Properties

Label 400.4.a.k.1.1
Level $400$
Weight $4$
Character 400.1
Self dual yes
Analytic conductor $23.601$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, 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("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -6.00000 q^{7} -26.0000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -6.00000 q^{7} -26.0000 q^{9} +19.0000 q^{11} -12.0000 q^{13} +75.0000 q^{17} +91.0000 q^{19} -6.00000 q^{21} +174.000 q^{23} -53.0000 q^{27} -272.000 q^{29} +230.000 q^{31} +19.0000 q^{33} +182.000 q^{37} -12.0000 q^{39} +117.000 q^{41} +372.000 q^{43} -52.0000 q^{47} -307.000 q^{49} +75.0000 q^{51} +402.000 q^{53} +91.0000 q^{57} -312.000 q^{59} +170.000 q^{61} +156.000 q^{63} +763.000 q^{67} +174.000 q^{69} +52.0000 q^{71} +981.000 q^{73} -114.000 q^{77} -1054.00 q^{79} +649.000 q^{81} +351.000 q^{83} -272.000 q^{87} +799.000 q^{89} +72.0000 q^{91} +230.000 q^{93} -962.000 q^{97} -494.000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.192450 0.0962250 0.995360i \(-0.469323\pi\)
0.0962250 + 0.995360i \(0.469323\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −6.00000 −0.323970 −0.161985 0.986793i \(-0.551790\pi\)
−0.161985 + 0.986793i \(0.551790\pi\)
\(8\) 0 0
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) 19.0000 0.520792 0.260396 0.965502i \(-0.416147\pi\)
0.260396 + 0.965502i \(0.416147\pi\)
\(12\) 0 0
\(13\) −12.0000 −0.256015 −0.128008 0.991773i \(-0.540858\pi\)
−0.128008 + 0.991773i \(0.540858\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 75.0000 1.07001 0.535005 0.844849i \(-0.320310\pi\)
0.535005 + 0.844849i \(0.320310\pi\)
\(18\) 0 0
\(19\) 91.0000 1.09878 0.549390 0.835566i \(-0.314860\pi\)
0.549390 + 0.835566i \(0.314860\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.0623480
\(22\) 0 0
\(23\) 174.000 1.57746 0.788728 0.614742i \(-0.210740\pi\)
0.788728 + 0.614742i \(0.210740\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −53.0000 −0.377772
\(28\) 0 0
\(29\) −272.000 −1.74169 −0.870847 0.491554i \(-0.836429\pi\)
−0.870847 + 0.491554i \(0.836429\pi\)
\(30\) 0 0
\(31\) 230.000 1.33256 0.666278 0.745704i \(-0.267887\pi\)
0.666278 + 0.745704i \(0.267887\pi\)
\(32\) 0 0
\(33\) 19.0000 0.100227
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 182.000 0.808665 0.404333 0.914612i \(-0.367504\pi\)
0.404333 + 0.914612i \(0.367504\pi\)
\(38\) 0 0
\(39\) −12.0000 −0.0492702
\(40\) 0 0
\(41\) 117.000 0.445667 0.222833 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(42\) 0 0
\(43\) 372.000 1.31929 0.659645 0.751577i \(-0.270707\pi\)
0.659645 + 0.751577i \(0.270707\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −52.0000 −0.161383 −0.0806913 0.996739i \(-0.525713\pi\)
−0.0806913 + 0.996739i \(0.525713\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) 75.0000 0.205924
\(52\) 0 0
\(53\) 402.000 1.04187 0.520933 0.853597i \(-0.325584\pi\)
0.520933 + 0.853597i \(0.325584\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 91.0000 0.211460
\(58\) 0 0
\(59\) −312.000 −0.688457 −0.344228 0.938886i \(-0.611859\pi\)
−0.344228 + 0.938886i \(0.611859\pi\)
\(60\) 0 0
\(61\) 170.000 0.356824 0.178412 0.983956i \(-0.442904\pi\)
0.178412 + 0.983956i \(0.442904\pi\)
\(62\) 0 0
\(63\) 156.000 0.311971
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 763.000 1.39127 0.695636 0.718394i \(-0.255122\pi\)
0.695636 + 0.718394i \(0.255122\pi\)
\(68\) 0 0
\(69\) 174.000 0.303582
\(70\) 0 0
\(71\) 52.0000 0.0869192 0.0434596 0.999055i \(-0.486162\pi\)
0.0434596 + 0.999055i \(0.486162\pi\)
\(72\) 0 0
\(73\) 981.000 1.57284 0.786420 0.617692i \(-0.211932\pi\)
0.786420 + 0.617692i \(0.211932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −114.000 −0.168721
\(78\) 0 0
\(79\) −1054.00 −1.50107 −0.750533 0.660833i \(-0.770203\pi\)
−0.750533 + 0.660833i \(0.770203\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) 351.000 0.464184 0.232092 0.972694i \(-0.425443\pi\)
0.232092 + 0.972694i \(0.425443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −272.000 −0.335189
\(88\) 0 0
