Properties

Label 9450.2.a.em.1.1
Level $9450$
Weight $2$
Character 9450.1
Self dual yes
Analytic conductor $75.459$
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] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, 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("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.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: \(75.4586299101\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.16228\) of defining polynomial
Character \(\chi\) \(=\) 9450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{11} -4.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.32456 q^{17} +3.00000 q^{19} -2.00000 q^{22} +1.16228 q^{23} -4.16228 q^{26} -1.00000 q^{28} -1.83772 q^{29} -6.32456 q^{31} +1.00000 q^{32} +7.32456 q^{34} -7.48683 q^{37} +3.00000 q^{38} -4.00000 q^{41} -3.16228 q^{43} -2.00000 q^{44} +1.16228 q^{46} -10.4868 q^{47} +1.00000 q^{49} -4.16228 q^{52} -0.162278 q^{53} -1.00000 q^{56} -1.83772 q^{58} +0.324555 q^{59} -3.83772 q^{61} -6.32456 q^{62} +1.00000 q^{64} +3.48683 q^{67} +7.32456 q^{68} +10.3246 q^{71} +4.32456 q^{73} -7.48683 q^{74} +3.00000 q^{76} +2.00000 q^{77} -14.4868 q^{79} -4.00000 q^{82} +3.16228 q^{83} -3.16228 q^{86} -2.00000 q^{88} -5.00000 q^{89} +4.16228 q^{91} +1.16228 q^{92} -10.4868 q^{94} +11.4868 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 6 q^{19} - 4 q^{22} - 4 q^{23} - 2 q^{26} - 2 q^{28} - 10 q^{29} + 2 q^{32} + 2 q^{34} + 4 q^{37} + 6 q^{38} - 8 q^{41} - 4 q^{44} - 4 q^{46} - 2 q^{47} + 2 q^{49} - 2 q^{52} + 6 q^{53} - 2 q^{56} - 10 q^{58} - 12 q^{59} - 14 q^{61} + 2 q^{64} - 12 q^{67} + 2 q^{68} + 8 q^{71} - 4 q^{73} + 4 q^{74} + 6 q^{76} + 4 q^{77} - 10 q^{79} - 8 q^{82} - 4 q^{88} - 10 q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{94} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −4.16228 −1.15441 −0.577204 0.816600i \(-0.695856\pi\)
−0.577204 + 0.816600i \(0.695856\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.32456 1.77647 0.888233 0.459394i \(-0.151933\pi\)
0.888233 + 0.459394i \(0.151933\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 1.16228 0.242352 0.121176 0.992631i \(-0.461334\pi\)
0.121176 + 0.992631i \(0.461334\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.16228 −0.816290
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −1.83772 −0.341256 −0.170628 0.985335i \(-0.554580\pi\)
−0.170628 + 0.985335i \(0.554580\pi\)
\(30\) 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
\(34\) 7.32456 1.25615
\(35\) 0 0
\(36\) 0 0
\(37\) −7.48683 −1.23083 −0.615414 0.788204i \(-0.711011\pi\)
−0.615414 + 0.788204i \(0.711011\pi\)
\(38\) 3.00000 0.486664
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −3.16228 −0.482243 −0.241121 0.970495i \(-0.577515\pi\)
−0.241121 + 0.970495i \(0.577515\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.16228 0.171368
\(47\) −10.4868 −1.52966 −0.764831 0.644231i \(-0.777178\pi\)
−0.764831 + 0.644231i \(0.777178\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −4.16228 −0.577204
\(53\) −0.162278 −0.0222906 −0.0111453 0.999938i \(-0.503548\pi\)
−0.0111453 + 0.999938i \(0.503548\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −1.83772 −0.241305
\(59\) 0.324555 0.0422535 0.0211268 0.999777i \(-0.493275\pi\)
0.0211268 + 0.999777i \(0.493275\pi\)
\(60\) 0 0
\(61\) −3.83772 −0.491370 −0.245685 0.969350i \(-0.579013\pi\)
−0.245685 + 0.969350i \(0.579013\pi\)
\(62\) −6.32456 −0.803219
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.48683 0.425984 0.212992 0.977054i \(-0.431679\pi\)
0.212992 + 0.977054i \(0.431679\pi\)
\(68\) 7.32456 0.888233
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3246 1.22530 0.612650 0.790355i \(-0.290104\pi\)
0.612650 + 0.790355i \(0.290104\pi\)
\(72\) 0 0
\(73\) 4.32456 0.506151 0.253075 0.967447i \(-0.418558\pi\)
0.253075 + 0.967447i \(0.418558\pi\)
\(74\) −7.48683 −0.870327
\(75\) 0 0
\(76\) 3.00000 0.344124
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −14.4868 −1.62990 −0.814948 0.579534i \(-0.803235\pi\)
