Properties

Label 225.6.a.u.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
Analytic rank $1$
Dimension $2$
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] = [225,6,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(36.0863594579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{89}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.21699\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.216991 q^{2} -31.9529 q^{4} +59.2078 q^{7} +13.8772 q^{8} +O(q^{10})\) \(q-0.216991 q^{2} -31.9529 q^{4} +59.2078 q^{7} +13.8772 q^{8} +425.435 q^{11} -1044.23 q^{13} -12.8475 q^{14} +1019.48 q^{16} +948.379 q^{17} -328.377 q^{19} -92.3154 q^{22} +3367.10 q^{23} +226.587 q^{26} -1891.86 q^{28} -2299.48 q^{29} -5144.75 q^{31} -665.288 q^{32} -205.789 q^{34} -9211.93 q^{37} +71.2546 q^{38} -15170.2 q^{41} +10119.6 q^{43} -13593.9 q^{44} -730.628 q^{46} -4455.48 q^{47} -13301.4 q^{49} +33366.1 q^{52} -20578.6 q^{53} +821.637 q^{56} +498.965 q^{58} -44460.5 q^{59} -126.574 q^{61} +1116.36 q^{62} -32479.1 q^{64} -18917.5 q^{67} -30303.5 q^{68} +6571.05 q^{71} +5463.86 q^{73} +1998.90 q^{74} +10492.6 q^{76} +25189.1 q^{77} +15391.0 q^{79} +3291.79 q^{82} +99222.5 q^{83} -2195.87 q^{86} +5903.84 q^{88} -90797.1 q^{89} -61826.4 q^{91} -107589. q^{92} +966.798 q^{94} -116353. q^{97} +2886.29 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} + 21 q^{4} - 108 q^{7} + 207 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{2} + 21 q^{4} - 108 q^{7} + 207 q^{8} - 168 q^{11} - 1296 q^{13} - 1554 q^{14} + 1105 q^{16} + 576 q^{17} - 1336 q^{19} - 5562 q^{22} + 5904 q^{23} - 2094 q^{26} - 10746 q^{28} + 3552 q^{29} - 11648 q^{31} - 6057 q^{32} - 3638 q^{34} - 14688 q^{37} - 9216 q^{38} - 1812 q^{41} + 7560 q^{43} - 45018 q^{44} + 22652 q^{46} + 3240 q^{47} - 2150 q^{49} + 20034 q^{52} - 22176 q^{53} - 31470 q^{56} + 54432 q^{58} - 57336 q^{59} - 30140 q^{61} - 58824 q^{62} - 84911 q^{64} + 5184 q^{67} - 50022 q^{68} - 17424 q^{71} + 3456 q^{73} - 48474 q^{74} - 42864 q^{76} + 124416 q^{77} - 57520 q^{79} + 126414 q^{82} + 133992 q^{83} - 25788 q^{86} - 108702 q^{88} - 136764 q^{89} - 19728 q^{91} + 26748 q^{92} + 71896 q^{94} - 85536 q^{97} + 105669 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.216991 −0.0383589 −0.0191794 0.999816i \(-0.506105\pi\)
−0.0191794 + 0.999816i \(0.506105\pi\)
\(3\) 0 0
\(4\) −31.9529 −0.998529
\(5\) 0 0
\(6\) 0 0
\(7\) 59.2078 0.456703 0.228351 0.973579i \(-0.426667\pi\)
0.228351 + 0.973579i \(0.426667\pi\)
\(8\) 13.8772 0.0766613
\(9\) 0 0
\(10\) 0 0
\(11\) 425.435 1.06011 0.530056 0.847963i \(-0.322171\pi\)
0.530056 + 0.847963i \(0.322171\pi\)
\(12\) 0 0
\(13\) −1044.23 −1.71371 −0.856854 0.515560i \(-0.827584\pi\)
−0.856854 + 0.515560i \(0.827584\pi\)
\(14\) −12.8475 −0.0175186
\(15\) 0 0
\(16\) 1019.48 0.995588
\(17\) 948.379 0.795902 0.397951 0.917407i \(-0.369721\pi\)
0.397951 + 0.917407i \(0.369721\pi\)
\(18\) 0 0
\(19\) −328.377 −0.208684 −0.104342 0.994541i \(-0.533274\pi\)
−0.104342 + 0.994541i \(0.533274\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −92.3154 −0.0406647
\(23\) 3367.10 1.32720 0.663599 0.748088i \(-0.269028\pi\)
0.663599 + 0.748088i \(0.269028\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 226.587 0.0657359
\(27\) 0 0
\(28\) −1891.86 −0.456031
\(29\) −2299.48 −0.507732 −0.253866 0.967239i \(-0.581702\pi\)
−0.253866 + 0.967239i \(0.581702\pi\)
\(30\) 0 0
\(31\) −5144.75 −0.961524 −0.480762 0.876851i \(-0.659640\pi\)
−0.480762 + 0.876851i \(0.659640\pi\)
\(32\) −665.288 −0.114851
\(33\) 0 0
\(34\) −205.789 −0.0305299
\(35\) 0 0
\(36\) 0 0
\(37\) −9211.93 −1.10623 −0.553116 0.833104i \(-0.686561\pi\)
−0.553116 + 0.833104i \(0.686561\pi\)
\(38\) 71.2546 0.00800487
\(39\) 0 0
\(40\) 0 0
\(41\) −15170.2 −1.40939 −0.704695 0.709511i \(-0.748916\pi\)
−0.704695 + 0.709511i \(0.748916\pi\)
\(42\) 0 0
\(43\) 10119.6 0.834629 0.417315 0.908762i \(-0.362971\pi\)
0.417315 + 0.908762i \(0.362971\pi\)
\(44\) −13593.9 −1.05855
\(45\) 0 0
\(46\) −730.628 −0.0509098
\(47\) −4455.48 −0.294205 −0.147103 0.989121i \(-0.546995\pi\)
−0.147103 + 0.989121i \(0.546995\pi\)
\(48\) 0 0
\(49\) −13301.4 −0.791423
\(50\) 0 0
\(51\) 0 0
\(52\) 33366.1 1.71119
\(53\) −20578.6 −1.00630 −0.503148 0.864200i \(-0.667825\pi\)
−0.503148 + 0.864200i \(0.667825\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 821.637 0.0350114
\(57\) 0 0
\(58\) 498.965 0.0194760
\(59\) −44460.5 −1.66282 −0.831408 0.555663i \(-0.812465\pi\)
−0.831408 + 0.555663i \(0.812465\pi\)
