Properties

Label 7440.2.a.bs.1.2
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7440,2,Mod(1,7440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7440.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.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: \(59.4086991038\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.42801 q^{7} +1.00000 q^{9} -1.14399 q^{11} +1.57199 q^{13} -1.00000 q^{15} +4.67282 q^{17} -5.34565 q^{19} -2.42801 q^{21} +1.81681 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.95684 q^{29} -1.00000 q^{31} -1.14399 q^{33} +2.42801 q^{35} +3.57199 q^{37} +1.57199 q^{39} -3.14399 q^{41} -2.85601 q^{43} -1.00000 q^{45} -7.16246 q^{47} -1.10478 q^{49} +4.67282 q^{51} -3.81681 q^{53} +1.14399 q^{55} -5.34565 q^{57} +6.24482 q^{59} -2.00000 q^{61} -2.42801 q^{63} -1.57199 q^{65} +0.917641 q^{67} +1.81681 q^{69} +6.44648 q^{71} +6.71598 q^{73} +1.00000 q^{75} +2.77761 q^{77} -9.52884 q^{79} +1.00000 q^{81} +3.16246 q^{83} -4.67282 q^{85} +1.95684 q^{87} -15.2241 q^{89} -3.81681 q^{91} -1.00000 q^{93} +5.34565 q^{95} +10.7776 q^{97} -1.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.42801 −0.917700 −0.458850 0.888514i \(-0.651739\pi\)
−0.458850 + 0.888514i \(0.651739\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.14399 −0.344925 −0.172462 0.985016i \(-0.555172\pi\)
−0.172462 + 0.985016i \(0.555172\pi\)
\(12\) 0 0
\(13\) 1.57199 0.435992 0.217996 0.975950i \(-0.430048\pi\)
0.217996 + 0.975950i \(0.430048\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.67282 1.13333 0.566663 0.823950i \(-0.308234\pi\)
0.566663 + 0.823950i \(0.308234\pi\)
\(18\) 0 0
\(19\) −5.34565 −1.22638 −0.613188 0.789937i \(-0.710113\pi\)
−0.613188 + 0.789937i \(0.710113\pi\)
\(20\) 0 0
\(21\) −2.42801 −0.529835
\(22\) 0 0
\(23\) 1.81681 0.378831 0.189416 0.981897i \(-0.439341\pi\)
0.189416 + 0.981897i \(0.439341\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.95684 0.363377 0.181688 0.983356i \(-0.441844\pi\)
0.181688 + 0.983356i \(0.441844\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −1.14399 −0.199142
\(34\) 0 0
\(35\) 2.42801 0.410408
\(36\) 0 0
\(37\) 3.57199 0.587232 0.293616 0.955923i \(-0.405141\pi\)
0.293616 + 0.955923i \(0.405141\pi\)
\(38\) 0 0
\(39\) 1.57199 0.251720
\(40\) 0 0
\(41\) −3.14399 −0.491008 −0.245504 0.969396i \(-0.578953\pi\)
−0.245504 + 0.969396i \(0.578953\pi\)
\(42\) 0 0
\(43\) −2.85601 −0.435538 −0.217769 0.976000i \(-0.569878\pi\)
−0.217769 + 0.976000i \(0.569878\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −7.16246 −1.04475 −0.522376 0.852715i \(-0.674954\pi\)
−0.522376 + 0.852715i \(0.674954\pi\)
\(48\) 0 0
\(49\) −1.10478 −0.157826
\(50\) 0 0
\(51\) 4.67282 0.654326
\(52\) 0 0
\(53\) −3.81681 −0.524279 −0.262140 0.965030i \(-0.584428\pi\)
−0.262140 + 0.965030i \(0.584428\pi\)
\(54\) 0 0
\(55\) 1.14399 0.154255
\(56\) 0 0
\(57\) −5.34565 −0.708048
\(58\) 0 0
\(59\) 6.24482 0.813006 0.406503 0.913649i \(-0.366748\pi\)
0.406503 + 0.913649i \(0.366748\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −2.42801 −0.305900
\(64\) 0 0
\(65\) −1.57199 −0.194982
\(66\) 0 0
\(67\) 0.917641 0.112108 0.0560538 0.998428i \(-0.482148\pi\)
0.0560538 + 0.998428i \(0.482148\pi\)
\(68\) 0 0
\(69\) 1.81681 0.218718
\(70\) 0 0
\(71\) 6.44648 0.765056 0.382528 0.923944i \(-0.375054\pi\)
0.382528 + 0.923944i \(0.375054\pi\)
\(72\) 0 0
\(73\) 6.71598 0.786046 0.393023 0.919529i \(-0.371429\pi\)
0.393023 + 0.919529i \(0.371429\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 2.77761 0.316538
\(78\) 0 0
\(79\) −9.52884 −1.07208 −0.536039 0.844193i \(-0.680080\pi\)
−0.536039 + 0.844193i \(0.680080\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.16246 0.347125 0.173562 0.984823i \(-0.444472\pi\)
0.173562 + 0.984823i \(0.444472\pi\)
\(84\) 0 0
\(85\) −4.67282 −0.506839
\(86\) 0 0
\(87\) 1.95684 0.209796
\(88\) 0 0
\(89\) −15.2241 −1.61375 −0.806875 0.590722i \(-0.798843\pi\)
−0.806875 + 0.590722i \(0.798843\pi\)
\(90\) 0 0
\(91\) −3.81681 −0.400110
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 5.34565 0.548452
