Properties

Label 7056.2.b.k.1567.2
Level $7056$
Weight $2$
Character 7056.1567
Analytic conductor $56.342$
Analytic rank $1$
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] = [7056,2,Mod(1567,7056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, 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("7056.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.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: \(56.3424436662\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 7056.1567
Dual form 7056.2.b.k.1567.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.46410i q^{5} +O(q^{10})\) \(q+3.46410i q^{5} -3.46410i q^{11} +1.73205i q^{13} +5.00000 q^{19} -6.92820i q^{23} -7.00000 q^{25} -5.00000 q^{31} -11.0000 q^{37} -3.46410i q^{41} +8.66025i q^{43} +6.00000 q^{47} -12.0000 q^{53} +12.0000 q^{55} -12.0000 q^{59} +13.8564i q^{61} -6.00000 q^{65} -8.66025i q^{67} -3.46410i q^{71} -5.19615i q^{73} +12.1244i q^{79} -18.0000 q^{83} +6.92820i q^{89} +17.3205i q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{19} - 14 q^{25} - 10 q^{31} - 22 q^{37} + 12 q^{47} - 24 q^{53} + 24 q^{55} - 24 q^{59} - 12 q^{65} - 36 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 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
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.92820i − 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.46410i − 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 8.66025i 1.32068i 0.750968 + 0.660338i \(0.229587\pi\)
−0.750968 + 0.660338i \(0.770413\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) − 8.66025i − 1.05802i −0.848616 0.529009i \(-0.822564\pi\)
0.848616 0.529009i \(-0.177436\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3.46410i − 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) − 5.19615i − 0.608164i −0.952646 0.304082i \(-0.901650\pi\)
0.952646 0.304082i \(-0.0983496\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.1244i 1.36410i 0.731307 + 0.682048i \(0.238911\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 17.3205i 1.77705i
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 10.3923i − 1.03407i −0.855963 0.517036i \(-0.827035\pi\)
0.855963 0.517036i \(-0.172965\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 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
\(115\) 24.0000 2.23801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 6.92820i − 0.619677i
\(126\) 0 0
\(127\) 12.1244i 1.07586i 0.842989 + 0.537931i \(0.180794\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) − 3.46410i − 0.281905i −0.990016 0.140952i \(-0.954984\pi\)
0.990016 0.140952i \(-0.0450164\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 17.3205i − 1.39122i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 3.46410i − 0.271329i −0.990755 0.135665i \(-0.956683\pi\)
0.990755 0.135665i \(-0.0433170\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.92820i 0.526742i 0.964695 + 0.263371i \(0.0848343\pi\)
−0.964695 + 0.263371i \(0.915166\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 10.3923i − 0.776757i −0.921500 0.388379i \(-0.873035\pi\)
0.921500 0.388379i \(-0.126965\pi\)
\(180\) 0 0
\(181\) 5.19615i 0.386227i 0.981176 + 0.193113i \(0.0618586\pi\)
−0.981176 + 0.193113i \(0.938141\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 38.1051i − 2.80154i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 17.3205i − 1.25327i −0.779314 0.626634i \(-0.784432\pi\)
0.779314 0.626634i \(-0.215568\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 17.3205i − 1.19808i
\(210\) 0 0
\(211\) 3.46410i 0.238479i 0.992866 + 0.119239i \(0.0380456\pi\)
−0.992866 + 0.119239i \(0.961954\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −30.0000 −2.04598
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) − 8.66025i − 0.572286i −0.958187 0.286143i \(-0.907627\pi\)
0.958187 0.286143i \(-0.0923732\pi\)
\(230\) 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
\(235\) 20.7846i 1.35584i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 10.3923i − 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) − 13.8564i − 0.892570i −0.894891 0.446285i \(-0.852747\pi\)
0.894891 0.446285i \(-0.147253\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.66025i 0.551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 17.3205i − 1.08042i −0.841529 0.540212i \(-0.818344\pi\)
0.841529 0.540212i \(-0.181656\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.8564i 0.854423i 0.904152 + 0.427211i \(0.140504\pi\)
−0.904152 + 0.427211i \(0.859496\pi\)
\(264\) 0 0
\(265\) − 41.5692i − 2.55358i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.3205i 1.05605i 0.849229 + 0.528025i \(0.177067\pi\)
