Properties

Label 507.4.b.j.337.18
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
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] = [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: \( x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + 88861676 x^{4} + 98825392 x^{2} + 26460736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.18
Root \(4.48584i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.j.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.48584i q^{2} -3.00000 q^{3} -22.0945 q^{4} +13.3185i q^{5} -16.4575i q^{6} -21.4234i q^{7} -77.3200i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.48584i q^{2} -3.00000 q^{3} -22.0945 q^{4} +13.3185i q^{5} -16.4575i q^{6} -21.4234i q^{7} -77.3200i q^{8} +9.00000 q^{9} -73.0635 q^{10} +19.0520i q^{11} +66.2834 q^{12} +117.525 q^{14} -39.9556i q^{15} +247.410 q^{16} +71.7906 q^{17} +49.3726i q^{18} -102.134i q^{19} -294.266i q^{20} +64.2701i q^{21} -104.516 q^{22} +37.8302 q^{23} +231.960i q^{24} -52.3838 q^{25} -27.0000 q^{27} +473.338i q^{28} +40.8605 q^{29} +219.190 q^{30} -6.05542i q^{31} +738.690i q^{32} -57.1559i q^{33} +393.832i q^{34} +285.328 q^{35} -198.850 q^{36} +285.682i q^{37} +560.293 q^{38} +1029.79 q^{40} -342.705i q^{41} -352.576 q^{42} -306.458 q^{43} -420.943i q^{44} +119.867i q^{45} +207.530i q^{46} -346.863i q^{47} -742.229 q^{48} -115.961 q^{49} -287.369i q^{50} -215.372 q^{51} +398.219 q^{53} -148.118i q^{54} -253.745 q^{55} -1656.46 q^{56} +306.403i q^{57} +224.155i q^{58} +208.497i q^{59} +882.799i q^{60} +546.936 q^{61} +33.2191 q^{62} -192.810i q^{63} -2073.06 q^{64} +313.548 q^{66} -678.268i q^{67} -1586.18 q^{68} -113.490 q^{69} +1565.27i q^{70} +957.777i q^{71} -695.880i q^{72} -270.360i q^{73} -1567.21 q^{74} +157.151 q^{75} +2256.60i q^{76} +408.157 q^{77} -1032.86 q^{79} +3295.14i q^{80} +81.0000 q^{81} +1880.03 q^{82} -1065.90i q^{83} -1420.01i q^{84} +956.147i q^{85} -1681.18i q^{86} -122.582 q^{87} +1473.10 q^{88} -427.185i q^{89} -657.571 q^{90} -835.837 q^{92} +18.1663i q^{93} +1902.83 q^{94} +1360.28 q^{95} -2216.07i q^{96} +698.084i q^{97} -636.144i q^{98} +171.468i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.48584i 1.93954i 0.244025 + 0.969769i \(0.421532\pi\)
−0.244025 + 0.969769i \(0.578468\pi\)
\(3\) −3.00000 −0.577350
\(4\) −22.0945 −2.76181
\(5\) 13.3185i 1.19125i 0.803264 + 0.595624i \(0.203095\pi\)
−0.803264 + 0.595624i \(0.796905\pi\)
\(6\) − 16.4575i − 1.11979i
\(7\) − 21.4234i − 1.15675i −0.815770 0.578377i \(-0.803686\pi\)
0.815770 0.578377i \(-0.196314\pi\)
\(8\) − 77.3200i − 3.41709i
\(9\) 9.00000 0.333333
\(10\) −73.0635 −2.31047
\(11\) 19.0520i 0.522217i 0.965310 + 0.261108i \(0.0840880\pi\)
−0.965310 + 0.261108i \(0.915912\pi\)
\(12\) 66.2834 1.59453
\(13\) 0 0
\(14\) 117.525 2.24357
\(15\) − 39.9556i − 0.687767i
\(16\) 247.410 3.86578
\(17\) 71.7906 1.02422 0.512111 0.858919i \(-0.328864\pi\)
0.512111 + 0.858919i \(0.328864\pi\)
\(18\) 49.3726i 0.646513i
\(19\) − 102.134i − 1.23322i −0.787268 0.616611i \(-0.788505\pi\)
0.787268 0.616611i \(-0.211495\pi\)
\(20\) − 294.266i − 3.29000i
\(21\) 64.2701i 0.667852i
\(22\) −104.516 −1.01286
\(23\) 37.8302 0.342962 0.171481 0.985187i \(-0.445145\pi\)
0.171481 + 0.985187i \(0.445145\pi\)
\(24\) 231.960i 1.97286i
\(25\) −52.3838 −0.419070
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 473.338i 3.19473i
\(29\) 40.8605 0.261642 0.130821 0.991406i \(-0.458239\pi\)
0.130821 + 0.991406i \(0.458239\pi\)
\(30\) 219.190 1.33395
\(31\) − 6.05542i − 0.0350834i −0.999846 0.0175417i \(-0.994416\pi\)
0.999846 0.0175417i \(-0.00558399\pi\)
\(32\) 738.690i 4.08073i
\(33\) − 57.1559i − 0.301502i
\(34\) 393.832i 1.98652i
\(35\) 285.328 1.37798
\(36\) −198.850 −0.920603
\(37\) 285.682i 1.26935i 0.772781 + 0.634673i \(0.218865\pi\)
−0.772781 + 0.634673i \(0.781135\pi\)
\(38\) 560.293 2.39188
\(39\) 0 0
\(40\) 1029.79 4.07060
\(41\) − 342.705i − 1.30540i −0.757615 0.652702i \(-0.773636\pi\)
0.757615 0.652702i \(-0.226364\pi\)
\(42\) −352.576 −1.29532
\(43\) −306.458 −1.08685 −0.543424 0.839458i \(-0.682872\pi\)
−0.543424 + 0.839458i \(0.682872\pi\)
\(44\) − 420.943i − 1.44226i
\(45\) 119.867i 0.397082i
\(46\) 207.530i 0.665188i
\(47\) − 346.863i − 1.07649i −0.842788 0.538246i \(-0.819087\pi\)
0.842788 0.538246i \(-0.180913\pi\)
\(48\) −742.229 −2.23191
\(49\) −115.961 −0.338079
\(50\) − 287.369i − 0.812802i
\(51\) −215.372 −0.591335
\(52\) 0 0
\(53\) 398.219 1.03207 0.516034 0.856568i \(-0.327408\pi\)
0.516034 + 0.856568i \(0.327408\pi\)
\(54\) − 148.118i − 0.373264i
\(55\) −253.745 −0.622089
\(56\) −1656.46 −3.95274
\(57\) 306.403i 0.712001i
\(58\) 224.155i 0.507464i
\(59\) 208.497i 0.460067i 0.973183 + 0.230034i \(0.0738836\pi\)
−0.973183 + 0.230034i \(0.926116\pi\)
\(60\) 882.799i 1.89948i
\(61\) 546.936 1.14800 0.574000 0.818855i \(-0.305391\pi\)
0.574000 + 0.818855i \(0.305391\pi\)
\(62\) 33.2191 0.0680456
\(63\) − 192.810i − 0.385585i
\(64\) −2073.06 −4.04895
\(65\) 0 0
\(66\) 313.548 0.584774
\(67\) − 678.268i − 1.23677i −0.785875 0.618386i \(-0.787787\pi\)
0.785875 0.618386i \(-0.212213\pi\)
\(68\) −1586.18 −2.82871
