Properties

Label 1287.2.b.c.298.4
Level $1287$
Weight $2$
Character 1287.298
Analytic conductor $10.277$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,2,Mod(298,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1287.298");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.b (of order \(2\), degree \(1\), 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: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 23x^{12} + 201x^{10} + 835x^{8} + 1695x^{6} + 1565x^{4} + 511x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 429)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 298.4
Root \(-1.42819i\) of defining polynomial
Character \(\chi\) \(=\) 1287.298
Dual form 1287.2.b.c.298.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.42819i q^{2} -0.0397381 q^{4} -0.0606573i q^{5} +1.70646i q^{7} -2.79963i q^{8} +O(q^{10})\) \(q-1.42819i q^{2} -0.0397381 q^{4} -0.0606573i q^{5} +1.70646i q^{7} -2.79963i q^{8} -0.0866304 q^{10} +1.00000i q^{11} +(3.52912 - 0.738445i) q^{13} +2.43715 q^{14} -4.07790 q^{16} +3.75599 q^{17} +2.02597i q^{19} +0.00241041i q^{20} +1.42819 q^{22} +0.704722 q^{23} +4.99632 q^{25} +(-1.05464 - 5.04027i) q^{26} -0.0678114i q^{28} +0.0346842 q^{29} +1.85987i q^{31} +0.224760i q^{32} -5.36429i q^{34} +0.103509 q^{35} -8.82674i q^{37} +2.89348 q^{38} -0.169818 q^{40} -3.32960i q^{41} -5.29022 q^{43} -0.0397381i q^{44} -1.00648i q^{46} +6.04622i q^{47} +4.08800 q^{49} -7.13572i q^{50} +(-0.140241 + 0.0293444i) q^{52} +10.6694 q^{53} +0.0606573 q^{55} +4.77746 q^{56} -0.0495358i q^{58} +3.34547i q^{59} +3.26757 q^{61} +2.65625 q^{62} -7.83479 q^{64} +(-0.0447920 - 0.214067i) q^{65} -10.9701i q^{67} -0.149256 q^{68} -0.147831i q^{70} -1.96928i q^{71} -15.3057i q^{73} -12.6063 q^{74} -0.0805084i q^{76} -1.70646 q^{77} +10.5938 q^{79} +0.247354i q^{80} -4.75532 q^{82} -4.19820i q^{83} -0.227828i q^{85} +7.55546i q^{86} +2.79963 q^{88} +14.3643i q^{89} +(1.26012 + 6.02229i) q^{91} -0.0280043 q^{92} +8.63517 q^{94} +0.122890 q^{95} +5.86228i q^{97} -5.83846i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 18 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 18 q^{4} - 16 q^{14} + 34 q^{16} - 4 q^{17} + 6 q^{22} + 8 q^{23} - 26 q^{25} + 6 q^{26} + 24 q^{29} + 8 q^{35} + 32 q^{38} - 20 q^{40} + 32 q^{43} - 46 q^{49} + 4 q^{52} - 20 q^{53} + 12 q^{55} + 32 q^{56} - 20 q^{61} - 72 q^{62} - 58 q^{64} - 12 q^{65} + 20 q^{68} + 12 q^{77} + 12 q^{79} + 20 q^{82} - 30 q^{88} + 16 q^{91} + 24 q^{92} + 64 q^{94} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.42819i 1.00989i −0.863153 0.504943i \(-0.831514\pi\)
0.863153 0.504943i \(-0.168486\pi\)
\(3\) 0 0
\(4\) −0.0397381 −0.0198691
\(5\) 0.0606573i 0.0271268i −0.999908 0.0135634i \(-0.995683\pi\)
0.999908 0.0135634i \(-0.00431749\pi\)
\(6\) 0 0
\(7\) 1.70646i 0.644980i 0.946573 + 0.322490i \(0.104520\pi\)
−0.946573 + 0.322490i \(0.895480\pi\)
\(8\) 2.79963i 0.989820i
\(9\) 0 0
\(10\) −0.0866304 −0.0273949
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 3.52912 0.738445i 0.978802 0.204808i
\(14\) 2.43715 0.651356
\(15\) 0 0
\(16\) −4.07790 −1.01947
\(17\) 3.75599 0.910962 0.455481 0.890246i \(-0.349467\pi\)
0.455481 + 0.890246i \(0.349467\pi\)
\(18\) 0 0
\(19\) 2.02597i 0.464790i 0.972621 + 0.232395i \(0.0746562\pi\)
−0.972621 + 0.232395i \(0.925344\pi\)
\(20\) 0.00241041i 0.000538983i
\(21\) 0 0
\(22\) 1.42819 0.304492
\(23\) 0.704722 0.146945 0.0734723 0.997297i \(-0.476592\pi\)
0.0734723 + 0.997297i \(0.476592\pi\)
\(24\) 0 0
\(25\) 4.99632 0.999264
\(26\) −1.05464 5.04027i −0.206832 0.988478i
\(27\) 0 0
\(28\) 0.0678114i 0.0128151i
\(29\) 0.0346842 0.00644069 0.00322035 0.999995i \(-0.498975\pi\)
0.00322035 + 0.999995i \(0.498975\pi\)
\(30\) 0 0
\(31\) 1.85987i 0.334042i 0.985953 + 0.167021i \(0.0534148\pi\)
−0.985953 + 0.167021i \(0.946585\pi\)
\(32\) 0.224760i 0.0397323i
\(33\) 0 0
\(34\) 5.36429i 0.919967i
\(35\) 0.103509 0.0174962
\(36\) 0 0
\(37\) 8.82674i 1.45111i −0.688166 0.725553i \(-0.741584\pi\)
0.688166 0.725553i \(-0.258416\pi\)
\(38\) 2.89348 0.469385
\(39\) 0 0
\(40\) −0.169818 −0.0268506
\(41\) 3.32960i 0.519996i −0.965609 0.259998i \(-0.916278\pi\)
0.965609 0.259998i \(-0.0837220\pi\)
\(42\) 0 0
\(43\) −5.29022 −0.806751 −0.403376 0.915035i \(-0.632163\pi\)
−0.403376 + 0.915035i \(0.632163\pi\)
\(44\) 0.0397381i 0.00599075i
\(45\) 0 0
\(46\) 1.00648i 0.148397i
\(47\) 6.04622i 0.881931i 0.897524 + 0.440966i \(0.145364\pi\)
−0.897524 + 0.440966i \(0.854636\pi\)
\(48\) 0 0
\(49\) 4.08800 0.584001
\(50\) 7.13572i 1.00914i
\(51\) 0 0
\(52\) −0.140241 + 0.0293444i −0.0194479 + 0.00406934i
\(53\) 10.6694 1.46555 0.732775 0.680471i \(-0.238225\pi\)
0.732775 + 0.680471i \(0.238225\pi\)
\(54\) 0 0
\(55\) 0.0606573 0.00817903
\(56\) 4.77746 0.638414
\(57\) 0 0
\(58\) 0.0495358i 0.00650436i
