Properties

Label 900.2.k.h.307.2
Level $900$
Weight $2$
Character 900.307
Analytic conductor $7.187$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(307,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("900.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 307.2
Root \(-0.581861 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 900.307
Dual form 900.2.k.h.343.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.581861 - 1.28897i) q^{2} +(-1.32288 + 1.50000i) q^{4} +(-1.41421 + 1.41421i) q^{7} +(2.70318 + 0.832353i) q^{8} +O(q^{10})\) \(q+(-0.581861 - 1.28897i) q^{2} +(-1.32288 + 1.50000i) q^{4} +(-1.41421 + 1.41421i) q^{7} +(2.70318 + 0.832353i) q^{8} -5.29150i q^{11} +(-3.74166 + 3.74166i) q^{13} +(2.64575 + 1.00000i) q^{14} +(-0.500000 - 3.96863i) q^{16} +5.29150 q^{19} +(-6.82058 + 3.07892i) q^{22} +(2.82843 + 2.82843i) q^{23} +(7.00000 + 2.64575i) q^{26} +(-0.250492 - 3.99215i) q^{28} +8.00000i q^{29} +5.29150i q^{31} +(-4.82450 + 2.95367i) q^{32} +(3.74166 + 3.74166i) q^{37} +(-3.07892 - 6.82058i) q^{38} +2.00000 q^{41} +(5.65685 + 5.65685i) q^{43} +(7.93725 + 7.00000i) q^{44} +(2.00000 - 5.29150i) q^{46} +3.00000i q^{49} +(-0.662739 - 10.5622i) q^{52} +(-7.48331 + 7.48331i) q^{53} +(-5.00000 + 2.64575i) q^{56} +(10.3117 - 4.65489i) q^{58} +5.29150 q^{59} +6.00000 q^{61} +(6.82058 - 3.07892i) q^{62} +(6.61438 + 4.50000i) q^{64} +(8.48528 - 8.48528i) q^{67} +(-7.48331 + 7.48331i) q^{73} +(2.64575 - 7.00000i) q^{74} +(-7.00000 + 7.93725i) q^{76} +(7.48331 + 7.48331i) q^{77} -5.29150 q^{79} +(-1.16372 - 2.57794i) q^{82} +(-8.48528 - 8.48528i) q^{83} +(4.00000 - 10.5830i) q^{86} +(4.40440 - 14.3039i) q^{88} -6.00000i q^{89} -10.5830i q^{91} +(-7.98430 + 0.500983i) q^{92} +(7.48331 + 7.48331i) q^{97} +(3.86690 - 1.74558i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{16} + 56 q^{26} + 16 q^{41} + 16 q^{46} - 40 q^{56} + 48 q^{61} - 56 q^{76} + 32 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.581861 1.28897i −0.411438 0.911438i
\(3\) 0 0
\(4\) −1.32288 + 1.50000i −0.661438 + 0.750000i
\(5\) 0 0
\(6\) 0 0
\(7\) −1.41421 + 1.41421i −0.534522 + 0.534522i −0.921915 0.387392i \(-0.873376\pi\)
0.387392 + 0.921915i \(0.373376\pi\)
\(8\) 2.70318 + 0.832353i 0.955719 + 0.294281i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.29150i 1.59545i −0.603023 0.797724i \(-0.706037\pi\)
0.603023 0.797724i \(-0.293963\pi\)
\(12\) 0 0
\(13\) −3.74166 + 3.74166i −1.03775 + 1.03775i −0.0384901 + 0.999259i \(0.512255\pi\)
−0.999259 + 0.0384901i \(0.987745\pi\)
\(14\) 2.64575 + 1.00000i 0.707107 + 0.267261i
\(15\) 0 0
\(16\) −0.500000 3.96863i −0.125000 0.992157i
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 5.29150 1.21395 0.606977 0.794719i \(-0.292382\pi\)
0.606977 + 0.794719i \(0.292382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.82058 + 3.07892i −1.45415 + 0.656428i
\(23\) 2.82843 + 2.82843i 0.589768 + 0.589768i 0.937568 0.347801i \(-0.113071\pi\)
−0.347801 + 0.937568i \(0.613071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 7.00000 + 2.64575i 1.37281 + 0.518875i
\(27\) 0 0
\(28\) −0.250492 3.99215i −0.0473385 0.754445i
\(29\) 8.00000i 1.48556i 0.669534 + 0.742781i \(0.266494\pi\)
−0.669534 + 0.742781i \(0.733506\pi\)
\(30\) 0 0
\(31\) 5.29150i 0.950382i 0.879883 + 0.475191i \(0.157621\pi\)
−0.879883 + 0.475191i \(0.842379\pi\)
\(32\) −4.82450 + 2.95367i −0.852859 + 0.522141i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.74166 + 3.74166i 0.615125 + 0.615125i 0.944277 0.329152i \(-0.106763\pi\)
−0.329152 + 0.944277i \(0.606763\pi\)
\(38\) −3.07892 6.82058i −0.499467 1.10644i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 5.65685 + 5.65685i 0.862662 + 0.862662i 0.991647 0.128984i \(-0.0411717\pi\)
−0.128984 + 0.991647i \(0.541172\pi\)
\(44\) 7.93725 + 7.00000i 1.19659 + 1.05529i
\(45\) 0 0
\(46\) 2.00000 5.29150i 0.294884 0.780189i
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 3.00000i 0.428571i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.662739 10.5622i −0.0919053 1.46472i
\(53\) −7.48331 + 7.48331i −1.02791 + 1.02791i −0.0283132 + 0.999599i \(0.509014\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.00000 + 2.64575i −0.668153 + 0.353553i
\(57\) 0 0
\(58\) 10.3117 4.65489i 1.35400 0.611217i
\(59\) 5.29150 0.688895 0.344447 0.938806i \(-0.388066\pi\)
0.344447 + 0.938806i \(0.388066\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 6.82058 3.07892i 0.866214 0.391023i
\(63\) 0 0
\(64\) 6.61438 + 4.50000i 0.826797 + 0.562500i
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528 8.48528i 1.03664 1.03664i 0.0373395 0.999303i \(-0.488112\pi\)
0.999303 0.0373395i \(-0.0118883\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −7.48331 + 7.48331i −0.875856 + 0.875856i −0.993103 0.117247i \(-0.962593\pi\)
0.117247 + 0.993103i \(0.462593\pi\)
\(74\) 2.64575 7.00000i 0.307562 0.813733i
\(75\) 0 0
