Properties

Label 1305.2.a.j.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q-2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -2.23607 q^{8} -2.23607 q^{10} +2.00000 q^{11} +2.00000 q^{13} -4.47214 q^{14} -1.00000 q^{16} +4.47214 q^{17} +2.00000 q^{19} +3.00000 q^{20} -4.47214 q^{22} +2.00000 q^{23} +1.00000 q^{25} -4.47214 q^{26} +6.00000 q^{28} +1.00000 q^{29} -2.00000 q^{31} +6.70820 q^{32} -10.0000 q^{34} +2.00000 q^{35} -0.472136 q^{37} -4.47214 q^{38} -2.23607 q^{40} -2.00000 q^{41} +4.00000 q^{43} +6.00000 q^{44} -4.47214 q^{46} +4.94427 q^{47} -3.00000 q^{49} -2.23607 q^{50} +6.00000 q^{52} -2.00000 q^{53} +2.00000 q^{55} -4.47214 q^{56} -2.23607 q^{58} -8.00000 q^{59} -10.9443 q^{61} +4.47214 q^{62} -13.0000 q^{64} +2.00000 q^{65} -14.9443 q^{67} +13.4164 q^{68} -4.47214 q^{70} -4.00000 q^{71} +12.4721 q^{73} +1.05573 q^{74} +6.00000 q^{76} +4.00000 q^{77} +14.9443 q^{79} -1.00000 q^{80} +4.47214 q^{82} -2.94427 q^{83} +4.47214 q^{85} -8.94427 q^{86} -4.47214 q^{88} +6.00000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -11.0557 q^{94} +2.00000 q^{95} +9.41641 q^{97} +6.70820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} + 2 q^{5} + 4 q^{7} + 4 q^{11} + 4 q^{13} - 2 q^{16} + 4 q^{19} + 6 q^{20} + 4 q^{23} + 2 q^{25} + 12 q^{28} + 2 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{35} + 8 q^{37} - 4 q^{41} + 8 q^{43} + 12 q^{44} - 8 q^{47} - 6 q^{49} + 12 q^{52} - 4 q^{53} + 4 q^{55} - 16 q^{59} - 4 q^{61} - 26 q^{64} + 4 q^{65} - 12 q^{67} - 8 q^{71} + 16 q^{73} + 20 q^{74} + 12 q^{76} + 8 q^{77} + 12 q^{79} - 2 q^{80} + 12 q^{83} + 12 q^{89} + 8 q^{91} + 12 q^{92} - 40 q^{94} + 4 q^{95} - 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 0 0
\(4\) 3.00000 1.50000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −2.23607 −0.707107
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.47214 −1.19523
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) −4.47214 −0.953463
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.47214 −0.877058
\(27\) 0 0
\(28\) 6.00000 1.13389
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −0.472136 −0.0776187 −0.0388093 0.999247i \(-0.512356\pi\)
−0.0388093 + 0.999247i \(0.512356\pi\)
\(38\) −4.47214 −0.725476
\(39\) 0 0
\(40\) −2.23607 −0.353553
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −4.47214 −0.659380
\(47\) 4.94427 0.721196 0.360598 0.932721i \(-0.382573\pi\)
0.360598 + 0.932721i \(0.382573\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −2.23607 −0.316228
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −4.47214 −0.597614
\(57\) 0 0
\(58\) −2.23607 −0.293610
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.9443 −1.40127 −0.700635 0.713520i \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) 4.47214 0.567962
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −14.9443 −1.82573 −0.912867 0.408258i \(-0.866136\pi\)
−0.912867 + 0.408258i \(0.866136\pi\)
\(68\) 13.4164 1.62698
\(69\) 0 0
\(70\) −4.47214 −0.534522
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 12.4721 1.45975 0.729877 0.683579i \(-0.239578\pi\)
0.729877 + 0.683579i \(0.239578\pi\)
\(74\) 1.05573 0.122726
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 14.9443 1.68136 0.840681 0.541531i \(-0.182155\pi\)
0.840681 + 0.541531i \(0.182155\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 4.47214 0.493865
\(83\) −2.94427 −0.323176 −0.161588 0.986858i \(-0.551662\pi\)
−0.161588 + 0.986858i \(0.551662\pi\)
\(84\) 0 0
\(85\) 4.47214 0.485071
\(86\) −8.94427 −0.964486
\(87\) 0 0
\(88\) −4.47214 −0.476731
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −11.0557 −1.14031
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 9.41641 0.956091 0.478046 0.878335i \(-0.341345\pi\)
0.478046 + 0.878335i \(0.341345\pi\)
\(98\) 6.70820 0.677631
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) −14.9443 −1.48701 −0.743505 0.668730i \(-0.766838\pi\)
−0.743505 + 0.668730i \(0.766838\pi\)
\(102\) 0 0
\(103\) 2.94427 0.290108 0.145054 0.989424i \(-0.453664\pi\)
