Properties

Label 135.2.b.b.109.1
Level $135$
Weight $2$
Character 135.109
Analytic conductor $1.078$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,2,Mod(109,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 109.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 135.109
Dual form 135.2.b.b.109.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +(-2.12132 - 0.707107i) q^{5} -3.00000i q^{7} -2.82843i q^{8} +(-1.00000 + 3.00000i) q^{10} +4.24264 q^{11} +3.00000i q^{13} -4.24264 q^{14} -4.00000 q^{16} +2.82843i q^{17} -1.00000 q^{19} -6.00000i q^{22} +7.07107i q^{23} +(4.00000 + 3.00000i) q^{25} +4.24264 q^{26} +4.24264 q^{29} +2.00000 q^{31} +4.00000 q^{34} +(-2.12132 + 6.36396i) q^{35} -9.00000i q^{37} +1.41421i q^{38} +(-2.00000 + 6.00000i) q^{40} -4.24264 q^{41} +6.00000i q^{43} +10.0000 q^{46} +2.82843i q^{47} -2.00000 q^{49} +(4.24264 - 5.65685i) q^{50} -9.89949i q^{53} +(-9.00000 - 3.00000i) q^{55} -8.48528 q^{56} -6.00000i q^{58} +8.48528 q^{59} -13.0000 q^{61} -2.82843i q^{62} -8.00000 q^{64} +(2.12132 - 6.36396i) q^{65} +3.00000i q^{67} +(9.00000 + 3.00000i) q^{70} -12.7279 q^{71} +9.00000i q^{73} -12.7279 q^{74} -12.7279i q^{77} +5.00000 q^{79} +(8.48528 + 2.82843i) q^{80} +6.00000i q^{82} -1.41421i q^{83} +(2.00000 - 6.00000i) q^{85} +8.48528 q^{86} -12.0000i q^{88} +9.00000 q^{91} +4.00000 q^{94} +(2.12132 + 0.707107i) q^{95} -3.00000i q^{97} +2.82843i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{10} - 16 q^{16} - 4 q^{19} + 16 q^{25} + 8 q^{31} + 16 q^{34} - 8 q^{40} + 40 q^{46} - 8 q^{49} - 36 q^{55} - 52 q^{61} - 32 q^{64} + 36 q^{70} + 20 q^{79} + 8 q^{85} + 36 q^{91} + 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) −2.12132 0.707107i −0.948683 0.316228i
\(6\) 0 0
\(7\) 3.00000i 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) −1.00000 + 3.00000i −0.316228 + 0.948683i
\(11\) 4.24264 1.27920 0.639602 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) −4.24264 −1.13389
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 7.07107i 1.47442i 0.675664 + 0.737210i \(0.263857\pi\)
−0.675664 + 0.737210i \(0.736143\pi\)
\(24\) 0 0
\(25\) 4.00000 + 3.00000i 0.800000 + 0.600000i
\(26\) 4.24264 0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264 0.787839 0.393919 0.919145i \(-0.371119\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −2.12132 + 6.36396i −0.358569 + 1.07571i
\(36\) 0 0
\(37\) 9.00000i 1.47959i −0.672832 0.739795i \(-0.734922\pi\)
0.672832 0.739795i \(-0.265078\pi\)
\(38\) 1.41421i 0.229416i
\(39\) 0 0
\(40\) −2.00000 + 6.00000i −0.316228 + 0.948683i
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 10.0000 1.47442
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 4.24264 5.65685i 0.600000 0.800000i
\(51\) 0 0
\(52\) 0 0
\(53\) 9.89949i 1.35980i −0.733305 0.679900i \(-0.762023\pi\)
0.733305 0.679900i \(-0.237977\pi\)
\(54\) 0 0
\(55\) −9.00000 3.00000i −1.21356 0.404520i
\(56\) −8.48528 −1.13389
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 8.48528 1.10469 0.552345 0.833616i \(-0.313733\pi\)
0.552345 + 0.833616i \(0.313733\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 2.82843i 0.359211i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.12132 6.36396i 0.263117 0.789352i
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 9.00000 + 3.00000i 1.07571 + 0.358569i
\(71\) −12.7279 −1.51053 −0.755263 0.655422i \(-0.772491\pi\)
−0.755263 + 0.655422i \(0.772491\pi\)
\(72\) 0 0
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) −12.7279 −1.47959
\(75\) 0 0
\(76\) 0 0
\(77\) 12.7279i 1.45048i
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 8.48528 + 2.82843i 0.948683 + 0.316228i
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) 1.41421i 0.155230i −0.996983 0.0776151i \(-0.975269\pi\)
0.996983 0.0776151i \(-0.0247305\pi\)
\(84\) 0 0
\(85\) 2.00000 6.00000i 0.216930 0.650791i
\(86\) 8.48528 0.914991
\(87\) 0 0
\(88\) 12.0000i 1.27920i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 9.00000 0.943456
\(92\) 0 0
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 2.12132 + 0.707107i 0.217643 + 0.0725476i
\(96\) 0 0
\(97\) 3.00000i 0.304604i −0.988334 0.152302i \(-0.951331\pi\)
0.988334 0.152302i \(-0.0486686\pi\)
\(98\) 2.82843i 0.285714i
\(99\) 0 0
\(100\) 0 0
