Properties

Label 75.4.b.a.49.1
Level $75$
Weight $4$
Character 75.49
Analytic conductor $4.425$
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] = [75,4,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, 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("75.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(4.42514325043\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.4.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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} -20.0000i q^{7} -21.0000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} -20.0000i q^{7} -21.0000i q^{8} -9.00000 q^{9} -24.0000 q^{11} +3.00000i q^{12} +74.0000i q^{13} -60.0000 q^{14} -71.0000 q^{16} -54.0000i q^{17} +27.0000i q^{18} +124.000 q^{19} -60.0000 q^{21} +72.0000i q^{22} -120.000i q^{23} -63.0000 q^{24} +222.000 q^{26} +27.0000i q^{27} +20.0000i q^{28} +78.0000 q^{29} +200.000 q^{31} +45.0000i q^{32} +72.0000i q^{33} -162.000 q^{34} +9.00000 q^{36} +70.0000i q^{37} -372.000i q^{38} +222.000 q^{39} +330.000 q^{41} +180.000i q^{42} +92.0000i q^{43} +24.0000 q^{44} -360.000 q^{46} +24.0000i q^{47} +213.000i q^{48} -57.0000 q^{49} -162.000 q^{51} -74.0000i q^{52} +450.000i q^{53} +81.0000 q^{54} -420.000 q^{56} -372.000i q^{57} -234.000i q^{58} -24.0000 q^{59} -322.000 q^{61} -600.000i q^{62} +180.000i q^{63} -433.000 q^{64} +216.000 q^{66} +196.000i q^{67} +54.0000i q^{68} -360.000 q^{69} -288.000 q^{71} +189.000i q^{72} -430.000i q^{73} +210.000 q^{74} -124.000 q^{76} +480.000i q^{77} -666.000i q^{78} +520.000 q^{79} +81.0000 q^{81} -990.000i q^{82} +156.000i q^{83} +60.0000 q^{84} +276.000 q^{86} -234.000i q^{87} +504.000i q^{88} -1026.00 q^{89} +1480.00 q^{91} +120.000i q^{92} -600.000i q^{93} +72.0000 q^{94} +135.000 q^{96} +286.000i q^{97} +171.000i q^{98} +216.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 18 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 18 q^{6} - 18 q^{9} - 48 q^{11} - 120 q^{14} - 142 q^{16} + 248 q^{19} - 120 q^{21} - 126 q^{24} + 444 q^{26} + 156 q^{29} + 400 q^{31} - 324 q^{34} + 18 q^{36} + 444 q^{39} + 660 q^{41} + 48 q^{44} - 720 q^{46} - 114 q^{49} - 324 q^{51} + 162 q^{54} - 840 q^{56} - 48 q^{59} - 644 q^{61} - 866 q^{64} + 432 q^{66} - 720 q^{69} - 576 q^{71} + 420 q^{74} - 248 q^{76} + 1040 q^{79} + 162 q^{81} + 120 q^{84} + 552 q^{86} - 2052 q^{89} + 2960 q^{91} + 144 q^{94} + 270 q^{96} + 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.06066i −0.847791 0.530330i \(-0.822068\pi\)
0.847791 0.530330i \(-0.177932\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) −9.00000 −0.612372
\(7\) − 20.0000i − 1.07990i −0.841698 0.539949i \(-0.818443\pi\)
0.841698 0.539949i \(-0.181557\pi\)
\(8\) − 21.0000i − 0.928078i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 3.00000i 0.0721688i
\(13\) 74.0000i 1.57876i 0.613904 + 0.789381i \(0.289598\pi\)
−0.613904 + 0.789381i \(0.710402\pi\)
\(14\) −60.0000 −1.14541
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) − 54.0000i − 0.770407i −0.922832 0.385204i \(-0.874131\pi\)
0.922832 0.385204i \(-0.125869\pi\)
\(18\) 27.0000i 0.353553i
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) −60.0000 −0.623480
\(22\) 72.0000i 0.697748i
\(23\) − 120.000i − 1.08790i −0.839117 0.543951i \(-0.816928\pi\)
0.839117 0.543951i \(-0.183072\pi\)
\(24\) −63.0000 −0.535826
\(25\) 0 0
\(26\) 222.000 1.67453
\(27\) 27.0000i 0.192450i
\(28\) 20.0000i 0.134987i
\(29\) 78.0000 0.499456 0.249728 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) 0 0
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) 45.0000i 0.248592i
\(33\) 72.0000i 0.379806i
\(34\) −162.000 −0.817140
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) 70.0000i 0.311025i 0.987834 + 0.155513i \(0.0497029\pi\)
−0.987834 + 0.155513i \(0.950297\pi\)
\(38\) − 372.000i − 1.58806i
\(39\) 222.000 0.911499
\(40\) 0 0
\(41\) 330.000 1.25701 0.628504 0.777806i \(-0.283668\pi\)
0.628504 + 0.777806i \(0.283668\pi\)
\(42\) 180.000i 0.661300i
\(43\) 92.0000i 0.326276i 0.986603 + 0.163138i \(0.0521616\pi\)
−0.986603 + 0.163138i \(0.947838\pi\)
\(44\) 24.0000 0.0822304
\(45\) 0 0
\(46\) −360.000 −1.15389
\(47\) 24.0000i 0.0744843i 0.999306 + 0.0372421i \(0.0118573\pi\)
−0.999306 + 0.0372421i \(0.988143\pi\)
\(48\) 213.000i 0.640498i
\(49\) −57.0000 −0.166181
\(50\) 0 0
\(51\) −162.000 −0.444795
\(52\) − 74.0000i − 0.197345i
\(53\) 450.000i 1.16627i 0.812376 + 0.583134i \(0.198174\pi\)
−0.812376 + 0.583134i \(0.801826\pi\)
\(54\) 81.0000 0.204124
\(55\) 0 0
\(56\) −420.000 −1.00223
\(57\) − 372.000i − 0.864432i
\(58\) − 234.000i − 0.529754i
\(59\) −24.0000 −0.0529582 −0.0264791 0.999649i \(-0.508430\pi\)
−0.0264791 + 0.999649i \(0.508430\pi\)
\(60\) 0 0
\(61\) −322.000 −0.675867 −0.337933 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) − 600.000i − 1.22903i
\(63\) 180.000i 0.359966i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) 216.000 0.402845
\(67\) 196.000i 0.357391i 0.983904 + 0.178696i \(0.0571877\pi\)
−0.983904 + 0.178696i \(0.942812\pi\)
