Properties

Label 3840.2.f.l.769.12
Level $3840$
Weight $2$
Character 3840.769
Analytic conductor $30.663$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(769,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3840.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.f (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: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.180227832610816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.12
Root \(-0.450129 + 1.34067i\) of defining polynomial
Character \(\chi\) \(=\) 3840.769
Dual form 3840.2.f.l.769.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(2.22158 + 0.254102i) q^{5} -2.64265i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(2.22158 + 0.254102i) q^{5} -2.64265i q^{7} -1.00000 q^{9} -1.51363 q^{11} +3.87086i q^{13} +(-0.254102 + 2.22158i) q^{15} +3.31415i q^{17} +7.08582 q^{19} +2.64265 q^{21} +4.82778i q^{23} +(4.87086 + 1.12902i) q^{25} -1.00000i q^{27} -2.18513 q^{29} -7.36266 q^{31} -1.51363i q^{33} +(0.671502 - 5.87086i) q^{35} +7.87086i q^{37} -3.87086 q^{39} -8.72532 q^{41} -1.01641i q^{43} +(-2.22158 - 0.254102i) q^{45} -7.08582i q^{47} +0.0164068 q^{49} -3.31415 q^{51} -4.50820i q^{53} +(-3.36266 - 0.384617i) q^{55} +7.08582i q^{57} +6.79893 q^{59} +3.60104 q^{61} +2.64265i q^{63} +(-0.983593 + 8.59945i) q^{65} -1.01641i q^{67} -4.82778 q^{69} +6.72532 q^{71} +15.5146i q^{73} +(-1.12902 + 4.87086i) q^{75} +4.00000i q^{77} +7.36266 q^{79} +1.00000 q^{81} +7.74173i q^{83} +(-0.842131 + 7.36266i) q^{85} -2.18513i q^{87} +14.7581 q^{89} +10.2293 q^{91} -7.36266i q^{93} +(15.7417 + 1.80052i) q^{95} +11.1444i q^{97} +1.51363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} - 4 q^{25} - 32 q^{31} + 16 q^{39} + 8 q^{41} - 12 q^{49} + 16 q^{55} - 24 q^{65} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89} + 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.22158 + 0.254102i 0.993522 + 0.113638i
\(6\) 0 0
\(7\) 2.64265i 0.998827i −0.866364 0.499414i \(-0.833549\pi\)
0.866364 0.499414i \(-0.166451\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.51363 −0.456377 −0.228189 0.973617i \(-0.573280\pi\)
−0.228189 + 0.973617i \(0.573280\pi\)
\(12\) 0 0
\(13\) 3.87086i 1.07358i 0.843714 + 0.536792i \(0.180364\pi\)
−0.843714 + 0.536792i \(0.819636\pi\)
\(14\) 0 0
\(15\) −0.254102 + 2.22158i −0.0656088 + 0.573610i
\(16\) 0 0
\(17\) 3.31415i 0.803800i 0.915684 + 0.401900i \(0.131650\pi\)
−0.915684 + 0.401900i \(0.868350\pi\)
\(18\) 0 0
\(19\) 7.08582 1.62560 0.812799 0.582545i \(-0.197943\pi\)
0.812799 + 0.582545i \(0.197943\pi\)
\(20\) 0 0
\(21\) 2.64265 0.576673
\(22\) 0 0
\(23\) 4.82778i 1.00666i 0.864094 + 0.503331i \(0.167892\pi\)
−0.864094 + 0.503331i \(0.832108\pi\)
\(24\) 0 0
\(25\) 4.87086 + 1.12902i 0.974173 + 0.225803i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.18513 −0.405769 −0.202885 0.979203i \(-0.565032\pi\)
−0.202885 + 0.979203i \(0.565032\pi\)
\(30\) 0 0
\(31\) −7.36266 −1.32237 −0.661187 0.750222i \(-0.729947\pi\)
−0.661187 + 0.750222i \(0.729947\pi\)
\(32\) 0 0
\(33\) 1.51363i 0.263490i
\(34\) 0 0
\(35\) 0.671502 5.87086i 0.113504 0.992357i
\(36\) 0 0
\(37\) 7.87086i 1.29396i 0.762506 + 0.646981i \(0.223969\pi\)
−0.762506 + 0.646981i \(0.776031\pi\)
\(38\) 0 0
\(39\) −3.87086 −0.619834
\(40\) 0 0
\(41\) −8.72532 −1.36267 −0.681333 0.731973i \(-0.738600\pi\)
−0.681333 + 0.731973i \(0.738600\pi\)
\(42\) 0 0
\(43\) 1.01641i 0.155001i −0.996992 0.0775003i \(-0.975306\pi\)
0.996992 0.0775003i \(-0.0246939\pi\)
\(44\) 0 0
\(45\) −2.22158 0.254102i −0.331174 0.0378792i
\(46\) 0 0
\(47\) 7.08582i 1.03357i −0.856114 0.516786i \(-0.827128\pi\)
0.856114 0.516786i \(-0.172872\pi\)
\(48\) 0 0
\(49\) 0.0164068 0.00234382
\(50\) 0 0
\(51\) −3.31415 −0.464074
\(52\) 0 0
\(53\) 4.50820i 0.619249i −0.950859 0.309625i \(-0.899797\pi\)
0.950859 0.309625i \(-0.100203\pi\)
\(54\) 0 0
\(55\) −3.36266 0.384617i −0.453421 0.0518617i
\(56\) 0 0
\(57\) 7.08582i 0.938539i
\(58\) 0 0
\(59\) 6.79893 0.885145 0.442573 0.896733i \(-0.354066\pi\)
0.442573 + 0.896733i \(0.354066\pi\)
\(60\) 0 0
\(61\) 3.60104 0.461065 0.230533 0.973065i \(-0.425953\pi\)
0.230533 + 0.973065i \(0.425953\pi\)
\(62\) 0 0
\(63\) 2.64265i 0.332942i
\(64\) 0 0
\(65\) −0.983593 + 8.59945i −0.122000 + 1.06663i
\(66\) 0 0
\(67\) 1.01641i 0.124174i −0.998071 0.0620869i \(-0.980224\pi\)
0.998071 0.0620869i \(-0.0197756\pi\)
\(68\) 0 0
\(69\) −4.82778 −0.581197
\(70\) 0 0
\(71\) 6.72532 0.798149 0.399074 0.916919i \(-0.369331\pi\)
0.399074 + 0.916919i \(0.369331\pi\)
\(72\) 0 0
\(73\) 15.5146i 1.81585i 0.419132 + 0.907925i \(0.362334\pi\)
−0.419132 + 0.907925i \(0.637666\pi\)
\(74\) 0 0
\(75\) −1.12902 + 4.87086i −0.130368 + 0.562439i
