Properties

Label 495.6.a.d.1.3
Level $495$
Weight $6$
Character 495.1
Self dual yes
Analytic conductor $79.390$
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] = [495,6,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, 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("495.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(79.3899908074\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.03932\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.07863 q^{2} +33.2643 q^{4} +25.0000 q^{5} -39.3760 q^{7} +10.2141 q^{8} +O(q^{10})\) \(q+8.07863 q^{2} +33.2643 q^{4} +25.0000 q^{5} -39.3760 q^{7} +10.2141 q^{8} +201.966 q^{10} -121.000 q^{11} -220.765 q^{13} -318.105 q^{14} -981.943 q^{16} +200.343 q^{17} +350.345 q^{19} +831.608 q^{20} -977.515 q^{22} -1385.09 q^{23} +625.000 q^{25} -1783.48 q^{26} -1309.82 q^{28} -5506.64 q^{29} -2450.86 q^{31} -8259.61 q^{32} +1618.50 q^{34} -984.401 q^{35} -4060.46 q^{37} +2830.31 q^{38} +255.354 q^{40} -527.283 q^{41} -12078.9 q^{43} -4024.99 q^{44} -11189.6 q^{46} -563.023 q^{47} -15256.5 q^{49} +5049.15 q^{50} -7343.59 q^{52} +37203.0 q^{53} -3025.00 q^{55} -402.192 q^{56} -44486.2 q^{58} -2157.64 q^{59} -39938.0 q^{61} -19799.6 q^{62} -35304.2 q^{64} -5519.11 q^{65} -38473.2 q^{67} +6664.29 q^{68} -7952.62 q^{70} +13725.3 q^{71} -39736.3 q^{73} -32803.0 q^{74} +11654.0 q^{76} +4764.50 q^{77} +35672.9 q^{79} -24548.6 q^{80} -4259.73 q^{82} +79999.7 q^{83} +5008.58 q^{85} -97580.6 q^{86} -1235.91 q^{88} +37783.0 q^{89} +8692.83 q^{91} -46074.1 q^{92} -4548.46 q^{94} +8758.63 q^{95} -7616.35 q^{97} -123252. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.07863 1.42811 0.714057 0.700087i \(-0.246856\pi\)
0.714057 + 0.700087i \(0.246856\pi\)
\(3\) 0 0
\(4\) 33.2643 1.03951
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −39.3760 −0.303730 −0.151865 0.988401i \(-0.548528\pi\)
−0.151865 + 0.988401i \(0.548528\pi\)
\(8\) 10.2141 0.0564257
\(9\) 0 0
\(10\) 201.966 0.638672
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) −220.765 −0.362302 −0.181151 0.983455i \(-0.557982\pi\)
−0.181151 + 0.983455i \(0.557982\pi\)
\(14\) −318.105 −0.433761
\(15\) 0 0
\(16\) −981.943 −0.958928
\(17\) 200.343 0.168133 0.0840664 0.996460i \(-0.473209\pi\)
0.0840664 + 0.996460i \(0.473209\pi\)
\(18\) 0 0
\(19\) 350.345 0.222645 0.111322 0.993784i \(-0.464491\pi\)
0.111322 + 0.993784i \(0.464491\pi\)
\(20\) 831.608 0.464883
\(21\) 0 0
\(22\) −977.515 −0.430593
\(23\) −1385.09 −0.545957 −0.272978 0.962020i \(-0.588009\pi\)
−0.272978 + 0.962020i \(0.588009\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −1783.48 −0.517409
\(27\) 0 0
\(28\) −1309.82 −0.315730
\(29\) −5506.64 −1.21588 −0.607942 0.793982i \(-0.708005\pi\)
−0.607942 + 0.793982i \(0.708005\pi\)
\(30\) 0 0
\(31\) −2450.86 −0.458051 −0.229025 0.973420i \(-0.573554\pi\)
−0.229025 + 0.973420i \(0.573554\pi\)
\(32\) −8259.61 −1.42588
\(33\) 0 0
\(34\) 1618.50 0.240113
\(35\) −984.401 −0.135832
\(36\) 0 0
\(37\) −4060.46 −0.487609 −0.243804 0.969824i \(-0.578395\pi\)
−0.243804 + 0.969824i \(0.578395\pi\)
\(38\) 2830.31 0.317962
\(39\) 0 0
\(40\) 255.354 0.0252343
\(41\) −527.283 −0.0489874 −0.0244937 0.999700i \(-0.507797\pi\)
−0.0244937 + 0.999700i \(0.507797\pi\)
\(42\) 0 0
\(43\) −12078.9 −0.996218 −0.498109 0.867114i \(-0.665972\pi\)
−0.498109 + 0.867114i \(0.665972\pi\)
\(44\) −4024.99 −0.313424
\(45\) 0 0
\(46\) −11189.6 −0.779689
\(47\) −563.023 −0.0371776 −0.0185888 0.999827i \(-0.505917\pi\)
−0.0185888 + 0.999827i \(0.505917\pi\)
\(48\) 0 0
\(49\) −15256.5 −0.907748
\(50\) 5049.15 0.285623
\(51\) 0 0
\(52\) −7343.59 −0.376617
\(53\) 37203.0 1.81923 0.909617 0.415449i \(-0.136375\pi\)
0.909617 + 0.415449i \(0.136375\pi\)
\(54\) 0 0
\(55\) −3025.00 −0.134840
\(56\) −402.192 −0.0171381
\(57\) 0 0
\(58\) −44486.2 −1.73642
\(59\) −2157.64 −0.0806955 −0.0403477 0.999186i \(-0.512847\pi\)
−0.0403477 + 0.999186i \(0.512847\pi\)
\(60\) 0 0
\(61\) −39938.0 −1.37424 −0.687119 0.726545i \(-0.741125\pi\)
−0.687119 + 0.726545i \(0.741125\pi\)
\(62\) −19799.6 −0.654149
\(63\) 0 0
\(64\) −35304.2 −1.07740
\(65\) −5519.11 −0.162026
\(66\) 0 0
\(67\) −38473.2 −1.04706 −0.523529 0.852008i \(-0.675385\pi\)
−0.523529 + 0.852008i \(0.675385\pi\)
\(68\) 6664.29 0.174776
\(69\) 0 0
\(70\) −7952.62 −0.193984
\(71\) 13725.3 0.323129 0.161564 0.986862i \(-0.448346\pi\)
0.161564 + 0.986862i \(0.448346\pi\)
\(72\) 0 0
\(73\) −39736.3 −0.872731 −0.436366 0.899769i \(-0.643735\pi\)
−0.436366 + 0.899769i \(0.643735\pi\)
\(74\) −32803.0 −0.696361
