Properties

Label 1875.2.a.m.1.4
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: 8.8.5444000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.0898194\) 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)
 
\(f(q)\) \(=\) \(q-1.08982 q^{2} +1.00000 q^{3} -0.812294 q^{4} -1.08982 q^{6} +3.08724 q^{7} +3.06489 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.08982 q^{2} +1.00000 q^{3} -0.812294 q^{4} -1.08982 q^{6} +3.08724 q^{7} +3.06489 q^{8} +1.00000 q^{9} -1.14831 q^{11} -0.812294 q^{12} -4.07954 q^{13} -3.36454 q^{14} -1.71559 q^{16} -4.62758 q^{17} -1.08982 q^{18} -5.96899 q^{19} +3.08724 q^{21} +1.25145 q^{22} +2.32568 q^{23} +3.06489 q^{24} +4.44596 q^{26} +1.00000 q^{27} -2.50775 q^{28} -5.28417 q^{29} -0.589279 q^{31} -4.26010 q^{32} -1.14831 q^{33} +5.04322 q^{34} -0.812294 q^{36} -11.3997 q^{37} +6.50512 q^{38} -4.07954 q^{39} +9.49200 q^{41} -3.36454 q^{42} -2.42954 q^{43} +0.932764 q^{44} -2.53458 q^{46} -6.04998 q^{47} -1.71559 q^{48} +2.53108 q^{49} -4.62758 q^{51} +3.31379 q^{52} +3.24380 q^{53} -1.08982 q^{54} +9.46207 q^{56} -5.96899 q^{57} +5.75879 q^{58} +3.18640 q^{59} +13.7452 q^{61} +0.642208 q^{62} +3.08724 q^{63} +8.07392 q^{64} +1.25145 q^{66} -3.15873 q^{67} +3.75895 q^{68} +2.32568 q^{69} +6.46551 q^{71} +3.06489 q^{72} -7.20998 q^{73} +12.4236 q^{74} +4.84857 q^{76} -3.54511 q^{77} +4.44596 q^{78} -12.3374 q^{79} +1.00000 q^{81} -10.3446 q^{82} -12.3941 q^{83} -2.50775 q^{84} +2.64776 q^{86} -5.28417 q^{87} -3.51944 q^{88} +1.08404 q^{89} -12.5945 q^{91} -1.88914 q^{92} -0.589279 q^{93} +6.59339 q^{94} -4.26010 q^{96} -4.52132 q^{97} -2.75842 q^{98} -1.14831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24} + 6 q^{26} + 8 q^{27} - 16 q^{28} + 2 q^{29} - 22 q^{31} + 2 q^{32} + 2 q^{33} - 12 q^{34} + 4 q^{36} - 28 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} + 6 q^{42} - 20 q^{43} + 22 q^{44} - 2 q^{46} - 10 q^{47} - 16 q^{51} - 16 q^{52} - 44 q^{53} - 4 q^{54} + 30 q^{56} - 14 q^{57} - 8 q^{58} + 14 q^{59} - 20 q^{61} - 16 q^{62} - 8 q^{63} + 6 q^{64} - 12 q^{66} - 16 q^{67} + 2 q^{68} - 14 q^{69} + 16 q^{71} - 12 q^{72} - 24 q^{73} + 26 q^{74} - 16 q^{76} - 46 q^{77} + 6 q^{78} - 30 q^{79} + 8 q^{81} - 16 q^{82} - 12 q^{83} - 16 q^{84} + 32 q^{86} + 2 q^{87} - 32 q^{88} + 16 q^{89} - 12 q^{91} + 2 q^{92} - 22 q^{93} + 14 q^{94} + 2 q^{96} - 16 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08982 −0.770619 −0.385309 0.922787i \(-0.625905\pi\)
−0.385309 + 0.922787i \(0.625905\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.812294 −0.406147
\(5\) 0 0
\(6\) −1.08982 −0.444917
\(7\) 3.08724 1.16687 0.583434 0.812160i \(-0.301709\pi\)
0.583434 + 0.812160i \(0.301709\pi\)
\(8\) 3.06489 1.08360
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.14831 −0.346228 −0.173114 0.984902i \(-0.555383\pi\)
−0.173114 + 0.984902i \(0.555383\pi\)
\(12\) −0.812294 −0.234489
\(13\) −4.07954 −1.13146 −0.565731 0.824590i \(-0.691406\pi\)
−0.565731 + 0.824590i \(0.691406\pi\)
\(14\) −3.36454 −0.899211
\(15\) 0 0
\(16\) −1.71559 −0.428898
\(17\) −4.62758 −1.12235 −0.561176 0.827696i \(-0.689651\pi\)
−0.561176 + 0.827696i \(0.689651\pi\)
\(18\) −1.08982 −0.256873
\(19\) −5.96899 −1.36938 −0.684690 0.728834i \(-0.740062\pi\)
−0.684690 + 0.728834i \(0.740062\pi\)
\(20\) 0 0
\(21\) 3.08724 0.673692
\(22\) 1.25145 0.266810
\(23\) 2.32568 0.484939 0.242469 0.970159i \(-0.422043\pi\)
0.242469 + 0.970159i \(0.422043\pi\)
\(24\) 3.06489 0.625619
\(25\) 0 0
\(26\) 4.44596 0.871925
\(27\) 1.00000 0.192450
\(28\) −2.50775 −0.473920
\(29\) −5.28417 −0.981245 −0.490623 0.871372i \(-0.663231\pi\)
−0.490623 + 0.871372i \(0.663231\pi\)
\(30\) 0 0
\(31\) −0.589279 −0.105838 −0.0529188 0.998599i \(-0.516852\pi\)
−0.0529188 + 0.998599i \(0.516852\pi\)
\(32\) −4.26010 −0.753086
\(33\) −1.14831 −0.199895
\(34\) 5.04322 0.864906
\(35\) 0 0
\(36\) −0.812294 −0.135382
\(37\) −11.3997 −1.87409 −0.937047 0.349204i \(-0.886452\pi\)
−0.937047 + 0.349204i \(0.886452\pi\)
\(38\) 6.50512 1.05527
\(39\) −4.07954 −0.653250
\(40\) 0 0
\(41\) 9.49200 1.48240 0.741201 0.671283i \(-0.234257\pi\)
0.741201 + 0.671283i \(0.234257\pi\)
\(42\) −3.36454 −0.519160
\(43\) −2.42954 −0.370501 −0.185250 0.982691i \(-0.559310\pi\)
−0.185250 + 0.982691i \(0.559310\pi\)
\(44\) 0.932764 0.140619
\(45\) 0 0
\(46\) −2.53458 −0.373703
\(47\) −6.04998 −0.882480 −0.441240 0.897389i \(-0.645461\pi\)
−0.441240 + 0.897389i \(0.645461\pi\)
\(48\) −1.71559 −0.247624
\(49\) 2.53108 0.361583
\(50\) 0 0
\(51\) −4.62758 −0.647990
\(52\) 3.31379 0.459539
\(53\) 3.24380 0.445570 0.222785 0.974868i \(-0.428485\pi\)
0.222785 + 0.974868i \(0.428485\pi\)
\(54\) −1.08982 −0.148306
\(55\) 0 0
\(56\) 9.46207 1.26442
\(57\) −5.96899 −0.790612
\(58\) 5.75879 0.756166
\(59\) 3.18640 0.414833 0.207417 0.978253i \(-0.433494\pi\)
0.207417 + 0.978253i \(0.433494\pi\)
\(60\) 0 0
