Properties

Label 624.4.q.i.529.4
Level $624$
Weight $4$
Character 624.529
Analytic conductor $36.817$
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] = [624,4,Mod(289,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("624.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.4
Root \(-0.733051 - 1.26968i\) of defining polynomial
Character \(\chi\) \(=\) 624.529
Dual form 624.4.q.i.289.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{3} +9.85055 q^{5} +(14.9698 + 25.9285i) q^{7} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 2.59808i) q^{3} +9.85055 q^{5} +(14.9698 + 25.9285i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(23.4629 - 40.6389i) q^{11} +(3.71050 + 46.7251i) q^{13} +(-14.7758 + 25.5925i) q^{15} +(24.1308 + 41.7958i) q^{17} +(60.1501 + 104.183i) q^{19} -89.8188 q^{21} +(65.3485 - 113.187i) q^{23} -27.9667 q^{25} +27.0000 q^{27} +(97.4729 - 168.828i) q^{29} +32.0123 q^{31} +(70.3886 + 121.917i) q^{33} +(147.461 + 255.409i) q^{35} +(16.2125 - 28.0808i) q^{37} +(-126.961 - 60.4474i) q^{39} +(120.913 - 209.427i) q^{41} +(48.2044 + 83.4924i) q^{43} +(-44.3275 - 76.7774i) q^{45} -539.015 q^{47} +(-276.690 + 479.241i) q^{49} -144.785 q^{51} -152.277 q^{53} +(231.122 - 400.315i) q^{55} -360.901 q^{57} +(163.896 + 283.876i) q^{59} +(49.2090 + 85.2325i) q^{61} +(134.728 - 233.356i) q^{63} +(36.5504 + 460.267i) q^{65} +(-220.575 + 382.048i) q^{67} +(196.046 + 339.561i) q^{69} +(172.524 + 298.821i) q^{71} +773.839 q^{73} +(41.9501 - 72.6597i) q^{75} +1404.94 q^{77} +150.332 q^{79} +(-40.5000 + 70.1481i) q^{81} -337.966 q^{83} +(237.702 + 411.711i) q^{85} +(292.419 + 506.484i) q^{87} +(-84.9567 + 147.149i) q^{89} +(-1155.96 + 795.673i) q^{91} +(-48.0184 + 83.1703i) q^{93} +(592.511 + 1026.26i) q^{95} +(-107.101 - 185.504i) q^{97} -422.332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9} + 40 q^{11} - 60 q^{13} + 18 q^{15} - 98 q^{17} + 124 q^{19} + 84 q^{21} + 104 q^{23} - 116 q^{25} + 216 q^{27} - 194 q^{29} - 52 q^{31} + 120 q^{33} + 88 q^{35} - 102 q^{37} - 342 q^{39} + 1054 q^{41} + 450 q^{43} + 54 q^{45} + 192 q^{47} - 1070 q^{49} + 588 q^{51} + 524 q^{53} + 204 q^{55} - 744 q^{57} + 308 q^{59} + 928 q^{61} - 126 q^{63} + 2346 q^{65} - 1134 q^{67} + 312 q^{69} + 1064 q^{71} + 1904 q^{73} + 174 q^{75} + 5016 q^{77} + 1492 q^{79} - 324 q^{81} + 808 q^{83} + 1394 q^{85} - 582 q^{87} - 1620 q^{89} - 3278 q^{91} + 78 q^{93} + 2204 q^{95} - 2166 q^{97} - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 9.85055 0.881060 0.440530 0.897738i \(-0.354791\pi\)
0.440530 + 0.897738i \(0.354791\pi\)
\(6\) 0 0
\(7\) 14.9698 + 25.9285i 0.808293 + 1.40001i 0.914045 + 0.405613i \(0.132942\pi\)
−0.105752 + 0.994393i \(0.533725\pi\)
\(8\) 0 0
\(9\) −4.50000 7.79423i −0.166667 0.288675i
\(10\) 0 0
\(11\) 23.4629 40.6389i 0.643120 1.11392i −0.341612 0.939841i \(-0.610973\pi\)
0.984732 0.174076i \(-0.0556938\pi\)
\(12\) 0 0
\(13\) 3.71050 + 46.7251i 0.0791621 + 0.996862i
\(14\) 0 0
\(15\) −14.7758 + 25.5925i −0.254340 + 0.440530i
\(16\) 0 0
\(17\) 24.1308 + 41.7958i 0.344270 + 0.596292i 0.985221 0.171289i \(-0.0547932\pi\)
−0.640951 + 0.767582i \(0.721460\pi\)
\(18\) 0 0
\(19\) 60.1501 + 104.183i 0.726283 + 1.25796i 0.958444 + 0.285282i \(0.0920872\pi\)
−0.232161 + 0.972677i \(0.574579\pi\)
\(20\) 0 0
\(21\) −89.8188 −0.933337
\(22\) 0 0
\(23\) 65.3485 113.187i 0.592440 1.02614i −0.401463 0.915875i \(-0.631498\pi\)
0.993903 0.110260i \(-0.0351685\pi\)
\(24\) 0 0
\(25\) −27.9667 −0.223734
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 97.4729 168.828i 0.624147 1.08105i −0.364558 0.931181i \(-0.618780\pi\)
0.988705 0.149874i \(-0.0478867\pi\)
\(30\) 0 0
\(31\) 32.0123 0.185470 0.0927351 0.995691i \(-0.470439\pi\)
0.0927351 + 0.995691i \(0.470439\pi\)
\(32\) 0 0
\(33\) 70.3886 + 121.917i 0.371306 + 0.643120i
\(34\) 0 0
\(35\) 147.461 + 255.409i 0.712155 + 1.23349i
\(36\) 0 0
\(37\) 16.2125 28.0808i 0.0720355 0.124769i −0.827758 0.561086i \(-0.810384\pi\)
0.899793 + 0.436316i \(0.143717\pi\)
\(38\) 0 0
\(39\) −126.961 60.4474i −0.521283 0.248188i
\(40\) 0 0
\(41\) 120.913 209.427i 0.460570 0.797731i −0.538419 0.842677i \(-0.680978\pi\)
0.998989 + 0.0449461i \(0.0143116\pi\)
\(42\) 0 0
\(43\) 48.2044 + 83.4924i 0.170956 + 0.296104i 0.938754 0.344587i \(-0.111981\pi\)
−0.767799 + 0.640691i \(0.778648\pi\)
\(44\) 0 0
\(45\) −44.3275 76.7774i −0.146843 0.254340i
\(46\) 0 0
\(47\) −539.015 −1.67284 −0.836419 0.548090i \(-0.815355\pi\)
−0.836419 + 0.548090i \(0.815355\pi\)
\(48\) 0 0
\(49\) −276.690 + 479.241i −0.806676 + 1.39720i
\(50\) 0 0
\(51\) −144.785 −0.397528
\(52\) 0 0
\(53\) −152.277 −0.394657 −0.197328 0.980337i \(-0.563226\pi\)
−0.197328 + 0.980337i \(0.563226\pi\)
\(54\) 0 0
\(55\) 231.122 400.315i 0.566627 0.981427i
\(56\) 0 0
\(57\) −360.901 −0.838640
\(58\) 0 0
\(59\) 163.896 + 283.876i 0.361652 + 0.626399i 0.988233 0.152957i \(-0.0488796\pi\)
−0.626581 + 0.779356i \(0.715546\pi\)
\(60\) 0 0
\(61\) 49.2090 + 85.2325i 0.103288 + 0.178900i 0.913037 0.407876i \(-0.133730\pi\)
−0.809749 + 0.586776i \(0.800397\pi\)
\(62\) 0 0
\(63\) 134.728 233.356i 0.269431 0.466668i
\(64\) 0 0
\(65\) 36.5504 + 460.267i 0.0697465 + 0.878295i
\(66\) 0 0
\(67\) −220.575 + 382.048i −0.402202 + 0.696635i −0.993991 0.109458i \(-0.965089\pi\)
