Properties

Label 2300.2.c.b.1749.1
Level $2300$
Weight $2$
Character 2300.1749
Analytic conductor $18.366$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,2,Mod(1749,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, 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("2300.1749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (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: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1749
Dual form 2300.2.c.b.1749.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +4.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +4.00000i q^{7} -6.00000 q^{9} +2.00000 q^{11} -5.00000i q^{13} -4.00000i q^{17} +2.00000 q^{19} +12.0000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +7.00000 q^{29} -3.00000 q^{31} -6.00000i q^{33} -2.00000i q^{37} -15.0000 q^{39} -9.00000 q^{41} -8.00000i q^{43} -9.00000i q^{47} -9.00000 q^{49} -12.0000 q^{51} +2.00000i q^{53} -6.00000i q^{57} -2.00000 q^{61} -24.0000i q^{63} -14.0000i q^{67} +3.00000 q^{69} -3.00000 q^{71} -3.00000i q^{73} +8.00000i q^{77} +6.00000 q^{79} +9.00000 q^{81} +8.00000i q^{83} -21.0000i q^{87} -12.0000 q^{89} +20.0000 q^{91} +9.00000i q^{93} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} + 4 q^{11} + 4 q^{19} + 24 q^{21} + 14 q^{29} - 6 q^{31} - 30 q^{39} - 18 q^{41} - 18 q^{49} - 24 q^{51} - 4 q^{61} + 6 q^{69} - 6 q^{71} + 12 q^{79} + 18 q^{81} - 24 q^{89} + 40 q^{91} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) − 6.00000i − 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −15.0000 −2.40192
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.00000i − 1.31278i −0.754420 0.656392i \(-0.772082\pi\)
0.754420 0.656392i \(-0.227918\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) − 24.0000i − 3.02372i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 14.0000i − 1.71037i −0.518321 0.855186i \(-0.673443\pi\)
0.518321 0.855186i \(-0.326557\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) − 3.00000i − 0.351123i −0.984468 0.175562i \(-0.943826\pi\)
0.984468 0.175562i \(-0.0561742\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 21.0000i − 2.25144i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 20.0000 2.09657
\(92\) 0 0
\(93\) 9.00000i 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 2.00000i − 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) − 20.0000i − 1.88144i −0.339182 0.940721i \(-0.610150\pi\)
0.339182 0.940721i \(-0.389850\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 30.0000i 2.77350i
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 27.0000i 2.43451i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 17.0000i 1.50851i 0.656584 + 0.754253i \(0.272001\pi\)
−0.656584 + 0.754253i \(0.727999\pi\)
\(128\) 0 0
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 1.00000 0.0848189 0.0424094 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) −27.0000 −2.27381
\(142\) 0 0
\(143\) − 10.0000i − 0.836242i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 27.0000i 2.22692i
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 0 0
\(153\) 24.0000i 1.94029i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) − 5.00000i − 0.391630i −0.980641 0.195815i \(-0.937265\pi\)
0.980641 0.195815i \(-0.0627352\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25.0000 1.86859 0.934294 0.356504i \(-0.116031\pi\)
0.934294 + 0.356504i \(0.116031\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) −36.0000 −2.61861
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 0 0
\(193\) 17.0000i 1.22369i 0.790979 + 0.611843i \(0.209572\pi\)
−0.790979 + 0.611843i \(0.790428\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 9.00000i − 0.641223i −0.947211 0.320612i \(-0.896112\pi\)
0.947211 0.320612i \(-0.103888\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −42.0000 −2.96245
\(202\) 0 0
\(203\) 28.0000i 1.96521i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.00000i − 0.417029i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 9.00000i 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 12.0000i − 0.814613i
\(218\) 0 0
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 24.0000 1.57908
\(232\) 0 0
\(233\) 27.0000i 1.76883i 0.466702 + 0.884414i \(0.345442\pi\)
−0.466702 + 0.884414i \(0.654558\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 18.0000i − 1.16923i
\(238\) 0 0
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 10.0000i − 0.636285i
\(248\) 0 0
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.00000i − 0.436648i −0.975876 0.218324i \(-0.929941\pi\)
0.975876 0.218324i \(-0.0700590\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −42.0000 −2.59973
\(262\) 0 0
\(263\) − 10.0000i − 0.616626i −0.951285 0.308313i \(-0.900236\pi\)
0.951285 0.308313i \(-0.0997645\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 36.0000i 2.20316i
