Properties

Label 6525.2.a.by.1.1
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2175)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.59587\) 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)
 
\(f(q)\) \(=\) \(q-2.59587 q^{2} +4.73855 q^{4} -3.93826 q^{7} -7.10893 q^{8} +O(q^{10})\) \(q-2.59587 q^{2} +4.73855 q^{4} -3.93826 q^{7} -7.10893 q^{8} -2.24645 q^{11} +2.09464 q^{13} +10.2232 q^{14} +8.97678 q^{16} -6.95310 q^{17} +4.29591 q^{19} +5.83149 q^{22} +5.19955 q^{23} -5.43743 q^{26} -18.6617 q^{28} -1.00000 q^{29} +2.25035 q^{31} -9.08471 q^{32} +18.0494 q^{34} -10.5280 q^{37} -11.1516 q^{38} +2.66629 q^{41} +6.03681 q^{43} -10.6449 q^{44} -13.4974 q^{46} +7.74829 q^{47} +8.50991 q^{49} +9.92559 q^{52} +1.41309 q^{53} +27.9968 q^{56} +2.59587 q^{58} -5.76800 q^{59} -10.9192 q^{61} -5.84162 q^{62} +5.62918 q^{64} -11.7471 q^{67} -32.9476 q^{68} +4.55691 q^{71} +12.3564 q^{73} +27.3294 q^{74} +20.3564 q^{76} +8.84710 q^{77} +14.4098 q^{79} -6.92135 q^{82} +3.81272 q^{83} -15.6708 q^{86} +15.9699 q^{88} +2.07340 q^{89} -8.24926 q^{91} +24.6383 q^{92} -20.1136 q^{94} +12.0068 q^{97} -22.0906 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.59587 −1.83556 −0.917779 0.397090i \(-0.870020\pi\)
−0.917779 + 0.397090i \(0.870020\pi\)
\(3\) 0 0
\(4\) 4.73855 2.36928
\(5\) 0 0
\(6\) 0 0
\(7\) −3.93826 −1.48852 −0.744262 0.667888i \(-0.767198\pi\)
−0.744262 + 0.667888i \(0.767198\pi\)
\(8\) −7.10893 −2.51339
\(9\) 0 0
\(10\) 0 0
\(11\) −2.24645 −0.677329 −0.338665 0.940907i \(-0.609975\pi\)
−0.338665 + 0.940907i \(0.609975\pi\)
\(12\) 0 0
\(13\) 2.09464 0.580950 0.290475 0.956883i \(-0.406187\pi\)
0.290475 + 0.956883i \(0.406187\pi\)
\(14\) 10.2232 2.73227
\(15\) 0 0
\(16\) 8.97678 2.24420
\(17\) −6.95310 −1.68637 −0.843187 0.537620i \(-0.819324\pi\)
−0.843187 + 0.537620i \(0.819324\pi\)
\(18\) 0 0
\(19\) 4.29591 0.985549 0.492774 0.870157i \(-0.335983\pi\)
0.492774 + 0.870157i \(0.335983\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.83149 1.24328
\(23\) 5.19955 1.08418 0.542090 0.840320i \(-0.317633\pi\)
0.542090 + 0.840320i \(0.317633\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.43743 −1.06637
\(27\) 0 0
\(28\) −18.6617 −3.52672
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 2.25035 0.404175 0.202087 0.979367i \(-0.435227\pi\)
0.202087 + 0.979367i \(0.435227\pi\)
\(32\) −9.08471 −1.60596
\(33\) 0 0
\(34\) 18.0494 3.09544
\(35\) 0 0
\(36\) 0 0
\(37\) −10.5280 −1.73079 −0.865397 0.501086i \(-0.832934\pi\)
−0.865397 + 0.501086i \(0.832934\pi\)
\(38\) −11.1516 −1.80903
\(39\) 0 0
\(40\) 0 0
\(41\) 2.66629 0.416405 0.208202 0.978086i \(-0.433239\pi\)
0.208202 + 0.978086i \(0.433239\pi\)
\(42\) 0 0
\(43\) 6.03681 0.920605 0.460302 0.887762i \(-0.347741\pi\)
0.460302 + 0.887762i \(0.347741\pi\)
\(44\) −10.6449 −1.60478
\(45\) 0 0
\(46\) −13.4974 −1.99008
\(47\) 7.74829 1.13020 0.565102 0.825021i \(-0.308837\pi\)
0.565102 + 0.825021i \(0.308837\pi\)
\(48\) 0 0
\(49\) 8.50991 1.21570
\(50\) 0 0
\(51\) 0 0
\(52\) 9.92559 1.37643
\(53\) 1.41309 0.194103 0.0970513 0.995279i \(-0.469059\pi\)
0.0970513 + 0.995279i \(0.469059\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 27.9968 3.74124
\(57\) 0 0
\(58\) 2.59587 0.340855
\(59\) −5.76800 −0.750929 −0.375465 0.926837i \(-0.622517\pi\)
−0.375465 + 0.926837i \(0.622517\pi\)
\(60\) 0 0
\(61\) −10.9192 −1.39806 −0.699030 0.715092i \(-0.746385\pi\)
−0.699030 + 0.715092i \(0.746385\pi\)
\(62\) −5.84162 −0.741887
\(63\) 0 0
\(64\) 5.62918 0.703647
\(65\) 0 0
\(66\) 0 0
\(67\) −11.7471 −1.43514 −0.717569 0.696487i \(-0.754745\pi\)
−0.717569 + 0.696487i \(0.754745\pi\)
\(68\) −32.9476 −3.99549
\(69\) 0 0
\(70\) 0 0
\(71\) 4.55691 0.540806 0.270403 0.962747i \(-0.412843\pi\)
0.270403 + 0.962747i \(0.412843\pi\)
\(72\) 0 0
\(73\) 12.3564 1.44621 0.723103 0.690740i \(-0.242715\pi\)
0.723103 + 0.690740i \(0.242715\pi\)
\(74\) 27.3294 3.17698
\(75\) 0 0
\(76\) 20.3564 2.33504
\(77\) 8.84710 1.00822
\(78\) 0 0
\(79\) 14.4098 1.62122 0.810612 0.585584i \(-0.199135\pi\)
0.810612 + 0.585584i \(0.199135\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.92135 −0.764335
\(83\) 3.81272 0.418501 0.209250 0.977862i \(-0.432898\pi\)
