Properties

Label 176.6.a.i.1.2
Level $176$
Weight $6$
Character 176.1
Self dual yes
Analytic conductor $28.228$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,6,Mod(1,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("176.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 176.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: \(28.2275522871\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.04796\) of defining polynomial
Character \(\chi\) \(=\) 176.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.8394 q^{3} +75.2230 q^{5} +225.525 q^{7} +40.5643 q^{9} +O(q^{10})\) \(q-16.8394 q^{3} +75.2230 q^{5} +225.525 q^{7} +40.5643 q^{9} -121.000 q^{11} +455.465 q^{13} -1266.71 q^{15} +190.657 q^{17} +135.393 q^{19} -3797.69 q^{21} -2796.65 q^{23} +2533.51 q^{25} +3408.89 q^{27} -2608.58 q^{29} +1056.76 q^{31} +2037.56 q^{33} +16964.7 q^{35} +12536.8 q^{37} -7669.74 q^{39} +1130.09 q^{41} +14671.0 q^{43} +3051.37 q^{45} +16882.2 q^{47} +34054.4 q^{49} -3210.54 q^{51} +3313.02 q^{53} -9101.99 q^{55} -2279.93 q^{57} -11454.0 q^{59} -28227.5 q^{61} +9148.26 q^{63} +34261.4 q^{65} +51431.0 q^{67} +47093.8 q^{69} +16218.0 q^{71} -10168.8 q^{73} -42662.6 q^{75} -27288.5 q^{77} -60841.2 q^{79} -67260.7 q^{81} -45770.6 q^{83} +14341.8 q^{85} +43926.9 q^{87} -82267.9 q^{89} +102719. q^{91} -17795.1 q^{93} +10184.7 q^{95} +53097.0 q^{97} -4908.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 24 q^{5} - 84 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 24 q^{5} - 84 q^{7} - 7 q^{9} - 363 q^{11} + 486 q^{13} - 1654 q^{15} + 1086 q^{17} - 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 57 q^{25} + 2990 q^{27} - 3426 q^{29} + 4098 q^{31} + 4114 q^{33} + 24228 q^{35} + 17724 q^{37} + 6560 q^{39} + 5994 q^{41} + 26208 q^{43} + 18458 q^{45} + 17232 q^{47} + 48531 q^{49} + 22724 q^{51} + 50586 q^{53} - 2904 q^{55} + 20160 q^{57} + 3738 q^{59} + 18486 q^{61} + 12496 q^{63} - 7668 q^{65} + 47754 q^{67} + 35042 q^{69} - 39282 q^{71} + 15426 q^{73} + 21916 q^{75} + 10164 q^{77} - 125148 q^{79} - 86917 q^{81} + 143928 q^{83} - 104040 q^{85} + 19368 q^{87} - 106824 q^{89} + 109632 q^{91} - 16622 q^{93} + 22200 q^{95} + 9684 q^{97} + 847 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −16.8394 −1.08025 −0.540123 0.841586i \(-0.681622\pi\)
−0.540123 + 0.841586i \(0.681622\pi\)
\(4\) 0 0
\(5\) 75.2230 1.34563 0.672815 0.739810i \(-0.265085\pi\)
0.672815 + 0.739810i \(0.265085\pi\)
\(6\) 0 0
\(7\) 225.525 1.73960 0.869799 0.493406i \(-0.164248\pi\)
0.869799 + 0.493406i \(0.164248\pi\)
\(8\) 0 0
\(9\) 40.5643 0.166931
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) 455.465 0.747474 0.373737 0.927535i \(-0.378076\pi\)
0.373737 + 0.927535i \(0.378076\pi\)
\(14\) 0 0
\(15\) −1266.71 −1.45361
\(16\) 0 0
\(17\) 190.657 0.160003 0.0800017 0.996795i \(-0.474507\pi\)
0.0800017 + 0.996795i \(0.474507\pi\)
\(18\) 0 0
\(19\) 135.393 0.0860424 0.0430212 0.999074i \(-0.486302\pi\)
0.0430212 + 0.999074i \(0.486302\pi\)
\(20\) 0 0
\(21\) −3797.69 −1.87919
\(22\) 0 0
\(23\) −2796.65 −1.10235 −0.551173 0.834391i \(-0.685820\pi\)
−0.551173 + 0.834391i \(0.685820\pi\)
\(24\) 0 0
\(25\) 2533.51 0.810722
\(26\) 0 0
\(27\) 3408.89 0.899919
\(28\) 0 0
\(29\) −2608.58 −0.575983 −0.287991 0.957633i \(-0.592987\pi\)
−0.287991 + 0.957633i \(0.592987\pi\)
\(30\) 0 0
\(31\) 1056.76 0.197502 0.0987510 0.995112i \(-0.468515\pi\)
0.0987510 + 0.995112i \(0.468515\pi\)
\(32\) 0 0
\(33\) 2037.56 0.325706
\(34\) 0 0
\(35\) 16964.7 2.34086
\(36\) 0 0
\(37\) 12536.8 1.50550 0.752752 0.658304i \(-0.228726\pi\)
0.752752 + 0.658304i \(0.228726\pi\)
\(38\) 0 0
\(39\) −7669.74 −0.807456
\(40\) 0 0
\(41\) 1130.09 0.104991 0.0524954 0.998621i \(-0.483282\pi\)
0.0524954 + 0.998621i \(0.483282\pi\)
\(42\) 0 0
\(43\) 14671.0 1.21001 0.605005 0.796222i \(-0.293171\pi\)
0.605005 + 0.796222i \(0.293171\pi\)
\(44\) 0 0
\(45\) 3051.37 0.224628
\(46\) 0 0
\(47\) 16882.2 1.11477 0.557383 0.830256i \(-0.311806\pi\)
0.557383 + 0.830256i \(0.311806\pi\)
\(48\) 0 0
\(49\) 34054.4 2.02620
\(50\) 0 0
\(51\) −3210.54 −0.172843
\(52\) 0 0
\(53\) 3313.02 0.162007 0.0810035 0.996714i \(-0.474187\pi\)
0.0810035 + 0.996714i \(0.474187\pi\)
\(54\) 0 0
\(55\) −9101.99 −0.405723
\(56\) 0 0
\(57\) −2279.93 −0.0929469
\(58\) 0 0
\(59\) −11454.0 −0.428378 −0.214189 0.976792i \(-0.568711\pi\)
