Properties

Label 1875.2.a.i.1.2
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $6$
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] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.46840000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.13324\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.13324 q^{2} +1.00000 q^{3} +2.55073 q^{4} -2.13324 q^{6} +2.16876 q^{7} -1.17484 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.13324 q^{2} +1.00000 q^{3} +2.55073 q^{4} -2.13324 q^{6} +2.16876 q^{7} -1.17484 q^{8} +1.00000 q^{9} -2.50913 q^{11} +2.55073 q^{12} -4.33379 q^{13} -4.62650 q^{14} -2.59524 q^{16} -6.77562 q^{17} -2.13324 q^{18} +6.83602 q^{19} +2.16876 q^{21} +5.35259 q^{22} -1.67843 q^{23} -1.17484 q^{24} +9.24504 q^{26} +1.00000 q^{27} +5.53193 q^{28} +4.44927 q^{29} +6.56295 q^{31} +7.88596 q^{32} -2.50913 q^{33} +14.4541 q^{34} +2.55073 q^{36} +7.97720 q^{37} -14.5829 q^{38} -4.33379 q^{39} +11.2249 q^{41} -4.62650 q^{42} -4.25487 q^{43} -6.40012 q^{44} +3.58050 q^{46} +4.98652 q^{47} -2.59524 q^{48} -2.29646 q^{49} -6.77562 q^{51} -11.0543 q^{52} -8.21338 q^{53} -2.13324 q^{54} -2.54795 q^{56} +6.83602 q^{57} -9.49138 q^{58} +3.67416 q^{59} +4.93960 q^{61} -14.0004 q^{62} +2.16876 q^{63} -11.6322 q^{64} +5.35259 q^{66} +11.5812 q^{67} -17.2828 q^{68} -1.67843 q^{69} -2.30251 q^{71} -1.17484 q^{72} -1.11599 q^{73} -17.0173 q^{74} +17.4368 q^{76} -5.44172 q^{77} +9.24504 q^{78} +7.78306 q^{79} +1.00000 q^{81} -23.9454 q^{82} +9.87708 q^{83} +5.53193 q^{84} +9.07667 q^{86} +4.44927 q^{87} +2.94783 q^{88} -1.24025 q^{89} -9.39897 q^{91} -4.28122 q^{92} +6.56295 q^{93} -10.6375 q^{94} +7.88596 q^{96} +15.0488 q^{97} +4.89892 q^{98} -2.50913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 6 q^{3} + 11 q^{4} - q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 6 q^{3} + 11 q^{4} - q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9} + 11 q^{12} - 4 q^{14} + 17 q^{16} - 2 q^{17} - q^{18} - 2 q^{19} + 2 q^{21} - 9 q^{22} + q^{23} - 6 q^{24} + 37 q^{26} + 6 q^{27} + 44 q^{28} + 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} + 11 q^{36} + 22 q^{37} - 27 q^{38} + 33 q^{41} - 4 q^{42} + 3 q^{43} - 11 q^{44} - 12 q^{46} + 6 q^{47} + 17 q^{48} + 4 q^{49} - 2 q^{51} + 33 q^{52} - 14 q^{53} - q^{54} - 30 q^{56} - 2 q^{57} + q^{58} - 8 q^{59} + 34 q^{61} - 31 q^{62} + 2 q^{63} + 12 q^{64} - 9 q^{66} - 2 q^{67} - 27 q^{68} + q^{69} - 3 q^{71} - 6 q^{72} + 36 q^{73} + 36 q^{74} + 27 q^{76} - 16 q^{77} + 37 q^{78} + 25 q^{79} + 6 q^{81} - 36 q^{82} + 12 q^{83} + 44 q^{84} - 30 q^{86} + 31 q^{87} - 56 q^{88} + 18 q^{89} + 28 q^{91} - 3 q^{92} - 2 q^{93} - 50 q^{94} - 33 q^{96} - 7 q^{97} - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.13324 −1.50843 −0.754216 0.656627i \(-0.771983\pi\)
−0.754216 + 0.656627i \(0.771983\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.55073 1.27536
\(5\) 0 0
\(6\) −2.13324 −0.870893
\(7\) 2.16876 0.819716 0.409858 0.912149i \(-0.365578\pi\)
0.409858 + 0.912149i \(0.365578\pi\)
\(8\) −1.17484 −0.415369
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.50913 −0.756532 −0.378266 0.925697i \(-0.623480\pi\)
−0.378266 + 0.925697i \(0.623480\pi\)
\(12\) 2.55073 0.736332
\(13\) −4.33379 −1.20198 −0.600989 0.799257i \(-0.705226\pi\)
−0.600989 + 0.799257i \(0.705226\pi\)
\(14\) −4.62650 −1.23648
\(15\) 0 0
\(16\) −2.59524 −0.648810
\(17\) −6.77562 −1.64333 −0.821665 0.569971i \(-0.806954\pi\)
−0.821665 + 0.569971i \(0.806954\pi\)
\(18\) −2.13324 −0.502810
\(19\) 6.83602 1.56829 0.784145 0.620578i \(-0.213102\pi\)
0.784145 + 0.620578i \(0.213102\pi\)
\(20\) 0 0
\(21\) 2.16876 0.473263
\(22\) 5.35259 1.14118
\(23\) −1.67843 −0.349977 −0.174989 0.984570i \(-0.555989\pi\)
−0.174989 + 0.984570i \(0.555989\pi\)
\(24\) −1.17484 −0.239813
\(25\) 0 0
\(26\) 9.24504 1.81310
\(27\) 1.00000 0.192450
\(28\) 5.53193 1.04544
\(29\) 4.44927 0.826209 0.413104 0.910684i \(-0.364444\pi\)
0.413104 + 0.910684i \(0.364444\pi\)
\(30\) 0 0
\(31\) 6.56295 1.17874 0.589371 0.807863i \(-0.299376\pi\)
0.589371 + 0.807863i \(0.299376\pi\)
\(32\) 7.88596 1.39405
\(33\) −2.50913 −0.436784
\(34\) 14.4541 2.47885
\(35\) 0 0
\(36\) 2.55073 0.425122
\(37\) 7.97720 1.31144 0.655722 0.755002i \(-0.272364\pi\)
0.655722 + 0.755002i \(0.272364\pi\)
\(38\) −14.5829 −2.36566
\(39\) −4.33379 −0.693962
\(40\) 0 0
\(41\) 11.2249 1.75303 0.876517 0.481371i \(-0.159861\pi\)
0.876517 + 0.481371i \(0.159861\pi\)
\(42\) −4.62650 −0.713885
\(43\) −4.25487 −0.648861 −0.324431 0.945909i \(-0.605173\pi\)
−0.324431 + 0.945909i \(0.605173\pi\)
\(44\) −6.40012 −0.964854
\(45\) 0 0
\(46\) 3.58050 0.527916
\(47\) 4.98652 0.727358 0.363679 0.931524i \(-0.381521\pi\)
0.363679 + 0.931524i \(0.381521\pi\)
\(48\) −2.59524 −0.374590
\(49\) −2.29646 −0.328066
\(50\) 0 0
\(51\) −6.77562 −0.948777
