Properties

Label 1440.2.x.b.127.1
Level $1440$
Weight $2$
Character 1440.127
Analytic conductor $11.498$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(127,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 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("1440.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.x (of order \(4\), degree \(2\), 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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 127.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1440.127
Dual form 1440.2.x.b.703.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 2.00000i) q^{5} +(-1.00000 + 1.00000i) q^{7} +O(q^{10})\) \(q+(-1.00000 + 2.00000i) q^{5} +(-1.00000 + 1.00000i) q^{7} +6.00000i q^{11} +(-1.00000 + 1.00000i) q^{13} +(-1.00000 - 1.00000i) q^{17} +4.00000 q^{19} +(-5.00000 - 5.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} -8.00000i q^{29} +2.00000i q^{31} +(-1.00000 - 3.00000i) q^{35} +(-5.00000 - 5.00000i) q^{37} -6.00000 q^{41} +(3.00000 + 3.00000i) q^{43} +(-7.00000 + 7.00000i) q^{47} +5.00000i q^{49} +(1.00000 - 1.00000i) q^{53} +(-12.0000 - 6.00000i) q^{55} -4.00000 q^{59} +2.00000 q^{61} +(-1.00000 - 3.00000i) q^{65} +(-7.00000 + 7.00000i) q^{67} +6.00000i q^{71} +(9.00000 - 9.00000i) q^{73} +(-6.00000 - 6.00000i) q^{77} -8.00000 q^{79} +(5.00000 + 5.00000i) q^{83} +(3.00000 - 1.00000i) q^{85} -2.00000i q^{91} +(-4.00000 + 8.00000i) q^{95} +(-3.00000 - 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} - 2 q^{13} - 2 q^{17} + 8 q^{19} - 10 q^{23} - 6 q^{25} - 2 q^{35} - 10 q^{37} - 12 q^{41} + 6 q^{43} - 14 q^{47} + 2 q^{53} - 24 q^{55} - 8 q^{59} + 4 q^{61} - 2 q^{65} - 14 q^{67} + 18 q^{73} - 12 q^{77} - 16 q^{79} + 10 q^{83} + 6 q^{85} - 8 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 0 0
\(7\) −1.00000 + 1.00000i −0.377964 + 0.377964i −0.870367 0.492403i \(-0.836119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.00000i −0.277350 + 0.277350i −0.832050 0.554700i \(-0.812833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.00000i −0.242536 0.242536i 0.575363 0.817898i \(-0.304861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.00000 5.00000i −1.04257 1.04257i −0.999053 0.0435195i \(-0.986143\pi\)
−0.0435195 0.999053i \(-0.513857\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.00000i 1.48556i −0.669534 0.742781i \(-0.733506\pi\)
0.669534 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) 2.00000i 0.359211i 0.983739 + 0.179605i \(0.0574821\pi\)
−0.983739 + 0.179605i \(0.942518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 3.00000i −0.169031 0.507093i
\(36\) 0 0
\(37\) −5.00000 5.00000i −0.821995 0.821995i 0.164399 0.986394i \(-0.447432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 3.00000 + 3.00000i 0.457496 + 0.457496i 0.897833 0.440337i \(-0.145141\pi\)
−0.440337 + 0.897833i \(0.645141\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.00000 + 7.00000i −1.02105 + 1.02105i −0.0212814 + 0.999774i \(0.506775\pi\)
−0.999774 + 0.0212814i \(0.993225\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.00000 1.00000i 0.137361 0.137361i −0.635083 0.772444i \(-0.719034\pi\)
0.772444 + 0.635083i \(0.219034\pi\)
\(54\) 0 0
\(55\) −12.0000 6.00000i −1.61808 0.809040i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 3.00000i −0.124035 0.372104i
\(66\) 0 0
\(67\) −7.00000 + 7.00000i −0.855186 + 0.855186i −0.990766 0.135580i \(-0.956710\pi\)
0.135580 + 0.990766i \(0.456710\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 0 0
\(73\) 9.00000 9.00000i 1.05337 1.05337i 0.0548772 0.998493i \(-0.482523\pi\)
0.998493 0.0548772i \(-0.0174767\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 6.00000i −0.683763 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.00000 + 5.00000i 0.548821 + 0.548821i 0.926100 0.377279i \(-0.123140\pi\)
