Properties

Label 7220.2.a.p.1.4
Level $7220$
Weight $2$
Character 7220.1
Self dual yes
Analytic conductor $57.652$
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] = [7220,2,Mod(1,7220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7220.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7220.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: \(57.6519902594\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.133593.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.52082\) of defining polynomial
Character \(\chi\) \(=\) 7220.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.52082 q^{3} +1.00000 q^{5} +2.72743 q^{7} +3.35453 q^{9} +O(q^{10})\) \(q+2.52082 q^{3} +1.00000 q^{5} +2.72743 q^{7} +3.35453 q^{9} +3.31421 q^{11} -3.24825 q^{13} +2.52082 q^{15} -2.35453 q^{17} +6.87535 q^{21} +2.14793 q^{23} +1.00000 q^{25} +0.893714 q^{27} +3.93403 q^{29} +10.1896 q^{31} +8.35453 q^{33} +2.72743 q^{35} +3.68579 q^{37} -8.18825 q^{39} +0.727427 q^{41} -2.37289 q^{43} +3.35453 q^{45} -11.0233 q^{47} +0.438860 q^{49} -5.93535 q^{51} +8.98296 q^{53} +3.31421 q^{55} +10.9757 q^{59} -8.45485 q^{61} +9.14925 q^{63} -3.24825 q^{65} +9.75071 q^{67} +5.41453 q^{69} -6.91699 q^{71} -2.48050 q^{73} +2.52082 q^{75} +9.03927 q^{77} -11.9990 q^{79} -7.81071 q^{81} +4.68711 q^{83} -2.35453 q^{85} +9.91699 q^{87} -8.54410 q^{89} -8.85936 q^{91} +25.6861 q^{93} -7.22989 q^{97} +11.1176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47} + 10 q^{49} + 12 q^{51} + 5 q^{53} + 2 q^{55} + 11 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} - 3 q^{69} + 14 q^{71} + 4 q^{73} - q^{75} - 22 q^{77} + 13 q^{79} + 24 q^{81} + 5 q^{83} - q^{85} - 2 q^{87} + 5 q^{89} - 46 q^{91} + 28 q^{93} - q^{97} - 24 q^{99}+O(q^{100}) \) 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
\(3\) 2.52082 1.45540 0.727698 0.685898i \(-0.240590\pi\)
0.727698 + 0.685898i \(0.240590\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.72743 1.03087 0.515435 0.856928i \(-0.327630\pi\)
0.515435 + 0.856928i \(0.327630\pi\)
\(8\) 0 0
\(9\) 3.35453 1.11818
\(10\) 0 0
\(11\) 3.31421 0.999273 0.499636 0.866235i \(-0.333467\pi\)
0.499636 + 0.866235i \(0.333467\pi\)
\(12\) 0 0
\(13\) −3.24825 −0.900902 −0.450451 0.892801i \(-0.648737\pi\)
−0.450451 + 0.892801i \(0.648737\pi\)
\(14\) 0 0
\(15\) 2.52082 0.650873
\(16\) 0 0
\(17\) −2.35453 −0.571058 −0.285529 0.958370i \(-0.592169\pi\)
−0.285529 + 0.958370i \(0.592169\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 6.87535 1.50033
\(22\) 0 0
\(23\) 2.14793 0.447873 0.223937 0.974604i \(-0.428109\pi\)
0.223937 + 0.974604i \(0.428109\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.893714 0.171995
\(28\) 0 0
\(29\) 3.93403 0.730532 0.365266 0.930903i \(-0.380978\pi\)
0.365266 + 0.930903i \(0.380978\pi\)
\(30\) 0 0
\(31\) 10.1896 1.83010 0.915050 0.403340i \(-0.132151\pi\)
0.915050 + 0.403340i \(0.132151\pi\)
\(32\) 0 0
\(33\) 8.35453 1.45434
\(34\) 0 0
\(35\) 2.72743 0.461019
\(36\) 0 0
\(37\) 3.68579 0.605940 0.302970 0.953000i \(-0.402022\pi\)
0.302970 + 0.953000i \(0.402022\pi\)
\(38\) 0 0
\(39\) −8.18825 −1.31117
\(40\) 0 0
\(41\) 0.727427 0.113605 0.0568025 0.998385i \(-0.481909\pi\)
0.0568025 + 0.998385i \(0.481909\pi\)
\(42\) 0 0
\(43\) −2.37289 −0.361863 −0.180931 0.983496i \(-0.557911\pi\)
−0.180931 + 0.983496i \(0.557911\pi\)
\(44\) 0 0
\(45\) 3.35453 0.500064
\(46\) 0 0
\(47\) −11.0233 −1.60791 −0.803955 0.594690i \(-0.797275\pi\)
−0.803955 + 0.594690i \(0.797275\pi\)
\(48\) 0 0
\(49\) 0.438860 0.0626942
\(50\) 0 0
\(51\) −5.93535 −0.831116
\(52\) 0 0
\(53\) 8.98296 1.23390 0.616952 0.787001i \(-0.288367\pi\)
0.616952 + 0.787001i \(0.288367\pi\)
\(54\) 0 0
\(55\) 3.31421 0.446888
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.9757 1.42891 0.714456 0.699681i \(-0.246674\pi\)
0.714456 + 0.699681i \(0.246674\pi\)
\(60\) 0 0
\(61\) −8.45485 −1.08253 −0.541267 0.840851i \(-0.682055\pi\)
−0.541267 + 0.840851i \(0.682055\pi\)
\(62\) 0 0
\(63\) 9.14925 1.15270
\(64\) 0 0
\(65\) −3.24825 −0.402895
\(66\) 0 0
\(67\) 9.75071 1.19124 0.595619 0.803267i \(-0.296907\pi\)
0.595619 + 0.803267i \(0.296907\pi\)
\(68\) 0 0
\(69\) 5.41453 0.651833
\(70\) 0 0
