Properties

Label 300.2.e.a.251.3
Level $300$
Weight $2$
Character 300.251
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,2,Mod(251,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("300.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.e (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: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.3
Root \(0.866025 - 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 300.251
Dual form 300.2.e.a.251.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 - 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 - 1.93649i) q^{4} +(1.50000 - 1.93649i) q^{6} +(-2.59808 - 1.11803i) q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+(0.866025 - 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 - 1.93649i) q^{4} +(1.50000 - 1.93649i) q^{6} +(-2.59808 - 1.11803i) q^{8} +3.00000 q^{9} +(-0.866025 - 3.35410i) q^{12} +(-3.50000 + 1.93649i) q^{16} -4.47214i q^{17} +(2.59808 - 3.35410i) q^{18} +7.74597i q^{19} -3.46410 q^{23} +(-4.50000 - 1.93649i) q^{24} +5.19615 q^{27} +7.74597i q^{31} +(-0.866025 + 5.59017i) q^{32} +(-5.00000 - 3.87298i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(8.66025 + 6.70820i) q^{38} +(-3.00000 + 3.87298i) q^{46} -10.3923 q^{47} +(-6.06218 + 3.35410i) q^{48} +7.00000 q^{49} -7.74597i q^{51} -4.47214i q^{53} +(4.50000 - 5.80948i) q^{54} +13.4164i q^{57} -2.00000 q^{61} +(8.66025 + 6.70820i) q^{62} +(5.50000 + 5.80948i) q^{64} +(-8.66025 + 2.23607i) q^{68} -6.00000 q^{69} +(-7.79423 - 3.35410i) q^{72} +(15.0000 - 3.87298i) q^{76} -7.74597i q^{79} +9.00000 q^{81} +3.46410 q^{83} +(1.73205 + 6.70820i) q^{92} +13.4164i q^{93} +(-9.00000 + 11.6190i) q^{94} +(-1.50000 + 9.68246i) q^{96} +(6.06218 - 7.82624i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 6 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 6 q^{6} + 12 q^{9} - 14 q^{16} - 18 q^{24} - 20 q^{34} - 6 q^{36} - 12 q^{46} + 28 q^{49} + 18 q^{54} - 8 q^{61} + 22 q^{64} - 24 q^{69} + 60 q^{76} + 36 q^{81} - 36 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 1.11803i 0.612372 0.790569i
\(3\) 1.73205 1.00000
\(4\) −0.500000 1.93649i −0.250000 0.968246i
\(5\) 0 0
\(6\) 1.50000 1.93649i 0.612372 0.790569i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −2.59808 1.11803i −0.918559 0.395285i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −0.866025 3.35410i −0.250000 0.968246i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.50000 + 1.93649i −0.875000 + 0.484123i
\(17\) 4.47214i 1.08465i −0.840168 0.542326i \(-0.817544\pi\)
0.840168 0.542326i \(-0.182456\pi\)
\(18\) 2.59808 3.35410i 0.612372 0.790569i
\(19\) 7.74597i 1.77705i 0.458831 + 0.888523i \(0.348268\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) −4.50000 1.93649i −0.918559 0.395285i
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.74597i 1.39122i 0.718421 + 0.695608i \(0.244865\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −0.866025 + 5.59017i −0.153093 + 0.988212i
\(33\) 0 0
\(34\) −5.00000 3.87298i −0.857493 0.664211i
\(35\) 0 0
\(36\) −1.50000 5.80948i −0.250000 0.968246i
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.66025 + 6.70820i 1.40488 + 1.08821i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 + 3.87298i −0.442326 + 0.571040i
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) −6.06218 + 3.35410i −0.875000 + 0.484123i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 7.74597i 1.08465i
