Properties

Label 3040.2.a.x.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $5$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.387268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) 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 1.3
Root \(2.42170\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.44292 q^{3} -1.00000 q^{5} -2.92540 q^{7} -0.917992 q^{9} +O(q^{10})\) \(q+1.44292 q^{3} -1.00000 q^{5} -2.92540 q^{7} -0.917992 q^{9} +4.55509 q^{11} -1.15462 q^{13} -1.44292 q^{15} +5.35892 q^{17} -1.00000 q^{19} -4.22111 q^{21} -6.10565 q^{23} +1.00000 q^{25} -5.65334 q^{27} -5.09212 q^{29} +1.54157 q^{31} +6.57262 q^{33} +2.92540 q^{35} -8.92953 q^{37} -1.66602 q^{39} -2.88583 q^{41} -7.86833 q^{43} +0.917992 q^{45} +7.58614 q^{47} +1.55797 q^{49} +7.73247 q^{51} +4.48008 q^{53} -4.55509 q^{55} -1.44292 q^{57} -2.09953 q^{59} +1.59013 q^{61} +2.68549 q^{63} +1.15462 q^{65} +3.73733 q^{67} -8.80994 q^{69} +3.73263 q^{71} -9.81735 q^{73} +1.44292 q^{75} -13.3255 q^{77} -4.60365 q^{79} -5.40331 q^{81} -9.89538 q^{83} -5.35892 q^{85} -7.34751 q^{87} +1.91688 q^{89} +3.37772 q^{91} +2.22435 q^{93} +1.00000 q^{95} -15.6391 q^{97} -4.18154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} - 12 q^{17} - 5 q^{19} - 10 q^{21} - 8 q^{23} + 5 q^{25} + 16 q^{27} - 6 q^{29} - 10 q^{31} - 18 q^{33} + 4 q^{35} - 6 q^{37} - 18 q^{39} - 8 q^{41} + 12 q^{43} - 7 q^{45} - 16 q^{47} + 7 q^{49} - 14 q^{51} - 18 q^{53} - 2 q^{55} - 4 q^{57} + 8 q^{59} + 2 q^{61} - 36 q^{63} + 4 q^{65} + 10 q^{67} - 22 q^{69} + 18 q^{71} - 28 q^{73} + 4 q^{75} - 28 q^{77} - 14 q^{79} + 25 q^{81} + 8 q^{83} + 12 q^{85} - 24 q^{87} - 30 q^{89} + 28 q^{91} - 24 q^{93} + 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) 1.44292 0.833068 0.416534 0.909120i \(-0.363245\pi\)
0.416534 + 0.909120i \(0.363245\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.92540 −1.10570 −0.552849 0.833282i \(-0.686459\pi\)
−0.552849 + 0.833282i \(0.686459\pi\)
\(8\) 0 0
\(9\) −0.917992 −0.305997
\(10\) 0 0
\(11\) 4.55509 1.37341 0.686706 0.726935i \(-0.259056\pi\)
0.686706 + 0.726935i \(0.259056\pi\)
\(12\) 0 0
\(13\) −1.15462 −0.320233 −0.160116 0.987098i \(-0.551187\pi\)
−0.160116 + 0.987098i \(0.551187\pi\)
\(14\) 0 0
\(15\) −1.44292 −0.372559
\(16\) 0 0
\(17\) 5.35892 1.29973 0.649864 0.760050i \(-0.274826\pi\)
0.649864 + 0.760050i \(0.274826\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.22111 −0.921121
\(22\) 0 0
\(23\) −6.10565 −1.27312 −0.636558 0.771229i \(-0.719642\pi\)
−0.636558 + 0.771229i \(0.719642\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65334 −1.08798
\(28\) 0 0
\(29\) −5.09212 −0.945584 −0.472792 0.881174i \(-0.656754\pi\)
−0.472792 + 0.881174i \(0.656754\pi\)
\(30\) 0 0
\(31\) 1.54157 0.276874 0.138437 0.990371i \(-0.455792\pi\)
0.138437 + 0.990371i \(0.455792\pi\)
\(32\) 0 0
\(33\) 6.57262 1.14415
\(34\) 0 0
\(35\) 2.92540 0.494483
\(36\) 0 0
\(37\) −8.92953 −1.46801 −0.734003 0.679147i \(-0.762350\pi\)
−0.734003 + 0.679147i \(0.762350\pi\)
\(38\) 0 0
\(39\) −1.66602 −0.266776
\(40\) 0 0
\(41\) −2.88583 −0.450691 −0.225346 0.974279i \(-0.572351\pi\)
−0.225346 + 0.974279i \(0.572351\pi\)
\(42\) 0 0
\(43\) −7.86833 −1.19991 −0.599954 0.800034i \(-0.704815\pi\)
−0.599954 + 0.800034i \(0.704815\pi\)
\(44\) 0 0
\(45\) 0.917992 0.136846
\(46\) 0 0
\(47\) 7.58614 1.10655 0.553276 0.832998i \(-0.313377\pi\)
0.553276 + 0.832998i \(0.313377\pi\)
\(48\) 0 0
\(49\) 1.55797 0.222567
\(50\) 0 0
\(51\) 7.73247 1.08276
\(52\) 0 0
\(53\) 4.48008 0.615387 0.307693 0.951486i \(-0.400443\pi\)
0.307693 + 0.951486i \(0.400443\pi\)
\(54\) 0 0
\(55\) −4.55509 −0.614209
\(56\) 0 0
\(57\) −1.44292 −0.191119
\(58\) 0 0
\(59\) −2.09953 −0.273336 −0.136668 0.990617i \(-0.543639\pi\)
−0.136668 + 0.990617i \(0.543639\pi\)
\(60\) 0 0
\(61\) 1.59013 0.203595 0.101797 0.994805i \(-0.467541\pi\)
0.101797 + 0.994805i \(0.467541\pi\)
\(62\) 0 0
\(63\) 2.68549 0.338341
\(64\) 0 0
\(65\) 1.15462 0.143213
\(66\) 0 0
\(67\) 3.73733 0.456588 0.228294 0.973592i \(-0.426685\pi\)
0.228294 + 0.973592i \(0.426685\pi\)
\(68\) 0 0
\(69\) −8.80994 −1.06059
\(70\) 0 0
