Properties

Label 2550.2.a.bo.1.2
Level $2550$
Weight $2$
Character 2550.1
Self dual yes
Analytic conductor $20.362$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2550,2,Mod(1,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.a (trivial)

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: yes
Analytic conductor: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 2550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.86464 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.86464 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.52311 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.86464 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +5.52311 q^{19} +2.86464 q^{21} -3.52311 q^{22} -8.11704 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +2.86464 q^{28} +7.91087 q^{29} +10.3878 q^{31} +1.00000 q^{32} -3.52311 q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.65847 q^{37} +5.52311 q^{38} +2.00000 q^{39} -10.7755 q^{41} +2.86464 q^{42} +1.52311 q^{43} -3.52311 q^{44} -8.11704 q^{46} +12.7755 q^{47} +1.00000 q^{48} +1.20617 q^{49} -1.00000 q^{51} +2.00000 q^{52} +0.270718 q^{53} +1.00000 q^{54} +2.86464 q^{56} +5.52311 q^{57} +7.91087 q^{58} -13.2524 q^{59} +0.929192 q^{61} +10.3878 q^{62} +2.86464 q^{63} +1.00000 q^{64} -3.52311 q^{66} -14.5048 q^{67} -1.00000 q^{68} -8.11704 q^{69} +11.1633 q^{71} +1.00000 q^{72} +4.98168 q^{73} +6.65847 q^{74} +5.52311 q^{76} -10.0925 q^{77} +2.00000 q^{78} -2.38776 q^{79} +1.00000 q^{81} -10.7755 q^{82} -1.52311 q^{83} +2.86464 q^{84} +1.52311 q^{86} +7.91087 q^{87} -3.52311 q^{88} -9.25240 q^{89} +5.72928 q^{91} -8.11704 q^{92} +10.3878 q^{93} +12.7755 q^{94} +1.00000 q^{96} +0.747604 q^{97} +1.20617 q^{98} -3.52311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{11} + 3 q^{12} + 6 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 4 q^{19} + 6 q^{21} + 2 q^{22} - 4 q^{23} + 3 q^{24} + 6 q^{26} + 3 q^{27} + 6 q^{28} - 4 q^{29} + 16 q^{31} + 3 q^{32} + 2 q^{33} - 3 q^{34} + 3 q^{36} + 10 q^{37} + 4 q^{38} + 6 q^{39} - 2 q^{41} + 6 q^{42} - 8 q^{43} + 2 q^{44} - 4 q^{46} + 8 q^{47} + 3 q^{48} + 11 q^{49} - 3 q^{51} + 6 q^{52} + 6 q^{53} + 3 q^{54} + 6 q^{56} + 4 q^{57} - 4 q^{58} - 22 q^{59} - 2 q^{61} + 16 q^{62} + 6 q^{63} + 3 q^{64} + 2 q^{66} - 8 q^{67} - 3 q^{68} - 4 q^{69} - 12 q^{71} + 3 q^{72} - 8 q^{73} + 10 q^{74} + 4 q^{76} + 20 q^{77} + 6 q^{78} + 8 q^{79} + 3 q^{81} - 2 q^{82} + 8 q^{83} + 6 q^{84} - 8 q^{86} - 4 q^{87} + 2 q^{88} - 10 q^{89} + 12 q^{91} - 4 q^{92} + 16 q^{93} + 8 q^{94} + 3 q^{96} + 20 q^{97} + 11 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.86464 1.08273 0.541366 0.840787i \(-0.317907\pi\)
0.541366 + 0.840787i \(0.317907\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.52311 −1.06226 −0.531129 0.847291i \(-0.678232\pi\)
−0.531129 + 0.847291i \(0.678232\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.86464 0.765607
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 5.52311 1.26709 0.633545 0.773706i \(-0.281599\pi\)
0.633545 + 0.773706i \(0.281599\pi\)
\(20\) 0 0
\(21\) 2.86464 0.625116
\(22\) −3.52311 −0.751131
\(23\) −8.11704 −1.69252 −0.846260 0.532771i \(-0.821151\pi\)
−0.846260 + 0.532771i \(0.821151\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 2.86464 0.541366
\(29\) 7.91087 1.46901 0.734506 0.678602i \(-0.237414\pi\)
0.734506 + 0.678602i \(0.237414\pi\)
\(30\) 0 0
\(31\) 10.3878 1.86570 0.932848 0.360270i \(-0.117316\pi\)
0.932848 + 0.360270i \(0.117316\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.52311 −0.613295
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.65847 1.09465 0.547323 0.836921i \(-0.315647\pi\)
0.547323 + 0.836921i \(0.315647\pi\)
\(38\) 5.52311 0.895967
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −10.7755 −1.68285 −0.841426 0.540372i \(-0.818283\pi\)
−0.841426 + 0.540372i \(0.818283\pi\)
\(42\) 2.86464 0.442024
\(43\) 1.52311 0.232273 0.116136 0.993233i \(-0.462949\pi\)
0.116136 + 0.993233i \(0.462949\pi\)
\(44\) −3.52311 −0.531129
\(45\) 0 0
\(46\) −8.11704 −1.19679
\(47\) 12.7755 1.86350 0.931750 0.363101i \(-0.118282\pi\)
0.931750 + 0.363101i \(0.118282\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.20617 0.172310
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 2.00000 0.277350
\(53\) 0.270718 0.0371860 0.0185930 0.999827i \(-0.494081\pi\)
0.0185930 + 0.999827i \(0.494081\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.86464 0.382804
\(57\) 5.52311 0.731554
\(58\) 7.91087 1.03875
\(59\) −13.2524 −1.72532 −0.862658 0.505789i \(-0.831202\pi\)
−0.862658 + 0.505789i \(0.831202\pi\)
\(60\) 0 0
\(61\) 0.929192 0.118971 0.0594854 0.998229i \(-0.481054\pi\)
0.0594854 + 0.998229i \(0.481054\pi\)
\(62\) 10.3878 1.31925
\(63\) 2.86464 0.360911
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.52311 −0.433665
