Properties

Label 5625.2.a.bc.1.6
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
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] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, 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("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.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: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.13366265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1875)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.31354\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.31354 q^{2} -0.274605 q^{4} -4.19091 q^{7} -2.98779 q^{8} +O(q^{10})\) \(q+1.31354 q^{2} -0.274605 q^{4} -4.19091 q^{7} -2.98779 q^{8} +0.167760 q^{11} +3.39519 q^{13} -5.50493 q^{14} -3.37538 q^{16} +4.57476 q^{17} +3.78258 q^{19} +0.220360 q^{22} +8.31374 q^{23} +4.45973 q^{26} +1.15084 q^{28} -2.74553 q^{29} -7.71771 q^{31} +1.54187 q^{32} +6.00914 q^{34} -2.07700 q^{37} +4.96858 q^{38} +1.28043 q^{41} -11.6226 q^{43} -0.0460679 q^{44} +10.9204 q^{46} -9.99490 q^{47} +10.5637 q^{49} -0.932338 q^{52} -1.07165 q^{53} +12.5216 q^{56} -3.60638 q^{58} +4.95536 q^{59} +2.36706 q^{61} -10.1375 q^{62} +8.77608 q^{64} -12.2369 q^{67} -1.25625 q^{68} -14.1746 q^{71} +8.67832 q^{73} -2.72823 q^{74} -1.03872 q^{76} -0.703068 q^{77} -2.64466 q^{79} +1.68189 q^{82} -14.3131 q^{83} -15.2668 q^{86} -0.501233 q^{88} +4.06867 q^{89} -14.2289 q^{91} -2.28300 q^{92} -13.1287 q^{94} +2.78196 q^{97} +13.8759 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8} - 12 q^{11} - 14 q^{13} - 16 q^{14} + 15 q^{16} - q^{17} + 16 q^{19} - 18 q^{22} - 4 q^{23} + 34 q^{26} + 21 q^{28} - 2 q^{29} + 13 q^{31} - 18 q^{32} - 37 q^{34} + 8 q^{37} - 24 q^{38} + 12 q^{41} - 20 q^{43} - 47 q^{44} + 33 q^{46} - 15 q^{47} + 30 q^{49} + q^{52} - 4 q^{53} - 60 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} + 4 q^{62} + 41 q^{64} - 19 q^{67} - 33 q^{68} - 21 q^{71} + 19 q^{73} + 9 q^{74} - q^{76} - 11 q^{77} + 10 q^{79} - 24 q^{82} - 27 q^{83} - 42 q^{86} - 53 q^{88} + 9 q^{89} - 12 q^{91} - 63 q^{92} + 14 q^{94} - 24 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.31354 0.928815 0.464408 0.885622i \(-0.346267\pi\)
0.464408 + 0.885622i \(0.346267\pi\)
\(3\) 0 0
\(4\) −0.274605 −0.137303
\(5\) 0 0
\(6\) 0 0
\(7\) −4.19091 −1.58401 −0.792007 0.610512i \(-0.790964\pi\)
−0.792007 + 0.610512i \(0.790964\pi\)
\(8\) −2.98779 −1.05634
\(9\) 0 0
\(10\) 0 0
\(11\) 0.167760 0.0505816 0.0252908 0.999680i \(-0.491949\pi\)
0.0252908 + 0.999680i \(0.491949\pi\)
\(12\) 0 0
\(13\) 3.39519 0.941658 0.470829 0.882225i \(-0.343955\pi\)
0.470829 + 0.882225i \(0.343955\pi\)
\(14\) −5.50493 −1.47126
\(15\) 0 0
\(16\) −3.37538 −0.843845
\(17\) 4.57476 1.10954 0.554771 0.832003i \(-0.312806\pi\)
0.554771 + 0.832003i \(0.312806\pi\)
\(18\) 0 0
\(19\) 3.78258 0.867784 0.433892 0.900965i \(-0.357140\pi\)
0.433892 + 0.900965i \(0.357140\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.220360 0.0469810
\(23\) 8.31374 1.73353 0.866767 0.498713i \(-0.166194\pi\)
0.866767 + 0.498713i \(0.166194\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.45973 0.874626
\(27\) 0 0
\(28\) 1.15084 0.217489
\(29\) −2.74553 −0.509833 −0.254916 0.966963i \(-0.582048\pi\)
−0.254916 + 0.966963i \(0.582048\pi\)
\(30\) 0 0
\(31\) −7.71771 −1.38614 −0.693070 0.720870i \(-0.743742\pi\)
−0.693070 + 0.720870i \(0.743742\pi\)
\(32\) 1.54187 0.272567
\(33\) 0 0
\(34\) 6.00914 1.03056
\(35\) 0 0
\(36\) 0 0
\(37\) −2.07700 −0.341457 −0.170728 0.985318i \(-0.554612\pi\)
−0.170728 + 0.985318i \(0.554612\pi\)
\(38\) 4.96858 0.806011
\(39\) 0 0
\(40\) 0 0
\(41\) 1.28043 0.199969 0.0999845 0.994989i \(-0.468121\pi\)
0.0999845 + 0.994989i \(0.468121\pi\)
\(42\) 0 0
\(43\) −11.6226 −1.77243 −0.886217 0.463270i \(-0.846676\pi\)
−0.886217 + 0.463270i \(0.846676\pi\)
\(44\) −0.0460679 −0.00694499
\(45\) 0 0
\(46\) 10.9204 1.61013
\(47\) −9.99490 −1.45791 −0.728953 0.684564i \(-0.759993\pi\)
−0.728953 + 0.684564i \(0.759993\pi\)
\(48\) 0 0
\(49\) 10.5637 1.50910
\(50\) 0 0
\(51\) 0 0
\(52\) −0.932338 −0.129292
\(53\) −1.07165 −0.147203 −0.0736015 0.997288i \(-0.523449\pi\)
−0.0736015 + 0.997288i \(0.523449\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 12.5216 1.67326
\(57\) 0 0
\(58\) −3.60638 −0.473540
\(59\) 4.95536 0.645133 0.322566 0.946547i \(-0.395454\pi\)
0.322566 + 0.946547i \(0.395454\pi\)
