Properties

Label 1148.2.k.a.729.2
Level $1148$
Weight $2$
Character 1148.729
Analytic conductor $9.167$
Analytic rank $0$
Dimension $8$
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] = [1148,2,Mod(337,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.k (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 729.2
Root \(1.22833i\) of defining polynomial
Character \(\chi\) \(=\) 1148.729
Dual form 1148.2.k.a.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.868559 + 0.868559i) q^{3} +4.37966i q^{5} +(-0.707107 + 0.707107i) q^{7} +1.49121i q^{9} +O(q^{10})\) \(q+(-0.868559 + 0.868559i) q^{3} +4.37966i q^{5} +(-0.707107 + 0.707107i) q^{7} +1.49121i q^{9} +(1.10701 - 1.10701i) q^{11} +(0.155874 - 0.155874i) q^{13} +(-3.80399 - 3.80399i) q^{15} +(5.27265 + 5.27265i) q^{17} +(5.24822 + 5.24822i) q^{19} -1.22833i q^{21} +5.28835 q^{23} -14.1814 q^{25} +(-3.90088 - 3.90088i) q^{27} +(-4.12690 + 4.12690i) q^{29} +1.03455 q^{31} +1.92300i q^{33} +(-3.09689 - 3.09689i) q^{35} -2.58579 q^{37} +0.270771i q^{39} +(-4.93089 - 4.08489i) q^{41} -8.71688i q^{43} -6.53099 q^{45} +(-2.00789 - 2.00789i) q^{47} -1.00000i q^{49} -9.15922 q^{51} +(5.98800 - 5.98800i) q^{53} +(4.84832 + 4.84832i) q^{55} -9.11678 q^{57} +6.40829 q^{59} -12.4391i q^{61} +(-1.05444 - 1.05444i) q^{63} +(0.682674 + 0.682674i) q^{65} +(-1.54565 - 1.54565i) q^{67} +(-4.59325 + 4.59325i) q^{69} +(0.123981 - 0.123981i) q^{71} +5.51668i q^{73} +(12.3174 - 12.3174i) q^{75} +1.56555i q^{77} +(-8.20809 + 8.20809i) q^{79} +2.30266 q^{81} -12.8108 q^{83} +(-23.0924 + 23.0924i) q^{85} -7.16891i q^{87} +(11.9756 - 11.9756i) q^{89} +0.220439i q^{91} +(-0.898571 + 0.898571i) q^{93} +(-22.9854 + 22.9854i) q^{95} +(6.20844 + 6.20844i) q^{97} +(1.65078 + 1.65078i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 12 q^{11} + 4 q^{13} - 8 q^{15} + 8 q^{17} + 8 q^{19} + 28 q^{23} - 4 q^{25} + 8 q^{27} - 16 q^{29} + 28 q^{31} - 8 q^{35} - 32 q^{37} - 4 q^{45} + 20 q^{47} - 20 q^{51} + 32 q^{53} - 4 q^{55} - 36 q^{57} - 20 q^{59} - 8 q^{63} - 4 q^{67} - 44 q^{69} + 8 q^{71} + 12 q^{75} - 12 q^{79} - 16 q^{81} - 64 q^{83} - 56 q^{85} + 4 q^{89} - 4 q^{93} - 52 q^{95} + 56 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) −0.868559 + 0.868559i −0.501463 + 0.501463i −0.911892 0.410429i \(-0.865379\pi\)
0.410429 + 0.911892i \(0.365379\pi\)
\(4\) 0 0
\(5\) 4.37966i 1.95864i 0.202309 + 0.979322i \(0.435156\pi\)
−0.202309 + 0.979322i \(0.564844\pi\)
\(6\) 0 0
\(7\) −0.707107 + 0.707107i −0.267261 + 0.267261i
\(8\) 0 0
\(9\) 1.49121i 0.497070i
\(10\) 0 0
\(11\) 1.10701 1.10701i 0.333775 0.333775i −0.520243 0.854018i \(-0.674159\pi\)
0.854018 + 0.520243i \(0.174159\pi\)
\(12\) 0 0
\(13\) 0.155874 0.155874i 0.0432316 0.0432316i −0.685161 0.728392i \(-0.740268\pi\)
0.728392 + 0.685161i \(0.240268\pi\)
\(14\) 0 0
\(15\) −3.80399 3.80399i −0.982187 0.982187i
\(16\) 0 0
\(17\) 5.27265 + 5.27265i 1.27881 + 1.27881i 0.941336 + 0.337470i \(0.109571\pi\)
0.337470 + 0.941336i \(0.390429\pi\)
\(18\) 0 0
\(19\) 5.24822 + 5.24822i 1.20402 + 1.20402i 0.972932 + 0.231092i \(0.0742299\pi\)
0.231092 + 0.972932i \(0.425770\pi\)
\(20\) 0 0
\(21\) 1.22833i 0.268043i
\(22\) 0 0
\(23\) 5.28835 1.10270 0.551349 0.834275i \(-0.314113\pi\)
0.551349 + 0.834275i \(0.314113\pi\)
\(24\) 0 0
\(25\) −14.1814 −2.83628
\(26\) 0 0
\(27\) −3.90088 3.90088i −0.750725 0.750725i
\(28\) 0 0
\(29\) −4.12690 + 4.12690i −0.766346 + 0.766346i −0.977461 0.211115i \(-0.932290\pi\)
0.211115 + 0.977461i \(0.432290\pi\)
\(30\) 0 0
\(31\) 1.03455 0.185811 0.0929056 0.995675i \(-0.470385\pi\)
0.0929056 + 0.995675i \(0.470385\pi\)
\(32\) 0 0
\(33\) 1.92300i 0.334752i
\(34\) 0 0
\(35\) −3.09689 3.09689i −0.523470 0.523470i
\(36\) 0 0
\(37\) −2.58579 −0.425101 −0.212550 0.977150i \(-0.568177\pi\)
−0.212550 + 0.977150i \(0.568177\pi\)
\(38\) 0 0
\(39\) 0.270771i 0.0433581i
\(40\) 0 0
\(41\) −4.93089 4.08489i −0.770076 0.637952i
\(42\) 0 0
\(43\) 8.71688i 1.32931i −0.747150 0.664656i \(-0.768578\pi\)
0.747150 0.664656i \(-0.231422\pi\)
\(44\) 0 0
\(45\) −6.53099 −0.973583
\(46\) 0 0
\(47\) −2.00789 2.00789i −0.292881 0.292881i 0.545336 0.838217i \(-0.316402\pi\)
−0.838217 + 0.545336i \(0.816402\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) −9.15922 −1.28255
\(52\) 0 0
\(53\) 5.98800 5.98800i 0.822515 0.822515i −0.163953 0.986468i \(-0.552425\pi\)
0.986468 + 0.163953i \(0.0524246\pi\)
\(54\) 0 0
\(55\) 4.84832 + 4.84832i 0.653747 + 0.653747i
\(56\) 0 0
\(57\) −9.11678 −1.20755
\(58\) 0 0
\(59\) 6.40829 0.834288 0.417144 0.908840i \(-0.363031\pi\)
0.417144 + 0.908840i \(0.363031\pi\)
\(60\) 0 0
\(61\) 12.4391i 1.59266i −0.604862 0.796330i \(-0.706772\pi\)
0.604862 0.796330i \(-0.293228\pi\)
\(62\) 0 0
\(63\) −1.05444 1.05444i −0.132848 0.132848i
\(64\) 0 0
\(65\) 0.682674 + 0.682674i 0.0846753 + 0.0846753i
\(66\) 0 0
\(67\) −1.54565 1.54565i −0.188832 0.188832i 0.606359 0.795191i \(-0.292629\pi\)