\(52\) 0 0
\(53\) 4.47214i 0.614295i −0.951662 0.307148i \(-0.900625\pi\)
0.951662 0.307148i \(-0.0993745\pi\)
\(54\) 4.50000 5.80948i 0.612372 0.790569i
\(55\) 0 0
\(56\) 0 0
\(57\) 13.4164i 1.77705i
\(58\) 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 \(0.540868\pi\)
\(62\) 8.66025 + 6.70820i 1.09985 + 0.851943i
\(63\) 0 0
\(64\) 5.50000 + 5.80948i 0.687500 + 0.726184i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −8.66025 + 2.23607i −1.05021 + 0.271163i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −7.79423 3.35410i −0.918559 0.395285i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 15.0000 3.87298i 1.72062 0.444262i
\(77\) 0 0
\(78\) 0 0
\(79\) 7.74597i 0.871489i −0.900070 0.435745i \(-0.856485\pi\)
0.900070 0.435745i \(-0.143515\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.73205 + 6.70820i 0.180579 + 0.699379i
\(93\) 13.4164i 1.39122i
\(94\) −9.00000 + 11.6190i −0.928279 + 1.19840i
\(95\) 0 0
\(96\) −1.50000 + 9.68246i −0.153093 + 0.988212i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 6.06218 7.82624i 0.612372 0.790569i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −8.66025 6.70820i −0.857493 0.664211i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −5.00000 3.87298i −0.485643 0.376177i
\(107\) 10.3923 1.00466 0.502331 0.864675i \(-0.332476\pi\)
0.502331 + 0.864675i \(0.332476\pi\)
\(108\) −2.59808 10.0623i −0.250000 0.968246i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.47214i 0.420703i 0.977626 + 0.210352i \(0.0674609\pi\)
−0.977626 + 0.210352i \(0.932539\pi\)
\(114\) 15.0000 + 11.6190i 1.40488 + 1.08821i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −1.73205 + 2.23607i −0.156813 + 0.202444i
\(123\) 0 0
\(124\) 15.0000 3.87298i 1.34704 0.347804i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.2583 1.11803i 0.995105 0.0988212i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −5.00000 + 11.6190i −0.428746 + 0.996317i
\(137\) 22.3607i 1.91040i −0.295958 0.955201i \(-0.595639\pi\)
0.295958 0.955201i \(-0.404361\pi\)
\(138\) −5.19615 + 6.70820i −0.442326 + 0.571040i
\(139\) 23.2379i 1.97101i −0.169638 0.985506i \(-0.554260\pi\)
0.169638 0.985506i \(-0.445740\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) 0 0
\(144\) −10.5000 + 5.80948i −0.875000 + 0.484123i
\(145\) 0 0
\(146\) 0 0
\(147\) 12.1244 1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 23.2379i 1.89107i −0.325515 0.945537i \(-0.605538\pi\)
0.325515 0.945537i \(-0.394462\pi\)
\(152\) 8.66025 20.1246i 0.702439 1.63232i
\(153\) 13.4164i 1.08465i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −8.66025 6.70820i −0.688973 0.533676i
\(159\) 7.74597i 0.614295i
\(160\) 0 0
\(161\) 0 0
\(162\) 7.79423 10.0623i 0.612372 0.790569i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 3.00000 3.87298i 0.232845 0.300602i
\(167\) −24.2487 −1.87642 −0.938211 0.346064i \(-0.887518\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 23.2379i 1.77705i
\(172\) 0 0
\(173\) 22.3607i 1.70005i −0.526742 0.850026i \(-0.676586\pi\)
0.526742 0.850026i \(-0.323414\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −3.46410 −0.256074
\(184\) 9.00000 + 3.87298i 0.663489 + 0.285520i
\(185\) 0 0
\(186\) 15.0000 + 11.6190i 1.09985 + 0.851943i
\(187\) 0 0
