Properties

Label 84.4.i.b.37.1
Level $84$
Weight $4$
Character 84.37
Analytic conductor $4.956$
Analytic rank $0$
Dimension $4$
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] = [84,4,Mod(25,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), 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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(-1.63746 + 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 84.37
Dual form 84.4.i.b.25.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{3} +(-4.91238 + 8.50848i) q^{5} +(-16.3248 + 8.74657i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{3} +(-4.91238 + 8.50848i) q^{5} +(-16.3248 + 8.74657i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(7.08762 + 12.2761i) q^{11} -26.1238 q^{13} -29.4743 q^{15} +(39.2990 + 68.0679i) q^{17} +(-36.5876 + 63.3716i) q^{19} +(-47.2114 - 29.2931i) q^{21} +(48.0000 - 83.1384i) q^{23} +(14.2371 + 24.6594i) q^{25} -27.0000 q^{27} +173.021 q^{29} +(-33.6238 - 58.2381i) q^{31} +(-21.2629 + 36.8284i) q^{33} +(5.77326 - 181.865i) q^{35} +(150.835 - 261.254i) q^{37} +(-39.1856 - 67.8715i) q^{39} +472.042 q^{41} -463.670 q^{43} +(-44.2114 - 76.5764i) q^{45} +(45.5980 - 78.9781i) q^{47} +(189.995 - 285.571i) q^{49} +(-117.897 + 204.204i) q^{51} +(81.6341 + 141.394i) q^{53} -139.268 q^{55} -219.526 q^{57} +(300.263 + 520.071i) q^{59} +(-285.846 + 495.099i) q^{61} +(5.28862 - 166.598i) q^{63} +(128.330 - 222.274i) q^{65} +(269.711 + 467.154i) q^{67} +288.000 q^{69} -1064.39 q^{71} +(221.340 + 383.372i) q^{73} +(-42.7114 + 73.9783i) q^{75} +(-223.078 - 138.412i) q^{77} +(22.8505 - 39.5782i) q^{79} +(-40.5000 - 70.1481i) q^{81} -686.464 q^{83} -772.206 q^{85} +(259.531 + 449.521i) q^{87} +(-330.072 + 571.702i) q^{89} +(426.464 - 228.493i) q^{91} +(100.871 - 174.714i) q^{93} +(-359.464 - 622.610i) q^{95} +658.320 q^{97} -127.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 3 q^{5} - 20 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 3 q^{5} - 20 q^{7} - 18 q^{9} + 51 q^{11} + 122 q^{13} + 18 q^{15} - 24 q^{17} - 169 q^{19} + 15 q^{21} + 192 q^{23} - 11 q^{25} - 108 q^{27} - 78 q^{29} + 92 q^{31} - 153 q^{33} - 294 q^{35} + 173 q^{37} + 183 q^{39} + 348 q^{41} - 994 q^{43} + 27 q^{45} - 180 q^{47} - 146 q^{49} + 72 q^{51} - 285 q^{53} + 666 q^{55} - 1014 q^{57} + 1269 q^{59} - 328 q^{61} + 225 q^{63} + 1374 q^{65} + 875 q^{67} + 1152 q^{69} - 2808 q^{71} + 1361 q^{73} + 33 q^{75} + 897 q^{77} + 182 q^{79} - 162 q^{81} - 798 q^{83} - 4176 q^{85} - 117 q^{87} - 822 q^{89} + 1955 q^{91} - 276 q^{93} + 510 q^{95} + 1682 q^{97} - 918 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.50000 + 2.59808i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −4.91238 + 8.50848i −0.439376 + 0.761022i −0.997641 0.0686406i \(-0.978134\pi\)
0.558265 + 0.829663i \(0.311467\pi\)
\(6\) 0 0
\(7\) −16.3248 + 8.74657i −0.881454 + 0.472270i
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 7.08762 + 12.2761i 0.194273 + 0.336490i 0.946662 0.322229i \(-0.104432\pi\)
−0.752389 + 0.658719i \(0.771099\pi\)
\(12\) 0 0
\(13\) −26.1238 −0.557341 −0.278670 0.960387i \(-0.589894\pi\)
−0.278670 + 0.960387i \(0.589894\pi\)
\(14\) 0 0
\(15\) −29.4743 −0.507348
\(16\) 0 0
\(17\) 39.2990 + 68.0679i 0.560671 + 0.971111i 0.997438 + 0.0715361i \(0.0227901\pi\)
−0.436767 + 0.899575i \(0.643877\pi\)
\(18\) 0 0
\(19\) −36.5876 + 63.3716i −0.441778 + 0.765181i −0.997822 0.0659715i \(-0.978985\pi\)
0.556044 + 0.831153i \(0.312319\pi\)
\(20\) 0 0
\(21\) −47.2114 29.2931i −0.490589 0.304394i
\(22\) 0 0
\(23\) 48.0000 83.1384i 0.435161 0.753720i −0.562148 0.827037i \(-0.690025\pi\)
0.997309 + 0.0733164i \(0.0233583\pi\)
\(24\) 0 0
\(25\) 14.2371 + 24.6594i 0.113897 + 0.197275i
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 173.021 1.10790 0.553951 0.832549i \(-0.313120\pi\)
0.553951 + 0.832549i \(0.313120\pi\)
\(30\) 0 0
\(31\) −33.6238 58.2381i −0.194807 0.337415i 0.752030 0.659128i \(-0.229075\pi\)
−0.946837 + 0.321713i \(0.895741\pi\)
\(32\) 0 0
\(33\) −21.2629 + 36.8284i −0.112163 + 0.194273i
\(34\) 0 0
\(35\) 5.77326 181.865i 0.0278817 0.878310i
\(36\) 0 0
\(37\) 150.835 261.254i 0.670193 1.16081i −0.307656 0.951498i \(-0.599545\pi\)
0.977849 0.209311i \(-0.0671221\pi\)
\(38\) 0 0
\(39\) −39.1856 67.8715i −0.160890 0.278670i
\(40\) 0 0
\(41\) 472.042 1.79806 0.899031 0.437886i \(-0.144273\pi\)
0.899031 + 0.437886i \(0.144273\pi\)
\(42\) 0 0
\(43\) −463.670 −1.64440 −0.822198 0.569201i \(-0.807253\pi\)
−0.822198 + 0.569201i \(0.807253\pi\)
\(44\) 0 0
\(45\) −44.2114 76.5764i −0.146459 0.253674i
\(46\) 0 0
\(47\) 45.5980 78.9781i 0.141514 0.245109i −0.786553 0.617523i \(-0.788136\pi\)
0.928067 + 0.372413i \(0.121470\pi\)
\(48\) 0 0
\(49\) 189.995 285.571i 0.553921 0.832569i
\(50\) 0 0
\(51\) −117.897 + 204.204i −0.323704 + 0.560671i
\(52\) 0 0
\(53\) 81.6341 + 141.394i 0.211572 + 0.366453i 0.952207 0.305455i \(-0.0988084\pi\)
−0.740635 + 0.671908i \(0.765475\pi\)
\(54\) 0 0
\(55\) −139.268 −0.341435
\(56\) 0 0
\(57\) −219.526 −0.510121
\(58\) 0 0
\(59\) 300.263 + 520.071i 0.662558 + 1.14758i 0.979941 + 0.199286i \(0.0638623\pi\)
−0.317384 + 0.948297i \(0.602804\pi\)
\(60\) 0 0
\(61\) −285.846 + 495.099i −0.599980 + 1.03920i 0.392844 + 0.919605i \(0.371491\pi\)
−0.992823 + 0.119590i \(0.961842\pi\)
\(62\) 0 0
