Properties

Label 9025.2.a.y.1.1
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
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] = [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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.37720\) 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)
 
\(f(q)\) \(=\) \(q-2.37720 q^{2} -1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} +0.726109 q^{7} -3.92498 q^{8} -1.37720 q^{9} +O(q^{10})\) \(q-2.37720 q^{2} -1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} +0.726109 q^{7} -3.92498 q^{8} -1.37720 q^{9} -0.273891 q^{11} -4.65109 q^{12} +5.95328 q^{13} -1.72611 q^{14} +2.02830 q^{16} +5.27389 q^{17} +3.27389 q^{18} -0.924984 q^{21} +0.651093 q^{22} -3.67939 q^{23} +5.00000 q^{24} -14.1522 q^{26} +5.57608 q^{27} +2.65109 q^{28} +2.27389 q^{29} -3.19887 q^{31} +3.02830 q^{32} +0.348907 q^{33} -12.5371 q^{34} -5.02830 q^{36} -8.12386 q^{37} -7.58383 q^{39} +9.43380 q^{41} +2.19887 q^{42} +9.81100 q^{43} -1.00000 q^{44} +8.74666 q^{46} +12.1599 q^{47} -2.58383 q^{48} -6.47277 q^{49} -6.71836 q^{51} +21.7360 q^{52} -5.69781 q^{53} -13.2555 q^{54} -2.84997 q^{56} -5.40550 q^{58} +4.20662 q^{59} -0.103312 q^{61} +7.60437 q^{62} -1.00000 q^{63} -11.2555 q^{64} -0.829422 q^{66} +11.7827 q^{67} +19.2555 q^{68} +4.68714 q^{69} -5.75441 q^{71} +5.40550 q^{72} +6.67939 q^{73} +19.3121 q^{74} -0.198875 q^{77} +18.0283 q^{78} -3.87826 q^{79} -2.97170 q^{81} -22.4260 q^{82} -0.488265 q^{83} -3.37720 q^{84} -23.3227 q^{86} -2.89669 q^{87} +1.07502 q^{88} +16.4338 q^{89} +4.32273 q^{91} -13.4338 q^{92} +4.07502 q^{93} -28.9066 q^{94} -3.85772 q^{96} +4.44447 q^{97} +15.3871 q^{98} +0.377203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} - 7 q^{14} - 6 q^{16} + 14 q^{17} + 8 q^{18} + 6 q^{21} - 5 q^{22} + 8 q^{23} + 15 q^{24} - 11 q^{26} + q^{27} + q^{28} + 5 q^{29} + q^{31} - 3 q^{32} + 8 q^{33} - 5 q^{34} - 3 q^{36} - 5 q^{37} - 11 q^{39} - q^{41} - 4 q^{42} - 5 q^{43} - 3 q^{44} + 12 q^{46} + 9 q^{47} + 4 q^{48} - 7 q^{49} - 18 q^{51} + 22 q^{52} - 31 q^{53} - 5 q^{54} + 9 q^{56} + q^{58} + 6 q^{59} + 3 q^{61} - 5 q^{62} - 3 q^{63} + q^{64} - q^{66} + 13 q^{67} + 23 q^{68} - q^{69} - 7 q^{71} - q^{72} + q^{73} - q^{74} + 10 q^{77} + 42 q^{78} + 18 q^{79} - 21 q^{81} - 34 q^{82} + 3 q^{83} - 5 q^{84} - 40 q^{86} - 12 q^{87} + 12 q^{88} + 20 q^{89} - 17 q^{91} - 11 q^{92} + 21 q^{93} - 45 q^{94} + 2 q^{96} + 13 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37720 −1.68094 −0.840468 0.541861i \(-0.817720\pi\)
−0.840468 + 0.541861i \(0.817720\pi\)
\(3\) −1.27389 −0.735481 −0.367741 0.929928i \(-0.619869\pi\)
−0.367741 + 0.929928i \(0.619869\pi\)
\(4\) 3.65109 1.82555
\(5\) 0 0
\(6\) 3.02830 1.23630
\(7\) 0.726109 0.274444 0.137222 0.990540i \(-0.456183\pi\)
0.137222 + 0.990540i \(0.456183\pi\)
\(8\) −3.92498 −1.38769
\(9\) −1.37720 −0.459068
\(10\) 0 0
\(11\) −0.273891 −0.0825811 −0.0412906 0.999147i \(-0.513147\pi\)
−0.0412906 + 0.999147i \(0.513147\pi\)
\(12\) −4.65109 −1.34266
\(13\) 5.95328 1.65114 0.825571 0.564298i \(-0.190853\pi\)
0.825571 + 0.564298i \(0.190853\pi\)
\(14\) −1.72611 −0.461322
\(15\) 0 0
\(16\) 2.02830 0.507074
\(17\) 5.27389 1.27911 0.639553 0.768747i \(-0.279119\pi\)
0.639553 + 0.768747i \(0.279119\pi\)
\(18\) 3.27389 0.771663
\(19\) 0 0
\(20\) 0 0
\(21\) −0.924984 −0.201848
\(22\) 0.651093 0.138814
\(23\) −3.67939 −0.767206 −0.383603 0.923498i \(-0.625317\pi\)
−0.383603 + 0.923498i \(0.625317\pi\)
\(24\) 5.00000 1.02062
\(25\) 0 0
\(26\) −14.1522 −2.77547
\(27\) 5.57608 1.07312
\(28\) 2.65109 0.501010
\(29\) 2.27389 0.422251 0.211125 0.977459i \(-0.432287\pi\)
0.211125 + 0.977459i \(0.432287\pi\)
\(30\) 0 0
\(31\) −3.19887 −0.574535 −0.287267 0.957850i \(-0.592747\pi\)
−0.287267 + 0.957850i \(0.592747\pi\)
\(32\) 3.02830 0.535332
\(33\) 0.348907 0.0607368
\(34\) −12.5371 −2.15010
\(35\) 0 0
\(36\) −5.02830 −0.838049
\(37\) −8.12386 −1.33555 −0.667777 0.744361i \(-0.732754\pi\)
−0.667777 + 0.744361i \(0.732754\pi\)
\(38\) 0 0
\(39\) −7.58383 −1.21438
\(40\) 0 0
\(41\) 9.43380 1.47331 0.736656 0.676268i \(-0.236404\pi\)
0.736656 + 0.676268i \(0.236404\pi\)
\(42\) 2.19887 0.339294
\(43\) 9.81100 1.49616 0.748082 0.663607i \(-0.230975\pi\)
0.748082 + 0.663607i \(0.230975\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 8.74666 1.28962
\(47\) 12.1599 1.77370 0.886852 0.462053i \(-0.152887\pi\)
0.886852 + 0.462053i \(0.152887\pi\)
\(48\) −2.58383 −0.372943
\(49\) −6.47277 −0.924681
\(50\) 0 0
\(51\) −6.71836 −0.940758
\(52\) 21.7360 3.01424
\(53\) −5.69781 −0.782655 −0.391327 0.920252i \(-0.627984\pi\)
−0.391327 + 0.920252i \(0.627984\pi\)
\(54\) −13.2555 −1.80384
\(55\) 0 0
\(56\) −2.84997 −0.380843
\(57\) 0 0
\(58\) −5.40550 −0.709777
