Properties

Label 4275.2.a.z.1.3
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
Dimension $3$
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] = [4275,2,Mod(1,4275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4275, 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("4275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.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: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +O(q^{10})\) \(q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +0.911854 q^{11} +1.55496 q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590 q^{17} -1.00000 q^{19} +0.225209 q^{22} -4.24698 q^{23} +0.384043 q^{26} -3.28083 q^{28} -5.00969 q^{29} +1.82908 q^{31} +2.84415 q^{32} -1.30798 q^{34} +6.29590 q^{37} -0.246980 q^{38} -4.18060 q^{41} +7.31767 q^{43} -1.76809 q^{44} -1.04892 q^{46} +2.04892 q^{47} -4.13706 q^{49} -3.01507 q^{52} +2.70171 q^{53} -1.64609 q^{56} -1.23729 q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747 q^{62} -6.57301 q^{64} -13.9976 q^{67} +10.2687 q^{68} +12.8780 q^{71} -2.80731 q^{73} +1.55496 q^{74} +1.93900 q^{76} +1.54288 q^{77} +1.59419 q^{79} -1.03252 q^{82} -12.2349 q^{83} +1.80731 q^{86} -0.887100 q^{88} -2.91723 q^{89} +2.63102 q^{91} +8.23490 q^{92} +0.506041 q^{94} +1.55496 q^{97} -1.02177 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8} - q^{11} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 3 q^{19} - q^{22} - 8 q^{23} - 9 q^{26} - 21 q^{28} + 7 q^{29} - 5 q^{31} - 27 q^{32} - 9 q^{34} + 5 q^{37} + 4 q^{38} - q^{41} + 5 q^{43} + 15 q^{44} + 6 q^{46} - 3 q^{47} - 7 q^{49} + 16 q^{52} - 19 q^{53} + 35 q^{56} - 21 q^{58} - 10 q^{59} - 17 q^{61} + 23 q^{62} + 49 q^{64} - q^{67} + 23 q^{68} + 19 q^{71} - q^{73} + 5 q^{74} - 4 q^{76} - 14 q^{77} + 18 q^{79} + 6 q^{82} - 13 q^{83} - 2 q^{86} - 46 q^{88} - 2 q^{89} - 7 q^{91} + q^{92} + 11 q^{94} + 5 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.246980 0.174641 0.0873205 0.996180i \(-0.472170\pi\)
0.0873205 + 0.996180i \(0.472170\pi\)
\(3\) 0 0
\(4\) −1.93900 −0.969501
\(5\) 0 0
\(6\) 0 0
\(7\) 1.69202 0.639524 0.319762 0.947498i \(-0.396397\pi\)
0.319762 + 0.947498i \(0.396397\pi\)
\(8\) −0.972853 −0.343955
\(9\) 0 0
\(10\) 0 0
\(11\) 0.911854 0.274934 0.137467 0.990506i \(-0.456104\pi\)
0.137467 + 0.990506i \(0.456104\pi\)
\(12\) 0 0
\(13\) 1.55496 0.431268 0.215634 0.976474i \(-0.430818\pi\)
0.215634 + 0.976474i \(0.430818\pi\)
\(14\) 0.417895 0.111687
\(15\) 0 0
\(16\) 3.63773 0.909432
\(17\) −5.29590 −1.28444 −0.642222 0.766519i \(-0.721987\pi\)
−0.642222 + 0.766519i \(0.721987\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0.225209 0.0480148
\(23\) −4.24698 −0.885556 −0.442778 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.384043 0.0753170
\(27\) 0 0
\(28\) −3.28083 −0.620019
\(29\) −5.00969 −0.930276 −0.465138 0.885238i \(-0.653995\pi\)
−0.465138 + 0.885238i \(0.653995\pi\)
\(30\) 0 0
\(31\) 1.82908 0.328513 0.164257 0.986418i \(-0.447477\pi\)
0.164257 + 0.986418i \(0.447477\pi\)
\(32\) 2.84415 0.502779
\(33\) 0 0
\(34\) −1.30798 −0.224316
\(35\) 0 0
\(36\) 0 0
\(37\) 6.29590 1.03504 0.517520 0.855671i \(-0.326855\pi\)
0.517520 + 0.855671i \(0.326855\pi\)
\(38\) −0.246980 −0.0400654
\(39\) 0 0
\(40\) 0 0
\(41\) −4.18060 −0.652901 −0.326450 0.945214i \(-0.605853\pi\)
−0.326450 + 0.945214i \(0.605853\pi\)
\(42\) 0 0
\(43\) 7.31767 1.11593 0.557967 0.829863i \(-0.311582\pi\)
0.557967 + 0.829863i \(0.311582\pi\)
\(44\) −1.76809 −0.266549
\(45\) 0 0
\(46\) −1.04892 −0.154654
\(47\) 2.04892 0.298865 0.149433 0.988772i \(-0.452255\pi\)
0.149433 + 0.988772i \(0.452255\pi\)
\(48\) 0 0
\(49\) −4.13706 −0.591009
\(50\) 0 0
\(51\) 0 0
\(52\) −3.01507 −0.418114
\(53\) 2.70171 0.371108 0.185554 0.982634i \(-0.440592\pi\)
0.185554 + 0.982634i \(0.440592\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.64609 −0.219968
\(57\) 0 0
\(58\) −1.23729 −0.162464
\(59\) −9.87800 −1.28601 −0.643003 0.765864i \(-0.722312\pi\)
−0.643003 + 0.765864i \(0.722312\pi\)
\(60\) 0 0
\(61\) 0.542877 0.0695082 0.0347541 0.999396i \(-0.488935\pi\)
0.0347541 + 0.999396i \(0.488935\pi\)
\(62\) 0.451747 0.0573719
\(63\) 0 0
\(64\) −6.57301 −0.821626
\(65\) 0 0
\(66\) 0 0
\(67\) −13.9976 −1.71008 −0.855040 0.518562i \(-0.826467\pi\)
−0.855040 + 0.518562i \(0.826467\pi\)
\(68\) 10.2687 1.24527
\(69\) 0 0
\(70\) 0 0
