Properties

Label 2304.4.a.bb.1.2
Level $2304$
Weight $4$
Character 2304.1
Self dual yes
Analytic conductor $135.940$
Analytic rank $1$
Dimension $2$
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] = [2304,4,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)

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: yes
Analytic conductor: \(135.940400653\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843 q^{5} -14.1421 q^{7} +O(q^{10})\) \(q+2.82843 q^{5} -14.1421 q^{7} -20.0000 q^{11} -39.5980 q^{13} +34.0000 q^{17} +52.0000 q^{19} +62.2254 q^{23} -117.000 q^{25} +200.818 q^{29} +110.309 q^{31} -40.0000 q^{35} +271.529 q^{37} +26.0000 q^{41} +252.000 q^{43} +345.068 q^{47} -143.000 q^{49} -681.651 q^{53} -56.5685 q^{55} -364.000 q^{59} -735.391 q^{61} -112.000 q^{65} +628.000 q^{67} -333.754 q^{71} +338.000 q^{73} +282.843 q^{77} +789.131 q^{79} -1036.00 q^{83} +96.1665 q^{85} -234.000 q^{89} +560.000 q^{91} +147.078 q^{95} -178.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 40 q^{11} + 68 q^{17} + 104 q^{19} - 234 q^{25} - 80 q^{35} + 52 q^{41} + 504 q^{43} - 286 q^{49} - 728 q^{59} - 224 q^{65} + 1256 q^{67} + 676 q^{73} - 2072 q^{83} - 468 q^{89} + 1120 q^{91} - 356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.82843 0.252982 0.126491 0.991968i \(-0.459628\pi\)
0.126491 + 0.991968i \(0.459628\pi\)
\(6\) 0 0
\(7\) −14.1421 −0.763604 −0.381802 0.924244i \(-0.624696\pi\)
−0.381802 + 0.924244i \(0.624696\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −20.0000 −0.548202 −0.274101 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) −39.5980 −0.844808 −0.422404 0.906408i \(-0.638814\pi\)
−0.422404 + 0.906408i \(0.638814\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 34.0000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 52.0000 0.627875 0.313937 0.949444i \(-0.398352\pi\)
0.313937 + 0.949444i \(0.398352\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 62.2254 0.564126 0.282063 0.959396i \(-0.408981\pi\)
0.282063 + 0.959396i \(0.408981\pi\)
\(24\) 0 0
\(25\) −117.000 −0.936000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 200.818 1.28590 0.642949 0.765909i \(-0.277711\pi\)
0.642949 + 0.765909i \(0.277711\pi\)
\(30\) 0 0
\(31\) 110.309 0.639097 0.319549 0.947570i \(-0.396469\pi\)
0.319549 + 0.947570i \(0.396469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −40.0000 −0.193178
\(36\) 0 0
\(37\) 271.529 1.20646 0.603231 0.797567i \(-0.293880\pi\)
0.603231 + 0.797567i \(0.293880\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 26.0000 0.0990370 0.0495185 0.998773i \(-0.484231\pi\)
0.0495185 + 0.998773i \(0.484231\pi\)
\(42\) 0 0
\(43\) 252.000 0.893713 0.446856 0.894606i \(-0.352544\pi\)
0.446856 + 0.894606i \(0.352544\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 345.068 1.07092 0.535461 0.844560i \(-0.320138\pi\)
0.535461 + 0.844560i \(0.320138\pi\)
\(48\) 0 0
\(49\) −143.000 −0.416910
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −681.651 −1.76664 −0.883320 0.468770i \(-0.844697\pi\)
−0.883320 + 0.468770i \(0.844697\pi\)
\(54\) 0 0
\(55\) −56.5685 −0.138685
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −364.000 −0.803199 −0.401600 0.915815i \(-0.631546\pi\)
−0.401600 + 0.915815i \(0.631546\pi\)
\(60\) 0 0
\(61\) −735.391 −1.54356 −0.771780 0.635889i \(-0.780633\pi\)
−0.771780 + 0.635889i \(0.780633\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −112.000 −0.213721
\(66\) 0 0
\(67\) 628.000 1.14511 0.572555 0.819866i \(-0.305952\pi\)
0.572555 + 0.819866i \(0.305952\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −333.754 −0.557878 −0.278939 0.960309i \(-0.589983\pi\)
−0.278939 + 0.960309i \(0.589983\pi\)
\(72\) 0 0
\(73\) 338.000 0.541917 0.270958 0.962591i \(-0.412659\pi\)
0.270958 + 0.962591i \(0.412659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 282.843 0.418609
\(78\) 0 0
\(79\) 789.131 1.12385 0.561925 0.827188i \(-0.310061\pi\)