\(76\) −7.00000 + 7.93725i −0.802955 + 0.910465i
\(77\) 7.48331 + 7.48331i 0.852803 + 0.852803i
\(78\) 0 0
\(79\) −5.29150 −0.595341 −0.297670 0.954669i \(-0.596210\pi\)
−0.297670 + 0.954669i \(0.596210\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.16372 2.57794i −0.128512 0.284685i
\(83\) −8.48528 8.48528i −0.931381 0.931381i 0.0664117 0.997792i \(-0.478845\pi\)
−0.997792 + 0.0664117i \(0.978845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 10.5830i 0.431331 1.14119i
\(87\) 0 0
\(88\) 4.40440 14.3039i 0.469510 1.52480i
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 10.5830i 1.10940i
\(92\) −7.98430 + 0.500983i −0.832421 + 0.0522311i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.48331 + 7.48331i 0.759815 + 0.759815i 0.976289 0.216473i \(-0.0694554\pi\)
−0.216473 + 0.976289i \(0.569455\pi\)
\(98\) 3.86690 1.74558i 0.390616 0.176330i
\(99\) 0 0
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) −4.24264 4.24264i −0.418040 0.418040i 0.466488 0.884528i \(-0.345519\pi\)
−0.884528 + 0.466488i \(0.845519\pi\)
\(104\) −13.2288 + 7.00000i −1.29719 + 0.686406i
\(105\) 0 0
\(106\) 14.0000 + 5.29150i 1.35980 + 0.513956i
\(107\) 2.82843 2.82843i 0.273434 0.273434i −0.557047 0.830481i \(-0.688066\pi\)
0.830481 + 0.557047i \(0.188066\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.31959 + 4.90538i 0.597145 + 0.463515i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −12.0000 10.5830i −1.11417 0.982607i
\(117\) 0 0
\(118\) −3.07892 6.82058i −0.283437 0.627885i
\(119\) 0 0
\(120\) 0 0
\(121\) −17.0000 −1.54545
\(122\) −3.49117 7.73381i −0.316075 0.700186i
\(123\) 0 0
\(124\) −7.93725 7.00000i −0.712786 0.628619i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.41421 + 1.41421i −0.125491 + 0.125491i −0.767063 0.641572i \(-0.778283\pi\)
0.641572 + 0.767063i \(0.278283\pi\)
\(128\) 1.95171 11.1441i 0.172508 0.985008i
\(129\) 0 0
\(130\) 0 0
\(131\) 15.8745i 1.38696i 0.720475 + 0.693481i \(0.243924\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(132\) 0 0
\(133\) −7.48331 + 7.48331i −0.648886 + 0.648886i
\(134\) −15.8745 6.00000i −1.37135 0.518321i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) −5.29150 −0.448819 −0.224410 0.974495i \(-0.572045\pi\)
−0.224410 + 0.974495i \(0.572045\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 19.7990 + 19.7990i 1.65567 + 1.65567i
\(144\) 0 0
\(145\) 0 0
\(146\) 14.0000 + 5.29150i 1.15865 + 0.437928i
\(147\) 0 0
\(148\) −10.5622 + 0.662739i −0.868210 + 0.0544768i
\(149\) 4.00000i 0.327693i −0.986486 0.163846i \(-0.947610\pi\)
0.986486 0.163846i \(-0.0523901\pi\)
\(150\) 0 0
\(151\) 15.8745i 1.29185i −0.763401 0.645925i \(-0.776472\pi\)
0.763401 0.645925i \(-0.223528\pi\)
\(152\) 14.3039 + 4.40440i 1.16020 + 0.357244i
\(153\) 0 0
\(154\) 5.29150 14.0000i 0.426401 1.12815i
\(155\) 0 0
\(156\) 0 0
\(157\) 3.74166 + 3.74166i 0.298617 + 0.298617i 0.840472 0.541855i \(-0.182278\pi\)
−0.541855 + 0.840472i \(0.682278\pi\)
\(158\) 3.07892 + 6.82058i 0.244946 + 0.542616i
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 5.65685 + 5.65685i 0.443079 + 0.443079i 0.893045 0.449966i \(-0.148564\pi\)
−0.449966 + 0.893045i \(0.648564\pi\)
\(164\) −2.64575 + 3.00000i −0.206598 + 0.234261i
\(165\) 0 0
\(166\) −6.00000 + 15.8745i −0.465690 + 1.23210i
\(167\) 8.48528 8.48528i 0.656611 0.656611i −0.297966 0.954577i \(-0.596308\pi\)
0.954577 + 0.297966i \(0.0963081\pi\)
\(168\) 0 0
\(169\) 15.0000i 1.15385i
\(170\) 0 0
\(171\) 0 0
\(172\) −15.9686 + 1.00197i −1.21759 + 0.0763992i
\(173\) 7.48331 7.48331i 0.568946 0.568946i −0.362887 0.931833i \(-0.618209\pi\)
0.931833 + 0.362887i \(0.118209\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −21.0000 + 2.64575i −1.58293 + 0.199431i
\(177\) 0 0
\(178\) −7.73381 + 3.49117i −0.579673 + 0.261674i
\(179\) −5.29150 −0.395505 −0.197753 0.980252i \(-0.563364\pi\)
−0.197753 + 0.980252i \(0.563364\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −13.6412 + 6.15784i −1.01115 + 0.456449i
\(183\) 0 0
\(184\) 5.29150 + 10.0000i 0.390095 + 0.737210i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.5830i 0.765759i 0.923798 + 0.382880i \(0.125068\pi\)
−0.923798 + 0.382880i \(0.874932\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 5.29150 14.0000i 0.379908 1.00514i
\(195\) 0 0
\(196\) −4.50000 3.96863i −0.321429 0.283473i
\(197\) −14.9666 14.9666i −1.06633 1.06633i −0.997638 0.0686902i \(-0.978118\pi\)
−0.0686902 0.997638i \(-0.521882\pi\)
\(198\) 0 0
\(199\) −5.29150 −0.375105 −0.187552 0.982255i \(-0.560055\pi\)
−0.187552 + 0.982255i \(0.560055\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.32744 5.15587i −0.163758 0.362766i
\(203\) −11.3137 11.3137i −0.794067 0.794067i
\(204\) 0 0
\(205\) 0 0
\(206\) −3.00000 + 7.93725i −0.209020 + 0.553015i
\(207\) 0 0
\(208\) 16.7201 + 12.9784i 1.15933 + 0.899891i
\(209\) 28.0000i 1.93680i
\(210\) 0 0
