Properties

Label 2960.2.a.bc.1.3
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $6$
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] = [2960,2,Mod(1,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.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: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.693982032.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.07986\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.07986 q^{3} +1.00000 q^{5} -3.77160 q^{7} -1.83390 q^{9} +O(q^{10})\) \(q-1.07986 q^{3} +1.00000 q^{5} -3.77160 q^{7} -1.83390 q^{9} -0.592024 q^{11} -3.63293 q^{13} -1.07986 q^{15} +4.24187 q^{17} -2.63998 q^{19} +4.07281 q^{21} -4.35915 q^{23} +1.00000 q^{25} +5.21995 q^{27} +0.768672 q^{29} -6.69522 q^{31} +0.639305 q^{33} -3.77160 q^{35} -1.00000 q^{37} +3.92306 q^{39} +7.71226 q^{41} -2.33160 q^{43} -1.83390 q^{45} +7.64640 q^{47} +7.22495 q^{49} -4.58064 q^{51} +0.432298 q^{53} -0.592024 q^{55} +2.85082 q^{57} +13.2065 q^{59} -1.39105 q^{61} +6.91672 q^{63} -3.63293 q^{65} -4.83940 q^{67} +4.70728 q^{69} -0.639305 q^{71} -8.23872 q^{73} -1.07986 q^{75} +2.23288 q^{77} +5.94066 q^{79} -0.135139 q^{81} +4.80252 q^{83} +4.24187 q^{85} -0.830060 q^{87} +17.4653 q^{89} +13.7019 q^{91} +7.22993 q^{93} -2.63998 q^{95} +7.77889 q^{97} +1.08571 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100}) \) 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
\(3\) −1.07986 −0.623459 −0.311730 0.950171i \(-0.600908\pi\)
−0.311730 + 0.950171i \(0.600908\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.77160 −1.42553 −0.712765 0.701403i \(-0.752557\pi\)
−0.712765 + 0.701403i \(0.752557\pi\)
\(8\) 0 0
\(9\) −1.83390 −0.611299
\(10\) 0 0
\(11\) −0.592024 −0.178502 −0.0892509 0.996009i \(-0.528447\pi\)
−0.0892509 + 0.996009i \(0.528447\pi\)
\(12\) 0 0
\(13\) −3.63293 −1.00759 −0.503796 0.863823i \(-0.668064\pi\)
−0.503796 + 0.863823i \(0.668064\pi\)
\(14\) 0 0
\(15\) −1.07986 −0.278819
\(16\) 0 0
\(17\) 4.24187 1.02881 0.514403 0.857549i \(-0.328014\pi\)
0.514403 + 0.857549i \(0.328014\pi\)
\(18\) 0 0
\(19\) −2.63998 −0.605653 −0.302827 0.953046i \(-0.597930\pi\)
−0.302827 + 0.953046i \(0.597930\pi\)
\(20\) 0 0
\(21\) 4.07281 0.888760
\(22\) 0 0
\(23\) −4.35915 −0.908945 −0.454473 0.890761i \(-0.650172\pi\)
−0.454473 + 0.890761i \(0.650172\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.21995 1.00458
\(28\) 0 0
\(29\) 0.768672 0.142739 0.0713694 0.997450i \(-0.477263\pi\)
0.0713694 + 0.997450i \(0.477263\pi\)
\(30\) 0 0
\(31\) −6.69522 −1.20250 −0.601249 0.799062i \(-0.705330\pi\)
−0.601249 + 0.799062i \(0.705330\pi\)
\(32\) 0 0
\(33\) 0.639305 0.111289
\(34\) 0 0
\(35\) −3.77160 −0.637516
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 3.92306 0.628193
\(40\) 0 0
\(41\) 7.71226 1.20445 0.602226 0.798325i \(-0.294281\pi\)
0.602226 + 0.798325i \(0.294281\pi\)
\(42\) 0 0
\(43\) −2.33160 −0.355566 −0.177783 0.984070i \(-0.556893\pi\)
−0.177783 + 0.984070i \(0.556893\pi\)
\(44\) 0 0
\(45\) −1.83390 −0.273381
\(46\) 0 0
\(47\) 7.64640 1.11534 0.557671 0.830062i \(-0.311695\pi\)
0.557671 + 0.830062i \(0.311695\pi\)
\(48\) 0 0
\(49\) 7.22495 1.03214
\(50\) 0 0
\(51\) −4.58064 −0.641418
\(52\) 0 0
\(53\) 0.432298 0.0593806 0.0296903 0.999559i \(-0.490548\pi\)
0.0296903 + 0.999559i \(0.490548\pi\)
\(54\) 0 0
\(55\) −0.592024 −0.0798285
\(56\) 0 0
\(57\) 2.85082 0.377600
\(58\) 0 0
\(59\) 13.2065 1.71934 0.859671 0.510848i \(-0.170669\pi\)
0.859671 + 0.510848i \(0.170669\pi\)
\(60\) 0 0
\(61\) −1.39105 −0.178106 −0.0890531 0.996027i \(-0.528384\pi\)
−0.0890531 + 0.996027i \(0.528384\pi\)
\(62\) 0 0
\(63\) 6.91672 0.871424
\(64\) 0 0
\(65\) −3.63293 −0.450609
\(66\) 0 0
\(67\) −4.83940 −0.591227 −0.295614 0.955308i \(-0.595524\pi\)
−0.295614 + 0.955308i \(0.595524\pi\)
\(68\) 0 0
\(69\) 4.70728 0.566690
\(70\) 0 0
\(71\) −0.639305 −0.0758715 −0.0379358 0.999280i \(-0.512078\pi\)
−0.0379358 + 0.999280i \(0.512078\pi\)
\(72\) 0 0
\(73\) −8.23872 −0.964269 −0.482134 0.876097i \(-0.660138\pi\)
−0.482134 + 0.876097i \(0.660138\pi\)
\(74\) 0 0
\(75\) −1.07986 −0.124692
\(76\) 0 0
\(77\) 2.23288 0.254460