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 7.36266 0.828364 0.414182 0.910194i \(-0.364068\pi\)
0.414182 + 0.910194i \(0.364068\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.74173i 0.849765i 0.905248 + 0.424883i \(0.139685\pi\)
−0.905248 + 0.424883i \(0.860315\pi\)
\(84\) 0 0
\(85\) −0.842131 + 7.36266i −0.0913420 + 0.798593i
\(86\) 0 0
\(87\) 2.18513i 0.234271i
\(88\) 0 0
\(89\) 14.7581 1.56436 0.782180 0.623053i \(-0.214108\pi\)
0.782180 + 0.623053i \(0.214108\pi\)
\(90\) 0 0
\(91\) 10.2293 1.07233
\(92\) 0 0
\(93\) 7.36266i 0.763472i
\(94\) 0 0
\(95\) 15.7417 + 1.80052i 1.61507 + 0.184729i
\(96\) 0 0
\(97\) 11.1444i 1.13154i 0.824563 + 0.565769i \(0.191421\pi\)
−0.824563 + 0.565769i \(0.808579\pi\)
\(98\) 0 0
\(99\) 1.51363 0.152126
\(100\) 0 0
\(101\) −13.3295 −1.32633 −0.663167 0.748471i \(-0.730788\pi\)
−0.663167 + 0.748471i \(0.730788\pi\)
\(102\) 0 0
\(103\) 0.958386i 0.0944326i −0.998885 0.0472163i \(-0.984965\pi\)
0.998885 0.0472163i \(-0.0150350\pi\)
\(104\) 0 0
\(105\) 5.87086 + 0.671502i 0.572938 + 0.0655318i
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 0.769233 0.0736792 0.0368396 0.999321i \(-0.488271\pi\)
0.0368396 + 0.999321i \(0.488271\pi\)
\(110\) 0 0
\(111\) −7.87086 −0.747069
\(112\) 0 0
\(113\) 14.4585i 1.36014i 0.733146 + 0.680071i \(0.238051\pi\)
−0.733146 + 0.680071i \(0.761949\pi\)
\(114\) 0 0
\(115\) −1.22675 + 10.7253i −0.114395 + 1.00014i
\(116\) 0 0
\(117\) 3.87086i 0.357862i
\(118\) 0 0
\(119\) 8.75814 0.802857
\(120\) 0 0
\(121\) −8.70892 −0.791720
\(122\) 0 0
\(123\) 8.72532i 0.786736i
\(124\) 0 0
\(125\) 10.5341 + 3.74590i 0.942203 + 0.335043i
\(126\) 0 0
\(127\) 11.5290i 1.02303i 0.859274 + 0.511516i \(0.170916\pi\)
−0.859274 + 0.511516i \(0.829084\pi\)
\(128\) 0 0
\(129\) 1.01641 0.0894896
\(130\) 0 0
\(131\) −7.37270 −0.644156 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(132\) 0 0
\(133\) 18.7253i 1.62369i
\(134\) 0 0
\(135\) 0.254102 2.22158i 0.0218696 0.191203i
\(136\) 0 0
\(137\) 3.88792i 0.332167i 0.986112 + 0.166084i \(0.0531122\pi\)
−0.986112 + 0.166084i \(0.946888\pi\)
\(138\) 0 0
\(139\) 14.6291 1.24083 0.620414 0.784275i \(-0.286965\pi\)
0.620414 + 0.784275i \(0.286965\pi\)
\(140\) 0 0
\(141\) 7.08582 0.596733
\(142\) 0 0
\(143\) 5.85907i 0.489960i
\(144\) 0 0
\(145\) −4.85446 0.555246i −0.403141 0.0461107i
\(146\) 0 0
\(147\) 0.0164068i 0.00135321i
\(148\) 0 0
\(149\) 11.0715 0.907010 0.453505 0.891254i \(-0.350173\pi\)
0.453505 + 0.891254i \(0.350173\pi\)
\(150\) 0 0
\(151\) −0.637339 −0.0518659 −0.0259329 0.999664i \(-0.508256\pi\)
−0.0259329 + 0.999664i \(0.508256\pi\)
\(152\) 0 0
\(153\) 3.31415i 0.267933i
\(154\) 0 0
\(155\) −16.3568 1.87086i −1.31381 0.150271i
\(156\) 0 0
\(157\) 0.129135i 0.0103061i 0.999987 + 0.00515306i \(0.00164028\pi\)
−0.999987 + 0.00515306i \(0.998360\pi\)
\(158\) 0 0
\(159\) 4.50820 0.357524
\(160\) 0 0
\(161\) 12.7581 1.00548
\(162\) 0 0
\(163\) 19.4835i 1.52606i −0.646362 0.763031i \(-0.723710\pi\)
0.646362 0.763031i \(-0.276290\pi\)
\(164\) 0 0
\(165\) 0.384617 3.36266i 0.0299424 0.261783i
\(166\) 0 0
\(167\) 1.80052i 0.139328i −0.997571 0.0696641i \(-0.977807\pi\)
0.997571 0.0696641i \(-0.0221928\pi\)
\(168\) 0 0
\(169\) −1.98359 −0.152584
\(170\) 0 0
\(171\) −7.08582 −0.541866
\(172\) 0 0
\(173\) 23.2335i 1.76641i −0.468985 0.883206i \(-0.655380\pi\)
0.468985 0.883206i \(-0.344620\pi\)
\(174\) 0 0
\(175\) 2.98359 12.8720i 0.225538 0.973031i
\(176\) 0 0
\(177\) 6.79893i 0.511039i
\(178\) 0 0
\(179\) −2.85664 −0.213515 −0.106757 0.994285i \(-0.534047\pi\)
−0.106757 + 0.994285i \(0.534047\pi\)
\(180\) 0 0
\(181\) 5.28530 0.392853 0.196427 0.980519i \(-0.437066\pi\)
0.196427 + 0.980519i \(0.437066\pi\)
\(182\) 0 0
\(183\) 3.60104i 0.266196i
\(184\) 0 0
\(185\) −2.00000 + 17.4858i −0.147043 + 1.28558i
\(186\) 0 0
\(187\) 5.01641i 0.366836i
\(188\) 0 0
\(189\) −2.64265 −0.192224
\(190\) 0 0
\(191\) −5.96719 −0.431770 −0.215885 0.976419i \(-0.569264\pi\)
−0.215885 + 0.976419i \(0.569264\pi\)
\(192\) 0 0
\(193\) 14.9409i 1.07547i 0.843115 + 0.537733i \(0.180719\pi\)
−0.843115 + 0.537733i \(0.819281\pi\)
\(194\) 0 0
\(195\) −8.59945 0.983593i −0.615819 0.0704366i
\(196\) 0 0
\(197\) 3.23353i 0.230379i −0.993344 0.115190i \(-0.963252\pi\)
0.993344 0.115190i \(-0.0367476\pi\)
\(198\) 0 0
\(199\) −8.12080 −0.575668 −0.287834 0.957680i \(-0.592935\pi\)
−0.287834 + 0.957680i \(0.592935\pi\)
\(200\) 0 0
\(201\) 1.01641 0.0716918
\(202\) 0 0
\(203\) 5.77454i 0.405293i
\(204\) 0 0
\(205\) −19.3840 2.21712i −1.35384 0.154850i
\(206\) 0 0
\(207\) 4.82778i 0.335554i
\(208\) 0 0
\(209\) −10.7253 −0.741886
\(210\) 0 0
\(211\) 13.7141 0.944119 0.472059 0.881567i \(-0.343511\pi\)
0.472059 + 0.881567i \(0.343511\pi\)
\(212\) 0 0
\(213\) 6.72532i 0.460812i
\(214\) 0 0