\(61\) 13.7452 1.75989 0.879946 0.475074i \(-0.157579\pi\)
0.879946 + 0.475074i \(0.157579\pi\)
\(62\) 0.642208 0.0815605
\(63\) 3.08724 0.388956
\(64\) 8.07392 1.00924
\(65\) 0 0
\(66\) 1.25145 0.154043
\(67\) −3.15873 −0.385901 −0.192950 0.981209i \(-0.561806\pi\)
−0.192950 + 0.981209i \(0.561806\pi\)
\(68\) 3.75895 0.455840
\(69\) 2.32568 0.279979
\(70\) 0 0
\(71\) 6.46551 0.767315 0.383657 0.923475i \(-0.374664\pi\)
0.383657 + 0.923475i \(0.374664\pi\)
\(72\) 3.06489 0.361201
\(73\) −7.20998 −0.843864 −0.421932 0.906627i \(-0.638648\pi\)
−0.421932 + 0.906627i \(0.638648\pi\)
\(74\) 12.4236 1.44421
\(75\) 0 0
\(76\) 4.84857 0.556169
\(77\) −3.54511 −0.404003
\(78\) 4.44596 0.503406
\(79\) −12.3374 −1.38807 −0.694033 0.719944i \(-0.744168\pi\)
−0.694033 + 0.719944i \(0.744168\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.3446 −1.14237
\(83\) −12.3941 −1.36043 −0.680214 0.733013i \(-0.738113\pi\)
−0.680214 + 0.733013i \(0.738113\pi\)
\(84\) −2.50775 −0.273618
\(85\) 0 0
\(86\) 2.64776 0.285515
\(87\) −5.28417 −0.566522
\(88\) −3.51944 −0.375174
\(89\) 1.08404 0.114907 0.0574537 0.998348i \(-0.481702\pi\)
0.0574537 + 0.998348i \(0.481702\pi\)
\(90\) 0 0
\(91\) −12.5945 −1.32027
\(92\) −1.88914 −0.196956
\(93\) −0.589279 −0.0611054
\(94\) 6.59339 0.680056
\(95\) 0 0
\(96\) −4.26010 −0.434795
\(97\) −4.52132 −0.459071 −0.229535 0.973300i \(-0.573721\pi\)
−0.229535 + 0.973300i \(0.573721\pi\)
\(98\) −2.75842 −0.278642
\(99\) −1.14831 −0.115409
\(100\) 0 0
\(101\) −6.61332 −0.658050 −0.329025 0.944321i \(-0.606720\pi\)
−0.329025 + 0.944321i \(0.606720\pi\)
\(102\) 5.04322 0.499354
\(103\) −4.20634 −0.414463 −0.207231 0.978292i \(-0.566445\pi\)
−0.207231 + 0.978292i \(0.566445\pi\)
\(104\) −12.5034 −1.22606
\(105\) 0 0
\(106\) −3.53515 −0.343364
\(107\) 4.01195 0.387849 0.193925 0.981016i \(-0.437878\pi\)
0.193925 + 0.981016i \(0.437878\pi\)
\(108\) −0.812294 −0.0781630
\(109\) 9.09364 0.871013 0.435506 0.900186i \(-0.356569\pi\)
0.435506 + 0.900186i \(0.356569\pi\)
\(110\) 0 0
\(111\) −11.3997 −1.08201
\(112\) −5.29645 −0.500468
\(113\) −3.75465 −0.353208 −0.176604 0.984282i \(-0.556511\pi\)
−0.176604 + 0.984282i \(0.556511\pi\)
\(114\) 6.50512 0.609260
\(115\) 0 0
\(116\) 4.29230 0.398530
\(117\) −4.07954 −0.377154
\(118\) −3.47260 −0.319678
\(119\) −14.2865 −1.30964
\(120\) 0 0
\(121\) −9.68139 −0.880126
\(122\) −14.9798 −1.35621
\(123\) 9.49200 0.855865
\(124\) 0.478668 0.0429856
\(125\) 0 0
\(126\) −3.36454 −0.299737
\(127\) −11.3583 −1.00788 −0.503942 0.863738i \(-0.668117\pi\)
−0.503942 + 0.863738i \(0.668117\pi\)
\(128\) −0.278920 −0.0246532
\(129\) −2.42954 −0.213909
\(130\) 0 0
\(131\) −2.07849 −0.181599 −0.0907993 0.995869i \(-0.528942\pi\)
−0.0907993 + 0.995869i \(0.528942\pi\)
\(132\) 0.932764 0.0811867
\(133\) −18.4277 −1.59789
\(134\) 3.44245 0.297382
\(135\) 0 0
\(136\) −14.1830 −1.21618
\(137\) 19.4032 1.65773 0.828863 0.559452i \(-0.188988\pi\)
0.828863 + 0.559452i \(0.188988\pi\)
\(138\) −2.53458 −0.215757
\(139\) −17.1603 −1.45552 −0.727758 0.685834i \(-0.759438\pi\)
−0.727758 + 0.685834i \(0.759438\pi\)
\(140\) 0 0
\(141\) −6.04998 −0.509500
\(142\) −7.04624 −0.591307
\(143\) 4.68458 0.391744
\(144\) −1.71559 −0.142966
\(145\) 0 0
\(146\) 7.85758 0.650298
\(147\) 2.53108 0.208760
\(148\) 9.25988 0.761157
\(149\) 0.210127 0.0172143 0.00860714 0.999963i \(-0.497260\pi\)
0.00860714 + 0.999963i \(0.497260\pi\)
\(150\) 0 0
\(151\) −4.05924 −0.330336 −0.165168 0.986265i \(-0.552817\pi\)
−0.165168 + 0.986265i \(0.552817\pi\)
\(152\) −18.2943 −1.48386
\(153\) −4.62758 −0.374117
\(154\) 3.86353 0.311332
\(155\) 0 0
\(156\) 3.31379 0.265315
\(157\) −0.440336 −0.0351426 −0.0175713 0.999846i \(-0.505593\pi\)
−0.0175713 + 0.999846i \(0.505593\pi\)
\(158\) 13.4455 1.06967
\(159\) 3.24380 0.257250
\(160\) 0 0
\(161\) 7.17996 0.565860
\(162\) −1.08982 −0.0856243
\(163\) −3.24109 −0.253862 −0.126931 0.991912i \(-0.540513\pi\)
−0.126931 + 0.991912i \(0.540513\pi\)
\(164\) −7.71029 −0.602073
\(165\) 0 0
\(166\) 13.5073 1.04837
\(167\) 15.9605 1.23506 0.617530 0.786547i \(-0.288133\pi\)
0.617530 + 0.786547i \(0.288133\pi\)
\(168\) 9.46207 0.730015
\(169\) 3.64267 0.280205
\(170\) 0 0
\(171\) −5.96899 −0.456460
\(172\) 1.97350 0.150478
\(173\) −0.561789 −0.0427121 −0.0213560 0.999772i \(-0.506798\pi\)
−0.0213560 + 0.999772i \(0.506798\pi\)
\(174\) 5.75879 0.436573
\(175\) 0 0
\(176\) 1.97003 0.148497
\(177\) 3.18640 0.239504
\(178\) −1.18140 −0.0885499
\(179\) 15.9387 1.19131 0.595656 0.803239i \(-0.296892\pi\)
0.595656 + 0.803239i \(0.296892\pi\)
\(180\) 0 0
\(181\) −21.4875 −1.59715 −0.798577 0.601892i \(-0.794414\pi\)
−0.798577 + 0.601892i \(0.794414\pi\)
\(182\) 13.7258 1.01742
\(183\) 13.7452 1.01607
\(184\) 7.12797 0.525481
\(185\) 0 0
\(186\) 0.642208 0.0470890
\(187\) 5.31389 0.388590
\(188\) 4.91436 0.358417
\(189\) 3.08724 0.224564
\(190\) 0 0
\(191\) −18.3939 −1.33093 −0.665467 0.746427i \(-0.731768\pi\)
