Properties

Label 294.5.c.a.97.1
Level $294$
Weight $5$
Character 294.97
Analytic conductor $30.391$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,5,Mod(97,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.97");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 294.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.3907691467\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.1
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 294.97
Dual form 294.5.c.a.97.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{2} -5.19615i q^{3} +8.00000 q^{4} +23.9515i q^{5} +14.6969i q^{6} -22.6274 q^{8} -27.0000 q^{9} +O(q^{10})\) \(q-2.82843 q^{2} -5.19615i q^{3} +8.00000 q^{4} +23.9515i q^{5} +14.6969i q^{6} -22.6274 q^{8} -27.0000 q^{9} -67.7452i q^{10} +97.9706 q^{11} -41.5692i q^{12} +104.211i q^{13} +124.456 q^{15} +64.0000 q^{16} +107.676i q^{17} +76.3675 q^{18} -38.3935i q^{19} +191.612i q^{20} -277.103 q^{22} -1021.50 q^{23} +117.576i q^{24} +51.3238 q^{25} -294.755i q^{26} +140.296i q^{27} +621.603 q^{29} -352.014 q^{30} -1519.14i q^{31} -181.019 q^{32} -509.070i q^{33} -304.553i q^{34} -216.000 q^{36} -562.235 q^{37} +108.593i q^{38} +541.499 q^{39} -541.961i q^{40} +1023.20i q^{41} -3382.41 q^{43} +783.765 q^{44} -646.692i q^{45} +2889.23 q^{46} +3945.03i q^{47} -332.554i q^{48} -145.166 q^{50} +559.499 q^{51} +833.692i q^{52} +2190.60 q^{53} -396.817i q^{54} +2346.55i q^{55} -199.499 q^{57} -1758.16 q^{58} +2934.87i q^{59} +995.647 q^{60} -665.346i q^{61} +4296.76i q^{62} +512.000 q^{64} -2496.03 q^{65} +1439.87i q^{66} -5925.38 q^{67} +861.405i q^{68} +5307.86i q^{69} -4494.41 q^{71} +610.940 q^{72} +8968.93i q^{73} +1590.24 q^{74} -266.686i q^{75} -307.148i q^{76} -1531.59 q^{78} -10446.8 q^{79} +1532.90i q^{80} +729.000 q^{81} -2894.04i q^{82} -1269.28i q^{83} -2579.00 q^{85} +9566.89 q^{86} -3229.94i q^{87} -2216.82 q^{88} +3604.36i q^{89} +1829.12i q^{90} -8171.99 q^{92} -7893.66 q^{93} -11158.2i q^{94} +919.584 q^{95} +940.604i q^{96} +1950.53i q^{97} -2645.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{4} - 108 q^{9} + 324 q^{11} + 396 q^{15} + 256 q^{16} - 192 q^{22} - 624 q^{23} + 952 q^{25} + 2724 q^{29} - 288 q^{30} - 864 q^{36} - 2792 q^{37} - 1296 q^{39} - 632 q^{43} + 2592 q^{44} + 9792 q^{46} + 2112 q^{50} - 1224 q^{51} + 2076 q^{53} + 2664 q^{57} + 672 q^{58} + 3168 q^{60} + 2048 q^{64} + 1488 q^{65} - 29200 q^{67} - 9696 q^{71} - 1536 q^{74} - 9792 q^{78} - 7948 q^{79} + 2916 q^{81} + 1224 q^{85} + 36480 q^{86} - 1536 q^{88} - 4992 q^{92} - 22716 q^{93} - 6504 q^{95} - 8748 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 −0.707107
\(3\) − 5.19615i − 0.577350i
\(4\) 8.00000 0.500000
\(5\) 23.9515i 0.958062i 0.877798 + 0.479031i \(0.159012\pi\)
−0.877798 + 0.479031i \(0.840988\pi\)
\(6\) 14.6969i 0.408248i
\(7\) 0 0
\(8\) −22.6274 −0.353553
\(9\) −27.0000 −0.333333
\(10\) − 67.7452i − 0.677452i
\(11\) 97.9706 0.809674 0.404837 0.914389i \(-0.367328\pi\)
0.404837 + 0.914389i \(0.367328\pi\)
\(12\) − 41.5692i − 0.288675i
\(13\) 104.211i 0.616636i 0.951283 + 0.308318i \(0.0997661\pi\)
−0.951283 + 0.308318i \(0.900234\pi\)
\(14\) 0 0
\(15\) 124.456 0.553137
\(16\) 64.0000 0.250000
\(17\) 107.676i 0.372580i 0.982495 + 0.186290i \(0.0596464\pi\)
−0.982495 + 0.186290i \(0.940354\pi\)
\(18\) 76.3675 0.235702
\(19\) − 38.3935i − 0.106353i −0.998585 0.0531767i \(-0.983065\pi\)
0.998585 0.0531767i \(-0.0169346\pi\)
\(20\) 191.612i 0.479031i
\(21\) 0 0
\(22\) −277.103 −0.572526
\(23\) −1021.50 −1.93100 −0.965500 0.260404i \(-0.916144\pi\)
−0.965500 + 0.260404i \(0.916144\pi\)
\(24\) 117.576i 0.204124i
\(25\) 51.3238 0.0821181
\(26\) − 294.755i − 0.436027i
\(27\) 140.296i 0.192450i
\(28\) 0 0
\(29\) 621.603 0.739124 0.369562 0.929206i \(-0.379508\pi\)
0.369562 + 0.929206i \(0.379508\pi\)
\(30\) −352.014 −0.391127
\(31\) − 1519.14i − 1.58079i −0.612600 0.790393i \(-0.709876\pi\)
0.612600 0.790393i \(-0.290124\pi\)
\(32\) −181.019 −0.176777
\(33\) − 509.070i − 0.467466i
\(34\) − 304.553i − 0.263454i
\(35\) 0 0
\(36\) −216.000 −0.166667
\(37\) −562.235 −0.410691 −0.205345 0.978690i \(-0.565832\pi\)
−0.205345 + 0.978690i \(0.565832\pi\)
\(38\) 108.593i 0.0752031i
\(39\) 541.499 0.356015
\(40\) − 541.961i − 0.338726i
\(41\) 1023.20i 0.608684i 0.952563 + 0.304342i \(0.0984365\pi\)
−0.952563 + 0.304342i \(0.901563\pi\)
\(42\) 0 0
\(43\) −3382.41 −1.82932 −0.914658 0.404228i \(-0.867540\pi\)
−0.914658 + 0.404228i \(0.867540\pi\)
\(44\) 783.765 0.404837
\(45\) − 646.692i − 0.319354i
\(46\) 2889.23 1.36542
\(47\) 3945.03i 1.78589i 0.450165 + 0.892945i \(0.351365\pi\)
−0.450165 + 0.892945i \(0.648635\pi\)
\(48\) − 332.554i − 0.144338i
\(49\) 0 0
\(50\) −145.166 −0.0580663
\(51\) 559.499 0.215109
\(52\) 833.692i 0.308318i
\(53\) 2190.60 0.779851 0.389925 0.920846i \(-0.372501\pi\)
0.389925 + 0.920846i \(0.372501\pi\)
\(54\) − 396.817i − 0.136083i
\(55\) 2346.55i 0.775718i
\(56\) 0 0
\(57\) −199.499 −0.0614031
\(58\) −1758.16 −0.522639
\(59\) 2934.87i 0.843111i 0.906803 + 0.421556i \(0.138516\pi\)
−0.906803 + 0.421556i \(0.861484\pi\)
\(60\) 995.647 0.276569
\(61\) − 665.346i − 0.178809i −0.995995 0.0894043i \(-0.971504\pi\)
