Properties

Label 8046.2.a.m.1.1
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $1$
Dimension $12$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, 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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} - 2249 x^{3} + 761 x^{2} + 910 x - 375 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.25305\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.25305 q^{5} +2.10378 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.25305 q^{5} +2.10378 q^{7} +1.00000 q^{8} -4.25305 q^{10} +4.67709 q^{11} +0.547094 q^{13} +2.10378 q^{14} +1.00000 q^{16} +0.524967 q^{17} -8.11080 q^{19} -4.25305 q^{20} +4.67709 q^{22} -6.00153 q^{23} +13.0884 q^{25} +0.547094 q^{26} +2.10378 q^{28} -8.07750 q^{29} +4.87410 q^{31} +1.00000 q^{32} +0.524967 q^{34} -8.94750 q^{35} -10.1411 q^{37} -8.11080 q^{38} -4.25305 q^{40} +2.58901 q^{41} +7.72324 q^{43} +4.67709 q^{44} -6.00153 q^{46} +5.67930 q^{47} -2.57409 q^{49} +13.0884 q^{50} +0.547094 q^{52} +0.544241 q^{53} -19.8919 q^{55} +2.10378 q^{56} -8.07750 q^{58} -7.68172 q^{59} +3.68590 q^{61} +4.87410 q^{62} +1.00000 q^{64} -2.32682 q^{65} +0.0374969 q^{67} +0.524967 q^{68} -8.94750 q^{70} -13.0756 q^{71} +14.3053 q^{73} -10.1411 q^{74} -8.11080 q^{76} +9.83959 q^{77} -4.75832 q^{79} -4.25305 q^{80} +2.58901 q^{82} +7.02674 q^{83} -2.23271 q^{85} +7.72324 q^{86} +4.67709 q^{88} +15.5764 q^{89} +1.15097 q^{91} -6.00153 q^{92} +5.67930 q^{94} +34.4956 q^{95} +8.91148 q^{97} -2.57409 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) \) 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
\(5\) −4.25305 −1.90202 −0.951011 0.309158i \(-0.899953\pi\)
−0.951011 + 0.309158i \(0.899953\pi\)
\(6\) 0 0
\(7\) 2.10378 0.795156 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −4.25305 −1.34493
\(11\) 4.67709 1.41020 0.705098 0.709110i \(-0.250903\pi\)
0.705098 + 0.709110i \(0.250903\pi\)
\(12\) 0 0
\(13\) 0.547094 0.151736 0.0758682 0.997118i \(-0.475827\pi\)
0.0758682 + 0.997118i \(0.475827\pi\)
\(14\) 2.10378 0.562260
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.524967 0.127323 0.0636616 0.997972i \(-0.479722\pi\)
0.0636616 + 0.997972i \(0.479722\pi\)
\(18\) 0 0
\(19\) −8.11080 −1.86074 −0.930372 0.366617i \(-0.880516\pi\)
−0.930372 + 0.366617i \(0.880516\pi\)
\(20\) −4.25305 −0.951011
\(21\) 0 0
\(22\) 4.67709 0.997159
\(23\) −6.00153 −1.25141 −0.625703 0.780061i \(-0.715188\pi\)
−0.625703 + 0.780061i \(0.715188\pi\)
\(24\) 0 0
\(25\) 13.0884 2.61768
\(26\) 0.547094 0.107294
\(27\) 0 0
\(28\) 2.10378 0.397578
\(29\) −8.07750 −1.49995 −0.749977 0.661464i \(-0.769935\pi\)
−0.749977 + 0.661464i \(0.769935\pi\)
\(30\) 0 0
\(31\) 4.87410 0.875414 0.437707 0.899118i \(-0.355791\pi\)
0.437707 + 0.899118i \(0.355791\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.524967 0.0900311
\(35\) −8.94750 −1.51240
\(36\) 0 0
\(37\) −10.1411 −1.66718 −0.833591 0.552382i \(-0.813719\pi\)
−0.833591 + 0.552382i \(0.813719\pi\)
\(38\) −8.11080 −1.31574
\(39\) 0 0
\(40\) −4.25305 −0.672466
\(41\) 2.58901 0.404336 0.202168 0.979351i \(-0.435201\pi\)
0.202168 + 0.979351i \(0.435201\pi\)
\(42\) 0 0
\(43\) 7.72324 1.17778 0.588892 0.808212i \(-0.299564\pi\)
0.588892 + 0.808212i \(0.299564\pi\)
\(44\) 4.67709 0.705098
\(45\) 0 0
\(46\) −6.00153 −0.884878
\(47\) 5.67930 0.828412 0.414206 0.910183i \(-0.364059\pi\)
0.414206 + 0.910183i \(0.364059\pi\)
\(48\) 0 0
\(49\) −2.57409 −0.367727
\(50\) 13.0884 1.85098
\(51\) 0 0
\(52\) 0.547094 0.0758682
\(53\) 0.544241 0.0747573 0.0373786 0.999301i \(-0.488099\pi\)
0.0373786 + 0.999301i \(0.488099\pi\)
\(54\) 0 0
\(55\) −19.8919 −2.68222
\(56\) 2.10378 0.281130
\(57\) 0 0
\(58\) −8.07750 −1.06063
\(59\) −7.68172 −1.00007 −0.500037 0.866004i \(-0.666680\pi\)
−0.500037 + 0.866004i \(0.666680\pi\)
\(60\) 0 0
\(61\) 3.68590 0.471931 0.235966 0.971761i \(-0.424175\pi\)
0.235966 + 0.971761i \(0.424175\pi\)
\(62\) 4.87410 0.619011
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.32682 −0.288606
\(66\) 0 0
\(67\) 0.0374969 0.00458098 0.00229049 0.999997i \(-0.499271\pi\)
0.00229049 + 0.999997i \(0.499271\pi\)
\(68\) 0.524967 0.0636616
\(69\) 0 0
\(70\) −8.94750 −1.06943
\(71\) −13.0756 −1.55179 −0.775894 0.630863i \(-0.782701\pi\)
−0.775894 + 0.630863i \(0.782701\pi\)
\(72\) 0 0
\(73\) 14.3053 1.67431 0.837155 0.546965i \(-0.184217\pi\)
0.837155 + 0.546965i \(0.184217\pi\)
