Properties

Label 3042.2.a.r.1.1
Level $3042$
Weight $2$
Character 3042.1
Self dual yes
Analytic conductor $24.290$
Analytic rank $1$
Dimension $2$
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] = [3042,2,Mod(1,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.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: \(24.2904922949\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 234)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.16228\) of defining polynomial
Character \(\chi\) \(=\) 3042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.16228 q^{5} +3.16228 q^{7} -1.00000 q^{8} +4.16228 q^{10} +1.16228 q^{11} -3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} -5.16228 q^{19} -4.16228 q^{20} -1.16228 q^{22} -7.16228 q^{23} +12.3246 q^{25} +3.16228 q^{28} +1.83772 q^{29} -6.32456 q^{31} -1.00000 q^{32} -3.00000 q^{34} -13.1623 q^{35} +3.83772 q^{37} +5.16228 q^{38} +4.16228 q^{40} -3.00000 q^{41} +9.16228 q^{43} +1.16228 q^{44} +7.16228 q^{46} -4.83772 q^{47} +3.00000 q^{49} -12.3246 q^{50} +12.4868 q^{53} -4.83772 q^{55} -3.16228 q^{56} -1.83772 q^{58} +2.32456 q^{59} +0.162278 q^{61} +6.32456 q^{62} +1.00000 q^{64} -2.83772 q^{67} +3.00000 q^{68} +13.1623 q^{70} +7.16228 q^{71} -1.00000 q^{73} -3.83772 q^{74} -5.16228 q^{76} +3.67544 q^{77} -4.00000 q^{79} -4.16228 q^{80} +3.00000 q^{82} +3.48683 q^{83} -12.4868 q^{85} -9.16228 q^{86} -1.16228 q^{88} -12.0000 q^{89} -7.16228 q^{92} +4.83772 q^{94} +21.4868 q^{95} -4.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} + 6 q^{17} - 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 12 q^{25} + 10 q^{29} - 2 q^{32} - 6 q^{34} - 20 q^{35} + 14 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −4.16228 −1.86143 −0.930714 0.365749i \(-0.880813\pi\)
−0.930714 + 0.365749i \(0.880813\pi\)
\(6\) 0 0
\(7\) 3.16228 1.19523 0.597614 0.801784i \(-0.296115\pi\)
0.597614 + 0.801784i \(0.296115\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 4.16228 1.31623
\(11\) 1.16228 0.350440 0.175220 0.984529i \(-0.443936\pi\)
0.175220 + 0.984529i \(0.443936\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −3.16228 −0.845154
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −5.16228 −1.18431 −0.592154 0.805825i \(-0.701722\pi\)
−0.592154 + 0.805825i \(0.701722\pi\)
\(20\) −4.16228 −0.930714
\(21\) 0 0
\(22\) −1.16228 −0.247798
\(23\) −7.16228 −1.49344 −0.746719 0.665140i \(-0.768372\pi\)
−0.746719 + 0.665140i \(0.768372\pi\)
\(24\) 0 0
\(25\) 12.3246 2.46491
\(26\) 0 0
\(27\) 0 0
\(28\) 3.16228 0.597614
\(29\) 1.83772 0.341256 0.170628 0.985335i \(-0.445420\pi\)
0.170628 + 0.985335i \(0.445420\pi\)
\(30\) 0 0
\(31\) −6.32456 −1.13592 −0.567962 0.823055i \(-0.692268\pi\)
−0.567962 + 0.823055i \(0.692268\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −13.1623 −2.22483
\(36\) 0 0
\(37\) 3.83772 0.630918 0.315459 0.948939i \(-0.397842\pi\)
0.315459 + 0.948939i \(0.397842\pi\)
\(38\) 5.16228 0.837432
\(39\) 0 0
\(40\) 4.16228 0.658114
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 9.16228 1.39723 0.698617 0.715496i \(-0.253799\pi\)
0.698617 + 0.715496i \(0.253799\pi\)
\(44\) 1.16228 0.175220
\(45\) 0 0
\(46\) 7.16228 1.05602
\(47\) −4.83772 −0.705654 −0.352827 0.935689i \(-0.614780\pi\)
−0.352827 + 0.935689i \(0.614780\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) −12.3246 −1.74296
\(51\) 0 0
\(52\) 0 0
\(53\) 12.4868 1.71520 0.857599 0.514319i \(-0.171955\pi\)
0.857599 + 0.514319i \(0.171955\pi\)
\(54\) 0 0
\(55\) −4.83772 −0.652318
\(56\) −3.16228 −0.422577
\(57\) 0 0
\(58\) −1.83772 −0.241305
\(59\) 2.32456 0.302631 0.151316 0.988485i \(-0.451649\pi\)
0.151316 + 0.988485i \(0.451649\pi\)
\(60\) 0 0
\(61\) 0.162278 0.0207775 0.0103888 0.999946i \(-0.496693\pi\)
0.0103888 + 0.999946i \(0.496693\pi\)
\(62\) 6.32456 0.803219
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.83772 −0.346683 −0.173341 0.984862i \(-0.555456\pi\)
−0.173341 + 0.984862i \(0.555456\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 13.1623 1.57319
\(71\) 7.16228 0.850006 0.425003 0.905192i \(-0.360273\pi\)
0.425003 + 0.905192i \(0.360273\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) −3.83772 −0.446126
\(75\) 0 0
\(76\) −5.16228 −0.592154
