Properties

Label 150.4.c.c.49.1
Level $150$
Weight $4$
Character 150.49
Analytic conductor $8.850$
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] = [150,4,Mod(49,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, 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("150.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(8.85028650086\)
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: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.4.c.c.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} +4.00000i q^{7} +8.00000i q^{8} -9.00000 q^{9} -48.0000 q^{11} -12.0000i q^{12} +2.00000i q^{13} +8.00000 q^{14} +16.0000 q^{16} +114.000i q^{17} +18.0000i q^{18} -140.000 q^{19} -12.0000 q^{21} +96.0000i q^{22} +72.0000i q^{23} -24.0000 q^{24} +4.00000 q^{26} -27.0000i q^{27} -16.0000i q^{28} -210.000 q^{29} +272.000 q^{31} -32.0000i q^{32} -144.000i q^{33} +228.000 q^{34} +36.0000 q^{36} +334.000i q^{37} +280.000i q^{38} -6.00000 q^{39} -198.000 q^{41} +24.0000i q^{42} -268.000i q^{43} +192.000 q^{44} +144.000 q^{46} -216.000i q^{47} +48.0000i q^{48} +327.000 q^{49} -342.000 q^{51} -8.00000i q^{52} -78.0000i q^{53} -54.0000 q^{54} -32.0000 q^{56} -420.000i q^{57} +420.000i q^{58} -240.000 q^{59} +302.000 q^{61} -544.000i q^{62} -36.0000i q^{63} -64.0000 q^{64} -288.000 q^{66} -596.000i q^{67} -456.000i q^{68} -216.000 q^{69} -768.000 q^{71} -72.0000i q^{72} -478.000i q^{73} +668.000 q^{74} +560.000 q^{76} -192.000i q^{77} +12.0000i q^{78} +640.000 q^{79} +81.0000 q^{81} +396.000i q^{82} -348.000i q^{83} +48.0000 q^{84} -536.000 q^{86} -630.000i q^{87} -384.000i q^{88} -210.000 q^{89} -8.00000 q^{91} -288.000i q^{92} +816.000i q^{93} -432.000 q^{94} +96.0000 q^{96} +1534.00i q^{97} -654.000i q^{98} +432.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} - 96 q^{11} + 16 q^{14} + 32 q^{16} - 280 q^{19} - 24 q^{21} - 48 q^{24} + 8 q^{26} - 420 q^{29} + 544 q^{31} + 456 q^{34} + 72 q^{36} - 12 q^{39} - 396 q^{41} + 384 q^{44}+ \cdots + 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 6.00000 0.408248
\(7\) 4.00000i 0.215980i 0.994152 + 0.107990i \(0.0344414\pi\)
−0.994152 + 0.107990i \(0.965559\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −48.0000 −1.31569 −0.657843 0.753155i \(-0.728531\pi\)
−0.657843 + 0.753155i \(0.728531\pi\)
\(12\) − 12.0000i − 0.288675i
\(13\) 2.00000i 0.0426692i 0.999772 + 0.0213346i \(0.00679154\pi\)
−0.999772 + 0.0213346i \(0.993208\pi\)
\(14\) 8.00000 0.152721
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 114.000i 1.62642i 0.581974 + 0.813208i \(0.302281\pi\)
−0.581974 + 0.813208i \(0.697719\pi\)
\(18\) 18.0000i 0.235702i
\(19\) −140.000 −1.69043 −0.845216 0.534425i \(-0.820528\pi\)
−0.845216 + 0.534425i \(0.820528\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 96.0000i 0.930330i
\(23\) 72.0000i 0.652741i 0.945242 + 0.326370i \(0.105826\pi\)
−0.945242 + 0.326370i \(0.894174\pi\)
\(24\) −24.0000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.0301717
\(27\) − 27.0000i − 0.192450i
\(28\) − 16.0000i − 0.107990i
\(29\) −210.000 −1.34469 −0.672345 0.740238i \(-0.734713\pi\)
−0.672345 + 0.740238i \(0.734713\pi\)
\(30\) 0 0
\(31\) 272.000 1.57589 0.787946 0.615745i \(-0.211145\pi\)
0.787946 + 0.615745i \(0.211145\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 144.000i − 0.759612i
\(34\) 228.000 1.15005
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 334.000i 1.48403i 0.670381 + 0.742017i \(0.266131\pi\)
−0.670381 + 0.742017i \(0.733869\pi\)
\(38\) 280.000i 1.19532i
\(39\) −6.00000 −0.0246351
\(40\) 0 0
\(41\) −198.000 −0.754205 −0.377102 0.926172i \(-0.623080\pi\)
−0.377102 + 0.926172i \(0.623080\pi\)
\(42\) 24.0000i 0.0881733i
\(43\) − 268.000i − 0.950456i −0.879863 0.475228i \(-0.842366\pi\)
0.879863 0.475228i \(-0.157634\pi\)
\(44\) 192.000 0.657843
\(45\) 0 0
\(46\) 144.000 0.461557
\(47\) − 216.000i − 0.670358i −0.942154 0.335179i \(-0.891203\pi\)
0.942154 0.335179i \(-0.108797\pi\)
\(48\) 48.0000i 0.144338i
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) −342.000 −0.939011
\(52\) − 8.00000i − 0.0213346i
\(53\) − 78.0000i − 0.202153i −0.994879 0.101077i \(-0.967771\pi\)
0.994879 0.101077i \(-0.0322287\pi\)
\(54\) −54.0000 −0.136083
\(55\) 0 0
\(56\) −32.0000 −0.0763604
\(57\) − 420.000i − 0.975971i
\(58\) 420.000i 0.950840i
\(59\) −240.000 −0.529582 −0.264791 0.964306i \(-0.585303\pi\)
−0.264791 + 0.964306i \(0.585303\pi\)
\(60\) 0 0
\(61\) 302.000 0.633888 0.316944 0.948444i \(-0.397343\pi\)
0.316944 + 0.948444i \(0.397343\pi\)
\(62\) − 544.000i − 1.11432i
\(63\) − 36.0000i − 0.0719932i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −288.000 −0.537127
\(67\) − 596.000i − 1.08676i −0.839487 0.543381i \(-0.817144\pi\)
0.839487 0.543381i \(-0.182856\pi\)
\(68\) − 456.000i − 0.813208i
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) −768.000 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(72\) − 72.0000i − 0.117851i
\(73\) − 478.000i − 0.766379i −0.923670 0.383190i \(-0.874826\pi\)
