Properties

Label 92.2.b.b.91.2
Level $92$
Weight $2$
Character 92.91
Analytic conductor $0.735$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [92,2,Mod(91,92)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(92, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("92.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 92.b (of order \(2\), degree \(1\), 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: \(0.734623698596\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{3} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 91.2
Root \(1.33454 - 0.467979i\) of defining polynomial
Character \(\chi\) \(=\) 92.91
Dual form 92.2.b.b.91.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.33454 + 0.467979i) q^{2} +3.43410i q^{3} +(1.56199 - 1.24907i) q^{4} +(-1.60709 - 4.58295i) q^{6} +(-1.50000 + 2.39792i) q^{8} -8.79306 q^{9} +(4.28944 + 5.36404i) q^{12} +4.88325 q^{13} +(0.879635 - 3.90208i) q^{16} +(11.7347 - 4.11497i) q^{18} +4.79583i q^{23} +(-8.23469 - 5.15115i) q^{24} +5.00000 q^{25} +(-6.51690 + 2.28526i) q^{26} -19.8940i q^{27} +6.70287 q^{29} -0.309728i q^{31} +(0.652183 + 5.61913i) q^{32} +(-13.7347 + 10.9832i) q^{36} +16.7696i q^{39} -3.97345 q^{41} +(-2.24435 - 6.40023i) q^{46} -6.55848i q^{47} +(13.4001 + 3.02076i) q^{48} -7.00000 q^{49} +(-6.67270 + 2.33989i) q^{50} +(7.62760 - 6.09954i) q^{52} +(9.30996 + 26.5493i) q^{54} +(-8.94525 + 3.13680i) q^{58} -9.59166i q^{59} +(0.144946 + 0.413344i) q^{62} +(-3.50000 - 7.19375i) q^{64} -16.4694 q^{69} -14.0461i q^{71} +(13.1896 - 21.0850i) q^{72} -7.61268 q^{73} +17.1705i q^{75} +(-7.84782 - 22.3797i) q^{78} +41.9388 q^{81} +(5.30272 - 1.85949i) q^{82} +23.0183i q^{87} +(5.99034 + 7.49105i) q^{92} +1.06364 q^{93} +(3.06923 + 8.75255i) q^{94} +(-19.2967 + 2.23966i) q^{96} +(9.34178 - 3.27585i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{6} - 9 q^{8} - 18 q^{9} + 15 q^{12} + 21 q^{18} + 30 q^{25} - 27 q^{26} - 33 q^{36} + 39 q^{48} - 42 q^{49} + 3 q^{52} + 9 q^{54} - 15 q^{58} + 45 q^{62} - 21 q^{64} + 27 q^{72} - 51 q^{78} + 54 q^{81}+ \cdots - 57 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33454 + 0.467979i −0.943662 + 0.330911i
\(3\) 3.43410i 1.98268i 0.131319 + 0.991340i \(0.458079\pi\)
−0.131319 + 0.991340i \(0.541921\pi\)
\(4\) 1.56199 1.24907i 0.780996 0.624536i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.60709 4.58295i −0.656091 1.87098i
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.50000 + 2.39792i −0.530330 + 0.847791i
\(9\) −8.79306 −2.93102
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 4.28944 + 5.36404i 1.23826 + 1.54846i
\(13\) 4.88325 1.35437 0.677185 0.735812i \(-0.263199\pi\)
0.677185 + 0.735812i \(0.263199\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.879635 3.90208i 0.219909 0.975520i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 11.7347 4.11497i 2.76589 0.969907i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) −8.23469 5.15115i −1.68090 1.05147i
\(25\) 5.00000 1.00000
\(26\) −6.51690 + 2.28526i −1.27807 + 0.448176i
\(27\) 19.8940i 3.82860i
\(28\) 0 0
\(29\) 6.70287 1.24469 0.622346 0.782742i \(-0.286180\pi\)
0.622346 + 0.782742i \(0.286180\pi\)
\(30\) 0 0
\(31\) 0.309728i 0.0556288i −0.999613 0.0278144i \(-0.991145\pi\)
0.999613 0.0278144i \(-0.00885474\pi\)
\(32\) 0.652183 + 5.61913i 0.115291 + 0.993332i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −13.7347 + 10.9832i −2.28911 + 1.83053i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 16.7696i 2.68528i
\(40\) 0 0
\(41\) −3.97345 −0.620548 −0.310274 0.950647i \(-0.600421\pi\)