\(67\) −14.5048 −1.77204 −0.886021 0.463645i \(-0.846541\pi\)
−0.886021 + 0.463645i \(0.846541\pi\)
\(68\) −1.00000 −0.121268
\(69\) −8.11704 −0.977176
\(70\) 0 0
\(71\) 11.1633 1.32484 0.662418 0.749134i \(-0.269530\pi\)
0.662418 + 0.749134i \(0.269530\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.98168 0.583061 0.291531 0.956561i \(-0.405835\pi\)
0.291531 + 0.956561i \(0.405835\pi\)
\(74\) 6.65847 0.774032
\(75\) 0 0
\(76\) 5.52311 0.633545
\(77\) −10.0925 −1.15014
\(78\) 2.00000 0.226455
\(79\) −2.38776 −0.268643 −0.134322 0.990938i \(-0.542886\pi\)
−0.134322 + 0.990938i \(0.542886\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.7755 −1.18996
\(83\) −1.52311 −0.167184 −0.0835918 0.996500i \(-0.526639\pi\)
−0.0835918 + 0.996500i \(0.526639\pi\)
\(84\) 2.86464 0.312558
\(85\) 0 0
\(86\) 1.52311 0.164242
\(87\) 7.91087 0.848134
\(88\) −3.52311 −0.375565
\(89\) −9.25240 −0.980752 −0.490376 0.871511i \(-0.663141\pi\)
−0.490376 + 0.871511i \(0.663141\pi\)
\(90\) 0 0
\(91\) 5.72928 0.600592
\(92\) −8.11704 −0.846260
\(93\) 10.3878 1.07716
\(94\) 12.7755 1.31769
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0.747604 0.0759077 0.0379538 0.999279i \(-0.487916\pi\)
0.0379538 + 0.999279i \(0.487916\pi\)
\(98\) 1.20617 0.121841
\(99\) −3.52311 −0.354086
\(100\) 0 0
\(101\) −15.2524 −1.51767 −0.758835 0.651283i \(-0.774231\pi\)
−0.758835 + 0.651283i \(0.774231\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −7.52311 −0.741274 −0.370637 0.928778i \(-0.620861\pi\)
−0.370637 + 0.928778i \(0.620861\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0.270718 0.0262945
\(107\) 19.8217 1.91624 0.958120 0.286367i \(-0.0924477\pi\)
0.958120 + 0.286367i \(0.0924477\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.92919 −0.472131 −0.236065 0.971737i \(-0.575858\pi\)
−0.236065 + 0.971737i \(0.575858\pi\)
\(110\) 0 0
\(111\) 6.65847 0.631994
\(112\) 2.86464 0.270683
\(113\) 3.72928 0.350821 0.175411 0.984495i \(-0.443875\pi\)
0.175411 + 0.984495i \(0.443875\pi\)
\(114\) 5.52311 0.517287
\(115\) 0 0
\(116\) 7.91087 0.734506
\(117\) 2.00000 0.184900
\(118\) −13.2524 −1.21998
\(119\) −2.86464 −0.262601
\(120\) 0 0
\(121\) 1.41233 0.128394
\(122\) 0.929192 0.0841251
\(123\) −10.7755 −0.971595
\(124\) 10.3878 0.932848
\(125\) 0 0
\(126\) 2.86464 0.255202
\(127\) 2.20617 0.195766 0.0978829 0.995198i \(-0.468793\pi\)
0.0978829 + 0.995198i \(0.468793\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.52311 0.134103
\(130\) 0 0
\(131\) 3.52311 0.307816 0.153908 0.988085i \(-0.450814\pi\)
0.153908 + 0.988085i \(0.450814\pi\)
\(132\) −3.52311 −0.306648
\(133\) 15.8217 1.37192
\(134\) −14.5048 −1.25302
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 6.20617 0.530229 0.265114 0.964217i \(-0.414590\pi\)
0.265114 + 0.964217i \(0.414590\pi\)
\(138\) −8.11704 −0.690968
\(139\) −12.2341 −1.03768 −0.518840 0.854871i \(-0.673636\pi\)
−0.518840 + 0.854871i \(0.673636\pi\)
\(140\) 0 0
\(141\) 12.7755 1.07589
\(142\) 11.1633 0.936800
\(143\) −7.04623 −0.589235
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.98168 0.412287
\(147\) 1.20617 0.0994830
\(148\) 6.65847 0.547323
\(149\) −3.01832 −0.247271 −0.123635 0.992328i \(-0.539455\pi\)
−0.123635 + 0.992328i \(0.539455\pi\)
\(150\) 0 0
\(151\) −3.04623 −0.247899 −0.123949 0.992289i \(-0.539556\pi\)
−0.123949 + 0.992289i \(0.539556\pi\)
\(152\) 5.52311 0.447984
\(153\) −1.00000 −0.0808452
\(154\) −10.0925 −0.813273
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 11.7293 0.936099 0.468049 0.883702i \(-0.344957\pi\)
0.468049 + 0.883702i \(0.344957\pi\)
\(158\) −2.38776 −0.189960
\(159\) 0.270718 0.0214694
\(160\) 0 0
\(161\) −23.2524 −1.83255
\(162\) 1.00000 0.0785674
\(163\) −11.0462 −0.865207 −0.432604 0.901584i \(-0.642405\pi\)
−0.432604 + 0.901584i \(0.642405\pi\)
\(164\) −10.7755 −0.841426
\(165\) 0 0
\(166\) −1.52311 −0.118217
\(167\) −11.5756 −0.895747 −0.447873 0.894097i \(-0.647818\pi\)
−0.447873 + 0.894097i \(0.647818\pi\)
\(168\) 2.86464 0.221012
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 5.52311 0.422363
\(172\) 1.52311 0.116136
\(173\) 5.34153 0.406109 0.203054 0.979167i \(-0.434913\pi\)
0.203054 + 0.979167i \(0.434913\pi\)
\(174\) 7.91087 0.599721
\(175\) 0 0
\(176\) −3.52311 −0.265565
\(177\) −13.2524 −0.996111
\(178\) −9.25240 −0.693496
\(179\) 10.2341 0.764931 0.382465 0.923970i \(-0.375075\pi\)
0.382465 + 0.923970i \(0.375075\pi\)
\(180\) 0 0
\(181\) −1.88296 −0.139960 −0.0699798 0.997548i \(-0.522293\pi\)
−0.0699798 + 0.997548i \(0.522293\pi\)
\(182\) 5.72928 0.424683
\(183\) 0.929192 0.0686878
\(184\) −8.11704 −0.598396
\(185\) 0 0
\(186\) 10.3878 0.761667
\(187\) 3.52311 0.257636
\(188\) 12.7755 0.931750
\(189\) 2.86464 0.208372
\(190\) 0 0
\(191\) 0.775511 0.0561140 0.0280570 0.999606i \(-0.491068\pi\)