\(188\) 5.19615 + 20.1246i 0.378968 + 1.46774i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 9.52628 + 10.0623i 0.687500 + 0.726184i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.50000 13.5554i −0.250000 0.968246i
\(197\) 4.47214i 0.318626i 0.987228 + 0.159313i \(0.0509280\pi\)
−0.987228 + 0.159313i \(0.949072\pi\)
\(198\) 0 0
\(199\) 23.2379i 1.64729i 0.567105 + 0.823646i \(0.308063\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) −15.0000 + 3.87298i −1.05021 + 0.271163i
\(205\) 0 0
\(206\) 0 0
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 7.74597i 0.533254i −0.963800 0.266627i \(-0.914091\pi\)
0.963800 0.266627i \(-0.0859092\pi\)
\(212\) −8.66025 + 2.23607i −0.594789 + 0.153574i
\(213\) 0 0
\(214\) 9.00000 11.6190i 0.615227 0.794255i
\(215\) 0 0
\(216\) −13.5000 5.80948i −0.918559 0.395285i
\(217\) 0 0
\(218\) −12.1244 + 15.6525i −0.821165 + 1.06012i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5.00000 + 3.87298i 0.332595 + 0.257627i
\(227\) 24.2487 1.60944 0.804722 0.593652i \(-0.202314\pi\)
0.804722 + 0.593652i \(0.202314\pi\)
\(228\) 25.9808 6.70820i 1.72062 0.444262i
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.3607i 1.46490i 0.680823 + 0.732448i \(0.261622\pi\)
−0.680823 + 0.732448i \(0.738378\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.4164i 0.871489i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −9.52628 + 12.2984i −0.612372 + 0.790569i
\(243\) 15.5885 1.00000
\(244\) 1.00000 + 3.87298i 0.0640184 + 0.247942i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 8.66025 20.1246i 0.549927 1.27791i
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 8.50000 13.5554i 0.531250 0.847215i
\(257\) 31.3050i 1.95275i 0.216085 + 0.976375i \(0.430671\pi\)
−0.216085 + 0.976375i \(0.569329\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −31.1769 −1.92245 −0.961225 0.275764i \(-0.911069\pi\)
−0.961225 + 0.275764i \(0.911069\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 7.74597i 0.470534i 0.971931 + 0.235267i \(0.0755965\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 8.66025 + 15.6525i 0.525105 + 0.949071i
\(273\) 0 0
\(274\) −25.0000 19.3649i −1.51031 1.16988i
\(275\) 0 0
\(276\) 3.00000 + 11.6190i 0.180579 + 0.699379i
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −25.9808 20.1246i −1.55822 1.20699i
\(279\) 23.2379i 1.39122i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −15.5885 + 20.1246i −0.928279 + 1.19840i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2.59808 + 16.7705i −0.153093 + 0.988212i
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 31.3050i 1.82885i 0.404750 + 0.914427i \(0.367359\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 10.5000 13.5554i 0.612372 0.790569i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −25.9808 20.1246i −1.49502 1.15804i
\(303\) 0 0
\(304\) −15.0000 27.1109i −0.860309 1.55492i
\(305\) 0 0
\(306\) −15.0000 11.6190i −0.857493 0.664211i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −15.0000 + 3.87298i −0.843816 + 0.217872i
\(317\) 22.3607i 1.25590i 0.778253 + 0.627950i \(0.216106\pi\)
−0.778253 + 0.627950i \(0.783894\pi\)
\(318\) −8.66025 6.70820i −0.485643 0.376177i
\(319\) 0 0
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 34.6410 1.92748
\(324\) −4.50000 17.4284i −0.250000 0.968246i
\(325\) 0 0
\(326\) 0 0
\(327\) −24.2487 −1.34096
