Properties

Label 9216.2.a.d.1.1
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
Dimension $2$
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] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{5} -2.00000 q^{7} +O(q^{10})\) \(q-1.41421 q^{5} -2.00000 q^{7} -1.41421 q^{11} +1.41421 q^{13} -2.00000 q^{17} -4.24264 q^{19} +6.00000 q^{23} -3.00000 q^{25} -4.24264 q^{29} -8.00000 q^{31} +2.82843 q^{35} -4.24264 q^{37} +7.07107 q^{43} +8.00000 q^{47} -3.00000 q^{49} +7.07107 q^{53} +2.00000 q^{55} -4.24264 q^{59} -12.7279 q^{61} -2.00000 q^{65} -7.07107 q^{67} +10.0000 q^{71} +4.00000 q^{73} +2.82843 q^{77} -1.41421 q^{83} +2.82843 q^{85} -4.00000 q^{89} -2.82843 q^{91} +6.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 4 q^{17} + 12 q^{23} - 6 q^{25} - 16 q^{31} + 16 q^{47} - 6 q^{49} + 4 q^{55} - 4 q^{65} + 20 q^{71} + 8 q^{73} - 8 q^{89} + 12 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.24264 −0.787839 −0.393919 0.919145i \(-0.628881\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) −4.24264 −0.697486 −0.348743 0.937218i \(-0.613391\pi\)
−0.348743 + 0.937218i \(0.613391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 7.07107 1.07833 0.539164 0.842201i \(-0.318740\pi\)
0.539164 + 0.842201i \(0.318740\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.07107 0.971286 0.485643 0.874157i \(-0.338586\pi\)
0.485643 + 0.874157i \(0.338586\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.24264 −0.552345 −0.276172 0.961108i \(-0.589066\pi\)
−0.276172 + 0.961108i \(0.589066\pi\)
\(60\) 0 0
\(61\) −12.7279 −1.62964 −0.814822 0.579712i \(-0.803165\pi\)
−0.814822 + 0.579712i \(0.803165\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −7.07107 −0.863868 −0.431934 0.901905i \(-0.642169\pi\)
−0.431934 + 0.901905i \(0.642169\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.41421 −0.155230 −0.0776151 0.996983i \(-0.524731\pi\)
−0.0776151 + 0.996983i \(0.524731\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.5563 1.54791 0.773957 0.633238i \(-0.218274\pi\)
0.773957 + 0.633238i \(0.218274\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.89949 0.957020 0.478510 0.878082i \(-0.341177\pi\)
0.478510 + 0.878082i \(0.341177\pi\)
\(108\) 0 0
\(109\) −4.24264 −0.406371 −0.203186 0.979140i \(-0.565129\pi\)
−0.203186 + 0.979140i \(0.565129\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −8.48528 −0.791257
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.5563 −1.35916 −0.679582 0.733599i \(-0.737839\pi\)
−0.679582 + 0.733599i \(0.737839\pi\)
\(132\) 0 0
\(133\) 8.48528 0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) −4.24264 −0.359856 −0.179928 0.983680i \(-0.557586\pi\)
−0.179928 + 0.983680i \(0.557586\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.89949 −0.810998 −0.405499 0.914095i \(-0.632902\pi\)
−0.405499 + 0.914095i \(0.632902\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.3137 0.908739
\(156\) 0 0
\(157\) 21.2132 1.69300 0.846499 0.532390i \(-0.178706\pi\)
0.846499 + 0.532390i \(0.178706\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −1.41421 −0.110770 −0.0553849 0.998465i \(-0.517639\pi\)
−0.0553849 + 0.998465i \(0.517639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.41421 −0.107521 −0.0537603 0.998554i \(-0.517121\pi\)
−0.0537603 + 0.998554i \(0.517121\pi\)
\(174\) 0 0
\(175\) 6.00000 0.453557
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0416 −1.79696 −0.898478 0.439019i \(-0.855326\pi\)
−0.898478 + 0.439019i \(0.855326\pi\)
\(180\) 0 0
\(181\) −12.7279 −0.946059 −0.473029 0.881047i \(-0.656840\pi\)
−0.473029 + 0.881047i \(0.656840\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.0416 −1.71290 −0.856448 0.516234i \(-0.827334\pi\)
−0.856448 + 0.516234i \(0.827334\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.48528 0.595550
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 12.7279 0.876226 0.438113 0.898920i \(-0.355647\pi\)
0.438113 + 0.898920i \(0.355647\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.82843 −0.190261
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.2132 −1.40797 −0.703985 0.710215i \(-0.748598\pi\)