\(211\) 26.4575i 1.82141i 0.413057 + 0.910705i \(0.364461\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −1.32548 21.1245i −0.0910341 1.45083i
\(213\) 0 0
\(214\) −5.29150 2.00000i −0.361720 0.136717i
\(215\) 0 0
\(216\) 0 0
\(217\) −7.48331 7.48331i −0.508001 0.508001i
\(218\) 2.57794 1.16372i 0.174600 0.0788172i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.89949 9.89949i −0.662919 0.662919i 0.293148 0.956067i \(-0.405297\pi\)
−0.956067 + 0.293148i \(0.905297\pi\)
\(224\) 2.64575 11.0000i 0.176777 0.734968i
\(225\) 0 0
\(226\) 0 0
\(227\) −19.7990 + 19.7990i −1.31411 + 1.31411i −0.395744 + 0.918361i \(0.629513\pi\)
−0.918361 + 0.395744i \(0.870487\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.65882 + 21.6255i −0.437173 + 1.41978i
\(233\) −14.9666 + 14.9666i −0.980497 + 0.980497i −0.999813 0.0193169i \(-0.993851\pi\)
0.0193169 + 0.999813i \(0.493851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.00000 + 7.93725i −0.455661 + 0.516671i
\(237\) 0 0
\(238\) 0 0
\(239\) −10.5830 −0.684558 −0.342279 0.939598i \(-0.611199\pi\)
−0.342279 + 0.939598i \(0.611199\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 9.89164 + 21.9125i 0.635858 + 1.40859i
\(243\) 0 0
\(244\) −7.93725 + 9.00000i −0.508131 + 0.576166i
\(245\) 0 0
\(246\) 0 0
\(247\) −19.7990 + 19.7990i −1.25978 + 1.25978i
\(248\) −4.40440 + 14.3039i −0.279679 + 0.908298i
\(249\) 0 0
\(250\) 0 0
\(251\) 5.29150i 0.333997i −0.985957 0.166998i \(-0.946593\pi\)
0.985957 0.166998i \(-0.0534075\pi\)
\(252\) 0 0
\(253\) 14.9666 14.9666i 0.940944 0.940944i
\(254\) 2.64575 + 1.00000i 0.166009 + 0.0627456i
\(255\) 0 0
\(256\) −15.5000 + 3.96863i −0.968750 + 0.248039i
\(257\) 14.9666 + 14.9666i 0.933593 + 0.933593i 0.997928 0.0643356i \(-0.0204928\pi\)
−0.0643356 + 0.997928i \(0.520493\pi\)
\(258\) 0 0
\(259\) −10.5830 −0.657596
\(260\) 0 0
\(261\) 0 0
\(262\) 20.4617 9.23676i 1.26413 0.570649i
\(263\) −8.48528 8.48528i −0.523225 0.523225i 0.395319 0.918544i \(-0.370634\pi\)
−0.918544 + 0.395319i \(0.870634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 14.0000 + 5.29150i 0.858395 + 0.324443i
\(267\) 0 0
\(268\) 1.50295 + 23.9529i 0.0918073 + 1.46316i
\(269\) 24.0000i 1.46331i 0.681677 + 0.731653i \(0.261251\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) 15.8745i 0.964308i 0.876087 + 0.482154i \(0.160145\pi\)
−0.876087 + 0.482154i \(0.839855\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7083 18.7083i −1.12407 1.12407i −0.991123 0.132949i \(-0.957555\pi\)
−0.132949 0.991123i \(-0.542445\pi\)
\(278\) 3.07892 + 6.82058i 0.184661 + 0.409071i
\(279\) 0 0
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 19.7990 + 19.7990i 1.17693 + 1.17693i 0.980523 + 0.196405i \(0.0629267\pi\)
0.196405 + 0.980523i \(0.437073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 14.0000 37.0405i 0.827837 2.19025i
\(287\) −2.82843 + 2.82843i −0.166957 + 0.166957i
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) −1.32548 21.1245i −0.0775677 1.23622i
\(293\) 14.9666 14.9666i 0.874360 0.874360i −0.118584 0.992944i \(-0.537836\pi\)
0.992944 + 0.118584i \(0.0378355\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.00000 + 13.2288i 0.406867 + 0.768906i
\(297\) 0 0
\(298\) −5.15587 + 2.32744i −0.298672 + 0.134825i
\(299\) −21.1660 −1.22406
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −20.4617 + 9.23676i −1.17744 + 0.531516i
\(303\) 0 0
\(304\) −2.64575 21.0000i −0.151744 1.20443i
\(305\) 0 0
\(306\) 0 0
\(307\) 8.48528 8.48528i 0.484281 0.484281i −0.422215 0.906496i \(-0.638747\pi\)
0.906496 + 0.422215i \(0.138747\pi\)
\(308\) −21.1245 + 1.32548i −1.20368 + 0.0755261i
\(309\) 0 0
\(310\) 0 0
\(311\) 31.7490i 1.80032i −0.435558 0.900161i \(-0.643449\pi\)
0.435558 0.900161i \(-0.356551\pi\)
\(312\) 0 0
\(313\) 14.9666 14.9666i 0.845964 0.845964i −0.143663 0.989627i \(-0.545888\pi\)
0.989627 + 0.143663i \(0.0458881\pi\)
\(314\) 2.64575 7.00000i 0.149308 0.395033i
\(315\) 0 0
\(316\) 7.00000 7.93725i 0.393781 0.446505i
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 42.3320 2.37014
\(320\) 0 0
\(321\) 0 0
\(322\) 4.65489 + 10.3117i 0.259407 + 0.574651i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 10.5830i 0.221540 0.586138i
\(327\) 0 0
\(328\) 5.40636 + 1.66471i 0.298516 + 0.0919180i
\(329\) 0 0
\(330\) 0 0
\(331\) 5.29150i 0.290847i −0.989369 0.145424i \(-0.953545\pi\)
0.989369 0.145424i \(-0.0464545\pi\)
\(332\) 23.9529 1.50295i 1.31459 0.0824851i
\(333\) 0 0
\(334\) −15.8745 6.00000i −0.868614 0.328305i
\(335\) 0 0
\(336\) 0 0
\(337\) −7.48331 7.48331i −0.407642 0.407642i 0.473273 0.880916i \(-0.343072\pi\)
−0.880916 + 0.473273i \(0.843072\pi\)
\(338\) −19.3345 + 8.72791i −1.05166 + 0.474736i
\(339\) 0 0
\(340\) 0 0
\(341\) 28.0000 1.51629
\(342\) 0 0
\(343\) −14.1421 14.1421i −0.763604 0.763604i
\(344\) 10.5830 + 20.0000i 0.570597 + 1.07833i
\(345\) 0 0
\(346\) −14.0000 5.29150i −0.752645 0.284473i
\(347\) 8.48528 8.48528i 0.455514 0.455514i −0.441666 0.897180i \(-0.645612\pi\)
