Properties

Label 1904.2.a.h.1.1
Level $1904$
Weight $2$
Character 1904.1
Self dual yes
Analytic conductor $15.204$
Analytic rank $1$
Dimension $2$
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] = [1904,2,Mod(1,1904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1904, 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("1904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1904.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: \(15.2035165449\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 1904.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30278 q^{3} +1.30278 q^{5} -1.00000 q^{7} +2.30278 q^{9} +O(q^{10})\) \(q-2.30278 q^{3} +1.30278 q^{5} -1.00000 q^{7} +2.30278 q^{9} -0.605551 q^{13} -3.00000 q^{15} +1.00000 q^{17} +0.605551 q^{19} +2.30278 q^{21} -3.30278 q^{25} +1.60555 q^{27} -0.697224 q^{31} -1.30278 q^{35} +4.60555 q^{37} +1.39445 q^{39} -6.90833 q^{41} -3.69722 q^{43} +3.00000 q^{45} +2.60555 q^{47} +1.00000 q^{49} -2.30278 q^{51} -7.30278 q^{53} -1.39445 q^{57} +5.21110 q^{59} +2.90833 q^{61} -2.30278 q^{63} -0.788897 q^{65} +5.30278 q^{67} -13.8167 q^{71} +2.90833 q^{73} +7.60555 q^{75} -5.39445 q^{79} -10.6056 q^{81} -6.00000 q^{83} +1.30278 q^{85} -9.39445 q^{89} +0.605551 q^{91} +1.60555 q^{93} +0.788897 q^{95} -2.69722 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} - 2 q^{7} + q^{9} + 6 q^{13} - 6 q^{15} + 2 q^{17} - 6 q^{19} + q^{21} - 3 q^{25} - 4 q^{27} - 5 q^{31} + q^{35} + 2 q^{37} + 10 q^{39} - 3 q^{41} - 11 q^{43} + 6 q^{45} - 2 q^{47} + 2 q^{49} - q^{51} - 11 q^{53} - 10 q^{57} - 4 q^{59} - 5 q^{61} - q^{63} - 16 q^{65} + 7 q^{67} - 6 q^{71} - 5 q^{73} + 8 q^{75} - 18 q^{79} - 14 q^{81} - 12 q^{83} - q^{85} - 26 q^{89} - 6 q^{91} - 4 q^{93} + 16 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.30278 −1.32951 −0.664754 0.747062i \(-0.731464\pi\)
−0.664754 + 0.747062i \(0.731464\pi\)
\(4\) 0 0
\(5\) 1.30278 0.582619 0.291309 0.956629i \(-0.405909\pi\)
0.291309 + 0.956629i \(0.405909\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −0.605551 −0.167950 −0.0839749 0.996468i \(-0.526762\pi\)
−0.0839749 + 0.996468i \(0.526762\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 0.605551 0.138923 0.0694615 0.997585i \(-0.477872\pi\)
0.0694615 + 0.997585i \(0.477872\pi\)
\(20\) 0 0
\(21\) 2.30278 0.502507
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −3.30278 −0.660555
\(26\) 0 0
\(27\) 1.60555 0.308988
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −0.697224 −0.125225 −0.0626126 0.998038i \(-0.519943\pi\)
−0.0626126 + 0.998038i \(0.519943\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.30278 −0.220209
\(36\) 0 0
\(37\) 4.60555 0.757148 0.378574 0.925571i \(-0.376415\pi\)
0.378574 + 0.925571i \(0.376415\pi\)
\(38\) 0 0
\(39\) 1.39445 0.223290
\(40\) 0 0
\(41\) −6.90833 −1.07890 −0.539450 0.842018i \(-0.681368\pi\)
−0.539450 + 0.842018i \(0.681368\pi\)
\(42\) 0 0
\(43\) −3.69722 −0.563821 −0.281911 0.959441i \(-0.590968\pi\)
−0.281911 + 0.959441i \(0.590968\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.30278 −0.322453
\(52\) 0 0
\(53\) −7.30278 −1.00311 −0.501557 0.865125i \(-0.667239\pi\)
−0.501557 + 0.865125i \(0.667239\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.39445 −0.184699
\(58\) 0 0
\(59\) 5.21110 0.678428 0.339214 0.940709i \(-0.389839\pi\)
0.339214 + 0.940709i \(0.389839\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) 0 0
\(63\) −2.30278 −0.290122
\(64\) 0 0
\(65\) −0.788897 −0.0978507
\(66\) 0 0
\(67\) 5.30278 0.647837 0.323919 0.946085i \(-0.395000\pi\)
0.323919 + 0.946085i \(0.395000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.8167 −1.63974 −0.819868 0.572553i \(-0.805953\pi\)
−0.819868 + 0.572553i \(0.805953\pi\)
\(72\) 0 0
\(73\) 2.90833 0.340394 0.170197 0.985410i \(-0.445560\pi\)
0.170197 + 0.985410i \(0.445560\pi\)