\(59\) 4.20662 0.547656 0.273828 0.961779i \(-0.411710\pi\)
0.273828 + 0.961779i \(0.411710\pi\)
\(60\) 0 0
\(61\) −0.103312 −0.0132278 −0.00661389 0.999978i \(-0.502105\pi\)
−0.00661389 + 0.999978i \(0.502105\pi\)
\(62\) 7.60437 0.965756
\(63\) −1.00000 −0.125988
\(64\) −11.2555 −1.40693
\(65\) 0 0
\(66\) −0.829422 −0.102095
\(67\) 11.7827 1.43949 0.719743 0.694241i \(-0.244260\pi\)
0.719743 + 0.694241i \(0.244260\pi\)
\(68\) 19.2555 2.33507
\(69\) 4.68714 0.564265
\(70\) 0 0
\(71\) −5.75441 −0.682922 −0.341461 0.939896i \(-0.610922\pi\)
−0.341461 + 0.939896i \(0.610922\pi\)
\(72\) 5.40550 0.637044
\(73\) 6.67939 0.781763 0.390882 0.920441i \(-0.372170\pi\)
0.390882 + 0.920441i \(0.372170\pi\)
\(74\) 19.3121 2.24498
\(75\) 0 0
\(76\) 0 0
\(77\) −0.198875 −0.0226639
\(78\) 18.0283 2.04130
\(79\) −3.87826 −0.436339 −0.218169 0.975911i \(-0.570009\pi\)
−0.218169 + 0.975911i \(0.570009\pi\)
\(80\) 0 0
\(81\) −2.97170 −0.330189
\(82\) −22.4260 −2.47654
\(83\) −0.488265 −0.0535941 −0.0267970 0.999641i \(-0.508531\pi\)
−0.0267970 + 0.999641i \(0.508531\pi\)
\(84\) −3.37720 −0.368483
\(85\) 0 0
\(86\) −23.3227 −2.51495
\(87\) −2.89669 −0.310558
\(88\) 1.07502 0.114597
\(89\) 16.4338 1.74198 0.870989 0.491302i \(-0.163479\pi\)
0.870989 + 0.491302i \(0.163479\pi\)
\(90\) 0 0
\(91\) 4.32273 0.453146
\(92\) −13.4338 −1.40057
\(93\) 4.07502 0.422559
\(94\) −28.9066 −2.98148
\(95\) 0 0
\(96\) −3.85772 −0.393727
\(97\) 4.44447 0.451267 0.225634 0.974212i \(-0.427555\pi\)
0.225634 + 0.974212i \(0.427555\pi\)
\(98\) 15.3871 1.55433
\(99\) 0.377203 0.0379103
\(100\) 0 0
\(101\) 4.38495 0.436319 0.218160 0.975913i \(-0.429995\pi\)
0.218160 + 0.975913i \(0.429995\pi\)
\(102\) 15.9709 1.58136
\(103\) −3.33048 −0.328162 −0.164081 0.986447i \(-0.552466\pi\)
−0.164081 + 0.986447i \(0.552466\pi\)
\(104\) −23.3665 −2.29128
\(105\) 0 0
\(106\) 13.5449 1.31559
\(107\) −16.4904 −1.59419 −0.797093 0.603857i \(-0.793630\pi\)
−0.797093 + 0.603857i \(0.793630\pi\)
\(108\) 20.3588 1.95902
\(109\) −7.79045 −0.746190 −0.373095 0.927793i \(-0.621703\pi\)
−0.373095 + 0.927793i \(0.621703\pi\)
\(110\) 0 0
\(111\) 10.3489 0.982275
\(112\) 1.47277 0.139163
\(113\) 0.142282 0.0133848 0.00669238 0.999978i \(-0.497870\pi\)
0.00669238 + 0.999978i \(0.497870\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.30219 0.770839
\(117\) −8.19887 −0.757986
\(118\) −10.0000 −0.920575
\(119\) 3.82942 0.351043
\(120\) 0 0
\(121\) −10.9250 −0.993180
\(122\) 0.245594 0.0222351
\(123\) −12.0176 −1.08359
\(124\) −11.6794 −1.04884
\(125\) 0 0
\(126\) 2.37720 0.211778
\(127\) 15.1316 1.34271 0.671357 0.741135i \(-0.265712\pi\)
0.671357 + 0.741135i \(0.265712\pi\)
\(128\) 20.6999 1.82963
\(129\) −12.4981 −1.10040
\(130\) 0 0
\(131\) 5.58383 0.487861 0.243931 0.969793i \(-0.421563\pi\)
0.243931 + 0.969793i \(0.421563\pi\)
\(132\) 1.27389 0.110878
\(133\) 0 0
\(134\) −28.0099 −2.41968
\(135\) 0 0
\(136\) −20.6999 −1.77500
\(137\) 12.8294 1.09609 0.548046 0.836448i \(-0.315372\pi\)
0.548046 + 0.836448i \(0.315372\pi\)
\(138\) −11.1423 −0.948494
\(139\) −15.2477 −1.29329 −0.646647 0.762789i \(-0.723829\pi\)
−0.646647 + 0.762789i \(0.723829\pi\)
\(140\) 0 0
\(141\) −15.4904 −1.30453
\(142\) 13.6794 1.14795
\(143\) −1.63055 −0.136353
\(144\) −2.79338 −0.232781
\(145\) 0 0
\(146\) −15.8783 −1.31409
\(147\) 8.24559 0.680085
\(148\) −29.6610 −2.43812
\(149\) 13.8315 1.13312 0.566562 0.824019i \(-0.308273\pi\)
0.566562 + 0.824019i \(0.308273\pi\)
\(150\) 0 0
\(151\) 11.7077 0.952758 0.476379 0.879240i \(-0.341949\pi\)
0.476379 + 0.879240i \(0.341949\pi\)
\(152\) 0 0
\(153\) −7.26322 −0.587196
\(154\) 0.472765 0.0380965
\(155\) 0 0
\(156\) −27.6893 −2.21692
\(157\) −4.79045 −0.382320 −0.191160 0.981559i \(-0.561225\pi\)
−0.191160 + 0.981559i \(0.561225\pi\)
\(158\) 9.21942 0.733458
\(159\) 7.25839 0.575628
\(160\) 0 0
\(161\) −2.67164 −0.210555
\(162\) 7.06434 0.555027
\(163\) 12.8011 1.00266 0.501331 0.865256i \(-0.332844\pi\)
0.501331 + 0.865256i \(0.332844\pi\)
\(164\) 34.4437 2.68960
\(165\) 0 0
\(166\) 1.16071 0.0900882
\(167\) 20.9426 1.62059 0.810294 0.586024i \(-0.199308\pi\)
0.810294 + 0.586024i \(0.199308\pi\)
\(168\) 3.63055 0.280103
\(169\) 22.4415 1.72627
\(170\) 0 0
\(171\) 0 0
\(172\) 35.8209 2.73132
\(173\) −15.7282 −1.19580 −0.597898 0.801572i \(-0.703997\pi\)
−0.597898 + 0.801572i \(0.703997\pi\)
\(174\) 6.88601 0.522027
\(175\) 0 0
\(176\) −0.555531 −0.0418747
\(177\) −5.35878 −0.402791
\(178\) −39.0665 −2.92816
\(179\) −3.41325 −0.255118 −0.127559 0.991831i \(-0.540714\pi\)
−0.127559 + 0.991831i \(0.540714\pi\)
\(180\) 0 0
\(181\) −23.5109 −1.74755 −0.873777 0.486327i \(-0.838336\pi\)
−0.873777 + 0.486327i \(0.838336\pi\)
\(182\) −10.2760 −0.761709
\(183\) 0.131609 0.00972878
\(184\) 14.4415 1.06464
\(185\) 0 0
