Properties

Label 672.2.q.c.193.1
Level $672$
Weight $2$
Character 672.193
Analytic conductor $5.366$
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] = [672,2,Mod(193,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.q (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 672.193
Dual form 672.2.q.c.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(1.00000 - 1.73205i) q^{17} +(-1.50000 - 2.59808i) q^{19} +(2.00000 + 1.73205i) q^{21} +(-1.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +8.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.50000 + 4.33013i) q^{37} +(-2.50000 + 4.33013i) q^{39} +2.00000 q^{41} +7.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +3.00000 q^{57} +(-5.00000 + 8.66025i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.50000 + 0.866025i) q^{63} +(5.50000 - 9.52628i) q^{67} +2.00000 q^{69} +12.0000 q^{71} +(1.50000 - 2.59808i) q^{73} +(2.50000 + 4.33013i) q^{75} +(4.00000 + 3.46410i) q^{77} +(-8.50000 - 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} -16.0000 q^{83} +(-4.00000 + 6.92820i) q^{87} +(-6.00000 - 10.3923i) q^{89} +(2.50000 - 12.9904i) q^{91} +(0.500000 + 0.866025i) q^{93} -14.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{7} - q^{9} - 2 q^{11} + 10 q^{13} + 2 q^{17} - 3 q^{19} + 4 q^{21} - 2 q^{23} + 5 q^{25} + 2 q^{27} + 16 q^{29} + q^{31} - 2 q^{33} + 5 q^{37} - 5 q^{39} + 4 q^{41} + 14 q^{43} + 8 q^{47} - 13 q^{49} + 2 q^{51} + 2 q^{53} + 6 q^{57} - 10 q^{59} + 2 q^{61} - 5 q^{63} + 11 q^{67} + 4 q^{69} + 24 q^{71} + 3 q^{73} + 5 q^{75} + 8 q^{77} - 17 q^{79} - q^{81} - 32 q^{83} - 8 q^{87} - 12 q^{89} + 5 q^{91} + q^{93} - 28 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0.500000 2.59808i 0.188982 0.981981i
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) −1.00000 + 1.73205i −0.301511 + 0.522233i −0.976478 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 1.73205i 0.242536 0.420084i −0.718900 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) −1.50000 2.59808i −0.344124 0.596040i 0.641071 0.767482i \(-0.278491\pi\)
−0.985194 + 0.171442i \(0.945157\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) −1.00000 1.73205i −0.208514 0.361158i 0.742732 0.669588i \(-0.233529\pi\)
−0.951247 + 0.308431i \(0.900196\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 0.500000 0.866025i 0.0898027 0.155543i −0.817625 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) 0 0
\(33\) −1.00000 1.73205i −0.174078 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.50000 + 4.33013i 0.410997 + 0.711868i 0.994999 0.0998840i \(-0.0318472\pi\)
−0.584002 + 0.811752i \(0.698514\pi\)
\(38\) 0 0
\(39\) −2.50000 + 4.33013i −0.400320 + 0.693375i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 1.00000 + 1.73205i 0.140028 + 0.242536i
\(52\) 0 0
\(53\) 1.00000 1.73205i 0.137361 0.237915i −0.789136 0.614218i \(-0.789471\pi\)
0.926497 + 0.376303i \(0.122805\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) −5.00000 + 8.66025i −0.650945 + 1.12747i 0.331949 + 0.943297i \(0.392294\pi\)
−0.982894 + 0.184172i \(0.941040\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) −2.50000 + 0.866025i −0.314970 + 0.109109i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.50000 9.52628i 0.671932 1.16382i −0.305424 0.952217i \(-0.598798\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 1.50000 2.59808i 0.175562 0.304082i −0.764794 0.644275i \(-0.777159\pi\)
0.940356 + 0.340193i \(0.110493\pi\)
\(74\) 0 0
\(75\) 2.50000 + 4.33013i 0.288675 + 0.500000i
\(76\) 0 0
\(77\) 4.00000 + 3.46410i 0.455842 + 0.394771i
\(78\) 0 0
\(79\) −8.50000 14.7224i −0.956325 1.65640i −0.731307 0.682048i \(-0.761089\pi\)
−0.225018 0.974355i \(-0.572244\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.00000 + 6.92820i −0.428845 + 0.742781i
\(88\) 0 0
\(89\) −6.00000 10.3923i −0.635999 1.10158i −0.986303 0.164946i \(-0.947255\pi\)
0.350304 0.936636i \(-0.386078\pi\)
\(90\) 0 0
\(91\) 2.50000 12.9904i 0.262071 1.36176i
\(92\) 0 0
\(93\) 0.500000 + 0.866025i 0.0518476 + 0.0898027i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −9.00000 + 15.5885i −0.895533 + 1.55111i −0.0623905 + 0.998052i \(0.519872\pi\)
