Properties

Label 2366.2.a.x.1.1
Level $2366$
Weight $2$
Character 2366.1
Self dual yes
Analytic conductor $18.893$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2366,2,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, 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("2366.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.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: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1384.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 10x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.30590\) of defining polynomial
Character \(\chi\) \(=\) 2366.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.30590 q^{3} +1.00000 q^{4} -0.376939 q^{5} +3.30590 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.92896 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.30590 q^{3} +1.00000 q^{4} -0.376939 q^{5} +3.30590 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.92896 q^{9} +0.376939 q^{10} -5.68284 q^{11} -3.30590 q^{12} +1.00000 q^{14} +1.24612 q^{15} +1.00000 q^{16} +1.24612 q^{17} -7.92896 q^{18} -1.62306 q^{19} -0.376939 q^{20} +3.30590 q^{21} +5.68284 q^{22} +4.92896 q^{23} +3.30590 q^{24} -4.85792 q^{25} -16.2946 q^{27} -1.00000 q^{28} -4.61180 q^{29} -1.24612 q^{30} +5.68284 q^{31} -1.00000 q^{32} +18.7869 q^{33} -1.24612 q^{34} +0.376939 q^{35} +7.92896 q^{36} -6.92896 q^{37} +1.62306 q^{38} +0.376939 q^{40} +1.07104 q^{41} -3.30590 q^{42} -3.24612 q^{43} -5.68284 q^{44} -2.98873 q^{45} -4.92896 q^{46} -2.31716 q^{47} -3.30590 q^{48} +1.00000 q^{49} +4.85792 q^{50} -4.11955 q^{51} -6.00000 q^{53} +16.2946 q^{54} +2.14208 q^{55} +1.00000 q^{56} +5.36567 q^{57} +4.61180 q^{58} -11.4810 q^{59} +1.24612 q^{60} +5.30590 q^{61} -5.68284 q^{62} -7.92896 q^{63} +1.00000 q^{64} -18.7869 q^{66} -8.92896 q^{67} +1.24612 q^{68} -16.2946 q^{69} -0.376939 q^{70} -13.2236 q^{71} -7.92896 q^{72} +0.317163 q^{73} +6.92896 q^{74} +16.0598 q^{75} -1.62306 q^{76} +5.68284 q^{77} +2.17508 q^{79} -0.376939 q^{80} +30.0815 q^{81} -1.07104 q^{82} -9.74261 q^{83} +3.30590 q^{84} -0.469712 q^{85} +3.24612 q^{86} +15.2461 q^{87} +5.68284 q^{88} +9.24612 q^{89} +2.98873 q^{90} +4.92896 q^{92} -18.7869 q^{93} +2.31716 q^{94} +0.611795 q^{95} +3.30590 q^{96} +10.9290 q^{97} -1.00000 q^{98} -45.0590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} + 12 q^{9} + 2 q^{10} - 7 q^{11} + q^{12} + 3 q^{14} + 2 q^{15} + 3 q^{16} + 2 q^{17} - 12 q^{18} - 4 q^{19} - 2 q^{20} - q^{21} + 7 q^{22} + 3 q^{23} - q^{24} + 9 q^{25} - 17 q^{27} - 3 q^{28} + 8 q^{29} - 2 q^{30} + 7 q^{31} - 3 q^{32} + 21 q^{33} - 2 q^{34} + 2 q^{35} + 12 q^{36} - 9 q^{37} + 4 q^{38} + 2 q^{40} + 15 q^{41} + q^{42} - 8 q^{43} - 7 q^{44} + 12 q^{45} - 3 q^{46} - 17 q^{47} + q^{48} + 3 q^{49} - 9 q^{50} + 6 q^{51} - 18 q^{53} + 17 q^{54} + 30 q^{55} + 3 q^{56} - 4 q^{57} - 8 q^{58} - 10 q^{59} + 2 q^{60} + 5 q^{61} - 7 q^{62} - 12 q^{63} + 3 q^{64} - 21 q^{66} - 15 q^{67} + 2 q^{68} - 17 q^{69} - 2 q^{70} + 4 q^{71} - 12 q^{72} + 11 q^{73} + 9 q^{74} + 39 q^{75} - 4 q^{76} + 7 q^{77} - 7 q^{79} - 2 q^{80} + 23 q^{81} - 15 q^{82} - 10 q^{83} - q^{84} + 44 q^{85} + 8 q^{86} + 44 q^{87} + 7 q^{88} + 26 q^{89} - 12 q^{90} + 3 q^{92} - 21 q^{93} + 17 q^{94} - 20 q^{95} - q^{96} + 21 q^{97} - 3 q^{98} - 26 q^{99}+O(q^{100}) \) 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
\(3\) −3.30590 −1.90866 −0.954330 0.298753i \(-0.903429\pi\)
−0.954330 + 0.298753i \(0.903429\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.376939 −0.168572 −0.0842861 0.996442i \(-0.526861\pi\)
−0.0842861 + 0.996442i \(0.526861\pi\)
\(6\) 3.30590 1.34963
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 7.92896 2.64299
\(10\) 0.376939 0.119199
\(11\) −5.68284 −1.71344 −0.856720 0.515782i \(-0.827501\pi\)
−0.856720 + 0.515782i \(0.827501\pi\)
\(12\) −3.30590 −0.954330
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 1.24612 0.321747
\(16\) 1.00000 0.250000
\(17\) 1.24612 0.302229 0.151114 0.988516i \(-0.451714\pi\)
0.151114 + 0.988516i \(0.451714\pi\)
\(18\) −7.92896 −1.86887
\(19\) −1.62306 −0.372356 −0.186178 0.982516i \(-0.559610\pi\)
−0.186178 + 0.982516i \(0.559610\pi\)
\(20\) −0.376939 −0.0842861
\(21\) 3.30590 0.721406
\(22\) 5.68284 1.21158
\(23\) 4.92896 1.02776 0.513879 0.857862i \(-0.328208\pi\)
0.513879 + 0.857862i \(0.328208\pi\)
\(24\) 3.30590 0.674814
\(25\) −4.85792 −0.971583
\(26\) 0 0
\(27\) −16.2946 −3.13590
\(28\) −1.00000 −0.188982
\(29\) −4.61180 −0.856389 −0.428194 0.903687i \(-0.640850\pi\)
−0.428194 + 0.903687i \(0.640850\pi\)
\(30\) −1.24612 −0.227510
\(31\) 5.68284 1.02067 0.510334 0.859976i \(-0.329522\pi\)
0.510334 + 0.859976i \(0.329522\pi\)
\(32\) −1.00000 −0.176777
\(33\) 18.7869 3.27038
\(34\) −1.24612 −0.213708
\(35\) 0.376939 0.0637143
\(36\) 7.92896 1.32149
\(37\) −6.92896 −1.13911 −0.569557 0.821952i \(-0.692885\pi\)
−0.569557 + 0.821952i \(0.692885\pi\)
\(38\) 1.62306 0.263295
\(39\) 0 0
\(40\) 0.376939 0.0595993
\(41\) 1.07104 0.167269 0.0836343 0.996497i \(-0.473347\pi\)
0.0836343 + 0.996497i \(0.473347\pi\)
\(42\) −3.30590 −0.510111
\(43\) −3.24612 −0.495029 −0.247514 0.968884i \(-0.579614\pi\)
−0.247514 + 0.968884i \(0.579614\pi\)
\(44\) −5.68284 −0.856720