\(74\) −10.1411 −1.17888
\(75\) 0 0
\(76\) −8.11080 −0.930372
\(77\) 9.83959 1.12133
\(78\) 0 0
\(79\) −4.75832 −0.535353 −0.267676 0.963509i \(-0.586256\pi\)
−0.267676 + 0.963509i \(0.586256\pi\)
\(80\) −4.25305 −0.475505
\(81\) 0 0
\(82\) 2.58901 0.285909
\(83\) 7.02674 0.771284 0.385642 0.922648i \(-0.373980\pi\)
0.385642 + 0.922648i \(0.373980\pi\)
\(84\) 0 0
\(85\) −2.23271 −0.242172
\(86\) 7.72324 0.832818
\(87\) 0 0
\(88\) 4.67709 0.498579
\(89\) 15.5764 1.65110 0.825548 0.564332i \(-0.190866\pi\)
0.825548 + 0.564332i \(0.190866\pi\)
\(90\) 0 0
\(91\) 1.15097 0.120654
\(92\) −6.00153 −0.625703
\(93\) 0 0
\(94\) 5.67930 0.585776
\(95\) 34.4956 3.53917
\(96\) 0 0
\(97\) 8.91148 0.904823 0.452412 0.891809i \(-0.350564\pi\)
0.452412 + 0.891809i \(0.350564\pi\)
\(98\) −2.57409 −0.260022
\(99\) 0 0
\(100\) 13.0884 1.30884
\(101\) −10.7801 −1.07266 −0.536329 0.844009i \(-0.680190\pi\)
−0.536329 + 0.844009i \(0.680190\pi\)
\(102\) 0 0
\(103\) 1.83807 0.181110 0.0905552 0.995891i \(-0.471136\pi\)
0.0905552 + 0.995891i \(0.471136\pi\)
\(104\) 0.547094 0.0536469
\(105\) 0 0
\(106\) 0.544241 0.0528614
\(107\) −17.1840 −1.66124 −0.830619 0.556841i \(-0.812013\pi\)
−0.830619 + 0.556841i \(0.812013\pi\)
\(108\) 0 0
\(109\) −3.72734 −0.357015 −0.178507 0.983939i \(-0.557127\pi\)
−0.178507 + 0.983939i \(0.557127\pi\)
\(110\) −19.8919 −1.89662
\(111\) 0 0
\(112\) 2.10378 0.198789
\(113\) −20.5365 −1.93191 −0.965955 0.258712i \(-0.916702\pi\)
−0.965955 + 0.258712i \(0.916702\pi\)
\(114\) 0 0
\(115\) 25.5248 2.38020
\(116\) −8.07750 −0.749977
\(117\) 0 0
\(118\) −7.68172 −0.707160
\(119\) 1.10442 0.101242
\(120\) 0 0
\(121\) 10.8752 0.988652
\(122\) 3.68590 0.333706
\(123\) 0 0
\(124\) 4.87410 0.437707
\(125\) −34.4005 −3.07687
\(126\) 0 0
\(127\) −17.7968 −1.57921 −0.789606 0.613614i \(-0.789715\pi\)
−0.789606 + 0.613614i \(0.789715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.32682 −0.204075
\(131\) −19.4735 −1.70141 −0.850704 0.525645i \(-0.823824\pi\)
−0.850704 + 0.525645i \(0.823824\pi\)
\(132\) 0 0
\(133\) −17.0634 −1.47958
\(134\) 0.0374969 0.00323924
\(135\) 0 0
\(136\) 0.524967 0.0450156
\(137\) −8.78583 −0.750624 −0.375312 0.926899i \(-0.622464\pi\)
−0.375312 + 0.926899i \(0.622464\pi\)
\(138\) 0 0
\(139\) 2.73608 0.232071 0.116036 0.993245i \(-0.462981\pi\)
0.116036 + 0.993245i \(0.462981\pi\)
\(140\) −8.94750 −0.756202
\(141\) 0 0
\(142\) −13.0756 −1.09728
\(143\) 2.55881 0.213978
\(144\) 0 0
\(145\) 34.3540 2.85294
\(146\) 14.3053 1.18392
\(147\) 0 0
\(148\) −10.1411 −0.833591
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) 2.34994 0.191235 0.0956175 0.995418i \(-0.469517\pi\)
0.0956175 + 0.995418i \(0.469517\pi\)
\(152\) −8.11080 −0.657872
\(153\) 0 0
\(154\) 9.83959 0.792897
\(155\) −20.7298 −1.66506
\(156\) 0 0
\(157\) 5.88317 0.469528 0.234764 0.972052i \(-0.424568\pi\)
0.234764 + 0.972052i \(0.424568\pi\)
\(158\) −4.75832 −0.378552
\(159\) 0 0
\(160\) −4.25305 −0.336233
\(161\) −12.6259 −0.995063
\(162\) 0 0
\(163\) 0.559443 0.0438190 0.0219095 0.999760i \(-0.493025\pi\)
0.0219095 + 0.999760i \(0.493025\pi\)
\(164\) 2.58901 0.202168
\(165\) 0 0
\(166\) 7.02674 0.545380
\(167\) −12.8886 −0.997349 −0.498674 0.866789i \(-0.666180\pi\)
−0.498674 + 0.866789i \(0.666180\pi\)
\(168\) 0 0
\(169\) −12.7007 −0.976976
\(170\) −2.23271 −0.171241
\(171\) 0 0
\(172\) 7.72324 0.588892
\(173\) −15.4812 −1.17701 −0.588506 0.808493i \(-0.700284\pi\)
−0.588506 + 0.808493i \(0.700284\pi\)
\(174\) 0 0
\(175\) 27.5352 2.08147
\(176\) 4.67709 0.352549
\(177\) 0 0
\(178\) 15.5764 1.16750
\(179\) 18.5958 1.38991 0.694957 0.719051i \(-0.255423\pi\)
0.694957 + 0.719051i \(0.255423\pi\)
\(180\) 0 0
\(181\) 12.7960 0.951123 0.475561 0.879683i \(-0.342245\pi\)
0.475561 + 0.879683i \(0.342245\pi\)
\(182\) 1.15097 0.0853154
\(183\) 0 0
\(184\) −6.00153 −0.442439
\(185\) 43.1305 3.17102
\(186\) 0 0
\(187\) 2.45532 0.179551
\(188\) 5.67930 0.414206
\(189\) 0 0
\(190\) 34.4956 2.50257
\(191\) 5.71191 0.413300 0.206650 0.978415i \(-0.433744\pi\)
0.206650 + 0.978415i \(0.433744\pi\)
\(192\) 0 0
\(193\) 5.85587 0.421515 0.210758 0.977538i \(-0.432407\pi\)
0.210758 + 0.977538i \(0.432407\pi\)
\(194\) 8.91148 0.639807
\(195\) 0 0
\(196\) −2.57409 −0.183864
\(197\) −20.7385 −1.47756 −0.738778 0.673949i \(-0.764597\pi\)
−0.738778 + 0.673949i \(0.764597\pi\)
\(198\) 0 0
\(199\) −1.14505 −0.0811706 −0.0405853 0.999176i \(-0.512922\pi\)
