Properties

Label 1014.4.b.a.337.2
Level $1014$
Weight $4$
Character 1014.337
Analytic conductor $59.828$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,4,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, 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("1014.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.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: \(59.8279367458\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.4.b.a.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +16.0000i q^{5} -6.00000i q^{6} +28.0000i q^{7} -8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +16.0000i q^{5} -6.00000i q^{6} +28.0000i q^{7} -8.00000i q^{8} +9.00000 q^{9} -32.0000 q^{10} +34.0000i q^{11} +12.0000 q^{12} -56.0000 q^{14} -48.0000i q^{15} +16.0000 q^{16} -138.000 q^{17} +18.0000i q^{18} -108.000i q^{19} -64.0000i q^{20} -84.0000i q^{21} -68.0000 q^{22} +52.0000 q^{23} +24.0000i q^{24} -131.000 q^{25} -27.0000 q^{27} -112.000i q^{28} -190.000 q^{29} +96.0000 q^{30} +176.000i q^{31} +32.0000i q^{32} -102.000i q^{33} -276.000i q^{34} -448.000 q^{35} -36.0000 q^{36} +342.000i q^{37} +216.000 q^{38} +128.000 q^{40} -240.000i q^{41} +168.000 q^{42} +140.000 q^{43} -136.000i q^{44} +144.000i q^{45} +104.000i q^{46} +454.000i q^{47} -48.0000 q^{48} -441.000 q^{49} -262.000i q^{50} +414.000 q^{51} +198.000 q^{53} -54.0000i q^{54} -544.000 q^{55} +224.000 q^{56} +324.000i q^{57} -380.000i q^{58} -154.000i q^{59} +192.000i q^{60} +34.0000 q^{61} -352.000 q^{62} +252.000i q^{63} -64.0000 q^{64} +204.000 q^{66} +656.000i q^{67} +552.000 q^{68} -156.000 q^{69} -896.000i q^{70} -550.000i q^{71} -72.0000i q^{72} +614.000i q^{73} -684.000 q^{74} +393.000 q^{75} +432.000i q^{76} -952.000 q^{77} +8.00000 q^{79} +256.000i q^{80} +81.0000 q^{81} +480.000 q^{82} -762.000i q^{83} +336.000i q^{84} -2208.00i q^{85} +280.000i q^{86} +570.000 q^{87} +272.000 q^{88} -444.000i q^{89} -288.000 q^{90} -208.000 q^{92} -528.000i q^{93} -908.000 q^{94} +1728.00 q^{95} -96.0000i q^{96} -1022.00i q^{97} -882.000i q^{98} +306.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} - 64 q^{10} + 24 q^{12} - 112 q^{14} + 32 q^{16} - 276 q^{17} - 136 q^{22} + 104 q^{23} - 262 q^{25} - 54 q^{27} - 380 q^{29} + 192 q^{30} - 896 q^{35} - 72 q^{36} + 432 q^{38} + 256 q^{40} + 336 q^{42} + 280 q^{43} - 96 q^{48} - 882 q^{49} + 828 q^{51} + 396 q^{53} - 1088 q^{55} + 448 q^{56} + 68 q^{61} - 704 q^{62} - 128 q^{64} + 408 q^{66} + 1104 q^{68} - 312 q^{69} - 1368 q^{74} + 786 q^{75} - 1904 q^{77} + 16 q^{79} + 162 q^{81} + 960 q^{82} + 1140 q^{87} + 544 q^{88} - 576 q^{90} - 416 q^{92} - 1816 q^{94} + 3456 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) −3.00000 −0.577350
\(4\) −4.00000 −0.500000
\(5\) 16.0000i 1.43108i 0.698570 + 0.715542i \(0.253820\pi\)
−0.698570 + 0.715542i \(0.746180\pi\)
\(6\) − 6.00000i − 0.408248i
\(7\) 28.0000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 9.00000 0.333333
\(10\) −32.0000 −1.01193
\(11\) 34.0000i 0.931944i 0.884799 + 0.465972i \(0.154295\pi\)
−0.884799 + 0.465972i \(0.845705\pi\)
\(12\) 12.0000 0.288675
\(13\) 0 0
\(14\) −56.0000 −1.06904
\(15\) − 48.0000i − 0.826236i
\(16\) 16.0000 0.250000
\(17\) −138.000 −1.96882 −0.984409 0.175893i \(-0.943719\pi\)
−0.984409 + 0.175893i \(0.943719\pi\)
\(18\) 18.0000i 0.235702i
\(19\) − 108.000i − 1.30405i −0.758199 0.652024i \(-0.773920\pi\)
0.758199 0.652024i \(-0.226080\pi\)
\(20\) − 64.0000i − 0.715542i
\(21\) − 84.0000i − 0.872872i
\(22\) −68.0000 −0.658984
\(23\) 52.0000 0.471424 0.235712 0.971823i \(-0.424258\pi\)
0.235712 + 0.971823i \(0.424258\pi\)
\(24\) 24.0000i 0.204124i
\(25\) −131.000 −1.04800
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) − 112.000i − 0.755929i
\(29\) −190.000 −1.21662 −0.608312 0.793698i \(-0.708153\pi\)
−0.608312 + 0.793698i \(0.708153\pi\)
\(30\) 96.0000 0.584237
\(31\) 176.000i 1.01969i 0.860265 + 0.509847i \(0.170298\pi\)
−0.860265 + 0.509847i \(0.829702\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 102.000i − 0.538058i
\(34\) − 276.000i − 1.39216i
\(35\) −448.000 −2.16359
\(36\) −36.0000 −0.166667
\(37\) 342.000i 1.51958i 0.650169 + 0.759790i \(0.274698\pi\)
−0.650169 + 0.759790i \(0.725302\pi\)
\(38\) 216.000 0.922101
\(39\) 0 0
\(40\) 128.000 0.505964
\(41\) − 240.000i − 0.914188i −0.889418 0.457094i \(-0.848890\pi\)
0.889418 0.457094i \(-0.151110\pi\)
\(42\) 168.000 0.617213
\(43\) 140.000 0.496507 0.248253 0.968695i \(-0.420143\pi\)
0.248253 + 0.968695i \(0.420143\pi\)
\(44\) − 136.000i − 0.465972i
\(45\) 144.000i 0.477028i
\(46\) 104.000i 0.333347i
\(47\) 454.000i 1.40899i 0.709707 + 0.704497i \(0.248827\pi\)
−0.709707 + 0.704497i \(0.751173\pi\)
\(48\) −48.0000 −0.144338
\(49\) −441.000 −1.28571
\(50\) − 262.000i − 0.741048i
\(51\) 414.000 1.13670
\(52\) 0 0
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) − 54.0000i − 0.136083i
\(55\) −544.000 −1.33369
\(56\) 224.000 0.534522
\(57\) 324.000i 0.752892i
\(58\) − 380.000i − 0.860284i
\(59\) − 154.000i − 0.339815i −0.985460 0.169908i \(-0.945653\pi\)
0.985460 0.169908i \(-0.0543469\pi\)
\(60\) 192.000i 0.413118i
\(61\) 34.0000 0.0713648 0.0356824 0.999363i \(-0.488640\pi\)
0.0356824 + 0.999363i \(0.488640\pi\)