\(101\) −8.48528 −0.844317 −0.422159 0.906522i \(-0.638727\pi\)
−0.422159 + 0.906522i \(0.638727\pi\)
\(102\) 0 0
\(103\) 3.00000i 0.295599i 0.989017 + 0.147799i \(0.0472190\pi\)
−0.989017 + 0.147799i \(0.952781\pi\)
\(104\) 8.48528 0.832050
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) −4.24264 + 12.7279i −0.404520 + 1.21356i
\(111\) 0 0
\(112\) 12.0000i 1.13389i
\(113\) 1.41421i 0.133038i −0.997785 0.0665190i \(-0.978811\pi\)
0.997785 0.0665190i \(-0.0211893\pi\)
\(114\) 0 0
\(115\) 5.00000 15.0000i 0.466252 1.39876i
\(116\) 0 0
\(117\) 0 0
\(118\) 12.0000i 1.10469i
\(119\) 8.48528 0.777844
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 18.3848i 1.66448i
\(123\) 0 0
\(124\) 0 0
\(125\) −6.36396 9.19239i −0.569210 0.822192i
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 0 0
\(130\) −9.00000 3.00000i −0.789352 0.263117i
\(131\) 8.48528 0.741362 0.370681 0.928760i \(-0.379124\pi\)
0.370681 + 0.928760i \(0.379124\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 4.24264 0.366508
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) 1.41421i 0.120824i −0.998174 0.0604122i \(-0.980758\pi\)
0.998174 0.0604122i \(-0.0192415\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 18.0000i 1.51053i
\(143\) 12.7279i 1.06436i
\(144\) 0 0
\(145\) −9.00000 3.00000i −0.747409 0.249136i
\(146\) 12.7279 1.05337
\(147\) 0 0
\(148\) 0 0
\(149\) −16.9706 −1.39028 −0.695141 0.718873i \(-0.744658\pi\)
−0.695141 + 0.718873i \(0.744658\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 2.82843i 0.229416i
\(153\) 0 0
\(154\) −18.0000 −1.45048
\(155\) −4.24264 1.41421i −0.340777 0.113592i
\(156\) 0 0
\(157\) 6.00000i 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 7.07107i 0.562544i
\(159\) 0 0
\(160\) 0 0
\(161\) 21.2132 1.67183
\(162\) 0 0
\(163\) 9.00000i 0.704934i −0.935824 0.352467i \(-0.885343\pi\)
0.935824 0.352467i \(-0.114657\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 5.65685i 0.437741i −0.975754 0.218870i \(-0.929763\pi\)
0.975754 0.218870i \(-0.0702371\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) −8.48528 2.82843i −0.650791 0.216930i
\(171\) 0 0
\(172\) 0 0
\(173\) 14.1421i 1.07521i −0.843198 0.537603i \(-0.819330\pi\)
0.843198 0.537603i \(-0.180670\pi\)
\(174\) 0 0
\(175\) 9.00000 12.0000i 0.680336 0.907115i
\(176\) −16.9706 −1.27920
\(177\) 0 0
\(178\) 0 0
\(179\) −12.7279 −0.951330 −0.475665 0.879627i \(-0.657792\pi\)
−0.475665 + 0.879627i \(0.657792\pi\)
\(180\) 0 0
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 12.7279i 0.943456i
\(183\) 0 0
\(184\) 20.0000 1.47442
\(185\) −6.36396 + 19.0919i −0.467888 + 1.40366i
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) 0 0
\(190\) 1.00000 3.00000i 0.0725476 0.217643i
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) 3.00000i 0.215945i 0.994154 + 0.107972i \(0.0344358\pi\)
−0.994154 + 0.107972i \(0.965564\pi\)
\(194\) −4.24264 −0.304604
\(195\) 0 0
\(196\) 0 0
\(197\) 22.6274i 1.61214i −0.591822 0.806068i \(-0.701591\pi\)
0.591822 0.806068i \(-0.298409\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 8.48528 11.3137i 0.600000 0.800000i
\(201\) 0 0
\(202\) 12.0000i 0.844317i
\(203\) 12.7279i 0.893325i
\(204\) 0 0
\(205\) 9.00000 + 3.00000i 0.628587 + 0.209529i
\(206\) 4.24264 0.295599
\(207\) 0 0
\(208\) 12.0000i 0.832050i
\(209\) −4.24264 −0.293470
\(210\) 0 0
\(211\) −7.00000 −0.481900 −0.240950 0.970538i \(-0.577459\pi\)
−0.240950 + 0.970538i \(0.577459\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 10.0000 0.683586
\(215\) 4.24264 12.7279i 0.289346 0.868037i
\(216\) 0 0
\(217\) 6.00000i 0.407307i
\(218\) 11.3137i 0.766261i
\(219\) 0 0
\(220\) 0 0
\(221\) −8.48528 −0.570782
\(222\) 0 0
\(223\) 6.00000i 0.401790i 0.979613 + 0.200895i \(0.0643850\pi\)
−0.979613 + 0.200895i \(0.935615\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 24.0416i 1.59570i 0.602857 + 0.797850i \(0.294029\pi\)
−0.602857 + 0.797850i \(0.705971\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −21.2132 7.07107i −1.39876 0.466252i
\(231\) 0 0
\(232\) 12.0000i 0.787839i
\(233\) 19.7990i 1.29707i 0.761183 + 0.648537i \(0.224619\pi\)
−0.761183 + 0.648537i \(0.775381\pi\)
\(234\) 0 0
\(235\) 2.00000 6.00000i 0.130466 0.391397i
\(236\) 0 0
\(237\) 0 0
\(238\) 12.0000i 0.777844i
