Properties

Label 3344.2.a.bb.1.9
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $9$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-3.22348\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.22348 q^{3} -1.96048 q^{5} -3.83651 q^{7} +7.39085 q^{9} +O(q^{10})\) \(q+3.22348 q^{3} -1.96048 q^{5} -3.83651 q^{7} +7.39085 q^{9} -1.00000 q^{11} -4.08725 q^{13} -6.31957 q^{15} +6.84309 q^{17} +1.00000 q^{19} -12.3669 q^{21} +6.67929 q^{23} -1.15652 q^{25} +14.1538 q^{27} +3.43658 q^{29} +5.74398 q^{31} -3.22348 q^{33} +7.52140 q^{35} +4.55434 q^{37} -13.1752 q^{39} -0.970944 q^{41} +6.42713 q^{43} -14.4896 q^{45} -7.92096 q^{47} +7.71881 q^{49} +22.0586 q^{51} +13.4503 q^{53} +1.96048 q^{55} +3.22348 q^{57} +11.7764 q^{59} +9.76071 q^{61} -28.3551 q^{63} +8.01298 q^{65} -9.19392 q^{67} +21.5306 q^{69} +7.97888 q^{71} +4.01922 q^{73} -3.72802 q^{75} +3.83651 q^{77} -7.07742 q^{79} +23.4521 q^{81} -12.2436 q^{83} -13.4157 q^{85} +11.0778 q^{87} -7.57690 q^{89} +15.6808 q^{91} +18.5156 q^{93} -1.96048 q^{95} -17.3311 q^{97} -7.39085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.22348 1.86108 0.930540 0.366191i \(-0.119338\pi\)
0.930540 + 0.366191i \(0.119338\pi\)
\(4\) 0 0
\(5\) −1.96048 −0.876753 −0.438377 0.898791i \(-0.644446\pi\)
−0.438377 + 0.898791i \(0.644446\pi\)
\(6\) 0 0
\(7\) −3.83651 −1.45006 −0.725032 0.688715i \(-0.758175\pi\)
−0.725032 + 0.688715i \(0.758175\pi\)
\(8\) 0 0
\(9\) 7.39085 2.46362
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.08725 −1.13360 −0.566800 0.823855i \(-0.691819\pi\)
−0.566800 + 0.823855i \(0.691819\pi\)
\(14\) 0 0
\(15\) −6.31957 −1.63171
\(16\) 0 0
\(17\) 6.84309 1.65969 0.829847 0.557991i \(-0.188428\pi\)
0.829847 + 0.557991i \(0.188428\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −12.3669 −2.69868
\(22\) 0 0
\(23\) 6.67929 1.39273 0.696364 0.717689i \(-0.254800\pi\)
0.696364 + 0.717689i \(0.254800\pi\)
\(24\) 0 0
\(25\) −1.15652 −0.231304
\(26\) 0 0
\(27\) 14.1538 2.72391
\(28\) 0 0
\(29\) 3.43658 0.638157 0.319079 0.947728i \(-0.396627\pi\)
0.319079 + 0.947728i \(0.396627\pi\)
\(30\) 0 0
\(31\) 5.74398 1.03165 0.515825 0.856694i \(-0.327486\pi\)
0.515825 + 0.856694i \(0.327486\pi\)
\(32\) 0 0
\(33\) −3.22348 −0.561137
\(34\) 0 0
\(35\) 7.52140 1.27135
\(36\) 0 0
\(37\) 4.55434 0.748729 0.374364 0.927282i \(-0.377861\pi\)
0.374364 + 0.927282i \(0.377861\pi\)
\(38\) 0 0
\(39\) −13.1752 −2.10972
\(40\) 0 0
\(41\) −0.970944 −0.151636 −0.0758180 0.997122i \(-0.524157\pi\)
−0.0758180 + 0.997122i \(0.524157\pi\)
\(42\) 0 0
\(43\) 6.42713 0.980128 0.490064 0.871687i \(-0.336973\pi\)
0.490064 + 0.871687i \(0.336973\pi\)
\(44\) 0 0
\(45\) −14.4896 −2.15998
\(46\) 0 0
\(47\) −7.92096 −1.15539 −0.577695 0.816252i \(-0.696048\pi\)
−0.577695 + 0.816252i \(0.696048\pi\)
\(48\) 0 0
\(49\) 7.71881 1.10269
\(50\) 0 0
\(51\) 22.0586 3.08882
\(52\) 0 0
\(53\) 13.4503 1.84754 0.923771 0.382946i \(-0.125090\pi\)
0.923771 + 0.382946i \(0.125090\pi\)
\(54\) 0 0
\(55\) 1.96048 0.264351
\(56\) 0 0
\(57\) 3.22348 0.426961
\(58\) 0 0
\(59\) 11.7764 1.53315 0.766577 0.642153i \(-0.221958\pi\)
0.766577 + 0.642153i \(0.221958\pi\)
\(60\) 0 0
\(61\) 9.76071 1.24973 0.624865 0.780732i \(-0.285154\pi\)
0.624865 + 0.780732i \(0.285154\pi\)
\(62\) 0 0
\(63\) −28.3551 −3.57240
\(64\) 0 0
\(65\) 8.01298 0.993888
\(66\) 0 0
\(67\) −9.19392 −1.12322 −0.561608 0.827403i \(-0.689817\pi\)
−0.561608 + 0.827403i \(0.689817\pi\)
\(68\) 0 0
\(69\) 21.5306 2.59198
\(70\) 0 0
\(71\) 7.97888 0.946919 0.473460 0.880815i \(-0.343005\pi\)
0.473460 + 0.880815i \(0.343005\pi\)
\(72\) 0 0
\(73\) 4.01922 0.470414 0.235207 0.971945i \(-0.424423\pi\)
0.235207 + 0.971945i \(0.424423\pi\)
\(74\) 0 0
\(75\) −3.72802 −0.430475
\(76\) 0 0
\(77\) 3.83651 0.437211
\(78\) 0 0
\(79\) −7.07742 −0.796272 −0.398136 0.917326i \(-0.630343\pi\)
−0.398136 + 0.917326i \(0.630343\pi\)
\(80\) 0 0
\(81\) 23.4521 2.60579
\(82\) 0 0
\(83\) −12.2436 −1.34391 −0.671957 0.740590i \(-0.734546\pi\)
−0.671957 + 0.740590i \(0.734546\pi\)
\(84\) 0 0
