Properties

Label 3856.2.a.n.1.3
Level $3856$
Weight $2$
Character 3856.1
Self dual yes
Analytic conductor $30.790$
Analytic rank $1$
Dimension $12$
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] = [3856,2,Mod(1,3856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3856, 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("3856.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.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: \(30.7903150194\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.342147\) of defining polynomial
Character \(\chi\) \(=\) 3856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.18519 q^{3} -0.548903 q^{5} -1.82459 q^{7} +1.77508 q^{9} +O(q^{10})\) \(q-2.18519 q^{3} -0.548903 q^{5} -1.82459 q^{7} +1.77508 q^{9} -5.99218 q^{11} +3.70515 q^{13} +1.19946 q^{15} -1.64451 q^{17} +3.15821 q^{19} +3.98709 q^{21} +5.46015 q^{23} -4.69871 q^{25} +2.67670 q^{27} +7.24801 q^{29} +9.41700 q^{31} +13.0941 q^{33} +1.00152 q^{35} +1.27680 q^{37} -8.09648 q^{39} -5.81239 q^{41} -7.82887 q^{43} -0.974345 q^{45} -2.61568 q^{47} -3.67086 q^{49} +3.59356 q^{51} +8.81076 q^{53} +3.28912 q^{55} -6.90130 q^{57} +7.78270 q^{59} +1.03194 q^{61} -3.23879 q^{63} -2.03377 q^{65} +8.39801 q^{67} -11.9315 q^{69} -13.1691 q^{71} -13.3963 q^{73} +10.2676 q^{75} +10.9333 q^{77} +10.6362 q^{79} -11.1743 q^{81} -6.25636 q^{83} +0.902673 q^{85} -15.8383 q^{87} -3.94129 q^{89} -6.76039 q^{91} -20.5780 q^{93} -1.73355 q^{95} -2.22451 q^{97} -10.6366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) −2.18519 −1.26162 −0.630811 0.775936i \(-0.717278\pi\)
−0.630811 + 0.775936i \(0.717278\pi\)
\(4\) 0 0
\(5\) −0.548903 −0.245477 −0.122738 0.992439i \(-0.539168\pi\)
−0.122738 + 0.992439i \(0.539168\pi\)
\(6\) 0 0
\(7\) −1.82459 −0.689631 −0.344815 0.938671i \(-0.612058\pi\)
−0.344815 + 0.938671i \(0.612058\pi\)
\(8\) 0 0
\(9\) 1.77508 0.591692
\(10\) 0 0
\(11\) −5.99218 −1.80671 −0.903355 0.428894i \(-0.858904\pi\)
−0.903355 + 0.428894i \(0.858904\pi\)
\(12\) 0 0
\(13\) 3.70515 1.02762 0.513812 0.857903i \(-0.328233\pi\)
0.513812 + 0.857903i \(0.328233\pi\)
\(14\) 0 0
\(15\) 1.19946 0.309699
\(16\) 0 0
\(17\) −1.64451 −0.398851 −0.199426 0.979913i \(-0.563908\pi\)
−0.199426 + 0.979913i \(0.563908\pi\)
\(18\) 0 0
\(19\) 3.15821 0.724543 0.362271 0.932073i \(-0.382001\pi\)
0.362271 + 0.932073i \(0.382001\pi\)
\(20\) 0 0
\(21\) 3.98709 0.870054
\(22\) 0 0
\(23\) 5.46015 1.13852 0.569260 0.822158i \(-0.307230\pi\)
0.569260 + 0.822158i \(0.307230\pi\)
\(24\) 0 0
\(25\) −4.69871 −0.939741
\(26\) 0 0
\(27\) 2.67670 0.515130
\(28\) 0 0
\(29\) 7.24801 1.34592 0.672961 0.739678i \(-0.265022\pi\)
0.672961 + 0.739678i \(0.265022\pi\)
\(30\) 0 0
\(31\) 9.41700 1.69134 0.845672 0.533703i \(-0.179200\pi\)
0.845672 + 0.533703i \(0.179200\pi\)
\(32\) 0 0
\(33\) 13.0941 2.27939
\(34\) 0 0
\(35\) 1.00152 0.169288
\(36\) 0 0
\(37\) 1.27680 0.209904 0.104952 0.994477i \(-0.466531\pi\)
0.104952 + 0.994477i \(0.466531\pi\)
\(38\) 0 0
\(39\) −8.09648 −1.29647
\(40\) 0 0
\(41\) −5.81239 −0.907743 −0.453872 0.891067i \(-0.649958\pi\)
−0.453872 + 0.891067i \(0.649958\pi\)
\(42\) 0 0
\(43\) −7.82887 −1.19389 −0.596946 0.802281i \(-0.703619\pi\)
−0.596946 + 0.802281i \(0.703619\pi\)
\(44\) 0 0
\(45\) −0.974345 −0.145247
\(46\) 0 0
\(47\) −2.61568 −0.381537 −0.190768 0.981635i \(-0.561098\pi\)
−0.190768 + 0.981635i \(0.561098\pi\)
\(48\) 0 0
\(49\) −3.67086 −0.524409
\(50\) 0 0
\(51\) 3.59356 0.503200
\(52\) 0 0
\(53\) 8.81076 1.21025 0.605126 0.796130i \(-0.293123\pi\)
0.605126 + 0.796130i \(0.293123\pi\)
\(54\) 0 0
\(55\) 3.28912 0.443505
\(56\) 0 0
\(57\) −6.90130 −0.914100
\(58\) 0 0
\(59\) 7.78270 1.01322 0.506610 0.862175i \(-0.330898\pi\)
0.506610 + 0.862175i \(0.330898\pi\)
\(60\) 0 0
\(61\) 1.03194 0.132127 0.0660635 0.997815i \(-0.478956\pi\)
0.0660635 + 0.997815i \(0.478956\pi\)
\(62\) 0 0
\(63\) −3.23879 −0.408049
\(64\) 0 0
\(65\) −2.03377 −0.252258
\(66\) 0 0
\(67\) 8.39801 1.02598 0.512990 0.858395i \(-0.328538\pi\)
0.512990 + 0.858395i \(0.328538\pi\)
\(68\) 0 0
\(69\) −11.9315 −1.43638
\(70\) 0 0
\(71\) −13.1691 −1.56288 −0.781440 0.623981i \(-0.785514\pi\)
