Properties

Label 4100.2.a.c.1.2
Level $4100$
Weight $2$
Character 4100.1
Self dual yes
Analytic conductor $32.739$
Analytic rank $0$
Dimension $4$
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] = [4100,2,Mod(1,4100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4100.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.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: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} - 6x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.46810\) of defining polynomial
Character \(\chi\) \(=\) 4100.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.21551 q^{3} +5.06479 q^{7} +1.90849 q^{9} +O(q^{10})\) \(q-2.21551 q^{3} +5.06479 q^{7} +1.90849 q^{9} -2.55961 q^{11} +4.93620 q^{13} -2.68821 q^{17} +4.72069 q^{19} -11.2211 q^{21} +1.49482 q^{23} +2.41826 q^{27} +2.43102 q^{29} +3.19339 q^{31} +5.67085 q^{33} -2.90849 q^{37} -10.9362 q^{39} -1.00000 q^{41} +7.62441 q^{43} -5.15171 q^{47} +18.6521 q^{49} +5.95575 q^{51} +11.1192 q^{53} -10.4587 q^{57} +1.49482 q^{59} +1.06380 q^{61} +9.66608 q^{63} -10.5596 q^{67} -3.31179 q^{69} -12.6023 q^{71} -8.28490 q^{73} -12.9639 q^{77} -4.28967 q^{79} -11.0831 q^{81} -10.5606 q^{83} -5.38595 q^{87} +9.36722 q^{89} +25.0008 q^{91} -7.07498 q^{93} +3.37641 q^{97} -4.88499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 12 q^{9} + 4 q^{11} + 4 q^{17} + 6 q^{19} + 12 q^{23} + 10 q^{27} - 4 q^{29} - 8 q^{31} + 20 q^{33} - 16 q^{37} - 24 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{47} + 16 q^{49} - 4 q^{51} + 16 q^{53} - 4 q^{57} + 12 q^{59} + 24 q^{61} + 10 q^{63} - 28 q^{67} - 28 q^{69} - 2 q^{71} - 8 q^{73} - 8 q^{77} - 18 q^{79} + 28 q^{81} + 12 q^{83} - 44 q^{87} + 4 q^{89} + 36 q^{91} + 28 q^{93} - 16 q^{97} + 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.21551 −1.27913 −0.639563 0.768739i \(-0.720885\pi\)
−0.639563 + 0.768739i \(0.720885\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.06479 1.91431 0.957156 0.289574i \(-0.0935135\pi\)
0.957156 + 0.289574i \(0.0935135\pi\)
\(8\) 0 0
\(9\) 1.90849 0.636162
\(10\) 0 0
\(11\) −2.55961 −0.771753 −0.385876 0.922551i \(-0.626101\pi\)
−0.385876 + 0.922551i \(0.626101\pi\)
\(12\) 0 0
\(13\) 4.93620 1.36906 0.684528 0.728987i \(-0.260009\pi\)
0.684528 + 0.728987i \(0.260009\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.68821 −0.651986 −0.325993 0.945372i \(-0.605699\pi\)
−0.325993 + 0.945372i \(0.605699\pi\)
\(18\) 0 0
\(19\) 4.72069 1.08300 0.541500 0.840701i \(-0.317857\pi\)
0.541500 + 0.840701i \(0.317857\pi\)
\(20\) 0 0
\(21\) −11.2211 −2.44864
\(22\) 0 0
\(23\) 1.49482 0.311692 0.155846 0.987781i \(-0.450190\pi\)
0.155846 + 0.987781i \(0.450190\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.41826 0.465395
\(28\) 0 0
\(29\) 2.43102 0.451429 0.225715 0.974193i \(-0.427528\pi\)
0.225715 + 0.974193i \(0.427528\pi\)
\(30\) 0 0
\(31\) 3.19339 0.573549 0.286774 0.957998i \(-0.407417\pi\)
0.286774 + 0.957998i \(0.407417\pi\)
\(32\) 0 0
\(33\) 5.67085 0.987168
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.90849 −0.478152 −0.239076 0.971001i \(-0.576845\pi\)
−0.239076 + 0.971001i \(0.576845\pi\)
\(38\) 0 0
\(39\) −10.9362 −1.75119
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 7.62441 1.16271 0.581356 0.813650i \(-0.302523\pi\)
0.581356 + 0.813650i \(0.302523\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.15171 −0.751454 −0.375727 0.926730i \(-0.622607\pi\)
−0.375727 + 0.926730i \(0.622607\pi\)
\(48\) 0 0
\(49\) 18.6521 2.66459
\(50\) 0 0
\(51\) 5.95575 0.833972
\(52\) 0 0
\(53\) 11.1192 1.52734 0.763672 0.645605i \(-0.223395\pi\)
0.763672 + 0.645605i \(0.223395\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.4587 −1.38529
\(58\) 0 0
\(59\) 1.49482 0.194609 0.0973046 0.995255i \(-0.468978\pi\)
0.0973046 + 0.995255i \(0.468978\pi\)
\(60\) 0 0
\(61\) 1.06380 0.136206 0.0681029 0.997678i \(-0.478305\pi\)
0.0681029 + 0.997678i \(0.478305\pi\)
\(62\) 0 0
\(63\) 9.66608 1.21781
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.5596 −1.29006 −0.645031 0.764156i \(-0.723156\pi\)
−0.645031 + 0.764156i \(0.723156\pi\)
