Properties

Label 4012.2.a.h.1.12
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $1$
Dimension $15$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} - 249 x^{6} + 2736 x^{5} - 801 x^{4} - 900 x^{3} + 429 x^{2} - 36 x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-0.829690\) of defining polynomial
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.84628 q^{3} -2.98774 q^{5} -0.920155 q^{7} +0.408737 q^{9} +O(q^{10})\) \(q+1.84628 q^{3} -2.98774 q^{5} -0.920155 q^{7} +0.408737 q^{9} -1.28823 q^{11} +2.44378 q^{13} -5.51619 q^{15} +1.00000 q^{17} +5.97874 q^{19} -1.69886 q^{21} +0.336054 q^{23} +3.92657 q^{25} -4.78419 q^{27} +0.998551 q^{29} +2.84063 q^{31} -2.37843 q^{33} +2.74918 q^{35} -8.24001 q^{37} +4.51190 q^{39} +2.60887 q^{41} -10.5002 q^{43} -1.22120 q^{45} -0.00237639 q^{47} -6.15331 q^{49} +1.84628 q^{51} -5.66109 q^{53} +3.84889 q^{55} +11.0384 q^{57} -1.00000 q^{59} +11.4554 q^{61} -0.376102 q^{63} -7.30137 q^{65} -12.2645 q^{67} +0.620449 q^{69} +1.33343 q^{71} -8.50879 q^{73} +7.24953 q^{75} +1.18537 q^{77} -9.03414 q^{79} -10.0591 q^{81} -5.93365 q^{83} -2.98774 q^{85} +1.84360 q^{87} -12.5190 q^{89} -2.24866 q^{91} +5.24459 q^{93} -17.8629 q^{95} -0.539984 q^{97} -0.526547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 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
\(3\) 1.84628 1.06595 0.532974 0.846132i \(-0.321074\pi\)
0.532974 + 0.846132i \(0.321074\pi\)
\(4\) 0 0
\(5\) −2.98774 −1.33616 −0.668078 0.744091i \(-0.732883\pi\)
−0.668078 + 0.744091i \(0.732883\pi\)
\(6\) 0 0
\(7\) −0.920155 −0.347786 −0.173893 0.984765i \(-0.555635\pi\)
−0.173893 + 0.984765i \(0.555635\pi\)
\(8\) 0 0
\(9\) 0.408737 0.136246
\(10\) 0 0
\(11\) −1.28823 −0.388415 −0.194208 0.980960i \(-0.562214\pi\)
−0.194208 + 0.980960i \(0.562214\pi\)
\(12\) 0 0
\(13\) 2.44378 0.677783 0.338891 0.940825i \(-0.389948\pi\)
0.338891 + 0.940825i \(0.389948\pi\)
\(14\) 0 0
\(15\) −5.51619 −1.42427
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 5.97874 1.37162 0.685809 0.727782i \(-0.259449\pi\)
0.685809 + 0.727782i \(0.259449\pi\)
\(20\) 0 0
\(21\) −1.69886 −0.370722
\(22\) 0 0
\(23\) 0.336054 0.0700722 0.0350361 0.999386i \(-0.488845\pi\)
0.0350361 + 0.999386i \(0.488845\pi\)
\(24\) 0 0
\(25\) 3.92657 0.785313
\(26\) 0 0
\(27\) −4.78419 −0.920717
\(28\) 0 0
\(29\) 0.998551 0.185426 0.0927131 0.995693i \(-0.470446\pi\)
0.0927131 + 0.995693i \(0.470446\pi\)
\(30\) 0 0
\(31\) 2.84063 0.510192 0.255096 0.966916i \(-0.417893\pi\)
0.255096 + 0.966916i \(0.417893\pi\)
\(32\) 0 0
\(33\) −2.37843 −0.414031
\(34\) 0 0
\(35\) 2.74918 0.464696
\(36\) 0 0
\(37\) −8.24001 −1.35465 −0.677325 0.735684i \(-0.736861\pi\)
−0.677325 + 0.735684i \(0.736861\pi\)
\(38\) 0 0
\(39\) 4.51190 0.722481
\(40\) 0 0
\(41\) 2.60887 0.407437 0.203718 0.979030i \(-0.434697\pi\)
0.203718 + 0.979030i \(0.434697\pi\)
\(42\) 0 0
\(43\) −10.5002 −1.60126 −0.800632 0.599156i \(-0.795503\pi\)
−0.800632 + 0.599156i \(0.795503\pi\)
\(44\) 0 0
\(45\) −1.22120 −0.182046
\(46\) 0 0
\(47\) −0.00237639 −0.000346633 0 −0.000173316 1.00000i \(-0.500055\pi\)
−0.000173316 1.00000i \(0.500055\pi\)
\(48\) 0 0
\(49\) −6.15331 −0.879045
\(50\) 0 0
\(51\) 1.84628 0.258530
\(52\) 0 0
\(53\) −5.66109 −0.777610 −0.388805 0.921320i \(-0.627112\pi\)
−0.388805 + 0.921320i \(0.627112\pi\)
\(54\) 0 0
\(55\) 3.84889 0.518984
\(56\) 0 0
\(57\) 11.0384 1.46207
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 11.4554 1.46671 0.733356 0.679845i \(-0.237953\pi\)
0.733356 + 0.679845i \(0.237953\pi\)
\(62\) 0 0
\(63\) −0.376102 −0.0473844
\(64\) 0 0
\(65\) −7.30137 −0.905624
\(66\) 0 0
\(67\) −12.2645 −1.49835 −0.749174 0.662373i \(-0.769549\pi\)
−0.749174 + 0.662373i \(0.769549\pi\)
\(68\) 0 0
\(69\) 0.620449 0.0746933
\(70\) 0 0
\(71\) 1.33343 0.158249 0.0791244 0.996865i \(-0.474788\pi\)
0.0791244 + 0.996865i \(0.474788\pi\)
\(72\) 0 0
\(73\) −8.50879 −0.995878 −0.497939 0.867212i \(-0.665910\pi\)
−0.497939 + 0.867212i \(0.665910\pi\)
\(74\) 0 0
\(75\) 7.24953 0.837103
\(76\) 0 0
\(77\) 1.18537 0.135085
\(78\) 0 0
\(79\) −9.03414 −1.01642 −0.508210 0.861233i \(-0.669693\pi\)
