Properties

Label 825.4.c.b.199.2
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
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] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (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: \(48.6765757547\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.b.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} +15.0000 q^{6} +3.00000i q^{7} -45.0000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} +15.0000 q^{6} +3.00000i q^{7} -45.0000i q^{8} -9.00000 q^{9} -11.0000 q^{11} +51.0000i q^{12} -32.0000i q^{13} -15.0000 q^{14} +89.0000 q^{16} +33.0000i q^{17} -45.0000i q^{18} -47.0000 q^{19} +9.00000 q^{21} -55.0000i q^{22} -113.000i q^{23} -135.000 q^{24} +160.000 q^{26} +27.0000i q^{27} -51.0000i q^{28} +54.0000 q^{29} +178.000 q^{31} +85.0000i q^{32} +33.0000i q^{33} -165.000 q^{34} +153.000 q^{36} +19.0000i q^{37} -235.000i q^{38} -96.0000 q^{39} +139.000 q^{41} +45.0000i q^{42} +308.000i q^{43} +187.000 q^{44} +565.000 q^{46} +195.000i q^{47} -267.000i q^{48} +334.000 q^{49} +99.0000 q^{51} +544.000i q^{52} -152.000i q^{53} -135.000 q^{54} +135.000 q^{56} +141.000i q^{57} +270.000i q^{58} +625.000 q^{59} +320.000 q^{61} +890.000i q^{62} -27.0000i q^{63} +287.000 q^{64} -165.000 q^{66} +200.000i q^{67} -561.000i q^{68} -339.000 q^{69} -947.000 q^{71} +405.000i q^{72} +448.000i q^{73} -95.0000 q^{74} +799.000 q^{76} -33.0000i q^{77} -480.000i q^{78} +721.000 q^{79} +81.0000 q^{81} +695.000i q^{82} -142.000i q^{83} -153.000 q^{84} -1540.00 q^{86} -162.000i q^{87} +495.000i q^{88} -404.000 q^{89} +96.0000 q^{91} +1921.00i q^{92} -534.000i q^{93} -975.000 q^{94} +255.000 q^{96} +79.0000i q^{97} +1670.00i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4} + 30 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 34 q^{4} + 30 q^{6} - 18 q^{9} - 22 q^{11} - 30 q^{14} + 178 q^{16} - 94 q^{19} + 18 q^{21} - 270 q^{24} + 320 q^{26} + 108 q^{29} + 356 q^{31} - 330 q^{34} + 306 q^{36} - 192 q^{39} + 278 q^{41} + 374 q^{44} + 1130 q^{46} + 668 q^{49} + 198 q^{51} - 270 q^{54} + 270 q^{56} + 1250 q^{59} + 640 q^{61} + 574 q^{64} - 330 q^{66} - 678 q^{69} - 1894 q^{71} - 190 q^{74} + 1598 q^{76} + 1442 q^{79} + 162 q^{81} - 306 q^{84} - 3080 q^{86} - 808 q^{89} + 192 q^{91} - 1950 q^{94} + 510 q^{96} + 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000i 1.76777i 0.467707 + 0.883883i \(0.345080\pi\)
−0.467707 + 0.883883i \(0.654920\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −17.0000 −2.12500
\(5\) 0 0
\(6\) 15.0000 1.02062
\(7\) 3.00000i 0.161985i 0.996715 + 0.0809924i \(0.0258089\pi\)
−0.996715 + 0.0809924i \(0.974191\pi\)
\(8\) − 45.0000i − 1.98874i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 51.0000i 1.22687i
\(13\) − 32.0000i − 0.682708i −0.939935 0.341354i \(-0.889115\pi\)
0.939935 0.341354i \(-0.110885\pi\)
\(14\) −15.0000 −0.286351
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) 33.0000i 0.470804i 0.971898 + 0.235402i \(0.0756407\pi\)
−0.971898 + 0.235402i \(0.924359\pi\)
\(18\) − 45.0000i − 0.589256i
\(19\) −47.0000 −0.567502 −0.283751 0.958898i \(-0.591579\pi\)
−0.283751 + 0.958898i \(0.591579\pi\)
\(20\) 0 0
\(21\) 9.00000 0.0935220
\(22\) − 55.0000i − 0.533002i
\(23\) − 113.000i − 1.02444i −0.858854 0.512220i \(-0.828823\pi\)
0.858854 0.512220i \(-0.171177\pi\)
\(24\) −135.000 −1.14820
\(25\) 0 0
\(26\) 160.000 1.20687
\(27\) 27.0000i 0.192450i
\(28\) − 51.0000i − 0.344218i
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) 178.000 1.03128 0.515641 0.856805i \(-0.327554\pi\)
0.515641 + 0.856805i \(0.327554\pi\)
\(32\) 85.0000i 0.469563i
\(33\) 33.0000i 0.174078i
\(34\) −165.000 −0.832273
\(35\) 0 0
\(36\) 153.000 0.708333
\(37\) 19.0000i 0.0844211i 0.999109 + 0.0422106i \(0.0134400\pi\)
−0.999109 + 0.0422106i \(0.986560\pi\)
\(38\) − 235.000i − 1.00321i
\(39\) −96.0000 −0.394162
\(40\) 0 0
\(41\) 139.000 0.529467 0.264734 0.964322i \(-0.414716\pi\)
0.264734 + 0.964322i \(0.414716\pi\)
\(42\) 45.0000i 0.165325i
\(43\) 308.000i 1.09232i 0.837682 + 0.546158i \(0.183910\pi\)
−0.837682 + 0.546158i \(0.816090\pi\)
\(44\) 187.000 0.640712
\(45\) 0 0
\(46\) 565.000 1.81097
\(47\) 195.000i 0.605185i 0.953120 + 0.302592i \(0.0978520\pi\)
−0.953120 + 0.302592i \(0.902148\pi\)
\(48\) − 267.000i − 0.802878i
\(49\) 334.000 0.973761
\(50\) 0 0
\(51\) 99.0000 0.271819
\(52\) 544.000i 1.45075i
\(53\) − 152.000i − 0.393940i −0.980410 0.196970i \(-0.936890\pi\)
0.980410 0.196970i \(-0.0631101\pi\)
\(54\) −135.000 −0.340207
\(55\) 0 0
\(56\) 135.000 0.322145
\(57\) 141.000i 0.327647i
\(58\) 270.000i 0.611254i
\(59\) 625.000 1.37912 0.689560 0.724229i \(-0.257804\pi\)
0.689560 + 0.724229i \(0.257804\pi\)
\(60\) 0 0
\(61\) 320.000 0.671669 0.335834 0.941921i \(-0.390982\pi\)
0.335834 + 0.941921i \(0.390982\pi\)
\(62\) 890.000i 1.82307i
\(63\) − 27.0000i − 0.0539949i
\(64\) 287.000 0.560547
\(65\) 0 0
\(66\) −165.000 −0.307729