\(68\) 0 0
\(69\) −3.31179 −0.398693
\(70\) 0 0
\(71\) −12.6023 −1.49562 −0.747808 0.663915i \(-0.768894\pi\)
−0.747808 + 0.663915i \(0.768894\pi\)
\(72\) 0 0
\(73\) −8.28490 −0.969674 −0.484837 0.874604i \(-0.661121\pi\)
−0.484837 + 0.874604i \(0.661121\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.9639 −1.47737
\(78\) 0 0
\(79\) −4.28967 −0.482625 −0.241313 0.970447i \(-0.577578\pi\)
−0.241313 + 0.970447i \(0.577578\pi\)
\(80\) 0 0
\(81\) −11.0831 −1.23146
\(82\) 0 0
\(83\) −10.5606 −1.15918 −0.579588 0.814909i \(-0.696787\pi\)
−0.579588 + 0.814909i \(0.696787\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.38595 −0.577435
\(88\) 0 0
\(89\) 9.36722 0.992923 0.496462 0.868059i \(-0.334632\pi\)
0.496462 + 0.868059i \(0.334632\pi\)
\(90\) 0 0
\(91\) 25.0008 2.62080
\(92\) 0 0
\(93\) −7.07498 −0.733641
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.37641 0.342823 0.171411 0.985200i \(-0.445167\pi\)
0.171411 + 0.985200i \(0.445167\pi\)
\(98\) 0 0
\(99\) −4.88499 −0.490960
\(100\) 0 0
\(101\) −4.24799 −0.422691 −0.211345 0.977411i \(-0.567785\pi\)
−0.211345 + 0.977411i \(0.567785\pi\)
\(102\) 0 0
\(103\) −1.11923 −0.110281 −0.0551404 0.998479i \(-0.517561\pi\)
−0.0551404 + 0.998479i \(0.517561\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.322149 0.0311434 0.0155717 0.999879i \(-0.495043\pi\)
0.0155717 + 0.999879i \(0.495043\pi\)
\(108\) 0 0
\(109\) 3.23763 0.310109 0.155055 0.987906i \(-0.450445\pi\)
0.155055 + 0.987906i \(0.450445\pi\)
\(110\) 0 0
\(111\) 6.44378 0.611616
\(112\) 0 0
\(113\) −2.10187 −0.197727 −0.0988637 0.995101i \(-0.531521\pi\)
−0.0988637 + 0.995101i \(0.531521\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.42066 0.870941
\(118\) 0 0
\(119\) −13.6152 −1.24810
\(120\) 0 0
\(121\) −4.44838 −0.404398
\(122\) 0 0
\(123\) 2.21551 0.199766
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0658 1.15940 0.579700 0.814830i \(-0.303170\pi\)
0.579700 + 0.814830i \(0.303170\pi\)
\(128\) 0 0
\(129\) −16.8919 −1.48725
\(130\) 0 0
\(131\) 8.43102 0.736622 0.368311 0.929703i \(-0.379936\pi\)
0.368311 + 0.929703i \(0.379936\pi\)
\(132\) 0 0
\(133\) 23.9093 2.07320
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.5606 1.41487 0.707434 0.706779i \(-0.249853\pi\)
0.707434 + 0.706779i \(0.249853\pi\)
\(138\) 0 0
\(139\) −2.18303 −0.185162 −0.0925810 0.995705i \(-0.529512\pi\)
−0.0925810 + 0.995705i \(0.529512\pi\)
\(140\) 0 0
\(141\) 11.4137 0.961204
\(142\) 0 0
\(143\) −12.6348 −1.05657
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −41.3240 −3.40834
\(148\) 0 0
\(149\) 2.43102 0.199157 0.0995785 0.995030i \(-0.468251\pi\)
0.0995785 + 0.995030i \(0.468251\pi\)
\(150\) 0 0
\(151\) −0.343113 −0.0279221 −0.0139611 0.999903i \(-0.504444\pi\)
−0.0139611 + 0.999903i \(0.504444\pi\)
\(152\) 0 0
\(153\) −5.13040 −0.414769
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.6348 0.848746 0.424373 0.905487i \(-0.360494\pi\)
0.424373 + 0.905487i \(0.360494\pi\)
\(158\) 0 0
\(159\) −24.6348 −1.95366
\(160\) 0 0
\(161\) 7.57096 0.596675
\(162\) 0 0
\(163\) 0.173834 0.0136157 0.00680785 0.999977i \(-0.497833\pi\)
0.00680785 + 0.999977i \(0.497833\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.5596 −1.12666 −0.563328 0.826233i \(-0.690479\pi\)
−0.563328 + 0.826233i \(0.690479\pi\)
\(168\) 0 0
\(169\) 11.3661 0.874312
\(170\) 0 0
\(171\) 9.00937 0.688963
\(172\) 0 0
\(173\) 22.2592 1.69233 0.846167 0.532918i \(-0.178905\pi\)
0.846167 + 0.532918i \(0.178905\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.31179 −0.248930
\(178\) 0 0
\(179\) −17.9268 −1.33991 −0.669957 0.742400i \(-0.733688\pi\)
−0.669957 + 0.742400i \(0.733688\pi\)
\(180\) 0 0
\(181\) 20.9916 1.56030 0.780148 0.625595i \(-0.215144\pi\)
0.780148 + 0.625595i \(0.215144\pi\)
\(182\) 0 0
\(183\) −2.35686 −0.174224
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.88077 0.503172
\(188\) 0 0
\(189\) 12.2480 0.890910
\(190\) 0 0
\(191\) 6.45074 0.466759 0.233380 0.972386i \(-0.425022\pi\)
0.233380 + 0.972386i \(0.425022\pi\)
\(192\) 0 0
\(193\) −6.20374 −0.446555 −0.223278 0.974755i \(-0.571676\pi\)
