Properties

Label 1344.2.q.v.193.1
Level $1344$
Weight $2$
Character 1344.193
Analytic conductor $10.732$
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] = [1344,2,Mod(193,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, 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("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.q (of order \(3\), degree \(2\), 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: \(10.7318940317\)
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: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1344.193
Dual form 1344.2.q.v.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(1.50000 + 2.59808i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(1.50000 + 2.59808i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{11} +4.00000 q^{13} +3.00000 q^{15} +(-2.00000 - 3.46410i) q^{19} +(0.500000 - 2.59808i) q^{21} +(-2.00000 + 3.46410i) q^{25} -1.00000 q^{27} -9.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(-1.50000 - 2.59808i) q^{33} +(6.00000 + 5.19615i) q^{35} +(4.00000 + 6.92820i) q^{37} +(2.00000 - 3.46410i) q^{39} +10.0000 q^{43} +(1.50000 - 2.59808i) q^{45} +(3.00000 + 5.19615i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-1.50000 + 2.59808i) q^{53} +9.00000 q^{55} -4.00000 q^{57} +(1.50000 - 2.59808i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(-2.00000 - 1.73205i) q^{63} +(6.00000 + 10.3923i) q^{65} +(-5.00000 + 8.66025i) q^{67} -6.00000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(2.00000 + 3.46410i) q^{75} +(1.50000 - 7.79423i) q^{77} +(0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +9.00000 q^{83} +(-4.50000 + 7.79423i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(10.0000 - 3.46410i) q^{91} +(-0.500000 - 0.866025i) q^{93} +(6.00000 - 10.3923i) q^{95} -1.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 3 q^{5} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 3 q^{5} + 5 q^{7} - q^{9} + 3 q^{11} + 8 q^{13} + 6 q^{15} - 4 q^{19} + q^{21} - 4 q^{25} - 2 q^{27} - 18 q^{29} + q^{31} - 3 q^{33} + 12 q^{35} + 8 q^{37} + 4 q^{39} + 20 q^{43} + 3 q^{45} + 6 q^{47} + 11 q^{49} - 3 q^{53} + 18 q^{55} - 8 q^{57} + 3 q^{59} - 10 q^{61} - 4 q^{63} + 12 q^{65} - 10 q^{67} - 12 q^{71} - 2 q^{73} + 4 q^{75} + 3 q^{77} + q^{79} - q^{81} + 18 q^{83} - 9 q^{87} - 6 q^{89} + 20 q^{91} - q^{93} + 12 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) 1.50000 + 2.59808i 0.670820 + 1.16190i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 2.50000 0.866025i 0.944911 0.327327i
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 1.50000 2.59808i 0.452267 0.783349i −0.546259 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −2.00000 3.46410i −0.458831 0.794719i 0.540068 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) 0.500000 2.59808i 0.109109 0.566947i
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 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
\(33\) −1.50000 2.59808i −0.261116 0.452267i
\(34\) 0 0
\(35\) 6.00000 + 5.19615i 1.01419 + 0.878310i
\(36\) 0 0
\(37\) 4.00000 + 6.92820i 0.657596 + 1.13899i 0.981236 + 0.192809i \(0.0617599\pi\)
−0.323640 + 0.946180i \(0.604907\pi\)
\(38\) 0 0
\(39\) 2.00000 3.46410i 0.320256 0.554700i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 1.50000 2.59808i 0.223607 0.387298i
\(46\) 0 0
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.50000 + 2.59808i −0.206041 + 0.356873i −0.950464 0.310835i \(-0.899391\pi\)
0.744423 + 0.667708i \(0.232725\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 1.50000 2.59808i 0.195283 0.338241i −0.751710 0.659494i \(-0.770771\pi\)
0.946993 + 0.321253i \(0.104104\pi\)
\(60\) 0 0
\(61\) −5.00000 8.66025i −0.640184 1.10883i −0.985391 0.170305i \(-0.945525\pi\)
0.345207 0.938527i \(-0.387809\pi\)
\(62\) 0 0
\(63\) −2.00000 1.73205i −0.251976 0.218218i
\(64\) 0 0
\(65\) 6.00000 + 10.3923i 0.744208 + 1.28901i
\(66\) 0 0
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 2.00000 + 3.46410i 0.230940 + 0.400000i
\(76\) 0 0
\(77\) 1.50000 7.79423i 0.170941 0.888235i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.50000 + 7.79423i −0.482451 + 0.835629i
\(88\) 0 0
\(89\) −3.00000 5.19615i −0.317999 0.550791i 0.662071 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 0 0