\(77\) 3.67544 0.418856
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −4.16228 −0.465357
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) 3.48683 0.382730 0.191365 0.981519i \(-0.438709\pi\)
0.191365 + 0.981519i \(0.438709\pi\)
\(84\) 0 0
\(85\) −12.4868 −1.35439
\(86\) −9.16228 −0.987994
\(87\) 0 0
\(88\) −1.16228 −0.123899
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.16228 −0.746719
\(93\) 0 0
\(94\) 4.83772 0.498973
\(95\) 21.4868 2.20450
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 12.3246 1.23246
\(101\) −7.83772 −0.779883 −0.389941 0.920840i \(-0.627505\pi\)
−0.389941 + 0.920840i \(0.627505\pi\)
\(102\) 0 0
\(103\) −15.8114 −1.55794 −0.778971 0.627060i \(-0.784258\pi\)
−0.778971 + 0.627060i \(0.784258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.4868 −1.21283
\(107\) 17.8114 1.72189 0.860946 0.508696i \(-0.169872\pi\)
0.860946 + 0.508696i \(0.169872\pi\)
\(108\) 0 0
\(109\) 6.64911 0.636869 0.318435 0.947945i \(-0.396843\pi\)
0.318435 + 0.947945i \(0.396843\pi\)
\(110\) 4.83772 0.461259
\(111\) 0 0
\(112\) 3.16228 0.298807
\(113\) −6.67544 −0.627973 −0.313987 0.949427i \(-0.601665\pi\)
−0.313987 + 0.949427i \(0.601665\pi\)
\(114\) 0 0
\(115\) 29.8114 2.77993
\(116\) 1.83772 0.170628
\(117\) 0 0
\(118\) −2.32456 −0.213993
\(119\) 9.48683 0.869657
\(120\) 0 0
\(121\) −9.64911 −0.877192
\(122\) −0.162278 −0.0146919
\(123\) 0 0
\(124\) −6.32456 −0.567962
\(125\) −30.4868 −2.72683
\(126\) 0 0
\(127\) −18.3246 −1.62604 −0.813021 0.582235i \(-0.802178\pi\)
−0.813021 + 0.582235i \(0.802178\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −16.6491 −1.45464 −0.727320 0.686299i \(-0.759234\pi\)
−0.727320 + 0.686299i \(0.759234\pi\)
\(132\) 0 0
\(133\) −16.3246 −1.41552
\(134\) 2.83772 0.245142
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −6.32456 −0.536442 −0.268221 0.963357i \(-0.586436\pi\)
−0.268221 + 0.963357i \(0.586436\pi\)
\(140\) −13.1623 −1.11242
\(141\) 0 0
\(142\) −7.16228 −0.601045
\(143\) 0 0
\(144\) 0 0
\(145\) −7.64911 −0.635224
\(146\) 1.00000 0.0827606
\(147\) 0 0
\(148\) 3.83772 0.315459
\(149\) −0.486833 −0.0398829 −0.0199415 0.999801i \(-0.506348\pi\)
−0.0199415 + 0.999801i \(0.506348\pi\)
\(150\) 0 0
\(151\) 0.837722 0.0681729 0.0340864 0.999419i \(-0.489148\pi\)
0.0340864 + 0.999419i \(0.489148\pi\)
\(152\) 5.16228 0.418716
\(153\) 0 0
\(154\) −3.67544 −0.296176
\(155\) 26.3246 2.11444
\(156\) 0 0
\(157\) −10.4868 −0.836940 −0.418470 0.908231i \(-0.637434\pi\)
−0.418470 + 0.908231i \(0.637434\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 4.16228 0.329057
\(161\) −22.6491 −1.78500
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −3.48683 −0.270631
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 12.4868 0.957696
\(171\) 0 0
\(172\) 9.16228 0.698617
\(173\) −22.6491 −1.72198 −0.860990 0.508622i \(-0.830155\pi\)
−0.860990 + 0.508622i \(0.830155\pi\)
\(174\) 0 0
\(175\) 38.9737 2.94613
\(176\) 1.16228 0.0876100
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) −15.4868 −1.15754 −0.578770 0.815491i \(-0.696467\pi\)
−0.578770 + 0.815491i \(0.696467\pi\)
\(180\) 0 0
\(181\) 3.83772 0.285256 0.142628 0.989776i \(-0.454445\pi\)
0.142628 + 0.989776i \(0.454445\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 7.16228 0.528010
\(185\) −15.9737 −1.17441
\(186\) 0 0
\(187\) 3.48683 0.254982
\(188\) −4.83772 −0.352827
\(189\) 0 0
\(190\) −21.4868 −1.55882
\(191\) −14.3246 −1.03649 −0.518244 0.855233i \(-0.673414\pi\)
−0.518244 + 0.855233i \(0.673414\pi\)
\(192\) 0 0
\(193\) −19.9737 −1.43774 −0.718868 0.695147i \(-0.755339\pi\)
−0.718868 + 0.695147i \(0.755339\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −18.9737 −1.35182 −0.675909 0.736985i \(-0.736249\pi\)
−0.675909 + 0.736985i \(0.736249\pi\)
\(198\) 0 0
\(199\) −6.51317 −0.461706 −0.230853 0.972989i \(-0.574152\pi\)
−0.230853 + 0.972989i \(0.574152\pi\)
\(200\) −12.3246 −0.871478
\(201\) 0 0
\(202\) 7.83772 0.551460
\(203\) 5.81139 0.407879
\(204\) 0 0
\(205\) 12.4868 0.872118
\(206\) 15.8114 1.10163
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 12.4868 0.857599
\(213\) 0 0
\(214\) −17.8114 −1.21756
\(215\) −38.1359 −2.60085
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) −6.64911 −0.450335
