Properties

Label 961.2.a.h.1.1
Level $961$
Weight $2$
Character 961.1
Self dual yes
Analytic conductor $7.674$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,2,Mod(1,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.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: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.28825\) of defining polynomial
Character \(\chi\) \(=\) 961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.381966 q^{2} -2.28825 q^{3} -1.85410 q^{4} -2.23607 q^{5} -0.874032 q^{6} -1.00000 q^{7} -1.47214 q^{8} +2.23607 q^{9} -0.854102 q^{10} -4.24264 q^{11} +4.24264 q^{12} -6.86474 q^{13} -0.381966 q^{14} +5.11667 q^{15} +3.14590 q^{16} -0.540182 q^{17} +0.854102 q^{18} +1.00000 q^{19} +4.14590 q^{20} +2.28825 q^{21} -1.62054 q^{22} -6.86474 q^{23} +3.36861 q^{24} -2.62210 q^{26} +1.74806 q^{27} +1.85410 q^{28} +3.70246 q^{29} +1.95440 q^{30} +4.14590 q^{32} +9.70820 q^{33} -0.206331 q^{34} +2.23607 q^{35} -4.14590 q^{36} -4.24264 q^{37} +0.381966 q^{38} +15.7082 q^{39} +3.29180 q^{40} +7.47214 q^{41} +0.874032 q^{42} +0.206331 q^{43} +7.86629 q^{44} -5.00000 q^{45} -2.62210 q^{46} +3.70820 q^{47} -7.19859 q^{48} -6.00000 q^{49} +1.23607 q^{51} +12.7279 q^{52} -5.24419 q^{53} +0.667701 q^{54} +9.48683 q^{55} +1.47214 q^{56} -2.28825 q^{57} +1.41421 q^{58} -5.94427 q^{59} -9.48683 q^{60} -4.44897 q^{61} -2.23607 q^{63} -4.70820 q^{64} +15.3500 q^{65} +3.70820 q^{66} +6.00000 q^{67} +1.00155 q^{68} +15.7082 q^{69} +0.854102 q^{70} +7.47214 q^{71} -3.29180 q^{72} -4.24264 q^{73} -1.62054 q^{74} -1.85410 q^{76} +4.24264 q^{77} +6.00000 q^{78} +11.1074 q^{79} -7.03444 q^{80} -10.7082 q^{81} +2.85410 q^{82} -3.16228 q^{83} -4.24264 q^{84} +1.20788 q^{85} +0.0788114 q^{86} -8.47214 q^{87} +6.24574 q^{88} +5.86319 q^{89} -1.90983 q^{90} +6.86474 q^{91} +12.7279 q^{92} +1.41641 q^{94} -2.23607 q^{95} -9.48683 q^{96} -7.00000 q^{97} -2.29180 q^{98} -9.48683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 6 q^{4} - 4 q^{7} + 12 q^{8} + 10 q^{10} - 6 q^{14} + 26 q^{16} - 10 q^{18} + 4 q^{19} + 30 q^{20} - 6 q^{28} + 30 q^{32} + 12 q^{33} - 30 q^{36} + 6 q^{38} + 36 q^{39} + 40 q^{40} + 12 q^{41}+ \cdots - 36 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381966 0.270091 0.135045 0.990839i \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) −2.28825 −1.32112 −0.660560 0.750774i \(-0.729681\pi\)
−0.660560 + 0.750774i \(0.729681\pi\)
\(4\) −1.85410 −0.927051
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −0.874032 −0.356822
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.47214 −0.520479
\(9\) 2.23607 0.745356
\(10\) −0.854102 −0.270091
\(11\) −4.24264 −1.27920 −0.639602 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 4.24264 1.22474
\(13\) −6.86474 −1.90394 −0.951968 0.306198i \(-0.900943\pi\)
−0.951968 + 0.306198i \(0.900943\pi\)
\(14\) −0.381966 −0.102085
\(15\) 5.11667 1.32112
\(16\) 3.14590 0.786475
\(17\) −0.540182 −0.131013 −0.0655066 0.997852i \(-0.520866\pi\)
−0.0655066 + 0.997852i \(0.520866\pi\)
\(18\) 0.854102 0.201314
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 4.14590 0.927051
\(21\) 2.28825 0.499336
\(22\) −1.62054 −0.345501
\(23\) −6.86474 −1.43140 −0.715698 0.698410i \(-0.753891\pi\)
−0.715698 + 0.698410i \(0.753891\pi\)
\(24\) 3.36861 0.687614
\(25\) 0 0
\(26\) −2.62210 −0.514235
\(27\) 1.74806 0.336415
\(28\) 1.85410 0.350392
\(29\) 3.70246 0.687529 0.343765 0.939056i \(-0.388298\pi\)
0.343765 + 0.939056i \(0.388298\pi\)
\(30\) 1.95440 0.356822
\(31\) 0 0
\(32\) 4.14590 0.732898
\(33\) 9.70820 1.68998
\(34\) −0.206331 −0.0353855
\(35\) 2.23607 0.377964
\(36\) −4.14590 −0.690983
\(37\) −4.24264 −0.697486 −0.348743 0.937218i \(-0.613391\pi\)
−0.348743 + 0.937218i \(0.613391\pi\)
\(38\) 0.381966 0.0619631
\(39\) 15.7082 2.51533
\(40\) 3.29180 0.520479
\(41\) 7.47214 1.16695 0.583476 0.812131i \(-0.301692\pi\)
0.583476 + 0.812131i \(0.301692\pi\)
\(42\) 0.874032 0.134866
\(43\) 0.206331 0.0314652 0.0157326 0.999876i \(-0.494992\pi\)
0.0157326 + 0.999876i \(0.494992\pi\)
\(44\) 7.86629 1.18589
\(45\) −5.00000 −0.745356
\(46\) −2.62210 −0.386607
\(47\) 3.70820 0.540897 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) −7.19859 −1.03903
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.23607 0.173084
\(52\) 12.7279 1.76505
\(53\) −5.24419 −0.720345 −0.360173 0.932886i \(-0.617282\pi\)
−0.360173 + 0.932886i \(0.617282\pi\)
\(54\) 0.667701 0.0908626
\(55\) 9.48683 1.27920
\(56\) 1.47214 0.196722
\(57\) −2.28825 −0.303086
\(58\) 1.41421 0.185695
\(59\) −5.94427 −0.773878 −0.386939 0.922105i \(-0.626468\pi\)
−0.386939 + 0.922105i \(0.626468\pi\)
\(60\) −9.48683 −1.22474
\(61\) −4.44897 −0.569632 −0.284816 0.958582i \(-0.591933\pi\)
−0.284816 + 0.958582i \(0.591933\pi\)
\(62\) 0 0
\(63\) −2.23607 −0.281718
\(64\) −4.70820 −0.588525
\(65\) 15.3500 1.90394
\(66\) 3.70820 0.456448
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 1.00155 0.121456
\(69\) 15.7082 1.89105
\(70\) 0.854102 0.102085
\(71\) 7.47214 0.886779 0.443390 0.896329i \(-0.353776\pi\)
0.443390 + 0.896329i \(0.353776\pi\)
\(72\) −3.29180 −0.387942
\(73\) −4.24264 −0.496564 −0.248282 0.968688i \(-0.579866\pi\)