−0.377279 + 0.926100i \(0.623140\pi\)
\(84\) 0 0
\(85\) 3.00000 1.00000i 0.325396 0.108465i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 2.00000i 0.209657i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 + 8.00000i −0.410391 + 0.820783i
\(96\) 0 0
\(97\) −3.00000 3.00000i −0.304604 0.304604i 0.538208 0.842812i \(-0.319101\pi\)
−0.842812 + 0.538208i \(0.819101\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −3.00000 3.00000i −0.295599 0.295599i 0.543688 0.839287i \(-0.317027\pi\)
−0.839287 + 0.543688i \(0.817027\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 3.00000i 0.290021 0.290021i −0.547068 0.837088i \(-0.684256\pi\)
0.837088 + 0.547068i \(0.184256\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.00000 3.00000i 0.282216 0.282216i −0.551776 0.833992i \(-0.686050\pi\)
0.833992 + 0.551776i \(0.186050\pi\)
\(114\) 0 0
\(115\) 15.0000 5.00000i 1.39876 0.466252i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) −5.00000 + 5.00000i −0.443678 + 0.443678i −0.893246 0.449568i \(-0.851578\pi\)
0.449568 + 0.893246i \(0.351578\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000i 0.174741i −0.996176 0.0873704i \(-0.972154\pi\)
0.996176 0.0873704i \(-0.0278464\pi\)
\(132\) 0 0
\(133\) −4.00000 + 4.00000i −0.346844 + 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.0000 13.0000i −1.11066 1.11066i −0.993061 0.117604i \(-0.962479\pi\)
−0.117604 0.993061i \(-0.537521\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.00000 6.00000i −0.501745 0.501745i
\(144\) 0 0
\(145\) 16.0000 + 8.00000i 1.32873 + 0.664364i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) 18.0000i 1.46482i 0.680864 + 0.732410i \(0.261604\pi\)
−0.680864 + 0.732410i \(0.738396\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 2.00000i −0.321288 0.160644i
\(156\) 0 0
\(157\) 3.00000 + 3.00000i 0.239426 + 0.239426i 0.816612 0.577186i \(-0.195849\pi\)
−0.577186 + 0.816612i \(0.695849\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.0000 0.788110
\(162\) 0 0
\(163\) −1.00000 1.00000i −0.0783260 0.0783260i 0.666858 0.745184i \(-0.267639\pi\)
−0.745184 + 0.666858i \(0.767639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.00000 5.00000i 0.386912 0.386912i −0.486673 0.873584i \(-0.661790\pi\)
0.873584 + 0.486673i \(0.161790\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.00000 + 7.00000i −0.532200 + 0.532200i −0.921227 0.389026i \(-0.872811\pi\)
0.389026 + 0.921227i \(0.372811\pi\)
\(174\) 0 0
\(175\) 7.00000 + 1.00000i 0.529150 + 0.0755929i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.0000 5.00000i 1.10282 0.367607i
\(186\) 0 0
\(187\) 6.00000 6.00000i 0.438763 0.438763i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000i 0.723575i −0.932261 0.361787i \(-0.882167\pi\)
0.932261 0.361787i \(-0.117833\pi\)
\(192\) 0 0
\(193\) 1.00000 1.00000i 0.0719816 0.0719816i −0.670199 0.742181i \(-0.733791\pi\)
0.742181 + 0.670199i \(0.233791\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.00000 + 1.00000i 0.0712470 + 0.0712470i 0.741832 0.670585i \(-0.233957\pi\)
−0.670585 + 0.741832i \(0.733957\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.00000 + 8.00000i 0.561490 + 0.561490i
\(204\) 0 0
\(205\) 6.00000 12.0000i 0.419058 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000i 1.66011i
\(210\) 0 0
\(211\) 10.0000i 0.688428i 0.938891 + 0.344214i \(0.111855\pi\)
−0.938891 + 0.344214i \(0.888145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.00000 + 3.00000i −0.613795 + 0.204598i
\(216\) 0 0
\(217\) −2.00000 2.00000i −0.135769 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −19.0000 19.0000i −1.27233 1.27233i −0.944860 0.327474i \(-0.893803\pi\)
