Properties

Label 1550.2.b.e.249.2
Level $1550$
Weight $2$
Character 1550.249
Analytic conductor $12.377$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1550,2,Mod(249,1550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1550.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1550.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(12.3768123133\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 310)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 249.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1550.249
Dual form 1550.2.b.e.249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{12} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000i q^{22} +4.00000i q^{23} -2.00000 q^{24} -4.00000i q^{27} +4.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -2.00000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -2.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} -8.00000i q^{53} +4.00000 q^{54} -8.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} -1.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} +1.00000i q^{72} -6.00000i q^{73} +8.00000 q^{74} -4.00000 q^{76} +4.00000 q^{79} -11.0000 q^{81} +6.00000i q^{82} -6.00000i q^{83} +2.00000 q^{86} -8.00000i q^{87} -2.00000i q^{88} +6.00000 q^{89} -4.00000i q^{92} +2.00000i q^{93} +2.00000 q^{96} -2.00000i q^{97} +7.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{11} + 2 q^{16} + 8 q^{19} - 4 q^{24} + 8 q^{29} - 2 q^{31} - 4 q^{34} + 2 q^{36} + 12 q^{41} - 4 q^{44} - 8 q^{46} + 14 q^{49} + 8 q^{51} + 8 q^{54} - 16 q^{59} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 16 q^{74} - 8 q^{76} + 8 q^{79} - 22 q^{81} + 4 q^{86} + 12 q^{89} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1550\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(1427\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000i 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) − 8.00000i − 1.09888i −0.835532 0.549442i \(-0.814840\pi\)
0.835532 0.549442i \(-0.185160\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) − 8.00000i − 1.05963i
\(58\) 4.00000i 0.525226i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000i 0.662589i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) − 8.00000i − 0.857690i
\(88\) − 2.00000i − 0.213201i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 7.00000i 0.707107i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 4.00000i 0.396059i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −16.0000 −1.51865
\(112\) 0 0
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) − 8.00000i − 0.736460i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) − 12.0000i − 1.08200i
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) − 14.0000i − 1.15470i
\(148\) 8.00000i 0.657596i
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 4.00000i 0.318223i
\(159\) −16.0000 −1.26888
\(160\) 0 0
\(161\) 0 0
\(162\) − 11.0000i − 0.864242i
\(163\) − 24.0000i − 1.87983i −0.341415 0.939913i \(-0.610906\pi\)
0.341415 0.939913i \(-0.389094\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 2.00000i 0.152499i
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 16.0000i 1.20263i
\(178\) 6.00000i 0.449719i
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) −24.0000 −1.78391 −0.891953 0.452128i \(-0.850665\pi\)
−0.891953 + 0.452128i \(0.850665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 2.00000i 0.140720i
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) − 4.00000i − 0.278019i
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 8.00000i 0.549442i
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 18.0000i 1.21911i
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) − 16.0000i − 1.07385i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 8.00000i 0.529813i
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 4.00000i − 0.262613i
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 1.00000i 0.0635001i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0000i 0.623783i 0.950118 + 0.311891i \(0.100963\pi\)
−0.950118 + 0.311891i \(0.899037\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 20.0000i 1.23560i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 12.0000i − 0.734388i
\(268\) − 4.00000i − 0.244339i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) − 2.00000i − 0.119952i
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 24.0000i 1.42665i 0.700832 + 0.713326i \(0.252812\pi\)
−0.700832 + 0.713326i \(0.747188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 6.00000i 0.351123i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) 14.0000 0.816497
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) − 8.00000i − 0.464207i
\(298\) 14.0000i 0.810998i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 4.00000i 0.230174i
\(303\) − 4.00000i − 0.229794i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 16.0000i − 0.897235i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −16.0000 −0.893033
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) − 36.0000i − 1.99080i
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −30.0000 −1.64895 −0.824475 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 8.00000i 0.438397i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 13.0000i 0.707107i
\(339\) −28.0000 −1.52075
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) − 22.0000i − 1.18102i −0.807030 0.590511i \(-0.798926\pi\)
0.807030 0.590511i \(-0.201074\pi\)
\(348\) 8.00000i 0.428845i
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000i 0.106600i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) −16.0000 −0.850390
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) − 14.0000i − 0.739923i
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 24.0000i − 1.26141i
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) − 2.00000i − 0.103695i
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) − 8.00000i − 0.409316i
\(383\) − 16.0000i − 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 2.00000i 0.101666i