\(45\) −2.98873 −0.445534
\(46\) −4.92896 −0.726735
\(47\) −2.31716 −0.337993 −0.168997 0.985617i \(-0.554053\pi\)
−0.168997 + 0.985617i \(0.554053\pi\)
\(48\) −3.30590 −0.477165
\(49\) 1.00000 0.142857
\(50\) 4.85792 0.687013
\(51\) −4.11955 −0.576853
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 16.2946 2.21742
\(55\) 2.14208 0.288838
\(56\) 1.00000 0.133631
\(57\) 5.36567 0.710701
\(58\) 4.61180 0.605558
\(59\) −11.4810 −1.49470 −0.747348 0.664433i \(-0.768673\pi\)
−0.747348 + 0.664433i \(0.768673\pi\)
\(60\) 1.24612 0.160874
\(61\) 5.30590 0.679351 0.339675 0.940543i \(-0.389683\pi\)
0.339675 + 0.940543i \(0.389683\pi\)
\(62\) −5.68284 −0.721721
\(63\) −7.92896 −0.998955
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −18.7869 −2.31250
\(67\) −8.92896 −1.09085 −0.545423 0.838161i \(-0.683631\pi\)
−0.545423 + 0.838161i \(0.683631\pi\)
\(68\) 1.24612 0.151114
\(69\) −16.2946 −1.96164
\(70\) −0.376939 −0.0450528
\(71\) −13.2236 −1.56935 −0.784676 0.619906i \(-0.787171\pi\)
−0.784676 + 0.619906i \(0.787171\pi\)
\(72\) −7.92896 −0.934437
\(73\) 0.317163 0.0371212 0.0185606 0.999828i \(-0.494092\pi\)
0.0185606 + 0.999828i \(0.494092\pi\)
\(74\) 6.92896 0.805475
\(75\) 16.0598 1.85442
\(76\) −1.62306 −0.186178
\(77\) 5.68284 0.647619
\(78\) 0 0
\(79\) 2.17508 0.244716 0.122358 0.992486i \(-0.460954\pi\)
0.122358 + 0.992486i \(0.460954\pi\)
\(80\) −0.376939 −0.0421431
\(81\) 30.0815 3.34239
\(82\) −1.07104 −0.118277
\(83\) −9.74261 −1.06939 −0.534695 0.845045i \(-0.679574\pi\)
−0.534695 + 0.845045i \(0.679574\pi\)
\(84\) 3.30590 0.360703
\(85\) −0.469712 −0.0509474
\(86\) 3.24612 0.350038
\(87\) 15.2461 1.63456
\(88\) 5.68284 0.605792
\(89\) 9.24612 0.980087 0.490043 0.871698i \(-0.336981\pi\)
0.490043 + 0.871698i \(0.336981\pi\)
\(90\) 2.98873 0.315040
\(91\) 0 0
\(92\) 4.92896 0.513879
\(93\) −18.7869 −1.94811
\(94\) 2.31716 0.238997
\(95\) 0.611795 0.0627688
\(96\) 3.30590 0.337407
\(97\) 10.9290 1.10967 0.554834 0.831961i \(-0.312782\pi\)
0.554834 + 0.831961i \(0.312782\pi\)
\(98\) −1.00000 −0.101015
\(99\) −45.0590 −4.52860
\(100\) −4.85792 −0.485792
\(101\) 3.44798 0.343087 0.171543 0.985177i \(-0.445125\pi\)
0.171543 + 0.985177i \(0.445125\pi\)
\(102\) 4.11955 0.407896
\(103\) −7.24612 −0.713982 −0.356991 0.934108i \(-0.616197\pi\)
−0.356991 + 0.934108i \(0.616197\pi\)
\(104\) 0 0
\(105\) −1.24612 −0.121609
\(106\) 6.00000 0.582772
\(107\) −9.85792 −0.953001 −0.476500 0.879174i \(-0.658095\pi\)
−0.476500 + 0.879174i \(0.658095\pi\)
\(108\) −16.2946 −1.56795
\(109\) 11.2236 1.07502 0.537512 0.843256i \(-0.319364\pi\)
0.537512 + 0.843256i \(0.319364\pi\)
\(110\) −2.14208 −0.204240
\(111\) 22.9064 2.17418
\(112\) −1.00000 −0.0944911
\(113\) −4.92896 −0.463677 −0.231839 0.972754i \(-0.574474\pi\)
−0.231839 + 0.972754i \(0.574474\pi\)
\(114\) −5.36567 −0.502541
\(115\) −1.85792 −0.173252
\(116\) −4.61180 −0.428194
\(117\) 0 0
\(118\) 11.4810 1.05691
\(119\) −1.24612 −0.114232
\(120\) −1.24612 −0.113755
\(121\) 21.2946 1.93588
\(122\) −5.30590 −0.480373
\(123\) −3.54075 −0.319259
\(124\) 5.68284 0.510334
\(125\) 3.71583 0.332354
\(126\) 7.92896 0.706368
\(127\) −8.29463 −0.736030 −0.368015 0.929820i \(-0.619962\pi\)
−0.368015 + 0.929820i \(0.619962\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.7313 0.944842
\(130\) 0 0
\(131\) 16.3544 1.42889 0.714446 0.699691i \(-0.246679\pi\)
0.714446 + 0.699691i \(0.246679\pi\)
\(132\) 18.7869 1.63519
\(133\) 1.62306 0.140737
\(134\) 8.92896 0.771345
\(135\) 6.14208 0.528626
\(136\) −1.24612 −0.106854
\(137\) −13.2236 −1.12977 −0.564884 0.825170i \(-0.691079\pi\)
−0.564884 + 0.825170i \(0.691079\pi\)
\(138\) 16.2946 1.38709
\(139\) 1.62306 0.137666 0.0688331 0.997628i \(-0.478072\pi\)
0.0688331 + 0.997628i \(0.478072\pi\)
\(140\) 0.376939 0.0318572
\(141\) 7.66030 0.645114
\(142\) 13.2236 1.10970
\(143\) 0 0
\(144\) 7.92896 0.660747
\(145\) 1.73837 0.144363
\(146\) −0.317163 −0.0262486
\(147\) −3.30590 −0.272666
\(148\) −6.92896 −0.569557
\(149\) 9.54075 0.781609 0.390804 0.920474i \(-0.372197\pi\)
0.390804 + 0.920474i \(0.372197\pi\)
\(150\) −16.0598 −1.31128
\(151\) 2.75388 0.224107 0.112054 0.993702i \(-0.464257\pi\)
0.112054 + 0.993702i \(0.464257\pi\)
\(152\) 1.62306 0.131648
\(153\) 9.88045 0.798787
\(154\) −5.68284 −0.457936
\(155\) −2.14208 −0.172056
\(156\) 0 0
\(157\) 22.5295 1.79805 0.899024 0.437898i \(-0.144277\pi\)
0.899024 + 0.437898i \(0.144277\pi\)
\(158\) −2.17508 −0.173040
\(159\) 19.8354 1.57305
\(160\) 0.376939 0.0297996
\(161\) −4.92896 −0.388456
\(162\) −30.0815 −2.36343
\(163\) 8.46971 0.663399 0.331700 0.943385i \(-0.392378\pi\)
0.331700 + 0.943385i \(0.392378\pi\)
\(164\) 1.07104 0.0836343
\(165\) −7.08151 −0.551295
\(166\) 9.74261 0.756173
\(167\) −5.97747 −0.462550 −0.231275 0.972888i \(-0.574290\pi\)
−0.231275 + 0.972888i \(0.574290\pi\)
\(168\) −3.30590 −0.255056
\(169\) 0 0
\(170\) 0.469712 0.0360253
\(171\) −12.8692 −0.984131
\(172\) −3.24612 −0.247514
\(173\) 0.846651 0.0643697 0.0321848 0.999482i \(-0.489753\pi\)
0.0321848 + 0.999482i \(0.489753\pi\)
\(174\) −15.2461 −1.15581
\(175\) 4.85792 0.367224
\(176\) −5.68284 −0.428360
\(177\) 37.9549 2.85287
\(178\) −9.24612 −0.693026
\(179\) 18.7313 1.40005 0.700023 0.714120i \(-0.253173\pi\)
