Properties

Label 2175.2.c.e.349.1
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.e.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000 q^{4} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.00000 q^{4} +2.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} +4.00000 q^{16} -2.00000 q^{19} +2.00000 q^{21} -3.00000i q^{23} +1.00000i q^{27} +4.00000i q^{28} +1.00000 q^{29} +8.00000 q^{31} -3.00000i q^{33} -2.00000 q^{36} -1.00000i q^{37} -2.00000 q^{39} -3.00000 q^{41} +1.00000i q^{43} +6.00000 q^{44} -6.00000i q^{47} -4.00000i q^{48} +3.00000 q^{49} -4.00000i q^{52} +3.00000i q^{53} +2.00000i q^{57} +12.0000 q^{59} +8.00000 q^{61} -2.00000i q^{63} +8.00000 q^{64} +14.0000i q^{67} -3.00000 q^{69} -6.00000 q^{71} +7.00000i q^{73} -4.00000 q^{76} +6.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -9.00000i q^{83} +4.00000 q^{84} -1.00000i q^{87} +6.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} -8.00000i q^{93} +11.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{29} + 16 q^{31} - 4 q^{36} - 4 q^{39} - 6 q^{41} + 12 q^{44} + 6 q^{49} + 24 q^{59} + 16 q^{61} + 16 q^{64} - 6 q^{69} - 12 q^{71} - 8 q^{76} + 8 q^{79} + 2 q^{81} + 8 q^{84} + 12 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) − 3.00000i − 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 4.00000i − 0.554700i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 0 0
\(87\) − 1.00000i − 0.107211i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 6.00000i − 0.625543i
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000i 1.11688i 0.829545 + 0.558440i \(0.188600\pi\)
−0.829545 + 0.558440i \(0.811400\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 2.00000i 0.192450i
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 8.00000i 0.755929i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 3.00000i 0.270501i
\(124\) 16.0000 1.43684
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000i 0.443678i 0.975083 + 0.221839i \(0.0712060\pi\)
−0.975083 + 0.221839i \(0.928794\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) − 4.00000i − 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) − 6.00000i − 0.501745i
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.00000i − 0.247436i
\(148\) − 2.00000i − 0.164399i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 19.0000i 1.48819i 0.668071 + 0.744097i \(0.267120\pi\)
−0.668071 + 0.744097i \(0.732880\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000i 0.152499i
\(173\) − 21.0000i − 1.59660i −0.602260 0.798300i \(-0.705733\pi\)
0.602260 0.798300i \(-0.294267\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) − 8.00000i − 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) − 12.0000i − 0.875190i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) − 8.00000i − 0.577350i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 6.00000 0.428571
\(197\) 21.0000i 1.49619i 0.663593 + 0.748094i \(0.269031\pi\)
−0.663593 + 0.748094i \(0.730969\pi\)
\(198\) 0 0
\(199\) 1.00000 0.0708881 0.0354441 0.999372i \(-0.488715\pi\)
0.0354441 + 0.999372i \(0.488715\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.00000i 0.208514i
\(208\) − 8.00000i − 0.554700i
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 6.00000i 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 21.0000i − 1.39382i −0.717159 0.696909i \(-0.754558\pi\)
0.717159 0.696909i \(-0.245442\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) − 15.0000i − 0.982683i −0.870967 0.491341i \(-0.836507\pi\)
0.870967 0.491341i \(-0.163493\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 24.0000 1.56227
\(237\) − 4.00000i − 0.259828i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 16.0000 1.02430
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) − 9.00000i − 0.565825i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 3.00000i 0.187135i 0.995613 + 0.0935674i \(0.0298271\pi\)
−0.995613 + 0.0935674i \(0.970173\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 28.0000i 1.71037i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) − 4.00000i − 0.242091i
\(274\) 0 0
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 8.00000i 0.480673i 0.970690 + 0.240337i \(0.0772579\pi\)
−0.970690 + 0.240337i \(0.922742\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 10.0000i 0.594438i 0.954809 + 0.297219i \(0.0960592\pi\)
