Properties

Label 450.3.b.b.449.3
Level $450$
Weight $3$
Character 450.449
Analytic conductor $12.262$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,3,Mod(449,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.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: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.449
Dual form 450.3.b.b.449.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} -4.00000i q^{7} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} -4.00000i q^{7} +2.82843 q^{8} -16.9706i q^{11} -8.00000i q^{13} -5.65685i q^{14} +4.00000 q^{16} -12.7279 q^{17} +16.0000 q^{19} -24.0000i q^{22} +16.9706 q^{23} -11.3137i q^{26} -8.00000i q^{28} +4.24264i q^{29} +44.0000 q^{31} +5.65685 q^{32} -18.0000 q^{34} -34.0000i q^{37} +22.6274 q^{38} -46.6690i q^{41} +40.0000i q^{43} -33.9411i q^{44} +24.0000 q^{46} -84.8528 q^{47} +33.0000 q^{49} -16.0000i q^{52} -38.1838 q^{53} -11.3137i q^{56} +6.00000i q^{58} +33.9411i q^{59} +50.0000 q^{61} +62.2254 q^{62} +8.00000 q^{64} +8.00000i q^{67} -25.4558 q^{68} +50.9117i q^{71} +16.0000i q^{73} -48.0833i q^{74} +32.0000 q^{76} -67.8823 q^{77} +76.0000 q^{79} -66.0000i q^{82} -118.794 q^{83} +56.5685i q^{86} -48.0000i q^{88} +12.7279i q^{89} -32.0000 q^{91} +33.9411 q^{92} -120.000 q^{94} +176.000i q^{97} +46.6690 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 16 q^{16} + 64 q^{19} + 176 q^{31} - 72 q^{34} + 96 q^{46} + 132 q^{49} + 200 q^{61} + 32 q^{64} + 128 q^{76} + 304 q^{79} - 128 q^{91} - 480 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 0.571429i −0.958315 0.285714i \(-0.907769\pi\)
0.958315 0.285714i \(-0.0922308\pi\)
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) − 16.9706i − 1.54278i −0.636364 0.771389i \(-0.719562\pi\)
0.636364 0.771389i \(-0.280438\pi\)
\(12\) 0 0
\(13\) − 8.00000i − 0.615385i −0.951486 0.307692i \(-0.900443\pi\)
0.951486 0.307692i \(-0.0995567\pi\)
\(14\) − 5.65685i − 0.404061i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) −12.7279 −0.748701 −0.374351 0.927287i \(-0.622134\pi\)
−0.374351 + 0.927287i \(0.622134\pi\)
\(18\) 0 0
\(19\) 16.0000 0.842105 0.421053 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 24.0000i − 1.09091i
\(23\) 16.9706 0.737851 0.368925 0.929459i \(-0.379726\pi\)
0.368925 + 0.929459i \(0.379726\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 11.3137i − 0.435143i
\(27\) 0 0
\(28\) − 8.00000i − 0.285714i
\(29\) 4.24264i 0.146298i 0.997321 + 0.0731490i \(0.0233049\pi\)
−0.997321 + 0.0731490i \(0.976695\pi\)
\(30\) 0 0
\(31\) 44.0000 1.41935 0.709677 0.704527i \(-0.248841\pi\)
0.709677 + 0.704527i \(0.248841\pi\)
\(32\) 5.65685 0.176777
\(33\) 0 0
\(34\) −18.0000 −0.529412
\(35\) 0 0
\(36\) 0 0
\(37\) − 34.0000i − 0.918919i −0.888199 0.459459i \(-0.848043\pi\)
0.888199 0.459459i \(-0.151957\pi\)
\(38\) 22.6274 0.595458
\(39\) 0 0
\(40\) 0 0
\(41\) − 46.6690i − 1.13827i −0.822244 0.569135i \(-0.807278\pi\)
0.822244 0.569135i \(-0.192722\pi\)
\(42\) 0 0
\(43\) 40.0000i 0.930233i 0.885250 + 0.465116i \(0.153987\pi\)
−0.885250 + 0.465116i \(0.846013\pi\)
\(44\) − 33.9411i − 0.771389i
\(45\) 0 0
\(46\) 24.0000 0.521739
\(47\) −84.8528 −1.80538 −0.902690 0.430293i \(-0.858410\pi\)
−0.902690 + 0.430293i \(0.858410\pi\)
\(48\) 0 0
\(49\) 33.0000 0.673469
\(50\) 0 0
\(51\) 0 0
\(52\) − 16.0000i − 0.307692i
\(53\) −38.1838 −0.720448 −0.360224 0.932866i \(-0.617300\pi\)
−0.360224 + 0.932866i \(0.617300\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 11.3137i − 0.202031i
\(57\) 0 0
\(58\) 6.00000i 0.103448i
\(59\) 33.9411i 0.575273i 0.957740 + 0.287637i \(0.0928695\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(60\) 0 0
\(61\) 50.0000 0.819672 0.409836 0.912159i \(-0.365586\pi\)
0.409836 + 0.912159i \(0.365586\pi\)
\(62\) 62.2254 1.00364
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.119403i 0.998216 + 0.0597015i \(0.0190149\pi\)
−0.998216 + 0.0597015i \(0.980985\pi\)
\(68\) −25.4558 −0.374351
\(69\) 0 0
\(70\) 0 0
\(71\) 50.9117i 0.717066i 0.933517 + 0.358533i \(0.116723\pi\)
−0.933517 + 0.358533i \(0.883277\pi\)
\(72\) 0 0
\(73\) 16.0000i 0.219178i 0.993977 + 0.109589i \(0.0349535\pi\)
−0.993977 + 0.109589i \(0.965047\pi\)
\(74\) − 48.0833i − 0.649774i
\(75\) 0 0
\(76\) 32.0000 0.421053
\(77\) −67.8823 −0.881588
\(78\) 0 0
\(79\) 76.0000 0.962025 0.481013 0.876714i \(-0.340269\pi\)
0.481013 + 0.876714i \(0.340269\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 66.0000i − 0.804878i
\(83\) −118.794 −1.43125 −0.715626 0.698484i \(-0.753859\pi\)
−0.715626 + 0.698484i \(0.753859\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 56.5685i 0.657774i
\(87\) 0 0
\(88\) − 48.0000i − 0.545455i
\(89\) 12.7279i 0.143010i 0.997440 + 0.0715052i \(0.0227802\pi\)
−0.997440 + 0.0715052i \(0.977220\pi\)
