Properties

Label 1875.2.a.c.1.1
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
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] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.9719503790\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.61803 q^{12} -1.00000 q^{13} +1.23607 q^{14} +1.85410 q^{16} +0.236068 q^{17} -0.618034 q^{18} +6.70820 q^{19} -2.00000 q^{21} +1.85410 q^{22} +7.61803 q^{23} +2.23607 q^{24} +0.618034 q^{26} +1.00000 q^{27} +3.23607 q^{28} -1.38197 q^{29} -4.70820 q^{31} -5.61803 q^{32} -3.00000 q^{33} -0.145898 q^{34} -1.61803 q^{36} -2.00000 q^{37} -4.14590 q^{38} -1.00000 q^{39} -11.6180 q^{41} +1.23607 q^{42} -9.61803 q^{43} +4.85410 q^{44} -4.70820 q^{46} -9.23607 q^{47} +1.85410 q^{48} -3.00000 q^{49} +0.236068 q^{51} +1.61803 q^{52} +6.76393 q^{53} -0.618034 q^{54} -4.47214 q^{56} +6.70820 q^{57} +0.854102 q^{58} -13.9443 q^{59} -4.70820 q^{61} +2.90983 q^{62} -2.00000 q^{63} -0.236068 q^{64} +1.85410 q^{66} +9.18034 q^{67} -0.381966 q^{68} +7.61803 q^{69} -1.09017 q^{71} +2.23607 q^{72} +2.29180 q^{73} +1.23607 q^{74} -10.8541 q^{76} +6.00000 q^{77} +0.618034 q^{78} -15.8541 q^{79} +1.00000 q^{81} +7.18034 q^{82} +9.00000 q^{83} +3.23607 q^{84} +5.94427 q^{86} -1.38197 q^{87} -6.70820 q^{88} +11.1803 q^{89} +2.00000 q^{91} -12.3262 q^{92} -4.70820 q^{93} +5.70820 q^{94} -5.61803 q^{96} -2.85410 q^{97} +1.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 4 q^{21} - 3 q^{22} + 13 q^{23} - q^{26} + 2 q^{27} + 2 q^{28} - 5 q^{29} + 4 q^{31} - 9 q^{32} - 6 q^{33} - 7 q^{34} - q^{36} - 4 q^{37} - 15 q^{38} - 2 q^{39} - 21 q^{41} - 2 q^{42} - 17 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} - 3 q^{48} - 6 q^{49} - 4 q^{51} + q^{52} + 18 q^{53} + q^{54} - 5 q^{58} - 10 q^{59} + 4 q^{61} + 17 q^{62} - 4 q^{63} + 4 q^{64} - 3 q^{66} - 4 q^{67} - 3 q^{68} + 13 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{74} - 15 q^{76} + 12 q^{77} - q^{78} - 25 q^{79} + 2 q^{81} - 8 q^{82} + 18 q^{83} + 2 q^{84} - 6 q^{86} - 5 q^{87} + 4 q^{91} - 9 q^{92} + 4 q^{93} - 2 q^{94} - 9 q^{96} + q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.61803 −0.467086
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 1.23607 0.330353
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 0.236068 0.0572549 0.0286274 0.999590i \(-0.490886\pi\)
0.0286274 + 0.999590i \(0.490886\pi\)
\(18\) −0.618034 −0.145672
\(19\) 6.70820 1.53897 0.769484 0.638666i \(-0.220514\pi\)
0.769484 + 0.638666i \(0.220514\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 1.85410 0.395296
\(23\) 7.61803 1.58847 0.794235 0.607611i \(-0.207872\pi\)
0.794235 + 0.607611i \(0.207872\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) 0.618034 0.121206
\(27\) 1.00000 0.192450
\(28\) 3.23607 0.611559
\(29\) −1.38197 −0.256625 −0.128312 0.991734i \(-0.540956\pi\)
−0.128312 + 0.991734i \(0.540956\pi\)
\(30\) 0 0
\(31\) −4.70820 −0.845618 −0.422809 0.906219i \(-0.638956\pi\)
−0.422809 + 0.906219i \(0.638956\pi\)
\(32\) −5.61803 −0.993137
\(33\) −3.00000 −0.522233
\(34\) −0.145898 −0.0250213
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.14590 −0.672553
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −11.6180 −1.81443 −0.907216 0.420665i \(-0.861797\pi\)
−0.907216 + 0.420665i \(0.861797\pi\)
\(42\) 1.23607 0.190729
\(43\) −9.61803 −1.46674 −0.733368 0.679832i \(-0.762053\pi\)
−0.733368 + 0.679832i \(0.762053\pi\)
\(44\) 4.85410 0.731783
\(45\) 0 0
\(46\) −4.70820 −0.694187
\(47\) −9.23607 −1.34722 −0.673609 0.739087i \(-0.735257\pi\)
−0.673609 + 0.739087i \(0.735257\pi\)
\(48\) 1.85410 0.267617
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0.236068 0.0330561
\(52\) 1.61803 0.224381
\(53\) 6.76393 0.929098 0.464549 0.885548i \(-0.346217\pi\)
0.464549 + 0.885548i \(0.346217\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) −4.47214 −0.597614
\(57\) 6.70820 0.888523
\(58\) 0.854102 0.112149
\(59\) −13.9443 −1.81539 −0.907695 0.419631i \(-0.862159\pi\)
−0.907695 + 0.419631i \(0.862159\pi\)
\(60\) 0 0
\(61\) −4.70820 −0.602824 −0.301412 0.953494i \(-0.597458\pi\)
−0.301412 + 0.953494i \(0.597458\pi\)
\(62\) 2.90983 0.369549
\(63\) −2.00000 −0.251976
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 1.85410 0.228224
\(67\) 9.18034 1.12156 0.560779 0.827966i \(-0.310502\pi\)
0.560779 + 0.827966i \(0.310502\pi\)
\(68\) −0.381966 −0.0463202
\(69\) 7.61803 0.917104
\(70\) 0 0
\(71\) −1.09017 −0.129379 −0.0646897 0.997905i \(-0.520606\pi\)
−0.0646897 + 0.997905i \(0.520606\pi\)
\(72\) 2.23607 0.263523
\(73\) 2.29180 0.268234 0.134117 0.990965i \(-0.457180\pi\)
0.134117 + 0.990965i \(0.457180\pi\)
\(74\) 1.23607 0.143690
\(75\) 0 0
