Properties

Label 2225.2.a.i.1.2
Level $2225$
Weight $2$
Character 2225.1
Self dual yes
Analytic conductor $17.767$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2225,2,Mod(1,2225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2225, 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("2225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2225 = 5^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2225.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: \(17.7667144497\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 2225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.45589 q^{2} +2.19353 q^{3} +0.119606 q^{4} -3.19353 q^{6} -3.71135 q^{7} +2.73764 q^{8} +1.81156 q^{9} +O(q^{10})\) \(q-1.45589 q^{2} +2.19353 q^{3} +0.119606 q^{4} -3.19353 q^{6} -3.71135 q^{7} +2.73764 q^{8} +1.81156 q^{9} -1.80647 q^{11} +0.262360 q^{12} +6.78527 q^{13} +5.40330 q^{14} -4.22491 q^{16} +1.92293 q^{17} -2.63743 q^{18} -6.21801 q^{19} -8.14094 q^{21} +2.63002 q^{22} -1.39822 q^{23} +6.00509 q^{24} -9.87858 q^{26} -2.60687 q^{27} -0.443901 q^{28} -4.99310 q^{29} +1.92427 q^{31} +0.675706 q^{32} -3.96255 q^{33} -2.79958 q^{34} +0.216674 q^{36} -2.70709 q^{37} +9.05272 q^{38} +14.8837 q^{39} +2.91604 q^{41} +11.8523 q^{42} -0.211586 q^{43} -0.216066 q^{44} +2.03564 q^{46} +9.61114 q^{47} -9.26745 q^{48} +6.77411 q^{49} +4.21801 q^{51} +0.811561 q^{52} -10.9331 q^{53} +3.79531 q^{54} -10.1603 q^{56} -13.6394 q^{57} +7.26939 q^{58} -12.0984 q^{59} +10.1997 q^{61} -2.80152 q^{62} -6.72333 q^{63} +7.46606 q^{64} +5.76902 q^{66} -2.88548 q^{67} +0.229995 q^{68} -3.06702 q^{69} -3.85919 q^{71} +4.95940 q^{72} -11.5899 q^{73} +3.94121 q^{74} -0.743713 q^{76} +6.70445 q^{77} -21.6689 q^{78} -10.8158 q^{79} -11.1529 q^{81} -4.24542 q^{82} +5.59257 q^{83} -0.973708 q^{84} +0.308045 q^{86} -10.9525 q^{87} -4.94547 q^{88} -1.00000 q^{89} -25.1825 q^{91} -0.167235 q^{92} +4.22094 q^{93} -13.9927 q^{94} +1.48218 q^{96} -1.07064 q^{97} -9.86233 q^{98} -3.27254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{6} - 2 q^{7} + 9 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{6} - 2 q^{7} + 9 q^{8} - 4 q^{9} - 14 q^{11} + 3 q^{12} + 5 q^{13} - 5 q^{14} - 3 q^{16} + 3 q^{17} - 7 q^{18} - q^{19} - 4 q^{21} - 2 q^{22} + 3 q^{23} + 6 q^{24} - 9 q^{26} - q^{27} - 5 q^{28} - 10 q^{29} - 11 q^{31} + 2 q^{32} - 8 q^{34} - 9 q^{36} - 3 q^{37} - 2 q^{38} + 11 q^{39} - 3 q^{41} + 6 q^{42} - 9 q^{43} - 9 q^{44} - 4 q^{46} + 24 q^{47} - 21 q^{48} - 7 q^{51} - 8 q^{52} - 3 q^{53} + 17 q^{54} - 13 q^{56} - 19 q^{57} - 19 q^{58} - 22 q^{59} - 3 q^{61} + 24 q^{62} - 6 q^{63} - 11 q^{64} + 14 q^{66} + 9 q^{67} - 31 q^{68} + 7 q^{69} + 16 q^{71} - 10 q^{72} - 3 q^{73} + 31 q^{74} - 24 q^{76} + 4 q^{77} - 16 q^{78} - 27 q^{79} - 8 q^{81} + 15 q^{82} - 6 q^{83} + 7 q^{84} + 15 q^{86} - 4 q^{87} - 30 q^{88} - 4 q^{89} - 29 q^{91} + 17 q^{92} + 2 q^{93} + 13 q^{94} + 12 q^{96} - 41 q^{97} - 22 q^{98} + 21 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.45589 −1.02947 −0.514734 0.857350i \(-0.672109\pi\)
−0.514734 + 0.857350i \(0.672109\pi\)
\(3\) 2.19353 1.26643 0.633217 0.773975i \(-0.281734\pi\)
0.633217 + 0.773975i \(0.281734\pi\)
\(4\) 0.119606 0.0598032
\(5\) 0 0
\(6\) −3.19353 −1.30375
\(7\) −3.71135 −1.40276 −0.701379 0.712789i \(-0.747432\pi\)
−0.701379 + 0.712789i \(0.747432\pi\)
\(8\) 2.73764 0.967902
\(9\) 1.81156 0.603854
\(10\) 0 0
\(11\) −1.80647 −0.544672 −0.272336 0.962202i \(-0.587796\pi\)
−0.272336 + 0.962202i \(0.587796\pi\)
\(12\) 0.262360 0.0757367
\(13\) 6.78527 1.88190 0.940948 0.338552i \(-0.109937\pi\)
0.940948 + 0.338552i \(0.109937\pi\)
\(14\) 5.40330 1.44409
\(15\) 0 0
\(16\) −4.22491 −1.05623
\(17\) 1.92293 0.466380 0.233190 0.972431i \(-0.425084\pi\)
0.233190 + 0.972431i \(0.425084\pi\)
\(18\) −2.63743 −0.621648
\(19\) −6.21801 −1.42651 −0.713255 0.700905i \(-0.752780\pi\)
−0.713255 + 0.700905i \(0.752780\pi\)
\(20\) 0 0
\(21\) −8.14094 −1.77650
\(22\) 2.63002 0.560722
\(23\) −1.39822 −0.291548 −0.145774 0.989318i \(-0.546567\pi\)
−0.145774 + 0.989318i \(0.546567\pi\)
\(24\) 6.00509 1.22578
\(25\) 0 0
\(26\) −9.87858 −1.93735
\(27\) −2.60687 −0.501693
\(28\) −0.443901 −0.0838894
\(29\) −4.99310 −0.927196 −0.463598 0.886046i \(-0.653442\pi\)
−0.463598 + 0.886046i \(0.653442\pi\)
\(30\) 0 0
\(31\) 1.92427 0.345609 0.172805 0.984956i \(-0.444717\pi\)
0.172805 + 0.984956i \(0.444717\pi\)
\(32\) 0.675706 0.119449
\(33\) −3.96255 −0.689791
\(34\) −2.79958 −0.480123
\(35\) 0 0
\(36\) 0.216674 0.0361124
\(37\) −2.70709 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(38\) 9.05272 1.46854
\(39\) 14.8837 2.38329
\(40\) 0 0
\(41\) 2.91604 0.455408 0.227704 0.973730i \(-0.426878\pi\)
0.227704 + 0.973730i \(0.426878\pi\)
\(42\) 11.8523 1.82885
\(43\) −0.211586 −0.0322666 −0.0161333 0.999870i \(-0.505136\pi\)
−0.0161333 + 0.999870i \(0.505136\pi\)
