Properties

Label 4014.2.a.p.1.3
Level $4014$
Weight $2$
Character 4014.1
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $3$
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] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, 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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.473.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.33006\) of defining polynomial
Character \(\chi\) \(=\) 4014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.42917 q^{5} +0.669941 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.42917 q^{5} +0.669941 q^{7} +1.00000 q^{8} +2.42917 q^{10} +0.330059 q^{11} +1.57083 q^{13} +0.669941 q^{14} +1.00000 q^{16} +3.23094 q^{17} +1.90089 q^{19} +2.42917 q^{20} +0.330059 q^{22} +2.52829 q^{23} +0.900885 q^{25} +1.57083 q^{26} +0.669941 q^{28} +5.33006 q^{29} -6.56100 q^{31} +1.00000 q^{32} +3.23094 q^{34} +1.62740 q^{35} +7.38664 q^{37} +1.90089 q^{38} +2.42917 q^{40} -12.2211 q^{41} +5.56100 q^{43} +0.330059 q^{44} +2.52829 q^{46} -3.08929 q^{47} -6.55118 q^{49} +0.900885 q^{50} +1.57083 q^{52} -4.95746 q^{53} +0.801770 q^{55} +0.669941 q^{56} +5.33006 q^{58} -4.28752 q^{59} -7.79195 q^{61} -6.56100 q^{62} +1.00000 q^{64} +3.81581 q^{65} +11.2875 q^{67} +3.23094 q^{68} +1.62740 q^{70} +9.66012 q^{71} +10.5512 q^{73} +7.38664 q^{74} +1.90089 q^{76} +0.221120 q^{77} +14.7919 q^{79} +2.42917 q^{80} -12.2211 q^{82} +8.27770 q^{83} +7.84852 q^{85} +5.56100 q^{86} +0.330059 q^{88} -11.2450 q^{89} +1.05236 q^{91} +2.52829 q^{92} -3.08929 q^{94} +4.61758 q^{95} +9.22112 q^{97} -6.55118 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 9 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 9 q^{7} + 3 q^{8} + q^{10} - 6 q^{11} + 11 q^{13} + 9 q^{14} + 3 q^{16} + 2 q^{17} + 5 q^{19} + q^{20} - 6 q^{22} + 2 q^{23} + 2 q^{25} + 11 q^{26} + 9 q^{28} + 9 q^{29} - 5 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{37} + 5 q^{38} + q^{40} - 8 q^{41} + 2 q^{43} - 6 q^{44} + 2 q^{46} + 11 q^{47} + 16 q^{49} + 2 q^{50} + 11 q^{52} - 3 q^{53} + q^{55} + 9 q^{56} + 9 q^{58} + 6 q^{59} - q^{61} - 5 q^{62} + 3 q^{64} - 13 q^{65} + 15 q^{67} + 2 q^{68} + 15 q^{71} - 4 q^{73} + 4 q^{74} + 5 q^{76} - 28 q^{77} + 22 q^{79} + q^{80} - 8 q^{82} - 15 q^{83} - 10 q^{85} + 2 q^{86} - 6 q^{88} - 3 q^{89} + 36 q^{91} + 2 q^{92} + 11 q^{94} - 12 q^{95} - q^{97} + 16 q^{98}+O(q^{100}) \) 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
\(5\) 2.42917 1.08636 0.543180 0.839616i \(-0.317220\pi\)
0.543180 + 0.839616i \(0.317220\pi\)
\(6\) 0 0
\(7\) 0.669941 0.253214 0.126607 0.991953i \(-0.459591\pi\)
0.126607 + 0.991953i \(0.459591\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.42917 0.768172
\(11\) 0.330059 0.0995165 0.0497582 0.998761i \(-0.484155\pi\)
0.0497582 + 0.998761i \(0.484155\pi\)
\(12\) 0 0
\(13\) 1.57083 0.435669 0.217834 0.975986i \(-0.430101\pi\)
0.217834 + 0.975986i \(0.430101\pi\)
\(14\) 0.669941 0.179049
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.23094 0.783619 0.391809 0.920046i \(-0.371849\pi\)
0.391809 + 0.920046i \(0.371849\pi\)
\(18\) 0 0
\(19\) 1.90089 0.436093 0.218046 0.975938i \(-0.430032\pi\)
0.218046 + 0.975938i \(0.430032\pi\)
\(20\) 2.42917 0.543180
\(21\) 0 0
\(22\) 0.330059 0.0703688
\(23\) 2.52829 0.527185 0.263592 0.964634i \(-0.415093\pi\)
0.263592 + 0.964634i \(0.415093\pi\)
\(24\) 0 0
\(25\) 0.900885 0.180177
\(26\) 1.57083 0.308064
\(27\) 0 0
\(28\) 0.669941 0.126607
\(29\) 5.33006 0.989767 0.494884 0.868959i \(-0.335211\pi\)
0.494884 + 0.868959i \(0.335211\pi\)
\(30\) 0 0
\(31\) −6.56100 −1.17839 −0.589195 0.807991i \(-0.700555\pi\)
−0.589195 + 0.807991i \(0.700555\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.23094 0.554102
\(35\) 1.62740 0.275081
\(36\) 0 0
\(37\) 7.38664 1.21436 0.607178 0.794566i \(-0.292302\pi\)
0.607178 + 0.794566i \(0.292302\pi\)
\(38\) 1.90089 0.308364
\(39\) 0 0
\(40\) 2.42917 0.384086
\(41\) −12.2211 −1.90862 −0.954309 0.298821i \(-0.903407\pi\)
−0.954309 + 0.298821i \(0.903407\pi\)
\(42\) 0 0
\(43\) 5.56100 0.848045 0.424022 0.905652i \(-0.360618\pi\)
0.424022 + 0.905652i \(0.360618\pi\)
\(44\) 0.330059 0.0497582
\(45\) 0 0
\(46\) 2.52829 0.372776
\(47\) −3.08929 −0.450619 −0.225310 0.974287i \(-0.572339\pi\)
−0.225310 + 0.974287i \(0.572339\pi\)
\(48\) 0 0
\(49\) −6.55118 −0.935883
\(50\) 0.900885 0.127404
\(51\) 0 0
\(52\) 1.57083 0.217834
\(53\) −4.95746 −0.680960 −0.340480 0.940252i \(-0.610590\pi\)
−0.340480 + 0.940252i \(0.610590\pi\)
