Properties

Label 6525.2.a.bw.1.7
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
Dimension $7$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1305)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.57501\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.57501 q^{2} +4.63069 q^{4} +2.24826 q^{7} +6.77405 q^{8} +O(q^{10})\) \(q+2.57501 q^{2} +4.63069 q^{4} +2.24826 q^{7} +6.77405 q^{8} -3.75744 q^{11} +1.55119 q^{13} +5.78930 q^{14} +8.18188 q^{16} +3.55119 q^{17} -1.18909 q^{19} -9.67546 q^{22} +3.67991 q^{23} +3.99434 q^{26} +10.4110 q^{28} +1.00000 q^{29} +5.18909 q^{31} +7.52034 q^{32} +9.14436 q^{34} -0.969518 q^{37} -3.06193 q^{38} +11.3809 q^{41} +0.884393 q^{43} -17.3995 q^{44} +9.47581 q^{46} -11.2251 q^{47} -1.94533 q^{49} +7.18308 q^{52} +10.6567 q^{53} +15.2298 q^{56} +2.57501 q^{58} +8.49652 q^{59} -10.8743 q^{61} +13.3620 q^{62} +3.00121 q^{64} +4.05179 q^{67} +16.4445 q^{68} +6.37818 q^{71} +3.26017 q^{73} -2.49652 q^{74} -5.50631 q^{76} -8.44770 q^{77} +6.32579 q^{79} +29.3060 q^{82} +9.03930 q^{83} +2.27732 q^{86} -25.4531 q^{88} +1.46717 q^{89} +3.48748 q^{91} +17.0405 q^{92} -28.9047 q^{94} -12.3993 q^{97} -5.00924 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 13 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 13 q^{4} - 10 q^{7} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} - 9 q^{22} + 11 q^{23} - 3 q^{26} - 25 q^{28} + 7 q^{29} + 18 q^{31} + q^{32} - q^{34} - 13 q^{37} + 12 q^{38} + 13 q^{41} - 9 q^{43} + 37 q^{44} - 8 q^{46} + 2 q^{47} + 21 q^{49} + q^{52} + 5 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 14 q^{61} - 8 q^{62} + 8 q^{64} - 14 q^{67} + 27 q^{68} + 8 q^{71} - 3 q^{73} + 34 q^{74} + 4 q^{76} - 28 q^{77} + 4 q^{79} + 20 q^{82} + 17 q^{83} - 4 q^{86} - 26 q^{88} + 20 q^{89} + 12 q^{91} + 60 q^{92} - 21 q^{94} - 13 q^{97} - 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.57501 1.82081 0.910404 0.413720i \(-0.135771\pi\)
0.910404 + 0.413720i \(0.135771\pi\)
\(3\) 0 0
\(4\) 4.63069 2.31534
\(5\) 0 0
\(6\) 0 0
\(7\) 2.24826 0.849762 0.424881 0.905249i \(-0.360316\pi\)
0.424881 + 0.905249i \(0.360316\pi\)
\(8\) 6.77405 2.39499
\(9\) 0 0
\(10\) 0 0
\(11\) −3.75744 −1.13291 −0.566456 0.824092i \(-0.691686\pi\)
−0.566456 + 0.824092i \(0.691686\pi\)
\(12\) 0 0
\(13\) 1.55119 0.430223 0.215112 0.976589i \(-0.430988\pi\)
0.215112 + 0.976589i \(0.430988\pi\)
\(14\) 5.78930 1.54725
\(15\) 0 0
\(16\) 8.18188 2.04547
\(17\) 3.55119 0.861291 0.430645 0.902521i \(-0.358286\pi\)
0.430645 + 0.902521i \(0.358286\pi\)
\(18\) 0 0
\(19\) −1.18909 −0.272796 −0.136398 0.990654i \(-0.543553\pi\)
−0.136398 + 0.990654i \(0.543553\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −9.67546 −2.06281
\(23\) 3.67991 0.767314 0.383657 0.923476i \(-0.374664\pi\)
0.383657 + 0.923476i \(0.374664\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.99434 0.783354
\(27\) 0 0
\(28\) 10.4110 1.96749
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.18909 0.931988 0.465994 0.884788i \(-0.345697\pi\)
0.465994 + 0.884788i \(0.345697\pi\)
\(32\) 7.52034 1.32942
\(33\) 0 0
\(34\) 9.14436 1.56825
\(35\) 0 0
\(36\) 0 0
\(37\) −0.969518 −0.159388 −0.0796939 0.996819i \(-0.525394\pi\)
−0.0796939 + 0.996819i \(0.525394\pi\)
\(38\) −3.06193 −0.496710
\(39\) 0 0
\(40\) 0 0
\(41\) 11.3809 1.77740 0.888700 0.458489i \(-0.151609\pi\)
0.888700 + 0.458489i \(0.151609\pi\)
\(42\) 0 0
\(43\) 0.884393 0.134869 0.0674343 0.997724i \(-0.478519\pi\)
0.0674343 + 0.997724i \(0.478519\pi\)
\(44\) −17.3995 −2.62308
\(45\) 0 0
\(46\) 9.47581 1.39713
\(47\) −11.2251 −1.63734 −0.818672 0.574262i \(-0.805289\pi\)
−0.818672 + 0.574262i \(0.805289\pi\)
\(48\) 0 0
\(49\) −1.94533 −0.277904
\(50\) 0 0
\(51\) 0 0
\(52\) 7.18308 0.996115
\(53\) 10.6567 1.46381 0.731906 0.681406i \(-0.238631\pi\)
0.731906 + 0.681406i \(0.238631\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 15.2298 2.03517
\(57\) 0 0
\(58\) 2.57501 0.338116
\(59\) 8.49652 1.10615 0.553076 0.833131i \(-0.313454\pi\)
0.553076 + 0.833131i \(0.313454\pi\)
\(60\) 0 0
\(61\) −10.8743 −1.39231 −0.696154 0.717892i \(-0.745107\pi\)
−0.696154 + 0.717892i \(0.745107\pi\)
\(62\) 13.3620 1.69697
\(63\) 0 0
\(64\) 3.00121 0.375151
\(65\) 0 0
\(66\) 0 0
\(67\) 4.05179 0.495005 0.247502 0.968887i \(-0.420390\pi\)