\(69\) −113.490 −0.198009
\(70\) 1565.27i 2.67264i
\(71\) 957.777i 1.60095i 0.599368 + 0.800474i \(0.295419\pi\)
−0.599368 + 0.800474i \(0.704581\pi\)
\(72\) − 695.880i − 1.13903i
\(73\) − 270.360i − 0.433469i −0.976231 0.216734i \(-0.930459\pi\)
0.976231 0.216734i \(-0.0695405\pi\)
\(74\) −1567.21 −2.46195
\(75\) 157.151 0.241950
\(76\) 2256.60i 3.40592i
\(77\) 408.157 0.604076
\(78\) 0 0
\(79\) −1032.86 −1.47096 −0.735482 0.677544i \(-0.763044\pi\)
−0.735482 + 0.677544i \(0.763044\pi\)
\(80\) 3295.14i 4.60510i
\(81\) 81.0000 0.111111
\(82\) 1880.03 2.53188
\(83\) − 1065.90i − 1.40961i −0.709400 0.704806i \(-0.751034\pi\)
0.709400 0.704806i \(-0.248966\pi\)
\(84\) − 1420.01i − 1.84448i
\(85\) 956.147i 1.22010i
\(86\) − 1681.18i − 2.10798i
\(87\) −122.582 −0.151059
\(88\) 1473.10 1.78446
\(89\) − 427.185i − 0.508782i −0.967101 0.254391i \(-0.918125\pi\)
0.967101 0.254391i \(-0.0818750\pi\)
\(90\) −657.571 −0.770156
\(91\) 0 0
\(92\) −835.837 −0.947196
\(93\) 18.1663i 0.0202554i
\(94\) 1902.83 2.08790
\(95\) 1360.28 1.46907
\(96\) − 2216.07i − 2.35601i
\(97\) 698.084i 0.730718i 0.930867 + 0.365359i \(0.119054\pi\)
−0.930867 + 0.365359i \(0.880946\pi\)
\(98\) − 636.144i − 0.655717i
\(99\) 171.468i 0.174072i
\(100\) 1157.39 1.15739
\(101\) −88.7364 −0.0874218 −0.0437109 0.999044i \(-0.513918\pi\)
−0.0437109 + 0.999044i \(0.513918\pi\)
\(102\) − 1181.50i − 1.14692i
\(103\) 1427.61 1.36570 0.682849 0.730560i \(-0.260741\pi\)
0.682849 + 0.730560i \(0.260741\pi\)
\(104\) 0 0
\(105\) −855.985 −0.795577
\(106\) 2184.57i 2.00173i
\(107\) −15.0233 −0.0135735 −0.00678673 0.999977i \(-0.502160\pi\)
−0.00678673 + 0.999977i \(0.502160\pi\)
\(108\) 596.551 0.531510
\(109\) 2053.56i 1.80455i 0.431163 + 0.902274i \(0.358104\pi\)
−0.431163 + 0.902274i \(0.641896\pi\)
\(110\) − 1392.00i − 1.20657i
\(111\) − 857.046i − 0.732857i
\(112\) − 5300.35i − 4.47175i
\(113\) 717.456 0.597280 0.298640 0.954366i \(-0.403467\pi\)
0.298640 + 0.954366i \(0.403467\pi\)
\(114\) −1680.88 −1.38095
\(115\) 503.843i 0.408553i
\(116\) −902.792 −0.722605
\(117\) 0 0
\(118\) −1143.78 −0.892318
\(119\) − 1538.00i − 1.18477i
\(120\) −3089.37 −2.35016
\(121\) 968.023 0.727290
\(122\) 3000.41i 2.22659i
\(123\) 1028.12i 0.753675i
\(124\) 133.791i 0.0968937i
\(125\) 967.143i 0.692031i
\(126\) 1057.73 0.747856
\(127\) 117.640 0.0821955 0.0410978 0.999155i \(-0.486914\pi\)
0.0410978 + 0.999155i \(0.486914\pi\)
\(128\) − 5462.96i − 3.77236i
\(129\) 919.375 0.627492
\(130\) 0 0
\(131\) −262.376 −0.174991 −0.0874957 0.996165i \(-0.527886\pi\)
−0.0874957 + 0.996165i \(0.527886\pi\)
\(132\) 1262.83i 0.832690i
\(133\) −2188.06 −1.42653
\(134\) 3720.87 2.39876
\(135\) − 359.601i − 0.229256i
\(136\) − 5550.85i − 3.49986i
\(137\) 1317.41i 0.821563i 0.911734 + 0.410782i \(0.134744\pi\)
−0.911734 + 0.410782i \(0.865256\pi\)
\(138\) − 622.591i − 0.384047i
\(139\) −6.43478 −0.00392656 −0.00196328 0.999998i \(-0.500625\pi\)
−0.00196328 + 0.999998i \(0.500625\pi\)
\(140\) −6304.18 −3.80572
\(141\) 1040.59i 0.621513i
\(142\) −5254.22 −3.10510
\(143\) 0 0
\(144\) 2226.69 1.28859
\(145\) 544.203i 0.311680i
\(146\) 1483.15 0.840729
\(147\) 347.883 0.195190
\(148\) − 6311.99i − 3.50569i
\(149\) 548.312i 0.301473i 0.988574 + 0.150736i \(0.0481645\pi\)
−0.988574 + 0.150736i \(0.951836\pi\)
\(150\) 862.107i 0.469272i
\(151\) − 1485.90i − 0.800799i −0.916341 0.400399i \(-0.868871\pi\)
0.916341 0.400399i \(-0.131129\pi\)
\(152\) −7897.03 −4.21404
\(153\) 646.116 0.341408
\(154\) 2239.09i 1.17163i
\(155\) 80.6494 0.0417930
\(156\) 0 0
\(157\) 2132.36 1.08396 0.541978 0.840393i \(-0.317676\pi\)
0.541978 + 0.840393i \(0.317676\pi\)
\(158\) − 5666.13i − 2.85299i
\(159\) −1194.66 −0.595865
\(160\) −9838.28 −4.86115
\(161\) − 810.450i − 0.396723i
\(162\) 444.353i 0.215504i
\(163\) − 480.633i − 0.230957i −0.993310 0.115479i \(-0.963160\pi\)
0.993310 0.115479i \(-0.0368402\pi\)
\(164\) 7571.88i 3.60527i
\(165\) 761.234 0.359163
\(166\) 5847.36 2.73400
\(167\) 2919.71i 1.35290i 0.736490 + 0.676448i \(0.236482\pi\)
−0.736490 + 0.676448i \(0.763518\pi\)
\(168\) 4969.37 2.28211
\(169\) 0 0
\(170\) −5245.27 −2.36643
\(171\) − 919.209i − 0.411074i
\(172\) 6771.04 3.00167
\(173\) 2339.62 1.02820 0.514098 0.857731i \(-0.328127\pi\)
0.514098 + 0.857731i \(0.328127\pi\)
\(174\) − 672.464i − 0.292985i
\(175\) 1122.24i 0.484761i
\(176\) 4713.64i 2.01877i
\(177\) − 625.490i − 0.265620i
\(178\) 2343.47 0.986802
\(179\) 4558.71 1.90354 0.951772 0.306808i \(-0.0992608\pi\)
0.951772 + 0.306808i \(0.0992608\pi\)
\(180\) − 2648.40i − 1.09667i
\(181\) 3524.56 1.44739 0.723696 0.690118i \(-0.242442\pi\)
0.723696 + 0.690118i \(0.242442\pi\)
\(182\) 0 0
\(183\) −1640.81 −0.662798
\(184\) − 2925.03i − 1.17193i
\(185\) −3804.87 −1.51211
\(186\) −99.6573 −0.0392862
\(187\) 1367.75i 0.534866i
\(188\) 7663.75i 2.97306i
\(189\) 578.431i 0.222617i
\(190\) 7462.29i 2.84932i
\(191\) 3343.27 1.26655 0.633273 0.773929i \(-0.281711\pi\)
0.633273 + 0.773929i \(0.281711\pi\)
\(192\) 6219.18 2.33766
\(193\) − 3341.33i − 1.24619i −0.782147 0.623094i \(-0.785876\pi\)
0.782147 0.623094i \(-0.214124\pi\)
\(194\) −3829.58 −1.41726
\(195\) 0 0
