Properties

Label 1512.2.r.d.505.2
Level $1512$
Weight $2$
Character 1512.505
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(505,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.2
Root \(-0.577806 + 2.22188i\) of defining polynomial
Character \(\chi\) \(=\) 1512.505
Dual form 1512.2.r.d.1009.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.81197 - 3.13842i) q^{5} +(-0.500000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(-1.81197 - 3.13842i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(1.95863 - 3.39245i) q^{11} +(-2.53644 - 4.39324i) q^{13} -1.03225 q^{17} -2.50895 q^{19} +(2.47895 + 4.29366i) q^{23} +(-4.06644 + 7.04328i) q^{25} +(-4.60288 + 7.97242i) q^{29} +(0.422194 + 0.731261i) q^{31} +3.62393 q^{35} +4.84439 q^{37} +(-2.07362 - 3.59161i) q^{41} +(2.20174 - 3.81352i) q^{43} +(-3.93758 + 6.82008i) q^{47} +(-0.500000 - 0.866025i) q^{49} -12.2786 q^{53} -14.1959 q^{55} +(-5.60288 - 9.70447i) q^{59} +(-0.208348 + 0.360870i) q^{61} +(-9.19188 + 15.9208i) q^{65} +(-5.02507 - 8.70368i) q^{67} +5.05162 q^{71} +7.20723 q^{73} +(1.95863 + 3.39245i) q^{77} +(-7.56570 + 13.1042i) q^{79} +(0.932821 - 1.61569i) q^{83} +(1.87040 + 3.23963i) q^{85} +0.669401 q^{89} +5.07288 q^{91} +(4.54612 + 7.87412i) q^{95} +(-7.63513 + 13.2244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} - 4 q^{7} + 6 q^{11} - 3 q^{13} + 16 q^{17} - 4 q^{19} + 5 q^{23} - 14 q^{25} - q^{29} + 11 q^{31} + 8 q^{35} + 54 q^{37} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{55} - 9 q^{59} - 7 q^{61} + 9 q^{65} - 12 q^{67} + 24 q^{71} + 26 q^{73} + 6 q^{77} - 22 q^{79} + 6 q^{83} - 11 q^{85} + 28 q^{89} + 6 q^{91} + 23 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) −1.81197 3.13842i −0.810336 1.40354i −0.912629 0.408788i \(-0.865951\pi\)
0.102294 0.994754i \(-0.467382\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.95863 3.39245i 0.590550 1.02286i −0.403609 0.914932i \(-0.632244\pi\)
0.994158 0.107930i \(-0.0344224\pi\)
\(12\) 0 0
\(13\) −2.53644 4.39324i −0.703481 1.21847i −0.967237 0.253876i \(-0.918295\pi\)
0.263755 0.964590i \(-0.415039\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.03225 −0.250357 −0.125178 0.992134i \(-0.539950\pi\)
−0.125178 + 0.992134i \(0.539950\pi\)
\(18\) 0 0
\(19\) −2.50895 −0.575592 −0.287796 0.957692i \(-0.592922\pi\)
−0.287796 + 0.957692i \(0.592922\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.47895 + 4.29366i 0.516896 + 0.895290i 0.999807 + 0.0196209i \(0.00624592\pi\)
−0.482912 + 0.875669i \(0.660421\pi\)
\(24\) 0 0
\(25\) −4.06644 + 7.04328i −0.813288 + 1.40866i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.60288 + 7.97242i −0.854733 + 1.48044i 0.0221599 + 0.999754i \(0.492946\pi\)
−0.876893 + 0.480686i \(0.840388\pi\)
\(30\) 0 0
\(31\) 0.422194 + 0.731261i 0.0758282 + 0.131338i 0.901446 0.432891i \(-0.142507\pi\)
−0.825618 + 0.564230i \(0.809173\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.62393 0.612556
\(36\) 0 0
\(37\) 4.84439 0.796412 0.398206 0.917296i \(-0.369633\pi\)
0.398206 + 0.917296i \(0.369633\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.07362 3.59161i −0.323844 0.560915i 0.657433 0.753513i \(-0.271642\pi\)
−0.981278 + 0.192598i \(0.938309\pi\)
\(42\) 0 0
\(43\) 2.20174 3.81352i 0.335762 0.581557i −0.647869 0.761752i \(-0.724340\pi\)
0.983631 + 0.180195i \(0.0576729\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.93758 + 6.82008i −0.574355 + 0.994812i 0.421757 + 0.906709i \(0.361414\pi\)
−0.996111 + 0.0881025i \(0.971920\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.2786 −1.68660 −0.843300 0.537443i \(-0.819390\pi\)
−0.843300 + 0.537443i \(0.819390\pi\)
\(54\) 0 0
\(55\) −14.1959 −1.91417
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.60288 9.70447i −0.729432 1.26341i −0.957123 0.289681i \(-0.906451\pi\)
0.227691 0.973733i \(-0.426882\pi\)
\(60\) 0 0
\(61\) −0.208348 + 0.360870i −0.0266763 + 0.0462047i −0.879055 0.476720i \(-0.841826\pi\)
0.852379 + 0.522924i \(0.175159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −9.19188 + 15.9208i −1.14011 + 1.97473i
\(66\) 0 0
\(67\) −5.02507 8.70368i −0.613910 1.06332i −0.990575 0.136974i \(-0.956262\pi\)
0.376665 0.926350i \(-0.377071\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.05162 0.599517 0.299759 0.954015i \(-0.403094\pi\)
0.299759 + 0.954015i \(0.403094\pi\)
\(72\) 0 0
\(73\) 7.20723 0.843543 0.421771 0.906702i \(-0.361408\pi\)
0.421771 + 0.906702i \(0.361408\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.95863 + 3.39245i 0.223207 + 0.386606i
\(78\) 0 0
\(79\) −7.56570 + 13.1042i −0.851208 + 1.47433i 0.0289116 + 0.999582i \(0.490796\pi\)