−0.0405853 + 0.999176i \(0.512922\pi\)
\(200\) 13.0884 0.925491
\(201\) 0 0
\(202\) −10.7801 −0.758484
\(203\) −16.9933 −1.19270
\(204\) 0 0
\(205\) −11.0112 −0.769055
\(206\) 1.83807 0.128064
\(207\) 0 0
\(208\) 0.547094 0.0379341
\(209\) −37.9349 −2.62401
\(210\) 0 0
\(211\) 3.55721 0.244888 0.122444 0.992475i \(-0.460927\pi\)
0.122444 + 0.992475i \(0.460927\pi\)
\(212\) 0.544241 0.0373786
\(213\) 0 0
\(214\) −17.1840 −1.17467
\(215\) −32.8473 −2.24017
\(216\) 0 0
\(217\) 10.2540 0.696090
\(218\) −3.72734 −0.252447
\(219\) 0 0
\(220\) −19.8919 −1.34111
\(221\) 0.287206 0.0193196
\(222\) 0 0
\(223\) −5.36433 −0.359222 −0.179611 0.983738i \(-0.557484\pi\)
−0.179611 + 0.983738i \(0.557484\pi\)
\(224\) 2.10378 0.140565
\(225\) 0 0
\(226\) −20.5365 −1.36607
\(227\) −11.7698 −0.781188 −0.390594 0.920563i \(-0.627730\pi\)
−0.390594 + 0.920563i \(0.627730\pi\)
\(228\) 0 0
\(229\) −23.6187 −1.56077 −0.780384 0.625301i \(-0.784976\pi\)
−0.780384 + 0.625301i \(0.784976\pi\)
\(230\) 25.5248 1.68306
\(231\) 0 0
\(232\) −8.07750 −0.530314
\(233\) −27.4243 −1.79662 −0.898312 0.439358i \(-0.855206\pi\)
−0.898312 + 0.439358i \(0.855206\pi\)
\(234\) 0 0
\(235\) −24.1544 −1.57566
\(236\) −7.68172 −0.500037
\(237\) 0 0
\(238\) 1.10442 0.0715888
\(239\) 19.9867 1.29283 0.646416 0.762985i \(-0.276267\pi\)
0.646416 + 0.762985i \(0.276267\pi\)
\(240\) 0 0
\(241\) 1.69539 0.109209 0.0546047 0.998508i \(-0.482610\pi\)
0.0546047 + 0.998508i \(0.482610\pi\)
\(242\) 10.8752 0.699082
\(243\) 0 0
\(244\) 3.68590 0.235966
\(245\) 10.9477 0.699425
\(246\) 0 0
\(247\) −4.43736 −0.282343
\(248\) 4.87410 0.309505
\(249\) 0 0
\(250\) −34.4005 −2.17568
\(251\) 25.2458 1.59350 0.796752 0.604307i \(-0.206550\pi\)
0.796752 + 0.604307i \(0.206550\pi\)
\(252\) 0 0
\(253\) −28.0697 −1.76473
\(254\) −17.7968 −1.11667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0313 1.37428 0.687138 0.726527i \(-0.258867\pi\)
0.687138 + 0.726527i \(0.258867\pi\)
\(258\) 0 0
\(259\) −21.3346 −1.32567
\(260\) −2.32682 −0.144303
\(261\) 0 0
\(262\) −19.4735 −1.20308
\(263\) 28.4131 1.75203 0.876013 0.482287i \(-0.160194\pi\)
0.876013 + 0.482287i \(0.160194\pi\)
\(264\) 0 0
\(265\) −2.31468 −0.142190
\(266\) −17.0634 −1.04622
\(267\) 0 0
\(268\) 0.0374969 0.00229049
\(269\) −16.6698 −1.01638 −0.508189 0.861245i \(-0.669685\pi\)
−0.508189 + 0.861245i \(0.669685\pi\)
\(270\) 0 0
\(271\) 8.61463 0.523302 0.261651 0.965163i \(-0.415733\pi\)
0.261651 + 0.965163i \(0.415733\pi\)
\(272\) 0.524967 0.0318308
\(273\) 0 0
\(274\) −8.78583 −0.530771
\(275\) 61.2157 3.69145
\(276\) 0 0
\(277\) 4.78762 0.287660 0.143830 0.989602i \(-0.454058\pi\)
0.143830 + 0.989602i \(0.454058\pi\)
\(278\) 2.73608 0.164099
\(279\) 0 0
\(280\) −8.94750 −0.534715
\(281\) −1.93881 −0.115660 −0.0578298 0.998326i \(-0.518418\pi\)
−0.0578298 + 0.998326i \(0.518418\pi\)
\(282\) 0 0
\(283\) −28.3283 −1.68394 −0.841972 0.539521i \(-0.818606\pi\)
−0.841972 + 0.539521i \(0.818606\pi\)
\(284\) −13.0756 −0.775894
\(285\) 0 0
\(286\) 2.55881 0.151305
\(287\) 5.44672 0.321510
\(288\) 0 0
\(289\) −16.7244 −0.983789
\(290\) 34.3540 2.01734
\(291\) 0 0
\(292\) 14.3053 0.837155
\(293\) −24.7952 −1.44855 −0.724275 0.689511i \(-0.757826\pi\)
−0.724275 + 0.689511i \(0.757826\pi\)
\(294\) 0 0
\(295\) 32.6707 1.90216
\(296\) −10.1411 −0.589438
\(297\) 0 0
\(298\) −1.00000 −0.0579284
\(299\) −3.28340 −0.189884
\(300\) 0 0
\(301\) 16.2480 0.936521
\(302\) 2.34994 0.135224
\(303\) 0 0
\(304\) −8.11080 −0.465186
\(305\) −15.6763 −0.897623
\(306\) 0 0
\(307\) −15.6307 −0.892093 −0.446046 0.895010i \(-0.647168\pi\)
−0.446046 + 0.895010i \(0.647168\pi\)
\(308\) 9.83959 0.560663
\(309\) 0 0
\(310\) −20.7298 −1.17737
\(311\) 0.803622 0.0455692 0.0227846 0.999740i \(-0.492747\pi\)
0.0227846 + 0.999740i \(0.492747\pi\)
\(312\) 0 0
\(313\) −20.3580 −1.15070 −0.575350 0.817907i \(-0.695134\pi\)
−0.575350 + 0.817907i \(0.695134\pi\)
\(314\) 5.88317 0.332006
\(315\) 0 0
\(316\) −4.75832 −0.267676
\(317\) −23.4979 −1.31977 −0.659886 0.751365i \(-0.729396\pi\)
−0.659886 + 0.751365i \(0.729396\pi\)
\(318\) 0 0
\(319\) −37.7792 −2.11523
\(320\) −4.25305 −0.237753
\(321\) 0 0
\(322\) −12.6259 −0.703616
\(323\) −4.25790 −0.236916
\(324\) 0 0
\(325\) 7.16059 0.397198
\(326\) 0.559443 0.0309847
\(327\) 0 0
\(328\) 2.58901 0.142954
\(329\) 11.9480 0.658716
\(330\) 0 0
\(331\) 3.55426 0.195360 0.0976799 0.995218i \(-0.468858\pi\)
