Properties

Label 2028.4.b.d.337.2
Level $2028$
Weight $4$
Character 2028.337
Analytic conductor $119.656$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2028,4,Mod(337,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2028.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(119.655873492\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2028.337
Dual form 2028.4.b.d.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +2.00000i q^{5} -32.0000i q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +2.00000i q^{5} -32.0000i q^{7} +9.00000 q^{9} -68.0000i q^{11} +6.00000i q^{15} +14.0000 q^{17} -4.00000i q^{19} -96.0000i q^{21} -72.0000 q^{23} +121.000 q^{25} +27.0000 q^{27} +102.000 q^{29} +136.000i q^{31} -204.000i q^{33} +64.0000 q^{35} -386.000i q^{37} -250.000i q^{41} +140.000 q^{43} +18.0000i q^{45} -296.000i q^{47} -681.000 q^{49} +42.0000 q^{51} +526.000 q^{53} +136.000 q^{55} -12.0000i q^{57} +332.000i q^{59} -410.000 q^{61} -288.000i q^{63} -596.000i q^{67} -216.000 q^{69} +880.000i q^{71} +506.000i q^{73} +363.000 q^{75} -2176.00 q^{77} -640.000 q^{79} +81.0000 q^{81} -1380.00i q^{83} +28.0000i q^{85} +306.000 q^{87} +1450.00i q^{89} +408.000i q^{93} +8.00000 q^{95} +446.000i q^{97} -612.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 18 q^{9} + 28 q^{17} - 144 q^{23} + 242 q^{25} + 54 q^{27} + 204 q^{29} + 128 q^{35} + 280 q^{43} - 1362 q^{49} + 84 q^{51} + 1052 q^{53} + 272 q^{55} - 820 q^{61} - 432 q^{69} + 726 q^{75} - 4352 q^{77} - 1280 q^{79} + 162 q^{81} + 612 q^{87} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 2.00000i 0.178885i 0.995992 + 0.0894427i \(0.0285086\pi\)
−0.995992 + 0.0894427i \(0.971491\pi\)
\(6\) 0 0
\(7\) − 32.0000i − 1.72784i −0.503631 0.863919i \(-0.668003\pi\)
0.503631 0.863919i \(-0.331997\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) − 68.0000i − 1.86389i −0.362602 0.931944i \(-0.618111\pi\)
0.362602 0.931944i \(-0.381889\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 6.00000i 0.103280i
\(16\) 0 0
\(17\) 14.0000 0.199735 0.0998676 0.995001i \(-0.468158\pi\)
0.0998676 + 0.995001i \(0.468158\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.0482980i −0.999708 0.0241490i \(-0.992312\pi\)
0.999708 0.0241490i \(-0.00768762\pi\)
\(20\) 0 0
\(21\) − 96.0000i − 0.997567i
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) 121.000 0.968000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 102.000 0.653135 0.326568 0.945174i \(-0.394108\pi\)
0.326568 + 0.945174i \(0.394108\pi\)
\(30\) 0 0
\(31\) 136.000i 0.787946i 0.919122 + 0.393973i \(0.128900\pi\)
−0.919122 + 0.393973i \(0.871100\pi\)
\(32\) 0 0
\(33\) − 204.000i − 1.07612i
\(34\) 0 0
\(35\) 64.0000 0.309085
\(36\) 0 0
\(37\) − 386.000i − 1.71508i −0.514416 0.857541i \(-0.671991\pi\)
0.514416 0.857541i \(-0.328009\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 250.000i − 0.952279i −0.879370 0.476140i \(-0.842036\pi\)
0.879370 0.476140i \(-0.157964\pi\)
\(42\) 0 0
\(43\) 140.000 0.496507 0.248253 0.968695i \(-0.420143\pi\)
0.248253 + 0.968695i \(0.420143\pi\)
\(44\) 0 0
\(45\) 18.0000i 0.0596285i
\(46\) 0 0
\(47\) − 296.000i − 0.918639i −0.888271 0.459320i \(-0.848093\pi\)
0.888271 0.459320i \(-0.151907\pi\)
\(48\) 0 0
\(49\) −681.000 −1.98542
\(50\) 0 0
\(51\) 42.0000 0.115317
\(52\) 0 0
\(53\) 526.000 1.36324 0.681619 0.731707i \(-0.261276\pi\)
0.681619 + 0.731707i \(0.261276\pi\)
\(54\) 0 0
\(55\) 136.000 0.333422
\(56\) 0 0
\(57\) − 12.0000i − 0.0278849i
\(58\) 0 0
\(59\) 332.000i 0.732588i 0.930499 + 0.366294i \(0.119374\pi\)
−0.930499 + 0.366294i \(0.880626\pi\)
\(60\) 0 0
\(61\) −410.000 −0.860576 −0.430288 0.902692i \(-0.641588\pi\)
−0.430288 + 0.902692i \(0.641588\pi\)
\(62\) 0 0
\(63\) − 288.000i − 0.575946i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 596.000i − 1.08676i −0.839487 0.543381i \(-0.817144\pi\)
0.839487 0.543381i \(-0.182856\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 880.000i 1.47094i 0.677557 + 0.735470i \(0.263039\pi\)
−0.677557 + 0.735470i \(0.736961\pi\)
\(72\) 0 0
\(73\) 506.000i 0.811272i 0.914035 + 0.405636i \(0.132950\pi\)
−0.914035 + 0.405636i \(0.867050\pi\)
\(74\) 0 0
\(75\) 363.000 0.558875
\(76\) 0 0
\(77\) −2176.00 −3.22050
\(78\) 0 0