\(388\) 2.00000i 0.101535i
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) − 7.00000i − 0.353553i
\(393\) − 40.0000i − 2.01773i
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 38.0000 1.89763 0.948815 0.315833i \(-0.102284\pi\)
0.948815 + 0.315833i \(0.102284\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.0000i − 0.793091i
\(408\) − 4.00000i − 0.198030i
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 8.00000i 0.394132i
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000i 0.195881i
\(418\) 8.00000i 0.391293i
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) 0 0
\(424\) −8.00000 −0.388514
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000i 0.386695i
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 16.0000i 0.765384i
\(438\) − 12.0000i − 0.573382i
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 28.0000i 1.33032i 0.746701 + 0.665160i \(0.231637\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(444\) 16.0000 0.759326
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) − 28.0000i − 1.32435i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 14.0000i 0.658505i
\(453\) − 8.00000i − 0.375873i
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) − 4.00000i − 0.186908i
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 0 0
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) 8.00000i 0.368230i
\(473\) − 4.00000i − 0.183920i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) 8.00000i 0.366295i
\(478\) − 12.0000i − 0.548867i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 2.00000i − 0.0910975i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 0 0
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 12.0000i 0.541002i
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) − 12.0000i − 0.537733i
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 14.0000i 0.624851i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) − 26.0000i − 1.15470i
\(508\) − 16.0000i − 0.709885i
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) − 16.0000i − 0.706417i
\(514\) −10.0000 −0.441081
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) − 38.0000i − 1.66162i −0.556553 0.830812i \(-0.687876\pi\)
0.556553 0.830812i \(-0.312124\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) − 2.00000i − 0.0871214i
\(528\) − 4.00000i − 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 28.0000i 1.20829i
\(538\) 0 0
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 48.0000i 2.05988i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 32.0000i − 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) − 18.0000i − 0.768922i
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) − 8.00000i − 0.340503i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 24.0000i 1.01691i 0.861088 + 0.508456i \(0.169784\pi\)
−0.861088 + 0.508456i \(0.830216\pi\)
\(558\) 1.00000i 0.0423334i
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 10.0000i 0.421825i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000i 1.24892i 0.781058 + 0.624458i \(0.214680\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(578\) 13.0000i 0.540729i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 0 0
\(582\) − 4.00000i − 0.165805i
\(583\) − 16.0000i − 0.662652i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) − 2.00000i − 0.0825488i −0.999148 0.0412744i \(-0.986858\pi\)
0.999148 0.0412744i \(-0.0131418\pi\)
\(588\) 14.0000i 0.577350i
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) − 8.00000i − 0.328798i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 8.00000 0.328244
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 32.0000i 1.30967i
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) − 24.0000i − 0.974130i −0.873366 0.487065i \(-0.838067\pi\)
0.873366 0.487065i \(-0.161933\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000i 0.0808452i
\(613\) 44.0000i 1.77714i 0.458738 + 0.888572i \(0.348302\pi\)
−0.458738 + 0.888572i \(0.651698\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) − 38.0000i − 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 8.00000i 0.320771i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) − 16.0000i − 0.638978i
\(628\) − 6.00000i − 0.239426i
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) − 4.00000i − 0.159111i
\(633\) 32.0000i 1.27189i
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 16.0000 0.634441
\(637\) 0 0
\(638\) 8.00000i 0.316723i
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) − 16.0000i − 0.631470i
\(643\) 2.00000i 0.0788723i 0.999222 + 0.0394362i \(0.0125562\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 11.0000i 0.432121i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000i 0.939913i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 36.0000 1.40771
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 30.0000i − 1.16598i
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 16.0000i 0.619522i
\(668\) − 12.0000i − 0.464294i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) − 48.0000i − 1.84479i −0.386248 0.922395i \(-0.626229\pi\)
0.386248 0.922395i \(-0.373771\pi\)
\(678\) − 28.0000i − 1.07533i
\(679\) 0 0
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) − 2.00000i − 0.0765840i
\(683\) 48.0000i 1.83667i 0.395805 + 0.918334i \(0.370466\pi\)
−0.395805 + 0.918334i \(0.629534\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) − 2.00000i − 0.0762493i
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) − 2.00000i − 0.0760286i
\(693\) 0 0
\(694\) 22.0000 0.835109
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) 12.0000i 0.454532i
\(698\) − 10.0000i − 0.378506i
\(699\) 52.0000 1.96682
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) − 32.0000i − 1.20690i
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) − 16.0000i − 0.601317i
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) − 6.00000i − 0.224860i
\(713\) − 4.00000i − 0.149801i
\(714\) 0 0
\(715\) 0 0
\(716\) 14.0000 0.523205