−0.327474 0.944860i \(-0.606197\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.0000 + 13.0000i −0.862840 + 0.862840i −0.991667 0.128827i \(-0.958879\pi\)
0.128827 + 0.991667i \(0.458879\pi\)
\(228\) 0 0
\(229\) 16.0000i 1.05731i 0.848837 + 0.528655i \(0.177303\pi\)
−0.848837 + 0.528655i \(0.822697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.0000 + 13.0000i −0.851658 + 0.851658i −0.990337 0.138679i \(-0.955714\pi\)
0.138679 + 0.990337i \(0.455714\pi\)
\(234\) 0 0
\(235\) −7.00000 21.0000i −0.456630 1.36989i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.0000 5.00000i −0.638877 0.319438i
\(246\) 0 0
\(247\) −4.00000 + 4.00000i −0.254514 + 0.254514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000i 1.13615i −0.822977 0.568075i \(-0.807688\pi\)
0.822977 0.568075i \(-0.192312\pi\)
\(252\) 0 0
\(253\) 30.0000 30.0000i 1.88608 1.88608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.0000 + 11.0000i 0.686161 + 0.686161i 0.961381 0.275220i \(-0.0887507\pi\)
−0.275220 + 0.961381i \(0.588751\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 9.00000i −0.554964 0.554964i 0.372906 0.927869i \(-0.378362\pi\)
−0.927869 + 0.372906i \(0.878362\pi\)
\(264\) 0 0
\(265\) 1.00000 + 3.00000i 0.0614295 + 0.184289i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.00000i 0.243884i 0.992537 + 0.121942i \(0.0389122\pi\)
−0.992537 + 0.121942i \(0.961088\pi\)
\(270\) 0 0
\(271\) 22.0000i 1.33640i −0.743980 0.668202i \(-0.767064\pi\)
0.743980 0.668202i \(-0.232936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.0000 18.0000i 1.44725 1.08544i
\(276\) 0 0
\(277\) −9.00000 9.00000i −0.540758 0.540758i 0.382993 0.923751i \(-0.374893\pi\)
−0.923751 + 0.382993i \(0.874893\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 15.0000 + 15.0000i 0.891657 + 0.891657i 0.994679 0.103022i \(-0.0328511\pi\)
−0.103022 + 0.994679i \(0.532851\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 6.00000i 0.354169 0.354169i
\(288\) 0 0
\(289\) 15.0000i 0.882353i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.0000 17.0000i 0.993151 0.993151i −0.00682610 0.999977i \(-0.502173\pi\)
0.999977 + 0.00682610i \(0.00217283\pi\)
\(294\) 0 0
\(295\) 4.00000 8.00000i 0.232889 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 + 4.00000i −0.114520 + 0.229039i
\(306\) 0 0
\(307\) −7.00000 + 7.00000i −0.399511 + 0.399511i −0.878061 0.478549i \(-0.841163\pi\)
0.478549 + 0.878061i \(0.341163\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.0000i 1.24751i 0.781622 + 0.623753i \(0.214393\pi\)
−0.781622 + 0.623753i \(0.785607\pi\)
\(312\) 0 0
\(313\) −15.0000 + 15.0000i −0.847850 + 0.847850i −0.989865 0.142014i \(-0.954642\pi\)
0.142014 + 0.989865i \(0.454642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.0000 + 25.0000i 1.40414 + 1.40414i 0.786318 + 0.617822i \(0.211985\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) 48.0000 2.68748
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 4.00000i −0.222566 0.222566i
\(324\) 0 0
\(325\) 7.00000 + 1.00000i 0.388290 + 0.0554700i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.0000i 0.771845i
\(330\) 0 0
\(331\) 18.0000i 0.989369i 0.869072 + 0.494685i \(0.164716\pi\)
−0.869072 + 0.494685i \(0.835284\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.00000 21.0000i −0.382451 1.14735i
\(336\) 0 0
\(337\) 1.00000 + 1.00000i 0.0544735 + 0.0544735i 0.733819 0.679345i \(-0.237736\pi\)
−0.679345 + 0.733819i \(0.737736\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −12.0000 12.0000i −0.647939 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 + 9.00000i −0.483145 + 0.483145i −0.906135 0.422989i \(-0.860981\pi\)
0.422989 + 0.906135i \(0.360981\pi\)
\(348\) 0 0
\(349\) 32.0000i 1.71292i 0.516213 + 0.856460i \(0.327341\pi\)
