Properties

Label 9025.2.a.s.1.2
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $2$
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] = [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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 361)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} -2.85410 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} +0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} -2.85410 q^{9} -1.61803 q^{11} +0.236068 q^{12} -1.00000 q^{13} -4.85410 q^{14} -4.85410 q^{16} -0.763932 q^{17} -4.61803 q^{18} -1.14590 q^{21} -2.61803 q^{22} -5.38197 q^{23} -0.854102 q^{24} -1.61803 q^{26} -2.23607 q^{27} -1.85410 q^{28} +3.61803 q^{29} +8.85410 q^{31} -3.38197 q^{32} -0.618034 q^{33} -1.23607 q^{34} -1.76393 q^{36} +8.85410 q^{37} -0.381966 q^{39} +3.00000 q^{41} -1.85410 q^{42} +0.145898 q^{43} -1.00000 q^{44} -8.70820 q^{46} -3.00000 q^{47} -1.85410 q^{48} +2.00000 q^{49} -0.291796 q^{51} -0.618034 q^{52} -6.32624 q^{53} -3.61803 q^{54} +6.70820 q^{56} +5.85410 q^{58} -0.326238 q^{59} -10.2361 q^{61} +14.3262 q^{62} +8.56231 q^{63} +4.23607 q^{64} -1.00000 q^{66} -7.00000 q^{67} -0.472136 q^{68} -2.05573 q^{69} +7.47214 q^{71} +6.38197 q^{72} +2.70820 q^{73} +14.3262 q^{74} +4.85410 q^{77} -0.618034 q^{78} -13.4164 q^{79} +7.70820 q^{81} +4.85410 q^{82} -8.47214 q^{83} -0.708204 q^{84} +0.236068 q^{86} +1.38197 q^{87} +3.61803 q^{88} +7.76393 q^{89} +3.00000 q^{91} -3.32624 q^{92} +3.38197 q^{93} -4.85410 q^{94} -1.29180 q^{96} +13.8541 q^{97} +3.23607 q^{98} +4.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} - 7 q^{18} - 9 q^{21} - 3 q^{22} - 13 q^{23} + 5 q^{24} - q^{26} + 3 q^{28} + 5 q^{29} + 11 q^{31} - 9 q^{32} + q^{33} + 2 q^{34} - 8 q^{36} + 11 q^{37} - 3 q^{39} + 6 q^{41} + 3 q^{42} + 7 q^{43} - 2 q^{44} - 4 q^{46} - 6 q^{47} + 3 q^{48} + 4 q^{49} - 14 q^{51} + q^{52} + 3 q^{53} - 5 q^{54} + 5 q^{58} + 15 q^{59} - 16 q^{61} + 13 q^{62} - 3 q^{63} + 4 q^{64} - 2 q^{66} - 14 q^{67} + 8 q^{68} - 22 q^{69} + 6 q^{71} + 15 q^{72} - 8 q^{73} + 13 q^{74} + 3 q^{77} + q^{78} + 2 q^{81} + 3 q^{82} - 8 q^{83} + 12 q^{84} - 4 q^{86} + 5 q^{87} + 5 q^{88} + 20 q^{89} + 6 q^{91} + 9 q^{92} + 9 q^{93} - 3 q^{94} - 16 q^{96} + 21 q^{97} + 2 q^{98} + 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0.381966 0.220528 0.110264 0.993902i \(-0.464830\pi\)
0.110264 + 0.993902i \(0.464830\pi\)
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −2.23607 −0.790569
\(9\) −2.85410 −0.951367
\(10\) 0 0
\(11\) −1.61803 −0.487856 −0.243928 0.969793i \(-0.578436\pi\)
−0.243928 + 0.969793i \(0.578436\pi\)
\(12\) 0.236068 0.0681470
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −4.85410 −1.29731
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) −4.61803 −1.08848
\(19\) 0 0
\(20\) 0 0
\(21\) −1.14590 −0.250055
\(22\) −2.61803 −0.558167
\(23\) −5.38197 −1.12222 −0.561109 0.827742i \(-0.689625\pi\)
−0.561109 + 0.827742i \(0.689625\pi\)
\(24\) −0.854102 −0.174343
\(25\) 0 0
\(26\) −1.61803 −0.317323
\(27\) −2.23607 −0.430331
\(28\) −1.85410 −0.350392
\(29\) 3.61803 0.671852 0.335926 0.941888i \(-0.390951\pi\)
0.335926 + 0.941888i \(0.390951\pi\)
\(30\) 0 0
\(31\) 8.85410 1.59024 0.795122 0.606450i \(-0.207407\pi\)
0.795122 + 0.606450i \(0.207407\pi\)
\(32\) −3.38197 −0.597853
\(33\) −0.618034 −0.107586
\(34\) −1.23607 −0.211984
\(35\) 0 0
\(36\) −1.76393 −0.293989
\(37\) 8.85410 1.45561 0.727803 0.685787i \(-0.240542\pi\)
0.727803 + 0.685787i \(0.240542\pi\)
\(38\) 0 0
\(39\) −0.381966 −0.0611635
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −1.85410 −0.286094
\(43\) 0.145898 0.0222492 0.0111246 0.999938i \(-0.496459\pi\)
0.0111246 + 0.999938i \(0.496459\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.70820 −1.28395
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −1.85410 −0.267617
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −0.291796 −0.0408596
\(52\) −0.618034 −0.0857059
\(53\) −6.32624 −0.868976 −0.434488 0.900678i \(-0.643071\pi\)
−0.434488 + 0.900678i \(0.643071\pi\)
\(54\) −3.61803 −0.492352
\(55\) 0 0
\(56\) 6.70820 0.896421
\(57\) 0 0
\(58\) 5.85410 0.768681
\(59\) −0.326238 −0.0424726 −0.0212363 0.999774i \(-0.506760\pi\)
−0.0212363 + 0.999774i \(0.506760\pi\)
\(60\) 0 0
\(61\) −10.2361 −1.31059 −0.655297 0.755371i \(-0.727457\pi\)
−0.655297 + 0.755371i \(0.727457\pi\)
\(62\) 14.3262 1.81943
\(63\) 8.56231 1.07875
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −0.472136 −0.0572549
\(69\) −2.05573 −0.247481
\(70\) 0 0
\(71\) 7.47214 0.886779 0.443390 0.896329i \(-0.353776\pi\)
0.443390 + 0.896329i \(0.353776\pi\)
\(72\) 6.38197 0.752122
\(73\) 2.70820 0.316971 0.158486 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(74\) 14.3262 1.66539
\(75\) 0 0
\(76\) 0 0