−0.781440 + 0.623981i \(0.785514\pi\)
\(72\) 0 0
\(73\) −13.3963 −1.56792 −0.783959 0.620813i \(-0.786803\pi\)
−0.783959 + 0.620813i \(0.786803\pi\)
\(74\) 0 0
\(75\) 10.2676 1.18560
\(76\) 0 0
\(77\) 10.9333 1.24596
\(78\) 0 0
\(79\) 10.6362 1.19666 0.598332 0.801248i \(-0.295830\pi\)
0.598332 + 0.801248i \(0.295830\pi\)
\(80\) 0 0
\(81\) −11.1743 −1.24159
\(82\) 0 0
\(83\) −6.25636 −0.686725 −0.343362 0.939203i \(-0.611566\pi\)
−0.343362 + 0.939203i \(0.611566\pi\)
\(84\) 0 0
\(85\) 0.902673 0.0979087
\(86\) 0 0
\(87\) −15.8383 −1.69805
\(88\) 0 0
\(89\) −3.94129 −0.417776 −0.208888 0.977940i \(-0.566984\pi\)
−0.208888 + 0.977940i \(0.566984\pi\)
\(90\) 0 0
\(91\) −6.76039 −0.708681
\(92\) 0 0
\(93\) −20.5780 −2.13384
\(94\) 0 0
\(95\) −1.73355 −0.177858
\(96\) 0 0
\(97\) −2.22451 −0.225865 −0.112933 0.993603i \(-0.536024\pi\)
−0.112933 + 0.993603i \(0.536024\pi\)
\(98\) 0 0
\(99\) −10.6366 −1.06902
\(100\) 0 0
\(101\) −4.24027 −0.421923 −0.210962 0.977494i \(-0.567659\pi\)
−0.210962 + 0.977494i \(0.567659\pi\)
\(102\) 0 0
\(103\) −0.164858 −0.0162439 −0.00812196 0.999967i \(-0.502585\pi\)
−0.00812196 + 0.999967i \(0.502585\pi\)
\(104\) 0 0
\(105\) −2.18852 −0.213578
\(106\) 0 0
\(107\) −9.28536 −0.897650 −0.448825 0.893620i \(-0.648157\pi\)
−0.448825 + 0.893620i \(0.648157\pi\)
\(108\) 0 0
\(109\) 12.1952 1.16809 0.584044 0.811722i \(-0.301470\pi\)
0.584044 + 0.811722i \(0.301470\pi\)
\(110\) 0 0
\(111\) −2.79005 −0.264820
\(112\) 0 0
\(113\) 13.1834 1.24019 0.620093 0.784528i \(-0.287095\pi\)
0.620093 + 0.784528i \(0.287095\pi\)
\(114\) 0 0
\(115\) −2.99709 −0.279480
\(116\) 0 0
\(117\) 6.57693 0.608037
\(118\) 0 0
\(119\) 3.00055 0.275060
\(120\) 0 0
\(121\) 24.9062 2.26420
\(122\) 0 0
\(123\) 12.7012 1.14523
\(124\) 0 0
\(125\) 5.32365 0.476161
\(126\) 0 0
\(127\) 11.7563 1.04321 0.521603 0.853188i \(-0.325334\pi\)
0.521603 + 0.853188i \(0.325334\pi\)
\(128\) 0 0
\(129\) 17.1076 1.50624
\(130\) 0 0
\(131\) −14.0839 −1.23051 −0.615257 0.788326i \(-0.710948\pi\)
−0.615257 + 0.788326i \(0.710948\pi\)
\(132\) 0 0
\(133\) −5.76244 −0.499667
\(134\) 0 0
\(135\) −1.46925 −0.126453
\(136\) 0 0
\(137\) −12.3993 −1.05935 −0.529673 0.848202i \(-0.677685\pi\)
−0.529673 + 0.848202i \(0.677685\pi\)
\(138\) 0 0
\(139\) 10.5456 0.894464 0.447232 0.894418i \(-0.352410\pi\)
0.447232 + 0.894418i \(0.352410\pi\)
\(140\) 0 0
\(141\) 5.71578 0.481356
\(142\) 0 0
\(143\) −22.2019 −1.85662
\(144\) 0 0
\(145\) −3.97845 −0.330393
\(146\) 0 0
\(147\) 8.02156 0.661607
\(148\) 0 0
\(149\) −9.30604 −0.762380 −0.381190 0.924497i \(-0.624486\pi\)
−0.381190 + 0.924497i \(0.624486\pi\)
\(150\) 0 0
\(151\) −10.2870 −0.837143 −0.418572 0.908184i \(-0.637469\pi\)
−0.418572 + 0.908184i \(0.637469\pi\)
\(152\) 0 0
\(153\) −2.91912 −0.235997
\(154\) 0 0
\(155\) −5.16902 −0.415186
\(156\) 0 0
\(157\) −4.23848 −0.338268 −0.169134 0.985593i \(-0.554097\pi\)
−0.169134 + 0.985593i \(0.554097\pi\)
\(158\) 0 0
\(159\) −19.2532 −1.52688
\(160\) 0 0
\(161\) −9.96254 −0.785159
\(162\) 0 0
\(163\) 6.42582 0.503309 0.251655 0.967817i \(-0.419025\pi\)
0.251655 + 0.967817i \(0.419025\pi\)
\(164\) 0 0
\(165\) −7.18738 −0.559536
\(166\) 0 0
\(167\) −3.96260 −0.306636 −0.153318 0.988177i \(-0.548996\pi\)
−0.153318 + 0.988177i \(0.548996\pi\)
\(168\) 0 0
\(169\) 0.728137 0.0560105
\(170\) 0 0
\(171\) 5.60606 0.428706
\(172\) 0 0
\(173\) −8.45221 −0.642610 −0.321305 0.946976i \(-0.604121\pi\)
−0.321305 + 0.946976i \(0.604121\pi\)
\(174\) 0 0
\(175\) 8.57322 0.648075
\(176\) 0 0
\(177\) −17.0067 −1.27830
\(178\) 0 0
\(179\) 10.5847 0.791138 0.395569 0.918436i \(-0.370547\pi\)
0.395569 + 0.918436i \(0.370547\pi\)
\(180\) 0 0
\(181\) −19.4914 −1.44879 −0.724394 0.689387i \(-0.757880\pi\)
−0.724394 + 0.689387i \(0.757880\pi\)
\(182\) 0 0
\(183\) −2.25500 −0.166694
\(184\) 0 0
\(185\) −0.700837 −0.0515265
\(186\) 0 0
\(187\) 9.85417 0.720608
\(188\) 0 0
\(189\) −4.88388 −0.355250
\(190\) 0 0
\(191\) 2.21537 0.160298 0.0801491 0.996783i \(-0.474460\pi\)
0.0801491 + 0.996783i \(0.474460\pi\)
\(192\) 0 0
\(193\) −8.84755 −0.636861 −0.318430 0.947946i \(-0.603156\pi\)
−0.318430 + 0.947946i \(0.603156\pi\)