0.923670 0.383190i \(-0.125174\pi\)
\(74\) 668.000 1.04937
\(75\) 0 0
\(76\) 560.000 0.845216
\(77\) − 192.000i − 0.284161i
\(78\) 12.0000i 0.0174196i
\(79\) 640.000 0.911464 0.455732 0.890117i \(-0.349378\pi\)
0.455732 + 0.890117i \(0.349378\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 396.000i 0.533303i
\(83\) − 348.000i − 0.460216i −0.973165 0.230108i \(-0.926092\pi\)
0.973165 0.230108i \(-0.0739080\pi\)
\(84\) 48.0000 0.0623480
\(85\) 0 0
\(86\) −536.000 −0.672074
\(87\) − 630.000i − 0.776357i
\(88\) − 384.000i − 0.465165i
\(89\) −210.000 −0.250112 −0.125056 0.992150i \(-0.539911\pi\)
−0.125056 + 0.992150i \(0.539911\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.00921569
\(92\) − 288.000i − 0.326370i
\(93\) 816.000i 0.909841i
\(94\) −432.000 −0.474015
\(95\) 0 0
\(96\) 96.0000 0.102062
\(97\) 1534.00i 1.60571i 0.596173 + 0.802856i \(0.296687\pi\)
−0.596173 + 0.802856i \(0.703313\pi\)
\(98\) − 654.000i − 0.674122i
\(99\) 432.000 0.438562
\(100\) 0 0
\(101\) 1722.00 1.69649 0.848245 0.529605i \(-0.177660\pi\)
0.848245 + 0.529605i \(0.177660\pi\)
\(102\) 684.000i 0.663981i
\(103\) 1052.00i 1.00638i 0.864177 + 0.503188i \(0.167840\pi\)
−0.864177 + 0.503188i \(0.832160\pi\)
\(104\) −16.0000 −0.0150859
\(105\) 0 0
\(106\) −156.000 −0.142944
\(107\) 564.000i 0.509570i 0.966998 + 0.254785i \(0.0820046\pi\)
−0.966998 + 0.254785i \(0.917995\pi\)
\(108\) 108.000i 0.0962250i
\(109\) 610.000 0.536031 0.268016 0.963415i \(-0.413632\pi\)
0.268016 + 0.963415i \(0.413632\pi\)
\(110\) 0 0
\(111\) −1002.00 −0.856807
\(112\) 64.0000i 0.0539949i
\(113\) 1302.00i 1.08391i 0.840407 + 0.541955i \(0.182316\pi\)
−0.840407 + 0.541955i \(0.817684\pi\)
\(114\) −840.000 −0.690116
\(115\) 0 0
\(116\) 840.000 0.672345
\(117\) − 18.0000i − 0.0142231i
\(118\) 480.000i 0.374471i
\(119\) −456.000 −0.351273
\(120\) 0 0
\(121\) 973.000 0.731029
\(122\) − 604.000i − 0.448226i
\(123\) − 594.000i − 0.435440i
\(124\) −1088.00 −0.787946
\(125\) 0 0
\(126\) −72.0000 −0.0509069
\(127\) 124.000i 0.0866395i 0.999061 + 0.0433198i \(0.0137934\pi\)
−0.999061 + 0.0433198i \(0.986207\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 804.000 0.548746
\(130\) 0 0
\(131\) 192.000 0.128054 0.0640272 0.997948i \(-0.479606\pi\)
0.0640272 + 0.997948i \(0.479606\pi\)
\(132\) 576.000i 0.379806i
\(133\) − 560.000i − 0.365099i
\(134\) −1192.00 −0.768456
\(135\) 0 0
\(136\) −912.000 −0.575025
\(137\) 2514.00i 1.56778i 0.620901 + 0.783889i \(0.286767\pi\)
−0.620901 + 0.783889i \(0.713233\pi\)
\(138\) 432.000i 0.266480i
\(139\) −1340.00 −0.817679 −0.408839 0.912606i \(-0.634066\pi\)
−0.408839 + 0.912606i \(0.634066\pi\)
\(140\) 0 0
\(141\) 648.000 0.387032
\(142\) 1536.00i 0.907734i
\(143\) − 96.0000i − 0.0561393i
\(144\) −144.000 −0.0833333
\(145\) 0 0
\(146\) −956.000 −0.541912
\(147\) 981.000i 0.550418i
\(148\) − 1336.00i − 0.742017i
\(149\) −1410.00 −0.775246 −0.387623 0.921818i \(-0.626704\pi\)
−0.387623 + 0.921818i \(0.626704\pi\)
\(150\) 0 0
\(151\) −2128.00 −1.14685 −0.573424 0.819258i \(-0.694385\pi\)
−0.573424 + 0.819258i \(0.694385\pi\)
\(152\) − 1120.00i − 0.597658i
\(153\) − 1026.00i − 0.542138i
\(154\) −384.000 −0.200932
\(155\) 0 0
\(156\) 24.0000 0.0123176
\(157\) − 3026.00i − 1.53822i −0.639114 0.769112i \(-0.720699\pi\)
0.639114 0.769112i \(-0.279301\pi\)
\(158\) − 1280.00i − 0.644502i
\(159\) 234.000 0.116713
\(160\) 0 0
\(161\) −288.000 −0.140979
\(162\) − 162.000i − 0.0785674i
\(163\) 2612.00i 1.25514i 0.778561 + 0.627569i \(0.215950\pi\)
−0.778561 + 0.627569i \(0.784050\pi\)
\(164\) 792.000 0.377102
\(165\) 0 0
\(166\) −696.000 −0.325422
\(167\) 24.0000i 0.0111208i 0.999985 + 0.00556041i \(0.00176994\pi\)
−0.999985 + 0.00556041i \(0.998230\pi\)
\(168\) − 96.0000i − 0.0440867i
\(169\) 2193.00 0.998179
\(170\) 0 0
\(171\) 1260.00 0.563477
\(172\) 1072.00i 0.475228i
\(173\) 1962.00i 0.862243i 0.902294 + 0.431122i \(0.141882\pi\)
−0.902294 + 0.431122i \(0.858118\pi\)
\(174\) −1260.00 −0.548968
\(175\) 0 0
\(176\) −768.000 −0.328921
\(177\) − 720.000i − 0.305754i
\(178\) 420.000i 0.176856i
\(179\) 120.000 0.0501074 0.0250537 0.999686i \(-0.492024\pi\)
0.0250537 + 0.999686i \(0.492024\pi\)
\(180\) 0 0
\(181\) 902.000 0.370415 0.185208 0.982699i \(-0.440704\pi\)
0.185208 + 0.982699i \(0.440704\pi\)
\(182\) 16.0000i 0.00651648i
\(183\) 906.000i 0.365975i
\(184\) −576.000 −0.230779
\(185\) 0 0
\(186\) 1632.00 0.643355
\(187\) − 5472.00i − 2.13985i
\(188\) 864.000i 0.335179i
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) −168.000 −0.0636443 −0.0318221 0.999494i \(-0.510131\pi\)
−0.0318221 + 0.999494i \(0.510131\pi\)
\(192\) − 192.000i − 0.0721688i
\(193\) − 1318.00i − 0.491563i −0.969325 0.245782i \(-0.920955\pi\)
0.969325 0.245782i \(-0.0790446\pi\)
\(194\) 3068.00 1.13541
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) 4014.00i 1.45170i 0.687851 + 0.725852i \(0.258554\pi\)
−0.687851 + 0.725852i \(0.741446\pi\)
\(198\) − 864.000i − 0.310110i
\(199\) −2000.00 −0.712443 −0.356222 0.934401i \(-0.615935\pi\)
