Properties

Label 7728.2.a.cd.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.24197.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.27460\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.39532 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.56912 q^{11} +5.05307 q^{13} +2.39532 q^{15} -1.44840 q^{17} -1.98008 q^{19} -1.00000 q^{21} +1.00000 q^{23} +0.737570 q^{25} +1.00000 q^{27} +3.68985 q^{29} +3.44840 q^{31} +4.56912 q^{33} -2.39532 q^{35} -3.99759 q^{37} +5.05307 q^{39} +1.77848 q^{41} +1.25467 q^{43} +2.39532 q^{45} -2.87928 q^{47} +1.00000 q^{49} -1.44840 q^{51} -10.4128 q^{53} +10.9445 q^{55} -1.98008 q^{57} +14.1315 q^{59} +8.37540 q^{61} -1.00000 q^{63} +12.1037 q^{65} +7.73517 q^{67} +1.00000 q^{69} +0.946925 q^{71} +3.86711 q^{73} +0.737570 q^{75} -4.56912 q^{77} -4.23904 q^{79} +1.00000 q^{81} -10.8281 q^{83} -3.46938 q^{85} +3.68985 q^{87} +0.262430 q^{89} -5.05307 q^{91} +3.44840 q^{93} -4.74292 q^{95} -7.20695 q^{97} +4.56912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.39532 1.07122 0.535610 0.844465i \(-0.320082\pi\)
0.535610 + 0.844465i \(0.320082\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.56912 1.37764 0.688821 0.724931i \(-0.258129\pi\)
0.688821 + 0.724931i \(0.258129\pi\)
\(12\) 0 0
\(13\) 5.05307 1.40147 0.700735 0.713421i \(-0.252855\pi\)
0.700735 + 0.713421i \(0.252855\pi\)
\(14\) 0 0
\(15\) 2.39532 0.618470
\(16\) 0 0
\(17\) −1.44840 −0.351288 −0.175644 0.984454i \(-0.556201\pi\)
−0.175644 + 0.984454i \(0.556201\pi\)
\(18\) 0 0
\(19\) −1.98008 −0.454261 −0.227130 0.973864i \(-0.572934\pi\)
−0.227130 + 0.973864i \(0.572934\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0.737570 0.147514
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.68985 0.685187 0.342594 0.939484i \(-0.388695\pi\)
0.342594 + 0.939484i \(0.388695\pi\)
\(30\) 0 0
\(31\) 3.44840 0.619350 0.309675 0.950842i \(-0.399780\pi\)
0.309675 + 0.950842i \(0.399780\pi\)
\(32\) 0 0
\(33\) 4.56912 0.795382
\(34\) 0 0
\(35\) −2.39532 −0.404883
\(36\) 0 0
\(37\) −3.99759 −0.657201 −0.328600 0.944469i \(-0.606577\pi\)
−0.328600 + 0.944469i \(0.606577\pi\)
\(38\) 0 0
\(39\) 5.05307 0.809140
\(40\) 0 0
\(41\) 1.77848 0.277751 0.138876 0.990310i \(-0.455651\pi\)
0.138876 + 0.990310i \(0.455651\pi\)
\(42\) 0 0
\(43\) 1.25467 0.191336 0.0956680 0.995413i \(-0.469501\pi\)
0.0956680 + 0.995413i \(0.469501\pi\)
\(44\) 0 0
\(45\) 2.39532 0.357074
\(46\) 0 0
\(47\) −2.87928 −0.419986 −0.209993 0.977703i \(-0.567344\pi\)
−0.209993 + 0.977703i \(0.567344\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.44840 −0.202816
\(52\) 0 0
\(53\) −10.4128 −1.43031 −0.715157 0.698964i \(-0.753645\pi\)
−0.715157 + 0.698964i \(0.753645\pi\)
\(54\) 0 0
\(55\) 10.9445 1.47576
\(56\) 0 0
\(57\) −1.98008 −0.262267
\(58\) 0 0
\(59\) 14.1315 1.83977 0.919885 0.392188i \(-0.128282\pi\)
0.919885 + 0.392188i \(0.128282\pi\)
\(60\) 0 0
\(61\) 8.37540 1.07236 0.536180 0.844104i \(-0.319867\pi\)
0.536180 + 0.844104i \(0.319867\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 12.1037 1.50128
\(66\) 0 0
\(67\) 7.73517 0.945001 0.472500 0.881330i \(-0.343352\pi\)
0.472500 + 0.881330i \(0.343352\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0.946925 0.112379 0.0561897 0.998420i \(-0.482105\pi\)
0.0561897 + 0.998420i \(0.482105\pi\)
\(72\) 0 0
\(73\) 3.86711 0.452611 0.226305 0.974056i \(-0.427335\pi\)
0.226305 + 0.974056i \(0.427335\pi\)
\(74\) 0 0
\(75\) 0.737570 0.0851673
\(76\) 0 0
\(77\) −4.56912 −0.520700
\(78\) 0 0
\(79\) −4.23904 −0.476930 −0.238465 0.971151i \(-0.576644\pi\)
−0.238465 + 0.971151i \(0.576644\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.8281 −1.18854 −0.594269 0.804267i \(-0.702558\pi\)
−0.594269 + 0.804267i \(0.702558\pi\)
\(84\) 0 0
\(85\) −3.46938 −0.376307
\(86\) 0 0
\(87\) 3.68985 0.395593
\(88\) 0 0
\(89\) 0.262430 0.0278175 0.0139087 0.999903i \(-0.495573\pi\)
0.0139087 + 0.999903i \(0.495573\pi\)
\(90\) 0 0
\(91\) −5.05307 −0.529706
\(92\) 0 0
\(93\) 3.44840 0.357582
\(94\) 0 0