\(194\) 0 0
\(195\) 4.44418 0.318254
\(196\) 0 0
\(197\) 6.19744 0.441549 0.220775 0.975325i \(-0.429141\pi\)
0.220775 + 0.975325i \(0.429141\pi\)
\(198\) 0 0
\(199\) −20.2303 −1.43409 −0.717043 0.697028i \(-0.754505\pi\)
−0.717043 + 0.697028i \(0.754505\pi\)
\(200\) 0 0
\(201\) −18.3513 −1.29440
\(202\) 0 0
\(203\) −13.2247 −0.928190
\(204\) 0 0
\(205\) 3.19044 0.222830
\(206\) 0 0
\(207\) 9.69219 0.673653
\(208\) 0 0
\(209\) −18.9245 −1.30904
\(210\) 0 0
\(211\) −1.96502 −0.135278 −0.0676388 0.997710i \(-0.521547\pi\)
−0.0676388 + 0.997710i \(0.521547\pi\)
\(212\) 0 0
\(213\) 28.7769 1.97176
\(214\) 0 0
\(215\) 4.29729 0.293073
\(216\) 0 0
\(217\) −17.1822 −1.16640
\(218\) 0 0
\(219\) 29.2735 1.97812
\(220\) 0 0
\(221\) −6.09314 −0.409869
\(222\) 0 0
\(223\) −21.4906 −1.43911 −0.719557 0.694433i \(-0.755655\pi\)
−0.719557 + 0.694433i \(0.755655\pi\)
\(224\) 0 0
\(225\) −8.34056 −0.556038
\(226\) 0 0
\(227\) −20.7806 −1.37925 −0.689627 0.724165i \(-0.742226\pi\)
−0.689627 + 0.724165i \(0.742226\pi\)
\(228\) 0 0
\(229\) 28.4761 1.88176 0.940878 0.338746i \(-0.110003\pi\)
0.940878 + 0.338746i \(0.110003\pi\)
\(230\) 0 0
\(231\) −23.8914 −1.57194
\(232\) 0 0
\(233\) −2.19220 −0.143616 −0.0718079 0.997418i \(-0.522877\pi\)
−0.0718079 + 0.997418i \(0.522877\pi\)
\(234\) 0 0
\(235\) 1.43576 0.0936584
\(236\) 0 0
\(237\) −23.2421 −1.50974
\(238\) 0 0
\(239\) −25.2449 −1.63296 −0.816478 0.577377i \(-0.804076\pi\)
−0.816478 + 0.577377i \(0.804076\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157
\(242\) 0 0
\(243\) 16.3880 1.05129
\(244\) 0 0
\(245\) 2.01495 0.128730
\(246\) 0 0
\(247\) 11.7016 0.744557
\(248\) 0 0
\(249\) 13.6714 0.866388
\(250\) 0 0
\(251\) −21.2167 −1.33919 −0.669593 0.742728i \(-0.733532\pi\)
−0.669593 + 0.742728i \(0.733532\pi\)
\(252\) 0 0
\(253\) −32.7182 −2.05698
\(254\) 0 0
\(255\) −1.97252 −0.123524
\(256\) 0 0
\(257\) 19.1130 1.19224 0.596119 0.802896i \(-0.296708\pi\)
0.596119 + 0.802896i \(0.296708\pi\)
\(258\) 0 0
\(259\) −2.32963 −0.144756
\(260\) 0 0
\(261\) 12.8658 0.796372
\(262\) 0 0
\(263\) 12.7792 0.787999 0.394000 0.919111i \(-0.371091\pi\)
0.394000 + 0.919111i \(0.371091\pi\)
\(264\) 0 0
\(265\) −4.83625 −0.297089
\(266\) 0 0
\(267\) 8.61249 0.527076
\(268\) 0 0
\(269\) 1.65545 0.100934 0.0504672 0.998726i \(-0.483929\pi\)
0.0504672 + 0.998726i \(0.483929\pi\)
\(270\) 0 0
\(271\) 17.0226 1.03405 0.517023 0.855971i \(-0.327040\pi\)
0.517023 + 0.855971i \(0.327040\pi\)
\(272\) 0 0
\(273\) 14.7728 0.894088
\(274\) 0 0
\(275\) 28.1555 1.69784
\(276\) 0 0
\(277\) 16.8245 1.01089 0.505443 0.862860i \(-0.331329\pi\)
0.505443 + 0.862860i \(0.331329\pi\)
\(278\) 0 0
\(279\) 16.7159 1.00076
\(280\) 0 0
\(281\) −16.1664 −0.964405 −0.482203 0.876060i \(-0.660163\pi\)
−0.482203 + 0.876060i \(0.660163\pi\)
\(282\) 0 0
\(283\) −2.53878 −0.150915 −0.0754573 0.997149i \(-0.524042\pi\)
−0.0754573 + 0.997149i \(0.524042\pi\)
\(284\) 0 0
\(285\) 3.78814 0.224390
\(286\) 0 0
\(287\) 10.6052 0.626008
\(288\) 0 0
\(289\) −14.2956 −0.840918
\(290\) 0 0
\(291\) 4.86100 0.284957
\(292\) 0 0
\(293\) −22.8971 −1.33767 −0.668833 0.743413i \(-0.733206\pi\)
−0.668833 + 0.743413i \(0.733206\pi\)
\(294\) 0 0
\(295\) −4.27194 −0.248722
\(296\) 0 0
\(297\) −16.0392 −0.930691
\(298\) 0 0
\(299\) 20.2307 1.16997
\(300\) 0 0
\(301\) 14.2845 0.823345
\(302\) 0 0
\(303\) 9.26583 0.532308
\(304\) 0 0
\(305\) −0.566437 −0.0324341
\(306\) 0 0
\(307\) 4.71512 0.269106 0.134553 0.990906i \(-0.457040\pi\)
0.134553 + 0.990906i \(0.457040\pi\)
\(308\) 0 0
\(309\) 0.360246 0.0204937
\(310\) 0 0
\(311\) −4.63908 −0.263058 −0.131529 0.991312i \(-0.541989\pi\)
−0.131529 + 0.991312i \(0.541989\pi\)
\(312\) 0 0
\(313\) −5.04057 −0.284910 −0.142455 0.989801i \(-0.545500\pi\)
−0.142455 + 0.989801i \(0.545500\pi\)
\(314\) 0 0
\(315\) 1.77778 0.100167
\(316\) 0 0
\(317\) −28.0992 −1.57821 −0.789105 0.614259i \(-0.789455\pi\)
−0.789105 + 0.614259i \(0.789455\pi\)
\(318\) 0 0
\(319\) −43.4314 −2.43169
\(320\) 0 0
\(321\) 20.2903 1.13250
\(322\) 0 0
\(323\) −5.19369 −0.288985
\(324\) 0 0
\(325\) −17.4094 −0.965700
\(326\) 0 0
\(327\) −26.6489 −1.47369