0.561925 + 0.827188i \(0.310061\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1036.00 −1.37007 −0.685035 0.728510i \(-0.740213\pi\)
−0.685035 + 0.728510i \(0.740213\pi\)
\(84\) 0 0
\(85\) 96.1665 0.122714
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −234.000 −0.278696 −0.139348 0.990243i \(-0.544501\pi\)
−0.139348 + 0.990243i \(0.544501\pi\)
\(90\) 0 0
\(91\) 560.000 0.645098
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 147.078 0.158841
\(96\) 0 0
\(97\) −178.000 −0.186321 −0.0931606 0.995651i \(-0.529697\pi\)
−0.0931606 + 0.995651i \(0.529697\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 257.387 0.253574 0.126787 0.991930i \(-0.459534\pi\)
0.126787 + 0.991930i \(0.459534\pi\)
\(102\) 0 0
\(103\) −1886.56 −1.80474 −0.902371 0.430961i \(-0.858175\pi\)
−0.902371 + 0.430961i \(0.858175\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1404.00 −1.26850 −0.634251 0.773127i \(-0.718692\pi\)
−0.634251 + 0.773127i \(0.718692\pi\)
\(108\) 0 0
\(109\) 39.5980 0.0347963 0.0173982 0.999849i \(-0.494462\pi\)
0.0173982 + 0.999849i \(0.494462\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1378.00 −1.14718 −0.573590 0.819143i \(-0.694450\pi\)
−0.573590 + 0.819143i \(0.694450\pi\)
\(114\) 0 0
\(115\) 176.000 0.142714
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −480.833 −0.370402
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −684.479 −0.489774
\(126\) 0 0
\(127\) 1790.39 1.25096 0.625480 0.780241i \(-0.284903\pi\)
0.625480 + 0.780241i \(0.284903\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1572.00 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −735.391 −0.479447
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2854.00 −1.77981 −0.889904 0.456148i \(-0.849229\pi\)
−0.889904 + 0.456148i \(0.849229\pi\)
\(138\) 0 0
\(139\) 1964.00 1.19845 0.599224 0.800581i \(-0.295476\pi\)
0.599224 + 0.800581i \(0.295476\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 791.960 0.463126
\(144\) 0 0
\(145\) 568.000 0.325309
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1507.55 0.828882 0.414441 0.910076i \(-0.363977\pi\)
0.414441 + 0.910076i \(0.363977\pi\)
\(150\) 0 0
\(151\) −2265.57 −1.22099 −0.610495 0.792020i \(-0.709029\pi\)
−0.610495 + 0.792020i \(0.709029\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 312.000 0.161680
\(156\) 0 0
\(157\) 3529.88 1.79436 0.897181 0.441663i \(-0.145611\pi\)
0.897181 + 0.441663i \(0.145611\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −880.000 −0.430768
\(162\) 0 0
\(163\) 2932.00 1.40891 0.704454 0.709750i \(-0.251192\pi\)
0.704454 + 0.709750i \(0.251192\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3676.96 −1.70378 −0.851890 0.523720i \(-0.824544\pi\)
−0.851890 + 0.523720i \(0.824544\pi\)
\(168\) 0 0
\(169\) −629.000 −0.286299
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1445.33 −0.635180 −0.317590 0.948228i \(-0.602874\pi\)
−0.317590 + 0.948228i \(0.602874\pi\)
\(174\) 0 0
\(175\) 1654.63 0.714733
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1308.00 0.546170 0.273085 0.961990i \(-0.411956\pi\)
0.273085 + 0.961990i \(0.411956\pi\)
\(180\) 0 0
\(181\) −1996.87 −0.820034 −0.410017 0.912078i \(-0.634477\pi\)
−0.410017 + 0.912078i \(0.634477\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 768.000 0.305213
\(186\) 0 0
\(187\) −680.000 −0.265917
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 939.038 0.355740 0.177870 0.984054i \(-0.443079\pi\)
0.177870 + 0.984054i \(0.443079\pi\)
\(192\) 0 0
\(193\) −2490.00 −0.928674 −0.464337 0.885659i \(-0.653707\pi\)
−0.464337 + 0.885659i \(0.653707\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2723.78 0.985081 0.492540 0.870290i \(-0.336068\pi\)
0.492540 + 0.870290i \(0.336068\pi\)
\(198\) 0 0
\(199\) −2158.09 −0.768758 −0.384379 0.923175i \(-0.625584\pi\)
−0.384379 + 0.923175i \(0.625584\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2840.00 −0.981916
\(204\) 0 0
\(205\) 73.5391 0.0250546
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1040.00 −0.344202
\(210\) 0 0