\(60\) 0 0
\(61\) −126.574 −0.00435531 −0.00217766 0.999998i \(-0.500693\pi\)
−0.00217766 + 0.999998i \(0.500693\pi\)
\(62\) 1116.36 0.0368830
\(63\) 0 0
\(64\) −32479.1 −0.991182
\(65\) 0 0
\(66\) 0 0
\(67\) −18917.5 −0.514845 −0.257422 0.966299i \(-0.582873\pi\)
−0.257422 + 0.966299i \(0.582873\pi\)
\(68\) −30303.5 −0.794731
\(69\) 0 0
\(70\) 0 0
\(71\) 6571.05 0.154699 0.0773497 0.997004i \(-0.475354\pi\)
0.0773497 + 0.997004i \(0.475354\pi\)
\(72\) 0 0
\(73\) 5463.86 0.120003 0.0600015 0.998198i \(-0.480889\pi\)
0.0600015 + 0.998198i \(0.480889\pi\)
\(74\) 1998.90 0.0424338
\(75\) 0 0
\(76\) 10492.6 0.208377
\(77\) 25189.1 0.484156
\(78\) 0 0
\(79\) 15391.0 0.277460 0.138730 0.990330i \(-0.455698\pi\)
0.138730 + 0.990330i \(0.455698\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3291.79 0.0540626
\(83\) 99222.5 1.58094 0.790469 0.612502i \(-0.209837\pi\)
0.790469 + 0.612502i \(0.209837\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2195.87 −0.0320154
\(87\) 0 0
\(88\) 5903.84 0.0812695
\(89\) −90797.1 −1.21506 −0.607529 0.794297i \(-0.707839\pi\)
−0.607529 + 0.794297i \(0.707839\pi\)
\(90\) 0 0
\(91\) −61826.4 −0.782655
\(92\) −107589. −1.32525
\(93\) 0 0
\(94\) 966.798 0.0112854
\(95\) 0 0
\(96\) 0 0
\(97\) −116353. −1.25559 −0.627796 0.778378i \(-0.716043\pi\)
−0.627796 + 0.778378i \(0.716043\pi\)
\(98\) 2886.29 0.0303581
\(99\) 0 0
\(100\) 0 0
\(101\) −145908. −1.42323 −0.711615 0.702569i \(-0.752036\pi\)
−0.711615 + 0.702569i \(0.752036\pi\)
\(102\) 0 0
\(103\) −116865. −1.08540 −0.542702 0.839925i \(-0.682599\pi\)
−0.542702 + 0.839925i \(0.682599\pi\)
\(104\) −14490.9 −0.131375
\(105\) 0 0
\(106\) 4465.36 0.0386004
\(107\) 186064. 1.57110 0.785549 0.618800i \(-0.212381\pi\)
0.785549 + 0.618800i \(0.212381\pi\)
\(108\) 0 0
\(109\) −132641. −1.06933 −0.534663 0.845065i \(-0.679562\pi\)
−0.534663 + 0.845065i \(0.679562\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 60361.3 0.454688
\(113\) 29607.7 0.218126 0.109063 0.994035i \(-0.465215\pi\)
0.109063 + 0.994035i \(0.465215\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 73475.1 0.506985
\(117\) 0 0
\(118\) 9647.51 0.0637837
\(119\) 56151.4 0.363491
\(120\) 0 0
\(121\) 19943.9 0.123836
\(122\) 27.4653 0.000167065 0
\(123\) 0 0
\(124\) 164390. 0.960110
\(125\) 0 0
\(126\) 0 0
\(127\) 189319. 1.04156 0.520780 0.853691i \(-0.325641\pi\)
0.520780 + 0.853691i \(0.325641\pi\)
\(128\) 28336.9 0.152872
\(129\) 0 0
\(130\) 0 0
\(131\) 107363. 0.546607 0.273304 0.961928i \(-0.411884\pi\)
0.273304 + 0.961928i \(0.411884\pi\)
\(132\) 0 0
\(133\) −19442.5 −0.0953064
\(134\) 4104.91 0.0197489
\(135\) 0 0
\(136\) 13160.8 0.0610149
\(137\) −173859. −0.791399 −0.395700 0.918380i \(-0.629498\pi\)
−0.395700 + 0.918380i \(0.629498\pi\)
\(138\) 0 0
\(139\) −30334.0 −0.133166 −0.0665830 0.997781i \(-0.521210\pi\)
−0.0665830 + 0.997781i \(0.521210\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1425.86 −0.00593410
\(143\) −444251. −1.81672
\(144\) 0 0
\(145\) 0 0
\(146\) −1185.61 −0.00460318
\(147\) 0 0
\(148\) 294348. 1.10460
\(149\) 296104. 1.09264 0.546322 0.837575i \(-0.316028\pi\)
0.546322 + 0.837575i \(0.316028\pi\)
\(150\) 0 0
\(151\) 240567. 0.858605 0.429303 0.903161i \(-0.358759\pi\)
0.429303 + 0.903161i \(0.358759\pi\)
\(152\) −4556.94 −0.0159980
\(153\) 0 0
\(154\) −5465.79 −0.0185717
\(155\) 0 0
\(156\) 0 0
\(157\) −110903. −0.359081 −0.179541 0.983751i \(-0.557461\pi\)
−0.179541 + 0.983751i \(0.557461\pi\)
\(158\) −3339.71 −0.0106430
\(159\) 0 0
\(160\) 0 0
\(161\) 199358. 0.606135
\(162\) 0 0
\(163\) 211488. 0.623472 0.311736 0.950169i \(-0.399090\pi\)
0.311736 + 0.950169i \(0.399090\pi\)
\(164\) 484731. 1.40732
\(165\) 0 0
\(166\) −21530.3 −0.0606430
\(167\) −74281.4 −0.206105 −0.103053 0.994676i \(-0.532861\pi\)
−0.103053 + 0.994676i \(0.532861\pi\)
\(168\) 0 0
\(169\) 719117. 1.93679
\(170\) 0 0
\(171\) 0 0
\(172\) −323352. −0.833401
\(173\) 386465. 0.981736 0.490868 0.871234i \(-0.336680\pi\)
0.490868 + 0.871234i \(0.336680\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 433723. 1.05543
\(177\) 0 0
\(178\) 19702.1 0.0466083
\(179\) 255550. 0.596133 0.298067 0.954545i \(-0.403658\pi\)
0.298067 + 0.954545i \(0.403658\pi\)
\(180\) 0 0
\(181\) −425741. −0.965937 −0.482969 0.875638i \(-0.660441\pi\)
−0.482969 + 0.875638i \(0.660441\pi\)
\(182\) 13415.7 0.0300218
\(183\) 0 0
\(184\) 46725.8 0.101745
\(185\) 0 0
\(186\) 0 0
\(187\) 403473. 0.843745
\(188\) 142366. 0.293772
\(189\) 0 0
\(190\) 0 0
\(191\) −380425. −0.754546 −0.377273 0.926102i \(-0.623138\pi\)
