Properties

Label 1078.6.a.f.1.1
Level $1078$
Weight $6$
Character 1078.1
Self dual yes
Analytic conductor $172.894$
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] = [1078,6,Mod(1,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1078.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1078.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: \(172.893757758\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1078.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +29.0000 q^{3} +16.0000 q^{4} +31.0000 q^{5} +116.000 q^{6} +64.0000 q^{8} +598.000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +29.0000 q^{3} +16.0000 q^{4} +31.0000 q^{5} +116.000 q^{6} +64.0000 q^{8} +598.000 q^{9} +124.000 q^{10} +121.000 q^{11} +464.000 q^{12} -112.000 q^{13} +899.000 q^{15} +256.000 q^{16} +1142.00 q^{17} +2392.00 q^{18} +612.000 q^{19} +496.000 q^{20} +484.000 q^{22} -1941.00 q^{23} +1856.00 q^{24} -2164.00 q^{25} -448.000 q^{26} +10295.0 q^{27} +1192.00 q^{29} +3596.00 q^{30} +1037.00 q^{31} +1024.00 q^{32} +3509.00 q^{33} +4568.00 q^{34} +9568.00 q^{36} +8083.00 q^{37} +2448.00 q^{38} -3248.00 q^{39} +1984.00 q^{40} +10444.0 q^{41} +58.0000 q^{43} +1936.00 q^{44} +18538.0 q^{45} -7764.00 q^{46} -8656.00 q^{47} +7424.00 q^{48} -8656.00 q^{50} +33118.0 q^{51} -1792.00 q^{52} -20318.0 q^{53} +41180.0 q^{54} +3751.00 q^{55} +17748.0 q^{57} +4768.00 q^{58} +21351.0 q^{59} +14384.0 q^{60} -47044.0 q^{61} +4148.00 q^{62} +4096.00 q^{64} -3472.00 q^{65} +14036.0 q^{66} +48093.0 q^{67} +18272.0 q^{68} -56289.0 q^{69} -24967.0 q^{71} +38272.0 q^{72} +42288.0 q^{73} +32332.0 q^{74} -62756.0 q^{75} +9792.00 q^{76} -12992.0 q^{78} -72410.0 q^{79} +7936.00 q^{80} +153241. q^{81} +41776.0 q^{82} +15806.0 q^{83} +35402.0 q^{85} +232.000 q^{86} +34568.0 q^{87} +7744.00 q^{88} +114761. q^{89} +74152.0 q^{90} -31056.0 q^{92} +30073.0 q^{93} -34624.0 q^{94} +18972.0 q^{95} +29696.0 q^{96} +5159.00 q^{97} +72358.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 29.0000 1.86035 0.930175 0.367115i \(-0.119655\pi\)
0.930175 + 0.367115i \(0.119655\pi\)
\(4\) 16.0000 0.500000
\(5\) 31.0000 0.554545 0.277272 0.960791i \(-0.410570\pi\)
0.277272 + 0.960791i \(0.410570\pi\)
\(6\) 116.000 1.31547
\(7\) 0 0
\(8\) 64.0000 0.353553
\(9\) 598.000 2.46091
\(10\) 124.000 0.392122
\(11\) 121.000 0.301511
\(12\) 464.000 0.930175
\(13\) −112.000 −0.183806 −0.0919030 0.995768i \(-0.529295\pi\)
−0.0919030 + 0.995768i \(0.529295\pi\)
\(14\) 0 0
\(15\) 899.000 1.03165
\(16\) 256.000 0.250000
\(17\) 1142.00 0.958393 0.479197 0.877708i \(-0.340928\pi\)
0.479197 + 0.877708i \(0.340928\pi\)
\(18\) 2392.00 1.74012
\(19\) 612.000 0.388926 0.194463 0.980910i \(-0.437704\pi\)
0.194463 + 0.980910i \(0.437704\pi\)
\(20\) 496.000 0.277272
\(21\) 0 0
\(22\) 484.000 0.213201
\(23\) −1941.00 −0.765078 −0.382539 0.923939i \(-0.624950\pi\)
−0.382539 + 0.923939i \(0.624950\pi\)
\(24\) 1856.00 0.657733
\(25\) −2164.00 −0.692480
\(26\) −448.000 −0.129970
\(27\) 10295.0 2.71780
\(28\) 0 0
\(29\) 1192.00 0.263197 0.131599 0.991303i \(-0.457989\pi\)
0.131599 + 0.991303i \(0.457989\pi\)
\(30\) 3596.00 0.729485
\(31\) 1037.00 0.193809 0.0969046 0.995294i \(-0.469106\pi\)
0.0969046 + 0.995294i \(0.469106\pi\)
\(32\) 1024.00 0.176777
\(33\) 3509.00 0.560917
\(34\) 4568.00 0.677686
\(35\) 0 0
\(36\) 9568.00 1.23045
\(37\) 8083.00 0.970663 0.485331 0.874330i \(-0.338699\pi\)
0.485331 + 0.874330i \(0.338699\pi\)
\(38\) 2448.00 0.275012
\(39\) −3248.00 −0.341944
\(40\) 1984.00 0.196061
\(41\) 10444.0 0.970303 0.485151 0.874430i \(-0.338765\pi\)
0.485151 + 0.874430i \(0.338765\pi\)
\(42\) 0 0
\(43\) 58.0000 0.00478362 0.00239181 0.999997i \(-0.499239\pi\)
0.00239181 + 0.999997i \(0.499239\pi\)
\(44\) 1936.00 0.150756
\(45\) 18538.0 1.36468
\(46\) −7764.00 −0.540992
\(47\) −8656.00 −0.571574 −0.285787 0.958293i \(-0.592255\pi\)
−0.285787 + 0.958293i \(0.592255\pi\)
\(48\) 7424.00 0.465088
\(49\) 0 0
\(50\) −8656.00 −0.489657
\(51\) 33118.0 1.78295
\(52\) −1792.00 −0.0919030
\(53\) −20318.0 −0.993554 −0.496777 0.867878i \(-0.665483\pi\)
−0.496777 + 0.867878i \(0.665483\pi\)
\(54\) 41180.0 1.92177
\(55\) 3751.00 0.167202
\(56\) 0 0
\(57\) 17748.0 0.723540
\(58\) 4768.00 0.186109
\(59\) 21351.0 0.798524 0.399262 0.916837i \(-0.369266\pi\)
0.399262 + 0.916837i \(0.369266\pi\)
\(60\) 14384.0 0.515824
\(61\) −47044.0 −1.61875 −0.809375 0.587293i \(-0.800194\pi\)
−0.809375 + 0.587293i \(0.800194\pi\)
\(62\) 4148.00 0.137044
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −3472.00 −0.101929
\(66\) 14036.0 0.396628
\(67\) 48093.0 1.30887 0.654433 0.756120i \(-0.272908\pi\)
0.654433 + 0.756120i \(0.272908\pi\)
\(68\) 18272.0 0.479197
\(69\) −56289.0 −1.42331
\(70\) 0 0
\(71\) −24967.0 −0.587788 −0.293894 0.955838i \(-0.594951\pi\)
−0.293894 + 0.955838i \(0.594951\pi\)
\(72\) 38272.0 0.870061
\(73\) 42288.0 0.928774 0.464387 0.885632i \(-0.346275\pi\)
0.464387 + 0.885632i \(0.346275\pi\)
\(74\) 32332.0 0.686362
