Properties

Label 507.4.b.k.337.7
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,4,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), 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: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.7
Root \(-0.100291i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.34727i q^{2} +3.00000 q^{3} +2.49032 q^{4} -15.3991i q^{5} -7.04181i q^{6} -10.1317i q^{7} -24.6236i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.34727i q^{2} +3.00000 q^{3} +2.49032 q^{4} -15.3991i q^{5} -7.04181i q^{6} -10.1317i q^{7} -24.6236i q^{8} +9.00000 q^{9} -36.1458 q^{10} +15.0669i q^{11} +7.47096 q^{12} -23.7819 q^{14} -46.1972i q^{15} -37.8757 q^{16} -90.8352 q^{17} -21.1254i q^{18} -114.640i q^{19} -38.3486i q^{20} -30.3952i q^{21} +35.3661 q^{22} -75.7635 q^{23} -73.8709i q^{24} -112.132 q^{25} +27.0000 q^{27} -25.2313i q^{28} +214.817 q^{29} -108.437 q^{30} +284.476i q^{31} -108.084i q^{32} +45.2007i q^{33} +213.215i q^{34} -156.019 q^{35} +22.4129 q^{36} +358.878i q^{37} -269.091 q^{38} -379.181 q^{40} -313.154i q^{41} -71.3458 q^{42} +296.702 q^{43} +37.5214i q^{44} -138.592i q^{45} +177.837i q^{46} -316.691i q^{47} -113.627 q^{48} +240.348 q^{49} +263.203i q^{50} -272.506 q^{51} +163.911 q^{53} -63.3763i q^{54} +232.016 q^{55} -249.480 q^{56} -343.920i q^{57} -504.233i q^{58} +254.149i q^{59} -115.046i q^{60} -935.247 q^{61} +667.742 q^{62} -91.1856i q^{63} -556.709 q^{64} +106.098 q^{66} +240.494i q^{67} -226.209 q^{68} -227.291 q^{69} +366.220i q^{70} -947.455i q^{71} -221.613i q^{72} -430.712i q^{73} +842.384 q^{74} -336.395 q^{75} -285.490i q^{76} +152.654 q^{77} -496.620 q^{79} +583.251i q^{80} +81.0000 q^{81} -735.058 q^{82} -392.527i q^{83} -75.6938i q^{84} +1398.78i q^{85} -696.439i q^{86} +644.451 q^{87} +371.002 q^{88} +979.895i q^{89} -325.312 q^{90} -188.675 q^{92} +853.428i q^{93} -743.360 q^{94} -1765.35 q^{95} -324.253i q^{96} +553.356i q^{97} -564.162i q^{98} +135.602i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.34727i − 0.829886i −0.909848 0.414943i \(-0.863802\pi\)
0.909848 0.414943i \(-0.136198\pi\)
\(3\) 3.00000 0.577350
\(4\) 2.49032 0.311290
\(5\) − 15.3991i − 1.37734i −0.725077 0.688668i \(-0.758196\pi\)
0.725077 0.688668i \(-0.241804\pi\)
\(6\) − 7.04181i − 0.479135i
\(7\) − 10.1317i − 0.547062i −0.961863 0.273531i \(-0.911808\pi\)
0.961863 0.273531i \(-0.0881917\pi\)
\(8\) − 24.6236i − 1.08822i
\(9\) 9.00000 0.333333
\(10\) −36.1458 −1.14303
\(11\) 15.0669i 0.412985i 0.978448 + 0.206493i \(0.0662050\pi\)
−0.978448 + 0.206493i \(0.933795\pi\)
\(12\) 7.47096 0.179723
\(13\) 0 0
\(14\) −23.7819 −0.453999
\(15\) − 46.1972i − 0.795205i
\(16\) −37.8757 −0.591809
\(17\) −90.8352 −1.29593 −0.647964 0.761671i \(-0.724379\pi\)
−0.647964 + 0.761671i \(0.724379\pi\)
\(18\) − 21.1254i − 0.276629i
\(19\) − 114.640i − 1.38422i −0.721792 0.692110i \(-0.756681\pi\)
0.721792 0.692110i \(-0.243319\pi\)
\(20\) − 38.3486i − 0.428751i
\(21\) − 30.3952i − 0.315847i
\(22\) 35.3661 0.342731
\(23\) −75.7635 −0.686860 −0.343430 0.939178i \(-0.611589\pi\)
−0.343430 + 0.939178i \(0.611589\pi\)
\(24\) − 73.8709i − 0.628284i
\(25\) −112.132 −0.897052
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 25.2313i − 0.170295i
\(29\) 214.817 1.37553 0.687767 0.725931i \(-0.258591\pi\)
0.687767 + 0.725931i \(0.258591\pi\)
\(30\) −108.437 −0.659929
\(31\) 284.476i 1.64817i 0.566463 + 0.824087i \(0.308311\pi\)
−0.566463 + 0.824087i \(0.691689\pi\)
\(32\) − 108.084i − 0.597087i
\(33\) 45.2007i 0.238437i
\(34\) 213.215i 1.07547i
\(35\) −156.019 −0.753488
\(36\) 22.4129 0.103763
\(37\) 358.878i 1.59457i 0.603602 + 0.797286i \(0.293732\pi\)
−0.603602 + 0.797286i \(0.706268\pi\)
\(38\) −269.091 −1.14874
\(39\) 0 0
\(40\) −379.181 −1.49884
\(41\) − 313.154i − 1.19284i −0.802672 0.596421i \(-0.796589\pi\)
0.802672 0.596421i \(-0.203411\pi\)
\(42\) −71.3458 −0.262116
\(43\) 296.702 1.05225 0.526123 0.850409i \(-0.323645\pi\)
0.526123 + 0.850409i \(0.323645\pi\)
\(44\) 37.5214i 0.128558i
\(45\) − 138.592i − 0.459112i
\(46\) 177.837i 0.570015i
\(47\) − 316.691i − 0.982854i −0.870919 0.491427i \(-0.836475\pi\)
0.870919 0.491427i \(-0.163525\pi\)
\(48\) −113.627 −0.341681
\(49\) 240.348 0.700723
\(50\) 263.203i 0.744451i
\(51\) −272.506 −0.748204
\(52\) 0 0
\(53\) 163.911 0.424810 0.212405 0.977182i \(-0.431870\pi\)
0.212405 + 0.977182i \(0.431870\pi\)
\(54\) − 63.3763i − 0.159712i
\(55\) 232.016 0.568819
\(56\) −249.480 −0.595324
\(57\) − 343.920i − 0.799180i
\(58\) − 504.233i − 1.14154i
\(59\) 254.149i 0.560803i 0.959883 + 0.280401i \(0.0904675\pi\)
−0.959883 + 0.280401i \(0.909532\pi\)
\(60\) − 115.046i − 0.247539i
\(61\) −935.247 −1.96305 −0.981526 0.191330i \(-0.938720\pi\)
−0.981526 + 0.191330i \(0.938720\pi\)
\(62\) 667.742 1.36780
\(63\) − 91.1856i − 0.182354i
\(64\) −556.709 −1.08732
\(65\) 0 0
\(66\) 106.098 0.197876
\(67\) 240.494i 0.438522i 0.975666 + 0.219261i \(0.0703646\pi\)
−0.975666 + 0.219261i \(0.929635\pi\)
\(68\) −226.209 −0.403409
\(69\) −227.291 −0.396559
\(70\) 366.220i 0.625309i
\(71\) − 947.455i − 1.58369i −0.610720 0.791847i \(-0.709120\pi\)
0.610720 0.791847i \(-0.290880\pi\)
