Properties

Label 414.2.a.g.1.1
Level $414$
Weight $2$
Character 414.1
Self dual yes
Analytic conductor $3.306$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [414,2,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.64575 q^{5} +2.00000 q^{7} +1.00000 q^{8} -1.64575 q^{10} +1.64575 q^{11} +5.29150 q^{13} +2.00000 q^{14} +1.00000 q^{16} +3.29150 q^{17} +0.354249 q^{19} -1.64575 q^{20} +1.64575 q^{22} -1.00000 q^{23} -2.29150 q^{25} +5.29150 q^{26} +2.00000 q^{28} -9.29150 q^{29} -1.29150 q^{31} +1.00000 q^{32} +3.29150 q^{34} -3.29150 q^{35} +6.93725 q^{37} +0.354249 q^{38} -1.64575 q^{40} -6.00000 q^{41} +0.354249 q^{43} +1.64575 q^{44} -1.00000 q^{46} -6.00000 q^{47} -3.00000 q^{49} -2.29150 q^{50} +5.29150 q^{52} +1.64575 q^{53} -2.70850 q^{55} +2.00000 q^{56} -9.29150 q^{58} +0.354249 q^{61} -1.29150 q^{62} +1.00000 q^{64} -8.70850 q^{65} -14.9373 q^{67} +3.29150 q^{68} -3.29150 q^{70} +6.00000 q^{71} -7.29150 q^{73} +6.93725 q^{74} +0.354249 q^{76} +3.29150 q^{77} +8.58301 q^{79} -1.64575 q^{80} -6.00000 q^{82} +13.6458 q^{83} -5.41699 q^{85} +0.354249 q^{86} +1.64575 q^{88} +10.5830 q^{91} -1.00000 q^{92} -6.00000 q^{94} -0.583005 q^{95} -1.29150 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{19} + 2 q^{20} - 2 q^{22} - 2 q^{23} + 6 q^{25} + 4 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.64575 −0.736002 −0.368001 0.929825i \(-0.619958\pi\)
−0.368001 + 0.929825i \(0.619958\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.64575 −0.520432
\(11\) 1.64575 0.496213 0.248106 0.968733i \(-0.420192\pi\)
0.248106 + 0.968733i \(0.420192\pi\)
\(12\) 0 0
\(13\) 5.29150 1.46760 0.733799 0.679366i \(-0.237745\pi\)
0.733799 + 0.679366i \(0.237745\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.29150 0.798307 0.399153 0.916884i \(-0.369304\pi\)
0.399153 + 0.916884i \(0.369304\pi\)
\(18\) 0 0
\(19\) 0.354249 0.0812702 0.0406351 0.999174i \(-0.487062\pi\)
0.0406351 + 0.999174i \(0.487062\pi\)
\(20\) −1.64575 −0.368001
\(21\) 0 0
\(22\) 1.64575 0.350875
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 5.29150 1.03775
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −9.29150 −1.72539 −0.862694 0.505726i \(-0.831225\pi\)
−0.862694 + 0.505726i \(0.831225\pi\)
\(30\) 0 0
\(31\) −1.29150 −0.231961 −0.115980 0.993252i \(-0.537001\pi\)
−0.115980 + 0.993252i \(0.537001\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.29150 0.564488
\(35\) −3.29150 −0.556365
\(36\) 0 0
\(37\) 6.93725 1.14048 0.570239 0.821479i \(-0.306851\pi\)
0.570239 + 0.821479i \(0.306851\pi\)
\(38\) 0.354249 0.0574667
\(39\) 0 0
\(40\) −1.64575 −0.260216
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 0.354249 0.0540224 0.0270112 0.999635i \(-0.491401\pi\)
0.0270112 + 0.999635i \(0.491401\pi\)
\(44\) 1.64575 0.248106
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −2.29150 −0.324067
\(51\) 0 0
\(52\) 5.29150 0.733799
\(53\) 1.64575 0.226061 0.113031 0.993592i \(-0.463944\pi\)
0.113031 + 0.993592i \(0.463944\pi\)
\(54\) 0 0
\(55\) −2.70850 −0.365214
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −9.29150 −1.22003
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0.354249 0.0453569 0.0226784 0.999743i \(-0.492781\pi\)
0.0226784 + 0.999743i \(0.492781\pi\)
\(62\) −1.29150 −0.164021
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.70850 −1.08016
\(66\) 0 0
\(67\) −14.9373 −1.82488 −0.912438 0.409215i \(-0.865803\pi\)
−0.912438 + 0.409215i \(0.865803\pi\)
\(68\) 3.29150 0.399153
\(69\) 0 0
\(70\) −3.29150 −0.393410
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −7.29150 −0.853406 −0.426703 0.904392i \(-0.640325\pi\)
−0.426703 + 0.904392i \(0.640325\pi\)
\(74\) 6.93725 0.806439
\(75\) 0 0
\(76\) 0.354249 0.0406351
\(77\) 3.29150 0.375102
\(78\) 0 0
\(79\) 8.58301 0.965664 0.482832 0.875713i \(-0.339608\pi\)
0.482832 + 0.875713i \(0.339608\pi\)
\(80\) −1.64575 −0.184001
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 13.6458 1.49782 0.748908 0.662674i \(-0.230579\pi\)
0.748908 + 0.662674i \(0.230579\pi\)
\(84\) 0 0
\(85\) −5.41699 −0.587556
\(86\) 0.354249 0.0381996
\(87\) 0 0
\(88\) 1.64575 0.175438
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 10.5830 1.10940
