Properties

Label 3675.2.a.bs.1.3
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 735)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.517638 q^{2} -1.00000 q^{3} -1.73205 q^{4} -0.517638 q^{6} -1.93185 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.517638 q^{2} -1.00000 q^{3} -1.73205 q^{4} -0.517638 q^{6} -1.93185 q^{8} +1.00000 q^{9} +3.46410 q^{11} +1.73205 q^{12} +4.00000 q^{13} +2.46410 q^{16} +4.00000 q^{17} +0.517638 q^{18} -5.27792 q^{19} +1.79315 q^{22} -3.48477 q^{23} +1.93185 q^{24} +2.07055 q^{26} -1.00000 q^{27} -4.92820 q^{29} +7.34847 q^{31} +5.13922 q^{32} -3.46410 q^{33} +2.07055 q^{34} -1.73205 q^{36} -10.5558 q^{37} -2.73205 q^{38} -4.00000 q^{39} -8.48528 q^{41} -6.00000 q^{44} -1.80385 q^{46} +6.00000 q^{47} -2.46410 q^{48} -4.00000 q^{51} -6.92820 q^{52} +7.34847 q^{53} -0.517638 q^{54} +5.27792 q^{57} -2.55103 q^{58} +0.757875 q^{59} -0.656339 q^{61} +3.80385 q^{62} -2.26795 q^{64} -1.79315 q^{66} +12.6264 q^{67} -6.92820 q^{68} +3.48477 q^{69} -6.39230 q^{71} -1.93185 q^{72} -2.92820 q^{73} -5.46410 q^{74} +9.14162 q^{76} -2.07055 q^{78} -4.53590 q^{79} +1.00000 q^{81} -4.39230 q^{82} +6.00000 q^{83} +4.92820 q^{87} -6.69213 q^{88} +15.4548 q^{89} +6.03579 q^{92} -7.34847 q^{93} +3.10583 q^{94} -5.13922 q^{96} +18.9282 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{9} + 16 q^{13} - 4 q^{16} + 16 q^{17} - 4 q^{27} + 8 q^{29} - 4 q^{38} - 16 q^{39} - 24 q^{44} - 28 q^{46} + 24 q^{47} + 4 q^{48} - 16 q^{51} + 36 q^{62} - 16 q^{64} + 16 q^{71} + 16 q^{73} - 8 q^{74} - 32 q^{79} + 4 q^{81} + 24 q^{82} + 24 q^{83} - 8 q^{87} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.517638 0.366025 0.183013 0.983111i \(-0.441415\pi\)
0.183013 + 0.983111i \(0.441415\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.73205 −0.866025
\(5\) 0 0
\(6\) −0.517638 −0.211325
\(7\) 0 0
\(8\) −1.93185 −0.683013
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 1.73205 0.500000
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0.517638 0.122008
\(19\) −5.27792 −1.21084 −0.605419 0.795907i \(-0.706994\pi\)
−0.605419 + 0.795907i \(0.706994\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.79315 0.382301
\(23\) −3.48477 −0.726624 −0.363312 0.931668i \(-0.618354\pi\)
−0.363312 + 0.931668i \(0.618354\pi\)
\(24\) 1.93185 0.394338
\(25\) 0 0
\(26\) 2.07055 0.406069
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.92820 −0.915144 −0.457572 0.889172i \(-0.651281\pi\)
−0.457572 + 0.889172i \(0.651281\pi\)
\(30\) 0 0
\(31\) 7.34847 1.31982 0.659912 0.751343i \(-0.270594\pi\)
0.659912 + 0.751343i \(0.270594\pi\)
\(32\) 5.13922 0.908494
\(33\) −3.46410 −0.603023
\(34\) 2.07055 0.355097
\(35\) 0 0
\(36\) −1.73205 −0.288675
\(37\) −10.5558 −1.73537 −0.867684 0.497116i \(-0.834392\pi\)
−0.867684 + 0.497116i \(0.834392\pi\)
\(38\) −2.73205 −0.443197
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −8.48528 −1.32518 −0.662589 0.748983i \(-0.730542\pi\)
−0.662589 + 0.748983i \(0.730542\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −1.80385 −0.265963
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −2.46410 −0.355662
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −6.92820 −0.960769
\(53\) 7.34847 1.00939 0.504695 0.863298i \(-0.331605\pi\)
0.504695 + 0.863298i \(0.331605\pi\)
\(54\) −0.517638 −0.0704416
\(55\) 0 0
\(56\) 0 0
\(57\) 5.27792 0.699077
\(58\) −2.55103 −0.334966
\(59\) 0.757875 0.0986669 0.0493334 0.998782i \(-0.484290\pi\)
0.0493334 + 0.998782i \(0.484290\pi\)
\(60\) 0 0
\(61\) −0.656339 −0.0840356 −0.0420178 0.999117i \(-0.513379\pi\)
−0.0420178 + 0.999117i \(0.513379\pi\)
\(62\) 3.80385 0.483089
\(63\) 0 0
\(64\) −2.26795 −0.283494
\(65\) 0 0
\(66\) −1.79315 −0.220722
\(67\) 12.6264 1.54256 0.771279 0.636497i \(-0.219617\pi\)
0.771279 + 0.636497i \(0.219617\pi\)
\(68\) −6.92820 −0.840168
\(69\) 3.48477 0.419517
\(70\) 0 0
\(71\) −6.39230 −0.758627 −0.379314 0.925268i \(-0.623840\pi\)
−0.379314 + 0.925268i \(0.623840\pi\)
\(72\) −1.93185 −0.227671
\(73\) −2.92820 −0.342720 −0.171360 0.985208i \(-0.554816\pi\)
−0.171360 + 0.985208i \(0.554816\pi\)
\(74\) −5.46410 −0.635189
\(75\) 0 0
\(76\) 9.14162 1.04862
\(77\) 0 0
\(78\) −2.07055 −0.234444
\(79\) −4.53590 −0.510328 −0.255164 0.966898i \(-0.582130\pi\)
−0.255164 + 0.966898i \(0.582130\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −4.39230 −0.485049
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.92820 0.528359
