Properties

Label 1260.2.f.a.629.7
Level $1260$
Weight $2$
Character 1260.629
Analytic conductor $10.061$
Analytic rank $0$
Dimension $8$
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] = [1260,2,Mod(629,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.629");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.f (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 629.7
Root \(-0.420861 - 1.68014i\) of defining polynomial
Character \(\chi\) \(=\) 1260.629
Dual form 1260.2.f.a.629.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.95522 - 1.08495i) q^{5} +(-2.37608 - 1.16372i) q^{7} +O(q^{10})\) \(q+(1.95522 - 1.08495i) q^{5} +(-2.37608 - 1.16372i) q^{7} -3.74166i q^{11} -0.841723 q^{13} +3.36028i q^{17} -4.55066i q^{19} -7.64575 q^{23} +(2.64575 - 4.24264i) q^{25} -1.41421i q^{29} +0.979531i q^{31} +(-5.90834 + 0.302606i) q^{35} -2.32744i q^{37} -10.3460 q^{41} -10.8127i q^{43} +7.91094i q^{47} +(4.29150 + 5.53019i) q^{49} +4.35425 q^{53} +(-4.05952 - 7.31575i) q^{55} +1.38527 q^{59} +(-1.64575 + 0.913230i) q^{65} -13.1402i q^{67} -3.74166i q^{71} -8.66259 q^{73} +(-4.35425 + 8.89047i) q^{77} +14.5830 q^{79} -3.14944i q^{83} +(3.64575 + 6.57008i) q^{85} -3.91044 q^{89} +(2.00000 + 0.979531i) q^{91} +(-4.93725 - 8.89753i) q^{95} +14.8000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{23} - 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} + 32 q^{79} + 8 q^{85} + 16 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.95522 1.08495i 0.874400 0.485206i
\(6\) 0 0
\(7\) −2.37608 1.16372i −0.898073 0.439846i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.74166i 1.12815i −0.825723 0.564076i \(-0.809232\pi\)
0.825723 0.564076i \(-0.190768\pi\)
\(12\) 0 0
\(13\) −0.841723 −0.233452 −0.116726 0.993164i \(-0.537240\pi\)
−0.116726 + 0.993164i \(0.537240\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.36028i 0.814988i 0.913208 + 0.407494i \(0.133597\pi\)
−0.913208 + 0.407494i \(0.866403\pi\)
\(18\) 0 0
\(19\) 4.55066i 1.04399i −0.852948 0.521996i \(-0.825187\pi\)
0.852948 0.521996i \(-0.174813\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.64575 −1.59425 −0.797125 0.603815i \(-0.793647\pi\)
−0.797125 + 0.603815i \(0.793647\pi\)
\(24\) 0 0
\(25\) 2.64575 4.24264i 0.529150 0.848528i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421i 0.262613i −0.991342 0.131306i \(-0.958083\pi\)
0.991342 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) 0.979531i 0.175929i 0.996124 + 0.0879645i \(0.0280362\pi\)
−0.996124 + 0.0879645i \(0.971964\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.90834 + 0.302606i −0.998691 + 0.0511497i
\(36\) 0 0
\(37\) 2.32744i 0.382629i −0.981529 0.191315i \(-0.938725\pi\)
0.981529 0.191315i \(-0.0612751\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.3460 −1.61578 −0.807890 0.589333i \(-0.799390\pi\)
−0.807890 + 0.589333i \(0.799390\pi\)
\(42\) 0 0
\(43\) 10.8127i 1.64893i −0.565916 0.824463i \(-0.691478\pi\)
0.565916 0.824463i \(-0.308522\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.91094i 1.15393i 0.816769 + 0.576965i \(0.195763\pi\)
−0.816769 + 0.576965i \(0.804237\pi\)
\(48\) 0 0
\(49\) 4.29150 + 5.53019i 0.613072 + 0.790027i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.35425 0.598102 0.299051 0.954237i \(-0.403330\pi\)
0.299051 + 0.954237i \(0.403330\pi\)
\(54\) 0 0
\(55\) −4.05952 7.31575i −0.547386 0.986456i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.38527 0.180346 0.0901732 0.995926i \(-0.471258\pi\)
0.0901732 + 0.995926i \(0.471258\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.64575 + 0.913230i −0.204130 + 0.113272i
\(66\) 0 0
\(67\) 13.1402i 1.60533i −0.596432 0.802664i \(-0.703415\pi\)
0.596432 0.802664i \(-0.296585\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.74166i 0.444053i −0.975041 0.222027i \(-0.928733\pi\)
0.975041 0.222027i \(-0.0712672\pi\)
\(72\) 0 0
\(73\) −8.66259 −1.01388 −0.506940 0.861981i \(-0.669223\pi\)
−0.506940 + 0.861981i \(0.669223\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.35425 + 8.89047i −0.496213 + 1.01316i
\(78\) 0 0
\(79\) 14.5830 1.64072 0.820358 0.571850i \(-0.193774\pi\)
0.820358 + 0.571850i \(0.193774\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.14944i 0.345696i −0.984949 0.172848i \(-0.944703\pi\)
0.984949 0.172848i \(-0.0552969\pi\)
\(84\) 0 0