\(44\) −0.216066 −0.0325731
\(45\) 0 0
\(46\) 2.03564 0.300139
\(47\) 9.61114 1.40193 0.700964 0.713197i \(-0.252753\pi\)
0.700964 + 0.713197i \(0.252753\pi\)
\(48\) −9.26745 −1.33764
\(49\) 6.77411 0.967730
\(50\) 0 0
\(51\) 4.21801 0.590639
\(52\) 0.811561 0.112543
\(53\) −10.9331 −1.50178 −0.750889 0.660428i \(-0.770375\pi\)
−0.750889 + 0.660428i \(0.770375\pi\)
\(54\) 3.79531 0.516477
\(55\) 0 0
\(56\) −10.1603 −1.35773
\(57\) −13.6394 −1.80658
\(58\) 7.26939 0.954518
\(59\) −12.0984 −1.57508 −0.787539 0.616265i \(-0.788645\pi\)
−0.787539 + 0.616265i \(0.788645\pi\)
\(60\) 0 0
\(61\) 10.1997 1.30594 0.652971 0.757383i \(-0.273522\pi\)
0.652971 + 0.757383i \(0.273522\pi\)
\(62\) −2.80152 −0.355793
\(63\) −6.72333 −0.847061
\(64\) 7.46606 0.933258
\(65\) 0 0
\(66\) 5.76902 0.710117
\(67\) −2.88548 −0.352518 −0.176259 0.984344i \(-0.556400\pi\)
−0.176259 + 0.984344i \(0.556400\pi\)
\(68\) 0.229995 0.0278910
\(69\) −3.06702 −0.369226
\(70\) 0 0
\(71\) −3.85919 −0.458002 −0.229001 0.973426i \(-0.573546\pi\)
−0.229001 + 0.973426i \(0.573546\pi\)
\(72\) 4.95940 0.584471
\(73\) −11.5899 −1.35650 −0.678250 0.734832i \(-0.737261\pi\)
−0.678250 + 0.734832i \(0.737261\pi\)
\(74\) 3.94121 0.458156
\(75\) 0 0
\(76\) −0.743713 −0.0853098
\(77\) 6.70445 0.764043
\(78\) −21.6689 −2.45352
\(79\) −10.8158 −1.21688 −0.608438 0.793602i \(-0.708203\pi\)
−0.608438 + 0.793602i \(0.708203\pi\)
\(80\) 0 0
\(81\) −11.1529 −1.23921
\(82\) −4.24542 −0.468828
\(83\) 5.59257 0.613864 0.306932 0.951731i \(-0.400698\pi\)
0.306932 + 0.951731i \(0.400698\pi\)
\(84\) −0.973708 −0.106240
\(85\) 0 0
\(86\) 0.308045 0.0332174
\(87\) −10.9525 −1.17423
\(88\) −4.94547 −0.527189
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −25.1825 −2.63984
\(92\) −0.167235 −0.0174355
\(93\) 4.22094 0.437691
\(94\) −13.9927 −1.44324
\(95\) 0 0
\(96\) 1.48218 0.151274
\(97\) −1.07064 −0.108707 −0.0543536 0.998522i \(-0.517310\pi\)
−0.0543536 + 0.998522i \(0.517310\pi\)
\(98\) −9.86233 −0.996246
\(99\) −3.27254 −0.328902
\(100\) 0 0
\(101\) −13.8603 −1.37915 −0.689576 0.724213i \(-0.742203\pi\)
−0.689576 + 0.724213i \(0.742203\pi\)
\(102\) −6.14094 −0.608044
\(103\) −2.85216 −0.281032 −0.140516 0.990078i \(-0.544876\pi\)
−0.140516 + 0.990078i \(0.544876\pi\)
\(104\) 18.5756 1.82149
\(105\) 0 0
\(106\) 15.9174 1.54603
\(107\) 1.48908 0.143954 0.0719772 0.997406i \(-0.477069\pi\)
0.0719772 + 0.997406i \(0.477069\pi\)
\(108\) −0.311798 −0.0300028
\(109\) 9.40330 0.900673 0.450337 0.892859i \(-0.351304\pi\)
0.450337 + 0.892859i \(0.351304\pi\)
\(110\) 0 0
\(111\) −5.93807 −0.563616
\(112\) 15.6801 1.48163
\(113\) −12.6841 −1.19322 −0.596609 0.802532i \(-0.703485\pi\)
−0.596609 + 0.802532i \(0.703485\pi\)
\(114\) 19.8574 1.85981
\(115\) 0 0
\(116\) −0.597207 −0.0554492
\(117\) 12.2919 1.13639
\(118\) 17.6139 1.62149
\(119\) −7.13668 −0.654218
\(120\) 0 0
\(121\) −7.73666 −0.703332
\(122\) −14.8497 −1.34442
\(123\) 6.39641 0.576744
\(124\) 0.230155 0.0206685
\(125\) 0 0
\(126\) 9.78841 0.872021
\(127\) −1.25413 −0.111286 −0.0556429 0.998451i \(-0.517721\pi\)
−0.0556429 + 0.998451i \(0.517721\pi\)
\(128\) −12.2212 −1.08021
\(129\) −0.464120 −0.0408635
\(130\) 0 0
\(131\) −2.27240 −0.198541 −0.0992704 0.995060i \(-0.531651\pi\)
−0.0992704 + 0.995060i \(0.531651\pi\)
\(132\) −0.473946 −0.0412517
\(133\) 23.0772 2.00105
\(134\) 4.20093 0.362905
\(135\) 0 0
\(136\) 5.26430 0.451410
\(137\) 15.5836 1.33140 0.665700 0.746219i \(-0.268133\pi\)
0.665700 + 0.746219i \(0.268133\pi\)
\(138\) 4.46524 0.380106
\(139\) −21.5004 −1.82364 −0.911819 0.410593i \(-0.865322\pi\)
−0.911819 + 0.410593i \(0.865322\pi\)
\(140\) 0 0
\(141\) 21.0823 1.77545
\(142\) 5.61854 0.471498
\(143\) −12.2574 −1.02502
\(144\) −7.65368 −0.637806
\(145\) 0 0
\(146\) 16.8736 1.39647
\(147\) 14.8592 1.22557
\(148\) −0.323785 −0.0266149
\(149\) −17.0028 −1.39292 −0.696461 0.717595i \(-0.745243\pi\)
−0.696461 + 0.717595i \(0.745243\pi\)
\(150\) 0 0
\(151\) −17.7866 −1.44745 −0.723727 0.690087i \(-0.757572\pi\)
−0.723727 + 0.690087i \(0.757572\pi\)
\(152\) −17.0227 −1.38072
\(153\) 3.48351 0.281625
\(154\) −9.76092 −0.786557
\(155\) 0 0
\(156\) 1.78018 0.142529
\(157\) 8.13061 0.648893 0.324447 0.945904i \(-0.394822\pi\)
0.324447 + 0.945904i \(0.394822\pi\)
\(158\) 15.7466 1.25273
\(159\) −23.9821 −1.90190
\(160\) 0 0
\(161\) 5.18926 0.408971
\(162\) 16.2374 1.27573
\(163\) −23.0135 −1.80255 −0.901276 0.433244i \(-0.857369\pi\)
−0.901276 + 0.433244i \(0.857369\pi\)
\(164\) 0.348776 0.0272349
\(165\) 0 0
\(166\) −8.14214 −0.631953
\(167\) −5.88974 −0.455762 −0.227881 0.973689i \(-0.573180\pi\)
−0.227881 + 0.973689i \(0.573180\pi\)
\(168\) −22.2870 −1.71948
\(169\) 33.0399 2.54153
\(170\) 0 0
\(171\) −11.2643 −0.861403
\(172\) −0.0253070 −0.00192964
\(173\) 15.0723 1.14593 0.572964 0.819581i \(-0.305794\pi\)
0.572964 + 0.819581i \(0.305794\pi\)