0.591789 + 0.806093i \(0.298422\pi\)
\(68\) 0 0
\(69\) 196.046 + 339.561i 0.342045 + 0.592440i
\(70\) 0 0
\(71\) 172.524 + 298.821i 0.288379 + 0.499486i 0.973423 0.229015i \(-0.0735505\pi\)
−0.685044 + 0.728501i \(0.740217\pi\)
\(72\) 0 0
\(73\) 773.839 1.24070 0.620349 0.784326i \(-0.286991\pi\)
0.620349 + 0.784326i \(0.286991\pi\)
\(74\) 0 0
\(75\) 41.9501 72.6597i 0.0645864 0.111867i
\(76\) 0 0
\(77\) 1404.94 2.07932
\(78\) 0 0
\(79\) 150.332 0.214097 0.107049 0.994254i \(-0.465860\pi\)
0.107049 + 0.994254i \(0.465860\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −337.966 −0.446947 −0.223473 0.974710i \(-0.571740\pi\)
−0.223473 + 0.974710i \(0.571740\pi\)
\(84\) 0 0
\(85\) 237.702 + 411.711i 0.303322 + 0.525369i
\(86\) 0 0
\(87\) 292.419 + 506.484i 0.360351 + 0.624147i
\(88\) 0 0
\(89\) −84.9567 + 147.149i −0.101184 + 0.175256i −0.912173 0.409806i \(-0.865596\pi\)
0.810989 + 0.585062i \(0.198930\pi\)
\(90\) 0 0
\(91\) −1155.96 + 795.673i −1.33163 + 0.916584i
\(92\) 0 0
\(93\) −48.0184 + 83.1703i −0.0535406 + 0.0927351i
\(94\) 0 0
\(95\) 592.511 + 1026.26i 0.639899 + 1.10834i
\(96\) 0 0
\(97\) −107.101 185.504i −0.112107 0.194176i 0.804512 0.593936i \(-0.202427\pi\)
−0.916620 + 0.399760i \(0.869093\pi\)
\(98\) 0 0
\(99\) −422.332 −0.428747
\(100\) 0 0
\(101\) −797.556 + 1381.41i −0.785741 + 1.36094i 0.142815 + 0.989749i \(0.454385\pi\)
−0.928555 + 0.371194i \(0.878949\pi\)
\(102\) 0 0
\(103\) 1570.30 1.50219 0.751096 0.660193i \(-0.229525\pi\)
0.751096 + 0.660193i \(0.229525\pi\)
\(104\) 0 0
\(105\) −884.764 −0.822325
\(106\) 0 0
\(107\) −3.36651 + 5.83096i −0.00304161 + 0.00526823i −0.867542 0.497364i \(-0.834302\pi\)
0.864501 + 0.502632i \(0.167635\pi\)
\(108\) 0 0
\(109\) −542.422 −0.476648 −0.238324 0.971186i \(-0.576598\pi\)
−0.238324 + 0.971186i \(0.576598\pi\)
\(110\) 0 0
\(111\) 48.6374 + 84.2425i 0.0415897 + 0.0720355i
\(112\) 0 0
\(113\) 721.411 + 1249.52i 0.600572 + 1.04022i 0.992735 + 0.120325i \(0.0383938\pi\)
−0.392162 + 0.919896i \(0.628273\pi\)
\(114\) 0 0
\(115\) 643.719 1114.95i 0.521975 0.904087i
\(116\) 0 0
\(117\) 347.489 239.183i 0.274576 0.188996i
\(118\) 0 0
\(119\) −722.467 + 1251.35i −0.556542 + 0.963958i
\(120\) 0 0
\(121\) −435.513 754.330i −0.327207 0.566739i
\(122\) 0 0
\(123\) 362.738 + 628.280i 0.265910 + 0.460570i
\(124\) 0 0
\(125\) −1506.81 −1.07818
\(126\) 0 0
\(127\) −1246.42 + 2158.86i −0.870881 + 1.50841i −0.00979442 + 0.999952i \(0.503118\pi\)
−0.861087 + 0.508458i \(0.830216\pi\)
\(128\) 0 0
\(129\) −289.226 −0.197403
\(130\) 0 0
\(131\) −744.561 −0.496585 −0.248292 0.968685i \(-0.579869\pi\)
−0.248292 + 0.968685i \(0.579869\pi\)
\(132\) 0 0
\(133\) −1800.87 + 3119.20i −1.17410 + 2.03360i
\(134\) 0 0
\(135\) 265.965 0.169560
\(136\) 0 0
\(137\) 111.083 + 192.402i 0.0692736 + 0.119985i 0.898582 0.438806i \(-0.144599\pi\)
−0.829308 + 0.558792i \(0.811265\pi\)
\(138\) 0 0
\(139\) −388.573 673.028i −0.237110 0.410687i 0.722774 0.691085i \(-0.242867\pi\)
−0.959884 + 0.280398i \(0.909534\pi\)
\(140\) 0 0
\(141\) 808.522 1400.40i 0.482907 0.836419i
\(142\) 0 0
\(143\) 1985.91 + 945.514i 1.16133 + 0.552922i
\(144\) 0 0
\(145\) 960.161 1663.05i 0.549911 0.952473i
\(146\) 0 0
\(147\) −830.070 1437.72i −0.465735 0.806676i
\(148\) 0 0
\(149\) −889.147 1540.05i −0.488871 0.846750i 0.511047 0.859553i \(-0.329258\pi\)
−0.999918 + 0.0128032i \(0.995925\pi\)
\(150\) 0 0
\(151\) −1166.00 −0.628394 −0.314197 0.949358i \(-0.601735\pi\)
−0.314197 + 0.949358i \(0.601735\pi\)
\(152\) 0 0
\(153\) 217.177 376.162i 0.114757 0.198764i
\(154\) 0 0
\(155\) 315.338 0.163410
\(156\) 0 0
\(157\) 517.628 0.263129 0.131564 0.991308i \(-0.458000\pi\)
0.131564 + 0.991308i \(0.458000\pi\)
\(158\) 0 0
\(159\) 228.415 395.626i 0.113928 0.197328i
\(160\) 0 0
\(161\) 3913.02 1.91546
\(162\) 0 0
\(163\) 305.094 + 528.438i 0.146606 + 0.253929i 0.929971 0.367633i \(-0.119832\pi\)
−0.783365 + 0.621562i \(0.786498\pi\)
\(164\) 0 0
\(165\) 693.366 + 1200.95i 0.327142 + 0.566627i
\(166\) 0 0
\(167\) 1491.51 2583.36i 0.691115 1.19705i −0.280358 0.959895i \(-0.590453\pi\)
0.971473 0.237150i \(-0.0762133\pi\)
\(168\) 0 0
\(169\) −2169.46 + 346.747i −0.987467 + 0.157827i
\(170\) 0 0
\(171\) 541.351 937.647i 0.242094 0.419320i
\(172\) 0 0
\(173\) −489.106 847.157i −0.214948 0.372301i 0.738308 0.674463i \(-0.235625\pi\)
−0.953257 + 0.302162i \(0.902292\pi\)
\(174\) 0 0
\(175\) −418.657 725.134i −0.180843 0.313229i
\(176\) 0 0
\(177\) −983.377 −0.417599
\(178\) 0 0
\(179\) 926.471 1604.69i 0.386858 0.670058i −0.605167 0.796099i \(-0.706894\pi\)
0.992025 + 0.126040i \(0.0402269\pi\)
\(180\) 0 0
\(181\) 852.777 0.350201 0.175101 0.984551i \(-0.443975\pi\)
0.175101 + 0.984551i \(0.443975\pi\)
\(182\) 0 0
\(183\) −295.254 −0.119267
\(184\) 0 0
\(185\) 159.702 276.612i 0.0634676 0.109929i
\(186\) 0 0
\(187\) 2264.71 0.885627
\(188\) 0 0
\(189\) 404.185 + 700.068i 0.155556 + 0.269431i
\(190\) 0 0
\(191\) 2220.65 + 3846.27i 0.841258 + 1.45710i 0.888831 + 0.458234i \(0.151518\pi\)
−0.0475730 + 0.998868i \(0.515149\pi\)
\(192\) 0 0
\(193\) 1241.21 2149.85i 0.462925 0.801810i −0.536180 0.844104i \(-0.680133\pi\)
0.999105 + 0.0422935i \(0.0134665\pi\)
\(194\) 0 0