0.247502 + 0.968887i \(0.420390\pi\)
\(68\) 16.4445 1.99418
\(69\) 0 0
\(70\) 0 0
\(71\) 6.37818 0.756951 0.378476 0.925611i \(-0.376448\pi\)
0.378476 + 0.925611i \(0.376448\pi\)
\(72\) 0 0
\(73\) 3.26017 0.381573 0.190787 0.981632i \(-0.438896\pi\)
0.190787 + 0.981632i \(0.438896\pi\)
\(74\) −2.49652 −0.290215
\(75\) 0 0
\(76\) −5.50631 −0.631617
\(77\) −8.44770 −0.962705
\(78\) 0 0
\(79\) 6.32579 0.711707 0.355854 0.934542i \(-0.384190\pi\)
0.355854 + 0.934542i \(0.384190\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 29.3060 3.23630
\(83\) 9.03930 0.992193 0.496096 0.868268i \(-0.334766\pi\)
0.496096 + 0.868268i \(0.334766\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.27732 0.245570
\(87\) 0 0
\(88\) −25.4531 −2.71331
\(89\) 1.46717 0.155520 0.0777599 0.996972i \(-0.475223\pi\)
0.0777599 + 0.996972i \(0.475223\pi\)
\(90\) 0 0
\(91\) 3.48748 0.365588
\(92\) 17.0405 1.77660
\(93\) 0 0
\(94\) −28.9047 −2.98129
\(95\) 0 0
\(96\) 0 0
\(97\) −12.3993 −1.25896 −0.629478 0.777018i \(-0.716731\pi\)
−0.629478 + 0.777018i \(0.716731\pi\)
\(98\) −5.00924 −0.506010
\(99\) 0 0
\(100\) 0 0
\(101\) 3.45196 0.343483 0.171742 0.985142i \(-0.445061\pi\)
0.171742 + 0.985142i \(0.445061\pi\)
\(102\) 0 0
\(103\) −18.1975 −1.79306 −0.896528 0.442988i \(-0.853918\pi\)
−0.896528 + 0.442988i \(0.853918\pi\)
\(104\) 10.5078 1.03038
\(105\) 0 0
\(106\) 27.4412 2.66532
\(107\) −8.07919 −0.781044 −0.390522 0.920594i \(-0.627706\pi\)
−0.390522 + 0.920594i \(0.627706\pi\)
\(108\) 0 0
\(109\) −14.0114 −1.34205 −0.671026 0.741434i \(-0.734146\pi\)
−0.671026 + 0.741434i \(0.734146\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 18.3950 1.73816
\(113\) 0.678491 0.0638270 0.0319135 0.999491i \(-0.489840\pi\)
0.0319135 + 0.999491i \(0.489840\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.63069 0.429948
\(117\) 0 0
\(118\) 21.8786 2.01409
\(119\) 7.98400 0.731892
\(120\) 0 0
\(121\) 3.11836 0.283488
\(122\) −28.0014 −2.53513
\(123\) 0 0
\(124\) 24.0291 2.15787
\(125\) 0 0
\(126\) 0 0
\(127\) 8.39928 0.745315 0.372658 0.927969i \(-0.378447\pi\)
0.372658 + 0.927969i \(0.378447\pi\)
\(128\) −7.31254 −0.646343
\(129\) 0 0
\(130\) 0 0
\(131\) 7.34298 0.641559 0.320779 0.947154i \(-0.396055\pi\)
0.320779 + 0.947154i \(0.396055\pi\)
\(132\) 0 0
\(133\) −2.67339 −0.231812
\(134\) 10.4334 0.901308
\(135\) 0 0
\(136\) 24.0559 2.06278
\(137\) −1.80353 −0.154086 −0.0770429 0.997028i \(-0.524548\pi\)
−0.0770429 + 0.997028i \(0.524548\pi\)
\(138\) 0 0
\(139\) 0.707451 0.0600052 0.0300026 0.999550i \(-0.490448\pi\)
0.0300026 + 0.999550i \(0.490448\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.4239 1.37826
\(143\) −5.82851 −0.487405
\(144\) 0 0
\(145\) 0 0
\(146\) 8.39496 0.694772
\(147\) 0 0
\(148\) −4.48953 −0.369037
\(149\) 16.9930 1.39212 0.696062 0.717982i \(-0.254934\pi\)
0.696062 + 0.717982i \(0.254934\pi\)
\(150\) 0 0
\(151\) 7.86646 0.640163 0.320082 0.947390i \(-0.396290\pi\)
0.320082 + 0.947390i \(0.396290\pi\)
\(152\) −8.05497 −0.653344
\(153\) 0 0
\(154\) −21.7529 −1.75290
\(155\) 0 0
\(156\) 0 0
\(157\) −9.85591 −0.786588 −0.393294 0.919413i \(-0.628664\pi\)
−0.393294 + 0.919413i \(0.628664\pi\)
\(158\) 16.2890 1.29588
\(159\) 0 0
\(160\) 0 0
\(161\) 8.27340 0.652035
\(162\) 0 0
\(163\) −16.5786 −1.29853 −0.649267 0.760560i \(-0.724924\pi\)
−0.649267 + 0.760560i \(0.724924\pi\)
\(164\) 52.7014 4.11529
\(165\) 0 0
\(166\) 23.2763 1.80659
\(167\) −4.74360 −0.367071 −0.183536 0.983013i \(-0.558754\pi\)
−0.183536 + 0.983013i \(0.558754\pi\)
\(168\) 0 0
\(169\) −10.5938 −0.814908
\(170\) 0 0
\(171\) 0 0
\(172\) 4.09534 0.312267
\(173\) −19.8201 −1.50689 −0.753446 0.657510i \(-0.771610\pi\)
−0.753446 + 0.657510i \(0.771610\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −30.7429 −2.31734
\(177\) 0 0
\(178\) 3.77798 0.283172
\(179\) −7.42036 −0.554624 −0.277312 0.960780i \(-0.589443\pi\)
−0.277312 + 0.960780i \(0.589443\pi\)
\(180\) 0 0
\(181\) 19.8411 1.47478 0.737390 0.675468i \(-0.236058\pi\)
0.737390 + 0.675468i \(0.236058\pi\)
\(182\) 8.98031 0.665665
\(183\) 0 0
\(184\) 24.9279 1.83771
\(185\) 0 0
\(186\) 0 0
\(187\) −13.3434 −0.975766
\(188\) −51.9797 −3.79101
\(189\) 0 0
\(190\) 0 0
\(191\) −5.66409 −0.409839 −0.204920 0.978779i \(-0.565693\pi\)
−0.204920 + 0.978779i \(0.565693\pi\)