\(174\) 15.9456 1.20883
\(175\) 0 0
\(176\) 7.63218 0.575297
\(177\) −26.5382 −1.99473
\(178\) 1.45589 0.109123
\(179\) 4.65562 0.347977 0.173989 0.984748i \(-0.444334\pi\)
0.173989 + 0.984748i \(0.444334\pi\)
\(180\) 0 0
\(181\) 9.62070 0.715101 0.357551 0.933894i \(-0.383612\pi\)
0.357551 + 0.933894i \(0.383612\pi\)
\(182\) 36.6629 2.71763
\(183\) 22.3734 1.65389
\(184\) −3.82781 −0.282190
\(185\) 0 0
\(186\) −6.14521 −0.450588
\(187\) −3.47373 −0.254024
\(188\) 1.14955 0.0838397
\(189\) 9.67501 0.703754
\(190\) 0 0
\(191\) 13.3629 0.966907 0.483454 0.875370i \(-0.339382\pi\)
0.483454 + 0.875370i \(0.339382\pi\)
\(192\) 16.3770 1.18191
\(193\) −0.449680 −0.0323687 −0.0161843 0.999869i \(-0.505152\pi\)
−0.0161843 + 0.999869i \(0.505152\pi\)
\(194\) 1.55873 0.111911
\(195\) 0 0
\(196\) 0.810226 0.0578733
\(197\) −8.56942 −0.610546 −0.305273 0.952265i \(-0.598748\pi\)
−0.305273 + 0.952265i \(0.598748\pi\)
\(198\) 4.76444 0.338594
\(199\) −8.67704 −0.615099 −0.307550 0.951532i \(-0.599509\pi\)
−0.307550 + 0.951532i \(0.599509\pi\)
\(200\) 0 0
\(201\) −6.32938 −0.446440
\(202\) 20.1790 1.41979
\(203\) 18.5311 1.30063
\(204\) 0.504501 0.0353221
\(205\) 0 0
\(206\) 4.15242 0.289313
\(207\) −2.53295 −0.176052
\(208\) −28.6671 −1.98771
\(209\) 11.2327 0.776980
\(210\) 0 0
\(211\) 25.1953 1.73452 0.867258 0.497858i \(-0.165880\pi\)
0.867258 + 0.497858i \(0.165880\pi\)
\(212\) −1.30767 −0.0898111
\(213\) −8.46524 −0.580029
\(214\) −2.16793 −0.148196
\(215\) 0 0
\(216\) −7.13668 −0.485590
\(217\) −7.14163 −0.484806
\(218\) −13.6901 −0.927214
\(219\) −25.4228 −1.71792
\(220\) 0 0
\(221\) 13.0476 0.877679
\(222\) 8.64515 0.580225
\(223\) −16.2438 −1.08777 −0.543884 0.839161i \(-0.683047\pi\)
−0.543884 + 0.839161i \(0.683047\pi\)
\(224\) −2.50778 −0.167558
\(225\) 0 0
\(226\) 18.4666 1.22838
\(227\) 13.1941 0.875726 0.437863 0.899042i \(-0.355736\pi\)
0.437863 + 0.899042i \(0.355736\pi\)
\(228\) −1.63135 −0.108039
\(229\) 2.36998 0.156613 0.0783063 0.996929i \(-0.475049\pi\)
0.0783063 + 0.996929i \(0.475049\pi\)
\(230\) 0 0
\(231\) 14.7064 0.967610
\(232\) −13.6693 −0.897435
\(233\) 9.67877 0.634077 0.317039 0.948413i \(-0.397312\pi\)
0.317039 + 0.948413i \(0.397312\pi\)
\(234\) −17.8957 −1.16988
\(235\) 0 0
\(236\) −1.44705 −0.0941946
\(237\) −23.7248 −1.54109
\(238\) 10.3902 0.673497
\(239\) −23.8871 −1.54513 −0.772564 0.634936i \(-0.781026\pi\)
−0.772564 + 0.634936i \(0.781026\pi\)
\(240\) 0 0
\(241\) −15.7221 −1.01275 −0.506376 0.862313i \(-0.669015\pi\)
−0.506376 + 0.862313i \(0.669015\pi\)
\(242\) 11.2637 0.724058
\(243\) −16.6436 −1.06769
\(244\) 1.21995 0.0780995
\(245\) 0 0
\(246\) −9.31244 −0.593740
\(247\) −42.1909 −2.68454
\(248\) 5.26796 0.334516
\(249\) 12.2674 0.777418
\(250\) 0 0
\(251\) 14.2933 0.902183 0.451091 0.892478i \(-0.351035\pi\)
0.451091 + 0.892478i \(0.351035\pi\)
\(252\) −0.804153 −0.0506569
\(253\) 2.52584 0.158798
\(254\) 1.82587 0.114565
\(255\) 0 0
\(256\) 2.86049 0.178781
\(257\) 8.51369 0.531070 0.265535 0.964101i \(-0.414451\pi\)
0.265535 + 0.964101i \(0.414451\pi\)
\(258\) 0.675706 0.0420676
\(259\) 10.0469 0.624286
\(260\) 0 0
\(261\) −9.04531 −0.559891
\(262\) 3.30836 0.204391
\(263\) 19.3754 1.19474 0.597370 0.801966i \(-0.296212\pi\)
0.597370 + 0.801966i \(0.296212\pi\)
\(264\) −10.8480 −0.667650
\(265\) 0 0
\(266\) −33.5978 −2.06001
\(267\) −2.19353 −0.134242
\(268\) −0.345122 −0.0210817
\(269\) −7.39853 −0.451096 −0.225548 0.974232i \(-0.572417\pi\)
−0.225548 + 0.974232i \(0.572417\pi\)
\(270\) 0 0
\(271\) −27.2477 −1.65518 −0.827590 0.561333i \(-0.810289\pi\)
−0.827590 + 0.561333i \(0.810289\pi\)
\(272\) −8.12422 −0.492603
\(273\) −55.2385 −3.34319
\(274\) −22.6880 −1.37063
\(275\) 0 0
\(276\) −0.366835 −0.0220809
\(277\) −28.4582 −1.70989 −0.854943 0.518722i \(-0.826408\pi\)
−0.854943 + 0.518722i \(0.826408\pi\)
\(278\) 31.3021 1.87738
\(279\) 3.48593 0.208697
\(280\) 0 0
\(281\) 22.1815 1.32323 0.661617 0.749842i \(-0.269870\pi\)
0.661617 + 0.749842i \(0.269870\pi\)
\(282\) −30.6934 −1.82777
\(283\) −0.247445 −0.0147091 −0.00735455 0.999973i \(-0.502341\pi\)
−0.00735455 + 0.999973i \(0.502341\pi\)
\(284\) −0.461584 −0.0273899
\(285\) 0 0
\(286\) 17.8454 1.05522
\(287\) −10.8224 −0.638828
\(288\) 1.22408 0.0721297
\(289\) −13.3023 −0.782490
\(290\) 0 0
\(291\) −2.34848 −0.137671
\(292\) −1.38623 −0.0811229
\(293\) 11.3844 0.665085 0.332542 0.943088i \(-0.392094\pi\)
0.332542 + 0.943088i \(0.392094\pi\)
\(294\) −21.6333 −1.26168
\(295\) 0 0
\(296\) −7.41103 −0.430757
\(297\) 4.70925 0.273258
\(298\) 24.7541 1.43397
\(299\) −9.48727 −0.548663
\(300\) 0 0
\(301\) 0.785269 0.0452622
\(302\) 25.8953 1.49011
\(303\) −30.4030 −1.74660
\(304\) 26.2705 1.50672
\(305\) 0 0
\(306\) −5.07160 −0.289924
\(307\) 6.37110 0.363618 0.181809 0.983334i \(-0.441805\pi\)
0.181809 + 0.983334i \(0.441805\pi\)