\(62\) −352.000 −0.721033
\(63\) 252.000i 0.503953i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 204.000 0.380465
\(67\) 656.000i 1.19617i 0.801434 + 0.598083i \(0.204071\pi\)
−0.801434 + 0.598083i \(0.795929\pi\)
\(68\) 552.000 0.984409
\(69\) −156.000 −0.272177
\(70\) − 896.000i − 1.52989i
\(71\) − 550.000i − 0.919338i −0.888090 0.459669i \(-0.847968\pi\)
0.888090 0.459669i \(-0.152032\pi\)
\(72\) − 72.0000i − 0.117851i
\(73\) 614.000i 0.984428i 0.870474 + 0.492214i \(0.163812\pi\)
−0.870474 + 0.492214i \(0.836188\pi\)
\(74\) −684.000 −1.07451
\(75\) 393.000 0.605063
\(76\) 432.000i 0.652024i
\(77\) −952.000 −1.40897
\(78\) 0 0
\(79\) 8.00000 0.0113933 0.00569665 0.999984i \(-0.498187\pi\)
0.00569665 + 0.999984i \(0.498187\pi\)
\(80\) 256.000i 0.357771i
\(81\) 81.0000 0.111111
\(82\) 480.000 0.646428
\(83\) − 762.000i − 1.00772i −0.863787 0.503858i \(-0.831914\pi\)
0.863787 0.503858i \(-0.168086\pi\)
\(84\) 336.000i 0.436436i
\(85\) − 2208.00i − 2.81754i
\(86\) 280.000i 0.351083i
\(87\) 570.000 0.702419
\(88\) 272.000 0.329492
\(89\) − 444.000i − 0.528808i −0.964412 0.264404i \(-0.914825\pi\)
0.964412 0.264404i \(-0.0851752\pi\)
\(90\) −288.000 −0.337310
\(91\) 0 0
\(92\) −208.000 −0.235712
\(93\) − 528.000i − 0.588721i
\(94\) −908.000 −0.996309
\(95\) 1728.00 1.86620
\(96\) − 96.0000i − 0.102062i
\(97\) − 1022.00i − 1.06978i −0.844923 0.534889i \(-0.820354\pi\)
0.844923 0.534889i \(-0.179646\pi\)
\(98\) − 882.000i − 0.909137i
\(99\) 306.000i 0.310648i
\(100\) 524.000 0.524000
\(101\) 1190.00 1.17237 0.586185 0.810177i \(-0.300629\pi\)
0.586185 + 0.810177i \(0.300629\pi\)
\(102\) 828.000i 0.803767i
\(103\) 224.000 0.214285 0.107143 0.994244i \(-0.465830\pi\)
0.107143 + 0.994244i \(0.465830\pi\)
\(104\) 0 0
\(105\) 1344.00 1.24915
\(106\) 396.000i 0.362858i
\(107\) −640.000 −0.578235 −0.289117 0.957294i \(-0.593362\pi\)
−0.289117 + 0.957294i \(0.593362\pi\)
\(108\) 108.000 0.0962250
\(109\) 1934.00i 1.69948i 0.527200 + 0.849741i \(0.323242\pi\)
−0.527200 + 0.849741i \(0.676758\pi\)
\(110\) − 1088.00i − 0.943061i
\(111\) − 1026.00i − 0.877330i
\(112\) 448.000i 0.377964i
\(113\) −418.000 −0.347983 −0.173992 0.984747i \(-0.555667\pi\)
−0.173992 + 0.984747i \(0.555667\pi\)
\(114\) −648.000 −0.532375
\(115\) 832.000i 0.674647i
\(116\) 760.000 0.608312
\(117\) 0 0
\(118\) 308.000 0.240286
\(119\) − 3864.00i − 2.97657i
\(120\) −384.000 −0.292119
\(121\) 175.000 0.131480
\(122\) 68.0000i 0.0504625i
\(123\) 720.000i 0.527807i
\(124\) − 704.000i − 0.509847i
\(125\) − 96.0000i − 0.0686920i
\(126\) −504.000 −0.356348
\(127\) 1040.00 0.726654 0.363327 0.931662i \(-0.381641\pi\)
0.363327 + 0.931662i \(0.381641\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −420.000 −0.286658
\(130\) 0 0
\(131\) −568.000 −0.378827 −0.189414 0.981897i \(-0.560659\pi\)
−0.189414 + 0.981897i \(0.560659\pi\)
\(132\) 408.000i 0.269029i
\(133\) 3024.00 1.97153
\(134\) −1312.00 −0.845817
\(135\) − 432.000i − 0.275412i
\(136\) 1104.00i 0.696082i
\(137\) 528.000i 0.329271i 0.986355 + 0.164635i \(0.0526447\pi\)
−0.986355 + 0.164635i \(0.947355\pi\)
\(138\) − 312.000i − 0.192458i
\(139\) −1556.00 −0.949483 −0.474742 0.880125i \(-0.657459\pi\)
−0.474742 + 0.880125i \(0.657459\pi\)
\(140\) 1792.00 1.08180
\(141\) − 1362.00i − 0.813483i
\(142\) 1100.00 0.650070
\(143\) 0 0
\(144\) 144.000 0.0833333
\(145\) − 3040.00i − 1.74109i
\(146\) −1228.00 −0.696096
\(147\) 1323.00 0.742307
\(148\) − 1368.00i − 0.759790i
\(149\) − 1524.00i − 0.837926i −0.908003 0.418963i \(-0.862394\pi\)
0.908003 0.418963i \(-0.137606\pi\)
\(150\) 786.000i 0.427844i
\(151\) − 3024.00i − 1.62973i −0.579649 0.814866i \(-0.696810\pi\)
0.579649 0.814866i \(-0.303190\pi\)
\(152\) −864.000 −0.461050
\(153\) −1242.00 −0.656273
\(154\) − 1904.00i − 0.996290i
\(155\) −2816.00 −1.45927
\(156\) 0 0
\(157\) 2198.00 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 16.0000i 0.00805628i
\(159\) −594.000 −0.296272
\(160\) −512.000 −0.252982
\(161\) 1456.00i 0.712726i
\(162\) 162.000i 0.0785674i
\(163\) − 268.000i − 0.128781i −0.997925 0.0643907i \(-0.979490\pi\)
0.997925 0.0643907i \(-0.0205104\pi\)
\(164\) 960.000i 0.457094i
\(165\) 1632.00 0.770006
\(166\) 1524.00 0.712562
\(167\) 702.000i 0.325284i 0.986685 + 0.162642i \(0.0520015\pi\)
−0.986685 + 0.162642i \(0.947998\pi\)
\(168\) −672.000 −0.308607
\(169\) 0 0
\(170\) 4416.00 1.99230
\(171\) − 972.000i − 0.434682i
\(172\) −560.000 −0.248253
\(173\) −2066.00 −0.907948 −0.453974 0.891015i \(-0.649994\pi\)
−0.453974 + 0.891015i \(0.649994\pi\)
\(174\) 1140.00i 0.496685i
\(175\) − 3668.00i − 1.58443i
\(176\) 544.000i 0.232986i
\(177\) 462.000i 0.196192i
\(178\) 888.000 0.373924
\(179\) 276.000 0.115247 0.0576235 0.998338i \(-0.481648\pi\)
0.0576235 + 0.998338i \(0.481648\pi\)
\(180\) − 576.000i − 0.238514i
\(181\) 3474.00 1.42663 0.713316 0.700843i \(-0.247192\pi\)
0.713316 + 0.700843i \(0.247192\pi\)
\(182\) 0 0
\(183\) −102.000 −0.0412025
\(184\) − 416.000i − 0.166674i
\(185\) −5472.00 −2.17465
\(186\) 1056.00 0.416289
\(187\) − 4692.00i − 1.83483i
\(188\) − 1816.00i − 0.704497i
\(189\) − 756.000i − 0.290957i
\(190\) 3456.00i 1.31960i
\(191\) −3920.00 −1.48503 −0.742516 0.669828i \(-0.766368\pi\)
−0.742516 + 0.669828i \(0.766368\pi\)