\(239\) −4.24264 −0.274434 −0.137217 0.990541i \(-0.543816\pi\)
−0.137217 + 0.990541i \(0.543816\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 9.89949i 0.636364i
\(243\) 0 0
\(244\) 0 0
\(245\) 4.24264 + 1.41421i 0.271052 + 0.0903508i
\(246\) 0 0
\(247\) 3.00000i 0.190885i
\(248\) 5.65685i 0.359211i
\(249\) 0 0
\(250\) −13.0000 + 9.00000i −0.822192 + 0.569210i
\(251\) −12.7279 −0.803379 −0.401690 0.915776i \(-0.631577\pi\)
−0.401690 + 0.915776i \(0.631577\pi\)
\(252\) 0 0
\(253\) 30.0000i 1.88608i
\(254\) 25.4558 1.59724
\(255\) 0 0
\(256\) 0 0
\(257\) 19.7990i 1.23503i 0.786560 + 0.617514i \(0.211860\pi\)
−0.786560 + 0.617514i \(0.788140\pi\)
\(258\) 0 0
\(259\) −27.0000 −1.67770
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 2.82843i 0.174408i 0.996190 + 0.0872041i \(0.0277932\pi\)
−0.996190 + 0.0872041i \(0.972207\pi\)
\(264\) 0 0
\(265\) −7.00000 + 21.0000i −0.430007 + 1.29002i
\(266\) 4.24264 0.260133
\(267\) 0 0
\(268\) 0 0
\(269\) 25.4558 1.55207 0.776035 0.630690i \(-0.217228\pi\)
0.776035 + 0.630690i \(0.217228\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) 11.3137i 0.685994i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 16.9706 + 12.7279i 1.02336 + 0.767523i
\(276\) 0 0
\(277\) 12.0000i 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 18.3848i 1.10265i
\(279\) 0 0
\(280\) 18.0000 + 6.00000i 1.07571 + 0.358569i
\(281\) −8.48528 −0.506189 −0.253095 0.967442i \(-0.581448\pi\)
−0.253095 + 0.967442i \(0.581448\pi\)
\(282\) 0 0
\(283\) 6.00000i 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 18.0000 1.06436
\(287\) 12.7279i 0.751305i
\(288\) 0 0
\(289\) 9.00000 0.529412
\(290\) −4.24264 + 12.7279i −0.249136 + 0.747409i
\(291\) 0 0
\(292\) 0 0
\(293\) 7.07107i 0.413096i 0.978436 + 0.206548i \(0.0662230\pi\)
−0.978436 + 0.206548i \(0.933777\pi\)
\(294\) 0 0
\(295\) −18.0000 6.00000i −1.04800 0.349334i
\(296\) −25.4558 −1.47959
\(297\) 0 0
\(298\) 24.0000i 1.39028i
\(299\) −21.2132 −1.22679
\(300\) 0 0
\(301\) 18.0000 1.03750
\(302\) 1.41421i 0.0813788i
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 27.5772 + 9.19239i 1.57906 + 0.526355i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 + 6.00000i −0.113592 + 0.340777i
\(311\) 8.48528 0.481156 0.240578 0.970630i \(-0.422663\pi\)
0.240578 + 0.970630i \(0.422663\pi\)
\(312\) 0 0
\(313\) 21.0000i 1.18699i −0.804838 0.593495i \(-0.797748\pi\)
0.804838 0.593495i \(-0.202252\pi\)
\(314\) −8.48528 −0.478852
\(315\) 0 0
\(316\) 0 0
\(317\) 11.3137i 0.635441i 0.948184 + 0.317721i \(0.102917\pi\)
−0.948184 + 0.317721i \(0.897083\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 16.9706 + 5.65685i 0.948683 + 0.316228i
\(321\) 0 0
\(322\) 30.0000i 1.67183i
\(323\) 2.82843i 0.157378i
\(324\) 0 0
\(325\) −9.00000 + 12.0000i −0.499230 + 0.665640i
\(326\) −12.7279 −0.704934
\(327\) 0 0
\(328\) 12.0000i 0.662589i
\(329\) 8.48528 0.467809
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 2.12132 6.36396i 0.115900 0.347700i
\(336\) 0 0
\(337\) 33.0000i 1.79762i −0.438334 0.898812i \(-0.644431\pi\)
0.438334 0.898812i \(-0.355569\pi\)
\(338\) 5.65685i 0.307692i
\(339\) 0 0
\(340\) 0 0
\(341\) 8.48528 0.459504
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 16.9706 0.914991
\(345\) 0 0
\(346\) −20.0000 −1.07521
\(347\) 26.8701i 1.44246i −0.692696 0.721230i \(-0.743577\pi\)
0.692696 0.721230i \(-0.256423\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) −16.9706 12.7279i −0.907115 0.680336i
\(351\) 0 0
\(352\) 0 0
\(353\) 28.2843i 1.50542i 0.658352 + 0.752710i \(0.271254\pi\)
−0.658352 + 0.752710i \(0.728746\pi\)
\(354\) 0 0
\(355\) 27.0000 + 9.00000i 1.43301 + 0.477670i
\(356\) 0 0
\(357\) 0 0
\(358\) 18.0000i 0.951330i
\(359\) −12.7279 −0.671754 −0.335877 0.941906i \(-0.609033\pi\)
−0.335877 + 0.941906i \(0.609033\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 1.41421i 0.0743294i
\(363\) 0 0
\(364\) 0 0
\(365\) 6.36396 19.0919i 0.333105 0.999315i
\(366\) 0 0
\(367\) 15.0000i 0.782994i 0.920179 + 0.391497i \(0.128043\pi\)
−0.920179 + 0.391497i \(0.871957\pi\)
\(368\) 28.2843i 1.47442i
\(369\) 0 0
\(370\) 27.0000 + 9.00000i 1.40366 + 0.467888i
\(371\) −29.6985 −1.54187
\(372\) 0 0
\(373\) 21.0000i 1.08734i 0.839299 + 0.543669i \(0.182965\pi\)
−0.839299 + 0.543669i \(0.817035\pi\)
\(374\) 16.9706 0.877527