\(75\) −62756.0 −1.28826
\(76\) 9792.00 0.194463
\(77\) 0 0
\(78\) −12992.0 −0.241791
\(79\) −72410.0 −1.30536 −0.652681 0.757633i \(-0.726356\pi\)
−0.652681 + 0.757633i \(0.726356\pi\)
\(80\) 7936.00 0.138636
\(81\) 153241. 2.59515
\(82\) 41776.0 0.686108
\(83\) 15806.0 0.251841 0.125921 0.992040i \(-0.459812\pi\)
0.125921 + 0.992040i \(0.459812\pi\)
\(84\) 0 0
\(85\) 35402.0 0.531472
\(86\) 232.000 0.00338253
\(87\) 34568.0 0.489639
\(88\) 7744.00 0.106600
\(89\) 114761. 1.53575 0.767873 0.640602i \(-0.221315\pi\)
0.767873 + 0.640602i \(0.221315\pi\)
\(90\) 74152.0 0.964976
\(91\) 0 0
\(92\) −31056.0 −0.382539
\(93\) 30073.0 0.360553
\(94\) −34624.0 −0.404164
\(95\) 18972.0 0.215677
\(96\) 29696.0 0.328867
\(97\) 5159.00 0.0556719 0.0278360 0.999613i \(-0.491138\pi\)
0.0278360 + 0.999613i \(0.491138\pi\)
\(98\) 0 0
\(99\) 72358.0 0.741991
\(100\) −34624.0 −0.346240
\(101\) 61426.0 0.599168 0.299584 0.954070i \(-0.403152\pi\)
0.299584 + 0.954070i \(0.403152\pi\)
\(102\) 132472. 1.26073
\(103\) −185896. −1.72654 −0.863271 0.504741i \(-0.831588\pi\)
−0.863271 + 0.504741i \(0.831588\pi\)
\(104\) −7168.00 −0.0649852
\(105\) 0 0
\(106\) −81272.0 −0.702548
\(107\) −23970.0 −0.202399 −0.101200 0.994866i \(-0.532268\pi\)
−0.101200 + 0.994866i \(0.532268\pi\)
\(108\) 164720. 1.35890
\(109\) −56326.0 −0.454091 −0.227045 0.973884i \(-0.572907\pi\)
−0.227045 + 0.973884i \(0.572907\pi\)
\(110\) 15004.0 0.118229
\(111\) 234407. 1.80577
\(112\) 0 0
\(113\) −261903. −1.92950 −0.964749 0.263171i \(-0.915231\pi\)
−0.964749 + 0.263171i \(0.915231\pi\)
\(114\) 70992.0 0.511620
\(115\) −60171.0 −0.424270
\(116\) 19072.0 0.131599
\(117\) −66976.0 −0.452329
\(118\) 85404.0 0.564642
\(119\) 0 0
\(120\) 57536.0 0.364743
\(121\) 14641.0 0.0909091
\(122\) −188176. −1.14463
\(123\) 302876. 1.80510
\(124\) 16592.0 0.0969046
\(125\) −163959. −0.938556
\(126\) 0 0
\(127\) 87404.0 0.480864 0.240432 0.970666i \(-0.422711\pi\)
0.240432 + 0.970666i \(0.422711\pi\)
\(128\) 16384.0 0.0883883
\(129\) 1682.00 0.00889922
\(130\) −13888.0 −0.0720744
\(131\) −265122. −1.34979 −0.674897 0.737912i \(-0.735812\pi\)
−0.674897 + 0.737912i \(0.735812\pi\)
\(132\) 56144.0 0.280458
\(133\) 0 0
\(134\) 192372. 0.925507
\(135\) 319145. 1.50714
\(136\) 73088.0 0.338843
\(137\) 245857. 1.11913 0.559566 0.828786i \(-0.310968\pi\)
0.559566 + 0.828786i \(0.310968\pi\)
\(138\) −225156. −1.00644
\(139\) 363594. 1.59617 0.798086 0.602544i \(-0.205846\pi\)
0.798086 + 0.602544i \(0.205846\pi\)
\(140\) 0 0
\(141\) −251024. −1.06333
\(142\) −99868.0 −0.415629
\(143\) −13552.0 −0.0554196
\(144\) 153088. 0.615226
\(145\) 36952.0 0.145955
\(146\) 169152. 0.656742
\(147\) 0 0
\(148\) 129328. 0.485331
\(149\) −55750.0 −0.205721 −0.102861 0.994696i \(-0.532800\pi\)
−0.102861 + 0.994696i \(0.532800\pi\)
\(150\) −251024. −0.910934
\(151\) 65642.0 0.234282 0.117141 0.993115i \(-0.462627\pi\)
0.117141 + 0.993115i \(0.462627\pi\)
\(152\) 39168.0 0.137506
\(153\) 682916. 2.35852
\(154\) 0 0
\(155\) 32147.0 0.107476
\(156\) −51968.0 −0.170972
\(157\) 275367. 0.891585 0.445793 0.895136i \(-0.352922\pi\)
0.445793 + 0.895136i \(0.352922\pi\)
\(158\) −289640. −0.923030
\(159\) −589222. −1.84836
\(160\) 31744.0 0.0980306
\(161\) 0 0
\(162\) 612964. 1.83505
\(163\) 291940. 0.860646 0.430323 0.902675i \(-0.358400\pi\)
0.430323 + 0.902675i \(0.358400\pi\)
\(164\) 167104. 0.485151
\(165\) 108779. 0.311054
\(166\) 63224.0 0.178079
\(167\) 337344. 0.936013 0.468006 0.883725i \(-0.344972\pi\)
0.468006 + 0.883725i \(0.344972\pi\)
\(168\) 0 0
\(169\) −358749. −0.966215
\(170\) 141608. 0.375808
\(171\) 365976. 0.957111
\(172\) 928.000 0.00239181
\(173\) 116742. 0.296560 0.148280 0.988945i \(-0.452626\pi\)
0.148280 + 0.988945i \(0.452626\pi\)
\(174\) 138272. 0.346227
\(175\) 0 0
\(176\) 30976.0 0.0753778
\(177\) 619179. 1.48554
\(178\) 459044. 1.08594
\(179\) −19107.0 −0.0445718 −0.0222859 0.999752i \(-0.507094\pi\)
−0.0222859 + 0.999752i \(0.507094\pi\)
\(180\) 296608. 0.682341
\(181\) 16177.0 0.0367030 0.0183515 0.999832i \(-0.494158\pi\)
0.0183515 + 0.999832i \(0.494158\pi\)
\(182\) 0 0
\(183\) −1.36428e6 −3.01144
\(184\) −124224. −0.270496
\(185\) 250573. 0.538276
\(186\) 120292. 0.254950
\(187\) 138182. 0.288966
\(188\) −138496. −0.285787
\(189\) 0 0
\(190\) 75888.0 0.152507
\(191\) 685333. 1.35931 0.679655 0.733532i \(-0.262130\pi\)
0.679655 + 0.733532i \(0.262130\pi\)
\(192\) 118784. 0.232544
\(193\) −309292. −0.597689 −0.298845 0.954302i \(-0.596601\pi\)
−0.298845 + 0.954302i \(0.596601\pi\)
\(194\) 20636.0 0.0393660
\(195\) −100688. −0.189623
\(196\) 0 0
\(197\) −120930. −0.222008 −0.111004 0.993820i \(-0.535407\pi\)
−0.111004 + 0.993820i \(0.535407\pi\)
\(198\) 289432. 0.524667
\(199\) −915536. −1.63886 −0.819432 0.573177i \(-0.805711\pi\)
−0.819432 + 0.573177i \(0.805711\pi\)
\(200\) −138496. −0.244829
\(201\) 1.39470e6 2.43495
\(202\) 245704. 0.423676
\(203\) 0 0
\(204\) 529888. 0.891474
\(205\) 323764. 0.538076
\(206\) −743584. −1.22085
