Properties

Label 1336.2.a.d.1.6
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} - 127 x^{3} - 652 x^{2} - 48 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.295455\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.295455 q^{3} -3.76247 q^{5} +1.37107 q^{7} -2.91271 q^{9} +O(q^{10})\) \(q-0.295455 q^{3} -3.76247 q^{5} +1.37107 q^{7} -2.91271 q^{9} +4.17758 q^{11} -2.50320 q^{13} +1.11164 q^{15} -5.93630 q^{17} +0.213920 q^{19} -0.405091 q^{21} +6.49944 q^{23} +9.15616 q^{25} +1.74694 q^{27} +6.78130 q^{29} -6.95934 q^{31} -1.23429 q^{33} -5.15862 q^{35} +6.45079 q^{37} +0.739585 q^{39} +3.06028 q^{41} -3.86093 q^{43} +10.9590 q^{45} +4.00956 q^{47} -5.12016 q^{49} +1.75391 q^{51} +12.1008 q^{53} -15.7180 q^{55} -0.0632038 q^{57} +13.3920 q^{59} +1.28385 q^{61} -3.99353 q^{63} +9.41822 q^{65} -2.37090 q^{67} -1.92029 q^{69} +8.31287 q^{71} +2.88459 q^{73} -2.70524 q^{75} +5.72776 q^{77} +13.8904 q^{79} +8.22198 q^{81} +8.20305 q^{83} +22.3351 q^{85} -2.00357 q^{87} -1.96072 q^{89} -3.43207 q^{91} +2.05617 q^{93} -0.804867 q^{95} -14.6668 q^{97} -12.1681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 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
\(3\) −0.295455 −0.170581 −0.0852906 0.996356i \(-0.527182\pi\)
−0.0852906 + 0.996356i \(0.527182\pi\)
\(4\) 0 0
\(5\) −3.76247 −1.68263 −0.841313 0.540548i \(-0.818217\pi\)
−0.841313 + 0.540548i \(0.818217\pi\)
\(6\) 0 0
\(7\) 1.37107 0.518217 0.259108 0.965848i \(-0.416571\pi\)
0.259108 + 0.965848i \(0.416571\pi\)
\(8\) 0 0
\(9\) −2.91271 −0.970902
\(10\) 0 0
\(11\) 4.17758 1.25959 0.629794 0.776762i \(-0.283139\pi\)
0.629794 + 0.776762i \(0.283139\pi\)
\(12\) 0 0
\(13\) −2.50320 −0.694264 −0.347132 0.937816i \(-0.612844\pi\)
−0.347132 + 0.937816i \(0.612844\pi\)
\(14\) 0 0
\(15\) 1.11164 0.287024
\(16\) 0 0
\(17\) −5.93630 −1.43976 −0.719882 0.694097i \(-0.755804\pi\)
−0.719882 + 0.694097i \(0.755804\pi\)
\(18\) 0 0
\(19\) 0.213920 0.0490766 0.0245383 0.999699i \(-0.492188\pi\)
0.0245383 + 0.999699i \(0.492188\pi\)
\(20\) 0 0
\(21\) −0.405091 −0.0883981
\(22\) 0 0
\(23\) 6.49944 1.35523 0.677613 0.735418i \(-0.263014\pi\)
0.677613 + 0.735418i \(0.263014\pi\)
\(24\) 0 0
\(25\) 9.15616 1.83123
\(26\) 0 0
\(27\) 1.74694 0.336199
\(28\) 0 0
\(29\) 6.78130 1.25926 0.629628 0.776897i \(-0.283207\pi\)
0.629628 + 0.776897i \(0.283207\pi\)
\(30\) 0 0
\(31\) −6.95934 −1.24993 −0.624967 0.780651i \(-0.714888\pi\)
−0.624967 + 0.780651i \(0.714888\pi\)
\(32\) 0 0
\(33\) −1.23429 −0.214862
\(34\) 0 0
\(35\) −5.15862 −0.871965
\(36\) 0 0
\(37\) 6.45079 1.06050 0.530251 0.847840i \(-0.322098\pi\)
0.530251 + 0.847840i \(0.322098\pi\)
\(38\) 0 0
\(39\) 0.739585 0.118428
\(40\) 0 0
\(41\) 3.06028 0.477936 0.238968 0.971027i \(-0.423191\pi\)
0.238968 + 0.971027i \(0.423191\pi\)
\(42\) 0 0
\(43\) −3.86093 −0.588787 −0.294393 0.955684i \(-0.595118\pi\)
−0.294393 + 0.955684i \(0.595118\pi\)
\(44\) 0 0
\(45\) 10.9590 1.63367
\(46\) 0 0
\(47\) 4.00956 0.584854 0.292427 0.956288i \(-0.405537\pi\)
0.292427 + 0.956288i \(0.405537\pi\)
\(48\) 0 0
\(49\) −5.12016 −0.731451
\(50\) 0 0
\(51\) 1.75391 0.245597
\(52\) 0 0
\(53\) 12.1008 1.66217 0.831086 0.556143i \(-0.187719\pi\)
0.831086 + 0.556143i \(0.187719\pi\)
\(54\) 0 0
\(55\) −15.7180 −2.11941
\(56\) 0 0
\(57\) −0.0632038 −0.00837155
\(58\) 0 0
\(59\) 13.3920 1.74349 0.871744 0.489961i \(-0.162989\pi\)
0.871744 + 0.489961i \(0.162989\pi\)
\(60\) 0 0
\(61\) 1.28385 0.164380 0.0821902 0.996617i \(-0.473809\pi\)
0.0821902 + 0.996617i \(0.473809\pi\)
\(62\) 0 0
\(63\) −3.99353 −0.503138
\(64\) 0 0
\(65\) 9.41822 1.16819
\(66\) 0 0
\(67\) −2.37090 −0.289652 −0.144826 0.989457i \(-0.546262\pi\)
−0.144826 + 0.989457i \(0.546262\pi\)
\(68\) 0 0
\(69\) −1.92029 −0.231176
\(70\) 0 0
\(71\) 8.31287 0.986557 0.493278 0.869872i \(-0.335798\pi\)
0.493278 + 0.869872i \(0.335798\pi\)
\(72\) 0 0
\(73\) 2.88459 0.337616 0.168808 0.985649i \(-0.446008\pi\)
0.168808 + 0.985649i \(0.446008\pi\)
\(74\) 0 0
\(75\) −2.70524 −0.312374
\(76\) 0 0
\(77\) 5.72776 0.652739
\(78\) 0 0
\(79\) 13.8904 1.56279 0.781397 0.624034i \(-0.214507\pi\)