\(59\) 3.34547i 0.435543i 0.976000 + 0.217771i \(0.0698787\pi\)
−0.976000 + 0.217771i \(0.930121\pi\)
\(60\) 0 0
\(61\) 3.26757 0.418369 0.209185 0.977876i \(-0.432919\pi\)
0.209185 + 0.977876i \(0.432919\pi\)
\(62\) 2.65625 0.337344
\(63\) 0 0
\(64\) −7.83479 −0.979349
\(65\) −0.0447920 0.214067i −0.00555577 0.0265517i
\(66\) 0 0
\(67\) 10.9701i 1.34021i −0.742268 0.670103i \(-0.766250\pi\)
0.742268 0.670103i \(-0.233750\pi\)
\(68\) −0.149256 −0.0181000
\(69\) 0 0
\(70\) 0.147831i 0.0176692i
\(71\) 1.96928i 0.233711i −0.993149 0.116855i \(-0.962719\pi\)
0.993149 0.116855i \(-0.0372814\pi\)
\(72\) 0 0
\(73\) 15.3057i 1.79140i −0.444659 0.895700i \(-0.646675\pi\)
0.444659 0.895700i \(-0.353325\pi\)
\(74\) −12.6063 −1.46545
\(75\) 0 0
\(76\) 0.0805084i 0.00923494i
\(77\) −1.70646 −0.194469
\(78\) 0 0
\(79\) 10.5938 1.19190 0.595951 0.803021i \(-0.296775\pi\)
0.595951 + 0.803021i \(0.296775\pi\)
\(80\) 0.247354i 0.0276550i
\(81\) 0 0
\(82\) −4.75532 −0.525137
\(83\) 4.19820i 0.460812i −0.973095 0.230406i \(-0.925994\pi\)
0.973095 0.230406i \(-0.0740055\pi\)
\(84\) 0 0
\(85\) 0.227828i 0.0247114i
\(86\) 7.55546i 0.814726i
\(87\) 0 0
\(88\) 2.79963 0.298442
\(89\) 14.3643i 1.52261i 0.648393 + 0.761305i \(0.275441\pi\)
−0.648393 + 0.761305i \(0.724559\pi\)
\(90\) 0 0
\(91\) 1.26012 + 6.02229i 0.132097 + 0.631308i
\(92\) −0.0280043 −0.00291965
\(93\) 0 0
\(94\) 8.63517 0.890650
\(95\) 0.122890 0.0126082
\(96\) 0 0
\(97\) 5.86228i 0.595224i 0.954687 + 0.297612i \(0.0961902\pi\)
−0.954687 + 0.297612i \(0.903810\pi\)
\(98\) 5.83846i 0.589774i
\(99\) 0 0
\(100\) −0.198544 −0.0198544
\(101\) −12.3706 −1.23092 −0.615460 0.788168i \(-0.711030\pi\)
−0.615460 + 0.788168i \(0.711030\pi\)
\(102\) 0 0
\(103\) −10.0788 −0.993094 −0.496547 0.868010i \(-0.665399\pi\)
−0.496547 + 0.868010i \(0.665399\pi\)
\(104\) −2.06738 9.88025i −0.202723 0.968838i
\(105\) 0 0
\(106\) 15.2379i 1.48004i
\(107\) −6.17793 −0.597243 −0.298622 0.954372i \(-0.596527\pi\)
−0.298622 + 0.954372i \(0.596527\pi\)
\(108\) 0 0
\(109\) 4.64928i 0.445320i 0.974896 + 0.222660i \(0.0714740\pi\)
−0.974896 + 0.222660i \(0.928526\pi\)
\(110\) 0.0866304i 0.00825988i
\(111\) 0 0
\(112\) 6.95876i 0.657541i
\(113\) −8.99146 −0.845846 −0.422923 0.906166i \(-0.638996\pi\)
−0.422923 + 0.906166i \(0.638996\pi\)
\(114\) 0 0
\(115\) 0.0427465i 0.00398613i
\(116\) −0.00137828 −0.000127971
\(117\) 0 0
\(118\) 4.77798 0.439848
\(119\) 6.40944i 0.587552i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 4.66672i 0.422505i
\(123\) 0 0
\(124\) 0.0739076i 0.00663710i
\(125\) 0.606350i 0.0542336i
\(126\) 0 0
\(127\) −1.74493 −0.154837 −0.0774187 0.996999i \(-0.524668\pi\)
−0.0774187 + 0.996999i \(0.524668\pi\)
\(128\) 11.6391i 1.02876i
\(129\) 0 0
\(130\) −0.305729 + 0.0639717i −0.0268142 + 0.00561069i
\(131\) 13.8961 1.21411 0.607054 0.794661i \(-0.292351\pi\)
0.607054 + 0.794661i \(0.292351\pi\)
\(132\) 0 0
\(133\) −3.45724 −0.299780
\(134\) −15.6674 −1.35345
\(135\) 0 0
\(136\) 10.5154i 0.901689i
\(137\) 3.08818i 0.263841i 0.991260 + 0.131921i \(0.0421144\pi\)
−0.991260 + 0.131921i \(0.957886\pi\)
\(138\) 0 0
\(139\) 1.92887 0.163605 0.0818023 0.996649i \(-0.473932\pi\)
0.0818023 + 0.996649i \(0.473932\pi\)
\(140\) −0.00411325 −0.000347633
\(141\) 0 0
\(142\) −2.81252 −0.236021
\(143\) 0.738445 + 3.52912i 0.0617519 + 0.295120i
\(144\) 0 0
\(145\) 0.00210385i 0.000174715i
\(146\) −21.8595 −1.80911
\(147\) 0 0
\(148\) 0.350758i 0.0288321i
\(149\) 9.05192i 0.741563i −0.928720 0.370781i \(-0.879090\pi\)
0.928720 0.370781i \(-0.120910\pi\)
\(150\) 0 0
\(151\) 2.44462i 0.198940i −0.995041 0.0994702i \(-0.968285\pi\)
0.995041 0.0994702i \(-0.0317148\pi\)
\(152\) 5.67198 0.460059
\(153\) 0 0
\(154\) 2.43715i 0.196391i
\(155\) 0.112814 0.00906147
\(156\) 0 0
\(157\) −8.42793 −0.672622 −0.336311 0.941751i \(-0.609179\pi\)
−0.336311 + 0.941751i \(0.609179\pi\)
\(158\) 15.1301i 1.20368i
\(159\) 0 0
\(160\) 0.0136333 0.00107781
\(161\) 1.20258i 0.0947763i
\(162\) 0 0
\(163\) 17.3969i 1.36263i 0.731990 + 0.681316i \(0.238592\pi\)
−0.731990 + 0.681316i \(0.761408\pi\)
\(164\) 0.132312i 0.0103318i
\(165\) 0 0
\(166\) −5.99585 −0.465368
\(167\) 1.05104i 0.0813320i 0.999173 + 0.0406660i \(0.0129480\pi\)
−0.999173 + 0.0406660i \(0.987052\pi\)
\(168\) 0 0
\(169\) 11.9094 5.21212i 0.916108 0.400933i
\(170\) −0.325383 −0.0249557
\(171\) 0 0
\(172\) 0.210223 0.0160294
\(173\) 18.6654 1.41910 0.709552 0.704653i \(-0.248897\pi\)
0.709552 + 0.704653i \(0.248897\pi\)
\(174\) 0 0
\(175\) 8.52601i 0.644505i
\(176\) 4.07790i 0.307383i
\(177\) 0 0
\(178\) 20.5150 1.53766
\(179\) −24.2443 −1.81210 −0.906051 0.423168i \(-0.860918\pi\)
−0.906051 + 0.423168i \(0.860918\pi\)
\(180\) 0 0
\(181\) −20.9251 −1.55535 −0.777675 0.628666i \(-0.783601\pi\)
−0.777675 + 0.628666i \(0.783601\pi\)
\(182\) 8.60100 1.79970i 0.637549 0.133403i
\(183\) 0 0