\(85\) −13.4157 −1.45514
\(86\) 0 0
\(87\) 11.0778 1.18766
\(88\) 0 0
\(89\) −7.57690 −0.803150 −0.401575 0.915826i \(-0.631537\pi\)
−0.401575 + 0.915826i \(0.631537\pi\)
\(90\) 0 0
\(91\) 15.6808 1.64379
\(92\) 0 0
\(93\) 18.5156 1.91998
\(94\) 0 0
\(95\) −1.96048 −0.201141
\(96\) 0 0
\(97\) −17.3311 −1.75971 −0.879853 0.475245i \(-0.842359\pi\)
−0.879853 + 0.475245i \(0.842359\pi\)
\(98\) 0 0
\(99\) −7.39085 −0.742808
\(100\) 0 0
\(101\) −5.21441 −0.518853 −0.259426 0.965763i \(-0.583533\pi\)
−0.259426 + 0.965763i \(0.583533\pi\)
\(102\) 0 0
\(103\) −4.55783 −0.449096 −0.224548 0.974463i \(-0.572091\pi\)
−0.224548 + 0.974463i \(0.572091\pi\)
\(104\) 0 0
\(105\) 24.2451 2.36608
\(106\) 0 0
\(107\) −6.15529 −0.595054 −0.297527 0.954713i \(-0.596162\pi\)
−0.297527 + 0.954713i \(0.596162\pi\)
\(108\) 0 0
\(109\) 4.26716 0.408720 0.204360 0.978896i \(-0.434489\pi\)
0.204360 + 0.978896i \(0.434489\pi\)
\(110\) 0 0
\(111\) 14.6808 1.39344
\(112\) 0 0
\(113\) 17.1642 1.61467 0.807336 0.590092i \(-0.200909\pi\)
0.807336 + 0.590092i \(0.200909\pi\)
\(114\) 0 0
\(115\) −13.0946 −1.22108
\(116\) 0 0
\(117\) −30.2083 −2.79276
\(118\) 0 0
\(119\) −26.2536 −2.40666
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.12982 −0.282207
\(124\) 0 0
\(125\) 12.0697 1.07955
\(126\) 0 0
\(127\) −2.39705 −0.212703 −0.106352 0.994329i \(-0.533917\pi\)
−0.106352 + 0.994329i \(0.533917\pi\)
\(128\) 0 0
\(129\) 20.7177 1.82410
\(130\) 0 0
\(131\) −6.81445 −0.595381 −0.297691 0.954662i \(-0.596216\pi\)
−0.297691 + 0.954662i \(0.596216\pi\)
\(132\) 0 0
\(133\) −3.83651 −0.332668
\(134\) 0 0
\(135\) −27.7483 −2.38819
\(136\) 0 0
\(137\) 20.9984 1.79401 0.897007 0.442016i \(-0.145736\pi\)
0.897007 + 0.442016i \(0.145736\pi\)
\(138\) 0 0
\(139\) −0.323311 −0.0274229 −0.0137115 0.999906i \(-0.504365\pi\)
−0.0137115 + 0.999906i \(0.504365\pi\)
\(140\) 0 0
\(141\) −25.5331 −2.15027
\(142\) 0 0
\(143\) 4.08725 0.341793
\(144\) 0 0
\(145\) −6.73735 −0.559506
\(146\) 0 0
\(147\) 24.8815 2.05219
\(148\) 0 0
\(149\) 12.4092 1.01660 0.508301 0.861180i \(-0.330274\pi\)
0.508301 + 0.861180i \(0.330274\pi\)
\(150\) 0 0
\(151\) −6.41414 −0.521975 −0.260987 0.965342i \(-0.584048\pi\)
−0.260987 + 0.965342i \(0.584048\pi\)
\(152\) 0 0
\(153\) 50.5762 4.08885
\(154\) 0 0
\(155\) −11.2610 −0.904502
\(156\) 0 0
\(157\) −10.5680 −0.843421 −0.421710 0.906731i \(-0.638570\pi\)
−0.421710 + 0.906731i \(0.638570\pi\)
\(158\) 0 0
\(159\) 43.3568 3.43842
\(160\) 0 0
\(161\) −25.6252 −2.01955
\(162\) 0 0
\(163\) 9.57594 0.750046 0.375023 0.927016i \(-0.377635\pi\)
0.375023 + 0.927016i \(0.377635\pi\)
\(164\) 0 0
\(165\) 6.31957 0.491978
\(166\) 0 0
\(167\) −11.0810 −0.857474 −0.428737 0.903429i \(-0.641041\pi\)
−0.428737 + 0.903429i \(0.641041\pi\)
\(168\) 0 0
\(169\) 3.70565 0.285050
\(170\) 0 0
\(171\) 7.39085 0.565192
\(172\) 0 0
\(173\) 6.79409 0.516545 0.258273 0.966072i \(-0.416847\pi\)
0.258273 + 0.966072i \(0.416847\pi\)
\(174\) 0 0
\(175\) 4.43700 0.335406
\(176\) 0 0
\(177\) 37.9610 2.85332
\(178\) 0 0
\(179\) 9.70621 0.725476 0.362738 0.931891i \(-0.381842\pi\)
0.362738 + 0.931891i \(0.381842\pi\)
\(180\) 0 0
\(181\) 5.42537 0.403264 0.201632 0.979461i \(-0.435375\pi\)
0.201632 + 0.979461i \(0.435375\pi\)
\(182\) 0 0
\(183\) 31.4635 2.32585
\(184\) 0 0
\(185\) −8.92869 −0.656450
\(186\) 0 0
\(187\) −6.84309 −0.500416
\(188\) 0 0
\(189\) −54.3013 −3.94984
\(190\) 0 0
\(191\) 8.12045 0.587575 0.293787 0.955871i \(-0.405084\pi\)
0.293787 + 0.955871i \(0.405084\pi\)
\(192\) 0 0
\(193\) −10.0898 −0.726283 −0.363142 0.931734i \(-0.618296\pi\)
−0.363142 + 0.931734i \(0.618296\pi\)
\(194\) 0 0
\(195\) 25.8297 1.84970
\(196\) 0 0
\(197\) 24.2693 1.72912 0.864559 0.502532i \(-0.167598\pi\)
0.864559 + 0.502532i \(0.167598\pi\)
\(198\) 0 0
\(199\) −3.49321 −0.247627 −0.123814 0.992305i \(-0.539512\pi\)
−0.123814 + 0.992305i \(0.539512\pi\)
\(200\) 0 0
\(201\) −29.6365 −2.09039
\(202\) 0 0
\(203\) −13.1845 −0.925369
\(204\) 0 0
\(205\) 1.90352 0.132947
\(206\) 0 0
\(207\) 49.3656 3.43115