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 12.7279i 0.655521i
\(378\) 0 0
\(379\) 35.0000 1.79783 0.898915 0.438124i \(-0.144357\pi\)
0.898915 + 0.438124i \(0.144357\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 24.0000i 1.22795i
\(383\) 5.65685i 0.289052i −0.989501 0.144526i \(-0.953834\pi\)
0.989501 0.144526i \(-0.0461657\pi\)
\(384\) 0 0
\(385\) −9.00000 + 27.0000i −0.458682 + 1.37605i
\(386\) 4.24264 0.215945
\(387\) 0 0
\(388\) 0 0
\(389\) −33.9411 −1.72088 −0.860442 0.509549i \(-0.829812\pi\)
−0.860442 + 0.509549i \(0.829812\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 5.65685i 0.285714i
\(393\) 0 0
\(394\) −32.0000 −1.61214
\(395\) −10.6066 3.53553i −0.533676 0.177892i
\(396\) 0 0
\(397\) 36.0000i 1.80679i −0.428811 0.903394i \(-0.641067\pi\)
0.428811 0.903394i \(-0.358933\pi\)
\(398\) 15.5563i 0.779769i
\(399\) 0 0
\(400\) −16.0000 12.0000i −0.800000 0.600000i
\(401\) −4.24264 −0.211867 −0.105934 0.994373i \(-0.533783\pi\)
−0.105934 + 0.994373i \(0.533783\pi\)
\(402\) 0 0
\(403\) 6.00000i 0.298881i
\(404\) 0 0
\(405\) 0 0
\(406\) −18.0000 −0.893325
\(407\) 38.1838i 1.89270i
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 4.24264 12.7279i 0.209529 0.628587i
\(411\) 0 0
\(412\) 0 0
\(413\) 25.4558i 1.25260i
\(414\) 0 0
\(415\) −1.00000 + 3.00000i −0.0490881 + 0.147264i
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000i 0.293470i
\(419\) 33.9411 1.65813 0.829066 0.559150i \(-0.188873\pi\)
0.829066 + 0.559150i \(0.188873\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 9.89949i 0.481900i
\(423\) 0 0
\(424\) −28.0000 −1.35980
\(425\) −8.48528 + 11.3137i −0.411597 + 0.548795i
\(426\) 0 0
\(427\) 39.0000i 1.88734i
\(428\) 0 0
\(429\) 0 0
\(430\) −18.0000 6.00000i −0.868037 0.289346i
\(431\) 12.7279 0.613082 0.306541 0.951857i \(-0.400828\pi\)
0.306541 + 0.951857i \(0.400828\pi\)
\(432\) 0 0
\(433\) 36.0000i 1.73005i −0.501729 0.865025i \(-0.667303\pi\)
0.501729 0.865025i \(-0.332697\pi\)
\(434\) −8.48528 −0.407307
\(435\) 0 0
\(436\) 0 0
\(437\) 7.07107i 0.338255i
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −8.48528 + 25.4558i −0.404520 + 1.21356i
\(441\) 0 0
\(442\) 12.0000i 0.570782i
\(443\) 14.1421i 0.671913i −0.941877 0.335957i \(-0.890940\pi\)
0.941877 0.335957i \(-0.109060\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 8.48528 0.401790
\(447\) 0 0
\(448\) 24.0000i 1.13389i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 34.0000 1.59570
\(455\) −19.0919 6.36396i −0.895041 0.298347i
\(456\) 0 0
\(457\) 12.0000i 0.561336i −0.959805 0.280668i \(-0.909444\pi\)
0.959805 0.280668i \(-0.0905560\pi\)
\(458\) 22.6274i 1.05731i
\(459\) 0 0
\(460\) 0 0
\(461\) 16.9706 0.790398 0.395199 0.918596i \(-0.370676\pi\)
0.395199 + 0.918596i \(0.370676\pi\)
\(462\) 0 0
\(463\) 15.0000i 0.697109i −0.937288 0.348555i \(-0.886673\pi\)
0.937288 0.348555i \(-0.113327\pi\)
\(464\) −16.9706 −0.787839
\(465\) 0 0
\(466\) 28.0000 1.29707
\(467\) 22.6274i 1.04707i −0.852004 0.523536i \(-0.824613\pi\)
0.852004 0.523536i \(-0.175387\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) −8.48528 2.82843i −0.391397 0.130466i
\(471\) 0 0
\(472\) 24.0000i 1.10469i
\(473\) 25.4558i 1.17046i
\(474\) 0 0
\(475\) −4.00000 3.00000i −0.183533 0.137649i
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000i 0.274434i
\(479\) 4.24264 0.193851 0.0969256 0.995292i \(-0.469099\pi\)
0.0969256 + 0.995292i \(0.469099\pi\)
\(480\) 0 0
\(481\) 27.0000 1.23109
\(482\) 7.07107i 0.322078i
\(483\) 0 0
\(484\) 0 0
\(485\) −2.12132 + 6.36396i −0.0963242 + 0.288973i
\(486\) 0 0
\(487\) 9.00000i 0.407829i 0.978989 + 0.203914i \(0.0653664\pi\)
−0.978989 + 0.203914i \(0.934634\pi\)
\(488\) 36.7696i 1.66448i
\(489\) 0 0
\(490\) 2.00000 6.00000i 0.0903508 0.271052i
\(491\) 21.2132 0.957338 0.478669 0.877995i \(-0.341119\pi\)
0.478669 + 0.877995i \(0.341119\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) −4.24264 −0.190885
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 38.1838i 1.71278i
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000i 0.803379i
\(503\) 35.3553i 1.57642i −0.615409 0.788208i \(-0.711009\pi\)
0.615409 0.788208i \(-0.288991\pi\)
\(504\) 0 0
\(505\) 18.0000 + 6.00000i 0.800989 + 0.266996i
\(506\) 42.4264 1.88608
\(507\) 0 0