\(308\) 0.801895 0.0456922
\(309\) −6.25629 −0.355908
\(310\) 0 0
\(311\) −30.5996 −1.73514 −0.867572 0.497312i \(-0.834321\pi\)
−0.867572 + 0.497312i \(0.834321\pi\)
\(312\) 40.7461 2.30680
\(313\) 19.2316 1.08704 0.543518 0.839398i \(-0.317092\pi\)
0.543518 + 0.839398i \(0.317092\pi\)
\(314\) −11.8372 −0.668014
\(315\) 0 0
\(316\) −1.29364 −0.0727730
\(317\) 9.70429 0.545047 0.272524 0.962149i \(-0.412142\pi\)
0.272524 + 0.962149i \(0.412142\pi\)
\(318\) 34.9152 1.95795
\(319\) 9.01990 0.505018
\(320\) 0 0
\(321\) 3.26633 0.182309
\(322\) −7.55498 −0.421023
\(323\) −11.9568 −0.665296
\(324\) −1.33396 −0.0741089
\(325\) 0 0
\(326\) 33.5050 1.85567
\(327\) 20.6264 1.14064
\(328\) 7.98306 0.440791
\(329\) −35.6703 −1.96657
\(330\) 0 0
\(331\) −28.9275 −1.59000 −0.795001 0.606608i \(-0.792530\pi\)
−0.795001 + 0.606608i \(0.792530\pi\)
\(332\) 0.668906 0.0367110
\(333\) −4.90405 −0.268740
\(334\) 8.57480 0.469192
\(335\) 0 0
\(336\) 34.3947 1.87639
\(337\) −2.30865 −0.125760 −0.0628802 0.998021i \(-0.520029\pi\)
−0.0628802 + 0.998021i \(0.520029\pi\)
\(338\) −48.1023 −2.61642
\(339\) −27.8229 −1.51113
\(340\) 0 0
\(341\) −3.47614 −0.188244
\(342\) 16.3996 0.886786
\(343\) 0.838363 0.0452673
\(344\) −0.579246 −0.0312309
\(345\) 0 0
\(346\) −21.9436 −1.17970
\(347\) 34.4649 1.85017 0.925085 0.379759i \(-0.123993\pi\)
0.925085 + 0.379759i \(0.123993\pi\)
\(348\) −1.30999 −0.0702228
\(349\) 19.3154 1.03393 0.516965 0.856007i \(-0.327062\pi\)
0.516965 + 0.856007i \(0.327062\pi\)
\(350\) 0 0
\(351\) −17.6883 −0.944133
\(352\) −1.22064 −0.0650605
\(353\) −1.41183 −0.0751441 −0.0375721 0.999294i \(-0.511962\pi\)
−0.0375721 + 0.999294i \(0.511962\pi\)
\(354\) 38.6366 2.05351
\(355\) 0 0
\(356\) −0.119606 −0.00633912
\(357\) −15.6545 −0.828524
\(358\) −6.77806 −0.358231
\(359\) −17.2200 −0.908834 −0.454417 0.890789i \(-0.650152\pi\)
−0.454417 + 0.890789i \(0.650152\pi\)
\(360\) 0 0
\(361\) 19.6636 1.03493
\(362\) −14.0067 −0.736173
\(363\) −16.9706 −0.890724
\(364\) −3.01199 −0.157871
\(365\) 0 0
\(366\) −32.5731 −1.70262
\(367\) 20.4193 1.06588 0.532939 0.846154i \(-0.321087\pi\)
0.532939 + 0.846154i \(0.321087\pi\)
\(368\) 5.90733 0.307941
\(369\) 5.28258 0.275000
\(370\) 0 0
\(371\) 40.5766 2.10663
\(372\) 0.504851 0.0261753
\(373\) 0.00410346 0.000212469 0 0.000106235 1.00000i \(-0.499966\pi\)
0.000106235 1.00000i \(0.499966\pi\)
\(374\) 5.05736 0.261510
\(375\) 0 0
\(376\) 26.3118 1.35693
\(377\) −33.8795 −1.74489
\(378\) −14.0857 −0.724492
\(379\) 33.3456 1.71285 0.856424 0.516273i \(-0.172681\pi\)
0.856424 + 0.516273i \(0.172681\pi\)
\(380\) 0 0
\(381\) −2.75096 −0.140936
\(382\) −19.4549 −0.995399
\(383\) −34.6473 −1.77039 −0.885196 0.465218i \(-0.845976\pi\)
−0.885196 + 0.465218i \(0.845976\pi\)
\(384\) −26.8074 −1.36801
\(385\) 0 0
\(386\) 0.654683 0.0333225
\(387\) −0.383301 −0.0194843
\(388\) −0.128056 −0.00650104
\(389\) 30.7954 1.56139 0.780696 0.624912i \(-0.214865\pi\)
0.780696 + 0.624912i \(0.214865\pi\)
\(390\) 0 0
\(391\) −2.68868 −0.135972
\(392\) 18.5451 0.936668
\(393\) −4.98458 −0.251439
\(394\) 12.4761 0.628537
\(395\) 0 0
\(396\) −0.391416 −0.0196694
\(397\) −30.1664 −1.51401 −0.757004 0.653411i \(-0.773338\pi\)
−0.757004 + 0.653411i \(0.773338\pi\)
\(398\) 12.6328 0.633224
\(399\) 50.6205 2.53419
\(400\) 0 0
\(401\) −7.51947 −0.375504 −0.187752 0.982216i \(-0.560120\pi\)
−0.187752 + 0.982216i \(0.560120\pi\)
\(402\) 9.21486 0.459596
\(403\) 13.0567 0.650400
\(404\) −1.65778 −0.0824777
\(405\) 0 0
\(406\) −26.9792 −1.33896
\(407\) 4.89028 0.242402
\(408\) 11.5474 0.571681
\(409\) 32.8936 1.62648 0.813241 0.581927i \(-0.197701\pi\)
0.813241 + 0.581927i \(0.197701\pi\)
\(410\) 0 0
\(411\) 34.1831 1.68613
\(412\) −0.341136 −0.0168066
\(413\) 44.9014 2.20945
\(414\) 3.68769 0.181240
\(415\) 0 0
\(416\) 4.58484 0.224790
\(417\) −47.1616 −2.30952
\(418\) −16.3535 −0.799875
\(419\) 15.0851 0.736957 0.368479 0.929636i \(-0.379879\pi\)
0.368479 + 0.929636i \(0.379879\pi\)
\(420\) 0 0
\(421\) −23.0575 −1.12375 −0.561876 0.827221i \(-0.689920\pi\)
−0.561876 + 0.827221i \(0.689920\pi\)
\(422\) −36.6815 −1.78563
\(423\) 17.4112 0.846560
\(424\) −29.9309 −1.45357
\(425\) 0 0
\(426\) 12.3244 0.597120
\(427\) −37.8548 −1.83192
\(428\) 0.178103 0.00860893
\(429\) −26.8870 −1.29811
\(430\) 0 0
\(431\) −12.8238 −0.617703 −0.308851 0.951110i \(-0.599945\pi\)
−0.308851 + 0.951110i \(0.599945\pi\)
\(432\) 11.0138 0.529901
\(433\) 28.7231 1.38034 0.690171 0.723646i \(-0.257535\pi\)
0.690171 + 0.723646i \(0.257535\pi\)
\(434\) 10.3974 0.499092
\(435\) 0 0
\(436\) 1.12469 0.0538631
\(437\) 8.69411 0.415896
\(438\) 37.0128 1.76854
\(439\) −23.6489 −1.12870 −0.564349 0.825536i \(-0.690873\pi\)
−0.564349 + 0.825536i \(0.690873\pi\)
\(440\) 0 0
\(441\) 12.2717 0.584367
\(442\) −18.9959 −0.903541
\(443\) 35.5043 1.68686 0.843430 0.537239i \(-0.180533\pi\)
