Properties

Label 2450.2.a.bl.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2450,2,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.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: \(19.5633484952\)
Analytic rank: \(1\)
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: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 2450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{8} +3.00000 q^{9} +4.89898 q^{11} -2.44949 q^{12} +0.449490 q^{13} +1.00000 q^{16} -2.00000 q^{17} -3.00000 q^{18} -6.44949 q^{19} -4.89898 q^{22} -6.89898 q^{23} +2.44949 q^{24} -0.449490 q^{26} -2.89898 q^{29} +0.898979 q^{31} -1.00000 q^{32} -12.0000 q^{33} +2.00000 q^{34} +3.00000 q^{36} +2.00000 q^{37} +6.44949 q^{38} -1.10102 q^{39} +10.8990 q^{41} +8.89898 q^{43} +4.89898 q^{44} +6.89898 q^{46} +0.898979 q^{47} -2.44949 q^{48} +4.89898 q^{51} +0.449490 q^{52} -1.10102 q^{53} +15.7980 q^{57} +2.89898 q^{58} +6.44949 q^{59} -8.44949 q^{61} -0.898979 q^{62} +1.00000 q^{64} +12.0000 q^{66} -8.00000 q^{67} -2.00000 q^{68} +16.8990 q^{69} -10.8990 q^{71} -3.00000 q^{72} +6.89898 q^{73} -2.00000 q^{74} -6.44949 q^{76} +1.10102 q^{78} -2.89898 q^{79} -9.00000 q^{81} -10.8990 q^{82} -2.44949 q^{83} -8.89898 q^{86} +7.10102 q^{87} -4.89898 q^{88} +10.0000 q^{89} -6.89898 q^{92} -2.20204 q^{93} -0.898979 q^{94} +2.44949 q^{96} +3.79796 q^{97} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9} - 4 q^{13} + 2 q^{16} - 4 q^{17} - 6 q^{18} - 8 q^{19} - 4 q^{23} + 4 q^{26} + 4 q^{29} - 8 q^{31} - 2 q^{32} - 24 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{37} + 8 q^{38} - 12 q^{39} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} - 4 q^{52} - 12 q^{53} + 12 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 24 q^{66} - 16 q^{67} - 4 q^{68} + 24 q^{69} - 12 q^{71} - 6 q^{72} + 4 q^{73} - 4 q^{74} - 8 q^{76} + 12 q^{78} + 4 q^{79} - 18 q^{81} - 12 q^{82} - 8 q^{86} + 24 q^{87} + 20 q^{89} - 4 q^{92} - 24 q^{93} + 8 q^{94} - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.44949 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.44949 1.00000
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) −2.44949 −0.707107
\(13\) 0.449490 0.124666 0.0623330 0.998055i \(-0.480146\pi\)
0.0623330 + 0.998055i \(0.480146\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −3.00000 −0.707107
\(19\) −6.44949 −1.47961 −0.739807 0.672819i \(-0.765083\pi\)
−0.739807 + 0.672819i \(0.765083\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.89898 −1.04447
\(23\) −6.89898 −1.43854 −0.719268 0.694732i \(-0.755523\pi\)
−0.719268 + 0.694732i \(0.755523\pi\)
\(24\) 2.44949 0.500000
\(25\) 0 0
\(26\) −0.449490 −0.0881522
\(27\) 0 0
\(28\) 0 0
\(29\) −2.89898 −0.538327 −0.269163 0.963095i \(-0.586747\pi\)
−0.269163 + 0.963095i \(0.586747\pi\)
\(30\) 0 0
\(31\) 0.898979 0.161461 0.0807307 0.996736i \(-0.474275\pi\)
0.0807307 + 0.996736i \(0.474275\pi\)
\(32\) −1.00000 −0.176777
\(33\) −12.0000 −2.08893
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 6.44949 1.04625
\(39\) −1.10102 −0.176304
\(40\) 0 0
\(41\) 10.8990 1.70213 0.851067 0.525057i \(-0.175956\pi\)
0.851067 + 0.525057i \(0.175956\pi\)
\(42\) 0 0
\(43\) 8.89898 1.35708 0.678541 0.734563i \(-0.262613\pi\)
0.678541 + 0.734563i \(0.262613\pi\)
\(44\) 4.89898 0.738549
\(45\) 0 0
\(46\) 6.89898 1.01720
\(47\) 0.898979 0.131130 0.0655648 0.997848i \(-0.479115\pi\)
0.0655648 + 0.997848i \(0.479115\pi\)
\(48\) −2.44949 −0.353553
\(49\) 0 0
\(50\) 0 0
\(51\) 4.89898 0.685994
\(52\) 0.449490 0.0623330
\(53\) −1.10102 −0.151237 −0.0756184 0.997137i \(-0.524093\pi\)
−0.0756184 + 0.997137i \(0.524093\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 15.7980 2.09249
\(58\) 2.89898 0.380655
\(59\) 6.44949 0.839652 0.419826 0.907605i \(-0.362091\pi\)
0.419826 + 0.907605i \(0.362091\pi\)
\(60\) 0 0
\(61\) −8.44949 −1.08185 −0.540923 0.841072i \(-0.681925\pi\)
−0.540923 + 0.841072i \(0.681925\pi\)
\(62\) −0.898979 −0.114171
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 12.0000 1.47710
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 16.8990 2.03440
\(70\) 0 0
\(71\) −10.8990 −1.29347 −0.646735 0.762714i \(-0.723866\pi\)
−0.646735 + 0.762714i \(0.723866\pi\)
\(72\) −3.00000 −0.353553
\(73\) 6.89898 0.807464 0.403732 0.914877i \(-0.367713\pi\)
0.403732 + 0.914877i \(0.367713\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −6.44949 −0.739807
\(77\) 0 0
\(78\) 1.10102 0.124666
\(79\) −2.89898 −0.326161 −0.163080 0.986613i \(-0.552143\pi\)
−0.163080 + 0.986613i \(0.552143\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) −10.8990 −1.20359
