Properties

Label 2100.2.x.b.1693.3
Level $2100$
Weight $2$
Character 2100.1693
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1357,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1357");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.x (of order \(4\), degree \(2\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1693.3
Root \(1.28897 + 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1693
Dual form 2100.2.x.b.1357.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +(-1.87083 + 1.87083i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{3} +(-1.87083 + 1.87083i) q^{7} -1.00000i q^{9} -2.00000 q^{11} +(1.41421 - 1.41421i) q^{13} +(2.82843 + 2.82843i) q^{17} -5.29150 q^{19} +2.64575i q^{21} +(-3.74166 - 3.74166i) q^{23} +(-0.707107 - 0.707107i) q^{27} +6.00000i q^{29} +5.29150i q^{31} +(-1.41421 + 1.41421i) q^{33} -2.00000i q^{39} +10.5830i q^{41} +(3.74166 + 3.74166i) q^{43} +(-5.65685 - 5.65685i) q^{47} -7.00000i q^{49} +4.00000 q^{51} +(-3.74166 - 3.74166i) q^{53} +(-3.74166 + 3.74166i) q^{57} -10.5830 q^{59} +10.5830i q^{61} +(1.87083 + 1.87083i) q^{63} +(-3.74166 + 3.74166i) q^{67} -5.29150 q^{69} -2.00000 q^{71} +(-9.89949 + 9.89949i) q^{73} +(3.74166 - 3.74166i) q^{77} +4.00000i q^{79} -1.00000 q^{81} +(4.24264 + 4.24264i) q^{87} +5.29150i q^{91} +(3.74166 + 3.74166i) q^{93} +(-1.41421 - 1.41421i) q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{11} + 32 q^{51} - 16 q^{71} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.707107 0.707107i 0.408248 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.87083 + 1.87083i −0.707107 + 0.707107i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.41421 1.41421i 0.392232 0.392232i −0.483250 0.875482i \(-0.660544\pi\)
0.875482 + 0.483250i \(0.160544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 + 2.82843i 0.685994 + 0.685994i 0.961344 0.275350i \(-0.0887937\pi\)
−0.275350 + 0.961344i \(0.588794\pi\)
\(18\) 0 0
\(19\) −5.29150 −1.21395 −0.606977 0.794719i \(-0.707618\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 2.64575i 0.577350i
\(22\) 0 0
\(23\) −3.74166 3.74166i −0.780189 0.780189i 0.199673 0.979863i \(-0.436012\pi\)
−0.979863 + 0.199673i \(0.936012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 5.29150i 0.950382i 0.879883 + 0.475191i \(0.157621\pi\)
−0.879883 + 0.475191i \(0.842379\pi\)
\(32\) 0 0
\(33\) −1.41421 + 1.41421i −0.246183 + 0.246183i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 2.00000i 0.320256i
\(40\) 0 0
\(41\) 10.5830i 1.65279i 0.563093 + 0.826394i \(0.309611\pi\)
−0.563093 + 0.826394i \(0.690389\pi\)
\(42\) 0 0
\(43\) 3.74166 + 3.74166i 0.570597 + 0.570597i 0.932295 0.361698i \(-0.117803\pi\)
−0.361698 + 0.932295i \(0.617803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.65685 5.65685i −0.825137 0.825137i 0.161703 0.986840i \(-0.448301\pi\)
−0.986840 + 0.161703i \(0.948301\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −3.74166 3.74166i −0.513956 0.513956i 0.401780 0.915736i \(-0.368392\pi\)
−0.915736 + 0.401780i \(0.868392\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.74166 + 3.74166i −0.495595 + 0.495595i
\(58\) 0 0
\(59\) −10.5830 −1.37779 −0.688895 0.724861i \(-0.741904\pi\)
−0.688895 + 0.724861i \(0.741904\pi\)
\(60\) 0 0
\(61\) 10.5830i 1.35501i 0.735516 + 0.677507i \(0.236940\pi\)
−0.735516 + 0.677507i \(0.763060\pi\)
\(62\) 0 0
\(63\) 1.87083 + 1.87083i 0.235702 + 0.235702i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.74166 + 3.74166i −0.457116 + 0.457116i −0.897708 0.440592i \(-0.854769\pi\)
0.440592 + 0.897708i \(0.354769\pi\)
\(68\) 0 0
\(69\) −5.29150 −0.637022
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −9.89949 + 9.89949i −1.15865 + 1.15865i −0.173882 + 0.984767i \(0.555631\pi\)
−0.984767 + 0.173882i \(0.944369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.74166 3.74166i 0.426401 0.426401i