−0.310274 + 0.950647i \(0.600421\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.24435 6.40023i −0.330911 0.943662i
\(47\) 6.55848i 0.956652i −0.878182 0.478326i \(-0.841244\pi\)
0.878182 0.478326i \(-0.158756\pi\)
\(48\) 13.4001 + 3.02076i 1.93415 + 0.436009i
\(49\) −7.00000 −1.00000
\(50\) −6.67270 + 2.33989i −0.943662 + 0.330911i
\(51\) 0 0
\(52\) 7.62760 6.09954i 1.05776 0.845854i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 9.30996 + 26.5493i 1.26692 + 3.61290i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −8.94525 + 3.13680i −1.17457 + 0.411882i
\(59\) 9.59166i 1.24873i −0.781133 0.624364i \(-0.785358\pi\)
0.781133 0.624364i \(-0.214642\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0.144946 + 0.413344i 0.0184082 + 0.0524948i
\(63\) 0 0
\(64\) −3.50000 7.19375i −0.437500 0.899218i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −16.4694 −1.98268
\(70\) 0 0
\(71\) 14.0461i 1.66697i −0.552542 0.833485i \(-0.686342\pi\)
0.552542 0.833485i \(-0.313658\pi\)
\(72\) 13.1896 21.0850i 1.55441 2.48489i
\(73\) −7.61268 −0.890997 −0.445498 0.895283i \(-0.646973\pi\)
−0.445498 + 0.895283i \(0.646973\pi\)
\(74\) 0 0
\(75\) 17.1705i 1.98268i
\(76\) 0 0
\(77\) 0 0
\(78\) −7.84782 22.3797i −0.888590 2.53400i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 41.9388 4.65986
\(82\) 5.30272 1.85949i 0.585588 0.205346i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23.0183i 2.46783i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.99034 + 7.49105i 0.624536 + 0.780996i
\(93\) 1.06364 0.110294
\(94\) 3.06923 + 8.75255i 0.316567 + 0.902756i
\(95\) 0 0
\(96\) −19.2967 + 2.23966i −1.96946 + 0.228585i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9.34178 3.27585i 0.943662 0.330911i
\(99\) 0 0
\(100\) 7.80996 6.24536i 0.780996 0.624536i
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −7.32488 + 11.7096i −0.718264 + 1.14822i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −24.8490 31.0742i −2.39110 2.99012i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.4698 8.37237i 0.972099 0.777355i
\(117\) −42.9388 −3.96969
\(118\) 4.48870 + 12.8005i 0.413218 + 1.17838i
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 13.6452i 1.23035i
\(124\) −0.386873 0.483792i −0.0347422 0.0434458i
\(125\) 0 0
\(126\) 0 0
\(127\) 20.9143i 1.85585i 0.372769 + 0.927924i \(0.378408\pi\)
−0.372769 + 0.927924i \(0.621592\pi\)
\(128\) 8.03741 + 7.96241i 0.710413 + 0.703785i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.81465i 0.245917i −0.992412 0.122958i \(-0.960762\pi\)
0.992412 0.122958i \(-0.0392382\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 21.9790 7.70732i 1.87098 0.656091i
\(139\) 10.9218i 0.926372i 0.886261 + 0.463186i \(0.153294\pi\)
−0.886261 + 0.463186i \(0.846706\pi\)
\(140\) 0 0
\(141\) 22.5225 1.89674
\(142\) 6.57330 + 18.7451i 0.551619 + 1.57306i
\(143\) 0 0
\(144\) −7.73469 + 34.3112i −0.644557 + 2.85927i
\(145\) 0 0
\(146\) 10.1594 3.56257i 0.840800 0.294841i
\(147\) 24.0387i 1.98268i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −8.03544 22.9147i −0.656091 1.87098i
\(151\) 20.2949i 1.65157i −0.563982 0.825787i \(-0.690731\pi\)
0.563982 0.825787i \(-0.309269\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 20.9464 + 26.1940i 1.67706 + 2.09720i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −55.9689 + 19.6265i −4.39733 + 1.54200i
\(163\) 9.68285i 0.758420i 0.925311 + 0.379210i \(0.123804\pi\)
−0.925311 + 0.379210i \(0.876196\pi\)
\(164\) −6.20649 + 4.96312i −0.484645 + 0.387555i
\(165\) 0 0
\(166\) 0 0
