Properties

Label 5625.2.a.d.1.2
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.61803 q^{4} +1.61803 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+0.618034 q^{2} -1.61803 q^{4} +1.61803 q^{7} -2.23607 q^{8} +0.763932 q^{11} +4.85410 q^{13} +1.00000 q^{14} +1.85410 q^{16} -0.763932 q^{17} -5.85410 q^{19} +0.472136 q^{22} +8.23607 q^{23} +3.00000 q^{26} -2.61803 q^{28} +1.38197 q^{29} -3.00000 q^{31} +5.61803 q^{32} -0.472136 q^{34} -4.23607 q^{37} -3.61803 q^{38} +5.23607 q^{41} -1.85410 q^{43} -1.23607 q^{44} +5.09017 q^{46} -1.61803 q^{47} -4.38197 q^{49} -7.85410 q^{52} +5.47214 q^{53} -3.61803 q^{56} +0.854102 q^{58} +4.14590 q^{59} -4.70820 q^{61} -1.85410 q^{62} -0.236068 q^{64} -9.23607 q^{67} +1.23607 q^{68} +4.38197 q^{71} +9.00000 q^{73} -2.61803 q^{74} +9.47214 q^{76} +1.23607 q^{77} +3.09017 q^{79} +3.23607 q^{82} -1.76393 q^{83} -1.14590 q^{86} -1.70820 q^{88} -8.94427 q^{89} +7.85410 q^{91} -13.3262 q^{92} -1.00000 q^{94} -2.85410 q^{97} -2.70820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + q^{7} + 6 q^{11} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 6 q^{17} - 5 q^{19} - 8 q^{22} + 12 q^{23} + 6 q^{26} - 3 q^{28} + 5 q^{29} - 6 q^{31} + 9 q^{32} + 8 q^{34} - 4 q^{37} - 5 q^{38} + 6 q^{41} + 3 q^{43} + 2 q^{44} - q^{46} - q^{47} - 11 q^{49} - 9 q^{52} + 2 q^{53} - 5 q^{56} - 5 q^{58} + 15 q^{59} + 4 q^{61} + 3 q^{62} + 4 q^{64} - 14 q^{67} - 2 q^{68} + 11 q^{71} + 18 q^{73} - 3 q^{74} + 10 q^{76} - 2 q^{77} - 5 q^{79} + 2 q^{82} - 8 q^{83} - 9 q^{86} + 10 q^{88} + 9 q^{91} - 11 q^{92} - 2 q^{94} + q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61803 0.611559 0.305780 0.952102i \(-0.401083\pi\)
0.305780 + 0.952102i \(0.401083\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) 0.763932 0.230334 0.115167 0.993346i \(-0.463260\pi\)
0.115167 + 0.993346i \(0.463260\pi\)
\(12\) 0 0
\(13\) 4.85410 1.34629 0.673143 0.739512i \(-0.264944\pi\)
0.673143 + 0.739512i \(0.264944\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 0 0
\(19\) −5.85410 −1.34302 −0.671512 0.740994i \(-0.734355\pi\)
−0.671512 + 0.740994i \(0.734355\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.472136 0.100660
\(23\) 8.23607 1.71734 0.858669 0.512530i \(-0.171292\pi\)
0.858669 + 0.512530i \(0.171292\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) −2.61803 −0.494762
\(29\) 1.38197 0.256625 0.128312 0.991734i \(-0.459044\pi\)
0.128312 + 0.991734i \(0.459044\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −0.472136 −0.0809706
\(35\) 0 0
\(36\) 0 0
\(37\) −4.23607 −0.696405 −0.348203 0.937419i \(-0.613208\pi\)
−0.348203 + 0.937419i \(0.613208\pi\)
\(38\) −3.61803 −0.586923
\(39\) 0 0
\(40\) 0 0
\(41\) 5.23607 0.817736 0.408868 0.912593i \(-0.365924\pi\)
0.408868 + 0.912593i \(0.365924\pi\)
\(42\) 0 0
\(43\) −1.85410 −0.282748 −0.141374 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(44\) −1.23607 −0.186344
\(45\) 0 0
\(46\) 5.09017 0.750505
\(47\) −1.61803 −0.236015 −0.118007 0.993013i \(-0.537651\pi\)
−0.118007 + 0.993013i \(0.537651\pi\)
\(48\) 0 0
\(49\) −4.38197 −0.625995
\(50\) 0 0
\(51\) 0 0
\(52\) −7.85410 −1.08917
\(53\) 5.47214 0.751656 0.375828 0.926690i \(-0.377358\pi\)
0.375828 + 0.926690i \(0.377358\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.61803 −0.483480
\(57\) 0 0
\(58\) 0.854102 0.112149
\(59\) 4.14590 0.539750 0.269875 0.962895i \(-0.413018\pi\)
0.269875 + 0.962895i \(0.413018\pi\)
\(60\) 0 0
\(61\) −4.70820 −0.602824 −0.301412 0.953494i \(-0.597458\pi\)
−0.301412 + 0.953494i \(0.597458\pi\)
\(62\) −1.85410 −0.235471
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) −9.23607 −1.12837 −0.564183 0.825650i \(-0.690809\pi\)
−0.564183 + 0.825650i \(0.690809\pi\)
\(68\) 1.23607 0.149895
\(69\) 0 0
\(70\) 0 0
\(71\) 4.38197 0.520044 0.260022 0.965603i \(-0.416270\pi\)
0.260022 + 0.965603i \(0.416270\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) −2.61803 −0.304340
\(75\) 0 0
\(76\) 9.47214 1.08653
\(77\) 1.23607 0.140863
\(78\) 0 0
\(79\) 3.09017 0.347671 0.173836 0.984775i \(-0.444384\pi\)
0.173836 + 0.984775i \(0.444384\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.23607 0.357364
\(83\) −1.76393 −0.193617 −0.0968083 0.995303i \(-0.530863\pi\)
−0.0968083 + 0.995303i \(0.530863\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.14590 −0.123565