\(76\) −10.8541 −1.24505
\(77\) 6.00000 0.683763
\(78\) 0.618034 0.0699786
\(79\) −15.8541 −1.78373 −0.891863 0.452306i \(-0.850602\pi\)
−0.891863 + 0.452306i \(0.850602\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.18034 0.792936
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 3.23607 0.353084
\(85\) 0 0
\(86\) 5.94427 0.640987
\(87\) −1.38197 −0.148162
\(88\) −6.70820 −0.715097
\(89\) 11.1803 1.18511 0.592557 0.805529i \(-0.298119\pi\)
0.592557 + 0.805529i \(0.298119\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −12.3262 −1.28510
\(93\) −4.70820 −0.488218
\(94\) 5.70820 0.588756
\(95\) 0 0
\(96\) −5.61803 −0.573388
\(97\) −2.85410 −0.289790 −0.144895 0.989447i \(-0.546284\pi\)
−0.144895 + 0.989447i \(0.546284\pi\)
\(98\) 1.85410 0.187293
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −11.6180 −1.15604 −0.578019 0.816023i \(-0.696174\pi\)
−0.578019 + 0.816023i \(0.696174\pi\)
\(102\) −0.145898 −0.0144461
\(103\) 12.4164 1.22343 0.611713 0.791080i \(-0.290481\pi\)
0.611713 + 0.791080i \(0.290481\pi\)
\(104\) −2.23607 −0.219265
\(105\) 0 0
\(106\) −4.18034 −0.406031
\(107\) −7.85410 −0.759285 −0.379642 0.925133i \(-0.623953\pi\)
−0.379642 + 0.925133i \(0.623953\pi\)
\(108\) −1.61803 −0.155695
\(109\) −10.8541 −1.03963 −0.519817 0.854278i \(-0.674000\pi\)
−0.519817 + 0.854278i \(0.674000\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −3.70820 −0.350392
\(113\) −8.23607 −0.774784 −0.387392 0.921915i \(-0.626624\pi\)
−0.387392 + 0.921915i \(0.626624\pi\)
\(114\) −4.14590 −0.388299
\(115\) 0 0
\(116\) 2.23607 0.207614
\(117\) −1.00000 −0.0924500
\(118\) 8.61803 0.793354
\(119\) −0.472136 −0.0432806
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.90983 0.263444
\(123\) −11.6180 −1.04756
\(124\) 7.61803 0.684120
\(125\) 0 0
\(126\) 1.23607 0.110118
\(127\) −17.6525 −1.56640 −0.783202 0.621767i \(-0.786415\pi\)
−0.783202 + 0.621767i \(0.786415\pi\)
\(128\) 11.3820 1.00603
\(129\) −9.61803 −0.846821
\(130\) 0 0
\(131\) 8.18034 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(132\) 4.85410 0.422495
\(133\) −13.4164 −1.16335
\(134\) −5.67376 −0.490138
\(135\) 0 0
\(136\) 0.527864 0.0452640
\(137\) 20.5623 1.75676 0.878378 0.477966i \(-0.158626\pi\)
0.878378 + 0.477966i \(0.158626\pi\)
\(138\) −4.70820 −0.400789
\(139\) −13.4164 −1.13796 −0.568982 0.822350i \(-0.692663\pi\)
−0.568982 + 0.822350i \(0.692663\pi\)
\(140\) 0 0
\(141\) −9.23607 −0.777817
\(142\) 0.673762 0.0565409
\(143\) 3.00000 0.250873
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) −1.41641 −0.117223
\(147\) −3.00000 −0.247436
\(148\) 3.23607 0.266003
\(149\) −1.90983 −0.156459 −0.0782297 0.996935i \(-0.524927\pi\)
−0.0782297 + 0.996935i \(0.524927\pi\)
\(150\) 0 0
\(151\) −4.38197 −0.356599 −0.178300 0.983976i \(-0.557060\pi\)
−0.178300 + 0.983976i \(0.557060\pi\)
\(152\) 15.0000 1.21666
\(153\) 0.236068 0.0190850
\(154\) −3.70820 −0.298816
\(155\) 0 0
\(156\) 1.61803 0.129546
\(157\) 3.85410 0.307591 0.153795 0.988103i \(-0.450850\pi\)
0.153795 + 0.988103i \(0.450850\pi\)
\(158\) 9.79837 0.779517
\(159\) 6.76393 0.536415
\(160\) 0 0
\(161\) −15.2361 −1.20077
\(162\) −0.618034 −0.0485573
\(163\) −15.2705 −1.19608 −0.598039 0.801467i \(-0.704053\pi\)
−0.598039 + 0.801467i \(0.704053\pi\)
\(164\) 18.7984 1.46791
\(165\) 0 0
\(166\) −5.56231 −0.431719
\(167\) −6.79837 −0.526074 −0.263037 0.964786i \(-0.584724\pi\)
−0.263037 + 0.964786i \(0.584724\pi\)
\(168\) −4.47214 −0.345033
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 6.70820 0.512989
\(172\) 15.5623 1.18661
\(173\) 12.0902 0.919199 0.459599 0.888126i \(-0.347993\pi\)
0.459599 + 0.888126i \(0.347993\pi\)
\(174\) 0.854102 0.0647493
\(175\) 0 0
\(176\) −5.56231 −0.419275
\(177\) −13.9443 −1.04812
\(178\) −6.90983 −0.517914
\(179\) 15.6525 1.16992 0.584960 0.811062i \(-0.301110\pi\)
0.584960 + 0.811062i \(0.301110\pi\)
\(180\) 0 0
\(181\) −3.52786 −0.262224 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(182\) −1.23607 −0.0916235
\(183\) −4.70820 −0.348040
\(184\) 17.0344 1.25580
\(185\) 0 0
\(186\) 2.90983 0.213359
\(187\) −0.708204 −0.0517890
\(188\) 14.9443 1.08992
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 1.67376 0.121109 0.0605546 0.998165i \(-0.480713\pi\)
0.0605546 + 0.998165i \(0.480713\pi\)
\(192\) −0.236068 −0.0170367
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 1.76393 0.126643
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) 11.0902 0.790142 0.395071 0.918651i \(-0.370720\pi\)
0.395071 + 0.918651i \(0.370720\pi\)
\(198\) 1.85410 0.131765
\(199\) −1.70820 −0.121091 −0.0605457 0.998165i \(-0.519284\pi\)
−0.0605457 + 0.998165i \(0.519284\pi\)
\(200\) 0 0