−0.833143 + 0.553058i \(0.813461\pi\)
\(102\) 0 0
\(103\) 0.500000 + 0.866025i 0.0492665 + 0.0853320i 0.889607 0.456727i \(-0.150978\pi\)
−0.840341 + 0.542059i \(0.817645\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.00000 1.73205i −0.0966736 0.167444i 0.813632 0.581380i \(-0.197487\pi\)
−0.910306 + 0.413936i \(0.864154\pi\)
\(108\) 0 0
\(109\) −0.500000 + 0.866025i −0.0478913 + 0.0829502i −0.888977 0.457951i \(-0.848583\pi\)
0.841086 + 0.540901i \(0.181917\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.50000 4.33013i −0.231125 0.400320i
\(118\) 0 0
\(119\) −4.00000 3.46410i −0.366679 0.317554i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) −1.00000 + 1.73205i −0.0901670 + 0.156174i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) −3.50000 + 6.06218i −0.308158 + 0.533745i
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) −7.50000 + 2.59808i −0.650332 + 0.225282i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 + 3.46410i −0.170872 + 0.295958i −0.938725 0.344668i \(-0.887992\pi\)
0.767853 + 0.640626i \(0.221325\pi\)
\(138\) 0 0
\(139\) −1.00000 −0.0848189 −0.0424094 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −5.00000 + 8.66025i −0.418121 + 0.724207i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.50000 4.33013i 0.453632 0.357143i
\(148\) 0 0
\(149\) −6.00000 10.3923i −0.491539 0.851371i 0.508413 0.861113i \(-0.330232\pi\)
−0.999953 + 0.00974235i \(0.996899\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.00000 5.19615i 0.239426 0.414698i −0.721124 0.692806i \(-0.756374\pi\)
0.960550 + 0.278108i \(0.0897074\pi\)
\(158\) 0 0
\(159\) 1.00000 + 1.73205i 0.0793052 + 0.137361i
\(160\) 0 0
\(161\) −5.00000 + 1.73205i −0.394055 + 0.136505i
\(162\) 0 0
\(163\) −2.00000 3.46410i −0.156652 0.271329i 0.777007 0.629492i \(-0.216737\pi\)
−0.933659 + 0.358162i \(0.883403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −1.50000 + 2.59808i −0.114708 + 0.198680i
\(172\) 0 0
\(173\) 3.00000 + 5.19615i 0.228086 + 0.395056i 0.957241 0.289292i \(-0.0934200\pi\)
−0.729155 + 0.684349i \(0.760087\pi\)
\(174\) 0 0
\(175\) −10.0000 8.66025i −0.755929 0.654654i
\(176\) 0 0
\(177\) −5.00000 8.66025i −0.375823 0.650945i
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) −15.0000 −1.11494 −0.557471 0.830197i \(-0.688228\pi\)
−0.557471 + 0.830197i \(0.688228\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 + 3.46410i 0.146254 + 0.253320i
\(188\) 0 0
\(189\) 0.500000 2.59808i 0.0363696 0.188982i
\(190\) 0 0
\(191\) −6.00000 10.3923i −0.434145 0.751961i 0.563081 0.826402i \(-0.309616\pi\)
−0.997225 + 0.0744412i \(0.976283\pi\)
\(192\) 0 0
\(193\) −11.5000 + 19.9186i −0.827788 + 1.43377i 0.0719816 + 0.997406i \(0.477068\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −2.00000 + 3.46410i −0.141776 + 0.245564i −0.928166 0.372168i \(-0.878615\pi\)
0.786389 + 0.617731i \(0.211948\pi\)
\(200\) 0 0
\(201\) 5.50000 + 9.52628i 0.387940 + 0.671932i
\(202\) 0 0
\(203\) 4.00000 20.7846i 0.280745 1.45879i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 + 1.73205i −0.0695048 + 0.120386i
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) −6.00000 + 10.3923i −0.411113 + 0.712069i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 1.73205i −0.135769 0.117579i
\(218\) 0 0
\(219\) 1.50000 + 2.59808i 0.101361 + 0.175562i
\(220\) 0 0
\(221\) 5.00000 8.66025i 0.336336 0.582552i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −13.0000 + 22.5167i −0.862840 + 1.49448i 0.00633544 + 0.999980i \(0.497983\pi\)
−0.869176 + 0.494503i \(0.835350\pi\)
\(228\) 0 0
\(229\) −6.50000 11.2583i −0.429532 0.743971i 0.567300 0.823511i \(-0.307988\pi\)
−0.996832 + 0.0795401i \(0.974655\pi\)
\(230\) 0 0
\(231\) −5.00000 + 1.73205i −0.328976 + 0.113961i
\(232\) 0 0
\(233\) 6.00000 + 10.3923i 0.393073 + 0.680823i 0.992853 0.119342i \(-0.0380786\pi\)
−0.599780 + 0.800165i \(0.704745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 17.0000 1.10427
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) 11.0000 19.0526i 0.708572 1.22728i −0.256814 0.966461i \(-0.582673\pi\)