\(186\) −9.68714 −0.710296
\(187\) −1.44447 −0.105630
\(188\) 44.3969 3.23798
\(189\) 4.04884 0.294510
\(190\) 0 0
\(191\) 12.4650 0.901937 0.450968 0.892540i \(-0.351079\pi\)
0.450968 + 0.892540i \(0.351079\pi\)
\(192\) 14.3382 1.03477
\(193\) −19.2993 −1.38919 −0.694596 0.719400i \(-0.744417\pi\)
−0.694596 + 0.719400i \(0.744417\pi\)
\(194\) −10.5654 −0.758552
\(195\) 0 0
\(196\) −23.6327 −1.68805
\(197\) 6.63055 0.472407 0.236203 0.971704i \(-0.424097\pi\)
0.236203 + 0.971704i \(0.424097\pi\)
\(198\) −0.896688 −0.0637248
\(199\) −23.0849 −1.63644 −0.818222 0.574902i \(-0.805040\pi\)
−0.818222 + 0.574902i \(0.805040\pi\)
\(200\) 0 0
\(201\) −15.0099 −1.05871
\(202\) −10.4239 −0.733425
\(203\) 1.65109 0.115884
\(204\) −24.5294 −1.71740
\(205\) 0 0
\(206\) 7.91723 0.551620
\(207\) 5.06727 0.352199
\(208\) 12.0750 0.837252
\(209\) 0 0
\(210\) 0 0
\(211\) 7.54778 0.519611 0.259805 0.965661i \(-0.416342\pi\)
0.259805 + 0.965661i \(0.416342\pi\)
\(212\) −20.8032 −1.42877
\(213\) 7.33048 0.502276
\(214\) 39.2010 2.67973
\(215\) 0 0
\(216\) −21.8860 −1.48915
\(217\) −2.32273 −0.157677
\(218\) 18.5195 1.25430
\(219\) −8.50881 −0.574972
\(220\) 0 0
\(221\) 31.3969 2.11199
\(222\) −24.6015 −1.65114
\(223\) 1.09344 0.0732221 0.0366111 0.999330i \(-0.488344\pi\)
0.0366111 + 0.999330i \(0.488344\pi\)
\(224\) 2.19887 0.146918
\(225\) 0 0
\(226\) −0.338233 −0.0224989
\(227\) 20.1316 1.33618 0.668091 0.744080i \(-0.267112\pi\)
0.668091 + 0.744080i \(0.267112\pi\)
\(228\) 0 0
\(229\) −5.51656 −0.364545 −0.182272 0.983248i \(-0.558345\pi\)
−0.182272 + 0.983248i \(0.558345\pi\)
\(230\) 0 0
\(231\) 0.253344 0.0166688
\(232\) −8.92498 −0.585954
\(233\) −18.1805 −1.19104 −0.595520 0.803340i \(-0.703054\pi\)
−0.595520 + 0.803340i \(0.703054\pi\)
\(234\) 19.4904 1.27413
\(235\) 0 0
\(236\) 15.3588 0.999771
\(237\) 4.94048 0.320919
\(238\) −9.10331 −0.590080
\(239\) 21.9164 1.41766 0.708828 0.705381i \(-0.249224\pi\)
0.708828 + 0.705381i \(0.249224\pi\)
\(240\) 0 0
\(241\) 28.1882 1.81576 0.907881 0.419228i \(-0.137699\pi\)
0.907881 + 0.419228i \(0.137699\pi\)
\(242\) 25.9709 1.66947
\(243\) −12.9426 −0.830269
\(244\) −0.377203 −0.0241479
\(245\) 0 0
\(246\) 28.5683 1.82145
\(247\) 0 0
\(248\) 12.5555 0.797277
\(249\) 0.621996 0.0394174
\(250\) 0 0
\(251\) 9.00987 0.568698 0.284349 0.958721i \(-0.408223\pi\)
0.284349 + 0.958721i \(0.408223\pi\)
\(252\) −3.65109 −0.229997
\(253\) 1.00775 0.0633567
\(254\) −35.9709 −2.25702
\(255\) 0 0
\(256\) −26.6970 −1.66856
\(257\) −6.86064 −0.427955 −0.213978 0.976839i \(-0.568642\pi\)
−0.213978 + 0.976839i \(0.568642\pi\)
\(258\) 29.7106 1.84970
\(259\) −5.89881 −0.366534
\(260\) 0 0
\(261\) −3.13161 −0.193842
\(262\) −13.2739 −0.820064
\(263\) −9.25547 −0.570717 −0.285358 0.958421i \(-0.592113\pi\)
−0.285358 + 0.958421i \(0.592113\pi\)
\(264\) −1.36945 −0.0842840
\(265\) 0 0
\(266\) 0 0
\(267\) −20.9349 −1.28119
\(268\) 43.0197 2.62785
\(269\) −0.498939 −0.0304208 −0.0152104 0.999884i \(-0.504842\pi\)
−0.0152104 + 0.999884i \(0.504842\pi\)
\(270\) 0 0
\(271\) 3.71061 0.225403 0.112702 0.993629i \(-0.464050\pi\)
0.112702 + 0.993629i \(0.464050\pi\)
\(272\) 10.6970 0.648602
\(273\) −5.50669 −0.333280
\(274\) −30.4981 −1.84246
\(275\) 0 0
\(276\) 17.1132 1.03009
\(277\) 4.58675 0.275591 0.137796 0.990461i \(-0.455998\pi\)
0.137796 + 0.990461i \(0.455998\pi\)
\(278\) 36.2469 2.17395
\(279\) 4.40550 0.263750
\(280\) 0 0
\(281\) −27.2653 −1.62651 −0.813257 0.581905i \(-0.802308\pi\)
−0.813257 + 0.581905i \(0.802308\pi\)
\(282\) 36.8238 2.19283
\(283\) −10.2661 −0.610259 −0.305129 0.952311i \(-0.598700\pi\)
−0.305129 + 0.952311i \(0.598700\pi\)
\(284\) −21.0099 −1.24671
\(285\) 0 0
\(286\) 3.87614 0.229201
\(287\) 6.84997 0.404341
\(288\) −4.17058 −0.245754
\(289\) 10.8139 0.636113
\(290\) 0 0
\(291\) −5.66177 −0.331899
\(292\) 24.3871 1.42715
\(293\) 1.87051 0.109277 0.0546383 0.998506i \(-0.482599\pi\)
0.0546383 + 0.998506i \(0.482599\pi\)
\(294\) −19.6015 −1.14318
\(295\) 0 0
\(296\) 31.8860 1.85334
\(297\) −1.52723 −0.0886192
\(298\) −32.8804 −1.90471
\(299\) −21.9044 −1.26677
\(300\) 0 0
\(301\) 7.12386 0.410612
\(302\) −27.8315 −1.60153
\(303\) −5.58595 −0.320904
\(304\) 0 0
\(305\) 0 0
\(306\) 17.2661 0.987040
\(307\) −0.227171 −0.0129653 −0.00648266 0.999979i \(-0.502064\pi\)
−0.00648266 + 0.999979i \(0.502064\pi\)
\(308\) −0.726109 −0.0413739
\(309\) 4.24267 0.241357
\(310\) 0 0
\(311\) 20.9554 1.18827 0.594136 0.804365i \(-0.297494\pi\)
0.594136 + 0.804365i \(0.297494\pi\)
\(312\) 29.7664 1.68519
\(313\) −11.2349 −0.635035 −0.317518 0.948252i \(-0.602849\pi\)
−0.317518 + 0.948252i \(0.602849\pi\)
\(314\) 11.3879 0.642655
\(315\) 0 0
\(316\) −14.1599 −0.796557
\(317\) −18.6228 −1.04596 −0.522980 0.852345i \(-0.675180\pi\)
