Properties

Label 7728.2.a.cd.1.4
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 1.4
Root \(0.509552\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.41546 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.67510 q^{11} -4.66537 q^{13} +4.41546 q^{15} +6.24991 q^{17} +0.694209 q^{19} -1.00000 q^{21} +1.00000 q^{23} +14.4963 q^{25} +1.00000 q^{27} +5.60012 q^{29} -4.24991 q^{31} +1.67510 q^{33} -4.41546 q^{35} +9.26901 q^{37} -4.66537 q^{39} -5.15582 q^{41} -4.20376 q^{43} +4.41546 q^{45} +1.92501 q^{47} +1.00000 q^{49} +6.24991 q^{51} -1.84066 q^{53} +7.39636 q^{55} +0.694209 q^{57} -9.39779 q^{59} +7.72125 q^{61} -1.00000 q^{63} -20.5998 q^{65} +8.22728 q^{67} +1.00000 q^{69} +10.6654 q^{71} -11.9117 q^{73} +14.4963 q^{75} -1.67510 q^{77} -0.581012 q^{79} +1.00000 q^{81} -6.95032 q^{83} +27.5962 q^{85} +5.60012 q^{87} -13.4963 q^{89} +4.66537 q^{91} -4.24991 q^{93} +3.06525 q^{95} +10.0999 q^{97} +1.67510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(5\) 4.41546 1.97465 0.987327 0.158699i \(-0.0507301\pi\)
0.987327 + 0.158699i \(0.0507301\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.67510 0.505063 0.252531 0.967589i \(-0.418737\pi\)
0.252531 + 0.967589i \(0.418737\pi\)
\(12\) 0 0
\(13\) −4.66537 −1.29394 −0.646970 0.762515i \(-0.723964\pi\)
−0.646970 + 0.762515i \(0.723964\pi\)
\(14\) 0 0
\(15\) 4.41546 1.14007
\(16\) 0 0
\(17\) 6.24991 1.51583 0.757913 0.652356i \(-0.226219\pi\)
0.757913 + 0.652356i \(0.226219\pi\)
\(18\) 0 0
\(19\) 0.694209 0.159262 0.0796312 0.996824i \(-0.474626\pi\)
0.0796312 + 0.996824i \(0.474626\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 14.4963 2.89926
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.60012 1.03992 0.519958 0.854192i \(-0.325948\pi\)
0.519958 + 0.854192i \(0.325948\pi\)
\(30\) 0 0
\(31\) −4.24991 −0.763306 −0.381653 0.924306i \(-0.624645\pi\)
−0.381653 + 0.924306i \(0.624645\pi\)
\(32\) 0 0
\(33\) 1.67510 0.291598
\(34\) 0 0
\(35\) −4.41546 −0.746349
\(36\) 0 0
\(37\) 9.26901 1.52382 0.761908 0.647685i \(-0.224263\pi\)
0.761908 + 0.647685i \(0.224263\pi\)
\(38\) 0 0
\(39\) −4.66537 −0.747057
\(40\) 0 0
\(41\) −5.15582 −0.805203 −0.402602 0.915375i \(-0.631894\pi\)
−0.402602 + 0.915375i \(0.631894\pi\)
\(42\) 0 0
\(43\) −4.20376 −0.641068 −0.320534 0.947237i \(-0.603862\pi\)
−0.320534 + 0.947237i \(0.603862\pi\)
\(44\) 0 0
\(45\) 4.41546 0.658218
\(46\) 0 0
\(47\) 1.92501 0.280792 0.140396 0.990095i \(-0.455162\pi\)
0.140396 + 0.990095i \(0.455162\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.24991 0.875162
\(52\) 0 0
\(53\) −1.84066 −0.252833 −0.126417 0.991977i \(-0.540348\pi\)
−0.126417 + 0.991977i \(0.540348\pi\)
\(54\) 0 0
\(55\) 7.39636 0.997324
\(56\) 0 0
\(57\) 0.694209 0.0919502
\(58\) 0 0
\(59\) −9.39779 −1.22349 −0.611744 0.791056i \(-0.709532\pi\)
−0.611744 + 0.791056i \(0.709532\pi\)
\(60\) 0 0
\(61\) 7.72125 0.988605 0.494302 0.869290i \(-0.335424\pi\)
0.494302 + 0.869290i \(0.335424\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −20.5998 −2.55508
\(66\) 0 0
\(67\) 8.22728 1.00512 0.502561 0.864542i \(-0.332391\pi\)
0.502561 + 0.864542i \(0.332391\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.6654 1.26575 0.632873 0.774255i \(-0.281875\pi\)
0.632873 + 0.774255i \(0.281875\pi\)
\(72\) 0 0
\(73\) −11.9117 −1.39416 −0.697082 0.716991i \(-0.745519\pi\)
−0.697082 + 0.716991i \(0.745519\pi\)
\(74\) 0 0
\(75\) 14.4963 1.67389
\(76\) 0 0
\(77\) −1.67510 −0.190896
\(78\) 0 0
\(79\) −0.581012 −0.0653689 −0.0326845 0.999466i \(-0.510406\pi\)
−0.0326845 + 0.999466i \(0.510406\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.95032 −0.762897 −0.381449 0.924390i \(-0.624575\pi\)
−0.381449 + 0.924390i \(0.624575\pi\)
\(84\) 0 0
\(85\) 27.5962 2.99323
\(86\) 0 0
\(87\) 5.60012 0.600396
\(88\) 0 0
\(89\) −13.4963 −1.43060 −0.715302 0.698816i \(-0.753711\pi\)
−0.715302 + 0.698816i \(0.753711\pi\)
\(90\) 0 0
\(91\) 4.66537 0.489064
\(92\) 0 0
\(93\) −4.24991 −0.440695
\(94\) 0 0
\(95\) 3.06525 0.314488
\(96\) 0 0