\(74\) 0 0
\(75\) 7.60555 0.878213
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.39445 −0.606923 −0.303461 0.952844i \(-0.598142\pi\)
−0.303461 + 0.952844i \(0.598142\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 1.30278 0.141306
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.39445 −0.995810 −0.497905 0.867232i \(-0.665897\pi\)
−0.497905 + 0.867232i \(0.665897\pi\)
\(90\) 0 0
\(91\) 0.605551 0.0634790
\(92\) 0 0
\(93\) 1.60555 0.166488
\(94\) 0 0
\(95\) 0.788897 0.0809392
\(96\) 0 0
\(97\) −2.69722 −0.273862 −0.136931 0.990581i \(-0.543724\pi\)
−0.136931 + 0.990581i \(0.543724\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.81665 −0.180764 −0.0903819 0.995907i \(-0.528809\pi\)
−0.0903819 + 0.995907i \(0.528809\pi\)
\(102\) 0 0
\(103\) −7.21110 −0.710531 −0.355266 0.934765i \(-0.615610\pi\)
−0.355266 + 0.934765i \(0.615610\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 8.60555 0.831930 0.415965 0.909381i \(-0.363444\pi\)
0.415965 + 0.909381i \(0.363444\pi\)
\(108\) 0 0
\(109\) −20.4222 −1.95609 −0.978046 0.208388i \(-0.933178\pi\)
−0.978046 + 0.208388i \(0.933178\pi\)
\(110\) 0 0
\(111\) −10.6056 −1.00663
\(112\) 0 0
\(113\) −2.60555 −0.245110 −0.122555 0.992462i \(-0.539109\pi\)
−0.122555 + 0.992462i \(0.539109\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.39445 −0.128917
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 15.9083 1.43441
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) −4.09167 −0.363077 −0.181539 0.983384i \(-0.558108\pi\)
−0.181539 + 0.983384i \(0.558108\pi\)
\(128\) 0 0
\(129\) 8.51388 0.749605
\(130\) 0 0
\(131\) 6.78890 0.593149 0.296574 0.955010i \(-0.404156\pi\)
0.296574 + 0.955010i \(0.404156\pi\)
\(132\) 0 0
\(133\) −0.605551 −0.0525080
\(134\) 0 0
\(135\) 2.09167 0.180023
\(136\) 0 0
\(137\) −16.3028 −1.39284 −0.696420 0.717634i \(-0.745225\pi\)
−0.696420 + 0.717634i \(0.745225\pi\)
\(138\) 0 0
\(139\) −17.9083 −1.51896 −0.759482 0.650528i \(-0.774548\pi\)
−0.759482 + 0.650528i \(0.774548\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.30278 −0.189930
\(148\) 0 0
\(149\) −14.7250 −1.20632 −0.603159 0.797621i \(-0.706091\pi\)
−0.603159 + 0.797621i \(0.706091\pi\)
\(150\) 0 0
\(151\) −14.1194 −1.14902 −0.574511 0.818497i \(-0.694808\pi\)
−0.574511 + 0.818497i \(0.694808\pi\)
\(152\) 0 0
\(153\) 2.30278 0.186168
\(154\) 0 0
\(155\) −0.908327 −0.0729586
\(156\) 0 0
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) 0 0
\(159\) 16.8167 1.33365
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.39445 −0.422526 −0.211263 0.977429i \(-0.567758\pi\)
−0.211263 + 0.977429i \(0.567758\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.72498 −0.675159 −0.337580 0.941297i \(-0.609608\pi\)
−0.337580 + 0.941297i \(0.609608\pi\)
\(168\) 0 0
\(169\) −12.6333 −0.971793
\(170\) 0 0
\(171\) 1.39445 0.106636
\(172\) 0 0
\(173\) 13.6972 1.04138 0.520690 0.853746i \(-0.325675\pi\)
0.520690 + 0.853746i \(0.325675\pi\)
\(174\) 0 0
\(175\) 3.30278 0.249666
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 0.908327 0.0678915 0.0339458 0.999424i \(-0.489193\pi\)
0.0339458 + 0.999424i \(0.489193\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −6.69722 −0.495073
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.60555 −0.116787
\(190\) 0 0
\(191\) 11.7250 0.848390 0.424195 0.905571i \(-0.360557\pi\)
0.424195 + 0.905571i \(0.360557\pi\)
\(192\) 0 0
\(193\) 9.81665 0.706618 0.353309 0.935507i \(-0.385056\pi\)
0.353309 + 0.935507i \(0.385056\pi\)
\(194\) 0 0
\(195\) 1.81665 0.130093
\(196\) 0 0
\(197\) 2.60555 0.185638 0.0928189 0.995683i \(-0.470412\pi\)
0.0928189 + 0.995683i \(0.470412\pi\)
\(198\) 0 0
\(199\) 4.11943 0.292019 0.146009 0.989283i \(-0.453357\pi\)
0.146009 + 0.989283i \(0.453357\pi\)
\(200\) 0 0
\(201\) −12.2111 −0.861305