0.145054 + 0.989424i \(0.453664\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0 0
\(106\) 4.47214 0.434372
\(107\) 10.9443 1.05802 0.529011 0.848615i \(-0.322563\pi\)
0.529011 + 0.848615i \(0.322563\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −4.47214 −0.426401
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −4.47214 −0.420703 −0.210352 0.977626i \(-0.567461\pi\)
−0.210352 + 0.977626i \(0.567461\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 17.8885 1.64677
\(119\) 8.94427 0.819920
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 24.4721 2.21560
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.94427 0.793676 0.396838 0.917889i \(-0.370108\pi\)
0.396838 + 0.917889i \(0.370108\pi\)
\(128\) 15.6525 1.38350
\(129\) 0 0
\(130\) −4.47214 −0.392232
\(131\) −6.94427 −0.606724 −0.303362 0.952875i \(-0.598109\pi\)
−0.303362 + 0.952875i \(0.598109\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 33.4164 2.88674
\(135\) 0 0
\(136\) −10.0000 −0.857493
\(137\) −3.52786 −0.301406 −0.150703 0.988579i \(-0.548154\pi\)
−0.150703 + 0.988579i \(0.548154\pi\)
\(138\) 0 0
\(139\) 17.8885 1.51729 0.758643 0.651506i \(-0.225863\pi\)
0.758643 + 0.651506i \(0.225863\pi\)
\(140\) 6.00000 0.507093
\(141\) 0 0
\(142\) 8.94427 0.750587
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −27.8885 −2.30807
\(147\) 0 0
\(148\) −1.41641 −0.116428
\(149\) 19.8885 1.62933 0.814666 0.579930i \(-0.196920\pi\)
0.814666 + 0.579930i \(0.196920\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −4.47214 −0.362738
\(153\) 0 0
\(154\) −8.94427 −0.720750
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 16.4721 1.31462 0.657310 0.753620i \(-0.271694\pi\)
0.657310 + 0.753620i \(0.271694\pi\)
\(158\) −33.4164 −2.65847
\(159\) 0 0
\(160\) 6.70820 0.530330
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 6.58359 0.510986
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −10.0000 −0.766965
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) −14.9443 −1.13619 −0.568096 0.822962i \(-0.692320\pi\)
−0.568096 + 0.822962i \(0.692320\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −13.4164 −1.00560
\(179\) 21.8885 1.63603 0.818013 0.575199i \(-0.195076\pi\)
0.818013 + 0.575199i \(0.195076\pi\)
\(180\) 0 0
\(181\) 15.8885 1.18099 0.590493 0.807043i \(-0.298933\pi\)
0.590493 + 0.807043i \(0.298933\pi\)
\(182\) −8.94427 −0.662994
\(183\) 0 0
\(184\) −4.47214 −0.329690
\(185\) −0.472136 −0.0347121
\(186\) 0 0
\(187\) 8.94427 0.654070
\(188\) 14.8328 1.08179
\(189\) 0 0
\(190\) −4.47214 −0.324443
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) 8.47214 0.609838 0.304919 0.952378i \(-0.401371\pi\)
0.304919 + 0.952378i \(0.401371\pi\)
\(194\) −21.0557 −1.51171
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 2.94427 0.209771 0.104885 0.994484i \(-0.466552\pi\)
0.104885 + 0.994484i \(0.466552\pi\)
\(198\) 0 0
\(199\) −13.8885 −0.984533 −0.492266 0.870445i \(-0.663831\pi\)
−0.492266 + 0.870445i \(0.663831\pi\)
\(200\) −2.23607 −0.158114
\(201\) 0 0
\(202\) 33.4164 2.35117
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −6.58359 −0.458701
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −18.9443 −1.30418 −0.652089 0.758143i \(-0.726107\pi\)
−0.652089 + 0.758143i \(0.726107\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −24.4721 −1.67288
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −4.47214 −0.302891
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) 8.94427 0.601657
\(222\) 0 0
\(223\) −5.05573 −0.338557 −0.169278 0.985568i \(-0.554144\pi\)
−0.169278 + 0.985568i \(0.554144\pi\)
\(224\) 13.4164 0.896421
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −11.8885 −0.789070 −0.394535 0.918881i \(-0.629094\pi\)
−0.394535 + 0.918881i \(0.629094\pi\)
\(228\) 0 0
\(229\) 22.9443 1.51620 0.758100 0.652138i \(-0.226128\pi\)
0.758100 + 0.652138i \(0.226128\pi\)
\(230\) −4.47214 −0.294884
\(231\) 0 0
\(232\) −2.23607 −0.146805
\(233\) 1.05573 0.0691630 0.0345815 0.999402i \(-0.488990\pi\)