\(219\) 0 0
\(220\) −4.83772 −0.326159
\(221\) 0 0
\(222\) 0 0
\(223\) −1.67544 −0.112196 −0.0560980 0.998425i \(-0.517866\pi\)
−0.0560980 + 0.998425i \(0.517866\pi\)
\(224\) −3.16228 −0.211289
\(225\) 0 0
\(226\) 6.67544 0.444044
\(227\) 15.4868 1.02790 0.513949 0.857821i \(-0.328182\pi\)
0.513949 + 0.857821i \(0.328182\pi\)
\(228\) 0 0
\(229\) 22.3246 1.47525 0.737624 0.675212i \(-0.235948\pi\)
0.737624 + 0.675212i \(0.235948\pi\)
\(230\) −29.8114 −1.96570
\(231\) 0 0
\(232\) −1.83772 −0.120652
\(233\) 16.6491 1.09072 0.545360 0.838202i \(-0.316393\pi\)
0.545360 + 0.838202i \(0.316393\pi\)
\(234\) 0 0
\(235\) 20.1359 1.31352
\(236\) 2.32456 0.151316
\(237\) 0 0
\(238\) −9.48683 −0.614940
\(239\) 21.4868 1.38987 0.694934 0.719074i \(-0.255434\pi\)
0.694934 + 0.719074i \(0.255434\pi\)
\(240\) 0 0
\(241\) 13.3246 0.858310 0.429155 0.903231i \(-0.358811\pi\)
0.429155 + 0.903231i \(0.358811\pi\)
\(242\) 9.64911 0.620268
\(243\) 0 0
\(244\) 0.162278 0.0103888
\(245\) −12.4868 −0.797754
\(246\) 0 0
\(247\) 0 0
\(248\) 6.32456 0.401610
\(249\) 0 0
\(250\) 30.4868 1.92816
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −8.32456 −0.523360
\(254\) 18.3246 1.14978
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.9737 1.37068 0.685340 0.728223i \(-0.259654\pi\)
0.685340 + 0.728223i \(0.259654\pi\)
\(258\) 0 0
\(259\) 12.1359 0.754091
\(260\) 0 0
\(261\) 0 0
\(262\) 16.6491 1.02859
\(263\) 2.51317 0.154969 0.0774843 0.996994i \(-0.475311\pi\)
0.0774843 + 0.996994i \(0.475311\pi\)
\(264\) 0 0
\(265\) −51.9737 −3.19272
\(266\) 16.3246 1.00092
\(267\) 0 0
\(268\) −2.83772 −0.173341
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −6.32456 −0.384189 −0.192095 0.981376i \(-0.561528\pi\)
−0.192095 + 0.981376i \(0.561528\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) 14.3246 0.863803
\(276\) 0 0
\(277\) 10.8114 0.649593 0.324797 0.945784i \(-0.394704\pi\)
0.324797 + 0.945784i \(0.394704\pi\)
\(278\) 6.32456 0.379322
\(279\) 0 0
\(280\) 13.1623 0.786597
\(281\) 0.675445 0.0402937 0.0201468 0.999797i \(-0.493587\pi\)
0.0201468 + 0.999797i \(0.493587\pi\)
\(282\) 0 0
\(283\) −2.83772 −0.168685 −0.0843425 0.996437i \(-0.526879\pi\)
−0.0843425 + 0.996437i \(0.526879\pi\)
\(284\) 7.16228 0.425003
\(285\) 0 0
\(286\) 0 0
\(287\) −9.48683 −0.559990
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 7.64911 0.449171
\(291\) 0 0
\(292\) −1.00000 −0.0585206
\(293\) −1.83772 −0.107361 −0.0536804 0.998558i \(-0.517095\pi\)
−0.0536804 + 0.998558i \(0.517095\pi\)
\(294\) 0 0
\(295\) −9.67544 −0.563326
\(296\) −3.83772 −0.223063
\(297\) 0 0
\(298\) 0.486833 0.0282015
\(299\) 0 0
\(300\) 0 0
\(301\) 28.9737 1.67001
\(302\) −0.837722 −0.0482055
\(303\) 0 0
\(304\) −5.16228 −0.296077
\(305\) −0.675445 −0.0386758
\(306\) 0 0
\(307\) 11.4868 0.655588 0.327794 0.944749i \(-0.393695\pi\)
0.327794 + 0.944749i \(0.393695\pi\)
\(308\) 3.67544 0.209428
\(309\) 0 0
\(310\) −26.3246 −1.49513
\(311\) −21.4868 −1.21841 −0.609203 0.793014i \(-0.708511\pi\)
−0.609203 + 0.793014i \(0.708511\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 10.4868 0.591806
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −12.4868 −0.701330 −0.350665 0.936501i \(-0.614044\pi\)
−0.350665 + 0.936501i \(0.614044\pi\)
\(318\) 0 0
\(319\) 2.13594 0.119590
\(320\) −4.16228 −0.232678
\(321\) 0 0
\(322\) 22.6491 1.26219
\(323\) −15.4868 −0.861710
\(324\) 0 0
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) −15.2982 −0.843418
\(330\) 0 0
\(331\) −10.9737 −0.603167 −0.301584 0.953440i \(-0.597515\pi\)
−0.301584 + 0.953440i \(0.597515\pi\)
\(332\) 3.48683 0.191365
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 11.8114 0.645325
\(336\) 0 0
\(337\) 11.0000 0.599208 0.299604 0.954064i \(-0.403145\pi\)
0.299604 + 0.954064i \(0.403145\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −12.4868 −0.677194
\(341\) −7.35089 −0.398073
\(342\) 0 0
\(343\) −12.6491 −0.682988
\(344\) −9.16228 −0.493997
\(345\) 0 0
\(346\) 22.6491 1.21762
\(347\) 15.4868 0.831377 0.415688 0.909507i \(-0.363541\pi\)
0.415688 + 0.909507i \(0.363541\pi\)
\(348\) 0 0
\(349\) −5.35089 −0.286427 −0.143213 0.989692i \(-0.545743\pi\)
−0.143213 + 0.989692i \(0.545743\pi\)