\(195\) −1250.64 595.440i −0.459281 0.218669i
\(196\) 0 0
\(197\) −630.115 + 1091.39i −0.227888 + 0.394713i −0.957182 0.289487i \(-0.906515\pi\)
0.729294 + 0.684200i \(0.239849\pi\)
\(198\) 0 0
\(199\) −2760.48 4781.30i −0.983344 1.70320i −0.649077 0.760722i \(-0.724845\pi\)
−0.334266 0.942479i \(-0.608489\pi\)
\(200\) 0 0
\(201\) −661.726 1146.14i −0.232212 0.402202i
\(202\) 0 0
\(203\) 5836.60 2.01798
\(204\) 0 0
\(205\) 1191.06 2062.97i 0.405790 0.702849i
\(206\) 0 0
\(207\) −1176.27 −0.394960
\(208\) 0 0
\(209\) 5645.18 1.86835
\(210\) 0 0
\(211\) −2263.90 + 3921.18i −0.738640 + 1.27936i 0.214468 + 0.976731i \(0.431198\pi\)
−0.953108 + 0.302631i \(0.902135\pi\)
\(212\) 0 0
\(213\) −1035.15 −0.332991
\(214\) 0 0
\(215\) 474.839 + 822.446i 0.150622 + 0.260885i
\(216\) 0 0
\(217\) 479.217 + 830.029i 0.149914 + 0.259659i
\(218\) 0 0
\(219\) −1160.76 + 2010.49i −0.358159 + 0.620349i
\(220\) 0 0
\(221\) −1863.37 + 1282.60i −0.567168 + 0.390393i
\(222\) 0 0
\(223\) 2240.59 3880.81i 0.672829 1.16537i −0.304270 0.952586i \(-0.598412\pi\)
0.977099 0.212787i \(-0.0682542\pi\)
\(224\) 0 0
\(225\) 125.850 + 217.979i 0.0372890 + 0.0645864i
\(226\) 0 0
\(227\) −2879.88 4988.09i −0.842044 1.45846i −0.888163 0.459528i \(-0.848019\pi\)
0.0461191 0.998936i \(-0.485315\pi\)
\(228\) 0 0
\(229\) −4635.08 −1.33753 −0.668766 0.743473i \(-0.733177\pi\)
−0.668766 + 0.743473i \(0.733177\pi\)
\(230\) 0 0
\(231\) −2107.41 + 3650.14i −0.600248 + 1.03966i
\(232\) 0 0
\(233\) 5886.33 1.65505 0.827524 0.561431i \(-0.189749\pi\)
0.827524 + 0.561431i \(0.189749\pi\)
\(234\) 0 0
\(235\) −5309.59 −1.47387
\(236\) 0 0
\(237\) −225.498 + 390.574i −0.0618046 + 0.107049i
\(238\) 0 0
\(239\) −2135.84 −0.578060 −0.289030 0.957320i \(-0.593333\pi\)
−0.289030 + 0.957320i \(0.593333\pi\)
\(240\) 0 0
\(241\) −2346.46 4064.19i −0.627173 1.08630i −0.988116 0.153708i \(-0.950879\pi\)
0.360943 0.932588i \(-0.382455\pi\)
\(242\) 0 0
\(243\) −121.500 210.444i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) −2725.55 + 4720.79i −0.710730 + 1.23102i
\(246\) 0 0
\(247\) −4644.77 + 3197.09i −1.19652 + 0.823587i
\(248\) 0 0
\(249\) 506.949 878.062i 0.129022 0.223473i
\(250\) 0 0
\(251\) −1951.44 3379.99i −0.490732 0.849973i 0.509211 0.860642i \(-0.329937\pi\)
−0.999943 + 0.0106687i \(0.996604\pi\)
\(252\) 0 0
\(253\) −3066.53 5311.38i −0.762020 1.31986i
\(254\) 0 0
\(255\) −1426.21 −0.350246
\(256\) 0 0
\(257\) 2065.42 3577.40i 0.501312 0.868297i −0.498687 0.866782i \(-0.666184\pi\)
0.999999 0.00151510i \(-0.000482272\pi\)
\(258\) 0 0
\(259\) 970.790 0.232903
\(260\) 0 0
\(261\) −1754.51 −0.416098
\(262\) 0 0
\(263\) 3176.09 5501.14i 0.744661 1.28979i −0.205692 0.978617i \(-0.565944\pi\)
0.950353 0.311174i \(-0.100722\pi\)
\(264\) 0 0
\(265\) −1500.01 −0.347716
\(266\) 0 0
\(267\) −254.870 441.448i −0.0584187 0.101184i
\(268\) 0 0
\(269\) −90.7619 157.204i −0.0205719 0.0356316i 0.855556 0.517710i \(-0.173215\pi\)
−0.876128 + 0.482078i \(0.839882\pi\)
\(270\) 0 0
\(271\) 1730.18 2996.75i 0.387825 0.671733i −0.604331 0.796733i \(-0.706560\pi\)
0.992157 + 0.125000i \(0.0398930\pi\)
\(272\) 0 0
\(273\) −333.273 4196.79i −0.0738849 0.930408i
\(274\) 0 0
\(275\) −656.180 + 1136.54i −0.143888 + 0.249221i
\(276\) 0 0
\(277\) 3218.97 + 5575.42i 0.698228 + 1.20937i 0.969080 + 0.246745i \(0.0793611\pi\)
−0.270853 + 0.962621i \(0.587306\pi\)
\(278\) 0 0
\(279\) −144.055 249.511i −0.0309117 0.0535406i
\(280\) 0 0
\(281\) −2974.26 −0.631421 −0.315711 0.948855i \(-0.602243\pi\)
−0.315711 + 0.948855i \(0.602243\pi\)
\(282\) 0 0
\(283\) 1517.86 2629.01i 0.318825 0.552221i −0.661418 0.750017i \(-0.730045\pi\)
0.980243 + 0.197797i \(0.0633785\pi\)
\(284\) 0 0
\(285\) −3555.07 −0.738891
\(286\) 0 0
\(287\) 7240.15 1.48910
\(288\) 0 0
\(289\) 1291.91 2237.65i 0.262957 0.455455i
\(290\) 0 0
\(291\) 642.604 0.129451
\(292\) 0 0
\(293\) 977.872 + 1693.72i 0.194976 + 0.337708i 0.946893 0.321550i \(-0.104204\pi\)
−0.751917 + 0.659258i \(0.770871\pi\)
\(294\) 0 0
\(295\) 1614.47 + 2796.34i 0.318637 + 0.551895i
\(296\) 0 0
\(297\) 633.498 1097.25i 0.123769 0.214373i
\(298\) 0 0
\(299\) 5531.15 + 2633.43i 1.06981 + 0.509349i
\(300\) 0 0
\(301\) −1443.22 + 2499.73i −0.276365 + 0.478678i
\(302\) 0 0
\(303\) −2392.67 4144.22i −0.453648 0.785741i
\(304\) 0 0
\(305\) 484.735 + 839.586i 0.0910029 + 0.157622i
\(306\) 0 0
\(307\) 1027.56 0.191029 0.0955147 0.995428i \(-0.469550\pi\)
0.0955147 + 0.995428i \(0.469550\pi\)
\(308\) 0 0
\(309\) −2355.44 + 4079.75i −0.433646 + 0.751096i
\(310\) 0 0
\(311\) −3405.61 −0.620947 −0.310474 0.950582i \(-0.600488\pi\)
−0.310474 + 0.950582i \(0.600488\pi\)
\(312\) 0 0
\(313\) 4813.20 0.869196 0.434598 0.900625i \(-0.356890\pi\)
0.434598 + 0.900625i \(0.356890\pi\)
\(314\) 0 0
\(315\) 1327.15 2298.69i 0.237385 0.411163i
\(316\) 0 0
\(317\) 1141.33 0.202219 0.101110 0.994875i \(-0.467761\pi\)
0.101110 + 0.994875i \(0.467761\pi\)
\(318\) 0 0
\(319\) −4573.99 7922.38i −0.802803 1.39050i
\(320\) 0 0
\(321\) −10.0995 17.4929i −0.00175608 0.00304161i
\(322\) 0 0
\(323\) −2902.94 + 5028.04i −0.500074 + 0.866154i
\(324\) 0 0
\(325\) −103.771 1306.75i −0.0177113 0.223032i
\(326\) 0 0
\(327\) 813.633 1409.25i 0.137596 0.238324i
\(328\) 0 0
\(329\) −8068.94 13975.8i −1.35214 2.34198i