\(63\) 5.28862 166.598i 0.0105762 0.333166i
\(64\) 0 0
\(65\) 128.330 222.274i 0.244882 0.424148i
\(66\) 0 0
\(67\) 269.711 + 467.154i 0.491798 + 0.851820i 0.999955 0.00944469i \(-0.00300638\pi\)
−0.508157 + 0.861264i \(0.669673\pi\)
\(68\) 0 0
\(69\) 288.000 0.502480
\(70\) 0 0
\(71\) −1064.39 −1.77916 −0.889578 0.456783i \(-0.849002\pi\)
−0.889578 + 0.456783i \(0.849002\pi\)
\(72\) 0 0
\(73\) 221.340 + 383.372i 0.354875 + 0.614662i 0.987097 0.160126i \(-0.0511900\pi\)
−0.632221 + 0.774788i \(0.717857\pi\)
\(74\) 0 0
\(75\) −42.7114 + 73.9783i −0.0657585 + 0.113897i
\(76\) 0 0
\(77\) −223.078 138.412i −0.330157 0.204851i
\(78\) 0 0
\(79\) 22.8505 39.5782i 0.0325428 0.0563658i −0.849295 0.527918i \(-0.822973\pi\)
0.881838 + 0.471552i \(0.156306\pi\)
\(80\) 0 0
\(81\) −40.5000 70.1481i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −686.464 −0.907822 −0.453911 0.891047i \(-0.649972\pi\)
−0.453911 + 0.891047i \(0.649972\pi\)
\(84\) 0 0
\(85\) −772.206 −0.985382
\(86\) 0 0
\(87\) 259.531 + 449.521i 0.319824 + 0.553951i
\(88\) 0 0
\(89\) −330.072 + 571.702i −0.393119 + 0.680902i −0.992859 0.119293i \(-0.961937\pi\)
0.599740 + 0.800195i \(0.295271\pi\)
\(90\) 0 0
\(91\) 426.464 228.493i 0.491270 0.263215i
\(92\) 0 0
\(93\) 100.871 174.714i 0.112472 0.194807i
\(94\) 0 0
\(95\) −359.464 622.610i −0.388213 0.672405i
\(96\) 0 0
\(97\) 658.320 0.689095 0.344548 0.938769i \(-0.388032\pi\)
0.344548 + 0.938769i \(0.388032\pi\)
\(98\) 0 0
\(99\) −127.577 −0.129515
\(100\) 0 0
\(101\) 901.949 + 1562.22i 0.888586 + 1.53908i 0.841547 + 0.540184i \(0.181645\pi\)
0.0470394 + 0.998893i \(0.485021\pi\)
\(102\) 0 0
\(103\) 537.454 930.898i 0.514145 0.890525i −0.485721 0.874114i \(-0.661443\pi\)
0.999865 0.0164107i \(-0.00522392\pi\)
\(104\) 0 0
\(105\) 481.160 257.799i 0.447204 0.239605i
\(106\) 0 0
\(107\) 829.676 1437.04i 0.749605 1.29835i −0.198407 0.980120i \(-0.563577\pi\)
0.948012 0.318235i \(-0.103090\pi\)
\(108\) 0 0
\(109\) −299.464 518.687i −0.263151 0.455791i 0.703927 0.710273i \(-0.251428\pi\)
−0.967077 + 0.254482i \(0.918095\pi\)
\(110\) 0 0
\(111\) 905.011 0.773872
\(112\) 0 0
\(113\) −781.650 −0.650720 −0.325360 0.945590i \(-0.605486\pi\)
−0.325360 + 0.945590i \(0.605486\pi\)
\(114\) 0 0
\(115\) 471.588 + 816.815i 0.382398 + 0.662333i
\(116\) 0 0
\(117\) 117.557 203.615i 0.0928901 0.160890i
\(118\) 0 0
\(119\) −1236.91 767.460i −0.952833 0.591201i
\(120\) 0 0
\(121\) 565.031 978.663i 0.424516 0.735284i
\(122\) 0 0
\(123\) 708.062 + 1226.40i 0.519056 + 0.899031i
\(124\) 0 0
\(125\) −1507.85 −1.07893
\(126\) 0 0
\(127\) 25.3722 0.0177277 0.00886385 0.999961i \(-0.497179\pi\)
0.00886385 + 0.999961i \(0.497179\pi\)
\(128\) 0 0
\(129\) −695.505 1204.65i −0.474696 0.822198i
\(130\) 0 0
\(131\) 237.789 411.862i 0.158593 0.274691i −0.775768 0.631018i \(-0.782638\pi\)
0.934362 + 0.356326i \(0.115971\pi\)
\(132\) 0 0
\(133\) 42.9995 1354.54i 0.0280341 0.883111i
\(134\) 0 0
\(135\) 132.634 229.729i 0.0845580 0.146459i
\(136\) 0 0
\(137\) −529.268 916.720i −0.330062 0.571683i 0.652462 0.757821i \(-0.273736\pi\)
−0.982524 + 0.186138i \(0.940403\pi\)
\(138\) 0 0
\(139\) 2580.99 1.57494 0.787470 0.616352i \(-0.211390\pi\)
0.787470 + 0.616352i \(0.211390\pi\)
\(140\) 0 0
\(141\) 273.588 0.163406
\(142\) 0 0
\(143\) −185.155 320.699i −0.108276 0.187540i
\(144\) 0 0
\(145\) −849.943 + 1472.14i −0.486786 + 0.843138i
\(146\) 0 0
\(147\) 1026.93 + 65.2648i 0.576188 + 0.0366187i
\(148\) 0 0
\(149\) 50.9377 88.2266i 0.0280066 0.0485088i −0.851682 0.524058i \(-0.824417\pi\)
0.879689 + 0.475549i \(0.157751\pi\)
\(150\) 0 0
\(151\) 889.211 + 1540.16i 0.479225 + 0.830042i 0.999716 0.0238249i \(-0.00758441\pi\)
−0.520491 + 0.853867i \(0.674251\pi\)
\(152\) 0 0
\(153\) −707.382 −0.373781
\(154\) 0 0
\(155\) 660.690 0.342374
\(156\) 0 0
\(157\) −699.629 1211.79i −0.355646 0.615997i 0.631582 0.775309i \(-0.282406\pi\)
−0.987228 + 0.159312i \(0.949073\pi\)
\(158\) 0 0
\(159\) −244.902 + 424.183i −0.122151 + 0.211572i
\(160\) 0 0
\(161\) −56.4119 + 1777.05i −0.0276142 + 0.869883i
\(162\) 0 0
\(163\) −1807.71 + 3131.05i −0.868656 + 1.50456i −0.00528615 + 0.999986i \(0.501683\pi\)
−0.863370 + 0.504571i \(0.831651\pi\)
\(164\) 0 0
\(165\) −208.902 361.830i −0.0985638 0.170718i
\(166\) 0 0
\(167\) −1076.72 −0.498917 −0.249459 0.968385i \(-0.580253\pi\)
−0.249459 + 0.968385i \(0.580253\pi\)
\(168\) 0 0
\(169\) −1514.55 −0.689372
\(170\) 0 0
\(171\) −329.289 570.345i −0.147259 0.255060i
\(172\) 0 0
\(173\) −936.815 + 1622.61i −0.411704 + 0.713091i −0.995076 0.0991132i \(-0.968399\pi\)
0.583373 + 0.812205i \(0.301733\pi\)
\(174\) 0 0
\(175\) −448.103 278.033i −0.193562 0.120099i
\(176\) 0 0
\(177\) −900.789 + 1560.21i −0.382528 + 0.662558i
\(178\) 0 0
\(179\) 1346.15 + 2331.59i 0.562099 + 0.973583i 0.997313 + 0.0732565i \(0.0233392\pi\)
−0.435215 + 0.900327i \(0.643328\pi\)
\(180\) 0 0
\(181\) 461.670 0.189589 0.0947947 0.995497i \(-0.469781\pi\)
0.0947947 + 0.995497i \(0.469781\pi\)
\(182\) 0 0
\(183\) −1715.07 −0.692797
\(184\) 0 0
\(185\) 1481.92 + 2566.76i 0.588934 + 1.02006i
\(186\) 0 0
\(187\) −557.073 + 964.879i −0.217846 + 0.377321i
\(188\) 0 0
\(189\) 440.768 236.157i 0.169636 0.0908885i
\(190\) 0 0
\(191\) 1882.41 3260.43i 0.713124 1.23517i −0.250555 0.968102i \(-0.580613\pi\)
0.963679 0.267064i \(-0.0860535\pi\)
\(192\) 0 0