\(89\) 799.000 0.951616 0.475808 0.879549i \(-0.342156\pi\)
0.475808 + 0.879549i \(0.342156\pi\)
\(90\) 0 0
\(91\) 72.0000 0.0829412
\(92\) 0 0
\(93\) 230.000 0.256450
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −962.000 −1.00697 −0.503486 0.864003i \(-0.667949\pi\)
−0.503486 + 0.864003i \(0.667949\pi\)
\(98\) 0 0
\(99\) −494.000 −0.501504
\(100\) 0 0
\(101\) 486.000 0.478800 0.239400 0.970921i \(-0.423049\pi\)
0.239400 + 0.970921i \(0.423049\pi\)
\(102\) 0 0
\(103\) −1188.00 −1.13648 −0.568238 0.822864i \(-0.692375\pi\)
−0.568238 + 0.822864i \(0.692375\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1325.00 1.19713 0.598563 0.801075i \(-0.295738\pi\)
0.598563 + 0.801075i \(0.295738\pi\)
\(108\) 0 0
\(109\) 126.000 0.110721 0.0553606 0.998466i \(-0.482369\pi\)
0.0553606 + 0.998466i \(0.482369\pi\)
\(110\) 0 0
\(111\) 182.000 0.155628
\(112\) 0 0
\(113\) −183.000 −0.152347 −0.0761734 0.997095i \(-0.524270\pi\)
−0.0761734 + 0.997095i \(0.524270\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 312.000 0.246533
\(118\) 0 0
\(119\) −450.000 −0.346651
\(120\) 0 0
\(121\) −970.000 −0.728775
\(122\) 0 0
\(123\) 117.000 0.0857686
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −902.000 −0.630233 −0.315116 0.949053i \(-0.602044\pi\)
−0.315116 + 0.949053i \(0.602044\pi\)
\(128\) 0 0
\(129\) 372.000 0.253897
\(130\) 0 0
\(131\) −2068.00 −1.37925 −0.689626 0.724166i \(-0.742225\pi\)
−0.689626 + 0.724166i \(0.742225\pi\)
\(132\) 0 0
\(133\) −546.000 −0.355971
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1339.00 0.835025 0.417513 0.908671i \(-0.362902\pi\)
0.417513 + 0.908671i \(0.362902\pi\)
\(138\) 0 0
\(139\) 2939.00 1.79340 0.896700 0.442638i \(-0.145957\pi\)
0.896700 + 0.442638i \(0.145957\pi\)
\(140\) 0 0
\(141\) −52.0000 −0.0310581
\(142\) 0 0
\(143\) −228.000 −0.133331
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −307.000 −0.172251
\(148\) 0 0
\(149\) −208.000 −0.114363 −0.0571813 0.998364i \(-0.518211\pi\)
−0.0571813 + 0.998364i \(0.518211\pi\)
\(150\) 0 0
\(151\) 2678.00 1.44326 0.721631 0.692278i \(-0.243393\pi\)
0.721631 + 0.692278i \(0.243393\pi\)
\(152\) 0 0
\(153\) −1950.00 −1.03038
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1482.00 −0.753353 −0.376677 0.926345i \(-0.622933\pi\)
−0.376677 + 0.926345i \(0.622933\pi\)
\(158\) 0 0
\(159\) 402.000 0.200507
\(160\) 0 0
\(161\) −1044.00 −0.511048
\(162\) 0 0
\(163\) −1469.00 −0.705895 −0.352948 0.935643i \(-0.614821\pi\)
−0.352948 + 0.935643i \(0.614821\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4004.00 −1.85532 −0.927661 0.373423i \(-0.878184\pi\)
−0.927661 + 0.373423i \(0.878184\pi\)
\(168\) 0 0
\(169\) −2053.00 −0.934456
\(170\) 0 0
\(171\) −2366.00 −1.05809
\(172\) 0 0
\(173\) −3224.00 −1.41686 −0.708428 0.705783i \(-0.750595\pi\)
−0.708428 + 0.705783i \(0.750595\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −312.000 −0.132494
\(178\) 0 0
\(179\) 4191.00 1.75000 0.875000 0.484123i \(-0.160861\pi\)
0.875000 + 0.484123i \(0.160861\pi\)
\(180\) 0 0
\(181\) −3718.00 −1.52683 −0.763416 0.645907i \(-0.776480\pi\)
−0.763416 + 0.645907i \(0.776480\pi\)
\(182\) 0 0
\(183\) 170.000 0.0686708
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1425.00 0.557253
\(188\) 0 0
\(189\) 318.000 0.122387
\(190\) 0 0
\(191\) −870.000 −0.329586 −0.164793 0.986328i \(-0.552696\pi\)
−0.164793 + 0.986328i \(0.552696\pi\)
\(192\) 0 0
\(193\) −2197.00 −0.819396 −0.409698 0.912221i \(-0.634366\pi\)
−0.409698 + 0.912221i \(0.634366\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2314.00 0.836882 0.418441 0.908244i \(-0.362577\pi\)
0.418441 + 0.908244i \(0.362577\pi\)
\(198\) 0 0
\(199\) 252.000 0.0897679 0.0448839 0.998992i \(-0.485708\pi\)
0.0448839 + 0.998992i \(0.485708\pi\)
\(200\) 0 0
\(201\) 763.000 0.267751
\(202\) 0 0
\(203\) 1632.00 0.564256
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4524.00 −1.51903
\(208\) 0 0
\(209\) 1729.00 0.572237
\(210\) 0 0
\(211\) 741.000 0.241766 0.120883 0.992667i \(-0.461427\pi\)
0.120883 + 0.992667i \(0.461427\pi\)
\(212\) 0 0
\(213\) 52.0000 0.0167276