\(85\) 3.64575 + 6.57008i 0.395437 + 0.712626i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.91044 −0.414505 −0.207253 0.978287i \(-0.566452\pi\)
−0.207253 + 0.978287i \(0.566452\pi\)
\(90\) 0 0
\(91\) 2.00000 + 0.979531i 0.209657 + 0.102683i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.93725 8.89753i −0.506552 0.912867i
\(96\) 0 0
\(97\) 14.8000 1.50271 0.751357 0.659896i \(-0.229400\pi\)
0.751357 + 0.659896i \(0.229400\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3460 1.02947 0.514735 0.857350i \(-0.327890\pi\)
0.514735 + 0.857350i \(0.327890\pi\)
\(102\) 0 0
\(103\) −3.36689 −0.331750 −0.165875 0.986147i \(-0.553045\pi\)
−0.165875 + 0.986147i \(0.553045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.06275 −0.102740 −0.0513698 0.998680i \(-0.516359\pi\)
−0.0513698 + 0.998680i \(0.516359\pi\)
\(108\) 0 0
\(109\) −13.8745 −1.32894 −0.664468 0.747316i \(-0.731342\pi\)
−0.664468 + 0.747316i \(0.731342\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.9373 −1.02889 −0.514445 0.857523i \(-0.672002\pi\)
−0.514445 + 0.857523i \(0.672002\pi\)
\(114\) 0 0
\(115\) −14.9491 + 8.29529i −1.39401 + 0.773539i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.91044 7.98430i 0.358469 0.731919i
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.569951 11.1658i 0.0509780 0.998700i
\(126\) 0 0
\(127\) 4.65489i 0.413054i −0.978441 0.206527i \(-0.933784\pi\)
0.978441 0.206527i \(-0.0662162\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.43560 −0.562281 −0.281141 0.959667i \(-0.590713\pi\)
−0.281141 + 0.959667i \(0.590713\pi\)
\(132\) 0 0
\(133\) −5.29570 + 10.8127i −0.459196 + 0.937582i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.2288 1.21564 0.607822 0.794073i \(-0.292043\pi\)
0.607822 + 0.794073i \(0.292043\pi\)
\(138\) 0 0
\(139\) 21.1412i 1.79318i −0.442866 0.896588i \(-0.646038\pi\)
0.442866 0.896588i \(-0.353962\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.14944i 0.263369i
\(144\) 0 0
\(145\) −1.53436 2.76510i −0.127421 0.229629i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.41618i 0.197941i 0.995090 + 0.0989706i \(0.0315550\pi\)
−0.995090 + 0.0989706i \(0.968445\pi\)
\(150\) 0 0
\(151\) 5.29150 0.430616 0.215308 0.976546i \(-0.430924\pi\)
0.215308 + 0.976546i \(0.430924\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.06275 + 1.91520i 0.0853618 + 0.153832i
\(156\) 0 0
\(157\) 18.4651 1.47367 0.736837 0.676070i \(-0.236318\pi\)
0.736837 + 0.676070i \(0.236318\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.1669 + 8.89753i 1.43175 + 0.701223i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.4821i 0.888509i 0.895901 + 0.444255i \(0.146531\pi\)
−0.895901 + 0.444255i \(0.853469\pi\)
\(168\) 0 0
\(169\) −12.2915 −0.945500
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.3725i 1.54890i −0.632638 0.774448i \(-0.718028\pi\)
0.632638 0.774448i \(-0.281972\pi\)
\(174\) 0 0
\(175\) −11.2238 + 7.00193i −0.848437 + 0.529296i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.0534i 1.05040i 0.850979 + 0.525200i \(0.176010\pi\)
−0.850979 + 0.525200i \(0.823990\pi\)
\(180\) 0 0
\(181\) 18.5496i 1.37878i −0.724389 0.689392i \(-0.757878\pi\)
0.724389 0.689392i \(-0.242122\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.52517 4.55066i −0.185654 0.334571i
\(186\) 0 0
\(187\) 12.5730 0.919431
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.56812i 0.402895i 0.979499 + 0.201447i \(0.0645645\pi\)
−0.979499 + 0.201447i \(0.935435\pi\)
\(192\) 0 0
\(193\) 6.98233i 0.502599i 0.967909 + 0.251300i \(0.0808579\pi\)
−0.967909 + 0.251300i \(0.919142\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.8118 1.48278 0.741388 0.671076i \(-0.234168\pi\)
0.741388 + 0.671076i \(0.234168\pi\)
\(198\) 0 0
\(199\) 26.6714i 1.89069i 0.326076 + 0.945343i \(0.394273\pi\)
−0.326076 + 0.945343i \(0.605727\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.64575 + 3.36028i −0.115509 + 0.235846i
\(204\) 0 0
\(205\) −20.2288 + 11.2250i −1.41284 + 0.783986i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −17.0270 −1.17778
\(210\) 0 0
\(211\) 5.29150 0.364282 0.182141 0.983272i \(-0.441697\pi\)
0.182141 + 0.983272i \(0.441697\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.7313 21.1412i −0.800068 1.44182i
\(216\) 0 0
\(217\) 1.13990 2.32744i 0.0773816 0.157997i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.82843i 0.190261i