\(71\) 12.8780 1.52834 0.764169 0.645016i \(-0.223149\pi\)
0.764169 + 0.645016i \(0.223149\pi\)
\(72\) 0 0
\(73\) −2.80731 −0.328571 −0.164286 0.986413i \(-0.552532\pi\)
−0.164286 + 0.986413i \(0.552532\pi\)
\(74\) 1.55496 0.180760
\(75\) 0 0
\(76\) 1.93900 0.222419
\(77\) 1.54288 0.175827
\(78\) 0 0
\(79\) 1.59419 0.179360 0.0896800 0.995971i \(-0.471416\pi\)
0.0896800 + 0.995971i \(0.471416\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.03252 −0.114023
\(83\) −12.2349 −1.34295 −0.671477 0.741025i \(-0.734340\pi\)
−0.671477 + 0.741025i \(0.734340\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.80731 0.194888
\(87\) 0 0
\(88\) −0.887100 −0.0945652
\(89\) −2.91723 −0.309226 −0.154613 0.987975i \(-0.549413\pi\)
−0.154613 + 0.987975i \(0.549413\pi\)
\(90\) 0 0
\(91\) 2.63102 0.275806
\(92\) 8.23490 0.858547
\(93\) 0 0
\(94\) 0.506041 0.0521941
\(95\) 0 0
\(96\) 0 0
\(97\) 1.55496 0.157882 0.0789410 0.996879i \(-0.474846\pi\)
0.0789410 + 0.996879i \(0.474846\pi\)
\(98\) −1.02177 −0.103214
\(99\) 0 0
\(100\) 0 0
\(101\) 16.6015 1.65191 0.825955 0.563737i \(-0.190637\pi\)
0.825955 + 0.563737i \(0.190637\pi\)
\(102\) 0 0
\(103\) −4.84548 −0.477439 −0.238720 0.971089i \(-0.576728\pi\)
−0.238720 + 0.971089i \(0.576728\pi\)
\(104\) −1.51275 −0.148337
\(105\) 0 0
\(106\) 0.667267 0.0648107
\(107\) 4.46681 0.431823 0.215912 0.976413i \(-0.430728\pi\)
0.215912 + 0.976413i \(0.430728\pi\)
\(108\) 0 0
\(109\) 18.8267 1.80327 0.901635 0.432498i \(-0.142368\pi\)
0.901635 + 0.432498i \(0.142368\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.15511 0.581603
\(113\) −20.0368 −1.88491 −0.942453 0.334337i \(-0.891488\pi\)
−0.942453 + 0.334337i \(0.891488\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.71379 0.901903
\(117\) 0 0
\(118\) −2.43967 −0.224589
\(119\) −8.96077 −0.821433
\(120\) 0 0
\(121\) −10.1685 −0.924411
\(122\) 0.134079 0.0121390
\(123\) 0 0
\(124\) −3.54660 −0.318494
\(125\) 0 0
\(126\) 0 0
\(127\) −17.8702 −1.58573 −0.792863 0.609399i \(-0.791411\pi\)
−0.792863 + 0.609399i \(0.791411\pi\)
\(128\) −7.31170 −0.646269
\(129\) 0 0
\(130\) 0 0
\(131\) −7.44265 −0.650267 −0.325134 0.945668i \(-0.605409\pi\)
−0.325134 + 0.945668i \(0.605409\pi\)
\(132\) 0 0
\(133\) −1.69202 −0.146717
\(134\) −3.45712 −0.298650
\(135\) 0 0
\(136\) 5.15213 0.441791
\(137\) −5.68664 −0.485843 −0.242921 0.970046i \(-0.578106\pi\)
−0.242921 + 0.970046i \(0.578106\pi\)
\(138\) 0 0
\(139\) 3.61596 0.306701 0.153351 0.988172i \(-0.450994\pi\)
0.153351 + 0.988172i \(0.450994\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.18060 0.266910
\(143\) 1.41789 0.118570
\(144\) 0 0
\(145\) 0 0
\(146\) −0.693349 −0.0573820
\(147\) 0 0
\(148\) −12.2078 −1.00347
\(149\) −3.29052 −0.269570 −0.134785 0.990875i \(-0.543034\pi\)
−0.134785 + 0.990875i \(0.543034\pi\)
\(150\) 0 0
\(151\) −10.2131 −0.831133 −0.415566 0.909563i \(-0.636417\pi\)
−0.415566 + 0.909563i \(0.636417\pi\)
\(152\) 0.972853 0.0789088
\(153\) 0 0
\(154\) 0.381059 0.0307066
\(155\) 0 0
\(156\) 0 0
\(157\) 14.3448 1.14484 0.572420 0.819960i \(-0.306005\pi\)
0.572420 + 0.819960i \(0.306005\pi\)
\(158\) 0.393732 0.0313236
\(159\) 0 0
\(160\) 0 0
\(161\) −7.18598 −0.566335
\(162\) 0 0
\(163\) −19.5308 −1.52977 −0.764885 0.644167i \(-0.777204\pi\)
−0.764885 + 0.644167i \(0.777204\pi\)
\(164\) 8.10620 0.632988
\(165\) 0 0
\(166\) −3.02177 −0.234535
\(167\) −11.8823 −0.919481 −0.459741 0.888053i \(-0.652058\pi\)
−0.459741 + 0.888053i \(0.652058\pi\)
\(168\) 0 0
\(169\) −10.5821 −0.814008
\(170\) 0 0
\(171\) 0 0
\(172\) −14.1890 −1.08190
\(173\) −15.4722 −1.17633 −0.588164 0.808741i \(-0.700149\pi\)
−0.588164 + 0.808741i \(0.700149\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.31708 0.250034
\(177\) 0 0
\(178\) −0.720497 −0.0540035
\(179\) −2.16421 −0.161761 −0.0808803 0.996724i \(-0.525773\pi\)
−0.0808803 + 0.996724i \(0.525773\pi\)
\(180\) 0 0
\(181\) 16.8974 1.25597 0.627986 0.778225i \(-0.283879\pi\)
0.627986 + 0.778225i \(0.283879\pi\)
\(182\) 0.649809 0.0481670
\(183\) 0 0
\(184\) 4.13169 0.304592
\(185\) 0 0
\(186\) 0 0
\(187\) −4.82908 −0.353138
\(188\) −3.97285 −0.289750
\(189\) 0 0
\(190\) 0 0
\(191\) −5.92394 −0.428641 −0.214320 0.976763i \(-0.568754\pi\)
−0.214320 + 0.976763i \(0.568754\pi\)
\(192\) 0 0
\(193\) −4.43535 −0.319264 −0.159632 0.987177i \(-0.551031\pi\)
−0.159632 + 0.987177i \(0.551031\pi\)