−0.356222 + 0.934401i \(0.615935\pi\)
\(200\) 0 0
\(201\) 1788.00 0.627442
\(202\) − 3444.00i − 1.19960i
\(203\) − 840.000i − 0.290426i
\(204\) 1368.00 0.469506
\(205\) 0 0
\(206\) 2104.00 0.711615
\(207\) − 648.000i − 0.217580i
\(208\) 32.0000i 0.0106673i
\(209\) 6720.00 2.22408
\(210\) 0 0
\(211\) −3868.00 −1.26201 −0.631005 0.775779i \(-0.717357\pi\)
−0.631005 + 0.775779i \(0.717357\pi\)
\(212\) 312.000i 0.101077i
\(213\) − 2304.00i − 0.741162i
\(214\) 1128.00 0.360320
\(215\) 0 0
\(216\) 216.000 0.0680414
\(217\) 1088.00i 0.340361i
\(218\) − 1220.00i − 0.379031i
\(219\) 1434.00 0.442469
\(220\) 0 0
\(221\) −228.000 −0.0693979
\(222\) 2004.00i 0.605854i
\(223\) − 3148.00i − 0.945317i −0.881246 0.472658i \(-0.843294\pi\)
0.881246 0.472658i \(-0.156706\pi\)
\(224\) 128.000 0.0381802
\(225\) 0 0
\(226\) 2604.00 0.766440
\(227\) − 2556.00i − 0.747347i −0.927560 0.373673i \(-0.878098\pi\)
0.927560 0.373673i \(-0.121902\pi\)
\(228\) 1680.00i 0.487986i
\(229\) 610.000 0.176026 0.0880130 0.996119i \(-0.471948\pi\)
0.0880130 + 0.996119i \(0.471948\pi\)
\(230\) 0 0
\(231\) 576.000 0.164061
\(232\) − 1680.00i − 0.475420i
\(233\) − 2058.00i − 0.578644i −0.957232 0.289322i \(-0.906570\pi\)
0.957232 0.289322i \(-0.0934298\pi\)
\(234\) −36.0000 −0.0100572
\(235\) 0 0
\(236\) 960.000 0.264791
\(237\) 1920.00i 0.526234i
\(238\) 912.000i 0.248387i
\(239\) −4920.00 −1.33158 −0.665792 0.746138i \(-0.731906\pi\)
−0.665792 + 0.746138i \(0.731906\pi\)
\(240\) 0 0
\(241\) −1438.00 −0.384356 −0.192178 0.981360i \(-0.561555\pi\)
−0.192178 + 0.981360i \(0.561555\pi\)
\(242\) − 1946.00i − 0.516916i
\(243\) 243.000i 0.0641500i
\(244\) −1208.00 −0.316944
\(245\) 0 0
\(246\) −1188.00 −0.307903
\(247\) − 280.000i − 0.0721294i
\(248\) 2176.00i 0.557162i
\(249\) 1044.00 0.265706
\(250\) 0 0
\(251\) 792.000 0.199166 0.0995829 0.995029i \(-0.468249\pi\)
0.0995829 + 0.995029i \(0.468249\pi\)
\(252\) 144.000i 0.0359966i
\(253\) − 3456.00i − 0.858802i
\(254\) 248.000 0.0612634
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 2166.00i − 0.525725i −0.964833 0.262863i \(-0.915333\pi\)
0.964833 0.262863i \(-0.0846666\pi\)
\(258\) − 1608.00i − 0.388022i
\(259\) −1336.00 −0.320521
\(260\) 0 0
\(261\) 1890.00 0.448230
\(262\) − 384.000i − 0.0905481i
\(263\) 3192.00i 0.748392i 0.927350 + 0.374196i \(0.122081\pi\)
−0.927350 + 0.374196i \(0.877919\pi\)
\(264\) 1152.00 0.268563
\(265\) 0 0
\(266\) −1120.00 −0.258164
\(267\) − 630.000i − 0.144402i
\(268\) 2384.00i 0.543381i
\(269\) −5490.00 −1.24435 −0.622177 0.782877i \(-0.713752\pi\)
−0.622177 + 0.782877i \(0.713752\pi\)
\(270\) 0 0
\(271\) −6328.00 −1.41845 −0.709223 0.704985i \(-0.750954\pi\)
−0.709223 + 0.704985i \(0.750954\pi\)
\(272\) 1824.00i 0.406604i
\(273\) − 24.0000i − 0.00532068i
\(274\) 5028.00 1.10859
\(275\) 0 0
\(276\) 864.000 0.188430
\(277\) 574.000i 0.124507i 0.998060 + 0.0622533i \(0.0198287\pi\)
−0.998060 + 0.0622533i \(0.980171\pi\)
\(278\) 2680.00i 0.578186i
\(279\) −2448.00 −0.525297
\(280\) 0 0
\(281\) 4242.00 0.900557 0.450278 0.892888i \(-0.351325\pi\)
0.450278 + 0.892888i \(0.351325\pi\)
\(282\) − 1296.00i − 0.273673i
\(283\) − 628.000i − 0.131911i −0.997823 0.0659553i \(-0.978991\pi\)
0.997823 0.0659553i \(-0.0210095\pi\)
\(284\) 3072.00 0.641865
\(285\) 0 0
\(286\) −192.000 −0.0396965
\(287\) − 792.000i − 0.162893i
\(288\) 288.000i 0.0589256i
\(289\) −8083.00 −1.64523
\(290\) 0 0
\(291\) −4602.00 −0.927058
\(292\) 1912.00i 0.383190i
\(293\) − 558.000i − 0.111258i −0.998451 0.0556292i \(-0.982284\pi\)
0.998451 0.0556292i \(-0.0177165\pi\)
\(294\) 1962.00 0.389205
\(295\) 0 0
\(296\) −2672.00 −0.524685
\(297\) 1296.00i 0.253204i
\(298\) 2820.00i 0.548182i
\(299\) −144.000 −0.0278520
\(300\) 0 0
\(301\) 1072.00 0.205279
\(302\) 4256.00i 0.810945i
\(303\) 5166.00i 0.979468i
\(304\) −2240.00 −0.422608
\(305\) 0 0
\(306\) −2052.00 −0.383350
\(307\) 6964.00i 1.29465i 0.762216 + 0.647323i \(0.224112\pi\)
−0.762216 + 0.647323i \(0.775888\pi\)
\(308\) 768.000i 0.142081i
\(309\) −3156.00 −0.581031
\(310\) 0 0
\(311\) 2832.00 0.516360 0.258180 0.966097i \(-0.416877\pi\)
0.258180 + 0.966097i \(0.416877\pi\)
\(312\) − 48.0000i − 0.00870982i
\(313\) 8642.00i 1.56062i 0.625392 + 0.780311i \(0.284939\pi\)
−0.625392 + 0.780311i \(0.715061\pi\)
\(314\) −6052.00 −1.08769
\(315\) 0 0
\(316\) −2560.00 −0.455732
\(317\) 2214.00i 0.392273i 0.980577 + 0.196137i \(0.0628396\pi\)
−0.980577 + 0.196137i \(0.937160\pi\)
\(318\) − 468.000i − 0.0825287i
\(319\) 10080.0 1.76919
\(320\) 0 0
\(321\) −1692.00 −0.294200
\(322\) 576.000i 0.0996870i
\(323\) − 15960.0i − 2.74934i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 5224.00 0.887517
\(327\) 1830.00i 0.309478i
\(328\) − 1584.00i − 0.266652i
\(329\) 864.000 0.144784
\(330\) 0 0
\(331\) 10772.0 1.78877 0.894385 0.447299i \(-0.147614\pi\)
0.894385 + 0.447299i \(0.147614\pi\)
\(332\) 1392.00i 0.230108i
\(333\) − 3006.00i − 0.494678i
\(334\) 48.0000 0.00786360
\(335\) 0 0
\(336\) −192.000 −0.0311740
