Properties

Label 4028.2.a.e.1.16
Level $4028$
Weight $2$
Character 4028.1
Self dual yes
Analytic conductor $32.164$
Analytic rank $0$
Dimension $19$
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] = [4028,2,Mod(1,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, 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("4028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + 16076 x^{11} - 42255 x^{10} - 38701 x^{9} + 93907 x^{8} + 49522 x^{7} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(2.42761\) of defining polynomial
Character \(\chi\) \(=\) 4028.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.42761 q^{3} +2.44173 q^{5} +4.41683 q^{7} +2.89329 q^{9} +O(q^{10})\) \(q+2.42761 q^{3} +2.44173 q^{5} +4.41683 q^{7} +2.89329 q^{9} -5.93135 q^{11} +4.07880 q^{13} +5.92756 q^{15} +2.12648 q^{17} +1.00000 q^{19} +10.7223 q^{21} -5.18265 q^{23} +0.962033 q^{25} -0.259048 q^{27} +2.70980 q^{29} +3.32365 q^{31} -14.3990 q^{33} +10.7847 q^{35} -7.04829 q^{37} +9.90174 q^{39} -0.996946 q^{41} +12.7651 q^{43} +7.06463 q^{45} +6.09666 q^{47} +12.5084 q^{49} +5.16226 q^{51} +1.00000 q^{53} -14.4827 q^{55} +2.42761 q^{57} -10.2555 q^{59} -4.30700 q^{61} +12.7792 q^{63} +9.95932 q^{65} +10.7266 q^{67} -12.5814 q^{69} -2.87319 q^{71} -6.21984 q^{73} +2.33544 q^{75} -26.1978 q^{77} +2.86596 q^{79} -9.30874 q^{81} +13.7266 q^{83} +5.19228 q^{85} +6.57834 q^{87} -4.25663 q^{89} +18.0154 q^{91} +8.06851 q^{93} +2.44173 q^{95} -3.67893 q^{97} -17.1611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 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
\(3\) 2.42761 1.40158 0.700791 0.713367i \(-0.252831\pi\)
0.700791 + 0.713367i \(0.252831\pi\)
\(4\) 0 0
\(5\) 2.44173 1.09197 0.545987 0.837794i \(-0.316155\pi\)
0.545987 + 0.837794i \(0.316155\pi\)
\(6\) 0 0
\(7\) 4.41683 1.66941 0.834703 0.550701i \(-0.185640\pi\)
0.834703 + 0.550701i \(0.185640\pi\)
\(8\) 0 0
\(9\) 2.89329 0.964430
\(10\) 0 0
\(11\) −5.93135 −1.78837 −0.894184 0.447699i \(-0.852243\pi\)
−0.894184 + 0.447699i \(0.852243\pi\)
\(12\) 0 0
\(13\) 4.07880 1.13126 0.565628 0.824660i \(-0.308634\pi\)
0.565628 + 0.824660i \(0.308634\pi\)
\(14\) 0 0
\(15\) 5.92756 1.53049
\(16\) 0 0
\(17\) 2.12648 0.515746 0.257873 0.966179i \(-0.416978\pi\)
0.257873 + 0.966179i \(0.416978\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 10.7223 2.33981
\(22\) 0 0
\(23\) −5.18265 −1.08066 −0.540328 0.841454i \(-0.681700\pi\)
−0.540328 + 0.841454i \(0.681700\pi\)
\(24\) 0 0
\(25\) 0.962033 0.192407
\(26\) 0 0
\(27\) −0.259048 −0.0498538
\(28\) 0 0
\(29\) 2.70980 0.503198 0.251599 0.967832i \(-0.419044\pi\)
0.251599 + 0.967832i \(0.419044\pi\)
\(30\) 0 0
\(31\) 3.32365 0.596944 0.298472 0.954418i \(-0.403523\pi\)
0.298472 + 0.954418i \(0.403523\pi\)
\(32\) 0 0
\(33\) −14.3990 −2.50654
\(34\) 0 0
\(35\) 10.7847 1.82295
\(36\) 0 0
\(37\) −7.04829 −1.15873 −0.579365 0.815068i \(-0.696700\pi\)
−0.579365 + 0.815068i \(0.696700\pi\)
\(38\) 0 0
\(39\) 9.90174 1.58555
\(40\) 0 0
\(41\) −0.996946 −0.155697 −0.0778484 0.996965i \(-0.524805\pi\)
−0.0778484 + 0.996965i \(0.524805\pi\)
\(42\) 0 0
\(43\) 12.7651 1.94665 0.973327 0.229424i \(-0.0736843\pi\)
0.973327 + 0.229424i \(0.0736843\pi\)
\(44\) 0 0
\(45\) 7.06463 1.05313
\(46\) 0 0
\(47\) 6.09666 0.889289 0.444644 0.895707i \(-0.353330\pi\)
0.444644 + 0.895707i \(0.353330\pi\)
\(48\) 0 0
\(49\) 12.5084 1.78691
\(50\) 0 0
\(51\) 5.16226 0.722860
\(52\) 0 0
\(53\) 1.00000 0.137361
\(54\) 0 0
\(55\) −14.4827 −1.95285
\(56\) 0 0
\(57\) 2.42761 0.321545
\(58\) 0 0
\(59\) −10.2555 −1.33515 −0.667576 0.744542i \(-0.732668\pi\)
−0.667576 + 0.744542i \(0.732668\pi\)
\(60\) 0 0
\(61\) −4.30700 −0.551454 −0.275727 0.961236i \(-0.588919\pi\)
−0.275727 + 0.961236i \(0.588919\pi\)
\(62\) 0 0
\(63\) 12.7792 1.61003
\(64\) 0 0
\(65\) 9.95932 1.23530
\(66\) 0 0
\(67\) 10.7266 1.31047 0.655234 0.755426i \(-0.272570\pi\)
0.655234 + 0.755426i \(0.272570\pi\)
\(68\) 0 0
\(69\) −12.5814 −1.51463
\(70\) 0 0
\(71\) −2.87319 −0.340985 −0.170492 0.985359i \(-0.554536\pi\)
−0.170492 + 0.985359i \(0.554536\pi\)
\(72\) 0 0