0.209250 + 0.977862i \(0.432898\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −15.6708 −1.68982
\(87\) 0 0
\(88\) 15.9699 1.70239
\(89\) 2.07340 0.219780 0.109890 0.993944i \(-0.464950\pi\)
0.109890 + 0.993944i \(0.464950\pi\)
\(90\) 0 0
\(91\) −8.24926 −0.864757
\(92\) 24.6383 2.56872
\(93\) 0 0
\(94\) −20.1136 −2.07456
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0068 1.21911 0.609553 0.792745i \(-0.291349\pi\)
0.609553 + 0.792745i \(0.291349\pi\)
\(98\) −22.0906 −2.23149
\(99\) 0 0
\(100\) 0 0
\(101\) 15.8642 1.57855 0.789276 0.614039i \(-0.210456\pi\)
0.789276 + 0.614039i \(0.210456\pi\)
\(102\) 0 0
\(103\) −10.3119 −1.01606 −0.508031 0.861339i \(-0.669627\pi\)
−0.508031 + 0.861339i \(0.669627\pi\)
\(104\) −14.8907 −1.46015
\(105\) 0 0
\(106\) −3.66820 −0.356287
\(107\) −4.47645 −0.432755 −0.216378 0.976310i \(-0.569424\pi\)
−0.216378 + 0.976310i \(0.569424\pi\)
\(108\) 0 0
\(109\) 13.4783 1.29098 0.645492 0.763767i \(-0.276652\pi\)
0.645492 + 0.763767i \(0.276652\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −35.3529 −3.34054
\(113\) 11.4525 1.07736 0.538679 0.842511i \(-0.318923\pi\)
0.538679 + 0.842511i \(0.318923\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.73855 −0.439964
\(117\) 0 0
\(118\) 14.9730 1.37838
\(119\) 27.3831 2.51021
\(120\) 0 0
\(121\) −5.95347 −0.541225
\(122\) 28.3449 2.56622
\(123\) 0 0
\(124\) 10.6634 0.957602
\(125\) 0 0
\(126\) 0 0
\(127\) −3.23499 −0.287059 −0.143529 0.989646i \(-0.545845\pi\)
−0.143529 + 0.989646i \(0.545845\pi\)
\(128\) 3.55679 0.314378
\(129\) 0 0
\(130\) 0 0
\(131\) 13.3567 1.16698 0.583489 0.812121i \(-0.301687\pi\)
0.583489 + 0.812121i \(0.301687\pi\)
\(132\) 0 0
\(133\) −16.9184 −1.46701
\(134\) 30.4940 2.63428
\(135\) 0 0
\(136\) 49.4291 4.23851
\(137\) −4.35642 −0.372194 −0.186097 0.982531i \(-0.559584\pi\)
−0.186097 + 0.982531i \(0.559584\pi\)
\(138\) 0 0
\(139\) −18.4171 −1.56212 −0.781061 0.624455i \(-0.785321\pi\)
−0.781061 + 0.624455i \(0.785321\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.8292 −0.992681
\(143\) −4.70551 −0.393495
\(144\) 0 0
\(145\) 0 0
\(146\) −32.0756 −2.65460
\(147\) 0 0
\(148\) −49.8876 −4.10073
\(149\) −17.6427 −1.44535 −0.722674 0.691189i \(-0.757087\pi\)
−0.722674 + 0.691189i \(0.757087\pi\)
\(150\) 0 0
\(151\) 4.02381 0.327453 0.163726 0.986506i \(-0.447649\pi\)
0.163726 + 0.986506i \(0.447649\pi\)
\(152\) −30.5393 −2.47707
\(153\) 0 0
\(154\) −22.9659 −1.85065
\(155\) 0 0
\(156\) 0 0
\(157\) −7.70119 −0.614622 −0.307311 0.951609i \(-0.599429\pi\)
−0.307311 + 0.951609i \(0.599429\pi\)
\(158\) −37.4059 −2.97585
\(159\) 0 0
\(160\) 0 0
\(161\) −20.4772 −1.61383
\(162\) 0 0
\(163\) −13.6829 −1.07173 −0.535866 0.844303i \(-0.680015\pi\)
−0.535866 + 0.844303i \(0.680015\pi\)
\(164\) 12.6344 0.986578
\(165\) 0 0
\(166\) −9.89734 −0.768183
\(167\) 24.9607 1.93151 0.965757 0.259449i \(-0.0835408\pi\)
0.965757 + 0.259449i \(0.0835408\pi\)
\(168\) 0 0
\(169\) −8.61246 −0.662497
\(170\) 0 0
\(171\) 0 0
\(172\) 28.6057 2.18117
\(173\) −19.9187 −1.51439 −0.757194 0.653191i \(-0.773430\pi\)
−0.757194 + 0.653191i \(0.773430\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.1659 −1.52006
\(177\) 0 0
\(178\) −5.38227 −0.403418
\(179\) −10.8816 −0.813329 −0.406664 0.913578i \(-0.633308\pi\)
−0.406664 + 0.913578i \(0.633308\pi\)
\(180\) 0 0
\(181\) −13.9372 −1.03594 −0.517972 0.855398i \(-0.673313\pi\)
−0.517972 + 0.855398i \(0.673313\pi\)
\(182\) 21.4140 1.58731
\(183\) 0 0
\(184\) −36.9632 −2.72497
\(185\) 0 0
\(186\) 0 0
\(187\) 15.6198 1.14223
\(188\) 36.7157 2.67777
\(189\) 0 0
\(190\) 0 0
\(191\) −13.3458 −0.965670 −0.482835 0.875711i \(-0.660393\pi\)
−0.482835 + 0.875711i \(0.660393\pi\)
\(192\) 0 0
\(193\) 15.6337 1.12534 0.562668 0.826683i \(-0.309775\pi\)
0.562668 + 0.826683i \(0.309775\pi\)
\(194\) −31.1681 −2.23774
\(195\) 0 0
\(196\) 40.3247 2.88033
\(197\) 20.1140 1.43306 0.716530 0.697556i \(-0.245729\pi\)
0.716530 + 0.697556i \(0.245729\pi\)
\(198\) 0 0
\(199\) −9.27232 −0.657297 −0.328649 0.944452i \(-0.606593\pi\)
−0.328649 + 0.944452i \(0.606593\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −41.1816 −2.89752
\(203\) 3.93826 0.276412
\(204\) 0 0
\(205\) 0 0
\(206\) 26.7684 1.86504
\(207\) 0 0
\(208\) 18.8032 1.30376
\(209\) −9.65053 −0.667541