−0.214189 + 0.976792i \(0.568711\pi\)
\(60\) 0 0
\(61\) −28227.5 −0.971286 −0.485643 0.874157i \(-0.661415\pi\)
−0.485643 + 0.874157i \(0.661415\pi\)
\(62\) 0 0
\(63\) 9148.26 0.290394
\(64\) 0 0
\(65\) 34261.4 1.00582
\(66\) 0 0
\(67\) 51431.0 1.39971 0.699855 0.714285i \(-0.253248\pi\)
0.699855 + 0.714285i \(0.253248\pi\)
\(68\) 0 0
\(69\) 47093.8 1.19081
\(70\) 0 0
\(71\) 16218.0 0.381814 0.190907 0.981608i \(-0.438857\pi\)
0.190907 + 0.981608i \(0.438857\pi\)
\(72\) 0 0
\(73\) −10168.8 −0.223337 −0.111669 0.993745i \(-0.535620\pi\)
−0.111669 + 0.993745i \(0.535620\pi\)
\(74\) 0 0
\(75\) −42662.6 −0.875779
\(76\) 0 0
\(77\) −27288.5 −0.524509
\(78\) 0 0
\(79\) −60841.2 −1.09681 −0.548404 0.836214i \(-0.684764\pi\)
−0.548404 + 0.836214i \(0.684764\pi\)
\(80\) 0 0
\(81\) −67260.7 −1.13907
\(82\) 0 0
\(83\) −45770.6 −0.729275 −0.364638 0.931150i \(-0.618807\pi\)
−0.364638 + 0.931150i \(0.618807\pi\)
\(84\) 0 0
\(85\) 14341.8 0.215306
\(86\) 0 0
\(87\) 43926.9 0.622203
\(88\) 0 0
\(89\) −82267.9 −1.10092 −0.550460 0.834862i \(-0.685548\pi\)
−0.550460 + 0.834862i \(0.685548\pi\)
\(90\) 0 0
\(91\) 102719. 1.30031
\(92\) 0 0
\(93\) −17795.1 −0.213351
\(94\) 0 0
\(95\) 10184.7 0.115781
\(96\) 0 0
\(97\) 53097.0 0.572981 0.286491 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(98\) 0 0
\(99\) −4908.28 −0.0503317
\(100\) 0 0
\(101\) 186821. 1.82231 0.911153 0.412069i \(-0.135194\pi\)
0.911153 + 0.412069i \(0.135194\pi\)
\(102\) 0 0
\(103\) −34290.5 −0.318479 −0.159240 0.987240i \(-0.550904\pi\)
−0.159240 + 0.987240i \(0.550904\pi\)
\(104\) 0 0
\(105\) −285674. −2.52870
\(106\) 0 0
\(107\) 224117. 1.89241 0.946206 0.323565i \(-0.104881\pi\)
0.946206 + 0.323565i \(0.104881\pi\)
\(108\) 0 0
\(109\) 162229. 1.30786 0.653931 0.756554i \(-0.273118\pi\)
0.653931 + 0.756554i \(0.273118\pi\)
\(110\) 0 0
\(111\) −211112. −1.62632
\(112\) 0 0
\(113\) 92225.0 0.679442 0.339721 0.940526i \(-0.389667\pi\)
0.339721 + 0.940526i \(0.389667\pi\)
\(114\) 0 0
\(115\) −210372. −1.48335
\(116\) 0 0
\(117\) 18475.6 0.124777
\(118\) 0 0
\(119\) 42997.8 0.278342
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 0 0
\(123\) −19029.9 −0.113416
\(124\) 0 0
\(125\) −44493.9 −0.254698
\(126\) 0 0
\(127\) −138299. −0.760868 −0.380434 0.924808i \(-0.624225\pi\)
−0.380434 + 0.924808i \(0.624225\pi\)
\(128\) 0 0
\(129\) −247051. −1.30711
\(130\) 0 0
\(131\) 54420.4 0.277066 0.138533 0.990358i \(-0.455761\pi\)
0.138533 + 0.990358i \(0.455761\pi\)
\(132\) 0 0
\(133\) 30534.5 0.149679
\(134\) 0 0
\(135\) 256427. 1.21096
\(136\) 0 0
\(137\) 40555.1 0.184605 0.0923025 0.995731i \(-0.470577\pi\)
0.0923025 + 0.995731i \(0.470577\pi\)
\(138\) 0 0
\(139\) −140537. −0.616955 −0.308477 0.951232i \(-0.599819\pi\)
−0.308477 + 0.951232i \(0.599819\pi\)
\(140\) 0 0
\(141\) −284285. −1.20422
\(142\) 0 0
\(143\) −55111.2 −0.225372
\(144\) 0 0
\(145\) −196225. −0.775060
\(146\) 0 0
\(147\) −573454. −2.18880
\(148\) 0 0
\(149\) 176073. 0.649722 0.324861 0.945762i \(-0.394683\pi\)
0.324861 + 0.945762i \(0.394683\pi\)
\(150\) 0 0
\(151\) −409241. −1.46062 −0.730309 0.683117i \(-0.760624\pi\)
−0.730309 + 0.683117i \(0.760624\pi\)
\(152\) 0 0
\(153\) 7733.85 0.0267096
\(154\) 0 0
\(155\) 79492.6 0.265765
\(156\) 0 0
\(157\) 14294.5 0.0462829 0.0231414 0.999732i \(-0.492633\pi\)
0.0231414 + 0.999732i \(0.492633\pi\)
\(158\) 0 0
\(159\) −55789.1 −0.175007
\(160\) 0 0
\(161\) −630713. −1.91764
\(162\) 0 0
\(163\) 418474. 1.23367 0.616836 0.787091i \(-0.288414\pi\)
0.616836 + 0.787091i \(0.288414\pi\)
\(164\) 0 0
\(165\) 153272. 0.438281
\(166\) 0 0
\(167\) 139747. 0.387749 0.193875 0.981026i \(-0.437894\pi\)
0.193875 + 0.981026i \(0.437894\pi\)
\(168\) 0 0
\(169\) −163845. −0.441282
\(170\) 0 0
\(171\) 5492.13 0.0143632
\(172\) 0 0
\(173\) −687104. −1.74545 −0.872725 0.488213i \(-0.837649\pi\)
−0.872725 + 0.488213i \(0.837649\pi\)
\(174\) 0 0
\(175\) 571368. 1.41033
\(176\) 0 0
\(177\) 192878. 0.462754
\(178\) 0 0
\(179\) −35496.4 −0.0828042 −0.0414021 0.999143i \(-0.513182\pi\)
−0.0414021 + 0.999143i \(0.513182\pi\)
\(180\) 0 0
\(181\) 260469. 0.590963 0.295481 0.955349i \(-0.404520\pi\)
0.295481 + 0.955349i \(0.404520\pi\)
\(182\) 0 0
\(183\) 475333. 1.04923
\(184\) 0 0
\(185\) 943056. 2.02585
\(186\) 0 0
\(187\) −23069.4 −0.0482429
\(188\) 0 0
\(189\) 768789. 1.56550
\(190\) 0 0
