Properties

Label 3072.2.d.j.1537.6
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3072,2,Mod(1537,3072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3072.1537");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.6
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.j.1537.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +0.331821i q^{5} +3.08239 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +0.331821i q^{5} +3.08239 q^{7} -1.00000 q^{9} +3.69552i q^{11} +4.64047i q^{13} -0.331821 q^{15} +6.52395 q^{17} -0.867091i q^{19} +3.08239i q^{21} +4.00000 q^{23} +4.88989 q^{25} -1.00000i q^{27} -4.89443i q^{29} -6.14386 q^{31} -3.69552 q^{33} +1.02280i q^{35} +3.64725i q^{37} -4.64047 q^{39} -3.92856 q^{41} -3.92856i q^{43} -0.331821i q^{45} +1.65685 q^{47} +2.50114 q^{49} +6.52395i q^{51} -0.564862i q^{53} -1.22625 q^{55} +0.867091 q^{57} +6.59539i q^{59} -14.8052i q^{61} -3.08239 q^{63} -1.53981 q^{65} +13.9864i q^{67} +4.00000i q^{69} +7.49207 q^{71} -5.62408 q^{73} +4.88989i q^{75} +11.3910i q^{77} -14.9040 q^{79} +1.00000 q^{81} +9.35237i q^{83} +2.16478i q^{85} +4.89443 q^{87} +18.1094 q^{89} +14.3037i q^{91} -6.14386i q^{93} +0.287719 q^{95} -17.0479 q^{97} -3.69552i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 8 q^{9} + 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{39} - 32 q^{47} + 8 q^{49} + 32 q^{55} - 16 q^{63} - 16 q^{65} + 32 q^{71} + 16 q^{73} - 48 q^{79} + 8 q^{81} + 16 q^{89} - 64 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0.331821i 0.148395i 0.997244 + 0.0741975i \(0.0236395\pi\)
−0.997244 + 0.0741975i \(0.976360\pi\)
\(6\) 0 0
\(7\) 3.08239 1.16503 0.582517 0.812818i \(-0.302068\pi\)
0.582517 + 0.812818i \(0.302068\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.69552i 1.11424i 0.830432 + 0.557120i \(0.188094\pi\)
−0.830432 + 0.557120i \(0.811906\pi\)
\(12\) 0 0
\(13\) 4.64047i 1.28703i 0.765432 + 0.643517i \(0.222525\pi\)
−0.765432 + 0.643517i \(0.777475\pi\)
\(14\) 0 0
\(15\) −0.331821 −0.0856759
\(16\) 0 0
\(17\) 6.52395 1.58229 0.791145 0.611629i \(-0.209486\pi\)
0.791145 + 0.611629i \(0.209486\pi\)
\(18\) 0 0
\(19\) − 0.867091i − 0.198924i −0.995041 0.0994622i \(-0.968288\pi\)
0.995041 0.0994622i \(-0.0317122\pi\)
\(20\) 0 0
\(21\) 3.08239i 0.672633i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 4.88989 0.977979
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 4.89443i − 0.908873i −0.890779 0.454436i \(-0.849841\pi\)
0.890779 0.454436i \(-0.150159\pi\)
\(30\) 0 0
\(31\) −6.14386 −1.10347 −0.551735 0.834020i \(-0.686034\pi\)
−0.551735 + 0.834020i \(0.686034\pi\)
\(32\) 0 0
\(33\) −3.69552 −0.643307
\(34\) 0 0
\(35\) 1.02280i 0.172885i
\(36\) 0 0
\(37\) 3.64725i 0.599605i 0.954001 + 0.299802i \(0.0969208\pi\)
−0.954001 + 0.299802i \(0.903079\pi\)
\(38\) 0 0
\(39\) −4.64047 −0.743069
\(40\) 0 0
\(41\) −3.92856 −0.613538 −0.306769 0.951784i \(-0.599248\pi\)
−0.306769 + 0.951784i \(0.599248\pi\)
\(42\) 0 0
\(43\) − 3.92856i − 0.599100i −0.954081 0.299550i \(-0.903164\pi\)
0.954081 0.299550i \(-0.0968365\pi\)
\(44\) 0 0
\(45\) − 0.331821i − 0.0494650i
\(46\) 0 0
\(47\) 1.65685 0.241677 0.120839 0.992672i \(-0.461442\pi\)
0.120839 + 0.992672i \(0.461442\pi\)
\(48\) 0 0
\(49\) 2.50114 0.357306
\(50\) 0 0
\(51\) 6.52395i 0.913535i
\(52\) 0 0
\(53\) − 0.564862i − 0.0775897i −0.999247 0.0387949i \(-0.987648\pi\)
0.999247 0.0387949i \(-0.0123519\pi\)
\(54\) 0 0
\(55\) −1.22625 −0.165348
\(56\) 0 0
\(57\) 0.867091 0.114849
\(58\) 0 0
\(59\) 6.59539i 0.858646i 0.903151 + 0.429323i \(0.141248\pi\)
−0.903151 + 0.429323i \(0.858752\pi\)
\(60\) 0 0
\(61\) − 14.8052i − 1.89562i −0.318839 0.947809i \(-0.603293\pi\)
0.318839 0.947809i \(-0.396707\pi\)
\(62\) 0 0
\(63\) −3.08239 −0.388345
\(64\) 0 0
\(65\) −1.53981 −0.190989
\(66\) 0 0
\(67\) 13.9864i 1.70871i 0.519687 + 0.854357i \(0.326049\pi\)
−0.519687 + 0.854357i \(0.673951\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 7.49207 0.889145 0.444573 0.895743i \(-0.353356\pi\)
0.444573 + 0.895743i \(0.353356\pi\)
\(72\) 0 0
\(73\) −5.62408 −0.658248 −0.329124 0.944287i \(-0.606753\pi\)
−0.329124 + 0.944287i \(0.606753\pi\)
\(74\) 0 0
\(75\) 4.88989i 0.564636i
\(76\) 0 0
\(77\) 11.3910i 1.29813i
\(78\) 0 0
\(79\) −14.9040 −1.67683 −0.838417 0.545029i \(-0.816519\pi\)
−0.838417 + 0.545029i \(0.816519\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.35237i 1.02656i 0.858222 + 0.513278i \(0.171569\pi\)
