Properties

Label 448.6.a.ba.1.3
Level $448$
Weight $6$
Character 448.1
Self dual yes
Analytic conductor $71.852$
Analytic rank $1$
Dimension $3$
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] = [448,6,Mod(1,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, 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("448.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.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: \(71.8519512762\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.367637.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 107x + 282 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-11.0251\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+20.0502 q^{3} +33.3202 q^{5} -49.0000 q^{7} +159.011 q^{9} +O(q^{10})\) \(q+20.0502 q^{3} +33.3202 q^{5} -49.0000 q^{7} +159.011 q^{9} -660.274 q^{11} +145.779 q^{13} +668.076 q^{15} +435.001 q^{17} -1527.81 q^{19} -982.460 q^{21} -206.539 q^{23} -2014.77 q^{25} -1684.00 q^{27} +2039.43 q^{29} +1151.33 q^{31} -13238.6 q^{33} -1632.69 q^{35} -6757.90 q^{37} +2922.90 q^{39} -15327.5 q^{41} -6258.55 q^{43} +5298.27 q^{45} -12602.1 q^{47} +2401.00 q^{49} +8721.86 q^{51} -10856.1 q^{53} -22000.4 q^{55} -30632.8 q^{57} +52563.4 q^{59} +46785.0 q^{61} -7791.53 q^{63} +4857.39 q^{65} -43838.4 q^{67} -4141.16 q^{69} -34624.8 q^{71} -60821.1 q^{73} -40396.5 q^{75} +32353.4 q^{77} +22685.6 q^{79} -72404.2 q^{81} -38812.4 q^{83} +14494.3 q^{85} +40891.0 q^{87} +5983.43 q^{89} -7143.18 q^{91} +23084.5 q^{93} -50906.7 q^{95} +67354.0 q^{97} -104991. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 8 q^{3} + 14 q^{5} - 147 q^{7} + 151 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 8 q^{3} + 14 q^{5} - 147 q^{7} + 151 q^{9} - 600 q^{11} + 974 q^{13} + 120 q^{15} + 718 q^{17} + 1056 q^{19} + 392 q^{21} - 3760 q^{23} - 147 q^{25} + 2872 q^{27} + 9134 q^{29} + 1448 q^{31} - 14872 q^{33} - 686 q^{35} + 4998 q^{37} + 824 q^{39} - 23186 q^{41} - 29880 q^{43} + 28350 q^{45} - 10840 q^{47} + 7203 q^{49} - 3776 q^{51} + 28006 q^{53} - 14952 q^{55} - 53504 q^{57} + 17456 q^{59} + 92294 q^{61} - 7399 q^{63} - 95300 q^{65} - 56024 q^{67} + 63480 q^{69} - 77064 q^{71} - 46346 q^{73} - 50768 q^{75} + 29400 q^{77} + 4376 q^{79} - 121973 q^{81} - 107128 q^{83} + 94348 q^{85} - 82720 q^{87} + 29814 q^{89} - 47726 q^{91} + 108432 q^{93} - 205304 q^{95} - 156482 q^{97} - 83128 q^{99}+O(q^{100}) \) 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
\(3\) 20.0502 1.28622 0.643111 0.765773i \(-0.277644\pi\)
0.643111 + 0.765773i \(0.277644\pi\)
\(4\) 0 0
\(5\) 33.3202 0.596049 0.298025 0.954558i \(-0.403672\pi\)
0.298025 + 0.954558i \(0.403672\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 159.011 0.654365
\(10\) 0 0
\(11\) −660.274 −1.64529 −0.822645 0.568555i \(-0.807503\pi\)
−0.822645 + 0.568555i \(0.807503\pi\)
\(12\) 0 0
\(13\) 145.779 0.239242 0.119621 0.992820i \(-0.461832\pi\)
0.119621 + 0.992820i \(0.461832\pi\)
\(14\) 0 0
\(15\) 668.076 0.766652
\(16\) 0 0
\(17\) 435.001 0.365063 0.182532 0.983200i \(-0.441571\pi\)
0.182532 + 0.983200i \(0.441571\pi\)
\(18\) 0 0
\(19\) −1527.81 −0.970921 −0.485461 0.874259i \(-0.661348\pi\)
−0.485461 + 0.874259i \(0.661348\pi\)
\(20\) 0 0
\(21\) −982.460 −0.486146
\(22\) 0 0
\(23\) −206.539 −0.0814110 −0.0407055 0.999171i \(-0.512961\pi\)
−0.0407055 + 0.999171i \(0.512961\pi\)
\(24\) 0 0
\(25\) −2014.77 −0.644725
\(26\) 0 0
\(27\) −1684.00 −0.444563
\(28\) 0 0
\(29\) 2039.43 0.450312 0.225156 0.974323i \(-0.427711\pi\)
0.225156 + 0.974323i \(0.427711\pi\)
\(30\) 0 0
\(31\) 1151.33 0.215178 0.107589 0.994195i \(-0.465687\pi\)
0.107589 + 0.994195i \(0.465687\pi\)
\(32\) 0 0
\(33\) −13238.6 −2.11621
\(34\) 0 0
\(35\) −1632.69 −0.225286
\(36\) 0 0
\(37\) −6757.90 −0.811536 −0.405768 0.913976i \(-0.632996\pi\)
−0.405768 + 0.913976i \(0.632996\pi\)
\(38\) 0 0
\(39\) 2922.90 0.307718
\(40\) 0 0
\(41\) −15327.5 −1.42400 −0.712002 0.702177i \(-0.752212\pi\)
−0.712002 + 0.702177i \(0.752212\pi\)
\(42\) 0 0
\(43\) −6258.55 −0.516182 −0.258091 0.966121i \(-0.583093\pi\)
−0.258091 + 0.966121i \(0.583093\pi\)
\(44\) 0 0
\(45\) 5298.27 0.390034
\(46\) 0 0
\(47\) −12602.1 −0.832142 −0.416071 0.909332i \(-0.636593\pi\)
−0.416071 + 0.909332i \(0.636593\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 8721.86 0.469552
\(52\) 0 0