\(68\) 54.0000i 0.0963009i
\(69\) −360.000 −0.628100
\(70\) 0 0
\(71\) −288.000 −0.481399 −0.240699 0.970600i \(-0.577377\pi\)
−0.240699 + 0.970600i \(0.577377\pi\)
\(72\) 189.000i 0.309359i
\(73\) − 430.000i − 0.689420i −0.938709 0.344710i \(-0.887977\pi\)
0.938709 0.344710i \(-0.112023\pi\)
\(74\) 210.000 0.329892
\(75\) 0 0
\(76\) −124.000 −0.187155
\(77\) 480.000i 0.710404i
\(78\) − 666.000i − 0.966790i
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 990.000i − 1.33326i
\(83\) 156.000i 0.206304i 0.994666 + 0.103152i \(0.0328928\pi\)
−0.994666 + 0.103152i \(0.967107\pi\)
\(84\) 60.0000 0.0779350
\(85\) 0 0
\(86\) 276.000 0.346068
\(87\) − 234.000i − 0.288361i
\(88\) 504.000i 0.610529i
\(89\) −1026.00 −1.22198 −0.610988 0.791640i \(-0.709227\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 1480.00 1.70490
\(92\) 120.000i 0.135988i
\(93\) − 600.000i − 0.669001i
\(94\) 72.0000 0.0790025
\(95\) 0 0
\(96\) 135.000 0.143525
\(97\) 286.000i 0.299370i 0.988734 + 0.149685i \(0.0478260\pi\)
−0.988734 + 0.149685i \(0.952174\pi\)
\(98\) 171.000i 0.176261i
\(99\) 216.000 0.219281
\(100\) 0 0
\(101\) −1734.00 −1.70831 −0.854156 0.520017i \(-0.825925\pi\)
−0.854156 + 0.520017i \(0.825925\pi\)
\(102\) 486.000i 0.471776i
\(103\) 452.000i 0.432397i 0.976349 + 0.216198i \(0.0693658\pi\)
−0.976349 + 0.216198i \(0.930634\pi\)
\(104\) 1554.00 1.46521
\(105\) 0 0
\(106\) 1350.00 1.23702
\(107\) 1404.00i 1.26850i 0.773127 + 0.634251i \(0.218692\pi\)
−0.773127 + 0.634251i \(0.781308\pi\)
\(108\) − 27.0000i − 0.0240563i
\(109\) 1474.00 1.29526 0.647631 0.761954i \(-0.275760\pi\)
0.647631 + 0.761954i \(0.275760\pi\)
\(110\) 0 0
\(111\) 210.000 0.179570
\(112\) 1420.00i 1.19801i
\(113\) 1086.00i 0.904091i 0.891995 + 0.452046i \(0.149306\pi\)
−0.891995 + 0.452046i \(0.850694\pi\)
\(114\) −1116.00 −0.916868
\(115\) 0 0
\(116\) −78.0000 −0.0624321
\(117\) − 666.000i − 0.526254i
\(118\) 72.0000i 0.0561707i
\(119\) −1080.00 −0.831962
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 966.000i 0.716865i
\(123\) − 990.000i − 0.725734i
\(124\) −200.000 −0.144843
\(125\) 0 0
\(126\) 540.000 0.381802
\(127\) − 1244.00i − 0.869190i −0.900626 0.434595i \(-0.856891\pi\)
0.900626 0.434595i \(-0.143109\pi\)
\(128\) 1659.00i 1.14560i
\(129\) 276.000 0.188376
\(130\) 0 0
\(131\) 2328.00 1.55266 0.776329 0.630327i \(-0.217079\pi\)
0.776329 + 0.630327i \(0.217079\pi\)
\(132\) − 72.0000i − 0.0474757i
\(133\) − 2480.00i − 1.61687i
\(134\) 588.000 0.379071
\(135\) 0 0
\(136\) −1134.00 −0.714998
\(137\) − 2118.00i − 1.32082i −0.750903 0.660412i \(-0.770382\pi\)
0.750903 0.660412i \(-0.229618\pi\)
\(138\) 1080.00i 0.666201i
\(139\) −2324.00 −1.41812 −0.709062 0.705147i \(-0.750881\pi\)
−0.709062 + 0.705147i \(0.750881\pi\)
\(140\) 0 0
\(141\) 72.0000 0.0430035
\(142\) 864.000i 0.510600i
\(143\) − 1776.00i − 1.03858i
\(144\) 639.000 0.369792
\(145\) 0 0
\(146\) −1290.00 −0.731241
\(147\) 171.000i 0.0959445i
\(148\) − 70.0000i − 0.0388781i
\(149\) −258.000 −0.141854 −0.0709268 0.997482i \(-0.522596\pi\)
−0.0709268 + 0.997482i \(0.522596\pi\)
\(150\) 0 0
\(151\) −808.000 −0.435458 −0.217729 0.976009i \(-0.569865\pi\)
−0.217729 + 0.976009i \(0.569865\pi\)
\(152\) − 2604.00i − 1.38955i
\(153\) 486.000i 0.256802i
\(154\) 1440.00 0.753497
\(155\) 0 0
\(156\) −222.000 −0.113937
\(157\) − 2378.00i − 1.20882i −0.796673 0.604411i \(-0.793408\pi\)
0.796673 0.604411i \(-0.206592\pi\)
\(158\) − 1560.00i − 0.785487i
\(159\) 1350.00 0.673346
\(160\) 0 0
\(161\) −2400.00 −1.17482
\(162\) − 243.000i − 0.117851i
\(163\) − 52.0000i − 0.0249874i −0.999922 0.0124937i \(-0.996023\pi\)
0.999922 0.0124937i \(-0.00397698\pi\)
\(164\) −330.000 −0.157126
\(165\) 0 0
\(166\) 468.000 0.218818
\(167\) 3720.00i 1.72373i 0.507141 + 0.861863i \(0.330702\pi\)
−0.507141 + 0.861863i \(0.669298\pi\)
\(168\) 1260.00i 0.578638i
\(169\) −3279.00 −1.49249
\(170\) 0 0
\(171\) −1116.00 −0.499080
\(172\) − 92.0000i − 0.0407845i
\(173\) 426.000i 0.187215i 0.995609 + 0.0936075i \(0.0298399\pi\)
−0.995609 + 0.0936075i \(0.970160\pi\)
\(174\) −702.000 −0.305853
\(175\) 0 0
\(176\) 1704.00 0.729795
\(177\) 72.0000i 0.0305754i
\(178\) 3078.00i 1.29610i
\(179\) 1440.00 0.601289 0.300644 0.953736i \(-0.402798\pi\)
0.300644 + 0.953736i \(0.402798\pi\)
\(180\) 0 0
\(181\) −3130.00 −1.28537 −0.642683 0.766133i \(-0.722179\pi\)
−0.642683 + 0.766133i \(0.722179\pi\)
\(182\) − 4440.00i − 1.80832i
\(183\) 966.000i 0.390212i
\(184\) −2520.00 −1.00966
\(185\) 0 0
\(186\) −1800.00 −0.709583
\(187\) 1296.00i 0.506807i
\(188\) − 24.0000i − 0.00931053i
\(189\) 540.000 0.207827
\(190\) 0 0
\(191\) 3576.00 1.35471 0.677357 0.735655i \(-0.263125\pi\)
0.677357 + 0.735655i \(0.263125\pi\)
\(192\) 1299.00i 0.488267i
\(193\) 2666.00i 0.994315i 0.867660 + 0.497158i \(0.165623\pi\)
−0.867660 + 0.497158i \(0.834377\pi\)
\(194\) 858.000 0.317530
\(195\) 0 0
\(196\) 57.0000 0.0207726
\(197\) 2718.00i 0.982992i 0.870880 + 0.491496i \(0.163550\pi\)
−0.870880 + 0.491496i \(0.836450\pi\)
\(198\) − 648.000i − 0.232583i