−0.814948 + 0.579534i \(0.803235\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) 3.16228 0.347105 0.173553 0.984825i \(-0.444475\pi\)
0.173553 + 0.984825i \(0.444475\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −3.16228 −0.340997
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) −5.00000 −0.529999 −0.264999 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 4.16228 0.436325
\(92\) 1.16228 0.121176
\(93\) 0 0
\(94\) −10.4868 −1.08163
\(95\) 0 0
\(96\) 0 0
\(97\) 11.4868 1.16631 0.583156 0.812360i \(-0.301818\pi\)
0.583156 + 0.812360i \(0.301818\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) −15.4868 −1.54100 −0.770499 0.637442i \(-0.779993\pi\)
−0.770499 + 0.637442i \(0.779993\pi\)
\(102\) 0 0
\(103\) 6.32456 0.623177 0.311588 0.950217i \(-0.399139\pi\)
0.311588 + 0.950217i \(0.399139\pi\)
\(104\) −4.16228 −0.408145
\(105\) 0 0
\(106\) −0.162278 −0.0157618
\(107\) 8.32456 0.804765 0.402383 0.915472i \(-0.368182\pi\)
0.402383 + 0.915472i \(0.368182\pi\)
\(108\) 0 0
\(109\) 15.4868 1.48337 0.741685 0.670749i \(-0.234027\pi\)
0.741685 + 0.670749i \(0.234027\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −14.6491 −1.37807 −0.689036 0.724727i \(-0.741966\pi\)
−0.689036 + 0.724727i \(0.741966\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.83772 −0.170628
\(117\) 0 0
\(118\) 0.324555 0.0298777
\(119\) −7.32456 −0.671441
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −3.83772 −0.347451
\(123\) 0 0
\(124\) −6.32456 −0.567962
\(125\) 0 0
\(126\) 0 0
\(127\) 4.48683 0.398142 0.199071 0.979985i \(-0.436208\pi\)
0.199071 + 0.979985i \(0.436208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 5.48683 0.479387 0.239693 0.970849i \(-0.422953\pi\)
0.239693 + 0.970849i \(0.422953\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 3.48683 0.301216
\(135\) 0 0
\(136\) 7.32456 0.628075
\(137\) −2.51317 −0.214714 −0.107357 0.994221i \(-0.534239\pi\)
−0.107357 + 0.994221i \(0.534239\pi\)
\(138\) 0 0
\(139\) 12.3246 1.04536 0.522678 0.852530i \(-0.324933\pi\)
0.522678 + 0.852530i \(0.324933\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.3246 0.866417
\(143\) 8.32456 0.696134
\(144\) 0 0
\(145\) 0 0
\(146\) 4.32456 0.357903
\(147\) 0 0
\(148\) −7.48683 −0.615414
\(149\) −18.4868 −1.51450 −0.757250 0.653125i \(-0.773458\pi\)
−0.757250 + 0.653125i \(0.773458\pi\)
\(150\) 0 0
\(151\) −8.16228 −0.664237 −0.332118 0.943238i \(-0.607763\pi\)
−0.332118 + 0.943238i \(0.607763\pi\)
\(152\) 3.00000 0.243332
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) −24.1623 −1.92836 −0.964180 0.265249i \(-0.914546\pi\)
−0.964180 + 0.265249i \(0.914546\pi\)
\(158\) −14.4868 −1.15251
\(159\) 0 0
\(160\) 0 0
\(161\) −1.16228 −0.0916003
\(162\) 0 0
\(163\) −0.324555 −0.0254211 −0.0127106 0.999919i \(-0.504046\pi\)
−0.0127106 + 0.999919i \(0.504046\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 3.16228 0.245440
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 4.32456 0.332658
\(170\) 0 0
\(171\) 0 0
\(172\) −3.16228 −0.241121
\(173\) 12.8377 0.976034 0.488017 0.872834i \(-0.337720\pi\)
0.488017 + 0.872834i \(0.337720\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −5.00000 −0.374766
\(179\) −12.6754 −0.947407 −0.473704 0.880684i \(-0.657083\pi\)
−0.473704 + 0.880684i \(0.657083\pi\)
\(180\) 0 0
\(181\) −9.83772 −0.731232 −0.365616 0.930766i \(-0.619142\pi\)
−0.365616 + 0.930766i \(0.619142\pi\)
\(182\) 4.16228 0.308529
\(183\) 0 0
\(184\) 1.16228 0.0856842
\(185\) 0 0
\(186\) 0 0
\(187\) −14.6491 −1.07125
\(188\) −10.4868 −0.764831
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 14.6754 1.05636 0.528181 0.849132i \(-0.322874\pi\)
0.528181 + 0.849132i \(0.322874\pi\)
\(194\) 11.4868 0.824707
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −12.8114 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(198\) 0 0
\(199\) −24.1359 −1.71095 −0.855476 0.517843i \(-0.826735\pi\)
−0.855476 + 0.517843i \(0.826735\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −15.4868 −1.08965
\(203\) 1.83772 0.128983
