Properties

Label 1875.2.a.k.1.1
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.44400625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.1
Root \(-2.44028\) 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)
 
\(f(q)\) \(=\) \(q-2.44028 q^{2} +1.00000 q^{3} +3.95498 q^{4} -2.44028 q^{6} +3.44028 q^{7} -4.77071 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.44028 q^{2} +1.00000 q^{3} +3.95498 q^{4} -2.44028 q^{6} +3.44028 q^{7} -4.77071 q^{8} +1.00000 q^{9} -3.26656 q^{11} +3.95498 q^{12} +3.23204 q^{13} -8.39527 q^{14} +3.73192 q^{16} +5.05337 q^{17} -2.44028 q^{18} -3.08119 q^{19} +3.44028 q^{21} +7.97134 q^{22} -1.54765 q^{23} -4.77071 q^{24} -7.88709 q^{26} +1.00000 q^{27} +13.6063 q^{28} -3.12218 q^{29} +7.44212 q^{31} +0.434479 q^{32} -3.26656 q^{33} -12.3317 q^{34} +3.95498 q^{36} +5.75838 q^{37} +7.51899 q^{38} +3.23204 q^{39} +5.41962 q^{41} -8.39527 q^{42} -2.53106 q^{43} -12.9192 q^{44} +3.77670 q^{46} -7.07162 q^{47} +3.73192 q^{48} +4.83555 q^{49} +5.05337 q^{51} +12.7827 q^{52} +10.1515 q^{53} -2.44028 q^{54} -16.4126 q^{56} -3.08119 q^{57} +7.61901 q^{58} +1.73056 q^{59} +7.83611 q^{61} -18.1609 q^{62} +3.44028 q^{63} -8.52409 q^{64} +7.97134 q^{66} +1.84910 q^{67} +19.9860 q^{68} -1.54765 q^{69} -0.713969 q^{71} -4.77071 q^{72} +1.88027 q^{73} -14.0521 q^{74} -12.1861 q^{76} -11.2379 q^{77} -7.88709 q^{78} -13.3332 q^{79} +1.00000 q^{81} -13.2254 q^{82} -3.95747 q^{83} +13.6063 q^{84} +6.17649 q^{86} -3.12218 q^{87} +15.5838 q^{88} +8.53392 q^{89} +11.1191 q^{91} -6.12092 q^{92} +7.44212 q^{93} +17.2567 q^{94} +0.434479 q^{96} +10.6528 q^{97} -11.8001 q^{98} -3.26656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9} + 3 q^{11} + 10 q^{12} + 6 q^{13} - 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} + 6 q^{21} + 16 q^{22} + 13 q^{23} + 3 q^{24} - 28 q^{26} + 6 q^{27} + 7 q^{28} - 3 q^{29} - 11 q^{31} + 16 q^{32} + 3 q^{33} + 15 q^{34} + 10 q^{36} + 21 q^{37} - 9 q^{38} + 6 q^{39} - q^{41} - 22 q^{42} + 2 q^{43} + 9 q^{44} + 19 q^{46} + 14 q^{47} + 18 q^{48} - 14 q^{49} + 13 q^{51} + 13 q^{52} + 23 q^{53} - 35 q^{56} + 11 q^{57} + 22 q^{58} + 9 q^{59} + 11 q^{61} - 23 q^{62} + 6 q^{63} - 23 q^{64} + 16 q^{66} + 8 q^{67} + 50 q^{68} + 13 q^{69} - 8 q^{71} + 3 q^{72} + 13 q^{73} - 22 q^{74} - 26 q^{76} - 13 q^{77} - 28 q^{78} - 5 q^{79} + 6 q^{81} - 13 q^{82} - 20 q^{83} + 7 q^{84} - 37 q^{86} - 3 q^{87} + 28 q^{88} - 4 q^{89} + 34 q^{91} + 61 q^{92} - 11 q^{93} + 41 q^{94} + 16 q^{96} - 7 q^{97} - 41 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44028 −1.72554 −0.862770 0.505596i \(-0.831273\pi\)
−0.862770 + 0.505596i \(0.831273\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.95498 1.97749
\(5\) 0 0
\(6\) −2.44028 −0.996241
\(7\) 3.44028 1.30030 0.650152 0.759804i \(-0.274705\pi\)
0.650152 + 0.759804i \(0.274705\pi\)
\(8\) −4.77071 −1.68670
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.26656 −0.984906 −0.492453 0.870339i \(-0.663900\pi\)
−0.492453 + 0.870339i \(0.663900\pi\)
\(12\) 3.95498 1.14171
\(13\) 3.23204 0.896406 0.448203 0.893932i \(-0.352064\pi\)
0.448203 + 0.893932i \(0.352064\pi\)
\(14\) −8.39527 −2.24373
\(15\) 0 0
\(16\) 3.73192 0.932980
\(17\) 5.05337 1.22562 0.612811 0.790229i \(-0.290039\pi\)
0.612811 + 0.790229i \(0.290039\pi\)
\(18\) −2.44028 −0.575180
\(19\) −3.08119 −0.706874 −0.353437 0.935458i \(-0.614987\pi\)
−0.353437 + 0.935458i \(0.614987\pi\)
\(20\) 0 0
\(21\) 3.44028 0.750731
\(22\) 7.97134 1.69950
\(23\) −1.54765 −0.322707 −0.161354 0.986897i \(-0.551586\pi\)
−0.161354 + 0.986897i \(0.551586\pi\)
\(24\) −4.77071 −0.973817
\(25\) 0 0
\(26\) −7.88709 −1.54679
\(27\) 1.00000 0.192450
\(28\) 13.6063 2.57134
\(29\) −3.12218 −0.579775 −0.289887 0.957061i \(-0.593618\pi\)
−0.289887 + 0.957061i \(0.593618\pi\)
\(30\) 0 0
\(31\) 7.44212 1.33664 0.668322 0.743872i \(-0.267013\pi\)
0.668322 + 0.743872i \(0.267013\pi\)
\(32\) 0.434479 0.0768057
\(33\) −3.26656 −0.568636
\(34\) −12.3317 −2.11486
\(35\) 0 0
\(36\) 3.95498 0.659164
\(37\) 5.75838 0.946673 0.473336 0.880882i \(-0.343050\pi\)
0.473336 + 0.880882i \(0.343050\pi\)
\(38\) 7.51899 1.21974
\(39\) 3.23204 0.517540
\(40\) 0 0
\(41\) 5.41962 0.846403 0.423201 0.906036i \(-0.360906\pi\)
0.423201 + 0.906036i \(0.360906\pi\)
\(42\) −8.39527 −1.29542
\(43\) −2.53106 −0.385982 −0.192991 0.981201i \(-0.561819\pi\)
−0.192991 + 0.981201i \(0.561819\pi\)
\(44\) −12.9192 −1.94764
\(45\) 0 0
\(46\) 3.77670 0.556844
\(47\) −7.07162 −1.03150 −0.515751 0.856739i \(-0.672487\pi\)
−0.515751 + 0.856739i \(0.672487\pi\)
\(48\) 3.73192 0.538656
\(49\) 4.83555 0.690793
\(50\) 0 0
\(51\) 5.05337 0.707614
\(52\) 12.7827 1.77263
\(53\) 10.1515 1.39442 0.697211 0.716866i \(-0.254424\pi\)
0.697211 + 0.716866i \(0.254424\pi\)
\(54\) −2.44028 −0.332080
\(55\) 0 0
\(56\) −16.4126 −2.19323
\(57\) −3.08119 −0.408114
\(58\) 7.61901 1.00042