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −0.583005 −0.0598151
\(96\) 0 0
\(97\) −1.29150 −0.131132 −0.0655661 0.997848i \(-0.520885\pi\)
−0.0655661 + 0.997848i \(0.520885\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −2.29150 −0.229150
\(101\) −9.29150 −0.924539 −0.462270 0.886739i \(-0.652965\pi\)
−0.462270 + 0.886739i \(0.652965\pi\)
\(102\) 0 0
\(103\) −6.70850 −0.661008 −0.330504 0.943805i \(-0.607219\pi\)
−0.330504 + 0.943805i \(0.607219\pi\)
\(104\) 5.29150 0.518875
\(105\) 0 0
\(106\) 1.64575 0.159849
\(107\) 13.6458 1.31918 0.659592 0.751624i \(-0.270729\pi\)
0.659592 + 0.751624i \(0.270729\pi\)
\(108\) 0 0
\(109\) −18.2288 −1.74600 −0.872999 0.487722i \(-0.837828\pi\)
−0.872999 + 0.487722i \(0.837828\pi\)
\(110\) −2.70850 −0.258245
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −6.58301 −0.619277 −0.309639 0.950854i \(-0.600208\pi\)
−0.309639 + 0.950854i \(0.600208\pi\)
\(114\) 0 0
\(115\) 1.64575 0.153467
\(116\) −9.29150 −0.862694
\(117\) 0 0
\(118\) 0 0
\(119\) 6.58301 0.603463
\(120\) 0 0
\(121\) −8.29150 −0.753773
\(122\) 0.354249 0.0320722
\(123\) 0 0
\(124\) −1.29150 −0.115980
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −10.5830 −0.939090 −0.469545 0.882909i \(-0.655582\pi\)
−0.469545 + 0.882909i \(0.655582\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −8.70850 −0.763786
\(131\) 21.8745 1.91118 0.955592 0.294692i \(-0.0952170\pi\)
0.955592 + 0.294692i \(0.0952170\pi\)
\(132\) 0 0
\(133\) 0.708497 0.0614345
\(134\) −14.9373 −1.29038
\(135\) 0 0
\(136\) 3.29150 0.282244
\(137\) −21.8745 −1.86887 −0.934433 0.356140i \(-0.884093\pi\)
−0.934433 + 0.356140i \(0.884093\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −3.29150 −0.278183
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 8.70850 0.728241
\(144\) 0 0
\(145\) 15.2915 1.26989
\(146\) −7.29150 −0.603449
\(147\) 0 0
\(148\) 6.93725 0.570239
\(149\) −10.3542 −0.848253 −0.424127 0.905603i \(-0.639419\pi\)
−0.424127 + 0.905603i \(0.639419\pi\)
\(150\) 0 0
\(151\) 17.2915 1.40716 0.703581 0.710615i \(-0.251583\pi\)
0.703581 + 0.710615i \(0.251583\pi\)
\(152\) 0.354249 0.0287334
\(153\) 0 0
\(154\) 3.29150 0.265237
\(155\) 2.12549 0.170724
\(156\) 0 0
\(157\) 3.64575 0.290963 0.145481 0.989361i \(-0.453527\pi\)
0.145481 + 0.989361i \(0.453527\pi\)
\(158\) 8.58301 0.682827
\(159\) 0 0
\(160\) −1.64575 −0.130108
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −0.708497 −0.0554938 −0.0277469 0.999615i \(-0.508833\pi\)
−0.0277469 + 0.999615i \(0.508833\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 13.6458 1.05912
\(167\) 19.1660 1.48311 0.741555 0.670892i \(-0.234089\pi\)
0.741555 + 0.670892i \(0.234089\pi\)
\(168\) 0 0
\(169\) 15.0000 1.15385
\(170\) −5.41699 −0.415465
\(171\) 0 0
\(172\) 0.354249 0.0270112
\(173\) 15.8745 1.20692 0.603458 0.797395i \(-0.293789\pi\)
0.603458 + 0.797395i \(0.293789\pi\)
\(174\) 0 0
\(175\) −4.58301 −0.346443
\(176\) 1.64575 0.124053
\(177\) 0 0
\(178\) 0 0
\(179\) −15.2915 −1.14294 −0.571470 0.820623i \(-0.693627\pi\)
−0.571470 + 0.820623i \(0.693627\pi\)
\(180\) 0 0
\(181\) 15.6458 1.16294 0.581470 0.813568i \(-0.302478\pi\)
0.581470 + 0.813568i \(0.302478\pi\)
\(182\) 10.5830 0.784465
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −11.4170 −0.839394
\(186\) 0 0
\(187\) 5.41699 0.396130
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −0.583005 −0.0422956
\(191\) −3.29150 −0.238165 −0.119082 0.992884i \(-0.537995\pi\)
−0.119082 + 0.992884i \(0.537995\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −1.29150 −0.0927245
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) −2.29150 −0.162034
\(201\) 0 0
\(202\) −9.29150 −0.653748
\(203\) −18.5830 −1.30427
\(204\) 0 0
\(205\) 9.87451 0.689666
\(206\) −6.70850 −0.467403
\(207\) 0 0
\(208\) 5.29150 0.366900
\(209\) 0.583005 0.0403273
\(210\) 0 0
\(211\) 11.2915 0.777339 0.388670 0.921377i \(-0.372935\pi\)
0.388670 + 0.921377i \(0.372935\pi\)
\(212\) 1.64575 0.113031
\(213\) 0 0
\(214\) 13.6458 0.932804
\(215\) −0.583005 −0.0397606
\(216\) 0 0
\(217\) −2.58301 −0.175346
\(218\) −18.2288 −1.23461
\(219\) 0 0
\(220\) −2.70850 −0.182607
\(221\) 17.4170 1.17159
\(222\) 0 0