\(90\) 0 0
\(91\) −32.0000 −0.351648
\(92\) 33.9411 0.368925
\(93\) 0 0
\(94\) −120.000 −1.27660
\(95\) 0 0
\(96\) 0 0
\(97\) 176.000i 1.81443i 0.420664 + 0.907216i \(0.361797\pi\)
−0.420664 + 0.907216i \(0.638203\pi\)
\(98\) 46.6690 0.476215
\(99\) 0 0
\(100\) 0 0
\(101\) − 29.6985i − 0.294044i −0.989133 0.147022i \(-0.953031\pi\)
0.989133 0.147022i \(-0.0469689\pi\)
\(102\) 0 0
\(103\) 28.0000i 0.271845i 0.990719 + 0.135922i \(0.0433998\pi\)
−0.990719 + 0.135922i \(0.956600\pi\)
\(104\) − 22.6274i − 0.217571i
\(105\) 0 0
\(106\) −54.0000 −0.509434
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −56.0000 −0.513761 −0.256881 0.966443i \(-0.582695\pi\)
−0.256881 + 0.966443i \(0.582695\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 16.0000i − 0.142857i
\(113\) 156.978 1.38918 0.694592 0.719404i \(-0.255585\pi\)
0.694592 + 0.719404i \(0.255585\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.48528i 0.0731490i
\(117\) 0 0
\(118\) 48.0000i 0.406780i
\(119\) 50.9117i 0.427829i
\(120\) 0 0
\(121\) −167.000 −1.38017
\(122\) 70.7107 0.579596
\(123\) 0 0
\(124\) 88.0000 0.709677
\(125\) 0 0
\(126\) 0 0
\(127\) 92.0000i 0.724409i 0.932099 + 0.362205i \(0.117976\pi\)
−0.932099 + 0.362205i \(0.882024\pi\)
\(128\) 11.3137 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 169.706i 1.29546i 0.761869 + 0.647731i \(0.224282\pi\)
−0.761869 + 0.647731i \(0.775718\pi\)
\(132\) 0 0
\(133\) − 64.0000i − 0.481203i
\(134\) 11.3137i 0.0844307i
\(135\) 0 0
\(136\) −36.0000 −0.264706
\(137\) 156.978 1.14582 0.572911 0.819617i \(-0.305814\pi\)
0.572911 + 0.819617i \(0.305814\pi\)
\(138\) 0 0
\(139\) −152.000 −1.09353 −0.546763 0.837288i \(-0.684140\pi\)
−0.546763 + 0.837288i \(0.684140\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 72.0000i 0.507042i
\(143\) −135.765 −0.949402
\(144\) 0 0
\(145\) 0 0
\(146\) 22.6274i 0.154982i
\(147\) 0 0
\(148\) − 68.0000i − 0.459459i
\(149\) 275.772i 1.85082i 0.378972 + 0.925408i \(0.376278\pi\)
−0.378972 + 0.925408i \(0.623722\pi\)
\(150\) 0 0
\(151\) −148.000 −0.980132 −0.490066 0.871685i \(-0.663027\pi\)
−0.490066 + 0.871685i \(0.663027\pi\)
\(152\) 45.2548 0.297729
\(153\) 0 0
\(154\) −96.0000 −0.623377
\(155\) 0 0
\(156\) 0 0
\(157\) − 82.0000i − 0.522293i −0.965299 0.261146i \(-0.915899\pi\)
0.965299 0.261146i \(-0.0841006\pi\)
\(158\) 107.480 0.680255
\(159\) 0 0
\(160\) 0 0
\(161\) − 67.8823i − 0.421629i
\(162\) 0 0
\(163\) − 56.0000i − 0.343558i −0.985135 0.171779i \(-0.945048\pi\)
0.985135 0.171779i \(-0.0549515\pi\)
\(164\) − 93.3381i − 0.569135i
\(165\) 0 0
\(166\) −168.000 −1.01205
\(167\) 33.9411 0.203240 0.101620 0.994823i \(-0.467597\pi\)
0.101620 + 0.994823i \(0.467597\pi\)
\(168\) 0 0
\(169\) 105.000 0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 80.0000i 0.465116i
\(173\) 173.948 1.00548 0.502741 0.864437i \(-0.332325\pi\)
0.502741 + 0.864437i \(0.332325\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 67.8823i − 0.385695i
\(177\) 0 0
\(178\) 18.0000i 0.101124i
\(179\) − 203.647i − 1.13769i −0.822444 0.568846i \(-0.807390\pi\)
0.822444 0.568846i \(-0.192610\pi\)
\(180\) 0 0
\(181\) −232.000 −1.28177 −0.640884 0.767638i \(-0.721432\pi\)
−0.640884 + 0.767638i \(0.721432\pi\)
\(182\) −45.2548 −0.248653
\(183\) 0 0
\(184\) 48.0000 0.260870
\(185\) 0 0
\(186\) 0 0
\(187\) 216.000i 1.15508i
\(188\) −169.706 −0.902690
\(189\) 0 0
\(190\) 0 0
\(191\) 33.9411i 0.177702i 0.996045 + 0.0888511i \(0.0283195\pi\)
−0.996045 + 0.0888511i \(0.971680\pi\)
\(192\) 0 0
\(193\) − 206.000i − 1.06736i −0.845687 0.533679i \(-0.820809\pi\)
0.845687 0.533679i \(-0.179191\pi\)
\(194\) 248.902i 1.28300i
\(195\) 0 0
\(196\) 66.0000 0.336735
\(197\) 165.463 0.839914 0.419957 0.907544i \(-0.362045\pi\)
0.419957 + 0.907544i \(0.362045\pi\)
\(198\) 0 0
\(199\) −20.0000 −0.100503 −0.0502513 0.998737i \(-0.516002\pi\)
−0.0502513 + 0.998737i \(0.516002\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 42.0000i − 0.207921i
\(203\) 16.9706 0.0835988
\(204\) 0 0
\(205\) 0 0
\(206\) 39.5980i 0.192223i
\(207\) 0 0
\(208\) − 32.0000i − 0.153846i
\(209\) − 271.529i − 1.29918i
\(210\) 0 0
\(211\) 296.000 1.40284 0.701422 0.712746i \(-0.252549\pi\)
0.701422 + 0.712746i \(0.252549\pi\)
\(212\) −76.3675 −0.360224
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 176.000i − 0.811060i
\(218\) −79.1960 −0.363284
\(219\) 0 0
\(220\) 0 0
\(221\) 101.823i 0.460739i
\(222\) 0 0
\(223\) 436.000i 1.95516i 0.210571 + 0.977578i \(0.432468\pi\)
−0.210571 + 0.977578i \(0.567532\pi\)
\(224\) − 22.6274i − 0.101015i
\(225\) 0 0
\(226\) 222.000 0.982301
\(227\) 16.9706 0.0747602 0.0373801 0.999301i \(-0.488099\pi\)
0.0373801 + 0.999301i \(0.488099\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.0349345 −0.0174672 0.999847i \(-0.505560\pi\)