\(199\) 3832.00 1.36504 0.682521 0.730866i \(-0.260884\pi\)
0.682521 + 0.730866i \(0.260884\pi\)
\(200\) 0 0
\(201\) 588.000 0.206340
\(202\) 5202.00i 1.81194i
\(203\) − 1560.00i − 0.539362i
\(204\) 162.000 0.0555994
\(205\) 0 0
\(206\) 1356.00 0.458626
\(207\) 1080.00i 0.362634i
\(208\) − 5254.00i − 1.75144i
\(209\) −2976.00 −0.984948
\(210\) 0 0
\(211\) 1100.00 0.358896 0.179448 0.983767i \(-0.442569\pi\)
0.179448 + 0.983767i \(0.442569\pi\)
\(212\) − 450.000i − 0.145784i
\(213\) 864.000i 0.277936i
\(214\) 4212.00 1.34545
\(215\) 0 0
\(216\) 567.000 0.178609
\(217\) − 4000.00i − 1.25133i
\(218\) − 4422.00i − 1.37383i
\(219\) −1290.00 −0.398037
\(220\) 0 0
\(221\) 3996.00 1.21629
\(222\) − 630.000i − 0.190463i
\(223\) 1964.00i 0.589772i 0.955532 + 0.294886i \(0.0952817\pi\)
−0.955532 + 0.294886i \(0.904718\pi\)
\(224\) 900.000 0.268454
\(225\) 0 0
\(226\) 3258.00 0.958933
\(227\) − 660.000i − 0.192977i −0.995334 0.0964884i \(-0.969239\pi\)
0.995334 0.0964884i \(-0.0307611\pi\)
\(228\) 372.000i 0.108054i
\(229\) 1906.00 0.550009 0.275004 0.961443i \(-0.411321\pi\)
0.275004 + 0.961443i \(0.411321\pi\)
\(230\) 0 0
\(231\) 1440.00 0.410152
\(232\) − 1638.00i − 0.463534i
\(233\) − 1458.00i − 0.409943i −0.978768 0.204972i \(-0.934290\pi\)
0.978768 0.204972i \(-0.0657102\pi\)
\(234\) −1998.00 −0.558177
\(235\) 0 0
\(236\) 24.0000 0.00661978
\(237\) − 1560.00i − 0.427565i
\(238\) 3240.00i 0.882429i
\(239\) −1176.00 −0.318281 −0.159140 0.987256i \(-0.550872\pi\)
−0.159140 + 0.987256i \(0.550872\pi\)
\(240\) 0 0
\(241\) 866.000 0.231469 0.115734 0.993280i \(-0.463078\pi\)
0.115734 + 0.993280i \(0.463078\pi\)
\(242\) 2265.00i 0.601652i
\(243\) − 243.000i − 0.0641500i
\(244\) 322.000 0.0844834
\(245\) 0 0
\(246\) −2970.00 −0.769757
\(247\) 9176.00i 2.36379i
\(248\) − 4200.00i − 1.07540i
\(249\) 468.000 0.119110
\(250\) 0 0
\(251\) 432.000 0.108636 0.0543179 0.998524i \(-0.482702\pi\)
0.0543179 + 0.998524i \(0.482702\pi\)
\(252\) − 180.000i − 0.0449958i
\(253\) 2880.00i 0.715668i
\(254\) −3732.00 −0.921915
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) − 2526.00i − 0.613103i −0.951854 0.306552i \(-0.900825\pi\)
0.951854 0.306552i \(-0.0991752\pi\)
\(258\) − 828.000i − 0.199802i
\(259\) 1400.00 0.335876
\(260\) 0 0
\(261\) −702.000 −0.166485
\(262\) − 6984.00i − 1.64684i
\(263\) 5448.00i 1.27733i 0.769484 + 0.638666i \(0.220513\pi\)
−0.769484 + 0.638666i \(0.779487\pi\)
\(264\) 1512.00 0.352489
\(265\) 0 0
\(266\) −7440.00 −1.71495
\(267\) 3078.00i 0.705508i
\(268\) − 196.000i − 0.0446739i
\(269\) 2574.00 0.583418 0.291709 0.956507i \(-0.405776\pi\)
0.291709 + 0.956507i \(0.405776\pi\)
\(270\) 0 0
\(271\) −3184.00 −0.713706 −0.356853 0.934161i \(-0.616150\pi\)
−0.356853 + 0.934161i \(0.616150\pi\)
\(272\) 3834.00i 0.854671i
\(273\) − 4440.00i − 0.984326i
\(274\) −6354.00 −1.40095
\(275\) 0 0
\(276\) 360.000 0.0785125
\(277\) − 3962.00i − 0.859399i −0.902972 0.429699i \(-0.858620\pi\)
0.902972 0.429699i \(-0.141380\pi\)
\(278\) 6972.00i 1.50415i
\(279\) −1800.00 −0.386248
\(280\) 0 0
\(281\) −8286.00 −1.75908 −0.879540 0.475825i \(-0.842149\pi\)
−0.879540 + 0.475825i \(0.842149\pi\)
\(282\) − 216.000i − 0.0456121i
\(283\) − 2716.00i − 0.570493i −0.958454 0.285246i \(-0.907925\pi\)
0.958454 0.285246i \(-0.0920754\pi\)
\(284\) 288.000 0.0601748
\(285\) 0 0
\(286\) −5328.00 −1.10158
\(287\) − 6600.00i − 1.35744i
\(288\) − 405.000i − 0.0828641i
\(289\) 1997.00 0.406473
\(290\) 0 0
\(291\) 858.000 0.172841
\(292\) 430.000i 0.0861776i
\(293\) 6018.00i 1.19992i 0.800032 + 0.599958i \(0.204816\pi\)
−0.800032 + 0.599958i \(0.795184\pi\)
\(294\) 513.000 0.101765
\(295\) 0 0
\(296\) 1470.00 0.288655
\(297\) − 648.000i − 0.126602i
\(298\) 774.000i 0.150458i
\(299\) 8880.00 1.71754
\(300\) 0 0
\(301\) 1840.00 0.352345
\(302\) 2424.00i 0.461873i
\(303\) 5202.00i 0.986294i
\(304\) −8804.00 −1.66100
\(305\) 0 0
\(306\) 1458.00 0.272380
\(307\) − 9236.00i − 1.71702i −0.512793 0.858512i \(-0.671389\pi\)
0.512793 0.858512i \(-0.328611\pi\)
\(308\) − 480.000i − 0.0888004i
\(309\) 1356.00 0.249644
\(310\) 0 0
\(311\) 1536.00 0.280060 0.140030 0.990147i \(-0.455280\pi\)
0.140030 + 0.990147i \(0.455280\pi\)
\(312\) − 4662.00i − 0.845942i
\(313\) − 7342.00i − 1.32586i −0.748681 0.662930i \(-0.769313\pi\)
0.748681 0.662930i \(-0.230687\pi\)
\(314\) −7134.00 −1.28215
\(315\) 0 0
\(316\) −520.000 −0.0925705
\(317\) 3894.00i 0.689933i 0.938615 + 0.344967i \(0.112110\pi\)
−0.938615 + 0.344967i \(0.887890\pi\)
\(318\) − 4050.00i − 0.714191i
\(319\) −1872.00 −0.328564
\(320\) 0 0
\(321\) 4212.00 0.732370
\(322\) 7200.00i 1.24609i
\(323\) − 6696.00i − 1.15348i
\(324\) −81.0000 −0.0138889
\(325\) 0 0
\(326\) −156.000 −0.0265032
\(327\) − 4422.00i − 0.747820i
\(328\) − 6930.00i − 1.16660i
\(329\) 480.000 0.0804354
\(330\) 0 0
\(331\) 3692.00 0.613084 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(332\) − 156.000i − 0.0257880i
\(333\) − 630.000i − 0.103675i
\(334\) 11160.0 1.82829
\(335\) 0 0
\(336\) 4260.00 0.691673