\(71\) 3.73263 0.442982 0.221491 0.975162i \(-0.428908\pi\)
0.221491 + 0.975162i \(0.428908\pi\)
\(72\) 0 0
\(73\) −9.81735 −1.14903 −0.574517 0.818493i \(-0.694810\pi\)
−0.574517 + 0.818493i \(0.694810\pi\)
\(74\) 0 0
\(75\) 1.44292 0.166614
\(76\) 0 0
\(77\) −13.3255 −1.51858
\(78\) 0 0
\(79\) −4.60365 −0.517951 −0.258976 0.965884i \(-0.583385\pi\)
−0.258976 + 0.965884i \(0.583385\pi\)
\(80\) 0 0
\(81\) −5.40331 −0.600368
\(82\) 0 0
\(83\) −9.89538 −1.08616 −0.543079 0.839681i \(-0.682742\pi\)
−0.543079 + 0.839681i \(0.682742\pi\)
\(84\) 0 0
\(85\) −5.35892 −0.581256
\(86\) 0 0
\(87\) −7.34751 −0.787736
\(88\) 0 0
\(89\) 1.91688 0.203189 0.101595 0.994826i \(-0.467606\pi\)
0.101595 + 0.994826i \(0.467606\pi\)
\(90\) 0 0
\(91\) 3.37772 0.354081
\(92\) 0 0
\(93\) 2.22435 0.230655
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −15.6391 −1.58791 −0.793955 0.607976i \(-0.791982\pi\)
−0.793955 + 0.607976i \(0.791982\pi\)
\(98\) 0 0
\(99\) −4.18154 −0.420261
\(100\) 0 0
\(101\) −8.55509 −0.851264 −0.425632 0.904896i \(-0.639948\pi\)
−0.425632 + 0.904896i \(0.639948\pi\)
\(102\) 0 0
\(103\) −9.01793 −0.888563 −0.444282 0.895887i \(-0.646541\pi\)
−0.444282 + 0.895887i \(0.646541\pi\)
\(104\) 0 0
\(105\) 4.22111 0.411938
\(106\) 0 0
\(107\) 19.4780 1.88301 0.941503 0.337004i \(-0.109414\pi\)
0.941503 + 0.337004i \(0.109414\pi\)
\(108\) 0 0
\(109\) 8.16328 0.781900 0.390950 0.920412i \(-0.372146\pi\)
0.390950 + 0.920412i \(0.372146\pi\)
\(110\) 0 0
\(111\) −12.8846 −1.22295
\(112\) 0 0
\(113\) −14.5893 −1.37244 −0.686221 0.727393i \(-0.740732\pi\)
−0.686221 + 0.727393i \(0.740732\pi\)
\(114\) 0 0
\(115\) 6.10565 0.569355
\(116\) 0 0
\(117\) 1.05993 0.0979905
\(118\) 0 0
\(119\) −15.6770 −1.43711
\(120\) 0 0
\(121\) 9.74887 0.886261
\(122\) 0 0
\(123\) −4.16402 −0.375457
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −9.99514 −0.886925 −0.443462 0.896293i \(-0.646250\pi\)
−0.443462 + 0.896293i \(0.646250\pi\)
\(128\) 0 0
\(129\) −11.3533 −0.999605
\(130\) 0 0
\(131\) 4.11817 0.359806 0.179903 0.983684i \(-0.442422\pi\)
0.179903 + 0.983684i \(0.442422\pi\)
\(132\) 0 0
\(133\) 2.92540 0.253664
\(134\) 0 0
\(135\) 5.65334 0.486562
\(136\) 0 0
\(137\) −11.7513 −1.00398 −0.501989 0.864874i \(-0.667398\pi\)
−0.501989 + 0.864874i \(0.667398\pi\)
\(138\) 0 0
\(139\) −12.8455 −1.08954 −0.544772 0.838584i \(-0.683384\pi\)
−0.544772 + 0.838584i \(0.683384\pi\)
\(140\) 0 0
\(141\) 10.9462 0.921834
\(142\) 0 0
\(143\) −5.25939 −0.439812
\(144\) 0 0
\(145\) 5.09212 0.422878
\(146\) 0 0
\(147\) 2.24801 0.185413
\(148\) 0 0
\(149\) −22.9907 −1.88348 −0.941738 0.336348i \(-0.890808\pi\)
−0.941738 + 0.336348i \(0.890808\pi\)
\(150\) 0 0
\(151\) 1.76768 0.143852 0.0719260 0.997410i \(-0.477085\pi\)
0.0719260 + 0.997410i \(0.477085\pi\)
\(152\) 0 0
\(153\) −4.91945 −0.397714
\(154\) 0 0
\(155\) −1.54157 −0.123822
\(156\) 0 0
\(157\) −6.57660 −0.524870 −0.262435 0.964950i \(-0.584525\pi\)
−0.262435 + 0.964950i \(0.584525\pi\)
\(158\) 0 0
\(159\) 6.46439 0.512659
\(160\) 0 0
\(161\) 17.8615 1.40768
\(162\) 0 0
\(163\) 3.01353 0.236038 0.118019 0.993011i \(-0.462346\pi\)
0.118019 + 0.993011i \(0.462346\pi\)
\(164\) 0 0
\(165\) −6.57262 −0.511678
\(166\) 0 0
\(167\) 6.45533 0.499529 0.249764 0.968307i \(-0.419647\pi\)
0.249764 + 0.968307i \(0.419647\pi\)
\(168\) 0 0
\(169\) −11.6669 −0.897451
\(170\) 0 0
\(171\) 0.917992 0.0702006
\(172\) 0 0
\(173\) −11.6824 −0.888195 −0.444097 0.895979i \(-0.646476\pi\)
−0.444097 + 0.895979i \(0.646476\pi\)
\(174\) 0 0
\(175\) −2.92540 −0.221139
\(176\) 0 0
\(177\) −3.02945 −0.227707
\(178\) 0 0
\(179\) 15.5646 1.16336 0.581678 0.813419i \(-0.302396\pi\)
0.581678 + 0.813419i \(0.302396\pi\)
\(180\) 0 0
\(181\) −0.603651 −0.0448690 −0.0224345 0.999748i \(-0.507142\pi\)
−0.0224345 + 0.999748i \(0.507142\pi\)
\(182\) 0 0
\(183\) 2.29442 0.169608
\(184\) 0 0
\(185\) 8.92953 0.656512
\(186\) 0 0
\(187\) 24.4104 1.78506
\(188\) 0 0
\(189\) 16.5383 1.20298
\(190\) 0 0
\(191\) 3.09611 0.224026 0.112013 0.993707i \(-0.464270\pi\)
0.112013 + 0.993707i \(0.464270\pi\)
\(192\) 0 0
\(193\) −20.0144 −1.44067 −0.720335 0.693626i \(-0.756012\pi\)
−0.720335 + 0.693626i \(0.756012\pi\)
