Properties

Label 520.2.a.f.1.1
Level $520$
Weight $2$
Character 520.1
Self dual yes
Analytic conductor $4.152$
Analytic rank $0$
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] = [520,2,Mod(1,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, 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("520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} -4.44949 q^{11} -1.00000 q^{13} -2.44949 q^{15} +6.89898 q^{17} +0.449490 q^{19} -4.89898 q^{21} +6.44949 q^{23} +1.00000 q^{25} +4.00000 q^{29} -4.44949 q^{31} +10.8990 q^{33} +2.00000 q^{35} +4.89898 q^{37} +2.44949 q^{39} +10.8990 q^{41} +11.3485 q^{43} +3.00000 q^{45} -2.00000 q^{47} -3.00000 q^{49} -16.8990 q^{51} +1.10102 q^{53} -4.44949 q^{55} -1.10102 q^{57} +9.34847 q^{59} -5.79796 q^{61} +6.00000 q^{63} -1.00000 q^{65} -5.10102 q^{67} -15.7980 q^{69} -3.55051 q^{71} -14.6969 q^{73} -2.44949 q^{75} -8.89898 q^{77} +4.89898 q^{79} -9.00000 q^{81} +2.00000 q^{83} +6.89898 q^{85} -9.79796 q^{87} +6.00000 q^{89} -2.00000 q^{91} +10.8990 q^{93} +0.449490 q^{95} +7.79796 q^{97} -13.3485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} + 12 q^{33} + 4 q^{35} + 12 q^{41} + 8 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 24 q^{51} + 12 q^{53} - 4 q^{55} - 12 q^{57} + 4 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{77} - 18 q^{81} + 4 q^{83} + 4 q^{85} + 12 q^{89} - 4 q^{91} + 12 q^{93} - 4 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) −2.44949 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −4.44949 −1.34157 −0.670786 0.741651i \(-0.734043\pi\)
−0.670786 + 0.741651i \(0.734043\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.44949 −0.632456
\(16\) 0 0
\(17\) 6.89898 1.67325 0.836624 0.547777i \(-0.184526\pi\)
0.836624 + 0.547777i \(0.184526\pi\)
\(18\) 0 0
\(19\) 0.449490 0.103120 0.0515600 0.998670i \(-0.483581\pi\)
0.0515600 + 0.998670i \(0.483581\pi\)
\(20\) 0 0
\(21\) −4.89898 −1.06904
\(22\) 0 0
\(23\) 6.44949 1.34481 0.672406 0.740183i \(-0.265261\pi\)
0.672406 + 0.740183i \(0.265261\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −4.44949 −0.799152 −0.399576 0.916700i \(-0.630843\pi\)
−0.399576 + 0.916700i \(0.630843\pi\)
\(32\) 0 0
\(33\) 10.8990 1.89727
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 4.89898 0.805387 0.402694 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(38\) 0 0
\(39\) 2.44949 0.392232
\(40\) 0 0
\(41\) 10.8990 1.70213 0.851067 0.525057i \(-0.175956\pi\)
0.851067 + 0.525057i \(0.175956\pi\)
\(42\) 0 0
\(43\) 11.3485 1.73063 0.865313 0.501232i \(-0.167120\pi\)
0.865313 + 0.501232i \(0.167120\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −16.8990 −2.36633
\(52\) 0 0
\(53\) 1.10102 0.151237 0.0756184 0.997137i \(-0.475907\pi\)
0.0756184 + 0.997137i \(0.475907\pi\)
\(54\) 0 0
\(55\) −4.44949 −0.599969
\(56\) 0 0
\(57\) −1.10102 −0.145834
\(58\) 0 0
\(59\) 9.34847 1.21707 0.608534 0.793528i \(-0.291758\pi\)
0.608534 + 0.793528i \(0.291758\pi\)
\(60\) 0 0
\(61\) −5.79796 −0.742353 −0.371176 0.928562i \(-0.621045\pi\)
−0.371176 + 0.928562i \(0.621045\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −5.10102 −0.623189 −0.311594 0.950215i \(-0.600863\pi\)
−0.311594 + 0.950215i \(0.600863\pi\)
\(68\) 0 0
\(69\) −15.7980 −1.90185
\(70\) 0 0
\(71\) −3.55051 −0.421368 −0.210684 0.977554i \(-0.567569\pi\)
−0.210684 + 0.977554i \(0.567569\pi\)
\(72\) 0 0
\(73\) −14.6969 −1.72015 −0.860073 0.510171i \(-0.829582\pi\)
−0.860073 + 0.510171i \(0.829582\pi\)
\(74\) 0 0
\(75\) −2.44949 −0.282843
\(76\) 0 0
\(77\) −8.89898 −1.01413
\(78\) 0 0
\(79\) 4.89898 0.551178 0.275589 0.961276i \(-0.411127\pi\)
0.275589 + 0.961276i \(0.411127\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 6.89898 0.748299
\(86\) 0 0
\(87\) −9.79796 −1.05045
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 10.8990 1.13017
\(94\) 0 0