\(96\) 0 0
\(97\) 10.7776 1.09430 0.547150 0.837035i \(-0.315713\pi\)
0.547150 + 0.837035i \(0.315713\pi\)
\(98\) 0 0
\(99\) −1.14399 −0.114975
\(100\) 0 0
\(101\) −3.51037 −0.349294 −0.174647 0.984631i \(-0.555879\pi\)
−0.174647 + 0.984631i \(0.555879\pi\)
\(102\) 0 0
\(103\) −18.2633 −1.79954 −0.899768 0.436369i \(-0.856264\pi\)
−0.899768 + 0.436369i \(0.856264\pi\)
\(104\) 0 0
\(105\) 2.42801 0.236949
\(106\) 0 0
\(107\) 4.96080 0.479578 0.239789 0.970825i \(-0.422922\pi\)
0.239789 + 0.970825i \(0.422922\pi\)
\(108\) 0 0
\(109\) 6.30644 0.604048 0.302024 0.953300i \(-0.402338\pi\)
0.302024 + 0.953300i \(0.402338\pi\)
\(110\) 0 0
\(111\) 3.57199 0.339039
\(112\) 0 0
\(113\) −3.43196 −0.322852 −0.161426 0.986885i \(-0.551609\pi\)
−0.161426 + 0.986885i \(0.551609\pi\)
\(114\) 0 0
\(115\) −1.81681 −0.169418
\(116\) 0 0
\(117\) 1.57199 0.145331
\(118\) 0 0
\(119\) −11.3456 −1.04005
\(120\) 0 0
\(121\) −9.69129 −0.881027
\(122\) 0 0
\(123\) −3.14399 −0.283484
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.8353 −1.22768 −0.613841 0.789429i \(-0.710377\pi\)
−0.613841 + 0.789429i \(0.710377\pi\)
\(128\) 0 0
\(129\) −2.85601 −0.251458
\(130\) 0 0
\(131\) −9.79213 −0.855542 −0.427771 0.903887i \(-0.640701\pi\)
−0.427771 + 0.903887i \(0.640701\pi\)
\(132\) 0 0
\(133\) 12.9793 1.12545
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 15.3641 1.31265 0.656323 0.754480i \(-0.272111\pi\)
0.656323 + 0.754480i \(0.272111\pi\)
\(138\) 0 0
\(139\) 19.1809 1.62691 0.813453 0.581631i \(-0.197585\pi\)
0.813453 + 0.581631i \(0.197585\pi\)
\(140\) 0 0
\(141\) −7.16246 −0.603188
\(142\) 0 0
\(143\) −1.79834 −0.150385
\(144\) 0 0
\(145\) −1.95684 −0.162507
\(146\) 0 0
\(147\) −1.10478 −0.0911210
\(148\) 0 0
\(149\) −11.5473 −0.945992 −0.472996 0.881064i \(-0.656828\pi\)
−0.472996 + 0.881064i \(0.656828\pi\)
\(150\) 0 0
\(151\) −5.61515 −0.456954 −0.228477 0.973549i \(-0.573375\pi\)
−0.228477 + 0.973549i \(0.573375\pi\)
\(152\) 0 0
\(153\) 4.67282 0.377775
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −16.1233 −1.28678 −0.643388 0.765540i \(-0.722472\pi\)
−0.643388 + 0.765540i \(0.722472\pi\)
\(158\) 0 0
\(159\) −3.81681 −0.302693
\(160\) 0 0
\(161\) −4.41123 −0.347653
\(162\) 0 0
\(163\) −6.71598 −0.526036 −0.263018 0.964791i \(-0.584718\pi\)
−0.263018 + 0.964791i \(0.584718\pi\)
\(164\) 0 0
\(165\) 1.14399 0.0890592
\(166\) 0 0
\(167\) −2.16472 −0.167511 −0.0837555 0.996486i \(-0.526691\pi\)
−0.0837555 + 0.996486i \(0.526691\pi\)
\(168\) 0 0
\(169\) −10.5288 −0.809911
\(170\) 0 0
\(171\) −5.34565 −0.408792
\(172\) 0 0
\(173\) −14.8560 −1.12948 −0.564741 0.825268i \(-0.691024\pi\)
−0.564741 + 0.825268i \(0.691024\pi\)
\(174\) 0 0
\(175\) −2.42801 −0.183540
\(176\) 0 0
\(177\) 6.24482 0.469389
\(178\) 0 0
\(179\) −20.0369 −1.49763 −0.748816 0.662778i \(-0.769377\pi\)
−0.748816 + 0.662778i \(0.769377\pi\)
\(180\) 0 0
\(181\) −0.287973 −0.0214049 −0.0107024 0.999943i \(-0.503407\pi\)
−0.0107024 + 0.999943i \(0.503407\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −3.57199 −0.262618
\(186\) 0 0
\(187\) −5.34565 −0.390912
\(188\) 0 0
\(189\) −2.42801 −0.176612
\(190\) 0 0
\(191\) −10.2448 −0.741289 −0.370644 0.928775i \(-0.620863\pi\)
−0.370644 + 0.928775i \(0.620863\pi\)
\(192\) 0 0
\(193\) 17.8353 1.28381 0.641906 0.766783i \(-0.278144\pi\)
0.641906 + 0.766783i \(0.278144\pi\)
\(194\) 0 0
\(195\) −1.57199 −0.112573
\(196\) 0 0
\(197\) 3.32718 0.237051 0.118526 0.992951i \(-0.462183\pi\)
0.118526 + 0.992951i \(0.462183\pi\)
\(198\) 0 0
\(199\) 5.65209 0.400666 0.200333 0.979728i \(-0.435798\pi\)
0.200333 + 0.979728i \(0.435798\pi\)
\(200\) 0 0
\(201\) 0.917641 0.0647254
\(202\) 0 0
\(203\) −4.75123 −0.333471
\(204\) 0 0
\(205\) 3.14399 0.219586
\(206\) 0 0
\(207\) 1.81681 0.126277
\(208\) 0 0
\(209\) 6.11535 0.423008
\(210\) 0 0
\(211\) 22.6050 1.55619 0.778096 0.628146i \(-0.216186\pi\)
0.778096 + 0.628146i \(0.216186\pi\)
\(212\) 0 0
\(213\) 6.44648 0.441705
\(214\) 0 0
\(215\) 2.85601 0.194779
\(216\) 0 0
\(217\) 2.42801 0.164824
\(218\) 0 0