\(201\) 9.18034 0.647531
\(202\) 7.18034 0.505207
\(203\) 2.76393 0.193990
\(204\) −0.381966 −0.0267430
\(205\) 0 0
\(206\) −7.67376 −0.534656
\(207\) 7.61803 0.529490
\(208\) −1.85410 −0.128559
\(209\) −20.1246 −1.39205
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −10.9443 −0.751656
\(213\) −1.09017 −0.0746972
\(214\) 4.85410 0.331820
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) 9.41641 0.639227
\(218\) 6.70820 0.454337
\(219\) 2.29180 0.154865
\(220\) 0 0
\(221\) −0.236068 −0.0158797
\(222\) 1.23607 0.0829595
\(223\) −16.8541 −1.12863 −0.564317 0.825558i \(-0.690860\pi\)
−0.564317 + 0.825558i \(0.690860\pi\)
\(224\) 11.2361 0.750741
\(225\) 0 0
\(226\) 5.09017 0.338593
\(227\) 10.2361 0.679392 0.339696 0.940535i \(-0.389676\pi\)
0.339696 + 0.940535i \(0.389676\pi\)
\(228\) −10.8541 −0.718830
\(229\) 6.18034 0.408408 0.204204 0.978928i \(-0.434539\pi\)
0.204204 + 0.978928i \(0.434539\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) −3.09017 −0.202880
\(233\) −12.1803 −0.797961 −0.398980 0.916959i \(-0.630636\pi\)
−0.398980 + 0.916959i \(0.630636\pi\)
\(234\) 0.618034 0.0404021
\(235\) 0 0
\(236\) 22.5623 1.46868
\(237\) −15.8541 −1.02983
\(238\) 0.291796 0.0189143
\(239\) 23.6180 1.52772 0.763862 0.645380i \(-0.223301\pi\)
0.763862 + 0.645380i \(0.223301\pi\)
\(240\) 0 0
\(241\) −8.32624 −0.536340 −0.268170 0.963372i \(-0.586419\pi\)
−0.268170 + 0.963372i \(0.586419\pi\)
\(242\) 1.23607 0.0794575
\(243\) 1.00000 0.0641500
\(244\) 7.61803 0.487695
\(245\) 0 0
\(246\) 7.18034 0.457802
\(247\) −6.70820 −0.426833
\(248\) −10.5279 −0.668520
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 27.9787 1.76600 0.883000 0.469372i \(-0.155520\pi\)
0.883000 + 0.469372i \(0.155520\pi\)
\(252\) 3.23607 0.203853
\(253\) −22.8541 −1.43683
\(254\) 10.9098 0.684544
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −20.2148 −1.26096 −0.630482 0.776204i \(-0.717143\pi\)
−0.630482 + 0.776204i \(0.717143\pi\)
\(258\) 5.94427 0.370074
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −1.38197 −0.0855415
\(262\) −5.05573 −0.312344
\(263\) 25.5066 1.57280 0.786401 0.617716i \(-0.211942\pi\)
0.786401 + 0.617716i \(0.211942\pi\)
\(264\) −6.70820 −0.412861
\(265\) 0 0
\(266\) 8.29180 0.508403
\(267\) 11.1803 0.684226
\(268\) −14.8541 −0.907359
\(269\) −29.4721 −1.79695 −0.898474 0.439027i \(-0.855323\pi\)
−0.898474 + 0.439027i \(0.855323\pi\)
\(270\) 0 0
\(271\) 15.4164 0.936480 0.468240 0.883601i \(-0.344888\pi\)
0.468240 + 0.883601i \(0.344888\pi\)
\(272\) 0.437694 0.0265391
\(273\) 2.00000 0.121046
\(274\) −12.7082 −0.767731
\(275\) 0 0
\(276\) −12.3262 −0.741952
\(277\) −30.9443 −1.85926 −0.929631 0.368493i \(-0.879874\pi\)
−0.929631 + 0.368493i \(0.879874\pi\)
\(278\) 8.29180 0.497309
\(279\) −4.70820 −0.281873
\(280\) 0 0
\(281\) 8.18034 0.487998 0.243999 0.969775i \(-0.421541\pi\)
0.243999 + 0.969775i \(0.421541\pi\)
\(282\) 5.70820 0.339919
\(283\) 15.7082 0.933756 0.466878 0.884322i \(-0.345379\pi\)
0.466878 + 0.884322i \(0.345379\pi\)
\(284\) 1.76393 0.104670
\(285\) 0 0
\(286\) −1.85410 −0.109635
\(287\) 23.2361 1.37158
\(288\) −5.61803 −0.331046
\(289\) −16.9443 −0.996722
\(290\) 0 0
\(291\) −2.85410 −0.167310
\(292\) −3.70820 −0.217006
\(293\) 9.32624 0.544845 0.272422 0.962178i \(-0.412175\pi\)
0.272422 + 0.962178i \(0.412175\pi\)
\(294\) 1.85410 0.108133
\(295\) 0 0
\(296\) −4.47214 −0.259938
\(297\) −3.00000 −0.174078
\(298\) 1.18034 0.0683753
\(299\) −7.61803 −0.440562
\(300\) 0 0
\(301\) 19.2361 1.10875
\(302\) 2.70820 0.155840
\(303\) −11.6180 −0.667439
\(304\) 12.4377 0.713351
\(305\) 0 0
\(306\) −0.145898 −0.00834044
\(307\) 2.14590 0.122473 0.0612364 0.998123i \(-0.480496\pi\)
0.0612364 + 0.998123i \(0.480496\pi\)
\(308\) −9.70820 −0.553176
\(309\) 12.4164 0.706345
\(310\) 0 0
\(311\) −22.4721 −1.27428 −0.637139 0.770749i \(-0.719882\pi\)
−0.637139 + 0.770749i \(0.719882\pi\)
\(312\) −2.23607 −0.126592
\(313\) 15.7082 0.887880 0.443940 0.896056i \(-0.353580\pi\)
0.443940 + 0.896056i \(0.353580\pi\)
\(314\) −2.38197 −0.134422
\(315\) 0 0
\(316\) 25.6525 1.44306
\(317\) 0.437694 0.0245833 0.0122917 0.999924i \(-0.496087\pi\)
0.0122917 + 0.999924i \(0.496087\pi\)
\(318\) −4.18034 −0.234422
\(319\) 4.14590 0.232126
\(320\) 0 0
\(321\) −7.85410 −0.438373
\(322\) 9.41641 0.524756
\(323\) 1.58359 0.0881134
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) 9.43769 0.522706
\(327\) −10.8541 −0.600233
\(328\) −25.9787 −1.43443
\(329\) 18.4721 1.01840
\(330\) 0 0
\(331\) 29.6869 1.63174 0.815870 0.578235i \(-0.196258\pi\)
0.815870 + 0.578235i \(0.196258\pi\)
\(332\) −14.5623 −0.799210
\(333\) −2.00000 −0.109599
\(334\) 4.20163 0.229903