\(194\) 0.384043 0.0275727
\(195\) 0 0
\(196\) 8.02177 0.572984
\(197\) −16.4722 −1.17359 −0.586797 0.809734i \(-0.699612\pi\)
−0.586797 + 0.809734i \(0.699612\pi\)
\(198\) 0 0
\(199\) −24.4131 −1.73060 −0.865300 0.501255i \(-0.832872\pi\)
−0.865300 + 0.501255i \(0.832872\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.10023 0.288491
\(203\) −8.47650 −0.594934
\(204\) 0 0
\(205\) 0 0
\(206\) −1.19673 −0.0833804
\(207\) 0 0
\(208\) 5.65651 0.392209
\(209\) −0.911854 −0.0630743
\(210\) 0 0
\(211\) −4.34050 −0.298813 −0.149406 0.988776i \(-0.547736\pi\)
−0.149406 + 0.988776i \(0.547736\pi\)
\(212\) −5.23862 −0.359790
\(213\) 0 0
\(214\) 1.10321 0.0754140
\(215\) 0 0
\(216\) 0 0
\(217\) 3.09485 0.210092
\(218\) 4.64981 0.314925
\(219\) 0 0
\(220\) 0 0
\(221\) −8.23490 −0.553939
\(222\) 0 0
\(223\) 26.2379 1.75702 0.878509 0.477725i \(-0.158539\pi\)
0.878509 + 0.477725i \(0.158539\pi\)
\(224\) 4.81236 0.321540
\(225\) 0 0
\(226\) −4.94869 −0.329182
\(227\) −14.6853 −0.974699 −0.487349 0.873207i \(-0.662036\pi\)
−0.487349 + 0.873207i \(0.662036\pi\)
\(228\) 0 0
\(229\) −21.6407 −1.43006 −0.715029 0.699095i \(-0.753587\pi\)
−0.715029 + 0.699095i \(0.753587\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.87369 0.319973
\(233\) −27.1183 −1.77658 −0.888289 0.459286i \(-0.848105\pi\)
−0.888289 + 0.459286i \(0.848105\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 19.1535 1.24678
\(237\) 0 0
\(238\) −2.21313 −0.143456
\(239\) 11.5308 0.745865 0.372933 0.927858i \(-0.378352\pi\)
0.372933 + 0.927858i \(0.378352\pi\)
\(240\) 0 0
\(241\) 11.8194 0.761354 0.380677 0.924708i \(-0.375691\pi\)
0.380677 + 0.924708i \(0.375691\pi\)
\(242\) −2.51142 −0.161440
\(243\) 0 0
\(244\) −1.05264 −0.0673883
\(245\) 0 0
\(246\) 0 0
\(247\) −1.55496 −0.0989396
\(248\) −1.77943 −0.112994
\(249\) 0 0
\(250\) 0 0
\(251\) 9.66487 0.610041 0.305021 0.952346i \(-0.401337\pi\)
0.305021 + 0.952346i \(0.401337\pi\)
\(252\) 0 0
\(253\) −3.87263 −0.243470
\(254\) −4.41358 −0.276933
\(255\) 0 0
\(256\) 11.3402 0.708761
\(257\) −14.1860 −0.884897 −0.442449 0.896794i \(-0.645890\pi\)
−0.442449 + 0.896794i \(0.645890\pi\)
\(258\) 0 0
\(259\) 10.6528 0.661932
\(260\) 0 0
\(261\) 0 0
\(262\) −1.83818 −0.113563
\(263\) 21.7942 1.34389 0.671943 0.740603i \(-0.265460\pi\)
0.671943 + 0.740603i \(0.265460\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.417895 −0.0256228
\(267\) 0 0
\(268\) 27.1414 1.65792
\(269\) 24.7265 1.50760 0.753800 0.657104i \(-0.228219\pi\)
0.753800 + 0.657104i \(0.228219\pi\)
\(270\) 0 0
\(271\) −13.2295 −0.803636 −0.401818 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(272\) −19.2650 −1.16811
\(273\) 0 0
\(274\) −1.40449 −0.0848481
\(275\) 0 0
\(276\) 0 0
\(277\) −0.560335 −0.0336673 −0.0168336 0.999858i \(-0.505359\pi\)
−0.0168336 + 0.999858i \(0.505359\pi\)
\(278\) 0.893068 0.0535626
\(279\) 0 0
\(280\) 0 0
\(281\) −27.6039 −1.64671 −0.823355 0.567527i \(-0.807900\pi\)
−0.823355 + 0.567527i \(0.807900\pi\)
\(282\) 0 0
\(283\) −15.9608 −0.948769 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(284\) −24.9705 −1.48172
\(285\) 0 0
\(286\) 0.350191 0.0207072
\(287\) −7.07367 −0.417546
\(288\) 0 0
\(289\) 11.0465 0.649796
\(290\) 0 0
\(291\) 0 0
\(292\) 5.44339 0.318550
\(293\) −25.3327 −1.47995 −0.739977 0.672632i \(-0.765164\pi\)
−0.739977 + 0.672632i \(0.765164\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.12498 −0.356007
\(297\) 0 0
\(298\) −0.812691 −0.0470779
\(299\) −6.60388 −0.381912
\(300\) 0 0
\(301\) 12.3817 0.713666
\(302\) −2.52243 −0.145150
\(303\) 0 0
\(304\) −3.63773 −0.208638
\(305\) 0 0
\(306\) 0 0
\(307\) −11.5574 −0.659613 −0.329806 0.944049i \(-0.606983\pi\)
−0.329806 + 0.944049i \(0.606983\pi\)
\(308\) −2.99164 −0.170464
\(309\) 0 0
\(310\) 0 0
\(311\) 18.3424 1.04010 0.520052 0.854135i \(-0.325913\pi\)
0.520052 + 0.854135i \(0.325913\pi\)
\(312\) 0 0
\(313\) −18.9119 −1.06896 −0.534481 0.845181i \(-0.679493\pi\)
−0.534481 + 0.845181i \(0.679493\pi\)
\(314\) 3.54288 0.199936
\(315\) 0 0
\(316\) −3.09113 −0.173890
\(317\) −20.2784 −1.13895 −0.569475 0.822008i \(-0.692854\pi\)
−0.569475 + 0.822008i \(0.692854\pi\)
\(318\) 0 0
\(319\) −4.56810 −0.255765
\(320\) 0 0
\(321\) 0 0
\(322\) −1.77479 −0.0989052
\(323\) 5.29590 0.294672
\(324\) 0 0
\(325\) 0 0
\(326\) −4.82371 −0.267160
\(327\) 0 0
\(328\) 4.06711 0.224569
