Properties

Label 4014.2.a.z.1.5
Level $4014$
Weight $2$
Character 4014.1
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 12x^{6} + 46x^{5} + 54x^{4} - 148x^{3} - 98x^{2} + 126x + 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 446)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.07351\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.701327 q^{5} +3.38691 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.701327 q^{5} +3.38691 q^{7} +1.00000 q^{8} -0.701327 q^{10} -3.84509 q^{11} -4.26701 q^{13} +3.38691 q^{14} +1.00000 q^{16} +6.75189 q^{17} +7.87608 q^{19} -0.701327 q^{20} -3.84509 q^{22} -1.63713 q^{23} -4.50814 q^{25} -4.26701 q^{26} +3.38691 q^{28} +2.14701 q^{29} -2.31098 q^{31} +1.00000 q^{32} +6.75189 q^{34} -2.37533 q^{35} +5.66056 q^{37} +7.87608 q^{38} -0.701327 q^{40} -1.47115 q^{41} +6.07202 q^{43} -3.84509 q^{44} -1.63713 q^{46} +10.2475 q^{47} +4.47115 q^{49} -4.50814 q^{50} -4.26701 q^{52} +7.27873 q^{53} +2.69667 q^{55} +3.38691 q^{56} +2.14701 q^{58} -8.21453 q^{59} +7.24775 q^{61} -2.31098 q^{62} +1.00000 q^{64} +2.99257 q^{65} +9.74666 q^{67} +6.75189 q^{68} -2.37533 q^{70} -13.1548 q^{71} -1.79989 q^{73} +5.66056 q^{74} +7.87608 q^{76} -13.0230 q^{77} -9.88457 q^{79} -0.701327 q^{80} -1.47115 q^{82} +15.7196 q^{83} -4.73528 q^{85} +6.07202 q^{86} -3.84509 q^{88} +12.7445 q^{89} -14.4520 q^{91} -1.63713 q^{92} +10.2475 q^{94} -5.52370 q^{95} +0.294371 q^{97} +4.47115 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8} - 4 q^{10} - 2 q^{11} + 10 q^{13} + 4 q^{14} + 8 q^{16} - 8 q^{17} + 16 q^{19} - 4 q^{20} - 2 q^{22} + 4 q^{23} + 24 q^{25} + 10 q^{26} + 4 q^{28} - 8 q^{29} + 12 q^{31} + 8 q^{32} - 8 q^{34} + 24 q^{35} + 16 q^{37} + 16 q^{38} - 4 q^{40} - 16 q^{41} + 12 q^{43} - 2 q^{44} + 4 q^{46} + 4 q^{47} + 40 q^{49} + 24 q^{50} + 10 q^{52} + 8 q^{53} + 8 q^{55} + 4 q^{56} - 8 q^{58} + 26 q^{61} + 12 q^{62} + 8 q^{64} + 24 q^{65} - 10 q^{67} - 8 q^{68} + 24 q^{70} - 8 q^{71} + 8 q^{73} + 16 q^{74} + 16 q^{76} + 56 q^{77} + 16 q^{79} - 4 q^{80} - 16 q^{82} + 44 q^{83} - 18 q^{85} + 12 q^{86} - 2 q^{88} + 4 q^{91} + 4 q^{92} + 4 q^{94} + 48 q^{95} + 12 q^{97} + 40 q^{98}+O(q^{100}) \) 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
\(5\) −0.701327 −0.313643 −0.156821 0.987627i \(-0.550125\pi\)
−0.156821 + 0.987627i \(0.550125\pi\)
\(6\) 0 0
\(7\) 3.38691 1.28013 0.640066 0.768320i \(-0.278907\pi\)
0.640066 + 0.768320i \(0.278907\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −0.701327 −0.221779
\(11\) −3.84509 −1.15934 −0.579670 0.814852i \(-0.696818\pi\)
−0.579670 + 0.814852i \(0.696818\pi\)
\(12\) 0 0
\(13\) −4.26701 −1.18346 −0.591728 0.806138i \(-0.701554\pi\)
−0.591728 + 0.806138i \(0.701554\pi\)
\(14\) 3.38691 0.905190
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.75189 1.63757 0.818787 0.574097i \(-0.194647\pi\)
0.818787 + 0.574097i \(0.194647\pi\)
\(18\) 0 0
\(19\) 7.87608 1.80690 0.903448 0.428698i \(-0.141028\pi\)
0.903448 + 0.428698i \(0.141028\pi\)
\(20\) −0.701327 −0.156821
\(21\) 0 0
\(22\) −3.84509 −0.819777
\(23\) −1.63713 −0.341365 −0.170682 0.985326i \(-0.554597\pi\)
−0.170682 + 0.985326i \(0.554597\pi\)
\(24\) 0 0
\(25\) −4.50814 −0.901628
\(26\) −4.26701 −0.836830
\(27\) 0 0
\(28\) 3.38691 0.640066
\(29\) 2.14701 0.398691 0.199345 0.979929i \(-0.436118\pi\)
0.199345 + 0.979929i \(0.436118\pi\)
\(30\) 0 0
\(31\) −2.31098 −0.415064 −0.207532 0.978228i \(-0.566543\pi\)
−0.207532 + 0.978228i \(0.566543\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.75189 1.15794
\(35\) −2.37533 −0.401504
\(36\) 0 0
\(37\) 5.66056 0.930590 0.465295 0.885156i \(-0.345948\pi\)
0.465295 + 0.885156i \(0.345948\pi\)
\(38\) 7.87608 1.27767
\(39\) 0 0
\(40\) −0.701327 −0.110889
\(41\) −1.47115 −0.229756 −0.114878 0.993380i \(-0.536648\pi\)
−0.114878 + 0.993380i \(0.536648\pi\)
\(42\) 0 0
\(43\) 6.07202 0.925974 0.462987 0.886365i \(-0.346778\pi\)
0.462987 + 0.886365i \(0.346778\pi\)
\(44\) −3.84509 −0.579670
\(45\) 0 0
\(46\) −1.63713 −0.241381
\(47\) 10.2475 1.49476 0.747378 0.664400i \(-0.231313\pi\)
0.747378 + 0.664400i \(0.231313\pi\)
\(48\) 0 0
\(49\) 4.47115 0.638736
\(50\) −4.50814 −0.637547
\(51\) 0 0
\(52\) −4.26701 −0.591728
\(53\) 7.27873 0.999810 0.499905 0.866080i \(-0.333368\pi\)
0.499905 + 0.866080i \(0.333368\pi\)
\(54\) 0 0
\(55\) 2.69667 0.363618
\(56\) 3.38691 0.452595
\(57\) 0 0
\(58\) 2.14701 0.281917
\(59\) −8.21453 −1.06944 −0.534720 0.845029i \(-0.679583\pi\)
−0.534720 + 0.845029i \(0.679583\pi\)
\(60\) 0 0
