Properties

Label 3150.3.c.f.449.7
Level $3150$
Weight $3$
Character 3150.449
Analytic conductor $85.831$
Analytic rank $0$
Dimension $16$
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] = [3150,3,Mod(449,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.c (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: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.9671731157401600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.7
Root \(0.359610 + 0.796626i\) of defining polynomial
Character \(\chi\) \(=\) 3150.449
Dual form 3150.3.c.f.449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +11.2717i q^{11} +22.9496i q^{13} -3.74166i q^{14} +4.00000 q^{16} +8.36199 q^{17} -3.57092 q^{19} -15.9406i q^{22} -3.92441 q^{23} -32.4556i q^{26} +5.29150i q^{28} +43.5101i q^{29} +46.3702 q^{31} -5.65685 q^{32} -11.8256 q^{34} +27.9648i q^{37} +5.05005 q^{38} -32.8306i q^{41} +65.5572i q^{43} +22.5434i q^{44} +5.54995 q^{46} -5.65248 q^{47} -7.00000 q^{49} +45.8992i q^{52} +24.7975 q^{53} -7.48331i q^{56} -61.5325i q^{58} -58.6564i q^{59} +28.2418 q^{61} -65.5774 q^{62} +8.00000 q^{64} -10.7504i q^{67} +16.7240 q^{68} -20.8535i q^{71} +113.896i q^{73} -39.5482i q^{74} -7.14185 q^{76} -29.8222 q^{77} -133.236 q^{79} +46.4295i q^{82} +30.9830 q^{83} -92.7119i q^{86} -31.8812i q^{88} -79.6261i q^{89} -60.7189 q^{91} -7.84882 q^{92} +7.99382 q^{94} +26.1819i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(7\) 2.64575i 0.377964i
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 11.2717i 1.02470i 0.858776 + 0.512351i \(0.171225\pi\)
−0.858776 + 0.512351i \(0.828775\pi\)
\(12\) 0 0
\(13\) 22.9496i 1.76535i 0.469980 + 0.882677i \(0.344261\pi\)
−0.469980 + 0.882677i \(0.655739\pi\)
\(14\) − 3.74166i − 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 8.36199 0.491882 0.245941 0.969285i \(-0.420903\pi\)
0.245941 + 0.969285i \(0.420903\pi\)
\(18\) 0 0
\(19\) −3.57092 −0.187943 −0.0939717 0.995575i \(-0.529956\pi\)
−0.0939717 + 0.995575i \(0.529956\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 15.9406i − 0.724573i
\(23\) −3.92441 −0.170626 −0.0853132 0.996354i \(-0.527189\pi\)
−0.0853132 + 0.996354i \(0.527189\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 32.4556i − 1.24829i
\(27\) 0 0
\(28\) 5.29150i 0.188982i
\(29\) 43.5101i 1.50035i 0.661241 + 0.750173i \(0.270030\pi\)
−0.661241 + 0.750173i \(0.729970\pi\)
\(30\) 0 0
\(31\) 46.3702 1.49581 0.747907 0.663804i \(-0.231059\pi\)
0.747907 + 0.663804i \(0.231059\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) −11.8256 −0.347813
\(35\) 0 0
\(36\) 0 0
\(37\) 27.9648i 0.755805i 0.925845 + 0.377902i \(0.123355\pi\)
−0.925845 + 0.377902i \(0.876645\pi\)
\(38\) 5.05005 0.132896
\(39\) 0 0
\(40\) 0 0
\(41\) − 32.8306i − 0.800748i −0.916352 0.400374i \(-0.868880\pi\)
0.916352 0.400374i \(-0.131120\pi\)
\(42\) 0 0
\(43\) 65.5572i 1.52459i 0.647232 + 0.762293i \(0.275927\pi\)
−0.647232 + 0.762293i \(0.724073\pi\)
\(44\) 22.5434i 0.512351i
\(45\) 0 0
\(46\) 5.54995 0.120651
\(47\) −5.65248 −0.120266 −0.0601328 0.998190i \(-0.519152\pi\)
−0.0601328 + 0.998190i \(0.519152\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 45.8992i 0.882677i
\(53\) 24.7975 0.467878 0.233939 0.972251i \(-0.424838\pi\)
0.233939 + 0.972251i \(0.424838\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 7.48331i − 0.133631i
\(57\) 0 0
\(58\) − 61.5325i − 1.06091i
\(59\) − 58.6564i − 0.994176i −0.867700 0.497088i \(-0.834403\pi\)
0.867700 0.497088i \(-0.165597\pi\)
\(60\) 0 0
\(61\) 28.2418 0.462980 0.231490 0.972837i \(-0.425640\pi\)
0.231490 + 0.972837i \(0.425640\pi\)
\(62\) −65.5774 −1.05770
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.7504i − 0.160454i −0.996777 0.0802271i \(-0.974435\pi\)
0.996777 0.0802271i \(-0.0255646\pi\)
\(68\) 16.7240 0.245941
\(69\) 0 0
\(70\) 0 0
\(71\) − 20.8535i − 0.293711i −0.989158 0.146856i \(-0.953085\pi\)
0.989158 0.146856i \(-0.0469153\pi\)
\(72\) 0 0
\(73\) 113.896i 1.56022i 0.625642 + 0.780110i \(0.284837\pi\)
−0.625642 + 0.780110i \(0.715163\pi\)
\(74\) − 39.5482i − 0.534435i
\(75\) 0 0
\(76\) −7.14185 −0.0939717
\(77\) −29.8222 −0.387301
\(78\) 0 0
\(79\) −133.236 −1.68654 −0.843268 0.537494i \(-0.819371\pi\)
−0.843268 + 0.537494i \(0.819371\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 46.4295i 0.566214i
\(83\) 30.9830 0.373289 0.186644 0.982428i \(-0.440239\pi\)