−0.248282 + 0.968688i \(0.579866\pi\)
\(74\) −1.62054 −0.188384
\(75\) 0 0
\(76\) −1.85410 −0.212680
\(77\) 4.24264 0.483494
\(78\) 6.00000 0.679366
\(79\) 11.1074 1.24968 0.624839 0.780754i \(-0.285165\pi\)
0.624839 + 0.780754i \(0.285165\pi\)
\(80\) −7.03444 −0.786475
\(81\) −10.7082 −1.18980
\(82\) 2.85410 0.315183
\(83\) −3.16228 −0.347105 −0.173553 0.984825i \(-0.555525\pi\)
−0.173553 + 0.984825i \(0.555525\pi\)
\(84\) −4.24264 −0.462910
\(85\) 1.20788 0.131013
\(86\) 0.0788114 0.00849845
\(87\) −8.47214 −0.908308
\(88\) 6.24574 0.665799
\(89\) 5.86319 0.621496 0.310748 0.950492i \(-0.399420\pi\)
0.310748 + 0.950492i \(0.399420\pi\)
\(90\) −1.90983 −0.201314
\(91\) 6.86474 0.719620
\(92\) 12.7279 1.32698
\(93\) 0 0
\(94\) 1.41641 0.146091
\(95\) −2.23607 −0.229416
\(96\) −9.48683 −0.968246
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −2.29180 −0.231506
\(99\) −9.48683 −0.953463
\(100\) 0 0
\(101\) −2.23607 −0.222497 −0.111249 0.993793i \(-0.535485\pi\)
−0.111249 + 0.993793i \(0.535485\pi\)
\(102\) 0.472136 0.0467484
\(103\) −2.70820 −0.266847 −0.133424 0.991059i \(-0.542597\pi\)
−0.133424 + 0.991059i \(0.542597\pi\)
\(104\) 10.1058 0.990958
\(105\) −5.11667 −0.499336
\(106\) −2.00310 −0.194559
\(107\) −7.47214 −0.722359 −0.361179 0.932496i \(-0.617626\pi\)
−0.361179 + 0.932496i \(0.617626\pi\)
\(108\) −3.24109 −0.311874
\(109\) −14.7082 −1.40879 −0.704395 0.709808i \(-0.748782\pi\)
−0.704395 + 0.709808i \(0.748782\pi\)
\(110\) 3.62365 0.345501
\(111\) 9.70820 0.921462
\(112\) −3.14590 −0.297259
\(113\) −14.2361 −1.33922 −0.669608 0.742714i \(-0.733538\pi\)
−0.669608 + 0.742714i \(0.733538\pi\)
\(114\) −0.874032 −0.0818606
\(115\) 15.3500 1.43140
\(116\) −6.86474 −0.637375
\(117\) −15.3500 −1.41911
\(118\) −2.27051 −0.209017
\(119\) 0.540182 0.0495184
\(120\) −7.53244 −0.687614
\(121\) 7.00000 0.636364
\(122\) −1.69936 −0.153852
\(123\) −17.0981 −1.54168
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) −0.854102 −0.0760895
\(127\) −9.48683 −0.841820 −0.420910 0.907102i \(-0.638289\pi\)
−0.420910 + 0.907102i \(0.638289\pi\)
\(128\) −10.0902 −0.891853
\(129\) −0.472136 −0.0415693
\(130\) 5.86319 0.514235
\(131\) −19.4164 −1.69642 −0.848210 0.529661i \(-0.822319\pi\)
−0.848210 + 0.529661i \(0.822319\pi\)
\(132\) −18.0000 −1.56670
\(133\) −1.00000 −0.0867110
\(134\) 2.29180 0.197981
\(135\) −3.90879 −0.336415
\(136\) 0.795221 0.0681896
\(137\) −6.86474 −0.586494 −0.293247 0.956037i \(-0.594736\pi\)
−0.293247 + 0.956037i \(0.594736\pi\)
\(138\) 6.00000 0.510754
\(139\) −1.00155 −0.0849505 −0.0424752 0.999098i \(-0.513524\pi\)
−0.0424752 + 0.999098i \(0.513524\pi\)
\(140\) −4.14590 −0.350392
\(141\) −8.48528 −0.714590
\(142\) 2.85410 0.239511
\(143\) 29.1246 2.43552
\(144\) 7.03444 0.586203
\(145\) −8.27895 −0.687529
\(146\) −1.62054 −0.134117
\(147\) 13.7295 1.13239
\(148\) 7.86629 0.646605
\(149\) 13.4164 1.09911 0.549557 0.835456i \(-0.314796\pi\)
0.549557 + 0.835456i \(0.314796\pi\)
\(150\) 0 0
\(151\) −3.03476 −0.246965 −0.123483 0.992347i \(-0.539406\pi\)
−0.123483 + 0.992347i \(0.539406\pi\)
\(152\) −1.47214 −0.119406
\(153\) −1.20788 −0.0976515
\(154\) 1.62054 0.130587
\(155\) 0 0
\(156\) −29.1246 −2.33184
\(157\) −4.70820 −0.375756 −0.187878 0.982192i \(-0.560161\pi\)
−0.187878 + 0.982192i \(0.560161\pi\)
\(158\) 4.24264 0.337526
\(159\) 12.0000 0.951662
\(160\) −9.27051 −0.732898
\(161\) 6.86474 0.541017
\(162\) −4.09017 −0.321354
\(163\) −12.4164 −0.972528 −0.486264 0.873812i \(-0.661641\pi\)
−0.486264 + 0.873812i \(0.661641\pi\)
\(164\) −13.8541 −1.08182
\(165\) −21.7082 −1.68998
\(166\) −1.20788 −0.0937499
\(167\) 7.40492 0.573010 0.286505 0.958079i \(-0.407507\pi\)
0.286505 + 0.958079i \(0.407507\pi\)
\(168\) −3.36861 −0.259894
\(169\) 34.1246 2.62497
\(170\) 0.461370 0.0353855
\(171\) 2.23607 0.170996
\(172\) −0.382559 −0.0291698
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −3.23607 −0.245326
\(175\) 0 0
\(176\) −13.3469 −1.00606
\(177\) 13.6020 1.02239
\(178\) 2.23954 0.167860
\(179\) −16.4304 −1.22806 −0.614032 0.789281i \(-0.710454\pi\)
−0.614032 + 0.789281i \(0.710454\pi\)
\(180\) 9.27051 0.690983
\(181\) −18.1784 −1.35119 −0.675597 0.737271i \(-0.736114\pi\)
−0.675597 + 0.737271i \(0.736114\pi\)
\(182\) 2.62210 0.194363
\(183\) 10.1803 0.752552
\(184\) 10.1058 0.745011
\(185\) 9.48683 0.697486
\(186\) 0 0
\(187\) 2.29180 0.167593
\(188\) −6.87539 −0.501439
\(189\) −1.74806 −0.127153
\(190\) −0.854102 −0.0619631
\(191\) 14.2361 1.03009 0.515043 0.857164i \(-0.327776\pi\)
0.515043 + 0.857164i \(0.327776\pi\)
\(192\) 10.7735 0.777512
\(193\) −7.29180 −0.524875 −0.262437 0.964949i \(-0.584526\pi\)
−0.262437 + 0.964949i \(0.584526\pi\)
\(194\) −2.67376 −0.191965
\(195\) −35.1246 −2.51533
\(196\) 11.1246 0.794615
\(197\) 7.40492 0.527579 0.263789 0.964580i \(-0.415028\pi\)
0.263789 + 0.964580i \(0.415028\pi\)
\(198\) −3.62365 −0.257521
\(199\) 14.7310 1.04425 0.522127 0.852868i \(-0.325139\pi\)
0.522127 + 0.852868i \(0.325139\pi\)
\(200\) 0 0
\(201\) −13.7295 −0.968402
\(202\) −0.854102 −0.0600944
\(203\) −3.70246 −0.259862
\(204\) −2.29180 −0.160458
\(205\) −16.7082 −1.16695