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −15.0278 −1.03455 −0.517277 0.855818i \(-0.673054\pi\)
−0.517277 + 0.855818i \(0.673054\pi\)
\(212\) 0 0
\(213\) 31.8167 2.18004
\(214\) 0 0
\(215\) −4.81665 −0.328493
\(216\) 0 0
\(217\) 0.697224 0.0473307
\(218\) 0 0
\(219\) −6.69722 −0.452556
\(220\) 0 0
\(221\) −0.605551 −0.0407338
\(222\) 0 0
\(223\) −27.0278 −1.80991 −0.904956 0.425505i \(-0.860097\pi\)
−0.904956 + 0.425505i \(0.860097\pi\)
\(224\) 0 0
\(225\) −7.60555 −0.507037
\(226\) 0 0
\(227\) 12.5139 0.830575 0.415288 0.909690i \(-0.363681\pi\)
0.415288 + 0.909690i \(0.363681\pi\)
\(228\) 0 0
\(229\) 25.2111 1.66600 0.832998 0.553276i \(-0.186622\pi\)
0.832998 + 0.553276i \(0.186622\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.81665 0.512086 0.256043 0.966665i \(-0.417581\pi\)
0.256043 + 0.966665i \(0.417581\pi\)
\(234\) 0 0
\(235\) 3.39445 0.221429
\(236\) 0 0
\(237\) 12.4222 0.806909
\(238\) 0 0
\(239\) −7.69722 −0.497892 −0.248946 0.968517i \(-0.580084\pi\)
−0.248946 + 0.968517i \(0.580084\pi\)
\(240\) 0 0
\(241\) 26.5139 1.70791 0.853955 0.520348i \(-0.174198\pi\)
0.853955 + 0.520348i \(0.174198\pi\)
\(242\) 0 0
\(243\) 19.6056 1.25770
\(244\) 0 0
\(245\) 1.30278 0.0832313
\(246\) 0 0
\(247\) −0.366692 −0.0233321
\(248\) 0 0
\(249\) 13.8167 0.875595
\(250\) 0 0
\(251\) −25.8167 −1.62953 −0.814766 0.579789i \(-0.803135\pi\)
−0.814766 + 0.579789i \(0.803135\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 0 0
\(257\) −11.2111 −0.699329 −0.349665 0.936875i \(-0.613704\pi\)
−0.349665 + 0.936875i \(0.613704\pi\)
\(258\) 0 0
\(259\) −4.60555 −0.286175
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.4222 −0.642661 −0.321330 0.946967i \(-0.604130\pi\)
−0.321330 + 0.946967i \(0.604130\pi\)
\(264\) 0 0
\(265\) −9.51388 −0.584433
\(266\) 0 0
\(267\) 21.6333 1.32394
\(268\) 0 0
\(269\) 0.788897 0.0480999 0.0240500 0.999711i \(-0.492344\pi\)
0.0240500 + 0.999711i \(0.492344\pi\)
\(270\) 0 0
\(271\) −13.2111 −0.802517 −0.401259 0.915965i \(-0.631427\pi\)
−0.401259 + 0.915965i \(0.631427\pi\)
\(272\) 0 0
\(273\) −1.39445 −0.0843959
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.42221 0.385873 0.192936 0.981211i \(-0.438199\pi\)
0.192936 + 0.981211i \(0.438199\pi\)
\(278\) 0 0
\(279\) −1.60555 −0.0961218
\(280\) 0 0
\(281\) 7.30278 0.435647 0.217824 0.975988i \(-0.430104\pi\)
0.217824 + 0.975988i \(0.430104\pi\)
\(282\) 0 0
\(283\) −8.51388 −0.506098 −0.253049 0.967454i \(-0.581433\pi\)
−0.253049 + 0.967454i \(0.581433\pi\)
\(284\) 0 0
\(285\) −1.81665 −0.107609
\(286\) 0 0
\(287\) 6.90833 0.407786
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.21110 0.364101
\(292\) 0 0
\(293\) 21.6333 1.26383 0.631916 0.775037i \(-0.282269\pi\)
0.631916 + 0.775037i \(0.282269\pi\)
\(294\) 0 0
\(295\) 6.78890 0.395265
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.69722 0.213104
\(302\) 0 0
\(303\) 4.18335 0.240327
\(304\) 0 0
\(305\) 3.78890 0.216952
\(306\) 0 0
\(307\) −13.2111 −0.753997 −0.376999 0.926214i \(-0.623044\pi\)
−0.376999 + 0.926214i \(0.623044\pi\)
\(308\) 0 0
\(309\) 16.6056 0.944657
\(310\) 0 0
\(311\) 27.1194 1.53780 0.768901 0.639368i \(-0.220804\pi\)
0.768901 + 0.639368i \(0.220804\pi\)
\(312\) 0 0
\(313\) −6.33053 −0.357823 −0.178911 0.983865i \(-0.557258\pi\)
−0.178911 + 0.983865i \(0.557258\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −19.8167 −1.10606
\(322\) 0 0
\(323\) 0.605551 0.0336938
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 47.0278 2.60064
\(328\) 0 0
\(329\) −2.60555 −0.143649
\(330\) 0 0
\(331\) −9.30278 −0.511327 −0.255663 0.966766i \(-0.582294\pi\)
−0.255663 + 0.966766i \(0.582294\pi\)
\(332\) 0 0
\(333\) 10.6056 0.581181
\(334\) 0 0
\(335\) 6.90833 0.377442
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) 14.7889 0.791632 0.395816 0.918330i \(-0.370462\pi\)
0.395816 + 0.918330i \(0.370462\pi\)
\(350\) 0 0