\(268\) 0 0
\(269\) −13.0000 −0.792624 −0.396312 0.918116i \(-0.629710\pi\)
−0.396312 + 0.918116i \(0.629710\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) − 60.0000i − 3.63137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.0000i 0.660926i 0.943819 + 0.330463i \(0.107205\pi\)
−0.943819 + 0.330463i \(0.892795\pi\)
\(278\) 0 0
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) − 26.0000i − 1.54554i −0.634686 0.772770i \(-0.718871\pi\)
0.634686 0.772770i \(-0.281129\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 36.0000i − 2.12501i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 16.0000i 0.934730i 0.884064 + 0.467365i \(0.154797\pi\)
−0.884064 + 0.467365i \(0.845203\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.0000i 1.04447i
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) 0 0
\(309\) −24.0000 −1.36531
\(310\) 0 0
\(311\) 5.00000 0.283524 0.141762 0.989901i \(-0.454723\pi\)
0.141762 + 0.989901i \(0.454723\pi\)
\(312\) 0 0
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 14.0000 0.783850
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 36.0000 1.98474
\(330\) 0 0
\(331\) 11.0000 0.604615 0.302307 0.953211i \(-0.402243\pi\)
0.302307 + 0.953211i \(0.402243\pi\)
\(332\) 0 0
\(333\) 12.0000i 0.657596i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 4.00000i − 0.217894i −0.994048 0.108947i \(-0.965252\pi\)
0.994048 0.108947i \(-0.0347479\pi\)
\(338\) 0 0
\(339\) −60.0000 −3.25875
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 45.0000 2.40192
\(352\) 0 0
\(353\) − 31.0000i − 1.64996i −0.565159 0.824982i \(-0.691185\pi\)
0.565159 0.824982i \(-0.308815\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 48.0000i − 2.54043i
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 21.0000i 1.10221i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) 0 0
\(369\) 54.0000 2.81113
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) − 34.0000i − 1.76045i −0.474554 0.880227i \(-0.657390\pi\)
0.474554 0.880227i \(-0.342610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 35.0000i − 1.80259i
\(378\) 0 0
\(379\) 32.0000 1.64373 0.821865 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 0 0
\(381\) 51.0000 2.61281
\(382\) 0 0
\(383\) − 36.0000i − 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 48.0000i 2.43998i
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 45.0000i 2.26995i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.0000i 1.85698i 0.371361 + 0.928488i \(0.378891\pi\)
−0.371361 + 0.928488i \(0.621109\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 15.0000i 0.747203i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.00000i − 0.198273i
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 3.00000i − 0.146911i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 54.0000i 2.62557i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) 0 0
\(429\) −30.0000 −1.44841
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000i 0.0956730i
\(438\) 0 0
\(439\) 13.0000 0.620456 0.310228 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) 54.0000 2.57143
\(442\) 0 0
\(443\) 29.0000i 1.37783i 0.724841 + 0.688916i \(0.241913\pi\)
−0.724841 + 0.688916i \(0.758087\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 42.0000i 1.98653i
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) − 39.0000i − 1.83238i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 40.0000i 1.87112i 0.353166 + 0.935561i \(0.385105\pi\)
−0.353166 + 0.935561i \(0.614895\pi\)
\(458\) 0 0
\(459\) 36.0000 1.68034
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.0000i − 1.01804i −0.860755 0.509019i \(-0.830008\pi\)
0.860755 0.509019i \(-0.169992\pi\)
\(468\) 0 0
\(469\) 56.0000 2.58584
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) 12.0000i 0.546019i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 31.0000i − 1.40474i −0.711810 0.702372i \(-0.752124\pi\)
0.711810 0.702372i \(-0.247876\pi\)
\(488\) 0 0
\(489\) −15.0000 −0.678323
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) − 28.0000i − 1.26106i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 12.0000i − 0.538274i
\(498\) 0 0
\(499\) −37.0000 −1.65635 −0.828174 0.560471i \(-0.810620\pi\)
−0.828174 + 0.560471i \(0.810620\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.0000i 1.59882i
\(508\) 0 0
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) 18.0000i 0.794719i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 18.0000i − 0.791639i
\(518\) 0 0
\(519\) −54.0000 −2.37034
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 45.0000i 1.94917i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 75.0000i − 3.23649i
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −15.0000 −0.644900 −0.322450 0.946586i \(-0.604506\pi\)
−0.322450 + 0.946586i \(0.604506\pi\)