\(223\) −19.8745 −1.33090 −0.665448 0.746444i \(-0.731759\pi\)
−0.665448 + 0.746444i \(0.731759\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −6.58301 −0.437895
\(227\) 7.06275 0.468771 0.234385 0.972144i \(-0.424692\pi\)
0.234385 + 0.972144i \(0.424692\pi\)
\(228\) 0 0
\(229\) 9.06275 0.598883 0.299442 0.954115i \(-0.403200\pi\)
0.299442 + 0.954115i \(0.403200\pi\)
\(230\) 1.64575 0.108518
\(231\) 0 0
\(232\) −9.29150 −0.610017
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 9.87451 0.644142
\(236\) 0 0
\(237\) 0 0
\(238\) 6.58301 0.426713
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 5.29150 0.340856 0.170428 0.985370i \(-0.445485\pi\)
0.170428 + 0.985370i \(0.445485\pi\)
\(242\) −8.29150 −0.532998
\(243\) 0 0
\(244\) 0.354249 0.0226784
\(245\) 4.93725 0.315430
\(246\) 0 0
\(247\) 1.87451 0.119272
\(248\) −1.29150 −0.0820105
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 25.6458 1.61875 0.809373 0.587295i \(-0.199807\pi\)
0.809373 + 0.587295i \(0.199807\pi\)
\(252\) 0 0
\(253\) −1.64575 −0.103467
\(254\) −10.5830 −0.664037
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.5830 −1.53345 −0.766723 0.641978i \(-0.778114\pi\)
−0.766723 + 0.641978i \(0.778114\pi\)
\(258\) 0 0
\(259\) 13.8745 0.862120
\(260\) −8.70850 −0.540078
\(261\) 0 0
\(262\) 21.8745 1.35141
\(263\) 30.5830 1.88583 0.942914 0.333035i \(-0.108073\pi\)
0.942914 + 0.333035i \(0.108073\pi\)
\(264\) 0 0
\(265\) −2.70850 −0.166382
\(266\) 0.708497 0.0434408
\(267\) 0 0
\(268\) −14.9373 −0.912438
\(269\) 11.4170 0.696106 0.348053 0.937475i \(-0.386843\pi\)
0.348053 + 0.937475i \(0.386843\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 3.29150 0.199577
\(273\) 0 0
\(274\) −21.8745 −1.32149
\(275\) −3.77124 −0.227415
\(276\) 0 0
\(277\) 5.29150 0.317936 0.158968 0.987284i \(-0.449183\pi\)
0.158968 + 0.987284i \(0.449183\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −3.29150 −0.196705
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −21.5203 −1.27925 −0.639623 0.768689i \(-0.720910\pi\)
−0.639623 + 0.768689i \(0.720910\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 8.70850 0.514944
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −6.16601 −0.362706
\(290\) 15.2915 0.897948
\(291\) 0 0
\(292\) −7.29150 −0.426703
\(293\) 32.2288 1.88282 0.941412 0.337259i \(-0.109500\pi\)
0.941412 + 0.337259i \(0.109500\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.93725 0.403220
\(297\) 0 0
\(298\) −10.3542 −0.599806
\(299\) −5.29150 −0.306015
\(300\) 0 0
\(301\) 0.708497 0.0408371
\(302\) 17.2915 0.995014
\(303\) 0 0
\(304\) 0.354249 0.0203176
\(305\) −0.583005 −0.0333828
\(306\) 0 0
\(307\) 29.8745 1.70503 0.852514 0.522704i \(-0.175077\pi\)
0.852514 + 0.522704i \(0.175077\pi\)
\(308\) 3.29150 0.187551
\(309\) 0 0
\(310\) 2.12549 0.120720
\(311\) −13.1660 −0.746576 −0.373288 0.927716i \(-0.621770\pi\)
−0.373288 + 0.927716i \(0.621770\pi\)
\(312\) 0 0
\(313\) 5.29150 0.299093 0.149547 0.988755i \(-0.452219\pi\)
0.149547 + 0.988755i \(0.452219\pi\)
\(314\) 3.64575 0.205742
\(315\) 0 0
\(316\) 8.58301 0.482832
\(317\) −21.2915 −1.19585 −0.597925 0.801552i \(-0.704008\pi\)
−0.597925 + 0.801552i \(0.704008\pi\)
\(318\) 0 0
\(319\) −15.2915 −0.856160
\(320\) −1.64575 −0.0920003
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 1.16601 0.0648786
\(324\) 0 0
\(325\) −12.1255 −0.672601
\(326\) −0.708497 −0.0392400
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −25.8745 −1.42219 −0.711096 0.703095i \(-0.751801\pi\)
−0.711096 + 0.703095i \(0.751801\pi\)
\(332\) 13.6458 0.748908
\(333\) 0 0
\(334\) 19.1660 1.04872
\(335\) 24.5830 1.34311
\(336\) 0 0
\(337\) 15.1660 0.826145 0.413073 0.910698i \(-0.364456\pi\)
0.413073 + 0.910698i \(0.364456\pi\)
\(338\) 15.0000 0.815892
\(339\) 0 0
\(340\) −5.41699 −0.293778
\(341\) −2.12549 −0.115102
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0.354249 0.0190998
\(345\) 0 0
\(346\) 15.8745 0.853419
\(347\) −15.2915 −0.820891 −0.410445 0.911885i \(-0.634627\pi\)
−0.410445 + 0.911885i \(0.634627\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −4.58301 −0.244972
\(351\) 0 0
\(352\) 1.64575 0.0877188
\(353\) 6.58301 0.350378 0.175189 0.984535i \(-0.443946\pi\)