0.843430 + 0.537239i \(0.180533\pi\)
\(444\) −0.710230 −0.0337060
\(445\) 0 0
\(446\) 23.6492 1.11982
\(447\) −37.2960 −1.76404
\(448\) −27.7092 −1.30913
\(449\) 32.7779 1.54688 0.773442 0.633867i \(-0.218533\pi\)
0.773442 + 0.633867i \(0.218533\pi\)
\(450\) 0 0
\(451\) −5.26774 −0.248048
\(452\) −1.51710 −0.0713582
\(453\) −39.0154 −1.83310
\(454\) −19.2092 −0.901531
\(455\) 0 0
\(456\) −37.3397 −1.74859
\(457\) −28.7506 −1.34490 −0.672448 0.740145i \(-0.734757\pi\)
−0.672448 + 0.740145i \(0.734757\pi\)
\(458\) −3.45042 −0.161228
\(459\) −5.01285 −0.233980
\(460\) 0 0
\(461\) −11.6097 −0.540717 −0.270358 0.962760i \(-0.587142\pi\)
−0.270358 + 0.962760i \(0.587142\pi\)
\(462\) −21.4108 −0.996123
\(463\) 18.7448 0.871145 0.435573 0.900154i \(-0.356546\pi\)
0.435573 + 0.900154i \(0.356546\pi\)
\(464\) 21.0954 0.979329
\(465\) 0 0
\(466\) −14.0912 −0.652762
\(467\) 0.634016 0.0293387 0.0146694 0.999892i \(-0.495330\pi\)
0.0146694 + 0.999892i \(0.495330\pi\)
\(468\) 1.47019 0.0679597
\(469\) 10.7090 0.494497
\(470\) 0 0
\(471\) 17.8347 0.821780
\(472\) −33.1211 −1.52452
\(473\) 0.382224 0.0175747
\(474\) 34.5406 1.58650
\(475\) 0 0
\(476\) −0.853592 −0.0391243
\(477\) −19.8060 −0.906854
\(478\) 34.7769 1.59066
\(479\) 29.2698 1.33737 0.668686 0.743545i \(-0.266857\pi\)
0.668686 + 0.743545i \(0.266857\pi\)
\(480\) 0 0
\(481\) −18.3683 −0.837523
\(482\) 22.8896 1.04259
\(483\) 11.3828 0.517935
\(484\) −0.925353 −0.0420615
\(485\) 0 0
\(486\) 24.2312 1.09915
\(487\) −8.96526 −0.406255 −0.203127 0.979152i \(-0.565111\pi\)
−0.203127 + 0.979152i \(0.565111\pi\)
\(488\) 27.9232 1.26402
\(489\) −50.4806 −2.28281
\(490\) 0 0
\(491\) 21.4601 0.968479 0.484240 0.874935i \(-0.339096\pi\)
0.484240 + 0.874935i \(0.339096\pi\)
\(492\) 0.765050 0.0344911
\(493\) −9.60141 −0.432426
\(494\) 61.4251 2.76365
\(495\) 0 0
\(496\) −8.12986 −0.365041
\(497\) 14.3228 0.642465
\(498\) −17.8600 −0.800326
\(499\) 21.8821 0.979576 0.489788 0.871841i \(-0.337074\pi\)
0.489788 + 0.871841i \(0.337074\pi\)
\(500\) 0 0
\(501\) −12.9193 −0.577192
\(502\) −20.8094 −0.928768
\(503\) 8.91940 0.397696 0.198848 0.980030i \(-0.436280\pi\)
0.198848 + 0.980030i \(0.436280\pi\)
\(504\) −18.4061 −0.819872
\(505\) 0 0
\(506\) −3.67733 −0.163477
\(507\) 72.4739 3.21868
\(508\) −0.150001 −0.00665524
\(509\) −5.76572 −0.255561 −0.127780 0.991802i \(-0.540785\pi\)
−0.127780 + 0.991802i \(0.540785\pi\)
\(510\) 0 0
\(511\) 43.0143 1.90284
\(512\) 20.2778 0.896159
\(513\) 16.2096 0.715670
\(514\) −12.3950 −0.546719
\(515\) 0 0
\(516\) −0.0555116 −0.00244376
\(517\) −17.3623 −0.763591
\(518\) −14.6272 −0.642682
\(519\) 33.0616 1.45124
\(520\) 0 0
\(521\) 14.8063 0.648676 0.324338 0.945941i \(-0.394858\pi\)
0.324338 + 0.945941i \(0.394858\pi\)
\(522\) 13.1689 0.576389
\(523\) −34.3182 −1.50063 −0.750315 0.661081i \(-0.770098\pi\)
−0.750315 + 0.661081i \(0.770098\pi\)
\(524\) −0.271794 −0.0118734
\(525\) 0 0
\(526\) −28.2084 −1.22995
\(527\) 3.70024 0.161185
\(528\) 16.7414 0.728576
\(529\) −21.0450 −0.915000
\(530\) 0 0
\(531\) −21.9170 −0.951117
\(532\) 2.76018 0.119669
\(533\) 19.7861 0.857031
\(534\) 3.19353 0.138197
\(535\) 0 0
\(536\) −7.89941 −0.341203
\(537\) 10.2122 0.440690
\(538\) 10.7714 0.464389
\(539\) −12.2372 −0.527095
\(540\) 0 0
\(541\) −19.0816 −0.820384 −0.410192 0.911999i \(-0.634538\pi\)
−0.410192 + 0.911999i \(0.634538\pi\)
\(542\) 39.6696 1.70395
\(543\) 21.1033 0.905628
\(544\) 1.29934 0.0557086
\(545\) 0 0
\(546\) 80.4210 3.44170
\(547\) 16.2497 0.694789 0.347394 0.937719i \(-0.387067\pi\)
0.347394 + 0.937719i \(0.387067\pi\)
\(548\) 1.86390 0.0796220
\(549\) 18.4774 0.788598
\(550\) 0 0
\(551\) 31.0472 1.32265
\(552\) −8.39641 −0.357375
\(553\) 40.1413 1.70698
\(554\) 41.4319 1.76027
\(555\) 0 0
\(556\) −2.57158 −0.109059
\(557\) −34.7023 −1.47038 −0.735191 0.677860i \(-0.762908\pi\)
−0.735191 + 0.677860i \(0.762908\pi\)
\(558\) −5.07512 −0.214847
\(559\) −1.43567 −0.0607223
\(560\) 0 0
\(561\) −7.61972 −0.321705
\(562\) −32.2937 −1.36223
\(563\) −11.6228 −0.489842 −0.244921 0.969543i \(-0.578762\pi\)
−0.244921 + 0.969543i \(0.578762\pi\)
\(564\) 2.52157 0.106177
\(565\) 0 0
\(566\) 0.360253 0.0151425
\(567\) 41.3924 1.73832
\(568\) −10.5651 −0.443301
\(569\) 42.6106 1.78633 0.893164 0.449732i \(-0.148480\pi\)
0.893164 + 0.449732i \(0.148480\pi\)
\(570\) 0 0
\(571\) −3.41985 −0.143116 −0.0715582 0.997436i \(-0.522797\pi\)
−0.0715582 + 0.997436i \(0.522797\pi\)
\(572\) −1.46606 −0.0612992
\(573\) 29.3119 1.22452
\(574\) 15.7562 0.657652
\(575\) 0 0
\(576\) 13.5252 0.563551
\(577\) −6.58744 −0.274239 −0.137119 0.990555i \(-0.543784\pi\)
−0.137119 + 0.990555i \(0.543784\pi\)
\(578\) 19.3667 0.805548
\(579\) −0.986386 −0.0409928
\(580\) 0 0
\(581\) −20.7560 −0.861103
\(582\) 3.41913 0.141727
\(583\) 19.7504 0.817977
\(584\) −31.7291 −1.31296
\(585\) 0 0
\(586\) −16.5744 −0.684683