−0.795191 + 0.606359i \(0.792629\pi\)
\(68\) 0 0
\(69\) −4.59325 + 4.59325i −0.552962 + 0.552962i
\(70\) 0 0
\(71\) 0.123981 0.123981i 0.0147138 0.0147138i −0.699712 0.714425i \(-0.746688\pi\)
0.714425 + 0.699712i \(0.246688\pi\)
\(72\) 0 0
\(73\) 5.51668i 0.645679i 0.946454 + 0.322839i \(0.104637\pi\)
−0.946454 + 0.322839i \(0.895363\pi\)
\(74\) 0 0
\(75\) 12.3174 12.3174i 1.42229 1.42229i
\(76\) 0 0
\(77\) 1.56555i 0.178410i
\(78\) 0 0
\(79\) −8.20809 + 8.20809i −0.923482 + 0.923482i −0.997274 0.0737917i \(-0.976490\pi\)
0.0737917 + 0.997274i \(0.476490\pi\)
\(80\) 0 0
\(81\) 2.30266 0.255852
\(82\) 0 0
\(83\) −12.8108 −1.40617 −0.703087 0.711104i \(-0.748195\pi\)
−0.703087 + 0.711104i \(0.748195\pi\)
\(84\) 0 0
\(85\) −23.0924 + 23.0924i −2.50473 + 2.50473i
\(86\) 0 0
\(87\) 7.16891i 0.768588i
\(88\) 0 0
\(89\) 11.9756 11.9756i 1.26942 1.26942i 0.323026 0.946390i \(-0.395300\pi\)
0.946390 0.323026i \(-0.104700\pi\)
\(90\) 0 0
\(91\) 0.220439i 0.0231083i
\(92\) 0 0
\(93\) −0.898571 + 0.898571i −0.0931775 + 0.0931775i
\(94\) 0 0
\(95\) −22.9854 + 22.9854i −2.35825 + 2.35825i
\(96\) 0 0
\(97\) 6.20844 + 6.20844i 0.630371 + 0.630371i 0.948161 0.317790i \(-0.102941\pi\)
−0.317790 + 0.948161i \(0.602941\pi\)
\(98\) 0 0
\(99\) 1.65078 + 1.65078i 0.165910 + 0.165910i
\(100\) 0 0
\(101\) 2.82012 + 2.82012i 0.280613 + 0.280613i 0.833353 0.552741i \(-0.186418\pi\)
−0.552741 + 0.833353i \(0.686418\pi\)
\(102\) 0 0
\(103\) 6.30859i 0.621604i 0.950475 + 0.310802i \(0.100598\pi\)
−0.950475 + 0.310802i \(0.899402\pi\)
\(104\) 0 0
\(105\) 5.37966 0.525001
\(106\) 0 0
\(107\) 17.5426 1.69591 0.847956 0.530067i \(-0.177833\pi\)
0.847956 + 0.530067i \(0.177833\pi\)
\(108\) 0 0
\(109\) −10.6850 10.6850i −1.02344 1.02344i −0.999719 0.0237172i \(-0.992450\pi\)
−0.0237172 0.999719i \(-0.507550\pi\)
\(110\) 0 0
\(111\) 2.24591 2.24591i 0.213172 0.213172i
\(112\) 0 0
\(113\) −3.42733 −0.322416 −0.161208 0.986920i \(-0.551539\pi\)
−0.161208 + 0.986920i \(0.551539\pi\)
\(114\) 0 0
\(115\) 23.1612i 2.15979i
\(116\) 0 0
\(117\) 0.232440 + 0.232440i 0.0214891 + 0.0214891i
\(118\) 0 0
\(119\) −7.45666 −0.683551
\(120\) 0 0
\(121\) 8.54907i 0.777188i
\(122\) 0 0
\(123\) 7.83074 0.734808i 0.706074 0.0662554i
\(124\) 0 0
\(125\) 40.2115i 3.59663i
\(126\) 0 0
\(127\) 21.0195 1.86518 0.932591 0.360934i \(-0.117542\pi\)
0.932591 + 0.360934i \(0.117542\pi\)
\(128\) 0 0
\(129\) 7.57112 + 7.57112i 0.666600 + 0.666600i
\(130\) 0 0
\(131\) 4.79121i 0.418610i −0.977850 0.209305i \(-0.932880\pi\)
0.977850 0.209305i \(-0.0671202\pi\)
\(132\) 0 0
\(133\) −7.42210 −0.643578
\(134\) 0 0
\(135\) 17.0845 17.0845i 1.47040 1.47040i
\(136\) 0 0
\(137\) 11.4447 + 11.4447i 0.977783 + 0.977783i 0.999759 0.0219757i \(-0.00699563\pi\)
−0.0219757 + 0.999759i \(0.506996\pi\)
\(138\) 0 0
\(139\) 13.7262 1.16424 0.582122 0.813101i \(-0.302223\pi\)
0.582122 + 0.813101i \(0.302223\pi\)
\(140\) 0 0
\(141\) 3.48794 0.293738
\(142\) 0 0
\(143\) 0.345107i 0.0288593i
\(144\) 0 0
\(145\) −18.0744 18.0744i −1.50100 1.50100i
\(146\) 0 0
\(147\) 0.868559 + 0.868559i 0.0716376 + 0.0716376i
\(148\) 0 0
\(149\) −5.91658 5.91658i −0.484705 0.484705i 0.421925 0.906631i \(-0.361354\pi\)
−0.906631 + 0.421925i \(0.861354\pi\)
\(150\) 0 0
\(151\) −4.68164 + 4.68164i −0.380986 + 0.380986i −0.871457 0.490471i \(-0.836825\pi\)
0.490471 + 0.871457i \(0.336825\pi\)
\(152\) 0 0
\(153\) −7.86263 + 7.86263i −0.635656 + 0.635656i
\(154\) 0 0
\(155\) 4.53099i 0.363938i
\(156\) 0 0
\(157\) −6.83898 + 6.83898i −0.545810 + 0.545810i −0.925226 0.379416i \(-0.876125\pi\)
0.379416 + 0.925226i \(0.376125\pi\)
\(158\) 0 0
\(159\) 10.4019i 0.824921i
\(160\) 0 0
\(161\) −3.73943 + 3.73943i −0.294708 + 0.294708i
\(162\) 0 0
\(163\) 0.0562608 0.00440669 0.00220334 0.999998i \(-0.499299\pi\)
0.00220334 + 0.999998i \(0.499299\pi\)
\(164\) 0 0
\(165\) −8.42210 −0.655660
\(166\) 0 0
\(167\) 17.3830 17.3830i 1.34514 1.34514i 0.454277 0.890860i \(-0.349898\pi\)
0.890860 0.454277i \(-0.150102\pi\)
\(168\) 0 0
\(169\) 12.9514i 0.996262i
\(170\) 0 0
\(171\) −7.82620 + 7.82620i −0.598484 + 0.598484i
\(172\) 0 0
\(173\) 22.1109i 1.68106i 0.541767 + 0.840528i \(0.317755\pi\)
−0.541767 + 0.840528i \(0.682245\pi\)
\(174\) 0 0
\(175\) 10.0278 10.0278i 0.758029 0.758029i
\(176\) 0 0
\(177\) −5.56598 + 5.56598i −0.418364 + 0.418364i
\(178\) 0 0
\(179\) −8.88021 8.88021i −0.663738 0.663738i 0.292521 0.956259i \(-0.405506\pi\)
−0.956259 + 0.292521i \(0.905506\pi\)
\(180\) 0 0
\(181\) 2.09192 + 2.09192i 0.155491 + 0.155491i 0.780565 0.625074i \(-0.214931\pi\)
−0.625074 + 0.780565i \(0.714931\pi\)
\(182\) 0 0
\(183\) 10.8041 + 10.8041i 0.798660 + 0.798660i
\(184\) 0 0
\(185\) 11.3249i 0.832621i
\(186\) 0 0
\(187\) 11.6737 0.853668
\(188\) 0 0
\(189\) 5.51668 0.401279
\(190\) 0 0
\(191\) −7.26811 7.26811i −0.525902 0.525902i 0.393446 0.919348i \(-0.371283\pi\)
−0.919348 + 0.393446i \(0.871283\pi\)
\(192\) 0 0