\(73\) −6.21984 −0.727977 −0.363988 0.931404i \(-0.618585\pi\)
−0.363988 + 0.931404i \(0.618585\pi\)
\(74\) 0 0
\(75\) 2.33544 0.269673
\(76\) 0 0
\(77\) −26.1978 −2.98551
\(78\) 0 0
\(79\) 2.86596 0.322446 0.161223 0.986918i \(-0.448456\pi\)
0.161223 + 0.986918i \(0.448456\pi\)
\(80\) 0 0
\(81\) −9.30874 −1.03430
\(82\) 0 0
\(83\) 13.7266 1.50669 0.753346 0.657625i \(-0.228439\pi\)
0.753346 + 0.657625i \(0.228439\pi\)
\(84\) 0 0
\(85\) 5.19228 0.563181
\(86\) 0 0
\(87\) 6.57834 0.705272
\(88\) 0 0
\(89\) −4.25663 −0.451202 −0.225601 0.974220i \(-0.572435\pi\)
−0.225601 + 0.974220i \(0.572435\pi\)
\(90\) 0 0
\(91\) 18.0154 1.88853
\(92\) 0 0
\(93\) 8.06851 0.836666
\(94\) 0 0
\(95\) 2.44173 0.250516
\(96\) 0 0
\(97\) −3.67893 −0.373539 −0.186769 0.982404i \(-0.559802\pi\)
−0.186769 + 0.982404i \(0.559802\pi\)
\(98\) 0 0
\(99\) −17.1611 −1.72476
\(100\) 0 0
\(101\) −5.31667 −0.529028 −0.264514 0.964382i \(-0.585212\pi\)
−0.264514 + 0.964382i \(0.585212\pi\)
\(102\) 0 0
\(103\) 0.122603 0.0120804 0.00604020 0.999982i \(-0.498077\pi\)
0.00604020 + 0.999982i \(0.498077\pi\)
\(104\) 0 0
\(105\) 26.1810 2.55501
\(106\) 0 0
\(107\) 5.47416 0.529207 0.264603 0.964357i \(-0.414759\pi\)
0.264603 + 0.964357i \(0.414759\pi\)
\(108\) 0 0
\(109\) −5.58156 −0.534617 −0.267308 0.963611i \(-0.586134\pi\)
−0.267308 + 0.963611i \(0.586134\pi\)
\(110\) 0 0
\(111\) −17.1105 −1.62406
\(112\) 0 0
\(113\) 0.813631 0.0765400 0.0382700 0.999267i \(-0.487815\pi\)
0.0382700 + 0.999267i \(0.487815\pi\)
\(114\) 0 0
\(115\) −12.6546 −1.18005
\(116\) 0 0
\(117\) 11.8012 1.09102
\(118\) 0 0
\(119\) 9.39229 0.860990
\(120\) 0 0
\(121\) 24.1809 2.19826
\(122\) 0 0
\(123\) −2.42020 −0.218222
\(124\) 0 0
\(125\) −9.85962 −0.881871
\(126\) 0 0
\(127\) 3.54992 0.315004 0.157502 0.987519i \(-0.449656\pi\)
0.157502 + 0.987519i \(0.449656\pi\)
\(128\) 0 0
\(129\) 30.9886 2.72839
\(130\) 0 0
\(131\) −4.68249 −0.409111 −0.204556 0.978855i \(-0.565575\pi\)
−0.204556 + 0.978855i \(0.565575\pi\)
\(132\) 0 0
\(133\) 4.41683 0.382988
\(134\) 0 0
\(135\) −0.632524 −0.0544390
\(136\) 0 0
\(137\) 2.80852 0.239948 0.119974 0.992777i \(-0.461719\pi\)
0.119974 + 0.992777i \(0.461719\pi\)
\(138\) 0 0
\(139\) −16.5884 −1.40701 −0.703503 0.710692i \(-0.748382\pi\)
−0.703503 + 0.710692i \(0.748382\pi\)
\(140\) 0 0
\(141\) 14.8003 1.24641
\(142\) 0 0
\(143\) −24.1928 −2.02310
\(144\) 0 0
\(145\) 6.61660 0.549478
\(146\) 0 0
\(147\) 30.3655 2.50451
\(148\) 0 0
\(149\) −8.80927 −0.721684 −0.360842 0.932627i \(-0.617511\pi\)
−0.360842 + 0.932627i \(0.617511\pi\)
\(150\) 0 0
\(151\) 16.1537 1.31457 0.657286 0.753641i \(-0.271704\pi\)
0.657286 + 0.753641i \(0.271704\pi\)
\(152\) 0 0
\(153\) 6.15252 0.497401
\(154\) 0 0
\(155\) 8.11544 0.651847
\(156\) 0 0
\(157\) 1.77649 0.141779 0.0708897 0.997484i \(-0.477416\pi\)
0.0708897 + 0.997484i \(0.477416\pi\)
\(158\) 0 0
\(159\) 2.42761 0.192522
\(160\) 0 0
\(161\) −22.8909 −1.80405
\(162\) 0 0
\(163\) −7.66129 −0.600079 −0.300039 0.953927i \(-0.597000\pi\)
−0.300039 + 0.953927i \(0.597000\pi\)
\(164\) 0 0
\(165\) −35.1584 −2.73708
\(166\) 0 0
\(167\) −11.4881 −0.888979 −0.444489 0.895784i \(-0.646615\pi\)
−0.444489 + 0.895784i \(0.646615\pi\)
\(168\) 0 0
\(169\) 3.63663 0.279741
\(170\) 0 0
\(171\) 2.89329 0.221255
\(172\) 0 0
\(173\) 1.21148 0.0921069 0.0460534 0.998939i \(-0.485336\pi\)
0.0460534 + 0.998939i \(0.485336\pi\)
\(174\) 0 0
\(175\) 4.24914 0.321204
\(176\) 0 0
\(177\) −24.8964 −1.87132
\(178\) 0 0
\(179\) −14.0305 −1.04869 −0.524345 0.851506i \(-0.675690\pi\)
−0.524345 + 0.851506i \(0.675690\pi\)
\(180\) 0 0
\(181\) −6.94037 −0.515874 −0.257937 0.966162i \(-0.583043\pi\)
−0.257937 + 0.966162i \(0.583043\pi\)
\(182\) 0 0
\(183\) −10.4557 −0.772908
\(184\) 0 0
\(185\) −17.2100 −1.26530
\(186\) 0 0
\(187\) −12.6129 −0.922344
\(188\) 0 0
\(189\) −1.14417 −0.0832261
\(190\) 0 0
\(191\) −24.2393 −1.75389 −0.876947 0.480586i \(-0.840424\pi\)
−0.876947 + 0.480586i \(0.840424\pi\)
\(192\) 0 0
\(193\) 15.7990 1.13724 0.568620 0.822600i \(-0.307478\pi\)
0.568620 + 0.822600i \(0.307478\pi\)
\(194\) 0 0
\(195\) 24.1774 1.73138
\(196\) 0 0
