Properties

Label 1152.2.k.b.865.1
Level $1152$
Weight $2$
Character 1152.865
Analytic conductor $9.199$
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] = [1152,2,Mod(289,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.k (of order \(4\), degree \(2\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 865.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1152.865
Dual form 1152.2.k.b.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{5} -2.00000i q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.00000i) q^{5} -2.00000i q^{7} +(1.00000 + 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{13} +2.00000 q^{17} +(-3.00000 + 3.00000i) q^{19} -6.00000i q^{23} -3.00000i q^{25} +(3.00000 - 3.00000i) q^{29} -8.00000 q^{31} +(-2.00000 + 2.00000i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(-5.00000 - 5.00000i) q^{43} -8.00000 q^{47} +3.00000 q^{49} +(-5.00000 - 5.00000i) q^{53} -2.00000i q^{55} +(-3.00000 - 3.00000i) q^{59} +(9.00000 - 9.00000i) q^{61} -2.00000 q^{65} +(5.00000 - 5.00000i) q^{67} +10.0000i q^{71} -4.00000i q^{73} +(2.00000 - 2.00000i) q^{77} +(-1.00000 + 1.00000i) q^{83} +(-2.00000 - 2.00000i) q^{85} -4.00000i q^{89} +(-2.00000 - 2.00000i) q^{91} +6.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{11} + 2 q^{13} + 4 q^{17} - 6 q^{19} + 6 q^{29} - 16 q^{31} - 4 q^{35} - 6 q^{37} - 10 q^{43} - 16 q^{47} + 6 q^{49} - 10 q^{53} - 6 q^{59} + 18 q^{61} - 4 q^{65} + 10 q^{67} + 4 q^{77} - 2 q^{83} - 4 q^{85} - 4 q^{91} + 12 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) −1.00000 1.00000i −0.447214 0.447214i 0.447214 0.894427i \(-0.352416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 + 1.00000i 0.301511 + 0.301511i 0.841605 0.540094i \(-0.181611\pi\)
−0.540094 + 0.841605i \(0.681611\pi\)
\(12\) 0 0
\(13\) 1.00000 1.00000i 0.277350 0.277350i −0.554700 0.832050i \(-0.687167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −3.00000 + 3.00000i −0.688247 + 0.688247i −0.961844 0.273597i \(-0.911786\pi\)
0.273597 + 0.961844i \(0.411786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 3.00000i 0.557086 0.557086i −0.371391 0.928477i \(-0.621119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 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
\(35\) −2.00000 + 2.00000i −0.338062 + 0.338062i
\(36\) 0 0
\(37\) −3.00000 3.00000i −0.493197 0.493197i 0.416115 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −5.00000 5.00000i −0.762493 0.762493i 0.214280 0.976772i \(-0.431260\pi\)
−0.976772 + 0.214280i \(0.931260\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.00000 5.00000i −0.686803 0.686803i 0.274721 0.961524i \(-0.411414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 2.00000i 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 3.00000i −0.390567 0.390567i 0.484323 0.874889i \(-0.339066\pi\)
−0.874889 + 0.484323i \(0.839066\pi\)
\(60\) 0 0
\(61\) 9.00000 9.00000i 1.15233 1.15233i 0.166248 0.986084i \(-0.446835\pi\)
0.986084 0.166248i \(-0.0531652\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 5.00000 5.00000i 0.610847 0.610847i −0.332320 0.943167i \(-0.607831\pi\)
0.943167 + 0.332320i \(0.107831\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000i 1.18678i 0.804914 + 0.593391i \(0.202211\pi\)
−0.804914 + 0.593391i \(0.797789\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 2.00000i 0.227921 0.227921i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.00000 + 1.00000i −0.109764 + 0.109764i −0.759856 0.650092i \(-0.774731\pi\)
0.650092 + 0.759856i \(0.274731\pi\)
\(84\) 0 0
\(85\) −2.00000 2.00000i −0.216930 0.216930i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000i 0.423999i −0.977270 0.212000i \(-0.932002\pi\)
0.977270 0.212000i \(-0.0679975\pi\)
\(90\) 0 0
\(91\) −2.00000 2.00000i −0.209657 0.209657i
