Properties

Label 2352.2.q.be.1537.2
Level $2352$
Weight $2$
Character 2352.1537
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
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] = [2352,2,Mod(961,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (of order \(3\), 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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1537.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1537
Dual form 2352.2.q.be.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(-0.292893 - 0.507306i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(-0.292893 - 0.507306i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-0.414214 + 0.717439i) q^{11} -1.41421 q^{13} -0.585786 q^{15} +(-1.12132 + 1.94218i) q^{17} +(3.41421 + 5.91359i) q^{19} +(2.41421 + 4.18154i) q^{23} +(2.32843 - 4.03295i) q^{25} -1.00000 q^{27} +8.48528 q^{29} +(2.58579 - 4.47871i) q^{31} +(0.414214 + 0.717439i) q^{33} +(-0.828427 - 1.43488i) q^{37} +(-0.707107 + 1.22474i) q^{39} -0.585786 q^{41} +8.00000 q^{43} +(-0.292893 + 0.507306i) q^{45} +(3.41421 + 5.91359i) q^{47} +(1.12132 + 1.94218i) q^{51} +(6.65685 - 11.5300i) q^{53} +0.485281 q^{55} +6.82843 q^{57} +(2.58579 - 4.47871i) q^{59} +(-6.94975 - 12.0373i) q^{61} +(0.414214 + 0.717439i) q^{65} +(-4.00000 + 6.92820i) q^{67} +4.82843 q^{69} +0.828427 q^{71} +(5.53553 - 9.58783i) q^{73} +(-2.32843 - 4.03295i) q^{75} +(-1.17157 - 2.02922i) q^{79} +(-0.500000 + 0.866025i) q^{81} -15.3137 q^{83} +1.31371 q^{85} +(4.24264 - 7.34847i) q^{87} +(5.36396 + 9.29065i) q^{89} +(-2.58579 - 4.47871i) q^{93} +(2.00000 - 3.46410i) q^{95} -7.75736 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9} + 4 q^{11} - 8 q^{15} + 4 q^{17} + 8 q^{19} + 4 q^{23} - 2 q^{25} - 4 q^{27} + 16 q^{31} - 4 q^{33} + 8 q^{37} - 8 q^{41} + 32 q^{43} - 4 q^{45} + 8 q^{47} - 4 q^{51} + 4 q^{53} - 32 q^{55} + 16 q^{57} + 16 q^{59} - 8 q^{61} - 4 q^{65} - 16 q^{67} + 8 q^{69} - 8 q^{71} + 8 q^{73} + 2 q^{75} - 16 q^{79} - 2 q^{81} - 16 q^{83} - 40 q^{85} - 4 q^{89} - 16 q^{93} + 8 q^{95} - 48 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) −0.292893 0.507306i −0.130986 0.226874i 0.793071 0.609129i \(-0.208481\pi\)
−0.924057 + 0.382255i \(0.875148\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) −0.414214 + 0.717439i −0.124890 + 0.216316i −0.921690 0.387927i \(-0.873191\pi\)
0.796800 + 0.604243i \(0.206524\pi\)
\(12\) 0 0
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −1.12132 + 1.94218i −0.271960 + 0.471049i −0.969364 0.245630i \(-0.921005\pi\)
0.697404 + 0.716679i \(0.254339\pi\)
\(18\) 0 0
\(19\) 3.41421 + 5.91359i 0.783274 + 1.35667i 0.930025 + 0.367497i \(0.119785\pi\)
−0.146750 + 0.989174i \(0.546881\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.41421 + 4.18154i 0.503398 + 0.871911i 0.999992 + 0.00392850i \(0.00125049\pi\)
−0.496594 + 0.867983i \(0.665416\pi\)
\(24\) 0 0
\(25\) 2.32843 4.03295i 0.465685 0.806591i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.48528 1.57568 0.787839 0.615882i \(-0.211200\pi\)
0.787839 + 0.615882i \(0.211200\pi\)
\(30\) 0 0
\(31\) 2.58579 4.47871i 0.464421 0.804401i −0.534754 0.845008i \(-0.679596\pi\)
0.999175 + 0.0406069i \(0.0129291\pi\)
\(32\) 0 0
\(33\) 0.414214 + 0.717439i 0.0721053 + 0.124890i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.828427 1.43488i −0.136193 0.235892i 0.789860 0.613287i \(-0.210153\pi\)
−0.926052 + 0.377395i \(0.876820\pi\)
\(38\) 0 0
\(39\) −0.707107 + 1.22474i −0.113228 + 0.196116i
\(40\) 0 0
\(41\) −0.585786 −0.0914845 −0.0457422 0.998953i \(-0.514565\pi\)
−0.0457422 + 0.998953i \(0.514565\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −0.292893 + 0.507306i −0.0436619 + 0.0756247i
\(46\) 0 0
\(47\) 3.41421 + 5.91359i 0.498014 + 0.862586i 0.999997 0.00229145i \(-0.000729392\pi\)
−0.501983 + 0.864877i \(0.667396\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.12132 + 1.94218i 0.157016 + 0.271960i
\(52\) 0 0
\(53\) 6.65685 11.5300i 0.914389 1.58377i 0.106596 0.994302i \(-0.466005\pi\)
0.807793 0.589466i \(-0.200662\pi\)
\(54\) 0 0
\(55\) 0.485281 0.0654353
\(56\) 0 0
\(57\) 6.82843 0.904447
\(58\) 0 0
\(59\) 2.58579 4.47871i 0.336641 0.583079i −0.647158 0.762356i \(-0.724043\pi\)
0.983799 + 0.179277i \(0.0573759\pi\)
\(60\) 0 0
\(61\) −6.94975 12.0373i −0.889824 1.54122i −0.840083 0.542458i \(-0.817494\pi\)
−0.0497412 0.998762i \(-0.515840\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.414214 + 0.717439i 0.0513769 + 0.0889873i
\(66\) 0 0
\(67\) −4.00000 + 6.92820i −0.488678 + 0.846415i −0.999915 0.0130248i \(-0.995854\pi\)
0.511237 + 0.859440i \(0.329187\pi\)
\(68\) 0 0
\(69\) 4.82843 0.581274
\(70\) 0 0
\(71\) 0.828427 0.0983162 0.0491581 0.998791i \(-0.484346\pi\)
0.0491581 + 0.998791i \(0.484346\pi\)
\(72\) 0 0
\(73\) 5.53553 9.58783i 0.647885 1.12217i −0.335742 0.941954i \(-0.608987\pi\)
0.983627 0.180216i \(-0.0576797\pi\)
\(74\) 0 0