\(87\) 0 0
\(88\) −1.70820 −0.182095
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) 0 0
\(91\) 7.85410 0.823334
\(92\) −13.3262 −1.38936
\(93\) 0 0
\(94\) −1.00000 −0.103142
\(95\) 0 0
\(96\) 0 0
\(97\) −2.85410 −0.289790 −0.144895 0.989447i \(-0.546284\pi\)
−0.144895 + 0.989447i \(0.546284\pi\)
\(98\) −2.70820 −0.273570
\(99\) 0 0
\(100\) 0 0
\(101\) 7.47214 0.743505 0.371753 0.928332i \(-0.378757\pi\)
0.371753 + 0.928332i \(0.378757\pi\)
\(102\) 0 0
\(103\) 11.5623 1.13927 0.569634 0.821899i \(-0.307085\pi\)
0.569634 + 0.821899i \(0.307085\pi\)
\(104\) −10.8541 −1.06433
\(105\) 0 0
\(106\) 3.38197 0.328486
\(107\) 10.4164 1.00699 0.503496 0.863998i \(-0.332047\pi\)
0.503496 + 0.863998i \(0.332047\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) 10.1459 0.954446 0.477223 0.878782i \(-0.341643\pi\)
0.477223 + 0.878782i \(0.341643\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.23607 −0.207614
\(117\) 0 0
\(118\) 2.56231 0.235879
\(119\) −1.23607 −0.113310
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) −2.90983 −0.263444
\(123\) 0 0
\(124\) 4.85410 0.435911
\(125\) 0 0
\(126\) 0 0
\(127\) 15.8885 1.40988 0.704940 0.709267i \(-0.250974\pi\)
0.704940 + 0.709267i \(0.250974\pi\)
\(128\) −11.3820 −1.00603
\(129\) 0 0
\(130\) 0 0
\(131\) 17.7984 1.55505 0.777526 0.628851i \(-0.216475\pi\)
0.777526 + 0.628851i \(0.216475\pi\)
\(132\) 0 0
\(133\) −9.47214 −0.821338
\(134\) −5.70820 −0.493114
\(135\) 0 0
\(136\) 1.70820 0.146477
\(137\) 5.94427 0.507853 0.253927 0.967223i \(-0.418278\pi\)
0.253927 + 0.967223i \(0.418278\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.70820 0.227267
\(143\) 3.70820 0.310096
\(144\) 0 0
\(145\) 0 0
\(146\) 5.56231 0.460340
\(147\) 0 0
\(148\) 6.85410 0.563404
\(149\) −13.9443 −1.14236 −0.571180 0.820825i \(-0.693514\pi\)
−0.571180 + 0.820825i \(0.693514\pi\)
\(150\) 0 0
\(151\) −5.56231 −0.452654 −0.226327 0.974051i \(-0.572672\pi\)
−0.226327 + 0.974051i \(0.572672\pi\)
\(152\) 13.0902 1.06175
\(153\) 0 0
\(154\) 0.763932 0.0615594
\(155\) 0 0
\(156\) 0 0
\(157\) 9.18034 0.732671 0.366335 0.930483i \(-0.380612\pi\)
0.366335 + 0.930483i \(0.380612\pi\)
\(158\) 1.90983 0.151938
\(159\) 0 0
\(160\) 0 0
\(161\) 13.3262 1.05025
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) −8.47214 −0.661563
\(165\) 0 0
\(166\) −1.09017 −0.0846136
\(167\) −5.56231 −0.430424 −0.215212 0.976567i \(-0.569044\pi\)
−0.215212 + 0.976567i \(0.569044\pi\)
\(168\) 0 0
\(169\) 10.5623 0.812485
\(170\) 0 0
\(171\) 0 0
\(172\) 3.00000 0.228748
\(173\) −16.8885 −1.28401 −0.642006 0.766700i \(-0.721898\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41641 0.106766
\(177\) 0 0
\(178\) −5.52786 −0.414331
\(179\) 9.47214 0.707981 0.353990 0.935249i \(-0.384825\pi\)
0.353990 + 0.935249i \(0.384825\pi\)
\(180\) 0 0
\(181\) 13.7082 1.01892 0.509461 0.860494i \(-0.329845\pi\)
0.509461 + 0.860494i \(0.329845\pi\)
\(182\) 4.85410 0.359810
\(183\) 0 0
\(184\) −18.4164 −1.35768
\(185\) 0 0
\(186\) 0 0
\(187\) −0.583592 −0.0426765
\(188\) 2.61803 0.190940
\(189\) 0 0
\(190\) 0 0
\(191\) 24.1803 1.74963 0.874814 0.484459i \(-0.160984\pi\)
0.874814 + 0.484459i \(0.160984\pi\)
\(192\) 0 0
\(193\) 5.70820 0.410886 0.205443 0.978669i \(-0.434137\pi\)
0.205443 + 0.978669i \(0.434137\pi\)
\(194\) −1.76393 −0.126643
\(195\) 0 0
\(196\) 7.09017 0.506441
\(197\) −9.70820 −0.691681 −0.345840 0.938293i \(-0.612406\pi\)
−0.345840 + 0.938293i \(0.612406\pi\)
\(198\) 0 0
\(199\) 2.56231 0.181637 0.0908185 0.995867i \(-0.471052\pi\)
0.0908185 + 0.995867i \(0.471052\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.61803 0.324924
\(203\) 2.23607 0.156941
\(204\) 0 0
\(205\) 0 0
\(206\) 7.14590 0.497878
\(207\) 0 0
\(208\) 9.00000 0.624038
\(209\) −4.47214 −0.309344
\(210\) 0 0
\(211\) 13.1803 0.907372 0.453686 0.891162i \(-0.350109\pi\)
0.453686 + 0.891162i \(0.350109\pi\)
\(212\) −8.85410 −0.608102
\(213\) 0 0
\(214\) 6.43769 0.440072
\(215\) 0 0
\(216\) 0 0
\(217\) −4.85410 −0.329518
\(218\) 6.18034 0.418585
\(219\) 0 0
\(220\) 0 0
\(221\) −3.70820 −0.249441
\(222\) 0 0
\(223\) −22.1803 −1.48531 −0.742653 0.669677i \(-0.766433\pi\)
−0.742653 + 0.669677i \(0.766433\pi\)
\(224\) 9.09017 0.607363
\(225\) 0 0
\(226\) 6.27051 0.417108
\(227\) 19.2361 1.27674 0.638371 0.769729i \(-0.279608\pi\)
0.638371 + 0.769729i \(0.279608\pi\)