\(206\) −1.03444 −0.0720730
\(207\) −15.3500 −1.06690
\(208\) −21.5958 −1.49740
\(209\) −4.24264 −0.293470
\(210\) −1.95440 −0.134866
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 9.72327 0.667797
\(213\) −17.0981 −1.17154
\(214\) −2.85410 −0.195102
\(215\) −0.461370 −0.0314652
\(216\) −2.57339 −0.175097
\(217\) 0 0
\(218\) −5.61803 −0.380501
\(219\) 9.70820 0.656020
\(220\) −17.5896 −1.18589
\(221\) 3.70820 0.249441
\(222\) 3.70820 0.248878
\(223\) 21.8021 1.45998 0.729988 0.683460i \(-0.239526\pi\)
0.729988 + 0.683460i \(0.239526\pi\)
\(224\) −4.14590 −0.277009
\(225\) 0 0
\(226\) −5.43769 −0.361710
\(227\) 15.7082 1.04259 0.521295 0.853377i \(-0.325449\pi\)
0.521295 + 0.853377i \(0.325449\pi\)
\(228\) 4.24264 0.280976
\(229\) −4.24264 −0.280362 −0.140181 0.990126i \(-0.544768\pi\)
−0.140181 + 0.990126i \(0.544768\pi\)
\(230\) 5.86319 0.386607
\(231\) −9.70820 −0.638753
\(232\) −5.45052 −0.357844
\(233\) −4.52786 −0.296630 −0.148315 0.988940i \(-0.547385\pi\)
−0.148315 + 0.988940i \(0.547385\pi\)
\(234\) −5.86319 −0.383288
\(235\) −8.29180 −0.540897
\(236\) 11.0213 0.717425
\(237\) −25.4164 −1.65097
\(238\) 0.206331 0.0133745
\(239\) 16.3516 1.05770 0.528848 0.848717i \(-0.322624\pi\)
0.528848 + 0.848717i \(0.322624\pi\)
\(240\) 16.0965 1.03903
\(241\) −14.7310 −0.948909 −0.474454 0.880280i \(-0.657355\pi\)
−0.474454 + 0.880280i \(0.657355\pi\)
\(242\) 2.67376 0.171876
\(243\) 19.2588 1.23545
\(244\) 8.24885 0.528078
\(245\) 13.4164 0.857143
\(246\) −6.53089 −0.416394
\(247\) −6.86474 −0.436793
\(248\) 0 0
\(249\) 7.23607 0.458567
\(250\) 4.27051 0.270091
\(251\) 23.7565 1.49950 0.749748 0.661723i \(-0.230175\pi\)
0.749748 + 0.661723i \(0.230175\pi\)
\(252\) 4.14590 0.261167
\(253\) 29.1246 1.83105
\(254\) −3.62365 −0.227368
\(255\) −2.76393 −0.173084
\(256\) 5.56231 0.347644
\(257\) 23.9443 1.49360 0.746801 0.665047i \(-0.231589\pi\)
0.746801 + 0.665047i \(0.231589\pi\)
\(258\) −0.180340 −0.0112275
\(259\) 4.24264 0.263625
\(260\) −28.4605 −1.76505
\(261\) 8.27895 0.512454
\(262\) −7.41641 −0.458187
\(263\) −2.70091 −0.166545 −0.0832725 0.996527i \(-0.526537\pi\)
−0.0832725 + 0.996527i \(0.526537\pi\)
\(264\) −14.2918 −0.879599
\(265\) 11.7264 0.720345
\(266\) −0.381966 −0.0234198
\(267\) −13.4164 −0.821071
\(268\) −11.1246 −0.679544
\(269\) −14.8098 −0.902972 −0.451486 0.892278i \(-0.649106\pi\)
−0.451486 + 0.892278i \(0.649106\pi\)
\(270\) −1.49302 −0.0908626
\(271\) 28.0779 1.70561 0.852807 0.522227i \(-0.174899\pi\)
0.852807 + 0.522227i \(0.174899\pi\)
\(272\) −1.69936 −0.103039
\(273\) −15.7082 −0.950704
\(274\) −2.62210 −0.158407
\(275\) 0 0
\(276\) −29.1246 −1.75310
\(277\) −19.5927 −1.17721 −0.588604 0.808421i \(-0.700322\pi\)
−0.588604 + 0.808421i \(0.700322\pi\)
\(278\) −0.382559 −0.0229443
\(279\) 0 0
\(280\) −3.29180 −0.196722
\(281\) −31.3607 −1.87082 −0.935411 0.353563i \(-0.884970\pi\)
−0.935411 + 0.353563i \(0.884970\pi\)
\(282\) −3.24109 −0.193004
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) −13.8541 −0.822090
\(285\) 5.11667 0.303086
\(286\) 11.1246 0.657812
\(287\) −7.47214 −0.441066
\(288\) 9.27051 0.546270
\(289\) −16.7082 −0.982836
\(290\) −3.16228 −0.185695
\(291\) 16.0177 0.938975
\(292\) 7.86629 0.460340
\(293\) −15.7082 −0.917683 −0.458842 0.888518i \(-0.651735\pi\)
−0.458842 + 0.888518i \(0.651735\pi\)
\(294\) 5.24419 0.305848
\(295\) 13.2918 0.773878
\(296\) 6.24574 0.363026
\(297\) −7.41641 −0.430344
\(298\) 5.12461 0.296861
\(299\) 47.1246 2.72529
\(300\) 0 0
\(301\) −0.206331 −0.0118927
\(302\) −1.15917 −0.0667030
\(303\) 5.11667 0.293945
\(304\) 3.14590 0.180430
\(305\) 9.94820 0.569632
\(306\) −0.461370 −0.0263748
\(307\) −16.1246 −0.920280 −0.460140 0.887846i \(-0.652201\pi\)
−0.460140 + 0.887846i \(0.652201\pi\)
\(308\) −7.86629 −0.448223
\(309\) 6.19704 0.352537
\(310\) 0 0
\(311\) 16.5279 0.937209 0.468605 0.883408i \(-0.344757\pi\)
0.468605 + 0.883408i \(0.344757\pi\)
\(312\) −23.1246 −1.30917
\(313\) 21.0069 1.18738 0.593689 0.804694i \(-0.297671\pi\)
0.593689 + 0.804694i \(0.297671\pi\)
\(314\) −1.79837 −0.101488
\(315\) 5.00000 0.281718
\(316\) −20.5942 −1.15851
\(317\) 21.7639 1.22238 0.611192 0.791482i \(-0.290690\pi\)
0.611192 + 0.791482i \(0.290690\pi\)
\(318\) 4.58359 0.257035
\(319\) −15.7082 −0.879491
\(320\) 10.5279 0.588525
\(321\) 17.0981 0.954322
\(322\) 2.62210 0.146124
\(323\) −0.540182 −0.0300565
\(324\) 19.8541 1.10301
\(325\) 0 0
\(326\) −4.74265 −0.262671
\(327\) 33.6560 1.86118
\(328\) −11.0000 −0.607373
\(329\) −3.70820 −0.204440
\(330\) −8.29180 −0.456448
\(331\) 25.6321 1.40887 0.704433 0.709770i \(-0.251201\pi\)
0.704433 + 0.709770i \(0.251201\pi\)
\(332\) 5.86319 0.321784
\(333\) −9.48683 −0.519875
\(334\) 2.82843 0.154765
\(335\) −13.4164 −0.733017
\(336\) 7.19859 0.392715
\(337\) 5.86319 0.319388 0.159694 0.987167i \(-0.448949\pi\)
0.159694 + 0.987167i \(0.448949\pi\)
\(338\) 13.0344 0.708980
\(339\) 32.5756 1.76926
\(340\) −2.23954 −0.121456
\(341\) 0 0
\(342\) 0.854102 0.0461845
\(343\) 13.0000 0.701934
\(344\) −0.303747 −0.0163770
\(345\) −35.1246 −1.89105
\(346\) 6.87539 0.369623
\(347\) −25.8384 −1.38708 −0.693539 0.720419i \(-0.743950\pi\)