−0.377273 + 0.926102i \(0.623138\pi\)
\(192\) 0 0
\(193\) −226896. −0.438463 −0.219231 0.975673i \(-0.570355\pi\)
−0.219231 + 0.975673i \(0.570355\pi\)
\(194\) 25247.5 0.0481631
\(195\) 0 0
\(196\) 425020. 0.790258
\(197\) 68875.2 0.126444 0.0632219 0.997999i \(-0.479862\pi\)
0.0632219 + 0.997999i \(0.479862\pi\)
\(198\) 0 0
\(199\) −597116. −1.06887 −0.534437 0.845209i \(-0.679476\pi\)
−0.534437 + 0.845209i \(0.679476\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 31660.6 0.0545935
\(203\) −136147. −0.231883
\(204\) 0 0
\(205\) 0 0
\(206\) 25358.6 0.0416349
\(207\) 0 0
\(208\) −1.06457e6 −1.70615
\(209\) −139703. −0.221228
\(210\) 0 0
\(211\) 170817. 0.264134 0.132067 0.991241i \(-0.457839\pi\)
0.132067 + 0.991241i \(0.457839\pi\)
\(212\) 657546. 1.00482
\(213\) 0 0
\(214\) −40374.1 −0.0602655
\(215\) 0 0
\(216\) 0 0
\(217\) −304609. −0.439131
\(218\) 28781.8 0.0410182
\(219\) 0 0
\(220\) 0 0
\(221\) −990323. −1.36394
\(222\) 0 0
\(223\) −233029. −0.313796 −0.156898 0.987615i \(-0.550149\pi\)
−0.156898 + 0.987615i \(0.550149\pi\)
\(224\) −39390.2 −0.0524527
\(225\) 0 0
\(226\) −6424.58 −0.00836708
\(227\) −174347. −0.224569 −0.112284 0.993676i \(-0.535817\pi\)
−0.112284 + 0.993676i \(0.535817\pi\)
\(228\) 0 0
\(229\) 322529. 0.406424 0.203212 0.979135i \(-0.434862\pi\)
0.203212 + 0.979135i \(0.434862\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −31910.3 −0.0389234
\(233\) 908890. 1.09679 0.548393 0.836221i \(-0.315240\pi\)
0.548393 + 0.836221i \(0.315240\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.42064e6 1.66037
\(237\) 0 0
\(238\) −12184.3 −0.0139431
\(239\) −569964. −0.645436 −0.322718 0.946495i \(-0.604596\pi\)
−0.322718 + 0.946495i \(0.604596\pi\)
\(240\) 0 0
\(241\) −752854. −0.834965 −0.417482 0.908685i \(-0.637087\pi\)
−0.417482 + 0.908685i \(0.637087\pi\)
\(242\) −4327.64 −0.00475021
\(243\) 0 0
\(244\) 4044.40 0.00434891
\(245\) 0 0
\(246\) 0 0
\(247\) 342900. 0.357623
\(248\) −71394.7 −0.0737117
\(249\) 0 0
\(250\) 0 0
\(251\) −616625. −0.617784 −0.308892 0.951097i \(-0.599958\pi\)
−0.308892 + 0.951097i \(0.599958\pi\)
\(252\) 0 0
\(253\) 1.43248e6 1.40698
\(254\) −41080.4 −0.0399531
\(255\) 0 0
\(256\) 1.03318e6 0.985318
\(257\) −1.34559e6 −1.27080 −0.635402 0.772181i \(-0.719166\pi\)
−0.635402 + 0.772181i \(0.719166\pi\)
\(258\) 0 0
\(259\) −545418. −0.505219
\(260\) 0 0
\(261\) 0 0
\(262\) −23296.7 −0.0209672
\(263\) −2.06966e6 −1.84505 −0.922527 0.385932i \(-0.873880\pi\)
−0.922527 + 0.385932i \(0.873880\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4218.83 0.00365585
\(267\) 0 0
\(268\) 604469. 0.514087
\(269\) −121636. −0.102490 −0.0512448 0.998686i \(-0.516319\pi\)
−0.0512448 + 0.998686i \(0.516319\pi\)
\(270\) 0 0
\(271\) −1.32515e6 −1.09608 −0.548040 0.836452i \(-0.684626\pi\)
−0.548040 + 0.836452i \(0.684626\pi\)
\(272\) 966855. 0.792390
\(273\) 0 0
\(274\) 37725.7 0.0303572
\(275\) 0 0
\(276\) 0 0
\(277\) −790419. −0.618953 −0.309477 0.950907i \(-0.600154\pi\)
−0.309477 + 0.950907i \(0.600154\pi\)
\(278\) 6582.20 0.00510810
\(279\) 0 0
\(280\) 0 0
\(281\) −52617.8 −0.0397527 −0.0198763 0.999802i \(-0.506327\pi\)
−0.0198763 + 0.999802i \(0.506327\pi\)
\(282\) 0 0
\(283\) 1.07971e6 0.801386 0.400693 0.916212i \(-0.368769\pi\)
0.400693 + 0.916212i \(0.368769\pi\)
\(284\) −209964. −0.154472
\(285\) 0 0
\(286\) 96398.2 0.0696874
\(287\) −898193. −0.643672
\(288\) 0 0
\(289\) −520435. −0.366540
\(290\) 0 0
\(291\) 0 0
\(292\) −174586. −0.119826
\(293\) 1.10695e6 0.753286 0.376643 0.926359i \(-0.377078\pi\)
0.376643 + 0.926359i \(0.377078\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −127836. −0.0848052
\(297\) 0 0
\(298\) −64251.7 −0.0419126
\(299\) −3.51601e6 −2.27443
\(300\) 0 0
\(301\) 599161. 0.381178
\(302\) −52200.7 −0.0329351
\(303\) 0 0
\(304\) −334774. −0.207763
\(305\) 0 0
\(306\) 0 0
\(307\) 2.18684e6 1.32425 0.662126 0.749393i \(-0.269654\pi\)
0.662126 + 0.749393i \(0.269654\pi\)
\(308\) −804864. −0.483443
\(309\) 0 0
\(310\) 0 0
\(311\) −1.43710e6 −0.842533 −0.421267 0.906937i \(-0.638414\pi\)
−0.421267 + 0.906937i \(0.638414\pi\)
\(312\) 0 0
\(313\) −1.60637e6 −0.926799 −0.463400 0.886149i \(-0.653371\pi\)
−0.463400 + 0.886149i \(0.653371\pi\)
\(314\) 24064.8 0.0137739
\(315\) 0 0
\(316\) −491788. −0.277052
\(317\) 2.71344e6 1.51660 0.758302 0.651904i \(-0.226029\pi\)
0.758302 + 0.651904i \(0.226029\pi\)
\(318\) 0 0
\(319\) −978279. −0.538253
\(320\) 0 0
\(321\) 0 0
\(322\) −43258.9 −0.0232507