0.965387 0.260822i \(-0.0839937\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −7.50000 12.9904i −0.477214 0.826558i
\(248\) 0 0
\(249\) 8.00000 13.8564i 0.506979 0.878114i
\(250\) 0 0
\(251\) −22.0000 −1.38863 −0.694314 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.00000 + 13.8564i 0.499026 + 0.864339i 0.999999 0.00112398i \(-0.000357774\pi\)
−0.500973 + 0.865463i \(0.667024\pi\)
\(258\) 0 0
\(259\) 12.5000 4.33013i 0.776712 0.269061i
\(260\) 0 0
\(261\) −4.00000 6.92820i −0.247594 0.428845i
\(262\) 0 0
\(263\) −4.00000 + 6.92820i −0.246651 + 0.427211i −0.962594 0.270947i \(-0.912663\pi\)
0.715944 + 0.698158i \(0.245997\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) 9.00000 15.5885i 0.548740 0.950445i −0.449622 0.893219i \(-0.648441\pi\)
0.998361 0.0572259i \(-0.0182255\pi\)
\(270\) 0 0
\(271\) 8.00000 + 13.8564i 0.485965 + 0.841717i 0.999870 0.0161307i \(-0.00513477\pi\)
−0.513905 + 0.857847i \(0.671801\pi\)
\(272\) 0 0
\(273\) 10.0000 + 8.66025i 0.605228 + 0.524142i
\(274\) 0 0
\(275\) 5.00000 + 8.66025i 0.301511 + 0.522233i
\(276\) 0 0
\(277\) −8.50000 + 14.7224i −0.510716 + 0.884585i 0.489207 + 0.872167i \(0.337286\pi\)
−0.999923 + 0.0124177i \(0.996047\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −15.5000 + 26.8468i −0.921379 + 1.59588i −0.124096 + 0.992270i \(0.539603\pi\)
−0.797283 + 0.603606i \(0.793730\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00000 5.19615i 0.0590281 0.306719i
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 7.00000 12.1244i 0.410347 0.710742i
\(292\) 0 0
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 + 1.73205i −0.0580259 + 0.100504i
\(298\) 0 0
\(299\) −5.00000 8.66025i −0.289157 0.500835i
\(300\) 0 0
\(301\) 3.50000 18.1865i 0.201737 1.04825i
\(302\) 0 0
\(303\) −9.00000 15.5885i −0.517036 0.895533i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 15.0000 25.9808i 0.850572 1.47323i −0.0301210 0.999546i \(-0.509589\pi\)
0.880693 0.473688i \(-0.157077\pi\)
\(312\) 0 0
\(313\) 8.50000 + 14.7224i 0.480448 + 0.832161i 0.999748 0.0224310i \(-0.00714060\pi\)
−0.519300 + 0.854592i \(0.673807\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.0000 17.3205i −0.561656 0.972817i −0.997352 0.0727229i \(-0.976831\pi\)
0.435696 0.900094i \(-0.356502\pi\)
\(318\) 0 0
\(319\) −8.00000 + 13.8564i −0.447914 + 0.775810i
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 12.5000 21.6506i 0.693375 1.20096i
\(326\) 0 0
\(327\) −0.500000 0.866025i −0.0276501 0.0478913i
\(328\) 0 0
\(329\) 20.0000 6.92820i 1.10264 0.381964i
\(330\) 0 0
\(331\) 7.50000 + 12.9904i 0.412237 + 0.714016i 0.995134 0.0985303i \(-0.0314141\pi\)
−0.582897 + 0.812546i \(0.698081\pi\)
\(332\) 0 0
\(333\) 2.50000 4.33013i 0.136999 0.237289i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 6.00000 10.3923i 0.325875 0.564433i
\(340\) 0 0
\(341\) 1.00000 + 1.73205i 0.0541530 + 0.0937958i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.0000 25.9808i 0.805242 1.39472i −0.110885 0.993833i \(-0.535369\pi\)
0.916127 0.400887i \(-0.131298\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −12.0000 + 20.7846i −0.638696 + 1.10625i 0.347024 + 0.937856i \(0.387192\pi\)
−0.985719 + 0.168397i \(0.946141\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.00000 1.73205i 0.264628 0.0916698i
\(358\) 0 0
\(359\) −16.0000 27.7128i −0.844448 1.46263i −0.886100 0.463494i \(-0.846596\pi\)
0.0416523 0.999132i \(-0.486738\pi\)
\(360\) 0 0
\(361\) 5.00000 8.66025i 0.263158 0.455803i
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.50000 + 14.7224i −0.443696 + 0.768505i −0.997960 0.0638362i \(-0.979666\pi\)
0.554264 + 0.832341i \(0.313000\pi\)
\(368\) 0 0
\(369\) −1.00000 1.73205i −0.0520579 0.0901670i
\(370\) 0 0
\(371\) −4.00000 3.46410i −0.207670 0.179847i
\(372\) 0 0
\(373\) 10.5000 + 18.1865i 0.543669 + 0.941663i 0.998689 + 0.0511818i \(0.0162988\pi\)
−0.455020 + 0.890481i \(0.650368\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 40.0000 2.06010
\(378\) 0 0
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) −5.50000 + 9.52628i −0.281774 + 0.488046i