\(79\) −640.000 −0.911464 −0.455732 0.890117i \(-0.650622\pi\)
−0.455732 + 0.890117i \(0.650622\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 1380.00i − 1.82500i −0.409081 0.912498i \(-0.634151\pi\)
0.409081 0.912498i \(-0.365849\pi\)
\(84\) 0 0
\(85\) 28.0000i 0.0357297i
\(86\) 0 0
\(87\) 306.000 0.377088
\(88\) 0 0
\(89\) 1450.00i 1.72696i 0.504381 + 0.863481i \(0.331721\pi\)
−0.504381 + 0.863481i \(0.668279\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 408.000i 0.454921i
\(94\) 0 0
\(95\) 8.00000 0.00863982
\(96\) 0 0
\(97\) 446.000i 0.466850i 0.972375 + 0.233425i \(0.0749933\pi\)
−0.972375 + 0.233425i \(0.925007\pi\)
\(98\) 0 0
\(99\) − 612.000i − 0.621296i
\(100\) 0 0
\(101\) 610.000 0.600963 0.300482 0.953788i \(-0.402853\pi\)
0.300482 + 0.953788i \(0.402853\pi\)
\(102\) 0 0
\(103\) 1352.00 1.29336 0.646682 0.762760i \(-0.276156\pi\)
0.646682 + 0.762760i \(0.276156\pi\)
\(104\) 0 0
\(105\) 192.000 0.178450
\(106\) 0 0
\(107\) −732.000 −0.661356 −0.330678 0.943744i \(-0.607277\pi\)
−0.330678 + 0.943744i \(0.607277\pi\)
\(108\) 0 0
\(109\) 1514.00i 1.33041i 0.746660 + 0.665206i \(0.231656\pi\)
−0.746660 + 0.665206i \(0.768344\pi\)
\(110\) 0 0
\(111\) − 1158.00i − 0.990203i
\(112\) 0 0
\(113\) −1518.00 −1.26373 −0.631865 0.775079i \(-0.717710\pi\)
−0.631865 + 0.775079i \(0.717710\pi\)
\(114\) 0 0
\(115\) − 144.000i − 0.116766i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 448.000i − 0.345110i
\(120\) 0 0
\(121\) −3293.00 −2.47408
\(122\) 0 0
\(123\) − 750.000i − 0.549799i
\(124\) 0 0
\(125\) 492.000i 0.352047i
\(126\) 0 0
\(127\) 96.0000 0.0670758 0.0335379 0.999437i \(-0.489323\pi\)
0.0335379 + 0.999437i \(0.489323\pi\)
\(128\) 0 0
\(129\) 420.000 0.286658
\(130\) 0 0
\(131\) −2548.00 −1.69939 −0.849694 0.527276i \(-0.823213\pi\)
−0.849694 + 0.527276i \(0.823213\pi\)
\(132\) 0 0
\(133\) −128.000 −0.0834512
\(134\) 0 0
\(135\) 54.0000i 0.0344265i
\(136\) 0 0
\(137\) − 230.000i − 0.143432i −0.997425 0.0717162i \(-0.977152\pi\)
0.997425 0.0717162i \(-0.0228476\pi\)
\(138\) 0 0
\(139\) 516.000 0.314867 0.157434 0.987530i \(-0.449678\pi\)
0.157434 + 0.987530i \(0.449678\pi\)
\(140\) 0 0
\(141\) − 888.000i − 0.530377i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 204.000i 0.116836i
\(146\) 0 0
\(147\) −2043.00 −1.14628
\(148\) 0 0
\(149\) 1842.00i 1.01277i 0.862308 + 0.506384i \(0.169018\pi\)
−0.862308 + 0.506384i \(0.830982\pi\)
\(150\) 0 0
\(151\) − 528.000i − 0.284556i −0.989827 0.142278i \(-0.954557\pi\)
0.989827 0.142278i \(-0.0454428\pi\)
\(152\) 0 0
\(153\) 126.000 0.0665784
\(154\) 0 0
\(155\) −272.000 −0.140952
\(156\) 0 0
\(157\) −1306.00 −0.663886 −0.331943 0.943299i \(-0.607704\pi\)
−0.331943 + 0.943299i \(0.607704\pi\)
\(158\) 0 0
\(159\) 1578.00 0.787066
\(160\) 0 0
\(161\) 2304.00i 1.12783i
\(162\) 0 0
\(163\) − 3772.00i − 1.81255i −0.422687 0.906276i \(-0.638913\pi\)
0.422687 0.906276i \(-0.361087\pi\)
\(164\) 0 0
\(165\) 408.000 0.192502
\(166\) 0 0
\(167\) 384.000i 0.177933i 0.996035 + 0.0889665i \(0.0283564\pi\)
−0.996035 + 0.0889665i \(0.971644\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 36.0000i − 0.0160993i
\(172\) 0 0
\(173\) −1462.00 −0.642508 −0.321254 0.946993i \(-0.604104\pi\)
−0.321254 + 0.946993i \(0.604104\pi\)
\(174\) 0 0
\(175\) − 3872.00i − 1.67255i
\(176\) 0 0
\(177\) 996.000i 0.422960i
\(178\) 0 0
\(179\) 1332.00 0.556192 0.278096 0.960553i \(-0.410297\pi\)
0.278096 + 0.960553i \(0.410297\pi\)
\(180\) 0 0
\(181\) −2030.00 −0.833639 −0.416820 0.908989i \(-0.636855\pi\)
−0.416820 + 0.908989i \(0.636855\pi\)
\(182\) 0 0
\(183\) −1230.00 −0.496854
\(184\) 0 0
\(185\) 772.000 0.306803
\(186\) 0 0
\(187\) − 952.000i − 0.372284i
\(188\) 0 0
\(189\) − 864.000i − 0.332522i
\(190\) 0 0
\(191\) −16.0000 −0.00606136 −0.00303068 0.999995i \(-0.500965\pi\)
−0.00303068 + 0.999995i \(0.500965\pi\)
\(192\) 0 0
\(193\) − 2078.00i − 0.775014i −0.921867 0.387507i \(-0.873336\pi\)
0.921867 0.387507i \(-0.126664\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3486.00i − 1.26075i −0.776292 0.630374i \(-0.782902\pi\)
0.776292 0.630374i \(-0.217098\pi\)
\(198\) 0 0
\(199\) −568.000 −0.202334 −0.101167 0.994869i \(-0.532258\pi\)
−0.101167 + 0.994869i \(0.532258\pi\)