\(335\) 0 0
\(336\) −3.70820 −0.202299
\(337\) −10.8197 −0.589384 −0.294692 0.955592i \(-0.595217\pi\)
−0.294692 + 0.955592i \(0.595217\pi\)
\(338\) 7.41641 0.403399
\(339\) −8.23607 −0.447322
\(340\) 0 0
\(341\) 14.1246 0.764891
\(342\) −4.14590 −0.224184
\(343\) 20.0000 1.07990
\(344\) −21.5066 −1.15956
\(345\) 0 0
\(346\) −7.47214 −0.401705
\(347\) −21.2705 −1.14186 −0.570930 0.820998i \(-0.693417\pi\)
−0.570930 + 0.820998i \(0.693417\pi\)
\(348\) 2.23607 0.119866
\(349\) 2.76393 0.147950 0.0739749 0.997260i \(-0.476432\pi\)
0.0739749 + 0.997260i \(0.476432\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 16.8541 0.898327
\(353\) −14.6180 −0.778039 −0.389020 0.921229i \(-0.627186\pi\)
−0.389020 + 0.921229i \(0.627186\pi\)
\(354\) 8.61803 0.458043
\(355\) 0 0
\(356\) −18.0902 −0.958777
\(357\) −0.472136 −0.0249881
\(358\) −9.67376 −0.511274
\(359\) 6.05573 0.319609 0.159805 0.987149i \(-0.448914\pi\)
0.159805 + 0.987149i \(0.448914\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 2.18034 0.114596
\(363\) −2.00000 −0.104973
\(364\) −3.23607 −0.169616
\(365\) 0 0
\(366\) 2.90983 0.152099
\(367\) −21.4721 −1.12084 −0.560418 0.828210i \(-0.689360\pi\)
−0.560418 + 0.828210i \(0.689360\pi\)
\(368\) 14.1246 0.736296
\(369\) −11.6180 −0.604811
\(370\) 0 0
\(371\) −13.5279 −0.702332
\(372\) 7.61803 0.394977
\(373\) −9.41641 −0.487563 −0.243782 0.969830i \(-0.578388\pi\)
−0.243782 + 0.969830i \(0.578388\pi\)
\(374\) 0.437694 0.0226326
\(375\) 0 0
\(376\) −20.6525 −1.06507
\(377\) 1.38197 0.0711749
\(378\) 1.23607 0.0635765
\(379\) −11.3820 −0.584652 −0.292326 0.956319i \(-0.594429\pi\)
−0.292326 + 0.956319i \(0.594429\pi\)
\(380\) 0 0
\(381\) −17.6525 −0.904364
\(382\) −1.03444 −0.0529266
\(383\) 22.9443 1.17240 0.586199 0.810167i \(-0.300624\pi\)
0.586199 + 0.810167i \(0.300624\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) 6.79837 0.346028
\(387\) −9.61803 −0.488912
\(388\) 4.61803 0.234445
\(389\) −30.6525 −1.55414 −0.777071 0.629413i \(-0.783295\pi\)
−0.777071 + 0.629413i \(0.783295\pi\)
\(390\) 0 0
\(391\) 1.79837 0.0909477
\(392\) −6.70820 −0.338815
\(393\) 8.18034 0.412644
\(394\) −6.85410 −0.345305
\(395\) 0 0
\(396\) 4.85410 0.243928
\(397\) −11.4721 −0.575770 −0.287885 0.957665i \(-0.592952\pi\)
−0.287885 + 0.957665i \(0.592952\pi\)
\(398\) 1.05573 0.0529189
\(399\) −13.4164 −0.671660
\(400\) 0 0
\(401\) 2.72949 0.136304 0.0681521 0.997675i \(-0.478290\pi\)
0.0681521 + 0.997675i \(0.478290\pi\)
\(402\) −5.67376 −0.282982
\(403\) 4.70820 0.234532
\(404\) 18.7984 0.935254
\(405\) 0 0
\(406\) −1.70820 −0.0847767
\(407\) 6.00000 0.297409
\(408\) 0.527864 0.0261332
\(409\) 35.1246 1.73680 0.868400 0.495864i \(-0.165149\pi\)
0.868400 + 0.495864i \(0.165149\pi\)
\(410\) 0 0
\(411\) 20.5623 1.01426
\(412\) −20.0902 −0.989772
\(413\) 27.8885 1.37231
\(414\) −4.70820 −0.231396
\(415\) 0 0
\(416\) 5.61803 0.275447
\(417\) −13.4164 −0.657004
\(418\) 12.4377 0.608348
\(419\) −15.3262 −0.748736 −0.374368 0.927280i \(-0.622140\pi\)
−0.374368 + 0.927280i \(0.622140\pi\)
\(420\) 0 0
\(421\) 14.3607 0.699897 0.349948 0.936769i \(-0.386199\pi\)
0.349948 + 0.936769i \(0.386199\pi\)
\(422\) 1.85410 0.0902563
\(423\) −9.23607 −0.449073
\(424\) 15.1246 0.734516
\(425\) 0 0
\(426\) 0.673762 0.0326439
\(427\) 9.41641 0.455692
\(428\) 12.7082 0.614274
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 34.2361 1.64909 0.824547 0.565794i \(-0.191430\pi\)
0.824547 + 0.565794i \(0.191430\pi\)
\(432\) 1.85410 0.0892055
\(433\) −5.47214 −0.262974 −0.131487 0.991318i \(-0.541975\pi\)
−0.131487 + 0.991318i \(0.541975\pi\)
\(434\) −5.81966 −0.279353
\(435\) 0 0
\(436\) 17.5623 0.841082
\(437\) 51.1033 2.44460
\(438\) −1.41641 −0.0676786
\(439\) −2.96556 −0.141538 −0.0707692 0.997493i \(-0.522545\pi\)
−0.0707692 + 0.997493i \(0.522545\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0.145898 0.00693966
\(443\) 7.41641 0.352364 0.176182 0.984358i \(-0.443625\pi\)
0.176182 + 0.984358i \(0.443625\pi\)
\(444\) 3.23607 0.153577
\(445\) 0 0
\(446\) 10.4164 0.493231
\(447\) −1.90983 −0.0903319
\(448\) 0.472136 0.0223063
\(449\) −21.5066 −1.01496 −0.507479 0.861664i \(-0.669423\pi\)
−0.507479 + 0.861664i \(0.669423\pi\)
\(450\) 0 0
\(451\) 34.8541 1.64122
\(452\) 13.3262 0.626814
\(453\) −4.38197 −0.205883
\(454\) −6.32624 −0.296905
\(455\) 0 0
\(456\) 15.0000 0.702439
\(457\) 25.8885 1.21102 0.605508 0.795840i \(-0.292970\pi\)
0.605508 + 0.795840i \(0.292970\pi\)
\(458\) −3.81966 −0.178481
\(459\) 0.236068 0.0110187
\(460\) 0 0
\(461\) 3.18034 0.148123 0.0740616 0.997254i \(-0.476404\pi\)
0.0740616 + 0.997254i \(0.476404\pi\)
\(462\) −3.70820 −0.172521