\(97\) 10.0999 1.02549 0.512746 0.858540i \(-0.328628\pi\)
0.512746 + 0.858540i \(0.328628\pi\)
\(98\) 0 0
\(99\) 1.67510 0.168354
\(100\) 0 0
\(101\) −13.7830 −1.37146 −0.685729 0.727857i \(-0.740516\pi\)
−0.685729 + 0.727857i \(0.740516\pi\)
\(102\) 0 0
\(103\) −16.5553 −1.63125 −0.815623 0.578584i \(-0.803605\pi\)
−0.815623 + 0.578584i \(0.803605\pi\)
\(104\) 0 0
\(105\) −4.41546 −0.430905
\(106\) 0 0
\(107\) 1.33463 0.129024 0.0645118 0.997917i \(-0.479451\pi\)
0.0645118 + 0.997917i \(0.479451\pi\)
\(108\) 0 0
\(109\) 4.00794 0.383891 0.191945 0.981406i \(-0.438520\pi\)
0.191945 + 0.981406i \(0.438520\pi\)
\(110\) 0 0
\(111\) 9.26901 0.879776
\(112\) 0 0
\(113\) −9.06525 −0.852787 −0.426394 0.904538i \(-0.640216\pi\)
−0.426394 + 0.904538i \(0.640216\pi\)
\(114\) 0 0
\(115\) 4.41546 0.411744
\(116\) 0 0
\(117\) −4.66537 −0.431314
\(118\) 0 0
\(119\) −6.24991 −0.572928
\(120\) 0 0
\(121\) −8.19403 −0.744911
\(122\) 0 0
\(123\) −5.15582 −0.464884
\(124\) 0 0
\(125\) 41.9305 3.75038
\(126\) 0 0
\(127\) 21.4977 1.90761 0.953807 0.300419i \(-0.0971267\pi\)
0.953807 + 0.300419i \(0.0971267\pi\)
\(128\) 0 0
\(129\) −4.20376 −0.370121
\(130\) 0 0
\(131\) −1.17529 −0.102685 −0.0513426 0.998681i \(-0.516350\pi\)
−0.0513426 + 0.998681i \(0.516350\pi\)
\(132\) 0 0
\(133\) −0.694209 −0.0601956
\(134\) 0 0
\(135\) 4.41546 0.380022
\(136\) 0 0
\(137\) 15.8562 1.35469 0.677345 0.735666i \(-0.263131\pi\)
0.677345 + 0.735666i \(0.263131\pi\)
\(138\) 0 0
\(139\) 1.93438 0.164072 0.0820361 0.996629i \(-0.473858\pi\)
0.0820361 + 0.996629i \(0.473858\pi\)
\(140\) 0 0
\(141\) 1.92501 0.162115
\(142\) 0 0
\(143\) −7.81498 −0.653521
\(144\) 0 0
\(145\) 24.7271 2.05347
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 2.61301 0.214067 0.107033 0.994255i \(-0.465865\pi\)
0.107033 + 0.994255i \(0.465865\pi\)
\(150\) 0 0
\(151\) 9.26281 0.753797 0.376898 0.926255i \(-0.376991\pi\)
0.376898 + 0.926255i \(0.376991\pi\)
\(152\) 0 0
\(153\) 6.24991 0.505275
\(154\) 0 0
\(155\) −18.7653 −1.50727
\(156\) 0 0
\(157\) −7.77504 −0.620516 −0.310258 0.950652i \(-0.600415\pi\)
−0.310258 + 0.950652i \(0.600415\pi\)
\(158\) 0 0
\(159\) −1.84066 −0.145973
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 7.12077 0.557742 0.278871 0.960329i \(-0.410040\pi\)
0.278871 + 0.960329i \(0.410040\pi\)
\(164\) 0 0
\(165\) 7.39636 0.575805
\(166\) 0 0
\(167\) 11.9867 0.927562 0.463781 0.885950i \(-0.346493\pi\)
0.463781 + 0.885950i \(0.346493\pi\)
\(168\) 0 0
\(169\) 8.76567 0.674282
\(170\) 0 0
\(171\) 0.694209 0.0530875
\(172\) 0 0
\(173\) −2.09993 −0.159655 −0.0798275 0.996809i \(-0.525437\pi\)
−0.0798275 + 0.996809i \(0.525437\pi\)
\(174\) 0 0
\(175\) −14.4963 −1.09582
\(176\) 0 0
\(177\) −9.39779 −0.706381
\(178\) 0 0
\(179\) 15.9726 1.19385 0.596924 0.802298i \(-0.296389\pi\)
0.596924 + 0.802298i \(0.296389\pi\)
\(180\) 0 0
\(181\) −2.89970 −0.215533 −0.107767 0.994176i \(-0.534370\pi\)
−0.107767 + 0.994176i \(0.534370\pi\)
\(182\) 0 0
\(183\) 7.72125 0.570771
\(184\) 0 0
\(185\) 40.9270 3.00901
\(186\) 0 0
\(187\) 10.4692 0.765587
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 21.5486 1.55921 0.779603 0.626275i \(-0.215421\pi\)
0.779603 + 0.626275i \(0.215421\pi\)
\(192\) 0 0
\(193\) 4.58722 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(194\) 0 0
\(195\) −20.5998 −1.47518
\(196\) 0 0
\(197\) −11.8656 −0.845389 −0.422695 0.906272i \(-0.638916\pi\)
−0.422695 + 0.906272i \(0.638916\pi\)
\(198\) 0 0
\(199\) 2.52866 0.179252 0.0896259 0.995976i \(-0.471433\pi\)
0.0896259 + 0.995976i \(0.471433\pi\)
\(200\) 0 0
\(201\) 8.22728 0.580307
\(202\) 0 0
\(203\) −5.60012 −0.393051
\(204\) 0 0
\(205\) −22.7653 −1.59000
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 1.16287 0.0804376
\(210\) 0 0
\(211\) 21.9797 1.51314 0.756572 0.653911i \(-0.226873\pi\)
0.756572 + 0.653911i \(0.226873\pi\)
\(212\) 0 0
\(213\) 10.6654 0.730779
\(214\) 0 0
\(215\) −18.5615 −1.26589
\(216\) 0 0
\(217\) 4.24991 0.288503
\(218\) 0 0
\(219\) −11.9117 −0.804921
\(220\) 0 0