0.0280570 + 0.999606i \(0.491068\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.2986 −1.02924 −0.514619 0.857419i \(-0.672067\pi\)
−0.514619 + 0.857419i \(0.672067\pi\)
\(194\) 0.747604 0.0536748
\(195\) 0 0
\(196\) 1.20617 0.0861548
\(197\) 7.61224 0.542350 0.271175 0.962530i \(-0.412588\pi\)
0.271175 + 0.962530i \(0.412588\pi\)
\(198\) −3.52311 −0.250377
\(199\) 2.38776 0.169263 0.0846317 0.996412i \(-0.473029\pi\)
0.0846317 + 0.996412i \(0.473029\pi\)
\(200\) 0 0
\(201\) −14.5048 −1.02309
\(202\) −15.2524 −1.07315
\(203\) 22.6618 1.59055
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −7.52311 −0.524160
\(207\) −8.11704 −0.564173
\(208\) 2.00000 0.138675
\(209\) −19.4586 −1.34598
\(210\) 0 0
\(211\) −1.18785 −0.0817746 −0.0408873 0.999164i \(-0.513018\pi\)
−0.0408873 + 0.999164i \(0.513018\pi\)
\(212\) 0.270718 0.0185930
\(213\) 11.1633 0.764894
\(214\) 19.8217 1.35499
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 29.7572 2.02005
\(218\) −4.92919 −0.333847
\(219\) 4.98168 0.336631
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 6.65847 0.446887
\(223\) −14.0279 −0.939378 −0.469689 0.882832i \(-0.655634\pi\)
−0.469689 + 0.882832i \(0.655634\pi\)
\(224\) 2.86464 0.191402
\(225\) 0 0
\(226\) 3.72928 0.248068
\(227\) 10.5048 0.697228 0.348614 0.937266i \(-0.386652\pi\)
0.348614 + 0.937266i \(0.386652\pi\)
\(228\) 5.52311 0.365777
\(229\) 4.50479 0.297685 0.148843 0.988861i \(-0.452445\pi\)
0.148843 + 0.988861i \(0.452445\pi\)
\(230\) 0 0
\(231\) −10.0925 −0.664035
\(232\) 7.91087 0.519374
\(233\) 18.7755 1.23002 0.615012 0.788518i \(-0.289151\pi\)
0.615012 + 0.788518i \(0.289151\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −13.2524 −0.862658
\(237\) −2.38776 −0.155101
\(238\) −2.86464 −0.185687
\(239\) 12.2341 0.791356 0.395678 0.918389i \(-0.370510\pi\)
0.395678 + 0.918389i \(0.370510\pi\)
\(240\) 0 0
\(241\) 10.7755 0.694112 0.347056 0.937844i \(-0.387181\pi\)
0.347056 + 0.937844i \(0.387181\pi\)
\(242\) 1.41233 0.0907883
\(243\) 1.00000 0.0641500
\(244\) 0.929192 0.0594854
\(245\) 0 0
\(246\) −10.7755 −0.687021
\(247\) 11.0462 0.702855
\(248\) 10.3878 0.659623
\(249\) −1.52311 −0.0965235
\(250\) 0 0
\(251\) −22.3632 −1.41155 −0.705776 0.708435i \(-0.749401\pi\)
−0.705776 + 0.708435i \(0.749401\pi\)
\(252\) 2.86464 0.180455
\(253\) 28.5972 1.79789
\(254\) 2.20617 0.138427
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.79383 −0.610922 −0.305461 0.952205i \(-0.598811\pi\)
−0.305461 + 0.952205i \(0.598811\pi\)
\(258\) 1.52311 0.0948250
\(259\) 19.0741 1.18521
\(260\) 0 0
\(261\) 7.91087 0.489671
\(262\) 3.52311 0.217659
\(263\) 8.77551 0.541121 0.270561 0.962703i \(-0.412791\pi\)
0.270561 + 0.962703i \(0.412791\pi\)
\(264\) −3.52311 −0.216833
\(265\) 0 0
\(266\) 15.8217 0.970093
\(267\) −9.25240 −0.566237
\(268\) −14.5048 −0.886021
\(269\) −20.4523 −1.24700 −0.623500 0.781824i \(-0.714290\pi\)
−0.623500 + 0.781824i \(0.714290\pi\)
\(270\) 0 0
\(271\) −22.5048 −1.36707 −0.683534 0.729918i \(-0.739558\pi\)
−0.683534 + 0.729918i \(0.739558\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 5.72928 0.346752
\(274\) 6.20617 0.374928
\(275\) 0 0
\(276\) −8.11704 −0.488588
\(277\) −29.9388 −1.79885 −0.899423 0.437078i \(-0.856013\pi\)
−0.899423 + 0.437078i \(0.856013\pi\)
\(278\) −12.2341 −0.733751
\(279\) 10.3878 0.621899
\(280\) 0 0
\(281\) −24.1695 −1.44183 −0.720916 0.693022i \(-0.756279\pi\)
−0.720916 + 0.693022i \(0.756279\pi\)
\(282\) 12.7755 0.760771
\(283\) −8.23407 −0.489465 −0.244732 0.969591i \(-0.578700\pi\)
−0.244732 + 0.969591i \(0.578700\pi\)
\(284\) 11.1633 0.662418
\(285\) 0 0
\(286\) −7.04623 −0.416652
\(287\) −30.8680 −1.82208
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0.747604 0.0438253
\(292\) 4.98168 0.291531
\(293\) 16.9171 0.988309 0.494155 0.869374i \(-0.335478\pi\)
0.494155 + 0.869374i \(0.335478\pi\)
\(294\) 1.20617 0.0703451
\(295\) 0 0
\(296\) 6.65847 0.387016
\(297\) −3.52311 −0.204432
\(298\) −3.01832 −0.174847
\(299\) −16.2341 −0.938841
\(300\) 0 0
\(301\) 4.36318 0.251489
\(302\) −3.04623 −0.175291
\(303\) −15.2524 −0.876227
\(304\) 5.52311 0.316772
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 6.47689 0.369655 0.184828 0.982771i \(-0.440827\pi\)
0.184828 + 0.982771i \(0.440827\pi\)
\(308\) −10.0925 −0.575071
\(309\) −7.52311 −0.427975
\(310\) 0 0
\(311\) −14.3878 −0.815855 −0.407927 0.913014i \(-0.633748\pi\)
−0.407927 + 0.913014i \(0.633748\pi\)
\(312\) 2.00000 0.113228
\(313\) −22.7110 −1.28370 −0.641850 0.766830i \(-0.721833\pi\)
−0.641850 + 0.766830i \(0.721833\pi\)
\(314\) 11.7293 0.661922
\(315\) 0 0
\(316\) −2.38776 −0.134322
\(317\) 16.2095 0.910416 0.455208 0.890385i \(-0.349565\pi\)
0.455208 + 0.890385i \(0.349565\pi\)
\(318\) 0.270718 0.0151811
\(319\) −27.8709 −1.56047
\(320\) 0 0
\(321\) 19.8217 1.10634