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1380.00 −0.431707
\(218\) 0 0
\(219\) 981.000 0.302693
\(220\) 0 0
\(221\) −900.000 −0.273939
\(222\) 0 0
\(223\) −5092.00 −1.52908 −0.764542 0.644574i \(-0.777035\pi\)
−0.764542 + 0.644574i \(0.777035\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5876.00 1.71808 0.859039 0.511910i \(-0.171062\pi\)
0.859039 + 0.511910i \(0.171062\pi\)
\(228\) 0 0
\(229\) −604.000 −0.174295 −0.0871473 0.996195i \(-0.527775\pi\)
−0.0871473 + 0.996195i \(0.527775\pi\)
\(230\) 0 0
\(231\) −114.000 −0.0324703
\(232\) 0 0
\(233\) −278.000 −0.0781647 −0.0390824 0.999236i \(-0.512443\pi\)
−0.0390824 + 0.999236i \(0.512443\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1054.00 −0.288880
\(238\) 0 0
\(239\) −2496.00 −0.675535 −0.337767 0.941230i \(-0.609672\pi\)
−0.337767 + 0.941230i \(0.609672\pi\)
\(240\) 0 0
\(241\) −2567.00 −0.686120 −0.343060 0.939313i \(-0.611463\pi\)
−0.343060 + 0.939313i \(0.611463\pi\)
\(242\) 0 0
\(243\) 2080.00 0.549103
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1092.00 −0.281305
\(248\) 0 0
\(249\) 351.000 0.0893322
\(250\) 0 0
\(251\) −5395.00 −1.35669 −0.678345 0.734743i \(-0.737303\pi\)
−0.678345 + 0.734743i \(0.737303\pi\)
\(252\) 0 0
\(253\) 3306.00 0.821527
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1490.00 0.361648 0.180824 0.983515i \(-0.442123\pi\)
0.180824 + 0.983515i \(0.442123\pi\)
\(258\) 0 0
\(259\) −1092.00 −0.261983
\(260\) 0 0
\(261\) 7072.00 1.67719
\(262\) 0 0
\(263\) −3330.00 −0.780748 −0.390374 0.920656i \(-0.627654\pi\)
−0.390374 + 0.920656i \(0.627654\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 799.000 0.183139
\(268\) 0 0
\(269\) 6096.00 1.38171 0.690854 0.722994i \(-0.257235\pi\)
0.690854 + 0.722994i \(0.257235\pi\)
\(270\) 0 0
\(271\) 6006.00 1.34627 0.673134 0.739521i \(-0.264948\pi\)
0.673134 + 0.739521i \(0.264948\pi\)
\(272\) 0 0
\(273\) 72.0000 0.0159620
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6976.00 −1.51317 −0.756583 0.653897i \(-0.773133\pi\)
−0.756583 + 0.653897i \(0.773133\pi\)
\(278\) 0 0
\(279\) −5980.00 −1.28320
\(280\) 0 0
\(281\) 3998.00 0.848757 0.424378 0.905485i \(-0.360493\pi\)
0.424378 + 0.905485i \(0.360493\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.00273064 −0.00136532 0.999999i \(-0.500435\pi\)
−0.00136532 + 0.999999i \(0.500435\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −702.000 −0.144382
\(288\) 0 0
\(289\) 712.000 0.144922
\(290\) 0 0
\(291\) −962.000 −0.193792
\(292\) 0 0
\(293\) 4466.00 0.890466 0.445233 0.895415i \(-0.353121\pi\)
0.445233 + 0.895415i \(0.353121\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1007.00 −0.196741
\(298\) 0 0
\(299\) −2088.00 −0.403853
\(300\) 0 0
\(301\) −2232.00 −0.427410
\(302\) 0 0
\(303\) 486.000 0.0921451
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9041.00 1.68077 0.840386 0.541988i \(-0.182328\pi\)
0.840386 + 0.541988i \(0.182328\pi\)
\(308\) 0 0
\(309\) −1188.00 −0.218715
\(310\) 0 0
\(311\) 346.000 0.0630864 0.0315432 0.999502i \(-0.489958\pi\)
0.0315432 + 0.999502i \(0.489958\pi\)
\(312\) 0 0
\(313\) 10646.0 1.92252 0.961258 0.275650i \(-0.0888932\pi\)
0.961258 + 0.275650i \(0.0888932\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8116.00 1.43798 0.718990 0.695020i \(-0.244604\pi\)
0.718990 + 0.695020i \(0.244604\pi\)
\(318\) 0 0
\(319\) −5168.00 −0.907061
\(320\) 0 0
\(321\) 1325.00 0.230387
\(322\) 0 0
\(323\) 6825.00 1.17571
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 126.000 0.0213083
\(328\) 0 0
\(329\) 312.000 0.0522830
\(330\) 0 0
\(331\) 3007.00 0.499334 0.249667 0.968332i \(-0.419679\pi\)
0.249667 + 0.968332i \(0.419679\pi\)
\(332\) 0 0
\(333\) −4732.00 −0.778715
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −83.0000 −0.0134163 −0.00670816 0.999978i \(-0.502135\pi\)
−0.00670816 + 0.999978i \(0.502135\pi\)
\(338\) 0 0
\(339\) −183.000 −0.0293192
\(340\) 0 0
\(341\) 4370.00 0.693985
\(342\) 0 0
\(343\) 3900.00 0.613936
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6189.00 0.957472 0.478736 0.877959i \(-0.341095\pi\)
0.478736 + 0.877959i \(0.341095\pi\)
