Properties

Label 2352.3.m.e.1471.1
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 1471.1
Root \(-1.27069 + 2.20090i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.e.1471.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} -9.08276 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -9.08276 q^{5} -3.00000 q^{9} +8.80360i q^{11} +15.7318i q^{15} +2.91724 q^{17} +17.6072i q^{19} -29.5882i q^{23} +57.4966 q^{25} +5.19615i q^{27} +48.4966 q^{29} +38.3918i q^{31} +15.2483 q^{33} -20.4966 q^{37} -26.9172 q^{41} +27.7128i q^{43} +27.2483 q^{45} -62.9272i q^{47} -5.05280i q^{51} +84.4966 q^{53} -79.9610i q^{55} +30.4966 q^{57} -42.1426i q^{59} -102.497 q^{61} +52.8216i q^{67} -51.2483 q^{69} +64.8026i q^{71} -30.4966 q^{73} -99.5870i q^{75} -149.816i q^{79} +9.00000 q^{81} +0.573396i q^{83} -26.4966 q^{85} -83.9985i q^{87} +10.0759 q^{89} +66.4966 q^{93} -159.922i q^{95} -113.503 q^{97} -26.4108i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{5} - 12 q^{9}+O(q^{10}) \) 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 \(-0.862601\pi\)
−0.908276 + 0.418371i \(0.862601\pi\)
\(6\) 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 \(0.131047\pi\)
−0.916444 + 0.400164i \(0.868953\pi\)
\(12\) 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 \(-0.472655\pi\)
0.0858011 + 0.996312i \(0.472655\pi\)
\(18\) 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 \(-0.777597\pi\)
0.765680 0.643222i \(-0.222403\pi\)
\(24\) 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 \(-0.606462\pi\)
−0.328259 + 0.944588i \(0.606462\pi\)
\(42\) 0 0
\(43\) 27.7128i 0.644484i 0.946657 + 0.322242i \(0.104436\pi\)
−0.946657 + 0.322242i \(0.895564\pi\)
\(44\) 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 \(-0.817525\pi\)
−0.840136 + 0.542376i \(0.817525\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 52.8216i 0.788382i 0.919028 + 0.394191i \(0.128975\pi\)
−0.919028 + 0.394191i \(0.871025\pi\)
\(68\) 0 0
\(69\) −51.2483 −0.742729
\(70\) 0 0
\(71\) 64.8026i 0.912713i 0.889797 + 0.456357i \(0.150846\pi\)
−0.889797 + 0.456357i \(0.849154\pi\)
\(72\) 0 0
\(73\) −30.4966 −0.417761 −0.208881 0.977941i \(-0.566982\pi\)
−0.208881 + 0.977941i \(0.566982\pi\)
\(74\) 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 \(-0.602896\pi\)
0.317658 0.948205i \(-0.397104\pi\)
\(80\) 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 \(-0.481972\pi\)
0.0566063 + 0.998397i \(0.481972\pi\)
\(90\) 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 \(-0.698933\pi\)
−0.585069 + 0.810983i \(0.698933\pi\)
\(98\) 0 0
\(99\) − 26.4108i − 0.266776i
\(100\) 0 0
\(101\) 142.076 1.40669 0.703346 0.710848i \(-0.251688\pi\)
0.703346 + 0.710848i \(0.251688\pi\)
\(102\) 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 \(0.190929\pi\)
−0.825437 + 0.564494i \(0.809071\pi\)
\(108\) 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 \(-0.950976\pi\)
0.988163 0.153406i \(-0.0490243\pi\)
\(128\) 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 \(-0.709359\pi\)
0.611314 0.791388i \(-0.290641\pi\)
\(152\) 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 \(-0.582508\pi\)
−0.256315 + 0.966593i \(0.582508\pi\)
\(158\) 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 \(0.364790\pi\)
−0.412115 + 0.911132i \(0.635210\pi\)
\(164\) 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 \(-0.704048\pi\)
−0.598027 + 0.801476i \(0.704048\pi\)
\(174\) 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 \(-0.949348\pi\)
0.987366 0.158457i \(-0.0506519\pi\)
\(180\) 0 0
\(181\) −22.0137 −0.121623 −0.0608113 0.998149i \(-0.519369\pi\)
−0.0608113 + 0.998149i \(0.519369\pi\)
\(182\) 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 \(-0.880443\pi\)
0.930288 0.366830i \(-0.119557\pi\)
\(192\) 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 \(0.0805422\pi\)
−0.968158 + 0.250339i \(0.919458\pi\)
\(212\) 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 \(-0.472876\pi\)