\(194\) 0 0
\(195\) 1.66602 0.119306
\(196\) 0 0
\(197\) 15.9790 1.13846 0.569228 0.822180i \(-0.307242\pi\)
0.569228 + 0.822180i \(0.307242\pi\)
\(198\) 0 0
\(199\) −4.38254 −0.310670 −0.155335 0.987862i \(-0.549646\pi\)
−0.155335 + 0.987862i \(0.549646\pi\)
\(200\) 0 0
\(201\) 5.39266 0.380369
\(202\) 0 0
\(203\) 14.8965 1.04553
\(204\) 0 0
\(205\) 2.88583 0.201555
\(206\) 0 0
\(207\) 5.60494 0.389570
\(208\) 0 0
\(209\) −4.55509 −0.315082
\(210\) 0 0
\(211\) 0.759249 0.0522689 0.0261344 0.999658i \(-0.491680\pi\)
0.0261344 + 0.999658i \(0.491680\pi\)
\(212\) 0 0
\(213\) 5.38588 0.369034
\(214\) 0 0
\(215\) 7.86833 0.536615
\(216\) 0 0
\(217\) −4.50970 −0.306138
\(218\) 0 0
\(219\) −14.1656 −0.957224
\(220\) 0 0
\(221\) −6.18749 −0.416216
\(222\) 0 0
\(223\) 12.9078 0.864368 0.432184 0.901785i \(-0.357743\pi\)
0.432184 + 0.901785i \(0.357743\pi\)
\(224\) 0 0
\(225\) −0.917992 −0.0611995
\(226\) 0 0
\(227\) 22.7792 1.51191 0.755955 0.654624i \(-0.227173\pi\)
0.755955 + 0.654624i \(0.227173\pi\)
\(228\) 0 0
\(229\) 10.4181 0.688449 0.344225 0.938887i \(-0.388142\pi\)
0.344225 + 0.938887i \(0.388142\pi\)
\(230\) 0 0
\(231\) −19.2275 −1.26508
\(232\) 0 0
\(233\) −13.2432 −0.867589 −0.433794 0.901012i \(-0.642826\pi\)
−0.433794 + 0.901012i \(0.642826\pi\)
\(234\) 0 0
\(235\) −7.58614 −0.494865
\(236\) 0 0
\(237\) −6.64268 −0.431489
\(238\) 0 0
\(239\) 24.6568 1.59491 0.797457 0.603376i \(-0.206178\pi\)
0.797457 + 0.603376i \(0.206178\pi\)
\(240\) 0 0
\(241\) 24.1951 1.55854 0.779271 0.626687i \(-0.215589\pi\)
0.779271 + 0.626687i \(0.215589\pi\)
\(242\) 0 0
\(243\) 9.16348 0.587837
\(244\) 0 0
\(245\) −1.55797 −0.0995348
\(246\) 0 0
\(247\) 1.15462 0.0734665
\(248\) 0 0
\(249\) −14.2782 −0.904844
\(250\) 0 0
\(251\) 23.0140 1.45263 0.726316 0.687361i \(-0.241231\pi\)
0.726316 + 0.687361i \(0.241231\pi\)
\(252\) 0 0
\(253\) −27.8118 −1.74851
\(254\) 0 0
\(255\) −7.73247 −0.484226
\(256\) 0 0
\(257\) 9.58128 0.597664 0.298832 0.954306i \(-0.403403\pi\)
0.298832 + 0.954306i \(0.403403\pi\)
\(258\) 0 0
\(259\) 26.1224 1.62317
\(260\) 0 0
\(261\) 4.67453 0.289346
\(262\) 0 0
\(263\) −20.5381 −1.26643 −0.633215 0.773976i \(-0.718265\pi\)
−0.633215 + 0.773976i \(0.718265\pi\)
\(264\) 0 0
\(265\) −4.48008 −0.275209
\(266\) 0 0
\(267\) 2.76590 0.169270
\(268\) 0 0
\(269\) −27.0932 −1.65190 −0.825949 0.563744i \(-0.809360\pi\)
−0.825949 + 0.563744i \(0.809360\pi\)
\(270\) 0 0
\(271\) 6.05357 0.367728 0.183864 0.982952i \(-0.441139\pi\)
0.183864 + 0.982952i \(0.441139\pi\)
\(272\) 0 0
\(273\) 4.87376 0.294973
\(274\) 0 0
\(275\) 4.55509 0.274682
\(276\) 0 0
\(277\) 3.69253 0.221863 0.110931 0.993828i \(-0.464617\pi\)
0.110931 + 0.993828i \(0.464617\pi\)
\(278\) 0 0
\(279\) −1.41515 −0.0847226
\(280\) 0 0
\(281\) −22.4212 −1.33754 −0.668768 0.743471i \(-0.733178\pi\)
−0.668768 + 0.743471i \(0.733178\pi\)
\(282\) 0 0
\(283\) −9.02152 −0.536274 −0.268137 0.963381i \(-0.586408\pi\)
−0.268137 + 0.963381i \(0.586408\pi\)
\(284\) 0 0
\(285\) 1.44292 0.0854710
\(286\) 0 0
\(287\) 8.44222 0.498328
\(288\) 0 0
\(289\) 11.7180 0.689294
\(290\) 0 0
\(291\) −22.5659 −1.32284
\(292\) 0 0
\(293\) 8.55922 0.500035 0.250017 0.968241i \(-0.419564\pi\)
0.250017 + 0.968241i \(0.419564\pi\)
\(294\) 0 0
\(295\) 2.09953 0.122240
\(296\) 0 0
\(297\) −25.7515 −1.49425
\(298\) 0 0
\(299\) 7.04968 0.407694
\(300\) 0 0
\(301\) 23.0180 1.32674
\(302\) 0 0
\(303\) −12.3443 −0.709161
\(304\) 0 0
\(305\) −1.59013 −0.0910503
\(306\) 0 0
\(307\) 1.07318 0.0612498 0.0306249 0.999531i \(-0.490250\pi\)
0.0306249 + 0.999531i \(0.490250\pi\)
\(308\) 0 0
\(309\) −13.0121 −0.740234
\(310\) 0 0
\(311\) 9.99150 0.566566 0.283283 0.959036i \(-0.408576\pi\)
0.283283 + 0.959036i \(0.408576\pi\)
\(312\) 0 0
\(313\) 24.6935 1.39576 0.697878 0.716216i \(-0.254128\pi\)
0.697878 + 0.716216i \(0.254128\pi\)
\(314\) 0 0
\(315\) −2.68549 −0.151310
\(316\) 0 0
\(317\) −9.23375 −0.518619 −0.259310 0.965794i \(-0.583495\pi\)
−0.259310 + 0.965794i \(0.583495\pi\)
\(318\) 0 0
\(319\) −23.1951 −1.29868
\(320\) 0 0
\(321\) 28.1051 1.56867
\(322\) 0 0
\(323\) −5.35892 −0.298178
\(324\) 0 0
\(325\) −1.15462 −0.0640466