\(221\) −29.1581 −1.96139
\(222\) 0 0
\(223\) −0.0302726 −0.00202720 −0.00101360 0.999999i \(-0.500323\pi\)
−0.00101360 + 0.999999i \(0.500323\pi\)
\(224\) 0 0
\(225\) 14.4963 0.966419
\(226\) 0 0
\(227\) −21.2393 −1.40970 −0.704852 0.709355i \(-0.748986\pi\)
−0.704852 + 0.709355i \(0.748986\pi\)
\(228\) 0 0
\(229\) 17.4360 1.15220 0.576102 0.817378i \(-0.304573\pi\)
0.576102 + 0.817378i \(0.304573\pi\)
\(230\) 0 0
\(231\) −1.67510 −0.110214
\(232\) 0 0
\(233\) −11.2082 −0.734272 −0.367136 0.930167i \(-0.619662\pi\)
−0.367136 + 0.930167i \(0.619662\pi\)
\(234\) 0 0
\(235\) 8.49982 0.554467
\(236\) 0 0
\(237\) −0.581012 −0.0377408
\(238\) 0 0
\(239\) 18.5654 1.20090 0.600449 0.799663i \(-0.294989\pi\)
0.600449 + 0.799663i \(0.294989\pi\)
\(240\) 0 0
\(241\) −13.6556 −0.879637 −0.439818 0.898087i \(-0.644957\pi\)
−0.439818 + 0.898087i \(0.644957\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 4.41546 0.282093
\(246\) 0 0
\(247\) −3.23874 −0.206076
\(248\) 0 0
\(249\) −6.95032 −0.440459
\(250\) 0 0
\(251\) −0.604526 −0.0381574 −0.0190787 0.999818i \(-0.506073\pi\)
−0.0190787 + 0.999818i \(0.506073\pi\)
\(252\) 0 0
\(253\) 1.67510 0.105313
\(254\) 0 0
\(255\) 27.5962 1.72814
\(256\) 0 0
\(257\) 14.1602 0.883291 0.441645 0.897190i \(-0.354395\pi\)
0.441645 + 0.897190i \(0.354395\pi\)
\(258\) 0 0
\(259\) −9.26901 −0.575948
\(260\) 0 0
\(261\) 5.60012 0.346639
\(262\) 0 0
\(263\) −20.5788 −1.26895 −0.634473 0.772945i \(-0.718783\pi\)
−0.634473 + 0.772945i \(0.718783\pi\)
\(264\) 0 0
\(265\) −8.12734 −0.499259
\(266\) 0 0
\(267\) −13.4963 −0.825960
\(268\) 0 0
\(269\) −0.496655 −0.0302816 −0.0151408 0.999885i \(-0.504820\pi\)
−0.0151408 + 0.999885i \(0.504820\pi\)
\(270\) 0 0
\(271\) −24.0547 −1.46122 −0.730609 0.682797i \(-0.760764\pi\)
−0.730609 + 0.682797i \(0.760764\pi\)
\(272\) 0 0
\(273\) 4.66537 0.282361
\(274\) 0 0
\(275\) 24.2828 1.46431
\(276\) 0 0
\(277\) 23.2402 1.39637 0.698183 0.715919i \(-0.253992\pi\)
0.698183 + 0.715919i \(0.253992\pi\)
\(278\) 0 0
\(279\) −4.24991 −0.254435
\(280\) 0 0
\(281\) −15.0697 −0.898985 −0.449492 0.893284i \(-0.648395\pi\)
−0.449492 + 0.893284i \(0.648395\pi\)
\(282\) 0 0
\(283\) −9.22691 −0.548483 −0.274241 0.961661i \(-0.588427\pi\)
−0.274241 + 0.961661i \(0.588427\pi\)
\(284\) 0 0
\(285\) 3.06525 0.181570
\(286\) 0 0
\(287\) 5.15582 0.304338
\(288\) 0 0
\(289\) 22.0614 1.29773
\(290\) 0 0
\(291\) 10.0999 0.592069
\(292\) 0 0
\(293\) −12.3881 −0.723718 −0.361859 0.932233i \(-0.617858\pi\)
−0.361859 + 0.932233i \(0.617858\pi\)
\(294\) 0 0
\(295\) −41.4956 −2.41596
\(296\) 0 0
\(297\) 1.67510 0.0971994
\(298\) 0 0
\(299\) −4.66537 −0.269805
\(300\) 0 0
\(301\) 4.20376 0.242301
\(302\) 0 0
\(303\) −13.7830 −0.791811
\(304\) 0 0
\(305\) 34.0929 1.95215
\(306\) 0 0
\(307\) −0.823281 −0.0469871 −0.0234936 0.999724i \(-0.507479\pi\)
−0.0234936 + 0.999724i \(0.507479\pi\)
\(308\) 0 0
\(309\) −16.5553 −0.941800
\(310\) 0 0
\(311\) 30.6632 1.73875 0.869375 0.494152i \(-0.164521\pi\)
0.869375 + 0.494152i \(0.164521\pi\)
\(312\) 0 0
\(313\) 13.1323 0.742282 0.371141 0.928577i \(-0.378967\pi\)
0.371141 + 0.928577i \(0.378967\pi\)
\(314\) 0 0
\(315\) −4.41546 −0.248783
\(316\) 0 0
\(317\) −15.2269 −0.855229 −0.427614 0.903961i \(-0.640646\pi\)
−0.427614 + 0.903961i \(0.640646\pi\)
\(318\) 0 0
\(319\) 9.38078 0.525223
\(320\) 0 0
\(321\) 1.33463 0.0744918
\(322\) 0 0
\(323\) 4.33874 0.241414
\(324\) 0 0
\(325\) −67.6305 −3.75147
\(326\) 0 0
\(327\) 4.00794 0.221639
\(328\) 0 0
\(329\) −1.92501 −0.106129
\(330\) 0 0
\(331\) −25.5674 −1.40531 −0.702655 0.711530i \(-0.748002\pi\)
−0.702655 + 0.711530i \(0.748002\pi\)
\(332\) 0 0
\(333\) 9.26901 0.507939
\(334\) 0 0
\(335\) 36.3272 1.98477
\(336\) 0 0
\(337\) 16.2890 0.887318 0.443659 0.896196i \(-0.353680\pi\)
0.443659 + 0.896196i \(0.353680\pi\)
\(338\) 0 0
\(339\) −9.06525 −0.492357
\(340\) 0 0
\(341\) −7.11904 −0.385518
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 4.41546 0.237720
\(346\) 0 0
\(347\) −21.6729 −1.16346 −0.581732 0.813380i \(-0.697625\pi\)