\(192\) 192.000 0.0721688
\(193\) 2186.00i 0.815294i 0.913140 + 0.407647i \(0.133651\pi\)
−0.913140 + 0.407647i \(0.866349\pi\)
\(194\) 2044.00 0.756447
\(195\) 0 0
\(196\) 1764.00 0.642857
\(197\) 1368.00i 0.494751i 0.968920 + 0.247376i \(0.0795681\pi\)
−0.968920 + 0.247376i \(0.920432\pi\)
\(198\) −612.000 −0.219661
\(199\) 1072.00 0.381870 0.190935 0.981603i \(-0.438848\pi\)
0.190935 + 0.981603i \(0.438848\pi\)
\(200\) 1048.00i 0.370524i
\(201\) − 1968.00i − 0.690607i
\(202\) 2380.00i 0.828991i
\(203\) − 5320.00i − 1.83936i
\(204\) −1656.00 −0.568349
\(205\) 3840.00 1.30828
\(206\) 448.000i 0.151523i
\(207\) 468.000 0.157141
\(208\) 0 0
\(209\) 3672.00 1.21530
\(210\) 2688.00i 0.883284i
\(211\) 5444.00 1.77621 0.888105 0.459640i \(-0.152022\pi\)
0.888105 + 0.459640i \(0.152022\pi\)
\(212\) −792.000 −0.256579
\(213\) 1650.00i 0.530780i
\(214\) − 1280.00i − 0.408874i
\(215\) 2240.00i 0.710543i
\(216\) 216.000i 0.0680414i
\(217\) −4928.00 −1.54163
\(218\) −3868.00 −1.20172
\(219\) − 1842.00i − 0.568360i
\(220\) 2176.00 0.666845
\(221\) 0 0
\(222\) 2052.00 0.620366
\(223\) − 96.0000i − 0.0288280i −0.999896 0.0144140i \(-0.995412\pi\)
0.999896 0.0144140i \(-0.00458827\pi\)
\(224\) −896.000 −0.267261
\(225\) −1179.00 −0.349333
\(226\) − 836.000i − 0.246061i
\(227\) − 198.000i − 0.0578930i −0.999581 0.0289465i \(-0.990785\pi\)
0.999581 0.0289465i \(-0.00921525\pi\)
\(228\) − 1296.00i − 0.376446i
\(229\) 5922.00i 1.70889i 0.519538 + 0.854447i \(0.326104\pi\)
−0.519538 + 0.854447i \(0.673896\pi\)
\(230\) −1664.00 −0.477047
\(231\) 2856.00 0.813468
\(232\) 1520.00i 0.430142i
\(233\) 5114.00 1.43789 0.718947 0.695065i \(-0.244624\pi\)
0.718947 + 0.695065i \(0.244624\pi\)
\(234\) 0 0
\(235\) −7264.00 −2.01639
\(236\) 616.000i 0.169908i
\(237\) −24.0000 −0.00657792
\(238\) 7728.00 2.10476
\(239\) 5226.00i 1.41440i 0.707013 + 0.707200i \(0.250042\pi\)
−0.707013 + 0.707200i \(0.749958\pi\)
\(240\) − 768.000i − 0.206559i
\(241\) − 762.000i − 0.203671i −0.994801 0.101836i \(-0.967528\pi\)
0.994801 0.101836i \(-0.0324716\pi\)
\(242\) 350.000i 0.0929705i
\(243\) −243.000 −0.0641500
\(244\) −136.000 −0.0356824
\(245\) − 7056.00i − 1.83996i
\(246\) −1440.00 −0.373216
\(247\) 0 0
\(248\) 1408.00 0.360516
\(249\) 2286.00i 0.581805i
\(250\) 192.000 0.0485726
\(251\) −3240.00 −0.814769 −0.407384 0.913257i \(-0.633559\pi\)
−0.407384 + 0.913257i \(0.633559\pi\)
\(252\) − 1008.00i − 0.251976i
\(253\) 1768.00i 0.439341i
\(254\) 2080.00i 0.513822i
\(255\) 6624.00i 1.62671i
\(256\) 256.000 0.0625000
\(257\) 1386.00 0.336406 0.168203 0.985752i \(-0.446204\pi\)
0.168203 + 0.985752i \(0.446204\pi\)
\(258\) − 840.000i − 0.202698i
\(259\) −9576.00 −2.29739
\(260\) 0 0
\(261\) −1710.00 −0.405542
\(262\) − 1136.00i − 0.267871i
\(263\) 3300.00 0.773714 0.386857 0.922140i \(-0.373561\pi\)
0.386857 + 0.922140i \(0.373561\pi\)
\(264\) −816.000 −0.190232
\(265\) 3168.00i 0.734372i
\(266\) 6048.00i 1.39409i
\(267\) 1332.00i 0.305307i
\(268\) − 2624.00i − 0.598083i
\(269\) 4290.00 0.972364 0.486182 0.873858i \(-0.338389\pi\)
0.486182 + 0.873858i \(0.338389\pi\)
\(270\) 864.000 0.194746
\(271\) 2452.00i 0.549625i 0.961498 + 0.274813i \(0.0886158\pi\)
−0.961498 + 0.274813i \(0.911384\pi\)
\(272\) −2208.00 −0.492205
\(273\) 0 0
\(274\) −1056.00 −0.232830
\(275\) − 4454.00i − 0.976677i
\(276\) 624.000 0.136088
\(277\) 42.0000 0.00911024 0.00455512 0.999990i \(-0.498550\pi\)
0.00455512 + 0.999990i \(0.498550\pi\)
\(278\) − 3112.00i − 0.671386i
\(279\) 1584.00i 0.339898i
\(280\) 3584.00i 0.764946i
\(281\) − 2288.00i − 0.485732i −0.970060 0.242866i \(-0.921912\pi\)
0.970060 0.242866i \(-0.0780875\pi\)
\(282\) 2724.00 0.575219
\(283\) −1156.00 −0.242816 −0.121408 0.992603i \(-0.538741\pi\)
−0.121408 + 0.992603i \(0.538741\pi\)
\(284\) 2200.00i 0.459669i
\(285\) −5184.00 −1.07745
\(286\) 0 0
\(287\) 6720.00 1.38212
\(288\) 288.000i 0.0589256i
\(289\) 14131.0 2.87625
\(290\) 6080.00 1.23114
\(291\) 3066.00i 0.617636i
\(292\) − 2456.00i − 0.492214i
\(293\) − 8684.00i − 1.73148i −0.500491 0.865742i \(-0.666847\pi\)
0.500491 0.865742i \(-0.333153\pi\)
\(294\) 2646.00i 0.524891i
\(295\) 2464.00 0.486304
\(296\) 2736.00 0.537253
\(297\) − 918.000i − 0.179353i
\(298\) 3048.00 0.592503
\(299\) 0 0
\(300\) −1572.00 −0.302532
\(301\) 3920.00i 0.750648i
\(302\) 6048.00 1.15240
\(303\) −3570.00 −0.676868
\(304\) − 1728.00i − 0.326012i
\(305\) 544.000i 0.102129i
\(306\) − 2484.00i − 0.464055i
\(307\) − 7552.00i − 1.40396i −0.712197 0.701979i \(-0.752300\pi\)
0.712197 0.701979i \(-0.247700\pi\)
\(308\) 3808.00 0.704484
\(309\) −672.000 −0.123718
\(310\) − 5632.00i − 1.03186i
\(311\) −2652.00 −0.483541 −0.241770 0.970334i \(-0.577728\pi\)
−0.241770 + 0.970334i \(0.577728\pi\)
\(312\) 0 0
\(313\) −4426.00 −0.799273 −0.399636 0.916674i \(-0.630864\pi\)
−0.399636 + 0.916674i \(0.630864\pi\)
\(314\) 4396.00i 0.790066i
\(315\) −4032.00 −0.721198
\(316\) −32.0000 −0.00569665
\(317\) 4944.00i 0.875971i 0.898982 + 0.437985i \(0.144308\pi\)
−0.898982 + 0.437985i \(0.855692\pi\)
\(318\) − 1188.00i − 0.209496i
\(319\) − 6460.00i − 1.13383i
\(320\) − 1024.00i − 0.178885i
\(321\) 1920.00 0.333844
\(322\) −2912.00 −0.503973
\(323\) 14904.0i 2.56743i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 536.000 0.0910623