\(192\) 0 0
\(193\) −14.7896 −1.06458 −0.532289 0.846563i \(-0.678668\pi\)
−0.532289 + 0.846563i \(0.678668\pi\)
\(194\) −31.9283 −2.29232
\(195\) 0 0
\(196\) −9.00820 −0.643443
\(197\) 27.1349 1.93328 0.966640 0.256139i \(-0.0824504\pi\)
0.966640 + 0.256139i \(0.0824504\pi\)
\(198\) 0 0
\(199\) −9.53523 −0.675934 −0.337967 0.941158i \(-0.609739\pi\)
−0.337967 + 0.941158i \(0.609739\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.88885 0.625417
\(203\) 2.24826 0.157797
\(204\) 0 0
\(205\) 0 0
\(206\) −46.8588 −3.26481
\(207\) 0 0
\(208\) 12.6917 0.880009
\(209\) 4.46794 0.309054
\(210\) 0 0
\(211\) 21.5727 1.48513 0.742564 0.669775i \(-0.233609\pi\)
0.742564 + 0.669775i \(0.233609\pi\)
\(212\) 49.3479 3.38923
\(213\) 0 0
\(214\) −20.8040 −1.42213
\(215\) 0 0
\(216\) 0 0
\(217\) 11.6664 0.791969
\(218\) −36.0796 −2.44362
\(219\) 0 0
\(220\) 0 0
\(221\) 5.50858 0.370547
\(222\) 0 0
\(223\) −22.5140 −1.50765 −0.753824 0.657076i \(-0.771793\pi\)
−0.753824 + 0.657076i \(0.771793\pi\)
\(224\) 16.9077 1.12969
\(225\) 0 0
\(226\) 1.74712 0.116217
\(227\) 8.38063 0.556242 0.278121 0.960546i \(-0.410288\pi\)
0.278121 + 0.960546i \(0.410288\pi\)
\(228\) 0 0
\(229\) −11.5492 −0.763192 −0.381596 0.924329i \(-0.624625\pi\)
−0.381596 + 0.924329i \(0.624625\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.77405 0.444738
\(233\) −8.85319 −0.579991 −0.289996 0.957028i \(-0.593654\pi\)
−0.289996 + 0.957028i \(0.593654\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 39.3447 2.56112
\(237\) 0 0
\(238\) 20.5589 1.33264
\(239\) −13.9536 −0.902584 −0.451292 0.892376i \(-0.649037\pi\)
−0.451292 + 0.892376i \(0.649037\pi\)
\(240\) 0 0
\(241\) −24.2138 −1.55975 −0.779874 0.625937i \(-0.784717\pi\)
−0.779874 + 0.625937i \(0.784717\pi\)
\(242\) 8.02983 0.516177
\(243\) 0 0
\(244\) −50.3554 −3.22367
\(245\) 0 0
\(246\) 0 0
\(247\) −1.84451 −0.117363
\(248\) 35.1512 2.23210
\(249\) 0 0
\(250\) 0 0
\(251\) 10.1141 0.638397 0.319198 0.947688i \(-0.396586\pi\)
0.319198 + 0.947688i \(0.396586\pi\)
\(252\) 0 0
\(253\) −13.8270 −0.869299
\(254\) 21.6282 1.35708
\(255\) 0 0
\(256\) −24.8323 −1.55202
\(257\) −18.2290 −1.13709 −0.568546 0.822652i \(-0.692494\pi\)
−0.568546 + 0.822652i \(0.692494\pi\)
\(258\) 0 0
\(259\) −2.17973 −0.135442
\(260\) 0 0
\(261\) 0 0
\(262\) 18.9083 1.16816
\(263\) 4.26573 0.263036 0.131518 0.991314i \(-0.458015\pi\)
0.131518 + 0.991314i \(0.458015\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.88401 −0.422086
\(267\) 0 0
\(268\) 18.7626 1.14611
\(269\) −8.10868 −0.494395 −0.247197 0.968965i \(-0.579510\pi\)
−0.247197 + 0.968965i \(0.579510\pi\)
\(270\) 0 0
\(271\) 26.9259 1.63563 0.817815 0.575481i \(-0.195185\pi\)
0.817815 + 0.575481i \(0.195185\pi\)
\(272\) 29.0554 1.76174
\(273\) 0 0
\(274\) −4.64410 −0.280561
\(275\) 0 0
\(276\) 0 0
\(277\) −29.1380 −1.75073 −0.875367 0.483459i \(-0.839380\pi\)
−0.875367 + 0.483459i \(0.839380\pi\)
\(278\) 1.82169 0.109258
\(279\) 0 0
\(280\) 0 0
\(281\) −21.6594 −1.29209 −0.646047 0.763298i \(-0.723579\pi\)
−0.646047 + 0.763298i \(0.723579\pi\)
\(282\) 0 0
\(283\) 16.1608 0.960660 0.480330 0.877088i \(-0.340517\pi\)
0.480330 + 0.877088i \(0.340517\pi\)
\(284\) 29.5354 1.75260
\(285\) 0 0
\(286\) −15.0085 −0.887471
\(287\) 25.5872 1.51037
\(288\) 0 0
\(289\) −4.38903 −0.258178
\(290\) 0 0
\(291\) 0 0
\(292\) 15.0968 0.883473
\(293\) 16.5348 0.965975 0.482988 0.875627i \(-0.339552\pi\)
0.482988 + 0.875627i \(0.339552\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.56756 −0.381732
\(297\) 0 0
\(298\) 43.7573 2.53479
\(299\) 5.70825 0.330117
\(300\) 0 0
\(301\) 1.98834 0.114606
\(302\) 20.2562 1.16561
\(303\) 0 0
\(304\) −9.72901 −0.557997
\(305\) 0 0
\(306\) 0 0
\(307\) 25.3923 1.44922 0.724608 0.689161i \(-0.242021\pi\)
0.724608 + 0.689161i \(0.242021\pi\)
\(308\) −39.1187 −2.22899
\(309\) 0 0
\(310\) 0 0
\(311\) −16.8950 −0.958030 −0.479015 0.877807i \(-0.659006\pi\)
−0.479015 + 0.877807i \(0.659006\pi\)
\(312\) 0 0
\(313\) 22.1978 1.25469 0.627347 0.778740i \(-0.284141\pi\)
0.627347 + 0.778740i \(0.284141\pi\)
\(314\) −25.3791 −1.43223
\(315\) 0 0
\(316\) 29.2927 1.64785
\(317\) −26.1161 −1.46683 −0.733414 0.679783i \(-0.762074\pi\)
−0.733414 + 0.679783i \(0.762074\pi\)
\(318\) 0 0
\(319\) −3.75744 −0.210376
\(320\) 0 0
\(321\) 0 0