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −19.2427 −1.32472 −0.662360 0.749186i \(-0.730445\pi\)
−0.662360 + 0.749186i \(0.730445\pi\)
\(212\) 0 0
\(213\) 25.7198 1.76229
\(214\) 0 0
\(215\) −12.6003 −0.859330
\(216\) 0 0
\(217\) −22.0368 −1.49596
\(218\) 0 0
\(219\) 12.9559 0.875478
\(220\) 0 0
\(221\) −27.9695 −1.88143
\(222\) 0 0
\(223\) 15.1209 1.01257 0.506286 0.862366i \(-0.331018\pi\)
0.506286 + 0.862366i \(0.331018\pi\)
\(224\) 0 0
\(225\) −8.54767 −0.569844
\(226\) 0 0
\(227\) −26.7270 −1.77393 −0.886966 0.461835i \(-0.847191\pi\)
−0.886966 + 0.461835i \(0.847191\pi\)
\(228\) 0 0
\(229\) 4.06443 0.268585 0.134292 0.990942i \(-0.457124\pi\)
0.134292 + 0.990942i \(0.457124\pi\)
\(230\) 0 0
\(231\) 12.3669 0.813684
\(232\) 0 0
\(233\) −7.23599 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(234\) 0 0
\(235\) 15.5289 1.01299
\(236\) 0 0
\(237\) −22.8139 −1.48193
\(238\) 0 0
\(239\) −23.8455 −1.54243 −0.771217 0.636572i \(-0.780352\pi\)
−0.771217 + 0.636572i \(0.780352\pi\)
\(240\) 0 0
\(241\) −9.90120 −0.637792 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(242\) 0 0
\(243\) 33.1360 2.12567
\(244\) 0 0
\(245\) −15.1326 −0.966784
\(246\) 0 0
\(247\) −4.08725 −0.260066
\(248\) 0 0
\(249\) −39.4672 −2.50113
\(250\) 0 0
\(251\) −2.21310 −0.139690 −0.0698448 0.997558i \(-0.522250\pi\)
−0.0698448 + 0.997558i \(0.522250\pi\)
\(252\) 0 0
\(253\) −6.67929 −0.419923
\(254\) 0 0
\(255\) −43.2454 −2.70813
\(256\) 0 0
\(257\) −23.9604 −1.49461 −0.747303 0.664484i \(-0.768652\pi\)
−0.747303 + 0.664484i \(0.768652\pi\)
\(258\) 0 0
\(259\) −17.4728 −1.08570
\(260\) 0 0
\(261\) 25.3993 1.57217
\(262\) 0 0
\(263\) −7.95541 −0.490552 −0.245276 0.969453i \(-0.578879\pi\)
−0.245276 + 0.969453i \(0.578879\pi\)
\(264\) 0 0
\(265\) −26.3691 −1.61984
\(266\) 0 0
\(267\) −24.4240 −1.49473
\(268\) 0 0
\(269\) −13.9571 −0.850979 −0.425490 0.904963i \(-0.639898\pi\)
−0.425490 + 0.904963i \(0.639898\pi\)
\(270\) 0 0
\(271\) 15.9194 0.967033 0.483516 0.875335i \(-0.339359\pi\)
0.483516 + 0.875335i \(0.339359\pi\)
\(272\) 0 0
\(273\) 50.5468 3.05923
\(274\) 0 0
\(275\) 1.15652 0.0697408
\(276\) 0 0
\(277\) −19.5440 −1.17428 −0.587141 0.809485i \(-0.699747\pi\)
−0.587141 + 0.809485i \(0.699747\pi\)
\(278\) 0 0
\(279\) 42.4529 2.54159
\(280\) 0 0
\(281\) 19.4707 1.16152 0.580761 0.814074i \(-0.302755\pi\)
0.580761 + 0.814074i \(0.302755\pi\)
\(282\) 0 0
\(283\) 4.48498 0.266604 0.133302 0.991075i \(-0.457442\pi\)
0.133302 + 0.991075i \(0.457442\pi\)
\(284\) 0 0
\(285\) −6.31957 −0.374339
\(286\) 0 0
\(287\) 3.72504 0.219882
\(288\) 0 0
\(289\) 29.8279 1.75458
\(290\) 0 0
\(291\) −55.8665 −3.27495
\(292\) 0 0
\(293\) 30.2958 1.76990 0.884951 0.465685i \(-0.154192\pi\)
0.884951 + 0.465685i \(0.154192\pi\)
\(294\) 0 0
\(295\) −23.0873 −1.34420
\(296\) 0 0
\(297\) −14.1538 −0.821288
\(298\) 0 0
\(299\) −27.2999 −1.57880
\(300\) 0 0
\(301\) −24.6577 −1.42125
\(302\) 0 0
\(303\) −16.8086 −0.965626
\(304\) 0 0
\(305\) −19.1357 −1.09571
\(306\) 0 0
\(307\) 4.31088 0.246035 0.123018 0.992404i \(-0.460743\pi\)
0.123018 + 0.992404i \(0.460743\pi\)
\(308\) 0 0
\(309\) −14.6921 −0.835804
\(310\) 0 0
\(311\) −18.7973 −1.06590 −0.532948 0.846148i \(-0.678916\pi\)
−0.532948 + 0.846148i \(0.678916\pi\)
\(312\) 0 0
\(313\) 12.5192 0.707627 0.353814 0.935316i \(-0.384885\pi\)
0.353814 + 0.935316i \(0.384885\pi\)
\(314\) 0 0
\(315\) 55.5895 3.13211
\(316\) 0 0
\(317\) 17.1197 0.961541 0.480770 0.876847i \(-0.340357\pi\)
0.480770 + 0.876847i \(0.340357\pi\)
\(318\) 0 0
\(319\) −3.43658 −0.192412
\(320\) 0 0
\(321\) −19.8415 −1.10744
\(322\) 0 0
\(323\) 6.84309 0.380760
\(324\) 0 0
\(325\) 4.72699 0.262206
\(326\) 0 0
\(327\) 13.7551 0.760660
\(328\) 0 0
\(329\) 30.3888 1.67539
\(330\) 0 0
\(331\) 10.9080 0.599556 0.299778 0.954009i \(-0.403087\pi\)
0.299778 + 0.954009i \(0.403087\pi\)
\(332\) 0 0
\(333\) 33.6604 1.84458
\(334\) 0 0
\(335\) 18.0245 0.984783
\(336\) 0 0
\(337\) 20.5700 1.12052 0.560260 0.828317i \(-0.310701\pi\)
0.560260 + 0.828317i \(0.310701\pi\)