−0.508210 + 0.861233i \(0.669693\pi\)
\(80\) 0 0
\(81\) −10.0591 −1.11768
\(82\) 0 0
\(83\) −5.93365 −0.651302 −0.325651 0.945490i \(-0.605584\pi\)
−0.325651 + 0.945490i \(0.605584\pi\)
\(84\) 0 0
\(85\) −2.98774 −0.324065
\(86\) 0 0
\(87\) 1.84360 0.197655
\(88\) 0 0
\(89\) −12.5190 −1.32701 −0.663506 0.748171i \(-0.730932\pi\)
−0.663506 + 0.748171i \(0.730932\pi\)
\(90\) 0 0
\(91\) −2.24866 −0.235723
\(92\) 0 0
\(93\) 5.24459 0.543838
\(94\) 0 0
\(95\) −17.8629 −1.83270
\(96\) 0 0
\(97\) −0.539984 −0.0548271 −0.0274135 0.999624i \(-0.508727\pi\)
−0.0274135 + 0.999624i \(0.508727\pi\)
\(98\) 0 0
\(99\) −0.526547 −0.0529200
\(100\) 0 0
\(101\) −2.27122 −0.225994 −0.112997 0.993595i \(-0.536045\pi\)
−0.112997 + 0.993595i \(0.536045\pi\)
\(102\) 0 0
\(103\) 1.08582 0.106989 0.0534943 0.998568i \(-0.482964\pi\)
0.0534943 + 0.998568i \(0.482964\pi\)
\(104\) 0 0
\(105\) 5.07575 0.495342
\(106\) 0 0
\(107\) −9.95470 −0.962357 −0.481179 0.876623i \(-0.659791\pi\)
−0.481179 + 0.876623i \(0.659791\pi\)
\(108\) 0 0
\(109\) 14.7596 1.41371 0.706856 0.707358i \(-0.250113\pi\)
0.706856 + 0.707358i \(0.250113\pi\)
\(110\) 0 0
\(111\) −15.2133 −1.44399
\(112\) 0 0
\(113\) −18.4972 −1.74007 −0.870035 0.492991i \(-0.835904\pi\)
−0.870035 + 0.492991i \(0.835904\pi\)
\(114\) 0 0
\(115\) −1.00404 −0.0936273
\(116\) 0 0
\(117\) 0.998864 0.0923450
\(118\) 0 0
\(119\) −0.920155 −0.0843505
\(120\) 0 0
\(121\) −9.34047 −0.849133
\(122\) 0 0
\(123\) 4.81669 0.434306
\(124\) 0 0
\(125\) 3.20714 0.286855
\(126\) 0 0
\(127\) 2.39775 0.212766 0.106383 0.994325i \(-0.466073\pi\)
0.106383 + 0.994325i \(0.466073\pi\)
\(128\) 0 0
\(129\) −19.3863 −1.70687
\(130\) 0 0
\(131\) −7.39946 −0.646494 −0.323247 0.946315i \(-0.604774\pi\)
−0.323247 + 0.946315i \(0.604774\pi\)
\(132\) 0 0
\(133\) −5.50137 −0.477029
\(134\) 0 0
\(135\) 14.2939 1.23022
\(136\) 0 0
\(137\) 3.23075 0.276022 0.138011 0.990431i \(-0.455929\pi\)
0.138011 + 0.990431i \(0.455929\pi\)
\(138\) 0 0
\(139\) 22.1902 1.88215 0.941076 0.338196i \(-0.109817\pi\)
0.941076 + 0.338196i \(0.109817\pi\)
\(140\) 0 0
\(141\) −0.00438748 −0.000369493 0
\(142\) 0 0
\(143\) −3.14815 −0.263261
\(144\) 0 0
\(145\) −2.98341 −0.247758
\(146\) 0 0
\(147\) −11.3607 −0.937016
\(148\) 0 0
\(149\) 0.187452 0.0153567 0.00767833 0.999971i \(-0.497556\pi\)
0.00767833 + 0.999971i \(0.497556\pi\)
\(150\) 0 0
\(151\) 16.8102 1.36800 0.683999 0.729483i \(-0.260239\pi\)
0.683999 + 0.729483i \(0.260239\pi\)
\(152\) 0 0
\(153\) 0.408737 0.0330444
\(154\) 0 0
\(155\) −8.48705 −0.681696
\(156\) 0 0
\(157\) −6.69664 −0.534450 −0.267225 0.963634i \(-0.586107\pi\)
−0.267225 + 0.963634i \(0.586107\pi\)
\(158\) 0 0
\(159\) −10.4519 −0.828892
\(160\) 0 0
\(161\) −0.309222 −0.0243701
\(162\) 0 0
\(163\) 19.7340 1.54568 0.772841 0.634599i \(-0.218835\pi\)
0.772841 + 0.634599i \(0.218835\pi\)
\(164\) 0 0
\(165\) 7.10611 0.553210
\(166\) 0 0
\(167\) −19.9146 −1.54104 −0.770519 0.637416i \(-0.780003\pi\)
−0.770519 + 0.637416i \(0.780003\pi\)
\(168\) 0 0
\(169\) −7.02794 −0.540610
\(170\) 0 0
\(171\) 2.44373 0.186877
\(172\) 0 0
\(173\) −25.0069 −1.90124 −0.950618 0.310363i \(-0.899549\pi\)
−0.950618 + 0.310363i \(0.899549\pi\)
\(174\) 0 0
\(175\) −3.61305 −0.273121
\(176\) 0 0
\(177\) −1.84628 −0.138775
\(178\) 0 0
\(179\) −8.80769 −0.658318 −0.329159 0.944275i \(-0.606765\pi\)
−0.329159 + 0.944275i \(0.606765\pi\)
\(180\) 0 0
\(181\) 16.1467 1.20017 0.600087 0.799935i \(-0.295133\pi\)
0.600087 + 0.799935i \(0.295133\pi\)
\(182\) 0 0
\(183\) 21.1498 1.56344
\(184\) 0 0
\(185\) 24.6190 1.81002
\(186\) 0 0
\(187\) −1.28823 −0.0942046
\(188\) 0 0
\(189\) 4.40219 0.320213
\(190\) 0 0
\(191\) −3.35701 −0.242905 −0.121453 0.992597i \(-0.538755\pi\)
−0.121453 + 0.992597i \(0.538755\pi\)
\(192\) 0 0
\(193\) −23.5565 −1.69563 −0.847815 0.530292i \(-0.822082\pi\)
−0.847815 + 0.530292i \(0.822082\pi\)
\(194\) 0 0
\(195\) −13.4804 −0.965348
\(196\) 0 0
\(197\) −23.9599 −1.70707 −0.853536 0.521033i \(-0.825547\pi\)
−0.853536 + 0.521033i \(0.825547\pi\)
\(198\) 0 0
\(199\) 7.03038 0.498370 0.249185 0.968456i \(-0.419837\pi\)