−0.223278 + 0.974755i \(0.571676\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.1850 −1.01064 −0.505320 0.862932i \(-0.668626\pi\)
−0.505320 + 0.862932i \(0.668626\pi\)
\(198\) 0 0
\(199\) 18.7099 1.32631 0.663155 0.748482i \(-0.269217\pi\)
0.663155 + 0.748482i \(0.269217\pi\)
\(200\) 0 0
\(201\) 23.3949 1.65015
\(202\) 0 0
\(203\) 12.3126 0.864176
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.85285 0.198286
\(208\) 0 0
\(209\) −12.0831 −0.835808
\(210\) 0 0
\(211\) 14.6984 1.01188 0.505940 0.862569i \(-0.331146\pi\)
0.505940 + 0.862569i \(0.331146\pi\)
\(212\) 0 0
\(213\) 27.9205 1.91308
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.1738 1.09795
\(218\) 0 0
\(219\) 18.3553 1.24033
\(220\) 0 0
\(221\) −13.2695 −0.892605
\(222\) 0 0
\(223\) −20.1109 −1.34672 −0.673361 0.739314i \(-0.735150\pi\)
−0.673361 + 0.739314i \(0.735150\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.71149 0.445457 0.222729 0.974880i \(-0.428504\pi\)
0.222729 + 0.974880i \(0.428504\pi\)
\(228\) 0 0
\(229\) 12.7624 0.843361 0.421680 0.906745i \(-0.361440\pi\)
0.421680 + 0.906745i \(0.361440\pi\)
\(230\) 0 0
\(231\) 28.7217 1.88975
\(232\) 0 0
\(233\) 29.0750 1.90477 0.952383 0.304906i \(-0.0986249\pi\)
0.952383 + 0.304906i \(0.0986249\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.50380 0.617338
\(238\) 0 0
\(239\) 26.0971 1.68808 0.844041 0.536279i \(-0.180171\pi\)
0.844041 + 0.536279i \(0.180171\pi\)
\(240\) 0 0
\(241\) 1.06380 0.0685255 0.0342627 0.999413i \(-0.489092\pi\)
0.0342627 + 0.999413i \(0.489092\pi\)
\(242\) 0 0
\(243\) 17.3000 1.10980
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 23.3023 1.48269
\(248\) 0 0
\(249\) 23.3971 1.48273
\(250\) 0 0
\(251\) −7.82815 −0.494108 −0.247054 0.969002i \(-0.579463\pi\)
−0.247054 + 0.969002i \(0.579463\pi\)
\(252\) 0 0
\(253\) −3.82617 −0.240549
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.49482 −0.467514 −0.233757 0.972295i \(-0.575102\pi\)
−0.233757 + 0.972295i \(0.575102\pi\)
\(258\) 0 0
\(259\) −14.7309 −0.915332
\(260\) 0 0
\(261\) 4.63957 0.287182
\(262\) 0 0
\(263\) −15.5919 −0.961439 −0.480720 0.876874i \(-0.659625\pi\)
−0.480720 + 0.876874i \(0.659625\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −20.7532 −1.27007
\(268\) 0 0
\(269\) 8.80860 0.537070 0.268535 0.963270i \(-0.413461\pi\)
0.268535 + 0.963270i \(0.413461\pi\)
\(270\) 0 0
\(271\) −19.8816 −1.20772 −0.603860 0.797090i \(-0.706372\pi\)
−0.603860 + 0.797090i \(0.706372\pi\)
\(272\) 0 0
\(273\) −55.3896 −3.35233
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.08232 0.425535 0.212768 0.977103i \(-0.431752\pi\)
0.212768 + 0.977103i \(0.431752\pi\)
\(278\) 0 0
\(279\) 6.09453 0.364870
\(280\) 0 0
\(281\) −31.2695 −1.86538 −0.932692 0.360675i \(-0.882547\pi\)
−0.932692 + 0.360675i \(0.882547\pi\)
\(282\) 0 0
\(283\) 2.18303 0.129768 0.0648838 0.997893i \(-0.479332\pi\)
0.0648838 + 0.997893i \(0.479332\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.06479 −0.298965
\(288\) 0 0
\(289\) −9.77354 −0.574914
\(290\) 0 0
\(291\) −7.48048 −0.438514
\(292\) 0 0
\(293\) 33.6798 1.96760 0.983798 0.179278i \(-0.0573762\pi\)
0.983798 + 0.179278i \(0.0573762\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −6.18982 −0.359170
\(298\) 0 0
\(299\) 7.37874 0.426723
\(300\) 0 0
\(301\) 38.6160 2.22579
\(302\) 0 0
\(303\) 9.41147 0.540675
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.7532 0.727862 0.363931 0.931426i \(-0.381434\pi\)
0.363931 + 0.931426i \(0.381434\pi\)
\(308\) 0 0
\(309\) 2.47966 0.141063
\(310\) 0 0
\(311\) 2.18204 0.123732 0.0618660 0.998084i \(-0.480295\pi\)
0.0618660 + 0.998084i \(0.480295\pi\)
\(312\) 0 0
\(313\) 29.3042 1.65637 0.828187 0.560452i \(-0.189373\pi\)
0.828187 + 0.560452i \(0.189373\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.5511 1.60359 0.801794 0.597601i \(-0.203879\pi\)
0.801794 + 0.597601i \(0.203879\pi\)
\(318\) 0 0
\(319\) −6.22247 −0.348392
\(320\) 0 0
\(321\) −0.713725 −0.0398363
\(322\) 0 0
\(323\) −12.6902 −0.706101
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −7.17301 −0.396669
\(328\) 0 0
\(329\) −26.0923 −1.43852
