Properties

Label 1344.4.b.e.895.6
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 895.6
Root \(-8.49618i\) of defining polynomial
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.e.895.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +7.52281i q^{5} +(4.86375 + 17.8702i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +7.52281i q^{5} +(4.86375 + 17.8702i) q^{7} +9.00000 q^{9} -29.4068i q^{11} -5.35686i q^{13} -22.5684i q^{15} +66.3364i q^{17} -22.3894 q^{19} +(-14.5913 - 53.6106i) q^{21} +33.9994i q^{23} +68.4073 q^{25} -27.0000 q^{27} +133.266 q^{29} +323.973 q^{31} +88.2204i q^{33} +(-134.434 + 36.5891i) q^{35} -120.395 q^{37} +16.0706i q^{39} +140.290i q^{41} -19.0522i q^{43} +67.7053i q^{45} +376.658 q^{47} +(-295.688 + 173.832i) q^{49} -199.009i q^{51} +441.895 q^{53} +221.222 q^{55} +67.1681 q^{57} -241.815 q^{59} -130.894i q^{61} +(43.7738 + 160.832i) q^{63} +40.2987 q^{65} -627.240i q^{67} -101.998i q^{69} -808.143i q^{71} +417.527i q^{73} -205.222 q^{75} +(525.505 - 143.027i) q^{77} +214.825i q^{79} +81.0000 q^{81} -639.640 q^{83} -499.037 q^{85} -399.798 q^{87} +686.995i q^{89} +(95.7281 - 26.0545i) q^{91} -971.918 q^{93} -168.431i q^{95} +103.684i q^{97} -264.661i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{3} + 4 q^{7} + 72 q^{9}+O(q^{10}) \) 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
\(5\) 7.52281i 0.672861i 0.941708 + 0.336430i \(0.109220\pi\)
−0.941708 + 0.336430i \(0.890780\pi\)
\(6\) 0 0
\(7\) 4.86375 + 17.8702i 0.262618 + 0.964900i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 29.4068i 0.806044i −0.915190 0.403022i \(-0.867960\pi\)
0.915190 0.403022i \(-0.132040\pi\)
\(12\) 0 0
\(13\) 5.35686i 0.114287i −0.998366 0.0571433i \(-0.981801\pi\)
0.998366 0.0571433i \(-0.0181992\pi\)
\(14\) 0 0
\(15\) 22.5684i 0.388476i
\(16\) 0 0
\(17\) 66.3364i 0.946408i 0.880953 + 0.473204i \(0.156903\pi\)
−0.880953 + 0.473204i \(0.843097\pi\)
\(18\) 0 0
\(19\) −22.3894 −0.270341 −0.135170 0.990822i \(-0.543158\pi\)
−0.135170 + 0.990822i \(0.543158\pi\)
\(20\) 0 0
\(21\) −14.5913 53.6106i −0.151623 0.557085i
\(22\) 0 0
\(23\) 33.9994i 0.308233i 0.988053 + 0.154117i \(0.0492532\pi\)
−0.988053 + 0.154117i \(0.950747\pi\)
\(24\) 0 0
\(25\) 68.4073 0.547258
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 133.266 0.853341 0.426671 0.904407i \(-0.359686\pi\)
0.426671 + 0.904407i \(0.359686\pi\)
\(30\) 0 0
\(31\) 323.973 1.87701 0.938504 0.345270i \(-0.112212\pi\)
0.938504 + 0.345270i \(0.112212\pi\)
\(32\) 0 0
\(33\) 88.2204i 0.465370i
\(34\) 0 0
\(35\) −134.434 + 36.5891i −0.649243 + 0.176705i
\(36\) 0 0
\(37\) −120.395 −0.534940 −0.267470 0.963566i \(-0.586188\pi\)
−0.267470 + 0.963566i \(0.586188\pi\)
\(38\) 0 0
\(39\) 16.0706i 0.0659834i
\(40\) 0 0
\(41\) 140.290i 0.534379i 0.963644 + 0.267190i \(0.0860950\pi\)
−0.963644 + 0.267190i \(0.913905\pi\)
\(42\) 0 0
\(43\) 19.0522i 0.0675681i −0.999429 0.0337841i \(-0.989244\pi\)
0.999429 0.0337841i \(-0.0107558\pi\)
\(44\) 0 0
\(45\) 67.7053i 0.224287i
\(46\) 0 0
\(47\) 376.658 1.16896 0.584481 0.811407i \(-0.301298\pi\)
0.584481 + 0.811407i \(0.301298\pi\)
\(48\) 0 0
\(49\) −295.688 + 173.832i −0.862064 + 0.506800i
\(50\) 0 0
\(51\) 199.009i 0.546409i
\(52\) 0 0
\(53\) 441.895 1.14526 0.572632 0.819813i \(-0.305922\pi\)
0.572632 + 0.819813i \(0.305922\pi\)
\(54\) 0 0
\(55\) 221.222 0.542356
\(56\) 0 0
\(57\) 67.1681 0.156081
\(58\) 0 0
\(59\) −241.815 −0.533586 −0.266793 0.963754i \(-0.585964\pi\)
−0.266793 + 0.963754i \(0.585964\pi\)
\(60\) 0 0
\(61\) 130.894i 0.274743i −0.990520 0.137371i \(-0.956135\pi\)
0.990520 0.137371i \(-0.0438654\pi\)
\(62\) 0 0
\(63\) 43.7738 + 160.832i 0.0875394 + 0.321633i
\(64\) 0 0
\(65\) 40.2987 0.0768990
\(66\) 0 0
\(67\) 627.240i 1.14373i −0.820349 0.571863i \(-0.806221\pi\)
0.820349 0.571863i \(-0.193779\pi\)
\(68\) 0 0
\(69\) 101.998i 0.177959i
\(70\) 0 0
\(71\) 808.143i 1.35083i −0.737438 0.675415i \(-0.763965\pi\)
0.737438 0.675415i \(-0.236035\pi\)
\(72\) 0 0
\(73\) 417.527i 0.669423i 0.942321 + 0.334711i \(0.108639\pi\)
−0.942321 + 0.334711i \(0.891361\pi\)
\(74\) 0 0
\(75\) −205.222 −0.315960
\(76\) 0 0
\(77\) 525.505 143.027i 0.777752 0.211682i
\(78\) 0 0
\(79\) 214.825i 0.305946i 0.988230 + 0.152973i \(0.0488847\pi\)
−0.988230 + 0.152973i \(0.951115\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −639.640 −0.845899 −0.422949 0.906153i \(-0.639005\pi\)