0.0976799 + 0.995218i \(0.468858\pi\)
\(332\) 7.02674 0.385642
\(333\) 0 0
\(334\) −12.8886 −0.705232
\(335\) −0.159476 −0.00871312
\(336\) 0 0
\(337\) 23.3744 1.27328 0.636642 0.771159i \(-0.280323\pi\)
0.636642 + 0.771159i \(0.280323\pi\)
\(338\) −12.7007 −0.690826
\(339\) 0 0
\(340\) −2.23271 −0.121086
\(341\) 22.7966 1.23450
\(342\) 0 0
\(343\) −20.1418 −1.08756
\(344\) 7.72324 0.416409
\(345\) 0 0
\(346\) −15.4812 −0.832274
\(347\) 14.6186 0.784768 0.392384 0.919801i \(-0.371650\pi\)
0.392384 + 0.919801i \(0.371650\pi\)
\(348\) 0 0
\(349\) 19.9685 1.06889 0.534443 0.845204i \(-0.320521\pi\)
0.534443 + 0.845204i \(0.320521\pi\)
\(350\) 27.5352 1.47182
\(351\) 0 0
\(352\) 4.67709 0.249290
\(353\) −32.4912 −1.72933 −0.864667 0.502346i \(-0.832470\pi\)
−0.864667 + 0.502346i \(0.832470\pi\)
\(354\) 0 0
\(355\) 55.6112 2.95153
\(356\) 15.5764 0.825548
\(357\) 0 0
\(358\) 18.5958 0.982818
\(359\) −11.0126 −0.581222 −0.290611 0.956841i \(-0.593859\pi\)
−0.290611 + 0.956841i \(0.593859\pi\)
\(360\) 0 0
\(361\) 46.7850 2.46237
\(362\) 12.7960 0.672545
\(363\) 0 0
\(364\) 1.15097 0.0603271
\(365\) −60.8412 −3.18457
\(366\) 0 0
\(367\) −32.4535 −1.69406 −0.847031 0.531544i \(-0.821612\pi\)
−0.847031 + 0.531544i \(0.821612\pi\)
\(368\) −6.00153 −0.312852
\(369\) 0 0
\(370\) 43.1305 2.24225
\(371\) 1.14497 0.0594437
\(372\) 0 0
\(373\) 8.06417 0.417547 0.208773 0.977964i \(-0.433053\pi\)
0.208773 + 0.977964i \(0.433053\pi\)
\(374\) 2.45532 0.126962
\(375\) 0 0
\(376\) 5.67930 0.292888
\(377\) −4.41915 −0.227598
\(378\) 0 0
\(379\) 2.31863 0.119100 0.0595499 0.998225i \(-0.481033\pi\)
0.0595499 + 0.998225i \(0.481033\pi\)
\(380\) 34.4956 1.76959
\(381\) 0 0
\(382\) 5.71191 0.292247
\(383\) −32.0190 −1.63610 −0.818048 0.575150i \(-0.804944\pi\)
−0.818048 + 0.575150i \(0.804944\pi\)
\(384\) 0 0
\(385\) −41.8483 −2.13278
\(386\) 5.85587 0.298056
\(387\) 0 0
\(388\) 8.91148 0.452412
\(389\) −5.77502 −0.292805 −0.146403 0.989225i \(-0.546769\pi\)
−0.146403 + 0.989225i \(0.546769\pi\)
\(390\) 0 0
\(391\) −3.15061 −0.159333
\(392\) −2.57409 −0.130011
\(393\) 0 0
\(394\) −20.7385 −1.04479
\(395\) 20.2374 1.01825
\(396\) 0 0
\(397\) 1.60163 0.0803833 0.0401917 0.999192i \(-0.487203\pi\)
0.0401917 + 0.999192i \(0.487203\pi\)
\(398\) −1.14505 −0.0573963
\(399\) 0 0
\(400\) 13.0884 0.654421
\(401\) −32.9876 −1.64732 −0.823662 0.567081i \(-0.808072\pi\)
−0.823662 + 0.567081i \(0.808072\pi\)
\(402\) 0 0
\(403\) 2.66659 0.132832
\(404\) −10.7801 −0.536329
\(405\) 0 0
\(406\) −16.9933 −0.843364
\(407\) −47.4307 −2.35105
\(408\) 0 0
\(409\) −20.9325 −1.03504 −0.517522 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(410\) −11.0112 −0.543804
\(411\) 0 0
\(412\) 1.83807 0.0905552
\(413\) −16.1607 −0.795215
\(414\) 0 0
\(415\) −29.8850 −1.46700
\(416\) 0.547094 0.0268235
\(417\) 0 0
\(418\) −37.9349 −1.85546
\(419\) 18.6183 0.909562 0.454781 0.890603i \(-0.349718\pi\)
0.454781 + 0.890603i \(0.349718\pi\)
\(420\) 0 0
\(421\) −29.0112 −1.41392 −0.706960 0.707254i \(-0.749934\pi\)
−0.706960 + 0.707254i \(0.749934\pi\)
\(422\) 3.55721 0.173162
\(423\) 0 0
\(424\) 0.544241 0.0264307
\(425\) 6.87099 0.333292
\(426\) 0 0
\(427\) 7.75434 0.375259
\(428\) −17.1840 −0.830619
\(429\) 0 0
\(430\) −32.8473 −1.58404
\(431\) 20.0767 0.967059 0.483530 0.875328i \(-0.339355\pi\)
0.483530 + 0.875328i \(0.339355\pi\)
\(432\) 0 0
\(433\) −26.1410 −1.25626 −0.628129 0.778109i \(-0.716179\pi\)
−0.628129 + 0.778109i \(0.716179\pi\)
\(434\) 10.2540 0.492210
\(435\) 0 0
\(436\) −3.72734 −0.178507
\(437\) 48.6772 2.32855
\(438\) 0 0
\(439\) 1.54942 0.0739497 0.0369749 0.999316i \(-0.488228\pi\)
0.0369749 + 0.999316i \(0.488228\pi\)
\(440\) −19.8919 −0.948309
\(441\) 0 0
\(442\) 0.287206 0.0136610
\(443\) 7.35831 0.349604 0.174802 0.984604i \(-0.444071\pi\)
0.174802 + 0.984604i \(0.444071\pi\)
\(444\) 0 0
\(445\) −66.2472 −3.14042
\(446\) −5.36433 −0.254008
\(447\) 0 0
\(448\) 2.10378 0.0993945
\(449\) −5.07130 −0.239330 −0.119665 0.992814i \(-0.538182\pi\)
−0.119665 + 0.992814i \(0.538182\pi\)
\(450\) 0 0
\(451\) 12.1090 0.570193
\(452\) −20.5365 −0.965955
\(453\) 0 0
\(454\) −11.7698 −0.552383
\(455\) −4.89512 −0.229487
\(456\) 0 0
\(457\) 6.68190 0.312566 0.156283 0.987712i \(-0.450049\pi\)
0.156283 + 0.987712i \(0.450049\pi\)
\(458\) −23.6187 −1.10363
\(459\) 0 0
\(460\) 25.5248 1.19010
\(461\) 17.6016 0.819788 0.409894 0.912133i \(-0.365566\pi\)