\(191\) −392051. −0.777605 −0.388803 0.921321i \(-0.627111\pi\)
−0.388803 + 0.921321i \(0.627111\pi\)
\(192\) 0 0
\(193\) 15776.8 0.0304878 0.0152439 0.999884i \(-0.495148\pi\)
0.0152439 + 0.999884i \(0.495148\pi\)
\(194\) 0 0
\(195\) −576941. −1.08654
\(196\) 0 0
\(197\) 545551. 1.00154 0.500771 0.865580i \(-0.333050\pi\)
0.500771 + 0.865580i \(0.333050\pi\)
\(198\) 0 0
\(199\) 546514. 0.978293 0.489146 0.872202i \(-0.337308\pi\)
0.489146 + 0.872202i \(0.337308\pi\)
\(200\) 0 0
\(201\) −866065. −1.51203
\(202\) 0 0
\(203\) −588300. −1.00198
\(204\) 0 0
\(205\) 85008.5 0.141279
\(206\) 0 0
\(207\) −113444. −0.184016
\(208\) 0 0
\(209\) −16382.6 −0.0259428
\(210\) 0 0
\(211\) −537150. −0.830596 −0.415298 0.909686i \(-0.636323\pi\)
−0.415298 + 0.909686i \(0.636323\pi\)
\(212\) 0 0
\(213\) −273101. −0.412453
\(214\) 0 0
\(215\) 1.10360e6 1.62823
\(216\) 0 0
\(217\) 238325. 0.343574
\(218\) 0 0
\(219\) 171236. 0.241259
\(220\) 0 0
\(221\) 86837.3 0.119598
\(222\) 0 0
\(223\) 189640. 0.255368 0.127684 0.991815i \(-0.459246\pi\)
0.127684 + 0.991815i \(0.459246\pi\)
\(224\) 0 0
\(225\) 102770. 0.135335
\(226\) 0 0
\(227\) −363428. −0.468116 −0.234058 0.972223i \(-0.575201\pi\)
−0.234058 + 0.972223i \(0.575201\pi\)
\(228\) 0 0
\(229\) 504331. 0.635516 0.317758 0.948172i \(-0.397070\pi\)
0.317758 + 0.948172i \(0.397070\pi\)
\(230\) 0 0
\(231\) 459521. 0.566598
\(232\) 0 0
\(233\) −1.20159e6 −1.45000 −0.724999 0.688750i \(-0.758160\pi\)
−0.724999 + 0.688750i \(0.758160\pi\)
\(234\) 0 0
\(235\) 1.26993e6 1.50006
\(236\) 0 0
\(237\) 1.02453e6 1.18482
\(238\) 0 0
\(239\) 185929. 0.210549 0.105275 0.994443i \(-0.466428\pi\)
0.105275 + 0.994443i \(0.466428\pi\)
\(240\) 0 0
\(241\) 174842. 0.193911 0.0969556 0.995289i \(-0.469090\pi\)
0.0969556 + 0.995289i \(0.469090\pi\)
\(242\) 0 0
\(243\) 304267. 0.330552
\(244\) 0 0
\(245\) 2.56168e6 2.72652
\(246\) 0 0
\(247\) 61666.8 0.0643145
\(248\) 0 0
\(249\) 770748. 0.787797
\(250\) 0 0
\(251\) −447906. −0.448748 −0.224374 0.974503i \(-0.572034\pi\)
−0.224374 + 0.974503i \(0.572034\pi\)
\(252\) 0 0
\(253\) 338394. 0.332370
\(254\) 0 0
\(255\) −241506. −0.232583
\(256\) 0 0
\(257\) −1.14572e6 −1.08204 −0.541022 0.841009i \(-0.681962\pi\)
−0.541022 + 0.841009i \(0.681962\pi\)
\(258\) 0 0
\(259\) 2.82736e6 2.61897
\(260\) 0 0
\(261\) −105815. −0.0961496
\(262\) 0 0
\(263\) −443228. −0.395128 −0.197564 0.980290i \(-0.563303\pi\)
−0.197564 + 0.980290i \(0.563303\pi\)
\(264\) 0 0
\(265\) 249215. 0.218002
\(266\) 0 0
\(267\) 1.38534e6 1.18926
\(268\) 0 0
\(269\) −1.88722e6 −1.59016 −0.795082 0.606502i \(-0.792572\pi\)
−0.795082 + 0.606502i \(0.792572\pi\)
\(270\) 0 0
\(271\) −2.24203e6 −1.85446 −0.927230 0.374491i \(-0.877817\pi\)
−0.927230 + 0.374491i \(0.877817\pi\)
\(272\) 0 0
\(273\) −1.72972e6 −1.40465
\(274\) 0 0
\(275\) −306554. −0.244442
\(276\) 0 0
\(277\) 1.27824e6 1.00095 0.500474 0.865751i \(-0.333159\pi\)
0.500474 + 0.865751i \(0.333159\pi\)
\(278\) 0 0
\(279\) 42866.7 0.0329693
\(280\) 0 0
\(281\) 549325. 0.415015 0.207508 0.978233i \(-0.433465\pi\)
0.207508 + 0.978233i \(0.433465\pi\)
\(282\) 0 0
\(283\) 135813. 0.100803 0.0504016 0.998729i \(-0.483950\pi\)
0.0504016 + 0.998729i \(0.483950\pi\)
\(284\) 0 0
\(285\) −171504. −0.125072
\(286\) 0 0
\(287\) 254862. 0.182642
\(288\) 0 0
\(289\) −1.38351e6 −0.974399
\(290\) 0 0
\(291\) −894120. −0.618961
\(292\) 0 0
\(293\) −1.76403e6 −1.20043 −0.600215 0.799839i \(-0.704918\pi\)
−0.600215 + 0.799839i \(0.704918\pi\)
\(294\) 0 0
\(295\) −861606. −0.576439
\(296\) 0 0
\(297\) −412476. −0.271336
\(298\) 0 0
\(299\) −1.27377e6 −0.823976
\(300\) 0 0
\(301\) 3.30868e6 2.10493
\(302\) 0 0
\(303\) −3.14594e6 −1.96854
\(304\) 0 0
\(305\) −2.12336e6 −1.30699
\(306\) 0 0
\(307\) 1.93533e6 1.17195 0.585975 0.810329i \(-0.300712\pi\)
0.585975 + 0.810329i \(0.300712\pi\)
\(308\) 0 0
\(309\) 577431. 0.344036
\(310\) 0 0
\(311\) −2.98327e6 −1.74901 −0.874504 0.485019i \(-0.838813\pi\)
−0.874504 + 0.485019i \(0.838813\pi\)
\(312\) 0 0
\(313\) −10701.5 −0.00617426 −0.00308713 0.999995i \(-0.500983\pi\)
−0.00308713 + 0.999995i \(0.500983\pi\)
\(314\) 0 0
\(315\) 688160. 0.390762
\(316\) 0 0
\(317\) −2.43658e6 −1.36186 −0.680929 0.732349i \(-0.738424\pi\)
−0.680929 + 0.732349i \(0.738424\pi\)
\(318\) 0 0
\(319\) 315638. 0.173665
\(320\) 0 0
\(321\) −3.77399e6 −2.04427
\(322\) 0 0
\(323\) 25813.6 0.0137671