\(211\) −924.000 −0.301473 −0.150736 0.988574i \(-0.548164\pi\)
−0.150736 + 0.988574i \(0.548164\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 712.764 0.226093
\(216\) 0 0
\(217\) −1560.00 −0.488017
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1346.33 −0.409792
\(222\) 0 0
\(223\) 2276.88 0.683728 0.341864 0.939749i \(-0.388942\pi\)
0.341864 + 0.939749i \(0.388942\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −156.000 −0.0456127 −0.0228064 0.999740i \(-0.507260\pi\)
−0.0228064 + 0.999740i \(0.507260\pi\)
\(228\) 0 0
\(229\) 639.225 0.184459 0.0922296 0.995738i \(-0.470601\pi\)
0.0922296 + 0.995738i \(0.470601\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2826.00 −0.794581 −0.397291 0.917693i \(-0.630049\pi\)
−0.397291 + 0.917693i \(0.630049\pi\)
\(234\) 0 0
\(235\) 976.000 0.270924
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2466.39 0.667521 0.333760 0.942658i \(-0.391682\pi\)
0.333760 + 0.942658i \(0.391682\pi\)
\(240\) 0 0
\(241\) −3354.00 −0.896474 −0.448237 0.893915i \(-0.647948\pi\)
−0.448237 + 0.893915i \(0.647948\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −404.465 −0.105471
\(246\) 0 0
\(247\) −2059.09 −0.530433
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6396.00 1.60841 0.804207 0.594349i \(-0.202590\pi\)
0.804207 + 0.594349i \(0.202590\pi\)
\(252\) 0 0
\(253\) −1244.51 −0.309255
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6882.00 −1.67038 −0.835189 0.549962i \(-0.814642\pi\)
−0.835189 + 0.549962i \(0.814642\pi\)
\(258\) 0 0
\(259\) −3840.00 −0.921259
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −1928.00 −0.446929
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1434.01 0.325031 0.162515 0.986706i \(-0.448039\pi\)
0.162515 + 0.986706i \(0.448039\pi\)
\(270\) 0 0
\(271\) 5942.53 1.33204 0.666020 0.745934i \(-0.267997\pi\)
0.666020 + 0.745934i \(0.267997\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2340.00 0.513117
\(276\) 0 0
\(277\) 1103.09 0.239271 0.119635 0.992818i \(-0.461827\pi\)
0.119635 + 0.992818i \(0.461827\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6266.00 −1.33024 −0.665121 0.746735i \(-0.731620\pi\)
−0.665121 + 0.746735i \(0.731620\pi\)
\(282\) 0 0
\(283\) −8596.00 −1.80558 −0.902790 0.430082i \(-0.858485\pi\)
−0.902790 + 0.430082i \(0.858485\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −367.696 −0.0756250
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8397.60 −1.67438 −0.837189 0.546913i \(-0.815803\pi\)
−0.837189 + 0.546913i \(0.815803\pi\)
\(294\) 0 0
\(295\) −1029.55 −0.203195
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2464.00 −0.476578
\(300\) 0 0
\(301\) −3563.82 −0.682442
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2080.00 −0.390493
\(306\) 0 0
\(307\) −4940.00 −0.918374 −0.459187 0.888340i \(-0.651859\pi\)
−0.459187 + 0.888340i \(0.651859\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3382.80 0.616788 0.308394 0.951259i \(-0.400209\pi\)
0.308394 + 0.951259i \(0.400209\pi\)
\(312\) 0 0
\(313\) −3106.00 −0.560899 −0.280450 0.959869i \(-0.590484\pi\)
−0.280450 + 0.959869i \(0.590484\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6728.83 1.19220 0.596102 0.802909i \(-0.296715\pi\)
0.596102 + 0.802909i \(0.296715\pi\)
\(318\) 0 0
\(319\) −4016.37 −0.704932
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1768.00 0.304564
\(324\) 0 0
\(325\) 4632.96 0.790740
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4880.00 −0.817760
\(330\) 0 0
\(331\) 2908.00 0.482895 0.241447 0.970414i \(-0.422378\pi\)
0.241447 + 0.970414i \(0.422378\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1776.25 0.289693
\(336\) 0 0
\(337\) 4298.00 0.694739 0.347369 0.937728i \(-0.387075\pi\)
0.347369 + 0.937728i \(0.387075\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2206.17 −0.350355
\(342\) 0 0
\(343\) 6873.08 1.08196
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9996.00 1.54644 0.773218 0.634140i \(-0.218646\pi\)
0.773218 + 0.634140i \(0.218646\pi\)
\(348\) 0 0
