Properties

Label 3024.2.b.o.1567.1
Level $3024$
Weight $2$
Character 3024.1567
Analytic conductor $24.147$
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] = [3024,2,Mod(1567,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3024.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.o.1567.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q-1.73205i q^{5} +(2.00000 + 1.73205i) q^{7} -3.46410i q^{11} +3.46410i q^{13} -1.73205i q^{17} -2.00000 q^{19} +2.00000 q^{25} +6.00000 q^{29} +2.00000 q^{31} +(3.00000 - 3.46410i) q^{35} +1.00000 q^{37} -5.19615i q^{41} -1.73205i q^{43} -3.00000 q^{47} +(1.00000 + 6.92820i) q^{49} +12.0000 q^{53} -6.00000 q^{55} +3.00000 q^{59} -10.3923i q^{61} +6.00000 q^{65} -10.3923i q^{67} +13.8564i q^{71} +(6.00000 - 6.92820i) q^{77} +1.73205i q^{79} -9.00000 q^{83} -3.00000 q^{85} -6.92820i q^{89} +(-6.00000 + 6.92820i) q^{91} +3.46410i q^{95} +3.46410i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} - 4 q^{19} + 4 q^{25} + 12 q^{29} + 4 q^{31} + 6 q^{35} + 2 q^{37} - 6 q^{47} + 2 q^{49} + 24 q^{53} - 12 q^{55} + 6 q^{59} + 12 q^{65} + 12 q^{77} - 18 q^{83} - 6 q^{85} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(5\) 1.73205i 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205i 0.420084i −0.977692 0.210042i \(-0.932640\pi\)
0.977692 0.210042i \(-0.0673601\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 3.46410i 0.507093 0.585540i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19615i 0.811503i −0.913984 0.405751i \(-0.867010\pi\)
0.913984 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) 1.73205i 0.264135i −0.991241 0.132068i \(-0.957838\pi\)
0.991241 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 10.3923i 1.33060i −0.746577 0.665299i \(-0.768304\pi\)
0.746577 0.665299i \(-0.231696\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 10.3923i 1.26962i −0.772667 0.634811i \(-0.781078\pi\)
0.772667 0.634811i \(-0.218922\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i 0.569160 + 0.822226i \(0.307268\pi\)
−0.569160 + 0.822226i \(0.692732\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 6.92820i 0.683763 0.789542i
\(78\) 0 0
\(79\) 1.73205i 0.194871i 0.995242 + 0.0974355i \(0.0310640\pi\)
−0.995242 + 0.0974355i \(0.968936\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820i 0.734388i −0.930144 0.367194i \(-0.880318\pi\)
0.930144 0.367194i \(-0.119682\pi\)
\(90\) 0 0
\(91\) −6.00000 + 6.92820i −0.628971 + 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) 3.46410i 0.351726i 0.984415 + 0.175863i \(0.0562716\pi\)
−0.984415 + 0.175863i \(0.943728\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8564i 1.37876i −0.724398 0.689382i \(-0.757882\pi\)
0.724398 0.689382i \(-0.242118\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.92820i 0.669775i 0.942258 + 0.334887i \(0.108698\pi\)
−0.942258 + 0.334887i \(0.891302\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 3.46410i 0.275010 0.317554i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 8.66025i 0.768473i 0.923235 + 0.384237i \(0.125535\pi\)
−0.923235 + 0.384237i \(0.874465\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −4.00000 3.46410i −0.346844 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 15.5885i 1.26857i −0.773099 0.634285i \(-0.781294\pi\)
0.773099 0.634285i \(-0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.46410i 0.278243i
\(156\) 0 0
\(157\) 17.3205i 1.38233i −0.722698 0.691164i \(-0.757098\pi\)
0.722698 0.691164i \(-0.242902\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 22.5167i 1.76364i 0.471585 + 0.881820i \(0.343682\pi\)
−0.471585 + 0.881820i \(0.656318\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.7846i 1.58022i −0.612962 0.790112i \(-0.710022\pi\)
0.612962 0.790112i \(-0.289978\pi\)
\(174\) 0 0
\(175\) 4.00000 + 3.46410i 0.302372 + 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.3205i 1.29460i −0.762237 0.647298i \(-0.775899\pi\)