\(215\) 0.258271 2.25803i 0.0176139 0.153997i
\(216\) 0 0
\(217\) 19.4569i 1.32082i
\(218\) 0 0
\(219\) −15.5146 −1.04838
\(220\) 0 0
\(221\) −12.8286 −0.862947
\(222\) 0 0
\(223\) 9.84472i 0.659251i 0.944112 + 0.329626i \(0.106923\pi\)
−0.944112 + 0.329626i \(0.893077\pi\)
\(224\) 0 0
\(225\) −4.87086 1.12902i −0.324724 0.0752677i
\(226\) 0 0
\(227\) 5.70892i 0.378914i −0.981889 0.189457i \(-0.939327\pi\)
0.981889 0.189457i \(-0.0606728\pi\)
\(228\) 0 0
\(229\) 0.769233 0.0508324 0.0254162 0.999677i \(-0.491909\pi\)
0.0254162 + 0.999677i \(0.491909\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 18.4008i 1.20548i 0.797939 + 0.602739i \(0.205924\pi\)
−0.797939 + 0.602739i \(0.794076\pi\)
\(234\) 0 0
\(235\) 1.80052 15.7417i 0.117453 1.02688i
\(236\) 0 0
\(237\) 7.36266i 0.478256i
\(238\) 0 0
\(239\) −10.0328 −0.648969 −0.324484 0.945891i \(-0.605191\pi\)
−0.324484 + 0.945891i \(0.605191\pi\)
\(240\) 0 0
\(241\) 10.7581 0.692992 0.346496 0.938051i \(-0.387371\pi\)
0.346496 + 0.938051i \(0.387371\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0.0364490 + 0.00416898i 0.00232864 + 0.000266347i
\(246\) 0 0
\(247\) 27.4282i 1.74522i
\(248\) 0 0
\(249\) −7.74173 −0.490612
\(250\) 0 0
\(251\) −12.6580 −0.798966 −0.399483 0.916741i \(-0.630810\pi\)
−0.399483 + 0.916741i \(0.630810\pi\)
\(252\) 0 0
\(253\) 7.30749i 0.459418i
\(254\) 0 0
\(255\) −7.36266 0.842131i −0.461068 0.0527363i
\(256\) 0 0
\(257\) 13.3110i 0.830316i −0.909749 0.415158i \(-0.863726\pi\)
0.909749 0.415158i \(-0.136274\pi\)
\(258\) 0 0
\(259\) 20.7999 1.29244
\(260\) 0 0
\(261\) 2.18513 0.135256
\(262\) 0 0
\(263\) 18.4256i 1.13617i −0.822969 0.568087i \(-0.807684\pi\)
0.822969 0.568087i \(-0.192316\pi\)
\(264\) 0 0
\(265\) 1.14554 10.0153i 0.0703701 0.615238i
\(266\) 0 0
\(267\) 14.7581i 0.903183i
\(268\) 0 0
\(269\) −3.86940 −0.235921 −0.117961 0.993018i \(-0.537636\pi\)
−0.117961 + 0.993018i \(0.537636\pi\)
\(270\) 0 0
\(271\) −17.3955 −1.05670 −0.528350 0.849027i \(-0.677189\pi\)
−0.528350 + 0.849027i \(0.677189\pi\)
\(272\) 0 0
\(273\) 10.2293i 0.619108i
\(274\) 0 0
\(275\) −7.37270 1.70892i −0.444591 0.103052i
\(276\) 0 0
\(277\) 0.887271i 0.0533110i −0.999645 0.0266555i \(-0.991514\pi\)
0.999645 0.0266555i \(-0.00848571\pi\)
\(278\) 0 0
\(279\) 7.36266 0.440791
\(280\) 0 0
\(281\) 13.4835 0.804356 0.402178 0.915562i \(-0.368253\pi\)
0.402178 + 0.915562i \(0.368253\pi\)
\(282\) 0 0
\(283\) 28.4342i 1.69024i 0.534577 + 0.845120i \(0.320471\pi\)
−0.534577 + 0.845120i \(0.679529\pi\)
\(284\) 0 0
\(285\) −1.80052 + 15.7417i −0.106653 + 0.932460i
\(286\) 0 0
\(287\) 23.0580i 1.36107i
\(288\) 0 0
\(289\) 6.01641 0.353906
\(290\) 0 0
\(291\) −11.1444 −0.653294
\(292\) 0 0
\(293\) 7.99166i 0.466878i 0.972371 + 0.233439i \(0.0749979\pi\)
−0.972371 + 0.233439i \(0.925002\pi\)
\(294\) 0 0
\(295\) 15.1044 + 1.72762i 0.879412 + 0.100586i
\(296\) 0 0
\(297\) 1.51363i 0.0878299i
\(298\) 0 0
\(299\) −18.6877 −1.08074
\(300\) 0 0
\(301\) −2.68601 −0.154819
\(302\) 0 0
\(303\) 13.3295i 0.765760i
\(304\) 0 0
\(305\) 8.00000 + 0.915029i 0.458079 + 0.0523944i
\(306\) 0 0
\(307\) 17.4506i 0.995961i 0.867188 + 0.497980i \(0.165925\pi\)
−0.867188 + 0.497980i \(0.834075\pi\)
\(308\) 0 0
\(309\) 0.958386 0.0545207
\(310\) 0 0
\(311\) −21.4506 −1.21635 −0.608177 0.793801i \(-0.708099\pi\)
−0.608177 + 0.793801i \(0.708099\pi\)
\(312\) 0 0
\(313\) 7.73879i 0.437422i −0.975790 0.218711i \(-0.929815\pi\)
0.975790 0.218711i \(-0.0701853\pi\)
\(314\) 0 0
\(315\) −0.671502 + 5.87086i −0.0378348 + 0.330786i
\(316\) 0 0
\(317\) 11.2335i 0.630938i 0.948936 + 0.315469i \(0.102162\pi\)
−0.948936 + 0.315469i \(0.897838\pi\)
\(318\) 0 0
\(319\) 3.30749 0.185184
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 23.4835i 1.30665i
\(324\) 0 0
\(325\) −4.37027 + 18.8545i −0.242419 + 1.04586i
\(326\) 0 0
\(327\) 0.769233i 0.0425387i
\(328\) 0 0
\(329\) −18.7253 −1.03236
\(330\) 0 0
\(331\) 8.00084 0.439766 0.219883 0.975526i \(-0.429432\pi\)
0.219883 + 0.975526i \(0.429432\pi\)
\(332\) 0 0
\(333\) 7.87086i 0.431321i
\(334\) 0 0
\(335\) 0.258271 2.25803i 0.0141108 0.123369i
\(336\) 0 0
\(337\) 21.5692i 1.17495i 0.809243 + 0.587474i \(0.199877\pi\)
−0.809243 + 0.587474i \(0.800123\pi\)
\(338\) 0 0
\(339\) −14.4585 −0.785279
\(340\) 0 0
\(341\) 11.1444 0.603501
\(342\) 0 0
\(343\) 18.5419i 1.00117i
\(344\) 0 0
\(345\) −10.7253 1.22675i −0.577432 0.0660459i
\(346\) 0 0
\(347\) 21.7089i 1.16540i −0.812689 0.582698i \(-0.801997\pi\)
0.812689 0.582698i \(-0.198003\pi\)
\(348\) 0 0
\(349\) 24.7422 1.32442 0.662211 0.749318i \(-0.269618\pi\)
0.662211 + 0.749318i \(0.269618\pi\)
\(350\) 0 0
\(351\) 3.87086 0.206611
\(352\) 0 0
\(353\) 3.31415i 0.176394i 0.996103 + 0.0881972i \(0.0281106\pi\)
−0.996103 + 0.0881972i \(0.971889\pi\)