0.897180 + 0.441666i \(0.145612\pi\)
\(348\) 0 0
\(349\) 2.00000i 0.107058i −0.998566 0.0535288i \(-0.982953\pi\)
0.998566 0.0535288i \(-0.0170469\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 15.6294 + 25.5289i 0.833048 + 1.36069i
\(353\) 14.9666 14.9666i 0.796593 0.796593i −0.185963 0.982557i \(-0.559541\pi\)
0.982557 + 0.185963i \(0.0595406\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.00000 + 7.93725i 0.476999 + 0.420674i
\(357\) 0 0
\(358\) 3.07892 + 6.82058i 0.162726 + 0.360479i
\(359\) 10.5830 0.558550 0.279275 0.960211i \(-0.409906\pi\)
0.279275 + 0.960211i \(0.409906\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) −1.16372 2.57794i −0.0611639 0.135493i
\(363\) 0 0
\(364\) 15.8745 + 14.0000i 0.832050 + 0.733799i
\(365\) 0 0
\(366\) 0 0
\(367\) 12.7279 12.7279i 0.664392 0.664392i −0.292020 0.956412i \(-0.594327\pi\)
0.956412 + 0.292020i \(0.0943274\pi\)
\(368\) 9.81076 12.6392i 0.511421 0.658863i
\(369\) 0 0
\(370\) 0 0
\(371\) 21.1660i 1.09888i
\(372\) 0 0
\(373\) 18.7083 18.7083i 0.968678 0.968678i −0.0308458 0.999524i \(-0.509820\pi\)
0.999524 + 0.0308458i \(0.00982007\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −29.9333 29.9333i −1.54164 1.54164i
\(378\) 0 0
\(379\) −5.29150 −0.271806 −0.135903 0.990722i \(-0.543394\pi\)
−0.135903 + 0.990722i \(0.543394\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 13.6412 6.15784i 0.697942 0.315062i
\(383\) 5.65685 + 5.65685i 0.289052 + 0.289052i 0.836705 0.547653i \(-0.184479\pi\)
−0.547653 + 0.836705i \(0.684479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −21.1245 + 1.32548i −1.07243 + 0.0672909i
\(389\) 24.0000i 1.21685i 0.793612 + 0.608424i \(0.208198\pi\)
−0.793612 + 0.608424i \(0.791802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.49706 + 8.10954i −0.126120 + 0.409594i
\(393\) 0 0
\(394\) −10.5830 + 28.0000i −0.533164 + 1.41062i
\(395\) 0 0
\(396\) 0 0
\(397\) 3.74166 + 3.74166i 0.187788 + 0.187788i 0.794739 0.606951i \(-0.207608\pi\)
−0.606951 + 0.794739i \(0.707608\pi\)
\(398\) 3.07892 + 6.82058i 0.154332 + 0.341885i
\(399\) 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
\(403\) −19.7990 19.7990i −0.986258 0.986258i
\(404\) −5.29150 + 6.00000i −0.263262 + 0.298511i
\(405\) 0 0
\(406\) −8.00000 + 21.1660i −0.397033 + 1.05045i
\(407\) 19.7990 19.7990i 0.981399 0.981399i
\(408\) 0 0
\(409\) 10.0000i 0.494468i 0.968956 + 0.247234i \(0.0795217\pi\)
−0.968956 + 0.247234i \(0.920478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 11.9764 0.751475i 0.590037 0.0370225i
\(413\) −7.48331 + 7.48331i −0.368230 + 0.368230i
\(414\) 0 0
\(415\) 0 0
\(416\) 7.00000 29.1033i 0.343203 1.42690i
\(417\) 0 0
\(418\) −36.0911 + 16.2921i −1.76527 + 0.796873i
\(419\) 15.8745 0.775520 0.387760 0.921760i \(-0.373249\pi\)
0.387760 + 0.921760i \(0.373249\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 34.1029 15.3946i 1.66010 0.749397i
\(423\) 0 0
\(424\) −26.4575 + 14.0000i −1.28489 + 0.679900i
\(425\) 0 0
\(426\) 0 0
\(427\) −8.48528 + 8.48528i −0.410632 + 0.410632i
\(428\) 0.500983 + 7.98430i 0.0242159 + 0.385936i
\(429\) 0 0
\(430\) 0 0
\(431\) 10.5830i 0.509765i 0.966972 + 0.254883i \(0.0820369\pi\)
−0.966972 + 0.254883i \(0.917963\pi\)
\(432\) 0 0
\(433\) 7.48331 7.48331i 0.359625 0.359625i −0.504050 0.863675i \(-0.668157\pi\)
0.863675 + 0.504050i \(0.168157\pi\)
\(434\) −5.29150 + 14.0000i −0.254000 + 0.672022i
\(435\) 0 0
\(436\) −3.00000 2.64575i −0.143674 0.126709i
\(437\) 14.9666 + 14.9666i 0.715951 + 0.715951i
\(438\) 0 0
\(439\) −5.29150 −0.252550 −0.126275 0.991995i \(-0.540302\pi\)
−0.126275 + 0.991995i \(0.540302\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.82843 2.82843i −0.134383 0.134383i 0.636716 0.771099i \(-0.280292\pi\)
−0.771099 + 0.636716i \(0.780292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −7.00000 + 18.5203i −0.331460 + 0.876960i
\(447\) 0 0
\(448\) −15.7181 + 2.99018i −0.742611 + 0.141273i
\(449\) 22.0000i 1.03824i −0.854700 0.519122i \(-0.826259\pi\)
0.854700 0.519122i \(-0.173741\pi\)
\(450\) 0 0
\(451\) 10.5830i 0.498334i
\(452\) 0 0
\(453\) 0 0
\(454\) 37.0405 + 14.0000i 1.73840 + 0.657053i
\(455\) 0 0
\(456\) 0 0
\(457\) 7.48331 + 7.48331i 0.350055 + 0.350055i 0.860130 0.510075i \(-0.170382\pi\)
−0.510075 + 0.860130i \(0.670382\pi\)
\(458\) 18.0455 8.14605i 0.843213 0.380640i
\(459\) 0 0
\(460\) 0 0
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) −24.0416 24.0416i −1.11731 1.11731i −0.992135 0.125175i \(-0.960051\pi\)
−0.125175 0.992135i \(-0.539949\pi\)
\(464\) 31.7490 4.00000i 1.47391 0.185695i
\(465\) 0 0
\(466\) 28.0000 + 10.5830i 1.29707 + 0.490248i
\(467\) −25.4558 + 25.4558i −1.17796 + 1.17796i −0.197692 + 0.980264i \(0.563345\pi\)
−0.980264 + 0.197692i \(0.936655\pi\)
\(468\) 0 0
\(469\) 24.0000i 1.10822i
\(470\) 0 0
\(471\) 0 0
\(472\) 14.3039 + 4.40440i 0.658390 + 0.202729i