0.700023 + 0.714120i \(0.253173\pi\)
\(180\) −2.98873 −0.222767
\(181\) 17.6561 1.31236 0.656182 0.754602i \(-0.272170\pi\)
0.656182 + 0.754602i \(0.272170\pi\)
\(182\) 0 0
\(183\) −17.5408 −1.29665
\(184\) −4.92896 −0.363368
\(185\) 2.61180 0.192023
\(186\) 18.7869 1.37752
\(187\) −7.08151 −0.517851
\(188\) −2.31716 −0.168997
\(189\) 16.2946 1.18526
\(190\) −0.611795 −0.0443843
\(191\) 16.6118 1.20199 0.600994 0.799254i \(-0.294772\pi\)
0.600994 + 0.799254i \(0.294772\pi\)
\(192\) −3.30590 −0.238583
\(193\) 3.85792 0.277699 0.138849 0.990313i \(-0.455660\pi\)
0.138849 + 0.990313i \(0.455660\pi\)
\(194\) −10.9290 −0.784653
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.17508 −0.439956 −0.219978 0.975505i \(-0.570599\pi\)
−0.219978 + 0.975505i \(0.570599\pi\)
\(198\) 45.0590 3.20220
\(199\) 4.75388 0.336993 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(200\) 4.85792 0.343507
\(201\) 29.5182 2.08205
\(202\) −3.44798 −0.242599
\(203\) 4.61180 0.323685
\(204\) −4.11955 −0.288426
\(205\) −0.403717 −0.0281968
\(206\) 7.24612 0.504861
\(207\) 39.0815 2.71635
\(208\) 0 0
\(209\) 9.22359 0.638009
\(210\) 1.24612 0.0859906
\(211\) 27.7158 1.90804 0.954018 0.299748i \(-0.0969027\pi\)
0.954018 + 0.299748i \(0.0969027\pi\)
\(212\) −6.00000 −0.412082
\(213\) 43.7158 2.99536
\(214\) 9.85792 0.673873
\(215\) 1.22359 0.0834482
\(216\) 16.2946 1.10871
\(217\) −5.68284 −0.385776
\(218\) −11.2236 −0.760157
\(219\) −1.04851 −0.0708517
\(220\) 2.14208 0.144419
\(221\) 0 0
\(222\) −22.9064 −1.53738
\(223\) 8.29463 0.555450 0.277725 0.960661i \(-0.410420\pi\)
0.277725 + 0.960661i \(0.410420\pi\)
\(224\) 1.00000 0.0668153
\(225\) −38.5182 −2.56788
\(226\) 4.92896 0.327869
\(227\) −0.234856 −0.0155879 −0.00779397 0.999970i \(-0.502481\pi\)
−0.00779397 + 0.999970i \(0.502481\pi\)
\(228\) 5.36567 0.355350
\(229\) 19.4584 1.28585 0.642925 0.765929i \(-0.277721\pi\)
0.642925 + 0.765929i \(0.277721\pi\)
\(230\) 1.85792 0.122507
\(231\) −18.7869 −1.23609
\(232\) 4.61180 0.302779
\(233\) −20.1525 −1.32024 −0.660119 0.751161i \(-0.729494\pi\)
−0.660119 + 0.751161i \(0.729494\pi\)
\(234\) 0 0
\(235\) 0.873429 0.0569763
\(236\) −11.4810 −0.747348
\(237\) −7.19059 −0.467079
\(238\) 1.24612 0.0807741
\(239\) 12.9620 0.838439 0.419220 0.907885i \(-0.362304\pi\)
0.419220 + 0.907885i \(0.362304\pi\)
\(240\) 1.24612 0.0804368
\(241\) −8.73135 −0.562435 −0.281218 0.959644i \(-0.590738\pi\)
−0.281218 + 0.959644i \(0.590738\pi\)
\(242\) −21.2946 −1.36887
\(243\) −50.5625 −3.24358
\(244\) 5.30590 0.339675
\(245\) −0.376939 −0.0240818
\(246\) 3.54075 0.225750
\(247\) 0 0
\(248\) −5.68284 −0.360860
\(249\) 32.2081 2.04110
\(250\) −3.71583 −0.235010
\(251\) −5.04426 −0.318391 −0.159196 0.987247i \(-0.550890\pi\)
−0.159196 + 0.987247i \(0.550890\pi\)
\(252\) −7.92896 −0.499477
\(253\) −28.0105 −1.76100
\(254\) 8.29463 0.520451
\(255\) 1.55282 0.0972413
\(256\) 1.00000 0.0625000
\(257\) −23.2236 −1.44865 −0.724324 0.689460i \(-0.757848\pi\)
−0.724324 + 0.689460i \(0.757848\pi\)
\(258\) −10.7313 −0.668104
\(259\) 6.92896 0.430545
\(260\) 0 0
\(261\) −36.5667 −2.26342
\(262\) −16.3544 −1.01038
\(263\) −11.1040 −0.684704 −0.342352 0.939572i \(-0.611224\pi\)
−0.342352 + 0.939572i \(0.611224\pi\)
\(264\) −18.7869 −1.15625
\(265\) 2.26163 0.138931
\(266\) −1.62306 −0.0995163
\(267\) −30.5667 −1.87065
\(268\) −8.92896 −0.545423
\(269\) 11.5675 0.705285 0.352642 0.935758i \(-0.385283\pi\)
0.352642 + 0.935758i \(0.385283\pi\)
\(270\) −6.14208 −0.373795
\(271\) −26.1525 −1.58865 −0.794327 0.607490i \(-0.792176\pi\)
−0.794327 + 0.607490i \(0.792176\pi\)
\(272\) 1.24612 0.0755572
\(273\) 0 0
\(274\) 13.2236 0.798866
\(275\) 27.6067 1.66475
\(276\) −16.2946 −0.980822
\(277\) 23.2236 1.39537 0.697685 0.716405i \(-0.254213\pi\)
0.697685 + 0.716405i \(0.254213\pi\)
\(278\) −1.62306 −0.0973447
\(279\) 45.0590 2.69761
\(280\) −0.376939 −0.0225264
\(281\) −17.2236 −1.02747 −0.513737 0.857948i \(-0.671739\pi\)
−0.513737 + 0.857948i \(0.671739\pi\)
\(282\) −7.66030 −0.456165
\(283\) −19.7756 −1.17554 −0.587769 0.809029i \(-0.699994\pi\)
−0.587769 + 0.809029i \(0.699994\pi\)
\(284\) −13.2236 −0.784676
\(285\) −2.02253 −0.119804
\(286\) 0 0
\(287\) −1.07104 −0.0632216
\(288\) −7.92896 −0.467218
\(289\) −15.4472 −0.908658
\(290\) −1.73837 −0.102080
\(291\) −36.1300 −2.11798
\(292\) 0.317163 0.0185606
\(293\) 1.13082 0.0660630 0.0330315 0.999454i \(-0.489484\pi\)
0.0330315 + 0.999454i \(0.489484\pi\)
\(294\) 3.30590 0.192804
\(295\) 4.32763 0.251964
\(296\) 6.92896 0.402738
\(297\) 92.5997 5.37318
\(298\) −9.54075 −0.552681
\(299\) 0 0
\(300\) 16.0598 0.927212
\(301\) 3.24612 0.187103
\(302\) −2.75388 −0.158468
\(303\) −11.3987 −0.654837
\(304\) −1.62306 −0.0930889
\(305\) −2.00000 −0.114520
\(306\) −9.88045 −0.564828
\(307\) 30.9662 1.76733 0.883667 0.468116i \(-0.155067\pi\)
0.883667 + 0.468116i \(0.155067\pi\)
\(308\) 5.68284 0.323810
\(309\) 23.9549 1.36275
\(310\) 2.14208 0.121662
\(311\) 9.50776 0.539135 0.269568 0.962981i \(-0.413119\pi\)
0.269568 + 0.962981i \(0.413119\pi\)
\(312\) 0 0
\(313\) 3.38820 0.191513 0.0957563 0.995405i \(-0.469473\pi\)
0.0957563 + 0.995405i \(0.469473\pi\)
\(314\) −22.5295 −1.27141
\(315\) 2.98873 0.168396
