Properties

Label 60.2.h.b.59.1
Level $60$
Weight $2$
Character 60.59
Analytic conductor $0.479$
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] = [60,2,Mod(59,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("60.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.h (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: \(0.479102412128\)
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: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 59.1
Root \(-0.309017 + 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 60.59
Dual form 60.2.h.b.59.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.11803 - 0.866025i) q^{2} +1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(1.50000 - 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+(-1.11803 - 0.866025i) q^{2} +1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(1.50000 - 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +(-2.50000 - 1.93649i) q^{10} +(-3.35410 + 0.866025i) q^{12} +3.87298i q^{15} +(-3.50000 + 1.93649i) q^{16} -4.47214 q^{17} +(3.35410 + 2.59808i) q^{18} -7.74597i q^{19} +(1.11803 + 4.33013i) q^{20} -3.46410i q^{23} +(4.50000 + 1.93649i) q^{24} +5.00000 q^{25} -5.19615i q^{27} +(3.35410 - 4.33013i) q^{30} +7.74597i q^{31} +(5.59017 + 0.866025i) q^{32} +(5.00000 + 3.87298i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(-6.70820 + 8.66025i) q^{38} +(2.50000 - 5.80948i) q^{40} -6.70820 q^{45} +(-3.00000 + 3.87298i) q^{46} +10.3923i q^{47} +(-3.35410 - 6.06218i) q^{48} -7.00000 q^{49} +(-5.59017 - 4.33013i) q^{50} -7.74597i q^{51} +4.47214 q^{53} +(-4.50000 + 5.80948i) q^{54} +13.4164 q^{57} +(-7.50000 + 1.93649i) q^{60} -2.00000 q^{61} +(6.70820 - 8.66025i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(-2.23607 - 8.66025i) q^{68} +6.00000 q^{69} +(-3.35410 + 7.79423i) q^{72} +8.66025i q^{75} +(15.0000 - 3.87298i) q^{76} +7.74597i q^{79} +(-7.82624 + 4.33013i) q^{80} +9.00000 q^{81} +3.46410i q^{83} -10.0000 q^{85} +(7.50000 + 5.80948i) q^{90} +(6.70820 - 1.73205i) q^{92} -13.4164 q^{93} +(9.00000 - 11.6190i) q^{94} -17.3205i q^{95} +(-1.50000 + 9.68246i) q^{96} +(7.82624 + 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9} - 10 q^{10} - 14 q^{16} + 18 q^{24} + 20 q^{25} + 20 q^{34} - 6 q^{36} + 10 q^{40} - 12 q^{46} - 28 q^{49} - 18 q^{54} - 30 q^{60} - 8 q^{61} - 22 q^{64} + 24 q^{69} + 60 q^{76} + 36 q^{81} - 40 q^{85} + 30 q^{90} + 36 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11803 0.866025i −0.790569 0.612372i
\(3\) 1.73205i 1.00000i
\(4\) 0.500000 + 1.93649i 0.250000 + 0.968246i
\(5\) 2.23607 1.00000
\(6\) 1.50000 1.93649i 0.612372 0.790569i
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.11803 2.59808i 0.395285 0.918559i
\(9\) −3.00000 −1.00000
\(10\) −2.50000 1.93649i −0.790569 0.612372i
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −3.35410 + 0.866025i −0.968246 + 0.250000i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 3.87298i 1.00000i
\(16\) −3.50000 + 1.93649i −0.875000 + 0.484123i
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 3.35410 + 2.59808i 0.790569 + 0.612372i
\(19\) 7.74597i 1.77705i −0.458831 0.888523i \(-0.651732\pi\)
0.458831 0.888523i \(-0.348268\pi\)
\(20\) 1.11803 + 4.33013i 0.250000 + 0.968246i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410i 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 4.50000 + 1.93649i 0.918559 + 0.395285i
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 3.35410 4.33013i 0.612372 0.790569i
\(31\) 7.74597i 1.39122i 0.718421 + 0.695608i \(0.244865\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.59017 + 0.866025i 0.988212 + 0.153093i
\(33\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −6.70820 + 8.66025i −1.08821 + 1.40488i
\(39\) 0 0
\(40\) 2.50000 5.80948i 0.395285 0.918559i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −6.70820 −1.00000