−0.422949 + 0.906153i \(0.639005\pi\)
\(84\) 0 0
\(85\) −499.037 −0.636801
\(86\) 0 0
\(87\) −399.798 −0.492677
\(88\) 0 0
\(89\) 686.995i 0.818217i 0.912486 + 0.409109i \(0.134160\pi\)
−0.912486 + 0.409109i \(0.865840\pi\)
\(90\) 0 0
\(91\) 95.7281 26.0545i 0.110275 0.0300137i
\(92\) 0 0
\(93\) −971.918 −1.08369
\(94\) 0 0
\(95\) 168.431i 0.181902i
\(96\) 0 0
\(97\) 103.684i 0.108532i 0.998527 + 0.0542658i \(0.0172818\pi\)
−0.998527 + 0.0542658i \(0.982718\pi\)
\(98\) 0 0
\(99\) 264.661i 0.268681i
\(100\) 0 0
\(101\) 1985.67i 1.95626i 0.208000 + 0.978129i \(0.433305\pi\)
−0.208000 + 0.978129i \(0.566695\pi\)
\(102\) 0 0
\(103\) −855.145 −0.818058 −0.409029 0.912521i \(-0.634133\pi\)
−0.409029 + 0.912521i \(0.634133\pi\)
\(104\) 0 0
\(105\) 403.303 109.767i 0.374841 0.102021i
\(106\) 0 0
\(107\) 436.104i 0.394016i 0.980402 + 0.197008i \(0.0631225\pi\)
−0.980402 + 0.197008i \(0.936877\pi\)
\(108\) 0 0
\(109\) −1424.63 −1.25188 −0.625938 0.779873i \(-0.715284\pi\)
−0.625938 + 0.779873i \(0.715284\pi\)
\(110\) 0 0
\(111\) 361.184 0.308848
\(112\) 0 0
\(113\) −1217.43 −1.01351 −0.506753 0.862091i \(-0.669154\pi\)
−0.506753 + 0.862091i \(0.669154\pi\)
\(114\) 0 0
\(115\) −255.771 −0.207398
\(116\) 0 0
\(117\) 48.2117i 0.0380955i
\(118\) 0 0
\(119\) −1185.44 + 322.644i −0.913189 + 0.248544i
\(120\) 0 0
\(121\) 466.240 0.350293
\(122\) 0 0
\(123\) 420.869i 0.308524i
\(124\) 0 0
\(125\) 1454.97i 1.04109i
\(126\) 0 0
\(127\) 1188.92i 0.830705i −0.909660 0.415353i \(-0.863658\pi\)
0.909660 0.415353i \(-0.136342\pi\)
\(128\) 0 0
\(129\) 57.1565i 0.0390105i
\(130\) 0 0
\(131\) 1652.03 1.10182 0.550911 0.834564i \(-0.314280\pi\)
0.550911 + 0.834564i \(0.314280\pi\)
\(132\) 0 0
\(133\) −108.896 400.102i −0.0709963 0.260852i
\(134\) 0 0
\(135\) 203.116i 0.129492i
\(136\) 0 0
\(137\) −2528.15 −1.57660 −0.788300 0.615291i \(-0.789039\pi\)
−0.788300 + 0.615291i \(0.789039\pi\)
\(138\) 0 0
\(139\) 2522.86 1.53947 0.769734 0.638365i \(-0.220389\pi\)
0.769734 + 0.638365i \(0.220389\pi\)
\(140\) 0 0
\(141\) −1129.97 −0.674901
\(142\) 0 0
\(143\) −157.528 −0.0921200
\(144\) 0 0
\(145\) 1002.54i 0.574180i
\(146\) 0 0
\(147\) 887.063 521.497i 0.497713 0.292601i
\(148\) 0 0
\(149\) 2456.90 1.35085 0.675425 0.737428i \(-0.263960\pi\)
0.675425 + 0.737428i \(0.263960\pi\)
\(150\) 0 0
\(151\) 1439.28i 0.775677i 0.921727 + 0.387838i \(0.126778\pi\)
−0.921727 + 0.387838i \(0.873222\pi\)
\(152\) 0 0
\(153\) 597.028i 0.315469i
\(154\) 0 0
\(155\) 2437.19i 1.26296i
\(156\) 0 0
\(157\) 3258.27i 1.65629i 0.560512 + 0.828146i \(0.310604\pi\)
−0.560512 + 0.828146i \(0.689396\pi\)
\(158\) 0 0
\(159\) −1325.69 −0.661218
\(160\) 0 0
\(161\) −607.576 + 165.365i −0.297414 + 0.0809477i
\(162\) 0 0
\(163\) 1190.87i 0.572246i 0.958193 + 0.286123i \(0.0923666\pi\)
−0.958193 + 0.286123i \(0.907633\pi\)
\(164\) 0 0
\(165\) −663.666 −0.313129
\(166\) 0 0
\(167\) −502.205 −0.232706 −0.116353 0.993208i \(-0.537120\pi\)
−0.116353 + 0.993208i \(0.537120\pi\)
\(168\) 0 0
\(169\) 2168.30 0.986939
\(170\) 0 0
\(171\) −201.504 −0.0901135
\(172\) 0 0
\(173\) 1241.93i 0.545791i 0.962044 + 0.272896i \(0.0879813\pi\)
−0.962044 + 0.272896i \(0.912019\pi\)
\(174\) 0 0
\(175\) 332.716 + 1222.45i 0.143720 + 0.528049i
\(176\) 0 0
\(177\) 725.444 0.308066
\(178\) 0 0
\(179\) 2473.50i 1.03284i 0.856336 + 0.516419i \(0.172735\pi\)
−0.856336 + 0.516419i \(0.827265\pi\)
\(180\) 0 0
\(181\) 4292.14i 1.76261i 0.472550 + 0.881304i \(0.343334\pi\)
−0.472550 + 0.881304i \(0.656666\pi\)
\(182\) 0 0
\(183\) 392.683i 0.158623i
\(184\) 0 0
\(185\) 905.708i 0.359940i
\(186\) 0 0
\(187\) 1950.74 0.762847
\(188\) 0 0
\(189\) −131.321 482.495i −0.0505409 0.185695i
\(190\) 0 0
\(191\) 1252.21i 0.474383i −0.971463 0.237191i \(-0.923773\pi\)
0.971463 0.237191i \(-0.0762268\pi\)
\(192\) 0 0
\(193\) −901.321 −0.336158 −0.168079 0.985774i \(-0.553756\pi\)
−0.168079 + 0.985774i \(0.553756\pi\)
\(194\) 0 0
\(195\) −120.896 −0.0443976
\(196\) 0 0
\(197\) −982.225 −0.355232 −0.177616 0.984100i \(-0.556838\pi\)
−0.177616 + 0.984100i \(0.556838\pi\)
\(198\) 0 0
\(199\) 1900.42 0.676969 0.338485 0.940972i \(-0.390086\pi\)
0.338485 + 0.940972i \(0.390086\pi\)
\(200\) 0 0
\(201\) 1881.72i 0.660330i
\(202\) 0 0
\(203\) 648.174 + 2381.49i 0.224103 + 0.823389i
\(204\) 0 0
\(205\) −1055.37 −0.359563
\(206\) 0 0
\(207\) 305.995i 0.102744i
\(208\) 0 0
\(209\) 658.399i 0.217906i
\(210\) 0 0