\(219\) 6.71598 0.453824
\(220\) 0 0
\(221\) 7.34565 0.494122
\(222\) 0 0
\(223\) 4.69129 0.314152 0.157076 0.987586i \(-0.449793\pi\)
0.157076 + 0.987586i \(0.449793\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −1.90312 −0.126315 −0.0631573 0.998004i \(-0.520117\pi\)
−0.0631573 + 0.998004i \(0.520117\pi\)
\(228\) 0 0
\(229\) −21.1440 −1.39723 −0.698617 0.715496i \(-0.746201\pi\)
−0.698617 + 0.715496i \(0.746201\pi\)
\(230\) 0 0
\(231\) 2.77761 0.182753
\(232\) 0 0
\(233\) −26.7882 −1.75495 −0.877476 0.479621i \(-0.840774\pi\)
−0.877476 + 0.479621i \(0.840774\pi\)
\(234\) 0 0
\(235\) 7.16246 0.467227
\(236\) 0 0
\(237\) −9.52884 −0.618964
\(238\) 0 0
\(239\) −24.3249 −1.57345 −0.786724 0.617305i \(-0.788224\pi\)
−0.786724 + 0.617305i \(0.788224\pi\)
\(240\) 0 0
\(241\) 6.48963 0.418034 0.209017 0.977912i \(-0.432974\pi\)
0.209017 + 0.977912i \(0.432974\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.10478 0.0705820
\(246\) 0 0
\(247\) −8.40332 −0.534691
\(248\) 0 0
\(249\) 3.16246 0.200413
\(250\) 0 0
\(251\) 21.9216 1.38368 0.691839 0.722051i \(-0.256801\pi\)
0.691839 + 0.722051i \(0.256801\pi\)
\(252\) 0 0
\(253\) −2.07841 −0.130668
\(254\) 0 0
\(255\) −4.67282 −0.292624
\(256\) 0 0
\(257\) −7.73050 −0.482215 −0.241108 0.970498i \(-0.577511\pi\)
−0.241108 + 0.970498i \(0.577511\pi\)
\(258\) 0 0
\(259\) −8.67282 −0.538903
\(260\) 0 0
\(261\) 1.95684 0.121126
\(262\) 0 0
\(263\) −25.4689 −1.57048 −0.785240 0.619192i \(-0.787460\pi\)
−0.785240 + 0.619192i \(0.787460\pi\)
\(264\) 0 0
\(265\) 3.81681 0.234465
\(266\) 0 0
\(267\) −15.2241 −0.931699
\(268\) 0 0
\(269\) −3.75518 −0.228958 −0.114479 0.993426i \(-0.536520\pi\)
−0.114479 + 0.993426i \(0.536520\pi\)
\(270\) 0 0
\(271\) 13.6257 0.827703 0.413852 0.910344i \(-0.364183\pi\)
0.413852 + 0.910344i \(0.364183\pi\)
\(272\) 0 0
\(273\) −3.81681 −0.231004
\(274\) 0 0
\(275\) −1.14399 −0.0689850
\(276\) 0 0
\(277\) −19.3994 −1.16560 −0.582798 0.812617i \(-0.698042\pi\)
−0.582798 + 0.812617i \(0.698042\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −9.14399 −0.545485 −0.272742 0.962087i \(-0.587931\pi\)
−0.272742 + 0.962087i \(0.587931\pi\)
\(282\) 0 0
\(283\) −22.0616 −1.31143 −0.655714 0.755010i \(-0.727632\pi\)
−0.655714 + 0.755010i \(0.727632\pi\)
\(284\) 0 0
\(285\) 5.34565 0.316649
\(286\) 0 0
\(287\) 7.63362 0.450598
\(288\) 0 0
\(289\) 4.83528 0.284428
\(290\) 0 0
\(291\) 10.7776 0.631795
\(292\) 0 0
\(293\) −7.90312 −0.461705 −0.230853 0.972989i \(-0.574152\pi\)
−0.230853 + 0.972989i \(0.574152\pi\)
\(294\) 0 0
\(295\) −6.24482 −0.363587
\(296\) 0 0
\(297\) −1.14399 −0.0663808
\(298\) 0 0
\(299\) 2.85601 0.165168
\(300\) 0 0
\(301\) 6.93442 0.399693
\(302\) 0 0
\(303\) −3.51037 −0.201665
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 24.3865 1.39181 0.695907 0.718132i \(-0.255003\pi\)
0.695907 + 0.718132i \(0.255003\pi\)
\(308\) 0 0
\(309\) −18.2633 −1.03896
\(310\) 0 0
\(311\) −32.3681 −1.83542 −0.917712 0.397245i \(-0.869966\pi\)
−0.917712 + 0.397245i \(0.869966\pi\)
\(312\) 0 0
\(313\) 10.8313 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(314\) 0 0
\(315\) 2.42801 0.136803
\(316\) 0 0
\(317\) −15.3641 −0.862935 −0.431467 0.902129i \(-0.642004\pi\)
−0.431467 + 0.902129i \(0.642004\pi\)
\(318\) 0 0
\(319\) −2.23860 −0.125338
\(320\) 0 0
\(321\) 4.96080 0.276885
\(322\) 0 0
\(323\) −24.9793 −1.38988
\(324\) 0 0
\(325\) 1.57199 0.0871985
\(326\) 0 0
\(327\) 6.30644 0.348747
\(328\) 0 0
\(329\) 17.3905 0.958769
\(330\) 0 0
\(331\) −22.1417 −1.21702 −0.608510 0.793546i \(-0.708232\pi\)
−0.608510 + 0.793546i \(0.708232\pi\)
\(332\) 0 0
\(333\) 3.57199 0.195744
\(334\) 0 0
\(335\) −0.917641 −0.0501361
\(336\) 0 0
\(337\) 25.6873 1.39928 0.699639 0.714496i \(-0.253344\pi\)
0.699639 + 0.714496i \(0.253344\pi\)
\(338\) 0 0
\(339\) −3.43196 −0.186398
\(340\) 0 0
\(341\) 1.14399 0.0619503
\(342\) 0 0
\(343\) 19.6785 1.06254
\(344\) 0 0
\(345\) −1.81681 −0.0978138
\(346\) 0 0
\(347\) −17.2593 −0.926530 −0.463265 0.886220i \(-0.653322\pi\)
−0.463265 + 0.886220i \(0.653322\pi\)