\(95\) −4.74292 −0.486613
\(96\) 0 0
\(97\) −7.20695 −0.731755 −0.365877 0.930663i \(-0.619231\pi\)
−0.365877 + 0.930663i \(0.619231\pi\)
\(98\) 0 0
\(99\) 4.56912 0.459214
\(100\) 0 0
\(101\) −8.26684 −0.822582 −0.411291 0.911504i \(-0.634922\pi\)
−0.411291 + 0.911504i \(0.634922\pi\)
\(102\) 0 0
\(103\) 19.2619 1.89793 0.948966 0.315378i \(-0.102131\pi\)
0.948966 + 0.315378i \(0.102131\pi\)
\(104\) 0 0
\(105\) −2.39532 −0.233760
\(106\) 0 0
\(107\) 11.0531 1.06854 0.534271 0.845314i \(-0.320586\pi\)
0.534271 + 0.845314i \(0.320586\pi\)
\(108\) 0 0
\(109\) 12.9047 1.23604 0.618022 0.786161i \(-0.287934\pi\)
0.618022 + 0.786161i \(0.287934\pi\)
\(110\) 0 0
\(111\) −3.99759 −0.379435
\(112\) 0 0
\(113\) −1.25708 −0.118256 −0.0591281 0.998250i \(-0.518832\pi\)
−0.0591281 + 0.998250i \(0.518832\pi\)
\(114\) 0 0
\(115\) 2.39532 0.223365
\(116\) 0 0
\(117\) 5.05307 0.467157
\(118\) 0 0
\(119\) 1.44840 0.132774
\(120\) 0 0
\(121\) 9.87687 0.897897
\(122\) 0 0
\(123\) 1.77848 0.160360
\(124\) 0 0
\(125\) −10.2099 −0.913201
\(126\) 0 0
\(127\) −19.3385 −1.71601 −0.858007 0.513638i \(-0.828297\pi\)
−0.858007 + 0.513638i \(0.828297\pi\)
\(128\) 0 0
\(129\) 1.25467 0.110468
\(130\) 0 0
\(131\) −19.4659 −1.70075 −0.850373 0.526181i \(-0.823623\pi\)
−0.850373 + 0.526181i \(0.823623\pi\)
\(132\) 0 0
\(133\) 1.98008 0.171694
\(134\) 0 0
\(135\) 2.39532 0.206157
\(136\) 0 0
\(137\) 20.4980 1.75126 0.875632 0.482980i \(-0.160446\pi\)
0.875632 + 0.482980i \(0.160446\pi\)
\(138\) 0 0
\(139\) −21.0507 −1.78549 −0.892747 0.450558i \(-0.851225\pi\)
−0.892747 + 0.450558i \(0.851225\pi\)
\(140\) 0 0
\(141\) −2.87928 −0.242479
\(142\) 0 0
\(143\) 23.0881 1.93072
\(144\) 0 0
\(145\) 8.83837 0.733987
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −19.1159 −1.56604 −0.783018 0.621999i \(-0.786321\pi\)
−0.783018 + 0.621999i \(0.786321\pi\)
\(150\) 0 0
\(151\) −18.2542 −1.48550 −0.742751 0.669568i \(-0.766479\pi\)
−0.742751 + 0.669568i \(0.766479\pi\)
\(152\) 0 0
\(153\) −1.44840 −0.117096
\(154\) 0 0
\(155\) 8.26002 0.663461
\(156\) 0 0
\(157\) 6.63783 0.529756 0.264878 0.964282i \(-0.414668\pi\)
0.264878 + 0.964282i \(0.414668\pi\)
\(158\) 0 0
\(159\) −10.4128 −0.825792
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −21.1080 −1.65331 −0.826655 0.562710i \(-0.809759\pi\)
−0.826655 + 0.562710i \(0.809759\pi\)
\(164\) 0 0
\(165\) 10.9445 0.852030
\(166\) 0 0
\(167\) 1.01217 0.0783240 0.0391620 0.999233i \(-0.487531\pi\)
0.0391620 + 0.999233i \(0.487531\pi\)
\(168\) 0 0
\(169\) 12.5336 0.964120
\(170\) 0 0
\(171\) −1.98008 −0.151420
\(172\) 0 0
\(173\) 15.2069 1.15616 0.578081 0.815979i \(-0.303802\pi\)
0.578081 + 0.815979i \(0.303802\pi\)
\(174\) 0 0
\(175\) −0.737570 −0.0557551
\(176\) 0 0
\(177\) 14.1315 1.06219
\(178\) 0 0
\(179\) −18.1491 −1.35652 −0.678262 0.734820i \(-0.737267\pi\)
−0.678262 + 0.734820i \(0.737267\pi\)
\(180\) 0 0
\(181\) 10.5866 0.786899 0.393449 0.919346i \(-0.371282\pi\)
0.393449 + 0.919346i \(0.371282\pi\)
\(182\) 0 0
\(183\) 8.37540 0.619127
\(184\) 0 0
\(185\) −9.57553 −0.704007
\(186\) 0 0
\(187\) −6.61790 −0.483949
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 19.8004 1.43271 0.716354 0.697737i \(-0.245810\pi\)
0.716354 + 0.697737i \(0.245810\pi\)
\(192\) 0 0
\(193\) 22.4956 1.61927 0.809635 0.586934i \(-0.199665\pi\)
0.809635 + 0.586934i \(0.199665\pi\)
\(194\) 0 0
\(195\) 12.1037 0.866767
\(196\) 0 0
\(197\) 1.67338 0.119224 0.0596118 0.998222i \(-0.481014\pi\)
0.0596118 + 0.998222i \(0.481014\pi\)
\(198\) 0 0
\(199\) −5.82380 −0.412838 −0.206419 0.978464i \(-0.566181\pi\)
−0.206419 + 0.978464i \(0.566181\pi\)
\(200\) 0 0
\(201\) 7.73517 0.545596
\(202\) 0 0
\(203\) −3.68985 −0.258976
\(204\) 0 0
\(205\) 4.26002 0.297533
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −9.04721 −0.625808
\(210\) 0 0
\(211\) 14.2809 0.983138 0.491569 0.870839i \(-0.336424\pi\)
0.491569 + 0.870839i \(0.336424\pi\)
\(212\) 0 0
\(213\) 0.946925 0.0648822
\(214\) 0 0
\(215\) 3.00535 0.204963
\(216\) 0 0
\(217\) −3.44840 −0.234092
\(218\) 0 0