\(77\) 4.85410 0.553176
\(78\) −0.618034 −0.0699786
\(79\) −13.4164 −1.50946 −0.754732 0.656033i \(-0.772233\pi\)
−0.754732 + 0.656033i \(0.772233\pi\)
\(80\) 0 0
\(81\) 7.70820 0.856467
\(82\) 4.85410 0.536046
\(83\) −8.47214 −0.929938 −0.464969 0.885327i \(-0.653934\pi\)
−0.464969 + 0.885327i \(0.653934\pi\)
\(84\) −0.708204 −0.0772714
\(85\) 0 0
\(86\) 0.236068 0.0254559
\(87\) 1.38197 0.148162
\(88\) 3.61803 0.385684
\(89\) 7.76393 0.822975 0.411488 0.911415i \(-0.365009\pi\)
0.411488 + 0.911415i \(0.365009\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) −3.32624 −0.346784
\(93\) 3.38197 0.350694
\(94\) −4.85410 −0.500662
\(95\) 0 0
\(96\) −1.29180 −0.131843
\(97\) 13.8541 1.40667 0.703335 0.710858i \(-0.251693\pi\)
0.703335 + 0.710858i \(0.251693\pi\)
\(98\) 3.23607 0.326892
\(99\) 4.61803 0.464130
\(100\) 0 0
\(101\) −9.18034 −0.913478 −0.456739 0.889601i \(-0.650983\pi\)
−0.456739 + 0.889601i \(0.650983\pi\)
\(102\) −0.472136 −0.0467484
\(103\) 14.3262 1.41161 0.705803 0.708408i \(-0.250586\pi\)
0.705803 + 0.708408i \(0.250586\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) −10.2361 −0.994215
\(107\) 16.4164 1.58703 0.793517 0.608548i \(-0.208248\pi\)
0.793517 + 0.608548i \(0.208248\pi\)
\(108\) −1.38197 −0.132980
\(109\) 3.29180 0.315297 0.157648 0.987495i \(-0.449609\pi\)
0.157648 + 0.987495i \(0.449609\pi\)
\(110\) 0 0
\(111\) 3.38197 0.321002
\(112\) 14.5623 1.37601
\(113\) 6.76393 0.636297 0.318149 0.948041i \(-0.396939\pi\)
0.318149 + 0.948041i \(0.396939\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.23607 0.207614
\(117\) 2.85410 0.263862
\(118\) −0.527864 −0.0485938
\(119\) 2.29180 0.210089
\(120\) 0 0
\(121\) −8.38197 −0.761997
\(122\) −16.5623 −1.49948
\(123\) 1.14590 0.103322
\(124\) 5.47214 0.491412
\(125\) 0 0
\(126\) 13.8541 1.23422
\(127\) 5.76393 0.511466 0.255733 0.966747i \(-0.417683\pi\)
0.255733 + 0.966747i \(0.417683\pi\)
\(128\) 13.6180 1.20368
\(129\) 0.0557281 0.00490658
\(130\) 0 0
\(131\) 15.0902 1.31843 0.659217 0.751953i \(-0.270888\pi\)
0.659217 + 0.751953i \(0.270888\pi\)
\(132\) −0.381966 −0.0332459
\(133\) 0 0
\(134\) −11.3262 −0.978438
\(135\) 0 0
\(136\) 1.70820 0.146477
\(137\) −7.47214 −0.638388 −0.319194 0.947689i \(-0.603412\pi\)
−0.319194 + 0.947689i \(0.603412\pi\)
\(138\) −3.32624 −0.283148
\(139\) 14.7984 1.25518 0.627591 0.778543i \(-0.284041\pi\)
0.627591 + 0.778543i \(0.284041\pi\)
\(140\) 0 0
\(141\) −1.14590 −0.0965020
\(142\) 12.0902 1.01458
\(143\) 1.61803 0.135307
\(144\) 13.8541 1.15451
\(145\) 0 0
\(146\) 4.38197 0.362654
\(147\) 0.763932 0.0630081
\(148\) 5.47214 0.449807
\(149\) −1.90983 −0.156459 −0.0782297 0.996935i \(-0.524927\pi\)
−0.0782297 + 0.996935i \(0.524927\pi\)
\(150\) 0 0
\(151\) 21.0902 1.71629 0.858147 0.513404i \(-0.171616\pi\)
0.858147 + 0.513404i \(0.171616\pi\)
\(152\) 0 0
\(153\) 2.18034 0.176270
\(154\) 7.85410 0.632902
\(155\) 0 0
\(156\) −0.236068 −0.0189006
\(157\) 17.8541 1.42491 0.712456 0.701717i \(-0.247583\pi\)
0.712456 + 0.701717i \(0.247583\pi\)
\(158\) −21.7082 −1.72701
\(159\) −2.41641 −0.191634
\(160\) 0 0
\(161\) 16.1459 1.27248
\(162\) 12.4721 0.979904
\(163\) −1.76393 −0.138162 −0.0690809 0.997611i \(-0.522007\pi\)
−0.0690809 + 0.997611i \(0.522007\pi\)
\(164\) 1.85410 0.144781
\(165\) 0 0
\(166\) −13.7082 −1.06396
\(167\) 20.2361 1.56591 0.782957 0.622076i \(-0.213710\pi\)
0.782957 + 0.622076i \(0.213710\pi\)
\(168\) 2.56231 0.197686
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0.0901699 0.00687539
\(173\) −0.472136 −0.0358958 −0.0179479 0.999839i \(-0.505713\pi\)
−0.0179479 + 0.999839i \(0.505713\pi\)
\(174\) 2.23607 0.169516
\(175\) 0 0
\(176\) 7.85410 0.592025
\(177\) −0.124612 −0.00936640
\(178\) 12.5623 0.941585
\(179\) −12.2361 −0.914567 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 4.85410 0.359810
\(183\) −3.90983 −0.289023
\(184\) 12.0344 0.887191
\(185\) 0 0
\(186\) 5.47214 0.401236
\(187\) 1.23607 0.0903902
\(188\) −1.85410 −0.135224
\(189\) 6.70820 0.487950
\(190\) 0 0
\(191\) 14.2361 1.03009 0.515043 0.857164i \(-0.327776\pi\)
0.515043 + 0.857164i \(0.327776\pi\)
\(192\) 1.61803 0.116772
\(193\) 5.05573 0.363919 0.181960 0.983306i \(-0.441756\pi\)
0.181960 + 0.983306i \(0.441756\pi\)
\(194\) 22.4164 1.60940
\(195\) 0 0
\(196\) 1.23607 0.0882906
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 7.47214 0.531022
\(199\) −13.4164 −0.951064 −0.475532 0.879698i \(-0.657744\pi\)
−0.475532 + 0.879698i \(0.657744\pi\)
\(200\) 0 0
\(201\) −2.67376 −0.188593
\(202\) −14.8541 −1.04513
\(203\) −10.8541 −0.761809
\(204\) −0.180340 −0.0126263
\(205\) 0 0
\(206\) 23.1803 1.61505
\(207\) 15.3607 1.06764
\(208\) 4.85410 0.336571
\(209\) 0 0
\(210\) 0 0
\(211\) 3.85410 0.265327 0.132664 0.991161i \(-0.457647\pi\)