\(60\) 0 0
\(61\) 2.36706 0.303071 0.151536 0.988452i \(-0.451578\pi\)
0.151536 + 0.988452i \(0.451578\pi\)
\(62\) −10.1375 −1.28747
\(63\) 0 0
\(64\) 8.77608 1.09701
\(65\) 0 0
\(66\) 0 0
\(67\) −12.2369 −1.49497 −0.747486 0.664277i \(-0.768739\pi\)
−0.747486 + 0.664277i \(0.768739\pi\)
\(68\) −1.25625 −0.152343
\(69\) 0 0
\(70\) 0 0
\(71\) −14.1746 −1.68222 −0.841109 0.540866i \(-0.818097\pi\)
−0.841109 + 0.540866i \(0.818097\pi\)
\(72\) 0 0
\(73\) 8.67832 1.01572 0.507860 0.861439i \(-0.330437\pi\)
0.507860 + 0.861439i \(0.330437\pi\)
\(74\) −2.72823 −0.317150
\(75\) 0 0
\(76\) −1.03872 −0.119149
\(77\) −0.703068 −0.0801220
\(78\) 0 0
\(79\) −2.64466 −0.297547 −0.148774 0.988871i \(-0.547533\pi\)
−0.148774 + 0.988871i \(0.547533\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.68189 0.185734
\(83\) −14.3131 −1.57106 −0.785532 0.618821i \(-0.787611\pi\)
−0.785532 + 0.618821i \(0.787611\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −15.2668 −1.64626
\(87\) 0 0
\(88\) −0.501233 −0.0534316
\(89\) 4.06867 0.431278 0.215639 0.976473i \(-0.430817\pi\)
0.215639 + 0.976473i \(0.430817\pi\)
\(90\) 0 0
\(91\) −14.2289 −1.49160
\(92\) −2.28300 −0.238019
\(93\) 0 0
\(94\) −13.1287 −1.35412
\(95\) 0 0
\(96\) 0 0
\(97\) 2.78196 0.282466 0.141233 0.989976i \(-0.454893\pi\)
0.141233 + 0.989976i \(0.454893\pi\)
\(98\) 13.8759 1.40167
\(99\) 0 0
\(100\) 0 0
\(101\) 10.7004 1.06473 0.532366 0.846514i \(-0.321303\pi\)
0.532366 + 0.846514i \(0.321303\pi\)
\(102\) 0 0
\(103\) 3.61627 0.356322 0.178161 0.984001i \(-0.442985\pi\)
0.178161 + 0.984001i \(0.442985\pi\)
\(104\) −10.1441 −0.994714
\(105\) 0 0
\(106\) −1.40766 −0.136724
\(107\) 0.522034 0.0504669 0.0252335 0.999682i \(-0.491967\pi\)
0.0252335 + 0.999682i \(0.491967\pi\)
\(108\) 0 0
\(109\) 11.5033 1.10181 0.550907 0.834566i \(-0.314282\pi\)
0.550907 + 0.834566i \(0.314282\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 14.1459 1.33666
\(113\) −20.1421 −1.89481 −0.947404 0.320041i \(-0.896303\pi\)
−0.947404 + 0.320041i \(0.896303\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.753938 0.0700014
\(117\) 0 0
\(118\) 6.50908 0.599209
\(119\) −19.1724 −1.75753
\(120\) 0 0
\(121\) −10.9719 −0.997441
\(122\) 3.10924 0.281497
\(123\) 0 0
\(124\) 2.11932 0.190321
\(125\) 0 0
\(126\) 0 0
\(127\) −16.3345 −1.44945 −0.724726 0.689037i \(-0.758034\pi\)
−0.724726 + 0.689037i \(0.758034\pi\)
\(128\) 8.44401 0.746352
\(129\) 0 0
\(130\) 0 0
\(131\) −18.9147 −1.65259 −0.826294 0.563239i \(-0.809555\pi\)
−0.826294 + 0.563239i \(0.809555\pi\)
\(132\) 0 0
\(133\) −15.8524 −1.37458
\(134\) −16.0737 −1.38855
\(135\) 0 0
\(136\) −13.6684 −1.17206
\(137\) −13.5975 −1.16171 −0.580856 0.814006i \(-0.697282\pi\)
−0.580856 + 0.814006i \(0.697282\pi\)
\(138\) 0 0
\(139\) −3.36103 −0.285079 −0.142540 0.989789i \(-0.545527\pi\)
−0.142540 + 0.989789i \(0.545527\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −18.6190 −1.56247
\(143\) 0.569579 0.0476306
\(144\) 0 0
\(145\) 0 0
\(146\) 11.3993 0.943417
\(147\) 0 0
\(148\) 0.570355 0.0468829
\(149\) 6.33430 0.518926 0.259463 0.965753i \(-0.416454\pi\)
0.259463 + 0.965753i \(0.416454\pi\)
\(150\) 0 0
\(151\) 0.375265 0.0305386 0.0152693 0.999883i \(-0.495139\pi\)
0.0152693 + 0.999883i \(0.495139\pi\)
\(152\) −11.3016 −0.916678
\(153\) 0 0
\(154\) −0.923510 −0.0744185
\(155\) 0 0
\(156\) 0 0
\(157\) −10.5002 −0.838004 −0.419002 0.907985i \(-0.637620\pi\)
−0.419002 + 0.907985i \(0.637620\pi\)
\(158\) −3.47387 −0.276366
\(159\) 0 0
\(160\) 0 0
\(161\) −34.8421 −2.74594
\(162\) 0 0
\(163\) 13.8457 1.08448 0.542241 0.840223i \(-0.317576\pi\)
0.542241 + 0.840223i \(0.317576\pi\)
\(164\) −0.351612 −0.0274563
\(165\) 0 0
\(166\) −18.8008 −1.45923
\(167\) −1.08114 −0.0836611 −0.0418306 0.999125i \(-0.513319\pi\)
−0.0418306 + 0.999125i \(0.513319\pi\)
\(168\) 0 0
\(169\) −1.47265 −0.113281
\(170\) 0 0
\(171\) 0 0
\(172\) 3.19163 0.243360
\(173\) −18.2635 −1.38854 −0.694272 0.719712i \(-0.744274\pi\)
−0.694272 + 0.719712i \(0.744274\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.566255 −0.0426831
\(177\) 0 0
\(178\) 5.34437 0.400578
\(179\) 6.93140 0.518077 0.259039 0.965867i \(-0.416594\pi\)
0.259039 + 0.965867i \(0.416594\pi\)
\(180\) 0 0
\(181\) 13.1858 0.980091 0.490045 0.871697i \(-0.336980\pi\)
0.490045 + 0.871697i \(0.336980\pi\)
\(182\) −18.6903 −1.38542
\(183\) 0 0
\(184\) −24.8397 −1.83121