−0.693539 + 0.720419i \(0.743950\pi\)
\(348\) 15.7082 0.842048
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) −12.0000 −0.640513
\(352\) −17.5896 −0.937526
\(353\) −15.8902 −0.845750 −0.422875 0.906188i \(-0.638979\pi\)
−0.422875 + 0.906188i \(0.638979\pi\)
\(354\) 5.19548 0.276137
\(355\) −16.7082 −0.886779
\(356\) −10.8709 −0.576159
\(357\) −1.23607 −0.0654197
\(358\) −6.27585 −0.331689
\(359\) 9.76393 0.515321 0.257660 0.966236i \(-0.417048\pi\)
0.257660 + 0.966236i \(0.417048\pi\)
\(360\) 7.36068 0.387942
\(361\) −18.0000 −0.947368
\(362\) −6.94355 −0.364945
\(363\) −16.0177 −0.840712
\(364\) −12.7279 −0.667124
\(365\) 9.48683 0.496564
\(366\) 3.88854 0.203257
\(367\) 12.3153 0.642851 0.321426 0.946935i \(-0.395838\pi\)
0.321426 + 0.946935i \(0.395838\pi\)
\(368\) −21.5958 −1.12576
\(369\) 16.7082 0.869794
\(370\) 3.62365 0.188384
\(371\) 5.24419 0.272265
\(372\) 0 0
\(373\) −22.7082 −1.17579 −0.587893 0.808939i \(-0.700042\pi\)
−0.587893 + 0.808939i \(0.700042\pi\)
\(374\) 0.875388 0.0452652
\(375\) −25.5834 −1.32112
\(376\) −5.45898 −0.281525
\(377\) −25.4164 −1.30901
\(378\) −0.667701 −0.0343428
\(379\) −19.4164 −0.997354 −0.498677 0.866788i \(-0.666181\pi\)
−0.498677 + 0.866788i \(0.666181\pi\)
\(380\) 4.14590 0.212680
\(381\) 21.7082 1.11214
\(382\) 5.43769 0.278217
\(383\) 19.0525 0.973536 0.486768 0.873531i \(-0.338176\pi\)
0.486768 + 0.873531i \(0.338176\pi\)
\(384\) 23.0888 1.17824
\(385\) −9.48683 −0.483494
\(386\) −2.78522 −0.141764
\(387\) 0.461370 0.0234528
\(388\) 12.9787 0.658894
\(389\) −33.7835 −1.71289 −0.856446 0.516237i \(-0.827332\pi\)
−0.856446 + 0.516237i \(0.827332\pi\)
\(390\) −13.4164 −0.679366
\(391\) 3.70820 0.187532
\(392\) 8.83282 0.446125
\(393\) 44.4295 2.24117
\(394\) 2.82843 0.142494
\(395\) −24.8369 −1.24968
\(396\) 17.5896 0.883908
\(397\) 1.29180 0.0648334 0.0324167 0.999474i \(-0.489680\pi\)
0.0324167 + 0.999474i \(0.489680\pi\)
\(398\) 5.62675 0.282044
\(399\) 2.28825 0.114556
\(400\) 0 0
\(401\) 3.70246 0.184892 0.0924460 0.995718i \(-0.470531\pi\)
0.0924460 + 0.995718i \(0.470531\pi\)
\(402\) −5.24419 −0.261557
\(403\) 0 0
\(404\) 4.14590 0.206266
\(405\) 23.9443 1.18980
\(406\) −1.41421 −0.0701862
\(407\) 18.0000 0.892227
\(408\) −1.81966 −0.0900866
\(409\) 17.3531 0.858057 0.429028 0.903291i \(-0.358856\pi\)
0.429028 + 0.903291i \(0.358856\pi\)
\(410\) −6.38197 −0.315183
\(411\) 15.7082 0.774829
\(412\) 5.02129 0.247381
\(413\) 5.94427 0.292498
\(414\) −5.86319 −0.288160
\(415\) 7.07107 0.347105
\(416\) −28.4605 −1.39539
\(417\) 2.29180 0.112230
\(418\) −1.62054 −0.0792634
\(419\) −18.5967 −0.908511 −0.454255 0.890872i \(-0.650095\pi\)
−0.454255 + 0.890872i \(0.650095\pi\)
\(420\) 9.48683 0.462910
\(421\) −0.416408 −0.0202945 −0.0101472 0.999949i \(-0.503230\pi\)
−0.0101472 + 0.999949i \(0.503230\pi\)
\(422\) −1.90983 −0.0929691
\(423\) 8.29180 0.403161
\(424\) 7.72016 0.374924
\(425\) 0 0
\(426\) −6.53089 −0.316422
\(427\) 4.44897 0.215301
\(428\) 13.8541 0.669663
\(429\) −66.6443 −3.21762
\(430\) −0.176228 −0.00849845
\(431\) −0.111456 −0.00536866 −0.00268433 0.999996i \(-0.500854\pi\)
−0.00268433 + 0.999996i \(0.500854\pi\)
\(432\) 5.49923 0.264582
\(433\) −21.5958 −1.03783 −0.518913 0.854827i \(-0.673663\pi\)
−0.518913 + 0.854827i \(0.673663\pi\)
\(434\) 0 0
\(435\) 18.9443 0.908308
\(436\) 27.2705 1.30602
\(437\) −6.86474 −0.328385
\(438\) 3.70820 0.177185
\(439\) −25.0000 −1.19318 −0.596592 0.802544i \(-0.703479\pi\)
−0.596592 + 0.802544i \(0.703479\pi\)
\(440\) −13.9659 −0.665799
\(441\) −13.4164 −0.638877
\(442\) 1.41641 0.0673717
\(443\) 23.9443 1.13763 0.568813 0.822467i \(-0.307403\pi\)
0.568813 + 0.822467i \(0.307403\pi\)
\(444\) −18.0000 −0.854242
\(445\) −13.1105 −0.621496
\(446\) 8.32766 0.394326
\(447\) −30.7000 −1.45206
\(448\) 4.70820 0.222442
\(449\) 42.1900 1.99107 0.995534 0.0944042i \(-0.0300946\pi\)
0.995534 + 0.0944042i \(0.0300946\pi\)
\(450\) 0 0
\(451\) −31.7016 −1.49277
\(452\) 26.3951 1.24152
\(453\) 6.94427 0.326270
\(454\) 6.00000 0.281594
\(455\) −15.3500 −0.719620
\(456\) 3.36861 0.157750
\(457\) −11.3137 −0.529233 −0.264616 0.964354i \(-0.585245\pi\)
−0.264616 + 0.964354i \(0.585245\pi\)
\(458\) −1.62054 −0.0757231
\(459\) −0.944272 −0.0440748
\(460\) −28.4605 −1.32698
\(461\) 39.1065 1.82137 0.910686 0.413100i \(-0.135554\pi\)
0.910686 + 0.413100i \(0.135554\pi\)
\(462\) −3.70820 −0.172521
\(463\) −15.1437 −0.703787 −0.351893 0.936040i \(-0.614462\pi\)
−0.351893 + 0.936040i \(0.614462\pi\)
\(464\) 11.6476 0.540724
\(465\) 0 0
\(466\) −1.72949 −0.0801171
\(467\) 15.6525 0.724310 0.362155 0.932118i \(-0.382041\pi\)
0.362155 + 0.932118i \(0.382041\pi\)
\(468\) 28.4605 1.31559
\(469\) −6.00000 −0.277054
\(470\) −3.16718 −0.146091
\(471\) 10.7735 0.496418
\(472\) 8.75078 0.402787
\(473\) −0.875388 −0.0402504
\(474\) −9.70820 −0.445913
\(475\) 0 0
\(476\) −1.00155 −0.0459060
\(477\) −11.7264 −0.536914
\(478\) 6.24574 0.285674
\(479\) 25.4721 1.16385 0.581926 0.813242i \(-0.302299\pi\)
0.581926 + 0.813242i \(0.302299\pi\)
\(480\) 21.2132 0.968246
\(481\) 29.1246 1.32797
\(482\) −5.62675 −0.256291
\(483\) −15.7082 −0.714748
\(484\) −12.9787 −0.589942