\(354\) 0 0
\(355\) 14.9409 + 1.70892i 0.792979 + 0.0906998i
\(356\) 0 0
\(357\) 8.75814i 0.463530i
\(358\) 0 0
\(359\) −16.7581 −0.884461 −0.442230 0.896902i \(-0.645813\pi\)
−0.442230 + 0.896902i \(0.645813\pi\)
\(360\) 0 0
\(361\) 31.2088 1.64257
\(362\) 0 0
\(363\) 8.70892i 0.457100i
\(364\) 0 0
\(365\) −3.94229 + 34.4671i −0.206349 + 1.80409i
\(366\) 0 0
\(367\) 28.5324i 1.48938i −0.667411 0.744690i \(-0.732597\pi\)
0.667411 0.744690i \(-0.267403\pi\)
\(368\) 0 0
\(369\) 8.72532 0.454222
\(370\) 0 0
\(371\) −11.9136 −0.618523
\(372\) 0 0
\(373\) 37.5798i 1.94581i −0.231211 0.972904i \(-0.574269\pi\)
0.231211 0.972904i \(-0.425731\pi\)
\(374\) 0 0
\(375\) −3.74590 + 10.5341i −0.193437 + 0.543981i
\(376\) 0 0
\(377\) 8.45836i 0.435628i
\(378\) 0 0
\(379\) 6.74456 0.346445 0.173222 0.984883i \(-0.444582\pi\)
0.173222 + 0.984883i \(0.444582\pi\)
\(380\) 0 0
\(381\) −11.5290 −0.590648
\(382\) 0 0
\(383\) 21.8312i 1.11552i 0.830001 + 0.557762i \(0.188340\pi\)
−0.830001 + 0.557762i \(0.811660\pi\)
\(384\) 0 0
\(385\) −1.01641 + 8.88633i −0.0518009 + 0.452889i
\(386\) 0 0
\(387\) 1.01641i 0.0516669i
\(388\) 0 0
\(389\) −8.81344 −0.446859 −0.223429 0.974720i \(-0.571725\pi\)
−0.223429 + 0.974720i \(0.571725\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 7.37270i 0.371904i
\(394\) 0 0
\(395\) 16.3568 + 1.87086i 0.822998 + 0.0941334i
\(396\) 0 0
\(397\) 0.821644i 0.0412372i −0.999787 0.0206186i \(-0.993436\pi\)
0.999787 0.0206186i \(-0.00656356\pi\)
\(398\) 0 0
\(399\) 18.7253 0.937439
\(400\) 0 0
\(401\) −12.7253 −0.635472 −0.317736 0.948179i \(-0.602923\pi\)
−0.317736 + 0.948179i \(0.602923\pi\)
\(402\) 0 0
\(403\) 28.4999i 1.41968i
\(404\) 0 0
\(405\) 2.22158 + 0.254102i 0.110391 + 0.0126264i
\(406\) 0 0
\(407\) 11.9136i 0.590535i
\(408\) 0 0
\(409\) 2.25827 0.111664 0.0558321 0.998440i \(-0.482219\pi\)
0.0558321 + 0.998440i \(0.482219\pi\)
\(410\) 0 0
\(411\) −3.88792 −0.191777
\(412\) 0 0
\(413\) 17.9672i 0.884107i
\(414\) 0 0
\(415\) −1.96719 + 17.1989i −0.0965654 + 0.844261i
\(416\) 0 0
\(417\) 14.6291i 0.716392i
\(418\) 0 0
\(419\) −33.4579 −1.63453 −0.817263 0.576264i \(-0.804510\pi\)
−0.817263 + 0.576264i \(0.804510\pi\)
\(420\) 0 0
\(421\) −11.3398 −0.552669 −0.276335 0.961061i \(-0.589120\pi\)
−0.276335 + 0.961061i \(0.589120\pi\)
\(422\) 0 0
\(423\) 7.08582i 0.344524i
\(424\) 0 0
\(425\) −3.74173 + 16.1428i −0.181501 + 0.783040i
\(426\) 0 0
\(427\) 9.51627i 0.460525i
\(428\) 0 0
\(429\) 5.85907 0.282878
\(430\) 0 0
\(431\) 10.6597 0.513459 0.256730 0.966483i \(-0.417355\pi\)
0.256730 + 0.966483i \(0.417355\pi\)
\(432\) 0 0
\(433\) 26.5132i 1.27414i −0.770805 0.637072i \(-0.780146\pi\)
0.770805 0.637072i \(-0.219854\pi\)
\(434\) 0 0
\(435\) 0.555246 4.85446i 0.0266220 0.232753i
\(436\) 0 0
\(437\) 34.2088i 1.63643i
\(438\) 0 0
\(439\) 32.8789 1.56923 0.784613 0.619986i \(-0.212862\pi\)
0.784613 + 0.619986i \(0.212862\pi\)
\(440\) 0 0
\(441\) −0.0164068 −0.000781274
\(442\) 0 0
\(443\) 5.70892i 0.271239i −0.990761 0.135619i \(-0.956698\pi\)
0.990761 0.135619i \(-0.0433024\pi\)
\(444\) 0 0
\(445\) 32.7864 + 3.75007i 1.55423 + 0.177770i
\(446\) 0 0
\(447\) 11.0715i 0.523662i
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 13.2069 0.621890
\(452\) 0 0
\(453\) 0.637339i 0.0299448i
\(454\) 0 0
\(455\) 22.7253 + 2.59929i 1.06538 + 0.121857i
\(456\) 0 0
\(457\) 3.94229i 0.184413i 0.995740 + 0.0922064i \(0.0293920\pi\)
−0.995740 + 0.0922064i \(0.970608\pi\)
\(458\) 0 0
\(459\) 3.31415 0.154691
\(460\) 0 0
\(461\) 33.8969 1.57874 0.789369 0.613920i \(-0.210408\pi\)
0.789369 + 0.613920i \(0.210408\pi\)
\(462\) 0 0
\(463\) 22.8688i 1.06280i −0.847120 0.531402i \(-0.821665\pi\)
0.847120 0.531402i \(-0.178335\pi\)
\(464\) 0 0
\(465\) 1.87086 16.3568i 0.0867593 0.758527i
\(466\) 0 0
\(467\) 15.7417i 0.728440i −0.931313 0.364220i \(-0.881336\pi\)
0.931313 0.364220i \(-0.118664\pi\)
\(468\) 0 0
\(469\) −2.68601 −0.124028
\(470\) 0 0
\(471\) −0.129135 −0.00595024
\(472\) 0 0
\(473\) 1.53847i 0.0707388i
\(474\) 0 0
\(475\) 34.5140 + 8.00000i 1.58361 + 0.367065i
\(476\) 0 0
\(477\) 4.50820i 0.206416i
\(478\) 0 0
\(479\) −20.6925 −0.945465 −0.472732 0.881206i \(-0.656732\pi\)
−0.472732 + 0.881206i \(0.656732\pi\)
\(480\) 0 0
\(481\) −30.4671 −1.38918
\(482\) 0 0
\(483\) 12.7581i 0.580515i
\(484\) 0 0
\(485\) −2.83180 + 24.7581i −0.128586 + 1.12421i
\(486\) 0 0
\(487\) 30.8401i 1.39750i −0.715366 0.698750i \(-0.753740\pi\)
0.715366 0.698750i \(-0.246260\pi\)
\(488\) 0 0
\(489\) 19.4835 0.881072
\(490\) 0 0
\(491\) −10.9737 −0.495238 −0.247619 0.968858i \(-0.579648\pi\)
−0.247619 + 0.968858i \(0.579648\pi\)
\(492\) 0 0
\(493\) 7.24186i 0.326157i
\(494\) 0 0
\(495\) 3.36266 + 0.384617i 0.151140 + 0.0172872i
\(496\) 0 0