\(330\) 0 0
\(331\) −4.80761 −0.264250 −0.132125 0.991233i \(-0.542180\pi\)
−0.132125 + 0.991233i \(0.542180\pi\)
\(332\) 0 0
\(333\) −5.55080 −0.304182
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −16.3148 −0.888724 −0.444362 0.895847i \(-0.646570\pi\)
−0.444362 + 0.895847i \(0.646570\pi\)
\(338\) 0 0
\(339\) 4.65672 0.252918
\(340\) 0 0
\(341\) −8.17383 −0.442638
\(342\) 0 0
\(343\) 59.0156 3.18654
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −31.6034 −1.69656 −0.848281 0.529547i \(-0.822362\pi\)
−0.848281 + 0.529547i \(0.822362\pi\)
\(348\) 0 0
\(349\) 4.97311 0.266204 0.133102 0.991102i \(-0.457506\pi\)
0.133102 + 0.991102i \(0.457506\pi\)
\(350\) 0 0
\(351\) 11.9370 0.637151
\(352\) 0 0
\(353\) −1.72430 −0.0917750 −0.0458875 0.998947i \(-0.514612\pi\)
−0.0458875 + 0.998947i \(0.514612\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 30.1646 1.59648
\(358\) 0 0
\(359\) −7.13796 −0.376727 −0.188364 0.982099i \(-0.560318\pi\)
−0.188364 + 0.982099i \(0.560318\pi\)
\(360\) 0 0
\(361\) 3.28490 0.172889
\(362\) 0 0
\(363\) 9.85542 0.517276
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.38595 0.489943 0.244971 0.969530i \(-0.421221\pi\)
0.244971 + 0.969530i \(0.421221\pi\)
\(368\) 0 0
\(369\) −1.90849 −0.0993518
\(370\) 0 0
\(371\) 56.3166 2.92381
\(372\) 0 0
\(373\) −23.1212 −1.19717 −0.598585 0.801059i \(-0.704270\pi\)
−0.598585 + 0.801059i \(0.704270\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 5.99080 0.307727 0.153863 0.988092i \(-0.450828\pi\)
0.153863 + 0.988092i \(0.450828\pi\)
\(380\) 0 0
\(381\) −28.9474 −1.48302
\(382\) 0 0
\(383\) 29.5811 1.51153 0.755763 0.654845i \(-0.227266\pi\)
0.755763 + 0.654845i \(0.227266\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 14.5511 0.739672
\(388\) 0 0
\(389\) 22.8620 1.15915 0.579576 0.814918i \(-0.303218\pi\)
0.579576 + 0.814918i \(0.303218\pi\)
\(390\) 0 0
\(391\) −4.01839 −0.203219
\(392\) 0 0
\(393\) −18.6790 −0.942232
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.588531 0.0295375 0.0147688 0.999891i \(-0.495299\pi\)
0.0147688 + 0.999891i \(0.495299\pi\)
\(398\) 0 0
\(399\) −52.9713 −2.65188
\(400\) 0 0
\(401\) −0.0926758 −0.00462801 −0.00231400 0.999997i \(-0.500737\pi\)
−0.00231400 + 0.999997i \(0.500737\pi\)
\(402\) 0 0
\(403\) 15.7632 0.785220
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.44460 0.369015
\(408\) 0 0
\(409\) −1.29526 −0.0640463 −0.0320232 0.999487i \(-0.510195\pi\)
−0.0320232 + 0.999487i \(0.510195\pi\)
\(410\) 0 0
\(411\) −36.6902 −1.80979
\(412\) 0 0
\(413\) 7.57096 0.372543
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.83652 0.236846
\(418\) 0 0
\(419\) 1.75201 0.0855912 0.0427956 0.999084i \(-0.486374\pi\)
0.0427956 + 0.999084i \(0.486374\pi\)
\(420\) 0 0
\(421\) 36.7364 1.79042 0.895212 0.445641i \(-0.147024\pi\)
0.895212 + 0.445641i \(0.147024\pi\)
\(422\) 0 0
\(423\) −9.83196 −0.478046
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.38793 0.260740
\(428\) 0 0
\(429\) 27.9924 1.35149
\(430\) 0 0
\(431\) 12.3126 0.593078 0.296539 0.955021i \(-0.404168\pi\)
0.296539 + 0.955021i \(0.404168\pi\)
\(432\) 0 0
\(433\) 17.7632 0.853644 0.426822 0.904336i \(-0.359633\pi\)
0.426822 + 0.904336i \(0.359633\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.05659 0.337562
\(438\) 0 0
\(439\) 3.22230 0.153792 0.0768961 0.997039i \(-0.475499\pi\)
0.0768961 + 0.997039i \(0.475499\pi\)
\(440\) 0 0
\(441\) 35.5973 1.69511
\(442\) 0 0
\(443\) 35.8537 1.70346 0.851730 0.523982i \(-0.175554\pi\)
0.851730 + 0.523982i \(0.175554\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −5.38595 −0.254747
\(448\) 0 0
\(449\) −15.3437 −0.724113 −0.362057 0.932156i \(-0.617925\pi\)
−0.362057 + 0.932156i \(0.617925\pi\)
\(450\) 0 0
\(451\) 2.55961 0.120528
\(452\) 0 0
\(453\) 0.760170 0.0357159
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −27.5917 −1.29068 −0.645342 0.763894i \(-0.723285\pi\)
−0.645342 + 0.763894i \(0.723285\pi\)
\(458\) 0 0
\(459\) −6.50079 −0.303431
\(460\) 0 0
\(461\) −8.08116 −0.376377 −0.188189 0.982133i \(-0.560262\pi\)
−0.188189 + 0.982133i \(0.560262\pi\)