\(83\) −2.44949 −0.268866 −0.134433 0.990923i \(-0.542921\pi\)
−0.134433 + 0.990923i \(0.542921\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.89898 −0.959602
\(87\) 7.10102 0.761309
\(88\) −4.89898 −0.522233
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.89898 −0.719268
\(93\) −2.20204 −0.228341
\(94\) −0.898979 −0.0927227
\(95\) 0 0
\(96\) 2.44949 0.250000
\(97\) 3.79796 0.385624 0.192812 0.981236i \(-0.438239\pi\)
0.192812 + 0.981236i \(0.438239\pi\)
\(98\) 0 0
\(99\) 14.6969 1.47710
\(100\) 0 0
\(101\) −8.44949 −0.840756 −0.420378 0.907349i \(-0.638102\pi\)
−0.420378 + 0.907349i \(0.638102\pi\)
\(102\) −4.89898 −0.485071
\(103\) −3.10102 −0.305553 −0.152776 0.988261i \(-0.548821\pi\)
−0.152776 + 0.988261i \(0.548821\pi\)
\(104\) −0.449490 −0.0440761
\(105\) 0 0
\(106\) 1.10102 0.106941
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −2.89898 −0.277672 −0.138836 0.990315i \(-0.544336\pi\)
−0.138836 + 0.990315i \(0.544336\pi\)
\(110\) 0 0
\(111\) −4.89898 −0.464991
\(112\) 0 0
\(113\) 0.202041 0.0190064 0.00950321 0.999955i \(-0.496975\pi\)
0.00950321 + 0.999955i \(0.496975\pi\)
\(114\) −15.7980 −1.47961
\(115\) 0 0
\(116\) −2.89898 −0.269163
\(117\) 1.34847 0.124666
\(118\) −6.44949 −0.593724
\(119\) 0 0
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 8.44949 0.764981
\(123\) −26.6969 −2.40718
\(124\) 0.898979 0.0807307
\(125\) 0 0
\(126\) 0 0
\(127\) −5.10102 −0.452642 −0.226321 0.974053i \(-0.572670\pi\)
−0.226321 + 0.974053i \(0.572670\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −21.7980 −1.91920
\(130\) 0 0
\(131\) 1.55051 0.135469 0.0677344 0.997703i \(-0.478423\pi\)
0.0677344 + 0.997703i \(0.478423\pi\)
\(132\) −12.0000 −1.04447
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 17.7980 1.52058 0.760291 0.649582i \(-0.225056\pi\)
0.760291 + 0.649582i \(0.225056\pi\)
\(138\) −16.8990 −1.43854
\(139\) −6.44949 −0.547039 −0.273519 0.961867i \(-0.588188\pi\)
−0.273519 + 0.961867i \(0.588188\pi\)
\(140\) 0 0
\(141\) −2.20204 −0.185445
\(142\) 10.8990 0.914622
\(143\) 2.20204 0.184144
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −6.89898 −0.570964
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 15.7980 1.29422 0.647110 0.762397i \(-0.275978\pi\)
0.647110 + 0.762397i \(0.275978\pi\)
\(150\) 0 0
\(151\) −19.5959 −1.59469 −0.797347 0.603522i \(-0.793764\pi\)
−0.797347 + 0.603522i \(0.793764\pi\)
\(152\) 6.44949 0.523123
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −1.10102 −0.0881522
\(157\) −8.44949 −0.674343 −0.337171 0.941443i \(-0.609470\pi\)
−0.337171 + 0.941443i \(0.609470\pi\)
\(158\) 2.89898 0.230630
\(159\) 2.69694 0.213881
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −16.8990 −1.32363 −0.661815 0.749667i \(-0.730214\pi\)
−0.661815 + 0.749667i \(0.730214\pi\)
\(164\) 10.8990 0.851067
\(165\) 0 0
\(166\) 2.44949 0.190117
\(167\) −4.89898 −0.379094 −0.189547 0.981872i \(-0.560702\pi\)
−0.189547 + 0.981872i \(0.560702\pi\)
\(168\) 0 0
\(169\) −12.7980 −0.984458
\(170\) 0 0
\(171\) −19.3485 −1.47961
\(172\) 8.89898 0.678541
\(173\) −18.2474 −1.38733 −0.693664 0.720299i \(-0.744005\pi\)
−0.693664 + 0.720299i \(0.744005\pi\)
\(174\) −7.10102 −0.538327
\(175\) 0 0
\(176\) 4.89898 0.369274
\(177\) −15.7980 −1.18745
\(178\) −10.0000 −0.749532
\(179\) −5.79796 −0.433360 −0.216680 0.976243i \(-0.569523\pi\)
−0.216680 + 0.976243i \(0.569523\pi\)
\(180\) 0 0
\(181\) −14.2474 −1.05900 −0.529502 0.848309i \(-0.677621\pi\)
−0.529502 + 0.848309i \(0.677621\pi\)
\(182\) 0 0
\(183\) 20.6969 1.52996
\(184\) 6.89898 0.508600
\(185\) 0 0
\(186\) 2.20204 0.161461
\(187\) −9.79796 −0.716498
\(188\) 0.898979 0.0655648
\(189\) 0 0
\(190\) 0 0
\(191\) −16.6969 −1.20815 −0.604074 0.796928i \(-0.706457\pi\)
−0.604074 + 0.796928i \(0.706457\pi\)
\(192\) −2.44949 −0.176777
\(193\) 17.5959 1.26658 0.633291 0.773914i \(-0.281704\pi\)
0.633291 + 0.773914i \(0.281704\pi\)
\(194\) −3.79796 −0.272678
\(195\) 0 0
\(196\) 0 0
\(197\) 9.10102 0.648421 0.324210 0.945985i \(-0.394901\pi\)
0.324210 + 0.945985i \(0.394901\pi\)
\(198\) −14.6969 −1.04447
\(199\) −7.10102 −0.503378 −0.251689 0.967808i \(-0.580986\pi\)
−0.251689 + 0.967808i \(0.580986\pi\)
\(200\) 0 0
\(201\) 19.5959 1.38219
\(202\) 8.44949 0.594504
\(203\) 0 0
\(204\) 4.89898 0.342997
\(205\) 0 0
\(206\) 3.10102 0.216058
\(207\) −20.6969 −1.43854
\(208\) 0.449490 0.0311665
\(209\) −31.5959 −2.18554
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −1.10102 −0.0756184
\(213\) 26.6969 1.82924