\(75\) −2.32843 4.03295i −0.268864 0.465685i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.17157 2.02922i −0.131812 0.228306i 0.792563 0.609790i \(-0.208746\pi\)
−0.924375 + 0.381485i \(0.875413\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −15.3137 −1.68090 −0.840449 0.541891i \(-0.817709\pi\)
−0.840449 + 0.541891i \(0.817709\pi\)
\(84\) 0 0
\(85\) 1.31371 0.142492
\(86\) 0 0
\(87\) 4.24264 7.34847i 0.454859 0.787839i
\(88\) 0 0
\(89\) 5.36396 + 9.29065i 0.568579 + 0.984807i 0.996707 + 0.0810892i \(0.0258399\pi\)
−0.428128 + 0.903718i \(0.640827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.58579 4.47871i −0.268134 0.464421i
\(94\) 0 0
\(95\) 2.00000 3.46410i 0.205196 0.355409i
\(96\) 0 0
\(97\) −7.75736 −0.787641 −0.393820 0.919187i \(-0.628847\pi\)
−0.393820 + 0.919187i \(0.628847\pi\)
\(98\) 0 0
\(99\) 0.828427 0.0832601
\(100\) 0 0
\(101\) −1.70711 + 2.95680i −0.169863 + 0.294212i −0.938372 0.345628i \(-0.887666\pi\)
0.768508 + 0.639840i \(0.220999\pi\)
\(102\) 0 0
\(103\) 5.41421 + 9.37769i 0.533478 + 0.924012i 0.999235 + 0.0390989i \(0.0124487\pi\)
−0.465757 + 0.884913i \(0.654218\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.24264 + 12.5446i 0.700173 + 1.21273i 0.968406 + 0.249380i \(0.0802269\pi\)
−0.268233 + 0.963354i \(0.586440\pi\)
\(108\) 0 0
\(109\) 5.65685 9.79796i 0.541828 0.938474i −0.456971 0.889482i \(-0.651066\pi\)
0.998799 0.0489926i \(-0.0156011\pi\)
\(110\) 0 0
\(111\) −1.65685 −0.157262
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 1.41421 2.44949i 0.131876 0.228416i
\(116\) 0 0
\(117\) 0.707107 + 1.22474i 0.0653720 + 0.113228i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.15685 + 8.93193i 0.468805 + 0.811994i
\(122\) 0 0
\(123\) −0.292893 + 0.507306i −0.0264093 + 0.0457422i
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 15.3137 1.35887 0.679436 0.733735i \(-0.262225\pi\)
0.679436 + 0.733735i \(0.262225\pi\)
\(128\) 0 0
\(129\) 4.00000 6.92820i 0.352180 0.609994i
\(130\) 0 0
\(131\) −3.65685 6.33386i −0.319501 0.553392i 0.660883 0.750489i \(-0.270182\pi\)
−0.980384 + 0.197097i \(0.936849\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.292893 + 0.507306i 0.0252082 + 0.0436619i
\(136\) 0 0
\(137\) −2.24264 + 3.88437i −0.191602 + 0.331864i −0.945781 0.324804i \(-0.894702\pi\)
0.754179 + 0.656668i \(0.228035\pi\)
\(138\) 0 0
\(139\) 1.65685 0.140533 0.0702663 0.997528i \(-0.477615\pi\)
0.0702663 + 0.997528i \(0.477615\pi\)
\(140\) 0 0
\(141\) 6.82843 0.575057
\(142\) 0 0
\(143\) 0.585786 1.01461i 0.0489859 0.0848461i
\(144\) 0 0
\(145\) −2.48528 4.30463i −0.206391 0.357480i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 + 8.66025i 0.409616 + 0.709476i 0.994847 0.101391i \(-0.0323294\pi\)
−0.585231 + 0.810867i \(0.698996\pi\)
\(150\) 0 0
\(151\) 4.82843 8.36308i 0.392932 0.680578i −0.599903 0.800073i \(-0.704794\pi\)
0.992835 + 0.119495i \(0.0381275\pi\)
\(152\) 0 0
\(153\) 2.24264 0.181307
\(154\) 0 0
\(155\) −3.02944 −0.243330
\(156\) 0 0
\(157\) 6.94975 12.0373i 0.554650 0.960682i −0.443280 0.896383i \(-0.646185\pi\)
0.997931 0.0642994i \(-0.0204813\pi\)
\(158\) 0 0
\(159\) −6.65685 11.5300i −0.527923 0.914389i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.82843 11.8272i −0.534844 0.926376i −0.999171 0.0407127i \(-0.987037\pi\)
0.464327 0.885664i \(-0.346296\pi\)
\(164\) 0 0
\(165\) 0.242641 0.420266i 0.0188896 0.0327177i
\(166\) 0 0
\(167\) −1.17157 −0.0906590 −0.0453295 0.998972i \(-0.514434\pi\)
−0.0453295 + 0.998972i \(0.514434\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 3.41421 5.91359i 0.261091 0.452224i
\(172\) 0 0
\(173\) 1.70711 + 2.95680i 0.129789 + 0.224801i 0.923595 0.383370i \(-0.125237\pi\)
−0.793806 + 0.608171i \(0.791903\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.58579 4.47871i −0.194360 0.336641i
\(178\) 0 0
\(179\) 8.89949 15.4144i 0.665179 1.15212i −0.314057 0.949404i \(-0.601688\pi\)
0.979237 0.202721i \(-0.0649783\pi\)
\(180\) 0 0
\(181\) −9.89949 −0.735824 −0.367912 0.929861i \(-0.619927\pi\)
−0.367912 + 0.929861i \(0.619927\pi\)
\(182\) 0 0
\(183\) −13.8995 −1.02748
\(184\) 0 0
\(185\) −0.485281 + 0.840532i −0.0356786 + 0.0617971i
\(186\) 0 0
\(187\) −0.928932 1.60896i −0.0679302 0.117659i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.58579 + 13.1390i 0.548888 + 0.950702i 0.998351 + 0.0574033i \(0.0182821\pi\)
−0.449463 + 0.893299i \(0.648385\pi\)
\(192\) 0 0
\(193\) −12.3137 + 21.3280i −0.886360 + 1.53522i −0.0422134 + 0.999109i \(0.513441\pi\)
−0.844147 + 0.536112i \(0.819892\pi\)
\(194\) 0 0
\(195\) 0.828427 0.0593249
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 2.82843 4.89898i 0.200502 0.347279i −0.748188 0.663486i \(-0.769076\pi\)
0.948690 + 0.316207i \(0.102409\pi\)
\(200\) 0 0
\(201\) 4.00000 + 6.92820i 0.282138 + 0.488678i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.171573 + 0.297173i 0.0119832 + 0.0207555i
\(206\) 0 0
\(207\) 2.41421 4.18154i 0.167799 0.290637i