\(508\) 0 0
\(509\) −29.6985 −1.31636 −0.658181 0.752860i \(-0.728674\pi\)
−0.658181 + 0.752860i \(0.728674\pi\)
\(510\) 0 0
\(511\) 27.0000 1.19441
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 28.0000 1.23503
\(515\) 2.12132 6.36396i 0.0934765 0.280430i
\(516\) 0 0
\(517\) 12.0000i 0.527759i
\(518\) 38.1838i 1.67770i
\(519\) 0 0
\(520\) −18.0000 6.00000i −0.789352 0.263117i
\(521\) 12.7279 0.557620 0.278810 0.960346i \(-0.410060\pi\)
0.278810 + 0.960346i \(0.410060\pi\)
\(522\) 0 0
\(523\) 9.00000i 0.393543i −0.980449 0.196771i \(-0.936954\pi\)
0.980449 0.196771i \(-0.0630456\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 5.65685i 0.246416i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) 29.6985 + 9.89949i 1.29002 + 0.430007i
\(531\) 0 0
\(532\) 0 0
\(533\) 12.7279i 0.551308i
\(534\) 0 0
\(535\) 5.00000 15.0000i 0.216169 0.648507i
\(536\) 8.48528 0.366508
\(537\) 0 0
\(538\) 36.0000i 1.55207i
\(539\) −8.48528 −0.365487
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 26.8701i 1.15417i
\(543\) 0 0
\(544\) 0 0
\(545\) −16.9706 5.65685i −0.726939 0.242313i
\(546\) 0 0
\(547\) 3.00000i 0.128271i −0.997941 0.0641354i \(-0.979571\pi\)
0.997941 0.0641354i \(-0.0204289\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 18.0000 24.0000i 0.767523 1.02336i
\(551\) −4.24264 −0.180743
\(552\) 0 0
\(553\) 15.0000i 0.637865i
\(554\) −16.9706 −0.721010
\(555\) 0 0
\(556\) 0 0
\(557\) 19.7990i 0.838910i 0.907776 + 0.419455i \(0.137779\pi\)
−0.907776 + 0.419455i \(0.862221\pi\)
\(558\) 0 0
\(559\) −18.0000 −0.761319
\(560\) 8.48528 25.4558i 0.358569 1.07571i
\(561\) 0 0
\(562\) 12.0000i 0.506189i
\(563\) 11.3137i 0.476816i 0.971165 + 0.238408i \(0.0766255\pi\)
−0.971165 + 0.238408i \(0.923374\pi\)
\(564\) 0 0
\(565\) −1.00000 + 3.00000i −0.0420703 + 0.126211i
\(566\) −8.48528 −0.356663
\(567\) 0 0
\(568\) 36.0000i 1.51053i
\(569\) 42.4264 1.77861 0.889304 0.457317i \(-0.151190\pi\)
0.889304 + 0.457317i \(0.151190\pi\)
\(570\) 0 0
\(571\) −25.0000 −1.04622 −0.523109 0.852266i \(-0.675228\pi\)
−0.523109 + 0.852266i \(0.675228\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 18.0000 0.751305
\(575\) −21.2132 + 28.2843i −0.884652 + 1.17954i
\(576\) 0 0
\(577\) 45.0000i 1.87337i 0.350167 + 0.936687i \(0.386125\pi\)
−0.350167 + 0.936687i \(0.613875\pi\)
\(578\) 12.7279i 0.529412i
\(579\) 0 0
\(580\) 0 0
\(581\) −4.24264 −0.176014
\(582\) 0 0
\(583\) 42.0000i 1.73946i
\(584\) 25.4558 1.05337
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) 26.8701i 1.10905i −0.832168 0.554523i \(-0.812901\pi\)
0.832168 0.554523i \(-0.187099\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) −8.48528 + 25.4558i −0.349334 + 1.04800i
\(591\) 0 0
\(592\) 36.0000i 1.47959i
\(593\) 9.89949i 0.406524i −0.979124 0.203262i \(-0.934846\pi\)
0.979124 0.203262i \(-0.0651542\pi\)
\(594\) 0 0
\(595\) −18.0000 6.00000i −0.737928 0.245976i
\(596\) 0 0
\(597\) 0 0
\(598\) 30.0000i 1.22679i
\(599\) −29.6985 −1.21345 −0.606724 0.794913i \(-0.707517\pi\)
−0.606724 + 0.794913i \(0.707517\pi\)
\(600\) 0 0
\(601\) −40.0000 −1.63163 −0.815817 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 25.4558i 1.03750i
\(603\) 0 0
\(604\) 0 0
\(605\) −14.8492 4.94975i −0.603708 0.201236i
\(606\) 0 0
\(607\) 39.0000i 1.58296i 0.611194 + 0.791481i \(0.290689\pi\)
−0.611194 + 0.791481i \(0.709311\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 13.0000 39.0000i 0.526355 1.57906i
\(611\) −8.48528 −0.343278
\(612\) 0 0
\(613\) 9.00000i 0.363507i 0.983344 + 0.181753i \(0.0581772\pi\)
−0.983344 + 0.181753i \(0.941823\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −36.0000 −1.45048
\(617\) 7.07107i 0.284670i 0.989819 + 0.142335i \(0.0454611\pi\)
−0.989819 + 0.142335i \(0.954539\pi\)
\(618\) 0 0
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000i 0.481156i
\(623\) 0 0
\(624\) 0 0
\(625\) 7.00000 + 24.0000i 0.280000 + 0.960000i
\(626\) −29.6985 −1.18699
\(627\) 0 0
\(628\) 0 0
\(629\) 25.4558 1.01499
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 14.1421i 0.562544i
\(633\) 0 0
\(634\) 16.0000 0.635441
\(635\) 12.7279 38.1838i 0.505092 1.51528i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 25.4558i 1.00781i
\(639\) 0 0
\(640\) 8.00000 24.0000i 0.316228 0.948683i