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.89898 0.196344
\(219\) −16.8990 −1.14193
\(220\) 0 0
\(221\) −0.898979 −0.0604719
\(222\) 4.89898 0.328798
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −0.202041 −0.0134396
\(227\) 7.34847 0.487735 0.243868 0.969809i \(-0.421584\pi\)
0.243868 + 0.969809i \(0.421584\pi\)
\(228\) 15.7980 1.04625
\(229\) −15.1464 −1.00090 −0.500452 0.865764i \(-0.666833\pi\)
−0.500452 + 0.865764i \(0.666833\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.89898 0.190327
\(233\) 10.2020 0.668358 0.334179 0.942510i \(-0.391541\pi\)
0.334179 + 0.942510i \(0.391541\pi\)
\(234\) −1.34847 −0.0881522
\(235\) 0 0
\(236\) 6.44949 0.419826
\(237\) 7.10102 0.461261
\(238\) 0 0
\(239\) −25.7980 −1.66873 −0.834366 0.551211i \(-0.814166\pi\)
−0.834366 + 0.551211i \(0.814166\pi\)
\(240\) 0 0
\(241\) −20.6969 −1.33321 −0.666604 0.745412i \(-0.732253\pi\)
−0.666604 + 0.745412i \(0.732253\pi\)
\(242\) −13.0000 −0.835672
\(243\) 22.0454 1.41421
\(244\) −8.44949 −0.540923
\(245\) 0 0
\(246\) 26.6969 1.70213
\(247\) −2.89898 −0.184458
\(248\) −0.898979 −0.0570853
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 1.55051 0.0978673 0.0489337 0.998802i \(-0.484418\pi\)
0.0489337 + 0.998802i \(0.484418\pi\)
\(252\) 0 0
\(253\) −33.7980 −2.12486
\(254\) 5.10102 0.320066
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.6969 −1.29104 −0.645520 0.763744i \(-0.723359\pi\)
−0.645520 + 0.763744i \(0.723359\pi\)
\(258\) 21.7980 1.35708
\(259\) 0 0
\(260\) 0 0
\(261\) −8.69694 −0.538327
\(262\) −1.55051 −0.0957908
\(263\) −9.79796 −0.604168 −0.302084 0.953281i \(-0.597682\pi\)
−0.302084 + 0.953281i \(0.597682\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) 0 0
\(267\) −24.4949 −1.49906
\(268\) −8.00000 −0.488678
\(269\) 15.1464 0.923494 0.461747 0.887012i \(-0.347223\pi\)
0.461747 + 0.887012i \(0.347223\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −17.7980 −1.07521
\(275\) 0 0
\(276\) 16.8990 1.01720
\(277\) −5.10102 −0.306491 −0.153245 0.988188i \(-0.548972\pi\)
−0.153245 + 0.988188i \(0.548972\pi\)
\(278\) 6.44949 0.386815
\(279\) 2.69694 0.161461
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 2.20204 0.131130
\(283\) −28.2474 −1.67914 −0.839568 0.543254i \(-0.817192\pi\)
−0.839568 + 0.543254i \(0.817192\pi\)
\(284\) −10.8990 −0.646735
\(285\) 0 0
\(286\) −2.20204 −0.130209
\(287\) 0 0
\(288\) −3.00000 −0.176777
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −9.30306 −0.545355
\(292\) 6.89898 0.403732
\(293\) 6.24745 0.364980 0.182490 0.983208i \(-0.441584\pi\)
0.182490 + 0.983208i \(0.441584\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −15.7980 −0.915151
\(299\) −3.10102 −0.179337
\(300\) 0 0
\(301\) 0 0
\(302\) 19.5959 1.12762
\(303\) 20.6969 1.18901
\(304\) −6.44949 −0.369904
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −4.24745 −0.242415 −0.121207 0.992627i \(-0.538677\pi\)
−0.121207 + 0.992627i \(0.538677\pi\)
\(308\) 0 0
\(309\) 7.59592 0.432117
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 1.10102 0.0623330
\(313\) −17.5959 −0.994580 −0.497290 0.867584i \(-0.665672\pi\)
−0.497290 + 0.867584i \(0.665672\pi\)
\(314\) 8.44949 0.476832
\(315\) 0 0
\(316\) −2.89898 −0.163080
\(317\) 26.4949 1.48810 0.744051 0.668123i \(-0.232902\pi\)
0.744051 + 0.668123i \(0.232902\pi\)
\(318\) −2.69694 −0.151237
\(319\) −14.2020 −0.795162
\(320\) 0 0
\(321\) 19.5959 1.09374
\(322\) 0 0
\(323\) 12.8990 0.717718
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 16.8990 0.935948
\(327\) 7.10102 0.392687
\(328\) −10.8990 −0.601795
\(329\) 0 0
\(330\) 0 0
\(331\) 10.6969 0.587957 0.293978 0.955812i \(-0.405021\pi\)
0.293978 + 0.955812i \(0.405021\pi\)
\(332\) −2.44949 −0.134433
\(333\) 6.00000 0.328798
\(334\) 4.89898 0.268060
\(335\) 0 0
\(336\) 0 0
\(337\) −29.5959 −1.61219 −0.806096 0.591785i \(-0.798424\pi\)
−0.806096 + 0.591785i \(0.798424\pi\)
\(338\) 12.7980 0.696117
\(339\) −0.494897 −0.0268791
\(340\) 0 0
\(341\) 4.40408 0.238494
\(342\) 19.3485 1.04625
\(343\) 0 0
\(344\) −8.89898 −0.479801
\(345\) 0 0
\(346\) 18.2474 0.980989
\(347\) 19.1010 1.02540 0.512698 0.858569i \(-0.328646\pi\)
0.512698 + 0.858569i \(0.328646\pi\)
\(348\) 7.10102 0.380655
\(349\) 3.55051 0.190054 0.0950272 0.995475i \(-0.469706\pi\)
0.0950272 + 0.995475i \(0.469706\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.89898 −0.261116
\(353\) −13.1010 −0.697297 −0.348648 0.937254i \(-0.613359\pi\)
−0.348648 + 0.937254i \(0.613359\pi\)
\(354\) 15.7980 0.839652
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 5.79796 0.306432