−0.880119 + 0.474753i \(0.842537\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.932821 1.61569i 0.102390 0.177345i −0.810279 0.586045i \(-0.800684\pi\)
0.912669 + 0.408699i \(0.134018\pi\)
\(84\) 0 0
\(85\) 1.87040 + 3.23963i 0.202873 + 0.351387i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.669401 0.0709564 0.0354782 0.999370i \(-0.488705\pi\)
0.0354782 + 0.999370i \(0.488705\pi\)
\(90\) 0 0
\(91\) 5.07288 0.531782
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.54612 + 7.87412i 0.466423 + 0.807868i
\(96\) 0 0
\(97\) −7.63513 + 13.2244i −0.775230 + 1.34274i 0.159436 + 0.987208i \(0.449032\pi\)
−0.934665 + 0.355529i \(0.884301\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.87840 + 10.1817i −0.584923 + 1.01312i 0.409962 + 0.912103i \(0.365542\pi\)
−0.994885 + 0.101014i \(0.967791\pi\)
\(102\) 0 0
\(103\) −5.51538 9.55293i −0.543447 0.941278i −0.998703 0.0509171i \(-0.983786\pi\)
0.455256 0.890361i \(-0.349548\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.842907 0.0814869 0.0407434 0.999170i \(-0.487027\pi\)
0.0407434 + 0.999170i \(0.487027\pi\)
\(108\) 0 0
\(109\) −17.1875 −1.64627 −0.823133 0.567849i \(-0.807776\pi\)
−0.823133 + 0.567849i \(0.807776\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.54538 + 7.87284i 0.427594 + 0.740614i 0.996659 0.0816784i \(-0.0260280\pi\)
−0.569065 + 0.822293i \(0.692695\pi\)
\(114\) 0 0
\(115\) 8.98353 15.5599i 0.837718 1.45097i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.516124 0.893953i 0.0473130 0.0819486i
\(120\) 0 0
\(121\) −2.17248 3.76284i −0.197498 0.342076i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3533 1.01547
\(126\) 0 0
\(127\) −14.4859 −1.28541 −0.642706 0.766113i \(-0.722188\pi\)
−0.642706 + 0.766113i \(0.722188\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.03476 + 6.98840i 0.352518 + 0.610580i 0.986690 0.162613i \(-0.0519921\pi\)
−0.634172 + 0.773192i \(0.718659\pi\)
\(132\) 0 0
\(133\) 1.25447 2.17281i 0.108777 0.188407i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.85792 17.0744i 0.842219 1.45877i −0.0457961 0.998951i \(-0.514582\pi\)
0.888015 0.459815i \(-0.152084\pi\)
\(138\) 0 0
\(139\) −8.35960 14.4792i −0.709052 1.22811i −0.965209 0.261479i \(-0.915790\pi\)
0.256157 0.966635i \(-0.417543\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −19.8718 −1.66176
\(144\) 0 0
\(145\) 33.3610 2.77048
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.16439 + 15.8732i 0.750776 + 1.30038i 0.947447 + 0.319912i \(0.103653\pi\)
−0.196671 + 0.980469i \(0.563013\pi\)
\(150\) 0 0
\(151\) 7.23100 12.5245i 0.588450 1.01923i −0.405985 0.913880i \(-0.633072\pi\)
0.994436 0.105346i \(-0.0335951\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.53000 2.65004i 0.122893 0.212856i
\(156\) 0 0
\(157\) −1.92387 3.33225i −0.153542 0.265942i 0.778985 0.627042i \(-0.215735\pi\)
−0.932527 + 0.361100i \(0.882401\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.95789 −0.390737
\(162\) 0 0
\(163\) 13.0322 1.02076 0.510382 0.859948i \(-0.329504\pi\)
0.510382 + 0.859948i \(0.329504\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.04538 5.27476i −0.235659 0.408173i 0.723805 0.690005i \(-0.242391\pi\)
−0.959464 + 0.281831i \(0.909058\pi\)
\(168\) 0 0
\(169\) −6.36704 + 11.0280i −0.489772 + 0.848310i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.89855 10.2166i 0.448458 0.776752i −0.549828 0.835278i \(-0.685307\pi\)
0.998286 + 0.0585258i \(0.0186400\pi\)
\(174\) 0 0
\(175\) −4.06644 7.04328i −0.307394 0.532422i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.06148 −0.0793389 −0.0396694 0.999213i \(-0.512630\pi\)
−0.0396694 + 0.999213i \(0.512630\pi\)
\(180\) 0 0
\(181\) −16.0384 −1.19212 −0.596062 0.802938i \(-0.703269\pi\)
−0.596062 + 0.802938i \(0.703269\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.77786 15.2037i −0.645361 1.11780i
\(186\) 0 0
\(187\) −2.02179 + 3.50185i −0.147848 + 0.256081i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.9676 20.7285i 0.865944 1.49986i −0.000163629 1.00000i \(-0.500052\pi\)
0.866107 0.499858i \(-0.166615\pi\)
\(192\) 0 0
\(193\) −6.60707 11.4438i −0.475587 0.823741i 0.524022 0.851705i \(-0.324431\pi\)
−0.999609 + 0.0279638i \(0.991098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.0403 1.78405 0.892023 0.451990i \(-0.149286\pi\)
0.892023 + 0.451990i \(0.149286\pi\)
\(198\) 0 0
\(199\) 8.28159 0.587066 0.293533 0.955949i \(-0.405169\pi\)
0.293533 + 0.955949i \(0.405169\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.60288 7.97242i −0.323059 0.559554i
\(204\) 0 0