−0.0174672 + 0.999847i \(0.505560\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 12.0000i 0.0517241i
\(233\) 12.7279 0.0546263 0.0273131 0.999627i \(-0.491305\pi\)
0.0273131 + 0.999627i \(0.491305\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 67.8823i 0.287637i
\(237\) 0 0
\(238\) 72.0000i 0.302521i
\(239\) 135.765i 0.568052i 0.958817 + 0.284026i \(0.0916703\pi\)
−0.958817 + 0.284026i \(0.908330\pi\)
\(240\) 0 0
\(241\) 32.0000 0.132780 0.0663900 0.997794i \(-0.478852\pi\)
0.0663900 + 0.997794i \(0.478852\pi\)
\(242\) −236.174 −0.975924
\(243\) 0 0
\(244\) 100.000 0.409836
\(245\) 0 0
\(246\) 0 0
\(247\) − 128.000i − 0.518219i
\(248\) 124.451 0.501818
\(249\) 0 0
\(250\) 0 0
\(251\) − 50.9117i − 0.202835i −0.994844 0.101418i \(-0.967662\pi\)
0.994844 0.101418i \(-0.0323379\pi\)
\(252\) 0 0
\(253\) − 288.000i − 1.13834i
\(254\) 130.108i 0.512235i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −182.434 −0.709858 −0.354929 0.934893i \(-0.615495\pi\)
−0.354929 + 0.934893i \(0.615495\pi\)
\(258\) 0 0
\(259\) −136.000 −0.525097
\(260\) 0 0
\(261\) 0 0
\(262\) 240.000i 0.916031i
\(263\) −373.352 −1.41959 −0.709795 0.704408i \(-0.751213\pi\)
−0.709795 + 0.704408i \(0.751213\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 90.5097i − 0.340262i
\(267\) 0 0
\(268\) 16.0000i 0.0597015i
\(269\) − 343.654i − 1.27752i −0.769404 0.638762i \(-0.779447\pi\)
0.769404 0.638762i \(-0.220553\pi\)
\(270\) 0 0
\(271\) 380.000 1.40221 0.701107 0.713056i \(-0.252690\pi\)
0.701107 + 0.713056i \(0.252690\pi\)
\(272\) −50.9117 −0.187175
\(273\) 0 0
\(274\) 222.000 0.810219
\(275\) 0 0
\(276\) 0 0
\(277\) − 328.000i − 1.18412i −0.805896 0.592058i \(-0.798316\pi\)
0.805896 0.592058i \(-0.201684\pi\)
\(278\) −214.960 −0.773239
\(279\) 0 0
\(280\) 0 0
\(281\) − 284.257i − 1.01159i −0.862654 0.505795i \(-0.831199\pi\)
0.862654 0.505795i \(-0.168801\pi\)
\(282\) 0 0
\(283\) 208.000i 0.734982i 0.930027 + 0.367491i \(0.119783\pi\)
−0.930027 + 0.367491i \(0.880217\pi\)
\(284\) 101.823i 0.358533i
\(285\) 0 0
\(286\) −192.000 −0.671329
\(287\) −186.676 −0.650440
\(288\) 0 0
\(289\) −127.000 −0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 32.0000i 0.109589i
\(293\) 436.992 1.49144 0.745720 0.666259i \(-0.232106\pi\)
0.745720 + 0.666259i \(0.232106\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 96.1665i − 0.324887i
\(297\) 0 0
\(298\) 390.000i 1.30872i
\(299\) − 135.765i − 0.454062i
\(300\) 0 0
\(301\) 160.000 0.531561
\(302\) −209.304 −0.693058
\(303\) 0 0
\(304\) 64.0000 0.210526
\(305\) 0 0
\(306\) 0 0
\(307\) − 520.000i − 1.69381i −0.531743 0.846906i \(-0.678463\pi\)
0.531743 0.846906i \(-0.321537\pi\)
\(308\) −135.765 −0.440794
\(309\) 0 0
\(310\) 0 0
\(311\) 373.352i 1.20049i 0.799816 + 0.600245i \(0.204930\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(312\) 0 0
\(313\) 94.0000i 0.300319i 0.988662 + 0.150160i \(0.0479788\pi\)
−0.988662 + 0.150160i \(0.952021\pi\)
\(314\) − 115.966i − 0.369317i
\(315\) 0 0
\(316\) 152.000 0.481013
\(317\) 335.169 1.05731 0.528657 0.848835i \(-0.322696\pi\)
0.528657 + 0.848835i \(0.322696\pi\)
\(318\) 0 0
\(319\) 72.0000 0.225705
\(320\) 0 0
\(321\) 0 0
\(322\) − 96.0000i − 0.298137i
\(323\) −203.647 −0.630485
\(324\) 0 0
\(325\) 0 0
\(326\) − 79.1960i − 0.242932i
\(327\) 0 0
\(328\) − 132.000i − 0.402439i
\(329\) 339.411i 1.03165i
\(330\) 0 0
\(331\) 536.000 1.61934 0.809668 0.586889i \(-0.199647\pi\)
0.809668 + 0.586889i \(0.199647\pi\)
\(332\) −237.588 −0.715626
\(333\) 0 0
\(334\) 48.0000 0.143713
\(335\) 0 0
\(336\) 0 0
\(337\) − 208.000i − 0.617211i −0.951190 0.308605i \(-0.900138\pi\)
0.951190 0.308605i \(-0.0998622\pi\)
\(338\) 148.492 0.439327
\(339\) 0 0
\(340\) 0 0
\(341\) − 746.705i − 2.18975i
\(342\) 0 0
\(343\) − 328.000i − 0.956268i
\(344\) 113.137i 0.328887i
\(345\) 0 0
\(346\) 246.000 0.710983
\(347\) 288.500 0.831411 0.415705 0.909499i \(-0.363535\pi\)
0.415705 + 0.909499i \(0.363535\pi\)
\(348\) 0 0
\(349\) 238.000 0.681948 0.340974 0.940073i \(-0.389243\pi\)
0.340974 + 0.940073i \(0.389243\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 96.0000i − 0.272727i
\(353\) 224.860 0.636997 0.318499 0.947923i \(-0.396821\pi\)
0.318499 + 0.947923i \(0.396821\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 25.4558i 0.0715052i
\(357\) 0 0
\(358\) − 288.000i − 0.804469i
\(359\) − 560.029i − 1.55997i −0.625799 0.779984i \(-0.715227\pi\)
0.625799 0.779984i \(-0.284773\pi\)
\(360\) 0 0
\(361\) −105.000 −0.290859
\(362\) −328.098 −0.906347
\(363\) 0 0
\(364\) −64.0000 −0.175824
\(365\) 0 0
\(366\) 0 0
\(367\) 284.000i 0.773842i 0.922113 + 0.386921i \(0.126461\pi\)
−0.922113 + 0.386921i \(0.873539\pi\)
\(368\) 67.8823 0.184463
\(369\) 0 0
\(370\) 0 0