\(219\) 3.86711 0.261315
\(220\) 0 0
\(221\) −7.31886 −0.492320
\(222\) 0 0
\(223\) 20.0031 1.33950 0.669752 0.742585i \(-0.266400\pi\)
0.669752 + 0.742585i \(0.266400\pi\)
\(224\) 0 0
\(225\) 0.737570 0.0491714
\(226\) 0 0
\(227\) −18.4547 −1.22488 −0.612441 0.790517i \(-0.709812\pi\)
−0.612441 + 0.790517i \(0.709812\pi\)
\(228\) 0 0
\(229\) −17.2299 −1.13859 −0.569293 0.822135i \(-0.692783\pi\)
−0.569293 + 0.822135i \(0.692783\pi\)
\(230\) 0 0
\(231\) −4.56912 −0.300626
\(232\) 0 0
\(233\) −16.2844 −1.06682 −0.533412 0.845856i \(-0.679090\pi\)
−0.533412 + 0.845856i \(0.679090\pi\)
\(234\) 0 0
\(235\) −6.89679 −0.449897
\(236\) 0 0
\(237\) −4.23904 −0.275355
\(238\) 0 0
\(239\) 26.1539 1.69175 0.845877 0.533378i \(-0.179078\pi\)
0.845877 + 0.533378i \(0.179078\pi\)
\(240\) 0 0
\(241\) 8.67527 0.558823 0.279412 0.960171i \(-0.409861\pi\)
0.279412 + 0.960171i \(0.409861\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.39532 0.153032
\(246\) 0 0
\(247\) −10.0055 −0.636633
\(248\) 0 0
\(249\) −10.8281 −0.686202
\(250\) 0 0
\(251\) −9.22888 −0.582522 −0.291261 0.956644i \(-0.594075\pi\)
−0.291261 + 0.956644i \(0.594075\pi\)
\(252\) 0 0
\(253\) 4.56912 0.287258
\(254\) 0 0
\(255\) −3.46938 −0.217261
\(256\) 0 0
\(257\) 17.7606 1.10787 0.553937 0.832559i \(-0.313125\pi\)
0.553937 + 0.832559i \(0.313125\pi\)
\(258\) 0 0
\(259\) 3.99759 0.248398
\(260\) 0 0
\(261\) 3.68985 0.228396
\(262\) 0 0
\(263\) 10.2721 0.633403 0.316702 0.948525i \(-0.397425\pi\)
0.316702 + 0.948525i \(0.397425\pi\)
\(264\) 0 0
\(265\) −24.9421 −1.53218
\(266\) 0 0
\(267\) 0.262430 0.0160604
\(268\) 0 0
\(269\) −17.5312 −1.06889 −0.534447 0.845202i \(-0.679480\pi\)
−0.534447 + 0.845202i \(0.679480\pi\)
\(270\) 0 0
\(271\) −21.1602 −1.28539 −0.642695 0.766123i \(-0.722184\pi\)
−0.642695 + 0.766123i \(0.722184\pi\)
\(272\) 0 0
\(273\) −5.05307 −0.305826
\(274\) 0 0
\(275\) 3.37005 0.203222
\(276\) 0 0
\(277\) 2.92941 0.176011 0.0880055 0.996120i \(-0.471951\pi\)
0.0880055 + 0.996120i \(0.471951\pi\)
\(278\) 0 0
\(279\) 3.44840 0.206450
\(280\) 0 0
\(281\) −26.8655 −1.60266 −0.801332 0.598220i \(-0.795875\pi\)
−0.801332 + 0.598220i \(0.795875\pi\)
\(282\) 0 0
\(283\) 22.0584 1.31124 0.655619 0.755092i \(-0.272408\pi\)
0.655619 + 0.755092i \(0.272408\pi\)
\(284\) 0 0
\(285\) −4.74292 −0.280946
\(286\) 0 0
\(287\) −1.77848 −0.104980
\(288\) 0 0
\(289\) −14.9021 −0.876597
\(290\) 0 0
\(291\) −7.20695 −0.422479
\(292\) 0 0
\(293\) 23.7537 1.38771 0.693854 0.720116i \(-0.255911\pi\)
0.693854 + 0.720116i \(0.255911\pi\)
\(294\) 0 0
\(295\) 33.8496 1.97080
\(296\) 0 0
\(297\) 4.56912 0.265127
\(298\) 0 0
\(299\) 5.05307 0.292227
\(300\) 0 0
\(301\) −1.25467 −0.0723182
\(302\) 0 0
\(303\) −8.26684 −0.474918
\(304\) 0 0
\(305\) 20.0618 1.14873
\(306\) 0 0
\(307\) −9.61015 −0.548480 −0.274240 0.961661i \(-0.588426\pi\)
−0.274240 + 0.961661i \(0.588426\pi\)
\(308\) 0 0
\(309\) 19.2619 1.09577
\(310\) 0 0
\(311\) −13.5642 −0.769155 −0.384577 0.923093i \(-0.625653\pi\)
−0.384577 + 0.923093i \(0.625653\pi\)
\(312\) 0 0
\(313\) 1.23168 0.0696189 0.0348095 0.999394i \(-0.488918\pi\)
0.0348095 + 0.999394i \(0.488918\pi\)
\(314\) 0 0
\(315\) −2.39532 −0.134961
\(316\) 0 0
\(317\) 16.0584 0.901931 0.450965 0.892541i \(-0.351080\pi\)
0.450965 + 0.892541i \(0.351080\pi\)
\(318\) 0 0
\(319\) 16.8594 0.943942
\(320\) 0 0
\(321\) 11.0531 0.616922
\(322\) 0 0
\(323\) 2.86794 0.159576
\(324\) 0 0
\(325\) 3.72700 0.206737
\(326\) 0 0
\(327\) 12.9047 0.713630
\(328\) 0 0
\(329\) 2.87928 0.158740
\(330\) 0 0
\(331\) 12.5424 0.689391 0.344696 0.938714i \(-0.387982\pi\)
0.344696 + 0.938714i \(0.387982\pi\)
\(332\) 0 0
\(333\) −3.99759 −0.219067
\(334\) 0 0
\(335\) 18.5282 1.01230
\(336\) 0 0
\(337\) 9.62661 0.524395 0.262197 0.965014i \(-0.415553\pi\)
0.262197 + 0.965014i \(0.415553\pi\)
\(338\) 0 0
\(339\) −1.25708 −0.0682752
\(340\) 0 0
\(341\) 15.7561 0.853243
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.39532 0.128960
\(346\) 0 0
\(347\) 9.94199 0.533714 0.266857 0.963736i \(-0.414015\pi\)