\(473\) 29.9333 29.9333i 1.37633 1.37633i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 6.15784 + 13.6412i 0.281653 + 0.623932i
\(479\) 42.3320 1.93420 0.967100 0.254398i \(-0.0818772\pi\)
0.967100 + 0.254398i \(0.0818772\pi\)
\(480\) 0 0
\(481\) −28.0000 −1.27669
\(482\) −8.14605 18.0455i −0.371043 0.821952i
\(483\) 0 0
\(484\) 22.4889 25.5000i 1.02222 1.15909i
\(485\) 0 0
\(486\) 0 0
\(487\) −7.07107 + 7.07107i −0.320421 + 0.320421i −0.848928 0.528508i \(-0.822752\pi\)
0.528508 + 0.848928i \(0.322752\pi\)
\(488\) 16.2191 + 4.99412i 0.734204 + 0.226073i
\(489\) 0 0
\(490\) 0 0
\(491\) 15.8745i 0.716407i 0.933644 + 0.358203i \(0.116611\pi\)
−0.933644 + 0.358203i \(0.883389\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 37.0405 + 14.0000i 1.66653 + 0.629890i
\(495\) 0 0
\(496\) 21.0000 2.64575i 0.942928 0.118798i
\(497\) 0 0
\(498\) 0 0
\(499\) 15.8745 0.710641 0.355320 0.934745i \(-0.384372\pi\)
0.355320 + 0.934745i \(0.384372\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.82058 + 3.07892i −0.304417 + 0.137419i
\(503\) −11.3137 11.3137i −0.504453 0.504453i 0.408365 0.912819i \(-0.366099\pi\)
−0.912819 + 0.408365i \(0.866099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −28.0000 10.5830i −1.24475 0.470472i
\(507\) 0 0
\(508\) −0.250492 3.99215i −0.0111138 0.177123i
\(509\) 36.0000i 1.59567i 0.602875 + 0.797836i \(0.294022\pi\)
−0.602875 + 0.797836i \(0.705978\pi\)
\(510\) 0 0
\(511\) 21.1660i 0.936329i
\(512\) 14.1343 + 17.6698i 0.624653 + 0.780903i
\(513\) 0 0
\(514\) 10.5830 28.0000i 0.466796 1.23503i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 6.15784 + 13.6412i 0.270560 + 0.599358i
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) −23.8118 21.0000i −1.04022 0.917389i
\(525\) 0 0
\(526\) −6.00000 + 15.8745i −0.261612 + 0.692161i
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000i 0.304348i
\(530\) 0 0
\(531\) 0 0
\(532\) −1.32548 21.1245i −0.0574667 0.915862i
\(533\) −7.48331 + 7.48331i −0.324138 + 0.324138i
\(534\) 0 0
\(535\) 0 0
\(536\) 30.0000 15.8745i 1.29580 0.685674i
\(537\) 0 0
\(538\) 30.9352 13.9647i 1.33371 0.602059i
\(539\) 15.8745 0.683763
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 20.4617 9.23676i 0.878906 0.396753i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.65685 + 5.65685i −0.241870 + 0.241870i −0.817623 0.575754i \(-0.804709\pi\)
0.575754 + 0.817623i \(0.304709\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 42.3320i 1.80340i
\(552\) 0 0
\(553\) 7.48331 7.48331i 0.318223 0.318223i
\(554\) −13.2288 + 35.0000i −0.562036 + 1.48701i
\(555\) 0 0
\(556\) 7.00000 7.93725i 0.296866 0.336615i
\(557\) −22.4499 22.4499i −0.951235 0.951235i 0.0476304 0.998865i \(-0.484833\pi\)
−0.998865 + 0.0476304i \(0.984833\pi\)
\(558\) 0 0
\(559\) −42.3320 −1.79045
\(560\) 0 0
\(561\) 0 0
\(562\) 15.1284 + 33.5132i 0.638152 + 1.41367i
\(563\) 2.82843 + 2.82843i 0.119204 + 0.119204i 0.764192 0.644988i \(-0.223138\pi\)
−0.644988 + 0.764192i \(0.723138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.0000 37.0405i 0.588464 1.55693i
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000i 0.251533i 0.992060 + 0.125767i \(0.0401390\pi\)
−0.992060 + 0.125767i \(0.959861\pi\)
\(570\) 0 0
\(571\) 5.29150i 0.221442i −0.993852 0.110721i \(-0.964684\pi\)
0.993852 0.110721i \(-0.0353161\pi\)
\(572\) −55.8901 + 3.50688i −2.33688 + 0.146630i
\(573\) 0 0
\(574\) 5.29150 + 2.00000i 0.220863 + 0.0834784i
\(575\) 0 0
\(576\) 0 0
\(577\) −22.4499 22.4499i −0.934603 0.934603i 0.0633857 0.997989i \(-0.479810\pi\)
−0.997989 + 0.0633857i \(0.979810\pi\)
\(578\) −21.9125 + 9.89164i −0.911438 + 0.411438i
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 39.5980 + 39.5980i 1.63998 + 1.63998i
\(584\) −26.4575 + 14.0000i −1.09482 + 0.579324i
\(585\) 0 0
\(586\) −28.0000 10.5830i −1.15667 0.437180i
\(587\) 19.7990 19.7990i 0.817192 0.817192i −0.168508 0.985700i \(-0.553895\pi\)
0.985700 + 0.168508i \(0.0538950\pi\)
\(588\) 0 0
\(589\) 28.0000i 1.15372i
\(590\) 0 0
\(591\) 0 0
\(592\) 12.9784 16.7201i 0.533410 0.687191i
\(593\) −14.9666 + 14.9666i −0.614606 + 0.614606i −0.944143 0.329537i \(-0.893107\pi\)
0.329537 + 0.944143i \(0.393107\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 + 5.29150i 0.245770 + 0.216748i
\(597\) 0 0
\(598\) 12.3157 + 27.2823i 0.503625 + 1.11566i
\(599\) −31.7490 −1.29723 −0.648615 0.761117i \(-0.724651\pi\)
−0.648615 + 0.761117i \(0.724651\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 9.30978 + 20.6235i 0.379438 + 0.840550i
\(603\) 0 0
\(604\) 23.8118 + 21.0000i 0.968887 + 0.854478i
\(605\) 0 0
\(606\) 0 0
\(607\) 24.0416 24.0416i 0.975820 0.975820i −0.0238948 0.999714i \(-0.507607\pi\)
0.999714 + 0.0238948i \(0.00760667\pi\)
\(608\) −25.5289 + 15.6294i −1.03533 + 0.633855i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −11.2250 + 11.2250i −0.453372 + 0.453372i −0.896472 0.443100i \(-0.853879\pi\)