0.762237 0.647298i \(-0.224101\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.73205i 0.127343i
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.92820i 0.501307i −0.968077 0.250654i \(-0.919354\pi\)
0.968077 0.250654i \(-0.0806455\pi\)
\(192\) 0 0
\(193\) 1.00000 0.0719816 0.0359908 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 + 10.3923i 0.842235 + 0.729397i
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) 24.2487i 1.66935i −0.550743 0.834675i \(-0.685655\pi\)
0.550743 0.834675i \(-0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 4.00000 + 3.46410i 0.271538 + 0.235159i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 13.8564i 0.915657i 0.889041 + 0.457829i \(0.151373\pi\)
−0.889041 + 0.457829i \(0.848627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 5.19615i 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.2487i 1.56852i −0.620433 0.784259i \(-0.713043\pi\)
0.620433 0.784259i \(-0.286957\pi\)
\(240\) 0 0
\(241\) 24.2487i 1.56200i 0.624533 + 0.780998i \(0.285289\pi\)
−0.624533 + 0.780998i \(0.714711\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.0000 1.73205i 0.766652 0.110657i
\(246\) 0 0
\(247\) 6.92820i 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.7846i 1.29651i 0.761424 + 0.648254i \(0.224501\pi\)
−0.761424 + 0.648254i \(0.775499\pi\)
\(258\) 0 0
\(259\) 2.00000 + 1.73205i 0.124274 + 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.3205i 1.06803i 0.845476 + 0.534014i \(0.179317\pi\)
−0.845476 + 0.534014i \(0.820683\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.19615i 0.316815i −0.987374 0.158408i \(-0.949364\pi\)
0.987374 0.158408i \(-0.0506360\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.92820i 0.417786i
\(276\) 0 0
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.00000 10.3923i 0.531253 0.613438i
\(288\) 0 0
\(289\) 14.0000 0.823529
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.9808i 1.51781i −0.651200 0.758906i \(-0.725734\pi\)
0.651200 0.758906i \(-0.274266\pi\)
\(294\) 0 0
\(295\) 5.19615i 0.302532i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.00000 3.46410i 0.172917 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −18.0000 −1.03068
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 0 0
\(313\) 13.8564i 0.783210i 0.920133 + 0.391605i \(0.128080\pi\)
−0.920133 + 0.391605i \(0.871920\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.46410i 0.192748i
\(324\) 0 0
\(325\) 6.92820i 0.384308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 5.19615i −0.330791 0.286473i
\(330\) 0 0
\(331\) 12.1244i 0.666415i 0.942854 + 0.333207i \(0.108131\pi\)
−0.942854 + 0.333207i \(0.891869\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −18.0000 −0.983445
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.92820i 0.375183i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.2487i 1.30174i 0.759190 + 0.650870i \(0.225596\pi\)
−0.759190 + 0.650870i \(0.774404\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.1244i 0.645314i 0.946516 + 0.322657i \(0.104576\pi\)
−0.946516 + 0.322657i \(0.895424\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.92820i 0.365657i −0.983145 0.182828i \(-0.941475\pi\)
0.983145 0.182828i \(-0.0585252\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.0000 + 20.7846i 1.24602 + 1.07908i
\(372\) 0 0
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.7846i 1.07046i
\(378\) 0 0
\(379\) 12.1244i 0.622786i 0.950281 + 0.311393i \(0.100796\pi\)
−0.950281 + 0.311393i \(0.899204\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) −12.0000 10.3923i −0.611577 0.529641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) 27.7128i 1.39087i 0.718591 + 0.695433i \(0.244787\pi\)
−0.718591 + 0.695433i \(0.755213\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) 6.92820i 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.46410i 0.171709i
\(408\) 0 0