\(463\) 26.6869 1.24025 0.620123 0.784505i \(-0.287083\pi\)
0.620123 + 0.784505i \(0.287083\pi\)
\(464\) −2.56231 −0.118952
\(465\) 0 0
\(466\) 7.52786 0.348722
\(467\) 16.4164 0.759661 0.379830 0.925056i \(-0.375982\pi\)
0.379830 + 0.925056i \(0.375982\pi\)
\(468\) 1.61803 0.0747936
\(469\) −18.3607 −0.847817
\(470\) 0 0
\(471\) 3.85410 0.177588
\(472\) −31.1803 −1.43519
\(473\) 28.8541 1.32671
\(474\) 9.79837 0.450054
\(475\) 0 0
\(476\) 0.763932 0.0350148
\(477\) 6.76393 0.309699
\(478\) −14.5967 −0.667640
\(479\) −19.7984 −0.904611 −0.452305 0.891863i \(-0.649398\pi\)
−0.452305 + 0.891863i \(0.649398\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 5.14590 0.234389
\(483\) −15.2361 −0.693265
\(484\) 3.23607 0.147094
\(485\) 0 0
\(486\) −0.618034 −0.0280346
\(487\) 14.3820 0.651709 0.325855 0.945420i \(-0.394348\pi\)
0.325855 + 0.945420i \(0.394348\pi\)
\(488\) −10.5279 −0.476574
\(489\) −15.2705 −0.690556
\(490\) 0 0
\(491\) 6.67376 0.301183 0.150591 0.988596i \(-0.451882\pi\)
0.150591 + 0.988596i \(0.451882\pi\)
\(492\) 18.7984 0.847496
\(493\) −0.326238 −0.0146930
\(494\) 4.14590 0.186533
\(495\) 0 0
\(496\) −8.72949 −0.391966
\(497\) 2.18034 0.0978016
\(498\) −5.56231 −0.249253
\(499\) −15.0000 −0.671492 −0.335746 0.941953i \(-0.608988\pi\)
−0.335746 + 0.941953i \(0.608988\pi\)
\(500\) 0 0
\(501\) −6.79837 −0.303729
\(502\) −17.2918 −0.771771
\(503\) −33.0344 −1.47293 −0.736466 0.676474i \(-0.763507\pi\)
−0.736466 + 0.676474i \(0.763507\pi\)
\(504\) −4.47214 −0.199205
\(505\) 0 0
\(506\) 14.1246 0.627916
\(507\) −12.0000 −0.532939
\(508\) 28.5623 1.26725
\(509\) −2.88854 −0.128032 −0.0640162 0.997949i \(-0.520391\pi\)
−0.0640162 + 0.997949i \(0.520391\pi\)
\(510\) 0 0
\(511\) −4.58359 −0.202766
\(512\) −18.7082 −0.826794
\(513\) 6.70820 0.296174
\(514\) 12.4934 0.551061
\(515\) 0 0
\(516\) 15.5623 0.685092
\(517\) 27.7082 1.21861
\(518\) −2.47214 −0.108619
\(519\) 12.0902 0.530700
\(520\) 0 0
\(521\) 28.9098 1.26656 0.633281 0.773922i \(-0.281708\pi\)
0.633281 + 0.773922i \(0.281708\pi\)
\(522\) 0.854102 0.0373830
\(523\) −18.5623 −0.811673 −0.405836 0.913946i \(-0.633020\pi\)
−0.405836 + 0.913946i \(0.633020\pi\)
\(524\) −13.2361 −0.578220
\(525\) 0 0
\(526\) −15.7639 −0.687340
\(527\) −1.11146 −0.0484158
\(528\) −5.56231 −0.242068
\(529\) 35.0344 1.52324
\(530\) 0 0
\(531\) −13.9443 −0.605130
\(532\) 21.7082 0.941170
\(533\) 11.6180 0.503233
\(534\) −6.90983 −0.299018
\(535\) 0 0
\(536\) 20.5279 0.886669
\(537\) 15.6525 0.675454
\(538\) 18.2148 0.785295
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −39.7082 −1.70719 −0.853595 0.520938i \(-0.825582\pi\)
−0.853595 + 0.520938i \(0.825582\pi\)
\(542\) −9.52786 −0.409257
\(543\) −3.52786 −0.151395
\(544\) −1.32624 −0.0568620
\(545\) 0 0
\(546\) −1.23607 −0.0528988
\(547\) 11.2918 0.482802 0.241401 0.970425i \(-0.422393\pi\)
0.241401 + 0.970425i \(0.422393\pi\)
\(548\) −33.2705 −1.42125
\(549\) −4.70820 −0.200941
\(550\) 0 0
\(551\) −9.27051 −0.394937
\(552\) 17.0344 0.725034
\(553\) 31.7082 1.34837
\(554\) 19.1246 0.812527
\(555\) 0 0
\(556\) 21.7082 0.920633
\(557\) −6.34752 −0.268953 −0.134477 0.990917i \(-0.542935\pi\)
−0.134477 + 0.990917i \(0.542935\pi\)
\(558\) 2.90983 0.123183
\(559\) 9.61803 0.406799
\(560\) 0 0
\(561\) −0.708204 −0.0299004
\(562\) −5.05573 −0.213263
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 14.9443 0.629267
\(565\) 0 0
\(566\) −9.70820 −0.408066
\(567\) −2.00000 −0.0839921
\(568\) −2.43769 −0.102283
\(569\) −4.14590 −0.173805 −0.0869025 0.996217i \(-0.527697\pi\)
−0.0869025 + 0.996217i \(0.527697\pi\)
\(570\) 0 0
\(571\) 2.12461 0.0889122 0.0444561 0.999011i \(-0.485845\pi\)
0.0444561 + 0.999011i \(0.485845\pi\)
\(572\) −4.85410 −0.202960
\(573\) 1.67376 0.0699224
\(574\) −14.3607 −0.599403
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) 37.2705 1.55159 0.775796 0.630984i \(-0.217349\pi\)
0.775796 + 0.630984i \(0.217349\pi\)
\(578\) 10.4721 0.435583
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 1.76393 0.0731173
\(583\) −20.2918 −0.840400
\(584\) 5.12461 0.212058
\(585\) 0 0
\(586\) −5.76393 −0.238106
\(587\) −23.3050 −0.961898 −0.480949 0.876748i \(-0.659708\pi\)
−0.480949 + 0.876748i \(0.659708\pi\)
\(588\) 4.85410 0.200180
\(589\) −31.5836 −1.30138
\(590\) 0 0
\(591\) 11.0902 0.456189
\(592\) −3.70820 −0.152406
\(593\) 15.3820 0.631662 0.315831 0.948816i \(-0.397717\pi\)
0.315831 + 0.948816i \(0.397717\pi\)
\(594\) 1.85410 0.0760747
\(595\) 0 0
\(596\) 3.09017 0.126578
\(597\) −1.70820 −0.0699121
\(598\) 4.70820 0.192533
\(599\) 5.72949 0.234101 0.117050 0.993126i \(-0.462656\pi\)
0.117050 + 0.993126i \(0.462656\pi\)
\(600\) 0 0