\(587\) 3.93893 0.162577 0.0812884 0.996691i \(-0.474097\pi\)
0.0812884 + 0.996691i \(0.474097\pi\)
\(588\) 1.77725 0.0732927
\(589\) −11.9651 −0.493014
\(590\) 0 0
\(591\) −18.7973 −0.773216
\(592\) 11.4372 0.470065
\(593\) 20.9067 0.858537 0.429269 0.903177i \(-0.358771\pi\)
0.429269 + 0.903177i \(0.358771\pi\)
\(594\) −6.85613 −0.281310
\(595\) 0 0
\(596\) −2.03364 −0.0833011
\(597\) −19.0333 −0.778982
\(598\) 13.8124 0.564830
\(599\) 8.80091 0.359595 0.179798 0.983704i \(-0.442456\pi\)
0.179798 + 0.983704i \(0.442456\pi\)
\(600\) 0 0
\(601\) 3.04731 0.124303 0.0621513 0.998067i \(-0.480204\pi\)
0.0621513 + 0.998067i \(0.480204\pi\)
\(602\) −1.14326 −0.0465959
\(603\) −5.22723 −0.212869
\(604\) −2.12739 −0.0865623
\(605\) 0 0
\(606\) 44.2633 1.79807
\(607\) 37.2207 1.51074 0.755371 0.655298i \(-0.227457\pi\)
0.755371 + 0.655298i \(0.227457\pi\)
\(608\) −4.20154 −0.170395
\(609\) 40.6486 1.64716
\(610\) 0 0
\(611\) 65.2141 2.63828
\(612\) 0.416650 0.0168421
\(613\) −7.03831 −0.284275 −0.142137 0.989847i \(-0.545397\pi\)
−0.142137 + 0.989847i \(0.545397\pi\)
\(614\) −9.27560 −0.374333
\(615\) 0 0
\(616\) 18.3544 0.739519
\(617\) 41.9256 1.68786 0.843931 0.536452i \(-0.180236\pi\)
0.843931 + 0.536452i \(0.180236\pi\)
\(618\) 9.10845 0.366395
\(619\) −7.92233 −0.318425 −0.159213 0.987244i \(-0.550896\pi\)
−0.159213 + 0.987244i \(0.550896\pi\)
\(620\) 0 0
\(621\) 3.64497 0.146268
\(622\) 44.5496 1.78627
\(623\) 3.71135 0.148692
\(624\) −62.8821 −2.51730
\(625\) 0 0
\(626\) −27.9991 −1.11907
\(627\) 24.6392 0.983993
\(628\) 0.972472 0.0388059
\(629\) −5.20555 −0.207559
\(630\) 0 0
\(631\) 27.9812 1.11391 0.556956 0.830542i \(-0.311969\pi\)
0.556956 + 0.830542i \(0.311969\pi\)
\(632\) −29.6098 −1.17782
\(633\) 55.2666 2.19665
\(634\) −14.1283 −0.561108
\(635\) 0 0
\(636\) −2.86841 −0.113740
\(637\) 45.9641 1.82117
\(638\) −13.1320 −0.519899
\(639\) −6.99116 −0.276566
\(640\) 0 0
\(641\) −10.8328 −0.427868 −0.213934 0.976848i \(-0.568628\pi\)
−0.213934 + 0.976848i \(0.568628\pi\)
\(642\) −4.75541 −0.187681
\(643\) −33.1604 −1.30772 −0.653859 0.756616i \(-0.726851\pi\)
−0.653859 + 0.756616i \(0.726851\pi\)
\(644\) 0.620669 0.0244578
\(645\) 0 0
\(646\) 17.4078 0.684900
\(647\) −7.80677 −0.306916 −0.153458 0.988155i \(-0.549041\pi\)
−0.153458 + 0.988155i \(0.549041\pi\)
\(648\) −30.5327 −1.19944
\(649\) 21.8554 0.857901
\(650\) 0 0
\(651\) −15.6654 −0.613974
\(652\) −2.75255 −0.107798
\(653\) 9.79178 0.383182 0.191591 0.981475i \(-0.438635\pi\)
0.191591 + 0.981475i \(0.438635\pi\)
\(654\) −30.0297 −1.17425
\(655\) 0 0
\(656\) −12.3200 −0.481015
\(657\) −20.9959 −0.819127
\(658\) 51.9319 2.02452
\(659\) −33.0449 −1.28725 −0.643623 0.765342i \(-0.722570\pi\)
−0.643623 + 0.765342i \(0.722570\pi\)
\(660\) 0 0
\(661\) 21.9348 0.853165 0.426582 0.904449i \(-0.359717\pi\)
0.426582 + 0.904449i \(0.359717\pi\)
\(662\) 42.1152 1.63686
\(663\) 28.6203 1.11152
\(664\) 15.3104 0.594160
\(665\) 0 0
\(666\) 7.13974 0.276659
\(667\) 6.98143 0.270322
\(668\) −0.704451 −0.0272560
\(669\) −35.6313 −1.37758
\(670\) 0 0
\(671\) −18.4255 −0.711310
\(672\) −5.50088 −0.212201
\(673\) −12.6202 −0.486472 −0.243236 0.969967i \(-0.578209\pi\)
−0.243236 + 0.969967i \(0.578209\pi\)
\(674\) 3.36114 0.129466
\(675\) 0 0
\(676\) 3.95178 0.151991
\(677\) 43.1296 1.65761 0.828803 0.559541i \(-0.189023\pi\)
0.828803 + 0.559541i \(0.189023\pi\)
\(678\) 40.5069 1.55566
\(679\) 3.97353 0.152490
\(680\) 0 0
\(681\) 28.9417 1.10905
\(682\) 5.06087 0.193791
\(683\) 33.1031 1.26666 0.633328 0.773884i \(-0.281689\pi\)
0.633328 + 0.773884i \(0.281689\pi\)
\(684\) −1.34728 −0.0515146
\(685\) 0 0
\(686\) −1.22056 −0.0466012
\(687\) 5.19862 0.198340
\(688\) 0.893931 0.0340808
\(689\) −74.1841 −2.82619
\(690\) 0 0
\(691\) 11.4778 0.436637 0.218319 0.975878i \(-0.429943\pi\)
0.218319 + 0.975878i \(0.429943\pi\)
\(692\) 1.80275 0.0685301
\(693\) 12.1455 0.461370
\(694\) −50.1769 −1.90469
\(695\) 0 0
\(696\) −29.9840 −1.13654
\(697\) 5.60735 0.212393
\(698\) −28.1210 −1.06440
\(699\) 21.2306 0.803017
\(700\) 0 0
\(701\) 27.7156 1.04680 0.523401 0.852087i \(-0.324663\pi\)
0.523401 + 0.852087i \(0.324663\pi\)
\(702\) 25.7522 0.971955
\(703\) 16.8327 0.634857
\(704\) −13.4872 −0.508320
\(705\) 0 0
\(706\) 2.05546 0.0773584
\(707\) 51.4404 1.93462
\(708\) −3.17413 −0.119291
\(709\) 21.6735 0.813965 0.406983 0.913436i \(-0.366581\pi\)
0.406983 + 0.913436i \(0.366581\pi\)
\(710\) 0 0
\(711\) −19.5935 −0.734815
\(712\) −2.73764 −0.102597
\(713\) −2.69054 −0.100762
\(714\) 22.7912 0.852939
\(715\) 0 0
\(716\) 0.556842 0.0208101
\(717\) −52.3970 −1.95680
\(718\) 25.0703 0.935615
\(719\) 13.5991 0.507160 0.253580 0.967314i \(-0.418392\pi\)
0.253580 + 0.967314i \(0.418392\pi\)
\(720\) 0 0
\(721\) 10.5854 0.394219
\(722\) −28.6280 −1.06542
\(723\) −34.4869 −1.28258
\(724\) 1.15070 0.0427653
\(725\) 0 0