0.443100 + 0.896472i \(0.353879\pi\)
\(614\) −15.8745 6.00000i −0.640643 0.242140i
\(615\) 0 0
\(616\) 14.0000 + 26.4575i 0.564076 + 1.06600i
\(617\) 29.9333 + 29.9333i 1.20507 + 1.20507i 0.972605 + 0.232462i \(0.0746782\pi\)
0.232462 + 0.972605i \(0.425322\pi\)
\(618\) 0 0
\(619\) 5.29150 0.212683 0.106342 0.994330i \(-0.466086\pi\)
0.106342 + 0.994330i \(0.466086\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −40.9235 + 18.4735i −1.64088 + 0.740720i
\(623\) 8.48528 + 8.48528i 0.339956 + 0.339956i
\(624\) 0 0
\(625\) 0 0
\(626\) −28.0000 10.5830i −1.11911 0.422982i
\(627\) 0 0
\(628\) −10.5622 + 0.662739i −0.421479 + 0.0264461i
\(629\) 0 0
\(630\) 0 0
\(631\) 5.29150i 0.210651i −0.994438 0.105326i \(-0.966411\pi\)
0.994438 0.105326i \(-0.0335885\pi\)
\(632\) −14.3039 4.40440i −0.568978 0.175197i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −11.2250 11.2250i −0.444750 0.444750i
\(638\) −24.6314 54.5646i −0.975164 2.16023i
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 10.5830 12.0000i 0.417029 0.472866i
\(645\) 0 0
\(646\) 0 0
\(647\) −16.9706 + 16.9706i −0.667182 + 0.667182i −0.957063 0.289881i \(-0.906384\pi\)
0.289881 + 0.957063i \(0.406384\pi\)
\(648\) 0 0
\(649\) 28.0000i 1.09910i
\(650\) 0 0
\(651\) 0 0
\(652\) −15.9686 + 1.00197i −0.625378 + 0.0392400i
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.00000 7.93725i −0.0390434 0.309898i
\(657\) 0 0
\(658\) 0 0
\(659\) 15.8745 0.618383 0.309192 0.951000i \(-0.399942\pi\)
0.309192 + 0.951000i \(0.399942\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) −6.82058 + 3.07892i −0.265089 + 0.119666i
\(663\) 0 0
\(664\) −15.8745 30.0000i −0.616050 1.16423i
\(665\) 0 0
\(666\) 0 0
\(667\) −22.6274 + 22.6274i −0.876137 + 0.876137i
\(668\) 1.50295 + 23.9529i 0.0581509 + 0.926765i
\(669\) 0 0
\(670\) 0 0
\(671\) 31.7490i 1.22566i
\(672\) 0 0
\(673\) 7.48331 7.48331i 0.288461 0.288461i −0.548011 0.836471i \(-0.684615\pi\)
0.836471 + 0.548011i \(0.184615\pi\)
\(674\) −5.29150 + 14.0000i −0.203821 + 0.539260i
\(675\) 0 0
\(676\) 22.5000 + 19.8431i 0.865385 + 0.763197i
\(677\) 14.9666 + 14.9666i 0.575214 + 0.575214i 0.933581 0.358367i \(-0.116666\pi\)
−0.358367 + 0.933581i \(0.616666\pi\)
\(678\) 0 0
\(679\) −21.1660 −0.812277
\(680\) 0 0
\(681\) 0 0
\(682\) −16.2921 36.0911i −0.623857 1.38200i
\(683\) 8.48528 + 8.48528i 0.324680 + 0.324680i 0.850559 0.525879i \(-0.176264\pi\)
−0.525879 + 0.850559i \(0.676264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.0000 + 26.4575i −0.381802 + 1.01015i
\(687\) 0 0
\(688\) 19.6215 25.2784i 0.748063 0.963729i
\(689\) 56.0000i 2.13343i
\(690\) 0 0
\(691\) 37.0405i 1.40909i −0.709660 0.704544i \(-0.751152\pi\)
0.709660 0.704544i \(-0.248848\pi\)
\(692\) 1.32548 + 21.1245i 0.0503871 + 0.803032i
\(693\) 0 0
\(694\) −15.8745 6.00000i −0.602588 0.227757i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −2.57794 + 1.16372i −0.0975763 + 0.0440475i
\(699\) 0 0
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) 19.7990 + 19.7990i 0.746733 + 0.746733i
\(704\) 23.8118 35.0000i 0.897440 1.31911i
\(705\) 0 0
\(706\) −28.0000 10.5830i −1.05379 0.398297i
\(707\) −5.65685 + 5.65685i −0.212748 + 0.212748i
\(708\) 0 0
\(709\) 10.0000i 0.375558i −0.982211 0.187779i \(-0.939871\pi\)
0.982211 0.187779i \(-0.0601289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.99412 16.2191i 0.187162 0.607836i
\(713\) −14.9666 + 14.9666i −0.560505 + 0.560505i
\(714\) 0 0
\(715\) 0 0
\(716\) 7.00000 7.93725i 0.261602 0.296629i
\(717\) 0 0
\(718\) −6.15784 13.6412i −0.229808 0.509083i
\(719\) −31.7490 −1.18404 −0.592019 0.805924i \(-0.701669\pi\)
−0.592019 + 0.805924i \(0.701669\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) −5.23675 11.6007i −0.194892 0.431734i
\(723\) 0 0
\(724\) −2.64575 + 3.00000i −0.0983286 + 0.111494i
\(725\) 0 0
\(726\) 0 0
\(727\) −21.2132 + 21.2132i −0.786754 + 0.786754i −0.980961 0.194207i \(-0.937787\pi\)
0.194207 + 0.980961i \(0.437787\pi\)
\(728\) 8.80879 28.6078i 0.326476 1.06027i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −26.1916 + 26.1916i −0.967409 + 0.967409i −0.999485 0.0320765i \(-0.989788\pi\)
0.0320765 + 0.999485i \(0.489788\pi\)
\(734\) −23.8118 9.00000i −0.878908 0.332196i
\(735\) 0 0
\(736\) −22.0000 5.29150i −0.810931 0.195047i
\(737\) −44.8999 44.8999i −1.65391 1.65391i
\(738\) 0 0
\(739\) 26.4575 0.973255 0.486628 0.873609i \(-0.338227\pi\)
0.486628 + 0.873609i \(0.338227\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −27.2823 + 12.3157i −1.00156 + 0.452123i
\(743\) −33.9411 33.9411i −1.24518 1.24518i −0.957824 0.287355i \(-0.907224\pi\)
−0.287355 0.957824i \(-0.592776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −35.0000 13.2288i −1.28144 0.484339i
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000i 0.292314i
\(750\) 0 0
\(751\) 26.4575i 0.965448i 0.875772 + 0.482724i \(0.160353\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −21.1660 + 56.0000i −0.770821 + 2.03940i