\(601\) −11.2918 −0.460602 −0.230301 0.973119i \(-0.573971\pi\)
−0.230301 + 0.973119i \(0.573971\pi\)
\(602\) −11.8885 −0.484541
\(603\) 9.18034 0.373852
\(604\) 7.09017 0.288495
\(605\) 0 0
\(606\) 7.18034 0.291681
\(607\) −16.1459 −0.655342 −0.327671 0.944792i \(-0.606264\pi\)
−0.327671 + 0.944792i \(0.606264\pi\)
\(608\) −37.6869 −1.52841
\(609\) 2.76393 0.112000
\(610\) 0 0
\(611\) 9.23607 0.373651
\(612\) −0.381966 −0.0154401
\(613\) −46.1246 −1.86296 −0.931478 0.363798i \(-0.881480\pi\)
−0.931478 + 0.363798i \(0.881480\pi\)
\(614\) −1.32624 −0.0535226
\(615\) 0 0
\(616\) 13.4164 0.540562
\(617\) 20.7639 0.835924 0.417962 0.908464i \(-0.362744\pi\)
0.417962 + 0.908464i \(0.362744\pi\)
\(618\) −7.67376 −0.308684
\(619\) 0.729490 0.0293207 0.0146603 0.999893i \(-0.495333\pi\)
0.0146603 + 0.999893i \(0.495333\pi\)
\(620\) 0 0
\(621\) 7.61803 0.305701
\(622\) 13.8885 0.556880
\(623\) −22.3607 −0.895862
\(624\) −1.85410 −0.0742235
\(625\) 0 0
\(626\) −9.70820 −0.388018
\(627\) −20.1246 −0.803700
\(628\) −6.23607 −0.248846
\(629\) −0.472136 −0.0188253
\(630\) 0 0
\(631\) −15.2361 −0.606538 −0.303269 0.952905i \(-0.598078\pi\)
−0.303269 + 0.952905i \(0.598078\pi\)
\(632\) −35.4508 −1.41016
\(633\) −3.00000 −0.119239
\(634\) −0.270510 −0.0107433
\(635\) 0 0
\(636\) −10.9443 −0.433969
\(637\) 3.00000 0.118864
\(638\) −2.56231 −0.101443
\(639\) −1.09017 −0.0431265
\(640\) 0 0
\(641\) −7.67376 −0.303095 −0.151548 0.988450i \(-0.548426\pi\)
−0.151548 + 0.988450i \(0.548426\pi\)
\(642\) 4.85410 0.191576
\(643\) 17.0902 0.673971 0.336985 0.941510i \(-0.390593\pi\)
0.336985 + 0.941510i \(0.390593\pi\)
\(644\) 24.6525 0.971444
\(645\) 0 0
\(646\) −0.978714 −0.0385070
\(647\) 10.0344 0.394495 0.197247 0.980354i \(-0.436800\pi\)
0.197247 + 0.980354i \(0.436800\pi\)
\(648\) 2.23607 0.0878410
\(649\) 41.8328 1.64208
\(650\) 0 0
\(651\) 9.41641 0.369058
\(652\) 24.7082 0.967648
\(653\) −1.65248 −0.0646664 −0.0323332 0.999477i \(-0.510294\pi\)
−0.0323332 + 0.999477i \(0.510294\pi\)
\(654\) 6.70820 0.262312
\(655\) 0 0
\(656\) −21.5410 −0.841036
\(657\) 2.29180 0.0894115
\(658\) −11.4164 −0.445058
\(659\) 2.23607 0.0871048 0.0435524 0.999051i \(-0.486132\pi\)
0.0435524 + 0.999051i \(0.486132\pi\)
\(660\) 0 0
\(661\) −30.8885 −1.20143 −0.600713 0.799465i \(-0.705116\pi\)
−0.600713 + 0.799465i \(0.705116\pi\)
\(662\) −18.3475 −0.713097
\(663\) −0.236068 −0.00916812
\(664\) 20.1246 0.780986
\(665\) 0 0
\(666\) 1.23607 0.0478967
\(667\) −10.5279 −0.407641
\(668\) 11.0000 0.425603
\(669\) −16.8541 −0.651617
\(670\) 0 0
\(671\) 14.1246 0.545275
\(672\) 11.2361 0.433441
\(673\) 24.7771 0.955087 0.477543 0.878608i \(-0.341527\pi\)
0.477543 + 0.878608i \(0.341527\pi\)
\(674\) 6.68692 0.257570
\(675\) 0 0
\(676\) 19.4164 0.746785
\(677\) −9.11146 −0.350182 −0.175091 0.984552i \(-0.556022\pi\)
−0.175091 + 0.984552i \(0.556022\pi\)
\(678\) 5.09017 0.195487
\(679\) 5.70820 0.219061
\(680\) 0 0
\(681\) 10.2361 0.392247
\(682\) −8.72949 −0.334269
\(683\) 48.5967 1.85950 0.929751 0.368188i \(-0.120022\pi\)
0.929751 + 0.368188i \(0.120022\pi\)
\(684\) −10.8541 −0.415017
\(685\) 0 0
\(686\) −12.3607 −0.471933
\(687\) 6.18034 0.235795
\(688\) −17.8328 −0.679870
\(689\) −6.76393 −0.257685
\(690\) 0 0
\(691\) 3.90983 0.148737 0.0743685 0.997231i \(-0.476306\pi\)
0.0743685 + 0.997231i \(0.476306\pi\)
\(692\) −19.5623 −0.743647
\(693\) 6.00000 0.227921
\(694\) 13.1459 0.499011
\(695\) 0 0
\(696\) −3.09017 −0.117133
\(697\) −2.74265 −0.103885
\(698\) −1.70820 −0.0646565
\(699\) −12.1803 −0.460703
\(700\) 0 0
\(701\) −17.3475 −0.655207 −0.327603 0.944815i \(-0.606241\pi\)
−0.327603 + 0.944815i \(0.606241\pi\)
\(702\) 0.618034 0.0233262
\(703\) −13.4164 −0.506009
\(704\) 0.708204 0.0266914
\(705\) 0 0
\(706\) 9.03444 0.340016
\(707\) 23.2361 0.873882
\(708\) 22.5623 0.847943
\(709\) −29.7984 −1.11910 −0.559551 0.828796i \(-0.689026\pi\)
−0.559551 + 0.828796i \(0.689026\pi\)
\(710\) 0 0
\(711\) −15.8541 −0.594575
\(712\) 25.0000 0.936915
\(713\) −35.8673 −1.34324
\(714\) 0.291796 0.0109202
\(715\) 0 0
\(716\) −25.3262 −0.946486
\(717\) 23.6180 0.882032
\(718\) −3.74265 −0.139674
\(719\) 5.12461 0.191116 0.0955579 0.995424i \(-0.469536\pi\)
0.0955579 + 0.995424i \(0.469536\pi\)
\(720\) 0 0
\(721\) −24.8328 −0.924822
\(722\) −16.0689 −0.598022
\(723\) −8.32624 −0.309656
\(724\) 5.70820 0.212144
\(725\) 0 0
\(726\) 1.23607 0.0458748
\(727\) −34.5623 −1.28184 −0.640922 0.767606i \(-0.721448\pi\)
−0.640922 + 0.767606i \(0.721448\pi\)
\(728\) 4.47214 0.165748
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.27051 −0.0839778
\(732\) 7.61803 0.281571
\(733\) −16.8541 −0.622520 −0.311260 0.950325i \(-0.600751\pi\)