0.995995 0.0894043i \(-0.0284963\pi\)
\(62\) 4296.76i 1.11778i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) −2496.03 −0.590775
\(66\) 1439.87i 0.330548i
\(67\) −5925.38 −1.31998 −0.659989 0.751275i \(-0.729439\pi\)
−0.659989 + 0.751275i \(0.729439\pi\)
\(68\) 861.405i 0.186290i
\(69\) 5307.86i 1.11486i
\(70\) 0 0
\(71\) −4494.41 −0.891571 −0.445785 0.895140i \(-0.647076\pi\)
−0.445785 + 0.895140i \(0.647076\pi\)
\(72\) 610.940 0.117851
\(73\) 8968.93i 1.68304i 0.540225 + 0.841521i \(0.318339\pi\)
−0.540225 + 0.841521i \(0.681661\pi\)
\(74\) 1590.24 0.290402
\(75\) − 266.686i − 0.0474109i
\(76\) − 307.148i − 0.0531767i
\(77\) 0 0
\(78\) −1531.59 −0.251741
\(79\) −10446.8 −1.67390 −0.836951 0.547277i \(-0.815664\pi\)
−0.836951 + 0.547277i \(0.815664\pi\)
\(80\) 1532.90i 0.239515i
\(81\) 729.000 0.111111
\(82\) − 2894.04i − 0.430405i
\(83\) − 1269.28i − 0.184247i −0.995748 0.0921234i \(-0.970635\pi\)
0.995748 0.0921234i \(-0.0293654\pi\)
\(84\) 0 0
\(85\) −2579.00 −0.356954
\(86\) 9566.89 1.29352
\(87\) − 3229.94i − 0.426733i
\(88\) −2216.82 −0.286263
\(89\) 3604.36i 0.455038i 0.973774 + 0.227519i \(0.0730614\pi\)
−0.973774 + 0.227519i \(0.926939\pi\)
\(90\) 1829.12i 0.225817i
\(91\) 0 0
\(92\) −8171.99 −0.965500
\(93\) −7893.66 −0.912667
\(94\) − 11158.2i − 1.26282i
\(95\) 919.584 0.101893
\(96\) 940.604i 0.102062i
\(97\) 1950.53i 0.207304i 0.994614 + 0.103652i \(0.0330529\pi\)
−0.994614 + 0.103652i \(0.966947\pi\)
\(98\) 0 0
\(99\) −2645.21 −0.269891
\(100\) 410.590 0.0410590
\(101\) 11326.2i 1.11031i 0.831748 + 0.555153i \(0.187340\pi\)
−0.831748 + 0.555153i \(0.812660\pi\)
\(102\) −1582.50 −0.152105
\(103\) 16328.9i 1.53915i 0.638555 + 0.769576i \(0.279532\pi\)
−0.638555 + 0.769576i \(0.720468\pi\)
\(104\) − 2358.04i − 0.218014i
\(105\) 0 0
\(106\) −6195.95 −0.551438
\(107\) 4396.46 0.384004 0.192002 0.981395i \(-0.438502\pi\)
0.192002 + 0.981395i \(0.438502\pi\)
\(108\) 1122.37i 0.0962250i
\(109\) −14697.8 −1.23708 −0.618541 0.785753i \(-0.712276\pi\)
−0.618541 + 0.785753i \(0.712276\pi\)
\(110\) − 6637.03i − 0.548515i
\(111\) 2921.46i 0.237112i
\(112\) 0 0
\(113\) 2124.36 0.166368 0.0831842 0.996534i \(-0.473491\pi\)
0.0831842 + 0.996534i \(0.473491\pi\)
\(114\) 564.268 0.0434186
\(115\) − 24466.5i − 1.85002i
\(116\) 4972.82 0.369562
\(117\) − 2813.71i − 0.205545i
\(118\) − 8301.07i − 0.596170i
\(119\) 0 0
\(120\) −2816.11 −0.195563
\(121\) −5042.77 −0.344428
\(122\) 1881.88i 0.126437i
\(123\) 5316.69 0.351424
\(124\) − 12153.1i − 0.790393i
\(125\) 16199.0i 1.03674i
\(126\) 0 0
\(127\) 8498.07 0.526881 0.263441 0.964676i \(-0.415143\pi\)
0.263441 + 0.964676i \(0.415143\pi\)
\(128\) −1448.15 −0.0883883
\(129\) 17575.5i 1.05616i
\(130\) 7059.83 0.417741
\(131\) − 14472.3i − 0.843324i −0.906753 0.421662i \(-0.861447\pi\)
0.906753 0.421662i \(-0.138553\pi\)
\(132\) − 4072.56i − 0.233733i
\(133\) 0 0
\(134\) 16759.5 0.933366
\(135\) −3360.31 −0.184379
\(136\) − 2436.42i − 0.131727i
\(137\) 12855.2 0.684915 0.342458 0.939533i \(-0.388741\pi\)
0.342458 + 0.939533i \(0.388741\pi\)
\(138\) − 15012.9i − 0.788327i
\(139\) − 5009.46i − 0.259275i −0.991561 0.129638i \(-0.958619\pi\)
0.991561 0.129638i \(-0.0413814\pi\)
\(140\) 0 0
\(141\) 20499.0 1.03108
\(142\) 12712.1 0.630436
\(143\) 10209.7i 0.499274i
\(144\) −1728.00 −0.0833333
\(145\) 14888.3i 0.708126i
\(146\) − 25368.0i − 1.19009i
\(147\) 0 0
\(148\) −4497.88 −0.205345
\(149\) 25398.9 1.14404 0.572022 0.820238i \(-0.306159\pi\)
0.572022 + 0.820238i \(0.306159\pi\)
\(150\) 754.303i 0.0335246i
\(151\) −18308.8 −0.802981 −0.401491 0.915863i \(-0.631508\pi\)
−0.401491 + 0.915863i \(0.631508\pi\)
\(152\) 868.747i 0.0376016i
\(153\) − 2907.24i − 0.124193i
\(154\) 0 0
\(155\) 36385.6 1.51449
\(156\) 4331.99 0.178007
\(157\) − 35244.4i − 1.42985i −0.699200 0.714926i \(-0.746460\pi\)
0.699200 0.714926i \(-0.253540\pi\)
\(158\) 29548.1 1.18363
\(159\) − 11382.7i − 0.450247i
\(160\) − 4335.69i − 0.169363i
\(161\) 0 0
\(162\) −2061.92 −0.0785674
\(163\) 145.939 0.00549281 0.00274641 0.999996i \(-0.499126\pi\)
0.00274641 + 0.999996i \(0.499126\pi\)
\(164\) 8185.58i 0.304342i
\(165\) 12193.0 0.447861
\(166\) 3590.05i 0.130282i
\(167\) − 31314.1i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(168\) 0 0
\(169\) 17701.0 0.619760
\(170\) 7294.50 0.252405
\(171\) 1036.63i 0.0354511i
\(172\) −27059.3 −0.914658
\(173\) 40115.8i 1.34037i 0.742196 + 0.670183i \(0.233784\pi\)
−0.742196 + 0.670183i \(0.766216\pi\)
\(174\) 9135.66i 0.301746i
\(175\) 0 0
\(176\) 6270.12 0.202419
\(177\) 15250.0 0.486771
\(178\) − 10194.7i − 0.321760i
\(179\) −50154.5 −1.56532 −0.782661 0.622448i \(-0.786138\pi\)
−0.782661 + 0.622448i \(0.786138\pi\)
\(180\) − 5173.53i − 0.159677i
\(181\) − 63974.3i − 1.95276i −0.216069 0.976378i \(-0.569323\pi\)
0.216069 0.976378i \(-0.430677\pi\)
\(182\) 0 0
\(183\) −3457.24 −0.103235
\(184\) 23113.9 0.682711
\(185\) − 13466.4i − 0.393467i
\(186\) 22326.6 0.645353
\(187\) 10549.0i 0.301668i
\(188\) 31560.3i 0.892945i
\(189\) 0 0
\(190\) −2600.98 −0.0720492
\(191\) −42768.9 −1.17236 −0.586180 0.810181i \(-0.699369\pi\)