\(323\) −311425. −0.166092
\(324\) 0 0
\(325\) 0 0
\(326\) −45890.9 −0.0239157
\(327\) 0 0
\(328\) −210519. −0.108046
\(329\) −263799. −0.134364
\(330\) 0 0
\(331\) 380684. 0.190983 0.0954914 0.995430i \(-0.469558\pi\)
0.0954914 + 0.995430i \(0.469558\pi\)
\(332\) −3.17045e6 −1.57861
\(333\) 0 0
\(334\) 16118.4 0.00790596
\(335\) 0 0
\(336\) 0 0
\(337\) −341423. −0.163764 −0.0818820 0.996642i \(-0.526093\pi\)
−0.0818820 + 0.996642i \(0.526093\pi\)
\(338\) −156042. −0.0742932
\(339\) 0 0
\(340\) 0 0
\(341\) −2.18876e6 −1.01932
\(342\) 0 0
\(343\) −1.78265e6 −0.818148
\(344\) 140432. 0.0639838
\(345\) 0 0
\(346\) −83859.3 −0.0376583
\(347\) 1.59293e6 0.710187 0.355094 0.934831i \(-0.384449\pi\)
0.355094 + 0.934831i \(0.384449\pi\)
\(348\) 0 0
\(349\) −2.83351e6 −1.24526 −0.622632 0.782515i \(-0.713937\pi\)
−0.622632 + 0.782515i \(0.713937\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −283037. −0.121755
\(353\) 921269. 0.393505 0.196752 0.980453i \(-0.436961\pi\)
0.196752 + 0.980453i \(0.436961\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.90123e6 1.21327
\(357\) 0 0
\(358\) −55451.9 −0.0228670
\(359\) −1.08583e6 −0.444658 −0.222329 0.974972i \(-0.571366\pi\)
−0.222329 + 0.974972i \(0.571366\pi\)
\(360\) 0 0
\(361\) −2.36827e6 −0.956451
\(362\) 92381.7 0.0370523
\(363\) 0 0
\(364\) 1.97553e6 0.781503
\(365\) 0 0
\(366\) 0 0
\(367\) −1.80160e6 −0.698223 −0.349112 0.937081i \(-0.613517\pi\)
−0.349112 + 0.937081i \(0.613517\pi\)
\(368\) 3.43269e6 1.32134
\(369\) 0 0
\(370\) 0 0
\(371\) −1.21841e6 −0.459578
\(372\) 0 0
\(373\) −3.85933e6 −1.43628 −0.718140 0.695898i \(-0.755006\pi\)
−0.718140 + 0.695898i \(0.755006\pi\)
\(374\) −87549.9 −0.0323651
\(375\) 0 0
\(376\) −61829.5 −0.0225541
\(377\) 2.40118e6 0.870104
\(378\) 0 0
\(379\) 2.44044e6 0.872711 0.436356 0.899774i \(-0.356269\pi\)
0.436356 + 0.899774i \(0.356269\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 82548.6 0.0289435
\(383\) 1.66772e6 0.580934 0.290467 0.956885i \(-0.406189\pi\)
0.290467 + 0.956885i \(0.406189\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 49234.2 0.0168189
\(387\) 0 0
\(388\) 3.71782e6 1.25374
\(389\) 2.33427e6 0.782125 0.391063 0.920364i \(-0.372108\pi\)
0.391063 + 0.920364i \(0.372108\pi\)
\(390\) 0 0
\(391\) 3.19328e6 1.05632
\(392\) −184586. −0.0606715
\(393\) 0 0
\(394\) −14945.3 −0.00485024
\(395\) 0 0
\(396\) 0 0
\(397\) −3.31607e6 −1.05596 −0.527981 0.849256i \(-0.677051\pi\)
−0.527981 + 0.849256i \(0.677051\pi\)
\(398\) 129569. 0.0410008
\(399\) 0 0
\(400\) 0 0
\(401\) 1.84120e6 0.571795 0.285898 0.958260i \(-0.407708\pi\)
0.285898 + 0.958260i \(0.407708\pi\)
\(402\) 0 0
\(403\) 5.37229e6 1.64777
\(404\) 4.66218e6 1.42114
\(405\) 0 0
\(406\) 29542.6 0.00889476
\(407\) −3.91908e6 −1.17273
\(408\) 0 0
\(409\) 3.83672e6 1.13410 0.567050 0.823683i \(-0.308085\pi\)
0.567050 + 0.823683i \(0.308085\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.73418e6 1.08381
\(413\) −2.63241e6 −0.759413
\(414\) 0 0
\(415\) 0 0
\(416\) 694712. 0.196821
\(417\) 0 0
\(418\) 30314.2 0.00848605
\(419\) 4.37456e6 1.21731 0.608653 0.793437i \(-0.291710\pi\)
0.608653 + 0.793437i \(0.291710\pi\)
\(420\) 0 0
\(421\) 76116.4 0.0209302 0.0104651 0.999945i \(-0.496669\pi\)
0.0104651 + 0.999945i \(0.496669\pi\)
\(422\) −37065.7 −0.0101319
\(423\) 0 0
\(424\) −285573. −0.0771440
\(425\) 0 0
\(426\) 0 0
\(427\) −7494.16 −0.00198908
\(428\) −5.94529e6 −1.56879
\(429\) 0 0
\(430\) 0 0
\(431\) 4.87872e6 1.26506 0.632532 0.774534i \(-0.282016\pi\)
0.632532 + 0.774534i \(0.282016\pi\)
\(432\) 0 0
\(433\) −4.13136e6 −1.05894 −0.529472 0.848327i \(-0.677610\pi\)
−0.529472 + 0.848327i \(0.677610\pi\)
\(434\) 66097.4 0.0168446
\(435\) 0 0
\(436\) 4.23825e6 1.06775
\(437\) −1.10568e6 −0.276964
\(438\) 0 0
\(439\) −492071. −0.121862 −0.0609308 0.998142i \(-0.519407\pi\)
−0.0609308 + 0.998142i \(0.519407\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 214891. 0.0523193
\(443\) −1.70371e6 −0.412464 −0.206232 0.978503i \(-0.566120\pi\)
−0.206232 + 0.978503i \(0.566120\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 50565.1 0.0120369
\(447\) 0 0
\(448\) −1.92301e6 −0.452676
\(449\) −1.25545e6 −0.293888 −0.146944 0.989145i \(-0.546944\pi\)
−0.146944 + 0.989145i \(0.546944\pi\)
\(450\) 0 0
\(451\) −6.45393e6 −1.49411
\(452\) −946051. −0.217805
\(453\) 0 0
\(454\) 37831.6 0.00861421
\(455\) 0 0
\(456\) 0 0
\(457\) 4.71886e6 1.05693 0.528466 0.848955i \(-0.322767\pi\)
0.528466 + 0.848955i \(0.322767\pi\)
\(458\) −69985.7 −0.0155900