\(485\) 15.6525 0.710742
\(486\) 7.35621 0.333684
\(487\) −18.5911 −0.842443 −0.421222 0.906958i \(-0.638399\pi\)
−0.421222 + 0.906958i \(0.638399\pi\)
\(488\) 6.54949 0.296482
\(489\) 28.4118 1.28483
\(490\) 5.12461 0.231506
\(491\) 9.56564 0.431691 0.215846 0.976427i \(-0.430749\pi\)
0.215846 + 0.976427i \(0.430749\pi\)
\(492\) 31.7016 1.42922
\(493\) −2.00000 −0.0900755
\(494\) −2.62210 −0.117974
\(495\) 21.2132 0.953463
\(496\) 0 0
\(497\) −7.47214 −0.335171
\(498\) 2.76393 0.123855
\(499\) −18.3848 −0.823016 −0.411508 0.911406i \(-0.634998\pi\)
−0.411508 + 0.911406i \(0.634998\pi\)
\(500\) −20.7295 −0.927051
\(501\) −16.9443 −0.757014
\(502\) 9.07417 0.405000
\(503\) 2.88854 0.128794 0.0643969 0.997924i \(-0.479488\pi\)
0.0643969 + 0.997924i \(0.479488\pi\)
\(504\) 3.29180 0.146628
\(505\) 5.00000 0.222497
\(506\) 11.1246 0.494549
\(507\) −78.0855 −3.46790
\(508\) 17.5896 0.780410
\(509\) 4.78282 0.211995 0.105997 0.994366i \(-0.466196\pi\)
0.105997 + 0.994366i \(0.466196\pi\)
\(510\) −1.05573 −0.0467484
\(511\) 4.24264 0.187683
\(512\) 22.3050 0.985749
\(513\) 1.74806 0.0771789
\(514\) 9.14590 0.403408
\(515\) 6.05573 0.266847
\(516\) 0.875388 0.0385368
\(517\) −15.7326 −0.691918
\(518\) 1.62054 0.0712026
\(519\) −41.1884 −1.80797
\(520\) −22.5973 −0.990958
\(521\) 13.4164 0.587784 0.293892 0.955839i \(-0.405049\pi\)
0.293892 + 0.955839i \(0.405049\pi\)
\(522\) 3.16228 0.138409
\(523\) −29.0795 −1.27156 −0.635779 0.771871i \(-0.719321\pi\)
−0.635779 + 0.771871i \(0.719321\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) −1.03165 −0.0449823
\(527\) 0 0
\(528\) 30.5410 1.32913
\(529\) 24.1246 1.04890
\(530\) 4.47907 0.194559
\(531\) −13.2918 −0.576815
\(532\) 1.85410 0.0803855
\(533\) −51.2942 −2.22180
\(534\) −5.12461 −0.221764
\(535\) 16.7082 0.722359
\(536\) −8.83282 −0.381520
\(537\) 37.5967 1.62242
\(538\) −5.65685 −0.243884
\(539\) 25.4558 1.09646
\(540\) 7.24730 0.311874
\(541\) −27.8328 −1.19663 −0.598313 0.801262i \(-0.704162\pi\)
−0.598313 + 0.801262i \(0.704162\pi\)
\(542\) 10.7248 0.460670
\(543\) 41.5967 1.78509
\(544\) −2.23954 −0.0960194
\(545\) 32.8885 1.40879
\(546\) −6.00000 −0.256776
\(547\) −24.4164 −1.04397 −0.521985 0.852955i \(-0.674808\pi\)
−0.521985 + 0.852955i \(0.674808\pi\)
\(548\) 12.7279 0.543710
\(549\) −9.94820 −0.424579
\(550\) 0 0
\(551\) 3.70246 0.157730
\(552\) −23.1246 −0.984249
\(553\) −11.1074 −0.472334
\(554\) −7.48373 −0.317953
\(555\) −21.7082 −0.921462
\(556\) 1.85698 0.0787534
\(557\) 23.2951 0.987046 0.493523 0.869733i \(-0.335709\pi\)
0.493523 + 0.869733i \(0.335709\pi\)
\(558\) 0 0
\(559\) −1.41641 −0.0599077
\(560\) 7.03444 0.297259
\(561\) −5.24419 −0.221410
\(562\) −11.9787 −0.505292
\(563\) 14.2361 0.599979 0.299989 0.953943i \(-0.403017\pi\)
0.299989 + 0.953943i \(0.403017\pi\)
\(564\) 15.7326 0.662461
\(565\) 31.8328 1.33922
\(566\) 9.16718 0.385325
\(567\) 10.7082 0.449702
\(568\) −11.0000 −0.461550
\(569\) 30.7000 1.28701 0.643506 0.765441i \(-0.277479\pi\)
0.643506 + 0.765441i \(0.277479\pi\)
\(570\) 1.95440 0.0818606
\(571\) −6.86474 −0.287280 −0.143640 0.989630i \(-0.545881\pi\)
−0.143640 + 0.989630i \(0.545881\pi\)
\(572\) −54.0000 −2.25785
\(573\) −32.5756 −1.36087
\(574\) −2.85410 −0.119128
\(575\) 0 0
\(576\) −10.5279 −0.438661
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −6.38197 −0.265455
\(579\) 16.6854 0.693422
\(580\) 15.3500 0.637375
\(581\) 3.16228 0.131193
\(582\) 6.11822 0.253609
\(583\) 22.2492 0.921469
\(584\) 6.24574 0.258451
\(585\) 34.3237 1.41911
\(586\) −6.00000 −0.247858
\(587\) −41.1884 −1.70003 −0.850014 0.526760i \(-0.823407\pi\)
−0.850014 + 0.526760i \(0.823407\pi\)
\(588\) −25.4558 −1.04978
\(589\) 0 0
\(590\) 5.07701 0.209017
\(591\) −16.9443 −0.696994
\(592\) −13.3469 −0.548555
\(593\) −17.1803 −0.705512 −0.352756 0.935715i \(-0.614755\pi\)
−0.352756 + 0.935715i \(0.614755\pi\)
\(594\) −2.83282 −0.116232
\(595\) −1.20788 −0.0495184
\(596\) −24.8754 −1.01894
\(597\) −33.7082 −1.37958
\(598\) 18.0000 0.736075
\(599\) 23.9443 0.978336 0.489168 0.872189i \(-0.337300\pi\)
0.489168 + 0.872189i \(0.337300\pi\)
\(600\) 0 0
\(601\) −8.69161 −0.354538 −0.177269 0.984162i \(-0.556726\pi\)
−0.177269 + 0.984162i \(0.556726\pi\)
\(602\) −0.0788114 −0.00321211
\(603\) 13.4164 0.546358
\(604\) 5.62675 0.228949
\(605\) −15.6525 −0.636364
\(606\) 1.95440 0.0793919
\(607\) −25.1246 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(608\) 4.14590 0.168138
\(609\) 8.47214 0.343308
\(610\) 3.79988 0.153852
\(611\) −25.4558 −1.02983
\(612\) 2.23954 0.0905279
\(613\) 37.3584 1.50889 0.754447 0.656361i \(-0.227905\pi\)
0.754447 + 0.656361i \(0.227905\pi\)
\(614\) −6.15905 −0.248559
\(615\) 38.2325 1.54168
\(616\) −6.24574 −0.251648
\(617\) −47.1246 −1.89717 −0.948583 0.316529i \(-0.897482\pi\)
−0.948583 + 0.316529i \(0.897482\pi\)
\(618\) 2.36706 0.0952170
\(619\) 21.5958 0.868007 0.434003 0.900911i \(-0.357101\pi\)
0.434003 + 0.900911i \(0.357101\pi\)
\(620\) 0 0
\(621\) −12.0000 −0.481543
\(622\) 6.31308 0.253132
\(623\) −5.86319 −0.234904
\(624\) 49.4164 1.97824
\(625\) −25.0000 −1.00000
\(626\) 8.02391 0.320700
\(627\) 9.70820 0.387708