\(316\) 2.17508 0.122358
\(317\) 29.5408 1.65917 0.829587 0.558377i \(-0.188576\pi\)
0.829587 + 0.558377i \(0.188576\pi\)
\(318\) −19.8354 −1.11231
\(319\) 26.2081 1.46737
\(320\) −0.376939 −0.0210715
\(321\) 32.5893 1.81896
\(322\) 4.92896 0.274680
\(323\) −2.02253 −0.112537
\(324\) 30.0815 1.67119
\(325\) 0 0
\(326\) −8.46971 −0.469094
\(327\) −37.1040 −2.05186
\(328\) −1.07104 −0.0591384
\(329\) 2.31716 0.127749
\(330\) 7.08151 0.389824
\(331\) −0.175080 −0.00962329 −0.00481164 0.999988i \(-0.501532\pi\)
−0.00481164 + 0.999988i \(0.501532\pi\)
\(332\) −9.74261 −0.534695
\(333\) −54.9394 −3.01066
\(334\) 5.97747 0.327073
\(335\) 3.36567 0.183886
\(336\) 3.30590 0.180351
\(337\) −19.3761 −1.05549 −0.527743 0.849404i \(-0.676961\pi\)
−0.527743 + 0.849404i \(0.676961\pi\)
\(338\) 0 0
\(339\) 16.2946 0.885003
\(340\) −0.469712 −0.0254737
\(341\) −32.2946 −1.74885
\(342\) 12.8692 0.695886
\(343\) −1.00000 −0.0539949
\(344\) 3.24612 0.175019
\(345\) 6.14208 0.330679
\(346\) −0.846651 −0.0455162
\(347\) 24.4697 1.31360 0.656801 0.754064i \(-0.271909\pi\)
0.656801 + 0.754064i \(0.271909\pi\)
\(348\) 15.2461 0.817278
\(349\) 11.8622 0.634967 0.317484 0.948264i \(-0.397162\pi\)
0.317484 + 0.948264i \(0.397162\pi\)
\(350\) −4.85792 −0.259667
\(351\) 0 0
\(352\) 5.68284 0.302896
\(353\) 1.82492 0.0971307 0.0485653 0.998820i \(-0.484535\pi\)
0.0485653 + 0.998820i \(0.484535\pi\)
\(354\) −37.9549 −2.01728
\(355\) 4.98449 0.264549
\(356\) 9.24612 0.490043
\(357\) 4.11955 0.218030
\(358\) −18.7313 −0.989982
\(359\) −7.62731 −0.402554 −0.201277 0.979534i \(-0.564509\pi\)
−0.201277 + 0.979534i \(0.564509\pi\)
\(360\) 2.98873 0.157520
\(361\) −16.3657 −0.861351
\(362\) −17.6561 −0.927982
\(363\) −70.3979 −3.69493
\(364\) 0 0
\(365\) −0.119551 −0.00625760
\(366\) 17.5408 0.916870
\(367\) 17.2236 0.899064 0.449532 0.893264i \(-0.351591\pi\)
0.449532 + 0.893264i \(0.351591\pi\)
\(368\) 4.92896 0.256940
\(369\) 8.49224 0.442089
\(370\) −2.61180 −0.135781
\(371\) 6.00000 0.311504
\(372\) −18.7869 −0.974054
\(373\) −8.77641 −0.454425 −0.227213 0.973845i \(-0.572961\pi\)
−0.227213 + 0.973845i \(0.572961\pi\)
\(374\) 7.08151 0.366176
\(375\) −12.2842 −0.634352
\(376\) 2.31716 0.119499
\(377\) 0 0
\(378\) −16.2946 −0.838105
\(379\) −17.7384 −0.911159 −0.455579 0.890195i \(-0.650568\pi\)
−0.455579 + 0.890195i \(0.650568\pi\)
\(380\) 0.611795 0.0313844
\(381\) 27.4212 1.40483
\(382\) −16.6118 −0.849933
\(383\) 32.7643 1.67418 0.837090 0.547065i \(-0.184255\pi\)
0.837090 + 0.547065i \(0.184255\pi\)
\(384\) 3.30590 0.168703
\(385\) −2.14208 −0.109171
\(386\) −3.85792 −0.196363
\(387\) −25.7384 −1.30835
\(388\) 10.9290 0.554834
\(389\) −12.2616 −0.621690 −0.310845 0.950461i \(-0.600612\pi\)
−0.310845 + 0.950461i \(0.600612\pi\)
\(390\) 0 0
\(391\) 6.14208 0.310618
\(392\) −1.00000 −0.0505076
\(393\) −54.0660 −2.72727
\(394\) 6.17508 0.311096
\(395\) −0.819873 −0.0412523
\(396\) −45.0590 −2.26430
\(397\) −22.5850 −1.13351 −0.566755 0.823887i \(-0.691801\pi\)
−0.566755 + 0.823887i \(0.691801\pi\)
\(398\) −4.75388 −0.238290
\(399\) −5.36567 −0.268620
\(400\) −4.85792 −0.242896
\(401\) 27.8579 1.39116 0.695579 0.718450i \(-0.255148\pi\)
0.695579 + 0.718450i \(0.255148\pi\)
\(402\) −29.5182 −1.47224
\(403\) 0 0
\(404\) 3.44798 0.171543
\(405\) −11.3389 −0.563434
\(406\) −4.61180 −0.228880
\(407\) 39.3761 1.95180
\(408\) 4.11955 0.203948
\(409\) −16.7313 −0.827312 −0.413656 0.910433i \(-0.635748\pi\)
−0.413656 + 0.910433i \(0.635748\pi\)
\(410\) 0.403717 0.0199382
\(411\) 43.7158 2.15634
\(412\) −7.24612 −0.356991
\(413\) 11.4810 0.564942
\(414\) −39.0815 −1.92075
\(415\) 3.67237 0.180270
\(416\) 0 0
\(417\) −5.36567 −0.262758
\(418\) −9.22359 −0.451141
\(419\) −11.7756 −0.575276 −0.287638 0.957739i \(-0.592870\pi\)
−0.287638 + 0.957739i \(0.592870\pi\)
\(420\) −1.24612 −0.0608045
\(421\) −12.3172 −0.600302 −0.300151 0.953892i \(-0.597037\pi\)
−0.300151 + 0.953892i \(0.597037\pi\)
\(422\) −27.7158 −1.34919
\(423\) −18.3727 −0.893311
\(424\) 6.00000 0.291386
\(425\) −6.05356 −0.293641
\(426\) −43.7158 −2.11804
\(427\) −5.30590 −0.256770
\(428\) −9.85792 −0.476500
\(429\) 0 0
\(430\) −1.22359 −0.0590068
\(431\) −0.611795 −0.0294691 −0.0147346 0.999891i \(-0.504690\pi\)
−0.0147346 + 0.999891i \(0.504690\pi\)
\(432\) −16.2946 −0.783976
\(433\) 25.2011 1.21109 0.605543 0.795813i \(-0.292956\pi\)
0.605543 + 0.795813i \(0.292956\pi\)
\(434\) 5.68284 0.272785
\(435\) −5.74686 −0.275541
\(436\) 11.2236 0.537512
\(437\) −8.00000 −0.382692
\(438\) 1.04851 0.0500997
\(439\) 36.9394 1.76302 0.881511 0.472163i \(-0.156527\pi\)
0.881511 + 0.472163i \(0.156527\pi\)
\(440\) −2.14208 −0.102120
\(441\) 7.92896 0.377569
\(442\) 0 0
\(443\) 2.77641 0.131911 0.0659556 0.997823i \(-0.478990\pi\)
0.0659556 + 0.997823i \(0.478990\pi\)
\(444\) 22.9064 1.08709
\(445\) −3.48522 −0.165215
\(446\) −8.29463 −0.392762
\(447\) −31.5408 −1.49183
\(448\) −1.00000 −0.0472456
\(449\) −4.35016 −0.205297 −0.102648 0.994718i \(-0.532732\pi\)
−0.102648 + 0.994718i \(0.532732\pi\)
\(450\) 38.5182 1.81577
\(451\) −6.08655 −0.286605
\(452\) −4.92896 −0.231839
\(453\) −9.10404 −0.427745
\(454\) 0.234856 0.0110223
\(455\) 0 0
\(456\) −5.36567 −0.251271