\(75\) 0 0
\(76\) 11654.0 0.231441
\(77\) 4764.50 0.0915779
\(78\) 0 0
\(79\) 35672.9 0.643088 0.321544 0.946895i \(-0.395798\pi\)
0.321544 + 0.946895i \(0.395798\pi\)
\(80\) −24548.6 −0.428846
\(81\) 0 0
\(82\) −4259.73 −0.0699596
\(83\) 79999.7 1.27466 0.637328 0.770593i \(-0.280040\pi\)
0.637328 + 0.770593i \(0.280040\pi\)
\(84\) 0 0
\(85\) 5008.58 0.0751913
\(86\) −97580.6 −1.42271
\(87\) 0 0
\(88\) −1235.91 −0.0170130
\(89\) 37783.0 0.505616 0.252808 0.967516i \(-0.418646\pi\)
0.252808 + 0.967516i \(0.418646\pi\)
\(90\) 0 0
\(91\) 8692.83 0.110042
\(92\) −46074.1 −0.567528
\(93\) 0 0
\(94\) −4548.46 −0.0530939
\(95\) 8758.63 0.0995697
\(96\) 0 0
\(97\) −7616.35 −0.0821897 −0.0410948 0.999155i \(-0.513085\pi\)
−0.0410948 + 0.999155i \(0.513085\pi\)
\(98\) −123252. −1.29637
\(99\) 0 0
\(100\) 20790.2 0.207902
\(101\) −125963. −1.22868 −0.614341 0.789041i \(-0.710578\pi\)
−0.614341 + 0.789041i \(0.710578\pi\)
\(102\) 0 0
\(103\) −165220. −1.53451 −0.767257 0.641340i \(-0.778379\pi\)
−0.767257 + 0.641340i \(0.778379\pi\)
\(104\) −2254.92 −0.0204431
\(105\) 0 0
\(106\) 300550. 2.59807
\(107\) 13152.3 0.111056 0.0555279 0.998457i \(-0.482316\pi\)
0.0555279 + 0.998457i \(0.482316\pi\)
\(108\) 0 0
\(109\) 37970.9 0.306115 0.153057 0.988217i \(-0.451088\pi\)
0.153057 + 0.988217i \(0.451088\pi\)
\(110\) −24437.9 −0.192567
\(111\) 0 0
\(112\) 38665.0 0.291255
\(113\) 85189.7 0.627612 0.313806 0.949487i \(-0.398396\pi\)
0.313806 + 0.949487i \(0.398396\pi\)
\(114\) 0 0
\(115\) −34627.3 −0.244159
\(116\) −183175. −1.26392
\(117\) 0 0
\(118\) −17430.8 −0.115242
\(119\) −7888.73 −0.0510669
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −322645. −1.96257
\(123\) 0 0
\(124\) −81526.1 −0.476148
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −141237. −0.777033 −0.388517 0.921442i \(-0.627012\pi\)
−0.388517 + 0.921442i \(0.627012\pi\)
\(128\) −20902.2 −0.112763
\(129\) 0 0
\(130\) −44586.9 −0.231392
\(131\) −210469. −1.07155 −0.535773 0.844362i \(-0.679980\pi\)
−0.535773 + 0.844362i \(0.679980\pi\)
\(132\) 0 0
\(133\) −13795.2 −0.0676237
\(134\) −310811. −1.49532
\(135\) 0 0
\(136\) 2046.33 0.00948701
\(137\) −19565.1 −0.0890594 −0.0445297 0.999008i \(-0.514179\pi\)
−0.0445297 + 0.999008i \(0.514179\pi\)
\(138\) 0 0
\(139\) 132498. 0.581663 0.290831 0.956774i \(-0.406068\pi\)
0.290831 + 0.956774i \(0.406068\pi\)
\(140\) −32745.5 −0.141199
\(141\) 0 0
\(142\) 110882. 0.461465
\(143\) 26712.5 0.109238
\(144\) 0 0
\(145\) −137666. −0.543760
\(146\) −321015. −1.24636
\(147\) 0 0
\(148\) −135069. −0.506874
\(149\) 153316. 0.565748 0.282874 0.959157i \(-0.408712\pi\)
0.282874 + 0.959157i \(0.408712\pi\)
\(150\) 0 0
\(151\) 248151. 0.885672 0.442836 0.896603i \(-0.353972\pi\)
0.442836 + 0.896603i \(0.353972\pi\)
\(152\) 3578.47 0.0125629
\(153\) 0 0
\(154\) 38490.7 0.130784
\(155\) −61271.4 −0.204846
\(156\) 0 0
\(157\) −180692. −0.585047 −0.292523 0.956258i \(-0.594495\pi\)
−0.292523 + 0.956258i \(0.594495\pi\)
\(158\) 288188. 0.918403
\(159\) 0 0
\(160\) −206490. −0.637675
\(161\) 54539.4 0.165823
\(162\) 0 0
\(163\) −176045. −0.518985 −0.259493 0.965745i \(-0.583555\pi\)
−0.259493 + 0.965745i \(0.583555\pi\)
\(164\) −17539.7 −0.0509229
\(165\) 0 0
\(166\) 646288. 1.82035
\(167\) −293818. −0.815244 −0.407622 0.913151i \(-0.633642\pi\)
−0.407622 + 0.913151i \(0.633642\pi\)
\(168\) 0 0
\(169\) −322556. −0.868737
\(170\) 40462.5 0.107382
\(171\) 0 0
\(172\) −401795. −1.03558
\(173\) 20784.9 0.0527999 0.0263999 0.999651i \(-0.491596\pi\)
0.0263999 + 0.999651i \(0.491596\pi\)
\(174\) 0 0
\(175\) −24610.0 −0.0607459
\(176\) 118815. 0.289128
\(177\) 0 0
\(178\) 305235. 0.722078
\(179\) 229326. 0.534958 0.267479 0.963564i \(-0.413809\pi\)
0.267479 + 0.963564i \(0.413809\pi\)
\(180\) 0 0
\(181\) −90807.3 −0.206027 −0.103014 0.994680i \(-0.532849\pi\)
−0.103014 + 0.994680i \(0.532849\pi\)
\(182\) 70226.2 0.157152
\(183\) 0 0
\(184\) −14147.5 −0.0308060
\(185\) −101512. −0.218065
\(186\) 0 0
\(187\) −24241.5 −0.0506939
\(188\) −18728.6 −0.0386465
\(189\) 0 0
\(190\) 70757.8 0.142197
\(191\) 496794. 0.985356 0.492678 0.870212i \(-0.336018\pi\)
0.492678 + 0.870212i \(0.336018\pi\)
\(192\) 0 0
\(193\) 271362. 0.524391 0.262196 0.965015i \(-0.415553\pi\)
0.262196 + 0.965015i \(0.415553\pi\)
\(194\) −61529.7 −0.117376
\(195\) 0 0
\(196\) −507498. −0.943614
\(197\) 190972. 0.350594 0.175297 0.984516i \(-0.443911\pi\)
0.175297 + 0.984516i \(0.443911\pi\)
\(198\) 0 0
\(199\) 602637. 1.07876 0.539378 0.842064i \(-0.318660\pi\)
0.539378 + 0.842064i \(0.318660\pi\)