0.132664 + 0.991161i \(0.457647\pi\)
\(212\) −3.90983 −0.268528
\(213\) 2.85410 0.195560
\(214\) 26.5623 1.81576
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) −26.5623 −1.80317
\(218\) 5.32624 0.360738
\(219\) 1.03444 0.0699011
\(220\) 0 0
\(221\) 0.763932 0.0513876
\(222\) 5.47214 0.367266
\(223\) −11.6525 −0.780307 −0.390154 0.920750i \(-0.627578\pi\)
−0.390154 + 0.920750i \(0.627578\pi\)
\(224\) 10.1459 0.677901
\(225\) 0 0
\(226\) 10.9443 0.728002
\(227\) 16.4164 1.08960 0.544798 0.838568i \(-0.316606\pi\)
0.544798 + 0.838568i \(0.316606\pi\)
\(228\) 0 0
\(229\) −13.6180 −0.899905 −0.449953 0.893052i \(-0.648559\pi\)
−0.449953 + 0.893052i \(0.648559\pi\)
\(230\) 0 0
\(231\) 1.85410 0.121991
\(232\) −8.09017 −0.531146
\(233\) −4.52786 −0.296630 −0.148315 0.988940i \(-0.547385\pi\)
−0.148315 + 0.988940i \(0.547385\pi\)
\(234\) 4.61803 0.301890
\(235\) 0 0
\(236\) −0.201626 −0.0131247
\(237\) −5.12461 −0.332879
\(238\) 3.70820 0.240367
\(239\) −0.326238 −0.0211026 −0.0105513 0.999944i \(-0.503359\pi\)
−0.0105513 + 0.999944i \(0.503359\pi\)
\(240\) 0 0
\(241\) −3.18034 −0.204864 −0.102432 0.994740i \(-0.532662\pi\)
−0.102432 + 0.994740i \(0.532662\pi\)
\(242\) −13.5623 −0.871818
\(243\) 9.65248 0.619207
\(244\) −6.32624 −0.404996
\(245\) 0 0
\(246\) 1.85410 0.118213
\(247\) 0 0
\(248\) −19.7984 −1.25720
\(249\) −3.23607 −0.205077
\(250\) 0 0
\(251\) −25.3607 −1.60075 −0.800376 0.599498i \(-0.795367\pi\)
−0.800376 + 0.599498i \(0.795367\pi\)
\(252\) 5.29180 0.333352
\(253\) 8.70820 0.547480
\(254\) 9.32624 0.585180
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 20.3607 1.27006 0.635032 0.772486i \(-0.280987\pi\)
0.635032 + 0.772486i \(0.280987\pi\)
\(258\) 0.0901699 0.00561374
\(259\) −26.5623 −1.65050
\(260\) 0 0
\(261\) −10.3262 −0.639178
\(262\) 24.4164 1.50845
\(263\) 14.9443 0.921503 0.460752 0.887529i \(-0.347580\pi\)
0.460752 + 0.887529i \(0.347580\pi\)
\(264\) 1.38197 0.0850541
\(265\) 0 0
\(266\) 0 0
\(267\) 2.96556 0.181489
\(268\) −4.32624 −0.264267
\(269\) −30.3262 −1.84902 −0.924512 0.381154i \(-0.875527\pi\)
−0.924512 + 0.381154i \(0.875527\pi\)
\(270\) 0 0
\(271\) 1.14590 0.0696083 0.0348042 0.999394i \(-0.488919\pi\)
0.0348042 + 0.999394i \(0.488919\pi\)
\(272\) 3.70820 0.224843
\(273\) 1.14590 0.0693529
\(274\) −12.0902 −0.730394
\(275\) 0 0
\(276\) −1.27051 −0.0764757
\(277\) −11.4164 −0.685945 −0.342973 0.939345i \(-0.611434\pi\)
−0.342973 + 0.939345i \(0.611434\pi\)
\(278\) 23.9443 1.43608
\(279\) −25.2705 −1.51291
\(280\) 0 0
\(281\) 29.5066 1.76021 0.880107 0.474775i \(-0.157470\pi\)
0.880107 + 0.474775i \(0.157470\pi\)
\(282\) −1.85410 −0.110410
\(283\) −26.0344 −1.54759 −0.773793 0.633438i \(-0.781643\pi\)
−0.773793 + 0.633438i \(0.781643\pi\)
\(284\) 4.61803 0.274030
\(285\) 0 0
\(286\) 2.61803 0.154808
\(287\) −9.00000 −0.531253
\(288\) 9.65248 0.568778
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) 5.29180 0.310211
\(292\) 1.67376 0.0979495
\(293\) −10.1459 −0.592730 −0.296365 0.955075i \(-0.595774\pi\)
−0.296365 + 0.955075i \(0.595774\pi\)
\(294\) 1.23607 0.0720889
\(295\) 0 0
\(296\) −19.7984 −1.15076
\(297\) 3.61803 0.209940
\(298\) −3.09017 −0.179009
\(299\) 5.38197 0.311247
\(300\) 0 0
\(301\) −0.437694 −0.0252283
\(302\) 34.1246 1.96365
\(303\) −3.50658 −0.201448
\(304\) 0 0
\(305\) 0 0
\(306\) 3.52786 0.201675
\(307\) −17.3262 −0.988861 −0.494430 0.869217i \(-0.664623\pi\)
−0.494430 + 0.869217i \(0.664623\pi\)
\(308\) 3.00000 0.170941
\(309\) 5.47214 0.311299
\(310\) 0 0
\(311\) 12.6525 0.717456 0.358728 0.933442i \(-0.383211\pi\)
0.358728 + 0.933442i \(0.383211\pi\)
\(312\) 0.854102 0.0483540
\(313\) 11.3262 0.640197 0.320098 0.947384i \(-0.396284\pi\)
0.320098 + 0.947384i \(0.396284\pi\)
\(314\) 28.8885 1.63027
\(315\) 0 0
\(316\) −8.29180 −0.466450
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −3.90983 −0.219252
\(319\) −5.85410 −0.327767
\(320\) 0 0
\(321\) 6.27051 0.349986
\(322\) 26.1246 1.45587
\(323\) 0 0
\(324\) 4.76393 0.264663
\(325\) 0 0
\(326\) −2.85410 −0.158074
\(327\) 1.25735 0.0695318
\(328\) −6.70820 −0.370399
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −10.9443 −0.601552 −0.300776 0.953695i \(-0.597246\pi\)
−0.300776 + 0.953695i \(0.597246\pi\)
\(332\) −5.23607 −0.287367
\(333\) −25.2705 −1.38482
\(334\) 32.7426 1.79160
\(335\) 0 0
\(336\) 5.56231 0.303449
\(337\) −17.1246 −0.932837 −0.466419 0.884564i \(-0.654456\pi\)
−0.466419 + 0.884564i \(0.654456\pi\)
\(338\) −19.4164 −1.05611
\(339\) 2.58359 0.140321
\(340\) 0 0
\(341\) −14.3262 −0.775809
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) −0.326238 −0.0175896
\(345\) 0 0
\(346\) −0.763932 −0.0410692
\(347\) 25.4164 1.36442 0.682212 0.731154i \(-0.261018\pi\)
0.682212 + 0.731154i \(0.261018\pi\)
\(348\) 0.854102 0.0457847