\(328\) 0 0
\(329\) 4.77256 0.263120
\(330\) 0 0
\(331\) −29.6777 −1.63124 −0.815618 0.578590i \(-0.803603\pi\)
−0.815618 + 0.578590i \(0.803603\pi\)
\(332\) 0 0
\(333\) 2.26641 0.124199
\(334\) 0 0
\(335\) −4.60969 −0.251854
\(336\) 0 0
\(337\) −12.2225 −0.665799 −0.332900 0.942962i \(-0.608027\pi\)
−0.332900 + 0.942962i \(0.608027\pi\)
\(338\) 0 0
\(339\) −28.8082 −1.56465
\(340\) 0 0
\(341\) −56.4284 −3.05577
\(342\) 0 0
\(343\) 19.4700 1.05128
\(344\) 0 0
\(345\) 6.54923 0.352599
\(346\) 0 0
\(347\) 17.9815 0.965296 0.482648 0.875814i \(-0.339675\pi\)
0.482648 + 0.875814i \(0.339675\pi\)
\(348\) 0 0
\(349\) 26.2848 1.40699 0.703496 0.710699i \(-0.251621\pi\)
0.703496 + 0.710699i \(0.251621\pi\)
\(350\) 0 0
\(351\) 9.91756 0.529360
\(352\) 0 0
\(353\) −13.7610 −0.732424 −0.366212 0.930531i \(-0.619346\pi\)
−0.366212 + 0.930531i \(0.619346\pi\)
\(354\) 0 0
\(355\) 7.22853 0.383651
\(356\) 0 0
\(357\) −6.55679 −0.347022
\(358\) 0 0
\(359\) 1.32555 0.0699598 0.0349799 0.999388i \(-0.488863\pi\)
0.0349799 + 0.999388i \(0.488863\pi\)
\(360\) 0 0
\(361\) −9.02572 −0.475038
\(362\) 0 0
\(363\) −54.4249 −2.85657
\(364\) 0 0
\(365\) 7.35326 0.384887
\(366\) 0 0
\(367\) −6.05979 −0.316319 −0.158159 0.987414i \(-0.550556\pi\)
−0.158159 + 0.987414i \(0.550556\pi\)
\(368\) 0 0
\(369\) −10.3174 −0.537105
\(370\) 0 0
\(371\) −16.0760 −0.834627
\(372\) 0 0
\(373\) 20.4445 1.05858 0.529288 0.848442i \(-0.322459\pi\)
0.529288 + 0.848442i \(0.322459\pi\)
\(374\) 0 0
\(375\) −11.6332 −0.600736
\(376\) 0 0
\(377\) 26.8550 1.38310
\(378\) 0 0
\(379\) −35.4042 −1.81859 −0.909296 0.416150i \(-0.863379\pi\)
−0.909296 + 0.416150i \(0.863379\pi\)
\(380\) 0 0
\(381\) −25.6899 −1.31613
\(382\) 0 0
\(383\) −6.79327 −0.347120 −0.173560 0.984823i \(-0.555527\pi\)
−0.173560 + 0.984823i \(0.555527\pi\)
\(384\) 0 0
\(385\) −6.00131 −0.305855
\(386\) 0 0
\(387\) −13.8969 −0.706417
\(388\) 0 0
\(389\) 24.6869 1.25168 0.625838 0.779953i \(-0.284757\pi\)
0.625838 + 0.779953i \(0.284757\pi\)
\(390\) 0 0
\(391\) −8.97924 −0.454100
\(392\) 0 0
\(393\) 30.7760 1.55245
\(394\) 0 0
\(395\) −5.83823 −0.293753
\(396\) 0 0
\(397\) 14.3804 0.721730 0.360865 0.932618i \(-0.382482\pi\)
0.360865 + 0.932618i \(0.382482\pi\)
\(398\) 0 0
\(399\) 12.5921 0.630391
\(400\) 0 0
\(401\) 7.62276 0.380663 0.190331 0.981720i \(-0.439044\pi\)
0.190331 + 0.981720i \(0.439044\pi\)
\(402\) 0 0
\(403\) 34.8914 1.73806
\(404\) 0 0
\(405\) 6.13362 0.304782
\(406\) 0 0
\(407\) −7.65079 −0.379236
\(408\) 0 0
\(409\) 5.12423 0.253377 0.126688 0.991943i \(-0.459565\pi\)
0.126688 + 0.991943i \(0.459565\pi\)
\(410\) 0 0
\(411\) 27.0949 1.33649
\(412\) 0 0
\(413\) −14.2002 −0.698748
\(414\) 0 0
\(415\) 3.43413 0.168575
\(416\) 0 0
\(417\) −23.0441 −1.12848
\(418\) 0 0
\(419\) −13.3513 −0.652254 −0.326127 0.945326i \(-0.605744\pi\)
−0.326127 + 0.945326i \(0.605744\pi\)
\(420\) 0 0
\(421\) 27.8059 1.35518 0.677588 0.735442i \(-0.263025\pi\)
0.677588 + 0.735442i \(0.263025\pi\)
\(422\) 0 0
\(423\) −4.64304 −0.225752
\(424\) 0 0
\(425\) 7.72705 0.374817
\(426\) 0 0
\(427\) −1.88288 −0.0911188
\(428\) 0 0
\(429\) 48.5155 2.34235
\(430\) 0 0
\(431\) −18.5717 −0.894567 −0.447283 0.894392i \(-0.647608\pi\)
−0.447283 + 0.894392i \(0.647608\pi\)
\(432\) 0 0
\(433\) −7.19376 −0.345710 −0.172855 0.984947i \(-0.555299\pi\)
−0.172855 + 0.984947i \(0.555299\pi\)
\(434\) 0 0
\(435\) 8.69370 0.416831
\(436\) 0 0
\(437\) 17.2443 0.824906
\(438\) 0 0
\(439\) 16.2866 0.777315 0.388657 0.921382i \(-0.372939\pi\)
0.388657 + 0.921382i \(0.372939\pi\)
\(440\) 0 0
\(441\) −6.51607 −0.310289
\(442\) 0 0
\(443\) 37.0915 1.76227 0.881135 0.472864i \(-0.156780\pi\)
0.881135 + 0.472864i \(0.156780\pi\)
\(444\) 0 0
\(445\) 2.16338 0.102554
\(446\) 0 0
\(447\) 20.3355 0.961837
\(448\) 0 0
\(449\) −17.1059 −0.807275 −0.403638 0.914919i \(-0.632254\pi\)
−0.403638 + 0.914919i \(0.632254\pi\)
\(450\) 0 0
\(451\) 34.8289 1.64003
\(452\) 0 0
\(453\) 22.4791 1.05616
\(454\) 0 0
\(455\) 3.71079 0.173965
\(456\) 0 0
\(457\) −9.35857 −0.437776 −0.218888 0.975750i \(-0.570243\pi\)
−0.218888 + 0.975750i \(0.570243\pi\)
\(458\) 0 0
\(459\) −4.40184 −0.205460
