Properties

Label 4560.2.d.c.2431.1
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
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] = [4560,2,Mod(2431,4560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("4560.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (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: \(36.4117833217\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.c.2431.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -3.46410i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -3.46410i q^{7} +1.00000 q^{9} -3.46410i q^{11} -1.00000 q^{15} +6.00000 q^{17} +(-4.00000 - 1.73205i) q^{19} -3.46410i q^{21} -3.46410i q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.00000 q^{31} -3.46410i q^{33} +3.46410i q^{35} -6.92820i q^{41} -3.46410i q^{43} -1.00000 q^{45} +10.3923i q^{47} -5.00000 q^{49} +6.00000 q^{51} +6.92820i q^{53} +3.46410i q^{55} +(-4.00000 - 1.73205i) q^{57} -12.0000 q^{59} -2.00000 q^{61} -3.46410i q^{63} +4.00000 q^{67} -3.46410i q^{69} +10.0000 q^{73} +1.00000 q^{75} -12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -10.3923i q^{83} -6.00000 q^{85} -6.92820i q^{89} -8.00000 q^{93} +(4.00000 + 1.73205i) q^{95} +6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{15} + 12 q^{17} - 8 q^{19} + 2 q^{25} + 2 q^{27} - 16 q^{31} - 2 q^{45} - 10 q^{49} + 12 q^{51} - 8 q^{57} - 24 q^{59} - 4 q^{61} + 8 q^{67} + 20 q^{73} + 2 q^{75} - 24 q^{77} + 16 q^{79} + 2 q^{81} - 12 q^{85} - 16 q^{93} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.46410i 1.30931i −0.755929 0.654654i \(-0.772814\pi\)
0.755929 0.654654i \(-0.227186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 1.73205i −0.917663 0.397360i
\(20\) 0 0
\(21\) 3.46410i 0.755929i
\(22\) 0 0
\(23\) 3.46410i 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 3.46410i 0.603023i
\(34\) 0 0
\(35\) 3.46410i 0.585540i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.3923i 1.51587i 0.652328 + 0.757937i \(0.273792\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 6.92820i 0.951662i 0.879537 + 0.475831i \(0.157853\pi\)
−0.879537 + 0.475831i \(0.842147\pi\)
\(54\) 0 0
\(55\) 3.46410i 0.467099i
\(56\) 0 0
\(57\) −4.00000 1.73205i −0.529813 0.229416i
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 3.46410i 0.436436i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 3.46410i 0.417029i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.3923i 1.14070i −0.821401 0.570352i \(-0.806807\pi\)
0.821401 0.570352i \(-0.193193\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 4.00000 + 1.73205i 0.410391 + 0.177705i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 3.46410i 0.338062i
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i 0.748086 + 0.663602i \(0.230973\pi\)
−0.748086 + 0.663602i \(0.769027\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.92820i 0.651751i −0.945413 0.325875i \(-0.894341\pi\)
0.945413 0.325875i \(-0.105659\pi\)
\(114\) 0 0
\(115\) 3.46410i 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.7846i 1.90532i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 6.92820i 0.624695i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 3.46410i 0.304997i
\(130\) 0 0
\(131\) 3.46410i 0.302660i −0.988483 0.151330i \(-0.951644\pi\)
0.988483 0.151330i \(-0.0483556\pi\)
\(132\) 0 0
\(133\) −6.00000 + 13.8564i −0.520266 + 1.20150i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 10.3923i 0.881464i 0.897639 + 0.440732i \(0.145281\pi\)
−0.897639 + 0.440732i \(0.854719\pi\)
\(140\) 0 0
\(141\) 10.3923i 0.875190i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 6.92820i 0.549442i
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 17.3205i 1.35665i −0.734763 0.678323i \(-0.762707\pi\)
0.734763 0.678323i \(-0.237293\pi\)
\(164\) 0 0
\(165\) 3.46410i 0.269680i
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −4.00000 1.73205i −0.305888 0.132453i
\(172\) 0 0
\(173\) 20.7846i 1.58022i −0.612962 0.790112i \(-0.710022\pi\)
0.612962 0.790112i \(-0.289978\pi\)
\(174\) 0 0
\(175\) 3.46410i 0.261861i
