Properties

Label 2352.3.m.j.1471.2
Level $2352$
Weight $3$
Character 2352.1471
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
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] = [2352,3,Mod(1471,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), 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: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.2
Root \(-1.27069 - 2.20090i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.j.1471.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +9.08276 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +9.08276 q^{5} -3.00000 q^{9} -8.80360i q^{11} -15.7318i q^{15} -2.91724 q^{17} +17.6072i q^{19} +29.5882i q^{23} +57.4966 q^{25} +5.19615i q^{27} +48.4966 q^{29} +38.3918i q^{31} -15.2483 q^{33} -20.4966 q^{37} +26.9172 q^{41} -27.7128i q^{43} -27.2483 q^{45} -62.9272i q^{47} +5.05280i q^{51} +84.4966 q^{53} -79.9610i q^{55} +30.4966 q^{57} -42.1426i q^{59} +102.497 q^{61} -52.8216i q^{67} +51.2483 q^{69} -64.8026i q^{71} +30.4966 q^{73} -99.5870i q^{75} +149.816i q^{79} +9.00000 q^{81} +0.573396i q^{83} -26.4966 q^{85} -83.9985i q^{87} -10.0759 q^{89} +66.4966 q^{93} +159.922i q^{95} +113.503 q^{97} +26.4108i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{5} - 12 q^{9} - 36 q^{17} + 84 q^{25} + 48 q^{29} + 12 q^{33} + 64 q^{37} + 132 q^{41} - 36 q^{45} + 192 q^{53} - 24 q^{57} + 264 q^{61} + 132 q^{69} - 24 q^{73} + 36 q^{81} + 40 q^{85} + 276 q^{89} + 120 q^{93} + 600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 9.08276 1.81655 0.908276 0.418371i \(-0.137399\pi\)
0.908276 + 0.418371i \(0.137399\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 8.80360i − 0.800328i −0.916444 0.400164i \(-0.868953\pi\)
0.916444 0.400164i \(-0.131047\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) − 15.7318i − 1.04879i
\(16\) 0 0
\(17\) −2.91724 −0.171602 −0.0858011 0.996312i \(-0.527345\pi\)
−0.0858011 + 0.996312i \(0.527345\pi\)
\(18\) 0 0
\(19\) 17.6072i 0.926695i 0.886177 + 0.463348i \(0.153352\pi\)
−0.886177 + 0.463348i \(0.846648\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 29.5882i 1.28644i 0.765680 + 0.643222i \(0.222403\pi\)
−0.765680 + 0.643222i \(0.777597\pi\)
\(24\) 0 0
\(25\) 57.4966 2.29986
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 48.4966 1.67230 0.836148 0.548504i \(-0.184803\pi\)
0.836148 + 0.548504i \(0.184803\pi\)
\(30\) 0 0
\(31\) 38.3918i 1.23845i 0.785215 + 0.619223i \(0.212552\pi\)
−0.785215 + 0.619223i \(0.787448\pi\)
\(32\) 0 0
\(33\) −15.2483 −0.462069
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −20.4966 −0.553961 −0.276981 0.960875i \(-0.589334\pi\)
−0.276981 + 0.960875i \(0.589334\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 26.9172 0.656518 0.328259 0.944588i \(-0.393538\pi\)
0.328259 + 0.944588i \(0.393538\pi\)
\(42\) 0 0
\(43\) − 27.7128i − 0.644484i −0.946657 0.322242i \(-0.895564\pi\)
0.946657 0.322242i \(-0.104436\pi\)
\(44\) 0 0
\(45\) −27.2483 −0.605518
\(46\) 0 0
\(47\) − 62.9272i − 1.33888i −0.742867 0.669439i \(-0.766535\pi\)
0.742867 0.669439i \(-0.233465\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.05280i 0.0990746i
\(52\) 0 0
\(53\) 84.4966 1.59428 0.797138 0.603798i \(-0.206347\pi\)
0.797138 + 0.603798i \(0.206347\pi\)
\(54\) 0 0
\(55\) − 79.9610i − 1.45384i
\(56\) 0 0
\(57\) 30.4966 0.535028
\(58\) 0 0
\(59\) − 42.1426i − 0.714282i −0.934051 0.357141i \(-0.883752\pi\)
0.934051 0.357141i \(-0.116248\pi\)
\(60\) 0 0
\(61\) 102.497 1.68027 0.840136 0.542376i \(-0.182475\pi\)
0.840136 + 0.542376i \(0.182475\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 52.8216i − 0.788382i −0.919028 0.394191i \(-0.871025\pi\)
0.919028 0.394191i \(-0.128975\pi\)
\(68\) 0 0
\(69\) 51.2483 0.742729
\(70\) 0 0
\(71\) − 64.8026i − 0.912713i −0.889797 0.456357i \(-0.849154\pi\)
0.889797 0.456357i \(-0.150846\pi\)
\(72\) 0 0
\(73\) 30.4966 0.417761 0.208881 0.977941i \(-0.433018\pi\)
0.208881 + 0.977941i \(0.433018\pi\)
\(74\) 0 0
\(75\) − 99.5870i − 1.32783i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 149.816i 1.89641i 0.317658 + 0.948205i \(0.397104\pi\)