\(338\) 0 0
\(339\) 55.3285 3.00503
\(340\) 0 0
\(341\) −5.74398 −0.311054
\(342\) 0 0
\(343\) −2.75771 −0.148903
\(344\) 0 0
\(345\) −42.2103 −2.27252
\(346\) 0 0
\(347\) −19.2051 −1.03098 −0.515492 0.856895i \(-0.672391\pi\)
−0.515492 + 0.856895i \(0.672391\pi\)
\(348\) 0 0
\(349\) 31.3696 1.67918 0.839588 0.543223i \(-0.182796\pi\)
0.839588 + 0.543223i \(0.182796\pi\)
\(350\) 0 0
\(351\) −57.8503 −3.08782
\(352\) 0 0
\(353\) −29.0842 −1.54800 −0.773998 0.633188i \(-0.781746\pi\)
−0.773998 + 0.633188i \(0.781746\pi\)
\(354\) 0 0
\(355\) −15.6424 −0.830214
\(356\) 0 0
\(357\) −84.6280 −4.47899
\(358\) 0 0
\(359\) −30.2593 −1.59703 −0.798514 0.601977i \(-0.794380\pi\)
−0.798514 + 0.601977i \(0.794380\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 3.22348 0.169189
\(364\) 0 0
\(365\) −7.87960 −0.412437
\(366\) 0 0
\(367\) −2.28222 −0.119131 −0.0595654 0.998224i \(-0.518971\pi\)
−0.0595654 + 0.998224i \(0.518971\pi\)
\(368\) 0 0
\(369\) −7.17610 −0.373573
\(370\) 0 0
\(371\) −51.6022 −2.67905
\(372\) 0 0
\(373\) 10.7646 0.557368 0.278684 0.960383i \(-0.410102\pi\)
0.278684 + 0.960383i \(0.410102\pi\)
\(374\) 0 0
\(375\) 38.9066 2.00913
\(376\) 0 0
\(377\) −14.0462 −0.723415
\(378\) 0 0
\(379\) 22.4988 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(380\) 0 0
\(381\) −7.72684 −0.395858
\(382\) 0 0
\(383\) −4.04990 −0.206940 −0.103470 0.994633i \(-0.532995\pi\)
−0.103470 + 0.994633i \(0.532995\pi\)
\(384\) 0 0
\(385\) −7.52140 −0.383326
\(386\) 0 0
\(387\) 47.5019 2.41466
\(388\) 0 0
\(389\) −10.9676 −0.556078 −0.278039 0.960570i \(-0.589684\pi\)
−0.278039 + 0.960570i \(0.589684\pi\)
\(390\) 0 0
\(391\) 45.7070 2.31150
\(392\) 0 0
\(393\) −21.9663 −1.10805
\(394\) 0 0
\(395\) 13.8751 0.698134
\(396\) 0 0
\(397\) −26.2611 −1.31801 −0.659004 0.752140i \(-0.729022\pi\)
−0.659004 + 0.752140i \(0.729022\pi\)
\(398\) 0 0
\(399\) −12.3669 −0.619121
\(400\) 0 0
\(401\) −26.5194 −1.32431 −0.662157 0.749365i \(-0.730359\pi\)
−0.662157 + 0.749365i \(0.730359\pi\)
\(402\) 0 0
\(403\) −23.4771 −1.16948
\(404\) 0 0
\(405\) −45.9774 −2.28463
\(406\) 0 0
\(407\) −4.55434 −0.225750
\(408\) 0 0
\(409\) 14.3338 0.708763 0.354381 0.935101i \(-0.384691\pi\)
0.354381 + 0.935101i \(0.384691\pi\)
\(410\) 0 0
\(411\) 67.6880 3.33880
\(412\) 0 0
\(413\) −45.1802 −2.22317
\(414\) 0 0
\(415\) 24.0034 1.17828
\(416\) 0 0
\(417\) −1.04219 −0.0510362
\(418\) 0 0
\(419\) 0.708329 0.0346042 0.0173021 0.999850i \(-0.494492\pi\)
0.0173021 + 0.999850i \(0.494492\pi\)
\(420\) 0 0
\(421\) −2.40268 −0.117100 −0.0585498 0.998284i \(-0.518648\pi\)
−0.0585498 + 0.998284i \(0.518648\pi\)
\(422\) 0 0
\(423\) −58.5426 −2.84644
\(424\) 0 0
\(425\) −7.91417 −0.383894
\(426\) 0 0
\(427\) −37.4471 −1.81219
\(428\) 0 0
\(429\) 13.1752 0.636105
\(430\) 0 0
\(431\) −7.38612 −0.355777 −0.177888 0.984051i \(-0.556927\pi\)
−0.177888 + 0.984051i \(0.556927\pi\)
\(432\) 0 0
\(433\) 7.68213 0.369180 0.184590 0.982816i \(-0.440904\pi\)
0.184590 + 0.982816i \(0.440904\pi\)
\(434\) 0 0
\(435\) −21.7177 −1.04129
\(436\) 0 0
\(437\) 6.67929 0.319514
\(438\) 0 0
\(439\) −14.2370 −0.679493 −0.339747 0.940517i \(-0.610341\pi\)
−0.339747 + 0.940517i \(0.610341\pi\)
\(440\) 0 0
\(441\) 57.0485 2.71660
\(442\) 0 0
\(443\) 0.846194 0.0402039 0.0201019 0.999798i \(-0.493601\pi\)
0.0201019 + 0.999798i \(0.493601\pi\)
\(444\) 0 0
\(445\) 14.8544 0.704164
\(446\) 0 0
\(447\) 40.0009 1.89198
\(448\) 0 0
\(449\) 12.8778 0.607742 0.303871 0.952713i \(-0.401721\pi\)
0.303871 + 0.952713i \(0.401721\pi\)
\(450\) 0 0
\(451\) 0.970944 0.0457200
\(452\) 0 0
\(453\) −20.6759 −0.971437
\(454\) 0 0
\(455\) −30.7419 −1.44120
\(456\) 0 0
\(457\) 31.4303 1.47025 0.735124 0.677932i \(-0.237124\pi\)
0.735124 + 0.677932i \(0.237124\pi\)
\(458\) 0 0
\(459\) 96.8559 4.52085
\(460\) 0 0
\(461\) 19.1045 0.889788 0.444894 0.895583i \(-0.353241\pi\)
0.444894 + 0.895583i \(0.353241\pi\)
\(462\) 0 0
\(463\) 17.8763 0.830782 0.415391 0.909643i \(-0.363645\pi\)
0.415391 + 0.909643i \(0.363645\pi\)
\(464\) 0 0