\(184\) 1.97296i 0.145449i
\(185\) −0.535406 −0.0393638
\(186\) 0 0
\(187\) 3.75599i 0.274665i
\(188\) 0.240265i 0.0175231i
\(189\) 0 0
\(190\) 0.175511i 0.0127329i
\(191\) −2.77108 −0.200508 −0.100254 0.994962i \(-0.531966\pi\)
−0.100254 + 0.994962i \(0.531966\pi\)
\(192\) 0 0
\(193\) 8.50742i 0.612378i 0.951971 + 0.306189i \(0.0990539\pi\)
−0.951971 + 0.306189i \(0.900946\pi\)
\(194\) 8.37247 0.601108
\(195\) 0 0
\(196\) −0.162450 −0.0116035
\(197\) 14.4278i 1.02794i 0.857810 + 0.513968i \(0.171825\pi\)
−0.857810 + 0.513968i \(0.828175\pi\)
\(198\) 0 0
\(199\) −15.9351 −1.12961 −0.564806 0.825224i \(-0.691049\pi\)
−0.564806 + 0.825224i \(0.691049\pi\)
\(200\) 13.9879i 0.989092i
\(201\) 0 0
\(202\) 17.6676i 1.24309i
\(203\) 0.0591871i 0.00415412i
\(204\) 0 0
\(205\) −0.201965 −0.0141058
\(206\) 14.3945i 1.00291i
\(207\) 0 0
\(208\) −14.3914 + 3.01130i −0.997864 + 0.208796i
\(209\) −2.02597 −0.140139
\(210\) 0 0
\(211\) −16.6404 −1.14557 −0.572786 0.819705i \(-0.694138\pi\)
−0.572786 + 0.819705i \(0.694138\pi\)
\(212\) −0.423980 −0.0291191
\(213\) 0 0
\(214\) 8.82329i 0.603147i
\(215\) 0.320890i 0.0218845i
\(216\) 0 0
\(217\) −3.17378 −0.215450
\(218\) 6.64007 0.449722
\(219\) 0 0
\(220\) −0.00241041 −0.000162510
\(221\) 13.2554 2.77359i 0.891652 0.186572i
\(222\) 0 0
\(223\) 29.6492i 1.98546i 0.120372 + 0.992729i \(0.461591\pi\)
−0.120372 + 0.992729i \(0.538409\pi\)
\(224\) −0.383543 −0.0256265
\(225\) 0 0
\(226\) 12.8416i 0.854208i
\(227\) 2.17847i 0.144590i 0.997383 + 0.0722951i \(0.0230323\pi\)
−0.997383 + 0.0722951i \(0.976968\pi\)
\(228\) 0 0
\(229\) 2.89071i 0.191024i 0.995428 + 0.0955118i \(0.0304488\pi\)
−0.995428 + 0.0955118i \(0.969551\pi\)
\(230\) −0.0610503 −0.00402554
\(231\) 0 0
\(232\) 0.0971031i 0.00637513i
\(233\) 8.88961 0.582377 0.291189 0.956666i \(-0.405949\pi\)
0.291189 + 0.956666i \(0.405949\pi\)
\(234\) 0 0
\(235\) 0.366747 0.0239239
\(236\) 0.132943i 0.00865382i
\(237\) 0 0
\(238\) 9.15392 0.593361
\(239\) 28.4940i 1.84313i −0.388229 0.921563i \(-0.626913\pi\)
0.388229 0.921563i \(-0.373087\pi\)
\(240\) 0 0
\(241\) 15.6266i 1.00660i 0.864112 + 0.503299i \(0.167881\pi\)
−0.864112 + 0.503299i \(0.832119\pi\)
\(242\) 1.42819i 0.0918078i
\(243\) 0 0
\(244\) −0.129847 −0.00831260
\(245\) 0.247967i 0.0158420i
\(246\) 0 0
\(247\) 1.49607 + 7.14991i 0.0951926 + 0.454938i
\(248\) 5.20695 0.330641
\(249\) 0 0
\(250\) −0.865985 −0.0547697
\(251\) 20.7286 1.30838 0.654189 0.756331i \(-0.273010\pi\)
0.654189 + 0.756331i \(0.273010\pi\)
\(252\) 0 0
\(253\) 0.704722i 0.0443055i
\(254\) 2.49210i 0.156368i
\(255\) 0 0
\(256\) 0.953341 0.0595838
\(257\) −1.38511 −0.0864008 −0.0432004 0.999066i \(-0.513755\pi\)
−0.0432004 + 0.999066i \(0.513755\pi\)
\(258\) 0 0
\(259\) 15.0624 0.935935
\(260\) 0.00177995 + 0.00850662i 0.000110388 + 0.000527558i
\(261\) 0 0
\(262\) 19.8463i 1.22611i
\(263\) 17.4668 1.07705 0.538524 0.842610i \(-0.318982\pi\)
0.538524 + 0.842610i \(0.318982\pi\)
\(264\) 0 0
\(265\) 0.647174i 0.0397556i
\(266\) 4.93760i 0.302744i
\(267\) 0 0
\(268\) 0.435929i 0.0266286i
\(269\) −0.559556 −0.0341167 −0.0170584 0.999854i \(-0.505430\pi\)
−0.0170584 + 0.999854i \(0.505430\pi\)
\(270\) 0 0
\(271\) 13.2324i 0.803810i 0.915681 + 0.401905i \(0.131652\pi\)
−0.915681 + 0.401905i \(0.868348\pi\)
\(272\) −15.3166 −0.928702
\(273\) 0 0
\(274\) 4.41053 0.266450
\(275\) 4.99632i 0.301289i
\(276\) 0 0
\(277\) 18.4921 1.11108 0.555541 0.831489i \(-0.312511\pi\)
0.555541 + 0.831489i \(0.312511\pi\)
\(278\) 2.75480i 0.165222i
\(279\) 0 0
\(280\) 0.289787i 0.0173181i
\(281\) 25.8276i 1.54075i 0.637593 + 0.770373i \(0.279930\pi\)
−0.637593 + 0.770373i \(0.720070\pi\)
\(282\) 0 0
\(283\) −6.19965 −0.368531 −0.184265 0.982877i \(-0.558991\pi\)
−0.184265 + 0.982877i \(0.558991\pi\)
\(284\) 0.0782556i 0.00464362i
\(285\) 0 0
\(286\) 5.04027 1.05464i 0.298037 0.0623623i
\(287\) 5.68182 0.335387
\(288\) 0 0
\(289\) −2.89252 −0.170148
\(290\) −0.00300470 −0.000176442
\(291\) 0 0
\(292\) 0.608221i 0.0355934i
\(293\) 8.49366i 0.496205i −0.968734 0.248102i \(-0.920193\pi\)
0.968734 0.248102i \(-0.0798070\pi\)
\(294\) 0 0
\(295\) 0.202927 0.0118149
\(296\) −24.7116 −1.43633
\(297\) 0 0
\(298\) −12.9279 −0.748893
\(299\) 2.48705 0.520398i 0.143830 0.0300954i
\(300\) 0 0
\(301\) 9.02753i 0.520338i
\(302\) −3.49139 −0.200907
\(303\) 0 0
\(304\) 8.26171i 0.473842i
\(305\) 0.198202i 0.0113490i
\(306\) 0 0
\(307\) 10.6196i 0.606095i 0.952976 + 0.303047i \(0.0980040\pi\)
−0.952976 + 0.303047i \(0.901996\pi\)
\(308\) 0.0678114 0.00386391
\(309\) 0 0
\(310\) 0.161121i 0.00915105i
\(311\) −27.5052 −1.55968 −0.779838 0.625981i \(-0.784699\pi\)
−0.779838 + 0.625981i \(0.784699\pi\)
\(312\) 0 0
\(313\) 0.791246 0.0447239 0.0223619 0.999750i \(-0.492881\pi\)
0.0223619 + 0.999750i \(0.492881\pi\)
\(314\) 12.0367i 0.679271i
\(315\) 0 0