\(359\) 11.5959 0.612009 0.306005 0.952030i \(-0.401008\pi\)
0.306005 + 0.952030i \(0.401008\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) 14.2474 0.748829
\(363\) −31.8434 −1.67134
\(364\) 0 0
\(365\) 0 0
\(366\) −20.6969 −1.08185
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −6.89898 −0.359634
\(369\) 32.6969 1.70213
\(370\) 0 0
\(371\) 0 0
\(372\) −2.20204 −0.114171
\(373\) 24.6969 1.27876 0.639380 0.768891i \(-0.279191\pi\)
0.639380 + 0.768891i \(0.279191\pi\)
\(374\) 9.79796 0.506640
\(375\) 0 0
\(376\) −0.898979 −0.0463613
\(377\) −1.30306 −0.0671111
\(378\) 0 0
\(379\) 1.30306 0.0669338 0.0334669 0.999440i \(-0.489345\pi\)
0.0334669 + 0.999440i \(0.489345\pi\)
\(380\) 0 0
\(381\) 12.4949 0.640133
\(382\) 16.6969 0.854290
\(383\) 16.8990 0.863498 0.431749 0.901994i \(-0.357897\pi\)
0.431749 + 0.901994i \(0.357897\pi\)
\(384\) 2.44949 0.125000
\(385\) 0 0
\(386\) −17.5959 −0.895609
\(387\) 26.6969 1.35708
\(388\) 3.79796 0.192812
\(389\) 22.8990 1.16102 0.580512 0.814252i \(-0.302852\pi\)
0.580512 + 0.814252i \(0.302852\pi\)
\(390\) 0 0
\(391\) 13.7980 0.697793
\(392\) 0 0
\(393\) −3.79796 −0.191582
\(394\) −9.10102 −0.458503
\(395\) 0 0
\(396\) 14.6969 0.738549
\(397\) 17.3485 0.870695 0.435347 0.900263i \(-0.356626\pi\)
0.435347 + 0.900263i \(0.356626\pi\)
\(398\) 7.10102 0.355942
\(399\) 0 0
\(400\) 0 0
\(401\) 29.3939 1.46786 0.733930 0.679225i \(-0.237684\pi\)
0.733930 + 0.679225i \(0.237684\pi\)
\(402\) −19.5959 −0.977356
\(403\) 0.404082 0.0201288
\(404\) −8.44949 −0.420378
\(405\) 0 0
\(406\) 0 0
\(407\) 9.79796 0.485667
\(408\) −4.89898 −0.242536
\(409\) 14.4949 0.716727 0.358363 0.933582i \(-0.383335\pi\)
0.358363 + 0.933582i \(0.383335\pi\)
\(410\) 0 0
\(411\) −43.5959 −2.15043
\(412\) −3.10102 −0.152776
\(413\) 0 0
\(414\) 20.6969 1.01720
\(415\) 0 0
\(416\) −0.449490 −0.0220380
\(417\) 15.7980 0.773629
\(418\) 31.5959 1.54541
\(419\) 6.44949 0.315078 0.157539 0.987513i \(-0.449644\pi\)
0.157539 + 0.987513i \(0.449644\pi\)
\(420\) 0 0
\(421\) −23.7980 −1.15984 −0.579921 0.814673i \(-0.696917\pi\)
−0.579921 + 0.814673i \(0.696917\pi\)
\(422\) −12.0000 −0.584151
\(423\) 2.69694 0.131130
\(424\) 1.10102 0.0534703
\(425\) 0 0
\(426\) −26.6969 −1.29347
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) −5.39388 −0.260419
\(430\) 0 0
\(431\) 17.7980 0.857298 0.428649 0.903471i \(-0.358990\pi\)
0.428649 + 0.903471i \(0.358990\pi\)
\(432\) 0 0
\(433\) 19.7980 0.951429 0.475715 0.879600i \(-0.342190\pi\)
0.475715 + 0.879600i \(0.342190\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.89898 −0.138836
\(437\) 44.4949 2.12848
\(438\) 16.8990 0.807464
\(439\) −37.3939 −1.78471 −0.892356 0.451332i \(-0.850949\pi\)
−0.892356 + 0.451332i \(0.850949\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.898979 0.0427601
\(443\) −9.79796 −0.465515 −0.232758 0.972535i \(-0.574775\pi\)
−0.232758 + 0.972535i \(0.574775\pi\)
\(444\) −4.89898 −0.232495
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) −38.6969 −1.83030
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 53.3939 2.51422
\(452\) 0.202041 0.00950321
\(453\) 48.0000 2.25524
\(454\) −7.34847 −0.344881
\(455\) 0 0
\(456\) −15.7980 −0.739807
\(457\) −9.59592 −0.448878 −0.224439 0.974488i \(-0.572055\pi\)
−0.224439 + 0.974488i \(0.572055\pi\)
\(458\) 15.1464 0.707746
\(459\) 0 0
\(460\) 0 0
\(461\) −2.65153 −0.123494 −0.0617470 0.998092i \(-0.519667\pi\)
−0.0617470 + 0.998092i \(0.519667\pi\)
\(462\) 0 0
\(463\) −35.5959 −1.65428 −0.827141 0.561994i \(-0.810034\pi\)
−0.827141 + 0.561994i \(0.810034\pi\)
\(464\) −2.89898 −0.134582
\(465\) 0 0
\(466\) −10.2020 −0.472600
\(467\) −5.55051 −0.256847 −0.128423 0.991719i \(-0.540992\pi\)
−0.128423 + 0.991719i \(0.540992\pi\)
\(468\) 1.34847 0.0623330
\(469\) 0 0
\(470\) 0 0
\(471\) 20.6969 0.953665
\(472\) −6.44949 −0.296862
\(473\) 43.5959 2.00454
\(474\) −7.10102 −0.326161
\(475\) 0 0
\(476\) 0 0
\(477\) −3.30306 −0.151237
\(478\) 25.7980 1.17997
\(479\) −38.6969 −1.76811 −0.884054 0.467385i \(-0.845196\pi\)
−0.884054 + 0.467385i \(0.845196\pi\)
\(480\) 0 0
\(481\) 0.898979 0.0409899
\(482\) 20.6969 0.942720
\(483\) 0 0
\(484\) 13.0000 0.590909
\(485\) 0 0
\(486\) −22.0454 −1.00000
\(487\) −36.6969 −1.66290 −0.831449 0.555602i \(-0.812488\pi\)
−0.831449 + 0.555602i \(0.812488\pi\)
\(488\) 8.44949 0.382490
\(489\) 41.3939 1.87190
\(490\) 0 0
\(491\) −19.5959 −0.884351 −0.442176 0.896928i \(-0.645793\pi\)
−0.442176 + 0.896928i \(0.645793\pi\)
\(492\) −26.6969 −1.20359
\(493\) 5.79796 0.261127
\(494\) 2.89898 0.130431