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 14.3431 0.987423 0.493711 0.869626i \(-0.335640\pi\)
0.493711 + 0.869626i \(0.335640\pi\)
\(212\) 0 0
\(213\) 0.414214 0.717439i 0.0283814 0.0491581i
\(214\) 0 0
\(215\) −2.34315 4.05845i −0.159801 0.276784i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.53553 9.58783i −0.374057 0.647885i
\(220\) 0 0
\(221\) 1.58579 2.74666i 0.106672 0.184761i
\(222\) 0 0
\(223\) 13.6569 0.914531 0.457265 0.889330i \(-0.348829\pi\)
0.457265 + 0.889330i \(0.348829\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) −9.89949 + 17.1464i −0.657053 + 1.13805i 0.324322 + 0.945947i \(0.394864\pi\)
−0.981375 + 0.192102i \(0.938470\pi\)
\(228\) 0 0
\(229\) −7.53553 13.0519i −0.497962 0.862496i 0.502035 0.864847i \(-0.332585\pi\)
−0.999997 + 0.00235157i \(0.999251\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.2426 + 24.6690i 0.933066 + 1.61612i 0.778046 + 0.628207i \(0.216211\pi\)
0.155020 + 0.987911i \(0.450456\pi\)
\(234\) 0 0
\(235\) 2.00000 3.46410i 0.130466 0.225973i
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) 3.17157 0.205152 0.102576 0.994725i \(-0.467292\pi\)
0.102576 + 0.994725i \(0.467292\pi\)
\(240\) 0 0
\(241\) 10.9497 18.9655i 0.705335 1.22168i −0.261235 0.965275i \(-0.584130\pi\)
0.966570 0.256401i \(-0.0825369\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.82843 8.36308i −0.307225 0.532130i
\(248\) 0 0
\(249\) −7.65685 + 13.2621i −0.485233 + 0.840449i
\(250\) 0 0
\(251\) −8.48528 −0.535586 −0.267793 0.963476i \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 0.656854 1.13770i 0.0411338 0.0712458i
\(256\) 0 0
\(257\) 15.1213 + 26.1909i 0.943242 + 1.63374i 0.759233 + 0.650819i \(0.225574\pi\)
0.184009 + 0.982925i \(0.441093\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.24264 7.34847i −0.262613 0.454859i
\(262\) 0 0
\(263\) 12.4142 21.5020i 0.765493 1.32587i −0.174492 0.984659i \(-0.555828\pi\)
0.939985 0.341215i \(-0.110838\pi\)
\(264\) 0 0
\(265\) −7.79899 −0.479088
\(266\) 0 0
\(267\) 10.7279 0.656538
\(268\) 0 0
\(269\) −15.0208 + 26.0168i −0.915835 + 1.58627i −0.110162 + 0.993914i \(0.535137\pi\)
−0.805674 + 0.592360i \(0.798197\pi\)
\(270\) 0 0
\(271\) −6.58579 11.4069i −0.400058 0.692921i 0.593674 0.804705i \(-0.297677\pi\)
−0.993732 + 0.111784i \(0.964343\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.92893 + 3.34101i 0.116319 + 0.201470i
\(276\) 0 0
\(277\) 3.00000 5.19615i 0.180253 0.312207i −0.761714 0.647913i \(-0.775642\pi\)
0.941966 + 0.335707i \(0.108975\pi\)
\(278\) 0 0
\(279\) −5.17157 −0.309614
\(280\) 0 0
\(281\) −12.4853 −0.744809 −0.372405 0.928070i \(-0.621467\pi\)
−0.372405 + 0.928070i \(0.621467\pi\)
\(282\) 0 0
\(283\) −7.41421 + 12.8418i −0.440729 + 0.763365i −0.997744 0.0671373i \(-0.978613\pi\)
0.557014 + 0.830503i \(0.311947\pi\)
\(284\) 0 0
\(285\) −2.00000 3.46410i −0.118470 0.205196i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.98528 + 10.3668i 0.352075 + 0.609812i
\(290\) 0 0
\(291\) −3.87868 + 6.71807i −0.227372 + 0.393820i
\(292\) 0 0
\(293\) 1.07107 0.0625724 0.0312862 0.999510i \(-0.490040\pi\)
0.0312862 + 0.999510i \(0.490040\pi\)
\(294\) 0 0
\(295\) −3.02944 −0.176381
\(296\) 0 0
\(297\) 0.414214 0.717439i 0.0240351 0.0416300i
\(298\) 0 0
\(299\) −3.41421 5.91359i −0.197449 0.341992i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.70711 + 2.95680i 0.0980707 + 0.169863i
\(304\) 0 0
\(305\) −4.07107 + 7.05130i −0.233109 + 0.403756i
\(306\) 0 0
\(307\) 11.5147 0.657180 0.328590 0.944473i \(-0.393427\pi\)
0.328590 + 0.944473i \(0.393427\pi\)
\(308\) 0 0
\(309\) 10.8284 0.616008
\(310\) 0 0
\(311\) −13.0711 + 22.6398i −0.741192 + 1.28378i 0.210761 + 0.977538i \(0.432406\pi\)
−0.951953 + 0.306245i \(0.900927\pi\)
\(312\) 0 0
\(313\) 8.70711 + 15.0812i 0.492155 + 0.852437i 0.999959 0.00903532i \(-0.00287607\pi\)
−0.507804 + 0.861472i \(0.669543\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.6569 18.4582i −0.598549 1.03672i −0.993036 0.117815i \(-0.962411\pi\)
0.394487 0.918902i \(-0.370922\pi\)
\(318\) 0 0
\(319\) −3.51472 + 6.08767i −0.196786 + 0.340844i
\(320\) 0 0
\(321\) 14.4853 0.808490
\(322\) 0 0
\(323\) −15.3137 −0.852078
\(324\) 0 0
\(325\) −3.29289 + 5.70346i −0.182657 + 0.316371i
\(326\) 0 0
\(327\) −5.65685 9.79796i −0.312825 0.541828i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.34315 7.52255i −0.238721 0.413477i 0.721627 0.692282i \(-0.243395\pi\)
−0.960348 + 0.278806i \(0.910061\pi\)
\(332\) 0 0
\(333\) −0.828427 + 1.43488i −0.0453975 + 0.0786308i
\(334\) 0 0
\(335\) 4.68629 0.256039
\(336\) 0 0
\(337\) −16.9706 −0.924445 −0.462223 0.886764i \(-0.652948\pi\)
−0.462223 + 0.886764i \(0.652948\pi\)
\(338\) 0 0
\(339\) 3.00000 5.19615i 0.162938 0.282216i
\(340\) 0 0
\(341\) 2.14214 + 3.71029i 0.116003 + 0.200923i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.41421 2.44949i −0.0761387 0.131876i
\(346\) 0 0
\(347\) −2.07107 + 3.58719i −0.111181 + 0.192571i −0.916247 0.400615i \(-0.868797\pi\)
0.805066 + 0.593185i \(0.202130\pi\)