\(167\) 9.59166i 0.742225i 0.928588 + 0.371113i \(0.121024\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 10.8462 0.834321
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −10.7721 30.7189i −0.816631 2.32879i
\(175\) 0 0
\(176\) 0 0
\(177\) 32.9388 2.47583
\(178\) 0 0
\(179\) 17.7900i 1.32968i −0.746984 0.664842i \(-0.768499\pi\)
0.746984 0.664842i \(-0.231501\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −11.5000 7.19375i −0.847791 0.530330i
\(185\) 0 0
\(186\) −1.41947 + 0.497760i −0.104080 + 0.0364975i
\(187\) 0 0
\(188\) −8.19201 10.2443i −0.597464 0.747141i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 24.7041 12.0194i 1.78286 0.867423i
\(193\) −27.1457 −1.95399 −0.976995 0.213262i \(-0.931591\pi\)
−0.976995 + 0.213262i \(0.931591\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −10.9339 + 8.74351i −0.780996 + 0.624536i
\(197\) 28.0555 1.99887 0.999436 0.0335834i \(-0.0106919\pi\)
0.999436 + 0.0335834i \(0.0106919\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −7.50000 + 11.9896i −0.530330 + 0.847791i
\(201\) 0 0
\(202\) −8.00724 + 2.80787i −0.563387 + 0.197561i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 42.1700i 2.93102i
\(208\) 4.29548 19.0549i 0.297838 1.32122i
\(209\) 0 0
\(210\) 0 0
\(211\) 28.7750i 1.98095i 0.137686 + 0.990476i \(0.456034\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 48.2359 3.30507
\(214\) 0 0
\(215\) 0 0
\(216\) 47.7041 + 29.8410i 3.24585 + 2.03042i
\(217\) 0 0
\(218\) 0 0
\(219\) 26.1427i 1.76656i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 28.7750i 1.92692i −0.267860 0.963458i \(-0.586316\pi\)
0.267860 0.963458i \(-0.413684\pi\)
\(224\) 0 0
\(225\) −43.9653 −2.93102
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.0543 + 16.0729i −0.660098 + 1.05524i
\(233\) −9.43229 −0.617930 −0.308965 0.951073i \(-0.599983\pi\)
−0.308965 + 0.951073i \(0.599983\pi\)
\(234\) 57.3035 20.0944i 3.74604 1.31361i
\(235\) 0 0
\(236\) −11.9807 14.9821i −0.779876 0.975251i
\(237\) 0 0
\(238\) 0 0
\(239\) 27.1631i 1.75703i 0.477711 + 0.878517i \(0.341467\pi\)
−0.477711 + 0.878517i \(0.658533\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 14.6799 5.14777i 0.943662 0.330911i
\(243\) 84.3401i 5.41042i
\(244\) 0 0
\(245\) 0 0
\(246\) 6.38568 + 18.2101i 0.407136 + 1.16103i
\(247\) 0 0
\(248\) 0.742702 + 0.464592i 0.0471616 + 0.0295016i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −9.78747 27.9110i −0.614121 1.75129i
\(255\) 0 0
\(256\) −14.4525 6.86482i −0.903280 0.429051i
\(257\) −25.3261 −1.57980 −0.789899 0.613237i \(-0.789867\pi\)
−0.789899 + 0.613237i \(0.789867\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −58.9388 −3.64822
\(262\) 1.31720 + 3.75626i 0.0813766 + 0.232062i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.24402 0.0758493 0.0379247 0.999281i \(-0.487925\pi\)
0.0379247 + 0.999281i \(0.487925\pi\)
\(270\) 0 0
\(271\) 28.7750i 1.74796i −0.485965 0.873978i \(-0.661532\pi\)
0.485965 0.873978i \(-0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −25.7250 + 20.5714i −1.54846 + 1.23826i
\(277\) 29.8751 1.79502 0.897511 0.440992i \(-0.145373\pi\)
0.897511 + 0.440992i \(0.145373\pi\)
\(278\) −5.11115 14.5755i −0.306547 0.874182i
\(279\) 2.72346i 0.163049i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −30.0571 + 10.5400i −1.78988 + 0.627651i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −17.5446 21.9400i −1.04108 1.30190i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −5.73469 49.4094i −0.337920 2.91148i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −11.8909 + 9.50879i −0.695865 + 0.556460i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 11.2496 + 32.0806i 0.656091 + 1.87098i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.4193i 1.35437i