\(67\) 200.000i 0.364685i 0.983235 + 0.182342i \(0.0583679\pi\)
−0.983235 + 0.182342i \(0.941632\pi\)
\(68\) − 561.000i − 1.00046i
\(69\) −339.000 −0.591461
\(70\) 0 0
\(71\) −947.000 −1.58293 −0.791466 0.611213i \(-0.790682\pi\)
−0.791466 + 0.611213i \(0.790682\pi\)
\(72\) 405.000i 0.662913i
\(73\) 448.000i 0.718280i 0.933284 + 0.359140i \(0.116930\pi\)
−0.933284 + 0.359140i \(0.883070\pi\)
\(74\) −95.0000 −0.149237
\(75\) 0 0
\(76\) 799.000 1.20594
\(77\) − 33.0000i − 0.0488402i
\(78\) − 480.000i − 0.696786i
\(79\) 721.000 1.02682 0.513410 0.858143i \(-0.328382\pi\)
0.513410 + 0.858143i \(0.328382\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 695.000i 0.935975i
\(83\) − 142.000i − 0.187789i −0.995582 0.0938947i \(-0.970068\pi\)
0.995582 0.0938947i \(-0.0299317\pi\)
\(84\) −153.000 −0.198734
\(85\) 0 0
\(86\) −1540.00 −1.93096
\(87\) − 162.000i − 0.199635i
\(88\) 495.000i 0.599627i
\(89\) −404.000 −0.481168 −0.240584 0.970628i \(-0.577339\pi\)
−0.240584 + 0.970628i \(0.577339\pi\)
\(90\) 0 0
\(91\) 96.0000 0.110588
\(92\) 1921.00i 2.17694i
\(93\) − 534.000i − 0.595411i
\(94\) −975.000 −1.06983
\(95\) 0 0
\(96\) 255.000 0.271102
\(97\) 79.0000i 0.0826931i 0.999145 + 0.0413466i \(0.0131648\pi\)
−0.999145 + 0.0413466i \(0.986835\pi\)
\(98\) 1670.00i 1.72138i
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −545.000 −0.536926 −0.268463 0.963290i \(-0.586516\pi\)
−0.268463 + 0.963290i \(0.586516\pi\)
\(102\) 495.000i 0.480513i
\(103\) 1306.00i 1.24936i 0.780881 + 0.624680i \(0.214770\pi\)
−0.780881 + 0.624680i \(0.785230\pi\)
\(104\) −1440.00 −1.35773
\(105\) 0 0
\(106\) 760.000 0.696394
\(107\) 1938.00i 1.75097i 0.483247 + 0.875484i \(0.339457\pi\)
−0.483247 + 0.875484i \(0.660543\pi\)
\(108\) − 459.000i − 0.408956i
\(109\) 576.000 0.506154 0.253077 0.967446i \(-0.418557\pi\)
0.253077 + 0.967446i \(0.418557\pi\)
\(110\) 0 0
\(111\) 57.0000 0.0487405
\(112\) 267.000i 0.225260i
\(113\) 1104.00i 0.919076i 0.888158 + 0.459538i \(0.151985\pi\)
−0.888158 + 0.459538i \(0.848015\pi\)
\(114\) −705.000 −0.579204
\(115\) 0 0
\(116\) −918.000 −0.734777
\(117\) 288.000i 0.227569i
\(118\) 3125.00i 2.43796i
\(119\) −99.0000 −0.0762632
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1600.00i 1.18735i
\(123\) − 417.000i − 0.305688i
\(124\) −3026.00 −2.19147
\(125\) 0 0
\(126\) 135.000 0.0954504
\(127\) − 1739.00i − 1.21505i −0.794301 0.607525i \(-0.792163\pi\)
0.794301 0.607525i \(-0.207837\pi\)
\(128\) 2115.00i 1.46048i
\(129\) 924.000 0.630649
\(130\) 0 0
\(131\) 1818.00 1.21251 0.606257 0.795269i \(-0.292670\pi\)
0.606257 + 0.795269i \(0.292670\pi\)
\(132\) − 561.000i − 0.369915i
\(133\) − 141.000i − 0.0919267i
\(134\) −1000.00 −0.644678
\(135\) 0 0
\(136\) 1485.00 0.936307
\(137\) 870.000i 0.542548i 0.962502 + 0.271274i \(0.0874450\pi\)
−0.962502 + 0.271274i \(0.912555\pi\)
\(138\) − 1695.00i − 1.04557i
\(139\) 636.000 0.388092 0.194046 0.980992i \(-0.437839\pi\)
0.194046 + 0.980992i \(0.437839\pi\)
\(140\) 0 0
\(141\) 585.000 0.349403
\(142\) − 4735.00i − 2.79826i
\(143\) 352.000i 0.205844i
\(144\) −801.000 −0.463542
\(145\) 0 0
\(146\) −2240.00 −1.26975
\(147\) − 1002.00i − 0.562201i
\(148\) − 323.000i − 0.179395i
\(149\) 239.000 0.131407 0.0657035 0.997839i \(-0.479071\pi\)
0.0657035 + 0.997839i \(0.479071\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2115.00i 1.12861i
\(153\) − 297.000i − 0.156935i
\(154\) 165.000 0.0863382
\(155\) 0 0
\(156\) 1632.00 0.837593
\(157\) − 1874.00i − 0.952621i −0.879277 0.476310i \(-0.841974\pi\)
0.879277 0.476310i \(-0.158026\pi\)
\(158\) 3605.00i 1.81518i
\(159\) −456.000 −0.227441
\(160\) 0 0
\(161\) 339.000 0.165944
\(162\) 405.000i 0.196419i
\(163\) 1904.00i 0.914925i 0.889229 + 0.457463i \(0.151242\pi\)
−0.889229 + 0.457463i \(0.848758\pi\)
\(164\) −2363.00 −1.12512
\(165\) 0 0
\(166\) 710.000 0.331968
\(167\) − 1180.00i − 0.546773i −0.961904 0.273387i \(-0.911856\pi\)
0.961904 0.273387i \(-0.0881438\pi\)
\(168\) − 405.000i − 0.185991i
\(169\) 1173.00 0.533910
\(170\) 0 0
\(171\) 423.000 0.189167
\(172\) − 5236.00i − 2.32117i
\(173\) 3177.00i 1.39620i 0.716000 + 0.698101i \(0.245971\pi\)
−0.716000 + 0.698101i \(0.754029\pi\)
\(174\) 810.000 0.352908
\(175\) 0 0
\(176\) −979.000 −0.419289
\(177\) − 1875.00i − 0.796235i
\(178\) − 2020.00i − 0.850592i
\(179\) −1787.00 −0.746182 −0.373091 0.927795i \(-0.621702\pi\)
−0.373091 + 0.927795i \(0.621702\pi\)
\(180\) 0 0
\(181\) −835.000 −0.342901 −0.171450 0.985193i \(-0.554845\pi\)
−0.171450 + 0.985193i \(0.554845\pi\)
\(182\) 480.000i 0.195494i
\(183\) − 960.000i − 0.387788i
\(184\) −5085.00 −2.03734
\(185\) 0 0
\(186\) 2670.00 1.05255
\(187\) − 363.000i − 0.141953i
\(188\) − 3315.00i − 1.28602i
\(189\) −81.0000 −0.0311740
\(190\) 0 0
\(191\) 3613.00 1.36873 0.684365 0.729139i \(-0.260079\pi\)
0.684365 + 0.729139i \(0.260079\pi\)
\(192\) − 861.000i − 0.323632i
\(193\) − 4204.00i − 1.56793i −0.620805 0.783965i \(-0.713194\pi\)