0.249185 + 0.968456i \(0.419837\pi\)
\(200\) 0 0
\(201\) −22.6437 −1.59716
\(202\) 0 0
\(203\) −0.918821 −0.0644886
\(204\) 0 0
\(205\) −7.79461 −0.544399
\(206\) 0 0
\(207\) 0.137358 0.00954703
\(208\) 0 0
\(209\) −7.70198 −0.532757
\(210\) 0 0
\(211\) −10.8192 −0.744823 −0.372411 0.928068i \(-0.621469\pi\)
−0.372411 + 0.928068i \(0.621469\pi\)
\(212\) 0 0
\(213\) 2.46188 0.168685
\(214\) 0 0
\(215\) 31.3718 2.13954
\(216\) 0 0
\(217\) −2.61382 −0.177438
\(218\) 0 0
\(219\) −15.7096 −1.06155
\(220\) 0 0
\(221\) 2.44378 0.164386
\(222\) 0 0
\(223\) 20.7214 1.38761 0.693804 0.720164i \(-0.255934\pi\)
0.693804 + 0.720164i \(0.255934\pi\)
\(224\) 0 0
\(225\) 1.60493 0.106996
\(226\) 0 0
\(227\) −21.8150 −1.44791 −0.723956 0.689846i \(-0.757678\pi\)
−0.723956 + 0.689846i \(0.757678\pi\)
\(228\) 0 0
\(229\) 29.7000 1.96263 0.981316 0.192403i \(-0.0616282\pi\)
0.981316 + 0.192403i \(0.0616282\pi\)
\(230\) 0 0
\(231\) 2.18852 0.143994
\(232\) 0 0
\(233\) −17.7710 −1.16422 −0.582108 0.813112i \(-0.697772\pi\)
−0.582108 + 0.813112i \(0.697772\pi\)
\(234\) 0 0
\(235\) 0.00710004 0.000463155 0
\(236\) 0 0
\(237\) −16.6795 −1.08345
\(238\) 0 0
\(239\) −3.78169 −0.244617 −0.122309 0.992492i \(-0.539030\pi\)
−0.122309 + 0.992492i \(0.539030\pi\)
\(240\) 0 0
\(241\) −11.7821 −0.758952 −0.379476 0.925202i \(-0.623896\pi\)
−0.379476 + 0.925202i \(0.623896\pi\)
\(242\) 0 0
\(243\) −4.21940 −0.270675
\(244\) 0 0
\(245\) 18.3845 1.17454
\(246\) 0 0
\(247\) 14.6107 0.929659
\(248\) 0 0
\(249\) −10.9552 −0.694255
\(250\) 0 0
\(251\) 20.6042 1.30052 0.650261 0.759710i \(-0.274659\pi\)
0.650261 + 0.759710i \(0.274659\pi\)
\(252\) 0 0
\(253\) −0.432915 −0.0272171
\(254\) 0 0
\(255\) −5.51619 −0.345437
\(256\) 0 0
\(257\) 16.9272 1.05589 0.527945 0.849278i \(-0.322963\pi\)
0.527945 + 0.849278i \(0.322963\pi\)
\(258\) 0 0
\(259\) 7.58208 0.471128
\(260\) 0 0
\(261\) 0.408145 0.0252635
\(262\) 0 0
\(263\) 13.5433 0.835118 0.417559 0.908650i \(-0.362886\pi\)
0.417559 + 0.908650i \(0.362886\pi\)
\(264\) 0 0
\(265\) 16.9138 1.03901
\(266\) 0 0
\(267\) −23.1136 −1.41453
\(268\) 0 0
\(269\) 24.4540 1.49099 0.745493 0.666514i \(-0.232214\pi\)
0.745493 + 0.666514i \(0.232214\pi\)
\(270\) 0 0
\(271\) 0.620782 0.0377098 0.0188549 0.999822i \(-0.493998\pi\)
0.0188549 + 0.999822i \(0.493998\pi\)
\(272\) 0 0
\(273\) −4.15164 −0.251269
\(274\) 0 0
\(275\) −5.05831 −0.305028
\(276\) 0 0
\(277\) −26.4470 −1.58904 −0.794522 0.607236i \(-0.792278\pi\)
−0.794522 + 0.607236i \(0.792278\pi\)
\(278\) 0 0
\(279\) 1.16107 0.0695115
\(280\) 0 0
\(281\) 15.6953 0.936306 0.468153 0.883647i \(-0.344920\pi\)
0.468153 + 0.883647i \(0.344920\pi\)
\(282\) 0 0
\(283\) 16.0161 0.952059 0.476030 0.879429i \(-0.342075\pi\)
0.476030 + 0.879429i \(0.342075\pi\)
\(284\) 0 0
\(285\) −32.9799 −1.95356
\(286\) 0 0
\(287\) −2.40056 −0.141701
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −0.996960 −0.0584428
\(292\) 0 0
\(293\) 17.2126 1.00557 0.502785 0.864411i \(-0.332309\pi\)
0.502785 + 0.864411i \(0.332309\pi\)
\(294\) 0 0
\(295\) 2.98774 0.173953
\(296\) 0 0
\(297\) 6.16313 0.357621
\(298\) 0 0
\(299\) 0.821243 0.0474937
\(300\) 0 0
\(301\) 9.66181 0.556897
\(302\) 0 0
\(303\) −4.19329 −0.240898
\(304\) 0 0
\(305\) −34.2256 −1.95975
\(306\) 0 0
\(307\) −5.07638 −0.289724 −0.144862 0.989452i \(-0.546274\pi\)
−0.144862 + 0.989452i \(0.546274\pi\)
\(308\) 0 0
\(309\) 2.00472 0.114044
\(310\) 0 0
\(311\) −8.58385 −0.486745 −0.243373 0.969933i \(-0.578254\pi\)
−0.243373 + 0.969933i \(0.578254\pi\)
\(312\) 0 0
\(313\) 32.5346 1.83897 0.919483 0.393130i \(-0.128608\pi\)
0.919483 + 0.393130i \(0.128608\pi\)
\(314\) 0 0
\(315\) 1.12369 0.0633129
\(316\) 0 0
\(317\) −35.2574 −1.98025 −0.990126 0.140182i \(-0.955231\pi\)
−0.990126 + 0.140182i \(0.955231\pi\)
\(318\) 0 0
\(319\) −1.28636 −0.0720224
\(320\) 0 0
\(321\) −18.3791 −1.02582
\(322\) 0 0
\(323\) 5.97874 0.332666
\(324\) 0 0
\(325\) 9.59567 0.532272
\(326\) 0 0
\(327\) 27.2503 1.50694
\(328\) 0 0
\(329\) 0.00218665 0.000120554 0
\(330\) 0 0
\(331\) −0.876073 −0.0481533 −0.0240767 0.999710i \(-0.507665\pi\)