−0.581732 + 0.813380i \(0.697625\pi\)
\(348\) 0 0
\(349\) −21.5198 −1.15193 −0.575964 0.817475i \(-0.695373\pi\)
−0.575964 + 0.817475i \(0.695373\pi\)
\(350\) 0 0
\(351\) −4.66537 −0.249019
\(352\) 0 0
\(353\) −20.1852 −1.07435 −0.537174 0.843471i \(-0.680508\pi\)
−0.537174 + 0.843471i \(0.680508\pi\)
\(354\) 0 0
\(355\) 47.0925 2.49941
\(356\) 0 0
\(357\) −6.24991 −0.330780
\(358\) 0 0
\(359\) −23.2492 −1.22705 −0.613524 0.789676i \(-0.710249\pi\)
−0.613524 + 0.789676i \(0.710249\pi\)
\(360\) 0 0
\(361\) −18.5181 −0.974635
\(362\) 0 0
\(363\) −8.19403 −0.430075
\(364\) 0 0
\(365\) −52.5959 −2.75299
\(366\) 0 0
\(367\) −27.7171 −1.44682 −0.723409 0.690419i \(-0.757426\pi\)
−0.723409 + 0.690419i \(0.757426\pi\)
\(368\) 0 0
\(369\) −5.15582 −0.268401
\(370\) 0 0
\(371\) 1.84066 0.0955621
\(372\) 0 0
\(373\) −24.7303 −1.28048 −0.640242 0.768173i \(-0.721166\pi\)
−0.640242 + 0.768173i \(0.721166\pi\)
\(374\) 0 0
\(375\) 41.9305 2.16528
\(376\) 0 0
\(377\) −26.1266 −1.34559
\(378\) 0 0
\(379\) 3.11140 0.159822 0.0799109 0.996802i \(-0.474536\pi\)
0.0799109 + 0.996802i \(0.474536\pi\)
\(380\) 0 0
\(381\) 21.4977 1.10136
\(382\) 0 0
\(383\) 18.9503 0.968316 0.484158 0.874980i \(-0.339126\pi\)
0.484158 + 0.874980i \(0.339126\pi\)
\(384\) 0 0
\(385\) −7.39636 −0.376953
\(386\) 0 0
\(387\) −4.20376 −0.213689
\(388\) 0 0
\(389\) −20.1693 −1.02262 −0.511312 0.859395i \(-0.670840\pi\)
−0.511312 + 0.859395i \(0.670840\pi\)
\(390\) 0 0
\(391\) 6.24991 0.316071
\(392\) 0 0
\(393\) −1.17529 −0.0592854
\(394\) 0 0
\(395\) −2.56543 −0.129081
\(396\) 0 0
\(397\) 8.41634 0.422404 0.211202 0.977442i \(-0.432262\pi\)
0.211202 + 0.977442i \(0.432262\pi\)
\(398\) 0 0
\(399\) −0.694209 −0.0347539
\(400\) 0 0
\(401\) −5.86329 −0.292799 −0.146399 0.989226i \(-0.546768\pi\)
−0.146399 + 0.989226i \(0.546768\pi\)
\(402\) 0 0
\(403\) 19.8274 0.987673
\(404\) 0 0
\(405\) 4.41546 0.219406
\(406\) 0 0
\(407\) 15.5266 0.769623
\(408\) 0 0
\(409\) −26.2686 −1.29890 −0.649451 0.760404i \(-0.725001\pi\)
−0.649451 + 0.760404i \(0.725001\pi\)
\(410\) 0 0
\(411\) 15.8562 0.782130
\(412\) 0 0
\(413\) 9.39779 0.462435
\(414\) 0 0
\(415\) −30.6889 −1.50646
\(416\) 0 0
\(417\) 1.93438 0.0947271
\(418\) 0 0
\(419\) 0.800648 0.0391142 0.0195571 0.999809i \(-0.493774\pi\)
0.0195571 + 0.999809i \(0.493774\pi\)
\(420\) 0 0
\(421\) 23.3845 1.13969 0.569846 0.821751i \(-0.307003\pi\)
0.569846 + 0.821751i \(0.307003\pi\)
\(422\) 0 0
\(423\) 1.92501 0.0935973
\(424\) 0 0
\(425\) 90.6005 4.39477
\(426\) 0 0
\(427\) −7.72125 −0.373658
\(428\) 0 0
\(429\) −7.81498 −0.377311
\(430\) 0 0
\(431\) 33.4853 1.61293 0.806465 0.591281i \(-0.201378\pi\)
0.806465 + 0.591281i \(0.201378\pi\)
\(432\) 0 0
\(433\) 24.1373 1.15996 0.579982 0.814629i \(-0.303059\pi\)
0.579982 + 0.814629i \(0.303059\pi\)
\(434\) 0 0
\(435\) 24.7271 1.18557
\(436\) 0 0
\(437\) 0.694209 0.0332085
\(438\) 0 0
\(439\) 16.4469 0.784968 0.392484 0.919759i \(-0.371616\pi\)
0.392484 + 0.919759i \(0.371616\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.58542 −0.455417 −0.227709 0.973729i \(-0.573123\pi\)
−0.227709 + 0.973729i \(0.573123\pi\)
\(444\) 0 0
\(445\) −59.5923 −2.82495
\(446\) 0 0
\(447\) 2.61301 0.123591
\(448\) 0 0
\(449\) 4.17319 0.196945 0.0984725 0.995140i \(-0.468604\pi\)
0.0984725 + 0.995140i \(0.468604\pi\)
\(450\) 0 0
\(451\) −8.63653 −0.406678
\(452\) 0 0
\(453\) 9.26281 0.435205
\(454\) 0 0
\(455\) 20.5998 0.965731
\(456\) 0 0
\(457\) 8.02668 0.375472 0.187736 0.982220i \(-0.439885\pi\)
0.187736 + 0.982220i \(0.439885\pi\)
\(458\) 0 0
\(459\) 6.24991 0.291721
\(460\) 0 0
\(461\) 11.6192 0.541158 0.270579 0.962698i \(-0.412785\pi\)
0.270579 + 0.962698i \(0.412785\pi\)
\(462\) 0 0
\(463\) 35.8714 1.66708 0.833542 0.552456i \(-0.186309\pi\)
0.833542 + 0.552456i \(0.186309\pi\)
\(464\) 0 0
\(465\) −18.7653 −0.870220
\(466\) 0 0
\(467\) 13.1731 0.609579 0.304790 0.952420i \(-0.401414\pi\)
0.304790 + 0.952420i \(0.401414\pi\)
\(468\) 0 0
\(469\) −8.22728 −0.379900
\(470\) 0 0
\(471\) −7.77504 −0.358255