−0.665467 + 0.746427i \(0.731768\pi\)
\(192\) 8.07392 0.582685
\(193\) 2.02523 0.145780 0.0728898 0.997340i \(-0.476778\pi\)
0.0728898 + 0.997340i \(0.476778\pi\)
\(194\) 4.92742 0.353768
\(195\) 0 0
\(196\) −2.05598 −0.146856
\(197\) −11.5454 −0.822578 −0.411289 0.911505i \(-0.634921\pi\)
−0.411289 + 0.911505i \(0.634921\pi\)
\(198\) 1.25145 0.0889367
\(199\) 22.9779 1.62886 0.814431 0.580260i \(-0.197049\pi\)
0.814431 + 0.580260i \(0.197049\pi\)
\(200\) 0 0
\(201\) −3.15873 −0.222800
\(202\) 7.20733 0.507106
\(203\) −16.3135 −1.14498
\(204\) 3.75895 0.263179
\(205\) 0 0
\(206\) 4.58415 0.319393
\(207\) 2.32568 0.161646
\(208\) 6.99883 0.485282
\(209\) 6.85425 0.474118
\(210\) 0 0
\(211\) −6.88140 −0.473735 −0.236868 0.971542i \(-0.576121\pi\)
−0.236868 + 0.971542i \(0.576121\pi\)
\(212\) −2.63492 −0.180967
\(213\) 6.46551 0.443010
\(214\) −4.37230 −0.298884
\(215\) 0 0
\(216\) 3.06489 0.208540
\(217\) −1.81925 −0.123499
\(218\) −9.91043 −0.671219
\(219\) −7.20998 −0.487205
\(220\) 0 0
\(221\) 18.8784 1.26990
\(222\) 12.4236 0.833816
\(223\) −8.12631 −0.544178 −0.272089 0.962272i \(-0.587714\pi\)
−0.272089 + 0.962272i \(0.587714\pi\)
\(224\) −13.1520 −0.878753
\(225\) 0 0
\(226\) 4.09189 0.272188
\(227\) −25.6300 −1.70112 −0.850561 0.525876i \(-0.823738\pi\)
−0.850561 + 0.525876i \(0.823738\pi\)
\(228\) 4.84857 0.321105
\(229\) 1.23314 0.0814884 0.0407442 0.999170i \(-0.487027\pi\)
0.0407442 + 0.999170i \(0.487027\pi\)
\(230\) 0 0
\(231\) −3.54511 −0.233251
\(232\) −16.1954 −1.06328
\(233\) 2.99987 0.196528 0.0982641 0.995160i \(-0.468671\pi\)
0.0982641 + 0.995160i \(0.468671\pi\)
\(234\) 4.44596 0.290642
\(235\) 0 0
\(236\) −2.58829 −0.168483
\(237\) −12.3374 −0.801400
\(238\) 15.5697 1.00923
\(239\) 22.1341 1.43174 0.715869 0.698235i \(-0.246031\pi\)
0.715869 + 0.698235i \(0.246031\pi\)
\(240\) 0 0
\(241\) −13.4403 −0.865764 −0.432882 0.901451i \(-0.642503\pi\)
−0.432882 + 0.901451i \(0.642503\pi\)
\(242\) 10.5510 0.678242
\(243\) 1.00000 0.0641500
\(244\) −11.1651 −0.714774
\(245\) 0 0
\(246\) −10.3446 −0.659546
\(247\) 24.3507 1.54940
\(248\) −1.80608 −0.114686
\(249\) −12.3941 −0.785444
\(250\) 0 0
\(251\) 18.8799 1.19169 0.595843 0.803101i \(-0.296818\pi\)
0.595843 + 0.803101i \(0.296818\pi\)
\(252\) −2.50775 −0.157973
\(253\) −2.67060 −0.167899
\(254\) 12.3785 0.776694
\(255\) 0 0
\(256\) −15.8439 −0.990242
\(257\) −7.06320 −0.440590 −0.220295 0.975433i \(-0.570702\pi\)
−0.220295 + 0.975433i \(0.570702\pi\)
\(258\) 2.64776 0.164842
\(259\) −35.1936 −2.18682
\(260\) 0 0
\(261\) −5.28417 −0.327082
\(262\) 2.26518 0.139943
\(263\) −20.4356 −1.26011 −0.630057 0.776549i \(-0.716969\pi\)
−0.630057 + 0.776549i \(0.716969\pi\)
\(264\) −3.51944 −0.216607
\(265\) 0 0
\(266\) 20.0829 1.23136
\(267\) 1.08404 0.0663419
\(268\) 2.56582 0.156732
\(269\) 21.1923 1.29212 0.646060 0.763287i \(-0.276416\pi\)
0.646060 + 0.763287i \(0.276416\pi\)
\(270\) 0 0
\(271\) 9.95317 0.604612 0.302306 0.953211i \(-0.402244\pi\)
0.302306 + 0.953211i \(0.402244\pi\)
\(272\) 7.93903 0.481375
\(273\) −12.5945 −0.762257
\(274\) −21.1460 −1.27747
\(275\) 0 0
\(276\) −1.88914 −0.113713
\(277\) 18.7349 1.12567 0.562837 0.826568i \(-0.309710\pi\)
0.562837 + 0.826568i \(0.309710\pi\)
\(278\) 18.7016 1.12165
\(279\) −0.589279 −0.0352792
\(280\) 0 0
\(281\) 29.0916 1.73546 0.867730 0.497035i \(-0.165578\pi\)
0.867730 + 0.497035i \(0.165578\pi\)
\(282\) 6.59339 0.392630
\(283\) −7.18007 −0.426811 −0.213405 0.976964i \(-0.568455\pi\)
−0.213405 + 0.976964i \(0.568455\pi\)
\(284\) −5.25189 −0.311643
\(285\) 0 0
\(286\) −5.10534 −0.301885
\(287\) 29.3041 1.72977
\(288\) −4.26010 −0.251029
\(289\) 4.41447 0.259675
\(290\) 0 0
\(291\) −4.52132 −0.265045
\(292\) 5.85662 0.342733
\(293\) 1.79825 0.105055 0.0525276 0.998619i \(-0.483272\pi\)
0.0525276 + 0.998619i \(0.483272\pi\)
\(294\) −2.75842 −0.160874
\(295\) 0 0
\(296\) −34.9388 −2.03077
\(297\) −1.14831 −0.0666317
\(298\) −0.229001 −0.0132656
\(299\) −9.48773 −0.548689
\(300\) 0 0
\(301\) −7.50057 −0.432326
\(302\) 4.42384 0.254563
\(303\) −6.61332 −0.379926
\(304\) 10.2404 0.587324
\(305\) 0 0
\(306\) 5.04322 0.288302
\(307\) 5.98864 0.341790 0.170895 0.985289i \(-0.445334\pi\)
0.170895 + 0.985289i \(0.445334\pi\)
\(308\) 2.87967 0.164085
\(309\) −4.20634 −0.239290
\(310\) 0 0
\(311\) 23.8684 1.35345 0.676726 0.736235i \(-0.263398\pi\)
0.676726 + 0.736235i \(0.263398\pi\)
\(312\) −12.5034 −0.707863
\(313\) 5.75913 0.325525 0.162763 0.986665i \(-0.447959\pi\)
0.162763 + 0.986665i \(0.447959\pi\)
\(314\) 0.479887 0.0270816
\(315\) 0 0
\(316\) 10.0216 0.563758
\(317\) −12.2481 −0.687922 −0.343961 0.938984i \(-0.611769\pi\)
−0.343961 + 0.938984i \(0.611769\pi\)
\(318\) −3.53515 −0.198242
\(319\) 6.06786 0.339735
\(320\) 0 0
\(321\) 4.01195 0.223925
\(322\) −7.82486 −0.436062
\(323\) 27.6220 1.53693
\(324\) −0.812294 −0.0451274
\(325\) 0 0