−0.516213 + 0.856460i \(0.672659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.0000 15.0000i 0.798369 0.798369i −0.184469 0.982838i \(-0.559057\pi\)
0.982838 + 0.184469i \(0.0590565\pi\)
\(354\) 0 0
\(355\) −12.0000 6.00000i −0.636894 0.318447i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.00000 + 27.0000i 0.471082 + 1.41324i
\(366\) 0 0
\(367\) −13.0000 + 13.0000i −0.678594 + 0.678594i −0.959682 0.281088i \(-0.909305\pi\)
0.281088 + 0.959682i \(0.409305\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000i 0.103835i
\(372\) 0 0
\(373\) −21.0000 + 21.0000i −1.08734 + 1.08734i −0.0915371 + 0.995802i \(0.529178\pi\)
−0.995802 + 0.0915371i \(0.970822\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 + 8.00000i 0.412021 + 0.412021i
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −13.0000 13.0000i −0.664269 0.664269i 0.292114 0.956383i \(-0.405641\pi\)
−0.956383 + 0.292114i \(0.905641\pi\)
\(384\) 0 0
\(385\) 18.0000 6.00000i 0.917365 0.305788i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.0000i 0.608424i 0.952604 + 0.304212i \(0.0983931\pi\)
−0.952604 + 0.304212i \(0.901607\pi\)
\(390\) 0 0
\(391\) 10.0000i 0.505722i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 16.0000i 0.402524 0.805047i
\(396\) 0 0
\(397\) 19.0000 + 19.0000i 0.953583 + 0.953583i 0.998969 0.0453868i \(-0.0144520\pi\)
−0.0453868 + 0.998969i \(0.514452\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −2.00000 2.00000i −0.0996271 0.0996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.0000 30.0000i 1.48704 1.48704i
\(408\) 0 0
\(409\) 20.0000i 0.988936i 0.869196 + 0.494468i \(0.164637\pi\)
−0.869196 + 0.494468i \(0.835363\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.00000 4.00000i 0.196827 0.196827i
\(414\) 0 0
\(415\) −15.0000 + 5.00000i −0.736321 + 0.245440i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.00000 + 7.00000i −0.0485071 + 0.339550i
\(426\) 0 0
\(427\) −2.00000 + 2.00000i −0.0967868 + 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.0000i 1.83040i 0.403005 + 0.915198i \(0.367966\pi\)
−0.403005 + 0.915198i \(0.632034\pi\)
\(432\) 0 0
\(433\) 5.00000 5.00000i 0.240285 0.240285i −0.576683 0.816968i \(-0.695653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20.0000 20.0000i −0.956730 0.956730i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.0000 + 17.0000i 0.807694 + 0.807694i 0.984284 0.176590i \(-0.0565067\pi\)
−0.176590 + 0.984284i \(0.556507\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.00000i 0.188772i −0.995536 0.0943858i \(-0.969911\pi\)
0.995536 0.0943858i \(-0.0300887\pi\)
\(450\) 0 0
\(451\) 36.0000i 1.69517i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 + 2.00000i 0.187523 + 0.0937614i
\(456\) 0 0
\(457\) −15.0000 15.0000i −0.701670 0.701670i 0.263099 0.964769i \(-0.415256\pi\)
−0.964769 + 0.263099i \(0.915256\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 17.0000 + 17.0000i 0.790057 + 0.790057i 0.981503 0.191446i \(-0.0613177\pi\)
−0.191446 + 0.981503i \(0.561318\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.00000 + 5.00000i −0.231372 + 0.231372i −0.813265 0.581893i \(-0.802312\pi\)
0.581893 + 0.813265i \(0.302312\pi\)
\(468\) 0 0
\(469\) 14.0000i 0.646460i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −18.0000 + 18.0000i −0.827641 + 0.827641i
\(474\) 0 0
\(475\) −12.0000 16.0000i −0.550598 0.734130i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.00000 3.00000i 0.408669 0.136223i
\(486\) 0 0
\(487\) 15.0000 15.0000i 0.679715 0.679715i −0.280221 0.959936i \(-0.590408\pi\)
0.959936 + 0.280221i \(0.0904077\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000i 0.0902587i −0.998981 0.0451294i \(-0.985630\pi\)
0.998981 0.0451294i \(-0.0143700\pi\)
\(492\) 0 0