\(200\) 6383.84 0.0112851
\(201\) 0 0
\(202\) −1.01761e6 −1.75470
\(203\) 216830. 0.369300
\(204\) 0 0
\(205\) −13182.1 −0.0219078
\(206\) −1.33476e6 −2.19146
\(207\) 0 0
\(208\) 216778. 0.347422
\(209\) −42391.8 −0.0671299
\(210\) 0 0
\(211\) 918176. 1.41978 0.709888 0.704315i \(-0.248746\pi\)
0.709888 + 0.704315i \(0.248746\pi\)
\(212\) 1.23753e6 1.89111
\(213\) 0 0
\(214\) 106252. 0.158600
\(215\) −301971. −0.445522
\(216\) 0 0
\(217\) 96505.0 0.139123
\(218\) 306753. 0.437167
\(219\) 0 0
\(220\) −100625. −0.140168
\(221\) −44228.7 −0.0609149
\(222\) 0 0
\(223\) 959605. 1.29220 0.646101 0.763252i \(-0.276399\pi\)
0.646101 + 0.763252i \(0.276399\pi\)
\(224\) 325231. 0.433083
\(225\) 0 0
\(226\) 688217. 0.896301
\(227\) 289954. 0.373477 0.186739 0.982410i \(-0.440208\pi\)
0.186739 + 0.982410i \(0.440208\pi\)
\(228\) 0 0
\(229\) 111161. 0.140076 0.0700379 0.997544i \(-0.477688\pi\)
0.0700379 + 0.997544i \(0.477688\pi\)
\(230\) −279741. −0.348688
\(231\) 0 0
\(232\) −56245.6 −0.0686071
\(233\) 453216. 0.546909 0.273455 0.961885i \(-0.411834\pi\)
0.273455 + 0.961885i \(0.411834\pi\)
\(234\) 0 0
\(235\) −14075.6 −0.0166263
\(236\) −71772.5 −0.0838838
\(237\) 0 0
\(238\) −63730.1 −0.0729294
\(239\) −1.26387e6 −1.43123 −0.715614 0.698496i \(-0.753853\pi\)
−0.715614 + 0.698496i \(0.753853\pi\)
\(240\) 0 0
\(241\) 134350. 0.149003 0.0745013 0.997221i \(-0.476264\pi\)
0.0745013 + 0.997221i \(0.476264\pi\)
\(242\) 118279. 0.129829
\(243\) 0 0
\(244\) −1.32851e6 −1.42853
\(245\) −381413. −0.405957
\(246\) 0 0
\(247\) −77343.8 −0.0806646
\(248\) −25033.4 −0.0258458
\(249\) 0 0
\(250\) 126229. 0.127734
\(251\) 1.43370e6 1.43639 0.718197 0.695840i \(-0.244968\pi\)
0.718197 + 0.695840i \(0.244968\pi\)
\(252\) 0 0
\(253\) 167596. 0.164612
\(254\) −1.14100e6 −1.10969
\(255\) 0 0
\(256\) 960873. 0.916360
\(257\) 953935. 0.900920 0.450460 0.892797i \(-0.351260\pi\)
0.450460 + 0.892797i \(0.351260\pi\)
\(258\) 0 0
\(259\) 159885. 0.148101
\(260\) −183590. −0.168428
\(261\) 0 0
\(262\) −1.70030e6 −1.53029
\(263\) −322280. −0.287306 −0.143653 0.989628i \(-0.545885\pi\)
−0.143653 + 0.989628i \(0.545885\pi\)
\(264\) 0 0
\(265\) 930075. 0.813586
\(266\) −111446. −0.0965744
\(267\) 0 0
\(268\) −1.27978e6 −1.08843
\(269\) −808737. −0.681438 −0.340719 0.940165i \(-0.610671\pi\)
−0.340719 + 0.940165i \(0.610671\pi\)
\(270\) 0 0
\(271\) −909303. −0.752117 −0.376058 0.926596i \(-0.622721\pi\)
−0.376058 + 0.926596i \(0.622721\pi\)
\(272\) −196726. −0.161227
\(273\) 0 0
\(274\) −158059. −0.127187
\(275\) −75625.0 −0.0603023
\(276\) 0 0
\(277\) 236933. 0.185535 0.0927675 0.995688i \(-0.470429\pi\)
0.0927675 + 0.995688i \(0.470429\pi\)
\(278\) 1.07040e6 0.830681
\(279\) 0 0
\(280\) −10054.8 −0.00766441
\(281\) 450779. 0.340563 0.170282 0.985395i \(-0.445532\pi\)
0.170282 + 0.985395i \(0.445532\pi\)
\(282\) 0 0
\(283\) 683413. 0.507245 0.253622 0.967303i \(-0.418378\pi\)
0.253622 + 0.967303i \(0.418378\pi\)
\(284\) 456563. 0.335896
\(285\) 0 0
\(286\) 215801. 0.156005
\(287\) 20762.3 0.0148789
\(288\) 0 0
\(289\) −1.37972e6 −0.971731
\(290\) −1.11215e6 −0.776551
\(291\) 0 0
\(292\) −1.32180e6 −0.907213
\(293\) 1.81068e6 1.23218 0.616088 0.787678i \(-0.288717\pi\)
0.616088 + 0.787678i \(0.288717\pi\)
\(294\) 0 0
\(295\) −53941.0 −0.0360881
\(296\) −41474.1 −0.0275136
\(297\) 0 0
\(298\) 1.23859e6 0.807953
\(299\) 305779. 0.197801
\(300\) 0 0
\(301\) 475617. 0.302581
\(302\) 2.00472e6 1.26484
\(303\) 0 0
\(304\) −344019. −0.213500
\(305\) −998450. −0.614578
\(306\) 0 0
\(307\) −3.00860e6 −1.82187 −0.910937 0.412546i \(-0.864639\pi\)
−0.910937 + 0.412546i \(0.864639\pi\)
\(308\) 158488. 0.0951962
\(309\) 0 0
\(310\) −494989. −0.292544
\(311\) 2.04083e6 1.19648 0.598241 0.801316i \(-0.295866\pi\)
0.598241 + 0.801316i \(0.295866\pi\)
\(312\) 0 0
\(313\) −1.73940e6 −1.00355 −0.501774 0.864999i \(-0.667319\pi\)
−0.501774 + 0.864999i \(0.667319\pi\)
\(314\) −1.45975e6 −0.835513
\(315\) 0 0
\(316\) 1.18663e6 0.668497
\(317\) −106704. −0.0596392 −0.0298196 0.999555i \(-0.509493\pi\)
−0.0298196 + 0.999555i \(0.509493\pi\)
\(318\) 0 0
\(319\) 666304. 0.366603
\(320\) −882605. −0.481827
\(321\) 0 0
\(322\) 440604. 0.236815
\(323\) 70189.3 0.0374338
\(324\) 0 0
\(325\) −137978. −0.0724604
\(326\) −1.42220e6 −0.741170
\(327\) 0 0
\(328\) −5385.74 −0.00276415
\(329\) 22169.6 0.0112919
\(330\) 0 0
\(331\) 1.54759e6 0.776402 0.388201 0.921575i \(-0.373097\pi\)
0.388201 + 0.921575i \(0.373097\pi\)
\(332\) 2.66114e6 1.32502
\(333\) 0 0
\(334\) −2.37365e6 −1.16426