\(628\) 8.72949 0.348345
\(629\) 2.29180 0.0913799
\(630\) 1.90983 0.0760895
\(631\) 4.65530 0.185325 0.0926623 0.995698i \(-0.470462\pi\)
0.0926623 + 0.995698i \(0.470462\pi\)
\(632\) −16.3516 −0.650431
\(633\) 11.4412 0.454748
\(634\) 8.31308 0.330155
\(635\) 21.2132 0.841820
\(636\) −22.2492 −0.882239
\(637\) 41.1884 1.63194
\(638\) −6.00000 −0.237542
\(639\) 16.7082 0.660966
\(640\) 22.5623 0.891853
\(641\) 8.48528 0.335148 0.167574 0.985859i \(-0.446407\pi\)
0.167574 + 0.985859i \(0.446407\pi\)
\(642\) 6.53089 0.257754
\(643\) 23.6290 0.931836 0.465918 0.884828i \(-0.345724\pi\)
0.465918 + 0.884828i \(0.345724\pi\)
\(644\) −12.7279 −0.501550
\(645\) 1.05573 0.0415693
\(646\) −0.206331 −0.00811798
\(647\) −19.1313 −0.752129 −0.376064 0.926594i \(-0.622723\pi\)
−0.376064 + 0.926594i \(0.622723\pi\)
\(648\) 15.7639 0.619266
\(649\) 25.2194 0.989948
\(650\) 0 0
\(651\) 0 0
\(652\) 23.0213 0.901583
\(653\) −6.65248 −0.260331 −0.130166 0.991492i \(-0.541551\pi\)
−0.130166 + 0.991492i \(0.541551\pi\)
\(654\) 12.8554 0.502688
\(655\) 43.4164 1.69642
\(656\) 23.5066 0.917778
\(657\) −9.48683 −0.370117
\(658\) −1.41641 −0.0552173
\(659\) 47.1803 1.83789 0.918943 0.394391i \(-0.129045\pi\)
0.918943 + 0.394391i \(0.129045\pi\)
\(660\) 40.2492 1.56670
\(661\) −45.8328 −1.78269 −0.891345 0.453326i \(-0.850237\pi\)
−0.891345 + 0.453326i \(0.850237\pi\)
\(662\) 9.79058 0.380522
\(663\) −8.48528 −0.329541
\(664\) 4.65530 0.180661
\(665\) 2.23607 0.0867110
\(666\) −3.62365 −0.140413
\(667\) −25.4164 −0.984127
\(668\) −13.7295 −0.531209
\(669\) −49.8885 −1.92880
\(670\) −5.12461 −0.197981
\(671\) 18.8754 0.728676
\(672\) 9.48683 0.365963
\(673\) 7.45363 0.287316 0.143658 0.989627i \(-0.454113\pi\)
0.143658 + 0.989627i \(0.454113\pi\)
\(674\) 2.23954 0.0862638
\(675\) 0 0
\(676\) −63.2705 −2.43348
\(677\) 21.2132 0.815290 0.407645 0.913141i \(-0.366350\pi\)
0.407645 + 0.913141i \(0.366350\pi\)
\(678\) 12.4428 0.477862
\(679\) 7.00000 0.268635
\(680\) −1.77817 −0.0681896
\(681\) −35.9442 −1.37739
\(682\) 0 0
\(683\) −18.8197 −0.720114 −0.360057 0.932930i \(-0.617243\pi\)
−0.360057 + 0.932930i \(0.617243\pi\)
\(684\) −4.14590 −0.158522
\(685\) 15.3500 0.586494
\(686\) 4.96556 0.189586
\(687\) 9.70820 0.370391
\(688\) 0.649096 0.0247466
\(689\) 36.0000 1.37149
\(690\) −13.4164 −0.510754
\(691\) −21.5836 −0.821079 −0.410539 0.911843i \(-0.634660\pi\)
−0.410539 + 0.911843i \(0.634660\pi\)
\(692\) −33.3738 −1.26868
\(693\) 9.48683 0.360375
\(694\) −9.86939 −0.374637
\(695\) 2.23954 0.0849505
\(696\) 12.4721 0.472755
\(697\) −4.03631 −0.152886
\(698\) 2.29180 0.0867458
\(699\) 10.3609 0.391884
\(700\) 0 0
\(701\) 27.6525 1.04442 0.522210 0.852817i \(-0.325108\pi\)
0.522210 + 0.852817i \(0.325108\pi\)
\(702\) −4.58359 −0.172997
\(703\) −4.24264 −0.160014
\(704\) 19.9752 0.752844
\(705\) 18.9737 0.714590
\(706\) −6.06952 −0.228429
\(707\) 2.23607 0.0840960
\(708\) −25.2194 −0.947803
\(709\) −15.7326 −0.590849 −0.295425 0.955366i \(-0.595461\pi\)
−0.295425 + 0.955366i \(0.595461\pi\)
\(710\) −6.38197 −0.239511
\(711\) 24.8369 0.931455
\(712\) −8.63141 −0.323476
\(713\) 0 0
\(714\) −0.472136 −0.0176692
\(715\) −65.1246 −2.43552
\(716\) 30.4636 1.13848
\(717\) −37.4164 −1.39734
\(718\) 3.72949 0.139183
\(719\) 9.10427 0.339532 0.169766 0.985484i \(-0.445699\pi\)
0.169766 + 0.985484i \(0.445699\pi\)
\(720\) −15.7295 −0.586203
\(721\) 2.70820 0.100859
\(722\) −6.87539 −0.255875
\(723\) 33.7082 1.25362
\(724\) 33.7047 1.25262
\(725\) 0 0
\(726\) −6.11822 −0.227069
\(727\) −3.58359 −0.132908 −0.0664540 0.997789i \(-0.521169\pi\)
−0.0664540 + 0.997789i \(0.521169\pi\)
\(728\) −10.1058 −0.374547
\(729\) −11.9443 −0.442380
\(730\) 3.62365 0.134117
\(731\) −0.111456 −0.00412236
\(732\) −18.8754 −0.697654
\(733\) 12.4164 0.458610 0.229305 0.973355i \(-0.426355\pi\)
0.229305 + 0.973355i \(0.426355\pi\)
\(734\) 4.70401 0.173628
\(735\) −30.7000 −1.13239
\(736\) −28.4605 −1.04907
\(737\) −25.4558 −0.937678
\(738\) 6.38197 0.234923
\(739\) 25.6622 0.943998 0.471999 0.881599i \(-0.343533\pi\)
0.471999 + 0.881599i \(0.343533\pi\)
\(740\) −17.5896 −0.646605
\(741\) 15.7082 0.577055
\(742\) 2.00310 0.0735362
\(743\) 1.69936 0.0623433 0.0311717 0.999514i \(-0.490076\pi\)
0.0311717 + 0.999514i \(0.490076\pi\)
\(744\) 0 0
\(745\) −30.0000 −1.09911
\(746\) −8.67376 −0.317569
\(747\) −7.07107 −0.258717
\(748\) −4.24922 −0.155367
\(749\) 7.47214 0.273026
\(750\) −9.77198 −0.356822
\(751\) 29.5410 1.07797 0.538984 0.842316i \(-0.318808\pi\)
0.538984 + 0.842316i \(0.318808\pi\)
\(752\) 11.6656 0.425402
\(753\) −54.3607 −1.98101
\(754\) −9.70820 −0.353552
\(755\) 6.78593 0.246965
\(756\) 3.24109 0.117877
\(757\) 19.1800 0.697109 0.348554 0.937289i \(-0.386673\pi\)
0.348554 + 0.937289i \(0.386673\pi\)
\(758\) −7.41641 −0.269376
\(759\) −66.6443 −2.41903
\(760\) 3.29180 0.119406
\(761\) −2.23954 −0.0811832 −0.0405916 0.999176i \(-0.512924\pi\)
−0.0405916 + 0.999176i \(0.512924\pi\)
\(762\) 8.29180 0.300380
\(763\) 14.7082 0.532473
\(764\) −26.3951 −0.954942
\(765\) 2.70091 0.0976515
\(766\) 7.27740 0.262943
\(767\) 40.8059 1.47341
\(768\) −12.7279 −0.459279
\(769\) 13.8754 0.500359 0.250180 0.968199i \(-0.419510\pi\)