\(197\) −12.2980 −0.876194 −0.438097 0.898928i \(-0.644347\pi\)
−0.438097 + 0.898928i \(0.644347\pi\)
\(198\) 0 0
\(199\) −14.9880 −1.06247 −0.531236 0.847224i \(-0.678272\pi\)
−0.531236 + 0.847224i \(0.678272\pi\)
\(200\) 0 0
\(201\) 26.0401 1.83673
\(202\) 0 0
\(203\) 11.9687 0.840041
\(204\) 0 0
\(205\) −2.43427 −0.170017
\(206\) 0 0
\(207\) −14.9949 −1.04222
\(208\) 0 0
\(209\) −5.93135 −0.410280
\(210\) 0 0
\(211\) 14.3066 0.984908 0.492454 0.870339i \(-0.336100\pi\)
0.492454 + 0.870339i \(0.336100\pi\)
\(212\) 0 0
\(213\) −6.97498 −0.477918
\(214\) 0 0
\(215\) 31.1688 2.12569
\(216\) 0 0
\(217\) 14.6800 0.996542
\(218\) 0 0
\(219\) −15.0993 −1.02032
\(220\) 0 0
\(221\) 8.67348 0.583441
\(222\) 0 0
\(223\) −18.9453 −1.26867 −0.634337 0.773057i \(-0.718727\pi\)
−0.634337 + 0.773057i \(0.718727\pi\)
\(224\) 0 0
\(225\) 2.78344 0.185563
\(226\) 0 0
\(227\) 9.85882 0.654353 0.327177 0.944963i \(-0.393903\pi\)
0.327177 + 0.944963i \(0.393903\pi\)
\(228\) 0 0
\(229\) −11.4970 −0.759746 −0.379873 0.925039i \(-0.624032\pi\)
−0.379873 + 0.925039i \(0.624032\pi\)
\(230\) 0 0
\(231\) −63.5979 −4.18444
\(232\) 0 0
\(233\) −17.9156 −1.17369 −0.586844 0.809700i \(-0.699630\pi\)
−0.586844 + 0.809700i \(0.699630\pi\)
\(234\) 0 0
\(235\) 14.8864 0.971080
\(236\) 0 0
\(237\) 6.95744 0.451934
\(238\) 0 0
\(239\) 23.1255 1.49586 0.747931 0.663776i \(-0.231047\pi\)
0.747931 + 0.663776i \(0.231047\pi\)
\(240\) 0 0
\(241\) −26.7508 −1.72317 −0.861584 0.507615i \(-0.830527\pi\)
−0.861584 + 0.507615i \(0.830527\pi\)
\(242\) 0 0
\(243\) −21.8208 −1.39981
\(244\) 0 0
\(245\) 30.5421 1.95126
\(246\) 0 0
\(247\) 4.07880 0.259528
\(248\) 0 0
\(249\) 33.3229 2.11175
\(250\) 0 0
\(251\) 1.43342 0.0904765 0.0452382 0.998976i \(-0.485595\pi\)
0.0452382 + 0.998976i \(0.485595\pi\)
\(252\) 0 0
\(253\) 30.7401 1.93261
\(254\) 0 0
\(255\) 12.6048 0.789345
\(256\) 0 0
\(257\) −10.4300 −0.650609 −0.325304 0.945609i \(-0.605467\pi\)
−0.325304 + 0.945609i \(0.605467\pi\)
\(258\) 0 0
\(259\) −31.1311 −1.93439
\(260\) 0 0
\(261\) 7.84025 0.485299
\(262\) 0 0
\(263\) −14.6721 −0.904722 −0.452361 0.891835i \(-0.649418\pi\)
−0.452361 + 0.891835i \(0.649418\pi\)
\(264\) 0 0
\(265\) 2.44173 0.149994
\(266\) 0 0
\(267\) −10.3334 −0.632396
\(268\) 0 0
\(269\) −25.0014 −1.52436 −0.762182 0.647363i \(-0.775872\pi\)
−0.762182 + 0.647363i \(0.775872\pi\)
\(270\) 0 0
\(271\) 11.4404 0.694952 0.347476 0.937689i \(-0.387039\pi\)
0.347476 + 0.937689i \(0.387039\pi\)
\(272\) 0 0
\(273\) 43.7343 2.64692
\(274\) 0 0
\(275\) −5.70615 −0.344094
\(276\) 0 0
\(277\) 32.3604 1.94435 0.972173 0.234266i \(-0.0752686\pi\)
0.972173 + 0.234266i \(0.0752686\pi\)
\(278\) 0 0
\(279\) 9.61627 0.575711
\(280\) 0 0
\(281\) 29.3855 1.75299 0.876496 0.481410i \(-0.159875\pi\)
0.876496 + 0.481410i \(0.159875\pi\)
\(282\) 0 0
\(283\) 24.2416 1.44101 0.720506 0.693449i \(-0.243910\pi\)
0.720506 + 0.693449i \(0.243910\pi\)
\(284\) 0 0
\(285\) 5.92756 0.351118
\(286\) 0 0
\(287\) −4.40334 −0.259921
\(288\) 0 0
\(289\) −12.4781 −0.734006
\(290\) 0 0
\(291\) −8.93100 −0.523545
\(292\) 0 0
\(293\) 4.41460 0.257904 0.128952 0.991651i \(-0.458839\pi\)
0.128952 + 0.991651i \(0.458839\pi\)
\(294\) 0 0
\(295\) −25.0411 −1.45795
\(296\) 0 0
\(297\) 1.53650 0.0891569
\(298\) 0 0
\(299\) −21.1390 −1.22250
\(300\) 0 0
\(301\) 56.3811 3.24975
\(302\) 0 0
\(303\) −12.9068 −0.741476
\(304\) 0 0
\(305\) −10.5165 −0.602174
\(306\) 0 0
\(307\) −6.78207 −0.387073 −0.193537 0.981093i \(-0.561996\pi\)
−0.193537 + 0.981093i \(0.561996\pi\)
\(308\) 0 0
\(309\) 0.297632 0.0169317
\(310\) 0 0
\(311\) −1.57129 −0.0890997 −0.0445499 0.999007i \(-0.514185\pi\)
−0.0445499 + 0.999007i \(0.514185\pi\)
\(312\) 0 0
\(313\) −12.4530 −0.703888 −0.351944 0.936021i \(-0.614479\pi\)
−0.351944 + 0.936021i \(0.614479\pi\)
\(314\) 0 0
\(315\) 31.2033 1.75810
\(316\) 0 0
\(317\) −8.76645 −0.492373 −0.246186 0.969222i \(-0.579178\pi\)
−0.246186 + 0.969222i \(0.579178\pi\)
\(318\) 0 0
\(319\) −16.0728 −0.899903
\(320\) 0 0
\(321\) 13.2891 0.741726
\(322\) 0 0
\(323\) 2.12648 0.118320
\(324\) 0 0
\(325\) 3.92394 0.217661
\(326\) 0 0