\(348\) 0 0
\(349\) −30.7098 −1.64386 −0.821928 0.569591i \(-0.807101\pi\)
−0.821928 + 0.569591i \(0.807101\pi\)
\(350\) 0 0
\(351\) 1.57199 0.0839068
\(352\) 0 0
\(353\) −32.1338 −1.71031 −0.855155 0.518372i \(-0.826538\pi\)
−0.855155 + 0.518372i \(0.826538\pi\)
\(354\) 0 0
\(355\) −6.44648 −0.342144
\(356\) 0 0
\(357\) −11.3456 −0.600475
\(358\) 0 0
\(359\) 25.0594 1.32258 0.661291 0.750129i \(-0.270009\pi\)
0.661291 + 0.750129i \(0.270009\pi\)
\(360\) 0 0
\(361\) 9.57595 0.503997
\(362\) 0 0
\(363\) −9.69129 −0.508661
\(364\) 0 0
\(365\) −6.71598 −0.351530
\(366\) 0 0
\(367\) 1.71203 0.0893671 0.0446835 0.999001i \(-0.485772\pi\)
0.0446835 + 0.999001i \(0.485772\pi\)
\(368\) 0 0
\(369\) −3.14399 −0.163669
\(370\) 0 0
\(371\) 9.26724 0.481131
\(372\) 0 0
\(373\) −32.4482 −1.68010 −0.840051 0.542507i \(-0.817475\pi\)
−0.840051 + 0.542507i \(0.817475\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 3.07615 0.158430
\(378\) 0 0
\(379\) 1.51827 0.0779884 0.0389942 0.999239i \(-0.487585\pi\)
0.0389942 + 0.999239i \(0.487585\pi\)
\(380\) 0 0
\(381\) −13.8353 −0.708803
\(382\) 0 0
\(383\) −30.2201 −1.54418 −0.772088 0.635515i \(-0.780788\pi\)
−0.772088 + 0.635515i \(0.780788\pi\)
\(384\) 0 0
\(385\) −2.77761 −0.141560
\(386\) 0 0
\(387\) −2.85601 −0.145179
\(388\) 0 0
\(389\) 7.67678 0.389228 0.194614 0.980880i \(-0.437655\pi\)
0.194614 + 0.980880i \(0.437655\pi\)
\(390\) 0 0
\(391\) 8.48963 0.429339
\(392\) 0 0
\(393\) −9.79213 −0.493947
\(394\) 0 0
\(395\) 9.52884 0.479448
\(396\) 0 0
\(397\) −14.3664 −0.721028 −0.360514 0.932754i \(-0.617399\pi\)
−0.360514 + 0.932754i \(0.617399\pi\)
\(398\) 0 0
\(399\) 12.9793 0.649776
\(400\) 0 0
\(401\) 10.2448 0.511602 0.255801 0.966729i \(-0.417661\pi\)
0.255801 + 0.966729i \(0.417661\pi\)
\(402\) 0 0
\(403\) −1.57199 −0.0783066
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −4.08631 −0.202551
\(408\) 0 0
\(409\) −24.2880 −1.20096 −0.600481 0.799639i \(-0.705024\pi\)
−0.600481 + 0.799639i \(0.705024\pi\)
\(410\) 0 0
\(411\) 15.3641 0.757856
\(412\) 0 0
\(413\) −15.1625 −0.746096
\(414\) 0 0
\(415\) −3.16246 −0.155239
\(416\) 0 0
\(417\) 19.1809 0.939294
\(418\) 0 0
\(419\) 37.9938 1.85612 0.928059 0.372433i \(-0.121476\pi\)
0.928059 + 0.372433i \(0.121476\pi\)
\(420\) 0 0
\(421\) 31.5658 1.53842 0.769211 0.638995i \(-0.220650\pi\)
0.769211 + 0.638995i \(0.220650\pi\)
\(422\) 0 0
\(423\) −7.16246 −0.348251
\(424\) 0 0
\(425\) 4.67282 0.226665
\(426\) 0 0
\(427\) 4.85601 0.234999
\(428\) 0 0
\(429\) −1.79834 −0.0868246
\(430\) 0 0
\(431\) 14.8992 0.717668 0.358834 0.933401i \(-0.383174\pi\)
0.358834 + 0.933401i \(0.383174\pi\)
\(432\) 0 0
\(433\) −15.9753 −0.767725 −0.383862 0.923390i \(-0.625406\pi\)
−0.383862 + 0.923390i \(0.625406\pi\)
\(434\) 0 0
\(435\) −1.95684 −0.0938235
\(436\) 0 0
\(437\) −9.71203 −0.464589
\(438\) 0 0
\(439\) 32.0739 1.53080 0.765401 0.643553i \(-0.222540\pi\)
0.765401 + 0.643553i \(0.222540\pi\)
\(440\) 0 0
\(441\) −1.10478 −0.0526087
\(442\) 0 0
\(443\) 12.5944 0.598379 0.299189 0.954194i \(-0.403284\pi\)
0.299189 + 0.954194i \(0.403284\pi\)
\(444\) 0 0
\(445\) 15.2241 0.721691
\(446\) 0 0
\(447\) −11.5473 −0.546169
\(448\) 0 0
\(449\) −7.46721 −0.352399 −0.176200 0.984354i \(-0.556380\pi\)
−0.176200 + 0.984354i \(0.556380\pi\)
\(450\) 0 0
\(451\) 3.59668 0.169361
\(452\) 0 0
\(453\) −5.61515 −0.263823
\(454\) 0 0
\(455\) 3.81681 0.178935
\(456\) 0 0
\(457\) −20.1849 −0.944209 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(458\) 0 0
\(459\) 4.67282 0.218109
\(460\) 0 0
\(461\) 28.0017 1.30417 0.652084 0.758146i \(-0.273895\pi\)
0.652084 + 0.758146i \(0.273895\pi\)
\(462\) 0 0
\(463\) 38.4896 1.78876 0.894382 0.447303i \(-0.147615\pi\)
0.894382 + 0.447303i \(0.147615\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) 20.2985 0.939304 0.469652 0.882852i \(-0.344379\pi\)
0.469652 + 0.882852i \(0.344379\pi\)
\(468\) 0 0
\(469\) −2.22804 −0.102881
\(470\) 0 0
\(471\) −16.1233 −0.742920
\(472\) 0 0
\(473\) 3.26724 0.150228
\(474\) 0 0
\(475\) −5.34565 −0.245275
\(476\) 0 0