\(350\) −38.9737 −2.08323
\(351\) 0 0
\(352\) −1.16228 −0.0619496
\(353\) −29.3246 −1.56079 −0.780394 0.625288i \(-0.784982\pi\)
−0.780394 + 0.625288i \(0.784982\pi\)
\(354\) 0 0
\(355\) −29.8114 −1.58222
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 15.4868 0.818505
\(359\) −28.4605 −1.50209 −0.751044 0.660252i \(-0.770449\pi\)
−0.751044 + 0.660252i \(0.770449\pi\)
\(360\) 0 0
\(361\) 7.64911 0.402585
\(362\) −3.83772 −0.201706
\(363\) 0 0
\(364\) 0 0
\(365\) 4.16228 0.217864
\(366\) 0 0
\(367\) −25.4868 −1.33040 −0.665201 0.746664i \(-0.731654\pi\)
−0.665201 + 0.746664i \(0.731654\pi\)
\(368\) −7.16228 −0.373360
\(369\) 0 0
\(370\) 15.9737 0.830431
\(371\) 39.4868 2.05005
\(372\) 0 0
\(373\) −16.4868 −0.853656 −0.426828 0.904333i \(-0.640369\pi\)
−0.426828 + 0.904333i \(0.640369\pi\)
\(374\) −3.48683 −0.180300
\(375\) 0 0
\(376\) 4.83772 0.249486
\(377\) 0 0
\(378\) 0 0
\(379\) 17.6754 0.907927 0.453963 0.891020i \(-0.350010\pi\)
0.453963 + 0.891020i \(0.350010\pi\)
\(380\) 21.4868 1.10225
\(381\) 0 0
\(382\) 14.3246 0.732908
\(383\) −30.9737 −1.58268 −0.791340 0.611376i \(-0.790616\pi\)
−0.791340 + 0.611376i \(0.790616\pi\)
\(384\) 0 0
\(385\) −15.2982 −0.779670
\(386\) 19.9737 1.01663
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) −12.4868 −0.633108 −0.316554 0.948575i \(-0.602526\pi\)
−0.316554 + 0.948575i \(0.602526\pi\)
\(390\) 0 0
\(391\) −21.4868 −1.08664
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 18.9737 0.955879
\(395\) 16.6491 0.837708
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 6.51317 0.326476
\(399\) 0 0
\(400\) 12.3246 0.616228
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −7.83772 −0.389941
\(405\) 0 0
\(406\) −5.81139 −0.288414
\(407\) 4.46050 0.221099
\(408\) 0 0
\(409\) 14.6754 0.725654 0.362827 0.931857i \(-0.381812\pi\)
0.362827 + 0.931857i \(0.381812\pi\)
\(410\) −12.4868 −0.616681
\(411\) 0 0
\(412\) −15.8114 −0.778971
\(413\) 7.35089 0.361714
\(414\) 0 0
\(415\) −14.5132 −0.712423
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 30.9737 1.51316 0.756581 0.653900i \(-0.226868\pi\)
0.756581 + 0.653900i \(0.226868\pi\)
\(420\) 0 0
\(421\) −23.8377 −1.16178 −0.580890 0.813982i \(-0.697295\pi\)
−0.580890 + 0.813982i \(0.697295\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −12.4868 −0.606414
\(425\) 36.9737 1.79349
\(426\) 0 0
\(427\) 0.513167 0.0248339
\(428\) 17.8114 0.860946
\(429\) 0 0
\(430\) 38.1359 1.83908
\(431\) 9.48683 0.456965 0.228482 0.973548i \(-0.426624\pi\)
0.228482 + 0.973548i \(0.426624\pi\)
\(432\) 0 0
\(433\) −9.32456 −0.448110 −0.224055 0.974577i \(-0.571929\pi\)
−0.224055 + 0.974577i \(0.571929\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) 6.64911 0.318435
\(437\) 36.9737 1.76869
\(438\) 0 0
\(439\) −25.4868 −1.21642 −0.608210 0.793776i \(-0.708112\pi\)
−0.608210 + 0.793776i \(0.708112\pi\)
\(440\) 4.83772 0.230629
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 49.9473 2.36773
\(446\) 1.67544 0.0793346
\(447\) 0 0
\(448\) 3.16228 0.149404
\(449\) −7.35089 −0.346910 −0.173455 0.984842i \(-0.555493\pi\)
−0.173455 + 0.984842i \(0.555493\pi\)
\(450\) 0 0
\(451\) −3.48683 −0.164189
\(452\) −6.67544 −0.313987
\(453\) 0 0
\(454\) −15.4868 −0.726833
\(455\) 0 0
\(456\) 0 0
\(457\) −3.32456 −0.155516 −0.0777581 0.996972i \(-0.524776\pi\)
−0.0777581 + 0.996972i \(0.524776\pi\)
\(458\) −22.3246 −1.04316
\(459\) 0 0
\(460\) 29.8114 1.38996
\(461\) −18.4868 −0.861018 −0.430509 0.902586i \(-0.641666\pi\)
−0.430509 + 0.902586i \(0.641666\pi\)
\(462\) 0 0
\(463\) 15.1623 0.704651 0.352325 0.935878i \(-0.385391\pi\)
0.352325 + 0.935878i \(0.385391\pi\)
\(464\) 1.83772 0.0853141
\(465\) 0 0
\(466\) −16.6491 −0.771255
\(467\) −6.18861 −0.286375 −0.143187 0.989696i \(-0.545735\pi\)
−0.143187 + 0.989696i \(0.545735\pi\)
\(468\) 0 0
\(469\) −8.97367 −0.414365
\(470\) −20.1359 −0.928802
\(471\) 0 0
\(472\) −2.32456 −0.106996
\(473\) 10.6491 0.489647
\(474\) 0 0
\(475\) −63.6228 −2.91921
\(476\) 9.48683 0.434828
\(477\) 0 0
\(478\) −21.4868 −0.982785
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −13.3246 −0.606917
\(483\) 0 0
\(484\) −9.64911 −0.438596
\(485\) 16.6491 0.755997
\(486\) 0 0