\(460\) 0 0
\(461\) −15.8670 −0.738999 −0.369500 0.929231i \(-0.620471\pi\)
−0.369500 + 0.929231i \(0.620471\pi\)
\(462\) 0 0
\(463\) 6.99749 0.325201 0.162601 0.986692i \(-0.448012\pi\)
0.162601 + 0.986692i \(0.448012\pi\)
\(464\) 0 0
\(465\) 11.2953 0.523808
\(466\) 0 0
\(467\) 24.2782 1.12346 0.561729 0.827321i \(-0.310136\pi\)
0.561729 + 0.827321i \(0.310136\pi\)
\(468\) 0 0
\(469\) −15.3229 −0.707547
\(470\) 0 0
\(471\) 9.26191 0.426766
\(472\) 0 0
\(473\) 46.9120 2.15702
\(474\) 0 0
\(475\) −14.8395 −0.680882
\(476\) 0 0
\(477\) 15.6398 0.716096
\(478\) 0 0
\(479\) −8.93171 −0.408100 −0.204050 0.978960i \(-0.565411\pi\)
−0.204050 + 0.978960i \(0.565411\pi\)
\(480\) 0 0
\(481\) 4.73072 0.215702
\(482\) 0 0
\(483\) 21.7701 0.990574
\(484\) 0 0
\(485\) 1.22104 0.0554447
\(486\) 0 0
\(487\) −34.5963 −1.56771 −0.783855 0.620944i \(-0.786750\pi\)
−0.783855 + 0.620944i \(0.786750\pi\)
\(488\) 0 0
\(489\) −14.0417 −0.634987
\(490\) 0 0
\(491\) −34.3798 −1.55154 −0.775770 0.631016i \(-0.782638\pi\)
−0.775770 + 0.631016i \(0.782638\pi\)
\(492\) 0 0
\(493\) −11.9194 −0.536823
\(494\) 0 0
\(495\) 5.83845 0.262419
\(496\) 0 0
\(497\) 24.0281 1.07781
\(498\) 0 0
\(499\) −38.3126 −1.71511 −0.857554 0.514394i \(-0.828017\pi\)
−0.857554 + 0.514394i \(0.828017\pi\)
\(500\) 0 0
\(501\) 8.65906 0.386858
\(502\) 0 0
\(503\) 12.6073 0.562132 0.281066 0.959688i \(-0.409312\pi\)
0.281066 + 0.959688i \(0.409312\pi\)
\(504\) 0 0
\(505\) 2.32750 0.103572
\(506\) 0 0
\(507\) −1.59112 −0.0706641
\(508\) 0 0
\(509\) −19.0748 −0.845476 −0.422738 0.906252i \(-0.638931\pi\)
−0.422738 + 0.906252i \(0.638931\pi\)
\(510\) 0 0
\(511\) 24.4428 1.08128
\(512\) 0 0
\(513\) 8.45356 0.373234
\(514\) 0 0
\(515\) 0.0904909 0.00398750
\(516\) 0 0
\(517\) 15.6737 0.689326
\(518\) 0 0
\(519\) 18.4697 0.810731
\(520\) 0 0
\(521\) 11.5960 0.508028 0.254014 0.967201i \(-0.418249\pi\)
0.254014 + 0.967201i \(0.418249\pi\)
\(522\) 0 0
\(523\) 26.2727 1.14883 0.574413 0.818566i \(-0.305230\pi\)
0.574413 + 0.818566i \(0.305230\pi\)
\(524\) 0 0
\(525\) −18.7342 −0.817626
\(526\) 0 0
\(527\) −15.4863 −0.674594
\(528\) 0 0
\(529\) 6.81324 0.296228
\(530\) 0 0
\(531\) 13.8149 0.599515
\(532\) 0 0
\(533\) −21.5358 −0.932819
\(534\) 0 0
\(535\) 5.09676 0.220352
\(536\) 0 0
\(537\) −23.1296 −0.998118
\(538\) 0 0
\(539\) 21.9965 0.947455
\(540\) 0 0
\(541\) −15.1441 −0.651097 −0.325548 0.945525i \(-0.605549\pi\)
−0.325548 + 0.945525i \(0.605549\pi\)
\(542\) 0 0
\(543\) 42.5926 1.82782
\(544\) 0 0
\(545\) −6.69397 −0.286738
\(546\) 0 0
\(547\) −34.7331 −1.48508 −0.742540 0.669802i \(-0.766379\pi\)
−0.742540 + 0.669802i \(0.766379\pi\)
\(548\) 0 0
\(549\) 1.83178 0.0781785
\(550\) 0 0
\(551\) 22.8907 0.975178
\(552\) 0 0
\(553\) −19.4067 −0.825256
\(554\) 0 0
\(555\) 1.53146 0.0650071
\(556\) 0 0
\(557\) 34.1470 1.44685 0.723427 0.690401i \(-0.242566\pi\)
0.723427 + 0.690401i \(0.242566\pi\)
\(558\) 0 0
\(559\) −29.0071 −1.22687
\(560\) 0 0
\(561\) −21.5333 −0.909136
\(562\) 0 0
\(563\) 19.2900 0.812975 0.406488 0.913656i \(-0.366753\pi\)
0.406488 + 0.913656i \(0.366753\pi\)
\(564\) 0 0
\(565\) −7.23638 −0.304437
\(566\) 0 0
\(567\) 20.3886 0.856241
\(568\) 0 0
\(569\) −33.4263 −1.40130 −0.700652 0.713503i \(-0.747107\pi\)
−0.700652 + 0.713503i \(0.747107\pi\)
\(570\) 0 0
\(571\) −3.07936 −0.128867 −0.0644336 0.997922i \(-0.520524\pi\)
−0.0644336 + 0.997922i \(0.520524\pi\)
\(572\) 0 0
\(573\) −4.84101 −0.202236
\(574\) 0 0
\(575\) −25.6556 −1.06991
\(576\) 0 0
\(577\) −5.95435 −0.247883 −0.123941 0.992290i \(-0.539553\pi\)
−0.123941 + 0.992290i \(0.539553\pi\)
\(578\) 0 0
\(579\) 19.3336 0.803478
\(580\) 0 0
\(581\) 11.4153 0.473587
\(582\) 0 0
\(583\) −52.7957 −2.18657
\(584\) 0 0
\(585\) −3.61009 −0.149259
\(586\) 0 0
\(587\) −42.0432 −1.73531 −0.867655 0.497167i \(-0.834374\pi\)
−0.867655 + 0.497167i \(0.834374\pi\)
\(588\) 0 0
\(589\) 29.7409 1.22545
\(590\) 0 0
\(591\) −13.5426 −0.557068
\(592\) 0 0
\(593\) 40.9908 1.68329 0.841645 0.540031i \(-0.181587\pi\)
0.841645 + 0.540031i \(0.181587\pi\)
\(594\) 0 0
\(595\) −1.64701 −0.0675208
\(596\) 0 0
\(597\) 44.2071 1.80928
\(598\) 0 0