\(95\) 0.449490 0.0461167
\(96\) 0 0
\(97\) 7.79796 0.791763 0.395881 0.918302i \(-0.370439\pi\)
0.395881 + 0.918302i \(0.370439\pi\)
\(98\) 0 0
\(99\) −13.3485 −1.34157
\(100\) 0 0
\(101\) −15.7980 −1.57196 −0.785978 0.618255i \(-0.787840\pi\)
−0.785978 + 0.618255i \(0.787840\pi\)
\(102\) 0 0
\(103\) 11.3485 1.11820 0.559099 0.829101i \(-0.311147\pi\)
0.559099 + 0.829101i \(0.311147\pi\)
\(104\) 0 0
\(105\) −4.89898 −0.478091
\(106\) 0 0
\(107\) −12.2474 −1.18401 −0.592003 0.805936i \(-0.701663\pi\)
−0.592003 + 0.805936i \(0.701663\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 1.10102 0.103575 0.0517876 0.998658i \(-0.483508\pi\)
0.0517876 + 0.998658i \(0.483508\pi\)
\(114\) 0 0
\(115\) 6.44949 0.601418
\(116\) 0 0
\(117\) −3.00000 −0.277350
\(118\) 0 0
\(119\) 13.7980 1.26486
\(120\) 0 0
\(121\) 8.79796 0.799814
\(122\) 0 0
\(123\) −26.6969 −2.40718
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.65153 0.412757 0.206378 0.978472i \(-0.433832\pi\)
0.206378 + 0.978472i \(0.433832\pi\)
\(128\) 0 0
\(129\) −27.7980 −2.44747
\(130\) 0 0
\(131\) 19.5959 1.71210 0.856052 0.516890i \(-0.172910\pi\)
0.856052 + 0.516890i \(0.172910\pi\)
\(132\) 0 0
\(133\) 0.898979 0.0779514
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −16.8990 −1.43335 −0.716676 0.697406i \(-0.754338\pi\)
−0.716676 + 0.697406i \(0.754338\pi\)
\(140\) 0 0
\(141\) 4.89898 0.412568
\(142\) 0 0
\(143\) 4.44949 0.372085
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 7.34847 0.606092
\(148\) 0 0
\(149\) 11.7980 0.966526 0.483263 0.875475i \(-0.339451\pi\)
0.483263 + 0.875475i \(0.339451\pi\)
\(150\) 0 0
\(151\) −2.65153 −0.215779 −0.107889 0.994163i \(-0.534409\pi\)
−0.107889 + 0.994163i \(0.534409\pi\)
\(152\) 0 0
\(153\) 20.6969 1.67325
\(154\) 0 0
\(155\) −4.44949 −0.357392
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −2.69694 −0.213881
\(160\) 0 0
\(161\) 12.8990 1.01658
\(162\) 0 0
\(163\) −9.10102 −0.712847 −0.356423 0.934325i \(-0.616004\pi\)
−0.356423 + 0.934325i \(0.616004\pi\)
\(164\) 0 0
\(165\) 10.8990 0.848484
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.34847 0.103120
\(172\) 0 0
\(173\) 6.89898 0.524520 0.262260 0.964997i \(-0.415532\pi\)
0.262260 + 0.964997i \(0.415532\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) −22.8990 −1.72119
\(178\) 0 0
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 0 0
\(181\) 5.79796 0.430959 0.215479 0.976508i \(-0.430869\pi\)
0.215479 + 0.976508i \(0.430869\pi\)
\(182\) 0 0
\(183\) 14.2020 1.04985
\(184\) 0 0
\(185\) 4.89898 0.360180
\(186\) 0 0
\(187\) −30.6969 −2.24478
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.79796 −0.130096 −0.0650479 0.997882i \(-0.520720\pi\)
−0.0650479 + 0.997882i \(0.520720\pi\)
\(192\) 0 0
\(193\) 0.202041 0.0145432 0.00727162 0.999974i \(-0.497685\pi\)
0.00727162 + 0.999974i \(0.497685\pi\)
\(194\) 0 0
\(195\) 2.44949 0.175412
\(196\) 0 0
\(197\) −23.7980 −1.69553 −0.847767 0.530369i \(-0.822054\pi\)
−0.847767 + 0.530369i \(0.822054\pi\)
\(198\) 0 0
\(199\) 5.79796 0.411006 0.205503 0.978656i \(-0.434117\pi\)
0.205503 + 0.978656i \(0.434117\pi\)
\(200\) 0 0
\(201\) 12.4949 0.881322
\(202\) 0 0
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 10.8990 0.761218
\(206\) 0 0
\(207\) 19.3485 1.34481
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −25.7980 −1.77600 −0.888002 0.459839i \(-0.847907\pi\)
−0.888002 + 0.459839i \(0.847907\pi\)
\(212\) 0 0
\(213\) 8.69694 0.595904
\(214\) 0 0
\(215\) 11.3485 0.773959
\(216\) 0 0
\(217\) −8.89898 −0.604102
\(218\) 0 0
\(219\) 36.0000 2.43265
\(220\) 0 0
\(221\) −6.89898 −0.464076
\(222\) 0 0
\(223\) −17.5959 −1.17831 −0.589155 0.808020i \(-0.700539\pi\)
−0.589155 + 0.808020i \(0.700539\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 22.8990 1.51986 0.759929 0.650006i \(-0.225234\pi\)
0.759929 + 0.650006i \(0.225234\pi\)
