Properties

Label 9025.2.a.bn.1.2
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, 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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.7168.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.25928\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +2.24264 q^{6} +0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} +O(q^{10})\) \(q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +2.24264 q^{6} +0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} +2.00000 q^{11} +0.737669 q^{12} -4.29945 q^{13} -1.04322 q^{14} -3.00000 q^{16} -7.65685 q^{17} -0.216058 q^{18} -1.47534 q^{21} -2.51856 q^{22} +0.828427 q^{23} -5.41421 q^{24} +5.41421 q^{26} +5.03712 q^{27} -0.343146 q^{28} +8.59890 q^{29} +3.56178 q^{31} -2.30250 q^{32} -3.56178 q^{33} +9.64212 q^{34} -0.0710678 q^{36} -0.737669 q^{37} +7.65685 q^{39} +1.85786 q^{42} +4.82843 q^{43} -0.828427 q^{44} -1.04322 q^{46} -10.4853 q^{47} +5.34267 q^{48} -6.31371 q^{49} +13.6360 q^{51} +1.78089 q^{52} +7.86123 q^{53} -6.34315 q^{54} +2.51856 q^{56} -10.8284 q^{58} -7.12356 q^{59} -2.82843 q^{61} -4.48528 q^{62} +0.142136 q^{63} +8.89949 q^{64} +4.48528 q^{66} -6.81801 q^{67} +3.17157 q^{68} -1.47534 q^{69} +13.6360 q^{71} +0.521611 q^{72} +11.6569 q^{73} +0.928932 q^{74} +1.65685 q^{77} -9.64212 q^{78} +7.12356 q^{79} -9.48528 q^{81} -8.82843 q^{83} +0.611105 q^{84} -6.08034 q^{86} -15.3137 q^{87} +6.08034 q^{88} -15.7225 q^{89} -3.56178 q^{91} -0.343146 q^{92} -6.34315 q^{93} +13.2039 q^{94} +4.10051 q^{96} -4.29945 q^{97} +7.95073 q^{98} +0.343146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} - 8 q^{17} - 8 q^{23} - 16 q^{24} + 16 q^{26} - 24 q^{28} + 28 q^{36} + 8 q^{39} + 64 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{47} + 20 q^{49} - 48 q^{54} - 32 q^{58} + 16 q^{62} - 56 q^{63} - 4 q^{64} - 16 q^{66} + 24 q^{68} + 24 q^{73} + 32 q^{74} - 16 q^{77} - 4 q^{81} - 24 q^{83} - 16 q^{87} - 24 q^{92} - 48 q^{93} + 56 q^{96} + 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25928 −0.890446 −0.445223 0.895420i \(-0.646876\pi\)
−0.445223 + 0.895420i \(0.646876\pi\)
\(3\) −1.78089 −1.02820 −0.514099 0.857731i \(-0.671874\pi\)
−0.514099 + 0.857731i \(0.671874\pi\)
\(4\) −0.414214 −0.207107
\(5\) 0 0
\(6\) 2.24264 0.915554
\(7\) 0.828427 0.313116 0.156558 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 3.04017 1.07486
\(9\) 0.171573 0.0571910
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0.737669 0.212947
\(13\) −4.29945 −1.19245 −0.596227 0.802816i \(-0.703334\pi\)
−0.596227 + 0.802816i \(0.703334\pi\)
\(14\) −1.04322 −0.278813
\(15\) 0 0
\(16\) −3.00000 −0.750000
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) −0.216058 −0.0509254
\(19\) 0 0
\(20\) 0 0
\(21\) −1.47534 −0.321945
\(22\) −2.51856 −0.536959
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) −5.41421 −1.10517
\(25\) 0 0
\(26\) 5.41421 1.06181
\(27\) 5.03712 0.969394
\(28\) −0.343146 −0.0648485
\(29\) 8.59890 1.59678 0.798388 0.602143i \(-0.205686\pi\)
0.798388 + 0.602143i \(0.205686\pi\)
\(30\) 0 0
\(31\) 3.56178 0.639715 0.319857 0.947466i \(-0.396365\pi\)
0.319857 + 0.947466i \(0.396365\pi\)
\(32\) −2.30250 −0.407029
\(33\) −3.56178 −0.620027
\(34\) 9.64212 1.65361
\(35\) 0 0
\(36\) −0.0710678 −0.0118446
\(37\) −0.737669 −0.121272 −0.0606360 0.998160i \(-0.519313\pi\)
−0.0606360 + 0.998160i \(0.519313\pi\)
\(38\) 0 0
\(39\) 7.65685 1.22608
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.85786 0.286675
\(43\) 4.82843 0.736328 0.368164 0.929761i \(-0.379986\pi\)
0.368164 + 0.929761i \(0.379986\pi\)
\(44\) −0.828427 −0.124890
\(45\) 0 0
\(46\) −1.04322 −0.153815
\(47\) −10.4853 −1.52944 −0.764718 0.644365i \(-0.777122\pi\)
−0.764718 + 0.644365i \(0.777122\pi\)
\(48\) 5.34267 0.771148
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) 13.6360 1.90943
\(52\) 1.78089 0.246965
\(53\) 7.86123 1.07982 0.539912 0.841722i \(-0.318458\pi\)
0.539912 + 0.841722i \(0.318458\pi\)
\(54\) −6.34315 −0.863193
\(55\) 0 0
\(56\) 2.51856 0.336557
\(57\) 0 0
\(58\) −10.8284 −1.42184
\(59\) −7.12356 −0.927409 −0.463705 0.885990i \(-0.653480\pi\)
−0.463705 + 0.885990i \(0.653480\pi\)
\(60\) 0 0
\(61\) −2.82843 −0.362143 −0.181071 0.983470i \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) −4.48528 −0.569631
\(63\) 0.142136 0.0179074
\(64\) 8.89949 1.11244
\(65\) 0 0
\(66\) 4.48528 0.552100
\(67\) −6.81801 −0.832953 −0.416476 0.909147i \(-0.636735\pi\)
−0.416476 + 0.909147i \(0.636735\pi\)
\(68\) 3.17157 0.384610
\(69\) −1.47534 −0.177610
\(70\) 0 0
