Properties

Label 448.4.i.q.193.2
Level $448$
Weight $4$
Character 448.193
Analytic conductor $26.433$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,4,Mod(65,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (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: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 4 x^{10} - 116 x^{9} - 217 x^{8} - 2018 x^{7} + 4474 x^{6} - 105024 x^{5} + \cdots + 292052964 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.2
Root \(-1.98748 + 4.22573i\) of defining polynomial
Character \(\chi\) \(=\) 448.193
Dual form 448.4.i.q.65.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.78574 + 4.82504i) q^{3} +(-2.27739 - 3.94456i) q^{5} +(-3.15583 + 18.2494i) q^{7} +(-2.02069 - 3.49995i) q^{9} +O(q^{10})\) \(q+(-2.78574 + 4.82504i) q^{3} +(-2.27739 - 3.94456i) q^{5} +(-3.15583 + 18.2494i) q^{7} +(-2.02069 - 3.49995i) q^{9} +(32.8812 - 56.9520i) q^{11} -73.6042 q^{13} +25.3769 q^{15} +(-65.5016 + 113.452i) q^{17} +(13.2261 + 22.9083i) q^{19} +(-79.2628 - 66.0651i) q^{21} +(-41.0709 - 71.1369i) q^{23} +(52.1270 - 90.2866i) q^{25} -127.913 q^{27} -4.65118 q^{29} +(40.0340 - 69.3410i) q^{31} +(183.197 + 317.307i) q^{33} +(79.1729 - 29.1127i) q^{35} +(-133.861 - 231.854i) q^{37} +(205.042 - 355.144i) q^{39} +328.215 q^{41} +289.643 q^{43} +(-9.20383 + 15.9415i) q^{45} +(-59.6350 - 103.291i) q^{47} +(-323.082 - 115.184i) q^{49} +(-364.941 - 632.096i) q^{51} +(-75.1052 + 130.086i) q^{53} -299.534 q^{55} -147.378 q^{57} +(211.594 - 366.491i) q^{59} +(-251.152 - 435.008i) q^{61} +(70.2489 - 25.8312i) q^{63} +(167.626 + 290.336i) q^{65} +(195.116 - 337.950i) q^{67} +457.651 q^{69} +803.592 q^{71} +(184.228 - 319.092i) q^{73} +(290.424 + 503.030i) q^{75} +(935.572 + 779.794i) q^{77} +(-276.400 - 478.738i) q^{79} +(410.892 - 711.686i) q^{81} -62.3318 q^{83} +596.691 q^{85} +(12.9570 - 22.4421i) q^{87} +(-237.684 - 411.681i) q^{89} +(232.282 - 1343.23i) q^{91} +(223.049 + 386.332i) q^{93} +(60.2420 - 104.342i) q^{95} +11.8811 q^{97} -265.772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 10 q^{5} + 4 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} - 10 q^{5} + 4 q^{7} - 40 q^{9} + 42 q^{11} + 16 q^{13} - 52 q^{15} - 70 q^{17} + 158 q^{19} - 178 q^{21} + 158 q^{23} - 72 q^{25} - 492 q^{27} + 112 q^{29} - 2 q^{31} - 262 q^{33} + 418 q^{35} - 102 q^{37} + 280 q^{39} + 96 q^{41} - 368 q^{43} - 392 q^{45} - 766 q^{47} + 132 q^{49} - 394 q^{51} + 562 q^{53} + 372 q^{55} + 1316 q^{57} + 854 q^{59} + 106 q^{61} - 1864 q^{63} - 488 q^{65} - 906 q^{67} + 2788 q^{69} + 2976 q^{71} - 202 q^{73} + 2248 q^{75} - 590 q^{77} + 942 q^{79} - 674 q^{81} + 1232 q^{83} - 1996 q^{85} - 1656 q^{87} - 858 q^{89} + 4192 q^{91} - 1362 q^{93} + 2386 q^{95} + 3392 q^{97} - 11504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −2.78574 + 4.82504i −0.536116 + 0.928580i 0.462993 + 0.886362i \(0.346776\pi\)
−0.999108 + 0.0422178i \(0.986558\pi\)
\(4\) 0 0
\(5\) −2.27739 3.94456i −0.203696 0.352812i 0.746020 0.665923i \(-0.231962\pi\)
−0.949717 + 0.313111i \(0.898629\pi\)
\(6\) 0 0
\(7\) −3.15583 + 18.2494i −0.170399 + 0.985375i
\(8\) 0 0
\(9\) −2.02069 3.49995i −0.0748405 0.129628i
\(10\) 0 0
\(11\) 32.8812 56.9520i 0.901279 1.56106i 0.0754429 0.997150i \(-0.475963\pi\)
0.825836 0.563911i \(-0.190704\pi\)
\(12\) 0 0
\(13\) −73.6042 −1.57032 −0.785159 0.619294i \(-0.787419\pi\)
−0.785159 + 0.619294i \(0.787419\pi\)
\(14\) 0 0
\(15\) 25.3769 0.436819
\(16\) 0 0
\(17\) −65.5016 + 113.452i −0.934498 + 1.61860i −0.158973 + 0.987283i \(0.550818\pi\)
−0.775526 + 0.631316i \(0.782515\pi\)
\(18\) 0 0
\(19\) 13.2261 + 22.9083i 0.159699 + 0.276606i 0.934760 0.355280i \(-0.115614\pi\)
−0.775061 + 0.631886i \(0.782281\pi\)
\(20\) 0 0
\(21\) −79.2628 66.0651i −0.823646 0.686504i
\(22\) 0 0
\(23\) −41.0709 71.1369i −0.372342 0.644916i 0.617583 0.786506i \(-0.288112\pi\)
−0.989925 + 0.141590i \(0.954779\pi\)
\(24\) 0 0
\(25\) 52.1270 90.2866i 0.417016 0.722292i
\(26\) 0 0
\(27\) −127.913 −0.911739
\(28\) 0 0
\(29\) −4.65118 −0.0297828 −0.0148914 0.999889i \(-0.504740\pi\)
−0.0148914 + 0.999889i \(0.504740\pi\)
\(30\) 0 0
\(31\) 40.0340 69.3410i 0.231946 0.401742i −0.726435 0.687235i \(-0.758824\pi\)
0.958381 + 0.285493i \(0.0921574\pi\)
\(32\) 0 0
\(33\) 183.197 + 317.307i 0.966380 + 1.67382i
\(34\) 0 0
\(35\) 79.1729 29.1127i 0.382362 0.140598i
\(36\) 0 0
\(37\) −133.861 231.854i −0.594774 1.03018i −0.993579 0.113143i \(-0.963908\pi\)
0.398805 0.917036i \(-0.369425\pi\)
\(38\) 0 0
\(39\) 205.042 355.144i 0.841873 1.45817i
\(40\) 0 0
\(41\) 328.215 1.25021 0.625104 0.780541i \(-0.285057\pi\)
0.625104 + 0.780541i \(0.285057\pi\)
\(42\) 0 0
\(43\) 289.643 1.02721 0.513606 0.858026i \(-0.328309\pi\)
0.513606 + 0.858026i \(0.328309\pi\)
\(44\) 0 0
\(45\) −9.20383 + 15.9415i −0.0304895 + 0.0528093i
\(46\) 0 0
\(47\) −59.6350 103.291i −0.185078 0.320564i 0.758525 0.651644i \(-0.225920\pi\)
−0.943603 + 0.331080i \(0.892587\pi\)
\(48\) 0 0
\(49\) −323.082 115.184i −0.941929 0.335813i
\(50\) 0 0
\(51\) −364.941 632.096i −1.00200 1.73551i
\(52\) 0 0
\(53\) −75.1052 + 130.086i −0.194651 + 0.337145i −0.946786 0.321864i \(-0.895691\pi\)
0.752135 + 0.659009i \(0.229024\pi\)
\(54\) 0 0
\(55\) −299.534 −0.734348
\(56\) 0 0
\(57\) −147.378 −0.342468
\(58\) 0 0
\(59\) 211.594 366.491i 0.466901 0.808696i −0.532384 0.846503i \(-0.678704\pi\)