0.620805 0.783965i \(-0.286806\pi\)
\(194\) −395.000 −0.146182
\(195\) 0 0
\(196\) −5678.00 −2.06924
\(197\) − 4517.00i − 1.63362i −0.576908 0.816809i \(-0.695741\pi\)
0.576908 0.816809i \(-0.304259\pi\)
\(198\) 495.000i 0.177667i
\(199\) −4164.00 −1.48331 −0.741654 0.670783i \(-0.765958\pi\)
−0.741654 + 0.670783i \(0.765958\pi\)
\(200\) 0 0
\(201\) 600.000 0.210551
\(202\) − 2725.00i − 0.949160i
\(203\) 162.000i 0.0560107i
\(204\) −1683.00 −0.577616
\(205\) 0 0
\(206\) −6530.00 −2.20858
\(207\) 1017.00i 0.341480i
\(208\) − 2848.00i − 0.949391i
\(209\) 517.000 0.171108
\(210\) 0 0
\(211\) 4660.00 1.52042 0.760208 0.649680i \(-0.225097\pi\)
0.760208 + 0.649680i \(0.225097\pi\)
\(212\) 2584.00i 0.837122i
\(213\) 2841.00i 0.913907i
\(214\) −9690.00 −3.09530
\(215\) 0 0
\(216\) 1215.00 0.382733
\(217\) 534.000i 0.167052i
\(218\) 2880.00i 0.894762i
\(219\) 1344.00 0.414699
\(220\) 0 0
\(221\) 1056.00 0.321422
\(222\) 285.000i 0.0861619i
\(223\) − 3560.00i − 1.06904i −0.845157 0.534518i \(-0.820493\pi\)
0.845157 0.534518i \(-0.179507\pi\)
\(224\) −255.000 −0.0760621
\(225\) 0 0
\(226\) −5520.00 −1.62471
\(227\) − 4678.00i − 1.36780i −0.729577 0.683898i \(-0.760283\pi\)
0.729577 0.683898i \(-0.239717\pi\)
\(228\) − 2397.00i − 0.696251i
\(229\) 4447.00 1.28326 0.641629 0.767015i \(-0.278259\pi\)
0.641629 + 0.767015i \(0.278259\pi\)
\(230\) 0 0
\(231\) −99.0000 −0.0281979
\(232\) − 2430.00i − 0.687661i
\(233\) − 411.000i − 0.115560i −0.998329 0.0577801i \(-0.981598\pi\)
0.998329 0.0577801i \(-0.0184022\pi\)
\(234\) −1440.00 −0.402290
\(235\) 0 0
\(236\) −10625.0 −2.93063
\(237\) − 2163.00i − 0.592835i
\(238\) − 495.000i − 0.134815i
\(239\) 6380.00 1.72673 0.863364 0.504582i \(-0.168353\pi\)
0.863364 + 0.504582i \(0.168353\pi\)
\(240\) 0 0
\(241\) 7282.00 1.94637 0.973184 0.230027i \(-0.0738813\pi\)
0.973184 + 0.230027i \(0.0738813\pi\)
\(242\) 605.000i 0.160706i
\(243\) − 243.000i − 0.0641500i
\(244\) −5440.00 −1.42730
\(245\) 0 0
\(246\) 2085.00 0.540385
\(247\) 1504.00i 0.387438i
\(248\) − 8010.00i − 2.05095i
\(249\) −426.000 −0.108420
\(250\) 0 0
\(251\) −4728.00 −1.18896 −0.594480 0.804111i \(-0.702642\pi\)
−0.594480 + 0.804111i \(0.702642\pi\)
\(252\) 459.000i 0.114739i
\(253\) 1243.00i 0.308880i
\(254\) 8695.00 2.14792
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) 5418.00i 1.31504i 0.753437 + 0.657521i \(0.228395\pi\)
−0.753437 + 0.657521i \(0.771605\pi\)
\(258\) 4620.00i 1.11484i
\(259\) −57.0000 −0.0136749
\(260\) 0 0
\(261\) −486.000 −0.115259
\(262\) 9090.00i 2.14344i
\(263\) 3354.00i 0.786375i 0.919458 + 0.393187i \(0.128628\pi\)
−0.919458 + 0.393187i \(0.871372\pi\)
\(264\) 1485.00 0.346195
\(265\) 0 0
\(266\) 705.000 0.162505
\(267\) 1212.00i 0.277802i
\(268\) − 3400.00i − 0.774955i
\(269\) −1062.00 −0.240711 −0.120356 0.992731i \(-0.538403\pi\)
−0.120356 + 0.992731i \(0.538403\pi\)
\(270\) 0 0
\(271\) −4821.00 −1.08065 −0.540323 0.841458i \(-0.681698\pi\)
−0.540323 + 0.841458i \(0.681698\pi\)
\(272\) 2937.00i 0.654712i
\(273\) − 288.000i − 0.0638482i
\(274\) −4350.00 −0.959099
\(275\) 0 0
\(276\) 5763.00 1.25685
\(277\) 4.00000i 0 0.000867642i 1.00000 0.000433821i \(0.000138089\pi\)
−1.00000 0.000433821i \(0.999862\pi\)
\(278\) 3180.00i 0.686057i
\(279\) −1602.00 −0.343761
\(280\) 0 0
\(281\) 4647.00 0.986537 0.493268 0.869877i \(-0.335802\pi\)
0.493268 + 0.869877i \(0.335802\pi\)
\(282\) 2925.00i 0.617664i
\(283\) 4283.00i 0.899639i 0.893119 + 0.449820i \(0.148512\pi\)
−0.893119 + 0.449820i \(0.851488\pi\)
\(284\) 16099.0 3.36373
\(285\) 0 0
\(286\) −1760.00 −0.363885
\(287\) 417.000i 0.0857656i
\(288\) − 765.000i − 0.156521i
\(289\) 3824.00 0.778343
\(290\) 0 0
\(291\) 237.000 0.0477429
\(292\) − 7616.00i − 1.52634i
\(293\) 6811.00i 1.35803i 0.734124 + 0.679015i \(0.237593\pi\)
−0.734124 + 0.679015i \(0.762407\pi\)
\(294\) 5010.00 0.993841
\(295\) 0 0
\(296\) 855.000 0.167891
\(297\) − 297.000i − 0.0580259i
\(298\) 1195.00i 0.232297i
\(299\) −3616.00 −0.699394
\(300\) 0 0
\(301\) −924.000 −0.176938
\(302\) 6040.00i 1.15087i
\(303\) 1635.00i 0.309994i
\(304\) −4183.00 −0.789183
\(305\) 0 0
\(306\) 1485.00 0.277424
\(307\) − 460.000i − 0.0855166i −0.999085 0.0427583i \(-0.986385\pi\)
0.999085 0.0427583i \(-0.0136145\pi\)
\(308\) 561.000i 0.103786i
\(309\) 3918.00 0.721318
\(310\) 0 0
\(311\) 8328.00 1.51845 0.759224 0.650829i \(-0.225579\pi\)
0.759224 + 0.650829i \(0.225579\pi\)
\(312\) 4320.00i 0.783884i
\(313\) 5929.00i 1.07069i 0.844633 + 0.535346i \(0.179819\pi\)
−0.844633 + 0.535346i \(0.820181\pi\)
\(314\) 9370.00 1.68401
\(315\) 0 0
\(316\) −12257.0 −2.18199
\(317\) 5040.00i 0.892980i 0.894789 + 0.446490i \(0.147326\pi\)
−0.894789 + 0.446490i \(0.852674\pi\)
\(318\) − 2280.00i − 0.402063i
\(319\) −594.000 −0.104256
\(320\) 0 0
\(321\) 5814.00 1.01092
\(322\) 1695.00i 0.293350i
\(323\) − 1551.00i − 0.267183i
\(324\) −1377.00 −0.236111
\(325\) 0 0