\(210\) 0 0
\(211\) −0.228998 −0.0157649 −0.00788245 0.999969i \(-0.502509\pi\)
−0.00788245 + 0.999969i \(0.502509\pi\)
\(212\) 6.69600 0.459883
\(213\) 0 0
\(214\) 11.6203 0.794348
\(215\) 0 0
\(216\) 0 0
\(217\) −8.86247 −0.601624
\(218\) −34.9879 −2.36968
\(219\) 0 0
\(220\) 0 0
\(221\) −14.5643 −0.979699
\(222\) 0 0
\(223\) −25.1117 −1.68160 −0.840802 0.541342i \(-0.817916\pi\)
−0.840802 + 0.541342i \(0.817916\pi\)
\(224\) 35.7780 2.39052
\(225\) 0 0
\(226\) −29.7292 −1.97756
\(227\) 4.98706 0.331003 0.165501 0.986210i \(-0.447076\pi\)
0.165501 + 0.986210i \(0.447076\pi\)
\(228\) 0 0
\(229\) 1.60591 0.106122 0.0530608 0.998591i \(-0.483102\pi\)
0.0530608 + 0.998591i \(0.483102\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.10893 0.466724
\(233\) −6.65588 −0.436041 −0.218021 0.975944i \(-0.569960\pi\)
−0.218021 + 0.975944i \(0.569960\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −27.3320 −1.77916
\(237\) 0 0
\(238\) −71.0831 −4.60763
\(239\) −2.87227 −0.185792 −0.0928959 0.995676i \(-0.529612\pi\)
−0.0928959 + 0.995676i \(0.529612\pi\)
\(240\) 0 0
\(241\) 15.2768 0.984068 0.492034 0.870576i \(-0.336254\pi\)
0.492034 + 0.870576i \(0.336254\pi\)
\(242\) 15.4545 0.993450
\(243\) 0 0
\(244\) −51.7412 −3.31239
\(245\) 0 0
\(246\) 0 0
\(247\) 8.99840 0.572554
\(248\) −15.9976 −1.01585
\(249\) 0 0
\(250\) 0 0
\(251\) −10.8782 −0.686627 −0.343314 0.939221i \(-0.611549\pi\)
−0.343314 + 0.939221i \(0.611549\pi\)
\(252\) 0 0
\(253\) −11.6805 −0.734348
\(254\) 8.39761 0.526913
\(255\) 0 0
\(256\) −20.4913 −1.28071
\(257\) 18.0748 1.12748 0.563739 0.825953i \(-0.309363\pi\)
0.563739 + 0.825953i \(0.309363\pi\)
\(258\) 0 0
\(259\) 41.4621 2.57633
\(260\) 0 0
\(261\) 0 0
\(262\) −34.6722 −2.14206
\(263\) 18.4454 1.13739 0.568697 0.822547i \(-0.307448\pi\)
0.568697 + 0.822547i \(0.307448\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 43.9180 2.69279
\(267\) 0 0
\(268\) −55.6643 −3.40024
\(269\) −11.6702 −0.711544 −0.355772 0.934573i \(-0.615782\pi\)
−0.355772 + 0.934573i \(0.615782\pi\)
\(270\) 0 0
\(271\) 20.2501 1.23011 0.615054 0.788485i \(-0.289134\pi\)
0.615054 + 0.788485i \(0.289134\pi\)
\(272\) −62.4165 −3.78455
\(273\) 0 0
\(274\) 11.3087 0.683184
\(275\) 0 0
\(276\) 0 0
\(277\) 3.66952 0.220480 0.110240 0.993905i \(-0.464838\pi\)
0.110240 + 0.993905i \(0.464838\pi\)
\(278\) 47.8085 2.86737
\(279\) 0 0
\(280\) 0 0
\(281\) −22.8056 −1.36047 −0.680233 0.732996i \(-0.738121\pi\)
−0.680233 + 0.732996i \(0.738121\pi\)
\(282\) 0 0
\(283\) −2.38657 −0.141867 −0.0709335 0.997481i \(-0.522598\pi\)
−0.0709335 + 0.997481i \(0.522598\pi\)
\(284\) 21.5932 1.28132
\(285\) 0 0
\(286\) 12.2149 0.722282
\(287\) −10.5006 −0.619828
\(288\) 0 0
\(289\) 31.3456 1.84386
\(290\) 0 0
\(291\) 0 0
\(292\) 58.5514 3.42646
\(293\) −19.4843 −1.13828 −0.569142 0.822239i \(-0.692725\pi\)
−0.569142 + 0.822239i \(0.692725\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 74.8430 4.35016
\(297\) 0 0
\(298\) 45.7982 2.65302
\(299\) 10.8912 0.629855
\(300\) 0 0
\(301\) −23.7745 −1.37034
\(302\) −10.4453 −0.601059
\(303\) 0 0
\(304\) 38.5634 2.21176
\(305\) 0 0
\(306\) 0 0
\(307\) −0.527333 −0.0300965 −0.0150482 0.999887i \(-0.504790\pi\)
−0.0150482 + 0.999887i \(0.504790\pi\)
\(308\) 41.9225 2.38875
\(309\) 0 0
\(310\) 0 0
\(311\) 7.89624 0.447755 0.223877 0.974617i \(-0.428129\pi\)
0.223877 + 0.974617i \(0.428129\pi\)
\(312\) 0 0
\(313\) 10.0064 0.565597 0.282799 0.959179i \(-0.408737\pi\)
0.282799 + 0.959179i \(0.408737\pi\)
\(314\) 19.9913 1.12818
\(315\) 0 0
\(316\) 68.2814 3.84113
\(317\) −32.5401 −1.82763 −0.913817 0.406125i \(-0.866880\pi\)
−0.913817 + 0.406125i \(0.866880\pi\)
\(318\) 0 0
\(319\) 2.24645 0.125777
\(320\) 0 0
\(321\) 0 0
\(322\) 53.1562 2.96228
\(323\) −29.8699 −1.66200
\(324\) 0 0
\(325\) 0 0
\(326\) 35.5192 1.96723
\(327\) 0 0
\(328\) −18.9545 −1.04659
\(329\) −30.5148 −1.68233
\(330\) 0 0
\(331\) 22.9411 1.26096 0.630480 0.776206i \(-0.282858\pi\)
0.630480 + 0.776206i \(0.282858\pi\)
\(332\) 18.0668 0.991544
\(333\) 0 0
\(334\) −64.7947 −3.54541
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0039 0.871789 0.435894 0.899998i \(-0.356432\pi\)
0.435894 + 0.899998i \(0.356432\pi\)
\(338\) 22.3569 1.21605