\(193\) −0.804344 + 0.804344i −0.0578979 + 0.0578979i −0.735463 0.677565i \(-0.763035\pi\)
0.677565 + 0.735463i \(0.263035\pi\)
\(194\) 0 0
\(195\) −1.18589 −0.0849230
\(196\) 0 0
\(197\) 2.18212i 0.155470i 0.996974 + 0.0777349i \(0.0247688\pi\)
−0.996974 + 0.0777349i \(0.975231\pi\)
\(198\) 0 0
\(199\) −1.70711 1.70711i −0.121014 0.121014i 0.644006 0.765020i \(-0.277271\pi\)
−0.765020 + 0.644006i \(0.777271\pi\)
\(200\) 0 0
\(201\) 2.68498 0.189384
\(202\) 0 0
\(203\) 5.83632i 0.409629i
\(204\) 0 0
\(205\) 17.8904 21.5956i 1.24952 1.50830i
\(206\) 0 0
\(207\) 7.88604i 0.548118i
\(208\) 0 0
\(209\) 11.6196 0.803747
\(210\) 0 0
\(211\) −6.36627 6.36627i −0.438272 0.438272i 0.453158 0.891430i \(-0.350297\pi\)
−0.891430 + 0.453158i \(0.850297\pi\)
\(212\) 0 0
\(213\) 0.215369i 0.0147569i
\(214\) 0 0
\(215\) 38.1770 2.60365
\(216\) 0 0
\(217\) −0.731540 + 0.731540i −0.0496602 + 0.0496602i
\(218\) 0 0
\(219\) −4.79156 4.79156i −0.323784 0.323784i
\(220\) 0 0
\(221\) 1.64374 0.110570
\(222\) 0 0
\(223\) 16.4795 1.10355 0.551773 0.833994i \(-0.313951\pi\)
0.551773 + 0.833994i \(0.313951\pi\)
\(224\) 0 0
\(225\) 21.1475i 1.40983i
\(226\) 0 0
\(227\) 11.2174 + 11.2174i 0.744523 + 0.744523i 0.973445 0.228922i \(-0.0735201\pi\)
−0.228922 + 0.973445i \(0.573520\pi\)
\(228\) 0 0
\(229\) −8.73009 8.73009i −0.576900 0.576900i 0.357148 0.934048i \(-0.383749\pi\)
−0.934048 + 0.357148i \(0.883749\pi\)
\(230\) 0 0
\(231\) −1.35977 1.35977i −0.0894662 0.0894662i
\(232\) 0 0
\(233\) 16.3305 16.3305i 1.06985 1.06985i 0.0724782 0.997370i \(-0.476909\pi\)
0.997370 0.0724782i \(-0.0230908\pi\)
\(234\) 0 0
\(235\) 8.79387 8.79387i 0.573649 0.573649i
\(236\) 0 0
\(237\) 14.2584i 0.926184i
\(238\) 0 0
\(239\) −4.12956 + 4.12956i −0.267119 + 0.267119i −0.827938 0.560819i \(-0.810486\pi\)
0.560819 + 0.827938i \(0.310486\pi\)
\(240\) 0 0
\(241\) 10.2473i 0.660084i 0.943966 + 0.330042i \(0.107063\pi\)
−0.943966 + 0.330042i \(0.892937\pi\)
\(242\) 0 0
\(243\) 9.70264 9.70264i 0.622425 0.622425i
\(244\) 0 0
\(245\) 4.37966 0.279806
\(246\) 0 0
\(247\) 1.63612 0.104104
\(248\) 0 0
\(249\) 11.1270 11.1270i 0.705144 0.705144i
\(250\) 0 0
\(251\) 14.1273i 0.891709i 0.895105 + 0.445855i \(0.147100\pi\)
−0.895105 + 0.445855i \(0.852900\pi\)
\(252\) 0 0
\(253\) 5.85425 5.85425i 0.368053 0.368053i
\(254\) 0 0
\(255\) 40.1143i 2.51205i
\(256\) 0 0
\(257\) −7.25507 + 7.25507i −0.452559 + 0.452559i −0.896203 0.443644i \(-0.853685\pi\)
0.443644 + 0.896203i \(0.353685\pi\)
\(258\) 0 0
\(259\) 1.82843 1.82843i 0.113613 0.113613i
\(260\) 0 0
\(261\) −6.15407 6.15407i −0.380928 0.380928i
\(262\) 0 0
\(263\) −10.4488 10.4488i −0.644303 0.644303i 0.307307 0.951610i \(-0.400572\pi\)
−0.951610 + 0.307307i \(0.900572\pi\)
\(264\) 0 0
\(265\) 26.2254 + 26.2254i 1.61101 + 1.61101i
\(266\) 0 0
\(267\) 20.8031i 1.27313i
\(268\) 0 0
\(269\) −18.9583 −1.15591 −0.577955 0.816069i \(-0.696149\pi\)
−0.577955 + 0.816069i \(0.696149\pi\)
\(270\) 0 0
\(271\) 1.58116 0.0960489 0.0480245 0.998846i \(-0.484707\pi\)
0.0480245 + 0.998846i \(0.484707\pi\)
\(272\) 0 0
\(273\) −0.191464 0.191464i −0.0115879 0.0115879i
\(274\) 0 0
\(275\) −15.6989 + 15.6989i −0.946682 + 0.946682i
\(276\) 0 0
\(277\) 4.00642 0.240723 0.120361 0.992730i \(-0.461595\pi\)
0.120361 + 0.992730i \(0.461595\pi\)
\(278\) 0 0
\(279\) 1.54274i 0.0923612i
\(280\) 0 0
\(281\) −6.60198 6.60198i −0.393841 0.393841i 0.482213 0.876054i \(-0.339833\pi\)
−0.876054 + 0.482213i \(0.839833\pi\)
\(282\) 0 0
\(283\) −25.4135 −1.51068 −0.755338 0.655335i \(-0.772527\pi\)
−0.755338 + 0.655335i \(0.772527\pi\)
\(284\) 0 0
\(285\) 39.9284i 2.36515i
\(286\) 0 0
\(287\) 6.37512 0.598218i 0.376311 0.0353117i
\(288\) 0 0
\(289\) 38.6017i 2.27069i
\(290\) 0 0
\(291\) −10.7848 −0.632216
\(292\) 0 0
\(293\) −1.58587 1.58587i −0.0926473 0.0926473i 0.659264 0.751911i \(-0.270868\pi\)
−0.751911 + 0.659264i \(0.770868\pi\)
\(294\) 0 0
\(295\) 28.0661i 1.63407i
\(296\) 0 0
\(297\) −8.63661 −0.501147
\(298\) 0 0
\(299\) 0.824315 0.824315i 0.0476714 0.0476714i
\(300\) 0 0
\(301\) 6.16376 + 6.16376i 0.355273 + 0.355273i
\(302\) 0 0
\(303\) −4.89889 −0.281434
\(304\) 0 0
\(305\) 54.4789 3.11945
\(306\) 0 0
\(307\) 9.98715i 0.569997i −0.958528 0.284998i \(-0.908007\pi\)
0.958528 0.284998i \(-0.0919931\pi\)
\(308\) 0 0
\(309\) −5.47939 5.47939i −0.311711 0.311711i
\(310\) 0 0
\(311\) −8.19335 8.19335i −0.464602 0.464602i 0.435558 0.900160i \(-0.356551\pi\)
−0.900160 + 0.435558i \(0.856551\pi\)
\(312\) 0 0
\(313\) 15.8900 + 15.8900i 0.898156 + 0.898156i 0.995273 0.0971172i \(-0.0309622\pi\)
−0.0971172 + 0.995273i \(0.530962\pi\)
\(314\) 0 0
\(315\) 4.61811 4.61811i 0.260201 0.260201i
\(316\) 0 0
\(317\) −6.85893 + 6.85893i −0.385236 + 0.385236i −0.872984 0.487748i \(-0.837818\pi\)
0.487748 + 0.872984i \(0.337818\pi\)
\(318\) 0 0
\(319\) 9.13702i 0.511575i
\(320\) 0 0
\(321\) −15.2368 + 15.2368i −0.850437 + 0.850437i
\(322\) 0 0
\(323\) 55.3441i 3.07943i