\(349\) −20.9787 −1.12296 −0.561482 0.827489i \(-0.689769\pi\)
−0.561482 + 0.827489i \(0.689769\pi\)
\(350\) 0 0
\(351\) 2.23607 0.119352
\(352\) 5.47214 0.291666
\(353\) −24.4508 −1.30139 −0.650694 0.759340i \(-0.725522\pi\)
−0.650694 + 0.759340i \(0.725522\pi\)
\(354\) −0.201626 −0.0107163
\(355\) 0 0
\(356\) 4.79837 0.254313
\(357\) 0.875388 0.0463305
\(358\) −19.7984 −1.04638
\(359\) 7.03444 0.371264 0.185632 0.982619i \(-0.440567\pi\)
0.185632 + 0.982619i \(0.440567\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −19.4164 −1.02050
\(363\) −3.20163 −0.168042
\(364\) 1.85410 0.0971813
\(365\) 0 0
\(366\) −6.32624 −0.330678
\(367\) 15.9443 0.832284 0.416142 0.909300i \(-0.363382\pi\)
0.416142 + 0.909300i \(0.363382\pi\)
\(368\) 26.1246 1.36184
\(369\) −8.56231 −0.445736
\(370\) 0 0
\(371\) 18.9787 0.985326
\(372\) 2.09017 0.108370
\(373\) −5.47214 −0.283336 −0.141668 0.989914i \(-0.545247\pi\)
−0.141668 + 0.989914i \(0.545247\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 6.70820 0.345949
\(377\) −3.61803 −0.186338
\(378\) 10.8541 0.558275
\(379\) 15.1246 0.776899 0.388450 0.921470i \(-0.373011\pi\)
0.388450 + 0.921470i \(0.373011\pi\)
\(380\) 0 0
\(381\) 2.20163 0.112793
\(382\) 23.0344 1.17854
\(383\) 0.381966 0.0195176 0.00975878 0.999952i \(-0.496894\pi\)
0.00975878 + 0.999952i \(0.496894\pi\)
\(384\) 5.20163 0.265444
\(385\) 0 0
\(386\) 8.18034 0.416368
\(387\) −0.416408 −0.0211672
\(388\) 8.56231 0.434685
\(389\) 9.27051 0.470034 0.235017 0.971991i \(-0.424485\pi\)
0.235017 + 0.971991i \(0.424485\pi\)
\(390\) 0 0
\(391\) 4.11146 0.207925
\(392\) −4.47214 −0.225877
\(393\) 5.76393 0.290752
\(394\) −4.85410 −0.244546
\(395\) 0 0
\(396\) 2.85410 0.143424
\(397\) 11.4721 0.575770 0.287885 0.957665i \(-0.407048\pi\)
0.287885 + 0.957665i \(0.407048\pi\)
\(398\) −21.7082 −1.08813
\(399\) 0 0
\(400\) 0 0
\(401\) 35.8885 1.79219 0.896094 0.443864i \(-0.146393\pi\)
0.896094 + 0.443864i \(0.146393\pi\)
\(402\) −4.32624 −0.215773
\(403\) −8.85410 −0.441054
\(404\) −5.67376 −0.282280
\(405\) 0 0
\(406\) −17.5623 −0.871603
\(407\) −14.3262 −0.710125
\(408\) 0.652476 0.0323024
\(409\) 8.29180 0.410003 0.205001 0.978762i \(-0.434280\pi\)
0.205001 + 0.978762i \(0.434280\pi\)
\(410\) 0 0
\(411\) −2.85410 −0.140782
\(412\) 8.85410 0.436210
\(413\) 0.978714 0.0481594
\(414\) 24.8541 1.22151
\(415\) 0 0
\(416\) 3.38197 0.165815
\(417\) 5.65248 0.276803
\(418\) 0 0
\(419\) −8.94427 −0.436956 −0.218478 0.975842i \(-0.570109\pi\)
−0.218478 + 0.975842i \(0.570109\pi\)
\(420\) 0 0
\(421\) 27.4721 1.33891 0.669455 0.742853i \(-0.266528\pi\)
0.669455 + 0.742853i \(0.266528\pi\)
\(422\) 6.23607 0.303567
\(423\) 8.56231 0.416314
\(424\) 14.1459 0.686986
\(425\) 0 0
\(426\) 4.61803 0.223744
\(427\) 30.7082 1.48607
\(428\) 10.1459 0.490420
\(429\) 0.618034 0.0298390
\(430\) 0 0
\(431\) −27.6525 −1.33197 −0.665986 0.745964i \(-0.731989\pi\)
−0.665986 + 0.745964i \(0.731989\pi\)
\(432\) 10.8541 0.522218
\(433\) −3.43769 −0.165205 −0.0826025 0.996583i \(-0.526323\pi\)
−0.0826025 + 0.996583i \(0.526323\pi\)
\(434\) −42.9787 −2.06304
\(435\) 0 0
\(436\) 2.03444 0.0974321
\(437\) 0 0
\(438\) 1.67376 0.0799754
\(439\) 34.5967 1.65121 0.825606 0.564247i \(-0.190833\pi\)
0.825606 + 0.564247i \(0.190833\pi\)
\(440\) 0 0
\(441\) −5.70820 −0.271819
\(442\) 1.23607 0.0587938
\(443\) 7.58359 0.360307 0.180154 0.983638i \(-0.442340\pi\)
0.180154 + 0.983638i \(0.442340\pi\)
\(444\) 2.09017 0.0991951
\(445\) 0 0
\(446\) −18.8541 −0.892768
\(447\) −0.729490 −0.0345037
\(448\) −12.7082 −0.600406
\(449\) −2.88854 −0.136319 −0.0681594 0.997674i \(-0.521713\pi\)
−0.0681594 + 0.997674i \(0.521713\pi\)
\(450\) 0 0
\(451\) −4.85410 −0.228571
\(452\) 4.18034 0.196627
\(453\) 8.05573 0.378491
\(454\) 26.5623 1.24663
\(455\) 0 0
\(456\) 0 0
\(457\) −19.7082 −0.921911 −0.460955 0.887423i \(-0.652493\pi\)
−0.460955 + 0.887423i \(0.652493\pi\)
\(458\) −22.0344 −1.02960
\(459\) 1.70820 0.0797321
\(460\) 0 0
\(461\) 20.9443 0.975472 0.487736 0.872991i \(-0.337823\pi\)
0.487736 + 0.872991i \(0.337823\pi\)
\(462\) 3.00000 0.139573
\(463\) −28.2705 −1.31384 −0.656921 0.753959i \(-0.728142\pi\)
−0.656921 + 0.753959i \(0.728142\pi\)
\(464\) −17.5623 −0.815310
\(465\) 0 0
\(466\) −7.32624 −0.339381
\(467\) 15.9443 0.737813 0.368906 0.929467i \(-0.379732\pi\)
0.368906 + 0.929467i \(0.379732\pi\)
\(468\) 1.76393 0.0815378
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) 6.81966 0.314233
\(472\) 0.729490 0.0335775
\(473\) −0.236068 −0.0108544
\(474\) −8.29180 −0.380855
\(475\) 0 0
\(476\) 1.41641 0.0649209
\(477\) 18.0557 0.826715
\(478\) −0.527864 −0.0241439
\(479\) 23.0902 1.05502 0.527508 0.849550i \(-0.323126\pi\)
0.527508 + 0.849550i \(0.323126\pi\)
\(480\) 0 0
\(481\) −8.85410 −0.403712