\(46\) −3.00000 + 3.87298i −0.442326 + 0.571040i
\(47\) 10.3923i 1.51587i 0.652328 + 0.757937i \(0.273792\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) −3.35410 6.06218i −0.484123 0.875000i
\(49\) −7.00000 −1.00000
\(50\) −5.59017 4.33013i −0.790569 0.612372i
\(51\) 7.74597i 1.08465i
\(52\) 0 0
\(53\) 4.47214 0.614295 0.307148 0.951662i \(-0.400625\pi\)
0.307148 + 0.951662i \(0.400625\pi\)
\(54\) −4.50000 + 5.80948i −0.612372 + 0.790569i
\(55\) 0 0
\(56\) 0 0
\(57\) 13.4164 1.77705
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −7.50000 + 1.93649i −0.968246 + 0.250000i
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 6.70820 8.66025i 0.851943 1.09985i
\(63\) 0 0
\(64\) −5.50000 5.80948i −0.687500 0.726184i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −2.23607 8.66025i −0.271163 1.05021i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.35410 + 7.79423i −0.395285 + 0.918559i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 15.0000 3.87298i 1.72062 0.444262i
\(77\) 0 0
\(78\) 0 0
\(79\) 7.74597i 0.871489i 0.900070 + 0.435745i \(0.143515\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −7.82624 + 4.33013i −0.875000 + 0.484123i
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 7.50000 + 5.80948i 0.790569 + 0.612372i
\(91\) 0 0
\(92\) 6.70820 1.73205i 0.699379 0.180579i
\(93\) −13.4164 −1.39122
\(94\) 9.00000 11.6190i 0.928279 1.19840i
\(95\) 17.3205i 1.77705i
\(96\) −1.50000 + 9.68246i −0.153093 + 0.988212i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 7.82624 + 6.06218i 0.790569 + 0.612372i
\(99\) 0 0
\(100\) 2.50000 + 9.68246i 0.250000 + 0.968246i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −6.70820 + 8.66025i −0.664211 + 0.857493i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −5.00000 3.87298i −0.485643 0.376177i
\(107\) 10.3923i 1.00466i −0.864675 0.502331i \(-0.832476\pi\)
0.864675 0.502331i \(-0.167524\pi\)
\(108\) 10.0623 2.59808i 0.968246 0.250000i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.47214 −0.420703 −0.210352 0.977626i \(-0.567461\pi\)
−0.210352 + 0.977626i \(0.567461\pi\)
\(114\) −15.0000 11.6190i −1.40488 1.08821i
\(115\) 7.74597i 0.722315i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 10.0623 + 4.33013i 0.918559 + 0.395285i
\(121\) −11.0000 −1.00000
\(122\) 2.23607 + 1.73205i 0.202444 + 0.156813i
\(123\) 0 0
\(124\) −15.0000 + 3.87298i −1.34704 + 0.347804i
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.11803 + 11.2583i 0.0988212 + 0.995105i
\(129\) 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
\(135\) 11.6190i 1.00000i
\(136\) −5.00000 + 11.6190i −0.428746 + 0.996317i
\(137\) −22.3607 −1.91040 −0.955201 0.295958i \(-0.904361\pi\)
−0.955201 + 0.295958i \(0.904361\pi\)
\(138\) −6.70820 5.19615i −0.571040 0.442326i
\(139\) 23.2379i 1.97101i 0.169638 + 0.985506i \(0.445740\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 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
\(147\) 12.1244i 1.00000i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 7.50000 9.68246i 0.612372 0.790569i
\(151\) 23.2379i 1.89107i −0.325515 0.945537i \(-0.605538\pi\)
0.325515 0.945537i \(-0.394462\pi\)
\(152\) −20.1246 8.66025i −1.63232 0.702439i
\(153\) 13.4164 1.08465
\(154\) 0 0
\(155\) 17.3205i 1.39122i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 6.70820 8.66025i 0.533676 0.688973i
\(159\) 7.74597i 0.614295i
\(160\) 12.5000 + 1.93649i 0.988212 + 0.153093i
\(161\) 0 0
\(162\) −10.0623 7.79423i −0.790569 0.612372i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 3.00000 3.87298i 0.232845 0.300602i
\(167\) 24.2487i 1.87642i 0.346064 + 0.938211i \(0.387518\pi\)
−0.346064 + 0.938211i \(0.612482\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 11.1803 + 8.66025i 0.857493 + 0.664211i
\(171\) 23.2379i 1.77705i
\(172\) 0 0
\(173\) 22.3607 1.70005 0.850026 0.526742i \(-0.176586\pi\)