\(222\) 0 0
\(223\) 18.7105 1.25294 0.626472 0.779444i \(-0.284498\pi\)
0.626472 + 0.779444i \(0.284498\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.9304i 1.38920i 0.719397 + 0.694599i \(0.244418\pi\)
−0.719397 + 0.694599i \(0.755582\pi\)
\(228\) 0 0
\(229\) 12.6724i 0.837419i 0.908120 + 0.418709i \(0.137517\pi\)
−0.908120 + 0.418709i \(0.862483\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.3542 1.07140 0.535701 0.844408i \(-0.320047\pi\)
0.535701 + 0.844408i \(0.320047\pi\)
\(234\) 0 0
\(235\) 8.58301 + 15.4676i 0.559894 + 1.00900i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.57205i 0.489795i −0.969549 0.244898i \(-0.921246\pi\)
0.969549 0.244898i \(-0.0787544\pi\)
\(240\) 0 0
\(241\) 14.6315i 0.942498i 0.882000 + 0.471249i \(0.156197\pi\)
−0.882000 + 0.471249i \(0.843803\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.3908 + 6.15664i 0.919396 + 0.393334i
\(246\) 0 0
\(247\) 3.83039i 0.243722i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.20614 −0.581086 −0.290543 0.956862i \(-0.593836\pi\)
−0.290543 + 0.956862i \(0.593836\pi\)
\(252\) 0 0
\(253\) 28.6078i 1.79856i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.5220i 1.46726i 0.679549 + 0.733630i \(0.262176\pi\)
−0.679549 + 0.733630i \(0.737824\pi\)
\(258\) 0 0
\(259\) −2.70850 + 5.53019i −0.168298 + 0.343629i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.35425 −0.268494 −0.134247 0.990948i \(-0.542862\pi\)
−0.134247 + 0.990948i \(0.542862\pi\)
\(264\) 0 0
\(265\) 8.51350 4.72416i 0.522980 0.290203i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.3964 0.938734 0.469367 0.883003i \(-0.344482\pi\)
0.469367 + 0.883003i \(0.344482\pi\)
\(270\) 0 0
\(271\) 12.0399i 0.731373i −0.930738 0.365686i \(-0.880834\pi\)
0.930738 0.365686i \(-0.119166\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.8745 9.89949i −0.957269 0.596962i
\(276\) 0 0
\(277\) 25.4558i 1.52949i 0.644331 + 0.764747i \(0.277136\pi\)
−0.644331 + 0.764747i \(0.722864\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.8661i 1.48339i 0.670738 + 0.741694i \(0.265977\pi\)
−0.670738 + 0.741694i \(0.734023\pi\)
\(282\) 0 0
\(283\) −15.3436 −0.912080 −0.456040 0.889959i \(-0.650733\pi\)
−0.456040 + 0.889959i \(0.650733\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.5830 + 12.0399i 1.45109 + 0.710694i
\(288\) 0 0
\(289\) 5.70850 0.335794
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 29.8209i 1.74215i −0.491147 0.871077i \(-0.663422\pi\)
0.491147 0.871077i \(-0.336578\pi\)
\(294\) 0 0
\(295\) 2.70850 1.50295i 0.157695 0.0875051i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.43560 0.372181
\(300\) 0 0
\(301\) −12.5830 + 25.6919i −0.725272 + 1.48086i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.5313 1.51422 0.757111 0.653286i \(-0.226610\pi\)
0.757111 + 0.653286i \(0.226610\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.0270 0.965513 0.482756 0.875755i \(-0.339636\pi\)
0.482756 + 0.875755i \(0.339636\pi\)
\(312\) 0 0
\(313\) 5.59388 0.316185 0.158092 0.987424i \(-0.449466\pi\)
0.158092 + 0.987424i \(0.449466\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.8118 1.84289 0.921446 0.388506i \(-0.127009\pi\)
0.921446 + 0.388506i \(0.127009\pi\)
\(318\) 0 0
\(319\) −5.29150 −0.296267
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 15.2915 0.850842
\(324\) 0 0
\(325\) −2.22699 + 3.57113i −0.123531 + 0.198091i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.20614 18.7970i 0.507551 1.03631i
\(330\) 0 0
\(331\) −19.8745 −1.09240 −0.546201 0.837654i \(-0.683926\pi\)
−0.546201 + 0.837654i \(0.683926\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −14.2565 25.6919i −0.778914 1.40370i
\(336\) 0 0
\(337\) 1.50295i 0.0818709i 0.999162 + 0.0409354i \(0.0130338\pi\)
−0.999162 + 0.0409354i \(0.986966\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.66507 0.198475
\(342\) 0 0
\(343\) −3.76135 18.1343i −0.203094 0.979159i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.0627 −0.701245 −0.350622 0.936517i \(-0.614030\pi\)
−0.350622 + 0.936517i \(0.614030\pi\)
\(348\) 0 0
\(349\) 24.0798i 1.28896i −0.764620 0.644482i \(-0.777073\pi\)
0.764620 0.644482i \(-0.222927\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.3725i 1.08432i −0.840275 0.542161i \(-0.817606\pi\)