\(196\) 2562.10 0.933709
\(197\) 3653.95i 1.32149i 0.750611 + 0.660745i \(0.229759\pi\)
−0.750611 + 0.660745i \(0.770241\pi\)
\(198\) −940.645 −0.337620
\(199\) −1126.55 −0.401301 −0.200651 0.979663i \(-0.564306\pi\)
−0.200651 + 0.979663i \(0.564306\pi\)
\(200\) 4050.31i 1.43200i
\(201\) 2034.81i 0.714050i
\(202\) − 486.794i − 0.169558i
\(203\) − 875.371i − 0.302655i
\(204\) 4758.53 1.63315
\(205\) 4564.33 1.55506
\(206\) 7831.66i 2.64882i
\(207\) 340.471 0.114321
\(208\) 0 0
\(209\) 1945.86 0.644009
\(210\) − 4695.80i − 1.54305i
\(211\) −74.5243 −0.0243150 −0.0121575 0.999926i \(-0.503870\pi\)
−0.0121575 + 0.999926i \(0.503870\pi\)
\(212\) −8798.44 −2.85037
\(213\) − 2873.33i − 0.924307i
\(214\) − 82.4156i − 0.0263262i
\(215\) − 4081.58i − 1.29471i
\(216\) 2087.64i 0.657620i
\(217\) −129.728 −0.0405829
\(218\) −11265.5 −3.49999
\(219\) 811.079i 0.250263i
\(220\) 5606.35 1.71809
\(221\) 0 0
\(222\) 4701.62 1.42140
\(223\) − 3178.62i − 0.954512i −0.878764 0.477256i \(-0.841631\pi\)
0.878764 0.477256i \(-0.158369\pi\)
\(224\) 15825.2 4.72039
\(225\) −471.454 −0.139690
\(226\) 3935.85i 1.15845i
\(227\) 1349.76i 0.394655i 0.980338 + 0.197327i \(0.0632262\pi\)
−0.980338 + 0.197327i \(0.936774\pi\)
\(228\) − 6769.81i − 1.96641i
\(229\) − 4821.46i − 1.39131i −0.718374 0.695657i \(-0.755113\pi\)
0.718374 0.695657i \(-0.244887\pi\)
\(230\) −2764.00 −0.792404
\(231\) −1224.47 −0.348763
\(232\) − 3159.34i − 0.894055i
\(233\) 2400.88 0.675050 0.337525 0.941317i \(-0.390410\pi\)
0.337525 + 0.941317i \(0.390410\pi\)
\(234\) 0 0
\(235\) 4619.71 1.28237
\(236\) − 4606.63i − 1.27062i
\(237\) 3098.59 0.849262
\(238\) 8437.21 2.29791
\(239\) 1880.85i 0.509047i 0.967067 + 0.254523i \(0.0819186\pi\)
−0.967067 + 0.254523i \(0.918081\pi\)
\(240\) − 9885.41i − 2.65875i
\(241\) 5435.01i 1.45270i 0.687327 + 0.726349i \(0.258784\pi\)
−0.687327 + 0.726349i \(0.741216\pi\)
\(242\) 5310.42i 1.41061i
\(243\) −243.000 −0.0641500
\(244\) −12084.3 −3.17056
\(245\) − 1544.43i − 0.402736i
\(246\) −5640.08 −1.46178
\(247\) 0 0
\(248\) −468.205 −0.119883
\(249\) 3197.70i 0.813840i
\(250\) −5305.59 −1.34222
\(251\) −1256.70 −0.316024 −0.158012 0.987437i \(-0.550508\pi\)
−0.158012 + 0.987437i \(0.550508\pi\)
\(252\) 4260.04i 1.06491i
\(253\) 720.739i 0.179101i
\(254\) 645.353i 0.159421i
\(255\) − 2868.44i − 0.704426i
\(256\) 13384.5 3.26769
\(257\) −5504.87 −1.33612 −0.668062 0.744105i \(-0.732876\pi\)
−0.668062 + 0.744105i \(0.732876\pi\)
\(258\) 5043.55i 1.21705i
\(259\) 6120.27 1.46832
\(260\) 0 0
\(261\) 367.745 0.0872139
\(262\) − 1439.35i − 0.339402i
\(263\) 2032.44 0.476522 0.238261 0.971201i \(-0.423423\pi\)
0.238261 + 0.971201i \(0.423423\pi\)
\(264\) −4419.29 −1.03026
\(265\) 5303.70i 1.22945i
\(266\) − 12003.4i − 2.76682i
\(267\) 1281.56i 0.293745i
\(268\) 14986.0i 3.41572i
\(269\) 5523.82 1.25202 0.626009 0.779816i \(-0.284687\pi\)
0.626009 + 0.779816i \(0.284687\pi\)
\(270\) 1972.71 0.444650
\(271\) 3918.69i 0.878389i 0.898392 + 0.439194i \(0.144736\pi\)
−0.898392 + 0.439194i \(0.855264\pi\)
\(272\) 17761.7 3.95941
\(273\) 0 0
\(274\) −7227.12 −1.59345
\(275\) − 998.013i − 0.218845i
\(276\) 2507.51 0.546864
\(277\) 2944.56 0.638705 0.319353 0.947636i \(-0.396535\pi\)
0.319353 + 0.947636i \(0.396535\pi\)
\(278\) − 35.3002i − 0.00761570i
\(279\) − 54.4988i − 0.0116945i
\(280\) − 22061.6i − 4.70869i
\(281\) − 5048.76i − 1.07183i −0.844273 0.535914i \(-0.819967\pi\)
0.844273 0.535914i \(-0.180033\pi\)
\(282\) −5708.50 −1.20545
\(283\) −4491.66 −0.943469 −0.471734 0.881741i \(-0.656372\pi\)
−0.471734 + 0.881741i \(0.656372\pi\)
\(284\) − 21161.6i − 4.42151i
\(285\) −4080.84 −0.848169
\(286\) 0 0
\(287\) −7341.90 −1.51003
\(288\) 6648.21i 1.36024i
\(289\) 240.893 0.0490318
\(290\) −2985.41 −0.604515
\(291\) − 2094.25i − 0.421880i
\(292\) 5973.45i 1.19716i
\(293\) − 4754.86i − 0.948061i −0.880508 0.474031i \(-0.842799\pi\)
0.880508 0.474031i \(-0.157201\pi\)
\(294\) 1908.43i 0.378578i
\(295\) −2776.88 −0.548054
\(296\) 22088.9 4.33747
\(297\) − 514.403i − 0.100501i
\(298\) −3007.95 −0.584718
\(299\) 0 0
\(300\) −3472.17 −0.668220
\(301\) 6565.38i 1.25722i
\(302\) 8151.40 1.55318
\(303\) 266.209 0.0504730
\(304\) − 25269.0i − 4.76736i
\(305\) 7284.40i 1.36755i
\(306\) 3544.49i 0.662173i
\(307\) − 1623.31i − 0.301783i −0.988550 0.150891i \(-0.951786\pi\)
0.988550 0.150891i \(-0.0482143\pi\)
\(308\) −9018.02 −1.66834
\(309\) −4282.84 −0.788486
\(310\) 442.430i 0.0810592i
\(311\) −6683.49 −1.21860 −0.609302 0.792938i \(-0.708550\pi\)
−0.609302 + 0.792938i \(0.708550\pi\)
\(312\) 0 0
\(313\) 1584.55 0.286147 0.143073 0.989712i \(-0.454301\pi\)
0.143073 + 0.989712i \(0.454301\pi\)
\(314\) 11697.8i 2.10237i
\(315\) 2567.95 0.459327
\(316\) 22820.6 4.06252
\(317\) 787.932i 0.139605i 0.997561 + 0.0698023i \(0.0222368\pi\)
−0.997561 + 0.0698023i \(0.977763\pi\)
\(318\) − 6553.70i − 1.15570i
\(319\) 778.474i 0.136634i
\(320\) − 27610.2i − 4.82330i
\(321\) 45.0700 0.00783664
\(322\) 4446.00 0.769459
\(323\) − 7332.28i − 1.26309i
\(324\) −1789.65 −0.306868
\(325\) 0 0
\(326\) 2636.67 0.447951
\(327\) − 6160.69i − 1.04186i
\(328\) −26498.0 −4.46069