\(59\) 1.73056 0.225300 0.112650 0.993635i \(-0.464066\pi\)
0.112650 + 0.993635i \(0.464066\pi\)
\(60\) 0 0
\(61\) 7.83611 1.00331 0.501656 0.865067i \(-0.332724\pi\)
0.501656 + 0.865067i \(0.332724\pi\)
\(62\) −18.1609 −2.30643
\(63\) 3.44028 0.433435
\(64\) −8.52409 −1.06551
\(65\) 0 0
\(66\) 7.97134 0.981204
\(67\) 1.84910 0.225903 0.112952 0.993600i \(-0.463969\pi\)
0.112952 + 0.993600i \(0.463969\pi\)
\(68\) 19.9860 2.42366
\(69\) −1.54765 −0.186315
\(70\) 0 0
\(71\) −0.713969 −0.0847325 −0.0423663 0.999102i \(-0.513490\pi\)
−0.0423663 + 0.999102i \(0.513490\pi\)
\(72\) −4.77071 −0.562234
\(73\) 1.88027 0.220069 0.110035 0.993928i \(-0.464904\pi\)
0.110035 + 0.993928i \(0.464904\pi\)
\(74\) −14.0521 −1.63352
\(75\) 0 0
\(76\) −12.1861 −1.39784
\(77\) −11.2379 −1.28068
\(78\) −7.88709 −0.893037
\(79\) −13.3332 −1.50011 −0.750053 0.661378i \(-0.769972\pi\)
−0.750053 + 0.661378i \(0.769972\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −13.2254 −1.46050
\(83\) −3.95747 −0.434389 −0.217195 0.976128i \(-0.569691\pi\)
−0.217195 + 0.976128i \(0.569691\pi\)
\(84\) 13.6063 1.48456
\(85\) 0 0
\(86\) 6.17649 0.666028
\(87\) −3.12218 −0.334733
\(88\) 15.5838 1.66124
\(89\) 8.53392 0.904593 0.452297 0.891868i \(-0.350605\pi\)
0.452297 + 0.891868i \(0.350605\pi\)
\(90\) 0 0
\(91\) 11.1191 1.16560
\(92\) −6.12092 −0.638150
\(93\) 7.44212 0.771712
\(94\) 17.2567 1.77990
\(95\) 0 0
\(96\) 0.434479 0.0443438
\(97\) 10.6528 1.08163 0.540815 0.841142i \(-0.318116\pi\)
0.540815 + 0.841142i \(0.318116\pi\)
\(98\) −11.8001 −1.19199
\(99\) −3.26656 −0.328302
\(100\) 0 0
\(101\) 1.76173 0.175299 0.0876496 0.996151i \(-0.472064\pi\)
0.0876496 + 0.996151i \(0.472064\pi\)
\(102\) −12.3317 −1.22102
\(103\) −15.9143 −1.56808 −0.784039 0.620712i \(-0.786844\pi\)
−0.784039 + 0.620712i \(0.786844\pi\)
\(104\) −15.4191 −1.51197
\(105\) 0 0
\(106\) −24.7726 −2.40613
\(107\) 15.7807 1.52558 0.762788 0.646649i \(-0.223830\pi\)
0.762788 + 0.646649i \(0.223830\pi\)
\(108\) 3.95498 0.380568
\(109\) −9.91980 −0.950144 −0.475072 0.879947i \(-0.657578\pi\)
−0.475072 + 0.879947i \(0.657578\pi\)
\(110\) 0 0
\(111\) 5.75838 0.546562
\(112\) 12.8389 1.21316
\(113\) −5.51386 −0.518700 −0.259350 0.965783i \(-0.583508\pi\)
−0.259350 + 0.965783i \(0.583508\pi\)
\(114\) 7.51899 0.704218
\(115\) 0 0
\(116\) −12.3482 −1.14650
\(117\) 3.23204 0.298802
\(118\) −4.22306 −0.388764
\(119\) 17.3850 1.59368
\(120\) 0 0
\(121\) −0.329568 −0.0299607
\(122\) −19.1223 −1.73126
\(123\) 5.41962 0.488671
\(124\) 29.4334 2.64320
\(125\) 0 0
\(126\) −8.39527 −0.747910
\(127\) 0.992089 0.0880336 0.0440168 0.999031i \(-0.485984\pi\)
0.0440168 + 0.999031i \(0.485984\pi\)
\(128\) 19.9322 1.76178
\(129\) −2.53106 −0.222847
\(130\) 0 0
\(131\) −12.6115 −1.10187 −0.550937 0.834547i \(-0.685729\pi\)
−0.550937 + 0.834547i \(0.685729\pi\)
\(132\) −12.9192 −1.12447
\(133\) −10.6002 −0.919152
\(134\) −4.51232 −0.389805
\(135\) 0 0
\(136\) −24.1082 −2.06726
\(137\) −8.24065 −0.704046 −0.352023 0.935991i \(-0.614506\pi\)
−0.352023 + 0.935991i \(0.614506\pi\)
\(138\) 3.77670 0.321494
\(139\) 8.06371 0.683955 0.341977 0.939708i \(-0.388903\pi\)
0.341977 + 0.939708i \(0.388903\pi\)
\(140\) 0 0
\(141\) −7.07162 −0.595538
\(142\) 1.74229 0.146209
\(143\) −10.5577 −0.882875
\(144\) 3.73192 0.310993
\(145\) 0 0
\(146\) −4.58840 −0.379739
\(147\) 4.83555 0.398829
\(148\) 22.7743 1.87204
\(149\) −19.1101 −1.56556 −0.782781 0.622298i \(-0.786199\pi\)
−0.782781 + 0.622298i \(0.786199\pi\)
\(150\) 0 0
\(151\) 1.58550 0.129026 0.0645132 0.997917i \(-0.479451\pi\)
0.0645132 + 0.997917i \(0.479451\pi\)
\(152\) 14.6995 1.19229
\(153\) 5.05337 0.408541
\(154\) 27.4237 2.20986
\(155\) 0 0
\(156\) 12.7827 1.02343
\(157\) 21.8510 1.74390 0.871948 0.489599i \(-0.162857\pi\)
0.871948 + 0.489599i \(0.162857\pi\)
\(158\) 32.5369 2.58849
\(159\) 10.1515 0.805070
\(160\) 0 0
\(161\) −5.32435 −0.419617
\(162\) −2.44028 −0.191727
\(163\) −10.1338 −0.793744 −0.396872 0.917874i \(-0.629904\pi\)
−0.396872 + 0.917874i \(0.629904\pi\)
\(164\) 21.4345 1.67375
\(165\) 0 0
\(166\) 9.65736 0.749556
\(167\) 1.51109 0.116931 0.0584657 0.998289i \(-0.481379\pi\)
0.0584657 + 0.998289i \(0.481379\pi\)
\(168\) −16.4126 −1.26626
\(169\) −2.55393 −0.196456
\(170\) 0 0
\(171\) −3.08119 −0.235625
\(172\) −10.0103 −0.763277
\(173\) 13.8553 1.05340 0.526700 0.850051i \(-0.323429\pi\)
0.526700 + 0.850051i \(0.323429\pi\)
\(174\) 7.61901 0.577595
\(175\) 0 0
\(176\) −12.1906 −0.918897
\(177\) 1.73056 0.130077
\(178\) −20.8252 −1.56091
\(179\) 14.4557 1.08047 0.540235 0.841514i \(-0.318335\pi\)
0.540235 + 0.841514i \(0.318335\pi\)
\(180\) 0 0
\(181\) 17.1898 1.27771 0.638854 0.769328i \(-0.279409\pi\)
0.638854 + 0.769328i \(0.279409\pi\)
\(182\) −27.1338 −2.01129
\(183\) 7.83611 0.579262
\(184\) 7.38338 0.544310
\(185\) 0 0
\(186\) −18.1609 −1.33162
\(187\) −16.5072 −1.20712
\(188\) −27.9681 −2.03978