\(327\) − 5802.00i − 0.981197i
\(328\) −1920.00 −0.323214
\(329\) −12712.0 −2.13020
\(330\) 3264.00i 0.544477i
\(331\) 6088.00i 1.01096i 0.862839 + 0.505478i \(0.168684\pi\)
−0.862839 + 0.505478i \(0.831316\pi\)
\(332\) 3048.00i 0.503858i
\(333\) 3078.00i 0.506527i
\(334\) −1404.00 −0.230010
\(335\) −10496.0 −1.71181
\(336\) − 1344.00i − 0.218218i
\(337\) −6638.00 −1.07298 −0.536491 0.843906i \(-0.680250\pi\)
−0.536491 + 0.843906i \(0.680250\pi\)
\(338\) 0 0
\(339\) 1254.00 0.200908
\(340\) 8832.00i 1.40877i
\(341\) −5984.00 −0.950298
\(342\) 1944.00 0.307367
\(343\) − 2744.00i − 0.431959i
\(344\) − 1120.00i − 0.175542i
\(345\) − 2496.00i − 0.389508i
\(346\) − 4132.00i − 0.642016i
\(347\) −2292.00 −0.354585 −0.177293 0.984158i \(-0.556734\pi\)
−0.177293 + 0.984158i \(0.556734\pi\)
\(348\) −2280.00 −0.351209
\(349\) − 9866.00i − 1.51322i −0.653865 0.756612i \(-0.726853\pi\)
0.653865 0.756612i \(-0.273147\pi\)
\(350\) 7336.00 1.12036
\(351\) 0 0
\(352\) −1088.00 −0.164746
\(353\) − 2368.00i − 0.357042i −0.983936 0.178521i \(-0.942869\pi\)
0.983936 0.178521i \(-0.0571313\pi\)
\(354\) −924.000 −0.138729
\(355\) 8800.00 1.31565
\(356\) 1776.00i 0.264404i
\(357\) 11592.0i 1.71853i
\(358\) 552.000i 0.0814919i
\(359\) 5070.00i 0.745360i 0.927960 + 0.372680i \(0.121561\pi\)
−0.927960 + 0.372680i \(0.878439\pi\)
\(360\) 1152.00 0.168655
\(361\) −4805.00 −0.700539
\(362\) 6948.00i 1.00878i
\(363\) −525.000 −0.0759101
\(364\) 0 0
\(365\) −9824.00 −1.40880
\(366\) − 204.000i − 0.0291346i
\(367\) −8584.00 −1.22093 −0.610465 0.792043i \(-0.709017\pi\)
−0.610465 + 0.792043i \(0.709017\pi\)
\(368\) 832.000 0.117856
\(369\) − 2160.00i − 0.304729i
\(370\) − 10944.0i − 1.53771i
\(371\) 5544.00i 0.775822i
\(372\) 2112.00i 0.294360i
\(373\) 4994.00 0.693243 0.346621 0.938005i \(-0.387329\pi\)
0.346621 + 0.938005i \(0.387329\pi\)
\(374\) 9384.00 1.29742
\(375\) 288.000i 0.0396593i
\(376\) 3632.00 0.498155
\(377\) 0 0
\(378\) 1512.00 0.205738
\(379\) − 1300.00i − 0.176191i −0.996112 0.0880957i \(-0.971922\pi\)
0.996112 0.0880957i \(-0.0280781\pi\)
\(380\) −6912.00 −0.933100
\(381\) −3120.00 −0.419534
\(382\) − 7840.00i − 1.05008i
\(383\) 4590.00i 0.612371i 0.951972 + 0.306185i \(0.0990528\pi\)
−0.951972 + 0.306185i \(0.900947\pi\)
\(384\) 384.000i 0.0510310i
\(385\) − 15232.0i − 2.01635i
\(386\) −4372.00 −0.576500
\(387\) 1260.00 0.165502
\(388\) 4088.00i 0.534889i
\(389\) −3510.00 −0.457491 −0.228746 0.973486i \(-0.573462\pi\)
−0.228746 + 0.973486i \(0.573462\pi\)
\(390\) 0 0
\(391\) −7176.00 −0.928148
\(392\) 3528.00i 0.454569i
\(393\) 1704.00 0.218716
\(394\) −2736.00 −0.349842
\(395\) 128.000i 0.0163048i
\(396\) − 1224.00i − 0.155324i
\(397\) 6230.00i 0.787594i 0.919197 + 0.393797i \(0.128839\pi\)
−0.919197 + 0.393797i \(0.871161\pi\)
\(398\) 2144.00i 0.270023i
\(399\) −9072.00 −1.13827
\(400\) −2096.00 −0.262000
\(401\) − 7500.00i − 0.933995i −0.884259 0.466998i \(-0.845336\pi\)
0.884259 0.466998i \(-0.154664\pi\)
\(402\) 3936.00 0.488333
\(403\) 0 0
\(404\) −4760.00 −0.586185
\(405\) 1296.00i 0.159009i
\(406\) 10640.0 1.30063
\(407\) −11628.0 −1.41616
\(408\) − 3312.00i − 0.401883i
\(409\) − 8254.00i − 0.997883i −0.866636 0.498941i \(-0.833722\pi\)
0.866636 0.498941i \(-0.166278\pi\)
\(410\) 7680.00i 0.925093i
\(411\) − 1584.00i − 0.190105i
\(412\) −896.000 −0.107143
\(413\) 4312.00 0.513752
\(414\) 936.000i 0.111116i
\(415\) 12192.0 1.44212
\(416\) 0 0
\(417\) 4668.00 0.548185
\(418\) 7344.00i 0.859346i
\(419\) −14808.0 −1.72653 −0.863267 0.504747i \(-0.831586\pi\)
−0.863267 + 0.504747i \(0.831586\pi\)
\(420\) −5376.00 −0.624576
\(421\) − 10354.0i − 1.19863i −0.800513 0.599315i \(-0.795440\pi\)
0.800513 0.599315i \(-0.204560\pi\)
\(422\) 10888.0i 1.25597i
\(423\) 4086.00i 0.469665i
\(424\) − 1584.00i − 0.181429i
\(425\) 18078.0 2.06332
\(426\) −3300.00 −0.375318
\(427\) 952.000i 0.107893i
\(428\) 2560.00 0.289117
\(429\) 0 0
\(430\) −4480.00 −0.502430
\(431\) 15486.0i 1.73071i 0.501163 + 0.865353i \(0.332906\pi\)
−0.501163 + 0.865353i \(0.667094\pi\)
\(432\) −432.000 −0.0481125
\(433\) 2018.00 0.223970 0.111985 0.993710i \(-0.464279\pi\)
0.111985 + 0.993710i \(0.464279\pi\)
\(434\) − 9856.00i − 1.09010i
\(435\) 9120.00i 1.00522i
\(436\) − 7736.00i − 0.849741i
\(437\) − 5616.00i − 0.614759i
\(438\) 3684.00 0.401891
\(439\) −8792.00 −0.955853 −0.477926 0.878400i \(-0.658611\pi\)
−0.477926 + 0.878400i \(0.658611\pi\)
\(440\) 4352.00i 0.471531i
\(441\) −3969.00 −0.428571
\(442\) 0 0
\(443\) −2760.00 −0.296008 −0.148004 0.988987i \(-0.547285\pi\)
−0.148004 + 0.988987i \(0.547285\pi\)
\(444\) 4104.00i 0.438665i
\(445\) 7104.00 0.756768
\(446\) 192.000 0.0203844
\(447\) 4572.00i 0.483777i
\(448\) − 1792.00i − 0.188982i
\(449\) 9532.00i 1.00188i 0.865483 + 0.500939i \(0.167012\pi\)
−0.865483 + 0.500939i \(0.832988\pi\)
\(450\) − 2358.00i − 0.247016i
\(451\) 8160.00 0.851972
\(452\) 1672.00 0.173992
\(453\) 9072.00i 0.940927i
\(454\) 396.000 0.0409366
\(455\) 0 0
\(456\) 2592.00 0.266188
\(457\) − 12862.0i − 1.31654i −0.752782 0.658270i \(-0.771288\pi\)
0.752782 0.658270i \(-0.228712\pi\)
\(458\) −11844.0 −1.20837
\(459\) 3726.00 0.378899
\(460\) − 3328.00i − 0.337323i