\(88\) −6.69213 −0.713384
\(89\) 15.4548 1.63821 0.819103 0.573646i \(-0.194471\pi\)
0.819103 + 0.573646i \(0.194471\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.03579 0.629275
\(93\) −7.34847 −0.762001
\(94\) 3.10583 0.320342
\(95\) 0 0
\(96\) −5.13922 −0.524519
\(97\) 18.9282 1.92187 0.960934 0.276778i \(-0.0892666\pi\)
0.960934 + 0.276778i \(0.0892666\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
\(100\) 0 0
\(101\) 6.96953 0.693494 0.346747 0.937959i \(-0.387286\pi\)
0.346747 + 0.937959i \(0.387286\pi\)
\(102\) −2.07055 −0.205015
\(103\) 13.8564 1.36531 0.682656 0.730740i \(-0.260825\pi\)
0.682656 + 0.730740i \(0.260825\pi\)
\(104\) −7.72741 −0.757735
\(105\) 0 0
\(106\) 3.80385 0.369462
\(107\) −15.5563 −1.50389 −0.751945 0.659226i \(-0.770884\pi\)
−0.751945 + 0.659226i \(0.770884\pi\)
\(108\) 1.73205 0.166667
\(109\) 15.8564 1.51877 0.759384 0.650643i \(-0.225500\pi\)
0.759384 + 0.650643i \(0.225500\pi\)
\(110\) 0 0
\(111\) 10.5558 1.00192
\(112\) 0 0
\(113\) 5.27792 0.496505 0.248252 0.968695i \(-0.420144\pi\)
0.248252 + 0.968695i \(0.420144\pi\)
\(114\) 2.73205 0.255880
\(115\) 0 0
\(116\) 8.53590 0.792538
\(117\) 4.00000 0.369800
\(118\) 0.392305 0.0361146
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.339746 −0.0307592
\(123\) 8.48528 0.765092
\(124\) −12.7279 −1.14300
\(125\) 0 0
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) −11.4524 −1.01226
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) 6.53590 0.564616
\(135\) 0 0
\(136\) −7.72741 −0.662620
\(137\) −10.1769 −0.869471 −0.434735 0.900558i \(-0.643158\pi\)
−0.434735 + 0.900558i \(0.643158\pi\)
\(138\) 1.80385 0.153554
\(139\) 0.378937 0.0321410 0.0160705 0.999871i \(-0.494884\pi\)
0.0160705 + 0.999871i \(0.494884\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −3.30890 −0.277677
\(143\) 13.8564 1.15873
\(144\) 2.46410 0.205342
\(145\) 0 0
\(146\) −1.51575 −0.125444
\(147\) 0 0
\(148\) 18.2832 1.50287
\(149\) 7.07180 0.579344 0.289672 0.957126i \(-0.406454\pi\)
0.289672 + 0.957126i \(0.406454\pi\)
\(150\) 0 0
\(151\) −19.4641 −1.58397 −0.791983 0.610543i \(-0.790951\pi\)
−0.791983 + 0.610543i \(0.790951\pi\)
\(152\) 10.1962 0.827017
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 6.92820 0.554700
\(157\) 5.07180 0.404773 0.202387 0.979306i \(-0.435130\pi\)
0.202387 + 0.979306i \(0.435130\pi\)
\(158\) −2.34795 −0.186793
\(159\) −7.34847 −0.582772
\(160\) 0 0
\(161\) 0 0
\(162\) 0.517638 0.0406695
\(163\) 22.4243 1.75641 0.878205 0.478284i \(-0.158741\pi\)
0.878205 + 0.478284i \(0.158741\pi\)
\(164\) 14.6969 1.14764
\(165\) 0 0
\(166\) 3.10583 0.241059
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −5.27792 −0.403612
\(172\) 0 0
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 2.55103 0.193393
\(175\) 0 0
\(176\) 8.53590 0.643418
\(177\) −0.757875 −0.0569654
\(178\) 8.00000 0.599625
\(179\) 10.3923 0.776757 0.388379 0.921500i \(-0.373035\pi\)
0.388379 + 0.921500i \(0.373035\pi\)
\(180\) 0 0
\(181\) 6.31319 0.469256 0.234628 0.972085i \(-0.424613\pi\)
0.234628 + 0.972085i \(0.424613\pi\)
\(182\) 0 0
\(183\) 0.656339 0.0485180
\(184\) 6.73205 0.496293
\(185\) 0 0
\(186\) −3.80385 −0.278912
\(187\) 13.8564 1.01328
\(188\) −10.3923 −0.757937
\(189\) 0 0
\(190\) 0 0
\(191\) −7.46410 −0.540083 −0.270042 0.962849i \(-0.587037\pi\)
−0.270042 + 0.962849i \(0.587037\pi\)
\(192\) 2.26795 0.163675
\(193\) 6.41473 0.461742 0.230871 0.972984i \(-0.425842\pi\)
0.230871 + 0.972984i \(0.425842\pi\)
\(194\) 9.79796 0.703452
\(195\) 0 0
\(196\) 0 0
\(197\) 22.8033 1.62467 0.812333 0.583193i \(-0.198197\pi\)
0.812333 + 0.583193i \(0.198197\pi\)
\(198\) 1.79315 0.127434
\(199\) −8.10634 −0.574643 −0.287322 0.957834i \(-0.592765\pi\)
−0.287322 + 0.957834i \(0.592765\pi\)
\(200\) 0 0
\(201\) −12.6264 −0.890597
\(202\) 3.60770 0.253837
\(203\) 0 0
\(204\) 6.92820 0.485071
\(205\) 0 0
\(206\) 7.17260 0.499739
\(207\) −3.48477 −0.242208
\(208\) 9.85641 0.683419
\(209\) −18.2832 −1.26468
\(210\) 0 0
\(211\) −26.9282 −1.85381 −0.926907 0.375291i \(-0.877543\pi\)
−0.926907 + 0.375291i \(0.877543\pi\)
\(212\) −12.7279 −0.874157
\(213\) 6.39230 0.437994
\(214\) −8.05256 −0.550462
\(215\) 0 0
\(216\) 1.93185 0.131446
\(217\) 0 0
\(218\) 8.20788 0.555908
\(219\) 2.92820 0.197870
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 5.46410 0.366726
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.73205 0.181733