\(465\) −36.2995 −1.68335
\(466\) 0 0
\(467\) 18.4325 0.852956 0.426478 0.904498i \(-0.359754\pi\)
0.426478 + 0.904498i \(0.359754\pi\)
\(468\) 0 0
\(469\) 35.2726 1.62874
\(470\) 0 0
\(471\) −34.0659 −1.56967
\(472\) 0 0
\(473\) −6.42713 −0.295520
\(474\) 0 0
\(475\) −1.15652 −0.0530648
\(476\) 0 0
\(477\) 99.4092 4.55163
\(478\) 0 0
\(479\) 15.2766 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(480\) 0 0
\(481\) −18.6147 −0.848759
\(482\) 0 0
\(483\) −82.6023 −3.75853
\(484\) 0 0
\(485\) 33.9773 1.54283
\(486\) 0 0
\(487\) −34.8012 −1.57699 −0.788496 0.615040i \(-0.789140\pi\)
−0.788496 + 0.615040i \(0.789140\pi\)
\(488\) 0 0
\(489\) 30.8679 1.39589
\(490\) 0 0
\(491\) 19.8170 0.894327 0.447164 0.894452i \(-0.352434\pi\)
0.447164 + 0.894452i \(0.352434\pi\)
\(492\) 0 0
\(493\) 23.5168 1.05915
\(494\) 0 0
\(495\) 14.4896 0.651259
\(496\) 0 0
\(497\) −30.6111 −1.37309
\(498\) 0 0
\(499\) 18.3775 0.822692 0.411346 0.911479i \(-0.365059\pi\)
0.411346 + 0.911479i \(0.365059\pi\)
\(500\) 0 0
\(501\) −35.7195 −1.59583
\(502\) 0 0
\(503\) −14.6934 −0.655148 −0.327574 0.944826i \(-0.606231\pi\)
−0.327574 + 0.944826i \(0.606231\pi\)
\(504\) 0 0
\(505\) 10.2227 0.454906
\(506\) 0 0
\(507\) 11.9451 0.530500
\(508\) 0 0
\(509\) −29.1430 −1.29174 −0.645870 0.763448i \(-0.723505\pi\)
−0.645870 + 0.763448i \(0.723505\pi\)
\(510\) 0 0
\(511\) −15.4198 −0.682131
\(512\) 0 0
\(513\) 14.1538 0.624907
\(514\) 0 0
\(515\) 8.93553 0.393747
\(516\) 0 0
\(517\) 7.92096 0.348363
\(518\) 0 0
\(519\) 21.9007 0.961332
\(520\) 0 0
\(521\) −2.48195 −0.108736 −0.0543680 0.998521i \(-0.517314\pi\)
−0.0543680 + 0.998521i \(0.517314\pi\)
\(522\) 0 0
\(523\) 0.967311 0.0422976 0.0211488 0.999776i \(-0.493268\pi\)
0.0211488 + 0.999776i \(0.493268\pi\)
\(524\) 0 0
\(525\) 14.3026 0.624217
\(526\) 0 0
\(527\) 39.3066 1.71222
\(528\) 0 0
\(529\) 21.6129 0.939691
\(530\) 0 0
\(531\) 87.0374 3.77710
\(532\) 0 0
\(533\) 3.96849 0.171895
\(534\) 0 0
\(535\) 12.0673 0.521715
\(536\) 0 0
\(537\) 31.2878 1.35017
\(538\) 0 0
\(539\) −7.71881 −0.332473
\(540\) 0 0
\(541\) −24.9523 −1.07278 −0.536391 0.843970i \(-0.680213\pi\)
−0.536391 + 0.843970i \(0.680213\pi\)
\(542\) 0 0
\(543\) 17.4886 0.750507
\(544\) 0 0
\(545\) −8.36568 −0.358346
\(546\) 0 0
\(547\) −39.1506 −1.67396 −0.836979 0.547234i \(-0.815681\pi\)
−0.836979 + 0.547234i \(0.815681\pi\)
\(548\) 0 0
\(549\) 72.1399 3.07886
\(550\) 0 0
\(551\) 3.43658 0.146403
\(552\) 0 0
\(553\) 27.1526 1.15465
\(554\) 0 0
\(555\) −28.7815 −1.22171
\(556\) 0 0
\(557\) −13.2091 −0.559686 −0.279843 0.960046i \(-0.590282\pi\)
−0.279843 + 0.960046i \(0.590282\pi\)
\(558\) 0 0
\(559\) −26.2693 −1.11107
\(560\) 0 0
\(561\) −22.0586 −0.931315
\(562\) 0 0
\(563\) −8.54591 −0.360167 −0.180084 0.983651i \(-0.557637\pi\)
−0.180084 + 0.983651i \(0.557637\pi\)
\(564\) 0 0
\(565\) −33.6501 −1.41567
\(566\) 0 0
\(567\) −89.9742 −3.77856
\(568\) 0 0
\(569\) −32.4177 −1.35902 −0.679511 0.733665i \(-0.737808\pi\)
−0.679511 + 0.733665i \(0.737808\pi\)
\(570\) 0 0
\(571\) 26.1261 1.09334 0.546672 0.837347i \(-0.315895\pi\)
0.546672 + 0.837347i \(0.315895\pi\)
\(572\) 0 0
\(573\) 26.1761 1.09352
\(574\) 0 0
\(575\) −7.72473 −0.322144
\(576\) 0 0
\(577\) 41.4296 1.72474 0.862369 0.506281i \(-0.168980\pi\)
0.862369 + 0.506281i \(0.168980\pi\)
\(578\) 0 0
\(579\) −32.5245 −1.35167
\(580\) 0 0
\(581\) 46.9729 1.94876
\(582\) 0 0
\(583\) −13.4503 −0.557055
\(584\) 0 0
\(585\) 59.2227 2.44856
\(586\) 0 0
\(587\) −22.4492 −0.926577 −0.463289 0.886207i \(-0.653331\pi\)
−0.463289 + 0.886207i \(0.653331\pi\)
\(588\) 0 0
\(589\) 5.74398 0.236677
\(590\) 0 0
\(591\) 78.2318 3.21802
\(592\) 0 0
\(593\) −9.79617 −0.402281 −0.201140 0.979562i \(-0.564465\pi\)
−0.201140 + 0.979562i \(0.564465\pi\)
\(594\) 0 0
\(595\) 51.4696 2.11005
\(596\) 0 0
\(597\) −11.2603 −0.460854
\(598\) 0 0
\(599\) 2.65509 0.108484 0.0542420 0.998528i \(-0.482726\pi\)
0.0542420 + 0.998528i \(0.482726\pi\)
\(600\) 0 0
\(601\) 25.1960 1.02777 0.513884 0.857860i \(-0.328206\pi\)
0.513884 + 0.857860i \(0.328206\pi\)