\(185\) 0 0
\(186\) 0 0
\(187\) 0.767463 0.0561224
\(188\) 2.74465 0.200174
\(189\) 0 0
\(190\) 0 0
\(191\) −23.3938 −1.69272 −0.846358 0.532614i \(-0.821210\pi\)
−0.846358 + 0.532614i \(0.821210\pi\)
\(192\) 0 0
\(193\) 9.06435 0.652466 0.326233 0.945289i \(-0.394221\pi\)
0.326233 + 0.945289i \(0.394221\pi\)
\(194\) 3.65423 0.262358
\(195\) 0 0
\(196\) −2.90084 −0.207203
\(197\) −13.2823 −0.946327 −0.473163 0.880975i \(-0.656888\pi\)
−0.473163 + 0.880975i \(0.656888\pi\)
\(198\) 0 0
\(199\) 23.9985 1.70121 0.850606 0.525804i \(-0.176235\pi\)
0.850606 + 0.525804i \(0.176235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.0555 0.988938
\(203\) 11.5063 0.807582
\(204\) 0 0
\(205\) 0 0
\(206\) 4.75013 0.330957
\(207\) 0 0
\(208\) −11.4601 −0.794613
\(209\) 0.634567 0.0438939
\(210\) 0 0
\(211\) −7.75260 −0.533711 −0.266855 0.963737i \(-0.585985\pi\)
−0.266855 + 0.963737i \(0.585985\pi\)
\(212\) 0.294282 0.0202114
\(213\) 0 0
\(214\) 0.685714 0.0468744
\(215\) 0 0
\(216\) 0 0
\(217\) 32.3442 2.19567
\(218\) 15.1101 1.02338
\(219\) 0 0
\(220\) 0 0
\(221\) 15.5322 1.04481
\(222\) 0 0
\(223\) 22.1577 1.48379 0.741895 0.670517i \(-0.233927\pi\)
0.741895 + 0.670517i \(0.233927\pi\)
\(224\) −6.46185 −0.431751
\(225\) 0 0
\(226\) −26.4575 −1.75993
\(227\) −11.9286 −0.791729 −0.395864 0.918309i \(-0.629555\pi\)
−0.395864 + 0.918309i \(0.629555\pi\)
\(228\) 0 0
\(229\) −7.73897 −0.511406 −0.255703 0.966755i \(-0.582307\pi\)
−0.255703 + 0.966755i \(0.582307\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.20308 0.538559
\(233\) −2.85168 −0.186820 −0.0934099 0.995628i \(-0.529777\pi\)
−0.0934099 + 0.995628i \(0.529777\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.36077 −0.0885784
\(237\) 0 0
\(238\) −25.1837 −1.63242
\(239\) 19.9126 1.28804 0.644020 0.765009i \(-0.277265\pi\)
0.644020 + 0.765009i \(0.277265\pi\)
\(240\) 0 0
\(241\) 6.61379 0.426032 0.213016 0.977049i \(-0.431671\pi\)
0.213016 + 0.977049i \(0.431671\pi\)
\(242\) −14.4120 −0.926439
\(243\) 0 0
\(244\) −0.650008 −0.0416125
\(245\) 0 0
\(246\) 0 0
\(247\) 12.8426 0.817155
\(248\) 23.0589 1.46424
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0644 0.761501 0.380750 0.924678i \(-0.375666\pi\)
0.380750 + 0.924678i \(0.375666\pi\)
\(252\) 0 0
\(253\) 1.39472 0.0876850
\(254\) −21.4561 −1.34627
\(255\) 0 0
\(256\) −6.46059 −0.403787
\(257\) −12.7951 −0.798136 −0.399068 0.916921i \(-0.630666\pi\)
−0.399068 + 0.916921i \(0.630666\pi\)
\(258\) 0 0
\(259\) 8.70451 0.540872
\(260\) 0 0
\(261\) 0 0
\(262\) −24.8453 −1.53495
\(263\) −10.5225 −0.648843 −0.324422 0.945913i \(-0.605170\pi\)
−0.324422 + 0.945913i \(0.605170\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −20.8229 −1.27673
\(267\) 0 0
\(268\) 3.36031 0.205264
\(269\) −6.80128 −0.414681 −0.207341 0.978269i \(-0.566481\pi\)
−0.207341 + 0.978269i \(0.566481\pi\)
\(270\) 0 0
\(271\) 5.66981 0.344416 0.172208 0.985061i \(-0.444910\pi\)
0.172208 + 0.985061i \(0.444910\pi\)
\(272\) −15.4415 −0.936281
\(273\) 0 0
\(274\) −17.8609 −1.07902
\(275\) 0 0
\(276\) 0 0
\(277\) −0.874982 −0.0525726 −0.0262863 0.999654i \(-0.508368\pi\)
−0.0262863 + 0.999654i \(0.508368\pi\)
\(278\) −4.41486 −0.264786
\(279\) 0 0
\(280\) 0 0
\(281\) 4.85806 0.289808 0.144904 0.989446i \(-0.453713\pi\)
0.144904 + 0.989446i \(0.453713\pi\)
\(282\) 0 0
\(283\) −11.4230 −0.679024 −0.339512 0.940602i \(-0.610262\pi\)
−0.339512 + 0.940602i \(0.610262\pi\)
\(284\) 3.89242 0.230973
\(285\) 0 0
\(286\) 0.748167 0.0442400
\(287\) −5.36615 −0.316754
\(288\) 0 0
\(289\) 3.92839 0.231082
\(290\) 0 0
\(291\) 0 0
\(292\) −2.38311 −0.139461
\(293\) 0.0808664 0.00472427 0.00236213 0.999997i \(-0.499248\pi\)
0.00236213 + 0.999997i \(0.499248\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.20564 0.360696
\(297\) 0 0
\(298\) 8.32038 0.481987
\(299\) 28.2268 1.63240
\(300\) 0 0
\(301\) 48.7093 2.80756
\(302\) 0.492927 0.0283647
\(303\) 0 0
\(304\) −12.7677 −0.732275
\(305\) 0 0
\(306\) 0 0
\(307\) 7.77807 0.443918 0.221959 0.975056i \(-0.428755\pi\)
0.221959 + 0.975056i \(0.428755\pi\)
\(308\) 0.193066 0.0110010
\(309\) 0 0
\(310\) 0 0
\(311\) 13.6181 0.772213 0.386106 0.922454i \(-0.373820\pi\)
0.386106 + 0.922454i \(0.373820\pi\)
\(312\) 0 0
\(313\) −4.85037 −0.274159 −0.137079 0.990560i \(-0.543772\pi\)
−0.137079 + 0.990560i \(0.543772\pi\)
\(314\) −13.7924 −0.778351
\(315\) 0 0
\(316\) 0.726237 0.0408540