0.409894 + 0.912133i \(0.365566\pi\)
\(462\) 0 0
\(463\) 12.1244 0.563467 0.281734 0.959493i \(-0.409091\pi\)
0.281734 + 0.959493i \(0.409091\pi\)
\(464\) −8.07750 −0.374988
\(465\) 0 0
\(466\) −27.4243 −1.27041
\(467\) 32.6155 1.50927 0.754633 0.656147i \(-0.227815\pi\)
0.754633 + 0.656147i \(0.227815\pi\)
\(468\) 0 0
\(469\) 0.0788855 0.00364259
\(470\) −24.1544 −1.11416
\(471\) 0 0
\(472\) −7.68172 −0.353580
\(473\) 36.1223 1.66090
\(474\) 0 0
\(475\) −106.158 −4.87084
\(476\) 1.10442 0.0506209
\(477\) 0 0
\(478\) 19.9867 0.914171
\(479\) −26.8635 −1.22742 −0.613712 0.789530i \(-0.710325\pi\)
−0.613712 + 0.789530i \(0.710325\pi\)
\(480\) 0 0
\(481\) −5.54812 −0.252972
\(482\) 1.69539 0.0772227
\(483\) 0 0
\(484\) 10.8752 0.494326
\(485\) −37.9009 −1.72099
\(486\) 0 0
\(487\) 35.2025 1.59518 0.797590 0.603201i \(-0.206108\pi\)
0.797590 + 0.603201i \(0.206108\pi\)
\(488\) 3.68590 0.166853
\(489\) 0 0
\(490\) 10.9477 0.494568
\(491\) −13.8169 −0.623548 −0.311774 0.950156i \(-0.600923\pi\)
−0.311774 + 0.950156i \(0.600923\pi\)
\(492\) 0 0
\(493\) −4.24042 −0.190979
\(494\) −4.43736 −0.199646
\(495\) 0 0
\(496\) 4.87410 0.218853
\(497\) −27.5082 −1.23391
\(498\) 0 0
\(499\) −29.5985 −1.32501 −0.662505 0.749057i \(-0.730507\pi\)
−0.662505 + 0.749057i \(0.730507\pi\)
\(500\) −34.4005 −1.53844
\(501\) 0 0
\(502\) 25.2458 1.12678
\(503\) 11.1481 0.497070 0.248535 0.968623i \(-0.420051\pi\)
0.248535 + 0.968623i \(0.420051\pi\)
\(504\) 0 0
\(505\) 45.8482 2.04022
\(506\) −28.0697 −1.24785
\(507\) 0 0
\(508\) −17.7968 −0.789606
\(509\) 22.5150 0.997962 0.498981 0.866613i \(-0.333708\pi\)
0.498981 + 0.866613i \(0.333708\pi\)
\(510\) 0 0
\(511\) 30.0953 1.33134
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 22.0313 0.971759
\(515\) −7.81740 −0.344476
\(516\) 0 0
\(517\) 26.5626 1.16822
\(518\) −21.3346 −0.937390
\(519\) 0 0
\(520\) −2.32682 −0.102038
\(521\) 41.1084 1.80099 0.900496 0.434864i \(-0.143203\pi\)
0.900496 + 0.434864i \(0.143203\pi\)
\(522\) 0 0
\(523\) 8.70738 0.380747 0.190374 0.981712i \(-0.439030\pi\)
0.190374 + 0.981712i \(0.439030\pi\)
\(524\) −19.4735 −0.850704
\(525\) 0 0
\(526\) 28.4131 1.23887
\(527\) 2.55874 0.111460
\(528\) 0 0
\(529\) 13.0184 0.566018
\(530\) −2.31468 −0.100543
\(531\) 0 0
\(532\) −17.0634 −0.739791
\(533\) 1.41643 0.0613525
\(534\) 0 0
\(535\) 73.0843 3.15971
\(536\) 0.0374969 0.00161962
\(537\) 0 0
\(538\) −16.6698 −0.718688
\(539\) −12.0393 −0.518567
\(540\) 0 0
\(541\) −13.3238 −0.572836 −0.286418 0.958105i \(-0.592465\pi\)
−0.286418 + 0.958105i \(0.592465\pi\)
\(542\) 8.61463 0.370030
\(543\) 0 0
\(544\) 0.524967 0.0225078
\(545\) 15.8526 0.679049
\(546\) 0 0
\(547\) −25.3966 −1.08588 −0.542940 0.839771i \(-0.682689\pi\)
−0.542940 + 0.839771i \(0.682689\pi\)
\(548\) −8.78583 −0.375312
\(549\) 0 0
\(550\) 61.2157 2.61025
\(551\) 65.5149 2.79103
\(552\) 0 0
\(553\) −10.0105 −0.425689
\(554\) 4.78762 0.203407
\(555\) 0 0
\(556\) 2.73608 0.116036
\(557\) −24.2298 −1.02665 −0.513325 0.858194i \(-0.671587\pi\)
−0.513325 + 0.858194i \(0.671587\pi\)
\(558\) 0 0
\(559\) 4.22534 0.178713
\(560\) −8.94750 −0.378101
\(561\) 0 0
\(562\) −1.93881 −0.0817837
\(563\) 2.60064 0.109604 0.0548019 0.998497i \(-0.482547\pi\)
0.0548019 + 0.998497i \(0.482547\pi\)
\(564\) 0 0
\(565\) 87.3426 3.67453
\(566\) −28.3283 −1.19073
\(567\) 0 0
\(568\) −13.0756 −0.548640
\(569\) −0.0803760 −0.00336954 −0.00168477 0.999999i \(-0.500536\pi\)
−0.00168477 + 0.999999i \(0.500536\pi\)
\(570\) 0 0
\(571\) −26.9549 −1.12803 −0.564013 0.825766i \(-0.690743\pi\)
−0.564013 + 0.825766i \(0.690743\pi\)
\(572\) 2.55881 0.106989
\(573\) 0 0
\(574\) 5.44672 0.227342
\(575\) −78.5506 −3.27579
\(576\) 0 0
\(577\) −18.8989 −0.786772 −0.393386 0.919373i \(-0.628696\pi\)
−0.393386 + 0.919373i \(0.628696\pi\)
\(578\) −16.7244 −0.695644
\(579\) 0 0
\(580\) 34.3540 1.42647
\(581\) 14.7827 0.613291
\(582\) 0 0
\(583\) 2.54546 0.105422
\(584\) 14.3053 0.591958
\(585\) 0 0
\(586\) −24.7952 −1.02428
\(587\) 6.24700 0.257841 0.128921 0.991655i \(-0.458849\pi\)
0.128921 + 0.991655i \(0.458849\pi\)
\(588\) 0 0
\(589\) −39.5328 −1.62892
\(590\) 32.6707 1.34503
\(591\) 0 0
\(592\) −10.1411 −0.416796
\(593\) 30.7455 1.26257 0.631283 0.775553i \(-0.282529\pi\)
0.631283 + 0.775553i \(0.282529\pi\)
\(594\) 0 0
\(595\) −4.69714 −0.192564
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) −3.28340 −0.134268