\(71\) −6.91699 −0.820896 −0.410448 0.911884i \(-0.634628\pi\)
−0.410448 + 0.911884i \(0.634628\pi\)
\(72\) 0 0
\(73\) −2.48050 −0.290320 −0.145160 0.989408i \(-0.546370\pi\)
−0.145160 + 0.989408i \(0.546370\pi\)
\(74\) 0 0
\(75\) 2.52082 0.291079
\(76\) 0 0
\(77\) 9.03927 1.03012
\(78\) 0 0
\(79\) −11.9990 −1.34999 −0.674994 0.737823i \(-0.735854\pi\)
−0.674994 + 0.737823i \(0.735854\pi\)
\(80\) 0 0
\(81\) −7.81071 −0.867856
\(82\) 0 0
\(83\) 4.68711 0.514477 0.257238 0.966348i \(-0.417187\pi\)
0.257238 + 0.966348i \(0.417187\pi\)
\(84\) 0 0
\(85\) −2.35453 −0.255385
\(86\) 0 0
\(87\) 9.91699 1.06321
\(88\) 0 0
\(89\) −8.54410 −0.905673 −0.452836 0.891594i \(-0.649588\pi\)
−0.452836 + 0.891594i \(0.649588\pi\)
\(90\) 0 0
\(91\) −8.85936 −0.928713
\(92\) 0 0
\(93\) 25.6861 2.66352
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.22989 −0.734084 −0.367042 0.930204i \(-0.619629\pi\)
−0.367042 + 0.930204i \(0.619629\pi\)
\(98\) 0 0
\(99\) 11.1176 1.11736
\(100\) 0 0
\(101\) −5.77635 −0.574768 −0.287384 0.957815i \(-0.592786\pi\)
−0.287384 + 0.957815i \(0.592786\pi\)
\(102\) 0 0
\(103\) 15.0576 1.48367 0.741836 0.670581i \(-0.233955\pi\)
0.741836 + 0.670581i \(0.233955\pi\)
\(104\) 0 0
\(105\) 6.87535 0.670966
\(106\) 0 0
\(107\) 15.1896 1.46843 0.734215 0.678917i \(-0.237550\pi\)
0.734215 + 0.678917i \(0.237550\pi\)
\(108\) 0 0
\(109\) 12.6211 1.20889 0.604443 0.796648i \(-0.293396\pi\)
0.604443 + 0.796648i \(0.293396\pi\)
\(110\) 0 0
\(111\) 9.29121 0.881882
\(112\) 0 0
\(113\) 6.10761 0.574555 0.287278 0.957847i \(-0.407250\pi\)
0.287278 + 0.957847i \(0.407250\pi\)
\(114\) 0 0
\(115\) 2.14793 0.200295
\(116\) 0 0
\(117\) −10.8964 −1.00737
\(118\) 0 0
\(119\) −6.42182 −0.588687
\(120\) 0 0
\(121\) −0.0159950 −0.00145409
\(122\) 0 0
\(123\) 1.83371 0.165340
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.12228 0.809472 0.404736 0.914434i \(-0.367364\pi\)
0.404736 + 0.914434i \(0.367364\pi\)
\(128\) 0 0
\(129\) −5.98164 −0.526654
\(130\) 0 0
\(131\) 22.2656 1.94535 0.972676 0.232169i \(-0.0745821\pi\)
0.972676 + 0.232169i \(0.0745821\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.893714 0.0769187
\(136\) 0 0
\(137\) −7.62711 −0.651628 −0.325814 0.945434i \(-0.605638\pi\)
−0.325814 + 0.945434i \(0.605638\pi\)
\(138\) 0 0
\(139\) 16.9183 1.43499 0.717496 0.696562i \(-0.245288\pi\)
0.717496 + 0.696562i \(0.245288\pi\)
\(140\) 0 0
\(141\) −27.7877 −2.34015
\(142\) 0 0
\(143\) −10.7654 −0.900246
\(144\) 0 0
\(145\) 3.93403 0.326704
\(146\) 0 0
\(147\) 1.10629 0.0912449
\(148\) 0 0
\(149\) −12.3962 −1.01553 −0.507767 0.861494i \(-0.669529\pi\)
−0.507767 + 0.861494i \(0.669529\pi\)
\(150\) 0 0
\(151\) 14.4549 1.17632 0.588160 0.808745i \(-0.299853\pi\)
0.588160 + 0.808745i \(0.299853\pi\)
\(152\) 0 0
\(153\) −7.89836 −0.638544
\(154\) 0 0
\(155\) 10.1896 0.818446
\(156\) 0 0
\(157\) −18.2128 −1.45354 −0.726772 0.686879i \(-0.758980\pi\)
−0.726772 + 0.686879i \(0.758980\pi\)
\(158\) 0 0
\(159\) 22.6444 1.79582
\(160\) 0 0
\(161\) 5.85831 0.461700
\(162\) 0 0
\(163\) 19.2642 1.50889 0.754446 0.656362i \(-0.227906\pi\)
0.754446 + 0.656362i \(0.227906\pi\)
\(164\) 0 0
\(165\) 8.35453 0.650400
\(166\) 0 0
\(167\) −5.68579 −0.439979 −0.219990 0.975502i \(-0.570602\pi\)
−0.219990 + 0.975502i \(0.570602\pi\)
\(168\) 0 0
\(169\) −2.44889 −0.188376
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.15521 0.696058 0.348029 0.937484i \(-0.386851\pi\)
0.348029 + 0.937484i \(0.386851\pi\)
\(174\) 0 0
\(175\) 2.72743 0.206174
\(176\) 0 0
\(177\) 27.6677 2.07963
\(178\) 0 0
\(179\) −12.1810 −0.910448 −0.455224 0.890377i \(-0.650441\pi\)
−0.455224 + 0.890377i \(0.650441\pi\)
\(180\) 0 0
\(181\) −15.4122 −1.14558 −0.572789 0.819703i \(-0.694138\pi\)
−0.572789 + 0.819703i \(0.694138\pi\)
\(182\) 0 0
\(183\) −21.3132 −1.57551
\(184\) 0 0
\(185\) 3.68579 0.270984
\(186\) 0 0
\(187\) −7.80342 −0.570643
\(188\) 0 0
\(189\) 2.43754 0.177305
\(190\) 0 0
\(191\) −16.2482 −1.17568 −0.587841 0.808977i \(-0.700022\pi\)
−0.587841 + 0.808977i \(0.700022\pi\)
\(192\) 0 0
\(193\) −0.851028 −0.0612583 −0.0306292 0.999531i \(-0.509751\pi\)
−0.0306292 + 0.999531i \(0.509751\pi\)
\(194\) 0 0
\(195\) −8.18825 −0.586372
\(196\) 0 0
\(197\) 19.9843 1.42382 0.711910 0.702270i \(-0.247830\pi\)