\(599\) 27.0862 1.10671 0.553356 0.832945i \(-0.313347\pi\)
0.553356 + 0.832945i \(0.313347\pi\)
\(600\) 0 0
\(601\) −3.03033 −0.123610 −0.0618049 0.998088i \(-0.519686\pi\)
−0.0618049 + 0.998088i \(0.519686\pi\)
\(602\) 0 0
\(603\) 14.9071 0.607064
\(604\) 0 0
\(605\) −13.6711 −0.555809
\(606\) 0 0
\(607\) −0.560838 −0.0227637 −0.0113818 0.999935i \(-0.503623\pi\)
−0.0113818 + 0.999935i \(0.503623\pi\)
\(608\) 0 0
\(609\) 28.8985 1.17103
\(610\) 0 0
\(611\) −9.69150 −0.392076
\(612\) 0 0
\(613\) 23.7388 0.958801 0.479401 0.877596i \(-0.340854\pi\)
0.479401 + 0.877596i \(0.340854\pi\)
\(614\) 0 0
\(615\) −6.97173 −0.281127
\(616\) 0 0
\(617\) −10.2411 −0.412290 −0.206145 0.978521i \(-0.566092\pi\)
−0.206145 + 0.978521i \(0.566092\pi\)
\(618\) 0 0
\(619\) 27.8348 1.11878 0.559388 0.828906i \(-0.311036\pi\)
0.559388 + 0.828906i \(0.311036\pi\)
\(620\) 0 0
\(621\) 14.6152 0.586486
\(622\) 0 0
\(623\) 7.19125 0.288111
\(624\) 0 0
\(625\) 20.5714 0.822855
\(626\) 0 0
\(627\) 41.3538 1.65151
\(628\) 0 0
\(629\) −2.09970 −0.0837204
\(630\) 0 0
\(631\) −23.0564 −0.917861 −0.458931 0.888472i \(-0.651767\pi\)
−0.458931 + 0.888472i \(0.651767\pi\)
\(632\) 0 0
\(633\) 4.29396 0.170669
\(634\) 0 0
\(635\) −6.45309 −0.256083
\(636\) 0 0
\(637\) −13.6011 −0.538895
\(638\) 0 0
\(639\) −23.3761 −0.924744
\(640\) 0 0
\(641\) 35.6143 1.40668 0.703340 0.710853i \(-0.251691\pi\)
0.703340 + 0.710853i \(0.251691\pi\)
\(642\) 0 0
\(643\) 28.2806 1.11528 0.557639 0.830084i \(-0.311707\pi\)
0.557639 + 0.830084i \(0.311707\pi\)
\(644\) 0 0
\(645\) −9.39042 −0.369747
\(646\) 0 0
\(647\) −14.5953 −0.573802 −0.286901 0.957960i \(-0.592625\pi\)
−0.286901 + 0.957960i \(0.592625\pi\)
\(648\) 0 0
\(649\) −46.6353 −1.83060
\(650\) 0 0
\(651\) 37.5464 1.47156
\(652\) 0 0
\(653\) −28.0718 −1.09853 −0.549267 0.835647i \(-0.685093\pi\)
−0.549267 + 0.835647i \(0.685093\pi\)
\(654\) 0 0
\(655\) 7.73068 0.302063
\(656\) 0 0
\(657\) −23.7794 −0.927724
\(658\) 0 0
\(659\) 2.33181 0.0908343 0.0454172 0.998968i \(-0.485538\pi\)
0.0454172 + 0.998968i \(0.485538\pi\)
\(660\) 0 0
\(661\) −21.3409 −0.830063 −0.415031 0.909807i \(-0.636229\pi\)
−0.415031 + 0.909807i \(0.636229\pi\)
\(662\) 0 0
\(663\) 13.3147 0.517100
\(664\) 0 0
\(665\) 3.16302 0.122657
\(666\) 0 0
\(667\) 39.5752 1.53236
\(668\) 0 0
\(669\) 46.9611 1.81562
\(670\) 0 0
\(671\) −6.18360 −0.238715
\(672\) 0 0
\(673\) −25.0948 −0.967333 −0.483667 0.875252i \(-0.660695\pi\)
−0.483667 + 0.875252i \(0.660695\pi\)
\(674\) 0 0
\(675\) −12.5770 −0.484089
\(676\) 0 0
\(677\) −7.78669 −0.299267 −0.149633 0.988742i \(-0.547809\pi\)
−0.149633 + 0.988742i \(0.547809\pi\)
\(678\) 0 0
\(679\) 4.05883 0.155764
\(680\) 0 0
\(681\) 45.4096 1.74010
\(682\) 0 0
\(683\) −2.68470 −0.102727 −0.0513636 0.998680i \(-0.516357\pi\)
−0.0513636 + 0.998680i \(0.516357\pi\)
\(684\) 0 0
\(685\) 6.80602 0.260045
\(686\) 0 0
\(687\) −62.2259 −2.37407
\(688\) 0 0
\(689\) 32.6452 1.24368
\(690\) 0 0
\(691\) 46.3260 1.76232 0.881162 0.472814i \(-0.156762\pi\)
0.881162 + 0.472814i \(0.156762\pi\)
\(692\) 0 0
\(693\) 19.4074 0.737227
\(694\) 0 0
\(695\) −5.78849 −0.219570
\(696\) 0 0
\(697\) 9.55851 0.362054
\(698\) 0 0
\(699\) 4.79039 0.181189
\(700\) 0 0
\(701\) 4.33233 0.163630 0.0818150 0.996648i \(-0.473928\pi\)
0.0818150 + 0.996648i \(0.473928\pi\)
\(702\) 0 0
\(703\) 4.03239 0.152084
\(704\) 0 0
\(705\) −3.13741 −0.118162
\(706\) 0 0
\(707\) 7.73677 0.290971
\(708\) 0 0
\(709\) −45.2491 −1.69936 −0.849682 0.527295i \(-0.823206\pi\)
−0.849682 + 0.527295i \(0.823206\pi\)
\(710\) 0 0
\(711\) 18.8800 0.708057
\(712\) 0 0
\(713\) 51.4182 1.92563
\(714\) 0 0
\(715\) 12.1867 0.455757
\(716\) 0 0
\(717\) 55.1650 2.06017
\(718\) 0 0
\(719\) −34.3560 −1.28126 −0.640632 0.767848i \(-0.721328\pi\)
−0.640632 + 0.767848i \(0.721328\pi\)
\(720\) 0 0
\(721\) 0.300798 0.0112023
\(722\) 0 0
\(723\) −2.18519 −0.0812683
\(724\) 0 0
\(725\) −34.0563 −1.26482
\(726\) 0 0
\(727\) 29.1579 1.08141 0.540703 0.841213i \(-0.318158\pi\)
0.540703 + 0.841213i \(0.318158\pi\)
\(728\) 0 0
\(729\) −2.28799 −0.0847405
\(730\) 0 0
\(731\) 12.8746 0.476185
\(732\) 0 0
\(733\) 36.8326 1.36044 0.680222 0.733007i \(-0.261884\pi\)