\(482\) −5.14590 −0.234389
\(483\) 6.16718 0.280617
\(484\) −5.18034 −0.235470
\(485\) 0 0
\(486\) 15.6180 0.708448
\(487\) 4.18034 0.189429 0.0947146 0.995504i \(-0.469806\pi\)
0.0947146 + 0.995504i \(0.469806\pi\)
\(488\) 22.8885 1.03612
\(489\) −0.673762 −0.0304686
\(490\) 0 0
\(491\) −21.2148 −0.957410 −0.478705 0.877976i \(-0.658894\pi\)
−0.478705 + 0.877976i \(0.658894\pi\)
\(492\) 0.708204 0.0319283
\(493\) −2.76393 −0.124481
\(494\) 0 0
\(495\) 0 0
\(496\) −42.9787 −1.92980
\(497\) −22.4164 −1.00551
\(498\) −5.23607 −0.234634
\(499\) 25.1246 1.12473 0.562366 0.826888i \(-0.309891\pi\)
0.562366 + 0.826888i \(0.309891\pi\)
\(500\) 0 0
\(501\) 7.72949 0.345328
\(502\) −41.0344 −1.83146
\(503\) −35.8328 −1.59771 −0.798853 0.601526i \(-0.794560\pi\)
−0.798853 + 0.601526i \(0.794560\pi\)
\(504\) −19.1459 −0.852826
\(505\) 0 0
\(506\) 14.0902 0.626384
\(507\) −4.58359 −0.203564
\(508\) 3.56231 0.158052
\(509\) −2.03444 −0.0901750 −0.0450875 0.998983i \(-0.514357\pi\)
−0.0450875 + 0.998983i \(0.514357\pi\)
\(510\) 0 0
\(511\) −8.12461 −0.359412
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) 32.9443 1.45311
\(515\) 0 0
\(516\) 0.0344419 0.00151622
\(517\) 4.85410 0.213483
\(518\) −42.9787 −1.88838
\(519\) −0.180340 −0.00791604
\(520\) 0 0
\(521\) −6.27051 −0.274716 −0.137358 0.990521i \(-0.543861\pi\)
−0.137358 + 0.990521i \(0.543861\pi\)
\(522\) −16.7082 −0.731298
\(523\) −4.41641 −0.193116 −0.0965580 0.995327i \(-0.530783\pi\)
−0.0965580 + 0.995327i \(0.530783\pi\)
\(524\) 9.32624 0.407419
\(525\) 0 0
\(526\) 24.1803 1.05431
\(527\) −6.76393 −0.294642
\(528\) 3.00000 0.130558
\(529\) 5.96556 0.259372
\(530\) 0 0
\(531\) 0.931116 0.0404070
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 4.79837 0.207646
\(535\) 0 0
\(536\) 15.6525 0.676084
\(537\) −4.67376 −0.201688
\(538\) −49.0689 −2.11551
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −29.8328 −1.28261 −0.641306 0.767285i \(-0.721607\pi\)
−0.641306 + 0.767285i \(0.721607\pi\)
\(542\) 1.85410 0.0796405
\(543\) −4.58359 −0.196701
\(544\) 2.58359 0.110771
\(545\) 0 0
\(546\) 1.85410 0.0793482
\(547\) −26.9230 −1.15114 −0.575572 0.817751i \(-0.695221\pi\)
−0.575572 + 0.817751i \(0.695221\pi\)
\(548\) −4.61803 −0.197273
\(549\) 29.2148 1.24686
\(550\) 0 0
\(551\) 0 0
\(552\) 4.59675 0.195651
\(553\) 40.2492 1.71157
\(554\) −18.4721 −0.784806
\(555\) 0 0
\(556\) 9.14590 0.387872
\(557\) 0.819660 0.0347301 0.0173651 0.999849i \(-0.494472\pi\)
0.0173651 + 0.999849i \(0.494472\pi\)
\(558\) −40.8885 −1.73095
\(559\) −0.145898 −0.00617083
\(560\) 0 0
\(561\) 0.472136 0.0199336
\(562\) 47.7426 2.01390
\(563\) −32.8328 −1.38374 −0.691869 0.722023i \(-0.743212\pi\)
−0.691869 + 0.722023i \(0.743212\pi\)
\(564\) −0.708204 −0.0298208
\(565\) 0 0
\(566\) −42.1246 −1.77063
\(567\) −23.1246 −0.971142
\(568\) −16.7082 −0.701061
\(569\) −16.9098 −0.708897 −0.354448 0.935076i \(-0.615331\pi\)
−0.354448 + 0.935076i \(0.615331\pi\)
\(570\) 0 0
\(571\) 6.67376 0.279288 0.139644 0.990202i \(-0.455404\pi\)
0.139644 + 0.990202i \(0.455404\pi\)
\(572\) 1.00000 0.0418121
\(573\) 5.43769 0.227163
\(574\) −14.5623 −0.607819
\(575\) 0 0
\(576\) −12.0902 −0.503757
\(577\) 12.1246 0.504754 0.252377 0.967629i \(-0.418788\pi\)
0.252377 + 0.967629i \(0.418788\pi\)
\(578\) −26.5623 −1.10485
\(579\) 1.93112 0.0802545
\(580\) 0 0
\(581\) 25.4164 1.05445
\(582\) 8.56231 0.354919
\(583\) 10.2361 0.423935
\(584\) −6.05573 −0.250588
\(585\) 0 0
\(586\) −16.4164 −0.678156
\(587\) 2.12461 0.0876921 0.0438461 0.999038i \(-0.486039\pi\)
0.0438461 + 0.999038i \(0.486039\pi\)
\(588\) 0.472136 0.0194706
\(589\) 0 0
\(590\) 0 0
\(591\) −1.14590 −0.0471359
\(592\) −42.9787 −1.76641
\(593\) −0.708204 −0.0290824 −0.0145412 0.999894i \(-0.504629\pi\)
−0.0145412 + 0.999894i \(0.504629\pi\)
\(594\) 5.85410 0.240197
\(595\) 0 0
\(596\) −1.18034 −0.0483486
\(597\) −5.12461 −0.209736
\(598\) 8.70820 0.356105
\(599\) −28.4164 −1.16106 −0.580531 0.814238i \(-0.697155\pi\)
−0.580531 + 0.814238i \(0.697155\pi\)
\(600\) 0 0
\(601\) −20.2918 −0.827720 −0.413860 0.910341i \(-0.635820\pi\)
−0.413860 + 0.910341i \(0.635820\pi\)
\(602\) −0.708204 −0.0288642
\(603\) 19.9787 0.813596
\(604\) 13.0344 0.530364
\(605\) 0 0
\(606\) −5.67376 −0.230481
\(607\) −6.27051 −0.254512 −0.127256 0.991870i \(-0.540617\pi\)
−0.127256 + 0.991870i \(0.540617\pi\)
\(608\) 0 0
\(609\) −4.14590 −0.168000
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 1.34752 0.0544704
\(613\) 19.9443 0.805542 0.402771 0.915301i \(-0.368047\pi\)
0.402771 + 0.915301i \(0.368047\pi\)
\(614\) −28.0344 −1.13138
\(615\) 0 0
\(616\) −10.8541 −0.437324
\(617\) −7.67376 −0.308934 −0.154467 0.987998i \(-0.549366\pi\)
−0.154467 + 0.987998i \(0.549366\pi\)
\(618\) 8.85410 0.356164