\(337\) 8998.00i 1.45446i 0.686395 + 0.727229i \(0.259192\pi\)
−0.686395 + 0.727229i \(0.740808\pi\)
\(338\) 9837.00i 1.58302i
\(339\) 3258.00 0.521977
\(340\) 0 0
\(341\) −4800.00 −0.762271
\(342\) 3348.00i 0.529354i
\(343\) − 5720.00i − 0.900440i
\(344\) 1932.00 0.302809
\(345\) 0 0
\(346\) 1278.00 0.198571
\(347\) − 5244.00i − 0.811276i −0.914034 0.405638i \(-0.867049\pi\)
0.914034 0.405638i \(-0.132951\pi\)
\(348\) 234.000i 0.0360452i
\(349\) −6302.00 −0.966585 −0.483293 0.875459i \(-0.660559\pi\)
−0.483293 + 0.875459i \(0.660559\pi\)
\(350\) 0 0
\(351\) −1998.00 −0.303833
\(352\) − 1080.00i − 0.163535i
\(353\) 3414.00i 0.514756i 0.966311 + 0.257378i \(0.0828586\pi\)
−0.966311 + 0.257378i \(0.917141\pi\)
\(354\) 216.000 0.0324301
\(355\) 0 0
\(356\) 1026.00 0.152747
\(357\) 3240.00i 0.480333i
\(358\) − 4320.00i − 0.637763i
\(359\) −4824.00 −0.709195 −0.354597 0.935019i \(-0.615382\pi\)
−0.354597 + 0.935019i \(0.615382\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 9390.00i 1.36334i
\(363\) 2265.00i 0.327498i
\(364\) −1480.00 −0.213113
\(365\) 0 0
\(366\) 2898.00 0.413882
\(367\) 3508.00i 0.498954i 0.968381 + 0.249477i \(0.0802587\pi\)
−0.968381 + 0.249477i \(0.919741\pi\)
\(368\) 8520.00i 1.20689i
\(369\) −2970.00 −0.419003
\(370\) 0 0
\(371\) 9000.00 1.25945
\(372\) 600.000i 0.0836251i
\(373\) 10802.0i 1.49948i 0.661732 + 0.749740i \(0.269822\pi\)
−0.661732 + 0.749740i \(0.730178\pi\)
\(374\) 3888.00 0.537550
\(375\) 0 0
\(376\) 504.000 0.0691272
\(377\) 5772.00i 0.788523i
\(378\) − 1620.00i − 0.220433i
\(379\) −1460.00 −0.197876 −0.0989382 0.995094i \(-0.531545\pi\)
−0.0989382 + 0.995094i \(0.531545\pi\)
\(380\) 0 0
\(381\) −3732.00 −0.501827
\(382\) − 10728.0i − 1.43689i
\(383\) − 4872.00i − 0.649994i −0.945715 0.324997i \(-0.894637\pi\)
0.945715 0.324997i \(-0.105363\pi\)
\(384\) 4977.00 0.661410
\(385\) 0 0
\(386\) 7998.00 1.05463
\(387\) − 828.000i − 0.108759i
\(388\) − 286.000i − 0.0374213i
\(389\) 14046.0 1.83075 0.915373 0.402606i \(-0.131896\pi\)
0.915373 + 0.402606i \(0.131896\pi\)
\(390\) 0 0
\(391\) −6480.00 −0.838127
\(392\) 1197.00i 0.154229i
\(393\) − 6984.00i − 0.896428i
\(394\) 8154.00 1.04262
\(395\) 0 0
\(396\) −216.000 −0.0274101
\(397\) 2734.00i 0.345631i 0.984954 + 0.172816i \(0.0552864\pi\)
−0.984954 + 0.172816i \(0.944714\pi\)
\(398\) − 11496.0i − 1.44785i
\(399\) −7440.00 −0.933498
\(400\) 0 0
\(401\) −15942.0 −1.98530 −0.992650 0.121019i \(-0.961384\pi\)
−0.992650 + 0.121019i \(0.961384\pi\)
\(402\) − 1764.00i − 0.218857i
\(403\) 14800.0i 1.82938i
\(404\) 1734.00 0.213539
\(405\) 0 0
\(406\) −4680.00 −0.572080
\(407\) − 1680.00i − 0.204606i
\(408\) 3402.00i 0.412804i
\(409\) −8714.00 −1.05350 −0.526748 0.850022i \(-0.676589\pi\)
−0.526748 + 0.850022i \(0.676589\pi\)
\(410\) 0 0
\(411\) −6354.00 −0.762578
\(412\) − 452.000i − 0.0540496i
\(413\) 480.000i 0.0571895i
\(414\) 3240.00 0.384631
\(415\) 0 0
\(416\) −3330.00 −0.392468
\(417\) 6972.00i 0.818754i
\(418\) 8928.00i 1.04470i
\(419\) −11976.0 −1.39634 −0.698169 0.715933i \(-0.746002\pi\)
−0.698169 + 0.715933i \(0.746002\pi\)
\(420\) 0 0
\(421\) 11054.0 1.27967 0.639833 0.768514i \(-0.279004\pi\)
0.639833 + 0.768514i \(0.279004\pi\)
\(422\) − 3300.00i − 0.380667i
\(423\) − 216.000i − 0.0248281i
\(424\) 9450.00 1.08239
\(425\) 0 0
\(426\) 2592.00 0.294795
\(427\) 6440.00i 0.729868i
\(428\) − 1404.00i − 0.158563i
\(429\) −5328.00 −0.599623
\(430\) 0 0
\(431\) 720.000 0.0804668 0.0402334 0.999190i \(-0.487190\pi\)
0.0402334 + 0.999190i \(0.487190\pi\)
\(432\) − 1917.00i − 0.213499i
\(433\) − 15622.0i − 1.73382i −0.498462 0.866912i \(-0.666102\pi\)
0.498462 0.866912i \(-0.333898\pi\)
\(434\) −12000.0 −1.32723
\(435\) 0 0
\(436\) −1474.00 −0.161908
\(437\) − 14880.0i − 1.62885i
\(438\) 3870.00i 0.422182i
\(439\) 9880.00 1.07414 0.537069 0.843538i \(-0.319531\pi\)
0.537069 + 0.843538i \(0.319531\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) − 11988.0i − 1.29007i
\(443\) − 16116.0i − 1.72843i −0.503123 0.864215i \(-0.667816\pi\)
0.503123 0.864215i \(-0.332184\pi\)
\(444\) −210.000 −0.0224463
\(445\) 0 0
\(446\) 5892.00 0.625548
\(447\) 774.000i 0.0818992i
\(448\) 8660.00i 0.913274i
\(449\) −9018.00 −0.947852 −0.473926 0.880565i \(-0.657164\pi\)
−0.473926 + 0.880565i \(0.657164\pi\)
\(450\) 0 0
\(451\) −7920.00 −0.826914
\(452\) − 1086.00i − 0.113011i
\(453\) 2424.00i 0.251412i
\(454\) −1980.00 −0.204683
\(455\) 0 0
\(456\) −7812.00 −0.802260
\(457\) 3670.00i 0.375657i 0.982202 + 0.187829i \(0.0601450\pi\)
−0.982202 + 0.187829i \(0.939855\pi\)
\(458\) − 5718.00i − 0.583372i
\(459\) 1458.00 0.148265
\(460\) 0 0
\(461\) 17562.0 1.77428 0.887141 0.461499i \(-0.152688\pi\)
0.887141 + 0.461499i \(0.152688\pi\)
\(462\) − 4320.00i − 0.435032i
\(463\) 1172.00i 0.117640i 0.998269 + 0.0588202i \(0.0187338\pi\)
−0.998269 + 0.0588202i \(0.981266\pi\)
\(464\) −5538.00 −0.554084
\(465\) 0 0
\(466\) −4374.00 −0.434810
\(467\) − 6876.00i − 0.681335i −0.940184 0.340667i \(-0.889347\pi\)
0.940184 0.340667i \(-0.110653\pi\)
\(468\) 666.000i 0.0657818i
\(469\) 3920.00 0.385946