\(52\) −11.0543 −1.53296
\(53\) −8.21338 −1.12820 −0.564098 0.825708i \(-0.690776\pi\)
−0.564098 + 0.825708i \(0.690776\pi\)
\(54\) −2.13324 −0.290298
\(55\) 0 0
\(56\) −2.54795 −0.340484
\(57\) 6.83602 0.905453
\(58\) −9.49138 −1.24628
\(59\) 3.67416 0.478335 0.239168 0.970978i \(-0.423125\pi\)
0.239168 + 0.970978i \(0.423125\pi\)
\(60\) 0 0
\(61\) 4.93960 0.632451 0.316226 0.948684i \(-0.397584\pi\)
0.316226 + 0.948684i \(0.397584\pi\)
\(62\) −14.0004 −1.77805
\(63\) 2.16876 0.273239
\(64\) −11.6322 −1.45402
\(65\) 0 0
\(66\) 5.35259 0.658859
\(67\) 11.5812 1.41487 0.707436 0.706778i \(-0.249852\pi\)
0.707436 + 0.706778i \(0.249852\pi\)
\(68\) −17.2828 −2.09584
\(69\) −1.67843 −0.202059
\(70\) 0 0
\(71\) −2.30251 −0.273257 −0.136629 0.990622i \(-0.543627\pi\)
−0.136629 + 0.990622i \(0.543627\pi\)
\(72\) −1.17484 −0.138456
\(73\) −1.11599 −0.130617 −0.0653084 0.997865i \(-0.520803\pi\)
−0.0653084 + 0.997865i \(0.520803\pi\)
\(74\) −17.0173 −1.97822
\(75\) 0 0
\(76\) 17.4368 2.00014
\(77\) −5.44172 −0.620141
\(78\) 9.24504 1.04679
\(79\) 7.78306 0.875663 0.437831 0.899057i \(-0.355747\pi\)
0.437831 + 0.899057i \(0.355747\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −23.9454 −2.64433
\(83\) 9.87708 1.08415 0.542075 0.840330i \(-0.317639\pi\)
0.542075 + 0.840330i \(0.317639\pi\)
\(84\) 5.53193 0.603583
\(85\) 0 0
\(86\) 9.07667 0.978763
\(87\) 4.44927 0.477012
\(88\) 2.94783 0.314240
\(89\) −1.24025 −0.131466 −0.0657329 0.997837i \(-0.520939\pi\)
−0.0657329 + 0.997837i \(0.520939\pi\)
\(90\) 0 0
\(91\) −9.39897 −0.985280
\(92\) −4.28122 −0.446348
\(93\) 6.56295 0.680546
\(94\) −10.6375 −1.09717
\(95\) 0 0
\(96\) 7.88596 0.804857
\(97\) 15.0488 1.52797 0.763985 0.645234i \(-0.223240\pi\)
0.763985 + 0.645234i \(0.223240\pi\)
\(98\) 4.89892 0.494866
\(99\) −2.50913 −0.252177
\(100\) 0 0
\(101\) −17.0211 −1.69366 −0.846830 0.531864i \(-0.821492\pi\)
−0.846830 + 0.531864i \(0.821492\pi\)
\(102\) 14.4541 1.43116
\(103\) 10.0859 0.993793 0.496897 0.867810i \(-0.334473\pi\)
0.496897 + 0.867810i \(0.334473\pi\)
\(104\) 5.09151 0.499264
\(105\) 0 0
\(106\) 17.5212 1.70180
\(107\) 5.67218 0.548351 0.274175 0.961680i \(-0.411595\pi\)
0.274175 + 0.961680i \(0.411595\pi\)
\(108\) 2.55073 0.245444
\(109\) 8.34557 0.799361 0.399680 0.916655i \(-0.369121\pi\)
0.399680 + 0.916655i \(0.369121\pi\)
\(110\) 0 0
\(111\) 7.97720 0.757163
\(112\) −5.62846 −0.531839
\(113\) −14.8224 −1.39438 −0.697188 0.716888i \(-0.745566\pi\)
−0.697188 + 0.716888i \(0.745566\pi\)
\(114\) −14.5829 −1.36581
\(115\) 0 0
\(116\) 11.3489 1.05372
\(117\) −4.33379 −0.400659
\(118\) −7.83788 −0.721536
\(119\) −14.6947 −1.34706
\(120\) 0 0
\(121\) −4.70425 −0.427659
\(122\) −10.5374 −0.954009
\(123\) 11.2249 1.01211
\(124\) 16.7403 1.50332
\(125\) 0 0
\(126\) −4.62650 −0.412162
\(127\) −0.502113 −0.0445553 −0.0222777 0.999752i \(-0.507092\pi\)
−0.0222777 + 0.999752i \(0.507092\pi\)
\(128\) 9.04239 0.799242
\(129\) −4.25487 −0.374620
\(130\) 0 0
\(131\) 6.05788 0.529280 0.264640 0.964347i \(-0.414747\pi\)
0.264640 + 0.964347i \(0.414747\pi\)
\(132\) −6.40012 −0.557059
\(133\) 14.8257 1.28555
\(134\) −24.7056 −2.13424
\(135\) 0 0
\(136\) 7.96027 0.682588
\(137\) −1.94014 −0.165757 −0.0828786 0.996560i \(-0.526411\pi\)
−0.0828786 + 0.996560i \(0.526411\pi\)
\(138\) 3.58050 0.304793
\(139\) 19.2023 1.62872 0.814358 0.580363i \(-0.197089\pi\)
0.814358 + 0.580363i \(0.197089\pi\)
\(140\) 0 0
\(141\) 4.98652 0.419940
\(142\) 4.91181 0.412190
\(143\) 10.8741 0.909335
\(144\) −2.59524 −0.216270
\(145\) 0 0
\(146\) 2.38068 0.197026
\(147\) −2.29646 −0.189409
\(148\) 20.3477 1.67257
\(149\) 9.49138 0.777564 0.388782 0.921330i \(-0.372896\pi\)
0.388782 + 0.921330i \(0.372896\pi\)
\(150\) 0 0
\(151\) −12.5747 −1.02332 −0.511659 0.859189i \(-0.670969\pi\)
−0.511659 + 0.859189i \(0.670969\pi\)
\(152\) −8.03123 −0.651419
\(153\) −6.77562 −0.547776
\(154\) 11.6085 0.935440
\(155\) 0 0
\(156\) −11.0543 −0.885055
\(157\) −18.7991 −1.50033 −0.750165 0.661250i \(-0.770026\pi\)
−0.750165 + 0.661250i \(0.770026\pi\)
\(158\) −16.6032 −1.32088
\(159\) −8.21338 −0.651364
\(160\) 0 0
\(161\) −3.64012 −0.286882
\(162\) −2.13324 −0.167603
\(163\) −7.29949 −0.571740 −0.285870 0.958268i \(-0.592283\pi\)
−0.285870 + 0.958268i \(0.592283\pi\)
\(164\) 28.6317 2.23576
\(165\) 0 0
\(166\) −21.0702 −1.63537
\(167\) 14.7991 1.14519 0.572594 0.819839i \(-0.305937\pi\)
0.572594 + 0.819839i \(0.305937\pi\)
\(168\) −2.54795 −0.196579
\(169\) 5.78176 0.444750
\(170\) 0 0
\(171\) 6.83602 0.522763
\(172\) −10.8530 −0.827535
\(173\) 2.29460 0.174455 0.0872276 0.996188i \(-0.472199\pi\)
0.0872276 + 0.996188i \(0.472199\pi\)
\(174\) −9.49138 −0.719540
\(175\) 0 0
\(176\) 6.51180 0.490845
\(177\) 3.67416 0.276167
\(178\) 2.64575 0.198307
\(179\) 13.6637 1.02127 0.510636 0.859797i \(-0.329410\pi\)