\(54\) 0 0
\(55\) 0.801770 0.108111
\(56\) 0.669941 0.0895247
\(57\) 0 0
\(58\) 5.33006 0.699871
\(59\) −4.28752 −0.558188 −0.279094 0.960264i \(-0.590034\pi\)
−0.279094 + 0.960264i \(0.590034\pi\)
\(60\) 0 0
\(61\) −7.79195 −0.997656 −0.498828 0.866701i \(-0.666236\pi\)
−0.498828 + 0.866701i \(0.666236\pi\)
\(62\) −6.56100 −0.833248
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.81581 0.473293
\(66\) 0 0
\(67\) 11.2875 1.37899 0.689495 0.724290i \(-0.257832\pi\)
0.689495 + 0.724290i \(0.257832\pi\)
\(68\) 3.23094 0.391809
\(69\) 0 0
\(70\) 1.62740 0.194512
\(71\) 9.66012 1.14645 0.573223 0.819400i \(-0.305693\pi\)
0.573223 + 0.819400i \(0.305693\pi\)
\(72\) 0 0
\(73\) 10.5512 1.23492 0.617461 0.786601i \(-0.288161\pi\)
0.617461 + 0.786601i \(0.288161\pi\)
\(74\) 7.38664 0.858679
\(75\) 0 0
\(76\) 1.90089 0.218046
\(77\) 0.221120 0.0251990
\(78\) 0 0
\(79\) 14.7919 1.66422 0.832112 0.554608i \(-0.187132\pi\)
0.832112 + 0.554608i \(0.187132\pi\)
\(80\) 2.42917 0.271590
\(81\) 0 0
\(82\) −12.2211 −1.34960
\(83\) 8.27770 0.908595 0.454298 0.890850i \(-0.349890\pi\)
0.454298 + 0.890850i \(0.349890\pi\)
\(84\) 0 0
\(85\) 7.84852 0.851292
\(86\) 5.56100 0.599658
\(87\) 0 0
\(88\) 0.330059 0.0351844
\(89\) −11.2450 −1.19197 −0.595983 0.802997i \(-0.703237\pi\)
−0.595983 + 0.802997i \(0.703237\pi\)
\(90\) 0 0
\(91\) 1.05236 0.110317
\(92\) 2.52829 0.263592
\(93\) 0 0
\(94\) −3.08929 −0.318636
\(95\) 4.61758 0.473754
\(96\) 0 0
\(97\) 9.22112 0.936263 0.468131 0.883659i \(-0.344927\pi\)
0.468131 + 0.883659i \(0.344927\pi\)
\(98\) −6.55118 −0.661769
\(99\) 0 0
\(100\) 0.900885 0.0900885
\(101\) −13.5086 −1.34416 −0.672080 0.740479i \(-0.734599\pi\)
−0.672080 + 0.740479i \(0.734599\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 1.57083 0.154032
\(105\) 0 0
\(106\) −4.95746 −0.481511
\(107\) −11.8158 −1.14228 −0.571139 0.820854i \(-0.693498\pi\)
−0.571139 + 0.820854i \(0.693498\pi\)
\(108\) 0 0
\(109\) −12.2777 −1.17599 −0.587995 0.808865i \(-0.700083\pi\)
−0.587995 + 0.808865i \(0.700083\pi\)
\(110\) 0.801770 0.0764458
\(111\) 0 0
\(112\) 0.669941 0.0633035
\(113\) −3.99018 −0.375364 −0.187682 0.982230i \(-0.560097\pi\)
−0.187682 + 0.982230i \(0.560097\pi\)
\(114\) 0 0
\(115\) 6.14165 0.572712
\(116\) 5.33006 0.494884
\(117\) 0 0
\(118\) −4.28752 −0.394698
\(119\) 2.16454 0.198423
\(120\) 0 0
\(121\) −10.8911 −0.990096
\(122\) −7.79195 −0.705450
\(123\) 0 0
\(124\) −6.56100 −0.589195
\(125\) −9.95746 −0.890623
\(126\) 0 0
\(127\) 2.50443 0.222232 0.111116 0.993807i \(-0.464558\pi\)
0.111116 + 0.993807i \(0.464558\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.81581 0.334669
\(131\) −7.85835 −0.686587 −0.343294 0.939228i \(-0.611543\pi\)
−0.343294 + 0.939228i \(0.611543\pi\)
\(132\) 0 0
\(133\) 1.27348 0.110425
\(134\) 11.2875 0.975093
\(135\) 0 0
\(136\) 3.23094 0.277051
\(137\) 22.5979 1.93067 0.965336 0.261011i \(-0.0840559\pi\)
0.965336 + 0.261011i \(0.0840559\pi\)
\(138\) 0 0
\(139\) 5.91071 0.501340 0.250670 0.968073i \(-0.419349\pi\)
0.250670 + 0.968073i \(0.419349\pi\)
\(140\) 1.62740 0.137541
\(141\) 0 0
\(142\) 9.66012 0.810659
\(143\) 0.518465 0.0433562
\(144\) 0 0
\(145\) 12.9476 1.07524
\(146\) 10.5512 0.873222
\(147\) 0 0
\(148\) 7.38664 0.607178
\(149\) 2.75923 0.226045 0.113023 0.993592i \(-0.463947\pi\)
0.113023 + 0.993592i \(0.463947\pi\)
\(150\) 0 0
\(151\) 4.07947 0.331982 0.165991 0.986127i \(-0.446918\pi\)
0.165991 + 0.986127i \(0.446918\pi\)
\(152\) 1.90089 0.154182
\(153\) 0 0
\(154\) 0.221120 0.0178184
\(155\) −15.9378 −1.28016
\(156\) 0 0
\(157\) −13.1688 −1.05098 −0.525491 0.850799i \(-0.676118\pi\)
−0.525491 + 0.850799i \(0.676118\pi\)
\(158\) 14.7919 1.17678
\(159\) 0 0
\(160\) 2.42917 0.192043
\(161\) 1.69380 0.133491
\(162\) 0 0
\(163\) 10.1132 0.792123 0.396062 0.918224i \(-0.370377\pi\)
0.396062 + 0.918224i \(0.370377\pi\)
\(164\) −12.2211 −0.954309
\(165\) 0 0
\(166\) 8.27770 0.642474
\(167\) −19.9575 −1.54435 −0.772177 0.635407i \(-0.780832\pi\)
−0.772177 + 0.635407i \(0.780832\pi\)
\(168\) 0 0
\(169\) −10.5325 −0.810193
\(170\) 7.84852 0.601954
\(171\) 0 0
\(172\) 5.56100 0.424022
\(173\) −13.4095 −1.01951 −0.509754 0.860320i \(-0.670263\pi\)
−0.509754 + 0.860320i \(0.670263\pi\)
\(174\) 0 0
\(175\) 0.603540 0.0456233
\(176\) 0.330059 0.0248791
\(177\) 0 0
\(178\) −11.2450 −0.842847
\(179\) −1.59372 −0.119120 −0.0595600 0.998225i \(-0.518970\pi\)
−0.0595600 + 0.998225i \(0.518970\pi\)
\(180\) 0 0
\(181\) 3.08929 0.229625 0.114813 0.993387i \(-0.463373\pi\)