\(78\) 0 0
\(79\) 5.94066 0.668377 0.334188 0.942506i \(-0.391538\pi\)
0.334188 + 0.942506i \(0.391538\pi\)
\(80\) 0 0
\(81\) −0.135139 −0.0150155
\(82\) 0 0
\(83\) 4.80252 0.527145 0.263573 0.964640i \(-0.415099\pi\)
0.263573 + 0.964640i \(0.415099\pi\)
\(84\) 0 0
\(85\) 4.24187 0.460096
\(86\) 0 0
\(87\) −0.830060 −0.0889918
\(88\) 0 0
\(89\) 17.4653 1.85132 0.925658 0.378362i \(-0.123513\pi\)
0.925658 + 0.378362i \(0.123513\pi\)
\(90\) 0 0
\(91\) 13.7019 1.43635
\(92\) 0 0
\(93\) 7.22993 0.749708
\(94\) 0 0
\(95\) −2.63998 −0.270856
\(96\) 0 0
\(97\) 7.77889 0.789826 0.394913 0.918718i \(-0.370775\pi\)
0.394913 + 0.918718i \(0.370775\pi\)
\(98\) 0 0
\(99\) 1.08571 0.109118
\(100\) 0 0
\(101\) 10.4320 1.03802 0.519009 0.854769i \(-0.326301\pi\)
0.519009 + 0.854769i \(0.326301\pi\)
\(102\) 0 0
\(103\) 6.30428 0.621179 0.310590 0.950544i \(-0.399473\pi\)
0.310590 + 0.950544i \(0.399473\pi\)
\(104\) 0 0
\(105\) 4.07281 0.397465
\(106\) 0 0
\(107\) −10.1260 −0.978915 −0.489457 0.872027i \(-0.662805\pi\)
−0.489457 + 0.872027i \(0.662805\pi\)
\(108\) 0 0
\(109\) −2.53997 −0.243285 −0.121643 0.992574i \(-0.538816\pi\)
−0.121643 + 0.992574i \(0.538816\pi\)
\(110\) 0 0
\(111\) 1.07986 0.102496
\(112\) 0 0
\(113\) −8.76075 −0.824142 −0.412071 0.911152i \(-0.635194\pi\)
−0.412071 + 0.911152i \(0.635194\pi\)
\(114\) 0 0
\(115\) −4.35915 −0.406493
\(116\) 0 0
\(117\) 6.66241 0.615940
\(118\) 0 0
\(119\) −15.9986 −1.46659
\(120\) 0 0
\(121\) −10.6495 −0.968137
\(122\) 0 0
\(123\) −8.32818 −0.750927
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.3305 −1.80404 −0.902022 0.431690i \(-0.857918\pi\)
−0.902022 + 0.431690i \(0.857918\pi\)
\(128\) 0 0
\(129\) 2.51781 0.221681
\(130\) 0 0
\(131\) 1.07705 0.0941022 0.0470511 0.998892i \(-0.485018\pi\)
0.0470511 + 0.998892i \(0.485018\pi\)
\(132\) 0 0
\(133\) 9.95694 0.863377
\(134\) 0 0
\(135\) 5.21995 0.449261
\(136\) 0 0
\(137\) −3.21099 −0.274333 −0.137167 0.990548i \(-0.543800\pi\)
−0.137167 + 0.990548i \(0.543800\pi\)
\(138\) 0 0
\(139\) 17.6116 1.49379 0.746897 0.664940i \(-0.231543\pi\)
0.746897 + 0.664940i \(0.231543\pi\)
\(140\) 0 0
\(141\) −8.25706 −0.695370
\(142\) 0 0
\(143\) 2.15078 0.179857
\(144\) 0 0
\(145\) 0.768672 0.0638347
\(146\) 0 0
\(147\) −7.80196 −0.643495
\(148\) 0 0
\(149\) 15.9003 1.30261 0.651303 0.758818i \(-0.274223\pi\)
0.651303 + 0.758818i \(0.274223\pi\)
\(150\) 0 0
\(151\) 16.5458 1.34648 0.673240 0.739424i \(-0.264902\pi\)
0.673240 + 0.739424i \(0.264902\pi\)
\(152\) 0 0
\(153\) −7.77915 −0.628907
\(154\) 0 0
\(155\) −6.69522 −0.537773
\(156\) 0 0
\(157\) −7.09573 −0.566301 −0.283150 0.959076i \(-0.591380\pi\)
−0.283150 + 0.959076i \(0.591380\pi\)
\(158\) 0 0
\(159\) −0.466822 −0.0370214
\(160\) 0 0
\(161\) 16.4410 1.29573
\(162\) 0 0
\(163\) 15.4729 1.21193 0.605964 0.795492i \(-0.292788\pi\)
0.605964 + 0.795492i \(0.292788\pi\)
\(164\) 0 0
\(165\) 0.639305 0.0497698
\(166\) 0 0
\(167\) 1.35479 0.104837 0.0524183 0.998625i \(-0.483307\pi\)
0.0524183 + 0.998625i \(0.483307\pi\)
\(168\) 0 0
\(169\) 0.198155 0.0152427
\(170\) 0 0
\(171\) 4.84145 0.370235
\(172\) 0 0
\(173\) 0.0106237 0.000807708 0 0.000403854 1.00000i \(-0.499871\pi\)
0.000403854 1.00000i \(0.499871\pi\)
\(174\) 0 0
\(175\) −3.77160 −0.285106
\(176\) 0 0
\(177\) −14.2612 −1.07194
\(178\) 0 0
\(179\) 17.3180 1.29441 0.647204 0.762317i \(-0.275938\pi\)
0.647204 + 0.762317i \(0.275938\pi\)
\(180\) 0 0
\(181\) −18.6673 −1.38753 −0.693763 0.720203i \(-0.744049\pi\)
−0.693763 + 0.720203i \(0.744049\pi\)
\(182\) 0 0
\(183\) 1.50215 0.111042
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −2.51129 −0.183644
\(188\) 0 0
\(189\) −19.6875 −1.43206
\(190\) 0 0
\(191\) 5.02878 0.363870 0.181935 0.983311i \(-0.441764\pi\)
0.181935 + 0.983311i \(0.441764\pi\)
\(192\) 0 0
\(193\) −7.84163 −0.564453 −0.282226 0.959348i \(-0.591073\pi\)
−0.282226 + 0.959348i \(0.591073\pi\)
\(194\) 0 0
\(195\) 3.92306 0.280936
\(196\) 0 0
\(197\) 19.9703 1.42283 0.711413 0.702774i \(-0.248055\pi\)
0.711413 + 0.702774i \(0.248055\pi\)
\(198\) 0 0
\(199\) −6.52822 −0.462773 −0.231386 0.972862i \(-0.574326\pi\)