\(339\) 0 0
\(340\) 0 0
\(341\) −5.05529 −0.273760
\(342\) 0 0
\(343\) −5.94642 −0.321076
\(344\) −42.9153 −2.31384
\(345\) 0 0
\(346\) 51.7063 2.77975
\(347\) −22.9990 −1.23465 −0.617326 0.786707i \(-0.711784\pi\)
−0.617326 + 0.786707i \(0.711784\pi\)
\(348\) 0 0
\(349\) 24.3108 1.30133 0.650664 0.759366i \(-0.274491\pi\)
0.650664 + 0.759366i \(0.274491\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 20.4083 1.08777
\(353\) 29.2685 1.55780 0.778902 0.627146i \(-0.215777\pi\)
0.778902 + 0.627146i \(0.215777\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.82490 0.520719
\(357\) 0 0
\(358\) 28.2472 1.49291
\(359\) −18.5090 −0.976868 −0.488434 0.872601i \(-0.662432\pi\)
−0.488434 + 0.872601i \(0.662432\pi\)
\(360\) 0 0
\(361\) −0.545179 −0.0286936
\(362\) 36.1792 1.90154
\(363\) 0 0
\(364\) −39.0896 −2.04885
\(365\) 0 0
\(366\) 0 0
\(367\) 0.896398 0.0467916 0.0233958 0.999726i \(-0.492552\pi\)
0.0233958 + 0.999726i \(0.492552\pi\)
\(368\) 46.6752 2.43311
\(369\) 0 0
\(370\) 0 0
\(371\) −5.56511 −0.288926
\(372\) 0 0
\(373\) 4.64559 0.240540 0.120270 0.992741i \(-0.461624\pi\)
0.120270 + 0.992741i \(0.461624\pi\)
\(374\) −40.5469 −2.09663
\(375\) 0 0
\(376\) −55.0821 −2.84064
\(377\) −2.09464 −0.107880
\(378\) 0 0
\(379\) −32.4863 −1.66871 −0.834356 0.551226i \(-0.814160\pi\)
−0.834356 + 0.551226i \(0.814160\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 34.6441 1.77254
\(383\) −9.48311 −0.484564 −0.242282 0.970206i \(-0.577896\pi\)
−0.242282 + 0.970206i \(0.577896\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −40.5830 −2.06562
\(387\) 0 0
\(388\) 56.8949 2.88840
\(389\) −33.9509 −1.72138 −0.860689 0.509132i \(-0.829967\pi\)
−0.860689 + 0.509132i \(0.829967\pi\)
\(390\) 0 0
\(391\) −36.1530 −1.82833
\(392\) −60.4964 −3.05553
\(393\) 0 0
\(394\) −52.2132 −2.63047
\(395\) 0 0
\(396\) 0 0
\(397\) −25.0892 −1.25919 −0.629596 0.776923i \(-0.716780\pi\)
−0.629596 + 0.776923i \(0.716780\pi\)
\(398\) 24.0698 1.20651
\(399\) 0 0
\(400\) 0 0
\(401\) 4.85814 0.242604 0.121302 0.992616i \(-0.461293\pi\)
0.121302 + 0.992616i \(0.461293\pi\)
\(402\) 0 0
\(403\) 4.71368 0.234805
\(404\) 75.1736 3.74003
\(405\) 0 0
\(406\) −10.2232 −0.507370
\(407\) 23.6506 1.17232
\(408\) 0 0
\(409\) 14.6604 0.724911 0.362456 0.932001i \(-0.381938\pi\)
0.362456 + 0.932001i \(0.381938\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −48.8635 −2.40733
\(413\) 22.7159 1.11778
\(414\) 0 0
\(415\) 0 0
\(416\) −19.0292 −0.932985
\(417\) 0 0
\(418\) 25.0515 1.22531
\(419\) 18.4123 0.899502 0.449751 0.893154i \(-0.351513\pi\)
0.449751 + 0.893154i \(0.351513\pi\)
\(420\) 0 0
\(421\) 29.1489 1.42063 0.710316 0.703883i \(-0.248552\pi\)
0.710316 + 0.703883i \(0.248552\pi\)
\(422\) 0.594451 0.0289374
\(423\) 0 0
\(424\) −10.0456 −0.487855
\(425\) 0 0
\(426\) 0 0
\(427\) 43.0027 2.08105
\(428\) −21.2119 −1.02532
\(429\) 0 0
\(430\) 0 0
\(431\) 2.05315 0.0988966 0.0494483 0.998777i \(-0.484254\pi\)
0.0494483 + 0.998777i \(0.484254\pi\)
\(432\) 0 0
\(433\) −18.7820 −0.902603 −0.451302 0.892371i \(-0.649040\pi\)
−0.451302 + 0.892371i \(0.649040\pi\)
\(434\) 23.0058 1.10432
\(435\) 0 0
\(436\) 63.8675 3.05870
\(437\) 22.3368 1.06851
\(438\) 0 0
\(439\) −33.7889 −1.61266 −0.806329 0.591467i \(-0.798549\pi\)
−0.806329 + 0.591467i \(0.798549\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 37.8070 1.79830
\(443\) −24.2962 −1.15435 −0.577173 0.816622i \(-0.695844\pi\)
−0.577173 + 0.816622i \(0.695844\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 65.1868 3.08668
\(447\) 0 0
\(448\) −22.1692 −1.04740
\(449\) 21.0976 0.995656 0.497828 0.867276i \(-0.334131\pi\)
0.497828 + 0.867276i \(0.334131\pi\)
\(450\) 0 0
\(451\) −5.98968 −0.282043
\(452\) 54.2682 2.55256
\(453\) 0 0
\(454\) −12.9458 −0.607575
\(455\) 0 0
\(456\) 0 0
\(457\) 8.70983 0.407429 0.203714 0.979030i \(-0.434699\pi\)
0.203714 + 0.979030i \(0.434699\pi\)
\(458\) −4.16874 −0.194793
\(459\) 0 0
\(460\) 0 0
\(461\) −29.5424 −1.37592 −0.687962 0.725746i \(-0.741495\pi\)
−0.687962 + 0.725746i \(0.741495\pi\)
\(462\) 0 0
\(463\) 37.3067 1.73379 0.866894 0.498493i \(-0.166113\pi\)
0.866894 + 0.498493i \(0.166113\pi\)
\(464\) −8.97678 −0.416737
\(465\) 0 0
\(466\) 17.2778 0.800379