\(205\) −7.51464 + 13.0157i −0.524845 + 0.909059i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.91410 + 8.51148i −0.339916 + 0.588751i
\(210\) 0 0
\(211\) −1.79752 3.11340i −0.123747 0.214335i 0.797496 0.603325i \(-0.206158\pi\)
−0.921242 + 0.388989i \(0.872824\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −15.9579 −1.08832
\(216\) 0 0
\(217\) −0.844387 −0.0573207
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.61823 + 4.53491i 0.176121 + 0.305051i
\(222\) 0 0
\(223\) −5.08601 + 8.80923i −0.340585 + 0.589910i −0.984541 0.175152i \(-0.943958\pi\)
0.643957 + 0.765062i \(0.277292\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.95054 12.0387i 0.461324 0.799036i −0.537704 0.843134i \(-0.680708\pi\)
0.999027 + 0.0440980i \(0.0140414\pi\)
\(228\) 0 0
\(229\) 2.84347 + 4.92504i 0.187902 + 0.325456i 0.944551 0.328366i \(-0.106498\pi\)
−0.756649 + 0.653822i \(0.773165\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.4775 −1.27601 −0.638006 0.770031i \(-0.720241\pi\)
−0.638006 + 0.770031i \(0.720241\pi\)
\(234\) 0 0
\(235\) 28.5390 1.86168
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.50000 + 4.33013i 0.161712 + 0.280093i 0.935483 0.353373i \(-0.114965\pi\)
−0.773771 + 0.633465i \(0.781632\pi\)
\(240\) 0 0
\(241\) −5.14080 + 8.90412i −0.331148 + 0.573565i −0.982737 0.185007i \(-0.940769\pi\)
0.651589 + 0.758572i \(0.274103\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.81197 + 3.13842i −0.115762 + 0.200506i
\(246\) 0 0
\(247\) 6.36379 + 11.0224i 0.404918 + 0.701339i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.829685 0.0523693 0.0261846 0.999657i \(-0.491664\pi\)
0.0261846 + 0.999657i \(0.491664\pi\)
\(252\) 0 0
\(253\) 19.4214 1.22101
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.87516 11.9081i −0.428860 0.742808i 0.567912 0.823089i \(-0.307751\pi\)
−0.996772 + 0.0802814i \(0.974418\pi\)
\(258\) 0 0
\(259\) −2.42219 + 4.19536i −0.150508 + 0.260687i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.9285 22.3929i 0.797208 1.38081i −0.124219 0.992255i \(-0.539643\pi\)
0.921427 0.388550i \(-0.127024\pi\)
\(264\) 0 0
\(265\) 22.2485 + 38.5355i 1.36671 + 2.36721i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.32410 −0.568500 −0.284250 0.958750i \(-0.591745\pi\)
−0.284250 + 0.958750i \(0.591745\pi\)
\(270\) 0 0
\(271\) 25.7421 1.56372 0.781861 0.623453i \(-0.214271\pi\)
0.781861 + 0.623453i \(0.214271\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.9293 + 27.5904i 0.960574 + 1.66376i
\(276\) 0 0
\(277\) 5.06570 8.77405i 0.304368 0.527181i −0.672752 0.739868i \(-0.734888\pi\)
0.977120 + 0.212686i \(0.0682213\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.47969 + 6.02699i −0.207581 + 0.359540i −0.950952 0.309339i \(-0.899892\pi\)
0.743371 + 0.668879i \(0.233226\pi\)
\(282\) 0 0
\(283\) 3.95920 + 6.85753i 0.235350 + 0.407638i 0.959374 0.282136i \(-0.0910430\pi\)
−0.724024 + 0.689774i \(0.757710\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.14723 0.244803
\(288\) 0 0
\(289\) −15.9345 −0.937321
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.63128 9.75367i −0.328983 0.569815i 0.653327 0.757076i \(-0.273373\pi\)
−0.982310 + 0.187260i \(0.940039\pi\)
\(294\) 0 0
\(295\) −20.3044 + 35.1683i −1.18217 + 2.04758i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.5754 21.7812i 0.727253 1.25964i
\(300\) 0 0
\(301\) 2.20174 + 3.81352i 0.126906 + 0.219808i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.51008 0.0864669
\(306\) 0 0
\(307\) 23.0142 1.31349 0.656744 0.754113i \(-0.271933\pi\)
0.656744 + 0.754113i \(0.271933\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.78832 11.7577i −0.384930 0.666719i 0.606829 0.794832i \(-0.292441\pi\)
−0.991760 + 0.128114i \(0.959108\pi\)
\(312\) 0 0
\(313\) 8.92362 15.4562i 0.504393 0.873634i −0.495595 0.868554i \(-0.665050\pi\)
0.999987 0.00507958i \(-0.00161689\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.69755 + 16.7966i −0.544669 + 0.943394i 0.453959 + 0.891022i \(0.350011\pi\)
−0.998628 + 0.0523711i \(0.983322\pi\)
\(318\) 0 0
\(319\) 18.0307 + 31.2301i 1.00952 + 1.74855i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.58986 0.144103
\(324\) 0 0
\(325\) 41.2571 2.28853
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.93758 6.82008i −0.217086 0.376003i
\(330\) 0 0
\(331\) −7.28729 + 12.6220i −0.400546 + 0.693765i −0.993792 0.111256i \(-0.964513\pi\)
0.593246 + 0.805021i \(0.297846\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −18.2105 + 31.5415i −0.994946 + 1.72330i
\(336\) 0 0
\(337\) −13.8962 24.0689i −0.756975 1.31112i −0.944387 0.328837i \(-0.893343\pi\)