0.510636 + 0.859797i \(0.329410\pi\)
\(180\) 0 0
\(181\) 8.71878 0.648062 0.324031 0.946046i \(-0.394962\pi\)
0.324031 + 0.946046i \(0.394962\pi\)
\(182\) 20.0503 1.48623
\(183\) 4.93960 0.365146
\(184\) 1.97189 0.145370
\(185\) 0 0
\(186\) −14.0004 −1.02656
\(187\) 17.0009 1.24323
\(188\) 12.7193 0.927647
\(189\) 2.16876 0.157754
\(190\) 0 0
\(191\) 11.6080 0.839925 0.419963 0.907541i \(-0.362043\pi\)
0.419963 + 0.907541i \(0.362043\pi\)
\(192\) −11.6322 −0.839481
\(193\) 14.8653 1.07002 0.535012 0.844844i \(-0.320307\pi\)
0.535012 + 0.844844i \(0.320307\pi\)
\(194\) −32.1027 −2.30484
\(195\) 0 0
\(196\) −5.85766 −0.418404
\(197\) −22.8292 −1.62651 −0.813256 0.581906i \(-0.802307\pi\)
−0.813256 + 0.581906i \(0.802307\pi\)
\(198\) 5.35259 0.380392
\(199\) 5.87697 0.416607 0.208304 0.978064i \(-0.433206\pi\)
0.208304 + 0.978064i \(0.433206\pi\)
\(200\) 0 0
\(201\) 11.5812 0.816876
\(202\) 36.3101 2.55477
\(203\) 9.64942 0.677256
\(204\) −17.2828 −1.21004
\(205\) 0 0
\(206\) −21.5157 −1.49907
\(207\) −1.67843 −0.116659
\(208\) 11.2472 0.779855
\(209\) −17.1525 −1.18646
\(210\) 0 0
\(211\) −24.6884 −1.69962 −0.849810 0.527090i \(-0.823283\pi\)
−0.849810 + 0.527090i \(0.823283\pi\)
\(212\) −20.9501 −1.43886
\(213\) −2.30251 −0.157765
\(214\) −12.1002 −0.827149
\(215\) 0 0
\(216\) −1.17484 −0.0799378
\(217\) 14.2335 0.966232
\(218\) −17.8031 −1.20578
\(219\) −1.11599 −0.0754116
\(220\) 0 0
\(221\) 29.3641 1.97525
\(222\) −17.0173 −1.14213
\(223\) 3.31060 0.221694 0.110847 0.993837i \(-0.464644\pi\)
0.110847 + 0.993837i \(0.464644\pi\)
\(224\) 17.1028 1.14273
\(225\) 0 0
\(226\) 31.6198 2.10332
\(227\) 2.70185 0.179328 0.0896641 0.995972i \(-0.471421\pi\)
0.0896641 + 0.995972i \(0.471421\pi\)
\(228\) 17.4368 1.15478
\(229\) −9.33473 −0.616856 −0.308428 0.951248i \(-0.599803\pi\)
−0.308428 + 0.951248i \(0.599803\pi\)
\(230\) 0 0
\(231\) −5.44172 −0.358039
\(232\) −5.22718 −0.343181
\(233\) −9.52491 −0.623998 −0.311999 0.950083i \(-0.600998\pi\)
−0.311999 + 0.950083i \(0.600998\pi\)
\(234\) 9.24504 0.604367
\(235\) 0 0
\(236\) 9.37179 0.610052
\(237\) 7.78306 0.505564
\(238\) 31.3474 2.03195
\(239\) 7.84648 0.507546 0.253773 0.967264i \(-0.418328\pi\)
0.253773 + 0.967264i \(0.418328\pi\)
\(240\) 0 0
\(241\) 1.42783 0.0919747 0.0459874 0.998942i \(-0.485357\pi\)
0.0459874 + 0.998942i \(0.485357\pi\)
\(242\) 10.0353 0.645095
\(243\) 1.00000 0.0641500
\(244\) 12.5996 0.806606
\(245\) 0 0
\(246\) −23.9454 −1.52670
\(247\) −29.6259 −1.88505
\(248\) −7.71042 −0.489612
\(249\) 9.87708 0.625935
\(250\) 0 0
\(251\) −17.5762 −1.10940 −0.554701 0.832050i \(-0.687167\pi\)
−0.554701 + 0.832050i \(0.687167\pi\)
\(252\) 5.53193 0.348479
\(253\) 4.21141 0.264769
\(254\) 1.07113 0.0672087
\(255\) 0 0
\(256\) 3.97476 0.248423
\(257\) 19.2890 1.20322 0.601608 0.798791i \(-0.294527\pi\)
0.601608 + 0.798791i \(0.294527\pi\)
\(258\) 9.07667 0.565089
\(259\) 17.3007 1.07501
\(260\) 0 0
\(261\) 4.44927 0.275403
\(262\) −12.9229 −0.798382
\(263\) 22.2432 1.37157 0.685786 0.727803i \(-0.259459\pi\)
0.685786 + 0.727803i \(0.259459\pi\)
\(264\) 2.94783 0.181426
\(265\) 0 0
\(266\) −31.6268 −1.93917
\(267\) −1.24025 −0.0759018
\(268\) 29.5406 1.80448
\(269\) 25.1785 1.53516 0.767581 0.640951i \(-0.221460\pi\)
0.767581 + 0.640951i \(0.221460\pi\)
\(270\) 0 0
\(271\) −5.70091 −0.346306 −0.173153 0.984895i \(-0.555395\pi\)
−0.173153 + 0.984895i \(0.555395\pi\)
\(272\) 17.5843 1.06621
\(273\) −9.39897 −0.568852
\(274\) 4.13879 0.250033
\(275\) 0 0
\(276\) −4.28122 −0.257699
\(277\) −19.1611 −1.15128 −0.575638 0.817704i \(-0.695246\pi\)
−0.575638 + 0.817704i \(0.695246\pi\)
\(278\) −40.9631 −2.45681
\(279\) 6.56295 0.392914
\(280\) 0 0
\(281\) 4.36015 0.260105 0.130052 0.991507i \(-0.458485\pi\)
0.130052 + 0.991507i \(0.458485\pi\)
\(282\) −10.6375 −0.633451
\(283\) 24.9081 1.48063 0.740317 0.672258i \(-0.234676\pi\)
0.740317 + 0.672258i \(0.234676\pi\)
\(284\) −5.87307 −0.348503
\(285\) 0 0
\(286\) −23.1970 −1.37167
\(287\) 24.3441 1.43699
\(288\) 7.88596 0.464684
\(289\) 28.9090 1.70053
\(290\) 0 0
\(291\) 15.0488 0.882174
\(292\) −2.84659 −0.166584
\(293\) 13.5651 0.792484 0.396242 0.918146i \(-0.370314\pi\)
0.396242 + 0.918146i \(0.370314\pi\)
\(294\) 4.89892 0.285711
\(295\) 0 0
\(296\) −9.37194 −0.544733
\(297\) −2.50913 −0.145595
\(298\) −20.2474 −1.17290
\(299\) 7.27397 0.420665
\(300\) 0 0
\(301\) −9.22780 −0.531882
\(302\) 26.8250 1.54360
\(303\) −17.0211 −0.977835
\(304\) −17.7411 −1.01752
\(305\) 0 0
\(306\) 14.4541 0.826283
\(307\) 11.9672 0.683007 0.341504 0.939880i \(-0.389064\pi\)
0.341504 + 0.939880i \(0.389064\pi\)
\(308\) −13.8803 −0.790906
\(309\) 10.0859 0.573767
\(310\) 0 0
\(311\) 7.66452 0.434615 0.217308 0.976103i \(-0.430273\pi\)
0.217308 + 0.976103i \(0.430273\pi\)
\(312\) 5.09151 0.288250