\(71\) 13.6360 1.61830 0.809149 0.587603i \(-0.199928\pi\)
0.809149 + 0.587603i \(0.199928\pi\)
\(72\) 0.521611 0.0614724
\(73\) 11.6569 1.36433 0.682166 0.731198i \(-0.261038\pi\)
0.682166 + 0.731198i \(0.261038\pi\)
\(74\) 0.928932 0.107986
\(75\) 0 0
\(76\) 0 0
\(77\) 1.65685 0.188816
\(78\) −9.64212 −1.09176
\(79\) 7.12356 0.801464 0.400732 0.916195i \(-0.368756\pi\)
0.400732 + 0.916195i \(0.368756\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) −8.82843 −0.969046 −0.484523 0.874779i \(-0.661007\pi\)
−0.484523 + 0.874779i \(0.661007\pi\)
\(84\) 0.611105 0.0666770
\(85\) 0 0
\(86\) −6.08034 −0.655660
\(87\) −15.3137 −1.64180
\(88\) 6.08034 0.648167
\(89\) −15.7225 −1.66658 −0.833289 0.552838i \(-0.813545\pi\)
−0.833289 + 0.552838i \(0.813545\pi\)
\(90\) 0 0
\(91\) −3.56178 −0.373376
\(92\) −0.343146 −0.0357754
\(93\) −6.34315 −0.657754
\(94\) 13.2039 1.36188
\(95\) 0 0
\(96\) 4.10051 0.418506
\(97\) −4.29945 −0.436543 −0.218272 0.975888i \(-0.570042\pi\)
−0.218272 + 0.975888i \(0.570042\pi\)
\(98\) 7.95073 0.803145
\(99\) 0.343146 0.0344874
\(100\) 0 0
\(101\) 0.485281 0.0482873 0.0241437 0.999708i \(-0.492314\pi\)
0.0241437 + 0.999708i \(0.492314\pi\)
\(102\) −17.1716 −1.70024
\(103\) 5.34267 0.526429 0.263215 0.964737i \(-0.415217\pi\)
0.263215 + 0.964737i \(0.415217\pi\)
\(104\) −13.0711 −1.28172
\(105\) 0 0
\(106\) −9.89949 −0.961524
\(107\) 13.9416 1.34778 0.673891 0.738830i \(-0.264622\pi\)
0.673891 + 0.738830i \(0.264622\pi\)
\(108\) −2.08644 −0.200768
\(109\) −18.6731 −1.78856 −0.894281 0.447505i \(-0.852313\pi\)
−0.894281 + 0.447505i \(0.852313\pi\)
\(110\) 0 0
\(111\) 1.31371 0.124692
\(112\) −2.48528 −0.234837
\(113\) 9.33657 0.878311 0.439155 0.898411i \(-0.355278\pi\)
0.439155 + 0.898411i \(0.355278\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.56178 −0.330703
\(117\) −0.737669 −0.0681975
\(118\) 8.97056 0.825807
\(119\) −6.34315 −0.581475
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 3.56178 0.322469
\(123\) 0 0
\(124\) −1.47534 −0.132489
\(125\) 0 0
\(126\) −0.178989 −0.0159456
\(127\) 17.5034 1.55317 0.776586 0.630011i \(-0.216950\pi\)
0.776586 + 0.630011i \(0.216950\pi\)
\(128\) −6.60195 −0.583536
\(129\) −8.59890 −0.757091
\(130\) 0 0
\(131\) 15.3137 1.33796 0.668982 0.743278i \(-0.266730\pi\)
0.668982 + 0.743278i \(0.266730\pi\)
\(132\) 1.47534 0.128412
\(133\) 0 0
\(134\) 8.58579 0.741699
\(135\) 0 0
\(136\) −23.2781 −1.99608
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 1.85786 0.158152
\(139\) 19.6569 1.66727 0.833636 0.552314i \(-0.186255\pi\)
0.833636 + 0.552314i \(0.186255\pi\)
\(140\) 0 0
\(141\) 18.6731 1.57256
\(142\) −17.1716 −1.44101
\(143\) −8.59890 −0.719076
\(144\) −0.514719 −0.0428932
\(145\) 0 0
\(146\) −14.6792 −1.21486
\(147\) 11.2440 0.927392
\(148\) 0.305553 0.0251163
\(149\) −14.8284 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(150\) 0 0
\(151\) −10.6853 −0.869561 −0.434781 0.900536i \(-0.643174\pi\)
−0.434781 + 0.900536i \(0.643174\pi\)
\(152\) 0 0
\(153\) −1.31371 −0.106207
\(154\) −2.08644 −0.168130
\(155\) 0 0
\(156\) −3.17157 −0.253929
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.97056 −0.713660
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) 0.686292 0.0540873
\(162\) 11.9446 0.938458
\(163\) −7.17157 −0.561721 −0.280860 0.959749i \(-0.590620\pi\)
−0.280860 + 0.959749i \(0.590620\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 11.1175 0.862882
\(167\) 16.8923 1.30716 0.653581 0.756857i \(-0.273266\pi\)
0.653581 + 0.756857i \(0.273266\pi\)
\(168\) −4.48528 −0.346047
\(169\) 5.48528 0.421945
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) −2.82411 −0.214713 −0.107357 0.994221i \(-0.534239\pi\)
−0.107357 + 0.994221i \(0.534239\pi\)
\(174\) 19.2842 1.46194
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 12.6863 0.953560
\(178\) 19.7990 1.48400
\(179\) 17.1978 1.28542 0.642712 0.766108i \(-0.277809\pi\)
0.642712 + 0.766108i \(0.277809\pi\)
\(180\) 0 0
\(181\) −10.0742 −0.748812 −0.374406 0.927265i \(-0.622153\pi\)
−0.374406 + 0.927265i \(0.622153\pi\)
\(182\) 4.48528 0.332471
\(183\) 5.03712 0.372355
\(184\) 2.51856 0.185671
\(185\) 0 0
\(186\) 7.98780 0.585694
\(187\) −15.3137 −1.11985
\(188\) 4.34315 0.316756
\(189\) 4.17289 0.303533
\(190\) 0 0
\(191\) −2.34315 −0.169544 −0.0847720 0.996400i \(-0.527016\pi\)
−0.0847720 + 0.996400i \(0.527016\pi\)
\(192\) −15.8490 −1.14381
\(193\) 20.0219 1.44121 0.720605 0.693346i \(-0.243864\pi\)
0.720605 + 0.693346i \(0.243864\pi\)
\(194\) 5.41421 0.388718
\(195\) 0 0
\(196\) 2.61522 0.186802
\(197\) −9.31371 −0.663574 −0.331787 0.943354i \(-0.607652\pi\)