0.266857 + 0.963736i \(0.414015\pi\)
\(348\) 0 0
\(349\) −12.7274 −0.681283 −0.340641 0.940193i \(-0.610644\pi\)
−0.340641 + 0.940193i \(0.610644\pi\)
\(350\) 0 0
\(351\) 5.05307 0.269713
\(352\) 0 0
\(353\) −1.67433 −0.0891159 −0.0445579 0.999007i \(-0.514188\pi\)
−0.0445579 + 0.999007i \(0.514188\pi\)
\(354\) 0 0
\(355\) 2.26819 0.120383
\(356\) 0 0
\(357\) 1.44840 0.0766573
\(358\) 0 0
\(359\) 36.9662 1.95100 0.975500 0.220001i \(-0.0706060\pi\)
0.975500 + 0.220001i \(0.0706060\pi\)
\(360\) 0 0
\(361\) −15.0793 −0.793647
\(362\) 0 0
\(363\) 9.87687 0.518401
\(364\) 0 0
\(365\) 9.26297 0.484846
\(366\) 0 0
\(367\) 22.5080 1.17491 0.587454 0.809258i \(-0.300130\pi\)
0.587454 + 0.809258i \(0.300130\pi\)
\(368\) 0 0
\(369\) 1.77848 0.0925838
\(370\) 0 0
\(371\) 10.4128 0.540608
\(372\) 0 0
\(373\) 23.5896 1.22142 0.610711 0.791853i \(-0.290884\pi\)
0.610711 + 0.791853i \(0.290884\pi\)
\(374\) 0 0
\(375\) −10.2099 −0.527237
\(376\) 0 0
\(377\) 18.6451 0.960270
\(378\) 0 0
\(379\) −6.93664 −0.356311 −0.178156 0.984002i \(-0.557013\pi\)
−0.178156 + 0.984002i \(0.557013\pi\)
\(380\) 0 0
\(381\) −19.3385 −0.990741
\(382\) 0 0
\(383\) 22.8281 1.16646 0.583230 0.812307i \(-0.301788\pi\)
0.583230 + 0.812307i \(0.301788\pi\)
\(384\) 0 0
\(385\) −10.9445 −0.557784
\(386\) 0 0
\(387\) 1.25467 0.0637787
\(388\) 0 0
\(389\) 16.1350 0.818077 0.409039 0.912517i \(-0.365864\pi\)
0.409039 + 0.912517i \(0.365864\pi\)
\(390\) 0 0
\(391\) −1.44840 −0.0732486
\(392\) 0 0
\(393\) −19.4659 −0.981926
\(394\) 0 0
\(395\) −10.1539 −0.510897
\(396\) 0 0
\(397\) 18.5687 0.931938 0.465969 0.884801i \(-0.345706\pi\)
0.465969 + 0.884801i \(0.345706\pi\)
\(398\) 0 0
\(399\) 1.98008 0.0991278
\(400\) 0 0
\(401\) −7.22928 −0.361013 −0.180506 0.983574i \(-0.557774\pi\)
−0.180506 + 0.983574i \(0.557774\pi\)
\(402\) 0 0
\(403\) 17.4250 0.868002
\(404\) 0 0
\(405\) 2.39532 0.119025
\(406\) 0 0
\(407\) −18.2655 −0.905387
\(408\) 0 0
\(409\) 17.7912 0.879717 0.439859 0.898067i \(-0.355029\pi\)
0.439859 + 0.898067i \(0.355029\pi\)
\(410\) 0 0
\(411\) 20.4980 1.01109
\(412\) 0 0
\(413\) −14.1315 −0.695368
\(414\) 0 0
\(415\) −25.9368 −1.27319
\(416\) 0 0
\(417\) −21.0507 −1.03086
\(418\) 0 0
\(419\) 16.7937 0.820426 0.410213 0.911990i \(-0.365454\pi\)
0.410213 + 0.911990i \(0.365454\pi\)
\(420\) 0 0
\(421\) −11.1194 −0.541925 −0.270963 0.962590i \(-0.587342\pi\)
−0.270963 + 0.962590i \(0.587342\pi\)
\(422\) 0 0
\(423\) −2.87928 −0.139995
\(424\) 0 0
\(425\) −1.06829 −0.0518199
\(426\) 0 0
\(427\) −8.37540 −0.405314
\(428\) 0 0
\(429\) 23.0881 1.11470
\(430\) 0 0
\(431\) −35.8516 −1.72691 −0.863456 0.504424i \(-0.831705\pi\)
−0.863456 + 0.504424i \(0.831705\pi\)
\(432\) 0 0
\(433\) 13.2199 0.635310 0.317655 0.948206i \(-0.397105\pi\)
0.317655 + 0.948206i \(0.397105\pi\)
\(434\) 0 0
\(435\) 8.83837 0.423767
\(436\) 0 0
\(437\) −1.98008 −0.0947199
\(438\) 0 0
\(439\) 28.2898 1.35020 0.675100 0.737726i \(-0.264100\pi\)
0.675100 + 0.737726i \(0.264100\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −23.7781 −1.12973 −0.564865 0.825183i \(-0.691072\pi\)
−0.564865 + 0.825183i \(0.691072\pi\)
\(444\) 0 0
\(445\) 0.628604 0.0297987
\(446\) 0 0
\(447\) −19.1159 −0.904152
\(448\) 0 0
\(449\) −2.97578 −0.140436 −0.0702179 0.997532i \(-0.522369\pi\)
−0.0702179 + 0.997532i \(0.522369\pi\)
\(450\) 0 0
\(451\) 8.12607 0.382642
\(452\) 0 0
\(453\) −18.2542 −0.857655
\(454\) 0 0
\(455\) −12.1037 −0.567432
\(456\) 0 0
\(457\) −19.4381 −0.909277 −0.454638 0.890676i \(-0.650231\pi\)
−0.454638 + 0.890676i \(0.650231\pi\)
\(458\) 0 0
\(459\) −1.44840 −0.0676054
\(460\) 0 0
\(461\) −4.92877 −0.229556 −0.114778 0.993391i \(-0.536616\pi\)
−0.114778 + 0.993391i \(0.536616\pi\)
\(462\) 0 0
\(463\) −3.27983 −0.152427 −0.0762133 0.997092i \(-0.524283\pi\)
−0.0762133 + 0.997092i \(0.524283\pi\)
\(464\) 0 0
\(465\) 8.26002 0.383049
\(466\) 0 0
\(467\) −3.04520 −0.140915 −0.0704575 0.997515i \(-0.522446\pi\)
−0.0704575 + 0.997515i \(0.522446\pi\)
\(468\) 0 0
\(469\) −7.73517 −0.357177
\(470\) 0 0