\(72\) − 221.613i − 0.362740i
\(73\) − 430.712i − 0.690562i −0.938499 0.345281i \(-0.887784\pi\)
0.938499 0.345281i \(-0.112216\pi\)
\(74\) 842.384 1.32331
\(75\) −336.395 −0.517913
\(76\) − 285.490i − 0.430894i
\(77\) 152.654 0.225929
\(78\) 0 0
\(79\) −496.620 −0.707268 −0.353634 0.935384i \(-0.615054\pi\)
−0.353634 + 0.935384i \(0.615054\pi\)
\(80\) 583.251i 0.815119i
\(81\) 81.0000 0.111111
\(82\) −735.058 −0.989922
\(83\) − 392.527i − 0.519102i −0.965729 0.259551i \(-0.916425\pi\)
0.965729 0.259551i \(-0.0835746\pi\)
\(84\) − 75.6938i − 0.0983199i
\(85\) 1398.78i 1.78493i
\(86\) − 696.439i − 0.873244i
\(87\) 644.451 0.794165
\(88\) 371.002 0.449419
\(89\) 979.895i 1.16706i 0.812090 + 0.583532i \(0.198330\pi\)
−0.812090 + 0.583532i \(0.801670\pi\)
\(90\) −325.312 −0.381010
\(91\) 0 0
\(92\) −188.675 −0.213813
\(93\) 853.428i 0.951574i
\(94\) −743.360 −0.815656
\(95\) −1765.35 −1.90654
\(96\) − 324.253i − 0.344729i
\(97\) 553.356i 0.579225i 0.957144 + 0.289613i \(0.0935265\pi\)
−0.957144 + 0.289613i \(0.906474\pi\)
\(98\) − 564.162i − 0.581520i
\(99\) 135.602i 0.137662i
\(100\) −279.243 −0.279243
\(101\) 763.951 0.752633 0.376317 0.926491i \(-0.377190\pi\)
0.376317 + 0.926491i \(0.377190\pi\)
\(102\) 639.645i 0.620924i
\(103\) −182.518 −0.174602 −0.0873010 0.996182i \(-0.527824\pi\)
−0.0873010 + 0.996182i \(0.527824\pi\)
\(104\) 0 0
\(105\) −468.058 −0.435027
\(106\) − 384.744i − 0.352544i
\(107\) 183.699 0.165971 0.0829855 0.996551i \(-0.473554\pi\)
0.0829855 + 0.996551i \(0.473554\pi\)
\(108\) 67.2386 0.0599078
\(109\) − 1774.42i − 1.55925i −0.626244 0.779627i \(-0.715408\pi\)
0.626244 0.779627i \(-0.284592\pi\)
\(110\) − 544.605i − 0.472055i
\(111\) 1076.63i 0.920627i
\(112\) 383.747i 0.323756i
\(113\) 417.288 0.347391 0.173695 0.984799i \(-0.444429\pi\)
0.173695 + 0.984799i \(0.444429\pi\)
\(114\) −807.273 −0.663228
\(115\) 1166.69i 0.946037i
\(116\) 534.963 0.428190
\(117\) 0 0
\(118\) 596.556 0.465402
\(119\) 920.318i 0.708953i
\(120\) −1137.54 −0.865358
\(121\) 1103.99 0.829443
\(122\) 2195.28i 1.62911i
\(123\) − 939.463i − 0.688687i
\(124\) 708.436i 0.513060i
\(125\) − 198.162i − 0.141793i
\(126\) −214.037 −0.151333
\(127\) 1951.69 1.36366 0.681828 0.731513i \(-0.261185\pi\)
0.681828 + 0.731513i \(0.261185\pi\)
\(128\) 442.072i 0.305266i
\(129\) 890.105 0.607514
\(130\) 0 0
\(131\) −1475.61 −0.984159 −0.492080 0.870550i \(-0.663763\pi\)
−0.492080 + 0.870550i \(0.663763\pi\)
\(132\) 112.564i 0.0742231i
\(133\) −1161.50 −0.757255
\(134\) 564.503 0.363923
\(135\) − 415.775i − 0.265068i
\(136\) 2236.69i 1.41026i
\(137\) − 1900.71i − 1.18532i −0.805453 0.592660i \(-0.798078\pi\)
0.805453 0.592660i \(-0.201922\pi\)
\(138\) 533.512i 0.329099i
\(139\) −2326.76 −1.41981 −0.709903 0.704299i \(-0.751261\pi\)
−0.709903 + 0.704299i \(0.751261\pi\)
\(140\) −388.538 −0.234553
\(141\) − 950.073i − 0.567451i
\(142\) −2223.93 −1.31428
\(143\) 0 0
\(144\) −340.882 −0.197270
\(145\) − 3307.98i − 1.89457i
\(146\) −1011.00 −0.573088
\(147\) 721.044 0.404563
\(148\) 893.721i 0.496374i
\(149\) 1370.92i 0.753761i 0.926262 + 0.376881i \(0.123003\pi\)
−0.926262 + 0.376881i \(0.876997\pi\)
\(150\) 789.609i 0.429809i
\(151\) 1177.57i 0.634628i 0.948320 + 0.317314i \(0.102781\pi\)
−0.948320 + 0.317314i \(0.897219\pi\)
\(152\) −2822.85 −1.50634
\(153\) −817.517 −0.431976
\(154\) − 358.320i − 0.187495i
\(155\) 4380.67 2.27009
\(156\) 0 0
\(157\) 1621.57 0.824302 0.412151 0.911116i \(-0.364778\pi\)
0.412151 + 0.911116i \(0.364778\pi\)
\(158\) 1165.70i 0.586951i
\(159\) 491.734 0.245264
\(160\) −1664.40 −0.822389
\(161\) 767.616i 0.375755i
\(162\) − 190.129i − 0.0922095i
\(163\) − 133.130i − 0.0639727i −0.999488 0.0319864i \(-0.989817\pi\)
0.999488 0.0319864i \(-0.0101833\pi\)
\(164\) − 779.854i − 0.371320i
\(165\) 696.049 0.328408
\(166\) −921.368 −0.430795
\(167\) − 490.724i − 0.227385i −0.993516 0.113693i \(-0.963732\pi\)
0.993516 0.113693i \(-0.0362679\pi\)
\(168\) −748.440 −0.343711
\(169\) 0 0
\(170\) 3283.31 1.48129
\(171\) − 1031.76i − 0.461407i
\(172\) 738.882 0.327554
\(173\) −2008.13 −0.882518 −0.441259 0.897380i \(-0.645468\pi\)
−0.441259 + 0.897380i \(0.645468\pi\)
\(174\) − 1512.70i − 0.659066i
\(175\) 1136.09i 0.490743i
\(176\) − 570.670i − 0.244408i
\(177\) 762.446i 0.323779i
\(178\) 2300.08 0.968529
\(179\) 2152.60 0.898843 0.449421 0.893320i \(-0.351630\pi\)
0.449421 + 0.893320i \(0.351630\pi\)
\(180\) − 345.138i − 0.142917i
\(181\) −834.690 −0.342774 −0.171387 0.985204i \(-0.554825\pi\)
−0.171387 + 0.985204i \(0.554825\pi\)
\(182\) 0 0
\(183\) −2805.74 −1.13337
\(184\) 1865.57i 0.747455i
\(185\) 5526.39 2.19626
\(186\) 2003.23 0.789697
\(187\) − 1368.60i − 0.535200i
\(188\) − 788.662i − 0.305953i
\(189\) − 273.557i − 0.105282i
\(190\) 4143.75i 1.58221i
\(191\) 4464.71 1.69139 0.845695 0.533667i \(-0.179186\pi\)
0.845695 + 0.533667i \(0.179186\pi\)
\(192\) −1670.13 −0.627766
\(193\) − 3299.84i − 1.23071i −0.788248 0.615357i \(-0.789012\pi\)
0.788248 0.615357i \(-0.210988\pi\)
\(194\) 1298.88 0.480691
\(195\) 0 0
\(196\) 598.543 0.218128
\(197\) − 2973.59i − 1.07543i −0.843127 0.537714i \(-0.819288\pi\)
0.843127 0.537714i \(-0.180712\pi\)