\(329\) −7430.97 −1.24524
\(330\) 4176.01i 0.696611i
\(331\) − 9493.56i − 1.57647i −0.615371 0.788237i \(-0.710994\pi\)
0.615371 0.788237i \(-0.289006\pi\)
\(332\) 23550.5i 3.89308i
\(333\) 2571.14i 0.423115i
\(334\) −16017.1 −2.62399
\(335\) 9033.55 1.47330
\(336\) 15901.1i 2.58177i
\(337\) 4123.06 0.666461 0.333230 0.942845i \(-0.391861\pi\)
0.333230 + 0.942845i \(0.391861\pi\)
\(338\) 0 0
\(339\) −2152.37 −0.344840
\(340\) − 21125.6i − 3.36969i
\(341\) 115.368 0.0183211
\(342\) 5042.63 0.797294
\(343\) − 4863.94i − 0.765680i
\(344\) 23695.4i 3.71386i
\(345\) − 1511.53i − 0.235878i
\(346\) 12834.8i 1.99423i
\(347\) 2320.76 0.359034 0.179517 0.983755i \(-0.442546\pi\)
0.179517 + 0.983755i \(0.442546\pi\)
\(348\) 2708.38 0.417196
\(349\) − 3818.85i − 0.585726i −0.956154 0.292863i \(-0.905392\pi\)
0.956154 0.292863i \(-0.0946079\pi\)
\(350\) −6156.41 −0.940212
\(351\) 0 0
\(352\) −14073.5 −2.13102
\(353\) − 6065.53i − 0.914549i −0.889326 0.457274i \(-0.848826\pi\)
0.889326 0.457274i \(-0.151174\pi\)
\(354\) 3431.34 0.515180
\(355\) −12756.2 −1.90712
\(356\) 9438.43i 1.40516i
\(357\) 4613.99i 0.684029i
\(358\) 25008.4i 3.69199i
\(359\) − 7406.02i − 1.08879i −0.838830 0.544394i \(-0.816760\pi\)
0.838830 0.544394i \(-0.183240\pi\)
\(360\) 9268.11 1.35687
\(361\) −3572.42 −0.520836
\(362\) 19335.2i 2.80727i
\(363\) −2904.07 −0.419901
\(364\) 0 0
\(365\) 3600.80 0.516368
\(366\) − 9001.22i − 1.28552i
\(367\) −4754.23 −0.676209 −0.338105 0.941109i \(-0.609786\pi\)
−0.338105 + 0.941109i \(0.609786\pi\)
\(368\) 9359.55 1.32582
\(369\) − 3084.35i − 0.435134i
\(370\) − 20872.9i − 2.93279i
\(371\) − 8531.20i − 1.19385i
\(372\) − 401.374i − 0.0559416i
\(373\) −4684.13 −0.650229 −0.325114 0.945675i \(-0.605403\pi\)
−0.325114 + 0.945675i \(0.605403\pi\)
\(374\) −7503.27 −1.03739
\(375\) − 2901.43i − 0.399544i
\(376\) −26819.4 −3.67847
\(377\) 0 0
\(378\) −3173.18 −0.431775
\(379\) 12103.9i 1.64047i 0.572030 + 0.820233i \(0.306156\pi\)
−0.572030 + 0.820233i \(0.693844\pi\)
\(380\) −30054.7 −4.05730
\(381\) −352.919 −0.0474556
\(382\) 18340.6i 2.45651i
\(383\) − 5601.56i − 0.747327i −0.927564 0.373663i \(-0.878102\pi\)
0.927564 0.373663i \(-0.121898\pi\)
\(384\) 16388.9i 2.17797i
\(385\) 5436.06i 0.719604i
\(386\) 18330.0 2.41703
\(387\) −2758.13 −0.362283
\(388\) − 15423.8i − 2.01810i
\(389\) −9450.46 −1.23177 −0.615883 0.787837i \(-0.711201\pi\)
−0.615883 + 0.787837i \(0.711201\pi\)
\(390\) 0 0
\(391\) 2715.85 0.351270
\(392\) 8966.11i 1.15525i
\(393\) 787.127 0.101031
\(394\) −20045.0 −2.56308
\(395\) − 13756.2i − 1.75228i
\(396\) − 3788.49i − 0.480754i
\(397\) 2723.98i 0.344364i 0.985065 + 0.172182i \(0.0550817\pi\)
−0.985065 + 0.172182i \(0.944918\pi\)
\(398\) − 6180.07i − 0.778339i
\(399\) 6564.18 0.823610
\(400\) −12960.2 −1.62003
\(401\) − 4680.46i − 0.582870i −0.956591 0.291435i \(-0.905867\pi\)
0.956591 0.291435i \(-0.0941328\pi\)
\(402\) −11162.6 −1.38493
\(403\) 0 0
\(404\) 1960.58 0.241442
\(405\) 1078.80i 0.132361i
\(406\) 4802.15 0.587011
\(407\) −5442.80 −0.662874
\(408\) 16652.6i 2.02065i
\(409\) − 3272.70i − 0.395659i −0.980236 0.197830i \(-0.936611\pi\)
0.980236 0.197830i \(-0.0633893\pi\)
\(410\) 25039.2i 3.01609i
\(411\) − 3952.24i − 0.474330i
\(412\) −31542.3 −3.77179
\(413\) 4466.71 0.532185
\(414\) 1867.77i 0.221729i
\(415\) 14196.2 1.67920
\(416\) 0 0
\(417\) 19.3043 0.00226700
\(418\) 10674.7i 1.24908i
\(419\) 1501.48 0.175065 0.0875323 0.996162i \(-0.472102\pi\)
0.0875323 + 0.996162i \(0.472102\pi\)
\(420\) 18912.5 2.19723
\(421\) − 16578.1i − 1.91916i −0.281436 0.959580i \(-0.590811\pi\)
0.281436 0.959580i \(-0.409189\pi\)
\(422\) − 408.828i − 0.0471598i
\(423\) − 3121.76i − 0.358831i
\(424\) − 30790.3i − 3.52667i
\(425\) −3760.66 −0.429221
\(426\) 15762.6 1.79273
\(427\) − 11717.2i − 1.32795i
\(428\) 331.933 0.0374873
\(429\) 0 0
\(430\) 22390.9 2.51113
\(431\) − 3776.55i − 0.422065i −0.977479 0.211032i \(-0.932317\pi\)
0.977479 0.211032i \(-0.0676826\pi\)
\(432\) −6680.06 −0.743969
\(433\) 709.953 0.0787948 0.0393974 0.999224i \(-0.487456\pi\)
0.0393974 + 0.999224i \(0.487456\pi\)
\(434\) − 711.665i − 0.0787120i
\(435\) − 1632.61i − 0.179949i
\(436\) − 45372.4i − 4.98382i
\(437\) − 3863.76i − 0.422949i
\(438\) −4449.45 −0.485395
\(439\) 16262.2 1.76800 0.884002 0.467484i \(-0.154839\pi\)
0.884002 + 0.467484i \(0.154839\pi\)
\(440\) 19619.5i 2.12574i
\(441\) −1043.65 −0.112693
\(442\) 0 0
\(443\) 2899.45 0.310964 0.155482 0.987839i \(-0.450307\pi\)
0.155482 + 0.987839i \(0.450307\pi\)
\(444\) 18936.0i 2.02401i
\(445\) 5689.49 0.606085
\(446\) 17437.4 1.85131
\(447\) − 1644.94i − 0.174055i
\(448\) 44412.0i 4.68363i
\(449\) − 7817.78i − 0.821701i −0.911703 0.410851i \(-0.865232\pi\)
0.911703 0.410851i \(-0.134768\pi\)
\(450\) − 2586.32i − 0.270934i
\(451\) 6529.20 0.681703
\(452\) −15851.8 −1.64957
\(453\) 4457.69i 0.462341i
\(454\) −7404.57 −0.765448
\(455\) 0 0
\(456\) 23691.1 2.43297
\(457\) 14452.9i 1.47939i 0.672943 + 0.739694i \(0.265030\pi\)
−0.672943 + 0.739694i \(0.734970\pi\)
\(458\) 26449.8 2.69851
\(459\) −1938.35 −0.197112
\(460\) − 11132.1i − 1.12834i