\(326\) −9520.00 −1.61737
\(327\) − 1728.00i − 0.292228i
\(328\) − 6255.00i − 1.05297i
\(329\) −585.000 −0.0980307
\(330\) 0 0
\(331\) 10396.0 1.72633 0.863166 0.504920i \(-0.168478\pi\)
0.863166 + 0.504920i \(0.168478\pi\)
\(332\) 2414.00i 0.399053i
\(333\) − 171.000i − 0.0281404i
\(334\) 5900.00 0.966568
\(335\) 0 0
\(336\) 801.000 0.130054
\(337\) − 7236.00i − 1.16964i −0.811162 0.584822i \(-0.801164\pi\)
0.811162 0.584822i \(-0.198836\pi\)
\(338\) 5865.00i 0.943828i
\(339\) 3312.00 0.530629
\(340\) 0 0
\(341\) −1958.00 −0.310943
\(342\) 2115.00i 0.334404i
\(343\) 2031.00i 0.319719i
\(344\) 13860.0 2.17233
\(345\) 0 0
\(346\) −15885.0 −2.46816
\(347\) 1468.00i 0.227108i 0.993532 + 0.113554i \(0.0362234\pi\)
−0.993532 + 0.113554i \(0.963777\pi\)
\(348\) 2754.00i 0.424224i
\(349\) −5690.00 −0.872718 −0.436359 0.899773i \(-0.643732\pi\)
−0.436359 + 0.899773i \(0.643732\pi\)
\(350\) 0 0
\(351\) 864.000 0.131387
\(352\) − 935.000i − 0.141579i
\(353\) − 5376.00i − 0.810582i −0.914188 0.405291i \(-0.867170\pi\)
0.914188 0.405291i \(-0.132830\pi\)
\(354\) 9375.00 1.40756
\(355\) 0 0
\(356\) 6868.00 1.02248
\(357\) 297.000i 0.0440306i
\(358\) − 8935.00i − 1.31908i
\(359\) −3734.00 −0.548950 −0.274475 0.961594i \(-0.588504\pi\)
−0.274475 + 0.961594i \(0.588504\pi\)
\(360\) 0 0
\(361\) −4650.00 −0.677941
\(362\) − 4175.00i − 0.606169i
\(363\) − 363.000i − 0.0524864i
\(364\) −1632.00 −0.235000
\(365\) 0 0
\(366\) 4800.00 0.685519
\(367\) − 10274.0i − 1.46130i −0.682750 0.730652i \(-0.739216\pi\)
0.682750 0.730652i \(-0.260784\pi\)
\(368\) − 10057.0i − 1.42461i
\(369\) −1251.00 −0.176489
\(370\) 0 0
\(371\) 456.000 0.0638122
\(372\) 9078.00i 1.26525i
\(373\) − 13662.0i − 1.89649i −0.317537 0.948246i \(-0.602856\pi\)
0.317537 0.948246i \(-0.397144\pi\)
\(374\) 1815.00 0.250940
\(375\) 0 0
\(376\) 8775.00 1.20355
\(377\) − 1728.00i − 0.236065i
\(378\) − 405.000i − 0.0551083i
\(379\) 7906.00 1.07151 0.535757 0.844372i \(-0.320026\pi\)
0.535757 + 0.844372i \(0.320026\pi\)
\(380\) 0 0
\(381\) −5217.00 −0.701509
\(382\) 18065.0i 2.41960i
\(383\) 3168.00i 0.422656i 0.977415 + 0.211328i \(0.0677788\pi\)
−0.977415 + 0.211328i \(0.932221\pi\)
\(384\) 6345.00 0.843208
\(385\) 0 0
\(386\) 21020.0 2.77174
\(387\) − 2772.00i − 0.364105i
\(388\) − 1343.00i − 0.175723i
\(389\) −10770.0 −1.40375 −0.701877 0.712298i \(-0.747655\pi\)
−0.701877 + 0.712298i \(0.747655\pi\)
\(390\) 0 0
\(391\) 3729.00 0.482311
\(392\) − 15030.0i − 1.93656i
\(393\) − 5454.00i − 0.700046i
\(394\) 22585.0 2.88786
\(395\) 0 0
\(396\) −1683.00 −0.213571
\(397\) 5670.00i 0.716799i 0.933568 + 0.358399i \(0.116677\pi\)
−0.933568 + 0.358399i \(0.883323\pi\)
\(398\) − 20820.0i − 2.62214i
\(399\) −423.000 −0.0530739
\(400\) 0 0
\(401\) 832.000 0.103611 0.0518056 0.998657i \(-0.483502\pi\)
0.0518056 + 0.998657i \(0.483502\pi\)
\(402\) 3000.00i 0.372205i
\(403\) − 5696.00i − 0.704064i
\(404\) 9265.00 1.14097
\(405\) 0 0
\(406\) −810.000 −0.0990139
\(407\) − 209.000i − 0.0254539i
\(408\) − 4455.00i − 0.540577i
\(409\) 5712.00 0.690563 0.345281 0.938499i \(-0.387783\pi\)
0.345281 + 0.938499i \(0.387783\pi\)
\(410\) 0 0
\(411\) 2610.00 0.313240
\(412\) − 22202.0i − 2.65489i
\(413\) 1875.00i 0.223396i
\(414\) −5085.00 −0.603657
\(415\) 0 0
\(416\) 2720.00 0.320574
\(417\) − 1908.00i − 0.224065i
\(418\) 2585.00i 0.302480i
\(419\) 4559.00 0.531555 0.265778 0.964034i \(-0.414371\pi\)
0.265778 + 0.964034i \(0.414371\pi\)
\(420\) 0 0
\(421\) 6855.00 0.793568 0.396784 0.917912i \(-0.370126\pi\)
0.396784 + 0.917912i \(0.370126\pi\)
\(422\) 23300.0i 2.68774i
\(423\) − 1755.00i − 0.201728i
\(424\) −6840.00 −0.783443
\(425\) 0 0
\(426\) −14205.0 −1.61557
\(427\) 960.000i 0.108800i
\(428\) − 32946.0i − 3.72081i
\(429\) 1056.00 0.118844
\(430\) 0 0
\(431\) 10770.0 1.20365 0.601824 0.798628i \(-0.294441\pi\)
0.601824 + 0.798628i \(0.294441\pi\)
\(432\) 2403.00i 0.267626i
\(433\) − 8498.00i − 0.943159i −0.881824 0.471579i \(-0.843684\pi\)
0.881824 0.471579i \(-0.156316\pi\)
\(434\) −2670.00 −0.295309
\(435\) 0 0
\(436\) −9792.00 −1.07558
\(437\) 5311.00i 0.581372i
\(438\) 6720.00i 0.733091i
\(439\) −9835.00 −1.06925 −0.534623 0.845091i \(-0.679546\pi\)
−0.534623 + 0.845091i \(0.679546\pi\)
\(440\) 0 0
\(441\) −3006.00 −0.324587
\(442\) 5280.00i 0.568199i
\(443\) 10745.0i 1.15239i 0.817311 + 0.576197i \(0.195464\pi\)
−0.817311 + 0.576197i \(0.804536\pi\)
\(444\) −969.000 −0.103574
\(445\) 0 0
\(446\) 17800.0 1.88981
\(447\) − 717.000i − 0.0758679i
\(448\) 861.000i 0.0908001i
\(449\) −8356.00 −0.878272 −0.439136 0.898421i \(-0.644715\pi\)
−0.439136 + 0.898421i \(0.644715\pi\)
\(450\) 0 0
\(451\) −1529.00 −0.159640
\(452\) − 18768.0i − 1.95304i
\(453\) − 3624.00i − 0.375873i
\(454\) 23390.0 2.41795
\(455\) 0 0
\(456\) 6345.00 0.651605
\(457\) − 7058.00i − 0.722449i −0.932479 0.361225i \(-0.882359\pi\)
0.932479 0.361225i \(-0.117641\pi\)
\(458\) 22235.0i 2.26850i
\(459\) −891.000 −0.0906064