−0.522980 + 0.852345i \(0.675180\pi\)
\(318\) −17.2547 −0.967594
\(319\) −0.622797 −0.0348699
\(320\) 0 0
\(321\) 21.0069 1.17249
\(322\) 6.35103 0.353929
\(323\) 0 0
\(324\) −10.8500 −0.602776
\(325\) 0 0
\(326\) −30.4309 −1.68541
\(327\) 9.92418 0.548809
\(328\) −37.0275 −2.04450
\(329\) 8.82942 0.486782
\(330\) 0 0
\(331\) 14.1054 0.775305 0.387652 0.921806i \(-0.373286\pi\)
0.387652 + 0.921806i \(0.373286\pi\)
\(332\) −1.78270 −0.0978385
\(333\) 11.1882 0.613110
\(334\) −49.7848 −2.72410
\(335\) 0 0
\(336\) −1.87614 −0.102352
\(337\) −22.9709 −1.25130 −0.625652 0.780102i \(-0.715167\pi\)
−0.625652 + 0.780102i \(0.715167\pi\)
\(338\) −53.3481 −2.90175
\(339\) −0.181252 −0.00984424
\(340\) 0 0
\(341\) 0.876142 0.0474457
\(342\) 0 0
\(343\) −9.78270 −0.528216
\(344\) −38.5080 −2.07621
\(345\) 0 0
\(346\) 37.3892 2.01006
\(347\) −3.93273 −0.211120 −0.105560 0.994413i \(-0.533664\pi\)
−0.105560 + 0.994413i \(0.533664\pi\)
\(348\) −10.5761 −0.566937
\(349\) −34.4252 −1.84274 −0.921371 0.388685i \(-0.872929\pi\)
−0.921371 + 0.388685i \(0.872929\pi\)
\(350\) 0 0
\(351\) 33.1960 1.77187
\(352\) −0.829422 −0.0442083
\(353\) −4.25547 −0.226496 −0.113248 0.993567i \(-0.536125\pi\)
−0.113248 + 0.993567i \(0.536125\pi\)
\(354\) 12.7389 0.677065
\(355\) 0 0
\(356\) 60.0013 3.18006
\(357\) −4.87826 −0.258185
\(358\) 8.11399 0.428837
\(359\) −20.2944 −1.07110 −0.535550 0.844504i \(-0.679896\pi\)
−0.535550 + 0.844504i \(0.679896\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 55.8903 2.93753
\(363\) 13.9172 0.730465
\(364\) 15.7827 0.827238
\(365\) 0 0
\(366\) −0.312860 −0.0163535
\(367\) 3.85289 0.201119 0.100560 0.994931i \(-0.467937\pi\)
0.100560 + 0.994931i \(0.467937\pi\)
\(368\) −7.46289 −0.389030
\(369\) −12.9922 −0.676350
\(370\) 0 0
\(371\) −4.13724 −0.214795
\(372\) 14.8783 0.771402
\(373\) −14.6356 −0.757802 −0.378901 0.925437i \(-0.623698\pi\)
−0.378901 + 0.925437i \(0.623698\pi\)
\(374\) 3.43380 0.177557
\(375\) 0 0
\(376\) −47.7274 −2.46135
\(377\) 13.5371 0.697197
\(378\) −9.62492 −0.495052
\(379\) 22.0099 1.13057 0.565286 0.824895i \(-0.308766\pi\)
0.565286 + 0.824895i \(0.308766\pi\)
\(380\) 0 0
\(381\) −19.2760 −0.987540
\(382\) −29.6319 −1.51610
\(383\) 3.08569 0.157671 0.0788357 0.996888i \(-0.474880\pi\)
0.0788357 + 0.996888i \(0.474880\pi\)
\(384\) −26.3695 −1.34566
\(385\) 0 0
\(386\) 45.8783 2.33514
\(387\) −13.5117 −0.686840
\(388\) 16.2272 0.823810
\(389\) 8.77203 0.444760 0.222380 0.974960i \(-0.428618\pi\)
0.222380 + 0.974960i \(0.428618\pi\)
\(390\) 0 0
\(391\) −19.4047 −0.981338
\(392\) 25.4055 1.28317
\(393\) −7.11319 −0.358813
\(394\) −15.7622 −0.794086
\(395\) 0 0
\(396\) 1.37720 0.0692070
\(397\) −1.59450 −0.0800257 −0.0400129 0.999199i \(-0.512740\pi\)
−0.0400129 + 0.999199i \(0.512740\pi\)
\(398\) 54.8775 2.75076
\(399\) 0 0
\(400\) 0 0
\(401\) 17.5526 0.876535 0.438268 0.898844i \(-0.355592\pi\)
0.438268 + 0.898844i \(0.355592\pi\)
\(402\) 35.6815 1.77963
\(403\) −19.0438 −0.948639
\(404\) 16.0099 0.796521
\(405\) 0 0
\(406\) −3.92498 −0.194794
\(407\) 2.22505 0.110292
\(408\) 26.3695 1.30548
\(409\) −36.6815 −1.81378 −0.906892 0.421363i \(-0.861552\pi\)
−0.906892 + 0.421363i \(0.861552\pi\)
\(410\) 0 0
\(411\) −16.3433 −0.806155
\(412\) −12.1599 −0.599076
\(413\) 3.05447 0.150301
\(414\) −12.0459 −0.592025
\(415\) 0 0
\(416\) 18.0283 0.883910
\(417\) 19.4239 0.951194
\(418\) 0 0
\(419\) −18.8187 −0.919356 −0.459678 0.888086i \(-0.652035\pi\)
−0.459678 + 0.888086i \(0.652035\pi\)
\(420\) 0 0
\(421\) 33.7819 1.64643 0.823215 0.567730i \(-0.192178\pi\)
0.823215 + 0.567730i \(0.192178\pi\)
\(422\) −17.9426 −0.873432
\(423\) −16.7467 −0.814250
\(424\) 22.3638 1.08608
\(425\) 0 0
\(426\) −17.4260 −0.844295
\(427\) −0.0750160 −0.00363028
\(428\) −60.2079 −2.91026
\(429\) 2.07714 0.100285
\(430\) 0 0
\(431\) 12.7651 0.614872 0.307436 0.951569i \(-0.400529\pi\)
0.307436 + 0.951569i \(0.400529\pi\)
\(432\) 11.3099 0.544150
\(433\) 16.0771 0.772618 0.386309 0.922369i \(-0.373750\pi\)
0.386309 + 0.922369i \(0.373750\pi\)
\(434\) 5.52161 0.265046
\(435\) 0 0
\(436\) −28.4437 −1.36220
\(437\) 0 0
\(438\) 20.2272 0.966492
\(439\) −1.36945 −0.0653604 −0.0326802 0.999466i \(-0.510404\pi\)
−0.0326802 + 0.999466i \(0.510404\pi\)
\(440\) 0 0
\(441\) 8.91431 0.424491
\(442\) −74.6369 −3.55012
\(443\) 4.62280 0.219636 0.109818 0.993952i \(-0.464973\pi\)
0.109818 + 0.993952i \(0.464973\pi\)
\(444\) 37.7848 1.79319
\(445\) 0 0
\(446\) −2.59933 −0.123082
\(447\) −17.6199 −0.833391
\(448\) −8.17270 −0.386124
\(449\) −23.2555 −1.09749 −0.548747 0.835989i \(-0.684895\pi\)
−0.548747 + 0.835989i \(0.684895\pi\)
\(450\) 0 0
\(451\) −2.58383 −0.121668
\(452\) 0.519485 0.0244345
\(453\) −14.9143 −0.700735
\(454\) −47.8569 −2.24604
\(455\) 0 0