\(326\) 3.53220 0.195631
\(327\) 9.09364 0.502880
\(328\) 29.0920 1.60633
\(329\) −18.6778 −1.02974
\(330\) 0 0
\(331\) −6.02899 −0.331383 −0.165692 0.986178i \(-0.552986\pi\)
−0.165692 + 0.986178i \(0.552986\pi\)
\(332\) 10.0676 0.552534
\(333\) −11.3997 −0.624698
\(334\) −17.3941 −0.951761
\(335\) 0 0
\(336\) −5.29645 −0.288945
\(337\) 5.53083 0.301284 0.150642 0.988588i \(-0.451866\pi\)
0.150642 + 0.988588i \(0.451866\pi\)
\(338\) −3.96985 −0.215931
\(339\) −3.75465 −0.203925
\(340\) 0 0
\(341\) 0.676675 0.0366440
\(342\) 6.50512 0.351757
\(343\) −13.7967 −0.744949
\(344\) −7.44626 −0.401476
\(345\) 0 0
\(346\) 0.612249 0.0329147
\(347\) −25.9445 −1.39277 −0.696387 0.717666i \(-0.745210\pi\)
−0.696387 + 0.717666i \(0.745210\pi\)
\(348\) 4.29230 0.230091
\(349\) −19.0025 −1.01718 −0.508591 0.861008i \(-0.669833\pi\)
−0.508591 + 0.861008i \(0.669833\pi\)
\(350\) 0 0
\(351\) −4.07954 −0.217750
\(352\) 4.89191 0.260740
\(353\) 8.19134 0.435981 0.217990 0.975951i \(-0.430050\pi\)
0.217990 + 0.975951i \(0.430050\pi\)
\(354\) −3.47260 −0.184566
\(355\) 0 0
\(356\) −0.880555 −0.0466693
\(357\) −14.2865 −0.756120
\(358\) −17.3703 −0.918048
\(359\) −26.7857 −1.41369 −0.706847 0.707367i \(-0.749883\pi\)
−0.706847 + 0.707367i \(0.749883\pi\)
\(360\) 0 0
\(361\) 16.6288 0.875202
\(362\) 23.4175 1.23080
\(363\) −9.68139 −0.508141
\(364\) 10.2305 0.536222
\(365\) 0 0
\(366\) −14.9798 −0.783006
\(367\) 18.8533 0.984135 0.492068 0.870557i \(-0.336241\pi\)
0.492068 + 0.870557i \(0.336241\pi\)
\(368\) −3.98992 −0.207989
\(369\) 9.49200 0.494134
\(370\) 0 0
\(371\) 10.0144 0.519921
\(372\) 0.478668 0.0248178
\(373\) −17.8477 −0.924118 −0.462059 0.886849i \(-0.652889\pi\)
−0.462059 + 0.886849i \(0.652889\pi\)
\(374\) −5.79118 −0.299455
\(375\) 0 0
\(376\) −18.5425 −0.956259
\(377\) 21.5570 1.11024
\(378\) −3.36454 −0.173053
\(379\) 12.5653 0.645438 0.322719 0.946495i \(-0.395403\pi\)
0.322719 + 0.946495i \(0.395403\pi\)
\(380\) 0 0
\(381\) −11.3583 −0.581902
\(382\) 20.0460 1.02564
\(383\) 7.53739 0.385143 0.192571 0.981283i \(-0.438317\pi\)
0.192571 + 0.981283i \(0.438317\pi\)
\(384\) −0.278920 −0.0142336
\(385\) 0 0
\(386\) −2.20714 −0.112340
\(387\) −2.42954 −0.123500
\(388\) 3.67264 0.186450
\(389\) −13.7331 −0.696296 −0.348148 0.937440i \(-0.613189\pi\)
−0.348148 + 0.937440i \(0.613189\pi\)
\(390\) 0 0
\(391\) −10.7623 −0.544272
\(392\) 7.75749 0.391812
\(393\) −2.07849 −0.104846
\(394\) 12.5824 0.633894
\(395\) 0 0
\(396\) 0.932764 0.0468732
\(397\) 13.0022 0.652562 0.326281 0.945273i \(-0.394204\pi\)
0.326281 + 0.945273i \(0.394204\pi\)
\(398\) −25.0418 −1.25523
\(399\) −18.4277 −0.922540
\(400\) 0 0
\(401\) −6.47047 −0.323120 −0.161560 0.986863i \(-0.551653\pi\)
−0.161560 + 0.986863i \(0.551653\pi\)
\(402\) 3.44245 0.171694
\(403\) 2.40399 0.119751
\(404\) 5.37196 0.267265
\(405\) 0 0
\(406\) 17.7788 0.882346
\(407\) 13.0903 0.648864
\(408\) −14.1830 −0.702164
\(409\) −1.11200 −0.0549850 −0.0274925 0.999622i \(-0.508752\pi\)
−0.0274925 + 0.999622i \(0.508752\pi\)
\(410\) 0 0
\(411\) 19.4032 0.957088
\(412\) 3.41678 0.168333
\(413\) 9.83718 0.484056
\(414\) −2.53458 −0.124568
\(415\) 0 0
\(416\) 17.3793 0.852088
\(417\) −17.1603 −0.840343
\(418\) −7.46989 −0.365364
\(419\) 11.7932 0.576136 0.288068 0.957610i \(-0.406987\pi\)
0.288068 + 0.957610i \(0.406987\pi\)
\(420\) 0 0
\(421\) −10.1257 −0.493494 −0.246747 0.969080i \(-0.579362\pi\)
−0.246747 + 0.969080i \(0.579362\pi\)
\(422\) 7.49949 0.365069
\(423\) −6.04998 −0.294160
\(424\) 9.94189 0.482821
\(425\) 0 0
\(426\) −7.04624 −0.341391
\(427\) 42.4348 2.05356
\(428\) −3.25888 −0.157524
\(429\) 4.68458 0.226173
\(430\) 0 0
\(431\) −5.69259 −0.274202 −0.137101 0.990557i \(-0.543779\pi\)
−0.137101 + 0.990557i \(0.543779\pi\)
\(432\) −1.71559 −0.0825415
\(433\) 33.0488 1.58822 0.794112 0.607771i \(-0.207936\pi\)
0.794112 + 0.607771i \(0.207936\pi\)
\(434\) 1.98265 0.0951704
\(435\) 0 0
\(436\) −7.38671 −0.353759
\(437\) −13.8820 −0.664065
\(438\) 7.85758 0.375449
\(439\) 11.4637 0.547131 0.273566 0.961853i \(-0.411797\pi\)
0.273566 + 0.961853i \(0.411797\pi\)
\(440\) 0 0
\(441\) 2.53108 0.120528
\(442\) −20.5740 −0.978607
\(443\) −17.8993 −0.850422 −0.425211 0.905094i \(-0.639800\pi\)
−0.425211 + 0.905094i \(0.639800\pi\)
\(444\) 9.25988 0.439454
\(445\) 0 0
\(446\) 8.85621 0.419353
\(447\) 0.210127 0.00993867
\(448\) 24.9262 1.17765
\(449\) −6.82040 −0.321874 −0.160937 0.986965i \(-0.551452\pi\)
−0.160937 + 0.986965i \(0.551452\pi\)
\(450\) 0 0
\(451\) −10.8998 −0.513249
\(452\) 3.04988 0.143454
\(453\) −4.05924 −0.190720
\(454\) 27.9321 1.31092
\(455\) 0 0
\(456\) −18.2943 −0.856710
\(457\) −2.76381 −0.129286 −0.0646429 0.997908i \(-0.520591\pi\)
−0.0646429 + 0.997908i \(0.520591\pi\)
\(458\) −1.34390 −0.0627965
\(459\) −4.62758 −0.215997
\(460\) 0 0
\(461\) −8.96463 −0.417525 −0.208762 0.977966i \(-0.566943\pi\)
−0.208762 + 0.977966i \(0.566943\pi\)