0.186644 + 0.982428i \(0.440239\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 92.7119i − 1.07805i
\(87\) 0 0
\(88\) − 31.8812i − 0.362287i
\(89\) − 79.6261i − 0.894675i −0.894365 0.447337i \(-0.852372\pi\)
0.894365 0.447337i \(-0.147628\pi\)
\(90\) 0 0
\(91\) −60.7189 −0.667241
\(92\) −7.84882 −0.0853132
\(93\) 0 0
\(94\) 7.99382 0.0850406
\(95\) 0 0
\(96\) 0 0
\(97\) 26.1819i 0.269917i 0.990851 + 0.134958i \(0.0430901\pi\)
−0.990851 + 0.134958i \(0.956910\pi\)
\(98\) 9.89949 0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) − 170.729i − 1.69038i −0.534465 0.845191i \(-0.679487\pi\)
0.534465 0.845191i \(-0.320513\pi\)
\(102\) 0 0
\(103\) − 159.260i − 1.54622i −0.634274 0.773108i \(-0.718701\pi\)
0.634274 0.773108i \(-0.281299\pi\)
\(104\) − 64.9113i − 0.624147i
\(105\) 0 0
\(106\) −35.0690 −0.330840
\(107\) −62.7625 −0.586565 −0.293283 0.956026i \(-0.594748\pi\)
−0.293283 + 0.956026i \(0.594748\pi\)
\(108\) 0 0
\(109\) −44.5696 −0.408895 −0.204448 0.978878i \(-0.565540\pi\)
−0.204448 + 0.978878i \(0.565540\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.5830i 0.0944911i
\(113\) −128.003 −1.13277 −0.566385 0.824141i \(-0.691659\pi\)
−0.566385 + 0.824141i \(0.691659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 87.0201i 0.750173i
\(117\) 0 0
\(118\) 82.9526i 0.702988i
\(119\) 22.1237i 0.185914i
\(120\) 0 0
\(121\) −6.05158 −0.0500130
\(122\) −39.9399 −0.327376
\(123\) 0 0
\(124\) 92.7405 0.747907
\(125\) 0 0
\(126\) 0 0
\(127\) 99.0952i 0.780277i 0.920756 + 0.390139i \(0.127573\pi\)
−0.920756 + 0.390139i \(0.872427\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 139.901i 1.06795i 0.845500 + 0.533975i \(0.179302\pi\)
−0.845500 + 0.533975i \(0.820698\pi\)
\(132\) 0 0
\(133\) − 9.44777i − 0.0710359i
\(134\) 15.2034i 0.113458i
\(135\) 0 0
\(136\) −23.6513 −0.173906
\(137\) 200.351 1.46242 0.731208 0.682154i \(-0.238957\pi\)
0.731208 + 0.682154i \(0.238957\pi\)
\(138\) 0 0
\(139\) 89.8593 0.646470 0.323235 0.946319i \(-0.395230\pi\)
0.323235 + 0.946319i \(0.395230\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 29.4913i 0.207685i
\(143\) −258.681 −1.80896
\(144\) 0 0
\(145\) 0 0
\(146\) − 161.073i − 1.10324i
\(147\) 0 0
\(148\) 55.9296i 0.377902i
\(149\) 154.434i 1.03647i 0.855239 + 0.518234i \(0.173411\pi\)
−0.855239 + 0.518234i \(0.826589\pi\)
\(150\) 0 0
\(151\) 293.829 1.94588 0.972942 0.231048i \(-0.0742153\pi\)
0.972942 + 0.231048i \(0.0742153\pi\)
\(152\) 10.1001 0.0664480
\(153\) 0 0
\(154\) 42.1749 0.273863
\(155\) 0 0
\(156\) 0 0
\(157\) 275.050i 1.75191i 0.482395 + 0.875954i \(0.339767\pi\)
−0.482395 + 0.875954i \(0.660233\pi\)
\(158\) 188.425 1.19256
\(159\) 0 0
\(160\) 0 0
\(161\) − 10.3830i − 0.0644907i
\(162\) 0 0
\(163\) 121.606i 0.746048i 0.927822 + 0.373024i \(0.121679\pi\)
−0.927822 + 0.373024i \(0.878321\pi\)
\(164\) − 65.6613i − 0.400374i
\(165\) 0 0
\(166\) −43.8165 −0.263955
\(167\) −155.389 −0.930475 −0.465238 0.885186i \(-0.654031\pi\)
−0.465238 + 0.885186i \(0.654031\pi\)
\(168\) 0 0
\(169\) −357.684 −2.11648
\(170\) 0 0
\(171\) 0 0
\(172\) 131.114i 0.762293i
\(173\) −191.015 −1.10413 −0.552067 0.833800i \(-0.686161\pi\)
−0.552067 + 0.833800i \(0.686161\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 45.0869i 0.256175i
\(177\) 0 0
\(178\) 112.608i 0.632631i
\(179\) − 321.028i − 1.79345i −0.442586 0.896726i \(-0.645939\pi\)
0.442586 0.896726i \(-0.354061\pi\)
\(180\) 0 0
\(181\) −21.3100 −0.117735 −0.0588675 0.998266i \(-0.518749\pi\)
−0.0588675 + 0.998266i \(0.518749\pi\)
\(182\) 85.8696 0.471811
\(183\) 0 0
\(184\) 11.0999 0.0603256
\(185\) 0 0
\(186\) 0 0
\(187\) 94.2540i 0.504032i
\(188\) −11.3050 −0.0601328
\(189\) 0 0
\(190\) 0 0
\(191\) − 44.6968i − 0.234015i −0.993131 0.117007i \(-0.962670\pi\)
0.993131 0.117007i \(-0.0373301\pi\)
\(192\) 0 0
\(193\) − 237.337i − 1.22973i −0.788634 0.614863i \(-0.789211\pi\)
0.788634 0.614863i \(-0.210789\pi\)
\(194\) − 37.0268i − 0.190860i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) 199.475 1.01256 0.506282 0.862368i \(-0.331020\pi\)
0.506282 + 0.862368i \(0.331020\pi\)
\(198\) 0 0
\(199\) −75.1828 −0.377803 −0.188902 0.981996i \(-0.560493\pi\)
−0.188902 + 0.981996i \(0.560493\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 241.447i 1.19528i
\(203\) −115.117 −0.567078
\(204\) 0 0
\(205\) 0 0
\(206\) 225.228i 1.09334i
\(207\) 0 0
\(208\) 91.7984i 0.441339i
\(209\) − 40.2504i − 0.192586i