\(460\) 0 0
\(461\) 646.000 0.0652651 0.0326326 0.999467i \(-0.489611\pi\)
0.0326326 + 0.999467i \(0.489611\pi\)
\(462\) − 495.000i − 0.0498474i
\(463\) 8982.00i 0.901574i 0.892631 + 0.450787i \(0.148857\pi\)
−0.892631 + 0.450787i \(0.851143\pi\)
\(464\) 4806.00 0.480847
\(465\) 0 0
\(466\) 2055.00 0.204283
\(467\) − 13476.0i − 1.33532i −0.744466 0.667661i \(-0.767296\pi\)
0.744466 0.667661i \(-0.232704\pi\)
\(468\) − 4896.00i − 0.483585i
\(469\) −600.000 −0.0590734
\(470\) 0 0
\(471\) −5622.00 −0.549996
\(472\) − 28125.0i − 2.74271i
\(473\) − 3388.00i − 0.329345i
\(474\) 10815.0 1.04799
\(475\) 0 0
\(476\) 1683.00 0.162059
\(477\) 1368.00i 0.131313i
\(478\) 31900.0i 3.05245i
\(479\) −12996.0 −1.23967 −0.619835 0.784732i \(-0.712801\pi\)
−0.619835 + 0.784732i \(0.712801\pi\)
\(480\) 0 0
\(481\) 608.000 0.0576350
\(482\) 36410.0i 3.44073i
\(483\) − 1017.00i − 0.0958077i
\(484\) −2057.00 −0.193182
\(485\) 0 0
\(486\) 1215.00 0.113402
\(487\) − 6026.00i − 0.560707i −0.959897 0.280353i \(-0.909548\pi\)
0.959897 0.280353i \(-0.0904516\pi\)
\(488\) − 14400.0i − 1.33577i
\(489\) 5712.00 0.528232
\(490\) 0 0
\(491\) 11698.0 1.07520 0.537600 0.843200i \(-0.319331\pi\)
0.537600 + 0.843200i \(0.319331\pi\)
\(492\) 7089.00i 0.649587i
\(493\) 1782.00i 0.162794i
\(494\) −7520.00 −0.684900
\(495\) 0 0
\(496\) 15842.0 1.43413
\(497\) − 2841.00i − 0.256411i
\(498\) − 2130.00i − 0.191662i
\(499\) 17052.0 1.52976 0.764882 0.644170i \(-0.222797\pi\)
0.764882 + 0.644170i \(0.222797\pi\)
\(500\) 0 0
\(501\) −3540.00 −0.315680
\(502\) − 23640.0i − 2.10180i
\(503\) 932.000i 0.0826160i 0.999146 + 0.0413080i \(0.0131525\pi\)
−0.999146 + 0.0413080i \(0.986848\pi\)
\(504\) −1215.00 −0.107382
\(505\) 0 0
\(506\) −6215.00 −0.546029
\(507\) − 3519.00i − 0.308253i
\(508\) 29563.0i 2.58198i
\(509\) −4384.00 −0.381763 −0.190882 0.981613i \(-0.561135\pi\)
−0.190882 + 0.981613i \(0.561135\pi\)
\(510\) 0 0
\(511\) −1344.00 −0.116350
\(512\) − 24475.0i − 2.11260i
\(513\) − 1269.00i − 0.109216i
\(514\) −27090.0 −2.32469
\(515\) 0 0
\(516\) −15708.0 −1.34013
\(517\) − 2145.00i − 0.182470i
\(518\) − 285.000i − 0.0241741i
\(519\) 9531.00 0.806097
\(520\) 0 0
\(521\) −2322.00 −0.195257 −0.0976283 0.995223i \(-0.531126\pi\)
−0.0976283 + 0.995223i \(0.531126\pi\)
\(522\) − 2430.00i − 0.203751i
\(523\) 9749.00i 0.815094i 0.913184 + 0.407547i \(0.133616\pi\)
−0.913184 + 0.407547i \(0.866384\pi\)
\(524\) −30906.0 −2.57659
\(525\) 0 0
\(526\) −16770.0 −1.39013
\(527\) 5874.00i 0.485532i
\(528\) 2937.00i 0.242077i
\(529\) −602.000 −0.0494781
\(530\) 0 0
\(531\) −5625.00 −0.459707
\(532\) 2397.00i 0.195344i
\(533\) − 4448.00i − 0.361471i
\(534\) −6060.00 −0.491090
\(535\) 0 0
\(536\) 9000.00 0.725263
\(537\) 5361.00i 0.430809i
\(538\) − 5310.00i − 0.425521i
\(539\) −3674.00 −0.293600
\(540\) 0 0
\(541\) 4208.00 0.334410 0.167205 0.985922i \(-0.446526\pi\)
0.167205 + 0.985922i \(0.446526\pi\)
\(542\) − 24105.0i − 1.91033i
\(543\) 2505.00i 0.197974i
\(544\) −2805.00 −0.221072
\(545\) 0 0
\(546\) 1440.00 0.112869
\(547\) 10179.0i 0.795654i 0.917461 + 0.397827i \(0.130236\pi\)
−0.917461 + 0.397827i \(0.869764\pi\)
\(548\) − 14790.0i − 1.15292i
\(549\) −2880.00 −0.223890
\(550\) 0 0
\(551\) −2538.00 −0.196229
\(552\) 15255.0i 1.17626i
\(553\) 2163.00i 0.166329i
\(554\) −20.0000 −0.00153379
\(555\) 0 0
\(556\) −10812.0 −0.824696
\(557\) − 2314.00i − 0.176028i −0.996119 0.0880138i \(-0.971948\pi\)
0.996119 0.0880138i \(-0.0280519\pi\)
\(558\) − 8010.00i − 0.607689i
\(559\) 9856.00 0.745732
\(560\) 0 0
\(561\) −1089.00 −0.0819565
\(562\) 23235.0i 1.74397i
\(563\) − 24330.0i − 1.82129i −0.413188 0.910646i \(-0.635585\pi\)
0.413188 0.910646i \(-0.364415\pi\)
\(564\) −9945.00 −0.742482
\(565\) 0 0
\(566\) −21415.0 −1.59035
\(567\) 243.000i 0.0179983i
\(568\) 42615.0i 3.14804i
\(569\) −3445.00 −0.253817 −0.126909 0.991914i \(-0.540505\pi\)
−0.126909 + 0.991914i \(0.540505\pi\)
\(570\) 0 0
\(571\) −13056.0 −0.956877 −0.478438 0.878121i \(-0.658797\pi\)
−0.478438 + 0.878121i \(0.658797\pi\)
\(572\) − 5984.00i − 0.437419i
\(573\) − 10839.0i − 0.790237i
\(574\) −2085.00 −0.151614
\(575\) 0 0
\(576\) −2583.00 −0.186849
\(577\) − 17347.0i − 1.25159i −0.779989 0.625793i \(-0.784775\pi\)
0.779989 0.625793i \(-0.215225\pi\)
\(578\) 19120.0i 1.37593i
\(579\) −12612.0 −0.905245
\(580\) 0 0
\(581\) 426.000 0.0304190
\(582\) 1185.00i 0.0843983i
\(583\) 1672.00i 0.118777i
\(584\) 20160.0 1.42847
\(585\) 0 0
\(586\) −34055.0 −2.40068
\(587\) 8379.00i 0.589162i 0.955626 + 0.294581i \(0.0951801\pi\)
−0.955626 + 0.294581i \(0.904820\pi\)
\(588\) 17034.0i 1.19468i
\(589\) −8366.00 −0.585255
\(590\) 0 0
\(591\) −13551.0 −0.943170
\(592\) 1691.00i 0.117398i
\(593\) − 1958.00i − 0.135591i −0.997699 0.0677955i \(-0.978403\pi\)
0.997699 0.0677955i \(-0.0215965\pi\)
\(594\) 1485.00 0.102576
\(595\) 0 0
\(596\) −4063.00 −0.279240
\(597\) 12492.0i 0.856388i
\(598\) − 18080.0i − 1.23636i