\(322\) 21.3041 1.18723
\(323\) −4.22270 −0.234957
\(324\) 0 0
\(325\) 0 0
\(326\) −42.6900 −2.36438
\(327\) 0 0
\(328\) 77.0948 4.25685
\(329\) −25.2369 −1.39135
\(330\) 0 0
\(331\) 24.3439 1.33806 0.669030 0.743236i \(-0.266710\pi\)
0.669030 + 0.743236i \(0.266710\pi\)
\(332\) 41.8582 2.29727
\(333\) 0 0
\(334\) −12.2148 −0.668366
\(335\) 0 0
\(336\) 0 0
\(337\) 23.5099 1.28067 0.640333 0.768098i \(-0.278796\pi\)
0.640333 + 0.768098i \(0.278796\pi\)
\(338\) −27.2792 −1.48379
\(339\) 0 0
\(340\) 0 0
\(341\) −19.4977 −1.05586
\(342\) 0 0
\(343\) −20.1114 −1.08591
\(344\) 5.99092 0.323009
\(345\) 0 0
\(346\) −51.0369 −2.74376
\(347\) −29.9492 −1.60776 −0.803879 0.594793i \(-0.797234\pi\)
−0.803879 + 0.594793i \(0.797234\pi\)
\(348\) 0 0
\(349\) −17.0927 −0.914951 −0.457476 0.889222i \(-0.651246\pi\)
−0.457476 + 0.889222i \(0.651246\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −28.2572 −1.50612
\(353\) −6.40124 −0.340704 −0.170352 0.985383i \(-0.554490\pi\)
−0.170352 + 0.985383i \(0.554490\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.79400 0.360082
\(357\) 0 0
\(358\) −19.1075 −1.00986
\(359\) 29.2509 1.54380 0.771902 0.635741i \(-0.219306\pi\)
0.771902 + 0.635741i \(0.219306\pi\)
\(360\) 0 0
\(361\) −17.5861 −0.925582
\(362\) 51.0911 2.68529
\(363\) 0 0
\(364\) 16.1494 0.846461
\(365\) 0 0
\(366\) 0 0
\(367\) 13.2919 0.693832 0.346916 0.937896i \(-0.387229\pi\)
0.346916 + 0.937896i \(0.387229\pi\)
\(368\) 30.1086 1.56952
\(369\) 0 0
\(370\) 0 0
\(371\) 23.9591 1.24389
\(372\) 0 0
\(373\) 19.7309 1.02163 0.510814 0.859691i \(-0.329344\pi\)
0.510814 + 0.859691i \(0.329344\pi\)
\(374\) −34.3594 −1.77668
\(375\) 0 0
\(376\) −76.0391 −3.92142
\(377\) 1.55119 0.0798905
\(378\) 0 0
\(379\) 25.4806 1.30885 0.654424 0.756128i \(-0.272911\pi\)
0.654424 + 0.756128i \(0.272911\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −14.5851 −0.746239
\(383\) −0.542784 −0.0277350 −0.0138675 0.999904i \(-0.504414\pi\)
−0.0138675 + 0.999904i \(0.504414\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −38.0833 −1.93839
\(387\) 0 0
\(388\) −57.4171 −2.91491
\(389\) −19.9301 −1.01050 −0.505249 0.862974i \(-0.668599\pi\)
−0.505249 + 0.862974i \(0.668599\pi\)
\(390\) 0 0
\(391\) 13.0681 0.660881
\(392\) −13.1777 −0.665576
\(393\) 0 0
\(394\) 69.8726 3.52013
\(395\) 0 0
\(396\) 0 0
\(397\) −9.40286 −0.471916 −0.235958 0.971763i \(-0.575823\pi\)
−0.235958 + 0.971763i \(0.575823\pi\)
\(398\) −24.5533 −1.23075
\(399\) 0 0
\(400\) 0 0
\(401\) 9.37578 0.468204 0.234102 0.972212i \(-0.424785\pi\)
0.234102 + 0.972212i \(0.424785\pi\)
\(402\) 0 0
\(403\) 8.04928 0.400963
\(404\) 15.9850 0.795282
\(405\) 0 0
\(406\) 5.78930 0.287318
\(407\) 3.64291 0.180572
\(408\) 0 0
\(409\) 19.7204 0.975111 0.487556 0.873092i \(-0.337889\pi\)
0.487556 + 0.873092i \(0.337889\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −84.2670 −4.15154
\(413\) 19.1024 0.939967
\(414\) 0 0
\(415\) 0 0
\(416\) 11.6655 0.571948
\(417\) 0 0
\(418\) 11.5050 0.562728
\(419\) −6.24026 −0.304857 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(420\) 0 0
\(421\) 6.31722 0.307882 0.153941 0.988080i \(-0.450803\pi\)
0.153941 + 0.988080i \(0.450803\pi\)
\(422\) 55.5500 2.70413
\(423\) 0 0
\(424\) 72.1891 3.50581
\(425\) 0 0
\(426\) 0 0
\(427\) −24.4482 −1.18313
\(428\) −37.4122 −1.80839
\(429\) 0 0
\(430\) 0 0
\(431\) −37.8006 −1.82079 −0.910396 0.413737i \(-0.864223\pi\)
−0.910396 + 0.413737i \(0.864223\pi\)
\(432\) 0 0
\(433\) −37.8447 −1.81870 −0.909350 0.416032i \(-0.863420\pi\)
−0.909350 + 0.416032i \(0.863420\pi\)
\(434\) 30.0412 1.44202
\(435\) 0 0
\(436\) −64.8825 −3.10731
\(437\) −4.37575 −0.209321
\(438\) 0 0
\(439\) −15.5318 −0.741291 −0.370646 0.928774i \(-0.620864\pi\)
−0.370646 + 0.928774i \(0.620864\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 14.1847 0.674696
\(443\) 5.24223 0.249066 0.124533 0.992215i \(-0.460257\pi\)
0.124533 + 0.992215i \(0.460257\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −57.9738 −2.74514
\(447\) 0 0
\(448\) 6.74749 0.318789
\(449\) −28.0123 −1.32198 −0.660992 0.750393i \(-0.729864\pi\)
−0.660992 + 0.750393i \(0.729864\pi\)
\(450\) 0 0
\(451\) −42.7631 −2.01364
\(452\) 3.14188 0.147781
\(453\) 0 0
\(454\) 21.5802 1.01281
\(455\) 0 0
\(456\) 0 0