\(457\) 29.6048 1.38485 0.692426 0.721488i \(-0.256542\pi\)
0.692426 + 0.721488i \(0.256542\pi\)
\(458\) −19.4584 −0.909233
\(459\) −20.3051 −0.947761
\(460\) −1.85792 −0.0866258
\(461\) −30.2349 −1.40818 −0.704089 0.710112i \(-0.748644\pi\)
−0.704089 + 0.710112i \(0.748644\pi\)
\(462\) 18.7869 0.874045
\(463\) −4.87343 −0.226487 −0.113244 0.993567i \(-0.536124\pi\)
−0.113244 + 0.993567i \(0.536124\pi\)
\(464\) −4.61180 −0.214097
\(465\) 7.08151 0.328397
\(466\) 20.1525 0.933549
\(467\) −15.8847 −0.735056 −0.367528 0.930012i \(-0.619796\pi\)
−0.367528 + 0.930012i \(0.619796\pi\)
\(468\) 0 0
\(469\) 8.92896 0.412301
\(470\) −0.873429 −0.0402883
\(471\) −74.4802 −3.43187
\(472\) 11.4810 0.528455
\(473\) 18.4472 0.848202
\(474\) 7.19059 0.330275
\(475\) 7.88470 0.361775
\(476\) −1.24612 −0.0571159
\(477\) −47.5738 −2.17825
\(478\) −12.9620 −0.592866
\(479\) 3.53029 0.161303 0.0806515 0.996742i \(-0.474300\pi\)
0.0806515 + 0.996742i \(0.474300\pi\)
\(480\) −1.24612 −0.0568774
\(481\) 0 0
\(482\) 8.73135 0.397702
\(483\) 16.2946 0.741431
\(484\) 21.2946 0.967938
\(485\) −4.11955 −0.187059
\(486\) 50.5625 2.29356
\(487\) 16.6118 0.752752 0.376376 0.926467i \(-0.377170\pi\)
0.376376 + 0.926467i \(0.377170\pi\)
\(488\) −5.30590 −0.240187
\(489\) −28.0000 −1.26620
\(490\) 0.376939 0.0170284
\(491\) −10.3812 −0.468496 −0.234248 0.972177i \(-0.575263\pi\)
−0.234248 + 0.972177i \(0.575263\pi\)
\(492\) −3.54075 −0.159629
\(493\) −5.74686 −0.258826
\(494\) 0 0
\(495\) 16.9845 0.763396
\(496\) 5.68284 0.255167
\(497\) 13.2236 0.593159
\(498\) −32.2081 −1.44328
\(499\) −9.80239 −0.438815 −0.219408 0.975633i \(-0.570412\pi\)
−0.219408 + 0.975633i \(0.570412\pi\)
\(500\) 3.71583 0.166177
\(501\) 19.7609 0.882852
\(502\) 5.04426 0.225136
\(503\) −12.3502 −0.550666 −0.275333 0.961349i \(-0.588788\pi\)
−0.275333 + 0.961349i \(0.588788\pi\)
\(504\) 7.92896 0.353184
\(505\) −1.29968 −0.0578349
\(506\) 28.0105 1.24522
\(507\) 0 0
\(508\) −8.29463 −0.368015
\(509\) −10.1153 −0.448353 −0.224176 0.974549i \(-0.571969\pi\)
−0.224176 + 0.974549i \(0.571969\pi\)
\(510\) −1.55282 −0.0687600
\(511\) −0.317163 −0.0140305
\(512\) −1.00000 −0.0441942
\(513\) 26.4472 1.16767
\(514\) 23.2236 1.02435
\(515\) 2.73135 0.120357
\(516\) 10.7313 0.472421
\(517\) 13.1681 0.579131
\(518\) −6.92896 −0.304441
\(519\) −2.79894 −0.122860
\(520\) 0 0
\(521\) −32.6118 −1.42875 −0.714374 0.699764i \(-0.753289\pi\)
−0.714374 + 0.699764i \(0.753289\pi\)
\(522\) 36.5667 1.60048
\(523\) 32.4099 1.41719 0.708594 0.705617i \(-0.249330\pi\)
0.708594 + 0.705617i \(0.249330\pi\)
\(524\) 16.3544 0.714446
\(525\) −16.0598 −0.700906
\(526\) 11.1040 0.484159
\(527\) 7.08151 0.308475
\(528\) 18.7869 0.817594
\(529\) 1.29463 0.0562883
\(530\) −2.26163 −0.0982391
\(531\) −91.0322 −3.95046
\(532\) 1.62306 0.0703686
\(533\) 0 0
\(534\) 30.5667 1.32275
\(535\) 3.71583 0.160650
\(536\) 8.92896 0.385672
\(537\) −61.9239 −2.67221
\(538\) −11.5675 −0.498712
\(539\) −5.68284 −0.244777
\(540\) 6.14208 0.264313
\(541\) −23.4542 −1.00837 −0.504187 0.863594i \(-0.668208\pi\)
−0.504187 + 0.863594i \(0.668208\pi\)
\(542\) 26.1525 1.12335
\(543\) −58.3691 −2.50486
\(544\) −1.24612 −0.0534270
\(545\) −4.23061 −0.181219
\(546\) 0 0
\(547\) 13.3882 0.572438 0.286219 0.958164i \(-0.407601\pi\)
0.286219 + 0.958164i \(0.407601\pi\)
\(548\) −13.2236 −0.564884
\(549\) 42.0702 1.79551
\(550\) −27.6067 −1.17716
\(551\) 7.48522 0.318881
\(552\) 16.2946 0.693546
\(553\) −2.17508 −0.0924938
\(554\) −23.2236 −0.986676
\(555\) −8.63433 −0.366507
\(556\) 1.62306 0.0688331
\(557\) 23.0485 0.976597 0.488298 0.872677i \(-0.337618\pi\)
0.488298 + 0.872677i \(0.337618\pi\)
\(558\) −45.0590 −1.90750
\(559\) 0 0
\(560\) 0.376939 0.0159286
\(561\) 23.4107 0.988402
\(562\) 17.2236 0.726533
\(563\) −0.858717 −0.0361906 −0.0180953 0.999836i \(-0.505760\pi\)
−0.0180953 + 0.999836i \(0.505760\pi\)
\(564\) 7.66030 0.322557
\(565\) 1.85792 0.0781632
\(566\) 19.7756 0.831231
\(567\) −30.0815 −1.26330
\(568\) 13.2236 0.554850
\(569\) 3.31014 0.138768 0.0693842 0.997590i \(-0.477897\pi\)
0.0693842 + 0.997590i \(0.477897\pi\)
\(570\) 2.02253 0.0847145
\(571\) 21.6273 0.905075 0.452537 0.891745i \(-0.350519\pi\)
0.452537 + 0.891745i \(0.350519\pi\)
\(572\) 0 0
\(573\) −54.9169 −2.29419
\(574\) 1.07104 0.0447044
\(575\) −23.9445 −0.998553
\(576\) 7.92896 0.330373
\(577\) −9.48522 −0.394875 −0.197438 0.980315i \(-0.563262\pi\)
−0.197438 + 0.980315i \(0.563262\pi\)
\(578\) 15.4472 0.642518
\(579\) −12.7539 −0.530033
\(580\) 1.73837 0.0721817
\(581\) 9.74261 0.404192
\(582\) 36.1300 1.49764
\(583\) 34.0970 1.41215
\(584\) −0.317163 −0.0131243
\(585\) 0 0
\(586\) −1.13082 −0.0467136
\(587\) 36.4204 1.50323 0.751615 0.659602i \(-0.229275\pi\)
0.751615 + 0.659602i \(0.229275\pi\)
\(588\) −3.30590 −0.136333
\(589\) −9.22359 −0.380051
\(590\) −4.32763 −0.178166
\(591\) 20.4142 0.839727
\(592\) −6.92896 −0.284778
\(593\) 21.4852 0.882292 0.441146 0.897435i \(-0.354572\pi\)
0.441146 + 0.897435i \(0.354572\pi\)
\(594\) −92.5997 −3.79941
\(595\) 0.469712 0.0192563
\(596\) 9.54075 0.390804
\(597\) −15.7158 −0.643206
\(598\) 0 0
\(599\) 1.89091 0.0772607 0.0386303 0.999254i \(-0.487701\pi\)
0.0386303 + 0.999254i \(0.487701\pi\)
\(600\) −16.0598 −0.655638