\(348\) 0 0
\(349\) −5362.00 −0.822411 −0.411205 0.911543i \(-0.634892\pi\)
−0.411205 + 0.911543i \(0.634892\pi\)
\(350\) 0 0
\(351\) 636.000 0.0967156
\(352\) 0 0
\(353\) 1690.00 0.254815 0.127407 0.991850i \(-0.459334\pi\)
0.127407 + 0.991850i \(0.459334\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −450.000 −0.0667130
\(358\) 0 0
\(359\) −1638.00 −0.240809 −0.120404 0.992725i \(-0.538419\pi\)
−0.120404 + 0.992725i \(0.538419\pi\)
\(360\) 0 0
\(361\) 1422.00 0.207319
\(362\) 0 0
\(363\) −970.000 −0.140253
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −7580.00 −1.07813 −0.539064 0.842265i \(-0.681222\pi\)
−0.539064 + 0.842265i \(0.681222\pi\)
\(368\) 0 0
\(369\) −3042.00 −0.429160
\(370\) 0 0
\(371\) −2412.00 −0.337533
\(372\) 0 0
\(373\) 5630.00 0.781529 0.390765 0.920491i \(-0.372211\pi\)
0.390765 + 0.920491i \(0.372211\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3264.00 0.445901
\(378\) 0 0
\(379\) 4385.00 0.594307 0.297153 0.954830i \(-0.403963\pi\)
0.297153 + 0.954830i \(0.403963\pi\)
\(380\) 0 0
\(381\) −902.000 −0.121288
\(382\) 0 0
\(383\) 12558.0 1.67541 0.837707 0.546119i \(-0.183896\pi\)
0.837707 + 0.546119i \(0.183896\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9672.00 −1.27043
\(388\) 0 0
\(389\) −6570.00 −0.856330 −0.428165 0.903701i \(-0.640840\pi\)
−0.428165 + 0.903701i \(0.640840\pi\)
\(390\) 0 0
\(391\) 13050.0 1.68789
\(392\) 0 0
\(393\) −2068.00 −0.265437
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1268.00 −0.160300 −0.0801500 0.996783i \(-0.525540\pi\)
−0.0801500 + 0.996783i \(0.525540\pi\)
\(398\) 0 0
\(399\) −546.000 −0.0685067
\(400\) 0 0
\(401\) −6299.00 −0.784432 −0.392216 0.919873i \(-0.628291\pi\)
−0.392216 + 0.919873i \(0.628291\pi\)
\(402\) 0 0
\(403\) −2760.00 −0.341155
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3458.00 0.421147
\(408\) 0 0
\(409\) −13459.0 −1.62715 −0.813575 0.581459i \(-0.802482\pi\)
−0.813575 + 0.581459i \(0.802482\pi\)
\(410\) 0 0
\(411\) 1339.00 0.160701
\(412\) 0 0
\(413\) 1872.00 0.223039
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2939.00 0.345140
\(418\) 0 0
\(419\) 9567.00 1.11546 0.557731 0.830022i \(-0.311672\pi\)
0.557731 + 0.830022i \(0.311672\pi\)
\(420\) 0 0
\(421\) 2708.00 0.313491 0.156746 0.987639i \(-0.449900\pi\)
0.156746 + 0.987639i \(0.449900\pi\)
\(422\) 0 0
\(423\) 1352.00 0.155405
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1020.00 −0.115600
\(428\) 0 0
\(429\) −228.000 −0.0256595
\(430\) 0 0
\(431\) −5126.00 −0.572879 −0.286439 0.958098i \(-0.592472\pi\)
−0.286439 + 0.958098i \(0.592472\pi\)
\(432\) 0 0
\(433\) −11445.0 −1.27023 −0.635117 0.772416i \(-0.719048\pi\)
−0.635117 + 0.772416i \(0.719048\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15834.0 1.73328
\(438\) 0 0
\(439\) 5096.00 0.554029 0.277015 0.960866i \(-0.410655\pi\)
0.277015 + 0.960866i \(0.410655\pi\)
\(440\) 0 0
\(441\) 7982.00 0.861894
\(442\) 0 0
\(443\) 13247.0 1.42073 0.710366 0.703833i \(-0.248530\pi\)
0.710366 + 0.703833i \(0.248530\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −208.000 −0.0220091
\(448\) 0 0
\(449\) 7449.00 0.782940 0.391470 0.920191i \(-0.371967\pi\)
0.391470 + 0.920191i \(0.371967\pi\)
\(450\) 0 0
\(451\) 2223.00 0.232100
\(452\) 0 0
\(453\) 2678.00 0.277756
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3497.00 −0.357949 −0.178975 0.983854i \(-0.557278\pi\)
−0.178975 + 0.983854i \(0.557278\pi\)
\(458\) 0 0
\(459\) −3975.00 −0.404220
\(460\) 0 0
\(461\) −13108.0 −1.32430 −0.662148 0.749373i \(-0.730355\pi\)
−0.662148 + 0.749373i \(0.730355\pi\)
\(462\) 0 0
\(463\) −5428.00 −0.544839 −0.272420 0.962179i \(-0.587824\pi\)
−0.272420 + 0.962179i \(0.587824\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1516.00 −0.150219 −0.0751093 0.997175i \(-0.523931\pi\)
−0.0751093 + 0.997175i \(0.523931\pi\)
\(468\) 0 0
\(469\) −4578.00 −0.450730
\(470\) 0 0
\(471\) −1482.00 −0.144983
\(472\) 0 0
\(473\) 7068.00 0.687076
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10452.0 −1.00328
\(478\) 0 0
\(479\) 14762.0 1.40813 0.704064 0.710137i \(-0.251367\pi\)