0.711910 + 0.702270i \(0.247830\pi\)
\(198\) 0 0
\(199\) −6.45722 −0.457740 −0.228870 0.973457i \(-0.573503\pi\)
−0.228870 + 0.973457i \(0.573503\pi\)
\(200\) 0 0
\(201\) 24.5798 1.73372
\(202\) 0 0
\(203\) 10.7298 0.753084
\(204\) 0 0
\(205\) 0.727427 0.0508057
\(206\) 0 0
\(207\) 7.20529 0.500802
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −10.5441 −0.725886 −0.362943 0.931811i \(-0.618228\pi\)
−0.362943 + 0.931811i \(0.618228\pi\)
\(212\) 0 0
\(213\) −17.4365 −1.19473
\(214\) 0 0
\(215\) −2.37289 −0.161830
\(216\) 0 0
\(217\) 27.7913 1.88660
\(218\) 0 0
\(219\) −6.25289 −0.422531
\(220\) 0 0
\(221\) 7.64811 0.514467
\(222\) 0 0
\(223\) −9.72638 −0.651327 −0.325663 0.945486i \(-0.605588\pi\)
−0.325663 + 0.945486i \(0.605588\pi\)
\(224\) 0 0
\(225\) 3.35453 0.223636
\(226\) 0 0
\(227\) −22.5798 −1.49867 −0.749336 0.662190i \(-0.769627\pi\)
−0.749336 + 0.662190i \(0.769627\pi\)
\(228\) 0 0
\(229\) −16.2239 −1.07211 −0.536053 0.844184i \(-0.680085\pi\)
−0.536053 + 0.844184i \(0.680085\pi\)
\(230\) 0 0
\(231\) 22.7864 1.49923
\(232\) 0 0
\(233\) −18.6628 −1.22264 −0.611320 0.791384i \(-0.709361\pi\)
−0.611320 + 0.791384i \(0.709361\pi\)
\(234\) 0 0
\(235\) −11.0233 −0.719079
\(236\) 0 0
\(237\) −30.2472 −1.96477
\(238\) 0 0
\(239\) 0.602780 0.0389906 0.0194953 0.999810i \(-0.493794\pi\)
0.0194953 + 0.999810i \(0.493794\pi\)
\(240\) 0 0
\(241\) −21.8816 −1.40952 −0.704759 0.709447i \(-0.748945\pi\)
−0.704759 + 0.709447i \(0.748945\pi\)
\(242\) 0 0
\(243\) −22.3705 −1.43507
\(244\) 0 0
\(245\) 0.438860 0.0280377
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 11.8154 0.748768
\(250\) 0 0
\(251\) 8.27257 0.522160 0.261080 0.965317i \(-0.415921\pi\)
0.261080 + 0.965317i \(0.415921\pi\)
\(252\) 0 0
\(253\) 7.11868 0.447548
\(254\) 0 0
\(255\) −5.93535 −0.371686
\(256\) 0 0
\(257\) −14.1909 −0.885203 −0.442602 0.896718i \(-0.645944\pi\)
−0.442602 + 0.896718i \(0.645944\pi\)
\(258\) 0 0
\(259\) 10.0527 0.624645
\(260\) 0 0
\(261\) 13.1968 0.816864
\(262\) 0 0
\(263\) −18.5431 −1.14341 −0.571707 0.820458i \(-0.693719\pi\)
−0.571707 + 0.820458i \(0.693719\pi\)
\(264\) 0 0
\(265\) 8.98296 0.551819
\(266\) 0 0
\(267\) −21.5381 −1.31811
\(268\) 0 0
\(269\) −6.66743 −0.406520 −0.203260 0.979125i \(-0.565154\pi\)
−0.203260 + 0.979125i \(0.565154\pi\)
\(270\) 0 0
\(271\) 22.0662 1.34043 0.670214 0.742168i \(-0.266202\pi\)
0.670214 + 0.742168i \(0.266202\pi\)
\(272\) 0 0
\(273\) −22.3328 −1.35165
\(274\) 0 0
\(275\) 3.31421 0.199855
\(276\) 0 0
\(277\) −24.4624 −1.46980 −0.734902 0.678173i \(-0.762772\pi\)
−0.734902 + 0.678173i \(0.762772\pi\)
\(278\) 0 0
\(279\) 34.1812 2.04638
\(280\) 0 0
\(281\) −18.6628 −1.11333 −0.556664 0.830738i \(-0.687919\pi\)
−0.556664 + 0.830738i \(0.687919\pi\)
\(282\) 0 0
\(283\) 4.49622 0.267273 0.133636 0.991030i \(-0.457335\pi\)
0.133636 + 0.991030i \(0.457335\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.98400 0.117112
\(288\) 0 0
\(289\) −11.4562 −0.673893
\(290\) 0 0
\(291\) −18.2252 −1.06838
\(292\) 0 0
\(293\) −1.27021 −0.0742063 −0.0371031 0.999311i \(-0.511813\pi\)
−0.0371031 + 0.999311i \(0.511813\pi\)
\(294\) 0 0
\(295\) 10.9757 0.639028
\(296\) 0 0
\(297\) 2.96196 0.171870
\(298\) 0 0
\(299\) −6.97699 −0.403490
\(300\) 0 0
\(301\) −6.47190 −0.373034
\(302\) 0 0
\(303\) −14.5611 −0.836516
\(304\) 0 0
\(305\) −8.45485 −0.484124
\(306\) 0 0
\(307\) 8.65275 0.493839 0.246919 0.969036i \(-0.420582\pi\)
0.246919 + 0.969036i \(0.420582\pi\)
\(308\) 0 0
\(309\) 37.9576 2.15933
\(310\) 0 0
\(311\) 23.8497 1.35239 0.676196 0.736721i \(-0.263627\pi\)
0.676196 + 0.736721i \(0.263627\pi\)
\(312\) 0 0
\(313\) −0.776351 −0.0438819 −0.0219410 0.999759i \(-0.506985\pi\)
−0.0219410 + 0.999759i \(0.506985\pi\)
\(314\) 0 0
\(315\) 9.14925 0.515502
\(316\) 0 0
\(317\) 13.8423 0.777462 0.388731 0.921351i \(-0.372914\pi\)
0.388731 + 0.921351i \(0.372914\pi\)
\(318\) 0 0
\(319\) 13.0382 0.730001
\(320\) 0 0
\(321\) 38.2902 2.13715
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.24825 −0.180180
\(326\) 0 0
\(327\) 31.8156 1.75941
\(328\) 0 0
\(329\) −30.0652 −1.65755
\(330\) 0 0
\(331\) −28.6993 −1.57746 −0.788728 0.614742i \(-0.789260\pi\)