\(228\) 0 0
\(229\) 8.29180 0.547937 0.273969 0.961739i \(-0.411664\pi\)
0.273969 + 0.961739i \(0.411664\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.09017 −0.202880
\(233\) 14.9443 0.979032 0.489516 0.871994i \(-0.337174\pi\)
0.489516 + 0.871994i \(0.337174\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.70820 −0.436667
\(237\) 0 0
\(238\) −0.763932 −0.0495184
\(239\) 29.4721 1.90639 0.953197 0.302350i \(-0.0977711\pi\)
0.953197 + 0.302350i \(0.0977711\pi\)
\(240\) 0 0
\(241\) 11.4721 0.738985 0.369493 0.929234i \(-0.379532\pi\)
0.369493 + 0.929234i \(0.379532\pi\)
\(242\) −6.43769 −0.413831
\(243\) 0 0
\(244\) 7.61803 0.487695
\(245\) 0 0
\(246\) 0 0
\(247\) −28.4164 −1.80809
\(248\) 6.70820 0.425971
\(249\) 0 0
\(250\) 0 0
\(251\) 6.81966 0.430453 0.215227 0.976564i \(-0.430951\pi\)
0.215227 + 0.976564i \(0.430951\pi\)
\(252\) 0 0
\(253\) 6.29180 0.395562
\(254\) 9.81966 0.616140
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 16.1459 1.00715 0.503577 0.863951i \(-0.332017\pi\)
0.503577 + 0.863951i \(0.332017\pi\)
\(258\) 0 0
\(259\) −6.85410 −0.425893
\(260\) 0 0
\(261\) 0 0
\(262\) 11.0000 0.679582
\(263\) −22.0902 −1.36214 −0.681069 0.732219i \(-0.738485\pi\)
−0.681069 + 0.732219i \(0.738485\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5.85410 −0.358938
\(267\) 0 0
\(268\) 14.9443 0.912867
\(269\) 17.2361 1.05090 0.525451 0.850824i \(-0.323897\pi\)
0.525451 + 0.850824i \(0.323897\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −1.41641 −0.0858823
\(273\) 0 0
\(274\) 3.67376 0.221940
\(275\) 0 0
\(276\) 0 0
\(277\) 11.2918 0.678458 0.339229 0.940704i \(-0.389834\pi\)
0.339229 + 0.940704i \(0.389834\pi\)
\(278\) 3.09017 0.185336
\(279\) 0 0
\(280\) 0 0
\(281\) 1.09017 0.0650341 0.0325170 0.999471i \(-0.489648\pi\)
0.0325170 + 0.999471i \(0.489648\pi\)
\(282\) 0 0
\(283\) 23.1459 1.37588 0.687940 0.725767i \(-0.258515\pi\)
0.687940 + 0.725767i \(0.258515\pi\)
\(284\) −7.09017 −0.420724
\(285\) 0 0
\(286\) 2.29180 0.135517
\(287\) 8.47214 0.500094
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) 0 0
\(292\) −14.5623 −0.852194
\(293\) −28.4721 −1.66336 −0.831680 0.555255i \(-0.812621\pi\)
−0.831680 + 0.555255i \(0.812621\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 9.47214 0.550557
\(297\) 0 0
\(298\) −8.61803 −0.499229
\(299\) 39.9787 2.31203
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) −3.43769 −0.197817
\(303\) 0 0
\(304\) −10.8541 −0.622525
\(305\) 0 0
\(306\) 0 0
\(307\) −4.76393 −0.271892 −0.135946 0.990716i \(-0.543407\pi\)
−0.135946 + 0.990716i \(0.543407\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) 29.5066 1.67316 0.836582 0.547841i \(-0.184550\pi\)
0.836582 + 0.547841i \(0.184550\pi\)
\(312\) 0 0
\(313\) 21.2361 1.20033 0.600167 0.799875i \(-0.295101\pi\)
0.600167 + 0.799875i \(0.295101\pi\)
\(314\) 5.67376 0.320189
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) −23.6525 −1.32846 −0.664228 0.747530i \(-0.731239\pi\)
−0.664228 + 0.747530i \(0.731239\pi\)
\(318\) 0 0
\(319\) 1.05573 0.0591094
\(320\) 0 0
\(321\) 0 0
\(322\) 8.23607 0.458978
\(323\) 4.47214 0.248836
\(324\) 0 0
\(325\) 0 0
\(326\) −6.79837 −0.376527
\(327\) 0 0
\(328\) −11.7082 −0.646477
\(329\) −2.61803 −0.144337
\(330\) 0 0
\(331\) 17.1246 0.941254 0.470627 0.882332i \(-0.344028\pi\)
0.470627 + 0.882332i \(0.344028\pi\)
\(332\) 2.85410 0.156639
\(333\) 0 0
\(334\) −3.43769 −0.188102
\(335\) 0 0
\(336\) 0 0
\(337\) −1.14590 −0.0624210 −0.0312105 0.999513i \(-0.509936\pi\)
−0.0312105 + 0.999513i \(0.509936\pi\)
\(338\) 6.52786 0.355069
\(339\) 0 0
\(340\) 0 0
\(341\) −2.29180 −0.124108
\(342\) 0 0
\(343\) −18.4164 −0.994393
\(344\) 4.14590 0.223532
\(345\) 0 0
\(346\) −10.4377 −0.561134
\(347\) −31.0902 −1.66901 −0.834504 0.551002i \(-0.814246\pi\)
−0.834504 + 0.551002i \(0.814246\pi\)
\(348\) 0 0
\(349\) 8.29180 0.443850 0.221925 0.975064i \(-0.428766\pi\)
0.221925 + 0.975064i \(0.428766\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.29180 0.228753
\(353\) 24.0902 1.28219 0.641095 0.767461i \(-0.278480\pi\)
0.641095 + 0.767461i \(0.278480\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.4721 0.767022
\(357\) 0 0
\(358\) 5.85410 0.309399
\(359\) 28.7426 1.51698 0.758489 0.651685i \(-0.225938\pi\)
0.758489 + 0.651685i \(0.225938\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) 8.47214 0.445286