0.781397 + 0.624034i \(0.214507\pi\)
\(80\) 0 0
\(81\) 8.22198 0.913553
\(82\) 0 0
\(83\) 8.20305 0.900401 0.450201 0.892927i \(-0.351352\pi\)
0.450201 + 0.892927i \(0.351352\pi\)
\(84\) 0 0
\(85\) 22.3351 2.42258
\(86\) 0 0
\(87\) −2.00357 −0.214805
\(88\) 0 0
\(89\) −1.96072 −0.207836 −0.103918 0.994586i \(-0.533138\pi\)
−0.103918 + 0.994586i \(0.533138\pi\)
\(90\) 0 0
\(91\) −3.43207 −0.359779
\(92\) 0 0
\(93\) 2.05617 0.213215
\(94\) 0 0
\(95\) −0.804867 −0.0825776
\(96\) 0 0
\(97\) −14.6668 −1.48919 −0.744593 0.667518i \(-0.767357\pi\)
−0.744593 + 0.667518i \(0.767357\pi\)
\(98\) 0 0
\(99\) −12.1681 −1.22294
\(100\) 0 0
\(101\) −11.0636 −1.10087 −0.550433 0.834880i \(-0.685537\pi\)
−0.550433 + 0.834880i \(0.685537\pi\)
\(102\) 0 0
\(103\) 9.38450 0.924682 0.462341 0.886702i \(-0.347010\pi\)
0.462341 + 0.886702i \(0.347010\pi\)
\(104\) 0 0
\(105\) 1.52414 0.148741
\(106\) 0 0
\(107\) −9.98190 −0.964987 −0.482493 0.875900i \(-0.660269\pi\)
−0.482493 + 0.875900i \(0.660269\pi\)
\(108\) 0 0
\(109\) 11.1199 1.06509 0.532547 0.846400i \(-0.321235\pi\)
0.532547 + 0.846400i \(0.321235\pi\)
\(110\) 0 0
\(111\) −1.90592 −0.180902
\(112\) 0 0
\(113\) −8.69056 −0.817539 −0.408770 0.912638i \(-0.634042\pi\)
−0.408770 + 0.912638i \(0.634042\pi\)
\(114\) 0 0
\(115\) −24.4539 −2.28034
\(116\) 0 0
\(117\) 7.29110 0.674062
\(118\) 0 0
\(119\) −8.13909 −0.746110
\(120\) 0 0
\(121\) 6.45216 0.586560
\(122\) 0 0
\(123\) −0.904177 −0.0815269
\(124\) 0 0
\(125\) −15.6374 −1.39865
\(126\) 0 0
\(127\) −9.42318 −0.836172 −0.418086 0.908407i \(-0.637299\pi\)
−0.418086 + 0.908407i \(0.637299\pi\)
\(128\) 0 0
\(129\) 1.14073 0.100436
\(130\) 0 0
\(131\) 1.72603 0.150804 0.0754018 0.997153i \(-0.475976\pi\)
0.0754018 + 0.997153i \(0.475976\pi\)
\(132\) 0 0
\(133\) 0.293300 0.0254323
\(134\) 0 0
\(135\) −6.57281 −0.565697
\(136\) 0 0
\(137\) 8.08826 0.691026 0.345513 0.938414i \(-0.387705\pi\)
0.345513 + 0.938414i \(0.387705\pi\)
\(138\) 0 0
\(139\) 13.4164 1.13796 0.568982 0.822350i \(-0.307337\pi\)
0.568982 + 0.822350i \(0.307337\pi\)
\(140\) 0 0
\(141\) −1.18465 −0.0997652
\(142\) 0 0
\(143\) −10.4573 −0.874486
\(144\) 0 0
\(145\) −25.5144 −2.11886
\(146\) 0 0
\(147\) 1.51278 0.124772
\(148\) 0 0
\(149\) 12.6352 1.03511 0.517557 0.855649i \(-0.326842\pi\)
0.517557 + 0.855649i \(0.326842\pi\)
\(150\) 0 0
\(151\) −17.8685 −1.45412 −0.727058 0.686576i \(-0.759113\pi\)
−0.727058 + 0.686576i \(0.759113\pi\)
\(152\) 0 0
\(153\) 17.2907 1.39787
\(154\) 0 0
\(155\) 26.1843 2.10317
\(156\) 0 0
\(157\) 6.80142 0.542812 0.271406 0.962465i \(-0.412511\pi\)
0.271406 + 0.962465i \(0.412511\pi\)
\(158\) 0 0
\(159\) −3.57525 −0.283536
\(160\) 0 0
\(161\) 8.91120 0.702301
\(162\) 0 0
\(163\) −8.77104 −0.687001 −0.343500 0.939153i \(-0.611613\pi\)
−0.343500 + 0.939153i \(0.611613\pi\)
\(164\) 0 0
\(165\) 4.64397 0.361532
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −6.73397 −0.517998
\(170\) 0 0
\(171\) −0.623086 −0.0476486
\(172\) 0 0
\(173\) 22.9419 1.74424 0.872122 0.489289i \(-0.162744\pi\)
0.872122 + 0.489289i \(0.162744\pi\)
\(174\) 0 0
\(175\) 12.5538 0.948975
\(176\) 0 0
\(177\) −3.95674 −0.297406
\(178\) 0 0
\(179\) 16.2901 1.21758 0.608788 0.793333i \(-0.291656\pi\)
0.608788 + 0.793333i \(0.291656\pi\)
\(180\) 0 0
\(181\) −11.9503 −0.888261 −0.444130 0.895962i \(-0.646487\pi\)
−0.444130 + 0.895962i \(0.646487\pi\)
\(182\) 0 0
\(183\) −0.379321 −0.0280402
\(184\) 0 0
\(185\) −24.2709 −1.78443
\(186\) 0 0
\(187\) −24.7993 −1.81351
\(188\) 0 0
\(189\) 2.39518 0.174224
\(190\) 0 0
\(191\) 3.67193 0.265691 0.132846 0.991137i \(-0.457589\pi\)
0.132846 + 0.991137i \(0.457589\pi\)
\(192\) 0 0
\(193\) −12.9092 −0.929223 −0.464611 0.885515i \(-0.653806\pi\)
−0.464611 + 0.885515i \(0.653806\pi\)
\(194\) 0 0
\(195\) −2.78266 −0.199271
\(196\) 0 0
\(197\) −14.9918 −1.06812 −0.534059 0.845447i \(-0.679334\pi\)
−0.534059 + 0.845447i \(0.679334\pi\)
\(198\) 0 0
\(199\) −24.5526 −1.74048 −0.870242 0.492624i \(-0.836038\pi\)
−0.870242 + 0.492624i \(0.836038\pi\)
\(200\) 0 0
\(201\) 0.700497 0.0494092