\(462\) 0 0
\(463\) 8.23622 0.382770 0.191385 0.981515i \(-0.438702\pi\)
0.191385 + 0.981515i \(0.438702\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.7333 1.09825 0.549123 0.835742i \(-0.314962\pi\)
0.549123 + 0.835742i \(0.314962\pi\)
\(468\) 0 0
\(469\) −53.4822 −2.46958
\(470\) 0 0
\(471\) −23.5614 −1.08565
\(472\) 0 0
\(473\) −19.5155 −0.897325
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 21.2209 0.971638
\(478\) 0 0
\(479\) −16.6987 −0.762985 −0.381492 0.924372i \(-0.624590\pi\)
−0.381492 + 0.924372i \(0.624590\pi\)
\(480\) 0 0
\(481\) −14.3569 −0.654617
\(482\) 0 0
\(483\) −16.7735 −0.763223
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.927003 −0.0420065 −0.0210033 0.999779i \(-0.506686\pi\)
−0.0210033 + 0.999779i \(0.506686\pi\)
\(488\) 0 0
\(489\) −0.385130 −0.0174162
\(490\) 0 0
\(491\) −26.8752 −1.21286 −0.606430 0.795137i \(-0.707399\pi\)
−0.606430 + 0.795137i \(0.707399\pi\)
\(492\) 0 0
\(493\) −6.53509 −0.294326
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −63.8279 −2.86307
\(498\) 0 0
\(499\) 18.2155 0.815438 0.407719 0.913107i \(-0.366324\pi\)
0.407719 + 0.913107i \(0.366324\pi\)
\(500\) 0 0
\(501\) 32.2570 1.44114
\(502\) 0 0
\(503\) −18.9591 −0.845346 −0.422673 0.906282i \(-0.638908\pi\)
−0.422673 + 0.906282i \(0.638908\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −25.1816 −1.11835
\(508\) 0 0
\(509\) 12.4218 0.550588 0.275294 0.961360i \(-0.411225\pi\)
0.275294 + 0.961360i \(0.411225\pi\)
\(510\) 0 0
\(511\) −41.9613 −1.85626
\(512\) 0 0
\(513\) 11.4159 0.504022
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.1864 0.579937
\(518\) 0 0
\(519\) −49.3154 −2.16471
\(520\) 0 0
\(521\) −7.17267 −0.314240 −0.157120 0.987579i \(-0.550221\pi\)
−0.157120 + 0.987579i \(0.550221\pi\)
\(522\) 0 0
\(523\) −35.6563 −1.55914 −0.779570 0.626315i \(-0.784563\pi\)
−0.779570 + 0.626315i \(0.784563\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.58448 −0.373946
\(528\) 0 0
\(529\) −20.7655 −0.902848
\(530\) 0 0
\(531\) 2.85285 0.123803
\(532\) 0 0
\(533\) −4.93620 −0.213811
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 39.7171 1.71392
\(538\) 0 0
\(539\) −47.7422 −2.05640
\(540\) 0 0
\(541\) 28.9731 1.24565 0.622826 0.782361i \(-0.285985\pi\)
0.622826 + 0.782361i \(0.285985\pi\)
\(542\) 0 0
\(543\) −46.5072 −1.99581
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −16.1178 −0.689148 −0.344574 0.938759i \(-0.611977\pi\)
−0.344574 + 0.938759i \(0.611977\pi\)
\(548\) 0 0
\(549\) 2.03025 0.0866489
\(550\) 0 0
\(551\) 11.4761 0.488898
\(552\) 0 0
\(553\) −21.7263 −0.923895
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.2675 1.15536 0.577681 0.816262i \(-0.303958\pi\)
0.577681 + 0.816262i \(0.303958\pi\)
\(558\) 0 0
\(559\) 37.6356 1.59182
\(560\) 0 0
\(561\) −15.2444 −0.643620
\(562\) 0 0
\(563\) −41.9173 −1.76660 −0.883302 0.468805i \(-0.844685\pi\)
−0.883302 + 0.468805i \(0.844685\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −56.1338 −2.35740
\(568\) 0 0
\(569\) 37.3762 1.56689 0.783446 0.621460i \(-0.213460\pi\)
0.783446 + 0.621460i \(0.213460\pi\)
\(570\) 0 0
\(571\) −26.3228 −1.10157 −0.550787 0.834646i \(-0.685673\pi\)
−0.550787 + 0.834646i \(0.685673\pi\)
\(572\) 0 0
\(573\) −14.2917 −0.597044
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.8262 0.575591 0.287795 0.957692i \(-0.407078\pi\)
0.287795 + 0.957692i \(0.407078\pi\)
\(578\) 0 0
\(579\) 13.7445 0.571200
\(580\) 0 0
\(581\) −53.4873 −2.21903
\(582\) 0 0
\(583\) −28.4609 −1.17873
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22.7004 −0.936945 −0.468472 0.883478i \(-0.655195\pi\)
−0.468472 + 0.883478i \(0.655195\pi\)
\(588\) 0 0
\(589\) 15.0750 0.621154
\(590\) 0 0
\(591\) 31.4270 1.29274
\(592\) 0 0
\(593\) −16.4167 −0.674152 −0.337076 0.941477i \(-0.609438\pi\)
−0.337076 + 0.941477i \(0.609438\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −41.4520 −1.69652
\(598\) 0 0
\(599\) −6.54909 −0.267588 −0.133794 0.991009i \(-0.542716\pi\)
−0.133794 + 0.991009i \(0.542716\pi\)
\(600\) 0 0
\(601\) −24.7783 −1.01073 −0.505365 0.862906i \(-0.668642\pi\)
−0.505365 + 0.862906i \(0.668642\pi\)