\(193\) −1463.10 2534.16i −0.545680 0.945145i −0.998564 0.0535755i \(-0.982938\pi\)
0.452884 0.891569i \(-0.350395\pi\)
\(194\) 0 0
\(195\) 769.978 0.282766
\(196\) 0 0
\(197\) 5045.55 1.82477 0.912387 0.409330i \(-0.134237\pi\)
0.912387 + 0.409330i \(0.134237\pi\)
\(198\) 0 0
\(199\) −2146.12 3717.20i −0.764496 1.32415i −0.940513 0.339759i \(-0.889654\pi\)
0.176016 0.984387i \(-0.443679\pi\)
\(200\) 0 0
\(201\) −809.134 + 1401.46i −0.283940 + 0.491798i
\(202\) 0 0
\(203\) −2824.52 + 1513.34i −0.976564 + 0.523229i
\(204\) 0 0
\(205\) −2318.85 + 4016.36i −0.790025 + 1.36836i
\(206\) 0 0
\(207\) 432.000 + 748.246i 0.145054 + 0.251240i
\(208\) 0 0
\(209\) −1037.28 −0.343301
\(210\) 0 0
\(211\) 3625.76 1.18297 0.591487 0.806315i \(-0.298541\pi\)
0.591487 + 0.806315i \(0.298541\pi\)
\(212\) 0 0
\(213\) −1596.59 2765.37i −0.513598 0.889578i
\(214\) 0 0
\(215\) 2277.72 3945.13i 0.722509 1.25142i
\(216\) 0 0
\(217\) 1058.28 + 656.629i 0.331064 + 0.205414i
\(218\) 0 0
\(219\) −664.020 + 1150.12i −0.204887 + 0.354875i
\(220\) 0 0
\(221\) −1026.64 1778.19i −0.312485 0.541239i
\(222\) 0 0
\(223\) −3145.00 −0.944417 −0.472208 0.881487i \(-0.656543\pi\)
−0.472208 + 0.881487i \(0.656543\pi\)
\(224\) 0 0
\(225\) −256.268 −0.0759313
\(226\) 0 0
\(227\) 772.873 + 1338.65i 0.225980 + 0.391408i 0.956613 0.291362i \(-0.0941084\pi\)
−0.730633 + 0.682770i \(0.760775\pi\)
\(228\) 0 0
\(229\) 2036.07 3526.58i 0.587544 1.01766i −0.407009 0.913424i \(-0.633428\pi\)
0.994553 0.104232i \(-0.0332383\pi\)
\(230\) 0 0
\(231\) 24.9892 787.191i 0.00711760 0.224214i
\(232\) 0 0
\(233\) 1507.86 2611.68i 0.423961 0.734322i −0.572362 0.820001i \(-0.693973\pi\)
0.996323 + 0.0856790i \(0.0273060\pi\)
\(234\) 0 0
\(235\) 447.989 + 775.940i 0.124356 + 0.215390i
\(236\) 0 0
\(237\) 137.103 0.0375772
\(238\) 0 0
\(239\) 4647.38 1.25780 0.628900 0.777486i \(-0.283505\pi\)
0.628900 + 0.777486i \(0.283505\pi\)
\(240\) 0 0
\(241\) −559.138 968.456i −0.149449 0.258854i 0.781575 0.623812i \(-0.214417\pi\)
−0.931024 + 0.364958i \(0.881083\pi\)
\(242\) 0 0
\(243\) 121.500 210.444i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 1496.45 + 3019.40i 0.390223 + 0.787357i
\(246\) 0 0
\(247\) 955.806 1655.50i 0.246221 0.426467i
\(248\) 0 0
\(249\) −1029.70 1783.49i −0.262066 0.453911i
\(250\) 0 0
\(251\) 883.518 0.222180 0.111090 0.993810i \(-0.464566\pi\)
0.111090 + 0.993810i \(0.464566\pi\)
\(252\) 0 0
\(253\) 1360.82 0.338159
\(254\) 0 0
\(255\) −1158.31 2006.25i −0.284455 0.492691i
\(256\) 0 0
\(257\) 2934.44 5082.61i 0.712240 1.23364i −0.251775 0.967786i \(-0.581014\pi\)
0.964015 0.265849i \(-0.0856523\pi\)
\(258\) 0 0
\(259\) −177.269 + 5584.20i −0.0425287 + 1.33971i
\(260\) 0 0
\(261\) −778.594 + 1348.56i −0.184650 + 0.319824i
\(262\) 0 0
\(263\) −2437.81 4222.41i −0.571565 0.989980i −0.996406 0.0847114i \(-0.973003\pi\)
0.424841 0.905268i \(-0.360330\pi\)
\(264\) 0 0
\(265\) −1604.07 −0.371839
\(266\) 0 0
\(267\) −1980.43 −0.453935
\(268\) 0 0
\(269\) 2763.55 + 4786.61i 0.626382 + 1.08493i 0.988272 + 0.152704i \(0.0487983\pi\)
−0.361890 + 0.932221i \(0.617868\pi\)
\(270\) 0 0
\(271\) 1612.37 2792.70i 0.361418 0.625994i −0.626777 0.779199i \(-0.715626\pi\)
0.988194 + 0.153205i \(0.0489594\pi\)
\(272\) 0 0
\(273\) 1233.34 + 765.246i 0.273425 + 0.169651i
\(274\) 0 0
\(275\) −201.815 + 349.554i −0.0442542 + 0.0766504i
\(276\) 0 0
\(277\) 979.022 + 1695.72i 0.212360 + 0.367818i 0.952453 0.304687i \(-0.0985517\pi\)
−0.740093 + 0.672505i \(0.765218\pi\)
\(278\) 0 0
\(279\) 605.228 0.129871
\(280\) 0 0
\(281\) 8531.16 1.81113 0.905563 0.424212i \(-0.139449\pi\)
0.905563 + 0.424212i \(0.139449\pi\)
\(282\) 0 0
\(283\) −3954.93 6850.14i −0.830728 1.43886i −0.897461 0.441093i \(-0.854591\pi\)
0.0667331 0.997771i \(-0.478742\pi\)
\(284\) 0 0
\(285\) 1078.39 1867.83i 0.224135 0.388213i
\(286\) 0 0
\(287\) −7705.96 + 4128.74i −1.58491 + 0.849171i
\(288\) 0 0
\(289\) −632.324 + 1095.22i −0.128704 + 0.222922i
\(290\) 0 0
\(291\) 987.480 + 1710.36i 0.198925 + 0.344548i
\(292\) 0 0
\(293\) −8321.41 −1.65919 −0.829594 0.558367i \(-0.811428\pi\)
−0.829594 + 0.558367i \(0.811428\pi\)
\(294\) 0 0
\(295\) −5900.02 −1.16445
\(296\) 0 0
\(297\) −191.366 331.455i −0.0373878 0.0647575i
\(298\) 0 0
\(299\) −1253.94 + 2171.89i −0.242533 + 0.420079i
\(300\) 0 0
\(301\) 7569.30 4055.52i 1.44946 0.776600i
\(302\) 0 0
\(303\) −2705.85 + 4686.66i −0.513026 + 0.888586i
\(304\) 0 0
\(305\) −2808.36 4864.22i −0.527234 0.913196i
\(306\) 0 0
\(307\) 2541.79 0.472533 0.236267 0.971688i \(-0.424076\pi\)
0.236267 + 0.971688i \(0.424076\pi\)
\(308\) 0 0
\(309\) 3224.72 0.593683
\(310\) 0 0
\(311\) 303.855 + 526.292i 0.0554020 + 0.0959590i 0.892396 0.451253i \(-0.149023\pi\)
−0.836994 + 0.547212i \(0.815689\pi\)
\(312\) 0 0
\(313\) −5513.64 + 9549.90i −0.995684 + 1.72458i −0.417471 + 0.908690i \(0.637084\pi\)
−0.578213 + 0.815886i \(0.696250\pi\)
\(314\) 0 0
\(315\) 1391.52 + 863.392i 0.248899 + 0.154434i
\(316\) 0 0
\(317\) −3630.80 + 6288.73i −0.643300 + 1.11423i 0.341391 + 0.939921i \(0.389102\pi\)
−0.984691 + 0.174307i \(0.944231\pi\)
\(318\) 0 0
\(319\) 1226.31 + 2124.02i 0.215235 + 0.372798i
\(320\) 0 0
\(321\) 4978.05 0.865570
\(322\) 0 0
\(323\) −5751.43 −0.990768
\(324\) 0 0
\(325\) −371.927 644.197i −0.0634794 0.109950i
\(326\) 0 0
\(327\) 898.392 1556.06i 0.151930 0.263151i