\(207\) −1.16072e6 −1.88279
\(208\) −28672.0 −0.0459515
\(209\) 74052.0 0.117266
\(210\) 0 0
\(211\) −134580. −0.208101 −0.104051 0.994572i \(-0.533180\pi\)
−0.104051 + 0.994572i \(0.533180\pi\)
\(212\) −325088. −0.496777
\(213\) −724043. −1.09349
\(214\) −95880.0 −0.143118
\(215\) 1798.00 0.00265273
\(216\) 658880. 0.960886
\(217\) 0 0
\(218\) −225304. −0.321091
\(219\) 1.22635e6 1.72785
\(220\) 60016.0 0.0836008
\(221\) −127904. −0.176158
\(222\) 937628. 1.27687
\(223\) −468839. −0.631337 −0.315669 0.948869i \(-0.602229\pi\)
−0.315669 + 0.948869i \(0.602229\pi\)
\(224\) 0 0
\(225\) −1.29407e6 −1.70413
\(226\) −1.04761e6 −1.36436
\(227\) −275022. −0.354244 −0.177122 0.984189i \(-0.556679\pi\)
−0.177122 + 0.984189i \(0.556679\pi\)
\(228\) 283968. 0.361770
\(229\) 642281. 0.809350 0.404675 0.914461i \(-0.367385\pi\)
0.404675 + 0.914461i \(0.367385\pi\)
\(230\) −240684. −0.300004
\(231\) 0 0
\(232\) 76288.0 0.0930543
\(233\) −1.50485e6 −1.81595 −0.907973 0.419029i \(-0.862371\pi\)
−0.907973 + 0.419029i \(0.862371\pi\)
\(234\) −267904. −0.319845
\(235\) −268336. −0.316964
\(236\) 341616. 0.399262
\(237\) −2.09989e6 −2.42843
\(238\) 0 0
\(239\) −304694. −0.345040 −0.172520 0.985006i \(-0.555191\pi\)
−0.172520 + 0.985006i \(0.555191\pi\)
\(240\) 230144. 0.257912
\(241\) −1.27181e6 −1.41052 −0.705260 0.708949i \(-0.749170\pi\)
−0.705260 + 0.708949i \(0.749170\pi\)
\(242\) 58564.0 0.0642824
\(243\) 1.94230e6 2.11009
\(244\) −752704. −0.809375
\(245\) 0 0
\(246\) 1.21150e6 1.27640
\(247\) −68544.0 −0.0714870
\(248\) 66368.0 0.0685219
\(249\) 458374. 0.468513
\(250\) −655836. −0.663659
\(251\) −629965. −0.631149 −0.315575 0.948901i \(-0.602197\pi\)
−0.315575 + 0.948901i \(0.602197\pi\)
\(252\) 0 0
\(253\) −234861. −0.230680
\(254\) 349616. 0.340022
\(255\) 1.02666e6 0.988725
\(256\) 65536.0 0.0625000
\(257\) −544086. −0.513848 −0.256924 0.966432i \(-0.582709\pi\)
−0.256924 + 0.966432i \(0.582709\pi\)
\(258\) 6728.00 0.00629270
\(259\) 0 0
\(260\) −55552.0 −0.0509643
\(261\) 712816. 0.647703
\(262\) −1.06049e6 −0.954449
\(263\) −1.98933e6 −1.77345 −0.886724 0.462300i \(-0.847024\pi\)
−0.886724 + 0.462300i \(0.847024\pi\)
\(264\) 224576. 0.198314
\(265\) −629858. −0.550970
\(266\) 0 0
\(267\) 3.32807e6 2.85703
\(268\) 769488. 0.654433
\(269\) −1.75446e6 −1.47830 −0.739149 0.673541i \(-0.764772\pi\)
−0.739149 + 0.673541i \(0.764772\pi\)
\(270\) 1.27658e6 1.06571
\(271\) 1.65824e6 1.37159 0.685795 0.727795i \(-0.259455\pi\)
0.685795 + 0.727795i \(0.259455\pi\)
\(272\) 292352. 0.239598
\(273\) 0 0
\(274\) 983428. 0.791346
\(275\) −261844. −0.208791
\(276\) −900624. −0.711657
\(277\) −42634.0 −0.0333854 −0.0166927 0.999861i \(-0.505314\pi\)
−0.0166927 + 0.999861i \(0.505314\pi\)
\(278\) 1.45438e6 1.12866
\(279\) 620126. 0.476946
\(280\) 0 0
\(281\) 319510. 0.241390 0.120695 0.992690i \(-0.461488\pi\)
0.120695 + 0.992690i \(0.461488\pi\)
\(282\) −1.00410e6 −0.751887
\(283\) 2.02735e6 1.50474 0.752371 0.658739i \(-0.228910\pi\)
0.752371 + 0.658739i \(0.228910\pi\)
\(284\) −399472. −0.293894
\(285\) 550188. 0.401235
\(286\) −54208.0 −0.0391876
\(287\) 0 0
\(288\) 612352. 0.435031
\(289\) −115693. −0.0814821
\(290\) 147808. 0.103206
\(291\) 149611. 0.103569
\(292\) 676608. 0.464387
\(293\) −718844. −0.489177 −0.244588 0.969627i \(-0.578653\pi\)
−0.244588 + 0.969627i \(0.578653\pi\)
\(294\) 0 0
\(295\) 661881. 0.442818
\(296\) 517312. 0.343181
\(297\) 1.24570e6 0.819446
\(298\) −223000. −0.145467
\(299\) 217392. 0.140626
\(300\) −1.00410e6 −0.644128
\(301\) 0 0
\(302\) 262568. 0.165663
\(303\) 1.78135e6 1.11466
\(304\) 156672. 0.0972316
\(305\) −1.45836e6 −0.897669
\(306\) 2.73166e6 1.66772
\(307\) 1.98142e6 1.19986 0.599930 0.800052i \(-0.295195\pi\)
0.599930 + 0.800052i \(0.295195\pi\)
\(308\) 0 0
\(309\) −5.39098e6 −3.21197
\(310\) 128588. 0.0759970
\(311\) 1.51030e6 0.885446 0.442723 0.896658i \(-0.354012\pi\)
0.442723 + 0.896658i \(0.354012\pi\)
\(312\) −207872. −0.120895
\(313\) −2.00092e6 −1.15443 −0.577216 0.816591i \(-0.695861\pi\)
−0.577216 + 0.816591i \(0.695861\pi\)
\(314\) 1.10147e6 0.630446
\(315\) 0 0
\(316\) −1.15856e6 −0.652681
\(317\) −259331. −0.144946 −0.0724730 0.997370i \(-0.523089\pi\)
−0.0724730 + 0.997370i \(0.523089\pi\)
\(318\) −2.35689e6 −1.30699
\(319\) 144232. 0.0793569
\(320\) 126976. 0.0693181
\(321\) −695130. −0.376533
\(322\) 0 0
\(323\) 698904. 0.372744
\(324\) 2.45186e6 1.29757
\(325\) 242368. 0.127282
\(326\) 1.16776e6 0.608569
\(327\) −1.63345e6 −0.844768
\(328\) 668416. 0.343054
\(329\) 0 0
\(330\) 435116. 0.219948
\(331\) 51203.0 0.0256877 0.0128439 0.999918i \(-0.495912\pi\)
0.0128439 + 0.999918i \(0.495912\pi\)
\(332\) 252896. 0.125921
\(333\) 4.83363e6 2.38871
\(334\) 1.34938e6 0.661861
\(335\) 1.49088e6 0.725824
\(336\) 0 0
\(337\) 266870. 0.128004 0.0640022 0.997950i \(-0.479614\pi\)
0.0640022 + 0.997950i \(0.479614\pi\)
\(338\) −1.43500e6 −0.683217
\(339\) −7.59519e6 −3.58954
\(340\) 566432. 0.265736
\(341\) 125477. 0.0584357