\(313\) −26.3840 −1.49131 −0.745655 0.666332i \(-0.767863\pi\)
−0.745655 + 0.666332i \(0.767863\pi\)
\(314\) 40.1030 2.26315
\(315\) 0 0
\(316\) 19.8525 1.11679
\(317\) 10.8880 0.611529 0.305765 0.952107i \(-0.401088\pi\)
0.305765 + 0.952107i \(0.401088\pi\)
\(318\) 17.5212 0.982537
\(319\) −11.1638 −0.625053
\(320\) 0 0
\(321\) 5.67218 0.316590
\(322\) 7.76526 0.432741
\(323\) −46.3183 −2.57722
\(324\) 2.55073 0.141707
\(325\) 0 0
\(326\) 15.5716 0.862431
\(327\) 8.34557 0.461511
\(328\) −13.1875 −0.728155
\(329\) 10.8146 0.596227
\(330\) 0 0
\(331\) −32.2137 −1.77062 −0.885312 0.464998i \(-0.846055\pi\)
−0.885312 + 0.464998i \(0.846055\pi\)
\(332\) 25.1938 1.38269
\(333\) 7.97720 0.437148
\(334\) −31.5701 −1.72744
\(335\) 0 0
\(336\) −5.62846 −0.307058
\(337\) 3.47169 0.189115 0.0945576 0.995519i \(-0.469856\pi\)
0.0945576 + 0.995519i \(0.469856\pi\)
\(338\) −12.3339 −0.670875
\(339\) −14.8224 −0.805043
\(340\) 0 0
\(341\) −16.4673 −0.891755
\(342\) −14.5829 −0.788553
\(343\) −20.1618 −1.08864
\(344\) 4.99879 0.269517
\(345\) 0 0
\(346\) −4.89494 −0.263154
\(347\) 10.9805 0.589465 0.294733 0.955580i \(-0.404769\pi\)
0.294733 + 0.955580i \(0.404769\pi\)
\(348\) 11.3489 0.608364
\(349\) −26.3158 −1.40865 −0.704325 0.709878i \(-0.748750\pi\)
−0.704325 + 0.709878i \(0.748750\pi\)
\(350\) 0 0
\(351\) −4.33379 −0.231321
\(352\) −19.7869 −1.05465
\(353\) 16.3668 0.871119 0.435559 0.900160i \(-0.356551\pi\)
0.435559 + 0.900160i \(0.356551\pi\)
\(354\) −7.83788 −0.416579
\(355\) 0 0
\(356\) −3.16353 −0.167667
\(357\) −14.6947 −0.777727
\(358\) −29.1480 −1.54052
\(359\) −17.4871 −0.922934 −0.461467 0.887157i \(-0.652677\pi\)
−0.461467 + 0.887157i \(0.652677\pi\)
\(360\) 0 0
\(361\) 27.7311 1.45953
\(362\) −18.5993 −0.977557
\(363\) −4.70425 −0.246909
\(364\) −23.9742 −1.25659
\(365\) 0 0
\(366\) −10.5374 −0.550798
\(367\) −8.29594 −0.433044 −0.216522 0.976278i \(-0.569471\pi\)
−0.216522 + 0.976278i \(0.569471\pi\)
\(368\) 4.35593 0.227068
\(369\) 11.2249 0.584345
\(370\) 0 0
\(371\) −17.8129 −0.924799
\(372\) 16.7403 0.867945
\(373\) 19.1295 0.990486 0.495243 0.868755i \(-0.335079\pi\)
0.495243 + 0.868755i \(0.335079\pi\)
\(374\) −36.2671 −1.87533
\(375\) 0 0
\(376\) −5.85836 −0.302122
\(377\) −19.2822 −0.993085
\(378\) −4.62650 −0.237962
\(379\) −27.0952 −1.39179 −0.695894 0.718145i \(-0.744992\pi\)
−0.695894 + 0.718145i \(0.744992\pi\)
\(380\) 0 0
\(381\) −0.502113 −0.0257240
\(382\) −24.7627 −1.26697
\(383\) −19.8026 −1.01187 −0.505933 0.862573i \(-0.668852\pi\)
−0.505933 + 0.862573i \(0.668852\pi\)
\(384\) 9.04239 0.461443
\(385\) 0 0
\(386\) −31.7112 −1.61406
\(387\) −4.25487 −0.216287
\(388\) 38.3853 1.94872
\(389\) −23.0130 −1.16680 −0.583402 0.812183i \(-0.698279\pi\)
−0.583402 + 0.812183i \(0.698279\pi\)
\(390\) 0 0
\(391\) 11.3724 0.575128
\(392\) 2.69798 0.136269
\(393\) 6.05788 0.305580
\(394\) 48.7002 2.45348
\(395\) 0 0
\(396\) −6.40012 −0.321618
\(397\) 20.5745 1.03260 0.516301 0.856407i \(-0.327309\pi\)
0.516301 + 0.856407i \(0.327309\pi\)
\(398\) −12.5370 −0.628423
\(399\) 14.8257 0.742214
\(400\) 0 0
\(401\) −13.1542 −0.656889 −0.328444 0.944523i \(-0.606524\pi\)
−0.328444 + 0.944523i \(0.606524\pi\)
\(402\) −24.7056 −1.23220
\(403\) −28.4425 −1.41682
\(404\) −43.4161 −2.16003
\(405\) 0 0
\(406\) −20.5846 −1.02159
\(407\) −20.0159 −0.992150
\(408\) 7.96027 0.394092
\(409\) 2.76531 0.136736 0.0683679 0.997660i \(-0.478221\pi\)
0.0683679 + 0.997660i \(0.478221\pi\)
\(410\) 0 0
\(411\) −1.94014 −0.0956999
\(412\) 25.7264 1.26745
\(413\) 7.96839 0.392099
\(414\) 3.58050 0.175972
\(415\) 0 0
\(416\) −34.1761 −1.67562
\(417\) 19.2023 0.940340
\(418\) 36.5904 1.78970
\(419\) −9.70852 −0.474292 −0.237146 0.971474i \(-0.576212\pi\)
−0.237146 + 0.971474i \(0.576212\pi\)
\(420\) 0 0
\(421\) −4.21745 −0.205546 −0.102773 0.994705i \(-0.532772\pi\)
−0.102773 + 0.994705i \(0.532772\pi\)
\(422\) 52.6664 2.56376
\(423\) 4.98652 0.242453
\(424\) 9.64942 0.468617
\(425\) 0 0
\(426\) 4.91181 0.237978
\(427\) 10.7128 0.518430
\(428\) 14.4682 0.699347
\(429\) 10.8741 0.525005
\(430\) 0 0
\(431\) 18.9785 0.914164 0.457082 0.889425i \(-0.348895\pi\)
0.457082 + 0.889425i \(0.348895\pi\)
\(432\) −2.59524 −0.124863
\(433\) −13.7054 −0.658640 −0.329320 0.944218i \(-0.606819\pi\)
−0.329320 + 0.944218i \(0.606819\pi\)
\(434\) −30.3635 −1.45750
\(435\) 0 0
\(436\) 21.2873 1.01948
\(437\) −11.4738 −0.548866
\(438\) 2.38068 0.113753
\(439\) −33.3843 −1.59335 −0.796673 0.604410i \(-0.793409\pi\)
−0.796673 + 0.604410i \(0.793409\pi\)
\(440\) 0 0
\(441\) −2.29646 −0.109355
\(442\) −62.6409 −2.97952
\(443\) −11.6290 −0.552512 −0.276256 0.961084i \(-0.589094\pi\)
−0.276256 + 0.961084i \(0.589094\pi\)
\(444\) 20.3477 0.965659
\(445\) 0 0
\(446\) −7.06231 −0.334410
\(447\) 9.49138 0.448927
\(448\) −25.2275 −1.19189
\(449\) −13.3509 −0.630069 −0.315034 0.949080i \(-0.602016\pi\)