−0.231386 + 0.972862i \(0.574326\pi\)
\(200\) 0 0
\(201\) 5.22589 0.368606
\(202\) 0 0
\(203\) −2.89912 −0.203478
\(204\) 0 0
\(205\) 7.71226 0.538647
\(206\) 0 0
\(207\) 7.99422 0.555637
\(208\) 0 0
\(209\) 1.56293 0.108110
\(210\) 0 0
\(211\) −0.514103 −0.0353923 −0.0176962 0.999843i \(-0.505633\pi\)
−0.0176962 + 0.999843i \(0.505633\pi\)
\(212\) 0 0
\(213\) 0.690361 0.0473028
\(214\) 0 0
\(215\) −2.33160 −0.159014
\(216\) 0 0
\(217\) 25.2517 1.71420
\(218\) 0 0
\(219\) 8.89669 0.601182
\(220\) 0 0
\(221\) −15.4104 −1.03662
\(222\) 0 0
\(223\) −5.80468 −0.388710 −0.194355 0.980931i \(-0.562261\pi\)
−0.194355 + 0.980931i \(0.562261\pi\)
\(224\) 0 0
\(225\) −1.83390 −0.122260
\(226\) 0 0
\(227\) −2.92840 −0.194365 −0.0971823 0.995267i \(-0.530983\pi\)
−0.0971823 + 0.995267i \(0.530983\pi\)
\(228\) 0 0
\(229\) 17.1595 1.13393 0.566967 0.823741i \(-0.308117\pi\)
0.566967 + 0.823741i \(0.308117\pi\)
\(230\) 0 0
\(231\) −2.41120 −0.158645
\(232\) 0 0
\(233\) −2.39737 −0.157057 −0.0785286 0.996912i \(-0.525022\pi\)
−0.0785286 + 0.996912i \(0.525022\pi\)
\(234\) 0 0
\(235\) 7.64640 0.498796
\(236\) 0 0
\(237\) −6.41510 −0.416706
\(238\) 0 0
\(239\) 23.6560 1.53018 0.765091 0.643922i \(-0.222694\pi\)
0.765091 + 0.643922i \(0.222694\pi\)
\(240\) 0 0
\(241\) −11.2110 −0.722163 −0.361082 0.932534i \(-0.617592\pi\)
−0.361082 + 0.932534i \(0.617592\pi\)
\(242\) 0 0
\(243\) −15.5139 −0.995217
\(244\) 0 0
\(245\) 7.22495 0.461585
\(246\) 0 0
\(247\) 9.59085 0.610251
\(248\) 0 0
\(249\) −5.18606 −0.328653
\(250\) 0 0
\(251\) 20.4734 1.29227 0.646133 0.763225i \(-0.276385\pi\)
0.646133 + 0.763225i \(0.276385\pi\)
\(252\) 0 0
\(253\) 2.58072 0.162248
\(254\) 0 0
\(255\) −4.58064 −0.286851
\(256\) 0 0
\(257\) 10.5889 0.660516 0.330258 0.943891i \(-0.392864\pi\)
0.330258 + 0.943891i \(0.392864\pi\)
\(258\) 0 0
\(259\) 3.77160 0.234356
\(260\) 0 0
\(261\) −1.40966 −0.0872560
\(262\) 0 0
\(263\) 10.3030 0.635310 0.317655 0.948206i \(-0.397105\pi\)
0.317655 + 0.948206i \(0.397105\pi\)
\(264\) 0 0
\(265\) 0.432298 0.0265558
\(266\) 0 0
\(267\) −18.8601 −1.15422
\(268\) 0 0
\(269\) −8.05589 −0.491176 −0.245588 0.969374i \(-0.578981\pi\)
−0.245588 + 0.969374i \(0.578981\pi\)
\(270\) 0 0
\(271\) 0.439883 0.0267210 0.0133605 0.999911i \(-0.495747\pi\)
0.0133605 + 0.999911i \(0.495747\pi\)
\(272\) 0 0
\(273\) −14.7962 −0.895508
\(274\) 0 0
\(275\) −0.592024 −0.0357004
\(276\) 0 0
\(277\) −25.3387 −1.52245 −0.761227 0.648486i \(-0.775403\pi\)
−0.761227 + 0.648486i \(0.775403\pi\)
\(278\) 0 0
\(279\) 12.2783 0.735085
\(280\) 0 0
\(281\) 9.36373 0.558594 0.279297 0.960205i \(-0.409899\pi\)
0.279297 + 0.960205i \(0.409899\pi\)
\(282\) 0 0
\(283\) 6.99025 0.415527 0.207764 0.978179i \(-0.433381\pi\)
0.207764 + 0.978179i \(0.433381\pi\)
\(284\) 0 0
\(285\) 2.85082 0.168868
\(286\) 0 0
\(287\) −29.0875 −1.71698
\(288\) 0 0
\(289\) 0.993477 0.0584398
\(290\) 0 0
\(291\) −8.40013 −0.492424
\(292\) 0 0
\(293\) 10.6724 0.623486 0.311743 0.950166i \(-0.399087\pi\)
0.311743 + 0.950166i \(0.399087\pi\)
\(294\) 0 0
\(295\) 13.2065 0.768913
\(296\) 0 0
\(297\) −3.09033 −0.179319
\(298\) 0 0
\(299\) 15.8365 0.915846
\(300\) 0 0
\(301\) 8.79387 0.506870
\(302\) 0 0
\(303\) −11.2651 −0.647162
\(304\) 0 0
\(305\) −1.39105 −0.0796516
\(306\) 0 0
\(307\) 17.9040 1.02184 0.510918 0.859630i \(-0.329306\pi\)
0.510918 + 0.859630i \(0.329306\pi\)
\(308\) 0 0
\(309\) −6.80776 −0.387280
\(310\) 0 0
\(311\) −10.6173 −0.602052 −0.301026 0.953616i \(-0.597329\pi\)
−0.301026 + 0.953616i \(0.597329\pi\)
\(312\) 0 0
\(313\) −18.4102 −1.04061 −0.520303 0.853982i \(-0.674181\pi\)
−0.520303 + 0.853982i \(0.674181\pi\)
\(314\) 0 0
\(315\) 6.91672 0.389713
\(316\) 0 0
\(317\) 30.1211 1.69177 0.845885 0.533365i \(-0.179073\pi\)
0.845885 + 0.533365i \(0.179073\pi\)
\(318\) 0 0
\(319\) −0.455072 −0.0254791
\(320\) 0 0
\(321\) 10.9347 0.610314
\(322\) 0 0
\(323\) −11.1985 −0.623099
\(324\) 0 0
\(325\) −3.63293 −0.201519
\(326\) 0 0
\(327\) 2.74282 0.151678
\(328\) 0 0
\(329\) −28.8391 −1.58995
\(330\) 0 0
\(331\) 28.8280 1.58453 0.792265 0.610177i \(-0.208902\pi\)
0.792265 + 0.610177i \(0.208902\pi\)