\(328\) 0 0
\(329\) −53.5890 + 1688.12i −0.00898011 + 0.282885i
\(330\) 0 0
\(331\) −3316.36 + 5744.11i −0.550706 + 0.953851i 0.447517 + 0.894275i \(0.352308\pi\)
−0.998224 + 0.0595761i \(0.981025\pi\)
\(332\) 0 0
\(333\) 1357.52 + 2351.29i 0.223398 + 0.386936i
\(334\) 0 0
\(335\) −5299.69 −0.864338
\(336\) 0 0
\(337\) 8104.59 1.31005 0.655023 0.755609i \(-0.272659\pi\)
0.655023 + 0.755609i \(0.272659\pi\)
\(338\) 0 0
\(339\) −1172.47 2030.78i −0.187847 0.325360i
\(340\) 0 0
\(341\) 476.625 825.539i 0.0756912 0.131101i
\(342\) 0 0
\(343\) −603.854 + 6323.68i −0.0950584 + 0.995472i
\(344\) 0 0
\(345\) −1414.76 + 2450.44i −0.220778 + 0.382398i
\(346\) 0 0
\(347\) 2968.64 + 5141.83i 0.459265 + 0.795470i 0.998922 0.0464146i \(-0.0147795\pi\)
−0.539657 + 0.841885i \(0.681446\pi\)
\(348\) 0 0
\(349\) 268.472 0.0411775 0.0205888 0.999788i \(-0.493446\pi\)
0.0205888 + 0.999788i \(0.493446\pi\)
\(350\) 0 0
\(351\) 705.341 0.107260
\(352\) 0 0
\(353\) −1331.40 2306.06i −0.200747 0.347703i 0.748023 0.663673i \(-0.231003\pi\)
−0.948769 + 0.315970i \(0.897670\pi\)
\(354\) 0 0
\(355\) 5228.69 9056.36i 0.781719 1.35398i
\(356\) 0 0
\(357\) 138.558 4364.77i 0.0205414 0.647081i
\(358\) 0 0
\(359\) 781.637 1353.83i 0.114911 0.199032i −0.802833 0.596204i \(-0.796675\pi\)
0.917744 + 0.397172i \(0.130008\pi\)
\(360\) 0 0
\(361\) 752.192 + 1302.83i 0.109665 + 0.189945i
\(362\) 0 0
\(363\) 3390.19 0.490189
\(364\) 0 0
\(365\) −4349.22 −0.623695
\(366\) 0 0
\(367\) −605.408 1048.60i −0.0861091 0.149145i 0.819754 0.572716i \(-0.194110\pi\)
−0.905863 + 0.423570i \(0.860777\pi\)
\(368\) 0 0
\(369\) −2124.19 + 3679.20i −0.299677 + 0.519056i
\(370\) 0 0
\(371\) −2569.37 1594.21i −0.359556 0.223093i
\(372\) 0 0
\(373\) 2385.93 4132.55i 0.331203 0.573661i −0.651545 0.758610i \(-0.725879\pi\)
0.982748 + 0.184949i \(0.0592121\pi\)
\(374\) 0 0
\(375\) −2261.77 3917.50i −0.311459 0.539464i
\(376\) 0 0
\(377\) −4519.95 −0.617479
\(378\) 0 0
\(379\) −4118.66 −0.558209 −0.279104 0.960261i \(-0.590038\pi\)
−0.279104 + 0.960261i \(0.590038\pi\)
\(380\) 0 0
\(381\) 38.0583 + 65.9189i 0.00511754 + 0.00886385i
\(382\) 0 0
\(383\) −1767.99 + 3062.25i −0.235875 + 0.408547i −0.959527 0.281618i \(-0.909129\pi\)
0.723652 + 0.690165i \(0.242462\pi\)
\(384\) 0 0
\(385\) 2273.52 1218.12i 0.300959 0.161250i
\(386\) 0 0
\(387\) 2086.52 3613.95i 0.274066 0.474696i
\(388\) 0 0
\(389\) −4803.23 8319.44i −0.626050 1.08435i −0.988337 0.152284i \(-0.951337\pi\)
0.362287 0.932067i \(-0.381996\pi\)
\(390\) 0 0
\(391\) 7545.41 0.975928
\(392\) 0 0
\(393\) 1426.73 0.183127
\(394\) 0 0
\(395\) 224.500 + 388.846i 0.0285971 + 0.0495316i
\(396\) 0 0
\(397\) −2683.54 + 4648.02i −0.339252 + 0.587601i −0.984292 0.176547i \(-0.943507\pi\)
0.645041 + 0.764148i \(0.276840\pi\)
\(398\) 0 0
\(399\) 3583.70 1920.10i 0.449648 0.240915i
\(400\) 0 0
\(401\) 7032.91 12181.4i 0.875828 1.51698i 0.0199492 0.999801i \(-0.493650\pi\)
0.855878 0.517177i \(-0.173017\pi\)
\(402\) 0 0
\(403\) 878.379 + 1521.40i 0.108574 + 0.188055i
\(404\) 0 0
\(405\) 795.805 0.0976392
\(406\) 0 0
\(407\) 4276.25 0.520801
\(408\) 0 0
\(409\) −1636.22 2834.02i −0.197814 0.342624i 0.750005 0.661432i \(-0.230051\pi\)
−0.947819 + 0.318808i \(0.896718\pi\)
\(410\) 0 0
\(411\) 1587.80 2750.16i 0.190561 0.330062i
\(412\) 0 0
\(413\) −9450.55 5863.75i −1.12598 0.698635i
\(414\) 0 0
\(415\) 3372.17 5840.77i 0.398876 0.690873i
\(416\) 0 0
\(417\) 3871.49 + 6705.61i 0.454646 + 0.787470i
\(418\) 0 0
\(419\) 8077.07 0.941744 0.470872 0.882202i \(-0.343939\pi\)
0.470872 + 0.882202i \(0.343939\pi\)
\(420\) 0 0
\(421\) 8051.68 0.932101 0.466051 0.884758i \(-0.345676\pi\)
0.466051 + 0.884758i \(0.345676\pi\)
\(422\) 0 0
\(423\) 410.382 + 710.803i 0.0471713 + 0.0817031i
\(424\) 0 0
\(425\) −1119.01 + 1938.18i −0.127718 + 0.221213i
\(426\) 0 0
\(427\) 335.939 10582.5i 0.0380732 1.19936i
\(428\) 0 0
\(429\) 555.466 962.096i 0.0625132 0.108276i
\(430\) 0 0
\(431\) 4950.11 + 8573.84i 0.553221 + 0.958207i 0.998040 + 0.0625863i \(0.0199349\pi\)
−0.444818 + 0.895621i \(0.646732\pi\)
\(432\) 0 0
\(433\) −511.795 −0.0568021 −0.0284010 0.999597i \(-0.509042\pi\)
−0.0284010 + 0.999597i \(0.509042\pi\)
\(434\) 0 0
\(435\) −5099.66 −0.562092
\(436\) 0 0
\(437\) 3512.41 + 6083.68i 0.384488 + 0.665954i
\(438\) 0 0
\(439\) 651.130 1127.79i 0.0707898 0.122612i −0.828458 0.560051i \(-0.810781\pi\)
0.899248 + 0.437440i \(0.144115\pi\)
\(440\) 0 0
\(441\) 1370.83 + 2765.93i 0.148022 + 0.298665i
\(442\) 0 0
\(443\) 4100.16 7101.69i 0.439739 0.761651i −0.557930 0.829888i \(-0.688404\pi\)
0.997669 + 0.0682372i \(0.0217375\pi\)
\(444\) 0 0
\(445\) −3242.88 5616.83i −0.345454 0.598344i
\(446\) 0 0
\(447\) 305.626 0.0323392
\(448\) 0 0
\(449\) −5788.12 −0.608370 −0.304185 0.952613i \(-0.598384\pi\)
−0.304185 + 0.952613i \(0.598384\pi\)
\(450\) 0 0
\(451\) 3345.65 + 5794.84i 0.349314 + 0.605030i
\(452\) 0 0
\(453\) −2667.63 + 4620.48i −0.276681 + 0.479225i
\(454\) 0 0
\(455\) −150.819 + 4751.01i −0.0155396 + 0.489518i
\(456\) 0 0
\(457\) −3629.59 + 6286.63i −0.371521 + 0.643493i −0.989800 0.142466i \(-0.954497\pi\)
0.618279 + 0.785959i \(0.287830\pi\)
\(458\) 0 0
\(459\) −1061.07 1837.83i −0.107901 0.186890i
\(460\) 0 0
\(461\) −5966.92 −0.602835 −0.301418 0.953492i \(-0.597460\pi\)
−0.301418 + 0.953492i \(0.597460\pi\)
\(462\) 0 0
