Properties

Label 1148.2.r.a.737.28
Level $1148$
Weight $2$
Character 1148.737
Analytic conductor $9.167$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1148,2,Mod(81,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.r (of order \(6\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 737.28
Character \(\chi\) \(=\) 1148.737
Dual form 1148.2.r.a.81.28

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.89464 - 1.67122i) q^{3} +(1.42784 - 2.47308i) q^{5} +(1.36322 + 2.26752i) q^{7} +(4.08596 - 7.07710i) q^{9} +O(q^{10})\) \(q+(2.89464 - 1.67122i) q^{3} +(1.42784 - 2.47308i) q^{5} +(1.36322 + 2.26752i) q^{7} +(4.08596 - 7.07710i) q^{9} +(-1.10288 + 0.636749i) q^{11} +3.41413i q^{13} -9.54492i q^{15} +(1.97022 - 1.13751i) q^{17} +(0.851130 + 0.491400i) q^{19} +(7.73555 + 4.28541i) q^{21} +(-3.90559 + 6.76468i) q^{23} +(-1.57743 - 2.73219i) q^{25} -17.2869i q^{27} -0.0172332i q^{29} +(-4.79997 - 8.31379i) q^{31} +(-2.12830 + 3.68632i) q^{33} +(7.55422 - 0.133706i) q^{35} +(-4.00389 + 6.93495i) q^{37} +(5.70576 + 9.88267i) q^{39} +(4.71859 + 4.32839i) q^{41} -2.35283 q^{43} +(-11.6682 - 20.2099i) q^{45} +(-3.58279 - 2.06853i) q^{47} +(-3.28328 + 6.18224i) q^{49} +(3.80205 - 6.58535i) q^{51} +(-1.38918 + 0.802043i) q^{53} +3.63669i q^{55} +3.28495 q^{57} +(-3.38451 - 5.86214i) q^{59} +(-3.36290 + 5.82471i) q^{61} +(21.6175 - 0.382618i) q^{63} +(8.44343 + 4.87481i) q^{65} +(-8.17149 + 4.71781i) q^{67} +26.1084i q^{69} -3.13702i q^{71} +(-5.62882 - 9.74940i) q^{73} +(-9.13220 - 5.27248i) q^{75} +(-2.94731 - 1.63278i) q^{77} +(1.10592 + 0.638505i) q^{79} +(-16.6323 - 28.8080i) q^{81} -3.83139 q^{83} -6.49669i q^{85} +(-0.0288004 - 0.0498838i) q^{87} +(11.9151 + 6.87919i) q^{89} +(-7.74160 + 4.65420i) q^{91} +(-27.7884 - 16.0436i) q^{93} +(2.43055 - 1.40328i) q^{95} -10.5399i q^{97} +10.4069i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{5} + 32 q^{9} - 6 q^{21} + 2 q^{23} - 24 q^{25} - 4 q^{31} + 10 q^{33} + 10 q^{37} + 10 q^{39} + 20 q^{41} + 8 q^{43} - 22 q^{45} - 4 q^{49} + 18 q^{51} + 28 q^{57} - 16 q^{59} + 16 q^{61} + 2 q^{73} + 34 q^{77} - 28 q^{81} - 48 q^{83} - 2 q^{87} - 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) 2.89464 1.67122i 1.67122 0.964880i 0.704266 0.709936i \(-0.251276\pi\)
0.966956 0.254944i \(-0.0820570\pi\)
\(4\) 0 0
\(5\) 1.42784 2.47308i 0.638548 1.10600i −0.347204 0.937790i \(-0.612869\pi\)
0.985752 0.168207i \(-0.0537979\pi\)
\(6\) 0 0
\(7\) 1.36322 + 2.26752i 0.515247 + 0.857041i
\(8\) 0 0
\(9\) 4.08596 7.07710i 1.36199 2.35903i
\(10\) 0 0
\(11\) −1.10288 + 0.636749i −0.332531 + 0.191987i −0.656964 0.753922i \(-0.728160\pi\)
0.324433 + 0.945909i \(0.394826\pi\)
\(12\) 0 0
\(13\) 3.41413i 0.946909i 0.880818 + 0.473454i \(0.156993\pi\)
−0.880818 + 0.473454i \(0.843007\pi\)
\(14\) 0 0
\(15\) 9.54492i 2.46449i
\(16\) 0 0
\(17\) 1.97022 1.13751i 0.477849 0.275886i −0.241671 0.970358i \(-0.577695\pi\)
0.719519 + 0.694472i \(0.244362\pi\)
\(18\) 0 0
\(19\) 0.851130 + 0.491400i 0.195263 + 0.112735i 0.594444 0.804137i \(-0.297372\pi\)
−0.399181 + 0.916872i \(0.630706\pi\)
\(20\) 0 0
\(21\) 7.73555 + 4.28541i 1.68804 + 0.935154i
\(22\) 0 0
\(23\) −3.90559 + 6.76468i −0.814372 + 1.41053i 0.0954064 + 0.995438i \(0.469585\pi\)
−0.909778 + 0.415095i \(0.863748\pi\)
\(24\) 0 0
\(25\) −1.57743 2.73219i −0.315486 0.546439i
\(26\) 0 0
\(27\) 17.2869i 3.32686i
\(28\) 0 0
\(29\) 0.0172332i 0.00320012i −0.999999 0.00160006i \(-0.999491\pi\)
0.999999 0.00160006i \(-0.000509315\pi\)
\(30\) 0 0
\(31\) −4.79997 8.31379i −0.862100 1.49320i −0.869898 0.493232i \(-0.835815\pi\)
0.00779754 0.999970i \(-0.497518\pi\)
\(32\) 0 0
\(33\) −2.12830 + 3.68632i −0.370489 + 0.641705i
\(34\) 0 0
\(35\) 7.55422 0.133706i 1.27690 0.0226004i
\(36\) 0 0
\(37\) −4.00389 + 6.93495i −0.658236 + 1.14010i 0.322836 + 0.946455i \(0.395364\pi\)
−0.981072 + 0.193644i \(0.937969\pi\)
\(38\) 0 0
\(39\) 5.70576 + 9.88267i 0.913653 + 1.58249i
\(40\) 0 0
\(41\) 4.71859 + 4.32839i 0.736920 + 0.675980i
\(42\) 0 0
\(43\) −2.35283 −0.358802 −0.179401 0.983776i \(-0.557416\pi\)
−0.179401 + 0.983776i \(0.557416\pi\)
\(44\) 0 0
\(45\) −11.6682 20.2099i −1.73939 3.01271i
\(46\) 0 0
\(47\) −3.58279 2.06853i −0.522604 0.301726i 0.215395 0.976527i \(-0.430896\pi\)
−0.737999 + 0.674801i \(0.764229\pi\)
\(48\) 0 0
\(49\) −3.28328 + 6.18224i −0.469040 + 0.883177i
\(50\) 0 0
\(51\) 3.80205 6.58535i 0.532394 0.922133i
\(52\) 0 0
\(53\) −1.38918 + 0.802043i −0.190819 + 0.110169i −0.592366 0.805669i \(-0.701806\pi\)
0.401547 + 0.915838i \(0.368473\pi\)
\(54\) 0 0
\(55\) 3.63669i 0.490371i
\(56\) 0 0
\(57\) 3.28495 0.435103
\(58\) 0 0
\(59\) −3.38451 5.86214i −0.440625 0.763185i 0.557111 0.830438i \(-0.311910\pi\)
−0.997736 + 0.0672529i \(0.978577\pi\)
\(60\) 0 0
\(61\) −3.36290 + 5.82471i −0.430575 + 0.745778i −0.996923 0.0783884i \(-0.975023\pi\)
0.566348 + 0.824166i \(0.308356\pi\)
\(62\) 0 0
\(63\) 21.6175 0.382618i 2.72355 0.0482054i
\(64\) 0 0
\(65\) 8.44343 + 4.87481i 1.04728 + 0.604646i
\(66\) 0 0
\(67\) −8.17149 + 4.71781i −0.998306 + 0.576372i −0.907747 0.419519i \(-0.862199\pi\)
−0.0905595 + 0.995891i \(0.528866\pi\)
\(68\) 0 0
\(69\) 26.1084i 3.14308i
\(70\) 0 0