\(61\) 7.24775 0.927979 0.463989 0.885841i \(-0.346418\pi\)
0.463989 + 0.885841i \(0.346418\pi\)
\(62\) −2.31098 −0.293495
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.99257 0.371183
\(66\) 0 0
\(67\) 9.74666 1.19074 0.595372 0.803450i \(-0.297005\pi\)
0.595372 + 0.803450i \(0.297005\pi\)
\(68\) 6.75189 0.818787
\(69\) 0 0
\(70\) −2.37533 −0.283906
\(71\) −13.1548 −1.56119 −0.780594 0.625038i \(-0.785083\pi\)
−0.780594 + 0.625038i \(0.785083\pi\)
\(72\) 0 0
\(73\) −1.79989 −0.210661 −0.105330 0.994437i \(-0.533590\pi\)
−0.105330 + 0.994437i \(0.533590\pi\)
\(74\) 5.66056 0.658027
\(75\) 0 0
\(76\) 7.87608 0.903448
\(77\) −13.0230 −1.48411
\(78\) 0 0
\(79\) −9.88457 −1.11210 −0.556050 0.831149i \(-0.687684\pi\)
−0.556050 + 0.831149i \(0.687684\pi\)
\(80\) −0.701327 −0.0784107
\(81\) 0 0
\(82\) −1.47115 −0.162462
\(83\) 15.7196 1.72546 0.862728 0.505669i \(-0.168754\pi\)
0.862728 + 0.505669i \(0.168754\pi\)
\(84\) 0 0
\(85\) −4.73528 −0.513614
\(86\) 6.07202 0.654762
\(87\) 0 0
\(88\) −3.84509 −0.409888
\(89\) 12.7445 1.35091 0.675456 0.737400i \(-0.263947\pi\)
0.675456 + 0.737400i \(0.263947\pi\)
\(90\) 0 0
\(91\) −14.4520 −1.51498
\(92\) −1.63713 −0.170682
\(93\) 0 0
\(94\) 10.2475 1.05695
\(95\) −5.52370 −0.566720
\(96\) 0 0
\(97\) 0.294371 0.0298888 0.0149444 0.999888i \(-0.495243\pi\)
0.0149444 + 0.999888i \(0.495243\pi\)
\(98\) 4.47115 0.451655
\(99\) 0 0
\(100\) −4.50814 −0.450814
\(101\) 16.3182 1.62372 0.811859 0.583853i \(-0.198456\pi\)
0.811859 + 0.583853i \(0.198456\pi\)
\(102\) 0 0
\(103\) 7.94961 0.783299 0.391649 0.920115i \(-0.371905\pi\)
0.391649 + 0.920115i \(0.371905\pi\)
\(104\) −4.26701 −0.418415
\(105\) 0 0
\(106\) 7.27873 0.706973
\(107\) 6.96523 0.673354 0.336677 0.941620i \(-0.390697\pi\)
0.336677 + 0.941620i \(0.390697\pi\)
\(108\) 0 0
\(109\) −11.3762 −1.08964 −0.544821 0.838552i \(-0.683402\pi\)
−0.544821 + 0.838552i \(0.683402\pi\)
\(110\) 2.69667 0.257117
\(111\) 0 0
\(112\) 3.38691 0.320033
\(113\) −1.07883 −0.101487 −0.0507437 0.998712i \(-0.516159\pi\)
−0.0507437 + 0.998712i \(0.516159\pi\)
\(114\) 0 0
\(115\) 1.14816 0.107067
\(116\) 2.14701 0.199345
\(117\) 0 0
\(118\) −8.21453 −0.756209
\(119\) 22.8680 2.09631
\(120\) 0 0
\(121\) 3.78474 0.344067
\(122\) 7.24775 0.656180
\(123\) 0 0
\(124\) −2.31098 −0.207532
\(125\) 6.66831 0.596432
\(126\) 0 0
\(127\) −18.2481 −1.61926 −0.809631 0.586940i \(-0.800333\pi\)
−0.809631 + 0.586940i \(0.800333\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.99257 0.262466
\(131\) −5.09782 −0.445398 −0.222699 0.974887i \(-0.571487\pi\)
−0.222699 + 0.974887i \(0.571487\pi\)
\(132\) 0 0
\(133\) 26.6756 2.31306
\(134\) 9.74666 0.841983
\(135\) 0 0
\(136\) 6.75189 0.578970
\(137\) −14.9155 −1.27432 −0.637159 0.770732i \(-0.719891\pi\)
−0.637159 + 0.770732i \(0.719891\pi\)
\(138\) 0 0
\(139\) −17.6022 −1.49300 −0.746500 0.665385i \(-0.768267\pi\)
−0.746500 + 0.665385i \(0.768267\pi\)
\(140\) −2.37533 −0.200752
\(141\) 0 0
\(142\) −13.1548 −1.10393
\(143\) 16.4071 1.37203
\(144\) 0 0
\(145\) −1.50576 −0.125046
\(146\) −1.79989 −0.148960
\(147\) 0 0
\(148\) 5.66056 0.465295
\(149\) 9.25428 0.758140 0.379070 0.925368i \(-0.376244\pi\)
0.379070 + 0.925368i \(0.376244\pi\)
\(150\) 0 0
\(151\) 20.0326 1.63023 0.815116 0.579297i \(-0.196673\pi\)
0.815116 + 0.579297i \(0.196673\pi\)
\(152\) 7.87608 0.638834
\(153\) 0 0
\(154\) −13.0230 −1.04942
\(155\) 1.62075 0.130182
\(156\) 0 0
\(157\) −13.5972 −1.08518 −0.542588 0.839999i \(-0.682555\pi\)
−0.542588 + 0.839999i \(0.682555\pi\)
\(158\) −9.88457 −0.786374
\(159\) 0 0
\(160\) −0.701327 −0.0554447
\(161\) −5.54480 −0.436992
\(162\) 0 0
\(163\) 17.6768 1.38455 0.692277 0.721632i \(-0.256608\pi\)
0.692277 + 0.721632i \(0.256608\pi\)
\(164\) −1.47115 −0.114878
\(165\) 0 0
\(166\) 15.7196 1.22008
\(167\) 6.94928 0.537752 0.268876 0.963175i \(-0.413348\pi\)
0.268876 + 0.963175i \(0.413348\pi\)
\(168\) 0 0
\(169\) 5.20740 0.400569
\(170\) −4.73528 −0.363180
\(171\) 0 0
\(172\) 6.07202 0.462987
\(173\) 2.88332 0.219215 0.109607 0.993975i \(-0.465041\pi\)
0.109607 + 0.993975i \(0.465041\pi\)
\(174\) 0 0
\(175\) −15.2687 −1.15420
\(176\) −3.84509 −0.289835
\(177\) 0 0
\(178\) 12.7445 0.955239
\(179\) −26.3697 −1.97096 −0.985482 0.169779i \(-0.945695\pi\)
−0.985482 + 0.169779i \(0.945695\pi\)
\(180\) 0 0
\(181\) 6.93387 0.515390 0.257695 0.966226i \(-0.417037\pi\)
0.257695 + 0.966226i \(0.417037\pi\)
\(182\) −14.4520 −1.07125
\(183\) 0 0
\(184\) −1.63713 −0.120691
\(185\) −3.96990 −0.291873
\(186\) 0 0