−0.317658 + 0.948205i \(0.602896\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 0.573396i 0.00690838i 0.999994 + 0.00345419i \(0.00109951\pi\)
−0.999994 + 0.00345419i \(0.998900\pi\)
\(84\) 0 0
\(85\) −26.4966 −0.311724
\(86\) 0 0
\(87\) − 83.9985i − 0.965500i
\(88\) 0 0
\(89\) −10.0759 −0.113213 −0.0566063 0.998397i \(-0.518028\pi\)
−0.0566063 + 0.998397i \(0.518028\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 66.4966 0.715017
\(94\) 0 0
\(95\) 159.922i 1.68339i
\(96\) 0 0
\(97\) 113.503 1.17014 0.585069 0.810983i \(-0.301067\pi\)
0.585069 + 0.810983i \(0.301067\pi\)
\(98\) 0 0
\(99\) 26.4108i 0.266776i
\(100\) 0 0
\(101\) −142.076 −1.40669 −0.703346 0.710848i \(-0.748312\pi\)
−0.703346 + 0.710848i \(0.748312\pi\)
\(102\) 0 0
\(103\) 185.604i 1.80198i 0.433837 + 0.900992i \(0.357160\pi\)
−0.433837 + 0.900992i \(0.642840\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 120.802i − 1.12899i −0.825437 0.564494i \(-0.809071\pi\)
0.825437 0.564494i \(-0.190929\pi\)
\(108\) 0 0
\(109\) −142.993 −1.31186 −0.655932 0.754820i \(-0.727724\pi\)
−0.655932 + 0.754820i \(0.727724\pi\)
\(110\) 0 0
\(111\) 35.5011i 0.319830i
\(112\) 0 0
\(113\) −114.993 −1.01764 −0.508819 0.860873i \(-0.669918\pi\)
−0.508819 + 0.860873i \(0.669918\pi\)
\(114\) 0 0
\(115\) 268.743i 2.33689i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 43.4966 0.359476
\(122\) 0 0
\(123\) − 46.6220i − 0.379041i
\(124\) 0 0
\(125\) 295.159 2.36127
\(126\) 0 0
\(127\) 38.9652i 0.306813i 0.988163 + 0.153406i \(0.0490243\pi\)
−0.988163 + 0.153406i \(0.950976\pi\)
\(128\) 0 0
\(129\) −48.0000 −0.372093
\(130\) 0 0
\(131\) − 187.061i − 1.42795i −0.700171 0.713975i \(-0.746893\pi\)
0.700171 0.713975i \(-0.253107\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 47.1954i 0.349596i
\(136\) 0 0
\(137\) −139.986 −1.02180 −0.510899 0.859641i \(-0.670687\pi\)
−0.510899 + 0.859641i \(0.670687\pi\)
\(138\) 0 0
\(139\) − 209.566i − 1.50767i −0.657063 0.753836i \(-0.728202\pi\)
0.657063 0.753836i \(-0.271798\pi\)
\(140\) 0 0
\(141\) −108.993 −0.773001
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 440.483 3.03781
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 84.4966 0.567091 0.283546 0.958959i \(-0.408489\pi\)
0.283546 + 0.958959i \(0.408489\pi\)
\(150\) 0 0
\(151\) 238.999i 1.58278i 0.611314 + 0.791388i \(0.290641\pi\)
−0.611314 + 0.791388i \(0.709359\pi\)
\(152\) 0 0
\(153\) 8.75171 0.0572007
\(154\) 0 0
\(155\) 348.704i 2.24970i
\(156\) 0 0
\(157\) 80.4829 0.512630 0.256315 0.966593i \(-0.417492\pi\)
0.256315 + 0.966593i \(0.417492\pi\)
\(158\) 0 0
\(159\) − 146.352i − 0.920455i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 297.029i − 1.82226i −0.412115 0.911132i \(-0.635210\pi\)
0.412115 0.911132i \(-0.364790\pi\)
\(164\) 0 0
\(165\) −138.497 −0.839373
\(166\) 0 0
\(167\) − 229.204i − 1.37248i −0.727375 0.686240i \(-0.759260\pi\)
0.727375 0.686240i \(-0.240740\pi\)
\(168\) 0 0
\(169\) −169.000 −1.00000
\(170\) 0 0
\(171\) − 52.8216i − 0.308898i
\(172\) 0 0
\(173\) 206.917 1.19605 0.598027 0.801476i \(-0.295952\pi\)
0.598027 + 0.801476i \(0.295952\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −72.9932 −0.412391
\(178\) 0 0
\(179\) 56.7276i 0.316914i 0.987366 + 0.158457i \(0.0506519\pi\)
−0.987366 + 0.158457i \(0.949348\pi\)
\(180\) 0 0
\(181\) 22.0137 0.121623 0.0608113 0.998149i \(-0.480631\pi\)
0.0608113 + 0.998149i \(0.480631\pi\)
\(182\) 0 0
\(183\) − 177.529i − 0.970105i
\(184\) 0 0
\(185\) −186.166 −1.00630
\(186\) 0 0
\(187\) 25.6822i 0.137338i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 140.129i 0.733660i 0.930288 + 0.366830i \(0.119557\pi\)
−0.930288 + 0.366830i \(0.880443\pi\)
\(192\) 0 0
\(193\) −149.490 −0.774558 −0.387279 0.921963i \(-0.626585\pi\)
−0.387279 + 0.921963i \(0.626585\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 106.510 0.540661 0.270331 0.962768i \(-0.412867\pi\)
0.270331 + 0.962768i \(0.412867\pi\)
\(198\) 0 0
\(199\) 111.998i 0.562804i 0.959590 + 0.281402i \(0.0907995\pi\)