\(71\) 3.13702i 0.372296i −0.982522 0.186148i \(-0.940400\pi\)
0.982522 0.186148i \(-0.0596003\pi\)
\(72\) 0 0
\(73\) −5.62882 9.74940i −0.658803 1.14108i −0.980926 0.194382i \(-0.937730\pi\)
0.322123 0.946698i \(-0.395604\pi\)
\(74\) 0 0
\(75\) −9.13220 5.27248i −1.05450 0.608813i
\(76\) 0 0
\(77\) −2.94731 1.63278i −0.335877 0.186072i
\(78\) 0 0
\(79\) 1.10592 + 0.638505i 0.124426 + 0.0718375i 0.560921 0.827869i \(-0.310447\pi\)
−0.436495 + 0.899707i \(0.643780\pi\)
\(80\) 0 0
\(81\) −16.6323 28.8080i −1.84803 3.20089i
\(82\) 0 0
\(83\) −3.83139 −0.420550 −0.210275 0.977642i \(-0.567436\pi\)
−0.210275 + 0.977642i \(0.567436\pi\)
\(84\) 0 0
\(85\) 6.49669i 0.704665i
\(86\) 0 0
\(87\) −0.0288004 0.0498838i −0.00308773 0.00534810i
\(88\) 0 0
\(89\) 11.9151 + 6.87919i 1.26300 + 0.729193i 0.973654 0.228032i \(-0.0732290\pi\)
0.289345 + 0.957225i \(0.406562\pi\)
\(90\) 0 0
\(91\) −7.74160 + 4.65420i −0.811540 + 0.487892i
\(92\) 0 0
\(93\) −27.7884 16.0436i −2.88152 1.66365i
\(94\) 0 0
\(95\) 2.43055 1.40328i 0.249369 0.143973i
\(96\) 0 0
\(97\) 10.5399i 1.07017i −0.844798 0.535085i \(-0.820280\pi\)
0.844798 0.535085i \(-0.179720\pi\)
\(98\) 0 0
\(99\) 10.4069i 1.04594i
\(100\) 0 0
\(101\) 16.0930 9.29128i 1.60131 0.924517i 0.610085 0.792336i \(-0.291135\pi\)
0.991226 0.132181i \(-0.0421979\pi\)
\(102\) 0 0
\(103\) −2.66295 + 4.61237i −0.262389 + 0.454470i −0.966876 0.255246i \(-0.917844\pi\)
0.704488 + 0.709716i \(0.251177\pi\)
\(104\) 0 0
\(105\) 21.6433 13.0118i 2.11217 1.26982i
\(106\) 0 0
\(107\) −4.41227 + 7.64228i −0.426550 + 0.738807i −0.996564 0.0828285i \(-0.973605\pi\)
0.570013 + 0.821635i \(0.306938\pi\)
\(108\) 0 0
\(109\) −12.7691 + 7.37222i −1.22305 + 0.706131i −0.965568 0.260151i \(-0.916228\pi\)
−0.257487 + 0.966282i \(0.582894\pi\)
\(110\) 0 0
\(111\) 26.7656i 2.54048i
\(112\) 0 0
\(113\) −5.57000 −0.523982 −0.261991 0.965070i \(-0.584379\pi\)
−0.261991 + 0.965070i \(0.584379\pi\)
\(114\) 0 0
\(115\) 11.1531 + 19.3177i 1.04003 + 1.80139i
\(116\) 0 0
\(117\) 24.1621 + 13.9500i 2.23379 + 1.28968i
\(118\) 0 0
\(119\) 5.26516 + 2.91684i 0.482656 + 0.267386i
\(120\) 0 0
\(121\) −4.68910 + 8.12176i −0.426282 + 0.738342i
\(122\) 0 0
\(123\) 20.8923 + 4.64332i 1.88380 + 0.418674i
\(124\) 0 0
\(125\) 5.26910 0.471283
\(126\) 0 0
\(127\) −11.7889 −1.04610 −0.523049 0.852302i \(-0.675206\pi\)
−0.523049 + 0.852302i \(0.675206\pi\)
\(128\) 0 0
\(129\) −6.81058 + 3.93209i −0.599638 + 0.346201i
\(130\) 0 0
\(131\) 3.27601 5.67422i 0.286227 0.495759i −0.686679 0.726960i \(-0.740932\pi\)
0.972906 + 0.231202i \(0.0742657\pi\)
\(132\) 0 0
\(133\) 0.0460158 + 2.59984i 0.00399007 + 0.225435i
\(134\) 0 0
\(135\) −42.7519 24.6828i −3.67950 2.12436i
\(136\) 0 0
\(137\) 11.7173 6.76496i 1.00107 0.577970i 0.0925076 0.995712i \(-0.470512\pi\)
0.908566 + 0.417742i \(0.137178\pi\)
\(138\) 0 0
\(139\) 12.1594 1.03135 0.515673 0.856785i \(-0.327542\pi\)
0.515673 + 0.856785i \(0.327542\pi\)
\(140\) 0 0
\(141\) −13.8279 −1.16452
\(142\) 0 0
\(143\) −2.17394 3.76538i −0.181794 0.314877i
\(144\) 0 0
\(145\) −0.0426191 0.0246061i −0.00353932 0.00204343i
\(146\) 0 0
\(147\) 0.827974 + 23.3824i 0.0682901 + 1.92855i
\(148\) 0 0
\(149\) −1.02593 0.592323i −0.0840477 0.0485250i 0.457387 0.889268i \(-0.348785\pi\)
−0.541435 + 0.840743i \(0.682119\pi\)
\(150\) 0 0
\(151\) 16.9980 9.81380i 1.38328 0.798635i 0.390731 0.920505i \(-0.372222\pi\)
0.992546 + 0.121869i \(0.0388889\pi\)
\(152\) 0 0
\(153\) 18.5912i 1.50301i
\(154\) 0 0
\(155\) −27.4143 −2.20197
\(156\) 0 0
\(157\) 0.469171 0.270876i 0.0374439 0.0216183i −0.481161 0.876632i \(-0.659785\pi\)
0.518605 + 0.855014i \(0.326451\pi\)
\(158\) 0 0
\(159\) −2.68078 + 4.64325i −0.212600 + 0.368234i
\(160\) 0 0
\(161\) −20.6632 + 0.365728i −1.62849 + 0.0288234i
\(162\) 0 0
\(163\) 12.7168 22.0262i 0.996057 1.72522i 0.421201 0.906967i \(-0.361609\pi\)
0.574856 0.818254i \(-0.305058\pi\)
\(164\) 0 0
\(165\) 6.07772 + 10.5269i 0.473150 + 0.819519i
\(166\) 0 0
\(167\) 6.26023i 0.484431i 0.970222 + 0.242216i \(0.0778742\pi\)
−0.970222 + 0.242216i \(0.922126\pi\)
\(168\) 0 0
\(169\) 1.34373 0.103364
\(170\) 0 0
\(171\) 6.95537 4.01569i 0.531891 0.307087i
\(172\) 0 0
\(173\) 9.34229 16.1813i 0.710281 1.23024i −0.254470 0.967081i \(-0.581901\pi\)
0.964751 0.263163i \(-0.0847657\pi\)
\(174\) 0 0
\(175\) 4.04492 7.30143i 0.305767 0.551936i
\(176\) 0 0
\(177\) −19.5939 11.3125i −1.47276 0.850301i
\(178\) 0 0
\(179\) −13.6712 + 7.89306i −1.02183 + 0.589955i −0.914634 0.404283i \(-0.867521\pi\)
−0.107198 + 0.994238i \(0.534188\pi\)
\(180\) 0 0
\(181\) 4.35013i 0.323342i 0.986845 + 0.161671i \(0.0516884\pi\)
−0.986845 + 0.161671i \(0.948312\pi\)
\(182\) 0 0
\(183\) 22.4806i 1.66181i
\(184\) 0 0
\(185\) 11.4338 + 19.8039i 0.840631 + 1.45601i
\(186\) 0 0
\(187\) −1.44861 + 2.50907i −0.105933 + 0.183481i
\(188\) 0 0
\(189\) 39.1983 23.5658i 2.85126 1.71416i
\(190\) 0 0
\(191\) 18.9422 + 10.9363i 1.37061 + 0.791323i 0.991005 0.133825i \(-0.0427259\pi\)
0.379607 + 0.925148i \(0.376059\pi\)
\(192\) 0 0
\(193\) 15.9608 9.21496i 1.14888 0.663308i 0.200268 0.979741i \(-0.435819\pi\)
0.948615 + 0.316434i \(0.102485\pi\)
\(194\) 0 0
\(195\) 32.5876 2.33365
\(196\) 0 0
\(197\) 19.4760 1.38761 0.693803 0.720165i \(-0.255934\pi\)