−0.331787 + 0.943354i \(0.607652\pi\)
\(198\) −0.432117 −0.0307092
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 12.1421 0.856440
\(202\) −0.611105 −0.0429972
\(203\) 7.12356 0.499976
\(204\) −5.64823 −0.395455
\(205\) 0 0
\(206\) −6.72792 −0.468757
\(207\) 0.142136 0.00987911
\(208\) 12.8984 0.894340
\(209\) 0 0
\(210\) 0 0
\(211\) −13.6360 −0.938743 −0.469371 0.883001i \(-0.655519\pi\)
−0.469371 + 0.883001i \(0.655519\pi\)
\(212\) −3.25623 −0.223639
\(213\) −24.2843 −1.66393
\(214\) −17.5563 −1.20013
\(215\) 0 0
\(216\) 15.3137 1.04197
\(217\) 2.95068 0.200305
\(218\) 23.5147 1.59262
\(219\) −20.7596 −1.40280
\(220\) 0 0
\(221\) 32.9203 2.21446
\(222\) −1.65433 −0.111031
\(223\) −0.305553 −0.0204613 −0.0102307 0.999948i \(-0.503257\pi\)
−0.0102307 + 0.999948i \(0.503257\pi\)
\(224\) −1.90746 −0.127447
\(225\) 0 0
\(226\) −11.7574 −0.782088
\(227\) 16.8923 1.12118 0.560589 0.828094i \(-0.310575\pi\)
0.560589 + 0.828094i \(0.310575\pi\)
\(228\) 0 0
\(229\) −4.48528 −0.296396 −0.148198 0.988958i \(-0.547347\pi\)
−0.148198 + 0.988958i \(0.547347\pi\)
\(230\) 0 0
\(231\) −2.95068 −0.194140
\(232\) 26.1421 1.71632
\(233\) 9.31371 0.610161 0.305081 0.952327i \(-0.401317\pi\)
0.305081 + 0.952327i \(0.401317\pi\)
\(234\) 0.928932 0.0607262
\(235\) 0 0
\(236\) 2.95068 0.192073
\(237\) −12.6863 −0.824063
\(238\) 7.98780 0.517772
\(239\) 9.65685 0.624650 0.312325 0.949975i \(-0.398892\pi\)
0.312325 + 0.949975i \(0.398892\pi\)
\(240\) 0 0
\(241\) 24.3214 1.56668 0.783339 0.621595i \(-0.213515\pi\)
0.783339 + 0.621595i \(0.213515\pi\)
\(242\) 8.81496 0.566647
\(243\) 1.78089 0.114244
\(244\) 1.17157 0.0750023
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 10.8284 0.687606
\(249\) 15.7225 0.996371
\(250\) 0 0
\(251\) 20.9706 1.32365 0.661825 0.749658i \(-0.269782\pi\)
0.661825 + 0.749658i \(0.269782\pi\)
\(252\) −0.0588745 −0.00370875
\(253\) 1.65685 0.104166
\(254\) −22.0416 −1.38301
\(255\) 0 0
\(256\) −9.48528 −0.592830
\(257\) −20.0219 −1.24893 −0.624466 0.781052i \(-0.714684\pi\)
−0.624466 + 0.781052i \(0.714684\pi\)
\(258\) 10.8284 0.674148
\(259\) −0.611105 −0.0379722
\(260\) 0 0
\(261\) 1.47534 0.0913212
\(262\) −19.2842 −1.19138
\(263\) 16.1421 0.995367 0.497683 0.867359i \(-0.334184\pi\)
0.497683 + 0.867359i \(0.334184\pi\)
\(264\) −10.8284 −0.666444
\(265\) 0 0
\(266\) 0 0
\(267\) 28.0000 1.71357
\(268\) 2.82411 0.172510
\(269\) 8.59890 0.524284 0.262142 0.965029i \(-0.415571\pi\)
0.262142 + 0.965029i \(0.415571\pi\)
\(270\) 0 0
\(271\) 30.9706 1.88133 0.940664 0.339340i \(-0.110204\pi\)
0.940664 + 0.339340i \(0.110204\pi\)
\(272\) 22.9706 1.39279
\(273\) 6.34315 0.383905
\(274\) −2.51856 −0.152152
\(275\) 0 0
\(276\) 0.611105 0.0367842
\(277\) 17.3137 1.04028 0.520140 0.854081i \(-0.325880\pi\)
0.520140 + 0.854081i \(0.325880\pi\)
\(278\) −24.7535 −1.48462
\(279\) 0.611105 0.0365859
\(280\) 0 0
\(281\) 14.2471 0.849912 0.424956 0.905214i \(-0.360289\pi\)
0.424956 + 0.905214i \(0.360289\pi\)
\(282\) −23.5147 −1.40028
\(283\) 9.79899 0.582489 0.291245 0.956649i \(-0.405931\pi\)
0.291245 + 0.956649i \(0.405931\pi\)
\(284\) −5.64823 −0.335161
\(285\) 0 0
\(286\) 10.8284 0.640298
\(287\) 0 0
\(288\) −0.395047 −0.0232784
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 7.65685 0.448853
\(292\) −4.82843 −0.282562
\(293\) 17.9355 1.04780 0.523901 0.851779i \(-0.324476\pi\)
0.523901 + 0.851779i \(0.324476\pi\)
\(294\) −14.1594 −0.825792
\(295\) 0 0
\(296\) −2.24264 −0.130351
\(297\) 10.0742 0.584567
\(298\) 18.6731 1.08171
\(299\) −3.56178 −0.205983
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 13.4558 0.774297
\(303\) −0.864233 −0.0496489
\(304\) 0 0
\(305\) 0 0
\(306\) 1.65433 0.0945716
\(307\) −3.25623 −0.185843 −0.0929214 0.995673i \(-0.529621\pi\)
−0.0929214 + 0.995673i \(0.529621\pi\)
\(308\) −0.686292 −0.0391051
\(309\) −9.51472 −0.541273
\(310\) 0 0
\(311\) −20.3431 −1.15355 −0.576777 0.816902i \(-0.695690\pi\)
−0.576777 + 0.816902i \(0.695690\pi\)
\(312\) 23.2781 1.31787
\(313\) −24.6274 −1.39202 −0.696012 0.718030i \(-0.745044\pi\)
−0.696012 + 0.718030i \(0.745044\pi\)
\(314\) 22.6670 1.27918
\(315\) 0 0
\(316\) −2.95068 −0.165989
\(317\) 5.77479 0.324345 0.162172 0.986762i \(-0.448150\pi\)
0.162172 + 0.986762i \(0.448150\pi\)
\(318\) 17.6299 0.988637
\(319\) 17.1978 0.962892
\(320\) 0 0
\(321\) −24.8284 −1.38579
\(322\) −0.864233 −0.0481618
\(323\) 0 0
\(324\) 3.92893 0.218274
\(325\) 0 0
\(326\) 9.03102 0.500182
\(327\) 33.2548 1.83900
\(328\) 0 0
\(329\) −8.68629 −0.478891
\(330\) 0 0