\(462\) 3.86353 0.179748
\(463\) −13.1617 −0.611675 −0.305837 0.952084i \(-0.598936\pi\)
−0.305837 + 0.952084i \(0.598936\pi\)
\(464\) 9.06547 0.420854
\(465\) 0 0
\(466\) −3.26932 −0.151448
\(467\) 21.8005 1.00880 0.504402 0.863469i \(-0.331713\pi\)
0.504402 + 0.863469i \(0.331713\pi\)
\(468\) 3.31379 0.153180
\(469\) −9.75179 −0.450296
\(470\) 0 0
\(471\) −0.440336 −0.0202896
\(472\) 9.76596 0.449515
\(473\) 2.78986 0.128278
\(474\) 13.4455 0.617574
\(475\) 0 0
\(476\) 11.6048 0.531905
\(477\) 3.24380 0.148523
\(478\) −24.1222 −1.10332
\(479\) 42.7490 1.95325 0.976626 0.214947i \(-0.0689578\pi\)
0.976626 + 0.214947i \(0.0689578\pi\)
\(480\) 0 0
\(481\) 46.5054 2.12046
\(482\) 14.6475 0.667174
\(483\) 7.17996 0.326699
\(484\) 7.86413 0.357460
\(485\) 0 0
\(486\) −1.08982 −0.0494352
\(487\) −11.3497 −0.514305 −0.257153 0.966371i \(-0.582784\pi\)
−0.257153 + 0.966371i \(0.582784\pi\)
\(488\) 42.1275 1.90702
\(489\) −3.24109 −0.146567
\(490\) 0 0
\(491\) 16.6546 0.751614 0.375807 0.926698i \(-0.377366\pi\)
0.375807 + 0.926698i \(0.377366\pi\)
\(492\) −7.71029 −0.347607
\(493\) 24.4529 1.10130
\(494\) −26.5379 −1.19400
\(495\) 0 0
\(496\) 1.01096 0.0453936
\(497\) 19.9606 0.895356
\(498\) 13.5073 0.605278
\(499\) −12.2321 −0.547584 −0.273792 0.961789i \(-0.588278\pi\)
−0.273792 + 0.961789i \(0.588278\pi\)
\(500\) 0 0
\(501\) 15.9605 0.713063
\(502\) −20.5756 −0.918336
\(503\) −1.30967 −0.0583951 −0.0291976 0.999574i \(-0.509295\pi\)
−0.0291976 + 0.999574i \(0.509295\pi\)
\(504\) 9.46207 0.421474
\(505\) 0 0
\(506\) 2.91048 0.129386
\(507\) 3.64267 0.161777
\(508\) 9.22625 0.409349
\(509\) 17.7318 0.785946 0.392973 0.919550i \(-0.371447\pi\)
0.392973 + 0.919550i \(0.371447\pi\)
\(510\) 0 0
\(511\) −22.2590 −0.984679
\(512\) 17.8248 0.787752
\(513\) −5.96899 −0.263537
\(514\) 7.69761 0.339527
\(515\) 0 0
\(516\) 1.97350 0.0868783
\(517\) 6.94725 0.305540
\(518\) 38.3546 1.68521
\(519\) −0.561789 −0.0246598
\(520\) 0 0
\(521\) 26.2616 1.15054 0.575270 0.817964i \(-0.304897\pi\)
0.575270 + 0.817964i \(0.304897\pi\)
\(522\) 5.75879 0.252055
\(523\) −2.55674 −0.111798 −0.0558991 0.998436i \(-0.517803\pi\)
−0.0558991 + 0.998436i \(0.517803\pi\)
\(524\) 1.68834 0.0737557
\(525\) 0 0
\(526\) 22.2711 0.971068
\(527\) 2.72693 0.118787
\(528\) 1.97003 0.0857346
\(529\) −17.5912 −0.764835
\(530\) 0 0
\(531\) 3.18640 0.138278
\(532\) 14.9687 0.648977
\(533\) −38.7230 −1.67728
\(534\) −1.18140 −0.0511243
\(535\) 0 0
\(536\) −9.68118 −0.418163
\(537\) 15.9387 0.687805
\(538\) −23.0958 −0.995732
\(539\) −2.90646 −0.125190
\(540\) 0 0
\(541\) 9.38916 0.403672 0.201836 0.979419i \(-0.435309\pi\)
0.201836 + 0.979419i \(0.435309\pi\)
\(542\) −10.8472 −0.465925
\(543\) −21.4875 −0.922118
\(544\) 19.7139 0.845228
\(545\) 0 0
\(546\) 13.7258 0.587409
\(547\) 34.8549 1.49029 0.745144 0.666903i \(-0.232381\pi\)
0.745144 + 0.666903i \(0.232381\pi\)
\(548\) −15.7611 −0.673280
\(549\) 13.7452 0.586631
\(550\) 0 0
\(551\) 31.5411 1.34370
\(552\) 7.12797 0.303387
\(553\) −38.0886 −1.61969
\(554\) −20.4177 −0.867465
\(555\) 0 0
\(556\) 13.9392 0.591153
\(557\) 14.1466 0.599411 0.299705 0.954032i \(-0.403112\pi\)
0.299705 + 0.954032i \(0.403112\pi\)
\(558\) 0.642208 0.0271868
\(559\) 9.91139 0.419207
\(560\) 0 0
\(561\) 5.31389 0.224353
\(562\) −31.7046 −1.33738
\(563\) −38.1854 −1.60932 −0.804661 0.593734i \(-0.797653\pi\)
−0.804661 + 0.593734i \(0.797653\pi\)
\(564\) 4.91436 0.206932
\(565\) 0 0
\(566\) 7.82498 0.328908
\(567\) 3.08724 0.129652
\(568\) 19.8161 0.831465
\(569\) 29.6301 1.24216 0.621080 0.783747i \(-0.286694\pi\)
0.621080 + 0.783747i \(0.286694\pi\)
\(570\) 0 0
\(571\) 32.6504 1.36638 0.683188 0.730242i \(-0.260593\pi\)
0.683188 + 0.730242i \(0.260593\pi\)
\(572\) −3.80525 −0.159106
\(573\) −18.3939 −0.768415
\(574\) −31.9362 −1.33299
\(575\) 0 0
\(576\) 8.07392 0.336413
\(577\) −22.6433 −0.942653 −0.471326 0.881959i \(-0.656225\pi\)
−0.471326 + 0.881959i \(0.656225\pi\)
\(578\) −4.81097 −0.200110
\(579\) 2.02523 0.0841659
\(580\) 0 0
\(581\) −38.2636 −1.58744
\(582\) 4.92742 0.204248
\(583\) −3.72488 −0.154269
\(584\) −22.0978 −0.914414
\(585\) 0 0
\(586\) −1.95977 −0.0809575
\(587\) −41.5667 −1.71564 −0.857820 0.513950i \(-0.828182\pi\)
−0.857820 + 0.513950i \(0.828182\pi\)
\(588\) −2.05598 −0.0847872
\(589\) 3.51740 0.144932
\(590\) 0 0
\(591\) −11.5454 −0.474916
\(592\) 19.5572 0.803795
\(593\) −2.09050 −0.0858465 −0.0429233 0.999078i \(-0.513667\pi\)
−0.0429233 + 0.999078i \(0.513667\pi\)
\(594\) 1.25145 0.0513476
\(595\) 0 0
\(596\) −0.170685 −0.00699152
\(597\) 22.9779 0.940424
\(598\) 10.3399 0.422830
\(599\) 17.9768 0.734511 0.367255 0.930120i \(-0.380298\pi\)
0.367255 + 0.930120i \(0.380298\pi\)
\(600\) 0 0
\(601\) −1.11000 −0.0452778 −0.0226389 0.999744i \(-0.507207\pi\)
−0.0226389 + 0.999744i \(0.507207\pi\)
\(602\) 8.17427 0.333158
\(603\) −3.15873 −0.128634
\(604\) 3.29730 0.134165