\(187\) −25.9617 −1.89850
\(188\) 10.2475 0.747378
\(189\) 0 0
\(190\) −5.52370 −0.400731
\(191\) 4.44243 0.321443 0.160722 0.987000i \(-0.448618\pi\)
0.160722 + 0.987000i \(0.448618\pi\)
\(192\) 0 0
\(193\) 23.6747 1.70414 0.852072 0.523424i \(-0.175346\pi\)
0.852072 + 0.523424i \(0.175346\pi\)
\(194\) 0.294371 0.0211346
\(195\) 0 0
\(196\) 4.47115 0.319368
\(197\) 1.49163 0.106274 0.0531372 0.998587i \(-0.483078\pi\)
0.0531372 + 0.998587i \(0.483078\pi\)
\(198\) 0 0
\(199\) 1.62560 0.115235 0.0576177 0.998339i \(-0.481650\pi\)
0.0576177 + 0.998339i \(0.481650\pi\)
\(200\) −4.50814 −0.318774
\(201\) 0 0
\(202\) 16.3182 1.14814
\(203\) 7.27175 0.510376
\(204\) 0 0
\(205\) 1.03176 0.0720613
\(206\) 7.94961 0.553876
\(207\) 0 0
\(208\) −4.26701 −0.295864
\(209\) −30.2842 −2.09480
\(210\) 0 0
\(211\) −11.4293 −0.786823 −0.393411 0.919363i \(-0.628705\pi\)
−0.393411 + 0.919363i \(0.628705\pi\)
\(212\) 7.27873 0.499905
\(213\) 0 0
\(214\) 6.96523 0.476133
\(215\) −4.25847 −0.290425
\(216\) 0 0
\(217\) −7.82707 −0.531336
\(218\) −11.3762 −0.770494
\(219\) 0 0
\(220\) 2.69667 0.181809
\(221\) −28.8104 −1.93800
\(222\) 0 0
\(223\) 1.00000 0.0669650
\(224\) 3.38691 0.226297
\(225\) 0 0
\(226\) −1.07883 −0.0717624
\(227\) −6.12257 −0.406369 −0.203185 0.979140i \(-0.565129\pi\)
−0.203185 + 0.979140i \(0.565129\pi\)
\(228\) 0 0
\(229\) 4.84326 0.320052 0.160026 0.987113i \(-0.448842\pi\)
0.160026 + 0.987113i \(0.448842\pi\)
\(230\) 1.14816 0.0757075
\(231\) 0 0
\(232\) 2.14701 0.140958
\(233\) −9.65575 −0.632569 −0.316285 0.948664i \(-0.602435\pi\)
−0.316285 + 0.948664i \(0.602435\pi\)
\(234\) 0 0
\(235\) −7.18686 −0.468819
\(236\) −8.21453 −0.534720
\(237\) 0 0
\(238\) 22.8680 1.48232
\(239\) 10.3110 0.666962 0.333481 0.942757i \(-0.391777\pi\)
0.333481 + 0.942757i \(0.391777\pi\)
\(240\) 0 0
\(241\) −15.6947 −1.01099 −0.505493 0.862831i \(-0.668689\pi\)
−0.505493 + 0.862831i \(0.668689\pi\)
\(242\) 3.78474 0.243292
\(243\) 0 0
\(244\) 7.24775 0.463989
\(245\) −3.13574 −0.200335
\(246\) 0 0
\(247\) −33.6073 −2.13838
\(248\) −2.31098 −0.146747
\(249\) 0 0
\(250\) 6.66831 0.421741
\(251\) −25.9448 −1.63762 −0.818812 0.574062i \(-0.805367\pi\)
−0.818812 + 0.574062i \(0.805367\pi\)
\(252\) 0 0
\(253\) 6.29491 0.395758
\(254\) −18.2481 −1.14499
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 1.15869 0.0722769 0.0361385 0.999347i \(-0.488494\pi\)
0.0361385 + 0.999347i \(0.488494\pi\)
\(258\) 0 0
\(259\) 19.1718 1.19128
\(260\) 2.99257 0.185591
\(261\) 0 0
\(262\) −5.09782 −0.314944
\(263\) 13.0580 0.805192 0.402596 0.915378i \(-0.368108\pi\)
0.402596 + 0.915378i \(0.368108\pi\)
\(264\) 0 0
\(265\) −5.10477 −0.313583
\(266\) 26.6756 1.63558
\(267\) 0 0
\(268\) 9.74666 0.595372
\(269\) 2.49787 0.152298 0.0761490 0.997096i \(-0.475738\pi\)
0.0761490 + 0.997096i \(0.475738\pi\)
\(270\) 0 0
\(271\) 21.4578 1.30347 0.651734 0.758448i \(-0.274042\pi\)
0.651734 + 0.758448i \(0.274042\pi\)
\(272\) 6.75189 0.409394
\(273\) 0 0
\(274\) −14.9155 −0.901079
\(275\) 17.3342 1.04529
\(276\) 0 0
\(277\) −0.158364 −0.00951516 −0.00475758 0.999989i \(-0.501514\pi\)
−0.00475758 + 0.999989i \(0.501514\pi\)
\(278\) −17.6022 −1.05571
\(279\) 0 0
\(280\) −2.37533 −0.141953
\(281\) −23.0387 −1.37437 −0.687186 0.726482i \(-0.741154\pi\)
−0.687186 + 0.726482i \(0.741154\pi\)
\(282\) 0 0
\(283\) 23.3254 1.38655 0.693276 0.720672i \(-0.256167\pi\)
0.693276 + 0.720672i \(0.256167\pi\)
\(284\) −13.1548 −0.780594
\(285\) 0 0
\(286\) 16.4071 0.970170
\(287\) −4.98267 −0.294118
\(288\) 0 0
\(289\) 28.5881 1.68165
\(290\) −1.50576 −0.0884212
\(291\) 0 0
\(292\) −1.79989 −0.105330
\(293\) −8.40583 −0.491074 −0.245537 0.969387i \(-0.578964\pi\)
−0.245537 + 0.969387i \(0.578964\pi\)
\(294\) 0 0
\(295\) 5.76107 0.335422
\(296\) 5.66056 0.329013
\(297\) 0 0
\(298\) 9.25428 0.536086
\(299\) 6.98565 0.403990
\(300\) 0 0
\(301\) 20.5654 1.18537
\(302\) 20.0326 1.15275
\(303\) 0 0
\(304\) 7.87608 0.451724
\(305\) −5.08304 −0.291054
\(306\) 0 0
\(307\) 8.38356 0.478475 0.239237 0.970961i \(-0.423103\pi\)
0.239237 + 0.970961i \(0.423103\pi\)
\(308\) −13.0230 −0.742053
\(309\) 0 0
\(310\) 1.62075 0.0920525
\(311\) −26.7140 −1.51481 −0.757406 0.652944i \(-0.773534\pi\)
−0.757406 + 0.652944i \(0.773534\pi\)
\(312\) 0 0
\(313\) −13.6102 −0.769292 −0.384646 0.923064i \(-0.625677\pi\)
−0.384646 + 0.923064i \(0.625677\pi\)
\(314\) −13.5972 −0.767335
\(315\) 0 0
\(316\) −9.88457 −0.556050
\(317\) 16.8160 0.944479 0.472239 0.881470i \(-0.343446\pi\)