\(330\) 0 0
\(331\) 3826.28 + 6627.32i 0.635382 + 1.10051i 0.986434 + 0.164158i \(0.0524908\pi\)
−0.351052 + 0.936356i \(0.614176\pi\)
\(332\) 0 0
\(333\) −291.825 −0.0480237
\(334\) 0 0
\(335\) −2172.79 + 3763.38i −0.354364 + 0.613777i
\(336\) 0 0
\(337\) −2503.69 −0.404702 −0.202351 0.979313i \(-0.564858\pi\)
−0.202351 + 0.979313i \(0.564858\pi\)
\(338\) 0 0
\(339\) −4328.47 −0.693481
\(340\) 0 0
\(341\) 751.100 1300.94i 0.119280 0.206598i
\(342\) 0 0
\(343\) −6298.69 −0.991537
\(344\) 0 0
\(345\) 1931.16 + 3344.86i 0.301362 + 0.521975i
\(346\) 0 0
\(347\) 2248.06 + 3893.75i 0.347787 + 0.602385i 0.985856 0.167594i \(-0.0535999\pi\)
−0.638069 + 0.769979i \(0.720267\pi\)
\(348\) 0 0
\(349\) −1788.81 + 3098.31i −0.274363 + 0.475211i −0.969974 0.243208i \(-0.921800\pi\)
0.695611 + 0.718418i \(0.255134\pi\)
\(350\) 0 0
\(351\) 100.183 + 1261.58i 0.0152348 + 0.191846i
\(352\) 0 0
\(353\) 4022.58 6967.32i 0.606517 1.05052i −0.385293 0.922794i \(-0.625900\pi\)
0.991810 0.127724i \(-0.0407672\pi\)
\(354\) 0 0
\(355\) 1699.46 + 2943.55i 0.254079 + 0.440077i
\(356\) 0 0
\(357\) −2167.40 3754.05i −0.321319 0.556542i
\(358\) 0 0
\(359\) 2172.90 0.319447 0.159724 0.987162i \(-0.448940\pi\)
0.159724 + 0.987162i \(0.448940\pi\)
\(360\) 0 0
\(361\) −3806.57 + 6593.17i −0.554974 + 0.961244i
\(362\) 0 0
\(363\) 2613.08 0.377826
\(364\) 0 0
\(365\) 7622.74 1.09313
\(366\) 0 0
\(367\) 3831.38 6636.15i 0.544949 0.943880i −0.453661 0.891175i \(-0.649882\pi\)
0.998610 0.0527056i \(-0.0167845\pi\)
\(368\) 0 0
\(369\) −2176.43 −0.307047
\(370\) 0 0
\(371\) −2279.55 3948.30i −0.318998 0.552521i
\(372\) 0 0
\(373\) −5271.27 9130.10i −0.731732 1.26740i −0.956143 0.292902i \(-0.905379\pi\)
0.224411 0.974495i \(-0.427954\pi\)
\(374\) 0 0
\(375\) 2260.21 3914.80i 0.311244 0.539091i
\(376\) 0 0
\(377\) 8250.17 + 3927.99i 1.12707 + 0.536610i
\(378\) 0 0
\(379\) 2737.77 4741.96i 0.371055 0.642686i −0.618673 0.785648i \(-0.712329\pi\)
0.989728 + 0.142962i \(0.0456628\pi\)
\(380\) 0 0
\(381\) −3739.26 6476.59i −0.502803 0.870881i
\(382\) 0 0
\(383\) 404.042 + 699.822i 0.0539050 + 0.0933661i 0.891719 0.452590i \(-0.149500\pi\)
−0.837814 + 0.545956i \(0.816167\pi\)
\(384\) 0 0
\(385\) 13839.4 1.83200
\(386\) 0 0
\(387\) 433.839 751.432i 0.0569852 0.0987013i
\(388\) 0 0
\(389\) −7060.26 −0.920230 −0.460115 0.887859i \(-0.652192\pi\)
−0.460115 + 0.887859i \(0.652192\pi\)
\(390\) 0 0
\(391\) 6307.65 0.815836
\(392\) 0 0
\(393\) 1116.84 1934.43i 0.143352 0.248292i
\(394\) 0 0
\(395\) 1480.85 0.188633
\(396\) 0 0
\(397\) 709.947 + 1229.66i 0.0897511 + 0.155453i 0.907406 0.420255i \(-0.138060\pi\)
−0.817655 + 0.575709i \(0.804726\pi\)
\(398\) 0 0
\(399\) −5402.61 9357.60i −0.677867 1.17410i
\(400\) 0 0
\(401\) −5335.37 + 9241.13i −0.664428 + 1.15082i 0.315012 + 0.949088i \(0.397991\pi\)
−0.979440 + 0.201735i \(0.935342\pi\)
\(402\) 0 0
\(403\) 118.782 + 1495.78i 0.0146822 + 0.184888i
\(404\) 0 0
\(405\) −398.947 + 690.997i −0.0489478 + 0.0847800i
\(406\) 0 0
\(407\) −760.782 1317.71i −0.0926550 0.160483i
\(408\) 0 0
\(409\) −3175.78 5500.61i −0.383942 0.665007i 0.607680 0.794182i \(-0.292100\pi\)
−0.991622 + 0.129175i \(0.958767\pi\)
\(410\) 0 0
\(411\) −666.500 −0.0799903
\(412\) 0 0
\(413\) −4906.98 + 8499.15i −0.584641 + 1.01263i
\(414\) 0 0
\(415\) −3329.15 −0.393787
\(416\) 0 0
\(417\) 2331.44 0.273791
\(418\) 0 0
\(419\) 2808.94 4865.22i 0.327507 0.567259i −0.654509 0.756054i \(-0.727125\pi\)
0.982017 + 0.188795i \(0.0604581\pi\)
\(420\) 0 0
\(421\) −1518.29 −0.175765 −0.0878825 0.996131i \(-0.528010\pi\)
−0.0878825 + 0.996131i \(0.528010\pi\)
\(422\) 0 0
\(423\) 2425.57 + 4201.20i 0.278806 + 0.482907i
\(424\) 0 0
\(425\) −674.860 1168.89i −0.0770248 0.133411i
\(426\) 0 0
\(427\) −1473.30 + 2551.83i −0.166974 + 0.289207i
\(428\) 0 0
\(429\) −5435.39 + 3741.28i −0.611709 + 0.421051i
\(430\) 0 0
\(431\) −2485.07 + 4304.26i −0.277730 + 0.481042i −0.970820 0.239809i \(-0.922915\pi\)
0.693091 + 0.720850i \(0.256249\pi\)
\(432\) 0 0
\(433\) 1649.36 + 2856.77i 0.183055 + 0.317061i 0.942919 0.333021i \(-0.108068\pi\)
−0.759864 + 0.650082i \(0.774735\pi\)
\(434\) 0 0
\(435\) 2880.48 + 4989.14i 0.317491 + 0.549911i
\(436\) 0 0
\(437\) 15722.9 1.72112
\(438\) 0 0
\(439\) −3024.24 + 5238.13i −0.328790 + 0.569481i −0.982272 0.187461i \(-0.939974\pi\)
0.653482 + 0.756942i \(0.273308\pi\)
\(440\) 0 0
\(441\) 4980.42 0.537784
\(442\) 0 0
\(443\) −6822.62 −0.731722 −0.365861 0.930670i \(-0.619225\pi\)
−0.365861 + 0.930670i \(0.619225\pi\)
\(444\) 0 0
\(445\) −836.869 + 1449.50i −0.0891493 + 0.154411i
\(446\) 0 0
\(447\) 5334.88 0.564500
\(448\) 0 0
\(449\) −205.205 355.426i −0.0215684 0.0373576i 0.855040 0.518563i \(-0.173533\pi\)
−0.876608 + 0.481205i \(0.840199\pi\)
\(450\) 0 0
\(451\) −5673.91 9827.51i −0.592404 1.02607i
\(452\) 0 0
\(453\) 1749.00 3029.35i 0.181402 0.314197i
\(454\) 0 0
\(455\) −11386.9 + 7837.81i −1.17324 + 0.807565i
\(456\) 0 0
\(457\) 8321.41 14413.1i 0.851771 1.47531i −0.0278385 0.999612i \(-0.508862\pi\)
0.879609 0.475697i \(-0.157804\pi\)
\(458\) 0 0
\(459\) 651.532 + 1128.49i 0.0662547 + 0.114757i
\(460\) 0 0
\(461\) 5864.85 + 10158.2i 0.592523 + 1.02628i 0.993891 + 0.110364i \(0.0352015\pi\)
−0.401368 + 0.915917i \(0.631465\pi\)