0.187412 0.982281i \(-0.439990\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.30769 0.179121
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.45604 + 12.9142i 0.400261 + 0.693273i 0.993757 0.111564i \(-0.0355861\pi\)
−0.593496 + 0.804837i \(0.702253\pi\)
\(348\) 0 0
\(349\) 10.9579 18.9796i 0.586562 1.01596i −0.408116 0.912930i \(-0.633814\pi\)
0.994679 0.103026i \(-0.0328525\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.73100 13.3905i 0.411480 0.712703i −0.583572 0.812061i \(-0.698346\pi\)
0.995052 + 0.0993578i \(0.0316788\pi\)
\(354\) 0 0
\(355\) −9.15337 15.8541i −0.485810 0.841448i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.56506 −0.399269 −0.199634 0.979870i \(-0.563975\pi\)
−0.199634 + 0.979870i \(0.563975\pi\)
\(360\) 0 0
\(361\) −12.7052 −0.668694
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.0593 22.6193i −0.683553 1.18395i
\(366\) 0 0
\(367\) 14.5046 25.1227i 0.757133 1.31139i −0.187174 0.982327i \(-0.559933\pi\)
0.944307 0.329066i \(-0.106734\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.13932 10.6336i 0.318737 0.552069i
\(372\) 0 0
\(373\) −0.655525 1.13540i −0.0339418 0.0587889i 0.848556 0.529106i \(-0.177473\pi\)
−0.882497 + 0.470317i \(0.844139\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 46.6997 2.40515
\(378\) 0 0
\(379\) −15.7015 −0.806531 −0.403265 0.915083i \(-0.632125\pi\)
−0.403265 + 0.915083i \(0.632125\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.4804 18.1525i −0.535522 0.927551i −0.999138 0.0415148i \(-0.986782\pi\)
0.463616 0.886036i \(-0.346552\pi\)
\(384\) 0 0
\(385\) 7.09795 12.2940i 0.361745 0.626561i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.80937 10.0621i 0.294547 0.510170i −0.680333 0.732904i \(-0.738165\pi\)
0.974879 + 0.222733i \(0.0714980\pi\)
\(390\) 0 0
\(391\) −2.55889 4.43212i −0.129409 0.224142i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 54.8351 2.75906
\(396\) 0 0
\(397\) −35.5217 −1.78278 −0.891390 0.453237i \(-0.850269\pi\)
−0.891390 + 0.453237i \(0.850269\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.45388 4.25024i −0.122541 0.212247i 0.798228 0.602355i \(-0.205771\pi\)
−0.920769 + 0.390108i \(0.872438\pi\)
\(402\) 0 0
\(403\) 2.14174 3.70960i 0.106687 0.184788i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.48837 16.4343i 0.470321 0.814620i
\(408\) 0 0
\(409\) −2.21561 3.83756i −0.109555 0.189755i 0.806035 0.591868i \(-0.201609\pi\)
−0.915590 + 0.402113i \(0.868276\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.2058 0.551399
\(414\) 0 0
\(415\) −6.76096 −0.331882
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.7719 + 30.7819i 0.868216 + 1.50379i 0.863818 + 0.503803i \(0.168066\pi\)
0.00439727 + 0.999990i \(0.498600\pi\)
\(420\) 0 0
\(421\) 1.81923 3.15100i 0.0886639 0.153570i −0.818283 0.574816i \(-0.805074\pi\)
0.906947 + 0.421246i \(0.138407\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.19757 7.27041i 0.203612 0.352667i
\(426\) 0 0
\(427\) −0.208348 0.360870i −0.0100827 0.0174637i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.9303 −0.526495 −0.263247 0.964728i \(-0.584794\pi\)
−0.263247 + 0.964728i \(0.584794\pi\)
\(432\) 0 0
\(433\) 15.3189 0.736180 0.368090 0.929790i \(-0.380012\pi\)
0.368090 + 0.929790i \(0.380012\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.21954 10.7726i −0.297521 0.515322i
\(438\) 0 0
\(439\) −5.30117 + 9.18189i −0.253011 + 0.438228i −0.964353 0.264618i \(-0.914754\pi\)
0.711343 + 0.702846i \(0.248087\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.0680 + 17.4383i −0.478347 + 0.828521i −0.999692 0.0248251i \(-0.992097\pi\)
0.521345 + 0.853346i \(0.325430\pi\)
\(444\) 0 0
\(445\) −1.21293 2.10086i −0.0574985 0.0995903i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.74616 0.129599 0.0647997 0.997898i \(-0.479359\pi\)
0.0647997 + 0.997898i \(0.479359\pi\)
\(450\) 0 0
\(451\) −16.2458 −0.764985
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.19188 15.9208i −0.430922 0.746379i
\(456\) 0 0
\(457\) −12.7715 + 22.1208i −0.597423 + 1.03477i 0.395777 + 0.918347i \(0.370475\pi\)
−0.993200 + 0.116421i \(0.962858\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.82316 + 15.2822i −0.410936 + 0.711761i −0.994992 0.0999516i \(-0.968131\pi\)
0.584057 + 0.811713i \(0.301465\pi\)
\(462\) 0 0
\(463\) −13.2501 22.9499i −0.615785 1.06657i −0.990246 0.139328i \(-0.955506\pi\)
0.374461 0.927242i \(-0.377828\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.1735 −0.933519 −0.466759 0.884384i \(-0.654579\pi\)
−0.466759 + 0.884384i \(0.654579\pi\)
\(468\) 0 0
\(469\) 10.0501 0.464072