\(331\) −17.8089 −0.978866 −0.489433 0.872041i \(-0.662796\pi\)
−0.489433 + 0.872041i \(0.662796\pi\)
\(332\) 3.65685 0.200696
\(333\) −0.126564 −0.00693567
\(334\) −21.2721 −1.16396
\(335\) 0 0
\(336\) 4.42602 0.241459
\(337\) 13.5095 0.735907 0.367954 0.929844i \(-0.380059\pi\)
0.367954 + 0.929844i \(0.380059\pi\)
\(338\) −6.90751 −0.375719
\(339\) −16.6274 −0.903077
\(340\) 0 0
\(341\) 7.12356 0.385763
\(342\) 0 0
\(343\) −11.0294 −0.595534
\(344\) 14.6792 0.791452
\(345\) 0 0
\(346\) 3.55635 0.191191
\(347\) −19.1716 −1.02918 −0.514592 0.857435i \(-0.672057\pi\)
−0.514592 + 0.857435i \(0.672057\pi\)
\(348\) 6.34315 0.340028
\(349\) −29.3137 −1.56913 −0.784563 0.620049i \(-0.787113\pi\)
−0.784563 + 0.620049i \(0.787113\pi\)
\(350\) 0 0
\(351\) −21.6569 −1.15596
\(352\) −4.60500 −0.245448
\(353\) −7.65685 −0.407533 −0.203767 0.979019i \(-0.565318\pi\)
−0.203767 + 0.979019i \(0.565318\pi\)
\(354\) −15.9756 −0.849093
\(355\) 0 0
\(356\) 6.51246 0.345160
\(357\) 11.2965 0.597872
\(358\) −21.6569 −1.14460
\(359\) −2.68629 −0.141777 −0.0708885 0.997484i \(-0.522583\pi\)
−0.0708885 + 0.997484i \(0.522583\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 12.6863 0.666777
\(363\) 12.4662 0.654308
\(364\) 1.47534 0.0773287
\(365\) 0 0
\(366\) −6.34315 −0.331562
\(367\) −3.17157 −0.165555 −0.0827774 0.996568i \(-0.526379\pi\)
−0.0827774 + 0.996568i \(0.526379\pi\)
\(368\) −2.48528 −0.129554
\(369\) 0 0
\(370\) 0 0
\(371\) 6.51246 0.338110
\(372\) 2.62742 0.136225
\(373\) 10.8119 0.559819 0.279910 0.960026i \(-0.409695\pi\)
0.279910 + 0.960026i \(0.409695\pi\)
\(374\) 19.2842 0.997165
\(375\) 0 0
\(376\) −31.8771 −1.64393
\(377\) −36.9706 −1.90408
\(378\) −5.25483 −0.270279
\(379\) −17.1978 −0.883392 −0.441696 0.897165i \(-0.645623\pi\)
−0.441696 + 0.897165i \(0.645623\pi\)
\(380\) 0 0
\(381\) −31.1716 −1.59697
\(382\) 2.95068 0.150970
\(383\) −32.6147 −1.66653 −0.833267 0.552871i \(-0.813532\pi\)
−0.833267 + 0.552871i \(0.813532\pi\)
\(384\) 11.7574 0.599990
\(385\) 0 0
\(386\) −25.2132 −1.28332
\(387\) 0.828427 0.0421113
\(388\) 1.78089 0.0904110
\(389\) −20.6274 −1.04585 −0.522926 0.852378i \(-0.675160\pi\)
−0.522926 + 0.852378i \(0.675160\pi\)
\(390\) 0 0
\(391\) −6.34315 −0.320787
\(392\) −19.1948 −0.969482
\(393\) −27.2720 −1.37569
\(394\) 11.7286 0.590877
\(395\) 0 0
\(396\) −0.142136 −0.00714258
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 5.03712 0.252488
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0742 −0.503084 −0.251542 0.967846i \(-0.580938\pi\)
−0.251542 + 0.967846i \(0.580938\pi\)
\(402\) −15.2904 −0.762613
\(403\) −15.3137 −0.762830
\(404\) −0.201010 −0.0100006
\(405\) 0 0
\(406\) −8.97056 −0.445202
\(407\) −1.47534 −0.0731298
\(408\) 41.4558 2.05237
\(409\) 12.7718 0.631524 0.315762 0.948838i \(-0.397740\pi\)
0.315762 + 0.948838i \(0.397740\pi\)
\(410\) 0 0
\(411\) −3.56178 −0.175690
\(412\) −2.21301 −0.109027
\(413\) −5.90135 −0.290387
\(414\) −0.178989 −0.00879681
\(415\) 0 0
\(416\) 9.89949 0.485363
\(417\) −35.0067 −1.71429
\(418\) 0 0
\(419\) −24.9706 −1.21989 −0.609946 0.792443i \(-0.708809\pi\)
−0.609946 + 0.792443i \(0.708809\pi\)
\(420\) 0 0
\(421\) −10.0742 −0.490988 −0.245494 0.969398i \(-0.578950\pi\)
−0.245494 + 0.969398i \(0.578950\pi\)
\(422\) 17.1716 0.835899
\(423\) −1.79899 −0.0874699
\(424\) 23.8995 1.16066
\(425\) 0 0
\(426\) 30.5807 1.48164
\(427\) −2.34315 −0.113393
\(428\) −5.77479 −0.279135
\(429\) 15.3137 0.739353
\(430\) 0 0
\(431\) 3.56178 0.171565 0.0857825 0.996314i \(-0.472661\pi\)
0.0857825 + 0.996314i \(0.472661\pi\)
\(432\) −15.1114 −0.727046
\(433\) −4.91056 −0.235986 −0.117993 0.993014i \(-0.537646\pi\)
−0.117993 + 0.993014i \(0.537646\pi\)
\(434\) −3.71573 −0.178361
\(435\) 0 0
\(436\) 7.73467 0.370423
\(437\) 0 0
\(438\) 26.1421 1.24912
\(439\) 2.95068 0.140828 0.0704141 0.997518i \(-0.477568\pi\)
0.0704141 + 0.997518i \(0.477568\pi\)
\(440\) 0 0
\(441\) −1.08326 −0.0515839
\(442\) −41.4558 −1.97185
\(443\) −20.1421 −0.956982 −0.478491 0.878093i \(-0.658816\pi\)
−0.478491 + 0.878093i \(0.658816\pi\)
\(444\) −0.544156 −0.0258245
\(445\) 0 0
\(446\) 0.384776 0.0182197
\(447\) 26.4078 1.24905
\(448\) 7.37258 0.348322
\(449\) −8.59890 −0.405807 −0.202904 0.979199i \(-0.565038\pi\)
−0.202904 + 0.979199i \(0.565038\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −3.86733 −0.181904
\(453\) 19.0294 0.894081
\(454\) −21.2721 −0.998348
\(455\) 0 0
\(456\) 0 0
\(457\) −5.31371 −0.248565 −0.124282 0.992247i \(-0.539663\pi\)
−0.124282 + 0.992247i \(0.539663\pi\)
\(458\) 5.64823 0.263924
\(459\) −38.5685 −1.80022
\(460\) 0 0