\(200\) 0 0
\(201\) − 1788.00i − 0.627442i
\(202\) 0 0
\(203\) − 3264.00i − 1.12851i
\(204\) 0 0
\(205\) 500.000 0.170349
\(206\) 0 0
\(207\) −648.000 −0.217580
\(208\) 0 0
\(209\) −272.000 −0.0900222
\(210\) 0 0
\(211\) 3804.00 1.24113 0.620564 0.784156i \(-0.286904\pi\)
0.620564 + 0.784156i \(0.286904\pi\)
\(212\) 0 0
\(213\) 2640.00i 0.849248i
\(214\) 0 0
\(215\) 280.000i 0.0888179i
\(216\) 0 0
\(217\) 4352.00 1.36144
\(218\) 0 0
\(219\) 1518.00i 0.468388i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 5912.00i − 1.77532i −0.460498 0.887661i \(-0.652329\pi\)
0.460498 0.887661i \(-0.347671\pi\)
\(224\) 0 0
\(225\) 1089.00 0.322667
\(226\) 0 0
\(227\) 3308.00i 0.967223i 0.875283 + 0.483612i \(0.160675\pi\)
−0.875283 + 0.483612i \(0.839325\pi\)
\(228\) 0 0
\(229\) − 6050.00i − 1.74583i −0.487872 0.872915i \(-0.662227\pi\)
0.487872 0.872915i \(-0.337773\pi\)
\(230\) 0 0
\(231\) −6528.00 −1.85935
\(232\) 0 0
\(233\) −4794.00 −1.34792 −0.673960 0.738768i \(-0.735408\pi\)
−0.673960 + 0.738768i \(0.735408\pi\)
\(234\) 0 0
\(235\) 592.000 0.164331
\(236\) 0 0
\(237\) −1920.00 −0.526234
\(238\) 0 0
\(239\) 4440.00i 1.20167i 0.799372 + 0.600836i \(0.205166\pi\)
−0.799372 + 0.600836i \(0.794834\pi\)
\(240\) 0 0
\(241\) 1330.00i 0.355489i 0.984077 + 0.177744i \(0.0568800\pi\)
−0.984077 + 0.177744i \(0.943120\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) − 1362.00i − 0.355163i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 4140.00i − 1.05366i
\(250\) 0 0
\(251\) 5116.00 1.28653 0.643265 0.765644i \(-0.277579\pi\)
0.643265 + 0.765644i \(0.277579\pi\)
\(252\) 0 0
\(253\) 4896.00i 1.21664i
\(254\) 0 0
\(255\) 84.0000i 0.0206286i
\(256\) 0 0
\(257\) −642.000 −0.155824 −0.0779122 0.996960i \(-0.524825\pi\)
−0.0779122 + 0.996960i \(0.524825\pi\)
\(258\) 0 0
\(259\) −12352.0 −2.96338
\(260\) 0 0
\(261\) 918.000 0.217712
\(262\) 0 0
\(263\) 4264.00 0.999732 0.499866 0.866103i \(-0.333383\pi\)
0.499866 + 0.866103i \(0.333383\pi\)
\(264\) 0 0
\(265\) 1052.00i 0.243864i
\(266\) 0 0
\(267\) 4350.00i 0.997062i
\(268\) 0 0
\(269\) −3850.00 −0.872634 −0.436317 0.899793i \(-0.643717\pi\)
−0.436317 + 0.899793i \(0.643717\pi\)
\(270\) 0 0
\(271\) 2936.00i 0.658115i 0.944310 + 0.329058i \(0.106731\pi\)
−0.944310 + 0.329058i \(0.893269\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 8228.00i − 1.80424i
\(276\) 0 0
\(277\) 2066.00 0.448137 0.224068 0.974573i \(-0.428066\pi\)
0.224068 + 0.974573i \(0.428066\pi\)
\(278\) 0 0
\(279\) 1224.00i 0.262649i
\(280\) 0 0
\(281\) − 214.000i − 0.0454312i −0.999742 0.0227156i \(-0.992769\pi\)
0.999742 0.0227156i \(-0.00723122\pi\)
\(282\) 0 0
\(283\) 2620.00 0.550328 0.275164 0.961397i \(-0.411268\pi\)
0.275164 + 0.961397i \(0.411268\pi\)
\(284\) 0 0
\(285\) 24.0000 0.00498820
\(286\) 0 0
\(287\) −8000.00 −1.64538
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 1338.00i 0.269536i
\(292\) 0 0
\(293\) − 1154.00i − 0.230094i −0.993360 0.115047i \(-0.963298\pi\)
0.993360 0.115047i \(-0.0367018\pi\)
\(294\) 0 0
\(295\) −664.000 −0.131049
\(296\) 0 0
\(297\) − 1836.00i − 0.358705i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 4480.00i − 0.857883i
\(302\) 0 0
\(303\) 1830.00 0.346966
\(304\) 0 0
\(305\) − 820.000i − 0.153944i
\(306\) 0 0
\(307\) − 4076.00i − 0.757751i −0.925448 0.378876i \(-0.876311\pi\)
0.925448 0.378876i \(-0.123689\pi\)
\(308\) 0 0
\(309\) 4056.00 0.746724
\(310\) 0 0
\(311\) 6456.00 1.17713 0.588563 0.808451i \(-0.299694\pi\)
0.588563 + 0.808451i \(0.299694\pi\)
\(312\) 0 0
\(313\) −5526.00 −0.997917 −0.498958 0.866626i \(-0.666284\pi\)
−0.498958 + 0.866626i \(0.666284\pi\)
\(314\) 0 0
\(315\) 576.000 0.103028
\(316\) 0 0
\(317\) 10458.0i 1.85293i 0.376377 + 0.926467i \(0.377170\pi\)
−0.376377 + 0.926467i \(0.622830\pi\)
\(318\) 0 0
\(319\) − 6936.00i − 1.21737i
\(320\) 0 0
\(321\) −2196.00 −0.381834
\(322\) 0 0
\(323\) − 56.0000i − 0.00964682i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4542.00i 0.768114i
\(328\) 0 0
\(329\) −9472.00 −1.58726
\(330\) 0 0
\(331\) − 2348.00i − 0.389903i −0.980813 0.194951i \(-0.937545\pi\)
0.980813 0.194951i \(-0.0624549\pi\)
\(332\) 0 0
\(333\) − 3474.00i − 0.571694i
\(334\) 0 0
\(335\) 1192.00 0.194406
\(336\) 0 0