\(210\) 0 0
\(211\) 177.079 0.839237 0.419619 0.907701i \(-0.362164\pi\)
0.419619 + 0.907701i \(0.362164\pi\)
\(212\) 49.5951 0.233939
\(213\) 0 0
\(214\) 88.7595 0.414764
\(215\) 0 0
\(216\) 0 0
\(217\) 122.684i 0.565364i
\(218\) 63.0309 0.289133
\(219\) 0 0
\(220\) 0 0
\(221\) 191.904i 0.868346i
\(222\) 0 0
\(223\) 257.329i 1.15394i 0.816765 + 0.576971i \(0.195765\pi\)
−0.816765 + 0.576971i \(0.804235\pi\)
\(224\) − 14.9666i − 0.0668153i
\(225\) 0 0
\(226\) 181.024 0.800990
\(227\) 111.108 0.489464 0.244732 0.969591i \(-0.421300\pi\)
0.244732 + 0.969591i \(0.421300\pi\)
\(228\) 0 0
\(229\) −100.914 −0.440671 −0.220336 0.975424i \(-0.570715\pi\)
−0.220336 + 0.975424i \(0.570715\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 123.065i − 0.530453i
\(233\) 33.7780 0.144970 0.0724850 0.997370i \(-0.476907\pi\)
0.0724850 + 0.997370i \(0.476907\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 117.313i − 0.497088i
\(237\) 0 0
\(238\) − 31.2877i − 0.131461i
\(239\) − 440.484i − 1.84303i −0.388342 0.921515i \(-0.626952\pi\)
0.388342 0.921515i \(-0.373048\pi\)
\(240\) 0 0
\(241\) −204.074 −0.846780 −0.423390 0.905947i \(-0.639160\pi\)
−0.423390 + 0.905947i \(0.639160\pi\)
\(242\) 8.55822 0.0353645
\(243\) 0 0
\(244\) 56.4835 0.231490
\(245\) 0 0
\(246\) 0 0
\(247\) − 81.9513i − 0.331787i
\(248\) −131.155 −0.528850
\(249\) 0 0
\(250\) 0 0
\(251\) 5.18183i 0.0206447i 0.999947 + 0.0103224i \(0.00328577\pi\)
−0.999947 + 0.0103224i \(0.996714\pi\)
\(252\) 0 0
\(253\) − 44.2348i − 0.174841i
\(254\) − 140.142i − 0.551739i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −236.312 −0.919501 −0.459750 0.888048i \(-0.652061\pi\)
−0.459750 + 0.888048i \(0.652061\pi\)
\(258\) 0 0
\(259\) −73.9878 −0.285667
\(260\) 0 0
\(261\) 0 0
\(262\) − 197.851i − 0.755155i
\(263\) −17.5835 −0.0668572 −0.0334286 0.999441i \(-0.510643\pi\)
−0.0334286 + 0.999441i \(0.510643\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 13.3612i 0.0502300i
\(267\) 0 0
\(268\) − 21.5009i − 0.0802271i
\(269\) − 19.9314i − 0.0740944i −0.999314 0.0370472i \(-0.988205\pi\)
0.999314 0.0370472i \(-0.0117952\pi\)
\(270\) 0 0
\(271\) −87.0972 −0.321392 −0.160696 0.987004i \(-0.551374\pi\)
−0.160696 + 0.987004i \(0.551374\pi\)
\(272\) 33.4480 0.122970
\(273\) 0 0
\(274\) −283.339 −1.03408
\(275\) 0 0
\(276\) 0 0
\(277\) 384.515i 1.38814i 0.719907 + 0.694071i \(0.244185\pi\)
−0.719907 + 0.694071i \(0.755815\pi\)
\(278\) −127.080 −0.457123
\(279\) 0 0
\(280\) 0 0
\(281\) − 246.478i − 0.877147i −0.898695 0.438573i \(-0.855484\pi\)
0.898695 0.438573i \(-0.144516\pi\)
\(282\) 0 0
\(283\) − 318.082i − 1.12396i −0.827150 0.561982i \(-0.810039\pi\)
0.827150 0.561982i \(-0.189961\pi\)
\(284\) − 41.7070i − 0.146856i
\(285\) 0 0
\(286\) 365.831 1.27913
\(287\) 86.8617 0.302654
\(288\) 0 0
\(289\) −219.077 −0.758052
\(290\) 0 0
\(291\) 0 0
\(292\) 227.792i 0.780110i
\(293\) 397.217 1.35569 0.677845 0.735205i \(-0.262914\pi\)
0.677845 + 0.735205i \(0.262914\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 79.0963i − 0.267217i
\(297\) 0 0
\(298\) − 218.402i − 0.732894i
\(299\) − 90.0636i − 0.301216i
\(300\) 0 0
\(301\) −173.448 −0.576239
\(302\) −415.536 −1.37595
\(303\) 0 0
\(304\) −14.2837 −0.0469858
\(305\) 0 0
\(306\) 0 0
\(307\) 450.564i 1.46764i 0.679346 + 0.733818i \(0.262263\pi\)
−0.679346 + 0.733818i \(0.737737\pi\)
\(308\) −59.6443 −0.193650
\(309\) 0 0
\(310\) 0 0
\(311\) 484.602i 1.55821i 0.626897 + 0.779103i \(0.284325\pi\)
−0.626897 + 0.779103i \(0.715675\pi\)
\(312\) 0 0
\(313\) − 364.372i − 1.16413i −0.813143 0.582064i \(-0.802245\pi\)
0.813143 0.582064i \(-0.197755\pi\)
\(314\) − 388.979i − 1.23879i
\(315\) 0 0
\(316\) −266.473 −0.843268
\(317\) −510.623 −1.61080 −0.805399 0.592733i \(-0.798049\pi\)
−0.805399 + 0.592733i \(0.798049\pi\)
\(318\) 0 0
\(319\) −490.433 −1.53741
\(320\) 0 0
\(321\) 0 0
\(322\) 14.6838i 0.0456018i
\(323\) −29.8600 −0.0924459
\(324\) 0 0
\(325\) 0 0
\(326\) − 171.977i − 0.527536i
\(327\) 0 0
\(328\) 92.8591i 0.283107i
\(329\) − 14.9551i − 0.0454561i
\(330\) 0 0
\(331\) 199.957 0.604101 0.302050 0.953292i \(-0.402329\pi\)
0.302050 + 0.953292i \(0.402329\pi\)
\(332\) 61.9660 0.186644
\(333\) 0 0
\(334\) 219.754 0.657945
\(335\) 0 0
\(336\) 0 0
\(337\) 197.218i 0.585217i 0.956232 + 0.292609i \(0.0945233\pi\)
−0.956232 + 0.292609i \(0.905477\pi\)
\(338\) 505.842 1.49657
\(339\) 0 0
\(340\) 0 0