\(493\) −8.00000 + 8.00000i −0.360302 + 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 6.00000i −0.269137 0.269137i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.0000 + 11.0000i 0.490466 + 0.490466i 0.908453 0.417987i \(-0.137264\pi\)
−0.417987 + 0.908453i \(0.637264\pi\)
\(504\) 0 0
\(505\) 6.00000 12.0000i 0.266996 0.533993i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 18.0000i 0.796273i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.00000 3.00000i 0.396587 0.132196i
\(516\) 0 0
\(517\) −42.0000 42.0000i −1.84716 1.84716i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −25.0000 25.0000i −1.09317 1.09317i −0.995188 0.0979859i \(-0.968760\pi\)
−0.0979859 0.995188i \(-0.531240\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 2.00000i 0.0871214 0.0871214i
\(528\) 0 0
\(529\) 27.0000i 1.17391i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 6.00000i 0.259889 0.259889i
\(534\) 0 0
\(535\) 3.00000 + 9.00000i 0.129701 + 0.389104i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.00000 4.00000i −0.342682 0.171341i
\(546\) 0 0
\(547\) −3.00000 + 3.00000i −0.128271 + 0.128271i −0.768328 0.640057i \(-0.778911\pi\)
0.640057 + 0.768328i \(0.278911\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 32.0000i 1.36325i
\(552\) 0 0
\(553\) 8.00000 8.00000i 0.340195 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15.0000 15.0000i −0.635570 0.635570i 0.313889 0.949460i \(-0.398368\pi\)
−0.949460 + 0.313889i \(0.898368\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.0000 15.0000i −0.632175 0.632175i 0.316438 0.948613i \(-0.397513\pi\)
−0.948613 + 0.316438i \(0.897513\pi\)
\(564\) 0 0
\(565\) 3.00000 + 9.00000i 0.126211 + 0.378633i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0000i 0.838444i 0.907884 + 0.419222i \(0.137697\pi\)
−0.907884 + 0.419222i \(0.862303\pi\)
\(570\) 0 0
\(571\) 26.0000i 1.08807i 0.839064 + 0.544033i \(0.183103\pi\)
−0.839064 + 0.544033i \(0.816897\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.00000 + 35.0000i −0.208514 + 1.45960i
\(576\) 0 0
\(577\) −3.00000 3.00000i −0.124892 0.124892i 0.641898 0.766790i \(-0.278147\pi\)
−0.766790 + 0.641898i \(0.778147\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −10.0000 −0.414870
\(582\) 0 0
\(583\) 6.00000 + 6.00000i 0.248495 + 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25.0000 + 25.0000i −1.03186 + 1.03186i −0.0323850 + 0.999475i \(0.510310\pi\)
−0.999475 + 0.0323850i \(0.989690\pi\)
\(588\) 0 0
\(589\) 8.00000i 0.329634i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.0000 23.0000i 0.944497 0.944497i −0.0540419 0.998539i \(-0.517210\pi\)
0.998539 + 0.0540419i \(0.0172104\pi\)
\(594\) 0 0
\(595\) −2.00000 + 4.00000i −0.0819920 + 0.163984i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25.0000 50.0000i 1.01639 2.03279i
\(606\) 0 0
\(607\) 15.0000 15.0000i 0.608831 0.608831i −0.333809 0.942641i \(-0.608334\pi\)
0.942641 + 0.333809i \(0.108334\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.0000i 0.566379i
\(612\) 0 0
\(613\) 3.00000 3.00000i 0.121169 0.121169i −0.643922 0.765091i \(-0.722694\pi\)
0.765091 + 0.643922i \(0.222694\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.0000 + 11.0000i 0.442843 + 0.442843i 0.892966 0.450123i \(-0.148620\pi\)
−0.450123 + 0.892966i \(0.648620\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0000i 0.398726i
\(630\) 0 0
\(631\) 10.0000i 0.398094i 0.979990 + 0.199047i \(0.0637846\pi\)
−0.979990 + 0.199047i \(0.936215\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.00000 15.0000i −0.198419 0.595257i
\(636\) 0 0
\(637\) −5.00000 5.00000i −0.198107 0.198107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 11.0000 + 11.0000i 0.433798 + 0.433798i 0.889918 0.456120i \(-0.150761\pi\)
−0.456120 + 0.889918i \(0.650761\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −31.0000 + 31.0000i −1.21874 + 1.21874i −0.250661 + 0.968075i \(0.580648\pi\)