\(337\) −5298.00 −0.856381 −0.428191 0.903688i \(-0.640849\pi\)
−0.428191 + 0.903688i \(0.640849\pi\)
\(338\) 0 0
\(339\) −4554.00 −0.729615
\(340\) 0 0
\(341\) 9248.00 1.46864
\(342\) 0 0
\(343\) 10816.0i 1.70265i
\(344\) 0 0
\(345\) − 432.000i − 0.0674148i
\(346\) 0 0
\(347\) 9876.00 1.52787 0.763936 0.645292i \(-0.223264\pi\)
0.763936 + 0.645292i \(0.223264\pi\)
\(348\) 0 0
\(349\) − 5370.00i − 0.823638i −0.911266 0.411819i \(-0.864894\pi\)
0.911266 0.411819i \(-0.135106\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 7330.00i − 1.10520i −0.833446 0.552601i \(-0.813635\pi\)
0.833446 0.552601i \(-0.186365\pi\)
\(354\) 0 0
\(355\) −1760.00 −0.263130
\(356\) 0 0
\(357\) − 1344.00i − 0.199249i
\(358\) 0 0
\(359\) 7488.00i 1.10084i 0.834888 + 0.550420i \(0.185532\pi\)
−0.834888 + 0.550420i \(0.814468\pi\)
\(360\) 0 0
\(361\) 6843.00 0.997667
\(362\) 0 0
\(363\) −9879.00 −1.42841
\(364\) 0 0
\(365\) −1012.00 −0.145125
\(366\) 0 0
\(367\) 1504.00 0.213919 0.106959 0.994263i \(-0.465889\pi\)
0.106959 + 0.994263i \(0.465889\pi\)
\(368\) 0 0
\(369\) − 2250.00i − 0.317426i
\(370\) 0 0
\(371\) − 16832.0i − 2.35546i
\(372\) 0 0
\(373\) 6702.00 0.930339 0.465169 0.885222i \(-0.345993\pi\)
0.465169 + 0.885222i \(0.345993\pi\)
\(374\) 0 0
\(375\) 1476.00i 0.203254i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5700.00i 0.772531i 0.922388 + 0.386266i \(0.126235\pi\)
−0.922388 + 0.386266i \(0.873765\pi\)
\(380\) 0 0
\(381\) 288.000 0.0387262
\(382\) 0 0
\(383\) − 2328.00i − 0.310588i −0.987868 0.155294i \(-0.950367\pi\)
0.987868 0.155294i \(-0.0496325\pi\)
\(384\) 0 0
\(385\) − 4352.00i − 0.576100i
\(386\) 0 0
\(387\) 1260.00 0.165502
\(388\) 0 0
\(389\) 11554.0 1.50594 0.752971 0.658054i \(-0.228620\pi\)
0.752971 + 0.658054i \(0.228620\pi\)
\(390\) 0 0
\(391\) −1008.00 −0.130375
\(392\) 0 0
\(393\) −7644.00 −0.981142
\(394\) 0 0
\(395\) − 1280.00i − 0.163048i
\(396\) 0 0
\(397\) 6486.00i 0.819957i 0.912095 + 0.409979i \(0.134464\pi\)
−0.912095 + 0.409979i \(0.865536\pi\)
\(398\) 0 0
\(399\) −384.000 −0.0481806
\(400\) 0 0
\(401\) 7698.00i 0.958653i 0.877637 + 0.479326i \(0.159119\pi\)
−0.877637 + 0.479326i \(0.840881\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 162.000i 0.0198762i
\(406\) 0 0
\(407\) −26248.0 −3.19672
\(408\) 0 0
\(409\) − 3338.00i − 0.403554i −0.979432 0.201777i \(-0.935328\pi\)
0.979432 0.201777i \(-0.0646716\pi\)
\(410\) 0 0
\(411\) − 690.000i − 0.0828107i
\(412\) 0 0
\(413\) 10624.0 1.26579
\(414\) 0 0
\(415\) 2760.00 0.326465
\(416\) 0 0
\(417\) 1548.00 0.181789
\(418\) 0 0
\(419\) −52.0000 −0.00606293 −0.00303146 0.999995i \(-0.500965\pi\)
−0.00303146 + 0.999995i \(0.500965\pi\)
\(420\) 0 0
\(421\) 5858.00i 0.678151i 0.940759 + 0.339075i \(0.110114\pi\)
−0.940759 + 0.339075i \(0.889886\pi\)
\(422\) 0 0
\(423\) − 2664.00i − 0.306213i
\(424\) 0 0
\(425\) 1694.00 0.193344
\(426\) 0 0
\(427\) 13120.0i 1.48694i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 8840.00i − 0.987953i −0.869475 0.493977i \(-0.835543\pi\)
0.869475 0.493977i \(-0.164457\pi\)
\(432\) 0 0
\(433\) −11346.0 −1.25925 −0.629624 0.776900i \(-0.716791\pi\)
−0.629624 + 0.776900i \(0.716791\pi\)
\(434\) 0 0
\(435\) 612.000i 0.0674555i
\(436\) 0 0
\(437\) 288.000i 0.0315261i
\(438\) 0 0
\(439\) −16456.0 −1.78907 −0.894535 0.446997i \(-0.852493\pi\)
−0.894535 + 0.446997i \(0.852493\pi\)
\(440\) 0 0
\(441\) −6129.00 −0.661808
\(442\) 0 0
\(443\) −3788.00 −0.406260 −0.203130 0.979152i \(-0.565111\pi\)
−0.203130 + 0.979152i \(0.565111\pi\)
\(444\) 0 0
\(445\) −2900.00 −0.308929
\(446\) 0 0
\(447\) 5526.00i 0.584722i
\(448\) 0 0
\(449\) 546.000i 0.0573883i 0.999588 + 0.0286941i \(0.00913488\pi\)
−0.999588 + 0.0286941i \(0.990865\pi\)
\(450\) 0 0
\(451\) −17000.0 −1.77494
\(452\) 0 0
\(453\) − 1584.00i − 0.164289i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3546.00i − 0.362965i −0.983394 0.181482i \(-0.941910\pi\)
0.983394 0.181482i \(-0.0580895\pi\)
\(458\) 0 0
\(459\) 378.000 0.0384391
\(460\) 0 0
\(461\) − 12918.0i − 1.30510i −0.757746 0.652550i \(-0.773699\pi\)
0.757746 0.652550i \(-0.226301\pi\)
\(462\) 0 0
\(463\) 18328.0i 1.83969i 0.392287 + 0.919843i \(0.371684\pi\)
−0.392287 + 0.919843i \(0.628316\pi\)
\(464\) 0 0
\(465\) −816.000 −0.0813787