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i −0.857209 0.514969i \(-0.827803\pi\)
0.857209 0.514969i \(-0.172197\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 0 0
\(189\) 3.46410i 0.251976i
\(190\) 0 0
\(191\) 17.3205i 1.25327i −0.779314 0.626634i \(-0.784432\pi\)
0.779314 0.626634i \(-0.215568\pi\)
\(192\) 0 0
\(193\) 20.7846i 1.49611i 0.663637 + 0.748054i \(0.269012\pi\)
−0.663637 + 0.748054i \(0.730988\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 10.3923i 0.736691i −0.929689 0.368345i \(-0.879924\pi\)
0.929689 0.368345i \(-0.120076\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 6.92820i 0.483887i
\(206\) 0 0
\(207\) 3.46410i 0.240772i
\(208\) 0 0
\(209\) −6.00000 + 13.8564i −0.415029 + 0.958468i
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.46410i 0.236250i
\(216\) 0 0
\(217\) 27.7128i 1.88127i
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 10.3923i 0.677919i
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 24.2487i 1.56852i 0.620433 + 0.784259i \(0.286957\pi\)
−0.620433 + 0.784259i \(0.713043\pi\)
\(240\) 0 0
\(241\) 13.8564i 0.892570i 0.894891 + 0.446285i \(0.147253\pi\)
−0.894891 + 0.446285i \(0.852747\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 5.00000 0.319438
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.3923i 0.658586i
\(250\) 0 0
\(251\) 3.46410i 0.218652i −0.994006 0.109326i \(-0.965131\pi\)
0.994006 0.109326i \(-0.0348693\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) 20.7846i 1.29651i −0.761424 0.648254i \(-0.775499\pi\)
0.761424 0.648254i \(-0.224501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.46410i 0.213606i −0.994280 0.106803i \(-0.965939\pi\)
0.994280 0.106803i \(-0.0340614\pi\)
\(264\) 0 0
\(265\) 6.92820i 0.425596i
\(266\) 0 0
\(267\) 6.92820i 0.423999i
\(268\) 0 0
\(269\) 13.8564i 0.844840i −0.906400 0.422420i \(-0.861181\pi\)
0.906400 0.422420i \(-0.138819\pi\)
\(270\) 0 0
\(271\) 3.46410i 0.210429i 0.994450 + 0.105215i \(0.0335529\pi\)
−0.994450 + 0.105215i \(0.966447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410i 0.208893i
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 20.7846i 1.23991i −0.784639 0.619953i \(-0.787152\pi\)
0.784639 0.619953i \(-0.212848\pi\)
\(282\) 0 0
\(283\) 31.1769i 1.85328i −0.375956 0.926638i \(-0.622686\pi\)
0.375956 0.926638i \(-0.377314\pi\)
\(284\) 0 0
\(285\) 4.00000 + 1.73205i 0.236940 + 0.102598i
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 6.92820i 0.406138i
\(292\) 0 0
\(293\) 20.7846i 1.21425i 0.794606 + 0.607125i \(0.207677\pi\)
−0.794606 + 0.607125i \(0.792323\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 3.46410i 0.201008i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 31.1769i 1.76788i −0.467600 0.883940i \(-0.654881\pi\)
0.467600 0.883940i \(-0.345119\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 3.46410i 0.195180i
\(316\) 0 0
\(317\) 6.92820i 0.389127i 0.980890 + 0.194563i \(0.0623290\pi\)
−0.980890 + 0.194563i \(0.937671\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −24.0000 10.3923i −1.33540 0.578243i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 13.8564i 0.766261i
\(328\) 0 0
\(329\) 36.0000 1.98474
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 6.92820i 0.377403i −0.982034 0.188702i \(-0.939572\pi\)
0.982034 0.188702i \(-0.0604279\pi\)
\(338\) 0 0
\(339\) 6.92820i 0.376288i
\(340\) 0 0
\(341\) 27.7128i 1.50073i
\(342\) 0 0
\(343\) 6.92820i 0.374088i
\(344\) 0 0
\(345\) 3.46410i 0.186501i
\(346\) 0 0
\(347\) 24.2487i 1.30174i −0.759190 0.650870i \(-0.774404\pi\)
0.759190 0.650870i \(-0.225596\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 20.7846i 1.10004i
\(358\) 0 0
\(359\) 10.3923i 0.548485i 0.961661 + 0.274242i \(0.0884271\pi\)
−0.961661 + 0.274242i \(0.911573\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 17.3205i 0.904123i −0.891987 0.452062i \(-0.850689\pi\)
0.891987 0.452062i \(-0.149311\pi\)
\(368\) 0 0
\(369\) 6.92820i 0.360668i
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) 13.8564i 0.717458i 0.933442 + 0.358729i \(0.116790\pi\)