\(342\) 1.46390e6 0.676780
\(343\) 0 0
\(344\) 3712.00 0.00169127
\(345\) −1.74496e6 −0.789292
\(346\) 466968. 0.209699
\(347\) 622800. 0.277667 0.138834 0.990316i \(-0.455665\pi\)
0.138834 + 0.990316i \(0.455665\pi\)
\(348\) 553088. 0.244820
\(349\) −2.43649e6 −1.07078 −0.535391 0.844604i \(-0.679836\pi\)
−0.535391 + 0.844604i \(0.679836\pi\)
\(350\) 0 0
\(351\) −1.15304e6 −0.499547
\(352\) 123904. 0.0533002
\(353\) −1.55957e6 −0.666144 −0.333072 0.942901i \(-0.608085\pi\)
−0.333072 + 0.942901i \(0.608085\pi\)
\(354\) 2.47672e6 1.05043
\(355\) −773977. −0.325955
\(356\) 1.83618e6 0.767873
\(357\) 0 0
\(358\) −76428.0 −0.0315170
\(359\) 1.91961e6 0.786098 0.393049 0.919518i \(-0.371420\pi\)
0.393049 + 0.919518i \(0.371420\pi\)
\(360\) 1.18643e6 0.482488
\(361\) −2.10156e6 −0.848736
\(362\) 64708.0 0.0259529
\(363\) 424589. 0.169123
\(364\) 0 0
\(365\) 1.31093e6 0.515047
\(366\) −5.45710e6 −2.12941
\(367\) −3.61225e6 −1.39995 −0.699975 0.714167i \(-0.746806\pi\)
−0.699975 + 0.714167i \(0.746806\pi\)
\(368\) −496896. −0.191270
\(369\) 6.24551e6 2.38782
\(370\) 1.00229e6 0.380619
\(371\) 0 0
\(372\) 481168. 0.180277
\(373\) 3.93968e6 1.46619 0.733093 0.680128i \(-0.238076\pi\)
0.733093 + 0.680128i \(0.238076\pi\)
\(374\) 552728. 0.204330
\(375\) −4.75481e6 −1.74604
\(376\) −553984. −0.202082
\(377\) −133504. −0.0483772
\(378\) 0 0
\(379\) −2.18829e6 −0.782540 −0.391270 0.920276i \(-0.627964\pi\)
−0.391270 + 0.920276i \(0.627964\pi\)
\(380\) 303552. 0.107839
\(381\) 2.53472e6 0.894575
\(382\) 2.74133e6 0.961177
\(383\) −768387. −0.267660 −0.133830 0.991004i \(-0.542728\pi\)
−0.133830 + 0.991004i \(0.542728\pi\)
\(384\) 475136. 0.164433
\(385\) 0 0
\(386\) −1.23717e6 −0.422630
\(387\) 34684.0 0.0117720
\(388\) 82544.0 0.0278360
\(389\) 324313. 0.108665 0.0543326 0.998523i \(-0.482697\pi\)
0.0543326 + 0.998523i \(0.482697\pi\)
\(390\) −402752. −0.134084
\(391\) −2.21662e6 −0.733246
\(392\) 0 0
\(393\) −7.68854e6 −2.51109
\(394\) −483720. −0.156983
\(395\) −2.24471e6 −0.723882
\(396\) 1.15773e6 0.370995
\(397\) −334758. −0.106599 −0.0532997 0.998579i \(-0.516974\pi\)
−0.0532997 + 0.998579i \(0.516974\pi\)
\(398\) −3.66214e6 −1.15885
\(399\) 0 0
\(400\) −553984. −0.173120
\(401\) −902022. −0.280128 −0.140064 0.990142i \(-0.544731\pi\)
−0.140064 + 0.990142i \(0.544731\pi\)
\(402\) 5.57879e6 1.72177
\(403\) −116144. −0.0356233
\(404\) 982816. 0.299584
\(405\) 4.75047e6 1.43913
\(406\) 0 0
\(407\) 978043. 0.292666
\(408\) 2.11955e6 0.630367
\(409\) 5.00457e6 1.47931 0.739654 0.672987i \(-0.234989\pi\)
0.739654 + 0.672987i \(0.234989\pi\)
\(410\) 1.29506e6 0.380477
\(411\) 7.12985e6 2.08198
\(412\) −2.97434e6 −0.863271
\(413\) 0 0
\(414\) −4.64287e6 −1.33133
\(415\) 489986. 0.139657
\(416\) −114688. −0.0324926
\(417\) 1.05442e7 2.96944
\(418\) 296208. 0.0829194
\(419\) 3.00124e6 0.835151 0.417576 0.908642i \(-0.362880\pi\)
0.417576 + 0.908642i \(0.362880\pi\)
\(420\) 0 0
\(421\) 4.56224e6 1.25451 0.627253 0.778816i \(-0.284179\pi\)
0.627253 + 0.778816i \(0.284179\pi\)
\(422\) −538320. −0.147150
\(423\) −5.17629e6 −1.40659
\(424\) −1.30035e6 −0.351274
\(425\) −2.47129e6 −0.663668
\(426\) −2.89617e6 −0.773215
\(427\) 0 0
\(428\) −383520. −0.101200
\(429\) −393008. −0.103100
\(430\) 7192.00 0.00187577
\(431\) 4.89783e6 1.27002 0.635009 0.772504i \(-0.280996\pi\)
0.635009 + 0.772504i \(0.280996\pi\)
\(432\) 2.63552e6 0.679449
\(433\) −6.72876e6 −1.72471 −0.862353 0.506307i \(-0.831010\pi\)
−0.862353 + 0.506307i \(0.831010\pi\)
\(434\) 0 0
\(435\) 1.07161e6 0.271527
\(436\) −901216. −0.227045
\(437\) −1.18789e6 −0.297559
\(438\) 4.90541e6 1.22177
\(439\) 3.35034e6 0.829711 0.414856 0.909887i \(-0.363832\pi\)
0.414856 + 0.909887i \(0.363832\pi\)
\(440\) 240064. 0.0591147
\(441\) 0 0
\(442\) −511616. −0.124563
\(443\) −7.12434e6 −1.72479 −0.862394 0.506238i \(-0.831036\pi\)
−0.862394 + 0.506238i \(0.831036\pi\)
\(444\) 3.75051e6 0.902886
\(445\) 3.55759e6 0.851640
\(446\) −1.87536e6 −0.446423
\(447\) −1.61675e6 −0.382714
\(448\) 0 0
\(449\) −2.70928e6 −0.634218 −0.317109 0.948389i \(-0.602712\pi\)
−0.317109 + 0.948389i \(0.602712\pi\)
\(450\) −5.17629e6 −1.20500
\(451\) 1.26372e6 0.292557
\(452\) −4.19045e6 −0.964749
\(453\) 1.90362e6 0.435847
\(454\) −1.10009e6 −0.250489
\(455\) 0 0
\(456\) 1.13587e6 0.255810
\(457\) 2.41361e6 0.540601 0.270301 0.962776i \(-0.412877\pi\)
0.270301 + 0.962776i \(0.412877\pi\)
\(458\) 2.56912e6 0.572297
\(459\) 1.17569e7 2.60472
\(460\) −962736. −0.212135
\(461\) 6.56065e6 1.43779 0.718894 0.695120i \(-0.244649\pi\)
0.718894 + 0.695120i \(0.244649\pi\)
\(462\) 0 0
\(463\) −4.72421e6 −1.02418 −0.512090 0.858932i \(-0.671129\pi\)
−0.512090 + 0.858932i \(0.671129\pi\)
\(464\) 305152. 0.0657993
\(465\) 932263. 0.199943
\(466\) −6.01939e6 −1.28407
\(467\) 2.28444e6 0.484716 0.242358 0.970187i \(-0.422079\pi\)
0.242358 + 0.970187i \(0.422079\pi\)
\(468\) −1.07162e6 −0.226165
\(469\) 0 0
\(470\) −1.07334e6 −0.224127
\(471\) 7.98564e6 1.65866