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −9.14162 −0.605419
\(229\) 13.4858 0.891167 0.445583 0.895240i \(-0.352996\pi\)
0.445583 + 0.895240i \(0.352996\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.52056 0.625055
\(233\) 5.27792 0.345768 0.172884 0.984942i \(-0.444691\pi\)
0.172884 + 0.984942i \(0.444691\pi\)
\(234\) 2.07055 0.135356
\(235\) 0 0
\(236\) −1.31268 −0.0854480
\(237\) 4.53590 0.294638
\(238\) 0 0
\(239\) −8.53590 −0.552141 −0.276071 0.961137i \(-0.589032\pi\)
−0.276071 + 0.961137i \(0.589032\pi\)
\(240\) 0 0
\(241\) −14.0406 −0.904435 −0.452217 0.891908i \(-0.649367\pi\)
−0.452217 + 0.891908i \(0.649367\pi\)
\(242\) 0.517638 0.0332750
\(243\) −1.00000 −0.0641500
\(244\) 1.13681 0.0727769
\(245\) 0 0
\(246\) 4.39230 0.280043
\(247\) −21.1117 −1.34330
\(248\) −14.1962 −0.901457
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −20.3538 −1.28472 −0.642360 0.766403i \(-0.722045\pi\)
−0.642360 + 0.766403i \(0.722045\pi\)
\(252\) 0 0
\(253\) −12.0716 −0.758934
\(254\) −1.46410 −0.0918659
\(255\) 0 0
\(256\) −1.39230 −0.0870191
\(257\) −3.46410 −0.216085 −0.108042 0.994146i \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.92820 −0.305048
\(262\) 2.92820 0.180905
\(263\) −7.62587 −0.470231 −0.235116 0.971967i \(-0.575547\pi\)
−0.235116 + 0.971967i \(0.575547\pi\)
\(264\) 6.69213 0.411872
\(265\) 0 0
\(266\) 0 0
\(267\) −15.4548 −0.945819
\(268\) −21.8695 −1.33589
\(269\) −30.9096 −1.88459 −0.942297 0.334779i \(-0.891338\pi\)
−0.942297 + 0.334779i \(0.891338\pi\)
\(270\) 0 0
\(271\) 1.69161 0.102758 0.0513791 0.998679i \(-0.483638\pi\)
0.0513791 + 0.998679i \(0.483638\pi\)
\(272\) 9.85641 0.597632
\(273\) 0 0
\(274\) −5.26795 −0.318248
\(275\) 0 0
\(276\) −6.03579 −0.363312
\(277\) 11.3137 0.679775 0.339887 0.940466i \(-0.389611\pi\)
0.339887 + 0.940466i \(0.389611\pi\)
\(278\) 0.196152 0.0117644
\(279\) 7.34847 0.439941
\(280\) 0 0
\(281\) 27.8564 1.66177 0.830887 0.556441i \(-0.187834\pi\)
0.830887 + 0.556441i \(0.187834\pi\)
\(282\) −3.10583 −0.184949
\(283\) 2.14359 0.127423 0.0637117 0.997968i \(-0.479706\pi\)
0.0637117 + 0.997968i \(0.479706\pi\)
\(284\) 11.0718 0.656990
\(285\) 0 0
\(286\) 7.17260 0.424125
\(287\) 0 0
\(288\) 5.13922 0.302831
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −18.9282 −1.10959
\(292\) 5.07180 0.296804
\(293\) 11.4641 0.669740 0.334870 0.942264i \(-0.391308\pi\)
0.334870 + 0.942264i \(0.391308\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 20.3923 1.18528
\(297\) −3.46410 −0.201008
\(298\) 3.66063 0.212055
\(299\) −13.9391 −0.806117
\(300\) 0 0
\(301\) 0 0
\(302\) −10.0754 −0.579772
\(303\) −6.96953 −0.400389
\(304\) −13.0053 −0.745906
\(305\) 0 0
\(306\) 2.07055 0.118366
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −13.8564 −0.788263
\(310\) 0 0
\(311\) −4.14110 −0.234821 −0.117410 0.993083i \(-0.537459\pi\)
−0.117410 + 0.993083i \(0.537459\pi\)
\(312\) 7.72741 0.437478
\(313\) −2.92820 −0.165512 −0.0827559 0.996570i \(-0.526372\pi\)
−0.0827559 + 0.996570i \(0.526372\pi\)
\(314\) 2.62536 0.148157
\(315\) 0 0
\(316\) 7.85641 0.441957
\(317\) 4.72311 0.265277 0.132638 0.991165i \(-0.457655\pi\)
0.132638 + 0.991165i \(0.457655\pi\)
\(318\) −3.80385 −0.213309
\(319\) −17.0718 −0.955837
\(320\) 0 0
\(321\) 15.5563 0.868271
\(322\) 0 0
\(323\) −21.1117 −1.17468
\(324\) −1.73205 −0.0962250
\(325\) 0 0
\(326\) 11.6077 0.642891
\(327\) −15.8564 −0.876861
\(328\) 16.3923 0.905114
\(329\) 0 0
\(330\) 0 0
\(331\) −8.78461 −0.482846 −0.241423 0.970420i \(-0.577614\pi\)
−0.241423 + 0.970420i \(0.577614\pi\)
\(332\) −10.3923 −0.570352
\(333\) −10.5558 −0.578456
\(334\) 9.79796 0.536120
\(335\) 0 0
\(336\) 0 0
\(337\) 17.7284 0.965730 0.482865 0.875695i \(-0.339596\pi\)
0.482865 + 0.875695i \(0.339596\pi\)
\(338\) 1.55291 0.0844674
\(339\) −5.27792 −0.286657
\(340\) 0 0
\(341\) 25.4558 1.37851
\(342\) −2.73205 −0.147732
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 8.28221 0.445254
\(347\) 31.0112 1.66477 0.832383 0.554200i \(-0.186976\pi\)
0.832383 + 0.554200i \(0.186976\pi\)
\(348\) −8.53590 −0.457572
\(349\) 17.6269 0.943546 0.471773 0.881720i \(-0.343614\pi\)
0.471773 + 0.881720i \(0.343614\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 17.8028 0.948891
\(353\) 11.4641 0.610173 0.305086 0.952325i \(-0.401315\pi\)
0.305086 + 0.952325i \(0.401315\pi\)
\(354\) −0.392305 −0.0208508
\(355\) 0 0
\(356\) −26.7685 −1.41873
\(357\) 0 0
\(358\) 5.37945 0.284313