\(602\) 0 0
\(603\) −20.1529 −0.820688
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.7982 0.641231 0.320615 0.947209i \(-0.396110\pi\)
0.320615 + 0.947209i \(0.396110\pi\)
\(608\) 0 0
\(609\) −27.2787 −1.10539
\(610\) 0 0
\(611\) −25.4299 −1.02878
\(612\) 0 0
\(613\) −39.3415 −1.58899 −0.794494 0.607272i \(-0.792264\pi\)
−0.794494 + 0.607272i \(0.792264\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.4075 1.06313 0.531563 0.847019i \(-0.321605\pi\)
0.531563 + 0.847019i \(0.321605\pi\)
\(618\) 0 0
\(619\) 33.9278 1.36367 0.681837 0.731504i \(-0.261181\pi\)
0.681837 + 0.731504i \(0.261181\pi\)
\(620\) 0 0
\(621\) 3.61487 0.145060
\(622\) 0 0
\(623\) 47.4430 1.90076
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 26.7703 1.06910
\(628\) 0 0
\(629\) 7.81861 0.311748
\(630\) 0 0
\(631\) −5.68017 −0.226124 −0.113062 0.993588i \(-0.536066\pi\)
−0.113062 + 0.993588i \(0.536066\pi\)
\(632\) 0 0
\(633\) −32.5644 −1.29432
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 92.0706 3.64797
\(638\) 0 0
\(639\) −24.0513 −0.951454
\(640\) 0 0
\(641\) 15.5802 0.615379 0.307690 0.951487i \(-0.400444\pi\)
0.307690 + 0.951487i \(0.400444\pi\)
\(642\) 0 0
\(643\) −22.9372 −0.904554 −0.452277 0.891877i \(-0.649388\pi\)
−0.452277 + 0.891877i \(0.649388\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.0546 −0.591858 −0.295929 0.955210i \(-0.595629\pi\)
−0.295929 + 0.955210i \(0.595629\pi\)
\(648\) 0 0
\(649\) −3.82617 −0.150190
\(650\) 0 0
\(651\) −35.8333 −1.40442
\(652\) 0 0
\(653\) 45.8652 1.79484 0.897422 0.441174i \(-0.145438\pi\)
0.897422 + 0.441174i \(0.145438\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −15.8116 −0.616870
\(658\) 0 0
\(659\) −4.37302 −0.170349 −0.0851744 0.996366i \(-0.527145\pi\)
−0.0851744 + 0.996366i \(0.527145\pi\)
\(660\) 0 0
\(661\) −25.4946 −0.991625 −0.495813 0.868430i \(-0.665130\pi\)
−0.495813 + 0.868430i \(0.665130\pi\)
\(662\) 0 0
\(663\) 29.3988 1.14175
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.63394 0.140707
\(668\) 0 0
\(669\) 44.5558 1.72263
\(670\) 0 0
\(671\) −2.72292 −0.105117
\(672\) 0 0
\(673\) −31.7540 −1.22403 −0.612013 0.790848i \(-0.709640\pi\)
−0.612013 + 0.790848i \(0.709640\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.8998 1.07228 0.536138 0.844131i \(-0.319883\pi\)
0.536138 + 0.844131i \(0.319883\pi\)
\(678\) 0 0
\(679\) 17.1008 0.656270
\(680\) 0 0
\(681\) −14.8694 −0.569796
\(682\) 0 0
\(683\) −8.11782 −0.310620 −0.155310 0.987866i \(-0.549638\pi\)
−0.155310 + 0.987866i \(0.549638\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −28.2752 −1.07876
\(688\) 0 0
\(689\) 54.8867 2.09102
\(690\) 0 0
\(691\) −26.6390 −1.01339 −0.506697 0.862124i \(-0.669134\pi\)
−0.506697 + 0.862124i \(0.669134\pi\)
\(692\) 0 0
\(693\) −24.7414 −0.939850
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.68821 0.101823
\(698\) 0 0
\(699\) −64.4159 −2.43643
\(700\) 0 0
\(701\) 43.1302 1.62900 0.814502 0.580160i \(-0.197010\pi\)
0.814502 + 0.580160i \(0.197010\pi\)
\(702\) 0 0
\(703\) −13.7301 −0.517839
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −21.5152 −0.809162
\(708\) 0 0
\(709\) 49.5175 1.85967 0.929835 0.367978i \(-0.119950\pi\)
0.929835 + 0.367978i \(0.119950\pi\)
\(710\) 0 0
\(711\) −8.18677 −0.307028
\(712\) 0 0
\(713\) 4.77354 0.178771
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −57.8184 −2.15927
\(718\) 0 0
\(719\) −16.4839 −0.614745 −0.307372 0.951589i \(-0.599450\pi\)
−0.307372 + 0.951589i \(0.599450\pi\)
\(720\) 0 0
\(721\) −5.66866 −0.211112
\(722\) 0 0
\(723\) −2.35686 −0.0876527
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −48.5947 −1.80228 −0.901139 0.433530i \(-0.857268\pi\)
−0.901139 + 0.433530i \(0.857268\pi\)
\(728\) 0 0
\(729\) −5.07896 −0.188110
\(730\) 0 0
\(731\) −20.4960 −0.758071
\(732\) 0 0
\(733\) −11.8166 −0.436457 −0.218229 0.975898i \(-0.570028\pi\)
−0.218229 + 0.975898i \(0.570028\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.0285 0.995609
\(738\) 0 0
\(739\) −5.68017 −0.208949 −0.104474 0.994528i \(-0.533316\pi\)
−0.104474 + 0.994528i \(0.533316\pi\)
\(740\) 0 0