\(337\) 1654.00i 0.267356i 0.991025 + 0.133678i \(0.0426789\pi\)
−0.991025 + 0.133678i \(0.957321\pi\)
\(338\) − 4386.00i − 0.705819i
\(339\) −3906.00 −0.625796
\(340\) 0 0
\(341\) −13056.0 −2.07338
\(342\) − 2520.00i − 0.398439i
\(343\) 2680.00i 0.421885i
\(344\) 2144.00 0.336037
\(345\) 0 0
\(346\) 3924.00 0.609698
\(347\) − 2196.00i − 0.339733i −0.985467 0.169867i \(-0.945666\pi\)
0.985467 0.169867i \(-0.0543337\pi\)
\(348\) 2520.00i 0.388179i
\(349\) −8270.00 −1.26843 −0.634216 0.773156i \(-0.718677\pi\)
−0.634216 + 0.773156i \(0.718677\pi\)
\(350\) 0 0
\(351\) 54.0000 0.00821170
\(352\) 1536.00i 0.232583i
\(353\) 10302.0i 1.55331i 0.629923 + 0.776657i \(0.283086\pi\)
−0.629923 + 0.776657i \(0.716914\pi\)
\(354\) −1440.00 −0.216201
\(355\) 0 0
\(356\) 840.000 0.125056
\(357\) − 1368.00i − 0.202807i
\(358\) − 240.000i − 0.0354313i
\(359\) 2280.00 0.335192 0.167596 0.985856i \(-0.446400\pi\)
0.167596 + 0.985856i \(0.446400\pi\)
\(360\) 0 0
\(361\) 12741.0 1.85756
\(362\) − 1804.00i − 0.261923i
\(363\) 2919.00i 0.422060i
\(364\) 32.0000 0.00460785
\(365\) 0 0
\(366\) 1812.00 0.258783
\(367\) 8764.00i 1.24653i 0.782010 + 0.623266i \(0.214195\pi\)
−0.782010 + 0.623266i \(0.785805\pi\)
\(368\) 1152.00i 0.163185i
\(369\) 1782.00 0.251402
\(370\) 0 0
\(371\) 312.000 0.0436610
\(372\) − 3264.00i − 0.454921i
\(373\) − 1318.00i − 0.182958i −0.995807 0.0914792i \(-0.970841\pi\)
0.995807 0.0914792i \(-0.0291595\pi\)
\(374\) −10944.0 −1.51310
\(375\) 0 0
\(376\) 1728.00 0.237007
\(377\) − 420.000i − 0.0573769i
\(378\) − 216.000i − 0.0293911i
\(379\) −1100.00 −0.149085 −0.0745425 0.997218i \(-0.523750\pi\)
−0.0745425 + 0.997218i \(0.523750\pi\)
\(380\) 0 0
\(381\) −372.000 −0.0500214
\(382\) 336.000i 0.0450033i
\(383\) − 3528.00i − 0.470685i −0.971912 0.235343i \(-0.924379\pi\)
0.971912 0.235343i \(-0.0756212\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −2636.00 −0.347588
\(387\) 2412.00i 0.316819i
\(388\) − 6136.00i − 0.802856i
\(389\) 9630.00 1.25517 0.627584 0.778549i \(-0.284044\pi\)
0.627584 + 0.778549i \(0.284044\pi\)
\(390\) 0 0
\(391\) −8208.00 −1.06163
\(392\) 2616.00i 0.337061i
\(393\) 576.000i 0.0739322i
\(394\) 8028.00 1.02651
\(395\) 0 0
\(396\) −1728.00 −0.219281
\(397\) 3094.00i 0.391142i 0.980690 + 0.195571i \(0.0626560\pi\)
−0.980690 + 0.195571i \(0.937344\pi\)
\(398\) 4000.00i 0.503774i
\(399\) 1680.00 0.210790
\(400\) 0 0
\(401\) −1638.00 −0.203985 −0.101992 0.994785i \(-0.532522\pi\)
−0.101992 + 0.994785i \(0.532522\pi\)
\(402\) − 3576.00i − 0.443668i
\(403\) 544.000i 0.0672421i
\(404\) −6888.00 −0.848245
\(405\) 0 0
\(406\) −1680.00 −0.205362
\(407\) − 16032.0i − 1.95252i
\(408\) − 2736.00i − 0.331991i
\(409\) 13750.0 1.66233 0.831166 0.556024i \(-0.187674\pi\)
0.831166 + 0.556024i \(0.187674\pi\)
\(410\) 0 0
\(411\) −7542.00 −0.905157
\(412\) − 4208.00i − 0.503188i
\(413\) − 960.000i − 0.114379i
\(414\) −1296.00 −0.153852
\(415\) 0 0
\(416\) 64.0000 0.00754293
\(417\) − 4020.00i − 0.472087i
\(418\) − 13440.0i − 1.57266i
\(419\) 12480.0 1.45510 0.727551 0.686053i \(-0.240658\pi\)
0.727551 + 0.686053i \(0.240658\pi\)
\(420\) 0 0
\(421\) 7262.00 0.840685 0.420342 0.907366i \(-0.361910\pi\)
0.420342 + 0.907366i \(0.361910\pi\)
\(422\) 7736.00i 0.892376i
\(423\) 1944.00i 0.223453i
\(424\) 624.000 0.0714720
\(425\) 0 0
\(426\) −4608.00 −0.524081
\(427\) 1208.00i 0.136907i
\(428\) − 2256.00i − 0.254785i
\(429\) 288.000 0.0324121
\(430\) 0 0
\(431\) 9792.00 1.09435 0.547174 0.837019i \(-0.315704\pi\)
0.547174 + 0.837019i \(0.315704\pi\)
\(432\) − 432.000i − 0.0481125i
\(433\) 1802.00i 0.199997i 0.994988 + 0.0999984i \(0.0318838\pi\)
−0.994988 + 0.0999984i \(0.968116\pi\)
\(434\) 2176.00 0.240671
\(435\) 0 0
\(436\) −2440.00 −0.268016
\(437\) − 10080.0i − 1.10341i
\(438\) − 2868.00i − 0.312873i
\(439\) 2320.00 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 456.000i 0.0490717i
\(443\) 11172.0i 1.19819i 0.800678 + 0.599095i \(0.204473\pi\)
−0.800678 + 0.599095i \(0.795527\pi\)
\(444\) 4008.00 0.428404
\(445\) 0 0
\(446\) −6296.00 −0.668440
\(447\) − 4230.00i − 0.447589i
\(448\) − 256.000i − 0.0269975i
\(449\) −6810.00 −0.715777 −0.357888 0.933764i \(-0.616503\pi\)
−0.357888 + 0.933764i \(0.616503\pi\)
\(450\) 0 0
\(451\) 9504.00 0.992297
\(452\) − 5208.00i − 0.541955i
\(453\) − 6384.00i − 0.662134i
\(454\) −5112.00 −0.528454
\(455\) 0 0
\(456\) 3360.00 0.345058
\(457\) − 17066.0i − 1.74686i −0.486952 0.873429i \(-0.661891\pi\)
0.486952 0.873429i \(-0.338109\pi\)
\(458\) − 1220.00i − 0.124469i
\(459\) 3078.00 0.313004
\(460\) 0 0
\(461\) −18918.0 −1.91128 −0.955639 0.294541i \(-0.904833\pi\)
−0.955639 + 0.294541i \(0.904833\pi\)
\(462\) − 1152.00i − 0.116008i
\(463\) 1052.00i 0.105595i 0.998605 + 0.0527976i \(0.0168138\pi\)
−0.998605 + 0.0527976i \(0.983186\pi\)
\(464\) −3360.00 −0.336173
\(465\) 0 0
\(466\) −4116.00 −0.409163
\(467\) − 11076.0i − 1.09751i −0.835984 0.548754i \(-0.815102\pi\)
0.835984 0.548754i \(-0.184898\pi\)
\(468\) 72.0000i 0.00711154i
\(469\) 2384.00 0.234718
\(470\) 0 0