\(456\) 0 0
\(457\) −35.8443 −1.67673 −0.838364 0.545111i \(-0.816487\pi\)
−0.838364 + 0.545111i \(0.816487\pi\)
\(458\) 13.1140 0.612776
\(459\) 29.4076 1.37263
\(460\) 0 0
\(461\) −14.8812 −0.693086 −0.346543 0.938034i \(-0.612645\pi\)
−0.346543 + 0.938034i \(0.612645\pi\)
\(462\) −0.602251 −0.0280193
\(463\) −29.9554 −1.39215 −0.696073 0.717971i \(-0.745071\pi\)
−0.696073 + 0.717971i \(0.745071\pi\)
\(464\) 4.61212 0.214112
\(465\) 0 0
\(466\) 43.2186 2.00206
\(467\) −6.73598 −0.311704 −0.155852 0.987780i \(-0.549812\pi\)
−0.155852 + 0.987780i \(0.549812\pi\)
\(468\) −29.9349 −1.38374
\(469\) 8.55553 0.395058
\(470\) 0 0
\(471\) 6.10251 0.281189
\(472\) −16.5109 −0.759977
\(473\) −2.68714 −0.123555
\(474\) −11.7445 −0.539444
\(475\) 0 0
\(476\) 13.9816 0.640845
\(477\) 7.84704 0.359291
\(478\) −52.0998 −2.38299
\(479\) 16.6978 0.762943 0.381471 0.924381i \(-0.375418\pi\)
0.381471 + 0.924381i \(0.375418\pi\)
\(480\) 0 0
\(481\) −48.3636 −2.20519
\(482\) −67.0091 −3.05218
\(483\) 3.40338 0.154859
\(484\) −39.8881 −1.81310
\(485\) 0 0
\(486\) 30.7672 1.39563
\(487\) −3.64042 −0.164963 −0.0824816 0.996593i \(-0.526285\pi\)
−0.0824816 + 0.996593i \(0.526285\pi\)
\(488\) 0.405499 0.0183561
\(489\) −16.3072 −0.737439
\(490\) 0 0
\(491\) −33.3249 −1.50393 −0.751965 0.659203i \(-0.770894\pi\)
−0.751965 + 0.659203i \(0.770894\pi\)
\(492\) −43.8775 −1.97815
\(493\) 11.9922 0.540104
\(494\) 0 0
\(495\) 0 0
\(496\) −6.48827 −0.291332
\(497\) −4.17833 −0.187424
\(498\) −1.47861 −0.0662582
\(499\) −37.9914 −1.70073 −0.850365 0.526193i \(-0.823619\pi\)
−0.850365 + 0.526193i \(0.823619\pi\)
\(500\) 0 0
\(501\) −26.6786 −1.19191
\(502\) −21.4183 −0.955945
\(503\) −42.1826 −1.88083 −0.940414 0.340032i \(-0.889562\pi\)
−0.940414 + 0.340032i \(0.889562\pi\)
\(504\) 3.92498 0.174833
\(505\) 0 0
\(506\) −2.39563 −0.106499
\(507\) −28.5881 −1.26964
\(508\) 55.2469 2.45119
\(509\) 21.9971 0.975003 0.487502 0.873122i \(-0.337908\pi\)
0.487502 + 0.873122i \(0.337908\pi\)
\(510\) 0 0
\(511\) 4.84997 0.214550
\(512\) 22.0643 0.975115
\(513\) 0 0
\(514\) 16.3091 0.719365
\(515\) 0 0
\(516\) −45.6319 −2.00883
\(517\) −3.33048 −0.146474
\(518\) 14.0227 0.616121
\(519\) 20.0360 0.879485
\(520\) 0 0
\(521\) −20.0977 −0.880496 −0.440248 0.897876i \(-0.645109\pi\)
−0.440248 + 0.897876i \(0.645109\pi\)
\(522\) 7.44447 0.325836
\(523\) 4.64817 0.203250 0.101625 0.994823i \(-0.467596\pi\)
0.101625 + 0.994823i \(0.467596\pi\)
\(524\) 20.3871 0.890614
\(525\) 0 0
\(526\) 22.0021 0.959338
\(527\) −16.8705 −0.734891
\(528\) 0.707686 0.0307981
\(529\) −9.46209 −0.411395
\(530\) 0 0
\(531\) −5.79338 −0.251411
\(532\) 0 0
\(533\) 56.1620 2.43265
\(534\) 49.7664 2.15360
\(535\) 0 0
\(536\) −46.2469 −1.99756
\(537\) 4.34811 0.187635
\(538\) 1.18608 0.0511355
\(539\) 1.77283 0.0763612
\(540\) 0 0
\(541\) 20.0673 0.862759 0.431380 0.902171i \(-0.358027\pi\)
0.431380 + 0.902171i \(0.358027\pi\)
\(542\) −8.82087 −0.378889
\(543\) 29.9504 1.28529
\(544\) 15.9709 0.684747
\(545\) 0 0
\(546\) 13.0905 0.560222
\(547\) 37.2010 1.59060 0.795300 0.606216i \(-0.207313\pi\)
0.795300 + 0.606216i \(0.207313\pi\)
\(548\) 46.8414 2.00097
\(549\) 0.142282 0.00607245
\(550\) 0 0
\(551\) 0 0
\(552\) −18.3969 −0.783026
\(553\) −2.81604 −0.119750
\(554\) −10.9036 −0.463251
\(555\) 0 0
\(556\) −55.6708 −2.36097
\(557\) 44.8393 1.89990 0.949951 0.312399i \(-0.101133\pi\)
0.949951 + 0.312399i \(0.101133\pi\)
\(558\) −10.4728 −0.443347
\(559\) 58.4076 2.47038
\(560\) 0 0
\(561\) 1.84010 0.0776889
\(562\) 64.8152 2.73407
\(563\) 21.9172 0.923701 0.461851 0.886958i \(-0.347186\pi\)
0.461851 + 0.886958i \(0.347186\pi\)
\(564\) −56.5569 −2.38147
\(565\) 0 0
\(566\) 24.4047 1.02581
\(567\) −2.15778 −0.0906183
\(568\) 22.5860 0.947685
\(569\) −9.90656 −0.415305 −0.207652 0.978203i \(-0.566582\pi\)
−0.207652 + 0.978203i \(0.566582\pi\)
\(570\) 0 0
\(571\) 17.6404 0.738229 0.369114 0.929384i \(-0.379661\pi\)
0.369114 + 0.929384i \(0.379661\pi\)
\(572\) −5.95328 −0.248919
\(573\) −15.8791 −0.663357
\(574\) −16.2838 −0.679671
\(575\) 0 0
\(576\) 15.5011 0.645878
\(577\) −12.7048 −0.528906 −0.264453 0.964399i \(-0.585191\pi\)
−0.264453 + 0.964399i \(0.585191\pi\)
\(578\) −25.7069 −1.06927
\(579\) 24.5851 1.02172
\(580\) 0 0
\(581\) −0.354534 −0.0147085
\(582\) 13.4592 0.557900
\(583\) 1.56058 0.0646325
\(584\) −26.2165 −1.08485
\(585\) 0 0
\(586\) −4.44659 −0.183687
\(587\) 15.0438 0.620924 0.310462 0.950586i \(-0.399516\pi\)
0.310462 + 0.950586i \(0.399516\pi\)
\(588\) 30.1054 1.24153
\(589\) 0 0
\(590\) 0 0
\(591\) −8.44659 −0.347446
\(592\) −16.4776 −0.677225
\(593\) −16.4231 −0.674417 −0.337208 0.941430i \(-0.609483\pi\)
−0.337208 + 0.941430i \(0.609483\pi\)
\(594\) 3.63055 0.148963
\(595\) 0 0
\(596\) 50.5003 2.06857