0.693803 + 0.720165i \(0.255934\pi\)
\(198\) 0 0
\(199\) 2.75202 1.58888i 0.195085 0.112633i −0.399276 0.916831i \(-0.630738\pi\)
0.594361 + 0.804198i \(0.297405\pi\)
\(200\) 0 0
\(201\) −15.7690 + 27.3127i −1.11226 + 1.92649i
\(202\) 0 0
\(203\) 0.0390765 0.0234925i 0.00274263 0.00164885i
\(204\) 0 0
\(205\) 17.4418 5.48924i 1.21819 0.383385i
\(206\) 0 0
\(207\) 31.9162 + 55.2805i 2.21833 + 3.84226i
\(208\) 0 0
\(209\) −1.25159 −0.0865745
\(210\) 0 0
\(211\) 20.5647i 1.41573i −0.706345 0.707867i \(-0.749657\pi\)
0.706345 0.707867i \(-0.250343\pi\)
\(212\) 0 0
\(213\) −5.24265 9.08054i −0.359221 0.622188i
\(214\) 0 0
\(215\) −3.35945 + 5.81874i −0.229113 + 0.396835i
\(216\) 0 0
\(217\) 12.3083 22.2175i 0.835541 1.50822i
\(218\) 0 0
\(219\) −32.5868 18.8140i −2.20201 1.27133i
\(220\) 0 0
\(221\) 3.88359 + 6.72658i 0.261239 + 0.452479i
\(222\) 0 0
\(223\) −22.3009 −1.49338 −0.746689 0.665173i \(-0.768358\pi\)
−0.746689 + 0.665173i \(0.768358\pi\)
\(224\) 0 0
\(225\) −25.7813 −1.71875
\(226\) 0 0
\(227\) 9.05944 5.23047i 0.601296 0.347159i −0.168255 0.985744i \(-0.553813\pi\)
0.769551 + 0.638585i \(0.220480\pi\)
\(228\) 0 0
\(229\) 17.9503 + 10.3636i 1.18619 + 0.684847i 0.957439 0.288637i \(-0.0932022\pi\)
0.228752 + 0.973485i \(0.426535\pi\)
\(230\) 0 0
\(231\) −11.2601 + 0.199298i −0.740862 + 0.0131129i
\(232\) 0 0
\(233\) −19.9415 11.5132i −1.30641 0.754255i −0.324913 0.945744i \(-0.605335\pi\)
−0.981495 + 0.191489i \(0.938668\pi\)
\(234\) 0 0
\(235\) −10.2313 + 5.90703i −0.667415 + 0.385332i
\(236\) 0 0
\(237\) 4.26833 0.277258
\(238\) 0 0
\(239\) 9.34289i 0.604341i 0.953254 + 0.302171i \(0.0977112\pi\)
−0.953254 + 0.302171i \(0.902289\pi\)
\(240\) 0 0
\(241\) 7.94995 + 13.7697i 0.512101 + 0.886986i 0.999902 + 0.0140303i \(0.00446612\pi\)
−0.487800 + 0.872955i \(0.662201\pi\)
\(242\) 0 0
\(243\) −51.3765 29.6622i −3.29580 1.90283i
\(244\) 0 0
\(245\) 10.6012 + 16.9471i 0.677287 + 1.08271i
\(246\) 0 0
\(247\) −1.67770 + 2.90587i −0.106750 + 0.184896i
\(248\) 0 0
\(249\) −11.0905 + 6.40311i −0.702833 + 0.405781i
\(250\) 0 0
\(251\) 10.4444 0.659247 0.329624 0.944112i \(-0.393078\pi\)
0.329624 + 0.944112i \(0.393078\pi\)
\(252\) 0 0
\(253\) 9.94752i 0.625395i
\(254\) 0 0
\(255\) −10.8574 18.8056i −0.679918 1.17765i
\(256\) 0 0
\(257\) −2.28526 1.31940i −0.142551 0.0823018i 0.427028 0.904238i \(-0.359560\pi\)
−0.569579 + 0.821936i \(0.692894\pi\)
\(258\) 0 0
\(259\) −21.1833 + 0.374933i −1.31627 + 0.0232972i
\(260\) 0 0
\(261\) −0.121961 0.0704141i −0.00754918 0.00435852i
\(262\) 0 0
\(263\) −12.7942 + 7.38672i −0.788922 + 0.455484i −0.839583 0.543232i \(-0.817201\pi\)
0.0506608 + 0.998716i \(0.483867\pi\)
\(264\) 0 0
\(265\) 4.58075i 0.281393i
\(266\) 0 0
\(267\) 45.9866 2.81434
\(268\) 0 0
\(269\) −3.59737 6.23082i −0.219335 0.379900i 0.735270 0.677775i \(-0.237056\pi\)
−0.954605 + 0.297875i \(0.903722\pi\)
\(270\) 0 0
\(271\) −12.8734 + 22.2973i −0.782002 + 1.35447i 0.148773 + 0.988871i \(0.452468\pi\)
−0.930774 + 0.365595i \(0.880866\pi\)
\(272\) 0 0
\(273\) −14.6309 + 26.4101i −0.885505 + 1.59841i
\(274\) 0 0
\(275\) 3.47944 + 2.00886i 0.209818 + 0.121139i
\(276\) 0 0
\(277\) 5.04779 + 8.74302i 0.303292 + 0.525317i 0.976880 0.213790i \(-0.0685809\pi\)
−0.673588 + 0.739107i \(0.735248\pi\)
\(278\) 0 0
\(279\) −78.4500 −4.69668
\(280\) 0 0
\(281\) 31.4886i 1.87845i 0.343300 + 0.939226i \(0.388455\pi\)
−0.343300 + 0.939226i \(0.611545\pi\)
\(282\) 0 0
\(283\) −10.9374 18.9441i −0.650161 1.12611i −0.983083 0.183158i \(-0.941368\pi\)
0.332922 0.942954i \(-0.391965\pi\)
\(284\) 0 0
\(285\) 4.69038 8.12397i 0.277834 0.481223i
\(286\) 0 0
\(287\) −3.38224 + 16.6000i −0.199647 + 0.979868i
\(288\) 0 0
\(289\) −5.91216 + 10.2402i −0.347774 + 0.602362i
\(290\) 0 0
\(291\) −17.6146 30.5094i −1.03259 1.78849i
\(292\) 0 0
\(293\) 2.76984i 0.161816i 0.996722 + 0.0809080i \(0.0257820\pi\)
−0.996722 + 0.0809080i \(0.974218\pi\)
\(294\) 0 0
\(295\) −19.3301 −1.12544
\(296\) 0 0
\(297\) 11.0074 + 19.0654i 0.638714 + 1.10628i
\(298\) 0 0
\(299\) −23.0955 13.3342i −1.33565 0.771136i
\(300\) 0 0
\(301\) −3.20741 5.33507i −0.184872 0.307509i
\(302\) 0 0
\(303\) 31.0556 53.7898i 1.78410 3.09015i
\(304\) 0 0
\(305\) 9.60334 + 16.6335i 0.549886 + 0.952430i
\(306\) 0 0
\(307\) −6.90088 −0.393854 −0.196927 0.980418i \(-0.563096\pi\)
−0.196927 + 0.980418i \(0.563096\pi\)
\(308\) 0 0
\(309\) 17.8015i 1.01269i
\(310\) 0 0
\(311\) −0.268231 + 0.154863i −0.0152100 + 0.00878148i −0.507586 0.861601i \(-0.669462\pi\)
0.492376 + 0.870383i \(0.336129\pi\)
\(312\) 0 0
\(313\) −6.57240 3.79458i −0.371494 0.214482i 0.302617 0.953112i \(-0.402140\pi\)
−0.674111 + 0.738630i \(0.735473\pi\)
\(314\) 0 0
\(315\) 29.9200 54.0082i 1.68580 3.04302i
\(316\) 0 0
\(317\) 16.0085 + 9.24253i 0.899129 + 0.519112i 0.876918 0.480641i \(-0.159596\pi\)
0.0222115 + 0.999753i \(0.492929\pi\)
\(318\) 0 0
\(319\) 0.0109732 + 0.0190061i 0.000614381 + 0.00106414i
\(320\) 0 0
\(321\) 29.4955i 1.64628i
\(322\) 0 0
\(323\) 2.23588 0.124408
\(324\) 0 0
\(325\) 9.32806 5.38556i 0.517427 0.298737i
\(326\) 0 0
\(327\) −24.6412 + 42.6799i −1.36266 + 2.36020i
\(328\) 0 0
\(329\) −0.193701 10.9439i −0.0106791 0.603357i