0.850026 + 0.526742i \(0.176586\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −3.35410 12.9904i −0.250000 0.968246i
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 3.46410i 0.256074i
\(184\) −9.00000 3.87298i −0.663489 0.285520i
\(185\) 0 0
\(186\) 15.0000 + 11.6190i 1.09985 + 0.851943i
\(187\) 0 0
\(188\) −20.1246 + 5.19615i −1.46774 + 0.378968i
\(189\) 0 0
\(190\) −15.0000 + 19.3649i −1.08821 + 1.40488i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 10.0623 9.52628i 0.726184 0.687500i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.50000 13.5554i −0.250000 0.968246i
\(197\) 4.47214 0.318626 0.159313 0.987228i \(-0.449072\pi\)
0.159313 + 0.987228i \(0.449072\pi\)
\(198\) 0 0
\(199\) 23.2379i 1.64729i −0.567105 0.823646i \(-0.691937\pi\)
0.567105 0.823646i \(-0.308063\pi\)
\(200\) 5.59017 12.9904i 0.395285 0.918559i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 15.0000 3.87298i 1.05021 0.271163i
\(205\) 0 0
\(206\) 0 0
\(207\) 10.3923i 0.722315i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 7.74597i 0.533254i −0.963800 0.266627i \(-0.914091\pi\)
0.963800 0.266627i \(-0.0859092\pi\)
\(212\) 2.23607 + 8.66025i 0.153574 + 0.594789i
\(213\) 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
\(218\) −15.6525 12.1244i −1.06012 0.821165i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 5.00000 + 3.87298i 0.332595 + 0.257627i
\(227\) 24.2487i 1.60944i −0.593652 0.804722i \(-0.702314\pi\)
0.593652 0.804722i \(-0.297686\pi\)
\(228\) 6.70820 + 25.9808i 0.444262 + 1.72062i
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −6.70820 + 8.66025i −0.442326 + 0.571040i
\(231\) 0 0
\(232\) 0 0
\(233\) −22.3607 −1.46490 −0.732448 0.680823i \(-0.761622\pi\)
−0.732448 + 0.680823i \(0.761622\pi\)
\(234\) 0 0
\(235\) 23.2379i 1.51587i
\(236\) 0 0
\(237\) −13.4164 −0.871489
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −7.50000 13.5554i −0.484123 0.875000i
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 12.2984 + 9.52628i 0.790569 + 0.612372i
\(243\) 15.5885i 1.00000i
\(244\) −1.00000 3.87298i −0.0640184 0.247942i
\(245\) −15.6525 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 20.1246 + 8.66025i 1.27791 + 0.549927i
\(249\) −6.00000 −0.380235
\(250\) −12.5000 9.68246i −0.790569 0.612372i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 17.3205i 1.08465i
\(256\) 8.50000 13.5554i 0.531250 0.847215i
\(257\) 31.3050 1.95275 0.976375 0.216085i \(-0.0693287\pi\)
0.976375 + 0.216085i \(0.0693287\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 31.1769i 1.92245i −0.275764 0.961225i \(-0.588931\pi\)
0.275764 0.961225i \(-0.411069\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −10.0623 + 12.9904i −0.612372 + 0.790569i
\(271\) 7.74597i 0.470534i 0.971931 + 0.235267i \(0.0755965\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 15.6525 8.66025i 0.949071 0.525105i
\(273\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 20.1246 25.9808i 1.20699 1.55822i
\(279\) 23.2379i 1.39122i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 20.1246 + 15.5885i 1.19840 + 0.928279i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 30.0000 1.77705
\(286\) 0 0
\(287\) 0 0
\(288\) −16.7705 2.59808i −0.988212 0.153093i
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −31.3050 −1.82885 −0.914427 0.404750i \(-0.867359\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) −10.5000 + 13.5554i −0.612372 + 0.790569i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −16.7705 + 4.33013i −0.968246 + 0.250000i
\(301\) 0 0
\(302\) −20.1246 + 25.9808i −1.15804 + 1.49502i
\(303\) 0 0
\(304\) 15.0000 + 27.1109i 0.860309 + 1.55492i
\(305\) −4.47214 −0.256074
\(306\) −15.0000 11.6190i −0.857493 0.664211i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 15.0000 19.3649i 0.851943 1.09985i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −15.0000 + 3.87298i −0.843816 + 0.217872i