−0.954809 + 0.297219i \(0.903941\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.00000i − 0.354169i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 11.0000 0.644831
\(292\) 14.0000i 0.819288i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000i 0.174078i
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 9.00000i 0.517036i
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) − 7.00000i − 0.399511i −0.979846 0.199756i \(-0.935985\pi\)
0.979846 0.199756i \(-0.0640148\pi\)
\(308\) 12.0000i 0.683763i
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) − 13.0000i − 0.718902i
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) − 18.0000i − 0.987878i
\(333\) 1.00000i 0.0547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 8.00000 0.436436
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.00000i 0.161048i 0.996753 + 0.0805242i \(0.0256594\pi\)
−0.996753 + 0.0805242i \(0.974341\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 0 0
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 2.00000i 0.104973i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) − 19.0000i − 0.991792i −0.868382 0.495896i \(-0.834840\pi\)
0.868382 0.495896i \(-0.165160\pi\)
\(368\) − 12.0000i − 0.625543i
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) − 16.0000i − 0.829561i
\(373\) − 32.0000i − 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.00000i − 0.103005i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 5.00000 0.256158
\(382\) 0 0
\(383\) 21.0000i 1.07305i 0.843884 + 0.536525i \(0.180263\pi\)
−0.843884 + 0.536525i \(0.819737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1.00000i − 0.0508329i
\(388\) 22.0000i 1.11688i
\(389\) −33.0000 −1.67317 −0.836583 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.00000i − 0.148704i
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) − 28.0000i − 1.37946i
\(413\) 24.0000i 1.18096i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.00000i 0.244851i
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.0000i 0.774294i
\(428\) − 24.0000i − 1.16008i
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000i 0.192450i
\(433\) − 5.00000i − 0.240285i −0.992757 0.120142i \(-0.961665\pi\)
0.992757 0.120142i \(-0.0383351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 26.0000 1.24517
\(437\) 6.00000i 0.287019i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 0 0
\(447\) 18.0000i 0.851371i
\(448\) 16.0000i 0.755929i
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) 19.0000i 0.892698i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 16.0000i − 0.748448i −0.927338 0.374224i \(-0.877909\pi\)
0.927338 0.374224i \(-0.122091\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) 34.0000i 1.58011i 0.613033 + 0.790057i \(0.289949\pi\)
−0.613033 + 0.790057i \(0.710051\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) 4.00000i 0.184900i
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 3.00000i 0.137940i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 3.00000i − 0.137361i
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) − 6.00000i − 0.273009i
\(484\) −4.00000 −0.181818
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 0 0
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 32.0000 1.43684
\(497\) − 12.0000i − 0.538274i
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.0000i 0.802580i 0.915951 + 0.401290i \(0.131438\pi\)
−0.915951 + 0.401290i \(0.868562\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 10.0000i 0.443678i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) − 2.00000i − 0.0883022i
\(514\) 0 0
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) − 18.0000i − 0.791639i
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −24.0000 −1.04844
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) − 12.0000i − 0.522233i
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) − 8.00000i − 0.346844i
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.00000i 0.258919i
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 7.00000i 0.300399i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000i 1.88130i 0.339372 + 0.940652i \(0.389785\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(548\) − 24.0000i − 1.02523i
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) − 27.0000i − 1.14403i −0.820244 0.572013i \(-0.806163\pi\)