\(324\) 0 0
\(325\) 1.15392e6 0.605994
\(326\) 0 0
\(327\) −2.73183e6 −1.41281
\(328\) 0 0
\(329\) 3.80734e6 1.93924
\(330\) 0 0
\(331\) −119576. −0.0599894 −0.0299947 0.999550i \(-0.509549\pi\)
−0.0299947 + 0.999550i \(0.509549\pi\)
\(332\) 0 0
\(333\) 508546. 0.251316
\(334\) 0 0
\(335\) 3.86880e6 1.88349
\(336\) 0 0
\(337\) −2.02195e6 −0.969830 −0.484915 0.874561i \(-0.661149\pi\)
−0.484915 + 0.874561i \(0.661149\pi\)
\(338\) 0 0
\(339\) −1.55301e6 −0.733965
\(340\) 0 0
\(341\) −127868. −0.0595491
\(342\) 0 0
\(343\) 3.88971e6 1.78518
\(344\) 0 0
\(345\) 3.54254e6 1.60238
\(346\) 0 0
\(347\) −3.01864e6 −1.34582 −0.672912 0.739723i \(-0.734957\pi\)
−0.672912 + 0.739723i \(0.734957\pi\)
\(348\) 0 0
\(349\) 2.40399e6 1.05650 0.528250 0.849089i \(-0.322848\pi\)
0.528250 + 0.849089i \(0.322848\pi\)
\(350\) 0 0
\(351\) 1.55263e6 0.672666
\(352\) 0 0
\(353\) 3.62981e6 1.55041 0.775206 0.631709i \(-0.217646\pi\)
0.775206 + 0.631709i \(0.217646\pi\)
\(354\) 0 0
\(355\) 1.21997e6 0.513780
\(356\) 0 0
\(357\) −724055. −0.300678
\(358\) 0 0
\(359\) −939181. −0.384603 −0.192302 0.981336i \(-0.561595\pi\)
−0.192302 + 0.981336i \(0.561595\pi\)
\(360\) 0 0
\(361\) −2.45777e6 −0.992597
\(362\) 0 0
\(363\) −246545. −0.0982042
\(364\) 0 0
\(365\) −764926. −0.300530
\(366\) 0 0
\(367\) 2.26697e6 0.878577 0.439288 0.898346i \(-0.355231\pi\)
0.439288 + 0.898346i \(0.355231\pi\)
\(368\) 0 0
\(369\) 45841.1 0.0175263
\(370\) 0 0
\(371\) 747167. 0.281827
\(372\) 0 0
\(373\) −4.55029e6 −1.69343 −0.846714 0.532048i \(-0.821423\pi\)
−0.846714 + 0.532048i \(0.821423\pi\)
\(374\) 0 0
\(375\) 749250. 0.275137
\(376\) 0 0
\(377\) −1.18812e6 −0.430532
\(378\) 0 0
\(379\) −618788. −0.221281 −0.110641 0.993860i \(-0.535290\pi\)
−0.110641 + 0.993860i \(0.535290\pi\)
\(380\) 0 0
\(381\) 2.32886e6 0.821924
\(382\) 0 0
\(383\) −2.23829e6 −0.779686 −0.389843 0.920881i \(-0.627471\pi\)
−0.389843 + 0.920881i \(0.627471\pi\)
\(384\) 0 0
\(385\) −2.05272e6 −0.705795
\(386\) 0 0
\(387\) 595120. 0.201989
\(388\) 0 0
\(389\) −4.60206e6 −1.54198 −0.770989 0.636848i \(-0.780238\pi\)
−0.770989 + 0.636848i \(0.780238\pi\)
\(390\) 0 0
\(391\) −533199. −0.176379
\(392\) 0 0
\(393\) −916405. −0.299299
\(394\) 0 0
\(395\) −4.57666e6 −1.47590
\(396\) 0 0
\(397\) −4.35532e6 −1.38690 −0.693448 0.720506i \(-0.743909\pi\)
−0.693448 + 0.720506i \(0.743909\pi\)
\(398\) 0 0
\(399\) −514181. −0.161690
\(400\) 0 0
\(401\) 3.62515e6 1.12581 0.562905 0.826522i \(-0.309684\pi\)
0.562905 + 0.826522i \(0.309684\pi\)
\(402\) 0 0
\(403\) 481316. 0.147628
\(404\) 0 0
\(405\) −5.05955e6 −1.53276
\(406\) 0 0
\(407\) −1.51695e6 −0.453927
\(408\) 0 0
\(409\) 4.13585e6 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(410\) 0 0
\(411\) −682922. −0.199419
\(412\) 0 0
\(413\) −2.58316e6 −0.745206
\(414\) 0 0
\(415\) −3.44300e6 −0.981335
\(416\) 0 0
\(417\) 2.36655e6 0.666463
\(418\) 0 0
\(419\) 2.46691e6 0.686464 0.343232 0.939251i \(-0.388478\pi\)
0.343232 + 0.939251i \(0.388478\pi\)
\(420\) 0 0
\(421\) 3.45258e6 0.949376 0.474688 0.880154i \(-0.342561\pi\)
0.474688 + 0.880154i \(0.342561\pi\)
\(422\) 0 0
\(423\) 684813. 0.186089
\(424\) 0 0
\(425\) 483029. 0.129718
\(426\) 0 0
\(427\) −6.36599e6 −1.68965
\(428\) 0 0
\(429\) 928038. 0.243457
\(430\) 0 0
\(431\) 3.65893e6 0.948770 0.474385 0.880318i \(-0.342671\pi\)
0.474385 + 0.880318i \(0.342671\pi\)
\(432\) 0 0
\(433\) −1.59716e6 −0.409381 −0.204690 0.978827i \(-0.565619\pi\)
−0.204690 + 0.978827i \(0.565619\pi\)
\(434\) 0 0
\(435\) 3.30431e6 0.837256
\(436\) 0 0
\(437\) −378647. −0.0948485
\(438\) 0 0
\(439\) 1.58464e6 0.392437 0.196219 0.980560i \(-0.437134\pi\)
0.196219 + 0.980560i \(0.437134\pi\)
\(440\) 0 0
\(441\) 1.38139e6 0.338237
\(442\) 0 0
\(443\) −2.29633e6 −0.555936 −0.277968 0.960590i \(-0.589661\pi\)
−0.277968 + 0.960590i \(0.589661\pi\)
\(444\) 0 0
\(445\) −6.18844e6 −1.48143
\(446\) 0 0
\(447\) −2.96496e6 −0.701859
\(448\) 0 0
\(449\) 3.31569e6 0.776171 0.388086 0.921623i \(-0.373136\pi\)
0.388086 + 0.921623i \(0.373136\pi\)
\(450\) 0 0
\(451\) −136740. −0.0316559
\(452\) 0 0
\(453\) 6.89136e6 1.57783
\(454\) 0 0
\(455\) 7.72680e6 1.74973
\(456\) 0 0
\(457\) 2.20892e6 0.494754 0.247377 0.968919i \(-0.420431\pi\)
0.247377 + 0.968919i \(0.420431\pi\)
\(458\) 0 0
\(459\) 649927. 0.143990
\(460\) 0 0
\(461\) −1.86064e6 −0.407764 −0.203882 0.978995i \(-0.565356\pi\)