\(457\) −1.90391 −0.0890610 −0.0445305 0.999008i \(-0.514179\pi\)
−0.0445305 + 0.999008i \(0.514179\pi\)
\(458\) −29.7393 −1.38963
\(459\) 0 0
\(460\) 0 0
\(461\) 30.5294 1.42190 0.710949 0.703244i \(-0.248266\pi\)
0.710949 + 0.703244i \(0.248266\pi\)
\(462\) 0 0
\(463\) −32.9230 −1.53006 −0.765030 0.643995i \(-0.777276\pi\)
−0.765030 + 0.643995i \(0.777276\pi\)
\(464\) 8.18188 0.379834
\(465\) 0 0
\(466\) −22.7971 −1.05605
\(467\) 4.42775 0.204892 0.102446 0.994739i \(-0.467333\pi\)
0.102446 + 0.994739i \(0.467333\pi\)
\(468\) 0 0
\(469\) 9.10947 0.420636
\(470\) 0 0
\(471\) 0 0
\(472\) 57.5558 2.64922
\(473\) −3.32305 −0.152794
\(474\) 0 0
\(475\) 0 0
\(476\) 36.9714 1.69458
\(477\) 0 0
\(478\) −35.9307 −1.64343
\(479\) −8.27071 −0.377898 −0.188949 0.981987i \(-0.560508\pi\)
−0.188949 + 0.981987i \(0.560508\pi\)
\(480\) 0 0
\(481\) −1.50391 −0.0685723
\(482\) −62.3508 −2.84000
\(483\) 0 0
\(484\) 14.4402 0.656371
\(485\) 0 0
\(486\) 0 0
\(487\) −39.9284 −1.80933 −0.904665 0.426124i \(-0.859879\pi\)
−0.904665 + 0.426124i \(0.859879\pi\)
\(488\) −73.6628 −3.33456
\(489\) 0 0
\(490\) 0 0
\(491\) −33.3896 −1.50685 −0.753427 0.657532i \(-0.771601\pi\)
−0.753427 + 0.657532i \(0.771601\pi\)
\(492\) 0 0
\(493\) 3.55119 0.159938
\(494\) −4.74964 −0.213696
\(495\) 0 0
\(496\) 42.4565 1.90635
\(497\) 14.3398 0.643229
\(498\) 0 0
\(499\) −3.12737 −0.140000 −0.0700002 0.997547i \(-0.522300\pi\)
−0.0700002 + 0.997547i \(0.522300\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 26.0439 1.16240
\(503\) 17.2563 0.769419 0.384710 0.923038i \(-0.374302\pi\)
0.384710 + 0.923038i \(0.374302\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −35.6048 −1.58283
\(507\) 0 0
\(508\) 38.8944 1.72566
\(509\) −4.71879 −0.209157 −0.104578 0.994517i \(-0.533349\pi\)
−0.104578 + 0.994517i \(0.533349\pi\)
\(510\) 0 0
\(511\) 7.32970 0.324247
\(512\) −49.3183 −2.17958
\(513\) 0 0
\(514\) −46.9398 −2.07043
\(515\) 0 0
\(516\) 0 0
\(517\) 42.1775 1.85496
\(518\) −5.61283 −0.246613
\(519\) 0 0
\(520\) 0 0
\(521\) 40.0385 1.75412 0.877060 0.480381i \(-0.159502\pi\)
0.877060 + 0.480381i \(0.159502\pi\)
\(522\) 0 0
\(523\) 5.38694 0.235555 0.117777 0.993040i \(-0.462423\pi\)
0.117777 + 0.993040i \(0.462423\pi\)
\(524\) 34.0030 1.48543
\(525\) 0 0
\(526\) 10.9843 0.478939
\(527\) 18.4275 0.802713
\(528\) 0 0
\(529\) −9.45826 −0.411229
\(530\) 0 0
\(531\) 0 0
\(532\) −12.3796 −0.536725
\(533\) 17.6540 0.764679
\(534\) 0 0
\(535\) 0 0
\(536\) 27.4470 1.18553
\(537\) 0 0
\(538\) −20.8799 −0.900198
\(539\) 7.30945 0.314840
\(540\) 0 0
\(541\) 35.9892 1.54730 0.773649 0.633614i \(-0.218429\pi\)
0.773649 + 0.633614i \(0.218429\pi\)
\(542\) 69.3345 2.97817
\(543\) 0 0
\(544\) 26.7062 1.14502
\(545\) 0 0
\(546\) 0 0
\(547\) −24.2745 −1.03790 −0.518951 0.854804i \(-0.673677\pi\)
−0.518951 + 0.854804i \(0.673677\pi\)
\(548\) −8.35157 −0.356761
\(549\) 0 0
\(550\) 0 0
\(551\) −1.18909 −0.0506570
\(552\) 0 0
\(553\) 14.2220 0.604782
\(554\) −75.0307 −3.18775
\(555\) 0 0
\(556\) 3.27598 0.138933
\(557\) 4.93097 0.208932 0.104466 0.994528i \(-0.466687\pi\)
0.104466 + 0.994528i \(0.466687\pi\)
\(558\) 0 0
\(559\) 1.37186 0.0580236
\(560\) 0 0
\(561\) 0 0
\(562\) −55.7733 −2.35266
\(563\) 34.7020 1.46252 0.731258 0.682101i \(-0.238933\pi\)
0.731258 + 0.682101i \(0.238933\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 41.6142 1.74918
\(567\) 0 0
\(568\) 43.2061 1.81289
\(569\) 19.9633 0.836903 0.418452 0.908239i \(-0.362573\pi\)
0.418452 + 0.908239i \(0.362573\pi\)
\(570\) 0 0
\(571\) −7.22542 −0.302375 −0.151187 0.988505i \(-0.548310\pi\)
−0.151187 + 0.988505i \(0.548310\pi\)
\(572\) −26.9900 −1.12851
\(573\) 0 0
\(574\) 65.8875 2.75009
\(575\) 0 0
\(576\) 0 0
\(577\) −17.6693 −0.735581 −0.367791 0.929909i \(-0.619886\pi\)
−0.367791 + 0.929909i \(0.619886\pi\)
\(578\) −11.3018 −0.470093
\(579\) 0 0
\(580\) 0 0
\(581\) 20.3227 0.843128
\(582\) 0 0
\(583\) −40.0420 −1.65837
\(584\) 22.0845 0.913864
\(585\) 0 0
\(586\) 42.5774 1.75886
\(587\) 13.1365 0.542203 0.271102 0.962551i \(-0.412612\pi\)
0.271102 + 0.962551i \(0.412612\pi\)
\(588\) 0 0
\(589\) −6.17031 −0.254243
\(590\) 0 0
\(591\) 0 0
\(592\) −7.93248 −0.326023
\(593\) 11.3675 0.466807 0.233403 0.972380i \(-0.425014\pi\)
0.233403 + 0.972380i \(0.425014\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 78.6894 3.22324