−0.586180 + 0.810181i \(0.699369\pi\)
\(192\) − 2660.43i − 0.0721688i
\(193\) −8772.58 −0.235512 −0.117756 0.993043i \(-0.537570\pi\)
−0.117756 + 0.993043i \(0.537570\pi\)
\(194\) − 5516.92i − 0.146586i
\(195\) 12969.7i 0.341084i
\(196\) 0 0
\(197\) −32825.0 −0.845808 −0.422904 0.906174i \(-0.638989\pi\)
−0.422904 + 0.906174i \(0.638989\pi\)
\(198\) 7481.77 0.190842
\(199\) − 26464.2i − 0.668271i −0.942525 0.334136i \(-0.891556\pi\)
0.942525 0.334136i \(-0.108444\pi\)
\(200\) −1161.33 −0.0290331
\(201\) 30789.2i 0.762090i
\(202\) − 32035.4i − 0.785106i
\(203\) 0 0
\(204\) 4475.99 0.107555
\(205\) −24507.2 −0.583157
\(206\) − 46185.0i − 1.08834i
\(207\) 27580.5 0.643666
\(208\) 6669.53i 0.154159i
\(209\) − 3761.44i − 0.0861115i
\(210\) 0 0
\(211\) 67056.3 1.50617 0.753087 0.657921i \(-0.228564\pi\)
0.753087 + 0.657921i \(0.228564\pi\)
\(212\) 17524.8 0.389925
\(213\) 23353.6i 0.514749i
\(214\) −12435.1 −0.271532
\(215\) − 81013.8i − 1.75260i
\(216\) − 3174.54i − 0.0680414i
\(217\) 0 0
\(218\) 41571.6 0.874749
\(219\) 46603.9 0.971705
\(220\) 18772.4i 0.387859i
\(221\) −11221.0 −0.229746
\(222\) − 8263.14i − 0.167664i
\(223\) − 56260.1i − 1.13133i −0.824634 0.565666i \(-0.808619\pi\)
0.824634 0.565666i \(-0.191381\pi\)
\(224\) 0 0
\(225\) −1385.74 −0.0273727
\(226\) −6008.59 −0.117640
\(227\) 61148.1i 1.18667i 0.804954 + 0.593337i \(0.202190\pi\)
−0.804954 + 0.593337i \(0.797810\pi\)
\(228\) −1595.99 −0.0307016
\(229\) − 33803.7i − 0.644605i −0.946637 0.322303i \(-0.895543\pi\)
0.946637 0.322303i \(-0.104457\pi\)
\(230\) 69201.6i 1.30816i
\(231\) 0 0
\(232\) −14065.3 −0.261320
\(233\) 8687.26 0.160019 0.0800094 0.996794i \(-0.474505\pi\)
0.0800094 + 0.996794i \(0.474505\pi\)
\(234\) 7958.37i 0.145342i
\(235\) −94489.6 −1.71099
\(236\) 23479.0i 0.421556i
\(237\) 54283.3i 0.966428i
\(238\) 0 0
\(239\) −15715.0 −0.275118 −0.137559 0.990494i \(-0.543926\pi\)
−0.137559 + 0.990494i \(0.543926\pi\)
\(240\) 7965.17 0.138284
\(241\) − 5478.82i − 0.0943307i −0.998887 0.0471654i \(-0.984981\pi\)
0.998887 0.0471654i \(-0.0150188\pi\)
\(242\) 14263.1 0.243547
\(243\) − 3788.00i − 0.0641500i
\(244\) − 5322.77i − 0.0894043i
\(245\) 0 0
\(246\) −15037.9 −0.248494
\(247\) 4001.05 0.0655813
\(248\) 34374.1i 0.558892i
\(249\) −6595.35 −0.106375
\(250\) − 45817.7i − 0.733083i
\(251\) 24350.9i 0.386516i 0.981148 + 0.193258i \(0.0619054\pi\)
−0.981148 + 0.193258i \(0.938095\pi\)
\(252\) 0 0
\(253\) −100077. −1.56348
\(254\) −24036.2 −0.372561
\(255\) 13400.9i 0.206088i
\(256\) 4096.00 0.0625000
\(257\) − 11606.7i − 0.175729i −0.996132 0.0878647i \(-0.971996\pi\)
0.996132 0.0878647i \(-0.0280043\pi\)
\(258\) − 49711.0i − 0.746815i
\(259\) 0 0
\(260\) −19968.2 −0.295388
\(261\) −16783.3 −0.246375
\(262\) 40933.8i 0.596320i
\(263\) 120668. 1.74453 0.872266 0.489031i \(-0.162649\pi\)
0.872266 + 0.489031i \(0.162649\pi\)
\(264\) 11518.9i 0.165274i
\(265\) 52468.3i 0.747145i
\(266\) 0 0
\(267\) 18728.8 0.262716
\(268\) −47403.1 −0.659989
\(269\) − 27246.6i − 0.376537i −0.982118 0.188269i \(-0.939712\pi\)
0.982118 0.188269i \(-0.0602876\pi\)
\(270\) 9504.39 0.130376
\(271\) 97043.3i 1.32138i 0.750660 + 0.660689i \(0.229736\pi\)
−0.750660 + 0.660689i \(0.770264\pi\)
\(272\) 6891.24i 0.0931450i
\(273\) 0 0
\(274\) −36359.9 −0.484308
\(275\) 5028.22 0.0664889
\(276\) 42462.9i 0.557432i
\(277\) 29752.8 0.387765 0.193882 0.981025i \(-0.437892\pi\)
0.193882 + 0.981025i \(0.437892\pi\)
\(278\) 14168.9i 0.183335i
\(279\) 41016.7i 0.526929i
\(280\) 0 0
\(281\) −122275. −1.54855 −0.774276 0.632849i \(-0.781886\pi\)
−0.774276 + 0.632849i \(0.781886\pi\)
\(282\) −57979.9 −0.729087
\(283\) − 60700.7i − 0.757916i −0.925414 0.378958i \(-0.876282\pi\)
0.925414 0.378958i \(-0.123718\pi\)
\(284\) −35955.3 −0.445785
\(285\) − 4778.30i − 0.0588280i
\(286\) − 28877.3i − 0.353040i
\(287\) 0 0
\(288\) 4887.52 0.0589256
\(289\) 71927.0 0.861184
\(290\) − 42110.6i − 0.500721i
\(291\) 10135.2 0.119687
\(292\) 71751.4i 0.841521i
\(293\) 29557.4i 0.344296i 0.985071 + 0.172148i \(0.0550707\pi\)
−0.985071 + 0.172148i \(0.944929\pi\)
\(294\) 0 0
\(295\) −70294.7 −0.807752
\(296\) 12721.9 0.145201
\(297\) 13744.9i 0.155822i
\(298\) −71839.0 −0.808961
\(299\) − 106452.i − 1.19072i
\(300\) − 2133.49i − 0.0237055i
\(301\) 0 0
\(302\) 51785.0 0.567794
\(303\) 58852.9 0.641036
\(304\) − 2457.19i − 0.0265883i
\(305\) 15936.1 0.171310
\(306\) 8222.92i 0.0878179i
\(307\) 107567.i 1.14131i 0.821191 + 0.570654i \(0.193310\pi\)
−0.821191 + 0.570654i \(0.806690\pi\)
\(308\) 0 0
\(309\) 84847.3 0.888630
\(310\) −102914. −1.07091
\(311\) 43371.8i 0.448421i 0.974541 + 0.224211i \(0.0719804\pi\)
−0.974541 + 0.224211i \(0.928020\pi\)
\(312\) −12252.7 −0.125870
\(313\) 107746.i 1.09980i 0.835230 + 0.549900i \(0.185334\pi\)
−0.835230 + 0.549900i \(0.814666\pi\)
\(314\) 99686.3i 1.01106i
\(315\) 0 0
\(316\) −83574.6 −0.836951
\(317\) −155994. −1.55235 −0.776174 0.630519i \(-0.782842\pi\)
−0.776174 + 0.630519i \(0.782842\pi\)
\(318\) 32195.1i 0.318373i
\(319\) 60898.8 0.598449
\(320\) 12263.2i 0.119758i
\(321\) − 22844.7i − 0.221705i
\(322\) 0 0
\(323\) 4134.05 0.0396251
\(324\) 5832.00 0.0555556