\(467\) −30.2144 −1.39816 −0.699079 0.715045i \(-0.746406\pi\)
−0.699079 + 0.715045i \(0.746406\pi\)
\(468\) 0 0
\(469\) 46.2632 2.13624
\(470\) 0 0
\(471\) 0 0
\(472\) 41.0043 1.88738
\(473\) −13.5614 −0.623553
\(474\) 0 0
\(475\) 0 0
\(476\) 129.756 5.94738
\(477\) 0 0
\(478\) 7.45605 0.341032
\(479\) −39.1511 −1.78886 −0.894430 0.447209i \(-0.852418\pi\)
−0.894430 + 0.447209i \(0.852418\pi\)
\(480\) 0 0
\(481\) −22.0524 −1.00551
\(482\) −39.6567 −1.80632
\(483\) 0 0
\(484\) −28.2108 −1.28231
\(485\) 0 0
\(486\) 0 0
\(487\) −23.9182 −1.08384 −0.541919 0.840430i \(-0.682302\pi\)
−0.541919 + 0.840430i \(0.682302\pi\)
\(488\) 77.6239 3.51387
\(489\) 0 0
\(490\) 0 0
\(491\) −4.40974 −0.199009 −0.0995044 0.995037i \(-0.531726\pi\)
−0.0995044 + 0.995037i \(0.531726\pi\)
\(492\) 0 0
\(493\) 6.95310 0.313152
\(494\) −23.3587 −1.05096
\(495\) 0 0
\(496\) 20.2009 0.907047
\(497\) −17.9463 −0.805002
\(498\) 0 0
\(499\) −37.0446 −1.65834 −0.829171 0.558995i \(-0.811187\pi\)
−0.829171 + 0.558995i \(0.811187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 28.2385 1.26034
\(503\) 12.6691 0.564886 0.282443 0.959284i \(-0.408855\pi\)
0.282443 + 0.959284i \(0.408855\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 30.3211 1.34794
\(507\) 0 0
\(508\) −15.3292 −0.680121
\(509\) 11.7078 0.518939 0.259469 0.965751i \(-0.416452\pi\)
0.259469 + 0.965751i \(0.416452\pi\)
\(510\) 0 0
\(511\) −48.6627 −2.15271
\(512\) 46.0793 2.03644
\(513\) 0 0
\(514\) −46.9200 −2.06955
\(515\) 0 0
\(516\) 0 0
\(517\) −17.4061 −0.765520
\(518\) −107.630 −4.72900
\(519\) 0 0
\(520\) 0 0
\(521\) −0.0559550 −0.00245143 −0.00122572 0.999999i \(-0.500390\pi\)
−0.00122572 + 0.999999i \(0.500390\pi\)
\(522\) 0 0
\(523\) −0.256669 −0.0112234 −0.00561168 0.999984i \(-0.501786\pi\)
−0.00561168 + 0.999984i \(0.501786\pi\)
\(524\) 63.2913 2.76489
\(525\) 0 0
\(526\) −47.8819 −2.08775
\(527\) −15.6469 −0.681590
\(528\) 0 0
\(529\) 4.03530 0.175448
\(530\) 0 0
\(531\) 0 0
\(532\) −80.1688 −3.47576
\(533\) 5.58493 0.241910
\(534\) 0 0
\(535\) 0 0
\(536\) 83.5095 3.60706
\(537\) 0 0
\(538\) 30.2943 1.30608
\(539\) −19.1171 −0.823430
\(540\) 0 0
\(541\) 1.88887 0.0812090 0.0406045 0.999175i \(-0.487072\pi\)
0.0406045 + 0.999175i \(0.487072\pi\)
\(542\) −52.5668 −2.25794
\(543\) 0 0
\(544\) 63.1669 2.70826
\(545\) 0 0
\(546\) 0 0
\(547\) −0.193341 −0.00826668 −0.00413334 0.999991i \(-0.501316\pi\)
−0.00413334 + 0.999991i \(0.501316\pi\)
\(548\) −20.6431 −0.881831
\(549\) 0 0
\(550\) 0 0
\(551\) −4.29591 −0.183012
\(552\) 0 0
\(553\) −56.7494 −2.41323
\(554\) −9.52561 −0.404704
\(555\) 0 0
\(556\) −87.2706 −3.70110
\(557\) 26.4282 1.11980 0.559899 0.828561i \(-0.310840\pi\)
0.559899 + 0.828561i \(0.310840\pi\)
\(558\) 0 0
\(559\) 12.6450 0.534825
\(560\) 0 0
\(561\) 0 0
\(562\) 59.2003 2.49722
\(563\) 6.18961 0.260861 0.130430 0.991457i \(-0.458364\pi\)
0.130430 + 0.991457i \(0.458364\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.19523 0.260405
\(567\) 0 0
\(568\) −32.3948 −1.35926
\(569\) 15.9034 0.666705 0.333353 0.942802i \(-0.391820\pi\)
0.333353 + 0.942802i \(0.391820\pi\)
\(570\) 0 0
\(571\) −16.8555 −0.705380 −0.352690 0.935740i \(-0.614733\pi\)
−0.352690 + 0.935740i \(0.614733\pi\)
\(572\) −22.2973 −0.932297
\(573\) 0 0
\(574\) 27.2581 1.13773
\(575\) 0 0
\(576\) 0 0
\(577\) −23.1237 −0.962650 −0.481325 0.876542i \(-0.659844\pi\)
−0.481325 + 0.876542i \(0.659844\pi\)
\(578\) −81.3692 −3.38451
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0155 −0.622948
\(582\) 0 0
\(583\) −3.17443 −0.131471
\(584\) −87.8408 −3.63488
\(585\) 0 0
\(586\) 50.5787 2.08939
\(587\) 7.80902 0.322313 0.161156 0.986929i \(-0.448478\pi\)
0.161156 + 0.986929i \(0.448478\pi\)
\(588\) 0 0
\(589\) 9.66730 0.398334
\(590\) 0 0
\(591\) 0 0
\(592\) −94.5077 −3.88424
\(593\) −19.5804 −0.804069 −0.402034 0.915625i \(-0.631697\pi\)
−0.402034 + 0.915625i \(0.631697\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −83.6009 −3.42443
\(597\) 0 0
\(598\) −28.2722 −1.15614
\(599\) −30.2322 −1.23525 −0.617627 0.786471i \(-0.711906\pi\)
−0.617627 + 0.786471i \(0.711906\pi\)
\(600\) 0 0
\(601\) 33.1180 1.35091 0.675457 0.737400i \(-0.263947\pi\)
0.675457 + 0.737400i \(0.263947\pi\)
\(602\) 61.7157 2.51534