0.472239 + 0.881470i \(0.343446\pi\)
\(318\) 0 0
\(319\) −8.25547 −0.462218
\(320\) −0.701327 −0.0392054
\(321\) 0 0
\(322\) −5.54480 −0.309000
\(323\) 53.1784 2.95893
\(324\) 0 0
\(325\) 19.2363 1.06704
\(326\) 17.6768 0.979028
\(327\) 0 0
\(328\) −1.47115 −0.0812309
\(329\) 34.7074 1.91348
\(330\) 0 0
\(331\) 19.3521 1.06369 0.531845 0.846842i \(-0.321499\pi\)
0.531845 + 0.846842i \(0.321499\pi\)
\(332\) 15.7196 0.862728
\(333\) 0 0
\(334\) 6.94928 0.380248
\(335\) −6.83559 −0.373468
\(336\) 0 0
\(337\) 2.23068 0.121513 0.0607563 0.998153i \(-0.480649\pi\)
0.0607563 + 0.998153i \(0.480649\pi\)
\(338\) 5.20740 0.283245
\(339\) 0 0
\(340\) −4.73528 −0.256807
\(341\) 8.88593 0.481200
\(342\) 0 0
\(343\) −8.56497 −0.462465
\(344\) 6.07202 0.327381
\(345\) 0 0
\(346\) 2.88332 0.155008
\(347\) −1.31333 −0.0705034 −0.0352517 0.999378i \(-0.511223\pi\)
−0.0352517 + 0.999378i \(0.511223\pi\)
\(348\) 0 0
\(349\) 3.42077 0.183109 0.0915547 0.995800i \(-0.470816\pi\)
0.0915547 + 0.995800i \(0.470816\pi\)
\(350\) −15.2687 −0.816144
\(351\) 0 0
\(352\) −3.84509 −0.204944
\(353\) 21.3404 1.13583 0.567916 0.823086i \(-0.307750\pi\)
0.567916 + 0.823086i \(0.307750\pi\)
\(354\) 0 0
\(355\) 9.22582 0.489655
\(356\) 12.7445 0.675456
\(357\) 0 0
\(358\) −26.3697 −1.39368
\(359\) 15.3455 0.809903 0.404951 0.914338i \(-0.367288\pi\)
0.404951 + 0.914338i \(0.367288\pi\)
\(360\) 0 0
\(361\) 43.0326 2.26487
\(362\) 6.93387 0.364436
\(363\) 0 0
\(364\) −14.4520 −0.757490
\(365\) 1.26231 0.0660723
\(366\) 0 0
\(367\) −27.1605 −1.41777 −0.708883 0.705326i \(-0.750800\pi\)
−0.708883 + 0.705326i \(0.750800\pi\)
\(368\) −1.63713 −0.0853412
\(369\) 0 0
\(370\) −3.96990 −0.206385
\(371\) 24.6524 1.27989
\(372\) 0 0
\(373\) −14.9138 −0.772208 −0.386104 0.922455i \(-0.626179\pi\)
−0.386104 + 0.922455i \(0.626179\pi\)
\(374\) −25.9617 −1.34245
\(375\) 0 0
\(376\) 10.2475 0.528476
\(377\) −9.16134 −0.471833
\(378\) 0 0
\(379\) −1.04923 −0.0538954 −0.0269477 0.999637i \(-0.508579\pi\)
−0.0269477 + 0.999637i \(0.508579\pi\)
\(380\) −5.52370 −0.283360
\(381\) 0 0
\(382\) 4.44243 0.227295
\(383\) −2.71233 −0.138593 −0.0692967 0.997596i \(-0.522076\pi\)
−0.0692967 + 0.997596i \(0.522076\pi\)
\(384\) 0 0
\(385\) 9.13337 0.465479
\(386\) 23.6747 1.20501
\(387\) 0 0
\(388\) 0.294371 0.0149444
\(389\) 3.46935 0.175903 0.0879516 0.996125i \(-0.471968\pi\)
0.0879516 + 0.996125i \(0.471968\pi\)
\(390\) 0 0
\(391\) −11.0537 −0.559010
\(392\) 4.47115 0.225827
\(393\) 0 0
\(394\) 1.49163 0.0751474
\(395\) 6.93231 0.348802
\(396\) 0 0
\(397\) −12.9785 −0.651373 −0.325687 0.945478i \(-0.605595\pi\)
−0.325687 + 0.945478i \(0.605595\pi\)
\(398\) 1.62560 0.0814838
\(399\) 0 0
\(400\) −4.50814 −0.225407
\(401\) −24.6893 −1.23292 −0.616462 0.787385i \(-0.711435\pi\)
−0.616462 + 0.787385i \(0.711435\pi\)
\(402\) 0 0
\(403\) 9.86098 0.491210
\(404\) 16.3182 0.811859
\(405\) 0 0
\(406\) 7.27175 0.360891
\(407\) −21.7654 −1.07887
\(408\) 0 0
\(409\) −2.79617 −0.138262 −0.0691309 0.997608i \(-0.522023\pi\)
−0.0691309 + 0.997608i \(0.522023\pi\)
\(410\) 1.03176 0.0509550
\(411\) 0 0
\(412\) 7.94961 0.391649
\(413\) −27.8219 −1.36902
\(414\) 0 0
\(415\) −11.0246 −0.541177
\(416\) −4.26701 −0.209208
\(417\) 0 0
\(418\) −30.2842 −1.48125
\(419\) −31.0300 −1.51591 −0.757956 0.652305i \(-0.773802\pi\)
−0.757956 + 0.652305i \(0.773802\pi\)
\(420\) 0 0
\(421\) 14.5370 0.708492 0.354246 0.935152i \(-0.384738\pi\)
0.354246 + 0.935152i \(0.384738\pi\)
\(422\) −11.4293 −0.556368
\(423\) 0 0
\(424\) 7.27873 0.353486
\(425\) −30.4385 −1.47648
\(426\) 0 0
\(427\) 24.5475 1.18793
\(428\) 6.96523 0.336677
\(429\) 0 0
\(430\) −4.25847 −0.205362
\(431\) 13.6708 0.658499 0.329249 0.944243i \(-0.393204\pi\)
0.329249 + 0.944243i \(0.393204\pi\)
\(432\) 0 0
\(433\) 15.5748 0.748478 0.374239 0.927332i \(-0.377904\pi\)
0.374239 + 0.927332i \(0.377904\pi\)
\(434\) −7.82707 −0.375712
\(435\) 0 0
\(436\) −11.3762 −0.544821
\(437\) −12.8941 −0.616810
\(438\) 0 0
\(439\) 10.0983 0.481964 0.240982 0.970530i \(-0.422531\pi\)
0.240982 + 0.970530i \(0.422531\pi\)
\(440\) 2.69667 0.128559
\(441\) 0 0
\(442\) −28.8104 −1.37037
\(443\) −17.6094 −0.836646 −0.418323 0.908298i \(-0.637382\pi\)
−0.418323 + 0.908298i \(0.637382\pi\)
\(444\) 0 0
\(445\) −8.93804 −0.423704
\(446\) 1.00000 0.0473514
\(447\) 0 0
\(448\) 3.38691 0.160016
\(449\) 0.274840 0.0129705 0.00648525 0.999979i \(-0.497936\pi\)
0.00648525 + 0.999979i \(0.497936\pi\)
\(450\) 0 0
\(451\) 5.65673 0.266365
\(452\) −1.07883 −0.0507437
\(453\) 0 0