−0.0240767 + 0.999710i \(0.507665\pi\)
\(332\) 0 0
\(333\) −3.36800 −0.184565
\(334\) 0 0
\(335\) 36.6431 2.00203
\(336\) 0 0
\(337\) 12.6057 0.686674 0.343337 0.939212i \(-0.388443\pi\)
0.343337 + 0.939212i \(0.388443\pi\)
\(338\) 0 0
\(339\) −34.1509 −1.85482
\(340\) 0 0
\(341\) −3.65938 −0.198167
\(342\) 0 0
\(343\) 12.1031 0.653505
\(344\) 0 0
\(345\) −1.85374 −0.0998019
\(346\) 0 0
\(347\) −0.0255483 −0.00137150 −0.000685751 1.00000i \(-0.500218\pi\)
−0.000685751 1.00000i \(0.500218\pi\)
\(348\) 0 0
\(349\) −17.7755 −0.951503 −0.475751 0.879580i \(-0.657824\pi\)
−0.475751 + 0.879580i \(0.657824\pi\)
\(350\) 0 0
\(351\) −11.6915 −0.624046
\(352\) 0 0
\(353\) 15.4502 0.822332 0.411166 0.911560i \(-0.365122\pi\)
0.411166 + 0.911560i \(0.365122\pi\)
\(354\) 0 0
\(355\) −3.98393 −0.211445
\(356\) 0 0
\(357\) −1.69886 −0.0899132
\(358\) 0 0
\(359\) 24.6396 1.30043 0.650214 0.759751i \(-0.274679\pi\)
0.650214 + 0.759751i \(0.274679\pi\)
\(360\) 0 0
\(361\) 16.7454 0.881335
\(362\) 0 0
\(363\) −17.2451 −0.905132
\(364\) 0 0
\(365\) 25.4220 1.33065
\(366\) 0 0
\(367\) 15.9405 0.832087 0.416043 0.909345i \(-0.363416\pi\)
0.416043 + 0.909345i \(0.363416\pi\)
\(368\) 0 0
\(369\) 1.06634 0.0555115
\(370\) 0 0
\(371\) 5.20908 0.270442
\(372\) 0 0
\(373\) 20.4515 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(374\) 0 0
\(375\) 5.92126 0.305773
\(376\) 0 0
\(377\) 2.44024 0.125679
\(378\) 0 0
\(379\) −17.6820 −0.908263 −0.454131 0.890935i \(-0.650050\pi\)
−0.454131 + 0.890935i \(0.650050\pi\)
\(380\) 0 0
\(381\) 4.42691 0.226798
\(382\) 0 0
\(383\) −14.5516 −0.743554 −0.371777 0.928322i \(-0.621251\pi\)
−0.371777 + 0.928322i \(0.621251\pi\)
\(384\) 0 0
\(385\) −3.54157 −0.180495
\(386\) 0 0
\(387\) −4.29182 −0.218166
\(388\) 0 0
\(389\) −10.1547 −0.514864 −0.257432 0.966296i \(-0.582876\pi\)
−0.257432 + 0.966296i \(0.582876\pi\)
\(390\) 0 0
\(391\) 0.336054 0.0169950
\(392\) 0 0
\(393\) −13.6615 −0.689129
\(394\) 0 0
\(395\) 26.9916 1.35810
\(396\) 0 0
\(397\) −26.4963 −1.32981 −0.664906 0.746927i \(-0.731528\pi\)
−0.664906 + 0.746927i \(0.731528\pi\)
\(398\) 0 0
\(399\) −10.1571 −0.508489
\(400\) 0 0
\(401\) −8.20993 −0.409984 −0.204992 0.978764i \(-0.565717\pi\)
−0.204992 + 0.978764i \(0.565717\pi\)
\(402\) 0 0
\(403\) 6.94188 0.345799
\(404\) 0 0
\(405\) 30.0541 1.49340
\(406\) 0 0
\(407\) 10.6150 0.526167
\(408\) 0 0
\(409\) −13.9582 −0.690187 −0.345093 0.938568i \(-0.612153\pi\)
−0.345093 + 0.938568i \(0.612153\pi\)
\(410\) 0 0
\(411\) 5.96486 0.294225
\(412\) 0 0
\(413\) 0.920155 0.0452779
\(414\) 0 0
\(415\) 17.7282 0.870242
\(416\) 0 0
\(417\) 40.9693 2.00628
\(418\) 0 0
\(419\) 5.23425 0.255710 0.127855 0.991793i \(-0.459191\pi\)
0.127855 + 0.991793i \(0.459191\pi\)
\(420\) 0 0
\(421\) 11.5396 0.562406 0.281203 0.959648i \(-0.409267\pi\)
0.281203 + 0.959648i \(0.409267\pi\)
\(422\) 0 0
\(423\) −0.000971321 0 −4.72272e−5 0
\(424\) 0 0
\(425\) 3.92657 0.190466
\(426\) 0 0
\(427\) −10.5407 −0.510101
\(428\) 0 0
\(429\) −5.81235 −0.280623
\(430\) 0 0
\(431\) −13.6113 −0.655631 −0.327816 0.944742i \(-0.606312\pi\)
−0.327816 + 0.944742i \(0.606312\pi\)
\(432\) 0 0
\(433\) 24.0180 1.15423 0.577116 0.816662i \(-0.304178\pi\)
0.577116 + 0.816662i \(0.304178\pi\)
\(434\) 0 0
\(435\) −5.50819 −0.264098
\(436\) 0 0
\(437\) 2.00918 0.0961122
\(438\) 0 0
\(439\) −20.6067 −0.983503 −0.491752 0.870735i \(-0.663643\pi\)
−0.491752 + 0.870735i \(0.663643\pi\)
\(440\) 0 0
\(441\) −2.51509 −0.119766
\(442\) 0 0
\(443\) 4.90401 0.232997 0.116498 0.993191i \(-0.462833\pi\)
0.116498 + 0.993191i \(0.462833\pi\)
\(444\) 0 0
\(445\) 37.4035 1.77310
\(446\) 0 0
\(447\) 0.346088 0.0163694
\(448\) 0 0
\(449\) 37.7661 1.78229 0.891147 0.453716i \(-0.149902\pi\)
0.891147 + 0.453716i \(0.149902\pi\)
\(450\) 0 0
\(451\) −3.36082 −0.158255
\(452\) 0 0
\(453\) 31.0363 1.45821
\(454\) 0 0
\(455\) 6.71839 0.314963
\(456\) 0 0
\(457\) 17.1073 0.800246 0.400123 0.916461i \(-0.368967\pi\)
0.400123 + 0.916461i \(0.368967\pi\)
\(458\) 0 0
\(459\) −4.78419 −0.223307
\(460\) 0 0
\(461\) 4.11911 0.191846 0.0959230 0.995389i \(-0.469420\pi\)