\(487\) −13.4868 −0.611147 −0.305573 0.952169i \(-0.598848\pi\)
−0.305573 + 0.952169i \(0.598848\pi\)
\(488\) −0.162278 −0.00734596
\(489\) 0 0
\(490\) 12.4868 0.564098
\(491\) −5.81139 −0.262264 −0.131132 0.991365i \(-0.541861\pi\)
−0.131132 + 0.991365i \(0.541861\pi\)
\(492\) 0 0
\(493\) 5.51317 0.248301
\(494\) 0 0
\(495\) 0 0
\(496\) −6.32456 −0.283981
\(497\) 22.6491 1.01595
\(498\) 0 0
\(499\) 12.6491 0.566252 0.283126 0.959083i \(-0.408629\pi\)
0.283126 + 0.959083i \(0.408629\pi\)
\(500\) −30.4868 −1.36341
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 0.188612 0.00840978 0.00420489 0.999991i \(-0.498662\pi\)
0.00420489 + 0.999991i \(0.498662\pi\)
\(504\) 0 0
\(505\) 32.6228 1.45169
\(506\) 8.32456 0.370072
\(507\) 0 0
\(508\) −18.3246 −0.813021
\(509\) 28.1623 1.24827 0.624136 0.781316i \(-0.285451\pi\)
0.624136 + 0.781316i \(0.285451\pi\)
\(510\) 0 0
\(511\) −3.16228 −0.139891
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −21.9737 −0.969217
\(515\) 65.8114 2.90000
\(516\) 0 0
\(517\) −5.62278 −0.247289
\(518\) −12.1359 −0.533223
\(519\) 0 0
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) −5.16228 −0.225731 −0.112865 0.993610i \(-0.536003\pi\)
−0.112865 + 0.993610i \(0.536003\pi\)
\(524\) −16.6491 −0.727320
\(525\) 0 0
\(526\) −2.51317 −0.109579
\(527\) −18.9737 −0.826506
\(528\) 0 0
\(529\) 28.2982 1.23036
\(530\) 51.9737 2.25759
\(531\) 0 0
\(532\) −16.3246 −0.707759
\(533\) 0 0
\(534\) 0 0
\(535\) −74.1359 −3.20518
\(536\) 2.83772 0.122571
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 3.48683 0.150189
\(540\) 0 0
\(541\) −15.5132 −0.666963 −0.333482 0.942757i \(-0.608223\pi\)
−0.333482 + 0.942757i \(0.608223\pi\)
\(542\) 6.32456 0.271663
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) −27.6754 −1.18549
\(546\) 0 0
\(547\) −33.8114 −1.44567 −0.722835 0.691020i \(-0.757161\pi\)
−0.722835 + 0.691020i \(0.757161\pi\)
\(548\) 3.00000 0.128154
\(549\) 0 0
\(550\) −14.3246 −0.610801
\(551\) −9.48683 −0.404153
\(552\) 0 0
\(553\) −12.6491 −0.537895
\(554\) −10.8114 −0.459332
\(555\) 0 0
\(556\) −6.32456 −0.268221
\(557\) 18.4868 0.783312 0.391656 0.920112i \(-0.371902\pi\)
0.391656 + 0.920112i \(0.371902\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −13.1623 −0.556208
\(561\) 0 0
\(562\) −0.675445 −0.0284919
\(563\) −40.6491 −1.71316 −0.856578 0.516018i \(-0.827414\pi\)
−0.856578 + 0.516018i \(0.827414\pi\)
\(564\) 0 0
\(565\) 27.7851 1.16893
\(566\) 2.83772 0.119278
\(567\) 0 0
\(568\) −7.16228 −0.300522
\(569\) −27.2982 −1.14440 −0.572200 0.820114i \(-0.693910\pi\)
−0.572200 + 0.820114i \(0.693910\pi\)
\(570\) 0 0
\(571\) −36.1359 −1.51224 −0.756121 0.654432i \(-0.772908\pi\)
−0.756121 + 0.654432i \(0.772908\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.48683 0.395973
\(575\) −88.2719 −3.68119
\(576\) 0 0
\(577\) 0.350889 0.0146077 0.00730386 0.999973i \(-0.497675\pi\)
0.00730386 + 0.999973i \(0.497675\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) −7.64911 −0.317612
\(581\) 11.0263 0.457449
\(582\) 0 0
\(583\) 14.5132 0.601074
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 1.83772 0.0759156
\(587\) −28.6491 −1.18248 −0.591238 0.806497i \(-0.701360\pi\)
−0.591238 + 0.806497i \(0.701360\pi\)
\(588\) 0 0
\(589\) 32.6491 1.34528
\(590\) 9.67544 0.398332
\(591\) 0 0
\(592\) 3.83772 0.157729
\(593\) −0.675445 −0.0277372 −0.0138686 0.999904i \(-0.504415\pi\)
−0.0138686 + 0.999904i \(0.504415\pi\)
\(594\) 0 0
\(595\) −39.4868 −1.61880
\(596\) −0.486833 −0.0199415
\(597\) 0 0
\(598\) 0 0
\(599\) 23.6228 0.965200 0.482600 0.875841i \(-0.339692\pi\)
0.482600 + 0.875841i \(0.339692\pi\)
\(600\) 0 0
\(601\) −34.2982 −1.39905 −0.699527 0.714606i \(-0.746606\pi\)
−0.699527 + 0.714606i \(0.746606\pi\)
\(602\) −28.9737 −1.18088
\(603\) 0 0
\(604\) 0.837722 0.0340864
\(605\) 40.1623 1.63283
\(606\) 0 0
\(607\) 17.2982 0.702113 0.351057 0.936354i \(-0.385822\pi\)
0.351057 + 0.936354i \(0.385822\pi\)
\(608\) 5.16228 0.209358
\(609\) 0 0
\(610\) 0.675445 0.0273480
\(611\) 0 0
\(612\) 0 0
\(613\) 20.4868 0.827455 0.413728 0.910401i \(-0.364227\pi\)
0.413728 + 0.910401i \(0.364227\pi\)
\(614\) −11.4868 −0.463571
\(615\) 0 0
\(616\) −3.67544 −0.148088
\(617\) 26.6228 1.07179 0.535896 0.844284i \(-0.319974\pi\)