−0.959590 + 0.281402i \(0.909201\pi\)
\(200\) 0 0
\(201\) −91.4897 −0.455173
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 244.483 1.19260
\(206\) 0 0
\(207\) − 88.7646i − 0.428815i
\(208\) 0 0
\(209\) 155.007 0.741660
\(210\) 0 0
\(211\) − 105.643i − 0.500679i −0.968158 0.250339i \(-0.919458\pi\)
0.968158 0.250339i \(-0.0805422\pi\)
\(212\) 0 0
\(213\) −112.241 −0.526955
\(214\) 0 0
\(215\) − 251.709i − 1.17074i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 52.8216i − 0.241195i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 255.770i − 1.14695i −0.819223 0.573476i \(-0.805595\pi\)
0.819223 0.573476i \(-0.194405\pi\)
\(224\) 0 0
\(225\) −172.490 −0.766621
\(226\) 0 0
\(227\) 81.4183i 0.358671i 0.983788 + 0.179335i \(0.0573947\pi\)
−0.983788 + 0.179335i \(0.942605\pi\)
\(228\) 0 0
\(229\) −38.9795 −0.170216 −0.0851080 0.996372i \(-0.527124\pi\)
−0.0851080 + 0.996372i \(0.527124\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −306.000 −1.31330 −0.656652 0.754193i \(-0.728028\pi\)
−0.656652 + 0.754193i \(0.728028\pi\)
\(234\) 0 0
\(235\) − 571.553i − 2.43214i
\(236\) 0 0
\(237\) 259.490 1.09489
\(238\) 0 0
\(239\) 243.789i 1.02004i 0.860163 + 0.510019i \(0.170362\pi\)
−0.860163 + 0.510019i \(0.829638\pi\)
\(240\) 0 0
\(241\) −357.476 −1.48330 −0.741652 0.670785i \(-0.765957\pi\)
−0.741652 + 0.670785i \(0.765957\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.993150 0.00398856
\(250\) 0 0
\(251\) 189.355i 0.754403i 0.926131 + 0.377201i \(0.123114\pi\)
−0.926131 + 0.377201i \(0.876886\pi\)
\(252\) 0 0
\(253\) 260.483 1.02958
\(254\) 0 0
\(255\) 45.8934i 0.179974i
\(256\) 0 0
\(257\) 207.910 0.808990 0.404495 0.914540i \(-0.367447\pi\)
0.404495 + 0.914540i \(0.367447\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −145.490 −0.557432
\(262\) 0 0
\(263\) 237.434i 0.902792i 0.892324 + 0.451396i \(0.149074\pi\)
−0.892324 + 0.451396i \(0.850926\pi\)
\(264\) 0 0
\(265\) 767.462 2.89608
\(266\) 0 0
\(267\) 17.4520i 0.0653633i
\(268\) 0 0
\(269\) 83.0691 0.308807 0.154403 0.988008i \(-0.450654\pi\)
0.154403 + 0.988008i \(0.450654\pi\)
\(270\) 0 0
\(271\) − 239.883i − 0.885178i −0.896725 0.442589i \(-0.854060\pi\)
0.896725 0.442589i \(-0.145940\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 506.177i − 1.84064i
\(276\) 0 0
\(277\) 78.4829 0.283332 0.141666 0.989915i \(-0.454754\pi\)
0.141666 + 0.989915i \(0.454754\pi\)
\(278\) 0 0
\(279\) − 115.175i − 0.412815i
\(280\) 0 0
\(281\) 380.979 1.35580 0.677899 0.735155i \(-0.262890\pi\)
0.677899 + 0.735155i \(0.262890\pi\)
\(282\) 0 0
\(283\) 49.3812i 0.174492i 0.996187 + 0.0872460i \(0.0278066\pi\)
−0.996187 + 0.0872460i \(0.972193\pi\)
\(284\) 0 0
\(285\) 276.993 0.971906
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −280.490 −0.970553
\(290\) 0 0
\(291\) − 196.594i − 0.675580i
\(292\) 0 0
\(293\) 217.924 0.743768 0.371884 0.928279i \(-0.378712\pi\)
0.371884 + 0.928279i \(0.378712\pi\)
\(294\) 0 0
\(295\) − 382.771i − 1.29753i
\(296\) 0 0
\(297\) 45.7449 0.154023
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 246.083i 0.812154i
\(304\) 0 0
\(305\) 930.952 3.05230
\(306\) 0 0
\(307\) 164.820i 0.536872i 0.963298 + 0.268436i \(0.0865068\pi\)
−0.963298 + 0.268436i \(0.913493\pi\)
\(308\) 0 0
\(309\) 321.476 1.04038
\(310\) 0 0
\(311\) − 353.912i − 1.13798i −0.822344 0.568990i \(-0.807334\pi\)
0.822344 0.568990i \(-0.192666\pi\)
\(312\) 0 0
\(313\) 265.986 0.849796 0.424898 0.905241i \(-0.360310\pi\)
0.424898 + 0.905241i \(0.360310\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −292.469 −0.922616 −0.461308 0.887240i \(-0.652620\pi\)
−0.461308 + 0.887240i \(0.652620\pi\)
\(318\) 0 0
\(319\) − 426.945i − 1.33838i
\(320\) 0 0
\(321\) −209.235 −0.651821
\(322\) 0 0
\(323\) − 51.3644i − 0.159023i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 247.671i 0.757405i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 538.632i 1.62729i 0.581364 + 0.813644i \(0.302519\pi\)
−0.581364 + 0.813644i \(0.697481\pi\)
\(332\) 0 0
\(333\) 61.4897 0.184654
\(334\) 0 0
\(335\) − 479.766i − 1.43214i