−0.858222 + 0.513278i \(0.828431\pi\)
\(84\) 0 0
\(85\) 2.16478i 0.234804i
\(86\) 0 0
\(87\) 4.89443 0.524738
\(88\) 0 0
\(89\) 18.1094 1.91959 0.959794 0.280705i \(-0.0905683\pi\)
0.959794 + 0.280705i \(0.0905683\pi\)
\(90\) 0 0
\(91\) 14.3037i 1.49944i
\(92\) 0 0
\(93\) − 6.14386i − 0.637089i
\(94\) 0 0
\(95\) 0.287719 0.0295194
\(96\) 0 0
\(97\) −17.0479 −1.73095 −0.865476 0.500951i \(-0.832984\pi\)
−0.865476 + 0.500951i \(0.832984\pi\)
\(98\) 0 0
\(99\) − 3.69552i − 0.371414i
\(100\) 0 0
\(101\) − 12.1535i − 1.20931i −0.796486 0.604657i \(-0.793310\pi\)
0.796486 0.604657i \(-0.206690\pi\)
\(102\) 0 0
\(103\) 17.1043 1.68534 0.842668 0.538433i \(-0.180984\pi\)
0.842668 + 0.538433i \(0.180984\pi\)
\(104\) 0 0
\(105\) −1.02280 −0.0998154
\(106\) 0 0
\(107\) 18.3752i 1.77640i 0.459462 + 0.888198i \(0.348042\pi\)
−0.459462 + 0.888198i \(0.651958\pi\)
\(108\) 0 0
\(109\) − 7.84482i − 0.751397i −0.926742 0.375699i \(-0.877403\pi\)
0.926742 0.375699i \(-0.122597\pi\)
\(110\) 0 0
\(111\) −3.64725 −0.346182
\(112\) 0 0
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) 1.32729i 0.123770i
\(116\) 0 0
\(117\) − 4.64047i − 0.429011i
\(118\) 0 0
\(119\) 20.1094 1.84342
\(120\) 0 0
\(121\) −2.65685 −0.241532
\(122\) 0 0
\(123\) − 3.92856i − 0.354226i
\(124\) 0 0
\(125\) 3.28168i 0.293522i
\(126\) 0 0
\(127\) 0.638213 0.0566322 0.0283161 0.999599i \(-0.490985\pi\)
0.0283161 + 0.999599i \(0.490985\pi\)
\(128\) 0 0
\(129\) 3.92856 0.345890
\(130\) 0 0
\(131\) − 9.51397i − 0.831240i −0.909538 0.415620i \(-0.863565\pi\)
0.909538 0.415620i \(-0.136435\pi\)
\(132\) 0 0
\(133\) − 2.67271i − 0.231754i
\(134\) 0 0
\(135\) 0.331821 0.0285586
\(136\) 0 0
\(137\) 13.9150 1.18884 0.594419 0.804156i \(-0.297382\pi\)
0.594419 + 0.804156i \(0.297382\pi\)
\(138\) 0 0
\(139\) − 0.0773278i − 0.00655886i −0.999995 0.00327943i \(-0.998956\pi\)
0.999995 0.00327943i \(-0.00104388\pi\)
\(140\) 0 0
\(141\) 1.65685i 0.139532i
\(142\) 0 0
\(143\) −17.1489 −1.43407
\(144\) 0 0
\(145\) 1.62408 0.134872
\(146\) 0 0
\(147\) 2.50114i 0.206291i
\(148\) 0 0
\(149\) − 13.9887i − 1.14600i −0.819556 0.572998i \(-0.805780\pi\)
0.819556 0.572998i \(-0.194220\pi\)
\(150\) 0 0
\(151\) −18.5828 −1.51225 −0.756123 0.654430i \(-0.772909\pi\)
−0.756123 + 0.654430i \(0.772909\pi\)
\(152\) 0 0
\(153\) −6.52395 −0.527430
\(154\) 0 0
\(155\) − 2.03866i − 0.163749i
\(156\) 0 0
\(157\) 5.30411i 0.423314i 0.977344 + 0.211657i \(0.0678860\pi\)
−0.977344 + 0.211657i \(0.932114\pi\)
\(158\) 0 0
\(159\) 0.564862 0.0447964
\(160\) 0 0
\(161\) 12.3296 0.971706
\(162\) 0 0
\(163\) 13.3060i 1.04221i 0.853493 + 0.521104i \(0.174480\pi\)
−0.853493 + 0.521104i \(0.825520\pi\)
\(164\) 0 0
\(165\) − 1.22625i − 0.0954636i
\(166\) 0 0
\(167\) −15.9310 −1.23278 −0.616389 0.787442i \(-0.711405\pi\)
−0.616389 + 0.787442i \(0.711405\pi\)
\(168\) 0 0
\(169\) −8.53392 −0.656455
\(170\) 0 0
\(171\) 0.867091i 0.0663081i
\(172\) 0 0
\(173\) 19.7229i 1.49950i 0.661721 + 0.749751i \(0.269827\pi\)
−0.661721 + 0.749751i \(0.730173\pi\)
\(174\) 0 0
\(175\) 15.0726 1.13938
\(176\) 0 0
\(177\) −6.59539 −0.495740
\(178\) 0 0
\(179\) 5.98642i 0.447446i 0.974653 + 0.223723i \(0.0718211\pi\)
−0.974653 + 0.223723i \(0.928179\pi\)
\(180\) 0 0
\(181\) 9.81204i 0.729323i 0.931140 + 0.364662i \(0.118815\pi\)
−0.931140 + 0.364662i \(0.881185\pi\)
\(182\) 0 0
\(183\) 14.8052 1.09444
\(184\) 0 0
\(185\) −1.21024 −0.0889784
\(186\) 0 0
\(187\) 24.1094i 1.76305i
\(188\) 0 0
\(189\) − 3.08239i − 0.224211i
\(190\) 0 0
\(191\) −19.2900 −1.39578 −0.697888 0.716207i \(-0.745877\pi\)
−0.697888 + 0.716207i \(0.745877\pi\)
\(192\) 0 0
\(193\) −0.155713 −0.0112084 −0.00560422 0.999984i \(-0.501784\pi\)
−0.00560422 + 0.999984i \(0.501784\pi\)
\(194\) 0 0
\(195\) − 1.53981i − 0.110268i
\(196\) 0 0
\(197\) 13.2014i 0.940557i 0.882518 + 0.470279i \(0.155847\pi\)
−0.882518 + 0.470279i \(0.844153\pi\)
\(198\) 0 0
\(199\) 23.1144 1.63854 0.819269 0.573409i \(-0.194380\pi\)
0.819269 + 0.573409i \(0.194380\pi\)
\(200\) 0 0
\(201\) −13.9864 −0.986526
\(202\) 0 0
\(203\) − 15.0866i − 1.05887i
\(204\) 0 0
\(205\) − 1.30358i − 0.0910459i
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 3.20435 0.221650
\(210\) 0 0
\(211\) − 3.39104i − 0.233449i −0.993164 0.116724i \(-0.962761\pi\)
0.993164 0.116724i \(-0.0372394\pi\)
\(212\) 0 0
\(213\) 7.49207i 0.513348i
\(214\) 0 0
\(215\) 1.30358 0.0889034
\(216\) 0 0
