Properties

Label 1344.3.m.a.127.1
Level $1344$
Weight $3$
Character 1344.127
Analytic conductor $36.621$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,3,Mod(127,1344)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1344.m (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: \(36.6213475300\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-0.895644 - 1.09445i\) of defining polynomial
Character \(\chi\) \(=\) 1344.127
Dual form 1344.3.m.a.127.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} -5.58258 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -5.58258 q^{5} +2.64575i q^{7} -3.00000 q^{9} -4.37780i q^{11} -17.1652 q^{13} +9.66930i q^{15} +0.747727 q^{17} -3.65480i q^{19} +4.58258 q^{21} +9.86001i q^{23} +6.16515 q^{25} +5.19615i q^{27} +2.00000 q^{29} +17.8926i q^{31} -7.58258 q^{33} -14.7701i q^{35} +7.49545 q^{37} +29.7309i q^{39} +76.5735 q^{41} -70.0448i q^{43} +16.7477 q^{45} +40.1232i q^{47} -7.00000 q^{49} -1.29510i q^{51} -49.8258 q^{53} +24.4394i q^{55} -6.33030 q^{57} +89.3834i q^{59} +120.991 q^{61} -7.93725i q^{63} +95.8258 q^{65} +23.3748i q^{67} +17.0780 q^{69} -66.0484i q^{71} -40.5045 q^{73} -10.6784i q^{75} +11.5826 q^{77} -95.5488i q^{79} +9.00000 q^{81} +115.650i q^{83} -4.17424 q^{85} -3.46410i q^{87} +34.9038 q^{89} -45.4147i q^{91} +30.9909 q^{93} +20.4032i q^{95} +52.5045 q^{97} +13.1334i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 12 q^{9} - 32 q^{13} - 52 q^{17} - 12 q^{25} + 8 q^{29} - 12 q^{33} - 80 q^{37} + 68 q^{41} + 12 q^{45} - 28 q^{49} - 16 q^{53} + 48 q^{57} + 264 q^{61} + 200 q^{65} - 60 q^{69} - 272 q^{73} + 28 q^{77} + 36 q^{81} - 200 q^{85} - 172 q^{89} - 96 q^{93} + 320 q^{97}+O(q^{100}) \) 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
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) −5.58258 −1.11652 −0.558258 0.829668i \(-0.688530\pi\)
−0.558258 + 0.829668i \(0.688530\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 4.37780i − 0.397982i −0.980001 0.198991i \(-0.936234\pi\)
0.980001 0.198991i \(-0.0637665\pi\)
\(12\) 0 0
\(13\) −17.1652 −1.32040 −0.660198 0.751091i \(-0.729528\pi\)
−0.660198 + 0.751091i \(0.729528\pi\)
\(14\) 0 0
\(15\) 9.66930i 0.644620i
\(16\) 0 0
\(17\) 0.747727 0.0439839 0.0219920 0.999758i \(-0.492999\pi\)
0.0219920 + 0.999758i \(0.492999\pi\)
\(18\) 0 0
\(19\) − 3.65480i − 0.192358i −0.995364 0.0961790i \(-0.969338\pi\)
0.995364 0.0961790i \(-0.0306621\pi\)
\(20\) 0 0
\(21\) 4.58258 0.218218
\(22\) 0 0
\(23\) 9.86001i 0.428696i 0.976757 + 0.214348i \(0.0687626\pi\)
−0.976757 + 0.214348i \(0.931237\pi\)
\(24\) 0 0
\(25\) 6.16515 0.246606
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.0689655 0.0344828 0.999405i \(-0.489022\pi\)
0.0344828 + 0.999405i \(0.489022\pi\)
\(30\) 0 0
\(31\) 17.8926i 0.577181i 0.957453 + 0.288590i \(0.0931866\pi\)
−0.957453 + 0.288590i \(0.906813\pi\)
\(32\) 0 0
\(33\) −7.58258 −0.229775
\(34\) 0 0
\(35\) − 14.7701i − 0.422003i
\(36\) 0 0
\(37\) 7.49545 0.202580 0.101290 0.994857i \(-0.467703\pi\)
0.101290 + 0.994857i \(0.467703\pi\)
\(38\) 0 0
\(39\) 29.7309i 0.762331i
\(40\) 0 0
\(41\) 76.5735 1.86765 0.933823 0.357735i \(-0.116451\pi\)
0.933823 + 0.357735i \(0.116451\pi\)
\(42\) 0 0
\(43\) − 70.0448i − 1.62895i −0.580199 0.814475i \(-0.697025\pi\)
0.580199 0.814475i \(-0.302975\pi\)
\(44\) 0 0
\(45\) 16.7477 0.372172
\(46\) 0 0
\(47\) 40.1232i 0.853686i 0.904326 + 0.426843i \(0.140374\pi\)
−0.904326 + 0.426843i \(0.859626\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) − 1.29510i − 0.0253941i
\(52\) 0 0
\(53\) −49.8258 −0.940109 −0.470054 0.882637i \(-0.655766\pi\)
−0.470054 + 0.882637i \(0.655766\pi\)
\(54\) 0 0
\(55\) 24.4394i 0.444353i
\(56\) 0 0
\(57\) −6.33030 −0.111058
\(58\) 0 0
\(59\) 89.3834i 1.51497i 0.652850 + 0.757487i \(0.273573\pi\)
−0.652850 + 0.757487i \(0.726427\pi\)
\(60\) 0 0
\(61\) 120.991 1.98346 0.991729 0.128351i \(-0.0409685\pi\)
0.991729 + 0.128351i \(0.0409685\pi\)
\(62\) 0 0
\(63\) − 7.93725i − 0.125988i
\(64\) 0 0
\(65\) 95.8258 1.47424
\(66\) 0 0
\(67\) 23.3748i 0.348878i 0.984668 + 0.174439i \(0.0558112\pi\)
−0.984668 + 0.174439i \(0.944189\pi\)
\(68\) 0 0
\(69\) 17.0780 0.247508
\(70\) 0 0
\(71\) − 66.0484i − 0.930260i −0.885242 0.465130i \(-0.846008\pi\)
0.885242 0.465130i \(-0.153992\pi\)
\(72\) 0 0