\(495\) 0 0
\(496\) 0.898979 0.0403654
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 25.7980 1.15488 0.577438 0.816435i \(-0.304053\pi\)
0.577438 + 0.816435i \(0.304053\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) −1.55051 −0.0692027
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 33.7980 1.50250
\(507\) 31.3485 1.39223
\(508\) −5.10102 −0.226321
\(509\) 36.4495 1.61560 0.807798 0.589460i \(-0.200659\pi\)
0.807798 + 0.589460i \(0.200659\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 20.6969 0.912903
\(515\) 0 0
\(516\) −21.7980 −0.959602
\(517\) 4.40408 0.193691
\(518\) 0 0
\(519\) 44.6969 1.96198
\(520\) 0 0
\(521\) −3.30306 −0.144710 −0.0723549 0.997379i \(-0.523051\pi\)
−0.0723549 + 0.997379i \(0.523051\pi\)
\(522\) 8.69694 0.380655
\(523\) −1.14643 −0.0501298 −0.0250649 0.999686i \(-0.507979\pi\)
−0.0250649 + 0.999686i \(0.507979\pi\)
\(524\) 1.55051 0.0677344
\(525\) 0 0
\(526\) 9.79796 0.427211
\(527\) −1.79796 −0.0783203
\(528\) −12.0000 −0.522233
\(529\) 24.5959 1.06939
\(530\) 0 0
\(531\) 19.3485 0.839652
\(532\) 0 0
\(533\) 4.89898 0.212198
\(534\) 24.4949 1.06000
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 14.2020 0.612863
\(538\) −15.1464 −0.653009
\(539\) 0 0
\(540\) 0 0
\(541\) −29.5959 −1.27243 −0.636214 0.771513i \(-0.719500\pi\)
−0.636214 + 0.771513i \(0.719500\pi\)
\(542\) 12.0000 0.515444
\(543\) 34.8990 1.49766
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 10.6969 0.457368 0.228684 0.973501i \(-0.426558\pi\)
0.228684 + 0.973501i \(0.426558\pi\)
\(548\) 17.7980 0.760291
\(549\) −25.3485 −1.08185
\(550\) 0 0
\(551\) 18.6969 0.796516
\(552\) −16.8990 −0.719268
\(553\) 0 0
\(554\) 5.10102 0.216722
\(555\) 0 0
\(556\) −6.44949 −0.273519
\(557\) −16.6969 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(558\) −2.69694 −0.114171
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 18.0000 0.759284
\(563\) −14.0454 −0.591943 −0.295972 0.955197i \(-0.595643\pi\)
−0.295972 + 0.955197i \(0.595643\pi\)
\(564\) −2.20204 −0.0927227
\(565\) 0 0
\(566\) 28.2474 1.18733
\(567\) 0 0
\(568\) 10.8990 0.457311
\(569\) 14.2020 0.595381 0.297690 0.954663i \(-0.403784\pi\)
0.297690 + 0.954663i \(0.403784\pi\)
\(570\) 0 0
\(571\) −20.8990 −0.874595 −0.437298 0.899317i \(-0.644064\pi\)
−0.437298 + 0.899317i \(0.644064\pi\)
\(572\) 2.20204 0.0920720
\(573\) 40.8990 1.70858
\(574\) 0 0
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) −46.4949 −1.93561 −0.967804 0.251705i \(-0.919009\pi\)
−0.967804 + 0.251705i \(0.919009\pi\)
\(578\) 13.0000 0.540729
\(579\) −43.1010 −1.79122
\(580\) 0 0
\(581\) 0 0
\(582\) 9.30306 0.385624
\(583\) −5.39388 −0.223392
\(584\) −6.89898 −0.285482
\(585\) 0 0
\(586\) −6.24745 −0.258080
\(587\) 33.1464 1.36810 0.684050 0.729435i \(-0.260217\pi\)
0.684050 + 0.729435i \(0.260217\pi\)
\(588\) 0 0
\(589\) −5.79796 −0.238901
\(590\) 0 0
\(591\) −22.2929 −0.917006
\(592\) 2.00000 0.0821995
\(593\) 1.10102 0.0452135 0.0226067 0.999744i \(-0.492803\pi\)
0.0226067 + 0.999744i \(0.492803\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.7980 0.647110
\(597\) 17.3939 0.711884
\(598\) 3.10102 0.126810
\(599\) −22.8990 −0.935627 −0.467813 0.883827i \(-0.654958\pi\)
−0.467813 + 0.883827i \(0.654958\pi\)
\(600\) 0 0
\(601\) −19.3939 −0.791093 −0.395546 0.918446i \(-0.629445\pi\)
−0.395546 + 0.918446i \(0.629445\pi\)
\(602\) 0 0
\(603\) −24.0000 −0.977356
\(604\) −19.5959 −0.797347
\(605\) 0 0
\(606\) −20.6969 −0.840756
\(607\) 25.3939 1.03071 0.515353 0.856978i \(-0.327661\pi\)
0.515353 + 0.856978i \(0.327661\pi\)
\(608\) 6.44949 0.261561
\(609\) 0 0
\(610\) 0 0
\(611\) 0.404082 0.0163474
\(612\) −6.00000 −0.242536
\(613\) −8.20204 −0.331277 −0.165639 0.986187i \(-0.552969\pi\)
−0.165639 + 0.986187i \(0.552969\pi\)
\(614\) 4.24745 0.171413
\(615\) 0 0
\(616\) 0 0
\(617\) −9.59592 −0.386317 −0.193159 0.981168i \(-0.561873\pi\)
−0.193159 + 0.981168i \(0.561873\pi\)
\(618\) −7.59592 −0.305553
\(619\) 46.4495 1.86696 0.933481 0.358626i \(-0.116755\pi\)
0.933481 + 0.358626i \(0.116755\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) −1.10102 −0.0440761
\(625\) 0 0
\(626\) 17.5959 0.703274
\(627\) 77.3939 3.09081
\(628\) −8.44949 −0.337171
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 6.49490 0.258558 0.129279 0.991608i \(-0.458734\pi\)
0.129279 + 0.991608i \(0.458734\pi\)
\(632\) 2.89898 0.115315
\(633\) −29.3939 −1.16830
\(634\) −26.4949 −1.05225
\(635\) 0 0
\(636\) 2.69694 0.106941
\(637\) 0 0
\(638\) 14.2020 0.562264
\(639\) −32.6969 −1.29347
\(640\) 0 0