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.24264 + 4.24264i 0.454859 + 0.454859i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 5.29150i 0.554700i
\(92\) 0 0
\(93\) 3.74166 + 3.74166i 0.387992 + 0.387992i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.41421 1.41421i −0.143592 0.143592i 0.631657 0.775248i \(-0.282375\pi\)
−0.775248 + 0.631657i \(0.782375\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 10.5830i 1.05305i 0.850160 + 0.526524i \(0.176505\pi\)
−0.850160 + 0.526524i \(0.823495\pi\)
\(102\) 0 0
\(103\) 5.65685 5.65685i 0.557386 0.557386i −0.371176 0.928562i \(-0.621045\pi\)
0.928562 + 0.371176i \(0.121045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.74166 3.74166i 0.361720 0.361720i −0.502726 0.864446i \(-0.667670\pi\)
0.864446 + 0.502726i \(0.167670\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.2250 11.2250i −1.05596 1.05596i −0.998339 0.0576178i \(-0.981650\pi\)
−0.0576178 0.998339i \(-0.518350\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.41421 1.41421i −0.130744 0.130744i
\(118\) 0 0
\(119\) −10.5830 −0.970143
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 7.48331 + 7.48331i 0.674748 + 0.674748i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.74166 3.74166i 0.332018 0.332018i −0.521334 0.853353i \(-0.674565\pi\)
0.853353 + 0.521334i \(0.174565\pi\)
\(128\) 0 0
\(129\) 5.29150 0.465891
\(130\) 0 0
\(131\) 10.5830i 0.924641i 0.886713 + 0.462321i \(0.152983\pi\)
−0.886713 + 0.462321i \(0.847017\pi\)
\(132\) 0 0
\(133\) 9.89949 9.89949i 0.858395 0.858395i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.2250 11.2250i 0.959014 0.959014i −0.0401784 0.999193i \(-0.512793\pi\)
0.999193 + 0.0401784i \(0.0127926\pi\)
\(138\) 0 0
\(139\) −15.8745 −1.34646 −0.673229 0.739434i \(-0.735093\pi\)
−0.673229 + 0.739434i \(0.735093\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −2.82843 + 2.82843i −0.236525 + 0.236525i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −4.94975 4.94975i −0.408248 0.408248i
\(148\) 0 0
\(149\) 6.00000i 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 2.82843 2.82843i 0.228665 0.228665i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.89949 + 9.89949i 0.790066 + 0.790066i 0.981505 0.191439i \(-0.0613154\pi\)
−0.191439 + 0.981505i \(0.561315\pi\)
\(158\) 0 0
\(159\) −5.29150 −0.419643
\(160\) 0 0
\(161\) 14.0000 1.10335
\(162\) 0 0
\(163\) 11.2250 + 11.2250i 0.879208 + 0.879208i 0.993453 0.114245i \(-0.0364449\pi\)
−0.114245 + 0.993453i \(0.536445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.48528 + 8.48528i 0.656611 + 0.656611i 0.954577 0.297966i \(-0.0963081\pi\)
−0.297966 + 0.954577i \(0.596308\pi\)
\(168\) 0 0
\(169\) 9.00000i 0.692308i
\(170\) 0 0
\(171\) 5.29150i 0.404651i
\(172\) 0 0
\(173\) 5.65685 5.65685i 0.430083 0.430083i −0.458574 0.888656i \(-0.651639\pi\)
0.888656 + 0.458574i \(0.151639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.48331 + 7.48331i −0.562480 + 0.562480i
\(178\) 0 0
\(179\) 18.0000i 1.34538i −0.739923 0.672692i \(-0.765138\pi\)
0.739923 0.672692i \(-0.234862\pi\)
\(180\) 0 0
\(181\) 10.5830i 0.786629i 0.919404 + 0.393314i \(0.128672\pi\)
−0.919404 + 0.393314i \(0.871328\pi\)
\(182\) 0 0
\(183\) 7.48331 + 7.48331i 0.553183 + 0.553183i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.65685 5.65685i −0.413670 0.413670i
\(188\) 0 0
\(189\) 2.64575 0.192450
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 7.48331 + 7.48331i 0.538661 + 0.538661i 0.923136 0.384475i \(-0.125617\pi\)
−0.384475 + 0.923136i \(0.625617\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.2250 11.2250i 0.799746 0.799746i −0.183309 0.983055i \(-0.558681\pi\)
0.983055 + 0.183309i \(0.0586809\pi\)
\(198\) 0 0
\(199\) −5.29150 −0.375105 −0.187552 0.982255i \(-0.560055\pi\)
−0.187552 + 0.982255i \(0.560055\pi\)