\(371\) 152.735i 0.411685i
\(372\) 0 0
\(373\) 190.000i 0.509383i 0.967022 + 0.254692i \(0.0819740\pi\)
−0.967022 + 0.254692i \(0.918026\pi\)
\(374\) 305.470i 0.816765i
\(375\) 0 0
\(376\) −240.000 −0.638298
\(377\) 33.9411 0.0900295
\(378\) 0 0
\(379\) 160.000 0.422164 0.211082 0.977468i \(-0.432301\pi\)
0.211082 + 0.977468i \(0.432301\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 48.0000i 0.125654i
\(383\) 271.529 0.708953 0.354477 0.935065i \(-0.384659\pi\)
0.354477 + 0.935065i \(0.384659\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 291.328i − 0.754736i
\(387\) 0 0
\(388\) 352.000i 0.907216i
\(389\) − 403.051i − 1.03612i −0.855344 0.518060i \(-0.826654\pi\)
0.855344 0.518060i \(-0.173346\pi\)
\(390\) 0 0
\(391\) −216.000 −0.552430
\(392\) 93.3381 0.238107
\(393\) 0 0
\(394\) 234.000 0.593909
\(395\) 0 0
\(396\) 0 0
\(397\) 146.000i 0.367758i 0.982949 + 0.183879i \(0.0588655\pi\)
−0.982949 + 0.183879i \(0.941135\pi\)
\(398\) −28.2843 −0.0710660
\(399\) 0 0
\(400\) 0 0
\(401\) − 326.683i − 0.814672i −0.913278 0.407336i \(-0.866458\pi\)
0.913278 0.407336i \(-0.133542\pi\)
\(402\) 0 0
\(403\) − 352.000i − 0.873449i
\(404\) − 59.3970i − 0.147022i
\(405\) 0 0
\(406\) 24.0000 0.0591133
\(407\) −576.999 −1.41769
\(408\) 0 0
\(409\) −368.000 −0.899756 −0.449878 0.893090i \(-0.648532\pi\)
−0.449878 + 0.893090i \(0.648532\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 56.0000i 0.135922i
\(413\) 135.765 0.328728
\(414\) 0 0
\(415\) 0 0
\(416\) − 45.2548i − 0.108786i
\(417\) 0 0
\(418\) − 384.000i − 0.918660i
\(419\) 390.323i 0.931558i 0.884901 + 0.465779i \(0.154226\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(420\) 0 0
\(421\) −40.0000 −0.0950119 −0.0475059 0.998871i \(-0.515127\pi\)
−0.0475059 + 0.998871i \(0.515127\pi\)
\(422\) 418.607 0.991960
\(423\) 0 0
\(424\) −108.000 −0.254717
\(425\) 0 0
\(426\) 0 0
\(427\) − 200.000i − 0.468384i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 152.735i 0.354374i 0.984177 + 0.177187i \(0.0566997\pi\)
−0.984177 + 0.177187i \(0.943300\pi\)
\(432\) 0 0
\(433\) − 542.000i − 1.25173i −0.779931 0.625866i \(-0.784746\pi\)
0.779931 0.625866i \(-0.215254\pi\)
\(434\) − 248.902i − 0.573506i
\(435\) 0 0
\(436\) −112.000 −0.256881
\(437\) 271.529 0.621348
\(438\) 0 0
\(439\) 4.00000 0.00911162 0.00455581 0.999990i \(-0.498550\pi\)
0.00455581 + 0.999990i \(0.498550\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 144.000i 0.325792i
\(443\) −322.441 −0.727857 −0.363929 0.931427i \(-0.618565\pi\)
−0.363929 + 0.931427i \(0.618565\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 616.597i 1.38250i
\(447\) 0 0
\(448\) − 32.0000i − 0.0714286i
\(449\) 216.375i 0.481904i 0.970537 + 0.240952i \(0.0774596\pi\)
−0.970537 + 0.240952i \(0.922540\pi\)
\(450\) 0 0
\(451\) −792.000 −1.75610
\(452\) 313.955 0.694592
\(453\) 0 0
\(454\) 24.0000 0.0528634
\(455\) 0 0
\(456\) 0 0
\(457\) − 400.000i − 0.875274i −0.899152 0.437637i \(-0.855816\pi\)
0.899152 0.437637i \(-0.144184\pi\)
\(458\) −11.3137 −0.0247024
\(459\) 0 0
\(460\) 0 0
\(461\) 301.227i 0.653422i 0.945124 + 0.326711i \(0.105940\pi\)
−0.945124 + 0.326711i \(0.894060\pi\)
\(462\) 0 0
\(463\) 604.000i 1.30454i 0.757989 + 0.652268i \(0.226182\pi\)
−0.757989 + 0.652268i \(0.773818\pi\)
\(464\) 16.9706i 0.0365745i
\(465\) 0 0
\(466\) 18.0000 0.0386266
\(467\) 356.382 0.763130 0.381565 0.924342i \(-0.375385\pi\)
0.381565 + 0.924342i \(0.375385\pi\)
\(468\) 0 0
\(469\) 32.0000 0.0682303
\(470\) 0 0
\(471\) 0 0
\(472\) 96.0000i 0.203390i
\(473\) 678.823 1.43514
\(474\) 0 0
\(475\) 0 0
\(476\) 101.823i 0.213915i
\(477\) 0 0
\(478\) 192.000i 0.401674i
\(479\) 526.087i 1.09830i 0.835723 + 0.549152i \(0.185049\pi\)
−0.835723 + 0.549152i \(0.814951\pi\)
\(480\) 0 0
\(481\) −272.000 −0.565489
\(482\) 45.2548 0.0938897
\(483\) 0 0
\(484\) −334.000 −0.690083
\(485\) 0 0
\(486\) 0 0
\(487\) 596.000i 1.22382i 0.790928 + 0.611910i \(0.209598\pi\)
−0.790928 + 0.611910i \(0.790402\pi\)
\(488\) 141.421 0.289798
\(489\) 0 0
\(490\) 0 0
\(491\) 271.529i 0.553012i 0.961012 + 0.276506i \(0.0891766\pi\)
−0.961012 + 0.276506i \(0.910823\pi\)
\(492\) 0 0
\(493\) − 54.0000i − 0.109533i
\(494\) − 181.019i − 0.366436i
\(495\) 0 0
\(496\) 176.000 0.354839
\(497\) 203.647 0.409752
\(498\) 0 0
\(499\) −224.000 −0.448898 −0.224449 0.974486i \(-0.572058\pi\)
−0.224449 + 0.974486i \(0.572058\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 72.0000i − 0.143426i
\(503\) −865.499 −1.72067 −0.860337 0.509726i \(-0.829747\pi\)
−0.860337 + 0.509726i \(0.829747\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) − 407.294i − 0.804928i
\(507\) 0 0
\(508\) 184.000i 0.362205i
\(509\) 479.418i 0.941883i 0.882164 + 0.470941i \(0.156086\pi\)
−0.882164 + 0.470941i \(0.843914\pi\)