0.0345815 + 0.999402i \(0.488990\pi\)
\(234\) 0 0
\(235\) 4.94427 0.322529
\(236\) −24.0000 −1.56227
\(237\) 0 0
\(238\) −20.0000 −1.29641
\(239\) −13.8885 −0.898375 −0.449188 0.893437i \(-0.648287\pi\)
−0.449188 + 0.893437i \(0.648287\pi\)
\(240\) 0 0
\(241\) −19.8885 −1.28113 −0.640567 0.767902i \(-0.721301\pi\)
−0.640567 + 0.767902i \(0.721301\pi\)
\(242\) 15.6525 1.00618
\(243\) 0 0
\(244\) −32.8328 −2.10191
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 4.47214 0.283981
\(249\) 0 0
\(250\) −2.23607 −0.141421
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −0.944272 −0.0586742
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 15.5279 0.959315
\(263\) −22.8328 −1.40793 −0.703966 0.710234i \(-0.748589\pi\)
−0.703966 + 0.710234i \(0.748589\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) −8.94427 −0.548408
\(267\) 0 0
\(268\) −44.8328 −2.73860
\(269\) −13.0557 −0.796022 −0.398011 0.917381i \(-0.630299\pi\)
−0.398011 + 0.917381i \(0.630299\pi\)
\(270\) 0 0
\(271\) −26.9443 −1.63675 −0.818374 0.574686i \(-0.805124\pi\)
−0.818374 + 0.574686i \(0.805124\pi\)
\(272\) −4.47214 −0.271163
\(273\) 0 0
\(274\) 7.88854 0.476564
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 6.94427 0.417241 0.208620 0.977997i \(-0.433103\pi\)
0.208620 + 0.977997i \(0.433103\pi\)
\(278\) −40.0000 −2.39904
\(279\) 0 0
\(280\) −4.47214 −0.267261
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −18.9443 −1.12612 −0.563060 0.826416i \(-0.690376\pi\)
−0.563060 + 0.826416i \(0.690376\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −8.94427 −0.528886
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) −2.23607 −0.131306
\(291\) 0 0
\(292\) 37.4164 2.18963
\(293\) −24.4721 −1.42968 −0.714839 0.699289i \(-0.753500\pi\)
−0.714839 + 0.699289i \(0.753500\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 1.05573 0.0613629
\(297\) 0 0
\(298\) −44.4721 −2.57620
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 8.94427 0.514685
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) −10.9443 −0.626667
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 12.0000 0.683763
\(309\) 0 0
\(310\) 4.47214 0.254000
\(311\) 32.8328 1.86178 0.930889 0.365302i \(-0.119034\pi\)
0.930889 + 0.365302i \(0.119034\pi\)
\(312\) 0 0
\(313\) 14.9443 0.844700 0.422350 0.906433i \(-0.361205\pi\)
0.422350 + 0.906433i \(0.361205\pi\)
\(314\) −36.8328 −2.07860
\(315\) 0 0
\(316\) 44.8328 2.52204
\(317\) −17.4164 −0.978203 −0.489101 0.872227i \(-0.662675\pi\)
−0.489101 + 0.872227i \(0.662675\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −13.0000 −0.726722
\(321\) 0 0
\(322\) −8.94427 −0.498445
\(323\) 8.94427 0.497673
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −35.7771 −1.98151
\(327\) 0 0
\(328\) 4.47214 0.246932
\(329\) 9.88854 0.545173
\(330\) 0 0
\(331\) −15.8885 −0.873313 −0.436657 0.899628i \(-0.643838\pi\)
−0.436657 + 0.899628i \(0.643838\pi\)
\(332\) −8.83282 −0.484764
\(333\) 0 0
\(334\) −22.3607 −1.22352
\(335\) −14.9443 −0.816493
\(336\) 0 0
\(337\) −10.3607 −0.564382 −0.282191 0.959358i \(-0.591061\pi\)
−0.282191 + 0.959358i \(0.591061\pi\)
\(338\) 20.1246 1.09463
\(339\) 0 0
\(340\) 13.4164 0.727607
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −8.94427 −0.482243
\(345\) 0 0
\(346\) 33.4164 1.79648
\(347\) 13.0557 0.700868 0.350434 0.936587i \(-0.386034\pi\)
0.350434 + 0.936587i \(0.386034\pi\)
\(348\) 0 0
\(349\) 27.8885 1.49284 0.746420 0.665475i \(-0.231771\pi\)
0.746420 + 0.665475i \(0.231771\pi\)
\(350\) −4.47214 −0.239046
\(351\) 0 0
\(352\) 13.4164 0.715097
\(353\) −13.0557 −0.694886 −0.347443 0.937701i \(-0.612950\pi\)
−0.347443 + 0.937701i \(0.612950\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −48.9443 −2.58679
\(359\) 10.9443 0.577617 0.288808 0.957387i \(-0.406741\pi\)
0.288808 + 0.957387i \(0.406741\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −35.5279 −1.86730
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 12.4721 0.652821
\(366\) 0 0