\(200\) 0 0
\(201\) 5.29150i 0.373234i
\(202\) 0 0
\(203\) −11.2250 11.2250i −0.787839 0.787839i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.74166 + 3.74166i −0.260063 + 0.260063i
\(208\) 0 0
\(209\) 10.5830 0.732042
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) −1.41421 + 1.41421i −0.0969003 + 0.0969003i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −9.89949 9.89949i −0.672022 0.672022i
\(218\) 0 0
\(219\) 14.0000i 0.946032i
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −11.3137 + 11.3137i −0.757622 + 0.757622i −0.975889 0.218267i \(-0.929960\pi\)
0.218267 + 0.975889i \(0.429960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.9706 + 16.9706i 1.12638 + 1.12638i 0.990762 + 0.135614i \(0.0433007\pi\)
0.135614 + 0.990762i \(0.456699\pi\)
\(228\) 0 0
\(229\) 10.5830 0.699345 0.349672 0.936872i \(-0.386293\pi\)
0.349672 + 0.936872i \(0.386293\pi\)
\(230\) 0 0
\(231\) 5.29150i 0.348155i
\(232\) 0 0
\(233\) −11.2250 11.2250i −0.735372 0.735372i 0.236306 0.971679i \(-0.424063\pi\)
−0.971679 + 0.236306i \(0.924063\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.82843 + 2.82843i 0.183726 + 0.183726i
\(238\) 0 0
\(239\) 6.00000i 0.388108i 0.980991 + 0.194054i \(0.0621637\pi\)
−0.980991 + 0.194054i \(0.937836\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −0.707107 + 0.707107i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −7.48331 + 7.48331i −0.476152 + 0.476152i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.5830i 0.667993i −0.942574 0.333997i \(-0.891603\pi\)
0.942574 0.333997i \(-0.108397\pi\)
\(252\) 0 0
\(253\) 7.48331 + 7.48331i 0.470472 + 0.470472i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.7990 19.7990i −1.23503 1.23503i −0.962009 0.273018i \(-0.911978\pi\)
−0.273018 0.962009i \(-0.588022\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −18.7083 18.7083i −1.15360 1.15360i −0.985825 0.167778i \(-0.946341\pi\)
−0.167778 0.985825i \(-0.553659\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.5830 0.645257 0.322629 0.946526i \(-0.395434\pi\)
0.322629 + 0.946526i \(0.395434\pi\)
\(270\) 0 0
\(271\) 26.4575i 1.60718i −0.595184 0.803590i \(-0.702921\pi\)
0.595184 0.803590i \(-0.297079\pi\)
\(272\) 0 0
\(273\) 3.74166 + 3.74166i 0.226455 + 0.226455i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.48331 + 7.48331i −0.449629 + 0.449629i −0.895231 0.445602i \(-0.852990\pi\)
0.445602 + 0.895231i \(0.352990\pi\)
\(278\) 0 0
\(279\) 5.29150 0.316794
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −2.82843 + 2.82843i −0.168133 + 0.168133i −0.786158 0.618026i \(-0.787933\pi\)
0.618026 + 0.786158i \(0.287933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −19.7990 19.7990i −1.16870 1.16870i
\(288\) 0 0
\(289\) 1.00000i 0.0588235i
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) −8.48528 + 8.48528i −0.495715 + 0.495715i −0.910101 0.414386i \(-0.863996\pi\)
0.414386 + 0.910101i \(0.363996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.41421 + 1.41421i 0.0820610 + 0.0820610i
\(298\) 0 0
\(299\) −10.5830 −0.612031
\(300\) 0 0
\(301\) −14.0000 −0.806947
\(302\) 0 0
\(303\) 7.48331 + 7.48331i 0.429905 + 0.429905i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.48528 + 8.48528i 0.484281 + 0.484281i 0.906496 0.422215i \(-0.138747\pi\)
−0.422215 + 0.906496i \(0.638747\pi\)
\(308\) 0 0
\(309\) 8.00000i 0.455104i
\(310\) 0 0
\(311\) 31.7490i 1.80032i −0.435558 0.900161i \(-0.643449\pi\)
0.435558 0.900161i \(-0.356551\pi\)
\(312\) 0 0
\(313\) 4.24264 4.24264i 0.239808 0.239808i −0.576962 0.816771i \(-0.695762\pi\)
0.816771 + 0.576962i \(0.195762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.7083 18.7083i 1.05076 1.05076i 0.0521213 0.998641i \(-0.483402\pi\)
0.998641 0.0521213i \(-0.0165983\pi\)
\(318\) 0 0
\(319\) 12.0000i 0.671871i
\(320\) 0 0
\(321\) 5.29150i 0.295343i