0.999285 + 0.0378067i \(0.0120371\pi\)
\(60\) 0 0
\(61\) −251.152 435.008i −0.527160 0.913068i −0.999499 0.0316507i \(-0.989924\pi\)
0.472339 0.881417i \(-0.343410\pi\)
\(62\) 0 0
\(63\) 70.2489 25.8312i 0.140485 0.0516576i
\(64\) 0 0
\(65\) 167.626 + 290.336i 0.319868 + 0.554027i
\(66\) 0 0
\(67\) 195.116 337.950i 0.355778 0.616226i −0.631472 0.775398i \(-0.717549\pi\)
0.987251 + 0.159172i \(0.0508824\pi\)
\(68\) 0 0
\(69\) 457.651 0.798475
\(70\) 0 0
\(71\) 803.592 1.34322 0.671611 0.740904i \(-0.265603\pi\)
0.671611 + 0.740904i \(0.265603\pi\)
\(72\) 0 0
\(73\) 184.228 319.092i 0.295373 0.511601i −0.679698 0.733492i \(-0.737889\pi\)
0.975072 + 0.221890i \(0.0712227\pi\)
\(74\) 0 0
\(75\) 290.424 + 503.030i 0.447138 + 0.774465i
\(76\) 0 0
\(77\) 935.572 + 779.794i 1.38465 + 1.15410i
\(78\) 0 0
\(79\) −276.400 478.738i −0.393638 0.681801i 0.599289 0.800533i \(-0.295450\pi\)
−0.992926 + 0.118733i \(0.962117\pi\)
\(80\) 0 0
\(81\) 410.892 711.686i 0.563638 0.976250i
\(82\) 0 0
\(83\) −62.3318 −0.0824313 −0.0412157 0.999150i \(-0.513123\pi\)
−0.0412157 + 0.999150i \(0.513123\pi\)
\(84\) 0 0
\(85\) 596.691 0.761415
\(86\) 0 0
\(87\) 12.9570 22.4421i 0.0159671 0.0276557i
\(88\) 0 0
\(89\) −237.684 411.681i −0.283084 0.490315i 0.689059 0.724705i \(-0.258024\pi\)
−0.972143 + 0.234390i \(0.924691\pi\)
\(90\) 0 0
\(91\) 232.282 1343.23i 0.267580 1.54735i
\(92\) 0 0
\(93\) 223.049 + 386.332i 0.248700 + 0.430761i
\(94\) 0 0
\(95\) 60.2420 104.342i 0.0650600 0.112687i
\(96\) 0 0
\(97\) 11.8811 0.0124365 0.00621824 0.999981i \(-0.498021\pi\)
0.00621824 + 0.999981i \(0.498021\pi\)
\(98\) 0 0
\(99\) −265.772 −0.269809
\(100\) 0 0
\(101\) 367.220 636.043i 0.361779 0.626620i −0.626474 0.779442i \(-0.715503\pi\)
0.988254 + 0.152822i \(0.0488360\pi\)
\(102\) 0 0
\(103\) −561.493 972.535i −0.537141 0.930356i −0.999056 0.0434320i \(-0.986171\pi\)
0.461915 0.886924i \(-0.347163\pi\)
\(104\) 0 0
\(105\) −80.0851 + 463.113i −0.0744334 + 0.430431i
\(106\) 0 0
\(107\) 202.334 + 350.452i 0.182807 + 0.316631i 0.942835 0.333259i \(-0.108148\pi\)
−0.760028 + 0.649890i \(0.774815\pi\)
\(108\) 0 0
\(109\) −972.773 + 1684.89i −0.854814 + 1.48058i 0.0220035 + 0.999758i \(0.492995\pi\)
−0.876817 + 0.480823i \(0.840338\pi\)
\(110\) 0 0
\(111\) 1491.61 1.27547
\(112\) 0 0
\(113\) −1661.80 −1.38344 −0.691722 0.722164i \(-0.743148\pi\)
−0.691722 + 0.722164i \(0.743148\pi\)
\(114\) 0 0
\(115\) −187.069 + 324.013i −0.151689 + 0.262734i
\(116\) 0 0
\(117\) 148.732 + 257.611i 0.117523 + 0.203557i
\(118\) 0 0
\(119\) −1863.72 1553.40i −1.43569 1.19664i
\(120\) 0 0
\(121\) −1496.85 2592.62i −1.12461 1.94788i
\(122\) 0 0
\(123\) −914.321 + 1583.65i −0.670257 + 1.16092i
\(124\) 0 0
\(125\) −1044.20 −0.747170
\(126\) 0 0
\(127\) 2187.49 1.52841 0.764205 0.644973i \(-0.223131\pi\)
0.764205 + 0.644973i \(0.223131\pi\)
\(128\) 0 0
\(129\) −806.870 + 1397.54i −0.550705 + 0.953849i
\(130\) 0 0
\(131\) −45.1460 78.1952i −0.0301101 0.0521523i 0.850578 0.525849i \(-0.176252\pi\)
−0.880688 + 0.473697i \(0.842919\pi\)
\(132\) 0 0
\(133\) −459.801 + 169.074i −0.299773 + 0.110230i
\(134\) 0 0
\(135\) 291.309 + 504.562i 0.185718 + 0.321673i
\(136\) 0 0
\(137\) −683.139 + 1183.23i −0.426018 + 0.737885i −0.996515 0.0834149i \(-0.973417\pi\)
0.570497 + 0.821300i \(0.306751\pi\)
\(138\) 0 0
\(139\) 213.957 0.130558 0.0652791 0.997867i \(-0.479206\pi\)
0.0652791 + 0.997867i \(0.479206\pi\)
\(140\) 0 0
\(141\) 664.511 0.396893
\(142\) 0 0
\(143\) −2420.20 + 4191.91i −1.41529 + 2.45136i
\(144\) 0 0
\(145\) 10.5926 + 18.3469i 0.00606665 + 0.0105077i
\(146\) 0 0
\(147\) 1455.79 1238.01i 0.816812 0.694621i
\(148\) 0 0
\(149\) −1115.35 1931.84i −0.613240 1.06216i −0.990690 0.136134i \(-0.956532\pi\)
0.377450 0.926030i \(-0.376801\pi\)
\(150\) 0 0
\(151\) −440.520 + 763.004i −0.237411 + 0.411208i −0.959971 0.280101i \(-0.909632\pi\)
0.722560 + 0.691309i \(0.242965\pi\)
\(152\) 0 0
\(153\) 529.435 0.279753
\(154\) 0 0
\(155\) −364.693 −0.188986
\(156\) 0 0
\(157\) −1080.82 + 1872.03i −0.549417 + 0.951618i 0.448897 + 0.893583i \(0.351817\pi\)
−0.998315 + 0.0580350i \(0.981516\pi\)
\(158\) 0 0
\(159\) −418.447 724.771i −0.208711 0.361498i
\(160\) 0 0
\(161\) 1427.82 525.024i 0.698931 0.257004i
\(162\) 0 0
\(163\) −1484.49 2571.22i −0.713340 1.23554i −0.963596 0.267361i \(-0.913848\pi\)
0.250257 0.968179i \(-0.419485\pi\)
\(164\) 0 0
\(165\) 834.424 1445.26i 0.393696 0.681901i
\(166\) 0 0
\(167\) −2216.74 −1.02716 −0.513582 0.858041i \(-0.671682\pi\)
−0.513582 + 0.858041i \(0.671682\pi\)
\(168\) 0 0
\(169\) 3220.58 1.46590
\(170\) 0 0
\(171\) 53.4518 92.5812i 0.0239038 0.0414027i
\(172\) 0 0
\(173\) −372.066 644.437i −0.163512 0.283212i 0.772614 0.634877i \(-0.218949\pi\)
−0.936126 + 0.351665i \(0.885616\pi\)
\(174\) 0 0
\(175\) 1483.17 + 1236.21i 0.640670 + 0.533995i
\(176\) 0 0
\(177\) 1178.89 + 2041.90i 0.500626 + 0.867110i
\(178\) 0 0
\(179\) −1456.26 + 2522.32i −0.608080 + 1.05323i 0.383476 + 0.923551i \(0.374727\pi\)
−0.991557 + 0.129675i \(0.958607\pi\)
\(180\) 0 0
\(181\) 2630.43 1.08021 0.540106 0.841597i \(-0.318384\pi\)
0.540106 + 0.841597i \(0.318384\pi\)
\(182\) 0 0
\(183\) 2798.58 1.13048
\(184\) 0 0
\(185\) −609.709 + 1056.05i −0.242306 + 0.419687i
\(186\) 0 0
\(187\) 4307.55 + 7460.89i 1.68449 + 2.91762i
\(188\) 0 0