\(477\) −3.81681 −0.174760
\(478\) 0 0
\(479\) 19.0594 0.870845 0.435422 0.900226i \(-0.356599\pi\)
0.435422 + 0.900226i \(0.356599\pi\)
\(480\) 0 0
\(481\) 5.61515 0.256029
\(482\) 0 0
\(483\) −4.41123 −0.200718
\(484\) 0 0
\(485\) −10.7776 −0.489386
\(486\) 0 0
\(487\) −13.0656 −0.592058 −0.296029 0.955179i \(-0.595662\pi\)
−0.296029 + 0.955179i \(0.595662\pi\)
\(488\) 0 0
\(489\) −6.71598 −0.303707
\(490\) 0 0
\(491\) −24.3170 −1.09741 −0.548706 0.836016i \(-0.684879\pi\)
−0.548706 + 0.836016i \(0.684879\pi\)
\(492\) 0 0
\(493\) 9.14399 0.411824
\(494\) 0 0
\(495\) 1.14399 0.0514184
\(496\) 0 0
\(497\) −15.6521 −0.702092
\(498\) 0 0
\(499\) 16.4033 0.734314 0.367157 0.930159i \(-0.380331\pi\)
0.367157 + 0.930159i \(0.380331\pi\)
\(500\) 0 0
\(501\) −2.16472 −0.0967125
\(502\) 0 0
\(503\) 0.837542 0.0373442 0.0186721 0.999826i \(-0.494056\pi\)
0.0186721 + 0.999826i \(0.494056\pi\)
\(504\) 0 0
\(505\) 3.51037 0.156209
\(506\) 0 0
\(507\) −10.5288 −0.467602
\(508\) 0 0
\(509\) −1.38880 −0.0615576 −0.0307788 0.999526i \(-0.509799\pi\)
−0.0307788 + 0.999526i \(0.509799\pi\)
\(510\) 0 0
\(511\) −16.3064 −0.721355
\(512\) 0 0
\(513\) −5.34565 −0.236016
\(514\) 0 0
\(515\) 18.2633 0.804777
\(516\) 0 0
\(517\) 8.19376 0.360361
\(518\) 0 0
\(519\) −14.8560 −0.652107
\(520\) 0 0
\(521\) 33.1025 1.45025 0.725124 0.688618i \(-0.241782\pi\)
0.725124 + 0.688618i \(0.241782\pi\)
\(522\) 0 0
\(523\) −1.62571 −0.0710875 −0.0355438 0.999368i \(-0.511316\pi\)
−0.0355438 + 0.999368i \(0.511316\pi\)
\(524\) 0 0
\(525\) −2.42801 −0.105967
\(526\) 0 0
\(527\) −4.67282 −0.203551
\(528\) 0 0
\(529\) −19.6992 −0.856487
\(530\) 0 0
\(531\) 6.24482 0.271002
\(532\) 0 0
\(533\) −4.94233 −0.214076
\(534\) 0 0
\(535\) −4.96080 −0.214474
\(536\) 0 0
\(537\) −20.0369 −0.864658
\(538\) 0 0
\(539\) 1.26386 0.0544382
\(540\) 0 0
\(541\) 26.5865 1.14304 0.571522 0.820587i \(-0.306353\pi\)
0.571522 + 0.820587i \(0.306353\pi\)
\(542\) 0 0
\(543\) −0.287973 −0.0123581
\(544\) 0 0
\(545\) −6.30644 −0.270138
\(546\) 0 0
\(547\) 19.9753 0.854083 0.427041 0.904232i \(-0.359556\pi\)
0.427041 + 0.904232i \(0.359556\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −10.4606 −0.445636
\(552\) 0 0
\(553\) 23.1361 0.983846
\(554\) 0 0
\(555\) −3.57199 −0.151623
\(556\) 0 0
\(557\) −29.2488 −1.23931 −0.619655 0.784874i \(-0.712728\pi\)
−0.619655 + 0.784874i \(0.712728\pi\)
\(558\) 0 0
\(559\) −4.48963 −0.189891
\(560\) 0 0
\(561\) −5.34565 −0.225693
\(562\) 0 0
\(563\) −0.960797 −0.0404928 −0.0202464 0.999795i \(-0.506445\pi\)
−0.0202464 + 0.999795i \(0.506445\pi\)
\(564\) 0 0
\(565\) 3.43196 0.144384
\(566\) 0 0
\(567\) −2.42801 −0.101967
\(568\) 0 0
\(569\) −3.87844 −0.162593 −0.0812963 0.996690i \(-0.525906\pi\)
−0.0812963 + 0.996690i \(0.525906\pi\)
\(570\) 0 0
\(571\) 6.20166 0.259531 0.129766 0.991545i \(-0.458577\pi\)
0.129766 + 0.991545i \(0.458577\pi\)
\(572\) 0 0
\(573\) −10.2448 −0.427983
\(574\) 0 0
\(575\) 1.81681 0.0757662
\(576\) 0 0
\(577\) 35.6336 1.48345 0.741724 0.670706i \(-0.234009\pi\)
0.741724 + 0.670706i \(0.234009\pi\)
\(578\) 0 0
\(579\) 17.8353 0.741209
\(580\) 0 0
\(581\) −7.67847 −0.318557
\(582\) 0 0
\(583\) 4.36638 0.180837
\(584\) 0 0
\(585\) −1.57199 −0.0649939
\(586\) 0 0
\(587\) 43.7569 1.80604 0.903020 0.429599i \(-0.141345\pi\)
0.903020 + 0.429599i \(0.141345\pi\)
\(588\) 0 0
\(589\) 5.34565 0.220264
\(590\) 0 0
\(591\) 3.32718 0.136862
\(592\) 0 0
\(593\) 17.3536 0.712625 0.356313 0.934367i \(-0.384034\pi\)
0.356313 + 0.934367i \(0.384034\pi\)
\(594\) 0 0
\(595\) 11.3456 0.465126
\(596\) 0 0
\(597\) 5.65209 0.231325
\(598\) 0 0
\(599\) −23.7552 −0.970610 −0.485305 0.874345i \(-0.661292\pi\)
−0.485305 + 0.874345i \(0.661292\pi\)
\(600\) 0 0
\(601\) −41.9506 −1.71120 −0.855601 0.517636i \(-0.826812\pi\)
−0.855601 + 0.517636i \(0.826812\pi\)
\(602\) 0 0
\(603\) 0.917641 0.0373692
\(604\) 0 0
\(605\) 9.69129 0.394007
\(606\) 0 0
\(607\) 4.25538 0.172721 0.0863603 0.996264i \(-0.472476\pi\)
0.0863603 + 0.996264i \(0.472476\pi\)
\(608\) 0 0
\(609\) −4.75123 −0.192530
\(610\) 0 0