\(327\) −13.5499 −0.749309
\(328\) 0 0
\(329\) 26.9279 1.48458
\(330\) 0 0
\(331\) 34.5910 1.90129 0.950647 0.310274i \(-0.100421\pi\)
0.950647 + 0.310274i \(0.100421\pi\)
\(332\) 0 0
\(333\) −20.3927 −1.11752
\(334\) 0 0
\(335\) 26.1915 1.43100
\(336\) 0 0
\(337\) 25.7342 1.40183 0.700916 0.713244i \(-0.252775\pi\)
0.700916 + 0.713244i \(0.252775\pi\)
\(338\) 0 0
\(339\) 1.97518 0.107277
\(340\) 0 0
\(341\) −19.7137 −1.06756
\(342\) 0 0
\(343\) 24.3297 1.31368
\(344\) 0 0
\(345\) −30.7205 −1.65393
\(346\) 0 0
\(347\) 15.3008 0.821392 0.410696 0.911772i \(-0.365286\pi\)
0.410696 + 0.911772i \(0.365286\pi\)
\(348\) 0 0
\(349\) 1.14302 0.0611842 0.0305921 0.999532i \(-0.490261\pi\)
0.0305921 + 0.999532i \(0.490261\pi\)
\(350\) 0 0
\(351\) −1.05660 −0.0563974
\(352\) 0 0
\(353\) −27.8086 −1.48010 −0.740051 0.672551i \(-0.765199\pi\)
−0.740051 + 0.672551i \(0.765199\pi\)
\(354\) 0 0
\(355\) −7.01554 −0.372346
\(356\) 0 0
\(357\) 22.8008 1.20675
\(358\) 0 0
\(359\) −26.9728 −1.42357 −0.711786 0.702396i \(-0.752113\pi\)
−0.711786 + 0.702396i \(0.752113\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 58.7017 3.08104
\(364\) 0 0
\(365\) −15.1871 −0.794931
\(366\) 0 0
\(367\) 11.2800 0.588814 0.294407 0.955680i \(-0.404878\pi\)
0.294407 + 0.955680i \(0.404878\pi\)
\(368\) 0 0
\(369\) −2.88445 −0.150159
\(370\) 0 0
\(371\) 4.41683 0.229310
\(372\) 0 0
\(373\) −28.1386 −1.45696 −0.728482 0.685065i \(-0.759774\pi\)
−0.728482 + 0.685065i \(0.759774\pi\)
\(374\) 0 0
\(375\) −23.9353 −1.23601
\(376\) 0 0
\(377\) 11.0527 0.569245
\(378\) 0 0
\(379\) −15.3518 −0.788567 −0.394284 0.918989i \(-0.629007\pi\)
−0.394284 + 0.918989i \(0.629007\pi\)
\(380\) 0 0
\(381\) 8.61782 0.441504
\(382\) 0 0
\(383\) −30.5658 −1.56184 −0.780919 0.624632i \(-0.785249\pi\)
−0.780919 + 0.624632i \(0.785249\pi\)
\(384\) 0 0
\(385\) −63.9678 −3.26010
\(386\) 0 0
\(387\) 36.9330 1.87741
\(388\) 0 0
\(389\) 28.4912 1.44456 0.722280 0.691601i \(-0.243094\pi\)
0.722280 + 0.691601i \(0.243094\pi\)
\(390\) 0 0
\(391\) −11.0208 −0.557345
\(392\) 0 0
\(393\) −11.3673 −0.573403
\(394\) 0 0
\(395\) 6.99790 0.352103
\(396\) 0 0
\(397\) −9.34029 −0.468776 −0.234388 0.972143i \(-0.575309\pi\)
−0.234388 + 0.972143i \(0.575309\pi\)
\(398\) 0 0
\(399\) 10.7223 0.536789
\(400\) 0 0
\(401\) −28.8059 −1.43850 −0.719250 0.694752i \(-0.755514\pi\)
−0.719250 + 0.694752i \(0.755514\pi\)
\(402\) 0 0
\(403\) 13.5565 0.675297
\(404\) 0 0
\(405\) −22.7294 −1.12943
\(406\) 0 0
\(407\) 41.8058 2.07224
\(408\) 0 0
\(409\) 29.8723 1.47709 0.738546 0.674203i \(-0.235513\pi\)
0.738546 + 0.674203i \(0.235513\pi\)
\(410\) 0 0
\(411\) 6.81800 0.336307
\(412\) 0 0
\(413\) −45.2968 −2.22891
\(414\) 0 0
\(415\) 33.5166 1.64527
\(416\) 0 0
\(417\) −40.2701 −1.97203
\(418\) 0 0
\(419\) 38.4272 1.87729 0.938646 0.344881i \(-0.112081\pi\)
0.938646 + 0.344881i \(0.112081\pi\)
\(420\) 0 0
\(421\) −3.10123 −0.151145 −0.0755724 0.997140i \(-0.524078\pi\)
−0.0755724 + 0.997140i \(0.524078\pi\)
\(422\) 0 0
\(423\) 17.6394 0.857657
\(424\) 0 0
\(425\) 2.04574 0.0992330
\(426\) 0 0
\(427\) −19.0233 −0.920601
\(428\) 0 0
\(429\) −58.7307 −2.83554
\(430\) 0 0
\(431\) 25.5812 1.23221 0.616103 0.787666i \(-0.288711\pi\)
0.616103 + 0.787666i \(0.288711\pi\)
\(432\) 0 0
\(433\) −27.8557 −1.33866 −0.669329 0.742966i \(-0.733418\pi\)
−0.669329 + 0.742966i \(0.733418\pi\)
\(434\) 0 0
\(435\) 16.0625 0.770139
\(436\) 0 0
\(437\) −5.18265 −0.247920
\(438\) 0 0
\(439\) 0.433274 0.0206790 0.0103395 0.999947i \(-0.496709\pi\)
0.0103395 + 0.999947i \(0.496709\pi\)
\(440\) 0 0
\(441\) 36.1904 1.72335
\(442\) 0 0
\(443\) 26.8878 1.27748 0.638738 0.769424i \(-0.279457\pi\)
0.638738 + 0.769424i \(0.279457\pi\)
\(444\) 0 0
\(445\) −10.3935 −0.492700
\(446\) 0 0
\(447\) −21.3855 −1.01150
\(448\) 0 0
\(449\) 34.1362 1.61099 0.805494 0.592604i \(-0.201900\pi\)
0.805494 + 0.592604i \(0.201900\pi\)
\(450\) 0 0
\(451\) 5.91323 0.278443
\(452\) 0 0
\(453\) 39.2149 1.84248
\(454\) 0 0
\(455\) 43.9887 2.06222
\(456\) 0 0
\(457\) −19.2474 −0.900354 −0.450177 0.892939i \(-0.648639\pi\)
−0.450177 + 0.892939i \(0.648639\pi\)