\(332\) 0 0
\(333\) 1.83390 0.100497
\(334\) 0 0
\(335\) −4.83940 −0.264405
\(336\) 0 0
\(337\) 33.3662 1.81757 0.908785 0.417264i \(-0.137011\pi\)
0.908785 + 0.417264i \(0.137011\pi\)
\(338\) 0 0
\(339\) 9.46041 0.513819
\(340\) 0 0
\(341\) 3.96373 0.214648
\(342\) 0 0
\(343\) −0.848422 −0.0458105
\(344\) 0 0
\(345\) 4.70728 0.253432
\(346\) 0 0
\(347\) 16.2626 0.873023 0.436512 0.899699i \(-0.356214\pi\)
0.436512 + 0.899699i \(0.356214\pi\)
\(348\) 0 0
\(349\) −13.1613 −0.704509 −0.352255 0.935904i \(-0.614585\pi\)
−0.352255 + 0.935904i \(0.614585\pi\)
\(350\) 0 0
\(351\) −18.9637 −1.01221
\(352\) 0 0
\(353\) −32.1014 −1.70858 −0.854292 0.519793i \(-0.826009\pi\)
−0.854292 + 0.519793i \(0.826009\pi\)
\(354\) 0 0
\(355\) −0.639305 −0.0339308
\(356\) 0 0
\(357\) 17.2763 0.914361
\(358\) 0 0
\(359\) −4.53580 −0.239390 −0.119695 0.992811i \(-0.538192\pi\)
−0.119695 + 0.992811i \(0.538192\pi\)
\(360\) 0 0
\(361\) −12.0305 −0.633184
\(362\) 0 0
\(363\) 11.5000 0.603594
\(364\) 0 0
\(365\) −8.23872 −0.431234
\(366\) 0 0
\(367\) −10.2661 −0.535887 −0.267944 0.963435i \(-0.586344\pi\)
−0.267944 + 0.963435i \(0.586344\pi\)
\(368\) 0 0
\(369\) −14.1435 −0.736280
\(370\) 0 0
\(371\) −1.63045 −0.0846489
\(372\) 0 0
\(373\) 20.3891 1.05571 0.527854 0.849335i \(-0.322997\pi\)
0.527854 + 0.849335i \(0.322997\pi\)
\(374\) 0 0
\(375\) −1.07986 −0.0557639
\(376\) 0 0
\(377\) −2.79253 −0.143822
\(378\) 0 0
\(379\) 31.8176 1.63436 0.817180 0.576383i \(-0.195536\pi\)
0.817180 + 0.576383i \(0.195536\pi\)
\(380\) 0 0
\(381\) 21.9542 1.12475
\(382\) 0 0
\(383\) −21.3238 −1.08960 −0.544798 0.838567i \(-0.683394\pi\)
−0.544798 + 0.838567i \(0.683394\pi\)
\(384\) 0 0
\(385\) 2.23288 0.113798
\(386\) 0 0
\(387\) 4.27592 0.217357
\(388\) 0 0
\(389\) 29.0717 1.47399 0.736996 0.675897i \(-0.236243\pi\)
0.736996 + 0.675897i \(0.236243\pi\)
\(390\) 0 0
\(391\) −18.4909 −0.935127
\(392\) 0 0
\(393\) −1.16307 −0.0586689
\(394\) 0 0
\(395\) 5.94066 0.298907
\(396\) 0 0
\(397\) −0.703118 −0.0352885 −0.0176442 0.999844i \(-0.505617\pi\)
−0.0176442 + 0.999844i \(0.505617\pi\)
\(398\) 0 0
\(399\) −10.7521 −0.538280
\(400\) 0 0
\(401\) −28.9306 −1.44472 −0.722361 0.691516i \(-0.756943\pi\)
−0.722361 + 0.691516i \(0.756943\pi\)
\(402\) 0 0
\(403\) 24.3233 1.21163
\(404\) 0 0
\(405\) −0.135139 −0.00671512
\(406\) 0 0
\(407\) 0.592024 0.0293455
\(408\) 0 0
\(409\) 6.42859 0.317873 0.158937 0.987289i \(-0.449193\pi\)
0.158937 + 0.987289i \(0.449193\pi\)
\(410\) 0 0
\(411\) 3.46743 0.171036
\(412\) 0 0
\(413\) −49.8097 −2.45097
\(414\) 0 0
\(415\) 4.80252 0.235746
\(416\) 0 0
\(417\) −19.0181 −0.931319
\(418\) 0 0
\(419\) −2.02413 −0.0988850 −0.0494425 0.998777i \(-0.515744\pi\)
−0.0494425 + 0.998777i \(0.515744\pi\)
\(420\) 0 0
\(421\) −34.5472 −1.68373 −0.841863 0.539691i \(-0.818541\pi\)
−0.841863 + 0.539691i \(0.818541\pi\)
\(422\) 0 0
\(423\) −14.0227 −0.681807
\(424\) 0 0
\(425\) 4.24187 0.205761
\(426\) 0 0
\(427\) 5.24650 0.253896
\(428\) 0 0
\(429\) −2.32255 −0.112134
\(430\) 0 0
\(431\) 15.7081 0.756631 0.378316 0.925677i \(-0.376503\pi\)
0.378316 + 0.925677i \(0.376503\pi\)
\(432\) 0 0
\(433\) −27.5812 −1.32547 −0.662734 0.748855i \(-0.730604\pi\)
−0.662734 + 0.748855i \(0.730604\pi\)
\(434\) 0 0
\(435\) −0.830060 −0.0397983
\(436\) 0 0
\(437\) 11.5081 0.550505
\(438\) 0 0
\(439\) −18.9495 −0.904411 −0.452205 0.891914i \(-0.649363\pi\)
−0.452205 + 0.891914i \(0.649363\pi\)
\(440\) 0 0
\(441\) −13.2498 −0.630943
\(442\) 0 0
\(443\) −14.3423 −0.681424 −0.340712 0.940168i \(-0.610668\pi\)
−0.340712 + 0.940168i \(0.610668\pi\)
\(444\) 0 0
\(445\) 17.4653 0.827933
\(446\) 0 0
\(447\) −17.1702 −0.812121
\(448\) 0 0
\(449\) −32.2535 −1.52214 −0.761069 0.648671i \(-0.775325\pi\)
−0.761069 + 0.648671i \(0.775325\pi\)
\(450\) 0 0
\(451\) −4.56584 −0.214997
\(452\) 0 0
\(453\) −17.8672 −0.839475
\(454\) 0 0
\(455\) 13.7019 0.642357
\(456\) 0 0
\(457\) 17.1417 0.801856 0.400928 0.916110i \(-0.368688\pi\)
0.400928 + 0.916110i \(0.368688\pi\)
\(458\) 0 0
\(459\) 22.1423 1.03352
\(460\) 0 0
\(461\) 2.53862 0.118235 0.0591177 0.998251i \(-0.481171\pi\)