\(466\) 0 0
\(467\) −11980.0 −1.18708 −0.593542 0.804803i \(-0.702271\pi\)
−0.593542 + 0.804803i \(0.702271\pi\)
\(468\) 0 0
\(469\) −19072.0 −1.87775
\(470\) 0 0
\(471\) −3918.00 −0.383295
\(472\) 0 0
\(473\) − 9520.00i − 0.925434i
\(474\) 0 0
\(475\) − 484.000i − 0.0467525i
\(476\) 0 0
\(477\) 4734.00 0.454413
\(478\) 0 0
\(479\) − 12344.0i − 1.17748i −0.808323 0.588739i \(-0.799625\pi\)
0.808323 0.588739i \(-0.200375\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 6912.00i 0.651153i
\(484\) 0 0
\(485\) −892.000 −0.0835126
\(486\) 0 0
\(487\) 80.0000i 0.00744383i 0.999993 + 0.00372192i \(0.00118473\pi\)
−0.999993 + 0.00372192i \(0.998815\pi\)
\(488\) 0 0
\(489\) − 11316.0i − 1.04648i
\(490\) 0 0
\(491\) 15660.0 1.43936 0.719680 0.694306i \(-0.244288\pi\)
0.719680 + 0.694306i \(0.244288\pi\)
\(492\) 0 0
\(493\) 1428.00 0.130454
\(494\) 0 0
\(495\) 1224.00 0.111141
\(496\) 0 0
\(497\) 28160.0 2.54155
\(498\) 0 0
\(499\) 60.0000i 0.00538270i 0.999996 + 0.00269135i \(0.000856685\pi\)
−0.999996 + 0.00269135i \(0.999143\pi\)
\(500\) 0 0
\(501\) 1152.00i 0.102730i
\(502\) 0 0
\(503\) 12248.0 1.08571 0.542854 0.839827i \(-0.317344\pi\)
0.542854 + 0.839827i \(0.317344\pi\)
\(504\) 0 0
\(505\) 1220.00i 0.107504i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 90.0000i 0.00783729i 0.999992 + 0.00391864i \(0.00124735\pi\)
−0.999992 + 0.00391864i \(0.998753\pi\)
\(510\) 0 0
\(511\) 16192.0 1.40175
\(512\) 0 0
\(513\) − 108.000i − 0.00929496i
\(514\) 0 0
\(515\) 2704.00i 0.231364i
\(516\) 0 0
\(517\) −20128.0 −1.71224
\(518\) 0 0
\(519\) −4386.00 −0.370952
\(520\) 0 0
\(521\) 9818.00 0.825594 0.412797 0.910823i \(-0.364552\pi\)
0.412797 + 0.910823i \(0.364552\pi\)
\(522\) 0 0
\(523\) −20252.0 −1.69323 −0.846614 0.532208i \(-0.821363\pi\)
−0.846614 + 0.532208i \(0.821363\pi\)
\(524\) 0 0
\(525\) − 11616.0i − 0.965645i
\(526\) 0 0
\(527\) 1904.00i 0.157381i
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 2988.00i 0.244196i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 1464.00i − 0.118307i
\(536\) 0 0
\(537\) 3996.00 0.321118
\(538\) 0 0
\(539\) 46308.0i 3.70061i
\(540\) 0 0
\(541\) − 12634.0i − 1.00403i −0.864860 0.502013i \(-0.832593\pi\)
0.864860 0.502013i \(-0.167407\pi\)
\(542\) 0 0
\(543\) −6090.00 −0.481302
\(544\) 0 0
\(545\) −3028.00 −0.237991
\(546\) 0 0
\(547\) 11756.0 0.918922 0.459461 0.888198i \(-0.348043\pi\)
0.459461 + 0.888198i \(0.348043\pi\)
\(548\) 0 0
\(549\) −3690.00 −0.286859
\(550\) 0 0
\(551\) − 408.000i − 0.0315452i
\(552\) 0 0
\(553\) 20480.0i 1.57486i
\(554\) 0 0
\(555\) 2316.00 0.177133
\(556\) 0 0
\(557\) 17622.0i 1.34052i 0.742128 + 0.670259i \(0.233817\pi\)
−0.742128 + 0.670259i \(0.766183\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 2856.00i − 0.214938i
\(562\) 0 0
\(563\) 23092.0 1.72862 0.864309 0.502961i \(-0.167756\pi\)
0.864309 + 0.502961i \(0.167756\pi\)
\(564\) 0 0
\(565\) − 3036.00i − 0.226063i
\(566\) 0 0
\(567\) − 2592.00i − 0.191982i
\(568\) 0 0
\(569\) 1302.00 0.0959274 0.0479637 0.998849i \(-0.484727\pi\)
0.0479637 + 0.998849i \(0.484727\pi\)
\(570\) 0 0
\(571\) −24868.0 −1.82258 −0.911290 0.411765i \(-0.864913\pi\)
−0.911290 + 0.411765i \(0.864913\pi\)
\(572\) 0 0
\(573\) −48.0000 −0.00349953
\(574\) 0 0
\(575\) −8712.00 −0.631853
\(576\) 0 0
\(577\) − 2562.00i − 0.184848i −0.995720 0.0924241i \(-0.970538\pi\)
0.995720 0.0924241i \(-0.0294616\pi\)
\(578\) 0 0
\(579\) − 6234.00i − 0.447455i
\(580\) 0 0
\(581\) −44160.0 −3.15330
\(582\) 0 0
\(583\) − 35768.0i − 2.54092i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 13484.0i − 0.948116i −0.880494 0.474058i \(-0.842789\pi\)
0.880494 0.474058i \(-0.157211\pi\)
\(588\) 0 0
\(589\) 544.000 0.0380562
\(590\) 0 0
\(591\) − 10458.0i − 0.727893i
\(592\) 0 0
\(593\) − 16974.0i − 1.17544i −0.809063 0.587722i \(-0.800025\pi\)
0.809063 0.587722i \(-0.199975\pi\)
\(594\) 0 0
\(595\) 896.000 0.0617352
\(596\) 0 0
\(597\) −1704.00 −0.116818
\(598\) 0 0
\(599\) 3864.00 0.263571 0.131785 0.991278i \(-0.457929\pi\)
0.131785 + 0.991278i \(0.457929\pi\)
\(600\) 0 0
\(601\) 17546.0 1.19088 0.595438 0.803401i \(-0.296979\pi\)
0.595438 + 0.803401i \(0.296979\pi\)
\(602\) 0 0
\(603\) − 5364.00i − 0.362254i
\(604\) 0 0