\(202\) 0 0
\(203\) 9.29765 0.652567
\(204\) 0 0
\(205\) −11.5142 −0.804187
\(206\) 0 0
\(207\) −18.9310 −1.31579
\(208\) 0 0
\(209\) 0.893668 0.0618163
\(210\) 0 0
\(211\) 21.6870 1.49300 0.746498 0.665388i \(-0.231734\pi\)
0.746498 + 0.665388i \(0.231734\pi\)
\(212\) 0 0
\(213\) −2.45608 −0.168288
\(214\) 0 0
\(215\) 14.5266 0.990708
\(216\) 0 0
\(217\) −9.54176 −0.647737
\(218\) 0 0
\(219\) −0.852269 −0.0575910
\(220\) 0 0
\(221\) 14.8598 0.999575
\(222\) 0 0
\(223\) 6.86021 0.459394 0.229697 0.973262i \(-0.426227\pi\)
0.229697 + 0.973262i \(0.426227\pi\)
\(224\) 0 0
\(225\) −26.6692 −1.77795
\(226\) 0 0
\(227\) −17.8395 −1.18405 −0.592023 0.805921i \(-0.701671\pi\)
−0.592023 + 0.805921i \(0.701671\pi\)
\(228\) 0 0
\(229\) 27.2832 1.80293 0.901464 0.432854i \(-0.142494\pi\)
0.901464 + 0.432854i \(0.142494\pi\)
\(230\) 0 0
\(231\) −1.69230 −0.111345
\(232\) 0 0
\(233\) 6.10321 0.399835 0.199917 0.979813i \(-0.435933\pi\)
0.199917 + 0.979813i \(0.435933\pi\)
\(234\) 0 0
\(235\) −15.0858 −0.984091
\(236\) 0 0
\(237\) −4.10400 −0.266583
\(238\) 0 0
\(239\) 13.9620 0.903130 0.451565 0.892238i \(-0.350866\pi\)
0.451565 + 0.892238i \(0.350866\pi\)
\(240\) 0 0
\(241\) −14.3180 −0.922304 −0.461152 0.887321i \(-0.652564\pi\)
−0.461152 + 0.887321i \(0.652564\pi\)
\(242\) 0 0
\(243\) −7.67005 −0.492034
\(244\) 0 0
\(245\) 19.2644 1.23076
\(246\) 0 0
\(247\) −0.535485 −0.0340721
\(248\) 0 0
\(249\) −2.42363 −0.153592
\(250\) 0 0
\(251\) −20.8463 −1.31580 −0.657902 0.753103i \(-0.728556\pi\)
−0.657902 + 0.753103i \(0.728556\pi\)
\(252\) 0 0
\(253\) 27.1519 1.70703
\(254\) 0 0
\(255\) −6.59903 −0.413247
\(256\) 0 0
\(257\) 8.62859 0.538237 0.269118 0.963107i \(-0.413268\pi\)
0.269118 + 0.963107i \(0.413268\pi\)
\(258\) 0 0
\(259\) 8.84450 0.549570
\(260\) 0 0
\(261\) −19.7519 −1.22261
\(262\) 0 0
\(263\) −4.91000 −0.302763 −0.151382 0.988475i \(-0.548372\pi\)
−0.151382 + 0.988475i \(0.548372\pi\)
\(264\) 0 0
\(265\) −45.5289 −2.79682
\(266\) 0 0
\(267\) 0.579305 0.0354529
\(268\) 0 0
\(269\) −18.2916 −1.11526 −0.557631 0.830089i \(-0.688289\pi\)
−0.557631 + 0.830089i \(0.688289\pi\)
\(270\) 0 0
\(271\) 27.3589 1.66193 0.830967 0.556322i \(-0.187788\pi\)
0.830967 + 0.556322i \(0.187788\pi\)
\(272\) 0 0
\(273\) 1.01402 0.0613716
\(274\) 0 0
\(275\) 38.2506 2.30660
\(276\) 0 0
\(277\) 11.4393 0.687320 0.343660 0.939094i \(-0.388333\pi\)
0.343660 + 0.939094i \(0.388333\pi\)
\(278\) 0 0
\(279\) 20.2705 1.21356
\(280\) 0 0
\(281\) 4.89756 0.292164 0.146082 0.989272i \(-0.453334\pi\)
0.146082 + 0.989272i \(0.453334\pi\)
\(282\) 0 0
\(283\) 0.710050 0.0422081 0.0211040 0.999777i \(-0.493282\pi\)
0.0211040 + 0.999777i \(0.493282\pi\)
\(284\) 0 0
\(285\) 0.237802 0.0140862
\(286\) 0 0
\(287\) 4.19587 0.247674
\(288\) 0 0
\(289\) 18.2396 1.07292
\(290\) 0 0
\(291\) 4.33338 0.254027
\(292\) 0 0
\(293\) 10.7416 0.627531 0.313766 0.949500i \(-0.398409\pi\)
0.313766 + 0.949500i \(0.398409\pi\)
\(294\) 0 0
\(295\) −50.3869 −2.93364
\(296\) 0 0
\(297\) 7.29798 0.423472
\(298\) 0 0
\(299\) −16.2694 −0.940884
\(300\) 0 0
\(301\) −5.29362 −0.305119
\(302\) 0 0
\(303\) 3.26879 0.187787
\(304\) 0 0
\(305\) −4.83045 −0.276591
\(306\) 0 0
\(307\) −5.00761 −0.285799 −0.142900 0.989737i \(-0.545643\pi\)
−0.142900 + 0.989737i \(0.545643\pi\)
\(308\) 0 0
\(309\) −2.77270 −0.157733
\(310\) 0 0
\(311\) −27.8277 −1.57796 −0.788982 0.614417i \(-0.789391\pi\)
−0.788982 + 0.614417i \(0.789391\pi\)
\(312\) 0 0
\(313\) 0.287419 0.0162459 0.00812294 0.999967i \(-0.497414\pi\)
0.00812294 + 0.999967i \(0.497414\pi\)
\(314\) 0 0
\(315\) 15.0255 0.846593
\(316\) 0 0
\(317\) −26.0857 −1.46512 −0.732558 0.680704i \(-0.761674\pi\)
−0.732558 + 0.680704i \(0.761674\pi\)
\(318\) 0 0
\(319\) 28.3294 1.58614
\(320\) 0 0
\(321\) 2.94921 0.164609
\(322\) 0 0
\(323\) −1.26989 −0.0706587
\(324\) 0 0
\(325\) −22.9197 −1.27136
\(326\) 0 0
\(327\) −3.28544 −0.181685
\(328\) 0 0
\(329\) 5.49740 0.303081
\(330\) 0 0
\(331\) 5.25813 0.289013 0.144506 0.989504i \(-0.453841\pi\)
0.144506 + 0.989504i \(0.453841\pi\)
\(332\) 0 0
\(333\) −18.7892 −1.02964
