Properties

Label 50.4.b.a.49.1
Level $50$
Weight $4$
Character 50.49
Analytic conductor $2.950$
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] = [50,4,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(2.95009550029\)
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 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.4.b.a.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} +4.00000i q^{7} +8.00000i q^{8} -37.0000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} -8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} +4.00000i q^{7} +8.00000i q^{8} -37.0000 q^{9} +12.0000 q^{11} +32.0000i q^{12} -58.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} -66.0000i q^{17} +74.0000i q^{18} +100.000 q^{19} +32.0000 q^{21} -24.0000i q^{22} +132.000i q^{23} +64.0000 q^{24} -116.000 q^{26} +80.0000i q^{27} -16.0000i q^{28} +90.0000 q^{29} +152.000 q^{31} -32.0000i q^{32} -96.0000i q^{33} -132.000 q^{34} +148.000 q^{36} +34.0000i q^{37} -200.000i q^{38} -464.000 q^{39} -438.000 q^{41} -64.0000i q^{42} +32.0000i q^{43} -48.0000 q^{44} +264.000 q^{46} +204.000i q^{47} -128.000i q^{48} +327.000 q^{49} -528.000 q^{51} +232.000i q^{52} +222.000i q^{53} +160.000 q^{54} -32.0000 q^{56} -800.000i q^{57} -180.000i q^{58} -420.000 q^{59} +902.000 q^{61} -304.000i q^{62} -148.000i q^{63} -64.0000 q^{64} -192.000 q^{66} +1024.00i q^{67} +264.000i q^{68} +1056.00 q^{69} +432.000 q^{71} -296.000i q^{72} +362.000i q^{73} +68.0000 q^{74} -400.000 q^{76} +48.0000i q^{77} +928.000i q^{78} +160.000 q^{79} -359.000 q^{81} +876.000i q^{82} +72.0000i q^{83} -128.000 q^{84} +64.0000 q^{86} -720.000i q^{87} +96.0000i q^{88} -810.000 q^{89} +232.000 q^{91} -528.000i q^{92} -1216.00i q^{93} +408.000 q^{94} -256.000 q^{96} -1106.00i q^{97} -654.000i q^{98} -444.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9} + 24 q^{11} + 16 q^{14} + 32 q^{16} + 200 q^{19} + 64 q^{21} + 128 q^{24} - 232 q^{26} + 180 q^{29} + 304 q^{31} - 264 q^{34} + 296 q^{36} - 928 q^{39} - 876 q^{41} - 96 q^{44} + 528 q^{46} + 654 q^{49} - 1056 q^{51} + 320 q^{54} - 64 q^{56} - 840 q^{59} + 1804 q^{61} - 128 q^{64} - 384 q^{66} + 2112 q^{69} + 864 q^{71} + 136 q^{74} - 800 q^{76} + 320 q^{79} - 718 q^{81} - 256 q^{84} + 128 q^{86} - 1620 q^{89} + 464 q^{91} + 816 q^{94} - 512 q^{96} - 888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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
\(3\) − 8.00000i − 1.53960i −0.638285 0.769800i \(-0.720356\pi\)
0.638285 0.769800i \(-0.279644\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −16.0000 −1.08866
\(7\) 4.00000i 0.215980i 0.994152 + 0.107990i \(0.0344414\pi\)
−0.994152 + 0.107990i \(0.965559\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −37.0000 −1.37037
\(10\) 0 0
\(11\) 12.0000 0.328921 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 32.0000i 0.769800i
\(13\) − 58.0000i − 1.23741i −0.785624 0.618704i \(-0.787658\pi\)
0.785624 0.618704i \(-0.212342\pi\)
\(14\) 8.00000 0.152721
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 66.0000i − 0.941609i −0.882238 0.470804i \(-0.843964\pi\)
0.882238 0.470804i \(-0.156036\pi\)
\(18\) 74.0000i 0.968998i
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 0 0
\(21\) 32.0000 0.332522
\(22\) − 24.0000i − 0.232583i
\(23\) 132.000i 1.19669i 0.801238 + 0.598346i \(0.204175\pi\)
−0.801238 + 0.598346i \(0.795825\pi\)
\(24\) 64.0000 0.544331
\(25\) 0 0
\(26\) −116.000 −0.874980
\(27\) 80.0000i 0.570222i
\(28\) − 16.0000i − 0.107990i
\(29\) 90.0000 0.576296 0.288148 0.957586i \(-0.406961\pi\)
0.288148 + 0.957586i \(0.406961\pi\)
\(30\) 0 0
\(31\) 152.000 0.880645 0.440323 0.897840i \(-0.354864\pi\)
0.440323 + 0.897840i \(0.354864\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 96.0000i − 0.506408i
\(34\) −132.000 −0.665818
\(35\) 0 0
\(36\) 148.000 0.685185
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) − 200.000i − 0.853797i
\(39\) −464.000 −1.90511
\(40\) 0 0
\(41\) −438.000 −1.66839 −0.834196 0.551467i \(-0.814068\pi\)
−0.834196 + 0.551467i \(0.814068\pi\)
\(42\) − 64.0000i − 0.235129i
\(43\) 32.0000i 0.113487i 0.998389 + 0.0567437i \(0.0180718\pi\)
−0.998389 + 0.0567437i \(0.981928\pi\)
\(44\) −48.0000 −0.164461
\(45\) 0 0
\(46\) 264.000 0.846189
\(47\) 204.000i 0.633116i 0.948573 + 0.316558i \(0.102527\pi\)
−0.948573 + 0.316558i \(0.897473\pi\)
\(48\) − 128.000i − 0.384900i
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) −528.000 −1.44970
\(52\) 232.000i 0.618704i
\(53\) 222.000i 0.575359i 0.957727 + 0.287680i \(0.0928838\pi\)
−0.957727 + 0.287680i \(0.907116\pi\)
\(54\) 160.000 0.403208
\(55\) 0 0
\(56\) −32.0000 −0.0763604
\(57\) − 800.000i − 1.85899i
\(58\) − 180.000i − 0.407503i
\(59\) −420.000 −0.926769 −0.463384 0.886157i \(-0.653365\pi\)
−0.463384 + 0.886157i \(0.653365\pi\)
\(60\) 0 0
\(61\) 902.000 1.89327 0.946633 0.322312i \(-0.104460\pi\)
0.946633 + 0.322312i \(0.104460\pi\)
\(62\) − 304.000i − 0.622710i
\(63\) − 148.000i − 0.295972i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −192.000 −0.358084
\(67\) 1024.00i 1.86719i 0.358334 + 0.933593i \(0.383345\pi\)
−0.358334 + 0.933593i \(0.616655\pi\)