\(317\) 22.3607 1.25590 0.627950 0.778253i \(-0.283894\pi\)
0.627950 + 0.778253i \(0.283894\pi\)
\(318\) 6.70820 8.66025i 0.376177 0.485643i
\(319\) 0 0
\(320\) −12.2984 12.9904i −0.687500 0.726184i
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 34.6410i 1.92748i
\(324\) 4.50000 + 17.4284i 0.250000 + 0.968246i
\(325\) 0 0
\(326\) 0 0
\(327\) 24.2487i 1.34096i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 23.2379i 1.27727i 0.769510 + 0.638635i \(0.220501\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −6.70820 + 1.73205i −0.368161 + 0.0950586i
\(333\) 0 0
\(334\) 21.0000 27.1109i 1.14907 1.48344i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −14.5344 11.2583i −0.790569 0.612372i
\(339\) 7.74597i 0.420703i
\(340\) −5.00000 19.3649i −0.271163 1.05021i
\(341\) 0 0
\(342\) 20.1246 25.9808i 1.08821 1.40488i
\(343\) 0 0
\(344\) 0 0
\(345\) 13.4164 0.722315
\(346\) −25.0000 19.3649i −1.34401 1.04106i
\(347\) 10.3923i 0.557888i −0.960307 0.278944i \(-0.910016\pi\)
0.960307 0.278944i \(-0.0899844\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.3050 1.66619 0.833097 0.553127i \(-0.186565\pi\)
0.833097 + 0.553127i \(0.186565\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −7.50000 + 17.4284i −0.395285 + 0.918559i
\(361\) −41.0000 −2.15789
\(362\) −24.5967 19.0526i −1.29278 1.00138i
\(363\) 19.0526i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) −3.00000 + 3.87298i −0.156813 + 0.202444i
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 6.70820 + 12.1244i 0.349689 + 0.632026i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −6.70820 25.9808i −0.347804 1.34704i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 19.3649i 1.00000i
\(376\) 27.0000 + 11.6190i 1.39242 + 0.599202i
\(377\) 0 0
\(378\) 0 0
\(379\) 38.7298i 1.98942i −0.102733 0.994709i \(-0.532759\pi\)
0.102733 0.994709i \(-0.467241\pi\)
\(380\) 33.5410 8.66025i 1.72062 0.444262i
\(381\) 0 0
\(382\) 0 0
\(383\) 38.1051i 1.94708i 0.228515 + 0.973540i \(0.426613\pi\)
−0.228515 + 0.973540i \(0.573387\pi\)
\(384\) −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
\(392\) −7.82624 + 18.1865i −0.395285 + 0.918559i
\(393\) 0 0
\(394\) −5.00000 3.87298i −0.251896 0.195118i
\(395\) 17.3205i 0.871489i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −20.1246 + 25.9808i −1.00876 + 1.30230i
\(399\) 0 0
\(400\) −17.5000 + 9.68246i −0.875000 + 0.484123i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 20.1246 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) −20.1246 8.66025i −0.996317 0.428746i
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 38.7298i 1.91040i
\(412\) 0 0
\(413\) 0 0
\(414\) 9.00000 11.6190i 0.442326 0.571040i
\(415\) 7.74597i 0.380235i
\(416\) 0 0
\(417\) −40.2492 −1.97101
\(418\) 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 \(0.123221\pi\)
\(422\) −6.70820 + 8.66025i −0.326550 + 0.421575i
\(423\) 31.1769i 1.51587i
\(424\) 5.00000 11.6190i 0.242821 0.564266i
\(425\) −22.3607 −1.08465
\(426\) 0 0
\(427\) 0 0
\(428\) 20.1246 5.19615i 0.972760 0.251166i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 10.0623 + 18.1865i 0.484123 + 0.875000i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.00000 + 27.1109i 0.335239 + 1.29838i
\(437\) −26.8328 −1.28359
\(438\) 0 0
\(439\) 38.7298i 1.84847i 0.381819 + 0.924237i \(0.375298\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 21.0000 1.00000
\(442\) 0 0
\(443\) 38.1051i 1.81043i −0.424955 0.905214i \(-0.639710\pi\)
0.424955 0.905214i \(-0.360290\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 16.7705 + 12.9904i 0.790569 + 0.612372i
\(451\) 0 0
\(452\) −2.23607 8.66025i −0.105176 0.407344i
\(453\) 40.2492 1.89107
\(454\) −21.0000 + 27.1109i −0.985579 + 1.27238i
\(455\) 0 0
\(456\) 15.0000 34.8569i 0.702439 1.63232i
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 29.0689 + 22.5167i 1.35830 + 1.05213i