\(458\) 0 0
\(459\) −0.550859 −0.0257119
\(460\) 0 0
\(461\) 31.4602 1.46525 0.732623 0.680634i \(-0.238296\pi\)
0.732623 + 0.680634i \(0.238296\pi\)
\(462\) 0 0
\(463\) 27.9080 1.29699 0.648497 0.761217i \(-0.275398\pi\)
0.648497 + 0.761217i \(0.275398\pi\)
\(464\) 0 0
\(465\) 19.7011 0.913617
\(466\) 0 0
\(467\) 30.2738 1.40090 0.700451 0.713700i \(-0.252982\pi\)
0.700451 + 0.713700i \(0.252982\pi\)
\(468\) 0 0
\(469\) 47.3778 2.18770
\(470\) 0 0
\(471\) 4.31263 0.198715
\(472\) 0 0
\(473\) −75.7140 −3.48133
\(474\) 0 0
\(475\) 0.962033 0.0441411
\(476\) 0 0
\(477\) 2.89329 0.132475
\(478\) 0 0
\(479\) 27.9319 1.27624 0.638121 0.769936i \(-0.279712\pi\)
0.638121 + 0.769936i \(0.279712\pi\)
\(480\) 0 0
\(481\) −28.7486 −1.31082
\(482\) 0 0
\(483\) −55.5701 −2.52853
\(484\) 0 0
\(485\) −8.98294 −0.407894
\(486\) 0 0
\(487\) −9.06821 −0.410920 −0.205460 0.978666i \(-0.565869\pi\)
−0.205460 + 0.978666i \(0.565869\pi\)
\(488\) 0 0
\(489\) −18.5986 −0.841059
\(490\) 0 0
\(491\) −24.4672 −1.10419 −0.552094 0.833782i \(-0.686171\pi\)
−0.552094 + 0.833782i \(0.686171\pi\)
\(492\) 0 0
\(493\) 5.76233 0.259522
\(494\) 0 0
\(495\) −41.9028 −1.88339
\(496\) 0 0
\(497\) −12.6904 −0.569242
\(498\) 0 0
\(499\) −13.2824 −0.594602 −0.297301 0.954784i \(-0.596087\pi\)
−0.297301 + 0.954784i \(0.596087\pi\)
\(500\) 0 0
\(501\) −27.8887 −1.24598
\(502\) 0 0
\(503\) 20.4328 0.911052 0.455526 0.890222i \(-0.349451\pi\)
0.455526 + 0.890222i \(0.349451\pi\)
\(504\) 0 0
\(505\) −12.9819 −0.577685
\(506\) 0 0
\(507\) 8.82832 0.392080
\(508\) 0 0
\(509\) 37.9047 1.68009 0.840047 0.542513i \(-0.182527\pi\)
0.840047 + 0.542513i \(0.182527\pi\)
\(510\) 0 0
\(511\) −27.4720 −1.21529
\(512\) 0 0
\(513\) −0.259048 −0.0114372
\(514\) 0 0
\(515\) 0.299362 0.0131915
\(516\) 0 0
\(517\) −36.1614 −1.59038
\(518\) 0 0
\(519\) 2.94099 0.129095
\(520\) 0 0
\(521\) 28.9888 1.27002 0.635011 0.772503i \(-0.280996\pi\)
0.635011 + 0.772503i \(0.280996\pi\)
\(522\) 0 0
\(523\) −42.6227 −1.86376 −0.931880 0.362766i \(-0.881832\pi\)
−0.931880 + 0.362766i \(0.881832\pi\)
\(524\) 0 0
\(525\) 10.3152 0.450194
\(526\) 0 0
\(527\) 7.06765 0.307872
\(528\) 0 0
\(529\) 3.85982 0.167818
\(530\) 0 0
\(531\) −29.6721 −1.28766
\(532\) 0 0
\(533\) −4.06635 −0.176133
\(534\) 0 0
\(535\) 13.3664 0.577880
\(536\) 0 0
\(537\) −34.0606 −1.46983
\(538\) 0 0
\(539\) −74.1916 −3.19566
\(540\) 0 0
\(541\) 20.2442 0.870368 0.435184 0.900342i \(-0.356683\pi\)
0.435184 + 0.900342i \(0.356683\pi\)
\(542\) 0 0
\(543\) −16.8485 −0.723039
\(544\) 0 0
\(545\) −13.6287 −0.583787
\(546\) 0 0
\(547\) −33.8092 −1.44558 −0.722788 0.691070i \(-0.757140\pi\)
−0.722788 + 0.691070i \(0.757140\pi\)
\(548\) 0 0
\(549\) −12.4614 −0.531839
\(550\) 0 0
\(551\) 2.70980 0.115441
\(552\) 0 0
\(553\) 12.6585 0.538293
\(554\) 0 0
\(555\) −41.7791 −1.77343
\(556\) 0 0
\(557\) −4.63096 −0.196220 −0.0981101 0.995176i \(-0.531280\pi\)
−0.0981101 + 0.995176i \(0.531280\pi\)
\(558\) 0 0
\(559\) 52.0662 2.20216
\(560\) 0 0
\(561\) −30.6191 −1.29274
\(562\) 0 0
\(563\) 8.58131 0.361659 0.180829 0.983514i \(-0.442122\pi\)
0.180829 + 0.983514i \(0.442122\pi\)
\(564\) 0 0
\(565\) 1.98667 0.0835797
\(566\) 0 0
\(567\) −41.1151 −1.72667
\(568\) 0 0
\(569\) −23.0341 −0.965641 −0.482821 0.875719i \(-0.660388\pi\)
−0.482821 + 0.875719i \(0.660388\pi\)
\(570\) 0 0
\(571\) −1.36276 −0.0570299 −0.0285149 0.999593i \(-0.509078\pi\)
−0.0285149 + 0.999593i \(0.509078\pi\)
\(572\) 0 0
\(573\) −58.8436 −2.45823
\(574\) 0 0
\(575\) −4.98587 −0.207925
\(576\) 0 0
\(577\) −20.5693 −0.856312 −0.428156 0.903705i \(-0.640837\pi\)
−0.428156 + 0.903705i \(0.640837\pi\)
\(578\) 0 0
\(579\) 38.3539 1.59393
\(580\) 0 0
\(581\) 60.6281 2.51528
\(582\) 0 0
\(583\) −5.93135 −0.245651
\(584\) 0 0
\(585\) 28.8152 1.19136
\(586\) 0 0
\(587\) −34.0370 −1.40486 −0.702428 0.711754i \(-0.747901\pi\)
−0.702428 + 0.711754i \(0.747901\pi\)
\(588\) 0 0
\(589\) 3.32365 0.136948
\(590\) 0 0
\(591\) −29.8547 −1.22806
\(592\) 0 0
\(593\) −12.1137 −0.497450 −0.248725 0.968574i \(-0.580011\pi\)
−0.248725 + 0.968574i \(0.580011\pi\)
\(594\) 0 0
\(595\) 22.9334 0.940178