\(471\) 6.63783 0.305855
\(472\) 0 0
\(473\) 5.73276 0.263593
\(474\) 0 0
\(475\) −1.46045 −0.0670098
\(476\) 0 0
\(477\) −10.4128 −0.476771
\(478\) 0 0
\(479\) −3.10080 −0.141679 −0.0708396 0.997488i \(-0.522568\pi\)
−0.0708396 + 0.997488i \(0.522568\pi\)
\(480\) 0 0
\(481\) −20.2001 −0.921047
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −17.2630 −0.783871
\(486\) 0 0
\(487\) −17.5634 −0.795872 −0.397936 0.917413i \(-0.630273\pi\)
−0.397936 + 0.917413i \(0.630273\pi\)
\(488\) 0 0
\(489\) −21.1080 −0.954538
\(490\) 0 0
\(491\) −32.6646 −1.47413 −0.737066 0.675821i \(-0.763789\pi\)
−0.737066 + 0.675821i \(0.763789\pi\)
\(492\) 0 0
\(493\) −5.34436 −0.240698
\(494\) 0 0
\(495\) 10.9445 0.491920
\(496\) 0 0
\(497\) −0.946925 −0.0424754
\(498\) 0 0
\(499\) 16.9084 0.756926 0.378463 0.925616i \(-0.376453\pi\)
0.378463 + 0.925616i \(0.376453\pi\)
\(500\) 0 0
\(501\) 1.01217 0.0452204
\(502\) 0 0
\(503\) 9.22582 0.411359 0.205679 0.978619i \(-0.434060\pi\)
0.205679 + 0.978619i \(0.434060\pi\)
\(504\) 0 0
\(505\) −19.8018 −0.881167
\(506\) 0 0
\(507\) 12.5336 0.556635
\(508\) 0 0
\(509\) −12.4362 −0.551226 −0.275613 0.961269i \(-0.588881\pi\)
−0.275613 + 0.961269i \(0.588881\pi\)
\(510\) 0 0
\(511\) −3.86711 −0.171071
\(512\) 0 0
\(513\) −1.98008 −0.0874225
\(514\) 0 0
\(515\) 46.1385 2.03310
\(516\) 0 0
\(517\) −13.1558 −0.578590
\(518\) 0 0
\(519\) 15.2069 0.667511
\(520\) 0 0
\(521\) −26.2123 −1.14838 −0.574191 0.818721i \(-0.694683\pi\)
−0.574191 + 0.818721i \(0.694683\pi\)
\(522\) 0 0
\(523\) −25.3309 −1.10764 −0.553822 0.832635i \(-0.686831\pi\)
−0.553822 + 0.832635i \(0.686831\pi\)
\(524\) 0 0
\(525\) −0.737570 −0.0321902
\(526\) 0 0
\(527\) −4.99465 −0.217570
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 14.1315 0.613257
\(532\) 0 0
\(533\) 8.98677 0.389260
\(534\) 0 0
\(535\) 26.4757 1.14464
\(536\) 0 0
\(537\) −18.1491 −0.783190
\(538\) 0 0
\(539\) 4.56912 0.196806
\(540\) 0 0
\(541\) −16.2940 −0.700534 −0.350267 0.936650i \(-0.613909\pi\)
−0.350267 + 0.936650i \(0.613909\pi\)
\(542\) 0 0
\(543\) 10.5866 0.454316
\(544\) 0 0
\(545\) 30.9109 1.32408
\(546\) 0 0
\(547\) −13.1461 −0.562087 −0.281044 0.959695i \(-0.590681\pi\)
−0.281044 + 0.959695i \(0.590681\pi\)
\(548\) 0 0
\(549\) 8.37540 0.357453
\(550\) 0 0
\(551\) −7.30617 −0.311253
\(552\) 0 0
\(553\) 4.23904 0.180262
\(554\) 0 0
\(555\) −9.57553 −0.406459
\(556\) 0 0
\(557\) 1.28424 0.0544150 0.0272075 0.999630i \(-0.491339\pi\)
0.0272075 + 0.999630i \(0.491339\pi\)
\(558\) 0 0
\(559\) 6.33996 0.268152
\(560\) 0 0
\(561\) −6.61790 −0.279408
\(562\) 0 0
\(563\) 1.02181 0.0430642 0.0215321 0.999768i \(-0.493146\pi\)
0.0215321 + 0.999768i \(0.493146\pi\)
\(564\) 0 0
\(565\) −3.01111 −0.126678
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −14.9943 −0.628592 −0.314296 0.949325i \(-0.601768\pi\)
−0.314296 + 0.949325i \(0.601768\pi\)
\(570\) 0 0
\(571\) 15.3179 0.641035 0.320517 0.947243i \(-0.396143\pi\)
0.320517 + 0.947243i \(0.396143\pi\)
\(572\) 0 0
\(573\) 19.8004 0.827174
\(574\) 0 0
\(575\) 0.737570 0.0307588
\(576\) 0 0
\(577\) −36.4091 −1.51573 −0.757865 0.652411i \(-0.773757\pi\)
−0.757865 + 0.652411i \(0.773757\pi\)
\(578\) 0 0
\(579\) 22.4956 0.934885
\(580\) 0 0
\(581\) 10.8281 0.449225
\(582\) 0 0
\(583\) −47.5775 −1.97046
\(584\) 0 0
\(585\) 12.1037 0.500428
\(586\) 0 0
\(587\) 32.1484 1.32691 0.663454 0.748217i \(-0.269090\pi\)
0.663454 + 0.748217i \(0.269090\pi\)
\(588\) 0 0
\(589\) −6.82809 −0.281346
\(590\) 0 0
\(591\) 1.67338 0.0688338
\(592\) 0 0
\(593\) 31.6719 1.30061 0.650305 0.759673i \(-0.274641\pi\)
0.650305 + 0.759673i \(0.274641\pi\)
\(594\) 0 0
\(595\) 3.46938 0.142231
\(596\) 0 0
\(597\) −5.82380 −0.238352
\(598\) 0 0
\(599\) 36.0114 1.47138 0.735692 0.677316i \(-0.236857\pi\)
0.735692 + 0.677316i \(0.236857\pi\)
\(600\) 0 0
\(601\) −33.4295 −1.36362 −0.681810 0.731530i \(-0.738807\pi\)
−0.681810 + 0.731530i \(0.738807\pi\)
\(602\) 0 0
\(603\) 7.73517 0.315000
\(604\) 0 0
\(605\) 23.6583 0.961846
\(606\) 0 0