\(322\) −23.2524 −1.29581
\(323\) −5.52311 −0.307314
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −11.0462 −0.611794
\(327\) −4.92919 −0.272585
\(328\) −10.7755 −0.594978
\(329\) 36.5972 2.01767
\(330\) 0 0
\(331\) −3.04623 −0.167436 −0.0837179 0.996489i \(-0.526679\pi\)
−0.0837179 + 0.996489i \(0.526679\pi\)
\(332\) −1.52311 −0.0835918
\(333\) 6.65847 0.364882
\(334\) −11.5756 −0.633389
\(335\) 0 0
\(336\) 2.86464 0.156279
\(337\) −8.44024 −0.459769 −0.229885 0.973218i \(-0.573835\pi\)
−0.229885 + 0.973218i \(0.573835\pi\)
\(338\) −9.00000 −0.489535
\(339\) 3.72928 0.202547
\(340\) 0 0
\(341\) −36.5972 −1.98185
\(342\) 5.52311 0.298656
\(343\) −16.5972 −0.896167
\(344\) 1.52311 0.0821208
\(345\) 0 0
\(346\) 5.34153 0.287162
\(347\) −8.54144 −0.458528 −0.229264 0.973364i \(-0.573632\pi\)
−0.229264 + 0.973364i \(0.573632\pi\)
\(348\) 7.91087 0.424067
\(349\) −9.28030 −0.496763 −0.248382 0.968662i \(-0.579899\pi\)
−0.248382 + 0.968662i \(0.579899\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −3.52311 −0.187783
\(353\) −15.0096 −0.798880 −0.399440 0.916759i \(-0.630795\pi\)
−0.399440 + 0.916759i \(0.630795\pi\)
\(354\) −13.2524 −0.704357
\(355\) 0 0
\(356\) −9.25240 −0.490376
\(357\) −2.86464 −0.151613
\(358\) 10.2341 0.540888
\(359\) −32.4681 −1.71360 −0.856802 0.515646i \(-0.827552\pi\)
−0.856802 + 0.515646i \(0.827552\pi\)
\(360\) 0 0
\(361\) 11.5048 0.605515
\(362\) −1.88296 −0.0989663
\(363\) 1.41233 0.0741284
\(364\) 5.72928 0.300296
\(365\) 0 0
\(366\) 0.929192 0.0485696
\(367\) 15.8742 0.828628 0.414314 0.910134i \(-0.364022\pi\)
0.414314 + 0.910134i \(0.364022\pi\)
\(368\) −8.11704 −0.423130
\(369\) −10.7755 −0.560951
\(370\) 0 0
\(371\) 0.775511 0.0402625
\(372\) 10.3878 0.538580
\(373\) 5.45856 0.282634 0.141317 0.989964i \(-0.454866\pi\)
0.141317 + 0.989964i \(0.454866\pi\)
\(374\) 3.52311 0.182176
\(375\) 0 0
\(376\) 12.7755 0.658847
\(377\) 15.8217 0.814861
\(378\) 2.86464 0.147341
\(379\) −21.5510 −1.10700 −0.553501 0.832849i \(-0.686708\pi\)
−0.553501 + 0.832849i \(0.686708\pi\)
\(380\) 0 0
\(381\) 2.20617 0.113025
\(382\) 0.775511 0.0396786
\(383\) 19.2803 0.985177 0.492589 0.870262i \(-0.336051\pi\)
0.492589 + 0.870262i \(0.336051\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.2986 −0.727781
\(387\) 1.52311 0.0774243
\(388\) 0.747604 0.0379538
\(389\) −6.06455 −0.307485 −0.153742 0.988111i \(-0.549133\pi\)
−0.153742 + 0.988111i \(0.549133\pi\)
\(390\) 0 0
\(391\) 8.11704 0.410496
\(392\) 1.20617 0.0609207
\(393\) 3.52311 0.177718
\(394\) 7.61224 0.383499
\(395\) 0 0
\(396\) −3.52311 −0.177043
\(397\) 6.65847 0.334179 0.167090 0.985942i \(-0.446563\pi\)
0.167090 + 0.985942i \(0.446563\pi\)
\(398\) 2.38776 0.119687
\(399\) 15.8217 0.792078
\(400\) 0 0
\(401\) 12.3265 0.615558 0.307779 0.951458i \(-0.400414\pi\)
0.307779 + 0.951458i \(0.400414\pi\)
\(402\) −14.5048 −0.723433
\(403\) 20.7755 1.03490
\(404\) −15.2524 −0.758835
\(405\) 0 0
\(406\) 22.6618 1.12469
\(407\) −23.4586 −1.16280
\(408\) −1.00000 −0.0495074
\(409\) −4.29862 −0.212553 −0.106277 0.994337i \(-0.533893\pi\)
−0.106277 + 0.994337i \(0.533893\pi\)
\(410\) 0 0
\(411\) 6.20617 0.306128
\(412\) −7.52311 −0.370637
\(413\) −37.9634 −1.86805
\(414\) −8.11704 −0.398931
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −12.2341 −0.599105
\(418\) −19.4586 −0.951749
\(419\) 24.7110 1.20721 0.603605 0.797284i \(-0.293731\pi\)
0.603605 + 0.797284i \(0.293731\pi\)
\(420\) 0 0
\(421\) −30.7755 −1.49991 −0.749953 0.661491i \(-0.769924\pi\)
−0.749953 + 0.661491i \(0.769924\pi\)
\(422\) −1.18785 −0.0578234
\(423\) 12.7755 0.621167
\(424\) 0.270718 0.0131472
\(425\) 0 0
\(426\) 11.1633 0.540862
\(427\) 2.66180 0.128814
\(428\) 19.8217 0.958120
\(429\) −7.04623 −0.340195
\(430\) 0 0
\(431\) −0.658473 −0.0317176 −0.0158588 0.999874i \(-0.505048\pi\)
−0.0158588 + 0.999874i \(0.505048\pi\)
\(432\) 1.00000 0.0481125
\(433\) −28.3911 −1.36439 −0.682194 0.731171i \(-0.738974\pi\)
−0.682194 + 0.731171i \(0.738974\pi\)
\(434\) 29.7572 1.42839
\(435\) 0 0
\(436\) −4.92919 −0.236065
\(437\) −44.8313 −2.14457
\(438\) 4.98168 0.238034
\(439\) 30.0804 1.43566 0.717829 0.696219i \(-0.245136\pi\)
0.717829 + 0.696219i \(0.245136\pi\)
\(440\) 0 0
\(441\) 1.20617 0.0574365
\(442\) −2.00000 −0.0951303
\(443\) 8.54144 0.405816 0.202908 0.979198i \(-0.434961\pi\)
0.202908 + 0.979198i \(0.434961\pi\)
\(444\) 6.65847 0.315997
\(445\) 0 0
\(446\) −14.0279 −0.664241
\(447\) −3.01832 −0.142762
\(448\) 2.86464 0.135342
\(449\) 23.3169 1.10039 0.550197 0.835035i \(-0.314552\pi\)
0.550197 + 0.835035i \(0.314552\pi\)
\(450\) 0 0
\(451\) 37.9634 1.78762
\(452\) 3.72928 0.175411
\(453\) −3.04623 −0.143124
\(454\) 10.5048 0.493014
\(455\) 0 0
\(456\) 5.52311 0.258644
\(457\) 10.2986 0.481749 0.240875 0.970556i \(-0.422566\pi\)
0.240875 + 0.970556i \(0.422566\pi\)