\(472\) 0 0
\(473\) −7.04174 −0.323779
\(474\) 0 0
\(475\) 10.0635 0.461743
\(476\) 0 0
\(477\) −1.84066 −0.0842778
\(478\) 0 0
\(479\) −5.23080 −0.239002 −0.119501 0.992834i \(-0.538129\pi\)
−0.119501 + 0.992834i \(0.538129\pi\)
\(480\) 0 0
\(481\) −43.2434 −1.97173
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 44.5959 2.02499
\(486\) 0 0
\(487\) −28.9779 −1.31311 −0.656557 0.754277i \(-0.727988\pi\)
−0.656557 + 0.754277i \(0.727988\pi\)
\(488\) 0 0
\(489\) 7.12077 0.322012
\(490\) 0 0
\(491\) 16.6912 0.753262 0.376631 0.926363i \(-0.377083\pi\)
0.376631 + 0.926363i \(0.377083\pi\)
\(492\) 0 0
\(493\) 35.0002 1.57633
\(494\) 0 0
\(495\) 7.39636 0.332441
\(496\) 0 0
\(497\) −10.6654 −0.478407
\(498\) 0 0
\(499\) −9.57213 −0.428507 −0.214254 0.976778i \(-0.568732\pi\)
−0.214254 + 0.976778i \(0.568732\pi\)
\(500\) 0 0
\(501\) 11.9867 0.535528
\(502\) 0 0
\(503\) 20.6348 0.920060 0.460030 0.887903i \(-0.347839\pi\)
0.460030 + 0.887903i \(0.347839\pi\)
\(504\) 0 0
\(505\) −60.8582 −2.70815
\(506\) 0 0
\(507\) 8.76567 0.389297
\(508\) 0 0
\(509\) 6.23665 0.276434 0.138217 0.990402i \(-0.455863\pi\)
0.138217 + 0.990402i \(0.455863\pi\)
\(510\) 0 0
\(511\) 11.9117 0.526945
\(512\) 0 0
\(513\) 0.694209 0.0306501
\(514\) 0 0
\(515\) −73.0994 −3.22115
\(516\) 0 0
\(517\) 3.22460 0.141818
\(518\) 0 0
\(519\) −2.09993 −0.0921769
\(520\) 0 0
\(521\) 12.6615 0.554709 0.277355 0.960768i \(-0.410542\pi\)
0.277355 + 0.960768i \(0.410542\pi\)
\(522\) 0 0
\(523\) −43.3446 −1.89533 −0.947663 0.319272i \(-0.896562\pi\)
−0.947663 + 0.319272i \(0.896562\pi\)
\(524\) 0 0
\(525\) −14.4963 −0.632670
\(526\) 0 0
\(527\) −26.5615 −1.15704
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.39779 −0.407829
\(532\) 0 0
\(533\) 24.0538 1.04189
\(534\) 0 0
\(535\) 5.89301 0.254777
\(536\) 0 0
\(537\) 15.9726 0.689268
\(538\) 0 0
\(539\) 1.67510 0.0721518
\(540\) 0 0
\(541\) 5.87439 0.252560 0.126280 0.991995i \(-0.459696\pi\)
0.126280 + 0.991995i \(0.459696\pi\)
\(542\) 0 0
\(543\) −2.89970 −0.124438
\(544\) 0 0
\(545\) 17.6969 0.758051
\(546\) 0 0
\(547\) −13.8580 −0.592524 −0.296262 0.955107i \(-0.595740\pi\)
−0.296262 + 0.955107i \(0.595740\pi\)
\(548\) 0 0
\(549\) 7.72125 0.329535
\(550\) 0 0
\(551\) 3.88765 0.165620
\(552\) 0 0
\(553\) 0.581012 0.0247071
\(554\) 0 0
\(555\) 40.9270 1.73725
\(556\) 0 0
\(557\) −18.5921 −0.787773 −0.393887 0.919159i \(-0.628870\pi\)
−0.393887 + 0.919159i \(0.628870\pi\)
\(558\) 0 0
\(559\) 19.6121 0.829503
\(560\) 0 0
\(561\) 10.4692 0.442012
\(562\) 0 0
\(563\) −5.09582 −0.214763 −0.107382 0.994218i \(-0.534247\pi\)
−0.107382 + 0.994218i \(0.534247\pi\)
\(564\) 0 0
\(565\) −40.0273 −1.68396
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −29.3028 −1.22844 −0.614218 0.789136i \(-0.710529\pi\)
−0.614218 + 0.789136i \(0.710529\pi\)
\(570\) 0 0
\(571\) 5.10736 0.213736 0.106868 0.994273i \(-0.465918\pi\)
0.106868 + 0.994273i \(0.465918\pi\)
\(572\) 0 0
\(573\) 21.5486 0.900208
\(574\) 0 0
\(575\) 14.4963 0.604537
\(576\) 0 0
\(577\) 24.7379 1.02985 0.514926 0.857235i \(-0.327819\pi\)
0.514926 + 0.857235i \(0.327819\pi\)
\(578\) 0 0
\(579\) 4.58722 0.190638
\(580\) 0 0
\(581\) 6.95032 0.288348
\(582\) 0 0
\(583\) −3.08329 −0.127697
\(584\) 0 0
\(585\) −20.5998 −0.851695
\(586\) 0 0
\(587\) −38.8319 −1.60276 −0.801382 0.598152i \(-0.795902\pi\)
−0.801382 + 0.598152i \(0.795902\pi\)
\(588\) 0 0
\(589\) −2.95032 −0.121566
\(590\) 0 0
\(591\) −11.8656 −0.488086
\(592\) 0 0
\(593\) 36.9162 1.51596 0.757982 0.652275i \(-0.226185\pi\)
0.757982 + 0.652275i \(0.226185\pi\)
\(594\) 0 0
\(595\) −27.5962 −1.13133
\(596\) 0 0
\(597\) 2.52866 0.103491
\(598\) 0 0
\(599\) −40.4224 −1.65161 −0.825807 0.563953i \(-0.809280\pi\)
−0.825807 + 0.563953i \(0.809280\pi\)
\(600\) 0 0
\(601\) 2.98464 0.121746 0.0608730 0.998146i \(-0.480612\pi\)
0.0608730 + 0.998146i \(0.480612\pi\)
\(602\) 0 0
\(603\) 8.22728 0.335041
\(604\) 0 0
\(605\) −36.1804 −1.47094
\(606\) 0 0
\(607\) 29.1893 1.18476 0.592378 0.805660i \(-0.298189\pi\)