\(472\) 1.36646e6 0.282321
\(473\) 7018.00 0.00144232
\(474\) −8.39956e6 −1.71716
\(475\) −1.32437e6 −0.269324
\(476\) 0 0
\(477\) −1.21502e7 −2.44504
\(478\) −1.21878e6 −0.243980
\(479\) −951544. −0.189492 −0.0947458 0.995501i \(-0.530204\pi\)
−0.0947458 + 0.995501i \(0.530204\pi\)
\(480\) 920576. 0.182371
\(481\) −905296. −0.178414
\(482\) −5.08723e6 −0.997388
\(483\) 0 0
\(484\) 234256. 0.0454545
\(485\) 159929. 0.0308726
\(486\) 7.76922e6 1.49206
\(487\) 3.51484e6 0.671558 0.335779 0.941941i \(-0.391000\pi\)
0.335779 + 0.941941i \(0.391000\pi\)
\(488\) −3.01082e6 −0.572314
\(489\) 8.46626e6 1.60110
\(490\) 0 0
\(491\) −5.78719e6 −1.08334 −0.541669 0.840592i \(-0.682207\pi\)
−0.541669 + 0.840592i \(0.682207\pi\)
\(492\) 4.84602e6 0.902552
\(493\) 1.36126e6 0.252246
\(494\) −274176. −0.0505489
\(495\) 2.24310e6 0.411467
\(496\) 265472. 0.0484523
\(497\) 0 0
\(498\) 1.83350e6 0.331289
\(499\) 1.02912e6 0.185019 0.0925095 0.995712i \(-0.470511\pi\)
0.0925095 + 0.995712i \(0.470511\pi\)
\(500\) −2.62334e6 −0.469278
\(501\) 9.78298e6 1.74131
\(502\) −2.51986e6 −0.446290
\(503\) 727370. 0.128184 0.0640922 0.997944i \(-0.479585\pi\)
0.0640922 + 0.997944i \(0.479585\pi\)
\(504\) 0 0
\(505\) 1.90421e6 0.332266
\(506\) −939444. −0.163115
\(507\) −1.04037e7 −1.79750
\(508\) 1.39846e6 0.240432
\(509\) 1.94630e6 0.332977 0.166489 0.986043i \(-0.446757\pi\)
0.166489 + 0.986043i \(0.446757\pi\)
\(510\) 4.10663e6 0.699134
\(511\) 0 0
\(512\) 262144. 0.0441942
\(513\) 6.30054e6 1.05702
\(514\) −2.17634e6 −0.363345
\(515\) −5.76278e6 −0.957445
\(516\) 26912.0 0.00444961
\(517\) −1.04738e6 −0.172336
\(518\) 0 0
\(519\) 3.38552e6 0.551705
\(520\) −222208. −0.0360372
\(521\) 1.03133e7 1.66457 0.832286 0.554346i \(-0.187032\pi\)
0.832286 + 0.554346i \(0.187032\pi\)
\(522\) 2.85126e6 0.457995
\(523\) 6.86840e6 1.09800 0.548998 0.835823i \(-0.315009\pi\)
0.548998 + 0.835823i \(0.315009\pi\)
\(524\) −4.24195e6 −0.674897
\(525\) 0 0
\(526\) −7.95734e6 −1.25402
\(527\) 1.18425e6 0.185746
\(528\) 898304. 0.140229
\(529\) −2.66886e6 −0.414655
\(530\) −2.51943e6 −0.389595
\(531\) 1.27679e7 1.96509
\(532\) 0 0
\(533\) −1.16973e6 −0.178347
\(534\) 1.33123e7 2.02022
\(535\) −743070. −0.112239
\(536\) 3.07795e6 0.462754
\(537\) −554103. −0.0829191
\(538\) −7.01783e6 −1.04532
\(539\) 0 0
\(540\) 5.10632e6 0.753570
\(541\) 1.00545e7 1.47695 0.738476 0.674280i \(-0.235546\pi\)
0.738476 + 0.674280i \(0.235546\pi\)
\(542\) 6.63296e6 0.969860
\(543\) 469133. 0.0682805
\(544\) 1.16941e6 0.169422
\(545\) −1.74611e6 −0.251814
\(546\) 0 0
\(547\) 9.85725e6 1.40860 0.704299 0.709903i \(-0.251261\pi\)
0.704299 + 0.709903i \(0.251261\pi\)
\(548\) 3.93371e6 0.559566
\(549\) −2.81323e7 −3.98359
\(550\) −1.04738e6 −0.147637
\(551\) 729504. 0.102364
\(552\) −3.60250e6 −0.503218
\(553\) 0 0
\(554\) −170536. −0.0236070
\(555\) 7.26662e6 1.00138
\(556\) 5.81750e6 0.798086
\(557\) −1.45892e7 −1.99247 −0.996237 0.0866757i \(-0.972376\pi\)
−0.996237 + 0.0866757i \(0.972376\pi\)
\(558\) 2.48050e6 0.337252
\(559\) −6496.00 −0.000879258 0
\(560\) 0 0
\(561\) 4.00728e6 0.537579
\(562\) 1.27804e6 0.170688
\(563\) 1.02413e7 1.36171 0.680855 0.732418i \(-0.261608\pi\)
0.680855 + 0.732418i \(0.261608\pi\)
\(564\) −4.01638e6 −0.531664
\(565\) −8.11899e6 −1.06999
\(566\) 8.10939e6 1.06401
\(567\) 0 0
\(568\) −1.59789e6 −0.207814
\(569\) −751816. −0.0973489 −0.0486744 0.998815i \(-0.515500\pi\)
−0.0486744 + 0.998815i \(0.515500\pi\)
\(570\) 2.20075e6 0.283716
\(571\) −7.01854e6 −0.900858 −0.450429 0.892812i \(-0.648729\pi\)
−0.450429 + 0.892812i \(0.648729\pi\)
\(572\) −216832. −0.0277098
\(573\) 1.98747e7 2.52879
\(574\) 0 0
\(575\) 4.20032e6 0.529801
\(576\) 2.44941e6 0.307613
\(577\) 3.36377e6 0.420617 0.210308 0.977635i \(-0.432553\pi\)
0.210308 + 0.977635i \(0.432553\pi\)
\(578\) −462772. −0.0576166
\(579\) −8.96947e6 −1.11191
\(580\) 591232. 0.0729773
\(581\) 0 0
\(582\) 598444. 0.0732346
\(583\) −2.45848e6 −0.299568
\(584\) 2.70643e6 0.328371
\(585\) −2.07626e6 −0.250837
\(586\) −2.87538e6 −0.345900
\(587\) −1.40585e7 −1.68401 −0.842006 0.539468i \(-0.818625\pi\)
−0.842006 + 0.539468i \(0.818625\pi\)
\(588\) 0 0
\(589\) 634644. 0.0753775
\(590\) 2.64752e6 0.313119
\(591\) −3.50697e6 −0.413013
\(592\) 2.06925e6 0.242666
\(593\) 5.39420e6 0.629927 0.314963 0.949104i \(-0.398008\pi\)
0.314963 + 0.949104i \(0.398008\pi\)
\(594\) 4.98278e6 0.579436
\(595\) 0 0
\(596\) −892000. −0.102861
\(597\) −2.65505e7 −3.04886
\(598\) 869568. 0.0994376
\(599\) −1.20204e7 −1.36883 −0.684417 0.729090i \(-0.739943\pi\)
−0.684417 + 0.729090i \(0.739943\pi\)
\(600\) −4.01638e6 −0.455467
\(601\) −1.64636e6 −0.185925 −0.0929626 0.995670i \(-0.529634\pi\)
−0.0929626 + 0.995670i \(0.529634\pi\)
\(602\) 0 0
\(603\) 2.87596e7 3.22099
\(604\) 1.05027e6 0.117141
\(605\) 453871. 0.0504132
\(606\) 7.12542e6 0.788186
\(607\) −4.88451e6 −0.538083 −0.269041 0.963129i \(-0.586707\pi\)
−0.269041 + 0.963129i \(0.586707\pi\)
\(608\) 626688. 0.0687531
\(609\) 0 0