\(599\) −23583.0 −1.60864 −0.804320 0.594196i \(-0.797470\pi\)
−0.804320 + 0.594196i \(0.797470\pi\)
\(600\) 0 0
\(601\) −15328.0 −1.04034 −0.520168 0.854064i \(-0.674131\pi\)
−0.520168 + 0.854064i \(0.674131\pi\)
\(602\) − 4620.00i − 0.312786i
\(603\) − 1800.00i − 0.121562i
\(604\) −20536.0 −1.38344
\(605\) 0 0
\(606\) −8175.00 −0.547998
\(607\) 160.000i 0.0106988i 0.999986 + 0.00534942i \(0.00170278\pi\)
−0.999986 + 0.00534942i \(0.998297\pi\)
\(608\) − 3995.00i − 0.266478i
\(609\) 486.000 0.0323378
\(610\) 0 0
\(611\) 6240.00 0.413164
\(612\) 5049.00i 0.333486i
\(613\) 5948.00i 0.391904i 0.980613 + 0.195952i \(0.0627798\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(614\) 2300.00 0.151173
\(615\) 0 0
\(616\) −1485.00 −0.0971304
\(617\) 334.000i 0.0217931i 0.999941 + 0.0108965i \(0.00346855\pi\)
−0.999941 + 0.0108965i \(0.996531\pi\)
\(618\) 19590.0i 1.27512i
\(619\) 7202.00 0.467646 0.233823 0.972279i \(-0.424876\pi\)
0.233823 + 0.972279i \(0.424876\pi\)
\(620\) 0 0
\(621\) 3051.00 0.197154
\(622\) 41640.0i 2.68426i
\(623\) − 1212.00i − 0.0779418i
\(624\) −8544.00 −0.548131
\(625\) 0 0
\(626\) −29645.0 −1.89274
\(627\) − 1551.00i − 0.0987894i
\(628\) 31858.0i 2.02432i
\(629\) −627.000 −0.0397458
\(630\) 0 0
\(631\) 10306.0 0.650199 0.325099 0.945680i \(-0.394602\pi\)
0.325099 + 0.945680i \(0.394602\pi\)
\(632\) − 32445.0i − 2.04208i
\(633\) − 13980.0i − 0.877812i
\(634\) −25200.0 −1.57858
\(635\) 0 0
\(636\) 7752.00 0.483313
\(637\) − 10688.0i − 0.664794i
\(638\) − 2970.00i − 0.184300i
\(639\) 8523.00 0.527644
\(640\) 0 0
\(641\) −1228.00 −0.0756678 −0.0378339 0.999284i \(-0.512046\pi\)
−0.0378339 + 0.999284i \(0.512046\pi\)
\(642\) 29070.0i 1.78707i
\(643\) 18454.0i 1.13181i 0.824470 + 0.565906i \(0.191473\pi\)
−0.824470 + 0.565906i \(0.808527\pi\)
\(644\) −5763.00 −0.352630
\(645\) 0 0
\(646\) 7755.00 0.472316
\(647\) 17647.0i 1.07230i 0.844124 + 0.536148i \(0.180121\pi\)
−0.844124 + 0.536148i \(0.819879\pi\)
\(648\) − 3645.00i − 0.220971i
\(649\) −6875.00 −0.415820
\(650\) 0 0
\(651\) 1602.00 0.0964475
\(652\) − 32368.0i − 1.94422i
\(653\) − 25918.0i − 1.55322i −0.629984 0.776608i \(-0.716939\pi\)
0.629984 0.776608i \(-0.283061\pi\)
\(654\) 8640.00 0.516591
\(655\) 0 0
\(656\) 12371.0 0.736290
\(657\) − 4032.00i − 0.239427i
\(658\) − 2925.00i − 0.173295i
\(659\) −12864.0 −0.760410 −0.380205 0.924902i \(-0.624147\pi\)
−0.380205 + 0.924902i \(0.624147\pi\)
\(660\) 0 0
\(661\) −11419.0 −0.671933 −0.335966 0.941874i \(-0.609063\pi\)
−0.335966 + 0.941874i \(0.609063\pi\)
\(662\) 51980.0i 3.05175i
\(663\) − 3168.00i − 0.185573i
\(664\) −6390.00 −0.373464
\(665\) 0 0
\(666\) 855.000 0.0497456
\(667\) − 6102.00i − 0.354228i
\(668\) 20060.0i 1.16189i
\(669\) −10680.0 −0.617209
\(670\) 0 0
\(671\) −3520.00 −0.202516
\(672\) 765.000i 0.0439145i
\(673\) − 15784.0i − 0.904054i −0.892004 0.452027i \(-0.850701\pi\)
0.892004 0.452027i \(-0.149299\pi\)
\(674\) 36180.0 2.06766
\(675\) 0 0
\(676\) −19941.0 −1.13456
\(677\) − 26050.0i − 1.47885i −0.673238 0.739426i \(-0.735097\pi\)
0.673238 0.739426i \(-0.264903\pi\)
\(678\) 16560.0i 0.938028i
\(679\) −237.000 −0.0133950
\(680\) 0 0
\(681\) −14034.0 −0.789698
\(682\) − 9790.00i − 0.549675i
\(683\) 15095.0i 0.845672i 0.906206 + 0.422836i \(0.138965\pi\)
−0.906206 + 0.422836i \(0.861035\pi\)
\(684\) −7191.00 −0.401981
\(685\) 0 0
\(686\) −10155.0 −0.565189
\(687\) − 13341.0i − 0.740889i
\(688\) 27412.0i 1.51900i
\(689\) −4864.00 −0.268946
\(690\) 0 0
\(691\) 15896.0 0.875126 0.437563 0.899188i \(-0.355842\pi\)
0.437563 + 0.899188i \(0.355842\pi\)
\(692\) − 54009.0i − 2.96693i
\(693\) 297.000i 0.0162801i
\(694\) −7340.00 −0.401473
\(695\) 0 0
\(696\) −7290.00 −0.397021
\(697\) 4587.00i 0.249275i
\(698\) − 28450.0i − 1.54276i
\(699\) −1233.00 −0.0667187
\(700\) 0 0
\(701\) 10529.0 0.567296 0.283648 0.958928i \(-0.408455\pi\)
0.283648 + 0.958928i \(0.408455\pi\)
\(702\) 4320.00i 0.232262i
\(703\) − 893.000i − 0.0479092i
\(704\) −3157.00 −0.169011
\(705\) 0 0
\(706\) 26880.0 1.43292
\(707\) − 1635.00i − 0.0869738i
\(708\) 31875.0i 1.69200i
\(709\) 16087.0 0.852130 0.426065 0.904693i \(-0.359900\pi\)
0.426065 + 0.904693i \(0.359900\pi\)
\(710\) 0 0
\(711\) −6489.00 −0.342274
\(712\) 18180.0i 0.956916i
\(713\) − 20114.0i − 1.05649i
\(714\) −1485.00 −0.0778358
\(715\) 0 0
\(716\) 30379.0 1.58564
\(717\) − 19140.0i − 0.996927i
\(718\) − 18670.0i − 0.970415i
\(719\) −24336.0 −1.26228 −0.631140 0.775669i \(-0.717413\pi\)
−0.631140 + 0.775669i \(0.717413\pi\)
\(720\) 0 0
\(721\) −3918.00 −0.202377
\(722\) − 23250.0i − 1.19844i
\(723\) − 21846.0i − 1.12374i
\(724\) 14195.0 0.728664
\(725\) 0 0
\(726\) 1815.00 0.0927837
\(727\) 13960.0i 0.712170i 0.934454 + 0.356085i \(0.115889\pi\)
−0.934454 + 0.356085i \(0.884111\pi\)
\(728\) − 4320.00i − 0.219931i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −10164.0 −0.514267
\(732\) 16320.0i 0.824050i
\(733\) − 9252.00i − 0.466208i −0.972452 0.233104i \(-0.925112\pi\)