\(461\) − 2891.46i − 0.292123i −0.989276 0.146061i \(-0.953340\pi\)
0.989276 0.146061i \(-0.0466597\pi\)
\(462\) − 6717.26i − 0.676440i
\(463\) 9223.83i 0.925848i 0.886398 + 0.462924i \(0.153200\pi\)
−0.886398 + 0.462924i \(0.846800\pi\)
\(464\) 10109.3 1.01145
\(465\) −241.948 −0.0241292
\(466\) 13170.8i 1.30929i
\(467\) 7988.25 0.791546 0.395773 0.918348i \(-0.370477\pi\)
0.395773 + 0.918348i \(0.370477\pi\)
\(468\) 0 0
\(469\) −14530.8 −1.43064
\(470\) 25343.0i 2.48720i
\(471\) −6397.09 −0.625822
\(472\) 16121.0 1.57209
\(473\) − 5838.64i − 0.567570i
\(474\) 16998.4i 1.64718i
\(475\) 5350.18i 0.516806i
\(476\) 33981.2i 3.27212i
\(477\) 3583.97 0.344023
\(478\) −10318.1 −0.987315
\(479\) − 6908.64i − 0.659006i −0.944155 0.329503i \(-0.893119\pi\)
0.944155 0.329503i \(-0.106881\pi\)
\(480\) 29514.8 2.80659
\(481\) 0 0
\(482\) −29815.6 −2.81756
\(483\) 2431.35i 0.229048i
\(484\) −21387.9 −2.00863
\(485\) −9297.47 −0.870466
\(486\) − 1333.06i − 0.124421i
\(487\) − 13455.0i − 1.25196i −0.779841 0.625978i \(-0.784700\pi\)
0.779841 0.625978i \(-0.215300\pi\)
\(488\) − 42289.1i − 3.92282i
\(489\) 1441.90i 0.133343i
\(490\) 8472.52 0.781121
\(491\) 1044.02 0.0959589 0.0479795 0.998848i \(-0.484722\pi\)
0.0479795 + 0.998848i \(0.484722\pi\)
\(492\) − 22715.7i − 2.08151i
\(493\) 2933.40 0.267979
\(494\) 0 0
\(495\) −2283.70 −0.207363
\(496\) − 1498.17i − 0.135625i
\(497\) 20518.8 1.85190
\(498\) −17542.1 −1.57847
\(499\) 16190.7i 1.45249i 0.687435 + 0.726246i \(0.258737\pi\)
−0.687435 + 0.726246i \(0.741263\pi\)
\(500\) − 21368.5i − 1.91126i
\(501\) − 8759.12i − 0.781095i
\(502\) − 6894.04i − 0.612940i
\(503\) −11854.8 −1.05086 −0.525429 0.850838i \(-0.676095\pi\)
−0.525429 + 0.850838i \(0.676095\pi\)
\(504\) −14908.1 −1.31758
\(505\) − 1181.84i − 0.104141i
\(506\) −3953.86 −0.347372
\(507\) 0 0
\(508\) −2599.19 −0.227008
\(509\) − 6647.07i − 0.578834i −0.957203 0.289417i \(-0.906539\pi\)
0.957203 0.289417i \(-0.0934614\pi\)
\(510\) 15735.8 1.36626
\(511\) −5792.02 −0.501416
\(512\) 29721.4i 2.56545i
\(513\) 2757.63i 0.237334i
\(514\) − 30198.8i − 2.59146i
\(515\) 19013.7i 1.62688i
\(516\) −20313.1 −1.73301
\(517\) 6608.42 0.562162
\(518\) 33574.8i 2.84786i
\(519\) −7018.86 −0.593630
\(520\) 0 0
\(521\) 16839.6 1.41604 0.708019 0.706193i \(-0.249589\pi\)
0.708019 + 0.706193i \(0.249589\pi\)
\(522\) 2017.39i 0.169155i
\(523\) 10414.1 0.870701 0.435351 0.900261i \(-0.356624\pi\)
0.435351 + 0.900261i \(0.356624\pi\)
\(524\) 5797.05 0.483293
\(525\) − 3366.71i − 0.279877i
\(526\) 11149.6i 0.924233i
\(527\) − 434.723i − 0.0359332i
\(528\) − 14140.9i − 1.16554i
\(529\) −10735.9 −0.882377
\(530\) −29095.3 −2.38456
\(531\) 1876.47i 0.153356i
\(532\) 48344.1 3.93981
\(533\) 0 0
\(534\) −7030.42 −0.569730
\(535\) − 200.089i − 0.0161694i
\(536\) −52443.7 −4.22616
\(537\) −13676.1 −1.09901
\(538\) 30302.8i 2.42834i
\(539\) − 2209.29i − 0.176550i
\(540\) 7945.19i 0.633160i
\(541\) − 5464.03i − 0.434227i −0.976146 0.217114i \(-0.930336\pi\)
0.976146 0.217114i \(-0.0696642\pi\)
\(542\) −21497.3 −1.70367
\(543\) −10573.7 −0.835653
\(544\) 53031.0i 4.17957i
\(545\) −27350.5 −2.14966
\(546\) 0 0
\(547\) 24902.3 1.94652 0.973261 0.229702i \(-0.0737753\pi\)
0.973261 + 0.229702i \(0.0737753\pi\)
\(548\) − 29107.5i − 2.26900i
\(549\) 4922.43 0.382667
\(550\) 5474.94 0.424459
\(551\) − 4173.26i − 0.322662i
\(552\) 8775.08i 0.676617i
\(553\) 22127.4i 1.70154i
\(554\) 16153.4i 1.23879i
\(555\) 11414.6 0.873014
\(556\) 142.173 0.0108444
\(557\) − 9817.71i − 0.746840i −0.927662 0.373420i \(-0.878185\pi\)
0.927662 0.373420i \(-0.121815\pi\)
\(558\) 298.972 0.0226819
\(559\) 0 0
\(560\) 70593.0 5.32696
\(561\) − 4103.26i − 0.308805i
\(562\) 27696.7 2.07885
\(563\) 23333.7 1.74671 0.873356 0.487083i \(-0.161939\pi\)
0.873356 + 0.487083i \(0.161939\pi\)
\(564\) − 22991.2i − 1.71650i
\(565\) 9555.48i 0.711508i
\(566\) − 24640.6i − 1.82989i
\(567\) − 1735.29i − 0.128528i
\(568\) 74055.4 5.47059
\(569\) −24543.9 −1.80832 −0.904160 0.427194i \(-0.859502\pi\)
−0.904160 + 0.427194i \(0.859502\pi\)
\(570\) − 22386.9i − 1.64506i
\(571\) 10562.5 0.774126 0.387063 0.922053i \(-0.373490\pi\)
0.387063 + 0.922053i \(0.373490\pi\)
\(572\) 0 0
\(573\) −10029.8 −0.731241
\(574\) − 40276.5i − 2.92876i
\(575\) −1981.69 −0.143725
\(576\) −18657.5 −1.34965
\(577\) 18922.5i 1.36526i 0.730765 + 0.682629i \(0.239163\pi\)
−0.730765 + 0.682629i \(0.760837\pi\)
\(578\) 1321.50i 0.0950991i
\(579\) 10024.0i 0.719486i
\(580\) − 12023.9i − 0.860801i
\(581\) −22835.2 −1.63057
\(582\) 11488.7 0.818253
\(583\) 7586.85i 0.538963i
\(584\) −20904.2 −1.48120
\(585\) 0 0
\(586\) 26084.4 1.83880
\(587\) 7989.65i 0.561786i 0.959739 + 0.280893i \(0.0906306\pi\)
−0.959739 + 0.280893i \(0.909369\pi\)
\(588\) −7686.29 −0.539077
\(589\) −618.466 −0.0432656
\(590\) − 15233.5i − 1.06297i
\(591\) − 10961.9i − 0.762962i
\(592\) 70680.4i 4.90701i
\(593\) − 25239.2i − 1.74781i −0.486097 0.873905i \(-0.661580\pi\)
0.486097 0.873905i \(-0.338420\pi\)
\(594\) 2821.93 0.194925
\(595\) 20483.9 1.41136
\(596\) − 12114.7i − 0.832610i
\(597\) 3379.65 0.231691
\(598\) 0 0
\(599\) −7412.19 −0.505599 −0.252800 0.967519i \(-0.581351\pi\)