\(607\) −0.510803 −0.0207329 −0.0103664 0.999946i \(-0.503300\pi\)
−0.0103664 + 0.999946i \(0.503300\pi\)
\(608\) 0 0
\(609\) −3.68985 −0.149520
\(610\) 0 0
\(611\) −14.5492 −0.588598
\(612\) 0 0
\(613\) −29.7444 −1.20137 −0.600683 0.799488i \(-0.705104\pi\)
−0.600683 + 0.799488i \(0.705104\pi\)
\(614\) 0 0
\(615\) 4.26002 0.171781
\(616\) 0 0
\(617\) −18.2312 −0.733959 −0.366980 0.930229i \(-0.619608\pi\)
−0.366980 + 0.930229i \(0.619608\pi\)
\(618\) 0 0
\(619\) −31.3889 −1.26163 −0.630814 0.775934i \(-0.717279\pi\)
−0.630814 + 0.775934i \(0.717279\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −0.262430 −0.0105140
\(624\) 0 0
\(625\) −28.1438 −1.12575
\(626\) 0 0
\(627\) −9.04721 −0.361311
\(628\) 0 0
\(629\) 5.79011 0.230867
\(630\) 0 0
\(631\) −42.9030 −1.70794 −0.853970 0.520322i \(-0.825812\pi\)
−0.853970 + 0.520322i \(0.825812\pi\)
\(632\) 0 0
\(633\) 14.2809 0.567615
\(634\) 0 0
\(635\) −46.3219 −1.83823
\(636\) 0 0
\(637\) 5.05307 0.200210
\(638\) 0 0
\(639\) 0.946925 0.0374598
\(640\) 0 0
\(641\) −19.8505 −0.784049 −0.392025 0.919955i \(-0.628225\pi\)
−0.392025 + 0.919955i \(0.628225\pi\)
\(642\) 0 0
\(643\) −26.5457 −1.04686 −0.523431 0.852068i \(-0.675348\pi\)
−0.523431 + 0.852068i \(0.675348\pi\)
\(644\) 0 0
\(645\) 3.00535 0.118336
\(646\) 0 0
\(647\) −30.0117 −1.17988 −0.589940 0.807447i \(-0.700849\pi\)
−0.589940 + 0.807447i \(0.700849\pi\)
\(648\) 0 0
\(649\) 64.5687 2.53454
\(650\) 0 0
\(651\) −3.44840 −0.135153
\(652\) 0 0
\(653\) 4.41083 0.172609 0.0863046 0.996269i \(-0.472494\pi\)
0.0863046 + 0.996269i \(0.472494\pi\)
\(654\) 0 0
\(655\) −46.6271 −1.82187
\(656\) 0 0
\(657\) 3.86711 0.150870
\(658\) 0 0
\(659\) 30.2147 1.17700 0.588499 0.808498i \(-0.299719\pi\)
0.588499 + 0.808498i \(0.299719\pi\)
\(660\) 0 0
\(661\) 18.8322 0.732488 0.366244 0.930519i \(-0.380644\pi\)
0.366244 + 0.930519i \(0.380644\pi\)
\(662\) 0 0
\(663\) −7.31886 −0.284241
\(664\) 0 0
\(665\) 4.74292 0.183923
\(666\) 0 0
\(667\) 3.68985 0.142871
\(668\) 0 0
\(669\) 20.0031 0.773363
\(670\) 0 0
\(671\) 38.2682 1.47733
\(672\) 0 0
\(673\) −17.0208 −0.656102 −0.328051 0.944660i \(-0.606392\pi\)
−0.328051 + 0.944660i \(0.606392\pi\)
\(674\) 0 0
\(675\) 0.737570 0.0283891
\(676\) 0 0
\(677\) 33.4116 1.28411 0.642056 0.766657i \(-0.278082\pi\)
0.642056 + 0.766657i \(0.278082\pi\)
\(678\) 0 0
\(679\) 7.20695 0.276577
\(680\) 0 0
\(681\) −18.4547 −0.707186
\(682\) 0 0
\(683\) 9.95082 0.380758 0.190379 0.981711i \(-0.439028\pi\)
0.190379 + 0.981711i \(0.439028\pi\)
\(684\) 0 0
\(685\) 49.0993 1.87599
\(686\) 0 0
\(687\) −17.2299 −0.657363
\(688\) 0 0
\(689\) −52.6169 −2.00454
\(690\) 0 0
\(691\) −21.7795 −0.828533 −0.414266 0.910156i \(-0.635962\pi\)
−0.414266 + 0.910156i \(0.635962\pi\)
\(692\) 0 0
\(693\) −4.56912 −0.173567
\(694\) 0 0
\(695\) −50.4231 −1.91266
\(696\) 0 0
\(697\) −2.57594 −0.0975707
\(698\) 0 0
\(699\) −16.2844 −0.615931
\(700\) 0 0
\(701\) 17.4292 0.658291 0.329146 0.944279i \(-0.393239\pi\)
0.329146 + 0.944279i \(0.393239\pi\)
\(702\) 0 0
\(703\) 7.91554 0.298540
\(704\) 0 0
\(705\) −6.89679 −0.259748
\(706\) 0 0
\(707\) 8.26684 0.310907
\(708\) 0 0
\(709\) 29.6266 1.11265 0.556325 0.830965i \(-0.312211\pi\)
0.556325 + 0.830965i \(0.312211\pi\)
\(710\) 0 0
\(711\) −4.23904 −0.158977
\(712\) 0 0
\(713\) 3.44840 0.129143
\(714\) 0 0
\(715\) 55.3035 2.06823
\(716\) 0 0
\(717\) 26.1539 0.976734
\(718\) 0 0
\(719\) 17.1492 0.639558 0.319779 0.947492i \(-0.396391\pi\)
0.319779 + 0.947492i \(0.396391\pi\)
\(720\) 0 0
\(721\) −19.2619 −0.717351
\(722\) 0 0
\(723\) 8.67527 0.322637
\(724\) 0 0
\(725\) 2.72152 0.101075
\(726\) 0 0
\(727\) −48.6199 −1.80321 −0.901606 0.432557i \(-0.857611\pi\)
−0.901606 + 0.432557i \(0.857611\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.81727 −0.0672141
\(732\) 0 0
\(733\) −39.3216 −1.45238 −0.726189 0.687495i \(-0.758710\pi\)
−0.726189 + 0.687495i \(0.758710\pi\)
\(734\) 0 0
\(735\) 2.39532 0.0883528
\(736\) 0 0
\(737\) 35.3429 1.30187
\(738\) 0 0
\(739\) −20.5818 −0.757115 −0.378557 0.925578i \(-0.623580\pi\)