\(189\) 3.44028 0.250244
\(190\) 0 0
\(191\) 10.5424 0.762821 0.381410 0.924406i \(-0.375438\pi\)
0.381410 + 0.924406i \(0.375438\pi\)
\(192\) −8.52409 −0.615173
\(193\) 0.682908 0.0491568 0.0245784 0.999698i \(-0.492176\pi\)
0.0245784 + 0.999698i \(0.492176\pi\)
\(194\) −25.9959 −1.86640
\(195\) 0 0
\(196\) 19.1245 1.36604
\(197\) 23.2972 1.65986 0.829928 0.557870i \(-0.188381\pi\)
0.829928 + 0.557870i \(0.188381\pi\)
\(198\) 7.97134 0.566498
\(199\) 10.1946 0.722679 0.361339 0.932434i \(-0.382320\pi\)
0.361339 + 0.932434i \(0.382320\pi\)
\(200\) 0 0
\(201\) 1.84910 0.130425
\(202\) −4.29913 −0.302486
\(203\) −10.7412 −0.753884
\(204\) 19.9860 1.39930
\(205\) 0 0
\(206\) 38.8353 2.70578
\(207\) −1.54765 −0.107569
\(208\) 12.0617 0.836329
\(209\) 10.0649 0.696205
\(210\) 0 0
\(211\) 18.2570 1.25686 0.628431 0.777865i \(-0.283697\pi\)
0.628431 + 0.777865i \(0.283697\pi\)
\(212\) 40.1492 2.75746
\(213\) −0.713969 −0.0489203
\(214\) −38.5093 −2.63244
\(215\) 0 0
\(216\) −4.77071 −0.324606
\(217\) 25.6030 1.73804
\(218\) 24.2071 1.63951
\(219\) 1.88027 0.127057
\(220\) 0 0
\(221\) 16.3327 1.09866
\(222\) −14.0521 −0.943115
\(223\) −17.2244 −1.15343 −0.576714 0.816946i \(-0.695665\pi\)
−0.576714 + 0.816946i \(0.695665\pi\)
\(224\) 1.49473 0.0998708
\(225\) 0 0
\(226\) 13.4554 0.895039
\(227\) 4.32879 0.287312 0.143656 0.989628i \(-0.454114\pi\)
0.143656 + 0.989628i \(0.454114\pi\)
\(228\) −12.1861 −0.807042
\(229\) 20.3147 1.34243 0.671216 0.741262i \(-0.265772\pi\)
0.671216 + 0.741262i \(0.265772\pi\)
\(230\) 0 0
\(231\) −11.2379 −0.739400
\(232\) 14.8950 0.977906
\(233\) 0.340956 0.0223368 0.0111684 0.999938i \(-0.496445\pi\)
0.0111684 + 0.999938i \(0.496445\pi\)
\(234\) −7.88709 −0.515595
\(235\) 0 0
\(236\) 6.84434 0.445529
\(237\) −13.3332 −0.866087
\(238\) −42.4244 −2.74997
\(239\) −7.16488 −0.463458 −0.231729 0.972780i \(-0.574438\pi\)
−0.231729 + 0.972780i \(0.574438\pi\)
\(240\) 0 0
\(241\) −12.5622 −0.809204 −0.404602 0.914493i \(-0.632590\pi\)
−0.404602 + 0.914493i \(0.632590\pi\)
\(242\) 0.804239 0.0516985
\(243\) 1.00000 0.0641500
\(244\) 30.9917 1.98404
\(245\) 0 0
\(246\) −13.2254 −0.843222
\(247\) −9.95854 −0.633646
\(248\) −35.5042 −2.25452
\(249\) −3.95747 −0.250795
\(250\) 0 0
\(251\) 17.0160 1.07404 0.537022 0.843568i \(-0.319549\pi\)
0.537022 + 0.843568i \(0.319549\pi\)
\(252\) 13.6063 0.857114
\(253\) 5.05549 0.317836
\(254\) −2.42098 −0.151906
\(255\) 0 0
\(256\) −31.5921 −1.97451
\(257\) −4.13200 −0.257747 −0.128874 0.991661i \(-0.541136\pi\)
−0.128874 + 0.991661i \(0.541136\pi\)
\(258\) 6.17649 0.384532
\(259\) 19.8105 1.23096
\(260\) 0 0
\(261\) −3.12218 −0.193258
\(262\) 30.7757 1.90133
\(263\) 1.03773 0.0639892 0.0319946 0.999488i \(-0.489814\pi\)
0.0319946 + 0.999488i \(0.489814\pi\)
\(264\) 15.5838 0.959118
\(265\) 0 0
\(266\) 25.8674 1.58603
\(267\) 8.53392 0.522267
\(268\) 7.31315 0.446722
\(269\) −15.7897 −0.962713 −0.481357 0.876525i \(-0.659856\pi\)
−0.481357 + 0.876525i \(0.659856\pi\)
\(270\) 0 0
\(271\) −13.7003 −0.832237 −0.416118 0.909310i \(-0.636610\pi\)
−0.416118 + 0.909310i \(0.636610\pi\)
\(272\) 18.8588 1.14348
\(273\) 11.1191 0.672960
\(274\) 20.1095 1.21486
\(275\) 0 0
\(276\) −6.12092 −0.368436
\(277\) −6.97989 −0.419381 −0.209691 0.977768i \(-0.567246\pi\)
−0.209691 + 0.977768i \(0.567246\pi\)
\(278\) −19.6777 −1.18019
\(279\) 7.44212 0.445548
\(280\) 0 0
\(281\) 7.95823 0.474748 0.237374 0.971418i \(-0.423713\pi\)
0.237374 + 0.971418i \(0.423713\pi\)
\(282\) 17.2567 1.02762
\(283\) 27.7952 1.65225 0.826126 0.563486i \(-0.190540\pi\)
0.826126 + 0.563486i \(0.190540\pi\)
\(284\) −2.82373 −0.167558
\(285\) 0 0
\(286\) 25.7637 1.52344
\(287\) 18.6450 1.10058
\(288\) 0.434479 0.0256019
\(289\) 8.53657 0.502151
\(290\) 0 0
\(291\) 10.6528 0.624479
\(292\) 7.43645 0.435185
\(293\) 14.2098 0.830146 0.415073 0.909788i \(-0.363756\pi\)
0.415073 + 0.909788i \(0.363756\pi\)
\(294\) −11.8001 −0.688196
\(295\) 0 0
\(296\) −27.4716 −1.59675
\(297\) −3.26656 −0.189545
\(298\) 46.6341 2.70144
\(299\) −5.00206 −0.289276
\(300\) 0 0
\(301\) −8.70755 −0.501895
\(302\) −3.86908 −0.222640
\(303\) 1.76173 0.101209
\(304\) −11.4988 −0.659500
\(305\) 0 0
\(306\) −12.3317 −0.704954
\(307\) 23.2911 1.32930 0.664648 0.747157i \(-0.268582\pi\)
0.664648 + 0.747157i \(0.268582\pi\)
\(308\) −44.4457 −2.53253
\(309\) −15.9143 −0.905330
\(310\) 0 0
\(311\) −7.54924 −0.428078 −0.214039 0.976825i \(-0.568662\pi\)
−0.214039 + 0.976825i \(0.568662\pi\)
\(312\) −15.4191 −0.872936
\(313\) 31.6565 1.78933 0.894667 0.446734i \(-0.147413\pi\)
0.894667 + 0.446734i \(0.147413\pi\)
\(314\) −53.3225 −3.00916
\(315\) 0 0
\(316\) −52.7327 −2.96645
\(317\) −25.8362 −1.45110 −0.725552 0.688167i \(-0.758416\pi\)
−0.725552 + 0.688167i \(0.758416\pi\)
\(318\) −24.7726 −1.38918
\(319\) 10.1988 0.571023
\(320\) 0 0
\(321\) 15.7807 0.880792
\(322\) 12.9929 0.724067