\(461\) −10.6863 −0.497710 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(462\) 3.71573 0.172871
\(463\) 32.8284 1.52567 0.762833 0.646595i \(-0.223808\pi\)
0.762833 + 0.646595i \(0.223808\pi\)
\(464\) −25.7967 −1.19758
\(465\) 0 0
\(466\) −11.7286 −0.543315
\(467\) −38.4853 −1.78089 −0.890443 0.455094i \(-0.849606\pi\)
−0.890443 + 0.455094i \(0.849606\pi\)
\(468\) 0.305553 0.0141242
\(469\) −5.64823 −0.260811
\(470\) 0 0
\(471\) 32.0560 1.47706
\(472\) −21.6569 −0.996838
\(473\) 9.65685 0.444023
\(474\) 15.9756 0.733783
\(475\) 0 0
\(476\) 2.62742 0.120427
\(477\) 1.34877 0.0617561
\(478\) −12.1607 −0.556217
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 3.17157 0.144611
\(482\) −30.6274 −1.39504
\(483\) −1.22221 −0.0556125
\(484\) 2.89949 0.131795
\(485\) 0 0
\(486\) −2.24264 −0.101728
\(487\) 29.6640 1.34421 0.672103 0.740458i \(-0.265391\pi\)
0.672103 + 0.740458i \(0.265391\pi\)
\(488\) −8.59890 −0.389254
\(489\) 12.7718 0.577560
\(490\) 0 0
\(491\) 32.2843 1.45697 0.728484 0.685062i \(-0.240225\pi\)
0.728484 + 0.685062i \(0.240225\pi\)
\(492\) 0 0
\(493\) −65.8405 −2.96531
\(494\) 0 0
\(495\) 0 0
\(496\) −10.6853 −0.479786
\(497\) 11.2965 0.506715
\(498\) −19.7990 −0.887214
\(499\) 1.02944 0.0460839 0.0230420 0.999734i \(-0.492665\pi\)
0.0230420 + 0.999734i \(0.492665\pi\)
\(500\) 0 0
\(501\) −30.0833 −1.34402
\(502\) −26.4078 −1.17864
\(503\) −2.48528 −0.110813 −0.0554066 0.998464i \(-0.517646\pi\)
−0.0554066 + 0.998464i \(0.517646\pi\)
\(504\) 0.432117 0.0192480
\(505\) 0 0
\(506\) −2.08644 −0.0927537
\(507\) −9.76869 −0.433843
\(508\) −7.25013 −0.321672
\(509\) −32.9203 −1.45917 −0.729583 0.683893i \(-0.760286\pi\)
−0.729583 + 0.683893i \(0.760286\pi\)
\(510\) 0 0
\(511\) 9.65685 0.427194
\(512\) 25.1485 1.11142
\(513\) 0 0
\(514\) 25.2132 1.11211
\(515\) 0 0
\(516\) 3.56178 0.156799
\(517\) −20.9706 −0.922284
\(518\) 0.769553 0.0338122
\(519\) 5.02944 0.220768
\(520\) 0 0
\(521\) −17.1978 −0.753450 −0.376725 0.926325i \(-0.622950\pi\)
−0.376725 + 0.926325i \(0.622950\pi\)
\(522\) −1.85786 −0.0813165
\(523\) −15.4169 −0.674135 −0.337067 0.941481i \(-0.609435\pi\)
−0.337067 + 0.941481i \(0.609435\pi\)
\(524\) −6.34315 −0.277102
\(525\) 0 0
\(526\) −20.3275 −0.886320
\(527\) −27.2720 −1.18799
\(528\) 10.6853 0.465020
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) −1.22221 −0.0530394
\(532\) 0 0
\(533\) 0 0
\(534\) −35.2598 −1.52584
\(535\) 0 0
\(536\) −20.7279 −0.895310
\(537\) −30.6274 −1.32167
\(538\) −10.8284 −0.466847
\(539\) −12.6274 −0.543901
\(540\) 0 0
\(541\) 22.1421 0.951965 0.475982 0.879455i \(-0.342093\pi\)
0.475982 + 0.879455i \(0.342093\pi\)
\(542\) −39.0006 −1.67522
\(543\) 17.9411 0.769927
\(544\) 17.6299 0.755877
\(545\) 0 0
\(546\) −7.98780 −0.341846
\(547\) −11.8551 −0.506889 −0.253444 0.967350i \(-0.581563\pi\)
−0.253444 + 0.967350i \(0.581563\pi\)
\(548\) −0.828427 −0.0353887
\(549\) −0.485281 −0.0207113
\(550\) 0 0
\(551\) 0 0
\(552\) −4.48528 −0.190906
\(553\) 5.90135 0.250951
\(554\) −21.8028 −0.926313
\(555\) 0 0
\(556\) −8.14214 −0.345303
\(557\) 28.6274 1.21298 0.606491 0.795090i \(-0.292576\pi\)
0.606491 + 0.795090i \(0.292576\pi\)
\(558\) −0.769553 −0.0325778
\(559\) −20.7596 −0.878037
\(560\) 0 0
\(561\) 27.2720 1.15143
\(562\) −17.9411 −0.756801
\(563\) 26.1023 1.10008 0.550040 0.835139i \(-0.314613\pi\)
0.550040 + 0.835139i \(0.314613\pi\)
\(564\) −7.73467 −0.325688
\(565\) 0 0
\(566\) −12.3397 −0.518675
\(567\) −7.85786 −0.329999
\(568\) 41.4558 1.73945
\(569\) −15.7225 −0.659120 −0.329560 0.944135i \(-0.606900\pi\)
−0.329560 + 0.944135i \(0.606900\pi\)
\(570\) 0 0
\(571\) 13.3137 0.557161 0.278581 0.960413i \(-0.410136\pi\)
0.278581 + 0.960413i \(0.410136\pi\)
\(572\) 3.56178 0.148926
\(573\) 4.17289 0.174325
\(574\) 0 0
\(575\) 0 0
\(576\) 1.52691 0.0636213
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −52.4206 −2.18041
\(579\) −35.6569 −1.48185
\(580\) 0 0
\(581\) −7.31371 −0.303424
\(582\) −9.64212 −0.399679
\(583\) 15.7225 0.651158
\(584\) 35.4388 1.46647
\(585\) 0 0
\(586\) −22.5858 −0.933010
\(587\) −18.4853 −0.762969 −0.381485 0.924375i \(-0.624587\pi\)
−0.381485 + 0.924375i \(0.624587\pi\)
\(588\) −4.65743 −0.192069
\(589\) 0 0
\(590\) 0 0
\(591\) 16.5867 0.682286
\(592\) 2.21301 0.0909541
\(593\) 9.31371 0.382468 0.191234 0.981544i \(-0.438751\pi\)
0.191234 + 0.981544i \(0.438751\pi\)
\(594\) −12.6863 −0.520525
\(595\) 0 0
\(596\) 6.14214 0.251592
\(597\) 7.12356 0.291548
\(598\) 4.48528 0.183417
\(599\) −34.3956 −1.40537 −0.702683 0.711503i \(-0.748015\pi\)
−0.702683 + 0.711503i \(0.748015\pi\)
\(600\) 0 0