\(596\) 0 0
\(597\) −36.3851 −1.48914
\(598\) 0 0
\(599\) 14.7127 0.601144 0.300572 0.953759i \(-0.402822\pi\)
0.300572 + 0.953759i \(0.402822\pi\)
\(600\) 0 0
\(601\) 8.49093 0.346352 0.173176 0.984891i \(-0.444597\pi\)
0.173176 + 0.984891i \(0.444597\pi\)
\(602\) 0 0
\(603\) 31.0353 1.26385
\(604\) 0 0
\(605\) 59.0431 2.40044
\(606\) 0 0
\(607\) −17.8445 −0.724285 −0.362142 0.932123i \(-0.617955\pi\)
−0.362142 + 0.932123i \(0.617955\pi\)
\(608\) 0 0
\(609\) 29.0554 1.17739
\(610\) 0 0
\(611\) 24.8671 1.00601
\(612\) 0 0
\(613\) 13.4422 0.542925 0.271463 0.962449i \(-0.412493\pi\)
0.271463 + 0.962449i \(0.412493\pi\)
\(614\) 0 0
\(615\) −5.90946 −0.238292
\(616\) 0 0
\(617\) −17.1603 −0.690849 −0.345425 0.938446i \(-0.612265\pi\)
−0.345425 + 0.938446i \(0.612265\pi\)
\(618\) 0 0
\(619\) 0.169156 0.00679897 0.00339949 0.999994i \(-0.498918\pi\)
0.00339949 + 0.999994i \(0.498918\pi\)
\(620\) 0 0
\(621\) 1.34255 0.0538748
\(622\) 0 0
\(623\) −18.8008 −0.753238
\(624\) 0 0
\(625\) −28.8847 −1.15539
\(626\) 0 0
\(627\) −14.3990 −0.575041
\(628\) 0 0
\(629\) −14.9880 −0.597611
\(630\) 0 0
\(631\) −3.91179 −0.155726 −0.0778629 0.996964i \(-0.524810\pi\)
−0.0778629 + 0.996964i \(0.524810\pi\)
\(632\) 0 0
\(633\) 34.7309 1.38043
\(634\) 0 0
\(635\) 8.66794 0.343977
\(636\) 0 0
\(637\) 51.0193 2.02146
\(638\) 0 0
\(639\) −8.31297 −0.328856
\(640\) 0 0
\(641\) −12.6148 −0.498255 −0.249127 0.968471i \(-0.580144\pi\)
−0.249127 + 0.968471i \(0.580144\pi\)
\(642\) 0 0
\(643\) 38.5038 1.51844 0.759220 0.650834i \(-0.225581\pi\)
0.759220 + 0.650834i \(0.225581\pi\)
\(644\) 0 0
\(645\) 75.6657 2.97933
\(646\) 0 0
\(647\) −37.0194 −1.45538 −0.727692 0.685904i \(-0.759407\pi\)
−0.727692 + 0.685904i \(0.759407\pi\)
\(648\) 0 0
\(649\) 60.8289 2.38774
\(650\) 0 0
\(651\) 35.6373 1.39673
\(652\) 0 0
\(653\) −14.7482 −0.577143 −0.288572 0.957458i \(-0.593180\pi\)
−0.288572 + 0.957458i \(0.593180\pi\)
\(654\) 0 0
\(655\) −11.4334 −0.446739
\(656\) 0 0
\(657\) −17.9958 −0.702083
\(658\) 0 0
\(659\) 3.52104 0.137160 0.0685801 0.997646i \(-0.478153\pi\)
0.0685801 + 0.997646i \(0.478153\pi\)
\(660\) 0 0
\(661\) −20.9393 −0.814443 −0.407222 0.913329i \(-0.633502\pi\)
−0.407222 + 0.913329i \(0.633502\pi\)
\(662\) 0 0
\(663\) 21.0558 0.817740
\(664\) 0 0
\(665\) 10.7847 0.418213
\(666\) 0 0
\(667\) −14.0439 −0.543784
\(668\) 0 0
\(669\) −45.9919 −1.77815
\(670\) 0 0
\(671\) 25.5463 0.986203
\(672\) 0 0
\(673\) 11.5325 0.444545 0.222273 0.974985i \(-0.428653\pi\)
0.222273 + 0.974985i \(0.428653\pi\)
\(674\) 0 0
\(675\) −0.249212 −0.00959219
\(676\) 0 0
\(677\) 4.31039 0.165662 0.0828309 0.996564i \(-0.473604\pi\)
0.0828309 + 0.996564i \(0.473604\pi\)
\(678\) 0 0
\(679\) −16.2492 −0.623587
\(680\) 0 0
\(681\) 23.9334 0.917129
\(682\) 0 0
\(683\) 49.7742 1.90456 0.952279 0.305230i \(-0.0987334\pi\)
0.952279 + 0.305230i \(0.0987334\pi\)
\(684\) 0 0
\(685\) 6.85765 0.262017
\(686\) 0 0
\(687\) −27.9103 −1.06485
\(688\) 0 0
\(689\) 4.07880 0.155390
\(690\) 0 0
\(691\) −25.4006 −0.966283 −0.483142 0.875542i \(-0.660504\pi\)
−0.483142 + 0.875542i \(0.660504\pi\)
\(692\) 0 0
\(693\) −75.7977 −2.87932
\(694\) 0 0
\(695\) −40.5043 −1.53641
\(696\) 0 0
\(697\) −2.11998 −0.0803000
\(698\) 0 0
\(699\) −43.4920 −1.64502
\(700\) 0 0
\(701\) 21.4730 0.811024 0.405512 0.914090i \(-0.367093\pi\)
0.405512 + 0.914090i \(0.367093\pi\)
\(702\) 0 0
\(703\) −7.04829 −0.265831
\(704\) 0 0
\(705\) 36.1383 1.36105
\(706\) 0 0
\(707\) −23.4828 −0.883162
\(708\) 0 0
\(709\) 22.6628 0.851121 0.425561 0.904930i \(-0.360077\pi\)
0.425561 + 0.904930i \(0.360077\pi\)
\(710\) 0 0
\(711\) 8.29206 0.310977
\(712\) 0 0
\(713\) −17.2253 −0.645092
\(714\) 0 0
\(715\) −59.0722 −2.20918
\(716\) 0 0
\(717\) 56.1396 2.09657
\(718\) 0 0
\(719\) 15.3632 0.572950 0.286475 0.958088i \(-0.407516\pi\)
0.286475 + 0.958088i \(0.407516\pi\)
\(720\) 0 0
\(721\) 0.541515 0.0201671
\(722\) 0 0
\(723\) −64.9404 −2.41516
\(724\) 0 0
\(725\) 2.60692 0.0968185
\(726\) 0 0
\(727\) 30.7985 1.14225 0.571127 0.820862i \(-0.306506\pi\)
0.571127 + 0.820862i \(0.306506\pi\)
\(728\) 0 0