\(92\) 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
\(101\) 11.0000 + 11.0000i 1.09454 + 1.09454i 0.995037 + 0.0995037i \(0.0317255\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.00000 7.00000i −0.676716 0.676716i 0.282540 0.959256i \(-0.408823\pi\)
−0.959256 + 0.282540i \(0.908823\pi\)
\(108\) 0 0
\(109\) −3.00000 + 3.00000i −0.287348 + 0.287348i −0.836031 0.548683i \(-0.815129\pi\)
0.548683 + 0.836031i \(0.315129\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −6.00000 + 6.00000i −0.559503 + 0.559503i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000i 0.366679i
\(120\) 0 0
\(121\) 9.00000i 0.818182i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.00000 + 8.00000i −0.715542 + 0.715542i
\(126\) 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
\(131\) 11.0000 11.0000i 0.961074 0.961074i −0.0381958 0.999270i \(-0.512161\pi\)
0.999270 + 0.0381958i \(0.0121611\pi\)
\(132\) 0 0
\(133\) 6.00000 + 6.00000i 0.520266 + 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.00000i 0.683486i −0.939793 0.341743i \(-0.888983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 0 0
\(139\) 3.00000 + 3.00000i 0.254457 + 0.254457i 0.822795 0.568338i \(-0.192414\pi\)
−0.568338 + 0.822795i \(0.692414\pi\)
\(140\) 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
\(149\) 7.00000 + 7.00000i 0.573462 + 0.573462i 0.933094 0.359632i \(-0.117098\pi\)
−0.359632 + 0.933094i \(0.617098\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i −0.913475 0.406894i \(-0.866612\pi\)
0.913475 0.406894i \(-0.133388\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 + 8.00000i 0.642575 + 0.642575i
\(156\) 0 0
\(157\) −15.0000 + 15.0000i −1.19713 + 1.19713i −0.222108 + 0.975022i \(0.571294\pi\)
−0.975022 + 0.222108i \(0.928706\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 1.00000 1.00000i 0.0783260 0.0783260i −0.666858 0.745184i \(-0.732361\pi\)
0.745184 + 0.666858i \(0.232361\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.00000 + 1.00000i −0.0760286 + 0.0760286i −0.744099 0.668070i \(-0.767121\pi\)
0.668070 + 0.744099i \(0.267121\pi\)
\(174\) 0 0
\(175\) −6.00000 −0.453557
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −17.0000 + 17.0000i −1.27064 + 1.27064i −0.324887 + 0.945753i \(0.605326\pi\)
−0.945753 + 0.324887i \(0.894674\pi\)
\(180\) 0 0
\(181\) 9.00000 + 9.00000i 0.668965 + 0.668965i 0.957476 0.288512i \(-0.0931604\pi\)
−0.288512 + 0.957476i \(0.593160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000i 0.441129i
\(186\) 0 0
\(187\) 2.00000 + 2.00000i 0.146254 + 0.146254i
\(188\) 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
\(197\) −17.0000 17.0000i −1.21120 1.21120i −0.970632 0.240567i \(-0.922666\pi\)
−0.240567 0.970632i \(-0.577334\pi\)
\(198\) 0 0
\(199\) 14.0000i 0.992434i 0.868199 + 0.496217i \(0.165278\pi\)
−0.868199 + 0.496217i \(0.834722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 6.00000i −0.421117 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 9.00000 9.00000i 0.619586 0.619586i −0.325840 0.945425i \(-0.605647\pi\)
0.945425 + 0.325840i \(0.105647\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000i 0.681994i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 2.00000i 0.134535 0.134535i
\(222\) 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
\(227\) 15.0000 15.0000i 0.995585 0.995585i −0.00440533 0.999990i \(-0.501402\pi\)
0.999990 + 0.00440533i \(0.00140226\pi\)
\(228\) 0 0
\(229\) −7.00000 7.00000i −0.462573 0.462573i 0.436925 0.899498i \(-0.356068\pi\)
−0.899498 + 0.436925i \(0.856068\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000i 0.262049i 0.991379 + 0.131024i \(0.0418266\pi\)
−0.991379 + 0.131024i \(0.958173\pi\)
\(234\) 0 0
\(235\) 8.00000 + 8.00000i 0.521862 + 0.521862i