\(363\) 0 0
\(364\) −12.7082 −0.666091
\(365\) 0 0
\(366\) 0 0
\(367\) 5.43769 0.283845 0.141923 0.989878i \(-0.454672\pi\)
0.141923 + 0.989878i \(0.454672\pi\)
\(368\) 15.2705 0.796030
\(369\) 0 0
\(370\) 0 0
\(371\) 8.85410 0.459682
\(372\) 0 0
\(373\) −5.27051 −0.272897 −0.136448 0.990647i \(-0.543569\pi\)
−0.136448 + 0.990647i \(0.543569\pi\)
\(374\) −0.360680 −0.0186503
\(375\) 0 0
\(376\) 3.61803 0.186586
\(377\) 6.70820 0.345490
\(378\) 0 0
\(379\) −34.5967 −1.77712 −0.888558 0.458765i \(-0.848292\pi\)
−0.888558 + 0.458765i \(0.848292\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 14.9443 0.764615
\(383\) −11.3607 −0.580504 −0.290252 0.956950i \(-0.593739\pi\)
−0.290252 + 0.956950i \(0.593739\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.52786 0.179564
\(387\) 0 0
\(388\) 4.61803 0.234445
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) −6.29180 −0.318190
\(392\) 9.79837 0.494893
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 0.0344419 0.00172859 0.000864294 1.00000i \(-0.499725\pi\)
0.000864294 1.00000i \(0.499725\pi\)
\(398\) 1.58359 0.0793783
\(399\) 0 0
\(400\) 0 0
\(401\) 22.5967 1.12843 0.564214 0.825629i \(-0.309179\pi\)
0.564214 + 0.825629i \(0.309179\pi\)
\(402\) 0 0
\(403\) −14.5623 −0.725400
\(404\) −12.0902 −0.601508
\(405\) 0 0
\(406\) 1.38197 0.0685858
\(407\) −3.23607 −0.160406
\(408\) 0 0
\(409\) 28.4164 1.40510 0.702550 0.711634i \(-0.252045\pi\)
0.702550 + 0.711634i \(0.252045\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −18.7082 −0.921687
\(413\) 6.70820 0.330089
\(414\) 0 0
\(415\) 0 0
\(416\) 27.2705 1.33705
\(417\) 0 0
\(418\) −2.76393 −0.135188
\(419\) 0.527864 0.0257878 0.0128939 0.999917i \(-0.495896\pi\)
0.0128939 + 0.999917i \(0.495896\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 8.14590 0.396536
\(423\) 0 0
\(424\) −12.2361 −0.594236
\(425\) 0 0
\(426\) 0 0
\(427\) −7.61803 −0.368663
\(428\) −16.8541 −0.814674
\(429\) 0 0
\(430\) 0 0
\(431\) −23.8328 −1.14799 −0.573993 0.818860i \(-0.694606\pi\)
−0.573993 + 0.818860i \(0.694606\pi\)
\(432\) 0 0
\(433\) −20.1459 −0.968150 −0.484075 0.875026i \(-0.660844\pi\)
−0.484075 + 0.875026i \(0.660844\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −16.1803 −0.774898
\(437\) −48.2148 −2.30643
\(438\) 0 0
\(439\) 5.97871 0.285348 0.142674 0.989770i \(-0.454430\pi\)
0.142674 + 0.989770i \(0.454430\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.29180 −0.109010
\(443\) 12.0557 0.572785 0.286392 0.958112i \(-0.407544\pi\)
0.286392 + 0.958112i \(0.407544\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −13.7082 −0.649102
\(447\) 0 0
\(448\) −0.381966 −0.0180462
\(449\) −20.3262 −0.959254 −0.479627 0.877472i \(-0.659228\pi\)
−0.479627 + 0.877472i \(0.659228\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −16.4164 −0.772163
\(453\) 0 0
\(454\) 11.8885 0.557957
\(455\) 0 0
\(456\) 0 0
\(457\) −5.41641 −0.253369 −0.126684 0.991943i \(-0.540434\pi\)
−0.126684 + 0.991943i \(0.540434\pi\)
\(458\) 5.12461 0.239457
\(459\) 0 0
\(460\) 0 0
\(461\) −23.1803 −1.07962 −0.539808 0.841788i \(-0.681503\pi\)
−0.539808 + 0.841788i \(0.681503\pi\)
\(462\) 0 0
\(463\) −16.1246 −0.749374 −0.374687 0.927151i \(-0.622250\pi\)
−0.374687 + 0.927151i \(0.622250\pi\)
\(464\) 2.56231 0.118952
\(465\) 0 0
\(466\) 9.23607 0.427853
\(467\) −28.4508 −1.31655 −0.658274 0.752778i \(-0.728713\pi\)
−0.658274 + 0.752778i \(0.728713\pi\)
\(468\) 0 0
\(469\) −14.9443 −0.690062
\(470\) 0 0
\(471\) 0 0
\(472\) −9.27051 −0.426710
\(473\) −1.41641 −0.0651265
\(474\) 0 0
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 18.2148 0.833125
\(479\) 4.14590 0.189431 0.0947155 0.995504i \(-0.469806\pi\)
0.0947155 + 0.995504i \(0.469806\pi\)
\(480\) 0 0
\(481\) −20.5623 −0.937560
\(482\) 7.09017 0.322948
\(483\) 0 0
\(484\) 16.8541 0.766096
\(485\) 0 0
\(486\) 0 0
\(487\) 9.58359 0.434274 0.217137 0.976141i \(-0.430328\pi\)
0.217137 + 0.976141i \(0.430328\pi\)
\(488\) 10.5279 0.476574
\(489\) 0 0
\(490\) 0 0
\(491\) −37.2492 −1.68103 −0.840517 0.541785i \(-0.817749\pi\)
−0.840517 + 0.541785i \(0.817749\pi\)
\(492\) 0 0
\(493\) −1.05573 −0.0475476
\(494\) −17.5623 −0.790165
\(495\) 0 0
\(496\) −5.56231 −0.249755
\(497\) 7.09017 0.318038
\(498\) 0 0
\(499\) −12.5623 −0.562366 −0.281183 0.959654i \(-0.590727\pi\)