−0.203882 + 0.978995i \(0.565356\pi\)
\(462\) 0 0
\(463\) −1.20592e6 −0.261437 −0.130718 0.991420i \(-0.541728\pi\)
−0.130718 + 0.991420i \(0.541728\pi\)
\(464\) 0 0
\(465\) −1.33861e6 −0.287091
\(466\) 0 0
\(467\) −2.29388e6 −0.486719 −0.243360 0.969936i \(-0.578250\pi\)
−0.243360 + 0.969936i \(0.578250\pi\)
\(468\) 0 0
\(469\) 1.15990e7 2.43493
\(470\) 0 0
\(471\) −240711. −0.0499969
\(472\) 0 0
\(473\) −1.77519e6 −0.364832
\(474\) 0 0
\(475\) 343019. 0.0697565
\(476\) 0 0
\(477\) 134390. 0.0270441
\(478\) 0 0
\(479\) 7.90892e6 1.57499 0.787496 0.616320i \(-0.211377\pi\)
0.787496 + 0.616320i \(0.211377\pi\)
\(480\) 0 0
\(481\) 5.71007e6 1.12533
\(482\) 0 0
\(483\) 1.06208e7 2.07152
\(484\) 0 0
\(485\) 3.99412e6 0.771021
\(486\) 0 0
\(487\) −3.48410e6 −0.665684 −0.332842 0.942983i \(-0.608008\pi\)
−0.332842 + 0.942983i \(0.608008\pi\)
\(488\) 0 0
\(489\) −7.04684e6 −1.33267
\(490\) 0 0
\(491\) 8.98096e6 1.68120 0.840599 0.541658i \(-0.182203\pi\)
0.840599 + 0.541658i \(0.182203\pi\)
\(492\) 0 0
\(493\) −497343. −0.0921592
\(494\) 0 0
\(495\) −369216. −0.0677279
\(496\) 0 0
\(497\) 3.65756e6 0.664203
\(498\) 0 0
\(499\) −6.67736e6 −1.20048 −0.600238 0.799821i \(-0.704928\pi\)
−0.600238 + 0.799821i \(0.704928\pi\)
\(500\) 0 0
\(501\) −2.35325e6 −0.418865
\(502\) 0 0
\(503\) −7.58428e6 −1.33658 −0.668289 0.743902i \(-0.732973\pi\)
−0.668289 + 0.743902i \(0.732973\pi\)
\(504\) 0 0
\(505\) 1.40532e7 2.45215
\(506\) 0 0
\(507\) 2.75905e6 0.476693
\(508\) 0 0
\(509\) −6.02580e6 −1.03091 −0.515454 0.856917i \(-0.672377\pi\)
−0.515454 + 0.856917i \(0.672377\pi\)
\(510\) 0 0
\(511\) −2.29331e6 −0.388518
\(512\) 0 0
\(513\) 461540. 0.0774312
\(514\) 0 0
\(515\) −2.57944e6 −0.428555
\(516\) 0 0
\(517\) −2.04274e6 −0.336114
\(518\) 0 0
\(519\) 1.15704e7 1.88551
\(520\) 0 0
\(521\) −4.58541e6 −0.740088 −0.370044 0.929014i \(-0.620657\pi\)
−0.370044 + 0.929014i \(0.620657\pi\)
\(522\) 0 0
\(523\) 4.88145e6 0.780359 0.390179 0.920739i \(-0.372413\pi\)
0.390179 + 0.920739i \(0.372413\pi\)
\(524\) 0 0
\(525\) −9.62148e6 −1.52350
\(526\) 0 0
\(527\) 201478. 0.0316010
\(528\) 0 0
\(529\) 1.38489e6 0.215168
\(530\) 0 0
\(531\) −464624. −0.0715098
\(532\) 0 0
\(533\) 514714. 0.0784780
\(534\) 0 0
\(535\) 1.68588e7 2.54649
\(536\) 0 0
\(537\) 597738. 0.0894489
\(538\) 0 0
\(539\) −4.12058e6 −0.610923
\(540\) 0 0
\(541\) 6.21940e6 0.913598 0.456799 0.889570i \(-0.348996\pi\)
0.456799 + 0.889570i \(0.348996\pi\)
\(542\) 0 0
\(543\) −4.38614e6 −0.638385
\(544\) 0 0
\(545\) 1.22034e7 1.75990
\(546\) 0 0
\(547\) 9.49047e6 1.35619 0.678093 0.734976i \(-0.262807\pi\)
0.678093 + 0.734976i \(0.262807\pi\)
\(548\) 0 0
\(549\) −1.14503e6 −0.162138
\(550\) 0 0
\(551\) −353184. −0.0495589
\(552\) 0 0
\(553\) −1.37212e7 −1.90800
\(554\) 0 0
\(555\) −1.58805e7 −2.18842
\(556\) 0 0
\(557\) −2.92907e6 −0.400029 −0.200014 0.979793i \(-0.564099\pi\)
−0.200014 + 0.979793i \(0.564099\pi\)
\(558\) 0 0
\(559\) 6.68213e6 0.904451
\(560\) 0 0
\(561\) 388475. 0.0521141
\(562\) 0 0
\(563\) 455079. 0.0605084 0.0302542 0.999542i \(-0.490368\pi\)
0.0302542 + 0.999542i \(0.490368\pi\)
\(564\) 0 0
\(565\) 6.93744e6 0.914278
\(566\) 0 0
\(567\) −1.51689e7 −1.98152
\(568\) 0 0
\(569\) −6.27664e6 −0.812730 −0.406365 0.913711i \(-0.633204\pi\)
−0.406365 + 0.913711i \(0.633204\pi\)
\(570\) 0 0
\(571\) 621794. 0.0798098 0.0399049 0.999203i \(-0.487294\pi\)
0.0399049 + 0.999203i \(0.487294\pi\)
\(572\) 0 0
\(573\) 6.60189e6 0.840005
\(574\) 0 0
\(575\) −7.08532e6 −0.893697
\(576\) 0 0
\(577\) 1.28776e7 1.61026 0.805130 0.593098i \(-0.202095\pi\)
0.805130 + 0.593098i \(0.202095\pi\)
\(578\) 0 0
\(579\) −265671. −0.0329343
\(580\) 0 0
\(581\) −1.03224e7 −1.26865
\(582\) 0 0
\(583\) −400875. −0.0488470
\(584\) 0 0
\(585\) 1.38979e6 0.167904
\(586\) 0 0
\(587\) −1.08775e7 −1.30296 −0.651482 0.758664i \(-0.725852\pi\)
−0.651482 + 0.758664i \(0.725852\pi\)
\(588\) 0 0
\(589\) 143078. 0.0169935
\(590\) 0 0
\(591\) −9.18673e6 −1.08191
\(592\) 0 0
\(593\) −7.50449e6 −0.876364 −0.438182 0.898886i \(-0.644378\pi\)
−0.438182 + 0.898886i \(0.644378\pi\)
\(594\) 0 0
\(595\) 3.23442e6 0.374545
\(596\) 0 0
\(597\) −9.20295e6 −1.05680
\(598\) 0 0
\(599\) 7.69438e6 0.876207 0.438104 0.898925i \(-0.355650\pi\)
0.438104 + 0.898925i \(0.355650\pi\)
\(600\) 0 0