\(351\) −0.972244 −0.0518945
\(352\) 0 0
\(353\) 27.6333 1.47077 0.735386 0.677648i \(-0.237001\pi\)
0.735386 + 0.677648i \(0.237001\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) 0 0
\(357\) 2.30278 0.121876
\(358\) 0 0
\(359\) 3.90833 0.206274 0.103137 0.994667i \(-0.467112\pi\)
0.103137 + 0.994667i \(0.467112\pi\)
\(360\) 0 0
\(361\) −18.6333 −0.980700
\(362\) 0 0
\(363\) 25.3305 1.32951
\(364\) 0 0
\(365\) 3.78890 0.198320
\(366\) 0 0
\(367\) 33.3305 1.73984 0.869920 0.493193i \(-0.164170\pi\)
0.869920 + 0.493193i \(0.164170\pi\)
\(368\) 0 0
\(369\) −15.9083 −0.828154
\(370\) 0 0
\(371\) 7.30278 0.379141
\(372\) 0 0
\(373\) 15.9361 0.825139 0.412570 0.910926i \(-0.364631\pi\)
0.412570 + 0.910926i \(0.364631\pi\)
\(374\) 0 0
\(375\) 24.9083 1.28626
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 21.2111 1.08954 0.544771 0.838585i \(-0.316617\pi\)
0.544771 + 0.838585i \(0.316617\pi\)
\(380\) 0 0
\(381\) 9.42221 0.482714
\(382\) 0 0
\(383\) 10.4222 0.532550 0.266275 0.963897i \(-0.414207\pi\)
0.266275 + 0.963897i \(0.414207\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.51388 −0.432785
\(388\) 0 0
\(389\) 15.1194 0.766586 0.383293 0.923627i \(-0.374790\pi\)
0.383293 + 0.923627i \(0.374790\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −15.6333 −0.788596
\(394\) 0 0
\(395\) −7.02776 −0.353605
\(396\) 0 0
\(397\) 24.9361 1.25151 0.625753 0.780021i \(-0.284792\pi\)
0.625753 + 0.780021i \(0.284792\pi\)
\(398\) 0 0
\(399\) 1.39445 0.0698098
\(400\) 0 0
\(401\) −9.39445 −0.469136 −0.234568 0.972100i \(-0.575368\pi\)
−0.234568 + 0.972100i \(0.575368\pi\)
\(402\) 0 0
\(403\) 0.422205 0.0210315
\(404\) 0 0
\(405\) −13.8167 −0.686555
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −19.3944 −0.958994 −0.479497 0.877544i \(-0.659181\pi\)
−0.479497 + 0.877544i \(0.659181\pi\)
\(410\) 0 0
\(411\) 37.5416 1.85179
\(412\) 0 0
\(413\) −5.21110 −0.256422
\(414\) 0 0
\(415\) −7.81665 −0.383704
\(416\) 0 0
\(417\) 41.2389 2.01948
\(418\) 0 0
\(419\) −0.513878 −0.0251046 −0.0125523 0.999921i \(-0.503996\pi\)
−0.0125523 + 0.999921i \(0.503996\pi\)
\(420\) 0 0
\(421\) 15.9361 0.776677 0.388339 0.921517i \(-0.373049\pi\)
0.388339 + 0.921517i \(0.373049\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −3.30278 −0.160208
\(426\) 0 0
\(427\) −2.90833 −0.140744
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.6333 −1.33105 −0.665525 0.746376i \(-0.731792\pi\)
−0.665525 + 0.746376i \(0.731792\pi\)
\(432\) 0 0
\(433\) −17.8167 −0.856214 −0.428107 0.903728i \(-0.640819\pi\)
−0.428107 + 0.903728i \(0.640819\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −2.90833 −0.138807 −0.0694034 0.997589i \(-0.522110\pi\)
−0.0694034 + 0.997589i \(0.522110\pi\)
\(440\) 0 0
\(441\) 2.30278 0.109656
\(442\) 0 0
\(443\) 6.78890 0.322550 0.161275 0.986909i \(-0.448439\pi\)
0.161275 + 0.986909i \(0.448439\pi\)
\(444\) 0 0
\(445\) −12.2389 −0.580178
\(446\) 0 0
\(447\) 33.9083 1.60381
\(448\) 0 0
\(449\) 18.2389 0.860745 0.430372 0.902651i \(-0.358382\pi\)
0.430372 + 0.902651i \(0.358382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 32.5139 1.52764
\(454\) 0 0
\(455\) 0.788897 0.0369841
\(456\) 0 0
\(457\) −13.9083 −0.650604 −0.325302 0.945610i \(-0.605466\pi\)
−0.325302 + 0.945610i \(0.605466\pi\)
\(458\) 0 0
\(459\) 1.60555 0.0749407
\(460\) 0 0
\(461\) −10.1833 −0.474286 −0.237143 0.971475i \(-0.576211\pi\)
−0.237143 + 0.971475i \(0.576211\pi\)
\(462\) 0 0
\(463\) 20.3028 0.943550 0.471775 0.881719i \(-0.343613\pi\)
0.471775 + 0.881719i \(0.343613\pi\)
\(464\) 0 0
\(465\) 2.09167 0.0969990
\(466\) 0 0
\(467\) −29.4500 −1.36278 −0.681391 0.731920i \(-0.738625\pi\)
−0.681391 + 0.731920i \(0.738625\pi\)
\(468\) 0 0
\(469\) −5.30278 −0.244859
\(470\) 0 0
\(471\) −16.6056 −0.765143
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) −16.8167 −0.769982