\(211\) 4281.82i 1.39703i −0.715597 0.698513i \(-0.753845\pi\)
0.715597 0.698513i \(-0.246155\pi\)
\(212\) 0 0
\(213\) 2424.43i 0.779902i
\(214\) 0 0
\(215\) 143.326 0.0454640
\(216\) 0 0
\(217\) 1575.72 + 5789.46i 0.492936 + 1.81112i
\(218\) 0 0
\(219\) 1252.58i 0.386491i
\(220\) 0 0
\(221\) 355.355 0.108162
\(222\) 0 0
\(223\) −1677.62 −0.503776 −0.251888 0.967756i \(-0.581051\pi\)
−0.251888 + 0.967756i \(0.581051\pi\)
\(224\) 0 0
\(225\) 615.665 0.182419
\(226\) 0 0
\(227\) −2852.74 −0.834110 −0.417055 0.908881i \(-0.636938\pi\)
−0.417055 + 0.908881i \(0.636938\pi\)
\(228\) 0 0
\(229\) 3156.40i 0.910833i −0.890278 0.455417i \(-0.849490\pi\)
0.890278 0.455417i \(-0.150510\pi\)
\(230\) 0 0
\(231\) −1576.52 + 429.082i −0.449035 + 0.122214i
\(232\) 0 0
\(233\) −3827.73 −1.07624 −0.538118 0.842870i \(-0.680864\pi\)
−0.538118 + 0.842870i \(0.680864\pi\)
\(234\) 0 0
\(235\) 2833.53i 0.786549i
\(236\) 0 0
\(237\) 644.475i 0.176638i
\(238\) 0 0
\(239\) 515.516i 0.139523i 0.997564 + 0.0697614i \(0.0222238\pi\)
−0.997564 + 0.0697614i \(0.977776\pi\)
\(240\) 0 0
\(241\) 3080.35i 0.823332i −0.911335 0.411666i \(-0.864947\pi\)
0.911335 0.411666i \(-0.135053\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −1307.71 2224.40i −0.341006 0.580049i
\(246\) 0 0
\(247\) 119.937i 0.0308963i
\(248\) 0 0
\(249\) 1918.92 0.488380
\(250\) 0 0
\(251\) −4782.52 −1.20267 −0.601335 0.798997i \(-0.705364\pi\)
−0.601335 + 0.798997i \(0.705364\pi\)
\(252\) 0 0
\(253\) 999.814 0.248450
\(254\) 0 0
\(255\) 1497.11 0.367657
\(256\) 0 0
\(257\) 6031.52i 1.46395i 0.681330 + 0.731976i \(0.261402\pi\)
−0.681330 + 0.731976i \(0.738598\pi\)
\(258\) 0 0
\(259\) −585.571 2151.48i −0.140485 0.516164i
\(260\) 0 0
\(261\) 1199.39 0.284447
\(262\) 0 0
\(263\) 5984.65i 1.40315i 0.712594 + 0.701577i \(0.247520\pi\)
−0.712594 + 0.701577i \(0.752480\pi\)
\(264\) 0 0
\(265\) 3324.30i 0.770603i
\(266\) 0 0
\(267\) 2060.98i 0.472398i
\(268\) 0 0
\(269\) 8138.44i 1.84464i 0.386422 + 0.922322i \(0.373711\pi\)
−0.386422 + 0.922322i \(0.626289\pi\)
\(270\) 0 0
\(271\) 1147.22 0.257154 0.128577 0.991700i \(-0.458959\pi\)
0.128577 + 0.991700i \(0.458959\pi\)
\(272\) 0 0
\(273\) −287.184 + 78.1634i −0.0636674 + 0.0173284i
\(274\) 0 0
\(275\) 2011.64i 0.441114i
\(276\) 0 0
\(277\) 122.946 0.0266682 0.0133341 0.999911i \(-0.495755\pi\)
0.0133341 + 0.999911i \(0.495755\pi\)
\(278\) 0 0
\(279\) 2915.75 0.625669
\(280\) 0 0
\(281\) −1308.64 −0.277817 −0.138909 0.990305i \(-0.544359\pi\)
−0.138909 + 0.990305i \(0.544359\pi\)
\(282\) 0 0
\(283\) −7443.18 −1.56343 −0.781715 0.623635i \(-0.785655\pi\)
−0.781715 + 0.623635i \(0.785655\pi\)
\(284\) 0 0
\(285\) 505.293i 0.105021i
\(286\) 0 0
\(287\) −2507.00 + 682.334i −0.515623 + 0.140338i
\(288\) 0 0
\(289\) 512.480 0.104311
\(290\) 0 0
\(291\) 311.053i 0.0626607i
\(292\) 0 0
\(293\) 121.333i 0.0241924i 0.999927 + 0.0120962i \(0.00385043\pi\)
−0.999927 + 0.0120962i \(0.996150\pi\)
\(294\) 0 0
\(295\) 1819.13i 0.359029i
\(296\) 0 0
\(297\) 793.984i 0.155123i
\(298\) 0 0
\(299\) 182.130 0.0352269
\(300\) 0 0
\(301\) 340.466 92.6651i 0.0651965 0.0177446i
\(302\) 0 0
\(303\) 5957.02i 1.12945i
\(304\) 0 0
\(305\) 984.694 0.184864
\(306\) 0 0
\(307\) −5928.91 −1.10222 −0.551109 0.834433i \(-0.685795\pi\)
−0.551109 + 0.834433i \(0.685795\pi\)
\(308\) 0 0
\(309\) 2565.44 0.472306
\(310\) 0 0
\(311\) 6039.69 1.10122 0.550610 0.834763i \(-0.314395\pi\)
0.550610 + 0.834763i \(0.314395\pi\)
\(312\) 0 0
\(313\) 6467.65i 1.16797i 0.811766 + 0.583983i \(0.198506\pi\)
−0.811766 + 0.583983i \(0.801494\pi\)
\(314\) 0 0
\(315\) −1209.91 + 329.302i −0.216414 + 0.0589018i
\(316\) 0 0
\(317\) 643.842 0.114075 0.0570375 0.998372i \(-0.481835\pi\)
0.0570375 + 0.998372i \(0.481835\pi\)
\(318\) 0 0
\(319\) 3918.93i 0.687830i
\(320\) 0 0
\(321\) 1308.31i 0.227485i
\(322\) 0 0
\(323\) 1485.23i 0.255853i
\(324\) 0 0
\(325\) 366.448i 0.0625443i
\(326\) 0 0
\(327\) 4273.88 0.722771
\(328\) 0 0
\(329\) 1831.97 + 6730.95i 0.306991 + 1.12793i
\(330\) 0 0
\(331\) 2157.62i 0.358288i −0.983823 0.179144i \(-0.942667\pi\)
0.983823 0.179144i \(-0.0573328\pi\)
\(332\) 0 0
\(333\) −1083.55 −0.178313
\(334\) 0 0
\(335\) 4718.61 0.769568
\(336\) 0 0
\(337\) −7841.90 −1.26758 −0.633791 0.773504i \(-0.718502\pi\)
−0.633791 + 0.773504i \(0.718502\pi\)
\(338\) 0 0
\(339\) 3652.29 0.585148
\(340\) 0 0
\(341\) 9527.00i 1.51295i
\(342\) 0 0
\(343\) −4544.57 4438.52i −0.715405 0.698710i