0.840275 0.542161i \(-0.182394\pi\)
\(354\) 0 0
\(355\) −4.05952 7.31575i −0.215457 0.388280i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.7142i 1.14603i 0.819545 + 0.573015i \(0.194227\pi\)
−0.819545 + 0.573015i \(0.805773\pi\)
\(360\) 0 0
\(361\) −1.70850 −0.0899209
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.9373 + 9.39851i −0.886536 + 0.491941i
\(366\) 0 0
\(367\) −15.3436 −0.800927 −0.400464 0.916313i \(-0.631151\pi\)
−0.400464 + 0.916313i \(0.631151\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −10.3460 5.06713i −0.537140 0.263073i
\(372\) 0 0
\(373\) 13.9647i 0.723063i 0.932360 + 0.361531i \(0.117746\pi\)
−0.932360 + 0.361531i \(0.882254\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.19038i 0.0613075i
\(378\) 0 0
\(379\) −13.2915 −0.682739 −0.341369 0.939929i \(-0.610891\pi\)
−0.341369 + 0.939929i \(0.610891\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.4821i 0.586706i 0.956004 + 0.293353i \(0.0947712\pi\)
−0.956004 + 0.293353i \(0.905229\pi\)
\(384\) 0 0
\(385\) 1.13225 + 22.1070i 0.0577047 + 1.12668i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.8661i 1.26076i 0.776286 + 0.630381i \(0.217101\pi\)
−0.776286 + 0.630381i \(0.782899\pi\)
\(390\) 0 0
\(391\) 25.6919i 1.29929i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 28.5129 15.8219i 1.43464 0.796085i
\(396\) 0 0
\(397\) 26.2860 1.31925 0.659627 0.751593i \(-0.270714\pi\)
0.659627 + 0.751593i \(0.270714\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.3494i 1.61545i −0.589557 0.807727i \(-0.700698\pi\)
0.589557 0.807727i \(-0.299302\pi\)
\(402\) 0 0
\(403\) 0.824494i 0.0410710i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.70850 −0.431664
\(408\) 0 0
\(409\) 3.57113i 0.176581i −0.996095 0.0882904i \(-0.971860\pi\)
0.996095 0.0882904i \(-0.0281404\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.29150 1.61206i −0.161964 0.0793245i
\(414\) 0 0
\(415\) −3.41699 6.15784i −0.167734 0.302276i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −28.5129 −1.39295 −0.696474 0.717582i \(-0.745249\pi\)
−0.696474 + 0.717582i \(0.745249\pi\)
\(420\) 0 0
\(421\) −7.87451 −0.383780 −0.191890 0.981416i \(-0.561462\pi\)
−0.191890 + 0.981416i \(0.561462\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.2565 + 8.89047i 0.691541 + 0.431251i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0534i 0.676928i 0.940979 + 0.338464i \(0.109907\pi\)
−0.940979 + 0.338464i \(0.890093\pi\)
\(432\) 0 0
\(433\) −4.50679 −0.216583 −0.108291 0.994119i \(-0.534538\pi\)
−0.108291 + 0.994119i \(0.534538\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 34.7932i 1.66438i
\(438\) 0 0
\(439\) 35.7727i 1.70734i −0.520815 0.853670i \(-0.674372\pi\)
0.520815 0.853670i \(-0.325628\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.35425 −0.206877 −0.103438 0.994636i \(-0.532984\pi\)
−0.103438 + 0.994636i \(0.532984\pi\)
\(444\) 0 0
\(445\) −7.64575 + 4.24264i −0.362443 + 0.201120i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.7299i 0.647954i −0.946065 0.323977i \(-0.894980\pi\)
0.946065 0.323977i \(-0.105020\pi\)
\(450\) 0 0
\(451\) 38.7113i 1.82285i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.97318 0.254710i 0.233146 0.0119410i
\(456\) 0 0
\(457\) 32.4382i 1.51739i −0.651444 0.758697i \(-0.725836\pi\)
0.651444 0.758697i \(-0.274164\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14.5018 −0.675418 −0.337709 0.941251i \(-0.609652\pi\)
−0.337709 + 0.941251i \(0.609652\pi\)
\(462\) 0 0
\(463\) 16.9706i 0.788689i 0.918963 + 0.394344i \(0.129028\pi\)
−0.918963 + 0.394344i \(0.870972\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.2098i 0.657552i −0.944408 0.328776i \(-0.893364\pi\)
0.944408 0.328776i \(-0.106636\pi\)
\(468\) 0 0
\(469\) −15.2915 + 31.2221i −0.706096 + 1.44170i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −40.4575 −1.86024
\(474\) 0 0
\(475\) −19.3068 12.0399i −0.885857 0.552429i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.0270 0.777984 0.388992 0.921241i \(-0.372823\pi\)
0.388992 + 0.921241i \(0.372823\pi\)
\(480\) 0 0
\(481\) 1.95906i 0.0893256i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.9373 16.0573i 1.31397 0.729126i
\(486\) 0 0
\(487\) 15.4676i 0.700904i −0.936581 0.350452i \(-0.886028\pi\)
0.936581 0.350452i \(-0.113972\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.1916i 1.18201i −0.806668 0.591005i \(-0.798731\pi\)