\(91\) 10.0000 3.46410i 1.04828 0.363137i
\(92\) 0 0
\(93\) −0.500000 0.866025i −0.0518476 0.0898027i
\(94\) 0 0
\(95\) 6.00000 10.3923i 0.615587 1.06623i
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 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
\(103\) −4.00000 6.92820i −0.394132 0.682656i 0.598858 0.800855i \(-0.295621\pi\)
−0.992990 + 0.118199i \(0.962288\pi\)
\(104\) 0 0
\(105\) 7.50000 2.59808i 0.731925 0.253546i
\(106\) 0 0
\(107\) −1.50000 2.59808i −0.145010 0.251166i 0.784366 0.620298i \(-0.212988\pi\)
−0.929377 + 0.369132i \(0.879655\pi\)
\(108\) 0 0
\(109\) 7.00000 12.1244i 0.670478 1.16130i −0.307290 0.951616i \(-0.599422\pi\)
0.977769 0.209687i \(-0.0672444\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 3.46410i −0.184900 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 0 0
\(129\) 5.00000 8.66025i 0.440225 0.762493i
\(130\) 0 0
\(131\) −4.50000 7.79423i −0.393167 0.680985i 0.599699 0.800226i \(-0.295287\pi\)
−0.992865 + 0.119241i \(0.961954\pi\)
\(132\) 0 0
\(133\) −8.00000 6.92820i −0.693688 0.600751i
\(134\) 0 0
\(135\) −1.50000 2.59808i −0.129099 0.223607i
\(136\) 0 0
\(137\) −9.00000 + 15.5885i −0.768922 + 1.33181i 0.169226 + 0.985577i \(0.445873\pi\)
−0.938148 + 0.346235i \(0.887460\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 6.00000 10.3923i 0.501745 0.869048i
\(144\) 0 0
\(145\) −13.5000 23.3827i −1.12111 1.94183i
\(146\) 0 0
\(147\) −1.00000 6.92820i −0.0824786 0.571429i
\(148\) 0 0
\(149\) 9.00000 + 15.5885i 0.737309 + 1.27706i 0.953703 + 0.300750i \(0.0972370\pi\)
−0.216394 + 0.976306i \(0.569430\pi\)
\(150\) 0 0
\(151\) 0.500000 0.866025i 0.0406894 0.0704761i −0.844963 0.534824i \(-0.820378\pi\)
0.885653 + 0.464348i \(0.153711\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) −2.00000 + 3.46410i −0.159617 + 0.276465i −0.934731 0.355357i \(-0.884359\pi\)
0.775113 + 0.631822i \(0.217693\pi\)
\(158\) 0 0
\(159\) 1.50000 + 2.59808i 0.118958 + 0.206041i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.00000 13.8564i −0.626608 1.08532i −0.988227 0.152992i \(-0.951109\pi\)
0.361619 0.932326i \(-0.382224\pi\)
\(164\) 0 0
\(165\) 4.50000 7.79423i 0.350325 0.606780i
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −2.00000 + 3.46410i −0.152944 + 0.264906i
\(172\) 0 0
\(173\) 9.00000 + 15.5885i 0.684257 + 1.18517i 0.973670 + 0.227964i \(0.0732068\pi\)
−0.289412 + 0.957205i \(0.593460\pi\)
\(174\) 0 0
\(175\) −2.00000 + 10.3923i −0.151186 + 0.785584i
\(176\) 0 0
\(177\) −1.50000 2.59808i −0.112747 0.195283i
\(178\) 0 0
\(179\) 6.00000 10.3923i 0.448461 0.776757i −0.549825 0.835280i \(-0.685306\pi\)
0.998286 + 0.0585225i \(0.0186389\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −12.0000 + 20.7846i −0.882258 + 1.52811i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2.50000 + 0.866025i −0.181848 + 0.0629941i
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) 9.50000 16.4545i 0.683825 1.18442i −0.289980 0.957033i \(-0.593649\pi\)
0.973805 0.227387i \(-0.0730182\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −10.0000 + 17.3205i −0.708881 + 1.22782i 0.256391 + 0.966573i \(0.417466\pi\)
−0.965272 + 0.261245i \(0.915867\pi\)
\(200\) 0 0
\(201\) 5.00000 + 8.66025i 0.352673 + 0.610847i
\(202\) 0 0
\(203\) −22.5000 + 7.79423i −1.57919 + 0.547048i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 0 0
\(213\) −3.00000 + 5.19615i −0.205557 + 0.356034i
\(214\) 0 0
\(215\) 15.0000 + 25.9808i 1.02299 + 1.77187i
\(216\) 0 0
\(217\) 0.500000 2.59808i 0.0339422 0.176369i
\(218\) 0 0
\(219\) 1.00000 + 1.73205i 0.0675737 + 0.117041i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −13.5000 + 23.3827i −0.896026 + 1.55196i −0.0634974 + 0.997982i \(0.520225\pi\)
−0.832529 + 0.553981i \(0.813108\pi\)
\(228\) 0 0
\(229\) −2.00000 3.46410i −0.132164 0.228914i 0.792347 0.610071i \(-0.208859\pi\)
−0.924510 + 0.381157i \(0.875526\pi\)
\(230\) 0 0
\(231\) −6.00000 5.19615i −0.394771 0.341882i
\(232\) 0 0
\(233\) 12.0000 + 20.7846i 0.786146 + 1.36165i 0.928312 + 0.371802i \(0.121260\pi\)
−0.142166 + 0.989843i \(0.545407\pi\)
\(234\) 0 0
\(235\) −9.00000 + 15.5885i −0.587095 + 1.01688i
\(236\) 0 0
\(237\) 1.00000 0.0649570