−0.378557 + 0.925578i \(0.623580\pi\)
\(740\) 0 0
\(741\) −10.0055 −0.367560
\(742\) 0 0
\(743\) 38.0313 1.39523 0.697616 0.716472i \(-0.254244\pi\)
0.697616 + 0.716472i \(0.254244\pi\)
\(744\) 0 0
\(745\) −45.7888 −1.67757
\(746\) 0 0
\(747\) −10.8281 −0.396179
\(748\) 0 0
\(749\) −11.0531 −0.403871
\(750\) 0 0
\(751\) 38.3146 1.39812 0.699060 0.715063i \(-0.253602\pi\)
0.699060 + 0.715063i \(0.253602\pi\)
\(752\) 0 0
\(753\) −9.22888 −0.336319
\(754\) 0 0
\(755\) −43.7246 −1.59130
\(756\) 0 0
\(757\) −25.4278 −0.924190 −0.462095 0.886830i \(-0.652902\pi\)
−0.462095 + 0.886830i \(0.652902\pi\)
\(758\) 0 0
\(759\) 4.56912 0.165849
\(760\) 0 0
\(761\) 46.3205 1.67912 0.839558 0.543271i \(-0.182814\pi\)
0.839558 + 0.543271i \(0.182814\pi\)
\(762\) 0 0
\(763\) −12.9047 −0.467180
\(764\) 0 0
\(765\) −3.46938 −0.125436
\(766\) 0 0
\(767\) 71.4078 2.57838
\(768\) 0 0
\(769\) 2.92806 0.105588 0.0527942 0.998605i \(-0.483187\pi\)
0.0527942 + 0.998605i \(0.483187\pi\)
\(770\) 0 0
\(771\) 17.7606 0.639631
\(772\) 0 0
\(773\) 0.848652 0.0305239 0.0152619 0.999884i \(-0.495142\pi\)
0.0152619 + 0.999884i \(0.495142\pi\)
\(774\) 0 0
\(775\) 2.54344 0.0913629
\(776\) 0 0
\(777\) 3.99759 0.143413
\(778\) 0 0
\(779\) −3.52152 −0.126171
\(780\) 0 0
\(781\) 4.32662 0.154818
\(782\) 0 0
\(783\) 3.68985 0.131864
\(784\) 0 0
\(785\) 15.8997 0.567486
\(786\) 0 0
\(787\) −31.3118 −1.11614 −0.558072 0.829793i \(-0.688459\pi\)
−0.558072 + 0.829793i \(0.688459\pi\)
\(788\) 0 0
\(789\) 10.2721 0.365695
\(790\) 0 0
\(791\) 1.25708 0.0446966
\(792\) 0 0
\(793\) 42.3215 1.50288
\(794\) 0 0
\(795\) −24.9421 −0.884606
\(796\) 0 0
\(797\) 4.66393 0.165205 0.0826025 0.996583i \(-0.473677\pi\)
0.0826025 + 0.996583i \(0.473677\pi\)
\(798\) 0 0
\(799\) 4.17034 0.147536
\(800\) 0 0
\(801\) 0.262430 0.00927250
\(802\) 0 0
\(803\) 17.6693 0.623535
\(804\) 0 0
\(805\) −2.39532 −0.0844240
\(806\) 0 0
\(807\) −17.5312 −0.617126
\(808\) 0 0
\(809\) 26.2106 0.921516 0.460758 0.887526i \(-0.347578\pi\)
0.460758 + 0.887526i \(0.347578\pi\)
\(810\) 0 0
\(811\) 35.2107 1.23642 0.618208 0.786015i \(-0.287859\pi\)
0.618208 + 0.786015i \(0.287859\pi\)
\(812\) 0 0
\(813\) −21.1602 −0.742120
\(814\) 0 0
\(815\) −50.5606 −1.77106
\(816\) 0 0
\(817\) −2.48435 −0.0869164
\(818\) 0 0
\(819\) −5.05307 −0.176569
\(820\) 0 0
\(821\) 30.9369 1.07970 0.539852 0.841760i \(-0.318480\pi\)
0.539852 + 0.841760i \(0.318480\pi\)
\(822\) 0 0
\(823\) 17.7770 0.619668 0.309834 0.950791i \(-0.399727\pi\)
0.309834 + 0.950791i \(0.399727\pi\)
\(824\) 0 0
\(825\) 3.37005 0.117330
\(826\) 0 0
\(827\) 3.40509 0.118406 0.0592032 0.998246i \(-0.481144\pi\)
0.0592032 + 0.998246i \(0.481144\pi\)
\(828\) 0 0
\(829\) 0.475680 0.0165210 0.00826052 0.999966i \(-0.497371\pi\)
0.00826052 + 0.999966i \(0.497371\pi\)
\(830\) 0 0
\(831\) 2.92941 0.101620
\(832\) 0 0
\(833\) −1.44840 −0.0501840
\(834\) 0 0
\(835\) 2.42447 0.0839023
\(836\) 0 0
\(837\) 3.44840 0.119194
\(838\) 0 0
\(839\) 15.4804 0.534442 0.267221 0.963635i \(-0.413895\pi\)
0.267221 + 0.963635i \(0.413895\pi\)
\(840\) 0 0
\(841\) −15.3850 −0.530519
\(842\) 0 0
\(843\) −26.8655 −0.925298
\(844\) 0 0
\(845\) 30.0219 1.03279
\(846\) 0 0
\(847\) −9.87687 −0.339373
\(848\) 0 0
\(849\) 22.0584 0.757043
\(850\) 0 0
\(851\) −3.99759 −0.137036
\(852\) 0 0
\(853\) 23.6583 0.810044 0.405022 0.914307i \(-0.367264\pi\)
0.405022 + 0.914307i \(0.367264\pi\)
\(854\) 0 0
\(855\) −4.74292 −0.162204
\(856\) 0 0
\(857\) −10.2346 −0.349608 −0.174804 0.984603i \(-0.555929\pi\)
−0.174804 + 0.984603i \(0.555929\pi\)
\(858\) 0 0
\(859\) 38.4043 1.31034 0.655169 0.755483i \(-0.272598\pi\)
0.655169 + 0.755483i \(0.272598\pi\)
\(860\) 0 0
\(861\) −1.77848 −0.0606103
\(862\) 0 0
\(863\) −33.5209 −1.14106 −0.570532 0.821275i \(-0.693263\pi\)
−0.570532 + 0.821275i \(0.693263\pi\)
\(864\) 0 0
\(865\) 36.4255 1.23851
\(866\) 0 0
\(867\) −14.9021 −0.506103
\(868\) 0 0
\(869\) −19.3687 −0.657038
\(870\) 0 0
\(871\) 39.0864 1.32439
\(872\) 0 0
\(873\) −7.20695 −0.243918
\(874\) 0 0