\(603\) 0 0
\(604\) 19.0670 0.775826
\(605\) 0 0
\(606\) 0 0
\(607\) −23.4711 −0.952661 −0.476330 0.879266i \(-0.658033\pi\)
−0.476330 + 0.879266i \(0.658033\pi\)
\(608\) −39.0271 −1.58276
\(609\) 0 0
\(610\) 0 0
\(611\) 16.2299 0.656592
\(612\) 0 0
\(613\) 24.1560 0.975650 0.487825 0.872941i \(-0.337790\pi\)
0.487825 + 0.872941i \(0.337790\pi\)
\(614\) 1.36889 0.0552439
\(615\) 0 0
\(616\) −62.8935 −2.53405
\(617\) 47.7509 1.92238 0.961189 0.275890i \(-0.0889725\pi\)
0.961189 + 0.275890i \(0.0889725\pi\)
\(618\) 0 0
\(619\) 0.779735 0.0313402 0.0156701 0.999877i \(-0.495012\pi\)
0.0156701 + 0.999877i \(0.495012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −20.4976 −0.821880
\(623\) −8.16558 −0.327147
\(624\) 0 0
\(625\) 0 0
\(626\) −25.9754 −1.03819
\(627\) 0 0
\(628\) −36.4925 −1.45621
\(629\) 73.2023 2.91877
\(630\) 0 0
\(631\) 17.8120 0.709083 0.354542 0.935040i \(-0.384637\pi\)
0.354542 + 0.935040i \(0.384637\pi\)
\(632\) −102.438 −4.07476
\(633\) 0 0
\(634\) 84.4700 3.35473
\(635\) 0 0
\(636\) 0 0
\(637\) 17.8252 0.706262
\(638\) −5.83149 −0.230871
\(639\) 0 0
\(640\) 0 0
\(641\) −29.7647 −1.17563 −0.587817 0.808994i \(-0.700013\pi\)
−0.587817 + 0.808994i \(0.700013\pi\)
\(642\) 0 0
\(643\) −41.4049 −1.63285 −0.816426 0.577451i \(-0.804048\pi\)
−0.816426 + 0.577451i \(0.804048\pi\)
\(644\) −97.0322 −3.82361
\(645\) 0 0
\(646\) 77.5384 3.05071
\(647\) −14.6074 −0.574275 −0.287137 0.957889i \(-0.592704\pi\)
−0.287137 + 0.957889i \(0.592704\pi\)
\(648\) 0 0
\(649\) 12.9575 0.508627
\(650\) 0 0
\(651\) 0 0
\(652\) −64.8374 −2.53923
\(653\) −30.5647 −1.19609 −0.598045 0.801462i \(-0.704056\pi\)
−0.598045 + 0.801462i \(0.704056\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 23.9347 0.934493
\(657\) 0 0
\(658\) 79.2125 3.08802
\(659\) −22.1893 −0.864371 −0.432186 0.901785i \(-0.642257\pi\)
−0.432186 + 0.901785i \(0.642257\pi\)
\(660\) 0 0
\(661\) −9.85731 −0.383405 −0.191702 0.981453i \(-0.561401\pi\)
−0.191702 + 0.981453i \(0.561401\pi\)
\(662\) −59.5523 −2.31457
\(663\) 0 0
\(664\) −27.1044 −1.05185
\(665\) 0 0
\(666\) 0 0
\(667\) −5.19955 −0.201327
\(668\) 118.277 4.57629
\(669\) 0 0
\(670\) 0 0
\(671\) 24.5294 0.946948
\(672\) 0 0
\(673\) −25.3057 −0.975463 −0.487732 0.872994i \(-0.662176\pi\)
−0.487732 + 0.872994i \(0.662176\pi\)
\(674\) −41.5441 −1.60022
\(675\) 0 0
\(676\) −40.8106 −1.56964
\(677\) 3.67306 0.141167 0.0705836 0.997506i \(-0.477514\pi\)
0.0705836 + 0.997506i \(0.477514\pi\)
\(678\) 0 0
\(679\) −47.2860 −1.81467
\(680\) 0 0
\(681\) 0 0
\(682\) 13.1229 0.502502
\(683\) −17.3807 −0.665053 −0.332527 0.943094i \(-0.607901\pi\)
−0.332527 + 0.943094i \(0.607901\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 15.4361 0.589354
\(687\) 0 0
\(688\) 54.1911 2.06602
\(689\) 2.95992 0.112764
\(690\) 0 0
\(691\) −28.5585 −1.08642 −0.543209 0.839598i \(-0.682791\pi\)
−0.543209 + 0.839598i \(0.682791\pi\)
\(692\) −94.3856 −3.58800
\(693\) 0 0
\(694\) 59.7025 2.26628
\(695\) 0 0
\(696\) 0 0
\(697\) −18.5390 −0.702214
\(698\) −63.1078 −2.38866
\(699\) 0 0
\(700\) 0 0
\(701\) −21.5401 −0.813557 −0.406779 0.913527i \(-0.633348\pi\)
−0.406779 + 0.913527i \(0.633348\pi\)
\(702\) 0 0
\(703\) −45.2274 −1.70578
\(704\) −12.6457 −0.476601
\(705\) 0 0
\(706\) −75.9772 −2.85944
\(707\) −62.4776 −2.34971
\(708\) 0 0
\(709\) −2.29948 −0.0863587 −0.0431793 0.999067i \(-0.513749\pi\)
−0.0431793 + 0.999067i \(0.513749\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −14.7396 −0.552391
\(713\) 11.7008 0.438199
\(714\) 0 0
\(715\) 0 0
\(716\) −51.5630 −1.92700
\(717\) 0 0
\(718\) 48.0470 1.79310
\(719\) −13.6814 −0.510232 −0.255116 0.966910i \(-0.582114\pi\)
−0.255116 + 0.966910i \(0.582114\pi\)
\(720\) 0 0
\(721\) 40.6110 1.51243
\(722\) 1.41521 0.0526688
\(723\) 0 0
\(724\) −66.0422 −2.45444
\(725\) 0 0
\(726\) 0 0
\(727\) 50.0451 1.85607 0.928035 0.372492i \(-0.121497\pi\)
0.928035 + 0.372492i \(0.121497\pi\)
\(728\) 58.6435 2.17347
\(729\) 0 0
\(730\) 0 0
\(731\) −41.9745 −1.55248
\(732\) 0 0
\(733\) 38.7734 1.43213 0.716065 0.698034i \(-0.245942\pi\)
0.716065 + 0.698034i \(0.245942\pi\)
\(734\) −2.32694 −0.0858887
\(735\) 0 0
\(736\) −47.2364 −1.74116
\(737\) 26.3893 0.972062
\(738\) 0 0
\(739\) −18.0707 −0.664742 −0.332371 0.943149i \(-0.607849\pi\)