\(349\) −3993.74 −0.612550 −0.306275 0.951943i \(-0.599083\pi\)
−0.306275 + 0.951943i \(0.599083\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6738.00 −1.01594 −0.507971 0.861374i \(-0.669604\pi\)
−0.507971 + 0.861374i \(0.669604\pi\)
\(354\) 0 0
\(355\) −944.000 −0.141133
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2132.63 0.313527 0.156763 0.987636i \(-0.449894\pi\)
0.156763 + 0.987636i \(0.449894\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 956.008 0.137095
\(366\) 0 0
\(367\) −7628.27 −1.08499 −0.542496 0.840058i \(-0.682521\pi\)
−0.542496 + 0.840058i \(0.682521\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9640.00 1.34901
\(372\) 0 0
\(373\) 8383.46 1.16375 0.581875 0.813278i \(-0.302319\pi\)
0.581875 + 0.813278i \(0.302319\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7952.00 −1.08634
\(378\) 0 0
\(379\) −12788.0 −1.73318 −0.866590 0.499020i \(-0.833693\pi\)
−0.866590 + 0.499020i \(0.833693\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2319.31 0.309429 0.154714 0.987959i \(-0.450554\pi\)
0.154714 + 0.987959i \(0.450554\pi\)
\(384\) 0 0
\(385\) 800.000 0.105901
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2684.18 0.349854 0.174927 0.984581i \(-0.444031\pi\)
0.174927 + 0.984581i \(0.444031\pi\)
\(390\) 0 0
\(391\) 2115.66 0.273641
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2232.00 0.284314
\(396\) 0 0
\(397\) −2206.17 −0.278903 −0.139452 0.990229i \(-0.544534\pi\)
−0.139452 + 0.990229i \(0.544534\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3582.00 −0.446076 −0.223038 0.974810i \(-0.571597\pi\)
−0.223038 + 0.974810i \(0.571597\pi\)
\(402\) 0 0
\(403\) −4368.00 −0.539915
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5430.58 −0.661385
\(408\) 0 0
\(409\) 5126.00 0.619717 0.309859 0.950783i \(-0.399718\pi\)
0.309859 + 0.950783i \(0.399718\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5147.74 0.613326
\(414\) 0 0
\(415\) −2930.25 −0.346603
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2924.00 −0.340923 −0.170462 0.985364i \(-0.554526\pi\)
−0.170462 + 0.985364i \(0.554526\pi\)
\(420\) 0 0
\(421\) −7314.31 −0.846741 −0.423370 0.905957i \(-0.639153\pi\)
−0.423370 + 0.905957i \(0.639153\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3978.00 −0.454027
\(426\) 0 0
\(427\) 10400.0 1.17867
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15844.8 −1.77081 −0.885405 0.464819i \(-0.846119\pi\)
−0.885405 + 0.464819i \(0.846119\pi\)
\(432\) 0 0
\(433\) −6274.00 −0.696326 −0.348163 0.937434i \(-0.613194\pi\)
−0.348163 + 0.937434i \(0.613194\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3235.72 0.354200
\(438\) 0 0
\(439\) 4596.19 0.499691 0.249846 0.968286i \(-0.419620\pi\)
0.249846 + 0.968286i \(0.419620\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5084.00 0.545255 0.272628 0.962120i \(-0.412107\pi\)
0.272628 + 0.962120i \(0.412107\pi\)
\(444\) 0 0
\(445\) −661.852 −0.0705051
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −14190.0 −1.49146 −0.745732 0.666246i \(-0.767900\pi\)
−0.745732 + 0.666246i \(0.767900\pi\)
\(450\) 0 0
\(451\) −520.000 −0.0542923
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1583.92 0.163198
\(456\) 0 0
\(457\) −6474.00 −0.662672 −0.331336 0.943513i \(-0.607499\pi\)
−0.331336 + 0.943513i \(0.607499\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6321.53 −0.638662 −0.319331 0.947643i \(-0.603458\pi\)
−0.319331 + 0.947643i \(0.603458\pi\)
\(462\) 0 0
\(463\) −11435.3 −1.14783 −0.573915 0.818915i \(-0.694576\pi\)
−0.573915 + 0.818915i \(0.694576\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3796.00 0.376141 0.188071 0.982156i \(-0.439777\pi\)
0.188071 + 0.982156i \(0.439777\pi\)
\(468\) 0 0
\(469\) −8881.26 −0.874411
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5040.00 −0.489935
\(474\) 0 0
\(475\) −6084.00 −0.587691
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10493.5 −1.00096 −0.500479 0.865749i \(-0.666843\pi\)
−0.500479 + 0.865749i \(0.666843\pi\)