\(217\) −18.9378 −1.28558
\(218\) 0 0
\(219\) − 5.62408i − 0.380040i
\(220\) 0 0
\(221\) 30.2741i 2.03646i
\(222\) 0 0
\(223\) −10.5326 −0.705316 −0.352658 0.935752i \(-0.614722\pi\)
−0.352658 + 0.935752i \(0.614722\pi\)
\(224\) 0 0
\(225\) −4.88989 −0.325993
\(226\) 0 0
\(227\) 23.4753i 1.55811i 0.626955 + 0.779055i \(0.284301\pi\)
−0.626955 + 0.779055i \(0.715699\pi\)
\(228\) 0 0
\(229\) − 3.13932i − 0.207452i −0.994606 0.103726i \(-0.966923\pi\)
0.994606 0.103726i \(-0.0330766\pi\)
\(230\) 0 0
\(231\) −11.3910 −0.749475
\(232\) 0 0
\(233\) 8.98414 0.588571 0.294285 0.955718i \(-0.404918\pi\)
0.294285 + 0.955718i \(0.404918\pi\)
\(234\) 0 0
\(235\) 0.549780i 0.0358637i
\(236\) 0 0
\(237\) − 14.9040i − 0.968121i
\(238\) 0 0
\(239\) 1.69870 0.109880 0.0549400 0.998490i \(-0.482503\pi\)
0.0549400 + 0.998490i \(0.482503\pi\)
\(240\) 0 0
\(241\) −14.3288 −0.923001 −0.461500 0.887140i \(-0.652689\pi\)
−0.461500 + 0.887140i \(0.652689\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0.829932i 0.0530224i
\(246\) 0 0
\(247\) 4.02371 0.256022
\(248\) 0 0
\(249\) −9.35237 −0.592683
\(250\) 0 0
\(251\) − 11.6955i − 0.738215i −0.929387 0.369107i \(-0.879663\pi\)
0.929387 0.369107i \(-0.120337\pi\)
\(252\) 0 0
\(253\) 14.7821i 0.929341i
\(254\) 0 0
\(255\) −2.16478 −0.135564
\(256\) 0 0
\(257\) 3.98642 0.248666 0.124333 0.992241i \(-0.460321\pi\)
0.124333 + 0.992241i \(0.460321\pi\)
\(258\) 0 0
\(259\) 11.2423i 0.698560i
\(260\) 0 0
\(261\) 4.89443i 0.302958i
\(262\) 0 0
\(263\) 11.8635 0.731534 0.365767 0.930706i \(-0.380807\pi\)
0.365767 + 0.930706i \(0.380807\pi\)
\(264\) 0 0
\(265\) 0.187433 0.0115139
\(266\) 0 0
\(267\) 18.1094i 1.10827i
\(268\) 0 0
\(269\) − 21.3880i − 1.30405i −0.758197 0.652026i \(-0.773919\pi\)
0.758197 0.652026i \(-0.226081\pi\)
\(270\) 0 0
\(271\) −12.0530 −0.732165 −0.366082 0.930582i \(-0.619301\pi\)
−0.366082 + 0.930582i \(0.619301\pi\)
\(272\) 0 0
\(273\) −14.3037 −0.865701
\(274\) 0 0
\(275\) 18.0707i 1.08970i
\(276\) 0 0
\(277\) − 4.81432i − 0.289265i −0.989485 0.144632i \(-0.953800\pi\)
0.989485 0.144632i \(-0.0461999\pi\)
\(278\) 0 0
\(279\) 6.14386 0.367823
\(280\) 0 0
\(281\) −18.5754 −1.10812 −0.554059 0.832477i \(-0.686922\pi\)
−0.554059 + 0.832477i \(0.686922\pi\)
\(282\) 0 0
\(283\) 1.40461i 0.0834956i 0.999128 + 0.0417478i \(0.0132926\pi\)
−0.999128 + 0.0417478i \(0.986707\pi\)
\(284\) 0 0
\(285\) 0.287719i 0.0170430i
\(286\) 0 0
\(287\) −12.1094 −0.714793
\(288\) 0 0
\(289\) 25.5619 1.50364
\(290\) 0 0
\(291\) − 17.0479i − 0.999365i
\(292\) 0 0
\(293\) − 27.8786i − 1.62868i −0.580386 0.814342i \(-0.697098\pi\)
0.580386 0.814342i \(-0.302902\pi\)
\(294\) 0 0
\(295\) −2.18849 −0.127419
\(296\) 0 0
\(297\) 3.69552 0.214436
\(298\) 0 0
\(299\) 18.5619i 1.07346i
\(300\) 0 0
\(301\) − 12.1094i − 0.697972i
\(302\) 0 0
\(303\) 12.1535 0.698198
\(304\) 0 0
\(305\) 4.91270 0.281300
\(306\) 0 0
\(307\) − 17.4548i − 0.996197i −0.867120 0.498099i \(-0.834032\pi\)
0.867120 0.498099i \(-0.165968\pi\)
\(308\) 0 0
\(309\) 17.1043i 0.973029i
\(310\) 0 0
\(311\) −0.466081 −0.0264290 −0.0132145 0.999913i \(-0.504206\pi\)
−0.0132145 + 0.999913i \(0.504206\pi\)
\(312\) 0 0
\(313\) −2.49886 −0.141244 −0.0706219 0.997503i \(-0.522498\pi\)
−0.0706219 + 0.997503i \(0.522498\pi\)
\(314\) 0 0
\(315\) − 1.02280i − 0.0576285i
\(316\) 0 0
\(317\) 10.7425i 0.603358i 0.953410 + 0.301679i \(0.0975470\pi\)
−0.953410 + 0.301679i \(0.902453\pi\)
\(318\) 0 0
\(319\) 18.0875 1.01270
\(320\) 0 0
\(321\) −18.3752 −1.02560
\(322\) 0 0
\(323\) − 5.65685i − 0.314756i
\(324\) 0 0
\(325\) 22.6914i 1.25869i
\(326\) 0 0
\(327\) 7.84482 0.433819
\(328\) 0 0
\(329\) 5.10707 0.281562
\(330\) 0 0
\(331\) 5.26810i 0.289561i 0.989464 + 0.144781i \(0.0462476\pi\)
−0.989464 + 0.144781i \(0.953752\pi\)
\(332\) 0 0
\(333\) − 3.64725i − 0.199868i
\(334\) 0 0
\(335\) −4.64099 −0.253565
\(336\) 0 0
\(337\) −27.4740 −1.49660 −0.748302 0.663359i \(-0.769130\pi\)
−0.748302 + 0.663359i \(0.769130\pi\)
\(338\) 0 0
\(339\) 3.65685i 0.198613i
\(340\) 0 0
\(341\) − 22.7047i − 1.22953i
\(342\) 0 0
\(343\) −13.8672 −0.748761
\(344\) 0 0
\(345\) −1.32729 −0.0714586
\(346\) 0 0
\(347\) − 8.08427i − 0.433986i −0.976173 0.216993i \(-0.930375\pi\)
0.976173 0.216993i \(-0.0696249\pi\)
\(348\) 0 0
\(349\) − 4.97454i − 0.266281i −0.991097 0.133140i \(-0.957494\pi\)
0.991097 0.133140i \(-0.0425062\pi\)
\(350\) 0 0
\(351\) 4.64047 0.247690
\(352\) 0 0