\(641\) 16.9706 0.670297 0.335148 0.942165i \(-0.391214\pi\)
0.335148 + 0.942165i \(0.391214\pi\)
\(642\) 0 0
\(643\) 42.0000i 1.65632i −0.560493 0.828159i \(-0.689388\pi\)
0.560493 0.828159i \(-0.310612\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 9.89949i 0.389189i −0.980884 0.194595i \(-0.937661\pi\)
0.980884 0.194595i \(-0.0623391\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 16.9706 + 12.7279i 0.665640 + 0.499230i
\(651\) 0 0
\(652\) 0 0
\(653\) 5.65685i 0.221370i −0.993856 0.110685i \(-0.964696\pi\)
0.993856 0.110685i \(-0.0353044\pi\)
\(654\) 0 0
\(655\) −18.0000 6.00000i −0.703318 0.234439i
\(656\) 16.9706 0.662589
\(657\) 0 0
\(658\) 12.0000i 0.467809i
\(659\) 16.9706 0.661079 0.330540 0.943792i \(-0.392769\pi\)
0.330540 + 0.943792i \(0.392769\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 24.0416i 0.934405i
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 2.12132 6.36396i 0.0822613 0.246784i
\(666\) 0 0
\(667\) 30.0000i 1.16160i
\(668\) 0 0
\(669\) 0 0
\(670\) −9.00000 3.00000i −0.347700 0.115900i
\(671\) −55.1543 −2.12921
\(672\) 0 0
\(673\) 15.0000i 0.578208i 0.957298 + 0.289104i \(0.0933573\pi\)
−0.957298 + 0.289104i \(0.906643\pi\)
\(674\) −46.6690 −1.79762
\(675\) 0 0
\(676\) 0 0
\(677\) 48.0833i 1.84799i −0.382405 0.923995i \(-0.624904\pi\)
0.382405 0.923995i \(-0.375096\pi\)
\(678\) 0 0
\(679\) −9.00000 −0.345388
\(680\) −16.9706 5.65685i −0.650791 0.216930i
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) 15.5563i 0.595247i 0.954683 + 0.297624i \(0.0961940\pi\)
−0.954683 + 0.297624i \(0.903806\pi\)
\(684\) 0 0
\(685\) −1.00000 + 3.00000i −0.0382080 + 0.114624i
\(686\) −21.2132 −0.809924
\(687\) 0 0
\(688\) 24.0000i 0.914991i
\(689\) 29.6985 1.13142
\(690\) 0 0
\(691\) 50.0000 1.90209 0.951045 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −38.0000 −1.44246
\(695\) 27.5772 + 9.19239i 1.04606 + 0.348687i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 7.07107i 0.267644i
\(699\) 0 0
\(700\) 0 0
\(701\) −38.1838 −1.44218 −0.721090 0.692841i \(-0.756359\pi\)
−0.721090 + 0.692841i \(0.756359\pi\)
\(702\) 0 0
\(703\) 9.00000i 0.339441i
\(704\) −33.9411 −1.27920
\(705\) 0 0
\(706\) 40.0000 1.50542
\(707\) 25.4558i 0.957366i
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 12.7279 38.1838i 0.477670 1.43301i
\(711\) 0 0
\(712\) 0 0
\(713\) 14.1421i 0.529627i
\(714\) 0 0
\(715\) 9.00000 27.0000i 0.336581 1.00974i
\(716\) 0 0
\(717\) 0 0
\(718\) 18.0000i 0.671754i
\(719\) 12.7279 0.474671 0.237336 0.971428i \(-0.423726\pi\)
0.237336 + 0.971428i \(0.423726\pi\)
\(720\) 0 0
\(721\) 9.00000 0.335178
\(722\) 25.4558i 0.947368i
\(723\) 0 0
\(724\) 0 0
\(725\) 16.9706 + 12.7279i 0.630271 + 0.472703i
\(726\) 0 0
\(727\) 42.0000i 1.55769i 0.627214 + 0.778847i \(0.284195\pi\)
−0.627214 + 0.778847i \(0.715805\pi\)
\(728\) 25.4558i 0.943456i
\(729\) 0 0
\(730\) −27.0000 9.00000i −0.999315 0.333105i
\(731\) −16.9706 −0.627679
\(732\) 0 0
\(733\) 24.0000i 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 21.2132 0.782994
\(735\) 0 0
\(736\) 0 0
\(737\) 12.7279i 0.468839i
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 42.0000i 1.54187i
\(743\) 14.1421i 0.518825i −0.965767 0.259412i \(-0.916471\pi\)
0.965767 0.259412i \(-0.0835289\pi\)
\(744\) 0 0
\(745\) 36.0000 + 12.0000i 1.31894 + 0.439646i
\(746\) 29.6985 1.08734
\(747\) 0 0
\(748\) 0 0
\(749\) 21.2132 0.775114
\(750\) 0 0
\(751\) −7.00000 −0.255434 −0.127717 0.991811i \(-0.540765\pi\)
−0.127717 + 0.991811i \(0.540765\pi\)
\(752\) 11.3137i 0.412568i
\(753\) 0 0
\(754\) 18.0000 0.655521
\(755\) 2.12132 + 0.707107i 0.0772028 + 0.0257343i
\(756\) 0 0
\(757\) 27.0000i 0.981332i −0.871348 0.490666i \(-0.836754\pi\)
0.871348 0.490666i \(-0.163246\pi\)
\(758\) 49.4975i 1.79783i
\(759\) 0 0
\(760\) 2.00000 6.00000i 0.0725476 0.217643i
\(761\) −4.24264 −0.153796 −0.0768978 0.997039i \(-0.524502\pi\)
−0.0768978 + 0.997039i \(0.524502\pi\)
\(762\) 0 0
\(763\) 24.0000i 0.868858i
\(764\) 0 0
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 25.4558i 0.919157i
\(768\) 0 0
\(769\) 11.0000 0.396670 0.198335 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 38.1838 + 12.7279i 1.37605 + 0.458682i
\(771\) 0 0
\(772\) 0 0
\(773\) 41.0122i 1.47511i 0.675289 + 0.737553i \(0.264019\pi\)
−0.675289 + 0.737553i \(0.735981\pi\)