\(323\) −15.5704 −0.866361
\(324\) 3.95498 0.219721
\(325\) 0 0
\(326\) 24.7295 1.36964
\(327\) −9.91980 −0.548566
\(328\) −25.8555 −1.42763
\(329\) −24.3284 −1.34127
\(330\) 0 0
\(331\) 20.1083 1.10525 0.552625 0.833430i \(-0.313626\pi\)
0.552625 + 0.833430i \(0.313626\pi\)
\(332\) −15.6517 −0.859001
\(333\) 5.75838 0.315558
\(334\) −3.68748 −0.201770
\(335\) 0 0
\(336\) 12.8389 0.700417
\(337\) −3.38378 −0.184326 −0.0921630 0.995744i \(-0.529378\pi\)
−0.0921630 + 0.995744i \(0.529378\pi\)
\(338\) 6.23232 0.338993
\(339\) −5.51386 −0.299472
\(340\) 0 0
\(341\) −24.3101 −1.31647
\(342\) 7.51899 0.406580
\(343\) −7.44633 −0.402064
\(344\) 12.0749 0.651037
\(345\) 0 0
\(346\) −33.8109 −1.81768
\(347\) 29.1964 1.56734 0.783672 0.621175i \(-0.213344\pi\)
0.783672 + 0.621175i \(0.213344\pi\)
\(348\) −12.3482 −0.661932
\(349\) −20.3979 −1.09187 −0.545937 0.837826i \(-0.683826\pi\)
−0.545937 + 0.837826i \(0.683826\pi\)
\(350\) 0 0
\(351\) 3.23204 0.172513
\(352\) −1.41925 −0.0756464
\(353\) 1.86346 0.0991821 0.0495910 0.998770i \(-0.484208\pi\)
0.0495910 + 0.998770i \(0.484208\pi\)
\(354\) −4.22306 −0.224453
\(355\) 0 0
\(356\) 33.7515 1.78883
\(357\) 17.3850 0.920113
\(358\) −35.2760 −1.86440
\(359\) 3.83795 0.202559 0.101280 0.994858i \(-0.467706\pi\)
0.101280 + 0.994858i \(0.467706\pi\)
\(360\) 0 0
\(361\) −9.50624 −0.500329
\(362\) −41.9480 −2.20474
\(363\) −0.329568 −0.0172978
\(364\) 43.9759 2.30497
\(365\) 0 0
\(366\) −19.1223 −0.999541
\(367\) −25.6281 −1.33777 −0.668887 0.743364i \(-0.733229\pi\)
−0.668887 + 0.743364i \(0.733229\pi\)
\(368\) −5.77570 −0.301079
\(369\) 5.41962 0.282134
\(370\) 0 0
\(371\) 34.9242 1.81317
\(372\) 29.4334 1.52605
\(373\) −21.4369 −1.10996 −0.554980 0.831863i \(-0.687274\pi\)
−0.554980 + 0.831863i \(0.687274\pi\)
\(374\) 40.2821 2.08294
\(375\) 0 0
\(376\) 33.7366 1.73983
\(377\) −10.0910 −0.519713
\(378\) −8.39527 −0.431806
\(379\) −25.8713 −1.32892 −0.664461 0.747323i \(-0.731339\pi\)
−0.664461 + 0.747323i \(0.731339\pi\)
\(380\) 0 0
\(381\) 0.992089 0.0508262
\(382\) −25.7264 −1.31628
\(383\) 14.5336 0.742632 0.371316 0.928507i \(-0.378907\pi\)
0.371316 + 0.928507i \(0.378907\pi\)
\(384\) 19.9322 1.01716
\(385\) 0 0
\(386\) −1.66649 −0.0848221
\(387\) −2.53106 −0.128661
\(388\) 42.1317 2.13891
\(389\) 23.1047 1.17145 0.585726 0.810509i \(-0.300809\pi\)
0.585726 + 0.810509i \(0.300809\pi\)
\(390\) 0 0
\(391\) −7.82084 −0.395517
\(392\) −23.0690 −1.16516
\(393\) −12.6115 −0.636167
\(394\) −56.8517 −2.86415
\(395\) 0 0
\(396\) −12.9192 −0.649214
\(397\) −13.9345 −0.699355 −0.349677 0.936870i \(-0.613709\pi\)
−0.349677 + 0.936870i \(0.613709\pi\)
\(398\) −24.8778 −1.24701
\(399\) −10.6002 −0.530673
\(400\) 0 0
\(401\) −27.5822 −1.37739 −0.688694 0.725052i \(-0.741816\pi\)
−0.688694 + 0.725052i \(0.741816\pi\)
\(402\) −4.51232 −0.225054
\(403\) 24.0532 1.19818
\(404\) 6.96763 0.346653
\(405\) 0 0
\(406\) 26.2115 1.30086
\(407\) −18.8101 −0.932383
\(408\) −24.1082 −1.19353
\(409\) 13.2125 0.653315 0.326658 0.945143i \(-0.394078\pi\)
0.326658 + 0.945143i \(0.394078\pi\)
\(410\) 0 0
\(411\) −8.24065 −0.406481
\(412\) −62.9406 −3.10086
\(413\) 5.95362 0.292959
\(414\) 3.77670 0.185615
\(415\) 0 0
\(416\) 1.40425 0.0688491
\(417\) 8.06371 0.394881
\(418\) −24.5612 −1.20133
\(419\) −8.84774 −0.432240 −0.216120 0.976367i \(-0.569340\pi\)
−0.216120 + 0.976367i \(0.569340\pi\)
\(420\) 0 0
\(421\) −37.4766 −1.82650 −0.913248 0.407404i \(-0.866434\pi\)
−0.913248 + 0.407404i \(0.866434\pi\)
\(422\) −44.5522 −2.16877
\(423\) −7.07162 −0.343834
\(424\) −48.4301 −2.35197
\(425\) 0 0
\(426\) 1.74229 0.0844140
\(427\) 26.9585 1.30461
\(428\) 62.4123 3.01681
\(429\) −10.5577 −0.509728
\(430\) 0 0
\(431\) −26.6245 −1.28246 −0.641228 0.767350i \(-0.721575\pi\)
−0.641228 + 0.767350i \(0.721575\pi\)
\(432\) 3.73192 0.179552
\(433\) −34.9972 −1.68186 −0.840928 0.541146i \(-0.817990\pi\)
−0.840928 + 0.541146i \(0.817990\pi\)
\(434\) −62.4785 −2.99907
\(435\) 0 0
\(436\) −39.2326 −1.87890
\(437\) 4.76861 0.228113
\(438\) −4.58840 −0.219242
\(439\) 20.0539 0.957122 0.478561 0.878054i \(-0.341159\pi\)
0.478561 + 0.878054i \(0.341159\pi\)
\(440\) 0 0
\(441\) 4.83555 0.230264
\(442\) −39.8564 −1.89577
\(443\) −4.14871 −0.197111 −0.0985556 0.995132i \(-0.531422\pi\)
−0.0985556 + 0.995132i \(0.531422\pi\)
\(444\) 22.7743 1.08082
\(445\) 0 0
\(446\) 42.0323 1.99029
\(447\) −19.1101 −0.903877
\(448\) −29.3253 −1.38549
\(449\) −8.34804 −0.393969 −0.196984 0.980407i \(-0.563115\pi\)
−0.196984 + 0.980407i \(0.563115\pi\)
\(450\) 0 0
\(451\) −17.7035 −0.833627
\(452\) −21.8072 −1.02573
\(453\) 1.58550 0.0744935
\(454\) −10.5635 −0.495768
\(455\) 0 0
\(456\) 14.6995 0.688366
\(457\) −20.5774 −0.962571 −0.481285 0.876564i \(-0.659830\pi\)
−0.481285 + 0.876564i \(0.659830\pi\)
\(458\) −49.5736 −2.31642
\(459\) 5.05337 0.235871
\(460\) 0 0