\(605\) 0 0
\(606\) 7.20733 0.292778
\(607\) 12.2310 0.496441 0.248220 0.968704i \(-0.420154\pi\)
0.248220 + 0.968704i \(0.420154\pi\)
\(608\) 25.4285 1.03126
\(609\) −16.3135 −0.661057
\(610\) 0 0
\(611\) 24.6812 0.998493
\(612\) 3.75895 0.151947
\(613\) 12.9776 0.524160 0.262080 0.965046i \(-0.415591\pi\)
0.262080 + 0.965046i \(0.415591\pi\)
\(614\) −6.52654 −0.263390
\(615\) 0 0
\(616\) −10.8654 −0.437779
\(617\) −20.4921 −0.824981 −0.412490 0.910962i \(-0.635341\pi\)
−0.412490 + 0.910962i \(0.635341\pi\)
\(618\) 4.58415 0.184402
\(619\) −41.0456 −1.64976 −0.824881 0.565307i \(-0.808758\pi\)
−0.824881 + 0.565307i \(0.808758\pi\)
\(620\) 0 0
\(621\) 2.32568 0.0933265
\(622\) −26.0122 −1.04300
\(623\) 3.34668 0.134082
\(624\) 6.99883 0.280177
\(625\) 0 0
\(626\) −6.27641 −0.250856
\(627\) 6.85425 0.273732
\(628\) 0.357682 0.0142731
\(629\) 52.7528 2.10339
\(630\) 0 0
\(631\) 14.0872 0.560804 0.280402 0.959883i \(-0.409532\pi\)
0.280402 + 0.959883i \(0.409532\pi\)
\(632\) −37.8128 −1.50411
\(633\) −6.88140 −0.273511
\(634\) 13.3482 0.530126
\(635\) 0 0
\(636\) −2.63492 −0.104481
\(637\) −10.3256 −0.409117
\(638\) −6.61287 −0.261806
\(639\) 6.46551 0.255772
\(640\) 0 0
\(641\) −28.4358 −1.12315 −0.561573 0.827427i \(-0.689803\pi\)
−0.561573 + 0.827427i \(0.689803\pi\)
\(642\) −4.37230 −0.172561
\(643\) 10.8408 0.427521 0.213761 0.976886i \(-0.431429\pi\)
0.213761 + 0.976886i \(0.431429\pi\)
\(644\) −5.83223 −0.229822
\(645\) 0 0
\(646\) −30.1029 −1.18438
\(647\) 34.6388 1.36179 0.680896 0.732380i \(-0.261590\pi\)
0.680896 + 0.732380i \(0.261590\pi\)
\(648\) 3.06489 0.120400
\(649\) −3.65897 −0.143627
\(650\) 0 0
\(651\) −1.81925 −0.0713020
\(652\) 2.63272 0.103105
\(653\) 4.15506 0.162600 0.0813000 0.996690i \(-0.474093\pi\)
0.0813000 + 0.996690i \(0.474093\pi\)
\(654\) −9.91043 −0.387528
\(655\) 0 0
\(656\) −16.2844 −0.635799
\(657\) −7.20998 −0.281288
\(658\) 20.3554 0.793536
\(659\) −4.01424 −0.156373 −0.0781863 0.996939i \(-0.524913\pi\)
−0.0781863 + 0.996939i \(0.524913\pi\)
\(660\) 0 0
\(661\) 19.6667 0.764944 0.382472 0.923967i \(-0.375073\pi\)
0.382472 + 0.923967i \(0.375073\pi\)
\(662\) 6.57051 0.255370
\(663\) 18.8784 0.733176
\(664\) −37.9866 −1.47416
\(665\) 0 0
\(666\) 12.4236 0.481404
\(667\) −12.2893 −0.475844
\(668\) −12.9646 −0.501616
\(669\) −8.12631 −0.314181
\(670\) 0 0
\(671\) −15.7837 −0.609324
\(672\) −13.1520 −0.507348
\(673\) 41.4463 1.59764 0.798819 0.601571i \(-0.205458\pi\)
0.798819 + 0.601571i \(0.205458\pi\)
\(674\) −6.02761 −0.232175
\(675\) 0 0
\(676\) −2.95892 −0.113804
\(677\) −18.1837 −0.698856 −0.349428 0.936963i \(-0.613624\pi\)
−0.349428 + 0.936963i \(0.613624\pi\)
\(678\) 4.09189 0.157148
\(679\) −13.9584 −0.535675
\(680\) 0 0
\(681\) −25.6300 −0.982144
\(682\) −0.737453 −0.0282385
\(683\) −28.4523 −1.08870 −0.544348 0.838859i \(-0.683223\pi\)
−0.544348 + 0.838859i \(0.683223\pi\)
\(684\) 4.84857 0.185390
\(685\) 0 0
\(686\) 15.0359 0.574072
\(687\) 1.23314 0.0470474
\(688\) 4.16809 0.158907
\(689\) −13.2332 −0.504145
\(690\) 0 0
\(691\) 24.4175 0.928884 0.464442 0.885603i \(-0.346255\pi\)
0.464442 + 0.885603i \(0.346255\pi\)
\(692\) 0.456338 0.0173474
\(693\) −3.54511 −0.134668
\(694\) 28.2748 1.07330
\(695\) 0 0
\(696\) −16.1954 −0.613885
\(697\) −43.9250 −1.66378
\(698\) 20.7093 0.783859
\(699\) 2.99987 0.113466
\(700\) 0 0
\(701\) −25.0371 −0.945638 −0.472819 0.881159i \(-0.656764\pi\)
−0.472819 + 0.881159i \(0.656764\pi\)
\(702\) 4.44596 0.167802
\(703\) 68.0445 2.56635
\(704\) −9.27136 −0.349428
\(705\) 0 0
\(706\) −8.92708 −0.335975
\(707\) −20.4170 −0.767858
\(708\) −2.58829 −0.0972738
\(709\) 36.5633 1.37316 0.686582 0.727052i \(-0.259110\pi\)
0.686582 + 0.727052i \(0.259110\pi\)
\(710\) 0 0
\(711\) −12.3374 −0.462688
\(712\) 3.32245 0.124514
\(713\) −1.37048 −0.0513248
\(714\) 15.5697 0.582680
\(715\) 0 0
\(716\) −12.9469 −0.483848
\(717\) 22.1341 0.826614
\(718\) 29.1915 1.08942
\(719\) −7.23571 −0.269847 −0.134923 0.990856i \(-0.543079\pi\)
−0.134923 + 0.990856i \(0.543079\pi\)
\(720\) 0 0
\(721\) −12.9860 −0.483624
\(722\) −18.1224 −0.674447
\(723\) −13.4403 −0.499849
\(724\) 17.4542 0.648679
\(725\) 0 0
\(726\) 10.5510 0.391583
\(727\) −13.3099 −0.493638 −0.246819 0.969062i \(-0.579385\pi\)
−0.246819 + 0.969062i \(0.579385\pi\)
\(728\) −38.6009 −1.43065
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 11.2429 0.415832
\(732\) −11.1651 −0.412675
\(733\) 28.7794 1.06299 0.531496 0.847061i \(-0.321630\pi\)
0.531496 + 0.847061i \(0.321630\pi\)
\(734\) −20.5467 −0.758393
\(735\) 0 0
\(736\) −9.90764 −0.365201
\(737\) 3.62720 0.133610
\(738\) −10.3446 −0.380789
\(739\) 40.3208 1.48322 0.741612 0.670829i \(-0.234061\pi\)
0.741612 + 0.670829i \(0.234061\pi\)
\(740\) 0 0
\(741\) 24.3507 0.894547
\(742\) −10.9139 −0.400661
\(743\) −27.8114 −1.02030 −0.510151 0.860085i \(-0.670411\pi\)
−0.510151 + 0.860085i \(0.670411\pi\)