\(189\) 403.673 2334.34i 0.155359 0.898405i
\(190\) 0 0
\(191\) 720.272 + 1247.55i 0.272864 + 0.472615i 0.969594 0.244719i \(-0.0786957\pi\)
−0.696730 + 0.717334i \(0.745362\pi\)
\(192\) 0 0
\(193\) 468.126 810.817i 0.174593 0.302404i −0.765427 0.643522i \(-0.777472\pi\)
0.940020 + 0.341119i \(0.110806\pi\)
\(194\) 0 0
\(195\) −1867.85 −0.685945
\(196\) 0 0
\(197\) −4679.21 −1.69228 −0.846141 0.532959i \(-0.821080\pi\)
−0.846141 + 0.532959i \(0.821080\pi\)
\(198\) 0 0
\(199\) −1172.71 + 2031.19i −0.417745 + 0.723556i −0.995712 0.0925044i \(-0.970513\pi\)
0.577967 + 0.816060i \(0.303846\pi\)
\(200\) 0 0
\(201\) 1087.08 + 1882.88i 0.381477 + 0.660738i
\(202\) 0 0
\(203\) 14.6783 84.8813i 0.00507496 0.0293473i
\(204\) 0 0
\(205\) −747.474 1294.66i −0.254663 0.441089i
\(206\) 0 0
\(207\) −165.984 + 287.492i −0.0557326 + 0.0965317i
\(208\) 0 0
\(209\) 1739.56 0.575732
\(210\) 0 0
\(211\) −3403.64 −1.11050 −0.555251 0.831683i \(-0.687378\pi\)
−0.555251 + 0.831683i \(0.687378\pi\)
\(212\) 0 0
\(213\) −2238.60 + 3877.36i −0.720123 + 1.24729i
\(214\) 0 0
\(215\) −659.631 1142.51i −0.209239 0.362413i
\(216\) 0 0
\(217\) 1139.09 + 949.426i 0.356344 + 0.297010i
\(218\) 0 0
\(219\) 1026.42 + 1777.82i 0.316709 + 0.548555i
\(220\) 0 0
\(221\) 4821.20 8350.56i 1.46746 2.54172i
\(222\) 0 0
\(223\) 367.463 0.110346 0.0551729 0.998477i \(-0.482429\pi\)
0.0551729 + 0.998477i \(0.482429\pi\)
\(224\) 0 0
\(225\) −421.331 −0.124839
\(226\) 0 0
\(227\) −1152.73 + 1996.58i −0.337045 + 0.583778i −0.983875 0.178855i \(-0.942761\pi\)
0.646831 + 0.762634i \(0.276094\pi\)
\(228\) 0 0
\(229\) 1433.78 + 2483.38i 0.413742 + 0.716622i 0.995295 0.0968860i \(-0.0308882\pi\)
−0.581553 + 0.813508i \(0.697555\pi\)
\(230\) 0 0
\(231\) −6368.80 + 2341.87i −1.81401 + 0.667030i
\(232\) 0 0
\(233\) 814.260 + 1410.34i 0.228944 + 0.396543i 0.957495 0.288448i \(-0.0931394\pi\)
−0.728551 + 0.684991i \(0.759806\pi\)
\(234\) 0 0
\(235\) −271.625 + 470.468i −0.0753994 + 0.130596i
\(236\) 0 0
\(237\) 3079.91 0.844142
\(238\) 0 0
\(239\) 4494.70 1.21648 0.608238 0.793755i \(-0.291877\pi\)
0.608238 + 0.793755i \(0.291877\pi\)
\(240\) 0 0
\(241\) 844.949 1463.49i 0.225842 0.391170i −0.730730 0.682667i \(-0.760820\pi\)
0.956572 + 0.291497i \(0.0941533\pi\)
\(242\) 0 0
\(243\) 562.447 + 974.187i 0.148481 + 0.257177i
\(244\) 0 0
\(245\) 281.433 + 1536.73i 0.0733883 + 0.400728i
\(246\) 0 0
\(247\) −973.496 1686.14i −0.250778 0.434359i
\(248\) 0 0
\(249\) 173.640 300.753i 0.0441928 0.0765441i
\(250\) 0 0
\(251\) −4402.52 −1.10711 −0.553555 0.832813i \(-0.686729\pi\)
−0.553555 + 0.832813i \(0.686729\pi\)
\(252\) 0 0
\(253\) −5401.85 −1.34234
\(254\) 0 0
\(255\) −1662.23 + 2879.06i −0.408207 + 0.707035i
\(256\) 0 0
\(257\) −2736.87 4740.40i −0.664286 1.15058i −0.979478 0.201549i \(-0.935402\pi\)
0.315192 0.949028i \(-0.397931\pi\)
\(258\) 0 0
\(259\) 4653.65 1711.19i 1.11646 0.410534i
\(260\) 0 0
\(261\) 9.39861 + 16.2789i 0.00222896 + 0.00386068i
\(262\) 0 0
\(263\) 2872.36 4975.07i 0.673449 1.16645i −0.303471 0.952841i \(-0.598145\pi\)
0.976920 0.213607i \(-0.0685213\pi\)
\(264\) 0 0
\(265\) 684.176 0.158598
\(266\) 0 0
\(267\) 2648.50 0.607063
\(268\) 0 0
\(269\) 1016.84 1761.22i 0.230475 0.399194i −0.727473 0.686136i \(-0.759305\pi\)
0.957948 + 0.286942i \(0.0926387\pi\)
\(270\) 0 0
\(271\) 772.577 + 1338.14i 0.173176 + 0.299950i 0.939529 0.342471i \(-0.111264\pi\)
−0.766353 + 0.642420i \(0.777930\pi\)
\(272\) 0 0
\(273\) 5834.08 + 4862.67i 1.29339 + 1.07803i
\(274\) 0 0
\(275\) −3428.00 5937.47i −0.751695 1.30197i
\(276\) 0 0
\(277\) −2000.97 + 3465.78i −0.434031 + 0.751763i −0.997216 0.0745674i \(-0.976242\pi\)
0.563185 + 0.826331i \(0.309576\pi\)
\(278\) 0 0
\(279\) −323.586 −0.0694359
\(280\) 0 0
\(281\) −6546.87 −1.38987 −0.694935 0.719072i \(-0.744567\pi\)
−0.694935 + 0.719072i \(0.744567\pi\)
\(282\) 0 0
\(283\) 2155.47 3733.38i 0.452753 0.784192i −0.545802 0.837914i \(-0.683775\pi\)
0.998556 + 0.0537219i \(0.0171084\pi\)
\(284\) 0 0
\(285\) 335.637 + 581.340i 0.0697594 + 0.120827i
\(286\) 0 0
\(287\) −1035.79 + 5989.72i −0.213034 + 1.23192i
\(288\) 0 0
\(289\) −6124.42 10607.8i −1.24657 2.15913i
\(290\) 0 0
\(291\) −33.0975 + 57.3266i −0.00666739 + 0.0115483i
\(292\) 0 0
\(293\) −357.486 −0.0712783 −0.0356392 0.999365i \(-0.511347\pi\)
−0.0356392 + 0.999365i \(0.511347\pi\)
\(294\) 0 0
\(295\) −1927.53 −0.380424
\(296\) 0 0
\(297\) −4205.95 + 7284.92i −0.821731 + 1.42328i
\(298\) 0 0
\(299\) 3022.99 + 5235.98i 0.584696 + 1.01272i
\(300\) 0 0
\(301\) −914.063 + 5285.81i −0.175036 + 1.01219i
\(302\) 0 0
\(303\) 2045.96 + 3543.70i 0.387911 + 0.671882i
\(304\) 0 0
\(305\) −1143.94 + 1981.37i −0.214761 + 0.371977i
\(306\) 0 0
\(307\) −4336.47 −0.806173 −0.403087 0.915162i \(-0.632063\pi\)
−0.403087 + 0.915162i \(0.632063\pi\)
\(308\) 0 0
\(309\) 6256.70 1.15188
\(310\) 0 0
\(311\) 465.530 806.321i 0.0848803 0.147017i −0.820460 0.571704i \(-0.806283\pi\)
0.905340 + 0.424687i \(0.139616\pi\)
\(312\) 0 0
\(313\) −3841.03 6652.85i −0.693635 1.20141i −0.970639 0.240542i \(-0.922675\pi\)
0.277004 0.960869i \(-0.410658\pi\)
\(314\) 0 0
\(315\) −261.877 218.273i −0.0468416 0.0390422i
\(316\) 0 0
\(317\) 2307.54 + 3996.77i 0.408846 + 0.708142i 0.994761 0.102230i \(-0.0325979\pi\)
−0.585914 + 0.810373i \(0.699265\pi\)
\(318\) 0 0
\(319\) −152.937 + 264.894i −0.0268426 + 0.0464928i