\(344\) 0 0
\(345\) 767.314 0.119741
\(346\) 0 0
\(347\) 4089.11i 0.632608i 0.948658 + 0.316304i \(0.102442\pi\)
−0.948658 + 0.316304i \(0.897558\pi\)
\(348\) 0 0
\(349\) 2627.48i 0.402997i 0.979489 + 0.201499i \(0.0645811\pi\)
−0.979489 + 0.201499i \(0.935419\pi\)
\(350\) 0 0
\(351\) 144.635i 0.0219945i
\(352\) 0 0
\(353\) 9452.79i 1.42527i −0.701534 0.712636i \(-0.747501\pi\)
0.701534 0.712636i \(-0.252499\pi\)
\(354\) 0 0
\(355\) 6079.51 0.908921
\(356\) 0 0
\(357\) 3556.33 967.932i 0.527230 0.143497i
\(358\) 0 0
\(359\) 12522.8i 1.84103i 0.390710 + 0.920514i \(0.372230\pi\)
−0.390710 + 0.920514i \(0.627770\pi\)
\(360\) 0 0
\(361\) −6357.72 −0.926916
\(362\) 0 0
\(363\) −1398.72 −0.202242
\(364\) 0 0
\(365\) −3140.98 −0.450428
\(366\) 0 0
\(367\) 9111.30 1.29593 0.647965 0.761670i \(-0.275620\pi\)
0.647965 + 0.761670i \(0.275620\pi\)
\(368\) 0 0
\(369\) 1262.61i 0.178126i
\(370\) 0 0
\(371\) 2149.27 + 7896.75i 0.300767 + 1.10506i
\(372\) 0 0
\(373\) 221.326 0.0307234 0.0153617 0.999882i \(-0.495110\pi\)
0.0153617 + 0.999882i \(0.495110\pi\)
\(374\) 0 0
\(375\) 4364.90i 0.601073i
\(376\) 0 0
\(377\) 713.888i 0.0975254i
\(378\) 0 0
\(379\) 7417.24i 1.00527i −0.864498 0.502636i \(-0.832364\pi\)
0.864498 0.502636i \(-0.167636\pi\)
\(380\) 0 0
\(381\) 3566.76i 0.479608i
\(382\) 0 0
\(383\) −6689.28 −0.892444 −0.446222 0.894922i \(-0.647231\pi\)
−0.446222 + 0.894922i \(0.647231\pi\)
\(384\) 0 0
\(385\) 1075.97 + 3953.28i 0.142432 + 0.523319i
\(386\) 0 0
\(387\) 171.470i 0.0225227i
\(388\) 0 0
\(389\) −13138.4 −1.71246 −0.856228 0.516598i \(-0.827198\pi\)
−0.856228 + 0.516598i \(0.827198\pi\)
\(390\) 0 0
\(391\) −2255.40 −0.291715
\(392\) 0 0
\(393\) −4956.09 −0.636137
\(394\) 0 0
\(395\) −1616.09 −0.205859
\(396\) 0 0
\(397\) 292.782i 0.0370134i 0.999829 + 0.0185067i \(0.00589120\pi\)
−0.999829 + 0.0185067i \(0.994109\pi\)
\(398\) 0 0
\(399\) 326.689 + 1200.31i 0.0409897 + 0.150603i
\(400\) 0 0
\(401\) 9071.46 1.12969 0.564847 0.825196i \(-0.308935\pi\)
0.564847 + 0.825196i \(0.308935\pi\)
\(402\) 0 0
\(403\) 1735.48i 0.214517i
\(404\) 0 0
\(405\) 609.348i 0.0747623i
\(406\) 0 0
\(407\) 3540.43i 0.431185i
\(408\) 0 0
\(409\) 16065.7i 1.94229i −0.238486 0.971146i \(-0.576651\pi\)
0.238486 0.971146i \(-0.423349\pi\)
\(410\) 0 0
\(411\) 7584.44 0.910251
\(412\) 0 0
\(413\) −1176.13 4321.27i −0.140129 0.514857i
\(414\) 0 0
\(415\) 4811.89i 0.569172i
\(416\) 0 0
\(417\) −7568.57 −0.888812
\(418\) 0 0
\(419\) 8693.06 1.01357 0.506783 0.862074i \(-0.330835\pi\)
0.506783 + 0.862074i \(0.330835\pi\)
\(420\) 0 0
\(421\) 999.478 0.115704 0.0578522 0.998325i \(-0.481575\pi\)
0.0578522 + 0.998325i \(0.481575\pi\)
\(422\) 0 0
\(423\) 3389.92 0.389654
\(424\) 0 0
\(425\) 4537.89i 0.517930i
\(426\) 0 0
\(427\) 2339.11 636.638i 0.265099 0.0721524i
\(428\) 0 0
\(429\) 472.584 0.0531855
\(430\) 0 0
\(431\) 3194.94i 0.357065i 0.983934 + 0.178532i \(0.0571349\pi\)
−0.983934 + 0.178532i \(0.942865\pi\)
\(432\) 0 0
\(433\) 14150.0i 1.57045i −0.619209 0.785226i \(-0.712547\pi\)
0.619209 0.785226i \(-0.287453\pi\)
\(434\) 0 0
\(435\) 3007.61i 0.331503i
\(436\) 0 0
\(437\) 761.225i 0.0833280i
\(438\) 0 0
\(439\) −4754.35 −0.516886 −0.258443 0.966027i \(-0.583209\pi\)
−0.258443 + 0.966027i \(0.583209\pi\)
\(440\) 0 0
\(441\) −2661.19 + 1564.49i −0.287355 + 0.168933i
\(442\) 0 0
\(443\) 8322.04i 0.892533i −0.894900 0.446266i \(-0.852753\pi\)
0.894900 0.446266i \(-0.147247\pi\)
\(444\) 0 0
\(445\) −5168.14 −0.550546
\(446\) 0 0
\(447\) −7370.69 −0.779914
\(448\) 0 0
\(449\) 15248.4 1.60270 0.801352 0.598193i \(-0.204114\pi\)
0.801352 + 0.598193i \(0.204114\pi\)
\(450\) 0 0
\(451\) 4125.47 0.430733
\(452\) 0 0
\(453\) 4317.85i 0.447837i
\(454\) 0 0
\(455\) 196.003 + 720.145i 0.0201951 + 0.0741998i
\(456\) 0 0
\(457\) 9050.07 0.926356 0.463178 0.886265i \(-0.346709\pi\)
0.463178 + 0.886265i \(0.346709\pi\)
\(458\) 0 0
\(459\) 1791.08i 0.182136i
\(460\) 0 0
\(461\) 1919.93i 0.193970i 0.995286 + 0.0969851i \(0.0309199\pi\)
−0.995286 + 0.0969851i \(0.969080\pi\)
\(462\) 0 0
\(463\) 7800.83i 0.783014i −0.920175 0.391507i \(-0.871954\pi\)
0.920175 0.391507i \(-0.128046\pi\)
\(464\) 0 0
\(465\) 7311.56i 0.729173i
\(466\) 0 0
\(467\) 4762.37 0.471898 0.235949 0.971765i \(-0.424180\pi\)
0.235949 + 0.971765i \(0.424180\pi\)
\(468\) 0 0
\(469\) 11208.9 3050.74i 1.10358 0.300363i
\(470\) 0 0
\(471\) 9774.80i 0.956261i