\(367\) 24.9443 1.30208 0.651040 0.759043i \(-0.274333\pi\)
0.651040 + 0.759043i \(0.274333\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) 1.05573 0.0548847
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −15.8885 −0.822678 −0.411339 0.911483i \(-0.634939\pi\)
−0.411339 + 0.911483i \(0.634939\pi\)
\(374\) −20.0000 −1.03418
\(375\) 0 0
\(376\) −11.0557 −0.570156
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −18.9443 −0.973102 −0.486551 0.873652i \(-0.661745\pi\)
−0.486551 + 0.873652i \(0.661745\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −22.3607 −1.14407
\(383\) 34.9443 1.78557 0.892784 0.450484i \(-0.148749\pi\)
0.892784 + 0.450484i \(0.148749\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −18.9443 −0.964238
\(387\) 0 0
\(388\) 28.2492 1.43414
\(389\) −24.8328 −1.25907 −0.629537 0.776971i \(-0.716755\pi\)
−0.629537 + 0.776971i \(0.716755\pi\)
\(390\) 0 0
\(391\) 8.94427 0.452331
\(392\) 6.70820 0.338815
\(393\) 0 0
\(394\) −6.58359 −0.331677
\(395\) 14.9443 0.751928
\(396\) 0 0
\(397\) −15.8885 −0.797423 −0.398712 0.917076i \(-0.630543\pi\)
−0.398712 + 0.917076i \(0.630543\pi\)
\(398\) 31.0557 1.55668
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −11.8885 −0.593686 −0.296843 0.954926i \(-0.595934\pi\)
−0.296843 + 0.954926i \(0.595934\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −44.8328 −2.23052
\(405\) 0 0
\(406\) −4.47214 −0.221948
\(407\) −0.944272 −0.0468058
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 4.47214 0.220863
\(411\) 0 0
\(412\) 8.83282 0.435162
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) −2.94427 −0.144529
\(416\) 13.4164 0.657794
\(417\) 0 0
\(418\) −8.94427 −0.437479
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −28.8328 −1.40523 −0.702613 0.711572i \(-0.747983\pi\)
−0.702613 + 0.711572i \(0.747983\pi\)
\(422\) 42.3607 2.06209
\(423\) 0 0
\(424\) 4.47214 0.217186
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) −21.8885 −1.05926
\(428\) 32.8328 1.58703
\(429\) 0 0
\(430\) −8.94427 −0.431331
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) −4.47214 −0.214917 −0.107459 0.994210i \(-0.534271\pi\)
−0.107459 + 0.994210i \(0.534271\pi\)
\(434\) 8.94427 0.429339
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −4.47214 −0.213201
\(441\) 0 0
\(442\) −20.0000 −0.951303
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 11.3050 0.535305
\(447\) 0 0
\(448\) −26.0000 −1.22838
\(449\) 26.9443 1.27158 0.635789 0.771863i \(-0.280675\pi\)
0.635789 + 0.771863i \(0.280675\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) −13.4164 −0.631055
\(453\) 0 0
\(454\) 26.5836 1.24763
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −23.8885 −1.11746 −0.558729 0.829350i \(-0.688711\pi\)
−0.558729 + 0.829350i \(0.688711\pi\)
\(458\) −51.3050 −2.39732
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) −6.94427 −0.323427 −0.161713 0.986838i \(-0.551702\pi\)
−0.161713 + 0.986838i \(0.551702\pi\)
\(462\) 0 0
\(463\) −20.8328 −0.968183 −0.484092 0.875017i \(-0.660850\pi\)
−0.484092 + 0.875017i \(0.660850\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −2.36068 −0.109356
\(467\) 41.8885 1.93837 0.969185 0.246333i \(-0.0792256\pi\)
0.969185 + 0.246333i \(0.0792256\pi\)
\(468\) 0 0
\(469\) −29.8885 −1.38012
\(470\) −11.0557 −0.509963
\(471\) 0 0
\(472\) 17.8885 0.823387
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 26.8328 1.22988
\(477\) 0 0
\(478\) 31.0557 1.42046
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 0 0
\(481\) −0.944272 −0.0430551
\(482\) 44.4721 2.02565
\(483\) 0 0
\(484\) −21.0000 −0.954545
\(485\) 9.41641 0.427577
\(486\) 0 0
\(487\) −7.88854 −0.357464 −0.178732 0.983898i \(-0.557200\pi\)
−0.178732 + 0.983898i \(0.557200\pi\)
\(488\) 24.4721 1.10780
\(489\) 0 0
\(490\) 6.70820 0.303046
\(491\) −6.94427 −0.313391 −0.156695 0.987647i \(-0.550084\pi\)
−0.156695 + 0.987647i \(0.550084\pi\)
\(492\) 0 0
\(493\) 4.47214 0.201415
\(494\) −8.94427 −0.402422
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) −40.2492 −1.79641