\(198\) 318.295 0.114244
\(199\) 5430.82 1.93457 0.967287 0.253684i \(-0.0816422\pi\)
0.967287 + 0.253684i \(0.0816422\pi\)
\(200\) 2761.08i 0.976191i
\(201\) 721.481i 0.253181i
\(202\) − 1793.20i − 0.624600i
\(203\) − 2176.47i − 0.752503i
\(204\) −678.626 −0.232909
\(205\) −4822.29 −1.64294
\(206\) 428.419i 0.144900i
\(207\) −681.872 −0.228953
\(208\) 0 0
\(209\) 1727.27 0.571663
\(210\) 1098.66i 0.361022i
\(211\) 1228.25 0.400742 0.200371 0.979720i \(-0.435785\pi\)
0.200371 + 0.979720i \(0.435785\pi\)
\(212\) 408.192 0.132239
\(213\) − 2842.37i − 0.914346i
\(214\) − 431.192i − 0.137737i
\(215\) − 4568.93i − 1.44930i
\(216\) − 664.838i − 0.209428i
\(217\) 2882.24 0.901654
\(218\) −4165.05 −1.29400
\(219\) − 1292.14i − 0.398696i
\(220\) 577.795 0.177068
\(221\) 0 0
\(222\) 2527.15 0.764015
\(223\) 685.256i 0.205776i 0.994693 + 0.102888i \(0.0328084\pi\)
−0.994693 + 0.102888i \(0.967192\pi\)
\(224\) −1095.08 −0.326644
\(225\) −1009.18 −0.299017
\(226\) − 979.487i − 0.288294i
\(227\) − 287.067i − 0.0839354i −0.999119 0.0419677i \(-0.986637\pi\)
0.999119 0.0419677i \(-0.0133627\pi\)
\(228\) − 856.470i − 0.248777i
\(229\) − 2302.23i − 0.664347i −0.943218 0.332174i \(-0.892218\pi\)
0.943218 0.332174i \(-0.107782\pi\)
\(230\) 2738.53 0.785102
\(231\) 457.961 0.130440
\(232\) − 5289.57i − 1.49688i
\(233\) 970.620 0.272908 0.136454 0.990646i \(-0.456429\pi\)
0.136454 + 0.990646i \(0.456429\pi\)
\(234\) 0 0
\(235\) −4876.75 −1.35372
\(236\) 632.912i 0.174572i
\(237\) −1489.86 −0.408341
\(238\) 2160.24 0.588350
\(239\) 5007.40i 1.35524i 0.735413 + 0.677619i \(0.236988\pi\)
−0.735413 + 0.677619i \(0.763012\pi\)
\(240\) 1749.75i 0.470609i
\(241\) − 540.092i − 0.144358i −0.997392 0.0721792i \(-0.977005\pi\)
0.997392 0.0721792i \(-0.0229953\pi\)
\(242\) − 2591.36i − 0.688343i
\(243\) 243.000 0.0641500
\(244\) −2329.07 −0.611078
\(245\) − 3701.14i − 0.965130i
\(246\) −2205.17 −0.571531
\(247\) 0 0
\(248\) 7004.83 1.79358
\(249\) − 1177.58i − 0.299704i
\(250\) −465.141 −0.117672
\(251\) 6087.80 1.53091 0.765455 0.643489i \(-0.222514\pi\)
0.765455 + 0.643489i \(0.222514\pi\)
\(252\) − 227.081i − 0.0567650i
\(253\) − 1141.52i − 0.283663i
\(254\) − 4581.14i − 1.13168i
\(255\) 4196.34i 1.03053i
\(256\) −3416.01 −0.833987
\(257\) 5096.34 1.23697 0.618484 0.785797i \(-0.287747\pi\)
0.618484 + 0.785797i \(0.287747\pi\)
\(258\) − 2089.32i − 0.504167i
\(259\) 3636.06 0.872330
\(260\) 0 0
\(261\) 1933.35 0.458511
\(262\) 3463.66i 0.816739i
\(263\) 3405.50 0.798450 0.399225 0.916853i \(-0.369279\pi\)
0.399225 + 0.916853i \(0.369279\pi\)
\(264\) 1113.00 0.259472
\(265\) − 2524.08i − 0.585106i
\(266\) 2726.36i 0.628435i
\(267\) 2939.68i 0.673804i
\(268\) 598.906i 0.136507i
\(269\) −2720.44 −0.616611 −0.308306 0.951287i \(-0.599762\pi\)
−0.308306 + 0.951287i \(0.599762\pi\)
\(270\) −975.937 −0.219976
\(271\) 6954.27i 1.55883i 0.626511 + 0.779413i \(0.284482\pi\)
−0.626511 + 0.779413i \(0.715518\pi\)
\(272\) 3440.45 0.766941
\(273\) 0 0
\(274\) −4461.49 −0.983680
\(275\) − 1689.47i − 0.370470i
\(276\) −566.026 −0.123445
\(277\) 6563.96 1.42379 0.711895 0.702286i \(-0.247837\pi\)
0.711895 + 0.702286i \(0.247837\pi\)
\(278\) 5461.53i 1.17828i
\(279\) 2560.28i 0.549392i
\(280\) 3841.76i 0.819961i
\(281\) − 652.800i − 0.138586i −0.997596 0.0692932i \(-0.977926\pi\)
0.997596 0.0692932i \(-0.0220744\pi\)
\(282\) −2230.08 −0.470919
\(283\) −6010.62 −1.26252 −0.631262 0.775570i \(-0.717463\pi\)
−0.631262 + 0.775570i \(0.717463\pi\)
\(284\) − 2359.47i − 0.492988i
\(285\) −5296.05 −1.10074
\(286\) 0 0
\(287\) −3172.80 −0.652558
\(288\) − 972.759i − 0.199029i
\(289\) 3338.04 0.679430
\(290\) −7764.73 −1.57228
\(291\) 1660.07i 0.334416i
\(292\) − 1072.61i − 0.214965i
\(293\) − 1912.72i − 0.381372i −0.981651 0.190686i \(-0.938929\pi\)
0.981651 0.190686i \(-0.0610713\pi\)
\(294\) − 1692.49i − 0.335741i
\(295\) 3913.66 0.772413
\(296\) 8836.87 1.73525
\(297\) 406.806i 0.0794791i
\(298\) 3217.93 0.625535
\(299\) 0 0
\(300\) −837.730 −0.161221
\(301\) − 3006.10i − 0.575644i
\(302\) 2764.06 0.526669
\(303\) 2291.85 0.434533
\(304\) 4342.07i 0.819194i
\(305\) 14401.9i 2.70378i
\(306\) 1918.93i 0.358491i
\(307\) − 1983.07i − 0.368665i −0.982864 0.184332i \(-0.940988\pi\)
0.982864 0.184332i \(-0.0590123\pi\)
\(308\) 380.157 0.0703294
\(309\) −547.553 −0.100807
\(310\) − 10282.6i − 1.88391i
\(311\) −1893.76 −0.345291 −0.172645 0.984984i \(-0.555231\pi\)
−0.172645 + 0.984984i \(0.555231\pi\)
\(312\) 0 0
\(313\) 4574.19 0.826034 0.413017 0.910723i \(-0.364475\pi\)
0.413017 + 0.910723i \(0.364475\pi\)
\(314\) − 3806.26i − 0.684076i
\(315\) −1404.17 −0.251163
\(316\) −1236.74 −0.220165
\(317\) − 7594.53i − 1.34559i −0.739830 0.672794i \(-0.765094\pi\)
0.739830 0.672794i \(-0.234906\pi\)
\(318\) − 1154.23i − 0.203541i
\(319\) 3236.62i 0.568076i
\(320\) 8572.81i 1.49761i
\(321\) 551.098 0.0958234
\(322\) 1801.80 0.311834
\(323\) 10413.3i 1.79385i
\(324\) 201.716 0.0345878
\(325\) 0 0
\(326\) −312.492 −0.0530901
\(327\) − 5323.27i − 0.900236i
\(328\) −7710.99 −1.29807
\(329\) −3208.63 −0.537682
\(330\) − 1633.82i − 0.272541i
\(331\) 1738.58i 0.288705i 0.989526 + 0.144352i \(0.0461099\pi\)
