pi\)
0.307643 + 0.951502i \(0.400460\pi\)
\(654\) 0 0
\(655\) 9940.74i 0.593003i
\(656\) 0 0
\(657\) 5883.88i 0.349394i
\(658\) 0 0
\(659\) 16592.3i 0.980797i −0.871498 0.490398i \(-0.836851\pi\)
0.871498 0.490398i \(-0.163149\pi\)
\(660\) 0 0
\(661\) 6908.93i 0.406545i 0.979122 + 0.203272i \(0.0651577\pi\)
−0.979122 + 0.203272i \(0.934842\pi\)
\(662\) 0 0
\(663\) 5138.50 0.301000
\(664\) 0 0
\(665\) 572.181 7176.84i 0.0333658 0.418505i
\(666\) 0 0
\(667\) 37468.0i 2.17506i
\(668\) 0 0
\(669\) −8329.90 −0.481394
\(670\) 0 0
\(671\) −109.142 −0.00627923
\(672\) 0 0
\(673\) −26133.0 −1.49681 −0.748404 0.663243i \(-0.769180\pi\)
−0.748404 + 0.663243i \(0.769180\pi\)
\(674\) 0 0
\(675\) 2868.48 0.163567
\(676\) 0 0
\(677\) 5448.04i 0.309284i 0.987971 + 0.154642i \(0.0494224\pi\)
−0.987971 + 0.154642i \(0.950578\pi\)
\(678\) 0 0
\(679\) 1310.39 16436.2i 0.0740623 0.928959i
\(680\) 0 0
\(681\) 7355.05 0.413871
\(682\) 0 0
\(683\) 3401.50i 0.190564i 0.995450 + 0.0952818i \(0.0303752\pi\)
−0.995450 + 0.0952818i \(0.969625\pi\)
\(684\) 0 0
\(685\) 1627.23i 0.0907639i
\(686\) 0 0
\(687\) 1086.99i 0.0603655i
\(688\) 0 0
\(689\) 731.004i 0.0404195i
\(690\) 0 0
\(691\) −2782.64 −0.153193 −0.0765967 0.997062i \(-0.524405\pi\)
−0.0765967 + 0.997062i \(0.524405\pi\)
\(692\) 0 0
\(693\) 279.926 3511.10i 0.0153442 0.192461i
\(694\) 0 0
\(695\) 5832.67i 0.318339i
\(696\) 0 0
\(697\) 17468.0 0.949281
\(698\) 0 0
\(699\) 19481.9 1.05418
\(700\) 0 0
\(701\) −7163.73 −0.385978 −0.192989 0.981201i \(-0.561818\pi\)
−0.192989 + 0.981201i \(0.561818\pi\)
\(702\) 0 0
\(703\) 1608.49 0.0862948
\(704\) 0 0
\(705\) 5188.27i 0.277165i
\(706\) 0 0
\(707\) 1006.98 12630.5i 0.0535664 0.671881i
\(708\) 0 0
\(709\) −30301.5 −1.60507 −0.802537 0.596602i \(-0.796517\pi\)
−0.802537 + 0.596602i \(0.796517\pi\)
\(710\) 0 0
\(711\) 5282.32i 0.278625i
\(712\) 0 0
\(713\) 16667.6i 0.875466i
\(714\) 0 0
\(715\) 3364.58i 0.175983i
\(716\) 0 0
\(717\) 816.889i 0.0425485i
\(718\) 0 0
\(719\) −4035.13 −0.209298 −0.104649 0.994509i \(-0.533372\pi\)
−0.104649 + 0.994509i \(0.533372\pi\)
\(720\) 0 0
\(721\) −7120.13 567.660i −0.367777 0.0293215i
\(722\) 0 0
\(723\) 17825.5i 0.916924i
\(724\) 0 0
\(725\) −20916.6 −1.07148
\(726\) 0 0
\(727\) 390.945 0.0199441 0.00997205 0.999950i \(-0.496826\pi\)
0.00997205 + 0.999950i \(0.496826\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −23504.1 −1.18923