\(228\) 0 0
\(229\) −18.8990 −1.24888 −0.624440 0.781073i \(-0.714673\pi\)
−0.624440 + 0.781073i \(0.714673\pi\)
\(230\) 0 0
\(231\) 21.7980 1.43420
\(232\) 0 0
\(233\) −25.5959 −1.67684 −0.838422 0.545021i \(-0.816522\pi\)
−0.838422 + 0.545021i \(0.816522\pi\)
\(234\) 0 0
\(235\) −2.00000 −0.130466
\(236\) 0 0
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −11.1464 −0.721003 −0.360501 0.932759i \(-0.617394\pi\)
−0.360501 + 0.932759i \(0.617394\pi\)
\(240\) 0 0
\(241\) −0.696938 −0.0448938 −0.0224469 0.999748i \(-0.507146\pi\)
−0.0224469 + 0.999748i \(0.507146\pi\)
\(242\) 0 0
\(243\) 22.0454 1.41421
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −0.449490 −0.0286003
\(248\) 0 0
\(249\) −4.89898 −0.310460
\(250\) 0 0
\(251\) −14.6969 −0.927663 −0.463831 0.885924i \(-0.653526\pi\)
−0.463831 + 0.885924i \(0.653526\pi\)
\(252\) 0 0
\(253\) −28.6969 −1.80416
\(254\) 0 0
\(255\) −16.8990 −1.05826
\(256\) 0 0
\(257\) −3.79796 −0.236910 −0.118455 0.992959i \(-0.537794\pi\)
−0.118455 + 0.992959i \(0.537794\pi\)
\(258\) 0 0
\(259\) 9.79796 0.608816
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 26.4495 1.63095 0.815473 0.578796i \(-0.196477\pi\)
0.815473 + 0.578796i \(0.196477\pi\)
\(264\) 0 0
\(265\) 1.10102 0.0676352
\(266\) 0 0
\(267\) −14.6969 −0.899438
\(268\) 0 0
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 3.55051 0.215678 0.107839 0.994168i \(-0.465607\pi\)
0.107839 + 0.994168i \(0.465607\pi\)
\(272\) 0 0
\(273\) 4.89898 0.296500
\(274\) 0 0
\(275\) −4.44949 −0.268314
\(276\) 0 0
\(277\) −22.4949 −1.35159 −0.675794 0.737091i \(-0.736199\pi\)
−0.675794 + 0.737091i \(0.736199\pi\)
\(278\) 0 0
\(279\) −13.3485 −0.799152
\(280\) 0 0
\(281\) −6.89898 −0.411559 −0.205779 0.978598i \(-0.565973\pi\)
−0.205779 + 0.978598i \(0.565973\pi\)
\(282\) 0 0
\(283\) −6.44949 −0.383382 −0.191691 0.981455i \(-0.561397\pi\)
−0.191691 + 0.981455i \(0.561397\pi\)
\(284\) 0 0
\(285\) −1.10102 −0.0652188
\(286\) 0 0
\(287\) 21.7980 1.28669
\(288\) 0 0
\(289\) 30.5959 1.79976
\(290\) 0 0
\(291\) −19.1010 −1.11972
\(292\) 0 0
\(293\) 28.4949 1.66469 0.832345 0.554258i \(-0.186998\pi\)
0.832345 + 0.554258i \(0.186998\pi\)
\(294\) 0 0
\(295\) 9.34847 0.544289
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.44949 −0.372984
\(300\) 0 0
\(301\) 22.6969 1.30823
\(302\) 0 0
\(303\) 38.6969 2.22308
\(304\) 0 0
\(305\) −5.79796 −0.331990
\(306\) 0 0
\(307\) 7.79796 0.445053 0.222527 0.974927i \(-0.428570\pi\)
0.222527 + 0.974927i \(0.428570\pi\)
\(308\) 0 0
\(309\) −27.7980 −1.58137
\(310\) 0 0
\(311\) 3.10102 0.175843 0.0879214 0.996127i \(-0.471978\pi\)
0.0879214 + 0.996127i \(0.471978\pi\)
\(312\) 0 0
\(313\) −22.4949 −1.27149 −0.635743 0.771900i \(-0.719306\pi\)
−0.635743 + 0.771900i \(0.719306\pi\)
\(314\) 0 0
\(315\) 6.00000 0.338062
\(316\) 0 0
\(317\) 26.6969 1.49945 0.749725 0.661750i \(-0.230186\pi\)
0.749725 + 0.661750i \(0.230186\pi\)
\(318\) 0 0
\(319\) −17.7980 −0.996494
\(320\) 0 0
\(321\) 30.0000 1.67444
\(322\) 0 0
\(323\) 3.10102 0.172545
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 24.4949 1.35457
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −3.55051 −0.195154 −0.0975768 0.995228i \(-0.531109\pi\)
−0.0975768 + 0.995228i \(0.531109\pi\)
\(332\) 0 0
\(333\) 14.6969 0.805387
\(334\) 0 0
\(335\) −5.10102 −0.278699
\(336\) 0 0
\(337\) −16.6969 −0.909540 −0.454770 0.890609i \(-0.650279\pi\)
−0.454770 + 0.890609i \(0.650279\pi\)
\(338\) 0 0
\(339\) −2.69694 −0.146478
\(340\) 0 0
\(341\) 19.7980 1.07212
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −15.7980 −0.850534
\(346\) 0 0
\(347\) 20.2474 1.08694 0.543470 0.839429i \(-0.317110\pi\)
0.543470 + 0.839429i \(0.317110\pi\)
\(348\) 0 0
\(349\) 20.6969 1.10788 0.553941 0.832556i \(-0.313123\pi\)
0.553941 + 0.832556i \(0.313123\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.4949 −1.51663 −0.758315 0.651888i \(-0.773977\pi\)