\(236\) 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 \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 3.00000i −0.191663 0.191663i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 + 21.0000i 1.32551 + 1.32551i 0.909243 + 0.416265i \(0.136661\pi\)
0.416265 + 0.909243i \(0.363339\pi\)
\(252\) 0 0
\(253\) 6.00000 6.00000i 0.377217 0.377217i
\(254\) 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
\(259\) −6.00000 + 6.00000i −0.372822 + 0.372822i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000i 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) 10.0000i 0.614295i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 3.00000i 0.182913 0.182913i −0.609711 0.792624i \(-0.708714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 3.00000i 0.180907 0.180907i
\(276\) 0 0
\(277\) −3.00000 3.00000i −0.180253 0.180253i 0.611213 0.791466i \(-0.290682\pi\)
−0.791466 + 0.611213i \(0.790682\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0000i 1.19310i 0.802576 + 0.596550i \(0.203462\pi\)
−0.802576 + 0.596550i \(0.796538\pi\)
\(282\) 0 0
\(283\) 15.0000 + 15.0000i 0.891657 + 0.891657i 0.994679 0.103022i \(-0.0328511\pi\)
−0.103022 + 0.994679i \(0.532851\pi\)
\(284\) 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
\(293\) 15.0000 + 15.0000i 0.876309 + 0.876309i 0.993151 0.116841i \(-0.0372769\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 6.00000i 0.349334i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 6.00000i −0.346989 0.346989i
\(300\) 0 0
\(301\) −10.0000 + 10.0000i −0.576390 + 0.576390i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −18.0000 −1.03068
\(306\) 0 0
\(307\) 5.00000 5.00000i 0.285365 0.285365i −0.549879 0.835244i \(-0.685326\pi\)
0.835244 + 0.549879i \(0.185326\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 30.0000i 1.70114i −0.525859 0.850572i \(-0.676256\pi\)
0.525859 0.850572i \(-0.323744\pi\)
\(312\) 0 0
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.00000 + 5.00000i −0.280828 + 0.280828i −0.833439 0.552611i \(-0.813631\pi\)
0.552611 + 0.833439i \(0.313631\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 + 6.00000i −0.333849 + 0.333849i
\(324\) 0 0
\(325\) −3.00000 3.00000i −0.166410 0.166410i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000i 0.882109i
\(330\) 0 0
\(331\) −1.00000 1.00000i −0.0549650 0.0549650i 0.679090 0.734055i \(-0.262375\pi\)
−0.734055 + 0.679090i \(0.762375\pi\)
\(332\) 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 \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.00000 8.00000i −0.433224 0.433224i
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.0000 + 13.0000i 0.697877 + 0.697877i 0.963952 0.266076i \(-0.0857271\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(348\) 0 0
\(349\) −3.00000 + 3.00000i −0.160586 + 0.160586i −0.782826 0.622240i \(-0.786223\pi\)
0.622240 + 0.782826i \(0.286223\pi\)
\(350\) 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
\(355\) 10.0000 10.0000i 0.530745 0.530745i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.0000i 1.37223i 0.727494 + 0.686114i \(0.240685\pi\)
−0.727494 + 0.686114i \(0.759315\pi\)
\(360\) 0 0
\(361\) 1.00000i 0.0526316i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 + 4.00000i −0.209370 + 0.209370i
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −10.0000 + 10.0000i −0.519174 + 0.519174i
\(372\) 0 0
\(373\) 5.00000 + 5.00000i 0.258890 + 0.258890i 0.824603 0.565712i \(-0.191399\pi\)
−0.565712 + 0.824603i \(0.691399\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) 3.00000 + 3.00000i 0.154100 + 0.154100i 0.779946 0.625847i \(-0.215246\pi\)
−0.625847 + 0.779946i \(0.715246\pi\)
\(380\) 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
\(389\) −13.0000 13.0000i −0.659126 0.659126i 0.296047 0.955173i \(-0.404331\pi\)