\(732\) 0 0
\(733\) 7225.92i 0.364114i 0.983288 + 0.182057i \(0.0582755\pi\)
−0.983288 + 0.182057i \(0.941724\pi\)
\(734\) 0 0
\(735\) 706.174 4400.60i 0.0354389 0.220842i
\(736\) 0 0
\(737\) −5850.28 −0.292399
\(738\) 0 0
\(739\) 3342.36i 0.166374i −0.996534 0.0831872i \(-0.973490\pi\)
0.996534 0.0831872i \(-0.0265100\pi\)
\(740\) 0 0
\(741\) 9898.03i 0.490706i
\(742\) 0 0
\(743\) 33993.8i 1.67848i 0.543761 + 0.839240i \(0.317000\pi\)
−0.543761 + 0.839240i \(0.683000\pi\)
\(744\) 0 0
\(745\) 10569.5i 0.519782i
\(746\) 0 0
\(747\) −3969.25 −0.194414
\(748\) 0 0
\(749\) −503.375 + 6313.81i −0.0245567 + 0.308013i
\(750\) 0 0
\(751\) 30957.8i 1.50422i 0.659039 + 0.752109i \(0.270963\pi\)
−0.659039 + 0.752109i \(0.729037\pi\)
\(752\) 0 0
\(753\) −11430.3 −0.553176
\(754\) 0 0
\(755\) −8212.61 −0.395877
\(756\) 0 0
\(757\) −25919.3 −1.24446 −0.622228 0.782836i \(-0.713772\pi\)
−0.622228 + 0.782836i \(0.713772\pi\)
\(758\) 0 0
\(759\) 12064.4 0.576959
\(760\) 0 0
\(761\) 29091.1i 1.38575i 0.721059 + 0.692873i \(0.243656\pi\)
−0.721059 + 0.692873i \(0.756344\pi\)
\(762\) 0 0
\(763\) 33373.6 + 2660.75i 1.58349 + 0.126246i
\(764\) 0 0
\(765\) −1816.32 −0.0858420
\(766\) 0 0
\(767\) 2730.83i 0.128559i
\(768\) 0 0
\(769\) 11075.2i 0.519352i 0.965696 + 0.259676i \(0.0836157\pi\)
−0.965696 + 0.259676i \(0.916384\pi\)
\(770\) 0 0
\(771\) 11865.3i 0.554237i
\(772\) 0 0
\(773\) 36140.2i 1.68160i −0.541350 0.840798i \(-0.682086\pi\)
0.541350 0.840798i \(-0.317914\pi\)
\(774\) 0 0
\(775\) 9304.75 0.431273
\(776\) 0 0
\(777\) 992.581 + 79.1346i 0.0458284 + 0.00365372i
\(778\) 0 0
\(779\) 33647.8i 1.54757i
\(780\) 0 0
\(781\) 10030.5 0.459566
\(782\) 0 0
\(783\) −5315.78 −0.242619
\(784\) 0 0
\(785\) 13851.6 0.629789
\(786\) 0 0
\(787\) −28453.9 −1.28878 −0.644392 0.764695i \(-0.722890\pi\)
−0.644392 + 0.764695i \(0.722890\pi\)
\(788\) 0 0
\(789\) 16519.6i 0.745391i
\(790\) 0 0
\(791\) 13057.1 + 1041.00i 0.586926 + 0.0467933i
\(792\) 0 0
\(793\) 189.865 0.00850226
\(794\) 0 0
\(795\) 258.390i 0.0115272i
\(796\) 0 0
\(797\) 22987.4i 1.02165i −0.859684 0.510826i \(-0.829340\pi\)
0.859684 0.510826i \(-0.170660\pi\)
\(798\) 0 0
\(799\) 18604.4i 0.823750i
\(800\) 0 0
\(801\) 7766.69i 0.342600i
\(802\) 0 0
\(803\) −13815.0 −0.607124
\(804\) 0 0
\(805\) 15217.6 + 1213.24i 0.666272 + 0.0531193i