−0.281183 + 0.959654i \(0.590727\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.21478 0.188115
\(503\) −10.5836 −0.471899 −0.235950 0.971765i \(-0.575820\pi\)
−0.235950 + 0.971765i \(0.575820\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.88854 0.172867
\(507\) 0 0
\(508\) −25.7082 −1.14062
\(509\) 4.67376 0.207161 0.103580 0.994621i \(-0.466970\pi\)
0.103580 + 0.994621i \(0.466970\pi\)
\(510\) 0 0
\(511\) 14.5623 0.644198
\(512\) 18.7082 0.826794
\(513\) 0 0
\(514\) 9.97871 0.440142
\(515\) 0 0
\(516\) 0 0
\(517\) −1.23607 −0.0543622
\(518\) −4.23607 −0.186122
\(519\) 0 0
\(520\) 0 0
\(521\) 15.3607 0.672964 0.336482 0.941690i \(-0.390763\pi\)
0.336482 + 0.941690i \(0.390763\pi\)
\(522\) 0 0
\(523\) 19.8541 0.868159 0.434080 0.900875i \(-0.357074\pi\)
0.434080 + 0.900875i \(0.357074\pi\)
\(524\) −28.7984 −1.25806
\(525\) 0 0
\(526\) −13.6525 −0.595276
\(527\) 2.29180 0.0998322
\(528\) 0 0
\(529\) 44.8328 1.94925
\(530\) 0 0
\(531\) 0 0
\(532\) 15.3262 0.664477
\(533\) 25.4164 1.10091
\(534\) 0 0
\(535\) 0 0
\(536\) 20.6525 0.892051
\(537\) 0 0
\(538\) 10.6525 0.459261
\(539\) −3.34752 −0.144188
\(540\) 0 0
\(541\) −13.1246 −0.564271 −0.282136 0.959375i \(-0.591043\pi\)
−0.282136 + 0.959375i \(0.591043\pi\)
\(542\) −4.94427 −0.212375
\(543\) 0 0
\(544\) −4.29180 −0.184009
\(545\) 0 0
\(546\) 0 0
\(547\) 34.7082 1.48402 0.742008 0.670391i \(-0.233874\pi\)
0.742008 + 0.670391i \(0.233874\pi\)
\(548\) −9.61803 −0.410862
\(549\) 0 0
\(550\) 0 0
\(551\) −8.09017 −0.344653
\(552\) 0 0
\(553\) 5.00000 0.212622
\(554\) 6.97871 0.296497
\(555\) 0 0
\(556\) −8.09017 −0.343100
\(557\) 9.23607 0.391345 0.195672 0.980669i \(-0.437311\pi\)
0.195672 + 0.980669i \(0.437311\pi\)
\(558\) 0 0
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 0 0
\(562\) 0.673762 0.0284209
\(563\) 9.61803 0.405352 0.202676 0.979246i \(-0.435036\pi\)
0.202676 + 0.979246i \(0.435036\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.3050 0.601282
\(567\) 0 0
\(568\) −9.79837 −0.411131
\(569\) −29.4721 −1.23554 −0.617768 0.786360i \(-0.711963\pi\)
−0.617768 + 0.786360i \(0.711963\pi\)
\(570\) 0 0
\(571\) 32.1246 1.34437 0.672187 0.740382i \(-0.265355\pi\)
0.672187 + 0.740382i \(0.265355\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 5.23607 0.218549
\(575\) 0 0
\(576\) 0 0
\(577\) −37.7771 −1.57268 −0.786340 0.617794i \(-0.788027\pi\)
−0.786340 + 0.617794i \(0.788027\pi\)
\(578\) −10.1459 −0.422014
\(579\) 0 0
\(580\) 0 0
\(581\) −2.85410 −0.118408
\(582\) 0 0
\(583\) 4.18034 0.173132
\(584\) −20.1246 −0.832762
\(585\) 0 0
\(586\) −17.5967 −0.726915
\(587\) 18.7082 0.772170 0.386085 0.922463i \(-0.373827\pi\)
0.386085 + 0.922463i \(0.373827\pi\)
\(588\) 0 0
\(589\) 17.5623 0.723642
\(590\) 0 0
\(591\) 0 0
\(592\) −7.85410 −0.322802
\(593\) −22.0902 −0.907135 −0.453567 0.891222i \(-0.649849\pi\)
−0.453567 + 0.891222i \(0.649849\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.5623 0.924188
\(597\) 0 0
\(598\) 24.7082 1.01039
\(599\) 0.527864 0.0215679 0.0107840 0.999942i \(-0.496567\pi\)
0.0107840 + 0.999942i \(0.496567\pi\)
\(600\) 0 0
\(601\) 36.2705 1.47950 0.739752 0.672879i \(-0.234943\pi\)
0.739752 + 0.672879i \(0.234943\pi\)
\(602\) −1.85410 −0.0755676
\(603\) 0 0
\(604\) 9.00000 0.366205
\(605\) 0 0
\(606\) 0 0
\(607\) 15.4377 0.626597 0.313298 0.949655i \(-0.398566\pi\)
0.313298 + 0.949655i \(0.398566\pi\)
\(608\) −32.8885 −1.33381
\(609\) 0 0
\(610\) 0 0
\(611\) −7.85410 −0.317743
\(612\) 0 0
\(613\) −31.9787 −1.29161 −0.645804 0.763503i \(-0.723478\pi\)
−0.645804 + 0.763503i \(0.723478\pi\)
\(614\) −2.94427 −0.118821
\(615\) 0 0
\(616\) −2.76393 −0.111362
\(617\) 9.76393 0.393081 0.196541 0.980496i \(-0.437029\pi\)
0.196541 + 0.980496i \(0.437029\pi\)
\(618\) 0 0
\(619\) 39.4721 1.58652 0.793260 0.608884i \(-0.208382\pi\)
0.793260 + 0.608884i \(0.208382\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.2361 0.731200
\(623\) −14.4721 −0.579814
\(624\) 0 0
\(625\) 0 0
\(626\) 13.1246 0.524565
\(627\) 0 0
\(628\) −14.8541 −0.592743
\(629\) 3.23607 0.129030
\(630\) 0 0
\(631\) −5.76393 −0.229459 −0.114729 0.993397i \(-0.536600\pi\)
−0.114729 + 0.993397i \(0.536600\pi\)
\(632\) −6.90983 −0.274858
\(633\) 0 0
\(634\) −14.6180 −0.580556
\(635\) 0 0
\(636\) 0 0
\(637\) −21.2705 −0.842768
\(638\) 0.652476 0.0258318
\(639\) 0 0
\(640\) 0 0