\(359\) 15.4641 0.816164 0.408082 0.912945i \(-0.366198\pi\)
0.408082 + 0.912945i \(0.366198\pi\)
\(360\) 0 0
\(361\) 8.85641 0.466127
\(362\) 3.26795 0.171760
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0.339746 0.0177588
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −8.58682 −0.447619
\(369\) −8.48528 −0.441726
\(370\) 0 0
\(371\) 0 0
\(372\) 12.7279 0.659912
\(373\) −7.17260 −0.371383 −0.185692 0.982608i \(-0.559453\pi\)
−0.185692 + 0.982608i \(0.559453\pi\)
\(374\) 7.17260 0.370887
\(375\) 0 0
\(376\) −11.5911 −0.597766
\(377\) −19.7128 −1.01526
\(378\) 0 0
\(379\) 11.4641 0.588871 0.294436 0.955671i \(-0.404868\pi\)
0.294436 + 0.955671i \(0.404868\pi\)
\(380\) 0 0
\(381\) 2.82843 0.144905
\(382\) −3.86370 −0.197684
\(383\) −23.8564 −1.21901 −0.609503 0.792784i \(-0.708631\pi\)
−0.609503 + 0.792784i \(0.708631\pi\)
\(384\) 11.4524 0.584428
\(385\) 0 0
\(386\) 3.32051 0.169009
\(387\) 0 0
\(388\) −32.7846 −1.66439
\(389\) 11.0718 0.561362 0.280681 0.959801i \(-0.409440\pi\)
0.280681 + 0.959801i \(0.409440\pi\)
\(390\) 0 0
\(391\) −13.9391 −0.704929
\(392\) 0 0
\(393\) −5.65685 −0.285351
\(394\) 11.8038 0.594669
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 23.7128 1.19011 0.595056 0.803684i \(-0.297130\pi\)
0.595056 + 0.803684i \(0.297130\pi\)
\(398\) −4.19615 −0.210334
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −6.53590 −0.325981
\(403\) 29.3939 1.46421
\(404\) −12.0716 −0.600584
\(405\) 0 0
\(406\) 0 0
\(407\) −36.5665 −1.81253
\(408\) 7.72741 0.382564
\(409\) 11.4152 0.564448 0.282224 0.959349i \(-0.408928\pi\)
0.282224 + 0.959349i \(0.408928\pi\)
\(410\) 0 0
\(411\) 10.1769 0.501989
\(412\) −24.0000 −1.18240
\(413\) 0 0
\(414\) −1.80385 −0.0886543
\(415\) 0 0
\(416\) 20.5569 1.00788
\(417\) −0.378937 −0.0185566
\(418\) −9.46410 −0.462904
\(419\) 21.1117 1.03137 0.515686 0.856778i \(-0.327537\pi\)
0.515686 + 0.856778i \(0.327537\pi\)
\(420\) 0 0
\(421\) 12.7846 0.623084 0.311542 0.950232i \(-0.399155\pi\)
0.311542 + 0.950232i \(0.399155\pi\)
\(422\) −13.9391 −0.678543
\(423\) 6.00000 0.291730
\(424\) −14.1962 −0.689426
\(425\) 0 0
\(426\) 3.30890 0.160317
\(427\) 0 0
\(428\) 26.9444 1.30241
\(429\) −13.8564 −0.668994
\(430\) 0 0
\(431\) 37.3205 1.79767 0.898833 0.438292i \(-0.144416\pi\)
0.898833 + 0.438292i \(0.144416\pi\)
\(432\) −2.46410 −0.118554
\(433\) −17.8564 −0.858124 −0.429062 0.903275i \(-0.641156\pi\)
−0.429062 + 0.903275i \(0.641156\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −27.4641 −1.31529
\(437\) 18.3923 0.879823
\(438\) 1.51575 0.0724253
\(439\) −13.0053 −0.620710 −0.310355 0.950621i \(-0.600448\pi\)
−0.310355 + 0.950621i \(0.600448\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.28221 0.393945
\(443\) −12.5249 −0.595074 −0.297537 0.954710i \(-0.596165\pi\)
−0.297537 + 0.954710i \(0.596165\pi\)
\(444\) −18.2832 −0.867684
\(445\) 0 0
\(446\) 4.14110 0.196087
\(447\) −7.07180 −0.334485
\(448\) 0 0
\(449\) 35.8564 1.69217 0.846084 0.533049i \(-0.178954\pi\)
0.846084 + 0.533049i \(0.178954\pi\)
\(450\) 0 0
\(451\) −29.3939 −1.38410
\(452\) −9.14162 −0.429986
\(453\) 19.4641 0.914503
\(454\) 2.07055 0.0971758
\(455\) 0 0
\(456\) −10.1962 −0.477479
\(457\) 21.8695 1.02301 0.511507 0.859279i \(-0.329087\pi\)
0.511507 + 0.859279i \(0.329087\pi\)
\(458\) 6.98076 0.326190
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −32.4254 −1.51020 −0.755100 0.655609i \(-0.772412\pi\)
−0.755100 + 0.655609i \(0.772412\pi\)
\(462\) 0 0
\(463\) −22.4243 −1.04215 −0.521074 0.853512i \(-0.674468\pi\)
−0.521074 + 0.853512i \(0.674468\pi\)
\(464\) −12.1436 −0.563752
\(465\) 0 0
\(466\) 2.73205 0.126560
\(467\) −9.85641 −0.456100 −0.228050 0.973649i \(-0.573235\pi\)
−0.228050 + 0.973649i \(0.573235\pi\)
\(468\) −6.92820 −0.320256
\(469\) 0 0
\(470\) 0 0
\(471\) −5.07180 −0.233696
\(472\) −1.46410 −0.0673907
\(473\) 0 0
\(474\) 2.34795 0.107845
\(475\) 0 0
\(476\) 0 0
\(477\) 7.34847 0.336463
\(478\) −4.41851 −0.202098
\(479\) 25.2528 1.15383 0.576914 0.816805i \(-0.304257\pi\)
0.576914 + 0.816805i \(0.304257\pi\)
\(480\) 0 0
\(481\) −42.2233 −1.92522
\(482\) −7.26795 −0.331046
\(483\) 0 0
\(484\) −1.73205 −0.0787296
\(485\) 0 0
\(486\) −0.517638 −0.0234805
\(487\) −15.4548 −0.700324 −0.350162 0.936689i \(-0.613874\pi\)
−0.350162 + 0.936689i \(0.613874\pi\)
\(488\) 1.26795 0.0573974
\(489\) −22.4243 −1.01406
\(490\) 0 0
\(491\) 41.3205 1.86477 0.932384 0.361469i \(-0.117725\pi\)