0.0591177 + 0.998251i \(0.481171\pi\)
\(462\) 0 0
\(463\) −28.8147 −1.33913 −0.669566 0.742753i \(-0.733520\pi\)
−0.669566 + 0.742753i \(0.733520\pi\)
\(464\) 0 0
\(465\) 7.22993 0.335280
\(466\) 0 0
\(467\) 21.5498 0.997206 0.498603 0.866831i \(-0.333847\pi\)
0.498603 + 0.866831i \(0.333847\pi\)
\(468\) 0 0
\(469\) 18.2523 0.842812
\(470\) 0 0
\(471\) 7.66241 0.353065
\(472\) 0 0
\(473\) 1.38036 0.0634692
\(474\) 0 0
\(475\) −2.63998 −0.121131
\(476\) 0 0
\(477\) −0.792789 −0.0362993
\(478\) 0 0
\(479\) 15.2747 0.697920 0.348960 0.937138i \(-0.386535\pi\)
0.348960 + 0.937138i \(0.386535\pi\)
\(480\) 0 0
\(481\) 3.63293 0.165647
\(482\) 0 0
\(483\) −17.7540 −0.807834
\(484\) 0 0
\(485\) 7.77889 0.353221
\(486\) 0 0
\(487\) −10.0338 −0.454677 −0.227338 0.973816i \(-0.573002\pi\)
−0.227338 + 0.973816i \(0.573002\pi\)
\(488\) 0 0
\(489\) −16.7086 −0.755588
\(490\) 0 0
\(491\) −30.0308 −1.35527 −0.677636 0.735397i \(-0.736996\pi\)
−0.677636 + 0.735397i \(0.736996\pi\)
\(492\) 0 0
\(493\) 3.26061 0.146850
\(494\) 0 0
\(495\) 1.08571 0.0487990
\(496\) 0 0
\(497\) 2.41120 0.108157
\(498\) 0 0
\(499\) 16.2377 0.726901 0.363451 0.931614i \(-0.381599\pi\)
0.363451 + 0.931614i \(0.381599\pi\)
\(500\) 0 0
\(501\) −1.46298 −0.0653613
\(502\) 0 0
\(503\) −24.4896 −1.09194 −0.545969 0.837806i \(-0.683838\pi\)
−0.545969 + 0.837806i \(0.683838\pi\)
\(504\) 0 0
\(505\) 10.4320 0.464216
\(506\) 0 0
\(507\) −0.213981 −0.00950321
\(508\) 0 0
\(509\) −11.7544 −0.521004 −0.260502 0.965473i \(-0.583888\pi\)
−0.260502 + 0.965473i \(0.583888\pi\)
\(510\) 0 0
\(511\) 31.0731 1.37459
\(512\) 0 0
\(513\) −13.7806 −0.608426
\(514\) 0 0
\(515\) 6.30428 0.277800
\(516\) 0 0
\(517\) −4.52685 −0.199091
\(518\) 0 0
\(519\) −0.0114722 −0.000503573 0
\(520\) 0 0
\(521\) 32.2637 1.41350 0.706748 0.707465i \(-0.250161\pi\)
0.706748 + 0.707465i \(0.250161\pi\)
\(522\) 0 0
\(523\) 29.6565 1.29679 0.648394 0.761305i \(-0.275441\pi\)
0.648394 + 0.761305i \(0.275441\pi\)
\(524\) 0 0
\(525\) 4.07281 0.177752
\(526\) 0 0
\(527\) −28.4003 −1.23714
\(528\) 0 0
\(529\) −3.99783 −0.173819
\(530\) 0 0
\(531\) −24.2194 −1.05103
\(532\) 0 0
\(533\) −28.0181 −1.21360
\(534\) 0 0
\(535\) −10.1260 −0.437784
\(536\) 0 0
\(537\) −18.7011 −0.807010
\(538\) 0 0
\(539\) −4.27734 −0.184238
\(540\) 0 0
\(541\) −6.79143 −0.291986 −0.145993 0.989286i \(-0.546638\pi\)
−0.145993 + 0.989286i \(0.546638\pi\)
\(542\) 0 0
\(543\) 20.1581 0.865066
\(544\) 0 0
\(545\) −2.53997 −0.108800
\(546\) 0 0
\(547\) 19.5412 0.835523 0.417762 0.908557i \(-0.362815\pi\)
0.417762 + 0.908557i \(0.362815\pi\)
\(548\) 0 0
\(549\) 2.55105 0.108876
\(550\) 0 0
\(551\) −2.02928 −0.0864501
\(552\) 0 0
\(553\) −22.4058 −0.952791
\(554\) 0 0
\(555\) 1.07986 0.0458376
\(556\) 0 0
\(557\) −27.2741 −1.15564 −0.577820 0.816164i \(-0.696096\pi\)
−0.577820 + 0.816164i \(0.696096\pi\)
\(558\) 0 0
\(559\) 8.47054 0.358266
\(560\) 0 0
\(561\) 2.71185 0.114494
\(562\) 0 0
\(563\) −12.0787 −0.509055 −0.254528 0.967066i \(-0.581920\pi\)
−0.254528 + 0.967066i \(0.581920\pi\)
\(564\) 0 0
\(565\) −8.76075 −0.368567
\(566\) 0 0
\(567\) 0.509691 0.0214050
\(568\) 0 0
\(569\) −21.9855 −0.921680 −0.460840 0.887483i \(-0.652452\pi\)
−0.460840 + 0.887483i \(0.652452\pi\)
\(570\) 0 0
\(571\) −37.4836 −1.56864 −0.784321 0.620356i \(-0.786988\pi\)
−0.784321 + 0.620356i \(0.786988\pi\)
\(572\) 0 0
\(573\) −5.43040 −0.226858
\(574\) 0 0
\(575\) −4.35915 −0.181789
\(576\) 0 0
\(577\) 11.1575 0.464493 0.232247 0.972657i \(-0.425392\pi\)
0.232247 + 0.972657i \(0.425392\pi\)
\(578\) 0 0
\(579\) 8.46788 0.351913
\(580\) 0 0
\(581\) −18.1132 −0.751461
\(582\) 0 0
\(583\) −0.255931 −0.0105996
\(584\) 0 0
\(585\) 6.66241 0.275457
\(586\) 0 0
\(587\) 22.9471 0.947127 0.473564 0.880760i \(-0.342967\pi\)
0.473564 + 0.880760i \(0.342967\pi\)
\(588\) 0 0
\(589\) 17.6753 0.728296
\(590\) 0 0
\(591\) −21.5652 −0.887074
\(592\) 0 0
\(593\) 11.7268 0.481563 0.240781 0.970579i \(-0.422596\pi\)
0.240781 + 0.970579i \(0.422596\pi\)
\(594\) 0 0
\(595\) −15.9986 −0.655880
\(596\) 0 0
\(597\) 7.04958 0.288520
\(598\) 0 0
\(599\) −32.9905 −1.34795 −0.673977 0.738752i \(-0.735415\pi\)