\(324\) 0 0
\(325\) −2.21051 + 2.21051i −0.122617 + 0.122617i
\(326\) 0 0
\(327\) 18.5611 1.02643
\(328\) 0 0
\(329\) 2.83958 0.156551
\(330\) 0 0
\(331\) 10.1612 10.1612i 0.558509 0.558509i −0.370374 0.928883i \(-0.620770\pi\)
0.928883 + 0.370374i \(0.120770\pi\)
\(332\) 0 0
\(333\) 3.85595i 0.211305i
\(334\) 0 0
\(335\) 6.76944 6.76944i 0.369854 0.369854i
\(336\) 0 0
\(337\) 3.07470i 0.167490i 0.996487 + 0.0837448i \(0.0266881\pi\)
−0.996487 + 0.0837448i \(0.973312\pi\)
\(338\) 0 0
\(339\) 2.97684 2.97684i 0.161680 0.161680i
\(340\) 0 0
\(341\) 1.14526 1.14526i 0.0620192 0.0620192i
\(342\) 0 0
\(343\) 0.707107 + 0.707107i 0.0381802 + 0.0381802i
\(344\) 0 0
\(345\) −20.1169 20.1169i −1.08306 1.08306i
\(346\) 0 0
\(347\) −19.7853 19.7853i −1.06213 1.06213i −0.997938 0.0641922i \(-0.979553\pi\)
−0.0641922 0.997938i \(-0.520447\pi\)
\(348\) 0 0
\(349\) 9.92170i 0.531096i −0.964098 0.265548i \(-0.914447\pi\)
0.964098 0.265548i \(-0.0855529\pi\)
\(350\) 0 0
\(351\) −1.21609 −0.0649101
\(352\) 0 0
\(353\) 8.01578 0.426637 0.213318 0.976983i \(-0.431573\pi\)
0.213318 + 0.976983i \(0.431573\pi\)
\(354\) 0 0
\(355\) 0.542994 + 0.542994i 0.0288191 + 0.0288191i
\(356\) 0 0
\(357\) 6.47655 6.47655i 0.342775 0.342775i
\(358\) 0 0
\(359\) −15.4833 −0.817178 −0.408589 0.912718i \(-0.633979\pi\)
−0.408589 + 0.912718i \(0.633979\pi\)
\(360\) 0 0
\(361\) 36.0876i 1.89935i
\(362\) 0 0
\(363\) −7.42537 7.42537i −0.389731 0.389731i
\(364\) 0 0
\(365\) −24.1612 −1.26465
\(366\) 0 0
\(367\) 17.5727i 0.917289i −0.888620 0.458645i \(-0.848335\pi\)
0.888620 0.458645i \(-0.151665\pi\)
\(368\) 0 0
\(369\) 6.09142 7.35300i 0.317107 0.382782i
\(370\) 0 0
\(371\) 8.46831i 0.439653i
\(372\) 0 0
\(373\) 11.1630 0.577997 0.288999 0.957329i \(-0.406678\pi\)
0.288999 + 0.957329i \(0.406678\pi\)
\(374\) 0 0
\(375\) 34.9261 + 34.9261i 1.80358 + 1.80358i
\(376\) 0 0
\(377\) 1.28655i 0.0662607i
\(378\) 0 0
\(379\) 25.3324 1.30124 0.650620 0.759404i \(-0.274509\pi\)
0.650620 + 0.759404i \(0.274509\pi\)
\(380\) 0 0
\(381\) −18.2567 + 18.2567i −0.935320 + 0.935320i
\(382\) 0 0
\(383\) −10.3746 10.3746i −0.530119 0.530119i 0.390489 0.920608i \(-0.372306\pi\)
−0.920608 + 0.390489i \(0.872306\pi\)
\(384\) 0 0
\(385\) −6.85656 −0.349443
\(386\) 0 0
\(387\) 12.9987 0.660761
\(388\) 0 0
\(389\) 9.08901i 0.460831i −0.973092 0.230416i \(-0.925991\pi\)
0.973092 0.230416i \(-0.0740086\pi\)
\(390\) 0 0
\(391\) 27.8836 + 27.8836i 1.41014 + 1.41014i
\(392\) 0 0
\(393\) 4.16145 + 4.16145i 0.209918 + 0.209918i
\(394\) 0 0
\(395\) −35.9486 35.9486i −1.80877 1.80877i
\(396\) 0 0
\(397\) −16.8991 + 16.8991i −0.848144 + 0.848144i −0.989901 0.141758i \(-0.954725\pi\)
0.141758 + 0.989901i \(0.454725\pi\)
\(398\) 0 0
\(399\) 6.44654 6.44654i 0.322730 0.322730i
\(400\) 0 0
\(401\) 17.2355i 0.860701i −0.902662 0.430350i \(-0.858390\pi\)
0.902662 0.430350i \(-0.141610\pi\)
\(402\) 0 0
\(403\) 0.161260 0.161260i 0.00803292 0.00803292i
\(404\) 0 0
\(405\) 10.0849i 0.501122i
\(406\) 0 0
\(407\) −2.86249 + 2.86249i −0.141888 + 0.141888i
\(408\) 0 0
\(409\) 6.14926 0.304061 0.152031 0.988376i \(-0.451419\pi\)
0.152031 + 0.988376i \(0.451419\pi\)
\(410\) 0 0
\(411\) −19.8807 −0.980644
\(412\) 0 0
\(413\) −4.53134 + 4.53134i −0.222973 + 0.222973i
\(414\) 0 0
\(415\) 56.1072i 2.75419i
\(416\) 0 0
\(417\) −11.9220 + 11.9220i −0.583825 + 0.583825i
\(418\) 0 0
\(419\) 18.6826i 0.912705i −0.889799 0.456352i \(-0.849156\pi\)
0.889799 0.456352i \(-0.150844\pi\)
\(420\) 0 0
\(421\) −14.0459 + 14.0459i −0.684554 + 0.684554i −0.961023 0.276469i \(-0.910836\pi\)
0.276469 + 0.961023i \(0.410836\pi\)
\(422\) 0 0
\(423\) 2.99418 2.99418i 0.145582 0.145582i
\(424\) 0 0
\(425\) −74.7737 74.7737i −3.62706 3.62706i
\(426\) 0 0
\(427\) 8.79576 + 8.79576i 0.425656 + 0.425656i
\(428\) 0 0
\(429\) 0.299746 + 0.299746i 0.0144719 + 0.0144719i
\(430\) 0 0
\(431\) 12.8378i 0.618374i 0.951001 + 0.309187i \(0.100057\pi\)
−0.951001 + 0.309187i \(0.899943\pi\)
\(432\) 0 0
\(433\) −5.64434 −0.271250 −0.135625 0.990760i \(-0.543304\pi\)
−0.135625 + 0.990760i \(0.543304\pi\)
\(434\) 0 0
\(435\) 31.3974 1.50539
\(436\) 0 0
\(437\) 27.7544 + 27.7544i 1.32767 + 1.32767i
\(438\) 0 0
\(439\) −9.47844 + 9.47844i −0.452381 + 0.452381i −0.896144 0.443763i \(-0.853643\pi\)
0.443763 + 0.896144i \(0.353643\pi\)
\(440\) 0 0
\(441\) 1.49121 0.0710100
\(442\) 0 0
\(443\) 9.52930i 0.452751i 0.974040 + 0.226375i \(0.0726876\pi\)
−0.974040 + 0.226375i \(0.927312\pi\)
\(444\) 0 0
\(445\) 52.4493 + 52.4493i 2.48633 + 2.48633i
\(446\) 0 0
\(447\) 10.2778 0.486123
\(448\) 0 0
\(449\) 22.0528i 1.04074i 0.853942 + 0.520368i \(0.174205\pi\)
−0.853942 + 0.520368i \(0.825795\pi\)
\(450\) 0 0
\(451\) −9.98054 + 0.936537i −0.469965 + 0.0440998i
\(452\) 0 0
\(453\) 8.13256i 0.382101i
\(454\) 0 0
\(455\) −0.965447 −0.0452608
\(456\) 0 0
\(457\) 18.2147 + 18.2147i 0.852048 + 0.852048i 0.990385 0.138337i \(-0.0441758\pi\)