0.114813 + 0.993387i \(0.463373\pi\)
\(182\) 1.05236 0.0780062
\(183\) 0 0
\(184\) 2.52829 0.186388
\(185\) 17.9434 1.31923
\(186\) 0 0
\(187\) 1.06640 0.0779830
\(188\) −3.08929 −0.225310
\(189\) 0 0
\(190\) 4.61758 0.334994
\(191\) −7.50443 −0.543001 −0.271501 0.962438i \(-0.587520\pi\)
−0.271501 + 0.962438i \(0.587520\pi\)
\(192\) 0 0
\(193\) 17.7069 1.27457 0.637284 0.770629i \(-0.280058\pi\)
0.637284 + 0.770629i \(0.280058\pi\)
\(194\) 9.22112 0.662038
\(195\) 0 0
\(196\) −6.55118 −0.467941
\(197\) −5.42917 −0.386813 −0.193406 0.981119i \(-0.561954\pi\)
−0.193406 + 0.981119i \(0.561954\pi\)
\(198\) 0 0
\(199\) −2.00982 −0.142473 −0.0712363 0.997459i \(-0.522694\pi\)
−0.0712363 + 0.997459i \(0.522694\pi\)
\(200\) 0.900885 0.0637022
\(201\) 0 0
\(202\) −13.5086 −0.950465
\(203\) 3.57083 0.250623
\(204\) 0 0
\(205\) −29.6872 −2.07345
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) 1.57083 0.108917
\(209\) 0.627404 0.0433984
\(210\) 0 0
\(211\) 1.19823 0.0824896 0.0412448 0.999149i \(-0.486868\pi\)
0.0412448 + 0.999149i \(0.486868\pi\)
\(212\) −4.95746 −0.340480
\(213\) 0 0
\(214\) −11.8158 −0.807712
\(215\) 13.5086 0.921282
\(216\) 0 0
\(217\) −4.39549 −0.298385
\(218\) −12.2777 −0.831551
\(219\) 0 0
\(220\) 0.801770 0.0540553
\(221\) 5.07525 0.341398
\(222\) 0 0
\(223\) 1.00000 0.0669650
\(224\) 0.669941 0.0447623
\(225\) 0 0
\(226\) −3.99018 −0.265423
\(227\) 1.93360 0.128337 0.0641687 0.997939i \(-0.479560\pi\)
0.0641687 + 0.997939i \(0.479560\pi\)
\(228\) 0 0
\(229\) 16.4563 1.08746 0.543731 0.839260i \(-0.317011\pi\)
0.543731 + 0.839260i \(0.317011\pi\)
\(230\) 6.14165 0.404969
\(231\) 0 0
\(232\) 5.33006 0.349936
\(233\) 7.17437 0.470008 0.235004 0.971994i \(-0.424490\pi\)
0.235004 + 0.971994i \(0.424490\pi\)
\(234\) 0 0
\(235\) −7.50443 −0.489535
\(236\) −4.28752 −0.279094
\(237\) 0 0
\(238\) 2.16454 0.140306
\(239\) 16.3997 1.06081 0.530404 0.847745i \(-0.322040\pi\)
0.530404 + 0.847745i \(0.322040\pi\)
\(240\) 0 0
\(241\) −2.82142 −0.181743 −0.0908717 0.995863i \(-0.528965\pi\)
−0.0908717 + 0.995863i \(0.528965\pi\)
\(242\) −10.8911 −0.700104
\(243\) 0 0
\(244\) −7.79195 −0.498828
\(245\) −15.9140 −1.01671
\(246\) 0 0
\(247\) 2.98596 0.189992
\(248\) −6.56100 −0.416624
\(249\) 0 0
\(250\) −9.95746 −0.629765
\(251\) −22.4661 −1.41805 −0.709024 0.705184i \(-0.750864\pi\)
−0.709024 + 0.705184i \(0.750864\pi\)
\(252\) 0 0
\(253\) 0.834484 0.0524635
\(254\) 2.50443 0.157142
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.7494 0.982421 0.491211 0.871041i \(-0.336555\pi\)
0.491211 + 0.871041i \(0.336555\pi\)
\(258\) 0 0
\(259\) 4.94861 0.307492
\(260\) 3.81581 0.236646
\(261\) 0 0
\(262\) −7.85835 −0.485490
\(263\) 9.13183 0.563093 0.281546 0.959548i \(-0.409153\pi\)
0.281546 + 0.959548i \(0.409153\pi\)
\(264\) 0 0
\(265\) −12.0425 −0.739767
\(266\) 1.27348 0.0780821
\(267\) 0 0
\(268\) 11.2875 0.689495
\(269\) −7.56100 −0.461002 −0.230501 0.973072i \(-0.574037\pi\)
−0.230501 + 0.973072i \(0.574037\pi\)
\(270\) 0 0
\(271\) −5.40531 −0.328349 −0.164175 0.986431i \(-0.552496\pi\)
−0.164175 + 0.986431i \(0.552496\pi\)
\(272\) 3.23094 0.195905
\(273\) 0 0
\(274\) 22.5979 1.36519
\(275\) 0.297345 0.0179306
\(276\) 0 0
\(277\) 12.6035 0.757273 0.378637 0.925545i \(-0.376393\pi\)
0.378637 + 0.925545i \(0.376393\pi\)
\(278\) 5.91071 0.354501
\(279\) 0 0
\(280\) 1.62740 0.0972560
\(281\) 20.9705 1.25100 0.625498 0.780225i \(-0.284896\pi\)
0.625498 + 0.780225i \(0.284896\pi\)
\(282\) 0 0
\(283\) −5.71669 −0.339822 −0.169911 0.985459i \(-0.554348\pi\)
−0.169911 + 0.985459i \(0.554348\pi\)
\(284\) 9.66012 0.573223
\(285\) 0 0
\(286\) 0.518465 0.0306575
\(287\) −8.18743 −0.483289
\(288\) 0 0
\(289\) −6.56100 −0.385941
\(290\) 12.9476 0.760312
\(291\) 0 0
\(292\) 10.5512 0.617461
\(293\) 9.68398 0.565744 0.282872 0.959158i \(-0.408713\pi\)
0.282872 + 0.959158i \(0.408713\pi\)
\(294\) 0 0
\(295\) −10.4151 −0.606393
\(296\) 7.38664 0.429340
\(297\) 0 0
\(298\) 2.75923 0.159838
\(299\) 3.97150 0.229678
\(300\) 0 0
\(301\) 3.72555 0.214737
\(302\) 4.07947 0.234747
\(303\) 0 0
\(304\) 1.90089 0.109023
\(305\) −18.9280 −1.08381
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0.221120 0.0125995
\(309\) 0 0
\(310\) −15.9378 −0.905207
\(311\) 28.1262 1.59489 0.797446 0.603391i \(-0.206184\pi\)
0.797446 + 0.603391i \(0.206184\pi\)
\(312\) 0 0
\(313\) −9.27770 −0.524406 −0.262203 0.965013i \(-0.584449\pi\)
−0.262203 + 0.965013i \(0.584449\pi\)
\(314\) −13.1688 −0.743156
\(315\) 0 0