−0.968075 + 0.250661i \(0.919352\pi\)
\(648\) 0 0
\(649\) 24.0000i 0.942082i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.00000 5.00000i 0.195665 0.195665i −0.602474 0.798139i \(-0.705818\pi\)
0.798139 + 0.602474i \(0.205818\pi\)
\(654\) 0 0
\(655\) 4.00000 + 2.00000i 0.156293 + 0.0781465i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 12.0000i −0.155113 0.465340i
\(666\) 0 0
\(667\) −40.0000 + 40.0000i −1.54881 + 1.54881i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000i 0.463255i
\(672\) 0 0
\(673\) 29.0000 29.0000i 1.11787 1.11787i 0.125814 0.992054i \(-0.459846\pi\)
0.992054 0.125814i \(-0.0401543\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.0000 15.0000i −0.576497 0.576497i 0.357439 0.933936i \(-0.383650\pi\)
−0.933936 + 0.357439i \(0.883650\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 13.0000 + 13.0000i 0.497431 + 0.497431i 0.910637 0.413206i \(-0.135591\pi\)
−0.413206 + 0.910637i \(0.635591\pi\)
\(684\) 0 0
\(685\) 39.0000 13.0000i 1.49011 0.496704i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.00000i 0.0761939i
\(690\) 0 0
\(691\) 10.0000i 0.380418i 0.981744 + 0.190209i \(0.0609166\pi\)
−0.981744 + 0.190209i \(0.939083\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 + 24.0000i −0.455186 + 0.910372i
\(696\) 0 0
\(697\) 6.00000 + 6.00000i 0.227266 + 0.227266i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −20.0000 20.0000i −0.754314 0.754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 6.00000i 0.225653 0.225653i
\(708\) 0 0
\(709\) 24.0000i 0.901339i −0.892691 0.450669i \(-0.851185\pi\)
0.892691 0.450669i \(-0.148815\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.0000 10.0000i 0.374503 0.374503i
\(714\) 0 0
\(715\) 18.0000 6.00000i 0.673162 0.224387i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −32.0000 + 24.0000i −1.18845 + 0.891338i
\(726\) 0 0
\(727\) 15.0000 15.0000i 0.556319 0.556319i −0.371938 0.928257i \(-0.621307\pi\)
0.928257 + 0.371938i \(0.121307\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.00000i 0.221918i
\(732\) 0 0
\(733\) −9.00000 + 9.00000i −0.332423 + 0.332423i −0.853506 0.521083i \(-0.825528\pi\)
0.521083 + 0.853506i \(0.325528\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −42.0000 42.0000i −1.54709 1.54709i
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.00000 9.00000i −0.330178 0.330178i 0.522476 0.852654i \(-0.325008\pi\)
−0.852654 + 0.522476i \(0.825008\pi\)
\(744\) 0 0
\(745\) −24.0000 12.0000i −0.879292 0.439646i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.00000i 0.219235i
\(750\) 0 0
\(751\) 26.0000i 0.948753i 0.880322 + 0.474377i \(0.157327\pi\)
−0.880322 + 0.474377i \(0.842673\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −36.0000 18.0000i −1.31017 0.655087i
\(756\) 0 0
\(757\) −1.00000 1.00000i −0.0363456 0.0363456i 0.688700 0.725046i \(-0.258182\pi\)
−0.725046 + 0.688700i \(0.758182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) 0 0
\(763\) −4.00000 4.00000i −0.144810 0.144810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.00000 4.00000i 0.144432 0.144432i
\(768\) 0 0
\(769\) 8.00000i 0.288487i −0.989542 0.144244i \(-0.953925\pi\)
0.989542 0.144244i \(-0.0460749\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −35.0000 + 35.0000i −1.25886 + 1.25886i −0.307226 + 0.951637i \(0.599401\pi\)
−0.951637 + 0.307226i \(0.900599\pi\)
\(774\) 0 0
\(775\) 8.00000 6.00000i 0.287368 0.215526i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.00000 + 3.00000i −0.321224 + 0.107075i
\(786\) 0 0
\(787\) 9.00000 9.00000i 0.320815 0.320815i −0.528265 0.849080i \(-0.677157\pi\)
0.849080 + 0.528265i \(0.177157\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000i 0.213335i