\(320\) 0 0
\(321\) −2254.60 −0.392023
\(322\) 0 0
\(323\) −3465.32 −0.596952
\(324\) 0 0
\(325\) −3836.77 + 6645.47i −0.654848 + 1.13423i
\(326\) 0 0
\(327\) −5419.78 9387.34i −0.916559 1.58753i
\(328\) 0 0
\(329\) 2073.20 762.336i 0.347413 0.127748i
\(330\) 0 0
\(331\) 1636.16 + 2833.91i 0.271696 + 0.470591i 0.969296 0.245896i \(-0.0790822\pi\)
−0.697600 + 0.716487i \(0.745749\pi\)
\(332\) 0 0
\(333\) −540.985 + 937.013i −0.0890264 + 0.154198i
\(334\) 0 0
\(335\) −1777.42 −0.289883
\(336\) 0 0
\(337\) 164.690 0.0266209 0.0133104 0.999911i \(-0.495763\pi\)
0.0133104 + 0.999911i \(0.495763\pi\)
\(338\) 0 0
\(339\) 4629.35 8018.26i 0.741686 1.28464i
\(340\) 0 0
\(341\) −2632.74 4560.04i −0.418096 0.724164i
\(342\) 0 0
\(343\) 3121.63 5532.54i 0.491405 0.870931i
\(344\) 0 0
\(345\) −1042.25 1805.23i −0.162646 0.281712i
\(346\) 0 0
\(347\) 877.357 1519.63i 0.135732 0.235095i −0.790145 0.612920i \(-0.789995\pi\)
0.925877 + 0.377825i \(0.123328\pi\)
\(348\) 0 0
\(349\) 4237.15 0.649884 0.324942 0.945734i \(-0.394655\pi\)
0.324942 + 0.945734i \(0.394655\pi\)
\(350\) 0 0
\(351\) 9414.97 1.43172
\(352\) 0 0
\(353\) −2312.34 + 4005.09i −0.348650 + 0.603879i −0.986010 0.166687i \(-0.946693\pi\)
0.637360 + 0.770566i \(0.280026\pi\)
\(354\) 0 0
\(355\) −1830.09 3169.81i −0.273609 0.473905i
\(356\) 0 0
\(357\) 12687.1 4665.17i 1.88087 0.691616i
\(358\) 0 0
\(359\) 1733.64 + 3002.76i 0.254870 + 0.441447i 0.964860 0.262764i \(-0.0846341\pi\)
−0.709990 + 0.704211i \(0.751301\pi\)
\(360\) 0 0
\(361\) 3079.64 5334.10i 0.448993 0.777678i
\(362\) 0 0
\(363\) 16679.4 2.41168
\(364\) 0 0
\(365\) −1678.24 −0.240666
\(366\) 0 0
\(367\) 4098.15 7098.20i 0.582892 1.00960i −0.412242 0.911074i \(-0.635254\pi\)
0.995135 0.0985250i \(-0.0314124\pi\)
\(368\) 0 0
\(369\) −663.222 1148.73i −0.0935662 0.162061i
\(370\) 0 0
\(371\) −2136.97 1781.15i −0.299046 0.249253i
\(372\) 0 0
\(373\) −3277.84 5677.39i −0.455014 0.788107i 0.543675 0.839296i \(-0.317032\pi\)
−0.998689 + 0.0511884i \(0.983699\pi\)
\(374\) 0 0
\(375\) 2908.88 5038.32i 0.400570 0.693807i
\(376\) 0 0
\(377\) 342.346 0.0467685
\(378\) 0 0
\(379\) −10883.3 −1.47503 −0.737517 0.675328i \(-0.764002\pi\)
−0.737517 + 0.675328i \(0.764002\pi\)
\(380\) 0 0
\(381\) −6093.77 + 10554.7i −0.819405 + 1.41925i
\(382\) 0 0
\(383\) −7232.35 12526.8i −0.964898 1.67125i −0.709890 0.704313i \(-0.751255\pi\)
−0.255008 0.966939i \(-0.582078\pi\)
\(384\) 0 0
\(385\) 945.277 5466.32i 0.125132 0.723608i
\(386\) 0 0
\(387\) −585.280 1013.73i −0.0768771 0.133155i
\(388\) 0 0
\(389\) −543.603 + 941.549i −0.0708529 + 0.122721i −0.899275 0.437383i \(-0.855905\pi\)
0.828422 + 0.560104i \(0.189239\pi\)
\(390\) 0 0
\(391\) 10760.8 1.39181
\(392\) 0 0
\(393\) 503.060 0.0645701
\(394\) 0 0
\(395\) −1258.94 + 2180.55i −0.160365 + 0.277760i
\(396\) 0 0
\(397\) −680.622 1178.87i −0.0860439 0.149032i 0.819792 0.572662i \(-0.194089\pi\)
−0.905836 + 0.423629i \(0.860756\pi\)
\(398\) 0 0
\(399\) 465.099 2689.56i 0.0583560 0.337459i
\(400\) 0 0
\(401\) 551.902 + 955.923i 0.0687299 + 0.119044i 0.898342 0.439296i \(-0.144772\pi\)
−0.829613 + 0.558339i \(0.811439\pi\)
\(402\) 0 0
\(403\) −2946.67 + 5103.79i −0.364229 + 0.630863i
\(404\) 0 0
\(405\) −3743.05 −0.459244
\(406\) 0 0
\(407\) −17606.1 −2.14423
\(408\) 0 0
\(409\) −3738.13 + 6474.63i −0.451928 + 0.782763i −0.998506 0.0546457i \(-0.982597\pi\)
0.546577 + 0.837409i \(0.315930\pi\)
\(410\) 0 0
\(411\) −3806.09 6592.35i −0.456790 0.791184i
\(412\) 0 0
\(413\) 6020.49 + 5018.04i 0.717310 + 0.597873i
\(414\) 0 0
\(415\) 141.954 + 245.871i 0.0167909 + 0.0290828i
\(416\) 0 0
\(417\) −596.028 + 1032.35i −0.0699943 + 0.121234i
\(418\) 0 0
\(419\) −1048.13 −0.122206 −0.0611030 0.998131i \(-0.519462\pi\)
−0.0611030 + 0.998131i \(0.519462\pi\)
\(420\) 0 0
\(421\) 1023.21 0.118452 0.0592259 0.998245i \(-0.481137\pi\)
0.0592259 + 0.998245i \(0.481137\pi\)
\(422\) 0 0
\(423\) −241.008 + 417.439i −0.0277027 + 0.0479824i
\(424\) 0 0
\(425\) 6828.80 + 11827.8i 0.779401 + 1.34996i
\(426\) 0 0
\(427\) 8731.24 3210.57i 0.989542 0.363865i
\(428\) 0 0
\(429\) −13484.1 23355.1i −1.51752 2.62843i
\(430\) 0 0
\(431\) −4049.26 + 7013.52i −0.452543 + 0.783827i −0.998543 0.0539578i \(-0.982816\pi\)
0.546000 + 0.837785i \(0.316150\pi\)
\(432\) 0 0
\(433\) 14919.0 1.65580 0.827901 0.560875i \(-0.189535\pi\)
0.827901 + 0.560875i \(0.189535\pi\)
\(434\) 0 0
\(435\) −118.032 −0.0130097
\(436\) 0 0
\(437\) 1086.41 1881.73i 0.118925 0.205984i
\(438\) 0 0
\(439\) 6719.81 + 11639.1i 0.730568 + 1.26538i 0.956641 + 0.291270i \(0.0940778\pi\)
−0.226073 + 0.974110i \(0.572589\pi\)
\(440\) 0 0
\(441\) 249.711 + 1363.52i 0.0269638 + 0.147232i
\(442\) 0 0
\(443\) 791.275 + 1370.53i 0.0848637 + 0.146988i 0.905333 0.424702i \(-0.139621\pi\)
−0.820469 + 0.571690i \(0.806288\pi\)
\(444\) 0 0
\(445\) −1082.60 + 1875.12i −0.115326 + 0.199751i
\(446\) 0 0
\(447\) 12428.3 1.31507
\(448\) 0 0
\(449\) −497.311 −0.0522707 −0.0261353 0.999658i \(-0.508320\pi\)
−0.0261353 + 0.999658i \(0.508320\pi\)
\(450\) 0 0
\(451\) 10792.1 18692.5i 1.12679 1.95165i
\(452\) 0 0
\(453\) −2454.35 4251.06i −0.254560 0.440910i
\(454\) 0 0
\(455\) −5827.46 + 2142.82i −0.600430 + 0.220784i
\(456\) 0 0
\(457\) −1170.79 2027.87i −0.119841 0.207571i 0.799863 0.600182i \(-0.204905\pi\)
−0.919705 + 0.392611i \(0.871572\pi\)