\(605\) − 6586.00i − 0.442577i
\(606\) 0 0
\(607\) 9296.00 0.621603 0.310801 0.950475i \(-0.399403\pi\)
0.310801 + 0.950475i \(0.399403\pi\)
\(608\) 0 0
\(609\) − 9792.00i − 0.651547i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 6914.00i 0.455553i 0.973713 + 0.227776i \(0.0731455\pi\)
−0.973713 + 0.227776i \(0.926854\pi\)
\(614\) 0 0
\(615\) 1500.00 0.0983510
\(616\) 0 0
\(617\) 25446.0i 1.66032i 0.557525 + 0.830160i \(0.311751\pi\)
−0.557525 + 0.830160i \(0.688249\pi\)
\(618\) 0 0
\(619\) − 11236.0i − 0.729585i −0.931089 0.364792i \(-0.881140\pi\)
0.931089 0.364792i \(-0.118860\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) 46400.0 2.98391
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) 0 0
\(627\) −816.000 −0.0519743
\(628\) 0 0
\(629\) − 5404.00i − 0.342562i
\(630\) 0 0
\(631\) − 29424.0i − 1.85634i −0.372156 0.928170i \(-0.621381\pi\)
0.372156 0.928170i \(-0.378619\pi\)
\(632\) 0 0
\(633\) 11412.0 0.716566
\(634\) 0 0
\(635\) 192.000i 0.0119989i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 7920.00i 0.490314i
\(640\) 0 0
\(641\) 5054.00 0.311421 0.155711 0.987803i \(-0.450233\pi\)
0.155711 + 0.987803i \(0.450233\pi\)
\(642\) 0 0
\(643\) 1132.00i 0.0694273i 0.999397 + 0.0347136i \(0.0110519\pi\)
−0.999397 + 0.0347136i \(0.988948\pi\)
\(644\) 0 0
\(645\) 840.000i 0.0512790i
\(646\) 0 0
\(647\) −1464.00 −0.0889579 −0.0444790 0.999010i \(-0.514163\pi\)
−0.0444790 + 0.999010i \(0.514163\pi\)
\(648\) 0 0
\(649\) 22576.0 1.36546
\(650\) 0 0
\(651\) 13056.0 0.786029
\(652\) 0 0
\(653\) 5494.00 0.329245 0.164622 0.986357i \(-0.447359\pi\)
0.164622 + 0.986357i \(0.447359\pi\)
\(654\) 0 0
\(655\) − 5096.00i − 0.303996i
\(656\) 0 0
\(657\) 4554.00i 0.270424i
\(658\) 0 0
\(659\) 11580.0 0.684511 0.342256 0.939607i \(-0.388809\pi\)
0.342256 + 0.939607i \(0.388809\pi\)
\(660\) 0 0
\(661\) − 1298.00i − 0.0763787i −0.999271 0.0381894i \(-0.987841\pi\)
0.999271 0.0381894i \(-0.0121590\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 256.000i − 0.0149282i
\(666\) 0 0
\(667\) −7344.00 −0.426328
\(668\) 0 0
\(669\) − 17736.0i − 1.02498i
\(670\) 0 0
\(671\) 27880.0i 1.60402i
\(672\) 0 0
\(673\) −16162.0 −0.925705 −0.462852 0.886435i \(-0.653174\pi\)
−0.462852 + 0.886435i \(0.653174\pi\)
\(674\) 0 0
\(675\) 3267.00 0.186292
\(676\) 0 0
\(677\) −9890.00 −0.561453 −0.280726 0.959788i \(-0.590575\pi\)
−0.280726 + 0.959788i \(0.590575\pi\)
\(678\) 0 0
\(679\) 14272.0 0.806641
\(680\) 0 0
\(681\) 9924.00i 0.558427i
\(682\) 0 0
\(683\) 27100.0i 1.51823i 0.650955 + 0.759116i \(0.274369\pi\)
−0.650955 + 0.759116i \(0.725631\pi\)
\(684\) 0 0
\(685\) 460.000 0.0256580
\(686\) 0 0
\(687\) − 18150.0i − 1.00796i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 11132.0i 0.612853i 0.951894 + 0.306426i \(0.0991333\pi\)
−0.951894 + 0.306426i \(0.900867\pi\)
\(692\) 0 0
\(693\) −19584.0 −1.07350
\(694\) 0 0
\(695\) 1032.00i 0.0563252i
\(696\) 0 0
\(697\) − 3500.00i − 0.190204i
\(698\) 0 0
\(699\) −14382.0 −0.778222
\(700\) 0 0
\(701\) −1862.00 −0.100323 −0.0501617 0.998741i \(-0.515974\pi\)
−0.0501617 + 0.998741i \(0.515974\pi\)
\(702\) 0 0
\(703\) −1544.00 −0.0828351
\(704\) 0 0
\(705\) 1776.00 0.0948766
\(706\) 0 0
\(707\) − 19520.0i − 1.03837i
\(708\) 0 0
\(709\) − 7938.00i − 0.420477i −0.977650 0.210238i \(-0.932576\pi\)
0.977650 0.210238i \(-0.0674240\pi\)
\(710\) 0 0
\(711\) −5760.00 −0.303821
\(712\) 0 0
\(713\) − 9792.00i − 0.514324i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13320.0i 0.693786i
\(718\) 0 0
\(719\) 24240.0 1.25730 0.628651 0.777688i \(-0.283608\pi\)
0.628651 + 0.777688i \(0.283608\pi\)
\(720\) 0 0
\(721\) − 43264.0i − 2.23472i
\(722\) 0 0
\(723\) 3990.00i 0.205242i
\(724\) 0 0
\(725\) 12342.0 0.632235
\(726\) 0 0
\(727\) 13720.0 0.699927 0.349963 0.936763i \(-0.386194\pi\)
0.349963 + 0.936763i \(0.386194\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1960.00 0.0991699
\(732\) 0 0
\(733\) − 21958.0i − 1.10646i −0.833028 0.553231i \(-0.813395\pi\)
0.833028 0.553231i \(-0.186605\pi\)
\(734\) 0 0
\(735\) − 4086.00i − 0.205054i
\(736\) 0 0
\(737\) −40528.0 −2.02560
\(738\) 0 0
\(739\) 13348.0i 0.664430i 0.943204 + 0.332215i \(0.107796\pi\)