\(461\) 28.6891 1.33618 0.668092 0.744079i \(-0.267111\pi\)
0.668092 + 0.744079i \(0.267111\pi\)
\(462\) 27.4237 1.27586
\(463\) −40.0797 −1.86266 −0.931331 0.364173i \(-0.881352\pi\)
−0.931331 + 0.364173i \(0.881352\pi\)
\(464\) −11.6517 −0.540918
\(465\) 0 0
\(466\) −0.832030 −0.0385430
\(467\) 8.14191 0.376763 0.188381 0.982096i \(-0.439676\pi\)
0.188381 + 0.982096i \(0.439676\pi\)
\(468\) 12.7827 0.590878
\(469\) 6.36142 0.293743
\(470\) 0 0
\(471\) 21.8510 1.00684
\(472\) −8.25601 −0.380014
\(473\) 8.26785 0.380156
\(474\) 32.5369 1.49447
\(475\) 0 0
\(476\) 68.7575 3.15149
\(477\) 10.1515 0.464807
\(478\) 17.4843 0.799715
\(479\) −28.0621 −1.28219 −0.641094 0.767462i \(-0.721519\pi\)
−0.641094 + 0.767462i \(0.721519\pi\)
\(480\) 0 0
\(481\) 18.6113 0.848603
\(482\) 30.6554 1.39631
\(483\) −5.32435 −0.242266
\(484\) −1.30344 −0.0592471
\(485\) 0 0
\(486\) −2.44028 −0.110693
\(487\) 15.2173 0.689560 0.344780 0.938684i \(-0.387954\pi\)
0.344780 + 0.938684i \(0.387954\pi\)
\(488\) −37.3838 −1.69229
\(489\) −10.1338 −0.458269
\(490\) 0 0
\(491\) 8.29230 0.374226 0.187113 0.982338i \(-0.440087\pi\)
0.187113 + 0.982338i \(0.440087\pi\)
\(492\) 21.4345 0.966343
\(493\) −15.7775 −0.710585
\(494\) 24.3016 1.09338
\(495\) 0 0
\(496\) 27.7734 1.24706
\(497\) −2.45625 −0.110178
\(498\) 9.65736 0.432756
\(499\) −24.4006 −1.09232 −0.546160 0.837681i \(-0.683911\pi\)
−0.546160 + 0.837681i \(0.683911\pi\)
\(500\) 0 0
\(501\) 1.51109 0.0675104
\(502\) −41.5240 −1.85331
\(503\) −10.4731 −0.466974 −0.233487 0.972360i \(-0.575014\pi\)
−0.233487 + 0.972360i \(0.575014\pi\)
\(504\) −16.4126 −0.731075
\(505\) 0 0
\(506\) −12.3368 −0.548439
\(507\) −2.55393 −0.113424
\(508\) 3.92369 0.174086
\(509\) −11.0599 −0.490220 −0.245110 0.969495i \(-0.578824\pi\)
−0.245110 + 0.969495i \(0.578824\pi\)
\(510\) 0 0
\(511\) 6.46867 0.286157
\(512\) 37.2293 1.64532
\(513\) −3.08119 −0.136038
\(514\) 10.0832 0.444753
\(515\) 0 0
\(516\) −10.0103 −0.440678
\(517\) 23.0999 1.01593
\(518\) −48.3432 −2.12408
\(519\) 13.8553 0.608181
\(520\) 0 0
\(521\) −41.4212 −1.81470 −0.907348 0.420380i \(-0.861897\pi\)
−0.907348 + 0.420380i \(0.861897\pi\)
\(522\) 7.61901 0.333475
\(523\) −13.6425 −0.596544 −0.298272 0.954481i \(-0.596410\pi\)
−0.298272 + 0.954481i \(0.596410\pi\)
\(524\) −49.8783 −2.17894
\(525\) 0 0
\(526\) −2.53235 −0.110416
\(527\) 37.6078 1.63822
\(528\) −12.1906 −0.530526
\(529\) −20.6048 −0.895860
\(530\) 0 0
\(531\) 1.73056 0.0751000
\(532\) −41.9235 −1.81762
\(533\) 17.5164 0.758721
\(534\) −20.8252 −0.901194
\(535\) 0 0
\(536\) −8.82151 −0.381031
\(537\) 14.4557 0.623810
\(538\) 38.5313 1.66120
\(539\) −15.7956 −0.680366
\(540\) 0 0
\(541\) 14.6598 0.630272 0.315136 0.949046i \(-0.397950\pi\)
0.315136 + 0.949046i \(0.397950\pi\)
\(542\) 33.4327 1.43606
\(543\) 17.1898 0.737685
\(544\) 2.19558 0.0941348
\(545\) 0 0
\(546\) −27.1338 −1.16122
\(547\) −26.6123 −1.13786 −0.568931 0.822385i \(-0.692643\pi\)
−0.568931 + 0.822385i \(0.692643\pi\)
\(548\) −32.5916 −1.39225
\(549\) 7.83611 0.334437
\(550\) 0 0
\(551\) 9.62005 0.409828
\(552\) 7.38338 0.314258
\(553\) −45.8701 −1.95060
\(554\) 17.0329 0.723659
\(555\) 0 0
\(556\) 31.8918 1.35251
\(557\) 10.3141 0.437020 0.218510 0.975835i \(-0.429880\pi\)
0.218510 + 0.975835i \(0.429880\pi\)
\(558\) −18.1609 −0.768811
\(559\) −8.18047 −0.345997
\(560\) 0 0
\(561\) −16.5072 −0.696933
\(562\) −19.4203 −0.819197
\(563\) 34.4031 1.44992 0.724958 0.688793i \(-0.241859\pi\)
0.724958 + 0.688793i \(0.241859\pi\)
\(564\) −27.9681 −1.17767
\(565\) 0 0
\(566\) −67.8281 −2.85103
\(567\) 3.44028 0.144478
\(568\) 3.40614 0.142918
\(569\) −12.2048 −0.511651 −0.255825 0.966723i \(-0.582347\pi\)
−0.255825 + 0.966723i \(0.582347\pi\)
\(570\) 0 0
\(571\) 42.0644 1.76034 0.880170 0.474658i \(-0.157428\pi\)
0.880170 + 0.474658i \(0.157428\pi\)
\(572\) −41.7553 −1.74588
\(573\) 10.5424 0.440415
\(574\) −45.4992 −1.89910
\(575\) 0 0
\(576\) −8.52409 −0.355170
\(577\) −17.5351 −0.729994 −0.364997 0.931009i \(-0.618930\pi\)
−0.364997 + 0.931009i \(0.618930\pi\)
\(578\) −20.8316 −0.866482
\(579\) 0.682908 0.0283807
\(580\) 0 0
\(581\) −13.6148 −0.564838
\(582\) −25.9959 −1.07756
\(583\) −33.1607 −1.37337
\(584\) −8.97024 −0.371191
\(585\) 0 0
\(586\) −34.6759 −1.43245
\(587\) 21.9119 0.904402 0.452201 0.891916i \(-0.350639\pi\)
0.452201 + 0.891916i \(0.350639\pi\)
\(588\) 19.1245 0.788681
\(589\) −22.9306 −0.944839
\(590\) 0 0
\(591\) 23.2972 0.958318
\(592\) 21.4898 0.883227
\(593\) 38.0061 1.56072 0.780361 0.625330i \(-0.215035\pi\)
0.780361 + 0.625330i \(0.215035\pi\)
\(594\) 7.97134 0.327068
\(595\) 0 0
\(596\) −75.5802 −3.09588
\(597\) 10.1946 0.417239
\(598\) 12.2064 0.499158
\(599\) 16.3154 0.666629 0.333314 0.942816i \(-0.391833\pi\)
0.333314 + 0.942816i \(0.391833\pi\)
\(600\) 0 0
\(601\) 2.31871 0.0945822 0.0472911 0.998881i \(-0.484941\pi\)
0.0472911 + 0.998881i \(0.484941\pi\)