\(741\) −51.6264 −1.89654
\(742\) 0 0
\(743\) 24.4980 0.898743 0.449372 0.893345i \(-0.351648\pi\)
0.449372 + 0.893345i \(0.351648\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −20.1548 −0.737424
\(748\) 0 0
\(749\) 1.63162 0.0596181
\(750\) 0 0
\(751\) −35.7897 −1.30598 −0.652992 0.757365i \(-0.726487\pi\)
−0.652992 + 0.757365i \(0.726487\pi\)
\(752\) 0 0
\(753\) 17.3433 0.632027
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −37.7169 −1.37084 −0.685421 0.728147i \(-0.740382\pi\)
−0.685421 + 0.728147i \(0.740382\pi\)
\(758\) 0 0
\(759\) 8.47691 0.307692
\(760\) 0 0
\(761\) −22.6533 −0.821181 −0.410590 0.911820i \(-0.634677\pi\)
−0.410590 + 0.911820i \(0.634677\pi\)
\(762\) 0 0
\(763\) 16.3979 0.593646
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.37874 0.266431
\(768\) 0 0
\(769\) −42.6292 −1.53725 −0.768624 0.639701i \(-0.779058\pi\)
−0.768624 + 0.639701i \(0.779058\pi\)
\(770\) 0 0
\(771\) 16.6049 0.598009
\(772\) 0 0
\(773\) −11.7635 −0.423105 −0.211552 0.977367i \(-0.567852\pi\)
−0.211552 + 0.977367i \(0.567852\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 32.6364 1.17082
\(778\) 0 0
\(779\) −4.72069 −0.169136
\(780\) 0 0
\(781\) 32.2570 1.15425
\(782\) 0 0
\(783\) 5.87884 0.210093
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −34.4517 −1.22807 −0.614036 0.789278i \(-0.710455\pi\)
−0.614036 + 0.789278i \(0.710455\pi\)
\(788\) 0 0
\(789\) 34.5441 1.22980
\(790\) 0 0
\(791\) −10.6455 −0.378512
\(792\) 0 0
\(793\) 5.25113 0.186473
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −41.2161 −1.45995 −0.729974 0.683475i \(-0.760468\pi\)
−0.729974 + 0.683475i \(0.760468\pi\)
\(798\) 0 0
\(799\) 13.8489 0.489937
\(800\) 0 0
\(801\) 17.8772 0.631660
\(802\) 0 0
\(803\) 21.2061 0.748349
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −19.5155 −0.686979
\(808\) 0 0
\(809\) 12.4123 0.436393 0.218196 0.975905i \(-0.429983\pi\)
0.218196 + 0.975905i \(0.429983\pi\)
\(810\) 0 0
\(811\) 3.91665 0.137532 0.0687660 0.997633i \(-0.478094\pi\)
0.0687660 + 0.997633i \(0.478094\pi\)
\(812\) 0 0
\(813\) 44.0479 1.54483
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 35.9924 1.25922
\(818\) 0 0
\(819\) 47.7137 1.66725
\(820\) 0 0
\(821\) 28.6326 0.999283 0.499642 0.866232i \(-0.333465\pi\)
0.499642 + 0.866232i \(0.333465\pi\)
\(822\) 0 0
\(823\) −43.8212 −1.52751 −0.763755 0.645506i \(-0.776647\pi\)
−0.763755 + 0.645506i \(0.776647\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.1924 −0.389198 −0.194599 0.980883i \(-0.562341\pi\)
−0.194599 + 0.980883i \(0.562341\pi\)
\(828\) 0 0
\(829\) 26.6278 0.924820 0.462410 0.886666i \(-0.346985\pi\)
0.462410 + 0.886666i \(0.346985\pi\)
\(830\) 0 0
\(831\) −15.6910 −0.544313
\(832\) 0 0
\(833\) −50.1408 −1.73727
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.72244 0.266927
\(838\) 0 0
\(839\) 9.43085 0.325589 0.162795 0.986660i \(-0.447949\pi\)
0.162795 + 0.986660i \(0.447949\pi\)
\(840\) 0 0
\(841\) −23.0901 −0.796212
\(842\) 0 0
\(843\) 69.2780 2.38606
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −22.5301 −0.774144
\(848\) 0 0
\(849\) −4.83652 −0.165989
\(850\) 0 0
\(851\) −4.34767 −0.149036
\(852\) 0 0
\(853\) −32.6045 −1.11636 −0.558179 0.829721i \(-0.688500\pi\)
−0.558179 + 0.829721i \(0.688500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.25499 0.145348 0.0726739 0.997356i \(-0.476847\pi\)
0.0726739 + 0.997356i \(0.476847\pi\)
\(858\) 0 0
\(859\) 24.5164 0.836487 0.418244 0.908335i \(-0.362646\pi\)
0.418244 + 0.908335i \(0.362646\pi\)
\(860\) 0 0
\(861\) 11.2211 0.382414
\(862\) 0 0
\(863\) 4.11643 0.140125 0.0700624 0.997543i \(-0.477680\pi\)
0.0700624 + 0.997543i \(0.477680\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.6534 0.735387
\(868\) 0 0
\(869\) 10.9799 0.372467
\(870\) 0 0
\(871\) −52.1244 −1.76617
\(872\) 0 0
\(873\) 6.44384 0.218091
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.17301 0.107145 0.0535725 0.998564i \(-0.482939\pi\)
0.0535725 + 0.998564i \(0.482939\pi\)
\(878\) 0 0
\(879\) −74.6180 −2.51680
\(880\) 0 0
\(881\) 16.0391 0.540371 0.270186 0.962808i \(-0.412915\pi\)