0.175189 + 0.984535i \(0.443946\pi\)
\(354\) 0 0
\(355\) −9.87451 −0.524084
\(356\) 0 0
\(357\) 0 0
\(358\) −15.2915 −0.808181
\(359\) 21.8745 1.15449 0.577246 0.816570i \(-0.304127\pi\)
0.577246 + 0.816570i \(0.304127\pi\)
\(360\) 0 0
\(361\) −18.8745 −0.993395
\(362\) 15.6458 0.822322
\(363\) 0 0
\(364\) 10.5830 0.554700
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −11.1660 −0.582861 −0.291431 0.956592i \(-0.594131\pi\)
−0.291431 + 0.956592i \(0.594131\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −11.4170 −0.593541
\(371\) 3.29150 0.170886
\(372\) 0 0
\(373\) −26.9373 −1.39476 −0.697379 0.716702i \(-0.745651\pi\)
−0.697379 + 0.716702i \(0.745651\pi\)
\(374\) 5.41699 0.280106
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −49.1660 −2.53218
\(378\) 0 0
\(379\) −33.5203 −1.72182 −0.860910 0.508757i \(-0.830105\pi\)
−0.860910 + 0.508757i \(0.830105\pi\)
\(380\) −0.583005 −0.0299075
\(381\) 0 0
\(382\) −3.29150 −0.168408
\(383\) −31.7490 −1.62230 −0.811149 0.584839i \(-0.801158\pi\)
−0.811149 + 0.584839i \(0.801158\pi\)
\(384\) 0 0
\(385\) −5.41699 −0.276076
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) −1.29150 −0.0655661
\(389\) 7.06275 0.358095 0.179048 0.983840i \(-0.442698\pi\)
0.179048 + 0.983840i \(0.442698\pi\)
\(390\) 0 0
\(391\) −3.29150 −0.166458
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −14.1255 −0.710731
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 14.0000 0.701757
\(399\) 0 0
\(400\) −2.29150 −0.114575
\(401\) −9.87451 −0.493109 −0.246555 0.969129i \(-0.579299\pi\)
−0.246555 + 0.969129i \(0.579299\pi\)
\(402\) 0 0
\(403\) −6.83399 −0.340425
\(404\) −9.29150 −0.462270
\(405\) 0 0
\(406\) −18.5830 −0.922259
\(407\) 11.4170 0.565919
\(408\) 0 0
\(409\) 15.1660 0.749911 0.374955 0.927043i \(-0.377658\pi\)
0.374955 + 0.927043i \(0.377658\pi\)
\(410\) 9.87451 0.487667
\(411\) 0 0
\(412\) −6.70850 −0.330504
\(413\) 0 0
\(414\) 0 0
\(415\) −22.4575 −1.10240
\(416\) 5.29150 0.259437
\(417\) 0 0
\(418\) 0.583005 0.0285157
\(419\) −16.9373 −0.827439 −0.413720 0.910404i \(-0.635771\pi\)
−0.413720 + 0.910404i \(0.635771\pi\)
\(420\) 0 0
\(421\) 22.2288 1.08336 0.541682 0.840584i \(-0.317788\pi\)
0.541682 + 0.840584i \(0.317788\pi\)
\(422\) 11.2915 0.549662
\(423\) 0 0
\(424\) 1.64575 0.0799247
\(425\) −7.54249 −0.365864
\(426\) 0 0
\(427\) 0.708497 0.0342866
\(428\) 13.6458 0.659592
\(429\) 0 0
\(430\) −0.583005 −0.0281150
\(431\) 33.8745 1.63168 0.815839 0.578279i \(-0.196276\pi\)
0.815839 + 0.578279i \(0.196276\pi\)
\(432\) 0 0
\(433\) −18.7085 −0.899073 −0.449537 0.893262i \(-0.648411\pi\)
−0.449537 + 0.893262i \(0.648411\pi\)
\(434\) −2.58301 −0.123988
\(435\) 0 0
\(436\) −18.2288 −0.872999
\(437\) −0.354249 −0.0169460
\(438\) 0 0
\(439\) 27.7490 1.32439 0.662194 0.749332i \(-0.269625\pi\)
0.662194 + 0.749332i \(0.269625\pi\)
\(440\) −2.70850 −0.129123
\(441\) 0 0
\(442\) 17.4170 0.828442
\(443\) −25.1660 −1.19567 −0.597837 0.801618i \(-0.703973\pi\)
−0.597837 + 0.801618i \(0.703973\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −19.8745 −0.941085
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −11.4170 −0.538801 −0.269401 0.963028i \(-0.586826\pi\)
−0.269401 + 0.963028i \(0.586826\pi\)
\(450\) 0 0
\(451\) −9.87451 −0.464972
\(452\) −6.58301 −0.309639
\(453\) 0 0
\(454\) 7.06275 0.331471
\(455\) −17.4170 −0.816521
\(456\) 0 0
\(457\) −0.125492 −0.00587027 −0.00293514 0.999996i \(-0.500934\pi\)
−0.00293514 + 0.999996i \(0.500934\pi\)
\(458\) 9.06275 0.423474
\(459\) 0 0
\(460\) 1.64575 0.0767336
\(461\) −25.7490 −1.19925 −0.599626 0.800281i \(-0.704684\pi\)
−0.599626 + 0.800281i \(0.704684\pi\)
\(462\) 0 0
\(463\) 21.1660 0.983668 0.491834 0.870689i \(-0.336327\pi\)
0.491834 + 0.870689i \(0.336327\pi\)
\(464\) −9.29150 −0.431347
\(465\) 0 0
\(466\) 0 0
\(467\) −13.6458 −0.631450 −0.315725 0.948851i \(-0.602248\pi\)
−0.315725 + 0.948851i \(0.602248\pi\)
\(468\) 0 0
\(469\) −29.8745 −1.37948
\(470\) 9.87451 0.455477
\(471\) 0 0
\(472\) 0 0
\(473\) 0.583005 0.0268066
\(474\) 0 0
\(475\) −0.811762 −0.0372462
\(476\) 6.58301 0.301732
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −42.5830 −1.94567 −0.972834 0.231506i \(-0.925635\pi\)