\(326\) 0 0
\(327\) 11.7789 0.651376
\(328\) 0 0
\(329\) −22.1925 −1.22351
\(330\) 0 0
\(331\) −8.22994 −0.452358 −0.226179 0.974086i \(-0.572623\pi\)
−0.226179 + 0.974086i \(0.572623\pi\)
\(332\) 0 0
\(333\) 8.19724 0.449206
\(334\) 0 0
\(335\) −3.73733 −0.204192
\(336\) 0 0
\(337\) 6.04627 0.329362 0.164681 0.986347i \(-0.447341\pi\)
0.164681 + 0.986347i \(0.447341\pi\)
\(338\) 0 0
\(339\) −21.0511 −1.14334
\(340\) 0 0
\(341\) 7.02198 0.380262
\(342\) 0 0
\(343\) 15.9201 0.859606
\(344\) 0 0
\(345\) 8.80994 0.474311
\(346\) 0 0
\(347\) 17.1938 0.923011 0.461506 0.887137i \(-0.347309\pi\)
0.461506 + 0.887137i \(0.347309\pi\)
\(348\) 0 0
\(349\) 7.79272 0.417135 0.208567 0.978008i \(-0.433120\pi\)
0.208567 + 0.978008i \(0.433120\pi\)
\(350\) 0 0
\(351\) 6.52743 0.348409
\(352\) 0 0
\(353\) −10.2758 −0.546925 −0.273462 0.961883i \(-0.588169\pi\)
−0.273462 + 0.961883i \(0.588169\pi\)
\(354\) 0 0
\(355\) −3.73263 −0.198108
\(356\) 0 0
\(357\) −22.6206 −1.19721
\(358\) 0 0
\(359\) 1.51424 0.0799183 0.0399591 0.999201i \(-0.487277\pi\)
0.0399591 + 0.999201i \(0.487277\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 14.0668 0.738316
\(364\) 0 0
\(365\) 9.81735 0.513864
\(366\) 0 0
\(367\) −2.02549 −0.105730 −0.0528648 0.998602i \(-0.516835\pi\)
−0.0528648 + 0.998602i \(0.516835\pi\)
\(368\) 0 0
\(369\) 2.64917 0.137910
\(370\) 0 0
\(371\) −13.1060 −0.680432
\(372\) 0 0
\(373\) 32.6834 1.69228 0.846142 0.532958i \(-0.178920\pi\)
0.846142 + 0.532958i \(0.178920\pi\)
\(374\) 0 0
\(375\) −1.44292 −0.0745119
\(376\) 0 0
\(377\) 5.87945 0.302807
\(378\) 0 0
\(379\) −5.64250 −0.289836 −0.144918 0.989444i \(-0.546292\pi\)
−0.144918 + 0.989444i \(0.546292\pi\)
\(380\) 0 0
\(381\) −14.4221 −0.738869
\(382\) 0 0
\(383\) 20.2873 1.03663 0.518317 0.855188i \(-0.326559\pi\)
0.518317 + 0.855188i \(0.326559\pi\)
\(384\) 0 0
\(385\) 13.3255 0.679129
\(386\) 0 0
\(387\) 7.22306 0.367169
\(388\) 0 0
\(389\) 32.6268 1.65424 0.827121 0.562024i \(-0.189977\pi\)
0.827121 + 0.562024i \(0.189977\pi\)
\(390\) 0 0
\(391\) −32.7197 −1.65471
\(392\) 0 0
\(393\) 5.94217 0.299743
\(394\) 0 0
\(395\) 4.60365 0.231635
\(396\) 0 0
\(397\) 6.23917 0.313135 0.156567 0.987667i \(-0.449957\pi\)
0.156567 + 0.987667i \(0.449957\pi\)
\(398\) 0 0
\(399\) 4.22111 0.211320
\(400\) 0 0
\(401\) −12.4252 −0.620484 −0.310242 0.950658i \(-0.600410\pi\)
−0.310242 + 0.950658i \(0.600410\pi\)
\(402\) 0 0
\(403\) −1.77992 −0.0886641
\(404\) 0 0
\(405\) 5.40331 0.268493
\(406\) 0 0
\(407\) −40.6748 −2.01618
\(408\) 0 0
\(409\) 35.4878 1.75476 0.877378 0.479799i \(-0.159291\pi\)
0.877378 + 0.479799i \(0.159291\pi\)
\(410\) 0 0
\(411\) −16.9561 −0.836383
\(412\) 0 0
\(413\) 6.14197 0.302227
\(414\) 0 0
\(415\) 9.89538 0.485745
\(416\) 0 0
\(417\) −18.5350 −0.907664
\(418\) 0 0
\(419\) −5.77992 −0.282368 −0.141184 0.989983i \(-0.545091\pi\)
−0.141184 + 0.989983i \(0.545091\pi\)
\(420\) 0 0
\(421\) −5.31824 −0.259195 −0.129598 0.991567i \(-0.541369\pi\)
−0.129598 + 0.991567i \(0.541369\pi\)
\(422\) 0 0
\(423\) −6.96402 −0.338602
\(424\) 0 0
\(425\) 5.35892 0.259946
\(426\) 0 0
\(427\) −4.65175 −0.225114
\(428\) 0 0
\(429\) −7.58885 −0.366393
\(430\) 0 0
\(431\) −34.6708 −1.67004 −0.835018 0.550223i \(-0.814543\pi\)
−0.835018 + 0.550223i \(0.814543\pi\)
\(432\) 0 0
\(433\) 35.4581 1.70401 0.852003 0.523537i \(-0.175388\pi\)
0.852003 + 0.523537i \(0.175388\pi\)
\(434\) 0 0
\(435\) 7.34751 0.352286
\(436\) 0 0
\(437\) 6.10565 0.292073
\(438\) 0 0
\(439\) 0.804934 0.0384174 0.0192087 0.999815i \(-0.493885\pi\)
0.0192087 + 0.999815i \(0.493885\pi\)
\(440\) 0 0
\(441\) −1.43020 −0.0681048
\(442\) 0 0
\(443\) −32.0901 −1.52465 −0.762323 0.647196i \(-0.775941\pi\)
−0.762323 + 0.647196i \(0.775941\pi\)
\(444\) 0 0
\(445\) −1.91688 −0.0908690
\(446\) 0 0
\(447\) −33.1737 −1.56906
\(448\) 0 0
\(449\) 10.0871 0.476041 0.238021 0.971260i \(-0.423501\pi\)
0.238021 + 0.971260i \(0.423501\pi\)
\(450\) 0 0
\(451\) −13.1452 −0.618985
\(452\) 0 0
\(453\) 2.55062 0.119839
\(454\) 0 0
\(455\) −3.37772 −0.158350
\(456\) 0 0
\(457\) −38.2058 −1.78719 −0.893595 0.448874i \(-0.851825\pi\)
−0.893595 + 0.448874i \(0.851825\pi\)