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.62479 14.9386i −0.396568 0.686876i
\(474\) 0 0
\(475\) 10.2025 17.6712i 0.468122 0.810811i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.92470 + 8.52984i −0.225015 + 0.389738i −0.956324 0.292309i \(-0.905577\pi\)
0.731309 + 0.682047i \(0.238910\pi\)
\(480\) 0 0
\(481\) −12.2875 21.2826i −0.560261 0.970401i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 55.3383 2.51278
\(486\) 0 0
\(487\) 10.0278 0.454403 0.227202 0.973848i \(-0.427042\pi\)
0.227202 + 0.973848i \(0.427042\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.610055 1.05665i −0.0275314 0.0476858i 0.851931 0.523653i \(-0.175431\pi\)
−0.879463 + 0.475968i \(0.842098\pi\)
\(492\) 0 0
\(493\) 4.75131 8.22951i 0.213988 0.370639i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.52581 + 4.37483i −0.113298 + 0.196238i
\(498\) 0 0
\(499\) −10.3222 17.8786i −0.462086 0.800356i 0.536979 0.843596i \(-0.319566\pi\)
−0.999065 + 0.0432393i \(0.986232\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.85094 −0.394644 −0.197322 0.980339i \(-0.563224\pi\)
−0.197322 + 0.980339i \(0.563224\pi\)
\(504\) 0 0
\(505\) 42.6059 1.89594
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.8460 22.2499i −0.569388 0.986209i −0.996627 0.0820702i \(-0.973847\pi\)
0.427238 0.904139i \(-0.359487\pi\)
\(510\) 0 0
\(511\) −3.60362 + 6.24165i −0.159415 + 0.276114i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −19.9874 + 34.6191i −0.880749 + 1.52550i
\(516\) 0 0
\(517\) 15.4245 + 26.7161i 0.678370 + 1.17497i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.6797 0.643129 0.321565 0.946888i \(-0.395791\pi\)
0.321565 + 0.946888i \(0.395791\pi\)
\(522\) 0 0
\(523\) 20.7922 0.909182 0.454591 0.890700i \(-0.349786\pi\)
0.454591 + 0.890700i \(0.349786\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.435809 0.754843i −0.0189841 0.0328815i
\(528\) 0 0
\(529\) −0.790345 + 1.36892i −0.0343628 + 0.0595181i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.5192 + 18.2198i −0.455637 + 0.789187i
\(534\) 0 0
\(535\) −1.52732 2.64539i −0.0660317 0.114370i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.91726 −0.168728
\(540\) 0 0
\(541\) −30.9593 −1.33104 −0.665521 0.746379i \(-0.731791\pi\)
−0.665521 + 0.746379i \(0.731791\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 31.1432 + 53.9416i 1.33403 + 2.31060i
\(546\) 0 0
\(547\) 5.80535 10.0552i 0.248219 0.429928i −0.714813 0.699316i \(-0.753488\pi\)
0.963032 + 0.269388i \(0.0868214\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.5484 20.0024i 0.491977 0.852130i
\(552\) 0 0
\(553\) −7.56570 13.1042i −0.321726 0.557246i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.0116 −0.805546 −0.402773 0.915300i \(-0.631954\pi\)
−0.402773 + 0.915300i \(0.631954\pi\)
\(558\) 0 0
\(559\) −22.3383 −0.944809
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.7959 18.6990i −0.454992 0.788069i 0.543696 0.839282i \(-0.317024\pi\)
−0.998688 + 0.0512136i \(0.983691\pi\)
\(564\) 0 0
\(565\) 16.4722 28.5306i 0.692989 1.20029i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.28858 16.0883i 0.389397 0.674456i −0.602971 0.797763i \(-0.706017\pi\)
0.992369 + 0.123307i \(0.0393500\pi\)
\(570\) 0 0
\(571\) 4.42902 + 7.67130i 0.185349 + 0.321034i 0.943694 0.330820i \(-0.107325\pi\)
−0.758345 + 0.651853i \(0.773992\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −40.3219 −1.68154
\(576\) 0 0
\(577\) −3.76684 −0.156816 −0.0784078 0.996921i \(-0.524984\pi\)
−0.0784078 + 0.996921i \(0.524984\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.932821 + 1.61569i 0.0386999 + 0.0670302i
\(582\) 0 0
\(583\) −24.0493 + 41.6546i −0.996021 + 1.72516i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.37616 14.5079i 0.345721 0.598806i −0.639763 0.768572i \(-0.720968\pi\)
0.985484 + 0.169766i \(0.0543010\pi\)
\(588\) 0 0
\(589\) −1.05926 1.83469i −0.0436461 0.0755973i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.3776 −1.24746 −0.623729 0.781640i \(-0.714383\pi\)
−0.623729 + 0.781640i \(0.714383\pi\)
\(594\) 0 0
\(595\) −3.74080 −0.153358
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.97578 + 15.5465i 0.366741 + 0.635213i 0.989054 0.147555i \(-0.0471403\pi\)
−0.622313 + 0.782768i \(0.713807\pi\)
\(600\) 0 0
\(601\) 2.66678 4.61900i 0.108780 0.188413i −0.806496 0.591239i \(-0.798639\pi\)
0.915276 + 0.402826i \(0.131972\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.87291 + 13.6363i −0.320079 + 0.554393i
\(606\) 0 0
\(607\) 9.11826 + 15.7933i 0.370099 + 0.641030i 0.989580 0.143981i \(-0.0459905\pi\)