\(382\) 0 0
\(383\) 11.0000 + 19.0526i 0.562074 + 0.973540i 0.997315 + 0.0732266i \(0.0233296\pi\)
−0.435242 + 0.900314i \(0.643337\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.50000 6.06218i −0.177915 0.308158i
\(388\) 0 0
\(389\) 12.0000 20.7846i 0.608424 1.05382i −0.383076 0.923717i \(-0.625135\pi\)
0.991500 0.130105i \(-0.0415314\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.50000 + 11.2583i 0.326226 + 0.565039i 0.981760 0.190126i \(-0.0608897\pi\)
−0.655534 + 0.755166i \(0.727556\pi\)
\(398\) 0 0
\(399\) 1.50000 7.79423i 0.0750939 0.390199i
\(400\) 0 0
\(401\) −5.00000 8.66025i −0.249688 0.432472i 0.713751 0.700399i \(-0.246995\pi\)
−0.963439 + 0.267927i \(0.913661\pi\)
\(402\) 0 0
\(403\) 2.50000 4.33013i 0.124534 0.215699i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 17.5000 30.3109i 0.865319 1.49878i −0.00141047 0.999999i \(-0.500449\pi\)
0.866730 0.498778i \(-0.166218\pi\)
\(410\) 0 0
\(411\) −2.00000 3.46410i −0.0986527 0.170872i
\(412\) 0 0
\(413\) 20.0000 + 17.3205i 0.984136 + 0.852286i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.500000 0.866025i 0.0244851 0.0424094i
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 0 0
\(423\) 4.00000 6.92820i 0.194487 0.336861i
\(424\) 0 0
\(425\) −5.00000 8.66025i −0.242536 0.420084i
\(426\) 0 0
\(427\) 5.00000 1.73205i 0.241967 0.0838198i
\(428\) 0 0
\(429\) −5.00000 8.66025i −0.241402 0.418121i
\(430\) 0 0
\(431\) −10.0000 + 17.3205i −0.481683 + 0.834300i −0.999779 0.0210230i \(-0.993308\pi\)
0.518096 + 0.855323i \(0.326641\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.00000 + 5.19615i −0.143509 + 0.248566i
\(438\) 0 0
\(439\) 12.0000 + 20.7846i 0.572729 + 0.991995i 0.996284 + 0.0861252i \(0.0274485\pi\)
−0.423556 + 0.905870i \(0.639218\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) −3.00000 5.19615i −0.142534 0.246877i 0.785916 0.618333i \(-0.212192\pi\)
−0.928450 + 0.371457i \(0.878858\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −2.00000 + 3.46410i −0.0941763 + 0.163118i
\(452\) 0 0
\(453\) −4.00000 6.92820i −0.187936 0.325515i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.5000 + 19.9186i 0.537947 + 0.931752i 0.999014 + 0.0443868i \(0.0141334\pi\)
−0.461067 + 0.887365i \(0.652533\pi\)
\(458\) 0 0
\(459\) 1.00000 1.73205i 0.0466760 0.0808452i
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) 9.00000 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.00000 10.3923i −0.277647 0.480899i 0.693153 0.720791i \(-0.256221\pi\)
−0.970799 + 0.239892i \(0.922888\pi\)
\(468\) 0 0
\(469\) −22.0000 19.0526i −1.01587 0.879765i
\(470\) 0 0
\(471\) 3.00000 + 5.19615i 0.138233 + 0.239426i
\(472\) 0 0
\(473\) −7.00000 + 12.1244i −0.321860 + 0.557478i
\(474\) 0 0
\(475\) −15.0000 −0.688247
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −9.00000 + 15.5885i −0.411220 + 0.712255i −0.995023 0.0996406i \(-0.968231\pi\)
0.583803 + 0.811895i \(0.301564\pi\)
\(480\) 0 0
\(481\) 12.5000 + 21.6506i 0.569951 + 0.987184i
\(482\) 0 0
\(483\) 1.00000 5.19615i 0.0455016 0.236433i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10.5000 + 18.1865i −0.475800 + 0.824110i −0.999616 0.0277214i \(-0.991175\pi\)
0.523815 + 0.851832i \(0.324508\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 0 0
\(493\) 8.00000 13.8564i 0.360302 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 31.1769i 0.269137 1.39848i
\(498\) 0 0
\(499\) 20.5000 + 35.5070i 0.917706 + 1.58951i 0.802890 + 0.596127i \(0.203294\pi\)
0.114816 + 0.993387i \(0.463372\pi\)
\(500\) 0 0
\(501\) −10.0000 + 17.3205i −0.446767 + 0.773823i
\(502\) 0 0
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.00000 + 10.3923i −0.266469 + 0.461538i
\(508\) 0 0
\(509\) 2.00000 + 3.46410i 0.0886484 + 0.153544i 0.906940 0.421260i \(-0.138412\pi\)
−0.818292 + 0.574803i \(0.805079\pi\)
\(510\) 0 0
\(511\) −6.00000 5.19615i −0.265424 0.229864i
\(512\) 0 0
\(513\) −1.50000 2.59808i −0.0662266 0.114708i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −16.0000 + 27.7128i −0.700973 + 1.21412i 0.267153 + 0.963654i \(0.413917\pi\)