\(53\) −10856.1 −0.530867 −0.265433 0.964129i \(-0.585515\pi\)
−0.265433 + 0.964129i \(0.585515\pi\)
\(54\) 0 0
\(55\) −22000.4 −0.980674
\(56\) 0 0
\(57\) −30632.8 −1.24882
\(58\) 0 0
\(59\) 52563.4 1.96586 0.982931 0.183973i \(-0.0588960\pi\)
0.982931 + 0.183973i \(0.0588960\pi\)
\(60\) 0 0
\(61\) 46785.0 1.60984 0.804919 0.593384i \(-0.202209\pi\)
0.804919 + 0.593384i \(0.202209\pi\)
\(62\) 0 0
\(63\) −7791.53 −0.247327
\(64\) 0 0
\(65\) 4857.39 0.142600
\(66\) 0 0
\(67\) −43838.4 −1.19307 −0.596537 0.802585i \(-0.703457\pi\)
−0.596537 + 0.802585i \(0.703457\pi\)
\(68\) 0 0
\(69\) −4141.16 −0.104713
\(70\) 0 0
\(71\) −34624.8 −0.815157 −0.407578 0.913170i \(-0.633627\pi\)
−0.407578 + 0.913170i \(0.633627\pi\)
\(72\) 0 0
\(73\) −60821.1 −1.33582 −0.667909 0.744243i \(-0.732810\pi\)
−0.667909 + 0.744243i \(0.732810\pi\)
\(74\) 0 0
\(75\) −40396.5 −0.829259
\(76\) 0 0
\(77\) 32353.4 0.621861
\(78\) 0 0
\(79\) 22685.6 0.408961 0.204480 0.978871i \(-0.434450\pi\)
0.204480 + 0.978871i \(0.434450\pi\)
\(80\) 0 0
\(81\) −72404.2 −1.22617
\(82\) 0 0
\(83\) −38812.4 −0.618408 −0.309204 0.950996i \(-0.600063\pi\)
−0.309204 + 0.950996i \(0.600063\pi\)
\(84\) 0 0
\(85\) 14494.3 0.217596
\(86\) 0 0
\(87\) 40891.0 0.579201
\(88\) 0 0
\(89\) 5983.43 0.0800710 0.0400355 0.999198i \(-0.487253\pi\)
0.0400355 + 0.999198i \(0.487253\pi\)
\(90\) 0 0
\(91\) −7143.18 −0.0904250
\(92\) 0 0
\(93\) 23084.5 0.276766
\(94\) 0 0
\(95\) −50906.7 −0.578717
\(96\) 0 0
\(97\) 67354.0 0.726832 0.363416 0.931627i \(-0.381610\pi\)
0.363416 + 0.931627i \(0.381610\pi\)
\(98\) 0 0
\(99\) −104991. −1.07662
\(100\) 0 0
\(101\) 99602.9 0.971558 0.485779 0.874082i \(-0.338536\pi\)
0.485779 + 0.874082i \(0.338536\pi\)
\(102\) 0 0
\(103\) 38290.5 0.355629 0.177815 0.984064i \(-0.443097\pi\)
0.177815 + 0.984064i \(0.443097\pi\)
\(104\) 0 0
\(105\) −32735.7 −0.289767
\(106\) 0 0
\(107\) −83934.5 −0.708730 −0.354365 0.935107i \(-0.615303\pi\)
−0.354365 + 0.935107i \(0.615303\pi\)
\(108\) 0 0
\(109\) 207724. 1.67464 0.837320 0.546713i \(-0.184121\pi\)
0.837320 + 0.546713i \(0.184121\pi\)
\(110\) 0 0
\(111\) −135497. −1.04381
\(112\) 0 0
\(113\) 193685. 1.42692 0.713462 0.700694i \(-0.247126\pi\)
0.713462 + 0.700694i \(0.247126\pi\)
\(114\) 0 0
\(115\) −6881.93 −0.0485250
\(116\) 0 0
\(117\) 23180.5 0.156552
\(118\) 0 0
\(119\) −21315.0 −0.137981
\(120\) 0 0
\(121\) 274911. 1.70698
\(122\) 0 0
\(123\) −307319. −1.83159
\(124\) 0 0
\(125\) −171258. −0.980337
\(126\) 0 0
\(127\) −269758. −1.48411 −0.742054 0.670340i \(-0.766148\pi\)
−0.742054 + 0.670340i \(0.766148\pi\)
\(128\) 0 0
\(129\) −125485. −0.663924
\(130\) 0 0
\(131\) −97453.8 −0.496159 −0.248079 0.968740i \(-0.579799\pi\)
−0.248079 + 0.968740i \(0.579799\pi\)
\(132\) 0 0
\(133\) 74862.5 0.366974
\(134\) 0 0
\(135\) −56111.2 −0.264981
\(136\) 0 0
\(137\) −23080.6 −0.105062 −0.0525310 0.998619i \(-0.516729\pi\)
−0.0525310 + 0.998619i \(0.516729\pi\)
\(138\) 0 0
\(139\) 244133. 1.07174 0.535871 0.844300i \(-0.319984\pi\)
0.535871 + 0.844300i \(0.319984\pi\)
\(140\) 0 0
\(141\) −252674. −1.07032
\(142\) 0 0
\(143\) −96254.3 −0.393623
\(144\) 0 0
\(145\) 67954.1 0.268408
\(146\) 0 0
\(147\) 48140.5 0.183746
\(148\) 0 0
\(149\) −497458. −1.83565 −0.917826 0.396982i \(-0.870058\pi\)
−0.917826 + 0.396982i \(0.870058\pi\)
\(150\) 0 0
\(151\) −536243. −1.91390 −0.956950 0.290252i \(-0.906261\pi\)
−0.956950 + 0.290252i \(0.906261\pi\)
\(152\) 0 0
\(153\) 69169.8 0.238885
\(154\) 0 0
\(155\) 38362.6 0.128256
\(156\) 0 0
\(157\) 87512.6 0.283349 0.141674 0.989913i \(-0.454751\pi\)
0.141674 + 0.989913i \(0.454751\pi\)
\(158\) 0 0
\(159\) −217668. −0.682812
\(160\) 0 0
\(161\) 10120.4 0.0307705
\(162\) 0 0
\(163\) −223968. −0.660262 −0.330131 0.943935i \(-0.607093\pi\)
−0.330131 + 0.943935i \(0.607093\pi\)
\(164\) 0 0
\(165\) −441114. −1.26136
\(166\) 0 0
\(167\) 41952.2 0.116403 0.0582015 0.998305i \(-0.481463\pi\)
0.0582015 + 0.998305i \(0.481463\pi\)
\(168\) 0 0
\(169\) −350041. −0.942763
\(170\) 0 0
\(171\) −242937. −0.635337
\(172\) 0 0
\(173\) 512961. 1.30307 0.651537 0.758617i \(-0.274124\pi\)
0.651537 + 0.758617i \(0.274124\pi\)
\(174\) 0 0
\(175\) 98723.5 0.243683
\(176\) 0 0
\(177\) 1.05391e6 2.52853
\(178\) 0 0
\(179\) 112735. 0.262982 0.131491 0.991317i \(-0.458024\pi\)
0.131491 + 0.991317i \(0.458024\pi\)
\(180\) 0 0