\(68\) 264.000i 0.470804i
\(69\) 1056.00 1.84243
\(70\) 0 0
\(71\) 432.000 0.722098 0.361049 0.932547i \(-0.382419\pi\)
0.361049 + 0.932547i \(0.382419\pi\)
\(72\) − 296.000i − 0.484499i
\(73\) 362.000i 0.580396i 0.956967 + 0.290198i \(0.0937211\pi\)
−0.956967 + 0.290198i \(0.906279\pi\)
\(74\) 68.0000 0.106822
\(75\) 0 0
\(76\) −400.000 −0.603726
\(77\) 48.0000i 0.0710404i
\(78\) 928.000i 1.34712i
\(79\) 160.000 0.227866 0.113933 0.993488i \(-0.463655\pi\)
0.113933 + 0.993488i \(0.463655\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 876.000i 1.17973i
\(83\) 72.0000i 0.0952172i 0.998866 + 0.0476086i \(0.0151600\pi\)
−0.998866 + 0.0476086i \(0.984840\pi\)
\(84\) −128.000 −0.166261
\(85\) 0 0
\(86\) 64.0000 0.0802476
\(87\) − 720.000i − 0.887266i
\(88\) 96.0000i 0.116291i
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 0 0
\(91\) 232.000 0.267255
\(92\) − 528.000i − 0.598346i
\(93\) − 1216.00i − 1.35584i
\(94\) 408.000 0.447681
\(95\) 0 0
\(96\) −256.000 −0.272166
\(97\) − 1106.00i − 1.15770i −0.815433 0.578852i \(-0.803501\pi\)
0.815433 0.578852i \(-0.196499\pi\)
\(98\) − 654.000i − 0.674122i
\(99\) −444.000 −0.450744
\(100\) 0 0
\(101\) −258.000 −0.254178 −0.127089 0.991891i \(-0.540563\pi\)
−0.127089 + 0.991891i \(0.540563\pi\)
\(102\) 1056.00i 1.02509i
\(103\) − 988.000i − 0.945151i −0.881290 0.472575i \(-0.843324\pi\)
0.881290 0.472575i \(-0.156676\pi\)
\(104\) 464.000 0.437490
\(105\) 0 0
\(106\) 444.000 0.406840
\(107\) 24.0000i 0.0216838i 0.999941 + 0.0108419i \(0.00345115\pi\)
−0.999941 + 0.0108419i \(0.996549\pi\)
\(108\) − 320.000i − 0.285111i
\(109\) −950.000 −0.834803 −0.417401 0.908722i \(-0.637059\pi\)
−0.417401 + 0.908722i \(0.637059\pi\)
\(110\) 0 0
\(111\) 272.000 0.232586
\(112\) 64.0000i 0.0539949i
\(113\) − 1038.00i − 0.864131i −0.901842 0.432066i \(-0.857785\pi\)
0.901842 0.432066i \(-0.142215\pi\)
\(114\) −1600.00 −1.31451
\(115\) 0 0
\(116\) −360.000 −0.288148
\(117\) 2146.00i 1.69571i
\(118\) 840.000i 0.655324i
\(119\) 264.000 0.203368
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) − 1804.00i − 1.33874i
\(123\) 3504.00i 2.56866i
\(124\) −608.000 −0.440323
\(125\) 0 0
\(126\) −296.000 −0.209284
\(127\) 124.000i 0.0866395i 0.999061 + 0.0433198i \(0.0137934\pi\)
−0.999061 + 0.0433198i \(0.986207\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 256.000 0.174725
\(130\) 0 0
\(131\) 132.000 0.0880374 0.0440187 0.999031i \(-0.485984\pi\)
0.0440187 + 0.999031i \(0.485984\pi\)
\(132\) 384.000i 0.253204i
\(133\) 400.000i 0.260785i
\(134\) 2048.00 1.32030
\(135\) 0 0
\(136\) 528.000 0.332909
\(137\) 1254.00i 0.782018i 0.920387 + 0.391009i \(0.127874\pi\)
−0.920387 + 0.391009i \(0.872126\pi\)
\(138\) − 2112.00i − 1.30279i
\(139\) 2860.00 1.74519 0.872597 0.488440i \(-0.162434\pi\)
0.872597 + 0.488440i \(0.162434\pi\)
\(140\) 0 0
\(141\) 1632.00 0.974746
\(142\) − 864.000i − 0.510600i
\(143\) − 696.000i − 0.407010i
\(144\) −592.000 −0.342593
\(145\) 0 0
\(146\) 724.000 0.410402
\(147\) − 2616.00i − 1.46778i
\(148\) − 136.000i − 0.0755347i
\(149\) −750.000 −0.412365 −0.206183 0.978514i \(-0.566104\pi\)
−0.206183 + 0.978514i \(0.566104\pi\)
\(150\) 0 0
\(151\) −448.000 −0.241442 −0.120721 0.992686i \(-0.538521\pi\)
−0.120721 + 0.992686i \(0.538521\pi\)
\(152\) 800.000i 0.426898i
\(153\) 2442.00i 1.29035i
\(154\) 96.0000 0.0502331
\(155\) 0 0
\(156\) 1856.00 0.952557
\(157\) − 2246.00i − 1.14172i −0.821047 0.570861i \(-0.806610\pi\)
0.821047 0.570861i \(-0.193390\pi\)
\(158\) − 320.000i − 0.161126i
\(159\) 1776.00 0.885824
\(160\) 0 0
\(161\) −528.000 −0.258461
\(162\) 718.000i 0.348219i
\(163\) − 568.000i − 0.272940i −0.990644 0.136470i \(-0.956424\pi\)
0.990644 0.136470i \(-0.0435757\pi\)
\(164\) 1752.00 0.834196
\(165\) 0 0
\(166\) 144.000 0.0673287
\(167\) 1524.00i 0.706172i 0.935591 + 0.353086i \(0.114868\pi\)
−0.935591 + 0.353086i \(0.885132\pi\)
\(168\) 256.000i 0.117564i
\(169\) −1167.00 −0.531179
\(170\) 0 0
\(171\) −3700.00 −1.65466
\(172\) − 128.000i − 0.0567437i
\(173\) 3702.00i 1.62692i 0.581618 + 0.813462i \(0.302420\pi\)
−0.581618 + 0.813462i \(0.697580\pi\)
\(174\) −1440.00 −0.627391
\(175\) 0 0
\(176\) 192.000 0.0822304
\(177\) 3360.00i 1.42685i
\(178\) 1620.00i 0.682158i
\(179\) −3180.00 −1.32785 −0.663923 0.747801i \(-0.731110\pi\)
−0.663923 + 0.747801i \(0.731110\pi\)
\(180\) 0 0
\(181\) −2098.00 −0.861564 −0.430782 0.902456i \(-0.641762\pi\)
−0.430782 + 0.902456i \(0.641762\pi\)
\(182\) − 464.000i − 0.188978i
\(183\) − 7216.00i − 2.91487i
\(184\) −1056.00 −0.423094
\(185\) 0 0
\(186\) −2432.00 −0.958725
\(187\) − 792.000i − 0.309715i
\(188\) − 816.000i − 0.316558i
\(189\) −320.000 −0.123156
\(190\) 0 0
\(191\) 4392.00 1.66384 0.831921 0.554894i \(-0.187241\pi\)
0.831921 + 0.554894i \(0.187241\pi\)
\(192\) 512.000i 0.192450i
\(193\) − 2158.00i − 0.804851i −0.915453 0.402425i \(-0.868167\pi\)
0.915453 0.402425i \(-0.131833\pi\)
\(194\) −2212.00 −0.818620
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) 1074.00i 0.388423i 0.980960 + 0.194212i \(0.0622148\pi\)
−0.980960 + 0.194212i \(0.937785\pi\)
\(198\) 888.000i 0.318724i