\(330\) 0 0
\(331\) 4.75436 + 2.74493i 0.261323 + 0.150875i 0.624938 0.780674i \(-0.285124\pi\)
−0.363615 + 0.931549i \(0.618458\pi\)
\(332\) 0 0
\(333\) 32.7195 + 56.6719i 1.79302 + 3.10560i
\(334\) 0 0
\(335\) 26.9450i 1.47217i
\(336\) 0 0
\(337\) −10.4452 −0.568986 −0.284493 0.958678i \(-0.591825\pi\)
−0.284493 + 0.958678i \(0.591825\pi\)
\(338\) 0 0
\(339\) −16.1232 + 9.30871i −0.875689 + 0.505580i
\(340\) 0 0
\(341\) 10.5876 + 6.11275i 0.573350 + 0.331024i
\(342\) 0 0
\(343\) −18.4942 + 0.982832i −0.998591 + 0.0530680i
\(344\) 0 0
\(345\) 64.5683 + 37.2786i 3.47624 + 2.00701i
\(346\) 0 0
\(347\) −21.4144 + 12.3636i −1.14958 + 0.663712i −0.948786 0.315920i \(-0.897687\pi\)
−0.200798 + 0.979633i \(0.564353\pi\)
\(348\) 0 0
\(349\) −3.15478 −0.168872 −0.0844359 0.996429i \(-0.526909\pi\)
−0.0844359 + 0.996429i \(0.526909\pi\)
\(350\) 0 0
\(351\) 59.0196 3.15023
\(352\) 0 0
\(353\) 0.697880 + 1.20876i 0.0371444 + 0.0643360i 0.884000 0.467487i \(-0.154841\pi\)
−0.846856 + 0.531823i \(0.821507\pi\)
\(354\) 0 0
\(355\) −7.75811 4.47915i −0.411758 0.237729i
\(356\) 0 0
\(357\) 20.1154 0.356032i 1.06462 0.0188432i
\(358\) 0 0
\(359\) 0.470898 0.815619i 0.0248530 0.0430467i −0.853331 0.521369i \(-0.825422\pi\)
0.878184 + 0.478322i \(0.158755\pi\)
\(360\) 0 0
\(361\) −9.01705 15.6180i −0.474582 0.822000i
\(362\) 0 0
\(363\) 31.3461i 1.64524i
\(364\) 0 0
\(365\) −32.1481 −1.68271
\(366\) 0 0
\(367\) −15.9467 27.6205i −0.832411 1.44178i −0.896121 0.443809i \(-0.853627\pi\)
0.0637108 0.997968i \(-0.479706\pi\)
\(368\) 0 0
\(369\) 49.9124 15.7083i 2.59834 0.817740i
\(370\) 0 0
\(371\) −3.71240 2.05663i −0.192738 0.106775i
\(372\) 0 0
\(373\) −3.36868 + 5.83472i −0.174424 + 0.302110i −0.939962 0.341280i \(-0.889140\pi\)
0.765538 + 0.643391i \(0.222473\pi\)
\(374\) 0 0
\(375\) 15.2522 8.80584i 0.787618 0.454731i
\(376\) 0 0
\(377\) 0.0588362 0.00303022
\(378\) 0 0
\(379\) 21.0763 1.08262 0.541308 0.840825i \(-0.317929\pi\)
0.541308 + 0.840825i \(0.317929\pi\)
\(380\) 0 0
\(381\) −34.1247 + 19.7019i −1.74826 + 1.00936i
\(382\) 0 0
\(383\) 8.45613 + 4.88215i 0.432088 + 0.249466i 0.700236 0.713912i \(-0.253078\pi\)
−0.268148 + 0.963378i \(0.586412\pi\)
\(384\) 0 0
\(385\) −8.24626 + 4.95760i −0.420269 + 0.252663i
\(386\) 0 0
\(387\) −9.61356 + 16.6512i −0.488685 + 0.846427i
\(388\) 0 0
\(389\) −8.89671 15.4095i −0.451081 0.781295i 0.547372 0.836889i \(-0.315628\pi\)
−0.998453 + 0.0555939i \(0.982295\pi\)
\(390\) 0 0
\(391\) 17.7705i 0.898695i
\(392\) 0 0
\(393\) 21.8998i 1.10470i
\(394\) 0 0
\(395\) 3.15816 1.82336i 0.158904 0.0917433i
\(396\) 0 0
\(397\) 22.2634 + 12.8538i 1.11737 + 0.645113i 0.940728 0.339162i \(-0.110144\pi\)
0.176641 + 0.984275i \(0.443477\pi\)
\(398\) 0 0
\(399\) 4.47811 + 7.44870i 0.224186 + 0.372901i
\(400\) 0 0
\(401\) 11.6874 20.2432i 0.583642 1.01090i −0.411401 0.911455i \(-0.634960\pi\)
0.995043 0.0994438i \(-0.0317064\pi\)
\(402\) 0 0
\(403\) 28.3844 16.3877i 1.41393 0.816330i
\(404\) 0 0
\(405\) −94.9928 −4.72023
\(406\) 0 0
\(407\) 10.1979i 0.505491i
\(408\) 0 0
\(409\) 8.07479 + 13.9860i 0.399273 + 0.691561i 0.993636 0.112636i \(-0.0359293\pi\)
−0.594363 + 0.804197i \(0.702596\pi\)
\(410\) 0 0
\(411\) 22.6115 39.1643i 1.11534 1.93183i
\(412\) 0 0
\(413\) 8.67869 15.6658i 0.427050 0.770863i
\(414\) 0 0
\(415\) −5.47060 + 9.47536i −0.268541 + 0.465127i
\(416\) 0 0
\(417\) 35.1971 20.3211i 1.72361 0.995126i
\(418\) 0 0
\(419\) −0.145809 −0.00712322 −0.00356161 0.999994i \(-0.501134\pi\)
−0.00356161 + 0.999994i \(0.501134\pi\)
\(420\) 0 0
\(421\) 29.9240i 1.45840i −0.684298 0.729202i \(-0.739891\pi\)
0.684298 0.729202i \(-0.260109\pi\)
\(422\) 0 0
\(423\) −29.2783 + 16.9038i −1.42356 + 0.821893i
\(424\) 0 0
\(425\) −6.21578 3.58868i −0.301509 0.174077i
\(426\) 0 0
\(427\) −17.7920 + 0.314909i −0.861015 + 0.0152395i
\(428\) 0 0
\(429\) −12.5856 7.26627i −0.607636 0.350819i
\(430\) 0 0
\(431\) −2.55062 4.41781i −0.122859 0.212798i 0.798035 0.602611i \(-0.205873\pi\)
−0.920894 + 0.389813i \(0.872540\pi\)
\(432\) 0 0
\(433\) −16.4540 −0.790727 −0.395363 0.918525i \(-0.629381\pi\)
−0.395363 + 0.918525i \(0.629381\pi\)
\(434\) 0 0
\(435\) −0.164489 −0.00788665
\(436\) 0 0
\(437\) −6.64833 + 3.83842i −0.318033 + 0.183616i
\(438\) 0 0
\(439\) 10.7398 + 6.20060i 0.512581 + 0.295939i 0.733894 0.679264i \(-0.237701\pi\)
−0.221313 + 0.975203i \(0.571034\pi\)
\(440\) 0 0
\(441\) 30.3369 + 48.4965i 1.44462 + 2.30936i
\(442\) 0 0
\(443\) −9.08072 + 15.7283i −0.431438 + 0.747272i −0.996997 0.0774351i \(-0.975327\pi\)
0.565559 + 0.824708i \(0.308660\pi\)
\(444\) 0 0
\(445\) 34.0257 19.6447i 1.61297 0.931249i
\(446\) 0 0
\(447\) −3.95961 −0.187283
\(448\) 0 0
\(449\) −0.727275 −0.0343222 −0.0171611 0.999853i \(-0.505463\pi\)
−0.0171611 + 0.999853i \(0.505463\pi\)
\(450\) 0 0
\(451\) −7.96014 1.76914i −0.374828 0.0833056i
\(452\) 0 0
\(453\) 32.8021 56.8148i 1.54118 2.66939i
\(454\) 0 0
\(455\) 0.456488 + 25.7911i 0.0214005 + 1.20910i
\(456\) 0 0
\(457\) −28.1671 16.2623i −1.31760 0.760719i −0.334261 0.942481i \(-0.608487\pi\)
−0.983343 + 0.181762i \(0.941820\pi\)
\(458\) 0 0
\(459\) −19.6639 34.0589i −0.917834 1.58974i
\(460\) 0 0
\(461\) 31.5168 1.46789 0.733943 0.679211i \(-0.237678\pi\)