0.704064 + 0.710137i \(0.251367\pi\)
\(480\) 0 0
\(481\) −2184.00 −0.207031
\(482\) 0 0
\(483\) −1044.00 −0.0983512
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3926.00 −0.365306 −0.182653 0.983177i \(-0.558468\pi\)
−0.182653 + 0.983177i \(0.558468\pi\)
\(488\) 0 0
\(489\) −1469.00 −0.135850
\(490\) 0 0
\(491\) −996.000 −0.0915455 −0.0457728 0.998952i \(-0.514575\pi\)
−0.0457728 + 0.998952i \(0.514575\pi\)
\(492\) 0 0
\(493\) −20400.0 −1.86363
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −312.000 −0.0281592
\(498\) 0 0
\(499\) −7804.00 −0.700110 −0.350055 0.936729i \(-0.613837\pi\)
−0.350055 + 0.936729i \(0.613837\pi\)
\(500\) 0 0
\(501\) −4004.00 −0.357057
\(502\) 0 0
\(503\) −16732.0 −1.48319 −0.741593 0.670850i \(-0.765930\pi\)
−0.741593 + 0.670850i \(0.765930\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2053.00 −0.179836
\(508\) 0 0
\(509\) 10426.0 0.907906 0.453953 0.891026i \(-0.350013\pi\)
0.453953 + 0.891026i \(0.350013\pi\)
\(510\) 0 0
\(511\) −5886.00 −0.509552
\(512\) 0 0
\(513\) −4823.00 −0.415089
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −988.000 −0.0840468
\(518\) 0 0
\(519\) −3224.00 −0.272674
\(520\) 0 0
\(521\) −2235.00 −0.187941 −0.0939704 0.995575i \(-0.529956\pi\)
−0.0939704 + 0.995575i \(0.529956\pi\)
\(522\) 0 0
\(523\) −10855.0 −0.907564 −0.453782 0.891113i \(-0.649926\pi\)
−0.453782 + 0.891113i \(0.649926\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17250.0 1.42585
\(528\) 0 0
\(529\) 18109.0 1.48837
\(530\) 0 0
\(531\) 8112.00 0.662958
\(532\) 0 0
\(533\) −1404.00 −0.114098
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4191.00 0.336788
\(538\) 0 0
\(539\) −5833.00 −0.466132
\(540\) 0 0
\(541\) 10608.0 0.843019 0.421510 0.906824i \(-0.361500\pi\)
0.421510 + 0.906824i \(0.361500\pi\)
\(542\) 0 0
\(543\) −3718.00 −0.293839
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −11583.0 −0.905399 −0.452700 0.891663i \(-0.649539\pi\)
−0.452700 + 0.891663i \(0.649539\pi\)
\(548\) 0 0
\(549\) −4420.00 −0.343608
\(550\) 0 0
\(551\) −24752.0 −1.91374
\(552\) 0 0
\(553\) 6324.00 0.486300
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17244.0 −1.31176 −0.655881 0.754864i \(-0.727703\pi\)
−0.655881 + 0.754864i \(0.727703\pi\)
\(558\) 0 0
\(559\) −4464.00 −0.337759
\(560\) 0 0
\(561\) 1425.00 0.107243
\(562\) 0 0
\(563\) −18416.0 −1.37858 −0.689291 0.724484i \(-0.742078\pi\)
−0.689291 + 0.724484i \(0.742078\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3894.00 −0.288417
\(568\) 0 0
\(569\) 11913.0 0.877713 0.438857 0.898557i \(-0.355384\pi\)
0.438857 + 0.898557i \(0.355384\pi\)
\(570\) 0 0
\(571\) −3900.00 −0.285832 −0.142916 0.989735i \(-0.545648\pi\)
−0.142916 + 0.989735i \(0.545648\pi\)
\(572\) 0 0
\(573\) −870.000 −0.0634289
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −24899.0 −1.79646 −0.898231 0.439523i \(-0.855147\pi\)
−0.898231 + 0.439523i \(0.855147\pi\)
\(578\) 0 0
\(579\) −2197.00 −0.157693
\(580\) 0 0
\(581\) −2106.00 −0.150381
\(582\) 0 0
\(583\) 7638.00 0.542596
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1751.00 0.123120 0.0615601 0.998103i \(-0.480392\pi\)
0.0615601 + 0.998103i \(0.480392\pi\)
\(588\) 0 0
\(589\) 20930.0 1.46419
\(590\) 0 0
\(591\) 2314.00 0.161058
\(592\) 0 0
\(593\) −10887.0 −0.753922 −0.376961 0.926229i \(-0.623031\pi\)
−0.376961 + 0.926229i \(0.623031\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 252.000 0.0172758
\(598\) 0 0
\(599\) 14650.0 0.999303 0.499652 0.866226i \(-0.333461\pi\)
0.499652 + 0.866226i \(0.333461\pi\)
\(600\) 0 0
\(601\) −4237.00 −0.287572 −0.143786 0.989609i \(-0.545928\pi\)
−0.143786 + 0.989609i \(0.545928\pi\)
\(602\) 0 0
\(603\) −19838.0 −1.33974
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11440.0 0.764968 0.382484 0.923962i \(-0.375069\pi\)
0.382484 + 0.923962i \(0.375069\pi\)
\(608\) 0 0
\(609\) 1632.00 0.108591
\(610\) 0 0
\(611\) 624.000 0.0413164
\(612\) 0 0
\(613\) 19370.0 1.27626 0.638130 0.769929i \(-0.279708\pi\)
0.638130 + 0.769929i \(0.279708\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21346.0 1.39280 0.696400 0.717654i \(-0.254784\pi\)