\(300\) 21.4472 + 26.8202i 1.23826 + 1.54846i
\(301\) 0 0
\(302\) 9.49758 + 27.0843i 0.546524 + 1.55853i
\(303\) 20.6046i 1.18370i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.7750i 1.64228i 0.570730 + 0.821138i \(0.306660\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.6508i 1.96486i 0.186621 + 0.982432i \(0.440246\pi\)
−0.186621 + 0.982432i \(0.559754\pi\)
\(312\) −40.2121 25.1544i −2.27656 1.42409i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 65.5080 52.3846i 3.63933 2.91025i
\(325\) 24.4163 1.35437
\(326\) −4.53137 12.9221i −0.250969 0.715692i
\(327\) 0 0
\(328\) 5.96017 9.52799i 0.329095 0.526095i
\(329\) 0 0
\(330\) 0 0
\(331\) 31.5264i 1.73285i −0.499310 0.866423i \(-0.666413\pi\)
0.499310 0.866423i \(-0.333587\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −4.48870 12.8005i −0.245610 0.700410i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −14.4746 + 5.07578i −0.787317 + 0.276086i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 24.0217 8.42362i 1.29142 0.452857i
\(347\) 9.59166i 0.514907i −0.966291 0.257454i \(-0.917117\pi\)
0.966291 0.257454i \(-0.0828835\pi\)
\(348\) 28.7516 + 35.9545i 1.54125 + 1.92736i
\(349\) 10.3421 0.553600 0.276800 0.960928i \(-0.410726\pi\)
0.276800 + 0.960928i \(0.410726\pi\)
\(350\) 0 0
\(351\) 97.1473i 5.18534i
\(352\) 0 0
\(353\) −30.7849 −1.63852 −0.819258 0.573425i \(-0.805614\pi\)
−0.819258 + 0.573425i \(0.805614\pi\)
\(354\) −43.9581 + 15.4146i −2.33635 + 0.819279i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 8.32533 + 23.7414i 0.440007 + 1.25477i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 37.7751i 1.98268i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 18.7137 + 4.21858i 0.975520 + 0.219909i
\(369\) 34.9388 1.81884
\(370\) 0 0
\(371\) 0 0
\(372\) 1.66139 1.32856i 0.0861392 0.0688827i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 15.7267 + 9.83772i 0.811041 + 0.507341i
\(377\) 32.7318 1.68577
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −71.8220 −3.67955
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −27.3437 + 27.6013i −1.39538 + 1.40852i
\(385\) 0 0
\(386\) 36.2270 12.7036i 1.84391 0.646597i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 10.5000 16.7854i 0.530330 0.847791i
\(393\) 9.66579 0.487574
\(394\) −37.4412 + 13.1294i −1.88626 + 0.661449i
\(395\) 0 0
\(396\) 0 0
\(397\) −39.6416 −1.98956 −0.994778 0.102061i \(-0.967456\pi\)
−0.994778 + 0.102061i \(0.967456\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.39818 19.5104i 0.219909 0.975520i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1.51248i 0.0753420i
\(404\) 9.37195 7.49444i 0.466272 0.372862i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 36.9122 1.82519 0.912595 0.408864i \(-0.134075\pi\)
0.912595 + 0.408864i \(0.134075\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 19.7347 + 56.2776i 0.969907 + 2.76589i
\(415\) 0 0
\(416\) 3.18478 + 27.4397i 0.156147 + 1.34534i
\(417\) −37.5065 −1.83670
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −13.4661 38.4014i −0.655519 1.86935i
\(423\) 57.6691i 2.80397i
\(424\) 0 0
\(425\) 0 0
\(426\) −64.3727 + 22.5734i −3.11887 + 1.09368i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −77.6279 17.4994i −3.73487 0.841942i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 12.2342 + 34.8885i 0.584575 + 1.66704i
\(439\) 41.5190i 1.98159i −0.135364 0.990796i \(-0.543220\pi\)
0.135364 0.990796i \(-0.456780\pi\)