−0.138337 + 0.990385i \(0.544176\pi\)
\(458\) 0 0
\(459\) 41.1360i 1.92006i
\(460\) 0 0
\(461\) −6.79545 −0.316496 −0.158248 0.987399i \(-0.550585\pi\)
−0.158248 + 0.987399i \(0.550585\pi\)
\(462\) 0 0
\(463\) −5.57774 + 5.57774i −0.259220 + 0.259220i −0.824737 0.565517i \(-0.808677\pi\)
0.565517 + 0.824737i \(0.308677\pi\)
\(464\) 0 0
\(465\) −3.93543 3.93543i −0.182501 0.182501i
\(466\) 0 0
\(467\) 36.3150 1.68046 0.840228 0.542234i \(-0.182421\pi\)
0.840228 + 0.542234i \(0.182421\pi\)
\(468\) 0 0
\(469\) 2.18589 0.100935
\(470\) 0 0
\(471\) 11.8801i 0.547407i
\(472\) 0 0
\(473\) −9.64965 9.64965i −0.443691 0.443691i
\(474\) 0 0
\(475\) −74.4272 74.4272i −3.41496 3.41496i
\(476\) 0 0
\(477\) 8.92936 + 8.92936i 0.408847 + 0.408847i
\(478\) 0 0
\(479\) −5.30292 + 5.30292i −0.242297 + 0.242297i −0.817800 0.575503i \(-0.804806\pi\)
0.575503 + 0.817800i \(0.304806\pi\)
\(480\) 0 0
\(481\) −0.403056 + 0.403056i −0.0183778 + 0.0183778i
\(482\) 0 0
\(483\) 6.49583i 0.295571i
\(484\) 0 0
\(485\) −27.1908 + 27.1908i −1.23467 + 1.23467i
\(486\) 0 0
\(487\) 3.16695i 0.143508i −0.997422 0.0717541i \(-0.977140\pi\)
0.997422 0.0717541i \(-0.0228597\pi\)
\(488\) 0 0
\(489\) −0.0488658 + 0.0488658i −0.00220979 + 0.00220979i
\(490\) 0 0
\(491\) −6.86591 −0.309854 −0.154927 0.987926i \(-0.549514\pi\)
−0.154927 + 0.987926i \(0.549514\pi\)
\(492\) 0 0
\(493\) −43.5194 −1.96002
\(494\) 0 0
\(495\) −7.22986 + 7.22986i −0.324958 + 0.324958i
\(496\) 0 0
\(497\) 0.175335i 0.00786487i
\(498\) 0 0
\(499\) 2.01551 2.01551i 0.0902265 0.0902265i −0.660553 0.750779i \(-0.729678\pi\)
0.750779 + 0.660553i \(0.229678\pi\)
\(500\) 0 0
\(501\) 30.1963i 1.34907i
\(502\) 0 0
\(503\) 12.2797 12.2797i 0.547525 0.547525i −0.378199 0.925724i \(-0.623457\pi\)
0.925724 + 0.378199i \(0.123457\pi\)
\(504\) 0 0
\(505\) −12.3512 + 12.3512i −0.549620 + 0.549620i
\(506\) 0 0
\(507\) −11.2491 11.2491i −0.499588 0.499588i
\(508\) 0 0
\(509\) −0.586823 0.586823i −0.0260105 0.0260105i 0.693982 0.719992i \(-0.255855\pi\)
−0.719992 + 0.693982i \(0.755855\pi\)
\(510\) 0 0
\(511\) −3.90088 3.90088i −0.172565 0.172565i
\(512\) 0 0
\(513\) 40.9454i 1.80778i
\(514\) 0 0
\(515\) −27.6295 −1.21750
\(516\) 0 0
\(517\) −4.44550 −0.195513
\(518\) 0 0
\(519\) −19.2046 19.2046i −0.842988 0.842988i
\(520\) 0 0
\(521\) 1.58336 1.58336i 0.0693684 0.0693684i −0.671571 0.740940i \(-0.734380\pi\)
0.740940 + 0.671571i \(0.234380\pi\)
\(522\) 0 0
\(523\) 30.4893 1.33320 0.666602 0.745414i \(-0.267748\pi\)
0.666602 + 0.745414i \(0.267748\pi\)
\(524\) 0 0
\(525\) 17.4194i 0.760247i
\(526\) 0 0
\(527\) 5.45484 + 5.45484i 0.237617 + 0.237617i
\(528\) 0 0
\(529\) 4.96666 0.215942
\(530\) 0 0
\(531\) 9.55610i 0.414699i
\(532\) 0 0
\(533\) −1.40532 + 0.131870i −0.0608713 + 0.00571194i
\(534\) 0 0
\(535\) 76.8308i 3.32169i
\(536\) 0 0
\(537\) 15.4260 0.665680
\(538\) 0 0
\(539\) −1.10701 1.10701i −0.0476822 0.0476822i
\(540\) 0 0
\(541\) 22.4848i 0.966696i −0.875428 0.483348i \(-0.839420\pi\)
0.875428 0.483348i \(-0.160580\pi\)
\(542\) 0 0
\(543\) −3.63391 −0.155946
\(544\) 0 0
\(545\) 46.7966 46.7966i 2.00455 2.00455i
\(546\) 0 0
\(547\) −11.2934 11.2934i −0.482870 0.482870i 0.423177 0.906047i \(-0.360915\pi\)
−0.906047 + 0.423177i \(0.860915\pi\)
\(548\) 0 0
\(549\) 18.5493 0.791664
\(550\) 0 0
\(551\) −43.3177 −1.84540
\(552\) 0 0
\(553\) 11.6080i 0.493622i
\(554\) 0 0
\(555\) 9.83632 + 9.83632i 0.417528 + 0.417528i
\(556\) 0 0
\(557\) −2.04517 2.04517i −0.0866566 0.0866566i 0.662450 0.749106i \(-0.269517\pi\)
−0.749106 + 0.662450i \(0.769517\pi\)
\(558\) 0 0
\(559\) −1.35873 1.35873i −0.0574682 0.0574682i
\(560\) 0 0
\(561\) −10.1393 + 10.1393i −0.428083 + 0.428083i
\(562\) 0 0
\(563\) 5.39793 5.39793i 0.227496 0.227496i −0.584150 0.811646i \(-0.698572\pi\)
0.811646 + 0.584150i \(0.198572\pi\)
\(564\) 0 0
\(565\) 15.0105i 0.631499i
\(566\) 0 0
\(567\) −1.62823 + 1.62823i −0.0683792 + 0.0683792i
\(568\) 0 0
\(569\) 23.2256i 0.973666i 0.873495 + 0.486833i \(0.161848\pi\)
−0.873495 + 0.486833i \(0.838152\pi\)
\(570\) 0 0
\(571\) −26.4191 + 26.4191i −1.10561 + 1.10561i −0.111884 + 0.993721i \(0.535688\pi\)
−0.993721 + 0.111884i \(0.964312\pi\)
\(572\) 0 0
\(573\) 12.6256 0.527441
\(574\) 0 0
\(575\) −74.9964 −3.12756
\(576\) 0 0
\(577\) 20.0881 20.0881i 0.836279 0.836279i −0.152088 0.988367i \(-0.548600\pi\)
0.988367 + 0.152088i \(0.0485996\pi\)
\(578\) 0 0
\(579\) 1.39724i 0.0580673i
\(580\) 0 0
\(581\) 9.05864 9.05864i 0.375816 0.375816i
\(582\) 0 0
\(583\) 13.2575i 0.549070i
\(584\) 0 0
\(585\) −1.01801 + 1.01801i −0.0420895 + 0.0420895i
\(586\) 0 0
\(587\) 24.7778 24.7778i 1.02269 1.02269i 0.0229544 0.999737i \(-0.492693\pi\)
0.999737 0.0229544i \(-0.00730725\pi\)
\(588\) 0 0
\(589\) 5.42956 + 5.42956i 0.223721 + 0.223721i
\(590\) 0 0
\(591\) −1.89530 1.89530i −0.0779623 0.0779623i
\(592\) 0 0
\(593\) 9.33118 + 9.33118i 0.383185 + 0.383185i 0.872248 0.489063i \(-0.162661\pi\)