0.932384 + 0.361469i \(0.117725\pi\)
\(492\) −14.6969 −0.662589
\(493\) −19.7128 −0.887820
\(494\) −10.9282 −0.491683
\(495\) 0 0
\(496\) 18.1074 0.813045
\(497\) 0 0
\(498\) −3.10583 −0.139176
\(499\) −32.2487 −1.44365 −0.721825 0.692075i \(-0.756697\pi\)
−0.721825 + 0.692075i \(0.756697\pi\)
\(500\) 0 0
\(501\) −18.9282 −0.845650
\(502\) −10.5359 −0.470240
\(503\) 23.8564 1.06370 0.531852 0.846837i \(-0.321496\pi\)
0.531852 + 0.846837i \(0.321496\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.24871 −0.277789
\(507\) −3.00000 −0.133235
\(508\) 4.89898 0.217357
\(509\) 3.03150 0.134369 0.0671844 0.997741i \(-0.478598\pi\)
0.0671844 + 0.997741i \(0.478598\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.1841 0.980408
\(513\) 5.27792 0.233026
\(514\) −1.79315 −0.0790925
\(515\) 0 0
\(516\) 0 0
\(517\) 20.7846 0.914106
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) 28.2843 1.23916 0.619578 0.784935i \(-0.287304\pi\)
0.619578 + 0.784935i \(0.287304\pi\)
\(522\) −2.55103 −0.111655
\(523\) −43.7128 −1.91143 −0.955714 0.294297i \(-0.904914\pi\)
−0.955714 + 0.294297i \(0.904914\pi\)
\(524\) −9.79796 −0.428026
\(525\) 0 0
\(526\) −3.94744 −0.172117
\(527\) 29.3939 1.28042
\(528\) −8.53590 −0.371477
\(529\) −10.8564 −0.472018
\(530\) 0 0
\(531\) 0.757875 0.0328890
\(532\) 0 0
\(533\) −33.9411 −1.47015
\(534\) −8.00000 −0.346194
\(535\) 0 0
\(536\) −24.3923 −1.05359
\(537\) −10.3923 −0.448461
\(538\) −16.0000 −0.689809
\(539\) 0 0
\(540\) 0 0
\(541\) 26.6410 1.14539 0.572693 0.819770i \(-0.305899\pi\)
0.572693 + 0.819770i \(0.305899\pi\)
\(542\) 0.875644 0.0376121
\(543\) −6.31319 −0.270925
\(544\) 20.5569 0.881368
\(545\) 0 0
\(546\) 0 0
\(547\) −10.0010 −0.427613 −0.213807 0.976876i \(-0.568586\pi\)
−0.213807 + 0.976876i \(0.568586\pi\)
\(548\) 17.6269 0.752984
\(549\) −0.656339 −0.0280119
\(550\) 0 0
\(551\) 26.0106 1.10809
\(552\) −6.73205 −0.286535
\(553\) 0 0
\(554\) 5.85641 0.248815
\(555\) 0 0
\(556\) −0.656339 −0.0278350
\(557\) −8.86422 −0.375589 −0.187795 0.982208i \(-0.560134\pi\)
−0.187795 + 0.982208i \(0.560134\pi\)
\(558\) 3.80385 0.161030
\(559\) 0 0
\(560\) 0 0
\(561\) −13.8564 −0.585018
\(562\) 14.4195 0.608251
\(563\) 4.14359 0.174632 0.0873158 0.996181i \(-0.472171\pi\)
0.0873158 + 0.996181i \(0.472171\pi\)
\(564\) 10.3923 0.437595
\(565\) 0 0
\(566\) 1.10961 0.0466402
\(567\) 0 0
\(568\) 12.3490 0.518152
\(569\) −15.8564 −0.664735 −0.332368 0.943150i \(-0.607847\pi\)
−0.332368 + 0.943150i \(0.607847\pi\)
\(570\) 0 0
\(571\) −15.7128 −0.657561 −0.328780 0.944406i \(-0.606638\pi\)
−0.328780 + 0.944406i \(0.606638\pi\)
\(572\) −24.0000 −1.00349
\(573\) 7.46410 0.311817
\(574\) 0 0
\(575\) 0 0
\(576\) −2.26795 −0.0944979
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −0.517638 −0.0215309
\(579\) −6.41473 −0.266587
\(580\) 0 0
\(581\) 0 0
\(582\) −9.79796 −0.406138
\(583\) 25.4558 1.05427
\(584\) 5.65685 0.234082
\(585\) 0 0
\(586\) 5.93426 0.245142
\(587\) −4.14359 −0.171024 −0.0855122 0.996337i \(-0.527253\pi\)
−0.0855122 + 0.996337i \(0.527253\pi\)
\(588\) 0 0
\(589\) −38.7846 −1.59809
\(590\) 0 0
\(591\) −22.8033 −0.938002
\(592\) −26.0106 −1.06903
\(593\) 25.3205 1.03979 0.519894 0.854231i \(-0.325971\pi\)
0.519894 + 0.854231i \(0.325971\pi\)
\(594\) −1.79315 −0.0735739
\(595\) 0 0
\(596\) −12.2487 −0.501727
\(597\) 8.10634 0.331771
\(598\) −7.21539 −0.295059
\(599\) 29.3205 1.19800 0.599002 0.800748i \(-0.295564\pi\)
0.599002 + 0.800748i \(0.295564\pi\)
\(600\) 0 0
\(601\) 25.1512 1.02594 0.512970 0.858406i \(-0.328545\pi\)
0.512970 + 0.858406i \(0.328545\pi\)
\(602\) 0 0
\(603\) 12.6264 0.514186
\(604\) 33.7128 1.37175
\(605\) 0 0
\(606\) −3.60770 −0.146553
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −27.1244 −1.10004
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) −6.92820 −0.280056
\(613\) −9.79796 −0.395736 −0.197868 0.980229i \(-0.563402\pi\)
−0.197868 + 0.980229i \(0.563402\pi\)
\(614\) −2.07055 −0.0835607
\(615\) 0 0
\(616\) 0 0
\(617\) −27.1475 −1.09292 −0.546458 0.837487i \(-0.684024\pi\)
−0.546458 + 0.837487i \(0.684024\pi\)
\(618\) −7.17260 −0.288524
\(619\) 20.7327 0.833319 0.416659 0.909063i \(-0.363201\pi\)
0.416659 + 0.909063i \(0.363201\pi\)
\(620\) 0 0
\(621\) 3.48477 0.139839
\(622\) −2.14359 −0.0859503
\(623\) 0 0
\(624\) −9.85641 −0.394572
\(625\) 0 0
\(626\) −1.51575 −0.0605815
\(627\) 18.2832 0.730162
\(628\) −8.78461 −0.350544
\(629\) −42.2233 −1.68355