\(204\) 0 0
\(205\) 0 0
\(206\) 6.32456 0.440653
\(207\) 0 0
\(208\) −4.16228 −0.288602
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 0.324555 0.0223433 0.0111717 0.999938i \(-0.496444\pi\)
0.0111717 + 0.999938i \(0.496444\pi\)
\(212\) −0.162278 −0.0111453
\(213\) 0 0
\(214\) 8.32456 0.569055
\(215\) 0 0
\(216\) 0 0
\(217\) 6.32456 0.429339
\(218\) 15.4868 1.04890
\(219\) 0 0
\(220\) 0 0
\(221\) −30.4868 −2.05077
\(222\) 0 0
\(223\) −0.837722 −0.0560980 −0.0280490 0.999607i \(-0.508929\pi\)
−0.0280490 + 0.999607i \(0.508929\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.6491 −0.974444
\(227\) −9.81139 −0.651205 −0.325602 0.945507i \(-0.605567\pi\)
−0.325602 + 0.945507i \(0.605567\pi\)
\(228\) 0 0
\(229\) 17.8377 1.17875 0.589375 0.807860i \(-0.299374\pi\)
0.589375 + 0.807860i \(0.299374\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.83772 −0.120652
\(233\) −15.4868 −1.01458 −0.507288 0.861777i \(-0.669352\pi\)
−0.507288 + 0.861777i \(0.669352\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.324555 0.0211268
\(237\) 0 0
\(238\) −7.32456 −0.474780
\(239\) −25.8114 −1.66960 −0.834800 0.550553i \(-0.814417\pi\)
−0.834800 + 0.550553i \(0.814417\pi\)
\(240\) 0 0
\(241\) −23.4868 −1.51292 −0.756460 0.654040i \(-0.773073\pi\)
−0.756460 + 0.654040i \(0.773073\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) −3.83772 −0.245685
\(245\) 0 0
\(246\) 0 0
\(247\) −12.4868 −0.794518
\(248\) −6.32456 −0.401610
\(249\) 0 0
\(250\) 0 0
\(251\) 3.67544 0.231992 0.115996 0.993250i \(-0.462994\pi\)
0.115996 + 0.993250i \(0.462994\pi\)
\(252\) 0 0
\(253\) −2.32456 −0.146144
\(254\) 4.48683 0.281529
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −19.0000 −1.18519 −0.592594 0.805502i \(-0.701896\pi\)
−0.592594 + 0.805502i \(0.701896\pi\)
\(258\) 0 0
\(259\) 7.48683 0.465209
\(260\) 0 0
\(261\) 0 0
\(262\) 5.48683 0.338978
\(263\) 16.3246 1.00662 0.503308 0.864107i \(-0.332116\pi\)
0.503308 + 0.864107i \(0.332116\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.00000 −0.183942
\(267\) 0 0
\(268\) 3.48683 0.212992
\(269\) −22.6491 −1.38094 −0.690470 0.723361i \(-0.742596\pi\)
−0.690470 + 0.723361i \(0.742596\pi\)
\(270\) 0 0
\(271\) −2.32456 −0.141207 −0.0706033 0.997504i \(-0.522492\pi\)
−0.0706033 + 0.997504i \(0.522492\pi\)
\(272\) 7.32456 0.444116
\(273\) 0 0
\(274\) −2.51317 −0.151826
\(275\) 0 0
\(276\) 0 0
\(277\) −8.32456 −0.500174 −0.250087 0.968223i \(-0.580459\pi\)
−0.250087 + 0.968223i \(0.580459\pi\)
\(278\) 12.3246 0.739178
\(279\) 0 0
\(280\) 0 0
\(281\) −8.83772 −0.527214 −0.263607 0.964630i \(-0.584912\pi\)
−0.263607 + 0.964630i \(0.584912\pi\)
\(282\) 0 0
\(283\) 21.6491 1.28691 0.643453 0.765486i \(-0.277501\pi\)
0.643453 + 0.765486i \(0.277501\pi\)
\(284\) 10.3246 0.612650
\(285\) 0 0
\(286\) 8.32456 0.492241
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 36.6491 2.15583
\(290\) 0 0
\(291\) 0 0
\(292\) 4.32456 0.253075
\(293\) 24.1359 1.41004 0.705018 0.709189i \(-0.250939\pi\)
0.705018 + 0.709189i \(0.250939\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.48683 −0.435163
\(297\) 0 0
\(298\) −18.4868 −1.07091
\(299\) −4.83772 −0.279773
\(300\) 0 0
\(301\) 3.16228 0.182271
\(302\) −8.16228 −0.469686
\(303\) 0 0
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) 0 0
\(307\) −6.64911 −0.379485 −0.189742 0.981834i \(-0.560765\pi\)
−0.189742 + 0.981834i \(0.560765\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) −5.16228 −0.291789 −0.145895 0.989300i \(-0.546606\pi\)
−0.145895 + 0.989300i \(0.546606\pi\)
\(314\) −24.1623 −1.36356
\(315\) 0 0
\(316\) −14.4868 −0.814948
\(317\) −21.8377 −1.22653 −0.613264 0.789878i \(-0.710144\pi\)
−0.613264 + 0.789878i \(0.710144\pi\)
\(318\) 0 0
\(319\) 3.67544 0.205785
\(320\) 0 0
\(321\) 0 0
\(322\) −1.16228 −0.0647712
\(323\) 21.9737 1.22265
\(324\) 0 0
\(325\) 0 0
\(326\) −0.324555 −0.0179755
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) 10.4868 0.578158
\(330\) 0 0
\(331\) −2.18861 −0.120297 −0.0601485 0.998189i \(-0.519157\pi\)
−0.0601485 + 0.998189i \(0.519157\pi\)