0.820244 0.572013i \(-0.193837\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −43.0000 −1.79949 −0.899747 0.436412i \(-0.856249\pi\)
−0.899747 + 0.436412i \(0.856249\pi\)
\(572\) − 12.0000i − 0.501745i
\(573\) − 15.0000i − 0.626634i
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 9.00000i 0.372742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 6.00000i − 0.247436i
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 21.0000 0.863825
\(592\) − 4.00000i − 0.164399i
\(593\) − 42.0000i − 1.72473i −0.506284 0.862367i \(-0.668981\pi\)
0.506284 0.862367i \(-0.331019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −36.0000 −1.47462
\(597\) − 1.00000i − 0.0409273i
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) − 14.0000i − 0.570124i
\(604\) −38.0000 −1.54620
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) 0 0
\(627\) 6.00000i 0.239617i
\(628\) 4.00000i 0.159617i
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 10.0000i 0.397464i
\(634\) 0 0
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) − 32.0000i − 1.26196i −0.775800 0.630978i \(-0.782654\pi\)
0.775800 0.630978i \(-0.217346\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) − 27.0000i − 1.06148i −0.847535 0.530740i \(-0.821914\pi\)
0.847535 0.530740i \(-0.178086\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 38.0000i 1.48819i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) − 7.00000i − 0.273096i
\(658\) 0 0
\(659\) −27.0000 −1.05177 −0.525885 0.850555i \(-0.676266\pi\)
−0.525885 + 0.850555i \(0.676266\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.00000i − 0.116160i
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 18.0000 0.692308
\(677\) 36.0000i 1.38359i 0.722093 + 0.691796i \(0.243180\pi\)
−0.722093 + 0.691796i \(0.756820\pi\)
\(678\) 0 0
\(679\) −22.0000 −0.844283
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) − 15.0000i − 0.573959i −0.957937 0.286980i \(-0.907349\pi\)
0.957937 0.286980i \(-0.0926512\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000i 0.534133i
\(688\) 4.00000i 0.152499i
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 42.0000i − 1.59660i
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −15.0000 −0.567352
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 2.00000i 0.0754314i
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) − 18.0000i − 0.676960i
\(708\) − 24.0000i − 0.901975i
\(709\) 1.00000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) − 24.0000i − 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 6.00000i 0.224074i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 7.00000i 0.260333i
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) 0 0
\(727\) 44.0000i 1.63187i 0.578144 + 0.815935i \(0.303777\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 16.0000i − 0.591377i
\(733\) − 50.0000i − 1.84679i −0.383849 0.923396i \(-0.625402\pi\)
0.383849 0.923396i \(-0.374598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.0000i 1.54709i
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 42.0000i 1.54083i 0.637542 + 0.770415i \(0.279951\pi\)
−0.637542 + 0.770415i \(0.720049\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9.00000i 0.329293i
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) − 24.0000i − 0.875190i
\(753\) − 12.0000i − 0.437304i
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 43.0000i − 1.56286i −0.623992 0.781431i \(-0.714490\pi\)
0.623992 0.781431i \(-0.285510\pi\)
\(758\) 0 0
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 0 0
\(763\) 26.0000i 0.941263i
\(764\) 30.0000 1.08536
\(765\) 0 0
\(766\) 0 0
\(767\) − 24.0000i − 0.866590i
\(768\) − 16.0000i − 0.577350i
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) − 28.0000i − 1.00774i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.00000i − 0.0717496i
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 1.00000i 0.0357371i
\(784\) 12.0000 0.428571
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 42.0000i 1.49619i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 16.0000i − 0.568177i
\(794\) 0 0
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 21.0000i 0.741074i
\(804\) 28.0000 0.987484
\(805\) 0 0
\(806\) 0 0
\(807\) − 18.0000i − 0.633630i