\(601\) 3.14770e6 0.355473 0.177737 0.984078i \(-0.443123\pi\)
0.177737 + 0.984078i \(0.443123\pi\)
\(602\) 0 0
\(603\) 2.08626e6 0.233655
\(604\) 0 0
\(605\) 1.10134e6 0.122330
\(606\) 0 0
\(607\) 4.57397e6 0.503874 0.251937 0.967744i \(-0.418932\pi\)
0.251937 + 0.967744i \(0.418932\pi\)
\(608\) 0 0
\(609\) 9.90660e6 1.08238
\(610\) 0 0
\(611\) 7.68923e6 0.833258
\(612\) 0 0
\(613\) −1.56075e7 −1.67758 −0.838790 0.544455i \(-0.816736\pi\)
−0.838790 + 0.544455i \(0.816736\pi\)
\(614\) 0 0
\(615\) −1.43149e6 −0.152616
\(616\) 0 0
\(617\) −1.18602e7 −1.25424 −0.627119 0.778924i \(-0.715766\pi\)
−0.627119 + 0.778924i \(0.715766\pi\)
\(618\) 0 0
\(619\) −7.91821e6 −0.830616 −0.415308 0.909681i \(-0.636326\pi\)
−0.415308 + 0.909681i \(0.636326\pi\)
\(620\) 0 0
\(621\) −9.53346e6 −0.992023
\(622\) 0 0
\(623\) −1.85534e7 −1.91516
\(624\) 0 0
\(625\) −1.12642e7 −1.15345
\(626\) 0 0
\(627\) 275872. 0.0280246
\(628\) 0 0
\(629\) 2.39022e6 0.240886
\(630\) 0 0
\(631\) 1.11561e7 1.11542 0.557709 0.830037i \(-0.311681\pi\)
0.557709 + 0.830037i \(0.311681\pi\)
\(632\) 0 0
\(633\) 9.04527e6 0.897248
\(634\) 0 0
\(635\) −1.04033e7 −1.02385
\(636\) 0 0
\(637\) 1.55106e7 1.51453
\(638\) 0 0
\(639\) 657872. 0.0637367
\(640\) 0 0
\(641\) 7.17389e6 0.689620 0.344810 0.938672i \(-0.387943\pi\)
0.344810 + 0.938672i \(0.387943\pi\)
\(642\) 0 0
\(643\) −7.14025e6 −0.681061 −0.340531 0.940233i \(-0.610607\pi\)
−0.340531 + 0.940233i \(0.610607\pi\)
\(644\) 0 0
\(645\) −1.85839e7 −1.75889
\(646\) 0 0
\(647\) −1.56897e7 −1.47351 −0.736756 0.676159i \(-0.763643\pi\)
−0.736756 + 0.676159i \(0.763643\pi\)
\(648\) 0 0
\(649\) 1.38594e6 0.129161
\(650\) 0 0
\(651\) −4.01324e6 −0.371145
\(652\) 0 0
\(653\) −5.04236e6 −0.462755 −0.231378 0.972864i \(-0.574323\pi\)
−0.231378 + 0.972864i \(0.574323\pi\)
\(654\) 0 0
\(655\) 4.09367e6 0.372829
\(656\) 0 0
\(657\) −412489. −0.0372820
\(658\) 0 0
\(659\) 9.10902e6 0.817068 0.408534 0.912743i \(-0.366040\pi\)
0.408534 + 0.912743i \(0.366040\pi\)
\(660\) 0 0
\(661\) 1.31308e7 1.16893 0.584464 0.811420i \(-0.301305\pi\)
0.584464 + 0.811420i \(0.301305\pi\)
\(662\) 0 0
\(663\) −1.46229e6 −0.129196
\(664\) 0 0
\(665\) 2.29690e6 0.201413
\(666\) 0 0
\(667\) 7.29528e6 0.634933
\(668\) 0 0
\(669\) −3.19341e6 −0.275861
\(670\) 0 0
\(671\) 3.41552e6 0.292854
\(672\) 0 0
\(673\) 1.55171e7 1.32061 0.660303 0.750999i \(-0.270428\pi\)
0.660303 + 0.750999i \(0.270428\pi\)
\(674\) 0 0
\(675\) 8.63644e6 0.729584
\(676\) 0 0
\(677\) −1.40356e7 −1.17695 −0.588476 0.808515i \(-0.700272\pi\)
−0.588476 + 0.808515i \(0.700272\pi\)
\(678\) 0 0
\(679\) 1.19747e7 0.996758
\(680\) 0 0
\(681\) 6.11990e6 0.505681
\(682\) 0 0
\(683\) −5.34969e6 −0.438810 −0.219405 0.975634i \(-0.570412\pi\)
−0.219405 + 0.975634i \(0.570412\pi\)
\(684\) 0 0
\(685\) 3.05068e6 0.248410
\(686\) 0 0
\(687\) −8.49261e6 −0.686514
\(688\) 0 0
\(689\) 1.50896e6 0.121096
\(690\) 0 0
\(691\) −1.31390e7 −1.04681 −0.523404 0.852084i \(-0.675338\pi\)
−0.523404 + 0.852084i \(0.675338\pi\)
\(692\) 0 0
\(693\) −1.10694e6 −0.0875569
\(694\) 0 0
\(695\) −1.05716e7 −0.830194
\(696\) 0 0
\(697\) 215458. 0.0167989
\(698\) 0 0
\(699\) 2.02341e7 1.56636
\(700\) 0 0
\(701\) −2.49888e7 −1.92066 −0.960330 0.278865i \(-0.910042\pi\)
−0.960330 + 0.278865i \(0.910042\pi\)
\(702\) 0 0
\(703\) 1.69740e6 0.129537
\(704\) 0 0
\(705\) −2.13848e7 −1.62044
\(706\) 0 0
\(707\) 4.21327e7 3.17008
\(708\) 0 0
\(709\) −8.86200e6 −0.662089 −0.331044 0.943615i \(-0.607401\pi\)
−0.331044 + 0.943615i \(0.607401\pi\)
\(710\) 0 0
\(711\) −2.46798e6 −0.183092
\(712\) 0 0
\(713\) −2.95538e6 −0.217716
\(714\) 0 0
\(715\) −4.14563e6 −0.303268
\(716\) 0 0
\(717\) −3.13093e6 −0.227445
\(718\) 0 0
\(719\) −2.58635e7 −1.86580 −0.932901 0.360132i \(-0.882732\pi\)
−0.932901 + 0.360132i \(0.882732\pi\)
\(720\) 0 0
\(721\) −7.73336e6 −0.554026
\(722\) 0 0
\(723\) −2.94423e6 −0.209472
\(724\) 0 0
\(725\) −6.60886e6 −0.466962
\(726\) 0 0
\(727\) 1.71871e7 1.20605 0.603026 0.797721i \(-0.293961\pi\)
0.603026 + 0.797721i \(0.293961\pi\)
\(728\) 0 0
\(729\) 1.12207e7 0.781988
\(730\) 0 0
\(731\) 2.79712e6 0.193606
\(732\) 0 0
\(733\) 1.85650e7 1.27625 0.638125 0.769932i \(-0.279710\pi\)
0.638125 + 0.769932i \(0.279710\pi\)
\(734\) 0 0
\(735\) −4.31370e7 −2.94531
\(736\) 0 0
\(737\) −6.22315e6 −0.422028
\(738\) 0 0
\(739\) −5.94724e6 −0.400594 −0.200297 0.979735i \(-0.564191\pi\)