\(510\) 0 0
\(511\) 64.0000 0.125245
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) −258.000 −0.501946
\(515\) 0 0
\(516\) 0 0
\(517\) 1440.00i 2.78530i
\(518\) −192.333 −0.371299
\(519\) 0 0
\(520\) 0 0
\(521\) 521.845i 1.00162i 0.865557 + 0.500811i \(0.166965\pi\)
−0.865557 + 0.500811i \(0.833035\pi\)
\(522\) 0 0
\(523\) 736.000i 1.40727i 0.710564 + 0.703633i \(0.248440\pi\)
−0.710564 + 0.703633i \(0.751560\pi\)
\(524\) 339.411i 0.647731i
\(525\) 0 0
\(526\) −528.000 −1.00380
\(527\) −560.029 −1.06267
\(528\) 0 0
\(529\) −241.000 −0.455577
\(530\) 0 0
\(531\) 0 0
\(532\) − 128.000i − 0.240602i
\(533\) −373.352 −0.700474
\(534\) 0 0
\(535\) 0 0
\(536\) 22.6274i 0.0422153i
\(537\) 0 0
\(538\) − 486.000i − 0.903346i
\(539\) − 560.029i − 1.03901i
\(540\) 0 0
\(541\) −808.000 −1.49353 −0.746765 0.665088i \(-0.768394\pi\)
−0.746765 + 0.665088i \(0.768394\pi\)
\(542\) 537.401 0.991515
\(543\) 0 0
\(544\) −72.0000 −0.132353
\(545\) 0 0
\(546\) 0 0
\(547\) 536.000i 0.979890i 0.871753 + 0.489945i \(0.162983\pi\)
−0.871753 + 0.489945i \(0.837017\pi\)
\(548\) 313.955 0.572911
\(549\) 0 0
\(550\) 0 0
\(551\) 67.8823i 0.123198i
\(552\) 0 0
\(553\) − 304.000i − 0.549729i
\(554\) − 463.862i − 0.837296i
\(555\) 0 0
\(556\) −304.000 −0.546763
\(557\) −165.463 −0.297061 −0.148531 0.988908i \(-0.547454\pi\)
−0.148531 + 0.988908i \(0.547454\pi\)
\(558\) 0 0
\(559\) 320.000 0.572451
\(560\) 0 0
\(561\) 0 0
\(562\) − 402.000i − 0.715302i
\(563\) 322.441 0.572719 0.286359 0.958122i \(-0.407555\pi\)
0.286359 + 0.958122i \(0.407555\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 294.156i 0.519711i
\(567\) 0 0
\(568\) 144.000i 0.253521i
\(569\) 156.978i 0.275883i 0.990440 + 0.137942i \(0.0440487\pi\)
−0.990440 + 0.137942i \(0.955951\pi\)
\(570\) 0 0
\(571\) 368.000 0.644483 0.322242 0.946657i \(-0.395564\pi\)
0.322242 + 0.946657i \(0.395564\pi\)
\(572\) −271.529 −0.474701
\(573\) 0 0
\(574\) −264.000 −0.459930
\(575\) 0 0
\(576\) 0 0
\(577\) − 142.000i − 0.246101i −0.992400 0.123050i \(-0.960732\pi\)
0.992400 0.123050i \(-0.0392676\pi\)
\(578\) −179.605 −0.310736
\(579\) 0 0
\(580\) 0 0
\(581\) 475.176i 0.817858i
\(582\) 0 0
\(583\) 648.000i 1.11149i
\(584\) 45.2548i 0.0774912i
\(585\) 0 0
\(586\) 618.000 1.05461
\(587\) 373.352 0.636035 0.318017 0.948085i \(-0.396983\pi\)
0.318017 + 0.948085i \(0.396983\pi\)
\(588\) 0 0
\(589\) 704.000 1.19525
\(590\) 0 0
\(591\) 0 0
\(592\) − 136.000i − 0.229730i
\(593\) −1107.33 −1.86733 −0.933667 0.358142i \(-0.883410\pi\)
−0.933667 + 0.358142i \(0.883410\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 551.543i 0.925408i
\(597\) 0 0
\(598\) − 192.000i − 0.321070i
\(599\) 797.616i 1.33158i 0.746139 + 0.665790i \(0.231905\pi\)
−0.746139 + 0.665790i \(0.768095\pi\)
\(600\) 0 0
\(601\) 158.000 0.262895 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(602\) 226.274 0.375871
\(603\) 0 0
\(604\) −296.000 −0.490066
\(605\) 0 0
\(606\) 0 0
\(607\) 332.000i 0.546952i 0.961879 + 0.273476i \(0.0881734\pi\)
−0.961879 + 0.273476i \(0.911827\pi\)
\(608\) 90.5097 0.148865
\(609\) 0 0
\(610\) 0 0
\(611\) 678.823i 1.11100i
\(612\) 0 0
\(613\) − 578.000i − 0.942904i −0.881892 0.471452i \(-0.843730\pi\)
0.881892 0.471452i \(-0.156270\pi\)
\(614\) − 735.391i − 1.19771i
\(615\) 0 0
\(616\) −192.000 −0.311688
\(617\) −55.1543 −0.0893911 −0.0446956 0.999001i \(-0.514232\pi\)
−0.0446956 + 0.999001i \(0.514232\pi\)
\(618\) 0 0
\(619\) −896.000 −1.44750 −0.723748 0.690064i \(-0.757582\pi\)
−0.723748 + 0.690064i \(0.757582\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 528.000i 0.848875i
\(623\) 50.9117 0.0817202
\(624\) 0 0
\(625\) 0 0
\(626\) 132.936i 0.212358i
\(627\) 0 0
\(628\) − 164.000i − 0.261146i
\(629\) 432.749i 0.687996i
\(630\) 0 0
\(631\) 20.0000 0.0316957 0.0158479 0.999874i \(-0.494955\pi\)
0.0158479 + 0.999874i \(0.494955\pi\)
\(632\) 214.960 0.340127
\(633\) 0 0
\(634\) 474.000 0.747634
\(635\) 0 0
\(636\) 0 0
\(637\) − 264.000i − 0.414443i
\(638\) 101.823 0.159598
\(639\) 0 0
\(640\) 0 0
\(641\) − 258.801i − 0.403746i −0.979412 0.201873i \(-0.935297\pi\)
0.979412 0.201873i \(-0.0647028\pi\)
\(642\) 0 0
\(643\) − 728.000i − 1.13219i −0.824339 0.566096i \(-0.808453\pi\)
0.824339 0.566096i \(-0.191547\pi\)
\(644\) − 135.765i − 0.210814i
\(645\) 0 0
\(646\) −288.000 −0.445820
\(647\) −458.205 −0.708200 −0.354100 0.935208i \(-0.615213\pi\)
−0.354100 + 0.935208i \(0.615213\pi\)
\(648\) 0 0
\(649\) 576.000 0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) − 112.000i − 0.171779i
\(653\) −301.227 −0.461298 −0.230649 0.973037i \(-0.574085\pi\)
−0.230649 + 0.973037i \(0.574085\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 186.676i − 0.284567i
\(657\) 0 0
\(658\) 480.000i 0.729483i