\(353\) 23.0799 1.22842 0.614210 0.789143i \(-0.289475\pi\)
0.614210 + 0.789143i \(0.289475\pi\)
\(354\) 0 0
\(355\) 2.48603i 0.131945i
\(356\) 0 0
\(357\) 20.1094i 1.06430i
\(358\) 0 0
\(359\) −29.2583 −1.54419 −0.772097 0.635505i \(-0.780792\pi\)
−0.772097 + 0.635505i \(0.780792\pi\)
\(360\) 0 0
\(361\) 18.2482 0.960429
\(362\) 0 0
\(363\) − 2.65685i − 0.139449i
\(364\) 0 0
\(365\) − 1.86619i − 0.0976808i
\(366\) 0 0
\(367\) 13.1698 0.687461 0.343730 0.939068i \(-0.388309\pi\)
0.343730 + 0.939068i \(0.388309\pi\)
\(368\) 0 0
\(369\) 3.92856 0.204513
\(370\) 0 0
\(371\) − 1.74113i − 0.0903947i
\(372\) 0 0
\(373\) − 15.2905i − 0.791714i −0.918312 0.395857i \(-0.870448\pi\)
0.918312 0.395857i \(-0.129552\pi\)
\(374\) 0 0
\(375\) −3.28168 −0.169465
\(376\) 0 0
\(377\) 22.7124 1.16975
\(378\) 0 0
\(379\) − 6.41459i − 0.329495i −0.986336 0.164748i \(-0.947319\pi\)
0.986336 0.164748i \(-0.0526810\pi\)
\(380\) 0 0
\(381\) 0.638213i 0.0326966i
\(382\) 0 0
\(383\) 34.0721 1.74100 0.870501 0.492167i \(-0.163795\pi\)
0.870501 + 0.492167i \(0.163795\pi\)
\(384\) 0 0
\(385\) −3.77979 −0.192636
\(386\) 0 0
\(387\) 3.92856i 0.199700i
\(388\) 0 0
\(389\) − 32.5185i − 1.64875i −0.566041 0.824377i \(-0.691526\pi\)
0.566041 0.824377i \(-0.308474\pi\)
\(390\) 0 0
\(391\) 26.0958 1.31972
\(392\) 0 0
\(393\) 9.51397 0.479916
\(394\) 0 0
\(395\) − 4.94548i − 0.248834i
\(396\) 0 0
\(397\) − 4.48926i − 0.225309i −0.993634 0.112655i \(-0.964065\pi\)
0.993634 0.112655i \(-0.0359354\pi\)
\(398\) 0 0
\(399\) 2.67271 0.133803
\(400\) 0 0
\(401\) −3.21024 −0.160312 −0.0801558 0.996782i \(-0.525542\pi\)
−0.0801558 + 0.996782i \(0.525542\pi\)
\(402\) 0 0
\(403\) − 28.5104i − 1.42020i
\(404\) 0 0
\(405\) 0.331821i 0.0164883i
\(406\) 0 0
\(407\) −13.4785 −0.668104
\(408\) 0 0
\(409\) −14.0456 −0.694511 −0.347255 0.937771i \(-0.612886\pi\)
−0.347255 + 0.937771i \(0.612886\pi\)
\(410\) 0 0
\(411\) 13.9150i 0.686375i
\(412\) 0 0
\(413\) 20.3296i 1.00035i
\(414\) 0 0
\(415\) −3.10332 −0.152336
\(416\) 0 0
\(417\) 0.0773278 0.00378676
\(418\) 0 0
\(419\) − 26.9437i − 1.31628i −0.752894 0.658142i \(-0.771343\pi\)
0.752894 0.658142i \(-0.228657\pi\)
\(420\) 0 0
\(421\) 5.72188i 0.278867i 0.990231 + 0.139434i \(0.0445282\pi\)
−0.990231 + 0.139434i \(0.955472\pi\)
\(422\) 0 0
\(423\) −1.65685 −0.0805590
\(424\) 0 0
\(425\) 31.9014 1.54745
\(426\) 0 0
\(427\) − 45.6356i − 2.20846i
\(428\) 0 0
\(429\) − 17.1489i − 0.827958i
\(430\) 0 0
\(431\) 27.9547 1.34653 0.673265 0.739401i \(-0.264891\pi\)
0.673265 + 0.739401i \(0.264891\pi\)
\(432\) 0 0
\(433\) −1.81151 −0.0870556 −0.0435278 0.999052i \(-0.513860\pi\)
−0.0435278 + 0.999052i \(0.513860\pi\)
\(434\) 0 0
\(435\) 1.62408i 0.0778685i
\(436\) 0 0
\(437\) − 3.46836i − 0.165914i
\(438\) 0 0
\(439\) 4.45985 0.212857 0.106429 0.994320i \(-0.466058\pi\)
0.106429 + 0.994320i \(0.466058\pi\)
\(440\) 0 0
\(441\) −2.50114 −0.119102
\(442\) 0 0
\(443\) − 2.62047i − 0.124502i −0.998061 0.0622512i \(-0.980172\pi\)
0.998061 0.0622512i \(-0.0198280\pi\)
\(444\) 0 0
\(445\) 6.00907i 0.284857i
\(446\) 0 0
\(447\) 13.9887 0.661642
\(448\) 0 0
\(449\) 27.9787 1.32040 0.660199 0.751091i \(-0.270472\pi\)
0.660199 + 0.751091i \(0.270472\pi\)
\(450\) 0 0
\(451\) − 14.5181i − 0.683629i
\(452\) 0 0
\(453\) − 18.5828i − 0.873095i
\(454\) 0 0
\(455\) −4.74628 −0.222509
\(456\) 0 0
\(457\) 14.3933 0.673291 0.336646 0.941631i \(-0.390708\pi\)
0.336646 + 0.941631i \(0.390708\pi\)
\(458\) 0 0
\(459\) − 6.52395i − 0.304512i
\(460\) 0 0
\(461\) − 7.78841i − 0.362743i −0.983415 0.181371i \(-0.941946\pi\)
0.983415 0.181371i \(-0.0580536\pi\)
\(462\) 0 0
\(463\) −15.6642 −0.727977 −0.363989 0.931403i \(-0.618585\pi\)
−0.363989 + 0.931403i \(0.618585\pi\)
\(464\) 0 0
\(465\) 2.03866 0.0945408
\(466\) 0 0
\(467\) − 5.27504i − 0.244100i −0.992524 0.122050i \(-0.961053\pi\)
0.992524 0.122050i \(-0.0389468\pi\)
\(468\) 0 0
\(469\) 43.1116i 1.99071i
\(470\) 0 0
\(471\) −5.30411 −0.244400
\(472\) 0 0
\(473\) 14.5181 0.667541
\(474\) 0 0
\(475\) − 4.23998i − 0.194544i
\(476\) 0 0
\(477\) 0.564862i 0.0258632i
\(478\) 0 0
\(479\) −11.9310 −0.545141 −0.272571 0.962136i \(-0.587874\pi\)
−0.272571 + 0.962136i \(0.587874\pi\)
\(480\) 0 0
\(481\) −16.9250 −0.771712
\(482\) 0 0
\(483\) 12.3296i 0.561015i
\(484\) 0 0
\(485\) − 5.65685i − 0.256865i
\(486\) 0 0
\(487\) −37.1782 −1.68470 −0.842352 0.538928i \(-0.818830\pi\)
−0.842352 + 0.538928i \(0.818830\pi\)
\(488\) 0 0