\(458\) 0 0
\(459\) −30.2958 −1.41408
\(460\) 0 0
\(461\) 35.1925 1.63908 0.819540 0.573023i \(-0.194229\pi\)
0.819540 + 0.573023i \(0.194229\pi\)
\(462\) 0 0
\(463\) 18.3358 0.852138 0.426069 0.904691i \(-0.359898\pi\)
0.426069 + 0.904691i \(0.359898\pi\)
\(464\) 0 0
\(465\) −2.22435 −0.103152
\(466\) 0 0
\(467\) −5.05941 −0.234122 −0.117061 0.993125i \(-0.537347\pi\)
−0.117061 + 0.993125i \(0.537347\pi\)
\(468\) 0 0
\(469\) −10.9332 −0.504848
\(470\) 0 0
\(471\) −9.48948 −0.437252
\(472\) 0 0
\(473\) −35.8409 −1.64797
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −4.11268 −0.188307
\(478\) 0 0
\(479\) −21.4868 −0.981757 −0.490879 0.871228i \(-0.663324\pi\)
−0.490879 + 0.871228i \(0.663324\pi\)
\(480\) 0 0
\(481\) 10.3102 0.470104
\(482\) 0 0
\(483\) 25.7726 1.17269
\(484\) 0 0
\(485\) 15.6391 0.710135
\(486\) 0 0
\(487\) −27.7776 −1.25872 −0.629362 0.777112i \(-0.716684\pi\)
−0.629362 + 0.777112i \(0.716684\pi\)
\(488\) 0 0
\(489\) 4.34827 0.196635
\(490\) 0 0
\(491\) −18.4624 −0.833199 −0.416599 0.909090i \(-0.636778\pi\)
−0.416599 + 0.909090i \(0.636778\pi\)
\(492\) 0 0
\(493\) −27.2883 −1.22900
\(494\) 0 0
\(495\) 4.18154 0.187946
\(496\) 0 0
\(497\) −10.9194 −0.489804
\(498\) 0 0
\(499\) 37.0366 1.65799 0.828994 0.559258i \(-0.188914\pi\)
0.828994 + 0.559258i \(0.188914\pi\)
\(500\) 0 0
\(501\) 9.31451 0.416141
\(502\) 0 0
\(503\) −18.5003 −0.824886 −0.412443 0.910984i \(-0.635324\pi\)
−0.412443 + 0.910984i \(0.635324\pi\)
\(504\) 0 0
\(505\) 8.55509 0.380697
\(506\) 0 0
\(507\) −16.8343 −0.747638
\(508\) 0 0
\(509\) −32.9570 −1.46079 −0.730397 0.683023i \(-0.760665\pi\)
−0.730397 + 0.683023i \(0.760665\pi\)
\(510\) 0 0
\(511\) 28.7197 1.27048
\(512\) 0 0
\(513\) 5.65334 0.249601
\(514\) 0 0
\(515\) 9.01793 0.397378
\(516\) 0 0
\(517\) 34.5556 1.51975
\(518\) 0 0
\(519\) −16.8567 −0.739927
\(520\) 0 0
\(521\) −26.8526 −1.17643 −0.588216 0.808704i \(-0.700170\pi\)
−0.588216 + 0.808704i \(0.700170\pi\)
\(522\) 0 0
\(523\) 38.2131 1.67094 0.835472 0.549534i \(-0.185195\pi\)
0.835472 + 0.549534i \(0.185195\pi\)
\(524\) 0 0
\(525\) −4.22111 −0.184224
\(526\) 0 0
\(527\) 8.26113 0.359861
\(528\) 0 0
\(529\) 14.2790 0.620825
\(530\) 0 0
\(531\) 1.92735 0.0836401
\(532\) 0 0
\(533\) 3.33203 0.144326
\(534\) 0 0
\(535\) −19.4780 −0.842106
\(536\) 0 0
\(537\) 22.4585 0.969154
\(538\) 0 0
\(539\) 7.09668 0.305676
\(540\) 0 0
\(541\) 10.4930 0.451129 0.225564 0.974228i \(-0.427577\pi\)
0.225564 + 0.974228i \(0.427577\pi\)
\(542\) 0 0
\(543\) −0.871017 −0.0373789
\(544\) 0 0
\(545\) −8.16328 −0.349676
\(546\) 0 0
\(547\) 38.0140 1.62536 0.812680 0.582710i \(-0.198008\pi\)
0.812680 + 0.582710i \(0.198008\pi\)
\(548\) 0 0
\(549\) −1.45972 −0.0622995
\(550\) 0 0
\(551\) 5.09212 0.216932
\(552\) 0 0
\(553\) 13.4675 0.572697
\(554\) 0 0
\(555\) 12.8846 0.546919
\(556\) 0 0
\(557\) −15.6925 −0.664914 −0.332457 0.943118i \(-0.607878\pi\)
−0.332457 + 0.943118i \(0.607878\pi\)
\(558\) 0 0
\(559\) 9.08490 0.384250
\(560\) 0 0
\(561\) 35.2221 1.48708
\(562\) 0 0
\(563\) 3.58794 0.151213 0.0756067 0.997138i \(-0.475911\pi\)
0.0756067 + 0.997138i \(0.475911\pi\)
\(564\) 0 0
\(565\) 14.5893 0.613775
\(566\) 0 0
\(567\) 15.8069 0.663825
\(568\) 0 0
\(569\) 28.1242 1.17903 0.589514 0.807758i \(-0.299319\pi\)
0.589514 + 0.807758i \(0.299319\pi\)
\(570\) 0 0
\(571\) 41.3658 1.73111 0.865553 0.500818i \(-0.166967\pi\)
0.865553 + 0.500818i \(0.166967\pi\)
\(572\) 0 0
\(573\) 4.46742 0.186629
\(574\) 0 0
\(575\) −6.10565 −0.254623
\(576\) 0 0
\(577\) −27.6480 −1.15100 −0.575500 0.817802i \(-0.695192\pi\)
−0.575500 + 0.817802i \(0.695192\pi\)
\(578\) 0 0
\(579\) −28.8791 −1.20018
\(580\) 0 0
\(581\) 28.9479 1.20096
\(582\) 0 0
\(583\) 20.4072 0.845180
\(584\) 0 0
\(585\) −1.05993 −0.0438227
\(586\) 0 0
\(587\) −25.2812 −1.04347 −0.521733 0.853109i \(-0.674714\pi\)
−0.521733 + 0.853109i \(0.674714\pi\)
\(588\) 0 0
\(589\) −1.54157 −0.0635192
\(590\) 0 0
\(591\) 23.0563 0.948411
\(592\) 0 0
\(593\) −29.9920 −1.23163 −0.615813 0.787893i \(-0.711172\pi\)
−0.615813 + 0.787893i \(0.711172\pi\)
\(594\) 0 0
\(595\) 15.6770 0.642693
\(596\) 0 0
\(597\) −6.32364 −0.258809