\(744\) −1.80608 −0.0662140
\(745\) 0 0
\(746\) 19.4507 0.712142
\(747\) −12.3941 −0.453476
\(748\) −4.31644 −0.157825
\(749\) 12.3859 0.452569
\(750\) 0 0
\(751\) −11.4124 −0.416443 −0.208221 0.978082i \(-0.566767\pi\)
−0.208221 + 0.978082i \(0.566767\pi\)
\(752\) 10.3793 0.378494
\(753\) 18.8799 0.688021
\(754\) −23.4932 −0.855573
\(755\) 0 0
\(756\) −2.50775 −0.0912060
\(757\) −13.7637 −0.500251 −0.250126 0.968213i \(-0.580472\pi\)
−0.250126 + 0.968213i \(0.580472\pi\)
\(758\) −13.6939 −0.497386
\(759\) −2.67060 −0.0969368
\(760\) 0 0
\(761\) −14.7409 −0.534359 −0.267179 0.963647i \(-0.586092\pi\)
−0.267179 + 0.963647i \(0.586092\pi\)
\(762\) 12.3785 0.448425
\(763\) 28.0743 1.01636
\(764\) 14.9412 0.540555
\(765\) 0 0
\(766\) −8.21440 −0.296798
\(767\) −12.9990 −0.469368
\(768\) −15.8439 −0.571717
\(769\) −2.43753 −0.0878997 −0.0439498 0.999034i \(-0.513994\pi\)
−0.0439498 + 0.999034i \(0.513994\pi\)
\(770\) 0 0
\(771\) −7.06320 −0.254375
\(772\) −1.64509 −0.0592079
\(773\) −25.3341 −0.911204 −0.455602 0.890184i \(-0.650576\pi\)
−0.455602 + 0.890184i \(0.650576\pi\)
\(774\) 2.64776 0.0951716
\(775\) 0 0
\(776\) −13.8574 −0.497450
\(777\) −35.1936 −1.26256
\(778\) 14.9666 0.536579
\(779\) −56.6577 −2.02997
\(780\) 0 0
\(781\) −7.42441 −0.265666
\(782\) 11.7289 0.419426
\(783\) −5.28417 −0.188841
\(784\) −4.34230 −0.155082
\(785\) 0 0
\(786\) 2.26518 0.0807963
\(787\) 38.6480 1.37765 0.688826 0.724927i \(-0.258127\pi\)
0.688826 + 0.724927i \(0.258127\pi\)
\(788\) 9.37829 0.334088
\(789\) −20.4356 −0.727527
\(790\) 0 0
\(791\) −11.5915 −0.412147
\(792\) −3.51944 −0.125058
\(793\) −56.0741 −1.99125
\(794\) −14.1700 −0.502876
\(795\) 0 0
\(796\) −18.6648 −0.661557
\(797\) −52.1020 −1.84555 −0.922774 0.385342i \(-0.874083\pi\)
−0.922774 + 0.385342i \(0.874083\pi\)
\(798\) 20.0829 0.710927
\(799\) 27.9968 0.990454
\(800\) 0 0
\(801\) 1.08404 0.0383025
\(802\) 7.05165 0.249002
\(803\) 8.27929 0.292170
\(804\) 2.56582 0.0904895
\(805\) 0 0
\(806\) −2.61991 −0.0922825
\(807\) 21.1923 0.746006
\(808\) −20.2691 −0.713065
\(809\) 31.7938 1.11781 0.558905 0.829231i \(-0.311222\pi\)
0.558905 + 0.829231i \(0.311222\pi\)
\(810\) 0 0
\(811\) −31.1213 −1.09282 −0.546409 0.837518i \(-0.684006\pi\)
−0.546409 + 0.837518i \(0.684006\pi\)
\(812\) 13.2514 0.465032
\(813\) 9.95317 0.349073
\(814\) −14.2661 −0.500027
\(815\) 0 0
\(816\) 7.93903 0.277922
\(817\) 14.5019 0.507356
\(818\) 1.21188 0.0423725
\(819\) −12.5945 −0.440089
\(820\) 0 0
\(821\) −47.0263 −1.64123 −0.820615 0.571482i \(-0.806369\pi\)
−0.820615 + 0.571482i \(0.806369\pi\)
\(822\) −21.1460 −0.737550
\(823\) −28.3425 −0.987959 −0.493979 0.869474i \(-0.664458\pi\)
−0.493979 + 0.869474i \(0.664458\pi\)
\(824\) −12.8920 −0.449113
\(825\) 0 0
\(826\) −10.7208 −0.373023
\(827\) 14.1680 0.492670 0.246335 0.969185i \(-0.420774\pi\)
0.246335 + 0.969185i \(0.420774\pi\)
\(828\) −1.88914 −0.0656521
\(829\) 1.51657 0.0526728 0.0263364 0.999653i \(-0.491616\pi\)
0.0263364 + 0.999653i \(0.491616\pi\)
\(830\) 0 0
\(831\) 18.7349 0.649908
\(832\) −32.9379 −1.14192
\(833\) −11.7128 −0.405823
\(834\) 18.7016 0.647584
\(835\) 0 0
\(836\) −5.56766 −0.192562
\(837\) −0.589279 −0.0203685
\(838\) −12.8525 −0.443981
\(839\) 10.4919 0.362220 0.181110 0.983463i \(-0.442031\pi\)
0.181110 + 0.983463i \(0.442031\pi\)
\(840\) 0 0
\(841\) −1.07758 −0.0371578
\(842\) 11.0351 0.380296
\(843\) 29.0916 1.00197
\(844\) 5.58972 0.192406
\(845\) 0 0
\(846\) 6.59339 0.226685
\(847\) −29.8888 −1.02699
\(848\) −5.56503 −0.191104
\(849\) −7.18007 −0.246419
\(850\) 0 0
\(851\) −26.5120 −0.908820
\(852\) −5.25189 −0.179927
\(853\) −14.8571 −0.508698 −0.254349 0.967112i \(-0.581861\pi\)
−0.254349 + 0.967112i \(0.581861\pi\)
\(854\) −46.2462 −1.58251
\(855\) 0 0
\(856\) 12.2962 0.420275
\(857\) −26.4088 −0.902109 −0.451054 0.892496i \(-0.648952\pi\)
−0.451054 + 0.892496i \(0.648952\pi\)
\(858\) −5.10534 −0.174294
\(859\) −9.13013 −0.311516 −0.155758 0.987795i \(-0.549782\pi\)
−0.155758 + 0.987795i \(0.549782\pi\)
\(860\) 0 0
\(861\) 29.3041 0.998682
\(862\) 6.20389 0.211305
\(863\) 12.4796 0.424810 0.212405 0.977182i \(-0.431870\pi\)
0.212405 + 0.977182i \(0.431870\pi\)
\(864\) −4.26010 −0.144932
\(865\) 0 0
\(866\) −36.0172 −1.22392
\(867\) 4.41447 0.149923
\(868\) 1.47776 0.0501586
\(869\) 14.1671 0.480587
\(870\) 0 0
\(871\) 12.8862 0.436632
\(872\) 27.8710 0.943832
\(873\) −4.52132 −0.153024
\(874\) 15.1289 0.511741
\(875\) 0 0
\(876\) 5.85662 0.197877
\(877\) −40.8121 −1.37813 −0.689064 0.724701i \(-0.741978\pi\)
−0.689064 + 0.724701i \(0.741978\pi\)
\(878\) −12.4933 −0.421629
\(879\) 1.79825 0.0606536
\(880\) 0 0
\(881\) −3.64199 −0.122702 −0.0613509 0.998116i \(-0.519541\pi\)
−0.0613509 + 0.998116i \(0.519541\pi\)
\(882\) −2.75842 −0.0928808
\(883\) −0.332855 −0.0112015 −0.00560073 0.999984i \(-0.501783\pi\)
−0.00560073 + 0.999984i \(0.501783\pi\)
\(884\) −15.3348 −0.515765