\(806\) 0 0
\(807\) 21398.5i 0.933409i
\(808\) 0 0
\(809\) 4688.53 0.203758 0.101879 0.994797i \(-0.467515\pi\)
0.101879 + 0.994797i \(0.467515\pi\)
\(810\) 0 0
\(811\) 26381.0 1.14225 0.571124 0.820864i \(-0.306508\pi\)
0.571124 + 0.820864i \(0.306508\pi\)
\(812\) 0 0
\(813\) 15051.1 0.649282
\(814\) 0 0
\(815\) 9124.43 0.392166
\(816\) 0 0
\(817\) 45274.7i 1.93875i
\(818\) 0 0
\(819\) −486.965 + 6107.97i −0.0207765 + 0.260598i
\(820\) 0 0
\(821\) 1244.52 0.0529039 0.0264520 0.999650i \(-0.491579\pi\)
0.0264520 + 0.999650i \(0.491579\pi\)
\(822\) 0 0
\(823\) 7926.32i 0.335716i −0.985811 0.167858i \(-0.946315\pi\)
0.985811 0.167858i \(-0.0536850\pi\)
\(824\) 0 0
\(825\) 6735.01i 0.284222i
\(826\) 0 0
\(827\) 32063.4i 1.34819i −0.738644 0.674095i \(-0.764534\pi\)
0.738644 0.674095i \(-0.235466\pi\)
\(828\) 0 0
\(829\) 41224.4i 1.72712i −0.504246 0.863560i \(-0.668230\pi\)
0.504246 0.863560i \(-0.331770\pi\)
\(830\) 0 0
\(831\) −18353.9 −0.766171
\(832\) 0 0
\(833\) 2532.24 15779.9i 0.105326 0.656353i
\(834\) 0 0
\(835\) 358.962i 0.0148771i
\(836\) 0 0
\(837\) 2364.73 0.0976546
\(838\) 0 0
\(839\) 25040.7 1.03039 0.515197 0.857072i \(-0.327719\pi\)
0.515197 + 0.857072i \(0.327719\pi\)
\(840\) 0 0
\(841\) 14373.1 0.589327
\(842\) 0 0
\(843\) 15880.3 0.648810
\(844\) 0 0
\(845\) 3662.77i 0.149116i
\(846\) 0 0
\(847\) 16328.6 + 1301.82i 0.662407 + 0.0528112i
\(848\) 0 0
\(849\) 1208.82 0.0488652
\(850\) 0 0
\(851\) 3410.59i 0.137384i
\(852\) 0 0
\(853\) 35611.4i 1.42944i 0.699411 + 0.714720i \(0.253446\pi\)
−0.699411 + 0.714720i \(0.746554\pi\)
\(854\) 0 0
\(855\) 3498.68i 0.139944i
\(856\) 0 0
\(857\) 20425.0i 0.814125i 0.913400 + 0.407063i \(0.133447\pi\)
−0.913400 + 0.407063i \(0.866553\pi\)
\(858\) 0 0
\(859\) 7823.05 0.310732 0.155366 0.987857i \(-0.450344\pi\)
0.155366 + 0.987857i \(0.450344\pi\)
\(860\) 0 0
\(861\) −1655.41 + 20763.7i −0.0655240 + 0.821863i
\(862\) 0 0
\(863\) 41712.4i 1.64532i −0.568536 0.822658i \(-0.692490\pi\)
0.568536 0.822658i \(-0.307510\pi\)
\(864\) 0 0
\(865\) −18986.5 −0.746314
\(866\) 0 0
\(867\) 8225.94 0.322223
\(868\) 0 0
\(869\) 12402.6 0.484152
\(870\) 0 0
\(871\) 10177.3 0.395916
\(872\) 0 0
\(873\) 8012.58i 0.310636i
\(874\) 0 0
\(875\) 1474.19 18490.6i 0.0569561 0.714397i
\(876\) 0 0
\(877\) 10968.2 0.422316 0.211158 0.977452i \(-0.432277\pi\)