−0.703985 + 0.710215i \(0.748598\pi\)
\(228\) 0 0
\(229\) −9.89949 −0.654177 −0.327089 0.944994i \(-0.606068\pi\)
−0.327089 + 0.944994i \(0.606068\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) −11.3137 −0.738025
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.24264 0.271052
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 29.6985 1.87455 0.937276 0.348589i \(-0.113339\pi\)
0.937276 + 0.348589i \(0.113339\pi\)
\(252\) 0 0
\(253\) −8.48528 −0.533465
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 8.48528 0.527250
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.24264 0.258678 0.129339 0.991600i \(-0.458714\pi\)
0.129339 + 0.991600i \(0.458714\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.24264 0.255841
\(276\) 0 0
\(277\) 4.24264 0.254916 0.127458 0.991844i \(-0.459318\pi\)
0.127458 + 0.991844i \(0.459318\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) 21.2132 1.26099 0.630497 0.776192i \(-0.282851\pi\)
0.630497 + 0.776192i \(0.282851\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.2132 1.23929 0.619644 0.784883i \(-0.287277\pi\)
0.619644 + 0.784883i \(0.287277\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.48528 0.490716
\(300\) 0 0
\(301\) −14.1421 −0.815139
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.0000 1.03068
\(306\) 0 0
\(307\) 7.07107 0.403567 0.201784 0.979430i \(-0.435326\pi\)
0.201784 + 0.979430i \(0.435326\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.07107 0.397151 0.198575 0.980086i \(-0.436369\pi\)
0.198575 + 0.980086i \(0.436369\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.48528 0.472134
\(324\) 0 0
\(325\) −4.24264 −0.235339
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 1.41421 0.0777322 0.0388661 0.999244i \(-0.487625\pi\)
0.0388661 + 0.999244i \(0.487625\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.3137 0.612672
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.3848 0.986947 0.493473 0.869761i \(-0.335727\pi\)
0.493473 + 0.869761i \(0.335727\pi\)
\(348\) 0 0
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −14.1421 −0.750587
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.65685 −0.296093
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −14.1421 −0.734223
\(372\) 0 0
\(373\) −7.07107 −0.366126 −0.183063 0.983101i \(-0.558601\pi\)
−0.183063 + 0.983101i \(0.558601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 4.24264 0.217930 0.108965 0.994046i \(-0.465246\pi\)
0.108965 + 0.994046i \(0.465246\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.3848 −0.932145 −0.466073 0.884746i \(-0.654331\pi\)
−0.466073 + 0.884746i \(0.654331\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.07107 0.354887 0.177443 0.984131i \(-0.443217\pi\)
0.177443 + 0.984131i \(0.443217\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −11.3137 −0.563576
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.48528 0.417533
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.24264 −0.207267 −0.103633 0.994616i \(-0.533047\pi\)
−0.103633 + 0.994616i \(0.533047\pi\)
\(420\) 0 0
\(421\) 12.7279 0.620321 0.310160 0.950684i \(-0.399617\pi\)
0.310160 + 0.950684i \(0.399617\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 25.4558 1.23189
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.4558 −1.21772
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −21.2132 −1.00787 −0.503935 0.863742i \(-0.668115\pi\)
−0.503935 + 0.863742i \(0.668115\pi\)
\(444\) 0 0
\(445\) 5.65685 0.268161
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.5563 0.724531 0.362266 0.932075i \(-0.382003\pi\)
0.362266 + 0.932075i \(0.382003\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.07107 −0.327210 −0.163605 0.986526i \(-0.552312\pi\)
−0.163605 + 0.986526i \(0.552312\pi\)
\(468\) 0 0
\(469\) 14.1421 0.653023
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 12.7279 0.583997
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.82843 0.128432
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.8701 1.21263 0.606314 0.795225i \(-0.292647\pi\)
0.606314 + 0.795225i \(0.292647\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −20.0000 −0.897123