−0.673977 + 0.738752i \(0.735415\pi\)
\(600\) 0 0
\(601\) 24.5572 1.00171 0.500855 0.865531i \(-0.333019\pi\)
0.500855 + 0.865531i \(0.333019\pi\)
\(602\) 0 0
\(603\) 8.87496 0.361416
\(604\) 0 0
\(605\) −10.6495 −0.432964
\(606\) 0 0
\(607\) −2.91430 −0.118288 −0.0591439 0.998249i \(-0.518837\pi\)
−0.0591439 + 0.998249i \(0.518837\pi\)
\(608\) 0 0
\(609\) 3.13065 0.126860
\(610\) 0 0
\(611\) −27.7788 −1.12381
\(612\) 0 0
\(613\) −6.36977 −0.257273 −0.128636 0.991692i \(-0.541060\pi\)
−0.128636 + 0.991692i \(0.541060\pi\)
\(614\) 0 0
\(615\) −8.32818 −0.335825
\(616\) 0 0
\(617\) 0.0123562 0.000497441 0 0.000248721 1.00000i \(-0.499921\pi\)
0.000248721 1.00000i \(0.499921\pi\)
\(618\) 0 0
\(619\) 9.07442 0.364732 0.182366 0.983231i \(-0.441624\pi\)
0.182366 + 0.983231i \(0.441624\pi\)
\(620\) 0 0
\(621\) −22.7545 −0.913107
\(622\) 0 0
\(623\) −65.8720 −2.63911
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.68775 −0.0674023
\(628\) 0 0
\(629\) −4.24187 −0.169135
\(630\) 0 0
\(631\) −25.1582 −1.00153 −0.500767 0.865582i \(-0.666949\pi\)
−0.500767 + 0.865582i \(0.666949\pi\)
\(632\) 0 0
\(633\) 0.555161 0.0220657
\(634\) 0 0
\(635\) −20.3305 −0.806793
\(636\) 0 0
\(637\) −26.2477 −1.03997
\(638\) 0 0
\(639\) 1.17242 0.0463801
\(640\) 0 0
\(641\) 38.3446 1.51452 0.757260 0.653113i \(-0.226538\pi\)
0.757260 + 0.653113i \(0.226538\pi\)
\(642\) 0 0
\(643\) −33.3835 −1.31652 −0.658259 0.752791i \(-0.728707\pi\)
−0.658259 + 0.752791i \(0.728707\pi\)
\(644\) 0 0
\(645\) 2.51781 0.0991387
\(646\) 0 0
\(647\) −36.7815 −1.44603 −0.723015 0.690832i \(-0.757244\pi\)
−0.723015 + 0.690832i \(0.757244\pi\)
\(648\) 0 0
\(649\) −7.81857 −0.306906
\(650\) 0 0
\(651\) −27.2684 −1.06873
\(652\) 0 0
\(653\) 10.9336 0.427867 0.213933 0.976848i \(-0.431372\pi\)
0.213933 + 0.976848i \(0.431372\pi\)
\(654\) 0 0
\(655\) 1.07705 0.0420838
\(656\) 0 0
\(657\) 15.1089 0.589456
\(658\) 0 0
\(659\) −9.74959 −0.379790 −0.189895 0.981804i \(-0.560815\pi\)
−0.189895 + 0.981804i \(0.560815\pi\)
\(660\) 0 0
\(661\) −8.93560 −0.347555 −0.173777 0.984785i \(-0.555597\pi\)
−0.173777 + 0.984785i \(0.555597\pi\)
\(662\) 0 0
\(663\) 16.6411 0.646288
\(664\) 0 0
\(665\) 9.95694 0.386114
\(666\) 0 0
\(667\) −3.35075 −0.129742
\(668\) 0 0
\(669\) 6.26826 0.242345
\(670\) 0 0
\(671\) 0.823537 0.0317923
\(672\) 0 0
\(673\) 41.5283 1.60080 0.800399 0.599468i \(-0.204621\pi\)
0.800399 + 0.599468i \(0.204621\pi\)
\(674\) 0 0
\(675\) 5.21995 0.200916
\(676\) 0 0
\(677\) −37.9150 −1.45719 −0.728595 0.684945i \(-0.759826\pi\)
−0.728595 + 0.684945i \(0.759826\pi\)
\(678\) 0 0
\(679\) −29.3388 −1.12592
\(680\) 0 0
\(681\) 3.16227 0.121178
\(682\) 0 0
\(683\) −33.5541 −1.28391 −0.641955 0.766742i \(-0.721877\pi\)
−0.641955 + 0.766742i \(0.721877\pi\)
\(684\) 0 0
\(685\) −3.21099 −0.122686
\(686\) 0 0
\(687\) −18.5299 −0.706961
\(688\) 0 0
\(689\) −1.57051 −0.0598315
\(690\) 0 0
\(691\) −1.03668 −0.0394370 −0.0197185 0.999806i \(-0.506277\pi\)
−0.0197185 + 0.999806i \(0.506277\pi\)
\(692\) 0 0
\(693\) −4.09486 −0.155551
\(694\) 0 0
\(695\) 17.6116 0.668045
\(696\) 0 0
\(697\) 32.7144 1.23915
\(698\) 0 0
\(699\) 2.58883 0.0979187
\(700\) 0 0
\(701\) 19.8334 0.749096 0.374548 0.927208i \(-0.377798\pi\)
0.374548 + 0.927208i \(0.377798\pi\)
\(702\) 0 0
\(703\) 2.63998 0.0995687
\(704\) 0 0
\(705\) −8.25706 −0.310979
\(706\) 0 0
\(707\) −39.3451 −1.47973
\(708\) 0 0
\(709\) −38.7110 −1.45382 −0.726911 0.686731i \(-0.759045\pi\)
−0.726911 + 0.686731i \(0.759045\pi\)
\(710\) 0 0
\(711\) −10.8946 −0.408578
\(712\) 0 0
\(713\) 29.1855 1.09300
\(714\) 0 0
\(715\) 2.15078 0.0804346
\(716\) 0 0
\(717\) −25.5453 −0.954006
\(718\) 0 0
\(719\) 21.7655 0.811715 0.405857 0.913936i \(-0.366973\pi\)
0.405857 + 0.913936i \(0.366973\pi\)
\(720\) 0 0
\(721\) −23.7772 −0.885510
\(722\) 0 0
\(723\) 12.1063 0.450239
\(724\) 0 0
\(725\) 0.768672 0.0285477
\(726\) 0 0
\(727\) 22.2072 0.823618 0.411809 0.911270i \(-0.364897\pi\)
0.411809 + 0.911270i \(0.364897\pi\)
\(728\) 0 0
\(729\) 17.1583 0.635493
\(730\) 0 0
\(731\) −9.89036 −0.365808
\(732\) 0 0
\(733\) −1.71719 −0.0634260 −0.0317130 0.999497i \(-0.510096\pi\)