−0.332371 + 0.943149i \(0.607849\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 14.4463 0.530341
\(743\) 20.9864 0.769917 0.384959 0.922934i \(-0.374216\pi\)
0.384959 + 0.922934i \(0.374216\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12.0594 −0.441525
\(747\) 0 0
\(748\) 74.0151 2.70626
\(749\) 17.6294 0.644166
\(750\) 0 0
\(751\) −20.3274 −0.741758 −0.370879 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(752\) 69.5547 2.53640
\(753\) 0 0
\(754\) 5.43743 0.198020
\(755\) 0 0
\(756\) 0 0
\(757\) 0.0654846 0.00238008 0.00119004 0.999999i \(-0.499621\pi\)
0.00119004 + 0.999999i \(0.499621\pi\)
\(758\) 84.3304 3.06302
\(759\) 0 0
\(760\) 0 0
\(761\) 28.8872 1.04716 0.523580 0.851976i \(-0.324596\pi\)
0.523580 + 0.851976i \(0.324596\pi\)
\(762\) 0 0
\(763\) −53.0810 −1.92166
\(764\) −63.2399 −2.28794
\(765\) 0 0
\(766\) 24.6169 0.889446
\(767\) −12.0819 −0.436252
\(768\) 0 0
\(769\) −10.0813 −0.363541 −0.181771 0.983341i \(-0.558183\pi\)
−0.181771 + 0.983341i \(0.558183\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 74.0809 2.66623
\(773\) 20.5353 0.738602 0.369301 0.929310i \(-0.379597\pi\)
0.369301 + 0.929310i \(0.379597\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −85.3556 −3.06409
\(777\) 0 0
\(778\) 88.1321 3.15969
\(779\) 11.4541 0.410387
\(780\) 0 0
\(781\) −10.2369 −0.366304
\(782\) 93.8485 3.35602
\(783\) 0 0
\(784\) 76.3916 2.72827
\(785\) 0 0
\(786\) 0 0
\(787\) −46.0601 −1.64186 −0.820932 0.571026i \(-0.806545\pi\)
−0.820932 + 0.571026i \(0.806545\pi\)
\(788\) 95.3110 3.39531
\(789\) 0 0
\(790\) 0 0
\(791\) −45.1029 −1.60367
\(792\) 0 0
\(793\) −22.8718 −0.812203
\(794\) 65.1284 2.31132
\(795\) 0 0
\(796\) −43.9374 −1.55732
\(797\) −37.4745 −1.32742 −0.663708 0.747992i \(-0.731018\pi\)
−0.663708 + 0.747992i \(0.731018\pi\)
\(798\) 0 0
\(799\) −53.8746 −1.90595
\(800\) 0 0
\(801\) 0 0
\(802\) −12.6111 −0.445314
\(803\) −27.7580 −0.979558
\(804\) 0 0
\(805\) 0 0
\(806\) −12.2361 −0.430999
\(807\) 0 0
\(808\) −112.778 −3.96751
\(809\) −29.9284 −1.05223 −0.526113 0.850414i \(-0.676351\pi\)
−0.526113 + 0.850414i \(0.676351\pi\)
\(810\) 0 0
\(811\) −49.1738 −1.72673 −0.863363 0.504583i \(-0.831646\pi\)
−0.863363 + 0.504583i \(0.831646\pi\)
\(812\) 18.6617 0.654896
\(813\) 0 0
\(814\) −61.3940 −2.15186
\(815\) 0 0
\(816\) 0 0
\(817\) 25.9336 0.907301
\(818\) −38.0566 −1.33062
\(819\) 0 0
\(820\) 0 0
\(821\) −33.9715 −1.18561 −0.592807 0.805345i \(-0.701980\pi\)
−0.592807 + 0.805345i \(0.701980\pi\)
\(822\) 0 0
\(823\) 33.8272 1.17914 0.589572 0.807716i \(-0.299297\pi\)
0.589572 + 0.807716i \(0.299297\pi\)
\(824\) 73.3067 2.55376
\(825\) 0 0
\(826\) −58.9676 −2.05174
\(827\) −35.1083 −1.22083 −0.610417 0.792080i \(-0.708998\pi\)
−0.610417 + 0.792080i \(0.708998\pi\)
\(828\) 0 0
\(829\) 15.6701 0.544245 0.272122 0.962263i \(-0.412274\pi\)
0.272122 + 0.962263i \(0.412274\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 11.7911 0.408784
\(833\) −59.1702 −2.05013
\(834\) 0 0
\(835\) 0 0
\(836\) −45.7296 −1.58159
\(837\) 0 0
\(838\) −47.7961 −1.65109
\(839\) 16.5131 0.570095 0.285047 0.958513i \(-0.407991\pi\)
0.285047 + 0.958513i \(0.407991\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −75.6669 −2.60765
\(843\) 0 0
\(844\) −1.08512 −0.0373514
\(845\) 0 0
\(846\) 0 0
\(847\) 23.4463 0.805626
\(848\) 12.6850 0.435604
\(849\) 0 0
\(850\) 0 0
\(851\) −54.7409 −1.87649
\(852\) 0 0
\(853\) −16.3926 −0.561271 −0.280636 0.959814i \(-0.590545\pi\)
−0.280636 + 0.959814i \(0.590545\pi\)
\(854\) −111.629 −3.81988
\(855\) 0 0
\(856\) 31.8228 1.08768
\(857\) −9.83745 −0.336041 −0.168020 0.985784i \(-0.553737\pi\)
−0.168020 + 0.985784i \(0.553737\pi\)
\(858\) 0 0
\(859\) 39.8257 1.35883 0.679417 0.733752i \(-0.262233\pi\)
0.679417 + 0.733752i \(0.262233\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.32971 −0.181530
\(863\) −15.6707 −0.533438 −0.266719 0.963774i \(-0.585940\pi\)
−0.266719 + 0.963774i \(0.585940\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 48.7556 1.65678
\(867\) 0 0
\(868\) −41.9953 −1.42541
\(869\) −32.3708 −1.09810
\(870\) 0 0
\(871\) −24.6060 −0.833744
\(872\) −95.8162 −3.24474
\(873\) 0 0
\(874\) −57.9834 −1.96132
\(875\) 0 0
\(876\) 0 0