\(470\) 0 0
\(471\) −7134.00 −0.697914
\(472\) 504.000i 0.0491493i
\(473\) − 2208.00i − 0.214638i
\(474\) −4680.00 −0.453501
\(475\) 0 0
\(476\) 1080.00 0.103995
\(477\) − 4050.00i − 0.388756i
\(478\) 3528.00i 0.337588i
\(479\) −2280.00 −0.217486 −0.108743 0.994070i \(-0.534683\pi\)
−0.108743 + 0.994070i \(0.534683\pi\)
\(480\) 0 0
\(481\) −5180.00 −0.491035
\(482\) − 2598.00i − 0.245510i
\(483\) 7200.00i 0.678284i
\(484\) 755.000 0.0709053
\(485\) 0 0
\(486\) −729.000 −0.0680414
\(487\) 3076.00i 0.286215i 0.989707 + 0.143108i \(0.0457095\pi\)
−0.989707 + 0.143108i \(0.954290\pi\)
\(488\) 6762.00i 0.627257i
\(489\) −156.000 −0.0144265
\(490\) 0 0
\(491\) −18912.0 −1.73826 −0.869131 0.494582i \(-0.835321\pi\)
−0.869131 + 0.494582i \(0.835321\pi\)
\(492\) 990.000i 0.0907168i
\(493\) − 4212.00i − 0.384785i
\(494\) 27528.0 2.50717
\(495\) 0 0
\(496\) −14200.0 −1.28548
\(497\) 5760.00i 0.519862i
\(498\) − 1404.00i − 0.126335i
\(499\) −9956.00 −0.893170 −0.446585 0.894741i \(-0.647360\pi\)
−0.446585 + 0.894741i \(0.647360\pi\)
\(500\) 0 0
\(501\) 11160.0 0.995194
\(502\) − 1296.00i − 0.115226i
\(503\) − 10656.0i − 0.944588i −0.881441 0.472294i \(-0.843426\pi\)
0.881441 0.472294i \(-0.156574\pi\)
\(504\) 3780.00 0.334077
\(505\) 0 0
\(506\) 8640.00 0.759081
\(507\) 9837.00i 0.861689i
\(508\) 1244.00i 0.108649i
\(509\) 2766.00 0.240866 0.120433 0.992721i \(-0.461572\pi\)
0.120433 + 0.992721i \(0.461572\pi\)
\(510\) 0 0
\(511\) −8600.00 −0.744504
\(512\) 8733.00i 0.753804i
\(513\) 3348.00i 0.288144i
\(514\) −7578.00 −0.650294
\(515\) 0 0
\(516\) −276.000 −0.0235469
\(517\) − 576.000i − 0.0489989i
\(518\) − 4200.00i − 0.356250i
\(519\) 1278.00 0.108089
\(520\) 0 0
\(521\) 10530.0 0.885466 0.442733 0.896654i \(-0.354009\pi\)
0.442733 + 0.896654i \(0.354009\pi\)
\(522\) 2106.00i 0.176585i
\(523\) 12692.0i 1.06115i 0.847637 + 0.530576i \(0.178024\pi\)
−0.847637 + 0.530576i \(0.821976\pi\)
\(524\) −2328.00 −0.194082
\(525\) 0 0
\(526\) 16344.0 1.35481
\(527\) − 10800.0i − 0.892705i
\(528\) − 5112.00i − 0.421347i
\(529\) −2233.00 −0.183529
\(530\) 0 0
\(531\) 216.000 0.0176527
\(532\) 2480.00i 0.202108i
\(533\) 24420.0i 1.98452i
\(534\) 9234.00 0.748304
\(535\) 0 0
\(536\) 4116.00 0.331687
\(537\) − 4320.00i − 0.347154i
\(538\) − 7722.00i − 0.618809i
\(539\) 1368.00 0.109321
\(540\) 0 0
\(541\) 18110.0 1.43920 0.719602 0.694386i \(-0.244324\pi\)
0.719602 + 0.694386i \(0.244324\pi\)
\(542\) 9552.00i 0.756999i
\(543\) 9390.00i 0.742106i
\(544\) 2430.00 0.191517
\(545\) 0 0
\(546\) −13320.0 −1.04404
\(547\) − 3620.00i − 0.282962i −0.989941 0.141481i \(-0.954814\pi\)
0.989941 0.141481i \(-0.0451864\pi\)
\(548\) 2118.00i 0.165103i
\(549\) 2898.00 0.225289
\(550\) 0 0
\(551\) 9672.00 0.747806
\(552\) 7560.00i 0.582926i
\(553\) − 10400.0i − 0.799734i
\(554\) −11886.0 −0.911530
\(555\) 0 0
\(556\) 2324.00 0.177265
\(557\) 14166.0i 1.07762i 0.842428 + 0.538809i \(0.181125\pi\)
−0.842428 + 0.538809i \(0.818875\pi\)
\(558\) 5400.00i 0.409678i
\(559\) −6808.00 −0.515112
\(560\) 0 0
\(561\) 3888.00 0.292605
\(562\) 24858.0i 1.86579i
\(563\) − 13404.0i − 1.00339i −0.865043 0.501697i \(-0.832709\pi\)
0.865043 0.501697i \(-0.167291\pi\)
\(564\) −72.0000 −0.00537544
\(565\) 0 0
\(566\) −8148.00 −0.605099
\(567\) − 1620.00i − 0.119989i
\(568\) 6048.00i 0.446775i
\(569\) 18654.0 1.37437 0.687185 0.726483i \(-0.258846\pi\)
0.687185 + 0.726483i \(0.258846\pi\)
\(570\) 0 0
\(571\) −7684.00 −0.563162 −0.281581 0.959537i \(-0.590859\pi\)
−0.281581 + 0.959537i \(0.590859\pi\)
\(572\) 1776.00i 0.129822i
\(573\) − 10728.0i − 0.782144i
\(574\) −19800.0 −1.43978
\(575\) 0 0
\(576\) 3897.00 0.281901
\(577\) 1726.00i 0.124531i 0.998060 + 0.0622654i \(0.0198325\pi\)
−0.998060 + 0.0622654i \(0.980167\pi\)
\(578\) − 5991.00i − 0.431129i
\(579\) 7998.00 0.574068
\(580\) 0 0
\(581\) 3120.00 0.222787
\(582\) − 2574.00i − 0.183326i
\(583\) − 10800.0i − 0.767222i
\(584\) −9030.00 −0.639836
\(585\) 0 0
\(586\) 18054.0 1.27270
\(587\) − 10596.0i − 0.745049i −0.928022 0.372524i \(-0.878492\pi\)
0.928022 0.372524i \(-0.121508\pi\)
\(588\) − 171.000i − 0.0119931i
\(589\) 24800.0 1.73492
\(590\) 0 0
\(591\) 8154.00 0.567531
\(592\) − 4970.00i − 0.345043i
\(593\) 2862.00i 0.198193i 0.995078 + 0.0990963i \(0.0315952\pi\)
−0.995078 + 0.0990963i \(0.968405\pi\)
\(594\) −1944.00 −0.134282
\(595\) 0 0
\(596\) 258.000 0.0177317
\(597\) − 11496.0i − 0.788107i
\(598\) − 26640.0i − 1.82172i
\(599\) 23592.0 1.60925 0.804627 0.593781i \(-0.202365\pi\)
0.804627 + 0.593781i \(0.202365\pi\)
\(600\) 0 0
\(601\) −9574.00 −0.649803 −0.324902 0.945748i \(-0.605331\pi\)
−0.324902 + 0.945748i \(0.605331\pi\)
\(602\) − 5520.00i − 0.373718i
\(603\) − 1764.00i − 0.119130i
\(604\) 808.000 0.0544322
\(605\) 0 0
\(606\) 15606.0 1.04612
\(607\) − 17444.0i − 1.16644i −0.812314 0.583221i \(-0.801792\pi\)
0.812314 0.583221i \(-0.198208\pi\)
\(608\) 5580.00i 0.372202i
\(609\) −4680.00 −0.311401
\(610\) 0 0
\(611\) −1776.00 −0.117593
\(612\) − 486.000i − 0.0321003i