\(181\) −282627. −0.641236 −0.320618 0.947209i \(-0.603890\pi\)
−0.320618 + 0.947209i \(0.603890\pi\)
\(182\) 0 0
\(183\) 938050. 2.07061
\(184\) 0 0
\(185\) −225175. −0.483716
\(186\) 0 0
\(187\) −287220. −0.600635
\(188\) 0 0
\(189\) 82516.1 0.168029
\(190\) 0 0
\(191\) −330464. −0.655452 −0.327726 0.944773i \(-0.606282\pi\)
−0.327726 + 0.944773i \(0.606282\pi\)
\(192\) 0 0
\(193\) 31480.1 0.0608336 0.0304168 0.999537i \(-0.490317\pi\)
0.0304168 + 0.999537i \(0.490317\pi\)
\(194\) 0 0
\(195\) 97391.7 0.183415
\(196\) 0 0
\(197\) −652849. −1.19853 −0.599263 0.800552i \(-0.704539\pi\)
−0.599263 + 0.800552i \(0.704539\pi\)
\(198\) 0 0
\(199\) −307902. −0.551163 −0.275582 0.961278i \(-0.588870\pi\)
−0.275582 + 0.961278i \(0.588870\pi\)
\(200\) 0 0
\(201\) −878969. −1.53456
\(202\) 0 0
\(203\) −99932.0 −0.170202
\(204\) 0 0
\(205\) −510715. −0.848777
\(206\) 0 0
\(207\) −32842.0 −0.0532725
\(208\) 0 0
\(209\) 1.00877e6 1.59745
\(210\) 0 0
\(211\) 171540. 0.265252 0.132626 0.991166i \(-0.457659\pi\)
0.132626 + 0.991166i \(0.457659\pi\)
\(212\) 0 0
\(213\) −694234. −1.04847
\(214\) 0 0
\(215\) −208536. −0.307670
\(216\) 0 0
\(217\) −56415.4 −0.0813295
\(218\) 0 0
\(219\) −1.21948e6 −1.71816
\(220\) 0 0
\(221\) 63414.1 0.0873384
\(222\) 0 0
\(223\) 225202. 0.303256 0.151628 0.988438i \(-0.451548\pi\)
0.151628 + 0.988438i \(0.451548\pi\)
\(224\) 0 0
\(225\) −320369. −0.421886
\(226\) 0 0
\(227\) 317578. 0.409059 0.204530 0.978860i \(-0.434434\pi\)
0.204530 + 0.978860i \(0.434434\pi\)
\(228\) 0 0
\(229\) −82887.7 −0.104448 −0.0522242 0.998635i \(-0.516631\pi\)
−0.0522242 + 0.998635i \(0.516631\pi\)
\(230\) 0 0
\(231\) 648693. 0.799851
\(232\) 0 0
\(233\) −562095. −0.678296 −0.339148 0.940733i \(-0.610139\pi\)
−0.339148 + 0.940733i \(0.610139\pi\)
\(234\) 0 0
\(235\) −419903. −0.495998
\(236\) 0 0
\(237\) 454850. 0.526014
\(238\) 0 0
\(239\) 623000. 0.705494 0.352747 0.935719i \(-0.385248\pi\)
0.352747 + 0.935719i \(0.385248\pi\)
\(240\) 0 0
\(241\) −578114. −0.641167 −0.320583 0.947220i \(-0.603879\pi\)
−0.320583 + 0.947220i \(0.603879\pi\)
\(242\) 0 0
\(243\) −1.04251e6 −1.13256
\(244\) 0 0
\(245\) 80001.7 0.0851499
\(246\) 0 0
\(247\) −222722. −0.232285
\(248\) 0 0
\(249\) −778196. −0.795409
\(250\) 0 0
\(251\) 1.59750e6 1.60050 0.800251 0.599665i \(-0.204700\pi\)
0.800251 + 0.599665i \(0.204700\pi\)
\(252\) 0 0
\(253\) 136373. 0.133945
\(254\) 0 0
\(255\) 290614. 0.279876
\(256\) 0 0
\(257\) 394458. 0.372536 0.186268 0.982499i \(-0.440361\pi\)
0.186268 + 0.982499i \(0.440361\pi\)
\(258\) 0 0
\(259\) 331137. 0.306732
\(260\) 0 0
\(261\) 324291. 0.294669
\(262\) 0 0
\(263\) −2.20714e6 −1.96762 −0.983810 0.179216i \(-0.942644\pi\)
−0.983810 + 0.179216i \(0.942644\pi\)
\(264\) 0 0
\(265\) −361728. −0.316423
\(266\) 0 0
\(267\) 119969. 0.102989
\(268\) 0 0
\(269\) 2.33693e6 1.96909 0.984545 0.175133i \(-0.0560355\pi\)
0.984545 + 0.175133i \(0.0560355\pi\)
\(270\) 0 0
\(271\) −1.62553e6 −1.34453 −0.672266 0.740310i \(-0.734679\pi\)
−0.672266 + 0.740310i \(0.734679\pi\)
\(272\) 0 0
\(273\) −143222. −0.116307
\(274\) 0 0
\(275\) 1.33030e6 1.06076
\(276\) 0 0
\(277\) 2.41218e6 1.88891 0.944454 0.328643i \(-0.106591\pi\)
0.944454 + 0.328643i \(0.106591\pi\)
\(278\) 0 0
\(279\) 183074. 0.140805
\(280\) 0 0
\(281\) 569517. 0.430270 0.215135 0.976584i \(-0.430981\pi\)
0.215135 + 0.976584i \(0.430981\pi\)
\(282\) 0 0
\(283\) 1.13388e6 0.841592 0.420796 0.907155i \(-0.361751\pi\)
0.420796 + 0.907155i \(0.361751\pi\)
\(284\) 0 0
\(285\) −1.02069e6 −0.744358
\(286\) 0 0
\(287\) 751047. 0.538223
\(288\) 0 0
\(289\) −1.23063e6 −0.866729
\(290\) 0 0
\(291\) 1.35046e6 0.934866
\(292\) 0 0
\(293\) 1.34094e6 0.912514 0.456257 0.889848i \(-0.349190\pi\)
0.456257 + 0.889848i \(0.349190\pi\)
\(294\) 0 0
\(295\) 1.75142e6 1.17175
\(296\) 0 0
\(297\) 1.11190e6 0.731435
\(298\) 0 0
\(299\) −30109.2 −0.0194769
\(300\) 0 0
\(301\) 306669. 0.195098
\(302\) 0 0
\(303\) 1.99706e6 1.24964
\(304\) 0 0
\(305\) 1.55889e6 0.959543
\(306\) 0 0
\(307\) 2.80842e6 1.70066 0.850328 0.526254i \(-0.176404\pi\)
0.850328 + 0.526254i \(0.176404\pi\)
\(308\) 0 0
\(309\) 767731. 0.457418
\(310\) 0 0
\(311\) 1.88370e6 1.10436 0.552181 0.833724i \(-0.313796\pi\)
0.552181 + 0.833724i \(0.313796\pi\)
\(312\) 0 0
\(313\) 1.26722e6 0.731126 0.365563 0.930787i \(-0.380876\pi\)