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 23.2379i 1.27727i 0.769510 + 0.638635i \(0.220501\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −1.73205 6.70820i −0.0950586 0.368161i
\(333\) 0 0
\(334\) −21.0000 + 27.1109i −1.14907 + 1.48344i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) −11.2583 + 14.5344i −0.612372 + 0.790569i
\(339\) 7.74597i 0.420703i
\(340\) 0 0
\(341\) 0 0
\(342\) 25.9808 + 20.1246i 1.40488 + 1.08821i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −25.0000 19.3649i −1.34401 1.04106i
\(347\) 10.3923 0.557888 0.278944 0.960307i \(-0.410016\pi\)
0.278944 + 0.960307i \(0.410016\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.3050i 1.66619i −0.553127 0.833097i \(-0.686565\pi\)
0.553127 0.833097i \(-0.313435\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −41.0000 −2.15789
\(362\) 19.0526 24.5967i 1.00138 1.29278i
\(363\) −19.0526 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) −3.00000 + 3.87298i −0.156813 + 0.202444i
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 12.1244 6.70820i 0.632026 0.349689i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 25.9808 6.70820i 1.34704 0.347804i
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 27.0000 + 11.6190i 1.39242 + 0.599202i
\(377\) 0 0
\(378\) 0 0
\(379\) 38.7298i 1.98942i 0.102733 + 0.994709i \(0.467241\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 38.1051 1.94708 0.973540 0.228515i \(-0.0733872\pi\)
0.973540 + 0.228515i \(0.0733872\pi\)
\(384\) 19.5000 1.93649i 0.995105 0.0988212i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 15.4919i 0.783461i
\(392\) −18.1865 7.82624i −0.918559 0.395285i
\(393\) 0 0
\(394\) 5.00000 + 3.87298i 0.251896 + 0.195118i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 25.9808 + 20.1246i 1.30230 + 1.00876i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −8.66025 + 20.1246i −0.428746 + 0.996317i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 38.7298i 1.91040i
\(412\) 0 0
\(413\) 0 0
\(414\) −9.00000 + 11.6190i −0.442326 + 0.571040i
\(415\) 0 0
\(416\) 0 0
\(417\) 40.2492i 1.97101i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −8.66025 6.70820i −0.421575 0.326550i
\(423\) −31.1769 −1.51587
\(424\) −5.00000 + 11.6190i −0.242821 + 0.564266i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −5.19615 20.1246i −0.251166 0.972760i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −18.1865 + 10.0623i −0.875000 + 0.484123i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.00000 + 27.1109i 0.335239 + 1.29838i
\(437\) 26.8328i 1.28359i
\(438\) 0 0
\(439\) 38.7298i 1.84847i −0.381819 0.924237i \(-0.624702\pi\)
0.381819 0.924237i \(-0.375298\pi\)
\(440\) 0 0
\(441\) 21.0000 1.00000
\(442\) 0 0
\(443\) −38.1051 −1.81043 −0.905214 0.424955i \(-0.860290\pi\)
−0.905214 + 0.424955i \(0.860290\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.66025 2.23607i 0.407344 0.105176i
\(453\) 40.2492i 1.89107i
\(454\) 21.0000 27.1109i 0.985579 1.27238i
\(455\) 0 0
\(456\) 15.0000 34.8569i 0.702439 1.63232i
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 22.5167 29.0689i 1.05213 1.35830i
\(459\) 23.2379i 1.08465i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 25.0000 + 19.3649i 1.15810 + 0.897062i
\(467\) 24.2487 1.12210 0.561048 0.827783i \(-0.310398\pi\)