\(334\) 0 0
\(335\) 8.92045 0.487376
\(336\) 0 0
\(337\) 22.2117 1.20995 0.604973 0.796246i \(-0.293184\pi\)
0.604973 + 0.796246i \(0.293184\pi\)
\(338\) 0 0
\(339\) 2.56767 0.139457
\(340\) 0 0
\(341\) −29.0732 −1.57440
\(342\) 0 0
\(343\) −16.6176 −0.897267
\(344\) 0 0
\(345\) 7.22504 0.388983
\(346\) 0 0
\(347\) −2.15577 −0.115728 −0.0578638 0.998324i \(-0.518429\pi\)
−0.0578638 + 0.998324i \(0.518429\pi\)
\(348\) 0 0
\(349\) 32.7297 1.75198 0.875990 0.482328i \(-0.160209\pi\)
0.875990 + 0.482328i \(0.160209\pi\)
\(350\) 0 0
\(351\) −4.37295 −0.233411
\(352\) 0 0
\(353\) −0.268480 −0.0142897 −0.00714487 0.999974i \(-0.502274\pi\)
−0.00714487 + 0.999974i \(0.502274\pi\)
\(354\) 0 0
\(355\) −31.2769 −1.66001
\(356\) 0 0
\(357\) 2.40474 0.127272
\(358\) 0 0
\(359\) −0.750307 −0.0395997 −0.0197998 0.999804i \(-0.506303\pi\)
−0.0197998 + 0.999804i \(0.506303\pi\)
\(360\) 0 0
\(361\) −18.9542 −0.997591
\(362\) 0 0
\(363\) −1.90633 −0.100056
\(364\) 0 0
\(365\) −10.8532 −0.568082
\(366\) 0 0
\(367\) 33.1730 1.73162 0.865809 0.500375i \(-0.166804\pi\)
0.865809 + 0.500375i \(0.166804\pi\)
\(368\) 0 0
\(369\) −8.91370 −0.464029
\(370\) 0 0
\(371\) 16.5911 0.861366
\(372\) 0 0
\(373\) −20.0638 −1.03886 −0.519431 0.854512i \(-0.673856\pi\)
−0.519431 + 0.854512i \(0.673856\pi\)
\(374\) 0 0
\(375\) 4.62015 0.238584
\(376\) 0 0
\(377\) −16.9750 −0.874255
\(378\) 0 0
\(379\) 5.28217 0.271327 0.135663 0.990755i \(-0.456683\pi\)
0.135663 + 0.990755i \(0.456683\pi\)
\(380\) 0 0
\(381\) 2.78413 0.142635
\(382\) 0 0
\(383\) 34.0695 1.74087 0.870435 0.492284i \(-0.163838\pi\)
0.870435 + 0.492284i \(0.163838\pi\)
\(384\) 0 0
\(385\) −21.5505 −1.09832
\(386\) 0 0
\(387\) 11.2458 0.571654
\(388\) 0 0
\(389\) −2.62732 −0.133211 −0.0666053 0.997779i \(-0.521217\pi\)
−0.0666053 + 0.997779i \(0.521217\pi\)
\(390\) 0 0
\(391\) −38.5826 −1.95121
\(392\) 0 0
\(393\) −0.509964 −0.0257243
\(394\) 0 0
\(395\) −52.2623 −2.62960
\(396\) 0 0
\(397\) 16.4252 0.824355 0.412177 0.911104i \(-0.364768\pi\)
0.412177 + 0.911104i \(0.364768\pi\)
\(398\) 0 0
\(399\) −0.0866571 −0.00433828
\(400\) 0 0
\(401\) 29.0646 1.45142 0.725708 0.688003i \(-0.241512\pi\)
0.725708 + 0.688003i \(0.241512\pi\)
\(402\) 0 0
\(403\) 17.4206 0.867784
\(404\) 0 0
\(405\) −30.9349 −1.53717
\(406\) 0 0
\(407\) 26.9487 1.33580
\(408\) 0 0
\(409\) 18.2105 0.900450 0.450225 0.892915i \(-0.351344\pi\)
0.450225 + 0.892915i \(0.351344\pi\)
\(410\) 0 0
\(411\) −2.38972 −0.117876
\(412\) 0 0
\(413\) 18.3614 0.903505
\(414\) 0 0
\(415\) −30.8637 −1.51504
\(416\) 0 0
\(417\) −3.96395 −0.194115
\(418\) 0 0
\(419\) −3.36841 −0.164558 −0.0822789 0.996609i \(-0.526220\pi\)
−0.0822789 + 0.996609i \(0.526220\pi\)
\(420\) 0 0
\(421\) −34.0408 −1.65905 −0.829524 0.558471i \(-0.811388\pi\)
−0.829524 + 0.558471i \(0.811388\pi\)
\(422\) 0 0
\(423\) −11.6787 −0.567836
\(424\) 0 0
\(425\) −54.3537 −2.63654
\(426\) 0 0
\(427\) 1.76025 0.0851847
\(428\) 0 0
\(429\) 3.08967 0.149171
\(430\) 0 0
\(431\) −25.3149 −1.21938 −0.609689 0.792641i \(-0.708705\pi\)
−0.609689 + 0.792641i \(0.708705\pi\)
\(432\) 0 0
\(433\) 12.7368 0.612093 0.306046 0.952017i \(-0.400994\pi\)
0.306046 + 0.952017i \(0.400994\pi\)
\(434\) 0 0
\(435\) 7.53837 0.361437
\(436\) 0 0
\(437\) 1.39036 0.0665099
\(438\) 0 0
\(439\) −6.94950 −0.331682 −0.165841 0.986153i \(-0.553034\pi\)
−0.165841 + 0.986153i \(0.553034\pi\)
\(440\) 0 0
\(441\) 14.9135 0.710168
\(442\) 0 0
\(443\) 8.73349 0.414941 0.207470 0.978241i \(-0.433477\pi\)
0.207470 + 0.978241i \(0.433477\pi\)
\(444\) 0 0
\(445\) 7.37714 0.349710
\(446\) 0 0
\(447\) −3.73313 −0.176571
\(448\) 0 0
\(449\) −26.3207 −1.24215 −0.621076 0.783750i \(-0.713304\pi\)
−0.621076 + 0.783750i \(0.713304\pi\)
\(450\) 0 0
\(451\) 12.7846 0.602002
\(452\) 0 0
\(453\) 5.27934 0.248045
\(454\) 0 0
\(455\) 12.9131 0.605374
\(456\) 0 0
\(457\) 7.50632 0.351131 0.175565 0.984468i \(-0.443825\pi\)
0.175565 + 0.984468i \(0.443825\pi\)
\(458\) 0 0
\(459\) −10.3704 −0.484047
\(460\) 0 0
\(461\) 27.8074 1.29512 0.647559 0.762015i \(-0.275790\pi\)
0.647559 + 0.762015i \(0.275790\pi\)