\(630\) 0 0
\(631\) −25.3205 −1.00799 −0.503997 0.863706i \(-0.668138\pi\)
−0.503997 + 0.863706i \(0.668138\pi\)
\(632\) 8.76268 0.348561
\(633\) 26.9282 1.07030
\(634\) 2.44486 0.0970979
\(635\) 0 0
\(636\) 12.7279 0.504695
\(637\) 0 0
\(638\) −8.83701 −0.349861
\(639\) −6.39230 −0.252876
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 8.05256 0.317809
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −10.9282 −0.429964
\(647\) −42.9282 −1.68768 −0.843841 0.536593i \(-0.819711\pi\)
−0.843841 + 0.536593i \(0.819711\pi\)
\(648\) −1.93185 −0.0758903
\(649\) 2.62536 0.103054
\(650\) 0 0
\(651\) 0 0
\(652\) −38.8401 −1.52110
\(653\) 3.96524 0.155172 0.0775859 0.996986i \(-0.475279\pi\)
0.0775859 + 0.996986i \(0.475279\pi\)
\(654\) −8.20788 −0.320954
\(655\) 0 0
\(656\) −20.9086 −0.816344
\(657\) −2.92820 −0.114240
\(658\) 0 0
\(659\) 10.3923 0.404827 0.202413 0.979300i \(-0.435122\pi\)
0.202413 + 0.979300i \(0.435122\pi\)
\(660\) 0 0
\(661\) −13.4858 −0.524537 −0.262268 0.964995i \(-0.584471\pi\)
−0.262268 + 0.964995i \(0.584471\pi\)
\(662\) −4.54725 −0.176734
\(663\) −16.0000 −0.621389
\(664\) −11.5911 −0.449822
\(665\) 0 0
\(666\) −5.46410 −0.211730
\(667\) 17.1736 0.664966
\(668\) −32.7846 −1.26847
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −2.27362 −0.0877723
\(672\) 0 0
\(673\) −21.8695 −0.843009 −0.421504 0.906826i \(-0.638498\pi\)
−0.421504 + 0.906826i \(0.638498\pi\)
\(674\) 9.17691 0.353482
\(675\) 0 0
\(676\) −5.19615 −0.199852
\(677\) −23.1769 −0.890761 −0.445381 0.895341i \(-0.646932\pi\)
−0.445381 + 0.895341i \(0.646932\pi\)
\(678\) −2.73205 −0.104924
\(679\) 0 0
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 13.1769 0.504570
\(683\) 42.3249 1.61952 0.809758 0.586764i \(-0.199598\pi\)
0.809758 + 0.586764i \(0.199598\pi\)
\(684\) 9.14162 0.349539
\(685\) 0 0
\(686\) 0 0
\(687\) −13.4858 −0.514515
\(688\) 0 0
\(689\) 29.3939 1.11982
\(690\) 0 0
\(691\) −10.9348 −0.415978 −0.207989 0.978131i \(-0.566692\pi\)
−0.207989 + 0.978131i \(0.566692\pi\)
\(692\) −27.7128 −1.05348
\(693\) 0 0
\(694\) 16.0526 0.609347
\(695\) 0 0
\(696\) −9.52056 −0.360876
\(697\) −33.9411 −1.28561
\(698\) 9.12436 0.345362
\(699\) −5.27792 −0.199629
\(700\) 0 0
\(701\) −11.0718 −0.418176 −0.209088 0.977897i \(-0.567050\pi\)
−0.209088 + 0.977897i \(0.567050\pi\)
\(702\) −2.07055 −0.0781480
\(703\) 55.7128 2.10125
\(704\) −7.85641 −0.296099
\(705\) 0 0
\(706\) 5.93426 0.223339
\(707\) 0 0
\(708\) 1.31268 0.0493334
\(709\) −10.9282 −0.410417 −0.205209 0.978718i \(-0.565787\pi\)
−0.205209 + 0.978718i \(0.565787\pi\)
\(710\) 0 0
\(711\) −4.53590 −0.170109
\(712\) −29.8564 −1.11892
\(713\) −25.6077 −0.959016
\(714\) 0 0
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) 8.53590 0.318779
\(718\) 8.00481 0.298737
\(719\) −14.6969 −0.548103 −0.274052 0.961715i \(-0.588364\pi\)
−0.274052 + 0.961715i \(0.588364\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 4.58441 0.170614
\(723\) 14.0406 0.522176
\(724\) −10.9348 −0.406388
\(725\) 0 0
\(726\) −0.517638 −0.0192114
\(727\) −23.7128 −0.879460 −0.439730 0.898130i \(-0.644926\pi\)
−0.439730 + 0.898130i \(0.644926\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −1.13681 −0.0420178
\(733\) 17.8564 0.659541 0.329771 0.944061i \(-0.393029\pi\)
0.329771 + 0.944061i \(0.393029\pi\)
\(734\) 2.07055 0.0764255
\(735\) 0 0
\(736\) −17.9090 −0.660133
\(737\) 43.7391 1.61115
\(738\) −4.39230 −0.161683
\(739\) −25.8564 −0.951143 −0.475572 0.879677i \(-0.657759\pi\)
−0.475572 + 0.879677i \(0.657759\pi\)
\(740\) 0 0
\(741\) 21.1117 0.775556
\(742\) 0 0
\(743\) 51.1619 1.87695 0.938474 0.345351i \(-0.112240\pi\)
0.938474 + 0.345351i \(0.112240\pi\)
\(744\) 14.1962 0.520456
\(745\) 0 0
\(746\) −3.71281 −0.135936
\(747\) 6.00000 0.219529
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −36.7846 −1.34229 −0.671145 0.741326i \(-0.734197\pi\)
−0.671145 + 0.741326i \(0.734197\pi\)
\(752\) 14.7846 0.539139
\(753\) 20.3538 0.741733
\(754\) −10.2041 −0.371612
\(755\) 0 0
\(756\) 0 0
\(757\) −34.2929 −1.24640 −0.623198 0.782064i \(-0.714167\pi\)
−0.623198 + 0.782064i \(0.714167\pi\)
\(758\) 5.93426 0.215542
\(759\) 12.0716 0.438171
\(760\) 0 0
\(761\) −18.0802 −0.655406 −0.327703 0.944781i \(-0.606274\pi\)
−0.327703 + 0.944781i \(0.606274\pi\)
\(762\) 1.46410 0.0530388
\(763\) 0 0
\(764\) 12.9282 0.467726
\(765\) 0 0
\(766\) −12.3490 −0.446187
\(767\) 3.03150 0.109461