\(461\) 6744.00i 0.681344i 0.940182 + 0.340672i \(0.110654\pi\)
−0.940182 + 0.340672i \(0.889346\pi\)
\(462\) 5712.00i 0.575208i
\(463\) − 9572.00i − 0.960796i −0.877051 0.480398i \(-0.840492\pi\)
0.877051 0.480398i \(-0.159508\pi\)
\(464\) −3040.00 −0.304156
\(465\) 8448.00 0.842509
\(466\) 10228.0i 1.01674i
\(467\) −9104.00 −0.902105 −0.451052 0.892498i \(-0.648951\pi\)
−0.451052 + 0.892498i \(0.648951\pi\)
\(468\) 0 0
\(469\) −18368.0 −1.80843
\(470\) − 14528.0i − 1.42580i
\(471\) −6594.00 −0.645086
\(472\) −1232.00 −0.120143
\(473\) 4760.00i 0.462717i
\(474\) − 48.0000i − 0.00465129i
\(475\) 14148.0i 1.36664i
\(476\) 15456.0i 1.48829i
\(477\) 1782.00 0.171053
\(478\) −10452.0 −1.00013
\(479\) − 18870.0i − 1.79998i −0.435906 0.899992i \(-0.643572\pi\)
0.435906 0.899992i \(-0.356428\pi\)
\(480\) 1536.00 0.146059
\(481\) 0 0
\(482\) 1524.00 0.144017
\(483\) − 4368.00i − 0.411493i
\(484\) −700.000 −0.0657400
\(485\) 16352.0 1.53094
\(486\) − 486.000i − 0.0453609i
\(487\) 1744.00i 0.162276i 0.996703 + 0.0811378i \(0.0258554\pi\)
−0.996703 + 0.0811378i \(0.974145\pi\)
\(488\) − 272.000i − 0.0252313i
\(489\) 804.000i 0.0743520i
\(490\) 14112.0 1.30105
\(491\) −13360.0 −1.22796 −0.613980 0.789322i \(-0.710432\pi\)
−0.613980 + 0.789322i \(0.710432\pi\)
\(492\) − 2880.00i − 0.263903i
\(493\) 26220.0 2.39531
\(494\) 0 0
\(495\) −4896.00 −0.444563
\(496\) 2816.00i 0.254924i
\(497\) 15400.0 1.38991
\(498\) −4572.00 −0.411398
\(499\) − 17368.0i − 1.55811i −0.626954 0.779057i \(-0.715698\pi\)
0.626954 0.779057i \(-0.284302\pi\)
\(500\) 384.000i 0.0343460i
\(501\) − 2106.00i − 0.187803i
\(502\) − 6480.00i − 0.576129i
\(503\) −5828.00 −0.516616 −0.258308 0.966063i \(-0.583165\pi\)
−0.258308 + 0.966063i \(0.583165\pi\)
\(504\) 2016.00 0.178174
\(505\) 19040.0i 1.67776i
\(506\) −3536.00 −0.310661
\(507\) 0 0
\(508\) −4160.00 −0.363327
\(509\) − 10744.0i − 0.935598i −0.883835 0.467799i \(-0.845047\pi\)
0.883835 0.467799i \(-0.154953\pi\)
\(510\) −13248.0 −1.15026
\(511\) −17192.0 −1.48832
\(512\) 512.000i 0.0441942i
\(513\) 2916.00i 0.250964i
\(514\) 2772.00i 0.237875i
\(515\) 3584.00i 0.306660i
\(516\) 1680.00 0.143329
\(517\) −15436.0 −1.31310
\(518\) − 19152.0i − 1.62450i
\(519\) 6198.00 0.524204
\(520\) 0 0
\(521\) −12234.0 −1.02875 −0.514377 0.857564i \(-0.671977\pi\)
−0.514377 + 0.857564i \(0.671977\pi\)
\(522\) − 3420.00i − 0.286761i
\(523\) 1812.00 0.151498 0.0757488 0.997127i \(-0.475865\pi\)
0.0757488 + 0.997127i \(0.475865\pi\)
\(524\) 2272.00 0.189414
\(525\) 11004.0i 0.914769i
\(526\) 6600.00i 0.547098i
\(527\) − 24288.0i − 2.00759i
\(528\) − 1632.00i − 0.134515i
\(529\) −9463.00 −0.777760
\(530\) −6336.00 −0.519280
\(531\) − 1386.00i − 0.113272i
\(532\) −12096.0 −0.985767
\(533\) 0 0
\(534\) −2664.00 −0.215885
\(535\) − 10240.0i − 0.827502i
\(536\) 5248.00 0.422909
\(537\) −828.000 −0.0665379
\(538\) 8580.00i 0.687565i
\(539\) − 14994.0i − 1.19821i
\(540\) 1728.00i 0.137706i
\(541\) 6098.00i 0.484609i 0.970200 + 0.242305i \(0.0779033\pi\)
−0.970200 + 0.242305i \(0.922097\pi\)
\(542\) −4904.00 −0.388644
\(543\) −10422.0 −0.823666
\(544\) − 4416.00i − 0.348041i
\(545\) −30944.0 −2.43210
\(546\) 0 0
\(547\) −18332.0 −1.43294 −0.716471 0.697616i \(-0.754244\pi\)
−0.716471 + 0.697616i \(0.754244\pi\)
\(548\) − 2112.00i − 0.164635i
\(549\) 306.000 0.0237883
\(550\) 8908.00 0.690615
\(551\) 20520.0i 1.58654i
\(552\) 1248.00i 0.0962290i
\(553\) 224.000i 0.0172250i
\(554\) 84.0000i 0.00644191i
\(555\) 16416.0 1.25553
\(556\) 6224.00 0.474742
\(557\) − 20004.0i − 1.52172i −0.648917 0.760859i \(-0.724778\pi\)
0.648917 0.760859i \(-0.275222\pi\)
\(558\) −3168.00 −0.240344
\(559\) 0 0
\(560\) −7168.00 −0.540899
\(561\) 14076.0i 1.05934i
\(562\) 4576.00 0.343464
\(563\) −10988.0 −0.822538 −0.411269 0.911514i \(-0.634914\pi\)
−0.411269 + 0.911514i \(0.634914\pi\)
\(564\) 5448.00i 0.406741i
\(565\) − 6688.00i − 0.497993i
\(566\) − 2312.00i − 0.171697i
\(567\) 2268.00i 0.167984i
\(568\) −4400.00 −0.325035
\(569\) −11062.0 −0.815014 −0.407507 0.913202i \(-0.633602\pi\)
−0.407507 + 0.913202i \(0.633602\pi\)
\(570\) − 10368.0i − 0.761873i
\(571\) 708.000 0.0518895 0.0259447 0.999663i \(-0.491741\pi\)
0.0259447 + 0.999663i \(0.491741\pi\)
\(572\) 0 0
\(573\) 11760.0 0.857384
\(574\) 13440.0i 0.977308i
\(575\) −6812.00 −0.494052
\(576\) −576.000 −0.0416667
\(577\) 2094.00i 0.151082i 0.997143 + 0.0755410i \(0.0240684\pi\)
−0.997143 + 0.0755410i \(0.975932\pi\)
\(578\) 28262.0i 2.03381i
\(579\) − 6558.00i − 0.470710i
\(580\) 12160.0i 0.870546i
\(581\) 21336.0 1.52352
\(582\) −6132.00 −0.436735
\(583\) 6732.00i 0.478235i
\(584\) 4912.00 0.348048
\(585\) 0 0
\(586\) 17368.0 1.22434
\(587\) 17854.0i 1.25539i 0.778460 + 0.627695i \(0.216001\pi\)
−0.778460 + 0.627695i \(0.783999\pi\)
\(588\) −5292.00 −0.371154
\(589\) 19008.0 1.32973
\(590\) 4928.00i 0.343869i
\(591\) − 4104.00i − 0.285645i
\(592\) 5472.00i 0.379895i
\(593\) 23948.0i 1.65839i 0.558958 + 0.829196i \(0.311201\pi\)
−0.558958 + 0.829196i \(0.688799\pi\)
\(594\) 1836.00 0.126822
\(595\) 61824.0 4.25973
\(596\) 6096.00i 0.418963i
\(597\) −3216.00 −0.220473
\(598\) 0 0
\(599\) −18068.0 −1.23245 −0.616226 0.787570i \(-0.711339\pi\)
−0.616226 + 0.787570i \(0.711339\pi\)