\(409\) 6.92820i 0.342578i −0.985221 0.171289i \(-0.945207\pi\)
0.985221 0.171289i \(-0.0547931\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.00000 + 5.19615i 0.295241 + 0.255686i
\(414\) 0 0
\(415\) 15.5885i 0.765207i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.46410i 0.168034i
\(426\) 0 0
\(427\) 18.0000 20.7846i 0.871081 1.00584i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.92820i 0.333720i 0.985981 + 0.166860i \(0.0533628\pi\)
−0.985981 + 0.166860i \(0.946637\pi\)
\(432\) 0 0
\(433\) 17.3205i 0.832370i −0.909280 0.416185i \(-0.863367\pi\)
0.909280 0.416185i \(-0.136633\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.2487i 1.15209i 0.817418 + 0.576046i \(0.195405\pi\)
−0.817418 + 0.576046i \(0.804595\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 12.0000 + 10.3923i 0.562569 + 0.487199i
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.4449i 1.37138i 0.727892 + 0.685692i \(0.240500\pi\)
−0.727892 + 0.685692i \(0.759500\pi\)
\(462\) 0 0
\(463\) 8.66025i 0.402476i 0.979542 + 0.201238i \(0.0644965\pi\)
−0.979542 + 0.201238i \(0.935504\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 18.0000 20.7846i 0.831163 0.959744i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 39.0000 1.78196 0.890978 0.454047i \(-0.150020\pi\)
0.890978 + 0.454047i \(0.150020\pi\)
\(480\) 0 0
\(481\) 3.46410i 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 17.3205i 0.784867i 0.919780 + 0.392434i \(0.128367\pi\)
−0.919780 + 0.392434i \(0.871633\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.3923i 0.468998i −0.972116 0.234499i \(-0.924655\pi\)
0.972116 0.234499i \(-0.0753450\pi\)
\(492\) 0 0
\(493\) 10.3923i 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 + 27.7128i −1.07655 + 1.24309i
\(498\) 0 0
\(499\) 15.5885i 0.697835i −0.937153 0.348918i \(-0.886549\pi\)
0.937153 0.348918i \(-0.113451\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −33.0000 −1.47140 −0.735699 0.677309i \(-0.763146\pi\)
−0.735699 + 0.677309i \(0.763146\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.4449i 1.30512i 0.757737 + 0.652560i \(0.226305\pi\)
−0.757737 + 0.652560i \(0.773695\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.8564i 0.610586i
\(516\) 0 0
\(517\) 10.3923i 0.457053i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.5167i 0.986473i 0.869895 + 0.493236i \(0.164186\pi\)
−0.869895 + 0.493236i \(0.835814\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.46410i 0.150899i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.0000 3.46410i 1.03375 0.149209i
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.1244i 0.519350i
\(546\) 0 0
\(547\) 5.19615i 0.222171i −0.993811 0.111086i \(-0.964567\pi\)
0.993811 0.111086i \(-0.0354328\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −3.00000 + 3.46410i −0.127573 + 0.147309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 10.3923i 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 12.1244i 0.507388i −0.967284 0.253694i \(-0.918354\pi\)
0.967284 0.253694i \(-0.0816457\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −18.0000 15.5885i −0.746766 0.646718i
\(582\) 0 0
\(583\) 41.5692i 1.72162i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.5167i 0.924648i 0.886711 + 0.462324i \(0.152984\pi\)
−0.886711 + 0.462324i \(0.847016\pi\)
\(594\) 0 0
\(595\) −6.00000 5.19615i −0.245976 0.213021i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.2487i 0.990775i −0.868672 0.495388i \(-0.835026\pi\)
0.868672 0.495388i \(-0.164974\pi\)
\(600\) 0 0
\(601\) 38.1051i 1.55434i −0.629291 0.777170i \(-0.716654\pi\)
0.629291 0.777170i \(-0.283346\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.73205i 0.0704179i
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.3923i 0.420428i
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000 13.8564i 0.480770 0.555145i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.73205i 0.0690614i
\(630\) 0 0