\(875\) 10.2099 0.345157
\(876\) 0 0
\(877\) −52.7392 −1.78088 −0.890438 0.455105i \(-0.849602\pi\)
−0.890438 + 0.455105i \(0.849602\pi\)
\(878\) 0 0
\(879\) 23.7537 0.801194
\(880\) 0 0
\(881\) 19.1890 0.646495 0.323247 0.946314i \(-0.395225\pi\)
0.323247 + 0.946314i \(0.395225\pi\)
\(882\) 0 0
\(883\) 33.2985 1.12059 0.560293 0.828295i \(-0.310689\pi\)
0.560293 + 0.828295i \(0.310689\pi\)
\(884\) 0 0
\(885\) 33.8496 1.13784
\(886\) 0 0
\(887\) 17.8750 0.600183 0.300092 0.953910i \(-0.402983\pi\)
0.300092 + 0.953910i \(0.402983\pi\)
\(888\) 0 0
\(889\) 19.3385 0.648592
\(890\) 0 0
\(891\) 4.56912 0.153071
\(892\) 0 0
\(893\) 5.70118 0.190783
\(894\) 0 0
\(895\) −43.4729 −1.45314
\(896\) 0 0
\(897\) 5.05307 0.168717
\(898\) 0 0
\(899\) 12.7241 0.424371
\(900\) 0 0
\(901\) 15.0819 0.502452
\(902\) 0 0
\(903\) −1.25467 −0.0417529
\(904\) 0 0
\(905\) 25.3584 0.842942
\(906\) 0 0
\(907\) −6.71166 −0.222857 −0.111428 0.993772i \(-0.535543\pi\)
−0.111428 + 0.993772i \(0.535543\pi\)
\(908\) 0 0
\(909\) −8.26684 −0.274194
\(910\) 0 0
\(911\) 14.5404 0.481744 0.240872 0.970557i \(-0.422567\pi\)
0.240872 + 0.970557i \(0.422567\pi\)
\(912\) 0 0
\(913\) −49.4749 −1.63738
\(914\) 0 0
\(915\) 20.0618 0.663222
\(916\) 0 0
\(917\) 19.4659 0.642821
\(918\) 0 0
\(919\) −25.3532 −0.836325 −0.418162 0.908372i \(-0.637326\pi\)
−0.418162 + 0.908372i \(0.637326\pi\)
\(920\) 0 0
\(921\) −9.61015 −0.316665
\(922\) 0 0
\(923\) 4.78488 0.157496
\(924\) 0 0
\(925\) −2.94851 −0.0969463
\(926\) 0 0
\(927\) 19.2619 0.632644
\(928\) 0 0
\(929\) 46.6378 1.53014 0.765069 0.643948i \(-0.222705\pi\)
0.765069 + 0.643948i \(0.222705\pi\)
\(930\) 0 0
\(931\) −1.98008 −0.0648944
\(932\) 0 0
\(933\) −13.5642 −0.444072
\(934\) 0 0
\(935\) −15.8520 −0.518416
\(936\) 0 0
\(937\) −34.8699 −1.13915 −0.569576 0.821939i \(-0.692892\pi\)
−0.569576 + 0.821939i \(0.692892\pi\)
\(938\) 0 0
\(939\) 1.23168 0.0401945
\(940\) 0 0
\(941\) −12.0243 −0.391982 −0.195991 0.980606i \(-0.562792\pi\)
−0.195991 + 0.980606i \(0.562792\pi\)
\(942\) 0 0
\(943\) 1.77848 0.0579152
\(944\) 0 0
\(945\) −2.39532 −0.0779198
\(946\) 0 0
\(947\) 18.4626 0.599953 0.299977 0.953947i \(-0.403021\pi\)
0.299977 + 0.953947i \(0.403021\pi\)
\(948\) 0 0
\(949\) 19.5408 0.634321
\(950\) 0 0
\(951\) 16.0584 0.520730
\(952\) 0 0
\(953\) 8.91601 0.288818 0.144409 0.989518i \(-0.453872\pi\)
0.144409 + 0.989518i \(0.453872\pi\)
\(954\) 0 0
\(955\) 47.4284 1.53475
\(956\) 0 0
\(957\) 16.8594 0.544985
\(958\) 0 0
\(959\) −20.4980 −0.661915
\(960\) 0 0
\(961\) −19.1086 −0.616405
\(962\) 0 0
\(963\) 11.0531 0.356180
\(964\) 0 0
\(965\) 53.8842 1.73459
\(966\) 0 0
\(967\) 11.1091 0.357244 0.178622 0.983918i \(-0.442836\pi\)
0.178622 + 0.983918i \(0.442836\pi\)
\(968\) 0 0
\(969\) 2.86794 0.0921314
\(970\) 0 0
\(971\) −18.9226 −0.607255 −0.303627 0.952791i \(-0.598198\pi\)
−0.303627 + 0.952791i \(0.598198\pi\)
\(972\) 0 0
\(973\) 21.0507 0.674853
\(974\) 0 0
\(975\) 3.72700 0.119359
\(976\) 0 0
\(977\) 17.0219 0.544579 0.272290 0.962215i \(-0.412219\pi\)
0.272290 + 0.962215i \(0.412219\pi\)
\(978\) 0 0
\(979\) 1.19907 0.0383225
\(980\) 0 0
\(981\) 12.9047 0.412014
\(982\) 0 0
\(983\) −37.7249 −1.20324 −0.601618 0.798784i \(-0.705477\pi\)
−0.601618 + 0.798784i \(0.705477\pi\)
\(984\) 0 0
\(985\) 4.00829 0.127715
\(986\) 0 0
\(987\) 2.87928 0.0916484
\(988\) 0 0
\(989\) 1.25467 0.0398963
\(990\) 0 0
\(991\) 51.0190 1.62067 0.810336 0.585965i \(-0.199285\pi\)
0.810336 + 0.585965i \(0.199285\pi\)
\(992\) 0 0
\(993\) 12.5424 0.398020
\(994\) 0 0
\(995\) −13.9499 −0.442241
\(996\) 0 0
\(997\) −43.6823 −1.38343 −0.691717 0.722169i \(-0.743145\pi\)
−0.691717 + 0.722169i \(0.743145\pi\)
\(998\) 0 0
\(999\) −3.99759 −0.126478
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 7728.2.a.cd.1.3 4
4.3 odd 2 483.2.a.i.1.4 4
12.11 even 2 1449.2.a.p.1.1 4
28.27 even 2 3381.2.a.w.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.i.1.4 4 4.3 odd 2
1449.2.a.p.1.1 4 12.11 even 2
3381.2.a.w.1.4 4 28.27 even 2
7728.2.a.cd.1.3 4 1.1 even 1 trivial