0.365563 + 0.930787i \(0.380876\pi\)
\(314\) 0 0
\(315\) −259615. −0.147419
\(316\) 0 0
\(317\) 2.88579e6 1.61293 0.806466 0.591280i \(-0.201377\pi\)
0.806466 + 0.591280i \(0.201377\pi\)
\(318\) 0 0
\(319\) −1.34658e6 −0.740894
\(320\) 0 0
\(321\) −1.68290e6 −0.911584
\(322\) 0 0
\(323\) −664597. −0.354447
\(324\) 0 0
\(325\) −293711. −0.154245
\(326\) 0 0
\(327\) 4.16492e6 2.15396
\(328\) 0 0
\(329\) 617501. 0.314520
\(330\) 0 0
\(331\) −2.48205e6 −1.24520 −0.622601 0.782539i \(-0.713924\pi\)
−0.622601 + 0.782539i \(0.713924\pi\)
\(332\) 0 0
\(333\) −1.07458e6 −0.531041
\(334\) 0 0
\(335\) −1.46070e6 −0.711131
\(336\) 0 0
\(337\) −2.52524e6 −1.21124 −0.605618 0.795756i \(-0.707074\pi\)
−0.605618 + 0.795756i \(0.707074\pi\)
\(338\) 0 0
\(339\) 3.88343e6 1.83534
\(340\) 0 0
\(341\) −760196. −0.354030
\(342\) 0 0
\(343\) −117649. −0.0539949
\(344\) 0 0
\(345\) −137984. −0.0624139
\(346\) 0 0
\(347\) −4.10373e6 −1.82960 −0.914799 0.403910i \(-0.867651\pi\)
−0.914799 + 0.403910i \(0.867651\pi\)
\(348\) 0 0
\(349\) 818617. 0.359764 0.179882 0.983688i \(-0.442428\pi\)
0.179882 + 0.983688i \(0.442428\pi\)
\(350\) 0 0
\(351\) −245493. −0.106358
\(352\) 0 0
\(353\) 141420. 0.0604054 0.0302027 0.999544i \(-0.490385\pi\)
0.0302027 + 0.999544i \(0.490385\pi\)
\(354\) 0 0
\(355\) −1.15370e6 −0.485874
\(356\) 0 0
\(357\) −427371. −0.177474
\(358\) 0 0
\(359\) 4.38324e6 1.79498 0.897488 0.441038i \(-0.145390\pi\)
0.897488 + 0.441038i \(0.145390\pi\)
\(360\) 0 0
\(361\) −141910. −0.0573120
\(362\) 0 0
\(363\) 5.51202e6 2.19555
\(364\) 0 0
\(365\) −2.02657e6 −0.796213
\(366\) 0 0
\(367\) 3.86104e6 1.49637 0.748186 0.663489i \(-0.230925\pi\)
0.748186 + 0.663489i \(0.230925\pi\)
\(368\) 0 0
\(369\) −2.43724e6 −0.931819
\(370\) 0 0
\(371\) 531951. 0.200649
\(372\) 0 0
\(373\) 2.73188e6 1.01669 0.508346 0.861153i \(-0.330257\pi\)
0.508346 + 0.861153i \(0.330257\pi\)
\(374\) 0 0
\(375\) −3.43376e6 −1.26093
\(376\) 0 0
\(377\) 297307. 0.107734
\(378\) 0 0
\(379\) −1.25353e6 −0.448266 −0.224133 0.974559i \(-0.571955\pi\)
−0.224133 + 0.974559i \(0.571955\pi\)
\(380\) 0 0
\(381\) −5.40871e6 −1.90889
\(382\) 0 0
\(383\) −1.26669e6 −0.441240 −0.220620 0.975360i \(-0.570808\pi\)
−0.220620 + 0.975360i \(0.570808\pi\)
\(384\) 0 0
\(385\) 1.07802e6 0.370660
\(386\) 0 0
\(387\) −995176. −0.337771
\(388\) 0 0
\(389\) −3.08766e6 −1.03456 −0.517279 0.855817i \(-0.673055\pi\)
−0.517279 + 0.855817i \(0.673055\pi\)
\(390\) 0 0
\(391\) −89844.8 −0.0297202
\(392\) 0 0
\(393\) −1.95397e6 −0.638170
\(394\) 0 0
\(395\) 755887. 0.243761
\(396\) 0 0
\(397\) 306775. 0.0976885 0.0488442 0.998806i \(-0.484446\pi\)
0.0488442 + 0.998806i \(0.484446\pi\)
\(398\) 0 0
\(399\) 1.50101e6 0.472009
\(400\) 0 0
\(401\) −3.07193e6 −0.954005 −0.477002 0.878902i \(-0.658277\pi\)
−0.477002 + 0.878902i \(0.658277\pi\)
\(402\) 0 0
\(403\) 167841. 0.0514795
\(404\) 0 0
\(405\) −2.41252e6 −0.730859
\(406\) 0 0
\(407\) 4.46207e6 1.33521
\(408\) 0 0
\(409\) 6.03016e6 1.78246 0.891231 0.453549i \(-0.149842\pi\)
0.891231 + 0.453549i \(0.149842\pi\)
\(410\) 0 0
\(411\) −462771. −0.135133
\(412\) 0 0
\(413\) −2.57560e6 −0.743026
\(414\) 0 0
\(415\) −1.29324e6 −0.368602
\(416\) 0 0
\(417\) 4.89492e6 1.37850
\(418\) 0 0
\(419\) 4.43568e6 1.23431 0.617156 0.786841i \(-0.288285\pi\)
0.617156 + 0.786841i \(0.288285\pi\)
\(420\) 0 0
\(421\) 5.03797e6 1.38532 0.692660 0.721264i \(-0.256438\pi\)
0.692660 + 0.721264i \(0.256438\pi\)
\(422\) 0 0
\(423\) −2.00386e6 −0.544525
\(424\) 0 0
\(425\) −876425. −0.235365
\(426\) 0 0
\(427\) −2.29247e6 −0.608462
\(428\) 0 0
\(429\) −1.92992e6 −0.506286
\(430\) 0 0
\(431\) −4.63743e6 −1.20250 −0.601248 0.799062i \(-0.705330\pi\)
−0.601248 + 0.799062i \(0.705330\pi\)
\(432\) 0 0
\(433\) −5.03341e6 −1.29016 −0.645079 0.764116i \(-0.723176\pi\)
−0.645079 + 0.764116i \(0.723176\pi\)
\(434\) 0 0
\(435\) 1.36249e6 0.345232
\(436\) 0 0
\(437\) 315552. 0.0790437
\(438\) 0 0
\(439\) 4.85818e6 1.20313 0.601565 0.798824i \(-0.294544\pi\)
0.601565 + 0.798824i \(0.294544\pi\)
\(440\) 0 0
\(441\) 381785. 0.0934807
\(442\) 0 0
\(443\) −2.47332e6 −0.598784 −0.299392 0.954130i \(-0.596784\pi\)
−0.299392 + 0.954130i \(0.596784\pi\)
\(444\) 0 0
\(445\) 199369. 0.0477263
\(446\) 0 0
\(447\) −9.97413e6 −2.36106
\(448\) 0 0
\(449\) 6.11819e6 1.43221 0.716106 0.697992i \(-0.245923\pi\)
0.716106 + 0.697992i \(0.245923\pi\)