0.680222 + 0.733007i \(0.261884\pi\)
\(734\) 0 0
\(735\) −4.40305 −0.162409
\(736\) 0 0
\(737\) −50.3224 −1.85365
\(738\) 0 0
\(739\) −24.4258 −0.898517 −0.449258 0.893402i \(-0.648312\pi\)
−0.449258 + 0.893402i \(0.648312\pi\)
\(740\) 0 0
\(741\) −25.5704 −0.939350
\(742\) 0 0
\(743\) 24.4245 0.896049 0.448024 0.894021i \(-0.352128\pi\)
0.448024 + 0.894021i \(0.352128\pi\)
\(744\) 0 0
\(745\) 5.10811 0.187147
\(746\) 0 0
\(747\) −11.1055 −0.406330
\(748\) 0 0
\(749\) 16.9420 0.619047
\(750\) 0 0
\(751\) 30.0598 1.09690 0.548449 0.836184i \(-0.315219\pi\)
0.548449 + 0.836184i \(0.315219\pi\)
\(752\) 0 0
\(753\) 46.3627 1.68955
\(754\) 0 0
\(755\) 5.64656 0.205499
\(756\) 0 0
\(757\) 5.79087 0.210473 0.105236 0.994447i \(-0.466440\pi\)
0.105236 + 0.994447i \(0.466440\pi\)
\(758\) 0 0
\(759\) 71.4956 2.59513
\(760\) 0 0
\(761\) 2.42691 0.0879754 0.0439877 0.999032i \(-0.485994\pi\)
0.0439877 + 0.999032i \(0.485994\pi\)
\(762\) 0 0
\(763\) −22.2512 −0.805549
\(764\) 0 0
\(765\) 1.60231 0.0579318
\(766\) 0 0
\(767\) 28.8361 1.04121
\(768\) 0 0
\(769\) 22.0750 0.796045 0.398022 0.917376i \(-0.369697\pi\)
0.398022 + 0.917376i \(0.369697\pi\)
\(770\) 0 0
\(771\) −41.7657 −1.50416
\(772\) 0 0
\(773\) 28.7802 1.03515 0.517576 0.855637i \(-0.326834\pi\)
0.517576 + 0.855637i \(0.326834\pi\)
\(774\) 0 0
\(775\) −44.2477 −1.58943
\(776\) 0 0
\(777\) 5.09070 0.182628
\(778\) 0 0
\(779\) −18.3567 −0.657699
\(780\) 0 0
\(781\) 78.9113 2.82367
\(782\) 0 0
\(783\) 19.4007 0.693325
\(784\) 0 0
\(785\) 2.32651 0.0830369
\(786\) 0 0
\(787\) 53.4388 1.90489 0.952443 0.304715i \(-0.0985613\pi\)
0.952443 + 0.304715i \(0.0985613\pi\)
\(788\) 0 0
\(789\) −27.9250 −0.994158
\(790\) 0 0
\(791\) −24.0543 −0.855271
\(792\) 0 0
\(793\) 3.82351 0.135777
\(794\) 0 0
\(795\) 10.5682 0.374814
\(796\) 0 0
\(797\) 10.4149 0.368914 0.184457 0.982841i \(-0.440947\pi\)
0.184457 + 0.982841i \(0.440947\pi\)
\(798\) 0 0
\(799\) 4.30151 0.152176
\(800\) 0 0
\(801\) −6.99609 −0.247195
\(802\) 0 0
\(803\) 80.2730 2.83277
\(804\) 0 0
\(805\) 5.46847 0.192738
\(806\) 0 0
\(807\) −3.61747 −0.127341
\(808\) 0 0
\(809\) −25.9467 −0.912237 −0.456118 0.889919i \(-0.650761\pi\)
−0.456118 + 0.889919i \(0.650761\pi\)
\(810\) 0 0
\(811\) 11.1712 0.392275 0.196137 0.980576i \(-0.437160\pi\)
0.196137 + 0.980576i \(0.437160\pi\)
\(812\) 0 0
\(813\) −37.1976 −1.30458
\(814\) 0 0
\(815\) −3.52715 −0.123551
\(816\) 0 0
\(817\) −24.7252 −0.865026
\(818\) 0 0
\(819\) −12.0002 −0.419321
\(820\) 0 0
\(821\) 18.3232 0.639484 0.319742 0.947505i \(-0.396404\pi\)
0.319742 + 0.947505i \(0.396404\pi\)
\(822\) 0 0
\(823\) −21.1230 −0.736301 −0.368150 0.929766i \(-0.620009\pi\)
−0.368150 + 0.929766i \(0.620009\pi\)
\(824\) 0 0
\(825\) −61.5252 −2.14203
\(826\) 0 0
\(827\) −47.5383 −1.65307 −0.826534 0.562887i \(-0.809690\pi\)
−0.826534 + 0.562887i \(0.809690\pi\)
\(828\) 0 0
\(829\) 15.8701 0.551190 0.275595 0.961274i \(-0.411125\pi\)
0.275595 + 0.961274i \(0.411125\pi\)
\(830\) 0 0
\(831\) −36.7648 −1.27536
\(832\) 0 0
\(833\) 6.03676 0.209161
\(834\) 0 0
\(835\) 2.17508 0.0752719
\(836\) 0 0
\(837\) 25.2065 0.871262
\(838\) 0 0
\(839\) −34.3114 −1.18456 −0.592281 0.805732i \(-0.701772\pi\)
−0.592281 + 0.805732i \(0.701772\pi\)
\(840\) 0 0
\(841\) 23.5337 0.811507
\(842\) 0 0
\(843\) 35.3267 1.21672
\(844\) 0 0
\(845\) −0.399676 −0.0137493
\(846\) 0 0
\(847\) −45.4437 −1.56146
\(848\) 0 0
\(849\) 5.54772 0.190397
\(850\) 0 0
\(851\) 6.97150 0.238980
\(852\) 0 0
\(853\) −45.8466 −1.56976 −0.784879 0.619649i \(-0.787275\pi\)
−0.784879 + 0.619649i \(0.787275\pi\)
\(854\) 0 0
\(855\) −3.07718 −0.105237
\(856\) 0 0
\(857\) 21.7254 0.742125 0.371063 0.928608i \(-0.378993\pi\)
0.371063 + 0.928608i \(0.378993\pi\)
\(858\) 0 0
\(859\) −52.5363 −1.79252 −0.896258 0.443534i \(-0.853725\pi\)
−0.896258 + 0.443534i \(0.853725\pi\)
\(860\) 0 0
\(861\) −23.1745 −0.789786
\(862\) 0 0
\(863\) −32.9971 −1.12323 −0.561617 0.827397i \(-0.689821\pi\)
−0.561617 + 0.827397i \(0.689821\pi\)
\(864\) 0 0
\(865\) 4.63944 0.157746
\(866\) 0 0
\(867\) 31.2387 1.06092
\(868\) 0 0
\(869\) −63.7339 −2.16202
\(870\) 0 0