−0.788728 + 0.614742i \(0.789260\pi\)
\(332\) 0 0
\(333\) 12.3641 0.677548
\(334\) 0 0
\(335\) 9.75071 0.532738
\(336\) 0 0
\(337\) 35.3256 1.92431 0.962153 0.272510i \(-0.0878537\pi\)
0.962153 + 0.272510i \(0.0878537\pi\)
\(338\) 0 0
\(339\) 15.3962 0.836205
\(340\) 0 0
\(341\) 33.7704 1.82877
\(342\) 0 0
\(343\) −17.8950 −0.966241
\(344\) 0 0
\(345\) 5.41453 0.291509
\(346\) 0 0
\(347\) 26.0062 1.39609 0.698044 0.716055i \(-0.254054\pi\)
0.698044 + 0.716055i \(0.254054\pi\)
\(348\) 0 0
\(349\) 2.44757 0.131015 0.0655077 0.997852i \(-0.479133\pi\)
0.0655077 + 0.997852i \(0.479133\pi\)
\(350\) 0 0
\(351\) −2.90300 −0.154951
\(352\) 0 0
\(353\) −4.85103 −0.258194 −0.129097 0.991632i \(-0.541208\pi\)
−0.129097 + 0.991632i \(0.541208\pi\)
\(354\) 0 0
\(355\) −6.91699 −0.367116
\(356\) 0 0
\(357\) −16.1882 −0.856773
\(358\) 0 0
\(359\) −7.34752 −0.387787 −0.193894 0.981023i \(-0.562112\pi\)
−0.193894 + 0.981023i \(0.562112\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −0.0403205 −0.00211628
\(364\) 0 0
\(365\) −2.48050 −0.129835
\(366\) 0 0
\(367\) 13.0357 0.680457 0.340228 0.940343i \(-0.389496\pi\)
0.340228 + 0.940343i \(0.389496\pi\)
\(368\) 0 0
\(369\) 2.44018 0.127031
\(370\) 0 0
\(371\) 24.5004 1.27200
\(372\) 0 0
\(373\) 28.5514 1.47833 0.739167 0.673522i \(-0.235219\pi\)
0.739167 + 0.673522i \(0.235219\pi\)
\(374\) 0 0
\(375\) 2.52082 0.130175
\(376\) 0 0
\(377\) −12.7787 −0.658137
\(378\) 0 0
\(379\) −13.2128 −0.678698 −0.339349 0.940661i \(-0.610207\pi\)
−0.339349 + 0.940661i \(0.610207\pi\)
\(380\) 0 0
\(381\) 22.9956 1.17810
\(382\) 0 0
\(383\) 37.6617 1.92442 0.962212 0.272300i \(-0.0877844\pi\)
0.962212 + 0.272300i \(0.0877844\pi\)
\(384\) 0 0
\(385\) 9.03927 0.460684
\(386\) 0 0
\(387\) −7.95995 −0.404627
\(388\) 0 0
\(389\) −25.6447 −1.30024 −0.650119 0.759833i \(-0.725281\pi\)
−0.650119 + 0.759833i \(0.725281\pi\)
\(390\) 0 0
\(391\) −5.05736 −0.255762
\(392\) 0 0
\(393\) 56.1275 2.83126
\(394\) 0 0
\(395\) −11.9990 −0.603733
\(396\) 0 0
\(397\) −33.2731 −1.66993 −0.834965 0.550303i \(-0.814512\pi\)
−0.834965 + 0.550303i \(0.814512\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.41349 −0.470087 −0.235044 0.971985i \(-0.575523\pi\)
−0.235044 + 0.971985i \(0.575523\pi\)
\(402\) 0 0
\(403\) −33.0982 −1.64874
\(404\) 0 0
\(405\) −7.81071 −0.388117
\(406\) 0 0
\(407\) 12.2155 0.605499
\(408\) 0 0
\(409\) 2.51090 0.124156 0.0620779 0.998071i \(-0.480227\pi\)
0.0620779 + 0.998071i \(0.480227\pi\)
\(410\) 0 0
\(411\) −19.2266 −0.948376
\(412\) 0 0
\(413\) 29.9354 1.47302
\(414\) 0 0
\(415\) 4.68711 0.230081
\(416\) 0 0
\(417\) 42.6480 2.08848
\(418\) 0 0
\(419\) −39.5638 −1.93282 −0.966409 0.257011i \(-0.917262\pi\)
−0.966409 + 0.257011i \(0.917262\pi\)
\(420\) 0 0
\(421\) 14.1578 0.690011 0.345006 0.938601i \(-0.387877\pi\)
0.345006 + 0.938601i \(0.387877\pi\)
\(422\) 0 0
\(423\) −36.9780 −1.79793
\(424\) 0 0
\(425\) −2.35453 −0.114212
\(426\) 0 0
\(427\) −23.0600 −1.11595
\(428\) 0 0
\(429\) −27.1376 −1.31022
\(430\) 0 0
\(431\) 6.52574 0.314334 0.157167 0.987572i \(-0.449764\pi\)
0.157167 + 0.987572i \(0.449764\pi\)
\(432\) 0 0
\(433\) 8.60146 0.413360 0.206680 0.978409i \(-0.433734\pi\)
0.206680 + 0.978409i \(0.433734\pi\)
\(434\) 0 0
\(435\) 9.91699 0.475483
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 11.5588 0.551670 0.275835 0.961205i \(-0.411046\pi\)
0.275835 + 0.961205i \(0.411046\pi\)
\(440\) 0 0
\(441\) 1.47217 0.0701033
\(442\) 0 0
\(443\) −11.3926 −0.541278 −0.270639 0.962681i \(-0.587235\pi\)
−0.270639 + 0.962681i \(0.587235\pi\)
\(444\) 0 0
\(445\) −8.54410 −0.405029
\(446\) 0 0
\(447\) −31.2485 −1.47800
\(448\) 0 0
\(449\) −17.2252 −0.812909 −0.406455 0.913671i \(-0.633235\pi\)
−0.406455 + 0.913671i \(0.633235\pi\)
\(450\) 0 0
\(451\) 2.41085 0.113522
\(452\) 0 0
\(453\) 36.4381 1.71201
\(454\) 0 0
\(455\) −8.85936 −0.415333
\(456\) 0 0
\(457\) −26.1885 −1.22505 −0.612524 0.790452i \(-0.709846\pi\)
−0.612524 + 0.790452i \(0.709846\pi\)
\(458\) 0 0
\(459\) −2.10428 −0.0982194
\(460\) 0 0
\(461\) 40.4332 1.88316 0.941580 0.336789i \(-0.109341\pi\)
0.941580 + 0.336789i \(0.109341\pi\)
\(462\) 0 0
\(463\) 9.48542 0.440825 0.220412 0.975407i \(-0.429260\pi\)
0.220412 + 0.975407i \(0.429260\pi\)