\(458\) 4.50479 0.210495
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −1.70138 −0.0792409 −0.0396205 0.999215i \(-0.512615\pi\)
−0.0396205 + 0.999215i \(0.512615\pi\)
\(462\) −10.0925 −0.469544
\(463\) 34.6252 1.60917 0.804584 0.593839i \(-0.202388\pi\)
0.804584 + 0.593839i \(0.202388\pi\)
\(464\) 7.91087 0.367253
\(465\) 0 0
\(466\) 18.7755 0.869759
\(467\) −18.9171 −0.875380 −0.437690 0.899126i \(-0.644203\pi\)
−0.437690 + 0.899126i \(0.644203\pi\)
\(468\) 2.00000 0.0924500
\(469\) −41.5510 −1.91865
\(470\) 0 0
\(471\) 11.7293 0.540457
\(472\) −13.2524 −0.609991
\(473\) −5.36611 −0.246734
\(474\) −2.38776 −0.109673
\(475\) 0 0
\(476\) −2.86464 −0.131301
\(477\) 0.270718 0.0123953
\(478\) 12.2341 0.559574
\(479\) −14.6218 −0.668088 −0.334044 0.942557i \(-0.608413\pi\)
−0.334044 + 0.942557i \(0.608413\pi\)
\(480\) 0 0
\(481\) 13.3169 0.607201
\(482\) 10.7755 0.490811
\(483\) −23.2524 −1.05802
\(484\) 1.41233 0.0641970
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 23.5110 1.06539 0.532694 0.846308i \(-0.321180\pi\)
0.532694 + 0.846308i \(0.321180\pi\)
\(488\) 0.929192 0.0420625
\(489\) −11.0462 −0.499528
\(490\) 0 0
\(491\) 19.8584 0.896196 0.448098 0.893984i \(-0.352102\pi\)
0.448098 + 0.893984i \(0.352102\pi\)
\(492\) −10.7755 −0.485798
\(493\) −7.91087 −0.356288
\(494\) 11.0462 0.496993
\(495\) 0 0
\(496\) 10.3878 0.466424
\(497\) 31.9787 1.43444
\(498\) −1.52311 −0.0682524
\(499\) −25.3728 −1.13584 −0.567920 0.823084i \(-0.692252\pi\)
−0.567920 + 0.823084i \(0.692252\pi\)
\(500\) 0 0
\(501\) −11.5756 −0.517160
\(502\) −22.3632 −0.998117
\(503\) −25.5631 −1.13980 −0.569901 0.821713i \(-0.693018\pi\)
−0.569901 + 0.821713i \(0.693018\pi\)
\(504\) 2.86464 0.127601
\(505\) 0 0
\(506\) 28.5972 1.27130
\(507\) −9.00000 −0.399704
\(508\) 2.20617 0.0978829
\(509\) 7.07414 0.313556 0.156778 0.987634i \(-0.449889\pi\)
0.156778 + 0.987634i \(0.449889\pi\)
\(510\) 0 0
\(511\) 14.2707 0.631299
\(512\) 1.00000 0.0441942
\(513\) 5.52311 0.243851
\(514\) −9.79383 −0.431987
\(515\) 0 0
\(516\) 1.52311 0.0670514
\(517\) −45.0096 −1.97952
\(518\) 19.0741 0.838069
\(519\) 5.34153 0.234467
\(520\) 0 0
\(521\) 0.141617 0.00620434 0.00310217 0.999995i \(-0.499013\pi\)
0.00310217 + 0.999995i \(0.499013\pi\)
\(522\) 7.91087 0.346249
\(523\) 9.49521 0.415196 0.207598 0.978214i \(-0.433435\pi\)
0.207598 + 0.978214i \(0.433435\pi\)
\(524\) 3.52311 0.153908
\(525\) 0 0
\(526\) 8.77551 0.382630
\(527\) −10.3878 −0.452498
\(528\) −3.52311 −0.153324
\(529\) 42.8863 1.86462
\(530\) 0 0
\(531\) −13.2524 −0.575105
\(532\) 15.8217 0.685959
\(533\) −21.5510 −0.933478
\(534\) −9.25240 −0.400390
\(535\) 0 0
\(536\) −14.5048 −0.626512
\(537\) 10.2341 0.441633
\(538\) −20.4523 −0.881762
\(539\) −4.24947 −0.183037
\(540\) 0 0
\(541\) −16.6218 −0.714628 −0.357314 0.933984i \(-0.616307\pi\)
−0.357314 + 0.933984i \(0.616307\pi\)
\(542\) −22.5048 −0.966664
\(543\) −1.88296 −0.0808057
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 5.72928 0.245191
\(547\) −8.77551 −0.375214 −0.187607 0.982244i \(-0.560073\pi\)
−0.187607 + 0.982244i \(0.560073\pi\)
\(548\) 6.20617 0.265114
\(549\) 0.929192 0.0396569
\(550\) 0 0
\(551\) 43.6926 1.86137
\(552\) −8.11704 −0.345484
\(553\) −6.84006 −0.290869
\(554\) −29.9388 −1.27198
\(555\) 0 0
\(556\) −12.2341 −0.518840
\(557\) −10.3632 −0.439102 −0.219551 0.975601i \(-0.570459\pi\)
−0.219551 + 0.975601i \(0.570459\pi\)
\(558\) 10.3878 0.439749
\(559\) 3.04623 0.128842
\(560\) 0 0
\(561\) 3.52311 0.148746
\(562\) −24.1695 −1.01953
\(563\) −33.1108 −1.39545 −0.697726 0.716364i \(-0.745805\pi\)
−0.697726 + 0.716364i \(0.745805\pi\)
\(564\) 12.7755 0.537946
\(565\) 0 0
\(566\) −8.23407 −0.346104
\(567\) 2.86464 0.120304
\(568\) 11.1633 0.468400
\(569\) −39.0096 −1.63537 −0.817683 0.575668i \(-0.804742\pi\)
−0.817683 + 0.575668i \(0.804742\pi\)
\(570\) 0 0
\(571\) 11.4586 0.479526 0.239763 0.970831i \(-0.422930\pi\)
0.239763 + 0.970831i \(0.422930\pi\)
\(572\) −7.04623 −0.294618
\(573\) 0.775511 0.0323974
\(574\) −30.8680 −1.28840
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 1.96336 0.0817356 0.0408678 0.999165i \(-0.486988\pi\)
0.0408678 + 0.999165i \(0.486988\pi\)
\(578\) 1.00000 0.0415945
\(579\) −14.2986 −0.594231
\(580\) 0 0
\(581\) −4.36318 −0.181015
\(582\) 0.747604 0.0309892
\(583\) −0.953771 −0.0395012
\(584\) 4.98168 0.206143
\(585\) 0 0
\(586\) 16.9171 0.698840
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 1.20617 0.0497415
\(589\) 57.3728 2.36400
\(590\) 0 0
\(591\) 7.61224 0.313126
\(592\) 6.65847 0.273662
\(593\) 31.3449 1.28718 0.643589 0.765371i \(-0.277444\pi\)
0.643589 + 0.765371i \(0.277444\pi\)
\(594\) −3.52311 −0.144555
\(595\) 0 0
\(596\) −3.01832 −0.123635
\(597\) 2.38776 0.0977243
\(598\) −16.2341 −0.663861
\(599\) −17.1387 −0.700268 −0.350134 0.936700i \(-0.613864\pi\)