\(459\) 23.2379i 1.08465i
\(460\) 15.0000 3.87298i 0.699379 0.180579i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) −30.0000 −1.39122
\(466\) 25.0000 + 19.3649i 1.15810 + 0.897062i
\(467\) 24.2487i 1.12210i −0.827783 0.561048i \(-0.810398\pi\)
0.827783 0.561048i \(-0.189602\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 20.1246 25.9808i 0.928279 1.19840i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 15.0000 + 11.6190i 0.688973 + 0.533676i
\(475\) 38.7298i 1.77705i
\(476\) 0 0
\(477\) −13.4164 −0.614295
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −3.35410 + 21.6506i −0.153093 + 0.988212i
\(481\) 0 0
\(482\) −2.23607 1.73205i −0.101850 0.0788928i
\(483\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −2.23607 + 5.19615i −0.101222 + 0.235219i
\(489\) 0 0
\(490\) 17.5000 + 13.5554i 0.790569 + 0.612372i
\(491\) 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
\(498\) 6.70820 + 5.19615i 0.300602 + 0.232845i
\(499\) 7.74597i 0.346757i −0.984855 0.173379i \(-0.944532\pi\)
0.984855 0.173379i \(-0.0554684\pi\)
\(500\) 5.59017 + 21.6506i 0.250000 + 0.968246i
\(501\) −42.0000 −1.87642
\(502\) 0 0
\(503\) 3.46410i 0.154457i −0.997013 0.0772283i \(-0.975393\pi\)
0.997013 0.0772283i \(-0.0246070\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 22.5167i 1.00000i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) −15.0000 + 19.3649i −0.664211 + 0.857493i
\(511\) 0 0
\(512\) −21.2426 + 7.79423i −0.938801 + 0.344459i
\(513\) −40.2492 −1.77705
\(514\) −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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −27.0000 + 34.8569i −1.17726 + 1.51983i
\(527\) 34.6410i 1.50899i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) −11.1803 8.66025i −0.485643 0.376177i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 23.2379i 1.00466i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 22.5000 5.80948i 0.968246 0.250000i
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 6.70820 8.66025i 0.288142 0.371990i
\(543\) 38.1051i 1.63525i
\(544\) −25.0000 3.87298i −1.07187 0.166053i
\(545\) 31.3050 1.34096
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −11.1803 43.3013i −0.477600 1.84974i
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 6.70820 15.5885i 0.285520 0.663489i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −45.0000 + 11.6190i −1.90843 + 0.492753i
\(557\) 22.3607 0.947452 0.473726 0.880672i \(-0.342909\pi\)
0.473726 + 0.880672i \(0.342909\pi\)
\(558\) −20.1246 + 25.9808i −0.851943 + 1.09985i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.1769i 1.31395i 0.753912 + 0.656975i \(0.228164\pi\)
−0.753912 + 0.656975i \(0.771836\pi\)
\(564\) −9.00000 34.8569i −0.378968 1.46774i
\(565\) −10.0000 −0.420703
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) −33.5410 25.9808i −1.40488 1.08821i
\(571\) 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
\(575\) 17.3205i 0.722315i
\(576\) 16.5000 + 17.4284i 0.687500 + 0.726184i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −3.35410 2.59808i −0.139512 0.108066i
\(579\) 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
\(587\) 45.0333i 1.85872i 0.369170 + 0.929362i \(0.379642\pi\)
−0.369170 + 0.929362i \(0.620358\pi\)
\(588\) 23.4787 6.06218i 0.968246 0.250000i
\(589\) 60.0000 2.47226
\(590\) 0 0
\(591\) 7.74597i 0.318626i
\(592\) 0 0
\(593\) −4.47214 −0.183649 −0.0918243 0.995775i \(-0.529270\pi\)
−0.0918243 + 0.995775i \(0.529270\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 40.2492 1.64729
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 22.5000 + 9.68246i 0.918559 + 0.395285i
\(601\) −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
\(605\) −24.5967 −1.00000
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 6.70820 43.3013i 0.272054 1.75610i
\(609\) 0 0
\(610\) 5.00000 + 3.87298i 0.202444 + 0.156813i
\(611\) 0 0