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.0322078 0.0557856i −0.849472 0.527633i \(-0.823079\pi\)
0.881680 + 0.471848i \(0.156413\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 19.5000 + 7.79423i 1.24581 + 0.497955i
\(246\) 0 0
\(247\) −8.00000 13.8564i −0.509028 0.881662i
\(248\) 0 0
\(249\) 4.50000 7.79423i 0.285176 0.493939i
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.00000 5.19615i −0.187135 0.324127i 0.757159 0.653231i \(-0.226587\pi\)
−0.944294 + 0.329104i \(0.893253\pi\)
\(258\) 0 0
\(259\) 16.0000 + 13.8564i 0.994192 + 0.860995i
\(260\) 0 0
\(261\) 4.50000 + 7.79423i 0.278543 + 0.482451i
\(262\) 0 0
\(263\) 3.00000 5.19615i 0.184988 0.320408i −0.758585 0.651575i \(-0.774109\pi\)
0.943572 + 0.331166i \(0.107442\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 10.5000 18.1865i 0.640196 1.10885i −0.345192 0.938532i \(-0.612186\pi\)
0.985389 0.170321i \(-0.0544803\pi\)
\(270\) 0 0
\(271\) −5.50000 9.52628i −0.334101 0.578680i 0.649211 0.760609i \(-0.275099\pi\)
−0.983312 + 0.181928i \(0.941766\pi\)
\(272\) 0 0
\(273\) 2.00000 10.3923i 0.121046 0.628971i
\(274\) 0 0
\(275\) 6.00000 + 10.3923i 0.361814 + 0.626680i
\(276\) 0 0
\(277\) 4.00000 6.92820i 0.240337 0.416275i −0.720473 0.693482i \(-0.756075\pi\)
0.960810 + 0.277207i \(0.0894088\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 7.00000 12.1244i 0.416107 0.720718i −0.579437 0.815017i \(-0.696728\pi\)
0.995544 + 0.0942988i \(0.0300609\pi\)
\(284\) 0 0
\(285\) −6.00000 10.3923i −0.355409 0.615587i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) −0.500000 + 0.866025i −0.0293105 + 0.0507673i
\(292\) 0 0
\(293\) −33.0000 −1.92788 −0.963940 0.266119i \(-0.914259\pi\)
−0.963940 + 0.266119i \(0.914259\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) −1.50000 + 2.59808i −0.0870388 + 0.150756i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 25.0000 8.66025i 1.44098 0.499169i
\(302\) 0 0
\(303\) 9.00000 + 15.5885i 0.517036 + 0.895533i
\(304\) 0 0
\(305\) 15.0000 25.9808i 0.858898 1.48765i
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −12.0000 + 20.7846i −0.680458 + 1.17859i 0.294384 + 0.955687i \(0.404886\pi\)
−0.974841 + 0.222900i \(0.928448\pi\)
\(312\) 0 0
\(313\) 15.5000 + 26.8468i 0.876112 + 1.51747i 0.855574 + 0.517681i \(0.173205\pi\)
0.0205381 + 0.999789i \(0.493462\pi\)
\(314\) 0 0
\(315\) 1.50000 7.79423i 0.0845154 0.439155i
\(316\) 0 0
\(317\) 4.50000 + 7.79423i 0.252745 + 0.437767i 0.964281 0.264883i \(-0.0853332\pi\)
−0.711535 + 0.702650i \(0.752000\pi\)
\(318\) 0 0
\(319\) −13.5000 + 23.3827i −0.755855 + 1.30918i
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −8.00000 + 13.8564i −0.443760 + 0.768615i
\(326\) 0 0
\(327\) −7.00000 12.1244i −0.387101 0.670478i
\(328\) 0 0
\(329\) 12.0000 + 10.3923i 0.661581 + 0.572946i
\(330\) 0 0
\(331\) 10.0000 + 17.3205i 0.549650 + 0.952021i 0.998298 + 0.0583130i \(0.0185721\pi\)
−0.448649 + 0.893708i \(0.648095\pi\)
\(332\) 0 0
\(333\) 4.00000 6.92820i 0.219199 0.379663i
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) 0 0
\(337\) −7.00000 −0.381314 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.50000 2.59808i −0.0812296 0.140694i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 10.3923i 0.322097 0.557888i −0.658824 0.752297i \(-0.728946\pi\)
0.980921 + 0.194409i \(0.0622790\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 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
\(355\) −9.00000 15.5885i −0.477670 0.827349i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 25.9808i −0.791670 1.37121i −0.924932 0.380131i \(-0.875879\pi\)
0.133263 0.991081i \(-0.457455\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 9.50000 16.4545i 0.495896 0.858917i −0.504093 0.863649i \(-0.668173\pi\)
0.999989 + 0.00473247i \(0.00150640\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.50000 + 7.79423i −0.0778761 + 0.404656i
\(372\) 0 0
\(373\) 4.00000 + 6.92820i 0.207112 + 0.358729i 0.950804 0.309794i \(-0.100260\pi\)
−0.743691 + 0.668523i \(0.766927\pi\)
\(374\) 0 0
\(375\) 1.50000 2.59808i 0.0774597 0.134164i
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 2.50000 4.33013i 0.128079 0.221839i
\(382\) 0 0
\(383\) −9.00000 15.5885i −0.459879 0.796533i 0.539076 0.842257i \(-0.318774\pi\)