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) −49.0000 −1.72062 −0.860311 0.509769i \(-0.829731\pi\)
−0.860311 + 0.509769i \(0.829731\pi\)
\(812\) 4.00000i 0.140372i
\(813\) − 2.00000i − 0.0701431i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 2.00000i − 0.0699711i
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 40.0000i 1.39431i 0.716919 + 0.697156i \(0.245552\pi\)
−0.716919 + 0.697156i \(0.754448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) − 16.0000i − 0.554700i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.00000i − 0.137442i
\(848\) 12.0000i 0.412082i
\(849\) 10.0000 0.343199
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 12.0000i 0.411113i
\(853\) − 17.0000i − 0.582069i −0.956713 0.291034i \(-0.906001\pi\)
0.956713 0.291034i \(-0.0939994\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 39.0000i − 1.33221i −0.745856 0.666107i \(-0.767959\pi\)
0.745856 0.666107i \(-0.232041\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 17.0000i − 0.577350i
\(868\) 32.0000i 1.08615i
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 0 0
\(873\) − 11.0000i − 0.372294i
\(874\) 0 0
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) − 28.0000i − 0.945493i −0.881199 0.472746i \(-0.843263\pi\)
0.881199 0.472746i \(-0.156737\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.00000 0.101073 0.0505363 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.0000i 1.41022i 0.709097 + 0.705111i \(0.249103\pi\)
−0.709097 + 0.705111i \(0.750897\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 8.00000i 0.267860i
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000i 0.200334i
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 2.00000i 0.0665558i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 19.0000i − 0.630885i −0.948945 0.315442i \(-0.897847\pi\)
0.948945 0.315442i \(-0.102153\pi\)
\(908\) − 42.0000i − 1.39382i
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) 33.0000 1.09334 0.546669 0.837349i \(-0.315895\pi\)
0.546669 + 0.837349i \(0.315895\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 27.0000i − 0.893570i
\(914\) 0 0
\(915\) 0 0
\(916\) −28.0000 −0.925146
\(917\) − 24.0000i − 0.792550i
\(918\) 0 0
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) 0 0
\(927\) 14.0000i 0.459820i
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) − 30.0000i − 0.982683i
\(933\) 21.0000i 0.687509i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 52.0000i − 1.69877i −0.527777 0.849383i \(-0.676974\pi\)
0.527777 0.849383i \(-0.323026\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) 9.00000i 0.293080i
\(944\) 48.0000 1.56227
\(945\) 0 0
\(946\) 0 0
\(947\) 30.0000i 0.974869i 0.873160 + 0.487435i \(0.162067\pi\)
−0.873160 + 0.487435i \(0.837933\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) − 3.00000i − 0.0969762i
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 41.0000i 1.31847i 0.751936 + 0.659236i \(0.229120\pi\)
−0.751936 + 0.659236i \(0.770880\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −57.0000 −1.82922 −0.914609 0.404341i \(-0.867501\pi\)
−0.914609 + 0.404341i \(0.867501\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) − 10.0000i − 0.320585i
\(974\) 0 0
\(975\) 0 0
\(976\) 32.0000 1.02430
\(977\) 57.0000i 1.82359i 0.410644 + 0.911796i \(0.365304\pi\)
−0.410644 + 0.911796i \(0.634696\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −13.0000 −0.415058
\(982\) 0 0
\(983\) − 30.0000i − 0.956851i −0.878128 0.478426i \(-0.841208\pi\)
0.878128 0.478426i \(-0.158792\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 12.0000i − 0.381964i
\(988\) 8.00000i 0.254514i
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) −18.0000 −0.570352
\(997\) − 37.0000i − 1.17180i −0.810383 0.585901i \(-0.800741\pi\)
0.810383 0.585901i \(-0.199259\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
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 2175.2.c.e.349.1 2
5.2 odd 4 2175.2.a.d.1.1 1
5.3 odd 4 435.2.a.c.1.1 1
5.4 even 2 inner 2175.2.c.e.349.2 2
15.2 even 4 6525.2.a.f.1.1 1
15.8 even 4 1305.2.a.d.1.1 1
20.3 even 4 6960.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.c.1.1 1 5.3 odd 4
1305.2.a.d.1.1 1 15.8 even 4
2175.2.a.d.1.1 1 5.2 odd 4
2175.2.c.e.349.1 2 1.1 even 1 trivial
2175.2.c.e.349.2 2 5.4 even 2 inner
6525.2.a.f.1.1 1 15.2 even 4
6960.2.a.b.1.1 1 20.3 even 4