\(601\) −2.35016 −0.0958651 −0.0479325 0.998851i \(-0.515263\pi\)
−0.0479325 + 0.998851i \(0.515263\pi\)
\(602\) −3.24612 −0.132302
\(603\) −70.7973 −2.88309
\(604\) 2.75388 0.112054
\(605\) −8.02678 −0.326335
\(606\) 11.3987 0.463039
\(607\) −28.3502 −1.15070 −0.575349 0.817908i \(-0.695134\pi\)
−0.575349 + 0.817908i \(0.695134\pi\)
\(608\) 1.62306 0.0658238
\(609\) −15.2461 −0.617804
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 9.88045 0.399393
\(613\) −21.3101 −0.860709 −0.430354 0.902660i \(-0.641611\pi\)
−0.430354 + 0.902660i \(0.641611\pi\)
\(614\) −30.9662 −1.24969
\(615\) 1.33465 0.0538182
\(616\) −5.68284 −0.228968
\(617\) −7.85792 −0.316348 −0.158174 0.987411i \(-0.550561\pi\)
−0.158174 + 0.987411i \(0.550561\pi\)
\(618\) −23.9549 −0.963609
\(619\) 32.3544 1.30043 0.650217 0.759749i \(-0.274678\pi\)
0.650217 + 0.759749i \(0.274678\pi\)
\(620\) −2.14208 −0.0860281
\(621\) −80.3156 −3.22295
\(622\) −9.50776 −0.381226
\(623\) −9.24612 −0.370438
\(624\) 0 0
\(625\) 22.8889 0.915558
\(626\) −3.38820 −0.135420
\(627\) −30.4922 −1.21774
\(628\) 22.5295 0.899024
\(629\) −8.63433 −0.344273
\(630\) −2.98873 −0.119074
\(631\) 27.9549 1.11287 0.556434 0.830892i \(-0.312169\pi\)
0.556434 + 0.830892i \(0.312169\pi\)
\(632\) −2.17508 −0.0865201
\(633\) −91.6257 −3.64179
\(634\) −29.5408 −1.17321
\(635\) 3.12657 0.124074
\(636\) 19.8354 0.786524
\(637\) 0 0
\(638\) −26.2081 −1.03759
\(639\) −104.849 −4.14777
\(640\) 0.376939 0.0148998
\(641\) −15.0260 −0.593490 −0.296745 0.954957i \(-0.595901\pi\)
−0.296745 + 0.954957i \(0.595901\pi\)
\(642\) −32.5893 −1.28620
\(643\) 9.27290 0.365687 0.182844 0.983142i \(-0.441470\pi\)
0.182844 + 0.983142i \(0.441470\pi\)
\(644\) −4.92896 −0.194228
\(645\) −4.04506 −0.159274
\(646\) 2.02253 0.0795754
\(647\) −20.1196 −0.790981 −0.395491 0.918470i \(-0.629425\pi\)
−0.395491 + 0.918470i \(0.629425\pi\)
\(648\) −30.0815 −1.18171
\(649\) 65.2445 2.56107
\(650\) 0 0
\(651\) 18.7869 0.736316
\(652\) 8.46971 0.331700
\(653\) 25.3121 0.990540 0.495270 0.868739i \(-0.335069\pi\)
0.495270 + 0.868739i \(0.335069\pi\)
\(654\) 37.1040 1.45088
\(655\) −6.16461 −0.240871
\(656\) 1.07104 0.0418171
\(657\) 2.51478 0.0981107
\(658\) −2.31716 −0.0903324
\(659\) −15.2461 −0.593905 −0.296952 0.954892i \(-0.595970\pi\)
−0.296952 + 0.954892i \(0.595970\pi\)
\(660\) −7.08151 −0.275647
\(661\) −29.4359 −1.14492 −0.572462 0.819931i \(-0.694012\pi\)
−0.572462 + 0.819931i \(0.694012\pi\)
\(662\) 0.175080 0.00680469
\(663\) 0 0
\(664\) 9.74261 0.378087
\(665\) −0.611795 −0.0237244
\(666\) 54.9394 2.12886
\(667\) −22.7313 −0.880161
\(668\) −5.97747 −0.231275
\(669\) −27.4212 −1.06016
\(670\) −3.36567 −0.130027
\(671\) −30.1525 −1.16403
\(672\) −3.30590 −0.127528
\(673\) −26.6448 −1.02708 −0.513541 0.858065i \(-0.671666\pi\)
−0.513541 + 0.858065i \(0.671666\pi\)
\(674\) 19.3761 0.746341
\(675\) 79.1580 3.04679
\(676\) 0 0
\(677\) 39.1188 1.50346 0.751728 0.659473i \(-0.229221\pi\)
0.751728 + 0.659473i \(0.229221\pi\)
\(678\) −16.2946 −0.625792
\(679\) −10.9290 −0.419415
\(680\) 0.469712 0.0180126
\(681\) 0.776410 0.0297521
\(682\) 32.2946 1.23663
\(683\) 12.0105 0.459568 0.229784 0.973242i \(-0.426198\pi\)
0.229784 + 0.973242i \(0.426198\pi\)
\(684\) −12.8692 −0.492066
\(685\) 4.98449 0.190447
\(686\) 1.00000 0.0381802
\(687\) −64.3276 −2.45425
\(688\) −3.24612 −0.123757
\(689\) 0 0
\(690\) −6.14208 −0.233825
\(691\) 50.2123 1.91017 0.955083 0.296337i \(-0.0957652\pi\)
0.955083 + 0.296337i \(0.0957652\pi\)
\(692\) 0.846651 0.0321848
\(693\) 45.0590 1.71165
\(694\) −24.4697 −0.928858
\(695\) −0.611795 −0.0232067
\(696\) −15.2461 −0.577903
\(697\) 1.33465 0.0505534
\(698\) −11.8622 −0.448990
\(699\) 66.6223 2.51989
\(700\) 4.85792 0.183612
\(701\) −31.6933 −1.19704 −0.598520 0.801108i \(-0.704244\pi\)
−0.598520 + 0.801108i \(0.704244\pi\)
\(702\) 0 0
\(703\) 11.2461 0.424156
\(704\) −5.68284 −0.214180
\(705\) −2.88747 −0.108748
\(706\) −1.82492 −0.0686818
\(707\) −3.44798 −0.129675
\(708\) 37.9549 1.42643
\(709\) 10.6448 0.399774 0.199887 0.979819i \(-0.435943\pi\)
0.199887 + 0.979819i \(0.435943\pi\)
\(710\) −4.98449 −0.187064
\(711\) 17.2461 0.646780
\(712\) −9.24612 −0.346513
\(713\) 28.0105 1.04900
\(714\) −4.11955 −0.154170
\(715\) 0 0
\(716\) 18.7313 0.700023
\(717\) −42.8509 −1.60030
\(718\) 7.62731 0.284649
\(719\) 19.5303 0.728357 0.364178 0.931329i \(-0.381350\pi\)
0.364178 + 0.931329i \(0.381350\pi\)
\(720\) −2.98873 −0.111384
\(721\) 7.24612 0.269860
\(722\) 16.3657 0.609067
\(723\) 28.8649 1.07350
\(724\) 17.6561 0.656182
\(725\) 22.4037 0.832053
\(726\) 70.3979 2.61271
\(727\) 29.1040 1.07941 0.539705 0.841855i \(-0.318536\pi\)
0.539705 + 0.841855i \(0.318536\pi\)
\(728\) 0 0
\(729\) 76.9099 2.84851
\(730\) 0.119551 0.00442479
\(731\) −4.04506 −0.149612
\(732\) −17.5408 −0.648325
\(733\) 43.9282 1.62252 0.811262 0.584683i \(-0.198781\pi\)
0.811262 + 0.584683i \(0.198781\pi\)
\(734\) −17.2236 −0.635734
\(735\) 1.24612 0.0459639
\(736\) −4.92896 −0.181684
\(737\) 50.7418 1.86910
\(738\) −8.49224 −0.312604
\(739\) −43.4852 −1.59963 −0.799815 0.600247i \(-0.795069\pi\)
−0.799815 + 0.600247i \(0.795069\pi\)
\(740\) 2.61180 0.0960115
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 30.4697 1.11783 0.558913 0.829227i \(-0.311219\pi\)