\(335\) −961829. −0.468259
\(336\) 0 0
\(337\) 1.22550e6 0.587814 0.293907 0.955834i \(-0.405044\pi\)
0.293907 + 0.955834i \(0.405044\pi\)
\(338\) −2.60581e6 −1.24066
\(339\) 0 0
\(340\) 166607. 0.0781621
\(341\) 296553. 0.138107
\(342\) 0 0
\(343\) 1.26253e6 0.579440
\(344\) −123375. −0.0562123
\(345\) 0 0
\(346\) 167914. 0.0754042
\(347\) −3.82684e6 −1.70615 −0.853074 0.521789i \(-0.825265\pi\)
−0.853074 + 0.521789i \(0.825265\pi\)
\(348\) 0 0
\(349\) 456968. 0.200827 0.100413 0.994946i \(-0.467983\pi\)
0.100413 + 0.994946i \(0.467983\pi\)
\(350\) −198815. −0.0867521
\(351\) 0 0
\(352\) 999413. 0.429920
\(353\) −4.32830e6 −1.84876 −0.924380 0.381474i \(-0.875417\pi\)
−0.924380 + 0.381474i \(0.875417\pi\)
\(354\) 0 0
\(355\) 343132. 0.144508
\(356\) 1.25683e6 0.525594
\(357\) 0 0
\(358\) 1.85264e6 0.763982
\(359\) 845380. 0.346191 0.173095 0.984905i \(-0.444623\pi\)
0.173095 + 0.984905i \(0.444623\pi\)
\(360\) 0 0
\(361\) −2.35336e6 −0.950429
\(362\) −733599. −0.294230
\(363\) 0 0
\(364\) 289161. 0.114390
\(365\) −993408. −0.390297
\(366\) 0 0
\(367\) 2.96016e6 1.14723 0.573615 0.819125i \(-0.305540\pi\)
0.573615 + 0.819125i \(0.305540\pi\)
\(368\) 1.36008e6 0.523534
\(369\) 0 0
\(370\) −820075. −0.311422
\(371\) −1.46491e6 −0.552555
\(372\) 0 0
\(373\) 1.02299e6 0.380713 0.190357 0.981715i \(-0.439036\pi\)
0.190357 + 0.981715i \(0.439036\pi\)
\(374\) −195839. −0.0723968
\(375\) 0 0
\(376\) −5750.80 −0.00209777
\(377\) 1.21567e6 0.440517
\(378\) 0 0
\(379\) −4.19402e6 −1.49980 −0.749899 0.661552i \(-0.769898\pi\)
−0.749899 + 0.661552i \(0.769898\pi\)
\(380\) 291350. 0.103504
\(381\) 0 0
\(382\) 4.01342e6 1.40720
\(383\) 1.64287e6 0.572276 0.286138 0.958188i \(-0.407628\pi\)
0.286138 + 0.958188i \(0.407628\pi\)
\(384\) 0 0
\(385\) 119113. 0.0409549
\(386\) 2.19223e6 0.748891
\(387\) 0 0
\(388\) −253353. −0.0854371
\(389\) −769820. −0.257938 −0.128969 0.991649i \(-0.541167\pi\)
−0.128969 + 0.991649i \(0.541167\pi\)
\(390\) 0 0
\(391\) −277493. −0.0917933
\(392\) −155832. −0.0512203
\(393\) 0 0
\(394\) 1.54279e6 0.500688
\(395\) 891822. 0.287598
\(396\) 0 0
\(397\) 4.86631e6 1.54961 0.774806 0.632199i \(-0.217847\pi\)
0.774806 + 0.632199i \(0.217847\pi\)
\(398\) 4.86848e6 1.54059
\(399\) 0 0
\(400\) −613714. −0.191786
\(401\) −717064. −0.222688 −0.111344 0.993782i \(-0.535516\pi\)
−0.111344 + 0.993782i \(0.535516\pi\)
\(402\) 0 0
\(403\) 541062. 0.165953
\(404\) −4.19007e6 −1.27723
\(405\) 0 0
\(406\) 1.75169e6 0.527402
\(407\) 491316. 0.147019
\(408\) 0 0
\(409\) −3.80091e6 −1.12352 −0.561758 0.827302i \(-0.689875\pi\)
−0.561758 + 0.827302i \(0.689875\pi\)
\(410\) −106493. −0.0312869
\(411\) 0 0
\(412\) −5.49595e6 −1.59514
\(413\) 84959.4 0.0245096
\(414\) 0 0
\(415\) 1.99999e6 0.570044
\(416\) 1.82343e6 0.516601
\(417\) 0 0
\(418\) −342468. −0.0958691
\(419\) 1.55888e6 0.433787 0.216894 0.976195i \(-0.430408\pi\)
0.216894 + 0.976195i \(0.430408\pi\)
\(420\) 0 0
\(421\) 4.13561e6 1.13719 0.568596 0.822617i \(-0.307487\pi\)
0.568596 + 0.822617i \(0.307487\pi\)
\(422\) 7.41761e6 2.02760
\(423\) 0 0
\(424\) 379997. 0.102651
\(425\) 125215. 0.0336266
\(426\) 0 0
\(427\) 1.57260e6 0.417397
\(428\) 437502. 0.115444
\(429\) 0 0
\(430\) −2.43952e6 −0.636257
\(431\) 5.25900e6 1.36367 0.681836 0.731505i \(-0.261182\pi\)
0.681836 + 0.731505i \(0.261182\pi\)
\(432\) 0 0
\(433\) −694549. −0.178026 −0.0890130 0.996030i \(-0.528371\pi\)
−0.0890130 + 0.996030i \(0.528371\pi\)
\(434\) 779629. 0.198684
\(435\) 0 0
\(436\) 1.26308e6 0.318210
\(437\) −485260. −0.121554
\(438\) 0 0
\(439\) −7.80969e6 −1.93407 −0.967036 0.254640i \(-0.918043\pi\)
−0.967036 + 0.254640i \(0.918043\pi\)
\(440\) −30897.8 −0.00760844
\(441\) 0 0
\(442\) −357307. −0.0869934
\(443\) 5.43876e6 1.31671 0.658356 0.752707i \(-0.271252\pi\)
0.658356 + 0.752707i \(0.271252\pi\)
\(444\) 0 0
\(445\) 944575. 0.226119
\(446\) 7.75229e6 1.84541
\(447\) 0 0
\(448\) 1.39014e6 0.327238
\(449\) −5.46277e6 −1.27878 −0.639392 0.768881i \(-0.720814\pi\)
−0.639392 + 0.768881i \(0.720814\pi\)
\(450\) 0 0
\(451\) 63801.3 0.0147703
\(452\) 2.83378e6 0.652409
\(453\) 0 0
\(454\) 2.34243e6 0.533368
\(455\) 217321. 0.0492122
\(456\) 0 0
\(457\) 3.07144e6 0.687941 0.343970 0.938980i \(-0.388228\pi\)
0.343970 + 0.938980i \(0.388228\pi\)
\(458\) 898028. 0.200044
\(459\) 0 0
\(460\) −1.15185e6 −0.253806
\(461\) −3.85641e6 −0.845143 −0.422572 0.906329i \(-0.638873\pi\)
−0.422572 + 0.906329i \(0.638873\pi\)
\(462\) 0 0
\(463\) 3.20594e6 0.695029 0.347514 0.937675i \(-0.387026\pi\)
0.347514 + 0.937675i \(0.387026\pi\)