\(598\) 0 0
\(599\) 23.4775 0.959266 0.479633 0.877469i \(-0.340770\pi\)
0.479633 + 0.877469i \(0.340770\pi\)
\(600\) 0 0
\(601\) −0.213065 −0.00869108 −0.00434554 0.999991i \(-0.501383\pi\)
−0.00434554 + 0.999991i \(0.501383\pi\)
\(602\) 0 0
\(603\) −3.43084 −0.139715
\(604\) 0 0
\(605\) −9.74887 −0.396348
\(606\) 0 0
\(607\) −32.3246 −1.31202 −0.656008 0.754754i \(-0.727756\pi\)
−0.656008 + 0.754754i \(0.727756\pi\)
\(608\) 0 0
\(609\) 21.4944 0.870997
\(610\) 0 0
\(611\) −8.75909 −0.354355
\(612\) 0 0
\(613\) 39.5596 1.59780 0.798898 0.601466i \(-0.205417\pi\)
0.798898 + 0.601466i \(0.205417\pi\)
\(614\) 0 0
\(615\) 4.16402 0.167909
\(616\) 0 0
\(617\) −37.8781 −1.52492 −0.762458 0.647037i \(-0.776008\pi\)
−0.762458 + 0.647037i \(0.776008\pi\)
\(618\) 0 0
\(619\) 6.77119 0.272157 0.136079 0.990698i \(-0.456550\pi\)
0.136079 + 0.990698i \(0.456550\pi\)
\(620\) 0 0
\(621\) 34.5173 1.38513
\(622\) 0 0
\(623\) −5.60765 −0.224666
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.57262 −0.262485
\(628\) 0 0
\(629\) −47.8526 −1.90801
\(630\) 0 0
\(631\) −28.8072 −1.14680 −0.573398 0.819277i \(-0.694375\pi\)
−0.573398 + 0.819277i \(0.694375\pi\)
\(632\) 0 0
\(633\) 1.09553 0.0435435
\(634\) 0 0
\(635\) 9.99514 0.396645
\(636\) 0 0
\(637\) −1.79885 −0.0712731
\(638\) 0 0
\(639\) −3.42653 −0.135551
\(640\) 0 0
\(641\) 4.50030 0.177751 0.0888756 0.996043i \(-0.471673\pi\)
0.0888756 + 0.996043i \(0.471673\pi\)
\(642\) 0 0
\(643\) 26.7345 1.05431 0.527154 0.849770i \(-0.323259\pi\)
0.527154 + 0.849770i \(0.323259\pi\)
\(644\) 0 0
\(645\) 11.3533 0.447037
\(646\) 0 0
\(647\) 12.9066 0.507411 0.253705 0.967282i \(-0.418351\pi\)
0.253705 + 0.967282i \(0.418351\pi\)
\(648\) 0 0
\(649\) −9.56357 −0.375403
\(650\) 0 0
\(651\) −6.50712 −0.255034
\(652\) 0 0
\(653\) 20.0704 0.785414 0.392707 0.919664i \(-0.371539\pi\)
0.392707 + 0.919664i \(0.371539\pi\)
\(654\) 0 0
\(655\) −4.11817 −0.160910
\(656\) 0 0
\(657\) 9.01225 0.351601
\(658\) 0 0
\(659\) −31.1534 −1.21356 −0.606782 0.794868i \(-0.707540\pi\)
−0.606782 + 0.794868i \(0.707540\pi\)
\(660\) 0 0
\(661\) 26.3854 1.02627 0.513137 0.858307i \(-0.328483\pi\)
0.513137 + 0.858307i \(0.328483\pi\)
\(662\) 0 0
\(663\) −8.92804 −0.346736
\(664\) 0 0
\(665\) −2.92540 −0.113442
\(666\) 0 0
\(667\) 31.0907 1.20384
\(668\) 0 0
\(669\) 18.6248 0.720077
\(670\) 0 0
\(671\) 7.24317 0.279619
\(672\) 0 0
\(673\) −19.0512 −0.734371 −0.367185 0.930148i \(-0.619679\pi\)
−0.367185 + 0.930148i \(0.619679\pi\)
\(674\) 0 0
\(675\) −5.65334 −0.217597
\(676\) 0 0
\(677\) −22.5780 −0.867744 −0.433872 0.900975i \(-0.642853\pi\)
−0.433872 + 0.900975i \(0.642853\pi\)
\(678\) 0 0
\(679\) 45.7506 1.75575
\(680\) 0 0
\(681\) 32.8685 1.25952
\(682\) 0 0
\(683\) −39.2754 −1.50283 −0.751415 0.659830i \(-0.770628\pi\)
−0.751415 + 0.659830i \(0.770628\pi\)
\(684\) 0 0
\(685\) 11.7513 0.448993
\(686\) 0 0
\(687\) 15.0325 0.573525
\(688\) 0 0
\(689\) −5.17278 −0.197067
\(690\) 0 0
\(691\) 21.3498 0.812185 0.406093 0.913832i \(-0.366891\pi\)
0.406093 + 0.913832i \(0.366891\pi\)
\(692\) 0 0
\(693\) 12.2327 0.464681
\(694\) 0 0
\(695\) 12.8455 0.487259
\(696\) 0 0
\(697\) −15.4649 −0.585776
\(698\) 0 0
\(699\) −19.1088 −0.722761
\(700\) 0 0
\(701\) 45.8535 1.73186 0.865932 0.500161i \(-0.166726\pi\)
0.865932 + 0.500161i \(0.166726\pi\)
\(702\) 0 0
\(703\) 8.92953 0.336783
\(704\) 0 0
\(705\) −10.9462 −0.412257
\(706\) 0 0
\(707\) 25.0271 0.941240
\(708\) 0 0
\(709\) 6.34908 0.238445 0.119222 0.992868i \(-0.461960\pi\)
0.119222 + 0.992868i \(0.461960\pi\)
\(710\) 0 0
\(711\) 4.22612 0.158492
\(712\) 0 0
\(713\) −9.41227 −0.352492
\(714\) 0 0
\(715\) 5.25939 0.196690
\(716\) 0 0
\(717\) 35.5776 1.32867
\(718\) 0 0
\(719\) 21.5313 0.802982 0.401491 0.915863i \(-0.368492\pi\)
0.401491 + 0.915863i \(0.368492\pi\)
\(720\) 0 0
\(721\) 26.3811 0.982482
\(722\) 0 0
\(723\) 34.9115 1.29837
\(724\) 0 0
\(725\) −5.09212 −0.189117
\(726\) 0 0
\(727\) −49.4459 −1.83385 −0.916924 0.399062i \(-0.869336\pi\)
−0.916924 + 0.399062i \(0.869336\pi\)
\(728\) 0 0
\(729\) 29.4321 1.09008
\(730\) 0 0
\(731\) −42.1657 −1.55955
\(732\) 0 0