\(597\) 0 0
\(598\) 14.6988 0.601079
\(599\) 8.98828 0.367251 0.183626 0.982996i \(-0.441217\pi\)
0.183626 + 0.982996i \(0.441217\pi\)
\(600\) 0 0
\(601\) −5.15704 −0.210360 −0.105180 0.994453i \(-0.533542\pi\)
−0.105180 + 0.994453i \(0.533542\pi\)
\(602\) 5.12001 0.208676
\(603\) 0 0
\(604\) 36.4271 1.48220
\(605\) 0 0
\(606\) 0 0
\(607\) 36.3436 1.47514 0.737570 0.675271i \(-0.235973\pi\)
0.737570 + 0.675271i \(0.235973\pi\)
\(608\) −8.94238 −0.362661
\(609\) 0 0
\(610\) 0 0
\(611\) −17.4122 −0.704423
\(612\) 0 0
\(613\) −18.9943 −0.767171 −0.383585 0.923505i \(-0.625311\pi\)
−0.383585 + 0.923505i \(0.625311\pi\)
\(614\) 65.3855 2.63874
\(615\) 0 0
\(616\) −57.2251 −2.30567
\(617\) −24.5060 −0.986574 −0.493287 0.869867i \(-0.664205\pi\)
−0.493287 + 0.869867i \(0.664205\pi\)
\(618\) 0 0
\(619\) 12.5314 0.503680 0.251840 0.967769i \(-0.418964\pi\)
0.251840 + 0.967769i \(0.418964\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −43.5049 −1.74439
\(623\) 3.29858 0.132155
\(624\) 0 0
\(625\) 0 0
\(626\) 57.1596 2.28456
\(627\) 0 0
\(628\) −45.6396 −1.82122
\(629\) −3.44294 −0.137279
\(630\) 0 0
\(631\) 17.8019 0.708681 0.354340 0.935116i \(-0.384705\pi\)
0.354340 + 0.935116i \(0.384705\pi\)
\(632\) 42.8512 1.70453
\(633\) 0 0
\(634\) −67.2493 −2.67081
\(635\) 0 0
\(636\) 0 0
\(637\) −3.01758 −0.119561
\(638\) −9.67546 −0.383055
\(639\) 0 0
\(640\) 0 0
\(641\) −20.3973 −0.805644 −0.402822 0.915278i \(-0.631971\pi\)
−0.402822 + 0.915278i \(0.631971\pi\)
\(642\) 0 0
\(643\) 23.6531 0.932787 0.466394 0.884577i \(-0.345553\pi\)
0.466394 + 0.884577i \(0.345553\pi\)
\(644\) 38.3115 1.50968
\(645\) 0 0
\(646\) −10.8735 −0.427812
\(647\) −17.5789 −0.691096 −0.345548 0.938401i \(-0.612307\pi\)
−0.345548 + 0.938401i \(0.612307\pi\)
\(648\) 0 0
\(649\) −31.9252 −1.25317
\(650\) 0 0
\(651\) 0 0
\(652\) −76.7702 −3.00655
\(653\) 10.4035 0.407121 0.203560 0.979062i \(-0.434749\pi\)
0.203560 + 0.979062i \(0.434749\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 93.1172 3.63562
\(657\) 0 0
\(658\) −64.9852 −2.53339
\(659\) 43.5638 1.69700 0.848502 0.529192i \(-0.177505\pi\)
0.848502 + 0.529192i \(0.177505\pi\)
\(660\) 0 0
\(661\) 18.2404 0.709469 0.354734 0.934967i \(-0.384571\pi\)
0.354734 + 0.934967i \(0.384571\pi\)
\(662\) 62.6857 2.43635
\(663\) 0 0
\(664\) 61.2327 2.37629
\(665\) 0 0
\(666\) 0 0
\(667\) 3.67991 0.142487
\(668\) −21.9661 −0.849895
\(669\) 0 0
\(670\) 0 0
\(671\) 40.8595 1.57736
\(672\) 0 0
\(673\) 25.8626 0.996929 0.498464 0.866910i \(-0.333897\pi\)
0.498464 + 0.866910i \(0.333897\pi\)
\(674\) 60.5383 2.33185
\(675\) 0 0
\(676\) −49.0566 −1.88679
\(677\) 30.3287 1.16563 0.582813 0.812606i \(-0.301952\pi\)
0.582813 + 0.812606i \(0.301952\pi\)
\(678\) 0 0
\(679\) −27.8768 −1.06981
\(680\) 0 0
\(681\) 0 0
\(682\) −50.2068 −1.92252
\(683\) 39.4637 1.51004 0.755018 0.655704i \(-0.227628\pi\)
0.755018 + 0.655704i \(0.227628\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −51.7871 −1.97724
\(687\) 0 0
\(688\) 7.23599 0.275870
\(689\) 16.5306 0.629766
\(690\) 0 0
\(691\) −24.5853 −0.935269 −0.467635 0.883922i \(-0.654894\pi\)
−0.467635 + 0.883922i \(0.654894\pi\)
\(692\) −91.7805 −3.48897
\(693\) 0 0
\(694\) −77.1196 −2.92742
\(695\) 0 0
\(696\) 0 0
\(697\) 40.4158 1.53086
\(698\) −44.0139 −1.66595
\(699\) 0 0
\(700\) 0 0
\(701\) −34.8635 −1.31678 −0.658388 0.752678i \(-0.728762\pi\)
−0.658388 + 0.752678i \(0.728762\pi\)
\(702\) 0 0
\(703\) 1.15285 0.0434804
\(704\) −11.2769 −0.425012
\(705\) 0 0
\(706\) −16.4833 −0.620356
\(707\) 7.76091 0.291879
\(708\) 0 0
\(709\) −24.4622 −0.918698 −0.459349 0.888256i \(-0.651917\pi\)
−0.459349 + 0.888256i \(0.651917\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.93868 0.372468
\(713\) 19.0954 0.715128
\(714\) 0 0
\(715\) 0 0
\(716\) −34.3613 −1.28414
\(717\) 0 0
\(718\) 75.3214 2.81097
\(719\) −27.1333 −1.01190 −0.505951 0.862562i \(-0.668858\pi\)
−0.505951 + 0.862562i \(0.668858\pi\)
\(720\) 0 0
\(721\) −40.9128 −1.52367
\(722\) −45.2843 −1.68531
\(723\) 0 0
\(724\) 91.8780 3.41462
\(725\) 0 0
\(726\) 0 0
\(727\) −33.9854 −1.26045 −0.630224 0.776413i \(-0.717037\pi\)
−0.630224 + 0.776413i \(0.717037\pi\)
\(728\) 23.6244 0.875578
\(729\) 0 0
\(730\) 0 0
\(731\) 3.14065 0.116161
\(732\) 0 0