−0.758315 + 0.651888i \(0.773977\pi\)
\(354\) 0 0
\(355\) −3.55051 −0.188442
\(356\) 0 0
\(357\) −33.7980 −1.78878
\(358\) 0 0
\(359\) −5.75255 −0.303608 −0.151804 0.988411i \(-0.548508\pi\)
−0.151804 + 0.988411i \(0.548508\pi\)
\(360\) 0 0
\(361\) −18.7980 −0.989366
\(362\) 0 0
\(363\) −21.5505 −1.13111
\(364\) 0 0
\(365\) −14.6969 −0.769273
\(366\) 0 0
\(367\) 34.9444 1.82408 0.912041 0.410099i \(-0.134506\pi\)
0.912041 + 0.410099i \(0.134506\pi\)
\(368\) 0 0
\(369\) 32.6969 1.70213
\(370\) 0 0
\(371\) 2.20204 0.114324
\(372\) 0 0
\(373\) 29.5959 1.53242 0.766209 0.642591i \(-0.222141\pi\)
0.766209 + 0.642591i \(0.222141\pi\)
\(374\) 0 0
\(375\) −2.44949 −0.126491
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −16.4495 −0.844954 −0.422477 0.906374i \(-0.638839\pi\)
−0.422477 + 0.906374i \(0.638839\pi\)
\(380\) 0 0
\(381\) −11.3939 −0.583726
\(382\) 0 0
\(383\) −25.5959 −1.30789 −0.653945 0.756542i \(-0.726887\pi\)
−0.653945 + 0.756542i \(0.726887\pi\)
\(384\) 0 0
\(385\) −8.89898 −0.453534
\(386\) 0 0
\(387\) 34.0454 1.73063
\(388\) 0 0
\(389\) 1.59592 0.0809163 0.0404581 0.999181i \(-0.487118\pi\)
0.0404581 + 0.999181i \(0.487118\pi\)
\(390\) 0 0
\(391\) 44.4949 2.25020
\(392\) 0 0
\(393\) −48.0000 −2.42128
\(394\) 0 0
\(395\) 4.89898 0.246494
\(396\) 0 0
\(397\) 30.6969 1.54064 0.770318 0.637660i \(-0.220098\pi\)
0.770318 + 0.637660i \(0.220098\pi\)
\(398\) 0 0
\(399\) −2.20204 −0.110240
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 4.44949 0.221645
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) −21.7980 −1.08048
\(408\) 0 0
\(409\) −6.89898 −0.341133 −0.170566 0.985346i \(-0.554560\pi\)
−0.170566 + 0.985346i \(0.554560\pi\)
\(410\) 0 0
\(411\) 34.2929 1.69154
\(412\) 0 0
\(413\) 18.6969 0.920016
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 41.3939 2.02707
\(418\) 0 0
\(419\) 38.6969 1.89047 0.945235 0.326392i \(-0.105833\pi\)
0.945235 + 0.326392i \(0.105833\pi\)
\(420\) 0 0
\(421\) −3.79796 −0.185101 −0.0925506 0.995708i \(-0.529502\pi\)
−0.0925506 + 0.995708i \(0.529502\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 6.89898 0.334650
\(426\) 0 0
\(427\) −11.5959 −0.561166
\(428\) 0 0
\(429\) −10.8990 −0.526208
\(430\) 0 0
\(431\) −32.0454 −1.54357 −0.771786 0.635882i \(-0.780637\pi\)
−0.771786 + 0.635882i \(0.780637\pi\)
\(432\) 0 0
\(433\) 11.7980 0.566974 0.283487 0.958976i \(-0.408509\pi\)
0.283487 + 0.958976i \(0.408509\pi\)
\(434\) 0 0
\(435\) −9.79796 −0.469776
\(436\) 0 0
\(437\) 2.89898 0.138677
\(438\) 0 0
\(439\) −17.7980 −0.849450 −0.424725 0.905322i \(-0.639629\pi\)
−0.424725 + 0.905322i \(0.639629\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) −22.0454 −1.04741 −0.523704 0.851900i \(-0.675450\pi\)
−0.523704 + 0.851900i \(0.675450\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) −28.8990 −1.36687
\(448\) 0 0
\(449\) −20.6969 −0.976749 −0.488374 0.872634i \(-0.662410\pi\)
−0.488374 + 0.872634i \(0.662410\pi\)
\(450\) 0 0
\(451\) −48.4949 −2.28354
\(452\) 0 0
\(453\) 6.49490 0.305157
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.6969 −1.52285 −0.761424 0.648254i \(-0.775499\pi\)
−0.761424 + 0.648254i \(0.775499\pi\)
\(462\) 0 0
\(463\) −22.8990 −1.06421 −0.532103 0.846680i \(-0.678598\pi\)
−0.532103 + 0.846680i \(0.678598\pi\)
\(464\) 0 0
\(465\) 10.8990 0.505428
\(466\) 0 0
\(467\) 22.9444 1.06174 0.530870 0.847453i \(-0.321865\pi\)
0.530870 + 0.847453i \(0.321865\pi\)
\(468\) 0 0
\(469\) −10.2020 −0.471086
\(470\) 0 0
\(471\) −24.4949 −1.12867
\(472\) 0 0
\(473\) −50.4949 −2.32176
\(474\) 0 0
\(475\) 0.449490 0.0206240
\(476\) 0 0
\(477\) 3.30306 0.151237
\(478\) 0 0
\(479\) −9.34847 −0.427142 −0.213571 0.976927i \(-0.568510\pi\)
−0.213571 + 0.976927i \(0.568510\pi\)
\(480\) 0 0
\(481\) −4.89898 −0.223374
\(482\) 0 0
\(483\) −31.5959 −1.43766