\(450\) 0 0
\(451\) 1.01203e7 2.34290
\(452\) 0 0
\(453\) −1.07518e7 −2.46170
\(454\) 0 0
\(455\) −238012. −0.0538978
\(456\) 0 0
\(457\) 650177. 0.145627 0.0728133 0.997346i \(-0.476802\pi\)
0.0728133 + 0.997346i \(0.476802\pi\)
\(458\) 0 0
\(459\) −732542. −0.162293
\(460\) 0 0
\(461\) −6.43756e6 −1.41081 −0.705405 0.708804i \(-0.749235\pi\)
−0.705405 + 0.708804i \(0.749235\pi\)
\(462\) 0 0
\(463\) 196947. 0.0426969 0.0213485 0.999772i \(-0.493204\pi\)
0.0213485 + 0.999772i \(0.493204\pi\)
\(464\) 0 0
\(465\) 769179. 0.164966
\(466\) 0 0
\(467\) −4.89949e6 −1.03958 −0.519790 0.854294i \(-0.673990\pi\)
−0.519790 + 0.854294i \(0.673990\pi\)
\(468\) 0 0
\(469\) 2.14808e6 0.450940
\(470\) 0 0
\(471\) 1.75465e6 0.364449
\(472\) 0 0
\(473\) 4.13236e6 0.849269
\(474\) 0 0
\(475\) 3.07817e6 0.625977
\(476\) 0 0
\(477\) −1.72624e6 −0.347381
\(478\) 0 0
\(479\) −4.52409e6 −0.900934 −0.450467 0.892793i \(-0.648742\pi\)
−0.450467 + 0.892793i \(0.648742\pi\)
\(480\) 0 0
\(481\) −985162. −0.194153
\(482\) 0 0
\(483\) 202917. 0.0395776
\(484\) 0 0
\(485\) 2.24425e6 0.433228
\(486\) 0 0
\(487\) 2.09327e6 0.399948 0.199974 0.979801i \(-0.435914\pi\)
0.199974 + 0.979801i \(0.435914\pi\)
\(488\) 0 0
\(489\) −4.49060e6 −0.849243
\(490\) 0 0
\(491\) −2.85600e6 −0.534631 −0.267315 0.963609i \(-0.586137\pi\)
−0.267315 + 0.963609i \(0.586137\pi\)
\(492\) 0 0
\(493\) 887154. 0.164392
\(494\) 0 0
\(495\) −3.49831e6 −0.641719
\(496\) 0 0
\(497\) 1.69661e6 0.308100
\(498\) 0 0
\(499\) −7.92264e6 −1.42436 −0.712178 0.701999i \(-0.752291\pi\)
−0.712178 + 0.701999i \(0.752291\pi\)
\(500\) 0 0
\(501\) 841151. 0.149720
\(502\) 0 0
\(503\) −2.28311e6 −0.402352 −0.201176 0.979555i \(-0.564476\pi\)
−0.201176 + 0.979555i \(0.564476\pi\)
\(504\) 0 0
\(505\) 3.31879e6 0.579096
\(506\) 0 0
\(507\) −7.01840e6 −1.21260
\(508\) 0 0
\(509\) −8.32454e6 −1.42418 −0.712092 0.702087i \(-0.752252\pi\)
−0.712092 + 0.702087i \(0.752252\pi\)
\(510\) 0 0
\(511\) 2.98023e6 0.504891
\(512\) 0 0
\(513\) 2.57283e6 0.431636
\(514\) 0 0
\(515\) 1.27584e6 0.211973
\(516\) 0 0
\(517\) 8.32082e6 1.36911
\(518\) 0 0
\(519\) 1.02850e7 1.67604
\(520\) 0 0
\(521\) 562984. 0.0908660 0.0454330 0.998967i \(-0.485533\pi\)
0.0454330 + 0.998967i \(0.485533\pi\)
\(522\) 0 0
\(523\) −5.35275e6 −0.855703 −0.427852 0.903849i \(-0.640729\pi\)
−0.427852 + 0.903849i \(0.640729\pi\)
\(524\) 0 0
\(525\) 1.97943e6 0.313430
\(526\) 0 0
\(527\) 500831. 0.0785534
\(528\) 0 0
\(529\) −6.39368e6 −0.993372
\(530\) 0 0
\(531\) 8.35814e6 1.28639
\(532\) 0 0
\(533\) −2.23443e6 −0.340682
\(534\) 0 0
\(535\) −2.79671e6 −0.422438
\(536\) 0 0
\(537\) 2.26036e6 0.338253
\(538\) 0 0
\(539\) −1.58532e6 −0.235041
\(540\) 0 0
\(541\) 1.24141e7 1.82357 0.911783 0.410673i \(-0.134706\pi\)
0.911783 + 0.410673i \(0.134706\pi\)
\(542\) 0 0
\(543\) −5.66673e6 −0.824771
\(544\) 0 0
\(545\) 6.92142e6 0.998168
\(546\) 0 0
\(547\) −7.19845e6 −1.02866 −0.514328 0.857593i \(-0.671959\pi\)
−0.514328 + 0.857593i \(0.671959\pi\)
\(548\) 0 0
\(549\) 7.43932e6 1.05342
\(550\) 0 0
\(551\) −3.11585e6 −0.437218
\(552\) 0 0
\(553\) −1.11159e6 −0.154573
\(554\) 0 0
\(555\) −4.51480e6 −0.622165
\(556\) 0 0
\(557\) 9.45495e6 1.29128 0.645641 0.763641i \(-0.276590\pi\)
0.645641 + 0.763641i \(0.276590\pi\)
\(558\) 0 0
\(559\) −912367. −0.123492
\(560\) 0 0
\(561\) −5.75882e6 −0.772549
\(562\) 0 0
\(563\) −8.80283e6 −1.17045 −0.585223 0.810873i \(-0.698993\pi\)
−0.585223 + 0.810873i \(0.698993\pi\)
\(564\) 0 0
\(565\) 6.45363e6 0.850517
\(566\) 0 0
\(567\) 3.54781e6 0.463449
\(568\) 0 0
\(569\) 8.86214e6 1.14751 0.573757 0.819025i \(-0.305485\pi\)
0.573757 + 0.819025i \(0.305485\pi\)
\(570\) 0 0
\(571\) 8.40188e6 1.07842 0.539208 0.842173i \(-0.318724\pi\)
0.539208 + 0.842173i \(0.318724\pi\)
\(572\) 0 0
\(573\) −6.62587e6 −0.843056
\(574\) 0 0
\(575\) 416128. 0.0524877
\(576\) 0 0
\(577\) −3.94823e6 −0.493700 −0.246850 0.969054i \(-0.579396\pi\)
−0.246850 + 0.969054i \(0.579396\pi\)
\(578\) 0 0
\(579\) 631183. 0.0782454
\(580\) 0 0
\(581\) 1.90181e6 0.233736
\(582\) 0 0
\(583\) 7.16803e6 0.873430
\(584\) 0 0
\(585\) 772377. 0.0933125
\(586\) 0 0
\(587\) −5.22309e6 −0.625652 −0.312826 0.949811i \(-0.601276\pi\)
−0.312826 + 0.949811i \(0.601276\pi\)
\(588\) 0 0
\(589\) −1.75901e6 −0.208920
\(590\) 0 0
\(591\) −1.30898e7 −1.54157
\(592\) 0 0