\(503\) 34.8328 1.55312 0.776559 0.630044i \(-0.216963\pi\)
0.776559 + 0.630044i \(0.216963\pi\)
\(504\) 0 0
\(505\) −14.9443 −0.665011
\(506\) −8.94427 −0.397621
\(507\) 0 0
\(508\) 26.8328 1.19051
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 24.9443 1.10347
\(512\) −11.1803 −0.494106
\(513\) 0 0
\(514\) 40.2492 1.77532
\(515\) 2.94427 0.129740
\(516\) 0 0
\(517\) 9.88854 0.434898
\(518\) 2.11146 0.0927721
\(519\) 0 0
\(520\) −4.47214 −0.196116
\(521\) 27.8885 1.22182 0.610910 0.791700i \(-0.290804\pi\)
0.610910 + 0.791700i \(0.290804\pi\)
\(522\) 0 0
\(523\) −21.0557 −0.920703 −0.460351 0.887737i \(-0.652277\pi\)
−0.460351 + 0.887737i \(0.652277\pi\)
\(524\) −20.8328 −0.910086
\(525\) 0 0
\(526\) 51.0557 2.22614
\(527\) −8.94427 −0.389619
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 4.47214 0.194257
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 10.9443 0.473162
\(536\) 33.4164 1.44337
\(537\) 0 0
\(538\) 29.1935 1.25862
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 35.8885 1.54297 0.771485 0.636248i \(-0.219515\pi\)
0.771485 + 0.636248i \(0.219515\pi\)
\(542\) 60.2492 2.58793
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) −10.5836 −0.452109
\(549\) 0 0
\(550\) −4.47214 −0.190693
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 29.8885 1.27099
\(554\) −15.5279 −0.659716
\(555\) 0 0
\(556\) 53.6656 2.27593
\(557\) −0.111456 −0.00472255 −0.00236127 0.999997i \(-0.500752\pi\)
−0.00236127 + 0.999997i \(0.500752\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −22.3607 −0.943228
\(563\) −35.7771 −1.50782 −0.753912 0.656975i \(-0.771836\pi\)
−0.753912 + 0.656975i \(0.771836\pi\)
\(564\) 0 0
\(565\) −4.47214 −0.188144
\(566\) 42.3607 1.78055
\(567\) 0 0
\(568\) 8.94427 0.375293
\(569\) 17.0557 0.715013 0.357507 0.933911i \(-0.383627\pi\)
0.357507 + 0.933911i \(0.383627\pi\)
\(570\) 0 0
\(571\) −10.1115 −0.423151 −0.211576 0.977362i \(-0.567859\pi\)
−0.211576 + 0.977362i \(0.567859\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 8.94427 0.373327
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) −7.52786 −0.313389 −0.156695 0.987647i \(-0.550084\pi\)
−0.156695 + 0.987647i \(0.550084\pi\)
\(578\) −6.70820 −0.279024
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) −5.88854 −0.244298
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) −27.8885 −1.15404
\(585\) 0 0
\(586\) 54.7214 2.26052
\(587\) −35.8885 −1.48128 −0.740639 0.671903i \(-0.765477\pi\)
−0.740639 + 0.671903i \(0.765477\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 17.8885 0.736460
\(591\) 0 0
\(592\) 0.472136 0.0194047
\(593\) 9.05573 0.371874 0.185937 0.982562i \(-0.440468\pi\)
0.185937 + 0.982562i \(0.440468\pi\)
\(594\) 0 0
\(595\) 8.94427 0.366679
\(596\) 59.6656 2.44400
\(597\) 0 0
\(598\) −8.94427 −0.365758
\(599\) −35.8885 −1.46637 −0.733183 0.680031i \(-0.761966\pi\)
−0.733183 + 0.680031i \(0.761966\pi\)
\(600\) 0 0
\(601\) −1.05573 −0.0430640 −0.0215320 0.999768i \(-0.506854\pi\)
−0.0215320 + 0.999768i \(0.506854\pi\)
\(602\) −17.8885 −0.729083
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 19.0557 0.773448 0.386724 0.922195i \(-0.373607\pi\)
0.386724 + 0.922195i \(0.373607\pi\)
\(608\) 13.4164 0.544107
\(609\) 0 0
\(610\) 24.4721 0.990848
\(611\) 9.88854 0.400048
\(612\) 0 0
\(613\) 13.0557 0.527316 0.263658 0.964616i \(-0.415071\pi\)
0.263658 + 0.964616i \(0.415071\pi\)
\(614\) −17.8885 −0.721923
\(615\) 0 0
\(616\) −8.94427 −0.360375
\(617\) 24.4721 0.985211 0.492606 0.870253i \(-0.336045\pi\)
0.492606 + 0.870253i \(0.336045\pi\)
\(618\) 0 0
\(619\) 18.9443 0.761435 0.380717 0.924691i \(-0.375677\pi\)
0.380717 + 0.924691i \(0.375677\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) −73.4164 −2.94373
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −33.4164 −1.33559
\(627\) 0 0
\(628\) 49.4164 1.97193
\(629\) −2.11146 −0.0841893
\(630\) 0 0
\(631\) 23.7771 0.946551 0.473275 0.880914i \(-0.343072\pi\)
0.473275 + 0.880914i \(0.343072\pi\)