0.696400 + 0.717654i \(0.254784\pi\)
\(618\) 0 0
\(619\) −7436.00 −0.482840 −0.241420 0.970421i \(-0.577613\pi\)
−0.241420 + 0.970421i \(0.577613\pi\)
\(620\) 0 0
\(621\) −9222.00 −0.595920
\(622\) 0 0
\(623\) −4794.00 −0.308295
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1729.00 0.110127
\(628\) 0 0
\(629\) 13650.0 0.865280
\(630\) 0 0
\(631\) −22490.0 −1.41888 −0.709440 0.704766i \(-0.751052\pi\)
−0.709440 + 0.704766i \(0.751052\pi\)
\(632\) 0 0
\(633\) 741.000 0.0465278
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3684.00 0.229145
\(638\) 0 0
\(639\) −1352.00 −0.0837000
\(640\) 0 0
\(641\) −16086.0 −0.991199 −0.495600 0.868551i \(-0.665052\pi\)
−0.495600 + 0.868551i \(0.665052\pi\)
\(642\) 0 0
\(643\) −2396.00 −0.146950 −0.0734751 0.997297i \(-0.523409\pi\)
−0.0734751 + 0.997297i \(0.523409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23244.0 1.41239 0.706195 0.708018i \(-0.250410\pi\)
0.706195 + 0.708018i \(0.250410\pi\)
\(648\) 0 0
\(649\) −5928.00 −0.358543
\(650\) 0 0
\(651\) −1380.00 −0.0830821
\(652\) 0 0
\(653\) −13598.0 −0.814902 −0.407451 0.913227i \(-0.633582\pi\)
−0.407451 + 0.913227i \(0.633582\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −25506.0 −1.51459
\(658\) 0 0
\(659\) −9751.00 −0.576396 −0.288198 0.957571i \(-0.593056\pi\)
−0.288198 + 0.957571i \(0.593056\pi\)
\(660\) 0 0
\(661\) 19104.0 1.12414 0.562072 0.827088i \(-0.310004\pi\)
0.562072 + 0.827088i \(0.310004\pi\)
\(662\) 0 0
\(663\) −900.000 −0.0527196
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −47328.0 −2.74745
\(668\) 0 0
\(669\) −5092.00 −0.294272
\(670\) 0 0
\(671\) 3230.00 0.185831
\(672\) 0 0
\(673\) −25402.0 −1.45494 −0.727470 0.686139i \(-0.759304\pi\)
−0.727470 + 0.686139i \(0.759304\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2224.00 0.126256 0.0631280 0.998005i \(-0.479892\pi\)
0.0631280 + 0.998005i \(0.479892\pi\)
\(678\) 0 0
\(679\) 5772.00 0.326228
\(680\) 0 0
\(681\) 5876.00 0.330644
\(682\) 0 0
\(683\) 8281.00 0.463929 0.231965 0.972724i \(-0.425485\pi\)
0.231965 + 0.972724i \(0.425485\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −604.000 −0.0335430
\(688\) 0 0
\(689\) −4824.00 −0.266734
\(690\) 0 0
\(691\) −481.000 −0.0264806 −0.0132403 0.999912i \(-0.504215\pi\)
−0.0132403 + 0.999912i \(0.504215\pi\)
\(692\) 0 0
\(693\) 2964.00 0.162472
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8775.00 0.476868
\(698\) 0 0
\(699\) −278.000 −0.0150428
\(700\) 0 0
\(701\) −16788.0 −0.904528 −0.452264 0.891884i \(-0.649383\pi\)
−0.452264 + 0.891884i \(0.649383\pi\)
\(702\) 0 0
\(703\) 16562.0 0.888546
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2916.00 −0.155117
\(708\) 0 0
\(709\) −23452.0 −1.24225 −0.621127 0.783710i \(-0.713325\pi\)
−0.621127 + 0.783710i \(0.713325\pi\)
\(710\) 0 0
\(711\) 27404.0 1.44547
\(712\) 0 0
\(713\) 40020.0 2.10205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2496.00 −0.130007
\(718\) 0 0
\(719\) 20886.0 1.08333 0.541666 0.840594i \(-0.317794\pi\)
0.541666 + 0.840594i \(0.317794\pi\)
\(720\) 0 0
\(721\) 7128.00 0.368184
\(722\) 0 0
\(723\) −2567.00 −0.132044
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 22576.0 1.15172 0.575858 0.817550i \(-0.304668\pi\)
0.575858 + 0.817550i \(0.304668\pi\)
\(728\) 0 0
\(729\) −15443.0 −0.784586
\(730\) 0 0
\(731\) 27900.0 1.41165
\(732\) 0 0
\(733\) −35308.0 −1.77917 −0.889584 0.456771i \(-0.849006\pi\)
−0.889584 + 0.456771i \(0.849006\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14497.0 0.724564
\(738\) 0 0
\(739\) −188.000 −0.00935818 −0.00467909 0.999989i \(-0.501489\pi\)
−0.00467909 + 0.999989i \(0.501489\pi\)
\(740\) 0 0
\(741\) −1092.00 −0.0541371
\(742\) 0 0
\(743\) 29870.0 1.47486 0.737432 0.675421i \(-0.236038\pi\)
0.737432 + 0.675421i \(0.236038\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9126.00 −0.446992
\(748\) 0 0
\(749\) −7950.00 −0.387833
\(750\) 0 0
\(751\) −22784.0 −1.10706 −0.553529 0.832830i \(-0.686719\pi\)
−0.553529 + 0.832830i \(0.686719\pi\)
\(752\) 0 0