−0.933442 + 0.358729i \(0.883210\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 3.46410i 0.176090i
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) 0 0
\(393\) 3.46410i 0.174741i
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) −6.00000 + 13.8564i −0.300376 + 0.693688i
\(400\) 0 0
\(401\) 6.92820i 0.345978i −0.984924 0.172989i \(-0.944657\pi\)
0.984924 0.172989i \(-0.0553425\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 13.8564i 0.685155i −0.939490 0.342578i \(-0.888700\pi\)
0.939490 0.342578i \(-0.111300\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 41.5692i 2.04549i
\(414\) 0 0
\(415\) 10.3923i 0.510138i
\(416\) 0 0
\(417\) 10.3923i 0.508913i
\(418\) 0 0
\(419\) 10.3923i 0.507697i 0.967244 + 0.253849i \(0.0816965\pi\)
−0.967244 + 0.253849i \(0.918303\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 10.3923i 0.505291i
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 6.92820i 0.335279i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 34.6410i 1.66474i 0.554220 + 0.832370i \(0.313017\pi\)
−0.554220 + 0.832370i \(0.686983\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 + 13.8564i −0.287019 + 0.662842i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) 10.3923i 0.493753i −0.969047 0.246877i \(-0.920596\pi\)
0.969047 0.246877i \(-0.0794043\pi\)
\(444\) 0 0
\(445\) 6.92820i 0.328428i
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 6.92820i 0.326962i −0.986546 0.163481i \(-0.947728\pi\)
0.986546 0.163481i \(-0.0522723\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 24.2487i 1.12693i 0.826139 + 0.563467i \(0.190533\pi\)
−0.826139 + 0.563467i \(0.809467\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 3.46410i 0.160300i 0.996783 + 0.0801498i \(0.0255399\pi\)
−0.996783 + 0.0801498i \(0.974460\pi\)
\(468\) 0 0
\(469\) 13.8564i 0.639829i
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −4.00000 1.73205i −0.183533 0.0794719i
\(476\) 0 0
\(477\) 6.92820i 0.317221i
\(478\) 0 0
\(479\) 24.2487i 1.10795i 0.832533 + 0.553976i \(0.186890\pi\)
−0.832533 + 0.553976i \(0.813110\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 6.92820i 0.314594i
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 17.3205i 0.783260i
\(490\) 0 0
\(491\) 38.1051i 1.71966i 0.510581 + 0.859830i \(0.329431\pi\)
−0.510581 + 0.859830i \(0.670569\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 3.46410i 0.155700i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 24.2487i 1.08552i 0.839887 + 0.542761i \(0.182621\pi\)
−0.839887 + 0.542761i \(0.817379\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 3.46410i 0.154457i −0.997013 0.0772283i \(-0.975393\pi\)
0.997013 0.0772283i \(-0.0246070\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) 13.8564i 0.614174i −0.951681 0.307087i \(-0.900646\pi\)
0.951681 0.307087i \(-0.0993543\pi\)
\(510\) 0 0
\(511\) 34.6410i 1.53243i
\(512\) 0 0
\(513\) −4.00000 1.73205i −0.176604 0.0764719i
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 36.0000 1.58328
\(518\) 0 0
\(519\) 20.7846i 0.912343i
\(520\) 0 0
\(521\) 20.7846i 0.910590i −0.890341 0.455295i \(-0.849534\pi\)
0.890341 0.455295i \(-0.150466\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 3.46410i 0.151186i
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 17.3205i 0.746047i
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 13.8564i 0.594635i
\(544\) 0 0
\(545\) 13.8564i 0.593543i
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 27.7128i 1.17847i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 20.7846i 0.877527i
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 6.92820i 0.291472i
\(566\) 0 0
\(567\) 3.46410i 0.145479i
\(568\) 0 0
\(569\) 34.6410i 1.45223i 0.687575 + 0.726113i \(0.258675\pi\)
−0.687575 + 0.726113i \(0.741325\pi\)
\(570\) 0 0
\(571\) 38.1051i 1.59465i 0.603550 + 0.797325i \(0.293752\pi\)
−0.603550 + 0.797325i \(0.706248\pi\)
\(572\) 0 0
\(573\) 17.3205i 0.723575i
\(574\) 0 0
\(575\) 3.46410i 0.144463i
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 20.7846i 0.863779i