\(73\) −40.5045 −0.554857 −0.277428 0.960746i \(-0.589482\pi\)
−0.277428 + 0.960746i \(0.589482\pi\)
\(74\) 0 0
\(75\) − 10.6784i − 0.142378i
\(76\) 0 0
\(77\) 11.5826 0.150423
\(78\) 0 0
\(79\) − 95.5488i − 1.20948i −0.796423 0.604740i \(-0.793277\pi\)
0.796423 0.604740i \(-0.206723\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 115.650i 1.39338i 0.717374 + 0.696688i \(0.245344\pi\)
−0.717374 + 0.696688i \(0.754656\pi\)
\(84\) 0 0
\(85\) −4.17424 −0.0491087
\(86\) 0 0
\(87\) − 3.46410i − 0.0398173i
\(88\) 0 0
\(89\) 34.9038 0.392177 0.196089 0.980586i \(-0.437176\pi\)
0.196089 + 0.980586i \(0.437176\pi\)
\(90\) 0 0
\(91\) − 45.4147i − 0.499063i
\(92\) 0 0
\(93\) 30.9909 0.333236
\(94\) 0 0
\(95\) 20.4032i 0.214771i
\(96\) 0 0
\(97\) 52.5045 0.541284 0.270642 0.962680i \(-0.412764\pi\)
0.270642 + 0.962680i \(0.412764\pi\)
\(98\) 0 0
\(99\) 13.1334i 0.132661i
\(100\) 0 0
\(101\) 113.583 1.12458 0.562290 0.826940i \(-0.309920\pi\)
0.562290 + 0.826940i \(0.309920\pi\)
\(102\) 0 0
\(103\) − 50.0230i − 0.485660i −0.970069 0.242830i \(-0.921924\pi\)
0.970069 0.242830i \(-0.0780758\pi\)
\(104\) 0 0
\(105\) −25.5826 −0.243644
\(106\) 0 0
\(107\) 160.533i 1.50031i 0.661265 + 0.750153i \(0.270020\pi\)
−0.661265 + 0.750153i \(0.729980\pi\)
\(108\) 0 0
\(109\) 205.303 1.88351 0.941757 0.336294i \(-0.109174\pi\)
0.941757 + 0.336294i \(0.109174\pi\)
\(110\) 0 0
\(111\) − 12.9825i − 0.116960i
\(112\) 0 0
\(113\) 179.321 1.58691 0.793457 0.608627i \(-0.208279\pi\)
0.793457 + 0.608627i \(0.208279\pi\)
\(114\) 0 0
\(115\) − 55.0442i − 0.478645i
\(116\) 0 0
\(117\) 51.4955 0.440132
\(118\) 0 0
\(119\) 1.97830i 0.0166244i
\(120\) 0 0
\(121\) 101.835 0.841610
\(122\) 0 0
\(123\) − 132.629i − 1.07829i
\(124\) 0 0
\(125\) 105.147 0.841176
\(126\) 0 0
\(127\) − 130.571i − 1.02812i −0.857754 0.514060i \(-0.828141\pi\)
0.857754 0.514060i \(-0.171859\pi\)
\(128\) 0 0
\(129\) −121.321 −0.940475
\(130\) 0 0
\(131\) 71.1890i 0.543428i 0.962378 + 0.271714i \(0.0875904\pi\)
−0.962378 + 0.271714i \(0.912410\pi\)
\(132\) 0 0
\(133\) 9.66970 0.0727045
\(134\) 0 0
\(135\) − 29.0079i − 0.214873i
\(136\) 0 0
\(137\) −99.8439 −0.728788 −0.364394 0.931245i \(-0.618724\pi\)
−0.364394 + 0.931245i \(0.618724\pi\)
\(138\) 0 0
\(139\) − 162.400i − 1.16834i −0.811630 0.584172i \(-0.801419\pi\)
0.811630 0.584172i \(-0.198581\pi\)
\(140\) 0 0
\(141\) 69.4955 0.492876
\(142\) 0 0
\(143\) 75.1456i 0.525494i
\(144\) 0 0
\(145\) −11.1652 −0.0770010
\(146\) 0 0
\(147\) 12.1244i 0.0824786i
\(148\) 0 0
\(149\) −293.165 −1.96755 −0.983776 0.179403i \(-0.942583\pi\)
−0.983776 + 0.179403i \(0.942583\pi\)
\(150\) 0 0
\(151\) − 120.449i − 0.797677i −0.917021 0.398839i \(-0.869413\pi\)
0.917021 0.398839i \(-0.130587\pi\)
\(152\) 0 0
\(153\) −2.24318 −0.0146613
\(154\) 0 0
\(155\) − 99.8868i − 0.644431i
\(156\) 0 0
\(157\) 163.982 1.04447 0.522235 0.852802i \(-0.325098\pi\)
0.522235 + 0.852802i \(0.325098\pi\)
\(158\) 0 0
\(159\) 86.3007i 0.542772i
\(160\) 0 0
\(161\) −26.0871 −0.162032
\(162\) 0 0
\(163\) 226.803i 1.39143i 0.718317 + 0.695716i \(0.244913\pi\)
−0.718317 + 0.695716i \(0.755087\pi\)
\(164\) 0 0
\(165\) 42.3303 0.256547
\(166\) 0 0
\(167\) − 226.803i − 1.35810i −0.734090 0.679052i \(-0.762391\pi\)
0.734090 0.679052i \(-0.237609\pi\)
\(168\) 0 0
\(169\) 125.642 0.743446
\(170\) 0 0
\(171\) 10.9644i 0.0641193i
\(172\) 0 0
\(173\) 108.922 0.629607 0.314803 0.949157i \(-0.398061\pi\)
0.314803 + 0.949157i \(0.398061\pi\)
\(174\) 0 0
\(175\) 16.3115i 0.0932083i
\(176\) 0 0
\(177\) 154.817 0.874670
\(178\) 0 0
\(179\) − 91.2506i − 0.509780i −0.966970 0.254890i \(-0.917961\pi\)
0.966970 0.254890i \(-0.0820393\pi\)
\(180\) 0 0
\(181\) −135.495 −0.748594 −0.374297 0.927309i \(-0.622116\pi\)
−0.374297 + 0.927309i \(0.622116\pi\)
\(182\) 0 0
\(183\) − 209.562i − 1.14515i
\(184\) 0 0
\(185\) −41.8439 −0.226183
\(186\) 0 0
\(187\) − 3.27340i − 0.0175048i
\(188\) 0 0
\(189\) −13.7477 −0.0727393
\(190\) 0 0
\(191\) 106.934i 0.559866i 0.960020 + 0.279933i \(0.0903123\pi\)
−0.960020 + 0.279933i \(0.909688\pi\)
\(192\) 0 0
\(193\) −116.505 −0.603650 −0.301825 0.953363i \(-0.597596\pi\)
−0.301825 + 0.953363i \(0.597596\pi\)
\(194\) 0 0
\(195\) − 165.975i − 0.851154i
\(196\) 0 0
\(197\) −38.8348 −0.197131 −0.0985656 0.995131i \(-0.531425\pi\)