\(597\) 29.4076 1.20357
\(598\) 52.0713 2.12935
\(599\) 19.1260 0.781466 0.390733 0.920504i \(-0.372222\pi\)
0.390733 + 0.920504i \(0.372222\pi\)
\(600\) 0 0
\(601\) −31.4124 −1.28134 −0.640670 0.767816i \(-0.721343\pi\)
−0.640670 + 0.767816i \(0.721343\pi\)
\(602\) −16.9349 −0.690213
\(603\) −16.2272 −0.660821
\(604\) 42.7459 1.73930
\(605\) 0 0
\(606\) 13.2789 0.539420
\(607\) 41.5315 1.68571 0.842855 0.538140i \(-0.180873\pi\)
0.842855 + 0.538140i \(0.180873\pi\)
\(608\) 0 0
\(609\) −2.10331 −0.0852305
\(610\) 0 0
\(611\) 72.3913 2.92864
\(612\) −26.5187 −1.07195
\(613\) 21.7274 0.877563 0.438781 0.898594i \(-0.355410\pi\)
0.438781 + 0.898594i \(0.355410\pi\)
\(614\) 0.540031 0.0217939
\(615\) 0 0
\(616\) 0.780579 0.0314504
\(617\) 33.6065 1.35295 0.676473 0.736467i \(-0.263507\pi\)
0.676473 + 0.736467i \(0.263507\pi\)
\(618\) −10.0857 −0.405706
\(619\) −27.6036 −1.10948 −0.554741 0.832023i \(-0.687183\pi\)
−0.554741 + 0.832023i \(0.687183\pi\)
\(620\) 0 0
\(621\) −20.5166 −0.823301
\(622\) −49.8152 −1.99741
\(623\) 11.9327 0.478075
\(624\) −15.3822 −0.615783
\(625\) 0 0
\(626\) 26.7077 1.06745
\(627\) 0 0
\(628\) −17.4904 −0.697942
\(629\) −42.8443 −1.70832
\(630\) 0 0
\(631\) 1.94048 0.0772495 0.0386247 0.999254i \(-0.487702\pi\)
0.0386247 + 0.999254i \(0.487702\pi\)
\(632\) 15.2221 0.605504
\(633\) −9.61505 −0.382164
\(634\) 44.2702 1.75819
\(635\) 0 0
\(636\) 26.5011 1.05084
\(637\) −38.5342 −1.52678
\(638\) 1.48052 0.0586142
\(639\) 7.92498 0.313508
\(640\) 0 0
\(641\) −1.01975 −0.0402775 −0.0201388 0.999797i \(-0.506411\pi\)
−0.0201388 + 0.999797i \(0.506411\pi\)
\(642\) −49.9378 −1.97089
\(643\) 36.9866 1.45861 0.729305 0.684189i \(-0.239844\pi\)
0.729305 + 0.684189i \(0.239844\pi\)
\(644\) −9.75441 −0.384377
\(645\) 0 0
\(646\) 0 0
\(647\) 24.1182 0.948186 0.474093 0.880475i \(-0.342776\pi\)
0.474093 + 0.880475i \(0.342776\pi\)
\(648\) 11.6639 0.458201
\(649\) −1.15215 −0.0452260
\(650\) 0 0
\(651\) 2.95891 0.115969
\(652\) 46.7381 1.83041
\(653\) 37.2603 1.45811 0.729054 0.684456i \(-0.239960\pi\)
0.729054 + 0.684456i \(0.239960\pi\)
\(654\) −23.5918 −0.922512
\(655\) 0 0
\(656\) 19.1345 0.747078
\(657\) −9.19887 −0.358882
\(658\) −20.9893 −0.818249
\(659\) −21.4386 −0.835130 −0.417565 0.908647i \(-0.637116\pi\)
−0.417565 + 0.908647i \(0.637116\pi\)
\(660\) 0 0
\(661\) −0.783503 −0.0304747 −0.0152374 0.999884i \(-0.504850\pi\)
−0.0152374 + 0.999884i \(0.504850\pi\)
\(662\) −33.5315 −1.30324
\(663\) −39.9963 −1.55333
\(664\) 1.91643 0.0743720
\(665\) 0 0
\(666\) −26.5966 −1.03060
\(667\) −8.36653 −0.323953
\(668\) 76.4634 2.95846
\(669\) −1.39292 −0.0538535
\(670\) 0 0
\(671\) 0.0282963 0.00109237
\(672\) −2.80113 −0.108056
\(673\) 50.1903 1.93469 0.967347 0.253454i \(-0.0815667\pi\)
0.967347 + 0.253454i \(0.0815667\pi\)
\(674\) 54.6065 2.10336
\(675\) 0 0
\(676\) 81.9362 3.15139
\(677\) 29.8804 1.14840 0.574198 0.818716i \(-0.305314\pi\)
0.574198 + 0.818716i \(0.305314\pi\)
\(678\) 0.430872 0.0165475
\(679\) 3.22717 0.123847
\(680\) 0 0
\(681\) −25.6455 −0.982736
\(682\) −2.08277 −0.0797532
\(683\) −12.3326 −0.471894 −0.235947 0.971766i \(-0.575819\pi\)
−0.235947 + 0.971766i \(0.575819\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 23.2555 0.887898
\(687\) 7.02750 0.268116
\(688\) 19.8996 0.758666
\(689\) −33.9207 −1.29227
\(690\) 0 0
\(691\) 3.62200 0.137787 0.0688936 0.997624i \(-0.478053\pi\)
0.0688936 + 0.997624i \(0.478053\pi\)
\(692\) −57.4252 −2.18298
\(693\) 0.273891 0.0104042
\(694\) 9.34891 0.354880
\(695\) 0 0
\(696\) 11.3695 0.430958
\(697\) 49.7528 1.88452
\(698\) 81.8358 3.09753
\(699\) 23.1599 0.875988
\(700\) 0 0
\(701\) 34.1209 1.28873 0.644365 0.764718i \(-0.277122\pi\)
0.644365 + 0.764718i \(0.277122\pi\)
\(702\) −78.9135 −2.97840
\(703\) 0 0
\(704\) 3.08277 0.116186
\(705\) 0 0
\(706\) 10.1161 0.380725
\(707\) 3.18396 0.119745
\(708\) −19.5654 −0.735313
\(709\) −17.1209 −0.642990 −0.321495 0.946911i \(-0.604185\pi\)
−0.321495 + 0.946911i \(0.604185\pi\)
\(710\) 0 0
\(711\) 5.34116 0.200309
\(712\) −64.5024 −2.41733
\(713\) 11.7699 0.440786
\(714\) 11.5966 0.433993
\(715\) 0 0
\(716\) −12.4621 −0.465730
\(717\) −27.9191 −1.04266
\(718\) 48.2440 1.80045
\(719\) −7.02750 −0.262081 −0.131041 0.991377i \(-0.541832\pi\)
−0.131041 + 0.991377i \(0.541832\pi\)
\(720\) 0 0
\(721\) −2.41830 −0.0900620
\(722\) 0 0
\(723\) −35.9087 −1.33546
\(724\) −85.8406 −3.19024
\(725\) 0 0
\(726\) −33.0841 −1.22787
\(727\) −11.8938 −0.441115 −0.220558 0.975374i \(-0.570788\pi\)
−0.220558 + 0.975374i \(0.570788\pi\)
\(728\) −16.9667 −0.628826
\(729\) 25.4026 0.940836
\(730\) 0 0
\(731\) 51.7421 1.91375
\(732\) 0.480515 0.0177604
\(733\) 20.7154 0.765142 0.382571 0.923926i \(-0.375039\pi\)
0.382571 + 0.923926i \(0.375039\pi\)