−0.955173 + 0.296047i \(0.904331\pi\)
\(390\) 0 0
\(391\) 12.0000i 0.606866i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.00000 5.00000i 0.250943 0.250943i −0.570414 0.821357i \(-0.693217\pi\)
0.821357 + 0.570414i \(0.193217\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −8.00000 + 8.00000i −0.398508 + 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000i 0.297409i
\(408\) 0 0
\(409\) 16.0000i 0.791149i −0.918434 0.395575i \(-0.870545\pi\)
0.918434 0.395575i \(-0.129455\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.00000 + 6.00000i −0.295241 + 0.295241i
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.00000 3.00000i 0.146560 0.146560i −0.630020 0.776579i \(-0.716953\pi\)
0.776579 + 0.630020i \(0.216953\pi\)
\(420\) 0 0
\(421\) 9.00000 + 9.00000i 0.438633 + 0.438633i 0.891552 0.452919i \(-0.149617\pi\)
−0.452919 + 0.891552i \(0.649617\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000i 0.291043i
\(426\) 0 0
\(427\) −18.0000 18.0000i −0.871081 0.871081i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18.0000 + 18.0000i 0.861057 + 0.861057i
\(438\) 0 0
\(439\) 14.0000i 0.668184i 0.942541 + 0.334092i \(0.108430\pi\)
−0.942541 + 0.334092i \(0.891570\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.0000 15.0000i −0.712672 0.712672i 0.254422 0.967093i \(-0.418115\pi\)
−0.967093 + 0.254422i \(0.918115\pi\)
\(444\) 0 0
\(445\) −4.00000 + 4.00000i −0.189618 + 0.189618i
\(446\) 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
\(455\) 4.00000i 0.187523i
\(456\) 0 0
\(457\) 32.0000i 1.49690i −0.663193 0.748448i \(-0.730799\pi\)
0.663193 0.748448i \(-0.269201\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.0000 11.0000i 0.512321 0.512321i −0.402916 0.915237i \(-0.632003\pi\)
0.915237 + 0.402916i \(0.132003\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.00000 + 5.00000i −0.231372 + 0.231372i −0.813265 0.581893i \(-0.802312\pi\)
0.581893 + 0.813265i \(0.302312\pi\)
\(468\) 0 0
\(469\) −10.0000 10.0000i −0.461757 0.461757i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000i 0.459800i
\(474\) 0 0
\(475\) 9.00000 + 9.00000i 0.412948 + 0.412948i
\(476\) 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
\(485\) 2.00000 + 2.00000i 0.0908153 + 0.0908153i
\(486\) 0 0
\(487\) 2.00000i 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −19.0000 19.0000i −0.857458 0.857458i 0.133580 0.991038i \(-0.457353\pi\)
−0.991038 + 0.133580i \(0.957353\pi\)
\(492\) 0 0
\(493\) 6.00000 6.00000i 0.270226 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.0000 0.897123
\(498\) 0 0
\(499\) −23.0000 + 23.0000i −1.02962 + 1.02962i −0.0300737 + 0.999548i \(0.509574\pi\)
−0.999548 + 0.0300737i \(0.990426\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000i 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 22.0000i 0.978987i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 23.0000 23.0000i 1.01946 1.01946i 0.0196502 0.999807i \(-0.493745\pi\)
0.999807 0.0196502i \(-0.00625524\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.00000 6.00000i 0.264392 0.264392i
\(516\) 0 0
\(517\) −8.00000 8.00000i −0.351840 0.351840i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 40.0000i 1.75243i −0.481919 0.876216i \(-0.660060\pi\)
0.481919 0.876216i \(-0.339940\pi\)
\(522\) 0 0
\(523\) −25.0000 25.0000i −1.09317 1.09317i −0.995188 0.0979859i \(-0.968760\pi\)
−0.0979859 0.995188i \(-0.531240\pi\)
\(524\) 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
\(535\) 14.0000i 0.605273i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.00000 + 3.00000i 0.129219 + 0.129219i
\(540\) 0 0
\(541\) 9.00000 9.00000i 0.386940 0.386940i −0.486654 0.873595i \(-0.661783\pi\)