\(199\) −2840.00 −1.01167 −0.505835 0.862630i \(-0.668815\pi\)
−0.505835 + 0.862630i \(0.668815\pi\)
\(200\) 0 0
\(201\) 8192.00 2.87472
\(202\) 516.000i 0.179731i
\(203\) 360.000i 0.124468i
\(204\) 2112.00 0.724851
\(205\) 0 0
\(206\) −1976.00 −0.668323
\(207\) − 4884.00i − 1.63991i
\(208\) − 928.000i − 0.309352i
\(209\) 1200.00 0.397157
\(210\) 0 0
\(211\) −2668.00 −0.870487 −0.435243 0.900313i \(-0.643338\pi\)
−0.435243 + 0.900313i \(0.643338\pi\)
\(212\) − 888.000i − 0.287680i
\(213\) − 3456.00i − 1.11174i
\(214\) 48.0000 0.0153328
\(215\) 0 0
\(216\) −640.000 −0.201604
\(217\) 608.000i 0.190202i
\(218\) 1900.00i 0.590295i
\(219\) 2896.00 0.893578
\(220\) 0 0
\(221\) −3828.00 −1.16515
\(222\) − 544.000i − 0.164463i
\(223\) 1772.00i 0.532116i 0.963957 + 0.266058i \(0.0857213\pi\)
−0.963957 + 0.266058i \(0.914279\pi\)
\(224\) 128.000 0.0381802
\(225\) 0 0
\(226\) −2076.00 −0.611033
\(227\) 2784.00i 0.814011i 0.913426 + 0.407006i \(0.133427\pi\)
−0.913426 + 0.407006i \(0.866573\pi\)
\(228\) 3200.00i 0.929496i
\(229\) −350.000 −0.100998 −0.0504992 0.998724i \(-0.516081\pi\)
−0.0504992 + 0.998724i \(0.516081\pi\)
\(230\) 0 0
\(231\) 384.000 0.109374
\(232\) 720.000i 0.203751i
\(233\) 1962.00i 0.551652i 0.961208 + 0.275826i \(0.0889513\pi\)
−0.961208 + 0.275826i \(0.911049\pi\)
\(234\) 4292.00 1.19905
\(235\) 0 0
\(236\) 1680.00 0.463384
\(237\) − 1280.00i − 0.350823i
\(238\) − 528.000i − 0.143803i
\(239\) 4320.00 1.16919 0.584597 0.811324i \(-0.301252\pi\)
0.584597 + 0.811324i \(0.301252\pi\)
\(240\) 0 0
\(241\) −478.000 −0.127762 −0.0638811 0.997958i \(-0.520348\pi\)
−0.0638811 + 0.997958i \(0.520348\pi\)
\(242\) 2374.00i 0.630605i
\(243\) 5032.00i 1.32841i
\(244\) −3608.00 −0.946633
\(245\) 0 0
\(246\) 7008.00 1.81632
\(247\) − 5800.00i − 1.49411i
\(248\) 1216.00i 0.311355i
\(249\) 576.000 0.146596
\(250\) 0 0
\(251\) 2652.00 0.666903 0.333452 0.942767i \(-0.391787\pi\)
0.333452 + 0.942767i \(0.391787\pi\)
\(252\) 592.000i 0.147986i
\(253\) 1584.00i 0.393617i
\(254\) 248.000 0.0612634
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 2334.00i 0.566502i 0.959046 + 0.283251i \(0.0914129\pi\)
−0.959046 + 0.283251i \(0.908587\pi\)
\(258\) − 512.000i − 0.123549i
\(259\) −136.000 −0.0326279
\(260\) 0 0
\(261\) −3330.00 −0.789739
\(262\) − 264.000i − 0.0622518i
\(263\) − 3948.00i − 0.925643i −0.886451 0.462822i \(-0.846837\pi\)
0.886451 0.462822i \(-0.153163\pi\)
\(264\) 768.000 0.179042
\(265\) 0 0
\(266\) 800.000 0.184403
\(267\) 6480.00i 1.48528i
\(268\) − 4096.00i − 0.933593i
\(269\) −1590.00 −0.360387 −0.180193 0.983631i \(-0.557672\pi\)
−0.180193 + 0.983631i \(0.557672\pi\)
\(270\) 0 0
\(271\) 4952.00 1.11001 0.555005 0.831847i \(-0.312716\pi\)
0.555005 + 0.831847i \(0.312716\pi\)
\(272\) − 1056.00i − 0.235402i
\(273\) − 1856.00i − 0.411466i
\(274\) 2508.00 0.552970
\(275\) 0 0
\(276\) −4224.00 −0.921213
\(277\) − 1646.00i − 0.357034i −0.983937 0.178517i \(-0.942870\pi\)
0.983937 0.178517i \(-0.0571300\pi\)
\(278\) − 5720.00i − 1.23404i
\(279\) −5624.00 −1.20681
\(280\) 0 0
\(281\) −1158.00 −0.245838 −0.122919 0.992417i \(-0.539226\pi\)
−0.122919 + 0.992417i \(0.539226\pi\)
\(282\) − 3264.00i − 0.689250i
\(283\) 6992.00i 1.46866i 0.678792 + 0.734331i \(0.262504\pi\)
−0.678792 + 0.734331i \(0.737496\pi\)
\(284\) −1728.00 −0.361049
\(285\) 0 0
\(286\) −1392.00 −0.287800
\(287\) − 1752.00i − 0.360339i
\(288\) 1184.00i 0.242250i
\(289\) 557.000 0.113373
\(290\) 0 0
\(291\) −8848.00 −1.78240
\(292\) − 1448.00i − 0.290198i
\(293\) − 258.000i − 0.0514421i −0.999669 0.0257210i \(-0.991812\pi\)
0.999669 0.0257210i \(-0.00818816\pi\)
\(294\) −5232.00 −1.03788
\(295\) 0 0
\(296\) −272.000 −0.0534111
\(297\) 960.000i 0.187558i
\(298\) 1500.00i 0.291586i
\(299\) 7656.00 1.48080
\(300\) 0 0
\(301\) −128.000 −0.0245110
\(302\) 896.000i 0.170725i
\(303\) 2064.00i 0.391332i
\(304\) 1600.00 0.301863
\(305\) 0 0
\(306\) 4884.00 0.912417
\(307\) 8944.00i 1.66274i 0.555720 + 0.831370i \(0.312443\pi\)
−0.555720 + 0.831370i \(0.687557\pi\)
\(308\) − 192.000i − 0.0355202i
\(309\) −7904.00 −1.45515
\(310\) 0 0
\(311\) 1392.00 0.253804 0.126902 0.991915i \(-0.459497\pi\)
0.126902 + 0.991915i \(0.459497\pi\)
\(312\) − 3712.00i − 0.673560i
\(313\) − 5878.00i − 1.06148i −0.847534 0.530742i \(-0.821913\pi\)
0.847534 0.530742i \(-0.178087\pi\)
\(314\) −4492.00 −0.807319
\(315\) 0 0
\(316\) −640.000 −0.113933
\(317\) − 10326.0i − 1.82955i −0.403969 0.914773i \(-0.632370\pi\)
0.403969 0.914773i \(-0.367630\pi\)
\(318\) − 3552.00i − 0.626372i
\(319\) 1080.00 0.189556
\(320\) 0 0
\(321\) 192.000 0.0333844
\(322\) 1056.00i 0.182760i
\(323\) − 6600.00i − 1.13695i
\(324\) 1436.00 0.246228
\(325\) 0 0
\(326\) −1136.00 −0.192998
\(327\) 7600.00i 1.28526i
\(328\) − 3504.00i − 0.589866i
\(329\) −816.000 −0.136740
\(330\) 0 0
\(331\) −4228.00 −0.702090 −0.351045 0.936359i \(-0.614174\pi\)
−0.351045 + 0.936359i \(0.614174\pi\)
\(332\) − 288.000i − 0.0476086i
\(333\) − 1258.00i − 0.207021i
\(334\) 3048.00 0.499339
\(335\) 0 0
\(336\) 512.000 0.0831306
\(337\) − 1106.00i − 0.178776i −0.995997 0.0893882i \(-0.971509\pi\)
0.995997 0.0893882i \(-0.0284912\pi\)
\(338\) 2334.00i 0.375600i