\(440\) 0 0
\(441\) 61.5514 2.93102
\(442\) 0 0
\(443\) 38.3946i 1.82418i 0.409988 + 0.912091i \(0.365533\pi\)
−0.409988 + 0.912091i \(0.634467\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.4661 + 38.4014i 0.637638 + 1.81836i
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 58.6734 20.5748i 2.76589 0.969907i
\(451\) 0 0
\(452\) 0 0
\(453\) 69.6947 3.27454
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −36.0024 −1.67680 −0.838399 0.545056i \(-0.816508\pi\)
−0.838399 + 0.545056i \(0.816508\pi\)
\(462\) 0 0
\(463\) 28.7750i 1.33729i −0.743583 0.668644i \(-0.766875\pi\)
0.743583 0.668644i \(-0.233125\pi\)
\(464\) 5.89608 26.1551i 0.273719 1.21422i
\(465\) 0 0
\(466\) 12.5878 4.41411i 0.583117 0.204480i
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −67.0700 + 53.6336i −3.10031 + 2.47922i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 23.0000 + 14.3875i 1.05866 + 0.662238i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −12.7118 36.2502i −0.581422 1.65805i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −17.1819 + 13.7398i −0.780996 + 0.624536i
\(485\) 0 0
\(486\) −39.4694 112.555i −1.79037 5.10561i
\(487\) 40.8995i 1.85333i 0.375884 + 0.926667i \(0.377339\pi\)
−0.375884 + 0.926667i \(0.622661\pi\)
\(488\) 0 0
\(489\) −33.2519 −1.50370
\(490\) 0 0
\(491\) 4.67301i 0.210890i 0.994425 + 0.105445i \(0.0336267\pi\)
−0.994425 + 0.105445i \(0.966373\pi\)
\(492\) −17.0439 21.3137i −0.768397 0.960897i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.20858 0.272448i −0.0542670 0.0122333i
\(497\) 0 0
\(498\) 0 0
\(499\) 11.5412i 0.516656i −0.966057 0.258328i \(-0.916828\pi\)
0.966057 0.258328i \(-0.0831715\pi\)
\(500\) 0 0
\(501\) −32.9388 −1.47160
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 37.2469i 1.65419i
\(508\) 26.1235 + 32.6680i 1.15904 + 1.44941i
\(509\) −41.4612 −1.83774 −0.918869 0.394564i \(-0.870896\pi\)
−0.918869 + 0.394564i \(0.870896\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.5000 + 2.39792i 0.994369 + 0.105974i
\(513\) 0 0
\(514\) 33.7987 11.8521i 1.49079 0.522772i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 61.8139i 2.71333i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 78.6561 27.5821i 3.44268 1.20724i
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) −3.51570 4.39645i −0.153584 0.192060i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 84.3401i 3.66005i
\(532\) 0 0
\(533\) −19.4033 −0.840452
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 61.0926 2.63634
\(538\) −1.66020 + 0.582176i −0.0715761 + 0.0250994i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.575595 −0.0247468 −0.0123734 0.999923i \(-0.503939\pi\)
−0.0123734 + 0.999923i \(0.503939\pi\)
\(542\) 13.4661 + 38.4014i 0.578418 + 1.64948i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.06340i 0.387523i −0.981049 0.193761i \(-0.937931\pi\)
0.981049 0.193761i \(-0.0620688\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 24.7041 39.4922i 1.05147 1.68090i
\(553\) 0 0
\(554\) −39.8695 + 13.9809i −1.69389 + 0.593993i
\(555\) 0 0
\(556\) 13.6421 + 17.0597i 0.578553 + 0.723493i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −1.27452 3.63456i −0.0539548 0.153863i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 35.1799 28.1322i 1.48134 1.18458i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 33.6815 + 21.0692i 1.41324 + 0.884044i
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9792i 1.00000i
\(576\) 30.7757 + 63.2551i 1.28232 + 2.63563i
\(577\) −32.6045 −1.35734 −0.678672 0.734441i \(-0.737444\pi\)
−0.678672 + 0.734441i \(0.737444\pi\)