\(316\) 14.7919 0.832112
\(317\) −10.4151 −0.584972 −0.292486 0.956270i \(-0.594483\pi\)
−0.292486 + 0.956270i \(0.594483\pi\)
\(318\) 0 0
\(319\) 1.75923 0.0984981
\(320\) 2.42917 0.135795
\(321\) 0 0
\(322\) 1.69380 0.0943921
\(323\) 6.14165 0.341731
\(324\) 0 0
\(325\) 1.41513 0.0784975
\(326\) 10.1132 0.560116
\(327\) 0 0
\(328\) −12.2211 −0.674798
\(329\) −2.06964 −0.114103
\(330\) 0 0
\(331\) −2.40628 −0.132261 −0.0661307 0.997811i \(-0.521065\pi\)
−0.0661307 + 0.997811i \(0.521065\pi\)
\(332\) 8.27770 0.454298
\(333\) 0 0
\(334\) −19.9575 −1.09202
\(335\) 27.4193 1.49808
\(336\) 0 0
\(337\) −27.2875 −1.48645 −0.743223 0.669044i \(-0.766704\pi\)
−0.743223 + 0.669044i \(0.766704\pi\)
\(338\) −10.5325 −0.572893
\(339\) 0 0
\(340\) 7.84852 0.425646
\(341\) −2.16552 −0.117269
\(342\) 0 0
\(343\) −9.07849 −0.490193
\(344\) 5.56100 0.299829
\(345\) 0 0
\(346\) −13.4095 −0.720901
\(347\) −30.2538 −1.62411 −0.812055 0.583580i \(-0.801651\pi\)
−0.812055 + 0.583580i \(0.801651\pi\)
\(348\) 0 0
\(349\) −24.3866 −1.30539 −0.652693 0.757622i \(-0.726361\pi\)
−0.652693 + 0.757622i \(0.726361\pi\)
\(350\) 0.603540 0.0322606
\(351\) 0 0
\(352\) 0.330059 0.0175922
\(353\) 8.18841 0.435825 0.217912 0.975968i \(-0.430075\pi\)
0.217912 + 0.975968i \(0.430075\pi\)
\(354\) 0 0
\(355\) 23.4661 1.24545
\(356\) −11.2450 −0.595983
\(357\) 0 0
\(358\) −1.59372 −0.0842305
\(359\) −23.9149 −1.26218 −0.631091 0.775709i \(-0.717392\pi\)
−0.631091 + 0.775709i \(0.717392\pi\)
\(360\) 0 0
\(361\) −15.3866 −0.809823
\(362\) 3.08929 0.162370
\(363\) 0 0
\(364\) 1.05236 0.0551587
\(365\) 25.6306 1.34157
\(366\) 0 0
\(367\) 6.54794 0.341799 0.170900 0.985288i \(-0.445333\pi\)
0.170900 + 0.985288i \(0.445333\pi\)
\(368\) 2.52829 0.131796
\(369\) 0 0
\(370\) 17.9434 0.932834
\(371\) −3.32121 −0.172429
\(372\) 0 0
\(373\) −21.7396 −1.12563 −0.562817 0.826582i \(-0.690282\pi\)
−0.562817 + 0.826582i \(0.690282\pi\)
\(374\) 1.06640 0.0551423
\(375\) 0 0
\(376\) −3.08929 −0.159318
\(377\) 8.37260 0.431211
\(378\) 0 0
\(379\) 0.514249 0.0264152 0.0132076 0.999913i \(-0.495796\pi\)
0.0132076 + 0.999913i \(0.495796\pi\)
\(380\) 4.61758 0.236877
\(381\) 0 0
\(382\) −7.50443 −0.383960
\(383\) 9.90089 0.505912 0.252956 0.967478i \(-0.418597\pi\)
0.252956 + 0.967478i \(0.418597\pi\)
\(384\) 0 0
\(385\) 0.537139 0.0273751
\(386\) 17.7069 0.901256
\(387\) 0 0
\(388\) 9.22112 0.468131
\(389\) 18.1515 0.920316 0.460158 0.887837i \(-0.347793\pi\)
0.460158 + 0.887837i \(0.347793\pi\)
\(390\) 0 0
\(391\) 8.16876 0.413112
\(392\) −6.55118 −0.330884
\(393\) 0 0
\(394\) −5.42917 −0.273518
\(395\) 35.9322 1.80795
\(396\) 0 0
\(397\) 24.4848 1.22886 0.614428 0.788973i \(-0.289387\pi\)
0.614428 + 0.788973i \(0.289387\pi\)
\(398\) −2.00982 −0.100743
\(399\) 0 0
\(400\) 0.900885 0.0450443
\(401\) 12.2309 0.610784 0.305392 0.952227i \(-0.401212\pi\)
0.305392 + 0.952227i \(0.401212\pi\)
\(402\) 0 0
\(403\) −10.3062 −0.513388
\(404\) −13.5086 −0.672080
\(405\) 0 0
\(406\) 3.57083 0.177217
\(407\) 2.43802 0.120848
\(408\) 0 0
\(409\) 14.6643 0.725105 0.362552 0.931963i \(-0.381905\pi\)
0.362552 + 0.931963i \(0.381905\pi\)
\(410\) −29.6872 −1.46615
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) −2.87239 −0.141341
\(414\) 0 0
\(415\) 20.1080 0.987061
\(416\) 1.57083 0.0770161
\(417\) 0 0
\(418\) 0.627404 0.0306873
\(419\) −17.6166 −0.860628 −0.430314 0.902679i \(-0.641597\pi\)
−0.430314 + 0.902679i \(0.641597\pi\)
\(420\) 0 0
\(421\) −27.0369 −1.31770 −0.658850 0.752275i \(-0.728957\pi\)
−0.658850 + 0.752275i \(0.728957\pi\)
\(422\) 1.19823 0.0583289
\(423\) 0 0
\(424\) −4.95746 −0.240756
\(425\) 2.91071 0.141190
\(426\) 0 0
\(427\) −5.22015 −0.252621
\(428\) −11.8158 −0.571139
\(429\) 0 0
\(430\) 13.5086 0.651445
\(431\) −4.15472 −0.200126 −0.100063 0.994981i \(-0.531904\pi\)
−0.100063 + 0.994981i \(0.531904\pi\)
\(432\) 0 0
\(433\) −32.7438 −1.57357 −0.786783 0.617229i \(-0.788255\pi\)
−0.786783 + 0.617229i \(0.788255\pi\)
\(434\) −4.39549 −0.210990
\(435\) 0 0
\(436\) −12.2777 −0.587995
\(437\) 4.80599 0.229901
\(438\) 0 0
\(439\) −2.29313 −0.109445 −0.0547225 0.998502i \(-0.517427\pi\)
−0.0547225 + 0.998502i \(0.517427\pi\)
\(440\) 0.801770 0.0382229
\(441\) 0 0
\(442\) 5.07525 0.241405
\(443\) −17.2113 −0.817733 −0.408867 0.912594i \(-0.634076\pi\)
−0.408867 + 0.912594i \(0.634076\pi\)
\(444\) 0 0
\(445\) −27.3160 −1.29490
\(446\) 1.00000 0.0473514
\(447\) 0 0
\(448\) 0.669941 0.0316517
\(449\) −28.3202 −1.33651 −0.668257 0.743930i \(-0.732959\pi\)