\(726\) 24.7072 0.916971
\(727\) −22.5512 −0.836379 −0.418189 0.908360i \(-0.637335\pi\)
−0.418189 + 0.908360i \(0.637335\pi\)
\(728\) −68.9406 −2.55511
\(729\) −3.04947 −0.112943
\(730\) 0 0
\(731\) −0.406866 −0.0150485
\(732\) 2.67600 0.0989078
\(733\) −11.3716 −0.420019 −0.210009 0.977699i \(-0.567349\pi\)
−0.210009 + 0.977699i \(0.567349\pi\)
\(734\) −29.7282 −1.09729
\(735\) 0 0
\(736\) −0.944782 −0.0348251
\(737\) 5.21254 0.192007
\(738\) −7.69084 −0.283104
\(739\) −51.8437 −1.90710 −0.953551 0.301232i \(-0.902602\pi\)
−0.953551 + 0.301232i \(0.902602\pi\)
\(740\) 0 0
\(741\) −92.5468 −3.39979
\(742\) −59.0749 −2.16871
\(743\) −20.9239 −0.767623 −0.383812 0.923411i \(-0.625389\pi\)
−0.383812 + 0.923411i \(0.625389\pi\)
\(744\) 11.5554 0.423642
\(745\) 0 0
\(746\) −0.00597418 −0.000218730 0
\(747\) 10.1313 0.370684
\(748\) −0.415480 −0.0151915
\(749\) −5.52648 −0.201933
\(750\) 0 0
\(751\) 27.2392 0.993972 0.496986 0.867759i \(-0.334440\pi\)
0.496986 + 0.867759i \(0.334440\pi\)
\(752\) −40.6062 −1.48075
\(753\) 31.3527 1.14255
\(754\) 49.3248 1.79630
\(755\) 0 0
\(756\) 1.15719 0.0420867
\(757\) 18.6306 0.677142 0.338571 0.940941i \(-0.390057\pi\)
0.338571 + 0.940941i \(0.390057\pi\)
\(758\) −48.5474 −1.76332
\(759\) 5.54049 0.201107
\(760\) 0 0
\(761\) −4.06378 −0.147312 −0.0736559 0.997284i \(-0.523467\pi\)
−0.0736559 + 0.997284i \(0.523467\pi\)
\(762\) 4.00509 0.145089
\(763\) −34.8989 −1.26343
\(764\) 1.59829 0.0578241
\(765\) 0 0
\(766\) 50.4425 1.82256
\(767\) −82.0909 −2.96413
\(768\) 6.27456 0.226414
\(769\) −32.7396 −1.18062 −0.590310 0.807177i \(-0.700994\pi\)
−0.590310 + 0.807177i \(0.700994\pi\)
\(770\) 0 0
\(771\) 18.6750 0.672564
\(772\) −0.0537846 −0.00193575
\(773\) −19.0463 −0.685049 −0.342524 0.939509i \(-0.611282\pi\)
−0.342524 + 0.939509i \(0.611282\pi\)
\(774\) 0.558043 0.0200584
\(775\) 0 0
\(776\) −2.93103 −0.105218
\(777\) 22.0382 0.790617
\(778\) −44.8347 −1.60740
\(779\) −18.1319 −0.649644
\(780\) 0 0
\(781\) 6.97152 0.249461
\(782\) 3.91441 0.139979
\(783\) 13.0164 0.465168
\(784\) −28.6200 −1.02214
\(785\) 0 0
\(786\) 7.25698 0.258848
\(787\) −36.9703 −1.31785 −0.658924 0.752210i \(-0.728988\pi\)
−0.658924 + 0.752210i \(0.728988\pi\)
\(788\) −1.02496 −0.0365126
\(789\) 42.5005 1.51306
\(790\) 0 0
\(791\) 47.0750 1.67379
\(792\) −8.95903 −0.318345
\(793\) 69.2079 2.45765
\(794\) 43.9188 1.55862
\(795\) 0 0
\(796\) −1.03783 −0.0367849
\(797\) 10.1404 0.359191 0.179596 0.983741i \(-0.442521\pi\)
0.179596 + 0.983741i \(0.442521\pi\)
\(798\) −73.6977 −2.60887
\(799\) 18.4816 0.653832
\(800\) 0 0
\(801\) −1.81156 −0.0640084
\(802\) 10.9475 0.386570
\(803\) 20.9369 0.738847
\(804\) −0.757034 −0.0266985
\(805\) 0 0
\(806\) −19.0091 −0.669566
\(807\) −16.2289 −0.571284
\(808\) −37.9445 −1.33488
\(809\) 0.167200 0.00587845 0.00293922 0.999996i \(-0.499064\pi\)
0.00293922 + 0.999996i \(0.499064\pi\)
\(810\) 0 0
\(811\) −41.1102 −1.44357 −0.721787 0.692115i \(-0.756679\pi\)
−0.721787 + 0.692115i \(0.756679\pi\)
\(812\) 2.21644 0.0777819
\(813\) −59.7686 −2.09618
\(814\) −7.11969 −0.249545
\(815\) 0 0
\(816\) −17.8207 −0.623849
\(817\) 1.31564 0.0460285
\(818\) −47.8893 −1.67441
\(819\) −45.6196 −1.59408
\(820\) 0 0
\(821\) −43.1698 −1.50664 −0.753318 0.657656i \(-0.771548\pi\)
−0.753318 + 0.657656i \(0.771548\pi\)
\(822\) −49.7668 −1.73582
\(823\) 8.46445 0.295052 0.147526 0.989058i \(-0.452869\pi\)
0.147526 + 0.989058i \(0.452869\pi\)
\(824\) −7.80818 −0.272011
\(825\) 0 0
\(826\) −65.3713 −2.27456
\(827\) −6.41763 −0.223163 −0.111581 0.993755i \(-0.535592\pi\)
−0.111581 + 0.993755i \(0.535592\pi\)
\(828\) −0.302957 −0.0105285
\(829\) −16.7935 −0.583261 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(830\) 0 0
\(831\) −62.4238 −2.16546
\(832\) 50.6592 1.75629
\(833\) 13.0262 0.451330
\(834\) 68.6620 2.37757
\(835\) 0 0
\(836\) 1.34350 0.0464658
\(837\) −5.01633 −0.173390
\(838\) −21.9623 −0.758673
\(839\) −7.92834 −0.273717 −0.136858 0.990591i \(-0.543701\pi\)
−0.136858 + 0.990591i \(0.543701\pi\)
\(840\) 0 0
\(841\) −4.06893 −0.140308
\(842\) 33.5691 1.15687
\(843\) 48.6556 1.67579
\(844\) 3.01352 0.103730
\(845\) 0 0
\(846\) −25.3487 −0.871505
\(847\) 28.7134 0.986605
\(848\) 46.1914 1.58622
\(849\) −0.542778 −0.0186281
\(850\) 0 0
\(851\) 3.78509 0.129751
\(852\) −1.01250 −0.0346875
\(853\) 27.1727 0.930375 0.465187 0.885212i \(-0.345987\pi\)
0.465187 + 0.885212i \(0.345987\pi\)
\(854\) 55.1123 1.88590
\(855\) 0 0
\(856\) 4.07656 0.139334
\(857\) 43.2642 1.47788 0.738938 0.673773i \(-0.235328\pi\)
0.738938 + 0.673773i \(0.235328\pi\)
\(858\) 39.1444 1.33637
\(859\) 18.4487 0.629463 0.314731 0.949181i \(-0.398086\pi\)
0.314731 + 0.949181i \(0.398086\pi\)
\(860\) 0 0
\(861\) −23.7393 −0.809033
\(862\) 18.6701 0.635905
\(863\) −1.54533 −0.0526037 −0.0263018 0.999654i \(-0.508373\pi\)
−0.0263018 + 0.999654i \(0.508373\pi\)