\(332\) 3.16228 0.173553
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −13.3246 −0.725835 −0.362917 0.931821i \(-0.618219\pi\)
−0.362917 + 0.931821i \(0.618219\pi\)
\(338\) 4.32456 0.235225
\(339\) 0 0
\(340\) 0 0
\(341\) 12.6491 0.684988
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −3.16228 −0.170499
\(345\) 0 0
\(346\) 12.8377 0.690160
\(347\) −23.0000 −1.23470 −0.617352 0.786687i \(-0.711795\pi\)
−0.617352 + 0.786687i \(0.711795\pi\)
\(348\) 0 0
\(349\) −10.8114 −0.578720 −0.289360 0.957220i \(-0.593443\pi\)
−0.289360 + 0.957220i \(0.593443\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −5.00000 −0.264999
\(357\) 0 0
\(358\) −12.6754 −0.669918
\(359\) −22.0000 −1.16112 −0.580558 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −9.83772 −0.517059
\(363\) 0 0
\(364\) 4.16228 0.218163
\(365\) 0 0
\(366\) 0 0
\(367\) −30.4605 −1.59003 −0.795013 0.606593i \(-0.792536\pi\)
−0.795013 + 0.606593i \(0.792536\pi\)
\(368\) 1.16228 0.0605879
\(369\) 0 0
\(370\) 0 0
\(371\) 0.162278 0.00842504
\(372\) 0 0
\(373\) 4.32456 0.223917 0.111958 0.993713i \(-0.464288\pi\)
0.111958 + 0.993713i \(0.464288\pi\)
\(374\) −14.6491 −0.757487
\(375\) 0 0
\(376\) −10.4868 −0.540817
\(377\) 7.64911 0.393949
\(378\) 0 0
\(379\) −4.83772 −0.248497 −0.124249 0.992251i \(-0.539652\pi\)
−0.124249 + 0.992251i \(0.539652\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.3246 1.34512 0.672561 0.740042i \(-0.265194\pi\)
0.672561 + 0.740042i \(0.265194\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.6754 0.746960
\(387\) 0 0
\(388\) 11.4868 0.583156
\(389\) 26.4868 1.34294 0.671468 0.741034i \(-0.265664\pi\)
0.671468 + 0.741034i \(0.265664\pi\)
\(390\) 0 0
\(391\) 8.51317 0.430529
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −12.8114 −0.645428
\(395\) 0 0
\(396\) 0 0
\(397\) 18.8114 0.944117 0.472058 0.881567i \(-0.343511\pi\)
0.472058 + 0.881567i \(0.343511\pi\)
\(398\) −24.1359 −1.20983
\(399\) 0 0
\(400\) 0 0
\(401\) −9.16228 −0.457542 −0.228771 0.973480i \(-0.573471\pi\)
−0.228771 + 0.973480i \(0.573471\pi\)
\(402\) 0 0
\(403\) 26.3246 1.31132
\(404\) −15.4868 −0.770499
\(405\) 0 0
\(406\) 1.83772 0.0912046
\(407\) 14.9737 0.742217
\(408\) 0 0
\(409\) 20.1359 0.995658 0.497829 0.867275i \(-0.334131\pi\)
0.497829 + 0.867275i \(0.334131\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.32456 0.311588
\(413\) −0.324555 −0.0159703
\(414\) 0 0
\(415\) 0 0
\(416\) −4.16228 −0.204072
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 20.1359 0.983705 0.491853 0.870678i \(-0.336320\pi\)
0.491853 + 0.870678i \(0.336320\pi\)
\(420\) 0 0
\(421\) −5.35089 −0.260786 −0.130393 0.991462i \(-0.541624\pi\)
−0.130393 + 0.991462i \(0.541624\pi\)
\(422\) 0.324555 0.0157991
\(423\) 0 0
\(424\) −0.162278 −0.00788090
\(425\) 0 0
\(426\) 0 0
\(427\) 3.83772 0.185720
\(428\) 8.32456 0.402383
\(429\) 0 0
\(430\) 0 0
\(431\) −30.9737 −1.49195 −0.745974 0.665975i \(-0.768016\pi\)
−0.745974 + 0.665975i \(0.768016\pi\)
\(432\) 0 0
\(433\) −30.3246 −1.45731 −0.728653 0.684884i \(-0.759853\pi\)
−0.728653 + 0.684884i \(0.759853\pi\)
\(434\) 6.32456 0.303588
\(435\) 0 0
\(436\) 15.4868 0.741685
\(437\) 3.48683 0.166798
\(438\) 0 0
\(439\) 33.8114 1.61373 0.806865 0.590736i \(-0.201163\pi\)
0.806865 + 0.590736i \(0.201163\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −30.4868 −1.45011
\(443\) −28.6228 −1.35991 −0.679955 0.733254i \(-0.738001\pi\)
−0.679955 + 0.733254i \(0.738001\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.837722 −0.0396673
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −25.1623 −1.18748 −0.593741 0.804656i \(-0.702349\pi\)
−0.593741 + 0.804656i \(0.702349\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) −14.6491 −0.689036
\(453\) 0 0
\(454\) −9.81139 −0.460471
\(455\) 0 0
\(456\) 0 0
\(457\) 30.6228 1.43247 0.716237 0.697858i \(-0.245863\pi\)
0.716237 + 0.697858i \(0.245863\pi\)
\(458\) 17.8377 0.833502
\(459\) 0 0
\(460\) 0 0
\(461\) 5.16228 0.240431 0.120216 0.992748i \(-0.461641\pi\)