−0.972834 + 0.231506i \(0.925635\pi\)
\(480\) 0 0
\(481\) 36.7085 1.67376
\(482\) 5.29150 0.241021
\(483\) 0 0
\(484\) −8.29150 −0.376886
\(485\) 2.12549 0.0965136
\(486\) 0 0
\(487\) 21.1660 0.959123 0.479562 0.877508i \(-0.340796\pi\)
0.479562 + 0.877508i \(0.340796\pi\)
\(488\) 0.354249 0.0160361
\(489\) 0 0
\(490\) 4.93725 0.223042
\(491\) 21.8745 0.987183 0.493591 0.869694i \(-0.335684\pi\)
0.493591 + 0.869694i \(0.335684\pi\)
\(492\) 0 0
\(493\) −30.5830 −1.37739
\(494\) 1.87451 0.0843381
\(495\) 0 0
\(496\) −1.29150 −0.0579902
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 31.0405 1.38956 0.694782 0.719220i \(-0.255501\pi\)
0.694782 + 0.719220i \(0.255501\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 25.6458 1.14463
\(503\) 2.12549 0.0947710 0.0473855 0.998877i \(-0.484911\pi\)
0.0473855 + 0.998877i \(0.484911\pi\)
\(504\) 0 0
\(505\) 15.2915 0.680463
\(506\) −1.64575 −0.0731626
\(507\) 0 0
\(508\) −10.5830 −0.469545
\(509\) 27.8745 1.23552 0.617758 0.786368i \(-0.288041\pi\)
0.617758 + 0.786368i \(0.288041\pi\)
\(510\) 0 0
\(511\) −14.5830 −0.645114
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −24.5830 −1.08431
\(515\) 11.0405 0.486503
\(516\) 0 0
\(517\) −9.87451 −0.434280
\(518\) 13.8745 0.609611
\(519\) 0 0
\(520\) −8.70850 −0.381893
\(521\) 28.4575 1.24675 0.623373 0.781925i \(-0.285762\pi\)
0.623373 + 0.781925i \(0.285762\pi\)
\(522\) 0 0
\(523\) −5.06275 −0.221378 −0.110689 0.993855i \(-0.535306\pi\)
−0.110689 + 0.993855i \(0.535306\pi\)
\(524\) 21.8745 0.955592
\(525\) 0 0
\(526\) 30.5830 1.33348
\(527\) −4.25098 −0.185176
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −2.70850 −0.117650
\(531\) 0 0
\(532\) 0.708497 0.0307173
\(533\) −31.7490 −1.37520
\(534\) 0 0
\(535\) −22.4575 −0.970923
\(536\) −14.9373 −0.645191
\(537\) 0 0
\(538\) 11.4170 0.492222
\(539\) −4.93725 −0.212663
\(540\) 0 0
\(541\) 20.5830 0.884933 0.442466 0.896785i \(-0.354104\pi\)
0.442466 + 0.896785i \(0.354104\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 3.29150 0.141122
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) 16.7085 0.714404 0.357202 0.934027i \(-0.383731\pi\)
0.357202 + 0.934027i \(0.383731\pi\)
\(548\) −21.8745 −0.934433
\(549\) 0 0
\(550\) −3.77124 −0.160806
\(551\) −3.29150 −0.140223
\(552\) 0 0
\(553\) 17.1660 0.729973
\(554\) 5.29150 0.224814
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 42.1033 1.78397 0.891986 0.452062i \(-0.149312\pi\)
0.891986 + 0.452062i \(0.149312\pi\)
\(558\) 0 0
\(559\) 1.87451 0.0792832
\(560\) −3.29150 −0.139091
\(561\) 0 0
\(562\) 0 0
\(563\) −45.3948 −1.91316 −0.956581 0.291468i \(-0.905856\pi\)
−0.956581 + 0.291468i \(0.905856\pi\)
\(564\) 0 0
\(565\) 10.8340 0.455789
\(566\) −21.5203 −0.904564
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 8.70850 0.365079 0.182540 0.983199i \(-0.441568\pi\)
0.182540 + 0.983199i \(0.441568\pi\)
\(570\) 0 0
\(571\) 18.9373 0.792499 0.396250 0.918143i \(-0.370311\pi\)
0.396250 + 0.918143i \(0.370311\pi\)
\(572\) 8.70850 0.364121
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 2.29150 0.0955623
\(576\) 0 0
\(577\) 28.7085 1.19515 0.597575 0.801813i \(-0.296131\pi\)
0.597575 + 0.801813i \(0.296131\pi\)
\(578\) −6.16601 −0.256472
\(579\) 0 0
\(580\) 15.2915 0.634945
\(581\) 27.2915 1.13224
\(582\) 0 0
\(583\) 2.70850 0.112174
\(584\) −7.29150 −0.301725
\(585\) 0 0
\(586\) 32.2288 1.33136
\(587\) 20.7085 0.854731 0.427366 0.904079i \(-0.359442\pi\)
0.427366 + 0.904079i \(0.359442\pi\)
\(588\) 0 0
\(589\) −0.457513 −0.0188515
\(590\) 0 0
\(591\) 0 0
\(592\) 6.93725 0.285119
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −10.8340 −0.444150
\(596\) −10.3542 −0.424127
\(597\) 0 0
\(598\) −5.29150 −0.216386
\(599\) −30.5830 −1.24959 −0.624794 0.780790i \(-0.714817\pi\)
−0.624794 + 0.780790i \(0.714817\pi\)
\(600\) 0 0
\(601\) −32.4575 −1.32397 −0.661985 0.749517i \(-0.730286\pi\)
−0.661985 + 0.749517i \(0.730286\pi\)
\(602\) 0.708497 0.0288762
\(603\) 0 0
\(604\) 17.2915 0.703581
\(605\) 13.6458 0.554779
\(606\) 0 0
\(607\) 35.8745 1.45610 0.728051 0.685523i \(-0.240427\pi\)
0.728051 + 0.685523i \(0.240427\pi\)
\(608\) 0.354249 0.0143667
\(609\) 0 0