\(871\) 31.1159 1.05432
\(872\) 0 0
\(873\) −3.94868 −0.133643
\(874\) 0 0
\(875\) −9.71348 −0.328376
\(876\) 0 0
\(877\) −21.4221 −0.723372 −0.361686 0.932300i \(-0.617799\pi\)
−0.361686 + 0.932300i \(0.617799\pi\)
\(878\) 0 0
\(879\) 50.0347 1.68763
\(880\) 0 0
\(881\) −39.1162 −1.31786 −0.658929 0.752205i \(-0.728990\pi\)
−0.658929 + 0.752205i \(0.728990\pi\)
\(882\) 0 0
\(883\) 2.84717 0.0958150 0.0479075 0.998852i \(-0.484745\pi\)
0.0479075 + 0.998852i \(0.484745\pi\)
\(884\) 0 0
\(885\) 9.33503 0.313794
\(886\) 0 0
\(887\) 46.9262 1.57563 0.787814 0.615913i \(-0.211213\pi\)
0.787814 + 0.615913i \(0.211213\pi\)
\(888\) 0 0
\(889\) −21.4505 −0.719428
\(890\) 0 0
\(891\) 66.9586 2.24320
\(892\) 0 0
\(893\) −8.26088 −0.276440
\(894\) 0 0
\(895\) −5.80997 −0.194206
\(896\) 0 0
\(897\) −44.2080 −1.47606
\(898\) 0 0
\(899\) 68.2546 2.27642
\(900\) 0 0
\(901\) −14.4893 −0.482710
\(902\) 0 0
\(903\) −31.2144 −1.03875
\(904\) 0 0
\(905\) 10.6989 0.355644
\(906\) 0 0
\(907\) −15.7164 −0.521856 −0.260928 0.965358i \(-0.584028\pi\)
−0.260928 + 0.965358i \(0.584028\pi\)
\(908\) 0 0
\(909\) −7.52681 −0.249649
\(910\) 0 0
\(911\) −0.0451186 −0.00149485 −0.000747423 1.00000i \(-0.500238\pi\)
−0.000747423 1.00000i \(0.500238\pi\)
\(912\) 0 0
\(913\) 37.4892 1.24071
\(914\) 0 0
\(915\) 1.23778 0.0409196
\(916\) 0 0
\(917\) 25.6973 0.848601
\(918\) 0 0
\(919\) 7.88987 0.260263 0.130131 0.991497i \(-0.458460\pi\)
0.130131 + 0.991497i \(0.458460\pi\)
\(920\) 0 0
\(921\) −10.3035 −0.339511
\(922\) 0 0
\(923\) −48.7933 −1.60605
\(924\) 0 0
\(925\) −5.99929 −0.197255
\(926\) 0 0
\(927\) −0.292635 −0.00961140
\(928\) 0 0
\(929\) 25.6276 0.840815 0.420408 0.907335i \(-0.361887\pi\)
0.420408 + 0.907335i \(0.361887\pi\)
\(930\) 0 0
\(931\) −11.5934 −0.379957
\(932\) 0 0
\(933\) 10.1373 0.331880
\(934\) 0 0
\(935\) −5.40898 −0.176893
\(936\) 0 0
\(937\) −57.8627 −1.89029 −0.945146 0.326649i \(-0.894081\pi\)
−0.945146 + 0.326649i \(0.894081\pi\)
\(938\) 0 0
\(939\) 11.0146 0.359449
\(940\) 0 0
\(941\) −14.1272 −0.460533 −0.230266 0.973128i \(-0.573960\pi\)
−0.230266 + 0.973128i \(0.573960\pi\)
\(942\) 0 0
\(943\) −31.7365 −1.03348
\(944\) 0 0
\(945\) 2.68077 0.0872056
\(946\) 0 0
\(947\) −22.5318 −0.732185 −0.366092 0.930578i \(-0.619305\pi\)
−0.366092 + 0.930578i \(0.619305\pi\)
\(948\) 0 0
\(949\) −49.6353 −1.61123
\(950\) 0 0
\(951\) 61.4023 1.99110
\(952\) 0 0
\(953\) −34.0142 −1.10183 −0.550914 0.834562i \(-0.685721\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(954\) 0 0
\(955\) −1.21602 −0.0393495
\(956\) 0 0
\(957\) 94.9061 3.06788
\(958\) 0 0
\(959\) 22.6237 0.730557
\(960\) 0 0
\(961\) 57.6799 1.86064
\(962\) 0 0
\(963\) −16.4822 −0.531133
\(964\) 0 0
\(965\) 4.85644 0.156334
\(966\) 0 0
\(967\) 62.1874 1.99981 0.999906 0.0137093i \(-0.00436394\pi\)
0.999906 + 0.0137093i \(0.00436394\pi\)
\(968\) 0 0
\(969\) 11.3492 0.364590
\(970\) 0 0
\(971\) −18.4346 −0.591594 −0.295797 0.955251i \(-0.595585\pi\)
−0.295797 + 0.955251i \(0.595585\pi\)
\(972\) 0 0
\(973\) −19.2414 −0.616850
\(974\) 0 0
\(975\) 38.0430 1.21835
\(976\) 0 0
\(977\) 25.2228 0.806949 0.403474 0.914991i \(-0.367802\pi\)
0.403474 + 0.914991i \(0.367802\pi\)
\(978\) 0 0
\(979\) 23.6169 0.754800
\(980\) 0 0
\(981\) 21.6474 0.691148
\(982\) 0 0
\(983\) −12.7405 −0.406358 −0.203179 0.979142i \(-0.565127\pi\)
−0.203179 + 0.979142i \(0.565127\pi\)
\(984\) 0 0
\(985\) −3.40179 −0.108390
\(986\) 0 0
\(987\) −10.4290 −0.331958
\(988\) 0 0
\(989\) −42.7468 −1.35927
\(990\) 0 0
\(991\) −15.6775 −0.498011 −0.249005 0.968502i \(-0.580104\pi\)
−0.249005 + 0.968502i \(0.580104\pi\)
\(992\) 0 0
\(993\) 64.8517 2.05801
\(994\) 0 0
\(995\) 11.1045 0.352035
\(996\) 0 0
\(997\) 28.1048 0.890088 0.445044 0.895509i \(-0.353188\pi\)
0.445044 + 0.895509i \(0.353188\pi\)
\(998\) 0 0
\(999\) 3.41759 0.108128
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 3856.2.a.n.1.3 12
4.3 odd 2 241.2.a.b.1.5 12
12.11 even 2 2169.2.a.h.1.8 12
20.19 odd 2 6025.2.a.h.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.b.1.5 12 4.3 odd 2
2169.2.a.h.1.8 12 12.11 even 2
3856.2.a.n.1.3 12 1.1 even 1 trivial
6025.2.a.h.1.8 12 20.19 odd 2