−0.619482 + 0.785011i \(0.712657\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 39.9497 1.61619
\(612\) 0 0
\(613\) 24.2030 0.977550 0.488775 0.872410i \(-0.337444\pi\)
0.488775 + 0.872410i \(0.337444\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.15635 5.46696i −0.127070 0.220092i 0.795470 0.605993i \(-0.207224\pi\)
−0.922540 + 0.385901i \(0.873891\pi\)
\(618\) 0 0
\(619\) −7.03450 + 12.1841i −0.282740 + 0.489721i −0.972059 0.234738i \(-0.924577\pi\)
0.689318 + 0.724459i \(0.257910\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.334701 + 0.579718i −0.0134095 + 0.0232259i
\(624\) 0 0
\(625\) −0.239656 0.415097i −0.00958625 0.0166039i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.00061 −0.199387
\(630\) 0 0
\(631\) −1.75345 −0.0698036 −0.0349018 0.999391i \(-0.511112\pi\)
−0.0349018 + 0.999391i \(0.511112\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 26.2479 + 45.4627i 1.04162 + 1.80413i
\(636\) 0 0
\(637\) −2.53644 + 4.39324i −0.100497 + 0.174066i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.6942 23.7191i 0.540889 0.936847i −0.457964 0.888971i \(-0.651421\pi\)
0.998853 0.0478765i \(-0.0152454\pi\)
\(642\) 0 0
\(643\) 21.3323 + 36.9486i 0.841263 + 1.45711i 0.888827 + 0.458242i \(0.151521\pi\)
−0.0475644 + 0.998868i \(0.515146\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −45.7615 −1.79907 −0.899536 0.436847i \(-0.856095\pi\)
−0.899536 + 0.436847i \(0.856095\pi\)
\(648\) 0 0
\(649\) −43.8959 −1.72306
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.388265 0.672494i −0.0151940 0.0263167i 0.858328 0.513101i \(-0.171503\pi\)
−0.873522 + 0.486784i \(0.838170\pi\)
\(654\) 0 0
\(655\) 14.6217 25.3255i 0.571316 0.989549i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.97934 + 10.3565i −0.232922 + 0.403433i −0.958667 0.284531i \(-0.908162\pi\)
0.725745 + 0.687964i \(0.241495\pi\)
\(660\) 0 0
\(661\) −6.31373 10.9357i −0.245576 0.425350i 0.716718 0.697364i \(-0.245644\pi\)
−0.962293 + 0.272014i \(0.912310\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −9.09225 −0.352582
\(666\) 0 0
\(667\) −45.6411 −1.76723
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.816155 + 1.41362i 0.0315073 + 0.0545723i
\(672\) 0 0
\(673\) −14.5735 + 25.2421i −0.561768 + 0.973011i 0.435574 + 0.900153i \(0.356545\pi\)
−0.997342 + 0.0728584i \(0.976788\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.92732 + 6.80232i −0.150939 + 0.261435i −0.931573 0.363554i \(-0.881563\pi\)
0.780634 + 0.624989i \(0.214896\pi\)
\(678\) 0 0
\(679\) −7.63513 13.2244i −0.293009 0.507507i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.87441 −0.339570 −0.169785 0.985481i \(-0.554307\pi\)
−0.169785 + 0.985481i \(0.554307\pi\)
\(684\) 0 0
\(685\) −71.4488 −2.72992
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 31.1440 + 53.9430i 1.18649 + 2.05506i
\(690\) 0 0
\(691\) 1.82008 3.15248i 0.0692392 0.119926i −0.829327 0.558763i \(-0.811276\pi\)
0.898567 + 0.438837i \(0.144609\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −30.2946 + 52.4718i −1.14914 + 1.99037i
\(696\) 0 0
\(697\) 2.14049 + 3.70743i 0.0810767 + 0.140429i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.1003 −0.683638 −0.341819 0.939766i \(-0.611043\pi\)
−0.341819 + 0.939766i \(0.611043\pi\)
\(702\) 0 0
\(703\) −12.1543 −0.458408
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.87840 10.1817i −0.221080 0.382922i
\(708\) 0 0
\(709\) −17.5624 + 30.4191i −0.659572 + 1.14241i 0.321155 + 0.947027i \(0.395929\pi\)
−0.980727 + 0.195385i \(0.937404\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.09319 + 3.62551i −0.0783906 + 0.135776i
\(714\) 0 0
\(715\) 36.0070 + 62.3660i 1.34659 + 2.33235i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.8092 0.887933 0.443966 0.896044i \(-0.353571\pi\)
0.443966 + 0.896044i \(0.353571\pi\)
\(720\) 0 0
\(721\) 11.0308 0.410807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −37.4346 64.8387i −1.39029 2.40805i
\(726\) 0 0
\(727\) −1.29251 + 2.23869i −0.0479364 + 0.0830283i −0.888998 0.457911i \(-0.848598\pi\)
0.841062 + 0.540939i \(0.181931\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.27274 + 3.93650i −0.0840603 + 0.145597i
\(732\) 0 0
\(733\) 11.7493 + 20.3505i 0.433972 + 0.751661i 0.997211 0.0746325i \(-0.0237784\pi\)
−0.563239 + 0.826294i \(0.690445\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −39.3691 −1.45018
\(738\) 0 0
\(739\) 16.9236 0.622544 0.311272 0.950321i \(-0.399245\pi\)
0.311272 + 0.950321i \(0.399245\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.39163 + 7.60652i 0.161113 + 0.279056i 0.935268 0.353940i \(-0.115158\pi\)
−0.774155 + 0.632996i \(0.781825\pi\)