\(612\) 6.70820 + 25.9808i 0.271163 + 1.05021i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.1935 1.98046 0.990228 0.139459i \(-0.0445365\pi\)
0.990228 + 0.139459i \(0.0445365\pi\)
\(618\) 0 0
\(619\) 23.2379i 0.934010i 0.884255 + 0.467005i \(0.154667\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) −33.5410 + 8.66025i −1.34704 + 0.347804i
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 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 \(0.719804\pi\)
\(632\) 20.1246 + 8.66025i 0.800514 + 0.344486i
\(633\) 13.4164 0.533254
\(634\) −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
\(640\) 2.50000 + 25.1744i 0.0988212 + 0.995105i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −20.1246 15.5885i −0.794255 0.615227i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30.0000 38.7298i 1.18033 1.52380i
\(647\) 24.2487i 0.953315i 0.879089 + 0.476658i \(0.158152\pi\)
−0.879089 + 0.476658i \(0.841848\pi\)
\(648\) 10.0623 23.3827i 0.395285 0.918559i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −49.1935 −1.92509 −0.962545 0.271122i \(-0.912605\pi\)
−0.962545 + 0.271122i \(0.912605\pi\)
\(654\) 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 \(0.359271\pi\)
\(662\) 20.1246 25.9808i 0.782165 1.00977i
\(663\) 0 0
\(664\) 9.00000 + 3.87298i 0.349268 + 0.150301i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −46.9574 + 12.1244i −1.81684 + 0.469105i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 6.50000 + 25.1744i 0.250000 + 0.968246i
\(677\) −31.3050 −1.20315 −0.601574 0.798817i \(-0.705459\pi\)
−0.601574 + 0.798817i \(0.705459\pi\)
\(678\) −6.70820 + 8.66025i −0.257627 + 0.332595i
\(679\) 0 0
\(680\) −11.1803 + 25.9808i −0.428746 + 0.996317i
\(681\) 42.0000 1.60944
\(682\) 0 0
\(683\) 38.1051i 1.45805i −0.684486 0.729026i \(-0.739973\pi\)
0.684486 0.729026i \(-0.260027\pi\)
\(684\) −45.0000 + 11.6190i −1.72062 + 0.444262i
\(685\) −50.0000 −1.91040
\(686\) 0 0
\(687\) 45.0333i 1.71813i
\(688\) 0 0
\(689\) 0 0
\(690\) −15.0000 11.6190i −0.571040 0.442326i
\(691\) 7.74597i 0.294670i −0.989087 0.147335i \(-0.952930\pi\)
0.989087 0.147335i \(-0.0470696\pi\)
\(692\) 11.1803 + 43.3013i 0.425013 + 1.64607i
\(693\) 0 0
\(694\) −9.00000 + 11.6190i −0.341635 + 0.441049i
\(695\) 51.9615i 1.97101i
\(696\) 0 0
\(697\) 0 0
\(698\) 38.0132 + 29.4449i 1.43882 + 1.11450i
\(699\) 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
\(705\) −40.2492 −1.51587
\(706\) −35.0000 27.1109i −1.31724 1.02033i
\(707\) 0 0
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 23.2379i 0.871489i
\(712\) 0 0
\(713\) 26.8328 1.00490
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 23.4787 12.9904i 0.875000 0.484123i
\(721\) 0 0
\(722\) 45.8394 + 35.5070i 1.70597 + 1.32144i
\(723\) 3.46410i 0.128831i
\(724\) 11.0000 + 42.6028i 0.408812 + 1.58332i
\(725\) 0 0
\(726\) −16.5000 + 21.3014i −0.612372 + 0.790569i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 6.70820 1.73205i 0.247942 0.0640184i
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 27.1109i 1.00000i
\(736\) 3.00000 19.3649i 0.110581 0.713800i
\(737\) 0 0
\(738\) 0 0
\(739\) 54.2218i 1.99458i 0.0735712 + 0.997290i \(0.476560\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.1769i 1.14377i −0.820334 0.571885i \(-0.806212\pi\)
0.820334 0.571885i \(-0.193788\pi\)
\(744\) −15.0000 + 34.8569i −0.549927 + 1.27791i
\(745\) 0 0
\(746\) 0 0
\(747\) 10.3923i 0.380235i
\(748\) 0 0
\(749\) 0 0
\(750\) 16.7705 21.6506i 0.612372 0.790569i
\(751\) 54.2218i 1.97858i −0.145962 0.989290i \(-0.546628\pi\)
0.145962 0.989290i \(-0.453372\pi\)
\(752\) −20.1246 36.3731i −0.733869 1.32639i
\(753\) 0 0
\(754\) 0 0
\(755\) 51.9615i 1.89107i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −33.5410 + 43.3013i −1.21826 + 1.57277i
\(759\) 0 0
\(760\) −45.0000 19.3649i −1.63232 0.702439i
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 30.0000 1.08465