\(792\) 0 0
\(793\) −2.00000 + 2.00000i −0.0710221 + 0.0710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.0000 + 17.0000i 0.602171 + 0.602171i 0.940888 0.338717i \(-0.109993\pi\)
−0.338717 + 0.940888i \(0.609993\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 54.0000 + 54.0000i 1.90562 + 1.90562i
\(804\) 0 0
\(805\) −10.0000 + 20.0000i −0.352454 + 0.704907i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 30.0000i 1.05344i −0.850038 0.526721i \(-0.823421\pi\)
0.850038 0.526721i \(-0.176579\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.00000 1.00000i 0.105085 0.0350285i
\(816\) 0 0
\(817\) 12.0000 + 12.0000i 0.419827 + 0.419827i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −3.00000 3.00000i −0.104573 0.104573i 0.652884 0.757458i \(-0.273559\pi\)
−0.757458 + 0.652884i \(0.773559\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.00000 7.00000i 0.243414 0.243414i −0.574847 0.818261i \(-0.694938\pi\)
0.818261 + 0.574847i \(0.194938\pi\)
\(828\) 0 0
\(829\) 36.0000i 1.25033i −0.780492 0.625166i \(-0.785031\pi\)
0.780492 0.625166i \(-0.214969\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.00000 5.00000i 0.173240 0.173240i
\(834\) 0 0
\(835\) 5.00000 + 15.0000i 0.173032 + 0.519096i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −22.0000 11.0000i −0.756823 0.378412i
\(846\) 0 0
\(847\) 25.0000 25.0000i 0.859010 0.859010i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 50.0000i 1.71398i
\(852\) 0 0
\(853\) 27.0000 27.0000i 0.924462 0.924462i −0.0728784 0.997341i \(-0.523219\pi\)
0.997341 + 0.0728784i \(0.0232185\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.00000 5.00000i −0.170797 0.170797i 0.616533 0.787329i \(-0.288537\pi\)
−0.787329 + 0.616533i \(0.788537\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.0000 25.0000i −0.851010 0.851010i 0.139248 0.990258i \(-0.455532\pi\)
−0.990258 + 0.139248i \(0.955532\pi\)
\(864\) 0 0
\(865\) −7.00000 21.0000i −0.238007 0.714021i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 48.0000i 1.62829i
\(870\) 0 0
\(871\) 14.0000i 0.474372i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.00000 + 13.0000i −0.304256 + 0.439480i
\(876\) 0 0
\(877\) 27.0000 + 27.0000i 0.911725 + 0.911725i 0.996408 0.0846827i \(-0.0269877\pi\)
−0.0846827 + 0.996408i \(0.526988\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) −17.0000 17.0000i −0.572096 0.572096i 0.360618 0.932714i \(-0.382566\pi\)
−0.932714 + 0.360618i \(0.882566\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.00000 + 7.00000i −0.235037 + 0.235037i −0.814791 0.579754i \(-0.803149\pi\)
0.579754 + 0.814791i \(0.303149\pi\)
\(888\) 0 0
\(889\) 10.0000i 0.335389i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −28.0000 + 28.0000i −0.936984 + 0.936984i
\(894\) 0 0
\(895\) −12.0000 + 24.0000i −0.401116 + 0.802232i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 20.0000i 0.332411 0.664822i
\(906\) 0 0
\(907\) −27.0000 + 27.0000i −0.896520 + 0.896520i −0.995127 0.0986062i \(-0.968562\pi\)
0.0986062 + 0.995127i \(0.468562\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 50.0000i 1.65657i −0.560304 0.828287i \(-0.689316\pi\)
0.560304 0.828287i \(-0.310684\pi\)
\(912\) 0 0
\(913\) −30.0000 + 30.0000i −0.992855 + 0.992855i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.00000 + 2.00000i 0.0660458 + 0.0660458i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.00000 6.00000i −0.197492 0.197492i
\(924\) 0 0
\(925\) −5.00000 + 35.0000i −0.164399 + 1.15079i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.0000i 0.393707i −0.980433 0.196854i \(-0.936928\pi\)
0.980433 0.196854i \(-0.0630724\pi\)
\(930\) 0 0
\(931\) 20.0000i 0.655474i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00000 + 18.0000i 0.196221 + 0.588663i