\(458\) 0 0
\(459\) 8378.54 14512.1i 0.852019 1.47574i
\(460\) 0 0
\(461\) 16842.4 1.70158 0.850792 0.525503i \(-0.176123\pi\)
0.850792 + 0.525503i \(0.176123\pi\)
\(462\) 0 0
\(463\) −4464.62 −0.448139 −0.224069 0.974573i \(-0.571934\pi\)
−0.224069 + 0.974573i \(0.571934\pi\)
\(464\) 0 0
\(465\) 1015.94 1759.66i 0.101318 0.175489i
\(466\) 0 0
\(467\) −8175.33 14160.1i −0.810084 1.40311i −0.912805 0.408396i \(-0.866088\pi\)
0.102721 0.994710i \(-0.467245\pi\)
\(468\) 0 0
\(469\) 5551.64 + 4627.25i 0.546590 + 0.455579i
\(470\) 0 0
\(471\) −6021.74 10430.0i −0.589102 1.02036i
\(472\) 0 0
\(473\) 9523.82 16495.7i 0.925805 1.60354i
\(474\) 0 0
\(475\) 2757.74 0.266387
\(476\) 0 0
\(477\) 607.058 0.0582711
\(478\) 0 0
\(479\) 3563.37 6171.93i 0.339905 0.588732i −0.644510 0.764596i \(-0.722938\pi\)
0.984415 + 0.175864i \(0.0562718\pi\)
\(480\) 0 0
\(481\) 9852.75 + 17065.5i 0.933984 + 1.61771i
\(482\) 0 0
\(483\) −1444.27 + 8351.87i −0.136059 + 0.786797i
\(484\) 0 0
\(485\) −27.0578 46.8655i −0.00253326 0.00438774i
\(486\) 0 0
\(487\) 6869.62 11898.5i 0.639204 1.10713i −0.346404 0.938085i \(-0.612597\pi\)
0.985608 0.169048i \(-0.0540693\pi\)
\(488\) 0 0
\(489\) 16541.6 1.52973
\(490\) 0 0
\(491\) −10840.2 −0.996362 −0.498181 0.867073i \(-0.665998\pi\)
−0.498181 + 0.867073i \(0.665998\pi\)
\(492\) 0 0
\(493\) 304.660 527.686i 0.0278320 0.0482065i
\(494\) 0 0
\(495\) 605.267 + 1048.35i 0.0549590 + 0.0951918i
\(496\) 0 0
\(497\) −2536.00 + 14665.1i −0.228883 + 1.32358i
\(498\) 0 0
\(499\) 8638.65 + 14962.6i 0.774989 + 1.34232i 0.934801 + 0.355172i \(0.115578\pi\)
−0.159812 + 0.987147i \(0.551089\pi\)
\(500\) 0 0
\(501\) 6175.26 10695.9i 0.550679 0.953804i
\(502\) 0 0
\(503\) −8368.03 −0.741774 −0.370887 0.928678i \(-0.620946\pi\)
−0.370887 + 0.928678i \(0.620946\pi\)
\(504\) 0 0
\(505\) −3345.21 −0.294772
\(506\) 0 0
\(507\) −8971.70 + 15539.4i −0.785892 + 1.36121i
\(508\) 0 0
\(509\) −6998.07 12121.0i −0.609398 1.05551i −0.991340 0.131323i \(-0.958078\pi\)
0.381941 0.924187i \(-0.375256\pi\)
\(510\) 0 0
\(511\) 5241.85 + 4369.05i 0.453788 + 0.378230i
\(512\) 0 0
\(513\) −1691.79 2930.27i −0.145603 0.252192i
\(514\) 0 0
\(515\) −2557.48 + 4429.69i −0.218827 + 0.379020i
\(516\) 0 0
\(517\) −7843.50 −0.667228
\(518\) 0 0
\(519\) 4145.92 0.350647
\(520\) 0 0
\(521\) −10463.1 + 18122.6i −0.879838 + 1.52392i −0.0283192 + 0.999599i \(0.509015\pi\)
−0.851518 + 0.524325i \(0.824318\pi\)
\(522\) 0 0
\(523\) 2297.44 + 3979.28i 0.192084 + 0.332699i 0.945941 0.324340i \(-0.105142\pi\)
−0.753857 + 0.657039i \(0.771809\pi\)
\(524\) 0 0
\(525\) −10096.5 + 3712.60i −0.839330 + 0.308630i
\(526\) 0 0
\(527\) 5244.59 + 9083.89i 0.433506 + 0.750855i
\(528\) 0 0
\(529\) 2709.86 4693.62i 0.222722 0.385766i
\(530\) 0 0
\(531\) −1710.27 −0.139772
\(532\) 0 0
\(533\) −24158.0 −1.96322
\(534\) 0 0
\(535\) 921.587 1596.23i 0.0744741 0.128993i
\(536\) 0 0
\(537\) −8113.55 14053.1i −0.652003 1.12930i
\(538\) 0 0
\(539\) −17183.3 + 14612.7i −1.37317 + 1.16775i
\(540\) 0 0
\(541\) 5490.24 + 9509.37i 0.436310 + 0.755711i 0.997402 0.0720427i \(-0.0229518\pi\)
−0.561092 + 0.827754i \(0.689618\pi\)
\(542\) 0 0
\(543\) −7327.70 + 12692.0i −0.579119 + 1.00306i
\(544\) 0 0
\(545\) 8861.54 0.696489
\(546\) 0 0
\(547\) 20353.1 1.59092 0.795462 0.606003i \(-0.207228\pi\)
0.795462 + 0.606003i \(0.207228\pi\)
\(548\) 0 0
\(549\) −1015.00 + 1758.04i −0.0789058 + 0.136669i
\(550\) 0 0
\(551\) −61.5169 106.550i −0.00475628 0.00823811i
\(552\) 0 0
\(553\) 9608.95 3533.31i 0.738905 0.271703i
\(554\) 0 0
\(555\) −3396.98 5883.74i −0.259809 0.450002i
\(556\) 0 0
\(557\) −5784.40 + 10018.9i −0.440024 + 0.762143i −0.997691 0.0679216i \(-0.978363\pi\)
0.557667 + 0.830065i \(0.311697\pi\)
\(558\) 0 0
\(559\) −21318.9 −1.61305
\(560\) 0 0
\(561\) −47998.8 −3.61232
\(562\) 0 0
\(563\) −4159.87 + 7205.11i −0.311399 + 0.539359i −0.978666 0.205460i \(-0.934131\pi\)
0.667266 + 0.744819i \(0.267464\pi\)
\(564\) 0 0
\(565\) 3784.57 + 6555.07i 0.281802 + 0.488096i
\(566\) 0 0
\(567\) 11691.1 + 9744.50i 0.865930 + 0.721747i
\(568\) 0 0
\(569\) −6839.67 11846.7i −0.503926 0.872826i −0.999990 0.00453945i \(-0.998555\pi\)
0.496064 0.868286i \(-0.334778\pi\)
\(570\) 0 0
\(571\) 2980.95 5163.16i 0.218475 0.378409i −0.735867 0.677126i \(-0.763225\pi\)
0.954342 + 0.298717i \(0.0965586\pi\)
\(572\) 0 0
\(573\) −8025.96 −0.585147
\(574\) 0 0
\(575\) −8563.61 −0.621091
\(576\) 0 0
\(577\) −9122.89 + 15801.3i −0.658217 + 1.14006i 0.322860 + 0.946447i \(0.395356\pi\)
−0.981077 + 0.193618i \(0.937978\pi\)
\(578\) 0 0
\(579\) 2608.15 + 4517.45i 0.187204 + 0.324247i
\(580\) 0 0
\(581\) 196.708 1137.52i 0.0140462 0.0812258i
\(582\) 0 0
\(583\) 4939.10 + 8554.78i 0.350869 + 0.607723i
\(584\) 0 0
\(585\) 677.441 1173.36i 0.0478782 0.0829274i
\(586\) 0 0
\(587\) 2476.93 0.174163 0.0870816 0.996201i \(-0.472246\pi\)
0.0870816 + 0.996201i \(0.472246\pi\)
\(588\) 0 0
\(589\) 2117.97 0.148166
\(590\) 0 0
\(591\) 13035.1 22577.4i 0.907260 1.57142i
\(592\) 0 0
\(593\) −563.058 975.245i −0.0389916 0.0675354i 0.845871 0.533387i \(-0.179081\pi\)
−0.884863 + 0.465852i \(0.845748\pi\)
\(594\) 0 0
\(595\) −1883.06 + 10889.3i −0.129744 + 0.750279i
\(596\) 0 0
\(597\) −6533.73 11316.8i −0.447919 0.775819i
\(598\) 0 0
\(599\) −3269.00 + 5662.08i −0.222985 + 0.386221i −0.955713 0.294301i \(-0.904913\pi\)