\(578\) −22.6872 + 7.95564i −0.943662 + 0.330911i
\(579\) 93.2211i 3.87414i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 11.4190 18.2546i 0.472522 0.755379i
\(585\) 0 0
\(586\) 0 0
\(587\) 25.2776i 1.04332i −0.853154 0.521660i \(-0.825313\pi\)
0.853154 0.521660i \(-0.174687\pi\)
\(588\) −30.0261 37.5483i −1.23826 1.54846i
\(589\) 0 0
\(590\) 0 0
\(591\) 96.3455i 3.96312i
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −10.9597 31.2539i −0.448176 1.27807i
\(599\) 9.59166i 0.391905i 0.980613 + 0.195952i \(0.0627798\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) −41.1734 25.7558i −1.68090 1.05147i
\(601\) 42.3711 1.72835 0.864176 0.503190i \(-0.167841\pi\)
0.864176 + 0.503190i \(0.167841\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −25.3498 31.7004i −1.03147 1.28987i
\(605\) 0 0
\(606\) −9.64252 27.4977i −0.391701 1.11702i
\(607\) 28.7750i 1.16794i −0.811775 0.583970i \(-0.801498\pi\)
0.811775 0.583970i \(-0.198502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0267i 1.29566i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −13.4661 38.4014i −0.543447 1.54975i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 95.4081 3.82860
\(622\) −16.2158 46.2428i −0.650195 1.85417i
\(623\) 0 0
\(624\) 65.4363 + 14.7511i 2.61955 + 0.590518i
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −98.8163 −3.92759
\(634\) −40.0362 + 14.0394i −1.59004 + 0.557574i
\(635\) 0 0
\(636\) 0 0
\(637\) −34.1828 −1.35437
\(638\) 0 0
\(639\) 123.509i 4.88592i
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.7677i 1.87794i −0.343996 0.938971i \(-0.611781\pi\)
0.343996 0.938971i \(-0.388219\pi\)
\(648\) −62.9081 + 100.566i −2.47126 + 3.95059i
\(649\) 0 0
\(650\) −32.5845 + 11.4263i −1.27807 + 0.448176i
\(651\) 0 0
\(652\) 12.0946 + 15.1245i 0.473661 + 0.592322i
\(653\) 12.1617 0.475925 0.237962 0.971274i \(-0.423520\pi\)
0.237962 + 0.971274i \(0.423520\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.49518 + 15.5047i −0.136464 + 0.605357i
\(657\) 66.9388 2.61153
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 14.7537 + 42.0732i 0.573418 + 1.63522i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 32.1458i 1.24469i
\(668\) 11.9807 + 14.9821i 0.463547 + 0.579675i
\(669\) 98.8163 3.82046
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −21.6868 −0.835966 −0.417983 0.908455i \(-0.637263\pi\)
−0.417983 + 0.908455i \(0.637263\pi\)
\(674\) 0 0
\(675\) 99.4699i 3.82860i
\(676\) 16.9416 13.5477i 0.651601 0.521064i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0239i 1.68453i −0.539066 0.842263i \(-0.681223\pi\)
0.539066 0.842263i \(-0.318777\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.7750i 1.09465i 0.836919 + 0.547326i \(0.184354\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −28.1158 + 22.4833i −1.06880 + 0.854687i
\(693\) 0 0
\(694\) 4.48870 + 12.8005i 0.170389 + 0.485899i
\(695\) 0 0
\(696\) −55.1961 34.5275i −2.09220 1.30876i
\(697\) 0 0
\(698\) −13.8019 + 4.83989i −0.522411 + 0.183192i
\(699\) 32.3915i 1.22516i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 45.4629 + 129.647i 1.71589 + 4.89321i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 41.0837 14.4067i 1.54621 0.542203i
\(707\) 0 0
\(708\) 51.4501 41.1429i 1.93361 1.54624i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.48540 0.0556288
\(714\) 0 0
\(715\) 0 0
\(716\) −22.2210 27.7878i −0.830436 1.03848i
\(717\) −93.2809 −3.48364
\(718\) 0 0
\(719\) 47.9583i 1.78854i 0.447524 + 0.894272i \(0.352306\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 25.3563 8.89160i 0.943662 0.330911i