0.0851080 + 0.996372i \(0.472876\pi\)
\(230\) 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 \(-0.829638\pi\)
0.860163 0.510019i \(-0.170362\pi\)
\(240\) 0 0
\(241\) 357.476 1.48330 0.741652 0.670785i \(-0.234043\pi\)
0.741652 + 0.670785i \(0.234043\pi\)
\(242\) 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 \(-0.632553\pi\)
−0.404495 + 0.914540i \(0.632553\pi\)
\(258\) 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 \(-0.850926\pi\)
0.892324 0.451396i \(-0.149074\pi\)
\(264\) 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 \(-0.549346\pi\)
−0.154403 + 0.988008i \(0.549346\pi\)
\(270\) 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 \(-0.621288\pi\)
−0.371884 + 0.928279i \(0.621288\pi\)
\(294\) 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 \(-0.639690\pi\)
−0.424898 + 0.905241i \(0.639690\pi\)
\(314\) 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 \(-0.697481\pi\)
0.581364 0.813644i \(-0.302519\pi\)
\(332\) 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 \(-0.851322\pi\)
0.892885 0.450285i \(-0.148678\pi\)
\(348\) 0 0
\(349\) 407.462 1.16751 0.583757 0.811928i \(-0.301582\pi\)
0.583757 + 0.811928i \(0.301582\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −154.076 −0.436476 −0.218238 0.975896i \(-0.570031\pi\)
−0.218238 + 0.975896i \(0.570031\pi\)
\(354\) 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 \(0.195354\pi\)
−0.817510 + 0.575914i \(0.804646\pi\)
\(360\) 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 \(0.0138289\pi\)
−0.999056 + 0.0434312i \(0.986171\pi\)
\(380\) 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 \(-0.160270\pi\)
0.875897 + 0.482498i \(0.160270\pi\)
\(398\) 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 \(-0.398064\pi\)
0.314796 + 0.949159i \(0.398064\pi\)
\(410\) 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 \(-0.887038\pi\)
0.937689 0.347477i \(-0.112962\pi\)
\(432\) 0 0
\(433\) −448.966 −1.03687 −0.518436 0.855116i \(-0.673486\pi\)
−0.518436 + 0.855116i \(0.673486\pi\)
\(434\) 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 \(-0.863935\pi\)
0.910021 0.414563i \(-0.136065\pi\)
\(444\) 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 \(-0.535941\pi\)
−0.112671 + 0.993632i \(0.535941\pi\)
\(462\) 0 0
\(463\) − 189.618i − 0.409542i −0.978810 0.204771i \(-0.934355\pi\)
0.978810 0.204771i \(-0.0656450\pi\)
\(464\) 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 \(-0.965407\pi\)
0.994100 0.108463i \(-0.0345930\pi\)
\(488\) 0 0
\(489\) 514.469 1.05208
\(490\) 0 0
\(491\) − 920.412i − 1.87457i −0.348569 0.937283i \(-0.613332\pi\)
0.348569 0.937283i \(-0.386668\pi\)
\(492\) 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 \(-0.890101\pi\)
0.940988 0.338439i \(-0.109899\pi\)
\(500\) 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 \(-0.556915\pi\)
−0.177854 + 0.984057i \(0.556915\pi\)
\(510\) 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 \(-0.819010\pi\)
−0.842657 + 0.538451i \(0.819010\pi\)
\(522\) 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 \(0.177652\pi\)
−0.848257 + 0.529584i \(0.822348\pi\)
\(548\) 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 \(-0.920748\pi\)
0.969165 0.246414i \(-0.0792523\pi\)
\(572\) 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 \(-0.335256\pi\)
0.494759 + 0.869030i \(0.335256\pi\)
\(578\) 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 \(-0.160228\pi\)
0.875961 + 0.482382i \(0.160228\pi\)
\(594\) 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 \(-0.735451\pi\)
0.674060 0.738677i \(-0.264549\pi\)
\(600\) 0 0
\(601\) −282.952 −0.470802 −0.235401 0.971898i \(-0.575640\pi\)
−0.235401 + 0.971898i \(0.575640\pi\)
\(602\) 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 \(0.167256\pi\)
−0.865098 + 0.501603i \(0.832744\pi\)
\(632\) 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 \(0.180148\pi\)
−0.844079 + 0.536219i \(0.819852\pi\)
\(660\) 0 0