0.561048 + 0.827783i \(0.310398\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) −15.0000 11.6190i −0.688973 0.533676i
\(475\) 0 0
\(476\) 0 0
\(477\) 13.4164i 0.614295i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.73205 2.23607i 0.0788928 0.101850i
\(483\) 0 0
\(484\) 5.50000 + 21.3014i 0.250000 + 0.968246i
\(485\) 0 0
\(486\) 13.5000 17.4284i 0.612372 0.790569i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 5.19615 + 2.23607i 0.235219 + 0.101222i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −15.0000 27.1109i −0.673520 1.21731i
\(497\) 0 0
\(498\) 5.19615 6.70820i 0.232845 0.300602i
\(499\) 7.74597i 0.346757i 0.984855 + 0.173379i \(0.0554684\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 0 0
\(501\) −42.0000 −1.87642
\(502\) 0 0
\(503\) −3.46410 −0.154457 −0.0772283 0.997013i \(-0.524607\pi\)
−0.0772283 + 0.997013i \(0.524607\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −22.5167 −1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −7.79423 21.2426i −0.344459 0.938801i
\(513\) 40.2492i 1.77705i
\(514\) 35.0000 + 27.1109i 1.54378 + 1.19581i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 38.7298i 1.70005i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −27.0000 + 34.8569i −1.17726 + 1.51983i
\(527\) 34.6410 1.50899
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 8.66025 + 6.70820i 0.371990 + 0.288142i
\(543\) 38.1051 1.63525
\(544\) 25.0000 + 3.87298i 1.07187 + 0.166053i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −43.3013 + 11.1803i −1.84974 + 0.477600i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 15.5885 + 6.70820i 0.663489 + 0.285520i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −45.0000 + 11.6190i −1.90843 + 0.492753i
\(557\) 22.3607i 0.947452i 0.880672 + 0.473726i \(0.157091\pi\)
−0.880672 + 0.473726i \(0.842909\pi\)
\(558\) 25.9808 + 20.1246i 1.09985 + 0.851943i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.1769 1.31395 0.656975 0.753912i \(-0.271836\pi\)
0.656975 + 0.753912i \(0.271836\pi\)
\(564\) 9.00000 + 34.8569i 0.378968 + 1.46774i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 38.7298i 1.62079i −0.585882 0.810397i \(-0.699252\pi\)
0.585882 0.810397i \(-0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 16.5000 + 17.4284i 0.687500 + 0.726184i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −2.59808 + 3.35410i −0.108066 + 0.139512i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 35.0000 + 27.1109i 1.44584 + 1.11994i
\(587\) −45.0333 −1.85872 −0.929362 0.369170i \(-0.879642\pi\)
−0.929362 + 0.369170i \(0.879642\pi\)
\(588\) −6.06218 23.4787i −0.250000 0.968246i
\(589\) −60.0000 −2.47226
\(590\) 0 0
\(591\) 7.74597i 0.318626i
\(592\) 0 0
\(593\) 4.47214i 0.183649i 0.995775 + 0.0918243i \(0.0292698\pi\)
−0.995775 + 0.0918243i \(0.970730\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 40.2492i 1.64729i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −45.0000 + 11.6190i −1.83102 + 0.472768i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −43.3013 6.70820i −1.75610 0.272054i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −25.9808 + 6.70820i −1.05021 + 0.271163i
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.1935i 1.98046i 0.139459 + 0.990228i \(0.455464\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) 0 0
\(619\) 23.2379i 0.934010i −0.884255 0.467005i \(-0.845333\pi\)