\(600\) − 3144.00i − 0.213922i
\(601\) 19942.0 1.35350 0.676748 0.736215i \(-0.263389\pi\)
0.676748 + 0.736215i \(0.263389\pi\)
\(602\) −7840.00 −0.530788
\(603\) 5904.00i 0.398722i
\(604\) 12096.0i 0.814866i
\(605\) 2800.00i 0.188159i
\(606\) − 7140.00i − 0.478618i
\(607\) 26376.0 1.76370 0.881852 0.471526i \(-0.156296\pi\)
0.881852 + 0.471526i \(0.156296\pi\)
\(608\) 3456.00 0.230525
\(609\) 15960.0i 1.06196i
\(610\) −1088.00 −0.0722161
\(611\) 0 0
\(612\) 4968.00 0.328136
\(613\) 19426.0i 1.27995i 0.768396 + 0.639975i \(0.221055\pi\)
−0.768396 + 0.639975i \(0.778945\pi\)
\(614\) 15104.0 0.992749
\(615\) −11520.0 −0.755335
\(616\) 7616.00i 0.498145i
\(617\) 8024.00i 0.523556i 0.965128 + 0.261778i \(0.0843088\pi\)
−0.965128 + 0.261778i \(0.915691\pi\)
\(618\) − 1344.00i − 0.0874816i
\(619\) − 20648.0i − 1.34073i −0.742031 0.670366i \(-0.766137\pi\)
0.742031 0.670366i \(-0.233863\pi\)
\(620\) 11264.0 0.729634
\(621\) −1404.00 −0.0907256
\(622\) − 5304.00i − 0.341915i
\(623\) 12432.0 0.799482
\(624\) 0 0
\(625\) −14839.0 −0.949696
\(626\) − 8852.00i − 0.565171i
\(627\) −11016.0 −0.701653
\(628\) −8792.00 −0.558661
\(629\) − 47196.0i − 2.99178i
\(630\) − 8064.00i − 0.509964i
\(631\) 12280.0i 0.774737i 0.921925 + 0.387369i \(0.126616\pi\)
−0.921925 + 0.387369i \(0.873384\pi\)
\(632\) − 64.0000i − 0.00402814i
\(633\) −16332.0 −1.02550
\(634\) −9888.00 −0.619405
\(635\) 16640.0i 1.03990i
\(636\) 2376.00 0.148136
\(637\) 0 0
\(638\) 12920.0 0.801736
\(639\) − 4950.00i − 0.306446i
\(640\) 2048.00 0.126491
\(641\) 15878.0 0.978383 0.489191 0.872176i \(-0.337292\pi\)
0.489191 + 0.872176i \(0.337292\pi\)
\(642\) 3840.00i 0.236063i
\(643\) 21520.0i 1.31985i 0.751330 + 0.659927i \(0.229413\pi\)
−0.751330 + 0.659927i \(0.770587\pi\)
\(644\) − 5824.00i − 0.356363i
\(645\) − 6720.00i − 0.410232i
\(646\) −29808.0 −1.81545
\(647\) −7312.00 −0.444304 −0.222152 0.975012i \(-0.571308\pi\)
−0.222152 + 0.975012i \(0.571308\pi\)
\(648\) − 648.000i − 0.0392837i
\(649\) 5236.00 0.316689
\(650\) 0 0
\(651\) 14784.0 0.890062
\(652\) 1072.00i 0.0643907i
\(653\) 3090.00 0.185178 0.0925889 0.995704i \(-0.470486\pi\)
0.0925889 + 0.995704i \(0.470486\pi\)
\(654\) 11604.0 0.693811
\(655\) − 9088.00i − 0.542134i
\(656\) − 3840.00i − 0.228547i
\(657\) 5526.00i 0.328143i
\(658\) − 25424.0i − 1.50628i
\(659\) −13428.0 −0.793749 −0.396875 0.917873i \(-0.629905\pi\)
−0.396875 + 0.917873i \(0.629905\pi\)
\(660\) −6528.00 −0.385003
\(661\) 22598.0i 1.32974i 0.746958 + 0.664872i \(0.231514\pi\)
−0.746958 + 0.664872i \(0.768486\pi\)
\(662\) −12176.0 −0.714854
\(663\) 0 0
\(664\) −6096.00 −0.356281
\(665\) 48384.0i 2.82143i
\(666\) −6156.00 −0.358168
\(667\) −9880.00 −0.573546
\(668\) − 2808.00i − 0.162642i
\(669\) 288.000i 0.0166438i
\(670\) − 20992.0i − 1.21044i
\(671\) 1156.00i 0.0665080i
\(672\) 2688.00 0.154303
\(673\) −6178.00 −0.353855 −0.176927 0.984224i \(-0.556616\pi\)
−0.176927 + 0.984224i \(0.556616\pi\)
\(674\) − 13276.0i − 0.758713i
\(675\) 3537.00 0.201688
\(676\) 0 0
\(677\) 22398.0 1.27153 0.635764 0.771883i \(-0.280685\pi\)
0.635764 + 0.771883i \(0.280685\pi\)
\(678\) 2508.00i 0.142064i
\(679\) 28616.0 1.61735
\(680\) −17664.0 −0.996152
\(681\) 594.000i 0.0334246i
\(682\) − 11968.0i − 0.671962i
\(683\) 11410.0i 0.639226i 0.947548 + 0.319613i \(0.103553\pi\)
−0.947548 + 0.319613i \(0.896447\pi\)
\(684\) 3888.00i 0.217341i
\(685\) −8448.00 −0.471214
\(686\) 5488.00 0.305441
\(687\) − 17766.0i − 0.986631i
\(688\) 2240.00 0.124127
\(689\) 0 0
\(690\) 4992.00 0.275423
\(691\) − 32488.0i − 1.78857i −0.447498 0.894285i \(-0.647685\pi\)
0.447498 0.894285i \(-0.352315\pi\)
\(692\) 8264.00 0.453974
\(693\) −8568.00 −0.469656
\(694\) − 4584.00i − 0.250729i
\(695\) − 24896.0i − 1.35879i
\(696\) − 4560.00i − 0.248342i
\(697\) 33120.0i 1.79987i
\(698\) 19732.0 1.07001
\(699\) −15342.0 −0.830168
\(700\) 14672.0i 0.792214i
\(701\) −5094.00 −0.274462 −0.137231 0.990539i \(-0.543820\pi\)
−0.137231 + 0.990539i \(0.543820\pi\)
\(702\) 0 0
\(703\) 36936.0 1.98160
\(704\) − 2176.00i − 0.116493i
\(705\) 21792.0 1.16416
\(706\) 4736.00 0.252467
\(707\) 33320.0i 1.77246i
\(708\) − 1848.00i − 0.0980962i
\(709\) 25418.0i 1.34639i 0.739463 + 0.673197i \(0.235079\pi\)
−0.739463 + 0.673197i \(0.764921\pi\)
\(710\) 17600.0i 0.930305i
\(711\) 72.0000 0.00379777
\(712\) −3552.00 −0.186962
\(713\) 9152.00i 0.480708i
\(714\) −23184.0 −1.21518
\(715\) 0 0
\(716\) −1104.00 −0.0576235
\(717\) − 15678.0i − 0.816605i
\(718\) −10140.0 −0.527049
\(719\) 20428.0 1.05958 0.529788 0.848130i \(-0.322271\pi\)
0.529788 + 0.848130i \(0.322271\pi\)
\(720\) 2304.00i 0.119257i
\(721\) 6272.00i 0.323969i
\(722\) − 9610.00i − 0.495356i
\(723\) 2286.00i 0.117590i
\(724\) −13896.0 −0.713316
\(725\) 24890.0 1.27502
\(726\) − 1050.00i − 0.0536765i
\(727\) 38336.0 1.95571 0.977857 0.209276i \(-0.0671107\pi\)
0.977857 + 0.209276i \(0.0671107\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 19648.0i − 0.996171i
\(731\) −19320.0 −0.977532
\(732\) 408.000 0.0206012
\(733\) 166.000i 0.00836473i 0.999991 + 0.00418237i \(0.00133129\pi\)
−0.999991 + 0.00418237i \(0.998669\pi\)
\(734\) − 17168.0i − 0.863328i
\(735\) 21168.0i 1.06230i
\(736\) 1664.00i 0.0833368i
\(737\) −22304.0 −1.11476
\(738\) 4320.00 0.215476