\(864\) −1.76148 −0.0599267
\(865\) 0 0
\(866\) −41.8175 −1.42102
\(867\) −29.1790 −0.990971
\(868\) −0.854185 −0.0289929
\(869\) 19.5385 0.662798
\(870\) 0 0
\(871\) −19.5788 −0.663401
\(872\) 25.7429 0.871763
\(873\) −1.93953 −0.0656433
\(874\) −12.6576 −0.428151
\(875\) 0 0
\(876\) −3.04073 −0.102737
\(877\) −47.7230 −1.61149 −0.805745 0.592262i \(-0.798235\pi\)
−0.805745 + 0.592262i \(0.798235\pi\)
\(878\) 34.4301 1.16196
\(879\) 24.9720 0.842286
\(880\) 0 0
\(881\) 28.2004 0.950095 0.475047 0.879960i \(-0.342431\pi\)
0.475047 + 0.879960i \(0.342431\pi\)
\(882\) −17.8662 −0.601587
\(883\) 11.2269 0.377816 0.188908 0.981995i \(-0.439505\pi\)
0.188908 + 0.981995i \(0.439505\pi\)
\(884\) 1.56058 0.0524879
\(885\) 0 0
\(886\) −51.6902 −1.73657
\(887\) −12.8249 −0.430619 −0.215309 0.976546i \(-0.569076\pi\)
−0.215309 + 0.976546i \(0.569076\pi\)
\(888\) −16.2563 −0.545525
\(889\) 4.65450 0.156107
\(890\) 0 0
\(891\) 20.1475 0.674965
\(892\) −1.94286 −0.0650519
\(893\) −59.7621 −1.99986
\(894\) 54.2988 1.81602
\(895\) 0 0
\(896\) 45.3570 1.51527
\(897\) −20.8106 −0.694845
\(898\) −47.7209 −1.59247
\(899\) −9.60807 −0.320447
\(900\) 0 0
\(901\) −21.0237 −0.700400
\(902\) 7.66923 0.255358
\(903\) 1.72251 0.0573215
\(904\) −34.7244 −1.15492
\(905\) 0 0
\(906\) 56.8020 1.88712
\(907\) 45.9112 1.52446 0.762229 0.647308i \(-0.224105\pi\)
0.762229 + 0.647308i \(0.224105\pi\)
\(908\) 1.57810 0.0523712
\(909\) −25.1088 −0.832806
\(910\) 0 0
\(911\) 55.7475 1.84700 0.923499 0.383600i \(-0.125316\pi\)
0.923499 + 0.383600i \(0.125316\pi\)
\(912\) 57.6251 1.90816
\(913\) −10.1028 −0.334355
\(914\) 41.8576 1.38453
\(915\) 0 0
\(916\) 0.283465 0.00936593
\(917\) 8.43368 0.278505
\(918\) 7.29814 0.240874
\(919\) −49.0675 −1.61859 −0.809293 0.587405i \(-0.800150\pi\)
−0.809293 + 0.587405i \(0.800150\pi\)
\(920\) 0 0
\(921\) 13.9752 0.460498
\(922\) 16.9024 0.556650
\(923\) −26.1856 −0.861911
\(924\) 1.75898 0.0578661
\(925\) 0 0
\(926\) −27.2903 −0.896815
\(927\) −5.16686 −0.169702
\(928\) −3.37387 −0.110753
\(929\) −0.564626 −0.0185248 −0.00926238 0.999957i \(-0.502948\pi\)
−0.00926238 + 0.999957i \(0.502948\pi\)
\(930\) 0 0
\(931\) −42.1215 −1.38048
\(932\) 1.15764 0.0379198
\(933\) −67.1210 −2.19744
\(934\) −0.923055 −0.0302033
\(935\) 0 0
\(936\) 33.6509 1.09991
\(937\) −21.4710 −0.701426 −0.350713 0.936483i \(-0.614061\pi\)
−0.350713 + 0.936483i \(0.614061\pi\)
\(938\) −15.5911 −0.509069
\(939\) 42.1851 1.37666
\(940\) 0 0
\(941\) 32.9258 1.07335 0.536676 0.843789i \(-0.319680\pi\)
0.536676 + 0.843789i \(0.319680\pi\)
\(942\) −25.9653 −0.845996
\(943\) −4.07725 −0.132773
\(944\) 51.1146 1.66364
\(945\) 0 0
\(946\) −0.556475 −0.0180926
\(947\) 46.4200 1.50845 0.754224 0.656617i \(-0.228013\pi\)
0.754224 + 0.656617i \(0.228013\pi\)
\(948\) −2.83764 −0.0921621
\(949\) −78.6408 −2.55279
\(950\) 0 0
\(951\) 21.2866 0.690266
\(952\) −19.5377 −0.633219
\(953\) −40.6651 −1.31727 −0.658636 0.752462i \(-0.728866\pi\)
−0.658636 + 0.752462i \(0.728866\pi\)
\(954\) 28.8353 0.933577
\(955\) 0 0
\(956\) −2.85705 −0.0924036
\(957\) 19.7854 0.639571
\(958\) −42.6135 −1.37678
\(959\) −57.8363 −1.86763
\(960\) 0 0
\(961\) −27.2972 −0.880554
\(962\) 26.7422 0.862202
\(963\) 2.69755 0.0869274
\(964\) −1.88047 −0.0605657
\(965\) 0 0
\(966\) −16.5721 −0.533197
\(967\) −13.7463 −0.442051 −0.221026 0.975268i \(-0.570940\pi\)
−0.221026 + 0.975268i \(0.570940\pi\)
\(968\) −21.1802 −0.680757
\(969\) −26.2276 −0.842553
\(970\) 0 0
\(971\) −10.4871 −0.336548 −0.168274 0.985740i \(-0.553819\pi\)
−0.168274 + 0.985740i \(0.553819\pi\)
\(972\) −1.99068 −0.0638512
\(973\) 79.7954 2.55812
\(974\) 13.0524 0.418226
\(975\) 0 0
\(976\) −43.0929 −1.37937
\(977\) −26.8721 −0.859716 −0.429858 0.902897i \(-0.641436\pi\)
−0.429858 + 0.902897i \(0.641436\pi\)
\(978\) 73.4941 2.35008
\(979\) 1.80647 0.0577351
\(980\) 0 0
\(981\) 17.0347 0.543875
\(982\) −31.2434 −0.997018
\(983\) −5.47408 −0.174596 −0.0872980 0.996182i \(-0.527823\pi\)
−0.0872980 + 0.996182i \(0.527823\pi\)
\(984\) 17.5111 0.558232
\(985\) 0 0
\(986\) 13.9786 0.445168
\(987\) −78.2437 −2.49052
\(988\) −5.04629 −0.160544
\(989\) 0.295843 0.00940725
\(990\) 0 0
\(991\) 25.1620 0.799296 0.399648 0.916669i \(-0.369132\pi\)
0.399648 + 0.916669i \(0.369132\pi\)
\(992\) 1.30024 0.0412827
\(993\) −63.4534 −2.01363
\(994\) −20.8524 −0.661397
\(995\) 0 0
\(996\) 1.46726 0.0464920
\(997\) −11.3414 −0.359186 −0.179593 0.983741i \(-0.557478\pi\)
−0.179593 + 0.983741i \(0.557478\pi\)
\(998\) −31.8578 −1.00844
\(999\) 7.05703 0.223274
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 2225.2.a.i.1.2 4
5.4 even 2 445.2.a.d.1.3 4
15.14 odd 2 4005.2.a.l.1.2 4
20.19 odd 2 7120.2.a.bc.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.d.1.3 4 5.4 even 2
2225.2.a.i.1.2 4 1.1 even 1 trivial
4005.2.a.l.1.2 4 15.14 odd 2
7120.2.a.bc.1.4 4 20.19 odd 2