0.592378 + 0.805660i \(0.298189\pi\)
\(608\) 0 0
\(609\) −5.60012 −0.226928
\(610\) 0 0
\(611\) −8.98090 −0.363328
\(612\) 0 0
\(613\) −5.88595 −0.237731 −0.118866 0.992910i \(-0.537926\pi\)
−0.118866 + 0.992910i \(0.537926\pi\)
\(614\) 0 0
\(615\) −22.7653 −0.917986
\(616\) 0 0
\(617\) −8.07326 −0.325017 −0.162509 0.986707i \(-0.551959\pi\)
−0.162509 + 0.986707i \(0.551959\pi\)
\(618\) 0 0
\(619\) −10.8589 −0.436457 −0.218228 0.975898i \(-0.570028\pi\)
−0.218228 + 0.975898i \(0.570028\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 13.4963 0.540717
\(624\) 0 0
\(625\) 112.661 4.50644
\(626\) 0 0
\(627\) 1.16287 0.0464406
\(628\) 0 0
\(629\) 57.9305 2.30984
\(630\) 0 0
\(631\) −23.1891 −0.923145 −0.461572 0.887103i \(-0.652715\pi\)
−0.461572 + 0.887103i \(0.652715\pi\)
\(632\) 0 0
\(633\) 21.9797 0.873614
\(634\) 0 0
\(635\) 94.9223 3.76688
\(636\) 0 0
\(637\) −4.66537 −0.184849
\(638\) 0 0
\(639\) 10.6654 0.421915
\(640\) 0 0
\(641\) −46.7138 −1.84509 −0.922543 0.385895i \(-0.873893\pi\)
−0.922543 + 0.385895i \(0.873893\pi\)
\(642\) 0 0
\(643\) −33.7524 −1.33106 −0.665532 0.746369i \(-0.731795\pi\)
−0.665532 + 0.746369i \(0.731795\pi\)
\(644\) 0 0
\(645\) −18.5615 −0.730860
\(646\) 0 0
\(647\) −18.9277 −0.744124 −0.372062 0.928208i \(-0.621349\pi\)
−0.372062 + 0.928208i \(0.621349\pi\)
\(648\) 0 0
\(649\) −15.7423 −0.617938
\(650\) 0 0
\(651\) 4.24991 0.166567
\(652\) 0 0
\(653\) −10.1696 −0.397967 −0.198984 0.980003i \(-0.563764\pi\)
−0.198984 + 0.980003i \(0.563764\pi\)
\(654\) 0 0
\(655\) −5.18943 −0.202768
\(656\) 0 0
\(657\) −11.9117 −0.464722
\(658\) 0 0
\(659\) 4.60754 0.179484 0.0897421 0.995965i \(-0.471396\pi\)
0.0897421 + 0.995965i \(0.471396\pi\)
\(660\) 0 0
\(661\) −31.1205 −1.21045 −0.605223 0.796056i \(-0.706916\pi\)
−0.605223 + 0.796056i \(0.706916\pi\)
\(662\) 0 0
\(663\) −29.1581 −1.13241
\(664\) 0 0
\(665\) −3.06525 −0.118865
\(666\) 0 0
\(667\) 5.60012 0.216837
\(668\) 0 0
\(669\) −0.0302726 −0.00117041
\(670\) 0 0
\(671\) 12.9339 0.499308
\(672\) 0 0
\(673\) −36.9447 −1.42411 −0.712057 0.702122i \(-0.752236\pi\)
−0.712057 + 0.702122i \(0.752236\pi\)
\(674\) 0 0
\(675\) 14.4963 0.557962
\(676\) 0 0
\(677\) 0.331401 0.0127368 0.00636838 0.999980i \(-0.497973\pi\)
0.00636838 + 0.999980i \(0.497973\pi\)
\(678\) 0 0
\(679\) −10.0999 −0.387600
\(680\) 0 0
\(681\) −21.2393 −0.813893
\(682\) 0 0
\(683\) 16.8856 0.646109 0.323055 0.946380i \(-0.395290\pi\)
0.323055 + 0.946380i \(0.395290\pi\)
\(684\) 0 0
\(685\) 70.0126 2.67504
\(686\) 0 0
\(687\) 17.4360 0.665225
\(688\) 0 0
\(689\) 8.58734 0.327152
\(690\) 0 0
\(691\) 11.1963 0.425929 0.212964 0.977060i \(-0.431688\pi\)
0.212964 + 0.977060i \(0.431688\pi\)
\(692\) 0 0
\(693\) −1.67510 −0.0636319
\(694\) 0 0
\(695\) 8.54119 0.323986
\(696\) 0 0
\(697\) −32.2234 −1.22055
\(698\) 0 0
\(699\) −11.2082 −0.423932
\(700\) 0 0
\(701\) 9.50609 0.359040 0.179520 0.983754i \(-0.442546\pi\)
0.179520 + 0.983754i \(0.442546\pi\)
\(702\) 0 0
\(703\) 6.43463 0.242687
\(704\) 0 0
\(705\) 8.49982 0.320122
\(706\) 0 0
\(707\) 13.7830 0.518362
\(708\) 0 0
\(709\) 36.2890 1.36286 0.681431 0.731882i \(-0.261358\pi\)
0.681431 + 0.731882i \(0.261358\pi\)
\(710\) 0 0
\(711\) −0.581012 −0.0217896
\(712\) 0 0
\(713\) −4.24991 −0.159160
\(714\) 0 0
\(715\) −34.5067 −1.29048
\(716\) 0 0
\(717\) 18.5654 0.693339
\(718\) 0 0
\(719\) 33.5772 1.25222 0.626109 0.779736i \(-0.284647\pi\)
0.626109 + 0.779736i \(0.284647\pi\)
\(720\) 0 0
\(721\) 16.5553 0.616553
\(722\) 0 0
\(723\) −13.6556 −0.507858
\(724\) 0 0
\(725\) 81.1809 3.01498
\(726\) 0 0
\(727\) −37.5410 −1.39232 −0.696159 0.717887i \(-0.745109\pi\)
−0.696159 + 0.717887i \(0.745109\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −26.2731 −0.971747
\(732\) 0 0
\(733\) 24.2222 0.894668 0.447334 0.894367i \(-0.352374\pi\)
0.447334 + 0.894367i \(0.352374\pi\)
\(734\) 0 0
\(735\) 4.41546 0.162867
\(736\) 0 0
\(737\) 13.7815 0.507650
\(738\) 0 0
\(739\) 19.4377 0.715028 0.357514 0.933908i \(-0.383624\pi\)
0.357514 + 0.933908i \(0.383624\pi\)