−0.0985656 + 0.995131i \(0.531425\pi\)
\(198\) 0 0
\(199\) 353.116i 1.77445i 0.461334 + 0.887227i \(0.347371\pi\)
−0.461334 + 0.887227i \(0.652629\pi\)
\(200\) 0 0
\(201\) 40.4864 0.201425
\(202\) 0 0
\(203\) 5.29150i 0.0260665i
\(204\) 0 0
\(205\) −427.477 −2.08525
\(206\) 0 0
\(207\) − 29.5800i − 0.142899i
\(208\) 0 0
\(209\) −16.0000 −0.0765550
\(210\) 0 0
\(211\) − 54.0592i − 0.256205i −0.991761 0.128102i \(-0.959111\pi\)
0.991761 0.128102i \(-0.0408886\pi\)
\(212\) 0 0
\(213\) −114.399 −0.537086
\(214\) 0 0
\(215\) 391.031i 1.81875i
\(216\) 0 0
\(217\) −47.3394 −0.218154
\(218\) 0 0
\(219\) 70.1559i 0.320347i
\(220\) 0 0
\(221\) −12.8348 −0.0580762
\(222\) 0 0
\(223\) 93.3400i 0.418565i 0.977855 + 0.209283i \(0.0671129\pi\)
−0.977855 + 0.209283i \(0.932887\pi\)
\(224\) 0 0
\(225\) −18.4955 −0.0822020
\(226\) 0 0
\(227\) − 170.233i − 0.749927i −0.927040 0.374964i \(-0.877655\pi\)
0.927040 0.374964i \(-0.122345\pi\)
\(228\) 0 0
\(229\) 227.844 0.994952 0.497476 0.867478i \(-0.334260\pi\)
0.497476 + 0.867478i \(0.334260\pi\)
\(230\) 0 0
\(231\) − 20.0616i − 0.0868468i
\(232\) 0 0
\(233\) 378.486 1.62440 0.812202 0.583376i \(-0.198268\pi\)
0.812202 + 0.583376i \(0.198268\pi\)
\(234\) 0 0
\(235\) − 223.991i − 0.953153i
\(236\) 0 0
\(237\) −165.495 −0.698293
\(238\) 0 0
\(239\) 205.535i 0.859977i 0.902834 + 0.429989i \(0.141482\pi\)
−0.902834 + 0.429989i \(0.858518\pi\)
\(240\) 0 0
\(241\) 40.4682 0.167918 0.0839589 0.996469i \(-0.473244\pi\)
0.0839589 + 0.996469i \(0.473244\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) 39.0780 0.159502
\(246\) 0 0
\(247\) 62.7352i 0.253989i
\(248\) 0 0
\(249\) 200.312 0.804466
\(250\) 0 0
\(251\) 332.856i 1.32612i 0.748567 + 0.663059i \(0.230742\pi\)
−0.748567 + 0.663059i \(0.769258\pi\)
\(252\) 0 0
\(253\) 43.1652 0.170613
\(254\) 0 0
\(255\) 7.23000i 0.0283529i
\(256\) 0 0
\(257\) −205.372 −0.799113 −0.399556 0.916709i \(-0.630836\pi\)
−0.399556 + 0.916709i \(0.630836\pi\)
\(258\) 0 0
\(259\) 19.8311i 0.0765680i
\(260\) 0 0
\(261\) −6.00000 −0.0229885
\(262\) 0 0
\(263\) − 45.4064i − 0.172648i −0.996267 0.0863240i \(-0.972488\pi\)
0.996267 0.0863240i \(-0.0275120\pi\)
\(264\) 0 0
\(265\) 278.156 1.04965
\(266\) 0 0
\(267\) − 60.4551i − 0.226424i
\(268\) 0 0
\(269\) 202.069 0.751186 0.375593 0.926785i \(-0.377439\pi\)
0.375593 + 0.926785i \(0.377439\pi\)
\(270\) 0 0
\(271\) 279.798i 1.03246i 0.856449 + 0.516232i \(0.172666\pi\)
−0.856449 + 0.516232i \(0.827334\pi\)
\(272\) 0 0
\(273\) −78.6606 −0.288134
\(274\) 0 0
\(275\) − 26.9898i − 0.0981448i
\(276\) 0 0
\(277\) −135.459 −0.489022 −0.244511 0.969647i \(-0.578627\pi\)
−0.244511 + 0.969647i \(0.578627\pi\)
\(278\) 0 0
\(279\) − 53.6778i − 0.192394i
\(280\) 0 0
\(281\) 20.8530 0.0742101 0.0371050 0.999311i \(-0.488186\pi\)
0.0371050 + 0.999311i \(0.488186\pi\)
\(282\) 0 0
\(283\) − 113.743i − 0.401920i −0.979599 0.200960i \(-0.935594\pi\)
0.979599 0.200960i \(-0.0644061\pi\)
\(284\) 0 0
\(285\) 35.3394 0.123998
\(286\) 0 0
\(287\) 202.594i 0.705904i
\(288\) 0 0
\(289\) −288.441 −0.998065
\(290\) 0 0
\(291\) − 90.9405i − 0.312510i
\(292\) 0 0
\(293\) 121.931 0.416147 0.208073 0.978113i \(-0.433281\pi\)
0.208073 + 0.978113i \(0.433281\pi\)
\(294\) 0 0
\(295\) − 498.990i − 1.69149i
\(296\) 0 0
\(297\) 22.7477 0.0765917
\(298\) 0 0
\(299\) − 169.248i − 0.566048i
\(300\) 0 0
\(301\) 185.321 0.615685
\(302\) 0 0
\(303\) − 196.731i − 0.649277i
\(304\) 0 0
\(305\) −675.441 −2.21456
\(306\) 0 0
\(307\) 337.131i 1.09815i 0.835775 + 0.549073i \(0.185019\pi\)
−0.835775 + 0.549073i \(0.814981\pi\)
\(308\) 0 0
\(309\) −86.6424 −0.280396
\(310\) 0 0
\(311\) 246.762i 0.793447i 0.917938 + 0.396724i \(0.129853\pi\)
−0.917938 + 0.396724i \(0.870147\pi\)
\(312\) 0 0
\(313\) 274.624 0.877394 0.438697 0.898635i \(-0.355440\pi\)
0.438697 + 0.898635i \(0.355440\pi\)
\(314\) 0 0
\(315\) 44.3103i 0.140668i
\(316\) 0 0
\(317\) 129.441 0.408331 0.204165 0.978936i \(-0.434552\pi\)
0.204165 + 0.978936i \(0.434552\pi\)
\(318\) 0 0
\(319\) − 8.75560i − 0.0274470i
\(320\) 0 0
\(321\) 278.051 0.866202
\(322\) 0 0
\(323\) − 2.73279i − 0.00846066i
\(324\) 0 0
\(325\) −105.826 −0.325618
\(326\) 0 0
\(327\) − 355.595i − 1.08745i
\(328\) 0 0
\(329\) −106.156 −0.322663
\(330\) 0 0
\(331\) 219.573i 0.663363i 0.943391 + 0.331682i \(0.107616\pi\)