−0.998954 + 0.0457244i \(0.985440\pi\)
\(384\) 0 0
\(385\) 22.5000 7.79423i 1.14671 0.397231i
\(386\) 0 0
\(387\) −5.00000 8.66025i −0.254164 0.440225i
\(388\) 0 0
\(389\) −3.00000 + 5.19615i −0.152106 + 0.263455i −0.932002 0.362454i \(-0.881939\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −9.00000 −0.453990
\(394\) 0 0
\(395\) −1.50000 + 2.59808i −0.0754732 + 0.130723i
\(396\) 0 0
\(397\) −2.00000 3.46410i −0.100377 0.173858i 0.811463 0.584404i \(-0.198672\pi\)
−0.911840 + 0.410546i \(0.865338\pi\)
\(398\) 0 0
\(399\) −10.0000 + 3.46410i −0.500626 + 0.173422i
\(400\) 0 0
\(401\) −12.0000 20.7846i −0.599251 1.03793i −0.992932 0.118686i \(-0.962132\pi\)
0.393680 0.919247i \(-0.371202\pi\)
\(402\) 0 0
\(403\) 2.00000 3.46410i 0.0996271 0.172559i
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 12.5000 21.6506i 0.618085 1.07056i −0.371750 0.928333i \(-0.621242\pi\)
0.989835 0.142222i \(-0.0454247\pi\)
\(410\) 0 0
\(411\) 9.00000 + 15.5885i 0.443937 + 0.768922i
\(412\) 0 0
\(413\) 1.50000 7.79423i 0.0738102 0.383529i
\(414\) 0 0
\(415\) 13.5000 + 23.3827i 0.662689 + 1.14781i
\(416\) 0 0
\(417\) −1.00000 + 1.73205i −0.0489702 + 0.0848189i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 3.00000 5.19615i 0.145865 0.252646i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −20.0000 17.3205i −0.967868 0.838198i
\(428\) 0 0
\(429\) −6.00000 10.3923i −0.289683 0.501745i
\(430\) 0 0
\(431\) −6.00000 + 10.3923i −0.289010 + 0.500580i −0.973574 0.228373i \(-0.926659\pi\)
0.684564 + 0.728953i \(0.259993\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) −27.0000 −1.29455
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −17.5000 30.3109i −0.835229 1.44666i −0.893843 0.448379i \(-0.852001\pi\)
0.0586141 0.998281i \(-0.481332\pi\)
\(440\) 0 0
\(441\) −6.50000 2.59808i −0.309524 0.123718i
\(442\) 0 0
\(443\) −16.5000 28.5788i −0.783939 1.35782i −0.929631 0.368492i \(-0.879874\pi\)
0.145692 0.989330i \(-0.453459\pi\)
\(444\) 0 0
\(445\) 9.00000 15.5885i 0.426641 0.738964i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.500000 0.866025i −0.0234920 0.0406894i
\(454\) 0 0
\(455\) 24.0000 + 20.7846i 1.12514 + 0.974398i
\(456\) 0 0
\(457\) 0.500000 + 0.866025i 0.0233890 + 0.0405110i 0.877483 0.479608i \(-0.159221\pi\)
−0.854094 + 0.520119i \(0.825888\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 1.50000 2.59808i 0.0695608 0.120483i
\(466\) 0 0
\(467\) 18.0000 + 31.1769i 0.832941 + 1.44270i 0.895696 + 0.444667i \(0.146678\pi\)
−0.0627555 + 0.998029i \(0.519989\pi\)
\(468\) 0 0
\(469\) −5.00000 + 25.9808i −0.230879 + 1.19968i
\(470\) 0 0
\(471\) 2.00000 + 3.46410i 0.0921551 + 0.159617i
\(472\) 0 0
\(473\) 15.0000 25.9808i 0.689701 1.19460i
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 9.00000 15.5885i 0.411220 0.712255i −0.583803 0.811895i \(-0.698436\pi\)
0.995023 + 0.0996406i \(0.0317693\pi\)
\(480\) 0 0
\(481\) 16.0000 + 27.7128i 0.729537 + 1.26360i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.50000 2.59808i −0.0681115 0.117973i
\(486\) 0 0
\(487\) −20.5000 + 35.5070i −0.928944 + 1.60898i −0.143851 + 0.989599i \(0.545949\pi\)
−0.785093 + 0.619378i \(0.787385\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.50000 7.79423i −0.202260 0.350325i
\(496\) 0 0
\(497\) −15.0000 + 5.19615i −0.672842 + 0.233079i
\(498\) 0 0
\(499\) 1.00000 + 1.73205i 0.0447661 + 0.0775372i 0.887540 0.460730i \(-0.152412\pi\)
−0.842774 + 0.538267i \(0.819079\pi\)
\(500\) 0 0
\(501\) 3.00000 5.19615i 0.134030 0.232147i
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −54.0000 −2.40297
\(506\) 0 0
\(507\) 1.50000 2.59808i 0.0666173 0.115385i
\(508\) 0 0
\(509\) −1.50000 2.59808i −0.0664863 0.115158i 0.830866 0.556473i \(-0.187846\pi\)
−0.897352 + 0.441315i \(0.854512\pi\)
\(510\) 0 0
\(511\) −1.00000 + 5.19615i −0.0442374 + 0.229864i
\(512\) 0 0
\(513\) 2.00000 + 3.46410i 0.0883022 + 0.152944i
\(514\) 0 0
\(515\) 12.0000 20.7846i 0.528783 0.915879i
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 9.00000 15.5885i 0.394297 0.682943i −0.598714 0.800963i \(-0.704321\pi\)
0.993011 + 0.118020i \(0.0376547\pi\)
\(522\) 0 0
\(523\) −2.00000 3.46410i −0.0874539 0.151475i 0.818980 0.573822i \(-0.194540\pi\)
