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Properties

Label 322.2.a.a.1.1
Level $322$
Weight $2$
Character 322.1
Self dual yes
Analytic conductor $2.571$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,2,Mod(1,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, 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("322.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(2.57118294509\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 322.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} +3.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{28} +2.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} +8.00000 q^{34} -2.00000 q^{35} -3.00000 q^{36} -10.0000 q^{37} +2.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +6.00000 q^{45} -1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} +2.00000 q^{53} +8.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} +10.0000 q^{61} +6.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} +8.00000 q^{67} -8.00000 q^{68} +2.00000 q^{70} -12.0000 q^{71} +3.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} -2.00000 q^{76} -4.00000 q^{77} -2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +2.00000 q^{83} +16.0000 q^{85} +8.00000 q^{86} +4.00000 q^{88} +12.0000 q^{89} -6.00000 q^{90} +4.00000 q^{91} +1.00000 q^{92} -6.00000 q^{94} +4.00000 q^{95} +12.0000 q^{97} -1.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 3.00000 0.707107
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) −2.00000 −0.338062
\(36\) −3.00000 −0.500000
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 6.00000 0.894427
\(46\) −1.00000 −0.147442
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 3.00000 0.353553
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 16.0000 1.73544
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −6.00000 −0.632456
\(91\) 4.00000 0.419314
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −1.00000 −0.101015
\(99\) 12.0000 1.20605
\(100\) −1.00000 −0.100000
\(101\) 8.00000 0.796030 0.398015 − 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) −20.0000 −1.97066 −0.985329 − 0.170664i \(-0.945409\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 − 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 2.00000 0.185695
\(117\) −12.0000 −1.10940
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 12.0000 1.07331
\(126\) 3.00000 0.267261
\(127\) −20.0000 −1.77471 −0.887357 − 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −16.0000 −1.33799
\(144\) −3.00000 −0.250000
\(145\) −4.00000 −0.332182
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 − 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 2.00000 0.162221
\(153\) 24.0000 1.94029
\(154\) 4.00000 0.322329
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 1.00000 0.0788110
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) −14.0000 −1.08335 −0.541676 − 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −16.0000 −1.22714
\(171\) 6.00000 0.458831
\(172\) −8.00000 −0.609994
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −12.0000 −0.899438
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 6.00000 0.447214
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 20.0000 1.47043
\(186\) 0 0
\(187\) 32.0000 2.34007
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −12.0000 −0.852803
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 20.0000 1.39347
\(207\) −3.00000 −0.208514
\(208\) 4.00000 0.277350
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) −18.0000 −1.21911
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) −32.0000 −2.15255
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 3.00000 0.200000
\(226\) 14.0000 0.931266
\(227\) 6.00000 0.398234 0.199117 − 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 12.0000 0.784465
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −22.0000 −1.38863 −0.694314 − 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) −3.00000 −0.188982
\(253\) −4.00000 −0.251478
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) −8.00000 −0.496139
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 16.0000 0.975537 0.487769 − 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 4.00000 0.239904
\(279\) 18.0000 1.07763
\(280\) 2.00000 0.119523
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −2.00000 −0.118888 −0.0594438 − 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) 6.00000 0.354169
\(288\) 3.00000 0.176777
\(289\) 47.0000 2.76471
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 22.0000 1.28525 0.642627 − 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) −20.0000 −1.14520
\(306\) −24.0000 −1.37199
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) −10.0000 −0.567048 −0.283524 − 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 18.0000 1.01580
\(315\) 6.00000 0.338062
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 16.0000 0.890264
\(324\) 9.00000 0.500000
\(325\) −4.00000 −0.221880
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 2.00000 0.109764
\(333\) 30.0000 1.64399
\(334\) 14.0000 0.766046
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 16.0000 0.867722
\(341\) 24.0000 1.29967
\(342\) −6.00000 −0.324443
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 − 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −6.00000 −0.316228
\(361\) −15.0000 −0.789474
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 1.00000 0.0521286
\(369\) −18.0000 −0.937043
\(370\) −20.0000 −1.03975
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −32.0000 −1.65468
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 10.0000 0.508987
\(387\) 24.0000 1.21999