\(602\) 21.2489 0.866040
\(603\) 1.84910 0.0753011
\(604\) 6.27064 0.255149
\(605\) 0 0
\(606\) −4.29913 −0.174640
\(607\) −32.2134 −1.30750 −0.653752 0.756709i \(-0.726806\pi\)
−0.653752 + 0.756709i \(0.726806\pi\)
\(608\) −1.33871 −0.0542920
\(609\) −10.7412 −0.435255
\(610\) 0 0
\(611\) −22.8557 −0.924644
\(612\) 19.9860 0.807886
\(613\) −25.9482 −1.04804 −0.524018 0.851707i \(-0.675568\pi\)
−0.524018 + 0.851707i \(0.675568\pi\)
\(614\) −56.8370 −2.29375
\(615\) 0 0
\(616\) 53.6128 2.16012
\(617\) −1.90374 −0.0766416 −0.0383208 0.999265i \(-0.512201\pi\)
−0.0383208 + 0.999265i \(0.512201\pi\)
\(618\) 38.8353 1.56218
\(619\) −19.8795 −0.799025 −0.399513 0.916728i \(-0.630821\pi\)
−0.399513 + 0.916728i \(0.630821\pi\)
\(620\) 0 0
\(621\) −1.54765 −0.0621050
\(622\) 18.4223 0.738666
\(623\) 29.3591 1.17625
\(624\) 12.0617 0.482855
\(625\) 0 0
\(626\) −77.2509 −3.08757
\(627\) 10.0649 0.401954
\(628\) 86.4201 3.44854
\(629\) 29.0993 1.16026
\(630\) 0 0
\(631\) −32.8982 −1.30966 −0.654828 0.755778i \(-0.727259\pi\)
−0.654828 + 0.755778i \(0.727259\pi\)
\(632\) 63.6090 2.53023
\(633\) 18.2570 0.725650
\(634\) 63.0476 2.50394
\(635\) 0 0
\(636\) 40.1492 1.59202
\(637\) 15.6287 0.619231
\(638\) −24.8880 −0.985324
\(639\) −0.713969 −0.0282442
\(640\) 0 0
\(641\) −40.7624 −1.61002 −0.805009 0.593263i \(-0.797839\pi\)
−0.805009 + 0.593263i \(0.797839\pi\)
\(642\) −38.5093 −1.51984
\(643\) −24.9947 −0.985695 −0.492847 0.870116i \(-0.664044\pi\)
−0.492847 + 0.870116i \(0.664044\pi\)
\(644\) −21.0577 −0.829790
\(645\) 0 0
\(646\) 37.9962 1.49494
\(647\) 32.0232 1.25896 0.629481 0.777016i \(-0.283268\pi\)
0.629481 + 0.777016i \(0.283268\pi\)
\(648\) −4.77071 −0.187411
\(649\) −5.65299 −0.221899
\(650\) 0 0
\(651\) 25.6030 1.00346
\(652\) −40.0792 −1.56962
\(653\) 44.6772 1.74835 0.874177 0.485607i \(-0.161401\pi\)
0.874177 + 0.485607i \(0.161401\pi\)
\(654\) 24.2071 0.946573
\(655\) 0 0
\(656\) 20.2256 0.789677
\(657\) 1.88027 0.0733564
\(658\) 59.3681 2.31441
\(659\) −24.0215 −0.935747 −0.467873 0.883796i \(-0.654980\pi\)
−0.467873 + 0.883796i \(0.654980\pi\)
\(660\) 0 0
\(661\) 37.8928 1.47386 0.736929 0.675970i \(-0.236275\pi\)
0.736929 + 0.675970i \(0.236275\pi\)
\(662\) −49.0699 −1.90715
\(663\) 16.3327 0.634309
\(664\) 18.8800 0.732684
\(665\) 0 0
\(666\) −14.0521 −0.544507
\(667\) 4.83204 0.187097
\(668\) 5.97633 0.231231
\(669\) −17.2244 −0.665932
\(670\) 0 0
\(671\) −25.5972 −0.988167
\(672\) 1.49473 0.0576604
\(673\) 18.4267 0.710299 0.355149 0.934810i \(-0.384430\pi\)
0.355149 + 0.934810i \(0.384430\pi\)
\(674\) 8.25737 0.318062
\(675\) 0 0
\(676\) −10.1008 −0.388491
\(677\) 22.8164 0.876904 0.438452 0.898755i \(-0.355527\pi\)
0.438452 + 0.898755i \(0.355527\pi\)
\(678\) 13.4554 0.516751
\(679\) 36.6487 1.40645
\(680\) 0 0
\(681\) 4.32879 0.165879
\(682\) 59.3236 2.27162
\(683\) −32.2064 −1.23234 −0.616172 0.787612i \(-0.711317\pi\)
−0.616172 + 0.787612i \(0.711317\pi\)
\(684\) −12.1861 −0.465946
\(685\) 0 0
\(686\) 18.1711 0.693777
\(687\) 20.3147 0.775053
\(688\) −9.44570 −0.360114
\(689\) 32.8102 1.24997
\(690\) 0 0
\(691\) 24.5769 0.934947 0.467474 0.884007i \(-0.345164\pi\)
0.467474 + 0.884007i \(0.345164\pi\)
\(692\) 54.7975 2.08309
\(693\) −11.2379 −0.426893
\(694\) −71.2475 −2.70452
\(695\) 0 0
\(696\) 14.8950 0.564594
\(697\) 27.3874 1.03737
\(698\) 49.7766 1.88407
\(699\) 0.340956 0.0128961
\(700\) 0 0
\(701\) −3.66355 −0.138370 −0.0691852 0.997604i \(-0.522040\pi\)
−0.0691852 + 0.997604i \(0.522040\pi\)
\(702\) −7.88709 −0.297679
\(703\) −17.7427 −0.669179
\(704\) 27.8445 1.04943
\(705\) 0 0
\(706\) −4.54737 −0.171143
\(707\) 6.06087 0.227942
\(708\) 6.84434 0.257226
\(709\) −26.6969 −1.00262 −0.501312 0.865267i \(-0.667149\pi\)
−0.501312 + 0.865267i \(0.667149\pi\)
\(710\) 0 0
\(711\) −13.3332 −0.500035
\(712\) −40.7129 −1.52578
\(713\) −11.5178 −0.431344
\(714\) −42.4244 −1.58769
\(715\) 0 0
\(716\) 57.1721 2.13662
\(717\) −7.16488 −0.267577
\(718\) −9.36569 −0.349524
\(719\) 17.8699 0.666433 0.333217 0.942850i \(-0.391866\pi\)
0.333217 + 0.942850i \(0.391866\pi\)
\(720\) 0 0
\(721\) −54.7495 −2.03898
\(722\) 23.1979 0.863337
\(723\) −12.5622 −0.467194
\(724\) 67.9853 2.52665
\(725\) 0 0
\(726\) 0.804239 0.0298481
\(727\) 34.0727 1.26369 0.631843 0.775097i \(-0.282299\pi\)
0.631843 + 0.775097i \(0.282299\pi\)
\(728\) −53.0461 −1.96602
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.7904 −0.473069
\(732\) 30.9917 1.14549
\(733\) 40.1806 1.48410 0.742052 0.670342i \(-0.233853\pi\)
0.742052 + 0.670342i \(0.233853\pi\)
\(734\) 62.5398 2.30838
\(735\) 0 0
\(736\) −0.672420 −0.0247857
\(737\) −6.04019 −0.222493
\(738\) −13.2254 −0.486834
\(739\) 15.4750 0.569256 0.284628 0.958638i \(-0.408130\pi\)
0.284628 + 0.958638i \(0.408130\pi\)
\(740\) 0 0
\(741\) −9.95854 −0.365836
\(742\) −85.2249 −3.12870
\(743\) 17.1140 0.627853 0.313926 0.949447i \(-0.398355\pi\)