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.3205i 0.714894i 0.933933 + 0.357447i \(0.116353\pi\)
−0.933933 + 0.357447i \(0.883647\pi\)
\(588\) 0 0
\(589\) 32.0000 + 13.8564i 1.31854 + 0.570943i
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 20.7846i 0.852086i
\(596\) 0 0
\(597\) 10.3923i 0.425329i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 13.8564i 0.565215i 0.959236 + 0.282607i \(0.0911993\pi\)
−0.959236 + 0.282607i \(0.908801\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 6.92820i 0.279372i
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i 0.977947 + 0.208851i \(0.0669724\pi\)
−0.977947 + 0.208851i \(0.933028\pi\)
\(620\) 0 0
\(621\) 3.46410i 0.139010i
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.00000 + 13.8564i −0.239617 + 0.553372i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 24.2487i 0.965326i −0.875806 0.482663i \(-0.839670\pi\)
0.875806 0.482663i \(-0.160330\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 48.4974i 1.91553i −0.287547 0.957767i \(-0.592840\pi\)
0.287547 0.957767i \(-0.407160\pi\)
\(642\) 0 0
\(643\) 31.1769i 1.22950i −0.788723 0.614749i \(-0.789257\pi\)
0.788723 0.614749i \(-0.210743\pi\)
\(644\) 0 0
\(645\) 3.46410i 0.136399i
\(646\) 0 0
\(647\) 24.2487i 0.953315i 0.879089 + 0.476658i \(0.158152\pi\)
−0.879089 + 0.476658i \(0.841848\pi\)
\(648\) 0 0
\(649\) 41.5692i 1.63173i
\(650\) 0 0
\(651\) 27.7128i 1.08615i
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 3.46410i 0.135354i
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 13.8564i 0.538952i −0.963007 0.269476i \(-0.913150\pi\)
0.963007 0.269476i \(-0.0868504\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 13.8564i 0.232670 0.537328i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 6.92820i 0.267460i
\(672\) 0 0
\(673\) 6.92820i 0.267063i −0.991045 0.133531i \(-0.957368\pi\)
0.991045 0.133531i \(-0.0426317\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 20.7846i 0.798817i −0.916773 0.399409i \(-0.869215\pi\)
0.916773 0.399409i \(-0.130785\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 3.46410i 0.131781i −0.997827 0.0658903i \(-0.979011\pi\)
0.997827 0.0658903i \(-0.0209887\pi\)
\(692\) 0 0
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) 10.3923i 0.394203i
\(696\) 0 0
\(697\) 41.5692i 1.57455i
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 10.3923i 0.391397i
\(706\) 0 0
\(707\) 20.7846i 0.781686i
\(708\) 0 0
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 27.7128i 1.03785i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.2487i 0.905585i
\(718\) 0 0
\(719\) 3.46410i 0.129189i −0.997912 0.0645946i \(-0.979425\pi\)
0.997912 0.0645946i \(-0.0205754\pi\)
\(720\) 0 0
\(721\) 55.4256i 2.06416i
\(722\) 0 0
\(723\) 13.8564i 0.515325i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.3205i 0.642382i −0.947014 0.321191i \(-0.895917\pi\)
0.947014 0.321191i \(-0.104083\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 20.7846i 0.768747i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 5.00000 0.184428
\(736\) 0 0
\(737\) 13.8564i 0.510407i
\(738\) 0 0
\(739\) 24.2487i 0.892003i 0.895032 + 0.446002i \(0.147152\pi\)
−0.895032 + 0.446002i \(0.852848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 10.3923i 0.380235i
\(748\) 0 0
\(749\) 41.5692i 1.51891i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 3.46410i 0.126239i
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 48.0000 1.73772
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 20.7846i 0.748539i
\(772\) 0 0
\(773\) 6.92820i 0.249190i −0.992208 0.124595i \(-0.960237\pi\)
0.992208 0.124595i \(-0.0397632\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 + 27.7128i −0.429945 + 0.992915i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) 3.46410i 0.123325i
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 6.92820i 0.245718i
\(796\) 0 0
\(797\) 34.6410i 1.22705i −0.789676 0.613524i \(-0.789751\pi\)
0.789676 0.613524i \(-0.210249\pi\)
\(798\) 0 0
\(799\) 62.3538i 2.20592i
\(800\) 0 0
\(801\) 6.92820i 0.244796i
\(802\) 0 0
\(803\) 34.6410i 1.22245i