0.120216 + 0.992748i \(0.461641\pi\)
\(462\) 0 0
\(463\) 20.4868 0.952104 0.476052 0.879417i \(-0.342067\pi\)
0.476052 + 0.879417i \(0.342067\pi\)
\(464\) −1.83772 −0.0853141
\(465\) 0 0
\(466\) −15.4868 −0.717414
\(467\) 10.3246 0.477763 0.238882 0.971049i \(-0.423219\pi\)
0.238882 + 0.971049i \(0.423219\pi\)
\(468\) 0 0
\(469\) −3.48683 −0.161007
\(470\) 0 0
\(471\) 0 0
\(472\) 0.324555 0.0149389
\(473\) 6.32456 0.290803
\(474\) 0 0
\(475\) 0 0
\(476\) −7.32456 −0.335720
\(477\) 0 0
\(478\) −25.8114 −1.18059
\(479\) 4.16228 0.190179 0.0950897 0.995469i \(-0.469686\pi\)
0.0950897 + 0.995469i \(0.469686\pi\)
\(480\) 0 0
\(481\) 31.1623 1.42088
\(482\) −23.4868 −1.06980
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 26.9737 1.22229 0.611147 0.791517i \(-0.290709\pi\)
0.611147 + 0.791517i \(0.290709\pi\)
\(488\) −3.83772 −0.173726
\(489\) 0 0
\(490\) 0 0
\(491\) 17.3246 0.781846 0.390923 0.920423i \(-0.372156\pi\)
0.390923 + 0.920423i \(0.372156\pi\)
\(492\) 0 0
\(493\) −13.4605 −0.606230
\(494\) −12.4868 −0.561809
\(495\) 0 0
\(496\) −6.32456 −0.283981
\(497\) −10.3246 −0.463120
\(498\) 0 0
\(499\) 4.83772 0.216566 0.108283 0.994120i \(-0.465465\pi\)
0.108283 + 0.994120i \(0.465465\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.67544 0.164043
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.32456 −0.103339
\(507\) 0 0
\(508\) 4.48683 0.199071
\(509\) 19.4868 0.863739 0.431869 0.901936i \(-0.357854\pi\)
0.431869 + 0.901936i \(0.357854\pi\)
\(510\) 0 0
\(511\) −4.32456 −0.191307
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −19.0000 −0.838054
\(515\) 0 0
\(516\) 0 0
\(517\) 20.9737 0.922421
\(518\) 7.48683 0.328953
\(519\) 0 0
\(520\) 0 0
\(521\) 31.3246 1.37235 0.686177 0.727435i \(-0.259288\pi\)
0.686177 + 0.727435i \(0.259288\pi\)
\(522\) 0 0
\(523\) 27.3246 1.19482 0.597410 0.801936i \(-0.296197\pi\)
0.597410 + 0.801936i \(0.296197\pi\)
\(524\) 5.48683 0.239693
\(525\) 0 0
\(526\) 16.3246 0.711784
\(527\) −46.3246 −2.01793
\(528\) 0 0
\(529\) −21.6491 −0.941266
\(530\) 0 0
\(531\) 0 0
\(532\) −3.00000 −0.130066
\(533\) 16.6491 0.721153
\(534\) 0 0
\(535\) 0 0
\(536\) 3.48683 0.150608
\(537\) 0 0
\(538\) −22.6491 −0.976472
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −17.8114 −0.765771 −0.382886 0.923796i \(-0.625070\pi\)
−0.382886 + 0.923796i \(0.625070\pi\)
\(542\) −2.32456 −0.0998482
\(543\) 0 0
\(544\) 7.32456 0.314038
\(545\) 0 0
\(546\) 0 0
\(547\) 33.2982 1.42373 0.711865 0.702317i \(-0.247851\pi\)
0.711865 + 0.702317i \(0.247851\pi\)
\(548\) −2.51317 −0.107357
\(549\) 0 0
\(550\) 0 0
\(551\) −5.51317 −0.234869
\(552\) 0 0
\(553\) 14.4868 0.616043
\(554\) −8.32456 −0.353676
\(555\) 0 0
\(556\) 12.3246 0.522678
\(557\) 1.18861 0.0503631 0.0251815 0.999683i \(-0.491984\pi\)
0.0251815 + 0.999683i \(0.491984\pi\)
\(558\) 0 0
\(559\) 13.1623 0.556705
\(560\) 0 0
\(561\) 0 0
\(562\) −8.83772 −0.372797
\(563\) −14.4605 −0.609437 −0.304719 0.952442i \(-0.598562\pi\)
−0.304719 + 0.952442i \(0.598562\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 21.6491 0.909980
\(567\) 0 0
\(568\) 10.3246 0.433209
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 39.2982 1.64458 0.822290 0.569069i \(-0.192696\pi\)
0.822290 + 0.569069i \(0.192696\pi\)
\(572\) 8.32456 0.348067
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 36.6491 1.52440
\(579\) 0 0
\(580\) 0 0
\(581\) −3.16228 −0.131193
\(582\) 0 0
\(583\) 0.324555 0.0134417
\(584\) 4.32456 0.178951
\(585\) 0 0
\(586\) 24.1359 0.997047
\(587\) 18.8377 0.777516 0.388758 0.921340i \(-0.372904\pi\)
0.388758 + 0.921340i \(0.372904\pi\)
\(588\) 0 0
\(589\) −18.9737 −0.781796
\(590\) 0 0
\(591\) 0 0
\(592\) −7.48683 −0.307707
\(593\) 0.675445 0.0277372 0.0138686 0.999904i \(-0.495585\pi\)
0.0138686 + 0.999904i \(0.495585\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.4868 −0.757250
\(597\) 0 0
\(598\) −4.83772 −0.197829
\(599\) −20.1359 −0.822732 −0.411366 0.911470i \(-0.634948\pi\)
−0.411366 + 0.911470i \(0.634948\pi\)
\(600\) 0 0
\(601\) −21.6754 −0.884160 −0.442080 0.896976i \(-0.645759\pi\)