−0.906434 + 0.422347i \(0.861206\pi\)
\(524\) 0 0
\(525\) 8.00000 + 6.92820i 0.349149 + 0.302372i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 4.50000 7.79423i 0.194552 0.336974i
\(536\) 0 0
\(537\) −6.00000 10.3923i −0.258919 0.448461i
\(538\) 0 0
\(539\) −3.00000 20.7846i −0.129219 0.895257i
\(540\) 0 0
\(541\) 13.0000 + 22.5167i 0.558914 + 0.968067i 0.997587 + 0.0694205i \(0.0221150\pi\)
−0.438674 + 0.898646i \(0.644552\pi\)
\(542\) 0 0
\(543\) −4.00000 + 6.92820i −0.171656 + 0.297318i
\(544\) 0 0
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) −5.00000 + 8.66025i −0.213395 + 0.369611i
\(550\) 0 0
\(551\) 18.0000 + 31.1769i 0.766826 + 1.32818i
\(552\) 0 0
\(553\) 2.00000 + 1.73205i 0.0850487 + 0.0736543i
\(554\) 0 0
\(555\) 12.0000 + 20.7846i 0.509372 + 0.882258i
\(556\) 0 0
\(557\) 1.50000 2.59808i 0.0635570 0.110084i −0.832496 0.554031i \(-0.813089\pi\)
0.896053 + 0.443947i \(0.146422\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.5000 33.7750i 0.821827 1.42345i −0.0824933 0.996592i \(-0.526288\pi\)
0.904320 0.426855i \(-0.140378\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.500000 + 2.59808i −0.0209980 + 0.109109i
\(568\) 0 0
\(569\) 18.0000 + 31.1769i 0.754599 + 1.30700i 0.945573 + 0.325409i \(0.105502\pi\)
−0.190974 + 0.981595i \(0.561165\pi\)
\(570\) 0 0
\(571\) −17.0000 + 29.4449i −0.711428 + 1.23223i 0.252893 + 0.967494i \(0.418618\pi\)
−0.964321 + 0.264735i \(0.914716\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −11.5000 + 19.9186i −0.478751 + 0.829222i −0.999703 0.0243645i \(-0.992244\pi\)
0.520952 + 0.853586i \(0.325577\pi\)
\(578\) 0 0
\(579\) −9.50000 16.4545i −0.394807 0.683825i
\(580\) 0 0
\(581\) 22.5000 7.79423i 0.933457 0.323359i
\(582\) 0 0
\(583\) 4.50000 + 7.79423i 0.186371 + 0.322804i
\(584\) 0 0
\(585\) 6.00000 10.3923i 0.248069 0.429669i
\(586\) 0 0
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −3.00000 + 5.19615i −0.123404 + 0.213741i
\(592\) 0 0
\(593\) −12.0000 20.7846i −0.492781 0.853522i 0.507184 0.861838i \(-0.330686\pi\)
−0.999965 + 0.00831589i \(0.997353\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.0000 + 17.3205i 0.409273 + 0.708881i
\(598\) 0 0
\(599\) 9.00000 15.5885i 0.367730 0.636927i −0.621480 0.783430i \(-0.713468\pi\)
0.989210 + 0.146503i \(0.0468017\pi\)
\(600\) 0 0
\(601\) 11.0000 0.448699 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) −3.00000 + 5.19615i −0.121967 + 0.211254i
\(606\) 0 0
\(607\) 3.50000 + 6.06218i 0.142061 + 0.246056i 0.928272 0.371901i \(-0.121294\pi\)
−0.786212 + 0.617957i \(0.787961\pi\)
\(608\) 0 0
\(609\) −4.50000 + 23.3827i −0.182349 + 0.947514i
\(610\) 0 0
\(611\) 12.0000 + 20.7846i 0.485468 + 0.840855i
\(612\) 0 0
\(613\) −8.00000 + 13.8564i −0.323117 + 0.559655i −0.981129 0.193352i \(-0.938064\pi\)
0.658012 + 0.753007i \(0.271397\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −17.0000 + 29.4449i −0.683288 + 1.18349i 0.290684 + 0.956819i \(0.406117\pi\)
−0.973972 + 0.226670i \(0.927216\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 10.3923i −0.480770 0.416359i
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) −6.00000 + 10.3923i −0.239617 + 0.415029i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) −7.00000 + 12.1244i −0.278225 + 0.481900i
\(634\) 0 0
\(635\) 7.50000 + 12.9904i 0.297628 + 0.515508i
\(636\) 0 0
\(637\) 22.0000 17.3205i 0.871672 0.686264i
\(638\) 0 0
\(639\) 3.00000 + 5.19615i 0.118678 + 0.205557i
\(640\) 0 0
\(641\) 15.0000 25.9808i 0.592464 1.02618i −0.401435 0.915888i \(-0.631488\pi\)
0.993899 0.110291i \(-0.0351782\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 30.0000 1.18125
\(646\) 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
\(649\) −4.50000 7.79423i −0.176640 0.305950i
\(650\) 0 0
\(651\) −2.00000 1.73205i −0.0783862 0.0678844i
\(652\) 0 0
\(653\) 1.50000 + 2.59808i 0.0586995 + 0.101671i 0.893882 0.448303i \(-0.147971\pi\)
−0.835182 + 0.549973i \(0.814638\pi\)
\(654\) 0 0
\(655\) 13.5000 23.3827i 0.527489 0.913637i
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 7.00000 12.1244i 0.272268 0.471583i −0.697174 0.716902i \(-0.745559\pi\)
0.969442 + 0.245319i \(0.0788928\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 31.1769i 0.232670 1.20899i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −9.50000 + 16.4545i −0.367291 + 0.636167i