\(341\) 522.672i 1.53276i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) − 185.424i − 0.539023i
\(345\) 0 0
\(346\) 270.136 0.780741
\(347\) −265.136 −0.764080 −0.382040 0.924146i \(-0.624778\pi\)
−0.382040 + 0.924146i \(0.624778\pi\)
\(348\) 0 0
\(349\) −77.8495 −0.223065 −0.111532 0.993761i \(-0.535576\pi\)
−0.111532 + 0.993761i \(0.535576\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 63.7625i − 0.181143i
\(353\) −272.107 −0.770842 −0.385421 0.922741i \(-0.625944\pi\)
−0.385421 + 0.922741i \(0.625944\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 159.252i − 0.447337i
\(357\) 0 0
\(358\) 454.002i 1.26816i
\(359\) 695.157i 1.93637i 0.250236 + 0.968185i \(0.419492\pi\)
−0.250236 + 0.968185i \(0.580508\pi\)
\(360\) 0 0
\(361\) −348.249 −0.964677
\(362\) 30.1369 0.0832512
\(363\) 0 0
\(364\) −121.438 −0.333621
\(365\) 0 0
\(366\) 0 0
\(367\) 161.491i 0.440031i 0.975496 + 0.220015i \(0.0706108\pi\)
−0.975496 + 0.220015i \(0.929389\pi\)
\(368\) −15.6976 −0.0426566
\(369\) 0 0
\(370\) 0 0
\(371\) 65.6081i 0.176841i
\(372\) 0 0
\(373\) 459.127i 1.23090i 0.788175 + 0.615451i \(0.211026\pi\)
−0.788175 + 0.615451i \(0.788974\pi\)
\(374\) − 133.295i − 0.356404i
\(375\) 0 0
\(376\) 15.9876 0.0425203
\(377\) −998.539 −2.64864
\(378\) 0 0
\(379\) 132.150 0.348680 0.174340 0.984685i \(-0.444221\pi\)
0.174340 + 0.984685i \(0.444221\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 63.2109i 0.165474i
\(383\) −178.047 −0.464876 −0.232438 0.972611i \(-0.574670\pi\)
−0.232438 + 0.972611i \(0.574670\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 335.646i 0.869548i
\(387\) 0 0
\(388\) 52.3639i 0.134958i
\(389\) 432.024i 1.11060i 0.831650 + 0.555301i \(0.187397\pi\)
−0.831650 + 0.555301i \(0.812603\pi\)
\(390\) 0 0
\(391\) −32.8159 −0.0839281
\(392\) 19.7990 0.0505076
\(393\) 0 0
\(394\) −282.100 −0.715990
\(395\) 0 0
\(396\) 0 0
\(397\) − 394.193i − 0.992929i −0.868057 0.496464i \(-0.834631\pi\)
0.868057 0.496464i \(-0.165369\pi\)
\(398\) 106.325 0.267147
\(399\) 0 0
\(400\) 0 0
\(401\) − 711.381i − 1.77402i −0.461752 0.887009i \(-0.652779\pi\)
0.461752 0.887009i \(-0.347221\pi\)
\(402\) 0 0
\(403\) 1064.18i 2.64064i
\(404\) − 341.457i − 0.845191i
\(405\) 0 0
\(406\) 162.800 0.400985
\(407\) −315.211 −0.774474
\(408\) 0 0
\(409\) 532.984 1.30314 0.651570 0.758588i \(-0.274111\pi\)
0.651570 + 0.758588i \(0.274111\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 318.521i − 0.773108i
\(413\) 155.190 0.375763
\(414\) 0 0
\(415\) 0 0
\(416\) − 129.823i − 0.312073i
\(417\) 0 0
\(418\) 56.9227i 0.136179i
\(419\) 537.395i 1.28257i 0.767305 + 0.641283i \(0.221597\pi\)
−0.767305 + 0.641283i \(0.778403\pi\)
\(420\) 0 0
\(421\) 41.5394 0.0986685 0.0493342 0.998782i \(-0.484290\pi\)
0.0493342 + 0.998782i \(0.484290\pi\)
\(422\) −250.428 −0.593430
\(423\) 0 0
\(424\) −70.1380 −0.165420
\(425\) 0 0
\(426\) 0 0
\(427\) 74.7206i 0.174990i
\(428\) −125.525 −0.293283
\(429\) 0 0
\(430\) 0 0
\(431\) 19.8004i 0.0459406i 0.999736 + 0.0229703i \(0.00731232\pi\)
−0.999736 + 0.0229703i \(0.992688\pi\)
\(432\) 0 0
\(433\) − 709.349i − 1.63822i −0.573637 0.819110i \(-0.694468\pi\)
0.573637 0.819110i \(-0.305532\pi\)
\(434\) − 173.502i − 0.399773i
\(435\) 0 0
\(436\) −89.1391 −0.204448
\(437\) 14.0138 0.0320681
\(438\) 0 0
\(439\) −148.273 −0.337751 −0.168875 0.985637i \(-0.554014\pi\)
−0.168875 + 0.985637i \(0.554014\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 271.394i − 0.614013i
\(443\) −119.132 −0.268921 −0.134461 0.990919i \(-0.542930\pi\)
−0.134461 + 0.990919i \(0.542930\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) − 363.918i − 0.815960i
\(447\) 0 0
\(448\) 21.1660i 0.0472456i
\(449\) 299.789i 0.667682i 0.942629 + 0.333841i \(0.108345\pi\)
−0.942629 + 0.333841i \(0.891655\pi\)
\(450\) 0 0
\(451\) 370.058 0.820527
\(452\) −256.006 −0.566385
\(453\) 0 0
\(454\) −157.131 −0.346103
\(455\) 0 0
\(456\) 0 0
\(457\) − 210.083i − 0.459701i −0.973226 0.229850i \(-0.926176\pi\)
0.973226 0.229850i \(-0.0738236\pi\)
\(458\) 142.713 0.311602
\(459\) 0 0
\(460\) 0 0
\(461\) 635.944i 1.37949i 0.724053 + 0.689744i \(0.242277\pi\)
−0.724053 + 0.689744i \(0.757723\pi\)
\(462\) 0 0
\(463\) − 630.339i − 1.36142i −0.732551 0.680712i \(-0.761671\pi\)
0.732551 0.680712i \(-0.238329\pi\)
\(464\) 174.040i 0.375087i
\(465\) 0 0
\(466\) −47.7693 −0.102509
\(467\) 169.232 0.362381 0.181190 0.983448i \(-0.442005\pi\)