−0.350134 + 0.936700i \(0.613864\pi\)
\(600\) 0 0
\(601\) 12.3757 0.504815 0.252407 0.967621i \(-0.418778\pi\)
0.252407 + 0.967621i \(0.418778\pi\)
\(602\) 4.36318 0.177830
\(603\) −14.5048 −0.590681
\(604\) −3.04623 −0.123949
\(605\) 0 0
\(606\) −15.2524 −0.619586
\(607\) 38.4523 1.56073 0.780365 0.625324i \(-0.215033\pi\)
0.780365 + 0.625324i \(0.215033\pi\)
\(608\) 5.52311 0.223992
\(609\) 22.6618 0.918302
\(610\) 0 0
\(611\) 25.5510 1.03368
\(612\) −1.00000 −0.0404226
\(613\) 2.77551 0.112102 0.0560509 0.998428i \(-0.482149\pi\)
0.0560509 + 0.998428i \(0.482149\pi\)
\(614\) 6.47689 0.261386
\(615\) 0 0
\(616\) −10.0925 −0.406637
\(617\) −40.2707 −1.62124 −0.810619 0.585574i \(-0.800869\pi\)
−0.810619 + 0.585574i \(0.800869\pi\)
\(618\) −7.52311 −0.302624
\(619\) 40.5972 1.63174 0.815871 0.578234i \(-0.196258\pi\)
0.815871 + 0.578234i \(0.196258\pi\)
\(620\) 0 0
\(621\) −8.11704 −0.325725
\(622\) −14.3878 −0.576896
\(623\) −26.5048 −1.06189
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −22.7110 −0.907713
\(627\) −19.4586 −0.777100
\(628\) 11.7293 0.468049
\(629\) −6.65847 −0.265491
\(630\) 0 0
\(631\) 17.3169 0.689377 0.344688 0.938717i \(-0.387985\pi\)
0.344688 + 0.938717i \(0.387985\pi\)
\(632\) −2.38776 −0.0949798
\(633\) −1.18785 −0.0472126
\(634\) 16.2095 0.643761
\(635\) 0 0
\(636\) 0.270718 0.0107347
\(637\) 2.41233 0.0955802
\(638\) −27.8709 −1.10342
\(639\) 11.1633 0.441612
\(640\) 0 0
\(641\) 21.6926 0.856808 0.428404 0.903587i \(-0.359076\pi\)
0.428404 + 0.903587i \(0.359076\pi\)
\(642\) 19.8217 0.782302
\(643\) 37.0096 1.45952 0.729758 0.683706i \(-0.239633\pi\)
0.729758 + 0.683706i \(0.239633\pi\)
\(644\) −23.2524 −0.916273
\(645\) 0 0
\(646\) −5.52311 −0.217304
\(647\) 16.6464 0.654438 0.327219 0.944949i \(-0.393889\pi\)
0.327219 + 0.944949i \(0.393889\pi\)
\(648\) 1.00000 0.0392837
\(649\) 46.6897 1.83273
\(650\) 0 0
\(651\) 29.7572 1.16628
\(652\) −11.0462 −0.432604
\(653\) 6.89255 0.269726 0.134863 0.990864i \(-0.456941\pi\)
0.134863 + 0.990864i \(0.456941\pi\)
\(654\) −4.92919 −0.192747
\(655\) 0 0
\(656\) −10.7755 −0.420713
\(657\) 4.98168 0.194354
\(658\) 36.5972 1.42671
\(659\) −14.2062 −0.553394 −0.276697 0.960957i \(-0.589240\pi\)
−0.276697 + 0.960957i \(0.589240\pi\)
\(660\) 0 0
\(661\) −28.3265 −1.10177 −0.550887 0.834580i \(-0.685711\pi\)
−0.550887 + 0.834580i \(0.685711\pi\)
\(662\) −3.04623 −0.118395
\(663\) −2.00000 −0.0776736
\(664\) −1.52311 −0.0591083
\(665\) 0 0
\(666\) 6.65847 0.258011
\(667\) −64.2128 −2.48633
\(668\) −11.5756 −0.447873
\(669\) −14.0279 −0.542350
\(670\) 0 0
\(671\) −3.27365 −0.126378
\(672\) 2.86464 0.110506
\(673\) −1.70138 −0.0655832 −0.0327916 0.999462i \(-0.510440\pi\)
−0.0327916 + 0.999462i \(0.510440\pi\)
\(674\) −8.44024 −0.325106
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 7.30488 0.280749 0.140375 0.990098i \(-0.455169\pi\)
0.140375 + 0.990098i \(0.455169\pi\)
\(678\) 3.72928 0.143222
\(679\) 2.14162 0.0821877
\(680\) 0 0
\(681\) 10.5048 0.402545
\(682\) −36.5972 −1.40138
\(683\) 15.0462 0.575728 0.287864 0.957671i \(-0.407055\pi\)
0.287864 + 0.957671i \(0.407055\pi\)
\(684\) 5.52311 0.211182
\(685\) 0 0
\(686\) −16.5972 −0.633686
\(687\) 4.50479 0.171869
\(688\) 1.52311 0.0580682
\(689\) 0.541436 0.0206271
\(690\) 0 0
\(691\) −9.78510 −0.372243 −0.186121 0.982527i \(-0.559592\pi\)
−0.186121 + 0.982527i \(0.559592\pi\)
\(692\) 5.34153 0.203054
\(693\) −10.0925 −0.383381
\(694\) −8.54144 −0.324228
\(695\) 0 0
\(696\) 7.91087 0.299861
\(697\) 10.7755 0.408152
\(698\) −9.28030 −0.351265
\(699\) 18.7755 0.710155
\(700\) 0 0
\(701\) 9.88629 0.373400 0.186700 0.982417i \(-0.440221\pi\)
0.186700 + 0.982417i \(0.440221\pi\)
\(702\) 2.00000 0.0754851
\(703\) 36.7755 1.38701
\(704\) −3.52311 −0.132782
\(705\) 0 0
\(706\) −15.0096 −0.564893
\(707\) −43.6926 −1.64323
\(708\) −13.2524 −0.498056
\(709\) −34.7143 −1.30372 −0.651861 0.758338i \(-0.726012\pi\)
−0.651861 + 0.758338i \(0.726012\pi\)
\(710\) 0 0
\(711\) −2.38776 −0.0895478
\(712\) −9.25240 −0.346748
\(713\) −84.3178 −3.15773
\(714\) −2.86464 −0.107206
\(715\) 0 0
\(716\) 10.2341 0.382465
\(717\) 12.2341 0.456890
\(718\) −32.4681 −1.21170
\(719\) 34.2095 1.27580 0.637899 0.770120i \(-0.279804\pi\)
0.637899 + 0.770120i \(0.279804\pi\)
\(720\) 0 0
\(721\) −21.5510 −0.802602
\(722\) 11.5048 0.428164
\(723\) 10.7755 0.400746
\(724\) −1.88296 −0.0699798
\(725\) 0 0
\(726\) 1.41233 0.0524167
\(727\) 35.8863 1.33095 0.665474 0.746421i \(-0.268229\pi\)
0.665474 + 0.746421i \(0.268229\pi\)
\(728\) 5.72928 0.212341
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.52311 −0.0563344
\(732\) 0.929192 0.0343439
\(733\) −20.3757 −0.752593 −0.376297 0.926499i \(-0.622803\pi\)
−0.376297 + 0.926499i \(0.622803\pi\)
\(734\) 15.8742 0.585928
\(735\) 0 0
\(736\) −8.11704 −0.299198