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) 2.00000 3.46410i 0.0769800 0.133333i
\(676\) 0 0
\(677\) −16.5000 28.5788i −0.634147 1.09837i −0.986695 0.162581i \(-0.948018\pi\)
0.352549 0.935793i \(-0.385315\pi\)
\(678\) 0 0
\(679\) −2.50000 + 0.866025i −0.0959412 + 0.0332350i
\(680\) 0 0
\(681\) 13.5000 + 23.3827i 0.517321 + 0.896026i
\(682\) 0 0
\(683\) 16.5000 28.5788i 0.631355 1.09354i −0.355920 0.934516i \(-0.615832\pi\)
0.987275 0.159022i \(-0.0508342\pi\)
\(684\) 0 0
\(685\) −54.0000 −2.06323
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) −6.00000 + 10.3923i −0.228582 + 0.395915i
\(690\) 0 0
\(691\) 4.00000 + 6.92820i 0.152167 + 0.263561i 0.932024 0.362397i \(-0.118041\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) −7.50000 + 2.59808i −0.284901 + 0.0986928i
\(694\) 0 0
\(695\) −3.00000 5.19615i −0.113796 0.197101i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 0 0
\(703\) 16.0000 27.7128i 0.603451 1.04521i
\(704\) 0 0
\(705\) 9.00000 + 15.5885i 0.338960 + 0.587095i
\(706\) 0 0
\(707\) −9.00000 + 46.7654i −0.338480 + 1.75879i
\(708\) 0 0
\(709\) −5.00000 8.66025i −0.187779 0.325243i 0.756730 0.653727i \(-0.226796\pi\)
−0.944509 + 0.328484i \(0.893462\pi\)
\(710\) 0 0
\(711\) 0.500000 0.866025i 0.0187515 0.0324785i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 0 0
\(717\) −12.0000 + 20.7846i −0.448148 + 0.776215i
\(718\) 0 0
\(719\) 9.00000 + 15.5885i 0.335643 + 0.581351i 0.983608 0.180319i \(-0.0577130\pi\)
−0.647965 + 0.761670i \(0.724380\pi\)
\(720\) 0 0
\(721\) −16.0000 13.8564i −0.595871 0.516040i
\(722\) 0 0
\(723\) −0.500000 0.866025i −0.0185952 0.0322078i
\(724\) 0 0
\(725\) 18.0000 31.1769i 0.668503 1.15788i
\(726\) 0 0
\(727\) −13.0000 −0.482143 −0.241072 0.970507i \(-0.577499\pi\)
−0.241072 + 0.970507i \(0.577499\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.00000 8.66025i −0.184679 0.319874i 0.758789 0.651336i \(-0.225791\pi\)
−0.943468 + 0.331463i \(0.892458\pi\)
\(734\) 0 0
\(735\) 16.5000 12.9904i 0.608612 0.479157i
\(736\) 0 0
\(737\) 15.0000 + 25.9808i 0.552532 + 0.957014i
\(738\) 0 0
\(739\) 25.0000 43.3013i 0.919640 1.59286i 0.119677 0.992813i \(-0.461814\pi\)
0.799962 0.600050i \(-0.204853\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) 42.0000 1.54083 0.770415 0.637542i \(-0.220049\pi\)
0.770415 + 0.637542i \(0.220049\pi\)
\(744\) 0 0
\(745\) −27.0000 + 46.7654i −0.989203 + 1.71335i
\(746\) 0 0
\(747\) −4.50000 7.79423i −0.164646 0.285176i
\(748\) 0 0
\(749\) −6.00000 5.19615i −0.219235 0.189863i
\(750\) 0 0
\(751\) 3.50000 + 6.06218i 0.127717 + 0.221212i 0.922792 0.385299i \(-0.125902\pi\)
−0.795075 + 0.606511i \(0.792568\pi\)
\(752\) 0 0
\(753\) −13.5000 + 23.3827i −0.491967 + 0.852112i
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 10.3923i −0.217500 0.376721i 0.736543 0.676391i \(-0.236457\pi\)
−0.954043 + 0.299670i \(0.903123\pi\)
\(762\) 0 0
\(763\) 7.00000 36.3731i 0.253417 1.31679i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000 10.3923i 0.216647 0.375244i
\(768\) 0 0
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 3.00000 5.19615i 0.107903 0.186893i −0.807018 0.590527i \(-0.798920\pi\)
0.914920 + 0.403634i \(0.132253\pi\)
\(774\) 0 0
\(775\) 2.00000 + 3.46410i 0.0718421 + 0.124434i
\(776\) 0 0
\(777\) 20.0000 6.92820i 0.717496 0.248548i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −9.00000 + 15.5885i −0.322045 + 0.557799i
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 25.0000 43.3013i 0.891154 1.54352i 0.0526599 0.998613i \(-0.483230\pi\)
0.838494 0.544911i \(-0.183437\pi\)
\(788\) 0 0
\(789\) −3.00000 5.19615i −0.106803 0.184988i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 34.6410i −0.710221 1.23014i
\(794\) 0 0
\(795\) −4.50000 + 7.79423i −0.159599 + 0.276433i
\(796\) 0 0
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3.00000 + 5.19615i −0.106000 + 0.183597i
\(802\) 0 0
\(803\) 3.00000 + 5.19615i 0.105868 + 0.183368i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.5000 18.1865i −0.369618 0.640196i
\(808\) 0 0
\(809\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −11.0000 −0.385787
\(814\) 0 0