\(484\) 0 0
\(485\) 7.79796 0.354087
\(486\) 0 0
\(487\) −6.89898 −0.312623 −0.156311 0.987708i \(-0.549960\pi\)
−0.156311 + 0.987708i \(0.549960\pi\)
\(488\) 0 0
\(489\) 22.2929 1.00812
\(490\) 0 0
\(491\) −34.2929 −1.54761 −0.773807 0.633421i \(-0.781650\pi\)
−0.773807 + 0.633421i \(0.781650\pi\)
\(492\) 0 0
\(493\) 27.5959 1.24286
\(494\) 0 0
\(495\) −13.3485 −0.599969
\(496\) 0 0
\(497\) −7.10102 −0.318524
\(498\) 0 0
\(499\) 6.65153 0.297763 0.148882 0.988855i \(-0.452433\pi\)
0.148882 + 0.988855i \(0.452433\pi\)
\(500\) 0 0
\(501\) −44.0908 −1.96983
\(502\) 0 0
\(503\) 26.9444 1.20139 0.600695 0.799478i \(-0.294890\pi\)
0.600695 + 0.799478i \(0.294890\pi\)
\(504\) 0 0
\(505\) −15.7980 −0.703000
\(506\) 0 0
\(507\) −2.44949 −0.108786
\(508\) 0 0
\(509\) 17.1010 0.757989 0.378995 0.925399i \(-0.376270\pi\)
0.378995 + 0.925399i \(0.376270\pi\)
\(510\) 0 0
\(511\) −29.3939 −1.30031
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.3485 0.500073
\(516\) 0 0
\(517\) 8.89898 0.391377
\(518\) 0 0
\(519\) −16.8990 −0.741783
\(520\) 0 0
\(521\) 25.7980 1.13023 0.565115 0.825012i \(-0.308832\pi\)
0.565115 + 0.825012i \(0.308832\pi\)
\(522\) 0 0
\(523\) −24.6515 −1.07794 −0.538968 0.842326i \(-0.681186\pi\)
−0.538968 + 0.842326i \(0.681186\pi\)
\(524\) 0 0
\(525\) −4.89898 −0.213809
\(526\) 0 0
\(527\) −30.6969 −1.33718
\(528\) 0 0
\(529\) 18.5959 0.808518
\(530\) 0 0
\(531\) 28.0454 1.21707
\(532\) 0 0
\(533\) −10.8990 −0.472087
\(534\) 0 0
\(535\) −12.2474 −0.529503
\(536\) 0 0
\(537\) −19.5959 −0.845626
\(538\) 0 0
\(539\) 13.3485 0.574959
\(540\) 0 0
\(541\) 14.8990 0.640557 0.320279 0.947323i \(-0.396223\pi\)
0.320279 + 0.947323i \(0.396223\pi\)
\(542\) 0 0
\(543\) −14.2020 −0.609468
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 15.3485 0.656253 0.328127 0.944634i \(-0.393583\pi\)
0.328127 + 0.944634i \(0.393583\pi\)
\(548\) 0 0
\(549\) −17.3939 −0.742353
\(550\) 0 0
\(551\) 1.79796 0.0765956
\(552\) 0 0
\(553\) 9.79796 0.416652
\(554\) 0 0
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) −30.6969 −1.30067 −0.650336 0.759647i \(-0.725372\pi\)
−0.650336 + 0.759647i \(0.725372\pi\)
\(558\) 0 0
\(559\) −11.3485 −0.479989
\(560\) 0 0
\(561\) 75.1918 3.17460
\(562\) 0 0
\(563\) −32.2474 −1.35907 −0.679534 0.733644i \(-0.737818\pi\)
−0.679534 + 0.733644i \(0.737818\pi\)
\(564\) 0 0
\(565\) 1.10102 0.0463203
\(566\) 0 0
\(567\) −18.0000 −0.755929
\(568\) 0 0
\(569\) 29.7980 1.24920 0.624598 0.780947i \(-0.285263\pi\)
0.624598 + 0.780947i \(0.285263\pi\)
\(570\) 0 0
\(571\) −20.4949 −0.857685 −0.428842 0.903379i \(-0.641078\pi\)
−0.428842 + 0.903379i \(0.641078\pi\)
\(572\) 0 0
\(573\) 4.40408 0.183983
\(574\) 0 0
\(575\) 6.44949 0.268962
\(576\) 0 0
\(577\) 36.8990 1.53612 0.768062 0.640375i \(-0.221221\pi\)
0.768062 + 0.640375i \(0.221221\pi\)
\(578\) 0 0
\(579\) −0.494897 −0.0205672
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −4.89898 −0.202895
\(584\) 0 0
\(585\) −3.00000 −0.124035
\(586\) 0 0
\(587\) −1.10102 −0.0454440 −0.0227220 0.999742i \(-0.507233\pi\)
−0.0227220 + 0.999742i \(0.507233\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 58.2929 2.39785
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 13.7980 0.565661
\(596\) 0 0
\(597\) −14.2020 −0.581251
\(598\) 0 0
\(599\) −19.1010 −0.780447 −0.390223 0.920720i \(-0.627602\pi\)
−0.390223 + 0.920720i \(0.627602\pi\)
\(600\) 0 0
\(601\) 5.59592 0.228262 0.114131 0.993466i \(-0.463592\pi\)
0.114131 + 0.993466i \(0.463592\pi\)
\(602\) 0 0
\(603\) −15.3031 −0.623189
\(604\) 0 0
\(605\) 8.79796 0.357688
\(606\) 0 0
\(607\) 26.4495 1.07355 0.536776 0.843725i \(-0.319642\pi\)
0.536776 + 0.843725i \(0.319642\pi\)
\(608\) 0 0
\(609\) −19.5959 −0.794067
\(610\) 0 0
\(611\) 2.00000 0.0809113
\(612\) 0 0
\(613\) −33.1918 −1.34061 −0.670303 0.742088i \(-0.733836\pi\)
−0.670303 + 0.742088i \(0.733836\pi\)