0.732728 + 0.680521i \(0.238247\pi\)
\(600\) 0 0
\(601\) −11848.0 −0.804143 −0.402072 0.915608i \(-0.631710\pi\)
−0.402072 + 0.915608i \(0.631710\pi\)
\(602\) 0 0
\(603\) −1577.08 −0.106507
\(604\) 0 0
\(605\) −6817.84 + 11808.8i −0.458156 + 0.793550i
\(606\) 0 0
\(607\) 4972.08 + 8611.90i 0.332472 + 0.575859i 0.982996 0.183627i \(-0.0587839\pi\)
−0.650524 + 0.759486i \(0.725451\pi\)
\(608\) 0 0
\(609\) 368.666 + 307.281i 0.0245305 + 0.0204460i
\(610\) 0 0
\(611\) 4389.39 + 7602.65i 0.290631 + 0.503388i
\(612\) 0 0
\(613\) 2778.87 4813.15i 0.183096 0.317131i −0.759838 0.650113i \(-0.774722\pi\)
0.942933 + 0.332982i \(0.108055\pi\)
\(614\) 0 0
\(615\) 8329.07 0.546115
\(616\) 0 0
\(617\) 18075.0 1.17937 0.589686 0.807633i \(-0.299252\pi\)
0.589686 + 0.807633i \(0.299252\pi\)
\(618\) 0 0
\(619\) 2121.53 3674.59i 0.137757 0.238602i −0.788890 0.614534i \(-0.789344\pi\)
0.926647 + 0.375932i \(0.122677\pi\)
\(620\) 0 0
\(621\) 5253.52 + 9099.37i 0.339479 + 0.587995i
\(622\) 0 0
\(623\) 8263.02 3038.40i 0.531382 0.195395i
\(624\) 0 0
\(625\) −4137.81 7166.90i −0.264820 0.458682i
\(626\) 0 0
\(627\) −4845.96 + 8393.45i −0.308659 + 0.534613i
\(628\) 0 0
\(629\) 35072.5 2.22326
\(630\) 0 0
\(631\) −29769.9 −1.87816 −0.939082 0.343692i \(-0.888322\pi\)
−0.939082 + 0.343692i \(0.888322\pi\)
\(632\) 0 0
\(633\) 9481.65 16422.7i 0.595358 1.03119i
\(634\) 0 0
\(635\) −4981.77 8628.68i −0.311331 0.539242i
\(636\) 0 0
\(637\) 23780.2 + 8478.02i 1.47913 + 0.527334i
\(638\) 0 0
\(639\) −1623.81 2812.53i −0.100527 0.174119i
\(640\) 0 0
\(641\) −4069.32 + 7048.27i −0.250746 + 0.434306i −0.963732 0.266874i \(-0.914009\pi\)
0.712985 + 0.701179i \(0.247343\pi\)
\(642\) 0 0
\(643\) −14788.0 −0.906971 −0.453486 0.891264i \(-0.649820\pi\)
−0.453486 + 0.891264i \(0.649820\pi\)
\(644\) 0 0
\(645\) 7350.24 0.448706
\(646\) 0 0
\(647\) −14177.5 + 24556.2i −0.861477 + 1.49212i 0.00902698 + 0.999959i \(0.497127\pi\)
−0.870504 + 0.492162i \(0.836207\pi\)
\(648\) 0 0
\(649\) −13914.9 24101.4i −0.841616 1.45772i
\(650\) 0 0
\(651\) −7754.23 + 2851.31i −0.466839 + 0.171662i
\(652\) 0 0
\(653\) 592.944 + 1027.01i 0.0355340 + 0.0615466i 0.883245 0.468911i \(-0.155353\pi\)
−0.847711 + 0.530458i \(0.822020\pi\)
\(654\) 0 0
\(655\) −205.630 + 356.162i −0.0122666 + 0.0212464i
\(656\) 0 0
\(657\) −1489.07 −0.0884236
\(658\) 0 0
\(659\) 7804.32 0.461325 0.230662 0.973034i \(-0.425911\pi\)
0.230662 + 0.973034i \(0.425911\pi\)
\(660\) 0 0
\(661\) −4344.92 + 7525.62i −0.255670 + 0.442833i −0.965077 0.261966i \(-0.915629\pi\)
0.709408 + 0.704799i \(0.248963\pi\)
\(662\) 0 0
\(663\) 26861.2 + 46525.0i 1.57346 + 2.72531i
\(664\) 0 0
\(665\) 1714.07 + 1428.67i 0.0999530 + 0.0833102i
\(666\) 0 0
\(667\) 191.028 + 330.870i 0.0110894 + 0.0192074i
\(668\) 0 0
\(669\) −1023.66 + 1773.02i −0.0591582 + 0.102465i
\(670\) 0 0
\(671\) −33032.8 −1.90047
\(672\) 0 0
\(673\) 31681.3 1.81460 0.907300 0.420483i \(-0.138139\pi\)
0.907300 + 0.420483i \(0.138139\pi\)
\(674\) 0 0
\(675\) −6667.74 + 11548.9i −0.380210 + 0.658542i
\(676\) 0 0
\(677\) 9283.75 + 16079.9i 0.527036 + 0.912853i 0.999504 + 0.0315052i \(0.0100301\pi\)
−0.472467 + 0.881348i \(0.656637\pi\)
\(678\) 0 0
\(679\) −37.4946 + 216.822i −0.00211916 + 0.0122546i
\(680\) 0 0
\(681\) −6422.39 11123.9i −0.361390 0.625946i
\(682\) 0 0
\(683\) 6943.95 12027.3i 0.389023 0.673808i −0.603295 0.797518i \(-0.706146\pi\)
0.992318 + 0.123710i \(0.0394793\pi\)
\(684\) 0 0
\(685\) 6223.10 0.347113
\(686\) 0 0
\(687\) −15976.6 −0.887255
\(688\) 0 0
\(689\) 5528.06 9574.88i 0.305664 0.529425i
\(690\) 0 0
\(691\) −5112.48 8855.08i −0.281459 0.487501i 0.690286 0.723537i \(-0.257485\pi\)
−0.971744 + 0.236036i \(0.924152\pi\)
\(692\) 0 0
\(693\) 838.730 4850.18i 0.0459751 0.265863i
\(694\) 0 0
\(695\) −487.264 843.965i −0.0265942 0.0460625i
\(696\) 0 0
\(697\) −21498.6 + 37236.7i −1.16832 + 2.02359i
\(698\) 0 0
\(699\) −9073.27 −0.490962
\(700\) 0 0
\(701\) 17766.2 0.957232 0.478616 0.878024i \(-0.341139\pi\)
0.478616 + 0.878024i \(0.341139\pi\)
\(702\) 0 0
\(703\) 3540.92 6133.05i 0.189969 0.329036i
\(704\) 0 0
\(705\) −1513.35 2621.20i −0.0808456 0.140029i
\(706\) 0 0
\(707\) 10448.5 + 8708.78i 0.555809 + 0.463264i
\(708\) 0 0
\(709\) −7584.77 13137.2i −0.401766 0.695879i 0.592173 0.805811i \(-0.298270\pi\)
−0.993939 + 0.109932i \(0.964937\pi\)
\(710\) 0 0
\(711\) −1117.04 + 1934.77i −0.0589201 + 0.102053i
\(712\) 0 0
\(713\) −6576.94 −0.345453
\(714\) 0 0
\(715\) 22047.0 1.15316
\(716\) 0 0
\(717\) −12521.1 + 21687.1i −0.652172 + 1.12960i
\(718\) 0 0
\(719\) 2855.78 + 4946.35i 0.148126 + 0.256562i 0.930535 0.366203i \(-0.119343\pi\)
−0.782409 + 0.622765i \(0.786009\pi\)
\(720\) 0 0
\(721\) 19520.2 7177.76i 1.00828 0.370754i
\(722\) 0 0
\(723\) 4707.61 + 8153.83i 0.242155 + 0.419425i
\(724\) 0 0
\(725\) −242.452 + 419.939i −0.0124199 + 0.0215119i
\(726\) 0 0
\(727\) 38137.1 1.94557 0.972783 0.231719i \(-0.0744349\pi\)
0.972783 + 0.231719i \(0.0744349\pi\)
\(728\) 0 0
\(729\) 15920.9 0.808864
\(730\) 0 0
\(731\) −18972.1 + 32860.6i −0.959929 + 1.66265i
\(732\) 0 0
\(733\) −14783.8 25606.3i −0.744956 1.29030i −0.950215 0.311594i \(-0.899137\pi\)
0.205260 0.978708i \(-0.434196\pi\)
\(734\) 0 0
\(735\) −8198.80 2923.01i −0.411452 0.146690i
\(736\) 0 0
\(737\) −12831.3 22224.4i −0.641311 1.11078i
\(738\) 0 0