\(815\) 24.0000 41.5692i 0.840683 1.45611i
\(816\) 0 0
\(817\) −20.0000 34.6410i −0.699711 1.21194i
\(818\) 0 0
\(819\) −8.00000 6.92820i −0.279543 0.242091i
\(820\) 0 0
\(821\) −1.50000 2.59808i −0.0523504 0.0906735i 0.838663 0.544651i \(-0.183338\pi\)
−0.891013 + 0.453978i \(0.850005\pi\)
\(822\) 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
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −15.0000 −0.521601 −0.260801 0.965393i \(-0.583986\pi\)
−0.260801 + 0.965393i \(0.583986\pi\)
\(828\) 0 0
\(829\) −2.00000 + 3.46410i −0.0694629 + 0.120313i −0.898665 0.438636i \(-0.855462\pi\)
0.829202 + 0.558949i \(0.188795\pi\)
\(830\) 0 0
\(831\) −4.00000 6.92820i −0.138758 0.240337i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 9.00000 + 15.5885i 0.311458 + 0.539461i
\(836\) 0 0
\(837\) −0.500000 + 0.866025i −0.0172825 + 0.0299342i
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 3.00000 5.19615i 0.103325 0.178965i
\(844\) 0 0
\(845\) 4.50000 + 7.79423i 0.154805 + 0.268130i
\(846\) 0 0
\(847\) 4.00000 + 3.46410i 0.137442 + 0.119028i
\(848\) 0 0
\(849\) −7.00000 12.1244i −0.240239 0.416107i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) 21.0000 36.3731i 0.717346 1.24248i −0.244701 0.969599i \(-0.578690\pi\)
0.962048 0.272882i \(-0.0879768\pi\)
\(858\) 0 0
\(859\) 25.0000 + 43.3013i 0.852989 + 1.47742i 0.878498 + 0.477746i \(0.158546\pi\)
−0.0255092 + 0.999675i \(0.508121\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.00000 5.19615i −0.102121 0.176879i 0.810437 0.585826i \(-0.199230\pi\)
−0.912558 + 0.408946i \(0.865896\pi\)
\(864\) 0 0
\(865\) −27.0000 + 46.7654i −0.918028 + 1.59007i
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −20.0000 + 34.6410i −0.677674 + 1.17377i
\(872\) 0 0
\(873\) 0.500000 + 0.866025i 0.0169224 + 0.0293105i
\(874\) 0 0
\(875\) 7.50000 2.59808i 0.253546 0.0878310i
\(876\) 0 0
\(877\) 16.0000 + 27.7128i 0.540282 + 0.935795i 0.998888 + 0.0471555i \(0.0150156\pi\)
−0.458606 + 0.888640i \(0.651651\pi\)
\(878\) 0 0
\(879\) −16.5000 + 28.5788i −0.556531 + 0.963940i
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 0 0
\(885\) 4.50000 7.79423i 0.151266 0.262000i
\(886\) 0 0
\(887\) 12.0000 + 20.7846i 0.402921 + 0.697879i 0.994077 0.108678i \(-0.0346618\pi\)
−0.591156 + 0.806557i \(0.701328\pi\)
\(888\) 0 0
\(889\) 12.5000 4.33013i 0.419237 0.145228i
\(890\) 0 0
\(891\) 1.50000 + 2.59808i 0.0502519 + 0.0870388i
\(892\) 0 0
\(893\) 12.0000 20.7846i 0.401565 0.695530i
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.50000 + 7.79423i −0.150083 + 0.259952i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 5.00000 25.9808i 0.166390 0.864586i
\(904\) 0 0
\(905\) −12.0000 20.7846i −0.398893 0.690904i
\(906\) 0 0
\(907\) 4.00000 6.92820i 0.132818 0.230047i −0.791944 0.610594i \(-0.790931\pi\)
0.924762 + 0.380547i \(0.124264\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) 13.5000 23.3827i 0.446785 0.773854i
\(914\) 0 0
\(915\) −15.0000 25.9808i −0.495885 0.858898i
\(916\) 0 0
\(917\) −18.0000 15.5885i −0.594412 0.514776i
\(918\) 0 0
\(919\) −4.00000 6.92820i −0.131948 0.228540i 0.792480 0.609898i \(-0.208790\pi\)
−0.924427 + 0.381358i \(0.875456\pi\)
\(920\) 0 0
\(921\) −4.00000 + 6.92820i −0.131804 + 0.228292i
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 0 0
\(927\) −4.00000 + 6.92820i −0.131377 + 0.227552i
\(928\) 0 0
\(929\) 3.00000 + 5.19615i 0.0984268 + 0.170480i 0.911034 0.412332i \(-0.135286\pi\)
−0.812607 + 0.582812i \(0.801952\pi\)
\(930\) 0 0
\(931\) −26.0000 10.3923i −0.852116 0.340594i
\(932\) 0 0
\(933\) 12.0000 + 20.7846i 0.392862 + 0.680458i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) 31.0000 1.01165
\(940\) 0 0
\(941\) −4.50000 + 7.79423i −0.146696 + 0.254085i −0.930004 0.367549i \(-0.880197\pi\)
0.783309 + 0.621633i \(0.213531\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −6.00000 5.19615i −0.195180 0.169031i
\(946\) 0 0
\(947\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(948\) 0 0
\(949\) −4.00000 + 6.92820i −0.129845 + 0.224899i
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 13.5000 + 23.3827i 0.436393 + 0.755855i
\(958\) 0 0
\(959\) −9.00000 + 46.7654i −0.290625 + 1.51013i
\(960\) 0 0
\(961\) 15.0000 + 25.9808i 0.483871 + 0.838089i