\(641\) 6.20204 0.244966 0.122483 0.992471i \(-0.460914\pi\)
0.122483 + 0.992471i \(0.460914\pi\)
\(642\) −19.5959 −0.773389
\(643\) 9.14643 0.360700 0.180350 0.983603i \(-0.442277\pi\)
0.180350 + 0.983603i \(0.442277\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.8990 −0.507504
\(647\) −22.2929 −0.876423 −0.438211 0.898872i \(-0.644388\pi\)
−0.438211 + 0.898872i \(0.644388\pi\)
\(648\) 9.00000 0.353553
\(649\) 31.5959 1.24025
\(650\) 0 0
\(651\) 0 0
\(652\) −16.8990 −0.661815
\(653\) −39.7980 −1.55741 −0.778707 0.627387i \(-0.784124\pi\)
−0.778707 + 0.627387i \(0.784124\pi\)
\(654\) −7.10102 −0.277672
\(655\) 0 0
\(656\) 10.8990 0.425534
\(657\) 20.6969 0.807464
\(658\) 0 0
\(659\) 7.10102 0.276616 0.138308 0.990389i \(-0.455834\pi\)
0.138308 + 0.990389i \(0.455834\pi\)
\(660\) 0 0
\(661\) −12.9444 −0.503478 −0.251739 0.967795i \(-0.581003\pi\)
−0.251739 + 0.967795i \(0.581003\pi\)
\(662\) −10.6969 −0.415748
\(663\) 2.20204 0.0855202
\(664\) 2.44949 0.0950586
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 20.0000 0.774403
\(668\) −4.89898 −0.189547
\(669\) −9.79796 −0.378811
\(670\) 0 0
\(671\) −41.3939 −1.59799
\(672\) 0 0
\(673\) 1.79796 0.0693062 0.0346531 0.999399i \(-0.488967\pi\)
0.0346531 + 0.999399i \(0.488967\pi\)
\(674\) 29.5959 1.13999
\(675\) 0 0
\(676\) −12.7980 −0.492229
\(677\) 31.5505 1.21258 0.606292 0.795242i \(-0.292656\pi\)
0.606292 + 0.795242i \(0.292656\pi\)
\(678\) 0.494897 0.0190064
\(679\) 0 0
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) −4.40408 −0.168641
\(683\) −35.5959 −1.36204 −0.681020 0.732265i \(-0.738463\pi\)
−0.681020 + 0.732265i \(0.738463\pi\)
\(684\) −19.3485 −0.739807
\(685\) 0 0
\(686\) 0 0
\(687\) 37.1010 1.41549
\(688\) 8.89898 0.339270
\(689\) −0.494897 −0.0188541
\(690\) 0 0
\(691\) 13.1464 0.500114 0.250057 0.968231i \(-0.419551\pi\)
0.250057 + 0.968231i \(0.419551\pi\)
\(692\) −18.2474 −0.693664
\(693\) 0 0
\(694\) −19.1010 −0.725065
\(695\) 0 0
\(696\) −7.10102 −0.269163
\(697\) −21.7980 −0.825657
\(698\) −3.55051 −0.134389
\(699\) −24.9898 −0.945201
\(700\) 0 0
\(701\) 40.6969 1.53710 0.768551 0.639788i \(-0.220978\pi\)
0.768551 + 0.639788i \(0.220978\pi\)
\(702\) 0 0
\(703\) −12.8990 −0.486494
\(704\) 4.89898 0.184637
\(705\) 0 0
\(706\) 13.1010 0.493063
\(707\) 0 0
\(708\) −15.7980 −0.593724
\(709\) 40.2929 1.51323 0.756615 0.653861i \(-0.226852\pi\)
0.756615 + 0.653861i \(0.226852\pi\)
\(710\) 0 0
\(711\) −8.69694 −0.326161
\(712\) −10.0000 −0.374766
\(713\) −6.20204 −0.232268
\(714\) 0 0
\(715\) 0 0
\(716\) −5.79796 −0.216680
\(717\) 63.1918 2.35994
\(718\) −11.5959 −0.432756
\(719\) −44.4949 −1.65938 −0.829690 0.558225i \(-0.811483\pi\)
−0.829690 + 0.558225i \(0.811483\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −22.5959 −0.840933
\(723\) 50.6969 1.88544
\(724\) −14.2474 −0.529502
\(725\) 0 0
\(726\) 31.8434 1.18182
\(727\) 6.69694 0.248376 0.124188 0.992259i \(-0.460367\pi\)
0.124188 + 0.992259i \(0.460367\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −17.7980 −0.658281
\(732\) 20.6969 0.764981
\(733\) 43.6413 1.61193 0.805965 0.591964i \(-0.201647\pi\)
0.805965 + 0.591964i \(0.201647\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 6.89898 0.254300
\(737\) −39.1918 −1.44365
\(738\) −32.6969 −1.20359
\(739\) −44.4949 −1.63677 −0.818386 0.574669i \(-0.805131\pi\)
−0.818386 + 0.574669i \(0.805131\pi\)
\(740\) 0 0
\(741\) 7.10102 0.260863
\(742\) 0 0
\(743\) −15.3031 −0.561415 −0.280707 0.959793i \(-0.590569\pi\)
−0.280707 + 0.959793i \(0.590569\pi\)
\(744\) 2.20204 0.0807307
\(745\) 0 0
\(746\) −24.6969 −0.904219
\(747\) −7.34847 −0.268866
\(748\) −9.79796 −0.358249
\(749\) 0 0
\(750\) 0 0
\(751\) −22.2020 −0.810164 −0.405082 0.914280i \(-0.632757\pi\)
−0.405082 + 0.914280i \(0.632757\pi\)
\(752\) 0.898979 0.0327824
\(753\) −3.79796 −0.138405
\(754\) 1.30306 0.0474547
\(755\) 0 0
\(756\) 0 0
\(757\) −32.2020 −1.17040 −0.585202 0.810888i \(-0.698985\pi\)
−0.585202 + 0.810888i \(0.698985\pi\)
\(758\) −1.30306 −0.0473293
\(759\) 82.7878 3.00501
\(760\) 0 0
\(761\) 30.8990 1.12009 0.560044 0.828463i \(-0.310784\pi\)
0.560044 + 0.828463i \(0.310784\pi\)
\(762\) −12.4949 −0.452642
\(763\) 0 0
\(764\) −16.6969 −0.604074
\(765\) 0 0
\(766\) −16.8990 −0.610585
\(767\) 2.89898 0.104676
\(768\) −2.44949 −0.0883883
\(769\) 11.3031 0.407599 0.203799 0.979013i \(-0.434671\pi\)
0.203799 + 0.979013i \(0.434671\pi\)
\(770\) 0 0
\(771\) 50.6969 1.82581
\(772\) 17.5959 0.633291
\(773\) 13.3485 0.480111 0.240056 0.970759i \(-0.422834\pi\)
0.240056 + 0.970759i \(0.422834\pi\)
\(774\) −26.6969 −0.959602