\(459\) 0 0
\(460\) 0 0
\(461\) 1.00772e6 0.220844 0.110422 0.993885i \(-0.464780\pi\)
0.110422 + 0.993885i \(0.464780\pi\)
\(462\) 0 0
\(463\) 7.52400e6 1.63116 0.815580 0.578645i \(-0.196418\pi\)
0.815580 + 0.578645i \(0.196418\pi\)
\(464\) −2.34428e6 −0.505492
\(465\) 0 0
\(466\) −197221. −0.0420714
\(467\) 4.77568e6 1.01331 0.506656 0.862148i \(-0.330881\pi\)
0.506656 + 0.862148i \(0.330881\pi\)
\(468\) 0 0
\(469\) −1.12006e6 −0.235131
\(470\) 0 0
\(471\) 0 0
\(472\) −616986. −0.127474
\(473\) 4.30525e6 0.884800
\(474\) 0 0
\(475\) 0 0
\(476\) −1.79420e6 −0.362956
\(477\) 0 0
\(478\) 123677. 0.0247582
\(479\) 8.09043e6 1.61114 0.805569 0.592501i \(-0.201860\pi\)
0.805569 + 0.592501i \(0.201860\pi\)
\(480\) 0 0
\(481\) 9.61935e6 1.89576
\(482\) 163362. 0.0320283
\(483\) 0 0
\(484\) −637266. −0.123654
\(485\) 0 0
\(486\) 0 0
\(487\) −605970. −0.115779 −0.0578894 0.998323i \(-0.518437\pi\)
−0.0578894 + 0.998323i \(0.518437\pi\)
\(488\) −1756.49 −0.000333884 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.91166e6 −1.29383 −0.646917 0.762560i \(-0.723942\pi\)
−0.646917 + 0.762560i \(0.723942\pi\)
\(492\) 0 0
\(493\) −2.18078e6 −0.404105
\(494\) −74406.0 −0.0137180
\(495\) 0 0
\(496\) −5.24498e6 −0.957282
\(497\) 389057. 0.0706517
\(498\) 0 0
\(499\) 4.20448e6 0.755895 0.377947 0.925827i \(-0.376630\pi\)
0.377947 + 0.925827i \(0.376630\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 133802. 0.0236975
\(503\) −7.22627e6 −1.27349 −0.636743 0.771076i \(-0.719719\pi\)
−0.636743 + 0.771076i \(0.719719\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −310835. −0.0539701
\(507\) 0 0
\(508\) −6.04929e6 −1.04003
\(509\) 2.81947e6 0.482362 0.241181 0.970480i \(-0.422465\pi\)
0.241181 + 0.970480i \(0.422465\pi\)
\(510\) 0 0
\(511\) 323503. 0.0548057
\(512\) −1.13097e6 −0.190667
\(513\) 0 0
\(514\) 291980. 0.0487466
\(515\) 0 0
\(516\) 0 0
\(517\) −1.89552e6 −0.311890
\(518\) 118351. 0.0193796
\(519\) 0 0
\(520\) 0 0
\(521\) 4.64191e6 0.749208 0.374604 0.927185i \(-0.377779\pi\)
0.374604 + 0.927185i \(0.377779\pi\)
\(522\) 0 0
\(523\) 1.98997e6 0.318121 0.159061 0.987269i \(-0.449154\pi\)
0.159061 + 0.987269i \(0.449154\pi\)
\(524\) −3.43055e6 −0.545803
\(525\) 0 0
\(526\) 449096. 0.0707742
\(527\) −4.87917e6 −0.765279
\(528\) 0 0
\(529\) 4.90099e6 0.761455
\(530\) 0 0
\(531\) 0 0
\(532\) 621243. 0.0951661
\(533\) 1.58411e7 2.41528
\(534\) 0 0
\(535\) 0 0
\(536\) −262521. −0.0394687
\(537\) 0 0
\(538\) 26393.8 0.00393139
\(539\) −5.65890e6 −0.838996
\(540\) 0 0
\(541\) −1.06167e7 −1.55954 −0.779768 0.626069i \(-0.784663\pi\)
−0.779768 + 0.626069i \(0.784663\pi\)
\(542\) 287545. 0.0420444
\(543\) 0 0
\(544\) −630945. −0.0914101
\(545\) 0 0
\(546\) 0 0
\(547\) −1.25513e7 −1.79358 −0.896789 0.442458i \(-0.854107\pi\)
−0.896789 + 0.442458i \(0.854107\pi\)
\(548\) 5.55530e6 0.790235
\(549\) 0 0
\(550\) 0 0
\(551\) 755096. 0.105955
\(552\) 0 0
\(553\) 911269. 0.126717
\(554\) 171513. 0.0237423
\(555\) 0 0
\(556\) 969261. 0.132970
\(557\) 3.98643e6 0.544435 0.272218 0.962236i \(-0.412243\pi\)
0.272218 + 0.962236i \(0.412243\pi\)
\(558\) 0 0
\(559\) −1.05672e7 −1.43031
\(560\) 0 0
\(561\) 0 0
\(562\) 11417.6 0.00152487
\(563\) 453419. 0.0602877 0.0301438 0.999546i \(-0.490403\pi\)
0.0301438 + 0.999546i \(0.490403\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −234287. −0.0307403
\(567\) 0 0
\(568\) 91187.6 0.0118595
\(569\) 3.96567e6 0.513494 0.256747 0.966479i \(-0.417349\pi\)
0.256747 + 0.966479i \(0.417349\pi\)
\(570\) 0 0
\(571\) 5.14150e6 0.659932 0.329966 0.943993i \(-0.392963\pi\)
0.329966 + 0.943993i \(0.392963\pi\)
\(572\) 1.41951e7 1.81405
\(573\) 0 0
\(574\) 194899. 0.0246905
\(575\) 0 0
\(576\) 0 0
\(577\) 8.48527e6 1.06103 0.530513 0.847677i \(-0.321999\pi\)
0.530513 + 0.847677i \(0.321999\pi\)
\(578\) 112929. 0.0140601
\(579\) 0 0
\(580\) 0 0
\(581\) 5.87474e6 0.722019
\(582\) 0 0
\(583\) −8.75485e6 −1.06679
\(584\) 75822.9 0.00919959
\(585\) 0 0
\(586\) −240198. −0.0288952
\(587\) −1.39429e7 −1.67016 −0.835079 0.550130i \(-0.814578\pi\)
−0.835079 + 0.550130i \(0.814578\pi\)
\(588\) 0 0
\(589\) 1.68942e6 0.200654
\(590\) 0 0
\(591\) 0 0
\(592\) −9.39140e6 −1.10135
\(593\) −1.32665e7 −1.54925 −0.774623 0.632423i \(-0.782060\pi\)
−0.774623 + 0.632423i \(0.782060\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.46138e6 −1.09104
\(597\) 0 0
\(598\) 762942. 0.0872445
\(599\) −1.62821e7 −1.85415 −0.927073 0.374880i \(-0.877684\pi\)
−0.927073 + 0.374880i \(0.877684\pi\)
\(600\) 0 0