−0.943391 + 0.331682i \(0.892384\pi\)
\(332\) 0 0
\(333\) −22.4864 −0.0675266
\(334\) 0 0
\(335\) − 130.492i − 0.389527i
\(336\) 0 0
\(337\) −132.955 −0.394524 −0.197262 0.980351i \(-0.563205\pi\)
−0.197262 + 0.980351i \(0.563205\pi\)
\(338\) 0 0
\(339\) − 310.593i − 0.916205i
\(340\) 0 0
\(341\) 78.3303 0.229708
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) −95.3394 −0.276346
\(346\) 0 0
\(347\) 453.282i 1.30629i 0.757234 + 0.653144i \(0.226550\pi\)
−0.757234 + 0.653144i \(0.773450\pi\)
\(348\) 0 0
\(349\) −525.615 −1.50606 −0.753030 0.657986i \(-0.771409\pi\)
−0.753030 + 0.657986i \(0.771409\pi\)
\(350\) 0 0
\(351\) − 89.1927i − 0.254110i
\(352\) 0 0
\(353\) −389.858 −1.10441 −0.552207 0.833707i \(-0.686214\pi\)
−0.552207 + 0.833707i \(0.686214\pi\)
\(354\) 0 0
\(355\) 368.720i 1.03865i
\(356\) 0 0
\(357\) 3.42652 0.00959808
\(358\) 0 0
\(359\) 692.718i 1.92958i 0.263030 + 0.964788i \(0.415278\pi\)
−0.263030 + 0.964788i \(0.584722\pi\)
\(360\) 0 0
\(361\) 347.642 0.962998
\(362\) 0 0
\(363\) − 176.383i − 0.485904i
\(364\) 0 0
\(365\) 226.120 0.619506
\(366\) 0 0
\(367\) − 556.863i − 1.51734i −0.651476 0.758669i \(-0.725850\pi\)
0.651476 0.758669i \(-0.274150\pi\)
\(368\) 0 0
\(369\) −229.720 −0.622549
\(370\) 0 0
\(371\) − 131.827i − 0.355328i
\(372\) 0 0
\(373\) −549.267 −1.47256 −0.736282 0.676674i \(-0.763420\pi\)
−0.736282 + 0.676674i \(0.763420\pi\)
\(374\) 0 0
\(375\) − 182.120i − 0.485653i
\(376\) 0 0
\(377\) −34.3303 −0.0910618
\(378\) 0 0
\(379\) − 232.667i − 0.613897i −0.951726 0.306948i \(-0.900692\pi\)
0.951726 0.306948i \(-0.0993079\pi\)
\(380\) 0 0
\(381\) −226.156 −0.593585
\(382\) 0 0
\(383\) 394.526i 1.03009i 0.857162 + 0.515047i \(0.172226\pi\)
−0.857162 + 0.515047i \(0.827774\pi\)
\(384\) 0 0
\(385\) −64.6606 −0.167950
\(386\) 0 0
\(387\) 210.135i 0.542983i
\(388\) 0 0
\(389\) −3.98182 −0.0102360 −0.00511802 0.999987i \(-0.501629\pi\)
−0.00511802 + 0.999987i \(0.501629\pi\)
\(390\) 0 0
\(391\) 7.37259i 0.0188557i
\(392\) 0 0
\(393\) 123.303 0.313748
\(394\) 0 0
\(395\) 533.409i 1.35040i
\(396\) 0 0
\(397\) 476.606 1.20052 0.600260 0.799805i \(-0.295064\pi\)
0.600260 + 0.799805i \(0.295064\pi\)
\(398\) 0 0
\(399\) − 16.7484i − 0.0419760i
\(400\) 0 0
\(401\) −157.441 −0.392621 −0.196310 0.980542i \(-0.562896\pi\)
−0.196310 + 0.980542i \(0.562896\pi\)
\(402\) 0 0
\(403\) − 307.129i − 0.762108i
\(404\) 0 0
\(405\) −50.2432 −0.124057
\(406\) 0 0
\(407\) − 32.8136i − 0.0806231i
\(408\) 0 0
\(409\) −112.156 −0.274220 −0.137110 0.990556i \(-0.543781\pi\)
−0.137110 + 0.990556i \(0.543781\pi\)
\(410\) 0 0
\(411\) 172.935i 0.420766i
\(412\) 0 0
\(413\) −236.486 −0.572606
\(414\) 0 0
\(415\) − 645.626i − 1.55573i
\(416\) 0 0
\(417\) −281.285 −0.674544
\(418\) 0 0
\(419\) 699.065i 1.66841i 0.551452 + 0.834207i \(0.314074\pi\)
−0.551452 + 0.834207i \(0.685926\pi\)
\(420\) 0 0
\(421\) −198.762 −0.472119 −0.236060 0.971739i \(-0.575856\pi\)
−0.236060 + 0.971739i \(0.575856\pi\)
\(422\) 0 0
\(423\) − 120.370i − 0.284562i
\(424\) 0 0
\(425\) 4.60985 0.0108467
\(426\) 0 0
\(427\) 320.112i 0.749676i
\(428\) 0 0
\(429\) 130.156 0.303394
\(430\) 0 0
\(431\) − 513.871i − 1.19228i −0.802882 0.596138i \(-0.796701\pi\)
0.802882 0.596138i \(-0.203299\pi\)
\(432\) 0 0
\(433\) −654.900 −1.51247 −0.756236 0.654299i \(-0.772964\pi\)
−0.756236 + 0.654299i \(0.772964\pi\)
\(434\) 0 0
\(435\) 19.3386i 0.0444566i
\(436\) 0 0
\(437\) 36.0364 0.0824631
\(438\) 0 0
\(439\) − 143.141i − 0.326061i −0.986621 0.163031i \(-0.947873\pi\)
0.986621 0.163031i \(-0.0521269\pi\)
\(440\) 0 0
\(441\) 21.0000 0.0476190
\(442\) 0 0
\(443\) 46.8690i 0.105799i 0.998600 + 0.0528996i \(0.0168463\pi\)
−0.998600 + 0.0528996i \(0.983154\pi\)
\(444\) 0 0
\(445\) −194.853 −0.437872
\(446\) 0 0
\(447\) 507.777i 1.13597i
\(448\) 0 0
\(449\) −292.955 −0.652460 −0.326230 0.945290i \(-0.605778\pi\)
−0.326230 + 0.945290i \(0.605778\pi\)
\(450\) 0 0
\(451\) − 335.224i − 0.743289i
\(452\) 0 0
\(453\) −208.624 −0.460539
\(454\) 0 0
\(455\) 253.531i 0.557211i
\(456\) 0 0
\(457\) −424.018 −0.927830 −0.463915 0.885880i \(-0.653556\pi\)
−0.463915 + 0.885880i \(0.653556\pi\)
\(458\) 0 0
\(459\) 3.88530i 0.00846471i
\(460\) 0 0
\(461\) 344.886 0.748125 0.374062 0.927404i \(-0.377965\pi\)
0.374062 + 0.927404i \(0.377965\pi\)