0.884255 0.467005i \(-0.154667\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 38.7298i 1.54181i 0.636950 + 0.770905i \(0.280196\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) −8.66025 + 20.1246i −0.344486 + 0.800514i
\(633\) 13.4164i 0.533254i
\(634\) 25.0000 + 19.3649i 0.992877 + 0.769079i
\(635\) 0 0
\(636\) −15.0000 + 3.87298i −0.594789 + 0.153574i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 15.5885 20.1246i 0.615227 0.794255i
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30.0000 38.7298i 1.18033 1.52380i
\(647\) −24.2487 −0.953315 −0.476658 0.879089i \(-0.658152\pi\)
−0.476658 + 0.879089i \(0.658152\pi\)
\(648\) −23.3827 10.0623i −0.918559 0.395285i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 49.1935i 1.92509i 0.271122 + 0.962545i \(0.412605\pi\)
−0.271122 + 0.962545i \(0.587395\pi\)
\(654\) −21.0000 + 27.1109i −0.821165 + 1.06012i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 25.9808 + 20.1246i 1.00977 + 0.782165i
\(663\) 0 0
\(664\) −9.00000 3.87298i −0.349268 0.150301i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 12.1244 + 46.9574i 0.469105 + 1.81684i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 6.50000 + 25.1744i 0.250000 + 0.968246i
\(677\) 31.3050i 1.20315i −0.798817 0.601574i \(-0.794541\pi\)
0.798817 0.601574i \(-0.205459\pi\)
\(678\) 8.66025 + 6.70820i 0.332595 + 0.257627i
\(679\) 0 0
\(680\) 0 0
\(681\) 42.0000 1.60944
\(682\) 0 0
\(683\) −38.1051 −1.45805 −0.729026 0.684486i \(-0.760027\pi\)
−0.729026 + 0.684486i \(0.760027\pi\)
\(684\) 45.0000 11.6190i 1.72062 0.444262i
\(685\) 0 0
\(686\) 0 0
\(687\) 45.0333 1.71813
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 7.74597i 0.294670i −0.989087 0.147335i \(-0.952930\pi\)
0.989087 0.147335i \(-0.0470696\pi\)
\(692\) −43.3013 + 11.1803i −1.64607 + 0.425013i
\(693\) 0 0
\(694\) 9.00000 11.6190i 0.341635 0.441049i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 29.4449 38.0132i 1.11450 1.43882i
\(699\) 38.7298i 1.46490i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −35.0000 27.1109i −1.31724 1.02033i
\(707\) 0 0
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 23.2379i 0.871489i
\(712\) 0 0
\(713\) 26.8328i 1.00490i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −35.5070 + 45.8394i −1.32144 + 1.70597i
\(723\) 3.46410 0.128831
\(724\) −11.0000 42.6028i −0.408812 1.58332i
\(725\) 0 0
\(726\) −16.5000 + 21.3014i −0.612372 + 0.790569i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 1.73205 + 6.70820i 0.0640184 + 0.247942i
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 3.00000 19.3649i 0.110581 0.713800i
\(737\) 0 0
\(738\) 0 0
\(739\) 54.2218i 1.99458i −0.0735712 0.997290i \(-0.523440\pi\)
0.0735712 0.997290i \(-0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1769 −1.14377 −0.571885 0.820334i \(-0.693788\pi\)
−0.571885 + 0.820334i \(0.693788\pi\)
\(744\) 15.0000 34.8569i 0.549927 1.27791i
\(745\) 0 0
\(746\) 0 0
\(747\) 10.3923 0.380235
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 54.2218i 1.97858i −0.145962 0.989290i \(-0.546628\pi\)
0.145962 0.989290i \(-0.453372\pi\)
\(752\) 36.3731 20.1246i 1.32639 0.733869i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 43.3013 + 33.5410i 1.57277 + 1.21826i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 33.0000 42.6028i 1.19234 1.53930i
\(767\) 0 0