\(478\) 0 0
\(479\) 3.51388 0.160553 0.0802766 0.996773i \(-0.474420\pi\)
0.0802766 + 0.996773i \(0.474420\pi\)
\(480\) 0 0
\(481\) −2.78890 −0.127163
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.51388 −0.159557
\(486\) 0 0
\(487\) 9.21110 0.417395 0.208697 0.977980i \(-0.433078\pi\)
0.208697 + 0.977980i \(0.433078\pi\)
\(488\) 0 0
\(489\) 12.4222 0.561752
\(490\) 0 0
\(491\) −14.3305 −0.646728 −0.323364 0.946275i \(-0.604814\pi\)
−0.323364 + 0.946275i \(0.604814\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.8167 0.619762
\(498\) 0 0
\(499\) −30.4222 −1.36188 −0.680942 0.732337i \(-0.738430\pi\)
−0.680942 + 0.732337i \(0.738430\pi\)
\(500\) 0 0
\(501\) 20.0917 0.897630
\(502\) 0 0
\(503\) 40.1472 1.79007 0.895037 0.445991i \(-0.147149\pi\)
0.895037 + 0.445991i \(0.147149\pi\)
\(504\) 0 0
\(505\) −2.36669 −0.105316
\(506\) 0 0
\(507\) 29.0917 1.29201
\(508\) 0 0
\(509\) 25.0278 1.10934 0.554668 0.832072i \(-0.312845\pi\)
0.554668 + 0.832072i \(0.312845\pi\)
\(510\) 0 0
\(511\) −2.90833 −0.128657
\(512\) 0 0
\(513\) 0.972244 0.0429256
\(514\) 0 0
\(515\) −9.39445 −0.413969
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −31.5416 −1.38452
\(520\) 0 0
\(521\) −36.3583 −1.59289 −0.796443 0.604714i \(-0.793287\pi\)
−0.796443 + 0.604714i \(0.793287\pi\)
\(522\) 0 0
\(523\) −25.2111 −1.10240 −0.551202 0.834372i \(-0.685831\pi\)
−0.551202 + 0.834372i \(0.685831\pi\)
\(524\) 0 0
\(525\) −7.60555 −0.331933
\(526\) 0 0
\(527\) −0.697224 −0.0303716
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 4.18335 0.181201
\(534\) 0 0
\(535\) 11.2111 0.484698
\(536\) 0 0
\(537\) −2.09167 −0.0902624
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0555 −1.29219 −0.646094 0.763258i \(-0.723598\pi\)
−0.646094 + 0.763258i \(0.723598\pi\)
\(542\) 0 0
\(543\) −32.2389 −1.38350
\(544\) 0 0
\(545\) −26.6056 −1.13966
\(546\) 0 0
\(547\) 16.7889 0.717841 0.358921 0.933368i \(-0.383145\pi\)
0.358921 + 0.933368i \(0.383145\pi\)
\(548\) 0 0
\(549\) 6.69722 0.285831
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.39445 0.229395
\(554\) 0 0
\(555\) −13.8167 −0.586484
\(556\) 0 0
\(557\) −9.63331 −0.408176 −0.204088 0.978953i \(-0.565423\pi\)
−0.204088 + 0.978953i \(0.565423\pi\)
\(558\) 0 0
\(559\) 2.23886 0.0946936
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.6333 −0.658865 −0.329433 0.944179i \(-0.606857\pi\)
−0.329433 + 0.944179i \(0.606857\pi\)
\(564\) 0 0
\(565\) −3.39445 −0.142806
\(566\) 0 0
\(567\) 10.6056 0.445391
\(568\) 0 0
\(569\) 30.1194 1.26267 0.631336 0.775509i \(-0.282507\pi\)
0.631336 + 0.775509i \(0.282507\pi\)
\(570\) 0 0
\(571\) −27.0278 −1.13108 −0.565538 0.824722i \(-0.691332\pi\)
−0.565538 + 0.824722i \(0.691332\pi\)
\(572\) 0 0
\(573\) −27.0000 −1.12794
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −22.2389 −0.925816 −0.462908 0.886406i \(-0.653194\pi\)
−0.462908 + 0.886406i \(0.653194\pi\)
\(578\) 0 0
\(579\) −22.6056 −0.939455
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.81665 −0.0751094
\(586\) 0 0
\(587\) 3.39445 0.140104 0.0700519 0.997543i \(-0.477683\pi\)
0.0700519 + 0.997543i \(0.477683\pi\)
\(588\) 0 0
\(589\) −0.422205 −0.0173967
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 22.1833 0.910961 0.455480 0.890246i \(-0.349468\pi\)
0.455480 + 0.890246i \(0.349468\pi\)
\(594\) 0 0
\(595\) −1.30278 −0.0534086
\(596\) 0 0
\(597\) −9.48612 −0.388241
\(598\) 0 0
\(599\) 38.9638 1.59202 0.796010 0.605284i \(-0.206940\pi\)
0.796010 + 0.605284i \(0.206940\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 12.2111 0.497275
\(604\) 0 0
\(605\) −14.3305 −0.582619
\(606\) 0 0
\(607\) 3.48612 0.141497 0.0707487 0.997494i \(-0.477461\pi\)
0.0707487 + 0.997494i \(0.477461\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.57779 −0.0638307
\(612\) 0 0
\(613\) −32.1472 −1.29841 −0.649206 0.760612i \(-0.724899\pi\)