\(734\) −9.15910 −0.338069
\(735\) 0 0
\(736\) −11.1423 −0.410710
\(737\) −3.22717 −0.118874
\(738\) 30.8852 1.13690
\(739\) 33.8620 1.24563 0.622816 0.782368i \(-0.285988\pi\)
0.622816 + 0.782368i \(0.285988\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.83505 0.361056
\(743\) −42.7381 −1.56791 −0.783955 0.620818i \(-0.786800\pi\)
−0.783955 + 0.620818i \(0.786800\pi\)
\(744\) −15.9944 −0.586382
\(745\) 0 0
\(746\) 34.7918 1.27382
\(747\) 0.672440 0.0246033
\(748\) −5.27389 −0.192833
\(749\) −11.9738 −0.437514
\(750\) 0 0
\(751\) −11.9581 −0.436358 −0.218179 0.975909i \(-0.570012\pi\)
−0.218179 + 0.975909i \(0.570012\pi\)
\(752\) 24.6639 0.899400
\(753\) −11.4776 −0.418267
\(754\) −32.1805 −1.17194
\(755\) 0 0
\(756\) 14.7827 0.537642
\(757\) 29.1103 1.05803 0.529015 0.848612i \(-0.322561\pi\)
0.529015 + 0.848612i \(0.322561\pi\)
\(758\) −52.3219 −1.90042
\(759\) −1.28376 −0.0465977
\(760\) 0 0
\(761\) −22.3014 −0.808425 −0.404212 0.914665i \(-0.632454\pi\)
−0.404212 + 0.914665i \(0.632454\pi\)
\(762\) 45.8230 1.65999
\(763\) −5.65672 −0.204787
\(764\) 45.5109 1.64653
\(765\) 0 0
\(766\) −7.33531 −0.265036
\(767\) 25.0432 0.904258
\(768\) 34.0091 1.22720
\(769\) 3.95891 0.142762 0.0713809 0.997449i \(-0.477259\pi\)
0.0713809 + 0.997449i \(0.477259\pi\)
\(770\) 0 0
\(771\) 8.73971 0.314753
\(772\) −70.4634 −2.53603
\(773\) −27.8139 −1.00040 −0.500199 0.865911i \(-0.666740\pi\)
−0.500199 + 0.865911i \(0.666740\pi\)
\(774\) 32.1201 1.15453
\(775\) 0 0
\(776\) −17.4445 −0.626220
\(777\) 7.51444 0.269579
\(778\) −20.8529 −0.747612
\(779\) 0 0
\(780\) 0 0
\(781\) 1.57608 0.0563965
\(782\) 46.1289 1.64957
\(783\) 12.6794 0.453124
\(784\) −13.1287 −0.468882
\(785\) 0 0
\(786\) 16.9095 0.603141
\(787\) 1.82460 0.0650398 0.0325199 0.999471i \(-0.489647\pi\)
0.0325199 + 0.999471i \(0.489647\pi\)
\(788\) 24.2087 0.862401
\(789\) 11.7905 0.419751
\(790\) 0 0
\(791\) 0.103312 0.00367336
\(792\) −1.48052 −0.0526078
\(793\) −0.615047 −0.0218410
\(794\) 3.79045 0.134518
\(795\) 0 0
\(796\) −84.2851 −2.98741
\(797\) 21.0360 0.745135 0.372567 0.928005i \(-0.378478\pi\)
0.372567 + 0.928005i \(0.378478\pi\)
\(798\) 0 0
\(799\) 64.1300 2.26876
\(800\) 0 0
\(801\) −22.6327 −0.799686
\(802\) −41.7261 −1.47340
\(803\) −1.82942 −0.0645589
\(804\) −54.8024 −1.93273
\(805\) 0 0
\(806\) 45.2710 1.59460
\(807\) 0.635593 0.0223739
\(808\) −17.2109 −0.605476
\(809\) −0.0819654 −0.00288175 −0.00144088 0.999999i \(-0.500459\pi\)
−0.00144088 + 0.999999i \(0.500459\pi\)
\(810\) 0 0
\(811\) 6.72531 0.236158 0.118079 0.993004i \(-0.462326\pi\)
0.118079 + 0.993004i \(0.462326\pi\)
\(812\) 6.02830 0.211552
\(813\) −4.72691 −0.165780
\(814\) −5.28939 −0.185393
\(815\) 0 0
\(816\) −13.6268 −0.477034
\(817\) 0 0
\(818\) 87.1994 3.04886
\(819\) −5.95328 −0.208024
\(820\) 0 0
\(821\) 30.9426 1.07990 0.539952 0.841696i \(-0.318442\pi\)
0.539952 + 0.841696i \(0.318442\pi\)
\(822\) 38.8513 1.35509
\(823\) 26.5908 0.926896 0.463448 0.886124i \(-0.346612\pi\)
0.463448 + 0.886124i \(0.346612\pi\)
\(824\) 13.0721 0.455388
\(825\) 0 0
\(826\) −7.26109 −0.252646
\(827\) 5.57900 0.194001 0.0970004 0.995284i \(-0.469075\pi\)
0.0970004 + 0.995284i \(0.469075\pi\)
\(828\) 18.5011 0.642956
\(829\) −18.9765 −0.659082 −0.329541 0.944141i \(-0.606894\pi\)
−0.329541 + 0.944141i \(0.606894\pi\)
\(830\) 0 0
\(831\) −5.84302 −0.202692
\(832\) −67.0069 −2.32305
\(833\) −34.1367 −1.18276
\(834\) −46.1746 −1.59890
\(835\) 0 0
\(836\) 0 0
\(837\) −17.8372 −0.616543
\(838\) 44.7360 1.54538
\(839\) −12.9143 −0.445852 −0.222926 0.974835i \(-0.571561\pi\)
−0.222926 + 0.974835i \(0.571561\pi\)
\(840\) 0 0
\(841\) −23.8294 −0.821704
\(842\) −80.3064 −2.76754
\(843\) 34.7331 1.19627
\(844\) 27.5577 0.948574
\(845\) 0 0
\(846\) 39.8102 1.36870
\(847\) −7.93273 −0.272572
\(848\) −11.5569 −0.396864
\(849\) 13.0779 0.448834
\(850\) 0 0
\(851\) 29.8908 1.02464
\(852\) 26.7643 0.916929
\(853\) 6.46077 0.221213 0.110606 0.993864i \(-0.464721\pi\)
0.110606 + 0.993864i \(0.464721\pi\)
\(854\) 0.178328 0.00610227
\(855\) 0 0
\(856\) 64.7245 2.21224
\(857\) 12.4055 0.423764 0.211882 0.977295i \(-0.432041\pi\)
0.211882 + 0.977295i \(0.432041\pi\)
\(858\) −4.93778 −0.168573
\(859\) 40.3425 1.37647 0.688234 0.725489i \(-0.258386\pi\)
0.688234 + 0.725489i \(0.258386\pi\)
\(860\) 0 0
\(861\) −8.72611 −0.297385
\(862\) −30.3452 −1.03356
\(863\) −1.83235 −0.0623738 −0.0311869 0.999514i \(-0.509929\pi\)
−0.0311869 + 0.999514i \(0.509929\pi\)
\(864\) 16.8860 0.574474
\(865\) 0 0
\(866\) −38.2186 −1.29872
\(867\) −13.7758 −0.467849
\(868\) −8.48052 −0.287847
\(869\) 1.06222 0.0360333
\(870\) 0 0
\(871\) 70.1457 2.37680
\(872\) 30.5774 1.03548
\(873\) −6.12094 −0.207162
\(874\) 0 0
\(875\) 0 0
\(876\) −31.0665 −1.04964