\(775\) 0 0
\(776\) −3.79796 −0.136339
\(777\) 0 0
\(778\) −22.8990 −0.820968
\(779\) −70.2929 −2.51850
\(780\) 0 0
\(781\) −53.3939 −1.91058
\(782\) −13.7980 −0.493414
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 3.79796 0.135469
\(787\) −45.5505 −1.62370 −0.811850 0.583866i \(-0.801539\pi\)
−0.811850 + 0.583866i \(0.801539\pi\)
\(788\) 9.10102 0.324210
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) −14.6969 −0.522233
\(793\) −3.79796 −0.134869
\(794\) −17.3485 −0.615674
\(795\) 0 0
\(796\) −7.10102 −0.251689
\(797\) −52.9444 −1.87539 −0.937693 0.347464i \(-0.887043\pi\)
−0.937693 + 0.347464i \(0.887043\pi\)
\(798\) 0 0
\(799\) −1.79796 −0.0636072
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) −29.3939 −1.03793
\(803\) 33.7980 1.19270
\(804\) 19.5959 0.691095
\(805\) 0 0
\(806\) −0.404082 −0.0142332
\(807\) −37.1010 −1.30602
\(808\) 8.44949 0.297252
\(809\) 8.40408 0.295472 0.147736 0.989027i \(-0.452801\pi\)
0.147736 + 0.989027i \(0.452801\pi\)
\(810\) 0 0
\(811\) 38.9444 1.36752 0.683761 0.729706i \(-0.260343\pi\)
0.683761 + 0.729706i \(0.260343\pi\)
\(812\) 0 0
\(813\) 29.3939 1.03089
\(814\) −9.79796 −0.343418
\(815\) 0 0
\(816\) 4.89898 0.171499
\(817\) −57.3939 −2.00796
\(818\) −14.4949 −0.506802
\(819\) 0 0
\(820\) 0 0
\(821\) 27.7980 0.970155 0.485078 0.874471i \(-0.338791\pi\)
0.485078 + 0.874471i \(0.338791\pi\)
\(822\) 43.5959 1.52058
\(823\) 39.1918 1.36614 0.683071 0.730352i \(-0.260644\pi\)
0.683071 + 0.730352i \(0.260644\pi\)
\(824\) 3.10102 0.108029
\(825\) 0 0
\(826\) 0 0
\(827\) 23.5959 0.820510 0.410255 0.911971i \(-0.365440\pi\)
0.410255 + 0.911971i \(0.365440\pi\)
\(828\) −20.6969 −0.719268
\(829\) −39.6413 −1.37680 −0.688400 0.725331i \(-0.741687\pi\)
−0.688400 + 0.725331i \(0.741687\pi\)
\(830\) 0 0
\(831\) 12.4949 0.433443
\(832\) 0.449490 0.0155833
\(833\) 0 0
\(834\) −15.7980 −0.547039
\(835\) 0 0
\(836\) −31.5959 −1.09277
\(837\) 0 0
\(838\) −6.44949 −0.222794
\(839\) −27.1010 −0.935631 −0.467816 0.883826i \(-0.654959\pi\)
−0.467816 + 0.883826i \(0.654959\pi\)
\(840\) 0 0
\(841\) −20.5959 −0.710204
\(842\) 23.7980 0.820132
\(843\) 44.0908 1.51857
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −2.69694 −0.0927227
\(847\) 0 0
\(848\) −1.10102 −0.0378092
\(849\) 69.1918 2.37466
\(850\) 0 0
\(851\) −13.7980 −0.472988
\(852\) 26.6969 0.914622
\(853\) −29.8434 −1.02182 −0.510909 0.859635i \(-0.670691\pi\)
−0.510909 + 0.859635i \(0.670691\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −25.1918 −0.860537 −0.430268 0.902701i \(-0.641581\pi\)
−0.430268 + 0.902701i \(0.641581\pi\)
\(858\) 5.39388 0.184144
\(859\) 29.6413 1.01135 0.505674 0.862724i \(-0.331244\pi\)
0.505674 + 0.862724i \(0.331244\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −17.7980 −0.606201
\(863\) 13.3939 0.455933 0.227966 0.973669i \(-0.426792\pi\)
0.227966 + 0.973669i \(0.426792\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −19.7980 −0.672762
\(867\) 31.8434 1.08146
\(868\) 0 0
\(869\) −14.2020 −0.481771
\(870\) 0 0
\(871\) −3.59592 −0.121843
\(872\) 2.89898 0.0981718
\(873\) 11.3939 0.385624
\(874\) −44.4949 −1.50506
\(875\) 0 0
\(876\) −16.8990 −0.570964
\(877\) 19.3939 0.654885 0.327442 0.944871i \(-0.393813\pi\)
0.327442 + 0.944871i \(0.393813\pi\)
\(878\) 37.3939 1.26198
\(879\) −15.3031 −0.516159
\(880\) 0 0
\(881\) −27.7980 −0.936537 −0.468269 0.883586i \(-0.655122\pi\)
−0.468269 + 0.883586i \(0.655122\pi\)
\(882\) 0 0
\(883\) 41.7980 1.40661 0.703307 0.710887i \(-0.251706\pi\)
0.703307 + 0.710887i \(0.251706\pi\)
\(884\) −0.898979 −0.0302360
\(885\) 0 0
\(886\) 9.79796 0.329169
\(887\) 26.6969 0.896395 0.448198 0.893934i \(-0.352066\pi\)
0.448198 + 0.893934i \(0.352066\pi\)
\(888\) 4.89898 0.164399
\(889\) 0 0
\(890\) 0 0
\(891\) −44.0908 −1.47710
\(892\) 4.00000 0.133930
\(893\) −5.79796 −0.194021
\(894\) 38.6969 1.29422
\(895\) 0 0
\(896\) 0 0
\(897\) 7.59592 0.253620
\(898\) 10.0000 0.333704
\(899\) −2.60612 −0.0869191
\(900\) 0 0
\(901\) 2.20204 0.0733606
\(902\) −53.3939 −1.77782
\(903\) 0 0
\(904\) −0.202041 −0.00671978
\(905\) 0 0
\(906\) −48.0000 −1.59469
\(907\) −22.2020 −0.737207 −0.368603 0.929587i \(-0.620164\pi\)
−0.368603 + 0.929587i \(0.620164\pi\)
\(908\) 7.34847 0.243868
\(909\) −25.3485 −0.840756
\(910\) 0 0
\(911\) 3.59592 0.119138 0.0595690 0.998224i \(-0.481027\pi\)
0.0595690 + 0.998224i \(0.481027\pi\)
\(912\) 15.7980 0.523123
\(913\) −12.0000 −0.397142
\(914\) 9.59592 0.317405
\(915\) 0 0
\(916\) −15.1464 −0.500452
\(917\) 0 0
\(918\) 0 0
\(919\) −17.1010 −0.564111 −0.282055 0.959398i \(-0.591016\pi\)