\(659\) − 1052.17i − 1.59662i −0.602244 0.798312i \(-0.705727\pi\)
0.602244 0.798312i \(-0.294273\pi\)
\(660\) 0 0
\(661\) 62.0000 0.0937973 0.0468986 0.998900i \(-0.485066\pi\)
0.0468986 + 0.998900i \(0.485066\pi\)
\(662\) 758.018 1.14504
\(663\) 0 0
\(664\) −336.000 −0.506024
\(665\) 0 0
\(666\) 0 0
\(667\) 72.0000i 0.107946i
\(668\) 67.8823 0.101620
\(669\) 0 0
\(670\) 0 0
\(671\) − 848.528i − 1.26457i
\(672\) 0 0
\(673\) 670.000i 0.995542i 0.867308 + 0.497771i \(0.165848\pi\)
−0.867308 + 0.497771i \(0.834152\pi\)
\(674\) − 294.156i − 0.436434i
\(675\) 0 0
\(676\) 210.000 0.310651
\(677\) −1294.01 −1.91138 −0.955691 0.294372i \(-0.904889\pi\)
−0.955691 + 0.294372i \(0.904889\pi\)
\(678\) 0 0
\(679\) 704.000 1.03682
\(680\) 0 0
\(681\) 0 0
\(682\) − 1056.00i − 1.54839i
\(683\) 560.029 0.819954 0.409977 0.912096i \(-0.365537\pi\)
0.409977 + 0.912096i \(0.365537\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) − 463.862i − 0.676184i
\(687\) 0 0
\(688\) 160.000i 0.232558i
\(689\) 305.470i 0.443353i
\(690\) 0 0
\(691\) −40.0000 −0.0578871 −0.0289436 0.999581i \(-0.509214\pi\)
−0.0289436 + 0.999581i \(0.509214\pi\)
\(692\) 347.897 0.502741
\(693\) 0 0
\(694\) 408.000 0.587896
\(695\) 0 0
\(696\) 0 0
\(697\) 594.000i 0.852224i
\(698\) 336.583 0.482210
\(699\) 0 0
\(700\) 0 0
\(701\) − 954.594i − 1.36176i −0.732395 0.680880i \(-0.761597\pi\)
0.732395 0.680880i \(-0.238403\pi\)
\(702\) 0 0
\(703\) − 544.000i − 0.773826i
\(704\) − 135.765i − 0.192847i
\(705\) 0 0
\(706\) 318.000 0.450425
\(707\) −118.794 −0.168025
\(708\) 0 0
\(709\) −968.000 −1.36530 −0.682652 0.730744i \(-0.739173\pi\)
−0.682652 + 0.730744i \(0.739173\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 36.0000i 0.0505618i
\(713\) 746.705 1.04727
\(714\) 0 0
\(715\) 0 0
\(716\) − 407.294i − 0.568846i
\(717\) 0 0
\(718\) − 792.000i − 1.10306i
\(719\) 1170.97i 1.62861i 0.580439 + 0.814304i \(0.302881\pi\)
−0.580439 + 0.814304i \(0.697119\pi\)
\(720\) 0 0
\(721\) 112.000 0.155340
\(722\) −148.492 −0.205668
\(723\) 0 0
\(724\) −464.000 −0.640884
\(725\) 0 0
\(726\) 0 0
\(727\) − 508.000i − 0.698762i −0.936981 0.349381i \(-0.886392\pi\)
0.936981 0.349381i \(-0.113608\pi\)
\(728\) −90.5097 −0.124326
\(729\) 0 0
\(730\) 0 0
\(731\) − 509.117i − 0.696466i
\(732\) 0 0
\(733\) 1144.00i 1.56071i 0.625337 + 0.780355i \(0.284961\pi\)
−0.625337 + 0.780355i \(0.715039\pi\)
\(734\) 401.637i 0.547189i
\(735\) 0 0
\(736\) 96.0000 0.130435
\(737\) 135.765 0.184212
\(738\) 0 0
\(739\) 304.000 0.411367 0.205683 0.978619i \(-0.434058\pi\)
0.205683 + 0.978619i \(0.434058\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 216.000i 0.291105i
\(743\) −848.528 −1.14203 −0.571015 0.820940i \(-0.693450\pi\)
−0.571015 + 0.820940i \(0.693450\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 268.701i 0.360188i
\(747\) 0 0
\(748\) 432.000i 0.577540i
\(749\) 0 0
\(750\) 0 0
\(751\) 188.000 0.250333 0.125166 0.992136i \(-0.460054\pi\)
0.125166 + 0.992136i \(0.460054\pi\)
\(752\) −339.411 −0.451345
\(753\) 0 0
\(754\) 48.0000 0.0636605
\(755\) 0 0
\(756\) 0 0
\(757\) − 1240.00i − 1.63804i −0.573761 0.819022i \(-0.694516\pi\)
0.573761 0.819022i \(-0.305484\pi\)
\(758\) 226.274 0.298515
\(759\) 0 0
\(760\) 0 0
\(761\) 156.978i 0.206278i 0.994667 + 0.103139i \(0.0328887\pi\)
−0.994667 + 0.103139i \(0.967111\pi\)
\(762\) 0 0
\(763\) 224.000i 0.293578i
\(764\) 67.8823i 0.0888511i
\(765\) 0 0
\(766\) 384.000 0.501305
\(767\) 271.529 0.354014
\(768\) 0 0
\(769\) 910.000 1.18336 0.591678 0.806175i \(-0.298466\pi\)
0.591678 + 0.806175i \(0.298466\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 412.000i − 0.533679i
\(773\) 1387.34 1.79475 0.897376 0.441266i \(-0.145471\pi\)
0.897376 + 0.441266i \(0.145471\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 497.803i 0.641499i
\(777\) 0 0
\(778\) − 570.000i − 0.732648i
\(779\) − 746.705i − 0.958543i
\(780\) 0 0
\(781\) 864.000 1.10627
\(782\) −305.470 −0.390627
\(783\) 0 0
\(784\) 132.000 0.168367
\(785\) 0 0
\(786\) 0 0
\(787\) − 1360.00i − 1.72808i −0.503422 0.864041i \(-0.667926\pi\)
0.503422 0.864041i \(-0.332074\pi\)
\(788\) 330.926 0.419957
\(789\) 0 0
\(790\) 0 0
\(791\) − 627.911i − 0.793819i
\(792\) 0 0
\(793\) − 400.000i − 0.504414i
\(794\) 206.475i 0.260044i
\(795\) 0 0
\(796\) −40.0000 −0.0502513
\(797\) −106.066 −0.133082 −0.0665408 0.997784i \(-0.521196\pi\)
−0.0665408 + 0.997784i \(0.521196\pi\)
\(798\) 0 0
\(799\) 1080.00 1.35169
\(800\) 0 0
\(801\) 0 0
\(802\) − 462.000i − 0.576060i
\(803\) 271.529 0.338143
\(804\) 0 0
\(805\) 0 0
\(806\) − 497.803i − 0.617622i
\(807\) 0 0
\(808\) − 84.0000i − 0.103960i
\(809\) − 1107.33i − 1.36876i −0.729124 0.684381i \(-0.760072\pi\)
0.729124 0.684381i \(-0.239928\pi\)
\(810\) 0 0
\(811\) −160.000 −0.197287 −0.0986436 0.995123i \(-0.531450\pi\)