\(542\) 0 0
\(543\) − 54.0000i − 2.31736i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21.0000i 0.897895i 0.893558 + 0.448948i \(0.148201\pi\)
−0.893558 + 0.448948i \(0.851799\pi\)
\(548\) 0 0
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 14.0000 0.596420
\(552\) 0 0
\(553\) 24.0000i 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.0000i − 1.18640i −0.805056 0.593199i \(-0.797865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 36.0000i 1.51186i
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) − 6.00000i − 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 33.0000i − 1.37381i −0.726748 0.686904i \(-0.758969\pi\)
0.726748 0.686904i \(-0.241031\pi\)
\(578\) 0 0
\(579\) 51.0000 2.11949
\(580\) 0 0
\(581\) −32.0000 −1.32758
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.0000i 0.784214i 0.919920 + 0.392107i \(0.128254\pi\)
−0.919920 + 0.392107i \(0.871746\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −27.0000 −1.11063
\(592\) 0 0
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 30.0000i − 1.22782i
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 13.0000 0.530281 0.265141 0.964210i \(-0.414582\pi\)
0.265141 + 0.964210i \(0.414582\pi\)
\(602\) 0 0
\(603\) 84.0000i 3.42074i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) 0 0
\(609\) 84.0000 3.40385
\(610\) 0 0
\(611\) −45.0000 −1.82051
\(612\) 0 0
\(613\) 20.0000i 0.807792i 0.914805 + 0.403896i \(0.132344\pi\)
−0.914805 + 0.403896i \(0.867656\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) − 48.0000i − 1.92308i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 12.0000i − 0.479234i
\(628\) 0 0
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) − 12.0000i − 0.476957i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 45.0000i 1.78296i
\(638\) 0 0
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) 10.0000i 0.394362i 0.980367 + 0.197181i \(0.0631786\pi\)
−0.980367 + 0.197181i \(0.936821\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 17.0000i − 0.668339i −0.942513 0.334169i \(-0.891544\pi\)
0.942513 0.334169i \(-0.108456\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −36.0000 −1.41095
\(652\) 0 0
\(653\) 3.00000i 0.117399i 0.998276 + 0.0586995i \(0.0186954\pi\)
−0.998276 + 0.0586995i \(0.981305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000i 0.702247i
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) 60.0000i 2.33021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.00000i 0.271041i
\(668\) 0 0
\(669\) −48.0000 −1.85579
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 43.0000i 1.65753i 0.559598 + 0.828764i \(0.310955\pi\)
−0.559598 + 0.828764i \(0.689045\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 9.00000i − 0.344375i −0.985064 0.172188i \(-0.944916\pi\)
0.985064 0.172188i \(-0.0550836\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 12.0000i − 0.457829i
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 0 0
\(693\) − 48.0000i − 1.82337i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) 81.0000 3.06370
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.00000i 0.300871i
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) 0 0
\(713\) − 3.00000i − 0.112351i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 9.00000i − 0.336111i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 6.00000i 0.223142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 30.0000i 1.11264i 0.830969 + 0.556319i \(0.187787\pi\)
−0.830969 + 0.556319i \(0.812213\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) 28.0000i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 28.0000i − 1.03139i
\(738\) 0 0
\(739\) 17.0000 0.625355 0.312678 0.949859i \(-0.398774\pi\)
0.312678 + 0.949859i \(0.398774\pi\)
\(740\) 0 0
\(741\) −30.0000 −1.10208
\(742\) 0 0
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 48.0000i − 1.75623i
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) − 24.0000i − 0.874609i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000i 1.52652i 0.646094 + 0.763258i \(0.276401\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −51.0000 −1.84875 −0.924374 0.381487i \(-0.875412\pi\)
−0.924374 + 0.381487i \(0.875412\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −21.0000 −0.756297
\(772\) 0 0
\(773\) − 52.0000i − 1.87031i −0.354239 0.935155i \(-0.615260\pi\)
0.354239 0.935155i \(-0.384740\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 24.0000i − 0.860995i
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 63.0000i 2.25144i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) 0 0
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 80.0000 2.84447
\(792\) 0 0
\(793\) 10.0000i 0.355110i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.0000i 0.354218i 0.984191 + 0.177109i \(0.0566745\pi\)