\(336\) 0 0
\(337\) 135.986 0.403520 0.201760 0.979435i \(-0.435334\pi\)
0.201760 + 0.979435i \(0.435334\pi\)
\(338\) 0 0
\(339\) 199.174i 0.587534i
\(340\) 0 0
\(341\) 337.986 0.991162
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 465.476 1.34921
\(346\) 0 0
\(347\) 312.498i 0.900570i 0.892885 + 0.450285i \(0.148678\pi\)
−0.892885 + 0.450285i \(0.851322\pi\)
\(348\) 0 0
\(349\) −407.462 −1.16751 −0.583757 0.811928i \(-0.698418\pi\)
−0.583757 + 0.811928i \(0.698418\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 154.076 0.436476 0.218238 0.975896i \(-0.429969\pi\)
0.218238 + 0.975896i \(0.429969\pi\)
\(354\) 0 0
\(355\) − 588.587i − 1.65799i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 413.506i − 1.15183i −0.817510 0.575914i \(-0.804646\pi\)
0.817510 0.575914i \(-0.195354\pi\)
\(360\) 0 0
\(361\) 50.9863 0.141236
\(362\) 0 0
\(363\) − 75.3383i − 0.207543i
\(364\) 0 0
\(365\) 276.993 0.758885
\(366\) 0 0
\(367\) − 639.688i − 1.74302i −0.490378 0.871510i \(-0.663141\pi\)
0.490378 0.871510i \(-0.336859\pi\)
\(368\) 0 0
\(369\) −80.7517 −0.218839
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −265.973 −0.713063 −0.356532 0.934283i \(-0.616041\pi\)
−0.356532 + 0.934283i \(0.616041\pi\)
\(374\) 0 0
\(375\) − 511.230i − 1.36328i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 32.9208i − 0.0868623i −0.999056 0.0434312i \(-0.986171\pi\)
0.999056 0.0434312i \(-0.0138289\pi\)
\(380\) 0 0
\(381\) 67.4897 0.177138
\(382\) 0 0
\(383\) 673.135i 1.75753i 0.477252 + 0.878766i \(0.341633\pi\)
−0.477252 + 0.878766i \(0.658367\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 83.1384i 0.214828i
\(388\) 0 0
\(389\) 236.524 0.608031 0.304015 0.952667i \(-0.401673\pi\)
0.304015 + 0.952667i \(0.401673\pi\)
\(390\) 0 0
\(391\) − 86.3158i − 0.220757i
\(392\) 0 0
\(393\) −324.000 −0.824427
\(394\) 0 0
\(395\) 1360.75i 3.44493i
\(396\) 0 0
\(397\) −695.462 −1.75179 −0.875897 0.482498i \(-0.839730\pi\)
−0.875897 + 0.482498i \(0.839730\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −184.014 −0.458887 −0.229444 0.973322i \(-0.573691\pi\)
−0.229444 + 0.973322i \(0.573691\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 81.7449 0.201839
\(406\) 0 0
\(407\) 180.444i 0.443351i
\(408\) 0 0
\(409\) −257.503 −0.629593 −0.314796 0.949159i \(-0.601936\pi\)
−0.314796 + 0.949159i \(0.601936\pi\)
\(410\) 0 0
\(411\) 242.463i 0.589935i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 5.20802i 0.0125494i
\(416\) 0 0
\(417\) −362.979 −0.870454
\(418\) 0 0
\(419\) − 361.366i − 0.862448i −0.902245 0.431224i \(-0.858082\pi\)
0.902245 0.431224i \(-0.141918\pi\)
\(420\) 0 0
\(421\) −191.476 −0.454812 −0.227406 0.973800i \(-0.573025\pi\)
−0.227406 + 0.973800i \(0.573025\pi\)
\(422\) 0 0
\(423\) 188.782i 0.446292i
\(424\) 0 0
\(425\) −167.731 −0.394662
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 299.525i 0.694954i 0.937689 + 0.347477i \(0.112962\pi\)
−0.937689 + 0.347477i \(0.887038\pi\)
\(432\) 0 0
\(433\) 448.966 1.03687 0.518436 0.855116i \(-0.326514\pi\)
0.518436 + 0.855116i \(0.326514\pi\)
\(434\) 0 0
\(435\) − 762.939i − 1.75388i
\(436\) 0 0
\(437\) −520.966 −1.19214
\(438\) 0 0
\(439\) − 486.647i − 1.10854i −0.832339 0.554268i \(-0.812998\pi\)
0.832339 0.554268i \(-0.187002\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 367.303i 0.829125i 0.910021 + 0.414563i \(0.136065\pi\)
−0.910021 + 0.414563i \(0.863935\pi\)
\(444\) 0 0
\(445\) −91.5171 −0.205656
\(446\) 0 0
\(447\) − 146.352i − 0.327410i
\(448\) 0 0
\(449\) 306.000 0.681514 0.340757 0.940151i \(-0.389317\pi\)
0.340757 + 0.940151i \(0.389317\pi\)
\(450\) 0 0
\(451\) − 236.969i − 0.525429i
\(452\) 0 0
\(453\) 413.959 0.913817
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 152.014 0.332634 0.166317 0.986072i \(-0.446813\pi\)
0.166317 + 0.986072i \(0.446813\pi\)
\(458\) 0 0
\(459\) − 15.1584i − 0.0330249i
\(460\) 0 0
\(461\) 103.883 0.225343 0.112671 0.993632i \(-0.464059\pi\)
0.112671 + 0.993632i \(0.464059\pi\)
\(462\) 0 0
\(463\) 189.618i 0.409542i 0.978810 + 0.204771i \(0.0656450\pi\)