\(601\) 1.32895e7 1.50080 0.750398 0.660987i \(-0.229862\pi\)
0.750398 + 0.660987i \(0.229862\pi\)
\(602\) −130012. −0.0146215
\(603\) 0 0
\(604\) −7.68681e6 −0.857342
\(605\) 0 0
\(606\) 0 0
\(607\) 1.57179e7 1.73151 0.865753 0.500472i \(-0.166840\pi\)
0.865753 + 0.500472i \(0.166840\pi\)
\(608\) 218465. 0.0239675
\(609\) 0 0
\(610\) 0 0
\(611\) 4.65254e6 0.504181
\(612\) 0 0
\(613\) 910487. 0.0978639 0.0489320 0.998802i \(-0.484418\pi\)
0.0489320 + 0.998802i \(0.484418\pi\)
\(614\) −474523. −0.0507968
\(615\) 0 0
\(616\) 349553. 0.0371160
\(617\) −4.73246e6 −0.500466 −0.250233 0.968186i \(-0.580507\pi\)
−0.250233 + 0.968186i \(0.580507\pi\)
\(618\) 0 0
\(619\) 6.54970e6 0.687060 0.343530 0.939142i \(-0.388377\pi\)
0.343530 + 0.939142i \(0.388377\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 311838. 0.0323186
\(623\) −5.37590e6 −0.554921
\(624\) 0 0
\(625\) 0 0
\(626\) 348568. 0.0355510
\(627\) 0 0
\(628\) 3.54366e6 0.358553
\(629\) −8.73640e6 −0.880452
\(630\) 0 0
\(631\) −1.89272e7 −1.89240 −0.946199 0.323585i \(-0.895112\pi\)
−0.946199 + 0.323585i \(0.895112\pi\)
\(632\) 213584. 0.0212704
\(633\) 0 0
\(634\) −588791. −0.0581752
\(635\) 0 0
\(636\) 0 0
\(637\) 1.38897e7 1.35627
\(638\) 212277. 0.0206468
\(639\) 0 0
\(640\) 0 0
\(641\) −5.61144e6 −0.539422 −0.269711 0.962941i \(-0.586928\pi\)
−0.269711 + 0.962941i \(0.586928\pi\)
\(642\) 0 0
\(643\) 4.43672e6 0.423190 0.211595 0.977357i \(-0.432134\pi\)
0.211595 + 0.977357i \(0.432134\pi\)
\(644\) −6.37008e6 −0.605243
\(645\) 0 0
\(646\) 67576.4 0.00637109
\(647\) 8.38503e6 0.787488 0.393744 0.919220i \(-0.371180\pi\)
0.393744 + 0.919220i \(0.371180\pi\)
\(648\) 0 0
\(649\) −1.89150e7 −1.76277
\(650\) 0 0
\(651\) 0 0
\(652\) −6.75766e6 −0.622555
\(653\) −1.33396e7 −1.22422 −0.612112 0.790771i \(-0.709680\pi\)
−0.612112 + 0.790771i \(0.709680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.54657e7 −1.40317
\(657\) 0 0
\(658\) 57242.0 0.00515406
\(659\) −1.16435e7 −1.04441 −0.522205 0.852820i \(-0.674890\pi\)
−0.522205 + 0.852820i \(0.674890\pi\)
\(660\) 0 0
\(661\) 1.52598e7 1.35845 0.679227 0.733928i \(-0.262315\pi\)
0.679227 + 0.733928i \(0.262315\pi\)
\(662\) −82604.7 −0.00732588
\(663\) 0 0
\(664\) 1.37693e6 0.121197
\(665\) 0 0
\(666\) 0 0
\(667\) −7.74257e6 −0.673861
\(668\) 2.37351e6 0.205802
\(669\) 0 0
\(670\) 0 0
\(671\) −53849.0 −0.00461712
\(672\) 0 0
\(673\) 8.38345e6 0.713485 0.356743 0.934203i \(-0.383887\pi\)
0.356743 + 0.934203i \(0.383887\pi\)
\(674\) 74085.6 0.00628180
\(675\) 0 0
\(676\) −2.29779e7 −1.93394
\(677\) 4.32488e6 0.362662 0.181331 0.983422i \(-0.441959\pi\)
0.181331 + 0.983422i \(0.441959\pi\)
\(678\) 0 0
\(679\) −6.88901e6 −0.573432
\(680\) 0 0
\(681\) 0 0
\(682\) 474940. 0.0391001
\(683\) 1.61800e7 1.32717 0.663585 0.748101i \(-0.269034\pi\)
0.663585 + 0.748101i \(0.269034\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 386819. 0.0313832
\(687\) 0 0
\(688\) 1.03168e7 0.830947
\(689\) 2.14887e7 1.72450
\(690\) 0 0
\(691\) 1.21864e6 0.0970913 0.0485456 0.998821i \(-0.484541\pi\)
0.0485456 + 0.998821i \(0.484541\pi\)
\(692\) −1.23487e7 −0.980292
\(693\) 0 0
\(694\) −345651. −0.0272420
\(695\) 0 0
\(696\) 0 0
\(697\) −1.43871e7 −1.12174
\(698\) 614845. 0.0477669
\(699\) 0 0
\(700\) 0 0
\(701\) 883759. 0.0679264 0.0339632 0.999423i \(-0.489187\pi\)
0.0339632 + 0.999423i \(0.489187\pi\)
\(702\) 0 0
\(703\) 3.02498e6 0.230852
\(704\) −1.38177e7 −1.05076
\(705\) 0 0
\(706\) −199907. −0.0150944
\(707\) −8.63888e6 −0.649993
\(708\) 0 0
\(709\) 1.76739e7 1.32043 0.660216 0.751076i \(-0.270465\pi\)
0.660216 + 0.751076i \(0.270465\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.26001e6 −0.0931480
\(713\) −1.73229e7 −1.27613
\(714\) 0 0
\(715\) 0 0
\(716\) −8.16557e6 −0.595256
\(717\) 0 0
\(718\) 235615. 0.0170566
\(719\) −2.08338e6 −0.150296 −0.0751480 0.997172i \(-0.523943\pi\)
−0.0751480 + 0.997172i \(0.523943\pi\)
\(720\) 0 0
\(721\) −6.91931e6 −0.495707
\(722\) 513892. 0.0366884
\(723\) 0 0
\(724\) 1.36037e7 0.964516
\(725\) 0 0
\(726\) 0 0
\(727\) 1.02846e7 0.721689 0.360845 0.932626i \(-0.382488\pi\)
0.360845 + 0.932626i \(0.382488\pi\)
\(728\) −857976. −0.0599993
\(729\) 0 0
\(730\) 0 0
\(731\) 9.59725e6 0.664283
\(732\) 0 0
\(733\) −1.72749e7 −1.18756 −0.593782 0.804626i \(-0.702366\pi\)
−0.593782 + 0.804626i \(0.702366\pi\)
\(734\) 390931. 0.0267831
\(735\) 0 0
\(736\) −2.24009e6 −0.152430
\(737\) −8.04816e6 −0.545793
\(738\) 0 0
\(739\) −6.26221e6 −0.421810 −0.210905 0.977507i \(-0.567641\pi\)