\(641\) −10.0902 −0.398538 −0.199269 0.979945i \(-0.563857\pi\)
−0.199269 + 0.979945i \(0.563857\pi\)
\(642\) 0 0
\(643\) −22.8328 −0.900438 −0.450219 0.892918i \(-0.648654\pi\)
−0.450219 + 0.892918i \(0.648654\pi\)
\(644\) −21.5623 −0.849674
\(645\) 0 0
\(646\) 2.76393 0.108745
\(647\) 30.5410 1.20069 0.600346 0.799741i \(-0.295030\pi\)
0.600346 + 0.799741i \(0.295030\pi\)
\(648\) 0 0
\(649\) 3.16718 0.124323
\(650\) 0 0
\(651\) 0 0
\(652\) 17.7984 0.697038
\(653\) 7.90983 0.309536 0.154768 0.987951i \(-0.450537\pi\)
0.154768 + 0.987951i \(0.450537\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.70820 0.379042
\(657\) 0 0
\(658\) −1.61803 −0.0630775
\(659\) −24.4721 −0.953299 −0.476650 0.879093i \(-0.658149\pi\)
−0.476650 + 0.879093i \(0.658149\pi\)
\(660\) 0 0
\(661\) −40.6869 −1.58254 −0.791269 0.611468i \(-0.790579\pi\)
−0.791269 + 0.611468i \(0.790579\pi\)
\(662\) 10.5836 0.411343
\(663\) 0 0
\(664\) 3.94427 0.153067
\(665\) 0 0
\(666\) 0 0
\(667\) 11.3820 0.440711
\(668\) 9.00000 0.348220
\(669\) 0 0
\(670\) 0 0
\(671\) −3.59675 −0.138851
\(672\) 0 0
\(673\) 10.1803 0.392423 0.196212 0.980562i \(-0.437136\pi\)
0.196212 + 0.980562i \(0.437136\pi\)
\(674\) −0.708204 −0.0272790
\(675\) 0 0
\(676\) −17.0902 −0.657314
\(677\) 8.38197 0.322145 0.161073 0.986943i \(-0.448505\pi\)
0.161073 + 0.986943i \(0.448505\pi\)
\(678\) 0 0
\(679\) −4.61803 −0.177224
\(680\) 0 0
\(681\) 0 0
\(682\) −1.41641 −0.0542371
\(683\) −4.52786 −0.173254 −0.0866270 0.996241i \(-0.527609\pi\)
−0.0866270 + 0.996241i \(0.527609\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −11.3820 −0.434565
\(687\) 0 0
\(688\) −3.43769 −0.131061
\(689\) 26.5623 1.01194
\(690\) 0 0
\(691\) 2.72949 0.103835 0.0519173 0.998651i \(-0.483467\pi\)
0.0519173 + 0.998651i \(0.483467\pi\)
\(692\) 27.3262 1.03879
\(693\) 0 0
\(694\) −19.2148 −0.729383
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 5.12461 0.193969
\(699\) 0 0
\(700\) 0 0
\(701\) −35.0132 −1.32243 −0.661214 0.750197i \(-0.729959\pi\)
−0.661214 + 0.750197i \(0.729959\pi\)
\(702\) 0 0
\(703\) 24.7984 0.935288
\(704\) −0.180340 −0.00679682
\(705\) 0 0
\(706\) 14.8885 0.560338
\(707\) 12.0902 0.454698
\(708\) 0 0
\(709\) 33.5410 1.25966 0.629830 0.776733i \(-0.283125\pi\)
0.629830 + 0.776733i \(0.283125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 20.0000 0.749532
\(713\) −24.7082 −0.925330
\(714\) 0 0
\(715\) 0 0
\(716\) −15.3262 −0.572768
\(717\) 0 0
\(718\) 17.7639 0.662944
\(719\) 36.7082 1.36899 0.684493 0.729020i \(-0.260024\pi\)
0.684493 + 0.729020i \(0.260024\pi\)
\(720\) 0 0
\(721\) 18.7082 0.696730
\(722\) 9.43769 0.351235
\(723\) 0 0
\(724\) −22.1803 −0.824326
\(725\) 0 0
\(726\) 0 0
\(727\) −4.43769 −0.164585 −0.0822925 0.996608i \(-0.526224\pi\)
−0.0822925 + 0.996608i \(0.526224\pi\)
\(728\) −17.5623 −0.650902
\(729\) 0 0
\(730\) 0 0
\(731\) 1.41641 0.0523877
\(732\) 0 0
\(733\) −26.9787 −0.996482 −0.498241 0.867039i \(-0.666020\pi\)
−0.498241 + 0.867039i \(0.666020\pi\)
\(734\) 3.36068 0.124045
\(735\) 0 0
\(736\) 46.2705 1.70555
\(737\) −7.05573 −0.259901
\(738\) 0 0
\(739\) −30.9787 −1.13957 −0.569785 0.821794i \(-0.692974\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.47214 0.200888
\(743\) −16.3607 −0.600215 −0.300108 0.953905i \(-0.597023\pi\)
−0.300108 + 0.953905i \(0.597023\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.25735 −0.119260
\(747\) 0 0
\(748\) 0.944272 0.0345260
\(749\) 16.8541 0.615835
\(750\) 0 0
\(751\) −40.8885 −1.49204 −0.746022 0.665921i \(-0.768039\pi\)
−0.746022 + 0.665921i \(0.768039\pi\)
\(752\) −3.00000 −0.109399
\(753\) 0 0
\(754\) 4.14590 0.150985
\(755\) 0 0
\(756\) 0 0
\(757\) −3.58359 −0.130248 −0.0651239 0.997877i \(-0.520744\pi\)
−0.0651239 + 0.997877i \(0.520744\pi\)
\(758\) −21.3820 −0.776628
\(759\) 0 0
\(760\) 0 0
\(761\) −37.4508 −1.35759 −0.678796 0.734327i \(-0.737498\pi\)
−0.678796 + 0.734327i \(0.737498\pi\)
\(762\) 0 0
\(763\) 16.1803 0.585768
\(764\) −39.1246 −1.41548
\(765\) 0 0
\(766\) −7.02129 −0.253689
\(767\) 20.1246 0.726658
\(768\) 0 0
\(769\) −13.4164 −0.483808 −0.241904 0.970300i \(-0.577772\pi\)
−0.241904 + 0.970300i \(0.577772\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9.23607 −0.332413
\(773\) 33.1591 1.19265 0.596324 0.802744i \(-0.296627\pi\)
0.596324 + 0.802744i \(0.296627\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.38197 0.229099