\(348\) 0 0
\(349\) 30.3848 1.62646 0.813230 0.581943i \(-0.197707\pi\)
0.813230 + 0.581943i \(0.197707\pi\)
\(350\) 0 0
\(351\) 1.41421 0.0754851
\(352\) 0 0
\(353\) 7.94975 13.7694i 0.423122 0.732869i −0.573121 0.819471i \(-0.694267\pi\)
0.996243 + 0.0866016i \(0.0276007\pi\)
\(354\) 0 0
\(355\) −0.242641 0.420266i −0.0128780 0.0223054i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.7279 30.7057i −0.935644 1.62058i −0.773482 0.633819i \(-0.781487\pi\)
−0.162162 0.986764i \(-0.551847\pi\)
\(360\) 0 0
\(361\) −13.8137 + 23.9260i −0.727037 + 1.25927i
\(362\) 0 0
\(363\) 10.3137 0.541329
\(364\) 0 0
\(365\) −6.48528 −0.339455
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 0 0
\(369\) 0.292893 + 0.507306i 0.0152474 + 0.0264093i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.34315 12.7187i −0.380214 0.658549i 0.610879 0.791724i \(-0.290816\pi\)
−0.991093 + 0.133175i \(0.957483\pi\)
\(374\) 0 0
\(375\) −2.82843 + 4.89898i −0.146059 + 0.252982i
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 0.686292 0.0352524 0.0176262 0.999845i \(-0.494389\pi\)
0.0176262 + 0.999845i \(0.494389\pi\)
\(380\) 0 0
\(381\) 7.65685 13.2621i 0.392273 0.679436i
\(382\) 0 0
\(383\) 12.4853 + 21.6251i 0.637968 + 1.10499i 0.985878 + 0.167465i \(0.0535582\pi\)
−0.347910 + 0.937528i \(0.613108\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 6.92820i −0.203331 0.352180i
\(388\) 0 0
\(389\) 7.07107 12.2474i 0.358517 0.620970i −0.629196 0.777247i \(-0.716616\pi\)
0.987713 + 0.156276i \(0.0499491\pi\)
\(390\) 0 0
\(391\) −10.8284 −0.547617
\(392\) 0 0
\(393\) −7.31371 −0.368928
\(394\) 0 0
\(395\) −0.686292 + 1.18869i −0.0345311 + 0.0598096i
\(396\) 0 0
\(397\) −3.63604 6.29780i −0.182488 0.316078i 0.760239 0.649643i \(-0.225082\pi\)
−0.942727 + 0.333565i \(0.891748\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.24264 + 10.8126i 0.311743 + 0.539954i 0.978740 0.205106i \(-0.0657540\pi\)
−0.666997 + 0.745060i \(0.732421\pi\)
\(402\) 0 0
\(403\) −3.65685 + 6.33386i −0.182161 + 0.315512i
\(404\) 0 0
\(405\) 0.585786 0.0291080
\(406\) 0 0
\(407\) 1.37258 0.0680364
\(408\) 0 0
\(409\) −8.12132 + 14.0665i −0.401573 + 0.695546i −0.993916 0.110141i \(-0.964870\pi\)
0.592343 + 0.805686i \(0.298203\pi\)
\(410\) 0 0
\(411\) 2.24264 + 3.88437i 0.110621 + 0.191602i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.48528 + 7.76874i 0.220174 + 0.381352i
\(416\) 0 0
\(417\) 0.828427 1.43488i 0.0405683 0.0702663i
\(418\) 0 0
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) 0 0
\(421\) 6.68629 0.325870 0.162935 0.986637i \(-0.447904\pi\)
0.162935 + 0.986637i \(0.447904\pi\)
\(422\) 0 0
\(423\) 3.41421 5.91359i 0.166005 0.287529i
\(424\) 0 0
\(425\) 5.22183 + 9.04447i 0.253296 + 0.438721i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.585786 1.01461i −0.0282820 0.0489859i
\(430\) 0 0
\(431\) −15.2426 + 26.4010i −0.734212 + 1.27169i 0.220856 + 0.975306i \(0.429115\pi\)
−0.955068 + 0.296386i \(0.904218\pi\)
\(432\) 0 0
\(433\) 26.3848 1.26797 0.633986 0.773345i \(-0.281418\pi\)
0.633986 + 0.773345i \(0.281418\pi\)
\(434\) 0 0
\(435\) −4.97056 −0.238320
\(436\) 0 0
\(437\) −16.4853 + 28.5533i −0.788598 + 1.36589i
\(438\) 0 0
\(439\) −1.65685 2.86976i −0.0790773 0.136966i 0.823775 0.566917i \(-0.191864\pi\)
−0.902852 + 0.429951i \(0.858531\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.2426 + 33.3292i 0.914245 + 1.58352i 0.808002 + 0.589179i \(0.200549\pi\)
0.106243 + 0.994340i \(0.466118\pi\)
\(444\) 0 0
\(445\) 3.14214 5.44234i 0.148952 0.257992i
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0.242641 0.420266i 0.0114255 0.0197896i
\(452\) 0 0
\(453\) −4.82843 8.36308i −0.226859 0.392932i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.3137 17.8639i −0.482455 0.835636i 0.517342 0.855779i \(-0.326921\pi\)
−0.999797 + 0.0201422i \(0.993588\pi\)
\(458\) 0 0
\(459\) 1.12132 1.94218i 0.0523388 0.0906534i
\(460\) 0 0
\(461\) 18.2426 0.849644 0.424822 0.905277i \(-0.360337\pi\)
0.424822 + 0.905277i \(0.360337\pi\)
\(462\) 0 0
\(463\) −20.9706 −0.974585 −0.487292 0.873239i \(-0.662015\pi\)
−0.487292 + 0.873239i \(0.662015\pi\)
\(464\) 0 0
\(465\) −1.51472 + 2.62357i −0.0702434 + 0.121665i
\(466\) 0 0
\(467\) −5.41421 9.37769i −0.250540 0.433948i 0.713135 0.701027i \(-0.247275\pi\)
−0.963675 + 0.267079i \(0.913941\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.94975 12.0373i −0.320227 0.554650i
\(472\) 0 0
\(473\) −3.31371 + 5.73951i −0.152364 + 0.263903i
\(474\) 0 0
\(475\) 31.7990 1.45904
\(476\) 0 0
\(477\) −13.3137 −0.609593
\(478\) 0 0
\(479\) −21.5563 + 37.3367i −0.984935 + 1.70596i −0.342709 + 0.939442i \(0.611344\pi\)
−0.642226 + 0.766515i \(0.721989\pi\)
\(480\) 0 0
\(481\) 1.17157 + 2.02922i 0.0534191 + 0.0925246i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.27208 + 3.93535i 0.103170 + 0.178695i
\(486\) 0 0
\(487\) −10.4853 + 18.1610i −0.475133 + 0.822955i −0.999594 0.0284792i \(-0.990934\pi\)