\(610\) −0.583005 −0.0236052
\(611\) −31.7490 −1.28443
\(612\) 0 0
\(613\) −2.93725 −0.118635 −0.0593173 0.998239i \(-0.518892\pi\)
−0.0593173 + 0.998239i \(0.518892\pi\)
\(614\) 29.8745 1.20564
\(615\) 0 0
\(616\) 3.29150 0.132618
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 12.3542 0.496559 0.248280 0.968688i \(-0.420135\pi\)
0.248280 + 0.968688i \(0.420135\pi\)
\(620\) 2.12549 0.0853618
\(621\) 0 0
\(622\) −13.1660 −0.527909
\(623\) 0 0
\(624\) 0 0
\(625\) −8.29150 −0.331660
\(626\) 5.29150 0.211491
\(627\) 0 0
\(628\) 3.64575 0.145481
\(629\) 22.8340 0.910451
\(630\) 0 0
\(631\) 23.8745 0.950429 0.475215 0.879870i \(-0.342370\pi\)
0.475215 + 0.879870i \(0.342370\pi\)
\(632\) 8.58301 0.341414
\(633\) 0 0
\(634\) −21.2915 −0.845594
\(635\) 17.4170 0.691172
\(636\) 0 0
\(637\) −15.8745 −0.628971
\(638\) −15.2915 −0.605396
\(639\) 0 0
\(640\) −1.64575 −0.0650540
\(641\) 28.4575 1.12400 0.562002 0.827136i \(-0.310031\pi\)
0.562002 + 0.827136i \(0.310031\pi\)
\(642\) 0 0
\(643\) 24.3542 0.960438 0.480219 0.877149i \(-0.340557\pi\)
0.480219 + 0.877149i \(0.340557\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 1.16601 0.0458761
\(647\) −13.1660 −0.517609 −0.258805 0.965930i \(-0.583329\pi\)
−0.258805 + 0.965930i \(0.583329\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −12.1255 −0.475601
\(651\) 0 0
\(652\) −0.708497 −0.0277469
\(653\) 15.8745 0.621217 0.310609 0.950538i \(-0.399467\pi\)
0.310609 + 0.950538i \(0.399467\pi\)
\(654\) 0 0
\(655\) −36.0000 −1.40664
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) −7.06275 −0.275126 −0.137563 0.990493i \(-0.543927\pi\)
−0.137563 + 0.990493i \(0.543927\pi\)
\(660\) 0 0
\(661\) 27.6458 1.07530 0.537648 0.843170i \(-0.319313\pi\)
0.537648 + 0.843170i \(0.319313\pi\)
\(662\) −25.8745 −1.00564
\(663\) 0 0
\(664\) 13.6458 0.529558
\(665\) −1.16601 −0.0452159
\(666\) 0 0
\(667\) 9.29150 0.359768
\(668\) 19.1660 0.741555
\(669\) 0 0
\(670\) 24.5830 0.949724
\(671\) 0.583005 0.0225067
\(672\) 0 0
\(673\) −12.7085 −0.489877 −0.244938 0.969539i \(-0.578768\pi\)
−0.244938 + 0.969539i \(0.578768\pi\)
\(674\) 15.1660 0.584173
\(675\) 0 0
\(676\) 15.0000 0.576923
\(677\) 7.06275 0.271443 0.135722 0.990747i \(-0.456665\pi\)
0.135722 + 0.990747i \(0.456665\pi\)
\(678\) 0 0
\(679\) −2.58301 −0.0991266
\(680\) −5.41699 −0.207732
\(681\) 0 0
\(682\) −2.12549 −0.0813893
\(683\) 33.8745 1.29617 0.648086 0.761567i \(-0.275570\pi\)
0.648086 + 0.761567i \(0.275570\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 0.354249 0.0135056
\(689\) 8.70850 0.331767
\(690\) 0 0
\(691\) 17.8745 0.679978 0.339989 0.940429i \(-0.389577\pi\)
0.339989 + 0.940429i \(0.389577\pi\)
\(692\) 15.8745 0.603458
\(693\) 0 0
\(694\) −15.2915 −0.580458
\(695\) 6.58301 0.249708
\(696\) 0 0
\(697\) −19.7490 −0.748047
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) −4.58301 −0.173221
\(701\) 19.0627 0.719990 0.359995 0.932954i \(-0.382778\pi\)
0.359995 + 0.932954i \(0.382778\pi\)
\(702\) 0 0
\(703\) 2.45751 0.0926869
\(704\) 1.64575 0.0620266
\(705\) 0 0
\(706\) 6.58301 0.247755
\(707\) −18.5830 −0.698886
\(708\) 0 0
\(709\) −38.9373 −1.46232 −0.731160 0.682206i \(-0.761021\pi\)
−0.731160 + 0.682206i \(0.761021\pi\)
\(710\) −9.87451 −0.370584
\(711\) 0 0
\(712\) 0 0
\(713\) 1.29150 0.0483672
\(714\) 0 0
\(715\) −14.3320 −0.535987
\(716\) −15.2915 −0.571470
\(717\) 0 0
\(718\) 21.8745 0.816349
\(719\) 19.7490 0.736514 0.368257 0.929724i \(-0.379955\pi\)
0.368257 + 0.929724i \(0.379955\pi\)
\(720\) 0 0
\(721\) −13.4170 −0.499675
\(722\) −18.8745 −0.702436
\(723\) 0 0
\(724\) 15.6458 0.581470
\(725\) 21.2915 0.790747
\(726\) 0 0
\(727\) 40.3320 1.49583 0.747916 0.663793i \(-0.231055\pi\)
0.747916 + 0.663793i \(0.231055\pi\)
\(728\) 10.5830 0.392232
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 1.16601 0.0431265
\(732\) 0 0
\(733\) −31.3948 −1.15959 −0.579796 0.814762i \(-0.696868\pi\)
−0.579796 + 0.814762i \(0.696868\pi\)
\(734\) −11.1660 −0.412145
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −24.5830 −0.905527
\(738\) 0 0
\(739\) −10.5830 −0.389302 −0.194651 0.980873i \(-0.562357\pi\)
−0.194651 + 0.980873i \(0.562357\pi\)
\(740\) −11.4170 −0.419697
\(741\) 0 0