−0.442080 + 0.896976i \(0.645759\pi\)
\(602\) 3.16228 0.128885
\(603\) 0 0
\(604\) −8.16228 −0.332118
\(605\) 0 0
\(606\) 0 0
\(607\) 20.8377 0.845777 0.422889 0.906182i \(-0.361016\pi\)
0.422889 + 0.906182i \(0.361016\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 0 0
\(611\) 43.6491 1.76585
\(612\) 0 0
\(613\) 11.8114 0.477057 0.238529 0.971135i \(-0.423335\pi\)
0.238529 + 0.971135i \(0.423335\pi\)
\(614\) −6.64911 −0.268336
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) −37.4868 −1.50916 −0.754582 0.656206i \(-0.772160\pi\)
−0.754582 + 0.656206i \(0.772160\pi\)
\(618\) 0 0
\(619\) 8.97367 0.360682 0.180341 0.983604i \(-0.442280\pi\)
0.180341 + 0.983604i \(0.442280\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 5.00000 0.200321
\(624\) 0 0
\(625\) 0 0
\(626\) −5.16228 −0.206326
\(627\) 0 0
\(628\) −24.1623 −0.964180
\(629\) −54.8377 −2.18652
\(630\) 0 0
\(631\) −23.8377 −0.948965 −0.474482 0.880265i \(-0.657365\pi\)
−0.474482 + 0.880265i \(0.657365\pi\)
\(632\) −14.4868 −0.576255
\(633\) 0 0
\(634\) −21.8377 −0.867287
\(635\) 0 0
\(636\) 0 0
\(637\) −4.16228 −0.164915
\(638\) 3.67544 0.145512
\(639\) 0 0
\(640\) 0 0
\(641\) 10.3246 0.407795 0.203898 0.978992i \(-0.434639\pi\)
0.203898 + 0.978992i \(0.434639\pi\)
\(642\) 0 0
\(643\) 2.35089 0.0927100 0.0463550 0.998925i \(-0.485239\pi\)
0.0463550 + 0.998925i \(0.485239\pi\)
\(644\) −1.16228 −0.0458002
\(645\) 0 0
\(646\) 21.9737 0.864542
\(647\) −1.13594 −0.0446586 −0.0223293 0.999751i \(-0.507108\pi\)
−0.0223293 + 0.999751i \(0.507108\pi\)
\(648\) 0 0
\(649\) −0.649111 −0.0254798
\(650\) 0 0
\(651\) 0 0
\(652\) −0.324555 −0.0127106
\(653\) −21.5132 −0.841875 −0.420938 0.907090i \(-0.638299\pi\)
−0.420938 + 0.907090i \(0.638299\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) 10.4868 0.408819
\(659\) −6.35089 −0.247396 −0.123698 0.992320i \(-0.539475\pi\)
−0.123698 + 0.992320i \(0.539475\pi\)
\(660\) 0 0
\(661\) 2.64911 0.103038 0.0515192 0.998672i \(-0.483594\pi\)
0.0515192 + 0.998672i \(0.483594\pi\)
\(662\) −2.18861 −0.0850628
\(663\) 0 0
\(664\) 3.16228 0.122720
\(665\) 0 0
\(666\) 0 0
\(667\) −2.13594 −0.0827041
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) 7.67544 0.296307
\(672\) 0 0
\(673\) 34.9737 1.34814 0.674068 0.738669i \(-0.264546\pi\)
0.674068 + 0.738669i \(0.264546\pi\)
\(674\) −13.3246 −0.513243
\(675\) 0 0
\(676\) 4.32456 0.166329
\(677\) −3.67544 −0.141259 −0.0706294 0.997503i \(-0.522501\pi\)
−0.0706294 + 0.997503i \(0.522501\pi\)
\(678\) 0 0
\(679\) −11.4868 −0.440824
\(680\) 0 0
\(681\) 0 0
\(682\) 12.6491 0.484359
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −3.16228 −0.120561
\(689\) 0.675445 0.0257324
\(690\) 0 0
\(691\) −23.2982 −0.886306 −0.443153 0.896446i \(-0.646140\pi\)
−0.443153 + 0.896446i \(0.646140\pi\)
\(692\) 12.8377 0.488017
\(693\) 0 0
\(694\) −23.0000 −0.873068
\(695\) 0 0
\(696\) 0 0
\(697\) −29.2982 −1.10975
\(698\) −10.8114 −0.409217
\(699\) 0 0
\(700\) 0 0
\(701\) −16.3246 −0.616570 −0.308285 0.951294i \(-0.599755\pi\)
−0.308285 + 0.951294i \(0.599755\pi\)
\(702\) 0 0
\(703\) −22.4605 −0.847114
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 15.4868 0.582442
\(708\) 0 0
\(709\) 6.32456 0.237524 0.118762 0.992923i \(-0.462108\pi\)
0.118762 + 0.992923i \(0.462108\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −5.00000 −0.187383
\(713\) −7.35089 −0.275293
\(714\) 0 0
\(715\) 0 0
\(716\) −12.6754 −0.473704
\(717\) 0 0
\(718\) −22.0000 −0.821033
\(719\) 1.83772 0.0685355 0.0342677 0.999413i \(-0.489090\pi\)
0.0342677 + 0.999413i \(0.489090\pi\)
\(720\) 0 0
\(721\) −6.32456 −0.235539
\(722\) −10.0000 −0.372161
\(723\) 0 0
\(724\) −9.83772 −0.365616
\(725\) 0 0
\(726\) 0 0
\(727\) 25.1623 0.933217 0.466609 0.884464i \(-0.345476\pi\)
0.466609 + 0.884464i \(0.345476\pi\)
\(728\) 4.16228 0.154264
\(729\) 0 0
\(730\) 0 0
\(731\) −23.1623 −0.856688
\(732\) 0 0
\(733\) −25.1359 −0.928417 −0.464209 0.885726i \(-0.653661\pi\)
−0.464209 + 0.885726i \(0.653661\pi\)
\(734\) −30.4605 −1.12432
\(735\) 0 0