\(744\) 0 0
\(745\) 33.2111 57.5233i 1.21676 2.10749i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.421453 + 0.729979i −0.0153996 + 0.0266728i
\(750\) 0 0
\(751\) −0.983876 1.70412i −0.0359021 0.0621843i 0.847516 0.530770i \(-0.178097\pi\)
−0.883418 + 0.468585i \(0.844764\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −52.4093 −1.90737
\(756\) 0 0
\(757\) −10.3423 −0.375899 −0.187949 0.982179i \(-0.560184\pi\)
−0.187949 + 0.982179i \(0.560184\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.88205 13.6521i −0.285724 0.494889i 0.687060 0.726600i \(-0.258901\pi\)
−0.972785 + 0.231711i \(0.925567\pi\)
\(762\) 0 0
\(763\) 8.59376 14.8848i 0.311115 0.538867i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.4227 + 49.2296i −1.02628 + 1.77758i
\(768\) 0 0
\(769\) 22.1895 + 38.4333i 0.800172 + 1.38594i 0.919502 + 0.393084i \(0.128592\pi\)
−0.119330 + 0.992855i \(0.538075\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.5222 0.917969 0.458984 0.888444i \(-0.348213\pi\)
0.458984 + 0.888444i \(0.348213\pi\)
\(774\) 0 0
\(775\) −6.86730 −0.246681
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.20259 + 9.01116i 0.186402 + 0.322858i
\(780\) 0 0
\(781\) 9.89427 17.1374i 0.354045 0.613223i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.97199 + 12.0758i −0.248841 + 0.431005i
\(786\) 0 0
\(787\) −12.2841 21.2767i −0.437882 0.758434i 0.559644 0.828733i \(-0.310938\pi\)
−0.997526 + 0.0702995i \(0.977605\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.09077 −0.323231
\(792\) 0 0
\(793\) 2.11385 0.0750650
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.6983 32.3864i −0.662328 1.14719i −0.980002 0.198986i \(-0.936235\pi\)
0.317674 0.948200i \(-0.397098\pi\)
\(798\) 0 0
\(799\) 4.06456 7.04002i 0.143794 0.249058i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.1163 24.4502i 0.498154 0.862828i
\(804\) 0 0
\(805\) 8.98353 + 15.5599i 0.316628 + 0.548415i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.6565 −0.444980 −0.222490 0.974935i \(-0.571418\pi\)
−0.222490 + 0.974935i \(0.571418\pi\)
\(810\) 0 0
\(811\) −42.2499 −1.48360 −0.741798 0.670624i \(-0.766027\pi\)
−0.741798 + 0.670624i \(0.766027\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −23.6140 40.9006i −0.827162 1.43269i
\(816\) 0 0
\(817\) −5.52404 + 9.56792i −0.193262 + 0.334739i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.7218 34.1591i 0.688295 1.19216i −0.284094 0.958797i \(-0.591693\pi\)
0.972389 0.233366i \(-0.0749740\pi\)
\(822\) 0 0
\(823\) 1.33208 + 2.30723i 0.0464334 + 0.0804249i 0.888308 0.459248i \(-0.151881\pi\)
−0.841875 + 0.539673i \(0.818548\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.13009 0.0740704 0.0370352 0.999314i \(-0.488209\pi\)
0.0370352 + 0.999314i \(0.488209\pi\)
\(828\) 0 0
\(829\) −7.61167 −0.264364 −0.132182 0.991225i \(-0.542198\pi\)
−0.132182 + 0.991225i \(0.542198\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.516124 + 0.893953i 0.0178826 + 0.0309736i
\(834\) 0 0
\(835\) −11.0363 + 19.1154i −0.381926 + 0.661515i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 19.9543 34.5618i 0.688898 1.19321i −0.283297 0.959032i \(-0.591428\pi\)
0.972195 0.234174i \(-0.0752385\pi\)
\(840\) 0 0
\(841\) −27.8730 48.2774i −0.961136 1.66474i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 46.1474 1.58752
\(846\) 0 0
\(847\) 4.34495 0.149294
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0090 + 20.8002i 0.411662 + 0.713020i
\(852\) 0 0
\(853\) 24.1004 41.7431i 0.825182 1.42926i −0.0765985 0.997062i \(-0.524406\pi\)
0.901780 0.432195i \(-0.142261\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.3367 35.2242i 0.694688 1.20324i −0.275598 0.961273i \(-0.588876\pi\)
0.970286 0.241962i \(-0.0777909\pi\)
\(858\) 0 0
\(859\) 5.05826 + 8.76116i 0.172586 + 0.298927i 0.939323 0.343034i \(-0.111455\pi\)
−0.766737 + 0.641961i \(0.778121\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16.5930 −0.564832 −0.282416 0.959292i \(-0.591136\pi\)
−0.282416 + 0.959292i \(0.591136\pi\)
\(864\) 0 0
\(865\) −42.7518 −1.45361
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 29.6368 + 51.3325i 1.00536 + 1.74134i
\(870\) 0 0
\(871\) −25.4916 + 44.1527i −0.863749 + 1.49606i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.67667 + 9.83228i −0.191906 + 0.332392i
\(876\) 0 0
\(877\) −18.9649 32.8482i −0.640399 1.10920i −0.985344 0.170581i \(-0.945436\pi\)
0.344945 0.938623i \(-0.387898\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −20.9737 −0.706623 −0.353312 0.935506i \(-0.614944\pi\)
−0.353312 + 0.935506i \(0.614944\pi\)
\(882\) 0 0