\(740\) 0 0
\(741\) −3.23874 −0.118978
\(742\) 0 0
\(743\) −35.7282 −1.31074 −0.655370 0.755308i \(-0.727487\pi\)
−0.655370 + 0.755308i \(0.727487\pi\)
\(744\) 0 0
\(745\) 11.5377 0.422707
\(746\) 0 0
\(747\) −6.95032 −0.254299
\(748\) 0 0
\(749\) −1.33463 −0.0487664
\(750\) 0 0
\(751\) −14.4795 −0.528363 −0.264182 0.964473i \(-0.585102\pi\)
−0.264182 + 0.964473i \(0.585102\pi\)
\(752\) 0 0
\(753\) −0.604526 −0.0220302
\(754\) 0 0
\(755\) 40.8996 1.48849
\(756\) 0 0
\(757\) 21.8034 0.792460 0.396230 0.918151i \(-0.370318\pi\)
0.396230 + 0.918151i \(0.370318\pi\)
\(758\) 0 0
\(759\) 1.67510 0.0608024
\(760\) 0 0
\(761\) −5.98196 −0.216846 −0.108423 0.994105i \(-0.534580\pi\)
−0.108423 + 0.994105i \(0.534580\pi\)
\(762\) 0 0
\(763\) −4.00794 −0.145097
\(764\) 0 0
\(765\) 27.5962 0.997744
\(766\) 0 0
\(767\) 43.8441 1.58312
\(768\) 0 0
\(769\) −16.0694 −0.579476 −0.289738 0.957106i \(-0.593568\pi\)
−0.289738 + 0.957106i \(0.593568\pi\)
\(770\) 0 0
\(771\) 14.1602 0.509968
\(772\) 0 0
\(773\) 36.5039 1.31295 0.656476 0.754347i \(-0.272046\pi\)
0.656476 + 0.754347i \(0.272046\pi\)
\(774\) 0 0
\(775\) −61.6079 −2.21302
\(776\) 0 0
\(777\) −9.26901 −0.332524
\(778\) 0 0
\(779\) −3.57921 −0.128239
\(780\) 0 0
\(781\) 17.8656 0.639282
\(782\) 0 0
\(783\) 5.60012 0.200132
\(784\) 0 0
\(785\) −34.3304 −1.22530
\(786\) 0 0
\(787\) 44.7402 1.59482 0.797408 0.603440i \(-0.206204\pi\)
0.797408 + 0.603440i \(0.206204\pi\)
\(788\) 0 0
\(789\) −20.5788 −0.732626
\(790\) 0 0
\(791\) 9.06525 0.322323
\(792\) 0 0
\(793\) −36.0225 −1.27920
\(794\) 0 0
\(795\) −8.12734 −0.288247
\(796\) 0 0
\(797\) −11.3919 −0.403521 −0.201761 0.979435i \(-0.564666\pi\)
−0.201761 + 0.979435i \(0.564666\pi\)
\(798\) 0 0
\(799\) 12.0312 0.425632
\(800\) 0 0
\(801\) −13.4963 −0.476868
\(802\) 0 0
\(803\) −19.9534 −0.704141
\(804\) 0 0
\(805\) −4.41546 −0.155625
\(806\) 0 0
\(807\) −0.496655 −0.0174831
\(808\) 0 0
\(809\) −23.4803 −0.825523 −0.412761 0.910839i \(-0.635436\pi\)
−0.412761 + 0.910839i \(0.635436\pi\)
\(810\) 0 0
\(811\) 0.320003 0.0112368 0.00561841 0.999984i \(-0.498212\pi\)
0.00561841 + 0.999984i \(0.498212\pi\)
\(812\) 0 0
\(813\) −24.0547 −0.843634
\(814\) 0 0
\(815\) 31.4415 1.10135
\(816\) 0 0
\(817\) −2.91829 −0.102098
\(818\) 0 0
\(819\) 4.66537 0.163021
\(820\) 0 0
\(821\) 50.4892 1.76208 0.881042 0.473038i \(-0.156843\pi\)
0.881042 + 0.473038i \(0.156843\pi\)
\(822\) 0 0
\(823\) 29.6259 1.03270 0.516348 0.856379i \(-0.327291\pi\)
0.516348 + 0.856379i \(0.327291\pi\)
\(824\) 0 0
\(825\) 24.2828 0.845418
\(826\) 0 0
\(827\) 3.13318 0.108951 0.0544757 0.998515i \(-0.482651\pi\)
0.0544757 + 0.998515i \(0.482651\pi\)
\(828\) 0 0
\(829\) −20.1070 −0.698345 −0.349172 0.937059i \(-0.613537\pi\)
−0.349172 + 0.937059i \(0.613537\pi\)
\(830\) 0 0
\(831\) 23.2402 0.806193
\(832\) 0 0
\(833\) 6.24991 0.216547
\(834\) 0 0
\(835\) 52.9270 1.83161
\(836\) 0 0
\(837\) −4.24991 −0.146898
\(838\) 0 0
\(839\) 6.63075 0.228919 0.114460 0.993428i \(-0.463486\pi\)
0.114460 + 0.993428i \(0.463486\pi\)
\(840\) 0 0
\(841\) 2.36131 0.0814244
\(842\) 0 0
\(843\) −15.0697 −0.519029
\(844\) 0 0
\(845\) 38.7045 1.33147
\(846\) 0 0
\(847\) 8.19403 0.281550
\(848\) 0 0
\(849\) −9.22691 −0.316667
\(850\) 0 0
\(851\) 9.26901 0.317738
\(852\) 0 0
\(853\) −36.1804 −1.23879 −0.619397 0.785078i \(-0.712623\pi\)
−0.619397 + 0.785078i \(0.712623\pi\)
\(854\) 0 0
\(855\) 3.06525 0.104829
\(856\) 0 0
\(857\) 12.6983 0.433764 0.216882 0.976198i \(-0.430411\pi\)
0.216882 + 0.976198i \(0.430411\pi\)
\(858\) 0 0
\(859\) −49.2759 −1.68127 −0.840636 0.541600i \(-0.817819\pi\)
−0.840636 + 0.541600i \(0.817819\pi\)
\(860\) 0 0
\(861\) 5.15582 0.175710
\(862\) 0 0
\(863\) 3.28011 0.111656 0.0558282 0.998440i \(-0.482220\pi\)
0.0558282 + 0.998440i \(0.482220\pi\)
\(864\) 0 0
\(865\) −9.27218 −0.315263
\(866\) 0 0
\(867\) 22.0614 0.749243
\(868\) 0 0
\(869\) −0.973255 −0.0330154
\(870\) 0 0
\(871\) −38.3833 −1.30057
\(872\) 0 0
\(873\) 10.0999 0.341831
\(874\) 0 0
\(875\) −41.9305 −1.41751