−0.849229 + 0.528025i \(0.822933\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.2487i 1.46225i
\(276\) 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
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −7.00000 −0.416107 −0.208053 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 20.7846i − 1.21425i −0.794606 0.607125i \(-0.792323\pi\)
0.794606 0.607125i \(-0.207677\pi\)
\(294\) 0 0
\(295\) − 41.5692i − 2.42025i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −48.0000 −2.74847
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) − 15.5885i − 0.881112i −0.897725 0.440556i \(-0.854781\pi\)
0.897725 0.440556i \(-0.145219\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 12.1244i − 0.672538i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.5885i 0.856819i 0.903585 + 0.428410i \(0.140926\pi\)
−0.903585 + 0.428410i \(0.859074\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 0 0
\(337\) −19.0000 −1.03500 −0.517498 0.855684i \(-0.673136\pi\)
−0.517498 + 0.855684i \(0.673136\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.3205i 0.937958i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.7128i − 1.48770i −0.668346 0.743851i \(-0.732997\pi\)
0.668346 0.743851i \(-0.267003\pi\)
\(348\) 0 0
\(349\) 13.8564i 0.741716i 0.928689 + 0.370858i \(0.120936\pi\)
−0.928689 + 0.370858i \(0.879064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 24.2487i − 1.29063i −0.763917 0.645314i \(-0.776726\pi\)
0.763917 0.645314i \(-0.223274\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7846i 1.09697i 0.836160 + 0.548485i \(0.184795\pi\)
−0.836160 + 0.548485i \(0.815205\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 0 0
\(367\) 11.0000 0.574195 0.287098 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.00000 −0.0517780 −0.0258890 0.999665i \(-0.508242\pi\)
−0.0258890 + 0.999665i \(0.508242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.66025i 0.444847i 0.974950 + 0.222424i \(0.0713968\pi\)
−0.974950 + 0.222424i \(0.928603\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) −42.0000 −2.11325
\(396\) 0 0
\(397\) 1.73205i 0.0869291i 0.999055 + 0.0434646i \(0.0138396\pi\)
−0.999055 + 0.0434646i \(0.986160\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) − 8.66025i − 0.431398i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 38.1051i 1.88880i
\(408\) 0 0
\(409\) − 22.5167i − 1.11338i −0.830721 0.556689i \(-0.812072\pi\)
0.830721 0.556689i \(-0.187928\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 62.3538i − 3.06083i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 7.00000 0.341159 0.170580 0.985344i \(-0.445436\pi\)
0.170580 + 0.985344i \(0.445436\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.2487i 1.16802i 0.811747 + 0.584010i \(0.198517\pi\)
−0.811747 + 0.584010i \(0.801483\pi\)
\(432\) 0 0
\(433\) 1.73205i 0.0832370i 0.999134 + 0.0416185i \(0.0132514\pi\)
−0.999134 + 0.0416185i \(0.986749\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 34.6410i − 1.65710i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 13.8564i − 0.658338i −0.944271 0.329169i \(-0.893231\pi\)
0.944271 0.329169i \(-0.106769\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 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
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0000 −0.888783 −0.444391 0.895833i \(-0.646580\pi\)
−0.444391 + 0.895833i \(0.646580\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 13.8564i − 0.645357i −0.946509 0.322679i \(-0.895417\pi\)
0.946509 0.322679i \(-0.104583\pi\)
\(462\) 0 0
\(463\) − 15.5885i − 0.724457i −0.932089 0.362229i \(-0.882016\pi\)
0.932089 0.362229i \(-0.117984\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) −35.0000 −1.60591
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) − 19.0526i − 0.868722i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) − 25.9808i − 1.17730i −0.808388 0.588650i \(-0.799659\pi\)
0.808388 0.588650i \(-0.200341\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 34.6410i 1.56333i 0.623700 + 0.781664i \(0.285629\pi\)
−0.623700 + 0.781664i \(0.714371\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 32.9090i − 1.47321i −0.676325 0.736604i \(-0.736428\pi\)
0.676325 0.736604i \(-0.263572\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.1769i 1.38189i 0.722906 + 0.690946i \(0.242806\pi\)
−0.722906 + 0.690946i \(0.757194\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 17.3205i − 0.763233i
\(516\) 0 0
\(517\) − 20.7846i − 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.8564i 0.607060i 0.952822 + 0.303530i \(0.0981653\pi\)