\(497\) 17.7727i 0.797213i
\(498\) 0 0
\(499\) 3.71729 0.166409 0.0832044 0.996533i \(-0.473485\pi\)
0.0832044 + 0.996533i \(0.473485\pi\)
\(500\) 0 0
\(501\) 1.80052 0.0804412
\(502\) 0 0
\(503\) 39.9451i 1.78107i 0.454919 + 0.890533i \(0.349668\pi\)
−0.454919 + 0.890533i \(0.650332\pi\)
\(504\) 0 0
\(505\) −29.6126 3.38705i −1.31774 0.150722i
\(506\) 0 0
\(507\) 1.98359i 0.0880945i
\(508\) 0 0
\(509\) −0.0728979 −0.00323114 −0.00161557 0.999999i \(-0.500514\pi\)
−0.00161557 + 0.999999i \(0.500514\pi\)
\(510\) 0 0
\(511\) 40.9997 1.81372
\(512\) 0 0
\(513\) 7.08582i 0.312846i
\(514\) 0 0
\(515\) 0.243528 2.12914i 0.0107311 0.0938209i
\(516\) 0 0
\(517\) 10.7253i 0.471699i
\(518\) 0 0
\(519\) 23.2335 1.01984
\(520\) 0 0
\(521\) 11.9672 0.524292 0.262146 0.965028i \(-0.415570\pi\)
0.262146 + 0.965028i \(0.415570\pi\)
\(522\) 0 0
\(523\) 16.0656i 0.702501i −0.936282 0.351250i \(-0.885757\pi\)
0.936282 0.351250i \(-0.114243\pi\)
\(524\) 0 0
\(525\) 12.8720 + 2.98359i 0.561779 + 0.130215i
\(526\) 0 0
\(527\) 24.4010i 1.06292i
\(528\) 0 0
\(529\) −0.307491 −0.0133692
\(530\) 0 0
\(531\) −6.79893 −0.295048
\(532\) 0 0
\(533\) 33.7745i 1.46294i
\(534\) 0 0
\(535\) −1.01641 + 8.88633i −0.0439431 + 0.384190i
\(536\) 0 0
\(537\) 2.85664i 0.123273i
\(538\) 0 0
\(539\) −0.0248338 −0.00106967
\(540\) 0 0
\(541\) −15.8559 −0.681698 −0.340849 0.940118i \(-0.610715\pi\)
−0.340849 + 0.940118i \(0.610715\pi\)
\(542\) 0 0
\(543\) 5.28530i 0.226814i
\(544\) 0 0
\(545\) 1.70892 + 0.195463i 0.0732019 + 0.00837274i
\(546\) 0 0
\(547\) 4.95078i 0.211680i 0.994383 + 0.105840i \(0.0337531\pi\)
−0.994383 + 0.105840i \(0.966247\pi\)
\(548\) 0 0
\(549\) −3.60104 −0.153688
\(550\) 0 0
\(551\) −15.4835 −0.659618
\(552\) 0 0
\(553\) 19.4569i 0.827393i
\(554\) 0 0
\(555\) −17.4858 2.00000i −0.742230 0.0848953i
\(556\) 0 0
\(557\) 1.26634i 0.0536565i −0.999640 0.0268283i \(-0.991459\pi\)
0.999640 0.0268283i \(-0.00854073\pi\)
\(558\) 0 0
\(559\) 3.93437 0.166406
\(560\) 0 0
\(561\) 5.01641 0.211793
\(562\) 0 0
\(563\) 5.70892i 0.240602i −0.992737 0.120301i \(-0.961614\pi\)
0.992737 0.120301i \(-0.0383860\pi\)
\(564\) 0 0
\(565\) −3.67393 + 32.1208i −0.154564 + 1.35133i
\(566\) 0 0
\(567\) 2.64265i 0.110981i
\(568\) 0 0
\(569\) −2.75814 −0.115627 −0.0578135 0.998327i \(-0.518413\pi\)
−0.0578135 + 0.998327i \(0.518413\pi\)
\(570\) 0 0
\(571\) 25.7735 1.07859 0.539294 0.842118i \(-0.318691\pi\)
0.539294 + 0.842118i \(0.318691\pi\)
\(572\) 0 0
\(573\) 5.96719i 0.249283i
\(574\) 0 0
\(575\) −5.45065 + 23.5155i −0.227308 + 0.980663i
\(576\) 0 0
\(577\) 32.7135i 1.36188i −0.732338 0.680941i \(-0.761571\pi\)
0.732338 0.680941i \(-0.238429\pi\)
\(578\) 0 0
\(579\) −14.9409 −0.620921
\(580\) 0 0
\(581\) 20.4587 0.848769
\(582\) 0 0
\(583\) 6.82376i 0.282611i
\(584\) 0 0
\(585\) 0.983593 8.59945i 0.0406666 0.355543i
\(586\) 0 0
\(587\) 43.4835i 1.79475i 0.441264 + 0.897377i \(0.354530\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(588\) 0 0
\(589\) −52.1705 −2.14965
\(590\) 0 0
\(591\) 3.23353 0.133009
\(592\) 0 0
\(593\) 7.83021i 0.321548i −0.986991 0.160774i \(-0.948601\pi\)
0.986991 0.160774i \(-0.0513991\pi\)
\(594\) 0 0
\(595\) 19.4569 + 2.22546i 0.797656 + 0.0912348i
\(596\) 0 0
\(597\) 8.12080i 0.332362i
\(598\) 0 0
\(599\) −32.7581 −1.33846 −0.669231 0.743055i \(-0.733376\pi\)
−0.669231 + 0.743055i \(0.733376\pi\)
\(600\) 0 0
\(601\) −17.8074 −0.726377 −0.363189 0.931716i \(-0.618312\pi\)
−0.363189 + 0.931716i \(0.618312\pi\)
\(602\) 0 0
\(603\) 1.01641i 0.0413913i
\(604\) 0 0
\(605\) −19.3476 2.21295i −0.786591 0.0899692i
\(606\) 0 0
\(607\) 3.41188i 0.138484i −0.997600 0.0692420i \(-0.977942\pi\)
0.997600 0.0692420i \(-0.0220581\pi\)
\(608\) 0 0
\(609\) −5.77454 −0.233996
\(610\) 0 0
\(611\) 27.4282 1.10963
\(612\) 0 0
\(613\) 36.6290i 1.47943i −0.672920 0.739716i \(-0.734960\pi\)
0.672920 0.739716i \(-0.265040\pi\)
\(614\) 0 0
\(615\) 2.21712 19.3840i 0.0894029 0.781640i
\(616\) 0 0
\(617\) 40.3979i 1.62636i −0.582012 0.813180i \(-0.697734\pi\)
0.582012 0.813180i \(-0.302266\pi\)
\(618\) 0 0
\(619\) 24.5172 0.985430 0.492715 0.870191i \(-0.336004\pi\)
0.492715 + 0.870191i \(0.336004\pi\)
\(620\) 0 0
\(621\) 4.82778 0.193732
\(622\) 0 0
\(623\) 39.0006i 1.56252i
\(624\) 0 0
\(625\) 22.4506 + 10.9986i 0.898026 + 0.439943i
\(626\) 0 0
\(627\) 10.7253i 0.428328i
\(628\) 0 0
\(629\) −26.0852 −1.04009
\(630\) 0 0
\(631\) −18.7805 −0.747640 −0.373820 0.927501i \(-0.621952\pi\)
−0.373820 + 0.927501i \(0.621952\pi\)
\(632\) 0 0
\(633\) 13.7141i 0.545087i
\(634\) 0 0
\(635\) −2.92953 + 25.6126i −0.116255 + 1.01640i
\(636\) 0 0
\(637\) 0.0635083i 0.00251629i
\(638\) 0 0
\(639\) −6.72532 −0.266050
\(640\) 0 0
\(641\) −15.5163 −0.612856 −0.306428 0.951894i \(-0.599134\pi\)