0.313926 + 0.949447i \(0.398355\pi\)
\(744\) −35.5042 −1.30165
\(745\) 0 0
\(746\) 52.3121 1.91528
\(747\) −3.95747 −0.144796
\(748\) −65.2855 −2.38707
\(749\) 54.2900 1.98371
\(750\) 0 0
\(751\) −11.1559 −0.407086 −0.203543 0.979066i \(-0.565246\pi\)
−0.203543 + 0.979066i \(0.565246\pi\)
\(752\) −26.3907 −0.962370
\(753\) 17.0160 0.620099
\(754\) 24.6249 0.896787
\(755\) 0 0
\(756\) 13.6063 0.494855
\(757\) −24.6773 −0.896911 −0.448456 0.893805i \(-0.648026\pi\)
−0.448456 + 0.893805i \(0.648026\pi\)
\(758\) 63.1334 2.29311
\(759\) 5.05549 0.183503
\(760\) 0 0
\(761\) −40.2478 −1.45898 −0.729490 0.683991i \(-0.760243\pi\)
−0.729490 + 0.683991i \(0.760243\pi\)
\(762\) −2.42098 −0.0877028
\(763\) −34.1269 −1.23548
\(764\) 41.6950 1.50847
\(765\) 0 0
\(766\) −35.4661 −1.28144
\(767\) 5.59324 0.201960
\(768\) −31.5921 −1.13998
\(769\) 22.1010 0.796983 0.398492 0.917172i \(-0.369534\pi\)
0.398492 + 0.917172i \(0.369534\pi\)
\(770\) 0 0
\(771\) −4.13200 −0.148810
\(772\) 2.70089 0.0972072
\(773\) −4.90033 −0.176253 −0.0881264 0.996109i \(-0.528088\pi\)
−0.0881264 + 0.996109i \(0.528088\pi\)
\(774\) 6.17649 0.222009
\(775\) 0 0
\(776\) −50.8215 −1.82439
\(777\) 19.8105 0.710697
\(778\) −56.3819 −2.02139
\(779\) −16.6989 −0.598301
\(780\) 0 0
\(781\) 2.33222 0.0834535
\(782\) 19.0851 0.682481
\(783\) −3.12218 −0.111578
\(784\) 18.0459 0.644496
\(785\) 0 0
\(786\) 30.7757 1.09773
\(787\) 3.73477 0.133130 0.0665652 0.997782i \(-0.478796\pi\)
0.0665652 + 0.997782i \(0.478796\pi\)
\(788\) 92.1400 3.28235
\(789\) 1.03773 0.0369442
\(790\) 0 0
\(791\) −18.9692 −0.674469
\(792\) 15.5838 0.553747
\(793\) 25.3266 0.899375
\(794\) 34.0042 1.20677
\(795\) 0 0
\(796\) 40.3196 1.42909
\(797\) −3.34404 −0.118452 −0.0592259 0.998245i \(-0.518863\pi\)
−0.0592259 + 0.998245i \(0.518863\pi\)
\(798\) 25.8674 0.915698
\(799\) −35.7355 −1.26423
\(800\) 0 0
\(801\) 8.53392 0.301531
\(802\) 67.3083 2.37674
\(803\) −6.14203 −0.216748
\(804\) 7.31315 0.257915
\(805\) 0 0
\(806\) −58.6966 −2.06750
\(807\) −15.7897 −0.555823
\(808\) −8.40473 −0.295677
\(809\) 34.2875 1.20549 0.602743 0.797935i \(-0.294074\pi\)
0.602743 + 0.797935i \(0.294074\pi\)
\(810\) 0 0
\(811\) −3.46750 −0.121760 −0.0608801 0.998145i \(-0.519391\pi\)
−0.0608801 + 0.998145i \(0.519391\pi\)
\(812\) −42.4812 −1.49080
\(813\) −13.7003 −0.480492
\(814\) 45.9020 1.60887
\(815\) 0 0
\(816\) 18.8588 0.660189
\(817\) 7.79867 0.272841
\(818\) −32.2422 −1.12732
\(819\) 11.1191 0.388534
\(820\) 0 0
\(821\) −14.6809 −0.512367 −0.256184 0.966628i \(-0.582465\pi\)
−0.256184 + 0.966628i \(0.582465\pi\)
\(822\) 20.1095 0.701400
\(823\) 20.8698 0.727475 0.363737 0.931502i \(-0.381501\pi\)
0.363737 + 0.931502i \(0.381501\pi\)
\(824\) 75.9223 2.64488
\(825\) 0 0
\(826\) −14.5285 −0.505512
\(827\) −45.2876 −1.57480 −0.787401 0.616441i \(-0.788574\pi\)
−0.787401 + 0.616441i \(0.788574\pi\)
\(828\) −6.12092 −0.212717
\(829\) −9.65729 −0.335411 −0.167706 0.985837i \(-0.553636\pi\)
−0.167706 + 0.985837i \(0.553636\pi\)
\(830\) 0 0
\(831\) −6.97989 −0.242130
\(832\) −27.5502 −0.955131
\(833\) 24.4358 0.846651
\(834\) −19.6777 −0.681384
\(835\) 0 0
\(836\) 39.8066 1.37674
\(837\) 7.44212 0.257237
\(838\) 21.5910 0.745848
\(839\) −25.1717 −0.869023 −0.434511 0.900666i \(-0.643079\pi\)
−0.434511 + 0.900666i \(0.643079\pi\)
\(840\) 0 0
\(841\) −19.2520 −0.663861
\(842\) 91.4534 3.15169
\(843\) 7.95823 0.274096
\(844\) 72.2060 2.48543
\(845\) 0 0
\(846\) 17.2567 0.593299
\(847\) −1.13381 −0.0389581
\(848\) 37.8848 1.30097
\(849\) 27.7952 0.953928
\(850\) 0 0
\(851\) −8.91196 −0.305498
\(852\) −2.82373 −0.0967395
\(853\) 3.20847 0.109856 0.0549280 0.998490i \(-0.482507\pi\)
0.0549280 + 0.998490i \(0.482507\pi\)
\(854\) −65.7863 −2.25116
\(855\) 0 0
\(856\) −75.2851 −2.57319
\(857\) −19.4569 −0.664634 −0.332317 0.943168i \(-0.607830\pi\)
−0.332317 + 0.943168i \(0.607830\pi\)
\(858\) 25.7637 0.879557
\(859\) 16.8332 0.574340 0.287170 0.957880i \(-0.407286\pi\)
0.287170 + 0.957880i \(0.407286\pi\)
\(860\) 0 0
\(861\) 18.6450 0.635421
\(862\) 64.9713 2.21293
\(863\) −3.81218 −0.129768 −0.0648841 0.997893i \(-0.520668\pi\)
−0.0648841 + 0.997893i \(0.520668\pi\)
\(864\) 0.434479 0.0147813
\(865\) 0 0
\(866\) 85.4030 2.90211
\(867\) 8.53657 0.289917
\(868\) 101.259 3.43697
\(869\) 43.5538 1.47746
\(870\) 0 0
\(871\) 5.97635 0.202501
\(872\) 47.3245 1.60261
\(873\) 10.6528 0.360543
\(874\) −11.6367 −0.393619
\(875\) 0 0
\(876\) 7.43645 0.251254
\(877\) 25.7451 0.869352 0.434676 0.900587i \(-0.356863\pi\)
0.434676 + 0.900587i \(0.356863\pi\)
\(878\) −48.9373 −1.65155
\(879\) 14.2098 0.479285
\(880\) 0 0
\(881\) 45.2537 1.52464 0.762318 0.647202i \(-0.224061\pi\)
0.762318 + 0.647202i \(0.224061\pi\)
\(882\) −11.8001 −0.397330
\(883\) −44.3978 −1.49410 −0.747052 0.664765i \(-0.768531\pi\)
−0.747052 + 0.664765i \(0.768531\pi\)
\(884\) 64.5955 2.17258