\(464\) 5.40721e6 1.16594
\(465\) 0 0
\(466\) 3.66136e6 0.781049
\(467\) 2.43952e6 0.517622 0.258811 0.965928i \(-0.416669\pi\)
0.258811 + 0.965928i \(0.416669\pi\)
\(468\) 0 0
\(469\) 1.51492e6 0.318023
\(470\) −113711. −0.0237443
\(471\) 0 0
\(472\) −22038.5 −0.00455330
\(473\) 1.46154e6 0.300371
\(474\) 0 0
\(475\) 218966. 0.0445289
\(476\) −262413. −0.0530846
\(477\) 0 0
\(478\) −1.02104e7 −2.04396
\(479\) −4.69186e6 −0.934342 −0.467171 0.884167i \(-0.654727\pi\)
−0.467171 + 0.884167i \(0.654727\pi\)
\(480\) 0 0
\(481\) 896406. 0.176662
\(482\) 1.08536e6 0.212793
\(483\) 0 0
\(484\) 487023. 0.0945010
\(485\) −190409. −0.0367564
\(486\) 0 0
\(487\) 5.73305e6 1.09538 0.547688 0.836683i \(-0.315508\pi\)
0.547688 + 0.836683i \(0.315508\pi\)
\(488\) −407932. −0.0775423
\(489\) 0 0
\(490\) −3.08130e6 −0.579754
\(491\) 693872. 0.129890 0.0649449 0.997889i \(-0.479313\pi\)
0.0649449 + 0.997889i \(0.479313\pi\)
\(492\) 0 0
\(493\) −1.10322e6 −0.204430
\(494\) −624832. −0.115198
\(495\) 0 0
\(496\) 2.40660e6 0.439238
\(497\) −540448. −0.0981438
\(498\) 0 0
\(499\) 1.23153e6 0.221407 0.110704 0.993853i \(-0.464690\pi\)
0.110704 + 0.993853i \(0.464690\pi\)
\(500\) 519755. 0.0929767
\(501\) 0 0
\(502\) 1.15823e7 2.05133
\(503\) 7.61551e6 1.34208 0.671041 0.741420i \(-0.265847\pi\)
0.671041 + 0.741420i \(0.265847\pi\)
\(504\) 0 0
\(505\) −3.14907e6 −0.549483
\(506\) 1.35395e6 0.235085
\(507\) 0 0
\(508\) −4.69816e6 −0.807734
\(509\) 1.55496e6 0.266026 0.133013 0.991114i \(-0.457535\pi\)
0.133013 + 0.991114i \(0.457535\pi\)
\(510\) 0 0
\(511\) 1.56466e6 0.265074
\(512\) 8.43141e6 1.42143
\(513\) 0 0
\(514\) 7.70649e6 1.28662
\(515\) −4.13051e6 −0.686255
\(516\) 0 0
\(517\) 68125.8 0.0112095
\(518\) 1.29165e6 0.211505
\(519\) 0 0
\(520\) −56373.0 −0.00914245
\(521\) 3.26766e6 0.527403 0.263701 0.964604i \(-0.415057\pi\)
0.263701 + 0.964604i \(0.415057\pi\)
\(522\) 0 0
\(523\) −8.35303e6 −1.33533 −0.667667 0.744460i \(-0.732707\pi\)
−0.667667 + 0.744460i \(0.732707\pi\)
\(524\) −7.00112e6 −1.11388
\(525\) 0 0
\(526\) −2.60358e6 −0.410305
\(527\) −491012. −0.0770133
\(528\) 0 0
\(529\) −4.51787e6 −0.701931
\(530\) 7.51374e6 1.16189
\(531\) 0 0
\(532\) −458888. −0.0702956
\(533\) 116405. 0.0177482
\(534\) 0 0
\(535\) 328807. 0.0496657
\(536\) −392970. −0.0590810
\(537\) 0 0
\(538\) −6.53349e6 −0.973172
\(539\) 1.84604e6 0.273696
\(540\) 0 0
\(541\) 4.72814e6 0.694540 0.347270 0.937765i \(-0.387109\pi\)
0.347270 + 0.937765i \(0.387109\pi\)
\(542\) −7.34593e6 −1.07411
\(543\) 0 0
\(544\) −1.65476e6 −0.239738
\(545\) 949272. 0.136899
\(546\) 0 0
\(547\) −3.84639e6 −0.549648 −0.274824 0.961495i \(-0.588620\pi\)
−0.274824 + 0.961495i \(0.588620\pi\)
\(548\) −650819. −0.0925782
\(549\) 0 0
\(550\) −610947. −0.0861185
\(551\) −1.92923e6 −0.270710
\(552\) 0 0
\(553\) −1.40466e6 −0.195325
\(554\) 1.91409e6 0.264965
\(555\) 0 0
\(556\) 4.40745e6 0.604644
\(557\) −1.34095e7 −1.83137 −0.915684 0.401900i \(-0.868350\pi\)
−0.915684 + 0.401900i \(0.868350\pi\)
\(558\) 0 0
\(559\) 2.66658e6 0.360932
\(560\) 966625. 0.130253
\(561\) 0 0
\(562\) 3.64168e6 0.486363
\(563\) −1.51163e6 −0.200990 −0.100495 0.994938i \(-0.532043\pi\)
−0.100495 + 0.994938i \(0.532043\pi\)
\(564\) 0 0
\(565\) 2.12974e6 0.280677
\(566\) 5.52105e6 0.724403
\(567\) 0 0
\(568\) 140192. 0.0182328
\(569\) −1.39221e7 −1.80271 −0.901353 0.433085i \(-0.857425\pi\)
−0.901353 + 0.433085i \(0.857425\pi\)
\(570\) 0 0
\(571\) −3.44073e6 −0.441632 −0.220816 0.975316i \(-0.570872\pi\)
−0.220816 + 0.975316i \(0.570872\pi\)
\(572\) 888574. 0.113554
\(573\) 0 0
\(574\) 167731. 0.0212488
\(575\) −865681. −0.109191
\(576\) 0 0
\(577\) 3.96778e6 0.496144 0.248072 0.968742i \(-0.420203\pi\)
0.248072 + 0.968742i \(0.420203\pi\)
\(578\) −1.11463e7 −1.38774
\(579\) 0 0
\(580\) −4.57937e6 −0.565244
\(581\) −3.15007e6 −0.387151
\(582\) 0 0
\(583\) −4.50156e6 −0.548519
\(584\) −405872. −0.0492445
\(585\) 0 0
\(586\) 1.46278e7 1.75969
\(587\) −3.64595e6 −0.436733 −0.218366 0.975867i \(-0.570073\pi\)
−0.218366 + 0.975867i \(0.570073\pi\)
\(588\) 0 0
\(589\) −858645. −0.101982
\(590\) −435770. −0.0515380
\(591\) 0 0
\(592\) 3.98714e6 0.467582
\(593\) 486759. 0.0568431 0.0284215 0.999596i \(-0.490952\pi\)
0.0284215 + 0.999596i \(0.490952\pi\)
\(594\) 0 0
\(595\) −197218. −0.0228378
\(596\) 5.09997e6 0.588101
\(597\) 0 0
\(598\) 2.47027e6 0.282483
\(599\) −1.37413e7 −1.56481 −0.782404 0.622771i \(-0.786007\pi\)
−0.782404 + 0.622771i \(0.786007\pi\)
\(600\) 0 0
\(601\) 7.84470e6 0.885911 0.442955 0.896544i \(-0.353930\pi\)