\(462\) 0 0
\(463\) −23.3349 −1.08446 −0.542232 0.840229i \(-0.682420\pi\)
−0.542232 + 0.840229i \(0.682420\pi\)
\(464\) 0 0
\(465\) −7.73629 −0.358762
\(466\) 0 0
\(467\) −20.3584 −0.942075 −0.471037 0.882113i \(-0.656120\pi\)
−0.471037 + 0.882113i \(0.656120\pi\)
\(468\) 0 0
\(469\) −3.25068 −0.150103
\(470\) 0 0
\(471\) −2.00952 −0.0925936
\(472\) 0 0
\(473\) −16.1294 −0.741628
\(474\) 0 0
\(475\) 1.95869 0.0898707
\(476\) 0 0
\(477\) −35.2461 −1.61381
\(478\) 0 0
\(479\) 21.9153 1.00134 0.500669 0.865639i \(-0.333087\pi\)
0.500669 + 0.865639i \(0.333087\pi\)
\(480\) 0 0
\(481\) −16.1476 −0.736269
\(482\) 0 0
\(483\) −2.63286 −0.119799
\(484\) 0 0
\(485\) 55.1833 2.50575
\(486\) 0 0
\(487\) −21.0923 −0.955781 −0.477891 0.878419i \(-0.658599\pi\)
−0.477891 + 0.878419i \(0.658599\pi\)
\(488\) 0 0
\(489\) 2.59145 0.117189
\(490\) 0 0
\(491\) 9.77787 0.441269 0.220635 0.975357i \(-0.429187\pi\)
0.220635 + 0.975357i \(0.429187\pi\)
\(492\) 0 0
\(493\) −40.2558 −1.81303
\(494\) 0 0
\(495\) 45.7819 2.05774
\(496\) 0 0
\(497\) 11.3976 0.511250
\(498\) 0 0
\(499\) 20.4686 0.916301 0.458151 0.888875i \(-0.348512\pi\)
0.458151 + 0.888875i \(0.348512\pi\)
\(500\) 0 0
\(501\) 0.295455 0.0132000
\(502\) 0 0
\(503\) 9.26662 0.413178 0.206589 0.978428i \(-0.433764\pi\)
0.206589 + 0.978428i \(0.433764\pi\)
\(504\) 0 0
\(505\) 41.6263 1.85235
\(506\) 0 0
\(507\) 1.98959 0.0883607
\(508\) 0 0
\(509\) 8.96456 0.397347 0.198674 0.980066i \(-0.436337\pi\)
0.198674 + 0.980066i \(0.436337\pi\)
\(510\) 0 0
\(511\) 3.95499 0.174958
\(512\) 0 0
\(513\) 0.373706 0.0164995
\(514\) 0 0
\(515\) −35.3089 −1.55589
\(516\) 0 0
\(517\) 16.7502 0.736675
\(518\) 0 0
\(519\) −6.77832 −0.297535
\(520\) 0 0
\(521\) 28.9311 1.26749 0.633747 0.773540i \(-0.281516\pi\)
0.633747 + 0.773540i \(0.281516\pi\)
\(522\) 0 0
\(523\) 2.22763 0.0974076 0.0487038 0.998813i \(-0.484491\pi\)
0.0487038 + 0.998813i \(0.484491\pi\)
\(524\) 0 0
\(525\) −3.70908 −0.161877
\(526\) 0 0
\(527\) 41.3127 1.79961
\(528\) 0 0
\(529\) 19.2427 0.836639
\(530\) 0 0
\(531\) −39.0069 −1.69276
\(532\) 0 0
\(533\) −7.66051 −0.331813
\(534\) 0 0
\(535\) 37.5566 1.62371
\(536\) 0 0
\(537\) −4.81299 −0.207696
\(538\) 0 0
\(539\) −21.3899 −0.921327
\(540\) 0 0
\(541\) 44.1483 1.89808 0.949041 0.315153i \(-0.102056\pi\)
0.949041 + 0.315153i \(0.102056\pi\)
\(542\) 0 0
\(543\) 3.53079 0.151521
\(544\) 0 0
\(545\) −41.8383 −1.79216
\(546\) 0 0
\(547\) −30.8553 −1.31928 −0.659639 0.751583i \(-0.729291\pi\)
−0.659639 + 0.751583i \(0.729291\pi\)
\(548\) 0 0
\(549\) −3.73948 −0.159597
\(550\) 0 0
\(551\) 1.45066 0.0618000
\(552\) 0 0
\(553\) 19.0448 0.809866
\(554\) 0 0
\(555\) 7.17096 0.304390
\(556\) 0 0
\(557\) −5.07702 −0.215120 −0.107560 0.994199i \(-0.534304\pi\)
−0.107560 + 0.994199i \(0.534304\pi\)
\(558\) 0 0
\(559\) 9.66470 0.408773
\(560\) 0 0
\(561\) 7.32710 0.309350
\(562\) 0 0
\(563\) 19.4482 0.819642 0.409821 0.912166i \(-0.365591\pi\)
0.409821 + 0.912166i \(0.365591\pi\)
\(564\) 0 0
\(565\) 32.6979 1.37561
\(566\) 0 0
\(567\) 11.2729 0.473418
\(568\) 0 0
\(569\) −29.7827 −1.24856 −0.624279 0.781202i \(-0.714607\pi\)
−0.624279 + 0.781202i \(0.714607\pi\)
\(570\) 0 0
\(571\) −17.4520 −0.730344 −0.365172 0.930940i \(-0.618990\pi\)
−0.365172 + 0.930940i \(0.618990\pi\)
\(572\) 0 0
\(573\) −1.08489 −0.0453220
\(574\) 0 0
\(575\) 59.5099 2.48173
\(576\) 0 0
\(577\) 30.6090 1.27427 0.637135 0.770752i \(-0.280119\pi\)
0.637135 + 0.770752i \(0.280119\pi\)
\(578\) 0 0
\(579\) 3.81409 0.158508
\(580\) 0 0
\(581\) 11.2470 0.466603
\(582\) 0 0
\(583\) 50.5520 2.09365
\(584\) 0 0
\(585\) −27.4325 −1.13419
\(586\) 0 0
\(587\) 19.7563 0.815431 0.407715 0.913109i \(-0.366326\pi\)
0.407715 + 0.913109i \(0.366326\pi\)
\(588\) 0 0
\(589\) −1.48874 −0.0613426
\(590\) 0 0
\(591\) 4.42940 0.182201
\(592\) 0 0
\(593\) 13.9688 0.573630 0.286815 0.957986i \(-0.407404\pi\)
0.286815 + 0.957986i \(0.407404\pi\)
\(594\) 0 0
\(595\) 30.6231 1.25542
\(596\) 0 0
\(597\) 7.25419 0.296894
\(598\) 0 0
\(599\) −30.6941 −1.25413 −0.627063 0.778968i \(-0.715743\pi\)