\(632\) −33.4164 −1.32923
\(633\) 0 0
\(634\) 38.9443 1.54667
\(635\) 8.94427 0.354943
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) −4.47214 −0.177054
\(639\) 0 0
\(640\) 15.6525 0.618718
\(641\) −0.111456 −0.00440225 −0.00220113 0.999998i \(-0.500701\pi\)
−0.00220113 + 0.999998i \(0.500701\pi\)
\(642\) 0 0
\(643\) 13.0557 0.514868 0.257434 0.966296i \(-0.417123\pi\)
0.257434 + 0.966296i \(0.417123\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −20.0000 −0.786889
\(647\) −26.9443 −1.05929 −0.529644 0.848220i \(-0.677675\pi\)
−0.529644 + 0.848220i \(0.677675\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) −4.47214 −0.175412
\(651\) 0 0
\(652\) 48.0000 1.87983
\(653\) −39.3050 −1.53812 −0.769061 0.639176i \(-0.779276\pi\)
−0.769061 + 0.639176i \(0.779276\pi\)
\(654\) 0 0
\(655\) −6.94427 −0.271335
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −22.1115 −0.861994
\(659\) 44.8328 1.74644 0.873219 0.487328i \(-0.162028\pi\)
0.873219 + 0.487328i \(0.162028\pi\)
\(660\) 0 0
\(661\) −19.8885 −0.773575 −0.386787 0.922169i \(-0.626415\pi\)
−0.386787 + 0.922169i \(0.626415\pi\)
\(662\) 35.5279 1.38083
\(663\) 0 0
\(664\) 6.58359 0.255493
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 30.0000 1.16073
\(669\) 0 0
\(670\) 33.4164 1.29099
\(671\) −21.8885 −0.844998
\(672\) 0 0
\(673\) 29.0557 1.12002 0.560008 0.828487i \(-0.310798\pi\)
0.560008 + 0.828487i \(0.310798\pi\)
\(674\) 23.1672 0.892367
\(675\) 0 0
\(676\) −27.0000 −1.03846
\(677\) −40.4721 −1.55547 −0.777735 0.628592i \(-0.783632\pi\)
−0.777735 + 0.628592i \(0.783632\pi\)
\(678\) 0 0
\(679\) 18.8328 0.722737
\(680\) −10.0000 −0.383482
\(681\) 0 0
\(682\) 8.94427 0.342494
\(683\) −25.7771 −0.986333 −0.493166 0.869935i \(-0.664161\pi\)
−0.493166 + 0.869935i \(0.664161\pi\)
\(684\) 0 0
\(685\) −3.52786 −0.134793
\(686\) 44.7214 1.70747
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −37.8885 −1.44135 −0.720674 0.693274i \(-0.756168\pi\)
−0.720674 + 0.693274i \(0.756168\pi\)
\(692\) −44.8328 −1.70429
\(693\) 0 0
\(694\) −29.1935 −1.10817
\(695\) 17.8885 0.678551
\(696\) 0 0
\(697\) −8.94427 −0.338788
\(698\) −62.3607 −2.36039
\(699\) 0 0
\(700\) 6.00000 0.226779
\(701\) −31.8885 −1.20441 −0.602207 0.798340i \(-0.705712\pi\)
−0.602207 + 0.798340i \(0.705712\pi\)
\(702\) 0 0
\(703\) −0.944272 −0.0356139
\(704\) −26.0000 −0.979912
\(705\) 0 0
\(706\) 29.1935 1.09871
\(707\) −29.8885 −1.12407
\(708\) 0 0
\(709\) 29.7771 1.11830 0.559151 0.829066i \(-0.311127\pi\)
0.559151 + 0.829066i \(0.311127\pi\)
\(710\) 8.94427 0.335673
\(711\) 0 0
\(712\) −13.4164 −0.502801
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 65.6656 2.45404
\(717\) 0 0
\(718\) −24.4721 −0.913292
\(719\) 31.7771 1.18509 0.592543 0.805539i \(-0.298124\pi\)
0.592543 + 0.805539i \(0.298124\pi\)
\(720\) 0 0
\(721\) 5.88854 0.219301
\(722\) 33.5410 1.24827
\(723\) 0 0
\(724\) 47.6656 1.77148
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −2.83282 −0.105063 −0.0525317 0.998619i \(-0.516729\pi\)
−0.0525317 + 0.998619i \(0.516729\pi\)
\(728\) −8.94427 −0.331497
\(729\) 0 0
\(730\) −27.8885 −1.03220
\(731\) 17.8885 0.661632
\(732\) 0 0
\(733\) −49.1935 −1.81700 −0.908502 0.417881i \(-0.862773\pi\)
−0.908502 + 0.417881i \(0.862773\pi\)
\(734\) −55.7771 −2.05877
\(735\) 0 0
\(736\) 13.4164 0.494535
\(737\) −29.8885 −1.10096
\(738\) 0 0
\(739\) 11.8885 0.437327 0.218664 0.975800i \(-0.429830\pi\)
0.218664 + 0.975800i \(0.429830\pi\)
\(740\) −1.41641 −0.0520682
\(741\) 0 0
\(742\) 8.94427 0.328355
\(743\) −16.9443 −0.621625 −0.310813 0.950471i \(-0.600601\pi\)
−0.310813 + 0.950471i \(0.600601\pi\)
\(744\) 0 0
\(745\) 19.8885 0.728660
\(746\) 35.5279 1.30077
\(747\) 0 0
\(748\) 26.8328 0.981105
\(749\) 21.8885 0.799790
\(750\) 0 0
\(751\) −6.00000 −0.218943 −0.109472 0.993990i \(-0.534916\pi\)
−0.109472 + 0.993990i \(0.534916\pi\)
\(752\) −4.94427 −0.180299
\(753\) 0 0
\(754\) −4.47214 −0.162866
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) 5.41641 0.196863 0.0984313 0.995144i \(-0.468618\pi\)