\(325\) 5348.53i 0.0506370i
\(326\) −412.776 −0.00388400
\(327\) 76371.9i 0.714230i
\(328\) − 23152.3i − 0.215202i
\(329\) 0 0
\(330\) −34487.0 −0.316685
\(331\) 40218.7 0.367090 0.183545 0.983011i \(-0.441243\pi\)
0.183545 + 0.983011i \(0.441243\pi\)
\(332\) − 10154.2i − 0.0921234i
\(333\) 15180.4 0.136897
\(334\) 88569.6i 0.793947i
\(335\) − 141922.i − 1.26462i
\(336\) 0 0
\(337\) 90476.4 0.796665 0.398332 0.917241i \(-0.369589\pi\)
0.398332 + 0.917241i \(0.369589\pi\)
\(338\) −50065.9 −0.438237
\(339\) − 11038.5i − 0.0960529i
\(340\) −20632.0 −0.178477
\(341\) − 148831.i − 1.27992i
\(342\) − 2932.02i − 0.0250677i
\(343\) 0 0
\(344\) 76535.1 0.646761
\(345\) −127131. −1.06811
\(346\) − 113465.i − 0.947782i
\(347\) 32074.5 0.266379 0.133190 0.991091i \(-0.457478\pi\)
0.133190 + 0.991091i \(0.457478\pi\)
\(348\) − 25839.6i − 0.213367i
\(349\) 116783.i 0.958805i 0.877595 + 0.479402i \(0.159147\pi\)
−0.877595 + 0.479402i \(0.840853\pi\)
\(350\) 0 0
\(351\) −14620.5 −0.118672
\(352\) −17734.6 −0.143132
\(353\) 124658.i 1.00039i 0.865912 + 0.500197i \(0.166739\pi\)
−0.865912 + 0.500197i \(0.833261\pi\)
\(354\) −43133.6 −0.344199
\(355\) − 107648.i − 0.854180i
\(356\) 28834.8i 0.227519i
\(357\) 0 0
\(358\) 141858. 1.10685
\(359\) 99442.1 0.771581 0.385790 0.922587i \(-0.373929\pi\)
0.385790 + 0.922587i \(0.373929\pi\)
\(360\) 14633.0i 0.112909i
\(361\) 128847. 0.988689
\(362\) 180947.i 1.38081i
\(363\) 26203.0i 0.198856i
\(364\) 0 0
\(365\) −214820. −1.61246
\(366\) 9778.56 0.0729983
\(367\) 88914.6i 0.660147i 0.943955 + 0.330074i \(0.107074\pi\)
−0.943955 + 0.330074i \(0.892926\pi\)
\(368\) −65375.9 −0.482750
\(369\) − 27626.3i − 0.202895i
\(370\) 38088.7i 0.278223i
\(371\) 0 0
\(372\) −63149.3 −0.456334
\(373\) −91352.9 −0.656606 −0.328303 0.944573i \(-0.606477\pi\)
−0.328303 + 0.944573i \(0.606477\pi\)
\(374\) − 29837.2i − 0.213312i
\(375\) 84172.5 0.598560
\(376\) − 89265.9i − 0.631408i
\(377\) 64778.2i 0.455770i
\(378\) 0 0
\(379\) 6328.51 0.0440578 0.0220289 0.999757i \(-0.492987\pi\)
0.0220289 + 0.999757i \(0.492987\pi\)
\(380\) 7356.68 0.0509465
\(381\) − 44157.2i − 0.304195i
\(382\) 120969. 0.828984
\(383\) 13436.0i 0.0915949i 0.998951 + 0.0457974i \(0.0145829\pi\)
−0.998951 + 0.0457974i \(0.985417\pi\)
\(384\) 7524.83i 0.0510310i
\(385\) 0 0
\(386\) 24812.6 0.166532
\(387\) 91325.0 0.609772
\(388\) 15604.2i 0.103652i
\(389\) 88296.3 0.583503 0.291752 0.956494i \(-0.405762\pi\)
0.291752 + 0.956494i \(0.405762\pi\)
\(390\) − 36683.9i − 0.241183i
\(391\) − 109990.i − 0.719451i
\(392\) 0 0
\(393\) −75200.2 −0.486894
\(394\) 92843.0 0.598077
\(395\) − 250218.i − 1.60370i
\(396\) −21161.6 −0.134946
\(397\) − 216587.i − 1.37421i −0.726560 0.687103i \(-0.758882\pi\)
0.726560 0.687103i \(-0.241118\pi\)
\(398\) 74852.1i 0.472539i
\(399\) 0 0
\(400\) 3284.72 0.0205295
\(401\) −5166.84 −0.0321319 −0.0160659 0.999871i \(-0.505114\pi\)
−0.0160659 + 0.999871i \(0.505114\pi\)
\(402\) − 87085.0i − 0.538879i
\(403\) 158311. 0.974769
\(404\) 90609.9i 0.555153i
\(405\) 17460.7i 0.106451i
\(406\) 0 0
\(407\) −55082.5 −0.332526
\(408\) −12660.0 −0.0760525
\(409\) 38910.7i 0.232607i 0.993214 + 0.116303i \(0.0371045\pi\)
−0.993214 + 0.116303i \(0.962896\pi\)
\(410\) 69316.7 0.412354
\(411\) − 66797.4i − 0.395436i
\(412\) 130631.i 0.769576i
\(413\) 0 0
\(414\) −78009.3 −0.455141
\(415\) 30401.1 0.176520
\(416\) − 18864.3i − 0.109007i
\(417\) −26029.9 −0.149693
\(418\) 10639.0i 0.0608900i
\(419\) − 263425.i − 1.50048i −0.661167 0.750239i \(-0.729938\pi\)
0.661167 0.750239i \(-0.270062\pi\)
\(420\) 0 0
\(421\) −16375.1 −0.0923889 −0.0461944 0.998932i \(-0.514709\pi\)
−0.0461944 + 0.998932i \(0.514709\pi\)
\(422\) −189664. −1.06503
\(423\) − 106516.i − 0.595297i
\(424\) −49567.6 −0.275719
\(425\) 5526.32i 0.0305955i
\(426\) − 66054.0i − 0.363982i
\(427\) 0 0
\(428\) 35171.7 0.192002
\(429\) 53050.9 0.288256
\(430\) 229142.i 1.23927i
\(431\) −250416. −1.34806 −0.674028 0.738705i \(-0.735437\pi\)
−0.674028 + 0.738705i \(0.735437\pi\)
\(432\) 8978.95i 0.0481125i
\(433\) 91495.1i 0.488002i 0.969775 + 0.244001i \(0.0784601\pi\)
−0.969775 + 0.244001i \(0.921540\pi\)
\(434\) 0 0
\(435\) 77362.1 0.408837
\(436\) −117582. −0.618541
\(437\) 39219.0i 0.205368i
\(438\) −131816. −0.687099
\(439\) − 8859.37i − 0.0459699i −0.999736 0.0229850i \(-0.992683\pi\)
0.999736 0.0229850i \(-0.00731699\pi\)
\(440\) − 53096.3i − 0.274258i
\(441\) 0 0
\(442\) 31737.9 0.162455
\(443\) 300209. 1.52974 0.764868 0.644187i \(-0.222804\pi\)
0.764868 + 0.644187i \(0.222804\pi\)
\(444\) 23371.7i 0.118556i
\(445\) −86329.9 −0.435954
\(446\) 159127.i 0.799973i
\(447\) − 131977.i − 0.660514i
\(448\) 0 0
\(449\) 116866. 0.579691 0.289846 0.957073i \(-0.406396\pi\)
0.289846 + 0.957073i \(0.406396\pi\)
\(450\) 3919.47 0.0193554
\(451\) 100243.i 0.492836i
\(452\) 16994.9 0.0831842
\(453\) 95135.2i 0.463601i
\(454\) − 172953.i − 0.839105i
\(455\) 0 0
\(456\) 4514.14 0.0217093
\(457\) 73745.8 0.353106 0.176553 0.984291i \(-0.443505\pi\)
0.176553 + 0.984291i \(0.443505\pi\)
\(458\) 95611.4i 0.455805i
\(459\) −15106.5 −0.0717030
\(460\) − 195732.i − 0.925008i
\(461\) 88219.1i 0.415108i 0.978224 + 0.207554i \(0.0665502\pi\)