0.524461 + 0.851435i \(0.324267\pi\)
\(488\) 0 0
\(489\) −13.6569 −0.617584
\(490\) 0 0
\(491\) −24.1421 −1.08952 −0.544760 0.838592i \(-0.683379\pi\)
−0.544760 + 0.838592i \(0.683379\pi\)
\(492\) 0 0
\(493\) −9.51472 + 16.4800i −0.428521 + 0.742221i
\(494\) 0 0
\(495\) −0.242641 0.420266i −0.0109059 0.0188896i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.1421 24.4949i −0.633089 1.09654i −0.986917 0.161232i \(-0.948453\pi\)
0.353828 0.935311i \(-0.384880\pi\)
\(500\) 0 0
\(501\) −0.585786 + 1.01461i −0.0261710 + 0.0453295i
\(502\) 0 0
\(503\) 14.6274 0.652204 0.326102 0.945335i \(-0.394265\pi\)
0.326102 + 0.945335i \(0.394265\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −5.50000 + 9.52628i −0.244264 + 0.423077i
\(508\) 0 0
\(509\) −18.5355 32.1045i −0.821573 1.42301i −0.904510 0.426452i \(-0.859763\pi\)
0.0829373 0.996555i \(-0.473570\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.41421 5.91359i −0.150741 0.261091i
\(514\) 0 0
\(515\) 3.17157 5.49333i 0.139756 0.242065i
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 3.41421 0.149867
\(520\) 0 0
\(521\) −15.3640 + 26.6112i −0.673107 + 1.16586i 0.303911 + 0.952700i \(0.401707\pi\)
−0.977018 + 0.213156i \(0.931626\pi\)
\(522\) 0 0
\(523\) 4.82843 + 8.36308i 0.211132 + 0.365692i 0.952069 0.305883i \(-0.0989516\pi\)
−0.740937 + 0.671575i \(0.765618\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.79899 + 10.0441i 0.252608 + 0.437530i
\(528\) 0 0
\(529\) −0.156854 + 0.271680i −0.00681975 + 0.0118122i
\(530\) 0 0
\(531\) −5.17157 −0.224427
\(532\) 0 0
\(533\) 0.828427 0.0358832
\(534\) 0 0
\(535\) 4.24264 7.34847i 0.183425 0.317702i
\(536\) 0 0
\(537\) −8.89949 15.4144i −0.384042 0.665179i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.3137 31.7203i −0.787368 1.36376i −0.927574 0.373639i \(-0.878110\pi\)
0.140206 0.990122i \(-0.455223\pi\)
\(542\) 0 0
\(543\) −4.94975 + 8.57321i −0.212414 + 0.367912i
\(544\) 0 0
\(545\) −6.62742 −0.283887
\(546\) 0 0
\(547\) −28.9706 −1.23869 −0.619346 0.785118i \(-0.712602\pi\)
−0.619346 + 0.785118i \(0.712602\pi\)
\(548\) 0 0
\(549\) −6.94975 + 12.0373i −0.296608 + 0.513740i
\(550\) 0 0
\(551\) 28.9706 + 50.1785i 1.23419 + 2.13768i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.485281 + 0.840532i 0.0205990 + 0.0356786i
\(556\) 0 0
\(557\) 11.9706 20.7336i 0.507209 0.878512i −0.492756 0.870167i \(-0.664011\pi\)
0.999965 0.00834436i \(-0.00265612\pi\)
\(558\) 0 0
\(559\) −11.3137 −0.478519
\(560\) 0 0
\(561\) −1.85786 −0.0784391
\(562\) 0 0
\(563\) 10.5858 18.3351i 0.446138 0.772733i −0.551993 0.833849i \(-0.686132\pi\)
0.998131 + 0.0611156i \(0.0194658\pi\)
\(564\) 0 0
\(565\) −1.75736 3.04384i −0.0739327 0.128055i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.585786 1.01461i −0.0245574 0.0425347i 0.853486 0.521117i \(-0.174484\pi\)
−0.878043 + 0.478582i \(0.841151\pi\)
\(570\) 0 0
\(571\) −8.14214 + 14.1026i −0.340738 + 0.590175i −0.984570 0.174992i \(-0.944010\pi\)
0.643832 + 0.765167i \(0.277343\pi\)
\(572\) 0 0
\(573\) 15.1716 0.633802
\(574\) 0 0
\(575\) 22.4853 0.937701
\(576\) 0 0
\(577\) 3.29289 5.70346i 0.137085 0.237438i −0.789307 0.613999i \(-0.789560\pi\)
0.926392 + 0.376561i \(0.122893\pi\)
\(578\) 0 0
\(579\) 12.3137 + 21.3280i 0.511740 + 0.886360i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 5.51472 + 9.55177i 0.228396 + 0.395594i
\(584\) 0 0
\(585\) 0.414214 0.717439i 0.0171256 0.0296624i
\(586\) 0 0
\(587\) 21.1716 0.873844 0.436922 0.899499i \(-0.356069\pi\)
0.436922 + 0.899499i \(0.356069\pi\)
\(588\) 0 0
\(589\) 35.3137 1.45508
\(590\) 0 0
\(591\) 1.00000 1.73205i 0.0411345 0.0712470i
\(592\) 0 0
\(593\) −7.46447 12.9288i −0.306529 0.530924i 0.671072 0.741392i \(-0.265834\pi\)
−0.977601 + 0.210469i \(0.932501\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.82843 4.89898i −0.115760 0.200502i
\(598\) 0 0
\(599\) −6.07107 + 10.5154i −0.248057 + 0.429648i −0.962987 0.269549i \(-0.913125\pi\)
0.714930 + 0.699196i \(0.246459\pi\)
\(600\) 0 0
\(601\) −3.75736 −0.153266 −0.0766329 0.997059i \(-0.524417\pi\)
−0.0766329 + 0.997059i \(0.524417\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 3.02082 5.23221i 0.122814 0.212719i
\(606\) 0 0
\(607\) −23.7990 41.2211i −0.965971 1.67311i −0.706983 0.707231i \(-0.749944\pi\)
−0.258988 0.965880i \(-0.583389\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.82843 8.36308i −0.195337 0.338334i
\(612\) 0 0
\(613\) 6.82843 11.8272i 0.275798 0.477695i −0.694538 0.719456i \(-0.744391\pi\)
0.970336 + 0.241760i \(0.0777247\pi\)
\(614\) 0 0
\(615\) 0.343146 0.0138370
\(616\) 0 0
\(617\) −29.4558 −1.18585 −0.592924 0.805259i \(-0.702026\pi\)
−0.592924 + 0.805259i \(0.702026\pi\)
\(618\) 0 0
\(619\) 8.14214 14.1026i 0.327260 0.566831i −0.654707 0.755883i \(-0.727208\pi\)
0.981967 + 0.189052i \(0.0605414\pi\)
\(620\) 0 0
\(621\) −2.41421 4.18154i −0.0968791 0.167799i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.98528 17.2950i −0.399411 0.691801i