\(480\) 0 0
\(481\) −10752.0 −1.01923
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −503.460 −0.0471360
\(486\) 0 0
\(487\) 15406.4 1.43354 0.716769 0.697311i \(-0.245620\pi\)
0.716769 + 0.697311i \(0.245620\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15452.0 −1.42024 −0.710121 0.704079i \(-0.751360\pi\)
−0.710121 + 0.704079i \(0.751360\pi\)
\(492\) 0 0
\(493\) 6827.82 0.623752
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4720.00 0.425998
\(498\) 0 0
\(499\) 52.0000 0.00466501 0.00233250 0.999997i \(-0.499258\pi\)
0.00233250 + 0.999997i \(0.499258\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12428.1 1.10167 0.550837 0.834613i \(-0.314309\pi\)
0.550837 + 0.834613i \(0.314309\pi\)
\(504\) 0 0
\(505\) 728.000 0.0641497
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16362.5 −1.42486 −0.712429 0.701744i \(-0.752405\pi\)
−0.712429 + 0.701744i \(0.752405\pi\)
\(510\) 0 0
\(511\) −4780.04 −0.413809
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5336.00 −0.456567
\(516\) 0 0
\(517\) −6901.36 −0.587082
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 714.000 0.0600401 0.0300201 0.999549i \(-0.490443\pi\)
0.0300201 + 0.999549i \(0.490443\pi\)
\(522\) 0 0
\(523\) 5980.00 0.499975 0.249988 0.968249i \(-0.419573\pi\)
0.249988 + 0.968249i \(0.419573\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3750.49 0.310008
\(528\) 0 0
\(529\) −8295.00 −0.681762
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1029.55 −0.0836673
\(534\) 0 0
\(535\) −3971.11 −0.320909
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2860.00 0.228551
\(540\) 0 0
\(541\) 13729.2 1.09106 0.545530 0.838091i \(-0.316328\pi\)
0.545530 + 0.838091i \(0.316328\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 112.000 0.00880285
\(546\) 0 0
\(547\) 18500.0 1.44607 0.723037 0.690809i \(-0.242745\pi\)
0.723037 + 0.690809i \(0.242745\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10442.6 0.807382
\(552\) 0 0
\(553\) −11160.0 −0.858176
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8765.30 −0.666782 −0.333391 0.942789i \(-0.608193\pi\)
−0.333391 + 0.942789i \(0.608193\pi\)
\(558\) 0 0
\(559\) −9978.69 −0.755015
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −268.000 −0.0200619 −0.0100310 0.999950i \(-0.503193\pi\)
−0.0100310 + 0.999950i \(0.503193\pi\)
\(564\) 0 0
\(565\) −3897.57 −0.290216
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13866.0 1.02160 0.510802 0.859698i \(-0.329348\pi\)
0.510802 + 0.859698i \(0.329348\pi\)
\(570\) 0 0
\(571\) −5140.00 −0.376712 −0.188356 0.982101i \(-0.560316\pi\)
−0.188356 + 0.982101i \(0.560316\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7280.37 −0.528022
\(576\) 0 0
\(577\) 9386.00 0.677200 0.338600 0.940930i \(-0.390047\pi\)
0.338600 + 0.940930i \(0.390047\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14651.3 1.04619
\(582\) 0 0
\(583\) 13633.0 0.968477
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8844.00 −0.621859 −0.310929 0.950433i \(-0.600640\pi\)
−0.310929 + 0.950433i \(0.600640\pi\)
\(588\) 0 0
\(589\) 5736.05 0.401273
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9406.00 0.651363 0.325681 0.945480i \(-0.394406\pi\)
0.325681 + 0.945480i \(0.394406\pi\)
\(594\) 0 0
\(595\) −1360.00 −0.0937051
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −23459.0 −1.60018 −0.800090 0.599880i \(-0.795215\pi\)
−0.800090 + 0.599880i \(0.795215\pi\)
\(600\) 0 0
\(601\) −1262.00 −0.0856540 −0.0428270 0.999083i \(-0.513636\pi\)
−0.0428270 + 0.999083i \(0.513636\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2633.27 −0.176955
\(606\) 0 0
\(607\) −16288.9 −1.08920 −0.544602 0.838695i \(-0.683319\pi\)
−0.544602 + 0.838695i \(0.683319\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13664.0 −0.904724
\(612\) 0 0
\(613\) 7138.95 0.470374 0.235187 0.971950i \(-0.424430\pi\)
0.235187 + 0.971950i \(0.424430\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −16874.0 −1.10101 −0.550504 0.834833i \(-0.685564\pi\)