−0.952822 + 0.303530i \(0.901835\pi\)
\(522\) 0 0
\(523\) −11.0000 −0.480996 −0.240498 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.00000 −0.214967 −0.107483 0.994207i \(-0.534279\pi\)
−0.107483 + 0.994207i \(0.534279\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 24.2487i − 1.03870i
\(546\) 0 0
\(547\) − 3.46410i − 0.148114i −0.997254 0.0740571i \(-0.976405\pi\)
0.997254 0.0740571i \(-0.0235947\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) −15.0000 −0.634432
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 46.7654i 1.95707i 0.206081 + 0.978535i \(0.433929\pi\)
−0.206081 + 0.978535i \(0.566071\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 48.4974i 2.02248i
\(576\) 0 0
\(577\) 12.1244i 0.504744i 0.967630 + 0.252372i \(0.0812106\pi\)
−0.967630 + 0.252372i \(0.918789\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 41.5692i 1.72162i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −25.0000 −1.03011
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.46410i 0.142254i 0.997467 + 0.0711268i \(0.0226595\pi\)
−0.997467 + 0.0711268i \(0.977341\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 48.4974i − 1.98155i −0.135514 0.990775i \(-0.543268\pi\)
0.135514 0.990775i \(-0.456732\pi\)
\(600\) 0 0
\(601\) 39.8372i 1.62499i 0.582967 + 0.812496i \(0.301892\pi\)
−0.582967 + 0.812496i \(0.698108\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.46410i − 0.140836i
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.3923i 0.420428i
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.46410i 0.137904i 0.997620 + 0.0689519i \(0.0219655\pi\)
−0.997620 + 0.0689519i \(0.978035\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −42.0000 −1.66672
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) 0 0
\(649\) 41.5692i 1.63173i
\(650\) 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
\(655\) − 20.7846i − 0.812122i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 13.8564i − 0.539769i −0.962893 0.269884i \(-0.913014\pi\)
0.962893 0.269884i \(-0.0869855\pi\)
\(660\) 0 0
\(661\) 25.9808i 1.01053i 0.862963 + 0.505267i \(0.168606\pi\)
−0.862963 + 0.505267i \(0.831394\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 6.92820i − 0.266272i −0.991098 0.133136i \(-0.957495\pi\)
0.991098 0.133136i \(-0.0425048\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.92820i 0.265100i 0.991176 + 0.132550i \(0.0423166\pi\)
−0.991176 + 0.132550i \(0.957683\pi\)
\(684\) 0 0
\(685\) − 41.5692i − 1.58828i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 20.7846i − 0.791831i
\(690\) 0 0
\(691\) −23.0000 −0.874961 −0.437481 0.899228i \(-0.644129\pi\)
−0.437481 + 0.899228i \(0.644129\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.2487i 0.919806i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) −55.0000 −2.07436
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 34.6410i 1.29732i
\(714\) 0 0
\(715\) 20.7846i 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 29.4449i 1.08757i 0.839224 + 0.543785i \(0.183009\pi\)
−0.839224 + 0.543785i \(0.816991\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) 53.6936i 1.97515i 0.157148 + 0.987575i \(0.449770\pi\)
−0.157148 + 0.987575i \(0.550230\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 3.46410i − 0.127086i −0.997979 0.0635428i \(-0.979760\pi\)
0.997979 0.0635428i \(-0.0202399\pi\)
\(744\) 0 0
\(745\) − 41.5692i − 1.52298i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 5.19615i − 0.189610i −0.995496 0.0948051i \(-0.969777\pi\)
0.995496 0.0948051i \(-0.0302228\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 41.5692i − 1.50688i −0.657515 0.753442i \(-0.728392\pi\)
0.657515 0.753442i \(-0.271608\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 20.7846i − 0.750489i
\(768\) 0 0
\(769\) 15.5885i 0.562134i 0.959688 + 0.281067i \(0.0906883\pi\)
−0.959688 + 0.281067i \(0.909312\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 45.0333i 1.61974i 0.586612 + 0.809868i \(0.300461\pi\)
−0.586612 + 0.809868i \(0.699539\pi\)
\(774\) 0 0
\(775\) 35.0000 1.25724
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 17.3205i − 0.620572i
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.6410i 1.22705i 0.789676 + 0.613524i \(0.210249\pi\)
−0.789676 + 0.613524i \(0.789751\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 43.3013i 1.51492i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 3.46410i 0.120751i 0.998176 + 0.0603755i \(0.0192298\pi\)
−0.998176 + 0.0603755i \(0.980770\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 38.1051i − 1.32504i −0.749042 0.662522i \(-0.769486\pi\)