0.535896 + 0.844284i \(0.319974\pi\)
\(618\) 0 0
\(619\) 24.6491 0.990731 0.495366 0.868685i \(-0.335034\pi\)
0.495366 + 0.868685i \(0.335034\pi\)
\(620\) 26.3246 1.05722
\(621\) 0 0
\(622\) 21.4868 0.861544
\(623\) −37.9473 −1.52033
\(624\) 0 0
\(625\) 65.2719 2.61088
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) −10.4868 −0.418470
\(629\) 11.5132 0.459060
\(630\) 0 0
\(631\) −25.2982 −1.00711 −0.503553 0.863964i \(-0.667974\pi\)
−0.503553 + 0.863964i \(0.667974\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 12.4868 0.495915
\(635\) 76.2719 3.02676
\(636\) 0 0
\(637\) 0 0
\(638\) −2.13594 −0.0845628
\(639\) 0 0
\(640\) 4.16228 0.164528
\(641\) −16.3509 −0.645821 −0.322911 0.946429i \(-0.604661\pi\)
−0.322911 + 0.946429i \(0.604661\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −22.6491 −0.892500
\(645\) 0 0
\(646\) 15.4868 0.609321
\(647\) 40.6491 1.59808 0.799041 0.601277i \(-0.205341\pi\)
0.799041 + 0.601277i \(0.205341\pi\)
\(648\) 0 0
\(649\) 2.70178 0.106054
\(650\) 0 0
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 69.2982 2.70771
\(656\) −3.00000 −0.117130
\(657\) 0 0
\(658\) 15.2982 0.596387
\(659\) 5.02633 0.195798 0.0978991 0.995196i \(-0.468788\pi\)
0.0978991 + 0.995196i \(0.468788\pi\)
\(660\) 0 0
\(661\) 26.4868 1.03022 0.515109 0.857125i \(-0.327751\pi\)
0.515109 + 0.857125i \(0.327751\pi\)
\(662\) 10.9737 0.426504
\(663\) 0 0
\(664\) −3.48683 −0.135315
\(665\) 67.9473 2.63488
\(666\) 0 0
\(667\) −13.1623 −0.509645
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) −11.8114 −0.456314
\(671\) 0.188612 0.00728127
\(672\) 0 0
\(673\) 14.6754 0.565697 0.282848 0.959165i \(-0.408721\pi\)
0.282848 + 0.959165i \(0.408721\pi\)
\(674\) −11.0000 −0.423704
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −12.6491 −0.485428
\(680\) 12.4868 0.478848
\(681\) 0 0
\(682\) 7.35089 0.281480
\(683\) 35.6228 1.36307 0.681534 0.731787i \(-0.261313\pi\)
0.681534 + 0.731787i \(0.261313\pi\)
\(684\) 0 0
\(685\) −12.4868 −0.477097
\(686\) 12.6491 0.482945
\(687\) 0 0
\(688\) 9.16228 0.349309
\(689\) 0 0
\(690\) 0 0
\(691\) 9.16228 0.348549 0.174275 0.984697i \(-0.444242\pi\)
0.174275 + 0.984697i \(0.444242\pi\)
\(692\) −22.6491 −0.860990
\(693\) 0 0
\(694\) −15.4868 −0.587872
\(695\) 26.3246 0.998547
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) 5.35089 0.202534
\(699\) 0 0
\(700\) 38.9737 1.47307
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −19.8114 −0.747201
\(704\) 1.16228 0.0438050
\(705\) 0 0
\(706\) 29.3246 1.10364
\(707\) −24.7851 −0.932138
\(708\) 0 0
\(709\) 45.4605 1.70730 0.853652 0.520843i \(-0.174382\pi\)
0.853652 + 0.520843i \(0.174382\pi\)
\(710\) 29.8114 1.11880
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 45.2982 1.69643
\(714\) 0 0
\(715\) 0 0
\(716\) −15.4868 −0.578770
\(717\) 0 0
\(718\) 28.4605 1.06214
\(719\) 38.3246 1.42926 0.714632 0.699500i \(-0.246594\pi\)
0.714632 + 0.699500i \(0.246594\pi\)
\(720\) 0 0
\(721\) −50.0000 −1.86210
\(722\) −7.64911 −0.284670
\(723\) 0 0
\(724\) 3.83772 0.142628
\(725\) 22.6491 0.841167
\(726\) 0 0
\(727\) −8.83772 −0.327773 −0.163886 0.986479i \(-0.552403\pi\)
−0.163886 + 0.986479i \(0.552403\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.16228 −0.154053
\(731\) 27.4868 1.01664
\(732\) 0 0
\(733\) 21.4605 0.792662 0.396331 0.918108i \(-0.370283\pi\)
0.396331 + 0.918108i \(0.370283\pi\)
\(734\) 25.4868 0.940736
\(735\) 0 0
\(736\) 7.16228 0.264005
\(737\) −3.29822 −0.121492
\(738\) 0 0
\(739\) −39.6228 −1.45755 −0.728774 0.684755i \(-0.759909\pi\)
−0.728774 + 0.684755i \(0.759909\pi\)
\(740\) −15.9737 −0.587204
\(741\) 0 0
\(742\) −39.4868 −1.44961
\(743\) −2.32456 −0.0852797 −0.0426398 0.999091i \(-0.513577\pi\)
−0.0426398 + 0.999091i \(0.513577\pi\)
\(744\) 0 0
\(745\) 2.02633 0.0742391
\(746\) 16.4868 0.603626
\(747\) 0 0
\(748\) 3.48683 0.127491
\(749\) 56.3246 2.05805
\(750\) 0 0
\(751\) 38.7851 1.41529 0.707643 0.706570i \(-0.249758\pi\)
0.707643 + 0.706570i \(0.249758\pi\)
\(752\) −4.83772 −0.176414
\(753\) 0 0
\(754\) 0 0
\(755\) −3.48683 −0.126899
\(756\) 0 0
\(757\) −26.6491 −0.968578 −0.484289 0.874908i \(-0.660922\pi\)
−0.484289 + 0.874908i \(0.660922\pi\)
\(758\) −17.6754 −0.642001