\(610\) −5.83346e6 −0.634748
\(611\) 969472. 0.105059
\(612\) 1.09267e7 1.17926
\(613\) 3.49011e6 0.375136 0.187568 0.982252i \(-0.439940\pi\)
0.187568 + 0.982252i \(0.439940\pi\)
\(614\) 7.92568e6 0.848429
\(615\) 9.38916e6 1.00101
\(616\) 0 0
\(617\) 9.12072e6 0.964531 0.482266 0.876025i \(-0.339814\pi\)
0.482266 + 0.876025i \(0.339814\pi\)
\(618\) −2.15639e7 −2.27121
\(619\) −1.46635e7 −1.53820 −0.769098 0.639131i \(-0.779294\pi\)
−0.769098 + 0.639131i \(0.779294\pi\)
\(620\) 514352. 0.0537380
\(621\) −1.99826e7 −2.07933
\(622\) 6.04120e6 0.626105
\(623\) 0 0
\(624\) −831488. −0.0854859
\(625\) 1.67977e6 0.172009
\(626\) −8.00368e6 −0.816307
\(627\) 2.14751e6 0.218155
\(628\) 4.40587e6 0.445793
\(629\) 9.23079e6 0.930277
\(630\) 0 0
\(631\) −1.63870e7 −1.63842 −0.819212 0.573491i \(-0.805589\pi\)
−0.819212 + 0.573491i \(0.805589\pi\)
\(632\) −4.63424e6 −0.461515
\(633\) −3.90282e6 −0.387141
\(634\) −1.03732e6 −0.102492
\(635\) 2.70952e6 0.266660
\(636\) −9.42755e6 −0.924179
\(637\) 0 0
\(638\) 576928. 0.0561138
\(639\) −1.49303e7 −1.44649
\(640\) 507904. 0.0490153
\(641\) −3.26835e6 −0.314184 −0.157092 0.987584i \(-0.550212\pi\)
−0.157092 + 0.987584i \(0.550212\pi\)
\(642\) −2.78052e6 −0.266249
\(643\) −8.32842e6 −0.794393 −0.397197 0.917734i \(-0.630017\pi\)
−0.397197 + 0.917734i \(0.630017\pi\)
\(644\) 0 0
\(645\) 52142.0 0.00493501
\(646\) 2.79562e6 0.263570
\(647\) −2.49694e6 −0.234503 −0.117251 0.993102i \(-0.537408\pi\)
−0.117251 + 0.993102i \(0.537408\pi\)
\(648\) 9.80742e6 0.917524
\(649\) 2.58347e6 0.240764
\(650\) 969472. 0.0900019
\(651\) 0 0
\(652\) 4.67104e6 0.430323
\(653\) −789105. −0.0724189 −0.0362094 0.999344i \(-0.511528\pi\)
−0.0362094 + 0.999344i \(0.511528\pi\)
\(654\) −6.53382e6 −0.597341
\(655\) −8.21878e6 −0.748521
\(656\) 2.67366e6 0.242576
\(657\) 2.52882e7 2.28562
\(658\) 0 0
\(659\) −8.31393e6 −0.745749 −0.372874 0.927882i \(-0.621628\pi\)
−0.372874 + 0.927882i \(0.621628\pi\)
\(660\) 1.74046e6 0.155527
\(661\) 4.33517e6 0.385925 0.192962 0.981206i \(-0.438190\pi\)
0.192962 + 0.981206i \(0.438190\pi\)
\(662\) 204812. 0.0181640
\(663\) −3.70922e6 −0.327716
\(664\) 1.01158e6 0.0890393
\(665\) 0 0
\(666\) 1.93345e7 1.68907
\(667\) −2.31367e6 −0.201366
\(668\) 5.39750e6 0.468006
\(669\) −1.35963e7 −1.17451
\(670\) 5.96353e6 0.513235
\(671\) −5.69232e6 −0.488071
\(672\) 0 0
\(673\) 7.29313e6 0.620693 0.310346 0.950624i \(-0.399555\pi\)
0.310346 + 0.950624i \(0.399555\pi\)
\(674\) 1.06748e6 0.0905128
\(675\) −2.22784e7 −1.88202
\(676\) −5.73998e6 −0.483108
\(677\) −1.55814e7 −1.30658 −0.653288 0.757109i \(-0.726611\pi\)
−0.653288 + 0.757109i \(0.726611\pi\)
\(678\) −3.03807e7 −2.53819
\(679\) 0 0
\(680\) 2.26573e6 0.187904
\(681\) −7.97564e6 −0.659019
\(682\) 501908. 0.0413203
\(683\) −2.16930e6 −0.177938 −0.0889690 0.996034i \(-0.528357\pi\)
−0.0889690 + 0.996034i \(0.528357\pi\)
\(684\) 5.85562e6 0.478556
\(685\) 7.62157e6 0.620609
\(686\) 0 0
\(687\) 1.86261e7 1.50567
\(688\) 14848.0 0.00119591
\(689\) 2.27562e6 0.182621
\(690\) −6.97984e6 −0.558113
\(691\) 1.32195e7 1.05322 0.526610 0.850107i \(-0.323463\pi\)
0.526610 + 0.850107i \(0.323463\pi\)
\(692\) 1.86787e6 0.148280
\(693\) 0 0
\(694\) 2.49120e6 0.196341
\(695\) 1.12714e7 0.885149
\(696\) 2.21235e6 0.173114
\(697\) 1.19270e7 0.929932
\(698\) −9.74596e6 −0.757157
\(699\) −4.36406e7 −3.37830
\(700\) 0 0
\(701\) −2.59395e7 −1.99373 −0.996866 0.0791122i \(-0.974791\pi\)
−0.996866 + 0.0791122i \(0.974791\pi\)
\(702\) −4.61216e6 −0.353233
\(703\) 4.94680e6 0.377516
\(704\) 495616. 0.0376889
\(705\) −7.78174e6 −0.589663
\(706\) −6.23828e6 −0.471035
\(707\) 0 0
\(708\) 9.90686e6 0.742768
\(709\) 3.57531e6 0.267115 0.133557 0.991041i \(-0.457360\pi\)
0.133557 + 0.991041i \(0.457360\pi\)
\(710\) −3.09591e6 −0.230485
\(711\) −4.33012e7 −3.21237
\(712\) 7.34470e6 0.542968
\(713\) −2.01282e6 −0.148279
\(714\) 0 0
\(715\) −420112. −0.0307326
\(716\) −305712. −0.0222859
\(717\) −8.83613e6 −0.641895
\(718\) 7.67843e6 0.555855
\(719\) 1.95814e6 0.141261 0.0706304 0.997503i \(-0.477499\pi\)
0.0706304 + 0.997503i \(0.477499\pi\)
\(720\) 4.74573e6 0.341171
\(721\) 0 0
\(722\) −8.40622e6 −0.600147
\(723\) −3.68824e7 −2.62406
\(724\) 258832. 0.0183515
\(725\) −2.57949e6 −0.182259
\(726\) 1.69836e6 0.119588
\(727\) −1.55360e7 −1.09019 −0.545095 0.838374i \(-0.683507\pi\)
−0.545095 + 0.838374i \(0.683507\pi\)
\(728\) 0 0
\(729\) 1.90893e7 1.33036
\(730\) 5.24371e6 0.364193
\(731\) 66236.0 0.00458459
\(732\) −2.18284e7 −1.50572
\(733\) 1.46002e7 1.00369 0.501844 0.864958i \(-0.332655\pi\)
0.501844 + 0.864958i \(0.332655\pi\)
\(734\) −1.44490e7 −0.989915
\(735\) 0 0
\(736\) −1.98758e6 −0.135248
\(737\) 5.81925e6 0.394638
\(738\) 2.49820e7 1.68845
\(739\) −2.06682e7 −1.39217 −0.696085 0.717959i \(-0.745077\pi\)
−0.696085 + 0.717959i \(0.745077\pi\)
\(740\) 4.00917e6 0.269138
\(741\) −1.98778e6 −0.132991
\(742\) 0 0
\(743\) 1.17065e7 0.777953 0.388976 0.921248i \(-0.372829\pi\)