\(619\) 10.1246 0.406943 0.203471 0.979081i \(-0.434778\pi\)
0.203471 + 0.979081i \(0.434778\pi\)
\(620\) 0 0
\(621\) 12.0344 0.482926
\(622\) 20.4721 0.820858
\(623\) −23.2918 −0.933166
\(624\) 1.85410 0.0742235
\(625\) 0 0
\(626\) 18.3262 0.732464
\(627\) 0 0
\(628\) 11.0344 0.440322
\(629\) −6.76393 −0.269696
\(630\) 0 0
\(631\) 29.3607 1.16883 0.584415 0.811455i \(-0.301324\pi\)
0.584415 + 0.811455i \(0.301324\pi\)
\(632\) 30.0000 1.19334
\(633\) 1.47214 0.0585122
\(634\) 29.1246 1.15669
\(635\) 0 0
\(636\) −1.49342 −0.0592180
\(637\) −2.00000 −0.0792429
\(638\) −9.47214 −0.375005
\(639\) −21.3262 −0.843653
\(640\) 0 0
\(641\) 39.5066 1.56042 0.780208 0.625520i \(-0.215113\pi\)
0.780208 + 0.625520i \(0.215113\pi\)
\(642\) 10.1459 0.400427
\(643\) 24.2918 0.957975 0.478987 0.877822i \(-0.341004\pi\)
0.478987 + 0.877822i \(0.341004\pi\)
\(644\) 9.97871 0.393216
\(645\) 0 0
\(646\) 0 0
\(647\) −7.47214 −0.293760 −0.146880 0.989154i \(-0.546923\pi\)
−0.146880 + 0.989154i \(0.546923\pi\)
\(648\) −17.2361 −0.677097
\(649\) 0.527864 0.0207205
\(650\) 0 0
\(651\) −10.1459 −0.397649
\(652\) −1.09017 −0.0426944
\(653\) 23.5623 0.922064 0.461032 0.887383i \(-0.347479\pi\)
0.461032 + 0.887383i \(0.347479\pi\)
\(654\) 2.03444 0.0795530
\(655\) 0 0
\(656\) −14.5623 −0.568563
\(657\) −7.72949 −0.301556
\(658\) 14.5623 0.567698
\(659\) 25.7771 1.00413 0.502066 0.864829i \(-0.332573\pi\)
0.502066 + 0.864829i \(0.332573\pi\)
\(660\) 0 0
\(661\) −5.41641 −0.210674 −0.105337 0.994437i \(-0.533592\pi\)
−0.105337 + 0.994437i \(0.533592\pi\)
\(662\) −17.7082 −0.688249
\(663\) 0.291796 0.0113324
\(664\) 18.9443 0.735180
\(665\) 0 0
\(666\) −40.8885 −1.58440
\(667\) −19.4721 −0.753964
\(668\) 12.5066 0.483894
\(669\) −4.45085 −0.172080
\(670\) 0 0
\(671\) 16.5623 0.639381
\(672\) 3.87539 0.149496
\(673\) 34.1246 1.31541 0.657704 0.753277i \(-0.271528\pi\)
0.657704 + 0.753277i \(0.271528\pi\)
\(674\) −27.7082 −1.06728
\(675\) 0 0
\(676\) −7.41641 −0.285246
\(677\) −30.7426 −1.18154 −0.590768 0.806842i \(-0.701175\pi\)
−0.590768 + 0.806842i \(0.701175\pi\)
\(678\) 4.18034 0.160545
\(679\) −41.5623 −1.59501
\(680\) 0 0
\(681\) 6.27051 0.240286
\(682\) −23.1803 −0.887621
\(683\) 9.65248 0.369342 0.184671 0.982800i \(-0.440878\pi\)
0.184671 + 0.982800i \(0.440878\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 24.2705 0.926652
\(687\) −5.20163 −0.198454
\(688\) −0.708204 −0.0270000
\(689\) 6.32624 0.241010
\(690\) 0 0
\(691\) −39.1803 −1.49049 −0.745245 0.666791i \(-0.767668\pi\)
−0.745245 + 0.666791i \(0.767668\pi\)
\(692\) −0.291796 −0.0110924
\(693\) −13.8541 −0.526274
\(694\) 41.1246 1.56107
\(695\) 0 0
\(696\) −3.09017 −0.117133
\(697\) −2.29180 −0.0868080
\(698\) −33.9443 −1.28481
\(699\) −1.72949 −0.0654153
\(700\) 0 0
\(701\) −39.6312 −1.49685 −0.748425 0.663220i \(-0.769189\pi\)
−0.748425 + 0.663220i \(0.769189\pi\)
\(702\) 3.61803 0.136554
\(703\) 0 0
\(704\) −6.85410 −0.258324
\(705\) 0 0
\(706\) −39.5623 −1.48895
\(707\) 27.5410 1.03579
\(708\) −0.0770143 −0.00289438
\(709\) 16.5836 0.622810 0.311405 0.950277i \(-0.399200\pi\)
0.311405 + 0.950277i \(0.399200\pi\)
\(710\) 0 0
\(711\) 38.2918 1.43605
\(712\) −17.3607 −0.650619
\(713\) −47.6525 −1.78460
\(714\) 1.41641 0.0530077
\(715\) 0 0
\(716\) −7.56231 −0.282617
\(717\) −0.124612 −0.00465371
\(718\) 11.3820 0.424771
\(719\) 47.0344 1.75409 0.877044 0.480409i \(-0.159512\pi\)
0.877044 + 0.480409i \(0.159512\pi\)
\(720\) 0 0
\(721\) −42.9787 −1.60061
\(722\) 0 0
\(723\) −1.21478 −0.0451782
\(724\) −7.41641 −0.275629
\(725\) 0 0
\(726\) −5.18034 −0.192260
\(727\) 16.0689 0.595962 0.297981 0.954572i \(-0.403687\pi\)
0.297981 + 0.954572i \(0.403687\pi\)
\(728\) −6.70820 −0.248623
\(729\) −19.4377 −0.719915
\(730\) 0 0
\(731\) −0.111456 −0.00412236
\(732\) −2.41641 −0.0893130
\(733\) 52.9574 1.95603 0.978014 0.208541i \(-0.0668715\pi\)
0.978014 + 0.208541i \(0.0668715\pi\)
\(734\) 25.7984 0.952235
\(735\) 0 0
\(736\) 18.2016 0.670921
\(737\) 11.3262 0.417207
\(738\) −13.8541 −0.509977
\(739\) −25.0000 −0.919640 −0.459820 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 30.7082 1.12733
\(743\) −3.36068 −0.123291 −0.0616457 0.998098i \(-0.519635\pi\)
−0.0616457 + 0.998098i \(0.519635\pi\)
\(744\) −7.56231 −0.277248
\(745\) 0 0
\(746\) −8.85410 −0.324172
\(747\) 24.1803 0.884712
\(748\) 0.763932 0.0279321
\(749\) −49.2492 −1.79953
\(750\) 0 0
\(751\) 17.1459 0.625663 0.312831 0.949809i \(-0.398723\pi\)
0.312831 + 0.949809i \(0.398723\pi\)
\(752\) 14.5623 0.531033
\(753\) −9.68692 −0.353011
\(754\) −5.85410 −0.213194
\(755\) 0 0
\(756\) 4.14590 0.150785
\(757\) −26.7426 −0.971978 −0.485989 0.873965i \(-0.661540\pi\)
−0.485989 + 0.873965i \(0.661540\pi\)
\(758\) 24.4721 0.888868
\(759\) 3.32624 0.120735
\(760\) 0 0