0.181190 + 0.983448i \(0.442005\pi\)
\(468\) 0 0
\(469\) 28.4430 0.0606460
\(470\) 0 0
\(471\) 0 0
\(472\) 165.905i 0.351494i
\(473\) −738.942 −1.56225
\(474\) 0 0
\(475\) 0 0
\(476\) 44.2475i 0.0929569i
\(477\) 0 0
\(478\) 622.939i 1.30322i
\(479\) 572.103i 1.19437i 0.802103 + 0.597185i \(0.203714\pi\)
−0.802103 + 0.597185i \(0.796286\pi\)
\(480\) 0 0
\(481\) −641.781 −1.33426
\(482\) 288.604 0.598764
\(483\) 0 0
\(484\) −12.1032 −0.0250065
\(485\) 0 0
\(486\) 0 0
\(487\) 468.392i 0.961791i 0.876778 + 0.480896i \(0.159688\pi\)
−0.876778 + 0.480896i \(0.840312\pi\)
\(488\) −79.8797 −0.163688
\(489\) 0 0
\(490\) 0 0
\(491\) − 294.928i − 0.600668i −0.953834 0.300334i \(-0.902902\pi\)
0.953834 0.300334i \(-0.0970981\pi\)
\(492\) 0 0
\(493\) 363.831i 0.737993i
\(494\) 115.897i 0.234609i
\(495\) 0 0
\(496\) 185.481 0.373953
\(497\) 55.1732 0.111012
\(498\) 0 0
\(499\) −610.716 −1.22388 −0.611940 0.790904i \(-0.709611\pi\)
−0.611940 + 0.790904i \(0.709611\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 7.32821i − 0.0145980i
\(503\) −58.6435 −0.116587 −0.0582937 0.998299i \(-0.518566\pi\)
−0.0582937 + 0.998299i \(0.518566\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 62.5575i 0.123631i
\(507\) 0 0
\(508\) 198.190i 0.390139i
\(509\) − 225.419i − 0.442866i −0.975176 0.221433i \(-0.928927\pi\)
0.975176 0.221433i \(-0.0710734\pi\)
\(510\) 0 0
\(511\) −301.341 −0.589708
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) 334.195 0.650185
\(515\) 0 0
\(516\) 0 0
\(517\) − 63.7132i − 0.123236i
\(518\) 104.635 0.201997
\(519\) 0 0
\(520\) 0 0
\(521\) − 6.65155i − 0.0127669i −0.999980 0.00638344i \(-0.997968\pi\)
0.999980 0.00638344i \(-0.00203193\pi\)
\(522\) 0 0
\(523\) 516.493i 0.987558i 0.869587 + 0.493779i \(0.164385\pi\)
−0.869587 + 0.493779i \(0.835615\pi\)
\(524\) 279.803i 0.533975i
\(525\) 0 0
\(526\) 24.8668 0.0472752
\(527\) 387.747 0.735764
\(528\) 0 0
\(529\) −513.599 −0.970887
\(530\) 0 0
\(531\) 0 0
\(532\) − 18.8955i − 0.0355179i
\(533\) 753.450 1.41360
\(534\) 0 0
\(535\) 0 0
\(536\) 30.4068i 0.0567292i
\(537\) 0 0
\(538\) 28.1872i 0.0523926i
\(539\) − 78.9020i − 0.146386i
\(540\) 0 0
\(541\) −70.6311 −0.130556 −0.0652782 0.997867i \(-0.520793\pi\)
−0.0652782 + 0.997867i \(0.520793\pi\)
\(542\) 123.174 0.227259
\(543\) 0 0
\(544\) −47.3026 −0.0869532
\(545\) 0 0
\(546\) 0 0
\(547\) − 707.126i − 1.29273i −0.763026 0.646367i \(-0.776287\pi\)
0.763026 0.646367i \(-0.223713\pi\)
\(548\) 400.702 0.731208
\(549\) 0 0
\(550\) 0 0
\(551\) − 155.371i − 0.281980i
\(552\) 0 0
\(553\) − 352.510i − 0.637451i
\(554\) − 543.787i − 0.981565i
\(555\) 0 0
\(556\) 179.719 0.323235
\(557\) 420.729 0.755348 0.377674 0.925939i \(-0.376724\pi\)
0.377674 + 0.925939i \(0.376724\pi\)
\(558\) 0 0
\(559\) −1504.51 −2.69143
\(560\) 0 0
\(561\) 0 0
\(562\) 348.573i 0.620236i
\(563\) −425.266 −0.755357 −0.377679 0.925937i \(-0.623278\pi\)
−0.377679 + 0.925937i \(0.623278\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 449.835i 0.794762i
\(567\) 0 0
\(568\) 58.9826i 0.103843i
\(569\) 565.501i 0.993851i 0.867793 + 0.496925i \(0.165538\pi\)
−0.867793 + 0.496925i \(0.834462\pi\)
\(570\) 0 0
\(571\) −655.541 −1.14806 −0.574029 0.818835i \(-0.694620\pi\)
−0.574029 + 0.818835i \(0.694620\pi\)
\(572\) −517.363 −0.904481
\(573\) 0 0
\(574\) −122.841 −0.214009
\(575\) 0 0
\(576\) 0 0
\(577\) 866.848i 1.50234i 0.660111 + 0.751168i \(0.270509\pi\)
−0.660111 + 0.751168i \(0.729491\pi\)
\(578\) 309.822 0.536024
\(579\) 0 0
\(580\) 0 0
\(581\) 81.9733i 0.141090i
\(582\) 0 0
\(583\) 279.511i 0.479435i
\(584\) − 322.147i − 0.551621i
\(585\) 0 0
\(586\) −561.750 −0.958617
\(587\) −72.2212 −0.123034 −0.0615172 0.998106i \(-0.519594\pi\)
−0.0615172 + 0.998106i \(0.519594\pi\)
\(588\) 0 0
\(589\) −165.585 −0.281128
\(590\) 0 0
\(591\) 0 0
\(592\) 111.859i 0.188951i
\(593\) −445.912 −0.751959 −0.375979 0.926628i \(-0.622694\pi\)
−0.375979 + 0.926628i \(0.622694\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 308.868i 0.518234i
\(597\) 0 0
\(598\) 127.369i 0.212992i
\(599\) − 624.774i − 1.04303i −0.853243 0.521514i \(-0.825367\pi\)
0.853243 0.521514i \(-0.174633\pi\)
\(600\) 0 0
\(601\) 366.209 0.609333 0.304667 0.952459i \(-0.401455\pi\)
0.304667 + 0.952459i \(0.401455\pi\)
\(602\) 245.293 0.407463
\(603\) 0 0
\(604\) 587.657 0.972942
\(605\) 0 0
\(606\) 0 0
\(607\) − 550.680i − 0.907215i −0.891201 0.453608i \(-0.850137\pi\)