\(462\) 0 0
\(463\) − 319.381i − 0.689807i −0.938638 0.344903i \(-0.887912\pi\)
0.938638 0.344903i \(-0.112088\pi\)
\(464\) 0 0
\(465\) −173.009 −0.372063
\(466\) 0 0
\(467\) 306.065i 0.655385i 0.944784 + 0.327692i \(0.106271\pi\)
−0.944784 + 0.327692i \(0.893729\pi\)
\(468\) 0 0
\(469\) −61.8439 −0.131863
\(470\) 0 0
\(471\) − 284.025i − 0.603025i
\(472\) 0 0
\(473\) −306.642 −0.648293
\(474\) 0 0
\(475\) − 22.5324i − 0.0474366i
\(476\) 0 0
\(477\) 149.477 0.313370
\(478\) 0 0
\(479\) − 355.769i − 0.742734i −0.928486 0.371367i \(-0.878889\pi\)
0.928486 0.371367i \(-0.121111\pi\)
\(480\) 0 0
\(481\) −128.661 −0.267486
\(482\) 0 0
\(483\) 45.1842i 0.0935491i
\(484\) 0 0
\(485\) −293.111 −0.604352
\(486\) 0 0
\(487\) 765.313i 1.57148i 0.618554 + 0.785742i \(0.287719\pi\)
−0.618554 + 0.785742i \(0.712281\pi\)
\(488\) 0 0
\(489\) 392.835 0.803343
\(490\) 0 0
\(491\) − 975.169i − 1.98609i −0.117749 0.993043i \(-0.537568\pi\)
0.117749 0.993043i \(-0.462432\pi\)
\(492\) 0 0
\(493\) 1.49545 0.00303338
\(494\) 0 0
\(495\) − 73.3182i − 0.148118i
\(496\) 0 0
\(497\) 174.748 0.351605
\(498\) 0 0
\(499\) − 472.358i − 0.946610i −0.880899 0.473305i \(-0.843061\pi\)
0.880899 0.473305i \(-0.156939\pi\)
\(500\) 0 0
\(501\) −392.835 −0.784101
\(502\) 0 0
\(503\) 513.165i 1.02021i 0.860113 + 0.510104i \(0.170393\pi\)
−0.860113 + 0.510104i \(0.829607\pi\)
\(504\) 0 0
\(505\) −634.083 −1.25561
\(506\) 0 0
\(507\) − 217.619i − 0.429229i
\(508\) 0 0
\(509\) −361.546 −0.710307 −0.355153 0.934808i \(-0.615571\pi\)
−0.355153 + 0.934808i \(0.615571\pi\)
\(510\) 0 0
\(511\) − 107.165i − 0.209716i
\(512\) 0 0
\(513\) 18.9909 0.0370193
\(514\) 0 0
\(515\) 279.257i 0.542247i
\(516\) 0 0
\(517\) 175.652 0.339751
\(518\) 0 0
\(519\) − 188.658i − 0.363504i
\(520\) 0 0
\(521\) 71.4265 0.137095 0.0685475 0.997648i \(-0.478164\pi\)
0.0685475 + 0.997648i \(0.478164\pi\)
\(522\) 0 0
\(523\) − 587.246i − 1.12284i −0.827531 0.561420i \(-0.810255\pi\)
0.827531 0.561420i \(-0.189745\pi\)
\(524\) 0 0
\(525\) 28.2523 0.0538139
\(526\) 0 0
\(527\) 13.3788i 0.0253867i
\(528\) 0 0
\(529\) 431.780 0.816220
\(530\) 0 0
\(531\) − 268.150i − 0.504991i
\(532\) 0 0
\(533\) −1314.40 −2.46603
\(534\) 0 0
\(535\) − 896.186i − 1.67511i
\(536\) 0 0
\(537\) −158.051 −0.294322
\(538\) 0 0
\(539\) 30.6446i 0.0568546i
\(540\) 0 0
\(541\) 436.505 0.806848 0.403424 0.915013i \(-0.367820\pi\)
0.403424 + 0.915013i \(0.367820\pi\)
\(542\) 0 0
\(543\) 234.685i 0.432201i
\(544\) 0 0
\(545\) −1146.12 −2.10297
\(546\) 0 0
\(547\) 25.2188i 0.0461039i 0.999734 + 0.0230519i \(0.00733831\pi\)
−0.999734 + 0.0230519i \(0.992662\pi\)
\(548\) 0 0
\(549\) −362.973 −0.661153
\(550\) 0 0
\(551\) − 7.30960i − 0.0132661i
\(552\) 0 0
\(553\) 252.798 0.457140
\(554\) 0 0
\(555\) 72.4758i 0.130587i
\(556\) 0 0
\(557\) −58.1742 −0.104442 −0.0522210 0.998636i \(-0.516630\pi\)
−0.0522210 + 0.998636i \(0.516630\pi\)
\(558\) 0 0
\(559\) 1202.33i 2.15086i
\(560\) 0 0
\(561\) −5.66970 −0.0101064
\(562\) 0 0
\(563\) − 565.396i − 1.00426i −0.864793 0.502128i \(-0.832551\pi\)
0.864793 0.502128i \(-0.167449\pi\)
\(564\) 0 0
\(565\) −1001.07 −1.77181
\(566\) 0 0
\(567\) 23.8118i 0.0419961i
\(568\) 0 0
\(569\) 573.056 1.00713 0.503564 0.863958i \(-0.332022\pi\)
0.503564 + 0.863958i \(0.332022\pi\)
\(570\) 0 0
\(571\) − 685.750i − 1.20096i −0.799639 0.600481i \(-0.794976\pi\)
0.799639 0.600481i \(-0.205024\pi\)
\(572\) 0 0
\(573\) 185.216 0.323239
\(574\) 0 0
\(575\) 60.7884i 0.105719i
\(576\) 0 0
\(577\) −278.900 −0.483362 −0.241681 0.970356i \(-0.577699\pi\)
−0.241681 + 0.970356i \(0.577699\pi\)
\(578\) 0 0
\(579\) 201.792i 0.348518i
\(580\) 0 0
\(581\) −305.982 −0.526647
\(582\) 0 0
\(583\) 218.127i 0.374146i
\(584\) 0 0
\(585\) −287.477 −0.491414
\(586\) 0 0
\(587\) 1169.74i 1.99274i 0.0851245 + 0.996370i \(0.472871\pi\)
−0.0851245 + 0.996370i \(0.527129\pi\)
\(588\) 0 0
\(589\) 65.3939 0.111025
\(590\) 0 0
\(591\) 67.2639i 0.113814i
\(592\) 0 0
\(593\) 969.023 1.63410 0.817052 0.576564i \(-0.195607\pi\)
0.817052 + 0.576564i \(0.195607\pi\)
\(594\) 0 0
\(595\) − 11.0440i − 0.0185614i
\(596\) 0 0
\(597\) 611.615 1.02448
\(598\) 0 0
\(599\) 656.488i 1.09597i 0.836487 + 0.547987i \(0.184606\pi\)
−0.836487 + 0.547987i \(0.815394\pi\)
\(600\) 0 0
\(601\) 312.955 0.520723 0.260362 0.965511i \(-0.416158\pi\)