\(768\) 1.39230 0.0502405
\(769\) −39.2934 −1.41696 −0.708478 0.705733i \(-0.750618\pi\)
−0.708478 + 0.705733i \(0.750618\pi\)
\(770\) 0 0
\(771\) 3.46410 0.124757
\(772\) −11.1106 −0.399881
\(773\) 37.8564 1.36160 0.680800 0.732469i \(-0.261632\pi\)
0.680800 + 0.732469i \(0.261632\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −36.5665 −1.31266
\(777\) 0 0
\(778\) 5.73118 0.205473
\(779\) 44.7846 1.60458
\(780\) 0 0
\(781\) −22.1436 −0.792360
\(782\) −7.21539 −0.258022
\(783\) 4.92820 0.176120
\(784\) 0 0
\(785\) 0 0
\(786\) −2.92820 −0.104446
\(787\) −45.8564 −1.63460 −0.817302 0.576209i \(-0.804531\pi\)
−0.817302 + 0.576209i \(0.804531\pi\)
\(788\) −39.4964 −1.40700
\(789\) 7.62587 0.271488
\(790\) 0 0
\(791\) 0 0
\(792\) −6.69213 −0.237795
\(793\) −2.62536 −0.0932291
\(794\) 12.2747 0.435611
\(795\) 0 0
\(796\) 14.0406 0.497656
\(797\) −10.1436 −0.359305 −0.179652 0.983730i \(-0.557497\pi\)
−0.179652 + 0.983730i \(0.557497\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 15.4548 0.546069
\(802\) 1.03528 0.0365569
\(803\) −10.1436 −0.357960
\(804\) 21.8695 0.771279
\(805\) 0 0
\(806\) 15.2154 0.535939
\(807\) 30.9096 1.08807
\(808\) −13.4641 −0.473665
\(809\) 21.7128 0.763382 0.381691 0.924290i \(-0.375342\pi\)
0.381691 + 0.924290i \(0.375342\pi\)
\(810\) 0 0
\(811\) 25.6317 0.900051 0.450026 0.893016i \(-0.351415\pi\)
0.450026 + 0.893016i \(0.351415\pi\)
\(812\) 0 0
\(813\) −1.69161 −0.0593275
\(814\) −18.9282 −0.663433
\(815\) 0 0
\(816\) −9.85641 −0.345043
\(817\) 0 0
\(818\) 5.90897 0.206602
\(819\) 0 0
\(820\) 0 0
\(821\) 18.7846 0.655587 0.327794 0.944749i \(-0.393695\pi\)
0.327794 + 0.944749i \(0.393695\pi\)
\(822\) 5.26795 0.183741
\(823\) −30.7066 −1.07036 −0.535182 0.844737i \(-0.679757\pi\)
−0.535182 + 0.844737i \(0.679757\pi\)
\(824\) −26.7685 −0.932526
\(825\) 0 0
\(826\) 0 0
\(827\) 8.18067 0.284470 0.142235 0.989833i \(-0.454571\pi\)
0.142235 + 0.989833i \(0.454571\pi\)
\(828\) 6.03579 0.209758
\(829\) 6.51626 0.226319 0.113160 0.993577i \(-0.463903\pi\)
0.113160 + 0.993577i \(0.463903\pi\)
\(830\) 0 0
\(831\) −11.3137 −0.392468
\(832\) −9.07180 −0.314508
\(833\) 0 0
\(834\) −0.196152 −0.00679220
\(835\) 0 0
\(836\) 31.6675 1.09524
\(837\) −7.34847 −0.254000
\(838\) 10.9282 0.377509
\(839\) 20.3538 0.702691 0.351345 0.936246i \(-0.385724\pi\)
0.351345 + 0.936246i \(0.385724\pi\)
\(840\) 0 0
\(841\) −4.71281 −0.162511
\(842\) 6.61780 0.228064
\(843\) −27.8564 −0.959426
\(844\) 46.6410 1.60545
\(845\) 0 0
\(846\) 3.10583 0.106781
\(847\) 0 0
\(848\) 18.1074 0.621810
\(849\) −2.14359 −0.0735679
\(850\) 0 0
\(851\) 36.7846 1.26096
\(852\) −11.0718 −0.379314
\(853\) 13.0718 0.447570 0.223785 0.974639i \(-0.428159\pi\)
0.223785 + 0.974639i \(0.428159\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 30.0526 1.02718
\(857\) −9.85641 −0.336688 −0.168344 0.985728i \(-0.553842\pi\)
−0.168344 + 0.985728i \(0.553842\pi\)
\(858\) −7.17260 −0.244869
\(859\) 3.41044 0.116363 0.0581813 0.998306i \(-0.481470\pi\)
0.0581813 + 0.998306i \(0.481470\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 19.3185 0.657991
\(863\) 15.9081 0.541517 0.270759 0.962647i \(-0.412725\pi\)
0.270759 + 0.962647i \(0.412725\pi\)
\(864\) −5.13922 −0.174840
\(865\) 0 0
\(866\) −9.24316 −0.314095
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −15.7128 −0.533021
\(870\) 0 0
\(871\) 50.5055 1.71132
\(872\) −30.6322 −1.03734
\(873\) 18.9282 0.640623
\(874\) 9.52056 0.322038
\(875\) 0 0
\(876\) −5.07180 −0.171360
\(877\) 33.9411 1.14611 0.573055 0.819517i \(-0.305758\pi\)
0.573055 + 0.819517i \(0.305758\pi\)
\(878\) −6.73205 −0.227196
\(879\) −11.4641 −0.386675
\(880\) 0 0
\(881\) 29.5969 0.997147 0.498573 0.866848i \(-0.333857\pi\)
0.498573 + 0.866848i \(0.333857\pi\)
\(882\) 0 0
\(883\) −29.3939 −0.989183 −0.494591 0.869126i \(-0.664682\pi\)
−0.494591 + 0.869126i \(0.664682\pi\)
\(884\) −27.7128 −0.932083
\(885\) 0 0
\(886\) −6.48334 −0.217812
\(887\) 13.7128 0.460431 0.230216 0.973140i \(-0.426057\pi\)
0.230216 + 0.973140i \(0.426057\pi\)
\(888\) −20.3923 −0.684321
\(889\) 0 0
\(890\) 0 0
\(891\) 3.46410 0.116052
\(892\) −13.8564 −0.463947
\(893\) −31.6675 −1.05971
\(894\) −3.66063 −0.122430
\(895\) 0 0
\(896\) 0 0
\(897\) 13.9391 0.465412
\(898\) 18.5606 0.619377
\(899\) −36.2147 −1.20783
\(900\) 0 0
\(901\) 29.3939 0.979252
\(902\) −15.2154 −0.506617
\(903\) 0 0
\(904\) −10.1962 −0.339119
\(905\) 0 0
\(906\) 10.0754 0.334731