\(613\) − 2374.00i − 0.156419i −0.996937 0.0782096i \(-0.975080\pi\)
0.996937 0.0782096i \(-0.0249203\pi\)
\(614\) −27708.0 −1.82118
\(615\) 0 0
\(616\) 10080.0 0.659310
\(617\) 12162.0i 0.793555i 0.917915 + 0.396778i \(0.129872\pi\)
−0.917915 + 0.396778i \(0.870128\pi\)
\(618\) − 4068.00i − 0.264788i
\(619\) −8804.00 −0.571668 −0.285834 0.958279i \(-0.592271\pi\)
−0.285834 + 0.958279i \(0.592271\pi\)
\(620\) 0 0
\(621\) 3240.00 0.209367
\(622\) − 4608.00i − 0.297048i
\(623\) 20520.0i 1.31961i
\(624\) −15762.0 −1.01119
\(625\) 0 0
\(626\) −22026.0 −1.40629
\(627\) 8928.00i 0.568660i
\(628\) 2378.00i 0.151103i
\(629\) 3780.00 0.239616
\(630\) 0 0
\(631\) −12688.0 −0.800478 −0.400239 0.916411i \(-0.631073\pi\)
−0.400239 + 0.916411i \(0.631073\pi\)
\(632\) − 10920.0i − 0.687301i
\(633\) − 3300.00i − 0.207209i
\(634\) 11682.0 0.731785
\(635\) 0 0
\(636\) −1350.00 −0.0841682
\(637\) − 4218.00i − 0.262360i
\(638\) 5616.00i 0.348495i
\(639\) 2592.00 0.160466
\(640\) 0 0
\(641\) −9150.00 −0.563812 −0.281906 0.959442i \(-0.590967\pi\)
−0.281906 + 0.959442i \(0.590967\pi\)
\(642\) − 12636.0i − 0.776796i
\(643\) 25292.0i 1.55120i 0.631227 + 0.775598i \(0.282552\pi\)
−0.631227 + 0.775598i \(0.717448\pi\)
\(644\) 2400.00 0.146853
\(645\) 0 0
\(646\) −20088.0 −1.22345
\(647\) 2736.00i 0.166249i 0.996539 + 0.0831246i \(0.0264900\pi\)
−0.996539 + 0.0831246i \(0.973510\pi\)
\(648\) − 1701.00i − 0.103120i
\(649\) 576.000 0.0348382
\(650\) 0 0
\(651\) −12000.0 −0.722453
\(652\) 52.0000i 0.00312343i
\(653\) 22218.0i 1.33148i 0.746183 + 0.665741i \(0.231884\pi\)
−0.746183 + 0.665741i \(0.768116\pi\)
\(654\) −13266.0 −0.793183
\(655\) 0 0
\(656\) −23430.0 −1.39449
\(657\) 3870.00i 0.229807i
\(658\) − 1440.00i − 0.0853147i
\(659\) −14520.0 −0.858299 −0.429149 0.903234i \(-0.641187\pi\)
−0.429149 + 0.903234i \(0.641187\pi\)
\(660\) 0 0
\(661\) −10618.0 −0.624799 −0.312400 0.949951i \(-0.601133\pi\)
−0.312400 + 0.949951i \(0.601133\pi\)
\(662\) − 11076.0i − 0.650273i
\(663\) − 11988.0i − 0.702225i
\(664\) 3276.00 0.191466
\(665\) 0 0
\(666\) −1890.00 −0.109964
\(667\) − 9360.00i − 0.543359i
\(668\) − 3720.00i − 0.215466i
\(669\) 5892.00 0.340505
\(670\) 0 0
\(671\) 7728.00 0.444614
\(672\) − 2700.00i − 0.154992i
\(673\) 1370.00i 0.0784690i 0.999230 + 0.0392345i \(0.0124919\pi\)
−0.999230 + 0.0392345i \(0.987508\pi\)
\(674\) 26994.0 1.54269
\(675\) 0 0
\(676\) 3279.00 0.186561
\(677\) 13758.0i 0.781038i 0.920595 + 0.390519i \(0.127704\pi\)
−0.920595 + 0.390519i \(0.872296\pi\)
\(678\) − 9774.00i − 0.553640i
\(679\) 5720.00 0.323289
\(680\) 0 0
\(681\) −1980.00 −0.111415
\(682\) 14400.0i 0.808511i
\(683\) 11988.0i 0.671608i 0.941932 + 0.335804i \(0.109008\pi\)
−0.941932 + 0.335804i \(0.890992\pi\)
\(684\) 1116.00 0.0623850
\(685\) 0 0
\(686\) −17160.0 −0.955061
\(687\) − 5718.00i − 0.317548i
\(688\) − 6532.00i − 0.361962i
\(689\) −33300.0 −1.84126
\(690\) 0 0
\(691\) 32996.0 1.81654 0.908268 0.418388i \(-0.137405\pi\)
0.908268 + 0.418388i \(0.137405\pi\)
\(692\) − 426.000i − 0.0234019i
\(693\) − 4320.00i − 0.236801i
\(694\) −15732.0 −0.860488
\(695\) 0 0
\(696\) −4914.00 −0.267622
\(697\) − 17820.0i − 0.968408i
\(698\) 18906.0i 1.02522i
\(699\) −4374.00 −0.236681
\(700\) 0 0
\(701\) −25902.0 −1.39558 −0.697792 0.716300i \(-0.745834\pi\)
−0.697792 + 0.716300i \(0.745834\pi\)
\(702\) 5994.00i 0.322263i
\(703\) 8680.00i 0.465679i
\(704\) 10392.0 0.556340
\(705\) 0 0
\(706\) 10242.0 0.545981
\(707\) 34680.0i 1.84480i
\(708\) − 72.0000i − 0.00382193i
\(709\) 27394.0 1.45106 0.725531 0.688189i \(-0.241594\pi\)
0.725531 + 0.688189i \(0.241594\pi\)
\(710\) 0 0
\(711\) −4680.00 −0.246855
\(712\) 21546.0i 1.13409i
\(713\) − 24000.0i − 1.26060i
\(714\) 9720.00 0.509470
\(715\) 0 0
\(716\) −1440.00 −0.0751611
\(717\) 3528.00i 0.183760i
\(718\) 14472.0i 0.752215i
\(719\) −34848.0 −1.80753 −0.903763 0.428033i \(-0.859207\pi\)
−0.903763 + 0.428033i \(0.859207\pi\)
\(720\) 0 0
\(721\) 9040.00 0.466945
\(722\) − 25551.0i − 1.31705i
\(723\) − 2598.00i − 0.133639i
\(724\) 3130.00 0.160671
\(725\) 0 0
\(726\) 6795.00 0.347364
\(727\) − 28028.0i − 1.42985i −0.699201 0.714925i \(-0.746461\pi\)
0.699201 0.714925i \(-0.253539\pi\)
\(728\) − 31080.0i − 1.58228i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 4968.00 0.251365
\(732\) − 966.000i − 0.0487765i
\(733\) 18002.0i 0.907120i 0.891226 + 0.453560i \(0.149846\pi\)
−0.891226 + 0.453560i \(0.850154\pi\)
\(734\) 10524.0 0.529221
\(735\) 0 0
\(736\) 5400.00 0.270444
\(737\) − 4704.00i − 0.235107i
\(738\) 8910.00i 0.444420i
\(739\) −15284.0 −0.760800 −0.380400 0.924822i \(-0.624214\pi\)
−0.380400 + 0.924822i \(0.624214\pi\)
\(740\) 0 0
\(741\) 27528.0 1.36473
\(742\) − 27000.0i − 1.33585i
\(743\) − 18768.0i − 0.926691i −0.886178 0.463345i \(-0.846649\pi\)
0.886178 0.463345i \(-0.153351\pi\)
\(744\) −12600.0 −0.620885
\(745\) 0 0
\(746\) 32406.0 1.59044
\(747\) − 1404.00i − 0.0687680i
\(748\) − 1296.00i − 0.0633509i
\(749\) 28080.0 1.36985
\(750\) 0 0
\(751\) 8696.00 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(752\) − 1704.00i − 0.0826310i