−0.649206 + 0.760612i \(0.724899\pi\)
\(614\) 0 0
\(615\) 20.7250 0.835712
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) 0.366692 0.0147386 0.00736930 0.999973i \(-0.497654\pi\)
0.00736930 + 0.999973i \(0.497654\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.39445 0.376381
\(624\) 0 0
\(625\) 2.42221 0.0968882
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.60555 0.183635
\(630\) 0 0
\(631\) −17.1194 −0.681514 −0.340757 0.940151i \(-0.610683\pi\)
−0.340757 + 0.940151i \(0.610683\pi\)
\(632\) 0 0
\(633\) 34.6056 1.37545
\(634\) 0 0
\(635\) −5.33053 −0.211536
\(636\) 0 0
\(637\) −0.605551 −0.0239928
\(638\) 0 0
\(639\) −31.8167 −1.25865
\(640\) 0 0
\(641\) 44.8444 1.77125 0.885624 0.464403i \(-0.153731\pi\)
0.885624 + 0.464403i \(0.153731\pi\)
\(642\) 0 0
\(643\) 23.1472 0.912836 0.456418 0.889766i \(-0.349132\pi\)
0.456418 + 0.889766i \(0.349132\pi\)
\(644\) 0 0
\(645\) 11.0917 0.436734
\(646\) 0 0
\(647\) 10.4222 0.409739 0.204870 0.978789i \(-0.434323\pi\)
0.204870 + 0.978789i \(0.434323\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.60555 −0.0629265
\(652\) 0 0
\(653\) −44.0555 −1.72403 −0.862013 0.506887i \(-0.830796\pi\)
−0.862013 + 0.506887i \(0.830796\pi\)
\(654\) 0 0
\(655\) 8.84441 0.345580
\(656\) 0 0
\(657\) 6.69722 0.261284
\(658\) 0 0
\(659\) −34.6972 −1.35161 −0.675806 0.737080i \(-0.736204\pi\)
−0.675806 + 0.737080i \(0.736204\pi\)
\(660\) 0 0
\(661\) −35.8167 −1.39311 −0.696553 0.717505i \(-0.745284\pi\)
−0.696553 + 0.717505i \(0.745284\pi\)
\(662\) 0 0
\(663\) 1.39445 0.0541559
\(664\) 0 0
\(665\) −0.788897 −0.0305921
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 62.2389 2.40629
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −41.8167 −1.61191 −0.805957 0.591974i \(-0.798349\pi\)
−0.805957 + 0.591974i \(0.798349\pi\)
\(674\) 0 0
\(675\) −5.30278 −0.204104
\(676\) 0 0
\(677\) 19.5778 0.752436 0.376218 0.926531i \(-0.377224\pi\)
0.376218 + 0.926531i \(0.377224\pi\)
\(678\) 0 0
\(679\) 2.69722 0.103510
\(680\) 0 0
\(681\) −28.8167 −1.10426
\(682\) 0 0
\(683\) 23.4500 0.897288 0.448644 0.893711i \(-0.351907\pi\)
0.448644 + 0.893711i \(0.351907\pi\)
\(684\) 0 0
\(685\) −21.2389 −0.811495
\(686\) 0 0
\(687\) −58.0555 −2.21496
\(688\) 0 0
\(689\) 4.42221 0.168473
\(690\) 0 0
\(691\) −47.5139 −1.80751 −0.903757 0.428047i \(-0.859202\pi\)
−0.903757 + 0.428047i \(0.859202\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −23.3305 −0.884978
\(696\) 0 0
\(697\) −6.90833 −0.261672
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 38.8444 1.46713 0.733567 0.679618i \(-0.237854\pi\)
0.733567 + 0.679618i \(0.237854\pi\)
\(702\) 0 0
\(703\) 2.78890 0.105185
\(704\) 0 0
\(705\) −7.81665 −0.294392
\(706\) 0 0
\(707\) 1.81665 0.0683223
\(708\) 0 0
\(709\) 33.8167 1.27001 0.635006 0.772508i \(-0.280998\pi\)
0.635006 + 0.772508i \(0.280998\pi\)
\(710\) 0 0
\(711\) −12.4222 −0.465869
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 17.7250 0.661952
\(718\) 0 0
\(719\) 25.6972 0.958345 0.479172 0.877721i \(-0.340937\pi\)
0.479172 + 0.877721i \(0.340937\pi\)
\(720\) 0 0
\(721\) 7.21110 0.268555
\(722\) 0 0
\(723\) −61.0555 −2.27068
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.42221 −0.238186 −0.119093 0.992883i \(-0.537999\pi\)
−0.119093 + 0.992883i \(0.537999\pi\)
\(728\) 0 0
\(729\) −13.3305 −0.493723
\(730\) 0 0
\(731\) −3.69722 −0.136747
\(732\) 0 0
\(733\) 20.2389 0.747539 0.373770 0.927522i \(-0.378065\pi\)
0.373770 + 0.927522i \(0.378065\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −50.3583 −1.85246 −0.926230 0.376959i \(-0.876970\pi\)
−0.926230 + 0.376959i \(0.876970\pi\)
\(740\) 0 0
\(741\) 0.844410 0.0310202
\(742\) 0 0
\(743\) 19.8167 0.727003 0.363501 0.931594i \(-0.381581\pi\)
0.363501 + 0.931594i \(0.381581\pi\)
\(744\) 0 0
\(745\) −19.1833 −0.702823
\(746\) 0 0