\(462\) 0 0
\(463\) 3564.93 0.357832 0.178916 0.983864i \(-0.442741\pi\)
0.178916 + 0.983864i \(0.442741\pi\)
\(464\) 0 0
\(465\) −473.008 + 819.273i −0.0471725 + 0.0817051i
\(466\) 0 0
\(467\) −1134.81 −0.112447 −0.0562233 0.998418i \(-0.517906\pi\)
−0.0562233 + 0.998418i \(0.517906\pi\)
\(468\) 0 0
\(469\) −13207.9 −1.30039
\(470\) 0 0
\(471\) −776.441 + 1344.84i −0.0759587 + 0.131564i
\(472\) 0 0
\(473\) 4524.05 0.439780
\(474\) 0 0
\(475\) −1682.20 2913.66i −0.162494 0.281448i
\(476\) 0 0
\(477\) 685.245 + 1186.88i 0.0657761 + 0.113928i
\(478\) 0 0
\(479\) −9686.30 + 16777.2i −0.923963 + 1.60035i −0.130743 + 0.991416i \(0.541736\pi\)
−0.793220 + 0.608935i \(0.791597\pi\)
\(480\) 0 0
\(481\) 1372.24 + 653.335i 0.130080 + 0.0619325i
\(482\) 0 0
\(483\) −5869.53 + 10166.3i −0.552946 + 0.957730i
\(484\) 0 0
\(485\) −1055.00 1827.31i −0.0987733 0.171080i
\(486\) 0 0
\(487\) −4522.98 7834.02i −0.420853 0.728939i 0.575170 0.818034i \(-0.304936\pi\)
−0.996023 + 0.0890946i \(0.971603\pi\)
\(488\) 0 0
\(489\) −1830.56 −0.169286
\(490\) 0 0
\(491\) 7201.66 12473.6i 0.661927 1.14649i −0.318181 0.948030i \(-0.603072\pi\)
0.980109 0.198462i \(-0.0635947\pi\)
\(492\) 0 0
\(493\) 9408.40 0.859499
\(494\) 0 0
\(495\) −4160.20 −0.377751
\(496\) 0 0
\(497\) −5165.31 + 8946.58i −0.466189 + 0.807463i
\(498\) 0 0
\(499\) 9319.75 0.836091 0.418045 0.908426i \(-0.362715\pi\)
0.418045 + 0.908426i \(0.362715\pi\)
\(500\) 0 0
\(501\) 4474.52 + 7750.09i 0.399015 + 0.691115i
\(502\) 0 0
\(503\) 1372.99 + 2378.09i 0.121707 + 0.210802i 0.920441 0.390882i \(-0.127830\pi\)
−0.798734 + 0.601684i \(0.794497\pi\)
\(504\) 0 0
\(505\) −7856.37 + 13607.6i −0.692285 + 1.19907i
\(506\) 0 0
\(507\) 2353.32 6156.55i 0.206143 0.539294i
\(508\) 0 0
\(509\) 591.055 1023.74i 0.0514697 0.0891481i −0.839143 0.543911i \(-0.816943\pi\)
0.890612 + 0.454763i \(0.150276\pi\)
\(510\) 0 0
\(511\) 11584.2 + 20064.5i 1.00285 + 1.73698i
\(512\) 0 0
\(513\) 1624.05 + 2812.94i 0.139773 + 0.242094i
\(514\) 0 0
\(515\) 15468.3 1.32352
\(516\) 0 0
\(517\) −12646.8 + 21905.0i −1.07584 + 1.86340i
\(518\) 0 0
\(519\) 2934.64 0.248201
\(520\) 0 0
\(521\) 10858.8 0.913115 0.456558 0.889694i \(-0.349082\pi\)
0.456558 + 0.889694i \(0.349082\pi\)
\(522\) 0 0
\(523\) −5080.87 + 8800.33i −0.424801 + 0.735777i −0.996402 0.0847546i \(-0.972989\pi\)
0.571601 + 0.820532i \(0.306323\pi\)
\(524\) 0 0
\(525\) 2511.94 0.208819
\(526\) 0 0
\(527\) 772.482 + 1337.98i 0.0638517 + 0.110594i
\(528\) 0 0
\(529\) −2457.36 4256.28i −0.201970 0.349821i
\(530\) 0 0
\(531\) 1475.06 2554.89i 0.120551 0.208800i
\(532\) 0 0
\(533\) 10234.1 + 4872.57i 0.831687 + 0.395975i
\(534\) 0 0
\(535\) −33.1620 + 57.4382i −0.00267984 + 0.00464162i
\(536\) 0 0
\(537\) 2779.41 + 4814.08i 0.223353 + 0.386858i
\(538\) 0 0
\(539\) 12983.9 + 22488.7i 1.03758 + 1.79714i
\(540\) 0 0
\(541\) −9573.04 −0.760771 −0.380386 0.924828i \(-0.624209\pi\)
−0.380386 + 0.924828i \(0.624209\pi\)
\(542\) 0 0
\(543\) −1279.16 + 2215.58i −0.101094 + 0.175101i
\(544\) 0 0
\(545\) −5343.15 −0.419955
\(546\) 0 0
\(547\) −15958.9 −1.24745 −0.623724 0.781645i \(-0.714381\pi\)
−0.623724 + 0.781645i \(0.714381\pi\)
\(548\) 0 0
\(549\) 442.881 767.092i 0.0344293 0.0596333i
\(550\) 0 0
\(551\) 23452.0 1.81323
\(552\) 0 0
\(553\) 2250.44 + 3897.88i 0.173053 + 0.299737i
\(554\) 0 0
\(555\) 479.105 + 829.835i 0.0366430 + 0.0634676i
\(556\) 0 0
\(557\) 1072.55 1857.70i 0.0815892 0.141317i −0.822344 0.568991i \(-0.807334\pi\)
0.903933 + 0.427675i \(0.140667\pi\)
\(558\) 0 0
\(559\) −3722.33 + 2562.15i −0.281642 + 0.193859i
\(560\) 0 0
\(561\) −3397.07 + 5883.90i −0.255658 + 0.442813i
\(562\) 0 0
\(563\) −11159.3 19328.5i −0.835362 1.44689i −0.893736 0.448594i \(-0.851925\pi\)
0.0583740 0.998295i \(-0.481408\pi\)
\(564\) 0 0
\(565\) 7106.29 + 12308.5i 0.529140 + 0.916497i
\(566\) 0 0
\(567\) −2425.11 −0.179621
\(568\) 0 0
\(569\) 9876.51 17106.6i 0.727671 1.26036i −0.230194 0.973145i \(-0.573936\pi\)
0.957865 0.287218i \(-0.0927304\pi\)
\(570\) 0 0
\(571\) 10640.6 0.779850 0.389925 0.920847i \(-0.372501\pi\)
0.389925 + 0.920847i \(0.372501\pi\)
\(572\) 0 0
\(573\) −13323.9 −0.971401
\(574\) 0 0
\(575\) −1827.59 + 3165.47i −0.132549 + 0.229581i
\(576\) 0 0
\(577\) 6547.89 0.472430 0.236215 0.971701i \(-0.424093\pi\)
0.236215 + 0.971701i \(0.424093\pi\)
\(578\) 0 0
\(579\) 3723.64 + 6449.54i 0.267270 + 0.462925i
\(580\) 0 0
\(581\) −5059.29 8762.94i −0.361264 0.625728i
\(582\) 0 0
\(583\) −3572.85 + 6188.35i −0.253812 + 0.439615i
\(584\) 0 0
\(585\) 3422.95 2356.09i 0.241917 0.166517i
\(586\) 0 0
\(587\) 2950.17 5109.84i 0.207439 0.359294i −0.743468 0.668771i \(-0.766821\pi\)
0.950907 + 0.309477i \(0.100154\pi\)
\(588\) 0 0
\(589\) 1925.54 + 3335.14i 0.134704 + 0.233314i
\(590\) 0 0
\(591\) −1890.35 3274.18i −0.131571 0.227888i
\(592\) 0 0
\(593\) −15261.5 −1.05686 −0.528428 0.848978i \(-0.677218\pi\)
−0.528428 + 0.848978i \(0.677218\pi\)
\(594\) 0 0
\(595\) −7116.70 + 12326.5i −0.490346 + 0.849305i
\(596\) 0 0
\(597\) 16562.9 1.13547
\(598\) 0 0
\(599\) 18900.6 1.28925 0.644623 0.764501i \(-0.277014\pi\)
0.644623 + 0.764501i \(0.277014\pi\)
\(600\) 0 0
\(601\) 9253.48 16027.5i 0.628049 1.08781i −0.359894 0.932993i \(-0.617187\pi\)