−0.252800 + 0.967519i \(0.581351\pi\)
\(600\) − 12150.9i − 0.826767i
\(601\) −21459.2 −1.45647 −0.728236 0.685327i \(-0.759659\pi\)
−0.728236 + 0.685327i \(0.759659\pi\)
\(602\) −36016.6 −2.43842
\(603\) − 6104.42i − 0.412257i
\(604\) 32830.1i 2.21165i
\(605\) 12892.7i 0.866382i
\(606\) 1460.38i 0.0978944i
\(607\) −18246.7 −1.22011 −0.610057 0.792358i \(-0.708853\pi\)
−0.610057 + 0.792358i \(0.708853\pi\)
\(608\) 75445.6 5.03244
\(609\) 2626.11i 0.174738i
\(610\) −39961.1 −2.65242
\(611\) 0 0
\(612\) −14275.6 −0.942902
\(613\) 24087.5i 1.58709i 0.608512 + 0.793545i \(0.291767\pi\)
−0.608512 + 0.793545i \(0.708233\pi\)
\(614\) 8905.23 0.585319
\(615\) −13693.0 −0.897813
\(616\) − 31558.7i − 2.06418i
\(617\) 14400.6i 0.939621i 0.882767 + 0.469811i \(0.155678\pi\)
−0.882767 + 0.469811i \(0.844322\pi\)
\(618\) − 23495.0i − 1.52930i
\(619\) 25635.8i 1.66460i 0.554322 + 0.832302i \(0.312978\pi\)
−0.554322 + 0.832302i \(0.687022\pi\)
\(620\) −1781.91 −0.115424
\(621\) −1021.41 −0.0660031
\(622\) − 36664.6i − 2.36353i
\(623\) −9151.76 −0.588535
\(624\) 0 0
\(625\) −19428.9 −1.24345
\(626\) 8692.58i 0.554993i
\(627\) −5837.58 −0.371819
\(628\) −47113.4 −2.99368
\(629\) 20509.3i 1.30009i
\(630\) 14087.4i 0.890881i
\(631\) − 2757.17i − 0.173948i −0.996211 0.0869741i \(-0.972280\pi\)
0.996211 0.0869741i \(-0.0277197\pi\)
\(632\) 79861.0i 5.02643i
\(633\) 223.573 0.0140383
\(634\) −4322.47 −0.270768
\(635\) 1566.79i 0.0979152i
\(636\) 26395.3 1.64566
\(637\) 0 0
\(638\) −4270.58 −0.265006
\(639\) 8620.00i 0.533649i
\(640\) 72758.7 4.49382
\(641\) −14317.0 −0.882196 −0.441098 0.897459i \(-0.645411\pi\)
−0.441098 + 0.897459i \(0.645411\pi\)
\(642\) 247.247i 0.0151995i
\(643\) 51.9171i 0.00318415i 0.999999 + 0.00159208i \(0.000506774\pi\)
−0.999999 + 0.00159208i \(0.999493\pi\)
\(644\) 17906.5i 1.09567i
\(645\) 12244.7i 0.747498i
\(646\) 40223.8 2.44982
\(647\) 9735.07 0.591538 0.295769 0.955260i \(-0.404424\pi\)
0.295769 + 0.955260i \(0.404424\pi\)
\(648\) − 6262.92i − 0.379677i
\(649\) −3972.27 −0.240255
\(650\) 0 0
\(651\) 389.183 0.0234305
\(652\) 10619.3i 0.637860i
\(653\) 10842.6 0.649778 0.324889 0.945752i \(-0.394673\pi\)
0.324889 + 0.945752i \(0.394673\pi\)
\(654\) 33796.6 2.02072
\(655\) − 3494.46i − 0.208458i
\(656\) − 84788.5i − 5.04640i
\(657\) − 2433.24i − 0.144490i
\(658\) − 40765.1i − 2.41518i
\(659\) 28940.9 1.71074 0.855369 0.518020i \(-0.173331\pi\)
0.855369 + 0.518020i \(0.173331\pi\)
\(660\) −16819.0 −0.991940
\(661\) − 3108.58i − 0.182919i −0.995809 0.0914597i \(-0.970847\pi\)
0.995809 0.0914597i \(-0.0291533\pi\)
\(662\) 52080.2 3.05763
\(663\) 0 0
\(664\) −82415.4 −4.81677
\(665\) − 29141.8i − 1.69935i
\(666\) −14104.8 −0.820648
\(667\) 1545.76 0.0897333
\(668\) − 64509.4i − 3.73644i
\(669\) 9535.86i 0.551088i
\(670\) 49556.6i 2.85752i
\(671\) 10420.2i 0.599505i
\(672\) −47475.7 −2.72532
\(673\) 9950.09 0.569908 0.284954 0.958541i \(-0.408022\pi\)
0.284954 + 0.958541i \(0.408022\pi\)
\(674\) 22618.4i 1.29263i
\(675\) 1414.36 0.0806501
\(676\) 0 0
\(677\) −3483.60 −0.197763 −0.0988816 0.995099i \(-0.531527\pi\)
−0.0988816 + 0.995099i \(0.531527\pi\)
\(678\) − 11807.6i − 0.668830i
\(679\) 14955.3 0.845261
\(680\) 73929.3 4.16920
\(681\) − 4049.28i − 0.227854i
\(682\) 632.889i 0.0355346i
\(683\) 16811.4i 0.941829i 0.882179 + 0.470915i \(0.156076\pi\)
−0.882179 + 0.470915i \(0.843924\pi\)
\(684\) 20309.4i 1.13531i
\(685\) −17546.0 −0.978685
\(686\) 26682.8 1.48506
\(687\) 14464.4i 0.803276i
\(688\) −75820.8 −4.20151
\(689\) 0 0
\(690\) 8292.01 0.457495
\(691\) − 3078.78i − 0.169497i −0.996402 0.0847484i \(-0.972991\pi\)
0.996402 0.0847484i \(-0.0270086\pi\)
\(692\) −51692.6 −2.83968
\(693\) 3673.42 0.201359
\(694\) 12731.3i 0.696361i
\(695\) − 85.7020i − 0.00467750i
\(696\) 9478.01i 0.516183i
\(697\) − 24603.0i − 1.33702i
\(698\) 20949.6 1.13604
\(699\) −7202.63 −0.389740
\(700\) − 24795.2i − 1.33882i
\(701\) 18608.6 1.00262 0.501311 0.865267i \(-0.332851\pi\)
0.501311 + 0.865267i \(0.332851\pi\)
\(702\) 0 0
\(703\) 29177.9 1.56539
\(704\) − 39495.9i − 2.11443i
\(705\) −13859.1 −0.740376
\(706\) 33274.5 1.77380
\(707\) 1901.03i 0.101126i
\(708\) 13819.9i 0.733591i
\(709\) − 19046.2i − 1.00888i −0.863448 0.504439i \(-0.831699\pi\)
0.863448 0.504439i \(-0.168301\pi\)
\(710\) − 69978.5i − 3.69894i
\(711\) −9295.77 −0.490322
\(712\) −33030.0 −1.73855
\(713\) − 229.078i − 0.0120323i
\(714\) −25311.6 −1.32670
\(715\) 0 0
\(716\) −100722. −5.25722
\(717\) − 5642.55i − 0.293898i
\(718\) 40628.2 2.11174
\(719\) 14013.4 0.726859 0.363430 0.931622i \(-0.381606\pi\)
0.363430 + 0.931622i \(0.381606\pi\)
\(720\) 29656.2i 1.53503i
\(721\) − 30584.3i − 1.57978i
\(722\) − 19597.7i − 1.01018i
\(723\) − 16305.0i − 0.838715i
\(724\) −77873.2 −3.99742
\(725\) −2140.43 −0.109646
\(726\) − 15931.3i − 0.814414i
\(727\) −2578.98 −0.131567 −0.0657834 0.997834i \(-0.520955\pi\)
−0.0657834 + 0.997834i \(0.520955\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 19753.4i 1.00152i
\(731\) −22000.8 −1.11317
\(732\) 36252.8 1.83052
\(733\) 5029.12i 0.253417i 0.991940 + 0.126709i \(0.0404413\pi\)
−0.991940 + 0.126709i \(0.959559\pi\)