\(717\) 24.0000i 0.896296i
\(718\) 8.00000i 0.298557i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) 4.00000i 0.148762i
\(724\) 24.0000 0.891953
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) 40.0000i 1.48352i 0.670667 + 0.741759i \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 8.00000i 0.294684i
\(738\) − 6.00000i − 0.220863i
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 6.00000i 0.219529i
\(748\) − 4.00000i − 0.146254i
\(749\) 0 0
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) − 28.0000i − 1.02038i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.0000i 0.436147i 0.975932 + 0.218074i \(0.0699773\pi\)
−0.975932 + 0.218074i \(0.930023\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 32.0000i 1.15924i
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) − 2.00000i − 0.0721688i
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 20.0000 0.720282
\(772\) 2.00000i 0.0719816i
\(773\) 28.0000i 1.00709i 0.863969 + 0.503545i \(0.167971\pi\)
−0.863969 + 0.503545i \(0.832029\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 36.0000i 1.29066i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) − 8.00000i − 0.286079i
\(783\) − 16.0000i − 0.571793i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 40.0000 1.42675
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) − 8.00000i − 0.284988i
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000i 0.0710669i
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 24.0000i 0.850124i 0.905164 + 0.425062i \(0.139748\pi\)
−0.905164 + 0.425062i \(0.860252\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 38.0000i 1.34183i
\(803\) − 12.0000i − 0.423471i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) − 2.00000i − 0.0703598i
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) 40.0000i 1.40286i
\(814\) 16.0000 0.560800
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) − 8.00000i − 0.279885i
\(818\) − 6.00000i − 0.209785i
\(819\) 0 0
\(820\) 0 0
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) 36.0000i 1.25564i
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) − 14.0000i − 0.486828i −0.969923 0.243414i \(-0.921733\pi\)
0.969923 0.243414i \(-0.0782673\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −32.0000 −1.11141 −0.555703 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14.0000i 0.485071i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 4.00000i 0.138260i
\(838\) 24.0000i 0.829066i
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) − 22.0000i − 0.758170i
\(843\) − 20.0000i − 0.688837i
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 8.00000i − 0.274721i
\(849\) 48.0000 1.64736
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 0 0
\(859\) −42.0000 −1.43302 −0.716511 0.697576i \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) − 26.0000i − 0.883006i
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) − 18.0000i − 0.609557i
\(873\) 2.00000i 0.0676897i
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) − 40.0000i − 1.34993i
\(879\) 36.0000 1.21425
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 26.0000i 0.874970i 0.899226 + 0.437485i \(0.144131\pi\)
−0.899226 + 0.437485i \(0.855869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 16.0000i 0.536925i
\(889\) 0 0
\(890\) 0 0
\(891\) −22.0000 −0.737028
\(892\) − 8.00000i − 0.267860i
\(893\) 0 0
\(894\) 28.0000 0.936460
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 6.00000i − 0.200223i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 12.0000i 0.399556i
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) − 32.0000i − 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) − 8.00000i − 0.264906i
\(913\) − 12.0000i − 0.397142i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) 8.00000i 0.264039i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) − 8.00000i − 0.263466i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 8.00000i 0.262754i
\(928\) 4.00000i 0.131306i
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) − 26.0000i − 0.851658i
\(933\) − 16.0000i − 0.523816i
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) − 14.0000i − 0.457360i −0.973502 0.228680i \(-0.926559\pi\)
0.973502 0.228680i \(-0.0734410\pi\)
\(938\) 0 0
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) 12.0000i 0.390981i
\(943\) 24.0000i 0.781548i
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) − 14.0000i − 0.454939i −0.973785 0.227469i \(-0.926955\pi\)
0.973785 0.227469i \(-0.0730452\pi\)
\(948\) 8.00000i 0.259828i
\(949\) 0 0
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) −8.00000 −0.259010
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) − 16.0000i − 0.517207i
\(958\) − 24.0000i − 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 8.00000i 0.257796i
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 0 0
\(977\) 34.0000i 1.08776i 0.839164 + 0.543878i \(0.183045\pi\)
−0.839164 + 0.543878i \(0.816955\pi\)
\(978\) − 48.0000i − 1.53487i
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 2.00000i 0.0638226i
\(983\) − 8.00000i − 0.255160i −0.991828 0.127580i \(-0.959279\pi\)
0.991828 0.127580i \(-0.0407210\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) − 1.00000i − 0.0317500i
\(993\) 60.0000i 1.90404i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) − 10.0000i − 0.316544i
\(999\) −32.0000 −1.01244
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 1550.2.b.e.249.2 2
5.2 odd 4 1550.2.a.a.1.1 1
5.3 odd 4 310.2.a.b.1.1 1
5.4 even 2 inner 1550.2.b.e.249.1 2
15.8 even 4 2790.2.a.h.1.1 1
20.3 even 4 2480.2.a.c.1.1 1
40.3 even 4 9920.2.a.bg.1.1 1
40.13 odd 4 9920.2.a.d.1.1 1
155.123 even 4 9610.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.a.b.1.1 1 5.3 odd 4
1550.2.a.a.1.1 1 5.2 odd 4
1550.2.b.e.249.1 2 5.4 even 2 inner
1550.2.b.e.249.2 2 1.1 even 1 trivial
2480.2.a.c.1.1 1 20.3 even 4
2790.2.a.h.1.1 1 15.8 even 4
9610.2.a.a.1.1 1 155.123 even 4
9920.2.a.d.1.1 1 40.13 odd 4
9920.2.a.bg.1.1 1 40.3 even 4