−0.627063 + 0.778968i \(0.715743\pi\)
\(600\) 0 0
\(601\) 15.6364 0.637822 0.318911 0.947785i \(-0.396683\pi\)
0.318911 + 0.947785i \(0.396683\pi\)
\(602\) 0 0
\(603\) 6.90575 0.281224
\(604\) 0 0
\(605\) −24.2761 −0.986962
\(606\) 0 0
\(607\) 19.8125 0.804165 0.402082 0.915604i \(-0.368287\pi\)
0.402082 + 0.915604i \(0.368287\pi\)
\(608\) 0 0
\(609\) −2.74704 −0.111316
\(610\) 0 0
\(611\) −10.0367 −0.406043
\(612\) 0 0
\(613\) 7.04779 0.284657 0.142329 0.989819i \(-0.454541\pi\)
0.142329 + 0.989819i \(0.454541\pi\)
\(614\) 0 0
\(615\) 3.40194 0.137179
\(616\) 0 0
\(617\) 12.7037 0.511433 0.255716 0.966752i \(-0.417689\pi\)
0.255716 + 0.966752i \(0.417689\pi\)
\(618\) 0 0
\(619\) 31.3477 1.25997 0.629985 0.776607i \(-0.283061\pi\)
0.629985 + 0.776607i \(0.283061\pi\)
\(620\) 0 0
\(621\) 11.3541 0.455626
\(622\) 0 0
\(623\) −2.68829 −0.107704
\(624\) 0 0
\(625\) 13.0544 0.522177
\(626\) 0 0
\(627\) −0.264039 −0.0105447
\(628\) 0 0
\(629\) −38.2938 −1.52687
\(630\) 0 0
\(631\) −20.0147 −0.796772 −0.398386 0.917218i \(-0.630430\pi\)
−0.398386 + 0.917218i \(0.630430\pi\)
\(632\) 0 0
\(633\) −6.40755 −0.254677
\(634\) 0 0
\(635\) 35.4544 1.40697
\(636\) 0 0
\(637\) 12.8168 0.507820
\(638\) 0 0
\(639\) −24.2130 −0.957850
\(640\) 0 0
\(641\) 26.0190 1.02769 0.513845 0.857883i \(-0.328221\pi\)
0.513845 + 0.857883i \(0.328221\pi\)
\(642\) 0 0
\(643\) −10.7960 −0.425754 −0.212877 0.977079i \(-0.568283\pi\)
−0.212877 + 0.977079i \(0.568283\pi\)
\(644\) 0 0
\(645\) −4.29197 −0.168996
\(646\) 0 0
\(647\) 43.0560 1.69271 0.846353 0.532623i \(-0.178794\pi\)
0.846353 + 0.532623i \(0.178794\pi\)
\(648\) 0 0
\(649\) 55.9461 2.19608
\(650\) 0 0
\(651\) 2.81916 0.110492
\(652\) 0 0
\(653\) 5.84832 0.228862 0.114431 0.993431i \(-0.463495\pi\)
0.114431 + 0.993431i \(0.463495\pi\)
\(654\) 0 0
\(655\) −6.49412 −0.253746
\(656\) 0 0
\(657\) −8.40197 −0.327792
\(658\) 0 0
\(659\) 41.1933 1.60466 0.802331 0.596880i \(-0.203593\pi\)
0.802331 + 0.596880i \(0.203593\pi\)
\(660\) 0 0
\(661\) 38.3584 1.49197 0.745985 0.665962i \(-0.231979\pi\)
0.745985 + 0.665962i \(0.231979\pi\)
\(662\) 0 0
\(663\) −4.39040 −0.170509
\(664\) 0 0
\(665\) −1.10353 −0.0427931
\(666\) 0 0
\(667\) 44.0746 1.70658
\(668\) 0 0
\(669\) −2.02689 −0.0783640
\(670\) 0 0
\(671\) 5.36339 0.207051
\(672\) 0 0
\(673\) −42.4250 −1.63536 −0.817682 0.575670i \(-0.804741\pi\)
−0.817682 + 0.575670i \(0.804741\pi\)
\(674\) 0 0
\(675\) 15.9953 0.615658
\(676\) 0 0
\(677\) −10.8745 −0.417940 −0.208970 0.977922i \(-0.567011\pi\)
−0.208970 + 0.977922i \(0.567011\pi\)
\(678\) 0 0
\(679\) −20.1092 −0.771722
\(680\) 0 0
\(681\) 5.27077 0.201976
\(682\) 0 0
\(683\) −40.9894 −1.56842 −0.784208 0.620498i \(-0.786930\pi\)
−0.784208 + 0.620498i \(0.786930\pi\)
\(684\) 0 0
\(685\) −30.4318 −1.16274
\(686\) 0 0
\(687\) −8.06098 −0.307546
\(688\) 0 0
\(689\) −30.2908 −1.15399
\(690\) 0 0
\(691\) −19.0390 −0.724280 −0.362140 0.932124i \(-0.617954\pi\)
−0.362140 + 0.932124i \(0.617954\pi\)
\(692\) 0 0
\(693\) −16.6833 −0.633746
\(694\) 0 0
\(695\) −50.4787 −1.91477
\(696\) 0 0
\(697\) −18.1667 −0.688114
\(698\) 0 0
\(699\) −1.80323 −0.0682043
\(700\) 0 0
\(701\) 16.8202 0.635292 0.317646 0.948209i \(-0.397108\pi\)
0.317646 + 0.948209i \(0.397108\pi\)
\(702\) 0 0
\(703\) 1.37995 0.0520459
\(704\) 0 0
\(705\) 4.45719 0.167868
\(706\) 0 0
\(707\) −15.1689 −0.570487
\(708\) 0 0
\(709\) 43.5192 1.63440 0.817200 0.576354i \(-0.195525\pi\)
0.817200 + 0.576354i \(0.195525\pi\)
\(710\) 0 0
\(711\) −40.4587 −1.51732
\(712\) 0 0
\(713\) −45.2318 −1.69394
\(714\) 0 0
\(715\) 39.3454 1.47143
\(716\) 0 0
\(717\) −4.12516 −0.154057
\(718\) 0 0
\(719\) 36.2606 1.35229 0.676145 0.736768i \(-0.263649\pi\)
0.676145 + 0.736768i \(0.263649\pi\)
\(720\) 0 0
\(721\) 12.8668 0.479186
\(722\) 0 0
\(723\) 4.23033 0.157328
\(724\) 0 0
\(725\) 62.0906 2.30599
\(726\) 0 0
\(727\) −9.97768 −0.370052 −0.185026 0.982734i \(-0.559237\pi\)
−0.185026 + 0.982734i \(0.559237\pi\)
\(728\) 0 0
\(729\) −22.3998 −0.829621
\(730\) 0 0
\(731\) 22.9196 0.847714
\(732\) 0 0
\(733\) −53.4062 −1.97260 −0.986301 0.164954i \(-0.947252\pi\)