\(498\) 0 0
\(499\) −32.5269 −1.45610 −0.728052 0.685522i \(-0.759574\pi\)
−0.728052 + 0.685522i \(0.759574\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −22.0000 −0.978987
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −32.5269 −1.44173 −0.720865 0.693075i \(-0.756255\pi\)
−0.720865 + 0.693075i \(0.756255\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.48528 −0.373906
\(516\) 0 0
\(517\) −11.3137 −0.497576
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 40.0000 1.75243 0.876216 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) 0 0
\(523\) 35.3553 1.54598 0.772991 0.634418i \(-0.218760\pi\)
0.772991 + 0.634418i \(0.218760\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −14.0000 −0.605273
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.24264 0.182743
\(540\) 0 0
\(541\) −12.7279 −0.547216 −0.273608 0.961841i \(-0.588217\pi\)
−0.273608 + 0.961841i \(0.588217\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −7.07107 −0.302337 −0.151169 0.988508i \(-0.548304\pi\)
−0.151169 + 0.988508i \(0.548304\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −35.3553 −1.49805 −0.749027 0.662540i \(-0.769479\pi\)
−0.749027 + 0.662540i \(0.769479\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 26.8701 1.13244 0.566219 0.824255i \(-0.308406\pi\)
0.566219 + 0.824255i \(0.308406\pi\)
\(564\) 0 0
\(565\) 8.48528 0.356978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −1.41421 −0.0591830 −0.0295915 0.999562i \(-0.509421\pi\)
−0.0295915 + 0.999562i \(0.509421\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18.0000 −0.750652
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.82843 0.117343
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.89949 0.408596 0.204298 0.978909i \(-0.434509\pi\)
0.204298 + 0.978909i \(0.434509\pi\)
\(588\) 0 0
\(589\) 33.9411 1.39852
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) −5.65685 −0.231908
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.7279 0.517464
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11.3137 0.457704
\(612\) 0 0
\(613\) 35.3553 1.42799 0.713994 0.700151i \(-0.246884\pi\)
0.713994 + 0.700151i \(0.246884\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 24.0416 0.966315 0.483157 0.875534i \(-0.339490\pi\)
0.483157 + 0.875534i \(0.339490\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.48528 0.338330
\(630\) 0 0
\(631\) 10.0000 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.3137 −0.448971
\(636\) 0 0
\(637\) −4.24264 −0.168100
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −29.6985 −1.17119 −0.585597 0.810602i \(-0.699140\pi\)
−0.585597 + 0.810602i \(0.699140\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.8701 1.05151 0.525753 0.850637i \(-0.323784\pi\)
0.525753 + 0.850637i \(0.323784\pi\)
\(654\) 0 0
\(655\) 22.0000 0.859611
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.0416 −0.936529 −0.468264 0.883588i \(-0.655121\pi\)
−0.468264 + 0.883588i \(0.655121\pi\)
\(660\) 0 0
\(661\) −12.7279 −0.495059 −0.247529 0.968880i \(-0.579619\pi\)
−0.247529 + 0.968880i \(0.579619\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) −25.4558 −0.985654
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.24264 0.163058 0.0815290 0.996671i \(-0.474020\pi\)
0.0815290 + 0.996671i \(0.474020\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.07107 −0.270567 −0.135283 0.990807i \(-0.543195\pi\)
−0.135283 + 0.990807i \(0.543195\pi\)
\(684\) 0 0
\(685\) −11.3137 −0.432275
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 12.7279 0.484193 0.242096 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.00000 0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −43.8406 −1.65584 −0.827919 0.560848i \(-0.810475\pi\)
−0.827919 + 0.560848i \(0.810475\pi\)
\(702\) 0 0
\(703\) 18.0000 0.678883
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31.1127 −1.17011
\(708\) 0 0
\(709\) −38.1838 −1.43402 −0.717011 0.697062i \(-0.754490\pi\)
−0.717011 + 0.697062i \(0.754490\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 2.82843 0.105777