\(733\) −53.0404 −1.95909 −0.979546 0.201220i \(-0.935509\pi\)
−0.979546 + 0.201220i \(0.935509\pi\)
\(734\) 0 0
\(735\) −2.24801 −0.0829192
\(736\) 0 0
\(737\) 17.0239 0.627084
\(738\) 0 0
\(739\) −26.3898 −0.970765 −0.485382 0.874302i \(-0.661320\pi\)
−0.485382 + 0.874302i \(0.661320\pi\)
\(740\) 0 0
\(741\) 1.66602 0.0612026
\(742\) 0 0
\(743\) −14.2042 −0.521102 −0.260551 0.965460i \(-0.583904\pi\)
−0.260551 + 0.965460i \(0.583904\pi\)
\(744\) 0 0
\(745\) 22.9907 0.842316
\(746\) 0 0
\(747\) 9.08388 0.332362
\(748\) 0 0
\(749\) −56.9808 −2.08203
\(750\) 0 0
\(751\) 31.4599 1.14799 0.573994 0.818860i \(-0.305393\pi\)
0.573994 + 0.818860i \(0.305393\pi\)
\(752\) 0 0
\(753\) 33.2073 1.21014
\(754\) 0 0
\(755\) −1.76768 −0.0643326
\(756\) 0 0
\(757\) 15.3981 0.559654 0.279827 0.960050i \(-0.409723\pi\)
0.279827 + 0.960050i \(0.409723\pi\)
\(758\) 0 0
\(759\) −40.1301 −1.45663
\(760\) 0 0
\(761\) 12.4838 0.452537 0.226269 0.974065i \(-0.427347\pi\)
0.226269 + 0.974065i \(0.427347\pi\)
\(762\) 0 0
\(763\) −23.8809 −0.864545
\(764\) 0 0
\(765\) 4.91945 0.177863
\(766\) 0 0
\(767\) 2.42416 0.0875312
\(768\) 0 0
\(769\) −23.4197 −0.844538 −0.422269 0.906471i \(-0.638766\pi\)
−0.422269 + 0.906471i \(0.638766\pi\)
\(770\) 0 0
\(771\) 13.8250 0.497895
\(772\) 0 0
\(773\) 52.0914 1.87360 0.936799 0.349868i \(-0.113774\pi\)
0.936799 + 0.349868i \(0.113774\pi\)
\(774\) 0 0
\(775\) 1.54157 0.0553747
\(776\) 0 0
\(777\) 37.6925 1.35221
\(778\) 0 0
\(779\) 2.88583 0.103396
\(780\) 0 0
\(781\) 17.0025 0.608397
\(782\) 0 0
\(783\) 28.7875 1.02878
\(784\) 0 0
\(785\) 6.57660 0.234729
\(786\) 0 0
\(787\) −8.88515 −0.316721 −0.158361 0.987381i \(-0.550621\pi\)
−0.158361 + 0.987381i \(0.550621\pi\)
\(788\) 0 0
\(789\) −29.6347 −1.05502
\(790\) 0 0
\(791\) 42.6794 1.51751
\(792\) 0 0
\(793\) −1.83598 −0.0651977
\(794\) 0 0
\(795\) −6.46439 −0.229268
\(796\) 0 0
\(797\) −23.0257 −0.815610 −0.407805 0.913069i \(-0.633706\pi\)
−0.407805 + 0.913069i \(0.633706\pi\)
\(798\) 0 0
\(799\) 40.6535 1.43822
\(800\) 0 0
\(801\) −1.75968 −0.0621754
\(802\) 0 0
\(803\) −44.7189 −1.57810
\(804\) 0 0
\(805\) −17.8615 −0.629534
\(806\) 0 0
\(807\) −39.0932 −1.37614
\(808\) 0 0
\(809\) 32.9398 1.15810 0.579051 0.815291i \(-0.303423\pi\)
0.579051 + 0.815291i \(0.303423\pi\)
\(810\) 0 0
\(811\) 28.7407 1.00922 0.504612 0.863346i \(-0.331636\pi\)
0.504612 + 0.863346i \(0.331636\pi\)
\(812\) 0 0
\(813\) 8.73479 0.306342
\(814\) 0 0
\(815\) −3.01353 −0.105559
\(816\) 0 0
\(817\) 7.86833 0.275278
\(818\) 0 0
\(819\) −3.10072 −0.108348
\(820\) 0 0
\(821\) 21.9542 0.766205 0.383103 0.923706i \(-0.374856\pi\)
0.383103 + 0.923706i \(0.374856\pi\)
\(822\) 0 0
\(823\) −31.2671 −1.08990 −0.544951 0.838468i \(-0.683452\pi\)
−0.544951 + 0.838468i \(0.683452\pi\)
\(824\) 0 0
\(825\) 6.57262 0.228829
\(826\) 0 0
\(827\) −16.9193 −0.588343 −0.294172 0.955753i \(-0.595044\pi\)
−0.294172 + 0.955753i \(0.595044\pi\)
\(828\) 0 0
\(829\) 2.71964 0.0944571 0.0472286 0.998884i \(-0.484961\pi\)
0.0472286 + 0.998884i \(0.484961\pi\)
\(830\) 0 0
\(831\) 5.32801 0.184827
\(832\) 0 0
\(833\) 8.34901 0.289276
\(834\) 0 0
\(835\) −6.45533 −0.223396
\(836\) 0 0
\(837\) −8.71500 −0.301234
\(838\) 0 0
\(839\) 35.7791 1.23523 0.617617 0.786479i \(-0.288098\pi\)
0.617617 + 0.786479i \(0.288098\pi\)
\(840\) 0 0
\(841\) −3.07026 −0.105871
\(842\) 0 0
\(843\) −32.3519 −1.11426
\(844\) 0 0
\(845\) 11.6669 0.401352
\(846\) 0 0
\(847\) −28.5193 −0.979936
\(848\) 0 0
\(849\) −13.0173 −0.446753
\(850\) 0 0
\(851\) 54.5206 1.86894
\(852\) 0 0
\(853\) −49.9150 −1.70906 −0.854528 0.519406i \(-0.826153\pi\)
−0.854528 + 0.519406i \(0.826153\pi\)
\(854\) 0 0
\(855\) −0.917992 −0.0313947
\(856\) 0 0
\(857\) −21.1297 −0.721778 −0.360889 0.932609i \(-0.617527\pi\)
−0.360889 + 0.932609i \(0.617527\pi\)
\(858\) 0 0
\(859\) −34.5319 −1.17821 −0.589106 0.808056i \(-0.700520\pi\)
−0.589106 + 0.808056i \(0.700520\pi\)
\(860\) 0 0
\(861\) 12.1814 0.415141
\(862\) 0 0
\(863\) −29.2411 −0.995380 −0.497690 0.867355i \(-0.665818\pi\)
−0.497690 + 0.867355i \(0.665818\pi\)
\(864\) 0 0
\(865\) 11.6824 0.397213
\(866\) 0 0
\(867\) 16.9081 0.574229
\(868\) 0 0
\(869\) −20.9701 −0.711360