0.442955 + 0.896544i \(0.353930\pi\)
\(602\) 3.84234e6 0.432120
\(603\) 0 0
\(604\) 8.25457e6 0.920666
\(605\) 366025. 0.0406558
\(606\) 0 0
\(607\) −3.41681e6 −0.376399 −0.188200 0.982131i \(-0.560265\pi\)
−0.188200 + 0.982131i \(0.560265\pi\)
\(608\) −2.89371e6 −0.317465
\(609\) 0 0
\(610\) −8.06611e6 −0.877687
\(611\) 124296. 0.0134695
\(612\) 0 0
\(613\) −1.32402e7 −1.42313 −0.711564 0.702621i \(-0.752013\pi\)
−0.711564 + 0.702621i \(0.752013\pi\)
\(614\) −2.43054e7 −2.60184
\(615\) 0 0
\(616\) 48665.3 0.00516735
\(617\) −1.72143e7 −1.82044 −0.910220 0.414126i \(-0.864087\pi\)
−0.910220 + 0.414126i \(0.864087\pi\)
\(618\) 0 0
\(619\) 1.60142e7 1.67988 0.839940 0.542679i \(-0.182590\pi\)
0.839940 + 0.542679i \(0.182590\pi\)
\(620\) −2.03815e6 −0.212940
\(621\) 0 0
\(622\) 1.64871e7 1.70871
\(623\) −1.48774e6 −0.153571
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) −1.40520e7 −1.43318
\(627\) 0 0
\(628\) −6.01061e6 −0.608162
\(629\) −813486. −0.0819830
\(630\) 0 0
\(631\) −7.75049e6 −0.774918 −0.387459 0.921887i \(-0.626647\pi\)
−0.387459 + 0.921887i \(0.626647\pi\)
\(632\) 364368. 0.0362867
\(633\) 0 0
\(634\) −862021. −0.0851716
\(635\) −3.53093e6 −0.347500
\(636\) 0 0
\(637\) 3.36810e6 0.328879
\(638\) 5.38282e6 0.523550
\(639\) 0 0
\(640\) −522556. −0.0504293
\(641\) −8.77648e6 −0.843675 −0.421838 0.906671i \(-0.638615\pi\)
−0.421838 + 0.906671i \(0.638615\pi\)
\(642\) 0 0
\(643\) −1.51011e7 −1.44040 −0.720198 0.693768i \(-0.755949\pi\)
−0.720198 + 0.693768i \(0.755949\pi\)
\(644\) 1.81422e6 0.172375
\(645\) 0 0
\(646\) 567034. 0.0534598
\(647\) −1.57910e7 −1.48303 −0.741516 0.670936i \(-0.765893\pi\)
−0.741516 + 0.670936i \(0.765893\pi\)
\(648\) 0 0
\(649\) 261075. 0.0243306
\(650\) −1.11467e6 −0.103482
\(651\) 0 0
\(652\) −5.85603e6 −0.539491
\(653\) −1.33260e7 −1.22297 −0.611484 0.791256i \(-0.709427\pi\)
−0.611484 + 0.791256i \(0.709427\pi\)
\(654\) 0 0
\(655\) −5.26173e6 −0.479210
\(656\) 517762. 0.0469754
\(657\) 0 0
\(658\) 179100. 0.0161262
\(659\) 882213. 0.0791334 0.0395667 0.999217i \(-0.487402\pi\)
0.0395667 + 0.999217i \(0.487402\pi\)
\(660\) 0 0
\(661\) −5.59700e6 −0.498255 −0.249127 0.968471i \(-0.580144\pi\)
−0.249127 + 0.968471i \(0.580144\pi\)
\(662\) 1.25024e7 1.10879
\(663\) 0 0
\(664\) 817128. 0.0719234
\(665\) −344880. −0.0302422
\(666\) 0 0
\(667\) 7.62720e6 0.663820
\(668\) −9.77367e6 −0.847454
\(669\) 0 0
\(670\) −7.77027e6 −0.668727
\(671\) 4.83250e6 0.414348
\(672\) 0 0
\(673\) 1.29320e6 0.110060 0.0550300 0.998485i \(-0.482475\pi\)
0.0550300 + 0.998485i \(0.482475\pi\)
\(674\) 9.90040e6 0.839466
\(675\) 0 0
\(676\) −1.07296e7 −0.903062
\(677\) −1.48798e7 −1.24775 −0.623873 0.781526i \(-0.714442\pi\)
−0.623873 + 0.781526i \(0.714442\pi\)
\(678\) 0 0
\(679\) 299902. 0.0249634
\(680\) 51158.4 0.00424272
\(681\) 0 0
\(682\) 2.39575e6 0.197233
\(683\) −8.62925e6 −0.707817 −0.353909 0.935280i \(-0.615148\pi\)
−0.353909 + 0.935280i \(0.615148\pi\)
\(684\) 0 0
\(685\) −489127. −0.0398286
\(686\) 1.01996e7 0.827506
\(687\) 0 0
\(688\) 1.18607e7 0.955302
\(689\) −8.21310e6 −0.659112
\(690\) 0 0
\(691\) −1.79734e7 −1.43198 −0.715988 0.698113i \(-0.754023\pi\)
−0.715988 + 0.698113i \(0.754023\pi\)
\(692\) 691396. 0.0548860
\(693\) 0 0
\(694\) −3.09157e7 −2.43658
\(695\) 3.31244e6 0.260127
\(696\) 0 0
\(697\) −105638. −0.00823639
\(698\) 3.69167e6 0.286804
\(699\) 0 0
\(700\) −818636. −0.0631460
\(701\) 2.13485e7 1.64086 0.820432 0.571744i \(-0.193733\pi\)
0.820432 + 0.571744i \(0.193733\pi\)
\(702\) 0 0
\(703\) −1.42256e6 −0.108563
\(704\) 4.27181e6 0.324848
\(705\) 0 0
\(706\) −3.49667e7 −2.64024
\(707\) 4.95992e6 0.373187
\(708\) 0 0
\(709\) 6.90604e6 0.515957 0.257979 0.966151i \(-0.416944\pi\)
0.257979 + 0.966151i \(0.416944\pi\)
\(710\) 2.77204e6 0.206373
\(711\) 0 0
\(712\) 385921. 0.0285298
\(713\) 3.39466e6 0.250076
\(714\) 0 0
\(715\) 667813. 0.0488528
\(716\) 7.62837e6 0.556095
\(717\) 0 0
\(718\) 6.82951e6 0.494400
\(719\) 1.71863e7 1.23983 0.619913 0.784670i \(-0.287168\pi\)
0.619913 + 0.784670i \(0.287168\pi\)
\(720\) 0 0
\(721\) 6.50573e6 0.466077
\(722\) −1.90119e7 −1.35732
\(723\) 0 0
\(724\) −3.02065e6 −0.214167
\(725\) −3.44165e6 −0.243177
\(726\) 0 0
\(727\) −2.43516e7 −1.70880 −0.854401 0.519615i \(-0.826075\pi\)
−0.854401 + 0.519615i \(0.826075\pi\)
\(728\) 88789.8 0.00620919
\(729\) 0 0
\(730\) −8.02538e6 −0.557389
\(731\) −2.41992e6 −0.167497
\(732\) 0 0
\(733\) −1.93102e7 −1.32747 −0.663737 0.747966i \(-0.731031\pi\)
−0.663737 + 0.747966i \(0.731031\pi\)