−0.311260 + 0.950325i \(0.600751\pi\)
\(734\) 13.2705 0.489823
\(735\) 0 0
\(736\) −42.7984 −1.57757
\(737\) −27.5410 −1.01449
\(738\) 7.18034 0.264312
\(739\) 11.7082 0.430693 0.215347 0.976538i \(-0.430912\pi\)
0.215347 + 0.976538i \(0.430912\pi\)
\(740\) 0 0
\(741\) −6.70820 −0.246432
\(742\) 8.36068 0.306930
\(743\) −25.4721 −0.934482 −0.467241 0.884130i \(-0.654752\pi\)
−0.467241 + 0.884130i \(0.654752\pi\)
\(744\) −10.5279 −0.385970
\(745\) 0 0
\(746\) 5.81966 0.213073
\(747\) 9.00000 0.329293
\(748\) 1.14590 0.0418982
\(749\) 15.7082 0.573965
\(750\) 0 0
\(751\) 28.7082 1.04758 0.523789 0.851848i \(-0.324518\pi\)
0.523789 + 0.851848i \(0.324518\pi\)
\(752\) −17.1246 −0.624470
\(753\) 27.9787 1.01960
\(754\) −0.854102 −0.0311046
\(755\) 0 0
\(756\) 3.23607 0.117695
\(757\) −1.27051 −0.0461775 −0.0230887 0.999733i \(-0.507350\pi\)
−0.0230887 + 0.999733i \(0.507350\pi\)
\(758\) 7.03444 0.255502
\(759\) −22.8541 −0.829551
\(760\) 0 0
\(761\) −19.1803 −0.695287 −0.347643 0.937627i \(-0.613018\pi\)
−0.347643 + 0.937627i \(0.613018\pi\)
\(762\) 10.9098 0.395221
\(763\) 21.7082 0.785890
\(764\) −2.70820 −0.0979794
\(765\) 0 0
\(766\) −14.1803 −0.512357
\(767\) 13.9443 0.503498
\(768\) −6.56231 −0.236797
\(769\) 26.3050 0.948581 0.474290 0.880368i \(-0.342705\pi\)
0.474290 + 0.880368i \(0.342705\pi\)
\(770\) 0 0
\(771\) −20.2148 −0.728018
\(772\) 17.7984 0.640577
\(773\) 29.7771 1.07101 0.535504 0.844533i \(-0.320122\pi\)
0.535504 + 0.844533i \(0.320122\pi\)
\(774\) 5.94427 0.213662
\(775\) 0 0
\(776\) −6.38197 −0.229099
\(777\) 4.00000 0.143499
\(778\) 18.9443 0.679185
\(779\) −77.9361 −2.79235
\(780\) 0 0
\(781\) 3.27051 0.117028
\(782\) −1.11146 −0.0397456
\(783\) −1.38197 −0.0493874
\(784\) −5.56231 −0.198654
\(785\) 0 0
\(786\) −5.05573 −0.180332
\(787\) 23.8541 0.850307 0.425153 0.905121i \(-0.360220\pi\)
0.425153 + 0.905121i \(0.360220\pi\)
\(788\) −17.9443 −0.639238
\(789\) 25.5066 0.908058
\(790\) 0 0
\(791\) 16.4721 0.585682
\(792\) −6.70820 −0.238366
\(793\) 4.70820 0.167193
\(794\) 7.09017 0.251621
\(795\) 0 0
\(796\) 2.76393 0.0979650
\(797\) 46.0132 1.62987 0.814935 0.579553i \(-0.196773\pi\)
0.814935 + 0.579553i \(0.196773\pi\)
\(798\) 8.29180 0.293526
\(799\) −2.18034 −0.0771349
\(800\) 0 0
\(801\) 11.1803 0.395038
\(802\) −1.68692 −0.0595671
\(803\) −6.87539 −0.242627
\(804\) −14.8541 −0.523864
\(805\) 0 0
\(806\) −2.90983 −0.102494
\(807\) −29.4721 −1.03747
\(808\) −25.9787 −0.913928
\(809\) 24.9230 0.876246 0.438123 0.898915i \(-0.355644\pi\)
0.438123 + 0.898915i \(0.355644\pi\)
\(810\) 0 0
\(811\) 37.7771 1.32653 0.663266 0.748383i \(-0.269170\pi\)
0.663266 + 0.748383i \(0.269170\pi\)
\(812\) −4.47214 −0.156941
\(813\) 15.4164 0.540677
\(814\) −3.70820 −0.129972
\(815\) 0 0
\(816\) 0.437694 0.0153224
\(817\) −64.5197 −2.25726
\(818\) −21.7082 −0.759010
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −11.9443 −0.416858 −0.208429 0.978038i \(-0.566835\pi\)
−0.208429 + 0.978038i \(0.566835\pi\)
\(822\) −12.7082 −0.443250
\(823\) −8.43769 −0.294120 −0.147060 0.989128i \(-0.546981\pi\)
−0.147060 + 0.989128i \(0.546981\pi\)
\(824\) 27.7639 0.967202
\(825\) 0 0
\(826\) −17.2361 −0.599720
\(827\) 2.02129 0.0702870 0.0351435 0.999382i \(-0.488811\pi\)
0.0351435 + 0.999382i \(0.488811\pi\)
\(828\) −12.3262 −0.428366
\(829\) −9.87539 −0.342986 −0.171493 0.985185i \(-0.554859\pi\)
−0.171493 + 0.985185i \(0.554859\pi\)
\(830\) 0 0
\(831\) −30.9443 −1.07344
\(832\) 0.236068 0.00818418
\(833\) −0.708204 −0.0245378
\(834\) 8.29180 0.287121
\(835\) 0 0
\(836\) 32.5623 1.12619
\(837\) −4.70820 −0.162739
\(838\) 9.47214 0.327210
\(839\) 48.2148 1.66456 0.832280 0.554356i \(-0.187035\pi\)
0.832280 + 0.554356i \(0.187035\pi\)
\(840\) 0 0
\(841\) −27.0902 −0.934144
\(842\) −8.87539 −0.305866
\(843\) 8.18034 0.281746
\(844\) 4.85410 0.167085
\(845\) 0 0
\(846\) 5.70820 0.196252
\(847\) 4.00000 0.137442
\(848\) 12.5410 0.430660
\(849\) 15.7082 0.539104
\(850\) 0 0
\(851\) −15.2361 −0.522286
\(852\) 1.76393 0.0604313
\(853\) 53.3951 1.82821 0.914107 0.405473i \(-0.132893\pi\)
0.914107 + 0.405473i \(0.132893\pi\)
\(854\) −5.81966 −0.199145
\(855\) 0 0
\(856\) −17.5623 −0.600267
\(857\) 26.9443 0.920399 0.460199 0.887816i \(-0.347778\pi\)
0.460199 + 0.887816i \(0.347778\pi\)
\(858\) −1.85410 −0.0632980
\(859\) −25.1246 −0.857241 −0.428620 0.903485i \(-0.641000\pi\)
−0.428620 + 0.903485i \(0.641000\pi\)
\(860\) 0 0
\(861\) 23.2361 0.791883
\(862\) −21.1591 −0.720680
\(863\) −45.0689 −1.53416 −0.767081 0.641550i \(-0.778292\pi\)
−0.767081 + 0.641550i \(0.778292\pi\)
\(864\) −5.61803 −0.191129
\(865\) 0 0
\(866\) 3.38197 0.114924
\(867\) −16.9443 −0.575458
\(868\) −15.2361 −0.517146