−0.282055 + 0.959398i \(0.591016\pi\)
\(920\) 0 0
\(921\) 10.4041 0.342826
\(922\) 2.65153 0.0873235
\(923\) −4.89898 −0.161252
\(924\) 0 0
\(925\) 0 0
\(926\) 35.5959 1.16975
\(927\) −9.30306 −0.305553
\(928\) 2.89898 0.0951637
\(929\) −40.2929 −1.32197 −0.660983 0.750401i \(-0.729860\pi\)
−0.660983 + 0.750401i \(0.729860\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.2020 0.334179
\(933\) 29.3939 0.962312
\(934\) 5.55051 0.181618
\(935\) 0 0
\(936\) −1.34847 −0.0440761
\(937\) 50.8990 1.66280 0.831399 0.555677i \(-0.187541\pi\)
0.831399 + 0.555677i \(0.187541\pi\)
\(938\) 0 0
\(939\) 43.1010 1.40655
\(940\) 0 0
\(941\) 24.4495 0.797031 0.398515 0.917162i \(-0.369526\pi\)
0.398515 + 0.917162i \(0.369526\pi\)
\(942\) −20.6969 −0.674343
\(943\) −75.1918 −2.44858
\(944\) 6.44949 0.209913
\(945\) 0 0
\(946\) −43.5959 −1.41743
\(947\) −44.0908 −1.43276 −0.716379 0.697711i \(-0.754202\pi\)
−0.716379 + 0.697711i \(0.754202\pi\)
\(948\) 7.10102 0.230630
\(949\) 3.10102 0.100663
\(950\) 0 0
\(951\) −64.8990 −2.10449
\(952\) 0 0
\(953\) 21.7980 0.706105 0.353053 0.935603i \(-0.385144\pi\)
0.353053 + 0.935603i \(0.385144\pi\)
\(954\) 3.30306 0.106941
\(955\) 0 0
\(956\) −25.7980 −0.834366
\(957\) 34.7878 1.12453
\(958\) 38.6969 1.25024
\(959\) 0 0
\(960\) 0 0
\(961\) −30.1918 −0.973930
\(962\) −0.898979 −0.0289843
\(963\) −24.0000 −0.773389
\(964\) −20.6969 −0.666604
\(965\) 0 0
\(966\) 0 0
\(967\) 32.2929 1.03847 0.519234 0.854632i \(-0.326217\pi\)
0.519234 + 0.854632i \(0.326217\pi\)
\(968\) −13.0000 −0.417836
\(969\) −31.5959 −1.01501
\(970\) 0 0
\(971\) 14.4495 0.463706 0.231853 0.972751i \(-0.425521\pi\)
0.231853 + 0.972751i \(0.425521\pi\)
\(972\) 22.0454 0.707107
\(973\) 0 0
\(974\) 36.6969 1.17585
\(975\) 0 0
\(976\) −8.44949 −0.270462
\(977\) 29.3939 0.940393 0.470197 0.882562i \(-0.344183\pi\)
0.470197 + 0.882562i \(0.344183\pi\)
\(978\) −41.3939 −1.32363
\(979\) 48.9898 1.56572
\(980\) 0 0
\(981\) −8.69694 −0.277672
\(982\) 19.5959 0.625331
\(983\) 42.6969 1.36182 0.680910 0.732367i \(-0.261584\pi\)
0.680910 + 0.732367i \(0.261584\pi\)
\(984\) 26.6969 0.851067
\(985\) 0 0
\(986\) −5.79796 −0.184645
\(987\) 0 0
\(988\) −2.89898 −0.0922288
\(989\) −61.3939 −1.95221
\(990\) 0 0
\(991\) 60.6969 1.92810 0.964051 0.265718i \(-0.0856089\pi\)
0.964051 + 0.265718i \(0.0856089\pi\)
\(992\) −0.898979 −0.0285426
\(993\) −26.2020 −0.831497
\(994\) 0 0
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −42.6515 −1.35079 −0.675394 0.737457i \(-0.736026\pi\)
−0.675394 + 0.737457i \(0.736026\pi\)
\(998\) −25.7980 −0.816620
\(999\) 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 2450.2.a.bl.1.1 2
5.2 odd 4 490.2.c.e.99.2 4
5.3 odd 4 490.2.c.e.99.3 4
5.4 even 2 2450.2.a.bq.1.2 2
7.6 odd 2 350.2.a.g.1.2 2
21.20 even 2 3150.2.a.bt.1.1 2
28.27 even 2 2800.2.a.bm.1.1 2
35.2 odd 12 490.2.i.f.459.1 8
35.3 even 12 490.2.i.c.79.2 8
35.12 even 12 490.2.i.c.459.2 8
35.13 even 4 70.2.c.a.29.4 yes 4
35.17 even 12 490.2.i.c.79.3 8
35.18 odd 12 490.2.i.f.79.1 8
35.23 odd 12 490.2.i.f.459.4 8
35.27 even 4 70.2.c.a.29.1 4
35.32 odd 12 490.2.i.f.79.4 8
35.33 even 12 490.2.i.c.459.3 8
35.34 odd 2 350.2.a.h.1.1 2
105.62 odd 4 630.2.g.g.379.4 4
105.83 odd 4 630.2.g.g.379.2 4
105.104 even 2 3150.2.a.bs.1.1 2
140.27 odd 4 560.2.g.e.449.3 4
140.83 odd 4 560.2.g.e.449.1 4
140.139 even 2 2800.2.a.bl.1.2 2
280.13 even 4 2240.2.g.j.449.2 4
280.27 odd 4 2240.2.g.i.449.2 4
280.83 odd 4 2240.2.g.i.449.4 4
280.237 even 4 2240.2.g.j.449.4 4
420.83 even 4 5040.2.t.t.1009.4 4
420.167 even 4 5040.2.t.t.1009.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.1 4 35.27 even 4
70.2.c.a.29.4 yes 4 35.13 even 4
350.2.a.g.1.2 2 7.6 odd 2
350.2.a.h.1.1 2 35.34 odd 2
490.2.c.e.99.2 4 5.2 odd 4
490.2.c.e.99.3 4 5.3 odd 4
490.2.i.c.79.2 8 35.3 even 12
490.2.i.c.79.3 8 35.17 even 12
490.2.i.c.459.2 8 35.12 even 12
490.2.i.c.459.3 8 35.33 even 12
490.2.i.f.79.1 8 35.18 odd 12
490.2.i.f.79.4 8 35.32 odd 12
490.2.i.f.459.1 8 35.2 odd 12
490.2.i.f.459.4 8 35.23 odd 12
560.2.g.e.449.1 4 140.83 odd 4
560.2.g.e.449.3 4 140.27 odd 4
630.2.g.g.379.2 4 105.83 odd 4
630.2.g.g.379.4 4 105.62 odd 4
2240.2.g.i.449.2 4 280.27 odd 4
2240.2.g.i.449.4 4 280.83 odd 4
2240.2.g.j.449.2 4 280.13 even 4
2240.2.g.j.449.4 4 280.237 even 4
2450.2.a.bl.1.1 2 1.1 even 1 trivial
2450.2.a.bq.1.2 2 5.4 even 2
2800.2.a.bl.1.2 2 140.139 even 2
2800.2.a.bm.1.1 2 28.27 even 2
3150.2.a.bs.1.1 2 105.104 even 2
3150.2.a.bt.1.1 2 21.20 even 2
5040.2.t.t.1009.3 4 420.167 even 4
5040.2.t.t.1009.4 4 420.83 even 4