−0.989526 + 0.144352i \(0.953890\pi\)
\(332\) − 977.519i − 0.161591i
\(333\) 3229.90i 0.531524i
\(334\) −1151.86 −0.188704
\(335\) 3703.38 0.603992
\(336\) 1151.24i 0.186921i
\(337\) 2710.61 0.438149 0.219074 0.975708i \(-0.429696\pi\)
0.219074 + 0.975708i \(0.429696\pi\)
\(338\) 0 0
\(339\) 1251.86 0.200566
\(340\) 3483.41i 0.555630i
\(341\) −4286.17 −0.680672
\(342\) −2421.82 −0.382915
\(343\) − 5910.33i − 0.930401i
\(344\) − 7305.87i − 1.14508i
\(345\) 3500.06i 0.546195i
\(346\) 4713.63i 0.732389i
\(347\) −2506.81 −0.387817 −0.193908 0.981020i \(-0.562116\pi\)
−0.193908 + 0.981020i \(0.562116\pi\)
\(348\) 1604.89 0.247216
\(349\) 7536.48i 1.15593i 0.816063 + 0.577963i \(0.196152\pi\)
−0.816063 + 0.577963i \(0.803848\pi\)
\(350\) 2666.70 0.407261
\(351\) 0 0
\(352\) 1628.50 0.246588
\(353\) 8992.88i 1.35593i 0.735095 + 0.677964i \(0.237138\pi\)
−0.735095 + 0.677964i \(0.762862\pi\)
\(354\) 1789.67 0.268700
\(355\) −14589.9 −2.18128
\(356\) 2440.25i 0.363295i
\(357\) 2760.96i 0.409314i
\(358\) − 5052.74i − 0.745937i
\(359\) − 4566.64i − 0.671359i −0.941976 0.335679i \(-0.891034\pi\)
0.941976 0.335679i \(-0.108966\pi\)
\(360\) −3412.63 −0.499615
\(361\) −6283.31 −0.916068
\(362\) 1959.24i 0.284463i
\(363\) 3311.97 0.478879
\(364\) 0 0
\(365\) −6632.57 −0.951136
\(366\) 6585.84i 0.940566i
\(367\) −6449.50 −0.917332 −0.458666 0.888609i \(-0.651673\pi\)
−0.458666 + 0.888609i \(0.651673\pi\)
\(368\) 2869.60 0.406490
\(369\) − 2818.39i − 0.397614i
\(370\) − 12971.9i − 1.82264i
\(371\) − 1660.71i − 0.232398i
\(372\) 2125.31i 0.296215i
\(373\) 7648.89 1.06178 0.530891 0.847440i \(-0.321857\pi\)
0.530891 + 0.847440i \(0.321857\pi\)
\(374\) −3212.49 −0.444154
\(375\) − 594.487i − 0.0818645i
\(376\) −7798.08 −1.06956
\(377\) 0 0
\(378\) −642.112 −0.0873722
\(379\) 10297.8i 1.39567i 0.716256 + 0.697837i \(0.245854\pi\)
−0.716256 + 0.697837i \(0.754146\pi\)
\(380\) −4396.28 −0.593486
\(381\) 5855.06 0.787307
\(382\) − 10479.9i − 1.40366i
\(383\) − 10258.0i − 1.36856i −0.729217 0.684282i \(-0.760116\pi\)
0.729217 0.684282i \(-0.239884\pi\)
\(384\) 1326.22i 0.176245i
\(385\) − 2350.73i − 0.311180i
\(386\) −7745.63 −1.02135
\(387\) 2670.31 0.350749
\(388\) 1378.03i 0.180307i
\(389\) 4771.57 0.621924 0.310962 0.950422i \(-0.399349\pi\)
0.310962 + 0.950422i \(0.399349\pi\)
\(390\) 0 0
\(391\) 6882.00 0.890122
\(392\) − 5918.24i − 0.762541i
\(393\) −4426.84 −0.568205
\(394\) −6979.82 −0.892483
\(395\) 7647.50i 0.974145i
\(396\) 337.693i 0.0428528i
\(397\) 2291.22i 0.289655i 0.989457 + 0.144827i \(0.0462627\pi\)
−0.989457 + 0.144827i \(0.953737\pi\)
\(398\) − 12747.6i − 1.60548i
\(399\) −3484.50 −0.437201
\(400\) 4247.07 0.530883
\(401\) 7534.63i 0.938308i 0.883116 + 0.469154i \(0.155441\pi\)
−0.883116 + 0.469154i \(0.844559\pi\)
\(402\) 1693.51 0.210111
\(403\) 0 0
\(404\) 1902.48 0.234287
\(405\) − 1247.33i − 0.153037i
\(406\) −5108.76 −0.624491
\(407\) −5407.18 −0.658535
\(408\) 6710.08i 0.814211i
\(409\) 1517.97i 0.183517i 0.995781 + 0.0917587i \(0.0292488\pi\)
−0.995781 + 0.0917587i \(0.970751\pi\)
\(410\) 11319.2i 1.36345i
\(411\) − 5702.14i − 0.684345i
\(412\) −454.528 −0.0543519
\(413\) 2574.97 0.306794
\(414\) 1600.54i 0.190005i
\(415\) −6044.56 −0.714978
\(416\) 0 0
\(417\) −6980.28 −0.819726
\(418\) − 4054.36i − 0.474415i
\(419\) 8080.97 0.942199 0.471100 0.882080i \(-0.343857\pi\)
0.471100 + 0.882080i \(0.343857\pi\)
\(420\) −1165.61 −0.135419
\(421\) − 8073.13i − 0.934585i −0.884103 0.467293i \(-0.845229\pi\)
0.884103 0.467293i \(-0.154771\pi\)
\(422\) − 2883.05i − 0.332570i
\(423\) − 2850.22i − 0.327618i
\(424\) − 4036.09i − 0.462287i
\(425\) 10185.5 1.16252
\(426\) −6671.80 −0.758803
\(427\) 9475.68i 1.07391i
\(428\) 457.470 0.0516651
\(429\) 0 0
\(430\) −10724.5 −1.20275
\(431\) − 2241.39i − 0.250496i −0.992125 0.125248i \(-0.960027\pi\)
0.992125 0.125248i \(-0.0399726\pi\)
\(432\) −1022.65 −0.113894
\(433\) −10237.9 −1.13626 −0.568130 0.822939i \(-0.692333\pi\)
−0.568130 + 0.822939i \(0.692333\pi\)
\(434\) − 6765.39i − 0.748270i
\(435\) − 9923.94i − 1.09383i
\(436\) − 4418.88i − 0.485380i
\(437\) 8685.52i 0.950766i
\(438\) −3033.00 −0.330872
\(439\) 5416.69 0.588894 0.294447 0.955668i \(-0.404865\pi\)
0.294447 + 0.955668i \(0.404865\pi\)
\(440\) − 5713.08i − 0.619001i
\(441\) 2163.13 0.233574
\(442\) 0 0
\(443\) −2537.44 −0.272138 −0.136069 0.990699i \(-0.543447\pi\)
−0.136069 + 0.990699i \(0.543447\pi\)
\(444\) 2681.16i 0.286582i
\(445\) 15089.5 1.60744
\(446\) 1608.48 0.170771
\(447\) 4112.77i 0.435184i
\(448\) 5640.43i 0.594833i
\(449\) 7790.61i 0.818846i 0.912345 + 0.409423i \(0.134270\pi\)
−0.912345 + 0.409423i \(0.865730\pi\)
\(450\) 2368.83i 0.248150i
\(451\) 4718.26 0.492626
\(452\) 1039.18 0.108139
\(453\) 3532.70i 0.366403i
\(454\) −673.825 −0.0696568
\(455\) 0 0
\(456\) −8468.55 −0.869685
\(457\) − 6599.71i − 0.675539i −0.941229 0.337770i \(-0.890328\pi\)
0.941229 0.337770i \(-0.109672\pi\)
\(458\) −5403.95 −0.551332
\(459\) −2452.55 −0.249401
\(460\) 2905.43i 0.294492i
\(461\) − 4482.57i − 0.452872i −0.974026 0.226436i \(-0.927293\pi\)
0.974026 0.226436i \(-0.0727074\pi\)
\(462\) − 1074.96i − 0.108250i