0.733943 + 0.679211i \(0.237678\pi\)
\(462\) 0 0
\(463\) 39.8842i 1.85358i 0.375583 + 0.926789i \(0.377442\pi\)
−0.375583 + 0.926789i \(0.622558\pi\)
\(464\) 0 0
\(465\) −79.3545 + 45.8154i −3.67998 + 2.12464i
\(466\) 0 0
\(467\) 9.76790 16.9185i 0.452005 0.782895i −0.546506 0.837455i \(-0.684042\pi\)
0.998510 + 0.0545604i \(0.0173757\pi\)
\(468\) 0 0
\(469\) −21.8372 12.0976i −1.00835 0.558615i
\(470\) 0 0
\(471\) 0.905388 1.56818i 0.0417181 0.0722578i
\(472\) 0 0
\(473\) 2.59489 1.49816i 0.119313 0.0688854i
\(474\) 0 0
\(475\) 3.10060i 0.142265i
\(476\) 0 0
\(477\) 13.1085i 0.600196i
\(478\) 0 0
\(479\) −15.8197 + 9.13352i −0.722822 + 0.417321i −0.815790 0.578348i \(-0.803698\pi\)
0.0929686 + 0.995669i \(0.470364\pi\)
\(480\) 0 0
\(481\) −23.6768 13.6698i −1.07957 0.623290i
\(482\) 0 0
\(483\) −59.2013 + 35.5914i −2.69375 + 1.61947i
\(484\) 0 0
\(485\) −26.0662 15.0493i −1.18360 0.683354i
\(486\) 0 0
\(487\) −2.37745 4.11786i −0.107732 0.186598i 0.807119 0.590389i \(-0.201026\pi\)
−0.914851 + 0.403791i \(0.867692\pi\)
\(488\) 0 0
\(489\) 85.0104i 3.84430i
\(490\) 0 0
\(491\) 23.6740 1.06839 0.534196 0.845361i \(-0.320614\pi\)
0.534196 + 0.845361i \(0.320614\pi\)
\(492\) 0 0
\(493\) −0.0196028 0.0339531i −0.000882867 0.00152917i
\(494\) 0 0
\(495\) 25.7372 + 14.8594i 1.15680 + 0.667880i
\(496\) 0 0
\(497\) 7.11325 4.27644i 0.319073 0.191824i
\(498\) 0 0
\(499\) 5.70632 + 3.29454i 0.255450 + 0.147484i 0.622257 0.782813i \(-0.286216\pi\)
−0.366807 + 0.930297i \(0.619549\pi\)
\(500\) 0 0
\(501\) 10.4622 + 18.1211i 0.467418 + 0.809592i
\(502\) 0 0
\(503\) 10.5303i 0.469522i 0.972053 + 0.234761i \(0.0754308\pi\)
−0.972053 + 0.234761i \(0.924569\pi\)
\(504\) 0 0
\(505\) 53.0657i 2.36139i
\(506\) 0 0
\(507\) 3.88963 2.24568i 0.172744 0.0997340i
\(508\) 0 0
\(509\) −31.8073 18.3640i −1.40983 0.813968i −0.414463 0.910066i \(-0.636030\pi\)
−0.995372 + 0.0960979i \(0.969364\pi\)
\(510\) 0 0
\(511\) 14.4336 26.0540i 0.638507 1.15256i
\(512\) 0 0
\(513\) 8.49477 14.7134i 0.375053 0.649611i
\(514\) 0 0
\(515\) 7.60452 + 13.1714i 0.335095 + 0.580402i
\(516\) 0 0
\(517\) 5.26852 0.231709
\(518\) 0 0
\(519\) 62.4522i 2.74135i
\(520\) 0 0
\(521\) −16.3222 + 9.42362i −0.715088 + 0.412856i −0.812942 0.582344i \(-0.802136\pi\)
0.0978540 + 0.995201i \(0.468802\pi\)
\(522\) 0 0
\(523\) −8.62436 + 14.9378i −0.377117 + 0.653185i −0.990641 0.136490i \(-0.956418\pi\)
0.613525 + 0.789676i \(0.289751\pi\)
\(524\) 0 0
\(525\) −0.493726 27.8950i −0.0215480 1.21744i
\(526\) 0 0
\(527\) −18.9140 10.9200i −0.823907 0.475683i
\(528\) 0 0
\(529\) −19.0073 32.9215i −0.826403 1.43137i
\(530\) 0 0
\(531\) −55.3159 −2.40050
\(532\) 0 0
\(533\) −14.7777 + 16.1099i −0.640092 + 0.697795i
\(534\) 0 0
\(535\) 12.6000 + 21.8238i 0.544746 + 0.943527i
\(536\) 0 0
\(537\) −26.3821 + 45.6951i −1.13847 + 1.97189i
\(538\) 0 0
\(539\) −0.315465 8.90890i −0.0135880 0.383733i
\(540\) 0 0
\(541\) 21.6967 37.5798i 0.932814 1.61568i 0.154327 0.988020i \(-0.450679\pi\)
0.778487 0.627661i \(-0.215988\pi\)
\(542\) 0 0
\(543\) 7.27003 + 12.5921i 0.311987 + 0.540377i
\(544\) 0 0
\(545\) 42.1053i 1.80359i
\(546\) 0 0
\(547\) 12.8757i 0.550527i −0.961369 0.275263i \(-0.911235\pi\)
0.961369 0.275263i \(-0.0887650\pi\)
\(548\) 0 0
\(549\) 27.4814 + 47.5991i 1.17288 + 2.03148i
\(550\) 0 0
\(551\) 0.00846838 0.0146677i 0.000360765 0.000624863i
\(552\) 0 0
\(553\) 0.0597910 + 3.37812i 0.00254257 + 0.143652i
\(554\) 0 0
\(555\) 66.1936 + 38.2169i 2.80976 + 1.62222i
\(556\) 0 0
\(557\) 30.3677 17.5328i 1.28672 0.742888i 0.308651 0.951175i \(-0.400122\pi\)
0.978068 + 0.208288i \(0.0667890\pi\)
\(558\) 0 0
\(559\) 8.03284i 0.339753i
\(560\) 0 0
\(561\) 9.68381i 0.408851i
\(562\) 0 0
\(563\) 34.6461 20.0029i 1.46016 0.843024i 0.461142 0.887326i \(-0.347440\pi\)
0.999018 + 0.0443026i \(0.0141066\pi\)
\(564\) 0 0
\(565\) −7.95305 + 13.7751i −0.334587 + 0.579522i
\(566\) 0 0
\(567\) 42.6492 76.9856i 1.79110 3.23309i
\(568\) 0 0
\(569\) −5.60021 + 9.69985i −0.234773 + 0.406639i −0.959207 0.282706i \(-0.908768\pi\)
0.724434 + 0.689345i \(0.242101\pi\)
\(570\) 0 0
\(571\) −24.8072 + 14.3224i −1.03815 + 0.599376i −0.919308 0.393538i \(-0.871251\pi\)
−0.118840 + 0.992913i \(0.537918\pi\)
\(572\) 0 0
\(573\) 73.1080 3.05413
\(574\) 0 0
\(575\) 24.6432 1.02769
\(576\) 0 0
\(577\) 7.53352 4.34948i 0.313625 0.181071i −0.334923 0.942246i \(-0.608710\pi\)
0.648547 + 0.761174i \(0.275377\pi\)
\(578\) 0 0
\(579\) 30.8005 53.3480i 1.28002 2.21707i
\(580\) 0 0
\(581\) −5.22302 8.68776i −0.216687 0.360429i
\(582\) 0 0
\(583\) 1.02140 1.76912i 0.0423021 0.0732693i
\(584\) 0 0
\(585\) 68.9991 39.8366i 2.85276 1.64704i
\(586\) 0 0
\(587\) 8.55700i 0.353185i −0.984284 0.176593i \(-0.943492\pi\)
0.984284 0.176593i \(-0.0565075\pi\)
\(588\) 0 0
\(589\) 9.43483i 0.388755i
\(590\) 0 0
\(591\) 56.3759 32.5487i 2.31900 1.33887i
\(592\) 0 0
\(593\) −28.5590 16.4886i −1.17278 0.677104i −0.218446 0.975849i \(-0.570099\pi\)
−0.954333 + 0.298745i \(0.903432\pi\)
\(594\) 0 0
\(595\) 14.7314 8.85640i 0.603928 0.363077i
\(596\) 0 0
\(597\) 5.31073 9.19846i 0.217354 0.376468i
\(598\) 0 0
\(599\) −13.8917 24.0612i −0.567600 0.983113i −0.996803 0.0799042i \(-0.974539\pi\)