−0.315034 + 0.949080i \(0.602016\pi\)
\(450\) 0 0
\(451\) −28.1647 −1.32623
\(452\) −37.8080 −1.77834
\(453\) −12.5747 −0.590813
\(454\) −5.76371 −0.270504
\(455\) 0 0
\(456\) −8.03123 −0.376097
\(457\) 23.3042 1.09013 0.545063 0.838395i \(-0.316506\pi\)
0.545063 + 0.838395i \(0.316506\pi\)
\(458\) 19.9132 0.930485
\(459\) −6.77562 −0.316259
\(460\) 0 0
\(461\) 5.03269 0.234396 0.117198 0.993109i \(-0.462609\pi\)
0.117198 + 0.993109i \(0.462609\pi\)
\(462\) 11.6085 0.540077
\(463\) 12.7759 0.593746 0.296873 0.954917i \(-0.404056\pi\)
0.296873 + 0.954917i \(0.404056\pi\)
\(464\) −11.5469 −0.536052
\(465\) 0 0
\(466\) 20.3190 0.941257
\(467\) −4.86284 −0.225025 −0.112513 0.993650i \(-0.535890\pi\)
−0.112513 + 0.993650i \(0.535890\pi\)
\(468\) −11.0543 −0.510987
\(469\) 25.1169 1.15979
\(470\) 0 0
\(471\) −18.7991 −0.866216
\(472\) −4.31655 −0.198685
\(473\) 10.6760 0.490884
\(474\) −16.6032 −0.762609
\(475\) 0 0
\(476\) −37.4823 −1.71800
\(477\) −8.21338 −0.376065
\(478\) −16.7385 −0.765599
\(479\) 4.79202 0.218953 0.109476 0.993989i \(-0.465083\pi\)
0.109476 + 0.993989i \(0.465083\pi\)
\(480\) 0 0
\(481\) −34.5715 −1.57633
\(482\) −3.04591 −0.138738
\(483\) −3.64012 −0.165631
\(484\) −11.9993 −0.545421
\(485\) 0 0
\(486\) −2.13324 −0.0967659
\(487\) −9.16949 −0.415509 −0.207755 0.978181i \(-0.566616\pi\)
−0.207755 + 0.978181i \(0.566616\pi\)
\(488\) −5.80325 −0.262701
\(489\) −7.29949 −0.330094
\(490\) 0 0
\(491\) 31.9054 1.43987 0.719935 0.694042i \(-0.244172\pi\)
0.719935 + 0.694042i \(0.244172\pi\)
\(492\) 28.6317 1.29081
\(493\) −30.1466 −1.35773
\(494\) 63.1992 2.84347
\(495\) 0 0
\(496\) −17.0324 −0.764778
\(497\) −4.99359 −0.223993
\(498\) −21.0702 −0.944179
\(499\) 9.07297 0.406162 0.203081 0.979162i \(-0.434905\pi\)
0.203081 + 0.979162i \(0.434905\pi\)
\(500\) 0 0
\(501\) 14.7991 0.661175
\(502\) 37.4944 1.67346
\(503\) −38.7485 −1.72771 −0.863856 0.503739i \(-0.831957\pi\)
−0.863856 + 0.503739i \(0.831957\pi\)
\(504\) −2.54795 −0.113495
\(505\) 0 0
\(506\) −8.98396 −0.399386
\(507\) 5.78176 0.256777
\(508\) −1.28075 −0.0568243
\(509\) 32.9367 1.45989 0.729947 0.683503i \(-0.239545\pi\)
0.729947 + 0.683503i \(0.239545\pi\)
\(510\) 0 0
\(511\) −2.42032 −0.107069
\(512\) −26.5639 −1.17397
\(513\) 6.83602 0.301818
\(514\) −41.1482 −1.81497
\(515\) 0 0
\(516\) −10.8530 −0.477777
\(517\) −12.5118 −0.550270
\(518\) −36.9065 −1.62158
\(519\) 2.29460 0.100722
\(520\) 0 0
\(521\) −17.7521 −0.777735 −0.388867 0.921294i \(-0.627134\pi\)
−0.388867 + 0.921294i \(0.627134\pi\)
\(522\) −9.49138 −0.415426
\(523\) 4.98678 0.218056 0.109028 0.994039i \(-0.465226\pi\)
0.109028 + 0.994039i \(0.465226\pi\)
\(524\) 15.4520 0.675025
\(525\) 0 0
\(526\) −47.4501 −2.06892
\(527\) −44.4681 −1.93706
\(528\) 6.51180 0.283390
\(529\) −20.1829 −0.877516
\(530\) 0 0
\(531\) 3.67416 0.159445
\(532\) 37.8164 1.63955
\(533\) −48.6463 −2.10711
\(534\) 2.64575 0.114493
\(535\) 0 0
\(536\) −13.6061 −0.587693
\(537\) 13.6637 0.589632
\(538\) −53.7120 −2.31569
\(539\) 5.76214 0.248193
\(540\) 0 0
\(541\) −3.49960 −0.150460 −0.0752298 0.997166i \(-0.523969\pi\)
−0.0752298 + 0.997166i \(0.523969\pi\)
\(542\) 12.1614 0.522378
\(543\) 8.71878 0.374159
\(544\) −53.4322 −2.29089
\(545\) 0 0
\(546\) 20.0503 0.858073
\(547\) 17.4640 0.746709 0.373354 0.927689i \(-0.378208\pi\)
0.373354 + 0.927689i \(0.378208\pi\)
\(548\) −4.94877 −0.211401
\(549\) 4.93960 0.210817
\(550\) 0 0
\(551\) 30.4153 1.29574
\(552\) 1.97189 0.0839291
\(553\) 16.8796 0.717795
\(554\) 40.8752 1.73662
\(555\) 0 0
\(556\) 48.9798 2.07721
\(557\) −6.02774 −0.255403 −0.127702 0.991813i \(-0.540760\pi\)
−0.127702 + 0.991813i \(0.540760\pi\)
\(558\) −14.0004 −0.592683
\(559\) 18.4397 0.779917
\(560\) 0 0
\(561\) 17.0009 0.717780
\(562\) −9.30126 −0.392350
\(563\) −9.19823 −0.387659 −0.193830 0.981035i \(-0.562091\pi\)
−0.193830 + 0.981035i \(0.562091\pi\)
\(564\) 12.7193 0.535577
\(565\) 0 0
\(566\) −53.1351 −2.23343
\(567\) 2.16876 0.0910795
\(568\) 2.70508 0.113503
\(569\) 9.06517 0.380032 0.190016 0.981781i \(-0.439146\pi\)
0.190016 + 0.981781i \(0.439146\pi\)
\(570\) 0 0
\(571\) −6.34688 −0.265609 −0.132804 0.991142i \(-0.542398\pi\)
−0.132804 + 0.991142i \(0.542398\pi\)
\(572\) 27.7368 1.15973
\(573\) 11.6080 0.484931
\(574\) −51.9320 −2.16760
\(575\) 0 0
\(576\) −11.6322 −0.484675
\(577\) −2.41418 −0.100504 −0.0502519 0.998737i \(-0.516002\pi\)
−0.0502519 + 0.998737i \(0.516002\pi\)
\(578\) −61.6700 −2.56513
\(579\) 14.8653 0.617779
\(580\) 0 0
\(581\) 21.4211 0.888695
\(582\) −32.1027 −1.33070
\(583\) 20.6085 0.853516
\(584\) 1.31111 0.0542541
\(585\) 0 0
\(586\) −28.9378 −1.19541
\(587\) −4.50976 −0.186138 −0.0930689 0.995660i \(-0.529668\pi\)
−0.0930689 + 0.995660i \(0.529668\pi\)
\(588\) −5.85766 −0.241566
\(589\) 44.8645 1.84861
\(590\) 0 0