\(317\) −22.3020 −1.25260 −0.626302 0.779580i \(-0.715432\pi\)
−0.626302 + 0.779580i \(0.715432\pi\)
\(318\) 0 0
\(319\) −0.460592 −0.0257882
\(320\) 0 0
\(321\) 0 0
\(322\) −45.7666 −2.55047
\(323\) 17.3044 0.962842
\(324\) 0 0
\(325\) 0 0
\(326\) 18.1870 1.00728
\(327\) 0 0
\(328\) −3.82565 −0.211236
\(329\) 41.8877 2.30934
\(330\) 0 0
\(331\) 0.144235 0.00792788 0.00396394 0.999992i \(-0.498738\pi\)
0.00396394 + 0.999992i \(0.498738\pi\)
\(332\) 3.93045 0.215711
\(333\) 0 0
\(334\) −1.42012 −0.0777057
\(335\) 0 0
\(336\) 0 0
\(337\) −10.1832 −0.554717 −0.277358 0.960767i \(-0.589459\pi\)
−0.277358 + 0.960767i \(0.589459\pi\)
\(338\) −1.93439 −0.105217
\(339\) 0 0
\(340\) 0 0
\(341\) −1.29472 −0.0701133
\(342\) 0 0
\(343\) −14.9351 −0.806418
\(344\) 34.7260 1.87230
\(345\) 0 0
\(346\) −23.9898 −1.28970
\(347\) 10.5782 0.567870 0.283935 0.958844i \(-0.408360\pi\)
0.283935 + 0.958844i \(0.408360\pi\)
\(348\) 0 0
\(349\) −29.1511 −1.56042 −0.780210 0.625517i \(-0.784888\pi\)
−0.780210 + 0.625517i \(0.784888\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.258665 0.0137869
\(353\) 3.81907 0.203268 0.101634 0.994822i \(-0.467593\pi\)
0.101634 + 0.994822i \(0.467593\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.11728 −0.0592156
\(357\) 0 0
\(358\) 9.10469 0.481198
\(359\) 10.9849 0.579759 0.289879 0.957063i \(-0.406385\pi\)
0.289879 + 0.957063i \(0.406385\pi\)
\(360\) 0 0
\(361\) −4.69207 −0.246951
\(362\) 17.3201 0.910323
\(363\) 0 0
\(364\) 3.90734 0.204800
\(365\) 0 0
\(366\) 0 0
\(367\) −4.07169 −0.212541 −0.106270 0.994337i \(-0.533891\pi\)
−0.106270 + 0.994337i \(0.533891\pi\)
\(368\) −28.0620 −1.46283
\(369\) 0 0
\(370\) 0 0
\(371\) 4.49120 0.233171
\(372\) 0 0
\(373\) −1.92973 −0.0999175 −0.0499588 0.998751i \(-0.515909\pi\)
−0.0499588 + 0.998751i \(0.515909\pi\)
\(374\) 1.00810 0.0521274
\(375\) 0 0
\(376\) 29.8627 1.54005
\(377\) −9.32162 −0.480088
\(378\) 0 0
\(379\) 14.6050 0.750210 0.375105 0.926982i \(-0.377607\pi\)
0.375105 + 0.926982i \(0.377607\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −30.7288 −1.57222
\(383\) −31.5290 −1.61106 −0.805530 0.592556i \(-0.798119\pi\)
−0.805530 + 0.592556i \(0.798119\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 11.9064 0.606021
\(387\) 0 0
\(388\) −0.763942 −0.0387833
\(389\) −14.6046 −0.740485 −0.370242 0.928935i \(-0.620725\pi\)
−0.370242 + 0.928935i \(0.620725\pi\)
\(390\) 0 0
\(391\) 38.0333 1.92343
\(392\) −31.5621 −1.59413
\(393\) 0 0
\(394\) −17.4469 −0.878962
\(395\) 0 0
\(396\) 0 0
\(397\) −15.2793 −0.766845 −0.383422 0.923573i \(-0.625255\pi\)
−0.383422 + 0.923573i \(0.625255\pi\)
\(398\) 31.5231 1.58011
\(399\) 0 0
\(400\) 0 0
\(401\) 15.7148 0.784760 0.392380 0.919803i \(-0.371652\pi\)
0.392380 + 0.919803i \(0.371652\pi\)
\(402\) 0 0
\(403\) −26.2031 −1.30527
\(404\) −2.93839 −0.146190
\(405\) 0 0
\(406\) 15.1140 0.750094
\(407\) −0.348438 −0.0172714
\(408\) 0 0
\(409\) 12.4187 0.614064 0.307032 0.951699i \(-0.400664\pi\)
0.307032 + 0.951699i \(0.400664\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.993047 −0.0489239
\(413\) −20.7674 −1.02190
\(414\) 0 0
\(415\) 0 0
\(416\) 5.23496 0.256665
\(417\) 0 0
\(418\) 0.833531 0.0407693
\(419\) 9.08687 0.443923 0.221961 0.975055i \(-0.428754\pi\)
0.221961 + 0.975055i \(0.428754\pi\)
\(420\) 0 0
\(421\) 5.15620 0.251298 0.125649 0.992075i \(-0.459899\pi\)
0.125649 + 0.992075i \(0.459899\pi\)
\(422\) −10.1834 −0.495719
\(423\) 0 0
\(424\) 3.20188 0.155497
\(425\) 0 0
\(426\) 0 0
\(427\) −9.92014 −0.480069
\(428\) −0.143353 −0.00692924
\(429\) 0 0
\(430\) 0 0
\(431\) −3.66953 −0.176755 −0.0883776 0.996087i \(-0.528168\pi\)
−0.0883776 + 0.996087i \(0.528168\pi\)
\(432\) 0 0
\(433\) 17.7340 0.852242 0.426121 0.904666i \(-0.359880\pi\)
0.426121 + 0.904666i \(0.359880\pi\)
\(434\) 42.4855 2.03937
\(435\) 0 0
\(436\) −3.15886 −0.151282
\(437\) 31.4474 1.50433
\(438\) 0 0
\(439\) −26.1823 −1.24961 −0.624806 0.780780i \(-0.714822\pi\)
−0.624806 + 0.780780i \(0.714822\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 20.4022 0.970433
\(443\) 19.2199 0.913166 0.456583 0.889681i \(-0.349073\pi\)
0.456583 + 0.889681i \(0.349073\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 29.1051 1.37817
\(447\) 0 0
\(448\) −36.7797 −1.73768
\(449\) −33.6407 −1.58760 −0.793802 0.608177i \(-0.791901\pi\)
−0.793802 + 0.608177i \(0.791901\pi\)
\(450\) 0 0