\(614\) 0 0
\(615\) −26.6969 −1.07652
\(616\) 0 0
\(617\) 23.3939 0.941802 0.470901 0.882186i \(-0.343929\pi\)
0.470901 + 0.882186i \(0.343929\pi\)
\(618\) 0 0
\(619\) 9.34847 0.375747 0.187873 0.982193i \(-0.439841\pi\)
0.187873 + 0.982193i \(0.439841\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.89898 0.195646
\(628\) 0 0
\(629\) 33.7980 1.34761
\(630\) 0 0
\(631\) −11.5505 −0.459819 −0.229909 0.973212i \(-0.573843\pi\)
−0.229909 + 0.973212i \(0.573843\pi\)
\(632\) 0 0
\(633\) 63.1918 2.51165
\(634\) 0 0
\(635\) 4.65153 0.184590
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −10.6515 −0.421368
\(640\) 0 0
\(641\) 23.3939 0.924003 0.462001 0.886879i \(-0.347132\pi\)
0.462001 + 0.886879i \(0.347132\pi\)
\(642\) 0 0
\(643\) −30.0000 −1.18308 −0.591542 0.806274i \(-0.701481\pi\)
−0.591542 + 0.806274i \(0.701481\pi\)
\(644\) 0 0
\(645\) −27.7980 −1.09454
\(646\) 0 0
\(647\) −38.9444 −1.53106 −0.765531 0.643399i \(-0.777524\pi\)
−0.765531 + 0.643399i \(0.777524\pi\)
\(648\) 0 0
\(649\) −41.5959 −1.63278
\(650\) 0 0
\(651\) 21.7980 0.854329
\(652\) 0 0
\(653\) −24.2020 −0.947099 −0.473550 0.880767i \(-0.657028\pi\)
−0.473550 + 0.880767i \(0.657028\pi\)
\(654\) 0 0
\(655\) 19.5959 0.765676
\(656\) 0 0
\(657\) −44.0908 −1.72015
\(658\) 0 0
\(659\) −14.6969 −0.572511 −0.286256 0.958153i \(-0.592411\pi\)
−0.286256 + 0.958153i \(0.592411\pi\)
\(660\) 0 0
\(661\) 20.2020 0.785768 0.392884 0.919588i \(-0.371477\pi\)
0.392884 + 0.919588i \(0.371477\pi\)
\(662\) 0 0
\(663\) 16.8990 0.656302
\(664\) 0 0
\(665\) 0.898979 0.0348609
\(666\) 0 0
\(667\) 25.7980 0.998901
\(668\) 0 0
\(669\) 43.1010 1.66638
\(670\) 0 0
\(671\) 25.7980 0.995919
\(672\) 0 0
\(673\) −10.8990 −0.420125 −0.210062 0.977688i \(-0.567367\pi\)
−0.210062 + 0.977688i \(0.567367\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11.3031 −0.434412 −0.217206 0.976126i \(-0.569694\pi\)
−0.217206 + 0.976126i \(0.569694\pi\)
\(678\) 0 0
\(679\) 15.5959 0.598516
\(680\) 0 0
\(681\) −56.0908 −2.14940
\(682\) 0 0
\(683\) 6.49490 0.248520 0.124260 0.992250i \(-0.460344\pi\)
0.124260 + 0.992250i \(0.460344\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) 46.2929 1.76618
\(688\) 0 0
\(689\) −1.10102 −0.0419455
\(690\) 0 0
\(691\) −29.3485 −1.11647 −0.558234 0.829683i \(-0.688521\pi\)
−0.558234 + 0.829683i \(0.688521\pi\)
\(692\) 0 0
\(693\) −26.6969 −1.01413
\(694\) 0 0
\(695\) −16.8990 −0.641015
\(696\) 0 0
\(697\) 75.1918 2.84809
\(698\) 0 0
\(699\) 62.6969 2.37142
\(700\) 0 0
\(701\) 41.1918 1.55579 0.777897 0.628392i \(-0.216286\pi\)
0.777897 + 0.628392i \(0.216286\pi\)
\(702\) 0 0
\(703\) 2.20204 0.0830516
\(704\) 0 0
\(705\) 4.89898 0.184506
\(706\) 0 0
\(707\) −31.5959 −1.18829
\(708\) 0 0
\(709\) 36.2929 1.36301 0.681503 0.731815i \(-0.261326\pi\)
0.681503 + 0.731815i \(0.261326\pi\)
\(710\) 0 0
\(711\) 14.6969 0.551178
\(712\) 0 0
\(713\) −28.6969 −1.07471
\(714\) 0 0
\(715\) 4.44949 0.166401
\(716\) 0 0
\(717\) 27.3031 1.01965
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 22.6969 0.845278
\(722\) 0 0
\(723\) 1.70714 0.0634894
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −26.0454 −0.965971 −0.482985 0.875628i \(-0.660448\pi\)
−0.482985 + 0.875628i \(0.660448\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 78.2929 2.89577
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) 7.34847 0.271052
\(736\) 0 0
\(737\) 22.6969 0.836052
\(738\) 0 0
\(739\) 18.2474 0.671243 0.335622 0.941997i \(-0.391054\pi\)
0.335622 + 0.941997i \(0.391054\pi\)
\(740\) 0 0
\(741\) 1.10102 0.0404470
\(742\) 0 0
\(743\) 23.3939 0.858238 0.429119 0.903248i \(-0.358824\pi\)
0.429119 + 0.903248i \(0.358824\pi\)
\(744\) 0 0
\(745\) 11.7980 0.432244
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −24.4949 −0.895024
\(750\) 0 0
\(751\) 40.8990 1.49242 0.746212 0.665708i \(-0.231870\pi\)