\(626\) 0 0
\(627\) −2.82843 + 4.89898i −0.112956 + 0.195646i
\(628\) 0 0
\(629\) 3.71573 0.148156
\(630\) 0 0
\(631\) −48.2843 −1.92217 −0.961083 0.276259i \(-0.910905\pi\)
−0.961083 + 0.276259i \(0.910905\pi\)
\(632\) 0 0
\(633\) 7.17157 12.4215i 0.285044 0.493711i
\(634\) 0 0
\(635\) −4.48528 7.76874i −0.177993 0.308293i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.414214 0.717439i −0.0163860 0.0283814i
\(640\) 0 0
\(641\) −15.4142 + 26.6982i −0.608825 + 1.05452i 0.382610 + 0.923910i \(0.375025\pi\)
−0.991434 + 0.130605i \(0.958308\pi\)
\(642\) 0 0
\(643\) −36.4853 −1.43884 −0.719420 0.694576i \(-0.755592\pi\)
−0.719420 + 0.694576i \(0.755592\pi\)
\(644\) 0 0
\(645\) −4.68629 −0.184523
\(646\) 0 0
\(647\) 4.58579 7.94282i 0.180286 0.312264i −0.761692 0.647939i \(-0.775631\pi\)
0.941978 + 0.335675i \(0.108964\pi\)
\(648\) 0 0
\(649\) 2.14214 + 3.71029i 0.0840862 + 0.145642i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.41421 + 2.44949i 0.0553425 + 0.0958559i 0.892369 0.451306i \(-0.149042\pi\)
−0.837027 + 0.547162i \(0.815708\pi\)
\(654\) 0 0
\(655\) −2.14214 + 3.71029i −0.0837002 + 0.144973i
\(656\) 0 0
\(657\) −11.0711 −0.431923
\(658\) 0 0
\(659\) −42.4853 −1.65499 −0.827496 0.561472i \(-0.810235\pi\)
−0.827496 + 0.561472i \(0.810235\pi\)
\(660\) 0 0
\(661\) −23.6777 + 41.0109i −0.920955 + 1.59514i −0.123013 + 0.992405i \(0.539256\pi\)
−0.797941 + 0.602735i \(0.794078\pi\)
\(662\) 0 0
\(663\) −1.58579 2.74666i −0.0615868 0.106672i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20.4853 + 35.4815i 0.793193 + 1.37385i
\(668\) 0 0
\(669\) 6.82843 11.8272i 0.264002 0.457265i
\(670\) 0 0
\(671\) 11.5147 0.444521
\(672\) 0 0
\(673\) −7.31371 −0.281923 −0.140961 0.990015i \(-0.545019\pi\)
−0.140961 + 0.990015i \(0.545019\pi\)
\(674\) 0 0
\(675\) −2.32843 + 4.03295i −0.0896212 + 0.155228i
\(676\) 0 0
\(677\) −14.7782 25.5965i −0.567971 0.983755i −0.996766 0.0803535i \(-0.974395\pi\)
0.428795 0.903402i \(-0.358938\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9.89949 + 17.1464i 0.379349 + 0.657053i
\(682\) 0 0
\(683\) −8.07107 + 13.9795i −0.308831 + 0.534911i −0.978107 0.208103i \(-0.933271\pi\)
0.669276 + 0.743014i \(0.266604\pi\)
\(684\) 0 0
\(685\) 2.62742 0.100388
\(686\) 0 0
\(687\) −15.0711 −0.574997
\(688\) 0 0
\(689\) −9.41421 + 16.3059i −0.358653 + 0.621205i
\(690\) 0 0
\(691\) 14.0000 + 24.2487i 0.532585 + 0.922464i 0.999276 + 0.0380440i \(0.0121127\pi\)
−0.466691 + 0.884420i \(0.654554\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.485281 0.840532i −0.0184078 0.0318832i
\(696\) 0 0
\(697\) 0.656854 1.13770i 0.0248801 0.0430936i
\(698\) 0 0
\(699\) 28.4853 1.07741
\(700\) 0 0
\(701\) −5.17157 −0.195328 −0.0976638 0.995219i \(-0.531137\pi\)
−0.0976638 + 0.995219i \(0.531137\pi\)
\(702\) 0 0
\(703\) 5.65685 9.79796i 0.213352 0.369537i
\(704\) 0 0
\(705\) −2.00000 3.46410i −0.0753244 0.130466i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 + 6.92820i 0.150223 + 0.260194i 0.931309 0.364229i \(-0.118667\pi\)
−0.781086 + 0.624423i \(0.785334\pi\)
\(710\) 0 0
\(711\) −1.17157 + 2.02922i −0.0439374 + 0.0761018i
\(712\) 0 0
\(713\) 24.9706 0.935155
\(714\) 0 0
\(715\) −0.686292 −0.0256658
\(716\) 0 0
\(717\) 1.58579 2.74666i 0.0592223 0.102576i
\(718\) 0 0
\(719\) −2.34315 4.05845i −0.0873846 0.151355i 0.819020 0.573765i \(-0.194518\pi\)
−0.906405 + 0.422410i \(0.861184\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −10.9497 18.9655i −0.407225 0.705335i
\(724\) 0 0
\(725\) 19.7574 34.2208i 0.733770 1.27093i
\(726\) 0 0
\(727\) −25.4558 −0.944105 −0.472052 0.881570i \(-0.656487\pi\)
−0.472052 + 0.881570i \(0.656487\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.97056 + 15.5375i −0.331788 + 0.574674i
\(732\) 0 0
\(733\) −5.87868 10.1822i −0.217134 0.376087i 0.736797 0.676114i \(-0.236338\pi\)
−0.953931 + 0.300027i \(0.903004\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.31371 5.73951i −0.122062 0.211418i
\(738\) 0 0
\(739\) 10.1421 17.5667i 0.373084 0.646201i −0.616954 0.786999i \(-0.711633\pi\)
0.990038 + 0.140798i \(0.0449668\pi\)
\(740\) 0 0
\(741\) −9.65685 −0.354753
\(742\) 0 0
\(743\) 11.1716 0.409845 0.204923 0.978778i \(-0.434306\pi\)
0.204923 + 0.978778i \(0.434306\pi\)
\(744\) 0 0
\(745\) 2.92893 5.07306i 0.107308 0.185863i
\(746\) 0 0
\(747\) 7.65685 + 13.2621i 0.280150 + 0.485233i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.8284 + 25.6836i 0.541097 + 0.937207i 0.998841 + 0.0481236i \(0.0153242\pi\)
−0.457744 + 0.889084i \(0.651343\pi\)
\(752\) 0 0
\(753\) −4.24264 + 7.34847i −0.154610 + 0.267793i
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) −11.3137 −0.411204 −0.205602 0.978636i \(-0.565915\pi\)
−0.205602 + 0.978636i \(0.565915\pi\)
\(758\) 0 0
\(759\) −2.00000 + 3.46410i −0.0725954 + 0.125739i
\(760\) 0 0
\(761\) 4.87868 + 8.45012i 0.176852 + 0.306317i 0.940801 0.338960i \(-0.110075\pi\)