−0.978224 + 0.207554i \(0.933450\pi\)
\(462\) 0 0
\(463\) 297880. 1.38957 0.694784 0.719219i \(-0.255500\pi\)
0.694784 + 0.719219i \(0.255500\pi\)
\(464\) 39782.6 0.184781
\(465\) − 189065.i − 0.874391i
\(466\) −24571.3 −0.113150
\(467\) 187470.i 0.859601i 0.902924 + 0.429801i \(0.141416\pi\)
−0.902924 + 0.429801i \(0.858584\pi\)
\(468\) − 22509.7i − 0.102773i
\(469\) 0 0
\(470\) 267257. 1.20985
\(471\) −183135. −0.825526
\(472\) − 66408.5i − 0.298085i
\(473\) −331376. −1.48115
\(474\) − 153536.i − 0.683368i
\(475\) − 1970.50i − 0.00873353i
\(476\) 0 0
\(477\) −59146.2 −0.259950
\(478\) 44448.8 0.194538
\(479\) − 120489.i − 0.525142i −0.964913 0.262571i \(-0.915430\pi\)
0.964913 0.262571i \(-0.0845703\pi\)
\(480\) −22528.9 −0.0977817
\(481\) − 58591.4i − 0.253247i
\(482\) 15496.4i 0.0667019i
\(483\) 0 0
\(484\) −40342.2 −0.172214
\(485\) −46718.1 −0.198610
\(486\) 10714.1i 0.0453609i
\(487\) −267394. −1.12744 −0.563721 0.825966i \(-0.690631\pi\)
−0.563721 + 0.825966i \(0.690631\pi\)
\(488\) 15055.1i 0.0632184i
\(489\) − 758.319i − 0.00317128i
\(490\) 0 0
\(491\) 178364. 0.739851 0.369926 0.929061i \(-0.379383\pi\)
0.369926 + 0.929061i \(0.379383\pi\)
\(492\) 42533.5 0.175712
\(493\) 66931.5i 0.275383i
\(494\) −11316.7 −0.0463730
\(495\) − 63356.7i − 0.258573i
\(496\) − 97224.7i − 0.395196i
\(497\) 0 0
\(498\) 18654.5 0.0752184
\(499\) 263747. 1.05922 0.529610 0.848241i \(-0.322338\pi\)
0.529610 + 0.848241i \(0.322338\pi\)
\(500\) 129592.i 0.518368i
\(501\) −162713. −0.648255
\(502\) − 68874.7i − 0.273308i
\(503\) − 480056.i − 1.89739i −0.316197 0.948694i \(-0.602406\pi\)
0.316197 0.948694i \(-0.397594\pi\)
\(504\) 0 0
\(505\) −271281. −1.06374
\(506\) 283060. 1.10555
\(507\) − 91976.9i − 0.357819i
\(508\) 67984.5 0.263441
\(509\) 60266.8i 0.232618i 0.993213 + 0.116309i \(0.0371062\pi\)
−0.993213 + 0.116309i \(0.962894\pi\)
\(510\) − 37903.3i − 0.145726i
\(511\) 0 0
\(512\) −11585.2 −0.0441942
\(513\) 5386.46 0.0204677
\(514\) 32828.8i 0.124259i
\(515\) −391101. −1.47460
\(516\) 140604.i 0.528078i
\(517\) 386497.i 1.44599i
\(518\) 0 0
\(519\) 208448. 0.773861
\(520\) 56478.6 0.208871
\(521\) 287316.i 1.05848i 0.848471 + 0.529242i \(0.177524\pi\)
−0.848471 + 0.529242i \(0.822476\pi\)
\(522\) 47470.3 0.174213
\(523\) 276444.i 1.01066i 0.862927 + 0.505328i \(0.168629\pi\)
−0.862927 + 0.505328i \(0.831371\pi\)
\(524\) − 115778.i − 0.421662i
\(525\) 0 0
\(526\) −341300. −1.23357
\(527\) 163574. 0.588969
\(528\) − 32580.5i − 0.116866i
\(529\) 763619. 2.72876
\(530\) − 148403.i − 0.528311i
\(531\) − 79241.5i − 0.281037i
\(532\) 0 0
\(533\) −106629. −0.375336
\(534\) −52973.0 −0.185768
\(535\) 105302.i 0.367900i
\(536\) 134076. 0.466683
\(537\) 260610.i 0.903739i
\(538\) 77065.1i 0.266252i
\(539\) 0 0
\(540\) −26882.5 −0.0921895
\(541\) −59664.0 −0.203853 −0.101927 0.994792i \(-0.532501\pi\)
−0.101927 + 0.994792i \(0.532501\pi\)
\(542\) − 274480.i − 0.934355i
\(543\) −332420. −1.12742
\(544\) − 19491.4i − 0.0658634i
\(545\) − 352034.i − 1.18520i
\(546\) 0 0
\(547\) 489585. 1.63627 0.818133 0.575030i \(-0.195010\pi\)
0.818133 + 0.575030i \(0.195010\pi\)
\(548\) 102841. 0.342458
\(549\) 17964.4i 0.0596028i
\(550\) −14222.0 −0.0470147
\(551\) − 23865.5i − 0.0786082i
\(552\) − 120103.i − 0.394164i
\(553\) 0 0
\(554\) −84153.6 −0.274191
\(555\) −69973.5 −0.227168
\(556\) − 40075.7i − 0.129638i
\(557\) 508534. 1.63912 0.819558 0.572996i \(-0.194219\pi\)
0.819558 + 0.572996i \(0.194219\pi\)
\(558\) − 116013.i − 0.372595i
\(559\) − 352486.i − 1.12802i
\(560\) 0 0
\(561\) 54814.4 0.174168
\(562\) 345846. 1.09499
\(563\) 119817.i 0.378009i 0.981976 + 0.189004i \(0.0605260\pi\)
−0.981976 + 0.189004i \(0.939474\pi\)
\(564\) 163992. 0.515542
\(565\) 50881.7i 0.159391i
\(566\) 171688.i 0.535927i
\(567\) 0 0
\(568\) 101697. 0.315218
\(569\) −97659.4 −0.301640 −0.150820 0.988561i \(-0.548191\pi\)
−0.150820 + 0.988561i \(0.548191\pi\)
\(570\) 13515.1i 0.0415976i
\(571\) 47651.4 0.146151 0.0730757 0.997326i \(-0.476719\pi\)
0.0730757 + 0.997326i \(0.476719\pi\)
\(572\) 81677.3i 0.249637i
\(573\) 222234.i 0.676863i
\(574\) 0 0
\(575\) −52427.2 −0.158570
\(576\) −13824.0 −0.0416667
\(577\) − 131457.i − 0.394849i −0.980318 0.197425i \(-0.936742\pi\)
0.980318 0.197425i \(-0.0632578\pi\)
\(578\) −203440. −0.608949
\(579\) 45583.7i 0.135973i
\(580\) 119107.i 0.354063i
\(581\) 0 0
\(582\) −28666.8 −0.0846316
\(583\) 214614. 0.631425
\(584\) − 202944.i − 0.595045i
\(585\) 67392.7 0.196925
\(586\) − 83601.0i − 0.243454i
\(587\) 135493.i 0.393224i 0.980481 + 0.196612i \(0.0629940\pi\)
−0.980481 + 0.196612i \(0.937006\pi\)
\(588\) 0 0
\(589\) −58325.0 −0.168122
\(590\) 198823. 0.571167
\(591\) 170564.i 0.488328i
\(592\) −35983.1 −0.102673
\(593\) − 188208.i − 0.535216i −0.963528 0.267608i \(-0.913767\pi\)
0.963528 0.267608i \(-0.0862332\pi\)
\(594\) − 38876.4i − 0.110183i
\(595\) 0 0
\(596\) 203191. 0.572022
\(597\) −137512. −0.385827
\(598\) 301091.i 0.841969i
\(599\) −78404.8 −0.218519 −0.109259 0.994013i \(-0.534848\pi\)
−0.109259 + 0.994013i \(0.534848\pi\)
\(600\) 6034.42i 0.0167623i
\(601\) 254898.i 0.705695i 0.935681 + 0.352848i \(0.114787\pi\)