−0.550504 + 0.834833i \(0.685564\pi\)
\(618\) 0 0
\(619\) 20748.0 1.34723 0.673613 0.739085i \(-0.264742\pi\)
0.673613 + 0.739085i \(0.264742\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3309.26 0.212813
\(624\) 0 0
\(625\) 12689.0 0.812096
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9231.99 0.585220
\(630\) 0 0
\(631\) 14840.8 0.936294 0.468147 0.883651i \(-0.344922\pi\)
0.468147 + 0.883651i \(0.344922\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5064.00 0.316470
\(636\) 0 0
\(637\) 5662.51 0.352209
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17758.0 −1.09423 −0.547113 0.837059i \(-0.684273\pi\)
−0.547113 + 0.837059i \(0.684273\pi\)
\(642\) 0 0
\(643\) −1148.00 −0.0704086 −0.0352043 0.999380i \(-0.511208\pi\)
−0.0352043 + 0.999380i \(0.511208\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26988.9 1.63994 0.819970 0.572406i \(-0.193990\pi\)
0.819970 + 0.572406i \(0.193990\pi\)
\(648\) 0 0
\(649\) 7280.00 0.440316
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21069.0 1.26262 0.631311 0.775530i \(-0.282517\pi\)
0.631311 + 0.775530i \(0.282517\pi\)
\(654\) 0 0
\(655\) −4446.29 −0.265238
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18356.0 −1.08505 −0.542525 0.840040i \(-0.682532\pi\)
−0.542525 + 0.840040i \(0.682532\pi\)
\(660\) 0 0
\(661\) −15250.9 −0.897414 −0.448707 0.893679i \(-0.648115\pi\)
−0.448707 + 0.893679i \(0.648115\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2080.00 −0.121292
\(666\) 0 0
\(667\) 12496.0 0.725408
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14707.8 0.846184
\(672\) 0 0
\(673\) −12082.0 −0.692016 −0.346008 0.938232i \(-0.612463\pi\)
−0.346008 + 0.938232i \(0.612463\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12742.1 0.723364 0.361682 0.932302i \(-0.382203\pi\)
0.361682 + 0.932302i \(0.382203\pi\)
\(678\) 0 0
\(679\) 2517.30 0.142276
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33508.0 −1.87723 −0.938615 0.344967i \(-0.887890\pi\)
−0.938615 + 0.344967i \(0.887890\pi\)
\(684\) 0 0
\(685\) −8072.33 −0.450260
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 26992.0 1.49247
\(690\) 0 0
\(691\) −364.000 −0.0200394 −0.0100197 0.999950i \(-0.503189\pi\)
−0.0100197 + 0.999950i \(0.503189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5555.03 0.303186
\(696\) 0 0
\(697\) 884.000 0.0480400
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3849.49 −0.207408 −0.103704 0.994608i \(-0.533070\pi\)
−0.103704 + 0.994608i \(0.533070\pi\)
\(702\) 0 0
\(703\) 14119.5 0.757507
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3640.00 −0.193630
\(708\) 0 0
\(709\) 23606.1 1.25041 0.625207 0.780459i \(-0.285014\pi\)
0.625207 + 0.780459i \(0.285014\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6864.00 0.360531
\(714\) 0 0
\(715\) 2240.00 0.117163
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15799.6 −0.819507 −0.409753 0.912196i \(-0.634385\pi\)
−0.409753 + 0.912196i \(0.634385\pi\)
\(720\) 0 0
\(721\) 26680.0 1.37811
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −23495.7 −1.20360
\(726\) 0 0
\(727\) −4607.51 −0.235052 −0.117526 0.993070i \(-0.537496\pi\)
−0.117526 + 0.993070i \(0.537496\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8568.00 0.433514
\(732\) 0 0
\(733\) −26219.5 −1.32120 −0.660600 0.750738i \(-0.729698\pi\)
−0.660600 + 0.750738i \(0.729698\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12560.0 −0.627752
\(738\) 0 0
\(739\) 27924.0 1.38999 0.694994 0.719016i \(-0.255407\pi\)
0.694994 + 0.719016i \(0.255407\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8937.83 −0.441315 −0.220658 0.975351i \(-0.570820\pi\)
−0.220658 + 0.975351i \(0.570820\pi\)
\(744\) 0 0
\(745\) 4264.00 0.209692
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19855.6 0.968633
\(750\) 0 0
\(751\) 14082.7 0.684270 0.342135 0.939651i \(-0.388850\pi\)
0.342135 + 0.939651i \(0.388850\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6408.00 −0.308889
\(756\) 0 0