\(339\) −8304.00 −1.33042
\(340\) 0 0
\(341\) 1824.00 0.289663
\(342\) 7400.00i 1.17002i
\(343\) 2680.00i 0.421885i
\(344\) −256.000 −0.0401238
\(345\) 0 0
\(346\) 7404.00 1.15041
\(347\) − 9336.00i − 1.44433i −0.691720 0.722165i \(-0.743147\pi\)
0.691720 0.722165i \(-0.256853\pi\)
\(348\) 2880.00i 0.443633i
\(349\) 11770.0 1.80525 0.902627 0.430424i \(-0.141636\pi\)
0.902627 + 0.430424i \(0.141636\pi\)
\(350\) 0 0
\(351\) 4640.00 0.705598
\(352\) − 384.000i − 0.0581456i
\(353\) 8322.00i 1.25477i 0.778707 + 0.627387i \(0.215876\pi\)
−0.778707 + 0.627387i \(0.784124\pi\)
\(354\) 6720.00 1.00894
\(355\) 0 0
\(356\) 3240.00 0.482359
\(357\) − 2112.00i − 0.313106i
\(358\) 6360.00i 0.938929i
\(359\) −10680.0 −1.57011 −0.785054 0.619427i \(-0.787365\pi\)
−0.785054 + 0.619427i \(0.787365\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 4196.00i 0.609218i
\(363\) 9496.00i 1.37303i
\(364\) −928.000 −0.133628
\(365\) 0 0
\(366\) −14432.0 −2.06113
\(367\) 5884.00i 0.836900i 0.908240 + 0.418450i \(0.137426\pi\)
−0.908240 + 0.418450i \(0.862574\pi\)
\(368\) 2112.00i 0.299173i
\(369\) 16206.0 2.28632
\(370\) 0 0
\(371\) −888.000 −0.124266
\(372\) 4864.00i 0.677921i
\(373\) − 2098.00i − 0.291234i −0.989341 0.145617i \(-0.953483\pi\)
0.989341 0.145617i \(-0.0465167\pi\)
\(374\) −1584.00 −0.219002
\(375\) 0 0
\(376\) −1632.00 −0.223840
\(377\) − 5220.00i − 0.713113i
\(378\) 640.000i 0.0870848i
\(379\) −3860.00 −0.523153 −0.261576 0.965183i \(-0.584242\pi\)
−0.261576 + 0.965183i \(0.584242\pi\)
\(380\) 0 0
\(381\) 992.000 0.133390
\(382\) − 8784.00i − 1.17651i
\(383\) − 9588.00i − 1.27917i −0.768718 0.639587i \(-0.779105\pi\)
0.768718 0.639587i \(-0.220895\pi\)
\(384\) 1024.00 0.136083
\(385\) 0 0
\(386\) −4316.00 −0.569116
\(387\) − 1184.00i − 0.155520i
\(388\) 4424.00i 0.578852i
\(389\) 13410.0 1.74785 0.873925 0.486060i \(-0.161566\pi\)
0.873925 + 0.486060i \(0.161566\pi\)
\(390\) 0 0
\(391\) 8712.00 1.12682
\(392\) 2616.00i 0.337061i
\(393\) − 1056.00i − 0.135542i
\(394\) 2148.00 0.274657
\(395\) 0 0
\(396\) 1776.00 0.225372
\(397\) 13114.0i 1.65787i 0.559348 + 0.828933i \(0.311052\pi\)
−0.559348 + 0.828933i \(0.688948\pi\)
\(398\) 5680.00i 0.715358i
\(399\) 3200.00 0.401505
\(400\) 0 0
\(401\) −5838.00 −0.727022 −0.363511 0.931590i \(-0.618422\pi\)
−0.363511 + 0.931590i \(0.618422\pi\)
\(402\) − 16384.0i − 2.03274i
\(403\) − 8816.00i − 1.08972i
\(404\) 1032.00 0.127089
\(405\) 0 0
\(406\) 720.000 0.0880123
\(407\) 408.000i 0.0496899i
\(408\) − 4224.00i − 0.512547i
\(409\) −9530.00 −1.15215 −0.576074 0.817398i \(-0.695416\pi\)
−0.576074 + 0.817398i \(0.695416\pi\)
\(410\) 0 0
\(411\) 10032.0 1.20400
\(412\) 3952.00i 0.472575i
\(413\) − 1680.00i − 0.200163i
\(414\) −9768.00 −1.15959
\(415\) 0 0
\(416\) −1856.00 −0.218745
\(417\) − 22880.0i − 2.68690i
\(418\) − 2400.00i − 0.280832i
\(419\) −7260.00 −0.846478 −0.423239 0.906018i \(-0.639107\pi\)
−0.423239 + 0.906018i \(0.639107\pi\)
\(420\) 0 0
\(421\) 12062.0 1.39636 0.698178 0.715924i \(-0.253994\pi\)
0.698178 + 0.715924i \(0.253994\pi\)
\(422\) 5336.00i 0.615527i
\(423\) − 7548.00i − 0.867604i
\(424\) −1776.00 −0.203420
\(425\) 0 0
\(426\) −6912.00 −0.786121
\(427\) 3608.00i 0.408907i
\(428\) − 96.0000i − 0.0108419i
\(429\) −5568.00 −0.626633
\(430\) 0 0
\(431\) −13608.0 −1.52082 −0.760411 0.649442i \(-0.775002\pi\)
−0.760411 + 0.649442i \(0.775002\pi\)
\(432\) 1280.00i 0.142556i
\(433\) − 3838.00i − 0.425964i −0.977056 0.212982i \(-0.931682\pi\)
0.977056 0.212982i \(-0.0683176\pi\)
\(434\) 1216.00 0.134493
\(435\) 0 0
\(436\) 3800.00 0.417401
\(437\) 13200.0i 1.44495i
\(438\) − 5792.00i − 0.631855i
\(439\) −7400.00 −0.804516 −0.402258 0.915526i \(-0.631775\pi\)
−0.402258 + 0.915526i \(0.631775\pi\)
\(440\) 0 0
\(441\) −12099.0 −1.30645
\(442\) 7656.00i 0.823889i
\(443\) 8352.00i 0.895746i 0.894097 + 0.447873i \(0.147818\pi\)
−0.894097 + 0.447873i \(0.852182\pi\)
\(444\) −1088.00 −0.116293
\(445\) 0 0
\(446\) 3544.00 0.376263
\(447\) 6000.00i 0.634878i
\(448\) − 256.000i − 0.0269975i
\(449\) −10770.0 −1.13200 −0.566000 0.824405i \(-0.691510\pi\)
−0.566000 + 0.824405i \(0.691510\pi\)
\(450\) 0 0
\(451\) −5256.00 −0.548770
\(452\) 4152.00i 0.432066i
\(453\) 3584.00i 0.371724i
\(454\) 5568.00 0.575593
\(455\) 0 0
\(456\) 6400.00 0.657253
\(457\) 6694.00i 0.685191i 0.939483 + 0.342595i \(0.111306\pi\)
−0.939483 + 0.342595i \(0.888694\pi\)
\(458\) 700.000i 0.0714167i
\(459\) 5280.00 0.536927
\(460\) 0 0
\(461\) −3018.00 −0.304907 −0.152454 0.988311i \(-0.548717\pi\)
−0.152454 + 0.988311i \(0.548717\pi\)
\(462\) − 768.000i − 0.0773389i
\(463\) 14492.0i 1.45464i 0.686296 + 0.727322i \(0.259235\pi\)
−0.686296 + 0.727322i \(0.740765\pi\)
\(464\) 1440.00 0.144074
\(465\) 0 0
\(466\) 3924.00 0.390077
\(467\) − 7776.00i − 0.770515i −0.922809 0.385257i \(-0.874113\pi\)
0.922809 0.385257i \(-0.125887\pi\)
\(468\) − 8584.00i − 0.847854i
\(469\) −4096.00 −0.403274
\(470\) 0 0
\(471\) −17968.0 −1.75780
\(472\) − 3360.00i − 0.327662i
\(473\) 384.000i 0.0373284i
\(474\) −2560.00 −0.248069
\(475\) 0 0
\(476\) −1056.00 −0.101684
\(477\) − 8214.00i − 0.788455i
\(478\) − 8640.00i − 0.826746i