0.873595 + 0.486654i \(0.161783\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 5.00000 5.00000i 0.213785 0.213785i −0.592088 0.805873i \(-0.701696\pi\)
0.805873 + 0.592088i \(0.201696\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 18.0000i 0.766826i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.0000 + 25.0000i −1.05928 + 1.05928i −0.0611558 + 0.998128i \(0.519479\pi\)
−0.998128 + 0.0611558i \(0.980521\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.0000 19.0000i 0.800755 0.800755i −0.182459 0.983213i \(-0.558406\pi\)
0.983213 + 0.182459i \(0.0584057\pi\)
\(564\) 0 0
\(565\) −6.00000 6.00000i −0.252422 0.252422i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000i 1.00613i −0.864248 0.503066i \(-0.832205\pi\)
0.864248 0.503066i \(-0.167795\pi\)
\(570\) 0 0
\(571\) −1.00000 1.00000i −0.0418487 0.0418487i 0.685873 0.727721i \(-0.259421\pi\)
−0.727721 + 0.685873i \(0.759421\pi\)
\(572\) 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
\(581\) 2.00000 + 2.00000i 0.0829740 + 0.0829740i
\(582\) 0 0
\(583\) 10.0000i 0.414158i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.00000 7.00000i −0.288921 0.288921i 0.547733 0.836653i \(-0.315491\pi\)
−0.836653 + 0.547733i \(0.815491\pi\)
\(588\) 0 0
\(589\) 24.0000 24.0000i 0.988903 0.988903i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) −4.00000 + 4.00000i −0.163984 + 0.163984i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.0000i 0.572024i −0.958226 0.286012i \(-0.907670\pi\)
0.958226 0.286012i \(-0.0923298\pi\)
\(600\) 0 0
\(601\) 20.0000i 0.815817i 0.913023 + 0.407909i \(0.133742\pi\)
−0.913023 + 0.407909i \(0.866258\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.00000 + 9.00000i −0.365902 + 0.365902i
\(606\) 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
\(611\) −8.00000 + 8.00000i −0.323645 + 0.323645i
\(612\) 0 0
\(613\) 25.0000 + 25.0000i 1.00974 + 1.00974i 0.999952 + 0.00978840i \(0.00311579\pi\)
0.00978840 + 0.999952i \(0.496884\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 0 0
\(619\) −17.0000 17.0000i −0.683288 0.683288i 0.277452 0.960740i \(-0.410510\pi\)
−0.960740 + 0.277452i \(0.910510\pi\)
\(620\) 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
\(629\) −6.00000 6.00000i −0.239236 0.239236i
\(630\) 0 0
\(631\) 10.0000i 0.398094i −0.979990 0.199047i \(-0.936215\pi\)
0.979990 0.199047i \(-0.0637846\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 8.00000i −0.317470 0.317470i
\(636\) 0 0
\(637\) 3.00000 3.00000i 0.118864 0.118864i
\(638\) 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
\(643\) 21.0000 21.0000i 0.828159 0.828159i −0.159103 0.987262i \(-0.550860\pi\)
0.987262 + 0.159103i \(0.0508601\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000i 1.65119i 0.564263 + 0.825595i \(0.309160\pi\)
−0.564263 + 0.825595i \(0.690840\pi\)
\(648\) 0 0
\(649\) 6.00000i 0.235521i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.0000 19.0000i 0.743527 0.743527i −0.229728 0.973255i \(-0.573784\pi\)
0.973255 + 0.229728i \(0.0737835\pi\)
\(654\) 0 0
\(655\) −22.0000 −0.859611
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −17.0000 + 17.0000i −0.662226 + 0.662226i −0.955904 0.293678i \(-0.905121\pi\)
0.293678 + 0.955904i \(0.405121\pi\)
\(660\) 0 0
\(661\) 9.00000 + 9.00000i 0.350059 + 0.350059i 0.860132 0.510072i \(-0.170381\pi\)
−0.510072 + 0.860132i \(0.670381\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000i 0.465340i
\(666\) 0 0
\(667\) −18.0000 18.0000i −0.696963 0.696963i
\(668\) 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
\(677\) 3.00000 + 3.00000i 0.115299 + 0.115299i 0.762402 0.647103i \(-0.224020\pi\)
−0.647103 + 0.762402i \(0.724020\pi\)
\(678\) 0 0
\(679\) 4.00000i 0.153506i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.00000 + 5.00000i 0.191320 + 0.191320i 0.796266 0.604946i \(-0.206805\pi\)