\(757\) 14871.9 0.714039 0.357019 0.934097i \(-0.383793\pi\)
0.357019 + 0.934097i \(0.383793\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15834.0 0.754247 0.377124 0.926163i \(-0.376913\pi\)
0.377124 + 0.926163i \(0.376913\pi\)
\(762\) 0 0
\(763\) −560.000 −0.0265706
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14413.7 0.678549
\(768\) 0 0
\(769\) −16666.0 −0.781523 −0.390762 0.920492i \(-0.627788\pi\)
−0.390762 + 0.920492i \(0.627788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30957.1 −1.44043 −0.720214 0.693752i \(-0.755956\pi\)
−0.720214 + 0.693752i \(0.755956\pi\)
\(774\) 0 0
\(775\) −12906.1 −0.598195
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1352.00 0.0621828
\(780\) 0 0
\(781\) 6675.09 0.305830
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9984.00 0.453942
\(786\) 0 0
\(787\) 20228.0 0.916201 0.458101 0.888900i \(-0.348530\pi\)
0.458101 + 0.888900i \(0.348530\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 19487.9 0.875991
\(792\) 0 0
\(793\) 29120.0 1.30401
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9008.54 0.400375 0.200187 0.979758i \(-0.435845\pi\)
0.200187 + 0.979758i \(0.435845\pi\)
\(798\) 0 0
\(799\) 11732.3 0.519474
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6760.00 −0.297080
\(804\) 0 0
\(805\) −2489.02 −0.108977
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9242.00 0.401646 0.200823 0.979628i \(-0.435638\pi\)
0.200823 + 0.979628i \(0.435638\pi\)
\(810\) 0 0
\(811\) 10972.0 0.475067 0.237533 0.971379i \(-0.423661\pi\)
0.237533 + 0.971379i \(0.423661\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8292.95 0.356429
\(816\) 0 0
\(817\) 13104.0 0.561139
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9336.64 −0.396895 −0.198448 0.980112i \(-0.563590\pi\)
−0.198448 + 0.980112i \(0.563590\pi\)
\(822\) 0 0
\(823\) −3566.65 −0.151064 −0.0755319 0.997143i \(-0.524065\pi\)
−0.0755319 + 0.997143i \(0.524065\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18876.0 −0.793691 −0.396846 0.917885i \(-0.629895\pi\)
−0.396846 + 0.917885i \(0.629895\pi\)
\(828\) 0 0
\(829\) −6974.90 −0.292218 −0.146109 0.989269i \(-0.546675\pi\)
−0.146109 + 0.989269i \(0.546675\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4862.00 −0.202231
\(834\) 0 0
\(835\) −10400.0 −0.431026
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −30077.5 −1.23765 −0.618826 0.785528i \(-0.712392\pi\)
−0.618826 + 0.785528i \(0.712392\pi\)
\(840\) 0 0
\(841\) 15939.0 0.653532
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1779.08 −0.0724287
\(846\) 0 0
\(847\) 13166.3 0.534121
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 16896.0 0.680596
\(852\) 0 0
\(853\) 41159.3 1.65213 0.826065 0.563575i \(-0.190574\pi\)
0.826065 + 0.563575i \(0.190574\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25194.0 1.00421 0.502107 0.864806i \(-0.332559\pi\)
0.502107 + 0.864806i \(0.332559\pi\)
\(858\) 0 0
\(859\) 9308.00 0.369715 0.184857 0.982765i \(-0.440818\pi\)
0.184857 + 0.982765i \(0.440818\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26802.2 −1.05719 −0.528596 0.848874i \(-0.677281\pi\)
−0.528596 + 0.848874i \(0.677281\pi\)
\(864\) 0 0
\(865\) −4088.00 −0.160689
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −15782.6 −0.616098
\(870\) 0 0
\(871\) −24867.5 −0.967399
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9680.00 0.373993
\(876\) 0 0
\(877\) −1436.84 −0.0553235 −0.0276617 0.999617i \(-0.508806\pi\)
−0.0276617 + 0.999617i \(0.508806\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42830.0 1.63789 0.818944 0.573873i \(-0.194560\pi\)
0.818944 + 0.573873i \(0.194560\pi\)
\(882\) 0 0
\(883\) −23964.0 −0.913310 −0.456655 0.889644i \(-0.650953\pi\)
−0.456655 + 0.889644i \(0.650953\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28239.0 −1.06897 −0.534483 0.845179i \(-0.679494\pi\)
−0.534483 + 0.845179i \(0.679494\pi\)
\(888\) 0 0
\(889\) −25320.0 −0.955237
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 17943.5 0.672405
\(894\) 0 0