\(753\) − 1296.00i − 0.0627209i
\(754\) 17316.0 0.836355
\(755\) 0 0
\(756\) −540.000 −0.0259783
\(757\) 38662.0i 1.85627i 0.372247 + 0.928134i \(0.378587\pi\)
−0.372247 + 0.928134i \(0.621413\pi\)
\(758\) 4380.00i 0.209880i
\(759\) 8640.00 0.413191
\(760\) 0 0
\(761\) 23874.0 1.13723 0.568615 0.822604i \(-0.307479\pi\)
0.568615 + 0.822604i \(0.307479\pi\)
\(762\) 11196.0i 0.532268i
\(763\) − 29480.0i − 1.39875i
\(764\) −3576.00 −0.169339
\(765\) 0 0
\(766\) −14616.0 −0.689422
\(767\) − 1776.00i − 0.0836084i
\(768\) − 4539.00i − 0.213264i
\(769\) −23618.0 −1.10753 −0.553763 0.832675i \(-0.686808\pi\)
−0.553763 + 0.832675i \(0.686808\pi\)
\(770\) 0 0
\(771\) −7578.00 −0.353975
\(772\) − 2666.00i − 0.124289i
\(773\) 11538.0i 0.536860i 0.963299 + 0.268430i \(0.0865049\pi\)
−0.963299 + 0.268430i \(0.913495\pi\)
\(774\) −2484.00 −0.115356
\(775\) 0 0
\(776\) 6006.00 0.277839
\(777\) − 4200.00i − 0.193918i
\(778\) − 42138.0i − 1.94180i
\(779\) 40920.0 1.88204
\(780\) 0 0
\(781\) 6912.00 0.316685
\(782\) 19440.0i 0.888968i
\(783\) 2106.00i 0.0961204i
\(784\) 4047.00 0.184357
\(785\) 0 0
\(786\) −20952.0 −0.950805
\(787\) 14884.0i 0.674152i 0.941478 + 0.337076i \(0.109438\pi\)
−0.941478 + 0.337076i \(0.890562\pi\)
\(788\) − 2718.00i − 0.122874i
\(789\) 16344.0 0.737467
\(790\) 0 0
\(791\) 21720.0 0.976327
\(792\) − 4536.00i − 0.203510i
\(793\) − 23828.0i − 1.06703i
\(794\) 8202.00 0.366597
\(795\) 0 0
\(796\) −3832.00 −0.170630
\(797\) 11334.0i 0.503728i 0.967763 + 0.251864i \(0.0810435\pi\)
−0.967763 + 0.251864i \(0.918957\pi\)
\(798\) 22320.0i 0.990125i
\(799\) 1296.00 0.0573832
\(800\) 0 0
\(801\) 9234.00 0.407325
\(802\) 47826.0i 2.10573i
\(803\) 10320.0i 0.453530i
\(804\) −588.000 −0.0257925
\(805\) 0 0
\(806\) 44400.0 1.94035
\(807\) − 7722.00i − 0.336837i
\(808\) 36414.0i 1.58545i
\(809\) −44730.0 −1.94391 −0.971955 0.235167i \(-0.924436\pi\)
−0.971955 + 0.235167i \(0.924436\pi\)
\(810\) 0 0
\(811\) −42748.0 −1.85091 −0.925453 0.378862i \(-0.876316\pi\)
−0.925453 + 0.378862i \(0.876316\pi\)
\(812\) 1560.00i 0.0674203i
\(813\) 9552.00i 0.412058i
\(814\) −5040.00 −0.217017
\(815\) 0 0
\(816\) 11502.0 0.493444
\(817\) 11408.0i 0.488513i
\(818\) 26142.0i 1.11740i
\(819\) −13320.0 −0.568301
\(820\) 0 0
\(821\) −31686.0 −1.34695 −0.673477 0.739208i \(-0.735200\pi\)
−0.673477 + 0.739208i \(0.735200\pi\)
\(822\) 19062.0i 0.808836i
\(823\) 11036.0i 0.467425i 0.972306 + 0.233713i \(0.0750875\pi\)
−0.972306 + 0.233713i \(0.924913\pi\)
\(824\) 9492.00 0.401298
\(825\) 0 0
\(826\) 1440.00 0.0606586
\(827\) − 25884.0i − 1.08836i −0.838968 0.544181i \(-0.816841\pi\)
0.838968 0.544181i \(-0.183159\pi\)
\(828\) − 1080.00i − 0.0453292i
\(829\) −15950.0 −0.668234 −0.334117 0.942532i \(-0.608438\pi\)
−0.334117 + 0.942532i \(0.608438\pi\)
\(830\) 0 0
\(831\) −11886.0 −0.496174
\(832\) − 32042.0i − 1.33516i
\(833\) 3078.00i 0.128027i
\(834\) 20916.0 0.868419
\(835\) 0 0
\(836\) 2976.00 0.123119
\(837\) 5400.00i 0.223000i
\(838\) 35928.0i 1.48104i
\(839\) −13800.0 −0.567853 −0.283927 0.958846i \(-0.591637\pi\)
−0.283927 + 0.958846i \(0.591637\pi\)
\(840\) 0 0
\(841\) −18305.0 −0.750543
\(842\) − 33162.0i − 1.35729i
\(843\) 24858.0i 1.01560i
\(844\) −1100.00 −0.0448620
\(845\) 0 0
\(846\) −648.000 −0.0263342
\(847\) 15100.0i 0.612565i
\(848\) − 31950.0i − 1.29383i
\(849\) −8148.00 −0.329374
\(850\) 0 0
\(851\) 8400.00 0.338365
\(852\) − 864.000i − 0.0347420i
\(853\) − 27862.0i − 1.11838i −0.829040 0.559189i \(-0.811113\pi\)
0.829040 0.559189i \(-0.188887\pi\)
\(854\) 19320.0 0.774141
\(855\) 0 0
\(856\) 29484.0 1.17727
\(857\) 7314.00i 0.291530i 0.989319 + 0.145765i \(0.0465644\pi\)
−0.989319 + 0.145765i \(0.953436\pi\)
\(858\) 15984.0i 0.635996i
\(859\) 28780.0 1.14314 0.571572 0.820552i \(-0.306334\pi\)
0.571572 + 0.820552i \(0.306334\pi\)
\(860\) 0 0
\(861\) −19800.0 −0.783719
\(862\) − 2160.00i − 0.0853479i
\(863\) − 32688.0i − 1.28935i −0.764455 0.644677i \(-0.776992\pi\)
0.764455 0.644677i \(-0.223008\pi\)
\(864\) −1215.00 −0.0478416
\(865\) 0 0
\(866\) −46866.0 −1.83900
\(867\) − 5991.00i − 0.234677i
\(868\) 4000.00i 0.156416i
\(869\) −12480.0 −0.487175
\(870\) 0 0
\(871\) −14504.0 −0.564236
\(872\) − 30954.0i − 1.20210i
\(873\) − 2574.00i − 0.0997900i
\(874\) −44640.0 −1.72766
\(875\) 0 0
\(876\) 1290.00 0.0497546
\(877\) − 36650.0i − 1.41115i −0.708633 0.705577i \(-0.750688\pi\)
0.708633 0.705577i \(-0.249312\pi\)
\(878\) − 29640.0i − 1.13930i
\(879\) 18054.0 0.692772
\(880\) 0 0
\(881\) −2646.00 −0.101187 −0.0505936 0.998719i \(-0.516111\pi\)
−0.0505936 + 0.998719i \(0.516111\pi\)
\(882\) − 1539.00i − 0.0587538i
\(883\) 10892.0i 0.415113i 0.978223 + 0.207557i \(0.0665511\pi\)
−0.978223 + 0.207557i \(0.933449\pi\)
\(884\) −3996.00 −0.152036
\(885\) 0 0
\(886\) −48348.0 −1.83328
\(887\) 43464.0i 1.64530i 0.568550 + 0.822648i \(0.307504\pi\)
−0.568550 + 0.822648i \(0.692496\pi\)
\(888\) − 4410.00i − 0.166655i
\(889\) −24880.0 −0.938637
\(890\) 0 0
\(891\) −1944.00 −0.0730937
\(892\) − 1964.00i − 0.0737215i