−0.943204 + 0.332215i \(0.892204\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 20304.0i − 1.00253i −0.865293 0.501266i \(-0.832868\pi\)
0.865293 0.501266i \(-0.167132\pi\)
\(744\) 0 0
\(745\) −3684.00 −0.181170
\(746\) 0 0
\(747\) − 12420.0i − 0.608332i
\(748\) 0 0
\(749\) 23424.0i 1.14272i
\(750\) 0 0
\(751\) 3952.00 0.192025 0.0960123 0.995380i \(-0.469391\pi\)
0.0960123 + 0.995380i \(0.469391\pi\)
\(752\) 0 0
\(753\) 15348.0 0.742779
\(754\) 0 0
\(755\) 1056.00 0.0509030
\(756\) 0 0
\(757\) −22386.0 −1.07481 −0.537406 0.843324i \(-0.680596\pi\)
−0.537406 + 0.843324i \(0.680596\pi\)
\(758\) 0 0
\(759\) 14688.0i 0.702425i
\(760\) 0 0
\(761\) 12458.0i 0.593433i 0.954966 + 0.296716i \(0.0958916\pi\)
−0.954966 + 0.296716i \(0.904108\pi\)
\(762\) 0 0
\(763\) 48448.0 2.29874
\(764\) 0 0
\(765\) 252.000i 0.0119099i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 28126.0i 1.31892i 0.751740 + 0.659460i \(0.229215\pi\)
−0.751740 + 0.659460i \(0.770785\pi\)
\(770\) 0 0
\(771\) −1926.00 −0.0899652
\(772\) 0 0
\(773\) 35778.0i 1.66474i 0.554219 + 0.832371i \(0.313017\pi\)
−0.554219 + 0.832371i \(0.686983\pi\)
\(774\) 0 0
\(775\) 16456.0i 0.762732i
\(776\) 0 0
\(777\) −37056.0 −1.71091
\(778\) 0 0
\(779\) −1000.00 −0.0459932
\(780\) 0 0
\(781\) 59840.0 2.74167
\(782\) 0 0
\(783\) 2754.00 0.125696
\(784\) 0 0
\(785\) − 2612.00i − 0.118760i
\(786\) 0 0
\(787\) 7252.00i 0.328470i 0.986421 + 0.164235i \(0.0525155\pi\)
−0.986421 + 0.164235i \(0.947484\pi\)
\(788\) 0 0
\(789\) 12792.0 0.577196
\(790\) 0 0
\(791\) 48576.0i 2.18352i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 3156.00i 0.140795i
\(796\) 0 0
\(797\) −4454.00 −0.197953 −0.0989766 0.995090i \(-0.531557\pi\)
−0.0989766 + 0.995090i \(0.531557\pi\)
\(798\) 0 0
\(799\) − 4144.00i − 0.183485i
\(800\) 0 0
\(801\) 13050.0i 0.575654i
\(802\) 0 0
\(803\) 34408.0 1.51212
\(804\) 0 0
\(805\) −4608.00 −0.201752
\(806\) 0 0
\(807\) −11550.0 −0.503816
\(808\) 0 0
\(809\) −34118.0 −1.48273 −0.741363 0.671104i \(-0.765820\pi\)
−0.741363 + 0.671104i \(0.765820\pi\)
\(810\) 0 0
\(811\) − 16428.0i − 0.711301i −0.934619 0.355650i \(-0.884259\pi\)
0.934619 0.355650i \(-0.115741\pi\)
\(812\) 0 0
\(813\) 8808.00i 0.379963i
\(814\) 0 0
\(815\) 7544.00 0.324239
\(816\) 0 0
\(817\) − 560.000i − 0.0239803i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18738.0i 0.796542i 0.917268 + 0.398271i \(0.130390\pi\)
−0.917268 + 0.398271i \(0.869610\pi\)
\(822\) 0 0
\(823\) −13928.0 −0.589914 −0.294957 0.955510i \(-0.595305\pi\)
−0.294957 + 0.955510i \(0.595305\pi\)
\(824\) 0 0
\(825\) − 24684.0i − 1.04168i
\(826\) 0 0
\(827\) 41804.0i 1.75776i 0.477043 + 0.878880i \(0.341709\pi\)
−0.477043 + 0.878880i \(0.658291\pi\)
\(828\) 0 0
\(829\) 43226.0 1.81098 0.905489 0.424369i \(-0.139504\pi\)
0.905489 + 0.424369i \(0.139504\pi\)
\(830\) 0 0
\(831\) 6198.00 0.258732
\(832\) 0 0
\(833\) −9534.00 −0.396559
\(834\) 0 0
\(835\) −768.000 −0.0318296
\(836\) 0 0
\(837\) 3672.00i 0.151640i
\(838\) 0 0
\(839\) 10240.0i 0.421364i 0.977555 + 0.210682i \(0.0675684\pi\)
−0.977555 + 0.210682i \(0.932432\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) 0 0
\(843\) − 642.000i − 0.0262297i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 105376.i 4.27481i
\(848\) 0 0
\(849\) 7860.00 0.317732
\(850\) 0 0
\(851\) 27792.0i 1.11950i
\(852\) 0 0
\(853\) − 25682.0i − 1.03087i −0.856928 0.515437i \(-0.827630\pi\)
0.856928 0.515437i \(-0.172370\pi\)
\(854\) 0 0
\(855\) 72.0000 0.00287994
\(856\) 0 0
\(857\) 21558.0 0.859285 0.429643 0.902999i \(-0.358640\pi\)
0.429643 + 0.902999i \(0.358640\pi\)
\(858\) 0 0
\(859\) −14060.0 −0.558465 −0.279232 0.960224i \(-0.590080\pi\)
−0.279232 + 0.960224i \(0.590080\pi\)
\(860\) 0 0
\(861\) −24000.0 −0.949963
\(862\) 0 0
\(863\) − 42008.0i − 1.65697i −0.560008 0.828487i \(-0.689202\pi\)
0.560008 0.828487i \(-0.310798\pi\)
\(864\) 0 0
\(865\) − 2924.00i − 0.114935i
\(866\) 0 0
\(867\) −14151.0 −0.554317
\(868\) 0 0
\(869\) 43520.0i 1.69887i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 4014.00i 0.155617i
\(874\) 0 0
\(875\) 15744.0 0.608279
\(876\) 0 0
\(877\) − 23734.0i − 0.913843i −0.889507 0.456921i \(-0.848952\pi\)
0.889507 0.456921i \(-0.151048\pi\)
\(878\) 0 0
\(879\) − 3462.00i − 0.132845i
\(880\) 0 0