0.250180 + 0.968199i \(0.419510\pi\)
\(770\) −3.62365 −0.130587
\(771\) −54.7904 −1.97323
\(772\) 13.5197 0.486586
\(773\) 26.9976 0.971035 0.485518 0.874227i \(-0.338631\pi\)
0.485518 + 0.874227i \(0.338631\pi\)
\(774\) 0.176228 0.00633437
\(775\) 0 0
\(776\) 10.3050 0.369926
\(777\) −9.70820 −0.348280
\(778\) −12.9041 −0.462636
\(779\) 7.47214 0.267717
\(780\) 65.1246 2.33184
\(781\) −31.7016 −1.13437
\(782\) 1.41641 0.0506506
\(783\) 6.47214 0.231295
\(784\) −18.8754 −0.674121
\(785\) 10.5279 0.375756
\(786\) 16.9706 0.605320
\(787\) 9.10427 0.324532 0.162266 0.986747i \(-0.448120\pi\)
0.162266 + 0.986747i \(0.448120\pi\)
\(788\) −13.7295 −0.489092
\(789\) 6.18034 0.220026
\(790\) −9.48683 −0.337526
\(791\) 14.2361 0.506176
\(792\) 13.9659 0.496257
\(793\) 30.5410 1.08454
\(794\) 0.493422 0.0175109
\(795\) −26.8328 −0.951662
\(796\) −27.3128 −0.968077
\(797\) −5.24419 −0.185759 −0.0928794 0.995677i \(-0.529607\pi\)
−0.0928794 + 0.995677i \(0.529607\pi\)
\(798\) 0.874032 0.0309404
\(799\) −2.00310 −0.0708647
\(800\) 0 0
\(801\) 13.1105 0.463236
\(802\) 1.41421 0.0499376
\(803\) 18.0000 0.635206
\(804\) 25.4558 0.897758
\(805\) −15.3500 −0.541017
\(806\) 0 0
\(807\) 33.8885 1.19293
\(808\) 3.29180 0.115805
\(809\) 22.0571 0.775487 0.387744 0.921767i \(-0.373255\pi\)
0.387744 + 0.921767i \(0.373255\pi\)
\(810\) 9.14590 0.321354
\(811\) −43.4164 −1.52456 −0.762278 0.647250i \(-0.775919\pi\)
−0.762278 + 0.647250i \(0.775919\pi\)
\(812\) 6.86474 0.240905
\(813\) −64.2492 −2.25332
\(814\) 6.87539 0.240982
\(815\) 27.7639 0.972528
\(816\) 3.88854 0.136126
\(817\) 0.206331 0.00721861
\(818\) 6.62830 0.231753
\(819\) 15.3500 0.536373
\(820\) 30.9787 1.08182
\(821\) −14.3485 −0.500765 −0.250382 0.968147i \(-0.580556\pi\)
−0.250382 + 0.968147i \(0.580556\pi\)
\(822\) 6.00000 0.209274
\(823\) −14.9374 −0.520684 −0.260342 0.965517i \(-0.583835\pi\)
−0.260342 + 0.965517i \(0.583835\pi\)
\(824\) 3.98684 0.138888
\(825\) 0 0
\(826\) 2.27051 0.0790011
\(827\) −48.0532 −1.67097 −0.835486 0.549512i \(-0.814814\pi\)
−0.835486 + 0.549512i \(0.814814\pi\)
\(828\) 28.4605 0.989071
\(829\) 5.83308 0.202591 0.101296 0.994856i \(-0.467701\pi\)
0.101296 + 0.994856i \(0.467701\pi\)
\(830\) 2.70091 0.0937499
\(831\) 44.8328 1.55523
\(832\) 32.3206 1.12051
\(833\) 3.24109 0.112297
\(834\) 0.875388 0.0303122
\(835\) −16.5579 −0.573010
\(836\) 7.86629 0.272061
\(837\) 0 0
\(838\) −7.10333 −0.245380
\(839\) 7.41641 0.256043 0.128021 0.991771i \(-0.459137\pi\)
0.128021 + 0.991771i \(0.459137\pi\)
\(840\) 7.53244 0.259894
\(841\) −15.2918 −0.527303
\(842\) −0.159054 −0.00548135
\(843\) 71.7609 2.47158
\(844\) 9.27051 0.319104
\(845\) −76.3050 −2.62497
\(846\) 3.16718 0.108890
\(847\) −7.00000 −0.240523
\(848\) −16.4977 −0.566533
\(849\) −54.9179 −1.88478
\(850\) 0 0
\(851\) 29.1246 0.998379
\(852\) 31.7016 1.08608
\(853\) −45.7082 −1.56502 −0.782510 0.622639i \(-0.786061\pi\)
−0.782510 + 0.622639i \(0.786061\pi\)
\(854\) 1.69936 0.0581508
\(855\) −5.00000 −0.170996
\(856\) 11.0000 0.375972
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −25.4558 −0.869048
\(859\) 3.65375 0.124664 0.0623322 0.998055i \(-0.480146\pi\)
0.0623322 + 0.998055i \(0.480146\pi\)
\(860\) 0.855427 0.0291698
\(861\) 17.0981 0.582701
\(862\) −0.0425725 −0.00145002
\(863\) −35.3252 −1.20249 −0.601243 0.799067i \(-0.705327\pi\)
−0.601243 + 0.799067i \(0.705327\pi\)
\(864\) 7.24730 0.246558
\(865\) −40.2492 −1.36851
\(866\) −8.24885 −0.280307
\(867\) 38.2325 1.29844
\(868\) 0 0
\(869\) −47.1246 −1.59859
\(870\) 7.23607 0.245326
\(871\) −41.1884 −1.39562
\(872\) 21.6525 0.733245
\(873\) −15.6525 −0.529756
\(874\) −2.62210 −0.0886937
\(875\) −11.1803 −0.377964
\(876\) −18.0000 −0.608164
\(877\) −15.2918 −0.516367 −0.258184 0.966096i \(-0.583124\pi\)
−0.258184 + 0.966096i \(0.583124\pi\)
\(878\) −9.54915 −0.322268
\(879\) 35.9442 1.21237
\(880\) 29.8446 1.00606
\(881\) 49.6737 1.67355 0.836775 0.547547i \(-0.184438\pi\)
0.836775 + 0.547547i \(0.184438\pi\)
\(882\) −5.12461 −0.172555
\(883\) 33.9411 1.14221 0.571105 0.820877i \(-0.306515\pi\)
0.571105 + 0.820877i \(0.306515\pi\)
\(884\) −6.87539 −0.231244
\(885\) −30.4149 −1.02239
\(886\) 9.14590 0.307262
\(887\) −19.3607 −0.650068 −0.325034 0.945702i \(-0.605376\pi\)
−0.325034 + 0.945702i \(0.605376\pi\)
\(888\) −14.2918 −0.479601
\(889\) 9.48683 0.318178
\(890\) −5.00776 −0.167860
\(891\) 45.4311 1.52200
\(892\) −40.4233 −1.35347
\(893\) 3.70820 0.124090
\(894\) −11.7264 −0.392188
\(895\) 36.7394 1.22806
\(896\) 10.0902 0.337089
\(897\) −107.833 −3.60043
\(898\) 16.1151 0.537769
\(899\) 0 0
\(900\) 0 0
\(901\) 2.83282 0.0943748
\(902\) −12.1089 −0.403183
\(903\) 0.472136 0.0157117
\(904\) 20.9574 0.697034
\(905\) 40.6482 1.35119
\(906\) 2.65248 0.0881226
\(907\) −55.5410 −1.84421 −0.922105 0.386941i \(-0.873532\pi\)
−0.922105 + 0.386941i \(0.873532\pi\)
\(908\) −29.1246 −0.966534
\(909\) −5.00000 −0.165840
\(910\) −5.86319 −0.194363
\(911\) −14.1908 −0.470164 −0.235082 0.971976i \(-0.575536\pi\)
−0.235082 + 0.971976i \(0.575536\pi\)
\(912\) −7.19859 −0.238369
\(913\) 13.4164 0.444018
\(914\) −4.32145 −0.142941