0.211158 + 0.977452i \(0.432277\pi\)
\(878\) 0 0
\(879\) 2702.20i 0.103689i
\(880\) 0 0
\(881\) 29482.5i 1.12746i 0.825960 + 0.563729i \(0.190634\pi\)
−0.825960 + 0.563729i \(0.809366\pi\)
\(882\) 0 0
\(883\) 39081.9i 1.48948i 0.667356 + 0.744739i \(0.267426\pi\)
−0.667356 + 0.744739i \(0.732574\pi\)
\(884\) 0 0
\(885\) 965.271i 0.0366636i
\(886\) 0 0
\(887\) −44545.5 −1.68624 −0.843118 0.537728i \(-0.819283\pi\)
−0.843118 + 0.537728i \(0.819283\pi\)
\(888\) 0 0
\(889\) 1887.69 23677.1i 0.0712160 0.893258i
\(890\) 0 0
\(891\) 1711.65i 0.0643573i
\(892\) 0 0
\(893\) 35836.7 1.34292
\(894\) 0 0
\(895\) 12970.7 0.484429
\(896\) 0 0
\(897\) −20987.5 −0.781219
\(898\) 0 0
\(899\) −17243.3 −0.639707
\(900\) 0 0
\(901\) 926.549i 0.0342595i
\(902\) 0 0
\(903\) 2227.43 27938.5i 0.0820867 1.02961i
\(904\) 0 0
\(905\) −2246.08 −0.0824995
\(906\) 0 0
\(907\) 12446.7i 0.455662i −0.973701 0.227831i \(-0.926837\pi\)
0.973701 0.227831i \(-0.0731634\pi\)
\(908\) 0 0
\(909\) 6157.33i 0.224671i
\(910\) 0 0
\(911\) 5652.41i 0.205568i −0.994704 0.102784i \(-0.967225\pi\)
0.994704 0.102784i \(-0.0327751\pi\)
\(912\) 0 0
\(913\) 9319.56i 0.337823i
\(914\) 0 0
\(915\) −67.1119 −0.00242475
\(916\) 0 0
\(917\) −42371.3 3378.10i −1.52587 0.121652i
\(918\) 0 0
\(919\) 27140.5i 0.974191i −0.873349 0.487096i \(-0.838056\pi\)
0.873349 0.487096i \(-0.161944\pi\)
\(920\) 0 0
\(921\) 8427.85 0.301528
\(922\) 0 0
\(923\) −17449.3 −0.622266
\(924\) 0 0
\(925\) 1903.97 0.0676782
\(926\) 0 0
\(927\) −3471.04 −0.122981
\(928\) 0 0
\(929\) 3257.14i 0.115031i 0.998345 + 0.0575153i \(0.0183178\pi\)
−0.998345 + 0.0575153i \(0.981682\pi\)
\(930\) 0 0
\(931\) 30396.1 + 4877.72i 1.07002 + 0.171709i
\(932\) 0 0
\(933\) −26513.2 −0.930334
\(934\) 0 0
\(935\) 4264.61i 0.149163i
\(936\) 0 0
\(937\) 20532.1i 0.715852i 0.933750 + 0.357926i \(0.116516\pi\)
−0.933750 + 0.357926i \(0.883484\pi\)
\(938\) 0 0
\(939\) 23532.1i 0.817828i
\(940\) 0 0
\(941\) 39190.7i 1.35768i −0.734284 0.678842i \(-0.762482\pi\)
0.734284 0.678842i \(-0.237518\pi\)
\(942\) 0 0
\(943\) −71345.8 −2.46378
\(944\) 0 0
\(945\) 172.129 2159.00i 0.00592523 0.0743198i
\(946\) 0 0
\(947\) 20071.9i 0.688754i 0.938831 + 0.344377i \(0.111910\pi\)
−0.938831 + 0.344377i \(0.888090\pi\)
\(948\) 0 0
\(949\) 24032.8 0.822064
\(950\) 0 0
\(951\) 17199.7 0.586477
\(952\) 0 0