\(329\) 3.46681 0.191132
\(330\) 0 0
\(331\) −4.77479 −0.262446 −0.131223 0.991353i \(-0.541890\pi\)
−0.131223 + 0.991353i \(0.541890\pi\)
\(332\) 23.7235 1.30200
\(333\) 0 0
\(334\) −2.93469 −0.160579
\(335\) 0 0
\(336\) 0 0
\(337\) 24.3967 1.32897 0.664487 0.747300i \(-0.268650\pi\)
0.664487 + 0.747300i \(0.268650\pi\)
\(338\) −2.61356 −0.142159
\(339\) 0 0
\(340\) 0 0
\(341\) 1.66786 0.0903196
\(342\) 0 0
\(343\) −18.8442 −1.01749
\(344\) −7.11901 −0.383832
\(345\) 0 0
\(346\) −3.82132 −0.205435
\(347\) 9.99761 0.536700 0.268350 0.963322i \(-0.413522\pi\)
0.268350 + 0.963322i \(0.413522\pi\)
\(348\) 0 0
\(349\) 21.9584 1.17541 0.587703 0.809077i \(-0.300033\pi\)
0.587703 + 0.809077i \(0.300033\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.59345 0.138231
\(353\) 36.8786 1.96285 0.981425 0.191848i \(-0.0614479\pi\)
0.981425 + 0.191848i \(0.0614479\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.65651 0.299795
\(357\) 0 0
\(358\) −0.534516 −0.0282500
\(359\) −16.5187 −0.871824 −0.435912 0.899989i \(-0.643574\pi\)
−0.435912 + 0.899989i \(0.643574\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.17331 0.219344
\(363\) 0 0
\(364\) −5.10156 −0.267394
\(365\) 0 0
\(366\) 0 0
\(367\) −6.83148 −0.356600 −0.178300 0.983976i \(-0.557060\pi\)
−0.178300 + 0.983976i \(0.557060\pi\)
\(368\) −15.4494 −0.805353
\(369\) 0 0
\(370\) 0 0
\(371\) 4.57135 0.237333
\(372\) 0 0
\(373\) −4.76271 −0.246604 −0.123302 0.992369i \(-0.539348\pi\)
−0.123302 + 0.992369i \(0.539348\pi\)
\(374\) −1.19269 −0.0616723
\(375\) 0 0
\(376\) −1.99330 −0.102796
\(377\) −7.78986 −0.401198
\(378\) 0 0
\(379\) 14.3773 0.738514 0.369257 0.929327i \(-0.379612\pi\)
0.369257 + 0.929327i \(0.379612\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.46309 −0.0748583
\(383\) 30.6708 1.56721 0.783603 0.621261i \(-0.213379\pi\)
0.783603 + 0.621261i \(0.213379\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.09544 −0.0557565
\(387\) 0 0
\(388\) −3.01507 −0.153067
\(389\) 12.9215 0.655148 0.327574 0.944825i \(-0.393769\pi\)
0.327574 + 0.944825i \(0.393769\pi\)
\(390\) 0 0
\(391\) 22.4916 1.13745
\(392\) 4.02475 0.203281
\(393\) 0 0
\(394\) −4.06829 −0.204958
\(395\) 0 0
\(396\) 0 0
\(397\) −29.8471 −1.49798 −0.748992 0.662579i \(-0.769462\pi\)
−0.748992 + 0.662579i \(0.769462\pi\)
\(398\) −6.02954 −0.302234
\(399\) 0 0
\(400\) 0 0
\(401\) 28.7101 1.43371 0.716856 0.697221i \(-0.245580\pi\)
0.716856 + 0.697221i \(0.245580\pi\)
\(402\) 0 0
\(403\) 2.84415 0.141677
\(404\) −32.1903 −1.60153
\(405\) 0 0
\(406\) −2.09352 −0.103900
\(407\) 5.74094 0.284568
\(408\) 0 0
\(409\) −13.6203 −0.673479 −0.336739 0.941598i \(-0.609324\pi\)
−0.336739 + 0.941598i \(0.609324\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9.39539 0.462878
\(413\) −16.7138 −0.822432
\(414\) 0 0
\(415\) 0 0
\(416\) 4.42253 0.216833
\(417\) 0 0
\(418\) −0.225209 −0.0110153
\(419\) 2.13946 0.104519 0.0522596 0.998634i \(-0.483358\pi\)
0.0522596 + 0.998634i \(0.483358\pi\)
\(420\) 0 0
\(421\) −15.0562 −0.733795 −0.366897 0.930261i \(-0.619580\pi\)
−0.366897 + 0.930261i \(0.619580\pi\)
\(422\) −1.07202 −0.0521849
\(423\) 0 0
\(424\) −2.62837 −0.127645
\(425\) 0 0
\(426\) 0 0
\(427\) 0.918559 0.0444522
\(428\) −8.66115 −0.418653
\(429\) 0 0
\(430\) 0 0
\(431\) −4.37435 −0.210705 −0.105353 0.994435i \(-0.533597\pi\)
−0.105353 + 0.994435i \(0.533597\pi\)
\(432\) 0 0
\(433\) −33.1564 −1.59340 −0.796698 0.604377i \(-0.793422\pi\)
−0.796698 + 0.604377i \(0.793422\pi\)
\(434\) 0.764365 0.0366907
\(435\) 0 0
\(436\) −36.5050 −1.74827
\(437\) 4.24698 0.203161
\(438\) 0 0
\(439\) 32.2368 1.53858 0.769290 0.638900i \(-0.220610\pi\)
0.769290 + 0.638900i \(0.220610\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.03385 −0.0967405
\(443\) −21.7362 −1.03272 −0.516358 0.856373i \(-0.672713\pi\)
−0.516358 + 0.856373i \(0.672713\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6.48022 0.306847
\(447\) 0 0
\(448\) −11.1217 −0.525450
\(449\) 24.9584 1.17786 0.588929 0.808185i \(-0.299550\pi\)
0.588929 + 0.808185i \(0.299550\pi\)
\(450\) 0 0
\(451\) −3.81210 −0.179505
\(452\) 38.8514 1.82742
\(453\) 0 0
\(454\) −3.62697 −0.170222
\(455\) 0 0
\(456\) 0 0
\(457\) 13.1347 0.614414 0.307207 0.951643i \(-0.400606\pi\)
0.307207 + 0.951643i \(0.400606\pi\)
\(458\) −5.34481 −0.249747
\(459\) 0 0
\(460\) 0 0
\(461\) 35.3726 1.64746 0.823732 0.566979i \(-0.191888\pi\)