\(593\) −9.33848e6 −1.09053 −0.545267 0.838262i \(-0.683572\pi\)
−0.545267 + 0.838262i \(0.683572\pi\)
\(594\) 0 0
\(595\) −710221. −0.0822434
\(596\) 0 0
\(597\) −6.17350e6 −0.708918
\(598\) 0 0
\(599\) −65978.1 −0.00751333 −0.00375667 0.999993i \(-0.501196\pi\)
−0.00375667 + 0.999993i \(0.501196\pi\)
\(600\) 0 0
\(601\) 4.69907e6 0.530671 0.265336 0.964156i \(-0.414517\pi\)
0.265336 + 0.964156i \(0.414517\pi\)
\(602\) 0 0
\(603\) −6.97077e6 −0.780706
\(604\) 0 0
\(605\) 9.16008e6 1.01744
\(606\) 0 0
\(607\) 2.04851e6 0.225666 0.112833 0.993614i \(-0.464008\pi\)
0.112833 + 0.993614i \(0.464008\pi\)
\(608\) 0 0
\(609\) −2.00366e6 −0.218917
\(610\) 0 0
\(611\) −1.83712e6 −0.199083
\(612\) 0 0
\(613\) 1.14241e7 1.22792 0.613962 0.789336i \(-0.289575\pi\)
0.613962 + 0.789336i \(0.289575\pi\)
\(614\) 0 0
\(615\) −1.02399e7 −1.09172
\(616\) 0 0
\(617\) −5.25878e6 −0.556125 −0.278062 0.960563i \(-0.589692\pi\)
−0.278062 + 0.960563i \(0.589692\pi\)
\(618\) 0 0
\(619\) −1.14617e7 −1.20233 −0.601163 0.799126i \(-0.705296\pi\)
−0.601163 + 0.799126i \(0.705296\pi\)
\(620\) 0 0
\(621\) 347813. 0.0361923
\(622\) 0 0
\(623\) −293188. −0.0302640
\(624\) 0 0
\(625\) 589799. 0.0603954
\(626\) 0 0
\(627\) 2.02260e7 2.05467
\(628\) 0 0
\(629\) −2.93969e6 −0.296262
\(630\) 0 0
\(631\) −1.24372e7 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(632\) 0 0
\(633\) 3.43941e6 0.341173
\(634\) 0 0
\(635\) −8.98840e6 −0.884602
\(636\) 0 0
\(637\) 350016. 0.0341774
\(638\) 0 0
\(639\) −5.50571e6 −0.533410
\(640\) 0 0
\(641\) 1.02997e7 0.990102 0.495051 0.868864i \(-0.335149\pi\)
0.495051 + 0.868864i \(0.335149\pi\)
\(642\) 0 0
\(643\) 1.23234e7 1.17545 0.587726 0.809060i \(-0.300023\pi\)
0.587726 + 0.809060i \(0.300023\pi\)
\(644\) 0 0
\(645\) −4.18119e6 −0.395731
\(646\) 0 0
\(647\) −4.45009e6 −0.417935 −0.208967 0.977923i \(-0.567010\pi\)
−0.208967 + 0.977923i \(0.567010\pi\)
\(648\) 0 0
\(649\) −3.47062e7 −3.23441
\(650\) 0 0
\(651\) −1.13114e6 −0.104608
\(652\) 0 0
\(653\) −9.78800e6 −0.898279 −0.449139 0.893462i \(-0.648269\pi\)
−0.449139 + 0.893462i \(0.648269\pi\)
\(654\) 0 0
\(655\) −3.24718e6 −0.295735
\(656\) 0 0
\(657\) −9.67121e6 −0.874112
\(658\) 0 0
\(659\) −7.70009e6 −0.690688 −0.345344 0.938476i \(-0.612238\pi\)
−0.345344 + 0.938476i \(0.612238\pi\)
\(660\) 0 0
\(661\) 6.47088e6 0.576049 0.288025 0.957623i \(-0.407001\pi\)
0.288025 + 0.957623i \(0.407001\pi\)
\(662\) 0 0
\(663\) 1.27147e6 0.112337
\(664\) 0 0
\(665\) 2.49443e6 0.218734
\(666\) 0 0
\(667\) −421222. −0.0366604
\(668\) 0 0
\(669\) 4.51534e6 0.390054
\(670\) 0 0
\(671\) −3.08909e7 −2.64865
\(672\) 0 0
\(673\) −1.26002e7 −1.07236 −0.536180 0.844104i \(-0.680133\pi\)
−0.536180 + 0.844104i \(0.680133\pi\)
\(674\) 0 0
\(675\) 3.39287e6 0.286621
\(676\) 0 0
\(677\) 1.92476e6 0.161401 0.0807004 0.996738i \(-0.474284\pi\)
0.0807004 + 0.996738i \(0.474284\pi\)
\(678\) 0 0
\(679\) −3.30034e6 −0.274717
\(680\) 0 0
\(681\) 6.36751e6 0.526141
\(682\) 0 0
\(683\) −2.96929e6 −0.243558 −0.121779 0.992557i \(-0.538860\pi\)
−0.121779 + 0.992557i \(0.538860\pi\)
\(684\) 0 0
\(685\) −769050. −0.0626222
\(686\) 0 0
\(687\) −1.66192e6 −0.134344
\(688\) 0 0
\(689\) −1.58260e6 −0.127006
\(690\) 0 0
\(691\) 1.44542e7 1.15159 0.575795 0.817594i \(-0.304692\pi\)
0.575795 + 0.817594i \(0.304692\pi\)
\(692\) 0 0
\(693\) 5.14454e6 0.406924
\(694\) 0 0
\(695\) 8.13456e6 0.638811
\(696\) 0 0
\(697\) −6.66747e6 −0.519852
\(698\) 0 0
\(699\) −1.12701e7 −0.872439
\(700\) 0 0
\(701\) 715005. 0.0549558 0.0274779 0.999622i \(-0.491252\pi\)
0.0274779 + 0.999622i \(0.491252\pi\)
\(702\) 0 0
\(703\) 1.03248e7 0.787937
\(704\) 0 0
\(705\) −8.41915e6 −0.637963
\(706\) 0 0
\(707\) −4.88054e6 −0.367214
\(708\) 0 0
\(709\) −1.75184e7 −1.30882 −0.654408 0.756141i \(-0.727082\pi\)
−0.654408 + 0.756141i \(0.727082\pi\)
\(710\) 0 0
\(711\) 3.60725e6 0.267610
\(712\) 0 0
\(713\) −237796. −0.0175178
\(714\) 0 0
\(715\) −3.20721e6 −0.234618
\(716\) 0 0
\(717\) 1.24913e7 0.907421
\(718\) 0 0
\(719\) 1.52406e7 1.09946 0.549731 0.835342i \(-0.314730\pi\)
0.549731 + 0.835342i \(0.314730\pi\)
\(720\) 0 0
\(721\) −1.87623e6 −0.134415
\(722\) 0 0
\(723\) −1.15913e7 −0.824682
\(724\) 0 0
\(725\) −4.10897e6 −0.290327
\(726\) 0 0
\(727\) 2.01811e7 1.41615 0.708073 0.706140i \(-0.249565\pi\)
0.708073 + 0.706140i \(0.249565\pi\)