\(472\) 0 0
\(473\) −560.263 −0.0544629
\(474\) 0 0
\(475\) −1531.59 −0.147946
\(476\) 0 0
\(477\) 3977.06 0.381755
\(478\) 0 0
\(479\) −8191.42 −0.781369 −0.390684 0.920525i \(-0.627762\pi\)
−0.390684 + 0.920525i \(0.627762\pi\)
\(480\) 0 0
\(481\) 644.938i 0.0611365i
\(482\) 0 0
\(483\) 1822.73 496.094i 0.171712 0.0467352i
\(484\) 0 0
\(485\) −779.999 −0.0730267
\(486\) 0 0
\(487\) 7326.15i 0.681683i −0.940121 0.340841i \(-0.889288\pi\)
0.940121 0.340841i \(-0.110712\pi\)
\(488\) 0 0
\(489\) 3572.61i 0.330386i
\(490\) 0 0
\(491\) 16260.3i 1.49454i 0.664523 + 0.747268i \(0.268635\pi\)
−0.664523 + 0.747268i \(0.731365\pi\)
\(492\) 0 0
\(493\) 8840.39i 0.807609i
\(494\) 0 0
\(495\) 1991.00 0.180785
\(496\) 0 0
\(497\) 14441.7 3930.61i 1.30342 0.354752i
\(498\) 0 0
\(499\) 14395.5i 1.29144i 0.763573 + 0.645721i \(0.223443\pi\)
−0.763573 + 0.645721i \(0.776557\pi\)
\(500\) 0 0
\(501\) 1506.62 0.134353
\(502\) 0 0
\(503\) −16722.5 −1.48235 −0.741174 0.671313i \(-0.765730\pi\)
−0.741174 + 0.671313i \(0.765730\pi\)
\(504\) 0 0
\(505\) −14937.9 −1.31629
\(506\) 0 0
\(507\) −6504.91 −0.569809
\(508\) 0 0
\(509\) 3391.00i 0.295292i 0.989040 + 0.147646i \(0.0471696\pi\)
−0.989040 + 0.147646i \(0.952830\pi\)
\(510\) 0 0
\(511\) −7461.29 + 2030.75i −0.645926 + 0.175803i
\(512\) 0 0
\(513\) 604.513 0.0520271
\(514\) 0 0
\(515\) 6433.10i 0.550439i
\(516\) 0 0
\(517\) 11076.3i 0.942235i
\(518\) 0 0
\(519\) 3725.78i 0.315113i
\(520\) 0 0
\(521\) 115.006i 0.00967084i −0.999988 0.00483542i \(-0.998461\pi\)
0.999988 0.00483542i \(-0.00153917\pi\)
\(522\) 0 0
\(523\) 9860.83 0.824443 0.412222 0.911084i \(-0.364753\pi\)
0.412222 + 0.911084i \(0.364753\pi\)
\(524\) 0 0
\(525\) −998.148 3667.35i −0.0829767 0.304869i
\(526\) 0 0
\(527\) 21491.2i 1.77642i
\(528\) 0 0
\(529\) 11011.0 0.904992
\(530\) 0 0
\(531\) −2176.33 −0.177862
\(532\) 0 0
\(533\) 751.512 0.0610724
\(534\) 0 0
\(535\) −3280.73 −0.265118
\(536\) 0 0
\(537\) 7420.49i 0.596309i
\(538\) 0 0
\(539\) 5111.86 + 8695.23i 0.408503 + 0.694861i
\(540\) 0 0
\(541\) 15529.6 1.23414 0.617069 0.786909i \(-0.288320\pi\)
0.617069 + 0.786909i \(0.288320\pi\)
\(542\) 0 0
\(543\) 12876.4i 1.01764i
\(544\) 0 0
\(545\) 10717.2i 0.842338i
\(546\) 0 0
\(547\) 9476.01i 0.740704i −0.928892 0.370352i \(-0.879237\pi\)
0.928892 0.370352i \(-0.120763\pi\)
\(548\) 0 0
\(549\) 1178.05i 0.0915809i
\(550\) 0 0
\(551\) −2983.74 −0.230693
\(552\) 0 0
\(553\) −3838.97 + 1044.86i −0.295207 + 0.0803469i
\(554\) 0 0
\(555\) 2717.12i 0.207812i
\(556\) 0 0
\(557\) 4458.02 0.339125 0.169562 0.985519i \(-0.445765\pi\)
0.169562 + 0.985519i \(0.445765\pi\)
\(558\) 0 0
\(559\) −102.060 −0.00772213
\(560\) 0 0
\(561\) −5852.22 −0.440430
\(562\) 0 0
\(563\) −8596.00 −0.643478 −0.321739 0.946828i \(-0.604267\pi\)
−0.321739 + 0.946828i \(0.604267\pi\)
\(564\) 0 0
\(565\) 9158.50i 0.681949i
\(566\) 0 0
\(567\) 393.964 + 1447.49i 0.0291798 + 0.107211i
\(568\) 0 0
\(569\) −4098.66 −0.301976 −0.150988 0.988536i \(-0.548246\pi\)
−0.150988 + 0.988536i \(0.548246\pi\)
\(570\) 0 0
\(571\) 86.4091i 0.00633294i 0.999995 + 0.00316647i \(0.00100792\pi\)
−0.999995 + 0.00316647i \(0.998992\pi\)
\(572\) 0 0
\(573\) 3756.64i 0.273885i
\(574\) 0 0
\(575\) 2325.81i 0.168683i
\(576\) 0 0
\(577\) 19490.5i 1.40624i −0.711071 0.703120i \(-0.751790\pi\)
0.711071 0.703120i \(-0.248210\pi\)
\(578\) 0 0
\(579\) 2703.96 0.194081
\(580\) 0 0
\(581\) −3111.05 11430.5i −0.222148 0.816207i
\(582\) 0 0
\(583\) 12994.7i 0.923133i
\(584\) 0 0
\(585\) 362.688 0.0256330
\(586\) 0 0
\(587\) −21546.8 −1.51504 −0.757522 0.652810i \(-0.773590\pi\)
−0.757522 + 0.652810i \(0.773590\pi\)
\(588\) 0 0
\(589\) −7253.54 −0.507431
\(590\) 0 0
\(591\) 2946.67 0.205093
\(592\) 0 0
\(593\) 8845.46i 0.612546i −0.951944 0.306273i \(-0.900918\pi\)
0.951944 0.306273i \(-0.0990820\pi\)
\(594\) 0 0
\(595\) −2427.19 8917.88i −0.167236 0.614450i
\(596\) 0 0
\(597\) −5701.25 −0.390848
\(598\) 0 0
\(599\) 20160.5i 1.37518i 0.726098 + 0.687592i \(0.241332\pi\)
−0.726098 + 0.687592i \(0.758668\pi\)
\(600\) 0 0
\(601\) 6467.44i 0.438956i 0.975618 + 0.219478i \(0.0704353\pi\)
−0.975618 + 0.219478i \(0.929565\pi\)
\(602\) 0 0
\(603\) 5645.16i 0.381242i
\(604\) 0 0
\(605\) 3507.44i 0.235699i
\(606\) 0 0
\(607\) 24606.0 1.64535 0.822673 0.568515i \(-0.192482\pi\)
0.822673 + 0.568515i \(0.192482\pi\)
\(608\) 0 0
\(609\) −1944.52 7144.47i −0.129386 0.475384i
\(610\) 0 0