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 12.7279 0.472703
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −14.1421 −0.523066
\(732\) 0 0
\(733\) −29.6985 −1.09694 −0.548469 0.836171i \(-0.684789\pi\)
−0.548469 + 0.836171i \(0.684789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) 32.5269 1.19652 0.598261 0.801301i \(-0.295859\pi\)
0.598261 + 0.801301i \(0.295859\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −46.0000 −1.68758 −0.843788 0.536676i \(-0.819680\pi\)
−0.843788 + 0.536676i \(0.819680\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −19.7990 −0.723439
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −14.1421 −0.514685
\(756\) 0 0
\(757\) 32.5269 1.18221 0.591105 0.806594i \(-0.298692\pi\)
0.591105 + 0.806594i \(0.298692\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 8.48528 0.307188
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.07107 −0.254329 −0.127164 0.991882i \(-0.540588\pi\)
−0.127164 + 0.991882i \(0.540588\pi\)
\(774\) 0 0
\(775\) 24.0000 0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −14.1421 −0.506045
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −30.0000 −1.07075
\(786\) 0 0
\(787\) −21.2132 −0.756169 −0.378085 0.925771i \(-0.623417\pi\)
−0.378085 + 0.925771i \(0.623417\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.3553 1.25235 0.626175 0.779682i \(-0.284619\pi\)
0.626175 + 0.779682i \(0.284619\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.65685 −0.199626
\(804\) 0 0
\(805\) 16.9706 0.598134
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) −55.1543 −1.93673 −0.968365 0.249537i \(-0.919722\pi\)
−0.968365 + 0.249537i \(0.919722\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.5563 −0.542920 −0.271460 0.962450i \(-0.587507\pi\)
−0.271460 + 0.962450i \(0.587507\pi\)
\(822\) 0 0
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.6690 1.62284 0.811421 0.584462i \(-0.198695\pi\)
0.811421 + 0.584462i \(0.198695\pi\)
\(828\) 0 0
\(829\) 32.5269 1.12971 0.564853 0.825191i \(-0.308933\pi\)
0.564853 + 0.825191i \(0.308933\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −2.82843 −0.0978818
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −11.0000 −0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 15.5563 0.535155
\(846\) 0 0
\(847\) 18.0000 0.618487
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −25.4558 −0.872615
\(852\) 0 0
\(853\) −7.07107 −0.242109 −0.121054 0.992646i \(-0.538628\pi\)
−0.121054 + 0.992646i \(0.538628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.00000 −0.273275 −0.136637 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) 4.24264 0.144757 0.0723785 0.997377i \(-0.476941\pi\)
0.0723785 + 0.997377i \(0.476941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −22.6274 −0.764946
\(876\) 0 0
\(877\) 7.07107 0.238773 0.119386 0.992848i \(-0.461907\pi\)
0.119386 + 0.992848i \(0.461907\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 29.6985 0.999434 0.499717 0.866189i \(-0.333437\pi\)
0.499717 + 0.866189i \(0.333437\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.00000 −0.0671534 −0.0335767 0.999436i \(-0.510690\pi\)
−0.0335767 + 0.999436i \(0.510690\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −33.9411 −1.13580
\(894\) 0 0
\(895\) 34.0000 1.13649
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 33.9411 1.13200
\(900\) 0 0
\(901\) −14.1421 −0.471143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −38.1838 −1.26787 −0.633936 0.773386i \(-0.718562\pi\)
−0.633936 + 0.773386i \(0.718562\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 2.00000 0.0661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 31.1127 1.02743
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 14.1421 0.465494
\(924\) 0 0
\(925\) 12.7279 0.418491
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 12.7279 0.417141
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −41.0122 −1.33696 −0.668480 0.743730i \(-0.733055\pi\)
−0.668480 + 0.743730i \(0.733055\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.07107 −0.229779 −0.114889 0.993378i \(-0.536651\pi\)
−0.114889 + 0.993378i \(0.536651\pi\)