0.558913 + 0.829227i \(0.311219\pi\)
\(744\) 18.7869 0.688760
\(745\) −3.59628 −0.131758
\(746\) 8.77641 0.321327
\(747\) −77.2488 −2.82638
\(748\) −7.08151 −0.258926
\(749\) 9.85792 0.360200
\(750\) 12.2842 0.448554
\(751\) 11.6603 0.425491 0.212745 0.977108i \(-0.431760\pi\)
0.212745 + 0.977108i \(0.431760\pi\)
\(752\) −2.31716 −0.0844983
\(753\) 16.6758 0.607701
\(754\) 0 0
\(755\) −1.03804 −0.0377783
\(756\) 16.2946 0.592630
\(757\) −8.02253 −0.291584 −0.145792 0.989315i \(-0.546573\pi\)
−0.145792 + 0.989315i \(0.546573\pi\)
\(758\) 17.7384 0.644286
\(759\) 92.5997 3.36116
\(760\) −0.611795 −0.0221921
\(761\) 0.831939 0.0301578 0.0150789 0.999886i \(-0.495200\pi\)
0.0150789 + 0.999886i \(0.495200\pi\)
\(762\) −27.4212 −0.993365
\(763\) −11.2236 −0.406321
\(764\) 16.6118 0.600994
\(765\) −3.72433 −0.134653
\(766\) −32.7643 −1.18382
\(767\) 0 0
\(768\) −3.30590 −0.119291
\(769\) 21.0710 0.759841 0.379921 0.925019i \(-0.375951\pi\)
0.379921 + 0.925019i \(0.375951\pi\)
\(770\) 2.14208 0.0771953
\(771\) 76.7748 2.76498
\(772\) 3.85792 0.138849
\(773\) −4.84665 −0.174322 −0.0871610 0.996194i \(-0.527779\pi\)
−0.0871610 + 0.996194i \(0.527779\pi\)
\(774\) 25.7384 0.925146
\(775\) −27.6067 −0.991664
\(776\) −10.9290 −0.392327
\(777\) −22.9064 −0.821763
\(778\) 12.2616 0.439601
\(779\) −1.73837 −0.0622834
\(780\) 0 0
\(781\) 75.1475 2.68899
\(782\) −6.14208 −0.219640
\(783\) 75.1475 2.68555
\(784\) 1.00000 0.0357143
\(785\) −8.49224 −0.303101
\(786\) 54.0660 1.92847
\(787\) −35.6005 −1.26902 −0.634511 0.772914i \(-0.718798\pi\)
−0.634511 + 0.772914i \(0.718798\pi\)
\(788\) −6.17508 −0.219978
\(789\) 36.7088 1.30687
\(790\) 0.819873 0.0291698
\(791\) 4.92896 0.175254
\(792\) 45.0590 1.60110
\(793\) 0 0
\(794\) 22.5850 0.801512
\(795\) −7.47673 −0.265172
\(796\) 4.75388 0.168497
\(797\) −24.5520 −0.869677 −0.434839 0.900508i \(-0.643195\pi\)
−0.434839 + 0.900508i \(0.643195\pi\)
\(798\) 5.36567 0.189943
\(799\) −2.88747 −0.102151
\(800\) 4.85792 0.171753
\(801\) 73.3121 2.59036
\(802\) −27.8579 −0.983697
\(803\) −1.80239 −0.0636049
\(804\) 29.5182 1.04103
\(805\) 1.85792 0.0654830
\(806\) 0 0
\(807\) −38.2411 −1.34615
\(808\) −3.44798 −0.121300
\(809\) 38.0970 1.33942 0.669710 0.742623i \(-0.266418\pi\)
0.669710 + 0.742623i \(0.266418\pi\)
\(810\) 11.3389 0.398408
\(811\) 36.3009 1.27470 0.637348 0.770576i \(-0.280032\pi\)
0.637348 + 0.770576i \(0.280032\pi\)
\(812\) 4.61180 0.161842
\(813\) 86.4576 3.03220
\(814\) −39.3761 −1.38013
\(815\) −3.19257 −0.111831
\(816\) −4.11955 −0.144213
\(817\) 5.26865 0.184327
\(818\) 16.7313 0.584998
\(819\) 0 0
\(820\) −0.403717 −0.0140984
\(821\) 34.4697 1.20300 0.601501 0.798872i \(-0.294570\pi\)
0.601501 + 0.798872i \(0.294570\pi\)
\(822\) −43.7158 −1.52476
\(823\) 3.04851 0.106264 0.0531322 0.998587i \(-0.483080\pi\)
0.0531322 + 0.998587i \(0.483080\pi\)
\(824\) 7.24612 0.252431
\(825\) −91.2651 −3.17744
\(826\) −11.4810 −0.399474
\(827\) 23.6708 0.823113 0.411557 0.911384i \(-0.364985\pi\)
0.411557 + 0.911384i \(0.364985\pi\)
\(828\) 39.0815 1.35818
\(829\) 9.83114 0.341450 0.170725 0.985319i \(-0.445389\pi\)
0.170725 + 0.985319i \(0.445389\pi\)
\(830\) −3.67237 −0.127470
\(831\) −76.7748 −2.66329
\(832\) 0 0
\(833\) 1.24612 0.0431756
\(834\) 5.36567 0.185798
\(835\) 2.25314 0.0779732
\(836\) 9.22359 0.319005
\(837\) −92.5997 −3.20071
\(838\) 11.7756 0.406782
\(839\) −25.6828 −0.886670 −0.443335 0.896356i \(-0.646205\pi\)
−0.443335 + 0.896356i \(0.646205\pi\)
\(840\) 1.24612 0.0429953
\(841\) −7.73135 −0.266598
\(842\) 12.3172 0.424477
\(843\) 56.9394 1.96110
\(844\) 27.7158 0.954018
\(845\) 0 0
\(846\) 18.3727 0.631666
\(847\) −21.2946 −0.731692
\(848\) −6.00000 −0.206041
\(849\) 65.3761 2.24370
\(850\) 6.05356 0.207635
\(851\) −34.1525 −1.17073
\(852\) 43.7158 1.49768
\(853\) −23.1743 −0.793472 −0.396736 0.917933i \(-0.629857\pi\)
−0.396736 + 0.917933i \(0.629857\pi\)
\(854\) 5.30590 0.181564
\(855\) 4.85090 0.165897
\(856\) 9.85792 0.336937
\(857\) −26.8199 −0.916149 −0.458075 0.888914i \(-0.651461\pi\)
−0.458075 + 0.888914i \(0.651461\pi\)
\(858\) 0 0
\(859\) 11.7756 0.401779 0.200889 0.979614i \(-0.435617\pi\)
0.200889 + 0.979614i \(0.435617\pi\)
\(860\) 1.22359 0.0417241
\(861\) 3.54075 0.120669
\(862\) 0.611795 0.0208378
\(863\) −8.91849 −0.303589 −0.151795 0.988412i \(-0.548505\pi\)
−0.151795 + 0.988412i \(0.548505\pi\)
\(864\) 16.2946 0.554355
\(865\) −0.319136 −0.0108509
\(866\) −25.2011 −0.856367
\(867\) 51.0668 1.73432
\(868\) −5.68284 −0.192888
\(869\) −12.3606 −0.419306
\(870\) 5.74686 0.194837
\(871\) 0 0
\(872\) −11.2236 −0.380079
\(873\) 86.6553 2.93284
\(874\) 8.00000 0.270604
\(875\) −3.71583 −0.125618
\(876\) −1.04851 −0.0354259
\(877\) −34.2496 −1.15653 −0.578263 0.815851i \(-0.696269\pi\)
−0.578263 + 0.815851i \(0.696269\pi\)
\(878\) −36.9394 −1.24665
\(879\) −3.73837 −0.126092
\(880\) 2.14208 0.0722096
\(881\) 1.41074 0.0475289 0.0237645 0.999718i \(-0.492435\pi\)
0.0237645 + 0.999718i \(0.492435\pi\)
\(882\) −7.92896 −0.266982
\(883\) 13.5078 0.454572 0.227286 0.973828i \(-0.427015\pi\)
0.227286 + 0.973828i \(0.427015\pi\)
\(884\) 0 0
\(885\) −14.3067 −0.480914
\(886\) −2.77641 −0.0932753