0.388976 + 0.921248i \(0.372829\pi\)
\(744\) 1.92467e6 0.127475
\(745\) −1.72825e6 −0.114082
\(746\) 1.57587e7 1.03675
\(747\) 9.45199e6 0.619757
\(748\) 2.21091e6 0.144483
\(749\) 0 0
\(750\) −1.90192e7 −1.23464
\(751\) −1.27607e7 −0.825610 −0.412805 0.910819i \(-0.635451\pi\)
−0.412805 + 0.910819i \(0.635451\pi\)
\(752\) −2.21594e6 −0.142894
\(753\) −1.82690e7 −1.17416
\(754\) −534016. −0.0342079
\(755\) 2.03490e6 0.129920
\(756\) 0 0
\(757\) −1.40869e7 −0.893458 −0.446729 0.894669i \(-0.647411\pi\)
−0.446729 + 0.894669i \(0.647411\pi\)
\(758\) −8.75316e6 −0.553339
\(759\) −6.81097e6 −0.429145
\(760\) 1.21421e6 0.0762534
\(761\) −2.33822e7 −1.46360 −0.731801 0.681518i \(-0.761320\pi\)
−0.731801 + 0.681518i \(0.761320\pi\)
\(762\) 1.01389e7 0.632560
\(763\) 0 0
\(764\) 1.09653e7 0.679655
\(765\) 2.11704e7 1.30790
\(766\) −3.07355e6 −0.189264
\(767\) −2.39131e6 −0.146774
\(768\) 1.90054e6 0.116272
\(769\) 1.09575e7 0.668185 0.334092 0.942540i \(-0.391570\pi\)
0.334092 + 0.942540i \(0.391570\pi\)
\(770\) 0 0
\(771\) −1.57785e7 −0.955938
\(772\) −4.94867e6 −0.298845
\(773\) −1.69336e7 −1.01930 −0.509648 0.860383i \(-0.670224\pi\)
−0.509648 + 0.860383i \(0.670224\pi\)
\(774\) 138736. 0.00832409
\(775\) −2.24407e6 −0.134209
\(776\) 330176. 0.0196830
\(777\) 0 0
\(778\) 1.29725e6 0.0768379
\(779\) 6.39173e6 0.377376
\(780\) −1.61101e6 −0.0948115
\(781\) −3.02101e6 −0.177225
\(782\) −8.86649e6 −0.518483
\(783\) 1.22716e7 0.715316
\(784\) 0 0
\(785\) 8.53638e6 0.494424
\(786\) −3.07542e7 −1.77561
\(787\) 7.51655e6 0.432595 0.216298 0.976327i \(-0.430602\pi\)
0.216298 + 0.976327i \(0.430602\pi\)
\(788\) −1.93488e6 −0.111004
\(789\) −5.76907e7 −3.29923
\(790\) −8.97884e6 −0.511862
\(791\) 0 0
\(792\) 4.63091e6 0.262333
\(793\) 5.26893e6 0.297536
\(794\) −1.33903e6 −0.0753771
\(795\) −1.82659e7 −1.02500
\(796\) −1.46486e7 −0.819432
\(797\) −3.93788e6 −0.219592 −0.109796 0.993954i \(-0.535020\pi\)
−0.109796 + 0.993954i \(0.535020\pi\)
\(798\) 0 0
\(799\) −9.88515e6 −0.547793
\(800\) −2.21594e6 −0.122414
\(801\) 6.86271e7 3.77932
\(802\) −3.60809e6 −0.198080
\(803\) 5.11685e6 0.280036
\(804\) 2.23152e7 1.21747
\(805\) 0 0
\(806\) −464576. −0.0251895
\(807\) −5.08793e7 −2.75015
\(808\) 3.93126e6 0.211838
\(809\) −1.73609e7 −0.932612 −0.466306 0.884624i \(-0.654415\pi\)
−0.466306 + 0.884624i \(0.654415\pi\)
\(810\) 1.90019e7 1.01762
\(811\) −2.70850e7 −1.44603 −0.723014 0.690833i \(-0.757244\pi\)
−0.723014 + 0.690833i \(0.757244\pi\)
\(812\) 0 0
\(813\) 4.80890e7 2.55164
\(814\) 3.91217e6 0.206946
\(815\) 9.05014e6 0.477267
\(816\) 8.47821e6 0.445737
\(817\) 35496.0 0.00186048
\(818\) 2.00183e7 1.04603
\(819\) 0 0
\(820\) 5.18022e6 0.269038
\(821\) 3.59384e7 1.86080 0.930402 0.366540i \(-0.119458\pi\)
0.930402 + 0.366540i \(0.119458\pi\)
\(822\) 2.85194e7 1.47218
\(823\) −505509. −0.0260153 −0.0130077 0.999915i \(-0.504141\pi\)
−0.0130077 + 0.999915i \(0.504141\pi\)
\(824\) −1.18973e7 −0.610425
\(825\) −7.59348e6 −0.388424
\(826\) 0 0
\(827\) −2.99955e7 −1.52508 −0.762539 0.646942i \(-0.776048\pi\)
−0.762539 + 0.646942i \(0.776048\pi\)
\(828\) −1.85715e7 −0.941393
\(829\) −2.96942e7 −1.50067 −0.750334 0.661059i \(-0.770107\pi\)
−0.750334 + 0.661059i \(0.770107\pi\)
\(830\) 1.95994e6 0.0987526
\(831\) −1.23639e6 −0.0621086
\(832\) −458752. −0.0229757
\(833\) 0 0
\(834\) 4.21769e7 2.09971
\(835\) 1.04577e7 0.519061
\(836\) 1.18483e6 0.0586329
\(837\) 1.06759e7 0.526734
\(838\) 1.20049e7 0.590541
\(839\) 1.41371e7 0.693356 0.346678 0.937984i \(-0.387310\pi\)
0.346678 + 0.937984i \(0.387310\pi\)
\(840\) 0 0
\(841\) −1.90903e7 −0.930727
\(842\) 1.82490e7 0.887070
\(843\) 9.26579e6 0.449069
\(844\) −2.15328e6 −0.104051
\(845\) −1.11212e7 −0.535810
\(846\) −2.07052e7 −0.994609
\(847\) 0 0
\(848\) −5.20141e6 −0.248388
\(849\) 5.87931e7 2.79935
\(850\) −9.88515e6 −0.469284
\(851\) −1.56891e7 −0.742633
\(852\) −1.15847e7 −0.546746
\(853\) 4.68539e6 0.220482 0.110241 0.993905i \(-0.464838\pi\)
0.110241 + 0.993905i \(0.464838\pi\)
\(854\) 0 0
\(855\) 1.13453e7 0.530761
\(856\) −1.53408e6 −0.0715589
\(857\) 4.12846e7 1.92015 0.960076 0.279740i \(-0.0902483\pi\)
0.960076 + 0.279740i \(0.0902483\pi\)
\(858\) −1.57203e6 −0.0729026
\(859\) 3.54805e6 0.164062 0.0820308 0.996630i \(-0.473859\pi\)
0.0820308 + 0.996630i \(0.473859\pi\)
\(860\) 28768.0 0.00132637
\(861\) 0 0
\(862\) 1.95913e7 0.898039
\(863\) −3.07605e7 −1.40594 −0.702970 0.711219i \(-0.748143\pi\)
−0.702970 + 0.711219i \(0.748143\pi\)
\(864\) 1.05421e7 0.480443
\(865\) 3.61900e6 0.164456
\(866\) −2.69150e7 −1.21955
\(867\) −3.35510e6 −0.151585
\(868\) 0 0
\(869\) −8.76161e6 −0.393581
\(870\) 4.28643e6 0.191998
\(871\) −5.38642e6 −0.240577
\(872\) −3.60486e6 −0.160545
\(873\) 3.08508e6 0.137003
\(874\) −4.75157e6 −0.210406
\(875\) 0 0
\(876\) 1.96216e7 0.863923
\(877\) 3.05535e7 1.34141 0.670706 0.741723i \(-0.265991\pi\)
0.670706 + 0.741723i \(0.265991\pi\)
\(878\) 1.34013e7 0.586695
\(879\) −2.08465e7 −0.910040
\(880\) 960256. 0.0418004