\(599\) −34.6147 −1.41432 −0.707158 0.707055i \(-0.750023\pi\)
−0.707158 + 0.707055i \(0.750023\pi\)
\(600\) 0 0
\(601\) 10.2891 0.419702 0.209851 0.977733i \(-0.432702\pi\)
0.209851 + 0.977733i \(0.432702\pi\)
\(602\) 16.2480 0.662220
\(603\) 0 0
\(604\) 2.34994 0.0956175
\(605\) −46.2526 −1.88044
\(606\) 0 0
\(607\) 38.5414 1.56435 0.782173 0.623062i \(-0.214111\pi\)
0.782173 + 0.623062i \(0.214111\pi\)
\(608\) −8.11080 −0.328936
\(609\) 0 0
\(610\) −15.6763 −0.634715
\(611\) 3.10711 0.125700
\(612\) 0 0
\(613\) 11.4761 0.463517 0.231759 0.972773i \(-0.425552\pi\)
0.231759 + 0.972773i \(0.425552\pi\)
\(614\) −15.6307 −0.630805
\(615\) 0 0
\(616\) 9.83959 0.396448
\(617\) −8.32005 −0.334952 −0.167476 0.985876i \(-0.553562\pi\)
−0.167476 + 0.985876i \(0.553562\pi\)
\(618\) 0 0
\(619\) −20.0989 −0.807841 −0.403921 0.914794i \(-0.632353\pi\)
−0.403921 + 0.914794i \(0.632353\pi\)
\(620\) −20.7298 −0.832528
\(621\) 0 0
\(622\) 0.803622 0.0322223
\(623\) 32.7694 1.31288
\(624\) 0 0
\(625\) 80.8647 3.23459
\(626\) −20.3580 −0.813668
\(627\) 0 0
\(628\) 5.88317 0.234764
\(629\) −5.32373 −0.212271
\(630\) 0 0
\(631\) −19.8280 −0.789339 −0.394670 0.918823i \(-0.629141\pi\)
−0.394670 + 0.918823i \(0.629141\pi\)
\(632\) −4.75832 −0.189276
\(633\) 0 0
\(634\) −23.4979 −0.933220
\(635\) 75.6907 3.00370
\(636\) 0 0
\(637\) −1.40827 −0.0557976
\(638\) −37.7792 −1.49569
\(639\) 0 0
\(640\) −4.25305 −0.168117
\(641\) 14.5984 0.576600 0.288300 0.957540i \(-0.406910\pi\)
0.288300 + 0.957540i \(0.406910\pi\)
\(642\) 0 0
\(643\) 6.63135 0.261515 0.130757 0.991414i \(-0.458259\pi\)
0.130757 + 0.991414i \(0.458259\pi\)
\(644\) −12.6259 −0.497532
\(645\) 0 0
\(646\) −4.25790 −0.167525
\(647\) 10.4522 0.410917 0.205458 0.978666i \(-0.434131\pi\)
0.205458 + 0.978666i \(0.434131\pi\)
\(648\) 0 0
\(649\) −35.9281 −1.41030
\(650\) 7.16059 0.280862
\(651\) 0 0
\(652\) 0.559443 0.0219095
\(653\) 9.33999 0.365502 0.182751 0.983159i \(-0.441500\pi\)
0.182751 + 0.983159i \(0.441500\pi\)
\(654\) 0 0
\(655\) 82.8217 3.23611
\(656\) 2.58901 0.101084
\(657\) 0 0
\(658\) 11.9480 0.465783
\(659\) 29.1535 1.13566 0.567830 0.823146i \(-0.307783\pi\)
0.567830 + 0.823146i \(0.307783\pi\)
\(660\) 0 0
\(661\) −24.9644 −0.971004 −0.485502 0.874236i \(-0.661363\pi\)
−0.485502 + 0.874236i \(0.661363\pi\)
\(662\) 3.55426 0.138140
\(663\) 0 0
\(664\) 7.02674 0.272690
\(665\) 72.5713 2.81420
\(666\) 0 0
\(667\) 48.4774 1.87705
\(668\) −12.8886 −0.498674
\(669\) 0 0
\(670\) −0.159476 −0.00616111
\(671\) 17.2393 0.665515
\(672\) 0 0
\(673\) 20.0978 0.774712 0.387356 0.921930i \(-0.373389\pi\)
0.387356 + 0.921930i \(0.373389\pi\)
\(674\) 23.3744 0.900348
\(675\) 0 0
\(676\) −12.7007 −0.488488
\(677\) 45.6993 1.75637 0.878184 0.478323i \(-0.158755\pi\)
0.878184 + 0.478323i \(0.158755\pi\)
\(678\) 0 0
\(679\) 18.7478 0.719475
\(680\) −2.23271 −0.0856206
\(681\) 0 0
\(682\) 22.7966 0.872927
\(683\) 35.2201 1.34766 0.673830 0.738886i \(-0.264648\pi\)
0.673830 + 0.738886i \(0.264648\pi\)
\(684\) 0 0
\(685\) 37.3666 1.42770
\(686\) −20.1418 −0.769018
\(687\) 0 0
\(688\) 7.72324 0.294446
\(689\) 0.297751 0.0113434
\(690\) 0 0
\(691\) −18.6521 −0.709558 −0.354779 0.934950i \(-0.615444\pi\)
−0.354779 + 0.934950i \(0.615444\pi\)
\(692\) −15.4812 −0.588506
\(693\) 0 0
\(694\) 14.6186 0.554915
\(695\) −11.6367 −0.441404
\(696\) 0 0
\(697\) 1.35915 0.0514813
\(698\) 19.9685 0.755817
\(699\) 0 0
\(700\) 27.5352 1.04073
\(701\) −46.3334 −1.74999 −0.874995 0.484133i \(-0.839135\pi\)
−0.874995 + 0.484133i \(0.839135\pi\)
\(702\) 0 0
\(703\) 82.2522 3.10220
\(704\) 4.67709 0.176274
\(705\) 0 0
\(706\) −32.4912 −1.22282
\(707\) −22.6790 −0.852931
\(708\) 0 0
\(709\) −5.62028 −0.211074 −0.105537 0.994415i \(-0.533656\pi\)
−0.105537 + 0.994415i \(0.533656\pi\)
\(710\) 55.6112 2.08705
\(711\) 0 0
\(712\) 15.5764 0.583750
\(713\) −29.2521 −1.09550
\(714\) 0 0
\(715\) −10.8827 −0.406991
\(716\) 18.5958 0.694957
\(717\) 0 0
\(718\) −11.0126 −0.410986
\(719\) 1.38058 0.0514870 0.0257435 0.999669i \(-0.491805\pi\)
0.0257435 + 0.999669i \(0.491805\pi\)
\(720\) 0 0
\(721\) 3.86690 0.144011
\(722\) 46.7850 1.74116
\(723\) 0 0
\(724\) 12.7960 0.475561
\(725\) −105.722 −3.92641
\(726\) 0 0
\(727\) 33.6994 1.24984 0.624920 0.780689i \(-0.285131\pi\)
0.624920 + 0.780689i \(0.285131\pi\)
\(728\) 1.15097 0.0426577
\(729\) 0 0
\(730\) −60.8412 −2.25183
\(731\) 4.05445 0.149959
\(732\) 0 0