−0.668257 + 0.743930i \(0.732959\pi\)
\(450\) 0 0
\(451\) −4.03369 −0.189939
\(452\) −3.99018 −0.187682
\(453\) 0 0
\(454\) 1.93360 0.0907483
\(455\) 2.55637 0.119844
\(456\) 0 0
\(457\) 27.5357 1.28807 0.644034 0.764997i \(-0.277260\pi\)
0.644034 + 0.764997i \(0.277260\pi\)
\(458\) 16.4563 0.768951
\(459\) 0 0
\(460\) 6.14165 0.286356
\(461\) 6.35295 0.295886 0.147943 0.988996i \(-0.452735\pi\)
0.147943 + 0.988996i \(0.452735\pi\)
\(462\) 0 0
\(463\) −10.6928 −0.496938 −0.248469 0.968640i \(-0.579927\pi\)
−0.248469 + 0.968640i \(0.579927\pi\)
\(464\) 5.33006 0.247442
\(465\) 0 0
\(466\) 7.17437 0.332346
\(467\) −28.2113 −1.30546 −0.652732 0.757589i \(-0.726377\pi\)
−0.652732 + 0.757589i \(0.726377\pi\)
\(468\) 0 0
\(469\) 7.56198 0.349180
\(470\) −7.50443 −0.346153
\(471\) 0 0
\(472\) −4.28752 −0.197349
\(473\) 1.83546 0.0843944
\(474\) 0 0
\(475\) 1.71248 0.0785739
\(476\) 2.16454 0.0992116
\(477\) 0 0
\(478\) 16.3997 0.750105
\(479\) −2.27770 −0.104071 −0.0520353 0.998645i \(-0.516571\pi\)
−0.0520353 + 0.998645i \(0.516571\pi\)
\(480\) 0 0
\(481\) 11.6031 0.529057
\(482\) −2.82142 −0.128512
\(483\) 0 0
\(484\) −10.8911 −0.495048
\(485\) 22.3997 1.01712
\(486\) 0 0
\(487\) −6.46610 −0.293007 −0.146504 0.989210i \(-0.546802\pi\)
−0.146504 + 0.989210i \(0.546802\pi\)
\(488\) −7.79195 −0.352725
\(489\) 0 0
\(490\) −15.9140 −0.718919
\(491\) 6.00885 0.271176 0.135588 0.990765i \(-0.456708\pi\)
0.135588 + 0.990765i \(0.456708\pi\)
\(492\) 0 0
\(493\) 17.2211 0.775600
\(494\) 2.98596 0.134345
\(495\) 0 0
\(496\) −6.56100 −0.294598
\(497\) 6.47171 0.290296
\(498\) 0 0
\(499\) 39.8616 1.78445 0.892225 0.451591i \(-0.149143\pi\)
0.892225 + 0.451591i \(0.149143\pi\)
\(500\) −9.95746 −0.445311
\(501\) 0 0
\(502\) −22.4661 −1.00271
\(503\) −19.1655 −0.854548 −0.427274 0.904122i \(-0.640526\pi\)
−0.427274 + 0.904122i \(0.640526\pi\)
\(504\) 0 0
\(505\) −32.8148 −1.46024
\(506\) 0.834484 0.0370973
\(507\) 0 0
\(508\) 2.50443 0.111116
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 7.06867 0.312700
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 15.7494 0.694677
\(515\) 34.0084 1.49859
\(516\) 0 0
\(517\) −1.01965 −0.0448440
\(518\) 4.94861 0.217430
\(519\) 0 0
\(520\) 3.81581 0.167334
\(521\) −29.2777 −1.28268 −0.641340 0.767257i \(-0.721621\pi\)
−0.641340 + 0.767257i \(0.721621\pi\)
\(522\) 0 0
\(523\) 23.4661 1.02610 0.513051 0.858358i \(-0.328515\pi\)
0.513051 + 0.858358i \(0.328515\pi\)
\(524\) −7.85835 −0.343294
\(525\) 0 0
\(526\) 9.13183 0.398167
\(527\) −21.1982 −0.923409
\(528\) 0 0
\(529\) −16.6078 −0.722076
\(530\) −12.0425 −0.523094
\(531\) 0 0
\(532\) 1.27348 0.0552124
\(533\) −19.1973 −0.831525
\(534\) 0 0
\(535\) −28.7027 −1.24092
\(536\) 11.2875 0.487547
\(537\) 0 0
\(538\) −7.56100 −0.325978
\(539\) −2.16227 −0.0931357
\(540\) 0 0
\(541\) 28.3768 1.22001 0.610007 0.792396i \(-0.291167\pi\)
0.610007 + 0.792396i \(0.291167\pi\)
\(542\) −5.40531 −0.232178
\(543\) 0 0
\(544\) 3.23094 0.138526
\(545\) −29.8247 −1.27755
\(546\) 0 0
\(547\) 8.11779 0.347092 0.173546 0.984826i \(-0.444478\pi\)
0.173546 + 0.984826i \(0.444478\pi\)
\(548\) 22.5979 0.965336
\(549\) 0 0
\(550\) 0.297345 0.0126788
\(551\) 10.1318 0.431630
\(552\) 0 0
\(553\) 9.90974 0.421405
\(554\) 12.6035 0.535473
\(555\) 0 0
\(556\) 5.91071 0.250670
\(557\) −8.25059 −0.349589 −0.174794 0.984605i \(-0.555926\pi\)
−0.174794 + 0.984605i \(0.555926\pi\)
\(558\) 0 0
\(559\) 8.73537 0.369467
\(560\) 1.62740 0.0687704
\(561\) 0 0
\(562\) 20.9705 0.884588
\(563\) −28.1220 −1.18520 −0.592601 0.805496i \(-0.701899\pi\)
−0.592601 + 0.805496i \(0.701899\pi\)
\(564\) 0 0
\(565\) −9.69283 −0.407780
\(566\) −5.71669 −0.240291
\(567\) 0 0
\(568\) 9.66012 0.405330
\(569\) −5.70266 −0.239068 −0.119534 0.992830i \(-0.538140\pi\)
−0.119534 + 0.992830i \(0.538140\pi\)
\(570\) 0 0
\(571\) 23.1884 0.970405 0.485202 0.874402i \(-0.338746\pi\)
0.485202 + 0.874402i \(0.338746\pi\)
\(572\) 0.518465 0.0216781
\(573\) 0 0
\(574\) −8.18743 −0.341737
\(575\) 2.27770 0.0949865
\(576\) 0 0
\(577\) −4.39970 −0.183162 −0.0915810 0.995798i \(-0.529192\pi\)
−0.0915810 + 0.995798i \(0.529192\pi\)
\(578\) −6.56100 −0.272902
\(579\) 0 0
\(580\) 12.9476 0.537621
\(581\) 5.54557 0.230069
\(582\) 0 0
\(583\) −1.63625 −0.0677667
\(584\) 10.5512 0.436611
\(585\) 0 0
\(586\) 9.68398 0.400042
\(587\) 5.19726 0.214514 0.107257 0.994231i \(-0.465793\pi\)
0.107257 + 0.994231i \(0.465793\pi\)
\(588\) 0 0
\(589\) −12.4717 −0.513888
\(590\) −10.4151 −0.428784
\(591\) 0 0