0.823732 + 0.566979i \(0.191888\pi\)
\(462\) 0 0
\(463\) 14.6963 0.682997 0.341498 0.939882i \(-0.389066\pi\)
0.341498 + 0.939882i \(0.389066\pi\)
\(464\) −18.2239 −0.846022
\(465\) 0 0
\(466\) −6.69766 −0.310263
\(467\) 33.2121 1.53687 0.768435 0.639927i \(-0.221036\pi\)
0.768435 + 0.639927i \(0.221036\pi\)
\(468\) 0 0
\(469\) −23.6843 −1.09364
\(470\) 0 0
\(471\) 0 0
\(472\) 9.60984 0.442329
\(473\) 6.67264 0.306809
\(474\) 0 0
\(475\) 0 0
\(476\) 17.3749 0.796379
\(477\) 0 0
\(478\) 2.84787 0.130259
\(479\) −24.6219 −1.12500 −0.562502 0.826796i \(-0.690161\pi\)
−0.562502 + 0.826796i \(0.690161\pi\)
\(480\) 0 0
\(481\) 9.78986 0.446379
\(482\) 2.91915 0.132964
\(483\) 0 0
\(484\) 19.7168 0.896217
\(485\) 0 0
\(486\) 0 0
\(487\) −29.5646 −1.33970 −0.669851 0.742496i \(-0.733642\pi\)
−0.669851 + 0.742496i \(0.733642\pi\)
\(488\) −0.528139 −0.0239077
\(489\) 0 0
\(490\) 0 0
\(491\) −36.2978 −1.63810 −0.819049 0.573724i \(-0.805498\pi\)
−0.819049 + 0.573724i \(0.805498\pi\)
\(492\) 0 0
\(493\) 26.5308 1.19489
\(494\) −0.384043 −0.0172789
\(495\) 0 0
\(496\) 6.65371 0.298760
\(497\) 21.7899 0.977409
\(498\) 0 0
\(499\) 7.84415 0.351152 0.175576 0.984466i \(-0.443821\pi\)
0.175576 + 0.984466i \(0.443821\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.38703 0.106538
\(503\) 20.4166 0.910330 0.455165 0.890407i \(-0.349580\pi\)
0.455165 + 0.890407i \(0.349580\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.956459 −0.0425198
\(507\) 0 0
\(508\) 34.6504 1.53736
\(509\) 14.7530 0.653916 0.326958 0.945039i \(-0.393976\pi\)
0.326958 + 0.945039i \(0.393976\pi\)
\(510\) 0 0
\(511\) −4.75004 −0.210129
\(512\) 17.4242 0.770048
\(513\) 0 0
\(514\) −3.50365 −0.154539
\(515\) 0 0
\(516\) 0 0
\(517\) 1.86831 0.0821683
\(518\) 2.63102 0.115600
\(519\) 0 0
\(520\) 0 0
\(521\) −37.1487 −1.62751 −0.813756 0.581206i \(-0.802581\pi\)
−0.813756 + 0.581206i \(0.802581\pi\)
\(522\) 0 0
\(523\) −16.3623 −0.715472 −0.357736 0.933823i \(-0.616451\pi\)
−0.357736 + 0.933823i \(0.616451\pi\)
\(524\) 14.4313 0.630434
\(525\) 0 0
\(526\) 5.38271 0.234698
\(527\) −9.68664 −0.421957
\(528\) 0 0
\(529\) −4.96316 −0.215790
\(530\) 0 0
\(531\) 0 0
\(532\) 3.28083 0.142242
\(533\) −6.50066 −0.281575
\(534\) 0 0
\(535\) 0 0
\(536\) 13.6176 0.588191
\(537\) 0 0
\(538\) 6.10693 0.263289
\(539\) −3.77240 −0.162489
\(540\) 0 0
\(541\) −2.08947 −0.0898335 −0.0449168 0.998991i \(-0.514302\pi\)
−0.0449168 + 0.998991i \(0.514302\pi\)
\(542\) −3.26742 −0.140348
\(543\) 0 0
\(544\) −15.0623 −0.645792
\(545\) 0 0
\(546\) 0 0
\(547\) −1.86054 −0.0795511 −0.0397756 0.999209i \(-0.512664\pi\)
−0.0397756 + 0.999209i \(0.512664\pi\)
\(548\) 11.0264 0.471025
\(549\) 0 0
\(550\) 0 0
\(551\) 5.00969 0.213420
\(552\) 0 0
\(553\) 2.69740 0.114705
\(554\) −0.138391 −0.00587968
\(555\) 0 0
\(556\) −7.01134 −0.297347
\(557\) −9.80061 −0.415265 −0.207633 0.978207i \(-0.566576\pi\)
−0.207633 + 0.978207i \(0.566576\pi\)
\(558\) 0 0
\(559\) 11.3787 0.481266
\(560\) 0 0
\(561\) 0 0
\(562\) −6.81759 −0.287583
\(563\) 38.0170 1.60222 0.801112 0.598514i \(-0.204242\pi\)
0.801112 + 0.598514i \(0.204242\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3.94198 −0.165694
\(567\) 0 0
\(568\) −12.5284 −0.525680
\(569\) 7.32975 0.307279 0.153640 0.988127i \(-0.450901\pi\)
0.153640 + 0.988127i \(0.450901\pi\)
\(570\) 0 0
\(571\) 37.8775 1.58513 0.792563 0.609791i \(-0.208746\pi\)
0.792563 + 0.609791i \(0.208746\pi\)
\(572\) −2.74930 −0.114954
\(573\) 0 0
\(574\) −1.74705 −0.0729206
\(575\) 0 0
\(576\) 0 0
\(577\) −28.6993 −1.19477 −0.597384 0.801955i \(-0.703793\pi\)
−0.597384 + 0.801955i \(0.703793\pi\)
\(578\) 2.72827 0.113481
\(579\) 0 0
\(580\) 0 0
\(581\) −20.7017 −0.858852
\(582\) 0 0
\(583\) 2.46357 0.102030
\(584\) 2.73110 0.113014
\(585\) 0 0
\(586\) −6.25667 −0.258461
\(587\) 3.72348 0.153684 0.0768422 0.997043i \(-0.475516\pi\)
0.0768422 + 0.997043i \(0.475516\pi\)
\(588\) 0 0
\(589\) −1.82908 −0.0753661
\(590\) 0 0
\(591\) 0 0
\(592\) 22.9028 0.941297
\(593\) 27.7399 1.13914 0.569570 0.821943i \(-0.307110\pi\)
0.569570 + 0.821943i \(0.307110\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.38032 0.261348
\(597\) 0 0
\(598\) −1.63102 −0.0666975
\(599\) 23.5579 0.962551 0.481276 0.876569i \(-0.340174\pi\)
0.481276 + 0.876569i \(0.340174\pi\)
\(600\) 0 0
\(601\) −7.36898 −0.300587 −0.150293 0.988641i \(-0.548022\pi\)