\(464\) 0 0
\(465\) 25.6861 1.19116
\(466\) 0 0
\(467\) −39.2462 −1.81610 −0.908048 0.418867i \(-0.862427\pi\)
−0.908048 + 0.418867i \(0.862427\pi\)
\(468\) 0 0
\(469\) 26.5943 1.22801
\(470\) 0 0
\(471\) −45.9113 −2.11548
\(472\) 0 0
\(473\) −7.86428 −0.361600
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 30.1336 1.37972
\(478\) 0 0
\(479\) 5.45354 0.249178 0.124589 0.992208i \(-0.460239\pi\)
0.124589 + 0.992208i \(0.460239\pi\)
\(480\) 0 0
\(481\) −11.9723 −0.545892
\(482\) 0 0
\(483\) 14.7677 0.671956
\(484\) 0 0
\(485\) −7.22989 −0.328292
\(486\) 0 0
\(487\) −25.0662 −1.13586 −0.567930 0.823077i \(-0.692256\pi\)
−0.567930 + 0.823077i \(0.692256\pi\)
\(488\) 0 0
\(489\) 48.5617 2.19604
\(490\) 0 0
\(491\) 29.6824 1.33955 0.669773 0.742566i \(-0.266391\pi\)
0.669773 + 0.742566i \(0.266391\pi\)
\(492\) 0 0
\(493\) −9.26281 −0.417176
\(494\) 0 0
\(495\) 11.1176 0.499701
\(496\) 0 0
\(497\) −18.8656 −0.846238
\(498\) 0 0
\(499\) −13.7104 −0.613761 −0.306881 0.951748i \(-0.599285\pi\)
−0.306881 + 0.951748i \(0.599285\pi\)
\(500\) 0 0
\(501\) −14.3328 −0.640344
\(502\) 0 0
\(503\) 36.6507 1.63417 0.817086 0.576516i \(-0.195588\pi\)
0.817086 + 0.576516i \(0.195588\pi\)
\(504\) 0 0
\(505\) −5.77635 −0.257044
\(506\) 0 0
\(507\) −6.17321 −0.274162
\(508\) 0 0
\(509\) 12.6504 0.560718 0.280359 0.959895i \(-0.409546\pi\)
0.280359 + 0.959895i \(0.409546\pi\)
\(510\) 0 0
\(511\) −6.76538 −0.299283
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 15.0576 0.663519
\(516\) 0 0
\(517\) −36.5335 −1.60674
\(518\) 0 0
\(519\) 23.0786 1.01304
\(520\) 0 0
\(521\) −16.3339 −0.715601 −0.357800 0.933798i \(-0.616473\pi\)
−0.357800 + 0.933798i \(0.616473\pi\)
\(522\) 0 0
\(523\) 9.13468 0.399432 0.199716 0.979854i \(-0.435998\pi\)
0.199716 + 0.979854i \(0.435998\pi\)
\(524\) 0 0
\(525\) 6.87535 0.300065
\(526\) 0 0
\(527\) −23.9917 −1.04509
\(528\) 0 0
\(529\) −18.3864 −0.799409
\(530\) 0 0
\(531\) 36.8183 1.59778
\(532\) 0 0
\(533\) −2.36286 −0.102347
\(534\) 0 0
\(535\) 15.1896 0.656702
\(536\) 0 0
\(537\) −30.7060 −1.32506
\(538\) 0 0
\(539\) 1.45447 0.0626486
\(540\) 0 0
\(541\) −2.91909 −0.125501 −0.0627507 0.998029i \(-0.519987\pi\)
−0.0627507 + 0.998029i \(0.519987\pi\)
\(542\) 0 0
\(543\) −38.8513 −1.66727
\(544\) 0 0
\(545\) 12.6211 0.540630
\(546\) 0 0
\(547\) 3.47945 0.148771 0.0743853 0.997230i \(-0.476301\pi\)
0.0743853 + 0.997230i \(0.476301\pi\)
\(548\) 0 0
\(549\) −28.3621 −1.21046
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −32.7263 −1.39166
\(554\) 0 0
\(555\) 9.29121 0.394390
\(556\) 0 0
\(557\) 15.9256 0.674789 0.337395 0.941363i \(-0.390454\pi\)
0.337395 + 0.941363i \(0.390454\pi\)
\(558\) 0 0
\(559\) 7.70775 0.326003
\(560\) 0 0
\(561\) −19.6710 −0.830511
\(562\) 0 0
\(563\) −19.6274 −0.827195 −0.413598 0.910460i \(-0.635728\pi\)
−0.413598 + 0.910460i \(0.635728\pi\)
\(564\) 0 0
\(565\) 6.10761 0.256949
\(566\) 0 0
\(567\) −21.3031 −0.894648
\(568\) 0 0
\(569\) 11.7544 0.492770 0.246385 0.969172i \(-0.420757\pi\)
0.246385 + 0.969172i \(0.420757\pi\)
\(570\) 0 0
\(571\) 32.4868 1.35953 0.679766 0.733429i \(-0.262081\pi\)
0.679766 + 0.733429i \(0.262081\pi\)
\(572\) 0 0
\(573\) −40.9589 −1.71108
\(574\) 0 0
\(575\) 2.14793 0.0895747
\(576\) 0 0
\(577\) 25.7287 1.07110 0.535551 0.844503i \(-0.320104\pi\)
0.535551 + 0.844503i \(0.320104\pi\)
\(578\) 0 0
\(579\) −2.14529 −0.0891551
\(580\) 0 0
\(581\) 12.7837 0.530359
\(582\) 0 0
\(583\) 29.7714 1.23301
\(584\) 0 0
\(585\) −10.8964 −0.450509
\(586\) 0 0
\(587\) −41.6558 −1.71932 −0.859659 0.510869i \(-0.829324\pi\)
−0.859659 + 0.510869i \(0.829324\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 50.3768 2.07222
\(592\) 0 0
\(593\) 17.6653 0.725428 0.362714 0.931900i \(-0.381850\pi\)
0.362714 + 0.931900i \(0.381850\pi\)
\(594\) 0 0
\(595\) −6.42182 −0.263269
\(596\) 0 0
\(597\) −16.2775 −0.666193
\(598\) 0 0
\(599\) 26.3851 1.07807 0.539033 0.842285i \(-0.318790\pi\)
0.539033 + 0.842285i \(0.318790\pi\)
\(600\) 0 0
\(601\) −9.46838 −0.386223 −0.193112 0.981177i \(-0.561858\pi\)
−0.193112 + 0.981177i \(0.561858\pi\)
\(602\) 0 0
\(603\) 32.7091 1.33202
\(604\) 0 0
\(605\) −0.0159950 −0.000650289 0
\(606\) 0 0