\(870\) 0 0
\(871\) −4.31519 −0.146215
\(872\) 0 0
\(873\) 14.3566 0.485897
\(874\) 0 0
\(875\) 2.92540 0.0988966
\(876\) 0 0
\(877\) −42.4819 −1.43451 −0.717256 0.696810i \(-0.754602\pi\)
−0.717256 + 0.696810i \(0.754602\pi\)
\(878\) 0 0
\(879\) 12.3502 0.416563
\(880\) 0 0
\(881\) 27.2758 0.918946 0.459473 0.888192i \(-0.348038\pi\)
0.459473 + 0.888192i \(0.348038\pi\)
\(882\) 0 0
\(883\) −8.38201 −0.282077 −0.141039 0.990004i \(-0.545044\pi\)
−0.141039 + 0.990004i \(0.545044\pi\)
\(884\) 0 0
\(885\) 3.02945 0.101834
\(886\) 0 0
\(887\) −44.0775 −1.47998 −0.739988 0.672620i \(-0.765169\pi\)
−0.739988 + 0.672620i \(0.765169\pi\)
\(888\) 0 0
\(889\) 29.2398 0.980670
\(890\) 0 0
\(891\) −24.6126 −0.824553
\(892\) 0 0
\(893\) −7.58614 −0.253861
\(894\) 0 0
\(895\) −15.5646 −0.520268
\(896\) 0 0
\(897\) 10.1721 0.339637
\(898\) 0 0
\(899\) −7.84985 −0.261807
\(900\) 0 0
\(901\) 24.0084 0.799836
\(902\) 0 0
\(903\) 33.2130 1.10526
\(904\) 0 0
\(905\) 0.603651 0.0200660
\(906\) 0 0
\(907\) −24.4179 −0.810784 −0.405392 0.914143i \(-0.632865\pi\)
−0.405392 + 0.914143i \(0.632865\pi\)
\(908\) 0 0
\(909\) 7.85351 0.260484
\(910\) 0 0
\(911\) 5.36495 0.177749 0.0888744 0.996043i \(-0.471673\pi\)
0.0888744 + 0.996043i \(0.471673\pi\)
\(912\) 0 0
\(913\) −45.0744 −1.49174
\(914\) 0 0
\(915\) −2.29442 −0.0758511
\(916\) 0 0
\(917\) −12.0473 −0.397836
\(918\) 0 0
\(919\) 12.8165 0.422777 0.211389 0.977402i \(-0.432201\pi\)
0.211389 + 0.977402i \(0.432201\pi\)
\(920\) 0 0
\(921\) 1.54851 0.0510252
\(922\) 0 0
\(923\) −4.30976 −0.141857
\(924\) 0 0
\(925\) −8.92953 −0.293601
\(926\) 0 0
\(927\) 8.27839 0.271898
\(928\) 0 0
\(929\) −42.4601 −1.39307 −0.696535 0.717523i \(-0.745276\pi\)
−0.696535 + 0.717523i \(0.745276\pi\)
\(930\) 0 0
\(931\) −1.55797 −0.0510603
\(932\) 0 0
\(933\) 14.4169 0.471988
\(934\) 0 0
\(935\) −24.4104 −0.798304
\(936\) 0 0
\(937\) 59.5062 1.94398 0.971991 0.235017i \(-0.0755146\pi\)
0.971991 + 0.235017i \(0.0755146\pi\)
\(938\) 0 0
\(939\) 35.6306 1.16276
\(940\) 0 0
\(941\) −52.9888 −1.72739 −0.863693 0.504018i \(-0.831854\pi\)
−0.863693 + 0.504018i \(0.831854\pi\)
\(942\) 0 0
\(943\) 17.6199 0.573782
\(944\) 0 0
\(945\) −16.5383 −0.537990
\(946\) 0 0
\(947\) 40.7246 1.32337 0.661685 0.749782i \(-0.269842\pi\)
0.661685 + 0.749782i \(0.269842\pi\)
\(948\) 0 0
\(949\) 11.3353 0.367959
\(950\) 0 0
\(951\) −13.3235 −0.432045
\(952\) 0 0
\(953\) −15.3515 −0.497284 −0.248642 0.968595i \(-0.579984\pi\)
−0.248642 + 0.968595i \(0.579984\pi\)
\(954\) 0 0
\(955\) −3.09611 −0.100188
\(956\) 0 0
\(957\) −33.4686 −1.08189
\(958\) 0 0
\(959\) 34.3772 1.11010
\(960\) 0 0
\(961\) −28.6236 −0.923341
\(962\) 0 0
\(963\) −17.8806 −0.576195
\(964\) 0 0
\(965\) 20.0144 0.644287
\(966\) 0 0
\(967\) −4.61690 −0.148470 −0.0742348 0.997241i \(-0.523651\pi\)
−0.0742348 + 0.997241i \(0.523651\pi\)
\(968\) 0 0
\(969\) −7.73247 −0.248403
\(970\) 0 0
\(971\) 11.9619 0.383876 0.191938 0.981407i \(-0.438523\pi\)
0.191938 + 0.981407i \(0.438523\pi\)
\(972\) 0 0
\(973\) 37.5783 1.20471
\(974\) 0 0
\(975\) −1.66602 −0.0533552
\(976\) 0 0
\(977\) −41.1459 −1.31637 −0.658187 0.752854i \(-0.728676\pi\)
−0.658187 + 0.752854i \(0.728676\pi\)
\(978\) 0 0
\(979\) 8.73158 0.279063
\(980\) 0 0
\(981\) −7.49383 −0.239259
\(982\) 0 0
\(983\) 59.4969 1.89766 0.948828 0.315793i \(-0.102271\pi\)
0.948828 + 0.315793i \(0.102271\pi\)
\(984\) 0 0
\(985\) −15.9790 −0.509133
\(986\) 0 0
\(987\) −32.0219 −1.01927
\(988\) 0 0
\(989\) 48.0412 1.52762
\(990\) 0 0
\(991\) 52.4403 1.66582 0.832910 0.553408i \(-0.186673\pi\)
0.832910 + 0.553408i \(0.186673\pi\)
\(992\) 0 0
\(993\) −11.8751 −0.376845
\(994\) 0 0
\(995\) 4.38254 0.138936
\(996\) 0 0
\(997\) −26.9834 −0.854574 −0.427287 0.904116i \(-0.640531\pi\)
−0.427287 + 0.904116i \(0.640531\pi\)
\(998\) 0 0
\(999\) 50.4816 1.59717
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 3040.2.a.x.1.3 yes 5
4.3 odd 2 3040.2.a.u.1.3 5
8.3 odd 2 6080.2.a.cl.1.3 5
8.5 even 2 6080.2.a.ci.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.3 5 4.3 odd 2
3040.2.a.x.1.3 yes 5 1.1 even 1 trivial
6080.2.a.ci.1.3 5 8.5 even 2
6080.2.a.cl.1.3 5 8.3 odd 2