−0.150293 + 0.988641i \(0.548022\pi\)
\(602\) 3.05802 0.124635
\(603\) 0 0
\(604\) 19.8033 0.805783
\(605\) 0 0
\(606\) 0 0
\(607\) 28.4198 1.15352 0.576762 0.816912i \(-0.304316\pi\)
0.576762 + 0.816912i \(0.304316\pi\)
\(608\) −2.84415 −0.115346
\(609\) 0 0
\(610\) 0 0
\(611\) 3.18598 0.128891
\(612\) 0 0
\(613\) 6.48129 0.261777 0.130888 0.991397i \(-0.458217\pi\)
0.130888 + 0.991397i \(0.458217\pi\)
\(614\) −2.85443 −0.115195
\(615\) 0 0
\(616\) −1.50099 −0.0604767
\(617\) 24.4650 0.984924 0.492462 0.870334i \(-0.336097\pi\)
0.492462 + 0.870334i \(0.336097\pi\)
\(618\) 0 0
\(619\) −22.2457 −0.894128 −0.447064 0.894502i \(-0.647530\pi\)
−0.447064 + 0.894502i \(0.647530\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 4.53020 0.181645
\(623\) −4.93602 −0.197757
\(624\) 0 0
\(625\) 0 0
\(626\) −4.67084 −0.186684
\(627\) 0 0
\(628\) −27.8146 −1.10992
\(629\) −33.3424 −1.32945
\(630\) 0 0
\(631\) −15.5888 −0.620581 −0.310290 0.950642i \(-0.600426\pi\)
−0.310290 + 0.950642i \(0.600426\pi\)
\(632\) −1.55091 −0.0616919
\(633\) 0 0
\(634\) −5.00836 −0.198907
\(635\) 0 0
\(636\) 0 0
\(637\) −6.43296 −0.254883
\(638\) −1.12823 −0.0446670
\(639\) 0 0
\(640\) 0 0
\(641\) −8.17496 −0.322892 −0.161446 0.986882i \(-0.551616\pi\)
−0.161446 + 0.986882i \(0.551616\pi\)
\(642\) 0 0
\(643\) 11.1836 0.441038 0.220519 0.975383i \(-0.429225\pi\)
0.220519 + 0.975383i \(0.429225\pi\)
\(644\) 13.9336 0.549062
\(645\) 0 0
\(646\) 1.30798 0.0514617
\(647\) −16.2121 −0.637362 −0.318681 0.947862i \(-0.603240\pi\)
−0.318681 + 0.947862i \(0.603240\pi\)
\(648\) 0 0
\(649\) −9.00730 −0.353567
\(650\) 0 0
\(651\) 0 0
\(652\) 37.8702 1.48311
\(653\) −24.7952 −0.970312 −0.485156 0.874427i \(-0.661237\pi\)
−0.485156 + 0.874427i \(0.661237\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −15.2079 −0.593769
\(657\) 0 0
\(658\) 0.856232 0.0333794
\(659\) 20.2868 0.790262 0.395131 0.918625i \(-0.370699\pi\)
0.395131 + 0.918625i \(0.370699\pi\)
\(660\) 0 0
\(661\) 20.4222 0.794332 0.397166 0.917747i \(-0.369994\pi\)
0.397166 + 0.917747i \(0.369994\pi\)
\(662\) −1.17928 −0.0458339
\(663\) 0 0
\(664\) 11.9028 0.461917
\(665\) 0 0
\(666\) 0 0
\(667\) 21.2760 0.823812
\(668\) 23.0398 0.891437
\(669\) 0 0
\(670\) 0 0
\(671\) 0.495024 0.0191102
\(672\) 0 0
\(673\) 28.7254 1.10728 0.553641 0.832755i \(-0.313238\pi\)
0.553641 + 0.832755i \(0.313238\pi\)
\(674\) 6.02549 0.232093
\(675\) 0 0
\(676\) 20.5187 0.789181
\(677\) −44.0062 −1.69130 −0.845648 0.533740i \(-0.820786\pi\)
−0.845648 + 0.533740i \(0.820786\pi\)
\(678\) 0 0
\(679\) 2.63102 0.100969
\(680\) 0 0
\(681\) 0 0
\(682\) 0.411927 0.0157735
\(683\) −8.48427 −0.324642 −0.162321 0.986738i \(-0.551898\pi\)
−0.162321 + 0.986738i \(0.551898\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −4.65412 −0.177695
\(687\) 0 0
\(688\) 26.6197 1.01487
\(689\) 4.20105 0.160047
\(690\) 0 0
\(691\) 20.2586 0.770673 0.385336 0.922776i \(-0.374085\pi\)
0.385336 + 0.922776i \(0.374085\pi\)
\(692\) 30.0006 1.14045
\(693\) 0 0
\(694\) 2.46921 0.0937297
\(695\) 0 0
\(696\) 0 0
\(697\) 22.1400 0.838614
\(698\) 5.42327 0.205274
\(699\) 0 0
\(700\) 0 0
\(701\) 23.4101 0.884188 0.442094 0.896969i \(-0.354236\pi\)
0.442094 + 0.896969i \(0.354236\pi\)
\(702\) 0 0
\(703\) −6.29590 −0.237454
\(704\) −5.99362 −0.225893
\(705\) 0 0
\(706\) 9.10826 0.342794
\(707\) 28.0901 1.05644
\(708\) 0 0
\(709\) 30.3472 1.13971 0.569857 0.821744i \(-0.306999\pi\)
0.569857 + 0.821744i \(0.306999\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.83804 0.106360
\(713\) −7.76809 −0.290917
\(714\) 0 0
\(715\) 0 0
\(716\) 4.19641 0.156827
\(717\) 0 0
\(718\) −4.07979 −0.152256
\(719\) 38.3230 1.42921 0.714604 0.699529i \(-0.246607\pi\)
0.714604 + 0.699529i \(0.246607\pi\)
\(720\) 0 0
\(721\) −8.19865 −0.305334
\(722\) 0.246980 0.00919163
\(723\) 0 0
\(724\) −32.7640 −1.21767
\(725\) 0 0
\(726\) 0 0
\(727\) 2.69069 0.0997923 0.0498961 0.998754i \(-0.484111\pi\)
0.0498961 + 0.998754i \(0.484111\pi\)
\(728\) −2.55960 −0.0948650
\(729\) 0 0
\(730\) 0 0
\(731\) −38.7536 −1.43335
\(732\) 0 0
\(733\) 18.9651 0.700491 0.350246 0.936658i \(-0.386098\pi\)
0.350246 + 0.936658i \(0.386098\pi\)
\(734\) −1.68724 −0.0622770
\(735\) 0 0
\(736\) −12.0790 −0.445240
\(737\) −12.7638 −0.470160
\(738\) 0 0
\(739\) 29.8278 1.09723 0.548616 0.836075i \(-0.315155\pi\)