0.749042 0.662522i \(-0.230514\pi\)
\(828\) 0 0
\(829\) 25.9808i 0.902349i 0.892436 + 0.451175i \(0.148995\pi\)
−0.892436 + 0.451175i \(0.851005\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 20.7846i − 0.719281i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 34.6410i 1.19169i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 76.2102i 2.61245i
\(852\) 0 0
\(853\) 5.19615i 0.177913i 0.996036 + 0.0889564i \(0.0283532\pi\)
−0.996036 + 0.0889564i \(0.971647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 13.8564i − 0.473326i −0.971592 0.236663i \(-0.923946\pi\)
0.971592 0.236663i \(-0.0760537\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 10.3923i − 0.353758i −0.984233 0.176879i \(-0.943400\pi\)
0.984233 0.176879i \(-0.0566002\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) 15.0000 0.508256
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.92820i 0.233417i 0.993166 + 0.116709i \(0.0372343\pi\)
−0.993166 + 0.116709i \(0.962766\pi\)
\(882\) 0 0
\(883\) − 46.7654i − 1.57378i −0.617093 0.786890i \(-0.711690\pi\)
0.617093 0.786890i \(-0.288310\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 30.0000 1.00391
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) − 32.9090i − 1.09272i −0.837549 0.546362i \(-0.816012\pi\)
0.837549 0.546362i \(-0.183988\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 41.5692i − 1.37725i −0.725118 0.688625i \(-0.758215\pi\)
0.725118 0.688625i \(-0.241785\pi\)
\(912\) 0 0
\(913\) 62.3538i 2.06361i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 43.3013i 1.42838i 0.699953 + 0.714189i \(0.253204\pi\)
−0.699953 + 0.714189i \(0.746796\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 77.0000 2.53174
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.3923i 0.340960i 0.985361 + 0.170480i \(0.0545319\pi\)
−0.985361 + 0.170480i \(0.945468\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 53.6936i 1.75409i 0.480406 + 0.877046i \(0.340489\pi\)
−0.480406 + 0.877046i \(0.659511\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13.8564i 0.451706i 0.974161 + 0.225853i \(0.0725169\pi\)
−0.974161 + 0.225853i \(0.927483\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.0333i 1.46339i 0.681634 + 0.731693i \(0.261270\pi\)
−0.681634 + 0.731693i \(0.738730\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) 60.0000 1.94155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 38.1051i − 1.22665i
\(966\) 0 0
\(967\) − 8.66025i − 0.278495i −0.990258 0.139247i \(-0.955532\pi\)
0.990258 0.139247i \(-0.0444684\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) − 83.1384i − 2.64901i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) − 39.8372i − 1.26547i −0.774369 0.632735i \(-0.781932\pi\)
0.774369 0.632735i \(-0.218068\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 55.4256i − 1.75711i
\(996\) 0 0
\(997\) − 19.0526i − 0.603401i −0.953403 0.301700i \(-0.902446\pi\)
0.953403 0.301700i \(-0.0975542\pi\)
\(998\) 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 7056.2.b.k.1567.2 2
3.2 odd 2 2352.2.b.d.1567.1 2
4.3 odd 2 7056.2.b.d.1567.2 2
7.4 even 3 1008.2.cs.a.271.1 2
7.5 odd 6 1008.2.cs.b.703.1 2
7.6 odd 2 7056.2.b.d.1567.1 2
12.11 even 2 2352.2.b.e.1567.1 2
21.2 odd 6 2352.2.bl.g.31.1 2
21.5 even 6 336.2.bl.d.31.1 2
21.11 odd 6 336.2.bl.h.271.1 yes 2
21.17 even 6 2352.2.bl.a.607.1 2
21.20 even 2 2352.2.b.e.1567.2 2
28.11 odd 6 1008.2.cs.b.271.1 2
28.19 even 6 1008.2.cs.a.703.1 2
28.27 even 2 inner 7056.2.b.k.1567.1 2
84.11 even 6 336.2.bl.d.271.1 yes 2
84.23 even 6 2352.2.bl.a.31.1 2
84.47 odd 6 336.2.bl.h.31.1 yes 2
84.59 odd 6 2352.2.bl.g.607.1 2
84.83 odd 2 2352.2.b.d.1567.2 2
168.5 even 6 1344.2.bl.e.703.1 2
168.11 even 6 1344.2.bl.e.1279.1 2
168.53 odd 6 1344.2.bl.a.1279.1 2
168.131 odd 6 1344.2.bl.a.703.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.d.31.1 2 21.5 even 6
336.2.bl.d.271.1 yes 2 84.11 even 6
336.2.bl.h.31.1 yes 2 84.47 odd 6
336.2.bl.h.271.1 yes 2 21.11 odd 6
1008.2.cs.a.271.1 2 7.4 even 3
1008.2.cs.a.703.1 2 28.19 even 6
1008.2.cs.b.271.1 2 28.11 odd 6
1008.2.cs.b.703.1 2 7.5 odd 6
1344.2.bl.a.703.1 2 168.131 odd 6
1344.2.bl.a.1279.1 2 168.53 odd 6
1344.2.bl.e.703.1 2 168.5 even 6
1344.2.bl.e.1279.1 2 168.11 even 6
2352.2.b.d.1567.1 2 3.2 odd 2
2352.2.b.d.1567.2 2 84.83 odd 2
2352.2.b.e.1567.1 2 12.11 even 2
2352.2.b.e.1567.2 2 21.20 even 2
2352.2.bl.a.31.1 2 84.23 even 6
2352.2.bl.a.607.1 2 21.17 even 6
2352.2.bl.g.31.1 2 21.2 odd 6
2352.2.bl.g.607.1 2 84.59 odd 6
7056.2.b.d.1567.1 2 7.6 odd 2
7056.2.b.d.1567.2 2 4.3 odd 2
7056.2.b.k.1567.1 2 28.27 even 2 inner
7056.2.b.k.1567.2 2 1.1 even 1 trivial