\(388\) 12.0000 0.609208
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 12.0000 0.603023
\(397\) 20.0000 1.00377 0.501886 − 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) 8.00000 0.398015
\(405\) −18.0000 −0.894427
\(406\) −2.00000 −0.0992583
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −20.0000 −0.985329
\(413\) 0 0
\(414\) 3.00000 0.147442
\(415\) −4.00000 −0.196352
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) −6.00000 −0.293119 −0.146560 − 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 − 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −4.00000 −0.194717
\(423\) −18.0000 −0.875190
\(424\) −2.00000 −0.0971286
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 − 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) −8.00000 −0.381385
\(441\) −3.00000 −0.142857
\(442\) 32.0000 1.52208
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 10.0000 0.473514
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −34.0000 −1.60456 −0.802280 − 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −3.00000 −0.141421
\(451\) −24.0000 −1.13012
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 − 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 22.0000 1.01804 0.509019 − 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) −12.0000 −0.554700
\(469\) 8.00000 0.369406
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) 32.0000 1.47136
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −8.00000 −0.366679
\(477\) −6.00000 −0.274721
\(478\) −12.0000 −0.548867
\(479\) 4.00000 0.182765 0.0913823 − 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) 8.00000 0.359937
\(495\) −24.0000 −1.07872
\(496\) −6.00000 −0.269408
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 22.0000 0.981908
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 3.00000 0.133631
\(505\) −16.0000 −0.711991
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) −20.0000 −0.887357
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 40.0000 1.76261
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 10.0000 0.439375
\(519\) 0 0
\(520\) 8.00000 0.350823
\(521\) 20.0000 0.876216 0.438108 − 0.898922i \(-0.355649\pi\)
0.438108 + 0.898922i \(0.355649\pi\)
\(522\) 6.00000 0.262613
\(523\) −34.0000 −1.48672 −0.743358 − 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 4.00000 0.173749
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) −16.0000 −0.689809
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 − 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) −36.0000 −1.54207
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −18.0000 −0.768922
\(549\) −30.0000 −1.28037
\(550\) −4.00000 −0.170561
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −18.0000 −0.762001
\(559\) −32.0000 −1.35346
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −38.0000 −1.60151 −0.800755 − 0.598993i \(-0.795568\pi\)
−0.800755 + 0.598993i \(0.795568\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) 2.00000 0.0840663
\(567\) 9.00000 0.377964
\(568\) 12.0000 0.503509
\(569\) −10.0000 −0.419222 −0.209611 − 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 − 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) −3.00000 −0.125000
\(577\) 30.0000 1.24892 0.624458 − 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −47.0000 −1.95494
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 2.00000 0.0829740
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) −6.00000 −0.248282
\(585\) 24.0000 0.992278
\(586\) −22.0000 −0.908812
\(587\) 44.0000 1.81607 0.908037 − 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) −32.0000 −1.30748 −0.653742 − 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 − 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 8.00000 0.326056
\(603\) −24.0000 −0.977356
\(604\) −20.0000 −0.813788
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) 24.0000 0.970936
\(612\) 24.0000 0.970143
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) 10.0000 0.400963
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 80.0000 3.18981
\(630\) −6.00000 −0.239046
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 40.0000 1.58735
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 8.00000 0.316723
\(639\) 36.0000 1.42414
\(640\) 2.00000 0.0790569
\(641\) 38.0000 1.50091 0.750455 − 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) 0 0
\(643\) −38.0000 −1.49857 −0.749287 − 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) −46.0000 −1.80845 −0.904223 − 0.427060i \(-0.859549\pi\)
−0.904223 + 0.427060i \(0.859549\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −18.0000 −0.702247
\(658\) −6.00000 −0.233904
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 − 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 4.00000 0.155113
\(666\) −30.0000 −1.16248
\(667\) 2.00000 0.0774403
\(668\) −14.0000 −0.541676
\(669\) 0 0
\(670\) 16.0000 0.618134
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 − 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) −16.0000 −0.613572
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 6.00000 0.229416
\(685\) 36.0000 1.37549
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 − 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 28.0000 1.06287
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) 16.0000 0.605609
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 8.00000 0.300871
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 32.0000 1.19673
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 42.0000 1.56634 0.783168 − 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 6.00000 0.223607
\(721\) −20.0000 −0.744839