−0.0317130 + 0.999497i \(0.510096\pi\)
\(734\) 0 0
\(735\) −7.80196 −0.287780
\(736\) 0 0
\(737\) 2.86504 0.105535
\(738\) 0 0
\(739\) 22.6254 0.832289 0.416145 0.909298i \(-0.363381\pi\)
0.416145 + 0.909298i \(0.363381\pi\)
\(740\) 0 0
\(741\) −10.3568 −0.380467
\(742\) 0 0
\(743\) −16.1746 −0.593388 −0.296694 0.954973i \(-0.595884\pi\)
−0.296694 + 0.954973i \(0.595884\pi\)
\(744\) 0 0
\(745\) 15.9003 0.582543
\(746\) 0 0
\(747\) −8.80732 −0.322243
\(748\) 0 0
\(749\) 38.1911 1.39547
\(750\) 0 0
\(751\) 6.30012 0.229895 0.114947 0.993372i \(-0.463330\pi\)
0.114947 + 0.993372i \(0.463330\pi\)
\(752\) 0 0
\(753\) −22.1084 −0.805675
\(754\) 0 0
\(755\) 16.5458 0.602164
\(756\) 0 0
\(757\) −1.40584 −0.0510963 −0.0255481 0.999674i \(-0.508133\pi\)
−0.0255481 + 0.999674i \(0.508133\pi\)
\(758\) 0 0
\(759\) −2.78682 −0.101155
\(760\) 0 0
\(761\) −5.57362 −0.202044 −0.101022 0.994884i \(-0.532211\pi\)
−0.101022 + 0.994884i \(0.532211\pi\)
\(762\) 0 0
\(763\) 9.57975 0.346810
\(764\) 0 0
\(765\) −7.77915 −0.281256
\(766\) 0 0
\(767\) −47.9783 −1.73240
\(768\) 0 0
\(769\) −8.14124 −0.293581 −0.146790 0.989168i \(-0.546894\pi\)
−0.146790 + 0.989168i \(0.546894\pi\)
\(770\) 0 0
\(771\) −11.4345 −0.411805
\(772\) 0 0
\(773\) 41.2184 1.48252 0.741261 0.671217i \(-0.234228\pi\)
0.741261 + 0.671217i \(0.234228\pi\)
\(774\) 0 0
\(775\) −6.69522 −0.240500
\(776\) 0 0
\(777\) −4.07281 −0.146111
\(778\) 0 0
\(779\) −20.3602 −0.729480
\(780\) 0 0
\(781\) 0.378484 0.0135432
\(782\) 0 0
\(783\) 4.01242 0.143392
\(784\) 0 0
\(785\) −7.09573 −0.253257
\(786\) 0 0
\(787\) 47.8596 1.70601 0.853006 0.521901i \(-0.174777\pi\)
0.853006 + 0.521901i \(0.174777\pi\)
\(788\) 0 0
\(789\) −11.1258 −0.396090
\(790\) 0 0
\(791\) 33.0420 1.17484
\(792\) 0 0
\(793\) 5.05360 0.179459
\(794\) 0 0
\(795\) −0.466822 −0.0165565
\(796\) 0 0
\(797\) 38.0039 1.34617 0.673084 0.739566i \(-0.264969\pi\)
0.673084 + 0.739566i \(0.264969\pi\)
\(798\) 0 0
\(799\) 32.4350 1.14747
\(800\) 0 0
\(801\) −32.0295 −1.13171
\(802\) 0 0
\(803\) 4.87752 0.172124
\(804\) 0 0
\(805\) 16.4410 0.579467
\(806\) 0 0
\(807\) 8.69926 0.306228
\(808\) 0 0
\(809\) −4.46341 −0.156925 −0.0784626 0.996917i \(-0.525001\pi\)
−0.0784626 + 0.996917i \(0.525001\pi\)
\(810\) 0 0
\(811\) 28.1300 0.987780 0.493890 0.869525i \(-0.335575\pi\)
0.493890 + 0.869525i \(0.335575\pi\)
\(812\) 0 0
\(813\) −0.475013 −0.0166594
\(814\) 0 0
\(815\) 15.4729 0.541991
\(816\) 0 0
\(817\) 6.15539 0.215350
\(818\) 0 0
\(819\) −25.1279 −0.878041
\(820\) 0 0
\(821\) −0.265657 −0.00927150 −0.00463575 0.999989i \(-0.501476\pi\)
−0.00463575 + 0.999989i \(0.501476\pi\)
\(822\) 0 0
\(823\) −23.7932 −0.829378 −0.414689 0.909963i \(-0.636110\pi\)
−0.414689 + 0.909963i \(0.636110\pi\)
\(824\) 0 0
\(825\) 0.639305 0.0222577
\(826\) 0 0
\(827\) −27.9323 −0.971301 −0.485651 0.874153i \(-0.661417\pi\)
−0.485651 + 0.874153i \(0.661417\pi\)
\(828\) 0 0
\(829\) 40.8474 1.41869 0.709345 0.704861i \(-0.248991\pi\)
0.709345 + 0.704861i \(0.248991\pi\)
\(830\) 0 0
\(831\) 27.3623 0.949187
\(832\) 0 0
\(833\) 30.6473 1.06187
\(834\) 0 0
\(835\) 1.35479 0.0468843
\(836\) 0 0
\(837\) −34.9487 −1.20800
\(838\) 0 0
\(839\) 15.6220 0.539331 0.269666 0.962954i \(-0.413087\pi\)
0.269666 + 0.962954i \(0.413087\pi\)
\(840\) 0 0
\(841\) −28.4091 −0.979626
\(842\) 0 0
\(843\) −10.1116 −0.348260
\(844\) 0 0
\(845\) 0.198155 0.00681675
\(846\) 0 0
\(847\) 40.1657 1.38011
\(848\) 0 0
\(849\) −7.54852 −0.259064
\(850\) 0 0
\(851\) 4.35915 0.149430
\(852\) 0 0
\(853\) 56.2878 1.92726 0.963629 0.267243i \(-0.0861128\pi\)
0.963629 + 0.267243i \(0.0861128\pi\)
\(854\) 0 0
\(855\) 4.84145 0.165574
\(856\) 0 0
\(857\) 23.8188 0.813635 0.406817 0.913509i \(-0.366639\pi\)
0.406817 + 0.913509i \(0.366639\pi\)
\(858\) 0 0
\(859\) −12.9856 −0.443063 −0.221532 0.975153i \(-0.571106\pi\)
−0.221532 + 0.975153i \(0.571106\pi\)
\(860\) 0 0
\(861\) 31.4106 1.07047
\(862\) 0 0
\(863\) 29.4615 1.00288 0.501440 0.865192i \(-0.332804\pi\)
0.501440 + 0.865192i \(0.332804\pi\)
\(864\) 0 0
\(865\) 0.0106237 0.000361218 0
\(866\) 0 0
\(867\) −1.07282 −0.0364349
\(868\) 0 0