\(881\) 37550.0 1.43597 0.717986 0.696057i \(-0.245064\pi\)
0.717986 + 0.696057i \(0.245064\pi\)
\(882\) 0 0
\(883\) −12556.0 −0.478531 −0.239266 0.970954i \(-0.576907\pi\)
−0.239266 + 0.970954i \(0.576907\pi\)
\(884\) 0 0
\(885\) −1992.00 −0.0756614
\(886\) 0 0
\(887\) 37368.0 1.41454 0.707269 0.706945i \(-0.249927\pi\)
0.707269 + 0.706945i \(0.249927\pi\)
\(888\) 0 0
\(889\) − 3072.00i − 0.115896i
\(890\) 0 0
\(891\) − 5508.00i − 0.207099i
\(892\) 0 0
\(893\) −1184.00 −0.0443685
\(894\) 0 0
\(895\) 2664.00i 0.0994946i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13872.0i 0.514635i
\(900\) 0 0
\(901\) 7364.00 0.272287
\(902\) 0 0
\(903\) − 13440.0i − 0.495299i
\(904\) 0 0
\(905\) − 4060.00i − 0.149126i
\(906\) 0 0
\(907\) −33364.0 −1.22143 −0.610713 0.791852i \(-0.709117\pi\)
−0.610713 + 0.791852i \(0.709117\pi\)
\(908\) 0 0
\(909\) 5490.00 0.200321
\(910\) 0 0
\(911\) 50432.0 1.83412 0.917062 0.398745i \(-0.130554\pi\)
0.917062 + 0.398745i \(0.130554\pi\)
\(912\) 0 0
\(913\) −93840.0 −3.40159
\(914\) 0 0
\(915\) − 2460.00i − 0.0888799i
\(916\) 0 0
\(917\) 81536.0i 2.93627i
\(918\) 0 0
\(919\) 11864.0 0.425851 0.212926 0.977068i \(-0.431701\pi\)
0.212926 + 0.977068i \(0.431701\pi\)
\(920\) 0 0
\(921\) − 12228.0i − 0.437488i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 46706.0i − 1.66020i
\(926\) 0 0
\(927\) 12168.0 0.431121
\(928\) 0 0
\(929\) 5950.00i 0.210133i 0.994465 + 0.105066i \(0.0335055\pi\)
−0.994465 + 0.105066i \(0.966495\pi\)
\(930\) 0 0
\(931\) 2724.00i 0.0958920i
\(932\) 0 0
\(933\) 19368.0 0.679614
\(934\) 0 0
\(935\) 1904.00 0.0665962
\(936\) 0 0
\(937\) −20806.0 −0.725403 −0.362701 0.931905i \(-0.618146\pi\)
−0.362701 + 0.931905i \(0.618146\pi\)
\(938\) 0 0
\(939\) −16578.0 −0.576148
\(940\) 0 0
\(941\) 22346.0i 0.774133i 0.922052 + 0.387066i \(0.126512\pi\)
−0.922052 + 0.387066i \(0.873488\pi\)
\(942\) 0 0
\(943\) 18000.0i 0.621591i
\(944\) 0 0
\(945\) 1728.00 0.0594834
\(946\) 0 0
\(947\) 31636.0i 1.08557i 0.839873 + 0.542783i \(0.182630\pi\)
−0.839873 + 0.542783i \(0.817370\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 31374.0i 1.06979i
\(952\) 0 0
\(953\) 23190.0 0.788245 0.394123 0.919058i \(-0.371049\pi\)
0.394123 + 0.919058i \(0.371049\pi\)
\(954\) 0 0
\(955\) − 32.0000i − 0.00108429i
\(956\) 0 0
\(957\) − 20808.0i − 0.702850i
\(958\) 0 0
\(959\) −7360.00 −0.247828
\(960\) 0 0
\(961\) 11295.0 0.379141
\(962\) 0 0
\(963\) −6588.00 −0.220452
\(964\) 0 0
\(965\) 4156.00 0.138639
\(966\) 0 0
\(967\) − 304.000i − 0.0101096i −0.999987 0.00505480i \(-0.998391\pi\)
0.999987 0.00505480i \(-0.00160900\pi\)
\(968\) 0 0
\(969\) − 168.000i − 0.00556960i
\(970\) 0 0
\(971\) −53372.0 −1.76394 −0.881972 0.471302i \(-0.843784\pi\)
−0.881972 + 0.471302i \(0.843784\pi\)
\(972\) 0 0
\(973\) − 16512.0i − 0.544039i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 13650.0i − 0.446983i −0.974706 0.223491i \(-0.928255\pi\)
0.974706 0.223491i \(-0.0717455\pi\)
\(978\) 0 0
\(979\) 98600.0 3.21887
\(980\) 0 0
\(981\) 13626.0i 0.443471i
\(982\) 0 0
\(983\) 34992.0i 1.13537i 0.823245 + 0.567686i \(0.192161\pi\)
−0.823245 + 0.567686i \(0.807839\pi\)
\(984\) 0 0
\(985\) 6972.00 0.225529
\(986\) 0 0
\(987\) −28416.0 −0.916405
\(988\) 0 0
\(989\) −10080.0 −0.324090
\(990\) 0 0
\(991\) 18096.0 0.580059 0.290029 0.957018i \(-0.406335\pi\)
0.290029 + 0.957018i \(0.406335\pi\)
\(992\) 0 0
\(993\) − 7044.00i − 0.225110i
\(994\) 0 0
\(995\) − 1136.00i − 0.0361946i
\(996\) 0 0
\(997\) −18914.0 −0.600815 −0.300407 0.953811i \(-0.597123\pi\)
−0.300407 + 0.953811i \(0.597123\pi\)
\(998\) 0 0
\(999\) − 10422.0i − 0.330068i
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 2028.4.b.d.337.2 2
13.5 odd 4 2028.4.a.b.1.1 1
13.8 odd 4 156.4.a.b.1.1 1
13.12 even 2 inner 2028.4.b.d.337.1 2
39.8 even 4 468.4.a.a.1.1 1
52.47 even 4 624.4.a.b.1.1 1
104.21 odd 4 2496.4.a.d.1.1 1
104.99 even 4 2496.4.a.m.1.1 1
156.47 odd 4 1872.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.b.1.1 1 13.8 odd 4
468.4.a.a.1.1 1 39.8 even 4
624.4.a.b.1.1 1 52.47 even 4
1872.4.a.i.1.1 1 156.47 odd 4
2028.4.a.b.1.1 1 13.5 odd 4
2028.4.b.d.337.1 2 13.12 even 2 inner
2028.4.b.d.337.2 2 1.1 even 1 trivial
2496.4.a.d.1.1 1 104.21 odd 4
2496.4.a.m.1.1 1 104.99 even 4