−0.986301 + 0.164954i \(0.947252\pi\)
\(734\) 0 0
\(735\) −5.69178 −0.209944
\(736\) 0 0
\(737\) −9.90464 −0.364842
\(738\) 0 0
\(739\) 14.0154 0.515564 0.257782 0.966203i \(-0.417008\pi\)
0.257782 + 0.966203i \(0.417008\pi\)
\(740\) 0 0
\(741\) 0.158212 0.00581207
\(742\) 0 0
\(743\) 1.50840 0.0553378 0.0276689 0.999617i \(-0.491192\pi\)
0.0276689 + 0.999617i \(0.491192\pi\)
\(744\) 0 0
\(745\) −47.5394 −1.74171
\(746\) 0 0
\(747\) −23.8931 −0.874202
\(748\) 0 0
\(749\) −13.6859 −0.500072
\(750\) 0 0
\(751\) 35.4689 1.29428 0.647139 0.762372i \(-0.275965\pi\)
0.647139 + 0.762372i \(0.275965\pi\)
\(752\) 0 0
\(753\) 6.15914 0.224452
\(754\) 0 0
\(755\) 67.2296 2.44673
\(756\) 0 0
\(757\) −40.2977 −1.46465 −0.732323 0.680957i \(-0.761564\pi\)
−0.732323 + 0.680957i \(0.761564\pi\)
\(758\) 0 0
\(759\) −8.02218 −0.291187
\(760\) 0 0
\(761\) −0.916494 −0.0332229 −0.0166114 0.999862i \(-0.505288\pi\)
−0.0166114 + 0.999862i \(0.505288\pi\)
\(762\) 0 0
\(763\) 15.2462 0.551950
\(764\) 0 0
\(765\) −65.0556 −2.35209
\(766\) 0 0
\(767\) −33.5229 −1.21044
\(768\) 0 0
\(769\) −28.8094 −1.03889 −0.519447 0.854503i \(-0.673862\pi\)
−0.519447 + 0.854503i \(0.673862\pi\)
\(770\) 0 0
\(771\) −2.54936 −0.0918131
\(772\) 0 0
\(773\) 31.3449 1.12740 0.563699 0.825980i \(-0.309378\pi\)
0.563699 + 0.825980i \(0.309378\pi\)
\(774\) 0 0
\(775\) −63.7208 −2.28892
\(776\) 0 0
\(777\) −2.61315 −0.0937464
\(778\) 0 0
\(779\) 0.654656 0.0234555
\(780\) 0 0
\(781\) 34.7277 1.24265
\(782\) 0 0
\(783\) 11.8465 0.423360
\(784\) 0 0
\(785\) −25.5901 −0.913350
\(786\) 0 0
\(787\) −13.3018 −0.474158 −0.237079 0.971490i \(-0.576190\pi\)
−0.237079 + 0.971490i \(0.576190\pi\)
\(788\) 0 0
\(789\) 1.45069 0.0516458
\(790\) 0 0
\(791\) −11.9154 −0.423662
\(792\) 0 0
\(793\) −3.21374 −0.114123
\(794\) 0 0
\(795\) 13.4517 0.477084
\(796\) 0 0
\(797\) 0.300180 0.0106329 0.00531647 0.999986i \(-0.498308\pi\)
0.00531647 + 0.999986i \(0.498308\pi\)
\(798\) 0 0
\(799\) −23.8019 −0.842052
\(800\) 0 0
\(801\) 5.71100 0.201788
\(802\) 0 0
\(803\) 12.0506 0.425257
\(804\) 0 0
\(805\) −33.5281 −1.18171
\(806\) 0 0
\(807\) 5.40436 0.190243
\(808\) 0 0
\(809\) −19.5410 −0.687024 −0.343512 0.939148i \(-0.611617\pi\)
−0.343512 + 0.939148i \(0.611617\pi\)
\(810\) 0 0
\(811\) 41.9263 1.47223 0.736115 0.676856i \(-0.236658\pi\)
0.736115 + 0.676856i \(0.236658\pi\)
\(812\) 0 0
\(813\) −8.08333 −0.283495
\(814\) 0 0
\(815\) 33.0007 1.15597
\(816\) 0 0
\(817\) −0.825931 −0.0288957
\(818\) 0 0
\(819\) 9.99662 0.349310
\(820\) 0 0
\(821\) −12.3744 −0.431869 −0.215935 0.976408i \(-0.569280\pi\)
−0.215935 + 0.976408i \(0.569280\pi\)
\(822\) 0 0
\(823\) 5.70767 0.198957 0.0994784 0.995040i \(-0.468283\pi\)
0.0994784 + 0.995040i \(0.468283\pi\)
\(824\) 0 0
\(825\) −11.3013 −0.393462
\(826\) 0 0
\(827\) −43.8063 −1.52329 −0.761647 0.647992i \(-0.775609\pi\)
−0.761647 + 0.647992i \(0.775609\pi\)
\(828\) 0 0
\(829\) −8.58825 −0.298282 −0.149141 0.988816i \(-0.547651\pi\)
−0.149141 + 0.988816i \(0.547651\pi\)
\(830\) 0 0
\(831\) −3.37980 −0.117244
\(832\) 0 0
\(833\) 30.3948 1.05312
\(834\) 0 0
\(835\) 3.76247 0.130206
\(836\) 0 0
\(837\) −12.1576 −0.420226
\(838\) 0 0
\(839\) 32.3818 1.11794 0.558972 0.829186i \(-0.311196\pi\)
0.558972 + 0.829186i \(0.311196\pi\)
\(840\) 0 0
\(841\) 16.9860 0.585725
\(842\) 0 0
\(843\) −1.44701 −0.0498377
\(844\) 0 0
\(845\) 25.3364 0.871597
\(846\) 0 0
\(847\) 8.84639 0.303965
\(848\) 0 0
\(849\) −0.209788 −0.00719991
\(850\) 0 0
\(851\) 41.9265 1.43722
\(852\) 0 0
\(853\) 33.2315 1.13783 0.568913 0.822398i \(-0.307364\pi\)
0.568913 + 0.822398i \(0.307364\pi\)
\(854\) 0 0
\(855\) 2.34434 0.0801748
\(856\) 0 0
\(857\) −5.16115 −0.176302 −0.0881508 0.996107i \(-0.528096\pi\)
−0.0881508 + 0.996107i \(0.528096\pi\)
\(858\) 0 0
\(859\) 20.3148 0.693133 0.346566 0.938025i \(-0.387348\pi\)
0.346566 + 0.938025i \(0.387348\pi\)
\(860\) 0 0
\(861\) −1.23969 −0.0422486
\(862\) 0 0
\(863\) −14.9284 −0.508169 −0.254085 0.967182i \(-0.581774\pi\)
−0.254085 + 0.967182i \(0.581774\pi\)
\(864\) 0 0
\(865\) −86.3183 −2.93491