\(736\) 1.16228 0.0428421
\(737\) −6.97367 −0.256878
\(738\) 0 0
\(739\) −6.64911 −0.244591 −0.122296 0.992494i \(-0.539026\pi\)
−0.122296 + 0.992494i \(0.539026\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.162278 0.00595740
\(743\) −39.2982 −1.44171 −0.720856 0.693085i \(-0.756251\pi\)
−0.720856 + 0.693085i \(0.756251\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.32456 0.158333
\(747\) 0 0
\(748\) −14.6491 −0.535625
\(749\) −8.32456 −0.304173
\(750\) 0 0
\(751\) −29.9473 −1.09279 −0.546397 0.837526i \(-0.684001\pi\)
−0.546397 + 0.837526i \(0.684001\pi\)
\(752\) −10.4868 −0.382415
\(753\) 0 0
\(754\) 7.64911 0.278564
\(755\) 0 0
\(756\) 0 0
\(757\) −53.1096 −1.93030 −0.965151 0.261694i \(-0.915719\pi\)
−0.965151 + 0.261694i \(0.915719\pi\)
\(758\) −4.83772 −0.175714
\(759\) 0 0
\(760\) 0 0
\(761\) −40.9473 −1.48434 −0.742170 0.670212i \(-0.766203\pi\)
−0.742170 + 0.670212i \(0.766203\pi\)
\(762\) 0 0
\(763\) −15.4868 −0.560661
\(764\) 0 0
\(765\) 0 0
\(766\) 26.3246 0.951145
\(767\) −1.35089 −0.0487778
\(768\) 0 0
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.6754 0.528181
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 11.4868 0.412353
\(777\) 0 0
\(778\) 26.4868 0.949599
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −20.6491 −0.738883
\(782\) 8.51317 0.304430
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 16.9737 0.605046 0.302523 0.953142i \(-0.402171\pi\)
0.302523 + 0.953142i \(0.402171\pi\)
\(788\) −12.8114 −0.456387
\(789\) 0 0
\(790\) 0 0
\(791\) 14.6491 0.520862
\(792\) 0 0
\(793\) 15.9737 0.567242
\(794\) 18.8114 0.667591
\(795\) 0 0
\(796\) −24.1359 −0.855476
\(797\) −2.51317 −0.0890209 −0.0445105 0.999009i \(-0.514173\pi\)
−0.0445105 + 0.999009i \(0.514173\pi\)
\(798\) 0 0
\(799\) −76.8114 −2.71739
\(800\) 0 0
\(801\) 0 0
\(802\) −9.16228 −0.323531
\(803\) −8.64911 −0.305220
\(804\) 0 0
\(805\) 0 0
\(806\) 26.3246 0.927243
\(807\) 0 0
\(808\) −15.4868 −0.544825
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 16.3246 0.573233 0.286616 0.958045i \(-0.407470\pi\)
0.286616 + 0.958045i \(0.407470\pi\)
\(812\) 1.83772 0.0644914
\(813\) 0 0
\(814\) 14.9737 0.524827
\(815\) 0 0
\(816\) 0 0
\(817\) −9.48683 −0.331902
\(818\) 20.1359 0.704037
\(819\) 0 0
\(820\) 0 0
\(821\) 43.4605 1.51678 0.758391 0.651800i \(-0.225986\pi\)
0.758391 + 0.651800i \(0.225986\pi\)
\(822\) 0 0
\(823\) 0.701779 0.0244625 0.0122312 0.999925i \(-0.496107\pi\)
0.0122312 + 0.999925i \(0.496107\pi\)
\(824\) 6.32456 0.220326
\(825\) 0 0
\(826\) −0.324555 −0.0112927
\(827\) 52.6228 1.82987 0.914937 0.403598i \(-0.132240\pi\)
0.914937 + 0.403598i \(0.132240\pi\)
\(828\) 0 0
\(829\) 32.1096 1.11521 0.557606 0.830105i \(-0.311720\pi\)
0.557606 + 0.830105i \(0.311720\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.16228 −0.144301
\(833\) 7.32456 0.253781
\(834\) 0 0
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 20.1359 0.695585
\(839\) −49.7851 −1.71877 −0.859385 0.511328i \(-0.829154\pi\)
−0.859385 + 0.511328i \(0.829154\pi\)
\(840\) 0 0
\(841\) −25.6228 −0.883544
\(842\) −5.35089 −0.184404
\(843\) 0 0
\(844\) 0.324555 0.0111717
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −0.162278 −0.00557264
\(849\) 0 0
\(850\) 0 0
\(851\) −8.70178 −0.298293
\(852\) 0 0
\(853\) 43.6228 1.49362 0.746808 0.665040i \(-0.231586\pi\)
0.746808 + 0.665040i \(0.231586\pi\)
\(854\) 3.83772 0.131324
\(855\) 0 0
\(856\) 8.32456 0.284527
\(857\) 57.5964 1.96746 0.983728 0.179661i \(-0.0575002\pi\)
0.983728 + 0.179661i \(0.0575002\pi\)
\(858\) 0 0
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −30.9737 −1.05497
\(863\) −8.18861 −0.278744 −0.139372 0.990240i \(-0.544508\pi\)
−0.139372 + 0.990240i \(0.544508\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −30.3246 −1.03047
\(867\) 0 0
\(868\) 6.32456 0.214669
\(869\) 28.9737 0.982864
\(870\) 0 0
\(871\) −14.5132 −0.491760
\(872\) 15.4868 0.524450
\(873\) 0 0
\(874\) 3.48683 0.117944
\(875\) 0 0
\(876\) 0 0
\(877\) 14.9737 0.505625 0.252812 0.967515i \(-0.418644\pi\)