\(471\) 9078.00 0.888094
\(472\) − 1920.00i − 0.187236i
\(473\) 12864.0i 1.25050i
\(474\) 3840.00 0.372103
\(475\) 0 0
\(476\) 1824.00 0.175636
\(477\) 702.000i 0.0673844i
\(478\) 9840.00i 0.941571i
\(479\) 9000.00 0.858498 0.429249 0.903186i \(-0.358778\pi\)
0.429249 + 0.903186i \(0.358778\pi\)
\(480\) 0 0
\(481\) −668.000 −0.0633226
\(482\) 2876.00i 0.271781i
\(483\) − 864.000i − 0.0813941i
\(484\) −3892.00 −0.365515
\(485\) 0 0
\(486\) 486.000 0.0453609
\(487\) 8764.00i 0.815472i 0.913100 + 0.407736i \(0.133682\pi\)
−0.913100 + 0.407736i \(0.866318\pi\)
\(488\) 2416.00i 0.224113i
\(489\) −7836.00 −0.724655
\(490\) 0 0
\(491\) 5592.00 0.513978 0.256989 0.966414i \(-0.417270\pi\)
0.256989 + 0.966414i \(0.417270\pi\)
\(492\) 2376.00i 0.217720i
\(493\) − 23940.0i − 2.18703i
\(494\) −560.000 −0.0510032
\(495\) 0 0
\(496\) 4352.00 0.393973
\(497\) − 3072.00i − 0.277260i
\(498\) − 2088.00i − 0.187883i
\(499\) −4700.00 −0.421645 −0.210823 0.977524i \(-0.567614\pi\)
−0.210823 + 0.977524i \(0.567614\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) − 1584.00i − 0.140831i
\(503\) − 11808.0i − 1.04671i −0.852116 0.523353i \(-0.824681\pi\)
0.852116 0.523353i \(-0.175319\pi\)
\(504\) 288.000 0.0254535
\(505\) 0 0
\(506\) −6912.00 −0.607265
\(507\) 6579.00i 0.576299i
\(508\) − 496.000i − 0.0433198i
\(509\) −1170.00 −0.101885 −0.0509424 0.998702i \(-0.516222\pi\)
−0.0509424 + 0.998702i \(0.516222\pi\)
\(510\) 0 0
\(511\) 1912.00 0.165522
\(512\) − 512.000i − 0.0441942i
\(513\) 3780.00i 0.325324i
\(514\) −4332.00 −0.371744
\(515\) 0 0
\(516\) −3216.00 −0.274373
\(517\) 10368.0i 0.881981i
\(518\) 2672.00i 0.226643i
\(519\) −5886.00 −0.497816
\(520\) 0 0
\(521\) −16638.0 −1.39909 −0.699543 0.714590i \(-0.746613\pi\)
−0.699543 + 0.714590i \(0.746613\pi\)
\(522\) − 3780.00i − 0.316947i
\(523\) 15692.0i 1.31198i 0.754771 + 0.655988i \(0.227748\pi\)
−0.754771 + 0.655988i \(0.772252\pi\)
\(524\) −768.000 −0.0640272
\(525\) 0 0
\(526\) 6384.00 0.529193
\(527\) 31008.0i 2.56305i
\(528\) − 2304.00i − 0.189903i
\(529\) 6983.00 0.573929
\(530\) 0 0
\(531\) 2160.00 0.176527
\(532\) 2240.00i 0.182549i
\(533\) − 396.000i − 0.0321814i
\(534\) −1260.00 −0.102108
\(535\) 0 0
\(536\) 4768.00 0.384228
\(537\) 360.000i 0.0289295i
\(538\) 10980.0i 0.879891i
\(539\) −15696.0 −1.25431
\(540\) 0 0
\(541\) −22018.0 −1.74977 −0.874887 0.484327i \(-0.839064\pi\)
−0.874887 + 0.484327i \(0.839064\pi\)
\(542\) 12656.0i 1.00299i
\(543\) 2706.00i 0.213859i
\(544\) 3648.00 0.287512
\(545\) 0 0
\(546\) −48.0000 −0.00376229
\(547\) 4564.00i 0.356751i 0.983963 + 0.178375i \(0.0570841\pi\)
−0.983963 + 0.178375i \(0.942916\pi\)
\(548\) − 10056.0i − 0.783889i
\(549\) −2718.00 −0.211296
\(550\) 0 0
\(551\) 29400.0 2.27311
\(552\) − 1728.00i − 0.133240i
\(553\) 2560.00i 0.196858i
\(554\) 1148.00 0.0880394
\(555\) 0 0
\(556\) 5360.00 0.408839
\(557\) 7734.00i 0.588331i 0.955755 + 0.294165i \(0.0950416\pi\)
−0.955755 + 0.294165i \(0.904958\pi\)
\(558\) 4896.00i 0.371441i
\(559\) 536.000 0.0405552
\(560\) 0 0
\(561\) 16416.0 1.23544
\(562\) − 8484.00i − 0.636790i
\(563\) − 20148.0i − 1.50824i −0.656739 0.754118i \(-0.728065\pi\)
0.656739 0.754118i \(-0.271935\pi\)
\(564\) −2592.00 −0.193516
\(565\) 0 0
\(566\) −1256.00 −0.0932749
\(567\) 324.000i 0.0239977i
\(568\) − 6144.00i − 0.453867i
\(569\) 24030.0 1.77046 0.885228 0.465156i \(-0.154002\pi\)
0.885228 + 0.465156i \(0.154002\pi\)
\(570\) 0 0
\(571\) 2372.00 0.173844 0.0869222 0.996215i \(-0.472297\pi\)
0.0869222 + 0.996215i \(0.472297\pi\)
\(572\) 384.000i 0.0280697i
\(573\) − 504.000i − 0.0367450i
\(574\) −1584.00 −0.115183
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) − 8546.00i − 0.616594i −0.951290 0.308297i \(-0.900241\pi\)
0.951290 0.308297i \(-0.0997590\pi\)
\(578\) 16166.0i 1.16335i
\(579\) 3954.00 0.283804
\(580\) 0 0
\(581\) 1392.00 0.0993974
\(582\) 9204.00i 0.655529i
\(583\) 3744.00i 0.265970i
\(584\) 3824.00 0.270956
\(585\) 0 0
\(586\) −1116.00 −0.0786716
\(587\) 15444.0i 1.08593i 0.839755 + 0.542966i \(0.182699\pi\)
−0.839755 + 0.542966i \(0.817301\pi\)
\(588\) − 3924.00i − 0.275209i
\(589\) −38080.0 −2.66394
\(590\) 0 0
\(591\) −12042.0 −0.838142
\(592\) 5344.00i 0.371009i
\(593\) 18342.0i 1.27018i 0.772439 + 0.635089i \(0.219037\pi\)
−0.772439 + 0.635089i \(0.780963\pi\)
\(594\) 2592.00 0.179042
\(595\) 0 0
\(596\) 5640.00 0.387623
\(597\) − 6000.00i − 0.411329i
\(598\) 288.000i 0.0196943i
\(599\) −24600.0 −1.67801 −0.839006 0.544123i \(-0.816863\pi\)
−0.839006 + 0.544123i \(0.816863\pi\)
\(600\) 0 0
\(601\) −8998.00 −0.610709 −0.305354 0.952239i \(-0.598775\pi\)
−0.305354 + 0.952239i \(0.598775\pi\)
\(602\) − 2144.00i − 0.145154i
\(603\) 5364.00i 0.362254i
\(604\) 8512.00 0.573424
\(605\) 0 0
\(606\) 10332.0 0.692589
\(607\) − 4076.00i − 0.272553i −0.990671 0.136277i \(-0.956486\pi\)
0.990671 0.136277i \(-0.0435136\pi\)
\(608\) 4480.00i 0.298829i
\(609\) 2520.00 0.167677
\(610\) 0 0
\(611\) 432.000 0.0286037
\(612\) 4104.00i 0.271069i