\(734\) − 26081.0i − 1.31153i
\(735\) 4633.30i 0.232520i
\(736\) 27944.8i 1.39954i
\(737\) 12922.3 0.645863
\(738\) 16920.2 0.843960
\(739\) 23856.4i 1.18751i 0.804644 + 0.593757i \(0.202356\pi\)
−0.804644 + 0.593757i \(0.797644\pi\)
\(740\) 84066.5 4.17614
\(741\) 0 0
\(742\) 46800.8 2.31551
\(743\) 4887.62i 0.241332i 0.992693 + 0.120666i \(0.0385030\pi\)
−0.992693 + 0.120666i \(0.961497\pi\)
\(744\) 1404.62 0.0692147
\(745\) −7302.72 −0.359129
\(746\) − 25696.4i − 1.26114i
\(747\) − 9593.10i − 0.469870i
\(748\) − 30219.8i − 1.47720i
\(749\) 321.851i 0.0157012i
\(750\) 15916.8 0.774932
\(751\) −24814.3 −1.20571 −0.602854 0.797851i \(-0.705970\pi\)
−0.602854 + 0.797851i \(0.705970\pi\)
\(752\) − 85817.2i − 4.16148i
\(753\) 3770.09 0.182456
\(754\) 0 0
\(755\) 19790.0 0.953949
\(756\) − 12780.1i − 0.614826i
\(757\) −6552.91 −0.314623 −0.157311 0.987549i \(-0.550283\pi\)
−0.157311 + 0.987549i \(0.550283\pi\)
\(758\) −66400.2 −3.18175
\(759\) − 2162.22i − 0.103404i
\(760\) − 105177.i − 5.01996i
\(761\) − 20351.4i − 0.969431i −0.874672 0.484715i \(-0.838923\pi\)
0.874672 0.484715i \(-0.161077\pi\)
\(762\) − 1936.06i − 0.0920420i
\(763\) 43994.3 2.08742
\(764\) −73867.7 −3.49796
\(765\) 8605.32i 0.406701i
\(766\) 30729.3 1.44947
\(767\) 0 0
\(768\) −40153.4 −1.88660
\(769\) − 22209.7i − 1.04148i −0.853714 0.520742i \(-0.825655\pi\)
0.853714 0.520742i \(-0.174345\pi\)
\(770\) −29821.4 −1.39570
\(771\) 16514.6 0.771412
\(772\) 73824.9i 3.44173i
\(773\) 28496.6i 1.32594i 0.748645 + 0.662971i \(0.230705\pi\)
−0.748645 + 0.662971i \(0.769295\pi\)
\(774\) − 15130.6i − 0.702661i
\(775\) 317.206i 0.0147024i
\(776\) 53975.9 2.49693
\(777\) −18360.8 −0.847735
\(778\) − 51843.7i − 2.38906i
\(779\) −35001.9 −1.60985
\(780\) 0 0
\(781\) −18247.5 −0.836041
\(782\) 14898.7i 0.681301i
\(783\) −1103.23 −0.0503530
\(784\) −28689.9 −1.30694
\(785\) 28400.0i 1.29126i
\(786\) 4318.05i 0.195954i
\(787\) 2394.07i 0.108436i 0.998529 + 0.0542181i \(0.0172666\pi\)
−0.998529 + 0.0542181i \(0.982733\pi\)
\(788\) − 80732.2i − 3.64970i
\(789\) −6097.31 −0.275120
\(790\) 75464.6 3.39862
\(791\) − 15370.3i − 0.690906i
\(792\) 13257.9 0.594821
\(793\) 0 0
\(794\) −14943.3 −0.667907
\(795\) − 15911.1i − 0.709822i
\(796\) 24890.5 1.10832
\(797\) 37996.7 1.68872 0.844362 0.535773i \(-0.179980\pi\)
0.844362 + 0.535773i \(0.179980\pi\)
\(798\) 36010.1i 1.59742i
\(799\) − 24901.5i − 1.10257i
\(800\) − 38695.4i − 1.71011i
\(801\) − 3844.67i − 0.169594i
\(802\) 25676.3 1.13050
\(803\) 5150.88 0.226364
\(804\) − 44957.9i − 1.97207i
\(805\) 10794.0 0.472595
\(806\) 0 0
\(807\) −16571.4 −0.722853
\(808\) 6861.10i 0.298729i
\(809\) −25993.4 −1.12964 −0.564821 0.825213i \(-0.691055\pi\)
−0.564821 + 0.825213i \(0.691055\pi\)
\(810\) −5918.14 −0.256719
\(811\) − 15524.4i − 0.672176i −0.941831 0.336088i \(-0.890896\pi\)
0.941831 0.336088i \(-0.109104\pi\)
\(812\) 19340.9i 0.835875i
\(813\) − 11756.1i − 0.507138i
\(814\) − 29858.3i − 1.28567i
\(815\) 6401.33 0.275127
\(816\) −53285.1 −2.28597
\(817\) 31299.9i 1.34033i
\(818\) 17953.5 0.767396
\(819\) 0 0
\(820\) −100847. −4.29477
\(821\) − 2029.40i − 0.0862684i −0.999069 0.0431342i \(-0.986266\pi\)
0.999069 0.0431342i \(-0.0137343\pi\)
\(822\) 21681.3 0.919980
\(823\) 42010.4 1.77933 0.889667 0.456610i \(-0.150936\pi\)
0.889667 + 0.456610i \(0.150936\pi\)
\(824\) − 110383.i − 4.66672i
\(825\) 2994.04i 0.126350i
\(826\) 24503.6i 1.03219i
\(827\) − 3941.24i − 0.165720i −0.996561 0.0828599i \(-0.973595\pi\)
0.996561 0.0828599i \(-0.0264054\pi\)
\(828\) −7522.53 −0.315732
\(829\) −11264.7 −0.471941 −0.235970 0.971760i \(-0.575827\pi\)
−0.235970 + 0.971760i \(0.575827\pi\)
\(830\) 77878.4i 3.25686i
\(831\) −8833.68 −0.368757
\(832\) 0 0
\(833\) −8324.92 −0.346268
\(834\) 105.901i 0.00439693i
\(835\) −38886.3 −1.61163
\(836\) −42992.7 −1.77863
\(837\) 163.496i 0.00675181i
\(838\) 8236.88i 0.339544i
\(839\) − 35221.1i − 1.44931i −0.689114 0.724653i \(-0.742000\pi\)
0.689114 0.724653i \(-0.258000\pi\)
\(840\) 66184.8i 2.71856i
\(841\) −22719.4 −0.931544
\(842\) 90944.8 3.72228
\(843\) 15146.3i 0.618820i
\(844\) 1646.57 0.0671533
\(845\) 0 0
\(846\) 17125.5 0.695966
\(847\) − 20738.3i − 0.841295i
\(848\) 98523.2 3.98974
\(849\) 13475.0 0.544712
\(850\) − 20630.4i − 0.832490i
\(851\) 10807.4i 0.435338i
\(852\) 63484.7i 2.55276i
\(853\) 37662.7i 1.51178i 0.654701 + 0.755888i \(0.272795\pi\)
−0.654701 + 0.755888i \(0.727205\pi\)
\(854\) 64278.8 2.57562
\(855\) 12242.5 0.489691
\(856\) 1161.60i 0.0463818i
\(857\) −800.368 −0.0319020 −0.0159510 0.999873i \(-0.505078\pi\)
−0.0159510 + 0.999873i \(0.505078\pi\)
\(858\) 0 0
\(859\) −8800.69 −0.349564 −0.174782 0.984607i \(-0.555922\pi\)
−0.174782 + 0.984607i \(0.555922\pi\)
\(860\) 90180.4i 3.57573i
\(861\) 22025.7 0.871816
\(862\) 20717.6 0.818611
\(863\) 17991.0i 0.709641i 0.934934 + 0.354821i \(0.115458\pi\)
−0.934934 + 0.354821i \(0.884542\pi\)
\(864\) − 19944.6i − 0.785336i
\(865\) 31160.3i 1.22484i
\(866\) 3894.69i 0.152826i
\(867\) −722.680 −0.0283085
\(868\) 2866.26 0.112082
\(869\) − 19678.1i − 0.768162i
\(870\) 8956.24 0.349017
\(871\) 0 0
\(872\) 158782. 6.16631