−0.604946 + 0.796266i \(0.706805\pi\)
\(684\) 0 0
\(685\) −8.00000 + 8.00000i −0.305664 + 0.305664i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) 9.00000 9.00000i 0.342376 0.342376i −0.514884 0.857260i \(-0.672165\pi\)
0.857260 + 0.514884i \(0.172165\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.00000i 0.227593i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.0000 31.0000i 1.17085 1.17085i 0.188847 0.982006i \(-0.439525\pi\)
0.982006 0.188847i \(-0.0604752\pi\)
\(702\) 0 0
\(703\) 18.0000 0.678883
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 22.0000 22.0000i 0.827395 0.827395i
\(708\) 0 0
\(709\) −27.0000 27.0000i −1.01401 1.01401i −0.999901 0.0141058i \(-0.995510\pi\)
−0.0141058 0.999901i \(-0.504490\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 48.0000i 1.79761i
\(714\) 0 0
\(715\) −2.00000 2.00000i −0.0747958 0.0747958i
\(716\) 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
\(725\) −9.00000 9.00000i −0.334252 0.334252i
\(726\) 0 0
\(727\) 2.00000i 0.0741759i −0.999312 0.0370879i \(-0.988192\pi\)
0.999312 0.0370879i \(-0.0118082\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.0000 10.0000i −0.369863 0.369863i
\(732\) 0 0
\(733\) 21.0000 21.0000i 0.775653 0.775653i −0.203436 0.979088i \(-0.565211\pi\)
0.979088 + 0.203436i \(0.0652108\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −23.0000 + 23.0000i −0.846069 + 0.846069i −0.989640 0.143571i \(-0.954141\pi\)
0.143571 + 0.989640i \(0.454141\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 46.0000i 1.68758i −0.536676 0.843788i \(-0.680320\pi\)
0.536676 0.843788i \(-0.319680\pi\)
\(744\) 0 0
\(745\) 14.0000i 0.512920i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.0000 + 14.0000i −0.511549 + 0.511549i
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.0000 + 10.0000i −0.363937 + 0.363937i
\(756\) 0 0
\(757\) −23.0000 23.0000i −0.835949 0.835949i 0.152374 0.988323i \(-0.451308\pi\)
−0.988323 + 0.152374i \(0.951308\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 6.00000 + 6.00000i 0.217215 + 0.217215i
\(764\) 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
\(773\) −5.00000 5.00000i −0.179838 0.179838i 0.611448 0.791285i \(-0.290588\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 0 0
\(775\) 24.0000i 0.862105i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −10.0000 + 10.0000i −0.357828 + 0.357828i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −15.0000 + 15.0000i −0.534692 + 0.534692i −0.921965 0.387273i \(-0.873417\pi\)
0.387273 + 0.921965i \(0.373417\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000i 0.426671i
\(792\) 0 0
\(793\) 18.0000i 0.639199i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.0000 + 25.0000i −0.885545 + 0.885545i −0.994091 0.108546i \(-0.965381\pi\)
0.108546 + 0.994091i \(0.465381\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.00000 4.00000i 0.141157 0.141157i
\(804\) 0 0
\(805\) 12.0000 + 12.0000i 0.422944 + 0.422944i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.0000i 0.562530i 0.959630 + 0.281265i \(0.0907540\pi\)
−0.959630 + 0.281265i \(0.909246\pi\)
\(810\) 0 0
\(811\) 39.0000 + 39.0000i 1.36948 + 1.36948i 0.861187 + 0.508288i \(0.169722\pi\)
0.508288 + 0.861187i \(0.330278\pi\)
\(812\) 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
\(821\) 11.0000 + 11.0000i 0.383903 + 0.383903i 0.872506 0.488603i \(-0.162493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) 0 0
\(823\) 34.0000i 1.18517i −0.805510 0.592583i \(-0.798108\pi\)
0.805510 0.592583i \(-0.201892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.0000 + 33.0000i 1.14752 + 1.14752i 0.987038 + 0.160484i \(0.0513055\pi\)
0.160484 + 0.987038i \(0.448695\pi\)
\(828\) 0 0
\(829\) −23.0000 + 23.0000i −0.798823 + 0.798823i −0.982910 0.184087i \(-0.941067\pi\)