\(936\) 0 0
\(937\) −3.00000 3.00000i −0.0980057 0.0980057i 0.656404 0.754410i \(-0.272077\pi\)
−0.754410 + 0.656404i \(0.772077\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 30.0000 + 30.0000i 0.976934 + 0.976934i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41.0000 + 41.0000i −1.33232 + 1.33232i −0.429031 + 0.903290i \(0.641145\pi\)
−0.903290 + 0.429031i \(0.858855\pi\)
\(948\) 0 0
\(949\) 18.0000i 0.584305i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −9.00000 + 9.00000i −0.291539 + 0.291539i −0.837688 0.546149i \(-0.816093\pi\)
0.546149 + 0.837688i \(0.316093\pi\)
\(954\) 0 0
\(955\) 20.0000 + 10.0000i 0.647185 + 0.323592i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 26.0000 0.839584
\(960\) 0 0
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.00000 + 3.00000i 0.0321911 + 0.0965734i
\(966\) 0 0
\(967\) −37.0000 + 37.0000i −1.18984 + 1.18984i −0.212728 + 0.977111i \(0.568235\pi\)
−0.977111 + 0.212728i \(0.931765\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26.0000i 0.834380i −0.908819 0.417190i \(-0.863015\pi\)
0.908819 0.417190i \(-0.136985\pi\)
\(972\) 0 0
\(973\) −12.0000 + 12.0000i −0.384702 + 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0000 + 27.0000i 0.863807 + 0.863807i 0.991778 0.127971i \(-0.0408466\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −41.0000 41.0000i −1.30770 1.30770i −0.923074 0.384623i \(-0.874331\pi\)
−0.384623 0.923074i \(-0.625669\pi\)
\(984\) 0 0
\(985\) −3.00000 + 1.00000i −0.0955879 + 0.0318626i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30.0000i 0.953945i
\(990\) 0 0
\(991\) 34.0000i 1.08005i 0.841650 + 0.540023i \(0.181584\pi\)
−0.841650 + 0.540023i \(0.818416\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 27.0000 + 27.0000i 0.855099 + 0.855099i 0.990756 0.135657i \(-0.0433146\pi\)
−0.135657 + 0.990756i \(0.543315\pi\)
\(998\) 0 0
\(999\) 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 1440.2.x.b.127.1 2
3.2 odd 2 160.2.n.b.127.1 yes 2
4.3 odd 2 1440.2.x.e.127.1 2
5.3 odd 4 1440.2.x.e.703.1 2
12.11 even 2 160.2.n.e.127.1 yes 2
15.2 even 4 800.2.n.c.543.1 2
15.8 even 4 160.2.n.e.63.1 yes 2
15.14 odd 2 800.2.n.h.607.1 2
20.3 even 4 inner 1440.2.x.b.703.1 2
24.5 odd 2 320.2.n.f.127.1 2
24.11 even 2 320.2.n.c.127.1 2
48.5 odd 4 1280.2.o.e.127.1 2
48.11 even 4 1280.2.o.n.127.1 2
48.29 odd 4 1280.2.o.k.127.1 2
48.35 even 4 1280.2.o.d.127.1 2
60.23 odd 4 160.2.n.b.63.1 2
60.47 odd 4 800.2.n.h.543.1 2
60.59 even 2 800.2.n.c.607.1 2
120.29 odd 2 1600.2.n.e.1407.1 2
120.53 even 4 320.2.n.c.63.1 2
120.59 even 2 1600.2.n.j.1407.1 2
120.77 even 4 1600.2.n.j.1343.1 2
120.83 odd 4 320.2.n.f.63.1 2
120.107 odd 4 1600.2.n.e.1343.1 2
240.53 even 4 1280.2.o.d.383.1 2
240.83 odd 4 1280.2.o.e.383.1 2
240.173 even 4 1280.2.o.n.383.1 2
240.203 odd 4 1280.2.o.k.383.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.n.b.63.1 2 60.23 odd 4
160.2.n.b.127.1 yes 2 3.2 odd 2
160.2.n.e.63.1 yes 2 15.8 even 4
160.2.n.e.127.1 yes 2 12.11 even 2
320.2.n.c.63.1 2 120.53 even 4
320.2.n.c.127.1 2 24.11 even 2
320.2.n.f.63.1 2 120.83 odd 4
320.2.n.f.127.1 2 24.5 odd 2
800.2.n.c.543.1 2 15.2 even 4
800.2.n.c.607.1 2 60.59 even 2
800.2.n.h.543.1 2 60.47 odd 4
800.2.n.h.607.1 2 15.14 odd 2
1280.2.o.d.127.1 2 48.35 even 4
1280.2.o.d.383.1 2 240.53 even 4
1280.2.o.e.127.1 2 48.5 odd 4
1280.2.o.e.383.1 2 240.83 odd 4
1280.2.o.k.127.1 2 48.29 odd 4
1280.2.o.k.383.1 2 240.203 odd 4
1280.2.o.n.127.1 2 48.11 even 4
1280.2.o.n.383.1 2 240.173 even 4
1440.2.x.b.127.1 2 1.1 even 1 trivial
1440.2.x.b.703.1 2 20.3 even 4 inner
1440.2.x.e.127.1 2 4.3 odd 2
1440.2.x.e.703.1 2 5.3 odd 4
1600.2.n.e.1343.1 2 120.107 odd 4
1600.2.n.e.1407.1 2 120.29 odd 2
1600.2.n.j.1343.1 2 120.77 even 4
1600.2.n.j.1407.1 2 120.59 even 2