\(454\) −6.12257 −0.287346
\(455\) 10.1356 0.475163
\(456\) 0 0
\(457\) −37.0426 −1.73278 −0.866389 0.499370i \(-0.833565\pi\)
−0.866389 + 0.499370i \(0.833565\pi\)
\(458\) 4.84326 0.226311
\(459\) 0 0
\(460\) 1.14816 0.0535333
\(461\) −12.2641 −0.571194 −0.285597 0.958350i \(-0.592192\pi\)
−0.285597 + 0.958350i \(0.592192\pi\)
\(462\) 0 0
\(463\) 0.156445 0.00727060 0.00363530 0.999993i \(-0.498843\pi\)
0.00363530 + 0.999993i \(0.498843\pi\)
\(464\) 2.14701 0.0996727
\(465\) 0 0
\(466\) −9.65575 −0.447294
\(467\) 15.7563 0.729116 0.364558 0.931181i \(-0.381220\pi\)
0.364558 + 0.931181i \(0.381220\pi\)
\(468\) 0 0
\(469\) 33.0111 1.52431
\(470\) −7.18686 −0.331505
\(471\) 0 0
\(472\) −8.21453 −0.378104
\(473\) −23.3475 −1.07352
\(474\) 0 0
\(475\) −35.5065 −1.62915
\(476\) 22.8680 1.04816
\(477\) 0 0
\(478\) 10.3110 0.471613
\(479\) −31.9259 −1.45873 −0.729365 0.684125i \(-0.760184\pi\)
−0.729365 + 0.684125i \(0.760184\pi\)
\(480\) 0 0
\(481\) −24.1537 −1.10131
\(482\) −15.6947 −0.714874
\(483\) 0 0
\(484\) 3.78474 0.172034
\(485\) −0.206450 −0.00937442
\(486\) 0 0
\(487\) −1.42661 −0.0646461 −0.0323230 0.999477i \(-0.510291\pi\)
−0.0323230 + 0.999477i \(0.510291\pi\)
\(488\) 7.24775 0.328090
\(489\) 0 0
\(490\) −3.13574 −0.141658
\(491\) 2.95757 0.133473 0.0667366 0.997771i \(-0.478741\pi\)
0.0667366 + 0.997771i \(0.478741\pi\)
\(492\) 0 0
\(493\) 14.4964 0.652886
\(494\) −33.6073 −1.51206
\(495\) 0 0
\(496\) −2.31098 −0.103766
\(497\) −44.5541 −1.99853
\(498\) 0 0
\(499\) −33.8099 −1.51354 −0.756769 0.653683i \(-0.773223\pi\)
−0.756769 + 0.653683i \(0.773223\pi\)
\(500\) 6.66831 0.298216
\(501\) 0 0
\(502\) −25.9448 −1.15797
\(503\) −31.5723 −1.40774 −0.703870 0.710329i \(-0.748546\pi\)
−0.703870 + 0.710329i \(0.748546\pi\)
\(504\) 0 0
\(505\) −11.4444 −0.509268
\(506\) 6.29491 0.279843
\(507\) 0 0
\(508\) −18.2481 −0.809631
\(509\) 1.20420 0.0533753 0.0266877 0.999644i \(-0.491504\pi\)
0.0266877 + 0.999644i \(0.491504\pi\)
\(510\) 0 0
\(511\) −6.09605 −0.269674
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 1.15869 0.0511075
\(515\) −5.57528 −0.245676
\(516\) 0 0
\(517\) −39.4027 −1.73293
\(518\) 19.1718 0.842361
\(519\) 0 0
\(520\) 2.99257 0.131233
\(521\) −32.4362 −1.42106 −0.710528 0.703669i \(-0.751544\pi\)
−0.710528 + 0.703669i \(0.751544\pi\)
\(522\) 0 0
\(523\) 22.4618 0.982186 0.491093 0.871107i \(-0.336597\pi\)
0.491093 + 0.871107i \(0.336597\pi\)
\(524\) −5.09782 −0.222699
\(525\) 0 0
\(526\) 13.0580 0.569357
\(527\) −15.6035 −0.679698
\(528\) 0 0
\(529\) −20.3198 −0.883470
\(530\) −5.10477 −0.221737
\(531\) 0 0
\(532\) 26.6756 1.15653
\(533\) 6.27744 0.271906
\(534\) 0 0
\(535\) −4.88490 −0.211193
\(536\) 9.74666 0.420992
\(537\) 0 0
\(538\) 2.49787 0.107691
\(539\) −17.1920 −0.740512
\(540\) 0 0
\(541\) −22.9994 −0.988821 −0.494410 0.869229i \(-0.664616\pi\)
−0.494410 + 0.869229i \(0.664616\pi\)
\(542\) 21.4578 0.921691
\(543\) 0 0
\(544\) 6.75189 0.289485
\(545\) 7.97843 0.341759
\(546\) 0 0
\(547\) 14.8746 0.635991 0.317996 0.948092i \(-0.396990\pi\)
0.317996 + 0.948092i \(0.396990\pi\)
\(548\) −14.9155 −0.637159
\(549\) 0 0
\(550\) 17.3342 0.739134
\(551\) 16.9101 0.720392
\(552\) 0 0
\(553\) −33.4781 −1.42364
\(554\) −0.158364 −0.00672824
\(555\) 0 0
\(556\) −17.6022 −0.746500
\(557\) −24.1336 −1.02257 −0.511287 0.859410i \(-0.670831\pi\)
−0.511287 + 0.859410i \(0.670831\pi\)
\(558\) 0 0
\(559\) −25.9094 −1.09585
\(560\) −2.37533 −0.100376
\(561\) 0 0
\(562\) −23.0387 −0.971828
\(563\) 14.6662 0.618107 0.309054 0.951045i \(-0.399988\pi\)
0.309054 + 0.951045i \(0.399988\pi\)
\(564\) 0 0
\(565\) 0.756609 0.0318308
\(566\) 23.3254 0.980440
\(567\) 0 0
\(568\) −13.1548 −0.551963
\(569\) 2.57588 0.107987 0.0539933 0.998541i \(-0.482805\pi\)
0.0539933 + 0.998541i \(0.482805\pi\)
\(570\) 0 0
\(571\) −13.2168 −0.553107 −0.276554 0.960998i \(-0.589192\pi\)
−0.276554 + 0.960998i \(0.589192\pi\)
\(572\) 16.4071 0.686014
\(573\) 0 0
\(574\) −4.98267 −0.207973
\(575\) 7.38040 0.307784
\(576\) 0 0
\(577\) 12.0810 0.502938 0.251469 0.967865i \(-0.419086\pi\)
0.251469 + 0.967865i \(0.419086\pi\)
\(578\) 28.5881 1.18911
\(579\) 0 0
\(580\) −1.50576 −0.0625232
\(581\) 53.2410 2.20881
\(582\) 0 0
\(583\) −27.9874 −1.15912
\(584\) −1.79989 −0.0744798
\(585\) 0 0
\(586\) −8.40583 −0.347241
\(587\) −23.7403 −0.979865 −0.489933 0.871760i \(-0.662979\pi\)
−0.489933 + 0.871760i \(0.662979\pi\)
\(588\) 0 0
\(589\) −18.2014 −0.749977
\(590\) 5.76107 0.237179
\(591\) 0 0
\(592\) 5.66056 0.232648