\(479\) 13680.0 1.30492 0.652458 0.757825i \(-0.273738\pi\)
0.652458 + 0.757825i \(0.273738\pi\)
\(480\) 0 0
\(481\) 1972.00 0.186934
\(482\) 956.000i 0.0903415i
\(483\) 4224.00i 0.397927i
\(484\) 4748.00 0.445905
\(485\) 0 0
\(486\) 10064.0 0.939326
\(487\) − 7916.00i − 0.736567i −0.929714 0.368284i \(-0.879946\pi\)
0.929714 0.368284i \(-0.120054\pi\)
\(488\) 7216.00i 0.669371i
\(489\) −4544.00 −0.420218
\(490\) 0 0
\(491\) 13932.0 1.28053 0.640267 0.768152i \(-0.278824\pi\)
0.640267 + 0.768152i \(0.278824\pi\)
\(492\) − 14016.0i − 1.28433i
\(493\) − 5940.00i − 0.542645i
\(494\) −11600.0 −1.05650
\(495\) 0 0
\(496\) 2432.00 0.220161
\(497\) 1728.00i 0.155959i
\(498\) − 1152.00i − 0.103659i
\(499\) 8260.00 0.741019 0.370509 0.928829i \(-0.379183\pi\)
0.370509 + 0.928829i \(0.379183\pi\)
\(500\) 0 0
\(501\) 12192.0 1.08722
\(502\) − 5304.00i − 0.471572i
\(503\) − 11148.0i − 0.988200i −0.869405 0.494100i \(-0.835498\pi\)
0.869405 0.494100i \(-0.164502\pi\)
\(504\) 1184.00 0.104642
\(505\) 0 0
\(506\) 3168.00 0.278330
\(507\) 9336.00i 0.817803i
\(508\) − 496.000i − 0.0433198i
\(509\) 9690.00 0.843815 0.421907 0.906639i \(-0.361361\pi\)
0.421907 + 0.906639i \(0.361361\pi\)
\(510\) 0 0
\(511\) −1448.00 −0.125354
\(512\) − 512.000i − 0.0441942i
\(513\) 8000.00i 0.688516i
\(514\) 4668.00 0.400577
\(515\) 0 0
\(516\) −1024.00 −0.0873626
\(517\) 2448.00i 0.208245i
\(518\) 272.000i 0.0230714i
\(519\) 29616.0 2.50481
\(520\) 0 0
\(521\) −16038.0 −1.34863 −0.674316 0.738443i \(-0.735562\pi\)
−0.674316 + 0.738443i \(0.735562\pi\)
\(522\) 6660.00i 0.558430i
\(523\) 992.000i 0.0829391i 0.999140 + 0.0414695i \(0.0132039\pi\)
−0.999140 + 0.0414695i \(0.986796\pi\)
\(524\) −528.000 −0.0440187
\(525\) 0 0
\(526\) −7896.00 −0.654528
\(527\) − 10032.0i − 0.829223i
\(528\) − 1536.00i − 0.126602i
\(529\) −5257.00 −0.432070
\(530\) 0 0
\(531\) 15540.0 1.27002
\(532\) − 1600.00i − 0.130392i
\(533\) 25404.0i 2.06448i
\(534\) 12960.0 1.05025
\(535\) 0 0
\(536\) −8192.00 −0.660150
\(537\) 25440.0i 2.04435i
\(538\) 3180.00i 0.254832i
\(539\) 3924.00 0.313578
\(540\) 0 0
\(541\) 7142.00 0.567576 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(542\) − 9904.00i − 0.784895i
\(543\) 16784.0i 1.32646i
\(544\) −2112.00 −0.166455
\(545\) 0 0
\(546\) −3712.00 −0.290950
\(547\) − 7616.00i − 0.595314i −0.954673 0.297657i \(-0.903795\pi\)
0.954673 0.297657i \(-0.0962051\pi\)
\(548\) − 5016.00i − 0.391009i
\(549\) −33374.0 −2.59448
\(550\) 0 0
\(551\) 9000.00 0.695849
\(552\) 8448.00i 0.651396i
\(553\) 640.000i 0.0492144i
\(554\) −3292.00 −0.252462
\(555\) 0 0
\(556\) −11440.0 −0.872597
\(557\) 10314.0i 0.784593i 0.919839 + 0.392296i \(0.128319\pi\)
−0.919839 + 0.392296i \(0.871681\pi\)
\(558\) 11248.0i 0.853344i
\(559\) 1856.00 0.140430
\(560\) 0 0
\(561\) −6336.00 −0.476838
\(562\) 2316.00i 0.173834i
\(563\) − 7128.00i − 0.533587i −0.963754 0.266793i \(-0.914036\pi\)
0.963754 0.266793i \(-0.0859641\pi\)
\(564\) −6528.00 −0.487373
\(565\) 0 0
\(566\) 13984.0 1.03850
\(567\) − 1436.00i − 0.106360i
\(568\) 3456.00i 0.255300i
\(569\) −2010.00 −0.148091 −0.0740453 0.997255i \(-0.523591\pi\)
−0.0740453 + 0.997255i \(0.523591\pi\)
\(570\) 0 0
\(571\) −23188.0 −1.69945 −0.849726 0.527224i \(-0.823233\pi\)
−0.849726 + 0.527224i \(0.823233\pi\)
\(572\) 2784.00i 0.203505i
\(573\) − 35136.0i − 2.56165i
\(574\) −3504.00 −0.254798
\(575\) 0 0
\(576\) 2368.00 0.171296
\(577\) − 22466.0i − 1.62092i −0.585793 0.810461i \(-0.699217\pi\)
0.585793 0.810461i \(-0.300783\pi\)
\(578\) − 1114.00i − 0.0801666i
\(579\) −17264.0 −1.23915
\(580\) 0 0
\(581\) −288.000 −0.0205650
\(582\) 17696.0i 1.26035i
\(583\) 2664.00i 0.189248i
\(584\) −2896.00 −0.205201
\(585\) 0 0
\(586\) −516.000 −0.0363750
\(587\) − 22776.0i − 1.60148i −0.599015 0.800738i \(-0.704441\pi\)
0.599015 0.800738i \(-0.295559\pi\)
\(588\) 10464.0i 0.733891i
\(589\) 15200.0 1.06334
\(590\) 0 0
\(591\) 8592.00 0.598016
\(592\) 544.000i 0.0377673i
\(593\) − 21198.0i − 1.46796i −0.679174 0.733978i \(-0.737662\pi\)
0.679174 0.733978i \(-0.262338\pi\)
\(594\) 1920.00 0.132624
\(595\) 0 0
\(596\) 3000.00 0.206183
\(597\) 22720.0i 1.55757i
\(598\) − 15312.0i − 1.04708i
\(599\) −15960.0 −1.08866 −0.544330 0.838871i \(-0.683216\pi\)
−0.544330 + 0.838871i \(0.683216\pi\)
\(600\) 0 0
\(601\) 5882.00 0.399221 0.199610 0.979875i \(-0.436032\pi\)
0.199610 + 0.979875i \(0.436032\pi\)
\(602\) 256.000i 0.0173319i
\(603\) − 37888.0i − 2.55874i
\(604\) 1792.00 0.120721
\(605\) 0 0
\(606\) 4128.00 0.276714
\(607\) − 8516.00i − 0.569446i −0.958610 0.284723i \(-0.908098\pi\)
0.958610 0.284723i \(-0.0919016\pi\)
\(608\) − 3200.00i − 0.213449i
\(609\) 2880.00 0.191631
\(610\) 0 0
\(611\) 11832.0 0.783423
\(612\) − 9768.00i − 0.645176i
\(613\) 8462.00i 0.557548i 0.960357 + 0.278774i \(0.0899280\pi\)
−0.960357 + 0.278774i \(0.910072\pi\)
\(614\) 17888.0 1.17573
\(615\) 0 0
\(616\) −384.000 −0.0251166
\(617\) 11094.0i 0.723870i 0.932203 + 0.361935i \(0.117884\pi\)
−0.932203 + 0.361935i \(0.882116\pi\)
\(618\) 15808.0i 1.02895i
\(619\) −2180.00 −0.141553 −0.0707767 0.997492i \(-0.522548\pi\)
−0.0707767 + 0.997492i \(0.522548\pi\)