−0.968125 + 0.250466i \(0.919416\pi\)
\(522\) 0 0
\(523\) 5.50000 + 9.52628i 0.240498 + 0.416555i 0.960856 0.277047i \(-0.0893559\pi\)
−0.720358 + 0.693602i \(0.756023\pi\)
\(524\) 0 0
\(525\) 12.5000 4.33013i 0.545545 0.188982i
\(526\) 0 0
\(527\) −1.00000 1.73205i −0.0435607 0.0754493i
\(528\) 0 0
\(529\) 9.50000 16.4545i 0.413043 0.715412i
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −6.00000 10.3923i −0.258919 0.448461i
\(538\) 0 0
\(539\) 11.0000 8.66025i 0.473804 0.373024i
\(540\) 0 0
\(541\) 13.5000 + 23.3827i 0.580410 + 1.00530i 0.995431 + 0.0954880i \(0.0304412\pi\)
−0.415020 + 0.909812i \(0.636226\pi\)
\(542\) 0 0
\(543\) 7.50000 12.9904i 0.321856 0.557471i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 0 0
\(549\) 1.00000 1.73205i 0.0426790 0.0739221i
\(550\) 0 0
\(551\) −12.0000 20.7846i −0.511217 0.885454i
\(552\) 0 0
\(553\) −42.5000 + 14.7224i −1.80728 + 0.626061i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.00000 + 15.5885i −0.381342 + 0.660504i −0.991254 0.131965i \(-0.957871\pi\)
0.609912 + 0.792469i \(0.291205\pi\)
\(558\) 0 0
\(559\) 35.0000 1.48034
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 10.0000 17.3205i 0.421450 0.729972i −0.574632 0.818412i \(-0.694855\pi\)
0.996082 + 0.0884397i \(0.0281881\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000 + 1.73205i 0.0839921 + 0.0727393i
\(568\) 0 0
\(569\) −21.0000 36.3731i −0.880366 1.52484i −0.850935 0.525271i \(-0.823964\pi\)
−0.0294311 0.999567i \(-0.509370\pi\)
\(570\) 0 0
\(571\) −11.5000 + 19.9186i −0.481260 + 0.833567i −0.999769 0.0215055i \(-0.993154\pi\)
0.518509 + 0.855072i \(0.326487\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −10.0000 −0.417029
\(576\) 0 0
\(577\) −3.50000 + 6.06218i −0.145707 + 0.252372i −0.929636 0.368478i \(-0.879879\pi\)
0.783930 + 0.620850i \(0.213212\pi\)
\(578\) 0 0
\(579\) −11.5000 19.9186i −0.477924 0.827788i
\(580\) 0 0
\(581\) −8.00000 + 41.5692i −0.331896 + 1.72458i
\(582\) 0 0
\(583\) 2.00000 + 3.46410i 0.0828315 + 0.143468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 9.00000 15.5885i 0.370211 0.641223i
\(592\) 0 0
\(593\) −15.0000 25.9808i −0.615976 1.06690i −0.990212 0.139569i \(-0.955428\pi\)
0.374236 0.927333i \(-0.377905\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.00000 3.46410i −0.0818546 0.141776i
\(598\) 0 0
\(599\) 18.0000 31.1769i 0.735460 1.27385i −0.219061 0.975711i \(-0.570299\pi\)
0.954521 0.298143i \(-0.0963673\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) −11.0000 −0.447955
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −16.5000 28.5788i −0.669714 1.15998i −0.977984 0.208680i \(-0.933083\pi\)
0.308270 0.951299i \(-0.400250\pi\)
\(608\) 0 0
\(609\) 16.0000 + 13.8564i 0.648353 + 0.561490i
\(610\) 0 0
\(611\) 20.0000 + 34.6410i 0.809113 + 1.40143i
\(612\) 0 0
\(613\) −13.0000 + 22.5167i −0.525065 + 0.909439i 0.474509 + 0.880251i \(0.342626\pi\)
−0.999574 + 0.0291886i \(0.990708\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −2.50000 + 4.33013i −0.100483 + 0.174042i −0.911884 0.410448i \(-0.865372\pi\)
0.811400 + 0.584491i \(0.198706\pi\)
\(620\) 0 0
\(621\) −1.00000 1.73205i −0.0401286 0.0695048i
\(622\) 0 0
\(623\) −30.0000 + 10.3923i −1.20192 + 0.416359i
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) −3.00000 + 5.19615i −0.119808 + 0.207514i
\(628\) 0 0
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) −10.0000 + 17.3205i −0.397464 + 0.688428i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −32.5000 12.9904i −1.28770 0.514698i
\(638\) 0 0
\(639\) −6.00000 10.3923i −0.237356 0.411113i
\(640\) 0 0
\(641\) −13.0000 + 22.5167i −0.513469 + 0.889355i 0.486409 + 0.873731i \(0.338307\pi\)
−0.999878 + 0.0156233i \(0.995027\pi\)
\(642\) 0 0
\(643\) 11.0000 0.433798 0.216899 0.976194i \(-0.430406\pi\)
0.216899 + 0.976194i \(0.430406\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000 15.5885i 0.353827 0.612845i −0.633090 0.774078i \(-0.718214\pi\)
0.986916 + 0.161233i \(0.0515470\pi\)
\(648\) 0 0
\(649\) −10.0000 17.3205i −0.392534 0.679889i
\(650\) 0 0
\(651\) 2.50000 0.866025i 0.0979827 0.0339422i