\(759\) 0 0
\(760\) −21.4868 −0.779409
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 21.0263 0.761204
\(764\) −14.3246 −0.518244
\(765\) 0 0
\(766\) 30.9737 1.11912
\(767\) 0 0
\(768\) 0 0
\(769\) 3.35089 0.120836 0.0604181 0.998173i \(-0.480757\pi\)
0.0604181 + 0.998173i \(0.480757\pi\)
\(770\) 15.2982 0.551310
\(771\) 0 0
\(772\) −19.9737 −0.718868
\(773\) −22.6491 −0.814632 −0.407316 0.913287i \(-0.633535\pi\)
−0.407316 + 0.913287i \(0.633535\pi\)
\(774\) 0 0
\(775\) −77.9473 −2.79995
\(776\) 4.00000 0.143592
\(777\) 0 0
\(778\) 12.4868 0.447675
\(779\) 15.4868 0.554873
\(780\) 0 0
\(781\) 8.32456 0.297876
\(782\) 21.4868 0.768368
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 43.6491 1.55790
\(786\) 0 0
\(787\) −9.02633 −0.321754 −0.160877 0.986974i \(-0.551432\pi\)
−0.160877 + 0.986974i \(0.551432\pi\)
\(788\) −18.9737 −0.675909
\(789\) 0 0
\(790\) −16.6491 −0.592349
\(791\) −21.1096 −0.750571
\(792\) 0 0
\(793\) 0 0
\(794\) −26.0000 −0.922705
\(795\) 0 0
\(796\) −6.51317 −0.230853
\(797\) 18.9737 0.672082 0.336041 0.941847i \(-0.390912\pi\)
0.336041 + 0.941847i \(0.390912\pi\)
\(798\) 0 0
\(799\) −14.5132 −0.513439
\(800\) −12.3246 −0.435739
\(801\) 0 0
\(802\) −15.0000 −0.529668
\(803\) −1.16228 −0.0410159
\(804\) 0 0
\(805\) 94.2719 3.32265
\(806\) 0 0
\(807\) 0 0
\(808\) 7.83772 0.275730
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 0 0
\(811\) 1.02633 0.0360395 0.0180197 0.999838i \(-0.494264\pi\)
0.0180197 + 0.999838i \(0.494264\pi\)
\(812\) 5.81139 0.203940
\(813\) 0 0
\(814\) −4.46050 −0.156340
\(815\) 66.5964 2.33277
\(816\) 0 0
\(817\) −47.2982 −1.65476
\(818\) −14.6754 −0.513115
\(819\) 0 0
\(820\) 12.4868 0.436059
\(821\) −10.6491 −0.371657 −0.185828 0.982582i \(-0.559497\pi\)
−0.185828 + 0.982582i \(0.559497\pi\)
\(822\) 0 0
\(823\) 53.2982 1.85786 0.928930 0.370256i \(-0.120730\pi\)
0.928930 + 0.370256i \(0.120730\pi\)
\(824\) 15.8114 0.550816
\(825\) 0 0
\(826\) −7.35089 −0.255770
\(827\) 6.97367 0.242498 0.121249 0.992622i \(-0.461310\pi\)
0.121249 + 0.992622i \(0.461310\pi\)
\(828\) 0 0
\(829\) 12.1623 0.422413 0.211207 0.977441i \(-0.432261\pi\)
0.211207 + 0.977441i \(0.432261\pi\)
\(830\) 14.5132 0.503759
\(831\) 0 0
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) 0 0
\(835\) 49.9473 1.72850
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −30.9737 −1.06997
\(839\) −4.64911 −0.160505 −0.0802526 0.996775i \(-0.525573\pi\)
−0.0802526 + 0.996775i \(0.525573\pi\)
\(840\) 0 0
\(841\) −25.6228 −0.883544
\(842\) 23.8377 0.821502
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) −30.5132 −1.04844
\(848\) 12.4868 0.428800
\(849\) 0 0
\(850\) −36.9737 −1.26819
\(851\) −27.4868 −0.942236
\(852\) 0 0
\(853\) 55.1359 1.88782 0.943909 0.330205i \(-0.107118\pi\)
0.943909 + 0.330205i \(0.107118\pi\)
\(854\) −0.513167 −0.0175602
\(855\) 0 0
\(856\) −17.8114 −0.608781
\(857\) 25.6491 0.876157 0.438078 0.898937i \(-0.355659\pi\)
0.438078 + 0.898937i \(0.355659\pi\)
\(858\) 0 0
\(859\) 28.5132 0.972857 0.486428 0.873720i \(-0.338299\pi\)
0.486428 + 0.873720i \(0.338299\pi\)
\(860\) −38.1359 −1.30042
\(861\) 0 0
\(862\) −9.48683 −0.323123
\(863\) −47.8114 −1.62752 −0.813759 0.581202i \(-0.802583\pi\)
−0.813759 + 0.581202i \(0.802583\pi\)
\(864\) 0 0
\(865\) 94.2719 3.20534
\(866\) 9.32456 0.316861
\(867\) 0 0
\(868\) −20.0000 −0.678844
\(869\) −4.64911 −0.157710
\(870\) 0 0
\(871\) 0 0
\(872\) −6.64911 −0.225167
\(873\) 0 0
\(874\) −36.9737 −1.25065
\(875\) −96.4078 −3.25918
\(876\) 0 0
\(877\) −45.1359 −1.52413 −0.762066 0.647499i \(-0.775815\pi\)
−0.762066 + 0.647499i \(0.775815\pi\)
\(878\) 25.4868 0.860139
\(879\) 0 0
\(880\) −4.83772 −0.163080
\(881\) −9.97367 −0.336021 −0.168011 0.985785i \(-0.553734\pi\)
−0.168011 + 0.985785i \(0.553734\pi\)
\(882\) 0 0
\(883\) −11.3509 −0.381988 −0.190994 0.981591i \(-0.561171\pi\)
−0.190994 + 0.981591i \(0.561171\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −38.3246 −1.28681 −0.643406 0.765525i \(-0.722479\pi\)
−0.643406 + 0.765525i \(0.722479\pi\)
\(888\) 0 0
\(889\) −57.9473 −1.94349
\(890\) −49.9473 −1.67424
\(891\) 0 0
\(892\) −1.67544 −0.0560980
\(893\) 24.9737 0.835712
\(894\) 0 0
\(895\) 64.4605 2.15468
\(896\) −3.16228 −0.105644