\(592\) 7.38664 0.303589
\(593\) −8.91071 −0.365919 −0.182959 0.983120i \(-0.558568\pi\)
−0.182959 + 0.983120i \(0.558568\pi\)
\(594\) 0 0
\(595\) 5.25805 0.215559
\(596\) 2.75923 0.113023
\(597\) 0 0
\(598\) 3.97150 0.162407
\(599\) 14.1080 0.576436 0.288218 0.957565i \(-0.406937\pi\)
0.288218 + 0.957565i \(0.406937\pi\)
\(600\) 0 0
\(601\) −22.2538 −0.907753 −0.453876 0.891065i \(-0.649959\pi\)
−0.453876 + 0.891065i \(0.649959\pi\)
\(602\) 3.72555 0.151842
\(603\) 0 0
\(604\) 4.07947 0.165991
\(605\) −26.4563 −1.07560
\(606\) 0 0
\(607\) 21.1491 0.858416 0.429208 0.903206i \(-0.358793\pi\)
0.429208 + 0.903206i \(0.358793\pi\)
\(608\) 1.90089 0.0770911
\(609\) 0 0
\(610\) −18.9280 −0.766372
\(611\) −4.85274 −0.196321
\(612\) 0 0
\(613\) −20.5750 −0.831018 −0.415509 0.909589i \(-0.636397\pi\)
−0.415509 + 0.909589i \(0.636397\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) 0.221120 0.00890918
\(617\) −16.7069 −0.672593 −0.336297 0.941756i \(-0.609174\pi\)
−0.336297 + 0.941756i \(0.609174\pi\)
\(618\) 0 0
\(619\) 29.6742 1.19271 0.596353 0.802723i \(-0.296616\pi\)
0.596353 + 0.802723i \(0.296616\pi\)
\(620\) −15.9378 −0.640078
\(621\) 0 0
\(622\) 28.1262 1.12776
\(623\) −7.53348 −0.301822
\(624\) 0 0
\(625\) −28.6928 −1.14771
\(626\) −9.27770 −0.370811
\(627\) 0 0
\(628\) −13.1688 −0.525491
\(629\) 23.8658 0.951592
\(630\) 0 0
\(631\) −44.7340 −1.78083 −0.890416 0.455148i \(-0.849586\pi\)
−0.890416 + 0.455148i \(0.849586\pi\)
\(632\) 14.7919 0.588392
\(633\) 0 0
\(634\) −10.4151 −0.413638
\(635\) 6.08368 0.241424
\(636\) 0 0
\(637\) −10.2908 −0.407735
\(638\) 1.75923 0.0696487
\(639\) 0 0
\(640\) 2.42917 0.0960215
\(641\) 18.3441 0.724548 0.362274 0.932072i \(-0.382000\pi\)
0.362274 + 0.932072i \(0.382000\pi\)
\(642\) 0 0
\(643\) 4.49557 0.177288 0.0886441 0.996063i \(-0.471747\pi\)
0.0886441 + 0.996063i \(0.471747\pi\)
\(644\) 1.69380 0.0667453
\(645\) 0 0
\(646\) 6.14165 0.241640
\(647\) −6.75923 −0.265733 −0.132866 0.991134i \(-0.542418\pi\)
−0.132866 + 0.991134i \(0.542418\pi\)
\(648\) 0 0
\(649\) −1.41513 −0.0555489
\(650\) 1.41513 0.0555061
\(651\) 0 0
\(652\) 10.1132 0.396062
\(653\) −46.3146 −1.81243 −0.906216 0.422816i \(-0.861042\pi\)
−0.906216 + 0.422816i \(0.861042\pi\)
\(654\) 0 0
\(655\) −19.0893 −0.745880
\(656\) −12.2211 −0.477155
\(657\) 0 0
\(658\) −2.06964 −0.0806831
\(659\) −9.36277 −0.364722 −0.182361 0.983232i \(-0.558374\pi\)
−0.182361 + 0.983232i \(0.558374\pi\)
\(660\) 0 0
\(661\) −25.8626 −1.00594 −0.502969 0.864305i \(-0.667759\pi\)
−0.502969 + 0.864305i \(0.667759\pi\)
\(662\) −2.40628 −0.0935229
\(663\) 0 0
\(664\) 8.27770 0.321237
\(665\) 3.09351 0.119961
\(666\) 0 0
\(667\) 13.4759 0.521790
\(668\) −19.9575 −0.772177
\(669\) 0 0
\(670\) 27.4193 1.05930
\(671\) −2.57180 −0.0992832
\(672\) 0 0
\(673\) 37.1818 1.43325 0.716627 0.697457i \(-0.245685\pi\)
0.716627 + 0.697457i \(0.245685\pi\)
\(674\) −27.2875 −1.05108
\(675\) 0 0
\(676\) −10.5325 −0.405096
\(677\) −9.18280 −0.352924 −0.176462 0.984307i \(-0.556465\pi\)
−0.176462 + 0.984307i \(0.556465\pi\)
\(678\) 0 0
\(679\) 6.17761 0.237075
\(680\) 7.84852 0.300977
\(681\) 0 0
\(682\) −2.16552 −0.0829219
\(683\) −26.0131 −0.995362 −0.497681 0.867360i \(-0.665815\pi\)
−0.497681 + 0.867360i \(0.665815\pi\)
\(684\) 0 0
\(685\) 54.8943 2.09740
\(686\) −9.07849 −0.346619
\(687\) 0 0
\(688\) 5.56100 0.212011
\(689\) −7.78731 −0.296673
\(690\) 0 0
\(691\) 12.6840 0.482521 0.241261 0.970460i \(-0.422439\pi\)
0.241261 + 0.970460i \(0.422439\pi\)
\(692\) −13.4095 −0.509754
\(693\) 0 0
\(694\) −30.2538 −1.14842
\(695\) 14.3581 0.544635
\(696\) 0 0
\(697\) −39.4858 −1.49563
\(698\) −24.3866 −0.923048
\(699\) 0 0
\(700\) 0.603540 0.0228117
\(701\) −39.8092 −1.50357 −0.751787 0.659406i \(-0.770808\pi\)
−0.751787 + 0.659406i \(0.770808\pi\)
\(702\) 0 0
\(703\) 14.0411 0.529572
\(704\) 0.330059 0.0124396
\(705\) 0 0
\(706\) 8.18841 0.308175
\(707\) −9.05000 −0.340360
\(708\) 0 0
\(709\) −28.7494 −1.07971 −0.539853 0.841759i \(-0.681520\pi\)
−0.539853 + 0.841759i \(0.681520\pi\)
\(710\) 23.4661 0.880667
\(711\) 0 0
\(712\) −11.2450 −0.421424
\(713\) −16.5881 −0.621230
\(714\) 0 0
\(715\) 1.25944 0.0471004
\(716\) −1.59372 −0.0595600
\(717\) 0 0
\(718\) −23.9149 −0.892497
\(719\) 1.88545 0.0703156 0.0351578 0.999382i \(-0.488807\pi\)
0.0351578 + 0.999382i \(0.488807\pi\)
\(720\) 0 0
\(721\) 9.37918 0.349299
\(722\) −15.3866 −0.572631
\(723\) 0 0
\(724\) 3.08929 0.114813
\(725\) 4.80177 0.178333
\(726\) 0 0