−0.306428 + 0.951894i \(0.599134\pi\)
\(642\) 0 0
\(643\) 17.4506i 0.688186i 0.938936 + 0.344093i \(0.111814\pi\)
−0.938936 + 0.344093i \(0.888186\pi\)
\(644\) 0 0
\(645\) 2.25803 + 0.258271i 0.0889099 + 0.0101694i
\(646\) 0 0
\(647\) 13.1403i 0.516600i 0.966065 + 0.258300i \(0.0831624\pi\)
−0.966065 + 0.258300i \(0.916838\pi\)
\(648\) 0 0
\(649\) −10.2911 −0.403960
\(650\) 0 0
\(651\) −19.4569 −0.762577
\(652\) 0 0
\(653\) 14.7993i 0.579141i −0.957157 0.289570i \(-0.906488\pi\)
0.957157 0.289570i \(-0.0935124\pi\)
\(654\) 0 0
\(655\) −16.3791 1.87342i −0.639983 0.0732004i
\(656\) 0 0
\(657\) 15.5146i 0.605284i
\(658\) 0 0
\(659\) 7.99614 0.311485 0.155743 0.987798i \(-0.450223\pi\)
0.155743 + 0.987798i \(0.450223\pi\)
\(660\) 0 0
\(661\) −0.915029 −0.0355905 −0.0177953 0.999842i \(-0.505665\pi\)
−0.0177953 + 0.999842i \(0.505665\pi\)
\(662\) 0 0
\(663\) 12.8286i 0.498223i
\(664\) 0 0
\(665\) 4.75814 41.5999i 0.184513 1.61317i
\(666\) 0 0
\(667\) 10.5494i 0.408473i
\(668\) 0 0
\(669\) −9.84472 −0.380619
\(670\) 0 0
\(671\) −5.45065 −0.210420
\(672\) 0 0
\(673\) 34.3978i 1.32594i −0.748647 0.662969i \(-0.769296\pi\)
0.748647 0.662969i \(-0.230704\pi\)
\(674\) 0 0
\(675\) 1.12902 4.87086i 0.0434559 0.187480i
\(676\) 0 0
\(677\) 40.1676i 1.54377i −0.635764 0.771884i \(-0.719315\pi\)
0.635764 0.771884i \(-0.280685\pi\)
\(678\) 0 0
\(679\) 29.4506 1.13021
\(680\) 0 0
\(681\) 5.70892 0.218766
\(682\) 0 0
\(683\) 33.2580i 1.27258i −0.771449 0.636291i \(-0.780468\pi\)
0.771449 0.636291i \(-0.219532\pi\)
\(684\) 0 0
\(685\) −0.987927 + 8.63734i −0.0377468 + 0.330016i
\(686\) 0 0
\(687\) 0.769233i 0.0293481i
\(688\) 0 0
\(689\) 17.4506 0.664817
\(690\) 0 0
\(691\) 50.2241 1.91062 0.955308 0.295611i \(-0.0955233\pi\)
0.955308 + 0.295611i \(0.0955233\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 32.4999 + 3.71729i 1.23279 + 0.141005i
\(696\) 0 0
\(697\) 28.9170i 1.09531i
\(698\) 0 0
\(699\) −18.4008 −0.695983
\(700\) 0 0
\(701\) −23.7543 −0.897188 −0.448594 0.893736i \(-0.648075\pi\)
−0.448594 + 0.893736i \(0.648075\pi\)
\(702\) 0 0
\(703\) 55.7715i 2.10346i
\(704\) 0 0
\(705\) 15.7417 + 1.80052i 0.592868 + 0.0678114i
\(706\) 0 0
\(707\) 35.2252i 1.32478i
\(708\) 0 0
\(709\) 36.3146 1.36382 0.681911 0.731435i \(-0.261149\pi\)
0.681911 + 0.731435i \(0.261149\pi\)
\(710\) 0 0
\(711\) −7.36266 −0.276121
\(712\) 0 0
\(713\) 35.5453i 1.33118i
\(714\) 0 0
\(715\) 1.48880 13.0164i 0.0556779 0.486786i
\(716\) 0 0
\(717\) 10.0328i 0.374682i
\(718\) 0 0
\(719\) −30.7253 −1.14586 −0.572931 0.819604i \(-0.694194\pi\)
−0.572931 + 0.819604i \(0.694194\pi\)
\(720\) 0 0
\(721\) −2.53268 −0.0943219
\(722\) 0 0
\(723\) 10.7581i 0.400099i
\(724\) 0 0
\(725\) −10.6435 2.46705i −0.395289 0.0916240i
\(726\) 0 0
\(727\) 5.47445i 0.203036i −0.994834 0.101518i \(-0.967630\pi\)
0.994834 0.101518i \(-0.0323700\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 3.36852 0.124589
\(732\) 0 0
\(733\) 17.1455i 0.633285i −0.948545 0.316643i \(-0.897444\pi\)
0.948545 0.316643i \(-0.102556\pi\)
\(734\) 0 0
\(735\) −0.00416898 + 0.0364490i −0.000153775 + 0.00134444i
\(736\) 0 0
\(737\) 1.53847i 0.0566701i
\(738\) 0 0
\(739\) 11.6019 0.426782 0.213391 0.976967i \(-0.431549\pi\)
0.213391 + 0.976967i \(0.431549\pi\)
\(740\) 0 0
\(741\) −27.4282 −1.00760
\(742\) 0 0
\(743\) 23.6613i 0.868048i −0.900901 0.434024i \(-0.857093\pi\)
0.900901 0.434024i \(-0.142907\pi\)
\(744\) 0 0
\(745\) 24.5962 + 2.81328i 0.901135 + 0.103071i
\(746\) 0 0
\(747\) 7.74173i 0.283255i
\(748\) 0 0
\(749\) 10.5706 0.386241
\(750\) 0 0
\(751\) −11.4283 −0.417024 −0.208512 0.978020i \(-0.566862\pi\)
−0.208512 + 0.978020i \(0.566862\pi\)
\(752\) 0 0
\(753\) 12.6580i 0.461283i
\(754\) 0 0
\(755\) −1.41590 0.161949i −0.0515299 0.00589392i
\(756\) 0 0
\(757\) 19.1784i 0.697049i 0.937300 + 0.348525i \(0.113317\pi\)
−0.937300 + 0.348525i \(0.886683\pi\)
\(758\) 0 0
\(759\) 7.30749 0.265245
\(760\) 0 0
\(761\) −4.03281 −0.146189 −0.0730947 0.997325i \(-0.523288\pi\)
−0.0730947 + 0.997325i \(0.523288\pi\)
\(762\) 0 0
\(763\) 2.03281i 0.0735928i
\(764\) 0 0
\(765\) 0.842131 7.36266i 0.0304473 0.266198i
\(766\) 0 0
\(767\) 26.3177i 0.950279i
\(768\) 0 0
\(769\) 2.95078 0.106408 0.0532039 0.998584i \(-0.483057\pi\)
0.0532039 + 0.998584i \(0.483057\pi\)
\(770\) 0 0
\(771\) 13.3110 0.479383
\(772\) 0 0
\(773\) 45.2663i 1.62812i −0.580783 0.814059i \(-0.697253\pi\)
0.580783 0.814059i \(-0.302747\pi\)
\(774\) 0 0
\(775\) −35.8625 8.31256i −1.28822 0.298596i
\(776\) 0 0
\(777\) 20.7999i 0.746193i
\(778\) 0 0
\(779\) −61.8260 −2.21515
\(780\) 0 0
\(781\) −10.1797 −0.364257
\(782\) 0 0
\(783\) 2.18513i 0.0780903i
\(784\) 0 0
\(785\) −0.0328135 + 0.286885i −0.00117116 + 0.0102394i
\(786\) 0 0
\(787\) 52.9997i 1.88924i −0.328171 0.944618i \(-0.606432\pi\)