−0.935681 + 0.352848i \(0.885213\pi\)
\(602\) 0 0
\(603\) 159985. 0.439993
\(604\) −146470. −0.401491
\(605\) − 120782.i − 0.329983i
\(606\) −166461. −0.453281
\(607\) − 565684.i − 1.53531i −0.640863 0.767655i \(-0.721423\pi\)
0.640863 0.767655i \(-0.278577\pi\)
\(608\) 6949.97i 0.0188008i
\(609\) 0 0
\(610\) −45074.0 −0.121134
\(611\) −411118. −1.10124
\(612\) − 23257.9i − 0.0620966i
\(613\) −414221. −1.10233 −0.551164 0.834397i \(-0.685816\pi\)
−0.551164 + 0.834397i \(0.685816\pi\)
\(614\) − 304246.i − 0.807026i
\(615\) 127343.i 0.336686i
\(616\) 0 0
\(617\) 22896.0 0.0601435 0.0300717 0.999548i \(-0.490426\pi\)
0.0300717 + 0.999548i \(0.490426\pi\)
\(618\) −239984. −0.628356
\(619\) 161155.i 0.420593i 0.977638 + 0.210296i \(0.0674429\pi\)
−0.977638 + 0.210296i \(0.932557\pi\)
\(620\) 291085. 0.757245
\(621\) − 143312.i − 0.371621i
\(622\) − 122674.i − 0.317082i
\(623\) 0 0
\(624\) 34655.9 0.0890037
\(625\) −355913. −0.911139
\(626\) − 304753.i − 0.777676i
\(627\) −19545.0 −0.0497165
\(628\) − 281956.i − 0.714926i
\(629\) − 60539.0i − 0.153015i
\(630\) 0 0
\(631\) 489038. 1.22824 0.614121 0.789212i \(-0.289511\pi\)
0.614121 + 0.789212i \(0.289511\pi\)
\(632\) 236385. 0.591814
\(633\) − 348435.i − 0.869590i
\(634\) 441217. 1.09768
\(635\) 203542.i 0.504785i
\(636\) − 91061.5i − 0.225123i
\(637\) 0 0
\(638\) −172248. −0.423168
\(639\) 121349. 0.297190
\(640\) − 34685.5i − 0.0846815i
\(641\) 119340. 0.290448 0.145224 0.989399i \(-0.453610\pi\)
0.145224 + 0.989399i \(0.453610\pi\)
\(642\) 64614.5i 0.156769i
\(643\) 324224.i 0.784194i 0.919924 + 0.392097i \(0.128250\pi\)
−0.919924 + 0.392097i \(0.871750\pi\)
\(644\) 0 0
\(645\) −420960. −1.01186
\(646\) −11692.9 −0.0280192
\(647\) − 460781.i − 1.10074i −0.834920 0.550372i \(-0.814486\pi\)
0.834920 0.550372i \(-0.185514\pi\)
\(648\) −16495.4 −0.0392837
\(649\) 287531.i 0.682645i
\(650\) − 15127.9i − 0.0358057i
\(651\) 0 0
\(652\) 1167.51 0.00274641
\(653\) −559124. −1.31124 −0.655620 0.755091i \(-0.727593\pi\)
−0.655620 + 0.755091i \(0.727593\pi\)
\(654\) − 216012.i − 0.505037i
\(655\) 346634. 0.807957
\(656\) 65484.6i 0.152171i
\(657\) − 242161.i − 0.561014i
\(658\) 0 0
\(659\) −184372. −0.424546 −0.212273 0.977210i \(-0.568087\pi\)
−0.212273 + 0.977210i \(0.568087\pi\)
\(660\) 97544.1 0.223930
\(661\) 79226.4i 0.181329i 0.995881 + 0.0906645i \(0.0288991\pi\)
−0.995881 + 0.0906645i \(0.971101\pi\)
\(662\) −113756. −0.259572
\(663\) 58306.2i 0.132644i
\(664\) 28720.4i 0.0651411i
\(665\) 0 0
\(666\) −42936.5 −0.0968007
\(667\) −634967. −1.42725
\(668\) − 250513.i − 0.561405i
\(669\) −292336. −0.653175
\(670\) 401416.i 0.894222i
\(671\) − 65184.4i − 0.144777i
\(672\) 0 0
\(673\) −482896. −1.06616 −0.533082 0.846064i \(-0.678966\pi\)
−0.533082 + 0.846064i \(0.678966\pi\)
\(674\) −255906. −0.563327
\(675\) 7200.53i 0.0158036i
\(676\) 141608. 0.309880
\(677\) 380329.i 0.829817i 0.909863 + 0.414908i \(0.136186\pi\)
−0.909863 + 0.414908i \(0.863814\pi\)
\(678\) 31221.6i 0.0679196i
\(679\) 0 0
\(680\) 58356.0 0.126202
\(681\) 317735. 0.685127
\(682\) 420956.i 0.905041i
\(683\) −342069. −0.733283 −0.366642 0.930362i \(-0.619493\pi\)
−0.366642 + 0.930362i \(0.619493\pi\)
\(684\) 8293.01i 0.0177256i
\(685\) 307901.i 0.656191i
\(686\) 0 0
\(687\) −175649. −0.372163
\(688\) −216474. −0.457329
\(689\) 228286.i 0.480884i
\(690\) 359582. 0.755266
\(691\) − 73062.6i − 0.153017i −0.997069 0.0765084i \(-0.975623\pi\)
0.997069 0.0765084i \(-0.0243772\pi\)
\(692\) 320927.i 0.670183i
\(693\) 0 0
\(694\) −90720.3 −0.188359
\(695\) 119984. 0.248402
\(696\) 73085.3i 0.150873i
\(697\) −110173. −0.226783
\(698\) − 330313.i − 0.677977i
\(699\) − 45140.3i − 0.0923869i
\(700\) 0 0
\(701\) −288287. −0.586664 −0.293332 0.956011i \(-0.594764\pi\)
−0.293332 + 0.956011i \(0.594764\pi\)
\(702\) 41352.9 0.0839135
\(703\) 21586.2i 0.0436783i
\(704\) 50160.9 0.101209
\(705\) 490982.i 0.987842i
\(706\) − 352586.i − 0.707386i
\(707\) 0 0
\(708\) 122000. 0.243385
\(709\) −310512. −0.617711 −0.308856 0.951109i \(-0.599946\pi\)
−0.308856 + 0.951109i \(0.599946\pi\)
\(710\) 304475.i 0.603996i
\(711\) 282064. 0.557968
\(712\) − 81557.3i − 0.160880i
\(713\) 1.55179e6i 3.05250i
\(714\) 0 0
\(715\) −244537. −0.478335
\(716\) −401236. −0.782661
\(717\) 81657.7i 0.158840i
\(718\) −281265. −0.545590
\(719\) − 10084.7i − 0.0195077i −0.999952 0.00975383i \(-0.996895\pi\)
0.999952 0.00975383i \(-0.00310479\pi\)
\(720\) − 41388.3i − 0.0798385i
\(721\) 0 0
\(722\) −364434. −0.699109
\(723\) −28468.8 −0.0544619
\(724\) − 511794.i − 0.976378i
\(725\) 31903.0 0.0606954
\(726\) − 74113.3i − 0.140612i
\(727\) 638552.i 1.20817i 0.796921 + 0.604084i \(0.206461\pi\)
−0.796921 + 0.604084i \(0.793539\pi\)
\(728\) 0 0
\(729\) −19683.0 −0.0370370
\(730\) 607602. 1.14018
\(731\) − 364203.i − 0.681567i
\(732\) −27657.9 −0.0516176
\(733\) 826084.i 1.53750i 0.639547 + 0.768752i \(0.279122\pi\)
−0.639547 + 0.768752i \(0.720878\pi\)
\(734\) − 251488.i − 0.466795i
\(735\) 0 0
\(736\) 184911. 0.341356
\(737\) −580513. −1.06875
\(738\) 78139.1i 0.143468i
\(739\) 769990. 1.40992 0.704962 0.709245i \(-0.250964\pi\)