\(755\) 0 0
\(756\) 0 0
\(757\) 33.6749 + 33.6749i 1.22394 + 1.22394i 0.966221 + 0.257715i \(0.0829694\pi\)
0.257715 + 0.966221i \(0.417031\pi\)
\(758\) 3.07892 + 6.82058i 0.111831 + 0.247734i
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) −2.82843 2.82843i −0.102396 0.102396i
\(764\) −15.8745 14.0000i −0.574320 0.506502i
\(765\) 0 0
\(766\) 4.00000 10.5830i 0.144526 0.382380i
\(767\) −19.7990 + 19.7990i −0.714900 + 0.714900i
\(768\) 0 0
\(769\) 14.0000i 0.504853i −0.967616 0.252426i \(-0.918771\pi\)
0.967616 0.252426i \(-0.0812286\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −14.9666 + 14.9666i −0.538312 + 0.538312i −0.923033 0.384721i \(-0.874298\pi\)
0.384721 + 0.923033i \(0.374298\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 14.0000 + 26.4575i 0.502571 + 0.949769i
\(777\) 0 0
\(778\) 30.9352 13.9647i 1.10908 0.500657i
\(779\) 10.5830 0.379176
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 11.9059 1.50000i 0.425210 0.0535714i
\(785\) 0 0
\(786\) 0 0
\(787\) 22.6274 22.6274i 0.806580 0.806580i −0.177534 0.984115i \(-0.556812\pi\)
0.984115 + 0.177534i \(0.0568121\pi\)
\(788\) 42.2489 2.65095i 1.50506 0.0944363i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −22.4499 + 22.4499i −0.797221 + 0.797221i
\(794\) 2.64575 7.00000i 0.0938942 0.248421i
\(795\) 0 0
\(796\) 7.00000 7.93725i 0.248108 0.281329i
\(797\) 22.4499 + 22.4499i 0.795218 + 0.795218i 0.982337 0.187119i \(-0.0599151\pi\)
−0.187119 + 0.982337i \(0.559915\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 5.81861 + 12.8897i 0.205462 + 0.455150i
\(803\) 39.5980 + 39.5980i 1.39738 + 1.39738i
\(804\) 0 0
\(805\) 0 0
\(806\) −14.0000 + 37.0405i −0.493129 + 1.30470i
\(807\) 0 0
\(808\) 10.8127 + 3.32941i 0.380390 + 0.117128i
\(809\) 30.0000i 1.05474i −0.849635 0.527372i \(-0.823177\pi\)
0.849635 0.527372i \(-0.176823\pi\)
\(810\) 0 0
\(811\) 37.0405i 1.30067i 0.759648 + 0.650334i \(0.225371\pi\)
−0.759648 + 0.650334i \(0.774629\pi\)
\(812\) 31.9372 2.00393i 1.12078 0.0703243i
\(813\) 0 0
\(814\) −37.0405 14.0000i −1.29827 0.490700i
\(815\) 0 0
\(816\) 0 0
\(817\) 29.9333 + 29.9333i 1.04723 + 1.04723i
\(818\) 12.8897 5.81861i 0.450677 0.203443i
\(819\) 0 0
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 0 0
\(823\) 1.41421 + 1.41421i 0.0492964 + 0.0492964i 0.731325 0.682029i \(-0.238902\pi\)
−0.682029 + 0.731325i \(0.738902\pi\)
\(824\) −7.93725 15.0000i −0.276507 0.522550i
\(825\) 0 0
\(826\) 14.0000 + 5.29150i 0.487122 + 0.184115i
\(827\) 14.1421 14.1421i 0.491770 0.491770i −0.417093 0.908864i \(-0.636951\pi\)
0.908864 + 0.417093i \(0.136951\pi\)
\(828\) 0 0
\(829\) 14.0000i 0.486240i 0.969996 + 0.243120i \(0.0781709\pi\)
−0.969996 + 0.243120i \(0.921829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −41.5862 + 7.91128i −1.44174 + 0.274274i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 42.0000 + 37.0405i 1.45260 + 1.28107i
\(837\) 0 0
\(838\) −9.23676 20.4617i −0.319078 0.706839i
\(839\) −31.7490 −1.09610 −0.548049 0.836446i \(-0.684629\pi\)
−0.548049 + 0.836446i \(0.684629\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) −19.7833 43.8249i −0.681777 1.51031i
\(843\) 0 0
\(844\) −39.6863 35.0000i −1.36606 1.20475i
\(845\) 0 0
\(846\) 0 0
\(847\) 24.0416 24.0416i 0.826080 0.826080i
\(848\) 33.4401 + 25.9568i 1.14834 + 0.891361i
\(849\) 0 0
\(850\) 0 0
\(851\) 21.1660i 0.725561i
\(852\) 0 0
\(853\) −11.2250 + 11.2250i −0.384336 + 0.384336i −0.872661 0.488326i \(-0.837608\pi\)
0.488326 + 0.872661i \(0.337608\pi\)
\(854\) 15.8745 + 6.00000i 0.543214 + 0.205316i
\(855\) 0 0
\(856\) 10.0000 5.29150i 0.341793 0.180860i
\(857\) 29.9333 + 29.9333i 1.02250 + 1.02250i 0.999741 + 0.0227597i \(0.00724526\pi\)
0.0227597 + 0.999741i \(0.492755\pi\)
\(858\) 0 0
\(859\) 5.29150 0.180544 0.0902719 0.995917i \(-0.471226\pi\)
0.0902719 + 0.995917i \(0.471226\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 13.6412 6.15784i 0.464619 0.209737i
\(863\) −2.82843 2.82843i −0.0962808 0.0962808i 0.657326 0.753607i \(-0.271688\pi\)
−0.753607 + 0.657326i \(0.771688\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −14.0000 5.29150i −0.475739 0.179813i
\(867\) 0 0
\(868\) 21.1245 1.32548i 0.717011 0.0449896i
\(869\) 28.0000i 0.949835i
\(870\) 0 0
\(871\) 63.4980i 2.15155i
\(872\) −1.66471 + 5.40636i −0.0563740 + 0.183083i
\(873\) 0 0
\(874\) 10.5830 28.0000i 0.357975 0.947114i
\(875\) 0 0
\(876\) 0 0
\(877\) −3.74166 3.74166i −0.126347 0.126347i 0.641106 0.767453i \(-0.278476\pi\)
−0.767453 + 0.641106i \(0.778476\pi\)
\(878\) 3.07892 + 6.82058i 0.103908 + 0.230183i
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −28.2843 28.2843i −0.951842 0.951842i 0.0470510 0.998892i \(-0.485018\pi\)
−0.998892 + 0.0470510i \(0.985018\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.00000 + 5.29150i −0.0671913 + 0.177772i
\(887\) 25.4558 25.4558i 0.854724 0.854724i −0.135987 0.990711i \(-0.543421\pi\)