\(601\) −17.1978 −0.701513 −0.350757 0.936467i \(-0.614076\pi\)
−0.350757 + 0.936467i \(0.614076\pi\)
\(602\) −5.03712 −0.205298
\(603\) −1.16979 −0.0476374
\(604\) 4.42602 0.180092
\(605\) 0 0
\(606\) 1.08831 0.0442096
\(607\) −33.2258 −1.34859 −0.674297 0.738460i \(-0.735553\pi\)
−0.674297 + 0.738460i \(0.735553\pi\)
\(608\) 0 0
\(609\) −12.6863 −0.514074
\(610\) 0 0
\(611\) 45.0810 1.82378
\(612\) 0.544156 0.0219962
\(613\) 14.2843 0.576936 0.288468 0.957489i \(-0.406854\pi\)
0.288468 + 0.957489i \(0.406854\pi\)
\(614\) 4.10051 0.165483
\(615\) 0 0
\(616\) 5.03712 0.202951
\(617\) −30.2843 −1.21920 −0.609599 0.792710i \(-0.708670\pi\)
−0.609599 + 0.792710i \(0.708670\pi\)
\(618\) 11.9817 0.481975
\(619\) −18.9706 −0.762491 −0.381246 0.924474i \(-0.624505\pi\)
−0.381246 + 0.924474i \(0.624505\pi\)
\(620\) 0 0
\(621\) 4.17289 0.167452
\(622\) 25.6177 1.02718
\(623\) −13.0249 −0.521832
\(624\) −22.9706 −0.919558
\(625\) 0 0
\(626\) 31.0128 1.23952
\(627\) 0 0
\(628\) 7.45584 0.297521
\(629\) 5.64823 0.225210
\(630\) 0 0
\(631\) 18.2843 0.727885 0.363943 0.931421i \(-0.381430\pi\)
0.363943 + 0.931421i \(0.381430\pi\)
\(632\) 21.6569 0.861463
\(633\) 24.2843 0.965213
\(634\) −7.27208 −0.288811
\(635\) 0 0
\(636\) 5.79899 0.229945
\(637\) 27.1455 1.07554
\(638\) −21.6569 −0.857403
\(639\) 2.33957 0.0925520
\(640\) 0 0
\(641\) −21.3707 −0.844092 −0.422046 0.906575i \(-0.638688\pi\)
−0.422046 + 0.906575i \(0.638688\pi\)
\(642\) 31.2659 1.23397
\(643\) −9.51472 −0.375224 −0.187612 0.982243i \(-0.560075\pi\)
−0.187612 + 0.982243i \(0.560075\pi\)
\(644\) −0.284271 −0.0112019
\(645\) 0 0
\(646\) 0 0
\(647\) 27.1716 1.06822 0.534112 0.845413i \(-0.320646\pi\)
0.534112 + 0.845413i \(0.320646\pi\)
\(648\) −28.8369 −1.13282
\(649\) −14.2471 −0.559249
\(650\) 0 0
\(651\) −5.25483 −0.205953
\(652\) 2.97056 0.116336
\(653\) −27.6569 −1.08230 −0.541148 0.840927i \(-0.682010\pi\)
−0.541148 + 0.840927i \(0.682010\pi\)
\(654\) −41.8772 −1.63753
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 10.9385 0.426426
\(659\) −20.1485 −0.784873 −0.392437 0.919779i \(-0.628368\pi\)
−0.392437 + 0.919779i \(0.628368\pi\)
\(660\) 0 0
\(661\) 2.95068 0.114768 0.0573840 0.998352i \(-0.481724\pi\)
0.0573840 + 0.998352i \(0.481724\pi\)
\(662\) 22.4264 0.871627
\(663\) −58.6274 −2.27690
\(664\) −26.8399 −1.04159
\(665\) 0 0
\(666\) 0.159380 0.00617583
\(667\) 7.12356 0.275826
\(668\) −6.99700 −0.270722
\(669\) 0.544156 0.0210383
\(670\) 0 0
\(671\) −5.65685 −0.218380
\(672\) 3.39697 0.131041
\(673\) −50.8557 −1.96034 −0.980172 0.198146i \(-0.936508\pi\)
−0.980172 + 0.198146i \(0.936508\pi\)
\(674\) −17.0122 −0.655285
\(675\) 0 0
\(676\) −2.27208 −0.0873876
\(677\) −2.21301 −0.0850528 −0.0425264 0.999095i \(-0.513541\pi\)
−0.0425264 + 0.999095i \(0.513541\pi\)
\(678\) 20.9386 0.804141
\(679\) −3.56178 −0.136689
\(680\) 0 0
\(681\) −30.0833 −1.15279
\(682\) −8.97056 −0.343501
\(683\) 11.8551 0.453624 0.226812 0.973939i \(-0.427170\pi\)
0.226812 + 0.973939i \(0.427170\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.8892 0.530290
\(687\) 7.98780 0.304753
\(688\) −14.4853 −0.552246
\(689\) −33.7990 −1.28764
\(690\) 0 0
\(691\) 16.6274 0.632537 0.316268 0.948670i \(-0.397570\pi\)
0.316268 + 0.948670i \(0.397570\pi\)
\(692\) 1.16979 0.0444686
\(693\) 0.284271 0.0107986
\(694\) 24.1424 0.916432
\(695\) 0 0
\(696\) −46.5563 −1.76471
\(697\) 0 0
\(698\) 36.9142 1.39722
\(699\) −16.5867 −0.627367
\(700\) 0 0
\(701\) −24.4853 −0.924796 −0.462398 0.886672i \(-0.653011\pi\)
−0.462398 + 0.886672i \(0.653011\pi\)
\(702\) 27.2720 1.02932
\(703\) 0 0
\(704\) 17.7990 0.670825
\(705\) 0 0
\(706\) 9.64212 0.362886
\(707\) 0.402020 0.0151195
\(708\) −5.25483 −0.197489
\(709\) −5.31371 −0.199561 −0.0997803 0.995009i \(-0.531814\pi\)
−0.0997803 + 0.995009i \(0.531814\pi\)
\(710\) 0 0
\(711\) 1.22221 0.0458365
\(712\) −47.7990 −1.79134
\(713\) 2.95068 0.110504
\(714\) −14.2254 −0.532372
\(715\) 0 0
\(716\) −7.12356 −0.266220
\(717\) −17.1978 −0.642264
\(718\) 3.38279 0.126245
\(719\) 31.9411 1.19120 0.595601 0.803280i \(-0.296914\pi\)
0.595601 + 0.803280i \(0.296914\pi\)
\(720\) 0 0
\(721\) 4.42602 0.164833
\(722\) 0 0
\(723\) −43.3137 −1.61085
\(724\) 4.17289 0.155084
\(725\) 0 0
\(726\) −15.6985 −0.582625
\(727\) −6.48528 −0.240526 −0.120263 0.992742i \(-0.538374\pi\)
−0.120263 + 0.992742i \(0.538374\pi\)
\(728\) −10.8284 −0.401328
\(729\) 25.2843 0.936454
\(730\) 0 0
\(731\) −36.9706 −1.36741
\(732\) −2.08644 −0.0771172
\(733\) 35.6569 1.31702 0.658508 0.752574i \(-0.271188\pi\)
0.658508 + 0.752574i \(0.271188\pi\)