0.806668 0.591005i \(-0.201269\pi\)
\(492\) 0 0
\(493\) 4.75216 0.214026
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.35425 + 8.89047i −0.195315 + 0.398792i
\(498\) 0 0
\(499\) 29.2915 1.31127 0.655634 0.755079i \(-0.272401\pi\)
0.655634 + 0.755079i \(0.272401\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.33981i 0.193503i 0.995309 + 0.0967514i \(0.0308452\pi\)
−0.995309 + 0.0967514i \(0.969155\pi\)
\(504\) 0 0
\(505\) 20.2288 11.2250i 0.900168 0.499505i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.57551 −0.335778 −0.167889 0.985806i \(-0.553695\pi\)
−0.167889 + 0.985806i \(0.553695\pi\)
\(510\) 0 0
\(511\) 20.5830 + 10.0808i 0.910539 + 0.445951i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.58301 + 3.65292i −0.290082 + 0.160967i
\(516\) 0 0
\(517\) 29.6000 1.30181
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −31.0381 −1.35980 −0.679902 0.733303i \(-0.737978\pi\)
−0.679902 + 0.733303i \(0.737978\pi\)
\(522\) 0 0
\(523\) −3.36689 −0.147224 −0.0736119 0.997287i \(-0.523453\pi\)
−0.0736119 + 0.997287i \(0.523453\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.29150 −0.143380
\(528\) 0 0
\(529\) 35.4575 1.54163
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.70850 0.377207
\(534\) 0 0
\(535\) −2.07790 + 1.15303i −0.0898354 + 0.0498498i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.6921 16.0573i 0.891271 0.691638i
\(540\) 0 0
\(541\) 11.8745 0.510525 0.255262 0.966872i \(-0.417838\pi\)
0.255262 + 0.966872i \(0.417838\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −27.1277 + 15.0532i −1.16202 + 0.644808i
\(546\) 0 0
\(547\) 25.4558i 1.08841i −0.838951 0.544207i \(-0.816831\pi\)
0.838951 0.544207i \(-0.183169\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.43560 −0.274166
\(552\) 0 0
\(553\) −34.6504 16.9706i −1.47348 0.721662i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.5203 0.742357 0.371179 0.928561i \(-0.378954\pi\)
0.371179 + 0.928561i \(0.378954\pi\)
\(558\) 0 0
\(559\) 9.10132i 0.384945i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 44.6632i 1.88233i 0.337947 + 0.941165i \(0.390267\pi\)
−0.337947 + 0.941165i \(0.609733\pi\)
\(564\) 0 0
\(565\) −21.3847 + 11.8664i −0.899662 + 0.499224i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 37.1818i 1.55874i 0.626563 + 0.779371i \(0.284461\pi\)
−0.626563 + 0.779371i \(0.715539\pi\)
\(570\) 0 0
\(571\) 1.41699 0.0592994 0.0296497 0.999560i \(-0.490561\pi\)
0.0296497 + 0.999560i \(0.490561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −20.2288 + 32.4382i −0.843597 + 1.35277i
\(576\) 0 0
\(577\) −20.1485 −0.838794 −0.419397 0.907803i \(-0.637759\pi\)
−0.419397 + 0.907803i \(0.637759\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.66507 + 7.48331i −0.152053 + 0.310460i
\(582\) 0 0
\(583\) 16.2921i 0.674750i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.6387i 0.439106i −0.975601 0.219553i \(-0.929540\pi\)
0.975601 0.219553i \(-0.0704599\pi\)
\(588\) 0 0
\(589\) 4.45751 0.183669
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.1894i 0.623752i −0.950123 0.311876i \(-0.899043\pi\)
0.950123 0.311876i \(-0.100957\pi\)
\(594\) 0 0
\(595\) −1.01684 19.8537i −0.0416864 0.813922i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.1916i 1.07016i −0.844801 0.535080i \(-0.820281\pi\)
0.844801 0.535080i \(-0.179719\pi\)
\(600\) 0 0
\(601\) 38.3643i 1.56491i −0.622705 0.782457i \(-0.713966\pi\)
0.622705 0.782457i \(-0.286034\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.86565 + 3.25486i −0.238473 + 0.132329i
\(606\) 0 0
\(607\) 13.6601 0.554447 0.277223 0.960805i \(-0.410586\pi\)
0.277223 + 0.960805i \(0.410586\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.65882i 0.269387i
\(612\) 0 0
\(613\) 12.3157i 0.497425i −0.968577 0.248713i \(-0.919992\pi\)
0.968577 0.248713i \(-0.0800075\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −29.5203 −1.18844 −0.594220 0.804302i \(-0.702539\pi\)
−0.594220 + 0.804302i \(0.702539\pi\)
\(618\) 0 0
\(619\) 23.1003i 0.928479i 0.885710 + 0.464240i \(0.153672\pi\)
−0.885710 + 0.464240i \(0.846328\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.29150 + 4.55066i 0.372256 + 0.182318i
\(624\) 0 0
\(625\) −11.0000 22.4499i −0.440000 0.897998i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.82087 0.311839