\(729\) −25.0463 −0.927641
\(730\) 0 0
\(731\) 27.1446 1.00398
\(732\) 0 0
\(733\) 42.5132 1.57026 0.785131 0.619329i \(-0.212595\pi\)
0.785131 + 0.619329i \(0.212595\pi\)
\(734\) 0 0
\(735\) 74.1443 2.73485
\(736\) 0 0
\(737\) −63.6234 −2.34360
\(738\) 0 0
\(739\) 23.0918 0.849445 0.424723 0.905324i \(-0.360372\pi\)
0.424723 + 0.905324i \(0.360372\pi\)
\(740\) 0 0
\(741\) 9.90174 0.363750
\(742\) 0 0
\(743\) −5.49231 −0.201493 −0.100747 0.994912i \(-0.532123\pi\)
−0.100747 + 0.994912i \(0.532123\pi\)
\(744\) 0 0
\(745\) −21.5098 −0.788059
\(746\) 0 0
\(747\) 39.7151 1.45310
\(748\) 0 0
\(749\) 24.1784 0.883461
\(750\) 0 0
\(751\) −13.2136 −0.482172 −0.241086 0.970504i \(-0.577504\pi\)
−0.241086 + 0.970504i \(0.577504\pi\)
\(752\) 0 0
\(753\) 3.47978 0.126810
\(754\) 0 0
\(755\) 39.4430 1.43548
\(756\) 0 0
\(757\) −34.2977 −1.24657 −0.623286 0.781994i \(-0.714203\pi\)
−0.623286 + 0.781994i \(0.714203\pi\)
\(758\) 0 0
\(759\) 74.6249 2.70871
\(760\) 0 0
\(761\) 2.66171 0.0964868 0.0482434 0.998836i \(-0.484638\pi\)
0.0482434 + 0.998836i \(0.484638\pi\)
\(762\) 0 0
\(763\) −24.6528 −0.892492
\(764\) 0 0
\(765\) 15.0228 0.543149
\(766\) 0 0
\(767\) −41.8302 −1.51040
\(768\) 0 0
\(769\) −18.5865 −0.670246 −0.335123 0.942174i \(-0.608778\pi\)
−0.335123 + 0.942174i \(0.608778\pi\)
\(770\) 0 0
\(771\) −25.3201 −0.911881
\(772\) 0 0
\(773\) −36.8044 −1.32376 −0.661881 0.749609i \(-0.730242\pi\)
−0.661881 + 0.749609i \(0.730242\pi\)
\(774\) 0 0
\(775\) 3.19746 0.114856
\(776\) 0 0
\(777\) −75.5741 −2.71121
\(778\) 0 0
\(779\) −0.996946 −0.0357193
\(780\) 0 0
\(781\) 17.0419 0.609806
\(782\) 0 0
\(783\) −0.701968 −0.0250863
\(784\) 0 0
\(785\) 4.33771 0.154819
\(786\) 0 0
\(787\) −31.0758 −1.10773 −0.553867 0.832605i \(-0.686848\pi\)
−0.553867 + 0.832605i \(0.686848\pi\)
\(788\) 0 0
\(789\) −35.6182 −1.26804
\(790\) 0 0
\(791\) 3.59367 0.127776
\(792\) 0 0
\(793\) −17.5674 −0.623836
\(794\) 0 0
\(795\) 5.92756 0.210229
\(796\) 0 0
\(797\) −13.3939 −0.474436 −0.237218 0.971456i \(-0.576236\pi\)
−0.237218 + 0.971456i \(0.576236\pi\)
\(798\) 0 0
\(799\) 12.9644 0.458647
\(800\) 0 0
\(801\) −12.3157 −0.435153
\(802\) 0 0
\(803\) 36.8920 1.30189
\(804\) 0 0
\(805\) −55.8933 −1.96998
\(806\) 0 0
\(807\) −60.6937 −2.13652
\(808\) 0 0
\(809\) 41.3412 1.45348 0.726740 0.686913i \(-0.241035\pi\)
0.726740 + 0.686913i \(0.241035\pi\)
\(810\) 0 0
\(811\) 40.0952 1.40793 0.703966 0.710234i \(-0.251411\pi\)
0.703966 + 0.710234i \(0.251411\pi\)
\(812\) 0 0
\(813\) 27.7727 0.974032
\(814\) 0 0
\(815\) −18.7068 −0.655270
\(816\) 0 0
\(817\) 12.7651 0.446593
\(818\) 0 0
\(819\) 52.1237 1.82135
\(820\) 0 0
\(821\) −27.4412 −0.957705 −0.478852 0.877895i \(-0.658947\pi\)
−0.478852 + 0.877895i \(0.658947\pi\)
\(822\) 0 0
\(823\) −30.3157 −1.05674 −0.528370 0.849014i \(-0.677197\pi\)
−0.528370 + 0.849014i \(0.677197\pi\)
\(824\) 0 0
\(825\) −13.8523 −0.482275
\(826\) 0 0
\(827\) −22.9775 −0.799006 −0.399503 0.916732i \(-0.630817\pi\)
−0.399503 + 0.916732i \(0.630817\pi\)
\(828\) 0 0
\(829\) 55.9198 1.94218 0.971088 0.238724i \(-0.0767290\pi\)
0.971088 + 0.238724i \(0.0767290\pi\)
\(830\) 0 0
\(831\) 78.5583 2.72516
\(832\) 0 0
\(833\) 26.5988 0.921594
\(834\) 0 0
\(835\) −28.0509 −0.970741
\(836\) 0 0
\(837\) −0.860983 −0.0297599
\(838\) 0 0
\(839\) 37.2795 1.28703 0.643515 0.765433i \(-0.277475\pi\)
0.643515 + 0.765433i \(0.277475\pi\)
\(840\) 0 0
\(841\) −21.6570 −0.746792
\(842\) 0 0
\(843\) 71.3365 2.45696
\(844\) 0 0
\(845\) 8.87966 0.305470
\(846\) 0 0
\(847\) 106.803 3.66979
\(848\) 0 0
\(849\) 58.8491 2.01969
\(850\) 0 0
\(851\) 36.5288 1.25219
\(852\) 0 0
\(853\) −44.5559 −1.52557 −0.762783 0.646654i \(-0.776168\pi\)
−0.762783 + 0.646654i \(0.776168\pi\)
\(854\) 0 0
\(855\) 7.06463 0.241605
\(856\) 0 0
\(857\) 37.2340 1.27189 0.635945 0.771735i \(-0.280611\pi\)
0.635945 + 0.771735i \(0.280611\pi\)
\(858\) 0 0
\(859\) −52.5366 −1.79253 −0.896263 0.443523i \(-0.853729\pi\)
−0.896263 + 0.443523i \(0.853729\pi\)
\(860\) 0 0
\(861\) −10.6896 −0.364300
\(862\) 0 0
\(863\) 47.6678 1.62263 0.811316 0.584608i \(-0.198752\pi\)
0.811316 + 0.584608i \(0.198752\pi\)