\(948\) 0 0
\(949\) 5.65685 0.183629
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −11.3137 −0.366103
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19.7990 −0.637352
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26.8701 0.862301 0.431151 0.902280i \(-0.358108\pi\)
0.431151 + 0.902280i \(0.358108\pi\)
\(972\) 0 0
\(973\) 8.48528 0.272026
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 5.65685 0.180794
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −34.0000 −1.08443 −0.542216 0.840239i \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(984\) 0 0
\(985\) 34.0000 1.08333
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 42.4264 1.34908
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −19.7990 −0.627670
\(996\) 0 0
\(997\) 52.3259 1.65718 0.828589 0.559857i \(-0.189144\pi\)
0.828589 + 0.559857i \(0.189144\pi\)
\(998\) 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 9216.2.a.d.1.1 2
3.2 odd 2 1024.2.a.b.1.1 2
4.3 odd 2 9216.2.a.s.1.1 2
8.3 odd 2 9216.2.a.s.1.2 2
8.5 even 2 inner 9216.2.a.d.1.2 2
12.11 even 2 1024.2.a.e.1.2 2
24.5 odd 2 1024.2.a.b.1.2 2
24.11 even 2 1024.2.a.e.1.1 2
32.3 odd 8 576.2.k.a.145.1 2
32.5 even 8 1152.2.k.b.865.1 2
32.11 odd 8 576.2.k.a.433.1 2
32.13 even 8 1152.2.k.b.289.1 2
32.19 odd 8 1152.2.k.a.289.1 2
32.21 even 8 144.2.k.a.37.1 2
32.27 odd 8 1152.2.k.a.865.1 2
32.29 even 8 144.2.k.a.109.1 2
48.5 odd 4 1024.2.b.e.513.2 2
48.11 even 4 1024.2.b.b.513.1 2
48.29 odd 4 1024.2.b.e.513.1 2
48.35 even 4 1024.2.b.b.513.2 2
96.5 odd 8 128.2.e.b.97.1 2
96.11 even 8 64.2.e.a.49.1 2
96.29 odd 8 16.2.e.a.13.1 yes 2
96.35 even 8 64.2.e.a.17.1 2
96.53 odd 8 16.2.e.a.5.1 2
96.59 even 8 128.2.e.a.97.1 2
96.77 odd 8 128.2.e.b.33.1 2
96.83 even 8 128.2.e.a.33.1 2
480.29 odd 8 400.2.l.c.301.1 2
480.53 even 8 400.2.q.a.149.1 2
480.107 odd 8 1600.2.q.a.49.1 2
480.149 odd 8 400.2.l.c.101.1 2
480.203 odd 8 1600.2.q.b.49.1 2
480.227 odd 8 1600.2.q.b.849.1 2
480.299 even 8 1600.2.l.a.1201.1 2
480.317 even 8 400.2.q.a.349.1 2
480.323 odd 8 1600.2.q.a.849.1 2
480.413 even 8 400.2.q.b.349.1 2
480.419 even 8 1600.2.l.a.401.1 2
480.437 even 8 400.2.q.b.149.1 2
672.53 odd 24 784.2.x.f.373.1 4
672.125 even 8 784.2.m.b.589.1 2
672.149 odd 24 784.2.x.f.165.1 4
672.221 odd 24 784.2.x.f.765.1 4
672.317 odd 24 784.2.x.f.557.1 4
672.341 even 24 784.2.x.c.165.1 4
672.437 even 24 784.2.x.c.373.1 4
672.509 even 24 784.2.x.c.557.1 4
672.605 even 24 784.2.x.c.765.1 4
672.629 even 8 784.2.m.b.197.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.2.e.a.5.1 2 96.53 odd 8
16.2.e.a.13.1 yes 2 96.29 odd 8
64.2.e.a.17.1 2 96.35 even 8
64.2.e.a.49.1 2 96.11 even 8
128.2.e.a.33.1 2 96.83 even 8
128.2.e.a.97.1 2 96.59 even 8
128.2.e.b.33.1 2 96.77 odd 8
128.2.e.b.97.1 2 96.5 odd 8
144.2.k.a.37.1 2 32.21 even 8
144.2.k.a.109.1 2 32.29 even 8
400.2.l.c.101.1 2 480.149 odd 8
400.2.l.c.301.1 2 480.29 odd 8
400.2.q.a.149.1 2 480.53 even 8
400.2.q.a.349.1 2 480.317 even 8
400.2.q.b.149.1 2 480.437 even 8
400.2.q.b.349.1 2 480.413 even 8
576.2.k.a.145.1 2 32.3 odd 8
576.2.k.a.433.1 2 32.11 odd 8
784.2.m.b.197.1 2 672.629 even 8
784.2.m.b.589.1 2 672.125 even 8
784.2.x.c.165.1 4 672.341 even 24
784.2.x.c.373.1 4 672.437 even 24
784.2.x.c.557.1 4 672.509 even 24
784.2.x.c.765.1 4 672.605 even 24
784.2.x.f.165.1 4 672.149 odd 24
784.2.x.f.373.1 4 672.53 odd 24
784.2.x.f.557.1 4 672.317 odd 24
784.2.x.f.765.1 4 672.221 odd 24
1024.2.a.b.1.1 2 3.2 odd 2
1024.2.a.b.1.2 2 24.5 odd 2
1024.2.a.e.1.1 2 24.11 even 2
1024.2.a.e.1.2 2 12.11 even 2
1024.2.b.b.513.1 2 48.11 even 4
1024.2.b.b.513.2 2 48.35 even 4
1024.2.b.e.513.1 2 48.29 odd 4
1024.2.b.e.513.2 2 48.5 odd 4
1152.2.k.a.289.1 2 32.19 odd 8
1152.2.k.a.865.1 2 32.27 odd 8
1152.2.k.b.289.1 2 32.13 even 8
1152.2.k.b.865.1 2 32.5 even 8
1600.2.l.a.401.1 2 480.419 even 8
1600.2.l.a.1201.1 2 480.299 even 8
1600.2.q.a.49.1 2 480.107 odd 8
1600.2.q.a.849.1 2 480.323 odd 8
1600.2.q.b.49.1 2 480.203 odd 8
1600.2.q.b.849.1 2 480.227 odd 8
9216.2.a.d.1.1 2 1.1 even 1 trivial
9216.2.a.d.1.2 2 8.5 even 2 inner
9216.2.a.s.1.1 2 4.3 odd 2
9216.2.a.s.1.2 2 8.3 odd 2