\(733\) 6.43137 0.237548 0.118774 0.992921i \(-0.462104\pi\)
0.118774 + 0.992921i \(0.462104\pi\)
\(734\) −32.4535 −1.19788
\(735\) 0 0
\(736\) −6.00153 −0.221220
\(737\) 0.175377 0.00646008
\(738\) 0 0
\(739\) −4.37703 −0.161012 −0.0805058 0.996754i \(-0.525654\pi\)
−0.0805058 + 0.996754i \(0.525654\pi\)
\(740\) 43.1305 1.58551
\(741\) 0 0
\(742\) 1.14497 0.0420330
\(743\) 29.5010 1.08229 0.541144 0.840930i \(-0.317991\pi\)
0.541144 + 0.840930i \(0.317991\pi\)
\(744\) 0 0
\(745\) 4.25305 0.155820
\(746\) 8.06417 0.295250
\(747\) 0 0
\(748\) 2.45532 0.0897753
\(749\) −36.1514 −1.32094
\(750\) 0 0
\(751\) 34.4108 1.25567 0.627834 0.778347i \(-0.283942\pi\)
0.627834 + 0.778347i \(0.283942\pi\)
\(752\) 5.67930 0.207103
\(753\) 0 0
\(754\) −4.41915 −0.160936
\(755\) −9.99439 −0.363733
\(756\) 0 0
\(757\) −20.1601 −0.732733 −0.366366 0.930471i \(-0.619398\pi\)
−0.366366 + 0.930471i \(0.619398\pi\)
\(758\) 2.31863 0.0842163
\(759\) 0 0
\(760\) 34.4956 1.25129
\(761\) −9.96886 −0.361371 −0.180686 0.983541i \(-0.557832\pi\)
−0.180686 + 0.983541i \(0.557832\pi\)
\(762\) 0 0
\(763\) −7.84152 −0.283882
\(764\) 5.71191 0.206650
\(765\) 0 0
\(766\) −32.0190 −1.15689
\(767\) −4.20262 −0.151748
\(768\) 0 0
\(769\) 23.3004 0.840233 0.420117 0.907470i \(-0.361989\pi\)
0.420117 + 0.907470i \(0.361989\pi\)
\(770\) −41.8483 −1.50811
\(771\) 0 0
\(772\) 5.85587 0.210758
\(773\) 24.2668 0.872816 0.436408 0.899749i \(-0.356250\pi\)
0.436408 + 0.899749i \(0.356250\pi\)
\(774\) 0 0
\(775\) 63.7942 2.29156
\(776\) 8.91148 0.319903
\(777\) 0 0
\(778\) −5.77502 −0.207044
\(779\) −20.9990 −0.752366
\(780\) 0 0
\(781\) −61.1558 −2.18833
\(782\) −3.15061 −0.112666
\(783\) 0 0
\(784\) −2.57409 −0.0919318
\(785\) −25.0214 −0.893052
\(786\) 0 0
\(787\) −29.1694 −1.03978 −0.519889 0.854234i \(-0.674027\pi\)
−0.519889 + 0.854234i \(0.674027\pi\)
\(788\) −20.7385 −0.738778
\(789\) 0 0
\(790\) 20.2374 0.720013
\(791\) −43.2043 −1.53617
\(792\) 0 0
\(793\) 2.01653 0.0716092
\(794\) 1.60163 0.0568396
\(795\) 0 0
\(796\) −1.14505 −0.0405853
\(797\) −13.1392 −0.465414 −0.232707 0.972547i \(-0.574758\pi\)
−0.232707 + 0.972547i \(0.574758\pi\)
\(798\) 0 0
\(799\) 2.98145 0.105476
\(800\) 13.0884 0.462746
\(801\) 0 0
\(802\) −32.9876 −1.16483
\(803\) 66.9072 2.36111
\(804\) 0 0
\(805\) 53.6987 1.89263
\(806\) 2.66659 0.0939265
\(807\) 0 0
\(808\) −10.7801 −0.379242
\(809\) −42.6320 −1.49886 −0.749432 0.662082i \(-0.769673\pi\)
−0.749432 + 0.662082i \(0.769673\pi\)
\(810\) 0 0
\(811\) 45.8586 1.61031 0.805156 0.593063i \(-0.202081\pi\)
0.805156 + 0.593063i \(0.202081\pi\)
\(812\) −16.9933 −0.596348
\(813\) 0 0
\(814\) −47.4307 −1.66245
\(815\) −2.37934 −0.0833446
\(816\) 0 0
\(817\) −62.6416 −2.19155
\(818\) −20.9325 −0.731886
\(819\) 0 0
\(820\) −11.0112 −0.384528
\(821\) 42.4539 1.48165 0.740825 0.671698i \(-0.234435\pi\)
0.740825 + 0.671698i \(0.234435\pi\)
\(822\) 0 0
\(823\) −8.96445 −0.312481 −0.156241 0.987719i \(-0.549937\pi\)
−0.156241 + 0.987719i \(0.549937\pi\)
\(824\) 1.83807 0.0640322
\(825\) 0 0
\(826\) −16.1607 −0.562302
\(827\) −16.7639 −0.582936 −0.291468 0.956581i \(-0.594144\pi\)
−0.291468 + 0.956581i \(0.594144\pi\)
\(828\) 0 0
\(829\) 33.0424 1.14761 0.573805 0.818992i \(-0.305467\pi\)
0.573805 + 0.818992i \(0.305467\pi\)
\(830\) −29.8850 −1.03733
\(831\) 0 0
\(832\) 0.547094 0.0189671
\(833\) −1.35131 −0.0468202
\(834\) 0 0
\(835\) 54.8158 1.89698
\(836\) −37.9349 −1.31201
\(837\) 0 0
\(838\) 18.6183 0.643157
\(839\) 16.0385 0.553712 0.276856 0.960911i \(-0.410708\pi\)
0.276856 + 0.960911i \(0.410708\pi\)
\(840\) 0 0
\(841\) 36.2460 1.24986
\(842\) −29.0112 −0.999792
\(843\) 0 0
\(844\) 3.55721 0.122444
\(845\) 54.0166 1.85823
\(846\) 0 0
\(847\) 22.8790 0.786132
\(848\) 0.544241 0.0186893
\(849\) 0 0
\(850\) 6.87099 0.235673
\(851\) 60.8620 2.08632
\(852\) 0 0
\(853\) −4.31797 −0.147845 −0.0739223 0.997264i \(-0.523552\pi\)
−0.0739223 + 0.997264i \(0.523552\pi\)
\(854\) 7.75434 0.265348
\(855\) 0 0
\(856\) −17.1840 −0.587336
\(857\) 32.3939 1.10656 0.553278 0.832997i \(-0.313377\pi\)
0.553278 + 0.832997i \(0.313377\pi\)
\(858\) 0 0
\(859\) 7.15934 0.244274 0.122137 0.992513i \(-0.461025\pi\)
0.122137 + 0.992513i \(0.461025\pi\)
\(860\) −32.8473 −1.12008
\(861\) 0 0
\(862\) 20.0767 0.683814
\(863\) −18.8898 −0.643016 −0.321508 0.946907i \(-0.604190\pi\)
−0.321508 + 0.946907i \(0.604190\pi\)
\(864\) 0 0
\(865\) 65.8422 2.23870
\(866\) −26.1410 −0.888308