\(593\) −19.0061 −0.780488 −0.390244 0.920711i \(-0.627609\pi\)
−0.390244 + 0.920711i \(0.627609\pi\)
\(594\) 0 0
\(595\) −16.0380 −0.657493
\(596\) 9.25428 0.379070
\(597\) 0 0
\(598\) 6.98565 0.285664
\(599\) −22.5395 −0.920937 −0.460469 0.887676i \(-0.652319\pi\)
−0.460469 + 0.887676i \(0.652319\pi\)
\(600\) 0 0
\(601\) −14.5353 −0.592908 −0.296454 0.955047i \(-0.595804\pi\)
−0.296454 + 0.955047i \(0.595804\pi\)
\(602\) 20.5654 0.838182
\(603\) 0 0
\(604\) 20.0326 0.815116
\(605\) −2.65434 −0.107914
\(606\) 0 0
\(607\) 27.5040 1.11635 0.558176 0.829723i \(-0.311502\pi\)
0.558176 + 0.829723i \(0.311502\pi\)
\(608\) 7.87608 0.319417
\(609\) 0 0
\(610\) −5.08304 −0.205806
\(611\) −43.7263 −1.76898
\(612\) 0 0
\(613\) 15.1418 0.611570 0.305785 0.952101i \(-0.401081\pi\)
0.305785 + 0.952101i \(0.401081\pi\)
\(614\) 8.38356 0.338333
\(615\) 0 0
\(616\) −13.0230 −0.524711
\(617\) −27.1620 −1.09350 −0.546751 0.837295i \(-0.684136\pi\)
−0.546751 + 0.837295i \(0.684136\pi\)
\(618\) 0 0
\(619\) −34.2116 −1.37508 −0.687540 0.726147i \(-0.741309\pi\)
−0.687540 + 0.726147i \(0.741309\pi\)
\(620\) 1.62075 0.0650909
\(621\) 0 0
\(622\) −26.7140 −1.07113
\(623\) 43.1644 1.72935
\(624\) 0 0
\(625\) 17.8640 0.714562
\(626\) −13.6102 −0.543972
\(627\) 0 0
\(628\) −13.5972 −0.542588
\(629\) 38.2195 1.52391
\(630\) 0 0
\(631\) 41.5717 1.65494 0.827472 0.561507i \(-0.189778\pi\)
0.827472 + 0.561507i \(0.189778\pi\)
\(632\) −9.88457 −0.393187
\(633\) 0 0
\(634\) 16.8160 0.667847
\(635\) 12.7979 0.507870
\(636\) 0 0
\(637\) −19.0785 −0.755917
\(638\) −8.25547 −0.326837
\(639\) 0 0
\(640\) −0.701327 −0.0277224
\(641\) −29.3818 −1.16051 −0.580256 0.814434i \(-0.697048\pi\)
−0.580256 + 0.814434i \(0.697048\pi\)
\(642\) 0 0
\(643\) 8.71200 0.343568 0.171784 0.985135i \(-0.445047\pi\)
0.171784 + 0.985135i \(0.445047\pi\)
\(644\) −5.54480 −0.218496
\(645\) 0 0
\(646\) 53.1784 2.09228
\(647\) −15.3152 −0.602102 −0.301051 0.953608i \(-0.597337\pi\)
−0.301051 + 0.953608i \(0.597337\pi\)
\(648\) 0 0
\(649\) 31.5856 1.23984
\(650\) 19.2363 0.754510
\(651\) 0 0
\(652\) 17.6768 0.692277
\(653\) 25.7650 1.00826 0.504131 0.863627i \(-0.331813\pi\)
0.504131 + 0.863627i \(0.331813\pi\)
\(654\) 0 0
\(655\) 3.57524 0.139696
\(656\) −1.47115 −0.0574389
\(657\) 0 0
\(658\) 34.7074 1.35304
\(659\) 34.9731 1.36236 0.681180 0.732116i \(-0.261467\pi\)
0.681180 + 0.732116i \(0.261467\pi\)
\(660\) 0 0
\(661\) −42.0894 −1.63709 −0.818544 0.574444i \(-0.805218\pi\)
−0.818544 + 0.574444i \(0.805218\pi\)
\(662\) 19.3521 0.752142
\(663\) 0 0
\(664\) 15.7196 0.610040
\(665\) −18.7083 −0.725476
\(666\) 0 0
\(667\) −3.51494 −0.136099
\(668\) 6.94928 0.268876
\(669\) 0 0
\(670\) −6.83559 −0.264082
\(671\) −27.8683 −1.07584
\(672\) 0 0
\(673\) −4.05744 −0.156403 −0.0782014 0.996938i \(-0.524918\pi\)
−0.0782014 + 0.996938i \(0.524918\pi\)
\(674\) 2.23068 0.0859224
\(675\) 0 0
\(676\) 5.20740 0.200285
\(677\) 19.3432 0.743421 0.371710 0.928349i \(-0.378771\pi\)
0.371710 + 0.928349i \(0.378771\pi\)
\(678\) 0 0
\(679\) 0.997008 0.0382617
\(680\) −4.73528 −0.181590
\(681\) 0 0
\(682\) 8.88593 0.340260
\(683\) −25.3201 −0.968847 −0.484423 0.874834i \(-0.660971\pi\)
−0.484423 + 0.874834i \(0.660971\pi\)
\(684\) 0 0
\(685\) 10.4606 0.399681
\(686\) −8.56497 −0.327012
\(687\) 0 0
\(688\) 6.07202 0.231493
\(689\) −31.0584 −1.18323
\(690\) 0 0
\(691\) 14.5336 0.552886 0.276443 0.961030i \(-0.410844\pi\)
0.276443 + 0.961030i \(0.410844\pi\)
\(692\) 2.88332 0.109607
\(693\) 0 0
\(694\) −1.31333 −0.0498534
\(695\) 12.3449 0.468269
\(696\) 0 0
\(697\) −9.93308 −0.376242
\(698\) 3.42077 0.129478
\(699\) 0 0
\(700\) −15.2687 −0.577101
\(701\) 28.6997 1.08397 0.541986 0.840388i \(-0.317673\pi\)
0.541986 + 0.840388i \(0.317673\pi\)
\(702\) 0 0
\(703\) 44.5830 1.68148
\(704\) −3.84509 −0.144917
\(705\) 0 0
\(706\) 21.3404 0.803155
\(707\) 55.2682 2.07857
\(708\) 0 0
\(709\) −46.4505 −1.74449 −0.872243 0.489073i \(-0.837335\pi\)
−0.872243 + 0.489073i \(0.837335\pi\)
\(710\) 9.22582 0.346239
\(711\) 0 0
\(712\) 12.7445 0.477620
\(713\) 3.78337 0.141688
\(714\) 0 0
\(715\) −11.5067 −0.430327
\(716\) −26.3697 −0.985482
\(717\) 0 0
\(718\) 15.3455 0.572688
\(719\) −43.6929 −1.62947 −0.814734 0.579835i \(-0.803117\pi\)
−0.814734 + 0.579835i \(0.803117\pi\)
\(720\) 0 0
\(721\) 26.9246 1.00273
\(722\) 43.0326 1.60151
\(723\) 0 0
\(724\) 6.93387 0.257695
\(725\) −9.67905 −0.359471
\(726\) 0 0
\(727\) −52.8123 −1.95870 −0.979350 0.202172i \(-0.935200\pi\)
−0.979350 + 0.202172i \(0.935200\pi\)