−0.978810 + 0.204771i \(0.934355\pi\)
\(464\) 0 0
\(465\) 603.973 1.29887
\(466\) 0 0
\(467\) − 187.061i − 0.400560i −0.979739 0.200280i \(-0.935815\pi\)
0.979739 0.200280i \(-0.0641852\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 139.400i − 0.295967i
\(472\) 0 0
\(473\) −243.973 −0.515798
\(474\) 0 0
\(475\) 1012.35i 2.13127i
\(476\) 0 0
\(477\) −253.490 −0.531425
\(478\) 0 0
\(479\) − 416.839i − 0.870228i −0.900375 0.435114i \(-0.856708\pi\)
0.900375 0.435114i \(-0.143292\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1030.92 2.12562
\(486\) 0 0
\(487\) 105.643i 0.216927i 0.994100 + 0.108463i \(0.0345930\pi\)
−0.994100 + 0.108463i \(0.965407\pi\)
\(488\) 0 0
\(489\) −514.469 −1.05208
\(490\) 0 0
\(491\) 920.412i 1.87457i 0.348569 + 0.937283i \(0.386668\pi\)
−0.348569 + 0.937283i \(0.613332\pi\)
\(492\) 0 0
\(493\) −141.476 −0.286970
\(494\) 0 0
\(495\) 239.883i 0.484612i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 337.762i 0.676877i 0.940988 + 0.338439i \(0.109899\pi\)
−0.940988 + 0.338439i \(0.890101\pi\)
\(500\) 0 0
\(501\) −396.993 −0.792401
\(502\) 0 0
\(503\) 612.501i 1.21770i 0.793287 + 0.608848i \(0.208368\pi\)
−0.793287 + 0.608848i \(0.791632\pi\)
\(504\) 0 0
\(505\) −1290.44 −2.55533
\(506\) 0 0
\(507\) 292.717i 0.577350i
\(508\) 0 0
\(509\) 181.055 0.355708 0.177854 0.984057i \(-0.443085\pi\)
0.177854 + 0.984057i \(0.443085\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −91.4897 −0.178343
\(514\) 0 0
\(515\) 1685.80i 3.27340i
\(516\) 0 0
\(517\) −553.986 −1.07154
\(518\) 0 0
\(519\) − 358.391i − 0.690542i
\(520\) 0 0
\(521\) 878.049 1.68531 0.842657 0.538451i \(-0.180990\pi\)
0.842657 + 0.538451i \(0.180990\pi\)
\(522\) 0 0
\(523\) − 456.067i − 0.872021i −0.899941 0.436011i \(-0.856391\pi\)
0.899941 0.436011i \(-0.143609\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 111.998i − 0.212520i
\(528\) 0 0
\(529\) −346.462 −0.654938
\(530\) 0 0
\(531\) 126.428i 0.238094i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 1097.21i − 2.05086i
\(536\) 0 0
\(537\) 98.2551 0.182970
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 558.483 1.03232 0.516158 0.856493i \(-0.327362\pi\)
0.516158 + 0.856493i \(0.327362\pi\)
\(542\) 0 0
\(543\) − 38.1288i − 0.0702189i
\(544\) 0 0
\(545\) −1298.77 −2.38307
\(546\) 0 0
\(547\) − 579.365i − 1.05917i −0.848257 0.529584i \(-0.822348\pi\)
0.848257 0.529584i \(-0.177652\pi\)
\(548\) 0 0
\(549\) −307.490 −0.560091
\(550\) 0 0
\(551\) 853.889i 1.54971i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 322.448i 0.580988i
\(556\) 0 0
\(557\) −15.4760 −0.0277846 −0.0138923 0.999903i \(-0.504422\pi\)
−0.0138923 + 0.999903i \(0.504422\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 44.4829 0.0792921
\(562\) 0 0
\(563\) − 715.278i − 1.27048i −0.772317 0.635238i \(-0.780902\pi\)
0.772317 0.635238i \(-0.219098\pi\)
\(564\) 0 0
\(565\) −1044.46 −1.84859
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −660.952 −1.16160 −0.580801 0.814045i \(-0.697261\pi\)
−0.580801 + 0.814045i \(0.697261\pi\)
\(570\) 0 0
\(571\) 281.405i 0.492828i 0.969165 + 0.246414i \(0.0792523\pi\)
−0.969165 + 0.246414i \(0.920748\pi\)
\(572\) 0 0
\(573\) 242.711 0.423579
\(574\) 0 0
\(575\) 1701.22i 2.95865i
\(576\) 0 0
\(577\) −570.952 −0.989518 −0.494759 0.869030i \(-0.664744\pi\)
−0.494759 + 0.869030i \(0.664744\pi\)
\(578\) 0 0
\(579\) 258.924i 0.447191i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 743.874i − 1.27594i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 455.541i 0.776050i 0.921649 + 0.388025i \(0.126843\pi\)
−0.921649 + 0.388025i \(0.873157\pi\)
\(588\) 0 0
\(589\) −675.973 −1.14766
\(590\) 0 0
\(591\) − 184.481i − 0.312151i
\(592\) 0 0
\(593\) −1038.89 −1.75192 −0.875961 0.482382i \(-0.839772\pi\)
−0.875961 + 0.482382i \(0.839772\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 193.986 0.324935
\(598\) 0 0
\(599\) 884.935i 1.47735i 0.674060 + 0.738677i \(0.264549\pi\)
−0.674060 + 0.738677i \(0.735451\pi\)
\(600\) 0 0
\(601\) 282.952 0.470802 0.235401 0.971898i \(-0.424360\pi\)