\(602\) 0 0
\(603\) −67.9509 −2.76717
\(604\) 0 0
\(605\) −1.96048 −0.0797048
\(606\) 0 0
\(607\) 33.4327 1.35699 0.678496 0.734605i \(-0.262632\pi\)
0.678496 + 0.734605i \(0.262632\pi\)
\(608\) 0 0
\(609\) −42.5000 −1.72219
\(610\) 0 0
\(611\) 32.3750 1.30975
\(612\) 0 0
\(613\) −17.3061 −0.698987 −0.349494 0.936939i \(-0.613646\pi\)
−0.349494 + 0.936939i \(0.613646\pi\)
\(614\) 0 0
\(615\) 6.13595 0.247425
\(616\) 0 0
\(617\) 14.5384 0.585294 0.292647 0.956221i \(-0.405464\pi\)
0.292647 + 0.956221i \(0.405464\pi\)
\(618\) 0 0
\(619\) 47.0886 1.89265 0.946325 0.323217i \(-0.104764\pi\)
0.946325 + 0.323217i \(0.104764\pi\)
\(620\) 0 0
\(621\) 94.5375 3.79366
\(622\) 0 0
\(623\) 29.0688 1.16462
\(624\) 0 0
\(625\) −17.8799 −0.715194
\(626\) 0 0
\(627\) −3.22348 −0.128734
\(628\) 0 0
\(629\) 31.1658 1.24266
\(630\) 0 0
\(631\) 18.2723 0.727411 0.363705 0.931514i \(-0.381512\pi\)
0.363705 + 0.931514i \(0.381512\pi\)
\(632\) 0 0
\(633\) −62.0284 −2.46541
\(634\) 0 0
\(635\) 4.69936 0.186488
\(636\) 0 0
\(637\) −31.5487 −1.25001
\(638\) 0 0
\(639\) 58.9707 2.33285
\(640\) 0 0
\(641\) 19.2261 0.759384 0.379692 0.925113i \(-0.376030\pi\)
0.379692 + 0.925113i \(0.376030\pi\)
\(642\) 0 0
\(643\) −18.4887 −0.729123 −0.364561 0.931179i \(-0.618781\pi\)
−0.364561 + 0.931179i \(0.618781\pi\)
\(644\) 0 0
\(645\) −40.6167 −1.59928
\(646\) 0 0
\(647\) 6.57015 0.258299 0.129150 0.991625i \(-0.458775\pi\)
0.129150 + 0.991625i \(0.458775\pi\)
\(648\) 0 0
\(649\) −11.7764 −0.462263
\(650\) 0 0
\(651\) −71.0354 −2.78410
\(652\) 0 0
\(653\) 7.56134 0.295898 0.147949 0.988995i \(-0.452733\pi\)
0.147949 + 0.988995i \(0.452733\pi\)
\(654\) 0 0
\(655\) 13.3596 0.522002
\(656\) 0 0
\(657\) 29.7055 1.15892
\(658\) 0 0
\(659\) 28.9806 1.12892 0.564461 0.825460i \(-0.309084\pi\)
0.564461 + 0.825460i \(0.309084\pi\)
\(660\) 0 0
\(661\) 35.3680 1.37566 0.687828 0.725873i \(-0.258564\pi\)
0.687828 + 0.725873i \(0.258564\pi\)
\(662\) 0 0
\(663\) −90.1591 −3.50149
\(664\) 0 0
\(665\) 7.52140 0.291667
\(666\) 0 0
\(667\) 22.9539 0.888780
\(668\) 0 0
\(669\) 48.7421 1.88448
\(670\) 0 0
\(671\) −9.76071 −0.376808
\(672\) 0 0
\(673\) 8.20026 0.316097 0.158048 0.987431i \(-0.449480\pi\)
0.158048 + 0.987431i \(0.449480\pi\)
\(674\) 0 0
\(675\) −16.3692 −0.630050
\(676\) 0 0
\(677\) −11.6685 −0.448455 −0.224228 0.974537i \(-0.571986\pi\)
−0.224228 + 0.974537i \(0.571986\pi\)
\(678\) 0 0
\(679\) 66.4909 2.55169
\(680\) 0 0
\(681\) −86.1540 −3.30143
\(682\) 0 0
\(683\) 6.25446 0.239320 0.119660 0.992815i \(-0.461820\pi\)
0.119660 + 0.992815i \(0.461820\pi\)
\(684\) 0 0
\(685\) −41.1669 −1.57291
\(686\) 0 0
\(687\) 13.1016 0.499858
\(688\) 0 0
\(689\) −54.9748 −2.09437
\(690\) 0 0
\(691\) −34.7580 −1.32226 −0.661128 0.750273i \(-0.729922\pi\)
−0.661128 + 0.750273i \(0.729922\pi\)
\(692\) 0 0
\(693\) 28.3551 1.07712
\(694\) 0 0
\(695\) 0.633845 0.0240431
\(696\) 0 0
\(697\) −6.64426 −0.251669
\(698\) 0 0
\(699\) −23.3251 −0.882236
\(700\) 0 0
\(701\) 26.6407 1.00621 0.503103 0.864227i \(-0.332192\pi\)
0.503103 + 0.864227i \(0.332192\pi\)
\(702\) 0 0
\(703\) 4.55434 0.171770
\(704\) 0 0
\(705\) 50.0571 1.88526
\(706\) 0 0
\(707\) 20.0051 0.752370
\(708\) 0 0
\(709\) −24.4732 −0.919110 −0.459555 0.888149i \(-0.651991\pi\)
−0.459555 + 0.888149i \(0.651991\pi\)
\(710\) 0 0
\(711\) −52.3081 −1.96171
\(712\) 0 0
\(713\) 38.3657 1.43681
\(714\) 0 0
\(715\) −8.01298 −0.299668
\(716\) 0 0
\(717\) −76.8654 −2.87059
\(718\) 0 0
\(719\) 9.82314 0.366341 0.183171 0.983081i \(-0.441364\pi\)
0.183171 + 0.983081i \(0.441364\pi\)
\(720\) 0 0
\(721\) 17.4862 0.651219
\(722\) 0 0
\(723\) −31.9164 −1.18698
\(724\) 0 0
\(725\) −3.97448 −0.147608
\(726\) 0 0
\(727\) −38.8971 −1.44262 −0.721308 0.692615i \(-0.756459\pi\)
−0.721308 + 0.692615i \(0.756459\pi\)
\(728\) 0 0
\(729\) 36.4570 1.35026
\(730\) 0 0
\(731\) 43.9814 1.62671
\(732\) 0 0
\(733\) −31.2696 −1.15497 −0.577485 0.816402i \(-0.695966\pi\)
−0.577485 + 0.816402i \(0.695966\pi\)
\(734\) 0 0
\(735\) −48.7796 −1.79926