0.270186 + 0.962808i \(0.412915\pi\)
\(882\) 0 0
\(883\) 36.5648 1.23050 0.615252 0.788330i \(-0.289054\pi\)
0.615252 + 0.788330i \(0.289054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44.6498 1.49919 0.749596 0.661896i \(-0.230248\pi\)
0.749596 + 0.661896i \(0.230248\pi\)
\(888\) 0 0
\(889\) 66.1755 2.21945
\(890\) 0 0
\(891\) 28.3686 0.950382
\(892\) 0 0
\(893\) −24.3196 −0.813825
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −16.3477 −0.545833
\(898\) 0 0
\(899\) 7.76319 0.258917
\(900\) 0 0
\(901\) −29.8908 −0.995807
\(902\) 0 0
\(903\) −85.5542 −2.84707
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 43.2692 1.43673 0.718365 0.695667i \(-0.244891\pi\)
0.718365 + 0.695667i \(0.244891\pi\)
\(908\) 0 0
\(909\) −8.10723 −0.268900
\(910\) 0 0
\(911\) −1.64116 −0.0543739 −0.0271870 0.999630i \(-0.508655\pi\)
−0.0271870 + 0.999630i \(0.508655\pi\)
\(912\) 0 0
\(913\) 27.0311 0.894598
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 42.7014 1.41012
\(918\) 0 0
\(919\) 6.67802 0.220288 0.110144 0.993916i \(-0.464869\pi\)
0.110144 + 0.993916i \(0.464869\pi\)
\(920\) 0 0
\(921\) −28.2548 −0.931027
\(922\) 0 0
\(923\) −62.2074 −2.04758
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.13603 −0.0701564
\(928\) 0 0
\(929\) −23.3505 −0.766104 −0.383052 0.923727i \(-0.625127\pi\)
−0.383052 + 0.923727i \(0.625127\pi\)
\(930\) 0 0
\(931\) 88.0508 2.88575
\(932\) 0 0
\(933\) −4.83433 −0.158269
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −45.9295 −1.50045 −0.750225 0.661182i \(-0.770055\pi\)
−0.750225 + 0.661182i \(0.770055\pi\)
\(938\) 0 0
\(939\) −64.9238 −2.11871
\(940\) 0 0
\(941\) −13.2488 −0.431899 −0.215949 0.976405i \(-0.569285\pi\)
−0.215949 + 0.976405i \(0.569285\pi\)
\(942\) 0 0
\(943\) −1.49482 −0.0486781
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.1017 0.718207 0.359104 0.933298i \(-0.383082\pi\)
0.359104 + 0.933298i \(0.383082\pi\)
\(948\) 0 0
\(949\) −40.8959 −1.32754
\(950\) 0 0
\(951\) −63.2552 −2.05119
\(952\) 0 0
\(953\) 18.2642 0.591635 0.295818 0.955244i \(-0.404408\pi\)
0.295818 + 0.955244i \(0.404408\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 13.7860 0.445637
\(958\) 0 0
\(959\) 83.8760 2.70850
\(960\) 0 0
\(961\) −20.8023 −0.671042
\(962\) 0 0
\(963\) 0.614817 0.0198122
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.0843 0.903132 0.451566 0.892238i \(-0.350866\pi\)
0.451566 + 0.892238i \(0.350866\pi\)
\(968\) 0 0
\(969\) 28.1152 0.903192
\(970\) 0 0
\(971\) −34.1505 −1.09594 −0.547972 0.836497i \(-0.684600\pi\)
−0.547972 + 0.836497i \(0.684600\pi\)
\(972\) 0 0
\(973\) −11.0566 −0.354458
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −41.4597 −1.32641 −0.663206 0.748437i \(-0.730805\pi\)
−0.663206 + 0.748437i \(0.730805\pi\)
\(978\) 0 0
\(979\) −23.9765 −0.766291
\(980\) 0 0
\(981\) 6.17898 0.197280
\(982\) 0 0
\(983\) 34.1240 1.08839 0.544193 0.838960i \(-0.316836\pi\)
0.544193 + 0.838960i \(0.316836\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 57.8078 1.84004
\(988\) 0 0
\(989\) 11.3971 0.362408
\(990\) 0 0
\(991\) −21.3535 −0.678315 −0.339158 0.940730i \(-0.610142\pi\)
−0.339158 + 0.940730i \(0.610142\pi\)
\(992\) 0 0
\(993\) 10.6513 0.338009
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.3560 0.613012 0.306506 0.951869i \(-0.400840\pi\)
0.306506 + 0.951869i \(0.400840\pi\)
\(998\) 0 0
\(999\) −7.03348 −0.222529
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 4100.2.a.c.1.2 4
5.2 odd 4 4100.2.d.c.1149.6 8
5.3 odd 4 4100.2.d.c.1149.3 8
5.4 even 2 164.2.a.a.1.3 4
15.14 odd 2 1476.2.a.g.1.1 4
20.19 odd 2 656.2.a.i.1.2 4
35.34 odd 2 8036.2.a.i.1.2 4
40.19 odd 2 2624.2.a.y.1.3 4
40.29 even 2 2624.2.a.v.1.2 4
60.59 even 2 5904.2.a.bp.1.1 4
205.204 even 2 6724.2.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.3 4 5.4 even 2
656.2.a.i.1.2 4 20.19 odd 2
1476.2.a.g.1.1 4 15.14 odd 2
2624.2.a.v.1.2 4 40.29 even 2
2624.2.a.y.1.3 4 40.19 odd 2
4100.2.a.c.1.2 4 1.1 even 1 trivial
4100.2.d.c.1149.3 8 5.3 odd 4
4100.2.d.c.1149.6 8 5.2 odd 4
5904.2.a.bp.1.1 4 60.59 even 2
6724.2.a.c.1.2 4 205.204 even 2
8036.2.a.i.1.2 4 35.34 odd 2