0.704962 + 0.709245i \(0.250964\pi\)
\(740\) − 107731.i − 0.196733i
\(741\) − 20790.1i − 0.0378634i
\(742\) 0 0
\(743\) −611777. −1.10819 −0.554097 0.832452i \(-0.686936\pi\)
−0.554097 + 0.832452i \(0.686936\pi\)
\(744\) 178613. 0.322677
\(745\) 608343.i 1.09606i
\(746\) 258385. 0.464290
\(747\) 34270.4i 0.0614156i
\(748\) 84392.3i 0.150834i
\(749\) 0 0
\(750\) −238076. −0.423246
\(751\) 370622. 0.657129 0.328565 0.944481i \(-0.393435\pi\)
0.328565 + 0.944481i \(0.393435\pi\)
\(752\) 252482.i 0.446473i
\(753\) 126531. 0.223155
\(754\) − 183220.i − 0.322278i
\(755\) − 438523.i − 0.769305i
\(756\) 0 0
\(757\) 298778. 0.521384 0.260692 0.965422i \(-0.416049\pi\)
0.260692 + 0.965422i \(0.416049\pi\)
\(758\) −17899.7 −0.0311536
\(759\) 520014.i 0.902676i
\(760\) −20807.8 −0.0360246
\(761\) − 803142.i − 1.38683i −0.720539 0.693414i \(-0.756106\pi\)
0.720539 0.693414i \(-0.243894\pi\)
\(762\) 124896.i 0.215098i
\(763\) 0 0
\(764\) −342151. −0.586180
\(765\) 69632.9 0.118985
\(766\) − 38002.6i − 0.0647674i
\(767\) −305847. −0.519893
\(768\) − 21283.4i − 0.0360844i
\(769\) − 452541.i − 0.765253i −0.923903 0.382626i \(-0.875020\pi\)
0.923903 0.382626i \(-0.124980\pi\)
\(770\) 0 0
\(771\) −60310.4 −0.101457
\(772\) −70180.7 −0.117756
\(773\) 1.04853e6i 1.75478i 0.479774 + 0.877392i \(0.340719\pi\)
−0.479774 + 0.877392i \(0.659281\pi\)
\(774\) −258306. −0.431174
\(775\) − 77967.8i − 0.129811i
\(776\) − 44135.4i − 0.0732932i
\(777\) 0 0
\(778\) −249740. −0.412599
\(779\) 39284.2 0.0647355
\(780\) 103758.i 0.170542i
\(781\) −440320. −0.721882
\(782\) 311100.i 0.508729i
\(783\) 87208.5i 0.142244i
\(784\) 0 0
\(785\) 844159. 1.36989
\(786\) 212698. 0.344286
\(787\) 195967.i 0.316398i 0.987407 + 0.158199i \(0.0505687\pi\)
−0.987407 + 0.158199i \(0.949431\pi\)
\(788\) −262600. −0.422904
\(789\) − 627007.i − 1.00721i
\(790\) 707722.i 1.13399i
\(791\) 0 0
\(792\) 59854.2 0.0954210
\(793\) 69336.7 0.110260
\(794\) 612601.i 0.971710i
\(795\) 272633. 0.431364
\(796\) − 211714.i − 0.334136i
\(797\) − 66453.8i − 0.104617i −0.998631 0.0523086i \(-0.983342\pi\)
0.998631 0.0523086i \(-0.0166579\pi\)
\(798\) 0 0
\(799\) −424784. −0.665387
\(800\) −9290.60 −0.0145166
\(801\) − 97317.6i − 0.151679i
\(802\) 14614.0 0.0227207
\(803\) 878691.i 1.36272i
\(804\) 246314.i 0.381045i
\(805\) 0 0
\(806\) −447772. −0.689266
\(807\) −141578. −0.217394
\(808\) − 256284.i − 0.392553i
\(809\) 82304.6 0.125755 0.0628777 0.998021i \(-0.479972\pi\)
0.0628777 + 0.998021i \(0.479972\pi\)
\(810\) − 49386.2i − 0.0752724i
\(811\) − 469097.i − 0.713215i −0.934254 0.356608i \(-0.883933\pi\)
0.934254 0.356608i \(-0.116067\pi\)
\(812\) 0 0
\(813\) 504252. 0.762898
\(814\) 155797. 0.235131
\(815\) 3495.45i 0.00526245i
\(816\) 35807.9 0.0537773
\(817\) 129863.i 0.194554i
\(818\) − 110056.i − 0.164478i
\(819\) 0 0
\(820\) −196057. −0.291578
\(821\) −5379.52 −0.00798099 −0.00399050 0.999992i \(-0.501270\pi\)
−0.00399050 + 0.999992i \(0.501270\pi\)
\(822\) 188932.i 0.279615i
\(823\) −969806. −1.43181 −0.715905 0.698198i \(-0.753986\pi\)
−0.715905 + 0.698198i \(0.753986\pi\)
\(824\) − 369480.i − 0.544172i
\(825\) − 26127.4i − 0.0383874i
\(826\) 0 0
\(827\) 838747. 1.22637 0.613183 0.789941i \(-0.289889\pi\)
0.613183 + 0.789941i \(0.289889\pi\)
\(828\) 220644. 0.321833
\(829\) − 883703.i − 1.28587i −0.765920 0.642935i \(-0.777716\pi\)
0.765920 0.642935i \(-0.222284\pi\)
\(830\) −85987.3 −0.124818
\(831\) − 154600.i − 0.223876i
\(832\) 53356.3i 0.0770795i
\(833\) 0 0
\(834\) 73623.7 0.105849
\(835\) 750020. 1.07572
\(836\) − 30091.5i − 0.0430558i
\(837\) 213129. 0.304222
\(838\) 745080.i 1.06100i
\(839\) 1.12626e6i 1.59998i 0.600011 + 0.799992i \(0.295163\pi\)
−0.600011 + 0.799992i \(0.704837\pi\)
\(840\) 0 0
\(841\) −320891. −0.453696
\(842\) 46315.8 0.0653288
\(843\) 635360.i 0.894056i
\(844\) 536451. 0.753087
\(845\) 423965.i 0.593768i
\(846\) 301272.i 0.420938i
\(847\) 0 0
\(848\) 140198. 0.194963
\(849\) −315410. −0.437583
\(850\) − 15630.8i − 0.0216343i
\(851\) 574323. 0.793043
\(852\) 186829.i 0.257374i
\(853\) − 805486.i − 1.10703i −0.832839 0.553516i \(-0.813286\pi\)
0.832839 0.553516i \(-0.186714\pi\)
\(854\) 0 0
\(855\) −24828.8 −0.0339643
\(856\) −99480.6 −0.135766
\(857\) − 980477.i − 1.33498i −0.744618 0.667491i \(-0.767368\pi\)
0.744618 0.667491i \(-0.232632\pi\)
\(858\) −150051. −0.203828
\(859\) 433780.i 0.587873i 0.955825 + 0.293936i \(0.0949654\pi\)
−0.955825 + 0.293936i \(0.905035\pi\)
\(860\) − 648111.i − 0.876299i
\(861\) 0 0
\(862\) 708284. 0.953220
\(863\) −303157. −0.407048 −0.203524 0.979070i \(-0.565240\pi\)
−0.203524 + 0.979070i \(0.565240\pi\)
\(864\) − 25396.3i − 0.0340207i
\(865\) −960836. −1.28415
\(866\) − 258787.i − 0.345070i
\(867\) − 373744.i − 0.497205i
\(868\) 0 0
\(869\) −1.02348e6 −1.35532
\(870\) −218813. −0.289091
\(871\) − 617493.i − 0.813946i
\(872\) 332573. 0.437375
\(873\) − 52664.2i − 0.0691015i
\(874\) − 110928.i − 0.145217i
\(875\) 0 0
\(876\) 372831. 0.485852
\(877\) 355000. 0.461562 0.230781 0.973006i \(-0.425872\pi\)
0.230781 + 0.973006i \(0.425872\pi\)
\(878\) 25058.1i 0.0325057i
\(879\) 153585. 0.198779
\(880\) 150179.i 0.193929i