\(736\) 0 0
\(737\) 9.19392 0.338662
\(738\) 0 0
\(739\) −20.2935 −0.746510 −0.373255 0.927729i \(-0.621758\pi\)
−0.373255 + 0.927729i \(0.621758\pi\)
\(740\) 0 0
\(741\) −13.1752 −0.484003
\(742\) 0 0
\(743\) −30.7900 −1.12958 −0.564788 0.825236i \(-0.691042\pi\)
−0.564788 + 0.825236i \(0.691042\pi\)
\(744\) 0 0
\(745\) −24.3280 −0.891308
\(746\) 0 0
\(747\) −90.4909 −3.31089
\(748\) 0 0
\(749\) 23.6148 0.862867
\(750\) 0 0
\(751\) −24.6425 −0.899218 −0.449609 0.893226i \(-0.648437\pi\)
−0.449609 + 0.893226i \(0.648437\pi\)
\(752\) 0 0
\(753\) −7.13389 −0.259973
\(754\) 0 0
\(755\) 12.5748 0.457643
\(756\) 0 0
\(757\) −18.2119 −0.661924 −0.330962 0.943644i \(-0.607373\pi\)
−0.330962 + 0.943644i \(0.607373\pi\)
\(758\) 0 0
\(759\) −21.5306 −0.781510
\(760\) 0 0
\(761\) −40.7014 −1.47542 −0.737712 0.675115i \(-0.764094\pi\)
−0.737712 + 0.675115i \(0.764094\pi\)
\(762\) 0 0
\(763\) −16.3710 −0.592670
\(764\) 0 0
\(765\) −99.1537 −3.58491
\(766\) 0 0
\(767\) −48.1330 −1.73798
\(768\) 0 0
\(769\) −18.3969 −0.663409 −0.331704 0.943383i \(-0.607624\pi\)
−0.331704 + 0.943383i \(0.607624\pi\)
\(770\) 0 0
\(771\) −77.2358 −2.78158
\(772\) 0 0
\(773\) −24.0262 −0.864162 −0.432081 0.901835i \(-0.642221\pi\)
−0.432081 + 0.901835i \(0.642221\pi\)
\(774\) 0 0
\(775\) −6.64303 −0.238625
\(776\) 0 0
\(777\) −56.3232 −2.02058
\(778\) 0 0
\(779\) −0.970944 −0.0347877
\(780\) 0 0
\(781\) −7.97888 −0.285507
\(782\) 0 0
\(783\) 48.6408 1.73828
\(784\) 0 0
\(785\) 20.7184 0.739472
\(786\) 0 0
\(787\) 47.4174 1.69025 0.845125 0.534569i \(-0.179526\pi\)
0.845125 + 0.534569i \(0.179526\pi\)
\(788\) 0 0
\(789\) −25.6441 −0.912956
\(790\) 0 0
\(791\) −65.8506 −2.34138
\(792\) 0 0
\(793\) −39.8945 −1.41670
\(794\) 0 0
\(795\) −85.0002 −3.01465
\(796\) 0 0
\(797\) 40.3765 1.43021 0.715105 0.699017i \(-0.246379\pi\)
0.715105 + 0.699017i \(0.246379\pi\)
\(798\) 0 0
\(799\) −54.2038 −1.91759
\(800\) 0 0
\(801\) −55.9997 −1.97865
\(802\) 0 0
\(803\) −4.01922 −0.141835
\(804\) 0 0
\(805\) 50.2376 1.77064
\(806\) 0 0
\(807\) −44.9905 −1.58374
\(808\) 0 0
\(809\) −34.4740 −1.21204 −0.606020 0.795449i \(-0.707235\pi\)
−0.606020 + 0.795449i \(0.707235\pi\)
\(810\) 0 0
\(811\) 19.8683 0.697670 0.348835 0.937184i \(-0.386577\pi\)
0.348835 + 0.937184i \(0.386577\pi\)
\(812\) 0 0
\(813\) 51.3158 1.79973
\(814\) 0 0
\(815\) −18.7734 −0.657605
\(816\) 0 0
\(817\) 6.42713 0.224857
\(818\) 0 0
\(819\) 115.894 4.04968
\(820\) 0 0
\(821\) −49.4349 −1.72529 −0.862644 0.505811i \(-0.831193\pi\)
−0.862644 + 0.505811i \(0.831193\pi\)
\(822\) 0 0
\(823\) −25.8830 −0.902224 −0.451112 0.892467i \(-0.648972\pi\)
−0.451112 + 0.892467i \(0.648972\pi\)
\(824\) 0 0
\(825\) 3.72802 0.129793
\(826\) 0 0
\(827\) 17.0775 0.593844 0.296922 0.954902i \(-0.404040\pi\)
0.296922 + 0.954902i \(0.404040\pi\)
\(828\) 0 0
\(829\) 1.94722 0.0676298 0.0338149 0.999428i \(-0.489234\pi\)
0.0338149 + 0.999428i \(0.489234\pi\)
\(830\) 0 0
\(831\) −62.9996 −2.18543
\(832\) 0 0
\(833\) 52.8205 1.83012
\(834\) 0 0
\(835\) 21.7241 0.751793
\(836\) 0 0
\(837\) 81.2993 2.81012
\(838\) 0 0
\(839\) 8.09414 0.279441 0.139720 0.990191i \(-0.455380\pi\)
0.139720 + 0.990191i \(0.455380\pi\)
\(840\) 0 0
\(841\) −17.1899 −0.592755
\(842\) 0 0
\(843\) 62.7633 2.16168
\(844\) 0 0
\(845\) −7.26484 −0.249918
\(846\) 0 0
\(847\) −3.83651 −0.131824
\(848\) 0 0
\(849\) 14.4573 0.496172
\(850\) 0 0
\(851\) 30.4197 1.04278
\(852\) 0 0
\(853\) 47.3021 1.61959 0.809797 0.586710i \(-0.199577\pi\)
0.809797 + 0.586710i \(0.199577\pi\)
\(854\) 0 0
\(855\) −14.4896 −0.495534
\(856\) 0 0
\(857\) −18.2751 −0.624265 −0.312133 0.950039i \(-0.601043\pi\)
−0.312133 + 0.950039i \(0.601043\pi\)
\(858\) 0 0
\(859\) 58.5032 1.99610 0.998051 0.0624014i \(-0.0198759\pi\)
0.998051 + 0.0624014i \(0.0198759\pi\)
\(860\) 0 0
\(861\) 12.0076 0.409218
\(862\) 0 0
\(863\) 27.5843 0.938981 0.469491 0.882938i \(-0.344438\pi\)
0.469491 + 0.882938i \(0.344438\pi\)
\(864\) 0 0
\(865\) −13.3197 −0.452883
\(866\) 0 0
\(867\) 96.1497 3.26542
\(868\) 0 0
\(869\) 7.07742 0.240085