\(489\) −13.3060 −0.601719
\(490\) 0 0
\(491\) − 25.0981i − 1.13266i −0.824179 0.566330i \(-0.808363\pi\)
0.824179 0.566330i \(-0.191637\pi\)
\(492\) 0 0
\(493\) − 31.9310i − 1.43810i
\(494\) 0 0
\(495\) 1.22625 0.0551159
\(496\) 0 0
\(497\) 23.0935 1.03588
\(498\) 0 0
\(499\) 32.8458i 1.47038i 0.677861 + 0.735190i \(0.262907\pi\)
−0.677861 + 0.735190i \(0.737093\pi\)
\(500\) 0 0
\(501\) − 15.9310i − 0.711744i
\(502\) 0 0
\(503\) 8.35327 0.372454 0.186227 0.982507i \(-0.440374\pi\)
0.186227 + 0.982507i \(0.440374\pi\)
\(504\) 0 0
\(505\) 4.03278 0.179456
\(506\) 0 0
\(507\) − 8.53392i − 0.379005i
\(508\) 0 0
\(509\) − 9.33786i − 0.413893i −0.978352 0.206947i \(-0.933647\pi\)
0.978352 0.206947i \(-0.0663527\pi\)
\(510\) 0 0
\(511\) −17.3356 −0.766882
\(512\) 0 0
\(513\) −0.867091 −0.0382830
\(514\) 0 0
\(515\) 5.67557i 0.250095i
\(516\) 0 0
\(517\) 6.12293i 0.269286i
\(518\) 0 0
\(519\) −19.7229 −0.865737
\(520\) 0 0
\(521\) 4.38287 0.192017 0.0960084 0.995381i \(-0.469392\pi\)
0.0960084 + 0.995381i \(0.469392\pi\)
\(522\) 0 0
\(523\) − 9.22869i − 0.403542i −0.979433 0.201771i \(-0.935330\pi\)
0.979433 0.201771i \(-0.0646697\pi\)
\(524\) 0 0
\(525\) 15.0726i 0.657821i
\(526\) 0 0
\(527\) −40.0822 −1.74601
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) − 6.59539i − 0.286215i
\(532\) 0 0
\(533\) − 18.2303i − 0.789644i
\(534\) 0 0
\(535\) −6.09728 −0.263608
\(536\) 0 0
\(537\) −5.98642 −0.258333
\(538\) 0 0
\(539\) 9.24301i 0.398125i
\(540\) 0 0
\(541\) − 36.9700i − 1.58947i −0.606959 0.794733i \(-0.707611\pi\)
0.606959 0.794733i \(-0.292389\pi\)
\(542\) 0 0
\(543\) −9.81204 −0.421075
\(544\) 0 0
\(545\) 2.60308 0.111504
\(546\) 0 0
\(547\) − 40.6789i − 1.73930i −0.493665 0.869652i \(-0.664343\pi\)
0.493665 0.869652i \(-0.335657\pi\)
\(548\) 0 0
\(549\) 14.8052i 0.631873i
\(550\) 0 0
\(551\) −4.24392 −0.180797
\(552\) 0 0
\(553\) −45.9401 −1.95357
\(554\) 0 0
\(555\) − 1.21024i − 0.0513717i
\(556\) 0 0
\(557\) 39.6184i 1.67868i 0.543603 + 0.839342i \(0.317060\pi\)
−0.543603 + 0.839342i \(0.682940\pi\)
\(558\) 0 0
\(559\) 18.2303 0.771061
\(560\) 0 0
\(561\) −24.1094 −1.01790
\(562\) 0 0
\(563\) − 20.8090i − 0.876993i −0.898733 0.438497i \(-0.855511\pi\)
0.898733 0.438497i \(-0.144489\pi\)
\(564\) 0 0
\(565\) 1.21342i 0.0510491i
\(566\) 0 0
\(567\) 3.08239 0.129448
\(568\) 0 0
\(569\) 15.7585 0.660632 0.330316 0.943870i \(-0.392845\pi\)
0.330316 + 0.943870i \(0.392845\pi\)
\(570\) 0 0
\(571\) − 24.6912i − 1.03329i −0.856199 0.516647i \(-0.827180\pi\)
0.856199 0.516647i \(-0.172820\pi\)
\(572\) 0 0
\(573\) − 19.2900i − 0.805851i
\(574\) 0 0
\(575\) 19.5596 0.815691
\(576\) 0 0
\(577\) −21.0924 −0.878090 −0.439045 0.898465i \(-0.644683\pi\)
−0.439045 + 0.898465i \(0.644683\pi\)
\(578\) 0 0
\(579\) − 0.155713i − 0.00647119i
\(580\) 0 0
\(581\) 28.8277i 1.19597i
\(582\) 0 0
\(583\) 2.08746 0.0864536
\(584\) 0 0
\(585\) 1.53981 0.0636631
\(586\) 0 0
\(587\) 14.3933i 0.594076i 0.954866 + 0.297038i \(0.0959988\pi\)
−0.954866 + 0.297038i \(0.904001\pi\)
\(588\) 0 0
\(589\) 5.32729i 0.219507i
\(590\) 0 0
\(591\) −13.2014 −0.543031
\(592\) 0 0
\(593\) 22.1931 0.911360 0.455680 0.890144i \(-0.349396\pi\)
0.455680 + 0.890144i \(0.349396\pi\)
\(594\) 0 0
\(595\) 6.67271i 0.273555i
\(596\) 0 0
\(597\) 23.1144i 0.946010i
\(598\) 0 0
\(599\) −37.4366 −1.52962 −0.764810 0.644256i \(-0.777167\pi\)
−0.764810 + 0.644256i \(0.777167\pi\)
\(600\) 0 0
\(601\) −1.68963 −0.0689215 −0.0344608 0.999406i \(-0.510971\pi\)
−0.0344608 + 0.999406i \(0.510971\pi\)
\(602\) 0 0
\(603\) − 13.9864i − 0.569571i
\(604\) 0 0
\(605\) − 0.881601i − 0.0358422i
\(606\) 0 0
\(607\) 36.8841 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(608\) 0 0
\(609\) 15.0866 0.611338
\(610\) 0 0
\(611\) 7.68857i 0.311046i
\(612\) 0 0
\(613\) 13.4598i 0.543637i 0.962349 + 0.271819i \(0.0876250\pi\)
−0.962349 + 0.271819i \(0.912375\pi\)
\(614\) 0 0
\(615\) 1.30358 0.0525654
\(616\) 0 0
\(617\) 20.3160 0.817891 0.408946 0.912559i \(-0.365897\pi\)
0.408946 + 0.912559i \(0.365897\pi\)
\(618\) 0 0
\(619\) 39.0279i 1.56867i 0.620340 + 0.784333i \(0.286994\pi\)
−0.620340 + 0.784333i \(0.713006\pi\)
\(620\) 0 0
\(621\) − 4.00000i − 0.160514i
\(622\) 0 0
\(623\) 55.8201 2.23639
\(624\) 0 0
\(625\) 23.3605 0.934422
\(626\) 0 0
\(627\) 3.20435i 0.127969i
\(628\) 0 0
\(629\) 23.7945i 0.948748i
\(630\) 0 0
\(631\) −0.629888 −0.0250755 −0.0125377 0.999921i \(-0.503991\pi\)