\(620\) 0 0
\(621\) −10560.0 −0.682380
\(622\) − 2784.00i − 0.179467i
\(623\) − 3240.00i − 0.208359i
\(624\) −7424.00 −0.476279
\(625\) 0 0
\(626\) −11756.0 −0.750582
\(627\) − 9600.00i − 0.611463i
\(628\) 8984.00i 0.570861i
\(629\) 2244.00 0.142248
\(630\) 0 0
\(631\) −26848.0 −1.69382 −0.846911 0.531734i \(-0.821541\pi\)
−0.846911 + 0.531734i \(0.821541\pi\)
\(632\) 1280.00i 0.0805628i
\(633\) 21344.0i 1.34020i
\(634\) −20652.0 −1.29368
\(635\) 0 0
\(636\) −7104.00 −0.442912
\(637\) − 18966.0i − 1.17969i
\(638\) − 2160.00i − 0.134036i
\(639\) −15984.0 −0.989542
\(640\) 0 0
\(641\) 26322.0 1.62193 0.810965 0.585095i \(-0.198943\pi\)
0.810965 + 0.585095i \(0.198943\pi\)
\(642\) − 384.000i − 0.0236063i
\(643\) − 10168.0i − 0.623619i −0.950145 0.311809i \(-0.899065\pi\)
0.950145 0.311809i \(-0.100935\pi\)
\(644\) 2112.00 0.129231
\(645\) 0 0
\(646\) −13200.0 −0.803943
\(647\) 23604.0i 1.43426i 0.696937 + 0.717132i \(0.254546\pi\)
−0.696937 + 0.717132i \(0.745454\pi\)
\(648\) − 2872.00i − 0.174109i
\(649\) −5040.00 −0.304834
\(650\) 0 0
\(651\) 4864.00 0.292834
\(652\) 2272.00i 0.136470i
\(653\) 16422.0i 0.984139i 0.870556 + 0.492069i \(0.163759\pi\)
−0.870556 + 0.492069i \(0.836241\pi\)
\(654\) 15200.0 0.908818
\(655\) 0 0
\(656\) −7008.00 −0.417098
\(657\) − 13394.0i − 0.795357i
\(658\) 1632.00i 0.0966899i
\(659\) 26100.0 1.54281 0.771405 0.636345i \(-0.219554\pi\)
0.771405 + 0.636345i \(0.219554\pi\)
\(660\) 0 0
\(661\) −3058.00 −0.179943 −0.0899716 0.995944i \(-0.528678\pi\)
−0.0899716 + 0.995944i \(0.528678\pi\)
\(662\) 8456.00i 0.496453i
\(663\) 30624.0i 1.79387i
\(664\) −576.000 −0.0336644
\(665\) 0 0
\(666\) −2516.00 −0.146386
\(667\) 11880.0i 0.689648i
\(668\) − 6096.00i − 0.353086i
\(669\) 14176.0 0.819246
\(670\) 0 0
\(671\) 10824.0 0.622736
\(672\) − 1024.00i − 0.0587822i
\(673\) 10802.0i 0.618702i 0.950948 + 0.309351i \(0.100112\pi\)
−0.950948 + 0.309351i \(0.899888\pi\)
\(674\) −2212.00 −0.126414
\(675\) 0 0
\(676\) 4668.00 0.265589
\(677\) 10674.0i 0.605960i 0.952997 + 0.302980i \(0.0979816\pi\)
−0.952997 + 0.302980i \(0.902018\pi\)
\(678\) 16608.0i 0.940747i
\(679\) 4424.00 0.250041
\(680\) 0 0
\(681\) 22272.0 1.25325
\(682\) − 3648.00i − 0.204823i
\(683\) − 28608.0i − 1.60272i −0.598185 0.801358i \(-0.704111\pi\)
0.598185 0.801358i \(-0.295889\pi\)
\(684\) 14800.0 0.827328
\(685\) 0 0
\(686\) 5360.00 0.298317
\(687\) 2800.00i 0.155497i
\(688\) 512.000i 0.0283718i
\(689\) 12876.0 0.711954
\(690\) 0 0
\(691\) −2428.00 −0.133669 −0.0668346 0.997764i \(-0.521290\pi\)
−0.0668346 + 0.997764i \(0.521290\pi\)
\(692\) − 14808.0i − 0.813462i
\(693\) − 1776.00i − 0.0973516i
\(694\) −18672.0 −1.02130
\(695\) 0 0
\(696\) 5760.00 0.313696
\(697\) 28908.0i 1.57097i
\(698\) − 23540.0i − 1.27651i
\(699\) 15696.0 0.849324
\(700\) 0 0
\(701\) −6618.00 −0.356574 −0.178287 0.983979i \(-0.557056\pi\)
−0.178287 + 0.983979i \(0.557056\pi\)
\(702\) − 9280.00i − 0.498933i
\(703\) 3400.00i 0.182409i
\(704\) −768.000 −0.0411152
\(705\) 0 0
\(706\) 16644.0 0.887259
\(707\) − 1032.00i − 0.0548972i
\(708\) − 13440.0i − 0.713427i
\(709\) −20510.0 −1.08642 −0.543208 0.839598i \(-0.682791\pi\)
−0.543208 + 0.839598i \(0.682791\pi\)
\(710\) 0 0
\(711\) −5920.00 −0.312261
\(712\) − 6480.00i − 0.341079i
\(713\) 20064.0i 1.05386i
\(714\) −4224.00 −0.221399
\(715\) 0 0
\(716\) 12720.0 0.663923
\(717\) − 34560.0i − 1.80009i
\(718\) 21360.0i 1.11023i
\(719\) −31680.0 −1.64321 −0.821603 0.570061i \(-0.806920\pi\)
−0.821603 + 0.570061i \(0.806920\pi\)
\(720\) 0 0
\(721\) 3952.00 0.204133
\(722\) − 6282.00i − 0.323811i
\(723\) 3824.00i 0.196703i
\(724\) 8392.00 0.430782
\(725\) 0 0
\(726\) 18992.0 0.970880
\(727\) − 13196.0i − 0.673195i −0.941649 0.336597i \(-0.890724\pi\)
0.941649 0.336597i \(-0.109276\pi\)
\(728\) 1856.00i 0.0944889i
\(729\) 30563.0 1.55276
\(730\) 0 0
\(731\) 2112.00 0.106861
\(732\) 28864.0i 1.45744i
\(733\) 8102.00i 0.408259i 0.978944 + 0.204130i \(0.0654364\pi\)
−0.978944 + 0.204130i \(0.934564\pi\)
\(734\) 11768.0 0.591778
\(735\) 0 0
\(736\) 4224.00 0.211547
\(737\) 12288.0i 0.614158i
\(738\) − 32412.0i − 1.61667i
\(739\) 12580.0 0.626201 0.313101 0.949720i \(-0.398632\pi\)
0.313101 + 0.949720i \(0.398632\pi\)
\(740\) 0 0
\(741\) −46400.0 −2.30033
\(742\) 1776.00i 0.0878693i
\(743\) 29892.0i 1.47595i 0.674828 + 0.737975i \(0.264218\pi\)
−0.674828 + 0.737975i \(0.735782\pi\)
\(744\) 9728.00 0.479363
\(745\) 0 0
\(746\) −4196.00 −0.205934
\(747\) − 2664.00i − 0.130483i
\(748\) 3168.00i 0.154858i
\(749\) −96.0000 −0.00468326
\(750\) 0 0
\(751\) −40408.0 −1.96339 −0.981697 0.190450i \(-0.939005\pi\)
−0.981697 + 0.190450i \(0.939005\pi\)
\(752\) 3264.00i 0.158279i
\(753\) − 21216.0i − 1.02676i
\(754\) −10440.0 −0.504247
\(755\) 0 0
\(756\) 1280.00 0.0615782
\(757\) − 32366.0i − 1.55398i −0.629513 0.776990i \(-0.716746\pi\)
0.629513 0.776990i \(-0.283254\pi\)
\(758\) 7720.00i 0.369925i
\(759\) 12672.0 0.606014
\(760\) 0 0
\(761\) −17238.0 −0.821126 −0.410563 0.911832i \(-0.634668\pi\)
−0.410563 + 0.911832i \(0.634668\pi\)
\(762\) − 1984.00i − 0.0943212i
\(763\) − 3800.00i − 0.180300i
\(764\) −17568.0 −0.831921
\(765\) 0 0