\(727\) −32.3670 −1.20043 −0.600213 0.799841i \(-0.704917\pi\)
−0.600213 + 0.799841i \(0.704917\pi\)
\(728\) 1.05236 0.0390031
\(729\) 0 0
\(730\) 25.6306 0.948633
\(731\) 17.9673 0.664544
\(732\) 0 0
\(733\) −22.4801 −0.830323 −0.415162 0.909748i \(-0.636275\pi\)
−0.415162 + 0.909748i \(0.636275\pi\)
\(734\) 6.54794 0.241689
\(735\) 0 0
\(736\) 2.52829 0.0931940
\(737\) 3.72555 0.137232
\(738\) 0 0
\(739\) 9.82044 0.361251 0.180625 0.983552i \(-0.442188\pi\)
0.180625 + 0.983552i \(0.442188\pi\)
\(740\) 17.9434 0.659613
\(741\) 0 0
\(742\) −3.32121 −0.121925
\(743\) 6.42032 0.235539 0.117769 0.993041i \(-0.462426\pi\)
0.117769 + 0.993041i \(0.462426\pi\)
\(744\) 0 0
\(745\) 6.70266 0.245566
\(746\) −21.7396 −0.795943
\(747\) 0 0
\(748\) 1.06640 0.0389915
\(749\) −7.91590 −0.289241
\(750\) 0 0
\(751\) −25.6601 −0.936351 −0.468175 0.883636i \(-0.655088\pi\)
−0.468175 + 0.883636i \(0.655088\pi\)
\(752\) −3.08929 −0.112655
\(753\) 0 0
\(754\) 8.37260 0.304912
\(755\) 9.90974 0.360652
\(756\) 0 0
\(757\) 3.17858 0.115528 0.0577638 0.998330i \(-0.481603\pi\)
0.0577638 + 0.998330i \(0.481603\pi\)
\(758\) 0.514249 0.0186784
\(759\) 0 0
\(760\) 4.61758 0.167497
\(761\) 43.1533 1.56431 0.782153 0.623086i \(-0.214121\pi\)
0.782153 + 0.623086i \(0.214121\pi\)
\(762\) 0 0
\(763\) −8.22534 −0.297777
\(764\) −7.50443 −0.271501
\(765\) 0 0
\(766\) 9.90089 0.357734
\(767\) −6.73495 −0.243185
\(768\) 0 0
\(769\) −48.4890 −1.74856 −0.874279 0.485424i \(-0.838665\pi\)
−0.874279 + 0.485424i \(0.838665\pi\)
\(770\) 0.537139 0.0193571
\(771\) 0 0
\(772\) 17.7069 0.637284
\(773\) 35.0925 1.26219 0.631095 0.775705i \(-0.282606\pi\)
0.631095 + 0.775705i \(0.282606\pi\)
\(774\) 0 0
\(775\) −5.91071 −0.212319
\(776\) 9.22112 0.331019
\(777\) 0 0
\(778\) 18.1515 0.650762
\(779\) −23.2309 −0.832335
\(780\) 0 0
\(781\) 3.18841 0.114090
\(782\) 8.16876 0.292114
\(783\) 0 0
\(784\) −6.55118 −0.233971
\(785\) −31.9892 −1.14174
\(786\) 0 0
\(787\) −19.3104 −0.688342 −0.344171 0.938907i \(-0.611840\pi\)
−0.344171 + 0.938907i \(0.611840\pi\)
\(788\) −5.42917 −0.193406
\(789\) 0 0
\(790\) 35.9322 1.27841
\(791\) −2.67318 −0.0950475
\(792\) 0 0
\(793\) −12.2398 −0.434648
\(794\) 24.4848 0.868932
\(795\) 0 0
\(796\) −2.00982 −0.0712363
\(797\) 27.4376 0.971890 0.485945 0.873989i \(-0.338476\pi\)
0.485945 + 0.873989i \(0.338476\pi\)
\(798\) 0 0
\(799\) −9.98133 −0.353114
\(800\) 0.900885 0.0318511
\(801\) 0 0
\(802\) 12.2309 0.431890
\(803\) 3.48251 0.122895
\(804\) 0 0
\(805\) 4.11455 0.145019
\(806\) −10.3062 −0.363020
\(807\) 0 0
\(808\) −13.5086 −0.475232
\(809\) −9.94666 −0.349706 −0.174853 0.984595i \(-0.555945\pi\)
−0.174853 + 0.984595i \(0.555945\pi\)
\(810\) 0 0
\(811\) 8.14587 0.286040 0.143020 0.989720i \(-0.454319\pi\)
0.143020 + 0.989720i \(0.454319\pi\)
\(812\) 3.57083 0.125311
\(813\) 0 0
\(814\) 2.43802 0.0854527
\(815\) 24.5666 0.860531
\(816\) 0 0
\(817\) 10.5708 0.369826
\(818\) 14.6643 0.512726
\(819\) 0 0
\(820\) −29.6872 −1.03672
\(821\) 31.3343 1.09357 0.546787 0.837272i \(-0.315851\pi\)
0.546787 + 0.837272i \(0.315851\pi\)
\(822\) 0 0
\(823\) −55.5685 −1.93699 −0.968497 0.249024i \(-0.919890\pi\)
−0.968497 + 0.249024i \(0.919890\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −2.87239 −0.0999431
\(827\) 32.2024 1.11979 0.559894 0.828564i \(-0.310842\pi\)
0.559894 + 0.828564i \(0.310842\pi\)
\(828\) 0 0
\(829\) 5.37681 0.186744 0.0933722 0.995631i \(-0.470235\pi\)
0.0933722 + 0.995631i \(0.470235\pi\)
\(830\) 20.1080 0.697958
\(831\) 0 0
\(832\) 1.57083 0.0544586
\(833\) −21.1665 −0.733375
\(834\) 0 0
\(835\) −48.4801 −1.67772
\(836\) 0.627404 0.0216992
\(837\) 0 0
\(838\) −17.6166 −0.608556
\(839\) −10.0042 −0.345384 −0.172692 0.984976i \(-0.555247\pi\)
−0.172692 + 0.984976i \(0.555247\pi\)
\(840\) 0 0
\(841\) −0.590474 −0.0203612
\(842\) −27.0369 −0.931754
\(843\) 0 0
\(844\) 1.19823 0.0412448
\(845\) −25.5853 −0.880161
\(846\) 0 0
\(847\) −7.29637 −0.250706
\(848\) −4.95746 −0.170240
\(849\) 0 0
\(850\) 2.91071 0.0998365
\(851\) 18.6755 0.640190
\(852\) 0 0
\(853\) −22.9519 −0.785857 −0.392928 0.919569i \(-0.628538\pi\)
−0.392928 + 0.919569i \(0.628538\pi\)
\(854\) −5.22015 −0.178630
\(855\) 0 0
\(856\) −11.8158 −0.403856
\(857\) 28.2908 0.966394 0.483197 0.875512i \(-0.339475\pi\)
0.483197 + 0.875512i \(0.339475\pi\)
\(858\) 0 0
\(859\) 18.5849 0.634107 0.317054 0.948408i \(-0.397306\pi\)
0.317054 + 0.948408i \(0.397306\pi\)
\(860\) 13.5086 0.460641
\(861\) 0 0
\(862\) −4.15472 −0.141510
\(863\) −44.1818 −1.50397 −0.751983 0.659182i \(-0.770903\pi\)