\(915\) −22.7639 −0.752552
\(916\) 7.86629 0.259909
\(917\) 19.4164 0.641186
\(918\) −0.360680 −0.0119042
\(919\) 24.5410 0.809534 0.404767 0.914420i \(-0.367353\pi\)
0.404767 + 0.914420i \(0.367353\pi\)
\(920\) −22.5973 −0.745011
\(921\) 36.8971 1.21580
\(922\) 14.9374 0.491936
\(923\) −51.2942 −1.68837
\(924\) 18.0000 0.592157
\(925\) 0 0
\(926\) −5.78437 −0.190086
\(927\) −6.05573 −0.198896
\(928\) 15.3500 0.503889
\(929\) −23.2951 −0.764288 −0.382144 0.924103i \(-0.624814\pi\)
−0.382144 + 0.924103i \(0.624814\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 8.39512 0.274991
\(933\) −37.8198 −1.23817
\(934\) 5.97871 0.195629
\(935\) −5.12461 −0.167593
\(936\) 22.5973 0.738616
\(937\) −56.2492 −1.83758 −0.918791 0.394744i \(-0.870833\pi\)
−0.918791 + 0.394744i \(0.870833\pi\)
\(938\) −2.29180 −0.0748298
\(939\) −48.0689 −1.56867
\(940\) 15.3738 0.501439
\(941\) −50.8329 −1.65710 −0.828552 0.559912i \(-0.810835\pi\)
−0.828552 + 0.559912i \(0.810835\pi\)
\(942\) 4.11512 0.134078
\(943\) −51.2942 −1.67037
\(944\) −18.7001 −0.608636
\(945\) 3.90879 0.127153
\(946\) −0.334369 −0.0108713
\(947\) −2.23954 −0.0727752 −0.0363876 0.999338i \(-0.511585\pi\)
−0.0363876 + 0.999338i \(0.511585\pi\)
\(948\) 47.1246 1.53054
\(949\) 29.1246 0.945425
\(950\) 0 0
\(951\) −49.8012 −1.61492
\(952\) −0.795221 −0.0257732
\(953\) −53.4550 −1.73158 −0.865788 0.500411i \(-0.833182\pi\)
−0.865788 + 0.500411i \(0.833182\pi\)
\(954\) −4.47907 −0.145015
\(955\) −31.8328 −1.03009
\(956\) −30.3175 −0.980537
\(957\) 35.9442 1.16191
\(958\) 9.72949 0.314346
\(959\) 6.86474 0.221674
\(960\) −24.0903 −0.777512
\(961\) 0 0
\(962\) 11.1246 0.358672
\(963\) −16.7082 −0.538414
\(964\) 27.3128 0.879687
\(965\) 16.3050 0.524875
\(966\) −6.00000 −0.193047
\(967\) 12.9041 0.414969 0.207485 0.978238i \(-0.433472\pi\)
0.207485 + 0.978238i \(0.433472\pi\)
\(968\) −10.3050 −0.331214
\(969\) 1.23607 0.0397082
\(970\) 5.97871 0.191965
\(971\) 18.1115 0.581224 0.290612 0.956841i \(-0.406141\pi\)
0.290612 + 0.956841i \(0.406141\pi\)
\(972\) −35.7078 −1.14533
\(973\) 1.00155 0.0321083
\(974\) −7.10117 −0.227536
\(975\) 0 0
\(976\) −13.9960 −0.448001
\(977\) 42.4853 1.35922 0.679612 0.733571i \(-0.262148\pi\)
0.679612 + 0.733571i \(0.262148\pi\)
\(978\) 10.8523 0.347020
\(979\) −24.8754 −0.795021
\(980\) −24.8754 −0.794615
\(981\) −32.8885 −1.05005
\(982\) 3.65375 0.116596
\(983\) 3.00465 0.0958336 0.0479168 0.998851i \(-0.484742\pi\)
0.0479168 + 0.998851i \(0.484742\pi\)
\(984\) 25.1707 0.802413
\(985\) −16.5579 −0.527579
\(986\) −0.763932 −0.0243286
\(987\) 8.48528 0.270089
\(988\) 12.7279 0.404929
\(989\) −1.41641 −0.0450391
\(990\) 8.10272 0.257521
\(991\) 4.44897 0.141326 0.0706631 0.997500i \(-0.477488\pi\)
0.0706631 + 0.997500i \(0.477488\pi\)
\(992\) 0 0
\(993\) −58.6525 −1.86128
\(994\) −2.85410 −0.0905266
\(995\) −32.9396 −1.04425
\(996\) −13.4164 −0.425115
\(997\) 48.6656 1.54126 0.770628 0.637285i \(-0.219943\pi\)
0.770628 + 0.637285i \(0.219943\pi\)
\(998\) −7.02236 −0.222289
\(999\) −7.41641 −0.234645
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 961.2.a.h.1.1 4
3.2 odd 2 8649.2.a.r.1.4 4
31.2 even 5 961.2.d.h.531.1 8
31.3 odd 30 961.2.g.i.846.1 16
31.4 even 5 961.2.d.j.388.2 8
31.5 even 3 961.2.c.h.521.2 8
31.6 odd 6 961.2.c.h.439.1 8
31.7 even 15 961.2.g.p.235.1 16
31.8 even 5 961.2.d.j.374.2 8
31.9 even 15 961.2.g.p.732.1 16
31.10 even 15 961.2.g.i.844.2 16
31.11 odd 30 961.2.g.p.338.1 16
31.12 odd 30 961.2.g.i.547.2 16
31.13 odd 30 961.2.g.i.448.2 16
31.14 even 15 961.2.g.p.816.2 16
31.15 odd 10 961.2.d.h.628.2 8
31.16 even 5 961.2.d.h.628.1 8
31.17 odd 30 961.2.g.p.816.1 16
31.18 even 15 961.2.g.i.448.1 16
31.19 even 15 961.2.g.i.547.1 16
31.20 even 15 961.2.g.p.338.2 16
31.21 odd 30 961.2.g.i.844.1 16
31.22 odd 30 961.2.g.p.732.2 16
31.23 odd 10 961.2.d.j.374.1 8
31.24 odd 30 961.2.g.p.235.2 16
31.25 even 3 961.2.c.h.439.2 8
31.26 odd 6 961.2.c.h.521.1 8
31.27 odd 10 961.2.d.j.388.1 8
31.28 even 15 961.2.g.i.846.2 16
31.29 odd 10 961.2.d.h.531.2 8
31.30 odd 2 inner 961.2.a.h.1.2 yes 4
93.92 even 2 8649.2.a.r.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
961.2.a.h.1.1 4 1.1 even 1 trivial
961.2.a.h.1.2 yes 4 31.30 odd 2 inner
961.2.c.h.439.1 8 31.6 odd 6
961.2.c.h.439.2 8 31.25 even 3
961.2.c.h.521.1 8 31.26 odd 6
961.2.c.h.521.2 8 31.5 even 3
961.2.d.h.531.1 8 31.2 even 5
961.2.d.h.531.2 8 31.29 odd 10
961.2.d.h.628.1 8 31.16 even 5
961.2.d.h.628.2 8 31.15 odd 10
961.2.d.j.374.1 8 31.23 odd 10
961.2.d.j.374.2 8 31.8 even 5
961.2.d.j.388.1 8 31.27 odd 10
961.2.d.j.388.2 8 31.4 even 5
961.2.g.i.448.1 16 31.18 even 15
961.2.g.i.448.2 16 31.13 odd 30
961.2.g.i.547.1 16 31.19 even 15
961.2.g.i.547.2 16 31.12 odd 30
961.2.g.i.844.1 16 31.21 odd 30
961.2.g.i.844.2 16 31.10 even 15
961.2.g.i.846.1 16 31.3 odd 30
961.2.g.i.846.2 16 31.28 even 15
961.2.g.p.235.1 16 31.7 even 15
961.2.g.p.235.2 16 31.24 odd 30
961.2.g.p.338.1 16 31.11 odd 30
961.2.g.p.338.2 16 31.20 even 15
961.2.g.p.732.1 16 31.9 even 15
961.2.g.p.732.2 16 31.22 odd 30
961.2.g.p.816.1 16 31.17 odd 30
961.2.g.p.816.2 16 31.14 even 15
8649.2.a.r.1.3 4 93.92 even 2
8649.2.a.r.1.4 4 3.2 odd 2