\(611\) 2017.70i 0.133597i
\(612\) 0 0
\(613\) 10251.2 0.675437 0.337719 0.941247i \(-0.390345\pi\)
0.337719 + 0.941247i \(0.390345\pi\)
\(614\) 0 0
\(615\) 3166.12 0.207594
\(616\) 0 0
\(617\) 19189.1 1.25206 0.626031 0.779798i \(-0.284678\pi\)
0.626031 + 0.779798i \(0.284678\pi\)
\(618\) 0 0
\(619\) 17050.4 1.10713 0.553565 0.832806i \(-0.313267\pi\)
0.553565 + 0.832806i \(0.313267\pi\)
\(620\) 0 0
\(621\) 917.984i 0.0593195i
\(622\) 0 0
\(623\) −12276.7 + 3341.37i −0.789498 + 0.214879i
\(624\) 0 0
\(625\) −2394.54 −0.153250
\(626\) 0 0
\(627\) 1975.20i 0.125808i
\(628\) 0 0
\(629\) 7986.56i 0.506272i
\(630\) 0 0
\(631\) 1257.65i 0.0793442i 0.999213 + 0.0396721i \(0.0126313\pi\)
−0.999213 + 0.0396721i \(0.987369\pi\)
\(632\) 0 0
\(633\) 12845.5i 0.806573i
\(634\) 0 0
\(635\) 8944.02 0.558949
\(636\) 0 0
\(637\) 931.196 + 1583.96i 0.0579205 + 0.0985223i
\(638\) 0 0
\(639\) 7273.29i 0.450277i
\(640\) 0 0
\(641\) −9903.08 −0.610215 −0.305108 0.952318i \(-0.598692\pi\)
−0.305108 + 0.952318i \(0.598692\pi\)
\(642\) 0 0
\(643\) −20327.6 −1.24672 −0.623361 0.781934i \(-0.714233\pi\)
−0.623361 + 0.781934i \(0.714233\pi\)
\(644\) 0 0
\(645\) −429.978 −0.0262486
\(646\) 0 0
\(647\) 21089.7 1.28149 0.640743 0.767755i \(-0.278626\pi\)
0.640743 + 0.767755i \(0.278626\pi\)
\(648\) 0 0
\(649\) 7110.99i 0.430094i
\(650\) 0 0
\(651\) −4727.17 17368.4i −0.284597 1.04565i
\(652\) 0 0
\(653\) 20130.9 1.20641 0.603204 0.797587i \(-0.293890\pi\)
0.603204 + 0.797587i \(0.293890\pi\)
\(654\) 0 0
\(655\) 12427.9i 0.741373i
\(656\) 0 0
\(657\) 3757.74i 0.223141i
\(658\) 0 0
\(659\) 31889.2i 1.88502i 0.334181 + 0.942509i \(0.391540\pi\)
−0.334181 + 0.942509i \(0.608460\pi\)
\(660\) 0 0
\(661\) 25390.7i 1.49408i −0.664780 0.747039i \(-0.731475\pi\)
0.664780 0.747039i \(-0.268525\pi\)
\(662\) 0 0
\(663\) −1066.06 −0.0624472
\(664\) 0 0
\(665\) 3009.90 819.207i 0.175517 0.0477707i
\(666\) 0 0
\(667\) 4530.97i 0.263028i
\(668\) 0 0
\(669\) 5032.87 0.290855
\(670\) 0 0
\(671\) −3849.18 −0.221455
\(672\) 0 0
\(673\) 28068.6 1.60767 0.803836 0.594851i \(-0.202789\pi\)
0.803836 + 0.594851i \(0.202789\pi\)
\(674\) 0 0
\(675\) −1847.00 −0.105320
\(676\) 0 0
\(677\) 31036.6i 1.76194i −0.473171 0.880971i \(-0.656891\pi\)
0.473171 0.880971i \(-0.343109\pi\)
\(678\) 0 0
\(679\) −1852.86 + 504.296i −0.104722 + 0.0285024i
\(680\) 0 0
\(681\) 8558.22 0.481574
\(682\) 0 0
\(683\) 11387.9i 0.637986i 0.947757 + 0.318993i \(0.103345\pi\)
−0.947757 + 0.318993i \(0.896655\pi\)
\(684\) 0 0
\(685\) 19018.8i 1.06083i
\(686\) 0 0
\(687\) 9469.20i 0.525870i
\(688\) 0 0
\(689\) 2367.17i 0.130888i
\(690\) 0 0
\(691\) −3707.61 −0.204116 −0.102058 0.994778i \(-0.532543\pi\)
−0.102058 + 0.994778i \(0.532543\pi\)
\(692\) 0 0
\(693\) 4729.55 1287.25i 0.259251 0.0705606i
\(694\) 0 0
\(695\) 18979.0i 1.03585i
\(696\) 0 0
\(697\) −9306.31 −0.505741
\(698\) 0 0
\(699\) 11483.2 0.621365
\(700\) 0 0
\(701\) −16000.1 −0.862079 −0.431039 0.902333i \(-0.641853\pi\)
−0.431039 + 0.902333i \(0.641853\pi\)
\(702\) 0 0
\(703\) 2695.56 0.144616
\(704\) 0 0
\(705\) 8500.59i 0.454114i
\(706\) 0 0
\(707\) −35484.4 + 9657.84i −1.88759 + 0.513749i
\(708\) 0 0
\(709\) −19839.3 −1.05089 −0.525444 0.850828i \(-0.676101\pi\)
−0.525444 + 0.850828i \(0.676101\pi\)
\(710\) 0 0
\(711\) 1933.43i 0.101982i
\(712\) 0 0
\(713\) 11014.9i 0.578556i
\(714\) 0 0
\(715\) 1185.05i 0.0619840i
\(716\) 0 0
\(717\) 1546.55i 0.0805536i
\(718\) 0 0
\(719\) 31744.9 1.64657 0.823286 0.567626i \(-0.192138\pi\)
0.823286 + 0.567626i \(0.192138\pi\)
\(720\) 0 0
\(721\) −4159.22 15281.6i −0.214837 0.789344i
\(722\) 0 0
\(723\) 9241.06i 0.475351i
\(724\) 0 0
\(725\) 9116.37 0.466998
\(726\) 0 0
\(727\) 13264.7 0.676697 0.338349 0.941021i \(-0.390132\pi\)
0.338349 + 0.941021i \(0.390132\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1263.85 0.0639470
\(732\) 0 0
\(733\) 15237.1i 0.767795i 0.923376 + 0.383897i \(0.125418\pi\)
−0.923376 + 0.383897i \(0.874582\pi\)
\(734\) 0 0
\(735\) 3923.13 + 6673.21i 0.196880 + 0.334891i
\(736\) 0 0
\(737\) −18445.1 −0.921893
\(738\) 0 0
\(739\) 9609.13i 0.478319i −0.970980 0.239159i \(-0.923128\pi\)
0.970980 0.239159i \(-0.0768719\pi\)
\(740\) 0 0
\(741\) 359.810i 0.0178380i
\(742\) 0 0
\(743\) 2760.49i 0.136302i −0.997675 0.0681510i \(-0.978290\pi\)
0.997675 0.0681510i \(-0.0217100\pi\)
\(744\) 0 0
\(745\) 18482.8i 0.908935i
\(746\) 0 0
\(747\) −5756.76 −0.281966