\(766\) −19176.0 −0.904513
\(767\) 24360.0i 1.14679i
\(768\) − 2048.00i − 0.0962250i
\(769\) −10850.0 −0.508792 −0.254396 0.967100i \(-0.581877\pi\)
−0.254396 + 0.967100i \(0.581877\pi\)
\(770\) 0 0
\(771\) 18672.0 0.872186
\(772\) 8632.00i 0.402425i
\(773\) 9102.00i 0.423514i 0.977322 + 0.211757i \(0.0679185\pi\)
−0.977322 + 0.211757i \(0.932081\pi\)
\(774\) −2368.00 −0.109969
\(775\) 0 0
\(776\) 8848.00 0.409310
\(777\) 1088.00i 0.0502340i
\(778\) − 26820.0i − 1.23592i
\(779\) −43800.0 −2.01450
\(780\) 0 0
\(781\) 5184.00 0.237514
\(782\) − 17424.0i − 0.796779i
\(783\) 7200.00i 0.328617i
\(784\) 5232.00 0.238338
\(785\) 0 0
\(786\) −2112.00 −0.0958429
\(787\) 25504.0i 1.15517i 0.816330 + 0.577585i \(0.196005\pi\)
−0.816330 + 0.577585i \(0.803995\pi\)
\(788\) − 4296.00i − 0.194212i
\(789\) −31584.0 −1.42512
\(790\) 0 0
\(791\) 4152.00 0.186635
\(792\) − 3552.00i − 0.159362i
\(793\) − 52316.0i − 2.34274i
\(794\) 26228.0 1.17229
\(795\) 0 0
\(796\) 11360.0 0.505835
\(797\) − 14166.0i − 0.629593i −0.949159 0.314796i \(-0.898064\pi\)
0.949159 0.314796i \(-0.101936\pi\)
\(798\) − 6400.00i − 0.283907i
\(799\) 13464.0 0.596148
\(800\) 0 0
\(801\) 29970.0 1.32202
\(802\) 11676.0i 0.514082i
\(803\) 4344.00i 0.190905i
\(804\) −32768.0 −1.43736
\(805\) 0 0
\(806\) −17632.0 −0.770547
\(807\) 12720.0i 0.554852i
\(808\) − 2064.00i − 0.0898654i
\(809\) −33210.0 −1.44327 −0.721633 0.692276i \(-0.756608\pi\)
−0.721633 + 0.692276i \(0.756608\pi\)
\(810\) 0 0
\(811\) 39212.0 1.69780 0.848902 0.528550i \(-0.177264\pi\)
0.848902 + 0.528550i \(0.177264\pi\)
\(812\) − 1440.00i − 0.0622341i
\(813\) − 39616.0i − 1.70897i
\(814\) 816.000 0.0351361
\(815\) 0 0
\(816\) −8448.00 −0.362425
\(817\) 3200.00i 0.137030i
\(818\) 19060.0i 0.814691i
\(819\) −8584.00 −0.366238
\(820\) 0 0
\(821\) 6222.00 0.264494 0.132247 0.991217i \(-0.457781\pi\)
0.132247 + 0.991217i \(0.457781\pi\)
\(822\) − 20064.0i − 0.851353i
\(823\) 31172.0i 1.32028i 0.751144 + 0.660138i \(0.229502\pi\)
−0.751144 + 0.660138i \(0.770498\pi\)
\(824\) 7904.00 0.334161
\(825\) 0 0
\(826\) −3360.00 −0.141537
\(827\) 264.000i 0.0111006i 0.999985 + 0.00555029i \(0.00176672\pi\)
−0.999985 + 0.00555029i \(0.998233\pi\)
\(828\) 19536.0i 0.819955i
\(829\) 29050.0 1.21707 0.608533 0.793528i \(-0.291758\pi\)
0.608533 + 0.793528i \(0.291758\pi\)
\(830\) 0 0
\(831\) −13168.0 −0.549691
\(832\) 3712.00i 0.154676i
\(833\) − 21582.0i − 0.897685i
\(834\) −45760.0 −1.89993
\(835\) 0 0
\(836\) −4800.00 −0.198578
\(837\) 12160.0i 0.502164i
\(838\) 14520.0i 0.598550i
\(839\) 21720.0 0.893752 0.446876 0.894596i \(-0.352537\pi\)
0.446876 + 0.894596i \(0.352537\pi\)
\(840\) 0 0
\(841\) −16289.0 −0.667883
\(842\) − 24124.0i − 0.987373i
\(843\) 9264.00i 0.378492i
\(844\) 10672.0 0.435243
\(845\) 0 0
\(846\) −15096.0 −0.613488
\(847\) − 4748.00i − 0.192613i
\(848\) 3552.00i 0.143840i
\(849\) 55936.0 2.26115
\(850\) 0 0
\(851\) −4488.00 −0.180783
\(852\) 13824.0i 0.555871i
\(853\) − 6658.00i − 0.267252i −0.991032 0.133626i \(-0.957338\pi\)
0.991032 0.133626i \(-0.0426620\pi\)
\(854\) 7216.00 0.289141
\(855\) 0 0
\(856\) −192.000 −0.00766638
\(857\) 13974.0i 0.556993i 0.960437 + 0.278496i \(0.0898360\pi\)
−0.960437 + 0.278496i \(0.910164\pi\)
\(858\) 11136.0i 0.443096i
\(859\) −23780.0 −0.944544 −0.472272 0.881453i \(-0.656566\pi\)
−0.472272 + 0.881453i \(0.656566\pi\)
\(860\) 0 0
\(861\) −14016.0 −0.554778
\(862\) 27216.0i 1.07538i
\(863\) − 12228.0i − 0.482324i −0.970485 0.241162i \(-0.922471\pi\)
0.970485 0.241162i \(-0.0775286\pi\)
\(864\) 2560.00 0.100802
\(865\) 0 0
\(866\) −7676.00 −0.301202
\(867\) − 4456.00i − 0.174549i
\(868\) − 2432.00i − 0.0951008i
\(869\) 1920.00 0.0749500
\(870\) 0 0
\(871\) 59392.0 2.31047
\(872\) − 7600.00i − 0.295147i
\(873\) 40922.0i 1.58648i
\(874\) 26400.0 1.02173
\(875\) 0 0
\(876\) −11584.0 −0.446789
\(877\) − 11606.0i − 0.446872i −0.974719 0.223436i \(-0.928273\pi\)
0.974719 0.223436i \(-0.0717274\pi\)
\(878\) 14800.0i 0.568879i
\(879\) −2064.00 −0.0792002
\(880\) 0 0
\(881\) −32958.0 −1.26037 −0.630183 0.776446i \(-0.717020\pi\)
−0.630183 + 0.776446i \(0.717020\pi\)
\(882\) 24198.0i 0.923797i
\(883\) 8072.00i 0.307638i 0.988099 + 0.153819i \(0.0491573\pi\)
−0.988099 + 0.153819i \(0.950843\pi\)
\(884\) 15312.0 0.582577
\(885\) 0 0
\(886\) 16704.0 0.633388
\(887\) − 15756.0i − 0.596431i −0.954498 0.298216i \(-0.903609\pi\)
0.954498 0.298216i \(-0.0963915\pi\)
\(888\) 2176.00i 0.0822317i
\(889\) −496.000 −0.0187124
\(890\) 0 0
\(891\) −4308.00 −0.161979
\(892\) − 7088.00i − 0.266058i
\(893\) 20400.0i 0.764457i
\(894\) 12000.0 0.448926
\(895\) 0 0
\(896\) −512.000 −0.0190901
\(897\) − 61248.0i − 2.27983i
\(898\) 21540.0i 0.800444i
\(899\) 13680.0 0.507512
\(900\) 0 0
\(901\) 14652.0 0.541763
\(902\) 10512.0i 0.388039i
\(903\) 1024.00i 0.0377371i
\(904\) 8304.00 0.305517
\(905\) 0 0
\(906\) 7168.00 0.262849
\(907\) − 18776.0i − 0.687372i −0.939085 0.343686i \(-0.888324\pi\)
0.939085 0.343686i \(-0.111676\pi\)
\(908\) − 11136.0i − 0.407006i
\(909\) 9546.00 0.348318
\(910\) 0 0
\(911\) −20568.0 −0.748022 −0.374011 0.927424i \(-0.622018\pi\)
−0.374011 + 0.927424i \(0.622018\pi\)