\(761\) 4.88854 0.177210 0.0886048 0.996067i \(-0.471759\pi\)
0.0886048 + 0.996067i \(0.471759\pi\)
\(762\) 3.56231 0.129049
\(763\) −9.87539 −0.357513
\(764\) 8.79837 0.318314
\(765\) 0 0
\(766\) 0.618034 0.0223305
\(767\) 0.326238 0.0117798
\(768\) 5.18034 0.186929
\(769\) 36.6312 1.32095 0.660477 0.750846i \(-0.270354\pi\)
0.660477 + 0.750846i \(0.270354\pi\)
\(770\) 0 0
\(771\) 7.77709 0.280085
\(772\) 3.12461 0.112457
\(773\) −35.9230 −1.29206 −0.646030 0.763312i \(-0.723572\pi\)
−0.646030 + 0.763312i \(0.723572\pi\)
\(774\) −0.673762 −0.0242179
\(775\) 0 0
\(776\) −30.9787 −1.11207
\(777\) −10.1459 −0.363982
\(778\) 15.0000 0.537776
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0902 −0.432620
\(782\) 6.65248 0.237892
\(783\) −8.09017 −0.289119
\(784\) −9.70820 −0.346722
\(785\) 0 0
\(786\) 9.32624 0.332656
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) −1.85410 −0.0660496
\(789\) 5.70820 0.203217
\(790\) 0 0
\(791\) −20.2918 −0.721493
\(792\) −10.3262 −0.366927
\(793\) 10.2361 0.363493
\(794\) 18.5623 0.658752
\(795\) 0 0
\(796\) −8.29180 −0.293895
\(797\) −20.2918 −0.718772 −0.359386 0.933189i \(-0.617014\pi\)
−0.359386 + 0.933189i \(0.617014\pi\)
\(798\) 0 0
\(799\) 2.29180 0.0810779
\(800\) 0 0
\(801\) −22.1591 −0.782952
\(802\) 58.0689 2.05048
\(803\) −4.38197 −0.154636
\(804\) −1.65248 −0.0582783
\(805\) 0 0
\(806\) −14.3262 −0.504620
\(807\) −11.5836 −0.407762
\(808\) 20.5279 0.722168
\(809\) −24.7984 −0.871864 −0.435932 0.899980i \(-0.643581\pi\)
−0.435932 + 0.899980i \(0.643581\pi\)
\(810\) 0 0
\(811\) −22.9787 −0.806892 −0.403446 0.915004i \(-0.632188\pi\)
−0.403446 + 0.915004i \(0.632188\pi\)
\(812\) −6.70820 −0.235412
\(813\) 0.437694 0.0153506
\(814\) −23.1803 −0.812470
\(815\) 0 0
\(816\) 1.41641 0.0495842
\(817\) 0 0
\(818\) 13.4164 0.469094
\(819\) −8.56231 −0.299191
\(820\) 0 0
\(821\) 52.0476 1.81647 0.908237 0.418457i \(-0.137429\pi\)
0.908237 + 0.418457i \(0.137429\pi\)
\(822\) −4.61803 −0.161072
\(823\) 25.5967 0.892247 0.446123 0.894972i \(-0.352804\pi\)
0.446123 + 0.894972i \(0.352804\pi\)
\(824\) −32.0344 −1.11597
\(825\) 0 0
\(826\) 1.58359 0.0551002
\(827\) 32.5967 1.13350 0.566750 0.823890i \(-0.308201\pi\)
0.566750 + 0.823890i \(0.308201\pi\)
\(828\) 9.49342 0.329919
\(829\) −4.67376 −0.162326 −0.0811632 0.996701i \(-0.525864\pi\)
−0.0811632 + 0.996701i \(0.525864\pi\)
\(830\) 0 0
\(831\) −4.36068 −0.151270
\(832\) −4.23607 −0.146859
\(833\) −1.52786 −0.0529374
\(834\) 9.14590 0.316697
\(835\) 0 0
\(836\) 0 0
\(837\) −19.7984 −0.684332
\(838\) −14.4721 −0.499932
\(839\) −15.2016 −0.524818 −0.262409 0.964957i \(-0.584517\pi\)
−0.262409 + 0.964957i \(0.584517\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) 44.4508 1.53188
\(843\) 11.2705 0.388177
\(844\) 2.38197 0.0819907
\(845\) 0 0
\(846\) 13.8541 0.476314
\(847\) 25.1459 0.864023
\(848\) 30.7082 1.05452
\(849\) −9.94427 −0.341287
\(850\) 0 0
\(851\) −47.6525 −1.63351
\(852\) 1.76393 0.0604313
\(853\) −30.7082 −1.05143 −0.525714 0.850661i \(-0.676202\pi\)
−0.525714 + 0.850661i \(0.676202\pi\)
\(854\) 49.6869 1.70025
\(855\) 0 0
\(856\) −36.7082 −1.25466
\(857\) −10.6180 −0.362705 −0.181353 0.983418i \(-0.558048\pi\)
−0.181353 + 0.983418i \(0.558048\pi\)
\(858\) 1.00000 0.0341394
\(859\) −48.5410 −1.65620 −0.828099 0.560582i \(-0.810578\pi\)
−0.828099 + 0.560582i \(0.810578\pi\)
\(860\) 0 0
\(861\) −3.43769 −0.117156
\(862\) −44.7426 −1.52394
\(863\) −27.0557 −0.920988 −0.460494 0.887663i \(-0.652328\pi\)
−0.460494 + 0.887663i \(0.652328\pi\)
\(864\) 7.56231 0.257275
\(865\) 0 0
\(866\) −5.56231 −0.189015
\(867\) −6.27051 −0.212958
\(868\) −16.4164 −0.557209
\(869\) 21.7082 0.736400
\(870\) 0 0
\(871\) 7.00000 0.237186
\(872\) −7.36068 −0.249264
\(873\) −39.5410 −1.33826
\(874\) 0 0
\(875\) 0 0
\(876\) 0.639320 0.0216006
\(877\) 11.8197 0.399122 0.199561 0.979885i \(-0.436048\pi\)
0.199561 + 0.979885i \(0.436048\pi\)
\(878\) 55.9787 1.88919
\(879\) −3.87539 −0.130714
\(880\) 0 0
\(881\) 32.4508 1.09330 0.546648 0.837362i \(-0.315903\pi\)
0.546648 + 0.837362i \(0.315903\pi\)
\(882\) −9.23607 −0.310995
\(883\) 50.9230 1.71369 0.856847 0.515570i \(-0.172420\pi\)
0.856847 + 0.515570i \(0.172420\pi\)
\(884\) 0.472136 0.0158797
\(885\) 0 0
\(886\) 12.2705 0.412236
\(887\) 27.3475 0.918240 0.459120 0.888374i \(-0.348165\pi\)
0.459120 + 0.888374i \(0.348165\pi\)
\(888\) −7.56231 −0.253774
\(889\) −17.2918 −0.579948
\(890\) 0 0
\(891\) −12.4721 −0.417832
\(892\) −7.20163 −0.241128
\(893\) 0 0
\(894\) −1.18034 −0.0394765
\(895\) 0 0
\(896\) −40.8541 −1.36484
\(897\) 2.05573 0.0686388
\(898\) −4.67376 −0.155965
\(899\) 32.0344 1.06841
\(900\) 0 0
\(901\) 4.83282 0.161004
\(902\) −7.85410 −0.261513
\(903\) −0.167184 −0.00556354
\(904\) −15.1246 −0.503037
\(905\) 0 0
\(906\) 13.0344 0.433040