\(867\) 0 0
\(868\) 10.2540 0.348045
\(869\) −22.2551 −0.754952
\(870\) 0 0
\(871\) 0.0205143 0.000695102 0
\(872\) −3.72734 −0.126224
\(873\) 0 0
\(874\) 48.6772 1.64653
\(875\) −72.3712 −2.44659
\(876\) 0 0
\(877\) −16.1749 −0.546189 −0.273094 0.961987i \(-0.588047\pi\)
−0.273094 + 0.961987i \(0.588047\pi\)
\(878\) 1.54942 0.0522904
\(879\) 0 0
\(880\) −19.8919 −0.670556
\(881\) −18.4187 −0.620540 −0.310270 0.950648i \(-0.600419\pi\)
−0.310270 + 0.950648i \(0.600419\pi\)
\(882\) 0 0
\(883\) 19.7770 0.665547 0.332774 0.943007i \(-0.392015\pi\)
0.332774 + 0.943007i \(0.392015\pi\)
\(884\) 0.287206 0.00965979
\(885\) 0 0
\(886\) 7.35831 0.247207
\(887\) 29.2805 0.983143 0.491571 0.870837i \(-0.336423\pi\)
0.491571 + 0.870837i \(0.336423\pi\)
\(888\) 0 0
\(889\) −37.4407 −1.25572
\(890\) −66.2472 −2.22061
\(891\) 0 0
\(892\) −5.36433 −0.179611
\(893\) −46.0637 −1.54146
\(894\) 0 0
\(895\) −79.0888 −2.64365
\(896\) 2.10378 0.0702825
\(897\) 0 0
\(898\) −5.07130 −0.169232
\(899\) −39.3705 −1.31308
\(900\) 0 0
\(901\) 0.285709 0.00951834
\(902\) 12.1090 0.403187
\(903\) 0 0
\(904\) −20.5365 −0.683033
\(905\) −54.4222 −1.80906
\(906\) 0 0
\(907\) 11.0299 0.366241 0.183120 0.983090i \(-0.441380\pi\)
0.183120 + 0.983090i \(0.441380\pi\)
\(908\) −11.7698 −0.390594
\(909\) 0 0
\(910\) −4.89512 −0.162272
\(911\) 46.2825 1.53341 0.766704 0.642001i \(-0.221895\pi\)
0.766704 + 0.642001i \(0.221895\pi\)
\(912\) 0 0
\(913\) 32.8647 1.08766
\(914\) 6.68190 0.221018
\(915\) 0 0
\(916\) −23.6187 −0.780384
\(917\) −40.9680 −1.35288
\(918\) 0 0
\(919\) 2.95357 0.0974293 0.0487146 0.998813i \(-0.484488\pi\)
0.0487146 + 0.998813i \(0.484488\pi\)
\(920\) 25.5248 0.841528
\(921\) 0 0
\(922\) 17.6016 0.579678
\(923\) −7.15358 −0.235463
\(924\) 0 0
\(925\) −132.731 −4.36416
\(926\) 12.1244 0.398432
\(927\) 0 0
\(928\) −8.07750 −0.265157
\(929\) 48.6223 1.59525 0.797623 0.603156i \(-0.206090\pi\)
0.797623 + 0.603156i \(0.206090\pi\)
\(930\) 0 0
\(931\) 20.8779 0.684246
\(932\) −27.4243 −0.898312
\(933\) 0 0
\(934\) 32.6155 1.06721
\(935\) −10.4426 −0.341509
\(936\) 0 0
\(937\) 12.4395 0.406380 0.203190 0.979139i \(-0.434869\pi\)
0.203190 + 0.979139i \(0.434869\pi\)
\(938\) 0.0788855 0.00257570
\(939\) 0 0
\(940\) −24.1544 −0.787828
\(941\) 14.0582 0.458283 0.229142 0.973393i \(-0.426408\pi\)
0.229142 + 0.973393i \(0.426408\pi\)
\(942\) 0 0
\(943\) −15.5380 −0.505988
\(944\) −7.68172 −0.250019
\(945\) 0 0
\(946\) 36.1223 1.17444
\(947\) 9.24375 0.300382 0.150191 0.988657i \(-0.452011\pi\)
0.150191 + 0.988657i \(0.452011\pi\)
\(948\) 0 0
\(949\) 7.82635 0.254054
\(950\) −106.158 −3.44421
\(951\) 0 0
\(952\) 1.10442 0.0357944
\(953\) −8.96912 −0.290538 −0.145269 0.989392i \(-0.546405\pi\)
−0.145269 + 0.989392i \(0.546405\pi\)
\(954\) 0 0
\(955\) −24.2931 −0.786105
\(956\) 19.9867 0.646416
\(957\) 0 0
\(958\) −26.8635 −0.867920
\(959\) −18.4835 −0.596863
\(960\) 0 0
\(961\) −7.24318 −0.233651
\(962\) −5.54812 −0.178879
\(963\) 0 0
\(964\) 1.69539 0.0546047
\(965\) −24.9053 −0.801731
\(966\) 0 0
\(967\) −20.7749 −0.668074 −0.334037 0.942560i \(-0.608411\pi\)
−0.334037 + 0.942560i \(0.608411\pi\)
\(968\) 10.8752 0.349541
\(969\) 0 0
\(970\) −37.9009 −1.21693
\(971\) 30.2800 0.971731 0.485865 0.874034i \(-0.338505\pi\)
0.485865 + 0.874034i \(0.338505\pi\)
\(972\) 0 0
\(973\) 5.75612 0.184533
\(974\) 35.2025 1.12796
\(975\) 0 0
\(976\) 3.68590 0.117983
\(977\) −3.60470 −0.115325 −0.0576623 0.998336i \(-0.518365\pi\)
−0.0576623 + 0.998336i \(0.518365\pi\)
\(978\) 0 0
\(979\) 72.8522 2.32837
\(980\) 10.9477 0.349713
\(981\) 0 0
\(982\) −13.8169 −0.440915
\(983\) −5.97607 −0.190607 −0.0953036 0.995448i \(-0.530382\pi\)
−0.0953036 + 0.995448i \(0.530382\pi\)
\(984\) 0 0
\(985\) 88.2018 2.81034
\(986\) −4.24042 −0.135043
\(987\) 0 0
\(988\) −4.43736 −0.141171
\(989\) −46.3513 −1.47389
\(990\) 0 0
\(991\) 28.9520 0.919691 0.459845 0.887999i \(-0.347905\pi\)
0.459845 + 0.887999i \(0.347905\pi\)
\(992\) 4.87410 0.154753
\(993\) 0 0
\(994\) −27.5082 −0.872509
\(995\) 4.86996 0.154388
\(996\) 0 0
\(997\) 7.17728 0.227307 0.113653 0.993520i \(-0.463745\pi\)
0.113653 + 0.993520i \(0.463745\pi\)
\(998\) −29.5985 −0.936924
\(999\) 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 8046.2.a.m.1.1 yes 12
3.2 odd 2 8046.2.a.l.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.l.1.12 12 3.2 odd 2
8046.2.a.m.1.1 yes 12 1.1 even 1 trivial