0.0984313 + 0.995144i \(0.468618\pi\)
\(758\) 42.3607 1.53861
\(759\) 0 0
\(760\) −4.47214 −0.162221
\(761\) −41.7771 −1.51442 −0.757209 0.653173i \(-0.773438\pi\)
−0.757209 + 0.653173i \(0.773438\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 30.0000 1.08536
\(765\) 0 0
\(766\) −78.1378 −2.82323
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 22.9443 0.827392 0.413696 0.910415i \(-0.364238\pi\)
0.413696 + 0.910415i \(0.364238\pi\)
\(770\) −8.94427 −0.322329
\(771\) 0 0
\(772\) 25.4164 0.914757
\(773\) −5.63932 −0.202832 −0.101416 0.994844i \(-0.532337\pi\)
−0.101416 + 0.994844i \(0.532337\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) −21.0557 −0.755857
\(777\) 0 0
\(778\) 55.5279 1.99077
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −20.0000 −0.715199
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 16.4721 0.587916
\(786\) 0 0
\(787\) 23.8885 0.851535 0.425767 0.904833i \(-0.360004\pi\)
0.425767 + 0.904833i \(0.360004\pi\)
\(788\) 8.83282 0.314656
\(789\) 0 0
\(790\) −33.4164 −1.18890
\(791\) −8.94427 −0.318022
\(792\) 0 0
\(793\) −21.8885 −0.777285
\(794\) 35.5279 1.26084
\(795\) 0 0
\(796\) −41.6656 −1.47680
\(797\) −13.4164 −0.475234 −0.237617 0.971359i \(-0.576366\pi\)
−0.237617 + 0.971359i \(0.576366\pi\)
\(798\) 0 0
\(799\) 22.1115 0.782247
\(800\) 6.70820 0.237171
\(801\) 0 0
\(802\) 26.5836 0.938699
\(803\) 24.9443 0.880264
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 8.94427 0.315049
\(807\) 0 0
\(808\) 33.4164 1.17559
\(809\) −35.8885 −1.26177 −0.630887 0.775875i \(-0.717309\pi\)
−0.630887 + 0.775875i \(0.717309\pi\)
\(810\) 0 0
\(811\) −6.11146 −0.214602 −0.107301 0.994227i \(-0.534221\pi\)
−0.107301 + 0.994227i \(0.534221\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) 2.11146 0.0740065
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 67.0820 2.34547
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −14.0000 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 0 0
\(823\) 11.0557 0.385378 0.192689 0.981260i \(-0.438279\pi\)
0.192689 + 0.981260i \(0.438279\pi\)
\(824\) −6.58359 −0.229350
\(825\) 0 0
\(826\) 35.7771 1.24484
\(827\) 5.88854 0.204765 0.102382 0.994745i \(-0.467353\pi\)
0.102382 + 0.994745i \(0.467353\pi\)
\(828\) 0 0
\(829\) −41.7771 −1.45098 −0.725489 0.688234i \(-0.758386\pi\)
−0.725489 + 0.688234i \(0.758386\pi\)
\(830\) 6.58359 0.228520
\(831\) 0 0
\(832\) −26.0000 −0.901388
\(833\) −13.4164 −0.464851
\(834\) 0 0
\(835\) 10.0000 0.346064
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) −8.94427 −0.308975
\(839\) −47.8885 −1.65330 −0.826648 0.562719i \(-0.809755\pi\)
−0.826648 + 0.562719i \(0.809755\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 64.4721 2.22186
\(843\) 0 0
\(844\) −56.8328 −1.95627
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −10.0000 −0.342997
\(851\) −0.944272 −0.0323692
\(852\) 0 0
\(853\) 36.4721 1.24878 0.624391 0.781112i \(-0.285347\pi\)
0.624391 + 0.781112i \(0.285347\pi\)
\(854\) 48.9443 1.67484
\(855\) 0 0
\(856\) −24.4721 −0.836440
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) −21.0557 −0.718412 −0.359206 0.933258i \(-0.616952\pi\)
−0.359206 + 0.933258i \(0.616952\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) −17.8885 −0.609286
\(863\) 17.0557 0.580584 0.290292 0.956938i \(-0.406248\pi\)
0.290292 + 0.956938i \(0.406248\pi\)
\(864\) 0 0
\(865\) −14.9443 −0.508120
\(866\) 10.0000 0.339814
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) 29.8885 1.01390
\(870\) 0 0
\(871\) −29.8885 −1.01273
\(872\) −4.47214 −0.151446
\(873\) 0 0
\(874\) −8.94427 −0.302545
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −26.9443 −0.909843 −0.454922 0.890531i \(-0.650333\pi\)
−0.454922 + 0.890531i \(0.650333\pi\)
\(878\) −17.8885 −0.603709
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −0.111456 −0.00375505 −0.00187753 0.999998i \(-0.500598\pi\)
−0.00187753 + 0.999998i \(0.500598\pi\)
\(882\) 0 0