\(753\) −5395.00 −0.261095
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 28496.0 1.36817 0.684085 0.729402i \(-0.260202\pi\)
0.684085 + 0.729402i \(0.260202\pi\)
\(758\) 0 0
\(759\) 3306.00 0.158103
\(760\) 0 0
\(761\) 7397.00 0.352354 0.176177 0.984359i \(-0.443627\pi\)
0.176177 + 0.984359i \(0.443627\pi\)
\(762\) 0 0
\(763\) −756.000 −0.0358703
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3744.00 0.176256
\(768\) 0 0
\(769\) −8883.00 −0.416553 −0.208276 0.978070i \(-0.566785\pi\)
−0.208276 + 0.978070i \(0.566785\pi\)
\(770\) 0 0
\(771\) 1490.00 0.0695993
\(772\) 0 0
\(773\) 33960.0 1.58015 0.790075 0.613010i \(-0.210041\pi\)
0.790075 + 0.613010i \(0.210041\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1092.00 −0.0504186
\(778\) 0 0
\(779\) 10647.0 0.489690
\(780\) 0 0
\(781\) 988.000 0.0452669
\(782\) 0 0
\(783\) 14416.0 0.657964
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −32084.0 −1.45320 −0.726602 0.687059i \(-0.758901\pi\)
−0.726602 + 0.687059i \(0.758901\pi\)
\(788\) 0 0
\(789\) −3330.00 −0.150255
\(790\) 0 0
\(791\) 1098.00 0.0493557
\(792\) 0 0
\(793\) −2040.00 −0.0913525
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4766.00 0.211820 0.105910 0.994376i \(-0.466224\pi\)
0.105910 + 0.994376i \(0.466224\pi\)
\(798\) 0 0
\(799\) −3900.00 −0.172681
\(800\) 0 0
\(801\) −20774.0 −0.916371
\(802\) 0 0
\(803\) 18639.0 0.819123
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6096.00 0.265910
\(808\) 0 0
\(809\) −31278.0 −1.35930 −0.679651 0.733535i \(-0.737869\pi\)
−0.679651 + 0.733535i \(0.737869\pi\)
\(810\) 0 0
\(811\) 29956.0 1.29704 0.648519 0.761199i \(-0.275389\pi\)
0.648519 + 0.761199i \(0.275389\pi\)
\(812\) 0 0
\(813\) 6006.00 0.259089
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 33852.0 1.44961
\(818\) 0 0
\(819\) −1872.00 −0.0798693
\(820\) 0 0
\(821\) 14642.0 0.622423 0.311212 0.950341i \(-0.399265\pi\)
0.311212 + 0.950341i \(0.399265\pi\)
\(822\) 0 0
\(823\) 20844.0 0.882839 0.441419 0.897301i \(-0.354475\pi\)
0.441419 + 0.897301i \(0.354475\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23751.0 0.998674 0.499337 0.866408i \(-0.333577\pi\)
0.499337 + 0.866408i \(0.333577\pi\)
\(828\) 0 0
\(829\) 11380.0 0.476772 0.238386 0.971171i \(-0.423382\pi\)
0.238386 + 0.971171i \(0.423382\pi\)
\(830\) 0 0
\(831\) −6976.00 −0.291209
\(832\) 0 0
\(833\) −23025.0 −0.957706
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −12190.0 −0.503403
\(838\) 0 0
\(839\) 29744.0 1.22393 0.611965 0.790885i \(-0.290379\pi\)
0.611965 + 0.790885i \(0.290379\pi\)
\(840\) 0 0
\(841\) 49595.0 2.03350
\(842\) 0 0
\(843\) 3998.00 0.163343
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5820.00 0.236101
\(848\) 0 0
\(849\) −13.0000 −0.000525511 0
\(850\) 0 0
\(851\) 31668.0 1.27563
\(852\) 0 0
\(853\) 37726.0 1.51432 0.757159 0.653230i \(-0.226587\pi\)
0.757159 + 0.653230i \(0.226587\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5429.00 −0.216396 −0.108198 0.994129i \(-0.534508\pi\)
−0.108198 + 0.994129i \(0.534508\pi\)
\(858\) 0 0
\(859\) −32149.0 −1.27696 −0.638481 0.769638i \(-0.720437\pi\)
−0.638481 + 0.769638i \(0.720437\pi\)
\(860\) 0 0
\(861\) −702.000 −0.0277864
\(862\) 0 0
\(863\) −29176.0 −1.15083 −0.575413 0.817863i \(-0.695159\pi\)
−0.575413 + 0.817863i \(0.695159\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 712.000 0.0278902
\(868\) 0 0
\(869\) −20026.0 −0.781744
\(870\) 0 0
\(871\) −9156.00 −0.356187
\(872\) 0 0
\(873\) 25012.0 0.969677
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20068.0 −0.772689 −0.386344 0.922355i \(-0.626262\pi\)
−0.386344 + 0.922355i \(0.626262\pi\)
\(878\) 0 0
\(879\) 4466.00 0.171370
\(880\) 0 0
\(881\) −36850.0 −1.40920 −0.704602 0.709603i \(-0.748874\pi\)
−0.704602 + 0.709603i \(0.748874\pi\)
\(882\) 0 0
\(883\) 30025.0 1.14431 0.572153 0.820147i \(-0.306108\pi\)
0.572153 + 0.820147i \(0.306108\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 156.000 0.00590526 0.00295263 0.999996i \(-0.499060\pi\)
0.00295263 + 0.999996i \(0.499060\pi\)
\(888\) 0 0
\(889\) 5412.00 0.204176
\(890\) 0 0