0.429202 0.903208i \(-0.358795\pi\)
\(600\) 0 0
\(601\) 3.12765i 0.127580i −0.997963 0.0637898i \(-0.979681\pi\)
0.997963 0.0637898i \(-0.0203187\pi\)
\(602\) 0 0
\(603\) 77.1072i 3.14005i
\(604\) 0 0
\(605\) 13.3905 + 23.1931i 0.544403 + 0.942933i
\(606\) 0 0
\(607\) −1.66790 + 2.88889i −0.0676980 + 0.117256i −0.897888 0.440225i \(-0.854899\pi\)
0.830190 + 0.557481i \(0.188232\pi\)
\(608\) 0 0
\(609\) 0.0738512 0.133308i 0.00299260 0.00540191i
\(610\) 0 0
\(611\) 7.06221 12.2321i 0.285706 0.494858i
\(612\) 0 0
\(613\) −10.7528 18.6244i −0.434302 0.752233i 0.562936 0.826500i \(-0.309672\pi\)
−0.997238 + 0.0742672i \(0.976338\pi\)
\(614\) 0 0
\(615\) 41.3141 45.0386i 1.66595 1.81613i
\(616\) 0 0
\(617\) 28.5261 1.14842 0.574210 0.818708i \(-0.305309\pi\)
0.574210 + 0.818708i \(0.305309\pi\)
\(618\) 0 0
\(619\) −13.4267 23.2556i −0.539663 0.934723i −0.998922 0.0464209i \(-0.985218\pi\)
0.459259 0.888302i \(-0.348115\pi\)
\(620\) 0 0
\(621\) 116.940 + 67.5154i 4.69265 + 2.70930i
\(622\) 0 0
\(623\) 0.644182 + 36.3956i 0.0258086 + 1.45816i
\(624\) 0 0
\(625\) 15.4106 26.6919i 0.616423 1.06768i
\(626\) 0 0
\(627\) −3.62291 + 2.09169i −0.144685 + 0.0835341i
\(628\) 0 0
\(629\) 18.2178i 0.726393i
\(630\) 0 0
\(631\) 22.7854 0.907072 0.453536 0.891238i \(-0.350162\pi\)
0.453536 + 0.891238i \(0.350162\pi\)
\(632\) 0 0
\(633\) −34.3682 59.5275i −1.36601 2.36601i
\(634\) 0 0
\(635\) −16.8327 + 29.1550i −0.667984 + 1.15698i
\(636\) 0 0
\(637\) −21.1069 11.2095i −0.836288 0.444138i
\(638\) 0 0
\(639\) −22.2010 12.8177i −0.878257 0.507062i
\(640\) 0 0
\(641\) −2.90828 + 1.67910i −0.114870 + 0.0663203i −0.556334 0.830959i \(-0.687793\pi\)
0.441464 + 0.897279i \(0.354459\pi\)
\(642\) 0 0
\(643\) 3.07260i 0.121171i −0.998163 0.0605857i \(-0.980703\pi\)
0.998163 0.0605857i \(-0.0192968\pi\)
\(644\) 0 0
\(645\) 22.4575i 0.884265i
\(646\) 0 0
\(647\) 18.2697 + 31.6440i 0.718255 + 1.24405i 0.961691 + 0.274136i \(0.0883920\pi\)
−0.243436 + 0.969917i \(0.578275\pi\)
\(648\) 0 0
\(649\) 7.46542 + 4.31016i 0.293043 + 0.169189i
\(650\) 0 0
\(651\) −1.50236 84.8816i −0.0588822 3.32677i
\(652\) 0 0
\(653\) 30.6308 + 17.6847i 1.19868 + 0.692056i 0.960260 0.279107i \(-0.0900384\pi\)
0.238416 + 0.971163i \(0.423372\pi\)
\(654\) 0 0
\(655\) −9.35522 16.2037i −0.365539 0.633132i
\(656\) 0 0
\(657\) −91.9965 −3.58913
\(658\) 0 0
\(659\) 39.9277i 1.55536i 0.628659 + 0.777681i \(0.283604\pi\)
−0.628659 + 0.777681i \(0.716396\pi\)
\(660\) 0 0
\(661\) −9.01545 15.6152i −0.350660 0.607362i 0.635705 0.771932i \(-0.280709\pi\)
−0.986365 + 0.164571i \(0.947376\pi\)
\(662\) 0 0
\(663\) 22.4832 + 12.9807i 0.873176 + 0.504128i
\(664\) 0 0
\(665\) 6.49532 + 3.59834i 0.251878 + 0.139538i
\(666\) 0 0
\(667\) 0.116577 + 0.0673056i 0.00451387 + 0.00260608i
\(668\) 0 0
\(669\) −64.5531 + 37.2697i −2.49577 + 1.44093i
\(670\) 0 0
\(671\) 8.56529i 0.330659i
\(672\) 0 0
\(673\) 3.02214i 0.116495i −0.998302 0.0582476i \(-0.981449\pi\)
0.998302 0.0582476i \(-0.0185513\pi\)
\(674\) 0 0
\(675\) −47.2311 + 27.2689i −1.81792 + 1.04958i
\(676\) 0 0
\(677\) 17.2286 29.8409i 0.662151 1.14688i −0.317899 0.948125i \(-0.602977\pi\)
0.980049 0.198754i \(-0.0636895\pi\)
\(678\) 0 0
\(679\) 23.8995 14.3682i 0.917180 0.551402i
\(680\) 0 0
\(681\) 17.4826 30.2807i 0.669933 1.16036i
\(682\) 0 0
\(683\) 9.61335 5.55027i 0.367844 0.212375i −0.304672 0.952457i \(-0.598547\pi\)
0.672516 + 0.740082i \(0.265213\pi\)
\(684\) 0 0
\(685\) 38.6370i 1.47625i
\(686\) 0 0
\(687\) 69.2796 2.64318
\(688\) 0 0
\(689\) −2.73828 4.74284i −0.104320 0.180688i
\(690\) 0 0
\(691\) −25.1725 14.5333i −0.957607 0.552875i −0.0621711 0.998066i \(-0.519802\pi\)
−0.895436 + 0.445191i \(0.853136\pi\)
\(692\) 0 0
\(693\) −23.5979 + 14.1869i −0.896410 + 0.538916i
\(694\) 0 0
\(695\) 17.3616 30.0712i 0.658564 1.14067i
\(696\) 0 0
\(697\) 14.2202 + 3.16045i 0.538629 + 0.119710i
\(698\) 0 0
\(699\) −76.9645 −2.91106
\(700\) 0 0
\(701\) −33.8934 −1.28014 −0.640068 0.768319i \(-0.721094\pi\)
−0.640068 + 0.768319i \(0.721094\pi\)
\(702\) 0 0
\(703\) −6.81567 + 3.93503i −0.257058 + 0.148412i
\(704\) 0 0
\(705\) −19.7439 + 34.1975i −0.743599 + 1.28795i
\(706\) 0 0
\(707\) 43.0064 + 23.8251i 1.61742 + 0.896034i
\(708\) 0 0
\(709\) 22.3457 + 12.9013i 0.839209 + 0.484518i 0.856995 0.515324i \(-0.172329\pi\)
−0.0177863 + 0.999842i \(0.505662\pi\)
\(710\) 0 0
\(711\) 9.03753 5.21782i 0.338934 0.195683i
\(712\) 0 0
\(713\) 74.9869 2.80828
\(714\) 0 0
\(715\) −12.4161 −0.464337
\(716\) 0 0
\(717\) 15.6140 + 27.0443i 0.583117 + 1.00999i
\(718\) 0 0
\(719\) −36.8802 21.2928i −1.37540 0.794086i −0.383796 0.923418i \(-0.625383\pi\)
−0.991601 + 0.129332i \(0.958717\pi\)
\(720\) 0 0
\(721\) −14.0888 + 0.249365i −0.524695 + 0.00928683i
\(722\) 0 0
\(723\) 46.0245 + 26.5723i 1.71167 + 0.988233i
\(724\) 0 0
\(725\) −0.0470843 + 0.0271841i −0.00174867 + 0.00100959i
\(726\) 0 0
\(727\) 18.7582i 0.695704i −0.937549 0.347852i \(-0.886911\pi\)
0.937549 0.347852i \(-0.113089\pi\)
\(728\) 0 0
\(729\) −98.4948 −3.64795
\(730\) 0 0
\(731\) −4.63558 + 2.67636i −0.171453 + 0.0989886i
\(732\) 0 0
\(733\) 7.63365 13.2219i 0.281955 0.488361i −0.689911 0.723894i \(-0.742350\pi\)
0.971866 + 0.235533i \(0.0756837\pi\)