\(962\) 0 0
\(963\) −1.50000 + 2.59808i −0.0483368 + 0.0837218i
\(964\) 0 0
\(965\) 57.0000 1.83489
\(966\) 0 0
\(967\) −1.00000 −0.0321578 −0.0160789 0.999871i \(-0.505118\pi\)
−0.0160789 + 0.999871i \(0.505118\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.5000 33.7750i −0.625785 1.08389i −0.988389 0.151948i \(-0.951445\pi\)
0.362604 0.931943i \(-0.381888\pi\)
\(972\) 0 0
\(973\) −5.00000 + 1.73205i −0.160293 + 0.0555270i
\(974\) 0 0
\(975\) 8.00000 + 13.8564i 0.256205 + 0.443760i
\(976\) 0 0
\(977\) −21.0000 + 36.3731i −0.671850 + 1.16368i 0.305530 + 0.952183i \(0.401167\pi\)
−0.977379 + 0.211495i \(0.932167\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) −18.0000 + 31.1769i −0.574111 + 0.994389i 0.422027 + 0.906583i \(0.361319\pi\)
−0.996138 + 0.0878058i \(0.972015\pi\)
\(984\) 0 0
\(985\) −9.00000 15.5885i −0.286764 0.496690i
\(986\) 0 0
\(987\) 15.0000 5.19615i 0.477455 0.165395i
\(988\) 0 0
\(989\) 0 0
\(990\) 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
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −60.0000 −1.90213
\(996\) 0 0
\(997\) 7.00000 12.1244i 0.221692 0.383982i −0.733630 0.679549i \(-0.762175\pi\)
0.955322 + 0.295567i \(0.0955086\pi\)
\(998\) 0 0
\(999\) −4.00000 6.92820i −0.126554 0.219199i
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 1344.2.q.v.193.1 2
4.3 odd 2 1344.2.q.j.193.1 2
7.2 even 3 inner 1344.2.q.v.961.1 2
7.3 odd 6 9408.2.a.db.1.1 1
7.4 even 3 9408.2.a.d.1.1 1
8.3 odd 2 336.2.q.d.193.1 2
8.5 even 2 42.2.e.b.25.1 2
24.5 odd 2 126.2.g.b.109.1 2
24.11 even 2 1008.2.s.n.865.1 2
28.3 even 6 9408.2.a.bm.1.1 1
28.11 odd 6 9408.2.a.bu.1.1 1
28.23 odd 6 1344.2.q.j.961.1 2
40.13 odd 4 1050.2.o.b.949.2 4
40.29 even 2 1050.2.i.e.151.1 2
40.37 odd 4 1050.2.o.b.949.1 4
56.3 even 6 2352.2.a.n.1.1 1
56.5 odd 6 294.2.e.f.79.1 2
56.11 odd 6 2352.2.a.m.1.1 1
56.13 odd 2 294.2.e.f.67.1 2
56.19 even 6 2352.2.q.m.961.1 2
56.27 even 2 2352.2.q.m.1537.1 2
56.37 even 6 42.2.e.b.37.1 yes 2
56.45 odd 6 294.2.a.a.1.1 1
56.51 odd 6 336.2.q.d.289.1 2
56.53 even 6 294.2.a.d.1.1 1
72.5 odd 6 1134.2.e.p.865.1 2
72.13 even 6 1134.2.e.a.865.1 2
72.29 odd 6 1134.2.h.a.109.1 2
72.61 even 6 1134.2.h.p.109.1 2
168.5 even 6 882.2.g.b.667.1 2
168.11 even 6 7056.2.a.g.1.1 1
168.53 odd 6 882.2.a.g.1.1 1
168.59 odd 6 7056.2.a.bz.1.1 1
168.101 even 6 882.2.a.k.1.1 1
168.107 even 6 1008.2.s.n.289.1 2
168.125 even 2 882.2.g.b.361.1 2
168.149 odd 6 126.2.g.b.37.1 2
280.37 odd 12 1050.2.o.b.499.2 4
280.93 odd 12 1050.2.o.b.499.1 4
280.109 even 6 7350.2.a.ce.1.1 1
280.149 even 6 1050.2.i.e.751.1 2
280.269 odd 6 7350.2.a.cw.1.1 1
504.149 odd 6 1134.2.h.a.541.1 2
504.205 even 6 1134.2.e.a.919.1 2
504.317 odd 6 1134.2.e.p.919.1 2
504.373 even 6 1134.2.h.p.541.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.b.25.1 2 8.5 even 2
42.2.e.b.37.1 yes 2 56.37 even 6
126.2.g.b.37.1 2 168.149 odd 6
126.2.g.b.109.1 2 24.5 odd 2
294.2.a.a.1.1 1 56.45 odd 6
294.2.a.d.1.1 1 56.53 even 6
294.2.e.f.67.1 2 56.13 odd 2
294.2.e.f.79.1 2 56.5 odd 6
336.2.q.d.193.1 2 8.3 odd 2
336.2.q.d.289.1 2 56.51 odd 6
882.2.a.g.1.1 1 168.53 odd 6
882.2.a.k.1.1 1 168.101 even 6
882.2.g.b.361.1 2 168.125 even 2
882.2.g.b.667.1 2 168.5 even 6
1008.2.s.n.289.1 2 168.107 even 6
1008.2.s.n.865.1 2 24.11 even 2
1050.2.i.e.151.1 2 40.29 even 2
1050.2.i.e.751.1 2 280.149 even 6
1050.2.o.b.499.1 4 280.93 odd 12
1050.2.o.b.499.2 4 280.37 odd 12
1050.2.o.b.949.1 4 40.37 odd 4
1050.2.o.b.949.2 4 40.13 odd 4
1134.2.e.a.865.1 2 72.13 even 6
1134.2.e.a.919.1 2 504.205 even 6
1134.2.e.p.865.1 2 72.5 odd 6
1134.2.e.p.919.1 2 504.317 odd 6
1134.2.h.a.109.1 2 72.29 odd 6
1134.2.h.a.541.1 2 504.149 odd 6
1134.2.h.p.109.1 2 72.61 even 6
1134.2.h.p.541.1 2 504.373 even 6
1344.2.q.j.193.1 2 4.3 odd 2
1344.2.q.j.961.1 2 28.23 odd 6
1344.2.q.v.193.1 2 1.1 even 1 trivial
1344.2.q.v.961.1 2 7.2 even 3 inner
2352.2.a.m.1.1 1 56.11 odd 6
2352.2.a.n.1.1 1 56.3 even 6
2352.2.q.m.961.1 2 56.19 even 6
2352.2.q.m.1537.1 2 56.27 even 2
7056.2.a.g.1.1 1 168.11 even 6
7056.2.a.bz.1.1 1 168.59 odd 6
7350.2.a.ce.1.1 1 280.109 even 6
7350.2.a.cw.1.1 1 280.269 odd 6
9408.2.a.d.1.1 1 7.4 even 3
9408.2.a.bm.1.1 1 28.3 even 6
9408.2.a.bu.1.1 1 28.11 odd 6
9408.2.a.db.1.1 1 7.3 odd 6