\(630\) 0 0
\(631\) 21.1660 0.842606 0.421303 0.906920i \(-0.361573\pi\)
0.421303 + 0.906920i \(0.361573\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.05034 9.10132i −0.200417 0.361175i
\(636\) 0 0
\(637\) −3.61226 4.65489i −0.143123 0.184433i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.6946i 1.09387i −0.837175 0.546935i \(-0.815795\pi\)
0.837175 0.546935i \(-0.184205\pi\)
\(642\) 0 0
\(643\) 5.83925 0.230277 0.115139 0.993349i \(-0.463269\pi\)
0.115139 + 0.993349i \(0.463269\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.29888i 0.247634i 0.992305 + 0.123817i \(0.0395136\pi\)
−0.992305 + 0.123817i \(0.960486\pi\)
\(648\) 0 0
\(649\) 5.18319i 0.203458i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.1033 −1.41283 −0.706415 0.707798i \(-0.749689\pi\)
−0.706415 + 0.707798i \(0.749689\pi\)
\(654\) 0 0
\(655\) −12.5830 + 6.98233i −0.491659 + 0.272822i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 38.5073i 1.50003i −0.661421 0.750015i \(-0.730046\pi\)
0.661421 0.750015i \(-0.269954\pi\)
\(660\) 0 0
\(661\) 11.0604i 0.430199i −0.976592 0.215099i \(-0.930992\pi\)
0.976592 0.215099i \(-0.0690076\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.37706 + 26.8868i 0.0534000 + 1.04263i
\(666\) 0 0
\(667\) 10.8127i 0.418670i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 45.5783i 1.75692i −0.477820 0.878458i \(-0.658573\pi\)
0.477820 0.878458i \(-0.341427\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.27841i 0.279732i 0.990170 + 0.139866i \(0.0446672\pi\)
−0.990170 + 0.139866i \(0.955333\pi\)
\(678\) 0 0
\(679\) −35.1660 17.2231i −1.34955 0.660962i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.5203 0.670394 0.335197 0.942148i \(-0.391197\pi\)
0.335197 + 0.942148i \(0.391197\pi\)
\(684\) 0 0
\(685\) 27.8203 15.4375i 1.06296 0.589838i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.66507 −0.139628
\(690\) 0 0
\(691\) 10.0808i 0.383494i −0.981444 0.191747i \(-0.938585\pi\)
0.981444 0.191747i \(-0.0614152\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22.9373 41.3357i −0.870060 1.56795i
\(696\) 0 0
\(697\) 34.7656i 1.31684i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.6965i 1.08385i 0.840426 + 0.541926i \(0.182305\pi\)
−0.840426 + 0.541926i \(0.817695\pi\)
\(702\) 0 0
\(703\) −10.5914 −0.399462
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −24.5830 12.0399i −0.924539 0.452807i
\(708\) 0 0
\(709\) 6.83399 0.256656 0.128328 0.991732i \(-0.459039\pi\)
0.128328 + 0.991732i \(0.459039\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.48925i 0.280475i
\(714\) 0 0
\(715\) 3.41699 + 6.15784i 0.127788 + 0.230290i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −48.3105 −1.80168 −0.900839 0.434154i \(-0.857047\pi\)
−0.900839 + 0.434154i \(0.857047\pi\)
\(720\) 0 0
\(721\) 8.00000 + 3.91813i 0.297936 + 0.145919i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 3.74166i −0.222834 0.138962i
\(726\) 0 0
\(727\) −34.6504 −1.28511 −0.642556 0.766239i \(-0.722126\pi\)
−0.642556 + 0.766239i \(0.722126\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 36.3338 1.34385
\(732\) 0 0
\(733\) −32.1252 −1.18657 −0.593286 0.804992i \(-0.702170\pi\)
−0.593286 + 0.804992i \(0.702170\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −49.1660 −1.81105
\(738\) 0 0
\(739\) 33.1660 1.22003 0.610016 0.792389i \(-0.291163\pi\)
0.610016 + 0.792389i \(0.291163\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.0627 −0.479226 −0.239613 0.970869i \(-0.577021\pi\)
−0.239613 + 0.970869i \(0.577021\pi\)
\(744\) 0 0
\(745\) 2.62144 + 4.72416i 0.0960423 + 0.173080i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.52517 + 1.23674i 0.0922677 + 0.0451895i
\(750\) 0 0
\(751\) −18.7085 −0.682683 −0.341341 0.939939i \(-0.610881\pi\)
−0.341341 + 0.939939i \(0.610881\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.3460 5.74103i 0.376531 0.208938i
\(756\) 0 0
\(757\) 6.15784i 0.223810i 0.993719 + 0.111905i \(0.0356953\pi\)
−0.993719 + 0.111905i \(0.964305\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.45150 0.342617 0.171308 0.985217i \(-0.445201\pi\)
0.171308 + 0.985217i \(0.445201\pi\)
\(762\) 0 0
\(763\) 32.9669 + 16.1461i 1.19348 + 0.584527i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.16601 −0.0421022
\(768\) 0 0