\(591\) −22.8292 −0.939067
\(592\) −20.7027 −0.850877
\(593\) 33.2824 1.36674 0.683371 0.730071i \(-0.260513\pi\)
0.683371 + 0.730071i \(0.260513\pi\)
\(594\) 5.35259 0.219620
\(595\) 0 0
\(596\) 24.2099 0.991678
\(597\) 5.87697 0.240528
\(598\) −15.5172 −0.634544
\(599\) 39.1061 1.59783 0.798915 0.601444i \(-0.205408\pi\)
0.798915 + 0.601444i \(0.205408\pi\)
\(600\) 0 0
\(601\) −28.8076 −1.17509 −0.587543 0.809193i \(-0.699905\pi\)
−0.587543 + 0.809193i \(0.699905\pi\)
\(602\) 19.6852 0.802307
\(603\) 11.5812 0.471624
\(604\) −32.0747 −1.30510
\(605\) 0 0
\(606\) 36.3101 1.47500
\(607\) 23.0933 0.937327 0.468663 0.883377i \(-0.344736\pi\)
0.468663 + 0.883377i \(0.344736\pi\)
\(608\) 53.9085 2.18628
\(609\) 9.64942 0.391014
\(610\) 0 0
\(611\) −21.6105 −0.874268
\(612\) −17.2828 −0.698615
\(613\) 18.2228 0.736013 0.368007 0.929823i \(-0.380040\pi\)
0.368007 + 0.929823i \(0.380040\pi\)
\(614\) −25.5291 −1.03027
\(615\) 0 0
\(616\) 6.39315 0.257587
\(617\) −15.4484 −0.621930 −0.310965 0.950421i \(-0.600652\pi\)
−0.310965 + 0.950421i \(0.600652\pi\)
\(618\) −21.5157 −0.865488
\(619\) 16.5042 0.663359 0.331680 0.943392i \(-0.392385\pi\)
0.331680 + 0.943392i \(0.392385\pi\)
\(620\) 0 0
\(621\) −1.67843 −0.0673531
\(622\) −16.3503 −0.655587
\(623\) −2.68980 −0.107765
\(624\) 11.2472 0.450249
\(625\) 0 0
\(626\) 56.2835 2.24954
\(627\) −17.1525 −0.685004
\(628\) −47.9514 −1.91347
\(629\) −54.0505 −2.15513
\(630\) 0 0
\(631\) −38.1237 −1.51768 −0.758841 0.651276i \(-0.774234\pi\)
−0.758841 + 0.651276i \(0.774234\pi\)
\(632\) −9.14386 −0.363723
\(633\) −24.6884 −0.981276
\(634\) −23.2267 −0.922450
\(635\) 0 0
\(636\) −20.9501 −0.830726
\(637\) 9.95240 0.394329
\(638\) 23.8151 0.942850
\(639\) −2.30251 −0.0910858
\(640\) 0 0
\(641\) 38.9299 1.53764 0.768819 0.639466i \(-0.220845\pi\)
0.768819 + 0.639466i \(0.220845\pi\)
\(642\) −12.1002 −0.477555
\(643\) −40.5346 −1.59853 −0.799265 0.600979i \(-0.794778\pi\)
−0.799265 + 0.600979i \(0.794778\pi\)
\(644\) −9.28496 −0.365879
\(645\) 0 0
\(646\) 98.8082 3.88755
\(647\) −31.8752 −1.25315 −0.626573 0.779363i \(-0.715543\pi\)
−0.626573 + 0.779363i \(0.715543\pi\)
\(648\) −1.17484 −0.0461521
\(649\) −9.21896 −0.361876
\(650\) 0 0
\(651\) 14.2335 0.557855
\(652\) −18.6190 −0.729177
\(653\) 19.5285 0.764210 0.382105 0.924119i \(-0.375199\pi\)
0.382105 + 0.924119i \(0.375199\pi\)
\(654\) −17.8031 −0.696158
\(655\) 0 0
\(656\) −29.1313 −1.13738
\(657\) −1.11599 −0.0435389
\(658\) −23.0701 −0.899367
\(659\) 47.6562 1.85642 0.928211 0.372055i \(-0.121347\pi\)
0.928211 + 0.372055i \(0.121347\pi\)
\(660\) 0 0
\(661\) 3.91738 0.152368 0.0761842 0.997094i \(-0.475726\pi\)
0.0761842 + 0.997094i \(0.475726\pi\)
\(662\) 68.7196 2.67086
\(663\) 29.3641 1.14041
\(664\) −11.6040 −0.450322
\(665\) 0 0
\(666\) −17.0173 −0.659408
\(667\) −7.46779 −0.289154
\(668\) 37.7485 1.46053
\(669\) 3.31060 0.127995
\(670\) 0 0
\(671\) −12.3941 −0.478470
\(672\) 17.1028 0.659754
\(673\) −28.1866 −1.08651 −0.543257 0.839566i \(-0.682809\pi\)
−0.543257 + 0.839566i \(0.682809\pi\)
\(674\) −7.40597 −0.285267
\(675\) 0 0
\(676\) 14.7477 0.567219
\(677\) 9.50611 0.365350 0.182675 0.983173i \(-0.441524\pi\)
0.182675 + 0.983173i \(0.441524\pi\)
\(678\) 31.6198 1.21435
\(679\) 32.6372 1.25250
\(680\) 0 0
\(681\) 2.70185 0.103535
\(682\) 35.1288 1.34515
\(683\) −7.49565 −0.286813 −0.143407 0.989664i \(-0.545806\pi\)
−0.143407 + 0.989664i \(0.545806\pi\)
\(684\) 17.4368 0.666714
\(685\) 0 0
\(686\) 43.0101 1.64213
\(687\) −9.33473 −0.356142
\(688\) 11.0424 0.420987
\(689\) 35.5951 1.35607
\(690\) 0 0
\(691\) −0.745188 −0.0283483 −0.0141741 0.999900i \(-0.504512\pi\)
−0.0141741 + 0.999900i \(0.504512\pi\)
\(692\) 5.85290 0.222494
\(693\) −5.44172 −0.206714
\(694\) −23.4241 −0.889167
\(695\) 0 0
\(696\) −5.22718 −0.198136
\(697\) −76.0556 −2.88081
\(698\) 56.1379 2.12485
\(699\) −9.52491 −0.360265
\(700\) 0 0
\(701\) 31.6488 1.19536 0.597679 0.801735i \(-0.296090\pi\)
0.597679 + 0.801735i \(0.296090\pi\)
\(702\) 9.24504 0.348931
\(703\) 54.5323 2.05672
\(704\) 29.1867 1.10002
\(705\) 0 0
\(706\) −34.9145 −1.31402
\(707\) −36.9147 −1.38832
\(708\) 9.37179 0.352214
\(709\) −16.0181 −0.601571 −0.300785 0.953692i \(-0.597249\pi\)
−0.300785 + 0.953692i \(0.597249\pi\)
\(710\) 0 0
\(711\) 7.78306 0.291888
\(712\) 1.45709 0.0546068
\(713\) −11.0155 −0.412532
\(714\) 31.3474 1.17315
\(715\) 0 0
\(716\) 34.8524 1.30250
\(717\) 7.84648 0.293032
\(718\) 37.3043 1.39218
\(719\) 14.3316 0.534478 0.267239 0.963630i \(-0.413889\pi\)
0.267239 + 0.963630i \(0.413889\pi\)
\(720\) 0 0
\(721\) 21.8739 0.814628
\(722\) −59.1573 −2.20161
\(723\) 1.42783 0.0531016
\(724\) 22.2393 0.826515
\(725\) 0 0
\(726\) 10.0353 0.372445
\(727\) 18.3551 0.680755 0.340377 0.940289i \(-0.389445\pi\)
0.340377 + 0.940289i \(0.389445\pi\)
\(728\) 11.0423 0.409254