\(891\) 12331.0 0.463641
\(892\) 0 0
\(893\) −4732.00 −0.177324
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2088.00 −0.0777216
\(898\) 0 0
\(899\) −62560.0 −2.32090
\(900\) 0 0
\(901\) 30150.0 1.11481
\(902\) 0 0
\(903\) −2232.00 −0.0822550
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −356.000 −0.0130328 −0.00651642 0.999979i \(-0.502074\pi\)
−0.00651642 + 0.999979i \(0.502074\pi\)
\(908\) 0 0
\(909\) −12636.0 −0.461067
\(910\) 0 0
\(911\) −8748.00 −0.318149 −0.159075 0.987267i \(-0.550851\pi\)
−0.159075 + 0.987267i \(0.550851\pi\)
\(912\) 0 0
\(913\) 6669.00 0.241743
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12408.0 0.446836
\(918\) 0 0
\(919\) −36974.0 −1.32716 −0.663580 0.748105i \(-0.730964\pi\)
−0.663580 + 0.748105i \(0.730964\pi\)
\(920\) 0 0
\(921\) 9041.00 0.323465
\(922\) 0 0
\(923\) −624.000 −0.0222527
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 30888.0 1.09439
\(928\) 0 0
\(929\) 44382.0 1.56741 0.783706 0.621132i \(-0.213327\pi\)
0.783706 + 0.621132i \(0.213327\pi\)
\(930\) 0 0
\(931\) −27937.0 −0.983457
\(932\) 0 0
\(933\) 346.000 0.0121410
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2445.00 0.0852451 0.0426226 0.999091i \(-0.486429\pi\)
0.0426226 + 0.999091i \(0.486429\pi\)
\(938\) 0 0
\(939\) 10646.0 0.369988
\(940\) 0 0
\(941\) −7076.00 −0.245134 −0.122567 0.992460i \(-0.539113\pi\)
−0.122567 + 0.992460i \(0.539113\pi\)
\(942\) 0 0
\(943\) 20358.0 0.703020
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1560.00 0.0535303 0.0267651 0.999642i \(-0.491479\pi\)
0.0267651 + 0.999642i \(0.491479\pi\)
\(948\) 0 0
\(949\) −11772.0 −0.402672
\(950\) 0 0
\(951\) 8116.00 0.276740
\(952\) 0 0
\(953\) 35087.0 1.19263 0.596317 0.802749i \(-0.296630\pi\)
0.596317 + 0.802749i \(0.296630\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −5168.00 −0.174564
\(958\) 0 0
\(959\) −8034.00 −0.270523
\(960\) 0 0
\(961\) 23109.0 0.775704
\(962\) 0 0
\(963\) −34450.0 −1.15279
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −9360.00 −0.311269 −0.155635 0.987815i \(-0.549742\pi\)
−0.155635 + 0.987815i \(0.549742\pi\)
\(968\) 0 0
\(969\) 6825.00 0.226265
\(970\) 0 0
\(971\) −22269.0 −0.735990 −0.367995 0.929828i \(-0.619956\pi\)
−0.367995 + 0.929828i \(0.619956\pi\)
\(972\) 0 0
\(973\) −17634.0 −0.581007
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −37249.0 −1.21976 −0.609878 0.792496i \(-0.708781\pi\)
−0.609878 + 0.792496i \(0.708781\pi\)
\(978\) 0 0
\(979\) 15181.0 0.495594
\(980\) 0 0
\(981\) −3276.00 −0.106620
\(982\) 0 0
\(983\) −17602.0 −0.571126 −0.285563 0.958360i \(-0.592181\pi\)
−0.285563 + 0.958360i \(0.592181\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 312.000 0.0100619
\(988\) 0 0
\(989\) 64728.0 2.08112
\(990\) 0 0
\(991\) 26402.0 0.846304 0.423152 0.906059i \(-0.360924\pi\)
0.423152 + 0.906059i \(0.360924\pi\)
\(992\) 0 0
\(993\) 3007.00 0.0960969
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −16836.0 −0.534806 −0.267403 0.963585i \(-0.586165\pi\)
−0.267403 + 0.963585i \(0.586165\pi\)
\(998\) 0 0
\(999\) −9646.00 −0.305491
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 400.4.a.k.1.1 1
4.3 odd 2 200.4.a.e.1.1 1
5.2 odd 4 400.4.c.m.49.1 2
5.3 odd 4 400.4.c.m.49.2 2
5.4 even 2 400.4.a.j.1.1 1
8.3 odd 2 1600.4.a.bf.1.1 1
8.5 even 2 1600.4.a.v.1.1 1
12.11 even 2 1800.4.a.w.1.1 1
20.3 even 4 200.4.c.g.49.1 2
20.7 even 4 200.4.c.g.49.2 2
20.19 odd 2 200.4.a.f.1.1 yes 1
40.19 odd 2 1600.4.a.w.1.1 1
40.29 even 2 1600.4.a.be.1.1 1
60.23 odd 4 1800.4.f.p.649.1 2
60.47 odd 4 1800.4.f.p.649.2 2
60.59 even 2 1800.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 1 4.3 odd 2
200.4.a.f.1.1 yes 1 20.19 odd 2
200.4.c.g.49.1 2 20.3 even 4
200.4.c.g.49.2 2 20.7 even 4
400.4.a.j.1.1 1 5.4 even 2
400.4.a.k.1.1 1 1.1 even 1 trivial
400.4.c.m.49.1 2 5.2 odd 4
400.4.c.m.49.2 2 5.3 odd 4
1600.4.a.v.1.1 1 8.5 even 2
1600.4.a.w.1.1 1 40.19 odd 2
1600.4.a.be.1.1 1 40.29 even 2
1600.4.a.bf.1.1 1 8.3 odd 2
1800.4.a.l.1.1 1 60.59 even 2
1800.4.a.w.1.1 1 12.11 even 2
1800.4.f.p.649.1 2 60.23 odd 4
1800.4.f.p.649.2 2 60.47 odd 4