0.260362 + 0.965511i \(0.416158\pi\)
\(602\) 0 0
\(603\) − 70.1244i − 0.116293i
\(604\) 0 0
\(605\) −568.501 −0.939671
\(606\) 0 0
\(607\) − 113.584i − 0.187124i −0.995613 0.0935618i \(-0.970175\pi\)
0.995613 0.0935618i \(-0.0298253\pi\)
\(608\) 0 0
\(609\) 9.16515 0.0150495
\(610\) 0 0
\(611\) − 688.721i − 1.12720i
\(612\) 0 0
\(613\) −10.0000 −0.0163132 −0.00815661 0.999967i \(-0.502596\pi\)
−0.00815661 + 0.999967i \(0.502596\pi\)
\(614\) 0 0
\(615\) 740.412i 1.20392i
\(616\) 0 0
\(617\) 984.323 1.59534 0.797668 0.603096i \(-0.206067\pi\)
0.797668 + 0.603096i \(0.206067\pi\)
\(618\) 0 0
\(619\) − 266.864i − 0.431120i −0.976491 0.215560i \(-0.930842\pi\)
0.976491 0.215560i \(-0.0691578\pi\)
\(620\) 0 0
\(621\) −51.2341 −0.0825026
\(622\) 0 0
\(623\) 92.3467i 0.148229i
\(624\) 0 0
\(625\) −741.120 −1.18579
\(626\) 0 0
\(627\) 27.7128i 0.0441991i
\(628\) 0 0
\(629\) 5.60455 0.00891026
\(630\) 0 0
\(631\) − 609.443i − 0.965837i −0.875665 0.482918i \(-0.839577\pi\)
0.875665 0.482918i \(-0.160423\pi\)
\(632\) 0 0
\(633\) −93.6333 −0.147920
\(634\) 0 0
\(635\) 728.924i 1.14791i
\(636\) 0 0
\(637\) 120.156 0.188628
\(638\) 0 0
\(639\) 198.145i 0.310087i
\(640\) 0 0
\(641\) 528.120 0.823900 0.411950 0.911207i \(-0.364848\pi\)
0.411950 + 0.911207i \(0.364848\pi\)
\(642\) 0 0
\(643\) − 812.666i − 1.26387i −0.775023 0.631933i \(-0.782262\pi\)
0.775023 0.631933i \(-0.217738\pi\)
\(644\) 0 0
\(645\) 677.285 1.05005
\(646\) 0 0
\(647\) − 1153.37i − 1.78265i −0.453369 0.891323i \(-0.649778\pi\)
0.453369 0.891323i \(-0.350222\pi\)
\(648\) 0 0
\(649\) 391.303 0.602932
\(650\) 0 0
\(651\) 81.9942i 0.125951i
\(652\) 0 0
\(653\) 994.000 1.52221 0.761103 0.648632i \(-0.224658\pi\)
0.761103 + 0.648632i \(0.224658\pi\)
\(654\) 0 0
\(655\) − 397.418i − 0.606745i
\(656\) 0 0
\(657\) 121.514 0.184952
\(658\) 0 0
\(659\) − 1223.14i − 1.85605i −0.372516 0.928026i \(-0.621505\pi\)
0.372516 0.928026i \(-0.378495\pi\)
\(660\) 0 0
\(661\) −24.0545 −0.0363911 −0.0181956 0.999834i \(-0.505792\pi\)
−0.0181956 + 0.999834i \(0.505792\pi\)
\(662\) 0 0
\(663\) 22.2306i 0.0335303i
\(664\) 0 0
\(665\) −53.9818 −0.0811757
\(666\) 0 0
\(667\) 19.7200i 0.0295652i
\(668\) 0 0
\(669\) 161.670 0.241659
\(670\) 0 0
\(671\) − 529.674i − 0.789380i
\(672\) 0 0
\(673\) 653.789 0.971455 0.485728 0.874110i \(-0.338555\pi\)
0.485728 + 0.874110i \(0.338555\pi\)
\(674\) 0 0
\(675\) 32.0351i 0.0474594i
\(676\) 0 0
\(677\) −366.069 −0.540722 −0.270361 0.962759i \(-0.587143\pi\)
−0.270361 + 0.962759i \(0.587143\pi\)
\(678\) 0 0
\(679\) 138.914i 0.204586i
\(680\) 0 0
\(681\) −294.853 −0.432971
\(682\) 0 0
\(683\) 971.912i 1.42300i 0.702684 + 0.711502i \(0.251985\pi\)
−0.702684 + 0.711502i \(0.748015\pi\)
\(684\) 0 0
\(685\) 557.386 0.813703
\(686\) 0 0
\(687\) − 394.637i − 0.574436i
\(688\) 0 0
\(689\) 855.267 1.24132
\(690\) 0 0
\(691\) 1028.20i 1.48799i 0.668184 + 0.743996i \(0.267072\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(692\) 0 0
\(693\) −34.7477 −0.0501410
\(694\) 0 0
\(695\) 906.610i 1.30447i
\(696\) 0 0
\(697\) 57.2561 0.0821464
\(698\) 0 0
\(699\) − 655.558i − 0.937851i
\(700\) 0 0
\(701\) 276.918 0.395033 0.197517 0.980300i \(-0.436712\pi\)
0.197517 + 0.980300i \(0.436712\pi\)
\(702\) 0 0
\(703\) − 27.3944i − 0.0389679i
\(704\) 0 0
\(705\) −387.964 −0.550303
\(706\) 0 0
\(707\) 300.511i 0.425051i
\(708\) 0 0
\(709\) −51.2485 −0.0722828 −0.0361414 0.999347i \(-0.511507\pi\)
−0.0361414 + 0.999347i \(0.511507\pi\)
\(710\) 0 0
\(711\) 286.647i 0.403160i
\(712\) 0 0
\(713\) −176.421 −0.247435
\(714\) 0 0
\(715\) − 419.506i − 0.586722i
\(716\) 0 0
\(717\) 355.996 0.496508
\(718\) 0 0
\(719\) 898.093i 1.24909i 0.780990 + 0.624543i \(0.214715\pi\)
−0.780990 + 0.624543i \(0.785285\pi\)
\(720\) 0 0
\(721\) 132.348 0.183562
\(722\) 0 0
\(723\) − 70.0929i − 0.0969474i
\(724\) 0 0
\(725\) 12.3303 0.0170073
\(726\) 0 0
\(727\) 107.100i 0.147318i 0.997283 + 0.0736590i \(0.0234677\pi\)
−0.997283 + 0.0736590i \(0.976532\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 52.3744i − 0.0716476i
\(732\) 0 0
\(733\) 541.368 0.738565 0.369283 0.929317i \(-0.379603\pi\)
0.369283 + 0.929317i \(0.379603\pi\)
\(734\) 0 0
\(735\) − 67.6851i − 0.0920886i
\(736\) 0 0
\(737\) 102.330 0.138847
\(738\) 0 0