\(734\) 0 0
\(735\) 59.0090 + 31.3387i 2.17658 + 1.15594i
\(736\) 0 0
\(737\) 6.00812 10.4064i 0.221312 0.383324i
\(738\) 0 0
\(739\) 13.4654 + 23.3227i 0.495331 + 0.857939i 0.999986 0.00538236i \(-0.00171327\pi\)
−0.504654 + 0.863322i \(0.668380\pi\)
\(740\) 0 0
\(741\) 11.2153i 0.412003i
\(742\) 0 0
\(743\) −38.2812 −1.40440 −0.702201 0.711979i \(-0.747799\pi\)
−0.702201 + 0.711979i \(0.747799\pi\)
\(744\) 0 0
\(745\) −2.92973 + 1.69148i −0.107337 + 0.0619710i
\(746\) 0 0
\(747\) −15.6549 + 27.1151i −0.572784 + 0.992091i
\(748\) 0 0
\(749\) −23.3439 + 0.413175i −0.852967 + 0.0150971i
\(750\) 0 0
\(751\) 23.9073 + 13.8029i 0.872389 + 0.503674i 0.868142 0.496317i \(-0.165315\pi\)
0.00424786 + 0.999991i \(0.498648\pi\)
\(752\) 0 0
\(753\) 30.2329 17.4550i 1.10175 0.636095i
\(754\) 0 0
\(755\) 56.0500i 2.03987i
\(756\) 0 0
\(757\) 23.3325i 0.848035i −0.905654 0.424017i \(-0.860620\pi\)
0.905654 0.424017i \(-0.139380\pi\)
\(758\) 0 0
\(759\) −16.6245 28.7945i −0.603431 1.04517i
\(760\) 0 0
\(761\) 14.3816 24.9097i 0.521332 0.902974i −0.478360 0.878164i \(-0.658769\pi\)
0.999692 0.0248103i \(-0.00789816\pi\)
\(762\) 0 0
\(763\) −34.1237 18.9042i −1.23536 0.684376i
\(764\) 0 0
\(765\) −45.9777 26.5453i −1.66233 0.959746i
\(766\) 0 0
\(767\) 20.0141 11.5551i 0.722667 0.417232i
\(768\) 0 0
\(769\) 16.1204 0.581318 0.290659 0.956827i \(-0.406126\pi\)
0.290659 + 0.956827i \(0.406126\pi\)
\(770\) 0 0
\(771\) −8.82003 −0.317645
\(772\) 0 0
\(773\) −43.4003 + 25.0572i −1.56100 + 0.901244i −0.563844 + 0.825881i \(0.690678\pi\)
−0.997156 + 0.0753630i \(0.975988\pi\)
\(774\) 0 0
\(775\) −15.1433 + 26.2289i −0.543962 + 0.942170i
\(776\) 0 0
\(777\) −60.6915 + 36.4873i −2.17729 + 1.30897i
\(778\) 0 0
\(779\) 1.88916 + 6.00274i 0.0676863 + 0.215070i
\(780\) 0 0
\(781\) 1.99749 + 3.45976i 0.0714759 + 0.123800i
\(782\) 0 0
\(783\) −0.297907 −0.0106463
\(784\) 0 0
\(785\) 1.54707i 0.0552172i
\(786\) 0 0
\(787\) 13.7910 + 23.8867i 0.491596 + 0.851470i 0.999953 0.00967687i \(-0.00308029\pi\)
−0.508357 + 0.861146i \(0.669747\pi\)
\(788\) 0 0
\(789\) −24.6897 + 42.7638i −0.878976 + 1.52243i
\(790\) 0 0
\(791\) −7.59312 12.6301i −0.269980 0.449074i
\(792\) 0 0
\(793\) −19.8863 11.4814i −0.706184 0.407715i
\(794\) 0 0
\(795\) 7.65544 + 13.2596i 0.271511 + 0.470270i
\(796\) 0 0
\(797\) −37.4626 −1.32699 −0.663496 0.748180i \(-0.730928\pi\)
−0.663496 + 0.748180i \(0.730928\pi\)
\(798\) 0 0
\(799\) −9.41185 −0.332967
\(800\) 0 0
\(801\) 97.3694 56.2163i 3.44038 1.98630i
\(802\) 0 0
\(803\) 12.4158 + 7.16828i 0.438145 + 0.252963i
\(804\) 0 0
\(805\) −28.5992 + 51.6241i −1.00799 + 1.81951i
\(806\) 0 0
\(807\) −20.8262 12.0240i −0.733116 0.423265i
\(808\) 0 0
\(809\) 22.1564 12.7920i 0.778977 0.449743i −0.0570906 0.998369i \(-0.518182\pi\)
0.836068 + 0.548626i \(0.184849\pi\)
\(810\) 0 0
\(811\) 6.33968 0.222616 0.111308 0.993786i \(-0.464496\pi\)
0.111308 + 0.993786i \(0.464496\pi\)
\(812\) 0 0
\(813\) 86.0570i 3.01815i
\(814\) 0 0
\(815\) −36.3150 62.8995i −1.27206 2.20327i
\(816\) 0 0
\(817\) −2.00256 1.15618i −0.0700607 0.0404496i
\(818\) 0 0
\(819\) 1.30631 + 73.8049i 0.0456461 + 2.57895i
\(820\) 0 0
\(821\) 14.9064 25.8187i 0.520238 0.901079i −0.479485 0.877550i \(-0.659177\pi\)
0.999723 0.0235286i \(-0.00749009\pi\)
\(822\) 0 0
\(823\) −20.5749 + 11.8789i −0.717195 + 0.414073i −0.813719 0.581258i \(-0.802561\pi\)
0.0965245 + 0.995331i \(0.469227\pi\)
\(824\) 0 0
\(825\) 13.4290 0.467537
\(826\) 0 0
\(827\) 15.7429i 0.547434i −0.961810 0.273717i \(-0.911747\pi\)
0.961810 0.273717i \(-0.0882531\pi\)
\(828\) 0 0
\(829\) 13.6466 + 23.6366i 0.473965 + 0.820932i 0.999556 0.0298060i \(-0.00948896\pi\)
−0.525591 + 0.850738i \(0.676156\pi\)
\(830\) 0 0
\(831\) 29.2231 + 16.8719i 1.01374 + 0.585281i
\(832\) 0 0
\(833\) 0.563555 + 15.9151i 0.0195260 + 0.551426i
\(834\) 0 0
\(835\) 15.4821 + 8.93859i 0.535780 + 0.309333i
\(836\) 0 0
\(837\) −143.719 + 82.9765i −4.96767 + 2.86809i
\(838\) 0 0
\(839\) 45.3952i 1.56722i 0.621255 + 0.783609i \(0.286623\pi\)
−0.621255 + 0.783609i \(0.713377\pi\)
\(840\) 0 0
\(841\) 28.9997 0.999990
\(842\) 0 0
\(843\) 52.6244 + 91.1481i 1.81248 + 3.13931i
\(844\) 0 0
\(845\) 1.91863 3.32317i 0.0660029 0.114320i
\(846\) 0 0
\(847\) −24.8085 + 0.439098i −0.852431 + 0.0150876i
\(848\) 0 0
\(849\) −63.3197 36.5577i −2.17313 1.25466i
\(850\) 0 0
\(851\) −31.2751 54.1701i −1.07210 1.85693i
\(852\) 0 0
\(853\) −19.9508 −0.683102 −0.341551 0.939863i \(-0.610952\pi\)
−0.341551 + 0.939863i \(0.610952\pi\)
\(854\) 0 0
\(855\) 22.9350i 0.784359i
\(856\) 0 0
\(857\) −4.97078 8.60964i −0.169798 0.294100i 0.768550 0.639789i \(-0.220978\pi\)
−0.938349 + 0.345690i \(0.887645\pi\)
\(858\) 0 0
\(859\) −4.58450 + 7.94059i −0.156421 + 0.270929i −0.933576 0.358381i \(-0.883329\pi\)
0.777154 + 0.629310i \(0.216662\pi\)
\(860\) 0 0
\(861\) 17.9519 + 53.7035i 0.611800 + 1.83021i
\(862\) 0 0
\(863\) 20.1781 34.9495i 0.686871 1.18970i −0.285974 0.958237i \(-0.592317\pi\)
0.972845 0.231458i \(-0.0743496\pi\)
\(864\) 0 0
\(865\) −26.6785 46.2086i −0.907097 1.57114i
\(866\) 0 0
\(867\) 39.5221i 1.34224i
\(868\) 0 0
\(869\) −1.62627 −0.0551674
\(870\) 0 0
\(871\) −16.1072 27.8985i −0.545772 0.945305i
\(872\) 0 0