\(733\) −17.2131 −0.635779 −0.317889 0.948128i \(-0.602974\pi\)
−0.317889 + 0.948128i \(0.602974\pi\)
\(734\) 34.2268 1.26333
\(735\) 0 0
\(736\) 27.6742 1.02008
\(737\) −15.2244 −0.560796
\(738\) 0 0
\(739\) −12.5603 −0.462039 −0.231019 0.972949i \(-0.574206\pi\)
−0.231019 + 0.972949i \(0.574206\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 61.6949 2.26489
\(743\) −43.0323 −1.57870 −0.789350 0.613943i \(-0.789582\pi\)
−0.789350 + 0.613943i \(0.789582\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 50.8073 1.86019
\(747\) 0 0
\(748\) −61.7891 −2.25923
\(749\) −18.1641 −0.663702
\(750\) 0 0
\(751\) −12.4133 −0.452969 −0.226485 0.974015i \(-0.572723\pi\)
−0.226485 + 0.974015i \(0.572723\pi\)
\(752\) −91.8421 −3.34914
\(753\) 0 0
\(754\) 3.99434 0.145465
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0439 −0.655815 −0.327907 0.944710i \(-0.606343\pi\)
−0.327907 + 0.944710i \(0.606343\pi\)
\(758\) 65.6127 2.38316
\(759\) 0 0
\(760\) 0 0
\(761\) −8.20477 −0.297423 −0.148711 0.988881i \(-0.547513\pi\)
−0.148711 + 0.988881i \(0.547513\pi\)
\(762\) 0 0
\(763\) −31.5013 −1.14043
\(764\) −26.2286 −0.948919
\(765\) 0 0
\(766\) −1.39768 −0.0505001
\(767\) 13.1797 0.475893
\(768\) 0 0
\(769\) −23.5524 −0.849320 −0.424660 0.905353i \(-0.639606\pi\)
−0.424660 + 0.905353i \(0.639606\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −68.4859 −2.46486
\(773\) 27.2326 0.979487 0.489743 0.871867i \(-0.337090\pi\)
0.489743 + 0.871867i \(0.337090\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −83.9933 −3.01518
\(777\) 0 0
\(778\) −51.3203 −1.83992
\(779\) −13.5330 −0.484868
\(780\) 0 0
\(781\) −23.9657 −0.857559
\(782\) 33.6504 1.20334
\(783\) 0 0
\(784\) −15.9164 −0.568444
\(785\) 0 0
\(786\) 0 0
\(787\) −20.2585 −0.722137 −0.361069 0.932539i \(-0.617588\pi\)
−0.361069 + 0.932539i \(0.617588\pi\)
\(788\) 125.653 4.47621
\(789\) 0 0
\(790\) 0 0
\(791\) 1.52542 0.0542378
\(792\) 0 0
\(793\) −16.8681 −0.599004
\(794\) −24.2125 −0.859269
\(795\) 0 0
\(796\) −44.1546 −1.56502
\(797\) −27.8574 −0.986761 −0.493381 0.869814i \(-0.664239\pi\)
−0.493381 + 0.869814i \(0.664239\pi\)
\(798\) 0 0
\(799\) −39.8624 −1.41023
\(800\) 0 0
\(801\) 0 0
\(802\) 24.1427 0.852510
\(803\) −12.2499 −0.432289
\(804\) 0 0
\(805\) 0 0
\(806\) 20.7270 0.730077
\(807\) 0 0
\(808\) 23.3838 0.822638
\(809\) −17.5060 −0.615477 −0.307739 0.951471i \(-0.599572\pi\)
−0.307739 + 0.951471i \(0.599572\pi\)
\(810\) 0 0
\(811\) 39.1690 1.37541 0.687704 0.725991i \(-0.258619\pi\)
0.687704 + 0.725991i \(0.258619\pi\)
\(812\) 10.4110 0.365354
\(813\) 0 0
\(814\) 9.38053 0.328787
\(815\) 0 0
\(816\) 0 0
\(817\) −1.05162 −0.0367917
\(818\) 50.7803 1.77549
\(819\) 0 0
\(820\) 0 0
\(821\) 19.8338 0.692203 0.346101 0.938197i \(-0.387505\pi\)
0.346101 + 0.938197i \(0.387505\pi\)
\(822\) 0 0
\(823\) −10.3727 −0.361568 −0.180784 0.983523i \(-0.557864\pi\)
−0.180784 + 0.983523i \(0.557864\pi\)
\(824\) −123.271 −4.29434
\(825\) 0 0
\(826\) 49.1889 1.71150
\(827\) 3.35272 0.116586 0.0582928 0.998300i \(-0.481434\pi\)
0.0582928 + 0.998300i \(0.481434\pi\)
\(828\) 0 0
\(829\) −43.0572 −1.49544 −0.747719 0.664015i \(-0.768851\pi\)
−0.747719 + 0.664015i \(0.768851\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.65545 0.161399
\(833\) −6.90823 −0.239356
\(834\) 0 0
\(835\) 0 0
\(836\) 20.6896 0.715566
\(837\) 0 0
\(838\) −16.0688 −0.555086
\(839\) 39.3736 1.35933 0.679664 0.733523i \(-0.262125\pi\)
0.679664 + 0.733523i \(0.262125\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 16.2669 0.560595
\(843\) 0 0
\(844\) 99.8965 3.43858
\(845\) 0 0
\(846\) 0 0
\(847\) 7.01089 0.240897
\(848\) 87.1919 2.99418
\(849\) 0 0
\(850\) 0 0
\(851\) −3.56774 −0.122301
\(852\) 0 0
\(853\) 45.2743 1.55016 0.775081 0.631862i \(-0.217709\pi\)
0.775081 + 0.631862i \(0.217709\pi\)
\(854\) −62.9544 −2.15426
\(855\) 0 0
\(856\) −54.7288 −1.87059
\(857\) 24.1619 0.825353 0.412677 0.910878i \(-0.364594\pi\)
0.412677 + 0.910878i \(0.364594\pi\)
\(858\) 0 0
\(859\) −22.3680 −0.763185 −0.381593 0.924331i \(-0.624624\pi\)
−0.381593 + 0.924331i \(0.624624\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −97.3371 −3.31531
\(863\) −2.98467 −0.101599 −0.0507997 0.998709i \(-0.516177\pi\)
−0.0507997 + 0.998709i \(0.516177\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −97.4505 −3.31150