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.8294 1.06629
\(732\) 12.5996 0.465694
\(733\) 0.665145 0.0245677 0.0122838 0.999925i \(-0.496090\pi\)
0.0122838 + 0.999925i \(0.496090\pi\)
\(734\) 17.6973 0.653218
\(735\) 0 0
\(736\) −13.2360 −0.487887
\(737\) −29.0588 −1.07040
\(738\) −23.9454 −0.881443
\(739\) −32.8332 −1.20779 −0.603895 0.797064i \(-0.706385\pi\)
−0.603895 + 0.797064i \(0.706385\pi\)
\(740\) 0 0
\(741\) −29.6259 −1.08833
\(742\) 37.9992 1.39500
\(743\) −29.6004 −1.08593 −0.542967 0.839754i \(-0.682699\pi\)
−0.542967 + 0.839754i \(0.682699\pi\)
\(744\) −7.71042 −0.282678
\(745\) 0 0
\(746\) −40.8078 −1.49408
\(747\) 9.87708 0.361383
\(748\) 43.3648 1.58557
\(749\) 12.3016 0.449492
\(750\) 0 0
\(751\) 24.9709 0.911199 0.455600 0.890185i \(-0.349425\pi\)
0.455600 + 0.890185i \(0.349425\pi\)
\(752\) −12.9412 −0.471917
\(753\) −17.5762 −0.640514
\(754\) 41.1337 1.49800
\(755\) 0 0
\(756\) 5.53193 0.201194
\(757\) −11.2173 −0.407700 −0.203850 0.979002i \(-0.565345\pi\)
−0.203850 + 0.979002i \(0.565345\pi\)
\(758\) 57.8007 2.09942
\(759\) 4.21141 0.152864
\(760\) 0 0
\(761\) 19.3105 0.700004 0.350002 0.936749i \(-0.386181\pi\)
0.350002 + 0.936749i \(0.386181\pi\)
\(762\) 1.07113 0.0388029
\(763\) 18.0996 0.655249
\(764\) 29.6089 1.07121
\(765\) 0 0
\(766\) 42.2438 1.52633
\(767\) −15.9231 −0.574948
\(768\) 3.97476 0.143427
\(769\) −19.3372 −0.697318 −0.348659 0.937250i \(-0.613363\pi\)
−0.348659 + 0.937250i \(0.613363\pi\)
\(770\) 0 0
\(771\) 19.2890 0.694677
\(772\) 37.9172 1.36467
\(773\) 7.79217 0.280265 0.140132 0.990133i \(-0.455247\pi\)
0.140132 + 0.990133i \(0.455247\pi\)
\(774\) 9.07667 0.326254
\(775\) 0 0
\(776\) −17.6799 −0.634671
\(777\) 17.3007 0.620658
\(778\) 49.0923 1.76004
\(779\) 76.7336 2.74926
\(780\) 0 0
\(781\) 5.77730 0.206728
\(782\) −24.2601 −0.867540
\(783\) 4.44927 0.159004
\(784\) 5.95987 0.212853
\(785\) 0 0
\(786\) −12.9229 −0.460946
\(787\) −9.07666 −0.323548 −0.161774 0.986828i \(-0.551722\pi\)
−0.161774 + 0.986828i \(0.551722\pi\)
\(788\) −58.2311 −2.07440
\(789\) 22.2432 0.791877
\(790\) 0 0
\(791\) −32.1463 −1.14299
\(792\) 2.94783 0.104747
\(793\) −21.4072 −0.760192
\(794\) −43.8903 −1.55761
\(795\) 0 0
\(796\) 14.9906 0.531326
\(797\) −22.9954 −0.814541 −0.407270 0.913308i \(-0.633519\pi\)
−0.407270 + 0.913308i \(0.633519\pi\)
\(798\) −31.6268 −1.11958
\(799\) −33.7867 −1.19529
\(800\) 0 0
\(801\) −1.24025 −0.0438219
\(802\) 28.0611 0.990872
\(803\) 2.80017 0.0988158
\(804\) 29.5406 1.04182
\(805\) 0 0
\(806\) 60.6747 2.13718
\(807\) 25.1785 0.886327
\(808\) 19.9970 0.703493
\(809\) −50.6773 −1.78172 −0.890860 0.454278i \(-0.849897\pi\)
−0.890860 + 0.454278i \(0.849897\pi\)
\(810\) 0 0
\(811\) 26.7614 0.939720 0.469860 0.882741i \(-0.344304\pi\)
0.469860 + 0.882741i \(0.344304\pi\)
\(812\) 24.6130 0.863749
\(813\) −5.70091 −0.199940
\(814\) 42.6987 1.49659
\(815\) 0 0
\(816\) 17.5843 0.615575
\(817\) −29.0864 −1.01760
\(818\) −5.89908 −0.206256
\(819\) −9.39897 −0.328427
\(820\) 0 0
\(821\) −0.832947 −0.0290700 −0.0145350 0.999894i \(-0.504627\pi\)
−0.0145350 + 0.999894i \(0.504627\pi\)
\(822\) 4.13879 0.144357
\(823\) −6.67034 −0.232514 −0.116257 0.993219i \(-0.537090\pi\)
−0.116257 + 0.993219i \(0.537090\pi\)
\(824\) −11.8493 −0.412791
\(825\) 0 0
\(826\) −16.9985 −0.591454
\(827\) 1.34470 0.0467599 0.0233799 0.999727i \(-0.492557\pi\)
0.0233799 + 0.999727i \(0.492557\pi\)
\(828\) −4.28122 −0.148783
\(829\) −24.7770 −0.860541 −0.430270 0.902700i \(-0.641582\pi\)
−0.430270 + 0.902700i \(0.641582\pi\)
\(830\) 0 0
\(831\) −19.1611 −0.664690
\(832\) 50.4115 1.74770
\(833\) 15.5600 0.539121
\(834\) −40.9631 −1.41844
\(835\) 0 0
\(836\) −43.7513 −1.51317
\(837\) 6.56295 0.226849
\(838\) 20.7106 0.715437
\(839\) −6.72534 −0.232184 −0.116092 0.993238i \(-0.537037\pi\)
−0.116092 + 0.993238i \(0.537037\pi\)
\(840\) 0 0
\(841\) −9.20399 −0.317379
\(842\) 8.99685 0.310052
\(843\) 4.36015 0.150171
\(844\) −62.9734 −2.16763
\(845\) 0 0
\(846\) −10.6375 −0.365723
\(847\) −10.2024 −0.350559
\(848\) 21.3157 0.731984
\(849\) 24.9081 0.854844
\(850\) 0 0
\(851\) −13.3892 −0.458975
\(852\) −5.87307 −0.201208
\(853\) 16.7244 0.572632 0.286316 0.958135i \(-0.407569\pi\)
0.286316 + 0.958135i \(0.407569\pi\)
\(854\) −22.8531 −0.782016
\(855\) 0 0
\(856\) −6.66391 −0.227768
\(857\) −12.9930 −0.443831 −0.221915 0.975066i \(-0.571231\pi\)
−0.221915 + 0.975066i \(0.571231\pi\)
\(858\) −23.1970 −0.791933
\(859\) −36.5682 −1.24769 −0.623845 0.781548i \(-0.714430\pi\)
−0.623845 + 0.781548i \(0.714430\pi\)
\(860\) 0 0
\(861\) 24.3441 0.829646
\(862\) −40.4859 −1.37895
\(863\) 46.3384 1.57738 0.788688 0.614794i \(-0.210761\pi\)
0.788688 + 0.614794i \(0.210761\pi\)
\(864\) 7.88596 0.268286
\(865\) 0 0
\(866\) 29.2370 0.993513
\(867\) 28.9090 0.981802
\(868\) 36.3058 1.23230