−0.210905 + 0.977507i \(0.567641\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 264384. 0.0176289
\(743\) 2.49470e7 1.65786 0.828928 0.559355i \(-0.188951\pi\)
0.828928 + 0.559355i \(0.188951\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 837437. 0.0550941
\(747\) 0 0
\(748\) −1.28922e7 −0.842503
\(749\) 1.10164e7 0.717524
\(750\) 0 0
\(751\) 2.89546e7 1.87334 0.936671 0.350211i \(-0.113890\pi\)
0.936671 + 0.350211i \(0.113890\pi\)
\(752\) −4.54229e6 −0.292907
\(753\) 0 0
\(754\) −521033. −0.0333762
\(755\) 0 0
\(756\) 0 0
\(757\) −8.35091e6 −0.529656 −0.264828 0.964296i \(-0.585315\pi\)
−0.264828 + 0.964296i \(0.585315\pi\)
\(758\) −529553. −0.0334762
\(759\) 0 0
\(760\) 0 0
\(761\) 2.05602e7 1.28696 0.643481 0.765462i \(-0.277489\pi\)
0.643481 + 0.765462i \(0.277489\pi\)
\(762\) 0 0
\(763\) −7.85336e6 −0.488364
\(764\) 1.21557e7 0.753436
\(765\) 0 0
\(766\) −361880. −0.0222840
\(767\) 4.64268e7 2.84958
\(768\) 0 0
\(769\) 6.86907e6 0.418873 0.209436 0.977822i \(-0.432837\pi\)
0.209436 + 0.977822i \(0.432837\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.24998e6 0.437818
\(773\) 1.55261e7 0.934574 0.467287 0.884106i \(-0.345232\pi\)
0.467287 + 0.884106i \(0.345232\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.61465e6 −0.0962553
\(777\) 0 0
\(778\) −506514. −0.0300015
\(779\) 4.98153e6 0.294116
\(780\) 0 0
\(781\) 2.79555e6 0.163999
\(782\) −692912. −0.0405192
\(783\) 0 0
\(784\) −1.35606e7 −0.787931
\(785\) 0 0
\(786\) 0 0
\(787\) −5.54797e6 −0.319299 −0.159649 0.987174i \(-0.551036\pi\)
−0.159649 + 0.987174i \(0.551036\pi\)
\(788\) −2.20076e6 −0.126258
\(789\) 0 0
\(790\) 0 0
\(791\) 1.75300e6 0.0996188
\(792\) 0 0
\(793\) 132172. 0.00746373
\(794\) 719557. 0.0405055
\(795\) 0 0
\(796\) 1.90796e7 1.06730
\(797\) 3.72345e6 0.207634 0.103817 0.994596i \(-0.466894\pi\)
0.103817 + 0.994596i \(0.466894\pi\)
\(798\) 0 0
\(799\) −4.22549e6 −0.234158
\(800\) 0 0
\(801\) 0 0
\(802\) −399523. −0.0219334
\(803\) 2.32452e6 0.127217
\(804\) 0 0
\(805\) 0 0
\(806\) −1.16574e6 −0.0632067
\(807\) 0 0
\(808\) −2.02479e6 −0.109107
\(809\) 3.24496e7 1.74317 0.871583 0.490249i \(-0.163094\pi\)
0.871583 + 0.490249i \(0.163094\pi\)
\(810\) 0 0
\(811\) −1.73709e7 −0.927409 −0.463704 0.885990i \(-0.653480\pi\)
−0.463704 + 0.885990i \(0.653480\pi\)
\(812\) 4.35030e6 0.231541
\(813\) 0 0
\(814\) 850403. 0.0449846
\(815\) 0 0
\(816\) 0 0
\(817\) −3.32305e6 −0.174173
\(818\) −832531. −0.0435028
\(819\) 0 0
\(820\) 0 0
\(821\) 3.70074e7 1.91615 0.958076 0.286513i \(-0.0924962\pi\)
0.958076 + 0.286513i \(0.0924962\pi\)
\(822\) 0 0
\(823\) −2.60977e7 −1.34308 −0.671541 0.740967i \(-0.734367\pi\)
−0.671541 + 0.740967i \(0.734367\pi\)
\(824\) −1.62176e6 −0.0832085
\(825\) 0 0
\(826\) 571207. 0.0291302
\(827\) −1.76833e7 −0.899083 −0.449541 0.893259i \(-0.648413\pi\)
−0.449541 + 0.893259i \(0.648413\pi\)
\(828\) 0 0
\(829\) 2.66610e7 1.34738 0.673689 0.739015i \(-0.264709\pi\)
0.673689 + 0.739015i \(0.264709\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.39155e7 1.69860
\(833\) −1.26148e7 −0.629895
\(834\) 0 0
\(835\) 0 0
\(836\) 4.46392e6 0.220902
\(837\) 0 0
\(838\) −949239. −0.0466945
\(839\) −6.39212e6 −0.313502 −0.156751 0.987638i \(-0.550102\pi\)
−0.156751 + 0.987638i \(0.550102\pi\)
\(840\) 0 0
\(841\) −1.52235e7 −0.742208
\(842\) −16516.5 −0.000802858 0
\(843\) 0 0
\(844\) −5.45810e6 −0.263746
\(845\) 0 0
\(846\) 0 0
\(847\) 1.18084e6 0.0565563
\(848\) −2.09795e7 −1.00186
\(849\) 0 0
\(850\) 0 0
\(851\) −3.10174e7 −1.46819
\(852\) 0 0
\(853\) 3.82139e7 1.79824 0.899122 0.437699i \(-0.144206\pi\)
0.899122 + 0.437699i \(0.144206\pi\)
\(854\) 1626.16 7.62990e−5 0
\(855\) 0 0
\(856\) 2.58204e6 0.120442
\(857\) −3.44766e7 −1.60351 −0.801755 0.597653i \(-0.796100\pi\)
−0.801755 + 0.597653i \(0.796100\pi\)
\(858\) 0 0
\(859\) 1.85135e7 0.856063 0.428032 0.903764i \(-0.359207\pi\)
0.428032 + 0.903764i \(0.359207\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.05864e6 −0.0485264
\(863\) −2.69763e6 −0.123298 −0.0616489 0.998098i \(-0.519636\pi\)
−0.0616489 + 0.998098i \(0.519636\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 896465. 0.0406199
\(867\) 0 0
\(868\) 9.73316e6 0.438485
\(869\) 6.54788e6 0.294138
\(870\) 0 0
\(871\) 1.97541e7 0.882293
\(872\) −1.84068e6 −0.0819760
\(873\) 0 0
\(874\) 239921. 0.0106240
\(875\) 0 0
\(876\) 0 0
\(877\) −2.83239e7 −1.24352 −0.621762 0.783207i \(-0.713583\pi\)
−0.621762 + 0.783207i \(0.713583\pi\)
\(878\) 106775. 0.00467447
\(879\) 0 0
\(880\) 0 0