0.0959230 + 0.995389i \(0.469420\pi\)
\(462\) 0 0
\(463\) 11.2807 0.524257 0.262129 0.965033i \(-0.415576\pi\)
0.262129 + 0.965033i \(0.415576\pi\)
\(464\) 0 0
\(465\) −15.6694 −0.726653
\(466\) 0 0
\(467\) 4.57635 0.211768 0.105884 0.994378i \(-0.466233\pi\)
0.105884 + 0.994378i \(0.466233\pi\)
\(468\) 0 0
\(469\) 11.2853 0.521105
\(470\) 0 0
\(471\) −12.3639 −0.569696
\(472\) 0 0
\(473\) 13.5266 0.621956
\(474\) 0 0
\(475\) 23.4759 1.07715
\(476\) 0 0
\(477\) −2.31390 −0.105946
\(478\) 0 0
\(479\) −15.7294 −0.718694 −0.359347 0.933204i \(-0.617001\pi\)
−0.359347 + 0.933204i \(0.617001\pi\)
\(480\) 0 0
\(481\) −20.1368 −0.918158
\(482\) 0 0
\(483\) −0.570909 −0.0259773
\(484\) 0 0
\(485\) 1.61333 0.0732575
\(486\) 0 0
\(487\) −39.3417 −1.78274 −0.891370 0.453276i \(-0.850255\pi\)
−0.891370 + 0.453276i \(0.850255\pi\)
\(488\) 0 0
\(489\) 36.4343 1.64762
\(490\) 0 0
\(491\) −17.6586 −0.796919 −0.398460 0.917186i \(-0.630455\pi\)
−0.398460 + 0.917186i \(0.630455\pi\)
\(492\) 0 0
\(493\) 0.998551 0.0449725
\(494\) 0 0
\(495\) 1.57318 0.0707093
\(496\) 0 0
\(497\) −1.22696 −0.0550367
\(498\) 0 0
\(499\) −32.4805 −1.45403 −0.727013 0.686623i \(-0.759092\pi\)
−0.727013 + 0.686623i \(0.759092\pi\)
\(500\) 0 0
\(501\) −36.7679 −1.64267
\(502\) 0 0
\(503\) −14.5928 −0.650659 −0.325330 0.945601i \(-0.605475\pi\)
−0.325330 + 0.945601i \(0.605475\pi\)
\(504\) 0 0
\(505\) 6.78579 0.301964
\(506\) 0 0
\(507\) −12.9755 −0.576263
\(508\) 0 0
\(509\) 19.1907 0.850612 0.425306 0.905050i \(-0.360167\pi\)
0.425306 + 0.905050i \(0.360167\pi\)
\(510\) 0 0
\(511\) 7.82940 0.346352
\(512\) 0 0
\(513\) −28.6034 −1.26287
\(514\) 0 0
\(515\) −3.24413 −0.142953
\(516\) 0 0
\(517\) 0.00306134 0.000134638 0
\(518\) 0 0
\(519\) −46.1696 −2.02662
\(520\) 0 0
\(521\) 36.0003 1.57720 0.788600 0.614906i \(-0.210806\pi\)
0.788600 + 0.614906i \(0.210806\pi\)
\(522\) 0 0
\(523\) 0.105819 0.00462716 0.00231358 0.999997i \(-0.499264\pi\)
0.00231358 + 0.999997i \(0.499264\pi\)
\(524\) 0 0
\(525\) −6.67069 −0.291133
\(526\) 0 0
\(527\) 2.84063 0.123740
\(528\) 0 0
\(529\) −22.8871 −0.995090
\(530\) 0 0
\(531\) −0.408737 −0.0177377
\(532\) 0 0
\(533\) 6.37550 0.276154
\(534\) 0 0
\(535\) 29.7420 1.28586
\(536\) 0 0
\(537\) −16.2614 −0.701733
\(538\) 0 0
\(539\) 7.92687 0.341435
\(540\) 0 0
\(541\) −4.64850 −0.199854 −0.0999272 0.994995i \(-0.531861\pi\)
−0.0999272 + 0.994995i \(0.531861\pi\)
\(542\) 0 0
\(543\) 29.8112 1.27932
\(544\) 0 0
\(545\) −44.0977 −1.88894
\(546\) 0 0
\(547\) 32.9851 1.41034 0.705170 0.709038i \(-0.250870\pi\)
0.705170 + 0.709038i \(0.250870\pi\)
\(548\) 0 0
\(549\) 4.68224 0.199833
\(550\) 0 0
\(551\) 5.97008 0.254334
\(552\) 0 0
\(553\) 8.31281 0.353497
\(554\) 0 0
\(555\) 45.4534 1.92939
\(556\) 0 0
\(557\) −40.4474 −1.71381 −0.856905 0.515474i \(-0.827616\pi\)
−0.856905 + 0.515474i \(0.827616\pi\)
\(558\) 0 0
\(559\) −25.6602 −1.08531
\(560\) 0 0
\(561\) −2.37843 −0.100417
\(562\) 0 0
\(563\) 13.0428 0.549687 0.274844 0.961489i \(-0.411374\pi\)
0.274844 + 0.961489i \(0.411374\pi\)
\(564\) 0 0
\(565\) 55.2647 2.32500
\(566\) 0 0
\(567\) 9.25597 0.388714
\(568\) 0 0
\(569\) −42.5410 −1.78341 −0.891705 0.452618i \(-0.850490\pi\)
−0.891705 + 0.452618i \(0.850490\pi\)
\(570\) 0 0
\(571\) −38.8720 −1.62674 −0.813371 0.581746i \(-0.802370\pi\)
−0.813371 + 0.581746i \(0.802370\pi\)
\(572\) 0 0
\(573\) −6.19798 −0.258924
\(574\) 0 0
\(575\) 1.31954 0.0550286
\(576\) 0 0
\(577\) −13.8267 −0.575615 −0.287808 0.957688i \(-0.592926\pi\)
−0.287808 + 0.957688i \(0.592926\pi\)
\(578\) 0 0
\(579\) −43.4917 −1.80745
\(580\) 0 0
\(581\) 5.45988 0.226514
\(582\) 0 0
\(583\) 7.29277 0.302036
\(584\) 0 0
\(585\) −2.98434 −0.123387
\(586\) 0 0
\(587\) −13.5809 −0.560542 −0.280271 0.959921i \(-0.590424\pi\)
−0.280271 + 0.959921i \(0.590424\pi\)
\(588\) 0 0
\(589\) 16.9834 0.699789
\(590\) 0 0
\(591\) −44.2366 −1.81965
\(592\) 0 0
\(593\) −4.64504 −0.190749 −0.0953744 0.995441i \(-0.530405\pi\)
−0.0953744 + 0.995441i \(0.530405\pi\)
\(594\) 0 0
\(595\) 2.74918 0.112705
\(596\) 0 0
\(597\) 12.9800 0.531237
\(598\) 0 0