\(739\) 1505.28 2607.23i 0.0749293 0.129781i −0.826126 0.563485i \(-0.809460\pi\)
0.901056 + 0.433704i \(0.142794\pi\)
\(740\) 0 0
\(741\) 10847.6 0.537783
\(742\) 0 0
\(743\) 10229.6 0.505100 0.252550 0.967584i \(-0.418731\pi\)
0.252550 + 0.967584i \(0.418731\pi\)
\(744\) 0 0
\(745\) −5080.17 + 8799.11i −0.249829 + 0.432717i
\(746\) 0 0
\(747\) 125.953 + 218.158i 0.00616921 + 0.0106854i
\(748\) 0 0
\(749\) −7034.08 + 2586.50i −0.343150 + 0.126180i
\(750\) 0 0
\(751\) −14630.9 25341.5i −0.710906 1.23132i −0.964518 0.264019i \(-0.914952\pi\)
0.253612 0.967306i \(-0.418381\pi\)
\(752\) 0 0
\(753\) 12264.3 21242.3i 0.593539 1.02804i
\(754\) 0 0
\(755\) 4012.95 0.193439
\(756\) 0 0
\(757\) −10735.1 −0.515420 −0.257710 0.966222i \(-0.582968\pi\)
−0.257710 + 0.966222i \(0.582968\pi\)
\(758\) 0 0
\(759\) 15048.1 26064.2i 0.719648 1.24647i
\(760\) 0 0
\(761\) −1827.51 3165.34i −0.0870527 0.150780i 0.819211 0.573492i \(-0.194412\pi\)
−0.906264 + 0.422712i \(0.861078\pi\)
\(762\) 0 0
\(763\) −27678.4 23069.7i −1.31327 1.09460i
\(764\) 0 0
\(765\) −1205.73 2088.39i −0.0569847 0.0987004i
\(766\) 0 0
\(767\) −15574.2 + 26975.3i −0.733183 + 1.26991i
\(768\) 0 0
\(769\) −24785.4 −1.16227 −0.581133 0.813808i \(-0.697390\pi\)
−0.581133 + 0.813808i \(0.697390\pi\)
\(770\) 0 0
\(771\) 30496.9 1.42454
\(772\) 0 0
\(773\) −4961.99 + 8594.42i −0.230880 + 0.399896i −0.958067 0.286543i \(-0.907494\pi\)
0.727187 + 0.686439i \(0.240827\pi\)
\(774\) 0 0
\(775\) −4173.71 7229.07i −0.193450 0.335066i
\(776\) 0 0
\(777\) −4707.26 + 27221.0i −0.217339 + 1.25682i
\(778\) 0 0
\(779\) 4341.00 + 7518.83i 0.199656 + 0.345815i
\(780\) 0 0
\(781\) 26423.1 45766.1i 1.21062 2.09685i
\(782\) 0 0
\(783\) 594.948 0.0271542
\(784\) 0 0
\(785\) 9845.76 0.447657
\(786\) 0 0
\(787\) −16491.9 + 28564.7i −0.746978 + 1.29380i 0.202288 + 0.979326i \(0.435162\pi\)
−0.949265 + 0.314477i \(0.898171\pi\)
\(788\) 0 0
\(789\) 16003.3 + 27718.5i 0.722094 + 1.25070i
\(790\) 0 0
\(791\) 5244.36 30326.9i 0.235737 1.36321i
\(792\) 0 0
\(793\) 18485.9 + 32018.5i 0.827809 + 1.43381i
\(794\) 0 0
\(795\) −1905.94 + 3301.18i −0.0850271 + 0.147271i
\(796\) 0 0
\(797\) −14832.3 −0.659207 −0.329603 0.944119i \(-0.606915\pi\)
−0.329603 + 0.944119i \(0.606915\pi\)
\(798\) 0 0
\(799\) 15624.8 0.691820
\(800\) 0 0
\(801\) −960.573 + 1663.76i −0.0423723 + 0.0733909i
\(802\) 0 0
\(803\) −12115.3 20984.3i −0.532427 0.922191i
\(804\) 0 0
\(805\) −5322.69 4436.43i −0.233044 0.194241i
\(806\) 0 0
\(807\) 5665.30 + 9812.58i 0.247123 + 0.428029i
\(808\) 0 0
\(809\) −2249.77 + 3896.72i −0.0977722 + 0.169346i −0.910762 0.412931i \(-0.864505\pi\)
0.812990 + 0.582278i \(0.197838\pi\)
\(810\) 0 0
\(811\) 24307.3 1.05246 0.526229 0.850343i \(-0.323605\pi\)
0.526229 + 0.850343i \(0.323605\pi\)
\(812\) 0 0
\(813\) −8608.79 −0.371370
\(814\) 0 0
\(815\) −6761.54 + 11711.3i −0.290609 + 0.503350i
\(816\) 0 0
\(817\) 3830.84 + 6635.21i 0.164044 + 0.284133i
\(818\) 0 0
\(819\) −5170.61 + 1901.29i −0.220605 + 0.0811189i
\(820\) 0 0
\(821\) 533.897 + 924.737i 0.0226957 + 0.0393100i 0.877150 0.480216i \(-0.159442\pi\)
−0.854454 + 0.519526i \(0.826108\pi\)
\(822\) 0 0
\(823\) −8752.96 + 15160.6i −0.370728 + 0.642120i −0.989678 0.143311i \(-0.954225\pi\)
0.618950 + 0.785431i \(0.287558\pi\)
\(824\) 0 0
\(825\) 38198.1 1.61198
\(826\) 0 0
\(827\) −32525.9 −1.36764 −0.683818 0.729653i \(-0.739682\pi\)
−0.683818 + 0.729653i \(0.739682\pi\)
\(828\) 0 0
\(829\) 17962.1 31111.2i 0.752532 1.30342i −0.194061 0.980990i \(-0.562166\pi\)
0.946592 0.322433i \(-0.104501\pi\)
\(830\) 0 0
\(831\) −11148.4 19309.5i −0.465382 0.806064i
\(832\) 0 0
\(833\) 34230.2 29109.5i 1.42378 1.21079i
\(834\) 0 0
\(835\) 5048.38 + 8744.06i 0.209229 + 0.362396i
\(836\) 0 0
\(837\) −5120.89 + 8869.65i −0.211474 + 0.366284i
\(838\) 0 0
\(839\) −11760.4 −0.483927 −0.241964 0.970285i \(-0.577791\pi\)
−0.241964 + 0.970285i \(0.577791\pi\)
\(840\) 0 0
\(841\) −24367.4 −0.999113
\(842\) 0 0
\(843\) 18237.9 31588.9i 0.745132 1.29061i
\(844\) 0 0
\(845\) −7334.53 12703.8i −0.298598 0.517187i
\(846\) 0 0
\(847\) 52037.6 19134.8i 2.11102 0.776244i
\(848\) 0 0
\(849\) 12009.1 + 20800.4i 0.485457 + 0.840836i
\(850\) 0 0
\(851\) −10995.6 + 19044.9i −0.442919 + 0.767158i
\(852\) 0 0
\(853\) −31670.4 −1.27125 −0.635624 0.771999i \(-0.719257\pi\)
−0.635624 + 0.771999i \(0.719257\pi\)
\(854\) 0 0
\(855\) −486.922 −0.0194765
\(856\) 0 0
\(857\) −10349.1 + 17925.2i −0.412508 + 0.714485i −0.995163 0.0982343i \(-0.968681\pi\)
0.582655 + 0.812720i \(0.302014\pi\)
\(858\) 0 0
\(859\) −11826.5 20484.1i −0.469749 0.813629i 0.529653 0.848215i \(-0.322322\pi\)
−0.999402 + 0.0345855i \(0.988989\pi\)
\(860\) 0 0
\(861\) −26015.2 21683.5i −1.02973 0.858273i
\(862\) 0 0
\(863\) −16195.0 28050.5i −0.638799 1.10643i −0.985697 0.168530i \(-0.946098\pi\)
0.346897 0.937903i \(-0.387235\pi\)
\(864\) 0 0
\(865\) −1694.68 + 2935.27i −0.0666137 + 0.115378i
\(866\) 0 0
\(867\) 68244.2 2.67323
\(868\) 0 0
\(869\) −36353.4 −1.41911
\(870\) 0 0
\(871\) −14361.3 + 24874.6i −0.558686 + 0.967672i
\(872\) 0 0
\(873\) −24.0080 41.5831i −0.000930753 0.00161211i
\(874\) 0 0
\(875\) 3295.32 19056.1i 0.127317 0.736243i
\(876\) 0 0
\(877\) 19202.3 + 33259.3i 0.739356 + 1.28060i 0.952786 + 0.303644i \(0.0982032\pi\)
−0.213430 + 0.976958i \(0.568463\pi\)
\(878\) 0 0