\(451\) 0.214805 0.0101148
\(452\) 5.53112 0.260162
\(453\) 0 0
\(454\) −15.6687 −0.735369
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0255 1.12387 0.561933 0.827183i \(-0.310058\pi\)
0.561933 + 0.827183i \(0.310058\pi\)
\(458\) −10.1655 −0.475001
\(459\) 0 0
\(460\) 0 0
\(461\) 14.4282 0.671990 0.335995 0.941864i \(-0.390927\pi\)
0.335995 + 0.941864i \(0.390927\pi\)
\(462\) 0 0
\(463\) 2.95599 0.137377 0.0686883 0.997638i \(-0.478119\pi\)
0.0686883 + 0.997638i \(0.478119\pi\)
\(464\) 9.26722 0.430220
\(465\) 0 0
\(466\) −3.74581 −0.173521
\(467\) 37.1933 1.72110 0.860551 0.509364i \(-0.170119\pi\)
0.860551 + 0.509364i \(0.170119\pi\)
\(468\) 0 0
\(469\) 51.2836 2.36806
\(470\) 0 0
\(471\) 0 0
\(472\) −14.8056 −0.681482
\(473\) −1.94982 −0.0896526
\(474\) 0 0
\(475\) 0 0
\(476\) 5.26483 0.241313
\(477\) 0 0
\(478\) 26.1561 1.19635
\(479\) 8.65394 0.395409 0.197704 0.980262i \(-0.436651\pi\)
0.197704 + 0.980262i \(0.436651\pi\)
\(480\) 0 0
\(481\) −7.05182 −0.321535
\(482\) 8.68750 0.395705
\(483\) 0 0
\(484\) 3.01293 0.136951
\(485\) 0 0
\(486\) 0 0
\(487\) −38.4645 −1.74299 −0.871496 0.490402i \(-0.836850\pi\)
−0.871496 + 0.490402i \(0.836850\pi\)
\(488\) −7.07229 −0.320148
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0853 −1.35773 −0.678865 0.734263i \(-0.737528\pi\)
−0.678865 + 0.734263i \(0.737528\pi\)
\(492\) 0 0
\(493\) −12.5601 −0.565680
\(494\) 16.8693 0.758986
\(495\) 0 0
\(496\) 26.0502 1.16969
\(497\) 59.4045 2.66466
\(498\) 0 0
\(499\) −19.6343 −0.878951 −0.439476 0.898254i \(-0.644836\pi\)
−0.439476 + 0.898254i \(0.644836\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 15.8472 0.707293
\(503\) −30.4802 −1.35904 −0.679522 0.733655i \(-0.737813\pi\)
−0.679522 + 0.733655i \(0.737813\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.83202 0.0814431
\(507\) 0 0
\(508\) 4.48554 0.199014
\(509\) −11.4371 −0.506942 −0.253471 0.967343i \(-0.581572\pi\)
−0.253471 + 0.967343i \(0.581572\pi\)
\(510\) 0 0
\(511\) −36.3700 −1.60892
\(512\) −25.3743 −1.12140
\(513\) 0 0
\(514\) −16.8069 −0.741321
\(515\) 0 0
\(516\) 0 0
\(517\) −1.67675 −0.0737433
\(518\) 11.4337 0.502370
\(519\) 0 0
\(520\) 0 0
\(521\) −16.6733 −0.730471 −0.365236 0.930915i \(-0.619012\pi\)
−0.365236 + 0.930915i \(0.619012\pi\)
\(522\) 0 0
\(523\) −2.54272 −0.111185 −0.0555926 0.998454i \(-0.517705\pi\)
−0.0555926 + 0.998454i \(0.517705\pi\)
\(524\) 5.19409 0.226905
\(525\) 0 0
\(526\) −13.8217 −0.602656
\(527\) −35.3066 −1.53798
\(528\) 0 0
\(529\) 46.1182 2.00514
\(530\) 0 0
\(531\) 0 0
\(532\) 4.35316 0.188734
\(533\) 4.34730 0.188302
\(534\) 0 0
\(535\) 0 0
\(536\) 36.5612 1.57921
\(537\) 0 0
\(538\) −8.93377 −0.385162
\(539\) 1.77217 0.0763327
\(540\) 0 0
\(541\) 26.3649 1.13352 0.566758 0.823884i \(-0.308197\pi\)
0.566758 + 0.823884i \(0.308197\pi\)
\(542\) 7.44753 0.319899
\(543\) 0 0
\(544\) 7.05370 0.302425
\(545\) 0 0
\(546\) 0 0
\(547\) 14.8319 0.634165 0.317082 0.948398i \(-0.397297\pi\)
0.317082 + 0.948398i \(0.397297\pi\)
\(548\) 3.73394 0.159506
\(549\) 0 0
\(550\) 0 0
\(551\) −10.3852 −0.442425
\(552\) 0 0
\(553\) 11.0835 0.471319
\(554\) −1.14933 −0.0488302
\(555\) 0 0
\(556\) 0.922957 0.0391421
\(557\) −18.7407 −0.794069 −0.397034 0.917804i \(-0.629961\pi\)
−0.397034 + 0.917804i \(0.629961\pi\)
\(558\) 0 0
\(559\) −39.4611 −1.66903
\(560\) 0 0
\(561\) 0 0
\(562\) 6.38127 0.269178
\(563\) 32.1447 1.35474 0.677368 0.735644i \(-0.263121\pi\)
0.677368 + 0.735644i \(0.263121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −15.0045 −0.630688
\(567\) 0 0
\(568\) 42.3508 1.77700
\(569\) 2.14141 0.0897725 0.0448862 0.998992i \(-0.485707\pi\)
0.0448862 + 0.998992i \(0.485707\pi\)
\(570\) 0 0
\(571\) 11.8476 0.495805 0.247902 0.968785i \(-0.420259\pi\)
0.247902 + 0.968785i \(0.420259\pi\)
\(572\) −0.156409 −0.00653981
\(573\) 0 0
\(574\) −7.04866 −0.294205
\(575\) 0 0
\(576\) 0 0
\(577\) −15.6985 −0.653539 −0.326769 0.945104i \(-0.605960\pi\)
−0.326769 + 0.945104i \(0.605960\pi\)
\(578\) 5.16011 0.214632
\(579\) 0 0
\(580\) 0 0
\(581\) 59.9847 2.48859
\(582\) 0 0
\(583\) −0.179781 −0.00744577
\(584\) −25.9290 −1.07295
\(585\) 0 0
\(586\) 0.106221 0.00438797
\(587\) 2.32845 0.0961055 0.0480527 0.998845i \(-0.484698\pi\)
0.0480527 + 0.998845i \(0.484698\pi\)
\(588\) 0 0
\(589\) −29.1929 −1.20287
\(590\) 0 0
\(591\) 0 0
\(592\) 7.01067 0.288137
\(593\) 34.0275 1.39734 0.698672 0.715442i \(-0.253775\pi\)