\(869\) 47.5623 1.61344
\(870\) 0 0
\(871\) −9.18034 −0.311064
\(872\) −24.2705 −0.821903
\(873\) −2.85410 −0.0965967
\(874\) −31.5836 −1.06833
\(875\) 0 0
\(876\) −3.70820 −0.125289
\(877\) 2.87539 0.0970950 0.0485475 0.998821i \(-0.484541\pi\)
0.0485475 + 0.998821i \(0.484541\pi\)
\(878\) 1.83282 0.0618545
\(879\) 9.32624 0.314566
\(880\) 0 0
\(881\) 3.90983 0.131726 0.0658628 0.997829i \(-0.479020\pi\)
0.0658628 + 0.997829i \(0.479020\pi\)
\(882\) 1.85410 0.0624309
\(883\) 53.7984 1.81046 0.905230 0.424923i \(-0.139699\pi\)
0.905230 + 0.424923i \(0.139699\pi\)
\(884\) 0.381966 0.0128469
\(885\) 0 0
\(886\) −4.58359 −0.153989
\(887\) −21.9230 −0.736102 −0.368051 0.929806i \(-0.619975\pi\)
−0.368051 + 0.929806i \(0.619975\pi\)
\(888\) −4.47214 −0.150075
\(889\) 35.3050 1.18409
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 27.2705 0.913084
\(893\) −61.9574 −2.07333
\(894\) 1.18034 0.0394765
\(895\) 0 0
\(896\) −22.7639 −0.760490
\(897\) −7.61803 −0.254359
\(898\) 13.2918 0.443553
\(899\) 6.50658 0.217007
\(900\) 0 0
\(901\) 1.59675 0.0531954
\(902\) −21.5410 −0.717238
\(903\) 19.2361 0.640136
\(904\) −18.4164 −0.612521
\(905\) 0 0
\(906\) 2.70820 0.0899740
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) −16.5623 −0.549639
\(909\) −11.6180 −0.385346
\(910\) 0 0
\(911\) −20.8885 −0.692068 −0.346034 0.938222i \(-0.612472\pi\)
−0.346034 + 0.938222i \(0.612472\pi\)
\(912\) 12.4377 0.411853
\(913\) −27.0000 −0.893570
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −16.3607 −0.540277
\(918\) −0.145898 −0.00481535
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) 2.14590 0.0707097
\(922\) −1.96556 −0.0647322
\(923\) 1.09017 0.0358834
\(924\) −9.70820 −0.319376
\(925\) 0 0
\(926\) −16.4934 −0.542007
\(927\) 12.4164 0.407808
\(928\) 7.76393 0.254864
\(929\) 19.5967 0.642948 0.321474 0.946918i \(-0.395822\pi\)
0.321474 + 0.946918i \(0.395822\pi\)
\(930\) 0 0
\(931\) −20.1246 −0.659558
\(932\) 19.7082 0.645564
\(933\) −22.4721 −0.735705
\(934\) −10.1459 −0.331984
\(935\) 0 0
\(936\) −2.23607 −0.0730882
\(937\) 16.4164 0.536301 0.268150 0.963377i \(-0.413588\pi\)
0.268150 + 0.963377i \(0.413588\pi\)
\(938\) 11.3475 0.370510
\(939\) 15.7082 0.512618
\(940\) 0 0
\(941\) 11.0213 0.359284 0.179642 0.983732i \(-0.442506\pi\)
0.179642 + 0.983732i \(0.442506\pi\)
\(942\) −2.38197 −0.0776086
\(943\) −88.5066 −2.88217
\(944\) −25.8541 −0.841479
\(945\) 0 0
\(946\) −17.8328 −0.579795
\(947\) 29.8328 0.969436 0.484718 0.874670i \(-0.338922\pi\)
0.484718 + 0.874670i \(0.338922\pi\)
\(948\) 25.6525 0.833154
\(949\) −2.29180 −0.0743948
\(950\) 0 0
\(951\) 0.437694 0.0141932
\(952\) −1.05573 −0.0342163
\(953\) −59.9443 −1.94179 −0.970893 0.239515i \(-0.923012\pi\)
−0.970893 + 0.239515i \(0.923012\pi\)
\(954\) −4.18034 −0.135344
\(955\) 0 0
\(956\) −38.2148 −1.23595
\(957\) 4.14590 0.134018
\(958\) 12.2361 0.395329
\(959\) −41.1246 −1.32798
\(960\) 0 0
\(961\) −8.83282 −0.284930
\(962\) −1.23607 −0.0398524
\(963\) −7.85410 −0.253095
\(964\) 13.4721 0.433908
\(965\) 0 0
\(966\) 9.41641 0.302968
\(967\) −8.58359 −0.276030 −0.138015 0.990430i \(-0.544072\pi\)
−0.138015 + 0.990430i \(0.544072\pi\)
\(968\) −4.47214 −0.143740
\(969\) 1.58359 0.0508723
\(970\) 0 0
\(971\) −5.88854 −0.188972 −0.0944862 0.995526i \(-0.530121\pi\)
−0.0944862 + 0.995526i \(0.530121\pi\)
\(972\) −1.61803 −0.0518985
\(973\) 26.8328 0.860221
\(974\) −8.88854 −0.284807
\(975\) 0 0
\(976\) −8.72949 −0.279424
\(977\) −6.34752 −0.203075 −0.101538 0.994832i \(-0.532376\pi\)
−0.101538 + 0.994832i \(0.532376\pi\)
\(978\) 9.43769 0.301784
\(979\) −33.5410 −1.07198
\(980\) 0 0
\(981\) −10.8541 −0.346545
\(982\) −4.12461 −0.131622
\(983\) −22.3820 −0.713874 −0.356937 0.934128i \(-0.616179\pi\)
−0.356937 + 0.934128i \(0.616179\pi\)
\(984\) −25.9787 −0.828171
\(985\) 0 0
\(986\) 0.201626 0.00642108
\(987\) 18.4721 0.587975
\(988\) 10.8541 0.345315
\(989\) −73.2705 −2.32987
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 26.4508 0.839815
\(993\) 29.6869 0.942086
\(994\) −1.34752 −0.0427409
\(995\) 0 0
\(996\) −14.5623 −0.461424
\(997\) −61.0689 −1.93407 −0.967035 0.254642i \(-0.918042\pi\)
−0.967035 + 0.254642i \(0.918042\pi\)
\(998\) 9.27051 0.293453
\(999\) −2.00000 −0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1875.2.a.c.1.1 yes 2
3.2 odd 2 5625.2.a.b.1.2 2
5.2 odd 4 1875.2.b.a.1249.2 4
5.3 odd 4 1875.2.b.a.1249.3 4
5.4 even 2 1875.2.a.b.1.2 2
15.14 odd 2 5625.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.2 2 5.4 even 2
1875.2.a.c.1.1 yes 2 1.1 even 1 trivial
1875.2.b.a.1249.2 4 5.2 odd 4
1875.2.b.a.1249.3 4 5.3 odd 4
5625.2.a.b.1.2 2 3.2 odd 2
5625.2.a.g.1.1 2 15.14 odd 2