\(881\) −4.21018e7 −1.82751 −0.913757 0.406262i \(-0.866832\pi\)
−0.913757 + 0.406262i \(0.866832\pi\)
\(882\) 0 0
\(883\) 57164.0 0.00246729 0.00123365 0.999999i \(-0.499607\pi\)
0.00123365 + 0.999999i \(0.499607\pi\)
\(884\) −2.04646e6 −0.0880792
\(885\) 1.91945e7 0.823796
\(886\) −2.84974e7 −1.21961
\(887\) 1.16106e7 0.495504 0.247752 0.968824i \(-0.420308\pi\)
0.247752 + 0.968824i \(0.420308\pi\)
\(888\) 1.50020e7 0.638437
\(889\) 0 0
\(890\) 1.42304e7 0.602200
\(891\) 1.85422e7 0.782467
\(892\) −7.50142e6 −0.315669
\(893\) −5.29747e6 −0.222300
\(894\) −6.46700e6 −0.270619
\(895\) −592317. −0.0247170
\(896\) 0 0
\(897\) 6.30437e6 0.261614
\(898\) −1.08371e7 −0.448460
\(899\) 1.23610e6 0.0510101
\(900\) −2.07052e7 −0.852064
\(901\) −2.32032e7 −0.952215
\(902\) 5.05490e6 0.206869
\(903\) 0 0
\(904\) −1.67618e7 −0.682181
\(905\) 501487. 0.0203535
\(906\) 7.61447e6 0.308191
\(907\) 2.09855e7 0.847034 0.423517 0.905888i \(-0.360795\pi\)
0.423517 + 0.905888i \(0.360795\pi\)
\(908\) −4.40035e6 −0.177122
\(909\) 3.67327e7 1.47450
\(910\) 0 0
\(911\) 4.74125e7 1.89277 0.946383 0.323047i \(-0.104707\pi\)
0.946383 + 0.323047i \(0.104707\pi\)
\(912\) 4.54349e6 0.180885
\(913\) 1.91253e6 0.0759330
\(914\) 9.65445e6 0.382263
\(915\) −4.22926e7 −1.66998
\(916\) 1.02765e7 0.404675
\(917\) 0 0
\(918\) 4.70276e7 1.84181
\(919\) 4.04326e7 1.57922 0.789610 0.613609i \(-0.210283\pi\)
0.789610 + 0.613609i \(0.210283\pi\)
\(920\) −3.85094e6 −0.150002
\(921\) 5.74612e7 2.23216
\(922\) 2.62426e7 1.01667
\(923\) 2.79630e6 0.108039
\(924\) 0 0
\(925\) −1.74916e7 −0.672164
\(926\) −1.88968e7 −0.724205
\(927\) −1.11166e8 −4.24885
\(928\) 1.22061e6 0.0465271
\(929\) −3.30757e7 −1.25739 −0.628694 0.777652i \(-0.716410\pi\)
−0.628694 + 0.777652i \(0.716410\pi\)
\(930\) 3.72905e6 0.141381
\(931\) 0 0
\(932\) −2.40776e7 −0.907973
\(933\) 4.37987e7 1.64724
\(934\) 9.13776e6 0.342746
\(935\) 4.28364e6 0.160245
\(936\) −4.28646e6 −0.159922
\(937\) 3.15132e7 1.17258 0.586292 0.810100i \(-0.300587\pi\)
0.586292 + 0.810100i \(0.300587\pi\)
\(938\) 0 0
\(939\) −5.80267e7 −2.14765
\(940\) −4.29338e6 −0.158482
\(941\) 9.54147e6 0.351270 0.175635 0.984455i \(-0.443802\pi\)
0.175635 + 0.984455i \(0.443802\pi\)
\(942\) 3.19426e7 1.17285
\(943\) −2.02718e7 −0.742358
\(944\) 5.46586e6 0.199631
\(945\) 0 0
\(946\) 28072.0 0.00101987
\(947\) 2.24208e7 0.812410 0.406205 0.913782i \(-0.366852\pi\)
0.406205 + 0.913782i \(0.366852\pi\)
\(948\) −3.35982e7 −1.21422
\(949\) −4.73626e6 −0.170714
\(950\) −5.29747e6 −0.190441
\(951\) −7.52060e6 −0.269650
\(952\) 0 0
\(953\) 1.68985e7 0.602720 0.301360 0.953510i \(-0.402559\pi\)
0.301360 + 0.953510i \(0.402559\pi\)
\(954\) −4.86007e7 −1.72891
\(955\) 2.12453e7 0.753798
\(956\) −4.87510e6 −0.172520
\(957\) 4.18273e6 0.147632
\(958\) −3.80618e6 −0.133991
\(959\) 0 0
\(960\) 3.68230e6 0.128956
\(961\) −2.75538e7 −0.962438
\(962\) −3.62118e6 −0.126157
\(963\) −1.43341e7 −0.498085
\(964\) −2.03489e7 −0.705260
\(965\) −9.58805e6 −0.331445
\(966\) 0 0
\(967\) −3.06946e7 −1.05559 −0.527796 0.849371i \(-0.676982\pi\)
−0.527796 + 0.849371i \(0.676982\pi\)
\(968\) 937024. 0.0321412
\(969\) 2.02682e7 0.693436
\(970\) 639716. 0.0218302
\(971\) −3.35664e7 −1.14250 −0.571251 0.820776i \(-0.693542\pi\)
−0.571251 + 0.820776i \(0.693542\pi\)
\(972\) 3.10769e7 1.05505
\(973\) 0 0
\(974\) 1.40594e7 0.474863
\(975\) 7.02867e6 0.236789
\(976\) −1.20433e7 −0.404687
\(977\) 2.47897e7 0.830873 0.415436 0.909622i \(-0.363629\pi\)
0.415436 + 0.909622i \(0.363629\pi\)
\(978\) 3.38650e7 1.13215
\(979\) 1.38861e7 0.463045
\(980\) 0 0
\(981\) −3.36829e7 −1.11747
\(982\) −2.31488e7 −0.766036
\(983\) −5.22606e6 −0.172501 −0.0862503 0.996274i \(-0.527488\pi\)
−0.0862503 + 0.996274i \(0.527488\pi\)
\(984\) 1.93841e7 0.638200
\(985\) −3.74883e6 −0.123113
\(986\) 5.44506e6 0.178365
\(987\) 0 0
\(988\) −1.09670e6 −0.0357435
\(989\) −112578. −0.00365985
\(990\) 8.97239e6 0.290951
\(991\) 2.40826e7 0.778967 0.389484 0.921033i \(-0.372653\pi\)
0.389484 + 0.921033i \(0.372653\pi\)
\(992\) 1.06189e6 0.0342610
\(993\) 1.48489e6 0.0477882
\(994\) 0 0
\(995\) −2.83816e7 −0.908823
\(996\) 7.33398e6 0.234256
\(997\) 1.32606e7 0.422499 0.211249 0.977432i \(-0.432247\pi\)
0.211249 + 0.977432i \(0.432247\pi\)
\(998\) 4.11650e6 0.130828
\(999\) 8.32145e7 2.63806
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 1078.6.a.f.1.1 1
7.6 odd 2 22.6.a.c.1.1 1
21.20 even 2 198.6.a.b.1.1 1
28.27 even 2 176.6.a.e.1.1 1
35.13 even 4 550.6.b.a.199.1 2
35.27 even 4 550.6.b.a.199.2 2
35.34 odd 2 550.6.a.c.1.1 1
56.13 odd 2 704.6.a.j.1.1 1
56.27 even 2 704.6.a.a.1.1 1
77.76 even 2 242.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.c.1.1 1 7.6 odd 2
176.6.a.e.1.1 1 28.27 even 2
198.6.a.b.1.1 1 21.20 even 2
242.6.a.a.1.1 1 77.76 even 2
550.6.a.c.1.1 1 35.34 odd 2
550.6.b.a.199.1 2 35.13 even 4
550.6.b.a.199.2 2 35.27 even 4
704.6.a.a.1.1 1 56.27 even 2
704.6.a.j.1.1 1 56.13 odd 2
1078.6.a.f.1.1 1 1.1 even 1 trivial