\(613\) − 4078.00i − 0.268693i −0.990934 0.134347i \(-0.957106\pi\)
0.990934 0.134347i \(-0.0428935\pi\)
\(614\) 13928.0 0.915453
\(615\) 0 0
\(616\) 1536.00 0.100466
\(617\) − 10086.0i − 0.658099i −0.944313 0.329049i \(-0.893272\pi\)
0.944313 0.329049i \(-0.106728\pi\)
\(618\) 6312.00i 0.410851i
\(619\) −8780.00 −0.570110 −0.285055 0.958511i \(-0.592012\pi\)
−0.285055 + 0.958511i \(0.592012\pi\)
\(620\) 0 0
\(621\) 1944.00 0.125620
\(622\) − 5664.00i − 0.365122i
\(623\) − 840.000i − 0.0540191i
\(624\) −96.0000 −0.00615878
\(625\) 0 0
\(626\) 17284.0 1.10353
\(627\) 20160.0i 1.28407i
\(628\) 12104.0i 0.769112i
\(629\) −38076.0 −2.41366
\(630\) 0 0
\(631\) 2792.00 0.176145 0.0880727 0.996114i \(-0.471929\pi\)
0.0880727 + 0.996114i \(0.471929\pi\)
\(632\) 5120.00i 0.322251i
\(633\) − 11604.0i − 0.728622i
\(634\) 4428.00 0.277379
\(635\) 0 0
\(636\) −936.000 −0.0583566
\(637\) 654.000i 0.0406788i
\(638\) − 20160.0i − 1.25101i
\(639\) 6912.00 0.427910
\(640\) 0 0
\(641\) 7602.00 0.468426 0.234213 0.972185i \(-0.424749\pi\)
0.234213 + 0.972185i \(0.424749\pi\)
\(642\) 3384.00i 0.208031i
\(643\) 24212.0i 1.48496i 0.669869 + 0.742479i \(0.266350\pi\)
−0.669869 + 0.742479i \(0.733650\pi\)
\(644\) 1152.00 0.0704894
\(645\) 0 0
\(646\) −31920.0 −1.94408
\(647\) − 9456.00i − 0.574581i −0.957844 0.287290i \(-0.907246\pi\)
0.957844 0.287290i \(-0.0927545\pi\)
\(648\) 648.000i 0.0392837i
\(649\) 11520.0 0.696764
\(650\) 0 0
\(651\) −3264.00 −0.196507
\(652\) − 10448.0i − 0.627569i
\(653\) − 9558.00i − 0.572792i −0.958111 0.286396i \(-0.907543\pi\)
0.958111 0.286396i \(-0.0924574\pi\)
\(654\) 3660.00 0.218834
\(655\) 0 0
\(656\) −3168.00 −0.188551
\(657\) 4302.00i 0.255460i
\(658\) − 1728.00i − 0.102378i
\(659\) 29280.0 1.73078 0.865392 0.501095i \(-0.167069\pi\)
0.865392 + 0.501095i \(0.167069\pi\)
\(660\) 0 0
\(661\) −29098.0 −1.71223 −0.856113 0.516789i \(-0.827127\pi\)
−0.856113 + 0.516789i \(0.827127\pi\)
\(662\) − 21544.0i − 1.26485i
\(663\) − 684.000i − 0.0400669i
\(664\) 2784.00 0.162711
\(665\) 0 0
\(666\) −6012.00 −0.349790
\(667\) − 15120.0i − 0.877734i
\(668\) − 96.0000i − 0.00556041i
\(669\) 9444.00 0.545779
\(670\) 0 0
\(671\) −14496.0 −0.833997
\(672\) 384.000i 0.0220433i
\(673\) − 11638.0i − 0.666585i −0.942823 0.333293i \(-0.891840\pi\)
0.942823 0.333293i \(-0.108160\pi\)
\(674\) 3308.00 0.189050
\(675\) 0 0
\(676\) −8772.00 −0.499090
\(677\) − 3426.00i − 0.194493i −0.995260 0.0972466i \(-0.968996\pi\)
0.995260 0.0972466i \(-0.0310035\pi\)
\(678\) 7812.00i 0.442505i
\(679\) −6136.00 −0.346801
\(680\) 0 0
\(681\) 7668.00 0.431481
\(682\) 26112.0i 1.46610i
\(683\) − 20148.0i − 1.12876i −0.825516 0.564379i \(-0.809116\pi\)
0.825516 0.564379i \(-0.190884\pi\)
\(684\) −5040.00 −0.281739
\(685\) 0 0
\(686\) 5360.00 0.298317
\(687\) 1830.00i 0.101629i
\(688\) − 4288.00i − 0.237614i
\(689\) 156.000 0.00862573
\(690\) 0 0
\(691\) −29428.0 −1.62011 −0.810053 0.586356i \(-0.800562\pi\)
−0.810053 + 0.586356i \(0.800562\pi\)
\(692\) − 7848.00i − 0.431122i
\(693\) 1728.00i 0.0947205i
\(694\) −4392.00 −0.240228
\(695\) 0 0
\(696\) 5040.00 0.274484
\(697\) − 22572.0i − 1.22665i
\(698\) 16540.0i 0.896917i
\(699\) 6174.00 0.334080
\(700\) 0 0
\(701\) 16242.0 0.875110 0.437555 0.899192i \(-0.355845\pi\)
0.437555 + 0.899192i \(0.355845\pi\)
\(702\) − 108.000i − 0.00580655i
\(703\) − 46760.0i − 2.50866i
\(704\) 3072.00 0.164461
\(705\) 0 0
\(706\) 20604.0 1.09836
\(707\) 6888.00i 0.366407i
\(708\) 2880.00i 0.152877i
\(709\) −2030.00 −0.107529 −0.0537646 0.998554i \(-0.517122\pi\)
−0.0537646 + 0.998554i \(0.517122\pi\)
\(710\) 0 0
\(711\) −5760.00 −0.303821
\(712\) − 1680.00i − 0.0884279i
\(713\) 19584.0i 1.02865i
\(714\) −2736.00 −0.143406
\(715\) 0 0
\(716\) −480.000 −0.0250537
\(717\) − 14760.0i − 0.768790i
\(718\) − 4560.00i − 0.237016i
\(719\) −6960.00 −0.361007 −0.180504 0.983574i \(-0.557773\pi\)
−0.180504 + 0.983574i \(0.557773\pi\)
\(720\) 0 0
\(721\) −4208.00 −0.217357
\(722\) − 25482.0i − 1.31349i
\(723\) − 4314.00i − 0.221908i
\(724\) −3608.00 −0.185208
\(725\) 0 0
\(726\) 5838.00 0.298441
\(727\) − 18596.0i − 0.948676i −0.880343 0.474338i \(-0.842687\pi\)
0.880343 0.474338i \(-0.157313\pi\)
\(728\) − 64.0000i − 0.00325824i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 30552.0 1.54584
\(732\) − 3624.00i − 0.182988i
\(733\) 21242.0i 1.07038i 0.844731 + 0.535192i \(0.179761\pi\)
−0.844731 + 0.535192i \(0.820239\pi\)
\(734\) 17528.0 0.881431
\(735\) 0 0
\(736\) 2304.00 0.115389
\(737\) 28608.0i 1.42984i
\(738\) − 3564.00i − 0.177768i
\(739\) 340.000 0.0169244 0.00846218 0.999964i \(-0.497306\pi\)
0.00846218 + 0.999964i \(0.497306\pi\)
\(740\) 0 0
\(741\) 840.000 0.0416440
\(742\) − 624.000i − 0.0308730i
\(743\) − 21888.0i − 1.08074i −0.841426 0.540372i \(-0.818284\pi\)
0.841426 0.540372i \(-0.181716\pi\)
\(744\) −6528.00 −0.321678
\(745\) 0 0
\(746\) −2636.00 −0.129371
\(747\) 3132.00i 0.153405i
\(748\) 21888.0i 1.06993i
\(749\) −2256.00 −0.110057
\(750\) 0 0
\(751\) 17792.0 0.864500 0.432250 0.901754i \(-0.357720\pi\)