−0.489063 + 0.872248i \(0.662661\pi\)
\(594\) 0 0
\(595\) 32.6576i 1.33883i
\(596\) 0 0
\(597\) 2.96545 0.121368
\(598\) 0 0
\(599\) −6.48082 −0.264799 −0.132400 0.991196i \(-0.542268\pi\)
−0.132400 + 0.991196i \(0.542268\pi\)
\(600\) 0 0
\(601\) 19.0624 + 19.0624i 0.777572 + 0.777572i 0.979417 0.201845i \(-0.0646938\pi\)
−0.201845 + 0.979417i \(0.564694\pi\)
\(602\) 0 0
\(603\) 2.30489 2.30489i 0.0938626 0.0938626i
\(604\) 0 0
\(605\) −37.4420 −1.52223
\(606\) 0 0
\(607\) 10.7243i 0.435285i −0.976029 0.217642i \(-0.930163\pi\)
0.976029 0.217642i \(-0.0698367\pi\)
\(608\) 0 0
\(609\) 5.06919 + 5.06919i 0.205414 + 0.205414i
\(610\) 0 0
\(611\) −0.625954 −0.0253234
\(612\) 0 0
\(613\) 32.8034i 1.32492i −0.749099 0.662458i \(-0.769513\pi\)
0.749099 0.662458i \(-0.230487\pi\)
\(614\) 0 0
\(615\) 3.21821 + 34.2960i 0.129771 + 1.38295i
\(616\) 0 0
\(617\) 23.6634i 0.952654i 0.879268 + 0.476327i \(0.158032\pi\)
−0.879268 + 0.476327i \(0.841968\pi\)
\(618\) 0 0
\(619\) 8.18699 0.329063 0.164531 0.986372i \(-0.447389\pi\)
0.164531 + 0.986372i \(0.447389\pi\)
\(620\) 0 0
\(621\) −20.6292 20.6292i −0.827823 0.827823i
\(622\) 0 0
\(623\) 16.9361i 0.678531i
\(624\) 0 0
\(625\) 105.206 4.20823
\(626\) 0 0
\(627\) −10.0923 + 10.0923i −0.403049 + 0.403049i
\(628\) 0 0
\(629\) −13.6340 13.6340i −0.543621 0.543621i
\(630\) 0 0
\(631\) 25.5192 1.01590 0.507952 0.861385i \(-0.330403\pi\)
0.507952 + 0.861385i \(0.330403\pi\)
\(632\) 0 0
\(633\) 11.0590 0.439555
\(634\) 0 0
\(635\) 92.0584i 3.65323i
\(636\) 0 0
\(637\) −0.155874 0.155874i −0.00617594 0.00617594i
\(638\) 0 0
\(639\) 0.184881 + 0.184881i 0.00731380 + 0.00731380i
\(640\) 0 0
\(641\) −16.8663 16.8663i −0.666180 0.666180i 0.290650 0.956830i \(-0.406129\pi\)
−0.956830 + 0.290650i \(0.906129\pi\)
\(642\) 0 0
\(643\) −31.2517 + 31.2517i −1.23245 + 1.23245i −0.269427 + 0.963021i \(0.586834\pi\)
−0.963021 + 0.269427i \(0.913166\pi\)
\(644\) 0 0
\(645\) −33.1590 + 33.1590i −1.30563 + 1.30563i
\(646\) 0 0
\(647\) 29.3661i 1.15450i 0.816567 + 0.577250i \(0.195874\pi\)
−0.816567 + 0.577250i \(0.804126\pi\)
\(648\) 0 0
\(649\) 7.09402 7.09402i 0.278465 0.278465i
\(650\) 0 0
\(651\) 1.27077i 0.0498054i
\(652\) 0 0
\(653\) −22.0154 + 22.0154i −0.861530 + 0.861530i −0.991516 0.129986i \(-0.958507\pi\)
0.129986 + 0.991516i \(0.458507\pi\)
\(654\) 0 0
\(655\) 20.9839 0.819908
\(656\) 0 0
\(657\) −8.22653 −0.320947
\(658\) 0 0
\(659\) 10.7540 10.7540i 0.418917 0.418917i −0.465913 0.884830i \(-0.654274\pi\)
0.884830 + 0.465913i \(0.154274\pi\)
\(660\) 0 0
\(661\) 31.0314i 1.20698i 0.797369 + 0.603492i \(0.206224\pi\)
−0.797369 + 0.603492i \(0.793776\pi\)
\(662\) 0 0
\(663\) −1.42768 + 1.42768i −0.0554466 + 0.0554466i
\(664\) 0 0
\(665\) 32.5063i 1.26054i
\(666\) 0 0
\(667\) −21.8245 + 21.8245i −0.845048 + 0.845048i
\(668\) 0 0
\(669\) −14.3134 + 14.3134i −0.553388 + 0.553388i
\(670\) 0 0
\(671\) −13.7702 13.7702i −0.531591 0.531591i
\(672\) 0 0
\(673\) −13.8292 13.8292i −0.533077 0.533077i 0.388410 0.921487i \(-0.373024\pi\)
−0.921487 + 0.388410i \(0.873024\pi\)
\(674\) 0 0
\(675\) 55.3201 + 55.3201i 2.12927 + 2.12927i
\(676\) 0 0
\(677\) 41.7263i 1.60367i 0.597544 + 0.801836i \(0.296143\pi\)
−0.597544 + 0.801836i \(0.703857\pi\)
\(678\) 0 0
\(679\) −8.78006 −0.336948
\(680\) 0 0
\(681\) −19.4859 −0.746701
\(682\) 0 0
\(683\) 0.539996 + 0.539996i 0.0206624 + 0.0206624i 0.717362 0.696700i \(-0.245349\pi\)
−0.696700 + 0.717362i \(0.745349\pi\)
\(684\) 0 0
\(685\) −50.1237 + 50.1237i −1.91513 + 1.91513i
\(686\) 0 0
\(687\) 15.1652 0.578588
\(688\) 0 0
\(689\) 1.86674i 0.0711172i
\(690\) 0 0
\(691\) −26.7208 26.7208i −1.01651 1.01651i −0.999861 0.0166475i \(-0.994701\pi\)
−0.0166475 0.999861i \(-0.505299\pi\)
\(692\) 0 0
\(693\) −2.33456 −0.0886825
\(694\) 0 0
\(695\) 60.1162i 2.28034i
\(696\) 0 0
\(697\) −4.46070 47.5371i −0.168961 1.80059i
\(698\) 0 0
\(699\) 28.3681i 1.07298i
\(700\) 0 0
\(701\) −28.3918 −1.07234 −0.536172 0.844109i \(-0.680130\pi\)
−0.536172 + 0.844109i \(0.680130\pi\)
\(702\) 0 0
\(703\) −13.5708 13.5708i −0.511831 0.511831i
\(704\) 0 0
\(705\) 15.2760i 0.575327i
\(706\) 0 0
\(707\) −3.98826 −0.149994
\(708\) 0 0
\(709\) −7.16891 + 7.16891i −0.269234 + 0.269234i −0.828792 0.559557i \(-0.810971\pi\)
0.559557 + 0.828792i \(0.310971\pi\)
\(710\) 0 0
\(711\) −12.2400 12.2400i −0.459035 0.459035i
\(712\) 0 0
\(713\) 5.47108 0.204894
\(714\) 0 0
\(715\) 1.51145 0.0565251
\(716\) 0 0
\(717\) 7.17353i 0.267901i
\(718\) 0 0
\(719\) −8.38409 8.38409i −0.312674 0.312674i 0.533271 0.845945i \(-0.320963\pi\)
−0.845945 + 0.533271i \(0.820963\pi\)
\(720\) 0 0
\(721\) −4.46085 4.46085i −0.166131 0.166131i
\(722\) 0 0
\(723\) −8.90035 8.90035i −0.331008 0.331008i
\(724\) 0 0
\(725\) 58.5253 58.5253i 2.17358 2.17358i
\(726\) 0 0
\(727\) −26.1718 + 26.1718i −0.970658 + 0.970658i −0.999582 0.0289234i \(-0.990792\pi\)
0.0289234 + 0.999582i \(0.490792\pi\)
\(728\) 0 0