\(463\) −8884.02 −0.891740 −0.445870 0.895098i \(-0.647106\pi\)
−0.445870 + 0.895098i \(0.647106\pi\)
\(464\) 0 0
\(465\) 991.035 + 1716.52i 0.0988347 + 0.171187i
\(466\) 0 0
\(467\) 2014.68 3489.53i 0.199632 0.345774i −0.748777 0.662822i \(-0.769359\pi\)
0.948409 + 0.317049i \(0.102692\pi\)
\(468\) 0 0
\(469\) −8488.96 5267.12i −0.835787 0.518578i
\(470\) 0 0
\(471\) 2098.89 3635.38i 0.205332 0.355646i
\(472\) 0 0
\(473\) −3286.32 5692.07i −0.319461 0.553323i
\(474\) 0 0
\(475\) −2083.61 −0.201269
\(476\) 0 0
\(477\) −1469.41 −0.141048
\(478\) 0 0
\(479\) 8720.33 + 15104.0i 0.831820 + 1.44075i 0.896593 + 0.442855i \(0.146034\pi\)
−0.0647733 + 0.997900i \(0.520632\pi\)
\(480\) 0 0
\(481\) −3940.38 + 6824.94i −0.373526 + 0.646966i
\(482\) 0 0
\(483\) −4701.53 + 2519.01i −0.442913 + 0.237306i
\(484\) 0 0
\(485\) −3233.91 + 5601.30i −0.302772 + 0.524417i
\(486\) 0 0
\(487\) 2136.14 + 3699.90i 0.198763 + 0.344268i 0.948128 0.317890i \(-0.102974\pi\)
−0.749364 + 0.662158i \(0.769641\pi\)
\(488\) 0 0
\(489\) −10846.3 −1.00304
\(490\) 0 0
\(491\) 2013.29 0.185048 0.0925240 0.995710i \(-0.470507\pi\)
0.0925240 + 0.995710i \(0.470507\pi\)
\(492\) 0 0
\(493\) 6799.54 + 11777.2i 0.621169 + 1.07590i
\(494\) 0 0
\(495\) 626.707 1085.49i 0.0569059 0.0985638i
\(496\) 0 0
\(497\) 17375.9 9309.78i 1.56824 0.840243i
\(498\) 0 0
\(499\) 2440.23 4226.60i 0.218917 0.379176i −0.735560 0.677460i \(-0.763081\pi\)
0.954477 + 0.298284i \(0.0964142\pi\)
\(500\) 0 0
\(501\) −1615.08 2797.40i −0.144025 0.249459i
\(502\) 0 0
\(503\) 12961.9 1.14899 0.574496 0.818508i \(-0.305198\pi\)
0.574496 + 0.818508i \(0.305198\pi\)
\(504\) 0 0
\(505\) −17722.8 −1.56170
\(506\) 0 0
\(507\) −2271.82 3934.91i −0.199004 0.344686i
\(508\) 0 0
\(509\) −4077.69 + 7062.77i −0.355089 + 0.615033i −0.987133 0.159900i \(-0.948883\pi\)
0.632044 + 0.774932i \(0.282216\pi\)
\(510\) 0 0
\(511\) −6966.51 4322.49i −0.603093 0.374199i
\(512\) 0 0
\(513\) 987.866 1711.03i 0.0850202 0.147259i
\(514\) 0 0
\(515\) 5280.35 + 9145.84i 0.451806 + 0.782551i
\(516\) 0 0
\(517\) 1292.73 0.109969
\(518\) 0 0
\(519\) −5620.89 −0.475394
\(520\) 0 0
\(521\) 166.009 + 287.536i 0.0139597 + 0.0241789i 0.872921 0.487862i \(-0.162223\pi\)
−0.858961 + 0.512041i \(0.828890\pi\)
\(522\) 0 0
\(523\) −3627.42 + 6282.87i −0.303281 + 0.525298i −0.976877 0.213802i \(-0.931415\pi\)
0.673596 + 0.739099i \(0.264749\pi\)
\(524\) 0 0
\(525\) 50.1965 1581.26i 0.00417286 0.131451i
\(526\) 0 0
\(527\) 2642.76 4577.40i 0.218445 0.378358i
\(528\) 0 0
\(529\) 1475.50 + 2555.64i 0.121271 + 0.210047i
\(530\) 0 0
\(531\) −5404.73 −0.441705
\(532\) 0 0
\(533\) −12331.5 −1.00213
\(534\) 0 0
\(535\) 8151.36 + 14118.6i 0.658718 + 1.14093i
\(536\) 0 0
\(537\) −4038.44 + 6994.78i −0.324528 + 0.562099i
\(538\) 0 0
\(539\) 4852.32 + 308.382i 0.387763 + 0.0246437i
\(540\) 0 0
\(541\) −2705.10 + 4685.36i −0.214974 + 0.372347i −0.953265 0.302137i \(-0.902300\pi\)
0.738290 + 0.674483i \(0.235633\pi\)
\(542\) 0 0
\(543\) 692.505 + 1199.45i 0.0547297 + 0.0947947i
\(544\) 0 0
\(545\) 5884.32 0.462489
\(546\) 0 0
\(547\) −18673.9 −1.45967 −0.729834 0.683625i \(-0.760403\pi\)
−0.729834 + 0.683625i \(0.760403\pi\)
\(548\) 0 0
\(549\) −2572.61 4455.89i −0.199993 0.346399i
\(550\) 0 0
\(551\) −6330.42 + 10964.6i −0.489446 + 0.847746i
\(552\) 0 0
\(553\) −26.8550 + 845.968i −0.00206508 + 0.0650528i
\(554\) 0 0
\(555\) −4445.75 + 7700.27i −0.340021 + 0.588934i
\(556\) 0 0
\(557\) 4104.74 + 7109.62i 0.312250 + 0.540833i 0.978849 0.204583i \(-0.0655839\pi\)
−0.666599 + 0.745417i \(0.732251\pi\)
\(558\) 0 0
\(559\) 12112.8 0.916489
\(560\) 0 0
\(561\) −3342.44 −0.251547
\(562\) 0 0
\(563\) −8761.10 15174.7i −0.655837 1.13594i −0.981683 0.190521i \(-0.938982\pi\)
0.325846 0.945423i \(-0.394351\pi\)
\(564\) 0 0
\(565\) 3839.76 6650.65i 0.285911 0.495212i
\(566\) 0 0
\(567\) 1274.71 + 790.914i 0.0944139 + 0.0585807i
\(568\) 0 0
\(569\) 4528.41 7843.44i 0.333640 0.577881i −0.649583 0.760291i \(-0.725056\pi\)
0.983223 + 0.182410i \(0.0583898\pi\)
\(570\) 0 0
\(571\) −5358.27 9280.80i −0.392709 0.680191i 0.600097 0.799927i \(-0.295129\pi\)
−0.992806 + 0.119736i \(0.961795\pi\)
\(572\) 0 0
\(573\) 11294.5 0.823444
\(574\) 0 0
\(575\) 2733.53 0.198254
\(576\) 0 0
\(577\) −1608.62 2786.21i −0.116062 0.201025i 0.802142 0.597133i \(-0.203694\pi\)
−0.918204 + 0.396109i \(0.870360\pi\)
\(578\) 0 0
\(579\) 4389.30 7602.49i 0.315048 0.545680i
\(580\) 0 0
\(581\) 11206.4 6004.21i 0.800203 0.428738i
\(582\) 0 0
\(583\) −1157.18 + 2004.30i −0.0822053 + 0.142384i
\(584\) 0 0
\(585\) 1154.97 + 2000.46i 0.0816274 + 0.141383i
\(586\) 0 0
\(587\) 3248.45 0.228412 0.114206 0.993457i \(-0.463568\pi\)
0.114206 + 0.993457i \(0.463568\pi\)
\(588\) 0 0
\(589\) 4920.85 0.344245
\(590\) 0 0
\(591\) 7568.32 + 13108.7i 0.526767 + 0.912387i
\(592\) 0 0
\(593\) 10555.2 18282.1i 0.730945 1.26603i −0.225535 0.974235i \(-0.572413\pi\)
0.956480 0.291799i \(-0.0942537\pi\)
\(594\) 0 0
\(595\) 12606.1 6754.15i 0.868569 0.465367i
\(596\) 0 0
\(597\) 6438.37 11151.6i 0.441382 0.764496i
\(598\) 0 0
\(599\) 7868.98 + 13629.5i 0.536758 + 0.929692i 0.999076 + 0.0429777i \(0.0136844\pi\)
−0.462318 + 0.886714i \(0.652982\pi\)
\(600\) 0 0
\(601\) −4856.96 −0.329650 −0.164825 0.986323i \(-0.552706\pi\)
−0.164825 + 0.986323i \(0.552706\pi\)
\(602\) 0 0
\(603\) −4854.80 −0.327866