\(742\) 3.29150 0.120835
\(743\) 19.7490 0.724521 0.362261 0.932077i \(-0.382005\pi\)
0.362261 + 0.932077i \(0.382005\pi\)
\(744\) 0 0
\(745\) 17.0405 0.624316
\(746\) −26.9373 −0.986243
\(747\) 0 0
\(748\) 5.41699 0.198065
\(749\) 27.2915 0.997210
\(750\) 0 0
\(751\) 42.4575 1.54930 0.774648 0.632392i \(-0.217927\pi\)
0.774648 + 0.632392i \(0.217927\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −49.1660 −1.79052
\(755\) −28.4575 −1.03567
\(756\) 0 0
\(757\) 5.77124 0.209759 0.104880 0.994485i \(-0.466554\pi\)
0.104880 + 0.994485i \(0.466554\pi\)
\(758\) −33.5203 −1.21751
\(759\) 0 0
\(760\) −0.583005 −0.0211478
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) −36.4575 −1.31985
\(764\) −3.29150 −0.119082
\(765\) 0 0
\(766\) −31.7490 −1.14714
\(767\) 0 0
\(768\) 0 0
\(769\) 25.0405 0.902984 0.451492 0.892275i \(-0.350892\pi\)
0.451492 + 0.892275i \(0.350892\pi\)
\(770\) −5.41699 −0.195215
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 1.64575 0.0591936 0.0295968 0.999562i \(-0.490578\pi\)
0.0295968 + 0.999562i \(0.490578\pi\)
\(774\) 0 0
\(775\) 2.95948 0.106308
\(776\) −1.29150 −0.0463622
\(777\) 0 0
\(778\) 7.06275 0.253212
\(779\) −2.12549 −0.0761537
\(780\) 0 0
\(781\) 9.87451 0.353338
\(782\) −3.29150 −0.117704
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) −44.3542 −1.58106 −0.790529 0.612424i \(-0.790194\pi\)
−0.790529 + 0.612424i \(0.790194\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −14.1255 −0.502562
\(791\) −13.1660 −0.468129
\(792\) 0 0
\(793\) 1.87451 0.0665657
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) 35.5203 1.25819 0.629096 0.777328i \(-0.283425\pi\)
0.629096 + 0.777328i \(0.283425\pi\)
\(798\) 0 0
\(799\) −19.7490 −0.698670
\(800\) −2.29150 −0.0810169
\(801\) 0 0
\(802\) −9.87451 −0.348681
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 3.29150 0.116010
\(806\) −6.83399 −0.240717
\(807\) 0 0
\(808\) −9.29150 −0.326874
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 29.8745 1.04904 0.524518 0.851399i \(-0.324246\pi\)
0.524518 + 0.851399i \(0.324246\pi\)
\(812\) −18.5830 −0.652136
\(813\) 0 0
\(814\) 11.4170 0.400165
\(815\) 1.16601 0.0408436
\(816\) 0 0
\(817\) 0.125492 0.00439041
\(818\) 15.1660 0.530267
\(819\) 0 0
\(820\) 9.87451 0.344833
\(821\) 8.12549 0.283582 0.141791 0.989897i \(-0.454714\pi\)
0.141791 + 0.989897i \(0.454714\pi\)
\(822\) 0 0
\(823\) −13.2915 −0.463313 −0.231656 0.972798i \(-0.574414\pi\)
−0.231656 + 0.972798i \(0.574414\pi\)
\(824\) −6.70850 −0.233702
\(825\) 0 0
\(826\) 0 0
\(827\) 14.8118 0.515055 0.257528 0.966271i \(-0.417092\pi\)
0.257528 + 0.966271i \(0.417092\pi\)
\(828\) 0 0
\(829\) −15.4170 −0.535454 −0.267727 0.963495i \(-0.586273\pi\)
−0.267727 + 0.963495i \(0.586273\pi\)
\(830\) −22.4575 −0.779512
\(831\) 0 0
\(832\) 5.29150 0.183450
\(833\) −9.87451 −0.342131
\(834\) 0 0
\(835\) −31.5425 −1.09157
\(836\) 0.583005 0.0201637
\(837\) 0 0
\(838\) −16.9373 −0.585088
\(839\) −43.7490 −1.51038 −0.755192 0.655504i \(-0.772456\pi\)
−0.755192 + 0.655504i \(0.772456\pi\)
\(840\) 0 0
\(841\) 57.3320 1.97697
\(842\) 22.2288 0.766054
\(843\) 0 0
\(844\) 11.2915 0.388670
\(845\) −24.6863 −0.849233
\(846\) 0 0
\(847\) −16.5830 −0.569799
\(848\) 1.64575 0.0565153
\(849\) 0 0
\(850\) −7.54249 −0.258705
\(851\) −6.93725 −0.237806
\(852\) 0 0
\(853\) 27.1660 0.930146 0.465073 0.885272i \(-0.346028\pi\)
0.465073 + 0.885272i \(0.346028\pi\)
\(854\) 0.708497 0.0242443
\(855\) 0 0
\(856\) 13.6458 0.466402
\(857\) −36.0000 −1.22974 −0.614868 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) 13.4170 0.457782 0.228891 0.973452i \(-0.426490\pi\)
0.228891 + 0.973452i \(0.426490\pi\)
\(860\) −0.583005 −0.0198803
\(861\) 0 0
\(862\) 33.8745 1.15377
\(863\) −10.8340 −0.368793 −0.184397 0.982852i \(-0.559033\pi\)
−0.184397 + 0.982852i \(0.559033\pi\)
\(864\) 0 0
\(865\) −26.1255 −0.888293
\(866\) −18.7085 −0.635741
\(867\) 0 0
\(868\) −2.58301 −0.0876729
\(869\) 14.1255 0.479175
\(870\) 0 0
\(871\) −79.0405 −2.67819
\(872\) −18.2288 −0.617304
\(873\) 0 0
\(874\) −0.354249 −0.0119826
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −13.2915 −0.448822 −0.224411 0.974495i \(-0.572046\pi\)
−0.224411 + 0.974495i \(0.572046\pi\)