\(866\) 0 0
\(867\) −5.38899 −0.183020
\(868\) 0 0
\(869\) 58.0283 1.96848
\(870\) 0 0
\(871\) 5.93486 0.201095
\(872\) 0 0
\(873\) 42.7200 1.44585
\(874\) 0 0
\(875\) −21.4400 −0.724805
\(876\) 0 0
\(877\) −10.7223 −0.362065 −0.181032 0.983477i \(-0.557944\pi\)
−0.181032 + 0.983477i \(0.557944\pi\)
\(878\) 0 0
\(879\) −3.17367 −0.107045
\(880\) 0 0
\(881\) 17.4841 0.589056 0.294528 0.955643i \(-0.404838\pi\)
0.294528 + 0.955643i \(0.404838\pi\)
\(882\) 0 0
\(883\) −40.7310 −1.37071 −0.685354 0.728210i \(-0.740353\pi\)
−0.685354 + 0.728210i \(0.740353\pi\)
\(884\) 0 0
\(885\) 14.8871 0.500424
\(886\) 0 0
\(887\) 35.3165 1.18581 0.592907 0.805271i \(-0.297980\pi\)
0.592907 + 0.805271i \(0.297980\pi\)
\(888\) 0 0
\(889\) −12.9199 −0.433318
\(890\) 0 0
\(891\) 34.3479 1.15070
\(892\) 0 0
\(893\) 0.857725 0.0287027
\(894\) 0 0
\(895\) −61.2908 −2.04873
\(896\) 0 0
\(897\) 4.80689 0.160497
\(898\) 0 0
\(899\) −47.1934 −1.57399
\(900\) 0 0
\(901\) −71.8339 −2.39314
\(902\) 0 0
\(903\) 1.56403 0.0520476
\(904\) 0 0
\(905\) 44.9627 1.49461
\(906\) 0 0
\(907\) −15.5500 −0.516331 −0.258165 0.966101i \(-0.583118\pi\)
−0.258165 + 0.966101i \(0.583118\pi\)
\(908\) 0 0
\(909\) 32.2249 1.06883
\(910\) 0 0
\(911\) 42.9269 1.42223 0.711115 0.703076i \(-0.248190\pi\)
0.711115 + 0.703076i \(0.248190\pi\)
\(912\) 0 0
\(913\) 34.2689 1.13413
\(914\) 0 0
\(915\) 1.42718 0.0471812
\(916\) 0 0
\(917\) 2.36651 0.0781490
\(918\) 0 0
\(919\) 27.2765 0.899770 0.449885 0.893086i \(-0.351465\pi\)
0.449885 + 0.893086i \(0.351465\pi\)
\(920\) 0 0
\(921\) 1.47952 0.0487520
\(922\) 0 0
\(923\) −20.8088 −0.684930
\(924\) 0 0
\(925\) 59.0644 1.94203
\(926\) 0 0
\(927\) −27.3343 −0.897776
\(928\) 0 0
\(929\) −43.1677 −1.41629 −0.708144 0.706068i \(-0.750467\pi\)
−0.708144 + 0.706068i \(0.750467\pi\)
\(930\) 0 0
\(931\) −1.09530 −0.0358972
\(932\) 0 0
\(933\) 8.22184 0.269171
\(934\) 0 0
\(935\) 93.3067 3.05146
\(936\) 0 0
\(937\) 3.06751 0.100211 0.0501056 0.998744i \(-0.484044\pi\)
0.0501056 + 0.998744i \(0.484044\pi\)
\(938\) 0 0
\(939\) −0.0849194 −0.00277124
\(940\) 0 0
\(941\) −13.5805 −0.442713 −0.221357 0.975193i \(-0.571048\pi\)
−0.221357 + 0.975193i \(0.571048\pi\)
\(942\) 0 0
\(943\) 19.8901 0.647711
\(944\) 0 0
\(945\) −9.01180 −0.293154
\(946\) 0 0
\(947\) −29.7877 −0.967971 −0.483986 0.875076i \(-0.660811\pi\)
−0.483986 + 0.875076i \(0.660811\pi\)
\(948\) 0 0
\(949\) −7.22072 −0.234395
\(950\) 0 0
\(951\) 7.70715 0.249921
\(952\) 0 0
\(953\) 32.9553 1.06753 0.533764 0.845633i \(-0.320777\pi\)
0.533764 + 0.845633i \(0.320777\pi\)
\(954\) 0 0
\(955\) −13.8155 −0.447059
\(956\) 0 0
\(957\) −8.37008 −0.270566
\(958\) 0 0
\(959\) 11.0896 0.358102
\(960\) 0 0
\(961\) 17.4324 0.562335
\(962\) 0 0
\(963\) 29.0743 0.936908
\(964\) 0 0
\(965\) 48.5704 1.56353
\(966\) 0 0
\(967\) 15.8616 0.510075 0.255037 0.966931i \(-0.417912\pi\)
0.255037 + 0.966931i \(0.417912\pi\)
\(968\) 0 0
\(969\) 0.375197 0.0120531
\(970\) 0 0
\(971\) 50.9434 1.63485 0.817425 0.576034i \(-0.195401\pi\)
0.817425 + 0.576034i \(0.195401\pi\)
\(972\) 0 0
\(973\) 18.3949 0.589712
\(974\) 0 0
\(975\) 6.77176 0.216870
\(976\) 0 0
\(977\) 25.7112 0.822575 0.411288 0.911506i \(-0.365079\pi\)
0.411288 + 0.911506i \(0.365079\pi\)
\(978\) 0 0
\(979\) −8.19106 −0.261787
\(980\) 0 0
\(981\) −32.3891 −1.03410
\(982\) 0 0
\(983\) −17.6514 −0.562992 −0.281496 0.959562i \(-0.590831\pi\)
−0.281496 + 0.959562i \(0.590831\pi\)
\(984\) 0 0
\(985\) 56.4060 1.79724
\(986\) 0 0
\(987\) −1.62424 −0.0517000
\(988\) 0 0
\(989\) −25.0939 −0.797939
\(990\) 0 0
\(991\) 10.2879 0.326807 0.163404 0.986559i \(-0.447753\pi\)
0.163404 + 0.986559i \(0.447753\pi\)
\(992\) 0 0
\(993\) −1.55354 −0.0493002
\(994\) 0 0
\(995\) 92.3782 2.92859
\(996\) 0 0
\(997\) −40.6279 −1.28670 −0.643350 0.765573i \(-0.722456\pi\)
−0.643350 + 0.765573i \(0.722456\pi\)
\(998\) 0 0
\(999\) 11.2691 0.356540
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 1336.2.a.d.1.6 12
4.3 odd 2 2672.2.a.p.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.d.1.6 12 1.1 even 1 trivial
2672.2.a.p.1.7 12 4.3 odd 2