\(768\) 14.7224 23.4787i 0.531250 0.847215i
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 54.2218i 1.95275i
\(772\) 0 0
\(773\) 4.47214i 0.160852i −0.996761 0.0804258i \(-0.974372\pi\)
0.996761 0.0804258i \(-0.0256280\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 17.3205 + 13.4164i 0.619380 + 0.479770i
\(783\) 0 0
\(784\) −24.5000 + 13.5554i −0.875000 + 0.484123i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 8.66025 2.23607i 0.308509 0.0796566i
\(789\) −54.0000 −1.92245
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 45.0000 11.6190i 1.59498 0.411823i
\(797\) 49.1935i 1.74252i −0.490819 0.871262i \(-0.663302\pi\)
0.490819 0.871262i \(-0.336698\pi\)
\(798\) 0 0
\(799\) 46.4758i 1.64420i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 23.2379i 0.815993i 0.912983 + 0.407997i \(0.133772\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) 13.4164i 0.470534i
\(814\) 0 0
\(815\) 0 0
\(816\) 15.0000 + 27.1109i 0.525105 + 0.949071i
\(817\) 0 0
\(818\) −22.5167 + 29.0689i −0.787277 + 1.01637i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −43.3013 33.5410i −1.51031 1.16988i
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.0333 −1.56596 −0.782981 0.622046i \(-0.786302\pi\)
−0.782981 + 0.622046i \(0.786302\pi\)
\(828\) 5.19615 + 20.1246i 0.180579 + 0.699379i
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 31.3050i 1.08465i
\(834\) −45.0000 34.8569i −1.55822 1.20699i
\(835\) 0 0
\(836\) 0 0
\(837\) 40.2492i 1.39122i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 32.9090 42.4853i 1.13412 1.46414i
\(843\) 0 0
\(844\) −15.0000 + 3.87298i −0.516321 + 0.133314i
\(845\) 0 0
\(846\) −27.0000 + 34.8569i −0.928279 + 1.19840i
\(847\) 0 0
\(848\) 8.66025 + 15.6525i 0.297394 + 0.537508i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −27.0000 11.6190i −0.922841 0.397128i
\(857\) 58.1378i 1.98595i −0.118331 0.992974i \(-0.537755\pi\)
0.118331 0.992974i \(-0.462245\pi\)
\(858\) 0 0
\(859\) 38.7298i 1.32144i 0.750630 + 0.660722i \(0.229750\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.1051 1.29711 0.648557 0.761166i \(-0.275373\pi\)
0.648557 + 0.761166i \(0.275373\pi\)
\(864\) −4.50000 + 29.0474i −0.153093 + 0.988212i
\(865\) 0 0
\(866\) 0 0
\(867\) −5.19615 −0.176471
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 36.3731 + 15.6525i 1.23175 + 0.530060i
\(873\) 0 0
\(874\) −30.0000 23.2379i −1.01477 0.786034i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) −43.3013 33.5410i −1.46135 1.13195i
\(879\) 54.2218i 1.82885i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 18.1865 23.4787i 0.612372 0.790569i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −33.0000 + 42.6028i −1.10866 + 1.43127i
\(887\) 58.8897 1.97732 0.988662 0.150160i \(-0.0479788\pi\)
0.988662 + 0.150160i \(0.0479788\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 80.4984i 2.69378i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) 5.00000 11.6190i 0.166298 0.386441i
\(905\) 0 0
\(906\) −45.0000 34.8569i −1.49502 1.15804i
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −12.1244 46.9574i −0.402361 1.55834i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −25.9808 46.9574i −0.860309 1.55492i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −13.0000 50.3488i −0.429532 1.66357i
\(917\) 0 0
\(918\) −25.9808 20.1246i −0.857493 0.664211i