0.987943 0.154819i \(-0.0494794\pi\)
\(602\) 0 0
\(603\) 3970.36 0.268135
\(604\) 0 0
\(605\) −4290.04 7430.56i −0.288289 0.499331i
\(606\) 0 0
\(607\) −3204.25 5549.92i −0.214261 0.371111i 0.738783 0.673944i \(-0.235401\pi\)
−0.953044 + 0.302833i \(0.902068\pi\)
\(608\) 0 0
\(609\) −8754.90 + 15163.9i −0.582539 + 1.00899i
\(610\) 0 0
\(611\) −2000.01 25185.5i −0.132425 1.66759i
\(612\) 0 0
\(613\) 1753.63 3037.38i 0.115544 0.200128i −0.802453 0.596715i \(-0.796472\pi\)
0.917997 + 0.396587i \(0.129806\pi\)
\(614\) 0 0
\(615\) 3573.17 + 6188.91i 0.234283 + 0.405790i
\(616\) 0 0
\(617\) −7253.89 12564.1i −0.473307 0.819792i 0.526226 0.850345i \(-0.323607\pi\)
−0.999533 + 0.0305526i \(0.990273\pi\)
\(618\) 0 0
\(619\) −4750.85 −0.308486 −0.154243 0.988033i \(-0.549294\pi\)
−0.154243 + 0.988033i \(0.549294\pi\)
\(620\) 0 0
\(621\) 1764.41 3056.05i 0.114015 0.197480i
\(622\) 0 0
\(623\) −5087.14 −0.327146
\(624\) 0 0
\(625\) −11347.0 −0.726209
\(626\) 0 0
\(627\) −8467.76 + 14666.6i −0.539346 + 0.934175i
\(628\) 0 0
\(629\) 1564.88 0.0991986
\(630\) 0 0
\(631\) 2412.60 + 4178.74i 0.152209 + 0.263634i 0.932039 0.362357i \(-0.118028\pi\)
−0.779830 + 0.625991i \(0.784695\pi\)
\(632\) 0 0
\(633\) −6791.69 11763.5i −0.426454 0.738640i
\(634\) 0 0
\(635\) −12277.9 + 21266.0i −0.767298 + 1.32900i
\(636\) 0 0
\(637\) −23419.2 11150.1i −1.45668 0.693539i
\(638\) 0 0
\(639\) 1552.72 2689.39i 0.0961262 0.166495i
\(640\) 0 0
\(641\) 2955.95 + 5119.85i 0.182142 + 0.315479i 0.942610 0.333897i \(-0.108364\pi\)
−0.760468 + 0.649376i \(0.775030\pi\)
\(642\) 0 0
\(643\) 11704.1 + 20272.2i 0.717833 + 1.24332i 0.961857 + 0.273554i \(0.0881991\pi\)
−0.244024 + 0.969769i \(0.578468\pi\)
\(644\) 0 0
\(645\) −2849.04 −0.173924
\(646\) 0 0
\(647\) −8199.33 + 14201.7i −0.498221 + 0.862944i −0.999998 0.00205298i \(-0.999347\pi\)
0.501777 + 0.864997i \(0.332680\pi\)
\(648\) 0 0
\(649\) 15381.9 0.930342
\(650\) 0 0
\(651\) −2875.30 −0.173106
\(652\) 0 0
\(653\) −13764.8 + 23841.3i −0.824897 + 1.42876i 0.0771005 + 0.997023i \(0.475434\pi\)
−0.901998 + 0.431741i \(0.857900\pi\)
\(654\) 0 0
\(655\) −7334.34 −0.437521
\(656\) 0 0
\(657\) −3482.28 6031.48i −0.206783 0.358159i
\(658\) 0 0
\(659\) −12089.9 20940.3i −0.714650 1.23781i −0.963094 0.269164i \(-0.913253\pi\)
0.248444 0.968646i \(-0.420081\pi\)
\(660\) 0 0
\(661\) −2262.52 + 3918.80i −0.133134 + 0.230595i −0.924883 0.380251i \(-0.875838\pi\)
0.791749 + 0.610847i \(0.209171\pi\)
\(662\) 0 0
\(663\) −537.224 6765.08i −0.0314692 0.396281i
\(664\) 0 0
\(665\) −17739.6 + 30725.8i −1.03445 + 1.79172i
\(666\) 0 0
\(667\) −12739.4 22065.3i −0.739539 1.28092i
\(668\) 0 0
\(669\) 6721.76 + 11642.4i 0.388458 + 0.672829i
\(670\) 0 0
\(671\) 4618.34 0.265706
\(672\) 0 0
\(673\) −1643.59 + 2846.78i −0.0941393 + 0.163054i −0.909249 0.416253i \(-0.863343\pi\)
0.815110 + 0.579307i \(0.196677\pi\)
\(674\) 0 0
\(675\) −755.102 −0.0430576
\(676\) 0 0
\(677\) −9724.21 −0.552041 −0.276020 0.961152i \(-0.589016\pi\)
−0.276020 + 0.961152i \(0.589016\pi\)
\(678\) 0 0
\(679\) 3206.55 5553.91i 0.181231 0.313902i
\(680\) 0 0
\(681\) 17279.3 0.972309
\(682\) 0 0
\(683\) 7274.33 + 12599.5i 0.407532 + 0.705867i 0.994613 0.103662i \(-0.0330560\pi\)
−0.587080 + 0.809529i \(0.699723\pi\)
\(684\) 0 0
\(685\) 1094.23 + 1895.26i 0.0610342 + 0.105714i
\(686\) 0 0
\(687\) 6952.62 12042.3i 0.386112 0.668766i
\(688\) 0 0
\(689\) −565.022 7115.13i −0.0312418 0.393418i
\(690\) 0 0
\(691\) 3364.48 5827.45i 0.185226 0.320820i −0.758427 0.651758i \(-0.774032\pi\)
0.943653 + 0.330938i \(0.107365\pi\)
\(692\) 0 0
\(693\) −6322.22 10950.4i −0.346553 0.600248i
\(694\) 0 0
\(695\) −3827.65 6629.69i −0.208908 0.361839i
\(696\) 0 0
\(697\) 11670.9 0.634241
\(698\) 0 0
\(699\) −8829.49 + 15293.1i −0.477771 + 0.827524i
\(700\) 0 0
\(701\) −29159.8 −1.57111 −0.785557 0.618789i \(-0.787624\pi\)
−0.785557 + 0.618789i \(0.787624\pi\)
\(702\) 0 0
\(703\) 3900.73 0.209273
\(704\) 0 0
\(705\) 7964.39 13794.7i 0.425470 0.736935i
\(706\) 0 0
\(707\) −47757.0 −2.54044
\(708\) 0 0
\(709\) 10244.5 + 17744.0i 0.542653 + 0.939903i 0.998751 + 0.0499730i \(0.0159135\pi\)
−0.456097 + 0.889930i \(0.650753\pi\)
\(710\) 0 0
\(711\) −676.495 1171.72i −0.0356829 0.0618046i
\(712\) 0 0
\(713\) 2091.96 3623.37i 0.109880 0.190318i
\(714\) 0 0
\(715\) 19562.3 + 9313.83i 1.02320 + 0.487157i
\(716\) 0 0
\(717\) 3203.77 5549.09i 0.166871 0.289030i
\(718\) 0 0
\(719\) −8995.06 15579.9i −0.466563 0.808111i 0.532707 0.846300i \(-0.321175\pi\)
−0.999271 + 0.0381883i \(0.987841\pi\)
\(720\) 0 0
\(721\) 23507.0 + 40715.3i 1.21421 + 2.10308i
\(722\) 0 0
\(723\) 14078.8 0.724197
\(724\) 0 0
\(725\) −2726.00 + 4721.57i −0.139643 + 0.241869i
\(726\) 0 0
\(727\) 37652.7 1.92086 0.960428 0.278528i \(-0.0898465\pi\)
0.960428 + 0.278528i \(0.0898465\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −2326.42 + 4029.48i −0.117710 + 0.203879i
\(732\) 0 0
\(733\) 4524.26 0.227977 0.113989 0.993482i \(-0.463637\pi\)
0.113989 + 0.993482i \(0.463637\pi\)
\(734\) 0 0
\(735\) −8176.64 14162.4i −0.410340 0.710730i
\(736\) 0 0
\(737\) 10350.7 + 17927.9i 0.517329 + 0.896040i
\(738\) 0 0
\(739\) 409.151 708.671i 0.0203665 0.0352759i −0.855663 0.517534i \(-0.826850\pi\)
0.876029 + 0.482258i \(0.160183\pi\)