0.891201 0.453608i \(-0.149863\pi\)
\(608\) 20.2002 0.0332240
\(609\) 0 0
\(610\) 0 0
\(611\) − 129.722i − 0.212311i
\(612\) 0 0
\(613\) − 380.620i − 0.620914i −0.950587 0.310457i \(-0.899518\pi\)
0.950587 0.310457i \(-0.100482\pi\)
\(614\) − 637.194i − 1.03778i
\(615\) 0 0
\(616\) 84.3498 0.136931
\(617\) −1097.16 −1.77821 −0.889106 0.457702i \(-0.848673\pi\)
−0.889106 + 0.457702i \(0.848673\pi\)
\(618\) 0 0
\(619\) 223.332 0.360795 0.180398 0.983594i \(-0.442262\pi\)
0.180398 + 0.983594i \(0.442262\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 685.330i − 1.10182i
\(623\) 210.671 0.338155
\(624\) 0 0
\(625\) 0 0
\(626\) 515.300i 0.823163i
\(627\) 0 0
\(628\) 550.099i 0.875954i
\(629\) 233.841i 0.371767i
\(630\) 0 0
\(631\) −724.190 −1.14769 −0.573843 0.818965i \(-0.694548\pi\)
−0.573843 + 0.818965i \(0.694548\pi\)
\(632\) 376.849 0.596280
\(633\) 0 0
\(634\) 722.130 1.13901
\(635\) 0 0
\(636\) 0 0
\(637\) − 160.647i − 0.252193i
\(638\) 693.577 1.08711
\(639\) 0 0
\(640\) 0 0
\(641\) 459.829i 0.717362i 0.933460 + 0.358681i \(0.116773\pi\)
−0.933460 + 0.358681i \(0.883227\pi\)
\(642\) 0 0
\(643\) − 816.842i − 1.27036i −0.772364 0.635180i \(-0.780926\pi\)
0.772364 0.635180i \(-0.219074\pi\)
\(644\) − 20.7660i − 0.0322454i
\(645\) 0 0
\(646\) 42.2284 0.0653691
\(647\) 726.332 1.12261 0.561307 0.827607i \(-0.310299\pi\)
0.561307 + 0.827607i \(0.310299\pi\)
\(648\) 0 0
\(649\) 661.158 1.01873
\(650\) 0 0
\(651\) 0 0
\(652\) 243.212i 0.373024i
\(653\) −763.701 −1.16953 −0.584763 0.811204i \(-0.698813\pi\)
−0.584763 + 0.811204i \(0.698813\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 131.323i − 0.200187i
\(657\) 0 0
\(658\) 21.1497i 0.0321423i
\(659\) 343.226i 0.520828i 0.965497 + 0.260414i \(0.0838591\pi\)
−0.965497 + 0.260414i \(0.916141\pi\)
\(660\) 0 0
\(661\) −541.334 −0.818962 −0.409481 0.912319i \(-0.634290\pi\)
−0.409481 + 0.912319i \(0.634290\pi\)
\(662\) −282.782 −0.427164
\(663\) 0 0
\(664\) −87.6331 −0.131978
\(665\) 0 0
\(666\) 0 0
\(667\) − 170.751i − 0.255999i
\(668\) −310.779 −0.465238
\(669\) 0 0
\(670\) 0 0
\(671\) 318.333i 0.474416i
\(672\) 0 0
\(673\) − 984.277i − 1.46252i −0.682098 0.731261i \(-0.738932\pi\)
0.682098 0.731261i \(-0.261068\pi\)
\(674\) − 278.909i − 0.413811i
\(675\) 0 0
\(676\) −715.369 −1.05824
\(677\) −1156.51 −1.70829 −0.854145 0.520034i \(-0.825919\pi\)
−0.854145 + 0.520034i \(0.825919\pi\)
\(678\) 0 0
\(679\) −69.2709 −0.102019
\(680\) 0 0
\(681\) 0 0
\(682\) − 739.170i − 1.08383i
\(683\) −110.138 −0.161256 −0.0806281 0.996744i \(-0.525693\pi\)
−0.0806281 + 0.996744i \(0.525693\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.1916i 0.0381802i
\(687\) 0 0
\(688\) 262.229i 0.381147i
\(689\) 569.094i 0.825971i
\(690\) 0 0
\(691\) 858.618 1.24257 0.621286 0.783584i \(-0.286610\pi\)
0.621286 + 0.783584i \(0.286610\pi\)
\(692\) −382.031 −0.552067
\(693\) 0 0
\(694\) 374.958 0.540286
\(695\) 0 0
\(696\) 0 0
\(697\) − 274.530i − 0.393873i
\(698\) 110.096 0.157730
\(699\) 0 0
\(700\) 0 0
\(701\) − 54.6278i − 0.0779285i −0.999241 0.0389642i \(-0.987594\pi\)
0.999241 0.0389642i \(-0.0124058\pi\)
\(702\) 0 0
\(703\) − 99.8601i − 0.142048i
\(704\) 90.1737i 0.128088i
\(705\) 0 0
\(706\) 384.818 0.545067
\(707\) 451.705 0.638904
\(708\) 0 0
\(709\) −98.3377 −0.138699 −0.0693496 0.997592i \(-0.522092\pi\)
−0.0693496 + 0.997592i \(0.522092\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 225.216i 0.316315i
\(713\) −181.976 −0.255225
\(714\) 0 0
\(715\) 0 0
\(716\) − 642.056i − 0.896726i
\(717\) 0 0
\(718\) − 983.100i − 1.36922i
\(719\) − 1302.52i − 1.81157i −0.423735 0.905786i \(-0.639281\pi\)
0.423735 0.905786i \(-0.360719\pi\)
\(720\) 0 0
\(721\) 421.363 0.584415
\(722\) 492.498 0.682130
\(723\) 0 0
\(724\) −42.6201 −0.0588675
\(725\) 0 0
\(726\) 0 0
\(727\) 29.4141i 0.0404596i 0.999795 + 0.0202298i \(0.00643978\pi\)
−0.999795 + 0.0202298i \(0.993560\pi\)
\(728\) 171.739 0.235905
\(729\) 0 0
\(730\) 0 0
\(731\) 548.189i 0.749916i
\(732\) 0 0
\(733\) − 13.8012i − 0.0188284i −0.999956 0.00941421i \(-0.997003\pi\)
0.999956 0.00941421i \(-0.00299668\pi\)
\(734\) − 228.383i − 0.311149i
\(735\) 0 0
\(736\) 22.1998 0.0301628
\(737\) 121.176 0.164418
\(738\) 0 0
\(739\) −25.7420 −0.0348335 −0.0174168 0.999848i \(-0.505544\pi\)
−0.0174168 + 0.999848i \(0.505544\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 92.7839i − 0.125046i