−0.0125377 + 0.999921i \(0.503991\pi\)
\(632\) 0 0
\(633\) 3.39104 0.134782
\(634\) 0 0
\(635\) 0.211773i 0.00840394i
\(636\) 0 0
\(637\) 11.6065i 0.459865i
\(638\) 0 0
\(639\) −7.49207 −0.296382
\(640\) 0 0
\(641\) 3.52166 0.139097 0.0695486 0.997579i \(-0.477844\pi\)
0.0695486 + 0.997579i \(0.477844\pi\)
\(642\) 0 0
\(643\) − 9.83765i − 0.387959i −0.981006 0.193980i \(-0.937860\pi\)
0.981006 0.193980i \(-0.0621396\pi\)
\(644\) 0 0
\(645\) 1.30358i 0.0513284i
\(646\) 0 0
\(647\) 30.7549 1.20910 0.604550 0.796567i \(-0.293353\pi\)
0.604550 + 0.796567i \(0.293353\pi\)
\(648\) 0 0
\(649\) −24.3734 −0.956739
\(650\) 0 0
\(651\) − 18.9378i − 0.742230i
\(652\) 0 0
\(653\) 9.22745i 0.361098i 0.983566 + 0.180549i \(0.0577874\pi\)
−0.983566 + 0.180549i \(0.942213\pi\)
\(654\) 0 0
\(655\) 3.15694 0.123352
\(656\) 0 0
\(657\) 5.62408 0.219416
\(658\) 0 0
\(659\) − 46.5619i − 1.81379i −0.421354 0.906896i \(-0.638445\pi\)
0.421354 0.906896i \(-0.361555\pi\)
\(660\) 0 0
\(661\) 27.9315i 1.08641i 0.839600 + 0.543205i \(0.182789\pi\)
−0.839600 + 0.543205i \(0.817211\pi\)
\(662\) 0 0
\(663\) −30.2741 −1.17575
\(664\) 0 0
\(665\) 0.886864 0.0343911
\(666\) 0 0
\(667\) − 19.5777i − 0.758052i
\(668\) 0 0
\(669\) − 10.5326i − 0.407214i
\(670\) 0 0
\(671\) 54.7131 2.11217
\(672\) 0 0
\(673\) 6.34315 0.244510 0.122255 0.992499i \(-0.460987\pi\)
0.122255 + 0.992499i \(0.460987\pi\)
\(674\) 0 0
\(675\) − 4.88989i − 0.188212i
\(676\) 0 0
\(677\) 31.4918i 1.21033i 0.796101 + 0.605164i \(0.206892\pi\)
−0.796101 + 0.605164i \(0.793108\pi\)
\(678\) 0 0
\(679\) −52.5483 −2.01662
\(680\) 0 0
\(681\) −23.4753 −0.899576
\(682\) 0 0
\(683\) − 28.0069i − 1.07166i −0.844327 0.535828i \(-0.820000\pi\)
0.844327 0.535828i \(-0.180000\pi\)
\(684\) 0 0
\(685\) 4.61729i 0.176418i
\(686\) 0 0
\(687\) 3.13932 0.119773
\(688\) 0 0
\(689\) 2.62122 0.0998606
\(690\) 0 0
\(691\) − 38.2264i − 1.45420i −0.686531 0.727101i \(-0.740867\pi\)
0.686531 0.727101i \(-0.259133\pi\)
\(692\) 0 0
\(693\) − 11.3910i − 0.432710i
\(694\) 0 0
\(695\) 0.0256590 0.000973301 0
\(696\) 0 0
\(697\) −25.6297 −0.970794
\(698\) 0 0
\(699\) 8.98414i 0.339811i
\(700\) 0 0
\(701\) 9.03731i 0.341335i 0.985329 + 0.170667i \(0.0545923\pi\)
−0.985329 + 0.170667i \(0.945408\pi\)
\(702\) 0 0
\(703\) 3.16250 0.119276
\(704\) 0 0
\(705\) −0.549780 −0.0207059
\(706\) 0 0
\(707\) − 37.4617i − 1.40889i
\(708\) 0 0
\(709\) − 39.9270i − 1.49949i −0.661726 0.749745i \(-0.730176\pi\)
0.661726 0.749745i \(-0.269824\pi\)
\(710\) 0 0
\(711\) 14.9040 0.558945
\(712\) 0 0
\(713\) −24.5754 −0.920357
\(714\) 0 0
\(715\) − 5.69038i − 0.212808i
\(716\) 0 0
\(717\) 1.69870i 0.0634393i
\(718\) 0 0
\(719\) 29.3036 1.09284 0.546420 0.837512i \(-0.315990\pi\)
0.546420 + 0.837512i \(0.315990\pi\)
\(720\) 0 0
\(721\) 52.7221 1.96348
\(722\) 0 0
\(723\) − 14.3288i − 0.532895i
\(724\) 0 0
\(725\) − 23.9332i − 0.888859i
\(726\) 0 0
\(727\) −12.6201 −0.468052 −0.234026 0.972230i \(-0.575190\pi\)
−0.234026 + 0.972230i \(0.575190\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 25.6297i − 0.947949i
\(732\) 0 0
\(733\) − 9.92140i − 0.366455i −0.983071 0.183228i \(-0.941345\pi\)
0.983071 0.183228i \(-0.0586545\pi\)
\(734\) 0 0
\(735\) −0.829932 −0.0306125
\(736\) 0 0
\(737\) −51.6871 −1.90392
\(738\) 0 0
\(739\) 20.0592i 0.737889i 0.929451 + 0.368945i \(0.120281\pi\)
−0.929451 + 0.368945i \(0.879719\pi\)
\(740\) 0 0
\(741\) 4.02371i 0.147815i
\(742\) 0 0
\(743\) 11.0969 0.407107 0.203554 0.979064i \(-0.434751\pi\)
0.203554 + 0.979064i \(0.434751\pi\)
\(744\) 0 0
\(745\) 4.64174 0.170060
\(746\) 0 0
\(747\) − 9.35237i − 0.342185i
\(748\) 0 0
\(749\) 56.6395i 2.06956i
\(750\) 0 0
\(751\) 37.3294 1.36217 0.681084 0.732205i \(-0.261509\pi\)
0.681084 + 0.732205i \(0.261509\pi\)
\(752\) 0 0
\(753\) 11.6955 0.426208
\(754\) 0 0
\(755\) − 6.16617i − 0.224410i
\(756\) 0 0
\(757\) − 26.7362i − 0.971745i −0.874030 0.485873i \(-0.838502\pi\)
0.874030 0.485873i \(-0.161498\pi\)
\(758\) 0 0
\(759\) −14.7821 −0.536555
\(760\) 0 0
\(761\) 27.1767 0.985155 0.492578 0.870269i \(-0.336055\pi\)
0.492578 + 0.870269i \(0.336055\pi\)
\(762\) 0 0
\(763\) − 24.1808i − 0.875404i
\(764\) 0 0
\(765\) − 2.16478i − 0.0782679i
\(766\) 0 0
\(767\) −30.6057 −1.10511
\(768\) 0 0
\(769\) −8.34877 −0.301064 −0.150532 0.988605i \(-0.548099\pi\)
−0.150532 + 0.988605i \(0.548099\pi\)
\(770\) 0 0
\(771\) 3.98642i 0.143568i
\(772\) 0 0