\(463\) − 3805.86i − 0.382016i −0.981589 0.191008i \(-0.938824\pi\)
0.981589 0.191008i \(-0.0611756\pi\)
\(464\) −8136.35 −0.814053
\(465\) 13142.0 1.31064
\(466\) − 2278.31i − 0.226482i
\(467\) −14778.8 −1.46441 −0.732207 0.681082i \(-0.761510\pi\)
−0.732207 + 0.681082i \(0.761510\pi\)
\(468\) 0 0
\(469\) 2436.62 0.239899
\(470\) 11447.0i 1.12343i
\(471\) 4864.71 0.475911
\(472\) 6258.06 0.610277
\(473\) 4470.37i 0.434562i
\(474\) 3497.11i 0.338877i
\(475\) 12854.8i 1.24172i
\(476\) 2291.89i 0.220690i
\(477\) 1475.20 0.141603
\(478\) 11753.7 1.12469
\(479\) 11171.2i 1.06560i 0.846240 + 0.532801i \(0.178861\pi\)
−0.846240 + 0.532801i \(0.821139\pi\)
\(480\) −4993.20 −0.474807
\(481\) 0 0
\(482\) −1267.74 −0.119801
\(483\) 2302.85i 0.216942i
\(484\) 2749.29 0.258197
\(485\) 8521.18 0.797787
\(486\) − 570.387i − 0.0532372i
\(487\) − 6046.57i − 0.562621i −0.959617 0.281310i \(-0.909231\pi\)
0.959617 0.281310i \(-0.0907690\pi\)
\(488\) 23029.2i 2.13623i
\(489\) − 399.390i − 0.0369347i
\(490\) −8687.57 −0.800948
\(491\) 1035.04 0.0951338 0.0475669 0.998868i \(-0.484853\pi\)
0.0475669 + 0.998868i \(0.484853\pi\)
\(492\) − 2339.56i − 0.214381i
\(493\) −19512.9 −1.78259
\(494\) 0 0
\(495\) 2088.15 0.189606
\(496\) − 10774.7i − 0.975404i
\(497\) −9599.37 −0.866379
\(498\) −2764.10 −0.248720
\(499\) 11698.3i 1.04947i 0.851265 + 0.524736i \(0.175836\pi\)
−0.851265 + 0.524736i \(0.824164\pi\)
\(500\) − 493.488i − 0.0441389i
\(501\) − 1472.17i − 0.131281i
\(502\) − 14289.7i − 1.27048i
\(503\) −13552.0 −1.20130 −0.600651 0.799511i \(-0.705092\pi\)
−0.600651 + 0.799511i \(0.705092\pi\)
\(504\) −2245.32 −0.198441
\(505\) − 11764.1i − 1.03663i
\(506\) −2679.46 −0.235408
\(507\) 0 0
\(508\) 4860.33 0.424492
\(509\) − 5076.46i − 0.442064i −0.975267 0.221032i \(-0.929058\pi\)
0.975267 0.221032i \(-0.0709425\pi\)
\(510\) 9849.94 0.855221
\(511\) −4363.86 −0.377781
\(512\) 11554.9i 0.997380i
\(513\) − 3095.28i − 0.266393i
\(514\) − 11962.5i − 1.02654i
\(515\) 2810.60i 0.240486i
\(516\) 2216.65 0.189113
\(517\) 4771.55 0.405904
\(518\) − 8534.81i − 0.723934i
\(519\) −6024.40 −0.509522
\(520\) 0 0
\(521\) −8493.73 −0.714236 −0.357118 0.934059i \(-0.616241\pi\)
−0.357118 + 0.934059i \(0.616241\pi\)
\(522\) − 4538.10i − 0.380512i
\(523\) −1384.53 −0.115758 −0.0578789 0.998324i \(-0.518434\pi\)
−0.0578789 + 0.998324i \(0.518434\pi\)
\(524\) −3674.75 −0.306359
\(525\) 3408.26i 0.283331i
\(526\) − 7993.64i − 0.662622i
\(527\) − 25840.4i − 2.13592i
\(528\) − 1712.01i − 0.141109i
\(529\) −6426.89 −0.528223
\(530\) −5924.71 −0.485571
\(531\) 2287.34i 0.186934i
\(532\) −2892.51 −0.235726
\(533\) 0 0
\(534\) 6900.24 0.559181
\(535\) − 2828.80i − 0.228598i
\(536\) 5921.82 0.477208
\(537\) 6457.80 0.518947
\(538\) 6385.62i 0.511717i
\(539\) 3621.30i 0.289388i
\(540\) − 1035.41i − 0.0825131i
\(541\) 5026.83i 0.399483i 0.979849 + 0.199742i \(0.0640103\pi\)
−0.979849 + 0.199742i \(0.935990\pi\)
\(542\) 16323.6 1.29365
\(543\) −2504.07 −0.197900
\(544\) 9817.87i 0.773782i
\(545\) −27324.5 −2.14762
\(546\) 0 0
\(547\) 1540.36 0.120404 0.0602021 0.998186i \(-0.480825\pi\)
0.0602021 + 0.998186i \(0.480825\pi\)
\(548\) − 4733.38i − 0.368978i
\(549\) −8417.23 −0.654351
\(550\) −3965.65 −0.307447
\(551\) − 24626.6i − 1.90404i
\(552\) 5596.72i 0.431544i
\(553\) 5031.63i 0.386920i
\(554\) − 15407.4i − 1.18158i
\(555\) 16579.2 1.26801
\(556\) −5794.38 −0.441972
\(557\) − 8550.51i − 0.650443i −0.945638 0.325221i \(-0.894561\pi\)
0.945638 0.325221i \(-0.105439\pi\)
\(558\) 6009.68 0.455932
\(559\) 0 0
\(560\) 5909.35 0.445921
\(561\) − 4105.81i − 0.308998i
\(562\) −1532.30 −0.115011
\(563\) 6569.35 0.491768 0.245884 0.969299i \(-0.420922\pi\)
0.245884 + 0.969299i \(0.420922\pi\)
\(564\) − 2365.99i − 0.176642i
\(565\) − 6425.85i − 0.478473i
\(566\) 14108.5i 1.04775i
\(567\) − 820.671i − 0.0607847i
\(568\) −23329.8 −1.72341
\(569\) −23766.1 −1.75102 −0.875508 0.483204i \(-0.839473\pi\)
−0.875508 + 0.483204i \(0.839473\pi\)
\(570\) 12431.3i 0.913488i
\(571\) 24971.7 1.83018 0.915091 0.403248i \(-0.132119\pi\)
0.915091 + 0.403248i \(0.132119\pi\)
\(572\) 0 0
\(573\) 13394.1 0.976524
\(574\) 7447.41i 0.541549i
\(575\) 8495.48 0.616150
\(576\) −5010.38 −0.362441
\(577\) 16643.4i 1.20082i 0.799693 + 0.600409i \(0.204996\pi\)
−0.799693 + 0.600409i \(0.795004\pi\)
\(578\) − 7835.28i − 0.563849i
\(579\) − 9899.53i − 0.710553i
\(580\) − 8237.93i − 0.589761i
\(581\) −3976.98 −0.283981
\(582\) 3896.63 0.277527
\(583\) 2469.64i 0.175441i
\(584\) −10605.7 −0.751484
\(585\) 0 0
\(586\) −4489.66 −0.316495
\(587\) 6720.67i 0.472558i 0.971685 + 0.236279i \(0.0759280\pi\)
−0.971685 + 0.236279i \(0.924072\pi\)
\(588\) 1795.63 0.125936
\(589\) 32612.3 2.28144
\(590\) − 9186.41i − 0.641014i
\(591\) − 8920.77i − 0.620899i
\(592\) − 13592.8i − 0.943681i
\(593\) 26349.8i 1.82471i 0.409395 + 0.912357i \(0.365740\pi\)
−0.409395 + 0.912357i \(0.634260\pi\)
\(594\) 954.884 0.0659586
\(595\) 14172.1 0.976466
\(596\) 3414.04i 0.234638i
\(597\) 16292.4 1.11693
\(598\) 0 0
\(599\) −6136.81 −0.418603 −0.209302 0.977851i \(-0.567119\pi\)
−0.209302 + 0.977851i \(0.567119\pi\)
\(600\) 8283.25i 0.563604i