0.184087 + 0.982910i \(0.441067\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 2.00000 2.00000i 0.0692129 0.0692129i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 14.0000i 0.483334i −0.970359 0.241667i \(-0.922306\pi\)
0.970359 0.241667i \(-0.0776941\pi\)
\(840\) 0 0
\(841\) 11.0000i 0.379310i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.0000 11.0000i 0.378412 0.378412i
\(846\) 0 0
\(847\) −18.0000 −0.618487
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −18.0000 + 18.0000i −0.617032 + 0.617032i
\(852\) 0 0
\(853\) 5.00000 + 5.00000i 0.171197 + 0.171197i 0.787505 0.616308i \(-0.211372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8.00000i 0.273275i −0.990621 0.136637i \(-0.956370\pi\)
0.990621 0.136637i \(-0.0436295\pi\)
\(858\) 0 0
\(859\) 3.00000 + 3.00000i 0.102359 + 0.102359i 0.756432 0.654073i \(-0.226941\pi\)
−0.654073 + 0.756432i \(0.726941\pi\)
\(860\) 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
\(871\) 10.0000i 0.338837i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 16.0000 + 16.0000i 0.540899 + 0.540899i
\(876\) 0 0
\(877\) 5.00000 5.00000i 0.168838 0.168838i −0.617630 0.786468i \(-0.711907\pi\)
0.786468 + 0.617630i \(0.211907\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 21.0000 21.0000i 0.706706 0.706706i −0.259135 0.965841i \(-0.583437\pi\)
0.965841 + 0.259135i \(0.0834374\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.00000i 0.0671534i 0.999436 + 0.0335767i \(0.0106898\pi\)
−0.999436 + 0.0335767i \(0.989310\pi\)
\(888\) 0 0
\(889\) 16.0000i 0.536623i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24.0000 24.0000i 0.803129 0.803129i
\(894\) 0 0
\(895\) 34.0000 1.13649
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −24.0000 + 24.0000i −0.800445 + 0.800445i
\(900\) 0 0
\(901\) −10.0000 10.0000i −0.333148 0.333148i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000i 0.598340i
\(906\) 0 0
\(907\) 27.0000 + 27.0000i 0.896520 + 0.896520i 0.995127 0.0986062i \(-0.0314384\pi\)
−0.0986062 + 0.995127i \(0.531438\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −2.00000 −0.0661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −22.0000 22.0000i −0.726504 0.726504i
\(918\) 0 0
\(919\) 26.0000i 0.857661i −0.903385 0.428830i \(-0.858926\pi\)
0.903385 0.428830i \(-0.141074\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.0000 + 10.0000i 0.329154 + 0.329154i
\(924\) 0 0
\(925\) −9.00000 + 9.00000i −0.295918 + 0.295918i
\(926\) 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
\(931\) −9.00000 + 9.00000i −0.294963 + 0.294963i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.00000i 0.130814i
\(936\) 0 0
\(937\) 28.0000i 0.914720i 0.889282 + 0.457360i \(0.151205\pi\)
−0.889282 + 0.457360i \(0.848795\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −29.0000 + 29.0000i −0.945373 + 0.945373i −0.998583 0.0532103i \(-0.983055\pi\)
0.0532103 + 0.998583i \(0.483055\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.00000 + 5.00000i −0.162478 + 0.162478i −0.783664 0.621185i \(-0.786651\pi\)
0.621185 + 0.783664i \(0.286651\pi\)
\(948\) 0 0
\(949\) −4.00000 4.00000i −0.129845 0.129845i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 0 0
\(955\) −8.00000 8.00000i −0.258874 0.258874i
\(956\) 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
\(965\) −14.0000 14.0000i −0.450676 0.450676i
\(966\) 0 0
\(967\) 2.00000i 0.0643157i −0.999483 0.0321578i \(-0.989762\pi\)
0.999483 0.0321578i \(-0.0102379\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.0000 19.0000i −0.609739 0.609739i 0.333139 0.942878i \(-0.391892\pi\)
−0.942878 + 0.333139i \(0.891892\pi\)
\(972\) 0 0