\(766\) 33.0000 42.6028i 1.19234 1.53930i
\(767\) 0 0
\(768\) 23.4787 + 14.7224i 0.847215 + 0.531250i
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 54.2218i 1.95275i
\(772\) 0 0
\(773\) 4.47214 0.160852 0.0804258 0.996761i \(-0.474372\pi\)
0.0804258 + 0.996761i \(0.474372\pi\)
\(774\) 0 0
\(775\) 38.7298i 1.39122i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 13.4164 17.3205i 0.479770 0.619380i
\(783\) 0 0
\(784\) 24.5000 13.5554i 0.875000 0.484123i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 2.23607 + 8.66025i 0.0796566 + 0.308509i
\(789\) 54.0000 1.92245
\(790\) 15.0000 19.3649i 0.533676 0.688973i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 17.3205i 0.614295i
\(796\) 45.0000 11.6190i 1.59498 0.411823i
\(797\) −49.1935 −1.74252 −0.871262 0.490819i \(-0.836698\pi\)
−0.871262 + 0.490819i \(0.836698\pi\)
\(798\) 0 0
\(799\) 46.4758i 1.64420i
\(800\) 27.9508 + 4.33013i 0.988212 + 0.153093i
\(801\) 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 \(\pi\)
\(810\) −22.5000 17.4284i −0.790569 0.612372i
\(811\) 23.2379i 0.815993i 0.912983 + 0.407997i \(0.133772\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) −13.4164 −0.470534
\(814\) 0 0
\(815\) 0 0
\(816\) 15.0000 + 27.1109i 0.525105 + 0.949071i
\(817\) 0 0
\(818\) −29.0689 22.5167i −1.01637 0.787277i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −33.5410 + 43.3013i −1.16988 + 1.51031i
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.0333i 1.56596i 0.622046 + 0.782981i \(0.286302\pi\)
−0.622046 + 0.782981i \(0.713698\pi\)
\(828\) −20.1246 + 5.19615i −0.699379 + 0.180579i
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 6.70820 8.66025i 0.232845 0.300602i
\(831\) 0 0
\(832\) 0 0
\(833\) 31.3050 1.08465
\(834\) 45.0000 + 34.8569i 1.55822 + 1.20699i
\(835\) 54.2218i 1.87642i
\(836\) 0 0
\(837\) 40.2492 1.39122
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) −42.4853 32.9090i −1.46414 1.13412i
\(843\) 0 0
\(844\) 15.0000 3.87298i 0.516321 0.133314i
\(845\) 29.0689 1.00000
\(846\) −27.0000 + 34.8569i −0.928279 + 1.19840i
\(847\) 0 0
\(848\) −15.6525 + 8.66025i −0.537508 + 0.297394i
\(849\) 0 0
\(850\) 25.0000 + 19.3649i 0.857493 + 0.664211i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 51.9615i 1.77705i
\(856\) −27.0000 11.6190i −0.922841 0.397128i
\(857\) −58.1378 −1.98595 −0.992974 0.118331i \(-0.962245\pi\)
−0.992974 + 0.118331i \(0.962245\pi\)
\(858\) 0 0
\(859\) 38.7298i 1.32144i −0.750630 0.660722i \(-0.770250\pi\)
0.750630 0.660722i \(-0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.1051i 1.29711i 0.761166 + 0.648557i \(0.224627\pi\)
−0.761166 + 0.648557i \(0.775373\pi\)
\(864\) 4.50000 29.0474i 0.153093 0.988212i
\(865\) 50.0000 1.70005
\(866\) 0 0
\(867\) 5.19615i 0.176471i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 15.6525 36.3731i 0.530060 1.23175i
\(873\) 0 0
\(874\) 30.0000 + 23.2379i 1.01477 + 0.786034i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 33.5410 43.3013i 1.13195 1.46135i
\(879\) 54.2218i 1.82885i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −23.4787 18.1865i −0.790569 0.612372i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −33.0000 + 42.6028i −1.10866 + 1.43127i
\(887\) 58.8897i 1.97732i −0.150160 0.988662i \(-0.547979\pi\)
0.150160 0.988662i \(-0.452021\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 80.4984 2.69378
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −7.50000 29.0474i −0.250000 0.968246i
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) −5.00000 + 11.6190i −0.166298 + 0.386441i
\(905\) 49.1935 1.63525
\(906\) −45.0000 34.8569i −1.49502 1.15804i
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 46.9574 12.1244i 1.55834 0.402361i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −46.9574 + 25.9808i −1.55492 + 0.860309i
\(913\) 0 0
\(914\) 0 0
\(915\) 7.74597i 0.256074i