\(895\) 3699.58 0.138171
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 22152.0 0.821814
\(900\) 0 0
\(901\) −23176.1 −0.856947
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5648.00 −0.207454
\(906\) 0 0
\(907\) −31972.0 −1.17047 −0.585233 0.810865i \(-0.698997\pi\)
−0.585233 + 0.810865i \(0.698997\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26858.7 0.976806 0.488403 0.872618i \(-0.337580\pi\)
0.488403 + 0.872618i \(0.337580\pi\)
\(912\) 0 0
\(913\) 20720.0 0.751075
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 22231.4 0.800596
\(918\) 0 0
\(919\) −40336.2 −1.44784 −0.723922 0.689882i \(-0.757662\pi\)
−0.723922 + 0.689882i \(0.757662\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13216.0 0.471300
\(924\) 0 0
\(925\) −31768.9 −1.12925
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13650.0 0.482069 0.241034 0.970517i \(-0.422513\pi\)
0.241034 + 0.970517i \(0.422513\pi\)
\(930\) 0 0
\(931\) −7436.00 −0.261767
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1923.33 −0.0672723
\(936\) 0 0
\(937\) 7098.00 0.247472 0.123736 0.992315i \(-0.460512\pi\)
0.123736 + 0.992315i \(0.460512\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −41326.1 −1.43166 −0.715831 0.698274i \(-0.753952\pi\)
−0.715831 + 0.698274i \(0.753952\pi\)
\(942\) 0 0
\(943\) 1617.86 0.0558693
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9900.00 0.339711 0.169856 0.985469i \(-0.445670\pi\)
0.169856 + 0.985469i \(0.445670\pi\)
\(948\) 0 0
\(949\) −13384.1 −0.457815
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 46938.0 1.59546 0.797729 0.603016i \(-0.206035\pi\)
0.797729 + 0.603016i \(0.206035\pi\)
\(954\) 0 0
\(955\) 2656.00 0.0899960
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 40361.7 1.35907
\(960\) 0 0
\(961\) −17623.0 −0.591554
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7042.78 −0.234938
\(966\) 0 0
\(967\) −6989.04 −0.232422 −0.116211 0.993225i \(-0.537075\pi\)
−0.116211 + 0.993225i \(0.537075\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 53052.0 1.75337 0.876684 0.481067i \(-0.159751\pi\)
0.876684 + 0.481067i \(0.159751\pi\)
\(972\) 0 0
\(973\) −27775.2 −0.915139
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41890.0 1.37173 0.685865 0.727729i \(-0.259424\pi\)
0.685865 + 0.727729i \(0.259424\pi\)
\(978\) 0 0
\(979\) 4680.00 0.152782
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10861.2 −0.352408 −0.176204 0.984354i \(-0.556382\pi\)
−0.176204 + 0.984354i \(0.556382\pi\)
\(984\) 0 0
\(985\) 7704.00 0.249208
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 15680.8 0.504166
\(990\) 0 0
\(991\) −330.926 −0.0106077 −0.00530384 0.999986i \(-0.501688\pi\)
−0.00530384 + 0.999986i \(0.501688\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6104.00 −0.194482
\(996\) 0 0
\(997\) −39948.7 −1.26900 −0.634498 0.772925i \(-0.718793\pi\)
−0.634498 + 0.772925i \(0.718793\pi\)
\(998\) 0 0
\(999\) 0 0
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 2304.4.a.bb.1.2 2
3.2 odd 2 768.4.a.h.1.1 2
4.3 odd 2 2304.4.a.bh.1.2 2
8.3 odd 2 inner 2304.4.a.bb.1.1 2
8.5 even 2 2304.4.a.bh.1.1 2
12.11 even 2 768.4.a.m.1.1 2
16.3 odd 4 1152.4.d.n.577.1 4
16.5 even 4 1152.4.d.n.577.4 4
16.11 odd 4 1152.4.d.n.577.3 4
16.13 even 4 1152.4.d.n.577.2 4
24.5 odd 2 768.4.a.m.1.2 2
24.11 even 2 768.4.a.h.1.2 2
48.5 odd 4 384.4.d.d.193.3 yes 4
48.11 even 4 384.4.d.d.193.1 4
48.29 odd 4 384.4.d.d.193.2 yes 4
48.35 even 4 384.4.d.d.193.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.d.193.1 4 48.11 even 4
384.4.d.d.193.2 yes 4 48.29 odd 4
384.4.d.d.193.3 yes 4 48.5 odd 4
384.4.d.d.193.4 yes 4 48.35 even 4
768.4.a.h.1.1 2 3.2 odd 2
768.4.a.h.1.2 2 24.11 even 2
768.4.a.m.1.1 2 12.11 even 2
768.4.a.m.1.2 2 24.5 odd 2
1152.4.d.n.577.1 4 16.3 odd 4
1152.4.d.n.577.2 4 16.13 even 4
1152.4.d.n.577.3 4 16.11 odd 4
1152.4.d.n.577.4 4 16.5 even 4
2304.4.a.bb.1.1 2 8.3 odd 2 inner
2304.4.a.bb.1.2 2 1.1 even 1 trivial
2304.4.a.bh.1.1 2 8.5 even 2
2304.4.a.bh.1.2 2 4.3 odd 2