\(973\) 6.00000 6.00000i 0.192351 0.192351i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 4.00000 4.00000i 0.127841 0.127841i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34.0000i 1.08443i 0.840239 + 0.542216i \(0.182414\pi\)
−0.840239 + 0.542216i \(0.817586\pi\)
\(984\) 0 0
\(985\) 34.0000i 1.08333i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0000 + 30.0000i −0.953945 + 0.953945i
\(990\) 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
\(995\) 14.0000 14.0000i 0.443830 0.443830i
\(996\) 0 0
\(997\) 37.0000 + 37.0000i 1.17180 + 1.17180i 0.981780 + 0.190022i \(0.0608559\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(998\) 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 1152.2.k.b.865.1 2
3.2 odd 2 128.2.e.b.97.1 2
4.3 odd 2 1152.2.k.a.865.1 2
8.3 odd 2 576.2.k.a.433.1 2
8.5 even 2 144.2.k.a.37.1 2
12.11 even 2 128.2.e.a.97.1 2
16.3 odd 4 1152.2.k.a.289.1 2
16.5 even 4 144.2.k.a.109.1 2
16.11 odd 4 576.2.k.a.145.1 2
16.13 even 4 inner 1152.2.k.b.289.1 2
24.5 odd 2 16.2.e.a.5.1 2
24.11 even 2 64.2.e.a.49.1 2
32.3 odd 8 9216.2.a.s.1.2 2
32.13 even 8 9216.2.a.d.1.1 2
32.19 odd 8 9216.2.a.s.1.1 2
32.29 even 8 9216.2.a.d.1.2 2
48.5 odd 4 16.2.e.a.13.1 yes 2
48.11 even 4 64.2.e.a.17.1 2
48.29 odd 4 128.2.e.b.33.1 2
48.35 even 4 128.2.e.a.33.1 2
96.5 odd 8 1024.2.b.e.513.2 2
96.11 even 8 1024.2.b.b.513.2 2
96.29 odd 8 1024.2.a.b.1.2 2
96.35 even 8 1024.2.a.e.1.1 2
96.53 odd 8 1024.2.b.e.513.1 2
96.59 even 8 1024.2.b.b.513.1 2
96.77 odd 8 1024.2.a.b.1.1 2
96.83 even 8 1024.2.a.e.1.2 2
120.29 odd 2 400.2.l.c.101.1 2
120.53 even 4 400.2.q.a.149.1 2
120.59 even 2 1600.2.l.a.1201.1 2
120.77 even 4 400.2.q.b.149.1 2
120.83 odd 4 1600.2.q.b.49.1 2
120.107 odd 4 1600.2.q.a.49.1 2
168.5 even 6 784.2.x.c.165.1 4
168.53 odd 6 784.2.x.f.373.1 4
168.101 even 6 784.2.x.c.373.1 4
168.125 even 2 784.2.m.b.197.1 2
168.149 odd 6 784.2.x.f.165.1 4
240.53 even 4 400.2.q.b.349.1 2
240.59 even 4 1600.2.l.a.401.1 2
240.107 odd 4 1600.2.q.b.849.1 2
240.149 odd 4 400.2.l.c.301.1 2
240.197 even 4 400.2.q.a.349.1 2
240.203 odd 4 1600.2.q.a.849.1 2
336.5 even 12 784.2.x.c.557.1 4
336.53 odd 12 784.2.x.f.765.1 4
336.101 even 12 784.2.x.c.765.1 4
336.149 odd 12 784.2.x.f.557.1 4
336.293 even 4 784.2.m.b.589.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.2.e.a.5.1 2 24.5 odd 2
16.2.e.a.13.1 yes 2 48.5 odd 4
64.2.e.a.17.1 2 48.11 even 4
64.2.e.a.49.1 2 24.11 even 2
128.2.e.a.33.1 2 48.35 even 4
128.2.e.a.97.1 2 12.11 even 2
128.2.e.b.33.1 2 48.29 odd 4
128.2.e.b.97.1 2 3.2 odd 2
144.2.k.a.37.1 2 8.5 even 2
144.2.k.a.109.1 2 16.5 even 4
400.2.l.c.101.1 2 120.29 odd 2
400.2.l.c.301.1 2 240.149 odd 4
400.2.q.a.149.1 2 120.53 even 4
400.2.q.a.349.1 2 240.197 even 4
400.2.q.b.149.1 2 120.77 even 4
400.2.q.b.349.1 2 240.53 even 4
576.2.k.a.145.1 2 16.11 odd 4
576.2.k.a.433.1 2 8.3 odd 2
784.2.m.b.197.1 2 168.125 even 2
784.2.m.b.589.1 2 336.293 even 4
784.2.x.c.165.1 4 168.5 even 6
784.2.x.c.373.1 4 168.101 even 6
784.2.x.c.557.1 4 336.5 even 12
784.2.x.c.765.1 4 336.101 even 12
784.2.x.f.165.1 4 168.149 odd 6
784.2.x.f.373.1 4 168.53 odd 6
784.2.x.f.557.1 4 336.149 odd 12
784.2.x.f.765.1 4 336.53 odd 12
1024.2.a.b.1.1 2 96.77 odd 8
1024.2.a.b.1.2 2 96.29 odd 8
1024.2.a.e.1.1 2 96.35 even 8
1024.2.a.e.1.2 2 96.83 even 8
1024.2.b.b.513.1 2 96.59 even 8
1024.2.b.b.513.2 2 96.11 even 8
1024.2.b.e.513.1 2 96.53 odd 8
1024.2.b.e.513.2 2 96.5 odd 8
1152.2.k.a.289.1 2 16.3 odd 4
1152.2.k.a.865.1 2 4.3 odd 2
1152.2.k.b.289.1 2 16.13 even 4 inner
1152.2.k.b.865.1 2 1.1 even 1 trivial
1600.2.l.a.401.1 2 240.59 even 4
1600.2.l.a.1201.1 2 120.59 even 2
1600.2.q.a.49.1 2 120.107 odd 4
1600.2.q.a.849.1 2 240.203 odd 4
1600.2.q.b.49.1 2 120.83 odd 4
1600.2.q.b.849.1 2 240.107 odd 4
9216.2.a.d.1.1 2 32.13 even 8
9216.2.a.d.1.2 2 32.29 even 8
9216.2.a.s.1.1 2 32.19 odd 8
9216.2.a.s.1.2 2 32.3 odd 8