\(652\) 0 0
\(653\) 7.00000 + 12.1244i 0.273931 + 0.474463i 0.969865 0.243643i \(-0.0783426\pi\)
−0.695934 + 0.718106i \(0.745009\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.00000 −0.117041
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −1.50000 + 2.59808i −0.0583432 + 0.101053i −0.893722 0.448622i \(-0.851915\pi\)
0.835379 + 0.549675i \(0.185248\pi\)
\(662\) 0 0
\(663\) 5.00000 + 8.66025i 0.194184 + 0.336336i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 13.8564i −0.309761 0.536522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −21.0000 −0.809491 −0.404745 0.914429i \(-0.632640\pi\)
−0.404745 + 0.914429i \(0.632640\pi\)
\(674\) 0 0
\(675\) 2.50000 4.33013i 0.0962250 0.166667i
\(676\) 0 0
\(677\) 6.00000 + 10.3923i 0.230599 + 0.399409i 0.957984 0.286820i \(-0.0925982\pi\)
−0.727386 + 0.686229i \(0.759265\pi\)
\(678\) 0 0
\(679\) −7.00000 + 36.3731i −0.268635 + 1.39587i
\(680\) 0 0
\(681\) −13.0000 22.5167i −0.498161 0.862840i
\(682\) 0 0
\(683\) 15.0000 25.9808i 0.573959 0.994126i −0.422195 0.906505i \(-0.638740\pi\)
0.996154 0.0876211i \(-0.0279265\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.0000 0.495981
\(688\) 0 0
\(689\) 5.00000 8.66025i 0.190485 0.329929i
\(690\) 0 0
\(691\) −2.50000 4.33013i −0.0951045 0.164726i 0.814548 0.580097i \(-0.196985\pi\)
−0.909652 + 0.415371i \(0.863652\pi\)
\(692\) 0 0
\(693\) 1.00000 5.19615i 0.0379869 0.197386i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.00000 3.46410i 0.0757554 0.131212i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 7.50000 12.9904i 0.282868 0.489942i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000 + 31.1769i 1.35392 + 1.17253i
\(708\) 0 0
\(709\) 11.0000 + 19.0526i 0.413114 + 0.715534i 0.995228 0.0975728i \(-0.0311079\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) −8.50000 + 14.7224i −0.318775 + 0.552134i
\(712\) 0 0
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −15.0000 + 25.9808i −0.560185 + 0.970269i
\(718\) 0 0
\(719\) 3.00000 + 5.19615i 0.111881 + 0.193784i 0.916529 0.399969i \(-0.130979\pi\)
−0.804648 + 0.593753i \(0.797646\pi\)
\(720\) 0 0
\(721\) 2.50000 0.866025i 0.0931049 0.0322525i
\(722\) 0 0
\(723\) 11.0000 + 19.0526i 0.409094 + 0.708572i
\(724\) 0 0
\(725\) 20.0000 34.6410i 0.742781 1.28654i
\(726\) 0 0
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.00000 12.1244i 0.258904 0.448435i
\(732\) 0 0
\(733\) 9.50000 + 16.4545i 0.350891 + 0.607760i 0.986406 0.164328i \(-0.0525456\pi\)
−0.635515 + 0.772088i \(0.719212\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.0000 + 19.0526i 0.405190 + 0.701810i
\(738\) 0 0
\(739\) 8.50000 14.7224i 0.312678 0.541573i −0.666264 0.745716i \(-0.732107\pi\)
0.978941 + 0.204143i \(0.0654407\pi\)
\(740\) 0 0
\(741\) 15.0000 0.551039
\(742\) 0 0
\(743\) −14.0000 −0.513610 −0.256805 0.966463i \(-0.582670\pi\)
−0.256805 + 0.966463i \(0.582670\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.00000 + 13.8564i 0.292705 + 0.506979i
\(748\) 0 0
\(749\) −5.00000 + 1.73205i −0.182696 + 0.0632878i
\(750\) 0 0
\(751\) −13.5000 23.3827i −0.492622 0.853246i 0.507342 0.861745i \(-0.330628\pi\)
−0.999964 + 0.00849853i \(0.997295\pi\)
\(752\) 0 0
\(753\) 11.0000 19.0526i 0.400862 0.694314i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) −2.00000 + 3.46410i −0.0725954 + 0.125739i
\(760\) 0 0
\(761\) −15.0000 25.9808i −0.543750 0.941802i −0.998684 0.0512772i \(-0.983671\pi\)
0.454935 0.890525i \(-0.349663\pi\)
\(762\) 0 0
\(763\) 2.00000 + 1.73205i 0.0724049 + 0.0627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25.0000 + 43.3013i −0.902698 + 1.56352i
\(768\) 0 0
\(769\) −9.00000 −0.324548 −0.162274 0.986746i \(-0.551883\pi\)
−0.162274 + 0.986746i \(0.551883\pi\)
\(770\) 0 0
\(771\) −16.0000 −0.576226
\(772\) 0 0
\(773\) −9.00000 + 15.5885i −0.323708 + 0.560678i −0.981250 0.192740i \(-0.938263\pi\)
0.657542 + 0.753418i \(0.271596\pi\)
\(774\) 0 0
\(775\) −2.50000 4.33013i −0.0898027 0.155543i
\(776\) 0 0
\(777\) −2.50000 + 12.9904i −0.0896870 + 0.466027i
\(778\) 0 0
\(779\) −3.00000 5.19615i −0.107486 0.186171i
\(780\) 0 0