\(878\) 27.7490 0.936484
\(879\) 0 0
\(880\) −2.70850 −0.0913034
\(881\) −31.7490 −1.06965 −0.534826 0.844962i \(-0.679623\pi\)
−0.534826 + 0.844962i \(0.679623\pi\)
\(882\) 0 0
\(883\) −7.29150 −0.245379 −0.122689 0.992445i \(-0.539152\pi\)
−0.122689 + 0.992445i \(0.539152\pi\)
\(884\) 17.4170 0.585797
\(885\) 0 0
\(886\) −25.1660 −0.845469
\(887\) −25.7490 −0.864567 −0.432284 0.901738i \(-0.642292\pi\)
−0.432284 + 0.901738i \(0.642292\pi\)
\(888\) 0 0
\(889\) −21.1660 −0.709885
\(890\) 0 0
\(891\) 0 0
\(892\) −19.8745 −0.665448
\(893\) −2.12549 −0.0711269
\(894\) 0 0
\(895\) 25.1660 0.841207
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −11.4170 −0.380990
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 5.41699 0.180466
\(902\) −9.87451 −0.328785
\(903\) 0 0
\(904\) −6.58301 −0.218947
\(905\) −25.7490 −0.855926
\(906\) 0 0
\(907\) −45.5203 −1.51148 −0.755738 0.654874i \(-0.772722\pi\)
−0.755738 + 0.654874i \(0.772722\pi\)
\(908\) 7.06275 0.234385
\(909\) 0 0
\(910\) −17.4170 −0.577368
\(911\) −3.29150 −0.109052 −0.0545262 0.998512i \(-0.517365\pi\)
−0.0545262 + 0.998512i \(0.517365\pi\)
\(912\) 0 0
\(913\) 22.4575 0.743235
\(914\) −0.125492 −0.00415091
\(915\) 0 0
\(916\) 9.06275 0.299442
\(917\) 43.7490 1.44472
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 1.64575 0.0542588
\(921\) 0 0
\(922\) −25.7490 −0.847999
\(923\) 31.7490 1.04503
\(924\) 0 0
\(925\) −15.8967 −0.522681
\(926\) 21.1660 0.695558
\(927\) 0 0
\(928\) −9.29150 −0.305009
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −1.06275 −0.0348301
\(932\) 0 0
\(933\) 0 0
\(934\) −13.6458 −0.446503
\(935\) −8.91503 −0.291553
\(936\) 0 0
\(937\) −28.5830 −0.933766 −0.466883 0.884319i \(-0.654623\pi\)
−0.466883 + 0.884319i \(0.654623\pi\)
\(938\) −29.8745 −0.975437
\(939\) 0 0
\(940\) 9.87451 0.322071
\(941\) 26.8118 0.874038 0.437019 0.899452i \(-0.356034\pi\)
0.437019 + 0.899452i \(0.356034\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0.583005 0.0189551
\(947\) −2.12549 −0.0690692 −0.0345346 0.999404i \(-0.510995\pi\)
−0.0345346 + 0.999404i \(0.510995\pi\)
\(948\) 0 0
\(949\) −38.5830 −1.25246
\(950\) −0.811762 −0.0263370
\(951\) 0 0
\(952\) 6.58301 0.213356
\(953\) 23.0405 0.746356 0.373178 0.927760i \(-0.378268\pi\)
0.373178 + 0.927760i \(0.378268\pi\)
\(954\) 0 0
\(955\) 5.41699 0.175290
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −42.5830 −1.37579
\(959\) −43.7490 −1.41273
\(960\) 0 0
\(961\) −29.3320 −0.946194
\(962\) 36.7085 1.18353
\(963\) 0 0
\(964\) 5.29150 0.170428
\(965\) 36.2065 1.16553
\(966\) 0 0
\(967\) 23.8745 0.767752 0.383876 0.923385i \(-0.374589\pi\)
0.383876 + 0.923385i \(0.374589\pi\)
\(968\) −8.29150 −0.266499
\(969\) 0 0
\(970\) 2.12549 0.0682454
\(971\) 20.2288 0.649172 0.324586 0.945856i \(-0.394775\pi\)
0.324586 + 0.945856i \(0.394775\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 21.1660 0.678203
\(975\) 0 0
\(976\) 0.354249 0.0113392
\(977\) 21.8745 0.699828 0.349914 0.936782i \(-0.386211\pi\)
0.349914 + 0.936782i \(0.386211\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 4.93725 0.157715
\(981\) 0 0
\(982\) 21.8745 0.698044
\(983\) 21.8745 0.697688 0.348844 0.937181i \(-0.386574\pi\)
0.348844 + 0.937181i \(0.386574\pi\)
\(984\) 0 0
\(985\) −9.87451 −0.314628
\(986\) −30.5830 −0.973961
\(987\) 0 0
\(988\) 1.87451 0.0596360
\(989\) −0.354249 −0.0112645
\(990\) 0 0
\(991\) −14.4575 −0.459258 −0.229629 0.973278i \(-0.573751\pi\)
−0.229629 + 0.973278i \(0.573751\pi\)
\(992\) −1.29150 −0.0410052
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) −23.0405 −0.730434
\(996\) 0 0
\(997\) −40.5830 −1.28528 −0.642638 0.766170i \(-0.722160\pi\)
−0.642638 + 0.766170i \(0.722160\pi\)
\(998\) 31.0405 0.982570
\(999\) 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 414.2.a.g.1.1 yes 2
3.2 odd 2 414.2.a.e.1.2 2
4.3 odd 2 3312.2.a.z.1.1 2
12.11 even 2 3312.2.a.v.1.2 2
23.22 odd 2 9522.2.a.bc.1.2 2
69.68 even 2 9522.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.2.a.e.1.2 2 3.2 odd 2
414.2.a.g.1.1 yes 2 1.1 even 1 trivial
3312.2.a.v.1.2 2 12.11 even 2
3312.2.a.z.1.1 2 4.3 odd 2
9522.2.a.bb.1.1 2 69.68 even 2
9522.2.a.bc.1.2 2 23.22 odd 2