\(769\) 38.3643i 1.38345i −0.722159 0.691727i \(-0.756850\pi\)
0.722159 0.691727i \(-0.243150\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 44.4524i 1.59884i −0.600772 0.799420i \(-0.705140\pi\)
0.600772 0.799420i \(-0.294860\pi\)
\(774\) 0 0
\(775\) 4.15580 + 2.59160i 0.149281 + 0.0930929i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 47.0813i 1.68686i
\(780\) 0 0
\(781\) −14.0000 −0.500959
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 36.1033 20.0338i 1.28858 0.715036i
\(786\) 0 0
\(787\) −29.6000 −1.05513 −0.527564 0.849516i \(-0.676894\pi\)
−0.527564 + 0.849516i \(0.676894\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 25.9878 + 12.7279i 0.924019 + 0.452553i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.0327i 0.567908i 0.958838 + 0.283954i \(0.0916464\pi\)
−0.958838 + 0.283954i \(0.908354\pi\)
\(798\) 0 0
\(799\) −26.5830 −0.940439
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 32.4125i 1.14381i
\(804\) 0 0
\(805\) 45.1737 2.31365i 1.59216 0.0815454i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.6887i 0.868007i −0.900911 0.434003i \(-0.857101\pi\)
0.900911 0.434003i \(-0.142899\pi\)
\(810\) 0 0
\(811\) 12.0399i 0.422778i −0.977402 0.211389i \(-0.932201\pi\)
0.977402 0.211389i \(-0.0677988\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −49.2050 −1.72147
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.89949i 0.345495i −0.984966 0.172747i \(-0.944736\pi\)
0.984966 0.172747i \(-0.0552644\pi\)
\(822\) 0 0
\(823\) 12.3157i 0.429297i −0.976691 0.214649i \(-0.931139\pi\)
0.976691 0.214649i \(-0.0688607\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.0627 1.28880 0.644399 0.764689i \(-0.277108\pi\)
0.644399 + 0.764689i \(0.277108\pi\)
\(828\) 0 0
\(829\) 27.6510i 0.960357i −0.877171 0.480179i \(-0.840572\pi\)
0.877171 0.480179i \(-0.159428\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.5830 + 14.4207i −0.643863 + 0.499646i
\(834\) 0 0
\(835\) 12.4575 + 22.4499i 0.431110 + 0.776912i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.82087 −0.270006 −0.135003 0.990845i \(-0.543104\pi\)
−0.135003 + 0.990845i \(0.543104\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −24.0326 + 13.3357i −0.826745 + 0.458762i
\(846\) 0 0
\(847\) 7.12824 + 3.49117i 0.244929 + 0.119958i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 17.7951i 0.610007i
\(852\) 0 0
\(853\) 13.4148 0.459312 0.229656 0.973272i \(-0.426240\pi\)
0.229656 + 0.973272i \(0.426240\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.2907i 0.829753i −0.909878 0.414877i \(-0.863825\pi\)
0.909878 0.414877i \(-0.136175\pi\)
\(858\) 0 0
\(859\) 31.8546i 1.08687i 0.839453 + 0.543433i \(0.182876\pi\)
−0.839453 + 0.543433i \(0.817124\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30.6863 1.04457 0.522286 0.852770i \(-0.325079\pi\)
0.522286 + 0.852770i \(0.325079\pi\)
\(864\) 0 0
\(865\) −22.1033 39.8328i −0.751534 1.35435i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 54.5646i 1.85098i
\(870\) 0 0
\(871\) 11.0604i 0.374767i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −14.3481 + 25.8676i −0.485056 + 0.874483i
\(876\) 0 0
\(877\) 16.9706i 0.573055i 0.958072 + 0.286528i \(0.0925010\pi\)
−0.958072 + 0.286528i \(0.907499\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27.3730 0.922221 0.461111 0.887343i \(-0.347451\pi\)
0.461111 + 0.887343i \(0.347451\pi\)
\(882\) 0 0
\(883\) 23.9529i 0.806079i −0.915183 0.403040i \(-0.867954\pi\)
0.915183 0.403040i \(-0.132046\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.4125i 1.08830i −0.838987 0.544152i \(-0.816852\pi\)
0.838987 0.544152i \(-0.183148\pi\)
\(888\) 0 0
\(889\) −5.41699 + 11.0604i −0.181680 + 0.370953i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 36.0000 1.20469
\(894\) 0 0
\(895\) 15.2473 + 27.4775i 0.509661 + 0.918470i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.38527 0.0462012
\(900\) 0 0
\(901\) 14.6315i 0.487446i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.1255 36.2686i −0.668994 1.20561i
\(906\) 0 0
\(907\) 37.9176i 1.25903i 0.776988 + 0.629516i \(0.216747\pi\)
−0.776988 + 0.629516i \(0.783253\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 32.6729i 1.08250i 0.840861 + 0.541252i \(0.182049\pi\)
−0.840861 + 0.541252i \(0.817951\pi\)
\(912\) 0 0
\(913\) −11.7841 −0.389997