\(729\) 23.7626i 0.880098i
\(730\) 0 0
\(731\) 45.9611 45.9611i 1.69993 1.69993i
\(732\) 0 0
\(733\) 41.1504i 1.51992i 0.649967 + 0.759962i \(0.274783\pi\)
−0.649967 + 0.759962i \(0.725217\pi\)
\(734\) 0 0
\(735\) −3.80399 + 3.80399i −0.140312 + 0.140312i
\(736\) 0 0
\(737\) −3.42210 −0.126055
\(738\) 0 0
\(739\) 8.37977 0.308255 0.154127 0.988051i \(-0.450743\pi\)
0.154127 + 0.988051i \(0.450743\pi\)
\(740\) 0 0
\(741\) −1.42107 + 1.42107i −0.0522042 + 0.0522042i
\(742\) 0 0
\(743\) 16.7799i 0.615596i −0.951452 0.307798i \(-0.900408\pi\)
0.951452 0.307798i \(-0.0995922\pi\)
\(744\) 0 0
\(745\) 25.9126 25.9126i 0.949365 0.949365i
\(746\) 0 0
\(747\) 19.1037i 0.698966i
\(748\) 0 0
\(749\) −12.4045 + 12.4045i −0.453251 + 0.453251i
\(750\) 0 0
\(751\) −12.1427 + 12.1427i −0.443092 + 0.443092i −0.893050 0.449958i \(-0.851439\pi\)
0.449958 + 0.893050i \(0.351439\pi\)
\(752\) 0 0
\(753\) −12.2704 12.2704i −0.447159 0.447159i
\(754\) 0 0
\(755\) −20.5040 20.5040i −0.746216 0.746216i
\(756\) 0 0
\(757\) −5.89100 5.89100i −0.214112 0.214112i 0.591900 0.806012i \(-0.298378\pi\)
−0.806012 + 0.591900i \(0.798378\pi\)
\(758\) 0 0
\(759\) 10.1695i 0.369130i
\(760\) 0 0
\(761\) 37.8289 1.37130 0.685648 0.727934i \(-0.259519\pi\)
0.685648 + 0.727934i \(0.259519\pi\)
\(762\) 0 0
\(763\) 15.1109 0.547050
\(764\) 0 0
\(765\) −34.4357 34.4357i −1.24502 1.24502i
\(766\) 0 0
\(767\) 0.998883 0.998883i 0.0360676 0.0360676i
\(768\) 0 0
\(769\) −21.4226 −0.772519 −0.386259 0.922390i \(-0.626233\pi\)
−0.386259 + 0.922390i \(0.626233\pi\)
\(770\) 0 0
\(771\) 12.6029i 0.453883i
\(772\) 0 0
\(773\) 18.4236 + 18.4236i 0.662653 + 0.662653i 0.956005 0.293352i \(-0.0947708\pi\)
−0.293352 + 0.956005i \(0.594771\pi\)
\(774\) 0 0
\(775\) −14.6714 −0.527014
\(776\) 0 0
\(777\) 3.17619i 0.113945i
\(778\) 0 0
\(779\) −4.44003 47.3168i −0.159081 1.69530i
\(780\) 0 0
\(781\) 0.274495i 0.00982222i
\(782\) 0 0
\(783\) 32.1971 1.15063
\(784\) 0 0
\(785\) −29.9524 29.9524i −1.06905 1.06905i
\(786\) 0 0
\(787\) 1.81107i 0.0645578i −0.999479 0.0322789i \(-0.989724\pi\)
0.999479 0.0322789i \(-0.0102765\pi\)
\(788\) 0 0
\(789\) 18.1509 0.646189
\(790\) 0 0
\(791\) 2.42349 2.42349i 0.0861694 0.0861694i
\(792\) 0 0
\(793\) −1.93892 1.93892i −0.0688532 0.0688532i
\(794\) 0 0
\(795\) −45.5566 −1.61573
\(796\) 0 0
\(797\) 32.9410 1.16683 0.583415 0.812174i \(-0.301716\pi\)
0.583415 + 0.812174i \(0.301716\pi\)
\(798\) 0 0
\(799\) 21.1738i 0.749075i
\(800\) 0 0
\(801\) 17.8582 + 17.8582i 0.630989 + 0.630989i
\(802\) 0 0
\(803\) 6.10701 + 6.10701i 0.215512 + 0.215512i
\(804\) 0 0
\(805\) −16.3774 16.3774i −0.577229 0.577229i
\(806\) 0 0
\(807\) 16.4664 16.4664i 0.579646 0.579646i
\(808\) 0 0
\(809\) 28.4128 28.4128i 0.998942 0.998942i −0.00105755 0.999999i \(-0.500337\pi\)
0.999999 + 0.00105755i \(0.000336628\pi\)
\(810\) 0 0
\(811\) 40.5163i 1.42272i 0.702829 + 0.711359i \(0.251920\pi\)
−0.702829 + 0.711359i \(0.748080\pi\)
\(812\) 0 0
\(813\) −1.37334 + 1.37334i −0.0481650 + 0.0481650i
\(814\) 0 0
\(815\) 0.246403i 0.00863113i
\(816\) 0 0
\(817\) 45.7481 45.7481i 1.60052 1.60052i
\(818\) 0 0
\(819\) −0.328720 −0.0114864
\(820\) 0 0
\(821\) 12.5128 0.436700 0.218350 0.975871i \(-0.429933\pi\)
0.218350 + 0.975871i \(0.429933\pi\)
\(822\) 0 0
\(823\) −16.7502 + 16.7502i −0.583874 + 0.583874i −0.935966 0.352091i \(-0.885471\pi\)
0.352091 + 0.935966i \(0.385471\pi\)
\(824\) 0 0
\(825\) 27.2709i 0.949452i
\(826\) 0 0
\(827\) 1.64710 1.64710i 0.0572754 0.0572754i −0.677889 0.735164i \(-0.737105\pi\)
0.735164 + 0.677889i \(0.237105\pi\)
\(828\) 0 0
\(829\) 1.13120i 0.0392883i 0.999807 + 0.0196442i \(0.00625334\pi\)
−0.999807 + 0.0196442i \(0.993747\pi\)
\(830\) 0 0
\(831\) −3.47982 + 3.47982i −0.120713 + 0.120713i
\(832\) 0 0
\(833\) 5.27265 5.27265i 0.182687 0.182687i
\(834\) 0 0
\(835\) 76.1317 + 76.1317i 2.63465 + 2.63465i
\(836\) 0 0
\(837\) −4.03567 4.03567i −0.139493 0.139493i
\(838\) 0 0
\(839\) 1.30070 + 1.30070i 0.0449053 + 0.0449053i 0.729203 0.684298i \(-0.239891\pi\)
−0.684298 + 0.729203i \(0.739891\pi\)
\(840\) 0 0
\(841\) 5.06259i 0.174572i
\(842\) 0 0
\(843\) 11.4684 0.394993
\(844\) 0 0
\(845\) −56.7228 −1.95132
\(846\) 0 0
\(847\) −6.04510 6.04510i −0.207712 0.207712i
\(848\) 0 0
\(849\) 22.0731 22.0731i 0.757548 0.757548i
\(850\) 0 0
\(851\) −13.6745 −0.468757
\(852\) 0 0
\(853\) 14.7835i 0.506177i 0.967443 + 0.253088i \(0.0814464\pi\)
−0.967443 + 0.253088i \(0.918554\pi\)
\(854\) 0 0
\(855\) −34.2761 34.2761i −1.17222 1.17222i
\(856\) 0 0
\(857\) 0.788037 0.0269188 0.0134594 0.999909i \(-0.495716\pi\)
0.0134594 + 0.999909i \(0.495716\pi\)
\(858\) 0 0
\(859\) 14.9245i 0.509217i −0.967044 0.254609i \(-0.918053\pi\)
0.967044 0.254609i \(-0.0819466\pi\)
\(860\) 0 0
\(861\) −5.01758 + 6.05676i −0.170999 + 0.206414i
\(862\) 0 0
\(863\) 36.5079i 1.24274i −0.783516 0.621371i \(-0.786576\pi\)
0.783516 0.621371i \(-0.213424\pi\)
\(864\) 0 0
\(865\) −96.8380 −3.29259