\(604\) 0 0
\(605\) 5551.29 + 9615.12i 0.373045 + 0.646132i
\(606\) 0 0
\(607\) 8629.29 14946.4i 0.577022 0.999431i −0.418797 0.908080i \(-0.637548\pi\)
0.995819 0.0913513i \(-0.0291186\pi\)
\(608\) 0 0
\(609\) −8168.55 5068.31i −0.543524 0.337239i
\(610\) 0 0
\(611\) −1191.19 + 2063.20i −0.0788714 + 0.136609i
\(612\) 0 0
\(613\) −8058.53 13957.8i −0.530964 0.919657i −0.999347 0.0361315i \(-0.988496\pi\)
0.468383 0.883526i \(-0.344837\pi\)
\(614\) 0 0
\(615\) −13913.1 −0.912243
\(616\) 0 0
\(617\) −17507.3 −1.14233 −0.571164 0.820836i \(-0.693508\pi\)
−0.571164 + 0.820836i \(0.693508\pi\)
\(618\) 0 0
\(619\) −4859.24 8416.46i −0.315524 0.546504i 0.664025 0.747711i \(-0.268847\pi\)
−0.979549 + 0.201207i \(0.935514\pi\)
\(620\) 0 0
\(621\) −1296.00 + 2244.74i −0.0837467 + 0.145054i
\(622\) 0 0
\(623\) 387.917 12219.9i 0.0249463 0.785842i
\(624\) 0 0
\(625\) 5627.47 9747.06i 0.360158 0.623812i
\(626\) 0 0
\(627\) −1555.92 2694.93i −0.0991025 0.171651i
\(628\) 0 0
\(629\) 23710.7 1.50303
\(630\) 0 0
\(631\) 5144.78 0.324580 0.162290 0.986743i \(-0.448112\pi\)
0.162290 + 0.986743i \(0.448112\pi\)
\(632\) 0 0
\(633\) 5438.63 + 9419.99i 0.341495 + 0.591487i
\(634\) 0 0
\(635\) −124.638 + 215.879i −0.00778913 + 0.0134912i
\(636\) 0 0
\(637\) −4963.38 + 7460.19i −0.308723 + 0.464024i
\(638\) 0 0
\(639\) 4789.76 8296.12i 0.296526 0.513598i
\(640\) 0 0
\(641\) −10843.0 18780.7i −0.668135 1.15724i −0.978425 0.206602i \(-0.933760\pi\)
0.310290 0.950642i \(-0.399574\pi\)
\(642\) 0 0
\(643\) −14171.1 −0.869132 −0.434566 0.900640i \(-0.643098\pi\)
−0.434566 + 0.900640i \(0.643098\pi\)
\(644\) 0 0
\(645\) 13666.3 0.834281
\(646\) 0 0
\(647\) −10026.6 17366.6i −0.609254 1.05526i −0.991364 0.131142i \(-0.958135\pi\)
0.382109 0.924117i \(-0.375198\pi\)
\(648\) 0 0
\(649\) −4256.30 + 7372.13i −0.257434 + 0.445888i
\(650\) 0 0
\(651\) −118.549 + 3734.44i −0.00713716 + 0.224830i
\(652\) 0 0
\(653\) −11820.6 + 20473.8i −0.708385 + 1.22696i 0.257072 + 0.966392i \(0.417242\pi\)
−0.965456 + 0.260566i \(0.916091\pi\)
\(654\) 0 0
\(655\) 2336.21 + 4046.44i 0.139364 + 0.241386i
\(656\) 0 0
\(657\) −3984.12 −0.236584
\(658\) 0 0
\(659\) 2367.34 0.139937 0.0699685 0.997549i \(-0.477710\pi\)
0.0699685 + 0.997549i \(0.477710\pi\)
\(660\) 0 0
\(661\) 194.991 + 337.733i 0.0114739 + 0.0198734i 0.871705 0.490030i \(-0.163014\pi\)
−0.860231 + 0.509904i \(0.829681\pi\)
\(662\) 0 0
\(663\) 3079.91 5334.57i 0.180413 0.312485i
\(664\) 0 0
\(665\) 11313.9 + 7019.88i 0.659749 + 0.409352i
\(666\) 0 0
\(667\) 8305.00 14384.7i 0.482115 0.835048i
\(668\) 0 0
\(669\) −4717.50 8170.96i −0.272630 0.472208i
\(670\) 0 0
\(671\) −8103.86 −0.466239
\(672\) 0 0
\(673\) 28764.5 1.64753 0.823766 0.566930i \(-0.191869\pi\)
0.823766 + 0.566930i \(0.191869\pi\)
\(674\) 0 0
\(675\) −384.402 665.805i −0.0219195 0.0379657i
\(676\) 0 0
\(677\) 881.061 1526.04i 0.0500176 0.0866331i −0.839933 0.542691i \(-0.817406\pi\)
0.889950 + 0.456058i \(0.150739\pi\)
\(678\) 0 0
\(679\) −10746.9 + 5758.04i −0.607406 + 0.325439i
\(680\) 0 0
\(681\) −2318.62 + 4015.96i −0.130469 + 0.225980i
\(682\) 0 0
\(683\) 1114.44 + 1930.26i 0.0624344 + 0.108140i 0.895553 0.444955i \(-0.146780\pi\)
−0.833119 + 0.553094i \(0.813447\pi\)
\(684\) 0 0
\(685\) 10399.9 0.580085
\(686\) 0 0
\(687\) 12216.4 0.678437
\(688\) 0 0
\(689\) −2132.59 3693.76i −0.117918 0.204239i
\(690\) 0 0
\(691\) −8672.14 + 15020.6i −0.477429 + 0.826932i −0.999665 0.0258690i \(-0.991765\pi\)
0.522236 + 0.852801i \(0.325098\pi\)
\(692\) 0 0
\(693\) 2082.67 1115.86i 0.114162 0.0611661i
\(694\) 0 0
\(695\) −12678.8 + 21960.3i −0.691992 + 1.19856i
\(696\) 0 0
\(697\) 18550.8 + 32130.9i 1.00812 + 1.74612i
\(698\) 0 0
\(699\) 9047.14 0.489548
\(700\) 0 0
\(701\) −23697.7 −1.27682 −0.638409 0.769697i \(-0.720407\pi\)
−0.638409 + 0.769697i \(0.720407\pi\)
\(702\) 0 0
\(703\) 11037.4 + 19117.3i 0.592153 + 1.02564i
\(704\) 0 0
\(705\) −1343.97 + 2327.82i −0.0717968 + 0.124356i
\(706\) 0 0
\(707\) −28388.2 17613.9i −1.51011 0.936972i
\(708\) 0 0
\(709\) 11395.5 19737.6i 0.603620 1.04550i −0.388648 0.921386i \(-0.627058\pi\)
0.992268 0.124114i \(-0.0396088\pi\)
\(710\) 0 0
\(711\) 205.654 + 356.204i 0.0108476 + 0.0187886i
\(712\) 0 0
\(713\) −6455.76 −0.339089
\(714\) 0 0
\(715\) 3638.21 0.190296
\(716\) 0 0
\(717\) 6971.08 + 12074.3i 0.363096 + 0.628900i
\(718\) 0 0
\(719\) −15922.4 + 27578.4i −0.825876 + 1.43046i 0.0753721 + 0.997155i \(0.475986\pi\)
−0.901248 + 0.433304i \(0.857348\pi\)
\(720\) 0 0
\(721\) −631.642 + 19897.5i −0.0326263 + 1.02777i
\(722\) 0 0
\(723\) 1677.41 2905.37i 0.0862845 0.149449i
\(724\) 0 0
\(725\) 2463.32 + 4266.59i 0.126187 + 0.218562i
\(726\) 0 0
\(727\) −22500.0 −1.14784 −0.573920 0.818912i \(-0.694578\pi\)
−0.573920 + 0.818912i \(0.694578\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −18221.8 31561.1i −0.921966 1.59689i
\(732\) 0 0
\(733\) 1611.38 2790.99i 0.0811972 0.140638i −0.822567 0.568668i \(-0.807459\pi\)
0.903764 + 0.428030i \(0.140792\pi\)
\(734\) 0 0
\(735\) −5599.96 + 8417.00i −0.281031 + 0.422402i
\(736\) 0 0
\(737\) −3823.23 + 6622.02i −0.191086 + 0.330971i
\(738\) 0 0
\(739\) −9179.77 15899.8i −0.456946 0.791454i 0.541851 0.840474i \(-0.317724\pi\)
−0.998798 + 0.0490199i \(0.984390\pi\)
\(740\) 0 0
\(741\) 5734.84 0.284311
\(742\) 0 0