\(774\) 0 0
\(775\) 8.00000 + 6.00000i 0.287368 + 0.215526i
\(776\) −8.48528 −0.304604
\(777\) 0 0
\(778\) 48.0000i 1.72088i
\(779\) 4.24264 0.152008
\(780\) 0 0
\(781\) −54.0000 −1.93227
\(782\) 28.2843i 1.01144i
\(783\) 0 0
\(784\) 8.00000 0.285714
\(785\) −4.24264 + 12.7279i −0.151426 + 0.454279i
\(786\) 0 0
\(787\) 15.0000i 0.534692i −0.963601 0.267346i \(-0.913853\pi\)
0.963601 0.267346i \(-0.0861467\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) −5.00000 + 15.0000i −0.177892 + 0.533676i
\(791\) −4.24264 −0.150851
\(792\) 0 0
\(793\) 39.0000i 1.38493i
\(794\) −50.9117 −1.80679
\(795\) 0 0
\(796\) 0 0
\(797\) 19.7990i 0.701316i 0.936504 + 0.350658i \(0.114042\pi\)
−0.936504 + 0.350658i \(0.885958\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 6.00000i 0.211867i
\(803\) 38.1838i 1.34748i
\(804\) 0 0
\(805\) −45.0000 15.0000i −1.58604 0.528681i
\(806\) 8.48528 0.298881
\(807\) 0 0
\(808\) 24.0000i 0.844317i
\(809\) −25.4558 −0.894980 −0.447490 0.894289i \(-0.647682\pi\)
−0.447490 + 0.894289i \(0.647682\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −54.0000 −1.89270
\(815\) −6.36396 + 19.0919i −0.222920 + 0.668759i
\(816\) 0 0
\(817\) 6.00000i 0.209913i
\(818\) 35.3553i 1.23617i
\(819\) 0 0
\(820\) 0 0
\(821\) −8.48528 −0.296138 −0.148069 0.988977i \(-0.547306\pi\)
−0.148069 + 0.988977i \(0.547306\pi\)
\(822\) 0 0
\(823\) 39.0000i 1.35945i 0.733465 + 0.679727i \(0.237902\pi\)
−0.733465 + 0.679727i \(0.762098\pi\)
\(824\) 8.48528 0.295599
\(825\) 0 0
\(826\) −36.0000 −1.25260
\(827\) 2.82843i 0.0983540i 0.998790 + 0.0491770i \(0.0156598\pi\)
−0.998790 + 0.0491770i \(0.984340\pi\)
\(828\) 0 0
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 4.24264 + 1.41421i 0.147264 + 0.0490881i
\(831\) 0 0
\(832\) 24.0000i 0.832050i
\(833\) 5.65685i 0.195998i
\(834\) 0 0
\(835\) −4.00000 + 12.0000i −0.138426 + 0.415277i
\(836\) 0 0
\(837\) 0 0
\(838\) 48.0000i 1.65813i
\(839\) 42.4264 1.46472 0.732361 0.680916i \(-0.238418\pi\)
0.732361 + 0.680916i \(0.238418\pi\)
\(840\) 0 0
\(841\) −11.0000 −0.379310
\(842\) 32.5269i 1.12095i
\(843\) 0 0
\(844\) 0 0
\(845\) −8.48528 2.82843i −0.291903 0.0973009i
\(846\) 0 0
\(847\) 21.0000i 0.721569i
\(848\) 39.5980i 1.35980i
\(849\) 0 0
\(850\) 16.0000 + 12.0000i 0.548795 + 0.411597i
\(851\) 63.6396 2.18154
\(852\) 0 0
\(853\) 33.0000i 1.12990i 0.825126 + 0.564949i \(0.191104\pi\)
−0.825126 + 0.564949i \(0.808896\pi\)
\(854\) 55.1543 1.88734
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) 26.8701i 0.917864i −0.888471 0.458932i \(-0.848232\pi\)
0.888471 0.458932i \(-0.151768\pi\)
\(858\) 0 0
\(859\) 47.0000 1.60362 0.801810 0.597580i \(-0.203871\pi\)
0.801810 + 0.597580i \(0.203871\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.0000i 0.613082i
\(863\) 28.2843i 0.962808i 0.876499 + 0.481404i \(0.159873\pi\)
−0.876499 + 0.481404i \(0.840127\pi\)
\(864\) 0 0
\(865\) −10.0000 + 30.0000i −0.340010 + 1.02003i
\(866\) −50.9117 −1.73005
\(867\) 0 0
\(868\) 0 0
\(869\) 21.2132 0.719609
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) 22.6274i 0.766261i
\(873\) 0 0
\(874\) −10.0000 −0.338255
\(875\) −27.5772 + 19.0919i −0.932278 + 0.645423i
\(876\) 0 0
\(877\) 15.0000i 0.506514i −0.967399 0.253257i \(-0.918498\pi\)
0.967399 0.253257i \(-0.0815018\pi\)
\(878\) 19.7990i 0.668184i
\(879\) 0 0
\(880\) 36.0000 + 12.0000i 1.21356 + 0.404520i
\(881\) −38.1838 −1.28644 −0.643222 0.765680i \(-0.722403\pi\)
−0.643222 + 0.765680i \(0.722403\pi\)
\(882\) 0 0
\(883\) 27.0000i 0.908622i 0.890843 + 0.454311i \(0.150115\pi\)
−0.890843 + 0.454311i \(0.849885\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 45.2548i 1.51951i 0.650210 + 0.759754i \(0.274681\pi\)
−0.650210 + 0.759754i \(0.725319\pi\)
\(888\) 0 0
\(889\) 54.0000 1.81110
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.82843i 0.0946497i
\(894\) 0 0
\(895\) 27.0000 + 9.00000i 0.902510 + 0.300837i
\(896\) 33.9411 1.13389
\(897\) 0 0
\(898\) 0 0
\(899\) 8.48528 0.283000
\(900\) 0 0
\(901\) 28.0000 0.932815
\(902\) 25.4558i 0.847587i
\(903\) 0 0
\(904\) −4.00000 −0.133038
\(905\) 2.12132 + 0.707107i 0.0705151 + 0.0235050i
\(906\) 0 0
\(907\) 3.00000i 0.0996134i −0.998759 0.0498067i \(-0.984139\pi\)