0.990711 + 0.135987i \(0.0434205\pi\)
\(888\) 0 0
\(889\) 4.00000i 0.134156i
\(890\) 0 0
\(891\) 0 0
\(892\) 27.9450 1.75344i 0.935669 0.0587096i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 13.0000 + 18.5203i 0.434300 + 0.618718i
\(897\) 0 0
\(898\) −28.3573 + 12.8009i −0.946295 + 0.427173i
\(899\) −42.3320 −1.41185
\(900\) 0 0
\(901\) 0 0
\(902\) −13.6412 + 6.15784i −0.454201 + 0.205034i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −11.3137 + 11.3137i −0.375666 + 0.375666i −0.869536 0.493870i \(-0.835582\pi\)
0.493870 + 0.869536i \(0.335582\pi\)
\(908\) −3.50688 55.8901i −0.116380 1.85478i
\(909\) 0 0
\(910\) 0 0
\(911\) 21.1660i 0.701261i −0.936514 0.350631i \(-0.885967\pi\)
0.936514 0.350631i \(-0.114033\pi\)
\(912\) 0 0
\(913\) −44.8999 + 44.8999i −1.48597 + 1.48597i
\(914\) 5.29150 14.0000i 0.175027 0.463079i
\(915\) 0 0
\(916\) −21.0000 18.5203i −0.693860 0.611927i
\(917\) −22.4499 22.4499i −0.741362 0.741362i
\(918\) 0 0
\(919\) 47.6235 1.57096 0.785478 0.618890i \(-0.212417\pi\)
0.785478 + 0.618890i \(0.212417\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −16.2921 36.0911i −0.536552 1.18860i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −17.0000 + 44.9778i −0.558655 + 1.47806i
\(927\) 0 0
\(928\) −23.6294 38.5960i −0.775673 1.26698i
\(929\) 22.0000i 0.721797i −0.932605 0.360898i \(-0.882470\pi\)
0.932605 0.360898i \(-0.117530\pi\)
\(930\) 0 0
\(931\) 15.8745i 0.520266i
\(932\) −2.65095 42.2489i −0.0868349 1.38391i
\(933\) 0 0
\(934\) 47.6235 + 18.0000i 1.55829 + 0.588978i
\(935\) 0 0
\(936\) 0 0
\(937\) 14.9666 + 14.9666i 0.488938 + 0.488938i 0.907971 0.419033i \(-0.137631\pi\)
−0.419033 + 0.907971i \(0.637631\pi\)
\(938\) 30.9352 13.9647i 1.01007 0.455962i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 5.65685 + 5.65685i 0.184213 + 0.184213i
\(944\) −2.64575 21.0000i −0.0861119 0.683492i
\(945\) 0 0
\(946\) −56.0000 21.1660i −1.82072 0.688166i
\(947\) 36.7696 36.7696i 1.19485 1.19485i 0.219161 0.975689i \(-0.429668\pi\)
0.975689 0.219161i \(-0.0703321\pi\)
\(948\) 0 0
\(949\) 56.0000i 1.81784i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14.0000 15.8745i 0.452792 0.513418i
\(957\) 0 0
\(958\) −24.6314 54.5646i −0.795803 1.76290i
\(959\) 0 0
\(960\) 0 0
\(961\) 3.00000 0.0967742
\(962\) 16.2921 + 36.0911i 0.525279 + 1.16362i
\(963\) 0 0
\(964\) −18.5203 + 21.0000i −0.596497 + 0.676364i
\(965\) 0 0
\(966\) 0 0
\(967\) 4.24264 4.24264i 0.136434 0.136434i −0.635591 0.772026i \(-0.719244\pi\)
0.772026 + 0.635591i \(0.219244\pi\)
\(968\) −45.9541 14.1500i −1.47702 0.454798i
\(969\) 0 0
\(970\) 0 0
\(971\) 15.8745i 0.509437i −0.967015 0.254719i \(-0.918017\pi\)
0.967015 0.254719i \(-0.0819828\pi\)
\(972\) 0 0
\(973\) 7.48331 7.48331i 0.239904 0.239904i
\(974\) 13.2288 + 5.00000i 0.423877 + 0.160210i
\(975\) 0 0
\(976\) −3.00000 23.8118i −0.0960277 0.762196i
\(977\) 14.9666 + 14.9666i 0.478825 + 0.478825i 0.904756 0.425931i \(-0.140053\pi\)
−0.425931 + 0.904756i \(0.640053\pi\)
\(978\) 0 0
\(979\) −31.7490 −1.01470
\(980\) 0 0
\(981\) 0 0
\(982\) 20.4617 9.23676i 0.652960 0.294757i
\(983\) 16.9706 + 16.9706i 0.541277 + 0.541277i 0.923903 0.382626i \(-0.124980\pi\)
−0.382626 + 0.923903i \(0.624980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −3.50688 55.8901i −0.111569 1.77810i
\(989\) 32.0000i 1.01754i
\(990\) 0 0
\(991\) 47.6235i 1.51281i −0.654103 0.756406i \(-0.726954\pi\)
0.654103 0.756406i \(-0.273046\pi\)
\(992\) −15.6294 25.5289i −0.496233 0.810542i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 11.2250 + 11.2250i 0.355498 + 0.355498i 0.862151 0.506652i \(-0.169117\pi\)
−0.506652 + 0.862151i \(0.669117\pi\)
\(998\) −9.23676 20.4617i −0.292384 0.647705i
\(999\) 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 900.2.k.h.307.2 8
3.2 odd 2 300.2.j.b.7.3 yes 8
4.3 odd 2 inner 900.2.k.h.307.1 8
5.2 odd 4 inner 900.2.k.h.343.4 8
5.3 odd 4 inner 900.2.k.h.343.1 8
5.4 even 2 inner 900.2.k.h.307.3 8
12.11 even 2 300.2.j.b.7.4 yes 8
15.2 even 4 300.2.j.b.43.1 yes 8
15.8 even 4 300.2.j.b.43.4 yes 8
15.14 odd 2 300.2.j.b.7.2 yes 8
20.3 even 4 inner 900.2.k.h.343.2 8
20.7 even 4 inner 900.2.k.h.343.3 8
20.19 odd 2 inner 900.2.k.h.307.4 8
60.23 odd 4 300.2.j.b.43.3 yes 8
60.47 odd 4 300.2.j.b.43.2 yes 8
60.59 even 2 300.2.j.b.7.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.j.b.7.1 8 60.59 even 2
300.2.j.b.7.2 yes 8 15.14 odd 2
300.2.j.b.7.3 yes 8 3.2 odd 2
300.2.j.b.7.4 yes 8 12.11 even 2
300.2.j.b.43.1 yes 8 15.2 even 4
300.2.j.b.43.2 yes 8 60.47 odd 4
300.2.j.b.43.3 yes 8 60.23 odd 4
300.2.j.b.43.4 yes 8 15.8 even 4
900.2.k.h.307.1 8 4.3 odd 2 inner
900.2.k.h.307.2 8 1.1 even 1 trivial
900.2.k.h.307.3 8 5.4 even 2 inner
900.2.k.h.307.4 8 20.19 odd 2 inner
900.2.k.h.343.1 8 5.3 odd 4 inner
900.2.k.h.343.2 8 20.3 even 4 inner
900.2.k.h.343.3 8 20.7 even 4 inner
900.2.k.h.343.4 8 5.2 odd 4 inner