\(885\) 0 0
\(886\) 10.1240 0.340124
\(887\) 38.9928 1.30925 0.654626 0.755953i \(-0.272826\pi\)
0.654626 + 0.755953i \(0.272826\pi\)
\(888\) −27.4716 −0.921886
\(889\) 3.41307 0.114471
\(890\) 0 0
\(891\) −3.26656 −0.109434
\(892\) −68.1220 −2.28089
\(893\) 21.7890 0.729142
\(894\) 46.6341 1.55968
\(895\) 0 0
\(896\) 68.5726 2.29085
\(897\) −5.00206 −0.167014
\(898\) 20.3716 0.679809
\(899\) −23.2356 −0.774952
\(900\) 0 0
\(901\) 51.2995 1.70903
\(902\) 43.2017 1.43846
\(903\) −8.70755 −0.289769
\(904\) 26.3050 0.874892
\(905\) 0 0
\(906\) −3.86908 −0.128541
\(907\) −11.0201 −0.365915 −0.182958 0.983121i \(-0.558567\pi\)
−0.182958 + 0.983121i \(0.558567\pi\)
\(908\) 17.1203 0.568156
\(909\) 1.76173 0.0584331
\(910\) 0 0
\(911\) −52.9227 −1.75341 −0.876704 0.481031i \(-0.840263\pi\)
−0.876704 + 0.481031i \(0.840263\pi\)
\(912\) −11.4988 −0.380762
\(913\) 12.9273 0.427832
\(914\) 50.2147 1.66096
\(915\) 0 0
\(916\) 80.3442 2.65465
\(917\) −43.3872 −1.43277
\(918\) −12.3317 −0.407005
\(919\) −47.6045 −1.57033 −0.785164 0.619288i \(-0.787421\pi\)
−0.785164 + 0.619288i \(0.787421\pi\)
\(920\) 0 0
\(921\) 23.2911 0.767469
\(922\) −70.0095 −2.30564
\(923\) −2.30757 −0.0759547
\(924\) −44.4457 −1.46216
\(925\) 0 0
\(926\) 97.8059 3.21410
\(927\) −15.9143 −0.522693
\(928\) −1.35652 −0.0445300
\(929\) 21.6887 0.711582 0.355791 0.934566i \(-0.384211\pi\)
0.355791 + 0.934566i \(0.384211\pi\)
\(930\) 0 0
\(931\) −14.8993 −0.488304
\(932\) 1.34848 0.0441708
\(933\) −7.54924 −0.247151
\(934\) −19.8686 −0.650119
\(935\) 0 0
\(936\) −15.4191 −0.503990
\(937\) −2.20089 −0.0718999 −0.0359500 0.999354i \(-0.511446\pi\)
−0.0359500 + 0.999354i \(0.511446\pi\)
\(938\) −15.5237 −0.506866
\(939\) 31.6565 1.03307
\(940\) 0 0
\(941\) −16.7898 −0.547331 −0.273666 0.961825i \(-0.588236\pi\)
−0.273666 + 0.961825i \(0.588236\pi\)
\(942\) −53.3225 −1.73734
\(943\) −8.38767 −0.273140
\(944\) 6.45832 0.210200
\(945\) 0 0
\(946\) −20.1759 −0.655975
\(947\) −36.6132 −1.18977 −0.594885 0.803811i \(-0.702802\pi\)
−0.594885 + 0.803811i \(0.702802\pi\)
\(948\) −52.7327 −1.71268
\(949\) 6.07711 0.197271
\(950\) 0 0
\(951\) −25.8362 −0.837796
\(952\) −82.9389 −2.68807
\(953\) −43.9277 −1.42296 −0.711479 0.702707i \(-0.751974\pi\)
−0.711479 + 0.702707i \(0.751974\pi\)
\(954\) −24.7726 −0.802044
\(955\) 0 0
\(956\) −28.3370 −0.916483
\(957\) 10.1988 0.329680
\(958\) 68.4794 2.21247
\(959\) −28.3502 −0.915475
\(960\) 0 0
\(961\) 24.3851 0.786616
\(962\) −45.4169 −1.46430
\(963\) 15.7807 0.508525
\(964\) −49.6834 −1.60019
\(965\) 0 0
\(966\) 12.9929 0.418040
\(967\) −24.4815 −0.787272 −0.393636 0.919266i \(-0.628783\pi\)
−0.393636 + 0.919266i \(0.628783\pi\)
\(968\) 1.57227 0.0505348
\(969\) −15.5704 −0.500194
\(970\) 0 0
\(971\) −19.6254 −0.629808 −0.314904 0.949123i \(-0.601972\pi\)
−0.314904 + 0.949123i \(0.601972\pi\)
\(972\) 3.95498 0.126856
\(973\) 27.7414 0.889349
\(974\) −37.1344 −1.18986
\(975\) 0 0
\(976\) 29.2438 0.936070
\(977\) −30.5238 −0.976541 −0.488271 0.872692i \(-0.662372\pi\)
−0.488271 + 0.872692i \(0.662372\pi\)
\(978\) 24.7295 0.790761
\(979\) −27.8766 −0.890939
\(980\) 0 0
\(981\) −9.91980 −0.316715
\(982\) −20.2356 −0.645743
\(983\) −10.3655 −0.330609 −0.165305 0.986243i \(-0.552861\pi\)
−0.165305 + 0.986243i \(0.552861\pi\)
\(984\) −25.8555 −0.824242
\(985\) 0 0
\(986\) 38.5017 1.22614
\(987\) −24.3284 −0.774380
\(988\) −39.3858 −1.25303
\(989\) 3.91718 0.124559
\(990\) 0 0
\(991\) −10.1087 −0.321113 −0.160557 0.987027i \(-0.551329\pi\)
−0.160557 + 0.987027i \(0.551329\pi\)
\(992\) 3.23344 0.102662
\(993\) 20.1083 0.638116
\(994\) 5.99396 0.190117
\(995\) 0 0
\(996\) −15.6517 −0.495944
\(997\) −21.3639 −0.676602 −0.338301 0.941038i \(-0.609852\pi\)
−0.338301 + 0.941038i \(0.609852\pi\)
\(998\) 59.5443 1.88484
\(999\) 5.75838 0.182187
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.k.1.1 6
3.2 odd 2 5625.2.a.q.1.6 6
5.2 odd 4 1875.2.b.f.1249.2 12
5.3 odd 4 1875.2.b.f.1249.11 12
5.4 even 2 1875.2.a.j.1.6 6
15.14 odd 2 5625.2.a.p.1.1 6
25.2 odd 20 375.2.i.d.274.2 24
25.9 even 10 75.2.g.c.31.3 12
25.11 even 5 375.2.g.c.226.1 12
25.12 odd 20 375.2.i.d.349.5 24
25.13 odd 20 375.2.i.d.349.2 24
25.14 even 10 75.2.g.c.46.3 yes 12
25.16 even 5 375.2.g.c.151.1 12
25.23 odd 20 375.2.i.d.274.5 24
75.14 odd 10 225.2.h.d.46.1 12
75.59 odd 10 225.2.h.d.181.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.31.3 12 25.9 even 10
75.2.g.c.46.3 yes 12 25.14 even 10
225.2.h.d.46.1 12 75.14 odd 10
225.2.h.d.181.1 12 75.59 odd 10
375.2.g.c.151.1 12 25.16 even 5
375.2.g.c.226.1 12 25.11 even 5
375.2.i.d.274.2 24 25.2 odd 20
375.2.i.d.274.5 24 25.23 odd 20
375.2.i.d.349.2 24 25.13 odd 20
375.2.i.d.349.5 24 25.12 odd 20
1875.2.a.j.1.6 6 5.4 even 2
1875.2.a.k.1.1 6 1.1 even 1 trivial
1875.2.b.f.1249.2 12 5.2 odd 4
1875.2.b.f.1249.11 12 5.3 odd 4
5625.2.a.p.1.1 6 15.14 odd 2
5625.2.a.q.1.6 6 3.2 odd 2