\(739\) − 25248.0i − 1.25678i −0.777897 0.628392i \(-0.783714\pi\)
0.777897 0.628392i \(-0.216286\pi\)
\(740\) 21888.0 1.08732
\(741\) 0 0
\(742\) −11088.0 −0.548589
\(743\) 4442.00i 0.219329i 0.993969 + 0.109664i \(0.0349776\pi\)
−0.993969 + 0.109664i \(0.965022\pi\)
\(744\) −4224.00 −0.208144
\(745\) 24384.0 1.19914
\(746\) 9988.00i 0.490197i
\(747\) − 6858.00i − 0.335905i
\(748\) 18768.0i 0.917414i
\(749\) − 17920.0i − 0.874209i
\(750\) −576.000 −0.0280434
\(751\) 19848.0 0.964399 0.482200 0.876061i \(-0.339838\pi\)
0.482200 + 0.876061i \(0.339838\pi\)
\(752\) 7264.00i 0.352248i
\(753\) 9720.00 0.470407
\(754\) 0 0
\(755\) 48384.0 2.33228
\(756\) 3024.00i 0.145479i
\(757\) −29166.0 −1.40034 −0.700169 0.713977i \(-0.746892\pi\)
−0.700169 + 0.713977i \(0.746892\pi\)
\(758\) 2600.00 0.124586
\(759\) − 5304.00i − 0.253653i
\(760\) − 13824.0i − 0.659802i
\(761\) − 6240.00i − 0.297240i −0.988894 0.148620i \(-0.952517\pi\)
0.988894 0.148620i \(-0.0474832\pi\)
\(762\) − 6240.00i − 0.296655i
\(763\) −54152.0 −2.56938
\(764\) 15680.0 0.742516
\(765\) − 19872.0i − 0.939181i
\(766\) −9180.00 −0.433012
\(767\) 0 0
\(768\) −768.000 −0.0360844
\(769\) 39750.0i 1.86401i 0.362449 + 0.932004i \(0.381941\pi\)
−0.362449 + 0.932004i \(0.618059\pi\)
\(770\) 30464.0 1.42577
\(771\) −4158.00 −0.194224
\(772\) − 8744.00i − 0.407647i
\(773\) − 9764.00i − 0.454317i −0.973858 0.227158i \(-0.927057\pi\)
0.973858 0.227158i \(-0.0729435\pi\)
\(774\) 2520.00i 0.117028i
\(775\) − 23056.0i − 1.06864i
\(776\) −8176.00 −0.378223
\(777\) 28728.0 1.32640
\(778\) − 7020.00i − 0.323495i
\(779\) −25920.0 −1.19214
\(780\) 0 0
\(781\) 18700.0 0.856772
\(782\) − 14352.0i − 0.656300i
\(783\) 5130.00 0.234140
\(784\) −7056.00 −0.321429
\(785\) 35168.0i 1.59898i
\(786\) 3408.00i 0.154656i
\(787\) − 36016.0i − 1.63130i −0.578547 0.815649i \(-0.696380\pi\)
0.578547 0.815649i \(-0.303620\pi\)
\(788\) − 5472.00i − 0.247376i
\(789\) −9900.00 −0.446704
\(790\) −256.000 −0.0115292
\(791\) − 11704.0i − 0.526102i
\(792\) 2448.00 0.109831
\(793\) 0 0
\(794\) −12460.0 −0.556913
\(795\) − 9504.00i − 0.423990i
\(796\) −4288.00 −0.190935
\(797\) 22290.0 0.990655 0.495328 0.868706i \(-0.335048\pi\)
0.495328 + 0.868706i \(0.335048\pi\)
\(798\) − 18144.0i − 0.804875i
\(799\) − 62652.0i − 2.77405i
\(800\) − 4192.00i − 0.185262i
\(801\) − 3996.00i − 0.176269i
\(802\) 15000.0 0.660434
\(803\) −20876.0 −0.917432
\(804\) 7872.00i 0.345304i
\(805\) −23296.0 −1.01997
\(806\) 0 0
\(807\) −12870.0 −0.561395
\(808\) − 9520.00i − 0.414496i
\(809\) −25578.0 −1.11159 −0.555794 0.831320i \(-0.687586\pi\)
−0.555794 + 0.831320i \(0.687586\pi\)
\(810\) −2592.00 −0.112437
\(811\) − 29900.0i − 1.29461i −0.762230 0.647306i \(-0.775895\pi\)
0.762230 0.647306i \(-0.224105\pi\)
\(812\) 21280.0i 0.919682i
\(813\) − 7356.00i − 0.317326i
\(814\) − 23256.0i − 1.00138i
\(815\) 4288.00 0.184297
\(816\) 6624.00 0.284174
\(817\) − 15120.0i − 0.647469i
\(818\) 16508.0 0.705610
\(819\) 0 0
\(820\) −15360.0 −0.654140
\(821\) − 16412.0i − 0.697665i −0.937185 0.348832i \(-0.886578\pi\)
0.937185 0.348832i \(-0.113422\pi\)
\(822\) 3168.00 0.134424
\(823\) −18552.0 −0.785762 −0.392881 0.919589i \(-0.628522\pi\)
−0.392881 + 0.919589i \(0.628522\pi\)
\(824\) − 1792.00i − 0.0757613i
\(825\) 13362.0i 0.563885i
\(826\) 8624.00i 0.363278i
\(827\) − 28662.0i − 1.20517i −0.798055 0.602585i \(-0.794137\pi\)
0.798055 0.602585i \(-0.205863\pi\)
\(828\) −1872.00 −0.0785706
\(829\) 3686.00 0.154427 0.0772136 0.997015i \(-0.475398\pi\)
0.0772136 + 0.997015i \(0.475398\pi\)
\(830\) 24384.0i 1.01974i
\(831\) −126.000 −0.00525980
\(832\) 0 0
\(833\) 60858.0 2.53134
\(834\) 9336.00i 0.387625i
\(835\) −11232.0 −0.465508
\(836\) −14688.0 −0.607650
\(837\) − 4752.00i − 0.196240i
\(838\) − 29616.0i − 1.22084i
\(839\) 13370.0i 0.550159i 0.961421 + 0.275080i \(0.0887042\pi\)
−0.961421 + 0.275080i \(0.911296\pi\)
\(840\) − 10752.0i − 0.441642i
\(841\) 11711.0 0.480175
\(842\) 20708.0 0.847559
\(843\) 6864.00i 0.280437i
\(844\) −21776.0 −0.888105
\(845\) 0 0
\(846\) −8172.00 −0.332103
\(847\) 4900.00i 0.198779i
\(848\) 3168.00 0.128290
\(849\) 3468.00 0.140190
\(850\) 36156.0i 1.45899i
\(851\) 17784.0i 0.716366i
\(852\) − 6600.00i − 0.265390i
\(853\) 11398.0i 0.457515i 0.973483 + 0.228757i \(0.0734662\pi\)
−0.973483 + 0.228757i \(0.926534\pi\)
\(854\) −1904.00 −0.0762922
\(855\) 15552.0 0.622067
\(856\) 5120.00i 0.204437i
\(857\) −7990.00 −0.318475 −0.159238 0.987240i \(-0.550904\pi\)
−0.159238 + 0.987240i \(0.550904\pi\)
\(858\) 0 0
\(859\) 7652.00 0.303938 0.151969 0.988385i \(-0.451439\pi\)
0.151969 + 0.988385i \(0.451439\pi\)
\(860\) − 8960.00i − 0.355271i
\(861\) −20160.0 −0.797969
\(862\) −30972.0 −1.22379
\(863\) 1022.00i 0.0403120i 0.999797 + 0.0201560i \(0.00641629\pi\)
−0.999797 + 0.0201560i \(0.993584\pi\)
\(864\) − 864.000i − 0.0340207i
\(865\) − 33056.0i − 1.29935i
\(866\) 4036.00i 0.158371i
\(867\) −42393.0 −1.66060
\(868\) 19712.0 0.770817
\(869\) 272.000i 0.0106179i
\(870\) −18240.0 −0.710798
\(871\) 0 0
\(872\) 15472.0 0.600858
\(873\) − 9198.00i − 0.356592i
\(874\) 11232.0 0.434700
\(875\) 2688.00 0.103853
\(876\) 7368.00i 0.284180i
\(877\) 15546.0i 0.598576i 0.954163 + 0.299288i \(0.0967491\pi\)