0.698672 + 0.715442i \(0.253775\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.73943 −0.0712499
\(597\) 0 0
\(598\) 37.0771 1.51619
\(599\) 2.93932 0.120097 0.0600486 0.998195i \(-0.480874\pi\)
0.0600486 + 0.998195i \(0.480874\pi\)
\(600\) 0 0
\(601\) 5.17826 0.211225 0.105613 0.994407i \(-0.466320\pi\)
0.105613 + 0.994407i \(0.466320\pi\)
\(602\) 63.9818 2.60770
\(603\) 0 0
\(604\) −0.103050 −0.00419303
\(605\) 0 0
\(606\) 0 0
\(607\) −1.45215 −0.0589409 −0.0294705 0.999566i \(-0.509382\pi\)
−0.0294705 + 0.999566i \(0.509382\pi\)
\(608\) 5.83227 0.236530
\(609\) 0 0
\(610\) 0 0
\(611\) −33.9346 −1.37285
\(612\) 0 0
\(613\) −41.1889 −1.66361 −0.831803 0.555072i \(-0.812691\pi\)
−0.831803 + 0.555072i \(0.812691\pi\)
\(614\) 10.2168 0.412318
\(615\) 0 0
\(616\) 2.10062 0.0846364
\(617\) −33.9582 −1.36711 −0.683553 0.729901i \(-0.739566\pi\)
−0.683553 + 0.729901i \(0.739566\pi\)
\(618\) 0 0
\(619\) −31.1879 −1.25355 −0.626774 0.779201i \(-0.715625\pi\)
−0.626774 + 0.779201i \(0.715625\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 17.8880 0.717243
\(623\) −17.0514 −0.683151
\(624\) 0 0
\(625\) 0 0
\(626\) −6.37117 −0.254643
\(627\) 0 0
\(628\) 2.88340 0.115060
\(629\) −9.50177 −0.378860
\(630\) 0 0
\(631\) 21.9274 0.872917 0.436459 0.899724i \(-0.356233\pi\)
0.436459 + 0.899724i \(0.356233\pi\)
\(632\) 7.90168 0.314312
\(633\) 0 0
\(634\) −29.2946 −1.16344
\(635\) 0 0
\(636\) 0 0
\(637\) 35.8658 1.42105
\(638\) −0.605007 −0.0239524
\(639\) 0 0
\(640\) 0 0
\(641\) 19.2185 0.759083 0.379542 0.925175i \(-0.376082\pi\)
0.379542 + 0.925175i \(0.376082\pi\)
\(642\) 0 0
\(643\) 13.6611 0.538739 0.269370 0.963037i \(-0.413185\pi\)
0.269370 + 0.963037i \(0.413185\pi\)
\(644\) 9.56782 0.377025
\(645\) 0 0
\(646\) 22.7301 0.894302
\(647\) −30.7060 −1.20718 −0.603588 0.797296i \(-0.706263\pi\)
−0.603588 + 0.797296i \(0.706263\pi\)
\(648\) 0 0
\(649\) 0.831313 0.0326319
\(650\) 0 0
\(651\) 0 0
\(652\) −3.80211 −0.148902
\(653\) −5.75846 −0.225346 −0.112673 0.993632i \(-0.535941\pi\)
−0.112673 + 0.993632i \(0.535941\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.32193 −0.168743
\(657\) 0 0
\(658\) 55.0212 2.14495
\(659\) −15.9750 −0.622298 −0.311149 0.950361i \(-0.600714\pi\)
−0.311149 + 0.950361i \(0.600714\pi\)
\(660\) 0 0
\(661\) 31.1613 1.21203 0.606017 0.795451i \(-0.292766\pi\)
0.606017 + 0.795451i \(0.292766\pi\)
\(662\) 0.189459 0.00736354
\(663\) 0 0
\(664\) 42.7645 1.65958
\(665\) 0 0
\(666\) 0 0
\(667\) −22.8256 −0.883812
\(668\) 0.296887 0.0114869
\(669\) 0 0
\(670\) 0 0
\(671\) 0.397099 0.0153298
\(672\) 0 0
\(673\) −19.1758 −0.739174 −0.369587 0.929196i \(-0.620501\pi\)
−0.369587 + 0.929196i \(0.620501\pi\)
\(674\) −13.3761 −0.515229
\(675\) 0 0
\(676\) 0.404398 0.0155538
\(677\) −25.2153 −0.969105 −0.484552 0.874762i \(-0.661017\pi\)
−0.484552 + 0.874762i \(0.661017\pi\)
\(678\) 0 0
\(679\) −11.6590 −0.447429
\(680\) 0 0
\(681\) 0 0
\(682\) −1.70068 −0.0651223
\(683\) 28.1160 1.07583 0.537914 0.843000i \(-0.319212\pi\)
0.537914 + 0.843000i \(0.319212\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −19.6179 −0.749013
\(687\) 0 0
\(688\) 39.2308 1.49566
\(689\) −3.63847 −0.138615
\(690\) 0 0
\(691\) 11.8688 0.451511 0.225756 0.974184i \(-0.427515\pi\)
0.225756 + 0.974184i \(0.427515\pi\)
\(692\) 5.01524 0.190651
\(693\) 0 0
\(694\) 13.8950 0.527446
\(695\) 0 0
\(696\) 0 0
\(697\) 5.85764 0.221874
\(698\) −38.2912 −1.44934
\(699\) 0 0
\(700\) 0 0
\(701\) −26.1748 −0.988608 −0.494304 0.869289i \(-0.664577\pi\)
−0.494304 + 0.869289i \(0.664577\pi\)
\(702\) 0 0
\(703\) −7.85642 −0.296311
\(704\) 1.47228 0.0554886
\(705\) 0 0
\(706\) 5.01651 0.188799
\(707\) −44.8444 −1.68655
\(708\) 0 0
\(709\) 19.8934 0.747111 0.373556 0.927608i \(-0.378138\pi\)
0.373556 + 0.927608i \(0.378138\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12.1563 −0.455578
\(713\) −64.1630 −2.40292
\(714\) 0 0
\(715\) 0 0
\(716\) −1.90340 −0.0711333
\(717\) 0 0
\(718\) 14.4291 0.538489
\(719\) −53.3070 −1.98802 −0.994008 0.109309i \(-0.965136\pi\)
−0.994008 + 0.109309i \(0.965136\pi\)
\(720\) 0 0
\(721\) −15.1555 −0.564419
\(722\) −6.16324 −0.229372
\(723\) 0 0
\(724\) −3.62088 −0.134569
\(725\) 0 0
\(726\) 0 0
\(727\) −21.0186 −0.779537 −0.389768 0.920913i \(-0.627445\pi\)
−0.389768 + 0.920913i \(0.627445\pi\)
\(728\) 42.5131 1.57564
\(729\) 0 0
\(730\) 0 0
\(731\) −53.1707 −1.96659