\(611\) −11.2593 −0.455504
\(612\) 0 0
\(613\) 11.7367 0.474041 0.237021 0.971505i \(-0.423829\pi\)
0.237021 + 0.971505i \(0.423829\pi\)
\(614\) 0 0
\(615\) 3.14399 0.126778
\(616\) 0 0
\(617\) −10.5759 −0.425772 −0.212886 0.977077i \(-0.568286\pi\)
−0.212886 + 0.977077i \(0.568286\pi\)
\(618\) 0 0
\(619\) −29.1994 −1.17362 −0.586811 0.809724i \(-0.699617\pi\)
−0.586811 + 0.809724i \(0.699617\pi\)
\(620\) 0 0
\(621\) 1.81681 0.0729061
\(622\) 0 0
\(623\) 36.9642 1.48094
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.11535 0.244224
\(628\) 0 0
\(629\) 16.6913 0.665526
\(630\) 0 0
\(631\) −19.6706 −0.783073 −0.391536 0.920163i \(-0.628056\pi\)
−0.391536 + 0.920163i \(0.628056\pi\)
\(632\) 0 0
\(633\) 22.6050 0.898467
\(634\) 0 0
\(635\) 13.8353 0.549036
\(636\) 0 0
\(637\) −1.73671 −0.0688110
\(638\) 0 0
\(639\) 6.44648 0.255019
\(640\) 0 0
\(641\) 40.3681 1.59444 0.797221 0.603687i \(-0.206302\pi\)
0.797221 + 0.603687i \(0.206302\pi\)
\(642\) 0 0
\(643\) −2.20166 −0.0868250 −0.0434125 0.999057i \(-0.513823\pi\)
−0.0434125 + 0.999057i \(0.513823\pi\)
\(644\) 0 0
\(645\) 2.85601 0.112455
\(646\) 0 0
\(647\) −41.0162 −1.61251 −0.806257 0.591566i \(-0.798510\pi\)
−0.806257 + 0.591566i \(0.798510\pi\)
\(648\) 0 0
\(649\) −7.14399 −0.280426
\(650\) 0 0
\(651\) 2.42801 0.0951611
\(652\) 0 0
\(653\) −5.00226 −0.195754 −0.0978768 0.995199i \(-0.531205\pi\)
−0.0978768 + 0.995199i \(0.531205\pi\)
\(654\) 0 0
\(655\) 9.79213 0.382610
\(656\) 0 0
\(657\) 6.71598 0.262015
\(658\) 0 0
\(659\) −11.5826 −0.451192 −0.225596 0.974221i \(-0.572433\pi\)
−0.225596 + 0.974221i \(0.572433\pi\)
\(660\) 0 0
\(661\) 48.2386 1.87626 0.938132 0.346278i \(-0.112554\pi\)
0.938132 + 0.346278i \(0.112554\pi\)
\(662\) 0 0
\(663\) 7.34565 0.285281
\(664\) 0 0
\(665\) −12.9793 −0.503314
\(666\) 0 0
\(667\) 3.55521 0.137658
\(668\) 0 0
\(669\) 4.69129 0.181376
\(670\) 0 0
\(671\) 2.28797 0.0883262
\(672\) 0 0
\(673\) 18.3417 0.707020 0.353510 0.935431i \(-0.384988\pi\)
0.353510 + 0.935431i \(0.384988\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 5.33508 0.205044 0.102522 0.994731i \(-0.467309\pi\)
0.102522 + 0.994731i \(0.467309\pi\)
\(678\) 0 0
\(679\) −26.1681 −1.00424
\(680\) 0 0
\(681\) −1.90312 −0.0729278
\(682\) 0 0
\(683\) −14.2695 −0.546007 −0.273004 0.962013i \(-0.588017\pi\)
−0.273004 + 0.962013i \(0.588017\pi\)
\(684\) 0 0
\(685\) −15.3641 −0.587033
\(686\) 0 0
\(687\) −21.1440 −0.806693
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 24.1233 0.917692 0.458846 0.888516i \(-0.348263\pi\)
0.458846 + 0.888516i \(0.348263\pi\)
\(692\) 0 0
\(693\) 2.77761 0.105513
\(694\) 0 0
\(695\) −19.1809 −0.727574
\(696\) 0 0
\(697\) −14.6913 −0.556472
\(698\) 0 0
\(699\) −26.7882 −1.01322
\(700\) 0 0
\(701\) −3.26724 −0.123402 −0.0617010 0.998095i \(-0.519653\pi\)
−0.0617010 + 0.998095i \(0.519653\pi\)
\(702\) 0 0
\(703\) −19.0946 −0.720167
\(704\) 0 0
\(705\) 7.16246 0.269754
\(706\) 0 0
\(707\) 8.52319 0.320548
\(708\) 0 0
\(709\) 49.1104 1.84438 0.922190 0.386736i \(-0.126398\pi\)
0.922190 + 0.386736i \(0.126398\pi\)
\(710\) 0 0
\(711\) −9.52884 −0.357359
\(712\) 0 0
\(713\) −1.81681 −0.0680401
\(714\) 0 0
\(715\) 1.79834 0.0672541
\(716\) 0 0
\(717\) −24.3249 −0.908431
\(718\) 0 0
\(719\) −11.4320 −0.426340 −0.213170 0.977015i \(-0.568379\pi\)
−0.213170 + 0.977015i \(0.568379\pi\)
\(720\) 0 0
\(721\) 44.3434 1.65143
\(722\) 0 0
\(723\) 6.48963 0.241352
\(724\) 0 0
\(725\) 1.95684 0.0726754
\(726\) 0 0
\(727\) 0.514319 0.0190750 0.00953752 0.999955i \(-0.496964\pi\)
0.00953752 + 0.999955i \(0.496964\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −13.3456 −0.493607
\(732\) 0 0
\(733\) 35.7859 1.32178 0.660891 0.750482i \(-0.270178\pi\)
0.660891 + 0.750482i \(0.270178\pi\)
\(734\) 0 0
\(735\) 1.10478 0.0407505
\(736\) 0 0
\(737\) −1.04977 −0.0386687
\(738\) 0 0
\(739\) −33.2857 −1.22443 −0.612217 0.790690i \(-0.709722\pi\)
−0.612217 + 0.790690i \(0.709722\pi\)
\(740\) 0 0
\(741\) −8.40332 −0.308704
\(742\) 0 0
\(743\) −38.0448 −1.39573 −0.697865 0.716229i \(-0.745866\pi\)