\(883\) 39.6536 1.33445 0.667225 0.744856i \(-0.267482\pi\)
0.667225 + 0.744856i \(0.267482\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.6160 + 20.1195i 0.390028 + 0.675548i 0.992453 0.122628i \(-0.0391321\pi\)
−0.602425 + 0.798175i \(0.705799\pi\)
\(888\) 0 0
\(889\) 7.24293 12.5451i 0.242920 0.420750i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.87917 17.1112i 0.330594 0.572605i
\(894\) 0 0
\(895\) 1.92337 + 3.33137i 0.0642911 + 0.111355i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.77322 −0.259251
\(900\) 0 0
\(901\) 12.6746 0.422252
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 29.0610 + 50.3352i 0.966021 + 1.67320i
\(906\) 0 0
\(907\) 21.1911 36.7041i 0.703640 1.21874i −0.263540 0.964649i \(-0.584890\pi\)
0.967180 0.254092i \(-0.0817766\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 9.66820 16.7458i 0.320322 0.554814i −0.660233 0.751061i \(-0.729542\pi\)
0.980554 + 0.196248i \(0.0628756\pi\)
\(912\) 0 0
\(913\) −3.65411 6.32910i −0.120933 0.209463i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.06951 −0.266479
\(918\) 0 0
\(919\) 28.4866 0.939685 0.469842 0.882750i \(-0.344311\pi\)
0.469842 + 0.882750i \(0.344311\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.8131 22.1930i −0.421749 0.730491i
\(924\) 0 0
\(925\) −19.6994 + 34.1204i −0.647712 + 1.12187i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −12.5930 + 21.8117i −0.413163 + 0.715620i −0.995234 0.0975184i \(-0.968910\pi\)
0.582070 + 0.813139i \(0.302243\pi\)
\(930\) 0 0
\(931\) 1.25447 + 2.17281i 0.0411137 + 0.0712110i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.6537 0.479227
\(936\) 0 0
\(937\) −17.4922 −0.571445 −0.285722 0.958312i \(-0.592234\pi\)
−0.285722 + 0.958312i \(0.592234\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15.9577 27.6396i −0.520207 0.901025i −0.999724 0.0234921i \(-0.992522\pi\)
0.479517 0.877532i \(-0.340812\pi\)
\(942\) 0 0
\(943\) 10.2808 17.8068i 0.334788 0.579869i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.04820 + 13.9399i −0.261531 + 0.452986i −0.966649 0.256105i \(-0.917561\pi\)
0.705118 + 0.709090i \(0.250894\pi\)
\(948\) 0 0
\(949\) −18.2807 31.6631i −0.593417 1.02783i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0358 −0.778595 −0.389298 0.921112i \(-0.627282\pi\)
−0.389298 + 0.921112i \(0.627282\pi\)
\(954\) 0 0
\(955\) −86.7394 −2.80682
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.85792 + 17.0744i 0.318329 + 0.551362i
\(960\) 0 0
\(961\) 15.1435 26.2293i 0.488500 0.846107i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −23.9436 + 41.4715i −0.770770 + 1.33501i
\(966\) 0 0
\(967\) 6.29484 + 10.9030i 0.202428 + 0.350616i 0.949310 0.314340i \(-0.101783\pi\)
−0.746882 + 0.664957i \(0.768450\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −30.7939 −0.988223 −0.494112 0.869398i \(-0.664507\pi\)
−0.494112 + 0.869398i \(0.664507\pi\)
\(972\) 0 0
\(973\) 16.7192 0.535993
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.78282 6.55204i −0.121023 0.209618i 0.799148 0.601134i \(-0.205284\pi\)
−0.920171 + 0.391516i \(0.871951\pi\)
\(978\) 0 0
\(979\) 1.31111 2.27091i 0.0419033 0.0725786i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.3823 23.1789i 0.426830 0.739292i −0.569759 0.821812i \(-0.692964\pi\)
0.996589 + 0.0825201i \(0.0262969\pi\)
\(984\) 0 0
\(985\) −45.3721 78.5868i −1.44568 2.50398i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.8320 0.694216
\(990\) 0 0
\(991\) −48.6851 −1.54653 −0.773266 0.634082i \(-0.781378\pi\)
−0.773266 + 0.634082i \(0.781378\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15.0060 25.9911i −0.475721 0.823973i
\(996\) 0 0
\(997\) −17.0635 + 29.5548i −0.540406 + 0.936011i 0.458474 + 0.888708i \(0.348396\pi\)
−0.998881 + 0.0473036i \(0.984937\pi\)
\(998\) 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 1512.2.r.d.505.2 8
3.2 odd 2 504.2.r.d.169.4 8
4.3 odd 2 3024.2.r.l.2017.2 8
9.2 odd 6 4536.2.a.x.1.2 4
9.4 even 3 inner 1512.2.r.d.1009.2 8
9.5 odd 6 504.2.r.d.337.4 yes 8
9.7 even 3 4536.2.a.ba.1.3 4
12.11 even 2 1008.2.r.m.673.1 8
36.7 odd 6 9072.2.a.cl.1.3 4
36.11 even 6 9072.2.a.ce.1.2 4
36.23 even 6 1008.2.r.m.337.1 8
36.31 odd 6 3024.2.r.l.1009.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.d.169.4 8 3.2 odd 2
504.2.r.d.337.4 yes 8 9.5 odd 6
1008.2.r.m.337.1 8 36.23 even 6
1008.2.r.m.673.1 8 12.11 even 2
1512.2.r.d.505.2 8 1.1 even 1 trivial
1512.2.r.d.1009.2 8 9.4 even 3 inner
3024.2.r.l.1009.2 8 36.31 odd 6
3024.2.r.l.2017.2 8 4.3 odd 2
4536.2.a.x.1.2 4 9.2 odd 6
4536.2.a.ba.1.3 4 9.7 even 3
9072.2.a.ce.1.2 4 36.11 even 6
9072.2.a.cl.1.3 4 36.7 odd 6