\(607\) 8.26529 0.335478 0.167739 0.985831i \(-0.446353\pi\)
0.167739 + 0.985831i \(0.446353\pi\)
\(608\) 0 0
\(609\) 27.0479 1.09604
\(610\) 0 0
\(611\) 35.8063 1.44857
\(612\) 0 0
\(613\) 6.80834 0.274986 0.137493 0.990503i \(-0.456095\pi\)
0.137493 + 0.990503i \(0.456095\pi\)
\(614\) 0 0
\(615\) 1.83371 0.0739425
\(616\) 0 0
\(617\) 6.01336 0.242089 0.121044 0.992647i \(-0.461376\pi\)
0.121044 + 0.992647i \(0.461376\pi\)
\(618\) 0 0
\(619\) 13.2299 0.531754 0.265877 0.964007i \(-0.414338\pi\)
0.265877 + 0.964007i \(0.414338\pi\)
\(620\) 0 0
\(621\) 1.91963 0.0770322
\(622\) 0 0
\(623\) −23.3034 −0.933631
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.67831 −0.346027
\(630\) 0 0
\(631\) −23.7864 −0.946921 −0.473460 0.880815i \(-0.656995\pi\)
−0.473460 + 0.880815i \(0.656995\pi\)
\(632\) 0 0
\(633\) −26.5798 −1.05645
\(634\) 0 0
\(635\) 9.12228 0.362007
\(636\) 0 0
\(637\) −1.42552 −0.0564813
\(638\) 0 0
\(639\) −23.2033 −0.917908
\(640\) 0 0
\(641\) −12.2848 −0.485219 −0.242610 0.970124i \(-0.578003\pi\)
−0.242610 + 0.970124i \(0.578003\pi\)
\(642\) 0 0
\(643\) 29.1552 1.14977 0.574885 0.818234i \(-0.305047\pi\)
0.574885 + 0.818234i \(0.305047\pi\)
\(644\) 0 0
\(645\) −5.98164 −0.235527
\(646\) 0 0
\(647\) −36.7499 −1.44479 −0.722394 0.691481i \(-0.756958\pi\)
−0.722394 + 0.691481i \(0.756958\pi\)
\(648\) 0 0
\(649\) 36.3757 1.42787
\(650\) 0 0
\(651\) 70.0569 2.74574
\(652\) 0 0
\(653\) 24.7707 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(654\) 0 0
\(655\) 22.2656 0.869987
\(656\) 0 0
\(657\) −8.32092 −0.324630
\(658\) 0 0
\(659\) −22.2312 −0.866005 −0.433002 0.901393i \(-0.642546\pi\)
−0.433002 + 0.901393i \(0.642546\pi\)
\(660\) 0 0
\(661\) −40.9369 −1.59226 −0.796130 0.605126i \(-0.793123\pi\)
−0.796130 + 0.605126i \(0.793123\pi\)
\(662\) 0 0
\(663\) 19.2795 0.748754
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.45001 0.327186
\(668\) 0 0
\(669\) −24.5185 −0.947938
\(670\) 0 0
\(671\) −28.0212 −1.08175
\(672\) 0 0
\(673\) 33.5992 1.29515 0.647577 0.762000i \(-0.275783\pi\)
0.647577 + 0.762000i \(0.275783\pi\)
\(674\) 0 0
\(675\) 0.893714 0.0343991
\(676\) 0 0
\(677\) −29.5883 −1.13717 −0.568585 0.822624i \(-0.692509\pi\)
−0.568585 + 0.822624i \(0.692509\pi\)
\(678\) 0 0
\(679\) −19.7190 −0.756745
\(680\) 0 0
\(681\) −56.9195 −2.18116
\(682\) 0 0
\(683\) −41.0567 −1.57099 −0.785495 0.618868i \(-0.787592\pi\)
−0.785495 + 0.618868i \(0.787592\pi\)
\(684\) 0 0
\(685\) −7.62711 −0.291417
\(686\) 0 0
\(687\) −40.8976 −1.56034
\(688\) 0 0
\(689\) −29.1789 −1.11163
\(690\) 0 0
\(691\) 1.22497 0.0466000 0.0233000 0.999729i \(-0.492583\pi\)
0.0233000 + 0.999729i \(0.492583\pi\)
\(692\) 0 0
\(693\) 30.3225 1.15186
\(694\) 0 0
\(695\) 16.9183 0.641748
\(696\) 0 0
\(697\) −1.71275 −0.0648751
\(698\) 0 0
\(699\) −47.0455 −1.77942
\(700\) 0 0
\(701\) 44.0201 1.66262 0.831309 0.555811i \(-0.187592\pi\)
0.831309 + 0.555811i \(0.187592\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −27.7877 −1.04655
\(706\) 0 0
\(707\) −15.7546 −0.592512
\(708\) 0 0
\(709\) 16.3251 0.613102 0.306551 0.951854i \(-0.400825\pi\)
0.306551 + 0.951854i \(0.400825\pi\)
\(710\) 0 0
\(711\) −40.2509 −1.50953
\(712\) 0 0
\(713\) 21.8864 0.819653
\(714\) 0 0
\(715\) −10.7654 −0.402602
\(716\) 0 0
\(717\) 1.51950 0.0567468
\(718\) 0 0
\(719\) 10.1715 0.379332 0.189666 0.981849i \(-0.439259\pi\)
0.189666 + 0.981849i \(0.439259\pi\)
\(720\) 0 0
\(721\) 41.0686 1.52947
\(722\) 0 0
\(723\) −55.1595 −2.05141
\(724\) 0 0
\(725\) 3.93403 0.146106
\(726\) 0 0
\(727\) 8.04296 0.298297 0.149148 0.988815i \(-0.452347\pi\)
0.149148 + 0.988815i \(0.452347\pi\)
\(728\) 0 0
\(729\) −32.9600 −1.22074
\(730\) 0 0
\(731\) 5.58706 0.206645
\(732\) 0 0
\(733\) 41.2325 1.52296 0.761479 0.648190i \(-0.224474\pi\)
0.761479 + 0.648190i \(0.224474\pi\)
\(734\) 0 0
\(735\) 1.10629 0.0408060
\(736\) 0 0
\(737\) 32.3159 1.19037
\(738\) 0 0
\(739\) 33.0506 1.21579 0.607893 0.794019i \(-0.292015\pi\)
0.607893 + 0.794019i \(0.292015\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.7867 −1.27620 −0.638099 0.769954i \(-0.720279\pi\)
−0.638099 + 0.769954i \(0.720279\pi\)
\(744\) 0 0
\(745\) −12.3962 −0.454161
\(746\) 0 0
\(747\) 15.7231 0.575276
\(748\) 0 0