\(739\) 241.423i 0.326688i 0.986569 + 0.163344i \(0.0522281\pi\)
−0.986569 + 0.163344i \(0.947772\pi\)
\(740\) 0 0
\(741\) 108.661 0.146640
\(742\) 0 0
\(743\) 593.753i 0.799129i 0.916705 + 0.399564i \(0.130839\pi\)
−0.916705 + 0.399564i \(0.869161\pi\)
\(744\) 0 0
\(745\) 1636.62 2.19680
\(746\) 0 0
\(747\) − 346.951i − 0.464459i
\(748\) 0 0
\(749\) −424.730 −0.567062
\(750\) 0 0
\(751\) 918.781i 1.22341i 0.791086 + 0.611705i \(0.209516\pi\)
−0.791086 + 0.611705i \(0.790484\pi\)
\(752\) 0 0
\(753\) 576.523 0.765634
\(754\) 0 0
\(755\) 672.417i 0.890619i
\(756\) 0 0
\(757\) 1407.80 1.85971 0.929855 0.367927i \(-0.119932\pi\)
0.929855 + 0.367927i \(0.119932\pi\)
\(758\) 0 0
\(759\) − 74.7642i − 0.0985036i
\(760\) 0 0
\(761\) −295.042 −0.387703 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(762\) 0 0
\(763\) 543.181i 0.711901i
\(764\) 0 0
\(765\) 12.5227 0.0163696
\(766\) 0 0
\(767\) − 1534.28i − 2.00037i
\(768\) 0 0
\(769\) 116.991 0.152134 0.0760669 0.997103i \(-0.475764\pi\)
0.0760669 + 0.997103i \(0.475764\pi\)
\(770\) 0 0
\(771\) 355.715i 0.461368i
\(772\) 0 0
\(773\) −105.408 −0.136363 −0.0681813 0.997673i \(-0.521720\pi\)
−0.0681813 + 0.997673i \(0.521720\pi\)
\(774\) 0 0
\(775\) 110.311i 0.142336i
\(776\) 0 0
\(777\) 34.3485 0.0442065
\(778\) 0 0
\(779\) − 279.861i − 0.359257i
\(780\) 0 0
\(781\) −289.147 −0.370227
\(782\) 0 0
\(783\) 10.3923i 0.0132724i
\(784\) 0 0
\(785\) −915.441 −1.16617
\(786\) 0 0
\(787\) − 542.148i − 0.688879i −0.938809 0.344439i \(-0.888069\pi\)
0.938809 0.344439i \(-0.111931\pi\)
\(788\) 0 0
\(789\) −78.6462 −0.0996783
\(790\) 0 0
\(791\) 474.439i 0.599797i
\(792\) 0 0
\(793\) −2076.83 −2.61895
\(794\) 0 0
\(795\) − 481.780i − 0.606013i
\(796\) 0 0
\(797\) 296.886 0.372504 0.186252 0.982502i \(-0.440366\pi\)
0.186252 + 0.982502i \(0.440366\pi\)
\(798\) 0 0
\(799\) 30.0012i 0.0375485i
\(800\) 0 0
\(801\) −104.711 −0.130726
\(802\) 0 0
\(803\) 177.321i 0.220823i
\(804\) 0 0
\(805\) 145.633 0.180911
\(806\) 0 0
\(807\) − 349.994i − 0.433697i
\(808\) 0 0
\(809\) 1066.83 1.31870 0.659349 0.751837i \(-0.270832\pi\)
0.659349 + 0.751837i \(0.270832\pi\)
\(810\) 0 0
\(811\) 9.82020i 0.0121088i 0.999982 + 0.00605438i \(0.00192718\pi\)
−0.999982 + 0.00605438i \(0.998073\pi\)
\(812\) 0 0
\(813\) 484.624 0.596094
\(814\) 0 0
\(815\) − 1266.15i − 1.55355i
\(816\) 0 0
\(817\) −256.000 −0.313341
\(818\) 0 0
\(819\) 136.244i 0.166354i
\(820\) 0 0
\(821\) 928.395 1.13081 0.565405 0.824813i \(-0.308720\pi\)
0.565405 + 0.824813i \(0.308720\pi\)
\(822\) 0 0
\(823\) − 396.291i − 0.481520i −0.970585 0.240760i \(-0.922603\pi\)
0.970585 0.240760i \(-0.0773966\pi\)
\(824\) 0 0
\(825\) −46.7477 −0.0566639
\(826\) 0 0
\(827\) 1094.28i 1.32320i 0.749858 + 0.661599i \(0.230122\pi\)
−0.749858 + 0.661599i \(0.769878\pi\)
\(828\) 0 0
\(829\) 107.074 0.129161 0.0645804 0.997913i \(-0.479429\pi\)
0.0645804 + 0.997913i \(0.479429\pi\)
\(830\) 0 0
\(831\) 234.622i 0.282337i
\(832\) 0 0
\(833\) −5.23409 −0.00628342
\(834\) 0 0
\(835\) 1266.15i 1.51634i
\(836\) 0 0
\(837\) −92.9727 −0.111079
\(838\) 0 0
\(839\) − 1110.28i − 1.32333i −0.749798 0.661667i \(-0.769849\pi\)
0.749798 0.661667i \(-0.230151\pi\)
\(840\) 0 0
\(841\) −837.000 −0.995244
\(842\) 0 0
\(843\) − 36.1185i − 0.0428452i
\(844\) 0 0
\(845\) −701.408 −0.830069
\(846\) 0 0
\(847\) 269.430i 0.318099i
\(848\) 0 0
\(849\) −197.009 −0.232048
\(850\) 0 0
\(851\) 73.9052i 0.0868451i
\(852\) 0 0
\(853\) −657.506 −0.770816 −0.385408 0.922746i \(-0.625939\pi\)
−0.385408 + 0.922746i \(0.625939\pi\)
\(854\) 0 0
\(855\) − 61.2096i − 0.0715902i
\(856\) 0 0
\(857\) 664.958 0.775914 0.387957 0.921677i \(-0.373181\pi\)
0.387957 + 0.921677i \(0.373181\pi\)
\(858\) 0 0
\(859\) 1071.96i 1.24792i 0.781456 + 0.623961i \(0.214477\pi\)
−0.781456 + 0.623961i \(0.785523\pi\)
\(860\) 0 0
\(861\) 350.904 0.407554
\(862\) 0 0
\(863\) 612.073i 0.709239i 0.935011 + 0.354619i \(0.115390\pi\)
−0.935011 + 0.354619i \(0.884610\pi\)
\(864\) 0 0
\(865\) −608.065 −0.702965
\(866\) 0 0
\(867\) 499.594i 0.576233i
\(868\) 0 0
\(869\) −418.294 −0.481351
\(870\) 0 0
\(871\) − 401.232i − 0.460657i
\(872\) 0 0
\(873\) −157.514 −0.180428
\(874\) 0 0
\(875\) 278.193i 0.317935i
\(876\) 0 0
\(877\) 1385.19 1.57947 0.789734 0.613449i \(-0.210218\pi\)
0.789734 + 0.613449i \(0.210218\pi\)
\(878\) 0 0