0.548616 + 0.836075i \(0.315155\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.12903 0.0414480
\(743\) −8.78448 −0.322271 −0.161136 0.986932i \(-0.551516\pi\)
−0.161136 + 0.986932i \(0.551516\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.17629 −0.0430671
\(747\) 0 0
\(748\) 9.36360 0.342367
\(749\) 7.55794 0.276161
\(750\) 0 0
\(751\) −18.1142 −0.660998 −0.330499 0.943806i \(-0.607217\pi\)
−0.330499 + 0.943806i \(0.607217\pi\)
\(752\) 7.45340 0.271798
\(753\) 0 0
\(754\) −1.92394 −0.0700656
\(755\) 0 0
\(756\) 0 0
\(757\) −0.222816 −0.00809840 −0.00404920 0.999992i \(-0.501289\pi\)
−0.00404920 + 0.999992i \(0.501289\pi\)
\(758\) 3.55091 0.128975
\(759\) 0 0
\(760\) 0 0
\(761\) 6.07798 0.220327 0.110163 0.993913i \(-0.464863\pi\)
0.110163 + 0.993913i \(0.464863\pi\)
\(762\) 0 0
\(763\) 31.8552 1.15323
\(764\) 11.4865 0.415568
\(765\) 0 0
\(766\) 7.57507 0.273698
\(767\) −15.3599 −0.554613
\(768\) 0 0
\(769\) −14.6267 −0.527453 −0.263726 0.964598i \(-0.584952\pi\)
−0.263726 + 0.964598i \(0.584952\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.60015 0.309526
\(773\) 4.05728 0.145930 0.0729651 0.997334i \(-0.476754\pi\)
0.0729651 + 0.997334i \(0.476754\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.51275 −0.0543044
\(777\) 0 0
\(778\) 3.19136 0.114416
\(779\) 4.18060 0.149786
\(780\) 0 0
\(781\) 11.7429 0.420192
\(782\) 5.55496 0.198645
\(783\) 0 0
\(784\) −15.0495 −0.537482
\(785\) 0 0
\(786\) 0 0
\(787\) 19.5657 0.697442 0.348721 0.937227i \(-0.386616\pi\)
0.348721 + 0.937227i \(0.386616\pi\)
\(788\) 31.9396 1.13780
\(789\) 0 0
\(790\) 0 0
\(791\) −33.9028 −1.20544
\(792\) 0 0
\(793\) 0.844150 0.0299767
\(794\) −7.37163 −0.261609
\(795\) 0 0
\(796\) 47.3370 1.67782
\(797\) −38.9051 −1.37809 −0.689046 0.724718i \(-0.741970\pi\)
−0.689046 + 0.724718i \(0.741970\pi\)
\(798\) 0 0
\(799\) −10.8509 −0.383876
\(800\) 0 0
\(801\) 0 0
\(802\) 7.09080 0.250385
\(803\) −2.55986 −0.0903355
\(804\) 0 0
\(805\) 0 0
\(806\) 0.702447 0.0247426
\(807\) 0 0
\(808\) −16.1508 −0.568183
\(809\) 22.5730 0.793625 0.396812 0.917900i \(-0.370116\pi\)
0.396812 + 0.917900i \(0.370116\pi\)
\(810\) 0 0
\(811\) −46.3605 −1.62794 −0.813968 0.580909i \(-0.802697\pi\)
−0.813968 + 0.580909i \(0.802697\pi\)
\(812\) 16.4359 0.576789
\(813\) 0 0
\(814\) 1.41789 0.0496972
\(815\) 0 0
\(816\) 0 0
\(817\) −7.31767 −0.256013
\(818\) −3.36393 −0.117617
\(819\) 0 0
\(820\) 0 0
\(821\) 17.8194 0.621901 0.310951 0.950426i \(-0.399353\pi\)
0.310951 + 0.950426i \(0.399353\pi\)
\(822\) 0 0
\(823\) −32.5394 −1.13425 −0.567126 0.823631i \(-0.691945\pi\)
−0.567126 + 0.823631i \(0.691945\pi\)
\(824\) 4.71394 0.164218
\(825\) 0 0
\(826\) −4.12797 −0.143630
\(827\) −39.8256 −1.38487 −0.692436 0.721479i \(-0.743463\pi\)
−0.692436 + 0.721479i \(0.743463\pi\)
\(828\) 0 0
\(829\) 38.7525 1.34593 0.672966 0.739674i \(-0.265020\pi\)
0.672966 + 0.739674i \(0.265020\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −10.2208 −0.354341
\(833\) 21.9095 0.759118
\(834\) 0 0
\(835\) 0 0
\(836\) 1.76809 0.0611505
\(837\) 0 0
\(838\) 0.528402 0.0182533
\(839\) −2.76749 −0.0955445 −0.0477723 0.998858i \(-0.515212\pi\)
−0.0477723 + 0.998858i \(0.515212\pi\)
\(840\) 0 0
\(841\) −3.90302 −0.134587
\(842\) −3.71858 −0.128151
\(843\) 0 0
\(844\) 8.41624 0.289699
\(845\) 0 0
\(846\) 0 0
\(847\) −17.2054 −0.591183
\(848\) 9.82808 0.337498
\(849\) 0 0
\(850\) 0 0
\(851\) −26.7385 −0.916586
\(852\) 0 0
\(853\) 24.1390 0.826503 0.413252 0.910617i \(-0.364393\pi\)
0.413252 + 0.910617i \(0.364393\pi\)
\(854\) 0.226865 0.00776317
\(855\) 0 0
\(856\) −4.34555 −0.148528
\(857\) 3.16229 0.108022 0.0540109 0.998540i \(-0.482799\pi\)
0.0540109 + 0.998540i \(0.482799\pi\)
\(858\) 0 0
\(859\) 4.86426 0.165967 0.0829833 0.996551i \(-0.473555\pi\)
0.0829833 + 0.996551i \(0.473555\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.08038 −0.0367978
\(863\) −4.54958 −0.154870 −0.0774348 0.996997i \(-0.524673\pi\)
−0.0774348 + 0.996997i \(0.524673\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −8.18896 −0.278272
\(867\) 0 0
\(868\) −6.00092 −0.203684
\(869\) 1.45367 0.0493122
\(870\) 0 0
\(871\) −21.7657 −0.737502
\(872\) −18.3156 −0.620245
\(873\) 0 0
\(874\) 1.04892 0.0354802
\(875\) 0 0
\(876\) 0 0
\(877\) −18.6595 −0.630086 −0.315043 0.949077i \(-0.602019\pi\)
−0.315043 + 0.949077i \(0.602019\pi\)
\(878\) 7.96184 0.268699