\(864\) 0 0
\(865\) 2.95810 0.100578
\(866\) 0 0
\(867\) −30.2920 −1.02877
\(868\) 0 0
\(869\) −16.9990 −0.576652
\(870\) 0 0
\(871\) 43.7519 1.48248
\(872\) 0 0
\(873\) −10.6442 −0.360252
\(874\) 0 0
\(875\) −43.5483 −1.47220
\(876\) 0 0
\(877\) 42.9010 1.44866 0.724332 0.689452i \(-0.242148\pi\)
0.724332 + 0.689452i \(0.242148\pi\)
\(878\) 0 0
\(879\) 10.7169 0.361473
\(880\) 0 0
\(881\) −55.0745 −1.85551 −0.927753 0.373195i \(-0.878262\pi\)
−0.927753 + 0.373195i \(0.878262\pi\)
\(882\) 0 0
\(883\) 11.1137 0.374006 0.187003 0.982359i \(-0.440123\pi\)
0.187003 + 0.982359i \(0.440123\pi\)
\(884\) 0 0
\(885\) −60.7901 −2.04344
\(886\) 0 0
\(887\) 5.75241 0.193147 0.0965735 0.995326i \(-0.469212\pi\)
0.0965735 + 0.995326i \(0.469212\pi\)
\(888\) 0 0
\(889\) 15.6794 0.525870
\(890\) 0 0
\(891\) 55.2134 1.84972
\(892\) 0 0
\(893\) 6.09666 0.204017
\(894\) 0 0
\(895\) −34.2587 −1.14514
\(896\) 0 0
\(897\) −51.3172 −1.71343
\(898\) 0 0
\(899\) 9.00642 0.300381
\(900\) 0 0
\(901\) 2.12648 0.0708432
\(902\) 0 0
\(903\) 136.871 4.55479
\(904\) 0 0
\(905\) −16.9465 −0.563320
\(906\) 0 0
\(907\) 6.42572 0.213362 0.106681 0.994293i \(-0.465978\pi\)
0.106681 + 0.994293i \(0.465978\pi\)
\(908\) 0 0
\(909\) −15.3827 −0.510211
\(910\) 0 0
\(911\) −0.672625 −0.0222850 −0.0111425 0.999938i \(-0.503547\pi\)
−0.0111425 + 0.999938i \(0.503547\pi\)
\(912\) 0 0
\(913\) −81.4173 −2.69452
\(914\) 0 0
\(915\) −25.5300 −0.843995
\(916\) 0 0
\(917\) −20.6818 −0.682973
\(918\) 0 0
\(919\) −17.9229 −0.591223 −0.295612 0.955308i \(-0.595523\pi\)
−0.295612 + 0.955308i \(0.595523\pi\)
\(920\) 0 0
\(921\) −16.4642 −0.542515
\(922\) 0 0
\(923\) −11.7192 −0.385741
\(924\) 0 0
\(925\) −6.78068 −0.222947
\(926\) 0 0
\(927\) 0.354725 0.0116507
\(928\) 0 0
\(929\) 3.10896 0.102002 0.0510008 0.998699i \(-0.483759\pi\)
0.0510008 + 0.998699i \(0.483759\pi\)
\(930\) 0 0
\(931\) 12.5084 0.409946
\(932\) 0 0
\(933\) −3.81448 −0.124881
\(934\) 0 0
\(935\) −30.7972 −1.00718
\(936\) 0 0
\(937\) 32.5833 1.06445 0.532225 0.846603i \(-0.321356\pi\)
0.532225 + 0.846603i \(0.321356\pi\)
\(938\) 0 0
\(939\) −30.2311 −0.986556
\(940\) 0 0
\(941\) −21.9322 −0.714968 −0.357484 0.933919i \(-0.616365\pi\)
−0.357484 + 0.933919i \(0.616365\pi\)
\(942\) 0 0
\(943\) 5.16682 0.168255
\(944\) 0 0
\(945\) −2.79375 −0.0908807
\(946\) 0 0
\(947\) −10.4704 −0.340241 −0.170121 0.985423i \(-0.554416\pi\)
−0.170121 + 0.985423i \(0.554416\pi\)
\(948\) 0 0
\(949\) −25.3695 −0.823528
\(950\) 0 0
\(951\) −21.2815 −0.690101
\(952\) 0 0
\(953\) −2.67948 −0.0867968 −0.0433984 0.999058i \(-0.513818\pi\)
−0.0433984 + 0.999058i \(0.513818\pi\)
\(954\) 0 0
\(955\) −59.1858 −1.91521
\(956\) 0 0
\(957\) −39.0184 −1.26129
\(958\) 0 0
\(959\) 12.4048 0.400571
\(960\) 0 0
\(961\) −19.9534 −0.643657
\(962\) 0 0
\(963\) 15.8383 0.510383
\(964\) 0 0
\(965\) 38.5770 1.24184
\(966\) 0 0
\(967\) 16.3168 0.524714 0.262357 0.964971i \(-0.415500\pi\)
0.262357 + 0.964971i \(0.415500\pi\)
\(968\) 0 0
\(969\) 5.16226 0.165836
\(970\) 0 0
\(971\) −27.9928 −0.898333 −0.449166 0.893448i \(-0.648279\pi\)
−0.449166 + 0.893448i \(0.648279\pi\)
\(972\) 0 0
\(973\) −73.2680 −2.34886
\(974\) 0 0
\(975\) 9.52580 0.305070
\(976\) 0 0
\(977\) −17.6814 −0.565677 −0.282838 0.959168i \(-0.591276\pi\)
−0.282838 + 0.959168i \(0.591276\pi\)
\(978\) 0 0
\(979\) 25.2475 0.806915
\(980\) 0 0
\(981\) −16.1491 −0.515601
\(982\) 0 0
\(983\) −7.18277 −0.229095 −0.114547 0.993418i \(-0.536542\pi\)
−0.114547 + 0.993418i \(0.536542\pi\)
\(984\) 0 0
\(985\) −30.0283 −0.956780
\(986\) 0 0
\(987\) 65.3704 2.08076
\(988\) 0 0
\(989\) −66.1568 −2.10366
\(990\) 0 0
\(991\) 33.5077 1.06441 0.532204 0.846616i \(-0.321364\pi\)
0.532204 + 0.846616i \(0.321364\pi\)
\(992\) 0 0
\(993\) 83.9735 2.66482
\(994\) 0 0
\(995\) −36.5967 −1.16019
\(996\) 0 0
\(997\) −50.9262 −1.61285 −0.806424 0.591338i \(-0.798600\pi\)
−0.806424 + 0.591338i \(0.798600\pi\)
\(998\) 0 0
\(999\) 1.82584 0.0577671
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 4028.2.a.e.1.16 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.a.e.1.16 19 1.1 even 1 trivial