0.235401 + 0.971898i \(0.424360\pi\)
\(602\) 0 0
\(603\) 158.465i 0.262794i
\(604\) 0 0
\(605\) 395.069 0.653007
\(606\) 0 0
\(607\) 185.341i 0.305340i 0.988277 + 0.152670i \(0.0487871\pi\)
−0.988277 + 0.152670i \(0.951213\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −849.945 −1.38653 −0.693267 0.720681i \(-0.743829\pi\)
−0.693267 + 0.720681i \(0.743829\pi\)
\(614\) 0 0
\(615\) − 423.457i − 0.688548i
\(616\) 0 0
\(617\) 37.0342 0.0600231 0.0300115 0.999550i \(-0.490446\pi\)
0.0300115 + 0.999550i \(0.490446\pi\)
\(618\) 0 0
\(619\) 554.830i 0.896332i 0.893950 + 0.448166i \(0.147923\pi\)
−0.893950 + 0.448166i \(0.852077\pi\)
\(620\) 0 0
\(621\) −153.745 −0.247576
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1243.44 1.98951
\(626\) 0 0
\(627\) − 268.480i − 0.428197i
\(628\) 0 0
\(629\) 59.7934 0.0950610
\(630\) 0 0
\(631\) − 633.023i − 1.00321i −0.865098 0.501603i \(-0.832744\pi\)
0.865098 0.501603i \(-0.167256\pi\)
\(632\) 0 0
\(633\) −182.979 −0.289067
\(634\) 0 0
\(635\) 353.912i 0.557341i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 194.408i 0.304238i
\(640\) 0 0
\(641\) −228.952 −0.357179 −0.178590 0.983924i \(-0.557153\pi\)
−0.178590 + 0.983924i \(0.557153\pi\)
\(642\) 0 0
\(643\) 724.284i 1.12641i 0.826316 + 0.563207i \(0.190433\pi\)
−0.826316 + 0.563207i \(0.809567\pi\)
\(644\) 0 0
\(645\) −435.973 −0.675927
\(646\) 0 0
\(647\) 739.503i 1.14297i 0.820612 + 0.571486i \(0.193633\pi\)
−0.820612 + 0.571486i \(0.806367\pi\)
\(648\) 0 0
\(649\) −371.007 −0.571659
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −195.476 −0.299351 −0.149675 0.988735i \(-0.547823\pi\)
−0.149675 + 0.988735i \(0.547823\pi\)
\(654\) 0 0
\(655\) − 1699.04i − 2.59395i
\(656\) 0 0
\(657\) −91.4897 −0.139254
\(658\) 0 0
\(659\) − 706.737i − 1.07244i −0.844079 0.536219i \(-0.819852\pi\)
0.844079 0.536219i \(-0.180148\pi\)
\(660\) 0 0
\(661\) 1210.47 1.83127 0.915635 0.402011i \(-0.131689\pi\)
0.915635 + 0.402011i \(0.131689\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1434.93i 2.15131i
\(668\) 0 0
\(669\) −443.007 −0.662193
\(670\) 0 0
\(671\) − 902.339i − 1.34477i
\(672\) 0 0
\(673\) 515.449 0.765897 0.382948 0.923770i \(-0.374909\pi\)
0.382948 + 0.923770i \(0.374909\pi\)
\(674\) 0 0
\(675\) 298.761i 0.442609i
\(676\) 0 0
\(677\) 179.856 0.265666 0.132833 0.991138i \(-0.457593\pi\)
0.132833 + 0.991138i \(0.457593\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 141.021 0.207079
\(682\) 0 0
\(683\) 753.609i 1.10338i 0.834049 + 0.551690i \(0.186017\pi\)
−0.834049 + 0.551690i \(0.813983\pi\)
\(684\) 0 0
\(685\) −1271.46 −1.85615
\(686\) 0 0
\(687\) 67.5144i 0.0982742i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 235.511i 0.340827i 0.985373 + 0.170413i \(0.0545103\pi\)
−0.985373 + 0.170413i \(0.945490\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1903.44i − 2.73876i
\(696\) 0 0
\(697\) −78.5240 −0.112660
\(698\) 0 0
\(699\) 530.008i 0.758237i
\(700\) 0 0
\(701\) −195.476 −0.278853 −0.139427 0.990232i \(-0.544526\pi\)
−0.139427 + 0.990232i \(0.544526\pi\)
\(702\) 0 0
\(703\) − 360.887i − 0.513353i
\(704\) 0 0
\(705\) −989.959 −1.40420
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −958.993 −1.35260 −0.676300 0.736626i \(-0.736418\pi\)
−0.676300 + 0.736626i \(0.736418\pi\)
\(710\) 0 0
\(711\) − 449.449i − 0.632137i
\(712\) 0 0
\(713\) −1135.95 −1.59319
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 422.255 0.588919
\(718\) 0 0
\(719\) − 549.574i − 0.764359i −0.924088 0.382180i \(-0.875174\pi\)
0.924088 0.382180i \(-0.124826\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 619.167i 0.856385i
\(724\) 0 0
\(725\) 2788.39 3.84605
\(726\) 0 0
\(727\) − 761.219i − 1.04707i −0.852005 0.523534i \(-0.824613\pi\)
0.852005 0.523534i \(-0.175387\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 80.8449i 0.110595i
\(732\) 0 0
\(733\) −476.027 −0.649423 −0.324712 0.945813i \(-0.605267\pi\)
−0.324712 + 0.945813i \(0.605267\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −465.021 −0.630964
\(738\) 0 0