\(728\) −14.4520 −0.535626
\(729\) 0 0
\(730\) 1.26231 0.0467201
\(731\) 40.9976 1.51635
\(732\) 0 0
\(733\) −34.2947 −1.26670 −0.633352 0.773864i \(-0.718321\pi\)
−0.633352 + 0.773864i \(0.718321\pi\)
\(734\) −27.1605 −1.00251
\(735\) 0 0
\(736\) −1.63713 −0.0603453
\(737\) −37.4768 −1.38048
\(738\) 0 0
\(739\) 1.66709 0.0613250 0.0306625 0.999530i \(-0.490238\pi\)
0.0306625 + 0.999530i \(0.490238\pi\)
\(740\) −3.96990 −0.145936
\(741\) 0 0
\(742\) 24.6524 0.905018
\(743\) −15.5597 −0.570831 −0.285416 0.958404i \(-0.592132\pi\)
−0.285416 + 0.958404i \(0.592132\pi\)
\(744\) 0 0
\(745\) −6.49027 −0.237785
\(746\) −14.9138 −0.546034
\(747\) 0 0
\(748\) −25.9617 −0.949252
\(749\) 23.5906 0.861981
\(750\) 0 0
\(751\) 44.1995 1.61286 0.806431 0.591328i \(-0.201396\pi\)
0.806431 + 0.591328i \(0.201396\pi\)
\(752\) 10.2475 0.373689
\(753\) 0 0
\(754\) −9.16134 −0.333636
\(755\) −14.0494 −0.511311
\(756\) 0 0
\(757\) −31.4100 −1.14162 −0.570808 0.821083i \(-0.693370\pi\)
−0.570808 + 0.821083i \(0.693370\pi\)
\(758\) −1.04923 −0.0381098
\(759\) 0 0
\(760\) −5.52370 −0.200366
\(761\) 40.2142 1.45776 0.728881 0.684640i \(-0.240041\pi\)
0.728881 + 0.684640i \(0.240041\pi\)
\(762\) 0 0
\(763\) −38.5302 −1.39489
\(764\) 4.44243 0.160722
\(765\) 0 0
\(766\) −2.71233 −0.0980003
\(767\) 35.0515 1.26564
\(768\) 0 0
\(769\) −24.4812 −0.882815 −0.441408 0.897307i \(-0.645521\pi\)
−0.441408 + 0.897307i \(0.645521\pi\)
\(770\) 9.13337 0.329144
\(771\) 0 0
\(772\) 23.6747 0.852072
\(773\) −0.486252 −0.0174893 −0.00874464 0.999962i \(-0.502784\pi\)
−0.00874464 + 0.999962i \(0.502784\pi\)
\(774\) 0 0
\(775\) 10.4182 0.374233
\(776\) 0.294371 0.0105673
\(777\) 0 0
\(778\) 3.46935 0.124382
\(779\) −11.5869 −0.415145
\(780\) 0 0
\(781\) 50.5814 1.80995
\(782\) −11.0537 −0.395280
\(783\) 0 0
\(784\) 4.47115 0.159684
\(785\) 9.53609 0.340358
\(786\) 0 0
\(787\) 31.7144 1.13050 0.565248 0.824921i \(-0.308780\pi\)
0.565248 + 0.824921i \(0.308780\pi\)
\(788\) 1.49163 0.0531372
\(789\) 0 0
\(790\) 6.93231 0.246641
\(791\) −3.65388 −0.129917
\(792\) 0 0
\(793\) −30.9262 −1.09822
\(794\) −12.9785 −0.460591
\(795\) 0 0
\(796\) 1.62560 0.0576177
\(797\) 23.9711 0.849101 0.424551 0.905404i \(-0.360432\pi\)
0.424551 + 0.905404i \(0.360432\pi\)
\(798\) 0 0
\(799\) 69.1902 2.44777
\(800\) −4.50814 −0.159387
\(801\) 0 0
\(802\) −24.6893 −0.871809
\(803\) 6.92073 0.244227
\(804\) 0 0
\(805\) 3.88872 0.137059
\(806\) 9.86098 0.347338
\(807\) 0 0
\(808\) 16.3182 0.574071
\(809\) −22.6055 −0.794766 −0.397383 0.917653i \(-0.630082\pi\)
−0.397383 + 0.917653i \(0.630082\pi\)
\(810\) 0 0
\(811\) −18.6409 −0.654572 −0.327286 0.944925i \(-0.606134\pi\)
−0.327286 + 0.944925i \(0.606134\pi\)
\(812\) 7.27175 0.255188
\(813\) 0 0
\(814\) −21.7654 −0.762876
\(815\) −12.3972 −0.434256
\(816\) 0 0
\(817\) 47.8237 1.67314
\(818\) −2.79617 −0.0977659
\(819\) 0 0
\(820\) 1.03176 0.0360306
\(821\) −40.3808 −1.40930 −0.704650 0.709555i \(-0.748896\pi\)
−0.704650 + 0.709555i \(0.748896\pi\)
\(822\) 0 0
\(823\) −30.8413 −1.07506 −0.537529 0.843245i \(-0.680642\pi\)
−0.537529 + 0.843245i \(0.680642\pi\)
\(824\) 7.94961 0.276938
\(825\) 0 0
\(826\) −27.8219 −0.968047
\(827\) 39.7181 1.38113 0.690567 0.723268i \(-0.257361\pi\)
0.690567 + 0.723268i \(0.257361\pi\)
\(828\) 0 0
\(829\) −14.4476 −0.501784 −0.250892 0.968015i \(-0.580724\pi\)
−0.250892 + 0.968015i \(0.580724\pi\)
\(830\) −11.0246 −0.382670
\(831\) 0 0
\(832\) −4.26701 −0.147932
\(833\) 30.1888 1.04598
\(834\) 0 0
\(835\) −4.87372 −0.168662
\(836\) −30.2842 −1.04740
\(837\) 0 0
\(838\) −31.0300 −1.07191
\(839\) −12.2534 −0.423035 −0.211518 0.977374i \(-0.567841\pi\)
−0.211518 + 0.977374i \(0.567841\pi\)
\(840\) 0 0
\(841\) −24.3903 −0.841046
\(842\) 14.5370 0.500979
\(843\) 0 0
\(844\) −11.4293 −0.393411
\(845\) −3.65209 −0.125636
\(846\) 0 0
\(847\) 12.8186 0.440451
\(848\) 7.27873 0.249953
\(849\) 0 0
\(850\) −30.4385 −1.04403
\(851\) −9.26706 −0.317671
\(852\) 0 0
\(853\) −19.2084 −0.657684 −0.328842 0.944385i \(-0.606658\pi\)
−0.328842 + 0.944385i \(0.606658\pi\)
\(854\) 24.5475 0.839997
\(855\) 0 0
\(856\) 6.96523 0.238066
\(857\) −23.6792 −0.808865 −0.404433 0.914568i \(-0.632531\pi\)
−0.404433 + 0.914568i \(0.632531\pi\)
\(858\) 0 0
\(859\) 17.3459 0.591834 0.295917 0.955214i \(-0.404375\pi\)
0.295917 + 0.955214i \(0.404375\pi\)
\(860\) −4.25847 −0.145213
\(861\) 0 0
\(862\) 13.6708 0.465629
\(863\) 3.48179 0.118522 0.0592608 0.998243i \(-0.481126\pi\)
0.0592608 + 0.998243i \(0.481126\pi\)
\(864\) 0 0