\(601\) 12493.7 0.847966 0.423983 0.905670i \(-0.360632\pi\)
0.423983 + 0.905670i \(0.360632\pi\)
\(602\) −7056.14 −0.477719
\(603\) 2164.44i 0.146174i
\(604\) 2932.51i 0.197553i
\(605\) − 17000.4i − 1.14242i
\(606\) − 5379.60i − 0.360613i
\(607\) 18696.6 1.25020 0.625099 0.780545i \(-0.285059\pi\)
0.625099 + 0.780545i \(0.285059\pi\)
\(608\) −12390.8 −0.826501
\(609\) − 6529.40i − 0.434458i
\(610\) 33805.3 2.24383
\(611\) 0 0
\(612\) −2035.88 −0.134470
\(613\) − 1238.64i − 0.0816123i −0.999167 0.0408062i \(-0.987007\pi\)
0.999167 0.0408062i \(-0.0129926\pi\)
\(614\) −4654.81 −0.305949
\(615\) −14466.9 −0.948553
\(616\) − 3758.89i − 0.245860i
\(617\) − 16549.5i − 1.07983i −0.841718 0.539917i \(-0.818456\pi\)
0.841718 0.539917i \(-0.181544\pi\)
\(618\) 1285.26i 0.0836579i
\(619\) 13945.4i 0.905513i 0.891634 + 0.452756i \(0.149559\pi\)
−0.891634 + 0.452756i \(0.850441\pi\)
\(620\) 10909.3 0.706656
\(621\) −2045.62 −0.132186
\(622\) 4445.17i 0.286552i
\(623\) 9928.03 0.638456
\(624\) 0 0
\(625\) −17068.0 −1.09235
\(626\) − 10736.9i − 0.685513i
\(627\) 5181.80 0.330050
\(628\) 4038.23 0.256597
\(629\) − 32598.8i − 2.06645i
\(630\) 3295.98i 0.208436i
\(631\) 15343.1i 0.967987i 0.875072 + 0.483994i \(0.160814\pi\)
−0.875072 + 0.483994i \(0.839186\pi\)
\(632\) 12228.6i 0.769663i
\(633\) 3684.76 0.231368
\(634\) −17826.4 −1.11668
\(635\) − 30054.2i − 1.87821i
\(636\) 1224.58 0.0763484
\(637\) 0 0
\(638\) 7597.23 0.471438
\(639\) − 8527.10i − 0.527898i
\(640\) 6807.51 0.420454
\(641\) −11067.7 −0.681979 −0.340989 0.940067i \(-0.610762\pi\)
−0.340989 + 0.940067i \(0.610762\pi\)
\(642\) − 1293.58i − 0.0795224i
\(643\) 25118.7i 1.54057i 0.637701 + 0.770284i \(0.279885\pi\)
−0.637701 + 0.770284i \(0.720115\pi\)
\(644\) 1911.61i 0.116969i
\(645\) − 13706.8i − 0.836751i
\(646\) 24442.9 1.48869
\(647\) 4447.03 0.270217 0.135109 0.990831i \(-0.456862\pi\)
0.135109 + 0.990831i \(0.456862\pi\)
\(648\) − 1994.51i − 0.120913i
\(649\) −3829.23 −0.231603
\(650\) 0 0
\(651\) 8646.71 0.520570
\(652\) − 331.537i − 0.0199141i
\(653\) −19426.2 −1.16418 −0.582088 0.813126i \(-0.697764\pi\)
−0.582088 + 0.813126i \(0.697764\pi\)
\(654\) −12495.1 −0.747093
\(655\) 22723.1i 1.35552i
\(656\) 11861.0i 0.705934i
\(657\) − 3876.41i − 0.230187i
\(658\) 7531.52i 0.446215i
\(659\) −14099.4 −0.833438 −0.416719 0.909035i \(-0.636820\pi\)
−0.416719 + 0.909035i \(0.636820\pi\)
\(660\) 1733.38 0.102230
\(661\) 2754.25i 0.162070i 0.996711 + 0.0810348i \(0.0258225\pi\)
−0.996711 + 0.0810348i \(0.974177\pi\)
\(662\) 4080.93 0.239592
\(663\) 0 0
\(664\) −9665.44 −0.564898
\(665\) 17886.0i 1.04299i
\(666\) 7581.45 0.441104
\(667\) −16275.3 −0.944800
\(668\) − 1222.06i − 0.0707828i
\(669\) 2055.77i 0.118805i
\(670\) − 8692.83i − 0.501244i
\(671\) − 14091.3i − 0.810712i
\(672\) −3285.25 −0.188588
\(673\) −11936.8 −0.683700 −0.341850 0.939754i \(-0.611054\pi\)
−0.341850 + 0.939754i \(0.611054\pi\)
\(674\) − 6362.53i − 0.363613i
\(675\) −3027.55 −0.172638
\(676\) 0 0
\(677\) −10255.2 −0.582186 −0.291093 0.956695i \(-0.594019\pi\)
−0.291093 + 0.956695i \(0.594019\pi\)
\(678\) − 2938.46i − 0.166447i
\(679\) 5606.46 0.316872
\(680\) 34443.0 1.94239
\(681\) − 861.202i − 0.0484601i
\(682\) 10060.8i 0.564880i
\(683\) 10605.7i 0.594169i 0.954851 + 0.297085i \(0.0960143\pi\)
−0.954851 + 0.297085i \(0.903986\pi\)
\(684\) − 2569.41i − 0.143631i
\(685\) −29269.2 −1.63258
\(686\) −13873.1 −0.772127
\(687\) − 6906.69i − 0.383561i
\(688\) −11237.8 −0.622728
\(689\) 0 0
\(690\) 8215.60 0.453279
\(691\) − 4660.14i − 0.256556i −0.991738 0.128278i \(-0.959055\pi\)
0.991738 0.128278i \(-0.0409449\pi\)
\(692\) −5000.90 −0.274719
\(693\) 1373.88 0.0753096
\(694\) 5884.15i 0.321844i
\(695\) 35829.9i 1.95555i
\(696\) − 15868.7i − 0.864227i
\(697\) 28445.4i 1.54584i
\(698\) 17690.2 0.959287
\(699\) 2911.86 0.157563
\(700\) 2829.22i 0.152764i
\(701\) 24016.9 1.29402 0.647008 0.762483i \(-0.276020\pi\)
0.647008 + 0.762483i \(0.276020\pi\)
\(702\) 0 0
\(703\) 41141.7 2.20724
\(704\) − 8387.88i − 0.449048i
\(705\) −14630.2 −0.781570
\(706\) 21108.7 1.12527
\(707\) − 7740.15i − 0.411737i
\(708\) 1898.74i 0.100789i
\(709\) − 7252.56i − 0.384169i −0.981378 0.192084i \(-0.938475\pi\)
0.981378 0.192084i \(-0.0615247\pi\)
\(710\) 34246.5i 1.81021i
\(711\) −4469.58 −0.235756
\(712\) 24128.6 1.27002
\(713\) − 21552.9i − 1.13207i
\(714\) 6480.71 0.339684
\(715\) 0 0
\(716\) 5360.66 0.279801
\(717\) 15022.2i 0.782447i
\(718\) −10719.1 −0.557151
\(719\) −10951.7 −0.568050 −0.284025 0.958817i \(-0.591670\pi\)
−0.284025 + 0.958817i \(0.591670\pi\)
\(720\) 5249.26i 0.271706i
\(721\) 1849.22i 0.0955182i
\(722\) 14748.6i 0.760231i
\(723\) − 1620.28i − 0.0833454i
\(724\) −2078.65 −0.106702
\(725\) −24087.7 −1.23393
\(726\) − 7774.08i − 0.397415i
\(727\) 27856.0 1.42108 0.710538 0.703658i \(-0.248451\pi\)
0.710538 + 0.703658i \(0.248451\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 15568.4i 0.789334i
\(731\) −26951.0 −1.36364
\(732\) −6987.20 −0.352806
\(733\) 31213.8i 1.57286i 0.617678 + 0.786431i \(0.288073\pi\)
−0.617678 + 0.786431i \(0.711927\pi\)
\(734\) 15138.7i 0.761281i
\(735\) − 11103.4i − 0.557218i
\(736\) 8188.85i 0.410116i