\(734\) 3.99390 0.147417
\(735\) 0 0
\(736\) −1.90746 −0.0703097
\(737\) −13.6360 −0.502289
\(738\) 0 0
\(739\) 1.37258 0.0504913 0.0252456 0.999681i \(-0.491963\pi\)
0.0252456 + 0.999681i \(0.491963\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.20101 −0.301069
\(743\) −6.81801 −0.250129 −0.125064 0.992149i \(-0.539914\pi\)
−0.125064 + 0.992149i \(0.539914\pi\)
\(744\) −19.2842 −0.706995
\(745\) 0 0
\(746\) −13.6152 −0.498489
\(747\) −1.51472 −0.0554207
\(748\) 6.34315 0.231928
\(749\) 11.5496 0.422012
\(750\) 0 0
\(751\) −6.51246 −0.237643 −0.118822 0.992916i \(-0.537912\pi\)
−0.118822 + 0.992916i \(0.537912\pi\)
\(752\) 31.4558 1.14708
\(753\) −37.3463 −1.36097
\(754\) 46.5563 1.69548
\(755\) 0 0
\(756\) −1.72847 −0.0628637
\(757\) 6.68629 0.243017 0.121509 0.992590i \(-0.461227\pi\)
0.121509 + 0.992590i \(0.461227\pi\)
\(758\) 21.6569 0.786612
\(759\) −2.95068 −0.107103
\(760\) 0 0
\(761\) −9.17157 −0.332469 −0.166235 0.986086i \(-0.553161\pi\)
−0.166235 + 0.986086i \(0.553161\pi\)
\(762\) 39.2537 1.42201
\(763\) −15.4693 −0.560028
\(764\) 0.970563 0.0351137
\(765\) 0 0
\(766\) 41.0711 1.48396
\(767\) 30.6274 1.10589
\(768\) 16.8923 0.609547
\(769\) 14.1421 0.509978 0.254989 0.966944i \(-0.417928\pi\)
0.254989 + 0.966944i \(0.417928\pi\)
\(770\) 0 0
\(771\) 35.6569 1.28415
\(772\) −8.29335 −0.298484
\(773\) −18.5466 −0.667074 −0.333537 0.942737i \(-0.608242\pi\)
−0.333537 + 0.942737i \(0.608242\pi\)
\(774\) −1.04322 −0.0374978
\(775\) 0 0
\(776\) −13.0711 −0.469224
\(777\) 1.08831 0.0390430
\(778\) 25.9757 0.931274
\(779\) 0 0
\(780\) 0 0
\(781\) 27.2720 0.975871
\(782\) 7.98780 0.285643
\(783\) 43.3137 1.54791
\(784\) 18.9411 0.676469
\(785\) 0 0
\(786\) 34.3431 1.22498
\(787\) −46.2507 −1.64866 −0.824330 0.566109i \(-0.808448\pi\)
−0.824330 + 0.566109i \(0.808448\pi\)
\(788\) 3.85786 0.137431
\(789\) −28.7474 −1.02343
\(790\) 0 0
\(791\) 7.73467 0.275013
\(792\) 1.04322 0.0370693
\(793\) 12.1607 0.431839
\(794\) −17.6299 −0.625663
\(795\) 0 0
\(796\) 1.65685 0.0587256
\(797\) 0.737669 0.0261296 0.0130648 0.999915i \(-0.495841\pi\)
0.0130648 + 0.999915i \(0.495841\pi\)
\(798\) 0 0
\(799\) 80.2843 2.84025
\(800\) 0 0
\(801\) −2.69755 −0.0953132
\(802\) 12.6863 0.447969
\(803\) 23.3137 0.822723
\(804\) −5.02944 −0.177375
\(805\) 0 0
\(806\) 19.2842 0.679259
\(807\) −15.3137 −0.539068
\(808\) 1.47534 0.0519022
\(809\) −13.3137 −0.468085 −0.234043 0.972226i \(-0.575196\pi\)
−0.234043 + 0.972226i \(0.575196\pi\)
\(810\) 0 0
\(811\) −33.7845 −1.18633 −0.593167 0.805079i \(-0.702123\pi\)
−0.593167 + 0.805079i \(0.702123\pi\)
\(812\) −2.95068 −0.103548
\(813\) −55.1552 −1.93438
\(814\) 1.85786 0.0651181
\(815\) 0 0
\(816\) −40.9081 −1.43207
\(817\) 0 0
\(818\) −16.0833 −0.562338
\(819\) −0.611105 −0.0213537
\(820\) 0 0
\(821\) 2.68629 0.0937522 0.0468761 0.998901i \(-0.485073\pi\)
0.0468761 + 0.998901i \(0.485073\pi\)
\(822\) 4.48528 0.156442
\(823\) −42.0833 −1.46693 −0.733465 0.679727i \(-0.762098\pi\)
−0.733465 + 0.679727i \(0.762098\pi\)
\(824\) 16.2426 0.565839
\(825\) 0 0
\(826\) 7.43146 0.258573
\(827\) 18.9787 0.659954 0.329977 0.943989i \(-0.392959\pi\)
0.329977 + 0.943989i \(0.392959\pi\)
\(828\) −0.0588745 −0.00204603
\(829\) 42.9945 1.49326 0.746631 0.665239i \(-0.231670\pi\)
0.746631 + 0.665239i \(0.231670\pi\)
\(830\) 0 0
\(831\) −30.8338 −1.06961
\(832\) −38.2629 −1.32653
\(833\) 48.3431 1.67499
\(834\) 44.0833 1.52648
\(835\) 0 0
\(836\) 0 0
\(837\) 17.9411 0.620136
\(838\) 31.4449 1.08625
\(839\) −31.4449 −1.08560 −0.542800 0.839862i \(-0.682636\pi\)
−0.542800 + 0.839862i \(0.682636\pi\)
\(840\) 0 0
\(841\) 44.9411 1.54969
\(842\) 12.6863 0.437198
\(843\) −25.3726 −0.873878
\(844\) 5.64823 0.194420
\(845\) 0 0
\(846\) 2.26543 0.0778872
\(847\) −5.79899 −0.199256
\(848\) −23.5837 −0.809868
\(849\) −17.4509 −0.598914
\(850\) 0 0
\(851\) −0.611105 −0.0209484
\(852\) 10.0589 0.344611
\(853\) −36.3431 −1.24437 −0.622183 0.782872i \(-0.713754\pi\)
−0.622183 + 0.782872i \(0.713754\pi\)
\(854\) 2.95068 0.100970
\(855\) 0 0
\(856\) 42.3848 1.44868
\(857\) −17.0712 −0.583142 −0.291571 0.956549i \(-0.594178\pi\)
−0.291571 + 0.956549i \(0.594178\pi\)
\(858\) −19.2842 −0.658353
\(859\) −44.2843 −1.51096 −0.755480 0.655172i \(-0.772596\pi\)
−0.755480 + 0.655172i \(0.772596\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4.48528 −0.152769
\(863\) 24.6269 0.838310 0.419155 0.907915i \(-0.362326\pi\)
0.419155 + 0.907915i \(0.362326\pi\)
\(864\) −11.5980 −0.394571
\(865\) 0 0
\(866\) 6.18377 0.210133
\(867\) −74.1339 −2.51772
\(868\) −1.22221 −0.0414845