0.972452 0.233104i \(-0.0748882\pi\)
\(734\) 51370.0 2.58324
\(735\) 0 0
\(736\) 9605.00 0.481039
\(737\) − 2200.00i − 0.109957i
\(738\) − 6255.00i − 0.311992i
\(739\) −28453.0 −1.41632 −0.708160 0.706052i \(-0.750474\pi\)
−0.708160 + 0.706052i \(0.750474\pi\)
\(740\) 0 0
\(741\) 4512.00 0.223688
\(742\) 2280.00i 0.112805i
\(743\) − 512.000i − 0.0252806i −0.999920 0.0126403i \(-0.995976\pi\)
0.999920 0.0126403i \(-0.00402363\pi\)
\(744\) −24030.0 −1.18412
\(745\) 0 0
\(746\) 68310.0 3.35256
\(747\) 1278.00i 0.0625965i
\(748\) 6171.00i 0.301650i
\(749\) −5814.00 −0.283630
\(750\) 0 0
\(751\) 772.000 0.0375109 0.0187554 0.999824i \(-0.494030\pi\)
0.0187554 + 0.999824i \(0.494030\pi\)
\(752\) 17355.0i 0.841585i
\(753\) 14184.0i 0.686446i
\(754\) 8640.00 0.417308
\(755\) 0 0
\(756\) 1377.00 0.0662447
\(757\) 8058.00i 0.386886i 0.981111 + 0.193443i \(0.0619655\pi\)
−0.981111 + 0.193443i \(0.938034\pi\)
\(758\) 39530.0i 1.89419i
\(759\) 3729.00 0.178332
\(760\) 0 0
\(761\) 18650.0 0.888386 0.444193 0.895931i \(-0.353490\pi\)
0.444193 + 0.895931i \(0.353490\pi\)
\(762\) − 26085.0i − 1.24010i
\(763\) 1728.00i 0.0819893i
\(764\) −61421.0 −2.90855
\(765\) 0 0
\(766\) −15840.0 −0.747157
\(767\) − 20000.0i − 0.941536i
\(768\) 24837.0i 1.16696i
\(769\) 7144.00 0.335005 0.167503 0.985872i \(-0.446430\pi\)
0.167503 + 0.985872i \(0.446430\pi\)
\(770\) 0 0
\(771\) 16254.0 0.759239
\(772\) 71468.0i 3.33185i
\(773\) 1904.00i 0.0885927i 0.999018 + 0.0442963i \(0.0141046\pi\)
−0.999018 + 0.0442963i \(0.985895\pi\)
\(774\) 13860.0 0.643653
\(775\) 0 0
\(776\) 3555.00 0.164455
\(777\) 171.000i 0.00789523i
\(778\) − 53850.0i − 2.48151i
\(779\) −6533.00 −0.300474
\(780\) 0 0
\(781\) 10417.0 0.477272
\(782\) 18645.0i 0.852614i
\(783\) 1458.00i 0.0665449i
\(784\) 29726.0 1.35414
\(785\) 0 0
\(786\) 27270.0 1.23752
\(787\) 7555.00i 0.342194i 0.985254 + 0.171097i \(0.0547311\pi\)
−0.985254 + 0.171097i \(0.945269\pi\)
\(788\) 76789.0i 3.47144i
\(789\) 10062.0 0.454014
\(790\) 0 0
\(791\) −3312.00 −0.148876
\(792\) − 4455.00i − 0.199876i
\(793\) − 10240.0i − 0.458554i
\(794\) −28350.0 −1.26713
\(795\) 0 0
\(796\) 70788.0 3.15203
\(797\) 24950.0i 1.10888i 0.832225 + 0.554438i \(0.187067\pi\)
−0.832225 + 0.554438i \(0.812933\pi\)
\(798\) − 2115.00i − 0.0938223i
\(799\) −6435.00 −0.284924
\(800\) 0 0
\(801\) 3636.00 0.160389
\(802\) 4160.00i 0.183160i
\(803\) − 4928.00i − 0.216570i
\(804\) −10200.0 −0.447421
\(805\) 0 0
\(806\) 28480.0 1.24462
\(807\) 3186.00i 0.138975i
\(808\) 24525.0i 1.06781i
\(809\) −19893.0 −0.864525 −0.432262 0.901748i \(-0.642285\pi\)
−0.432262 + 0.901748i \(0.642285\pi\)
\(810\) 0 0
\(811\) 34503.0 1.49391 0.746957 0.664872i \(-0.231514\pi\)
0.746957 + 0.664872i \(0.231514\pi\)
\(812\) − 2754.00i − 0.119023i
\(813\) 14463.0i 0.623911i
\(814\) 1045.00 0.0449966
\(815\) 0 0
\(816\) 8811.00 0.377998
\(817\) − 14476.0i − 0.619891i
\(818\) 28560.0i 1.22075i
\(819\) −864.000 −0.0368628
\(820\) 0 0
\(821\) 16890.0 0.717984 0.358992 0.933341i \(-0.383120\pi\)
0.358992 + 0.933341i \(0.383120\pi\)
\(822\) 13050.0i 0.553736i
\(823\) − 34692.0i − 1.46936i −0.678411 0.734682i \(-0.737331\pi\)
0.678411 0.734682i \(-0.262669\pi\)
\(824\) 58770.0 2.48465
\(825\) 0 0
\(826\) −9375.00 −0.394913
\(827\) 41424.0i 1.74178i 0.491476 + 0.870891i \(0.336457\pi\)
−0.491476 + 0.870891i \(0.663543\pi\)
\(828\) − 17289.0i − 0.725645i
\(829\) 18494.0 0.774817 0.387408 0.921908i \(-0.373370\pi\)
0.387408 + 0.921908i \(0.373370\pi\)
\(830\) 0 0
\(831\) 12.0000 0.000500933 0
\(832\) − 9184.00i − 0.382690i
\(833\) 11022.0i 0.458451i
\(834\) 9540.00 0.396095
\(835\) 0 0
\(836\) −8789.00 −0.363605
\(837\) 4806.00i 0.198470i
\(838\) 22795.0i 0.939666i
\(839\) −6680.00 −0.274874 −0.137437 0.990511i \(-0.543886\pi\)
−0.137437 + 0.990511i \(0.543886\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 34275.0i 1.40284i
\(843\) − 13941.0i − 0.569577i
\(844\) −79220.0 −3.23088
\(845\) 0 0
\(846\) 8775.00 0.356608
\(847\) 363.000i 0.0147259i
\(848\) − 13528.0i − 0.547822i
\(849\) 12849.0 0.519407
\(850\) 0 0
\(851\) 2147.00 0.0864844
\(852\) − 48297.0i − 1.94205i
\(853\) − 43358.0i − 1.74039i −0.492711 0.870193i \(-0.663994\pi\)
0.492711 0.870193i \(-0.336006\pi\)
\(854\) −4800.00 −0.192333
\(855\) 0 0
\(856\) 87210.0 3.48222
\(857\) 15585.0i 0.621206i 0.950540 + 0.310603i \(0.100531\pi\)
−0.950540 + 0.310603i \(0.899469\pi\)
\(858\) 5280.00i 0.210089i
\(859\) 17036.0 0.676672 0.338336 0.941025i \(-0.390136\pi\)
0.338336 + 0.941025i \(0.390136\pi\)
\(860\) 0 0
\(861\) 1251.00 0.0495168
\(862\) 53850.0i 2.12777i
\(863\) − 28064.0i − 1.10696i −0.832861 0.553482i \(-0.813299\pi\)
0.832861 0.553482i \(-0.186701\pi\)
\(864\) −2295.00 −0.0903675
\(865\) 0 0
\(866\) 42490.0 1.66729
\(867\) − 11472.0i − 0.449377i
\(868\) − 9078.00i − 0.354985i
\(869\) −7931.00 −0.309598
\(870\) 0 0
\(871\) 6400.00 0.248973
\(872\) − 25920.0i − 1.00661i