\(723\) 0 0
\(724\) 0 0
\(725\) 33.5144 1.24469
\(726\) 17.6780 + 50.4124i 0.656091 + 1.87098i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −163.816 −6.06727
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −26.9484 + 3.12776i −0.993332 + 0.115291i
\(737\) 0 0
\(738\) −46.6272 + 16.3506i −1.71637 + 0.601874i
\(739\) 52.1310i 1.91767i 0.283964 + 0.958835i \(0.408350\pi\)
−0.283964 + 0.958835i \(0.591650\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −1.59546 + 2.55051i −0.0584923 + 0.0935064i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −25.5917 5.76907i −0.933234 0.210376i
\(753\) 0 0
\(754\) −43.6819 + 15.3178i −1.59080 + 0.557841i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.2730 1.20614 0.603072 0.797687i \(-0.293943\pi\)
0.603072 + 0.797687i \(0.293943\pi\)
\(762\) 95.8493 33.6112i 3.47225 1.21760i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 46.8385i 1.69124i
\(768\) 23.5745 49.6313i 0.850671 1.79092i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 86.9724i 3.13223i
\(772\) −42.4013 + 33.9069i −1.52606 + 1.22034i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 1.54864i 0.0556288i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 133.347i 4.76542i
\(784\) −6.15745 + 27.3146i −0.219909 + 0.975520i
\(785\) 0 0
\(786\) −12.8994 + 4.52338i −0.460105 + 0.161344i
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 43.8225 35.0434i 1.56111 1.24837i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 52.9033 18.5514i 1.87747 0.658366i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.26092 + 28.0957i 0.115291 + 0.993332i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0.707809 + 2.01846i 0.0249315 + 0.0710974i
\(807\) 4.27210i 0.150385i
\(808\) −9.00000 + 14.3875i −0.316619 + 0.506150i
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 29.6680i 1.04178i 0.853622 + 0.520892i \(0.174401\pi\)
−0.853622 + 0.520892i \(0.825599\pi\)
\(812\) 0 0
\(813\) 98.8163 3.46564
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −49.2608 + 17.2741i −1.72236 + 0.603976i
\(819\) 0 0
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 0 0
\(823\) 22.1533i 0.772214i 0.922454 + 0.386107i \(0.126180\pi\)
−0.922454 + 0.386107i \(0.873820\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −52.6734 65.8693i −1.83053 2.28911i
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 102.594i 3.55895i
\(832\) −17.0914 35.1289i −0.592537 1.21788i
\(833\) 0 0
\(834\) 50.0539 17.5522i 1.73322 0.607784i
\(835\) 0 0
\(836\) 0 0
\(837\) −6.16172 −0.212980
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 15.9285 0.549258
\(842\) 0 0
\(843\) 0 0
\(844\) 35.9420 + 44.9463i 1.23718 + 1.54712i
\(845\) 0 0
\(846\) −26.9879 76.9617i −0.927864 2.64600i
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 75.3440 60.2501i 2.58124 2.06414i
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.1907 1.50952 0.754762 0.655998i \(-0.227752\pi\)
0.754762 + 0.655998i \(0.227752\pi\)
\(858\) 0 0
\(859\) 50.8921i 1.73642i 0.496201 + 0.868208i \(0.334728\pi\)
−0.496201 + 0.868208i \(0.665272\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.1878i 0.414877i 0.978248 + 0.207438i \(0.0665126\pi\)
−0.978248 + 0.207438i \(0.933487\pi\)
\(864\) 111.787 12.9745i 3.80307 0.441402i
\(865\) 0 0
\(866\) 0 0
\(867\) 58.3797i 1.98268i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −32.6542 40.8347i −1.10328 1.37968i
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 19.4300 + 55.4087i 0.655731 + 1.86995i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −82.1428 + 28.8048i −2.76589 + 0.969907i