\(748\) 0 0
\(749\) −7793.26 + 2121.10i −0.380186 + 0.103476i
\(750\) 0 0
\(751\) 3559.77i 0.172966i −0.996253 0.0864832i \(-0.972437\pi\)
0.996253 0.0864832i \(-0.0275629\pi\)
\(752\) 0 0
\(753\) 14347.6 0.694362
\(754\) 0 0
\(755\) −10827.5 −0.521922
\(756\) 0 0
\(757\) 18894.6 0.907180 0.453590 0.891210i \(-0.350143\pi\)
0.453590 + 0.891210i \(0.350143\pi\)
\(758\) 0 0
\(759\) −2999.44 −0.143442
\(760\) 0 0
\(761\) 22833.2i 1.08765i −0.839199 0.543825i \(-0.816976\pi\)
0.839199 0.543825i \(-0.183024\pi\)
\(762\) 0 0
\(763\) −6929.03 25458.4i −0.328765 1.20793i
\(764\) 0 0
\(765\) −4491.33 −0.212267
\(766\) 0 0
\(767\) 1295.37i 0.0609817i
\(768\) 0 0
\(769\) 25450.6i 1.19346i −0.802441 0.596731i \(-0.796466\pi\)
0.802441 0.596731i \(-0.203534\pi\)
\(770\) 0 0
\(771\) 18094.6i 0.845213i
\(772\) 0 0
\(773\) 19338.8i 0.899831i −0.893071 0.449915i \(-0.851454\pi\)
0.893071 0.449915i \(-0.148546\pi\)
\(774\) 0 0
\(775\) 22162.1 1.02721
\(776\) 0 0
\(777\) 1756.71 + 6454.44i 0.0811090 + 0.298007i
\(778\) 0 0
\(779\) 3140.99i 0.144464i
\(780\) 0 0
\(781\) −23764.9 −1.08883
\(782\) 0 0
\(783\) −3598.18 −0.164226
\(784\) 0 0
\(785\) −24511.3 −1.11445
\(786\) 0 0
\(787\) −4298.43 −0.194692 −0.0973459 0.995251i \(-0.531035\pi\)
−0.0973459 + 0.995251i \(0.531035\pi\)
\(788\) 0 0
\(789\) 17954.0i 0.810111i
\(790\) 0 0
\(791\) −5921.28 21755.7i −0.266165 0.977932i
\(792\) 0 0
\(793\) −701.183 −0.0313994
\(794\) 0 0
\(795\) 9972.89i 0.444908i
\(796\) 0 0
\(797\) 10181.8i 0.452519i 0.974067 + 0.226260i \(0.0726498\pi\)
−0.974067 + 0.226260i \(0.927350\pi\)
\(798\) 0 0
\(799\) 24986.1i 1.10632i
\(800\) 0 0
\(801\) 6182.95i 0.272739i
\(802\) 0 0
\(803\) 12278.1 0.539584
\(804\) 0 0
\(805\) −1244.01 4570.68i −0.0544665 0.200119i
\(806\) 0 0
\(807\) 24415.3i 1.06501i
\(808\) 0 0
\(809\) −29969.2 −1.30242 −0.651211 0.758896i \(-0.725739\pi\)
−0.651211 + 0.758896i \(0.725739\pi\)
\(810\) 0 0
\(811\) 3426.41 0.148357 0.0741785 0.997245i \(-0.476367\pi\)
0.0741785 + 0.997245i \(0.476367\pi\)
\(812\) 0 0
\(813\) −3441.66 −0.148468
\(814\) 0 0
\(815\) −8958.69 −0.385042
\(816\) 0 0
\(817\) 426.566i 0.0182664i
\(818\) 0 0
\(819\) 861.553 234.490i 0.0367584 0.0100046i
\(820\) 0 0
\(821\) 44007.6 1.87074 0.935368 0.353676i \(-0.115068\pi\)
0.935368 + 0.353676i \(0.115068\pi\)
\(822\) 0 0
\(823\) 19248.3i 0.815255i −0.913148 0.407627i \(-0.866356\pi\)
0.913148 0.407627i \(-0.133644\pi\)
\(824\) 0 0
\(825\) 6034.92i 0.254677i
\(826\) 0 0
\(827\) 39040.4i 1.64156i 0.571245 + 0.820779i \(0.306460\pi\)
−0.571245 + 0.820779i \(0.693540\pi\)
\(828\) 0 0
\(829\) 882.320i 0.0369653i 0.999829 + 0.0184827i \(0.00588355\pi\)
−0.999829 + 0.0184827i \(0.994116\pi\)
\(830\) 0 0
\(831\) −368.838 −0.0153969
\(832\) 0 0
\(833\) −11531.4 19614.9i −0.479640 0.815864i
\(834\) 0 0
\(835\) 3778.00i 0.156578i
\(836\) 0 0
\(837\) −8747.26 −0.361230
\(838\) 0 0
\(839\) −35277.8 −1.45164 −0.725819 0.687886i \(-0.758539\pi\)
−0.725819 + 0.687886i \(0.758539\pi\)
\(840\) 0 0
\(841\) −6629.15 −0.271809
\(842\) 0 0
\(843\) 3925.91 0.160398
\(844\) 0 0
\(845\) 16311.7i 0.664072i
\(846\) 0 0
\(847\) 2267.68 + 8331.80i 0.0919933 + 0.337998i
\(848\) 0 0
\(849\) 22329.5 0.902647
\(850\) 0 0
\(851\) 4093.35i 0.164886i
\(852\) 0 0
\(853\) 11073.8i 0.444501i −0.974990 0.222250i \(-0.928660\pi\)
0.974990 0.222250i \(-0.0713403\pi\)
\(854\) 0 0
\(855\) 1515.88i 0.0606339i
\(856\) 0 0
\(857\) 39961.2i 1.59282i 0.604757 + 0.796410i \(0.293270\pi\)
−0.604757 + 0.796410i \(0.706730\pi\)
\(858\) 0 0
\(859\) 36708.5 1.45807 0.729033 0.684479i \(-0.239970\pi\)
0.729033 + 0.684479i \(0.239970\pi\)
\(860\) 0 0
\(861\) 7521.01 2047.00i 0.297695 0.0810240i
\(862\) 0 0
\(863\) 35497.2i 1.40016i 0.714064 + 0.700081i \(0.246853\pi\)
−0.714064 + 0.700081i \(0.753147\pi\)
\(864\) 0 0
\(865\) −9342.78 −0.367242
\(866\) 0 0
\(867\) −1537.44 −0.0602240
\(868\) 0 0
\(869\) 6317.32 0.246606
\(870\) 0 0
\(871\) −3360.04 −0.130712
\(872\) 0 0
\(873\) 933.160i 0.0361772i
\(874\) 0 0
\(875\) −26000.5 + 7076.60i −1.00455 + 0.273409i
\(876\) 0 0
\(877\) −26876.6 −1.03485 −0.517423 0.855730i \(-0.673109\pi\)
−0.517423 + 0.855730i \(0.673109\pi\)
\(878\) 0 0
\(879\) 364.000i 0.0139675i
\(880\) 0 0
\(881\) 9934.42i 0.379908i −0.981793 0.189954i \(-0.939166\pi\)
0.981793 0.189954i \(-0.0608339\pi\)
\(882\) 0 0
\(883\) 20298.1i 0.773596i 0.922164 + 0.386798i \(0.126419\pi\)