\(316\) −0.420980 −0.0236820
\(317\) 10.4687i 0.587983i 0.955808 + 0.293991i \(0.0949837\pi\)
−0.955808 + 0.293991i \(0.905016\pi\)
\(318\) 0 0
\(319\) 0.0346842i 0.00194194i
\(320\) 0.475237i 0.0265666i
\(321\) 0 0
\(322\) 1.71751 0.0957133
\(323\) 7.60954i 0.423406i
\(324\) 0 0
\(325\) 17.6326 3.68951i 0.978082 0.204657i
\(326\) 24.8462 1.37610
\(327\) 0 0
\(328\) −9.32167 −0.514703
\(329\) −10.3176 −0.568828
\(330\) 0 0
\(331\) 21.1531i 1.16268i 0.813661 + 0.581339i \(0.197471\pi\)
−0.813661 + 0.581339i \(0.802529\pi\)
\(332\) 0.166829i 0.00915591i
\(333\) 0 0
\(334\) 1.50109 0.0821361
\(335\) −0.665414 −0.0363554
\(336\) 0 0
\(337\) 4.33688 0.236245 0.118123 0.992999i \(-0.462312\pi\)
0.118123 + 0.992999i \(0.462312\pi\)
\(338\) −7.44392 17.0089i −0.404896 0.925164i
\(339\) 0 0
\(340\) 0.00905347i 0.000490993i
\(341\) −1.85987 −0.100717
\(342\) 0 0
\(343\) 18.9212i 1.02165i
\(344\) 14.8107i 0.798539i
\(345\) 0 0
\(346\) 26.6578i 1.43313i
\(347\) 10.5906 0.568532 0.284266 0.958745i \(-0.408250\pi\)
0.284266 + 0.958745i \(0.408250\pi\)
\(348\) 0 0
\(349\) 24.3761i 1.30482i −0.757864 0.652412i \(-0.773757\pi\)
0.757864 0.652412i \(-0.226243\pi\)
\(350\) 12.1768 0.650877
\(351\) 0 0
\(352\) −0.224760 −0.0119797
\(353\) 21.4564i 1.14201i 0.820947 + 0.571004i \(0.193446\pi\)
−0.820947 + 0.571004i \(0.806554\pi\)
\(354\) 0 0
\(355\) −0.119451 −0.00633982
\(356\) 0.570810i 0.0302528i
\(357\) 0 0
\(358\) 34.6255i 1.83002i
\(359\) 33.2863i 1.75679i −0.477939 0.878393i \(-0.658616\pi\)
0.477939 0.878393i \(-0.341384\pi\)
\(360\) 0 0
\(361\) 14.8954 0.783970
\(362\) 29.8851i 1.57073i
\(363\) 0 0
\(364\) −0.0500750 0.239315i −0.00262464 0.0125435i
\(365\) −0.928404 −0.0485949
\(366\) 0 0
\(367\) 15.3463 0.801073 0.400536 0.916281i \(-0.368824\pi\)
0.400536 + 0.916281i \(0.368824\pi\)
\(368\) −2.87378 −0.149806
\(369\) 0 0
\(370\) 0.764663i 0.0397529i
\(371\) 18.2068i 0.945250i
\(372\) 0 0
\(373\) −22.1576 −1.14728 −0.573638 0.819109i \(-0.694469\pi\)
−0.573638 + 0.819109i \(0.694469\pi\)
\(374\) 5.36429 0.277381
\(375\) 0 0
\(376\) 16.9272 0.872954
\(377\) 0.122405 0.0256124i 0.00630416 0.00131910i
\(378\) 0 0
\(379\) 21.0884i 1.08324i −0.840624 0.541619i \(-0.817812\pi\)
0.840624 0.541619i \(-0.182188\pi\)
\(380\) −0.00488342 −0.000250514
\(381\) 0 0
\(382\) 3.95764i 0.202490i
\(383\) 6.90976i 0.353072i −0.984294 0.176536i \(-0.943511\pi\)
0.984294 0.176536i \(-0.0564892\pi\)
\(384\) 0 0
\(385\) 0.103509i 0.00527531i
\(386\) 12.1502 0.618431
\(387\) 0 0
\(388\) 0.232956i 0.0118265i
\(389\) −32.7232 −1.65913 −0.829565 0.558410i \(-0.811412\pi\)
−0.829565 + 0.558410i \(0.811412\pi\)
\(390\) 0 0
\(391\) 2.64693 0.133861
\(392\) 11.4449i 0.578056i
\(393\) 0 0
\(394\) 20.6056 1.03810
\(395\) 0.642594i 0.0323324i
\(396\) 0 0
\(397\) 10.1016i 0.506987i 0.967337 + 0.253494i \(0.0815797\pi\)
−0.967337 + 0.253494i \(0.918420\pi\)
\(398\) 22.7585i 1.14078i
\(399\) 0 0
\(400\) −20.3745 −1.01872
\(401\) 18.3377i 0.915739i 0.889019 + 0.457870i \(0.151387\pi\)
−0.889019 + 0.457870i \(0.848613\pi\)
\(402\) 0 0
\(403\) 1.37341 + 6.56369i 0.0684144 + 0.326961i
\(404\) 0.491584 0.0244572
\(405\) 0 0
\(406\) 0.0845306 0.00419518
\(407\) 8.82674 0.437525
\(408\) 0 0
\(409\) 5.60283i 0.277042i −0.990359 0.138521i \(-0.955765\pi\)
0.990359 0.138521i \(-0.0442349\pi\)
\(410\) 0.288445i 0.0142453i
\(411\) 0 0
\(412\) 0.400513 0.0197318
\(413\) −5.70890 −0.280916
\(414\) 0 0
\(415\) −0.254651 −0.0125003
\(416\) 0.165973 + 0.793204i 0.00813748 + 0.0388900i
\(417\) 0 0
\(418\) 2.89348i 0.141525i
\(419\) 24.6413 1.20380 0.601902 0.798570i \(-0.294410\pi\)
0.601902 + 0.798570i \(0.294410\pi\)
\(420\) 0 0
\(421\) 25.4002i 1.23793i 0.785419 + 0.618964i \(0.212447\pi\)
−0.785419 + 0.618964i \(0.787553\pi\)
\(422\) 23.7657i 1.15690i
\(423\) 0 0
\(424\) 29.8703i 1.45063i
\(425\) 18.7661 0.910292
\(426\) 0 0
\(427\) 5.57596i 0.269840i
\(428\) 0.245499 0.0118667
\(429\) 0 0
\(430\) 0.458294 0.0221009
\(431\) 33.6254i 1.61968i −0.586653 0.809838i \(-0.699555\pi\)
0.586653 0.809838i \(-0.300445\pi\)
\(432\) 0 0
\(433\) −9.72637 −0.467420 −0.233710 0.972306i \(-0.575087\pi\)
−0.233710 + 0.972306i \(0.575087\pi\)
\(434\) 4.53278i 0.217580i
\(435\) 0 0
\(436\) 0.184754i 0.00884809i
\(437\) 1.42775i 0.0682984i
\(438\) 0 0
\(439\) −17.0758 −0.814986 −0.407493 0.913208i \(-0.633597\pi\)
−0.407493 + 0.913208i \(0.633597\pi\)
\(440\) 0.169818i 0.00809576i
\(441\) 0 0
\(442\) −3.96123 18.9312i −0.188416 0.900466i
\(443\) −21.8637 −1.03878 −0.519389 0.854538i \(-0.673840\pi\)
−0.519389 + 0.854538i \(0.673840\pi\)
\(444\) 0 0
\(445\) 0.871298 0.0413035
\(446\) 42.3448 2.00509
\(447\) 0 0
\(448\) 13.3697i 0.631661i
\(449\) 8.95803i 0.422755i −0.977404 0.211378i \(-0.932205\pi\)
0.977404 0.211378i \(-0.0677950\pi\)
\(450\) 0 0
\(451\) 3.32960 0.156785
\(452\) 0.357304 0.0168062