−0.763949 + 0.645277i \(0.776742\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.656854 1.13770i −0.0237486 0.0411338i
\(766\) 0 0
\(767\) −3.65685 + 6.33386i −0.132041 + 0.228702i
\(768\) 0 0
\(769\) 15.0711 0.543477 0.271738 0.962371i \(-0.412401\pi\)
0.271738 + 0.962371i \(0.412401\pi\)
\(770\) 0 0
\(771\) 30.2426 1.08916
\(772\) 0 0
\(773\) 1.70711 2.95680i 0.0614004 0.106349i −0.833691 0.552231i \(-0.813777\pi\)
0.895092 + 0.445882i \(0.147110\pi\)
\(774\) 0 0
\(775\) −12.0416 20.8567i −0.432548 0.749195i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 3.46410i −0.0716574 0.124114i
\(780\) 0 0
\(781\) −0.343146 + 0.594346i −0.0122787 + 0.0212674i
\(782\) 0 0
\(783\) −8.48528 −0.303239
\(784\) 0 0
\(785\) −8.14214 −0.290605
\(786\) 0 0
\(787\) −5.31371 + 9.20361i −0.189413 + 0.328073i −0.945055 0.326912i \(-0.893992\pi\)
0.755642 + 0.654985i \(0.227325\pi\)
\(788\) 0 0
\(789\) −12.4142 21.5020i −0.441958 0.765493i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 9.82843 + 17.0233i 0.349018 + 0.604516i
\(794\) 0 0
\(795\) −3.89949 + 6.75412i −0.138301 + 0.239544i
\(796\) 0 0
\(797\) −43.2132 −1.53069 −0.765345 0.643620i \(-0.777432\pi\)
−0.765345 + 0.643620i \(0.777432\pi\)
\(798\) 0 0
\(799\) −15.3137 −0.541760
\(800\) 0 0
\(801\) 5.36396 9.29065i 0.189526 0.328269i
\(802\) 0 0
\(803\) 4.58579 + 7.94282i 0.161829 + 0.280296i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0208 + 26.0168i 0.528758 + 0.915835i
\(808\) 0 0
\(809\) 15.0000 25.9808i 0.527372 0.913435i −0.472119 0.881535i \(-0.656511\pi\)
0.999491 0.0319002i \(-0.0101559\pi\)
\(810\) 0 0
\(811\) 20.9706 0.736376 0.368188 0.929751i \(-0.379978\pi\)
0.368188 + 0.929751i \(0.379978\pi\)
\(812\) 0 0
\(813\) −13.1716 −0.461947
\(814\) 0 0
\(815\) −4.00000 + 6.92820i −0.140114 + 0.242684i
\(816\) 0 0
\(817\) 27.3137 + 47.3087i 0.955586 + 1.65512i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.00000 8.66025i −0.174501 0.302245i 0.765487 0.643451i \(-0.222498\pi\)
−0.939989 + 0.341206i \(0.889165\pi\)
\(822\) 0 0
\(823\) −1.17157 + 2.02922i −0.0408385 + 0.0707343i −0.885722 0.464216i \(-0.846336\pi\)
0.844884 + 0.534950i \(0.179670\pi\)
\(824\) 0 0
\(825\) 3.85786 0.134314
\(826\) 0 0
\(827\) 19.4558 0.676546 0.338273 0.941048i \(-0.390157\pi\)
0.338273 + 0.941048i \(0.390157\pi\)
\(828\) 0 0
\(829\) −1.63604 + 2.83370i −0.0568220 + 0.0984186i −0.893037 0.449983i \(-0.851430\pi\)
0.836215 + 0.548401i \(0.184763\pi\)
\(830\) 0 0
\(831\) −3.00000 5.19615i −0.104069 0.180253i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0.343146 + 0.594346i 0.0118750 + 0.0205682i
\(836\) 0 0
\(837\) −2.58579 + 4.47871i −0.0893779 + 0.154807i
\(838\) 0 0
\(839\) −49.1716 −1.69759 −0.848796 0.528721i \(-0.822672\pi\)
−0.848796 + 0.528721i \(0.822672\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) −6.24264 + 10.8126i −0.215008 + 0.372405i
\(844\) 0 0
\(845\) 3.22183 + 5.58037i 0.110834 + 0.191970i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.41421 + 12.8418i 0.254455 + 0.440729i
\(850\) 0 0
\(851\) 4.00000 6.92820i 0.137118 0.237496i
\(852\) 0 0
\(853\) 36.0416 1.23404 0.617021 0.786947i \(-0.288339\pi\)
0.617021 + 0.786947i \(0.288339\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 21.8492 37.8440i 0.746356 1.29273i −0.203203 0.979137i \(-0.565135\pi\)
0.949559 0.313590i \(-0.101532\pi\)
\(858\) 0 0
\(859\) −19.4142 33.6264i −0.662404 1.14732i −0.979982 0.199086i \(-0.936203\pi\)
0.317578 0.948232i \(-0.397131\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.7574 25.5605i −0.502346 0.870089i −0.999996 0.00271146i \(-0.999137\pi\)
0.497650 0.867378i \(-0.334196\pi\)
\(864\) 0 0
\(865\) 1.00000 1.73205i 0.0340010 0.0588915i
\(866\) 0 0
\(867\) 11.9706 0.406542
\(868\) 0 0
\(869\) 1.94113 0.0658482
\(870\) 0 0
\(871\) 5.65685 9.79796i 0.191675 0.331991i
\(872\) 0 0
\(873\) 3.87868 + 6.71807i 0.131273 + 0.227372i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −10.1421 17.5667i −0.342476 0.593185i 0.642416 0.766356i \(-0.277932\pi\)
−0.984892 + 0.173171i \(0.944599\pi\)
\(878\) 0 0
\(879\) 0.535534 0.927572i 0.0180631 0.0312862i
\(880\) 0 0
\(881\) −25.0711 −0.844666 −0.422333 0.906441i \(-0.638789\pi\)
−0.422333 + 0.906441i \(0.638789\pi\)
\(882\) 0 0
\(883\) −18.3431 −0.617296 −0.308648 0.951176i \(-0.599877\pi\)
−0.308648 + 0.951176i \(0.599877\pi\)
\(884\) 0 0
\(885\) −1.51472 + 2.62357i −0.0509167 + 0.0881903i
\(886\) 0 0
\(887\) 10.7279 + 18.5813i 0.360208 + 0.623899i 0.987995 0.154486i \(-0.0493723\pi\)
−0.627787 + 0.778386i \(0.716039\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.414214 0.717439i −0.0138767 0.0240351i
\(892\) 0 0
\(893\) −23.3137 + 40.3805i −0.780164 + 1.35128i
\(894\) 0 0
\(895\) −10.4264 −0.348516
\(896\) 0 0
\(897\) −6.82843 −0.227995
\(898\) 0 0
\(899\) 21.9411 38.0031i 0.731778 1.26748i
\(900\) 0 0
\(901\) 14.9289 + 25.8577i 0.497355 + 0.861444i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.89949 + 5.02207i 0.0963825 + 0.166939i
\(906\) 0 0