−0.697865 + 0.716229i \(0.745866\pi\)
\(744\) 0 0
\(745\) 11.5473 0.423061
\(746\) 0 0
\(747\) 3.16246 0.115708
\(748\) 0 0
\(749\) −12.0448 −0.440109
\(750\) 0 0
\(751\) 6.20166 0.226302 0.113151 0.993578i \(-0.463906\pi\)
0.113151 + 0.993578i \(0.463906\pi\)
\(752\) 0 0
\(753\) 21.9216 0.798867
\(754\) 0 0
\(755\) 5.61515 0.204356
\(756\) 0 0
\(757\) −0.522225 −0.0189806 −0.00949029 0.999955i \(-0.503021\pi\)
−0.00949029 + 0.999955i \(0.503021\pi\)
\(758\) 0 0
\(759\) −2.07841 −0.0754414
\(760\) 0 0
\(761\) −4.07219 −0.147617 −0.0738084 0.997272i \(-0.523515\pi\)
−0.0738084 + 0.997272i \(0.523515\pi\)
\(762\) 0 0
\(763\) −15.3121 −0.554335
\(764\) 0 0
\(765\) −4.67282 −0.168946
\(766\) 0 0
\(767\) 9.81681 0.354464
\(768\) 0 0
\(769\) −19.8089 −0.714327 −0.357164 0.934042i \(-0.616256\pi\)
−0.357164 + 0.934042i \(0.616256\pi\)
\(770\) 0 0
\(771\) −7.73050 −0.278407
\(772\) 0 0
\(773\) −40.3803 −1.45238 −0.726190 0.687494i \(-0.758711\pi\)
−0.726190 + 0.687494i \(0.758711\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −8.67282 −0.311136
\(778\) 0 0
\(779\) 16.8066 0.602160
\(780\) 0 0
\(781\) −7.37468 −0.263887
\(782\) 0 0
\(783\) 1.95684 0.0699319
\(784\) 0 0
\(785\) 16.1233 0.575464
\(786\) 0 0
\(787\) −19.1730 −0.683444 −0.341722 0.939801i \(-0.611010\pi\)
−0.341722 + 0.939801i \(0.611010\pi\)
\(788\) 0 0
\(789\) −25.4689 −0.906717
\(790\) 0 0
\(791\) 8.33282 0.296281
\(792\) 0 0
\(793\) −3.14399 −0.111646
\(794\) 0 0
\(795\) 3.81681 0.135368
\(796\) 0 0
\(797\) 39.0347 1.38268 0.691340 0.722530i \(-0.257021\pi\)
0.691340 + 0.722530i \(0.257021\pi\)
\(798\) 0 0
\(799\) −33.4689 −1.18404
\(800\) 0 0
\(801\) −15.2241 −0.537917
\(802\) 0 0
\(803\) −7.68299 −0.271127
\(804\) 0 0
\(805\) 4.41123 0.155475
\(806\) 0 0
\(807\) −3.75518 −0.132189
\(808\) 0 0
\(809\) −25.4627 −0.895220 −0.447610 0.894229i \(-0.647725\pi\)
−0.447610 + 0.894229i \(0.647725\pi\)
\(810\) 0 0
\(811\) 43.1440 1.51499 0.757495 0.652841i \(-0.226423\pi\)
0.757495 + 0.652841i \(0.226423\pi\)
\(812\) 0 0
\(813\) 13.6257 0.477875
\(814\) 0 0
\(815\) 6.71598 0.235251
\(816\) 0 0
\(817\) 15.2672 0.534133
\(818\) 0 0
\(819\) −3.81681 −0.133370
\(820\) 0 0
\(821\) 12.2527 0.427623 0.213811 0.976875i \(-0.431412\pi\)
0.213811 + 0.976875i \(0.431412\pi\)
\(822\) 0 0
\(823\) 33.7569 1.17669 0.588345 0.808610i \(-0.299780\pi\)
0.588345 + 0.808610i \(0.299780\pi\)
\(824\) 0 0
\(825\) −1.14399 −0.0398285
\(826\) 0 0
\(827\) 37.2778 1.29628 0.648138 0.761523i \(-0.275548\pi\)
0.648138 + 0.761523i \(0.275548\pi\)
\(828\) 0 0
\(829\) 26.3170 0.914028 0.457014 0.889460i \(-0.348919\pi\)
0.457014 + 0.889460i \(0.348919\pi\)
\(830\) 0 0
\(831\) −19.3994 −0.672957
\(832\) 0 0
\(833\) −5.16246 −0.178869
\(834\) 0 0
\(835\) 2.16472 0.0749132
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −28.1664 −0.972412 −0.486206 0.873844i \(-0.661620\pi\)
−0.486206 + 0.873844i \(0.661620\pi\)
\(840\) 0 0
\(841\) −25.1708 −0.867957
\(842\) 0 0
\(843\) −9.14399 −0.314936
\(844\) 0 0
\(845\) 10.5288 0.362203
\(846\) 0 0
\(847\) 23.5305 0.808519
\(848\) 0 0
\(849\) −22.0616 −0.757153
\(850\) 0 0
\(851\) 6.48963 0.222462
\(852\) 0 0
\(853\) −7.67056 −0.262635 −0.131318 0.991340i \(-0.541921\pi\)
−0.131318 + 0.991340i \(0.541921\pi\)
\(854\) 0 0
\(855\) 5.34565 0.182817
\(856\) 0 0
\(857\) −42.6992 −1.45858 −0.729288 0.684206i \(-0.760149\pi\)
−0.729288 + 0.684206i \(0.760149\pi\)
\(858\) 0 0
\(859\) 44.2465 1.50967 0.754836 0.655914i \(-0.227717\pi\)
0.754836 + 0.655914i \(0.227717\pi\)
\(860\) 0 0
\(861\) 7.63362 0.260153
\(862\) 0 0
\(863\) −22.8639 −0.778297 −0.389148 0.921175i \(-0.627231\pi\)
−0.389148 + 0.921175i \(0.627231\pi\)
\(864\) 0 0
\(865\) 14.8560 0.505120
\(866\) 0 0
\(867\) 4.83528 0.164215
\(868\) 0 0
\(869\) 10.9009 0.369786
\(870\) 0 0
\(871\) 1.44252 0.0488781
\(872\) 0 0
\(873\) 10.7776 0.364767
\(874\) 0 0
\(875\) 2.42801 0.0820816
\(876\) 0 0
\(877\) 31.0577 1.04874 0.524372 0.851490i \(-0.324300\pi\)
0.524372 + 0.851490i \(0.324300\pi\)
\(878\) 0 0
\(879\) −7.90312 −0.266566
\(880\) 0 0