\(879\) 995.863 1724.88i 0.0382134 0.0661876i
\(880\) 0 0
\(881\) 4438.83 0.169748 0.0848740 0.996392i \(-0.472951\pi\)
0.0848740 + 0.996392i \(0.472951\pi\)
\(882\) 0 0
\(883\) 29750.2 1.13383 0.566917 0.823775i \(-0.308136\pi\)
0.566917 + 0.823775i \(0.308136\pi\)
\(884\) 0 0
\(885\) 5369.59 9300.40i 0.203951 0.353254i
\(886\) 0 0
\(887\) 18748.4 + 32473.2i 0.709707 + 1.22925i 0.964966 + 0.262376i \(0.0845060\pi\)
−0.255259 + 0.966873i \(0.582161\pi\)
\(888\) 0 0
\(889\) −6903.33 + 39920.4i −0.260439 + 1.50606i
\(890\) 0 0
\(891\) −27021.3 46802.3i −1.01599 1.75975i
\(892\) 0 0
\(893\) 1577.48 2732.27i 0.0591134 0.102387i
\(894\) 0 0
\(895\) 13265.9 0.495454
\(896\) 0 0
\(897\) −33685.1 −1.25386
\(898\) 0 0
\(899\) −186.206 + 322.517i −0.00690801 + 0.0119650i
\(900\) 0 0
\(901\) −9839.02 17041.7i −0.363802 0.630123i
\(902\) 0 0
\(903\) −22957.9 19135.3i −0.846060 0.705186i
\(904\) 0 0
\(905\) −5990.53 10375.9i −0.220035 0.381112i
\(906\) 0 0
\(907\) 17782.3 30799.8i 0.650993 1.12755i −0.331889 0.943318i \(-0.607686\pi\)
0.982882 0.184235i \(-0.0589806\pi\)
\(908\) 0 0
\(909\) −2968.15 −0.108303
\(910\) 0 0
\(911\) −19554.7 −0.711170 −0.355585 0.934644i \(-0.615718\pi\)
−0.355585 + 0.934644i \(0.615718\pi\)
\(912\) 0 0
\(913\) −2049.55 + 3549.92i −0.0742936 + 0.128680i
\(914\) 0 0
\(915\) −6373.46 11039.2i −0.230273 0.398845i
\(916\) 0 0
\(917\) 1569.49 577.117i 0.0565203 0.0207831i
\(918\) 0 0
\(919\) 9949.36 + 17232.8i 0.357126 + 0.618561i 0.987479 0.157748i \(-0.0504232\pi\)
−0.630353 + 0.776309i \(0.717090\pi\)
\(920\) 0 0
\(921\) 12080.3 20923.6i 0.432202 0.748596i
\(922\) 0 0
\(923\) −59147.7 −2.10929
\(924\) 0 0
\(925\) −27911.1 −0.992120
\(926\) 0 0
\(927\) −2269.21 + 3930.39i −0.0803999 + 0.139257i
\(928\) 0 0
\(929\) 3504.98 + 6070.80i 0.123783 + 0.214399i 0.921257 0.388955i \(-0.127164\pi\)
−0.797473 + 0.603354i \(0.793831\pi\)
\(930\) 0 0
\(931\) −1634.44 8924.67i −0.0575367 0.314172i
\(932\) 0 0
\(933\) 2593.69 + 4492.40i 0.0910113 + 0.157636i
\(934\) 0 0
\(935\) 19620.0 33982.8i 0.686247 1.18861i
\(936\) 0 0
\(937\) −813.410 −0.0283596 −0.0141798 0.999899i \(-0.504514\pi\)
−0.0141798 + 0.999899i \(0.504514\pi\)
\(938\) 0 0
\(939\) 42800.4 1.48747
\(940\) 0 0
\(941\) −7123.15 + 12337.7i −0.246767 + 0.427414i −0.962627 0.270831i \(-0.912702\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(942\) 0 0
\(943\) −13480.1 23348.2i −0.465506 0.806279i
\(944\) 0 0
\(945\) −10127.3 + 3723.91i −0.348614 + 0.128189i
\(946\) 0 0
\(947\) −17053.1 29536.9i −0.585167 1.01354i −0.994855 0.101312i \(-0.967696\pi\)
0.409688 0.912226i \(-0.365638\pi\)
\(948\) 0 0
\(949\) −13560.0 + 23486.5i −0.463830 + 0.803377i
\(950\) 0 0
\(951\) −25712.8 −0.876756
\(952\) 0 0
\(953\) 10497.7 0.356826 0.178413 0.983956i \(-0.442904\pi\)
0.178413 + 0.983956i \(0.442904\pi\)
\(954\) 0 0
\(955\) 3280.68 5682.31i 0.111163 0.192540i
\(956\) 0 0
\(957\) −852.083 1475.85i −0.0287815 0.0498511i
\(958\) 0 0
\(959\) −19437.4 16200.9i −0.654501 0.545522i
\(960\) 0 0
\(961\) 11690.1 + 20247.8i 0.392402 + 0.679660i
\(962\) 0 0
\(963\) 817.709 1416.31i 0.0273627 0.0473937i
\(964\) 0 0
\(965\) −4264.42 −0.142256
\(966\) 0 0
\(967\) 47342.5 1.57439 0.787193 0.616707i \(-0.211534\pi\)
0.787193 + 0.616707i \(0.211534\pi\)
\(968\) 0 0
\(969\) 9653.48 16720.3i 0.320036 0.554318i
\(970\) 0 0
\(971\) −817.140 1415.33i −0.0270065 0.0467766i 0.852206 0.523206i \(-0.175264\pi\)
−0.879213 + 0.476429i \(0.841931\pi\)
\(972\) 0 0
\(973\) −675.211 + 3904.58i −0.0222469 + 0.128649i
\(974\) 0 0
\(975\) −21376.5 37025.1i −0.702148 1.21616i
\(976\) 0 0
\(977\) −12531.7 + 21705.6i −0.410364 + 0.710772i −0.994930 0.100575i \(-0.967932\pi\)
0.584565 + 0.811347i \(0.301265\pi\)
\(978\) 0 0
\(979\) −31261.4 −1.02055
\(980\) 0 0
\(981\) 7862.71 0.255899
\(982\) 0 0
\(983\) 29562.1 51203.1i 0.959191 1.66137i 0.234720 0.972063i \(-0.424583\pi\)
0.724472 0.689305i \(-0.242084\pi\)
\(984\) 0 0
\(985\) 10656.4 + 18457.4i 0.344711 + 0.597058i
\(986\) 0 0
\(987\) −2097.08 + 12126.9i −0.0676300 + 0.391089i
\(988\) 0 0
\(989\) −11895.9 20604.3i −0.382475 0.662466i
\(990\) 0 0
\(991\) 19487.5 33753.4i 0.624664 1.08195i −0.363942 0.931422i \(-0.618569\pi\)
0.988606 0.150528i \(-0.0480975\pi\)
\(992\) 0 0
\(993\) −18231.6 −0.582642
\(994\) 0 0
\(995\) 10682.9 0.340372
\(996\) 0 0
\(997\) 22151.9 38368.2i 0.703668 1.21879i −0.263503 0.964659i \(-0.584878\pi\)
0.967170 0.254129i \(-0.0817889\pi\)
\(998\) 0 0
\(999\) 17122.6 + 29657.3i 0.542279 + 0.939254i
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 448.4.i.q.193.2 12
4.3 odd 2 448.4.i.o.193.5 12
7.2 even 3 inner 448.4.i.q.65.2 12
8.3 odd 2 224.4.i.e.193.2 yes 12
8.5 even 2 224.4.i.c.193.5 yes 12
28.23 odd 6 448.4.i.o.65.5 12
56.3 even 6 1568.4.a.bj.1.2 6
56.11 odd 6 1568.4.a.be.1.5 6
56.37 even 6 224.4.i.c.65.5 12
56.45 odd 6 1568.4.a.bf.1.5 6
56.51 odd 6 224.4.i.e.65.2 yes 12
56.53 even 6 1568.4.a.bi.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.4.i.c.65.5 12 56.37 even 6
224.4.i.c.193.5 yes 12 8.5 even 2
224.4.i.e.65.2 yes 12 56.51 odd 6
224.4.i.e.193.2 yes 12 8.3 odd 2
448.4.i.o.65.5 12 28.23 odd 6
448.4.i.o.193.5 12 4.3 odd 2
448.4.i.q.65.2 12 7.2 even 3 inner
448.4.i.q.193.2 12 1.1 even 1 trivial
1568.4.a.be.1.5 6 56.11 odd 6
1568.4.a.bf.1.5 6 56.45 odd 6
1568.4.a.bi.1.2 6 56.53 even 6
1568.4.a.bj.1.2 6 56.3 even 6