\(739\) 686.681i 0.929203i 0.885520 + 0.464602i \(0.153802\pi\)
−0.885520 + 0.464602i \(0.846198\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 423.302i − 0.569719i −0.958569 0.284860i \(-0.908053\pi\)
0.958569 0.284860i \(-0.0919470\pi\)
\(744\) 0 0
\(745\) 767.462 1.03015
\(746\) 0 0
\(747\) − 1.72019i − 0.00230279i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 748.246i 0.996333i 0.867081 + 0.498166i \(0.165993\pi\)
−0.867081 + 0.498166i \(0.834007\pi\)
\(752\) 0 0
\(753\) 327.973 0.435555
\(754\) 0 0
\(755\) 2170.77i 2.87520i
\(756\) 0 0
\(757\) 1264.90 1.67093 0.835467 0.549540i \(-0.185197\pi\)
0.835467 + 0.549540i \(0.185197\pi\)
\(758\) 0 0
\(759\) − 451.170i − 0.594426i
\(760\) 0 0
\(761\) −28.1170 −0.0369475 −0.0184737 0.999829i \(-0.505881\pi\)
−0.0184737 + 0.999829i \(0.505881\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 79.4897 0.103908
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1334.98 −1.73599 −0.867997 0.496569i \(-0.834593\pi\)
−0.867997 + 0.496569i \(0.834593\pi\)
\(770\) 0 0
\(771\) − 360.111i − 0.467070i
\(772\) 0 0
\(773\) 969.869 1.25468 0.627341 0.778745i \(-0.284143\pi\)
0.627341 + 0.778745i \(0.284143\pi\)
\(774\) 0 0
\(775\) 2207.40i 2.84826i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 473.937i 0.608392i
\(780\) 0 0
\(781\) −570.497 −0.730469
\(782\) 0 0
\(783\) 251.996i 0.321833i
\(784\) 0 0
\(785\) 731.007 0.931219
\(786\) 0 0
\(787\) 1047.83i 1.33142i 0.746208 + 0.665712i \(0.231872\pi\)
−0.746208 + 0.665712i \(0.768128\pi\)
\(788\) 0 0
\(789\) 411.248 0.521227
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 1329.28i − 1.67206i
\(796\) 0 0
\(797\) −548.090 −0.687691 −0.343845 0.939026i \(-0.611730\pi\)
−0.343845 + 0.939026i \(0.611730\pi\)
\(798\) 0 0
\(799\) 183.574i 0.229754i
\(800\) 0 0
\(801\) 30.2277 0.0377375
\(802\) 0 0
\(803\) − 268.480i − 0.334346i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 143.880i − 0.178290i
\(808\) 0 0
\(809\) −203.048 −0.250986 −0.125493 0.992094i \(-0.540051\pi\)
−0.125493 + 0.992094i \(0.540051\pi\)
\(810\) 0 0
\(811\) 279.469i 0.344598i 0.985045 + 0.172299i \(0.0551196\pi\)
−0.985045 + 0.172299i \(0.944880\pi\)
\(812\) 0 0
\(813\) −415.490 −0.511057
\(814\) 0 0
\(815\) − 2697.84i − 3.31024i
\(816\) 0 0
\(817\) 487.945 0.597240
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1201.41 −1.46335 −0.731673 0.681655i \(-0.761260\pi\)
−0.731673 + 0.681655i \(0.761260\pi\)
\(822\) 0 0
\(823\) − 731.786i − 0.889168i −0.895737 0.444584i \(-0.853351\pi\)
0.895737 0.444584i \(-0.146649\pi\)
\(824\) 0 0
\(825\) −876.724 −1.06270
\(826\) 0 0
\(827\) 1164.93i 1.40862i 0.709892 + 0.704311i \(0.248744\pi\)
−0.709892 + 0.704311i \(0.751256\pi\)
\(828\) 0 0
\(829\) −22.0137 −0.0265545 −0.0132773 0.999912i \(-0.504226\pi\)
−0.0132773 + 0.999912i \(0.504226\pi\)
\(830\) 0 0
\(831\) − 135.936i − 0.163582i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 2081.81i − 2.49318i
\(836\) 0 0
\(837\) −199.490 −0.238339
\(838\) 0 0
\(839\) − 293.278i − 0.349557i −0.984608 0.174778i \(-0.944079\pi\)
0.984608 0.174778i \(-0.0559209\pi\)
\(840\) 0 0
\(841\) 1510.92 1.79657
\(842\) 0 0
\(843\) − 659.876i − 0.782771i
\(844\) 0 0
\(845\) −1534.99 −1.81655
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 85.5308 0.100743
\(850\) 0 0
\(851\) − 606.457i − 0.712640i
\(852\) 0 0
\(853\) −495.517 −0.580911 −0.290456 0.956888i \(-0.593807\pi\)
−0.290456 + 0.956888i \(0.593807\pi\)
\(854\) 0 0
\(855\) − 479.766i − 0.561130i
\(856\) 0 0
\(857\) −1029.99 −1.20186 −0.600930 0.799302i \(-0.705203\pi\)
−0.600930 + 0.799302i \(0.705203\pi\)
\(858\) 0 0
\(859\) 769.294i 0.895569i 0.894142 + 0.447784i \(0.147787\pi\)
−0.894142 + 0.447784i \(0.852213\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 170.972i 0.198113i 0.995082 + 0.0990567i \(0.0315825\pi\)
−0.995082 + 0.0990567i \(0.968417\pi\)
\(864\) 0 0
\(865\) 1879.38 2.17269
\(866\) 0 0
\(867\) 485.822i 0.560349i
\(868\) 0 0
\(869\) 1318.92 1.51775
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −340.510 −0.390046