\(631\) 29.4449i 1.17218i 0.810245 + 0.586091i \(0.199334\pi\)
−0.810245 + 0.586091i \(0.800666\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 15.0000 0.595257
\(636\) 0 0
\(637\) −24.0000 + 3.46410i −0.950915 + 0.137253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) 10.3923i 0.407934i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 31.1769i 1.21264i 0.795220 + 0.606321i \(0.207355\pi\)
−0.795220 + 0.606321i \(0.792645\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.00000 + 6.92820i −0.232670 + 0.268664i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.8564i 0.532545i −0.963898 0.266272i \(-0.914208\pi\)
0.963898 0.266272i \(-0.0857921\pi\)
\(678\) 0 0
\(679\) −6.00000 + 6.92820i −0.230259 + 0.265880i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.3923i 0.397650i −0.980035 0.198825i \(-0.936287\pi\)
0.980035 0.198825i \(-0.0637126\pi\)
\(684\) 0 0
\(685\) 10.3923i 0.397070i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 41.5692i 1.58366i
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.2487i 0.919806i
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 27.7128i 0.902613 1.04225i
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 20.7846i 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −21.0000 −0.783168 −0.391584 0.920142i \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(720\) 0 0
\(721\) 16.0000 + 13.8564i 0.595871 + 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 6.92820i 0.255899i −0.991781 0.127950i \(-0.959160\pi\)
0.991781 0.127950i \(-0.0408395\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) 45.0333i 1.65658i 0.560301 + 0.828289i \(0.310685\pi\)
−0.560301 + 0.828289i \(0.689315\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.1051i 1.39794i 0.715150 + 0.698971i \(0.246358\pi\)
−0.715150 + 0.698971i \(0.753642\pi\)
\(744\) 0 0
\(745\) 31.1769i 1.14223i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 + 13.8564i −0.438470 + 0.506302i
\(750\) 0 0
\(751\) 24.2487i 0.884848i −0.896806 0.442424i \(-0.854119\pi\)
0.896806 0.442424i \(-0.145881\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −27.0000 −0.982631
\(756\) 0 0
\(757\) 5.00000 0.181728 0.0908640 0.995863i \(-0.471037\pi\)
0.0908640 + 0.995863i \(0.471037\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.3013i 1.56967i 0.619705 + 0.784835i \(0.287252\pi\)
−0.619705 + 0.784835i \(0.712748\pi\)
\(762\) 0 0
\(763\) −14.0000 12.1244i −0.506834 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.3923i 0.375244i
\(768\) 0 0
\(769\) 34.6410i 1.24919i 0.780950 + 0.624593i \(0.214735\pi\)
−0.780950 + 0.624593i \(0.785265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.5885i 0.560678i −0.959901 0.280339i \(-0.909553\pi\)
0.959901 0.280339i \(-0.0904469\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.3923i 0.372343i
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −30.0000 −1.07075
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 + 10.3923i 0.426671 + 0.369508i
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 5.19615i 0.183827i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 39.0000 1.36611
\(816\) 0 0
\(817\) 3.46410i 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 43.3013i 1.50939i 0.656077 + 0.754694i \(0.272215\pi\)
−0.656077 + 0.754694i \(0.727785\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.46410i 0.120459i 0.998185 + 0.0602293i \(0.0191832\pi\)
−0.998185 + 0.0602293i \(0.980817\pi\)
\(828\) 0 0
\(829\) 17.3205i 0.601566i −0.953693 0.300783i \(-0.902752\pi\)
0.953693 0.300783i \(-0.0972480\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12.0000 1.73205i 0.415775 0.0600120i
\(834\) 0 0
\(835\) 25.9808i 0.899101i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −51.0000 −1.76072 −0.880358 0.474310i \(-0.842698\pi\)
−0.880358 + 0.474310i \(0.842698\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.73205i 0.0595844i
\(846\) 0 0