\(661\) −1210.47 −1.83127 −0.915635 0.402011i \(-0.868311\pi\)
−0.915635 + 0.402011i \(0.868311\pi\)
\(662\) 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 \(-0.542407\pi\)
−0.132833 + 0.991138i \(0.542407\pi\)
\(678\) 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 \(-0.813983\pi\)
0.834049 0.551690i \(-0.186017\pi\)
\(684\) 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 \(-0.394733\pi\)
0.324712 + 0.945813i \(0.394733\pi\)
\(734\) 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 \(-0.846198\pi\)
0.885520 0.464602i \(-0.153802\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 423.302i 0.569719i 0.958569 + 0.284860i \(0.0919470\pi\)
−0.958569 + 0.284860i \(0.908053\pi\)
\(744\) 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 \(-0.834007\pi\)
0.867081 0.498166i \(-0.165993\pi\)
\(752\) 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 \(-0.494119\pi\)
0.0184737 + 0.999829i \(0.494119\pi\)
\(762\) 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 \(-0.165407\pi\)
0.867997 + 0.496569i \(0.165407\pi\)
\(770\) 0 0
\(771\) 360.111i 0.467070i
\(772\) 0 0
\(773\) −969.869 −1.25468 −0.627341 0.778745i \(-0.715857\pi\)
−0.627341 + 0.778745i \(0.715857\pi\)
\(774\) 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 \(-0.388270\pi\)
0.343845 + 0.939026i \(0.388270\pi\)
\(798\) 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 \(0.146649\pi\)
−0.895737 + 0.444584i \(0.853351\pi\)
\(824\) 0 0
\(825\) 876.724 1.06270
\(826\) 0 0
\(827\) − 1164.93i − 1.40862i −0.709892 0.704311i \(-0.751256\pi\)
0.709892 0.704311i \(-0.248744\pi\)
\(828\) 0 0
\(829\) 22.0137 0.0265545 0.0132773 0.999912i \(-0.495774\pi\)
0.0132773 + 0.999912i \(0.495774\pi\)
\(830\) 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 \(-0.406193\pi\)
0.290456 + 0.956888i \(0.406193\pi\)
\(854\) 0 0
\(855\) 479.766i 0.561130i
\(856\) 0 0
\(857\) 1029.99 1.20186 0.600930 0.799302i \(-0.294797\pi\)
0.600930 + 0.799302i \(0.294797\pi\)
\(858\) 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 \(-0.968417\pi\)
0.995082 0.0990567i \(-0.0315825\pi\)
\(864\) 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 \(-0.733563\pi\)
−0.669667 + 0.742662i \(0.733563\pi\)
\(882\) 0 0
\(883\) 1453.16i 1.64570i 0.568257 + 0.822851i \(0.307618\pi\)
−0.568257 + 0.822851i \(0.692382\pi\)
\(884\) 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 \(0.130020\pi\)
−0.917729 + 0.397207i \(0.869980\pi\)
\(908\) 0 0
\(909\) −426.228 −0.468897
\(910\) 0 0
\(911\) − 1067.36i − 1.17164i −0.810442 0.585819i \(-0.800773\pi\)
0.810442 0.585819i \(-0.199227\pi\)
\(912\) 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 \(-0.932003\pi\)
0.977270 0.211998i \(-0.0679970\pi\)
\(920\) 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 \(-0.260911\pi\)
0.682458 + 0.730925i \(0.260911\pi\)
\(930\) 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 \(-0.520735\pi\)
−0.0650941 + 0.997879i \(0.520735\pi\)
\(938\) 0 0
\(939\) 460.702i 0.490630i
\(940\) 0 0
\(941\) 1160.79 1.23357 0.616787 0.787130i \(-0.288434\pi\)
0.616787 + 0.787130i \(0.288434\pi\)
\(942\) 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 \(0.0100252\pi\)
−0.999504 + 0.0314900i \(0.989975\pi\)
\(948\) 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 \(0.105528\pi\)
−0.945547 + 0.325485i \(0.894472\pi\)
\(968\) 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 \(0.110994\pi\)
−0.939819 + 0.341673i \(0.889006\pi\)
\(992\) 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 \(-0.890108\pi\)
−0.940996 + 0.338418i \(0.890108\pi\)
\(998\) 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.e.1471.1 4
4.3 odd 2 inner 2352.3.m.e.1471.3 yes 4
7.6 odd 2 2352.3.m.j.1471.4 yes 4
28.27 even 2 2352.3.m.j.1471.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.3.m.e.1471.1 4 1.1 even 1 trivial
2352.3.m.e.1471.3 yes 4 4.3 odd 2 inner
2352.3.m.j.1471.2 yes 4 28.27 even 2
2352.3.m.j.1471.4 yes 4 7.6 odd 2