0.746212 + 0.665708i \(0.231870\pi\)
\(752\) 0 0
\(753\) 36.0000 1.31191
\(754\) 0 0
\(755\) −2.65153 −0.0964991
\(756\) 0 0
\(757\) −22.4949 −0.817591 −0.408795 0.912626i \(-0.634051\pi\)
−0.408795 + 0.912626i \(0.634051\pi\)
\(758\) 0 0
\(759\) 70.2929 2.55147
\(760\) 0 0
\(761\) −49.5959 −1.79785 −0.898925 0.438102i \(-0.855651\pi\)
−0.898925 + 0.438102i \(0.855651\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 0 0
\(765\) 20.6969 0.748299
\(766\) 0 0
\(767\) −9.34847 −0.337554
\(768\) 0 0
\(769\) −19.3939 −0.699361 −0.349681 0.936869i \(-0.613710\pi\)
−0.349681 + 0.936869i \(0.613710\pi\)
\(770\) 0 0
\(771\) 9.30306 0.335042
\(772\) 0 0
\(773\) −44.0908 −1.58584 −0.792918 0.609328i \(-0.791439\pi\)
−0.792918 + 0.609328i \(0.791439\pi\)
\(774\) 0 0
\(775\) −4.44949 −0.159830
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) 4.89898 0.175524
\(780\) 0 0
\(781\) 15.7980 0.565295
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) 0 0
\(789\) −64.7878 −2.30651
\(790\) 0 0
\(791\) 2.20204 0.0782956
\(792\) 0 0
\(793\) 5.79796 0.205892
\(794\) 0 0
\(795\) −2.69694 −0.0956506
\(796\) 0 0
\(797\) 8.20204 0.290531 0.145266 0.989393i \(-0.453596\pi\)
0.145266 + 0.989393i \(0.453596\pi\)
\(798\) 0 0
\(799\) −13.7980 −0.488137
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 65.3939 2.30770
\(804\) 0 0
\(805\) 12.8990 0.454629
\(806\) 0 0
\(807\) −63.6867 −2.24188
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 9.84337 0.345647 0.172824 0.984953i \(-0.444711\pi\)
0.172824 + 0.984953i \(0.444711\pi\)
\(812\) 0 0
\(813\) −8.69694 −0.305015
\(814\) 0 0
\(815\) −9.10102 −0.318795
\(816\) 0 0
\(817\) 5.10102 0.178462
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 25.1918 0.879201 0.439601 0.898193i \(-0.355120\pi\)
0.439601 + 0.898193i \(0.355120\pi\)
\(822\) 0 0
\(823\) 25.5505 0.890635 0.445317 0.895373i \(-0.353091\pi\)
0.445317 + 0.895373i \(0.353091\pi\)
\(824\) 0 0
\(825\) 10.8990 0.379454
\(826\) 0 0
\(827\) −21.1918 −0.736912 −0.368456 0.929645i \(-0.620114\pi\)
−0.368456 + 0.929645i \(0.620114\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 55.1010 1.91143
\(832\) 0 0
\(833\) −20.6969 −0.717106
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.4495 −0.844090 −0.422045 0.906575i \(-0.638688\pi\)
−0.422045 + 0.906575i \(0.638688\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 16.8990 0.582032
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 17.5959 0.604603
\(848\) 0 0
\(849\) 15.7980 0.542185
\(850\) 0 0
\(851\) 31.5959 1.08309
\(852\) 0 0
\(853\) 10.6969 0.366256 0.183128 0.983089i \(-0.441378\pi\)
0.183128 + 0.983089i \(0.441378\pi\)
\(854\) 0 0
\(855\) 1.34847 0.0461167
\(856\) 0 0
\(857\) −23.3939 −0.799120 −0.399560 0.916707i \(-0.630837\pi\)
−0.399560 + 0.916707i \(0.630837\pi\)
\(858\) 0 0
\(859\) −1.30306 −0.0444599 −0.0222299 0.999753i \(-0.507077\pi\)
−0.0222299 + 0.999753i \(0.507077\pi\)
\(860\) 0 0
\(861\) −53.3939 −1.81966
\(862\) 0 0
\(863\) −35.3939 −1.20482 −0.602411 0.798186i \(-0.705793\pi\)
−0.602411 + 0.798186i \(0.705793\pi\)
\(864\) 0 0
\(865\) 6.89898 0.234572
\(866\) 0 0
\(867\) −74.9444 −2.54524
\(868\) 0 0
\(869\) −21.7980 −0.739445
\(870\) 0 0
\(871\) 5.10102 0.172841
\(872\) 0 0
\(873\) 23.3939 0.791763
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −21.5959 −0.729242 −0.364621 0.931156i \(-0.618802\pi\)
−0.364621 + 0.931156i \(0.618802\pi\)
\(878\) 0 0
\(879\) −69.7980 −2.35423
\(880\) 0 0
\(881\) −9.79796 −0.330102 −0.165051 0.986285i \(-0.552779\pi\)
−0.165051 + 0.986285i \(0.552779\pi\)
\(882\) 0 0
\(883\) 3.34847 0.112685 0.0563425 0.998412i \(-0.482056\pi\)
0.0563425 + 0.998412i \(0.482056\pi\)
\(884\) 0 0
\(885\) −22.8990 −0.769741
\(886\) 0 0
\(887\) −20.2474 −0.679843 −0.339921 0.940454i \(-0.610400\pi\)
−0.339921 + 0.940454i \(0.610400\pi\)
\(888\) 0 0
\(889\) 9.30306 0.312015