\(737\) −3623.49 −0.181103
\(738\) −6615.52 −0.329974
\(739\) 14423.1i 0.717946i 0.933348 + 0.358973i \(0.116873\pi\)
−0.933348 + 0.358973i \(0.883127\pi\)
\(740\) 13762.5 0.683674
\(741\) 0 0
\(742\) −3898.13 −0.192864
\(743\) − 13469.7i − 0.665079i −0.943089 0.332539i \(-0.892095\pi\)
0.943089 0.332539i \(-0.107905\pi\)
\(744\) 21014.5 1.03552
\(745\) 21111.0 1.03818
\(746\) − 17954.0i − 0.881158i
\(747\) − 3532.75i − 0.173034i
\(748\) − 3408.26i − 0.166602i
\(749\) − 1861.19i − 0.0907965i
\(750\) −1395.42 −0.0679381
\(751\) 32033.6 1.55649 0.778245 0.627961i \(-0.216110\pi\)
0.778245 + 0.627961i \(0.216110\pi\)
\(752\) 11994.9i 0.581661i
\(753\) 18263.4 0.883871
\(754\) 0 0
\(755\) 18133.4 0.874096
\(756\) − 681.244i − 0.0327733i
\(757\) −26097.2 −1.25300 −0.626498 0.779423i \(-0.715512\pi\)
−0.626498 + 0.779423i \(0.715512\pi\)
\(758\) 24171.6 1.15825
\(759\) − 3424.56i − 0.163773i
\(760\) 43469.3i 2.07473i
\(761\) 18238.2i 0.868772i 0.900727 + 0.434386i \(0.143035\pi\)
−0.900727 + 0.434386i \(0.856965\pi\)
\(762\) − 13743.4i − 0.653375i
\(763\) −17978.0 −0.853009
\(764\) 11118.6 0.526513
\(765\) 12589.0i 0.594976i
\(766\) −24078.4 −1.13575
\(767\) 0 0
\(768\) −10248.0 −0.481502
\(769\) 18817.0i 0.882390i 0.897411 + 0.441195i \(0.145445\pi\)
−0.897411 + 0.441195i \(0.854555\pi\)
\(770\) −5517.79 −0.258243
\(771\) 15289.0 0.714164
\(772\) − 8217.67i − 0.383109i
\(773\) 13795.6i 0.641904i 0.947095 + 0.320952i \(0.104003\pi\)
−0.947095 + 0.320952i \(0.895997\pi\)
\(774\) − 6267.95i − 0.291081i
\(775\) − 31898.7i − 1.47850i
\(776\) 13625.6 0.630325
\(777\) 10908.2 0.503640
\(778\) − 11200.2i − 0.516126i
\(779\) −35900.0 −1.65116
\(780\) 0 0
\(781\) 14275.2 0.654043
\(782\) − 16153.9i − 0.738699i
\(783\) 5800.05 0.264722
\(784\) −9103.36 −0.414694
\(785\) − 24970.7i − 1.13534i
\(786\) 10391.0i 0.471545i
\(787\) − 40545.4i − 1.83645i −0.396055 0.918227i \(-0.629621\pi\)
0.396055 0.918227i \(-0.370379\pi\)
\(788\) − 7405.19i − 0.334770i
\(789\) 10216.5 0.460985
\(790\) 17950.7 0.808429
\(791\) − 4227.85i − 0.190044i
\(792\) 3339.01 0.149806
\(793\) 0 0
\(794\) 5378.11 0.240380
\(795\) − 7572.25i − 0.337811i
\(796\) 13524.5 0.602214
\(797\) 31576.4 1.40338 0.701690 0.712482i \(-0.252429\pi\)
0.701690 + 0.712482i \(0.252429\pi\)
\(798\) 8179.07i 0.362827i
\(799\) 28766.7i 1.27371i
\(800\) 12119.7i 0.535619i
\(801\) 8819.05i 0.389021i
\(802\) 17685.8 0.778688
\(803\) 6489.50 0.285192
\(804\) 1796.72i 0.0788126i
\(805\) 11820.6 0.517541
\(806\) 0 0
\(807\) −8161.33 −0.356001
\(808\) − 18811.2i − 0.819031i
\(809\) 31130.1 1.35287 0.676437 0.736501i \(-0.263523\pi\)
0.676437 + 0.736501i \(0.263523\pi\)
\(810\) −2927.81 −0.127003
\(811\) 8733.71i 0.378153i 0.981962 + 0.189076i \(0.0605494\pi\)
−0.981962 + 0.189076i \(0.939451\pi\)
\(812\) − 5420.10i − 0.234247i
\(813\) 20862.8i 0.899988i
\(814\) 12692.1i 0.546509i
\(815\) −2050.08 −0.0881119
\(816\) 10321.4 0.442794
\(817\) − 34013.8i − 1.45654i
\(818\) 3563.08 0.152298
\(819\) 0 0
\(820\) −12009.0 −0.511431
\(821\) 36960.1i 1.57115i 0.618765 + 0.785576i \(0.287633\pi\)
−0.618765 + 0.785576i \(0.712367\pi\)
\(822\) −13384.5 −0.567928
\(823\) −20509.7 −0.868679 −0.434340 0.900749i \(-0.643018\pi\)
−0.434340 + 0.900749i \(0.643018\pi\)
\(824\) 4494.25i 0.190006i
\(825\) − 5068.42i − 0.213891i
\(826\) − 6044.15i − 0.254604i
\(827\) − 11533.4i − 0.484952i −0.970157 0.242476i \(-0.922040\pi\)
0.970157 0.242476i \(-0.0779596\pi\)
\(828\) −1698.08 −0.0712709
\(829\) −34096.4 −1.42849 −0.714244 0.699897i \(-0.753229\pi\)
−0.714244 + 0.699897i \(0.753229\pi\)
\(830\) 14188.2i 0.593350i
\(831\) 19691.9 0.822026
\(832\) 0 0
\(833\) −21832.1 −0.908087
\(834\) 16384.6i 0.680279i
\(835\) −7556.69 −0.313186
\(836\) 4301.45 0.177953
\(837\) 7680.85i 0.317191i
\(838\) − 18968.2i − 0.781917i
\(839\) − 1855.42i − 0.0763484i −0.999271 0.0381742i \(-0.987846\pi\)
0.999271 0.0381742i \(-0.0121542\pi\)
\(840\) 11525.3i 0.473405i
\(841\) 21757.3 0.892094
\(842\) −18949.8 −0.775599
\(843\) − 1958.40i − 0.0800129i
\(844\) 3058.75 0.124747
\(845\) 0 0
\(846\) −6690.24 −0.271885
\(847\) − 11185.3i − 0.453757i
\(848\) −6208.26 −0.251406
\(849\) −18031.9 −0.728918
\(850\) − 23908.1i − 0.964755i
\(851\) − 27189.9i − 1.09525i
\(852\) − 7078.40i − 0.284627i
\(853\) − 28668.7i − 1.15076i −0.817886 0.575380i \(-0.804854\pi\)
0.817886 0.575380i \(-0.195146\pi\)
\(854\) 22242.0 0.891224
\(855\) −15888.1 −0.635512
\(856\) − 4523.35i − 0.180613i
\(857\) −449.310 −0.0179091 −0.00895457 0.999960i \(-0.502850\pi\)
−0.00895457 + 0.999960i \(0.502850\pi\)
\(858\) 0 0
\(859\) 33466.9 1.32931 0.664654 0.747151i \(-0.268579\pi\)
0.664654 + 0.747151i \(0.268579\pi\)
\(860\) − 11378.1i − 0.451151i
\(861\) −9518.39 −0.376755
\(862\) −5261.14 −0.207883
\(863\) 25097.5i 0.989953i 0.868906 + 0.494976i \(0.164823\pi\)
−0.868906 + 0.494976i \(0.835177\pi\)
\(864\) − 2918.28i − 0.114910i
\(865\) 30923.4i 1.21552i
\(866\) 24031.0i 0.942965i
\(867\) 10014.1 0.392269
\(868\) 7177.69 0.280676
\(869\) − 7482.53i − 0.292091i
\(870\) −23294.2 −0.907755
\(871\) 0 0
\(872\) −43692.7 −1.69681
\(873\) 4980.21i 0.193075i
\(874\) 20387.3 0.789027