\(907\) −6.76646 −0.224677 −0.112338 0.993670i \(-0.535834\pi\)
−0.112338 + 0.993670i \(0.535834\pi\)
\(908\) −6.92820 −0.229920
\(909\) 6.96953 0.231165
\(910\) 0 0
\(911\) 9.60770 0.318317 0.159159 0.987253i \(-0.449122\pi\)
0.159159 + 0.987253i \(0.449122\pi\)
\(912\) 13.0053 0.430649
\(913\) 20.7846 0.687870
\(914\) 11.3205 0.374449
\(915\) 0 0
\(916\) −23.3581 −0.771773
\(917\) 0 0
\(918\) −2.07055 −0.0683384
\(919\) −9.07180 −0.299251 −0.149625 0.988743i \(-0.547807\pi\)
−0.149625 + 0.988743i \(0.547807\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −16.7846 −0.552772
\(923\) −25.5692 −0.841621
\(924\) 0 0
\(925\) 0 0
\(926\) −11.6077 −0.381453
\(927\) 13.8564 0.455104
\(928\) −25.3271 −0.831403
\(929\) 18.0802 0.593191 0.296596 0.955003i \(-0.404149\pi\)
0.296596 + 0.955003i \(0.404149\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9.14162 −0.299444
\(933\) 4.14110 0.135574
\(934\) −5.10205 −0.166944
\(935\) 0 0
\(936\) −7.72741 −0.252578
\(937\) −16.7846 −0.548329 −0.274165 0.961683i \(-0.588401\pi\)
−0.274165 + 0.961683i \(0.588401\pi\)
\(938\) 0 0
\(939\) 2.92820 0.0955583
\(940\) 0 0
\(941\) −0.203072 −0.00661996 −0.00330998 0.999995i \(-0.501054\pi\)
−0.00330998 + 0.999995i \(0.501054\pi\)
\(942\) −2.62536 −0.0855387
\(943\) 29.5692 0.962906
\(944\) 1.86748 0.0607813
\(945\) 0 0
\(946\) 0 0
\(947\) −18.3848 −0.597425 −0.298712 0.954343i \(-0.596557\pi\)
−0.298712 + 0.954343i \(0.596557\pi\)
\(948\) −7.85641 −0.255164
\(949\) −11.7128 −0.380214
\(950\) 0 0
\(951\) −4.72311 −0.153157
\(952\) 0 0
\(953\) −53.9160 −1.74651 −0.873255 0.487264i \(-0.837995\pi\)
−0.873255 + 0.487264i \(0.837995\pi\)
\(954\) 3.80385 0.123154
\(955\) 0 0
\(956\) 14.7846 0.478168
\(957\) 17.0718 0.551853
\(958\) 13.0718 0.422331
\(959\) 0 0
\(960\) 0 0
\(961\) 23.0000 0.741935
\(962\) −21.8564 −0.704679
\(963\) −15.5563 −0.501296
\(964\) 24.3190 0.783263
\(965\) 0 0
\(966\) 0 0
\(967\) −10.0010 −0.321611 −0.160806 0.986986i \(-0.551409\pi\)
−0.160806 + 0.986986i \(0.551409\pi\)
\(968\) −1.93185 −0.0620921
\(969\) 21.1117 0.678204
\(970\) 0 0
\(971\) 11.6654 0.374362 0.187181 0.982325i \(-0.440065\pi\)
0.187181 + 0.982325i \(0.440065\pi\)
\(972\) 1.73205 0.0555556
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −1.61729 −0.0517680
\(977\) −45.6338 −1.45995 −0.729977 0.683472i \(-0.760469\pi\)
−0.729977 + 0.683472i \(0.760469\pi\)
\(978\) −11.6077 −0.371173
\(979\) 53.5370 1.71105
\(980\) 0 0
\(981\) 15.8564 0.506256
\(982\) 21.3891 0.682553
\(983\) −24.7846 −0.790506 −0.395253 0.918572i \(-0.629343\pi\)
−0.395253 + 0.918572i \(0.629343\pi\)
\(984\) −16.3923 −0.522568
\(985\) 0 0
\(986\) −10.2041 −0.324965
\(987\) 0 0
\(988\) 36.5665 1.16333
\(989\) 0 0
\(990\) 0 0
\(991\) 12.7846 0.406117 0.203058 0.979167i \(-0.434912\pi\)
0.203058 + 0.979167i \(0.434912\pi\)
\(992\) 37.7654 1.19905
\(993\) 8.78461 0.278771
\(994\) 0 0
\(995\) 0 0
\(996\) 10.3923 0.329293
\(997\) 41.8564 1.32561 0.662803 0.748794i \(-0.269367\pi\)
0.662803 + 0.748794i \(0.269367\pi\)
\(998\) −16.6932 −0.528413
\(999\) 10.5558 0.333972
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 3675.2.a.bs.1.3 4
5.2 odd 4 735.2.d.f.589.6 yes 8
5.3 odd 4 735.2.d.f.589.3 8
5.4 even 2 3675.2.a.bu.1.2 4
7.6 odd 2 3675.2.a.bu.1.3 4
15.2 even 4 2205.2.d.t.1324.4 8
15.8 even 4 2205.2.d.t.1324.6 8
35.2 odd 12 735.2.q.c.214.3 8
35.3 even 12 735.2.q.d.79.3 8
35.12 even 12 735.2.q.d.214.3 8
35.13 even 4 735.2.d.f.589.4 yes 8
35.17 even 12 735.2.q.c.79.2 8
35.18 odd 12 735.2.q.c.79.3 8
35.23 odd 12 735.2.q.d.214.2 8
35.27 even 4 735.2.d.f.589.5 yes 8
35.32 odd 12 735.2.q.d.79.2 8
35.33 even 12 735.2.q.c.214.2 8
35.34 odd 2 inner 3675.2.a.bs.1.2 4
105.62 odd 4 2205.2.d.t.1324.3 8
105.83 odd 4 2205.2.d.t.1324.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
735.2.d.f.589.3 8 5.3 odd 4
735.2.d.f.589.4 yes 8 35.13 even 4
735.2.d.f.589.5 yes 8 35.27 even 4
735.2.d.f.589.6 yes 8 5.2 odd 4
735.2.q.c.79.2 8 35.17 even 12
735.2.q.c.79.3 8 35.18 odd 12
735.2.q.c.214.2 8 35.33 even 12
735.2.q.c.214.3 8 35.2 odd 12
735.2.q.d.79.2 8 35.32 odd 12
735.2.q.d.79.3 8 35.3 even 12
735.2.q.d.214.2 8 35.23 odd 12
735.2.q.d.214.3 8 35.12 even 12
2205.2.d.t.1324.3 8 105.62 odd 4
2205.2.d.t.1324.4 8 15.2 even 4
2205.2.d.t.1324.5 8 105.83 odd 4
2205.2.d.t.1324.6 8 15.8 even 4
3675.2.a.bs.1.2 4 35.34 odd 2 inner
3675.2.a.bs.1.3 4 1.1 even 1 trivial
3675.2.a.bu.1.2 4 5.4 even 2
3675.2.a.bu.1.3 4 7.6 odd 2