\(893\) 2976.00i 0.111521i
\(894\) 2322.00 0.0868672
\(895\) 0 0
\(896\) 33180.0 1.23713
\(897\) − 26640.0i − 0.991621i
\(898\) 27054.0i 1.00535i
\(899\) 15600.0 0.578742
\(900\) 0 0
\(901\) 24300.0 0.898502
\(902\) 23760.0i 0.877075i
\(903\) − 5520.00i − 0.203426i
\(904\) 22806.0 0.839067
\(905\) 0 0
\(906\) 7272.00 0.266662
\(907\) 14884.0i 0.544890i 0.962171 + 0.272445i \(0.0878323\pi\)
−0.962171 + 0.272445i \(0.912168\pi\)
\(908\) 660.000i 0.0241221i
\(909\) 15606.0 0.569437
\(910\) 0 0
\(911\) −1248.00 −0.0453876 −0.0226938 0.999742i \(-0.507224\pi\)
−0.0226938 + 0.999742i \(0.507224\pi\)
\(912\) 26412.0i 0.958979i
\(913\) − 3744.00i − 0.135716i
\(914\) 11010.0 0.398445
\(915\) 0 0
\(916\) −1906.00 −0.0687511
\(917\) − 46560.0i − 1.67671i
\(918\) − 4374.00i − 0.157259i
\(919\) 6640.00 0.238339 0.119169 0.992874i \(-0.461977\pi\)
0.119169 + 0.992874i \(0.461977\pi\)
\(920\) 0 0
\(921\) −27708.0 −0.991324
\(922\) − 52686.0i − 1.88191i
\(923\) − 21312.0i − 0.760014i
\(924\) −1440.00 −0.0512690
\(925\) 0 0
\(926\) 3516.00 0.124776
\(927\) − 4068.00i − 0.144132i
\(928\) 3510.00i 0.124161i
\(929\) −29946.0 −1.05758 −0.528792 0.848751i \(-0.677355\pi\)
−0.528792 + 0.848751i \(0.677355\pi\)
\(930\) 0 0
\(931\) −7068.00 −0.248812
\(932\) 1458.00i 0.0512429i
\(933\) − 4608.00i − 0.161693i
\(934\) −20628.0 −0.722665
\(935\) 0 0
\(936\) −13986.0 −0.488405
\(937\) − 45002.0i − 1.56900i −0.620130 0.784499i \(-0.712920\pi\)
0.620130 0.784499i \(-0.287080\pi\)
\(938\) − 11760.0i − 0.409358i
\(939\) −22026.0 −0.765486
\(940\) 0 0
\(941\) 6090.00 0.210976 0.105488 0.994421i \(-0.466360\pi\)
0.105488 + 0.994421i \(0.466360\pi\)
\(942\) 21402.0i 0.740249i
\(943\) − 39600.0i − 1.36750i
\(944\) 1704.00 0.0587505
\(945\) 0 0
\(946\) −6624.00 −0.227658
\(947\) − 56388.0i − 1.93491i −0.253035 0.967457i \(-0.581429\pi\)
0.253035 0.967457i \(-0.418571\pi\)
\(948\) 1560.00i 0.0534456i
\(949\) 31820.0 1.08843
\(950\) 0 0
\(951\) 11682.0 0.398333
\(952\) 22680.0i 0.772125i
\(953\) 10854.0i 0.368936i 0.982839 + 0.184468i \(0.0590561\pi\)
−0.982839 + 0.184468i \(0.940944\pi\)
\(954\) −12150.0 −0.412338
\(955\) 0 0
\(956\) 1176.00 0.0397851
\(957\) 5616.00i 0.189696i
\(958\) 6840.00i 0.230679i
\(959\) −42360.0 −1.42636
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 15540.0i 0.520821i
\(963\) − 12636.0i − 0.422834i
\(964\) −866.000 −0.0289336
\(965\) 0 0
\(966\) 21600.0 0.719429
\(967\) 42316.0i 1.40723i 0.710582 + 0.703615i \(0.248432\pi\)
−0.710582 + 0.703615i \(0.751568\pi\)
\(968\) 15855.0i 0.526445i
\(969\) −20088.0 −0.665964
\(970\) 0 0
\(971\) 24480.0 0.809063 0.404532 0.914524i \(-0.367435\pi\)
0.404532 + 0.914524i \(0.367435\pi\)
\(972\) 243.000i 0.00801875i
\(973\) 46480.0i 1.53143i
\(974\) 9228.00 0.303577
\(975\) 0 0
\(976\) 22862.0 0.749790
\(977\) 6906.00i 0.226144i 0.993587 + 0.113072i \(0.0360690\pi\)
−0.993587 + 0.113072i \(0.963931\pi\)
\(978\) 468.000i 0.0153016i
\(979\) 24624.0 0.803868
\(980\) 0 0
\(981\) −13266.0 −0.431754
\(982\) 56736.0i 1.84371i
\(983\) 6960.00i 0.225829i 0.993605 + 0.112914i \(0.0360186\pi\)
−0.993605 + 0.112914i \(0.963981\pi\)
\(984\) −20790.0 −0.673538
\(985\) 0 0
\(986\) −12636.0 −0.408126
\(987\) − 1440.00i − 0.0464394i
\(988\) − 9176.00i − 0.295473i
\(989\) 11040.0 0.354956
\(990\) 0 0
\(991\) 47792.0 1.53195 0.765975 0.642870i \(-0.222256\pi\)
0.765975 + 0.642870i \(0.222256\pi\)
\(992\) 9000.00i 0.288055i
\(993\) − 11076.0i − 0.353964i
\(994\) 17280.0 0.551397
\(995\) 0 0
\(996\) −468.000 −0.0148887
\(997\) − 9938.00i − 0.315687i −0.987464 0.157843i \(-0.949546\pi\)
0.987464 0.157843i \(-0.0504541\pi\)
\(998\) 29868.0i 0.947350i
\(999\) −1890.00 −0.0598568
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 75.4.b.a.49.1 2
3.2 odd 2 225.4.b.d.199.2 2
4.3 odd 2 1200.4.f.m.49.2 2
5.2 odd 4 15.4.a.b.1.1 1
5.3 odd 4 75.4.a.a.1.1 1
5.4 even 2 inner 75.4.b.a.49.2 2
15.2 even 4 45.4.a.b.1.1 1
15.8 even 4 225.4.a.g.1.1 1
15.14 odd 2 225.4.b.d.199.1 2
20.3 even 4 1200.4.a.o.1.1 1
20.7 even 4 240.4.a.f.1.1 1
20.19 odd 2 1200.4.f.m.49.1 2
35.27 even 4 735.4.a.i.1.1 1
40.27 even 4 960.4.a.l.1.1 1
40.37 odd 4 960.4.a.bi.1.1 1
45.2 even 12 405.4.e.k.271.1 2
45.7 odd 12 405.4.e.d.271.1 2
45.22 odd 12 405.4.e.d.136.1 2
45.32 even 12 405.4.e.k.136.1 2
55.32 even 4 1815.4.a.a.1.1 1
60.47 odd 4 720.4.a.r.1.1 1
105.62 odd 4 2205.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 5.2 odd 4
45.4.a.b.1.1 1 15.2 even 4
75.4.a.a.1.1 1 5.3 odd 4
75.4.b.a.49.1 2 1.1 even 1 trivial
75.4.b.a.49.2 2 5.4 even 2 inner
225.4.a.g.1.1 1 15.8 even 4
225.4.b.d.199.1 2 15.14 odd 2
225.4.b.d.199.2 2 3.2 odd 2
240.4.a.f.1.1 1 20.7 even 4
405.4.e.d.136.1 2 45.22 odd 12
405.4.e.d.271.1 2 45.7 odd 12
405.4.e.k.136.1 2 45.32 even 12
405.4.e.k.271.1 2 45.2 even 12
720.4.a.r.1.1 1 60.47 odd 4
735.4.a.i.1.1 1 35.27 even 4
960.4.a.l.1.1 1 40.27 even 4
960.4.a.bi.1.1 1 40.37 odd 4
1200.4.a.o.1.1 1 20.3 even 4
1200.4.f.m.49.1 2 20.19 odd 2
1200.4.f.m.49.2 2 4.3 odd 2
1815.4.a.a.1.1 1 55.32 even 4
2205.4.a.c.1.1 1 105.62 odd 4