\(869\) 14.2471 0.483301
\(870\) 0 0
\(871\) 29.3137 0.993257
\(872\) −56.7696 −1.92246
\(873\) −0.737669 −0.0249663
\(874\) 0 0
\(875\) 0 0
\(876\) 8.59890 0.290530
\(877\) 10.8119 0.365092 0.182546 0.983197i \(-0.441566\pi\)
0.182546 + 0.983197i \(0.441566\pi\)
\(878\) −3.71573 −0.125400
\(879\) −31.9411 −1.07735
\(880\) 0 0
\(881\) 4.48528 0.151113 0.0755565 0.997142i \(-0.475927\pi\)
0.0755565 + 0.997142i \(0.475927\pi\)
\(882\) 1.36413 0.0459326
\(883\) −3.85786 −0.129827 −0.0649137 0.997891i \(-0.520677\pi\)
−0.0649137 + 0.997891i \(0.520677\pi\)
\(884\) −13.6360 −0.458629
\(885\) 0 0
\(886\) 25.3646 0.852140
\(887\) −1.16979 −0.0392776 −0.0196388 0.999807i \(-0.506252\pi\)
−0.0196388 + 0.999807i \(0.506252\pi\)
\(888\) 3.99390 0.134026
\(889\) 14.5003 0.486323
\(890\) 0 0
\(891\) −18.9706 −0.635538
\(892\) 0.126564 0.00423768
\(893\) 0 0
\(894\) −33.2548 −1.11221
\(895\) 0 0
\(896\) −5.46924 −0.182714
\(897\) 6.34315 0.211791
\(898\) 10.8284 0.361349
\(899\) 30.6274 1.02148
\(900\) 0 0
\(901\) −60.1923 −2.00530
\(902\) 0 0
\(903\) −7.12356 −0.237057
\(904\) 28.3848 0.944064
\(905\) 0 0
\(906\) −23.9634 −0.796130
\(907\) −30.5283 −1.01367 −0.506837 0.862042i \(-0.669186\pi\)
−0.506837 + 0.862042i \(0.669186\pi\)
\(908\) −6.99700 −0.232204
\(909\) 0.0832611 0.00276160
\(910\) 0 0
\(911\) −26.6609 −0.883316 −0.441658 0.897183i \(-0.645610\pi\)
−0.441658 + 0.897183i \(0.645610\pi\)
\(912\) 0 0
\(913\) −17.6569 −0.584357
\(914\) 6.69145 0.221333
\(915\) 0 0
\(916\) 1.85786 0.0613856
\(917\) 12.6863 0.418938
\(918\) 48.5685 1.60300
\(919\) 0.284271 0.00937724 0.00468862 0.999989i \(-0.498508\pi\)
0.00468862 + 0.999989i \(0.498508\pi\)
\(920\) 0 0
\(921\) 5.79899 0.191083
\(922\) 13.4570 0.443184
\(923\) −58.6274 −1.92974
\(924\) 1.22221 0.0402078
\(925\) 0 0
\(926\) −41.3402 −1.35852
\(927\) 0.916658 0.0301070
\(928\) −19.7990 −0.649934
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −3.85786 −0.126369
\(933\) 36.2289 1.18608
\(934\) 48.4638 1.58578
\(935\) 0 0
\(936\) −2.24264 −0.0733030
\(937\) −10.9706 −0.358393 −0.179196 0.983813i \(-0.557350\pi\)
−0.179196 + 0.983813i \(0.557350\pi\)
\(938\) 7.11270 0.232238
\(939\) 43.8587 1.43128
\(940\) 0 0
\(941\) 2.95068 0.0961893 0.0480947 0.998843i \(-0.484685\pi\)
0.0480947 + 0.998843i \(0.484685\pi\)
\(942\) −40.3675 −1.31525
\(943\) 0 0
\(944\) 21.3707 0.695557
\(945\) 0 0
\(946\) −12.1607 −0.395378
\(947\) 4.14214 0.134601 0.0673007 0.997733i \(-0.478561\pi\)
0.0673007 + 0.997733i \(0.478561\pi\)
\(948\) 5.25483 0.170669
\(949\) −50.1181 −1.62690
\(950\) 0 0
\(951\) −10.2843 −0.333490
\(952\) −19.2842 −0.625006
\(953\) 1.34877 0.0436911 0.0218455 0.999761i \(-0.493046\pi\)
0.0218455 + 0.999761i \(0.493046\pi\)
\(954\) −1.69848 −0.0549905
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) −30.6274 −0.990044
\(958\) −12.5928 −0.406855
\(959\) 1.65685 0.0535026
\(960\) 0 0
\(961\) −18.3137 −0.590765
\(962\) −3.99390 −0.128768
\(963\) 2.39200 0.0770810
\(964\) −10.0742 −0.324469
\(965\) 0 0
\(966\) 1.53911 0.0495199
\(967\) −21.5147 −0.691867 −0.345933 0.938259i \(-0.612438\pi\)
−0.345933 + 0.938259i \(0.612438\pi\)
\(968\) −21.2812 −0.684004
\(969\) 0 0
\(970\) 0 0
\(971\) 7.73467 0.248217 0.124109 0.992269i \(-0.460393\pi\)
0.124109 + 0.992269i \(0.460393\pi\)
\(972\) −0.737669 −0.0236608
\(973\) 16.2843 0.522050
\(974\) −37.3553 −1.19694
\(975\) 0 0
\(976\) 8.48528 0.271607
\(977\) −47.9051 −1.53262 −0.766309 0.642472i \(-0.777909\pi\)
−0.766309 + 0.642472i \(0.777909\pi\)
\(978\) −16.0833 −0.514286
\(979\) −31.4449 −1.00498
\(980\) 0 0
\(981\) −3.20380 −0.102290
\(982\) −40.6549 −1.29735
\(983\) −33.8369 −1.07923 −0.539615 0.841912i \(-0.681430\pi\)
−0.539615 + 0.841912i \(0.681430\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 82.9117 2.64045
\(987\) 15.4693 0.492394
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 27.8832 0.885737 0.442869 0.896586i \(-0.353961\pi\)
0.442869 + 0.896586i \(0.353961\pi\)
\(992\) −8.20101 −0.260382
\(993\) 31.7157 1.00647
\(994\) −14.2254 −0.451202
\(995\) 0 0
\(996\) −6.51246 −0.206355
\(997\) 7.65685 0.242495 0.121248 0.992622i \(-0.461311\pi\)
0.121248 + 0.992622i \(0.461311\pi\)
\(998\) −1.29635 −0.0410352
\(999\) −3.71573 −0.117560
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 9025.2.a.bn.1.2 4
5.4 even 2 1805.2.a.m.1.3 yes 4
19.18 odd 2 inner 9025.2.a.bn.1.3 4
95.94 odd 2 1805.2.a.m.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.m.1.2 4 95.94 odd 2
1805.2.a.m.1.3 yes 4 5.4 even 2
9025.2.a.bn.1.2 4 1.1 even 1 trivial
9025.2.a.bn.1.3 4 19.18 odd 2 inner