\(781\) −12.0000 + 20.7846i −0.429394 + 0.743732i
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 20.0000 34.6410i 0.712923 1.23482i −0.250832 0.968031i \(-0.580704\pi\)
0.963755 0.266788i \(-0.0859624\pi\)
\(788\) 0 0
\(789\) −4.00000 6.92820i −0.142404 0.246651i
\(790\) 0 0
\(791\) −6.00000 + 31.1769i −0.213335 + 1.10852i
\(792\) 0 0
\(793\) 5.00000 + 8.66025i 0.177555 + 0.307535i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −6.00000 + 10.3923i −0.212000 + 0.367194i
\(802\) 0 0
\(803\) 3.00000 + 5.19615i 0.105868 + 0.183368i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000 + 15.5885i 0.316815 + 0.548740i
\(808\) 0 0
\(809\) −2.00000 + 3.46410i −0.0703163 + 0.121791i −0.899040 0.437867i \(-0.855734\pi\)
0.828724 + 0.559658i \(0.189068\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.5000 18.1865i −0.367348 0.636266i
\(818\) 0 0
\(819\) −12.5000 + 4.33013i −0.436785 + 0.151307i
\(820\) 0 0
\(821\) −8.00000 13.8564i −0.279202 0.483592i 0.691985 0.721912i \(-0.256737\pi\)
−0.971187 + 0.238320i \(0.923403\pi\)
\(822\) 0 0
\(823\) 20.0000 34.6410i 0.697156 1.20751i −0.272292 0.962215i \(-0.587782\pi\)
0.969448 0.245295i \(-0.0788849\pi\)
\(824\) 0 0
\(825\) −10.0000 −0.348155
\(826\) 0 0
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 0 0
\(829\) 13.5000 23.3827i 0.468874 0.812114i −0.530493 0.847690i \(-0.677993\pi\)
0.999367 + 0.0355753i \(0.0113264\pi\)
\(830\) 0 0
\(831\) −8.50000 14.7224i −0.294862 0.510716i
\(832\) 0 0
\(833\) −11.0000 + 8.66025i −0.381127 + 0.300060i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.500000 0.866025i 0.0172825 0.0299342i
\(838\) 0 0
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −9.00000 + 15.5885i −0.309976 + 0.536895i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 17.5000 6.06218i 0.601307 0.208299i
\(848\) 0 0
\(849\) −15.5000 26.8468i −0.531959 0.921379i
\(850\) 0 0
\(851\) 5.00000 8.66025i 0.171398 0.296870i
\(852\) 0 0
\(853\) 39.0000 1.33533 0.667667 0.744460i \(-0.267293\pi\)
0.667667 + 0.744460i \(0.267293\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.0000 + 20.7846i −0.409912 + 0.709989i −0.994880 0.101068i \(-0.967774\pi\)
0.584967 + 0.811057i \(0.301107\pi\)
\(858\) 0 0
\(859\) −2.00000 3.46410i −0.0682391 0.118194i 0.829887 0.557931i \(-0.188405\pi\)
−0.898126 + 0.439738i \(0.855071\pi\)
\(860\) 0 0
\(861\) 4.00000 + 3.46410i 0.136320 + 0.118056i
\(862\) 0 0
\(863\) 3.00000 + 5.19615i 0.102121 + 0.176879i 0.912558 0.408946i \(-0.134104\pi\)
−0.810437 + 0.585826i \(0.800770\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 34.0000 1.15337
\(870\) 0 0
\(871\) 27.5000 47.6314i 0.931802 1.61393i
\(872\) 0 0
\(873\) 7.00000 + 12.1244i 0.236914 + 0.410347i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.0000 + 32.9090i 0.641584 + 1.11126i 0.985079 + 0.172102i \(0.0550559\pi\)
−0.343495 + 0.939155i \(0.611611\pi\)
\(878\) 0 0
\(879\) 2.00000 3.46410i 0.0674583 0.116841i
\(880\) 0 0
\(881\) 58.0000 1.95407 0.977035 0.213080i \(-0.0683494\pi\)
0.977035 + 0.213080i \(0.0683494\pi\)
\(882\) 0 0
\(883\) −43.0000 −1.44707 −0.723533 0.690290i \(-0.757483\pi\)
−0.723533 + 0.690290i \(0.757483\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.00000 + 1.73205i 0.0335767 + 0.0581566i 0.882325 0.470640i \(-0.155977\pi\)
−0.848749 + 0.528796i \(0.822644\pi\)
\(888\) 0 0
\(889\) 5.50000 28.5788i 0.184464 0.958503i
\(890\) 0 0
\(891\) −1.00000 1.73205i −0.0335013 0.0580259i
\(892\) 0 0
\(893\) 12.0000 20.7846i 0.401565 0.695530i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 10.0000 0.333890
\(898\) 0 0
\(899\) 4.00000 6.92820i 0.133407 0.231069i
\(900\) 0 0
\(901\) −2.00000 3.46410i −0.0666297 0.115406i
\(902\) 0 0
\(903\) 14.0000 + 12.1244i 0.465891 + 0.403473i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.50000 6.06218i 0.116216 0.201291i −0.802049 0.597258i \(-0.796257\pi\)
0.918265 + 0.395966i \(0.129590\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −50.0000 −1.65657 −0.828287 0.560304i \(-0.810684\pi\)
−0.828287 + 0.560304i \(0.810684\pi\)
\(912\) 0 0