\(773\) − 20.7289i − 0.745567i −0.927918 0.372783i \(-0.878403\pi\)
0.927918 0.372783i \(-0.121597\pi\)
\(774\) 0 0
\(775\) −30.0428 −1.07917
\(776\) 0 0
\(777\) −11.2423 −0.403314
\(778\) 0 0
\(779\) 3.40642i 0.122048i
\(780\) 0 0
\(781\) 27.6871i 0.990722i
\(782\) 0 0
\(783\) −4.89443 −0.174913
\(784\) 0 0
\(785\) −1.76002 −0.0628177
\(786\) 0 0
\(787\) 21.7129i 0.773982i 0.922084 + 0.386991i \(0.126486\pi\)
−0.922084 + 0.386991i \(0.873514\pi\)
\(788\) 0 0
\(789\) 11.8635i 0.422351i
\(790\) 0 0
\(791\) 11.2719 0.400781
\(792\) 0 0
\(793\) 68.7032 2.43972
\(794\) 0 0
\(795\) 0.187433i 0.00664757i
\(796\) 0 0
\(797\) − 27.5528i − 0.975971i −0.872852 0.487985i \(-0.837732\pi\)
0.872852 0.487985i \(-0.162268\pi\)
\(798\) 0 0
\(799\) 10.8092 0.382403
\(800\) 0 0
\(801\) −18.1094 −0.639863
\(802\) 0 0
\(803\) − 20.7839i − 0.733447i
\(804\) 0 0
\(805\) 4.09121i 0.144196i
\(806\) 0 0
\(807\) 21.3880 0.752895
\(808\) 0 0
\(809\) −18.8535 −0.662854 −0.331427 0.943481i \(-0.607530\pi\)
−0.331427 + 0.943481i \(0.607530\pi\)
\(810\) 0 0
\(811\) 6.78796i 0.238357i 0.992873 + 0.119179i \(0.0380262\pi\)
−0.992873 + 0.119179i \(0.961974\pi\)
\(812\) 0 0
\(813\) − 12.0530i − 0.422716i
\(814\) 0 0
\(815\) −4.41522 −0.154658
\(816\) 0 0
\(817\) −3.40642 −0.119175
\(818\) 0 0
\(819\) − 14.3037i − 0.499813i
\(820\) 0 0
\(821\) 16.0169i 0.558995i 0.960146 + 0.279498i \(0.0901679\pi\)
−0.960146 + 0.279498i \(0.909832\pi\)
\(822\) 0 0
\(823\) 28.5609 0.995570 0.497785 0.867301i \(-0.334147\pi\)
0.497785 + 0.867301i \(0.334147\pi\)
\(824\) 0 0
\(825\) −18.0707 −0.629141
\(826\) 0 0
\(827\) 19.1388i 0.665521i 0.943011 + 0.332761i \(0.107980\pi\)
−0.943011 + 0.332761i \(0.892020\pi\)
\(828\) 0 0
\(829\) − 36.0907i − 1.25348i −0.779228 0.626741i \(-0.784389\pi\)
0.779228 0.626741i \(-0.215611\pi\)
\(830\) 0 0
\(831\) 4.81432 0.167007
\(832\) 0 0
\(833\) 16.3173 0.565361
\(834\) 0 0
\(835\) − 5.28624i − 0.182938i
\(836\) 0 0
\(837\) 6.14386i 0.212363i
\(838\) 0 0
\(839\) −18.8767 −0.651697 −0.325849 0.945422i \(-0.605650\pi\)
−0.325849 + 0.945422i \(0.605650\pi\)
\(840\) 0 0
\(841\) 5.04455 0.173950
\(842\) 0 0
\(843\) − 18.5754i − 0.639772i
\(844\) 0 0
\(845\) − 2.83174i − 0.0974147i
\(846\) 0 0
\(847\) −8.18947 −0.281393
\(848\) 0 0
\(849\) −1.40461 −0.0482062
\(850\) 0 0
\(851\) 14.5890i 0.500105i
\(852\) 0 0
\(853\) − 26.3018i − 0.900557i −0.892888 0.450279i \(-0.851325\pi\)
0.892888 0.450279i \(-0.148675\pi\)
\(854\) 0 0
\(855\) −0.287719 −0.00983979
\(856\) 0 0
\(857\) 24.8671 0.849444 0.424722 0.905324i \(-0.360372\pi\)
0.424722 + 0.905324i \(0.360372\pi\)
\(858\) 0 0
\(859\) − 15.5990i − 0.532231i −0.963941 0.266115i \(-0.914260\pi\)
0.963941 0.266115i \(-0.0857402\pi\)
\(860\) 0 0
\(861\) − 12.1094i − 0.412686i
\(862\) 0 0
\(863\) −43.8654 −1.49320 −0.746598 0.665275i \(-0.768314\pi\)
−0.746598 + 0.665275i \(0.768314\pi\)
\(864\) 0 0
\(865\) −6.54447 −0.222519
\(866\) 0 0
\(867\) 25.5619i 0.868126i
\(868\) 0 0
\(869\) − 55.0781i − 1.86840i
\(870\) 0 0
\(871\) −64.9035 −2.19917
\(872\) 0 0
\(873\) 17.0479 0.576984
\(874\) 0 0
\(875\) 10.1154i 0.341964i
\(876\) 0 0
\(877\) 45.5410i 1.53781i 0.639364 + 0.768905i \(0.279198\pi\)
−0.639364 + 0.768905i \(0.720802\pi\)
\(878\) 0 0
\(879\) 27.8786 0.940321
\(880\) 0 0
\(881\) −19.8589 −0.669064 −0.334532 0.942384i \(-0.608578\pi\)
−0.334532 + 0.942384i \(0.608578\pi\)
\(882\) 0 0
\(883\) 25.2740i 0.850537i 0.905067 + 0.425269i \(0.139820\pi\)
−0.905067 + 0.425269i \(0.860180\pi\)
\(884\) 0 0
\(885\) − 2.18849i − 0.0735653i
\(886\) 0 0
\(887\) 2.07012 0.0695079 0.0347539 0.999396i \(-0.488935\pi\)
0.0347539 + 0.999396i \(0.488935\pi\)
\(888\) 0 0
\(889\) 1.96722 0.0659785
\(890\) 0 0
\(891\) 3.69552i 0.123805i
\(892\) 0 0
\(893\) − 1.43664i − 0.0480754i
\(894\) 0 0
\(895\) −1.98642 −0.0663988
\(896\) 0 0
\(897\) −18.5619 −0.619763
\(898\) 0 0
\(899\) 30.0707i 1.00291i
\(900\) 0 0
\(901\) − 3.68513i − 0.122769i
\(902\) 0 0
\(903\) 12.1094 0.402974
\(904\) 0 0
\(905\) −3.25584 −0.108228
\(906\) 0 0
\(907\) − 8.10347i − 0.269071i −0.990909 0.134536i \(-0.957046\pi\)
0.990909 0.134536i \(-0.0429543\pi\)
\(908\) 0 0
\(909\) 12.1535i 0.403105i
\(910\) 0 0
\(911\) −6.23034 −0.206420 −0.103210 0.994660i \(-0.532911\pi\)
−0.103210 + 0.994660i \(0.532911\pi\)
\(912\) 0 0
\(913\) −34.5619 −1.14383
\(914\) 0 0
\(915\) 4.91270i 0.162409i
\(916\) 0 0
\(917\) − 29.3258i − 0.968423i
\(918\) 0 0