\(869\) −3.51701 −0.119306
\(870\) 0 0
\(871\) 17.5812 0.595716
\(872\) 0 0
\(873\) −14.2657 −0.482820
\(874\) 0 0
\(875\) −3.77160 −0.127503
\(876\) 0 0
\(877\) −10.2127 −0.344860 −0.172430 0.985022i \(-0.555162\pi\)
−0.172430 + 0.985022i \(0.555162\pi\)
\(878\) 0 0
\(879\) −11.5247 −0.388718
\(880\) 0 0
\(881\) 28.1715 0.949123 0.474561 0.880222i \(-0.342607\pi\)
0.474561 + 0.880222i \(0.342607\pi\)
\(882\) 0 0
\(883\) 30.8877 1.03945 0.519727 0.854332i \(-0.326034\pi\)
0.519727 + 0.854332i \(0.326034\pi\)
\(884\) 0 0
\(885\) −14.2612 −0.479386
\(886\) 0 0
\(887\) −46.9698 −1.57709 −0.788546 0.614975i \(-0.789166\pi\)
−0.788546 + 0.614975i \(0.789166\pi\)
\(888\) 0 0
\(889\) 76.6786 2.57172
\(890\) 0 0
\(891\) 0.0800056 0.00268029
\(892\) 0 0
\(893\) −20.1863 −0.675510
\(894\) 0 0
\(895\) 17.3180 0.578876
\(896\) 0 0
\(897\) −17.1012 −0.570993
\(898\) 0 0
\(899\) −5.14643 −0.171643
\(900\) 0 0
\(901\) 1.83375 0.0610911
\(902\) 0 0
\(903\) −9.49617 −0.316013
\(904\) 0 0
\(905\) −18.6673 −0.620521
\(906\) 0 0
\(907\) 34.8914 1.15855 0.579276 0.815132i \(-0.303335\pi\)
0.579276 + 0.815132i \(0.303335\pi\)
\(908\) 0 0
\(909\) −19.1311 −0.634539
\(910\) 0 0
\(911\) 1.21530 0.0402647 0.0201323 0.999797i \(-0.493591\pi\)
0.0201323 + 0.999797i \(0.493591\pi\)
\(912\) 0 0
\(913\) −2.84321 −0.0940964
\(914\) 0 0
\(915\) 1.50215 0.0496595
\(916\) 0 0
\(917\) −4.06220 −0.134146
\(918\) 0 0
\(919\) 14.6507 0.483282 0.241641 0.970366i \(-0.422314\pi\)
0.241641 + 0.970366i \(0.422314\pi\)
\(920\) 0 0
\(921\) −19.3339 −0.637073
\(922\) 0 0
\(923\) 2.32255 0.0764476
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) −11.5614 −0.379726
\(928\) 0 0
\(929\) 39.5478 1.29752 0.648760 0.760993i \(-0.275288\pi\)
0.648760 + 0.760993i \(0.275288\pi\)
\(930\) 0 0
\(931\) −19.0737 −0.625116
\(932\) 0 0
\(933\) 11.4652 0.375355
\(934\) 0 0
\(935\) −2.51129 −0.0821279
\(936\) 0 0
\(937\) 46.6343 1.52348 0.761739 0.647884i \(-0.224346\pi\)
0.761739 + 0.647884i \(0.224346\pi\)
\(938\) 0 0
\(939\) 19.8805 0.648776
\(940\) 0 0
\(941\) −22.0119 −0.717568 −0.358784 0.933421i \(-0.616809\pi\)
−0.358784 + 0.933421i \(0.616809\pi\)
\(942\) 0 0
\(943\) −33.6189 −1.09478
\(944\) 0 0
\(945\) −19.6875 −0.640436
\(946\) 0 0
\(947\) −7.46400 −0.242547 −0.121274 0.992619i \(-0.538698\pi\)
−0.121274 + 0.992619i \(0.538698\pi\)
\(948\) 0 0
\(949\) 29.9307 0.971590
\(950\) 0 0
\(951\) −32.5267 −1.05475
\(952\) 0 0
\(953\) −1.15089 −0.0372811 −0.0186405 0.999826i \(-0.505934\pi\)
−0.0186405 + 0.999826i \(0.505934\pi\)
\(954\) 0 0
\(955\) 5.02878 0.162728
\(956\) 0 0
\(957\) 0.491415 0.0158852
\(958\) 0 0
\(959\) 12.1106 0.391070
\(960\) 0 0
\(961\) 13.8260 0.446001
\(962\) 0 0
\(963\) 18.5700 0.598409
\(964\) 0 0
\(965\) −7.84163 −0.252431
\(966\) 0 0
\(967\) −54.8018 −1.76231 −0.881153 0.472832i \(-0.843232\pi\)
−0.881153 + 0.472832i \(0.843232\pi\)
\(968\) 0 0
\(969\) 12.0928 0.388477
\(970\) 0 0
\(971\) −28.9615 −0.929418 −0.464709 0.885463i \(-0.653841\pi\)
−0.464709 + 0.885463i \(0.653841\pi\)
\(972\) 0 0
\(973\) −66.4237 −2.12945
\(974\) 0 0
\(975\) 3.92306 0.125639
\(976\) 0 0
\(977\) 48.5806 1.55423 0.777116 0.629357i \(-0.216682\pi\)
0.777116 + 0.629357i \(0.216682\pi\)
\(978\) 0 0
\(979\) −10.3399 −0.330463
\(980\) 0 0
\(981\) 4.65804 0.148720
\(982\) 0 0
\(983\) −12.9133 −0.411871 −0.205935 0.978566i \(-0.566024\pi\)
−0.205935 + 0.978566i \(0.566024\pi\)
\(984\) 0 0
\(985\) 19.9703 0.636307
\(986\) 0 0
\(987\) 31.1423 0.991271
\(988\) 0 0
\(989\) 10.1638 0.323190
\(990\) 0 0
\(991\) −8.97819 −0.285202 −0.142601 0.989780i \(-0.545547\pi\)
−0.142601 + 0.989780i \(0.545547\pi\)
\(992\) 0 0
\(993\) −31.1303 −0.987890
\(994\) 0 0
\(995\) −6.52822 −0.206958
\(996\) 0 0
\(997\) 12.7518 0.403853 0.201926 0.979401i \(-0.435280\pi\)
0.201926 + 0.979401i \(0.435280\pi\)
\(998\) 0 0
\(999\) −5.21995 −0.165152
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 2960.2.a.bc.1.3 6
4.3 odd 2 1480.2.a.k.1.4 6
20.19 odd 2 7400.2.a.s.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1480.2.a.k.1.4 6 4.3 odd 2
2960.2.a.bc.1.3 6 1.1 even 1 trivial
7400.2.a.s.1.3 6 20.19 odd 2