\(732\) 0 0
\(733\) 24.4993 0.904901 0.452451 0.891789i \(-0.350550\pi\)
0.452451 + 0.891789i \(0.350550\pi\)
\(734\) −5.34834 −0.197411
\(735\) 0 0
\(736\) 12.8187 0.472505
\(737\) −2.05286 −0.0756182
\(738\) 0 0
\(739\) −11.3415 −0.417203 −0.208601 0.978001i \(-0.566891\pi\)
−0.208601 + 0.978001i \(0.566891\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.89938 0.216573
\(743\) −20.0546 −0.735732 −0.367866 0.929879i \(-0.619912\pi\)
−0.367866 + 0.929879i \(0.619912\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.53478 −0.0928049
\(747\) 0 0
\(748\) −0.210749 −0.00770576
\(749\) −2.18779 −0.0799403
\(750\) 0 0
\(751\) −10.7620 −0.392711 −0.196355 0.980533i \(-0.562911\pi\)
−0.196355 + 0.980533i \(0.562911\pi\)
\(752\) 33.7366 1.23025
\(753\) 0 0
\(754\) −12.2443 −0.445913
\(755\) 0 0
\(756\) 0 0
\(757\) −36.8528 −1.33944 −0.669718 0.742615i \(-0.733585\pi\)
−0.669718 + 0.742615i \(0.733585\pi\)
\(758\) 19.1843 0.696806
\(759\) 0 0
\(760\) 0 0
\(761\) 15.5973 0.565403 0.282701 0.959208i \(-0.408769\pi\)
0.282701 + 0.959208i \(0.408769\pi\)
\(762\) 0 0
\(763\) −48.2092 −1.74529
\(764\) 6.42406 0.232414
\(765\) 0 0
\(766\) −41.4148 −1.49638
\(767\) 16.8244 0.607494
\(768\) 0 0
\(769\) −10.4385 −0.376423 −0.188212 0.982128i \(-0.560269\pi\)
−0.188212 + 0.982128i \(0.560269\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.48912 −0.0895853
\(773\) −29.1248 −1.04755 −0.523773 0.851858i \(-0.675476\pi\)
−0.523773 + 0.851858i \(0.675476\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8.31193 −0.298381
\(777\) 0 0
\(778\) −19.1838 −0.687773
\(779\) 4.84332 0.173530
\(780\) 0 0
\(781\) −2.37794 −0.0850894
\(782\) 49.9584 1.78651
\(783\) 0 0
\(784\) −35.6565 −1.27345
\(785\) 0 0
\(786\) 0 0
\(787\) 50.9945 1.81776 0.908879 0.417060i \(-0.136940\pi\)
0.908879 + 0.417060i \(0.136940\pi\)
\(788\) 3.64740 0.129933
\(789\) 0 0
\(790\) 0 0
\(791\) 84.4135 3.00140
\(792\) 0 0
\(793\) 8.03664 0.285389
\(794\) −20.0700 −0.712257
\(795\) 0 0
\(796\) −6.59013 −0.233581
\(797\) 36.6176 1.29706 0.648532 0.761188i \(-0.275383\pi\)
0.648532 + 0.761188i \(0.275383\pi\)
\(798\) 0 0
\(799\) −45.7242 −1.61761
\(800\) 0 0
\(801\) 0 0
\(802\) 20.6421 0.728897
\(803\) 1.45588 0.0513768
\(804\) 0 0
\(805\) 0 0
\(806\) −34.4189 −1.21235
\(807\) 0 0
\(808\) −31.9706 −1.12472
\(809\) −34.4639 −1.21168 −0.605842 0.795585i \(-0.707164\pi\)
−0.605842 + 0.795585i \(0.707164\pi\)
\(810\) 0 0
\(811\) 35.3834 1.24248 0.621240 0.783620i \(-0.286629\pi\)
0.621240 + 0.783620i \(0.286629\pi\)
\(812\) −3.15968 −0.110883
\(813\) 0 0
\(814\) −0.457689 −0.0160420
\(815\) 0 0
\(816\) 0 0
\(817\) −43.9635 −1.53809
\(818\) 16.3125 0.570352
\(819\) 0 0
\(820\) 0 0
\(821\) 2.15613 0.0752493 0.0376246 0.999292i \(-0.488021\pi\)
0.0376246 + 0.999292i \(0.488021\pi\)
\(822\) 0 0
\(823\) −34.5492 −1.20431 −0.602154 0.798380i \(-0.705691\pi\)
−0.602154 + 0.798380i \(0.705691\pi\)
\(824\) −10.8047 −0.376398
\(825\) 0 0
\(826\) −27.2789 −0.949155
\(827\) −21.2543 −0.739083 −0.369542 0.929214i \(-0.620485\pi\)
−0.369542 + 0.929214i \(0.620485\pi\)
\(828\) 0 0
\(829\) 28.8519 1.00207 0.501035 0.865427i \(-0.332953\pi\)
0.501035 + 0.865427i \(0.332953\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 29.7965 1.03301
\(833\) 48.3263 1.67441
\(834\) 0 0
\(835\) 0 0
\(836\) −0.174255 −0.00602675
\(837\) 0 0
\(838\) 11.9360 0.412322
\(839\) −0.443198 −0.0153009 −0.00765045 0.999971i \(-0.502435\pi\)
−0.00765045 + 0.999971i \(0.502435\pi\)
\(840\) 0 0
\(841\) −21.4620 −0.740071
\(842\) 6.77289 0.233409
\(843\) 0 0
\(844\) 2.12890 0.0732799
\(845\) 0 0
\(846\) 0 0
\(847\) 45.9820 1.57996
\(848\) 3.61724 0.124217
\(849\) 0 0
\(850\) 0 0
\(851\) −17.2676 −0.591927
\(852\) 0 0
\(853\) −29.0351 −0.994142 −0.497071 0.867710i \(-0.665591\pi\)
−0.497071 + 0.867710i \(0.665591\pi\)
\(854\) −13.0305 −0.445895
\(855\) 0 0
\(856\) −1.55973 −0.0533104
\(857\) 44.1483 1.50808 0.754039 0.656830i \(-0.228103\pi\)
0.754039 + 0.656830i \(0.228103\pi\)
\(858\) 0 0
\(859\) 51.1393 1.74485 0.872426 0.488747i \(-0.162546\pi\)
0.872426 + 0.488747i \(0.162546\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4.82009 −0.164173
\(863\) −2.01515 −0.0685967 −0.0342983 0.999412i \(-0.510920\pi\)
−0.0342983 + 0.999412i \(0.510920\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 23.2944 0.791575
\(867\) 0 0
\(868\) −8.88188 −0.301471
\(869\) −0.443669 −0.0150504