\(734\) 2.39141e7 1.63838
\(735\) 0 0
\(736\) 1.14403e7 0.778472
\(737\) 4.65525e6 0.315700
\(738\) 0 0
\(739\) −9.23336e6 −0.621940 −0.310970 0.950420i \(-0.600654\pi\)
−0.310970 + 0.950420i \(0.600654\pi\)
\(740\) −3.37672e6 −0.226681
\(741\) 0 0
\(742\) −1.18345e7 −0.789112
\(743\) −8.02302e6 −0.533170 −0.266585 0.963811i \(-0.585895\pi\)
−0.266585 + 0.963811i \(0.585895\pi\)
\(744\) 0 0
\(745\) 3.83291e6 0.253010
\(746\) 8.26434e6 0.543702
\(747\) 0 0
\(748\) −806379. −0.0526969
\(749\) −517885. −0.0337309
\(750\) 0 0
\(751\) 1.19930e6 0.0775938 0.0387969 0.999247i \(-0.487647\pi\)
0.0387969 + 0.999247i \(0.487647\pi\)
\(752\) 552856. 0.0356507
\(753\) 0 0
\(754\) 9.82096e6 0.629109
\(755\) 6.20377e6 0.396085
\(756\) 0 0
\(757\) −1.40172e7 −0.889040 −0.444520 0.895769i \(-0.646626\pi\)
−0.444520 + 0.895769i \(0.646626\pi\)
\(758\) −3.38820e7 −2.14188
\(759\) 0 0
\(760\) 89461.9 0.00561829
\(761\) 1.98659e7 1.24350 0.621750 0.783215i \(-0.286422\pi\)
0.621750 + 0.783215i \(0.286422\pi\)
\(762\) 0 0
\(763\) −1.49514e6 −0.0929761
\(764\) 1.65255e7 1.02429
\(765\) 0 0
\(766\) 1.32721e7 0.817275
\(767\) 476331. 0.0292361
\(768\) 0 0
\(769\) −2.09738e7 −1.27897 −0.639486 0.768803i \(-0.720853\pi\)
−0.639486 + 0.768803i \(0.720853\pi\)
\(770\) 962267. 0.0584883
\(771\) 0 0
\(772\) 9.02668e6 0.545111
\(773\) −7.69041e6 −0.462914 −0.231457 0.972845i \(-0.574349\pi\)
−0.231457 + 0.972845i \(0.574349\pi\)
\(774\) 0 0
\(775\) −1.53178e6 −0.0916101
\(776\) −77794.4 −0.00463761
\(777\) 0 0
\(778\) −6.21909e6 −0.368365
\(779\) −184731. −0.0109068
\(780\) 0 0
\(781\) −1.66076e6 −0.0974270
\(782\) −2.24177e6 −0.131091
\(783\) 0 0
\(784\) 1.49810e7 0.870466
\(785\) −4.51731e6 −0.261641
\(786\) 0 0
\(787\) −1.39975e6 −0.0805589 −0.0402794 0.999188i \(-0.512825\pi\)
−0.0402794 + 0.999188i \(0.512825\pi\)
\(788\) 6.35256e6 0.364446
\(789\) 0 0
\(790\) 7.20470e6 0.410722
\(791\) −3.35443e6 −0.190624
\(792\) 0 0
\(793\) 8.81689e6 0.497889
\(794\) 3.93131e7 2.21302
\(795\) 0 0
\(796\) 2.00463e7 1.12138
\(797\) −1.22917e7 −0.685432 −0.342716 0.939439i \(-0.611347\pi\)
−0.342716 + 0.939439i \(0.611347\pi\)
\(798\) 0 0
\(799\) −112798. −0.00625078
\(800\) −5.16225e6 −0.285177
\(801\) 0 0
\(802\) −5.79290e6 −0.318024
\(803\) 4.80810e6 0.263138
\(804\) 0 0
\(805\) 1.36348e6 0.0741584
\(806\) 4.37104e6 0.236999
\(807\) 0 0
\(808\) −1.28660e6 −0.0693292
\(809\) 1.19607e7 0.642515 0.321258 0.946992i \(-0.395894\pi\)
0.321258 + 0.946992i \(0.395894\pi\)
\(810\) 0 0
\(811\) −1.15112e7 −0.614565 −0.307282 0.951618i \(-0.599420\pi\)
−0.307282 + 0.951618i \(0.599420\pi\)
\(812\) 7.21270e6 0.383891
\(813\) 0 0
\(814\) 3.96916e6 0.209961
\(815\) −4.40113e6 −0.232097
\(816\) 0 0
\(817\) −4.23177e6 −0.221803
\(818\) −3.07062e7 −1.60451
\(819\) 0 0
\(820\) −438493. −0.0227734
\(821\) −2.77684e7 −1.43778 −0.718890 0.695124i \(-0.755349\pi\)
−0.718890 + 0.695124i \(0.755349\pi\)
\(822\) 0 0
\(823\) −2.63447e7 −1.35580 −0.677898 0.735156i \(-0.737109\pi\)
−0.677898 + 0.735156i \(0.737109\pi\)
\(824\) −1.68758e6 −0.0865860
\(825\) 0 0
\(826\) 686356. 0.0350025
\(827\) 3.32551e7 1.69081 0.845406 0.534125i \(-0.179359\pi\)
0.845406 + 0.534125i \(0.179359\pi\)
\(828\) 0 0
\(829\) 2.84581e7 1.43820 0.719101 0.694905i \(-0.244554\pi\)
0.719101 + 0.694905i \(0.244554\pi\)
\(830\) 1.61572e7 0.814087
\(831\) 0 0
\(832\) 7.79391e6 0.390344
\(833\) −3.05654e6 −0.152622
\(834\) 0 0
\(835\) −7.34545e6 −0.364588
\(836\) −1.41013e6 −0.0697822
\(837\) 0 0
\(838\) 1.25936e7 0.619498
\(839\) 9.47931e6 0.464913 0.232457 0.972607i \(-0.425324\pi\)
0.232457 + 0.972607i \(0.425324\pi\)
\(840\) 0 0
\(841\) 9.81196e6 0.478372
\(842\) 3.34101e7 1.62404
\(843\) 0 0
\(844\) 3.05425e7 1.47587
\(845\) −8.06390e6 −0.388511
\(846\) 0 0
\(847\) −576505. −0.0276118
\(848\) −3.65312e7 −1.74451
\(849\) 0 0
\(850\) 1.01156e6 0.0480226
\(851\) 5.62411e6 0.266213
\(852\) 0 0
\(853\) 2.13145e7 1.00300 0.501501 0.865157i \(-0.332781\pi\)
0.501501 + 0.865157i \(0.332781\pi\)
\(854\) 1.27045e7 0.596090
\(855\) 0 0
\(856\) 134339. 0.00626640
\(857\) −2.21044e7 −1.02808 −0.514039 0.857767i \(-0.671851\pi\)
−0.514039 + 0.857767i \(0.671851\pi\)
\(858\) 0 0
\(859\) −7.07193e6 −0.327006 −0.163503 0.986543i \(-0.552279\pi\)
−0.163503 + 0.986543i \(0.552279\pi\)
\(860\) −1.00449e7 −0.463125
\(861\) 0 0
\(862\) 4.24855e7 1.94748
\(863\) 5.45080e6 0.249134 0.124567 0.992211i \(-0.460246\pi\)
0.124567 + 0.992211i \(0.460246\pi\)
\(864\) 0 0
\(865\) 519623. 0.0236128
\(866\) −5.61101e6 −0.254241
\(867\) 0 0