\(881\) −2.28203e7 −0.990564 −0.495282 0.868732i \(-0.664935\pi\)
−0.495282 + 0.868732i \(0.664935\pi\)
\(882\) 0 0
\(883\) 2.29055e7 0.988638 0.494319 0.869281i \(-0.335417\pi\)
0.494319 + 0.869281i \(0.335417\pi\)
\(884\) 3.16437e7 1.36194
\(885\) 0 0
\(886\) 369689. 0.0158216
\(887\) 1.99086e7 0.849634 0.424817 0.905279i \(-0.360338\pi\)
0.424817 + 0.905279i \(0.360338\pi\)
\(888\) 0 0
\(889\) 1.12091e7 0.475683
\(890\) 0 0
\(891\) 0 0
\(892\) 7.44595e6 0.313334
\(893\) 1.46308e6 0.0613958
\(894\) 0 0
\(895\) 0 0
\(896\) 1.67776e6 0.0698169
\(897\) 0 0
\(898\) 272420. 0.0112732
\(899\) 1.18303e7 0.488197
\(900\) 0 0
\(901\) −1.95163e7 −0.800913
\(902\) 1.40044e6 0.0573124
\(903\) 0 0
\(904\) 410871. 0.0167218
\(905\) 0 0
\(906\) 0 0
\(907\) −3.67681e7 −1.48407 −0.742033 0.670364i \(-0.766138\pi\)
−0.742033 + 0.670364i \(0.766138\pi\)
\(908\) 5.57089e6 0.224238
\(909\) 0 0
\(910\) 0 0
\(911\) −1.22729e7 −0.489948 −0.244974 0.969530i \(-0.578779\pi\)
−0.244974 + 0.969530i \(0.578779\pi\)
\(912\) 0 0
\(913\) 4.22127e7 1.67597
\(914\) −1.02395e6 −0.0405427
\(915\) 0 0
\(916\) −1.03057e7 −0.405826
\(917\) 6.35671e6 0.249637
\(918\) 0 0
\(919\) −1.49398e7 −0.583522 −0.291761 0.956491i \(-0.594241\pi\)
−0.291761 + 0.956491i \(0.594241\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −218665. −0.00847134
\(923\) −6.86167e6 −0.265110
\(924\) 0 0
\(925\) 0 0
\(926\) −1.63264e6 −0.0625694
\(927\) 0 0
\(928\) 1.52982e6 0.0583135
\(929\) −5.25145e6 −0.199637 −0.0998183 0.995006i \(-0.531826\pi\)
−0.0998183 + 0.995006i \(0.531826\pi\)
\(930\) 0 0
\(931\) 4.36788e6 0.165157
\(932\) −2.90417e7 −1.09517
\(933\) 0 0
\(934\) −1.03628e6 −0.0388695
\(935\) 0 0
\(936\) 0 0
\(937\) 3.30065e7 1.22815 0.614075 0.789248i \(-0.289529\pi\)
0.614075 + 0.789248i \(0.289529\pi\)
\(938\) 243043. 0.00901936
\(939\) 0 0
\(940\) 0 0
\(941\) 1.32700e7 0.488538 0.244269 0.969708i \(-0.421452\pi\)
0.244269 + 0.969708i \(0.421452\pi\)
\(942\) 0 0
\(943\) −5.10794e7 −1.87054
\(944\) −4.53267e7 −1.65548
\(945\) 0 0
\(946\) −934198. −0.0339399
\(947\) 7.08995e6 0.256903 0.128451 0.991716i \(-0.458999\pi\)
0.128451 + 0.991716i \(0.458999\pi\)
\(948\) 0 0
\(949\) −5.70551e6 −0.205650
\(950\) 0 0
\(951\) 0 0
\(952\) 779223. 0.0278657
\(953\) −6.20087e6 −0.221167 −0.110584 0.993867i \(-0.535272\pi\)
−0.110584 + 0.993867i \(0.535272\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.82120e7 0.644486
\(957\) 0 0
\(958\) −1.75555e6 −0.0618015
\(959\) −1.02938e7 −0.361434
\(960\) 0 0
\(961\) −2.16066e6 −0.0754708
\(962\) −2.08731e6 −0.0727191
\(963\) 0 0
\(964\) 2.40559e7 0.833736
\(965\) 0 0
\(966\) 0 0
\(967\) 3.19568e7 1.09900 0.549499 0.835494i \(-0.314819\pi\)
0.549499 + 0.835494i \(0.314819\pi\)
\(968\) 276765. 0.00949344
\(969\) 0 0
\(970\) 0 0
\(971\) 3.66750e7 1.24831 0.624154 0.781301i \(-0.285444\pi\)
0.624154 + 0.781301i \(0.285444\pi\)
\(972\) 0 0
\(973\) −1.79601e6 −0.0608173
\(974\) 131490. 0.00444114
\(975\) 0 0
\(976\) −129040. −0.00433610
\(977\) 2.13191e7 0.714549 0.357275 0.933999i \(-0.383706\pi\)
0.357275 + 0.933999i \(0.383706\pi\)
\(978\) 0 0
\(979\) −3.86283e7 −1.28810
\(980\) 0 0
\(981\) 0 0
\(982\) 1.49977e6 0.0496300
\(983\) −1.02603e7 −0.338670 −0.169335 0.985559i \(-0.554162\pi\)
−0.169335 + 0.985559i \(0.554162\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 473208. 0.0155010
\(987\) 0 0
\(988\) −1.09567e7 −0.357096
\(989\) 3.40738e7 1.10772
\(990\) 0 0
\(991\) 2.92940e7 0.947535 0.473767 0.880650i \(-0.342894\pi\)
0.473767 + 0.880650i \(0.342894\pi\)
\(992\) 3.42274e6 0.110432
\(993\) 0 0
\(994\) −84421.7 −0.00271012
\(995\) 0 0
\(996\) 0 0
\(997\) 1.30440e7 0.415596 0.207798 0.978172i \(-0.433370\pi\)
0.207798 + 0.978172i \(0.433370\pi\)
\(998\) −912333. −0.0289953
\(999\) 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 225.6.a.u.1.1 2
3.2 odd 2 75.6.a.f.1.2 2
5.2 odd 4 45.6.b.c.19.2 4
5.3 odd 4 45.6.b.c.19.3 4
5.4 even 2 225.6.a.i.1.2 2
15.2 even 4 15.6.b.a.4.3 yes 4
15.8 even 4 15.6.b.a.4.2 4
15.14 odd 2 75.6.a.j.1.1 2
20.3 even 4 720.6.f.h.289.1 4
20.7 even 4 720.6.f.h.289.4 4
60.23 odd 4 240.6.f.c.49.2 4
60.47 odd 4 240.6.f.c.49.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.6.b.a.4.2 4 15.8 even 4
15.6.b.a.4.3 yes 4 15.2 even 4
45.6.b.c.19.2 4 5.2 odd 4
45.6.b.c.19.3 4 5.3 odd 4
75.6.a.f.1.2 2 3.2 odd 2
75.6.a.j.1.1 2 15.14 odd 2
225.6.a.i.1.2 2 5.4 even 2
225.6.a.u.1.1 2 1.1 even 1 trivial
240.6.f.c.49.2 4 60.23 odd 4
240.6.f.c.49.3 4 60.47 odd 4
720.6.f.h.289.1 4 20.3 even 4
720.6.f.h.289.4 4 20.7 even 4