\(881\) 320002.i 0.412288i 0.978522 + 0.206144i \(0.0660915\pi\)
−0.978522 + 0.206144i \(0.933909\pi\)
\(882\) 0 0
\(883\) −1.08605e6 −1.39293 −0.696467 0.717589i \(-0.745246\pi\)
−0.696467 + 0.717589i \(0.745246\pi\)
\(884\) −89768.2 −0.114873
\(885\) 365262.i 0.466356i
\(886\) −849120. −1.08169
\(887\) − 1.28423e6i − 1.63229i −0.577848 0.816144i \(-0.696107\pi\)
0.577848 0.816144i \(-0.303893\pi\)
\(888\) − 66105.1i − 0.0838319i
\(889\) 0 0
\(890\) 244178. 0.308266
\(891\) 71420.5 0.0899638
\(892\) − 450080.i − 0.565666i
\(893\) 151464. 0.189935
\(894\) 373286.i 0.467054i
\(895\) − 1.20128e6i − 1.49967i
\(896\) 0 0
\(897\) −553140. −0.687465
\(898\) −330548. −0.409904
\(899\) − 944299.i − 1.16840i
\(900\) −11085.9 −0.0136863
\(901\) 235874.i 0.290557i
\(902\) − 283531.i − 0.348487i
\(903\) 0 0
\(904\) −48068.7 −0.0588201
\(905\) 1.53228e6 1.87086
\(906\) − 269083.i − 0.327816i
\(907\) 48435.5 0.0588775 0.0294388 0.999567i \(-0.490628\pi\)
0.0294388 + 0.999567i \(0.490628\pi\)
\(908\) 489185.i 0.593337i
\(909\) − 305808.i − 0.370102i
\(910\) 0 0
\(911\) 727675. 0.876801 0.438400 0.898780i \(-0.355545\pi\)
0.438400 + 0.898780i \(0.355545\pi\)
\(912\) −12767.9 −0.0153508
\(913\) − 124352.i − 0.149180i
\(914\) −208584. −0.249683
\(915\) − 82806.3i − 0.0989056i
\(916\) − 270430.i − 0.322303i
\(917\) 0 0
\(918\) 42727.5 0.0507017
\(919\) 87715.9 0.103860 0.0519299 0.998651i \(-0.483463\pi\)
0.0519299 + 0.998651i \(0.483463\pi\)
\(920\) 553613.i 0.654080i
\(921\) 558935. 0.658934
\(922\) − 249521.i − 0.293526i
\(923\) − 468369.i − 0.549775i
\(924\) 0 0
\(925\) −28856.1 −0.0337251
\(926\) −842533. −0.982573
\(927\) − 440879.i − 0.513051i
\(928\) −112522. −0.130660
\(929\) − 901913.i − 1.04504i −0.852627 0.522520i \(-0.824992\pi\)
0.852627 0.522520i \(-0.175008\pi\)
\(930\) 534757.i 0.618288i
\(931\) 0 0
\(932\) 69498.1 0.0800094
\(933\) 225366. 0.258896
\(934\) − 530244.i − 0.607830i
\(935\) −252666. −0.289017
\(936\) 63667.0i 0.0726712i
\(937\) − 1.24186e6i − 1.41447i −0.706978 0.707236i \(-0.749942\pi\)
0.706978 0.707236i \(-0.250058\pi\)
\(938\) 0 0
\(939\) 559867. 0.634970
\(940\) −755917. −0.855496
\(941\) 426245.i 0.481371i 0.970603 + 0.240685i \(0.0773722\pi\)
−0.970603 + 0.240685i \(0.922628\pi\)
\(942\) 517985. 0.583735
\(943\) − 1.04520e6i − 1.17537i
\(944\) 187832.i 0.210778i
\(945\) 0 0
\(946\) 937274. 1.04733
\(947\) 27330.4 0.0304752 0.0152376 0.999884i \(-0.495150\pi\)
0.0152376 + 0.999884i \(0.495150\pi\)
\(948\) 434266.i 0.483214i
\(949\) −934665. −1.03782
\(950\) 5573.42i 0.00617554i
\(951\) 810568.i 0.896248i
\(952\) 0 0
\(953\) 1.30093e6 1.43242 0.716208 0.697887i \(-0.245876\pi\)
0.716208 + 0.697887i \(0.245876\pi\)
\(954\) 167291. 0.183813
\(955\) − 1.02438e6i − 1.12319i
\(956\) −125720. −0.137559
\(957\) − 316439.i − 0.345515i
\(958\) 340794.i 0.371331i
\(959\) 0 0
\(960\) 63721.4 0.0691421
\(961\) −1.38425e6 −1.49888
\(962\) 165721.i 0.179072i
\(963\) −118705. −0.128001
\(964\) − 43830.6i − 0.0471654i
\(965\) − 210117.i − 0.225635i
\(966\) 0 0
\(967\) 463079. 0.495225 0.247612 0.968859i \(-0.420354\pi\)
0.247612 + 0.968859i \(0.420354\pi\)
\(968\) 114105. 0.121774
\(969\) − 21481.1i − 0.0228776i
\(970\) 132139. 0.140439
\(971\) − 63188.7i − 0.0670195i −0.999438 0.0335098i \(-0.989332\pi\)
0.999438 0.0335098i \(-0.0106685\pi\)
\(972\) − 30304.0i − 0.0320750i
\(973\) 0 0
\(974\) 756305. 0.797221
\(975\) 27791.8 0.0292353
\(976\) − 42582.2i − 0.0447021i
\(977\) 1.43899e6 1.50754 0.753772 0.657136i \(-0.228232\pi\)
0.753772 + 0.657136i \(0.228232\pi\)
\(978\) 2144.85i 0.00224243i
\(979\) 353121.i 0.368432i
\(980\) 0 0
\(981\) 396840. 0.412361
\(982\) −504490. −0.523154
\(983\) 400596.i 0.414572i 0.978280 + 0.207286i \(0.0664630\pi\)
−0.978280 + 0.207286i \(0.933537\pi\)
\(984\) −120303. −0.124247
\(985\) − 786209.i − 0.810336i
\(986\) − 189311.i − 0.194725i
\(987\) 0 0
\(988\) 32008.4 0.0327906
\(989\) 3.45512e6 3.53241
\(990\) 179200.i 0.182838i
\(991\) 198549. 0.202172 0.101086 0.994878i \(-0.467768\pi\)
0.101086 + 0.994878i \(0.467768\pi\)
\(992\) 274993.i 0.279446i
\(993\) − 208983.i − 0.211939i
\(994\) 0 0
\(995\) 633858. 0.640245
\(996\) −52762.8 −0.0531875
\(997\) 914786.i 0.920299i 0.887841 + 0.460150i \(0.152204\pi\)
−0.887841 + 0.460150i \(0.847796\pi\)
\(998\) −745988. −0.748981
\(999\) − 78879.5i − 0.0790375i
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 294.5.c.a.97.1 4
3.2 odd 2 882.5.c.a.685.3 4
7.2 even 3 294.5.g.c.31.2 4
7.3 odd 6 294.5.g.c.19.2 4
7.4 even 3 42.5.g.a.19.2 4
7.5 odd 6 42.5.g.a.31.2 yes 4
7.6 odd 2 inner 294.5.c.a.97.2 4
21.5 even 6 126.5.n.b.73.1 4
21.11 odd 6 126.5.n.b.19.1 4
21.20 even 2 882.5.c.a.685.4 4
28.11 odd 6 336.5.bh.d.145.1 4
28.19 even 6 336.5.bh.d.241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.5.g.a.19.2 4 7.4 even 3
42.5.g.a.31.2 yes 4 7.5 odd 6
126.5.n.b.19.1 4 21.11 odd 6
126.5.n.b.73.1 4 21.5 even 6
294.5.c.a.97.1 4 1.1 even 1 trivial
294.5.c.a.97.2 4 7.6 odd 2 inner
294.5.g.c.19.2 4 7.3 odd 6
294.5.g.c.31.2 4 7.2 even 3
336.5.bh.d.145.1 4 28.11 odd 6
336.5.bh.d.241.1 4 28.19 even 6
882.5.c.a.685.3 4 3.2 odd 2
882.5.c.a.685.4 4 21.20 even 2