\(777\) 0 0
\(778\) −9.27051 −0.332364
\(779\) −30.6525 −1.09824
\(780\) 0 0
\(781\) 3.34752 0.119784
\(782\) −3.88854 −0.139054
\(783\) 0 0
\(784\) −8.12461 −0.290165
\(785\) 0 0
\(786\) 0 0
\(787\) 34.1803 1.21840 0.609199 0.793018i \(-0.291491\pi\)
0.609199 + 0.793018i \(0.291491\pi\)
\(788\) 15.7082 0.559582
\(789\) 0 0
\(790\) 0 0
\(791\) 16.4164 0.583700
\(792\) 0 0
\(793\) −22.8541 −0.811573
\(794\) 0.0212862 0.000755420 0
\(795\) 0 0
\(796\) −4.14590 −0.146947
\(797\) 14.2361 0.504267 0.252134 0.967692i \(-0.418868\pi\)
0.252134 + 0.967692i \(0.418868\pi\)
\(798\) 0 0
\(799\) 1.23607 0.0437289
\(800\) 0 0
\(801\) 0 0
\(802\) 13.9656 0.493141
\(803\) 6.87539 0.242627
\(804\) 0 0
\(805\) 0 0
\(806\) −9.00000 −0.317011
\(807\) 0 0
\(808\) −16.7082 −0.587793
\(809\) 15.9787 0.561782 0.280891 0.959740i \(-0.409370\pi\)
0.280891 + 0.959740i \(0.409370\pi\)
\(810\) 0 0
\(811\) −1.29180 −0.0453611 −0.0226805 0.999743i \(-0.507220\pi\)
−0.0226805 + 0.999743i \(0.507220\pi\)
\(812\) −3.61803 −0.126968
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) 0 0
\(816\) 0 0
\(817\) 10.8541 0.379737
\(818\) 17.5623 0.614052
\(819\) 0 0
\(820\) 0 0
\(821\) −19.6869 −0.687078 −0.343539 0.939138i \(-0.611626\pi\)
−0.343539 + 0.939138i \(0.611626\pi\)
\(822\) 0 0
\(823\) −34.2918 −1.19534 −0.597668 0.801743i \(-0.703906\pi\)
−0.597668 + 0.801743i \(0.703906\pi\)
\(824\) −25.8541 −0.900670
\(825\) 0 0
\(826\) 4.14590 0.144254
\(827\) −30.0344 −1.04440 −0.522200 0.852823i \(-0.674889\pi\)
−0.522200 + 0.852823i \(0.674889\pi\)
\(828\) 0 0
\(829\) −29.1459 −1.01228 −0.506139 0.862452i \(-0.668928\pi\)
−0.506139 + 0.862452i \(0.668928\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.14590 −0.0397269
\(833\) 3.34752 0.115985
\(834\) 0 0
\(835\) 0 0
\(836\) 7.23607 0.250265
\(837\) 0 0
\(838\) 0.326238 0.0112697
\(839\) −4.14590 −0.143132 −0.0715661 0.997436i \(-0.522800\pi\)
−0.0715661 + 0.997436i \(0.522800\pi\)
\(840\) 0 0
\(841\) −27.0902 −0.934144
\(842\) 19.7771 0.681563
\(843\) 0 0
\(844\) −21.3262 −0.734079
\(845\) 0 0
\(846\) 0 0
\(847\) −16.8541 −0.579114
\(848\) 10.1459 0.348412
\(849\) 0 0
\(850\) 0 0
\(851\) −34.8885 −1.19596
\(852\) 0 0
\(853\) −47.3050 −1.61969 −0.809845 0.586643i \(-0.800449\pi\)
−0.809845 + 0.586643i \(0.800449\pi\)
\(854\) −4.70820 −0.161111
\(855\) 0 0
\(856\) −23.2918 −0.796097
\(857\) −40.6869 −1.38984 −0.694919 0.719088i \(-0.744560\pi\)
−0.694919 + 0.719088i \(0.744560\pi\)
\(858\) 0 0
\(859\) −28.4164 −0.969555 −0.484778 0.874637i \(-0.661099\pi\)
−0.484778 + 0.874637i \(0.661099\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −14.7295 −0.501688
\(863\) −41.5623 −1.41480 −0.707399 0.706815i \(-0.750131\pi\)
−0.707399 + 0.706815i \(0.750131\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −12.4508 −0.423097
\(867\) 0 0
\(868\) 7.85410 0.266586
\(869\) 2.36068 0.0800806
\(870\) 0 0
\(871\) −44.8328 −1.51910
\(872\) −22.3607 −0.757228
\(873\) 0 0
\(874\) −29.7984 −1.00795
\(875\) 0 0
\(876\) 0 0
\(877\) −30.5410 −1.03130 −0.515648 0.856800i \(-0.672449\pi\)
−0.515648 + 0.856800i \(0.672449\pi\)
\(878\) 3.69505 0.124702
\(879\) 0 0
\(880\) 0 0
\(881\) −4.36068 −0.146915 −0.0734575 0.997298i \(-0.523403\pi\)
−0.0734575 + 0.997298i \(0.523403\pi\)
\(882\) 0 0
\(883\) 47.4164 1.59569 0.797845 0.602863i \(-0.205974\pi\)
0.797845 + 0.602863i \(0.205974\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 7.45085 0.250316
\(887\) −5.88854 −0.197718 −0.0988590 0.995101i \(-0.531519\pi\)
−0.0988590 + 0.995101i \(0.531519\pi\)
\(888\) 0 0
\(889\) 25.7082 0.862225
\(890\) 0 0
\(891\) 0 0
\(892\) 35.8885 1.20164
\(893\) 9.47214 0.316973
\(894\) 0 0
\(895\) 0 0
\(896\) −18.4164 −0.615249
\(897\) 0 0
\(898\) −12.5623 −0.419210
\(899\) −4.14590 −0.138273
\(900\) 0 0
\(901\) −4.18034 −0.139267
\(902\) 2.47214 0.0823131
\(903\) 0 0
\(904\) −22.6869 −0.754556
\(905\) 0 0
\(906\) 0 0
\(907\) −47.2492 −1.56888 −0.784442 0.620202i \(-0.787051\pi\)
−0.784442 + 0.620202i \(0.787051\pi\)
\(908\) −31.1246 −1.03291
\(909\) 0 0
\(910\) 0 0
\(911\) 35.7639 1.18491 0.592456 0.805603i \(-0.298158\pi\)
0.592456 + 0.805603i \(0.298158\pi\)
\(912\) 0 0
\(913\) −1.34752 −0.0445965
\(914\) −3.34752 −0.110726
\(915\) 0 0
\(916\) −13.4164 −0.443291
\(917\) 28.7984 0.951006
\(918\) 0 0