\(879\) 0 0
\(880\) 0 0
\(881\) −19.4306 −0.654632 −0.327316 0.944915i \(-0.606144\pi\)
−0.327316 + 0.944915i \(0.606144\pi\)
\(882\) 0 0
\(883\) 25.6437 0.862979 0.431490 0.902118i \(-0.357988\pi\)
0.431490 + 0.902118i \(0.357988\pi\)
\(884\) 15.9675 0.537044
\(885\) 0 0
\(886\) −5.36839 −0.180354
\(887\) 31.0062 1.04109 0.520544 0.853835i \(-0.325729\pi\)
0.520544 + 0.853835i \(0.325729\pi\)
\(888\) 0 0
\(889\) −30.2368 −1.01411
\(890\) 0 0
\(891\) 0 0
\(892\) −50.8753 −1.70343
\(893\) −2.04892 −0.0685644
\(894\) 0 0
\(895\) 0 0
\(896\) −12.3716 −0.413305
\(897\) 0 0
\(898\) 6.16421 0.205702
\(899\) −9.16315 −0.305608
\(900\) 0 0
\(901\) −14.3080 −0.476668
\(902\) −0.941511 −0.0313489
\(903\) 0 0
\(904\) 19.4929 0.648324
\(905\) 0 0
\(906\) 0 0
\(907\) −17.7676 −0.589964 −0.294982 0.955503i \(-0.595314\pi\)
−0.294982 + 0.955503i \(0.595314\pi\)
\(908\) 28.4748 0.944971
\(909\) 0 0
\(910\) 0 0
\(911\) −41.7313 −1.38262 −0.691309 0.722559i \(-0.742966\pi\)
−0.691309 + 0.722559i \(0.742966\pi\)
\(912\) 0 0
\(913\) −11.1564 −0.369224
\(914\) 3.24400 0.107302
\(915\) 0 0
\(916\) 41.9614 1.38644
\(917\) −12.5931 −0.415862
\(918\) 0 0
\(919\) 30.4088 1.00309 0.501547 0.865130i \(-0.332764\pi\)
0.501547 + 0.865130i \(0.332764\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 8.73630 0.287715
\(923\) 20.0248 0.659123
\(924\) 0 0
\(925\) 0 0
\(926\) 3.62969 0.119279
\(927\) 0 0
\(928\) −14.2483 −0.467724
\(929\) 28.8219 0.945616 0.472808 0.881165i \(-0.343240\pi\)
0.472808 + 0.881165i \(0.343240\pi\)
\(930\) 0 0
\(931\) 4.13706 0.135587
\(932\) 52.5824 1.72239
\(933\) 0 0
\(934\) 8.20270 0.268401
\(935\) 0 0
\(936\) 0 0
\(937\) −50.4601 −1.64846 −0.824230 0.566255i \(-0.808392\pi\)
−0.824230 + 0.566255i \(0.808392\pi\)
\(938\) −5.84953 −0.190994
\(939\) 0 0
\(940\) 0 0
\(941\) −1.10082 −0.0358857 −0.0179428 0.999839i \(-0.505712\pi\)
−0.0179428 + 0.999839i \(0.505712\pi\)
\(942\) 0 0
\(943\) 17.7549 0.578180
\(944\) −35.9335 −1.16954
\(945\) 0 0
\(946\) 1.64801 0.0535813
\(947\) 13.9353 0.452836 0.226418 0.974030i \(-0.427299\pi\)
0.226418 + 0.974030i \(0.427299\pi\)
\(948\) 0 0
\(949\) −4.36526 −0.141702
\(950\) 0 0
\(951\) 0 0
\(952\) 8.71751 0.282536
\(953\) 41.3400 1.33913 0.669567 0.742751i \(-0.266480\pi\)
0.669567 + 0.742751i \(0.266480\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −22.3582 −0.723117
\(957\) 0 0
\(958\) −6.08111 −0.196472
\(959\) −9.62192 −0.310708
\(960\) 0 0
\(961\) −27.6544 −0.892079
\(962\) 2.41789 0.0779561
\(963\) 0 0
\(964\) −22.9178 −0.738133
\(965\) 0 0
\(966\) 0 0
\(967\) −5.26875 −0.169432 −0.0847158 0.996405i \(-0.526998\pi\)
−0.0847158 + 0.996405i \(0.526998\pi\)
\(968\) 9.89248 0.317956
\(969\) 0 0
\(970\) 0 0
\(971\) 5.15346 0.165382 0.0826911 0.996575i \(-0.473649\pi\)
0.0826911 + 0.996575i \(0.473649\pi\)
\(972\) 0 0
\(973\) 6.11828 0.196143
\(974\) −7.30186 −0.233967
\(975\) 0 0
\(976\) 1.97484 0.0632130
\(977\) −4.77612 −0.152802 −0.0764008 0.997077i \(-0.524343\pi\)
−0.0764008 + 0.997077i \(0.524343\pi\)
\(978\) 0 0
\(979\) −2.66009 −0.0850168
\(980\) 0 0
\(981\) 0 0
\(982\) −8.96482 −0.286079
\(983\) 28.9758 0.924186 0.462093 0.886832i \(-0.347099\pi\)
0.462093 + 0.886832i \(0.347099\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 6.55257 0.208676
\(987\) 0 0
\(988\) 3.01507 0.0959220
\(989\) −31.0780 −0.988222
\(990\) 0 0
\(991\) −38.5042 −1.22313 −0.611564 0.791195i \(-0.709459\pi\)
−0.611564 + 0.791195i \(0.709459\pi\)
\(992\) 5.20219 0.165170
\(993\) 0 0
\(994\) 5.38165 0.170696
\(995\) 0 0
\(996\) 0 0
\(997\) 10.1491 0.321427 0.160713 0.987001i \(-0.448621\pi\)
0.160713 + 0.987001i \(0.448621\pi\)
\(998\) 1.93735 0.0613256
\(999\) 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 4275.2.a.z.1.3 3
3.2 odd 2 475.2.a.h.1.1 yes 3
5.4 even 2 4275.2.a.bn.1.1 3
12.11 even 2 7600.2.a.bn.1.3 3
15.2 even 4 475.2.b.c.324.3 6
15.8 even 4 475.2.b.c.324.4 6
15.14 odd 2 475.2.a.d.1.3 3
57.56 even 2 9025.2.a.w.1.3 3
60.59 even 2 7600.2.a.bw.1.1 3
285.284 even 2 9025.2.a.be.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 3 15.14 odd 2
475.2.a.h.1.1 yes 3 3.2 odd 2
475.2.b.c.324.3 6 15.2 even 4
475.2.b.c.324.4 6 15.8 even 4
4275.2.a.z.1.3 3 1.1 even 1 trivial
4275.2.a.bn.1.1 3 5.4 even 2
7600.2.a.bn.1.3 3 12.11 even 2
7600.2.a.bw.1.1 3 60.59 even 2
9025.2.a.w.1.3 3 57.56 even 2
9025.2.a.be.1.1 3 285.284 even 2