\(847\) −2.00000 1.73205i −0.0687208 0.0595140i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 41.5692i 1.42330i 0.702533 + 0.711651i \(0.252052\pi\)
−0.702533 + 0.711651i \(0.747948\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.19615i 0.177497i 0.996054 + 0.0887486i \(0.0282868\pi\)
−0.996054 + 0.0887486i \(0.971713\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.5692i 1.41503i 0.706697 + 0.707516i \(0.250184\pi\)
−0.706697 + 0.707516i \(0.749816\pi\)
\(864\) 0 0
\(865\) −36.0000 −1.22404
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 21.0000 24.2487i 0.709930 0.819756i
\(876\) 0 0
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.4974i 1.63392i 0.576695 + 0.816960i \(0.304342\pi\)
−0.576695 + 0.816960i \(0.695658\pi\)
\(882\) 0 0
\(883\) 8.66025i 0.291441i −0.989326 0.145720i \(-0.953450\pi\)
0.989326 0.145720i \(-0.0465500\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15.0000 −0.503651 −0.251825 0.967773i \(-0.581031\pi\)
−0.251825 + 0.967773i \(0.581031\pi\)
\(888\) 0 0
\(889\) −15.0000 + 17.3205i −0.503084 + 0.580911i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −30.0000 −1.00279
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 20.7846i 0.692436i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) 19.0526i 0.632630i 0.948654 + 0.316315i \(0.102446\pi\)
−0.948654 + 0.316315i \(0.897554\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 31.1769i 1.03294i 0.856306 + 0.516469i \(0.172754\pi\)
−0.856306 + 0.516469i \(0.827246\pi\)
\(912\) 0 0
\(913\) 31.1769i 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 36.3731i 1.19984i −0.800061 0.599918i \(-0.795200\pi\)
0.800061 0.599918i \(-0.204800\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.66025i 0.284134i −0.989857 0.142067i \(-0.954625\pi\)
0.989857 0.142067i \(-0.0453748\pi\)
\(930\) 0 0
\(931\) −2.00000 13.8564i −0.0655474 0.454125i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.3923i 0.339865i
\(936\) 0 0
\(937\) 58.8897i 1.92384i 0.273325 + 0.961922i \(0.411877\pi\)
−0.273325 + 0.961922i \(0.588123\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 22.5167i 0.734022i 0.930217 + 0.367011i \(0.119619\pi\)
−0.930217 + 0.367011i \(0.880381\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.8564i 0.450273i −0.974327 0.225136i \(-0.927717\pi\)
0.974327 0.225136i \(-0.0722828\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 + 10.3923i 0.387500 + 0.335585i
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.73205i 0.0557567i
\(966\) 0 0
\(967\) 10.3923i 0.334194i 0.985940 + 0.167097i \(0.0534393\pi\)
−0.985940 + 0.167097i \(0.946561\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 28.0000 + 24.2487i 0.897639 + 0.777378i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.0000 −1.53566 −0.767828 0.640656i \(-0.778662\pi\)
−0.767828 + 0.640656i \(0.778662\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 31.1769i 0.993379i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 15.5885i 0.495184i −0.968864 0.247592i \(-0.920361\pi\)
0.968864 0.247592i \(-0.0796392\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 34.6410i 1.09819i
\(996\) 0 0
\(997\) 10.3923i 0.329128i −0.986366 0.164564i \(-0.947378\pi\)
0.986366 0.164564i \(-0.0526216\pi\)
\(998\) 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 3024.2.b.o.1567.1 yes 2
3.2 odd 2 3024.2.b.m.1567.2 yes 2
4.3 odd 2 3024.2.b.d.1567.1 yes 2
7.6 odd 2 3024.2.b.d.1567.2 yes 2
12.11 even 2 3024.2.b.b.1567.2 yes 2
21.20 even 2 3024.2.b.b.1567.1 2
28.27 even 2 inner 3024.2.b.o.1567.2 yes 2
84.83 odd 2 3024.2.b.m.1567.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.b.1567.1 2 21.20 even 2
3024.2.b.b.1567.2 yes 2 12.11 even 2
3024.2.b.d.1567.1 yes 2 4.3 odd 2
3024.2.b.d.1567.2 yes 2 7.6 odd 2
3024.2.b.m.1567.1 yes 2 84.83 odd 2
3024.2.b.m.1567.2 yes 2 3.2 odd 2
3024.2.b.o.1567.1 yes 2 1.1 even 1 trivial
3024.2.b.o.1567.2 yes 2 28.27 even 2 inner