\(747\) −13.8167 −0.505525
\(748\) 0 0
\(749\) −8.60555 −0.314440
\(750\) 0 0
\(751\) −11.3944 −0.415789 −0.207895 0.978151i \(-0.566661\pi\)
−0.207895 + 0.978151i \(0.566661\pi\)
\(752\) 0 0
\(753\) 59.4500 2.16648
\(754\) 0 0
\(755\) −18.3944 −0.669443
\(756\) 0 0
\(757\) −13.2750 −0.482489 −0.241244 0.970464i \(-0.577556\pi\)
−0.241244 + 0.970464i \(0.577556\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.1833 0.586646 0.293323 0.956013i \(-0.405239\pi\)
0.293323 + 0.956013i \(0.405239\pi\)
\(762\) 0 0
\(763\) 20.4222 0.739333
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) −3.15559 −0.113942
\(768\) 0 0
\(769\) 33.2666 1.19962 0.599812 0.800141i \(-0.295242\pi\)
0.599812 + 0.800141i \(0.295242\pi\)
\(770\) 0 0
\(771\) 25.8167 0.929764
\(772\) 0 0
\(773\) 4.18335 0.150465 0.0752323 0.997166i \(-0.476030\pi\)
0.0752323 + 0.997166i \(0.476030\pi\)
\(774\) 0 0
\(775\) 2.30278 0.0827181
\(776\) 0 0
\(777\) 10.6056 0.380472
\(778\) 0 0
\(779\) −4.18335 −0.149884
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.39445 0.335302
\(786\) 0 0
\(787\) 26.4222 0.941850 0.470925 0.882173i \(-0.343920\pi\)
0.470925 + 0.882173i \(0.343920\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 2.60555 0.0926427
\(792\) 0 0
\(793\) −1.76114 −0.0625400
\(794\) 0 0
\(795\) 21.9083 0.777008
\(796\) 0 0
\(797\) −28.4222 −1.00677 −0.503383 0.864063i \(-0.667912\pi\)
−0.503383 + 0.864063i \(0.667912\pi\)
\(798\) 0 0
\(799\) 2.60555 0.0921778
\(800\) 0 0
\(801\) −21.6333 −0.764375
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.81665 −0.0639492
\(808\) 0 0
\(809\) 33.3944 1.17409 0.587043 0.809556i \(-0.300292\pi\)
0.587043 + 0.809556i \(0.300292\pi\)
\(810\) 0 0
\(811\) −14.9083 −0.523502 −0.261751 0.965135i \(-0.584300\pi\)
−0.261751 + 0.965135i \(0.584300\pi\)
\(812\) 0 0
\(813\) 30.4222 1.06695
\(814\) 0 0
\(815\) −7.02776 −0.246172
\(816\) 0 0
\(817\) −2.23886 −0.0783278
\(818\) 0 0
\(819\) 1.39445 0.0487260
\(820\) 0 0
\(821\) −3.39445 −0.118467 −0.0592335 0.998244i \(-0.518866\pi\)
−0.0592335 + 0.998244i \(0.518866\pi\)
\(822\) 0 0
\(823\) 49.8722 1.73843 0.869217 0.494430i \(-0.164623\pi\)
0.869217 + 0.494430i \(0.164623\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.6333 0.543623 0.271812 0.962350i \(-0.412377\pi\)
0.271812 + 0.962350i \(0.412377\pi\)
\(828\) 0 0
\(829\) −19.3944 −0.673597 −0.336799 0.941577i \(-0.609344\pi\)
−0.336799 + 0.941577i \(0.609344\pi\)
\(830\) 0 0
\(831\) −14.7889 −0.513021
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −11.3667 −0.393361
\(836\) 0 0
\(837\) −1.11943 −0.0386931
\(838\) 0 0
\(839\) 10.4222 0.359814 0.179907 0.983684i \(-0.442420\pi\)
0.179907 + 0.983684i \(0.442420\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −16.8167 −0.579196
\(844\) 0 0
\(845\) −16.4584 −0.566185
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 19.6056 0.672861
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 12.4222 0.425328 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(854\) 0 0
\(855\) 1.81665 0.0621282
\(856\) 0 0
\(857\) 19.5416 0.667530 0.333765 0.942656i \(-0.391681\pi\)
0.333765 + 0.942656i \(0.391681\pi\)
\(858\) 0 0
\(859\) 27.4500 0.936581 0.468290 0.883575i \(-0.344870\pi\)
0.468290 + 0.883575i \(0.344870\pi\)
\(860\) 0 0
\(861\) −15.9083 −0.542154
\(862\) 0 0
\(863\) −8.09167 −0.275444 −0.137722 0.990471i \(-0.543978\pi\)
−0.137722 + 0.990471i \(0.543978\pi\)
\(864\) 0 0
\(865\) 17.8444 0.606728
\(866\) 0 0
\(867\) −2.30278 −0.0782064
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −3.21110 −0.108804
\(872\) 0 0
\(873\) −6.21110 −0.210214
\(874\) 0 0
\(875\) 10.8167 0.365670
\(876\) 0 0
\(877\) −25.3944 −0.857510 −0.428755 0.903421i \(-0.641048\pi\)
−0.428755 + 0.903421i \(0.641048\pi\)
\(878\) 0 0
\(879\) −49.8167 −1.68027
\(880\) 0 0
\(881\) −13.3028 −0.448182 −0.224091 0.974568i \(-0.571941\pi\)