−0.751983 + 0.659182i \(0.770903\pi\)
\(864\) 0 0
\(865\) −32.5741 −1.10755
\(866\) −32.7438 −1.11268
\(867\) 0 0
\(868\) −4.39549 −0.149193
\(869\) 4.88221 0.165618
\(870\) 0 0
\(871\) 17.7307 0.600783
\(872\) −12.2777 −0.415775
\(873\) 0 0
\(874\) 4.80599 0.162565
\(875\) −6.67091 −0.225518
\(876\) 0 0
\(877\) −21.0098 −0.709451 −0.354726 0.934970i \(-0.615426\pi\)
−0.354726 + 0.934970i \(0.615426\pi\)
\(878\) −2.29313 −0.0773894
\(879\) 0 0
\(880\) 0.801770 0.0270277
\(881\) −30.3660 −1.02306 −0.511529 0.859266i \(-0.670921\pi\)
−0.511529 + 0.859266i \(0.670921\pi\)
\(882\) 0 0
\(883\) 51.6535 1.73828 0.869140 0.494566i \(-0.164673\pi\)
0.869140 + 0.494566i \(0.164673\pi\)
\(884\) 5.07525 0.170699
\(885\) 0 0
\(886\) −17.2113 −0.578225
\(887\) −12.2725 −0.412070 −0.206035 0.978545i \(-0.566056\pi\)
−0.206035 + 0.978545i \(0.566056\pi\)
\(888\) 0 0
\(889\) 1.67782 0.0562722
\(890\) −27.3160 −0.915635
\(891\) 0 0
\(892\) 1.00000 0.0334825
\(893\) −5.87239 −0.196512
\(894\) 0 0
\(895\) −3.87141 −0.129407
\(896\) 0.669941 0.0223812
\(897\) 0 0
\(898\) −28.3202 −0.945058
\(899\) −34.9705 −1.16633
\(900\) 0 0
\(901\) −16.0173 −0.533613
\(902\) −4.03369 −0.134307
\(903\) 0 0
\(904\) −3.99018 −0.132711
\(905\) 7.50443 0.249456
\(906\) 0 0
\(907\) −5.68862 −0.188887 −0.0944437 0.995530i \(-0.530107\pi\)
−0.0944437 + 0.995530i \(0.530107\pi\)
\(908\) 1.93360 0.0641687
\(909\) 0 0
\(910\) 2.55637 0.0847428
\(911\) 28.4007 0.940956 0.470478 0.882412i \(-0.344081\pi\)
0.470478 + 0.882412i \(0.344081\pi\)
\(912\) 0 0
\(913\) 2.73213 0.0904202
\(914\) 27.5357 0.910802
\(915\) 0 0
\(916\) 16.4563 0.543731
\(917\) −5.26463 −0.173853
\(918\) 0 0
\(919\) 11.2342 0.370582 0.185291 0.982684i \(-0.440677\pi\)
0.185291 + 0.982684i \(0.440677\pi\)
\(920\) 6.14165 0.202484
\(921\) 0 0
\(922\) 6.35295 0.209223
\(923\) 15.1744 0.499470
\(924\) 0 0
\(925\) 6.65451 0.218799
\(926\) −10.6928 −0.351388
\(927\) 0 0
\(928\) 5.33006 0.174968
\(929\) −48.3048 −1.58483 −0.792415 0.609983i \(-0.791176\pi\)
−0.792415 + 0.609983i \(0.791176\pi\)
\(930\) 0 0
\(931\) −12.4530 −0.408132
\(932\) 7.17437 0.235004
\(933\) 0 0
\(934\) −28.2113 −0.923102
\(935\) 2.59047 0.0847176
\(936\) 0 0
\(937\) 32.0215 1.04610 0.523048 0.852303i \(-0.324795\pi\)
0.523048 + 0.852303i \(0.324795\pi\)
\(938\) 7.56198 0.246907
\(939\) 0 0
\(940\) −7.50443 −0.244767
\(941\) 19.1599 0.624595 0.312298 0.949984i \(-0.398901\pi\)
0.312298 + 0.949984i \(0.398901\pi\)
\(942\) 0 0
\(943\) −30.8985 −1.00619
\(944\) −4.28752 −0.139547
\(945\) 0 0
\(946\) 1.83546 0.0596759
\(947\) 20.8397 0.677198 0.338599 0.940931i \(-0.390047\pi\)
0.338599 + 0.940931i \(0.390047\pi\)
\(948\) 0 0
\(949\) 16.5741 0.538017
\(950\) 1.71248 0.0555601
\(951\) 0 0
\(952\) 2.16454 0.0701532
\(953\) −48.9280 −1.58493 −0.792467 0.609915i \(-0.791203\pi\)
−0.792467 + 0.609915i \(0.791203\pi\)
\(954\) 0 0
\(955\) −18.2296 −0.589894
\(956\) 16.3997 0.530404
\(957\) 0 0
\(958\) −2.27770 −0.0735891
\(959\) 15.1393 0.488873
\(960\) 0 0
\(961\) 12.0468 0.388605
\(962\) 11.6031 0.374100
\(963\) 0 0
\(964\) −2.82142 −0.0908717
\(965\) 43.0131 1.38464
\(966\) 0 0
\(967\) 5.88221 0.189159 0.0945796 0.995517i \(-0.469849\pi\)
0.0945796 + 0.995517i \(0.469849\pi\)
\(968\) −10.8911 −0.350052
\(969\) 0 0
\(970\) 22.3997 0.719211
\(971\) 3.95227 0.126834 0.0634172 0.997987i \(-0.479800\pi\)
0.0634172 + 0.997987i \(0.479800\pi\)
\(972\) 0 0
\(973\) 3.95983 0.126946
\(974\) −6.46610 −0.207187
\(975\) 0 0
\(976\) −7.79195 −0.249414
\(977\) −6.39085 −0.204461 −0.102231 0.994761i \(-0.532598\pi\)
−0.102231 + 0.994761i \(0.532598\pi\)
\(978\) 0 0
\(979\) −3.71151 −0.118620
\(980\) −15.9140 −0.508353
\(981\) 0 0
\(982\) 6.00885 0.191750
\(983\) −43.0650 −1.37356 −0.686780 0.726865i \(-0.740977\pi\)
−0.686780 + 0.726865i \(0.740977\pi\)
\(984\) 0 0
\(985\) −13.1884 −0.420218
\(986\) 17.2211 0.548432
\(987\) 0 0
\(988\) 2.98596 0.0949960
\(989\) 14.0598 0.447076
\(990\) 0 0
\(991\) −61.2057 −1.94426 −0.972131 0.234437i \(-0.924675\pi\)
−0.972131 + 0.234437i \(0.924675\pi\)
\(992\) −6.56100 −0.208312
\(993\) 0 0
\(994\) 6.47171 0.205270
\(995\) −4.88221 −0.154776
\(996\) 0 0
\(997\) 32.5119 1.02966 0.514831 0.857292i \(-0.327855\pi\)
0.514831 + 0.857292i \(0.327855\pi\)
\(998\) 39.8616 1.26180
\(999\) 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 4014.2.a.p.1.3 3
3.2 odd 2 1338.2.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.c.1.1 3 3.2 odd 2
4014.2.a.p.1.3 3 1.1 even 1 trivial