\(870\) 0 0
\(871\) 37.5779 1.27328
\(872\) 0 0
\(873\) −128.092 −4.33524
\(874\) 0 0
\(875\) −46.3056 −1.56542
\(876\) 0 0
\(877\) 10.7574 0.363252 0.181626 0.983368i \(-0.441864\pi\)
0.181626 + 0.983368i \(0.441864\pi\)
\(878\) 0 0
\(879\) 97.6581 3.29393
\(880\) 0 0
\(881\) −46.3314 −1.56094 −0.780472 0.625191i \(-0.785021\pi\)
−0.780472 + 0.625191i \(0.785021\pi\)
\(882\) 0 0
\(883\) −15.2331 −0.512636 −0.256318 0.966593i \(-0.582509\pi\)
−0.256318 + 0.966593i \(0.582509\pi\)
\(884\) 0 0
\(885\) −74.4217 −2.50166
\(886\) 0 0
\(887\) −23.6277 −0.793341 −0.396671 0.917961i \(-0.629835\pi\)
−0.396671 + 0.917961i \(0.629835\pi\)
\(888\) 0 0
\(889\) 9.19629 0.308434
\(890\) 0 0
\(891\) −23.4521 −0.785675
\(892\) 0 0
\(893\) −7.92096 −0.265065
\(894\) 0 0
\(895\) −19.0288 −0.636064
\(896\) 0 0
\(897\) −88.0009 −2.93827
\(898\) 0 0
\(899\) 19.7397 0.658355
\(900\) 0 0
\(901\) 92.0417 3.06635
\(902\) 0 0
\(903\) −79.4838 −2.64506
\(904\) 0 0
\(905\) −10.6363 −0.353563
\(906\) 0 0
\(907\) 16.7451 0.556013 0.278006 0.960579i \(-0.410326\pi\)
0.278006 + 0.960579i \(0.410326\pi\)
\(908\) 0 0
\(909\) −38.5389 −1.27825
\(910\) 0 0
\(911\) −3.40911 −0.112949 −0.0564744 0.998404i \(-0.517986\pi\)
−0.0564744 + 0.998404i \(0.517986\pi\)
\(912\) 0 0
\(913\) 12.2436 0.405205
\(914\) 0 0
\(915\) −61.6835 −2.03919
\(916\) 0 0
\(917\) 26.1437 0.863341
\(918\) 0 0
\(919\) 8.14196 0.268578 0.134289 0.990942i \(-0.457125\pi\)
0.134289 + 0.990942i \(0.457125\pi\)
\(920\) 0 0
\(921\) 13.8961 0.457891
\(922\) 0 0
\(923\) −32.6117 −1.07343
\(924\) 0 0
\(925\) −5.26718 −0.173184
\(926\) 0 0
\(927\) −33.6862 −1.10640
\(928\) 0 0
\(929\) 44.6605 1.46526 0.732632 0.680625i \(-0.238292\pi\)
0.732632 + 0.680625i \(0.238292\pi\)
\(930\) 0 0
\(931\) 7.71881 0.252974
\(932\) 0 0
\(933\) −60.5927 −1.98372
\(934\) 0 0
\(935\) 13.4157 0.438742
\(936\) 0 0
\(937\) 27.0422 0.883429 0.441714 0.897156i \(-0.354370\pi\)
0.441714 + 0.897156i \(0.354370\pi\)
\(938\) 0 0
\(939\) 40.3555 1.31695
\(940\) 0 0
\(941\) 23.5097 0.766395 0.383197 0.923666i \(-0.374823\pi\)
0.383197 + 0.923666i \(0.374823\pi\)
\(942\) 0 0
\(943\) −6.48521 −0.211188
\(944\) 0 0
\(945\) 106.457 3.46303
\(946\) 0 0
\(947\) −42.7025 −1.38765 −0.693823 0.720146i \(-0.744075\pi\)
−0.693823 + 0.720146i \(0.744075\pi\)
\(948\) 0 0
\(949\) −16.4276 −0.533262
\(950\) 0 0
\(951\) 55.1852 1.78950
\(952\) 0 0
\(953\) −25.2708 −0.818600 −0.409300 0.912400i \(-0.634227\pi\)
−0.409300 + 0.912400i \(0.634227\pi\)
\(954\) 0 0
\(955\) −15.9200 −0.515158
\(956\) 0 0
\(957\) −11.0778 −0.358093
\(958\) 0 0
\(959\) −80.5606 −2.60144
\(960\) 0 0
\(961\) 1.99332 0.0643006
\(962\) 0 0
\(963\) −45.4928 −1.46598
\(964\) 0 0
\(965\) 19.7809 0.636771
\(966\) 0 0
\(967\) −7.78722 −0.250420 −0.125210 0.992130i \(-0.539960\pi\)
−0.125210 + 0.992130i \(0.539960\pi\)
\(968\) 0 0
\(969\) 22.0586 0.708624
\(970\) 0 0
\(971\) −0.0683640 −0.00219390 −0.00109695 0.999999i \(-0.500349\pi\)
−0.00109695 + 0.999999i \(0.500349\pi\)
\(972\) 0 0
\(973\) 1.24039 0.0397650
\(974\) 0 0
\(975\) 15.2374 0.487987
\(976\) 0 0
\(977\) −28.2460 −0.903668 −0.451834 0.892102i \(-0.649230\pi\)
−0.451834 + 0.892102i \(0.649230\pi\)
\(978\) 0 0
\(979\) 7.57690 0.242159
\(980\) 0 0
\(981\) 31.5379 1.00693
\(982\) 0 0
\(983\) −60.3323 −1.92430 −0.962151 0.272518i \(-0.912143\pi\)
−0.962151 + 0.272518i \(0.912143\pi\)
\(984\) 0 0
\(985\) −47.5795 −1.51601
\(986\) 0 0
\(987\) 97.9579 3.11804
\(988\) 0 0
\(989\) 42.9286 1.36505
\(990\) 0 0
\(991\) −12.9051 −0.409944 −0.204972 0.978768i \(-0.565710\pi\)
−0.204972 + 0.978768i \(0.565710\pi\)
\(992\) 0 0
\(993\) 35.1617 1.11582
\(994\) 0 0
\(995\) 6.84837 0.217108
\(996\) 0 0
\(997\) −7.28960 −0.230864 −0.115432 0.993315i \(-0.536825\pi\)
−0.115432 + 0.993315i \(0.536825\pi\)
\(998\) 0 0
\(999\) 64.4613 2.03947
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 3344.2.a.bb.1.9 9
4.3 odd 2 1672.2.a.k.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.k.1.1 9 4.3 odd 2
3344.2.a.bb.1.9 9 1.1 even 1 trivial