\(912\) − 12800.0i − 0.464748i
\(913\) 864.000i 0.0313190i
\(914\) 13388.0 0.484503
\(915\) 0 0
\(916\) 1400.00 0.0504992
\(917\) 528.000i 0.0190143i
\(918\) − 10560.0i − 0.379664i
\(919\) 6280.00 0.225417 0.112708 0.993628i \(-0.464047\pi\)
0.112708 + 0.993628i \(0.464047\pi\)
\(920\) 0 0
\(921\) 71552.0 2.55996
\(922\) 6036.00i 0.215602i
\(923\) − 25056.0i − 0.893530i
\(924\) −1536.00 −0.0546869
\(925\) 0 0
\(926\) 28984.0 1.02859
\(927\) 36556.0i 1.29521i
\(928\) − 2880.00i − 0.101876i
\(929\) 20430.0 0.721514 0.360757 0.932660i \(-0.382518\pi\)
0.360757 + 0.932660i \(0.382518\pi\)
\(930\) 0 0
\(931\) 32700.0 1.15113
\(932\) − 7848.00i − 0.275826i
\(933\) − 11136.0i − 0.390757i
\(934\) −15552.0 −0.544836
\(935\) 0 0
\(936\) −17168.0 −0.599523
\(937\) − 8906.00i − 0.310508i −0.987875 0.155254i \(-0.950380\pi\)
0.987875 0.155254i \(-0.0496197\pi\)
\(938\) 8192.00i 0.285158i
\(939\) −47024.0 −1.63426
\(940\) 0 0
\(941\) −17418.0 −0.603412 −0.301706 0.953401i \(-0.597556\pi\)
−0.301706 + 0.953401i \(0.597556\pi\)
\(942\) 35936.0i 1.24295i
\(943\) − 57816.0i − 1.99655i
\(944\) −6720.00 −0.231692
\(945\) 0 0
\(946\) 768.000 0.0263952
\(947\) 2544.00i 0.0872956i 0.999047 + 0.0436478i \(0.0138979\pi\)
−0.999047 + 0.0436478i \(0.986102\pi\)
\(948\) 5120.00i 0.175411i
\(949\) 20996.0 0.718187
\(950\) 0 0
\(951\) −82608.0 −2.81677
\(952\) 2112.00i 0.0719016i
\(953\) 15402.0i 0.523525i 0.965132 + 0.261763i \(0.0843038\pi\)
−0.965132 + 0.261763i \(0.915696\pi\)
\(954\) −16428.0 −0.557522
\(955\) 0 0
\(956\) −17280.0 −0.584597
\(957\) − 8640.00i − 0.291841i
\(958\) − 27360.0i − 0.922716i
\(959\) −5016.00 −0.168900
\(960\) 0 0
\(961\) −6687.00 −0.224464
\(962\) − 3944.00i − 0.132183i
\(963\) − 888.000i − 0.0297148i
\(964\) 1912.00 0.0638811
\(965\) 0 0
\(966\) 8448.00 0.281377
\(967\) 49444.0i 1.64427i 0.569291 + 0.822136i \(0.307218\pi\)
−0.569291 + 0.822136i \(0.692782\pi\)
\(968\) − 9496.00i − 0.315303i
\(969\) −52800.0 −1.75044
\(970\) 0 0
\(971\) −25188.0 −0.832463 −0.416231 0.909259i \(-0.636649\pi\)
−0.416231 + 0.909259i \(0.636649\pi\)
\(972\) − 20128.0i − 0.664204i
\(973\) 11440.0i 0.376927i
\(974\) −15832.0 −0.520832
\(975\) 0 0
\(976\) 14432.0 0.473317
\(977\) − 2946.00i − 0.0964697i −0.998836 0.0482348i \(-0.984640\pi\)
0.998836 0.0482348i \(-0.0153596\pi\)
\(978\) 9088.00i 0.297139i
\(979\) −9720.00 −0.317316
\(980\) 0 0
\(981\) 35150.0 1.14399
\(982\) − 27864.0i − 0.905475i
\(983\) 15012.0i 0.487089i 0.969890 + 0.243544i \(0.0783102\pi\)
−0.969890 + 0.243544i \(0.921690\pi\)
\(984\) −28032.0 −0.908158
\(985\) 0 0
\(986\) −11880.0 −0.383708
\(987\) 6528.00i 0.210525i
\(988\) 23200.0i 0.747055i
\(989\) −4224.00 −0.135809
\(990\) 0 0
\(991\) −5128.00 −0.164376 −0.0821878 0.996617i \(-0.526191\pi\)
−0.0821878 + 0.996617i \(0.526191\pi\)
\(992\) − 4864.00i − 0.155678i
\(993\) 33824.0i 1.08094i
\(994\) 3456.00 0.110279
\(995\) 0 0
\(996\) −2304.00 −0.0732982
\(997\) 49714.0i 1.57920i 0.613625 + 0.789598i \(0.289711\pi\)
−0.613625 + 0.789598i \(0.710289\pi\)
\(998\) − 16520.0i − 0.523979i
\(999\) −2720.00 −0.0861431
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 50.4.b.a.49.1 2
3.2 odd 2 450.4.c.d.199.2 2
4.3 odd 2 400.4.c.c.49.2 2
5.2 odd 4 10.4.a.a.1.1 1
5.3 odd 4 50.4.a.c.1.1 1
5.4 even 2 inner 50.4.b.a.49.2 2
15.2 even 4 90.4.a.a.1.1 1
15.8 even 4 450.4.a.q.1.1 1
15.14 odd 2 450.4.c.d.199.1 2
20.3 even 4 400.4.a.b.1.1 1
20.7 even 4 80.4.a.f.1.1 1
20.19 odd 2 400.4.c.c.49.1 2
35.2 odd 12 490.4.e.i.361.1 2
35.12 even 12 490.4.e.a.361.1 2
35.13 even 4 2450.4.a.b.1.1 1
35.17 even 12 490.4.e.a.471.1 2
35.27 even 4 490.4.a.o.1.1 1
35.32 odd 12 490.4.e.i.471.1 2
40.3 even 4 1600.4.a.bx.1.1 1
40.13 odd 4 1600.4.a.d.1.1 1
40.27 even 4 320.4.a.b.1.1 1
40.37 odd 4 320.4.a.m.1.1 1
45.2 even 12 810.4.e.w.271.1 2
45.7 odd 12 810.4.e.c.271.1 2
45.22 odd 12 810.4.e.c.541.1 2
45.32 even 12 810.4.e.w.541.1 2
55.32 even 4 1210.4.a.b.1.1 1
60.47 odd 4 720.4.a.j.1.1 1
65.12 odd 4 1690.4.a.a.1.1 1
80.27 even 4 1280.4.d.g.641.1 2
80.37 odd 4 1280.4.d.j.641.2 2
80.67 even 4 1280.4.d.g.641.2 2
80.77 odd 4 1280.4.d.j.641.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.a.a.1.1 1 5.2 odd 4
50.4.a.c.1.1 1 5.3 odd 4
50.4.b.a.49.1 2 1.1 even 1 trivial
50.4.b.a.49.2 2 5.4 even 2 inner
80.4.a.f.1.1 1 20.7 even 4
90.4.a.a.1.1 1 15.2 even 4
320.4.a.b.1.1 1 40.27 even 4
320.4.a.m.1.1 1 40.37 odd 4
400.4.a.b.1.1 1 20.3 even 4
400.4.c.c.49.1 2 20.19 odd 2
400.4.c.c.49.2 2 4.3 odd 2
450.4.a.q.1.1 1 15.8 even 4
450.4.c.d.199.1 2 15.14 odd 2
450.4.c.d.199.2 2 3.2 odd 2
490.4.a.o.1.1 1 35.27 even 4
490.4.e.a.361.1 2 35.12 even 12
490.4.e.a.471.1 2 35.17 even 12
490.4.e.i.361.1 2 35.2 odd 12
490.4.e.i.471.1 2 35.32 odd 12
720.4.a.j.1.1 1 60.47 odd 4
810.4.e.c.271.1 2 45.7 odd 12
810.4.e.c.541.1 2 45.22 odd 12
810.4.e.w.271.1 2 45.2 even 12
810.4.e.w.541.1 2 45.32 even 12
1210.4.a.b.1.1 1 55.32 even 4
1280.4.d.g.641.1 2 80.27 even 4
1280.4.d.g.641.2 2 80.67 even 4
1280.4.d.j.641.1 2 80.77 odd 4
1280.4.d.j.641.2 2 80.37 odd 4
1600.4.a.d.1.1 1 40.13 odd 4
1600.4.a.bx.1.1 1 40.3 even 4
1690.4.a.a.1.1 1 65.12 odd 4
2450.4.a.b.1.1 1 35.13 even 4