−0.984191 + 0.177109i \(0.943325\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 72.0000 2.54399
\(802\) 0 0
\(803\) − 6.00000i − 0.211735i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 39.0000i 1.37287i
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) 0 0
\(813\) − 24.0000i − 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 0 0
\(819\) −120.000 −4.19314
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 45.0000i 1.56860i 0.620381 + 0.784301i \(0.286978\pi\)
−0.620381 + 0.784301i \(0.713022\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 33.0000 1.14476
\(832\) 0 0
\(833\) 36.0000i 1.24733i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 27.0000i − 0.933257i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) − 36.0000i − 1.23991i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 28.0000i − 0.962091i
\(848\) 0 0
\(849\) −78.0000 −2.67695
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 3.00000i − 0.102478i −0.998686 0.0512390i \(-0.983683\pi\)
0.998686 0.0512390i \(-0.0163170\pi\)
\(858\) 0 0
\(859\) −13.0000 −0.443554 −0.221777 0.975097i \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) 0 0
\(861\) −108.000 −3.68063
\(862\) 0 0
\(863\) 7.00000i 0.238283i 0.992877 + 0.119141i \(0.0380142\pi\)
−0.992877 + 0.119141i \(0.961986\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 3.00000i − 0.101885i
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −70.0000 −2.37186
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 6.00000i − 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) 0 0
\(879\) 48.0000 1.61900
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) 12.0000i 0.403832i 0.979403 + 0.201916i \(0.0647168\pi\)
−0.979403 + 0.201916i \(0.935283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.0000i 1.30949i 0.755849 + 0.654746i \(0.227224\pi\)
−0.755849 + 0.654746i \(0.772776\pi\)
\(888\) 0 0
\(889\) −68.0000 −2.28065
\(890\) 0 0
\(891\) 18.0000 0.603023
\(892\) 0 0
\(893\) − 18.0000i − 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 15.0000i − 0.500835i
\(898\) 0 0
\(899\) −21.0000 −0.700389
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) − 96.0000i − 3.19468i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 38.0000i 1.26177i 0.775877 + 0.630885i \(0.217308\pi\)
−0.775877 + 0.630885i \(0.782692\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −46.0000 −1.52405 −0.762024 0.647549i \(-0.775794\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(912\) 0 0
\(913\) 16.0000i 0.529523i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 60.0000i − 1.98137i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) 15.0000i 0.493731i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 48.0000i 1.57653i
\(928\) 0 0
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) − 15.0000i − 0.491078i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 0 0
\(939\) 48.0000 1.56642
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) − 9.00000i − 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.00000i 0.0974869i 0.998811 + 0.0487435i \(0.0155217\pi\)
−0.998811 + 0.0487435i \(0.984478\pi\)
\(948\) 0 0
\(949\) −15.0000 −0.486921
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) − 50.0000i − 1.61966i −0.586665 0.809829i \(-0.699560\pi\)
0.586665 0.809829i \(-0.300440\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 42.0000i − 1.35767i
\(958\) 0 0
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 13.0000i 0.418052i 0.977910 + 0.209026i \(0.0670293\pi\)
−0.977910 + 0.209026i \(0.932971\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 0 0
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.00000i 0.191370i 0.995412 + 0.0956851i \(0.0305042\pi\)
−0.995412 + 0.0956851i \(0.969496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 108.000i − 3.43768i
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) − 33.0000i − 1.04722i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) 0 0
\(999\) 18.0000 0.569495
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 2300.2.c.b.1749.1 2
5.2 odd 4 92.2.a.a.1.1 1
5.3 odd 4 2300.2.a.h.1.1 1
5.4 even 2 inner 2300.2.c.b.1749.2 2
15.2 even 4 828.2.a.c.1.1 1
20.3 even 4 9200.2.a.b.1.1 1
20.7 even 4 368.2.a.g.1.1 1
35.27 even 4 4508.2.a.d.1.1 1
40.27 even 4 1472.2.a.b.1.1 1
40.37 odd 4 1472.2.a.n.1.1 1
60.47 odd 4 3312.2.a.q.1.1 1
115.22 even 4 2116.2.a.a.1.1 1
460.367 odd 4 8464.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.a.a.1.1 1 5.2 odd 4
368.2.a.g.1.1 1 20.7 even 4
828.2.a.c.1.1 1 15.2 even 4
1472.2.a.b.1.1 1 40.27 even 4
1472.2.a.n.1.1 1 40.37 odd 4
2116.2.a.a.1.1 1 115.22 even 4
2300.2.a.h.1.1 1 5.3 odd 4
2300.2.c.b.1749.1 2 1.1 even 1 trivial
2300.2.c.b.1749.2 2 5.4 even 2 inner
3312.2.a.q.1.1 1 60.47 odd 4
4508.2.a.d.1.1 1 35.27 even 4
8464.2.a.s.1.1 1 460.367 odd 4
9200.2.a.b.1.1 1 20.3 even 4