\(743\) −567.032 −0.763165 −0.381583 0.924335i \(-0.624621\pi\)
−0.381583 + 0.924335i \(0.624621\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) − 649.303i − 0.870380i
\(747\) 0 0
\(748\) 188.508i 0.252016i
\(749\) − 166.054i − 0.221701i
\(750\) 0 0
\(751\) −328.985 −0.438062 −0.219031 0.975718i \(-0.570290\pi\)
−0.219031 + 0.975718i \(0.570290\pi\)
\(752\) −22.6099 −0.0300664
\(753\) 0 0
\(754\) 1412.15 1.87287
\(755\) 0 0
\(756\) 0 0
\(757\) − 323.996i − 0.428000i −0.976834 0.214000i \(-0.931351\pi\)
0.976834 0.214000i \(-0.0686493\pi\)
\(758\) −186.888 −0.246554
\(759\) 0 0
\(760\) 0 0
\(761\) 725.410i 0.953232i 0.879112 + 0.476616i \(0.158137\pi\)
−0.879112 + 0.476616i \(0.841863\pi\)
\(762\) 0 0
\(763\) − 117.920i − 0.154548i
\(764\) − 89.3937i − 0.117007i
\(765\) 0 0
\(766\) 251.797 0.328717
\(767\) 1346.14 1.75507
\(768\) 0 0
\(769\) 942.014 1.22499 0.612493 0.790476i \(-0.290167\pi\)
0.612493 + 0.790476i \(0.290167\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 474.675i − 0.614863i
\(773\) 707.864 0.915737 0.457868 0.889020i \(-0.348613\pi\)
0.457868 + 0.889020i \(0.348613\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) − 74.0537i − 0.0954300i
\(777\) 0 0
\(778\) − 610.974i − 0.785314i
\(779\) 117.236i 0.150495i
\(780\) 0 0
\(781\) 235.055 0.300966
\(782\) 46.4086 0.0593461
\(783\) 0 0
\(784\) −28.0000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 141.139i 0.179338i 0.995972 + 0.0896688i \(0.0285808\pi\)
−0.995972 + 0.0896688i \(0.971419\pi\)
\(788\) 398.950 0.506282
\(789\) 0 0
\(790\) 0 0
\(791\) − 338.664i − 0.428147i
\(792\) 0 0
\(793\) 648.137i 0.817323i
\(794\) 557.473i 0.702107i
\(795\) 0 0
\(796\) −150.366 −0.188902
\(797\) 1183.67 1.48515 0.742575 0.669763i \(-0.233604\pi\)
0.742575 + 0.669763i \(0.233604\pi\)
\(798\) 0 0
\(799\) −47.2660 −0.0591565
\(800\) 0 0
\(801\) 0 0
\(802\) 1006.05i 1.25442i
\(803\) −1283.80 −1.59876
\(804\) 0 0
\(805\) 0 0
\(806\) − 1504.98i − 1.86722i
\(807\) 0 0
\(808\) 482.893i 0.597640i
\(809\) 1300.44i 1.60747i 0.594988 + 0.803735i \(0.297157\pi\)
−0.594988 + 0.803735i \(0.702843\pi\)
\(810\) 0 0
\(811\) 180.853 0.223000 0.111500 0.993764i \(-0.464435\pi\)
0.111500 + 0.993764i \(0.464435\pi\)
\(812\) −230.234 −0.283539
\(813\) 0 0
\(814\) 445.776 0.547636
\(815\) 0 0
\(816\) 0 0
\(817\) − 234.100i − 0.286536i
\(818\) −753.754 −0.921459
\(819\) 0 0
\(820\) 0 0
\(821\) 1312.33i 1.59845i 0.601029 + 0.799227i \(0.294757\pi\)
−0.601029 + 0.799227i \(0.705243\pi\)
\(822\) 0 0
\(823\) − 120.713i − 0.146675i −0.997307 0.0733373i \(-0.976635\pi\)
0.997307 0.0733373i \(-0.0233650\pi\)
\(824\) 450.456i 0.546670i
\(825\) 0 0
\(826\) −219.472 −0.265705
\(827\) 573.888 0.693939 0.346970 0.937876i \(-0.387211\pi\)
0.346970 + 0.937876i \(0.387211\pi\)
\(828\) 0 0
\(829\) 1607.19 1.93871 0.969357 0.245657i \(-0.0790038\pi\)
0.969357 + 0.245657i \(0.0790038\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 183.597i 0.220669i
\(833\) −58.5339 −0.0702688
\(834\) 0 0
\(835\) 0 0
\(836\) − 80.5009i − 0.0962929i
\(837\) 0 0
\(838\) − 759.991i − 0.906911i
\(839\) 115.880i 0.138116i 0.997613 + 0.0690582i \(0.0219994\pi\)
−0.997613 + 0.0690582i \(0.978001\pi\)
\(840\) 0 0
\(841\) −1052.12 −1.25104
\(842\) −58.7456 −0.0697691
\(843\) 0 0
\(844\) 354.158 0.419619
\(845\) 0 0
\(846\) 0 0
\(847\) − 16.0110i − 0.0189031i
\(848\) 99.1902 0.116970
\(849\) 0 0
\(850\) 0 0
\(851\) − 109.745i − 0.128960i
\(852\) 0 0
\(853\) 93.1233i 0.109171i 0.998509 + 0.0545857i \(0.0173838\pi\)
−0.998509 + 0.0545857i \(0.982616\pi\)
\(854\) − 105.671i − 0.123736i
\(855\) 0 0
\(856\) 177.519 0.207382
\(857\) −347.355 −0.405315 −0.202658 0.979250i \(-0.564958\pi\)
−0.202658 + 0.979250i \(0.564958\pi\)
\(858\) 0 0
\(859\) 705.651 0.821479 0.410740 0.911753i \(-0.365271\pi\)
0.410740 + 0.911753i \(0.365271\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 28.0020i − 0.0324849i
\(863\) 1504.37 1.74318 0.871591 0.490235i \(-0.163089\pi\)
0.871591 + 0.490235i \(0.163089\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1003.17i 1.15840i
\(867\) 0 0
\(868\) 245.368i 0.282682i
\(869\) − 1501.80i − 1.72820i
\(870\) 0 0
\(871\) 246.718 0.283259
\(872\) 126.062 0.144566
\(873\) 0 0
\(874\) −19.8185 −0.0226756
\(875\) 0 0
\(876\) 0 0
\(877\) − 552.772i − 0.630298i −0.949042 0.315149i \(-0.897945\pi\)
0.949042 0.315149i \(-0.102055\pi\)
\(878\) 209.689 0.238826