\(865\) −2.02215 −0.0687551
\(866\) 15.5748 0.529254
\(867\) 0 0
\(868\) −7.82707 −0.265668
\(869\) 38.0071 1.28930
\(870\) 0 0
\(871\) −41.5891 −1.40919
\(872\) −11.3762 −0.385247
\(873\) 0 0
\(874\) −12.8941 −0.436151
\(875\) 22.5850 0.763511
\(876\) 0 0
\(877\) 43.1141 1.45586 0.727929 0.685652i \(-0.240483\pi\)
0.727929 + 0.685652i \(0.240483\pi\)
\(878\) 10.0983 0.340800
\(879\) 0 0
\(880\) 2.69667 0.0909046
\(881\) −37.3841 −1.25950 −0.629752 0.776796i \(-0.716843\pi\)
−0.629752 + 0.776796i \(0.716843\pi\)
\(882\) 0 0
\(883\) −2.74579 −0.0924031 −0.0462016 0.998932i \(-0.514712\pi\)
−0.0462016 + 0.998932i \(0.514712\pi\)
\(884\) −28.8104 −0.968999
\(885\) 0 0
\(886\) −17.6094 −0.591598
\(887\) 17.6940 0.594107 0.297054 0.954861i \(-0.403996\pi\)
0.297054 + 0.954861i \(0.403996\pi\)
\(888\) 0 0
\(889\) −61.8048 −2.07287
\(890\) −8.93804 −0.299604
\(891\) 0 0
\(892\) 1.00000 0.0334825
\(893\) 80.7103 2.70087
\(894\) 0 0
\(895\) 18.4938 0.618179
\(896\) 3.38691 0.113149
\(897\) 0 0
\(898\) 0.274840 0.00917153
\(899\) −4.96171 −0.165482
\(900\) 0 0
\(901\) 49.1452 1.63726
\(902\) 5.65673 0.188348
\(903\) 0 0
\(904\) −1.07883 −0.0358812
\(905\) −4.86291 −0.161648
\(906\) 0 0
\(907\) 36.2527 1.20375 0.601876 0.798590i \(-0.294420\pi\)
0.601876 + 0.798590i \(0.294420\pi\)
\(908\) −6.12257 −0.203185
\(909\) 0 0
\(910\) 10.1356 0.335991
\(911\) −37.4246 −1.23993 −0.619965 0.784629i \(-0.712853\pi\)
−0.619965 + 0.784629i \(0.712853\pi\)
\(912\) 0 0
\(913\) −60.4435 −2.00039
\(914\) −37.0426 −1.22526
\(915\) 0 0
\(916\) 4.84326 0.160026
\(917\) −17.2658 −0.570168
\(918\) 0 0
\(919\) 2.95028 0.0973208 0.0486604 0.998815i \(-0.484505\pi\)
0.0486604 + 0.998815i \(0.484505\pi\)
\(920\) 1.14816 0.0378538
\(921\) 0 0
\(922\) −12.2641 −0.403895
\(923\) 56.1317 1.84760
\(924\) 0 0
\(925\) −25.5186 −0.839046
\(926\) 0.156445 0.00514109
\(927\) 0 0
\(928\) 2.14701 0.0704792
\(929\) 28.6871 0.941194 0.470597 0.882348i \(-0.344039\pi\)
0.470597 + 0.882348i \(0.344039\pi\)
\(930\) 0 0
\(931\) 35.2152 1.15413
\(932\) −9.65575 −0.316285
\(933\) 0 0
\(934\) 15.7563 0.515563
\(935\) 18.2076 0.595452
\(936\) 0 0
\(937\) −37.2160 −1.21579 −0.607897 0.794016i \(-0.707986\pi\)
−0.607897 + 0.794016i \(0.707986\pi\)
\(938\) 33.0111 1.07785
\(939\) 0 0
\(940\) −7.18686 −0.234410
\(941\) 10.7583 0.350711 0.175355 0.984505i \(-0.443893\pi\)
0.175355 + 0.984505i \(0.443893\pi\)
\(942\) 0 0
\(943\) 2.40847 0.0784305
\(944\) −8.21453 −0.267360
\(945\) 0 0
\(946\) −23.3475 −0.759092
\(947\) −21.7826 −0.707840 −0.353920 0.935276i \(-0.615151\pi\)
−0.353920 + 0.935276i \(0.615151\pi\)
\(948\) 0 0
\(949\) 7.68014 0.249308
\(950\) −35.5065 −1.15198
\(951\) 0 0
\(952\) 22.8680 0.741158
\(953\) −53.2874 −1.72615 −0.863074 0.505078i \(-0.831464\pi\)
−0.863074 + 0.505078i \(0.831464\pi\)
\(954\) 0 0
\(955\) −3.11560 −0.100818
\(956\) 10.3110 0.333481
\(957\) 0 0
\(958\) −31.9259 −1.03148
\(959\) −50.5175 −1.63130
\(960\) 0 0
\(961\) −25.6594 −0.827722
\(962\) −24.1537 −0.778746
\(963\) 0 0
\(964\) −15.6947 −0.505493
\(965\) −16.6037 −0.534493
\(966\) 0 0
\(967\) −29.3472 −0.943743 −0.471872 0.881667i \(-0.656421\pi\)
−0.471872 + 0.881667i \(0.656421\pi\)
\(968\) 3.78474 0.121646
\(969\) 0 0
\(970\) −0.206450 −0.00662872
\(971\) 33.0902 1.06192 0.530958 0.847398i \(-0.321832\pi\)
0.530958 + 0.847398i \(0.321832\pi\)
\(972\) 0 0
\(973\) −59.6171 −1.91124
\(974\) −1.42661 −0.0457117
\(975\) 0 0
\(976\) 7.24775 0.231995
\(977\) 10.1556 0.324908 0.162454 0.986716i \(-0.448059\pi\)
0.162454 + 0.986716i \(0.448059\pi\)
\(978\) 0 0
\(979\) −49.0037 −1.56617
\(980\) −3.13574 −0.100168
\(981\) 0 0
\(982\) 2.95757 0.0943799
\(983\) −1.81725 −0.0579611 −0.0289806 0.999580i \(-0.509226\pi\)
−0.0289806 + 0.999580i \(0.509226\pi\)
\(984\) 0 0
\(985\) −1.04612 −0.0333322
\(986\) 14.4964 0.461660
\(987\) 0 0
\(988\) −33.6073 −1.06919
\(989\) −9.94067 −0.316095
\(990\) 0 0
\(991\) −1.50467 −0.0477974 −0.0238987 0.999714i \(-0.507608\pi\)
−0.0238987 + 0.999714i \(0.507608\pi\)
\(992\) −2.31098 −0.0733736
\(993\) 0 0
\(994\) −44.5541 −1.41317
\(995\) −1.14007 −0.0361428
\(996\) 0 0
\(997\) 59.3196 1.87867 0.939336 0.342999i \(-0.111443\pi\)
0.939336 + 0.342999i \(0.111443\pi\)
\(998\) −33.8099 −1.07023
\(999\) 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.z.1.5 8
3.2 odd 2 446.2.a.f.1.6 8
12.11 even 2 3568.2.a.n.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
446.2.a.f.1.6 8 3.2 odd 2
3568.2.a.n.1.3 8 12.11 even 2
4014.2.a.z.1.5 8 1.1 even 1 trivial