0.998759 0.0498067i \(-0.0158605\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) −9.00000 + 27.0000i −0.298347 + 0.895041i
\(911\) 42.4264 1.40565 0.702825 0.711363i \(-0.251922\pi\)
0.702825 + 0.711363i \(0.251922\pi\)
\(912\) 0 0
\(913\) 6.00000i 0.198571i
\(914\) −16.9706 −0.561336
\(915\) 0 0
\(916\) 0 0
\(917\) 25.4558i 0.840626i
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) −42.4264 14.1421i −1.39876 0.466252i
\(921\) 0 0
\(922\) 24.0000i 0.790398i
\(923\) 38.1838i 1.25683i
\(924\) 0 0
\(925\) 27.0000 36.0000i 0.887755 1.18367i
\(926\) −21.2132 −0.697109
\(927\) 0 0
\(928\) 0 0
\(929\) −46.6690 −1.53116 −0.765581 0.643340i \(-0.777548\pi\)
−0.765581 + 0.643340i \(0.777548\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 0 0
\(934\) −32.0000 −1.04707
\(935\) 8.48528 25.4558i 0.277498 0.832495i
\(936\) 0 0
\(937\) 9.00000i 0.294017i 0.989135 + 0.147009i \(0.0469645\pi\)
−0.989135 + 0.147009i \(0.953036\pi\)
\(938\) 12.7279i 0.415581i
\(939\) 0 0
\(940\) 0 0
\(941\) 21.2132 0.691531 0.345765 0.938321i \(-0.387619\pi\)
0.345765 + 0.938321i \(0.387619\pi\)
\(942\) 0 0
\(943\) 30.0000i 0.976934i
\(944\) −33.9411 −1.10469
\(945\) 0 0
\(946\) 36.0000 1.17046
\(947\) 36.7696i 1.19485i 0.801925 + 0.597425i \(0.203809\pi\)
−0.801925 + 0.597425i \(0.796191\pi\)
\(948\) 0 0
\(949\) −27.0000 −0.876457
\(950\) −4.24264 + 5.65685i −0.137649 + 0.183533i
\(951\) 0 0
\(952\) 24.0000i 0.777844i
\(953\) 5.65685i 0.183243i −0.995794 0.0916217i \(-0.970795\pi\)
0.995794 0.0916217i \(-0.0292051\pi\)
\(954\) 0 0
\(955\) −36.0000 12.0000i −1.16493 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) 6.00000i 0.193851i
\(959\) −4.24264 −0.137002
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 38.1838i 1.23109i
\(963\) 0 0
\(964\) 0 0
\(965\) 2.12132 6.36396i 0.0682877 0.204863i
\(966\) 0 0
\(967\) 33.0000i 1.06121i −0.847620 0.530604i \(-0.821965\pi\)
0.847620 0.530604i \(-0.178035\pi\)
\(968\) 19.7990i 0.636364i
\(969\) 0 0
\(970\) 9.00000 + 3.00000i 0.288973 + 0.0963242i
\(971\) 25.4558 0.816917 0.408458 0.912777i \(-0.366066\pi\)
0.408458 + 0.912777i \(0.366066\pi\)
\(972\) 0 0
\(973\) 39.0000i 1.25028i
\(974\) 12.7279 0.407829
\(975\) 0 0
\(976\) 52.0000 1.66448
\(977\) 1.41421i 0.0452447i −0.999744 0.0226224i \(-0.992798\pi\)
0.999744 0.0226224i \(-0.00720153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 30.0000i 0.957338i
\(983\) 52.3259i 1.66894i −0.551056 0.834469i \(-0.685775\pi\)
0.551056 0.834469i \(-0.314225\pi\)
\(984\) 0 0
\(985\) −16.0000 + 48.0000i −0.509802 + 1.52941i
\(986\) 16.9706 0.540453
\(987\) 0 0
\(988\) 0 0
\(989\) −42.4264 −1.34908
\(990\) 0 0
\(991\) 53.0000 1.68360 0.841800 0.539789i \(-0.181496\pi\)
0.841800 + 0.539789i \(0.181496\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 54.0000 1.71278
\(995\) −23.3345 7.77817i −0.739754 0.246585i
\(996\) 0 0
\(997\) 48.0000i 1.52018i −0.649821 0.760088i \(-0.725156\pi\)
0.649821 0.760088i \(-0.274844\pi\)
\(998\) 22.6274i 0.716258i
\(999\) 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 135.2.b.b.109.1 4
3.2 odd 2 inner 135.2.b.b.109.4 yes 4
4.3 odd 2 2160.2.f.k.1729.1 4
5.2 odd 4 675.2.a.m.1.2 2
5.3 odd 4 675.2.a.l.1.1 2
5.4 even 2 inner 135.2.b.b.109.3 yes 4
9.2 odd 6 405.2.j.f.109.3 8
9.4 even 3 405.2.j.f.379.4 8
9.5 odd 6 405.2.j.f.379.1 8
9.7 even 3 405.2.j.f.109.2 8
12.11 even 2 2160.2.f.k.1729.4 4
15.2 even 4 675.2.a.m.1.1 2
15.8 even 4 675.2.a.l.1.2 2
15.14 odd 2 inner 135.2.b.b.109.2 yes 4
20.19 odd 2 2160.2.f.k.1729.2 4
45.4 even 6 405.2.j.f.379.2 8
45.14 odd 6 405.2.j.f.379.3 8
45.29 odd 6 405.2.j.f.109.1 8
45.34 even 6 405.2.j.f.109.4 8
60.59 even 2 2160.2.f.k.1729.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.b.b.109.1 4 1.1 even 1 trivial
135.2.b.b.109.2 yes 4 15.14 odd 2 inner
135.2.b.b.109.3 yes 4 5.4 even 2 inner
135.2.b.b.109.4 yes 4 3.2 odd 2 inner
405.2.j.f.109.1 8 45.29 odd 6
405.2.j.f.109.2 8 9.7 even 3
405.2.j.f.109.3 8 9.2 odd 6
405.2.j.f.109.4 8 45.34 even 6
405.2.j.f.379.1 8 9.5 odd 6
405.2.j.f.379.2 8 45.4 even 6
405.2.j.f.379.3 8 45.14 odd 6
405.2.j.f.379.4 8 9.4 even 3
675.2.a.l.1.1 2 5.3 odd 4
675.2.a.l.1.2 2 15.8 even 4
675.2.a.m.1.1 2 15.2 even 4
675.2.a.m.1.2 2 5.2 odd 4
2160.2.f.k.1729.1 4 4.3 odd 2
2160.2.f.k.1729.2 4 20.19 odd 2
2160.2.f.k.1729.3 4 60.59 even 2
2160.2.f.k.1729.4 4 12.11 even 2