−0.0986436 + 0.995123i \(0.531450\pi\)
\(812\) 33.9411 0.0417994
\(813\) 0 0
\(814\) −816.000 −1.00246
\(815\) 0 0
\(816\) 0 0
\(817\) 640.000i 0.783354i
\(818\) −520.431 −0.636223
\(819\) 0 0
\(820\) 0 0
\(821\) − 436.992i − 0.532268i −0.963936 0.266134i \(-0.914254\pi\)
0.963936 0.266134i \(-0.0857464\pi\)
\(822\) 0 0
\(823\) − 332.000i − 0.403402i −0.979447 0.201701i \(-0.935353\pi\)
0.979447 0.201701i \(-0.0646470\pi\)
\(824\) 79.1960i 0.0961116i
\(825\) 0 0
\(826\) 192.000 0.232446
\(827\) 101.823 0.123124 0.0615619 0.998103i \(-0.480392\pi\)
0.0615619 + 0.998103i \(0.480392\pi\)
\(828\) 0 0
\(829\) −632.000 −0.762364 −0.381182 0.924500i \(-0.624483\pi\)
−0.381182 + 0.924500i \(0.624483\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 64.0000i − 0.0769231i
\(833\) −420.021 −0.504227
\(834\) 0 0
\(835\) 0 0
\(836\) − 543.058i − 0.649591i
\(837\) 0 0
\(838\) 552.000i 0.658711i
\(839\) 729.734i 0.869767i 0.900487 + 0.434883i \(0.143210\pi\)
−0.900487 + 0.434883i \(0.856790\pi\)
\(840\) 0 0
\(841\) 823.000 0.978597
\(842\) −56.5685 −0.0671835
\(843\) 0 0
\(844\) 592.000 0.701422
\(845\) 0 0
\(846\) 0 0
\(847\) 668.000i 0.788666i
\(848\) −152.735 −0.180112
\(849\) 0 0
\(850\) 0 0
\(851\) − 576.999i − 0.678025i
\(852\) 0 0
\(853\) − 446.000i − 0.522860i −0.965222 0.261430i \(-0.915806\pi\)
0.965222 0.261430i \(-0.0841941\pi\)
\(854\) − 282.843i − 0.331198i
\(855\) 0 0
\(856\) 0 0
\(857\) −428.507 −0.500008 −0.250004 0.968245i \(-0.580432\pi\)
−0.250004 + 0.968245i \(0.580432\pi\)
\(858\) 0 0
\(859\) −728.000 −0.847497 −0.423749 0.905780i \(-0.639286\pi\)
−0.423749 + 0.905780i \(0.639286\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 216.000i 0.250580i
\(863\) 916.410 1.06189 0.530945 0.847407i \(-0.321837\pi\)
0.530945 + 0.847407i \(0.321837\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) − 766.504i − 0.885108i
\(867\) 0 0
\(868\) − 352.000i − 0.405530i
\(869\) − 1289.76i − 1.48419i
\(870\) 0 0
\(871\) 64.0000 0.0734788
\(872\) −158.392 −0.181642
\(873\) 0 0
\(874\) 384.000 0.439359
\(875\) 0 0
\(876\) 0 0
\(877\) − 910.000i − 1.03763i −0.854887 0.518814i \(-0.826374\pi\)
0.854887 0.518814i \(-0.173626\pi\)
\(878\) 5.65685 0.00644289
\(879\) 0 0
\(880\) 0 0
\(881\) 929.138i 1.05464i 0.849667 + 0.527320i \(0.176803\pi\)
−0.849667 + 0.527320i \(0.823197\pi\)
\(882\) 0 0
\(883\) − 1064.00i − 1.20498i −0.798125 0.602492i \(-0.794175\pi\)
0.798125 0.602492i \(-0.205825\pi\)
\(884\) 203.647i 0.230370i
\(885\) 0 0
\(886\) −456.000 −0.514673
\(887\) −1391.59 −1.56887 −0.784434 0.620212i \(-0.787047\pi\)
−0.784434 + 0.620212i \(0.787047\pi\)
\(888\) 0 0
\(889\) 368.000 0.413948
\(890\) 0 0
\(891\) 0 0
\(892\) 872.000i 0.977578i
\(893\) −1357.65 −1.52032
\(894\) 0 0
\(895\) 0 0
\(896\) − 45.2548i − 0.0505076i
\(897\) 0 0
\(898\) 306.000i 0.340757i
\(899\) 186.676i 0.207649i
\(900\) 0 0
\(901\) 486.000 0.539401
\(902\) −1120.06 −1.24175
\(903\) 0 0
\(904\) 444.000 0.491150
\(905\) 0 0
\(906\) 0 0
\(907\) − 1768.00i − 1.94928i −0.223771 0.974642i \(-0.571837\pi\)
0.223771 0.974642i \(-0.428163\pi\)
\(908\) 33.9411 0.0373801
\(909\) 0 0
\(910\) 0 0
\(911\) 237.588i 0.260799i 0.991462 + 0.130399i \(0.0416260\pi\)
−0.991462 + 0.130399i \(0.958374\pi\)
\(912\) 0 0
\(913\) 2016.00i 2.20811i
\(914\) − 565.685i − 0.618912i
\(915\) 0 0
\(916\) −16.0000 −0.0174672
\(917\) 678.823 0.740264
\(918\) 0 0
\(919\) −380.000 −0.413493 −0.206746 0.978395i \(-0.566288\pi\)
−0.206746 + 0.978395i \(0.566288\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 426.000i 0.462039i
\(923\) 407.294 0.441271
\(924\) 0 0
\(925\) 0 0
\(926\) 854.185i 0.922446i
\(927\) 0 0
\(928\) 24.0000i 0.0258621i
\(929\) 666.095i 0.717002i 0.933529 + 0.358501i \(0.116712\pi\)
−0.933529 + 0.358501i \(0.883288\pi\)
\(930\) 0 0
\(931\) 528.000 0.567132
\(932\) 25.4558 0.0273131
\(933\) 0 0
\(934\) 504.000 0.539615
\(935\) 0 0
\(936\) 0 0
\(937\) − 178.000i − 0.189968i −0.995479 0.0949840i \(-0.969720\pi\)
0.995479 0.0949840i \(-0.0302800\pi\)
\(938\) 45.2548 0.0482461
\(939\) 0 0
\(940\) 0 0
\(941\) 436.992i 0.464391i 0.972669 + 0.232196i \(0.0745909\pi\)
−0.972669 + 0.232196i \(0.925409\pi\)
\(942\) 0 0
\(943\) − 792.000i − 0.839873i
\(944\) 135.765i 0.143818i
\(945\) 0 0
\(946\) 960.000 1.01480
\(947\) 1798.88 1.89956 0.949778 0.312924i \(-0.101309\pi\)
0.949778 + 0.312924i \(0.101309\pi\)
\(948\) 0 0
\(949\) 128.000 0.134879
\(950\) 0 0
\(951\) 0 0
\(952\) 144.000i 0.151261i
\(953\) −1310.98 −1.37563 −0.687815 0.725886i \(-0.741430\pi\)
−0.687815 + 0.725886i \(0.741430\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 271.529i 0.284026i
\(957\) 0 0
\(958\) 744.000i 0.776618i
\(959\) − 627.911i − 0.654756i
\(960\) 0 0
\(961\) 975.000 1.01457
\(962\) −384.666 −0.399861