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 − 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) −4.00000 −0.148250
\(729\) −27.0000 −1.00000
\(730\) 12.0000 0.444140
\(731\) 64.0000 2.36713
\(732\) 0 0
\(733\) −50.0000 −1.84679 −0.923396 − 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −32.0000 −1.17874
\(738\) 18.0000 0.662589
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 20.0000 0.735215
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 10.0000 0.366126
\(747\) −6.00000 −0.219529
\(748\) 32.0000 1.17004
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 − 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 40.0000 1.45575
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 − 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 18.0000 0.651644
\(764\) −16.0000 −0.578860
\(765\) −48.0000 −1.73544
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 0 0
\(769\) 16.0000 0.576975 0.288487 − 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −46.0000 −1.65451 −0.827253 − 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) −24.0000 −0.862662
\(775\) 6.00000 0.215526
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 − 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) −12.0000 −0.426401
\(793\) 40.0000 1.42044
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −22.0000 −0.779280 −0.389640 − 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 1.00000 0.0353553
\(801\) −36.0000 −1.27200
\(802\) 18.0000 0.635602
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) −8.00000 −0.281439
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 18.0000 0.632456
\(811\) 8.00000 0.280918 0.140459 − 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) −10.0000 −0.349642
\(819\) −12.0000 −0.419314
\(820\) −12.0000 −0.419058
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −12.0000 −0.418294 −0.209147 − 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 20.0000 0.696733
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −3.00000 −0.104257
\(829\) −12.0000 −0.416777 −0.208389 − 0.978046i \(-0.566822\pi\)
−0.208389 + 0.978046i \(0.566822\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) −8.00000 −0.277184
\(834\) 0 0
\(835\) 28.0000 0.968980
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) 6.00000 0.207267
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000 0.482472
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) −6.00000 −0.206406
\(846\) 18.0000 0.618853
\(847\) 5.00000 0.171802
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) −10.0000 −0.342796
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 − 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) −10.0000 −0.342193
\(855\) −12.0000 −0.410391
\(856\) 4.00000 0.136717
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 − 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −4.00000 −0.135926
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) −18.0000 −0.609557
\(873\) −36.0000 −1.21842
\(874\) 2.00000 0.0676510
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 10.0000 0.337484
\(879\) 0 0
\(880\) 8.00000 0.269680
\(881\) −24.0000 −0.808581 −0.404290 − 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 3.00000 0.101015
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −32.0000 −1.07628
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −54.0000 −1.81314 −0.906571 − 0.422053i \(-0.861310\pi\)
−0.906571 + 0.422053i \(0.861310\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 24.0000 0.804482
\(891\) −36.0000 −1.20605
\(892\) −10.0000 −0.334825
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −40.0000 −1.33705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) −12.0000 −0.400222
\(900\) 3.00000 0.100000
\(901\) −16.0000 −0.533037
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 6.00000 0.199117
\(909\) −24.0000 −0.796030
\(910\) 8.00000 0.265197
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 0 0
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 24.0000 0.790398
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −16.0000 −0.525793
\(927\) 60.0000 1.97066
\(928\) −2.00000 −0.0656532
\(929\) 54.0000 1.77168 0.885841 − 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −22.0000 −0.719862
\(935\) −64.0000 −2.09302
\(936\) 12.0000 0.392232
\(937\) −8.00000 −0.261349 −0.130674 − 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) −14.0000 −0.453504 −0.226752 − 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 6.00000 0.194257
\(955\) 32.0000 1.03550
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 40.0000 1.28965
\(963\) 12.0000 0.386695
\(964\) −8.00000 −0.257663
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −14.0000 −0.449281 −0.224641 − 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) −2.00000 −0.0638877
\(981\) −54.0000 −1.72409
\(982\) −36.0000 −1.14881
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 16.0000 0.509544
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −8.00000 −0.254385
\(990\) 24.0000 0.762770
\(991\) 52.0000 1.65183 0.825917 − 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) −40.0000 −1.26809
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 − 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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 322.2.a.a.1.1 ✓ 1
3.2 odd 2 2898.2.a.s.1.1 1
4.3 odd 2 2576.2.a.i.1.1 1
5.4 even 2 8050.2.a.o.1.1 1
7.6 odd 2 2254.2.a.d.1.1 1
23.22 odd 2 7406.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
2254.2.a.d.1.1 1 7.6 odd 2
2576.2.a.i.1.1 1 4.3 odd 2
2898.2.a.s.1.1 1 3.2 odd 2
7406.2.a.d.1.1 1 23.22 odd 2
8050.2.a.o.1.1 1 5.4 even 2