0.328171 0.944618i \(-0.393568\pi\)
\(788\) 0 0
\(789\) 18.4256 0.655970
\(790\) 0 0
\(791\) 38.2088 1.35855
\(792\) 0 0
\(793\) 13.9391i 0.494993i
\(794\) 0 0
\(795\) 10.0153 + 1.14554i 0.355208 + 0.0406282i
\(796\) 0 0
\(797\) 16.5738i 0.587075i 0.955948 + 0.293538i \(0.0948326\pi\)
−0.955948 + 0.293538i \(0.905167\pi\)
\(798\) 0 0
\(799\) 23.4835 0.830785
\(800\) 0 0
\(801\) −14.7581 −0.521453
\(802\) 0 0
\(803\) 23.4835i 0.828713i
\(804\) 0 0
\(805\) 28.3433 + 3.24186i 0.998969 + 0.114261i
\(806\) 0 0
\(807\) 3.86940i 0.136209i
\(808\) 0 0
\(809\) −37.5491 −1.32016 −0.660078 0.751197i \(-0.729477\pi\)
−0.660078 + 0.751197i \(0.729477\pi\)
\(810\) 0 0
\(811\) 32.1102 1.12754 0.563771 0.825931i \(-0.309350\pi\)
0.563771 + 0.825931i \(0.309350\pi\)
\(812\) 0 0
\(813\) 17.3955i 0.610086i
\(814\) 0 0
\(815\) 4.95078 43.2841i 0.173418 1.51618i
\(816\) 0 0
\(817\) 7.20207i 0.251969i
\(818\) 0 0
\(819\) −10.2293 −0.357442
\(820\) 0 0
\(821\) 29.3809 1.02540 0.512699 0.858568i \(-0.328646\pi\)
0.512699 + 0.858568i \(0.328646\pi\)
\(822\) 0 0
\(823\) 28.3866i 0.989495i 0.869037 + 0.494748i \(0.164740\pi\)
−0.869037 + 0.494748i \(0.835260\pi\)
\(824\) 0 0
\(825\) 1.70892 7.37270i 0.0594968 0.256684i
\(826\) 0 0
\(827\) 1.45065i 0.0504439i 0.999682 + 0.0252219i \(0.00802924\pi\)
−0.999682 + 0.0252219i \(0.991971\pi\)
\(828\) 0 0
\(829\) 37.4621 1.30111 0.650556 0.759458i \(-0.274536\pi\)
0.650556 + 0.759458i \(0.274536\pi\)
\(830\) 0 0
\(831\) 0.887271 0.0307791
\(832\) 0 0
\(833\) 0.0543744i 0.00188396i
\(834\) 0 0
\(835\) 0.457515 4.00000i 0.0158329 0.138426i
\(836\) 0 0
\(837\) 7.36266i 0.254491i
\(838\) 0 0
\(839\) −48.7581 −1.68332 −0.841659 0.540010i \(-0.818421\pi\)
−0.841659 + 0.540010i \(0.818421\pi\)
\(840\) 0 0
\(841\) −24.2252 −0.835351
\(842\) 0 0
\(843\) 13.4835i 0.464395i
\(844\) 0 0
\(845\) −4.40672 0.504034i −0.151596 0.0173393i
\(846\) 0 0
\(847\) 23.0146i 0.790791i
\(848\) 0 0
\(849\) −28.4342 −0.975861
\(850\) 0 0
\(851\) −37.9988 −1.30258
\(852\) 0 0
\(853\) 4.37073i 0.149651i 0.997197 + 0.0748255i \(0.0238400\pi\)
−0.997197 + 0.0748255i \(0.976160\pi\)
\(854\) 0 0
\(855\) −15.7417 1.80052i −0.538356 0.0615764i
\(856\) 0 0
\(857\) 20.5130i 0.700712i −0.936617 0.350356i \(-0.886061\pi\)
0.936617 0.350356i \(-0.113939\pi\)
\(858\) 0 0
\(859\) 10.1131 0.345054 0.172527 0.985005i \(-0.444807\pi\)
0.172527 + 0.985005i \(0.444807\pi\)
\(860\) 0 0
\(861\) −23.0580 −0.785813
\(862\) 0 0
\(863\) 13.2861i 0.452266i −0.974096 0.226133i \(-0.927392\pi\)
0.974096 0.226133i \(-0.0726083\pi\)
\(864\) 0 0
\(865\) 5.90368 51.6152i 0.200731 1.75497i
\(866\) 0 0
\(867\) 6.01641i 0.204328i
\(868\) 0 0
\(869\) −11.1444 −0.378047
\(870\) 0 0
\(871\) 3.93437 0.133311
\(872\) 0 0
\(873\) 11.1444i 0.377180i
\(874\) 0 0
\(875\) 9.89909 27.8381i 0.334650 0.941098i
\(876\) 0 0
\(877\) 33.6454i 1.13612i 0.822986 + 0.568062i \(0.192307\pi\)
−0.822986 + 0.568062i \(0.807693\pi\)
\(878\) 0 0
\(879\) −7.99166 −0.269552
\(880\) 0 0
\(881\) −32.7909 −1.10476 −0.552378 0.833594i \(-0.686279\pi\)
−0.552378 + 0.833594i \(0.686279\pi\)
\(882\) 0 0
\(883\) 33.4506i 1.12570i 0.826558 + 0.562852i \(0.190296\pi\)
−0.826558 + 0.562852i \(0.809704\pi\)
\(884\) 0 0
\(885\) −1.72762 + 15.1044i −0.0580733 + 0.507729i
\(886\) 0 0
\(887\) 34.8924i 1.17157i 0.810466 + 0.585785i \(0.199214\pi\)
−0.810466 + 0.585785i \(0.800786\pi\)
\(888\) 0 0
\(889\) 30.4671 1.02183
\(890\) 0 0
\(891\) −1.51363 −0.0507086
\(892\) 0 0
\(893\) 50.2088i 1.68017i
\(894\) 0 0
\(895\) −6.34625 0.725876i −0.212132 0.0242634i
\(896\) 0 0
\(897\) 18.6877i 0.623964i
\(898\) 0 0
\(899\) 16.0884 0.536578
\(900\) 0 0
\(901\) 14.9409 0.497752
\(902\) 0 0
\(903\) 2.68601i 0.0893847i
\(904\) 0 0
\(905\) 11.7417 + 1.34300i 0.390308 + 0.0446429i
\(906\) 0 0
\(907\) 30.9836i 1.02879i 0.857552 + 0.514397i \(0.171984\pi\)
−0.857552 + 0.514397i \(0.828016\pi\)
\(908\) 0 0
\(909\) 13.3295 0.442112
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 11.7181i 0.387814i
\(914\) 0 0
\(915\) −0.915029 + 8.00000i −0.0302499 + 0.264472i
\(916\) 0 0
\(917\) 19.4835i 0.643400i
\(918\) 0 0
\(919\) −15.6043 −0.514737 −0.257368 0.966313i \(-0.582855\pi\)
−0.257368 + 0.966313i \(0.582855\pi\)
\(920\) 0 0
\(921\) −17.4506 −0.575018
\(922\) 0 0
\(923\) 26.0328i 0.856880i
\(924\) 0 0
\(925\) −8.88633 + 38.3379i −0.292181 + 1.26054i
\(926\) 0 0
\(927\) 0.958386i 0.0314775i
\(928\) 0 0
\(929\) 38.9341 1.27739 0.638693 0.769461i \(-0.279475\pi\)
0.638693 + 0.769461i \(0.279475\pi\)
\(930\) 0 0
\(931\) 0.116255 0.00381011
\(932\) 0 0
\(933\) 21.4506i 0.702263i
\(934\) 0 0
\(935\) 1.27468 11.1444i 0.0416864 0.364460i
\(936\) 0 0
\(937\) 19.6027i 0.640393i −0.947351 0.320197i \(-0.896251\pi\)