\(737\) 51.1020 1.88237
\(738\) −10.7755 −0.396652
\(739\) −18.4769 −0.679683 −0.339842 0.940483i \(-0.610373\pi\)
−0.339842 + 0.940483i \(0.610373\pi\)
\(740\) 0 0
\(741\) 11.0462 0.405793
\(742\) 0.775511 0.0284699
\(743\) −20.7876 −0.762622 −0.381311 0.924447i \(-0.624527\pi\)
−0.381311 + 0.924447i \(0.624527\pi\)
\(744\) 10.3878 0.380834
\(745\) 0 0
\(746\) 5.45856 0.199852
\(747\) −1.52311 −0.0557278
\(748\) 3.52311 0.128818
\(749\) 56.7822 2.07478
\(750\) 0 0
\(751\) −18.8559 −0.688062 −0.344031 0.938958i \(-0.611792\pi\)
−0.344031 + 0.938958i \(0.611792\pi\)
\(752\) 12.7755 0.465875
\(753\) −22.3632 −0.814959
\(754\) 15.8217 0.576194
\(755\) 0 0
\(756\) 2.86464 0.104186
\(757\) −7.18785 −0.261247 −0.130623 0.991432i \(-0.541698\pi\)
−0.130623 + 0.991432i \(0.541698\pi\)
\(758\) −21.5510 −0.782768
\(759\) 28.5972 1.03801
\(760\) 0 0
\(761\) 4.09246 0.148351 0.0741757 0.997245i \(-0.476367\pi\)
0.0741757 + 0.997245i \(0.476367\pi\)
\(762\) 2.20617 0.0799210
\(763\) −14.1204 −0.511192
\(764\) 0.775511 0.0280570
\(765\) 0 0
\(766\) 19.2803 0.696626
\(767\) −26.5048 −0.957033
\(768\) 1.00000 0.0360844
\(769\) −53.2311 −1.91956 −0.959782 0.280746i \(-0.909418\pi\)
−0.959782 + 0.280746i \(0.909418\pi\)
\(770\) 0 0
\(771\) −9.79383 −0.352716
\(772\) −14.2986 −0.514619
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 1.52311 0.0547472
\(775\) 0 0
\(776\) 0.747604 0.0268374
\(777\) 19.0741 0.684281
\(778\) −6.06455 −0.217425
\(779\) −59.5144 −2.13232
\(780\) 0 0
\(781\) −39.3295 −1.40732
\(782\) 8.11704 0.290265
\(783\) 7.91087 0.282711
\(784\) 1.20617 0.0430774
\(785\) 0 0
\(786\) 3.52311 0.125665
\(787\) 11.2245 0.400110 0.200055 0.979785i \(-0.435888\pi\)
0.200055 + 0.979785i \(0.435888\pi\)
\(788\) 7.61224 0.271175
\(789\) 8.77551 0.312416
\(790\) 0 0
\(791\) 10.6831 0.379846
\(792\) −3.52311 −0.125188
\(793\) 1.85838 0.0659931
\(794\) 6.65847 0.236300
\(795\) 0 0
\(796\) 2.38776 0.0846317
\(797\) −8.27072 −0.292964 −0.146482 0.989213i \(-0.546795\pi\)
−0.146482 + 0.989213i \(0.546795\pi\)
\(798\) 15.8217 0.560083
\(799\) −12.7755 −0.451965
\(800\) 0 0
\(801\) −9.25240 −0.326917
\(802\) 12.3265 0.435265
\(803\) −17.5510 −0.619362
\(804\) −14.5048 −0.511545
\(805\) 0 0
\(806\) 20.7755 0.731786
\(807\) −20.4523 −0.719955
\(808\) −15.2524 −0.536577
\(809\) −34.3632 −1.20814 −0.604072 0.796929i \(-0.706456\pi\)
−0.604072 + 0.796929i \(0.706456\pi\)
\(810\) 0 0
\(811\) −29.9634 −1.05216 −0.526078 0.850436i \(-0.676338\pi\)
−0.526078 + 0.850436i \(0.676338\pi\)
\(812\) 22.6618 0.795273
\(813\) −22.5048 −0.789278
\(814\) −23.4586 −0.822222
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 8.41233 0.294310
\(818\) −4.29862 −0.150298
\(819\) 5.72928 0.200197
\(820\) 0 0
\(821\) −49.2278 −1.71806 −0.859031 0.511923i \(-0.828933\pi\)
−0.859031 + 0.511923i \(0.828933\pi\)
\(822\) 6.20617 0.216465
\(823\) 13.3694 0.466029 0.233015 0.972473i \(-0.425141\pi\)
0.233015 + 0.972473i \(0.425141\pi\)
\(824\) −7.52311 −0.262080
\(825\) 0 0
\(826\) −37.9634 −1.32091
\(827\) −26.7389 −0.929801 −0.464901 0.885363i \(-0.653910\pi\)
−0.464901 + 0.885363i \(0.653910\pi\)
\(828\) −8.11704 −0.282087
\(829\) −20.4490 −0.710222 −0.355111 0.934824i \(-0.615557\pi\)
−0.355111 + 0.934824i \(0.615557\pi\)
\(830\) 0 0
\(831\) −29.9388 −1.03856
\(832\) 2.00000 0.0693375
\(833\) −1.20617 −0.0417912
\(834\) −12.2341 −0.423631
\(835\) 0 0
\(836\) −19.4586 −0.672988
\(837\) 10.3878 0.359053
\(838\) 24.7110 0.853626
\(839\) −23.8097 −0.822001 −0.411001 0.911635i \(-0.634821\pi\)
−0.411001 + 0.911635i \(0.634821\pi\)
\(840\) 0 0
\(841\) 33.5819 1.15800
\(842\) −30.7755 −1.06059
\(843\) −24.1695 −0.832443
\(844\) −1.18785 −0.0408873
\(845\) 0 0
\(846\) 12.7755 0.439231
\(847\) 4.04583 0.139016
\(848\) 0.270718 0.00929650
\(849\) −8.23407 −0.282593
\(850\) 0 0
\(851\) −54.0471 −1.85271
\(852\) 11.1633 0.382447
\(853\) 23.9754 0.820903 0.410451 0.911882i \(-0.365371\pi\)
0.410451 + 0.911882i \(0.365371\pi\)
\(854\) 2.66180 0.0910849
\(855\) 0 0
\(856\) 19.8217 0.677493
\(857\) −1.58767 −0.0542336 −0.0271168 0.999632i \(-0.508633\pi\)
−0.0271168 + 0.999632i \(0.508633\pi\)
\(858\) −7.04623 −0.240554
\(859\) 39.4586 1.34631 0.673154 0.739502i \(-0.264939\pi\)
0.673154 + 0.739502i \(0.264939\pi\)
\(860\) 0 0
\(861\) −30.8680 −1.05198
\(862\) −0.658473 −0.0224277
\(863\) 54.0192 1.83883 0.919417 0.393284i \(-0.128661\pi\)
0.919417 + 0.393284i \(0.128661\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −28.3911 −0.964768
\(867\) 1.00000 0.0339618
\(868\) 29.7572 1.01002
\(869\) 8.41233 0.285369
\(870\) 0 0
\(871\) −29.0096 −0.982952
\(872\) −4.92919 −0.166923
\(873\) 0.747604 0.0253026
\(874\) −44.8313 −1.51644
\(875\) 0 0
\(876\) 4.98168 0.168315
\(877\) −32.6218 −1.10156 −0.550780 0.834650i \(-0.685670\pi\)
−0.550780 + 0.834650i \(0.685670\pi\)