\(866\) 0 0
\(867\) −33.5279 33.5279i −1.13867 1.13867i
\(868\) 0 0
\(869\) 18.1728i 0.616471i
\(870\) 0 0
\(871\) −0.481854 −0.0163270
\(872\) 0 0
\(873\) −9.25808 + 9.25808i −0.313339 + 0.313339i
\(874\) 0 0
\(875\) 28.4338 + 28.4338i 0.961239 + 0.961239i
\(876\) 0 0
\(877\) 20.2806 0.684829 0.342414 0.939549i \(-0.388755\pi\)
0.342414 + 0.939549i \(0.388755\pi\)
\(878\) 0 0
\(879\) 2.75484 0.0929184
\(880\) 0 0
\(881\) 55.5345i 1.87101i −0.353318 0.935503i \(-0.614947\pi\)
0.353318 0.935503i \(-0.385053\pi\)
\(882\) 0 0
\(883\) 11.6799 + 11.6799i 0.393060 + 0.393060i 0.875777 0.482717i \(-0.160350\pi\)
−0.482717 + 0.875777i \(0.660350\pi\)
\(884\) 0 0
\(885\) −24.3771 24.3771i −0.819427 0.819427i
\(886\) 0 0
\(887\) 7.12683 + 7.12683i 0.239296 + 0.239296i 0.816558 0.577263i \(-0.195879\pi\)
−0.577263 + 0.816558i \(0.695879\pi\)
\(888\) 0 0
\(889\) −14.8631 + 14.8631i −0.498491 + 0.498491i
\(890\) 0 0
\(891\) 2.54907 2.54907i 0.0853970 0.0853970i
\(892\) 0 0
\(893\) 21.0757i 0.705271i
\(894\) 0 0
\(895\) 38.8923 38.8923i 1.30003 1.30003i
\(896\) 0 0
\(897\) 1.43193i 0.0478108i
\(898\) 0 0
\(899\) −4.26950 + 4.26950i −0.142396 + 0.142396i
\(900\) 0 0
\(901\) 63.1453 2.10367
\(902\) 0 0
\(903\) −10.7072 −0.356313
\(904\) 0 0
\(905\) −9.16188 + 9.16188i −0.304551 + 0.304551i
\(906\) 0 0
\(907\) 13.2975i 0.441538i 0.975326 + 0.220769i \(0.0708567\pi\)
−0.975326 + 0.220769i \(0.929143\pi\)
\(908\) 0 0
\(909\) −4.20539 + 4.20539i −0.139484 + 0.139484i
\(910\) 0 0
\(911\) 18.3284i 0.607247i −0.952792 0.303623i \(-0.901804\pi\)
0.952792 0.303623i \(-0.0981964\pi\)
\(912\) 0 0
\(913\) −14.1817 + 14.1817i −0.469346 + 0.469346i
\(914\) 0 0
\(915\) −47.3182 + 47.3182i −1.56429 + 1.56429i
\(916\) 0 0
\(917\) 3.38790 + 3.38790i 0.111878 + 0.111878i
\(918\) 0 0
\(919\) 10.9667 + 10.9667i 0.361759 + 0.361759i 0.864460 0.502701i \(-0.167660\pi\)
−0.502701 + 0.864460i \(0.667660\pi\)
\(920\) 0 0
\(921\) 8.67443 + 8.67443i 0.285832 + 0.285832i
\(922\) 0 0
\(923\) 0.0386507i 0.00127220i
\(924\) 0 0
\(925\) 36.6701 1.20571
\(926\) 0 0
\(927\) −9.40743 −0.308981
\(928\) 0 0
\(929\) −3.36774 3.36774i −0.110492 0.110492i 0.649699 0.760191i \(-0.274895\pi\)
−0.760191 + 0.649699i \(0.774895\pi\)
\(930\) 0 0
\(931\) 5.24822 5.24822i 0.172003 0.172003i
\(932\) 0 0
\(933\) 14.2328 0.465961
\(934\) 0 0
\(935\) 51.1270i 1.67203i
\(936\) 0 0
\(937\) 4.36515 + 4.36515i 0.142603 + 0.142603i 0.774804 0.632201i \(-0.217848\pi\)
−0.632201 + 0.774804i \(0.717848\pi\)
\(938\) 0 0
\(939\) −27.6028 −0.900784
\(940\) 0 0
\(941\) 18.1833i 0.592758i 0.955070 + 0.296379i \(0.0957791\pi\)
−0.955070 + 0.296379i \(0.904221\pi\)
\(942\) 0 0
\(943\) −26.0763 21.6023i −0.849161 0.703468i
\(944\) 0 0
\(945\) 24.1612i 0.785963i
\(946\) 0 0
\(947\) −9.60761 −0.312205 −0.156103 0.987741i \(-0.549893\pi\)
−0.156103 + 0.987741i \(0.549893\pi\)
\(948\) 0 0
\(949\) 0.859905 + 0.859905i 0.0279137 + 0.0279137i
\(950\) 0 0
\(951\) 11.9148i 0.386363i
\(952\) 0 0
\(953\) 44.1635 1.43060 0.715299 0.698819i \(-0.246291\pi\)
0.715299 + 0.698819i \(0.246291\pi\)
\(954\) 0 0
\(955\) 31.8319 31.8319i 1.03005 1.03005i
\(956\) 0 0
\(957\) −7.93604 7.93604i −0.256536 0.256536i
\(958\) 0 0
\(959\) −16.1852 −0.522647
\(960\) 0 0
\(961\) −29.9297 −0.965474
\(962\) 0 0
\(963\) 26.1598i 0.842987i
\(964\) 0 0
\(965\) −3.52275 3.52275i −0.113401 0.113401i
\(966\) 0 0
\(967\) −17.2127 17.2127i −0.553523 0.553523i 0.373933 0.927456i \(-0.378009\pi\)
−0.927456 + 0.373933i \(0.878009\pi\)
\(968\) 0 0
\(969\) −48.0696 48.0696i −1.54422 1.54422i
\(970\) 0 0
\(971\) 8.80811 8.80811i 0.282666 0.282666i −0.551506 0.834171i \(-0.685946\pi\)
0.834171 + 0.551506i \(0.185946\pi\)
\(972\) 0 0
\(973\) −9.70591 + 9.70591i −0.311157 + 0.311157i
\(974\) 0 0
\(975\) 3.83992i 0.122976i
\(976\) 0 0
\(977\) 38.6771 38.6771i 1.23739 1.23739i 0.276326 0.961064i \(-0.410883\pi\)
0.961064 0.276326i \(-0.0891170\pi\)
\(978\) 0 0
\(979\) 26.5143i 0.847400i
\(980\) 0 0
\(981\) 15.9336 15.9336i 0.508719 0.508719i
\(982\) 0 0
\(983\) 33.8838 1.08072 0.540362 0.841433i \(-0.318287\pi\)
0.540362 + 0.841433i \(0.318287\pi\)
\(984\) 0 0
\(985\) −9.55696 −0.304510
\(986\) 0 0
\(987\) −2.46635 + 2.46635i −0.0785047 + 0.0785047i
\(988\) 0 0
\(989\) 46.0979i 1.46583i
\(990\) 0 0
\(991\) −9.42169 + 9.42169i −0.299290 + 0.299290i −0.840736 0.541446i \(-0.817877\pi\)
0.541446 + 0.840736i \(0.317877\pi\)
\(992\) 0 0
\(993\) 17.6512i 0.560143i
\(994\) 0 0
\(995\) 7.47655 7.47655i 0.237022 0.237022i
\(996\) 0 0
\(997\) 31.6502 31.6502i 1.00237 1.00237i 0.00237415 0.999997i \(-0.499244\pi\)
0.999997 0.00237415i \(-0.000755717\pi\)
\(998\) 0 0
\(999\) 10.0868 + 10.0868i 0.319134 + 0.319134i
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.k.a.729.2 yes 8
41.9 even 4 inner 1148.2.k.a.337.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.k.a.337.2 8 41.9 even 4 inner
1148.2.k.a.729.2 yes 8 1.1 even 1 trivial