\(873\) − 711.000i − 0.0275644i
\(874\) −26555.0 −1.02773
\(875\) 0 0
\(876\) −22848.0 −0.881236
\(877\) − 22654.0i − 0.872259i −0.899884 0.436130i \(-0.856349\pi\)
0.899884 0.436130i \(-0.143651\pi\)
\(878\) − 49175.0i − 1.89018i
\(879\) 20433.0 0.784059
\(880\) 0 0
\(881\) −22380.0 −0.855847 −0.427924 0.903815i \(-0.640755\pi\)
−0.427924 + 0.903815i \(0.640755\pi\)
\(882\) − 15030.0i − 0.573794i
\(883\) − 35174.0i − 1.34054i −0.742116 0.670271i \(-0.766178\pi\)
0.742116 0.670271i \(-0.233822\pi\)
\(884\) −17952.0 −0.683022
\(885\) 0 0
\(886\) −53725.0 −2.03716
\(887\) 30868.0i 1.16848i 0.811579 + 0.584242i \(0.198608\pi\)
−0.811579 + 0.584242i \(0.801392\pi\)
\(888\) − 2565.00i − 0.0969322i
\(889\) 5217.00 0.196820
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 60520.0i 2.27170i
\(893\) − 9165.00i − 0.343443i
\(894\) 3585.00 0.134117
\(895\) 0 0
\(896\) −6345.00 −0.236575
\(897\) 10848.0i 0.403795i
\(898\) − 41780.0i − 1.55258i
\(899\) 9612.00 0.356594
\(900\) 0 0
\(901\) 5016.00 0.185469
\(902\) − 7645.00i − 0.282207i
\(903\) 2772.00i 0.102155i
\(904\) 49680.0 1.82780
\(905\) 0 0
\(906\) 18120.0 0.664455
\(907\) 10070.0i 0.368654i 0.982865 + 0.184327i \(0.0590105\pi\)
−0.982865 + 0.184327i \(0.940990\pi\)
\(908\) 79526.0i 2.90657i
\(909\) 4905.00 0.178975
\(910\) 0 0
\(911\) 1885.00 0.0685542 0.0342771 0.999412i \(-0.489087\pi\)
0.0342771 + 0.999412i \(0.489087\pi\)
\(912\) 12549.0i 0.455635i
\(913\) 1562.00i 0.0566207i
\(914\) 35290.0 1.27712
\(915\) 0 0
\(916\) −75599.0 −2.72692
\(917\) 5454.00i 0.196409i
\(918\) − 4455.00i − 0.160171i
\(919\) −23703.0 −0.850805 −0.425403 0.905004i \(-0.639867\pi\)
−0.425403 + 0.905004i \(0.639867\pi\)
\(920\) 0 0
\(921\) −1380.00 −0.0493730
\(922\) 3230.00i 0.115374i
\(923\) 30304.0i 1.08068i
\(924\) 1683.00 0.0599206
\(925\) 0 0
\(926\) −44910.0 −1.59377
\(927\) − 11754.0i − 0.416453i
\(928\) 4590.00i 0.162364i
\(929\) −53804.0 −1.90016 −0.950082 0.312001i \(-0.899001\pi\)
−0.950082 + 0.312001i \(0.899001\pi\)
\(930\) 0 0
\(931\) −15698.0 −0.552611
\(932\) 6987.00i 0.245565i
\(933\) − 24984.0i − 0.876677i
\(934\) 67380.0 2.36054
\(935\) 0 0
\(936\) 12960.0 0.452576
\(937\) 1326.00i 0.0462311i 0.999733 + 0.0231155i \(0.00735856\pi\)
−0.999733 + 0.0231155i \(0.992641\pi\)
\(938\) − 3000.00i − 0.104428i
\(939\) 17787.0 0.618165
\(940\) 0 0
\(941\) −27109.0 −0.939137 −0.469569 0.882896i \(-0.655591\pi\)
−0.469569 + 0.882896i \(0.655591\pi\)
\(942\) − 28110.0i − 0.972265i
\(943\) − 15707.0i − 0.542408i
\(944\) 55625.0 1.91784
\(945\) 0 0
\(946\) 16940.0 0.582206
\(947\) 31143.0i 1.06865i 0.845279 + 0.534325i \(0.179434\pi\)
−0.845279 + 0.534325i \(0.820566\pi\)
\(948\) 36771.0i 1.25977i
\(949\) 14336.0 0.490375
\(950\) 0 0
\(951\) 15120.0 0.515562
\(952\) 4455.00i 0.151667i
\(953\) 879.000i 0.0298779i 0.999888 + 0.0149389i \(0.00475539\pi\)
−0.999888 + 0.0149389i \(0.995245\pi\)
\(954\) −6840.00 −0.232131
\(955\) 0 0
\(956\) −108460. −3.66930
\(957\) 1782.00i 0.0601921i
\(958\) − 64980.0i − 2.19145i
\(959\) −2610.00 −0.0878846
\(960\) 0 0
\(961\) 1893.00 0.0635427
\(962\) 3040.00i 0.101885i
\(963\) − 17442.0i − 0.583656i
\(964\) −123794. −4.13603
\(965\) 0 0
\(966\) 5085.00 0.169366
\(967\) − 14824.0i − 0.492976i −0.969146 0.246488i \(-0.920723\pi\)
0.969146 0.246488i \(-0.0792766\pi\)
\(968\) − 5445.00i − 0.180794i
\(969\) −4653.00 −0.154258
\(970\) 0 0
\(971\) 34089.0 1.12664 0.563320 0.826239i \(-0.309524\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(972\) 4131.00i 0.136319i
\(973\) 1908.00i 0.0628650i
\(974\) 30130.0 0.991199
\(975\) 0 0
\(976\) 28480.0 0.934040
\(977\) 33446.0i 1.09522i 0.836733 + 0.547611i \(0.184463\pi\)
−0.836733 + 0.547611i \(0.815537\pi\)
\(978\) 28560.0i 0.933792i
\(979\) 4444.00 0.145077
\(980\) 0 0
\(981\) −5184.00 −0.168718
\(982\) 58490.0i 1.90070i
\(983\) 52025.0i 1.68804i 0.536315 + 0.844018i \(0.319816\pi\)
−0.536315 + 0.844018i \(0.680184\pi\)
\(984\) −18765.0 −0.607933
\(985\) 0 0
\(986\) −8910.00 −0.287781
\(987\) 1755.00i 0.0565980i
\(988\) − 25568.0i − 0.823306i
\(989\) 34804.0 1.11901
\(990\) 0 0
\(991\) −41260.0 −1.32257 −0.661285 0.750135i \(-0.729989\pi\)
−0.661285 + 0.750135i \(0.729989\pi\)
\(992\) 15130.0i 0.484252i
\(993\) − 31188.0i − 0.996698i
\(994\) 14205.0 0.453275
\(995\) 0 0
\(996\) 7242.00 0.230393
\(997\) − 190.000i − 0.00603547i −0.999995 0.00301773i \(-0.999039\pi\)
0.999995 0.00301773i \(-0.000960576\pi\)
\(998\) 85260.0i 2.70427i
\(999\) −513.000 −0.0162468
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 825.4.c.b.199.2 2
5.2 odd 4 825.4.a.a.1.1 1
5.3 odd 4 825.4.a.j.1.1 yes 1
5.4 even 2 inner 825.4.c.b.199.1 2
15.2 even 4 2475.4.a.k.1.1 1
15.8 even 4 2475.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 5.2 odd 4
825.4.a.j.1.1 yes 1 5.3 odd 4
825.4.c.b.199.1 2 5.4 even 2 inner
825.4.c.b.199.2 2 1.1 even 1 trivial
2475.4.a.a.1.1 1 15.8 even 4
2475.4.a.k.1.1 1 15.2 even 4