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.2915 + 7.48925i 0.504970 + 0.247317i
\(918\) 0 0
\(919\) 16.1255 0.531931 0.265965 0.963983i \(-0.414309\pi\)
0.265965 + 0.963983i \(0.414309\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.14944i 0.103665i
\(924\) 0 0
\(925\) −9.87451 6.15784i −0.324672 0.202468i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48.0651 1.57697 0.788483 0.615057i \(-0.210867\pi\)
0.788483 + 0.615057i \(0.210867\pi\)
\(930\) 0 0
\(931\) 25.1660 19.5292i 0.824783 0.640043i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.5830 13.6412i 0.803950 0.446113i
\(936\) 0 0
\(937\) −27.9694 −0.913721 −0.456860 0.889538i \(-0.651026\pi\)
−0.456860 + 0.889538i \(0.651026\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.7313 −0.382430 −0.191215 0.981548i \(-0.561243\pi\)
−0.191215 + 0.981548i \(0.561243\pi\)
\(942\) 0 0
\(943\) 79.1032 2.57596
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.0627 0.424482 0.212241 0.977217i \(-0.431924\pi\)
0.212241 + 0.977217i \(0.431924\pi\)
\(948\) 0 0
\(949\) 7.29150 0.236692
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −44.8118 −1.45160 −0.725798 0.687908i \(-0.758529\pi\)
−0.725798 + 0.687908i \(0.758529\pi\)
\(954\) 0 0
\(955\) 6.04115 + 10.8869i 0.195487 + 0.352291i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −33.8086 16.5583i −1.09174 0.534696i
\(960\) 0 0
\(961\) 30.0405 0.969049
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7.57551 + 13.6520i 0.243864 + 0.439473i
\(966\) 0 0
\(967\) 39.4205i 1.26768i 0.773465 + 0.633839i \(0.218522\pi\)
−0.773465 + 0.633839i \(0.781478\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −34.0540 −1.09285 −0.546423 0.837510i \(-0.684011\pi\)
−0.546423 + 0.837510i \(0.684011\pi\)
\(972\) 0 0
\(973\) −24.6025 + 50.2332i −0.788720 + 1.61040i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.6863 −0.597827 −0.298913 0.954280i \(-0.596624\pi\)
−0.298913 + 0.954280i \(0.596624\pi\)
\(978\) 0 0
\(979\) 14.6315i 0.467625i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 41.0921i 1.31063i 0.755354 + 0.655317i \(0.227465\pi\)
−0.755354 + 0.655317i \(0.772535\pi\)
\(984\) 0 0
\(985\) 40.6915 22.5798i 1.29654 0.719452i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 82.6714i 2.62880i
\(990\) 0 0
\(991\) −38.4575 −1.22164 −0.610822 0.791768i \(-0.709161\pi\)
−0.610822 + 0.791768i \(0.709161\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 28.9373 + 52.1484i 0.917373 + 1.65322i
\(996\) 0 0
\(997\) 29.0565 0.920228 0.460114 0.887860i \(-0.347808\pi\)
0.460114 + 0.887860i \(0.347808\pi\)
\(998\) 0 0
\(999\) 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 1260.2.f.a.629.7 yes 8
3.2 odd 2 1260.2.f.b.629.2 yes 8
4.3 odd 2 5040.2.k.f.1889.7 8
5.2 odd 4 6300.2.d.f.3401.9 16
5.3 odd 4 6300.2.d.f.3401.7 16
5.4 even 2 1260.2.f.b.629.8 yes 8
7.6 odd 2 inner 1260.2.f.a.629.2 yes 8
12.11 even 2 5040.2.k.e.1889.2 8
15.2 even 4 6300.2.d.f.3401.10 16
15.8 even 4 6300.2.d.f.3401.8 16
15.14 odd 2 inner 1260.2.f.a.629.1 8
20.19 odd 2 5040.2.k.e.1889.8 8
21.20 even 2 1260.2.f.b.629.7 yes 8
28.27 even 2 5040.2.k.f.1889.2 8
35.13 even 4 6300.2.d.f.3401.5 16
35.27 even 4 6300.2.d.f.3401.11 16
35.34 odd 2 1260.2.f.b.629.1 yes 8
60.59 even 2 5040.2.k.f.1889.1 8
84.83 odd 2 5040.2.k.e.1889.7 8
105.62 odd 4 6300.2.d.f.3401.12 16
105.83 odd 4 6300.2.d.f.3401.6 16
105.104 even 2 inner 1260.2.f.a.629.8 yes 8
140.139 even 2 5040.2.k.e.1889.1 8
420.419 odd 2 5040.2.k.f.1889.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.f.a.629.1 8 15.14 odd 2 inner
1260.2.f.a.629.2 yes 8 7.6 odd 2 inner
1260.2.f.a.629.7 yes 8 1.1 even 1 trivial
1260.2.f.a.629.8 yes 8 105.104 even 2 inner
1260.2.f.b.629.1 yes 8 35.34 odd 2
1260.2.f.b.629.2 yes 8 3.2 odd 2
1260.2.f.b.629.7 yes 8 21.20 even 2
1260.2.f.b.629.8 yes 8 5.4 even 2
5040.2.k.e.1889.1 8 140.139 even 2
5040.2.k.e.1889.2 8 12.11 even 2
5040.2.k.e.1889.7 8 84.83 odd 2
5040.2.k.e.1889.8 8 20.19 odd 2
5040.2.k.f.1889.1 8 60.59 even 2
5040.2.k.f.1889.2 8 28.27 even 2
5040.2.k.f.1889.7 8 4.3 odd 2
5040.2.k.f.1889.8 8 420.419 odd 2
6300.2.d.f.3401.5 16 35.13 even 4
6300.2.d.f.3401.6 16 105.83 odd 4
6300.2.d.f.3401.7 16 5.3 odd 4
6300.2.d.f.3401.8 16 15.8 even 4
6300.2.d.f.3401.9 16 5.2 odd 4
6300.2.d.f.3401.10 16 15.2 even 4
6300.2.d.f.3401.11 16 35.27 even 4
6300.2.d.f.3401.12 16 105.62 odd 4