\(878\) 30.0804 1.01516
\(879\) 16.9171 0.570601
\(880\) 0 0
\(881\) −8.50479 −0.286534 −0.143267 0.989684i \(-0.545761\pi\)
−0.143267 + 0.989684i \(0.545761\pi\)
\(882\) 1.20617 0.0406138
\(883\) 24.0558 0.809543 0.404771 0.914418i \(-0.367351\pi\)
0.404771 + 0.914418i \(0.367351\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 8.54144 0.286955
\(887\) 39.5264 1.32717 0.663584 0.748102i \(-0.269035\pi\)
0.663584 + 0.748102i \(0.269035\pi\)
\(888\) 6.65847 0.223444
\(889\) 6.31988 0.211962
\(890\) 0 0
\(891\) −3.52311 −0.118029
\(892\) −14.0279 −0.469689
\(893\) 70.5606 2.36122
\(894\) −3.01832 −0.100948
\(895\) 0 0
\(896\) 2.86464 0.0957009
\(897\) −16.2341 −0.542040
\(898\) 23.3169 0.778097
\(899\) 82.1762 2.74073
\(900\) 0 0
\(901\) −0.270718 −0.00901893
\(902\) 37.9634 1.26404
\(903\) 4.36318 0.145197
\(904\) 3.72928 0.124034
\(905\) 0 0
\(906\) −3.04623 −0.101204
\(907\) 14.3998 0.478138 0.239069 0.971003i \(-0.423158\pi\)
0.239069 + 0.971003i \(0.423158\pi\)
\(908\) 10.5048 0.348614
\(909\) −15.2524 −0.505890
\(910\) 0 0
\(911\) 47.8097 1.58401 0.792003 0.610518i \(-0.209039\pi\)
0.792003 + 0.610518i \(0.209039\pi\)
\(912\) 5.52311 0.182889
\(913\) 5.36611 0.177592
\(914\) 10.2986 0.340648
\(915\) 0 0
\(916\) 4.50479 0.148843
\(917\) 10.0925 0.333282
\(918\) −1.00000 −0.0330049
\(919\) 57.1945 1.88667 0.943336 0.331838i \(-0.107669\pi\)
0.943336 + 0.331838i \(0.107669\pi\)
\(920\) 0 0
\(921\) 6.47689 0.213421
\(922\) −1.70138 −0.0560318
\(923\) 22.3265 0.734887
\(924\) −10.0925 −0.332017
\(925\) 0 0
\(926\) 34.6252 1.13785
\(927\) −7.52311 −0.247091
\(928\) 7.91087 0.259687
\(929\) 18.7197 0.614173 0.307087 0.951682i \(-0.400646\pi\)
0.307087 + 0.951682i \(0.400646\pi\)
\(930\) 0 0
\(931\) 6.66180 0.218332
\(932\) 18.7755 0.615012
\(933\) −14.3878 −0.471034
\(934\) −18.9171 −0.618987
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 25.2158 0.823763 0.411881 0.911237i \(-0.364872\pi\)
0.411881 + 0.911237i \(0.364872\pi\)
\(938\) −41.5510 −1.35669
\(939\) −22.7110 −0.741144
\(940\) 0 0
\(941\) 12.0891 0.394094 0.197047 0.980394i \(-0.436865\pi\)
0.197047 + 0.980394i \(0.436865\pi\)
\(942\) 11.7293 0.382161
\(943\) 87.4652 2.84826
\(944\) −13.2524 −0.431329
\(945\) 0 0
\(946\) −5.36611 −0.174467
\(947\) −29.7851 −0.967886 −0.483943 0.875100i \(-0.660796\pi\)
−0.483943 + 0.875100i \(0.660796\pi\)
\(948\) −2.38776 −0.0775507
\(949\) 9.96336 0.323424
\(950\) 0 0
\(951\) 16.2095 0.525629
\(952\) −2.86464 −0.0928435
\(953\) −25.5144 −0.826492 −0.413246 0.910619i \(-0.635605\pi\)
−0.413246 + 0.910619i \(0.635605\pi\)
\(954\) 0.270718 0.00876483
\(955\) 0 0
\(956\) 12.2341 0.395678
\(957\) −27.8709 −0.900938
\(958\) −14.6218 −0.472410
\(959\) 17.7784 0.574096
\(960\) 0 0
\(961\) 76.9055 2.48082
\(962\) 13.3169 0.429356
\(963\) 19.8217 0.638747
\(964\) 10.7755 0.347056
\(965\) 0 0
\(966\) −23.2524 −0.748134
\(967\) −31.5231 −1.01372 −0.506858 0.862030i \(-0.669193\pi\)
−0.506858 + 0.862030i \(0.669193\pi\)
\(968\) 1.41233 0.0453942
\(969\) −5.52311 −0.177428
\(970\) 0 0
\(971\) 36.8959 1.18404 0.592022 0.805921i \(-0.298330\pi\)
0.592022 + 0.805921i \(0.298330\pi\)
\(972\) 1.00000 0.0320750
\(973\) −35.0462 −1.12353
\(974\) 23.5110 0.753343
\(975\) 0 0
\(976\) 0.929192 0.0297427
\(977\) −36.2986 −1.16130 −0.580648 0.814155i \(-0.697201\pi\)
−0.580648 + 0.814155i \(0.697201\pi\)
\(978\) −11.0462 −0.353219
\(979\) 32.5972 1.04181
\(980\) 0 0
\(981\) −4.92919 −0.157377
\(982\) 19.8584 0.633706
\(983\) 3.21242 0.102460 0.0512302 0.998687i \(-0.483686\pi\)
0.0512302 + 0.998687i \(0.483686\pi\)
\(984\) −10.7755 −0.343511
\(985\) 0 0
\(986\) −7.91087 −0.251933
\(987\) 36.5972 1.16490
\(988\) 11.0462 0.351427
\(989\) −12.3632 −0.393126
\(990\) 0 0
\(991\) 24.2461 0.770204 0.385102 0.922874i \(-0.374166\pi\)
0.385102 + 0.922874i \(0.374166\pi\)
\(992\) 10.3878 0.329812
\(993\) −3.04623 −0.0966691
\(994\) 31.9787 1.01430
\(995\) 0 0
\(996\) −1.52311 −0.0482617
\(997\) 40.3386 1.27754 0.638768 0.769399i \(-0.279444\pi\)
0.638768 + 0.769399i \(0.279444\pi\)
\(998\) −25.3728 −0.803161
\(999\) 6.65847 0.210665
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 2550.2.a.bo.1.2 3
3.2 odd 2 7650.2.a.dl.1.2 3
5.2 odd 4 510.2.d.d.409.5 yes 6
5.3 odd 4 510.2.d.d.409.2 6
5.4 even 2 2550.2.a.bn.1.2 3
15.2 even 4 1530.2.d.i.919.2 6
15.8 even 4 1530.2.d.i.919.5 6
15.14 odd 2 7650.2.a.dm.1.2 3
20.3 even 4 4080.2.m.p.2449.2 6
20.7 even 4 4080.2.m.p.2449.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.d.d.409.2 6 5.3 odd 4
510.2.d.d.409.5 yes 6 5.2 odd 4
1530.2.d.i.919.2 6 15.2 even 4
1530.2.d.i.919.5 6 15.8 even 4
2550.2.a.bn.1.2 3 5.4 even 2
2550.2.a.bo.1.2 3 1.1 even 1 trivial
4080.2.m.p.2449.2 6 20.3 even 4
4080.2.m.p.2449.5 6 20.7 even 4
7650.2.a.dl.1.2 3 3.2 odd 2
7650.2.a.dm.1.2 3 15.14 odd 2