\(728\) 0 0
\(729\) −3.30825e6 −0.230558
\(730\) 0 0
\(731\) −2.72247e6 −0.188439
\(732\) 0 0
\(733\) −1.02074e6 −0.0701706 −0.0350853 0.999384i \(-0.511170\pi\)
−0.0350853 + 0.999384i \(0.511170\pi\)
\(734\) 0 0
\(735\) 1.60405e6 0.109522
\(736\) 0 0
\(737\) 2.89454e7 1.96295
\(738\) 0 0
\(739\) 1.36743e7 0.921070 0.460535 0.887642i \(-0.347658\pi\)
0.460535 + 0.887642i \(0.347658\pi\)
\(740\) 0 0
\(741\) −4.46563e6 −0.298770
\(742\) 0 0
\(743\) 1.13203e7 0.752289 0.376144 0.926561i \(-0.377250\pi\)
0.376144 + 0.926561i \(0.377250\pi\)
\(744\) 0 0
\(745\) −1.65754e7 −1.09414
\(746\) 0 0
\(747\) −6.17158e6 −0.404664
\(748\) 0 0
\(749\) 4.11279e6 0.267875
\(750\) 0 0
\(751\) −5.87893e6 −0.380363 −0.190181 0.981749i \(-0.560908\pi\)
−0.190181 + 0.981749i \(0.560908\pi\)
\(752\) 0 0
\(753\) 3.20302e7 2.05860
\(754\) 0 0
\(755\) −1.78677e7 −1.14078
\(756\) 0 0
\(757\) −2.35139e7 −1.49137 −0.745686 0.666298i \(-0.767878\pi\)
−0.745686 + 0.666298i \(0.767878\pi\)
\(758\) 0 0
\(759\) 2.73430e6 0.172283
\(760\) 0 0
\(761\) −1.30171e7 −0.814803 −0.407402 0.913249i \(-0.633565\pi\)
−0.407402 + 0.913249i \(0.633565\pi\)
\(762\) 0 0
\(763\) −1.01785e7 −0.632954
\(764\) 0 0
\(765\) 2.30475e6 0.142387
\(766\) 0 0
\(767\) 7.66265e6 0.470317
\(768\) 0 0
\(769\) 2.15837e6 0.131617 0.0658083 0.997832i \(-0.479037\pi\)
0.0658083 + 0.997832i \(0.479037\pi\)
\(770\) 0 0
\(771\) 7.90896e6 0.479163
\(772\) 0 0
\(773\) −2.08330e7 −1.25401 −0.627007 0.779013i \(-0.715720\pi\)
−0.627007 + 0.779013i \(0.715720\pi\)
\(774\) 0 0
\(775\) −2.31967e6 −0.138730
\(776\) 0 0
\(777\) 6.63937e6 0.394525
\(778\) 0 0
\(779\) 2.34174e7 1.38260
\(780\) 0 0
\(781\) 2.28618e7 1.34117
\(782\) 0 0
\(783\) −3.43440e6 −0.200192
\(784\) 0 0
\(785\) 2.91594e6 0.168890
\(786\) 0 0
\(787\) 1.77822e7 1.02341 0.511703 0.859162i \(-0.329015\pi\)
0.511703 + 0.859162i \(0.329015\pi\)
\(788\) 0 0
\(789\) −4.42537e7 −2.53079
\(790\) 0 0
\(791\) −9.49058e6 −0.539326
\(792\) 0 0
\(793\) 6.82029e6 0.385141
\(794\) 0 0
\(795\) −7.25273e6 −0.406990
\(796\) 0 0
\(797\) −1.03372e6 −0.0576447 −0.0288223 0.999585i \(-0.509176\pi\)
−0.0288223 + 0.999585i \(0.509176\pi\)
\(798\) 0 0
\(799\) −5.48191e6 −0.303784
\(800\) 0 0
\(801\) 951430. 0.0523957
\(802\) 0 0
\(803\) 4.01586e7 2.19781
\(804\) 0 0
\(805\) 337214. 0.0183407
\(806\) 0 0
\(807\) 4.68560e7 2.53269
\(808\) 0 0
\(809\) −3.22789e7 −1.73399 −0.866996 0.498315i \(-0.833952\pi\)
−0.866996 + 0.498315i \(0.833952\pi\)
\(810\) 0 0
\(811\) 1.57854e7 0.842757 0.421378 0.906885i \(-0.361546\pi\)
0.421378 + 0.906885i \(0.361546\pi\)
\(812\) 0 0
\(813\) −3.25922e7 −1.72937
\(814\) 0 0
\(815\) −7.46264e6 −0.393549
\(816\) 0 0
\(817\) 9.56184e6 0.501172
\(818\) 0 0
\(819\) −1.13584e6 −0.0591710
\(820\) 0 0
\(821\) 2.23707e7 1.15830 0.579150 0.815221i \(-0.303384\pi\)
0.579150 + 0.815221i \(0.303384\pi\)
\(822\) 0 0
\(823\) 3.49932e7 1.80087 0.900437 0.434986i \(-0.143247\pi\)
0.900437 + 0.434986i \(0.143247\pi\)
\(824\) 0 0
\(825\) 2.66727e7 1.36437
\(826\) 0 0
\(827\) −1.97157e7 −1.00242 −0.501208 0.865327i \(-0.667111\pi\)
−0.501208 + 0.865327i \(0.667111\pi\)
\(828\) 0 0
\(829\) 8.34998e6 0.421987 0.210994 0.977487i \(-0.432330\pi\)
0.210994 + 0.977487i \(0.432330\pi\)
\(830\) 0 0
\(831\) 4.83648e7 2.42955
\(832\) 0 0
\(833\) 1.04444e6 0.0521519
\(834\) 0 0
\(835\) 1.39786e6 0.0693819
\(836\) 0 0
\(837\) −1.93885e6 −0.0956600
\(838\) 0 0
\(839\) −3.00623e7 −1.47441 −0.737203 0.675671i \(-0.763854\pi\)
−0.737203 + 0.675671i \(0.763854\pi\)
\(840\) 0 0
\(841\) −1.63519e7 −0.797219
\(842\) 0 0
\(843\) 1.14189e7 0.553422
\(844\) 0 0
\(845\) −1.16634e7 −0.561934
\(846\) 0 0
\(847\) −1.34706e7 −0.645178
\(848\) 0 0
\(849\) 2.27346e7 1.08247
\(850\) 0 0
\(851\) 1.39577e6 0.0660680
\(852\) 0 0
\(853\) −1.83935e7 −0.865550 −0.432775 0.901502i \(-0.642466\pi\)
−0.432775 + 0.901502i \(0.642466\pi\)
\(854\) 0 0
\(855\) −8.09472e6 −0.378692
\(856\) 0 0
\(857\) 3.39540e7 1.57921 0.789603 0.613618i \(-0.210287\pi\)
0.789603 + 0.613618i \(0.210287\pi\)
\(858\) 0 0
\(859\) −2.03765e7 −0.942206 −0.471103 0.882078i \(-0.656144\pi\)
−0.471103 + 0.882078i \(0.656144\pi\)
\(860\) 0 0
\(861\) 1.50586e7 0.692274
\(862\) 0 0
\(863\) −3.05608e7 −1.39681 −0.698405 0.715702i \(-0.746107\pi\)
−0.698405 + 0.715702i \(0.746107\pi\)
\(864\) 0 0
\(865\) 1.70920e7 0.776697
\(866\) 0 0