\(868\) 3.21017e6 0.144620
\(869\) −4.31642e6 −0.193898
\(870\) 0 0
\(871\) 8.49351e6 0.379351
\(872\) 387840. 0.0172727
\(873\) 0 0
\(874\) −3.92023e6 −0.173593
\(875\) −615251. −0.0271664
\(876\) 0 0
\(877\) −2.02823e7 −0.890467 −0.445234 0.895414i \(-0.646879\pi\)
−0.445234 + 0.895414i \(0.646879\pi\)
\(878\) −6.30917e7 −2.76208
\(879\) 0 0
\(880\) 2.97038e6 0.129302
\(881\) 6.58178e6 0.285696 0.142848 0.989745i \(-0.454374\pi\)
0.142848 + 0.989745i \(0.454374\pi\)
\(882\) 0 0
\(883\) −1.63860e7 −0.707247 −0.353623 0.935388i \(-0.615051\pi\)
−0.353623 + 0.935388i \(0.615051\pi\)
\(884\) −1.47124e6 −0.0633217
\(885\) 0 0
\(886\) 4.39378e7 1.88042
\(887\) 2.92929e6 0.125013 0.0625063 0.998045i \(-0.480091\pi\)
0.0625063 + 0.998045i \(0.480091\pi\)
\(888\) 0 0
\(889\) 5.56136e6 0.236008
\(890\) 7.63087e6 0.322923
\(891\) 0 0
\(892\) 3.19206e7 1.34326
\(893\) −197252. −0.00827739
\(894\) 0 0
\(895\) 5.73314e6 0.239241
\(896\) 823047. 0.0342495
\(897\) 0 0
\(898\) −4.41317e7 −1.82625
\(899\) 1.34960e7 0.556936
\(900\) 0 0
\(901\) 7.45337e6 0.305873
\(902\) 515427. 0.0210936
\(903\) 0 0
\(904\) 870140. 0.0354134
\(905\) −2.27018e6 −0.0921381
\(906\) 0 0
\(907\) −7.88236e6 −0.318154 −0.159077 0.987266i \(-0.550852\pi\)
−0.159077 + 0.987266i \(0.550852\pi\)
\(908\) 9.64512e6 0.388234
\(909\) 0 0
\(910\) 1.75566e6 0.0702807
\(911\) 2.16713e7 0.865146 0.432573 0.901599i \(-0.357606\pi\)
0.432573 + 0.901599i \(0.357606\pi\)
\(912\) 0 0
\(913\) −9.67996e6 −0.384323
\(914\) 2.48130e7 0.982458
\(915\) 0 0
\(916\) 3.69769e6 0.145610
\(917\) 8.28745e6 0.325460
\(918\) 0 0
\(919\) 4.32037e7 1.68746 0.843728 0.536771i \(-0.180356\pi\)
0.843728 + 0.536771i \(0.180356\pi\)
\(920\) −353688. −0.0137769
\(921\) 0 0
\(922\) −3.11545e7 −1.20696
\(923\) −3.03006e6 −0.117070
\(924\) 0 0
\(925\) −2.53779e6 −0.0975217
\(926\) 2.58996e7 0.992580
\(927\) 0 0
\(928\) 4.54827e7 1.73371
\(929\) −3.70288e7 −1.40767 −0.703834 0.710364i \(-0.748530\pi\)
−0.703834 + 0.710364i \(0.748530\pi\)
\(930\) 0 0
\(931\) −5.34505e6 −0.202105
\(932\) 1.50759e7 0.568518
\(933\) 0 0
\(934\) 1.97080e7 0.739224
\(935\) −606038. −0.0226710
\(936\) 0 0
\(937\) 1.22598e7 0.456180 0.228090 0.973640i \(-0.426752\pi\)
0.228090 + 0.973640i \(0.426752\pi\)
\(938\) 1.22385e7 0.454173
\(939\) 0 0
\(940\) −468215. −0.0172833
\(941\) 1.25646e7 0.462565 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(942\) 0 0
\(943\) 730335. 0.0267450
\(944\) 2.11868e6 0.0773812
\(945\) 0 0
\(946\) 1.18073e7 0.428964
\(947\) −3.16122e7 −1.14546 −0.572730 0.819744i \(-0.694116\pi\)
−0.572730 + 0.819744i \(0.694116\pi\)
\(948\) 0 0
\(949\) 8.77237e6 0.316192
\(950\) 1.76894e6 0.0635924
\(951\) 0 0
\(952\) −80576.6 −0.00288148
\(953\) 2.84509e7 1.01476 0.507380 0.861722i \(-0.330614\pi\)
0.507380 + 0.861722i \(0.330614\pi\)
\(954\) 0 0
\(955\) 1.24199e7 0.440665
\(956\) −4.20419e7 −1.48778
\(957\) 0 0
\(958\) −3.79038e7 −1.33435
\(959\) 770395. 0.0270500
\(960\) 0 0
\(961\) −2.26225e7 −0.790190
\(962\) 7.24174e6 0.252293
\(963\) 0 0
\(964\) 4.46905e6 0.154890
\(965\) 6.78405e6 0.234515
\(966\) 0 0
\(967\) −4.73439e7 −1.62816 −0.814081 0.580751i \(-0.802759\pi\)
−0.814081 + 0.580751i \(0.802759\pi\)
\(968\) 149545. 0.00512961
\(969\) 0 0
\(970\) −1.53824e6 −0.0524923
\(971\) 4.33248e7 1.47465 0.737324 0.675539i \(-0.236089\pi\)
0.737324 + 0.675539i \(0.236089\pi\)
\(972\) 0 0
\(973\) −5.21723e6 −0.176668
\(974\) 4.63152e7 1.56432
\(975\) 0 0
\(976\) 3.92168e7 1.31780
\(977\) 4.64678e7 1.55745 0.778727 0.627363i \(-0.215866\pi\)
0.778727 + 0.627363i \(0.215866\pi\)
\(978\) 0 0
\(979\) −4.57174e6 −0.152449
\(980\) −1.26875e7 −0.421997
\(981\) 0 0
\(982\) 5.60553e6 0.185498
\(983\) 1.02443e7 0.338141 0.169071 0.985604i \(-0.445923\pi\)
0.169071 + 0.985604i \(0.445923\pi\)
\(984\) 0 0
\(985\) 4.77430e6 0.156790
\(986\) −8.91250e6 −0.291949
\(987\) 0 0
\(988\) −2.57279e6 −0.0838517
\(989\) 1.67303e7 0.543892
\(990\) 0 0
\(991\) −6.52638e6 −0.211100 −0.105550 0.994414i \(-0.533660\pi\)
−0.105550 + 0.994414i \(0.533660\pi\)
\(992\) 2.02431e7 0.653127
\(993\) 0 0
\(994\) −4.36608e6 −0.140161
\(995\) 1.50659e7 0.482434
\(996\) 0 0
\(997\) 1.08535e7 0.345804 0.172902 0.984939i \(-0.444686\pi\)
0.172902 + 0.984939i \(0.444686\pi\)
\(998\) 9.94904e6 0.316195
\(999\) 0 0
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 495.6.a.d.1.3 3
3.2 odd 2 165.6.a.b.1.1 3
15.14 odd 2 825.6.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.b.1.1 3 3.2 odd 2
495.6.a.d.1.3 3 1.1 even 1 trivial
825.6.a.i.1.3 3 15.14 odd 2