\(953\) 21364.2 0.726186 0.363093 0.931753i \(-0.381721\pi\)
0.363093 + 0.931753i \(0.381721\pi\)
\(954\) 0 0
\(955\) −11175.7 −0.378679
\(956\) 0 0
\(957\) 12481.1i 0.421586i
\(958\) 0 0
\(959\) 6935.90 + 552.972i 0.233547 + 0.0186198i
\(960\) 0 0
\(961\) −22120.3 −0.742517
\(962\) 0 0
\(963\) 3077.96i 0.102997i
\(964\) 0 0
\(965\) 3730.44i 0.124443i
\(966\) 0 0
\(967\) 437.986i 0.0145654i 0.999973 + 0.00728268i \(0.00231817\pi\)
−0.999973 + 0.00728268i \(0.997682\pi\)
\(968\) 0 0
\(969\) 12545.8i 0.415922i
\(970\) 0 0
\(971\) −43517.0 −1.43824 −0.719118 0.694888i \(-0.755454\pi\)
−0.719118 + 0.694888i \(0.755454\pi\)
\(972\) 0 0
\(973\) 24861.2 + 1982.08i 0.819129 + 0.0653059i
\(974\) 0 0
\(975\) 11716.3i 0.384845i
\(976\) 0 0
\(977\) 3333.85 0.109170 0.0545851 0.998509i \(-0.482616\pi\)
0.0545851 + 0.998509i \(0.482616\pi\)
\(978\) 0 0
\(979\) 18235.7 0.595317
\(980\) 0 0
\(981\) 16269.5 0.529506
\(982\) 0 0
\(983\) 10148.5 0.329284 0.164642 0.986353i \(-0.447353\pi\)
0.164642 + 0.986353i \(0.447353\pi\)
\(984\) 0 0
\(985\) 9556.03i 0.309117i
\(986\) 0 0
\(987\) 22114.5 + 1763.10i 0.713182 + 0.0568593i
\(988\) 0 0
\(989\) 95999.3 3.08655
\(990\) 0 0
\(991\) 38849.6i 1.24531i −0.782498 0.622653i \(-0.786055\pi\)
0.782498 0.622653i \(-0.213945\pi\)
\(992\) 0 0
\(993\) 18225.6i 0.582448i
\(994\) 0 0
\(995\) 12368.7i 0.394085i
\(996\) 0 0
\(997\) 32853.3i 1.04360i 0.853066 + 0.521802i \(0.174740\pi\)
−0.853066 + 0.521802i \(0.825260\pi\)
\(998\) 0 0
\(999\) 483.879 0.0153246
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 1344.4.b.f.895.4 8
4.3 odd 2 1344.4.b.e.895.4 8
7.6 odd 2 1344.4.b.e.895.5 8
8.3 odd 2 336.4.b.f.223.5 yes 8
8.5 even 2 336.4.b.e.223.5 yes 8
24.5 odd 2 1008.4.b.i.559.4 8
24.11 even 2 1008.4.b.k.559.4 8
28.27 even 2 inner 1344.4.b.f.895.5 8
56.13 odd 2 336.4.b.f.223.4 yes 8
56.27 even 2 336.4.b.e.223.4 8
168.83 odd 2 1008.4.b.i.559.5 8
168.125 even 2 1008.4.b.k.559.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.4.b.e.223.4 8 56.27 even 2
336.4.b.e.223.5 yes 8 8.5 even 2
336.4.b.f.223.4 yes 8 56.13 odd 2
336.4.b.f.223.5 yes 8 8.3 odd 2
1008.4.b.i.559.4 8 24.5 odd 2
1008.4.b.i.559.5 8 168.83 odd 2
1008.4.b.k.559.4 8 24.11 even 2
1008.4.b.k.559.5 8 168.125 even 2
1344.4.b.e.895.4 8 4.3 odd 2
1344.4.b.e.895.5 8 7.6 odd 2
1344.4.b.f.895.4 8 1.1 even 1 trivial
1344.4.b.f.895.5 8 28.27 even 2 inner