\(875\) −2007.73 −0.0775698
\(876\) − 3217.83i − 0.124110i
\(877\) − 27015.2i − 1.04018i −0.854112 0.520090i \(-0.825899\pi\)
0.854112 0.520090i \(-0.174101\pi\)
\(878\) − 12714.4i − 0.488715i
\(879\) − 5738.15i − 0.220185i
\(880\) −8787.79 −0.336632
\(881\) −48638.7 −1.86002 −0.930010 0.367534i \(-0.880202\pi\)
−0.930010 + 0.367534i \(0.880202\pi\)
\(882\) − 5077.46i − 0.193840i
\(883\) 25479.0 0.971048 0.485524 0.874223i \(-0.338629\pi\)
0.485524 + 0.874223i \(0.338629\pi\)
\(884\) 0 0
\(885\) 11741.0 0.445953
\(886\) 5956.05i 0.225843i
\(887\) 42359.7 1.60350 0.801748 0.597663i \(-0.203904\pi\)
0.801748 + 0.597663i \(0.203904\pi\)
\(888\) 26510.6 1.00184
\(889\) − 19774.0i − 0.746005i
\(890\) − 35419.1i − 1.33399i
\(891\) 1220.42i 0.0458873i
\(892\) 1706.51i 0.0640561i
\(893\) −36305.4 −1.36049
\(894\) 9653.78 0.361153
\(895\) − 33148.1i − 1.23801i
\(896\) 4478.96 0.167000
\(897\) 0 0
\(898\) 18286.7 0.679548
\(899\) 61110.3i 2.26712i
\(900\) −2513.19 −0.0930811
\(901\) −14888.9 −0.550524
\(902\) − 11075.0i − 0.408823i
\(903\) − 9018.31i − 0.332348i
\(904\) − 10275.1i − 0.378038i
\(905\) 12853.5i 0.472114i
\(906\) 8292.19 0.304072
\(907\) 5314.91 0.194574 0.0972871 0.995256i \(-0.468984\pi\)
0.0972871 + 0.995256i \(0.468984\pi\)
\(908\) − 714.890i − 0.0261283i
\(909\) 6875.56 0.250878
\(910\) 0 0
\(911\) 2471.71 0.0898919 0.0449459 0.998989i \(-0.485688\pi\)
0.0449459 + 0.998989i \(0.485688\pi\)
\(912\) 13026.2i 0.472962i
\(913\) 5914.17 0.214382
\(914\) −15491.3 −0.560620
\(915\) 43205.8i 1.56103i
\(916\) − 5733.29i − 0.206805i
\(917\) 14950.5i 0.538396i
\(918\) 5756.80i 0.206975i
\(919\) −9636.13 −0.345883 −0.172942 0.984932i \(-0.555327\pi\)
−0.172942 + 0.984932i \(0.555327\pi\)
\(920\) 28728.1 1.02950
\(921\) − 5949.22i − 0.212849i
\(922\) −10521.8 −0.375832
\(923\) 0 0
\(924\) 1140.47 0.0406047
\(925\) − 40241.5i − 1.43041i
\(926\) −8933.38 −0.317029
\(927\) −1642.66 −0.0582007
\(928\) − 23218.3i − 0.821314i
\(929\) 53249.1i 1.88057i 0.340394 + 0.940283i \(0.389439\pi\)
−0.340394 + 0.940283i \(0.610561\pi\)
\(930\) − 30847.8i − 1.08768i
\(931\) − 27553.5i − 0.969955i
\(932\) 2417.16 0.0849534
\(933\) −5681.28 −0.199354
\(934\) 34689.8i 1.21530i
\(935\) −21075.3 −0.737149
\(936\) 0 0
\(937\) −41122.6 −1.43374 −0.716871 0.697205i \(-0.754427\pi\)
−0.716871 + 0.697205i \(0.754427\pi\)
\(938\) − 5719.40i − 0.199088i
\(939\) 13722.6 0.476911
\(940\) −12144.7 −0.421399
\(941\) 8005.71i 0.277342i 0.990339 + 0.138671i \(0.0442831\pi\)
−0.990339 + 0.138671i \(0.955717\pi\)
\(942\) − 11418.8i − 0.394952i
\(943\) 23725.7i 0.819315i
\(944\) − 9626.07i − 0.331888i
\(945\) −4212.52 −0.145009
\(946\) 10493.2 0.360637
\(947\) − 17468.6i − 0.599424i −0.954030 0.299712i \(-0.903109\pi\)
0.954030 0.299712i \(-0.0968905\pi\)
\(948\) −3710.23 −0.127113
\(949\) 0 0
\(950\) 30173.6 1.03048
\(951\) − 22783.6i − 0.776875i
\(952\) 22661.6 0.771498
\(953\) −30681.4 −1.04288 −0.521442 0.853287i \(-0.674606\pi\)
−0.521442 + 0.853287i \(0.674606\pi\)
\(954\) − 3462.70i − 0.117515i
\(955\) − 68752.5i − 2.32961i
\(956\) 12470.0i 0.421872i
\(957\) 9709.87i 0.327979i
\(958\) 26221.8 0.884328
\(959\) −19257.5 −0.648444
\(960\) 25718.4i 0.864644i
\(961\) −51135.6 −1.71648
\(962\) 0 0
\(963\) 1653.29 0.0553237
\(964\) − 1345.00i − 0.0449373i
\(965\) −50814.5 −1.69511
\(966\) 5405.41 0.180037
\(967\) − 15616.0i − 0.519313i −0.965701 0.259656i \(-0.916391\pi\)
0.965701 0.259656i \(-0.0836093\pi\)
\(968\) − 27184.2i − 0.902617i
\(969\) 31240.0i 1.03568i
\(970\) − 20001.5i − 0.662072i
\(971\) 7185.98 0.237496 0.118748 0.992924i \(-0.462112\pi\)
0.118748 + 0.992924i \(0.462112\pi\)
\(972\) 605.148 0.0199693
\(973\) 23574.1i 0.776723i
\(974\) −14192.9 −0.466911
\(975\) 0 0
\(976\) 35423.2 1.16175
\(977\) − 33165.3i − 1.08603i −0.839723 0.543015i \(-0.817283\pi\)
0.839723 0.543015i \(-0.182717\pi\)
\(978\) −937.477 −0.0306516
\(979\) −14764.0 −0.481980
\(980\) − 9217.01i − 0.300435i
\(981\) − 15969.8i − 0.519752i
\(982\) − 2429.52i − 0.0789502i
\(983\) − 11658.7i − 0.378287i −0.981949 0.189143i \(-0.939429\pi\)
0.981949 0.189143i \(-0.0605711\pi\)
\(984\) −23133.0 −0.749444
\(985\) −45790.5 −1.48123
\(986\) 45802.1i 1.47935i
\(987\) −9625.89 −0.310431
\(988\) 0 0
\(989\) −22479.2 −0.722746
\(990\) − 4901.45i − 0.157352i
\(991\) −1957.32 −0.0627409 −0.0313704 0.999508i \(-0.509987\pi\)
−0.0313704 + 0.999508i \(0.509987\pi\)
\(992\) 30747.4 0.984104
\(993\) 5215.75i 0.166684i
\(994\) 22532.3i 0.718996i
\(995\) − 83629.5i − 2.66456i
\(996\) − 2932.56i − 0.0932948i
\(997\) −11434.8 −0.363233 −0.181617 0.983369i \(-0.558133\pi\)
−0.181617 + 0.983369i \(0.558133\pi\)
\(998\) 27459.0 0.870942
\(999\) 9689.70i 0.306876i
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 507.4.b.k.337.7 18
13.5 odd 4 507.4.a.o.1.4 9
13.8 odd 4 507.4.a.p.1.6 yes 9
13.12 even 2 inner 507.4.b.k.337.12 18
39.5 even 4 1521.4.a.bi.1.6 9
39.8 even 4 1521.4.a.bf.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.4 9 13.5 odd 4
507.4.a.p.1.6 yes 9 13.8 odd 4
507.4.b.k.337.7 18 1.1 even 1 trivial
507.4.b.k.337.12 18 13.12 even 2 inner
1521.4.a.bf.1.4 9 39.8 even 4
1521.4.a.bi.1.6 9 39.5 even 4