\(877\) 8.14419 0.275010 0.137505 0.990501i \(-0.456092\pi\)
0.137505 + 0.990501i \(0.456092\pi\)
\(878\) 3.25547 0.109867
\(879\) −2.38283 −0.0803709
\(880\) 0 0
\(881\) 12.1706 0.410037 0.205019 0.978758i \(-0.434275\pi\)
0.205019 + 0.978758i \(0.434275\pi\)
\(882\) −21.1911 −0.713542
\(883\) 46.7614 1.57364 0.786822 0.617179i \(-0.211725\pi\)
0.786822 + 0.617179i \(0.211725\pi\)
\(884\) 114.633 3.85553
\(885\) 0 0
\(886\) −10.9893 −0.369194
\(887\) 34.8804 1.17117 0.585584 0.810611i \(-0.300865\pi\)
0.585584 + 0.810611i \(0.300865\pi\)
\(888\) −40.6193 −1.36309
\(889\) 10.9872 0.368499
\(890\) 0 0
\(891\) 0.813922 0.0272674
\(892\) 3.99225 0.133670
\(893\) 0 0
\(894\) 41.8860 1.40088
\(895\) 0 0
\(896\) 15.0304 0.502131
\(897\) 27.9039 0.931683
\(898\) 55.2830 1.84482
\(899\) −7.27389 −0.242598
\(900\) 0 0
\(901\) −30.0496 −1.00110
\(902\) 6.14228 0.204516
\(903\) −9.07502 −0.301998
\(904\) −0.558455 −0.0185739
\(905\) 0 0
\(906\) 35.4543 1.17789
\(907\) 25.5080 0.846980 0.423490 0.905901i \(-0.360805\pi\)
0.423490 + 0.905901i \(0.360805\pi\)
\(908\) 73.5024 2.43926
\(909\) −6.03897 −0.200300
\(910\) 0 0
\(911\) 21.5032 0.712432 0.356216 0.934404i \(-0.384067\pi\)
0.356216 + 0.934404i \(0.384067\pi\)
\(912\) 0 0
\(913\) 0.133731 0.00442586
\(914\) 85.2093 2.81847
\(915\) 0 0
\(916\) −20.1415 −0.665493
\(917\) 4.05447 0.133890
\(918\) −69.9079 −2.30730
\(919\) 37.1386 1.22509 0.612544 0.790436i \(-0.290146\pi\)
0.612544 + 0.790436i \(0.290146\pi\)
\(920\) 0 0
\(921\) 0.289391 0.00953575
\(922\) 35.3756 1.16503
\(923\) −34.2576 −1.12760
\(924\) 0.924984 0.0304297
\(925\) 0 0
\(926\) 71.2101 2.34011
\(927\) 4.58675 0.150649
\(928\) 6.88601 0.226044
\(929\) −3.36170 −0.110294 −0.0551469 0.998478i \(-0.517563\pi\)
−0.0551469 + 0.998478i \(0.517563\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −66.3785 −2.17430
\(933\) −26.6949 −0.873951
\(934\) 16.0128 0.523955
\(935\) 0 0
\(936\) 32.1805 1.05185
\(937\) 10.0694 0.328953 0.164476 0.986381i \(-0.447407\pi\)
0.164476 + 0.986381i \(0.447407\pi\)
\(938\) −20.3382 −0.664067
\(939\) 14.3121 0.467056
\(940\) 0 0
\(941\) −20.8139 −0.678514 −0.339257 0.940694i \(-0.610176\pi\)
−0.339257 + 0.940694i \(0.610176\pi\)
\(942\) −14.5069 −0.472661
\(943\) −34.7106 −1.13033
\(944\) 8.53228 0.277702
\(945\) 0 0
\(946\) 6.38788 0.207688
\(947\) −30.4904 −0.990804 −0.495402 0.868664i \(-0.664979\pi\)
−0.495402 + 0.868664i \(0.664979\pi\)
\(948\) 18.0382 0.585852
\(949\) 39.7643 1.29080
\(950\) 0 0
\(951\) 23.7234 0.769284
\(952\) −15.0304 −0.487139
\(953\) −7.58383 −0.245664 −0.122832 0.992427i \(-0.539198\pi\)
−0.122832 + 0.992427i \(0.539198\pi\)
\(954\) −18.6540 −0.603946
\(955\) 0 0
\(956\) 80.0189 2.58800
\(957\) 0.793375 0.0256462
\(958\) −39.6941 −1.28246
\(959\) 9.31556 0.300815
\(960\) 0 0
\(961\) −20.7672 −0.669910
\(962\) 114.970 3.70678
\(963\) 22.7106 0.731839
\(964\) 102.918 3.31476
\(965\) 0 0
\(966\) −8.09052 −0.260308
\(967\) 54.6687 1.75803 0.879014 0.476797i \(-0.158202\pi\)
0.879014 + 0.476797i \(0.158202\pi\)
\(968\) 42.8804 1.37823
\(969\) 0 0
\(970\) 0 0
\(971\) 39.9632 1.28248 0.641239 0.767341i \(-0.278421\pi\)
0.641239 + 0.767341i \(0.278421\pi\)
\(972\) −47.2547 −1.51569
\(973\) −11.0715 −0.354936
\(974\) 8.65402 0.277293
\(975\) 0 0
\(976\) −0.209548 −0.00670747
\(977\) −15.2400 −0.487570 −0.243785 0.969829i \(-0.578389\pi\)
−0.243785 + 0.969829i \(0.578389\pi\)
\(978\) 38.7656 1.23959
\(979\) −4.50106 −0.143855
\(980\) 0 0
\(981\) 10.7290 0.342552
\(982\) 79.2199 2.52801
\(983\) 45.3609 1.44679 0.723394 0.690435i \(-0.242581\pi\)
0.723394 + 0.690435i \(0.242581\pi\)
\(984\) 47.1690 1.50369
\(985\) 0 0
\(986\) −28.5080 −0.907880
\(987\) −11.2477 −0.358019
\(988\) 0 0
\(989\) −36.0985 −1.14787
\(990\) 0 0
\(991\) 16.3537 0.519493 0.259747 0.965677i \(-0.416361\pi\)
0.259747 + 0.965677i \(0.416361\pi\)
\(992\) −9.68714 −0.307567
\(993\) −17.9688 −0.570222
\(994\) 9.93273 0.315047
\(995\) 0 0
\(996\) 2.27097 0.0719583
\(997\) −33.2037 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(998\) 90.3134 2.85882
\(999\) −45.2993 −1.43321
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.y.1.1 3
5.4 even 2 9025.2.a.bc.1.3 3
19.18 odd 2 475.2.a.g.1.3 yes 3
57.56 even 2 4275.2.a.ba.1.1 3
76.75 even 2 7600.2.a.bh.1.2 3
95.18 even 4 475.2.b.b.324.1 6
95.37 even 4 475.2.b.b.324.6 6
95.94 odd 2 475.2.a.e.1.1 3
285.284 even 2 4275.2.a.bm.1.3 3
380.379 even 2 7600.2.a.cc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.1 3 95.94 odd 2
475.2.a.g.1.3 yes 3 19.18 odd 2
475.2.b.b.324.1 6 95.18 even 4
475.2.b.b.324.6 6 95.37 even 4
4275.2.a.ba.1.1 3 57.56 even 2
4275.2.a.bm.1.3 3 285.284 even 2
7600.2.a.bh.1.2 3 76.75 even 2
7600.2.a.cc.1.2 3 380.379 even 2
9025.2.a.y.1.1 3 1.1 even 1 trivial
9025.2.a.bc.1.3 3 5.4 even 2