\(919\) 15.1680 0.500348 0.250174 0.968201i \(-0.419512\pi\)
0.250174 + 0.968201i \(0.419512\pi\)
\(920\) 0 0
\(921\) 17.4548 0.575155
\(922\) 0 0
\(923\) 34.7667i 1.14436i
\(924\) 0 0
\(925\) 17.8347i 0.586401i
\(926\) 0 0
\(927\) −17.1043 −0.561779
\(928\) 0 0
\(929\) −9.16951 −0.300842 −0.150421 0.988622i \(-0.548063\pi\)
−0.150421 + 0.988622i \(0.548063\pi\)
\(930\) 0 0
\(931\) − 2.16872i − 0.0710768i
\(932\) 0 0
\(933\) − 0.466081i − 0.0152588i
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 19.3183 0.631101 0.315550 0.948909i \(-0.397811\pi\)
0.315550 + 0.948909i \(0.397811\pi\)
\(938\) 0 0
\(939\) − 2.49886i − 0.0815472i
\(940\) 0 0
\(941\) 10.4385i 0.340285i 0.985419 + 0.170142i \(0.0544228\pi\)
−0.985419 + 0.170142i \(0.945577\pi\)
\(942\) 0 0
\(943\) −15.7142 −0.511726
\(944\) 0 0
\(945\) 1.02280 0.0332718
\(946\) 0 0
\(947\) − 36.2640i − 1.17842i −0.807979 0.589211i \(-0.799439\pi\)
0.807979 0.589211i \(-0.200561\pi\)
\(948\) 0 0
\(949\) − 26.0983i − 0.847188i
\(950\) 0 0
\(951\) −10.7425 −0.348349
\(952\) 0 0
\(953\) −47.9271 −1.55251 −0.776255 0.630419i \(-0.782883\pi\)
−0.776255 + 0.630419i \(0.782883\pi\)
\(954\) 0 0
\(955\) − 6.40083i − 0.207126i
\(956\) 0 0
\(957\) 18.0875i 0.584684i
\(958\) 0 0
\(959\) 42.8914 1.38504
\(960\) 0 0
\(961\) 6.74701 0.217646
\(962\) 0 0
\(963\) − 18.3752i − 0.592132i
\(964\) 0 0
\(965\) − 0.0516688i − 0.00166328i
\(966\) 0 0
\(967\) 20.7211 0.666346 0.333173 0.942866i \(-0.391881\pi\)
0.333173 + 0.942866i \(0.391881\pi\)
\(968\) 0 0
\(969\) 5.65685 0.181724
\(970\) 0 0
\(971\) − 38.0888i − 1.22233i −0.791504 0.611164i \(-0.790701\pi\)
0.791504 0.611164i \(-0.209299\pi\)
\(972\) 0 0
\(973\) − 0.238354i − 0.00764129i
\(974\) 0 0
\(975\) −22.6914 −0.726706
\(976\) 0 0
\(977\) 42.7363 1.36726 0.683628 0.729831i \(-0.260401\pi\)
0.683628 + 0.729831i \(0.260401\pi\)
\(978\) 0 0
\(979\) 66.9235i 2.13888i
\(980\) 0 0
\(981\) 7.84482i 0.250466i
\(982\) 0 0
\(983\) −42.0031 −1.33969 −0.669845 0.742501i \(-0.733639\pi\)
−0.669845 + 0.742501i \(0.733639\pi\)
\(984\) 0 0
\(985\) −4.38049 −0.139574
\(986\) 0 0
\(987\) 5.10707i 0.162560i
\(988\) 0 0
\(989\) − 15.7142i − 0.499684i
\(990\) 0 0
\(991\) −11.1918 −0.355518 −0.177759 0.984074i \(-0.556885\pi\)
−0.177759 + 0.984074i \(0.556885\pi\)
\(992\) 0 0
\(993\) −5.26810 −0.167178
\(994\) 0 0
\(995\) 7.66986i 0.243151i
\(996\) 0 0
\(997\) 52.3230i 1.65709i 0.559924 + 0.828544i \(0.310830\pi\)
−0.559924 + 0.828544i \(0.689170\pi\)
\(998\) 0 0
\(999\) 3.64725 0.115394
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 3072.2.d.j.1537.6 8
4.3 odd 2 3072.2.d.e.1537.2 8
8.3 odd 2 3072.2.d.e.1537.7 8
8.5 even 2 inner 3072.2.d.j.1537.3 8
16.3 odd 4 3072.2.a.s.1.2 4
16.5 even 4 3072.2.a.p.1.3 4
16.11 odd 4 3072.2.a.m.1.3 4
16.13 even 4 3072.2.a.j.1.2 4
32.3 odd 8 1536.2.j.e.385.4 8
32.5 even 8 1536.2.j.f.1153.2 yes 8
32.11 odd 8 1536.2.j.j.1153.1 yes 8
32.13 even 8 1536.2.j.i.385.3 yes 8
32.19 odd 8 1536.2.j.j.385.1 yes 8
32.21 even 8 1536.2.j.i.1153.3 yes 8
32.27 odd 8 1536.2.j.e.1153.4 yes 8
32.29 even 8 1536.2.j.f.385.2 yes 8
48.5 odd 4 9216.2.a.z.1.2 4
48.11 even 4 9216.2.a.bl.1.2 4
48.29 odd 4 9216.2.a.ba.1.3 4
48.35 even 4 9216.2.a.bm.1.3 4
96.5 odd 8 4608.2.k.bj.1153.2 8
96.11 even 8 4608.2.k.be.1153.3 8
96.29 odd 8 4608.2.k.bj.3457.2 8
96.35 even 8 4608.2.k.bh.3457.2 8
96.53 odd 8 4608.2.k.bc.1153.3 8
96.59 even 8 4608.2.k.bh.1153.2 8
96.77 odd 8 4608.2.k.bc.3457.3 8
96.83 even 8 4608.2.k.be.3457.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.j.e.385.4 8 32.3 odd 8
1536.2.j.e.1153.4 yes 8 32.27 odd 8
1536.2.j.f.385.2 yes 8 32.29 even 8
1536.2.j.f.1153.2 yes 8 32.5 even 8
1536.2.j.i.385.3 yes 8 32.13 even 8
1536.2.j.i.1153.3 yes 8 32.21 even 8
1536.2.j.j.385.1 yes 8 32.19 odd 8
1536.2.j.j.1153.1 yes 8 32.11 odd 8
3072.2.a.j.1.2 4 16.13 even 4
3072.2.a.m.1.3 4 16.11 odd 4
3072.2.a.p.1.3 4 16.5 even 4
3072.2.a.s.1.2 4 16.3 odd 4
3072.2.d.e.1537.2 8 4.3 odd 2
3072.2.d.e.1537.7 8 8.3 odd 2
3072.2.d.j.1537.3 8 8.5 even 2 inner
3072.2.d.j.1537.6 8 1.1 even 1 trivial
4608.2.k.bc.1153.3 8 96.53 odd 8
4608.2.k.bc.3457.3 8 96.77 odd 8
4608.2.k.be.1153.3 8 96.11 even 8
4608.2.k.be.3457.3 8 96.83 even 8
4608.2.k.bh.1153.2 8 96.59 even 8
4608.2.k.bh.3457.2 8 96.35 even 8
4608.2.k.bj.1153.2 8 96.5 odd 8
4608.2.k.bj.3457.2 8 96.29 odd 8
9216.2.a.z.1.2 4 48.5 odd 4
9216.2.a.ba.1.3 4 48.29 odd 4
9216.2.a.bl.1.2 4 48.11 even 4
9216.2.a.bm.1.3 4 48.35 even 4