\(919\) 23.2379i 0.766548i 0.923635 + 0.383274i \(0.125203\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 54.2218i 1.77705i
\(932\) 43.3013 11.1803i 1.41838 0.366224i
\(933\) 0 0
\(934\) 21.0000 27.1109i 0.687141 0.887095i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −58.8897 −1.91366 −0.956830 0.290650i \(-0.906129\pi\)
−0.956830 + 0.290650i \(0.906129\pi\)
\(948\) −25.9808 + 6.70820i −0.843816 + 0.217872i
\(949\) 0 0
\(950\) 0 0
\(951\) 38.7298i 1.25590i
\(952\) 0 0
\(953\) 58.1378i 1.88327i 0.336640 + 0.941634i \(0.390710\pi\)
−0.336640 + 0.941634i \(0.609290\pi\)
\(954\) −15.0000 11.6190i −0.485643 0.376177i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 31.1769 1.00466
\(964\) −1.00000 3.87298i −0.0322078 0.124740i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 28.5788 + 12.2984i 0.918559 + 0.395285i
\(969\) 60.0000 1.92748
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −7.79423 30.1869i −0.250000 0.968246i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 7.00000 3.87298i 0.224065 0.123971i
\(977\) 4.47214i 0.143076i −0.997438 0.0715382i \(-0.977209\pi\)
0.997438 0.0715382i \(-0.0227908\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −42.0000 −1.34096
\(982\) 0 0
\(983\) −3.46410 −0.110488 −0.0552438 0.998473i \(-0.517594\pi\)
−0.0552438 + 0.998473i \(0.517594\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 54.2218i 1.72241i −0.508257 0.861206i \(-0.669710\pi\)
0.508257 0.861206i \(-0.330290\pi\)
\(992\) −43.3013 6.70820i −1.37482 0.212986i
\(993\) 40.2492i 1.27727i
\(994\) 0 0
\(995\) 0 0
\(996\) −3.00000 11.6190i −0.0950586 0.368161i
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 8.66025 + 6.70820i 0.274136 + 0.212344i
\(999\) 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 300.2.e.a.251.3 4
3.2 odd 2 inner 300.2.e.a.251.2 4
4.3 odd 2 inner 300.2.e.a.251.1 4
5.2 odd 4 60.2.h.b.59.4 yes 4
5.3 odd 4 60.2.h.b.59.1 4
5.4 even 2 inner 300.2.e.a.251.2 4
12.11 even 2 inner 300.2.e.a.251.4 4
15.2 even 4 60.2.h.b.59.1 4
15.8 even 4 60.2.h.b.59.4 yes 4
15.14 odd 2 CM 300.2.e.a.251.3 4
20.3 even 4 60.2.h.b.59.2 yes 4
20.7 even 4 60.2.h.b.59.3 yes 4
20.19 odd 2 inner 300.2.e.a.251.4 4
40.3 even 4 960.2.o.a.959.3 4
40.13 odd 4 960.2.o.a.959.1 4
40.27 even 4 960.2.o.a.959.2 4
40.37 odd 4 960.2.o.a.959.4 4
60.23 odd 4 60.2.h.b.59.3 yes 4
60.47 odd 4 60.2.h.b.59.2 yes 4
60.59 even 2 inner 300.2.e.a.251.1 4
120.53 even 4 960.2.o.a.959.4 4
120.77 even 4 960.2.o.a.959.1 4
120.83 odd 4 960.2.o.a.959.2 4
120.107 odd 4 960.2.o.a.959.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.h.b.59.1 4 5.3 odd 4
60.2.h.b.59.1 4 15.2 even 4
60.2.h.b.59.2 yes 4 20.3 even 4
60.2.h.b.59.2 yes 4 60.47 odd 4
60.2.h.b.59.3 yes 4 20.7 even 4
60.2.h.b.59.3 yes 4 60.23 odd 4
60.2.h.b.59.4 yes 4 5.2 odd 4
60.2.h.b.59.4 yes 4 15.8 even 4
300.2.e.a.251.1 4 4.3 odd 2 inner
300.2.e.a.251.1 4 60.59 even 2 inner
300.2.e.a.251.2 4 3.2 odd 2 inner
300.2.e.a.251.2 4 5.4 even 2 inner
300.2.e.a.251.3 4 1.1 even 1 trivial
300.2.e.a.251.3 4 15.14 odd 2 CM
300.2.e.a.251.4 4 12.11 even 2 inner
300.2.e.a.251.4 4 20.19 odd 2 inner
960.2.o.a.959.1 4 40.13 odd 4
960.2.o.a.959.1 4 120.77 even 4
960.2.o.a.959.2 4 40.27 even 4
960.2.o.a.959.2 4 120.83 odd 4
960.2.o.a.959.3 4 40.3 even 4
960.2.o.a.959.3 4 120.107 odd 4
960.2.o.a.959.4 4 40.37 odd 4
960.2.o.a.959.4 4 120.53 even 4