\(897\) 0 0
\(898\) 7.35089 0.245302
\(899\) −11.6228 −0.387641
\(900\) 0 0
\(901\) 37.4605 1.24799
\(902\) 3.48683 0.116099
\(903\) 0 0
\(904\) 6.67544 0.222022
\(905\) −15.9737 −0.530983
\(906\) 0 0
\(907\) 36.2719 1.20439 0.602194 0.798350i \(-0.294293\pi\)
0.602194 + 0.798350i \(0.294293\pi\)
\(908\) 15.4868 0.513949
\(909\) 0 0
\(910\) 0 0
\(911\) 49.9473 1.65483 0.827414 0.561592i \(-0.189811\pi\)
0.827414 + 0.561592i \(0.189811\pi\)
\(912\) 0 0
\(913\) 4.05267 0.134124
\(914\) 3.32456 0.109967
\(915\) 0 0
\(916\) 22.3246 0.737624
\(917\) −52.6491 −1.73863
\(918\) 0 0
\(919\) 3.35089 0.110536 0.0552678 0.998472i \(-0.482399\pi\)
0.0552678 + 0.998472i \(0.482399\pi\)
\(920\) −29.8114 −0.982852
\(921\) 0 0
\(922\) 18.4868 0.608831
\(923\) 0 0
\(924\) 0 0
\(925\) 47.2982 1.55516
\(926\) −15.1623 −0.498263
\(927\) 0 0
\(928\) −1.83772 −0.0603262
\(929\) −28.9473 −0.949731 −0.474866 0.880058i \(-0.657503\pi\)
−0.474866 + 0.880058i \(0.657503\pi\)
\(930\) 0 0
\(931\) −15.4868 −0.507560
\(932\) 16.6491 0.545360
\(933\) 0 0
\(934\) 6.18861 0.202498
\(935\) −14.5132 −0.474631
\(936\) 0 0
\(937\) 14.2982 0.467103 0.233551 0.972344i \(-0.424965\pi\)
0.233551 + 0.972344i \(0.424965\pi\)
\(938\) 8.97367 0.293001
\(939\) 0 0
\(940\) 20.1359 0.656762
\(941\) 48.5964 1.58420 0.792099 0.610392i \(-0.208988\pi\)
0.792099 + 0.610392i \(0.208988\pi\)
\(942\) 0 0
\(943\) 21.4868 0.699708
\(944\) 2.32456 0.0756578
\(945\) 0 0
\(946\) −10.6491 −0.346232
\(947\) −18.9737 −0.616561 −0.308281 0.951295i \(-0.599754\pi\)
−0.308281 + 0.951295i \(0.599754\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 63.6228 2.06420
\(951\) 0 0
\(952\) −9.48683 −0.307470
\(953\) 27.2982 0.884276 0.442138 0.896947i \(-0.354220\pi\)
0.442138 + 0.896947i \(0.354220\pi\)
\(954\) 0 0
\(955\) 59.6228 1.92935
\(956\) 21.4868 0.694934
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 9.48683 0.306346
\(960\) 0 0
\(961\) 9.00000 0.290323
\(962\) 0 0
\(963\) 0 0
\(964\) 13.3246 0.429155
\(965\) 83.1359 2.67624
\(966\) 0 0
\(967\) 34.5132 1.10987 0.554934 0.831894i \(-0.312743\pi\)
0.554934 + 0.831894i \(0.312743\pi\)
\(968\) 9.64911 0.310134
\(969\) 0 0
\(970\) −16.6491 −0.534571
\(971\) 18.9737 0.608894 0.304447 0.952529i \(-0.401528\pi\)
0.304447 + 0.952529i \(0.401528\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) 13.4868 0.432146
\(975\) 0 0
\(976\) 0.162278 0.00519438
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 0 0
\(979\) −13.9473 −0.445759
\(980\) −12.4868 −0.398877
\(981\) 0 0
\(982\) 5.81139 0.185449
\(983\) −23.6228 −0.753450 −0.376725 0.926325i \(-0.622950\pi\)
−0.376725 + 0.926325i \(0.622950\pi\)
\(984\) 0 0
\(985\) 78.9737 2.51631
\(986\) −5.51317 −0.175575
\(987\) 0 0
\(988\) 0 0
\(989\) −65.6228 −2.08668
\(990\) 0 0
\(991\) 26.7851 0.850855 0.425428 0.904992i \(-0.360124\pi\)
0.425428 + 0.904992i \(0.360124\pi\)
\(992\) 6.32456 0.200805
\(993\) 0 0
\(994\) −22.6491 −0.718386
\(995\) 27.1096 0.859432
\(996\) 0 0
\(997\) −59.4605 −1.88313 −0.941566 0.336827i \(-0.890646\pi\)
−0.941566 + 0.336827i \(0.890646\pi\)
\(998\) −12.6491 −0.400401
\(999\) 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 3042.2.a.r.1.1 2
3.2 odd 2 3042.2.a.w.1.2 2
13.3 even 3 234.2.h.e.217.1 yes 4
13.5 odd 4 3042.2.b.j.1351.4 4
13.8 odd 4 3042.2.b.j.1351.1 4
13.9 even 3 234.2.h.e.55.1 yes 4
13.12 even 2 3042.2.a.x.1.2 2
39.5 even 4 3042.2.b.k.1351.1 4
39.8 even 4 3042.2.b.k.1351.4 4
39.29 odd 6 234.2.h.d.217.2 yes 4
39.35 odd 6 234.2.h.d.55.2 4
39.38 odd 2 3042.2.a.q.1.1 2
52.3 odd 6 1872.2.t.n.1153.1 4
52.35 odd 6 1872.2.t.n.289.1 4
156.35 even 6 1872.2.t.p.289.2 4
156.107 even 6 1872.2.t.p.1153.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.h.d.55.2 4 39.35 odd 6
234.2.h.d.217.2 yes 4 39.29 odd 6
234.2.h.e.55.1 yes 4 13.9 even 3
234.2.h.e.217.1 yes 4 13.3 even 3
1872.2.t.n.289.1 4 52.35 odd 6
1872.2.t.n.1153.1 4 52.3 odd 6
1872.2.t.p.289.2 4 156.35 even 6
1872.2.t.p.1153.2 4 156.107 even 6
3042.2.a.q.1.1 2 39.38 odd 2
3042.2.a.r.1.1 2 1.1 even 1 trivial
3042.2.a.w.1.2 2 3.2 odd 2
3042.2.a.x.1.2 2 13.12 even 2
3042.2.b.j.1351.1 4 13.8 odd 4
3042.2.b.j.1351.4 4 13.5 odd 4
3042.2.b.k.1351.1 4 39.5 even 4
3042.2.b.k.1351.4 4 39.8 even 4