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −919.959 −1.04898 −0.524492 0.851415i \(-0.675745\pi\)
−0.524492 + 0.851415i \(0.675745\pi\)
\(878\) 0 0
\(879\) − 377.456i − 0.429415i
\(880\) 0 0
\(881\) 1179.95 1.33933 0.669667 0.742662i \(-0.266437\pi\)
0.669667 + 0.742662i \(0.266437\pi\)
\(882\) 0 0
\(883\) − 1453.16i − 1.64570i −0.568257 0.822851i \(-0.692382\pi\)
0.568257 0.822851i \(-0.307618\pi\)
\(884\) 0 0
\(885\) −662.979 −0.749129
\(886\) 0 0
\(887\) 1017.87i 1.14755i 0.819015 + 0.573773i \(0.194521\pi\)
−0.819015 + 0.573773i \(0.805479\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 79.2324i − 0.0889253i
\(892\) 0 0
\(893\) 1107.97 1.24073
\(894\) 0 0
\(895\) 515.244i 0.575691i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1861.87i 2.07105i
\(900\) 0 0
\(901\) −246.497 −0.273581
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 199.945 0.220934
\(906\) 0 0
\(907\) − 720.533i − 0.794414i −0.917729 0.397207i \(-0.869980\pi\)
0.917729 0.397207i \(-0.130020\pi\)
\(908\) 0 0
\(909\) 426.228 0.468897
\(910\) 0 0
\(911\) 1067.36i 1.17164i 0.810442 + 0.585819i \(0.199227\pi\)
−0.810442 + 0.585819i \(0.800773\pi\)
\(912\) 0 0
\(913\) 5.04795 0.00552897
\(914\) 0 0
\(915\) − 1612.46i − 1.76225i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 389.652i 0.423996i 0.977270 + 0.211998i \(0.0679970\pi\)
−0.977270 + 0.211998i \(0.932003\pi\)
\(920\) 0 0
\(921\) 285.476 0.309963
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1178.48 −1.27404
\(926\) 0 0
\(927\) − 556.813i − 0.600661i
\(928\) 0 0
\(929\) −1268.01 −1.36492 −0.682458 0.730925i \(-0.739089\pi\)
−0.682458 + 0.730925i \(0.739089\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −612.993 −0.657013
\(934\) 0 0
\(935\) 233.265i 0.249482i
\(936\) 0 0
\(937\) 121.986 0.130188 0.0650941 0.997879i \(-0.479265\pi\)
0.0650941 + 0.997879i \(0.479265\pi\)
\(938\) 0 0
\(939\) − 460.702i − 0.490630i
\(940\) 0 0
\(941\) −1160.79 −1.23357 −0.616787 0.787130i \(-0.711566\pi\)
−0.616787 + 0.787130i \(0.711566\pi\)
\(942\) 0 0
\(943\) 796.433i 0.844574i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 59.6421i − 0.0629800i −0.999504 0.0314900i \(-0.989975\pi\)
0.999504 0.0314900i \(-0.0100252\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 506.571i 0.532672i
\(952\) 0 0
\(953\) 959.959 1.00730 0.503651 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(954\) 0 0
\(955\) 1272.76i 1.33273i
\(956\) 0 0
\(957\) −739.490 −0.772717
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −512.932 −0.533748
\(962\) 0 0
\(963\) 362.405i 0.376329i
\(964\) 0 0
\(965\) −1357.78 −1.40703
\(966\) 0 0
\(967\) − 629.488i − 0.650970i −0.945547 0.325485i \(-0.894472\pi\)
0.945547 0.325485i \(-0.105528\pi\)
\(968\) 0 0
\(969\) −88.9658 −0.0918119
\(970\) 0 0
\(971\) − 1048.98i − 1.08031i −0.841566 0.540154i \(-0.818366\pi\)
0.841566 0.540154i \(-0.181634\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −793.945 −0.812636 −0.406318 0.913732i \(-0.633187\pi\)
−0.406318 + 0.913732i \(0.633187\pi\)
\(978\) 0 0
\(979\) 88.7043i 0.0906071i
\(980\) 0 0
\(981\) 428.979 0.437288
\(982\) 0 0
\(983\) − 1271.87i − 1.29387i −0.762545 0.646935i \(-0.776050\pi\)
0.762545 0.646935i \(-0.223950\pi\)
\(984\) 0 0
\(985\) 967.408 0.982140
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 819.973 0.829093
\(990\) 0 0
\(991\) − 677.196i − 0.683346i −0.939819 0.341673i \(-0.889006\pi\)
0.939819 0.341673i \(-0.110994\pi\)
\(992\) 0 0
\(993\) 932.938 0.939515
\(994\) 0 0
\(995\) 1017.25i 1.02236i
\(996\) 0 0
\(997\) 1876.35 1.88199 0.940996 0.338418i \(-0.109892\pi\)
0.940996 + 0.338418i \(0.109892\pi\)
\(998\) 0 0
\(999\) − 106.503i − 0.106610i
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 2352.3.m.j.1471.2 yes 4
4.3 odd 2 inner 2352.3.m.j.1471.4 yes 4
7.6 odd 2 2352.3.m.e.1471.3 yes 4
28.27 even 2 2352.3.m.e.1471.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.3.m.e.1471.1 4 28.27 even 2
2352.3.m.e.1471.3 yes 4 7.6 odd 2
2352.3.m.j.1471.2 yes 4 1.1 even 1 trivial
2352.3.m.j.1471.4 yes 4 4.3 odd 2 inner