\(887\) −10.2616 −0.344552 −0.172276 0.985049i \(-0.555112\pi\)
−0.172276 + 0.985049i \(0.555112\pi\)
\(888\) −22.9064 −0.768689
\(889\) 8.29463 0.278193
\(890\) 3.48522 0.116825
\(891\) −170.948 −5.72698
\(892\) 8.29463 0.277725
\(893\) 3.76090 0.125854
\(894\) 31.5408 1.05488
\(895\) −7.06058 −0.236009
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 4.35016 0.145167
\(899\) −26.2081 −0.874088
\(900\) −38.5182 −1.28394
\(901\) −7.47673 −0.249086
\(902\) 6.08655 0.202660
\(903\) −10.7313 −0.357117
\(904\) 4.92896 0.163935
\(905\) −6.65526 −0.221228
\(906\) 9.10404 0.302461
\(907\) −47.9549 −1.59232 −0.796159 0.605088i \(-0.793138\pi\)
−0.796159 + 0.605088i \(0.793138\pi\)
\(908\) −0.234856 −0.00779397
\(909\) 27.3389 0.906774
\(910\) 0 0
\(911\) −23.4317 −0.776326 −0.388163 0.921591i \(-0.626890\pi\)
−0.388163 + 0.921591i \(0.626890\pi\)
\(912\) 5.36567 0.177675
\(913\) 55.3657 1.83234
\(914\) −29.6048 −0.979239
\(915\) 6.61180 0.218579
\(916\) 19.4584 0.642925
\(917\) −16.3544 −0.540070
\(918\) 20.3051 0.670168
\(919\) −41.2566 −1.36093 −0.680465 0.732781i \(-0.738222\pi\)
−0.680465 + 0.732781i \(0.738222\pi\)
\(920\) 1.85792 0.0612537
\(921\) −102.371 −3.37324
\(922\) 30.2349 0.995732
\(923\) 0 0
\(924\) −18.7869 −0.618043
\(925\) 33.6603 1.10674
\(926\) 4.87343 0.160151
\(927\) −57.4542 −1.88704
\(928\) 4.61180 0.151390
\(929\) −51.5842 −1.69242 −0.846212 0.532847i \(-0.821122\pi\)
−0.846212 + 0.532847i \(0.821122\pi\)
\(930\) −7.08151 −0.232212
\(931\) −1.62306 −0.0531937
\(932\) −20.1525 −0.660119
\(933\) −31.4317 −1.02903
\(934\) 15.8847 0.519763
\(935\) 2.66930 0.0872953
\(936\) 0 0
\(937\) 2.28417 0.0746205 0.0373102 0.999304i \(-0.488121\pi\)
0.0373102 + 0.999304i \(0.488121\pi\)
\(938\) −8.92896 −0.291541
\(939\) −11.2011 −0.365533
\(940\) 0.873429 0.0284881
\(941\) 6.51902 0.212514 0.106257 0.994339i \(-0.466113\pi\)
0.106257 + 0.994339i \(0.466113\pi\)
\(942\) 74.4802 2.42670
\(943\) 5.27912 0.171912
\(944\) −11.4810 −0.373674
\(945\) −6.14208 −0.199802
\(946\) −18.4472 −0.599770
\(947\) −23.9464 −0.778155 −0.389077 0.921205i \(-0.627206\pi\)
−0.389077 + 0.921205i \(0.627206\pi\)
\(948\) −7.19059 −0.233540
\(949\) 0 0
\(950\) −7.88470 −0.255813
\(951\) −97.6587 −3.16680
\(952\) 1.24612 0.0403870
\(953\) 6.28417 0.203564 0.101782 0.994807i \(-0.467546\pi\)
0.101782 + 0.994807i \(0.467546\pi\)
\(954\) 47.5738 1.54026
\(955\) −6.26163 −0.202622
\(956\) 12.9620 0.419220
\(957\) −86.6412 −2.80071
\(958\) −3.53029 −0.114058
\(959\) 13.2236 0.427012
\(960\) 1.24612 0.0402184
\(961\) 1.29463 0.0417623
\(962\) 0 0
\(963\) −78.1630 −2.51877
\(964\) −8.73135 −0.281218
\(965\) −1.45420 −0.0468123
\(966\) −16.2946 −0.524271
\(967\) 57.2671 1.84158 0.920792 0.390054i \(-0.127544\pi\)
0.920792 + 0.390054i \(0.127544\pi\)
\(968\) −21.2946 −0.684435
\(969\) 6.68628 0.214794
\(970\) 4.11955 0.132271
\(971\) −49.1188 −1.57630 −0.788148 0.615486i \(-0.788960\pi\)
−0.788148 + 0.615486i \(0.788960\pi\)
\(972\) −50.5625 −1.62179
\(973\) −1.62306 −0.0520329
\(974\) −16.6118 −0.532276
\(975\) 0 0
\(976\) 5.30590 0.169838
\(977\) −12.0660 −0.386025 −0.193013 0.981196i \(-0.561826\pi\)
−0.193013 + 0.981196i \(0.561826\pi\)
\(978\) 28.0000 0.895341
\(979\) −52.5442 −1.67932
\(980\) −0.376939 −0.0120409
\(981\) 88.9914 2.84128
\(982\) 10.3812 0.331277
\(983\) 54.2166 1.72924 0.864620 0.502426i \(-0.167559\pi\)
0.864620 + 0.502426i \(0.167559\pi\)
\(984\) 3.54075 0.112875
\(985\) 2.32763 0.0741644
\(986\) 5.74686 0.183017
\(987\) −7.66030 −0.243830
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) −16.9845 −0.539802
\(991\) 38.1751 1.21267 0.606336 0.795209i \(-0.292639\pi\)
0.606336 + 0.795209i \(0.292639\pi\)
\(992\) −5.68284 −0.180430
\(993\) 0.578798 0.0183676
\(994\) −13.2236 −0.419427
\(995\) −1.79192 −0.0568078
\(996\) 32.2081 1.02055
\(997\) 6.05978 0.191915 0.0959575 0.995385i \(-0.469409\pi\)
0.0959575 + 0.995385i \(0.469409\pi\)
\(998\) 9.80239 0.310289
\(999\) 112.905 3.57215
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 2366.2.a.x.1.1 3
13.5 odd 4 182.2.d.b.155.4 yes 6
13.8 odd 4 182.2.d.b.155.1 6
13.12 even 2 2366.2.a.bc.1.1 3
39.5 even 4 1638.2.c.i.883.2 6
39.8 even 4 1638.2.c.i.883.5 6
52.31 even 4 1456.2.k.b.337.6 6
52.47 even 4 1456.2.k.b.337.5 6
91.5 even 12 1274.2.n.n.753.4 12
91.18 odd 12 1274.2.n.m.961.3 12
91.31 even 12 1274.2.n.n.961.1 12
91.34 even 4 1274.2.d.l.883.3 6
91.44 odd 12 1274.2.n.m.753.6 12
91.47 even 12 1274.2.n.n.753.1 12
91.60 odd 12 1274.2.n.m.961.6 12
91.73 even 12 1274.2.n.n.961.4 12
91.83 even 4 1274.2.d.l.883.6 6
91.86 odd 12 1274.2.n.m.753.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.d.b.155.1 6 13.8 odd 4
182.2.d.b.155.4 yes 6 13.5 odd 4
1274.2.d.l.883.3 6 91.34 even 4
1274.2.d.l.883.6 6 91.83 even 4
1274.2.n.m.753.3 12 91.86 odd 12
1274.2.n.m.753.6 12 91.44 odd 12
1274.2.n.m.961.3 12 91.18 odd 12
1274.2.n.m.961.6 12 91.60 odd 12
1274.2.n.n.753.1 12 91.47 even 12
1274.2.n.n.753.4 12 91.5 even 12
1274.2.n.n.961.1 12 91.31 even 12
1274.2.n.n.961.4 12 91.73 even 12
1456.2.k.b.337.5 6 52.47 even 4
1456.2.k.b.337.6 6 52.31 even 4
1638.2.c.i.883.2 6 39.5 even 4
1638.2.c.i.883.5 6 39.8 even 4
2366.2.a.x.1.1 3 1.1 even 1 trivial
2366.2.a.bc.1.1 3 13.12 even 2