\(869\) −19.5287 −0.662467
\(870\) 0 0
\(871\) −50.1906 −1.70064
\(872\) −9.80472 −0.332030
\(873\) 15.0488 0.509323
\(874\) 24.4764 0.827926
\(875\) 0 0
\(876\) −2.84659 −0.0961773
\(877\) 44.2548 1.49438 0.747190 0.664611i \(-0.231403\pi\)
0.747190 + 0.664611i \(0.231403\pi\)
\(878\) 71.2169 2.40345
\(879\) 13.5651 0.457541
\(880\) 0 0
\(881\) −42.2495 −1.42342 −0.711712 0.702472i \(-0.752080\pi\)
−0.711712 + 0.702472i \(0.752080\pi\)
\(882\) 4.89892 0.164955
\(883\) −3.49289 −0.117545 −0.0587726 0.998271i \(-0.518719\pi\)
−0.0587726 + 0.998271i \(0.518719\pi\)
\(884\) 74.9000 2.51916
\(885\) 0 0
\(886\) 24.8076 0.833427
\(887\) −14.7275 −0.494501 −0.247250 0.968952i \(-0.579527\pi\)
−0.247250 + 0.968952i \(0.579527\pi\)
\(888\) −9.37194 −0.314502
\(889\) −1.08896 −0.0365227
\(890\) 0 0
\(891\) −2.50913 −0.0840591
\(892\) 8.44444 0.282741
\(893\) 34.0879 1.14071
\(894\) −20.2474 −0.677175
\(895\) 0 0
\(896\) 19.6108 0.655151
\(897\) 7.27397 0.242871
\(898\) 28.4808 0.950416
\(899\) 29.2004 0.973886
\(900\) 0 0
\(901\) 55.6508 1.85400
\(902\) 60.0823 2.00052
\(903\) −9.22780 −0.307082
\(904\) 17.4140 0.579180
\(905\) 0 0
\(906\) 26.8250 0.891200
\(907\) −7.33785 −0.243649 −0.121825 0.992552i \(-0.538875\pi\)
−0.121825 + 0.992552i \(0.538875\pi\)
\(908\) 6.89169 0.228709
\(909\) −17.0211 −0.564553
\(910\) 0 0
\(911\) −53.5731 −1.77496 −0.887479 0.460849i \(-0.847545\pi\)
−0.887479 + 0.460849i \(0.847545\pi\)
\(912\) −17.7411 −0.587466
\(913\) −24.7829 −0.820195
\(914\) −49.7136 −1.64438
\(915\) 0 0
\(916\) −23.8104 −0.786717
\(917\) 13.1381 0.433859
\(918\) 14.4541 0.477055
\(919\) −43.2016 −1.42509 −0.712545 0.701627i \(-0.752458\pi\)
−0.712545 + 0.701627i \(0.752458\pi\)
\(920\) 0 0
\(921\) 11.9672 0.394334
\(922\) −10.7359 −0.353570
\(923\) 9.97859 0.328449
\(924\) −13.8803 −0.456630
\(925\) 0 0
\(926\) −27.2541 −0.895625
\(927\) 10.0859 0.331264
\(928\) 35.0868 1.15178
\(929\) −14.8420 −0.486951 −0.243476 0.969907i \(-0.578288\pi\)
−0.243476 + 0.969907i \(0.578288\pi\)
\(930\) 0 0
\(931\) −15.6987 −0.514503
\(932\) −24.2955 −0.795824
\(933\) 7.66452 0.250925
\(934\) 10.3736 0.339435
\(935\) 0 0
\(936\) 5.09151 0.166421
\(937\) 3.75611 0.122707 0.0613534 0.998116i \(-0.480458\pi\)
0.0613534 + 0.998116i \(0.480458\pi\)
\(938\) −53.5805 −1.74947
\(939\) −26.3840 −0.861008
\(940\) 0 0
\(941\) 59.9208 1.95336 0.976682 0.214692i \(-0.0688747\pi\)
0.976682 + 0.214692i \(0.0688747\pi\)
\(942\) 40.1030 1.30663
\(943\) −18.8402 −0.613521
\(944\) −9.53532 −0.310348
\(945\) 0 0
\(946\) −22.7746 −0.740465
\(947\) 24.4212 0.793581 0.396791 0.917909i \(-0.370124\pi\)
0.396791 + 0.917909i \(0.370124\pi\)
\(948\) 19.8525 0.644779
\(949\) 4.83647 0.156998
\(950\) 0 0
\(951\) 10.8880 0.353067
\(952\) 17.2639 0.559528
\(953\) 36.6466 1.18710 0.593550 0.804797i \(-0.297726\pi\)
0.593550 + 0.804797i \(0.297726\pi\)
\(954\) 17.5212 0.567268
\(955\) 0 0
\(956\) 20.0142 0.647307
\(957\) −11.1638 −0.360875
\(958\) −10.2225 −0.330275
\(959\) −4.20770 −0.135874
\(960\) 0 0
\(961\) 12.0723 0.389431
\(962\) 73.7495 2.37778
\(963\) 5.67218 0.182784
\(964\) 3.64201 0.117301
\(965\) 0 0
\(966\) 7.76526 0.249843
\(967\) 47.4395 1.52555 0.762776 0.646663i \(-0.223836\pi\)
0.762776 + 0.646663i \(0.223836\pi\)
\(968\) 5.52674 0.177636
\(969\) −46.3183 −1.48796
\(970\) 0 0
\(971\) −5.68782 −0.182531 −0.0912655 0.995827i \(-0.529091\pi\)
−0.0912655 + 0.995827i \(0.529091\pi\)
\(972\) 2.55073 0.0818147
\(973\) 41.6452 1.33508
\(974\) 19.5608 0.626767
\(975\) 0 0
\(976\) −12.8194 −0.410340
\(977\) −2.29901 −0.0735518 −0.0367759 0.999324i \(-0.511709\pi\)
−0.0367759 + 0.999324i \(0.511709\pi\)
\(978\) 15.5716 0.497925
\(979\) 3.11194 0.0994581
\(980\) 0 0
\(981\) 8.34557 0.266454
\(982\) −68.0620 −2.17194
\(983\) −2.35879 −0.0752336 −0.0376168 0.999292i \(-0.511977\pi\)
−0.0376168 + 0.999292i \(0.511977\pi\)
\(984\) −13.1875 −0.420401
\(985\) 0 0
\(986\) 64.3100 2.04805
\(987\) 10.8146 0.344232
\(988\) −75.5676 −2.40413
\(989\) 7.14150 0.227087
\(990\) 0 0
\(991\) −7.76677 −0.246720 −0.123360 0.992362i \(-0.539367\pi\)
−0.123360 + 0.992362i \(0.539367\pi\)
\(992\) 51.7552 1.64323
\(993\) −32.2137 −1.02227
\(994\) 10.6526 0.337879
\(995\) 0 0
\(996\) 25.1938 0.798295
\(997\) 8.35947 0.264747 0.132374 0.991200i \(-0.457740\pi\)
0.132374 + 0.991200i \(0.457740\pi\)
\(998\) −19.3549 −0.612668
\(999\) 7.97720 0.252388
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 1875.2.a.i.1.2 6
3.2 odd 2 5625.2.a.r.1.5 6
5.2 odd 4 1875.2.b.e.1249.3 12
5.3 odd 4 1875.2.b.e.1249.10 12
5.4 even 2 1875.2.a.l.1.5 yes 6
15.14 odd 2 5625.2.a.o.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.2 6 1.1 even 1 trivial
1875.2.a.l.1.5 yes 6 5.4 even 2
1875.2.b.e.1249.3 12 5.2 odd 4
1875.2.b.e.1249.10 12 5.3 odd 4
5625.2.a.o.1.2 6 15.14 odd 2
5625.2.a.r.1.5 6 3.2 odd 2