\(873\) −74.5922 43.0658i −2.52456 1.45756i
\(874\) 0 0
\(875\) 7.18293 + 11.9478i 0.242827 + 0.403909i
\(876\) 0 0
\(877\) −7.63603 + 13.2260i −0.257850 + 0.446610i −0.965666 0.259787i \(-0.916348\pi\)
0.707815 + 0.706397i \(0.249681\pi\)
\(878\) 0 0
\(879\) 4.62902 + 8.01770i 0.156133 + 0.270430i
\(880\) 0 0
\(881\) −24.0412 −0.809967 −0.404984 0.914324i \(-0.632723\pi\)
−0.404984 + 0.914324i \(0.632723\pi\)
\(882\) 0 0
\(883\) 5.11357i 0.172085i 0.996291 + 0.0860426i \(0.0274221\pi\)
−0.996291 + 0.0860426i \(0.972578\pi\)
\(884\) 0 0
\(885\) −55.9537 + 32.3049i −1.88086 + 1.08592i
\(886\) 0 0
\(887\) 12.2220 + 7.05638i 0.410375 + 0.236930i 0.690951 0.722902i \(-0.257192\pi\)
−0.280576 + 0.959832i \(0.590525\pi\)
\(888\) 0 0
\(889\) −16.0709 26.7316i −0.539000 0.896550i
\(890\) 0 0
\(891\) 36.6869 + 21.1812i 1.22906 + 0.709597i
\(892\) 0 0
\(893\) −2.03295 3.52117i −0.0680300 0.117831i
\(894\) 0 0
\(895\) 45.0800i 1.50686i
\(896\) 0 0
\(897\) −89.1375 −2.97621
\(898\) 0 0
\(899\) −0.143273 + 0.0827187i −0.00477842 + 0.00275882i
\(900\) 0 0
\(901\) −1.82466 + 3.16040i −0.0607882 + 0.105288i
\(902\) 0 0
\(903\) −18.2004 10.0828i −0.605671 0.335536i
\(904\) 0 0
\(905\) 10.7582 + 6.21127i 0.357616 + 0.206470i
\(906\) 0 0
\(907\) −12.9635 22.4534i −0.430445 0.745552i 0.566467 0.824085i \(-0.308310\pi\)
−0.996912 + 0.0785324i \(0.974977\pi\)
\(908\) 0 0
\(909\) 151.855i 5.03672i
\(910\) 0 0
\(911\) −12.1565 −0.402763 −0.201382 0.979513i \(-0.564543\pi\)
−0.201382 + 0.979513i \(0.564543\pi\)
\(912\) 0 0
\(913\) 4.22557 2.43964i 0.139846 0.0807401i
\(914\) 0 0
\(915\) 55.5964 + 32.0986i 1.83796 + 1.06115i
\(916\) 0 0
\(917\) 17.3323 0.306773i 0.572363 0.0101305i
\(918\) 0 0
\(919\) −28.2671 16.3200i −0.932445 0.538347i −0.0448608 0.998993i \(-0.514284\pi\)
−0.887584 + 0.460646i \(0.847618\pi\)
\(920\) 0 0
\(921\) −19.9756 + 11.5329i −0.658217 + 0.380022i
\(922\) 0 0
\(923\) 10.7102 0.352530
\(924\) 0 0
\(925\) 25.2635 0.830659
\(926\) 0 0
\(927\) 21.7615 + 37.6920i 0.714740 + 1.23797i
\(928\) 0 0
\(929\) 25.0610 + 14.4690i 0.822224 + 0.474711i 0.851183 0.524870i \(-0.175886\pi\)
−0.0289590 + 0.999581i \(0.509219\pi\)
\(930\) 0 0
\(931\) −5.83245 + 3.64848i −0.191151 + 0.119574i
\(932\) 0 0
\(933\) −0.517621 + 0.896546i −0.0169461 + 0.0293516i
\(934\) 0 0
\(935\) 4.13676 + 7.16508i 0.135287 + 0.234323i
\(936\) 0 0
\(937\) 20.1555i 0.658452i 0.944251 + 0.329226i \(0.106788\pi\)
−0.944251 + 0.329226i \(0.893212\pi\)
\(938\) 0 0
\(939\) −25.3663 −0.827799
\(940\) 0 0
\(941\) 30.3519 + 52.5710i 0.989443 + 1.71376i 0.620230 + 0.784420i \(0.287039\pi\)
0.369213 + 0.929345i \(0.379627\pi\)
\(942\) 0 0
\(943\) −47.7090 + 15.0148i −1.55362 + 0.488950i
\(944\) 0 0
\(945\) −2.31135 130.589i −0.0751883 4.24805i
\(946\) 0 0
\(947\) 3.92380 6.79623i 0.127506 0.220848i −0.795203 0.606343i \(-0.792636\pi\)
0.922710 + 0.385495i \(0.125969\pi\)
\(948\) 0 0
\(949\) 33.2857 19.2175i 1.08050 0.623826i
\(950\) 0 0
\(951\) 61.7853 2.00353
\(952\) 0 0
\(953\) −15.6709 −0.507631 −0.253815 0.967253i \(-0.581686\pi\)
−0.253815 + 0.967253i \(0.581686\pi\)
\(954\) 0 0
\(955\) 54.0928 31.2305i 1.75040 1.01060i
\(956\) 0 0
\(957\) 0.0635269 + 0.0366773i 0.00205353 + 0.00118561i
\(958\) 0 0
\(959\) 31.3129 + 17.3470i 1.01114 + 0.560164i
\(960\) 0 0
\(961\) −30.5794 + 52.9652i −0.986434 + 1.70855i
\(962\) 0 0
\(963\) 36.0568 + 62.4521i 1.16191 + 2.01249i
\(964\) 0 0
\(965\) 52.6298i 1.69421i
\(966\) 0 0
\(967\) 14.7669i 0.474870i 0.971403 + 0.237435i \(0.0763067\pi\)
−0.971403 + 0.237435i \(0.923693\pi\)
\(968\) 0 0
\(969\) 6.47208 3.73666i 0.207913 0.120039i
\(970\) 0 0
\(971\) −8.46712 4.88849i −0.271723 0.156879i 0.357947 0.933742i \(-0.383477\pi\)
−0.629670 + 0.776862i \(0.716810\pi\)
\(972\) 0 0
\(973\) 16.5759 + 27.5717i 0.531399 + 0.883907i
\(974\) 0 0
\(975\) 18.0009 31.1785i 0.576491 0.998511i
\(976\) 0 0
\(977\) −5.95563 + 3.43848i −0.190537 + 0.110007i −0.592234 0.805766i \(-0.701754\pi\)
0.401697 + 0.915773i \(0.368421\pi\)
\(978\) 0 0
\(979\) −17.5213 −0.559982
\(980\) 0 0
\(981\) 120.491i 3.84697i
\(982\) 0 0
\(983\) −16.3611 28.3382i −0.521837 0.903848i −0.999677 0.0254015i \(-0.991914\pi\)
0.477840 0.878447i \(-0.341420\pi\)
\(984\) 0 0
\(985\) 27.8085 48.1657i 0.886052 1.53469i
\(986\) 0 0
\(987\) −18.8504 31.3549i −0.600014 0.998038i
\(988\) 0 0
\(989\) 9.18917 15.9161i 0.292199 0.506103i
\(990\) 0 0
\(991\) −36.8188 + 21.2573i −1.16959 + 0.675262i −0.953583 0.301131i \(-0.902636\pi\)
−0.216004 + 0.976392i \(0.569303\pi\)
\(992\) 0 0
\(993\) 18.3496 0.582306
\(994\) 0 0
\(995\) 9.07463i 0.287685i
\(996\) 0 0
\(997\) −30.9108 + 17.8463i −0.978954 + 0.565199i −0.901954 0.431832i \(-0.857867\pi\)
−0.0769995 + 0.997031i \(0.524534\pi\)
\(998\) 0 0
\(999\) 119.884 + 69.2148i 3.79295 + 2.18986i
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 1148.2.r.a.737.28 yes 56
7.4 even 3 inner 1148.2.r.a.81.1 56
41.40 even 2 inner 1148.2.r.a.737.1 yes 56
287.81 even 6 inner 1148.2.r.a.81.28 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.r.a.81.1 56 7.4 even 3 inner
1148.2.r.a.81.28 yes 56 287.81 even 6 inner
1148.2.r.a.737.1 yes 56 41.40 even 2 inner
1148.2.r.a.737.28 yes 56 1.1 even 1 trivial