\(599\) 9.78858 0.399950 0.199975 0.979801i \(-0.435914\pi\)
0.199975 + 0.979801i \(0.435914\pi\)
\(600\) 0 0
\(601\) 6.02445 0.245742 0.122871 0.992423i \(-0.460790\pi\)
0.122871 + 0.992423i \(0.460790\pi\)
\(602\) 0 0
\(603\) −5.01296 −0.204144
\(604\) 0 0
\(605\) 27.9069 1.13457
\(606\) 0 0
\(607\) −21.2650 −0.863121 −0.431561 0.902084i \(-0.642037\pi\)
−0.431561 + 0.902084i \(0.642037\pi\)
\(608\) 0 0
\(609\) −1.69640 −0.0687415
\(610\) 0 0
\(611\) −0.00580739 −0.000234942 0
\(612\) 0 0
\(613\) 23.7254 0.958258 0.479129 0.877744i \(-0.340953\pi\)
0.479129 + 0.877744i \(0.340953\pi\)
\(614\) 0 0
\(615\) −14.3910 −0.580301
\(616\) 0 0
\(617\) 18.4226 0.741666 0.370833 0.928700i \(-0.379072\pi\)
0.370833 + 0.928700i \(0.379072\pi\)
\(618\) 0 0
\(619\) 16.0814 0.646365 0.323183 0.946337i \(-0.395247\pi\)
0.323183 + 0.946337i \(0.395247\pi\)
\(620\) 0 0
\(621\) −1.60775 −0.0645166
\(622\) 0 0
\(623\) 11.5194 0.461516
\(624\) 0 0
\(625\) −29.2149 −1.16860
\(626\) 0 0
\(627\) −14.2200 −0.567892
\(628\) 0 0
\(629\) −8.24001 −0.328551
\(630\) 0 0
\(631\) 32.5260 1.29484 0.647419 0.762134i \(-0.275848\pi\)
0.647419 + 0.762134i \(0.275848\pi\)
\(632\) 0 0
\(633\) −19.9752 −0.793943
\(634\) 0 0
\(635\) −7.16385 −0.284289
\(636\) 0 0
\(637\) −15.0374 −0.595802
\(638\) 0 0
\(639\) 0.545022 0.0215607
\(640\) 0 0
\(641\) −19.9288 −0.787139 −0.393570 0.919295i \(-0.628760\pi\)
−0.393570 + 0.919295i \(0.628760\pi\)
\(642\) 0 0
\(643\) 1.39349 0.0549538 0.0274769 0.999622i \(-0.491253\pi\)
0.0274769 + 0.999622i \(0.491253\pi\)
\(644\) 0 0
\(645\) 57.9210 2.28064
\(646\) 0 0
\(647\) −26.8709 −1.05641 −0.528203 0.849118i \(-0.677134\pi\)
−0.528203 + 0.849118i \(0.677134\pi\)
\(648\) 0 0
\(649\) 1.28823 0.0505674
\(650\) 0 0
\(651\) −4.82583 −0.189139
\(652\) 0 0
\(653\) 1.64819 0.0644985 0.0322492 0.999480i \(-0.489733\pi\)
0.0322492 + 0.999480i \(0.489733\pi\)
\(654\) 0 0
\(655\) 22.1076 0.863817
\(656\) 0 0
\(657\) −3.47786 −0.135684
\(658\) 0 0
\(659\) 45.5279 1.77352 0.886758 0.462234i \(-0.152952\pi\)
0.886758 + 0.462234i \(0.152952\pi\)
\(660\) 0 0
\(661\) −16.6899 −0.649163 −0.324582 0.945858i \(-0.605223\pi\)
−0.324582 + 0.945858i \(0.605223\pi\)
\(662\) 0 0
\(663\) 4.51190 0.175227
\(664\) 0 0
\(665\) 16.4366 0.637386
\(666\) 0 0
\(667\) 0.335567 0.0129932
\(668\) 0 0
\(669\) 38.2574 1.47912
\(670\) 0 0
\(671\) −14.7571 −0.569693
\(672\) 0 0
\(673\) 13.3277 0.513747 0.256873 0.966445i \(-0.417308\pi\)
0.256873 + 0.966445i \(0.417308\pi\)
\(674\) 0 0
\(675\) −18.7854 −0.723051
\(676\) 0 0
\(677\) 28.1095 1.08034 0.540168 0.841557i \(-0.318361\pi\)
0.540168 + 0.841557i \(0.318361\pi\)
\(678\) 0 0
\(679\) 0.496869 0.0190681
\(680\) 0 0
\(681\) −40.2765 −1.54340
\(682\) 0 0
\(683\) −45.4010 −1.73722 −0.868610 0.495496i \(-0.834986\pi\)
−0.868610 + 0.495496i \(0.834986\pi\)
\(684\) 0 0
\(685\) −9.65263 −0.368808
\(686\) 0 0
\(687\) 54.8344 2.09206
\(688\) 0 0
\(689\) −13.8345 −0.527051
\(690\) 0 0
\(691\) −12.4877 −0.475056 −0.237528 0.971381i \(-0.576337\pi\)
−0.237528 + 0.971381i \(0.576337\pi\)
\(692\) 0 0
\(693\) 0.484505 0.0184048
\(694\) 0 0
\(695\) −66.2986 −2.51485
\(696\) 0 0
\(697\) 2.60887 0.0988179
\(698\) 0 0
\(699\) −32.8101 −1.24099
\(700\) 0 0
\(701\) 30.6614 1.15807 0.579033 0.815304i \(-0.303430\pi\)
0.579033 + 0.815304i \(0.303430\pi\)
\(702\) 0 0
\(703\) −49.2649 −1.85806
\(704\) 0 0
\(705\) 0.0131086 0.000493700 0
\(706\) 0 0
\(707\) 2.08987 0.0785977
\(708\) 0 0
\(709\) −22.9981 −0.863712 −0.431856 0.901942i \(-0.642141\pi\)
−0.431856 + 0.901942i \(0.642141\pi\)
\(710\) 0 0
\(711\) −3.69259 −0.138483
\(712\) 0 0
\(713\) 0.954606 0.0357503
\(714\) 0 0
\(715\) 9.40583 0.351758
\(716\) 0 0
\(717\) −6.98205 −0.260749
\(718\) 0 0
\(719\) −23.4124 −0.873135 −0.436568 0.899671i \(-0.643806\pi\)
−0.436568 + 0.899671i \(0.643806\pi\)
\(720\) 0 0
\(721\) −0.999119 −0.0372091
\(722\) 0 0
\(723\) −21.7530 −0.809004
\(724\) 0 0
\(725\) 3.92088 0.145618
\(726\) 0 0
\(727\) 12.3768 0.459031 0.229515 0.973305i \(-0.426286\pi\)
0.229515 + 0.973305i \(0.426286\pi\)
\(728\) 0 0
\(729\) 22.3873 0.829158
\(730\) 0 0