\(883\) 28.7750i 0.968355i 0.874970 + 0.484178i \(0.160881\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −17.9679 51.2391i −0.603642 1.72141i
\(887\) 29.0215i 0.974445i −0.873278 0.487223i \(-0.838010\pi\)
0.873278 0.487223i \(-0.161990\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −35.9420 44.9463i −1.20343 1.50491i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −80.4242 −2.68528
\(898\) −24.0217 + 8.42362i −0.801615 + 0.281100i
\(899\) 2.07607i 0.0692407i
\(900\) −68.6734 + 54.9159i −2.28911 + 1.83053i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −93.0104 + 32.6157i −3.09006 + 1.08358i
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −52.7584 −1.74988
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −98.8163 −3.25611
\(922\) 48.0466 16.8484i 1.58233 0.554871i
\(923\) 68.5909i 2.25770i
\(924\) 0 0
\(925\) 0 0
\(926\) 13.4661 + 38.4014i 0.442523 + 1.26195i
\(927\) 0 0
\(928\) 4.37150 + 37.6643i 0.143502 + 1.23639i
\(929\) 60.0845 1.97131 0.985653 0.168782i \(-0.0539833\pi\)
0.985653 + 0.168782i \(0.0539833\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −14.7332 + 11.7816i −0.482601 + 0.385920i
\(933\) −118.994 −3.89570
\(934\) 0 0
\(935\) 0 0
\(936\) 64.4081 102.964i 2.10525 3.36547i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 19.0560i 0.620548i
\(944\) −37.4275 8.43717i −1.21816 0.274606i
\(945\) 0 0
\(946\) 0 0
\(947\) 36.5362i 1.18727i −0.804735 0.593634i \(-0.797693\pi\)
0.804735 0.593634i \(-0.202307\pi\)
\(948\) 0 0
\(949\) −37.1746 −1.20674
\(950\) 0 0
\(951\) 103.023i 3.34075i
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 33.9287 + 42.4285i 1.09733 + 1.37224i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 30.9041 0.996905
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 62.1236i 1.99776i 0.0473194 + 0.998880i \(0.484932\pi\)
−0.0473194 + 0.998880i \(0.515068\pi\)
\(968\) 16.5000 26.3771i 0.530330 0.847791i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 105.347 + 131.739i 3.37900 + 4.22551i
\(973\) 0 0
\(974\) −19.1401 54.5820i −0.613288 1.74892i
\(975\) 83.8480i 2.68528i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 44.3760 15.5612i 1.41899 0.497592i
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −2.18687 6.23632i −0.0697859 0.199009i
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 32.7201 + 20.4678i 1.04308 + 0.652491i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 28.7750i 0.914068i −0.889449 0.457034i \(-0.848912\pi\)
0.889449 0.457034i \(-0.151088\pi\)
\(992\) 1.74040 0.201999i 0.0552578 0.00641349i
\(993\) 108.265 3.43568
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 5.40105 + 15.4022i 0.170967 + 0.487549i
\(999\) 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 92.2.b.b.91.2 yes 6
3.2 odd 2 828.2.e.b.91.5 6
4.3 odd 2 inner 92.2.b.b.91.1 6
8.3 odd 2 1472.2.c.c.1471.6 6
8.5 even 2 1472.2.c.c.1471.1 6
12.11 even 2 828.2.e.b.91.6 6
23.22 odd 2 CM 92.2.b.b.91.2 yes 6
69.68 even 2 828.2.e.b.91.5 6
92.91 even 2 inner 92.2.b.b.91.1 6
184.45 odd 2 1472.2.c.c.1471.1 6
184.91 even 2 1472.2.c.c.1471.6 6
276.275 odd 2 828.2.e.b.91.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.b.b.91.1 6 4.3 odd 2 inner
92.2.b.b.91.1 6 92.91 even 2 inner
92.2.b.b.91.2 yes 6 1.1 even 1 trivial
92.2.b.b.91.2 yes 6 23.22 odd 2 CM
828.2.e.b.91.5 6 3.2 odd 2
828.2.e.b.91.5 6 69.68 even 2
828.2.e.b.91.6 6 12.11 even 2
828.2.e.b.91.6 6 276.275 odd 2
1472.2.c.c.1471.1 6 8.5 even 2
1472.2.c.c.1471.1 6 184.45 odd 2
1472.2.c.c.1471.6 6 8.3 odd 2
1472.2.c.c.1471.6 6 184.91 even 2