\(879\) 0 0
\(880\) 0 0
\(881\) − 131.749i − 0.149545i −0.997201 0.0747726i \(-0.976177\pi\)
0.997201 0.0747726i \(-0.0238231\pi\)
\(882\) 0 0
\(883\) − 870.659i − 0.986024i −0.870022 0.493012i \(-0.835896\pi\)
0.870022 0.493012i \(-0.164104\pi\)
\(884\) 383.809i 0.434173i
\(885\) 0 0
\(886\) 168.478 0.190156
\(887\) 1488.49 1.67812 0.839060 0.544039i \(-0.183106\pi\)
0.839060 + 0.544039i \(0.183106\pi\)
\(888\) 0 0
\(889\) −262.181 −0.294917
\(890\) 0 0
\(891\) 0 0
\(892\) 514.658i 0.576971i
\(893\) 20.1846 0.0226031
\(894\) 0 0
\(895\) 0 0
\(896\) − 29.9333i − 0.0334077i
\(897\) 0 0
\(898\) − 423.966i − 0.472122i
\(899\) 2017.57i 2.24424i
\(900\) 0 0
\(901\) 207.357 0.230141
\(902\) −523.341 −0.580200
\(903\) 0 0
\(904\) 362.047 0.400495
\(905\) 0 0
\(906\) 0 0
\(907\) 217.860i 0.240198i 0.992762 + 0.120099i \(0.0383213\pi\)
−0.992762 + 0.120099i \(0.961679\pi\)
\(908\) 222.216 0.244732
\(909\) 0 0
\(910\) 0 0
\(911\) − 752.667i − 0.826198i −0.910686 0.413099i \(-0.864446\pi\)
0.910686 0.413099i \(-0.135554\pi\)
\(912\) 0 0
\(913\) 349.231i 0.382510i
\(914\) 297.102i 0.325057i
\(915\) 0 0
\(916\) −201.827 −0.220336
\(917\) −370.144 −0.403647
\(918\) 0 0
\(919\) 1765.94 1.92159 0.960793 0.277266i \(-0.0894282\pi\)
0.960793 + 0.277266i \(0.0894282\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 899.361i − 0.975446i
\(923\) 478.580 0.518505
\(924\) 0 0
\(925\) 0 0
\(926\) 891.434i 0.962672i
\(927\) 0 0
\(928\) − 246.130i − 0.265226i
\(929\) 1051.94i 1.13234i 0.824289 + 0.566169i \(0.191575\pi\)
−0.824289 + 0.566169i \(0.808425\pi\)
\(930\) 0 0
\(931\) 24.9965 0.0268490
\(932\) 67.5560 0.0724850
\(933\) 0 0
\(934\) −239.330 −0.256242
\(935\) 0 0
\(936\) 0 0
\(937\) − 934.930i − 0.997791i −0.866662 0.498896i \(-0.833739\pi\)
0.866662 0.498896i \(-0.166261\pi\)
\(938\) −40.2245 −0.0428832
\(939\) 0 0
\(940\) 0 0
\(941\) 962.012i 1.02233i 0.859483 + 0.511165i \(0.170786\pi\)
−0.859483 + 0.511165i \(0.829214\pi\)
\(942\) 0 0
\(943\) 128.841i 0.136629i
\(944\) − 234.625i − 0.248544i
\(945\) 0 0
\(946\) 1045.02 1.10467
\(947\) 1811.65 1.91304 0.956521 0.291665i \(-0.0942091\pi\)
0.956521 + 0.291665i \(0.0942091\pi\)
\(948\) 0 0
\(949\) −2613.87 −2.75434
\(950\) 0 0
\(951\) 0 0
\(952\) − 62.5754i − 0.0657305i
\(953\) 1140.52 1.19677 0.598385 0.801209i \(-0.295809\pi\)
0.598385 + 0.801209i \(0.295809\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) − 880.969i − 0.921515i
\(957\) 0 0
\(958\) − 809.076i − 0.844547i
\(959\) 530.079i 0.552742i
\(960\) 0 0
\(961\) 1189.20 1.23746
\(962\) 907.615 0.943467
\(963\) 0 0
\(964\) −408.148 −0.423390
\(965\) 0 0
\(966\) 0 0
\(967\) 535.438i 0.553710i 0.960912 + 0.276855i \(0.0892922\pi\)
−0.960912 + 0.276855i \(0.910708\pi\)
\(968\) 17.1164 0.0176823
\(969\) 0 0
\(970\) 0 0
\(971\) − 182.412i − 0.187860i −0.995579 0.0939301i \(-0.970057\pi\)
0.995579 0.0939301i \(-0.0299430\pi\)
\(972\) 0 0
\(973\) 237.745i 0.244343i
\(974\) − 662.407i − 0.680089i
\(975\) 0 0
\(976\) 112.967 0.115745
\(977\) 1146.82 1.17381 0.586906 0.809655i \(-0.300346\pi\)
0.586906 + 0.809655i \(0.300346\pi\)
\(978\) 0 0
\(979\) 897.522 0.916775
\(980\) 0 0
\(981\) 0 0
\(982\) 417.091i 0.424736i
\(983\) −198.963 −0.202403 −0.101202 0.994866i \(-0.532269\pi\)
−0.101202 + 0.994866i \(0.532269\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) − 514.534i − 0.521840i
\(987\) 0 0
\(988\) − 163.903i − 0.165893i
\(989\) − 257.273i − 0.260135i
\(990\) 0 0
\(991\) 1917.76 1.93517 0.967587 0.252539i \(-0.0812658\pi\)
0.967587 + 0.252539i \(0.0812658\pi\)
\(992\) −262.310 −0.264425
\(993\) 0 0
\(994\) −78.0267 −0.0784977
\(995\) 0 0
\(996\) 0 0
\(997\) − 1709.94i − 1.71509i −0.514410 0.857544i \(-0.671989\pi\)
0.514410 0.857544i \(-0.328011\pi\)
\(998\) 863.683 0.865414
\(999\) 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 3150.3.c.f.449.7 16
3.2 odd 2 inner 3150.3.c.f.449.14 16
5.2 odd 4 630.3.e.b.71.1 8
5.3 odd 4 3150.3.e.f.701.8 8
5.4 even 2 inner 3150.3.c.f.449.11 16
15.2 even 4 630.3.e.b.71.7 yes 8
15.8 even 4 3150.3.e.f.701.3 8
15.14 odd 2 inner 3150.3.c.f.449.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.3.e.b.71.1 8 5.2 odd 4
630.3.e.b.71.7 yes 8 15.2 even 4
3150.3.c.f.449.2 16 15.14 odd 2 inner
3150.3.c.f.449.7 16 1.1 even 1 trivial
3150.3.c.f.449.11 16 5.4 even 2 inner
3150.3.c.f.449.14 16 3.2 odd 2 inner
3150.3.e.f.701.3 8 15.8 even 4
3150.3.e.f.701.8 8 5.3 odd 4