\(731\) −10.5002 −0.388364
\(732\) 0 0
\(733\) −14.1601 −0.523015 −0.261508 0.965201i \(-0.584220\pi\)
−0.261508 + 0.965201i \(0.584220\pi\)
\(734\) 0 0
\(735\) 33.9428 1.25200
\(736\) 0 0
\(737\) 15.7995 0.581982
\(738\) 0 0
\(739\) −32.7321 −1.20407 −0.602036 0.798469i \(-0.705644\pi\)
−0.602036 + 0.798469i \(0.705644\pi\)
\(740\) 0 0
\(741\) 26.9755 0.990968
\(742\) 0 0
\(743\) −44.2542 −1.62353 −0.811765 0.583984i \(-0.801493\pi\)
−0.811765 + 0.583984i \(0.801493\pi\)
\(744\) 0 0
\(745\) −0.560057 −0.0205189
\(746\) 0 0
\(747\) −2.42530 −0.0887372
\(748\) 0 0
\(749\) 9.15987 0.334694
\(750\) 0 0
\(751\) −29.7398 −1.08522 −0.542611 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(752\) 0 0
\(753\) 38.0410 1.38629
\(754\) 0 0
\(755\) −50.2245 −1.82786
\(756\) 0 0
\(757\) 27.9286 1.01508 0.507541 0.861628i \(-0.330555\pi\)
0.507541 + 0.861628i \(0.330555\pi\)
\(758\) 0 0
\(759\) −0.799280 −0.0290120
\(760\) 0 0
\(761\) −6.86737 −0.248942 −0.124471 0.992223i \(-0.539723\pi\)
−0.124471 + 0.992223i \(0.539723\pi\)
\(762\) 0 0
\(763\) −13.5811 −0.491669
\(764\) 0 0
\(765\) −1.22120 −0.0441525
\(766\) 0 0
\(767\) −2.44378 −0.0882398
\(768\) 0 0
\(769\) 38.8947 1.40258 0.701290 0.712876i \(-0.252608\pi\)
0.701290 + 0.712876i \(0.252608\pi\)
\(770\) 0 0
\(771\) 31.2523 1.12552
\(772\) 0 0
\(773\) 28.4472 1.02317 0.511587 0.859232i \(-0.329058\pi\)
0.511587 + 0.859232i \(0.329058\pi\)
\(774\) 0 0
\(775\) 11.1539 0.400661
\(776\) 0 0
\(777\) 13.9986 0.502198
\(778\) 0 0
\(779\) 15.5977 0.558847
\(780\) 0 0
\(781\) −1.71776 −0.0614663
\(782\) 0 0
\(783\) −4.77725 −0.170725
\(784\) 0 0
\(785\) 20.0078 0.714109
\(786\) 0 0
\(787\) 18.0141 0.642134 0.321067 0.947056i \(-0.395958\pi\)
0.321067 + 0.947056i \(0.395958\pi\)
\(788\) 0 0
\(789\) 25.0048 0.890193
\(790\) 0 0
\(791\) 17.0203 0.605172
\(792\) 0 0
\(793\) 27.9944 0.994112
\(794\) 0 0
\(795\) 31.2276 1.10753
\(796\) 0 0
\(797\) −2.01310 −0.0713077 −0.0356538 0.999364i \(-0.511351\pi\)
−0.0356538 + 0.999364i \(0.511351\pi\)
\(798\) 0 0
\(799\) −0.00237639 −8.40708e−5 0
\(800\) 0 0
\(801\) −5.11699 −0.180800
\(802\) 0 0
\(803\) 10.9613 0.386814
\(804\) 0 0
\(805\) 0.923874 0.0325623
\(806\) 0 0
\(807\) 45.1488 1.58931
\(808\) 0 0
\(809\) 21.6206 0.760139 0.380070 0.924958i \(-0.375900\pi\)
0.380070 + 0.924958i \(0.375900\pi\)
\(810\) 0 0
\(811\) −11.9880 −0.420956 −0.210478 0.977599i \(-0.567502\pi\)
−0.210478 + 0.977599i \(0.567502\pi\)
\(812\) 0 0
\(813\) 1.14613 0.0401967
\(814\) 0 0
\(815\) −58.9599 −2.06527
\(816\) 0 0
\(817\) −62.7780 −2.19632
\(818\) 0 0
\(819\) −0.919110 −0.0321163
\(820\) 0 0
\(821\) −29.8778 −1.04274 −0.521372 0.853330i \(-0.674579\pi\)
−0.521372 + 0.853330i \(0.674579\pi\)
\(822\) 0 0
\(823\) −27.2418 −0.949591 −0.474795 0.880096i \(-0.657478\pi\)
−0.474795 + 0.880096i \(0.657478\pi\)
\(824\) 0 0
\(825\) −9.33905 −0.325144
\(826\) 0 0
\(827\) −54.7153 −1.90264 −0.951318 0.308210i \(-0.900270\pi\)
−0.951318 + 0.308210i \(0.900270\pi\)
\(828\) 0 0
\(829\) 37.1651 1.29080 0.645399 0.763845i \(-0.276691\pi\)
0.645399 + 0.763845i \(0.276691\pi\)
\(830\) 0 0
\(831\) −48.8284 −1.69384
\(832\) 0 0
\(833\) −6.15331 −0.213200
\(834\) 0 0
\(835\) 59.4996 2.05907
\(836\) 0 0
\(837\) −13.5901 −0.469743
\(838\) 0 0
\(839\) −44.6254 −1.54064 −0.770320 0.637658i \(-0.779903\pi\)
−0.770320 + 0.637658i \(0.779903\pi\)
\(840\) 0 0
\(841\) −28.0029 −0.965617
\(842\) 0 0
\(843\) 28.9780 0.998054
\(844\) 0 0
\(845\) 20.9976 0.722340
\(846\) 0 0
\(847\) 8.59468 0.295317
\(848\) 0 0
\(849\) 29.5702 1.01485
\(850\) 0 0
\(851\) −2.76909 −0.0949232
\(852\) 0 0
\(853\) 11.0605 0.378705 0.189352 0.981909i \(-0.439361\pi\)
0.189352 + 0.981909i \(0.439361\pi\)
\(854\) 0 0
\(855\) −7.30123 −0.249697
\(856\) 0 0
\(857\) 51.6770 1.76525 0.882626 0.470076i \(-0.155773\pi\)
0.882626 + 0.470076i \(0.155773\pi\)
\(858\) 0 0
\(859\) 49.4190 1.68616 0.843078 0.537791i \(-0.180741\pi\)
0.843078 + 0.537791i \(0.180741\pi\)
\(860\) 0 0
\(861\) −4.43210 −0.151046
\(862\) 0 0
\(863\) 7.90264 0.269009 0.134505 0.990913i \(-0.457056\pi\)
0.134505 + 0.990913i \(0.457056\pi\)