\(907\) −26.4721 −0.878993 −0.439496 0.898244i \(-0.644843\pi\)
−0.439496 + 0.898244i \(0.644843\pi\)
\(908\) 10.1459 0.336703
\(909\) 26.2016 0.869053
\(910\) 0 0
\(911\) −3.38197 −0.112050 −0.0560248 0.998429i \(-0.517843\pi\)
−0.0560248 + 0.998429i \(0.517843\pi\)
\(912\) 0 0
\(913\) 13.7082 0.453675
\(914\) −31.8885 −1.05478
\(915\) 0 0
\(916\) −8.41641 −0.278086
\(917\) −45.2705 −1.49496
\(918\) 2.76393 0.0912234
\(919\) 23.2918 0.768325 0.384163 0.923265i \(-0.374490\pi\)
0.384163 + 0.923265i \(0.374490\pi\)
\(920\) 0 0
\(921\) −6.61803 −0.218072
\(922\) 33.8885 1.11606
\(923\) −7.47214 −0.245948
\(924\) 1.14590 0.0376973
\(925\) 0 0
\(926\) −45.7426 −1.50320
\(927\) −40.8885 −1.34296
\(928\) −12.2361 −0.401669
\(929\) 16.3820 0.537475 0.268737 0.963213i \(-0.413394\pi\)
0.268737 + 0.963213i \(0.413394\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.79837 −0.0916638
\(933\) 4.83282 0.158219
\(934\) 25.7984 0.844149
\(935\) 0 0
\(936\) −6.38197 −0.208601
\(937\) −40.5623 −1.32511 −0.662556 0.749012i \(-0.730529\pi\)
−0.662556 + 0.749012i \(0.730529\pi\)
\(938\) 33.9787 1.10944
\(939\) 4.32624 0.141181
\(940\) 0 0
\(941\) 10.6869 0.348384 0.174192 0.984712i \(-0.444269\pi\)
0.174192 + 0.984712i \(0.444269\pi\)
\(942\) 11.0344 0.359522
\(943\) −16.1459 −0.525783
\(944\) 1.58359 0.0515415
\(945\) 0 0
\(946\) −0.381966 −0.0124188
\(947\) 32.6525 1.06106 0.530531 0.847665i \(-0.321992\pi\)
0.530531 + 0.847665i \(0.321992\pi\)
\(948\) −3.16718 −0.102865
\(949\) −2.70820 −0.0879120
\(950\) 0 0
\(951\) 6.87539 0.222950
\(952\) −5.12461 −0.166090
\(953\) 17.2918 0.560136 0.280068 0.959980i \(-0.409643\pi\)
0.280068 + 0.959980i \(0.409643\pi\)
\(954\) 29.2148 0.945863
\(955\) 0 0
\(956\) −0.201626 −0.00652105
\(957\) −2.23607 −0.0722818
\(958\) 37.3607 1.20707
\(959\) 22.4164 0.723864
\(960\) 0 0
\(961\) 47.3951 1.52887
\(962\) −14.3262 −0.461896
\(963\) −46.8541 −1.50985
\(964\) −1.96556 −0.0633064
\(965\) 0 0
\(966\) 9.97871 0.321060
\(967\) −6.54102 −0.210345 −0.105173 0.994454i \(-0.533539\pi\)
−0.105173 + 0.994454i \(0.533539\pi\)
\(968\) 18.7426 0.602411
\(969\) 0 0
\(970\) 0 0
\(971\) −23.5066 −0.754362 −0.377181 0.926140i \(-0.623107\pi\)
−0.377181 + 0.926140i \(0.623107\pi\)
\(972\) 5.96556 0.191345
\(973\) −44.3951 −1.42324
\(974\) 6.76393 0.216730
\(975\) 0 0
\(976\) 49.6869 1.59044
\(977\) 10.6393 0.340382 0.170191 0.985411i \(-0.445562\pi\)
0.170191 + 0.985411i \(0.445562\pi\)
\(978\) −1.09017 −0.0348598
\(979\) −12.5623 −0.401493
\(980\) 0 0
\(981\) −9.39512 −0.299963
\(982\) −34.3262 −1.09539
\(983\) 32.6180 1.04035 0.520177 0.854059i \(-0.325866\pi\)
0.520177 + 0.854059i \(0.325866\pi\)
\(984\) −2.56231 −0.0816833
\(985\) 0 0
\(986\) −4.47214 −0.142422
\(987\) 3.43769 0.109423
\(988\) 0 0
\(989\) −0.785218 −0.0249685
\(990\) 0 0
\(991\) −7.45085 −0.236684 −0.118342 0.992973i \(-0.537758\pi\)
−0.118342 + 0.992973i \(0.537758\pi\)
\(992\) −29.9443 −0.950732
\(993\) −4.18034 −0.132659
\(994\) −36.2705 −1.15043
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) −39.7082 −1.25757 −0.628786 0.777579i \(-0.716448\pi\)
−0.628786 + 0.777579i \(0.716448\pi\)
\(998\) 40.6525 1.28683
\(999\) −19.7984 −0.626393
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.s.1.2 2
5.4 even 2 361.2.a.c.1.1 2
15.14 odd 2 3249.2.a.o.1.2 2
19.18 odd 2 9025.2.a.n.1.1 2
20.19 odd 2 5776.2.a.bg.1.1 2
95.4 even 18 361.2.e.i.54.1 12
95.9 even 18 361.2.e.i.62.2 12
95.14 odd 18 361.2.e.j.234.2 12
95.24 even 18 361.2.e.i.234.1 12
95.29 odd 18 361.2.e.j.62.1 12
95.34 odd 18 361.2.e.j.54.2 12
95.44 even 18 361.2.e.i.245.1 12
95.49 even 6 361.2.c.g.292.2 4
95.54 even 18 361.2.e.i.28.1 12
95.59 odd 18 361.2.e.j.99.1 12
95.64 even 6 361.2.c.g.68.2 4
95.69 odd 6 361.2.c.d.68.1 4
95.74 even 18 361.2.e.i.99.2 12
95.79 odd 18 361.2.e.j.28.2 12
95.84 odd 6 361.2.c.d.292.1 4
95.89 odd 18 361.2.e.j.245.2 12
95.94 odd 2 361.2.a.f.1.2 yes 2
285.284 even 2 3249.2.a.i.1.1 2
380.379 even 2 5776.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
361.2.a.c.1.1 2 5.4 even 2
361.2.a.f.1.2 yes 2 95.94 odd 2
361.2.c.d.68.1 4 95.69 odd 6
361.2.c.d.292.1 4 95.84 odd 6
361.2.c.g.68.2 4 95.64 even 6
361.2.c.g.292.2 4 95.49 even 6
361.2.e.i.28.1 12 95.54 even 18
361.2.e.i.54.1 12 95.4 even 18
361.2.e.i.62.2 12 95.9 even 18
361.2.e.i.99.2 12 95.74 even 18
361.2.e.i.234.1 12 95.24 even 18
361.2.e.i.245.1 12 95.44 even 18
361.2.e.j.28.2 12 95.79 odd 18
361.2.e.j.54.2 12 95.34 odd 18
361.2.e.j.62.1 12 95.29 odd 18
361.2.e.j.99.1 12 95.59 odd 18
361.2.e.j.234.2 12 95.14 odd 18
361.2.e.j.245.2 12 95.89 odd 18
3249.2.a.i.1.1 2 285.284 even 2
3249.2.a.o.1.2 2 15.14 odd 2
5776.2.a.s.1.2 2 380.379 even 2
5776.2.a.bg.1.1 2 20.19 odd 2
9025.2.a.n.1.1 2 19.18 odd 2
9025.2.a.s.1.2 2 1.1 even 1 trivial