\(743\) 12890.4 0.636478 0.318239 0.948010i \(-0.396909\pi\)
0.318239 + 0.948010i \(0.396909\pi\)
\(744\) 0 0
\(745\) 500.450 + 866.805i 0.0246108 + 0.0426272i
\(746\) 0 0
\(747\) 3089.09 5350.46i 0.151304 0.262066i
\(748\) 0 0
\(749\) −975.075 + 30716.1i −0.0475680 + 1.49846i
\(750\) 0 0
\(751\) −7695.10 + 13328.3i −0.373899 + 0.647612i −0.990162 0.139929i \(-0.955313\pi\)
0.616263 + 0.787541i \(0.288646\pi\)
\(752\) 0 0
\(753\) 1325.28 + 2295.45i 0.0641378 + 0.111090i
\(754\) 0 0
\(755\) −17472.6 −0.842241
\(756\) 0 0
\(757\) 13974.6 0.670957 0.335479 0.942048i \(-0.391102\pi\)
0.335479 + 0.942048i \(0.391102\pi\)
\(758\) 0 0
\(759\) 2041.24 + 3535.52i 0.0976181 + 0.169080i
\(760\) 0 0
\(761\) 1687.57 2922.96i 0.0803868 0.139234i −0.823029 0.567999i \(-0.807718\pi\)
0.903416 + 0.428765i \(0.141051\pi\)
\(762\) 0 0
\(763\) 9425.40 + 5848.15i 0.447212 + 0.277480i
\(764\) 0 0
\(765\) 3474.93 6018.75i 0.164230 0.284455i
\(766\) 0 0
\(767\) −7843.99 13586.2i −0.369270 0.639595i
\(768\) 0 0
\(769\) −31253.9 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(770\) 0 0
\(771\) 17606.7 0.822423
\(772\) 0 0
\(773\) −11581.8 20060.2i −0.538897 0.933396i −0.998964 0.0455122i \(-0.985508\pi\)
0.460067 0.887884i \(-0.347825\pi\)
\(774\) 0 0
\(775\) 957.411 1658.29i 0.0443758 0.0768611i
\(776\) 0 0
\(777\) −14774.1 + 7915.74i −0.682133 + 0.365477i
\(778\) 0 0
\(779\) −17270.9 + 29914.0i −0.794343 + 1.37584i
\(780\) 0 0
\(781\) −7544.01 13066.6i −0.345641 0.598668i
\(782\) 0 0
\(783\) −4671.56 −0.213216
\(784\) 0 0
\(785\) 13747.4 0.625050
\(786\) 0 0
\(787\) 4951.47 + 8576.20i 0.224271 + 0.388448i 0.956100 0.293039i \(-0.0946667\pi\)
−0.731830 + 0.681488i \(0.761333\pi\)
\(788\) 0 0
\(789\) 7313.42 12667.2i 0.329993 0.571565i
\(790\) 0 0
\(791\) 12760.2 6836.75i 0.573580 0.307316i
\(792\) 0 0
\(793\) 7467.36 12933.8i 0.334393 0.579186i
\(794\) 0 0
\(795\) −2406.11 4167.50i −0.107341 0.185919i
\(796\) 0 0
\(797\) 16844.5 0.748638 0.374319 0.927300i \(-0.377877\pi\)
0.374319 + 0.927300i \(0.377877\pi\)
\(798\) 0 0
\(799\) 7167.83 0.317371
\(800\) 0 0
\(801\) −2970.65 5145.32i −0.131040 0.226967i
\(802\) 0 0
\(803\) −3137.55 + 5434.40i −0.137885 + 0.238824i
\(804\) 0 0
\(805\) −14842.9 9209.52i −0.649867 0.403221i
\(806\) 0 0
\(807\) −8290.66 + 14359.8i −0.361642 + 0.626382i
\(808\) 0 0
\(809\) 17215.3 + 29817.8i 0.748157 + 1.29585i 0.948705 + 0.316162i \(0.102394\pi\)
−0.200549 + 0.979684i \(0.564272\pi\)
\(810\) 0 0
\(811\) −62.4498 −0.00270396 −0.00135198 0.999999i \(-0.500430\pi\)
−0.00135198 + 0.999999i \(0.500430\pi\)
\(812\) 0 0
\(813\) 9674.20 0.417329
\(814\) 0 0
\(815\) −17760.3 30761.8i −0.763334 1.32213i
\(816\) 0 0
\(817\) 16964.6 29383.5i 0.726458 1.25826i
\(818\) 0 0
\(819\) −138.159 + 4352.18i −0.00589457 + 0.185687i
\(820\) 0 0
\(821\) −19475.8 + 33733.1i −0.827906 + 1.43398i 0.0717711 + 0.997421i \(0.477135\pi\)
−0.899678 + 0.436555i \(0.856198\pi\)
\(822\) 0 0
\(823\) 17741.2 + 30728.6i 0.751420 + 1.30150i 0.947135 + 0.320836i \(0.103964\pi\)
−0.195715 + 0.980661i \(0.562703\pi\)
\(824\) 0 0
\(825\) −1210.89 −0.0511003
\(826\) 0 0
\(827\) 7945.87 0.334105 0.167053 0.985948i \(-0.446575\pi\)
0.167053 + 0.985948i \(0.446575\pi\)
\(828\) 0 0
\(829\) −20318.2 35192.2i −0.851243 1.47440i −0.880087 0.474812i \(-0.842516\pi\)
0.0288439 0.999584i \(-0.490817\pi\)
\(830\) 0 0
\(831\) −2937.07 + 5087.15i −0.122606 + 0.212360i
\(832\) 0 0
\(833\) 26904.8 + 1709.90i 1.11908 + 0.0711217i
\(834\) 0 0
\(835\) 5289.26 9161.26i 0.219212 0.379687i
\(836\) 0 0
\(837\) 907.841 + 1572.43i 0.0374906 + 0.0649355i
\(838\) 0 0
\(839\) 27723.5 1.14079 0.570394 0.821371i \(-0.306791\pi\)
0.570394 + 0.821371i \(0.306791\pi\)
\(840\) 0 0
\(841\) 5547.19 0.227446
\(842\) 0 0
\(843\) 12796.7 + 22164.6i 0.522827 + 0.905563i
\(844\) 0 0
\(845\) 7440.04 12886.5i 0.302893 0.524627i
\(846\) 0 0
\(847\) −664.052 + 20918.5i −0.0269387 + 0.848605i
\(848\) 0 0
\(849\) 11864.8 20550.4i 0.479621 0.830728i
\(850\) 0 0
\(851\) −14480.2 25080.4i −0.583283 1.01028i
\(852\) 0 0
\(853\) 37599.1 1.50922 0.754612 0.656171i \(-0.227825\pi\)
0.754612 + 0.656171i \(0.227825\pi\)
\(854\) 0 0
\(855\) 6470.36 0.258809
\(856\) 0 0
\(857\) −12805.0 22178.9i −0.510397 0.884034i −0.999927 0.0120476i \(-0.996165\pi\)
0.489530 0.871986i \(-0.337168\pi\)
\(858\) 0 0
\(859\) 1989.70 3446.26i 0.0790309 0.136886i −0.823801 0.566879i \(-0.808151\pi\)
0.902832 + 0.429993i \(0.141484\pi\)
\(860\) 0 0
\(861\) −22285.7 13827.6i −0.882109 0.547319i
\(862\) 0 0
\(863\) 12410.2 21495.0i 0.489510 0.847856i −0.510417 0.859927i \(-0.670509\pi\)
0.999927 + 0.0120708i \(0.00384236\pi\)
\(864\) 0 0
\(865\) −9203.97 15941.7i −0.361786 0.626631i
\(866\) 0 0
\(867\) −3793.94 −0.148615
\(868\) 0 0
\(869\) 647.823 0.0252887
\(870\) 0 0
\(871\) −7045.87 12203.8i −0.274099 0.474754i
\(872\) 0 0
\(873\) −2962.44 + 5131.09i −0.114849 + 0.198925i
\(874\) 0 0
\(875\) 24615.2 13188.5i 0.951024 0.509545i
\(876\) 0 0
\(877\) −25009.7 + 43318.1i −0.962963 + 1.66790i −0.247971 + 0.968767i \(0.579764\pi\)
−0.714992 + 0.699133i \(0.753570\pi\)
\(878\) 0 0
\(879\) −12482.1 21619.7i −0.478966 0.829594i
\(880\) 0 0
\(881\) −10386.4 −0.397193 −0.198596 0.980081i \(-0.563638\pi\)
−0.198596 + 0.980081i \(0.563638\pi\)
\(882\) 0 0
\(883\) 36366.4 1.38599 0.692994 0.720944i \(-0.256291\pi\)