\(867\) 0 0
\(868\) 54.0236 1.83368
\(869\) −23.7688 −0.806301
\(870\) 0 0
\(871\) 6.28510 0.212963
\(872\) −94.9141 −3.21420
\(873\) 0 0
\(874\) −11.2676 −0.381133
\(875\) 0 0
\(876\) 0 0
\(877\) −29.5037 −0.996269 −0.498135 0.867100i \(-0.665981\pi\)
−0.498135 + 0.867100i \(0.665981\pi\)
\(878\) −39.9945 −1.34975
\(879\) 0 0
\(880\) 0 0
\(881\) 52.0937 1.75508 0.877541 0.479502i \(-0.159183\pi\)
0.877541 + 0.479502i \(0.159183\pi\)
\(882\) 0 0
\(883\) −22.2221 −0.747832 −0.373916 0.927463i \(-0.621985\pi\)
−0.373916 + 0.927463i \(0.621985\pi\)
\(884\) 25.5085 0.857944
\(885\) 0 0
\(886\) 13.4988 0.453502
\(887\) −38.6220 −1.29680 −0.648400 0.761300i \(-0.724562\pi\)
−0.648400 + 0.761300i \(0.724562\pi\)
\(888\) 0 0
\(889\) 18.8838 0.633341
\(890\) 0 0
\(891\) 0 0
\(892\) −104.255 −3.49072
\(893\) 13.3476 0.446662
\(894\) 0 0
\(895\) 0 0
\(896\) −16.4405 −0.549238
\(897\) 0 0
\(898\) −72.1321 −2.40708
\(899\) 5.18909 0.173066
\(900\) 0 0
\(901\) 37.8440 1.26077
\(902\) −110.116 −3.66645
\(903\) 0 0
\(904\) 4.59613 0.152865
\(905\) 0 0
\(906\) 0 0
\(907\) 29.7861 0.989030 0.494515 0.869169i \(-0.335346\pi\)
0.494515 + 0.869169i \(0.335346\pi\)
\(908\) 38.8080 1.28789
\(909\) 0 0
\(910\) 0 0
\(911\) 53.7163 1.77970 0.889850 0.456254i \(-0.150809\pi\)
0.889850 + 0.456254i \(0.150809\pi\)
\(912\) 0 0
\(913\) −33.9647 −1.12407
\(914\) −4.90258 −0.162163
\(915\) 0 0
\(916\) −53.4807 −1.76705
\(917\) 16.5089 0.545173
\(918\) 0 0
\(919\) 52.0358 1.71650 0.858251 0.513231i \(-0.171551\pi\)
0.858251 + 0.513231i \(0.171551\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 78.6137 2.58900
\(923\) 9.89379 0.325658
\(924\) 0 0
\(925\) 0 0
\(926\) −84.7770 −2.78595
\(927\) 0 0
\(928\) 7.52034 0.246867
\(929\) −52.5081 −1.72273 −0.861367 0.507982i \(-0.830391\pi\)
−0.861367 + 0.507982i \(0.830391\pi\)
\(930\) 0 0
\(931\) 2.31317 0.0758112
\(932\) −40.9963 −1.34288
\(933\) 0 0
\(934\) 11.4015 0.373068
\(935\) 0 0
\(936\) 0 0
\(937\) −46.1947 −1.50912 −0.754558 0.656233i \(-0.772149\pi\)
−0.754558 + 0.656233i \(0.772149\pi\)
\(938\) 23.4570 0.765898
\(939\) 0 0
\(940\) 0 0
\(941\) 29.8642 0.973545 0.486773 0.873529i \(-0.338174\pi\)
0.486773 + 0.873529i \(0.338174\pi\)
\(942\) 0 0
\(943\) 41.8807 1.36382
\(944\) 69.5175 2.26260
\(945\) 0 0
\(946\) −8.55690 −0.278209
\(947\) −11.7523 −0.381898 −0.190949 0.981600i \(-0.561157\pi\)
−0.190949 + 0.981600i \(0.561157\pi\)
\(948\) 0 0
\(949\) 5.05714 0.164162
\(950\) 0 0
\(951\) 0 0
\(952\) 54.0840 1.75287
\(953\) 17.1653 0.556040 0.278020 0.960575i \(-0.410322\pi\)
0.278020 + 0.960575i \(0.410322\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −64.6147 −2.08979
\(957\) 0 0
\(958\) −21.2972 −0.688081
\(959\) −4.05480 −0.130936
\(960\) 0 0
\(961\) −4.07332 −0.131397
\(962\) −3.87258 −0.124857
\(963\) 0 0
\(964\) −112.126 −3.61135
\(965\) 0 0
\(966\) 0 0
\(967\) −20.2076 −0.649831 −0.324916 0.945743i \(-0.605336\pi\)
−0.324916 + 0.945743i \(0.605336\pi\)
\(968\) 21.1239 0.678949
\(969\) 0 0
\(970\) 0 0
\(971\) 24.7827 0.795315 0.397658 0.917534i \(-0.369823\pi\)
0.397658 + 0.917534i \(0.369823\pi\)
\(972\) 0 0
\(973\) 1.59053 0.0509902
\(974\) −102.816 −3.29444
\(975\) 0 0
\(976\) −88.9720 −2.84792
\(977\) −38.5423 −1.23308 −0.616539 0.787324i \(-0.711466\pi\)
−0.616539 + 0.787324i \(0.711466\pi\)
\(978\) 0 0
\(979\) −5.51281 −0.176190
\(980\) 0 0
\(981\) 0 0
\(982\) −85.9787 −2.74369
\(983\) −13.0762 −0.417065 −0.208532 0.978015i \(-0.566869\pi\)
−0.208532 + 0.978015i \(0.566869\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.14436 0.291216
\(987\) 0 0
\(988\) −8.54135 −0.271737
\(989\) 3.25449 0.103487
\(990\) 0 0
\(991\) −36.8186 −1.16958 −0.584791 0.811184i \(-0.698823\pi\)
−0.584791 + 0.811184i \(0.698823\pi\)
\(992\) 39.0237 1.23900
\(993\) 0 0
\(994\) 36.9252 1.17120
\(995\) 0 0
\(996\) 0 0
\(997\) 18.5570 0.587707 0.293854 0.955850i \(-0.405062\pi\)
0.293854 + 0.955850i \(0.405062\pi\)
\(998\) −8.05302 −0.254914
\(999\) 0 0
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 6525.2.a.bw.1.7 7
3.2 odd 2 6525.2.a.bv.1.1 7
5.4 even 2 1305.2.a.s.1.1 7
15.14 odd 2 1305.2.a.t.1.7 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.1 7 5.4 even 2
1305.2.a.t.1.7 yes 7 15.14 odd 2
6525.2.a.bv.1.1 7 3.2 odd 2
6525.2.a.bw.1.7 7 1.1 even 1 trivial