\(907\) −3.51472 + 6.08767i −0.116704 + 0.202138i −0.918460 0.395514i \(-0.870566\pi\)
0.801755 + 0.597652i \(0.203900\pi\)
\(908\) 0 0
\(909\) 3.41421 0.113242
\(910\) 0 0
\(911\) 54.4853 1.80518 0.902589 0.430503i \(-0.141664\pi\)
0.902589 + 0.430503i \(0.141664\pi\)
\(912\) 0 0
\(913\) 6.34315 10.9867i 0.209927 0.363605i
\(914\) 0 0
\(915\) 4.07107 + 7.05130i 0.134585 + 0.233109i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −24.6274 42.6559i −0.812384 1.40709i −0.911192 0.411983i \(-0.864836\pi\)
0.0988080 0.995107i \(-0.468497\pi\)
\(920\) 0 0
\(921\) 5.75736 9.97204i 0.189711 0.328590i
\(922\) 0 0
\(923\) −1.17157 −0.0385628
\(924\) 0 0
\(925\) −7.71573 −0.253692
\(926\) 0 0
\(927\) 5.41421 9.37769i 0.177826 0.308004i
\(928\) 0 0
\(929\) −30.4350 52.7150i −0.998541 1.72952i −0.546036 0.837762i \(-0.683864\pi\)
−0.452505 0.891762i \(-0.649469\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13.0711 + 22.6398i 0.427928 + 0.741192i
\(934\) 0 0
\(935\) −0.544156 + 0.942506i −0.0177958 + 0.0308232i
\(936\) 0 0
\(937\) −6.10051 −0.199295 −0.0996474 0.995023i \(-0.531771\pi\)
−0.0996474 + 0.995023i \(0.531771\pi\)
\(938\) 0 0
\(939\) 17.4142 0.568291
\(940\) 0 0
\(941\) 21.9497 38.0181i 0.715541 1.23935i −0.247209 0.968962i \(-0.579513\pi\)
0.962750 0.270392i \(-0.0871532\pi\)
\(942\) 0 0
\(943\) −1.41421 2.44949i −0.0460531 0.0797664i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.0711 + 17.4436i 0.327266 + 0.566841i 0.981968 0.189046i \(-0.0605394\pi\)
−0.654703 + 0.755887i \(0.727206\pi\)
\(948\) 0 0
\(949\) −7.82843 + 13.5592i −0.254121 + 0.440151i
\(950\) 0 0
\(951\) −21.3137 −0.691144
\(952\) 0 0
\(953\) −7.37258 −0.238821 −0.119411 0.992845i \(-0.538101\pi\)
−0.119411 + 0.992845i \(0.538101\pi\)
\(954\) 0 0
\(955\) 4.44365 7.69663i 0.143793 0.249057i
\(956\) 0 0
\(957\) 3.51472 + 6.08767i 0.113615 + 0.196786i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.12742 + 3.68479i 0.0686264 + 0.118864i
\(962\) 0 0
\(963\) 7.24264 12.5446i 0.233391 0.404245i
\(964\) 0 0
\(965\) 14.4264 0.464402
\(966\) 0 0
\(967\) 34.6274 1.11354 0.556771 0.830666i \(-0.312040\pi\)
0.556771 + 0.830666i \(0.312040\pi\)
\(968\) 0 0
\(969\) −7.65685 + 13.2621i −0.245974 + 0.426039i
\(970\) 0 0
\(971\) −28.6274 49.5841i −0.918698 1.59123i −0.801396 0.598134i \(-0.795909\pi\)
−0.117302 0.993096i \(-0.537424\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 3.29289 + 5.70346i 0.105457 + 0.182657i
\(976\) 0 0
\(977\) −10.9289 + 18.9295i −0.349648 + 0.605607i −0.986187 0.165637i \(-0.947032\pi\)
0.636539 + 0.771244i \(0.280365\pi\)
\(978\) 0 0
\(979\) −8.88730 −0.284039
\(980\) 0 0
\(981\) −11.3137 −0.361219
\(982\) 0 0
\(983\) −7.31371 + 12.6677i −0.233271 + 0.404037i −0.958769 0.284187i \(-0.908276\pi\)
0.725498 + 0.688225i \(0.241610\pi\)
\(984\) 0 0
\(985\) −0.585786 1.01461i −0.0186647 0.0323282i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19.3137 + 33.4523i 0.614140 + 1.06372i
\(990\) 0 0
\(991\) 11.6569 20.1903i 0.370292 0.641365i −0.619318 0.785140i \(-0.712591\pi\)
0.989610 + 0.143775i \(0.0459242\pi\)
\(992\) 0 0
\(993\) −8.68629 −0.275651
\(994\) 0 0
\(995\) −3.31371 −0.105052
\(996\) 0 0
\(997\) 17.2929 29.9522i 0.547671 0.948595i −0.450762 0.892644i \(-0.648848\pi\)
0.998434 0.0559506i \(-0.0178189\pi\)
\(998\) 0 0
\(999\) 0.828427 + 1.43488i 0.0262103 + 0.0453975i
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 2352.2.q.be.1537.2 4
4.3 odd 2 1176.2.q.k.361.2 4
7.2 even 3 inner 2352.2.q.be.961.2 4
7.3 odd 6 2352.2.a.bd.1.2 2
7.4 even 3 2352.2.a.bb.1.1 2
7.5 odd 6 2352.2.q.bc.961.1 4
7.6 odd 2 2352.2.q.bc.1537.1 4
12.11 even 2 3528.2.s.bm.361.1 4
21.11 odd 6 7056.2.a.cg.1.2 2
21.17 even 6 7056.2.a.cx.1.1 2
28.3 even 6 1176.2.a.j.1.2 2
28.11 odd 6 1176.2.a.o.1.1 yes 2
28.19 even 6 1176.2.q.o.961.1 4
28.23 odd 6 1176.2.q.k.961.2 4
28.27 even 2 1176.2.q.o.361.1 4
56.3 even 6 9408.2.a.ee.1.1 2
56.11 odd 6 9408.2.a.dg.1.2 2
56.45 odd 6 9408.2.a.ds.1.1 2
56.53 even 6 9408.2.a.du.1.2 2
84.11 even 6 3528.2.a.bb.1.2 2
84.23 even 6 3528.2.s.bm.3313.1 4
84.47 odd 6 3528.2.s.bd.3313.2 4
84.59 odd 6 3528.2.a.bl.1.1 2
84.83 odd 2 3528.2.s.bd.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.j.1.2 2 28.3 even 6
1176.2.a.o.1.1 yes 2 28.11 odd 6
1176.2.q.k.361.2 4 4.3 odd 2
1176.2.q.k.961.2 4 28.23 odd 6
1176.2.q.o.361.1 4 28.27 even 2
1176.2.q.o.961.1 4 28.19 even 6
2352.2.a.bb.1.1 2 7.4 even 3
2352.2.a.bd.1.2 2 7.3 odd 6
2352.2.q.bc.961.1 4 7.5 odd 6
2352.2.q.bc.1537.1 4 7.6 odd 2
2352.2.q.be.961.2 4 7.2 even 3 inner
2352.2.q.be.1537.2 4 1.1 even 1 trivial
3528.2.a.bb.1.2 2 84.11 even 6
3528.2.a.bl.1.1 2 84.59 odd 6
3528.2.s.bd.361.2 4 84.83 odd 2
3528.2.s.bd.3313.2 4 84.47 odd 6
3528.2.s.bm.361.1 4 12.11 even 2
3528.2.s.bm.3313.1 4 84.23 even 6
7056.2.a.cg.1.2 2 21.11 odd 6
7056.2.a.cx.1.1 2 21.17 even 6
9408.2.a.dg.1.2 2 56.11 odd 6
9408.2.a.ds.1.1 2 56.45 odd 6
9408.2.a.du.1.2 2 56.53 even 6
9408.2.a.ee.1.1 2 56.3 even 6