−0.200297 + 0.979735i \(0.564191\pi\)
\(740\) 0 0
\(741\) −1.03843e6 −0.0694755
\(742\) 0 0
\(743\) 2.72654e7 1.81193 0.905963 0.423357i \(-0.139148\pi\)
0.905963 + 0.423357i \(0.139148\pi\)
\(744\) 0 0
\(745\) 1.32448e7 0.874285
\(746\) 0 0
\(747\) −1.85665e6 −0.121739
\(748\) 0 0
\(749\) 5.05440e7 3.29204
\(750\) 0 0
\(751\) −1.30069e7 −0.841541 −0.420770 0.907167i \(-0.638240\pi\)
−0.420770 + 0.907167i \(0.638240\pi\)
\(752\) 0 0
\(753\) 7.54245e6 0.484758
\(754\) 0 0
\(755\) −3.07844e7 −1.96545
\(756\) 0 0
\(757\) −9.22009e6 −0.584784 −0.292392 0.956299i \(-0.594451\pi\)
−0.292392 + 0.956299i \(0.594451\pi\)
\(758\) 0 0
\(759\) −5.69835e6 −0.359041
\(760\) 0 0
\(761\) 328083. 0.0205363 0.0102682 0.999947i \(-0.496731\pi\)
0.0102682 + 0.999947i \(0.496731\pi\)
\(762\) 0 0
\(763\) 3.65866e7 2.27516
\(764\) 0 0
\(765\) 581764. 0.0359412
\(766\) 0 0
\(767\) −5.21690e6 −0.320202
\(768\) 0 0
\(769\) 2.19214e6 0.133676 0.0668380 0.997764i \(-0.478709\pi\)
0.0668380 + 0.997764i \(0.478709\pi\)
\(770\) 0 0
\(771\) 1.92932e7 1.16887
\(772\) 0 0
\(773\) 2.18539e7 1.31547 0.657735 0.753249i \(-0.271515\pi\)
0.657735 + 0.753249i \(0.271515\pi\)
\(774\) 0 0
\(775\) 2.67730e6 0.160119
\(776\) 0 0
\(777\) −4.76109e7 −2.82914
\(778\) 0 0
\(779\) 153006. 0.00903367
\(780\) 0 0
\(781\) −1.96238e6 −0.115121
\(782\) 0 0
\(783\) −8.89237e6 −0.518338
\(784\) 0 0
\(785\) 1.07528e6 0.0622797
\(786\) 0 0
\(787\) −2.61010e7 −1.50217 −0.751087 0.660203i \(-0.770470\pi\)
−0.751087 + 0.660203i \(0.770470\pi\)
\(788\) 0 0
\(789\) 7.46368e6 0.426835
\(790\) 0 0
\(791\) 2.07990e7 1.18196
\(792\) 0 0
\(793\) −1.28566e7 −0.726011
\(794\) 0 0
\(795\) −4.19663e6 −0.235495
\(796\) 0 0
\(797\) 1.39846e7 0.779840 0.389920 0.920849i \(-0.372503\pi\)
0.389920 + 0.920849i \(0.372503\pi\)
\(798\) 0 0
\(799\) 3.21869e6 0.178366
\(800\) 0 0
\(801\) −3.33714e6 −0.183778
\(802\) 0 0
\(803\) 1.23042e6 0.0673388
\(804\) 0 0
\(805\) −4.74442e7 −2.58044
\(806\) 0 0
\(807\) 3.17796e7 1.71777
\(808\) 0 0
\(809\) 2.70989e7 1.45573 0.727865 0.685721i \(-0.240513\pi\)
0.727865 + 0.685721i \(0.240513\pi\)
\(810\) 0 0
\(811\) −1.99644e7 −1.06587 −0.532936 0.846156i \(-0.678911\pi\)
−0.532936 + 0.846156i \(0.678911\pi\)
\(812\) 0 0
\(813\) 3.77543e7 2.00327
\(814\) 0 0
\(815\) 3.14789e7 1.66007
\(816\) 0 0
\(817\) 1.98635e6 0.104112
\(818\) 0 0
\(819\) 4.16671e6 0.217062
\(820\) 0 0
\(821\) 3.18829e7 1.65082 0.825410 0.564533i \(-0.190944\pi\)
0.825410 + 0.564533i \(0.190944\pi\)
\(822\) 0 0
\(823\) 1.34203e7 0.690655 0.345328 0.938482i \(-0.387768\pi\)
0.345328 + 0.938482i \(0.387768\pi\)
\(824\) 0 0
\(825\) 5.16218e6 0.264057
\(826\) 0 0
\(827\) −1.19386e7 −0.607002 −0.303501 0.952831i \(-0.598156\pi\)
−0.303501 + 0.952831i \(0.598156\pi\)
\(828\) 0 0
\(829\) 2.59274e7 1.31031 0.655153 0.755497i \(-0.272604\pi\)
0.655153 + 0.755497i \(0.272604\pi\)
\(830\) 0 0
\(831\) −2.15247e7 −1.08127
\(832\) 0 0
\(833\) 6.49269e6 0.324199
\(834\) 0 0
\(835\) 1.05122e7 0.521768
\(836\) 0 0
\(837\) 3.60237e6 0.177736
\(838\) 0 0
\(839\) 3.21482e7 1.57671 0.788354 0.615222i \(-0.210933\pi\)
0.788354 + 0.615222i \(0.210933\pi\)
\(840\) 0 0
\(841\) −1.37064e7 −0.668244
\(842\) 0 0
\(843\) −9.25029e6 −0.448318
\(844\) 0 0
\(845\) −1.23249e7 −0.593803
\(846\) 0 0
\(847\) 3.30191e6 0.158145
\(848\) 0 0
\(849\) −2.28700e6 −0.108892
\(850\) 0 0
\(851\) −3.50610e7 −1.65959
\(852\) 0 0
\(853\) −5.64308e6 −0.265548 −0.132774 0.991146i \(-0.542388\pi\)
−0.132774 + 0.991146i \(0.542388\pi\)
\(854\) 0 0
\(855\) 413135. 0.0193275
\(856\) 0 0
\(857\) −1.77067e7 −0.823543 −0.411772 0.911287i \(-0.635090\pi\)
−0.411772 + 0.911287i \(0.635090\pi\)
\(858\) 0 0
\(859\) −1.57119e7 −0.726515 −0.363258 0.931689i \(-0.618336\pi\)
−0.363258 + 0.931689i \(0.618336\pi\)
\(860\) 0 0
\(861\) −4.29172e6 −0.197298
\(862\) 0 0
\(863\) −245263. −0.0112100 −0.00560500 0.999984i \(-0.501784\pi\)
−0.00560500 + 0.999984i \(0.501784\pi\)
\(864\) 0 0
\(865\) −5.16861e7 −2.34873
\(866\) 0 0
\(867\) 2.32974e7 1.05259
\(868\) 0 0
\(869\) 7.36179e6 0.330700
\(870\) 0 0
\(871\) 2.34250e7 1.04625
\(872\) 0 0
\(873\) 2.15384e6 0.0956486
\(874\) 0 0
\(875\) −1.00345e7 −0.443073
\(876\) 0 0
\(877\) 1.04352e7 0.458142 0.229071 0.973410i \(-0.426431\pi\)
0.229071 + 0.973410i \(0.426431\pi\)
\(878\) 0 0
\(879\) 2.97051e7 1.29676
\(880\) 0 0