\(453\) 0 0
\(454\) 3.11128 0.146020
\(455\) 0.365296 0.0764357i 0.0171253 0.00358336i
\(456\) 0 0
\(457\) 37.7837i 1.76744i −0.468011 0.883722i \(-0.655029\pi\)
0.468011 0.883722i \(-0.344971\pi\)
\(458\) 4.12850 0.192912
\(459\) 0 0
\(460\) 0.00169867i 7.92007e-5i
\(461\) 6.10348i 0.284268i 0.989847 + 0.142134i \(0.0453963\pi\)
−0.989847 + 0.142134i \(0.954604\pi\)
\(462\) 0 0
\(463\) 37.1501i 1.72651i 0.504768 + 0.863255i \(0.331578\pi\)
−0.504768 + 0.863255i \(0.668422\pi\)
\(464\) −0.141439 −0.00656612
\(465\) 0 0
\(466\) 12.6961i 0.588135i
\(467\) −10.3642 −0.479598 −0.239799 0.970823i \(-0.577081\pi\)
−0.239799 + 0.970823i \(0.577081\pi\)
\(468\) 0 0
\(469\) 18.7199 0.864406
\(470\) 0.523786i 0.0241604i
\(471\) 0 0
\(472\) 9.36608 0.431109
\(473\) 5.29022i 0.243245i
\(474\) 0 0
\(475\) 10.1224i 0.464448i
\(476\) 0.254699i 0.0116741i
\(477\) 0 0
\(478\) −40.6950 −1.86135
\(479\) 20.4038i 0.932273i 0.884713 + 0.466136i \(0.154354\pi\)
−0.884713 + 0.466136i \(0.845646\pi\)
\(480\) 0 0
\(481\) −6.51806 31.1506i −0.297198 1.42035i
\(482\) 22.3178 1.01655
\(483\) 0 0
\(484\) 0.0397381 0.00180628
\(485\) 0.355590 0.0161465
\(486\) 0 0
\(487\) 18.1762i 0.823644i 0.911264 + 0.411822i \(0.135107\pi\)
−0.911264 + 0.411822i \(0.864893\pi\)
\(488\) 9.14799i 0.414110i
\(489\) 0 0
\(490\) −0.354145 −0.0159987
\(491\) −19.2029 −0.866613 −0.433307 0.901247i \(-0.642653\pi\)
−0.433307 + 0.901247i \(0.642653\pi\)
\(492\) 0 0
\(493\) 0.130274 0.00586723
\(494\) 10.2115 2.13668i 0.459435 0.0961337i
\(495\) 0 0
\(496\) 7.58434i 0.340547i
\(497\) 3.36050 0.150739
\(498\) 0 0
\(499\) 6.46441i 0.289387i −0.989477 0.144693i \(-0.953780\pi\)
0.989477 0.144693i \(-0.0462196\pi\)
\(500\) 0.0240952i 0.00107757i
\(501\) 0 0
\(502\) 29.6045i 1.32131i
\(503\) 23.6756 1.05564 0.527821 0.849355i \(-0.323009\pi\)
0.527821 + 0.849355i \(0.323009\pi\)
\(504\) 0 0
\(505\) 0.750367i 0.0333909i
\(506\) 1.00648 0.0447435
\(507\) 0 0
\(508\) 0.0693402 0.00307648
\(509\) 29.5632i 1.31037i −0.755470 0.655183i \(-0.772591\pi\)
0.755470 0.655183i \(-0.227409\pi\)
\(510\) 0 0
\(511\) 26.1186 1.15542
\(512\) 21.9167i 0.968590i
\(513\) 0 0
\(514\) 1.97821i 0.0872550i
\(515\) 0.611353i 0.0269394i
\(516\) 0 0
\(517\) −6.04622 −0.265912
\(518\) 21.5121i 0.945187i
\(519\) 0 0
\(520\) −0.599309 + 0.125401i −0.0262814 + 0.00549921i
\(521\) −35.0790 −1.53684 −0.768420 0.639946i \(-0.778957\pi\)
−0.768420 + 0.639946i \(0.778957\pi\)
\(522\) 0 0
\(523\) 12.7792 0.558797 0.279399 0.960175i \(-0.409865\pi\)
0.279399 + 0.960175i \(0.409865\pi\)
\(524\) −0.552205 −0.0241232
\(525\) 0 0
\(526\) 24.9459i 1.08769i
\(527\) 6.98564i 0.304299i
\(528\) 0 0
\(529\) −22.5034 −0.978407
\(530\) −0.924290 −0.0401486
\(531\) 0 0
\(532\) 0.137384 0.00595635
\(533\) −2.45873 11.7506i −0.106499 0.508974i
\(534\) 0 0
\(535\) 0.374737i 0.0162013i
\(536\) −30.7121 −1.32656
\(537\) 0 0
\(538\) 0.799155i 0.0344540i
\(539\) 4.08800i 0.176083i
\(540\) 0 0
\(541\) 16.4144i 0.705711i −0.935678 0.352856i \(-0.885211\pi\)
0.935678 0.352856i \(-0.114789\pi\)
\(542\) 18.8984 0.811756
\(543\) 0 0
\(544\) 0.844196i 0.0361946i
\(545\) 0.282013 0.0120801
\(546\) 0 0
\(547\) −30.0865 −1.28640 −0.643202 0.765696i \(-0.722395\pi\)
−0.643202 + 0.765696i \(0.722395\pi\)
\(548\) 0.122719i 0.00524228i
\(549\) 0 0
\(550\) 7.13572 0.304268
\(551\) 0.0702692i 0.00299357i
\(552\) 0 0
\(553\) 18.0779i 0.768753i
\(554\) 26.4103i 1.12207i
\(555\) 0 0
\(556\) −0.0766497 −0.00325067
\(557\) 9.18200i 0.389054i −0.980897 0.194527i \(-0.937683\pi\)
0.980897 0.194527i \(-0.0623171\pi\)
\(558\) 0 0
\(559\) −18.6698 + 3.90654i −0.789650 + 0.165229i
\(560\) −0.422099 −0.0178369
\(561\) 0 0
\(562\) 36.8868 1.55598
\(563\) −15.8217 −0.666804 −0.333402 0.942785i \(-0.608197\pi\)
−0.333402 + 0.942785i \(0.608197\pi\)
\(564\) 0 0
\(565\) 0.545398i 0.0229451i
\(566\) 8.85430i 0.372174i
\(567\) 0 0
\(568\) −5.51327 −0.231332
\(569\) −35.8645 −1.50352 −0.751760 0.659437i \(-0.770795\pi\)
−0.751760 + 0.659437i \(0.770795\pi\)
\(570\) 0 0
\(571\) −0.130007 −0.00544063 −0.00272032 0.999996i \(-0.500866\pi\)
−0.00272032 + 0.999996i \(0.500866\pi\)
\(572\) −0.0293444 0.140241i −0.00122695 0.00586376i
\(573\) 0 0
\(574\) 8.11474i 0.338703i
\(575\) 3.52101 0.146836
\(576\) 0 0
\(577\) 24.3812i 1.01500i −0.861650 0.507502i \(-0.830569\pi\)
0.861650 0.507502i \(-0.169431\pi\)
\(578\) 4.13108i 0.171830i
\(579\) 0 0
\(580\) 0 8.36030e-5i 0 3.47143e-6i
\(581\) 7.16405 0.297215
\(582\) 0 0
\(583\) 10.6694i 0.441880i
\(584\) −42.8504 −1.77316
\(585\) 0 0
\(586\) −12.1306 −0.501110
\(587\) 6.61208i 0.272910i −0.990646 0.136455i \(-0.956429\pi\)
0.990646 0.136455i \(-0.0435709\pi\)
\(588\) 0 0
\(589\) −3.76804 −0.155259
\(590\) 0.289819i 0.0119317i
\(591\) 0 0
\(592\) 35.9945i 1.47937i
\(593\) 26.4128i 1.08464i −0.840170 0.542322i \(-0.817545\pi\)