0.947351 0.320197i \(-0.103749\pi\)
\(938\) 0 0
\(939\) 7.73879 0.252546
\(940\) 0 0
\(941\) 25.0476 0.816530 0.408265 0.912864i \(-0.366134\pi\)
0.408265 + 0.912864i \(0.366134\pi\)
\(942\) 0 0
\(943\) 42.1240i 1.37175i
\(944\) 0 0
\(945\) −5.87086 0.671502i −0.190979 0.0218439i
\(946\) 0 0
\(947\) 7.93437i 0.257832i −0.991655 0.128916i \(-0.958850\pi\)
0.991655 0.128916i \(-0.0411498\pi\)
\(948\) 0 0
\(949\) −60.0550 −1.94947
\(950\) 0 0
\(951\) −11.2335 −0.364272
\(952\) 0 0
\(953\) 11.4809i 0.371903i −0.982559 0.185952i \(-0.940463\pi\)
0.982559 0.185952i \(-0.0595368\pi\)
\(954\) 0 0
\(955\) −13.2566 1.51627i −0.428974 0.0490654i
\(956\) 0 0
\(957\) 3.30749i 0.106916i
\(958\) 0 0
\(959\) 10.2744 0.331778
\(960\) 0 0
\(961\) 23.2088 0.748670
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) −3.79650 + 33.1924i −0.122214 + 1.06850i
\(966\) 0 0
\(967\) 15.8993i 0.511285i −0.966771 0.255643i \(-0.917713\pi\)
0.966771 0.255643i \(-0.0822871\pi\)
\(968\) 0 0
\(969\) −23.4835 −0.754397
\(970\) 0 0
\(971\) 40.6600 1.30484 0.652421 0.757857i \(-0.273754\pi\)
0.652421 + 0.757857i \(0.273754\pi\)
\(972\) 0 0
\(973\) 38.6597i 1.23937i
\(974\) 0 0
\(975\) −18.8545 4.37027i −0.603826 0.139961i
\(976\) 0 0
\(977\) 26.5676i 0.849972i −0.905200 0.424986i \(-0.860279\pi\)
0.905200 0.424986i \(-0.139721\pi\)
\(978\) 0 0
\(979\) −22.3384 −0.713938
\(980\) 0 0
\(981\) −0.769233 −0.0245597
\(982\) 0 0
\(983\) 9.88057i 0.315141i 0.987508 + 0.157571i \(0.0503662\pi\)
−0.987508 + 0.157571i \(0.949634\pi\)
\(984\) 0 0
\(985\) 0.821644 7.18355i 0.0261798 0.228887i
\(986\) 0 0
\(987\) 18.7253i 0.596034i
\(988\) 0 0
\(989\) 4.90699 0.156033
\(990\) 0 0
\(991\) −53.0549 −1.68534 −0.842672 0.538427i \(-0.819019\pi\)
−0.842672 + 0.538427i \(0.819019\pi\)
\(992\) 0 0
\(993\) 8.00084i 0.253899i
\(994\) 0 0
\(995\) −18.0410 2.06351i −0.571939 0.0654176i
\(996\) 0 0
\(997\) 32.3051i 1.02311i 0.859250 + 0.511556i \(0.170931\pi\)
−0.859250 + 0.511556i \(0.829069\pi\)
\(998\) 0 0
\(999\) 7.87086 0.249023
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 3840.2.f.l.769.12 12
4.3 odd 2 3840.2.f.m.769.6 12
5.4 even 2 inner 3840.2.f.l.769.6 12
8.3 odd 2 3840.2.f.m.769.7 12
8.5 even 2 inner 3840.2.f.l.769.1 12
16.3 odd 4 480.2.d.b.49.3 6
16.5 even 4 120.2.d.a.109.1 6
16.11 odd 4 480.2.d.a.49.4 6
16.13 even 4 120.2.d.b.109.5 yes 6
20.19 odd 2 3840.2.f.m.769.12 12
40.19 odd 2 3840.2.f.m.769.1 12
40.29 even 2 inner 3840.2.f.l.769.7 12
48.5 odd 4 360.2.d.f.109.6 6
48.11 even 4 1440.2.d.e.1009.3 6
48.29 odd 4 360.2.d.e.109.2 6
48.35 even 4 1440.2.d.f.1009.4 6
80.3 even 4 2400.2.k.f.1201.8 12
80.13 odd 4 600.2.k.f.301.5 12
80.19 odd 4 480.2.d.a.49.3 6
80.27 even 4 2400.2.k.f.1201.11 12
80.29 even 4 120.2.d.a.109.2 yes 6
80.37 odd 4 600.2.k.f.301.7 12
80.43 even 4 2400.2.k.f.1201.2 12
80.53 odd 4 600.2.k.f.301.6 12
80.59 odd 4 480.2.d.b.49.4 6
80.67 even 4 2400.2.k.f.1201.5 12
80.69 even 4 120.2.d.b.109.6 yes 6
80.77 odd 4 600.2.k.f.301.8 12
240.29 odd 4 360.2.d.f.109.5 6
240.53 even 4 1800.2.k.u.901.7 12
240.59 even 4 1440.2.d.f.1009.3 6
240.77 even 4 1800.2.k.u.901.5 12
240.83 odd 4 7200.2.k.u.3601.4 12
240.107 odd 4 7200.2.k.u.3601.9 12
240.149 odd 4 360.2.d.e.109.1 6
240.173 even 4 1800.2.k.u.901.8 12
240.179 even 4 1440.2.d.e.1009.4 6
240.197 even 4 1800.2.k.u.901.6 12
240.203 odd 4 7200.2.k.u.3601.3 12
240.227 odd 4 7200.2.k.u.3601.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.d.a.109.1 6 16.5 even 4
120.2.d.a.109.2 yes 6 80.29 even 4
120.2.d.b.109.5 yes 6 16.13 even 4
120.2.d.b.109.6 yes 6 80.69 even 4
360.2.d.e.109.1 6 240.149 odd 4
360.2.d.e.109.2 6 48.29 odd 4
360.2.d.f.109.5 6 240.29 odd 4
360.2.d.f.109.6 6 48.5 odd 4
480.2.d.a.49.3 6 80.19 odd 4
480.2.d.a.49.4 6 16.11 odd 4
480.2.d.b.49.3 6 16.3 odd 4
480.2.d.b.49.4 6 80.59 odd 4
600.2.k.f.301.5 12 80.13 odd 4
600.2.k.f.301.6 12 80.53 odd 4
600.2.k.f.301.7 12 80.37 odd 4
600.2.k.f.301.8 12 80.77 odd 4
1440.2.d.e.1009.3 6 48.11 even 4
1440.2.d.e.1009.4 6 240.179 even 4
1440.2.d.f.1009.3 6 240.59 even 4
1440.2.d.f.1009.4 6 48.35 even 4
1800.2.k.u.901.5 12 240.77 even 4
1800.2.k.u.901.6 12 240.197 even 4
1800.2.k.u.901.7 12 240.53 even 4
1800.2.k.u.901.8 12 240.173 even 4
2400.2.k.f.1201.2 12 80.43 even 4
2400.2.k.f.1201.5 12 80.67 even 4
2400.2.k.f.1201.8 12 80.3 even 4
2400.2.k.f.1201.11 12 80.27 even 4
3840.2.f.l.769.1 12 8.5 even 2 inner
3840.2.f.l.769.6 12 5.4 even 2 inner
3840.2.f.l.769.7 12 40.29 even 2 inner
3840.2.f.l.769.12 12 1.1 even 1 trivial
3840.2.f.m.769.1 12 40.19 odd 2
3840.2.f.m.769.6 12 4.3 odd 2
3840.2.f.m.769.7 12 8.3 odd 2
3840.2.f.m.769.12 12 20.19 odd 2
7200.2.k.u.3601.3 12 240.203 odd 4
7200.2.k.u.3601.4 12 240.83 odd 4
7200.2.k.u.3601.9 12 240.107 odd 4
7200.2.k.u.3601.10 12 240.227 odd 4