\(881\) −17.8784 −0.602340 −0.301170 0.953570i \(-0.597377\pi\)
−0.301170 + 0.953570i \(0.597377\pi\)
\(882\) 0 0
\(883\) 22.2096 0.747411 0.373706 0.927547i \(-0.378087\pi\)
0.373706 + 0.927547i \(0.378087\pi\)
\(884\) 0 0
\(885\) −6.24482 −0.209917
\(886\) 0 0
\(887\) 4.10478 0.137825 0.0689126 0.997623i \(-0.478047\pi\)
0.0689126 + 0.997623i \(0.478047\pi\)
\(888\) 0 0
\(889\) 33.5922 1.12664
\(890\) 0 0
\(891\) −1.14399 −0.0383250
\(892\) 0 0
\(893\) 38.2880 1.28126
\(894\) 0 0
\(895\) 20.0369 0.669761
\(896\) 0 0
\(897\) 2.85601 0.0953595
\(898\) 0 0
\(899\) −1.95684 −0.0652644
\(900\) 0 0
\(901\) −17.8353 −0.594179
\(902\) 0 0
\(903\) 6.93442 0.230763
\(904\) 0 0
\(905\) 0.287973 0.00957255
\(906\) 0 0
\(907\) −8.62967 −0.286543 −0.143272 0.989683i \(-0.545762\pi\)
−0.143272 + 0.989683i \(0.545762\pi\)
\(908\) 0 0
\(909\) −3.51037 −0.116431
\(910\) 0 0
\(911\) −7.83528 −0.259594 −0.129797 0.991541i \(-0.541433\pi\)
−0.129797 + 0.991541i \(0.541433\pi\)
\(912\) 0 0
\(913\) −3.61781 −0.119732
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) 23.7753 0.785131
\(918\) 0 0
\(919\) −15.2514 −0.503098 −0.251549 0.967845i \(-0.580940\pi\)
−0.251549 + 0.967845i \(0.580940\pi\)
\(920\) 0 0
\(921\) 24.3865 0.803564
\(922\) 0 0
\(923\) 10.1338 0.333559
\(924\) 0 0
\(925\) 3.57199 0.117446
\(926\) 0 0
\(927\) −18.2633 −0.599845
\(928\) 0 0
\(929\) −4.15850 −0.136436 −0.0682181 0.997670i \(-0.521731\pi\)
−0.0682181 + 0.997670i \(0.521731\pi\)
\(930\) 0 0
\(931\) 5.90578 0.193554
\(932\) 0 0
\(933\) −32.3681 −1.05968
\(934\) 0 0
\(935\) 5.34565 0.174821
\(936\) 0 0
\(937\) −18.2175 −0.595139 −0.297569 0.954700i \(-0.596176\pi\)
−0.297569 + 0.954700i \(0.596176\pi\)
\(938\) 0 0
\(939\) 10.8313 0.353467
\(940\) 0 0
\(941\) −5.84150 −0.190427 −0.0952137 0.995457i \(-0.530353\pi\)
−0.0952137 + 0.995457i \(0.530353\pi\)
\(942\) 0 0
\(943\) −5.71203 −0.186009
\(944\) 0 0
\(945\) 2.42801 0.0789831
\(946\) 0 0
\(947\) −57.9276 −1.88240 −0.941198 0.337856i \(-0.890298\pi\)
−0.941198 + 0.337856i \(0.890298\pi\)
\(948\) 0 0
\(949\) 10.5575 0.342710
\(950\) 0 0
\(951\) −15.3641 −0.498216
\(952\) 0 0
\(953\) 22.9977 0.744970 0.372485 0.928038i \(-0.378506\pi\)
0.372485 + 0.928038i \(0.378506\pi\)
\(954\) 0 0
\(955\) 10.2448 0.331514
\(956\) 0 0
\(957\) −2.23860 −0.0723638
\(958\) 0 0
\(959\) −37.3042 −1.20461
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 4.96080 0.159859
\(964\) 0 0
\(965\) −17.8353 −0.574138
\(966\) 0 0
\(967\) −45.2672 −1.45570 −0.727848 0.685738i \(-0.759479\pi\)
−0.727848 + 0.685738i \(0.759479\pi\)
\(968\) 0 0
\(969\) −24.9793 −0.802450
\(970\) 0 0
\(971\) −7.59837 −0.243843 −0.121922 0.992540i \(-0.538906\pi\)
−0.121922 + 0.992540i \(0.538906\pi\)
\(972\) 0 0
\(973\) −46.5714 −1.49301
\(974\) 0 0
\(975\) 1.57199 0.0503441
\(976\) 0 0
\(977\) −19.1519 −0.612723 −0.306362 0.951915i \(-0.599112\pi\)
−0.306362 + 0.951915i \(0.599112\pi\)
\(978\) 0 0
\(979\) 17.4161 0.556623
\(980\) 0 0
\(981\) 6.30644 0.201349
\(982\) 0 0
\(983\) −4.20957 −0.134264 −0.0671322 0.997744i \(-0.521385\pi\)
−0.0671322 + 0.997744i \(0.521385\pi\)
\(984\) 0 0
\(985\) −3.32718 −0.106013
\(986\) 0 0
\(987\) 17.3905 0.553546
\(988\) 0 0
\(989\) −5.18883 −0.164995
\(990\) 0 0
\(991\) 37.3121 1.18526 0.592629 0.805476i \(-0.298090\pi\)
0.592629 + 0.805476i \(0.298090\pi\)
\(992\) 0 0
\(993\) −22.1417 −0.702646
\(994\) 0 0
\(995\) −5.65209 −0.179183
\(996\) 0 0
\(997\) −62.0034 −1.96367 −0.981833 0.189745i \(-0.939234\pi\)
−0.981833 + 0.189745i \(0.939234\pi\)
\(998\) 0 0
\(999\) 3.57199 0.113013
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 7440.2.a.bs.1.2 3
4.3 odd 2 465.2.a.e.1.2 3
12.11 even 2 1395.2.a.j.1.2 3
20.3 even 4 2325.2.c.k.1024.3 6
20.7 even 4 2325.2.c.k.1024.4 6
20.19 odd 2 2325.2.a.r.1.2 3
60.59 even 2 6975.2.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.e.1.2 3 4.3 odd 2
1395.2.a.j.1.2 3 12.11 even 2
2325.2.a.r.1.2 3 20.19 odd 2
2325.2.c.k.1024.3 6 20.3 even 4
2325.2.c.k.1024.4 6 20.7 even 4
6975.2.a.bf.1.2 3 60.59 even 2
7440.2.a.bs.1.2 3 1.1 even 1 trivial