0.840170 0.542322i \(-0.182455\pi\)
\(594\) 0 0
\(595\) 0.388779 0.0159384
\(596\) 0.359706i 0.0147342i
\(597\) 0 0
\(598\) −0.743229 3.55199i −0.0303929 0.145252i
\(599\) −21.4702 −0.877248 −0.438624 0.898671i \(-0.644534\pi\)
−0.438624 + 0.898671i \(0.644534\pi\)
\(600\) 0 0
\(601\) −9.57861 −0.390720 −0.195360 0.980732i \(-0.562587\pi\)
−0.195360 + 0.980732i \(0.562587\pi\)
\(602\) −12.8931 −0.525482
\(603\) 0 0
\(604\) 0.0971446i 0.00395276i
\(605\) 0.0606573i 0.00246607i
\(606\) 0 0
\(607\) −28.5549 −1.15901 −0.579504 0.814969i \(-0.696754\pi\)
−0.579504 + 0.814969i \(0.696754\pi\)
\(608\) −0.455357 −0.0184672
\(609\) 0 0
\(610\) −0.283070 −0.0114612
\(611\) 4.46480 + 21.3378i 0.180626 + 0.863236i
\(612\) 0 0
\(613\) 25.5390i 1.03151i −0.856736 0.515755i \(-0.827512\pi\)
0.856736 0.515755i \(-0.172488\pi\)
\(614\) 15.1669 0.612086
\(615\) 0 0
\(616\) 4.77746i 0.192489i
\(617\) 33.4193i 1.34541i −0.739910 0.672706i \(-0.765132\pi\)
0.739910 0.672706i \(-0.234868\pi\)
\(618\) 0 0
\(619\) 34.5887i 1.39024i −0.718895 0.695119i \(-0.755352\pi\)
0.718895 0.695119i \(-0.244648\pi\)
\(620\) −0.00448303 −0.000180043
\(621\) 0 0
\(622\) 39.2827i 1.57509i
\(623\) −24.5120 −0.982054
\(624\) 0 0
\(625\) 24.9448 0.997793
\(626\) 1.13005i 0.0451660i
\(627\) 0 0
\(628\) 0.334910 0.0133644
\(629\) 33.1532i 1.32190i
\(630\) 0 0
\(631\) 12.7021i 0.505662i 0.967510 + 0.252831i \(0.0813617\pi\)
−0.967510 + 0.252831i \(0.918638\pi\)
\(632\) 29.6589i 1.17977i
\(633\) 0 0
\(634\) 14.9514 0.593796
\(635\) 0.105843i 0.00420024i
\(636\) 0 0
\(637\) 14.4271 3.01877i 0.571621 0.119608i
\(638\) 0.0495358 0.00196114
\(639\) 0 0
\(640\) 0.705998 0.0279070
\(641\) −25.9825 −1.02625 −0.513124 0.858314i \(-0.671512\pi\)
−0.513124 + 0.858314i \(0.671512\pi\)
\(642\) 0 0
\(643\) 32.8848i 1.29685i 0.761278 + 0.648425i \(0.224572\pi\)
−0.761278 + 0.648425i \(0.775428\pi\)
\(644\) 0.0477881i 0.00188312i
\(645\) 0 0
\(646\) 10.8679 0.427592
\(647\) 2.75086 0.108148 0.0540738 0.998537i \(-0.482779\pi\)
0.0540738 + 0.998537i \(0.482779\pi\)
\(648\) 0 0
\(649\) −3.34547 −0.131321
\(650\) −5.26933 25.1828i −0.206680 0.987751i
\(651\) 0 0
\(652\) 0.691321i 0.0270742i
\(653\) −27.4851 −1.07557 −0.537787 0.843081i \(-0.680740\pi\)
−0.537787 + 0.843081i \(0.680740\pi\)
\(654\) 0 0
\(655\) 0.842900i 0.0329348i
\(656\) 13.5778i 0.530123i
\(657\) 0 0
\(658\) 14.7355i 0.574451i
\(659\) −17.2631 −0.672474 −0.336237 0.941777i \(-0.609154\pi\)
−0.336237 + 0.941777i \(0.609154\pi\)
\(660\) 0 0
\(661\) 2.44386i 0.0950551i 0.998870 + 0.0475276i \(0.0151342\pi\)
−0.998870 + 0.0475276i \(0.984866\pi\)
\(662\) 30.2107 1.17417
\(663\) 0 0
\(664\) −11.7534 −0.456122
\(665\) 0.209706i 0.00813207i
\(666\) 0 0
\(667\) 0.0244427 0.000946425
\(668\) 0.0417664i 0.00161599i
\(669\) 0 0
\(670\) 0.950340i 0.0367148i
\(671\) 3.26757i 0.126143i
\(672\) 0 0
\(673\) −26.7842 −1.03246 −0.516228 0.856451i \(-0.672664\pi\)
−0.516228 + 0.856451i \(0.672664\pi\)
\(674\) 6.19391i 0.238581i
\(675\) 0 0
\(676\) −0.473257 + 0.207120i −0.0182022 + 0.00796615i
\(677\) 25.7769 0.990687 0.495343 0.868697i \(-0.335042\pi\)
0.495343 + 0.868697i \(0.335042\pi\)
\(678\) 0 0
\(679\) −10.0037 −0.383908
\(680\) −0.637836 −0.0244599
\(681\) 0 0
\(682\) 2.65625i 0.101713i
\(683\) 12.8667i 0.492329i 0.969228 + 0.246165i \(0.0791704\pi\)
−0.969228 + 0.246165i \(0.920830\pi\)
\(684\) 0 0
\(685\) 0.187321 0.00715716
\(686\) 27.0231 1.03175
\(687\) 0 0
\(688\) 21.5730 0.822462
\(689\) 37.6535 7.87873i 1.43448 0.300156i
\(690\) 0 0
\(691\) 33.9707i 1.29230i −0.763208 0.646152i \(-0.776377\pi\)
0.763208 0.646152i \(-0.223623\pi\)
\(692\) −0.741728 −0.0281963
\(693\) 0 0
\(694\) 15.1254i 0.574152i
\(695\) 0.117000i 0.00443806i
\(696\) 0 0
\(697\) 12.5060i 0.473697i
\(698\) −34.8138 −1.31772
\(699\) 0 0
\(700\) 0.338807i 0.0128057i
\(701\) −18.4719 −0.697675 −0.348837 0.937183i \(-0.613423\pi\)
−0.348837 + 0.937183i \(0.613423\pi\)
\(702\) 0 0
\(703\) 17.8827 0.674460
\(704\) 7.83479i 0.295285i
\(705\) 0 0
\(706\) 30.6439 1.15330
\(707\) 21.1099i 0.793919i
\(708\) 0 0
\(709\) 33.8830i 1.27250i −0.771482 0.636251i \(-0.780484\pi\)
0.771482 0.636251i \(-0.219516\pi\)
\(710\) 0.170600i 0.00640249i
\(711\) 0 0
\(712\) 40.2147 1.50711
\(713\) 1.31069i 0.0490856i
\(714\) 0 0
\(715\) 0.214067 0.0447920i 0.00800565 0.00167513i
\(716\) 0.963422 0.0360048
\(717\) 0 0
\(718\) −47.5393 −1.77415
\(719\) 31.1501 1.16170 0.580851 0.814010i \(-0.302720\pi\)
0.580851 + 0.814010i \(0.302720\pi\)
\(720\) 0 0
\(721\) 17.1990i 0.640526i
\(722\) 21.2736i 0.791720i
\(723\) 0 0
\(724\) 0.831524 0.0309034
\(725\) 0.173293 0.00643595
\(726\) 0 0
\(727\) −1.50815 −0.0559341 −0.0279671 0.999609i \(-0.508903\pi\)
−0.0279671 + 0.999609i \(0.508903\pi\)
\(728\) 16.8602 3.52789i 0.624881 0.130752i
\(729\) 0 0
\(730\) 1.32594i 0.0490753i