\(877\) −21.2819 −0.718640 −0.359320 0.933214i \(-0.616991\pi\)
−0.359320 + 0.933214i \(0.616991\pi\)
\(878\) 87.7118 2.96013
\(879\) 0 0
\(880\) 0 0
\(881\) −29.1037 −0.980529 −0.490265 0.871574i \(-0.663100\pi\)
−0.490265 + 0.871574i \(0.663100\pi\)
\(882\) 0 0
\(883\) −4.03715 −0.135861 −0.0679305 0.997690i \(-0.521640\pi\)
−0.0679305 + 0.997690i \(0.521640\pi\)
\(884\) −69.0136 −2.32118
\(885\) 0 0
\(886\) 63.0698 2.11887
\(887\) 2.85498 0.0958608 0.0479304 0.998851i \(-0.484737\pi\)
0.0479304 + 0.998851i \(0.484737\pi\)
\(888\) 0 0
\(889\) 12.7402 0.427293
\(890\) 0 0
\(891\) 0 0
\(892\) −118.993 −3.98419
\(893\) 33.2859 1.11387
\(894\) 0 0
\(895\) 0 0
\(896\) −14.0076 −0.467960
\(897\) 0 0
\(898\) −54.7666 −1.82758
\(899\) −2.25035 −0.0750534
\(900\) 0 0
\(901\) −9.82535 −0.327330
\(902\) 15.5485 0.517707
\(903\) 0 0
\(904\) −81.4150 −2.70782
\(905\) 0 0
\(906\) 0 0
\(907\) −3.08757 −0.102521 −0.0512605 0.998685i \(-0.516324\pi\)
−0.0512605 + 0.998685i \(0.516324\pi\)
\(908\) 23.6314 0.784237
\(909\) 0 0
\(910\) 0 0
\(911\) −34.1306 −1.13080 −0.565398 0.824818i \(-0.691277\pi\)
−0.565398 + 0.824818i \(0.691277\pi\)
\(912\) 0 0
\(913\) −8.56508 −0.283463
\(914\) −22.6096 −0.747860
\(915\) 0 0
\(916\) 7.60970 0.251432
\(917\) −52.6021 −1.73707
\(918\) 0 0
\(919\) 16.9603 0.559468 0.279734 0.960078i \(-0.409754\pi\)
0.279734 + 0.960078i \(0.409754\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 76.6882 2.52559
\(923\) 9.54511 0.314181
\(924\) 0 0
\(925\) 0 0
\(926\) −96.8433 −3.18247
\(927\) 0 0
\(928\) 9.08471 0.298220
\(929\) −5.63831 −0.184987 −0.0924935 0.995713i \(-0.529484\pi\)
−0.0924935 + 0.995713i \(0.529484\pi\)
\(930\) 0 0
\(931\) 36.5578 1.19813
\(932\) −31.5392 −1.03310
\(933\) 0 0
\(934\) 78.4328 2.56640
\(935\) 0 0
\(936\) 0 0
\(937\) 32.0790 1.04798 0.523988 0.851726i \(-0.324444\pi\)
0.523988 + 0.851726i \(0.324444\pi\)
\(938\) −120.093 −3.92119
\(939\) 0 0
\(940\) 0 0
\(941\) −31.2028 −1.01718 −0.508592 0.861008i \(-0.669834\pi\)
−0.508592 + 0.861008i \(0.669834\pi\)
\(942\) 0 0
\(943\) 13.8635 0.451458
\(944\) −51.7781 −1.68523
\(945\) 0 0
\(946\) 35.2036 1.14457
\(947\) 27.7302 0.901110 0.450555 0.892749i \(-0.351226\pi\)
0.450555 + 0.892749i \(0.351226\pi\)
\(948\) 0 0
\(949\) 25.8822 0.840173
\(950\) 0 0
\(951\) 0 0
\(952\) −194.665 −6.30913
\(953\) −23.9481 −0.775755 −0.387877 0.921711i \(-0.626792\pi\)
−0.387877 + 0.921711i \(0.626792\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −13.6104 −0.440192
\(957\) 0 0
\(958\) 101.631 3.28356
\(959\) 17.1567 0.554020
\(960\) 0 0
\(961\) −25.9359 −0.836643
\(962\) 57.2453 1.84566
\(963\) 0 0
\(964\) 72.3902 2.33153
\(965\) 0 0
\(966\) 0 0
\(967\) −43.6821 −1.40472 −0.702361 0.711821i \(-0.747871\pi\)
−0.702361 + 0.711821i \(0.747871\pi\)
\(968\) 42.3228 1.36031
\(969\) 0 0
\(970\) 0 0
\(971\) −1.12406 −0.0360729 −0.0180365 0.999837i \(-0.505741\pi\)
−0.0180365 + 0.999837i \(0.505741\pi\)
\(972\) 0 0
\(973\) 72.5315 2.32525
\(974\) 62.0887 1.98945
\(975\) 0 0
\(976\) −98.0193 −3.13752
\(977\) 18.8819 0.604086 0.302043 0.953294i \(-0.402331\pi\)
0.302043 + 0.953294i \(0.402331\pi\)
\(978\) 0 0
\(979\) −4.65778 −0.148863
\(980\) 0 0
\(981\) 0 0
\(982\) 11.4471 0.365292
\(983\) 23.5807 0.752109 0.376054 0.926598i \(-0.377281\pi\)
0.376054 + 0.926598i \(0.377281\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −18.0494 −0.574809
\(987\) 0 0
\(988\) 42.6394 1.35654
\(989\) 31.3887 0.998102
\(990\) 0 0
\(991\) −31.3818 −0.996877 −0.498438 0.866925i \(-0.666093\pi\)
−0.498438 + 0.866925i \(0.666093\pi\)
\(992\) −20.4438 −0.649090
\(993\) 0 0
\(994\) 46.5864 1.47763
\(995\) 0 0
\(996\) 0 0
\(997\) 45.3442 1.43606 0.718032 0.696010i \(-0.245043\pi\)
0.718032 + 0.696010i \(0.245043\pi\)
\(998\) 96.1629 3.04398
\(999\) 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.by.1.1 8
3.2 odd 2 2175.2.a.bd.1.8 yes 8
5.4 even 2 6525.2.a.bz.1.8 8
15.2 even 4 2175.2.c.p.349.15 16
15.8 even 4 2175.2.c.p.349.2 16
15.14 odd 2 2175.2.a.bc.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.1 8 15.14 odd 2
2175.2.a.bd.1.8 yes 8 3.2 odd 2
2175.2.c.p.349.2 16 15.8 even 4
2175.2.c.p.349.15 16 15.2 even 4
6525.2.a.by.1.1 8 1.1 even 1 trivial
6525.2.a.bz.1.8 8 5.4 even 2