\(740\) 0 0
\(741\) −1339.12 16863.1i −0.0663885 0.836008i
\(742\) 0 0
\(743\) 19501.1 33776.9i 0.962888 1.66777i 0.247702 0.968836i \(-0.420324\pi\)
0.715185 0.698935i \(-0.246342\pi\)
\(744\) 0 0
\(745\) −8758.59 15170.3i −0.430725 0.746037i
\(746\) 0 0
\(747\) 1520.85 + 2634.19i 0.0744912 + 0.129022i
\(748\) 0 0
\(749\) −201.584 −0.00983407
\(750\) 0 0
\(751\) −11188.8 + 19379.7i −0.543658 + 0.941643i 0.455032 + 0.890475i \(0.349628\pi\)
−0.998690 + 0.0511678i \(0.983706\pi\)
\(752\) 0 0
\(753\) 11708.6 0.566649
\(754\) 0 0
\(755\) −11485.7 −0.553653
\(756\) 0 0
\(757\) 17256.3 29888.8i 0.828521 1.43504i −0.0706770 0.997499i \(-0.522516\pi\)
0.899198 0.437542i \(-0.144151\pi\)
\(758\) 0 0
\(759\) 18399.2 0.879905
\(760\) 0 0
\(761\) −9987.78 17299.3i −0.475764 0.824048i 0.523850 0.851810i \(-0.324495\pi\)
−0.999615 + 0.0277624i \(0.991162\pi\)
\(762\) 0 0
\(763\) −8119.95 14064.2i −0.385271 0.667309i
\(764\) 0 0
\(765\) 2139.32 3705.40i 0.101107 0.175123i
\(766\) 0 0
\(767\) −12656.0 + 8711.38i −0.595804 + 0.410104i
\(768\) 0 0
\(769\) 16532.4 28635.0i 0.775260 1.34279i −0.159388 0.987216i \(-0.550952\pi\)
0.934648 0.355574i \(-0.115715\pi\)
\(770\) 0 0
\(771\) 6196.25 + 10732.2i 0.289432 + 0.501312i
\(772\) 0 0
\(773\) 9282.01 + 16076.9i 0.431890 + 0.748055i 0.997036 0.0769359i \(-0.0245137\pi\)
−0.565146 + 0.824991i \(0.691180\pi\)
\(774\) 0 0
\(775\) −895.279 −0.0414960
\(776\) 0 0
\(777\) −1456.19 + 2522.19i −0.0672334 + 0.116452i
\(778\) 0 0
\(779\) 29091.6 1.33802
\(780\) 0 0
\(781\) 16191.7 0.741848
\(782\) 0 0
\(783\) 2631.77 4558.36i 0.120117 0.208049i
\(784\) 0 0
\(785\) 5098.91 0.231832
\(786\) 0 0
\(787\) 18218.4 + 31555.2i 0.825179 + 1.42925i 0.901782 + 0.432190i \(0.142259\pi\)
−0.0766032 + 0.997062i \(0.524407\pi\)
\(788\) 0 0
\(789\) 9528.26 + 16503.4i 0.429930 + 0.744661i
\(790\) 0 0
\(791\) −21598.8 + 37410.2i −0.970877 + 1.68161i
\(792\) 0 0
\(793\) −3799.90 + 2615.55i −0.170162 + 0.117126i
\(794\) 0 0
\(795\) 2250.01 3897.13i 0.100377 0.173858i
\(796\) 0 0
\(797\) −6182.80 10708.9i −0.274788 0.475947i 0.695294 0.718726i \(-0.255274\pi\)
−0.970082 + 0.242779i \(0.921941\pi\)
\(798\) 0 0
\(799\) −13006.9 22528.6i −0.575907 0.997501i
\(800\) 0 0
\(801\) 1529.22 0.0674561
\(802\) 0 0
\(803\) 18156.5 31448.0i 0.797918 1.38204i
\(804\) 0 0
\(805\) 38545.4 1.68763
\(806\) 0 0
\(807\) 544.571 0.0237544
\(808\) 0 0
\(809\) 4046.87 7009.39i 0.175872 0.304619i −0.764591 0.644516i \(-0.777059\pi\)
0.940463 + 0.339897i \(0.110392\pi\)
\(810\) 0 0
\(811\) −15984.7 −0.692105 −0.346052 0.938215i \(-0.612478\pi\)
−0.346052 + 0.938215i \(0.612478\pi\)
\(812\) 0 0
\(813\) 5190.53 + 8990.26i 0.223911 + 0.387825i
\(814\) 0 0
\(815\) 3005.34 + 5205.40i 0.129169 + 0.223727i
\(816\) 0 0
\(817\) −5798.99 + 10044.2i −0.248325 + 0.430111i
\(818\) 0 0
\(819\) 11403.5 + 5429.32i 0.486533 + 0.231643i
\(820\) 0 0
\(821\) 13430.9 23263.1i 0.570942 0.988900i −0.425528 0.904945i \(-0.639911\pi\)
0.996470 0.0839550i \(-0.0267552\pi\)
\(822\) 0 0
\(823\) −2602.97 4508.48i −0.110248 0.190955i 0.805622 0.592429i \(-0.201831\pi\)
−0.915870 + 0.401475i \(0.868498\pi\)
\(824\) 0 0
\(825\) −1968.54 3409.61i −0.0830737 0.143888i
\(826\) 0 0
\(827\) −46621.5 −1.96032 −0.980162 0.198200i \(-0.936491\pi\)
−0.980162 + 0.198200i \(0.936491\pi\)
\(828\) 0 0
\(829\) 20914.9 36225.6i 0.876241 1.51769i 0.0208053 0.999784i \(-0.493377\pi\)
0.855435 0.517910i \(-0.173290\pi\)
\(830\) 0 0
\(831\) −19313.8 −0.806244
\(832\) 0 0
\(833\) −26707.0 −1.11086
\(834\) 0 0
\(835\) 14692.1 25447.5i 0.608913 1.05467i
\(836\) 0 0
\(837\) 864.331 0.0356937
\(838\) 0 0
\(839\) 6342.54 + 10985.6i 0.260988 + 0.452044i 0.966505 0.256649i \(-0.0826185\pi\)
−0.705517 + 0.708693i \(0.749285\pi\)
\(840\) 0 0
\(841\) −6807.43 11790.8i −0.279119 0.483448i
\(842\) 0 0
\(843\) 4461.39 7727.35i 0.182276 0.315711i
\(844\) 0 0
\(845\) −21370.4 + 3415.64i −0.870017 + 0.139055i
\(846\) 0 0
\(847\) 13039.1 22584.3i 0.528959 0.916183i
\(848\) 0 0
\(849\) 4553.58 + 7887.03i 0.184074 + 0.318825i
\(850\) 0 0
\(851\) −2118.92 3670.08i −0.0853534 0.147836i
\(852\) 0 0
\(853\) −37493.3 −1.50498 −0.752488 0.658606i \(-0.771146\pi\)
−0.752488 + 0.658606i \(0.771146\pi\)
\(854\) 0 0
\(855\) 5332.60 9236.34i 0.213300 0.369446i
\(856\) 0 0
\(857\) −11826.3 −0.471386 −0.235693 0.971828i \(-0.575736\pi\)
−0.235693 + 0.971828i \(0.575736\pi\)
\(858\) 0 0
\(859\) 36498.7 1.44973 0.724866 0.688890i \(-0.241901\pi\)
0.724866 + 0.688890i \(0.241901\pi\)
\(860\) 0 0
\(861\) −10860.2 + 18810.5i −0.429867 + 0.744552i
\(862\) 0 0
\(863\) −2292.79 −0.0904372 −0.0452186 0.998977i \(-0.514398\pi\)
−0.0452186 + 0.998977i \(0.514398\pi\)
\(864\) 0 0
\(865\) −4817.96 8344.95i −0.189382 0.328020i
\(866\) 0 0
\(867\) 3875.72 + 6712.95i 0.151818 + 0.262957i
\(868\) 0 0
\(869\) 3527.22 6109.33i 0.137690 0.238487i
\(870\) 0 0
\(871\) −18669.6 8888.81i −0.726288 0.345793i
\(872\) 0 0
\(873\) −963.906 + 1669.53i −0.0373691 + 0.0647253i
\(874\) 0 0
\(875\) −22556.6 39069.2i −0.871488 1.50946i
\(876\) 0 0
\(877\) 10400.1 + 18013.6i 0.400442 + 0.693587i 0.993779 0.111368i \(-0.0355231\pi\)
−0.593337 + 0.804954i \(0.702190\pi\)
\(878\) 0 0
\(879\) −5867.23 −0.225139
\(880\) 0 0