\(919\) −1.78522 −0.0588889 −0.0294445 0.999566i \(-0.509374\pi\)
−0.0294445 + 0.999566i \(0.509374\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −14.3262 −0.471810
\(923\) 21.2705 0.700127
\(924\) 0 0
\(925\) 0 0
\(926\) −9.96556 −0.327489
\(927\) 0 0
\(928\) 7.76393 0.254864
\(929\) 36.6312 1.20183 0.600915 0.799313i \(-0.294803\pi\)
0.600915 + 0.799313i \(0.294803\pi\)
\(930\) 0 0
\(931\) 25.6525 0.840726
\(932\) −24.1803 −0.792053
\(933\) 0 0
\(934\) −17.5836 −0.575353
\(935\) 0 0
\(936\) 0 0
\(937\) −51.2705 −1.67493 −0.837467 0.546487i \(-0.815965\pi\)
−0.837467 + 0.546487i \(0.815965\pi\)
\(938\) −9.23607 −0.301568
\(939\) 0 0
\(940\) 0 0
\(941\) 19.5836 0.638407 0.319203 0.947686i \(-0.396585\pi\)
0.319203 + 0.947686i \(0.396585\pi\)
\(942\) 0 0
\(943\) 43.1246 1.40433
\(944\) 7.68692 0.250188
\(945\) 0 0
\(946\) −0.875388 −0.0284613
\(947\) −28.6525 −0.931080 −0.465540 0.885027i \(-0.654140\pi\)
−0.465540 + 0.885027i \(0.654140\pi\)
\(948\) 0 0
\(949\) 43.6869 1.41814
\(950\) 0 0
\(951\) 0 0
\(952\) 2.76393 0.0895796
\(953\) 34.7426 1.12542 0.562712 0.826653i \(-0.309758\pi\)
0.562712 + 0.826653i \(0.309758\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −47.6869 −1.54231
\(957\) 0 0
\(958\) 2.56231 0.0827843
\(959\) 9.61803 0.310583
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −12.7082 −0.409729
\(963\) 0 0
\(964\) −18.5623 −0.597852
\(965\) 0 0
\(966\) 0 0
\(967\) −39.8885 −1.28273 −0.641365 0.767236i \(-0.721631\pi\)
−0.641365 + 0.767236i \(0.721631\pi\)
\(968\) 23.2918 0.748627
\(969\) 0 0
\(970\) 0 0
\(971\) −3.38197 −0.108532 −0.0542662 0.998527i \(-0.517282\pi\)
−0.0542662 + 0.998527i \(0.517282\pi\)
\(972\) 0 0
\(973\) 8.09017 0.259359
\(974\) 5.92299 0.189785
\(975\) 0 0
\(976\) −8.72949 −0.279424
\(977\) −33.6525 −1.07664 −0.538319 0.842741i \(-0.680940\pi\)
−0.538319 + 0.842741i \(0.680940\pi\)
\(978\) 0 0
\(979\) −6.83282 −0.218378
\(980\) 0 0
\(981\) 0 0
\(982\) −23.0213 −0.734639
\(983\) 7.38197 0.235448 0.117724 0.993046i \(-0.462440\pi\)
0.117724 + 0.993046i \(0.462440\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.652476 −0.0207791
\(987\) 0 0
\(988\) 45.9787 1.46278
\(989\) −15.2705 −0.485574
\(990\) 0 0
\(991\) 29.3607 0.932673 0.466336 0.884607i \(-0.345574\pi\)
0.466336 + 0.884607i \(0.345574\pi\)
\(992\) −16.8541 −0.535118
\(993\) 0 0
\(994\) 4.38197 0.138988
\(995\) 0 0
\(996\) 0 0
\(997\) 10.8885 0.344844 0.172422 0.985023i \(-0.444841\pi\)
0.172422 + 0.985023i \(0.444841\pi\)
\(998\) −7.76393 −0.245763
\(999\) 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.d.1.2 2
3.2 odd 2 625.2.a.c.1.1 2
5.4 even 2 5625.2.a.f.1.1 2
12.11 even 2 10000.2.a.l.1.1 2
15.2 even 4 625.2.b.a.624.2 4
15.8 even 4 625.2.b.a.624.3 4
15.14 odd 2 625.2.a.b.1.2 2
25.4 even 10 225.2.h.b.91.1 4
25.19 even 10 225.2.h.b.136.1 4
60.59 even 2 10000.2.a.c.1.2 2
75.2 even 20 625.2.e.c.124.1 8
75.8 even 20 125.2.e.a.74.1 8
75.11 odd 10 625.2.d.b.501.1 4
75.14 odd 10 625.2.d.h.501.1 4
75.17 even 20 125.2.e.a.74.2 8
75.23 even 20 625.2.e.c.124.2 8
75.29 odd 10 25.2.d.a.16.1 yes 4
75.38 even 20 625.2.e.c.499.1 8
75.41 odd 10 625.2.d.b.126.1 4
75.44 odd 10 25.2.d.a.11.1 4
75.47 even 20 125.2.e.a.49.1 8
75.53 even 20 125.2.e.a.49.2 8
75.56 odd 10 125.2.d.a.51.1 4
75.59 odd 10 625.2.d.h.126.1 4
75.62 even 20 625.2.e.c.499.2 8
75.71 odd 10 125.2.d.a.76.1 4
300.119 even 10 400.2.u.b.161.1 4
300.179 even 10 400.2.u.b.241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.11.1 4 75.44 odd 10
25.2.d.a.16.1 yes 4 75.29 odd 10
125.2.d.a.51.1 4 75.56 odd 10
125.2.d.a.76.1 4 75.71 odd 10
125.2.e.a.49.1 8 75.47 even 20
125.2.e.a.49.2 8 75.53 even 20
125.2.e.a.74.1 8 75.8 even 20
125.2.e.a.74.2 8 75.17 even 20
225.2.h.b.91.1 4 25.4 even 10
225.2.h.b.136.1 4 25.19 even 10
400.2.u.b.161.1 4 300.119 even 10
400.2.u.b.241.1 4 300.179 even 10
625.2.a.b.1.2 2 15.14 odd 2
625.2.a.c.1.1 2 3.2 odd 2
625.2.b.a.624.2 4 15.2 even 4
625.2.b.a.624.3 4 15.8 even 4
625.2.d.b.126.1 4 75.41 odd 10
625.2.d.b.501.1 4 75.11 odd 10
625.2.d.h.126.1 4 75.59 odd 10
625.2.d.h.501.1 4 75.14 odd 10
625.2.e.c.124.1 8 75.2 even 20
625.2.e.c.124.2 8 75.23 even 20
625.2.e.c.499.1 8 75.38 even 20
625.2.e.c.499.2 8 75.62 even 20
5625.2.a.d.1.2 2 1.1 even 1 trivial
5625.2.a.f.1.1 2 5.4 even 2
10000.2.a.c.1.2 2 60.59 even 2
10000.2.a.l.1.1 2 12.11 even 2