\(913\) 16.0000 27.7128i 0.529523 0.917160i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.0000 5.19615i 0.495344 0.171592i
\(918\) 0 0
\(919\) 27.5000 + 47.6314i 0.907141 + 1.57121i 0.818017 + 0.575194i \(0.195074\pi\)
0.0891245 + 0.996020i \(0.471593\pi\)
\(920\) 0 0
\(921\) 11.5000 19.9186i 0.378938 0.656340i
\(922\) 0 0
\(923\) 60.0000 1.97492
\(924\) 0 0
\(925\) 25.0000 0.821995
\(926\) 0 0
\(927\) 0.500000 0.866025i 0.0164222 0.0284440i
\(928\) 0 0
\(929\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 0 0
\(931\) 3.00000 + 20.7846i 0.0983210 + 0.681188i
\(932\) 0 0
\(933\) 15.0000 + 25.9808i 0.491078 + 0.850572i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13.0000 −0.424691 −0.212346 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) −17.0000 −0.554774
\(940\) 0 0
\(941\) −1.00000 + 1.73205i −0.0325991 + 0.0564632i −0.881865 0.471503i \(-0.843712\pi\)
0.849266 + 0.527966i \(0.177045\pi\)
\(942\) 0 0
\(943\) −2.00000 3.46410i −0.0651290 0.112807i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.0000 + 29.4449i 0.552426 + 0.956830i 0.998099 + 0.0616337i \(0.0196311\pi\)
−0.445673 + 0.895196i \(0.647036\pi\)
\(948\) 0 0
\(949\) 7.50000 12.9904i 0.243460 0.421686i
\(950\) 0 0
\(951\) 20.0000 0.648544
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.00000 13.8564i −0.258603 0.447914i
\(958\) 0 0
\(959\) 8.00000 + 6.92820i 0.258333 + 0.223723i
\(960\) 0 0
\(961\) 15.0000 + 25.9808i 0.483871 + 0.838089i
\(962\) 0 0
\(963\) −1.00000 + 1.73205i −0.0322245 + 0.0558146i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −47.0000 −1.51142 −0.755709 0.654907i \(-0.772708\pi\)
−0.755709 + 0.654907i \(0.772708\pi\)
\(968\) 0 0
\(969\) 3.00000 5.19615i 0.0963739 0.166924i
\(970\) 0 0
\(971\) −3.00000 5.19615i −0.0962746 0.166752i 0.813865 0.581054i \(-0.197359\pi\)
−0.910140 + 0.414301i \(0.864026\pi\)
\(972\) 0 0
\(973\) −0.500000 + 2.59808i −0.0160293 + 0.0832905i
\(974\) 0 0
\(975\) 12.5000 + 21.6506i 0.400320 + 0.693375i
\(976\) 0 0
\(977\) 17.0000 29.4449i 0.543878 0.942025i −0.454798 0.890594i \(-0.650289\pi\)
0.998677 0.0514302i \(-0.0163780\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 0 0
\(983\) 3.00000 5.19615i 0.0956851 0.165732i −0.814209 0.580572i \(-0.802829\pi\)
0.909894 + 0.414840i \(0.136162\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.00000 + 20.7846i −0.127321 + 0.661581i
\(988\) 0 0
\(989\) −7.00000 12.1244i −0.222587 0.385532i
\(990\) 0 0
\(991\) 6.50000 11.2583i 0.206479 0.357633i −0.744124 0.668042i \(-0.767133\pi\)
0.950603 + 0.310409i \(0.100466\pi\)
\(992\) 0 0
\(993\) −15.0000 −0.476011
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18.5000 32.0429i 0.585901 1.01481i −0.408862 0.912596i \(-0.634074\pi\)
0.994762 0.102214i \(-0.0325925\pi\)
\(998\) 0 0
\(999\) 2.50000 + 4.33013i 0.0790965 + 0.136999i
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 672.2.q.c.193.1 2
3.2 odd 2 2016.2.s.g.865.1 2
4.3 odd 2 672.2.q.i.193.1 yes 2
7.2 even 3 inner 672.2.q.c.289.1 yes 2
7.3 odd 6 4704.2.a.i.1.1 1
7.4 even 3 4704.2.a.bb.1.1 1
8.3 odd 2 1344.2.q.f.193.1 2
8.5 even 2 1344.2.q.p.193.1 2
12.11 even 2 2016.2.s.f.865.1 2
21.2 odd 6 2016.2.s.g.289.1 2
28.3 even 6 4704.2.a.x.1.1 1
28.11 odd 6 4704.2.a.h.1.1 1
28.23 odd 6 672.2.q.i.289.1 yes 2
56.3 even 6 9408.2.a.y.1.1 1
56.11 odd 6 9408.2.a.cm.1.1 1
56.37 even 6 1344.2.q.p.961.1 2
56.45 odd 6 9408.2.a.cj.1.1 1
56.51 odd 6 1344.2.q.f.961.1 2
56.53 even 6 9408.2.a.s.1.1 1
84.23 even 6 2016.2.s.f.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.c.193.1 2 1.1 even 1 trivial
672.2.q.c.289.1 yes 2 7.2 even 3 inner
672.2.q.i.193.1 yes 2 4.3 odd 2
672.2.q.i.289.1 yes 2 28.23 odd 6
1344.2.q.f.193.1 2 8.3 odd 2
1344.2.q.f.961.1 2 56.51 odd 6
1344.2.q.p.193.1 2 8.5 even 2
1344.2.q.p.961.1 2 56.37 even 6
2016.2.s.f.289.1 2 84.23 even 6
2016.2.s.f.865.1 2 12.11 even 2
2016.2.s.g.289.1 2 21.2 odd 6
2016.2.s.g.865.1 2 3.2 odd 2
4704.2.a.h.1.1 1 28.11 odd 6
4704.2.a.i.1.1 1 7.3 odd 6
4704.2.a.x.1.1 1 28.3 even 6
4704.2.a.bb.1.1 1 7.4 even 3
9408.2.a.s.1.1 1 56.53 even 6
9408.2.a.y.1.1 1 56.3 even 6
9408.2.a.cj.1.1 1 56.45 odd 6
9408.2.a.cm.1.1 1 56.11 odd 6