\(890\) 0 0
\(891\) 40.0454 1.34157
\(892\) 0 0
\(893\) −0.898979 −0.0300832
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 15.7980 0.527478
\(898\) 0 0
\(899\) −17.7980 −0.593595
\(900\) 0 0
\(901\) 7.59592 0.253057
\(902\) 0 0
\(903\) −55.5959 −1.85012
\(904\) 0 0
\(905\) 5.79796 0.192731
\(906\) 0 0
\(907\) −19.8434 −0.658888 −0.329444 0.944175i \(-0.606861\pi\)
−0.329444 + 0.944175i \(0.606861\pi\)
\(908\) 0 0
\(909\) −47.3939 −1.57196
\(910\) 0 0
\(911\) 9.39388 0.311233 0.155617 0.987818i \(-0.450264\pi\)
0.155617 + 0.987818i \(0.450264\pi\)
\(912\) 0 0
\(913\) −8.89898 −0.294513
\(914\) 0 0
\(915\) 14.2020 0.469505
\(916\) 0 0
\(917\) 39.1918 1.29423
\(918\) 0 0
\(919\) −20.8990 −0.689394 −0.344697 0.938714i \(-0.612018\pi\)
−0.344697 + 0.938714i \(0.612018\pi\)
\(920\) 0 0
\(921\) −19.1010 −0.629400
\(922\) 0 0
\(923\) 3.55051 0.116866
\(924\) 0 0
\(925\) 4.89898 0.161077
\(926\) 0 0
\(927\) 34.0454 1.11820
\(928\) 0 0
\(929\) −4.69694 −0.154102 −0.0770508 0.997027i \(-0.524550\pi\)
−0.0770508 + 0.997027i \(0.524550\pi\)
\(930\) 0 0
\(931\) −1.34847 −0.0441943
\(932\) 0 0
\(933\) −7.59592 −0.248679
\(934\) 0 0
\(935\) −30.6969 −1.00390
\(936\) 0 0
\(937\) 39.3939 1.28694 0.643471 0.765471i \(-0.277494\pi\)
0.643471 + 0.765471i \(0.277494\pi\)
\(938\) 0 0
\(939\) 55.1010 1.79815
\(940\) 0 0
\(941\) −52.6969 −1.71787 −0.858936 0.512084i \(-0.828874\pi\)
−0.858936 + 0.512084i \(0.828874\pi\)
\(942\) 0 0
\(943\) 70.2929 2.28905
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −28.2020 −0.916443 −0.458222 0.888838i \(-0.651513\pi\)
−0.458222 + 0.888838i \(0.651513\pi\)
\(948\) 0 0
\(949\) 14.6969 0.477083
\(950\) 0 0
\(951\) −65.3939 −2.12054
\(952\) 0 0
\(953\) 21.1918 0.686471 0.343235 0.939249i \(-0.388477\pi\)
0.343235 + 0.939249i \(0.388477\pi\)
\(954\) 0 0
\(955\) −1.79796 −0.0581806
\(956\) 0 0
\(957\) 43.5959 1.40926
\(958\) 0 0
\(959\) −28.0000 −0.904167
\(960\) 0 0
\(961\) −11.2020 −0.361356
\(962\) 0 0
\(963\) −36.7423 −1.18401
\(964\) 0 0
\(965\) 0.202041 0.00650393
\(966\) 0 0
\(967\) 34.8990 1.12228 0.561138 0.827722i \(-0.310364\pi\)
0.561138 + 0.827722i \(0.310364\pi\)
\(968\) 0 0
\(969\) −7.59592 −0.244016
\(970\) 0 0
\(971\) −11.5959 −0.372131 −0.186065 0.982537i \(-0.559574\pi\)
−0.186065 + 0.982537i \(0.559574\pi\)
\(972\) 0 0
\(973\) −33.7980 −1.08351
\(974\) 0 0
\(975\) 2.44949 0.0784465
\(976\) 0 0
\(977\) −30.6969 −0.982082 −0.491041 0.871136i \(-0.663383\pi\)
−0.491041 + 0.871136i \(0.663383\pi\)
\(978\) 0 0
\(979\) −26.6969 −0.853238
\(980\) 0 0
\(981\) −30.0000 −0.957826
\(982\) 0 0
\(983\) 49.1010 1.56608 0.783040 0.621972i \(-0.213668\pi\)
0.783040 + 0.621972i \(0.213668\pi\)
\(984\) 0 0
\(985\) −23.7980 −0.758266
\(986\) 0 0
\(987\) 9.79796 0.311872
\(988\) 0 0
\(989\) 73.1918 2.32736
\(990\) 0 0
\(991\) −49.7980 −1.58188 −0.790942 0.611891i \(-0.790409\pi\)
−0.790942 + 0.611891i \(0.790409\pi\)
\(992\) 0 0
\(993\) 8.69694 0.275989
\(994\) 0 0
\(995\) 5.79796 0.183808
\(996\) 0 0
\(997\) 34.8990 1.10526 0.552631 0.833426i \(-0.313624\pi\)
0.552631 + 0.833426i \(0.313624\pi\)
\(998\) 0 0
\(999\) 0 0
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 520.2.a.f.1.1 2
3.2 odd 2 4680.2.a.z.1.2 2
4.3 odd 2 1040.2.a.k.1.2 2
5.2 odd 4 2600.2.d.h.1249.4 4
5.3 odd 4 2600.2.d.h.1249.1 4
5.4 even 2 2600.2.a.q.1.2 2
8.3 odd 2 4160.2.a.ba.1.1 2
8.5 even 2 4160.2.a.bg.1.2 2
12.11 even 2 9360.2.a.cc.1.1 2
13.12 even 2 6760.2.a.s.1.1 2
20.19 odd 2 5200.2.a.bv.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.a.f.1.1 2 1.1 even 1 trivial
1040.2.a.k.1.2 2 4.3 odd 2
2600.2.a.q.1.2 2 5.4 even 2
2600.2.d.h.1249.1 4 5.3 odd 4
2600.2.d.h.1249.4 4 5.2 odd 4
4160.2.a.ba.1.1 2 8.3 odd 2
4160.2.a.bg.1.2 2 8.5 even 2
4680.2.a.z.1.2 2 3.2 odd 2
5200.2.a.bv.1.1 2 20.19 odd 2
6760.2.a.s.1.1 2 13.12 even 2
9360.2.a.cc.1.1 2 12.11 even 2