−0.922164 + 0.386798i \(0.873581\pi\)
\(884\) 0 0
\(885\) 5457.38i 0.207286i
\(886\) 0 0
\(887\) 22739.8 0.860799 0.430399 0.902639i \(-0.358373\pi\)
0.430399 + 0.902639i \(0.358373\pi\)
\(888\) 0 0
\(889\) 21246.2 5782.61i 0.801548 0.218158i
\(890\) 0 0
\(891\) 2381.95i 0.0895604i
\(892\) 0 0
\(893\) −8433.13 −0.316018
\(894\) 0 0
\(895\) −18607.7 −0.694956
\(896\) 0 0
\(897\) −546.390 −0.0203383
\(898\) 0 0
\(899\) 43174.6 1.60173
\(900\) 0 0
\(901\) 29313.7i 1.08389i
\(902\) 0 0
\(903\) −1021.40 + 277.995i −0.0376412 + 0.0102449i
\(904\) 0 0
\(905\) −32288.9 −1.18599
\(906\) 0 0
\(907\) 11598.9i 0.424624i −0.977202 0.212312i \(-0.931901\pi\)
0.977202 0.212312i \(-0.0680993\pi\)
\(908\) 0 0
\(909\) 17871.1i 0.652086i
\(910\) 0 0
\(911\) 44200.3i 1.60749i −0.594975 0.803744i \(-0.702838\pi\)
0.594975 0.803744i \(-0.297162\pi\)
\(912\) 0 0
\(913\) 18809.8i 0.681831i
\(914\) 0 0
\(915\) −2954.08 −0.106731
\(916\) 0 0
\(917\) 8035.07 + 29522.1i 0.289358 + 1.06315i
\(918\) 0 0
\(919\) 48975.3i 1.75794i −0.476878 0.878969i \(-0.658232\pi\)
0.476878 0.878969i \(-0.341768\pi\)
\(920\) 0 0
\(921\) 17786.7 0.636366
\(922\) 0 0
\(923\) −4329.11 −0.154382
\(924\) 0 0
\(925\) −8235.88 −0.292750
\(926\) 0 0
\(927\) −7696.31 −0.272686
\(928\) 0 0
\(929\) 55123.2i 1.94675i −0.229212 0.973377i \(-0.573615\pi\)
0.229212 0.973377i \(-0.426385\pi\)
\(930\) 0 0
\(931\) 6620.26 3892.00i 0.233051 0.137009i
\(932\) 0 0
\(933\) −18119.1 −0.635789
\(934\) 0 0
\(935\) 14675.1i 0.513290i
\(936\) 0 0
\(937\) 41003.8i 1.42960i 0.699329 + 0.714800i \(0.253482\pi\)
−0.699329 + 0.714800i \(0.746518\pi\)
\(938\) 0 0
\(939\) 19402.9i 0.674325i
\(940\) 0 0
\(941\) 6567.48i 0.227517i 0.993508 + 0.113759i \(0.0362890\pi\)
−0.993508 + 0.113759i \(0.963711\pi\)
\(942\) 0 0
\(943\) −4769.77 −0.164714
\(944\) 0 0
\(945\) 3629.72 987.906i 0.124947 0.0340070i
\(946\) 0 0
\(947\) 27665.3i 0.949316i −0.880170 0.474658i \(-0.842572\pi\)
0.880170 0.474658i \(-0.157428\pi\)
\(948\) 0 0
\(949\) 2236.63 0.0765060
\(950\) 0 0
\(951\) −1931.53 −0.0658612
\(952\) 0 0
\(953\) 20375.2 0.692568 0.346284 0.938130i \(-0.387443\pi\)
0.346284 + 0.938130i \(0.387443\pi\)
\(954\) 0 0
\(955\) 9420.18 0.319193
\(956\) 0 0
\(957\) 11756.8i 0.397119i
\(958\) 0 0
\(959\) −12296.3 45178.5i −0.414044 1.52126i
\(960\) 0 0
\(961\) 75167.3 2.52316
\(962\) 0 0
\(963\) 3924.93i 0.131339i
\(964\) 0 0
\(965\) 6780.47i 0.226188i
\(966\) 0 0
\(967\) 32218.8i 1.07144i 0.844394 + 0.535722i \(0.179961\pi\)
−0.844394 + 0.535722i \(0.820039\pi\)
\(968\) 0 0
\(969\) 4455.69i 0.147717i
\(970\) 0 0
\(971\) 6664.98 0.220277 0.110139 0.993916i \(-0.464871\pi\)
0.110139 + 0.993916i \(0.464871\pi\)
\(972\) 0 0
\(973\) 12270.6 + 45084.0i 0.404292 + 1.48543i
\(974\) 0 0
\(975\) 1099.34i 0.0361099i
\(976\) 0 0
\(977\) −1037.07 −0.0339598 −0.0169799 0.999856i \(-0.505405\pi\)
−0.0169799 + 0.999856i \(0.505405\pi\)
\(978\) 0 0
\(979\) 20202.3 0.659519
\(980\) 0 0
\(981\) −12821.6 −0.417292
\(982\) 0 0
\(983\) −31143.8 −1.01051 −0.505256 0.862969i \(-0.668602\pi\)
−0.505256 + 0.862969i \(0.668602\pi\)
\(984\) 0 0
\(985\) 7389.09i 0.239021i
\(986\) 0 0
\(987\) −5495.92 20192.9i −0.177241 0.651212i
\(988\) 0 0
\(989\) 647.763 0.0208268
\(990\) 0 0
\(991\) 39874.3i 1.27815i −0.769144 0.639076i \(-0.779317\pi\)
0.769144 0.639076i \(-0.220683\pi\)
\(992\) 0 0
\(993\) 6472.85i 0.206858i
\(994\) 0 0
\(995\) 14296.5i 0.455506i
\(996\) 0 0
\(997\) 55124.7i 1.75107i −0.483154 0.875535i \(-0.660509\pi\)
0.483154 0.875535i \(-0.339491\pi\)
\(998\) 0 0
\(999\) 3250.66 0.102949
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.e.895.6 8
4.3 odd 2 1344.4.b.f.895.6 8
7.6 odd 2 1344.4.b.f.895.3 8
8.3 odd 2 336.4.b.e.223.3 8
8.5 even 2 336.4.b.f.223.3 yes 8
24.5 odd 2 1008.4.b.k.559.6 8
24.11 even 2 1008.4.b.i.559.6 8
28.27 even 2 inner 1344.4.b.e.895.3 8
56.13 odd 2 336.4.b.e.223.6 yes 8
56.27 even 2 336.4.b.f.223.6 yes 8
168.83 odd 2 1008.4.b.k.559.3 8
168.125 even 2 1008.4.b.i.559.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.4.b.e.223.3 8 8.3 odd 2
336.4.b.e.223.6 yes 8 56.13 odd 2
336.4.b.f.223.3 yes 8 8.5 even 2
336.4.b.f.223.6 yes 8 56.27 even 2
1008.4.b.i.559.3 8 168.125 even 2
1008.4.b.i.559.6 8 24.11 even 2
1008.4.b.k.559.3 8 168.83 odd 2
1008.4.b.k.559.6 8 24.5 odd 2
1344.4.b.e.895.3 8 28.27 even 2 inner
1344.4.b.e.895.6 8 1.1 even 1 trivial
1344.4.b.f.895.3 8 7.6 odd 2
1344.4.b.f.895.6 8 4.3 odd 2