\(749\) 41.4284 1.51376
\(750\) 0 0
\(751\) 43.4701 1.58625 0.793123 0.609062i \(-0.208454\pi\)
0.793123 + 0.609062i \(0.208454\pi\)
\(752\) 0 0
\(753\) 20.8537 0.759950
\(754\) 0 0
\(755\) 14.4549 0.526066
\(756\) 0 0
\(757\) −6.16989 −0.224248 −0.112124 0.993694i \(-0.535765\pi\)
−0.112124 + 0.993694i \(0.535765\pi\)
\(758\) 0 0
\(759\) 17.9449 0.651359
\(760\) 0 0
\(761\) −9.81318 −0.355728 −0.177864 0.984055i \(-0.556919\pi\)
−0.177864 + 0.984055i \(0.556919\pi\)
\(762\) 0 0
\(763\) 34.4232 1.24621
\(764\) 0 0
\(765\) −7.89836 −0.285566
\(766\) 0 0
\(767\) −35.6517 −1.28731
\(768\) 0 0
\(769\) −41.4903 −1.49618 −0.748090 0.663597i \(-0.769029\pi\)
−0.748090 + 0.663597i \(0.769029\pi\)
\(770\) 0 0
\(771\) −35.7727 −1.28832
\(772\) 0 0
\(773\) −41.7169 −1.50045 −0.750226 0.661181i \(-0.770055\pi\)
−0.750226 + 0.661181i \(0.770055\pi\)
\(774\) 0 0
\(775\) 10.1896 0.366020
\(776\) 0 0
\(777\) 25.3411 0.909107
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −22.9244 −0.820299
\(782\) 0 0
\(783\) 3.51590 0.125648
\(784\) 0 0
\(785\) −18.2128 −0.650044
\(786\) 0 0
\(787\) 13.0342 0.464621 0.232310 0.972642i \(-0.425371\pi\)
0.232310 + 0.972642i \(0.425371\pi\)
\(788\) 0 0
\(789\) −46.7437 −1.66412
\(790\) 0 0
\(791\) 16.6580 0.592292
\(792\) 0 0
\(793\) 27.4635 0.975256
\(794\) 0 0
\(795\) 22.6444 0.803115
\(796\) 0 0
\(797\) −33.1911 −1.17569 −0.587844 0.808974i \(-0.700023\pi\)
−0.587844 + 0.808974i \(0.700023\pi\)
\(798\) 0 0
\(799\) 25.9547 0.918210
\(800\) 0 0
\(801\) −28.6615 −1.01270
\(802\) 0 0
\(803\) −8.22090 −0.290109
\(804\) 0 0
\(805\) 5.85831 0.206478
\(806\) 0 0
\(807\) −16.8074 −0.591648
\(808\) 0 0
\(809\) −29.5271 −1.03812 −0.519058 0.854739i \(-0.673717\pi\)
−0.519058 + 0.854739i \(0.673717\pi\)
\(810\) 0 0
\(811\) −28.8194 −1.01199 −0.505993 0.862537i \(-0.668874\pi\)
−0.505993 + 0.862537i \(0.668874\pi\)
\(812\) 0 0
\(813\) 55.6250 1.95085
\(814\) 0 0
\(815\) 19.2642 0.674797
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −29.7190 −1.03847
\(820\) 0 0
\(821\) −18.8056 −0.656320 −0.328160 0.944622i \(-0.606428\pi\)
−0.328160 + 0.944622i \(0.606428\pi\)
\(822\) 0 0
\(823\) 7.56958 0.263859 0.131929 0.991259i \(-0.457883\pi\)
0.131929 + 0.991259i \(0.457883\pi\)
\(824\) 0 0
\(825\) 8.35453 0.290868
\(826\) 0 0
\(827\) −34.9367 −1.21487 −0.607434 0.794370i \(-0.707801\pi\)
−0.607434 + 0.794370i \(0.707801\pi\)
\(828\) 0 0
\(829\) 45.8376 1.59201 0.796003 0.605293i \(-0.206944\pi\)
0.796003 + 0.605293i \(0.206944\pi\)
\(830\) 0 0
\(831\) −61.6653 −2.13915
\(832\) 0 0
\(833\) −1.03331 −0.0358020
\(834\) 0 0
\(835\) −5.68579 −0.196765
\(836\) 0 0
\(837\) 9.10656 0.314769
\(838\) 0 0
\(839\) −14.0100 −0.483680 −0.241840 0.970316i \(-0.577751\pi\)
−0.241840 + 0.970316i \(0.577751\pi\)
\(840\) 0 0
\(841\) −13.5234 −0.466323
\(842\) 0 0
\(843\) −47.0455 −1.62033
\(844\) 0 0
\(845\) −2.44889 −0.0842444
\(846\) 0 0
\(847\) −0.0436252 −0.00149898
\(848\) 0 0
\(849\) 11.3342 0.388988
\(850\) 0 0
\(851\) 7.91680 0.271384
\(852\) 0 0
\(853\) −37.5515 −1.28574 −0.642869 0.765976i \(-0.722256\pi\)
−0.642869 + 0.765976i \(0.722256\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.06756 −0.138945 −0.0694726 0.997584i \(-0.522132\pi\)
−0.0694726 + 0.997584i \(0.522132\pi\)
\(858\) 0 0
\(859\) 12.6357 0.431125 0.215562 0.976490i \(-0.430842\pi\)
0.215562 + 0.976490i \(0.430842\pi\)
\(860\) 0 0
\(861\) 5.00132 0.170445
\(862\) 0 0
\(863\) 14.5147 0.494085 0.247042 0.969005i \(-0.420541\pi\)
0.247042 + 0.969005i \(0.420541\pi\)
\(864\) 0 0
\(865\) 9.15521 0.311286
\(866\) 0 0
\(867\) −28.8790 −0.980781
\(868\) 0 0
\(869\) −39.7671 −1.34901
\(870\) 0 0
\(871\) −31.6727 −1.07319
\(872\) 0 0
\(873\) −24.2529 −0.820836
\(874\) 0 0
\(875\) 2.72743 0.0922039
\(876\) 0 0
\(877\) −1.93508 −0.0653431 −0.0326715 0.999466i \(-0.510402\pi\)
−0.0326715 + 0.999466i \(0.510402\pi\)
\(878\) 0 0
\(879\) −3.20196 −0.108000
\(880\) 0 0
\(881\) 15.9037 0.535811 0.267905 0.963445i \(-0.413669\pi\)
0.267905 + 0.963445i \(0.413669\pi\)
\(882\) 0 0
\(883\) −35.0431 −1.17930 −0.589648 0.807660i \(-0.700733\pi\)
−0.589648 + 0.807660i \(0.700733\pi\)
\(884\) 0 0
\(885\) 27.6677 0.930040
\(886\) 0 0
\(887\) 16.1761 0.543141 0.271571 0.962419i \(-0.412457\pi\)