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 13.8564i 0.487769i
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 3.46410i 0.121491i
\(814\) 0 0
\(815\) 17.3205i 0.606711i
\(816\) 0 0
\(817\) −6.00000 + 13.8564i −0.209913 + 0.484774i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 31.1769i 1.08676i −0.839487 0.543379i \(-0.817144\pi\)
0.839487 0.543379i \(-0.182856\pi\)
\(824\) 0 0
\(825\) 3.46410i 0.120605i
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 13.8564i 0.481253i 0.970618 + 0.240626i \(0.0773529\pi\)
−0.970618 + 0.240626i \(0.922647\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −30.0000 −1.03944
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 20.7846i 0.715860i
\(844\) 0 0
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 3.46410i 0.119028i
\(848\) 0 0
\(849\) 31.1769i 1.06999i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 4.00000 + 1.73205i 0.136797 + 0.0592349i
\(856\) 0 0
\(857\) 20.7846i 0.709989i −0.934868 0.354994i \(-0.884483\pi\)
0.934868 0.354994i \(-0.115517\pi\)
\(858\) 0 0
\(859\) 10.3923i 0.354581i 0.984159 + 0.177290i \(0.0567332\pi\)
−0.984159 + 0.177290i \(0.943267\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 20.7846i 0.706698i
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 27.7128i 0.940093i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 6.92820i 0.234484i
\(874\) 0 0
\(875\) 3.46410i 0.117108i
\(876\) 0 0
\(877\) 13.8564i 0.467898i 0.972249 + 0.233949i \(0.0751648\pi\)
−0.972249 + 0.233949i \(0.924835\pi\)
\(878\) 0 0
\(879\) 20.7846i 0.701047i
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 24.2487i 0.816034i 0.912974 + 0.408017i \(0.133780\pi\)
−0.912974 + 0.408017i \(0.866220\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 55.4256i 1.85892i
\(890\) 0 0
\(891\) 3.46410i 0.116052i
\(892\) 0 0
\(893\) 18.0000 41.5692i 0.602347 1.39106i
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 41.5692i 1.38487i
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) 13.8564i 0.460603i
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 3.46410i 0.114270i 0.998366 + 0.0571351i \(0.0181966\pi\)
−0.998366 + 0.0571351i \(0.981803\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 20.0000 + 8.66025i 0.655474 + 0.283828i
\(932\) 0 0
\(933\) 31.1769i 1.02069i
\(934\) 0 0
\(935\) 20.7846i 0.679729i
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 41.5692i 1.35512i −0.735469 0.677559i \(-0.763038\pi\)
0.735469 0.677559i \(-0.236962\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 3.46410i 0.112687i
\(946\) 0 0
\(947\) 17.3205i 0.562841i 0.959585 + 0.281420i \(0.0908056\pi\)
−0.959585 + 0.281420i \(0.909194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 6.92820i 0.224662i
\(952\) 0 0
\(953\) 34.6410i 1.12213i 0.827771 + 0.561066i \(0.189609\pi\)
−0.827771 + 0.561066i \(0.810391\pi\)
\(954\) 0 0
\(955\) 17.3205i 0.560478i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 62.3538i 2.01351i
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 20.7846i 0.669080i
\(966\) 0 0
\(967\) 3.46410i 0.111398i −0.998448 0.0556990i \(-0.982261\pi\)
0.998448 0.0556990i \(-0.0177387\pi\)
\(968\) 0 0
\(969\) −24.0000 10.3923i −0.770991 0.333849i
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 36.0000 1.15411
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 34.6410i 1.10826i −0.832429 0.554132i \(-0.813050\pi\)
0.832429 0.554132i \(-0.186950\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 13.8564i 0.442401i
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 36.0000 1.14589
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) 10.3923i 0.329458i
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 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 4560.2.d.c.2431.1 yes 2
4.3 odd 2 4560.2.d.a.2431.2 yes 2
19.18 odd 2 4560.2.d.a.2431.1 2
76.75 even 2 inner 4560.2.d.c.2431.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4560.2.d.a.2431.1 2 19.18 odd 2
4560.2.d.a.2431.2 yes 2 4.3 odd 2
4560.2.d.c.2431.1 yes 2 1.1 even 1 trivial
4560.2.d.c.2431.2 yes 2 76.75 even 2 inner