0.432250 + 0.901754i \(0.357720\pi\)
\(752\) − 3456.00i − 0.167590i
\(753\) 2376.00i 0.114988i
\(754\) −840.000 −0.0405716
\(755\) 0 0
\(756\) −432.000 −0.0207827
\(757\) − 37346.0i − 1.79308i −0.442960 0.896541i \(-0.646072\pi\)
0.442960 0.896541i \(-0.353928\pi\)
\(758\) 2200.00i 0.105419i
\(759\) 10368.0 0.495829
\(760\) 0 0
\(761\) −11358.0 −0.541034 −0.270517 0.962715i \(-0.587195\pi\)
−0.270517 + 0.962715i \(0.587195\pi\)
\(762\) 744.000i 0.0353704i
\(763\) 2440.00i 0.115772i
\(764\) 672.000 0.0318221
\(765\) 0 0
\(766\) −7056.00 −0.332825
\(767\) − 480.000i − 0.0225969i
\(768\) 768.000i 0.0360844i
\(769\) 34270.0 1.60703 0.803516 0.595283i \(-0.202960\pi\)
0.803516 + 0.595283i \(0.202960\pi\)
\(770\) 0 0
\(771\) 6498.00 0.303528
\(772\) 5272.00i 0.245782i
\(773\) − 13278.0i − 0.617822i −0.951091 0.308911i \(-0.900035\pi\)
0.951091 0.308911i \(-0.0999645\pi\)
\(774\) 4824.00 0.224025
\(775\) 0 0
\(776\) −12272.0 −0.567705
\(777\) − 4008.00i − 0.185053i
\(778\) − 19260.0i − 0.887538i
\(779\) 27720.0 1.27493
\(780\) 0 0
\(781\) 36864.0 1.68899
\(782\) 16416.0i 0.750684i
\(783\) 5670.00i 0.258786i
\(784\) 5232.00 0.238338
\(785\) 0 0
\(786\) 1152.00 0.0522780
\(787\) 11164.0i 0.505659i 0.967511 + 0.252829i \(0.0813612\pi\)
−0.967511 + 0.252829i \(0.918639\pi\)
\(788\) − 16056.0i − 0.725852i
\(789\) −9576.00 −0.432084
\(790\) 0 0
\(791\) −5208.00 −0.234103
\(792\) 3456.00i 0.155055i
\(793\) 604.000i 0.0270475i
\(794\) 6188.00 0.276579
\(795\) 0 0
\(796\) 8000.00 0.356222
\(797\) 5094.00i 0.226397i 0.993572 + 0.113199i \(0.0361097\pi\)
−0.993572 + 0.113199i \(0.963890\pi\)
\(798\) − 3360.00i − 0.149051i
\(799\) 24624.0 1.09028
\(800\) 0 0
\(801\) 1890.00 0.0833706
\(802\) 3276.00i 0.144239i
\(803\) 22944.0i 1.00831i
\(804\) −7152.00 −0.313721
\(805\) 0 0
\(806\) 1088.00 0.0475474
\(807\) − 16470.0i − 0.718428i
\(808\) 13776.0i 0.599799i
\(809\) 8790.00 0.382002 0.191001 0.981590i \(-0.438827\pi\)
0.191001 + 0.981590i \(0.438827\pi\)
\(810\) 0 0
\(811\) 5852.00 0.253380 0.126690 0.991942i \(-0.459565\pi\)
0.126690 + 0.991942i \(0.459565\pi\)
\(812\) 3360.00i 0.145213i
\(813\) − 18984.0i − 0.818940i
\(814\) −32064.0 −1.38064
\(815\) 0 0
\(816\) −5472.00 −0.234753
\(817\) 37520.0i 1.60668i
\(818\) − 27500.0i − 1.17545i
\(819\) 72.0000 0.00307190
\(820\) 0 0
\(821\) −29478.0 −1.25309 −0.626546 0.779384i \(-0.715532\pi\)
−0.626546 + 0.779384i \(0.715532\pi\)
\(822\) 15084.0i 0.640042i
\(823\) 39332.0i 1.66589i 0.553356 + 0.832945i \(0.313347\pi\)
−0.553356 + 0.832945i \(0.686653\pi\)
\(824\) −8416.00 −0.355807
\(825\) 0 0
\(826\) −1920.00 −0.0808781
\(827\) − 6756.00i − 0.284074i −0.989861 0.142037i \(-0.954635\pi\)
0.989861 0.142037i \(-0.0453652\pi\)
\(828\) 2592.00i 0.108790i
\(829\) −3950.00 −0.165488 −0.0827438 0.996571i \(-0.526368\pi\)
−0.0827438 + 0.996571i \(0.526368\pi\)
\(830\) 0 0
\(831\) −1722.00 −0.0718839
\(832\) − 128.000i − 0.00533366i
\(833\) 37278.0i 1.55055i
\(834\) −8040.00 −0.333816
\(835\) 0 0
\(836\) −26880.0 −1.11204
\(837\) − 7344.00i − 0.303280i
\(838\) − 24960.0i − 1.02891i
\(839\) −12360.0 −0.508599 −0.254300 0.967126i \(-0.581845\pi\)
−0.254300 + 0.967126i \(0.581845\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) − 14524.0i − 0.594454i
\(843\) 12726.0i 0.519937i
\(844\) 15472.0 0.631005
\(845\) 0 0
\(846\) 3888.00 0.158005
\(847\) 3892.00i 0.157887i
\(848\) − 1248.00i − 0.0505383i
\(849\) 1884.00 0.0761587
\(850\) 0 0
\(851\) −24048.0 −0.968690
\(852\) 9216.00i 0.370581i
\(853\) − 35998.0i − 1.44496i −0.691394 0.722478i \(-0.743003\pi\)
0.691394 0.722478i \(-0.256997\pi\)
\(854\) 2416.00 0.0968077
\(855\) 0 0
\(856\) −4512.00 −0.180160
\(857\) 21594.0i 0.860720i 0.902657 + 0.430360i \(0.141613\pi\)
−0.902657 + 0.430360i \(0.858387\pi\)
\(858\) − 576.000i − 0.0229188i
\(859\) −9260.00 −0.367808 −0.183904 0.982944i \(-0.558874\pi\)
−0.183904 + 0.982944i \(0.558874\pi\)
\(860\) 0 0
\(861\) 2376.00 0.0940463
\(862\) − 19584.0i − 0.773821i
\(863\) 31632.0i 1.24770i 0.781544 + 0.623850i \(0.214433\pi\)
−0.781544 + 0.623850i \(0.785567\pi\)
\(864\) −864.000 −0.0340207
\(865\) 0 0
\(866\) 3604.00 0.141419
\(867\) − 24249.0i − 0.949872i
\(868\) − 4352.00i − 0.170180i
\(869\) −30720.0 −1.19920
\(870\) 0 0
\(871\) 1192.00 0.0463713
\(872\) 4880.00i 0.189516i
\(873\) − 13806.0i − 0.535237i
\(874\) −20160.0 −0.780231
\(875\) 0 0
\(876\) −5736.00 −0.221235
\(877\) 39694.0i 1.52836i 0.645003 + 0.764180i \(0.276856\pi\)
−0.645003 + 0.764180i \(0.723144\pi\)
\(878\) − 4640.00i − 0.178351i
\(879\) 1674.00 0.0642351
\(880\) 0 0
\(881\) 1242.00 0.0474961 0.0237480 0.999718i \(-0.492440\pi\)
0.0237480 + 0.999718i \(0.492440\pi\)
\(882\) 5886.00i 0.224707i
\(883\) − 2668.00i − 0.101682i −0.998707 0.0508411i \(-0.983810\pi\)
0.998707 0.0508411i \(-0.0161902\pi\)
\(884\) 912.000 0.0346990
\(885\) 0 0
\(886\) 22344.0 0.847248
\(887\) 4344.00i 0.164439i 0.996614 + 0.0822194i \(0.0262008\pi\)
−0.996614 + 0.0822194i \(0.973799\pi\)
\(888\) − 8016.00i − 0.302927i
\(889\) −496.000 −0.0187124