\(731\) −19.8700 −0.734920
\(732\) 0 0
\(733\) 4.42002i 0.163257i −0.996663 0.0816285i \(-0.973988\pi\)
0.996663 0.0816285i \(-0.0260121\pi\)
\(734\) 21.9176i 0.808992i
\(735\) 0 0
\(736\) 0.158393i 0.00583844i
\(737\) 10.9701 0.404087
\(738\) 0 0
\(739\) 13.7640i 0.506317i 0.967425 + 0.253159i \(0.0814695\pi\)
−0.967425 + 0.253159i \(0.918531\pi\)
\(740\) 0.0212760 0.000782122
\(741\) 0 0
\(742\) 26.0028 0.954594
\(743\) 52.1763i 1.91416i −0.289819 0.957081i \(-0.593595\pi\)
0.289819 0.957081i \(-0.406405\pi\)
\(744\) 0 0
\(745\) −0.549065 −0.0201162
\(746\) 31.6453i 1.15862i
\(747\) 0 0
\(748\) 0.149256i 0.00545734i
\(749\) 10.5424i 0.385210i
\(750\) 0 0
\(751\) −13.4236 −0.489836 −0.244918 0.969544i \(-0.578761\pi\)
−0.244918 + 0.969544i \(0.578761\pi\)
\(752\) 24.6559i 0.899106i
\(753\) 0 0
\(754\) −0.0365794 0.174818i −0.00133214 0.00636649i
\(755\) −0.148284 −0.00539661
\(756\) 0 0
\(757\) 12.9431 0.470426 0.235213 0.971944i \(-0.424421\pi\)
0.235213 + 0.971944i \(0.424421\pi\)
\(758\) −30.1183 −1.09395
\(759\) 0 0
\(760\) 0.344047i 0.0124799i
\(761\) 24.5952i 0.891576i 0.895139 + 0.445788i \(0.147077\pi\)
−0.895139 + 0.445788i \(0.852923\pi\)
\(762\) 0 0
\(763\) −7.93379 −0.287223
\(764\) 0.110117 0.00398391
\(765\) 0 0
\(766\) −9.86847 −0.356562
\(767\) 2.47044 + 11.8066i 0.0892025 + 0.426310i
\(768\) 0 0
\(769\) 48.0101i 1.73129i −0.500660 0.865644i \(-0.666909\pi\)
0.500660 0.865644i \(-0.333091\pi\)
\(770\) 0.147831 0.00532746
\(771\) 0 0
\(772\) 0.338069i 0.0121674i
\(773\) 20.9744i 0.754398i 0.926132 + 0.377199i \(0.123113\pi\)
−0.926132 + 0.377199i \(0.876887\pi\)
\(774\) 0 0
\(775\) 9.29249i 0.333796i
\(776\) 16.4122 0.589165
\(777\) 0 0
\(778\) 46.7350i 1.67553i
\(779\) 6.74568 0.241689
\(780\) 0 0
\(781\) 1.96928 0.0704665
\(782\) 3.78033i 0.135184i
\(783\) 0 0
\(784\) −16.6705 −0.595374
\(785\) 0.511215i 0.0182460i
\(786\) 0 0
\(787\) 42.1206i 1.50144i −0.660622 0.750719i \(-0.729707\pi\)
0.660622 0.750719i \(-0.270293\pi\)
\(788\) 0.573332i 0.0204241i
\(789\) 0 0
\(790\) −0.917749 −0.0326520
\(791\) 15.3435i 0.545554i
\(792\) 0 0
\(793\) 11.5316 2.41292i 0.409501 0.0856852i
\(794\) 14.4271 0.511999
\(795\) 0 0
\(796\) 0.633232 0.0224443
\(797\) 24.5853 0.870855 0.435428 0.900224i \(-0.356597\pi\)
0.435428 + 0.900224i \(0.356597\pi\)
\(798\) 0 0
\(799\) 22.7095i 0.803406i
\(800\) 1.12297i 0.0397030i
\(801\) 0 0
\(802\) 26.1897 0.924792
\(803\) 15.3057 0.540127
\(804\) 0 0
\(805\) 0.0729450 0.00257097
\(806\) 9.37423 1.96149i 0.330193 0.0690907i
\(807\) 0 0
\(808\) 34.6332i 1.21839i
\(809\) 23.9067 0.840516 0.420258 0.907405i \(-0.361940\pi\)
0.420258 + 0.907405i \(0.361940\pi\)
\(810\) 0 0
\(811\) 13.7011i 0.481110i −0.970636 0.240555i \(-0.922671\pi\)
0.970636 0.240555i \(-0.0773294\pi\)
\(812\) 0.00235198i 8.25384e-5i
\(813\) 0 0
\(814\) 12.6063i 0.441850i
\(815\) 1.05525 0.0369638
\(816\) 0 0
\(817\) 10.7178i 0.374970i
\(818\) −8.00193 −0.279781
\(819\) 0 0
\(820\) 0.00802569 0.000280269
\(821\) 35.5107i 1.23933i 0.784866 + 0.619666i \(0.212732\pi\)
−0.784866 + 0.619666i \(0.787268\pi\)
\(822\) 0 0
\(823\) 33.6096 1.17156 0.585778 0.810471i \(-0.300789\pi\)
0.585778 + 0.810471i \(0.300789\pi\)
\(824\) 28.2170i 0.982984i
\(825\) 0 0
\(826\) 8.15341i 0.283693i
\(827\) 33.8892i 1.17844i 0.807972 + 0.589221i \(0.200565\pi\)
−0.807972 + 0.589221i \(0.799435\pi\)
\(828\) 0 0
\(829\) −1.03923 −0.0360941 −0.0180470 0.999837i \(-0.505745\pi\)
−0.0180470 + 0.999837i \(0.505745\pi\)
\(830\) 0.363692i 0.0126239i
\(831\) 0 0
\(832\) −27.6499 + 5.78556i −0.958589 + 0.200578i
\(833\) 15.3545 0.532002
\(834\) 0 0
\(835\) 0.0637533 0.00220627
\(836\) 0.0805084 0.00278444
\(837\) 0 0
\(838\) 35.1925i 1.21570i
\(839\) 53.9422i 1.86229i 0.364646 + 0.931146i \(0.381190\pi\)
−0.364646 + 0.931146i \(0.618810\pi\)
\(840\) 0 0
\(841\) −28.9988 −0.999959
\(842\) 36.2764 1.25017
\(843\) 0 0
\(844\) 0.661258 0.0227615
\(845\) −0.316153 0.722392i −0.0108760 0.0248510i
\(846\) 0 0
\(847\) 1.70646i 0.0586346i
\(848\) −43.5085 −1.49409
\(849\) 0 0
\(850\) 26.8017i 0.919291i
\(851\) 6.22039i 0.213232i
\(852\) 0 0
\(853\) 20.4632i 0.700647i −0.936629 0.350324i \(-0.886072\pi\)
0.936629 0.350324i \(-0.113928\pi\)
\(854\) 7.96355 0.272507
\(855\) 0 0
\(856\) 17.2959i 0.591163i
\(857\) 24.3582 0.832059 0.416029 0.909351i \(-0.363421\pi\)
0.416029 + 0.909351i \(0.363421\pi\)
\(858\) 0 0
\(859\) 14.8900 0.508042 0.254021 0.967199i \(-0.418247\pi\)
0.254021 + 0.967199i \(0.418247\pi\)
\(860\) 0.0127516i 0.000434825i
\(861\) 0 0
\(862\) −48.0235 −1.63569
\(863\) 7.78442i 0.264985i 0.991184 + 0.132492i \(0.0422980\pi\)
−0.991184 + 0.132492i \(0.957702\pi\)
\(864\) 0 0
\(865\) 1.13219i 0.0384957i
\(866\) 13.8911i 0.472040i
\(867\) 0 0
\(868\) 0.126120 0.00428080
\(869\) 10.5938i 0.359372i
\(870\) 0 0
\(871\) −8.10078 38.7147i −0.274484 1.31180i