\(879\) − 211.191i − 0.240263i
\(880\) 0 0
\(881\) −623.325 −0.707520 −0.353760 0.935336i \(-0.615097\pi\)
−0.353760 + 0.935336i \(0.615097\pi\)
\(882\) 0 0
\(883\) − 636.791i − 0.721168i −0.932727 0.360584i \(-0.882577\pi\)
0.932727 0.360584i \(-0.117423\pi\)
\(884\) 0 0
\(885\) −864.276 −0.976583
\(886\) 0 0
\(887\) − 282.866i − 0.318902i −0.987206 0.159451i \(-0.949028\pi\)
0.987206 0.159451i \(-0.0509723\pi\)
\(888\) 0 0
\(889\) 345.459 0.388593
\(890\) 0 0
\(891\) − 39.4002i − 0.0442202i
\(892\) 0 0
\(893\) 146.642 0.164213
\(894\) 0 0
\(895\) 509.414i 0.569177i
\(896\) 0 0
\(897\) −293.147 −0.326808
\(898\) 0 0
\(899\) 35.7852i 0.0398056i
\(900\) 0 0
\(901\) −37.2561 −0.0413497
\(902\) 0 0
\(903\) − 320.986i − 0.355466i
\(904\) 0 0
\(905\) 756.414 0.835816
\(906\) 0 0
\(907\) 1419.65i 1.56521i 0.622517 + 0.782607i \(0.286110\pi\)
−0.622517 + 0.782607i \(0.713890\pi\)
\(908\) 0 0
\(909\) −340.748 −0.374860
\(910\) 0 0
\(911\) − 1170.46i − 1.28481i −0.766365 0.642405i \(-0.777937\pi\)
0.766365 0.642405i \(-0.222063\pi\)
\(912\) 0 0
\(913\) 506.294 0.554539
\(914\) 0 0
\(915\) 1169.90i 1.27858i
\(916\) 0 0
\(917\) −188.348 −0.205396
\(918\) 0 0
\(919\) 944.239i 1.02746i 0.857951 + 0.513732i \(0.171737\pi\)
−0.857951 + 0.513732i \(0.828263\pi\)
\(920\) 0 0
\(921\) 583.927 0.634014
\(922\) 0 0
\(923\) 1133.73i 1.22831i
\(924\) 0 0
\(925\) 46.2106 0.0499574
\(926\) 0 0
\(927\) 150.069i 0.161887i
\(928\) 0 0
\(929\) 1253.20 1.34897 0.674487 0.738286i \(-0.264365\pi\)
0.674487 + 0.738286i \(0.264365\pi\)
\(930\) 0 0
\(931\) 25.5836i 0.0274797i
\(932\) 0 0
\(933\) 427.405 0.458097
\(934\) 0 0
\(935\) 18.2740i 0.0195444i
\(936\) 0 0
\(937\) 470.936 0.502600 0.251300 0.967909i \(-0.419142\pi\)
0.251300 + 0.967909i \(0.419142\pi\)
\(938\) 0 0
\(939\) − 475.663i − 0.506564i
\(940\) 0 0
\(941\) 816.951 0.868173 0.434086 0.900871i \(-0.357071\pi\)
0.434086 + 0.900871i \(0.357071\pi\)
\(942\) 0 0
\(943\) 755.015i 0.800652i
\(944\) 0 0
\(945\) 76.7477 0.0812145
\(946\) 0 0
\(947\) − 994.889i − 1.05057i −0.850927 0.525284i \(-0.823959\pi\)
0.850927 0.525284i \(-0.176041\pi\)
\(948\) 0 0
\(949\) 695.267 0.732631
\(950\) 0 0
\(951\) − 224.198i − 0.235750i
\(952\) 0 0
\(953\) −1293.85 −1.35766 −0.678832 0.734293i \(-0.737514\pi\)
−0.678832 + 0.734293i \(0.737514\pi\)
\(954\) 0 0
\(955\) − 596.970i − 0.625099i
\(956\) 0 0
\(957\) −15.1652 −0.0158466
\(958\) 0 0
\(959\) − 264.162i − 0.275456i
\(960\) 0 0
\(961\) 640.855 0.666862
\(962\) 0 0
\(963\) − 481.598i − 0.500102i
\(964\) 0 0
\(965\) 650.395 0.673985
\(966\) 0 0
\(967\) − 1071.60i − 1.10817i −0.832460 0.554085i \(-0.813068\pi\)
0.832460 0.554085i \(-0.186932\pi\)
\(968\) 0 0
\(969\) −4.73334 −0.00488477
\(970\) 0 0
\(971\) − 1433.12i − 1.47593i −0.674842 0.737963i \(-0.735788\pi\)
0.674842 0.737963i \(-0.264212\pi\)
\(972\) 0 0
\(973\) 429.670 0.441593
\(974\) 0 0
\(975\) 183.296i 0.187995i
\(976\) 0 0
\(977\) −126.559 −0.129538 −0.0647692 0.997900i \(-0.520631\pi\)
−0.0647692 + 0.997900i \(0.520631\pi\)
\(978\) 0 0
\(979\) − 152.802i − 0.156080i
\(980\) 0 0
\(981\) −615.909 −0.627838
\(982\) 0 0
\(983\) 18.1480i 0.0184619i 0.999957 + 0.00923094i \(0.00293834\pi\)
−0.999957 + 0.00923094i \(0.997062\pi\)
\(984\) 0 0
\(985\) 216.798 0.220100
\(986\) 0 0
\(987\) 183.868i 0.186289i
\(988\) 0 0
\(989\) 690.642 0.698324
\(990\) 0 0
\(991\) − 643.338i − 0.649181i −0.945855 0.324590i \(-0.894774\pi\)
0.945855 0.324590i \(-0.105226\pi\)
\(992\) 0 0
\(993\) 380.312 0.382993
\(994\) 0 0
\(995\) − 1971.30i − 1.98120i
\(996\) 0 0
\(997\) −631.358 −0.633257 −0.316629 0.948550i \(-0.602551\pi\)
−0.316629 + 0.948550i \(0.602551\pi\)
\(998\) 0 0
\(999\) 38.9475i 0.0389865i
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 1344.3.m.a.127.1 4
4.3 odd 2 inner 1344.3.m.a.127.3 4
8.3 odd 2 336.3.m.c.127.2 4
8.5 even 2 336.3.m.c.127.4 yes 4
24.5 odd 2 1008.3.m.b.127.2 4
24.11 even 2 1008.3.m.b.127.1 4
56.13 odd 2 2352.3.m.f.1471.1 4
56.27 even 2 2352.3.m.f.1471.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.m.c.127.2 4 8.3 odd 2
336.3.m.c.127.4 yes 4 8.5 even 2
1008.3.m.b.127.1 4 24.11 even 2
1008.3.m.b.127.2 4 24.5 odd 2
1344.3.m.a.127.1 4 1.1 even 1 trivial
1344.3.m.a.127.3 4 4.3 odd 2 inner
2352.3.m.f.1471.1 4 56.13 odd 2
2352.3.m.f.1471.3 4 56.27 even 2