\(890\) 0 0
\(891\) −3888.00 −0.146187
\(892\) 12592.0i 0.472658i
\(893\) 30240.0i 1.13319i
\(894\) −8460.00 −0.316493
\(895\) 0 0
\(896\) −512.000 −0.0190901
\(897\) − 432.000i − 0.0160803i
\(898\) 13620.0i 0.506131i
\(899\) −57120.0 −2.11909
\(900\) 0 0
\(901\) 8892.00 0.328785
\(902\) − 19008.0i − 0.701660i
\(903\) 3216.00i 0.118518i
\(904\) −10416.0 −0.383220
\(905\) 0 0
\(906\) −12768.0 −0.468199
\(907\) − 4436.00i − 0.162398i −0.996698 0.0811990i \(-0.974125\pi\)
0.996698 0.0811990i \(-0.0258749\pi\)
\(908\) 10224.0i 0.373673i
\(909\) −15498.0 −0.565496
\(910\) 0 0
\(911\) 22752.0 0.827450 0.413725 0.910402i \(-0.364227\pi\)
0.413725 + 0.910402i \(0.364227\pi\)
\(912\) − 6720.00i − 0.243993i
\(913\) 16704.0i 0.605500i
\(914\) −34132.0 −1.23521
\(915\) 0 0
\(916\) −2440.00 −0.0880130
\(917\) 768.000i 0.0276571i
\(918\) − 6156.00i − 0.221327i
\(919\) 27160.0 0.974892 0.487446 0.873153i \(-0.337929\pi\)
0.487446 + 0.873153i \(0.337929\pi\)
\(920\) 0 0
\(921\) −20892.0 −0.747465
\(922\) 37836.0i 1.35148i
\(923\) − 1536.00i − 0.0547758i
\(924\) −2304.00 −0.0820303
\(925\) 0 0
\(926\) 2104.00 0.0746671
\(927\) − 9468.00i − 0.335458i
\(928\) 6720.00i 0.237710i
\(929\) 33030.0 1.16650 0.583250 0.812292i \(-0.301781\pi\)
0.583250 + 0.812292i \(0.301781\pi\)
\(930\) 0 0
\(931\) −45780.0 −1.61158
\(932\) 8232.00i 0.289322i
\(933\) 8496.00i 0.298121i
\(934\) −22152.0 −0.776055
\(935\) 0 0
\(936\) 144.000 0.00502862
\(937\) 29974.0i 1.04505i 0.852625 + 0.522523i \(0.175009\pi\)
−0.852625 + 0.522523i \(0.824991\pi\)
\(938\) − 4768.00i − 0.165971i
\(939\) −25926.0 −0.901026
\(940\) 0 0
\(941\) 13962.0 0.483686 0.241843 0.970315i \(-0.422248\pi\)
0.241843 + 0.970315i \(0.422248\pi\)
\(942\) − 18156.0i − 0.627977i
\(943\) − 14256.0i − 0.492300i
\(944\) −3840.00 −0.132396
\(945\) 0 0
\(946\) 25728.0 0.884238
\(947\) − 35196.0i − 1.20773i −0.797088 0.603863i \(-0.793627\pi\)
0.797088 0.603863i \(-0.206373\pi\)
\(948\) − 7680.00i − 0.263117i
\(949\) 956.000 0.0327008
\(950\) 0 0
\(951\) −6642.00 −0.226479
\(952\) − 3648.00i − 0.124194i
\(953\) − 28338.0i − 0.963230i −0.876383 0.481615i \(-0.840050\pi\)
0.876383 0.481615i \(-0.159950\pi\)
\(954\) 1404.00 0.0476480
\(955\) 0 0
\(956\) 19680.0 0.665792
\(957\) 30240.0i 1.02144i
\(958\) − 18000.0i − 0.607050i
\(959\) −10056.0 −0.338608
\(960\) 0 0
\(961\) 44193.0 1.48343
\(962\) 1336.00i 0.0447759i
\(963\) − 5076.00i − 0.169857i
\(964\) 5752.00 0.192178
\(965\) 0 0
\(966\) −1728.00 −0.0575543
\(967\) 17524.0i 0.582765i 0.956607 + 0.291383i \(0.0941153\pi\)
−0.956607 + 0.291383i \(0.905885\pi\)
\(968\) 7784.00i 0.258458i
\(969\) 47880.0 1.58733
\(970\) 0 0
\(971\) −26808.0 −0.886004 −0.443002 0.896521i \(-0.646087\pi\)
−0.443002 + 0.896521i \(0.646087\pi\)
\(972\) − 972.000i − 0.0320750i
\(973\) − 5360.00i − 0.176602i
\(974\) 17528.0 0.576626
\(975\) 0 0
\(976\) 4832.00 0.158472
\(977\) 10914.0i 0.357390i 0.983905 + 0.178695i \(0.0571875\pi\)
−0.983905 + 0.178695i \(0.942813\pi\)
\(978\) 15672.0i 0.512408i
\(979\) 10080.0 0.329069
\(980\) 0 0
\(981\) −5490.00 −0.178677
\(982\) − 11184.0i − 0.363438i
\(983\) 22272.0i 0.722652i 0.932440 + 0.361326i \(0.117676\pi\)
−0.932440 + 0.361326i \(0.882324\pi\)
\(984\) 4752.00 0.153951
\(985\) 0 0
\(986\) −47880.0 −1.54646
\(987\) 2592.00i 0.0835910i
\(988\) 1120.00i 0.0360647i
\(989\) 19296.0 0.620402
\(990\) 0 0
\(991\) 14072.0 0.451071 0.225536 0.974235i \(-0.427587\pi\)
0.225536 + 0.974235i \(0.427587\pi\)
\(992\) − 8704.00i − 0.278581i
\(993\) 32316.0i 1.03275i
\(994\) −6144.00 −0.196052
\(995\) 0 0
\(996\) −4176.00 −0.132853
\(997\) − 4826.00i − 0.153301i −0.997058 0.0766504i \(-0.975577\pi\)
0.997058 0.0766504i \(-0.0244225\pi\)
\(998\) 9400.00i 0.298148i
\(999\) 9018.00 0.285602
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 150.4.c.c.49.1 2
3.2 odd 2 450.4.c.j.199.2 2
4.3 odd 2 1200.4.f.r.49.1 2
5.2 odd 4 30.4.a.b.1.1 1
5.3 odd 4 150.4.a.b.1.1 1
5.4 even 2 inner 150.4.c.c.49.2 2
15.2 even 4 90.4.a.c.1.1 1
15.8 even 4 450.4.a.r.1.1 1
15.14 odd 2 450.4.c.j.199.1 2
20.3 even 4 1200.4.a.ba.1.1 1
20.7 even 4 240.4.a.b.1.1 1
20.19 odd 2 1200.4.f.r.49.2 2
35.27 even 4 1470.4.a.r.1.1 1
40.27 even 4 960.4.a.bg.1.1 1
40.37 odd 4 960.4.a.n.1.1 1
45.2 even 12 810.4.e.p.271.1 2
45.7 odd 12 810.4.e.i.271.1 2
45.22 odd 12 810.4.e.i.541.1 2
45.32 even 12 810.4.e.p.541.1 2
60.47 odd 4 720.4.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.b.1.1 1 5.2 odd 4
90.4.a.c.1.1 1 15.2 even 4
150.4.a.b.1.1 1 5.3 odd 4
150.4.c.c.49.1 2 1.1 even 1 trivial
150.4.c.c.49.2 2 5.4 even 2 inner
240.4.a.b.1.1 1 20.7 even 4
450.4.a.r.1.1 1 15.8 even 4
450.4.c.j.199.1 2 15.14 odd 2
450.4.c.j.199.2 2 3.2 odd 2
720.4.a.y.1.1 1 60.47 odd 4
810.4.e.i.271.1 2 45.7 odd 12
810.4.e.i.541.1 2 45.22 odd 12
810.4.e.p.271.1 2 45.2 even 12
810.4.e.p.541.1 2 45.32 even 12
960.4.a.n.1.1 1 40.37 odd 4
960.4.a.bg.1.1 1 40.27 even 4
1200.4.a.ba.1.1 1 20.3 even 4
1200.4.f.r.49.1 2 4.3 odd 2
1200.4.f.r.49.2 2 20.19 odd 2
1470.4.a.r.1.1 1 35.27 even 4