\(864\) 0 0
\(865\) 74.7139 2.54035
\(866\) 0 0
\(867\) 1.84628 0.0627028
\(868\) 0 0
\(869\) 11.6380 0.394793
\(870\) 0 0
\(871\) −29.9718 −1.01555
\(872\) 0 0
\(873\) −0.220712 −0.00746996
\(874\) 0 0
\(875\) −2.95106 −0.0997642
\(876\) 0 0
\(877\) 41.6026 1.40482 0.702410 0.711773i \(-0.252107\pi\)
0.702410 + 0.711773i \(0.252107\pi\)
\(878\) 0 0
\(879\) 31.7792 1.07189
\(880\) 0 0
\(881\) −8.40995 −0.283339 −0.141669 0.989914i \(-0.545247\pi\)
−0.141669 + 0.989914i \(0.545247\pi\)
\(882\) 0 0
\(883\) −6.86398 −0.230991 −0.115496 0.993308i \(-0.536846\pi\)
−0.115496 + 0.993308i \(0.536846\pi\)
\(884\) 0 0
\(885\) 5.51619 0.185425
\(886\) 0 0
\(887\) −19.9773 −0.670772 −0.335386 0.942081i \(-0.608867\pi\)
−0.335386 + 0.942081i \(0.608867\pi\)
\(888\) 0 0
\(889\) −2.20630 −0.0739970
\(890\) 0 0
\(891\) 12.9585 0.434125
\(892\) 0 0
\(893\) −0.0142079 −0.000475448 0
\(894\) 0 0
\(895\) 26.3151 0.879615
\(896\) 0 0
\(897\) 1.51624 0.0506258
\(898\) 0 0
\(899\) 2.83651 0.0946030
\(900\) 0 0
\(901\) −5.66109 −0.188598
\(902\) 0 0
\(903\) 17.8384 0.593624
\(904\) 0 0
\(905\) −48.2420 −1.60362
\(906\) 0 0
\(907\) 29.4888 0.979161 0.489580 0.871958i \(-0.337150\pi\)
0.489580 + 0.871958i \(0.337150\pi\)
\(908\) 0 0
\(909\) −0.928330 −0.0307908
\(910\) 0 0
\(911\) 16.1170 0.533981 0.266991 0.963699i \(-0.413971\pi\)
0.266991 + 0.963699i \(0.413971\pi\)
\(912\) 0 0
\(913\) 7.64389 0.252976
\(914\) 0 0
\(915\) −63.1900 −2.08900
\(916\) 0 0
\(917\) 6.80865 0.224841
\(918\) 0 0
\(919\) 23.6310 0.779515 0.389757 0.920918i \(-0.372559\pi\)
0.389757 + 0.920918i \(0.372559\pi\)
\(920\) 0 0
\(921\) −9.37240 −0.308831
\(922\) 0 0
\(923\) 3.25861 0.107258
\(924\) 0 0
\(925\) −32.3549 −1.06382
\(926\) 0 0
\(927\) 0.443813 0.0145767
\(928\) 0 0
\(929\) −44.3474 −1.45499 −0.727495 0.686113i \(-0.759315\pi\)
−0.727495 + 0.686113i \(0.759315\pi\)
\(930\) 0 0
\(931\) −36.7891 −1.20571
\(932\) 0 0
\(933\) −15.8482 −0.518845
\(934\) 0 0
\(935\) 3.84889 0.125872
\(936\) 0 0
\(937\) 18.6166 0.608179 0.304090 0.952643i \(-0.401648\pi\)
0.304090 + 0.952643i \(0.401648\pi\)
\(938\) 0 0
\(939\) 60.0679 1.96024
\(940\) 0 0
\(941\) 41.7179 1.35997 0.679983 0.733228i \(-0.261987\pi\)
0.679983 + 0.733228i \(0.261987\pi\)
\(942\) 0 0
\(943\) 0.876721 0.0285500
\(944\) 0 0
\(945\) −13.1526 −0.427854
\(946\) 0 0
\(947\) −21.6862 −0.704705 −0.352353 0.935867i \(-0.614618\pi\)
−0.352353 + 0.935867i \(0.614618\pi\)
\(948\) 0 0
\(949\) −20.7936 −0.674989
\(950\) 0 0
\(951\) −65.0949 −2.11085
\(952\) 0 0
\(953\) −23.0043 −0.745183 −0.372591 0.927996i \(-0.621531\pi\)
−0.372591 + 0.927996i \(0.621531\pi\)
\(954\) 0 0
\(955\) 10.0299 0.324559
\(956\) 0 0
\(957\) −2.37498 −0.0767722
\(958\) 0 0
\(959\) −2.97279 −0.0959964
\(960\) 0 0
\(961\) −22.9308 −0.739704
\(962\) 0 0
\(963\) −4.06886 −0.131117
\(964\) 0 0
\(965\) 70.3805 2.26563
\(966\) 0 0
\(967\) −13.2831 −0.427155 −0.213578 0.976926i \(-0.568512\pi\)
−0.213578 + 0.976926i \(0.568512\pi\)
\(968\) 0 0
\(969\) 11.0384 0.354605
\(970\) 0 0
\(971\) 25.2995 0.811901 0.405950 0.913895i \(-0.366941\pi\)
0.405950 + 0.913895i \(0.366941\pi\)
\(972\) 0 0
\(973\) −20.4185 −0.654586
\(974\) 0 0
\(975\) 17.7163 0.567374
\(976\) 0 0
\(977\) −44.7028 −1.43017 −0.715085 0.699037i \(-0.753612\pi\)
−0.715085 + 0.699037i \(0.753612\pi\)
\(978\) 0 0
\(979\) 16.1273 0.515432
\(980\) 0 0
\(981\) 6.03279 0.192612
\(982\) 0 0
\(983\) −41.1552 −1.31265 −0.656324 0.754479i \(-0.727889\pi\)
−0.656324 + 0.754479i \(0.727889\pi\)
\(984\) 0 0
\(985\) 71.5859 2.28092
\(986\) 0 0
\(987\) 0.00403716 0.000128504 0
\(988\) 0 0
\(989\) −3.52864 −0.112204
\(990\) 0 0
\(991\) −54.5372 −1.73243 −0.866215 0.499671i \(-0.833454\pi\)
−0.866215 + 0.499671i \(0.833454\pi\)
\(992\) 0 0
\(993\) −1.61747 −0.0513289
\(994\) 0 0
\(995\) −21.0049 −0.665900
\(996\) 0 0
\(997\) 42.5922 1.34891 0.674454 0.738317i \(-0.264379\pi\)
0.674454 + 0.738317i \(0.264379\pi\)
\(998\) 0 0
\(999\) 39.4217 1.24725
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 4012.2.a.h.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.h.1.12 15 1.1 even 1 trivial