\(872\) 13.0163 0.440787
\(873\) 0 0
\(874\) 2.03910 0.0689736
\(875\) 1.03471 0.0349796
\(876\) 0 0
\(877\) 26.9967i 0.911614i 0.890079 + 0.455807i \(0.150649\pi\)
−0.890079 + 0.455807i \(0.849351\pi\)
\(878\) 24.3876i 0.823042i
\(879\) 0 0
\(880\) −0.247354 −0.00833831
\(881\) 18.7865 0.632934 0.316467 0.948603i \(-0.397503\pi\)
0.316467 + 0.948603i \(0.397503\pi\)
\(882\) 0 0
\(883\) −34.5728 −1.16347 −0.581734 0.813379i \(-0.697625\pi\)
−0.581734 + 0.813379i \(0.697625\pi\)
\(884\) −0.526743 + 0.110217i −0.0177163 + 0.00370701i
\(885\) 0 0
\(886\) 31.2257i 1.04905i
\(887\) 32.8755 1.10385 0.551926 0.833893i \(-0.313893\pi\)
0.551926 + 0.833893i \(0.313893\pi\)
\(888\) 0 0
\(889\) 2.97765i 0.0998671i
\(890\) 1.24438i 0.0417118i
\(891\) 0 0
\(892\) 1.17820i 0.0394492i
\(893\) −12.2495 −0.409913
\(894\) 0 0
\(895\) 1.47059i 0.0491565i
\(896\) −19.8617 −0.663532
\(897\) 0 0
\(898\) −12.7938 −0.426935
\(899\) 0.0645080i 0.00215146i
\(900\) 0 0
\(901\) 40.0740 1.33506
\(902\) 4.75532i 0.158335i
\(903\) 0 0
\(904\) 25.1728i 0.837235i
\(905\) 1.26926i 0.0421916i
\(906\) 0 0
\(907\) 5.02982 0.167013 0.0835063 0.996507i \(-0.473388\pi\)
0.0835063 + 0.996507i \(0.473388\pi\)
\(908\) 0.0865684i 0.00287287i
\(909\) 0 0
\(910\) −0.109165 0.521713i −0.00361878 0.0172946i
\(911\) −22.8563 −0.757263 −0.378632 0.925547i \(-0.623605\pi\)
−0.378632 + 0.925547i \(0.623605\pi\)
\(912\) 0 0
\(913\) 4.19820 0.138940
\(914\) −53.9624 −1.78492
\(915\) 0 0
\(916\) 0.114871i 0.00379546i
\(917\) 23.7131i 0.783075i
\(918\) 0 0
\(919\) 2.37935 0.0784875 0.0392437 0.999230i \(-0.487505\pi\)
0.0392437 + 0.999230i \(0.487505\pi\)
\(920\) −0.119675 −0.00394555
\(921\) 0 0
\(922\) 8.71696 0.287078
\(923\) −1.45421 6.94984i −0.0478658 0.228757i
\(924\) 0 0
\(925\) 44.1012i 1.45004i
\(926\) 53.0575 1.74358
\(927\) 0 0
\(928\) 0.00779561i 0.000255903i
\(929\) 8.06120i 0.264480i −0.991218 0.132240i \(-0.957783\pi\)
0.991218 0.132240i \(-0.0422169\pi\)
\(930\) 0 0
\(931\) 8.28219i 0.271438i
\(932\) −0.353256 −0.0115713
\(933\) 0 0
\(934\) 14.8021i 0.484339i
\(935\) 0.227828 0.00745078
\(936\) 0 0
\(937\) −24.3236 −0.794618 −0.397309 0.917685i \(-0.630056\pi\)
−0.397309 + 0.917685i \(0.630056\pi\)
\(938\) 26.7357i 0.872951i
\(939\) 0 0
\(940\) −0.0145738 −0.000475346
\(941\) 17.0623i 0.556215i −0.960550 0.278107i \(-0.910293\pi\)
0.960550 0.278107i \(-0.0897071\pi\)
\(942\) 0 0
\(943\) 2.34644i 0.0764107i
\(944\) 13.6425i 0.444025i
\(945\) 0 0
\(946\) −7.55546 −0.245649
\(947\) 57.1623i 1.85753i 0.370673 + 0.928763i \(0.379127\pi\)
−0.370673 + 0.928763i \(0.620873\pi\)
\(948\) 0 0
\(949\) −11.3024 54.0158i −0.366893 1.75343i
\(950\) 14.4568 0.469039
\(951\) 0 0
\(952\) 17.9441 0.581571
\(953\) −2.88446 −0.0934370 −0.0467185 0.998908i \(-0.514876\pi\)
−0.0467185 + 0.998908i \(0.514876\pi\)
\(954\) 0 0
\(955\) 0.168086i 0.00543914i
\(956\) 1.13230i 0.0366212i
\(957\) 0 0
\(958\) 29.1406 0.941489
\(959\) −5.26985 −0.170172
\(960\) 0 0
\(961\) 27.5409 0.888416
\(962\) −44.4891 + 9.30905i −1.43439 + 0.300136i
\(963\) 0 0
\(964\) 0.620972i 0.0200002i
\(965\) 0.516037 0.0166118
\(966\) 0 0
\(967\) 44.8041i 1.44080i −0.693557 0.720402i \(-0.743957\pi\)
0.693557 0.720402i \(-0.256043\pi\)
\(968\) 2.79963i 0.0899837i
\(969\) 0 0
\(970\) 0.507851i 0.0163061i
\(971\) −36.2237 −1.16247 −0.581237 0.813735i \(-0.697431\pi\)
−0.581237 + 0.813735i \(0.697431\pi\)
\(972\) 0 0
\(973\) 3.29153i 0.105522i
\(974\) 25.9592 0.831786
\(975\) 0 0
\(976\) −13.3248 −0.426516
\(977\) 8.12311i 0.259881i −0.991522 0.129941i \(-0.958521\pi\)
0.991522 0.129941i \(-0.0414787\pi\)
\(978\) 0 0
\(979\) −14.3643 −0.459084
\(980\) 0.00985375i 0.000314767i
\(981\) 0 0
\(982\) 27.4254i 0.875180i
\(983\) 16.3355i 0.521020i 0.965471 + 0.260510i \(0.0838908\pi\)
−0.965471 + 0.260510i \(0.916109\pi\)
\(984\) 0 0
\(985\) 0.875149 0.0278846
\(986\) 0.186056i 0.00592523i
\(987\) 0 0
\(988\) −0.0594510 0.284124i −0.00189139 0.00903918i
\(989\) −3.72813 −0.118548
\(990\) 0 0
\(991\) 19.8188 0.629565 0.314783 0.949164i \(-0.398068\pi\)
0.314783 + 0.949164i \(0.398068\pi\)
\(992\) −0.418023 −0.0132722
\(993\) 0 0
\(994\) 4.79944i 0.152229i
\(995\) 0.966582i 0.0306427i
\(996\) 0 0
\(997\) 37.6595 1.19269 0.596344 0.802729i \(-0.296619\pi\)
0.596344 + 0.802729i \(0.296619\pi\)
\(998\) −9.23243 −0.292248
\(999\) 0 0
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 1287.2.b.c.298.4 14
3.2 odd 2 429.2.b.b.298.11 yes 14
13.12 even 2 inner 1287.2.b.c.298.11 14
39.5 even 4 5577.2.a.y.1.5 7
39.8 even 4 5577.2.a.x.1.3 7
39.38 odd 2 429.2.b.b.298.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.b.b.298.4 14 39.38 odd 2
429.2.b.b.298.11 yes 14 3.2 odd 2
1287.2.b.c.298.4 14 1.1 even 1 trivial
1287.2.b.c.298.11 14 13.12 even 2 inner
5577.2.a.x.1.3 7 39.8 even 4
5577.2.a.y.1.5 7 39.5 even 4