\(867\) −2.46744e7 −1.11481
\(868\) 0 0
\(869\) −1.49787e7 −0.672859
\(870\) 0 0
\(871\) −6.39073e6 −0.285434
\(872\) 0 0
\(873\) 1.07100e7 0.475613
\(874\) 0 0
\(875\) 8.39164e6 0.370533
\(876\) 0 0
\(877\) 2.99046e7 1.31292 0.656461 0.754360i \(-0.272052\pi\)
0.656461 + 0.754360i \(0.272052\pi\)
\(878\) 0 0
\(879\) 2.68861e7 1.17369
\(880\) 0 0
\(881\) 1.36000e7 0.590336 0.295168 0.955445i \(-0.404624\pi\)
0.295168 + 0.955445i \(0.404624\pi\)
\(882\) 0 0
\(883\) 1.56685e7 0.676279 0.338139 0.941096i \(-0.390203\pi\)
0.338139 + 0.941096i \(0.390203\pi\)
\(884\) 0 0
\(885\) 3.51163e7 1.50713
\(886\) 0 0
\(887\) −3.68184e7 −1.57129 −0.785644 0.618679i \(-0.787668\pi\)
−0.785644 + 0.618679i \(0.787668\pi\)
\(888\) 0 0
\(889\) 1.32182e7 0.560940
\(890\) 0 0
\(891\) 4.78066e7 2.01741
\(892\) 0 0
\(893\) 1.92535e7 0.807944
\(894\) 0 0
\(895\) 3.75635e6 0.156750
\(896\) 0 0
\(897\) −603695. −0.0250516
\(898\) 0 0
\(899\) 2.34806e6 0.0968971
\(900\) 0 0
\(901\) −4.72243e6 −0.193800
\(902\) 0 0
\(903\) 6.14877e6 0.250940
\(904\) 0 0
\(905\) −9.41719e6 −0.382208
\(906\) 0 0
\(907\) 2.91515e7 1.17664 0.588318 0.808630i \(-0.299790\pi\)
0.588318 + 0.808630i \(0.299790\pi\)
\(908\) 0 0
\(909\) 1.58379e7 0.635754
\(910\) 0 0
\(911\) −8.88578e6 −0.354731 −0.177366 0.984145i \(-0.556758\pi\)
−0.177366 + 0.984145i \(0.556758\pi\)
\(912\) 0 0
\(913\) 2.56268e7 1.01746
\(914\) 0 0
\(915\) 3.12560e7 1.23419
\(916\) 0 0
\(917\) 4.77524e6 0.187530
\(918\) 0 0
\(919\) −2.37925e6 −0.0929292 −0.0464646 0.998920i \(-0.514795\pi\)
−0.0464646 + 0.998920i \(0.514795\pi\)
\(920\) 0 0
\(921\) 5.63094e7 2.18742
\(922\) 0 0
\(923\) −5.04757e6 −0.195020
\(924\) 0 0
\(925\) 1.36156e7 0.523218
\(926\) 0 0
\(927\) 6.08859e6 0.232711
\(928\) 0 0
\(929\) −1.12549e7 −0.427859 −0.213930 0.976849i \(-0.568626\pi\)
−0.213930 + 0.976849i \(0.568626\pi\)
\(930\) 0 0
\(931\) −3.66826e6 −0.138703
\(932\) 0 0
\(933\) 3.77686e7 1.42045
\(934\) 0 0
\(935\) −9.57022e6 −0.358008
\(936\) 0 0
\(937\) −3.54517e7 −1.31913 −0.659566 0.751647i \(-0.729260\pi\)
−0.659566 + 0.751647i \(0.729260\pi\)
\(938\) 0 0
\(939\) 2.54081e7 0.940390
\(940\) 0 0
\(941\) −4.94573e7 −1.82077 −0.910387 0.413758i \(-0.864216\pi\)
−0.910387 + 0.413758i \(0.864216\pi\)
\(942\) 0 0
\(943\) 3.16573e6 0.115930
\(944\) 0 0
\(945\) 2.74945e6 0.100154
\(946\) 0 0
\(947\) −2.43591e7 −0.882645 −0.441323 0.897349i \(-0.645491\pi\)
−0.441323 + 0.897349i \(0.645491\pi\)
\(948\) 0 0
\(949\) −8.86645e6 −0.319584
\(950\) 0 0
\(951\) 5.78606e7 2.07459
\(952\) 0 0
\(953\) 358667. 0.0127926 0.00639631 0.999980i \(-0.497964\pi\)
0.00639631 + 0.999980i \(0.497964\pi\)
\(954\) 0 0
\(955\) −1.10111e7 −0.390682
\(956\) 0 0
\(957\) −2.69992e7 −0.952954
\(958\) 0 0
\(959\) 1.13095e6 0.0397097
\(960\) 0 0
\(961\) −2.73036e7 −0.953699
\(962\) 0 0
\(963\) −1.33465e7 −0.463768
\(964\) 0 0
\(965\) 1.04892e6 0.0362598
\(966\) 0 0
\(967\) −1.36902e7 −0.470809 −0.235404 0.971898i \(-0.575641\pi\)
−0.235404 + 0.971898i \(0.575641\pi\)
\(968\) 0 0
\(969\) −1.33253e7 −0.455898
\(970\) 0 0
\(971\) −3.29263e6 −0.112071 −0.0560357 0.998429i \(-0.517846\pi\)
−0.0560357 + 0.998429i \(0.517846\pi\)
\(972\) 0 0
\(973\) −1.19625e7 −0.405080
\(974\) 0 0
\(975\) −5.88897e6 −0.198394
\(976\) 0 0
\(977\) −2.04796e7 −0.686411 −0.343205 0.939260i \(-0.611513\pi\)
−0.343205 + 0.939260i \(0.611513\pi\)
\(978\) 0 0
\(979\) −3.95071e6 −0.131740
\(980\) 0 0
\(981\) 3.30304e7 1.09583
\(982\) 0 0
\(983\) 4.59364e6 0.151626 0.0758128 0.997122i \(-0.475845\pi\)
0.0758128 + 0.997122i \(0.475845\pi\)
\(984\) 0 0
\(985\) −2.17530e7 −0.714380
\(986\) 0 0
\(987\) 1.23810e7 0.404542
\(988\) 0 0
\(989\) 1.29264e6 0.0420229
\(990\) 0 0
\(991\) 5.37835e7 1.73966 0.869831 0.493350i \(-0.164228\pi\)
0.869831 + 0.493350i \(0.164228\pi\)
\(992\) 0 0
\(993\) −4.97656e7 −1.60161
\(994\) 0 0
\(995\) −1.02594e7 −0.328521
\(996\) 0 0
\(997\) 2.76551e7 0.881126 0.440563 0.897722i \(-0.354779\pi\)
0.440563 + 0.897722i \(0.354779\pi\)
\(998\) 0 0
\(999\) 1.13803e7 0.360779
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 448.6.a.ba.1.3 3
4.3 odd 2 448.6.a.bb.1.1 3
8.3 odd 2 224.6.a.e.1.3 3
8.5 even 2 224.6.a.f.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.6.a.e.1.3 3 8.3 odd 2
224.6.a.f.1.1 yes 3 8.5 even 2
448.6.a.ba.1.3 3 1.1 even 1 trivial
448.6.a.bb.1.1 3 4.3 odd 2