\(881\) −1.10430e7 −0.479344 −0.239672 0.970854i \(-0.577040\pi\)
−0.239672 + 0.970854i \(0.577040\pi\)
\(882\) 0 0
\(883\) −5.41498e6 −0.233720 −0.116860 0.993148i \(-0.537283\pi\)
−0.116860 + 0.993148i \(0.537283\pi\)
\(884\) 0 0
\(885\) 1.45089e7 0.622696
\(886\) 0 0
\(887\) 1.52663e7 0.651515 0.325757 0.945453i \(-0.394381\pi\)
0.325757 + 0.945453i \(0.394381\pi\)
\(888\) 0 0
\(889\) −3.11898e7 −1.32360
\(890\) 0 0
\(891\) 8.13854e6 0.343441
\(892\) 0 0
\(893\) 2.28573e6 0.0959170
\(894\) 0 0
\(895\) −2.67015e6 −0.111424
\(896\) 0 0
\(897\) 2.14496e7 0.890096
\(898\) 0 0
\(899\) −2.75664e6 −0.113758
\(900\) 0 0
\(901\) 631648. 0.0259217
\(902\) 0 0
\(903\) −5.57160e7 −2.27384
\(904\) 0 0
\(905\) 1.95933e7 0.795218
\(906\) 0 0
\(907\) −2.02437e7 −0.817094 −0.408547 0.912737i \(-0.633964\pi\)
−0.408547 + 0.912737i \(0.633964\pi\)
\(908\) 0 0
\(909\) 7.57825e6 0.304200
\(910\) 0 0
\(911\) 1.17158e7 0.467708 0.233854 0.972272i \(-0.424866\pi\)
0.233854 + 0.972272i \(0.424866\pi\)
\(912\) 0 0
\(913\) 5.53824e6 0.219885
\(914\) 0 0
\(915\) 3.57560e7 1.41187
\(916\) 0 0
\(917\) 1.22731e7 0.481984
\(918\) 0 0
\(919\) 1.95296e7 0.762788 0.381394 0.924413i \(-0.375444\pi\)
0.381394 + 0.924413i \(0.375444\pi\)
\(920\) 0 0
\(921\) −3.25898e7 −1.26599
\(922\) 0 0
\(923\) 7.38673e6 0.285396
\(924\) 0 0
\(925\) 3.17620e7 1.22055
\(926\) 0 0
\(927\) −1.39097e6 −0.0531641
\(928\) 0 0
\(929\) −4.29425e7 −1.63248 −0.816240 0.577713i \(-0.803945\pi\)
−0.816240 + 0.577713i \(0.803945\pi\)
\(930\) 0 0
\(931\) 4.61073e6 0.174339
\(932\) 0 0
\(933\) 5.02364e7 1.88936
\(934\) 0 0
\(935\) −1.73535e6 −0.0649171
\(936\) 0 0
\(937\) 2.27191e7 0.845361 0.422681 0.906279i \(-0.361089\pi\)
0.422681 + 0.906279i \(0.361089\pi\)
\(938\) 0 0
\(939\) 180207. 0.00666972
\(940\) 0 0
\(941\) 3.98095e6 0.146559 0.0732795 0.997311i \(-0.476653\pi\)
0.0732795 + 0.997311i \(0.476653\pi\)
\(942\) 0 0
\(943\) −3.16045e6 −0.115736
\(944\) 0 0
\(945\) 5.78306e7 2.10658
\(946\) 0 0
\(947\) 2.43639e7 0.882818 0.441409 0.897306i \(-0.354479\pi\)
0.441409 + 0.897306i \(0.354479\pi\)
\(948\) 0 0
\(949\) −4.63152e6 −0.166939
\(950\) 0 0
\(951\) 4.10304e7 1.47114
\(952\) 0 0
\(953\) 1.39017e7 0.495833 0.247916 0.968781i \(-0.420254\pi\)
0.247916 + 0.968781i \(0.420254\pi\)
\(954\) 0 0
\(955\) −2.94913e7 −1.04637
\(956\) 0 0
\(957\) −5.31515e6 −0.187601
\(958\) 0 0
\(959\) 9.14617e6 0.321139
\(960\) 0 0
\(961\) −2.75124e7 −0.960993
\(962\) 0 0
\(963\) 9.09116e6 0.315903
\(964\) 0 0
\(965\) 1.18678e6 0.0410253
\(966\) 0 0
\(967\) −5.16682e7 −1.77688 −0.888438 0.458998i \(-0.848209\pi\)
−0.888438 + 0.458998i \(0.848209\pi\)
\(968\) 0 0
\(969\) −434684. −0.0148718
\(970\) 0 0
\(971\) 1.45794e7 0.496240 0.248120 0.968729i \(-0.420187\pi\)
0.248120 + 0.968729i \(0.420187\pi\)
\(972\) 0 0
\(973\) −3.16945e7 −1.07325
\(974\) 0 0
\(975\) −1.94313e7 −0.654623
\(976\) 0 0
\(977\) −3.09921e6 −0.103876 −0.0519379 0.998650i \(-0.516540\pi\)
−0.0519379 + 0.998650i \(0.516540\pi\)
\(978\) 0 0
\(979\) 9.95442e6 0.331940
\(980\) 0 0
\(981\) 6.58071e6 0.218323
\(982\) 0 0
\(983\) −1.53445e7 −0.506489 −0.253245 0.967402i \(-0.581498\pi\)
−0.253245 + 0.967402i \(0.581498\pi\)
\(984\) 0 0
\(985\) 4.10380e7 1.34771
\(986\) 0 0
\(987\) −6.41133e7 −2.09486
\(988\) 0 0
\(989\) −4.10296e7 −1.33385
\(990\) 0 0
\(991\) −1.57747e7 −0.510244 −0.255122 0.966909i \(-0.582116\pi\)
−0.255122 + 0.966909i \(0.582116\pi\)
\(992\) 0 0
\(993\) 2.01359e6 0.0648033
\(994\) 0 0
\(995\) 4.11105e7 1.31642
\(996\) 0 0
\(997\) 1.85577e6 0.0591270 0.0295635 0.999563i \(-0.490588\pi\)
0.0295635 + 0.999563i \(0.490588\pi\)
\(998\) 0 0
\(999\) 4.27365e7 1.35483
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 176.6.a.i.1.2 3
4.3 odd 2 11.6.a.b.1.2 3
8.3 odd 2 704.6.a.q.1.2 3
8.5 even 2 704.6.a.t.1.2 3
12.11 even 2 99.6.a.g.1.2 3
20.3 even 4 275.6.b.b.199.3 6
20.7 even 4 275.6.b.b.199.4 6
20.19 odd 2 275.6.a.b.1.2 3
28.27 even 2 539.6.a.e.1.2 3
44.43 even 2 121.6.a.d.1.2 3
132.131 odd 2 1089.6.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 3 4.3 odd 2
99.6.a.g.1.2 3 12.11 even 2
121.6.a.d.1.2 3 44.43 even 2
176.6.a.i.1.2 3 1.1 even 1 trivial
275.6.a.b.1.2 3 20.19 odd 2
275.6.b.b.199.3 6 20.3 even 4
275.6.b.b.199.4 6 20.7 even 4
539.6.a.e.1.2 3 28.27 even 2
704.6.a.q.1.2 3 8.3 odd 2
704.6.a.t.1.2 3 8.5 even 2
1089.6.a.r.1.2 3 132.131 odd 2