\(873\) 6282.76i 0.243573i
\(874\) 21196.0 0.820325
\(875\) 20719.5 0.800510
\(876\) − 17920.4i − 0.691179i
\(877\) 44304.5i 1.70588i 0.522008 + 0.852940i \(0.325183\pi\)
−0.522008 + 0.852940i \(0.674817\pi\)
\(878\) 89212.0i 3.42911i
\(879\) 14264.6i 0.547363i
\(880\) −62778.8 −2.40486
\(881\) −10814.6 −0.413567 −0.206783 0.978387i \(-0.566300\pi\)
−0.206783 + 0.978387i \(0.566300\pi\)
\(882\) − 5725.30i − 0.218572i
\(883\) −14530.8 −0.553793 −0.276897 0.960900i \(-0.589306\pi\)
−0.276897 + 0.960900i \(0.589306\pi\)
\(884\) 0 0
\(885\) 8330.63 0.316419
\(886\) 15905.9i 0.603126i
\(887\) −3385.82 −0.128168 −0.0640838 0.997945i \(-0.520412\pi\)
−0.0640838 + 0.997945i \(0.520412\pi\)
\(888\) −66266.8 −2.50424
\(889\) − 2520.24i − 0.0950800i
\(890\) 31211.7i 1.17552i
\(891\) 1543.21i 0.0580241i
\(892\) 70229.9i 2.63618i
\(893\) −35426.6 −1.32755
\(894\) 9023.86 0.337587
\(895\) 60715.4i 2.26759i
\(896\) −117035. −4.36369
\(897\) 0 0
\(898\) 42887.1 1.59372
\(899\) − 247.428i − 0.00917929i
\(900\) 10416.5 0.385797
\(901\) 28588.4 1.05707
\(902\) 35818.2i 1.32219i
\(903\) − 19696.1i − 0.725854i
\(904\) − 55473.7i − 2.04096i
\(905\) 46942.0i 1.72420i
\(906\) −24454.2 −0.896729
\(907\) −38174.2 −1.39752 −0.698762 0.715354i \(-0.746265\pi\)
−0.698762 + 0.715354i \(0.746265\pi\)
\(908\) − 29822.2i − 1.08996i
\(909\) −798.628 −0.0291406
\(910\) 0 0
\(911\) −11699.5 −0.425489 −0.212745 0.977108i \(-0.568240\pi\)
−0.212745 + 0.977108i \(0.568240\pi\)
\(912\) 75807.0i 2.75244i
\(913\) 20307.5 0.736123
\(914\) −79286.6 −2.86933
\(915\) − 21853.2i − 0.789557i
\(916\) 106528.i 3.84254i
\(917\) 5620.97i 0.202422i
\(918\) − 10633.5i − 0.382306i
\(919\) 21615.6 0.775879 0.387939 0.921685i \(-0.373187\pi\)
0.387939 + 0.921685i \(0.373187\pi\)
\(920\) 38957.1 1.39606
\(921\) 4869.93i 0.174234i
\(922\) 15862.1 0.566583
\(923\) 0 0
\(924\) 27054.1 0.963218
\(925\) − 14965.1i − 0.531945i
\(926\) −50600.5 −1.79572
\(927\) 12848.5 0.455232
\(928\) 30183.3i 1.06769i
\(929\) 22325.9i 0.788471i 0.919009 + 0.394236i \(0.128991\pi\)
−0.919009 + 0.394236i \(0.871009\pi\)
\(930\) − 1327.29i − 0.0467995i
\(931\) 11843.6i 0.416926i
\(932\) −53046.1 −1.86436
\(933\) 20050.5 0.703561
\(934\) 43822.3i 1.53523i
\(935\) −18216.5 −0.637158
\(936\) 0 0
\(937\) −29401.5 −1.02509 −0.512543 0.858661i \(-0.671297\pi\)
−0.512543 + 0.858661i \(0.671297\pi\)
\(938\) − 79713.7i − 2.77478i
\(939\) −4753.65 −0.165207
\(940\) −102070. −3.54166
\(941\) 30280.7i 1.04902i 0.851406 + 0.524508i \(0.175751\pi\)
−0.851406 + 0.524508i \(0.824249\pi\)
\(942\) − 35093.4i − 1.21381i
\(943\) − 12964.6i − 0.447704i
\(944\) 51584.1i 1.77852i
\(945\) −7703.86 −0.265192
\(946\) 32029.8 1.10082
\(947\) − 25226.1i − 0.865617i −0.901486 0.432809i \(-0.857523\pi\)
0.901486 0.432809i \(-0.142477\pi\)
\(948\) −68461.7 −2.34550
\(949\) 0 0
\(950\) −29350.2 −1.00237
\(951\) − 2363.80i − 0.0806008i
\(952\) −118918. −4.04848
\(953\) 11893.7 0.404275 0.202137 0.979357i \(-0.435211\pi\)
0.202137 + 0.979357i \(0.435211\pi\)
\(954\) 19661.1i 0.667245i
\(955\) 44527.5i 1.50877i
\(956\) − 41556.4i − 1.40589i
\(957\) − 2335.42i − 0.0788855i
\(958\) 37899.7 1.27817
\(959\) 28223.4 0.950346
\(960\) 82830.5i 2.78473i
\(961\) 29754.3 0.998769
\(962\) 0 0
\(963\) −135.210 −0.00452449
\(964\) − 120084.i − 4.01207i
\(965\) 44501.7 1.48452
\(966\) −13338.0 −0.444247
\(967\) 8534.72i 0.283824i 0.989879 + 0.141912i \(0.0453250\pi\)
−0.989879 + 0.141912i \(0.954675\pi\)
\(968\) − 74847.5i − 2.48522i
\(969\) 21996.9i 0.729247i
\(970\) − 51004.4i − 1.68830i
\(971\) 25615.3 0.846585 0.423292 0.905993i \(-0.360874\pi\)
0.423292 + 0.905993i \(0.360874\pi\)
\(972\) 5368.95 0.177170
\(973\) 137.855i 0.00454206i
\(974\) 73811.8 2.42822
\(975\) 0 0
\(976\) 135317. 4.43791
\(977\) − 22995.5i − 0.753011i −0.926414 0.376506i \(-0.877126\pi\)
0.926414 0.376506i \(-0.122874\pi\)
\(978\) −7910.02 −0.258624
\(979\) 8138.72 0.265694
\(980\) 34123.4i 1.11228i
\(981\) 18482.1i 0.601516i
\(982\) 5727.31i 0.186116i
\(983\) 20592.1i 0.668144i 0.942548 + 0.334072i \(0.108423\pi\)
−0.942548 + 0.334072i \(0.891577\pi\)
\(984\) 79493.9 2.57538
\(985\) −48665.4 −1.57422
\(986\) 16092.2i 0.519756i
\(987\) 22292.9 0.718937
\(988\) 0 0
\(989\) −11593.4 −0.372748
\(990\) − 12528.0i − 0.402189i
\(991\) −11557.9 −0.370485 −0.185242 0.982693i \(-0.559307\pi\)
−0.185242 + 0.982693i \(0.559307\pi\)
\(992\) 4473.08 0.143166
\(993\) 28480.7i 0.910178i
\(994\) 112563.i 3.59183i
\(995\) − 15004.0i − 0.478049i
\(996\) − 70651.5i − 2.24767i
\(997\) −15481.5 −0.491778 −0.245889 0.969298i \(-0.579080\pi\)
−0.245889 + 0.969298i \(0.579080\pi\)
\(998\) −88819.4 −2.81716
\(999\) − 7713.41i − 0.244286i
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.j.337.18 18
13.5 odd 4 507.4.a.q.1.9 yes 9
13.8 odd 4 507.4.a.n.1.1 9
13.12 even 2 inner 507.4.b.j.337.1 18
39.5 even 4 1521.4.a.be.1.1 9
39.8 even 4 1521.4.a.bj.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.n.1.1 9 13.8 odd 4
507.4.a.q.1.9 yes 9 13.5 odd 4
507.4.b.j.337.1 18 13.12 even 2 inner
507.4.b.j.337.18 18 1.1 even 1 trivial
1521.4.a.be.1.1 9 39.5 even 4
1521.4.a.bj.1.9 9 39.8 even 4