−0.224091 + 0.974568i \(0.571941\pi\)
\(882\) 0 0
\(883\) 54.3305 1.82837 0.914184 0.405299i \(-0.132833\pi\)
0.914184 + 0.405299i \(0.132833\pi\)
\(884\) 0 0
\(885\) −15.6333 −0.525508
\(886\) 0 0
\(887\) 38.0917 1.27899 0.639497 0.768794i \(-0.279143\pi\)
0.639497 + 0.768794i \(0.279143\pi\)
\(888\) 0 0
\(889\) 4.09167 0.137230
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.57779 0.0527989
\(894\) 0 0
\(895\) 1.18335 0.0395549
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −7.30278 −0.243291
\(902\) 0 0
\(903\) −8.51388 −0.283324
\(904\) 0 0
\(905\) 18.2389 0.606280
\(906\) 0 0
\(907\) −34.8444 −1.15699 −0.578495 0.815686i \(-0.696360\pi\)
−0.578495 + 0.815686i \(0.696360\pi\)
\(908\) 0 0
\(909\) −4.18335 −0.138753
\(910\) 0 0
\(911\) −16.1833 −0.536178 −0.268089 0.963394i \(-0.586392\pi\)
−0.268089 + 0.963394i \(0.586392\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −8.72498 −0.288439
\(916\) 0 0
\(917\) −6.78890 −0.224189
\(918\) 0 0
\(919\) 15.3305 0.505708 0.252854 0.967505i \(-0.418631\pi\)
0.252854 + 0.967505i \(0.418631\pi\)
\(920\) 0 0
\(921\) 30.4222 1.00245
\(922\) 0 0
\(923\) 8.36669 0.275393
\(924\) 0 0
\(925\) −15.2111 −0.500138
\(926\) 0 0
\(927\) −16.6056 −0.545398
\(928\) 0 0
\(929\) 7.06392 0.231760 0.115880 0.993263i \(-0.463031\pi\)
0.115880 + 0.993263i \(0.463031\pi\)
\(930\) 0 0
\(931\) 0.605551 0.0198461
\(932\) 0 0
\(933\) −62.4500 −2.04452
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −21.2111 −0.692937 −0.346468 0.938062i \(-0.612619\pi\)
−0.346468 + 0.938062i \(0.612619\pi\)
\(938\) 0 0
\(939\) 14.5778 0.475728
\(940\) 0 0
\(941\) −32.7250 −1.06680 −0.533402 0.845862i \(-0.679087\pi\)
−0.533402 + 0.845862i \(0.679087\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.09167 −0.0680421
\(946\) 0 0
\(947\) 8.84441 0.287405 0.143702 0.989621i \(-0.454099\pi\)
0.143702 + 0.989621i \(0.454099\pi\)
\(948\) 0 0
\(949\) −1.76114 −0.0571691
\(950\) 0 0
\(951\) 13.8167 0.448036
\(952\) 0 0
\(953\) −0.513878 −0.0166461 −0.00832307 0.999965i \(-0.502649\pi\)
−0.00832307 + 0.999965i \(0.502649\pi\)
\(954\) 0 0
\(955\) 15.2750 0.494288
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.3028 0.526444
\(960\) 0 0
\(961\) −30.5139 −0.984319
\(962\) 0 0
\(963\) 19.8167 0.638583
\(964\) 0 0
\(965\) 12.7889 0.411689
\(966\) 0 0
\(967\) 1.51388 0.0486830 0.0243415 0.999704i \(-0.492251\pi\)
0.0243415 + 0.999704i \(0.492251\pi\)
\(968\) 0 0
\(969\) −1.39445 −0.0447961
\(970\) 0 0
\(971\) 20.8444 0.668929 0.334464 0.942408i \(-0.391445\pi\)
0.334464 + 0.942408i \(0.391445\pi\)
\(972\) 0 0
\(973\) 17.9083 0.574115
\(974\) 0 0
\(975\) −4.60555 −0.147496
\(976\) 0 0
\(977\) 28.1472 0.900508 0.450254 0.892900i \(-0.351333\pi\)
0.450254 + 0.892900i \(0.351333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −47.0278 −1.50148
\(982\) 0 0
\(983\) −33.9083 −1.08151 −0.540754 0.841181i \(-0.681861\pi\)
−0.540754 + 0.841181i \(0.681861\pi\)
\(984\) 0 0
\(985\) 3.39445 0.108156
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −14.0000 −0.444725 −0.222362 0.974964i \(-0.571377\pi\)
−0.222362 + 0.974964i \(0.571377\pi\)
\(992\) 0 0
\(993\) 21.4222 0.679813
\(994\) 0 0
\(995\) 5.36669 0.170136
\(996\) 0 0
\(997\) −47.1472 −1.49317 −0.746583 0.665292i \(-0.768307\pi\)
−0.746583 + 0.665292i \(0.768307\pi\)
\(998\) 0 0
\(999\) 7.39445 0.233950
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 1904.2.a.h.1.1 2
4.3 odd 2 476.2.a.d.1.2 2
8.3 odd 2 7616.2.a.q.1.1 2
8.5 even 2 7616.2.a.v.1.2 2
12.11 even 2 4284.2.a.n.1.1 2
28.27 even 2 3332.2.a.j.1.1 2
68.67 odd 2 8092.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.d.1.2 2 4.3 odd 2
1904.2.a.h.1.1 2 1.1 even 1 trivial
3332.2.a.j.1.1 2 28.27 even 2
4284.2.a.n.1.1 2 12.11 even 2
7616.2.a.q.1.1 2 8.3 odd 2
7616.2.a.v.1.2 2 8.5 even 2
8092.2.a.k.1.1 2 68.67 odd 2