0.692994 + 0.720944i \(0.256291\pi\)
\(884\) 0 0
\(885\) −8850.02 15328.7i −0.336147 0.582224i
\(886\) 0 0
\(887\) 3803.65 6588.12i 0.143984 0.249388i −0.785009 0.619484i \(-0.787342\pi\)
0.928994 + 0.370096i \(0.120675\pi\)
\(888\) 0 0
\(889\) −414.195 + 221.920i −0.0156261 + 0.00837227i
\(890\) 0 0
\(891\) 574.098 994.366i 0.0215858 0.0373878i
\(892\) 0 0
\(893\) 3336.65 + 5779.24i 0.125035 + 0.216568i
\(894\) 0 0
\(895\) −26451.1 −0.987891
\(896\) 0 0
\(897\) −7523.64 −0.280053
\(898\) 0 0
\(899\) −5817.61 10076.4i −0.215827 0.373823i
\(900\) 0 0
\(901\) −6416.28 + 11113.3i −0.237245 + 0.410920i
\(902\) 0 0
\(903\) 21890.5 + 13582.3i 0.806723 + 0.500545i
\(904\) 0 0
\(905\) −2267.90 + 3928.11i −0.0833011 + 0.144282i
\(906\) 0 0
\(907\) −12901.5 22346.1i −0.472314 0.818072i 0.527184 0.849751i \(-0.323248\pi\)
−0.999498 + 0.0316794i \(0.989914\pi\)
\(908\) 0 0
\(909\) −16235.1 −0.592391
\(910\) 0 0
\(911\) 8165.21 0.296954 0.148477 0.988916i \(-0.452563\pi\)
0.148477 + 0.988916i \(0.452563\pi\)
\(912\) 0 0
\(913\) −4865.40 8427.12i −0.176365 0.305473i
\(914\) 0 0
\(915\) 8425.08 14592.7i 0.304399 0.527234i
\(916\) 0 0
\(917\) −279.461 + 8803.38i −0.0100639 + 0.317026i
\(918\) 0 0
\(919\) −296.804 + 514.079i −0.0106536 + 0.0184526i −0.871303 0.490745i \(-0.836725\pi\)
0.860649 + 0.509198i \(0.170058\pi\)
\(920\) 0 0
\(921\) 3812.69 + 6603.77i 0.136409 + 0.236267i
\(922\) 0 0
\(923\) 27805.9 0.991596
\(924\) 0 0
\(925\) 8589.84 0.305332
\(926\) 0 0
\(927\) 4837.09 + 8378.08i 0.171382 + 0.296842i
\(928\) 0 0
\(929\) 16573.7 28706.5i 0.585323 1.01381i −0.409512 0.912305i \(-0.634301\pi\)
0.994835 0.101504i \(-0.0323655\pi\)
\(930\) 0 0
\(931\) 11145.6 + 22488.7i 0.392356 + 0.791661i
\(932\) 0 0
\(933\) −911.564 + 1578.87i −0.0319863 + 0.0554020i
\(934\) 0 0
\(935\) −5473.11 9479.70i −0.191433 0.331571i
\(936\) 0 0
\(937\) −24883.8 −0.867575 −0.433788 0.901015i \(-0.642823\pi\)
−0.433788 + 0.901015i \(0.642823\pi\)
\(938\) 0 0
\(939\) −33081.8 −1.14972
\(940\) 0 0
\(941\) 13065.1 + 22629.5i 0.452616 + 0.783954i 0.998548 0.0538758i \(-0.0171575\pi\)
−0.545932 + 0.837830i \(0.683824\pi\)
\(942\) 0 0
\(943\) 22658.0 39244.8i 0.782445 1.35523i
\(944\) 0 0
\(945\) −155.878 + 4910.36i −0.00536583 + 0.169031i
\(946\) 0 0
\(947\) −3984.35 + 6901.09i −0.136720 + 0.236806i −0.926253 0.376902i \(-0.876989\pi\)
0.789533 + 0.613708i \(0.210323\pi\)
\(948\) 0 0
\(949\) −5782.24 10015.1i −0.197786 0.342576i
\(950\) 0 0
\(951\) −21784.8 −0.742819
\(952\) 0 0
\(953\) −6742.36 −0.229178 −0.114589 0.993413i \(-0.536555\pi\)
−0.114589 + 0.993413i \(0.536555\pi\)
\(954\) 0 0
\(955\) 18494.2 + 32033.0i 0.626659 + 1.08541i
\(956\) 0 0
\(957\) −3678.92 + 6372.07i −0.124266 + 0.215235i
\(958\) 0 0
\(959\) 16658.3 + 10335.9i 0.560923 + 0.348034i
\(960\) 0 0
\(961\) 12634.4 21883.4i 0.424101 0.734564i
\(962\) 0 0
\(963\) 7467.08 + 12933.4i 0.249868 + 0.432785i
\(964\) 0 0
\(965\) 28749.2 0.959035
\(966\) 0 0
\(967\) −7202.50 −0.239521 −0.119760 0.992803i \(-0.538213\pi\)
−0.119760 + 0.992803i \(0.538213\pi\)
\(968\) 0 0
\(969\) −8627.14 14942.7i −0.286010 0.495384i
\(970\) 0 0
\(971\) −6852.86 + 11869.5i −0.226487 + 0.392286i −0.956764 0.290864i \(-0.906057\pi\)
0.730278 + 0.683150i \(0.239391\pi\)
\(972\) 0 0
\(973\) −42134.0 + 22574.8i −1.38824 + 0.743798i
\(974\) 0 0
\(975\) 1115.78 1932.59i 0.0366499 0.0634794i
\(976\) 0 0
\(977\) 28274.0 + 48972.0i 0.925860 + 1.60364i 0.790173 + 0.612884i \(0.209991\pi\)
0.135687 + 0.990752i \(0.456676\pi\)
\(978\) 0 0
\(979\) −9357.71 −0.305489
\(980\) 0 0
\(981\) 5390.35 0.175434
\(982\) 0 0
\(983\) −8681.87 15037.4i −0.281698 0.487914i 0.690105 0.723709i \(-0.257564\pi\)
−0.971803 + 0.235794i \(0.924231\pi\)
\(984\) 0 0
\(985\) −24785.6 + 42930.0i −0.801762 + 1.38869i
\(986\) 0 0
\(987\) −4466.26 + 2392.96i −0.144035 + 0.0771719i
\(988\) 0 0
\(989\) −22256.2 + 38548.8i −0.715576 + 1.23941i
\(990\) 0 0
\(991\) −24519.7 42469.4i −0.785968 1.36134i −0.928419 0.371535i \(-0.878832\pi\)
0.142451 0.989802i \(-0.454502\pi\)
\(992\) 0 0
\(993\) −19898.2 −0.635901
\(994\) 0 0
\(995\) 42170.3 1.34361
\(996\) 0 0
\(997\) 4558.30 + 7895.21i 0.144797 + 0.250796i 0.929297 0.369333i \(-0.120414\pi\)
−0.784500 + 0.620129i \(0.787080\pi\)
\(998\) 0 0
\(999\) −4072.55 + 7053.86i −0.128979 + 0.223398i
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 84.4.i.b.37.1 yes 4
3.2 odd 2 252.4.k.d.37.2 4
4.3 odd 2 336.4.q.h.289.1 4
7.2 even 3 588.4.a.g.1.2 2
7.3 odd 6 588.4.i.i.361.2 4
7.4 even 3 inner 84.4.i.b.25.1 4
7.5 odd 6 588.4.a.h.1.1 2
7.6 odd 2 588.4.i.i.373.2 4
21.2 odd 6 1764.4.a.x.1.1 2
21.5 even 6 1764.4.a.p.1.2 2
21.11 odd 6 252.4.k.d.109.2 4
21.17 even 6 1764.4.k.z.361.1 4
21.20 even 2 1764.4.k.z.1549.1 4
28.11 odd 6 336.4.q.h.193.1 4
28.19 even 6 2352.4.a.bp.1.1 2
28.23 odd 6 2352.4.a.cb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.b.25.1 4 7.4 even 3 inner
84.4.i.b.37.1 yes 4 1.1 even 1 trivial
252.4.k.d.37.2 4 3.2 odd 2
252.4.k.d.109.2 4 21.11 odd 6
336.4.q.h.193.1 4 28.11 odd 6
336.4.q.h.289.1 4 4.3 odd 2
588.4.a.g.1.2 2 7.2 even 3
588.4.a.h.1.1 2 7.5 odd 6
588.4.i.i.361.2 4 7.3 odd 6
588.4.i.i.373.2 4 7.6 odd 2
1764.4.a.p.1.2 2 21.5 even 6
1764.4.a.x.1.1 2 21.2 odd 6
1764.4.k.z.361.1 4 21.17 even 6
1764.4.k.z.1549.1 4 21.20 even 2
2352.4.a.bp.1.1 2 28.19 even 6
2352.4.a.cb.1.2 2 28.23 odd 6