\(916\) −13.0000 50.3488i −0.429532 1.66357i
\(917\) 0 0
\(918\) 20.1246 25.9808i 0.664211 0.857493i
\(919\) 23.2379i 0.766548i −0.923635 0.383274i \(-0.874797\pi\)
0.923635 0.383274i \(-0.125203\pi\)
\(920\) −20.1246 8.66025i −0.663489 0.285520i
\(921\) 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 \(\pi\)
\(930\) 33.5410 + 25.9808i 1.09985 + 0.851943i
\(931\) 54.2218i 1.77705i
\(932\) −11.1803 43.3013i −0.366224 1.41838i
\(933\) 0 0
\(934\) −21.0000 + 27.1109i −0.687141 + 0.887095i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −45.0000 + 11.6190i −1.46774 + 0.378968i
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 58.8897i 1.91366i 0.290650 + 0.956830i \(0.406129\pi\)
−0.290650 + 0.956830i \(0.593871\pi\)
\(948\) −6.70820 25.9808i −0.217872 0.843816i
\(949\) 0 0
\(950\) −33.5410 + 43.3013i −1.08821 + 1.40488i
\(951\) 38.7298i 1.25590i
\(952\) 0 0
\(953\) −58.1378 −1.88327 −0.941634 0.336640i \(-0.890710\pi\)
−0.941634 + 0.336640i \(0.890710\pi\)
\(954\) 15.0000 + 11.6190i 0.485643 + 0.376177i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 22.5000 21.3014i 0.726184 0.687500i
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 31.1769i 1.00466i
\(964\) 1.00000 + 3.87298i 0.0322078 + 0.124740i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −12.2984 + 28.5788i −0.395285 + 0.918559i
\(969\) −60.0000 −1.92748
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −30.1869 + 7.79423i −0.968246 + 0.250000i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 7.00000 3.87298i 0.224065 0.123971i
\(977\) −4.47214 −0.143076 −0.0715382 0.997438i \(-0.522791\pi\)
−0.0715382 + 0.997438i \(0.522791\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −7.82624 30.3109i −0.250000 0.968246i
\(981\) −42.0000 −1.34096
\(982\) 0 0
\(983\) 3.46410i 0.110488i −0.998473 0.0552438i \(-0.982406\pi\)
0.998473 0.0552438i \(-0.0175936\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 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 \(-0.330290\pi\)
\(992\) −6.70820 + 43.3013i −0.212986 + 1.37482i
\(993\) −40.2492 −1.27727
\(994\) 0 0
\(995\) 51.9615i 1.64729i
\(996\) −3.00000 11.6190i −0.0950586 0.368161i
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −6.70820 + 8.66025i −0.212344 + 0.274136i
\(999\) 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 60.2.h.b.59.1 4
3.2 odd 2 inner 60.2.h.b.59.4 yes 4
4.3 odd 2 inner 60.2.h.b.59.2 yes 4
5.2 odd 4 300.2.e.a.251.3 4
5.3 odd 4 300.2.e.a.251.2 4
5.4 even 2 inner 60.2.h.b.59.4 yes 4
8.3 odd 2 960.2.o.a.959.3 4
8.5 even 2 960.2.o.a.959.1 4
12.11 even 2 inner 60.2.h.b.59.3 yes 4
15.2 even 4 300.2.e.a.251.2 4
15.8 even 4 300.2.e.a.251.3 4
15.14 odd 2 CM 60.2.h.b.59.1 4
20.3 even 4 300.2.e.a.251.4 4
20.7 even 4 300.2.e.a.251.1 4
20.19 odd 2 inner 60.2.h.b.59.3 yes 4
24.5 odd 2 960.2.o.a.959.4 4
24.11 even 2 960.2.o.a.959.2 4
40.19 odd 2 960.2.o.a.959.2 4
40.29 even 2 960.2.o.a.959.4 4
60.23 odd 4 300.2.e.a.251.1 4
60.47 odd 4 300.2.e.a.251.4 4
60.59 even 2 inner 60.2.h.b.59.2 yes 4
120.29 odd 2 960.2.o.a.959.1 4
120.59 even 2 960.2.o.a.959.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.h.b.59.1 4 1.1 even 1 trivial
60.2.h.b.59.1 4 15.14 odd 2 CM
60.2.h.b.59.2 yes 4 4.3 odd 2 inner
60.2.h.b.59.2 yes 4 60.59 even 2 inner
60.2.h.b.59.3 yes 4 12.11 even 2 inner
60.2.h.b.59.3 yes 4 20.19 odd 2 inner
60.2.h.b.59.4 yes 4 3.2 odd 2 inner
60.2.h.b.59.4 yes 4 5.4 even 2 inner
300.2.e.a.251.1 4 20.7 even 4
300.2.e.a.251.1 4 60.23 odd 4
300.2.e.a.251.2 4 5.3 odd 4
300.2.e.a.251.2 4 15.2 even 4
300.2.e.a.251.3 4 5.2 odd 4
300.2.e.a.251.3 4 15.8 even 4
300.2.e.a.251.4 4 20.3 even 4
300.2.e.a.251.4 4 60.47 odd 4
960.2.o.a.959.1 4 8.5 even 2
960.2.o.a.959.1 4 120.29 odd 2
960.2.o.a.959.2 4 24.11 even 2
960.2.o.a.959.2 4 40.19 odd 2
960.2.o.a.959.3 4 8.3 odd 2
960.2.o.a.959.3 4 120.59 even 2
960.2.o.a.959.4 4 24.5 odd 2
960.2.o.a.959.4 4 40.29 even 2