\(885\) 0 0
\(886\) 19.5070 0.655351
\(887\) 1.14274 0.0383694 0.0191847 0.999816i \(-0.493893\pi\)
0.0191847 + 0.999816i \(0.493893\pi\)
\(888\) −34.9388 −1.17247
\(889\) −35.0658 −1.17607
\(890\) 0 0
\(891\) −1.14831 −0.0384698
\(892\) 6.60095 0.221016
\(893\) 36.1123 1.20845
\(894\) −0.229001 −0.00765892
\(895\) 0 0
\(896\) −0.861093 −0.0287671
\(897\) −9.48773 −0.316786
\(898\) 7.43300 0.248042
\(899\) 3.11385 0.103853
\(900\) 0 0
\(901\) −15.0109 −0.500086
\(902\) 11.8788 0.395520
\(903\) −7.50057 −0.249603
\(904\) −11.5076 −0.382737
\(905\) 0 0
\(906\) 4.42384 0.146972
\(907\) 18.4202 0.611632 0.305816 0.952091i \(-0.401071\pi\)
0.305816 + 0.952091i \(0.401071\pi\)
\(908\) 20.8191 0.690906
\(909\) −6.61332 −0.219350
\(910\) 0 0
\(911\) 18.2506 0.604670 0.302335 0.953202i \(-0.402234\pi\)
0.302335 + 0.953202i \(0.402234\pi\)
\(912\) 10.2404 0.339092
\(913\) 14.2323 0.471019
\(914\) 3.01206 0.0996300
\(915\) 0 0
\(916\) −1.00167 −0.0330963
\(917\) −6.41681 −0.211902
\(918\) 5.04322 0.166451
\(919\) 10.8382 0.357520 0.178760 0.983893i \(-0.442791\pi\)
0.178760 + 0.983893i \(0.442791\pi\)
\(920\) 0 0
\(921\) 5.98864 0.197333
\(922\) 9.76983 0.321752
\(923\) −26.3763 −0.868187
\(924\) 2.87967 0.0947342
\(925\) 0 0
\(926\) 14.3438 0.471368
\(927\) −4.20634 −0.138154
\(928\) 22.5111 0.738962
\(929\) −52.3491 −1.71752 −0.858759 0.512379i \(-0.828764\pi\)
−0.858759 + 0.512379i \(0.828764\pi\)
\(930\) 0 0
\(931\) −15.1080 −0.495144
\(932\) −2.43678 −0.0798193
\(933\) 23.8684 0.781416
\(934\) −23.7586 −0.777404
\(935\) 0 0
\(936\) −12.5034 −0.408685
\(937\) 0.00131225 4.28694e−5 0 2.14347e−5 1.00000i \(-0.499993\pi\)
2.14347e−5 1.00000i \(0.499993\pi\)
\(938\) 10.6277 0.347006
\(939\) 5.75913 0.187942
\(940\) 0 0
\(941\) −21.7222 −0.708124 −0.354062 0.935222i \(-0.615200\pi\)
−0.354062 + 0.935222i \(0.615200\pi\)
\(942\) 0.479887 0.0156356
\(943\) 22.0754 0.718874
\(944\) −5.46655 −0.177921
\(945\) 0 0
\(946\) −3.04044 −0.0988533
\(947\) 33.9266 1.10247 0.551234 0.834351i \(-0.314157\pi\)
0.551234 + 0.834351i \(0.314157\pi\)
\(948\) 10.0216 0.325486
\(949\) 29.4134 0.954800
\(950\) 0 0
\(951\) −12.2481 −0.397172
\(952\) −43.7865 −1.41913
\(953\) 8.30308 0.268963 0.134482 0.990916i \(-0.457063\pi\)
0.134482 + 0.990916i \(0.457063\pi\)
\(954\) −3.53515 −0.114455
\(955\) 0 0
\(956\) −17.9794 −0.581496
\(957\) 6.06786 0.196146
\(958\) −46.5887 −1.50521
\(959\) 59.9024 1.93435
\(960\) 0 0
\(961\) −30.6528 −0.988798
\(962\) −50.6825 −1.63407
\(963\) 4.01195 0.129283
\(964\) 10.9174 0.351627
\(965\) 0 0
\(966\) −7.82486 −0.251761
\(967\) −27.7915 −0.893714 −0.446857 0.894605i \(-0.647457\pi\)
−0.446857 + 0.894605i \(0.647457\pi\)
\(968\) −29.6724 −0.953707
\(969\) 27.6220 0.887345
\(970\) 0 0
\(971\) −31.0461 −0.996317 −0.498158 0.867086i \(-0.665990\pi\)
−0.498158 + 0.867086i \(0.665990\pi\)
\(972\) −0.812294 −0.0260543
\(973\) −52.9780 −1.69840
\(974\) 12.3692 0.396333
\(975\) 0 0
\(976\) −23.5811 −0.754814
\(977\) −44.3917 −1.42022 −0.710109 0.704092i \(-0.751354\pi\)
−0.710109 + 0.704092i \(0.751354\pi\)
\(978\) 3.53220 0.112947
\(979\) −1.24481 −0.0397842
\(980\) 0 0
\(981\) 9.09364 0.290338
\(982\) −18.1506 −0.579208
\(983\) 43.7857 1.39655 0.698274 0.715831i \(-0.253952\pi\)
0.698274 + 0.715831i \(0.253952\pi\)
\(984\) 29.0920 0.927418
\(985\) 0 0
\(986\) −26.6492 −0.848685
\(987\) −18.6778 −0.594520
\(988\) −19.7800 −0.629284
\(989\) −5.65033 −0.179670
\(990\) 0 0
\(991\) 34.4351 1.09387 0.546933 0.837176i \(-0.315795\pi\)
0.546933 + 0.837176i \(0.315795\pi\)
\(992\) 2.51039 0.0797049
\(993\) −6.02899 −0.191324
\(994\) −21.7535 −0.689978
\(995\) 0 0
\(996\) 10.0676 0.319006
\(997\) −19.3890 −0.614056 −0.307028 0.951700i \(-0.599335\pi\)
−0.307028 + 0.951700i \(0.599335\pi\)
\(998\) 13.3308 0.421979
\(999\) −11.3997 −0.360670
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.m.1.4 8
3.2 odd 2 5625.2.a.bd.1.5 8
5.2 odd 4 1875.2.b.h.1249.6 16
5.3 odd 4 1875.2.b.h.1249.11 16
5.4 even 2 1875.2.a.p.1.5 8
15.14 odd 2 5625.2.a.t.1.4 8
25.3 odd 20 375.2.i.c.49.3 16
25.4 even 10 375.2.g.d.76.2 16
25.6 even 5 375.2.g.e.301.3 16
25.8 odd 20 75.2.i.a.64.2 yes 16
25.17 odd 20 375.2.i.c.199.3 16
25.19 even 10 375.2.g.d.301.2 16
25.21 even 5 375.2.g.e.76.3 16
25.22 odd 20 75.2.i.a.34.2 16
75.8 even 20 225.2.m.b.64.3 16
75.47 even 20 225.2.m.b.109.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.2 16 25.22 odd 20
75.2.i.a.64.2 yes 16 25.8 odd 20
225.2.m.b.64.3 16 75.8 even 20
225.2.m.b.109.3 16 75.47 even 20
375.2.g.d.76.2 16 25.4 even 10
375.2.g.d.301.2 16 25.19 even 10
375.2.g.e.76.3 16 25.21 even 5
375.2.g.e.301.3 16 25.6 even 5
375.2.i.c.49.3 16 25.3 odd 20
375.2.i.c.199.3 16 25.17 odd 20
1875.2.a.m.1.4 8 1.1 even 1 trivial
1875.2.a.p.1.5 8 5.4 even 2
1875.2.b.h.1249.6 16 5.2 odd 4
1875.2.b.h.1249.11 16 5.3 odd 4
5625.2.a.t.1.4 8 15.14 odd 2
5625.2.a.bd.1.5 8 3.2 odd 2