\(870\) 0 0
\(871\) −41.5466 −1.40775
\(872\) −34.3694 −1.16390
\(873\) 0 0
\(874\) 41.3075 1.39725
\(875\) 0 0
\(876\) 0 0
\(877\) 5.02192 0.169578 0.0847891 0.996399i \(-0.472978\pi\)
0.0847891 + 0.996399i \(0.472978\pi\)
\(878\) −34.3916 −1.16066
\(879\) 0 0
\(880\) 0 0
\(881\) 42.2541 1.42358 0.711789 0.702393i \(-0.247885\pi\)
0.711789 + 0.702393i \(0.247885\pi\)
\(882\) 0 0
\(883\) −56.8687 −1.91379 −0.956893 0.290442i \(-0.906198\pi\)
−0.956893 + 0.290442i \(0.906198\pi\)
\(884\) −4.26522 −0.143455
\(885\) 0 0
\(886\) 25.2462 0.848162
\(887\) −17.1287 −0.575127 −0.287563 0.957762i \(-0.592845\pi\)
−0.287563 + 0.957762i \(0.592845\pi\)
\(888\) 0 0
\(889\) 68.4563 2.29595
\(890\) 0 0
\(891\) 0 0
\(892\) −6.08462 −0.203728
\(893\) −37.8065 −1.26515
\(894\) 0 0
\(895\) 0 0
\(896\) −35.3880 −1.18223
\(897\) 0 0
\(898\) −44.1885 −1.47459
\(899\) 21.1892 0.706700
\(900\) 0 0
\(901\) −4.90255 −0.163328
\(902\) 0.282155 0.00939474
\(903\) 0 0
\(904\) 60.1803 2.00157
\(905\) 0 0
\(906\) 0 0
\(907\) 57.9111 1.92291 0.961454 0.274967i \(-0.0886670\pi\)
0.961454 + 0.274967i \(0.0886670\pi\)
\(908\) 3.27565 0.108706
\(909\) 0 0
\(910\) 0 0
\(911\) −7.89167 −0.261463 −0.130731 0.991418i \(-0.541733\pi\)
−0.130731 + 0.991418i \(0.541733\pi\)
\(912\) 0 0
\(913\) −2.40117 −0.0794670
\(914\) 31.5585 1.04386
\(915\) 0 0
\(916\) 2.12516 0.0702173
\(917\) 79.2699 2.61772
\(918\) 0 0
\(919\) −3.66345 −0.120846 −0.0604230 0.998173i \(-0.519245\pi\)
−0.0604230 + 0.998173i \(0.519245\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 18.9521 0.624155
\(923\) −48.1256 −1.58407
\(924\) 0 0
\(925\) 0 0
\(926\) 3.88282 0.127597
\(927\) 0 0
\(928\) −4.23327 −0.138964
\(929\) 49.7787 1.63319 0.816594 0.577213i \(-0.195860\pi\)
0.816594 + 0.577213i \(0.195860\pi\)
\(930\) 0 0
\(931\) 39.9580 1.30957
\(932\) 0.783087 0.0256509
\(933\) 0 0
\(934\) 48.8550 1.59859
\(935\) 0 0
\(936\) 0 0
\(937\) 9.69363 0.316677 0.158338 0.987385i \(-0.449386\pi\)
0.158338 + 0.987385i \(0.449386\pi\)
\(938\) 67.3632 2.19949
\(939\) 0 0
\(940\) 0 0
\(941\) −27.4206 −0.893886 −0.446943 0.894562i \(-0.647487\pi\)
−0.446943 + 0.894562i \(0.647487\pi\)
\(942\) 0 0
\(943\) 10.6451 0.346653
\(944\) −16.7262 −0.544392
\(945\) 0 0
\(946\) −2.56117 −0.0832707
\(947\) −12.7622 −0.414716 −0.207358 0.978265i \(-0.566486\pi\)
−0.207358 + 0.978265i \(0.566486\pi\)
\(948\) 0 0
\(949\) 29.4646 0.956461
\(950\) 0 0
\(951\) 0 0
\(952\) 57.2830 1.85655
\(953\) 22.3816 0.725012 0.362506 0.931981i \(-0.381921\pi\)
0.362506 + 0.931981i \(0.381921\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −5.46811 −0.176851
\(957\) 0 0
\(958\) 11.3673 0.367262
\(959\) 56.9858 1.84017
\(960\) 0 0
\(961\) 28.5630 0.921386
\(962\) −9.26287 −0.298647
\(963\) 0 0
\(964\) −1.81618 −0.0584953
\(965\) 0 0
\(966\) 0 0
\(967\) 41.4419 1.33268 0.666342 0.745647i \(-0.267859\pi\)
0.666342 + 0.745647i \(0.267859\pi\)
\(968\) 32.7816 1.05364
\(969\) 0 0
\(970\) 0 0
\(971\) −40.7475 −1.30765 −0.653824 0.756646i \(-0.726836\pi\)
−0.653824 + 0.756646i \(0.726836\pi\)
\(972\) 0 0
\(973\) 14.0858 0.451569
\(974\) −50.5248 −1.61892
\(975\) 0 0
\(976\) −7.98974 −0.255745
\(977\) −21.8246 −0.698231 −0.349116 0.937080i \(-0.613518\pi\)
−0.349116 + 0.937080i \(0.613518\pi\)
\(978\) 0 0
\(979\) 0.682562 0.0218148
\(980\) 0 0
\(981\) 0 0
\(982\) −39.5183 −1.26108
\(983\) −15.8023 −0.504016 −0.252008 0.967725i \(-0.581091\pi\)
−0.252008 + 0.967725i \(0.581091\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −16.4983 −0.525412
\(987\) 0 0
\(988\) −3.52665 −0.112198
\(989\) −96.6275 −3.07257
\(990\) 0 0
\(991\) 7.78545 0.247313 0.123657 0.992325i \(-0.460538\pi\)
0.123657 + 0.992325i \(0.460538\pi\)
\(992\) −11.8997 −0.377817
\(993\) 0 0
\(994\) 78.0303 2.47497
\(995\) 0 0
\(996\) 0 0
\(997\) −20.9919 −0.664820 −0.332410 0.943135i \(-0.607862\pi\)
−0.332410 + 0.943135i \(0.607862\pi\)
\(998\) −25.7905 −0.816383
\(999\) 0 0
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 5625.2.a.bc.1.6 8
3.2 odd 2 1875.2.a.n.1.3 8
5.4 even 2 5625.2.a.u.1.3 8
15.2 even 4 1875.2.b.g.1249.5 16
15.8 even 4 1875.2.b.g.1249.12 16
15.14 odd 2 1875.2.a.o.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.n.1.3 8 3.2 odd 2
1875.2.a.o.1.6 yes 8 15.14 odd 2
1875.2.b.g.1249.5 16 15.2 even 4
1875.2.b.g.1249.12 16 15.8 even 4
5625.2.a.u.1.3 8 5.4 even 2
5625.2.a.bc.1.6 8 1.1 even 1 trivial