\(883\) 8.11146 0.272972 0.136486 0.990642i \(-0.456419\pi\)
0.136486 + 0.990642i \(0.456419\pi\)
\(884\) 26.8328 0.902485
\(885\) 0 0
\(886\) −53.6656 −1.80293
\(887\) −0.944272 −0.0317055 −0.0158528 0.999874i \(-0.505046\pi\)
−0.0158528 + 0.999874i \(0.505046\pi\)
\(888\) 0 0
\(889\) 17.8885 0.599963
\(890\) −13.4164 −0.449719
\(891\) 0 0
\(892\) −15.1672 −0.507835
\(893\) 9.88854 0.330908
\(894\) 0 0
\(895\) 21.8885 0.731653
\(896\) 31.3050 1.04583
\(897\) 0 0
\(898\) −60.2492 −2.01054
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) −8.94427 −0.297977
\(902\) 8.94427 0.297812
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 15.8885 0.528153
\(906\) 0 0
\(907\) −39.7771 −1.32078 −0.660388 0.750924i \(-0.729608\pi\)
−0.660388 + 0.750924i \(0.729608\pi\)
\(908\) −35.6656 −1.18361
\(909\) 0 0
\(910\) −8.94427 −0.296500
\(911\) −24.8328 −0.822748 −0.411374 0.911467i \(-0.634951\pi\)
−0.411374 + 0.911467i \(0.634951\pi\)
\(912\) 0 0
\(913\) −5.88854 −0.194882
\(914\) 53.4164 1.76686
\(915\) 0 0
\(916\) 68.8328 2.27430
\(917\) −13.8885 −0.458640
\(918\) 0 0
\(919\) −29.8885 −0.985932 −0.492966 0.870049i \(-0.664087\pi\)
−0.492966 + 0.870049i \(0.664087\pi\)
\(920\) −4.47214 −0.147442
\(921\) 0 0
\(922\) 15.5279 0.511383
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −0.472136 −0.0155237
\(926\) 46.5836 1.53083
\(927\) 0 0
\(928\) 6.70820 0.220208
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 3.16718 0.103745
\(933\) 0 0
\(934\) −93.6656 −3.06483
\(935\) 8.94427 0.292509
\(936\) 0 0
\(937\) 13.7771 0.450078 0.225039 0.974350i \(-0.427749\pi\)
0.225039 + 0.974350i \(0.427749\pi\)
\(938\) 66.8328 2.18217
\(939\) 0 0
\(940\) 14.8328 0.483793
\(941\) 3.88854 0.126763 0.0633815 0.997989i \(-0.479812\pi\)
0.0633815 + 0.997989i \(0.479812\pi\)
\(942\) 0 0
\(943\) −4.00000 −0.130258
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −17.8885 −0.581607
\(947\) 40.0000 1.29983 0.649913 0.760009i \(-0.274805\pi\)
0.649913 + 0.760009i \(0.274805\pi\)
\(948\) 0 0
\(949\) 24.9443 0.809725
\(950\) −4.47214 −0.145095
\(951\) 0 0
\(952\) −20.0000 −0.648204
\(953\) 2.94427 0.0953743 0.0476872 0.998862i \(-0.484815\pi\)
0.0476872 + 0.998862i \(0.484815\pi\)
\(954\) 0 0
\(955\) 10.0000 0.323592
\(956\) −41.6656 −1.34756
\(957\) 0 0
\(958\) −58.1378 −1.87835
\(959\) −7.05573 −0.227841
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 2.11146 0.0680761
\(963\) 0 0
\(964\) −59.6656 −1.92170
\(965\) 8.47214 0.272728
\(966\) 0 0
\(967\) 40.7214 1.30951 0.654755 0.755841i \(-0.272772\pi\)
0.654755 + 0.755841i \(0.272772\pi\)
\(968\) 15.6525 0.503090
\(969\) 0 0
\(970\) −21.0557 −0.676059
\(971\) −4.83282 −0.155092 −0.0775462 0.996989i \(-0.524709\pi\)
−0.0775462 + 0.996989i \(0.524709\pi\)
\(972\) 0 0
\(973\) 35.7771 1.14696
\(974\) 17.6393 0.565200
\(975\) 0 0
\(976\) 10.9443 0.350318
\(977\) −19.8885 −0.636291 −0.318145 0.948042i \(-0.603060\pi\)
−0.318145 + 0.948042i \(0.603060\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 15.5279 0.495514
\(983\) 48.9443 1.56108 0.780540 0.625106i \(-0.214944\pi\)
0.780540 + 0.625106i \(0.214944\pi\)
\(984\) 0 0
\(985\) 2.94427 0.0938123
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 2.11146 0.0670726 0.0335363 0.999437i \(-0.489323\pi\)
0.0335363 + 0.999437i \(0.489323\pi\)
\(992\) −13.4164 −0.425971
\(993\) 0 0
\(994\) 17.8885 0.567390
\(995\) −13.8885 −0.440296
\(996\) 0 0
\(997\) −36.4721 −1.15508 −0.577542 0.816361i \(-0.695988\pi\)
−0.577542 + 0.816361i \(0.695988\pi\)
\(998\) −44.7214 −1.41563
\(999\) 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 1305.2.a.j.1.1 2
3.2 odd 2 435.2.a.g.1.2 2
5.4 even 2 6525.2.a.y.1.2 2
12.11 even 2 6960.2.a.bp.1.1 2
15.2 even 4 2175.2.c.g.349.3 4
15.8 even 4 2175.2.c.g.349.2 4
15.14 odd 2 2175.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.g.1.2 2 3.2 odd 2
1305.2.a.j.1.1 2 1.1 even 1 trivial
2175.2.a.o.1.1 2 15.14 odd 2
2175.2.c.g.349.2 4 15.8 even 4
2175.2.c.g.349.3 4 15.2 even 4
6525.2.a.y.1.2 2 5.4 even 2
6960.2.a.bp.1.1 2 12.11 even 2