0.271571 + 0.962419i \(0.412457\pi\)
\(888\) 0 0
\(889\) 24.8804 0.834460
\(890\) 0 0
\(891\) −25.8863 −0.867225
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −12.1810 −0.407165
\(896\) 0 0
\(897\) −17.5877 −0.587238
\(898\) 0 0
\(899\) 40.0861 1.33695
\(900\) 0 0
\(901\) −21.1507 −0.704631
\(902\) 0 0
\(903\) −16.3145 −0.542912
\(904\) 0 0
\(905\) −15.4122 −0.512318
\(906\) 0 0
\(907\) −22.9951 −0.763539 −0.381770 0.924258i \(-0.624685\pi\)
−0.381770 + 0.924258i \(0.624685\pi\)
\(908\) 0 0
\(909\) −19.3770 −0.642693
\(910\) 0 0
\(911\) 23.0833 0.764783 0.382392 0.924000i \(-0.375101\pi\)
0.382392 + 0.924000i \(0.375101\pi\)
\(912\) 0 0
\(913\) 15.5341 0.514103
\(914\) 0 0
\(915\) −21.3132 −0.704592
\(916\) 0 0
\(917\) 60.7277 2.00541
\(918\) 0 0
\(919\) 12.5454 0.413835 0.206918 0.978358i \(-0.433657\pi\)
0.206918 + 0.978358i \(0.433657\pi\)
\(920\) 0 0
\(921\) 21.8120 0.718731
\(922\) 0 0
\(923\) 22.4681 0.739547
\(924\) 0 0
\(925\) 3.68579 0.121188
\(926\) 0 0
\(927\) 50.5113 1.65901
\(928\) 0 0
\(929\) −53.2988 −1.74868 −0.874338 0.485318i \(-0.838704\pi\)
−0.874338 + 0.485318i \(0.838704\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 60.1208 1.96827
\(934\) 0 0
\(935\) −7.80342 −0.255199
\(936\) 0 0
\(937\) −35.1205 −1.14734 −0.573668 0.819088i \(-0.694480\pi\)
−0.573668 + 0.819088i \(0.694480\pi\)
\(938\) 0 0
\(939\) −1.95704 −0.0638656
\(940\) 0 0
\(941\) −13.1580 −0.428937 −0.214469 0.976731i \(-0.568802\pi\)
−0.214469 + 0.976731i \(0.568802\pi\)
\(942\) 0 0
\(943\) 1.56246 0.0508807
\(944\) 0 0
\(945\) 2.43754 0.0792932
\(946\) 0 0
\(947\) −10.4502 −0.339586 −0.169793 0.985480i \(-0.554310\pi\)
−0.169793 + 0.985480i \(0.554310\pi\)
\(948\) 0 0
\(949\) 8.05728 0.261550
\(950\) 0 0
\(951\) 34.8940 1.13152
\(952\) 0 0
\(953\) −4.34233 −0.140662 −0.0703310 0.997524i \(-0.522406\pi\)
−0.0703310 + 0.997524i \(0.522406\pi\)
\(954\) 0 0
\(955\) −16.2482 −0.525781
\(956\) 0 0
\(957\) 32.8670 1.06244
\(958\) 0 0
\(959\) −20.8024 −0.671744
\(960\) 0 0
\(961\) 72.8272 2.34927
\(962\) 0 0
\(963\) 50.9539 1.64197
\(964\) 0 0
\(965\) −0.851028 −0.0273955
\(966\) 0 0
\(967\) −15.6820 −0.504299 −0.252149 0.967688i \(-0.581137\pi\)
−0.252149 + 0.967688i \(0.581137\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2.67735 −0.0859202 −0.0429601 0.999077i \(-0.513679\pi\)
−0.0429601 + 0.999077i \(0.513679\pi\)
\(972\) 0 0
\(973\) 46.1435 1.47929
\(974\) 0 0
\(975\) −8.18825 −0.262234
\(976\) 0 0
\(977\) −2.87156 −0.0918693 −0.0459347 0.998944i \(-0.514627\pi\)
−0.0459347 + 0.998944i \(0.514627\pi\)
\(978\) 0 0
\(979\) −28.3170 −0.905014
\(980\) 0 0
\(981\) 42.3380 1.35175
\(982\) 0 0
\(983\) 33.9089 1.08153 0.540764 0.841175i \(-0.318135\pi\)
0.540764 + 0.841175i \(0.318135\pi\)
\(984\) 0 0
\(985\) 19.9843 0.636752
\(986\) 0 0
\(987\) −75.7889 −2.41239
\(988\) 0 0
\(989\) −5.09680 −0.162069
\(990\) 0 0
\(991\) 40.1298 1.27477 0.637383 0.770547i \(-0.280017\pi\)
0.637383 + 0.770547i \(0.280017\pi\)
\(992\) 0 0
\(993\) −72.3458 −2.29582
\(994\) 0 0
\(995\) −6.45722 −0.204708
\(996\) 0 0
\(997\) 3.50738 0.111080 0.0555399 0.998456i \(-0.482312\pi\)
0.0555399 + 0.998456i \(0.482312\pi\)
\(998\) 0 0
\(999\) 3.29404 0.104219
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 7220.2.a.p.1.4 4
19.8 odd 6 380.2.i.c.121.4 8
19.12 odd 6 380.2.i.c.201.4 yes 8
19.18 odd 2 7220.2.a.r.1.1 4
57.8 even 6 3420.2.t.w.1261.3 8
57.50 even 6 3420.2.t.w.3241.3 8
76.27 even 6 1520.2.q.m.881.1 8
76.31 even 6 1520.2.q.m.961.1 8
95.8 even 12 1900.2.s.d.349.2 16
95.12 even 12 1900.2.s.d.49.2 16
95.27 even 12 1900.2.s.d.349.7 16
95.69 odd 6 1900.2.i.d.201.1 8
95.84 odd 6 1900.2.i.d.501.1 8
95.88 even 12 1900.2.s.d.49.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.i.c.121.4 8 19.8 odd 6
380.2.i.c.201.4 yes 8 19.12 odd 6
1520.2.q.m.881.1 8 76.27 even 6
1520.2.q.m.961.1 8 76.31 even 6
1900.2.i.d.201.1 8 95.69 odd 6
1900.2.i.d.501.1 8 95.84 odd 6
1900.2.s.d.49.2 16 95.12 even 12
1900.2.s.d.49.7 16 95.88 even 12
1900.2.s.d.349.2 16 95.8 even 12
1900.2.s.d.349.7 16 95.27 even 12
3420.2.t.w.1261.3 8 57.8 even 6
3420.2.t.w.3241.3 8 57.50 even 6
7220.2.a.p.1.4 4 1.1 even 1 trivial
7220.2.a.r.1.1 4 19.18 odd 2