\(876\) 0 0
\(877\) 7.64380 0.258113 0.129056 0.991637i \(-0.458805\pi\)
0.129056 + 0.991637i \(0.458805\pi\)
\(878\) 0 0
\(879\) −12.3881 −0.417839
\(880\) 0 0
\(881\) 5.21611 0.175735 0.0878676 0.996132i \(-0.471995\pi\)
0.0878676 + 0.996132i \(0.471995\pi\)
\(882\) 0 0
\(883\) 45.2052 1.52127 0.760637 0.649177i \(-0.224887\pi\)
0.760637 + 0.649177i \(0.224887\pi\)
\(884\) 0 0
\(885\) −41.4956 −1.39486
\(886\) 0 0
\(887\) 8.59600 0.288626 0.144313 0.989532i \(-0.453903\pi\)
0.144313 + 0.989532i \(0.453903\pi\)
\(888\) 0 0
\(889\) −21.4977 −0.721010
\(890\) 0 0
\(891\) 1.67510 0.0561181
\(892\) 0 0
\(893\) 1.33636 0.0447196
\(894\) 0 0
\(895\) 70.5263 2.35744
\(896\) 0 0
\(897\) −4.66537 −0.155772
\(898\) 0 0
\(899\) −23.8000 −0.793774
\(900\) 0 0
\(901\) −11.5039 −0.383251
\(902\) 0 0
\(903\) 4.20376 0.139892
\(904\) 0 0
\(905\) −12.8035 −0.425603
\(906\) 0 0
\(907\) −2.50429 −0.0831537 −0.0415769 0.999135i \(-0.513238\pi\)
−0.0415769 + 0.999135i \(0.513238\pi\)
\(908\) 0 0
\(909\) −13.7830 −0.457152
\(910\) 0 0
\(911\) −29.5776 −0.979951 −0.489975 0.871736i \(-0.662994\pi\)
−0.489975 + 0.871736i \(0.662994\pi\)
\(912\) 0 0
\(913\) −11.6425 −0.385311
\(914\) 0 0
\(915\) 34.0929 1.12708
\(916\) 0 0
\(917\) 1.17529 0.0388114
\(918\) 0 0
\(919\) −23.5583 −0.777117 −0.388558 0.921424i \(-0.627027\pi\)
−0.388558 + 0.921424i \(0.627027\pi\)
\(920\) 0 0
\(921\) −0.823281 −0.0271280
\(922\) 0 0
\(923\) −49.7579 −1.63780
\(924\) 0 0
\(925\) 134.366 4.41794
\(926\) 0 0
\(927\) −16.5553 −0.543749
\(928\) 0 0
\(929\) −37.9337 −1.24456 −0.622281 0.782794i \(-0.713794\pi\)
−0.622281 + 0.782794i \(0.713794\pi\)
\(930\) 0 0
\(931\) 0.694209 0.0227518
\(932\) 0 0
\(933\) 30.6632 1.00387
\(934\) 0 0
\(935\) 46.2265 1.51177
\(936\) 0 0
\(937\) −42.3490 −1.38348 −0.691741 0.722146i \(-0.743156\pi\)
−0.691741 + 0.722146i \(0.743156\pi\)
\(938\) 0 0
\(939\) 13.1323 0.428556
\(940\) 0 0
\(941\) −33.9735 −1.10750 −0.553752 0.832682i \(-0.686804\pi\)
−0.553752 + 0.832682i \(0.686804\pi\)
\(942\) 0 0
\(943\) −5.15582 −0.167896
\(944\) 0 0
\(945\) −4.41546 −0.143635
\(946\) 0 0
\(947\) 27.7471 0.901659 0.450829 0.892610i \(-0.351128\pi\)
0.450829 + 0.892610i \(0.351128\pi\)
\(948\) 0 0
\(949\) 55.5727 1.80397
\(950\) 0 0
\(951\) −15.2269 −0.493766
\(952\) 0 0
\(953\) −6.25582 −0.202646 −0.101323 0.994854i \(-0.532308\pi\)
−0.101323 + 0.994854i \(0.532308\pi\)
\(954\) 0 0
\(955\) 95.1472 3.07889
\(956\) 0 0
\(957\) 9.38078 0.303237
\(958\) 0 0
\(959\) −15.8562 −0.512024
\(960\) 0 0
\(961\) −12.9383 −0.417364
\(962\) 0 0
\(963\) 1.33463 0.0430079
\(964\) 0 0
\(965\) 20.2547 0.652021
\(966\) 0 0
\(967\) −43.1613 −1.38797 −0.693987 0.719988i \(-0.744147\pi\)
−0.693987 + 0.719988i \(0.744147\pi\)
\(968\) 0 0
\(969\) 4.33874 0.139381
\(970\) 0 0
\(971\) −6.69189 −0.214753 −0.107377 0.994218i \(-0.534245\pi\)
−0.107377 + 0.994218i \(0.534245\pi\)
\(972\) 0 0
\(973\) −1.93438 −0.0620135
\(974\) 0 0
\(975\) −67.6305 −2.16591
\(976\) 0 0
\(977\) −47.1871 −1.50965 −0.754825 0.655926i \(-0.772278\pi\)
−0.754825 + 0.655926i \(0.772278\pi\)
\(978\) 0 0
\(979\) −22.6077 −0.722545
\(980\) 0 0
\(981\) 4.00794 0.127964
\(982\) 0 0
\(983\) −18.4505 −0.588480 −0.294240 0.955732i \(-0.595066\pi\)
−0.294240 + 0.955732i \(0.595066\pi\)
\(984\) 0 0
\(985\) −52.3921 −1.66935
\(986\) 0 0
\(987\) −1.92501 −0.0612738
\(988\) 0 0
\(989\) −4.20376 −0.133672
\(990\) 0 0
\(991\) 24.3764 0.774342 0.387171 0.922008i \(-0.373452\pi\)
0.387171 + 0.922008i \(0.373452\pi\)
\(992\) 0 0
\(993\) −25.5674 −0.811356
\(994\) 0 0
\(995\) 11.1652 0.353960
\(996\) 0 0
\(997\) 59.5570 1.88619 0.943094 0.332525i \(-0.107901\pi\)
0.943094 + 0.332525i \(0.107901\pi\)
\(998\) 0 0
\(999\) 9.26901 0.293259
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.1.4 4
4.3 odd 2 483.2.a.i.1.2 4
12.11 even 2 1449.2.a.p.1.3 4
28.27 even 2 3381.2.a.w.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.i.1.2 4 4.3 odd 2
1449.2.a.p.1.3 4 12.11 even 2
3381.2.a.w.1.2 4 28.27 even 2
7728.2.a.cd.1.4 4 1.1 even 1 trivial