\(881\) 9627.95 16676.1i 0.368188 0.637721i −0.621094 0.783736i \(-0.713311\pi\)
0.989282 + 0.146015i \(0.0466448\pi\)
\(882\) 0 0
\(883\) −1744.49 −0.0664857 −0.0332429 0.999447i \(-0.510583\pi\)
−0.0332429 + 0.999447i \(0.510583\pi\)
\(884\) 0 0
\(885\) −9686.80 −0.367930
\(886\) 0 0
\(887\) 1485.35 2572.70i 0.0562268 0.0973877i −0.836542 0.547903i \(-0.815426\pi\)
0.892769 + 0.450515i \(0.148760\pi\)
\(888\) 0 0
\(889\) −74634.6 −2.81571
\(890\) 0 0
\(891\) 1900.49 + 3291.75i 0.0714578 + 0.123769i
\(892\) 0 0
\(893\) −32421.8 56156.2i −1.21495 2.10436i
\(894\) 0 0
\(895\) 9126.24 15807.1i 0.340845 0.590361i
\(896\) 0 0
\(897\) −15138.6 + 10420.2i −0.563503 + 0.387871i
\(898\) 0 0
\(899\) 3120.33 5404.57i 0.115761 0.200503i
\(900\) 0 0
\(901\) −3674.56 6364.52i −0.135868 0.235331i
\(902\) 0 0
\(903\) −4329.66 7499.19i −0.159559 0.276365i
\(904\) 0 0
\(905\) 8400.32 0.308548
\(906\) 0 0
\(907\) 16501.7 28581.7i 0.604111 1.04635i −0.388080 0.921626i \(-0.626862\pi\)
0.992191 0.124726i \(-0.0398051\pi\)
\(908\) 0 0
\(909\) 14356.0 0.523827
\(910\) 0 0
\(911\) −14977.0 −0.544686 −0.272343 0.962200i \(-0.587799\pi\)
−0.272343 + 0.962200i \(0.587799\pi\)
\(912\) 0 0
\(913\) −7929.66 + 13734.6i −0.287441 + 0.497862i
\(914\) 0 0
\(915\) −2908.41 −0.105081
\(916\) 0 0
\(917\) −11145.9 19305.3i −0.401386 0.695221i
\(918\) 0 0
\(919\) 5712.69 + 9894.67i 0.205054 + 0.355163i 0.950150 0.311794i \(-0.100930\pi\)
−0.745096 + 0.666957i \(0.767596\pi\)
\(920\) 0 0
\(921\) −1541.34 + 2669.68i −0.0551455 + 0.0955147i
\(922\) 0 0
\(923\) −13322.3 + 9169.99i −0.475090 + 0.327014i
\(924\) 0 0
\(925\) −453.410 + 785.329i −0.0161168 + 0.0279151i
\(926\) 0 0
\(927\) −7066.33 12239.2i −0.250365 0.433646i
\(928\) 0 0
\(929\) −5977.10 10352.6i −0.211090 0.365618i 0.740966 0.671542i \(-0.234368\pi\)
−0.952056 + 0.305924i \(0.901035\pi\)
\(930\) 0 0
\(931\) −66571.7 −2.34350
\(932\) 0 0
\(933\) 5108.42 8848.04i 0.179252 0.310474i
\(934\) 0 0
\(935\) 22308.7 0.780290
\(936\) 0 0
\(937\) 42546.4 1.48338 0.741692 0.670740i \(-0.234024\pi\)
0.741692 + 0.670740i \(0.234024\pi\)
\(938\) 0 0
\(939\) −7219.81 + 12505.1i −0.250915 + 0.434598i
\(940\) 0 0
\(941\) 20665.1 0.715903 0.357951 0.933740i \(-0.383475\pi\)
0.357951 + 0.933740i \(0.383475\pi\)
\(942\) 0 0
\(943\) −15802.9 27371.5i −0.545720 0.945215i
\(944\) 0 0
\(945\) 3981.44 + 6896.06i 0.137054 + 0.237385i
\(946\) 0 0
\(947\) −4746.87 + 8221.81i −0.162885 + 0.282126i −0.935902 0.352260i \(-0.885413\pi\)
0.773017 + 0.634385i \(0.218747\pi\)
\(948\) 0 0
\(949\) 2871.33 + 36157.7i 0.0982163 + 1.23681i
\(950\) 0 0
\(951\) −1711.99 + 2965.26i −0.0583756 + 0.101110i
\(952\) 0 0
\(953\) −26667.1 46188.7i −0.906433 1.56999i −0.818982 0.573819i \(-0.805461\pi\)
−0.0874512 0.996169i \(-0.527872\pi\)
\(954\) 0 0
\(955\) 21874.6 + 37887.9i 0.741199 + 1.28379i
\(956\) 0 0
\(957\) 27443.9 0.926997
\(958\) 0 0
\(959\) −3325.79 + 5760.44i −0.111987 + 0.193967i
\(960\) 0 0
\(961\) −28766.2 −0.965601
\(962\) 0 0
\(963\) 60.5972 0.00202774
\(964\) 0 0
\(965\) 12226.6 21177.2i 0.407865 0.706443i
\(966\) 0 0
\(967\) 42110.1 1.40038 0.700191 0.713956i \(-0.253098\pi\)
0.700191 + 0.713956i \(0.253098\pi\)
\(968\) 0 0
\(969\) −8708.83 15084.1i −0.288718 0.500074i
\(970\) 0 0
\(971\) −6913.86 11975.2i −0.228503 0.395779i 0.728862 0.684661i \(-0.240050\pi\)
−0.957365 + 0.288882i \(0.906716\pi\)
\(972\) 0 0
\(973\) 11633.7 20150.2i 0.383309 0.663911i
\(974\) 0 0
\(975\) 3550.69 + 1690.52i 0.116629 + 0.0555281i
\(976\) 0 0
\(977\) −566.943 + 981.973i −0.0185651 + 0.0321557i −0.875159 0.483836i \(-0.839243\pi\)
0.856594 + 0.515992i \(0.172576\pi\)
\(978\) 0 0
\(979\) 3986.65 + 6905.09i 0.130147 + 0.225421i
\(980\) 0 0
\(981\) 2440.90 + 4227.76i 0.0794413 + 0.137596i
\(982\) 0 0
\(983\) −26250.2 −0.851729 −0.425865 0.904787i \(-0.640030\pi\)
−0.425865 + 0.904787i \(0.640030\pi\)
\(984\) 0 0
\(985\) −6206.98 + 10750.8i −0.200783 + 0.347766i
\(986\) 0 0
\(987\) 48413.7 1.56132
\(988\) 0 0
\(989\) 12600.3 0.405124
\(990\) 0 0
\(991\) −14680.2 + 25426.8i −0.470566 + 0.815045i −0.999433 0.0336599i \(-0.989284\pi\)
0.528867 + 0.848705i \(0.322617\pi\)
\(992\) 0 0
\(993\) −22957.7 −0.733676
\(994\) 0 0
\(995\) −27192.3 47098.4i −0.866384 1.50062i
\(996\) 0 0
\(997\) −8417.95 14580.3i −0.267401 0.463153i 0.700789 0.713369i \(-0.252832\pi\)
−0.968190 + 0.250216i \(0.919498\pi\)
\(998\) 0 0
\(999\) 437.737 758.182i 0.0138632 0.0240118i
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 624.4.q.i.529.4 8
4.3 odd 2 39.4.e.c.22.3 yes 8
12.11 even 2 117.4.g.e.100.2 8
13.3 even 3 inner 624.4.q.i.289.4 8
52.3 odd 6 39.4.e.c.16.3 8
52.7 even 12 507.4.b.h.337.4 8
52.19 even 12 507.4.b.h.337.5 8
52.35 odd 6 507.4.a.m.1.2 4
52.43 odd 6 507.4.a.i.1.3 4
156.35 even 6 1521.4.a.v.1.3 4
156.95 even 6 1521.4.a.bb.1.2 4
156.107 even 6 117.4.g.e.55.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.3 8 52.3 odd 6
39.4.e.c.22.3 yes 8 4.3 odd 2
117.4.g.e.55.2 8 156.107 even 6
117.4.g.e.100.2 8 12.11 even 2
507.4.a.i.1.3 4 52.43 odd 6
507.4.a.m.1.2 4 52.35 odd 6
507.4.b.h.337.4 8 52.7 even 12
507.4.b.h.337.5 8 52.19 even 12
624.4.q.i.289.4 8 13.3 even 3 inner
624.4.q.i.529.4 8 1.1 even 1 trivial
1521.4.a.v.1.3 4 156.35 even 6
1521.4.a.bb.1.2 4 156.95 even 6