0.252812 + 0.967515i \(0.418644\pi\)
\(878\) 33.8114 1.14108
\(879\) 0 0
\(880\) 0 0
\(881\) 10.0263 0.337796 0.168898 0.985634i \(-0.445979\pi\)
0.168898 + 0.985634i \(0.445979\pi\)
\(882\) 0 0
\(883\) 54.9737 1.85001 0.925006 0.379954i \(-0.124060\pi\)
0.925006 + 0.379954i \(0.124060\pi\)
\(884\) −30.4868 −1.02538
\(885\) 0 0
\(886\) −28.6228 −0.961601
\(887\) −30.3246 −1.01820 −0.509099 0.860708i \(-0.670021\pi\)
−0.509099 + 0.860708i \(0.670021\pi\)
\(888\) 0 0
\(889\) −4.48683 −0.150484
\(890\) 0 0
\(891\) 0 0
\(892\) −0.837722 −0.0280490
\(893\) −31.4605 −1.05279
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −25.1623 −0.839676
\(899\) 11.6228 0.387641
\(900\) 0 0
\(901\) −1.18861 −0.0395984
\(902\) 8.00000 0.266371
\(903\) 0 0
\(904\) −14.6491 −0.487222
\(905\) 0 0
\(906\) 0 0
\(907\) −22.9737 −0.762828 −0.381414 0.924404i \(-0.624563\pi\)
−0.381414 + 0.924404i \(0.624563\pi\)
\(908\) −9.81139 −0.325602
\(909\) 0 0
\(910\) 0 0
\(911\) −45.2982 −1.50080 −0.750399 0.660986i \(-0.770138\pi\)
−0.750399 + 0.660986i \(0.770138\pi\)
\(912\) 0 0
\(913\) −6.32456 −0.209312
\(914\) 30.6228 1.01291
\(915\) 0 0
\(916\) 17.8377 0.589375
\(917\) −5.48683 −0.181191
\(918\) 0 0
\(919\) 23.2982 0.768537 0.384269 0.923221i \(-0.374454\pi\)
0.384269 + 0.923221i \(0.374454\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 5.16228 0.170011
\(923\) −42.9737 −1.41450
\(924\) 0 0
\(925\) 0 0
\(926\) 20.4868 0.673239
\(927\) 0 0
\(928\) −1.83772 −0.0603262
\(929\) −14.2982 −0.469109 −0.234555 0.972103i \(-0.575363\pi\)
−0.234555 + 0.972103i \(0.575363\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) −15.4868 −0.507288
\(933\) 0 0
\(934\) 10.3246 0.337830
\(935\) 0 0
\(936\) 0 0
\(937\) −12.1359 −0.396464 −0.198232 0.980155i \(-0.563520\pi\)
−0.198232 + 0.980155i \(0.563520\pi\)
\(938\) −3.48683 −0.113849
\(939\) 0 0
\(940\) 0 0
\(941\) 14.7851 0.481979 0.240989 0.970528i \(-0.422528\pi\)
0.240989 + 0.970528i \(0.422528\pi\)
\(942\) 0 0
\(943\) −4.64911 −0.151396
\(944\) 0.324555 0.0105634
\(945\) 0 0
\(946\) 6.32456 0.205629
\(947\) −46.0000 −1.49480 −0.747400 0.664375i \(-0.768698\pi\)
−0.747400 + 0.664375i \(0.768698\pi\)
\(948\) 0 0
\(949\) −18.0000 −0.584305
\(950\) 0 0
\(951\) 0 0
\(952\) −7.32456 −0.237390
\(953\) −24.7851 −0.802867 −0.401433 0.915888i \(-0.631488\pi\)
−0.401433 + 0.915888i \(0.631488\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −25.8114 −0.834800
\(957\) 0 0
\(958\) 4.16228 0.134477
\(959\) 2.51317 0.0811544
\(960\) 0 0
\(961\) 9.00000 0.290323
\(962\) 31.1623 1.00471
\(963\) 0 0
\(964\) −23.4868 −0.756460
\(965\) 0 0
\(966\) 0 0
\(967\) 22.4868 0.723128 0.361564 0.932347i \(-0.382243\pi\)
0.361564 + 0.932347i \(0.382243\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) −46.4605 −1.49099 −0.745494 0.666512i \(-0.767786\pi\)
−0.745494 + 0.666512i \(0.767786\pi\)
\(972\) 0 0
\(973\) −12.3246 −0.395107
\(974\) 26.9737 0.864292
\(975\) 0 0
\(976\) −3.83772 −0.122842
\(977\) −40.4605 −1.29445 −0.647223 0.762301i \(-0.724070\pi\)
−0.647223 + 0.762301i \(0.724070\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) 17.3246 0.552849
\(983\) 58.4342 1.86376 0.931880 0.362766i \(-0.118168\pi\)
0.931880 + 0.362766i \(0.118168\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −13.4605 −0.428670
\(987\) 0 0
\(988\) −12.4868 −0.397259
\(989\) −3.67544 −0.116872
\(990\) 0 0
\(991\) 58.4868 1.85790 0.928948 0.370211i \(-0.120715\pi\)
0.928948 + 0.370211i \(0.120715\pi\)
\(992\) −6.32456 −0.200805
\(993\) 0 0
\(994\) −10.3246 −0.327475
\(995\) 0 0
\(996\) 0 0
\(997\) 8.97367 0.284199 0.142099 0.989852i \(-0.454615\pi\)
0.142099 + 0.989852i \(0.454615\pi\)
\(998\) 4.83772 0.153135
\(999\) 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 9450.2.a.em.1.1 yes 2
3.2 odd 2 9450.2.a.ee.1.1 2
5.4 even 2 9450.2.a.eh.1.2 yes 2
15.14 odd 2 9450.2.a.ex.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9450.2.a.ee.1.1 2 3.2 odd 2
9450.2.a.eh.1.2 yes 2 5.4 even 2
9450.2.a.em.1.1 yes 2 1.1 even 1 trivial
9450.2.a.ex.1.2 yes 2 15.14 odd 2