\(963\) 0 0
\(964\) 64.0000 0.0663900
\(965\) 0 0
\(966\) 0 0
\(967\) 1700.00i 1.75801i 0.476808 + 0.879007i \(0.341794\pi\)
−0.476808 + 0.879007i \(0.658206\pi\)
\(968\) −472.347 −0.487962
\(969\) 0 0
\(970\) 0 0
\(971\) − 458.205i − 0.471890i −0.971766 0.235945i \(-0.924181\pi\)
0.971766 0.235945i \(-0.0758185\pi\)
\(972\) 0 0
\(973\) 608.000i 0.624872i
\(974\) 842.871i 0.865371i
\(975\) 0 0
\(976\) 200.000 0.204918
\(977\) 759.433 0.777311 0.388655 0.921383i \(-0.372940\pi\)
0.388655 + 0.921383i \(0.372940\pi\)
\(978\) 0 0
\(979\) 216.000 0.220633
\(980\) 0 0
\(981\) 0 0
\(982\) 384.000i 0.391039i
\(983\) 1052.17 1.07037 0.535186 0.844734i \(-0.320242\pi\)
0.535186 + 0.844734i \(0.320242\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) − 76.3675i − 0.0774519i
\(987\) 0 0
\(988\) − 256.000i − 0.259109i
\(989\) 678.823i 0.686373i
\(990\) 0 0
\(991\) −772.000 −0.779011 −0.389506 0.921024i \(-0.627354\pi\)
−0.389506 + 0.921024i \(0.627354\pi\)
\(992\) 248.902 0.250909
\(993\) 0 0
\(994\) 288.000 0.289738
\(995\) 0 0
\(996\) 0 0
\(997\) 194.000i 0.194584i 0.995256 + 0.0972919i \(0.0310180\pi\)
−0.995256 + 0.0972919i \(0.968982\pi\)
\(998\) −316.784 −0.317419
\(999\) 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 450.3.b.b.449.3 4
3.2 odd 2 inner 450.3.b.b.449.1 4
4.3 odd 2 3600.3.c.b.449.4 4
5.2 odd 4 450.3.d.f.251.2 2
5.3 odd 4 18.3.b.a.17.1 2
5.4 even 2 inner 450.3.b.b.449.2 4
12.11 even 2 3600.3.c.b.449.3 4
15.2 even 4 450.3.d.f.251.1 2
15.8 even 4 18.3.b.a.17.2 yes 2
15.14 odd 2 inner 450.3.b.b.449.4 4
20.3 even 4 144.3.e.b.17.2 2
20.7 even 4 3600.3.l.d.1601.2 2
20.19 odd 2 3600.3.c.b.449.2 4
35.3 even 12 882.3.s.d.863.2 4
35.13 even 4 882.3.b.a.197.1 2
35.18 odd 12 882.3.s.b.863.2 4
35.23 odd 12 882.3.s.b.557.1 4
35.33 even 12 882.3.s.d.557.1 4
40.3 even 4 576.3.e.f.449.1 2
40.13 odd 4 576.3.e.c.449.1 2
45.13 odd 12 162.3.d.b.107.2 4
45.23 even 12 162.3.d.b.107.1 4
45.38 even 12 162.3.d.b.53.2 4
45.43 odd 12 162.3.d.b.53.1 4
55.43 even 4 2178.3.c.d.485.2 2
60.23 odd 4 144.3.e.b.17.1 2
60.47 odd 4 3600.3.l.d.1601.1 2
60.59 even 2 3600.3.c.b.449.1 4
65.8 even 4 3042.3.d.a.3041.4 4
65.18 even 4 3042.3.d.a.3041.1 4
65.38 odd 4 3042.3.c.e.1691.2 2
80.3 even 4 2304.3.h.c.2177.4 4
80.13 odd 4 2304.3.h.f.2177.4 4
80.43 even 4 2304.3.h.c.2177.1 4
80.53 odd 4 2304.3.h.f.2177.1 4
105.23 even 12 882.3.s.b.557.2 4
105.38 odd 12 882.3.s.d.863.1 4
105.53 even 12 882.3.s.b.863.1 4
105.68 odd 12 882.3.s.d.557.2 4
105.83 odd 4 882.3.b.a.197.2 2
120.53 even 4 576.3.e.c.449.2 2
120.83 odd 4 576.3.e.f.449.2 2
165.98 odd 4 2178.3.c.d.485.1 2
180.23 odd 12 1296.3.q.f.593.1 4
180.43 even 12 1296.3.q.f.1025.1 4
180.83 odd 12 1296.3.q.f.1025.2 4
180.103 even 12 1296.3.q.f.593.2 4
195.8 odd 4 3042.3.d.a.3041.2 4
195.38 even 4 3042.3.c.e.1691.1 2
195.83 odd 4 3042.3.d.a.3041.3 4
240.53 even 4 2304.3.h.f.2177.3 4
240.83 odd 4 2304.3.h.c.2177.2 4
240.173 even 4 2304.3.h.f.2177.2 4
240.203 odd 4 2304.3.h.c.2177.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 5.3 odd 4
18.3.b.a.17.2 yes 2 15.8 even 4
144.3.e.b.17.1 2 60.23 odd 4
144.3.e.b.17.2 2 20.3 even 4
162.3.d.b.53.1 4 45.43 odd 12
162.3.d.b.53.2 4 45.38 even 12
162.3.d.b.107.1 4 45.23 even 12
162.3.d.b.107.2 4 45.13 odd 12
450.3.b.b.449.1 4 3.2 odd 2 inner
450.3.b.b.449.2 4 5.4 even 2 inner
450.3.b.b.449.3 4 1.1 even 1 trivial
450.3.b.b.449.4 4 15.14 odd 2 inner
450.3.d.f.251.1 2 15.2 even 4
450.3.d.f.251.2 2 5.2 odd 4
576.3.e.c.449.1 2 40.13 odd 4
576.3.e.c.449.2 2 120.53 even 4
576.3.e.f.449.1 2 40.3 even 4
576.3.e.f.449.2 2 120.83 odd 4
882.3.b.a.197.1 2 35.13 even 4
882.3.b.a.197.2 2 105.83 odd 4
882.3.s.b.557.1 4 35.23 odd 12
882.3.s.b.557.2 4 105.23 even 12
882.3.s.b.863.1 4 105.53 even 12
882.3.s.b.863.2 4 35.18 odd 12
882.3.s.d.557.1 4 35.33 even 12
882.3.s.d.557.2 4 105.68 odd 12
882.3.s.d.863.1 4 105.38 odd 12
882.3.s.d.863.2 4 35.3 even 12
1296.3.q.f.593.1 4 180.23 odd 12
1296.3.q.f.593.2 4 180.103 even 12
1296.3.q.f.1025.1 4 180.43 even 12
1296.3.q.f.1025.2 4 180.83 odd 12
2178.3.c.d.485.1 2 165.98 odd 4
2178.3.c.d.485.2 2 55.43 even 4
2304.3.h.c.2177.1 4 80.43 even 4
2304.3.h.c.2177.2 4 240.83 odd 4
2304.3.h.c.2177.3 4 240.203 odd 4
2304.3.h.c.2177.4 4 80.3 even 4
2304.3.h.f.2177.1 4 80.53 odd 4
2304.3.h.f.2177.2 4 240.173 even 4
2304.3.h.f.2177.3 4 240.53 even 4
2304.3.h.f.2177.4 4 80.13 odd 4
3042.3.c.e.1691.1 2 195.38 even 4
3042.3.c.e.1691.2 2 65.38 odd 4
3042.3.d.a.3041.1 4 65.18 even 4
3042.3.d.a.3041.2 4 195.8 odd 4
3042.3.d.a.3041.3 4 195.83 odd 4
3042.3.d.a.3041.4 4 65.8 even 4
3600.3.c.b.449.1 4 60.59 even 2
3600.3.c.b.449.2 4 20.19 odd 2
3600.3.c.b.449.3 4 12.11 even 2
3600.3.c.b.449.4 4 4.3 odd 2
3600.3.l.d.1601.1 2 60.47 odd 4
3600.3.l.d.1601.2 2 20.7 even 4