−0.954163 + 0.299288i \(0.903251\pi\)
\(878\) − 17584.0i − 0.675890i
\(879\) 26052.0i 0.999673i
\(880\) −8704.00 −0.333422
\(881\) −11310.0 −0.432513 −0.216256 0.976337i \(-0.569385\pi\)
−0.216256 + 0.976337i \(0.569385\pi\)
\(882\) − 7938.00i − 0.303046i
\(883\) −17260.0 −0.657809 −0.328904 0.944363i \(-0.606679\pi\)
−0.328904 + 0.944363i \(0.606679\pi\)
\(884\) 0 0
\(885\) −7392.00 −0.280768
\(886\) − 5520.00i − 0.209309i
\(887\) 832.000 0.0314947 0.0157474 0.999876i \(-0.494987\pi\)
0.0157474 + 0.999876i \(0.494987\pi\)
\(888\) −8208.00 −0.310183
\(889\) 29120.0i 1.09860i
\(890\) 14208.0i 0.535116i
\(891\) 2754.00i 0.103549i
\(892\) 384.000i 0.0144140i
\(893\) 49032.0 1.83739
\(894\) −9144.00 −0.342082
\(895\) 4416.00i 0.164928i
\(896\) 3584.00 0.133631
\(897\) 0 0
\(898\) −19064.0 −0.708434
\(899\) − 33440.0i − 1.24059i
\(900\) 4716.00 0.174667
\(901\) −27324.0 −1.01032
\(902\) 16320.0i 0.602435i
\(903\) − 11760.0i − 0.433387i
\(904\) 3344.00i 0.123031i
\(905\) 55584.0i 2.04163i
\(906\) −18144.0 −0.665336
\(907\) −31740.0 −1.16197 −0.580986 0.813913i \(-0.697333\pi\)
−0.580986 + 0.813913i \(0.697333\pi\)
\(908\) 792.000i 0.0289465i
\(909\) 10710.0 0.390790
\(910\) 0 0
\(911\) −23568.0 −0.857127 −0.428563 0.903512i \(-0.640980\pi\)
−0.428563 + 0.903512i \(0.640980\pi\)
\(912\) 5184.00i 0.188223i
\(913\) 25908.0 0.939134
\(914\) 25724.0 0.930935
\(915\) − 1632.00i − 0.0589642i
\(916\) − 23688.0i − 0.854447i
\(917\) − 15904.0i − 0.572733i
\(918\) 7452.00i 0.267922i
\(919\) −18864.0 −0.677112 −0.338556 0.940946i \(-0.609938\pi\)
−0.338556 + 0.940946i \(0.609938\pi\)
\(920\) 6656.00 0.238524
\(921\) 22656.0i 0.810576i
\(922\) −13488.0 −0.481783
\(923\) 0 0
\(924\) −11424.0 −0.406734
\(925\) − 44802.0i − 1.59252i
\(926\) 19144.0 0.679385
\(927\) 2016.00 0.0714284
\(928\) − 6080.00i − 0.215071i
\(929\) 19536.0i 0.689941i 0.938613 + 0.344971i \(0.112111\pi\)
−0.938613 + 0.344971i \(0.887889\pi\)
\(930\) 16896.0i 0.595744i
\(931\) 47628.0i 1.67663i
\(932\) −20456.0 −0.718947
\(933\) 7956.00 0.279172
\(934\) − 18208.0i − 0.637884i
\(935\) 75072.0 2.62579
\(936\) 0 0
\(937\) 18174.0 0.633638 0.316819 0.948486i \(-0.397385\pi\)
0.316819 + 0.948486i \(0.397385\pi\)
\(938\) − 36736.0i − 1.27876i
\(939\) 13278.0 0.461460
\(940\) 29056.0 1.00819
\(941\) − 51172.0i − 1.77275i −0.462966 0.886376i \(-0.653215\pi\)
0.462966 0.886376i \(-0.346785\pi\)
\(942\) − 13188.0i − 0.456145i
\(943\) − 12480.0i − 0.430970i
\(944\) − 2464.00i − 0.0849538i
\(945\) 12096.0 0.416384
\(946\) −9520.00 −0.327190
\(947\) 3726.00i 0.127855i 0.997955 + 0.0639275i \(0.0203626\pi\)
−0.997955 + 0.0639275i \(0.979637\pi\)
\(948\) 96.0000 0.00328896
\(949\) 0 0
\(950\) −28296.0 −0.966362
\(951\) − 14832.0i − 0.505742i
\(952\) −30912.0 −1.05238
\(953\) 40498.0 1.37656 0.688279 0.725447i \(-0.258367\pi\)
0.688279 + 0.725447i \(0.258367\pi\)
\(954\) 3564.00i 0.120953i
\(955\) − 62720.0i − 2.12521i
\(956\) − 20904.0i − 0.707200i
\(957\) 19380.0i 0.654615i
\(958\) 37740.0 1.27278
\(959\) −14784.0 −0.497810
\(960\) 3072.00i 0.103280i
\(961\) −1185.00 −0.0397771
\(962\) 0 0
\(963\) −5760.00 −0.192745
\(964\) 3048.00i 0.101836i
\(965\) −34976.0 −1.16675
\(966\) 8736.00 0.290969
\(967\) − 28568.0i − 0.950036i −0.879976 0.475018i \(-0.842442\pi\)
0.879976 0.475018i \(-0.157558\pi\)
\(968\) − 1400.00i − 0.0464852i
\(969\) − 44712.0i − 1.48231i
\(970\) 32704.0i 1.08254i
\(971\) −8676.00 −0.286742 −0.143371 0.989669i \(-0.545794\pi\)
−0.143371 + 0.989669i \(0.545794\pi\)
\(972\) 972.000 0.0320750
\(973\) − 43568.0i − 1.43548i
\(974\) −3488.00 −0.114746
\(975\) 0 0
\(976\) 544.000 0.0178412
\(977\) 2796.00i 0.0915578i 0.998952 + 0.0457789i \(0.0145770\pi\)
−0.998952 + 0.0457789i \(0.985423\pi\)
\(978\) −1608.00 −0.0525748
\(979\) 15096.0 0.492819
\(980\) 28224.0i 0.919982i
\(981\) 17406.0i 0.566494i
\(982\) − 26720.0i − 0.868299i
\(983\) − 406.000i − 0.0131733i −0.999978 0.00658667i \(-0.997903\pi\)
0.999978 0.00658667i \(-0.00209662\pi\)
\(984\) 5760.00 0.186608
\(985\) −21888.0 −0.708030
\(986\) 52440.0i 1.69374i
\(987\) 38136.0 1.22987
\(988\) 0 0
\(989\) 7280.00 0.234065
\(990\) − 9792.00i − 0.314354i
\(991\) −23232.0 −0.744691 −0.372346 0.928094i \(-0.621446\pi\)
−0.372346 + 0.928094i \(0.621446\pi\)
\(992\) −5632.00 −0.180258
\(993\) − 18264.0i − 0.583676i
\(994\) 30800.0i 0.982814i
\(995\) 17152.0i 0.546487i
\(996\) − 9144.00i − 0.290902i
\(997\) 6110.00 0.194088 0.0970440 0.995280i \(-0.469061\pi\)
0.0970440 + 0.995280i \(0.469061\pi\)
\(998\) 34736.0 1.10175
\(999\) − 9234.00i − 0.292443i
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 1014.4.b.a.337.2 2
13.5 odd 4 1014.4.a.i.1.1 1
13.8 odd 4 78.4.a.a.1.1 1
13.12 even 2 inner 1014.4.b.a.337.1 2
39.8 even 4 234.4.a.k.1.1 1
52.47 even 4 624.4.a.f.1.1 1
65.34 odd 4 1950.4.a.o.1.1 1
104.21 odd 4 2496.4.a.q.1.1 1
104.99 even 4 2496.4.a.g.1.1 1
156.47 odd 4 1872.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 13.8 odd 4
234.4.a.k.1.1 1 39.8 even 4
624.4.a.f.1.1 1 52.47 even 4
1014.4.a.i.1.1 1 13.5 odd 4
1014.4.b.a.337.1 2 13.12 even 2 inner
1014.4.b.a.337.2 2 1.1 even 1 trivial
1872.4.a.o.1.1 1 156.47 odd 4
1950.4.a.o.1.1 1 65.34 odd 4
2496.4.a.g.1.1 1 104.99 even 4
2496.4.a.q.1.1 1 104.21 odd 4