\(322\) 0 0
\(323\) −14.9666 14.9666i −0.832766 0.832766i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.41421 + 1.41421i 0.0782062 + 0.0782062i
\(328\) 0 0
\(329\) 21.1660 1.16692
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.4499 + 22.4499i −1.22293 + 1.22293i −0.256340 + 0.966587i \(0.582517\pi\)
−0.966587 + 0.256340i \(0.917483\pi\)
\(338\) 0 0
\(339\) −15.8745 −0.862185
\(340\) 0 0
\(341\) 10.5830i 0.573102i
\(342\) 0 0
\(343\) 13.0958 + 13.0958i 0.707107 + 0.707107i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.74166 + 3.74166i −0.200863 + 0.200863i −0.800370 0.599507i \(-0.795363\pi\)
0.599507 + 0.800370i \(0.295363\pi\)
\(348\) 0 0
\(349\) 31.7490 1.69949 0.849743 0.527197i \(-0.176757\pi\)
0.849743 + 0.527197i \(0.176757\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −16.9706 + 16.9706i −0.903252 + 0.903252i −0.995716 0.0924641i \(-0.970526\pi\)
0.0924641 + 0.995716i \(0.470526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7.48331 + 7.48331i −0.396059 + 0.396059i
\(358\) 0 0
\(359\) 6.00000i 0.316668i 0.987386 + 0.158334i \(0.0506123\pi\)
−0.987386 + 0.158334i \(0.949388\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) 0 0
\(363\) −4.94975 + 4.94975i −0.259794 + 0.259794i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 22.6274 + 22.6274i 1.18114 + 1.18114i 0.979449 + 0.201693i \(0.0646442\pi\)
0.201693 + 0.979449i \(0.435356\pi\)
\(368\) 0 0
\(369\) 10.5830 0.550929
\(370\) 0 0
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) −7.48331 7.48331i −0.387471 0.387471i 0.486313 0.873785i \(-0.338341\pi\)
−0.873785 + 0.486313i \(0.838341\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.48528 + 8.48528i 0.437014 + 0.437014i
\(378\) 0 0
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 0 0
\(381\) 5.29150i 0.271092i
\(382\) 0 0
\(383\) 19.7990 19.7990i 1.01168 1.01168i 0.0117502 0.999931i \(-0.496260\pi\)
0.999931 0.0117502i \(-0.00374028\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.74166 3.74166i 0.190199 0.190199i
\(388\) 0 0
\(389\) 2.00000i 0.101404i −0.998714 0.0507020i \(-0.983854\pi\)
0.998714 0.0507020i \(-0.0161459\pi\)
\(390\) 0 0
\(391\) 21.1660i 1.07041i
\(392\) 0 0
\(393\) 7.48331 + 7.48331i 0.377483 + 0.377483i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15.5563 15.5563i −0.780751 0.780751i 0.199207 0.979957i \(-0.436163\pi\)
−0.979957 + 0.199207i \(0.936163\pi\)
\(398\) 0 0
\(399\) 14.0000i 0.700877i
\(400\) 0 0
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 0 0
\(403\) 7.48331 + 7.48331i 0.372770 + 0.372770i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 21.1660 1.04659 0.523296 0.852151i \(-0.324702\pi\)
0.523296 + 0.852151i \(0.324702\pi\)
\(410\) 0 0
\(411\) 15.8745i 0.783032i
\(412\) 0 0
\(413\) 19.7990 19.7990i 0.974245 0.974245i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.2250 + 11.2250i −0.549689 + 0.549689i
\(418\) 0 0
\(419\) −21.1660 −1.03403 −0.517014 0.855977i \(-0.672956\pi\)
−0.517014 + 0.855977i \(0.672956\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) −5.65685 + 5.65685i −0.275046 + 0.275046i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −19.7990 19.7990i −0.958140 0.958140i
\(428\) 0 0
\(429\) 4.00000i 0.193122i
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 9.89949 9.89949i 0.475739 0.475739i −0.428027 0.903766i \(-0.640791\pi\)
0.903766 + 0.428027i \(0.140791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 19.7990 + 19.7990i 0.947114 + 0.947114i
\(438\) 0 0
\(439\) −5.29150 −0.252550 −0.126275 0.991995i \(-0.540302\pi\)
−0.126275 + 0.991995i \(0.540302\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −26.1916 26.1916i −1.24440 1.24440i −0.958157 0.286244i \(-0.907593\pi\)
−0.286244 0.958157i \(-0.592407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.24264 4.24264i −0.200670 0.200670i
\(448\) 0 0
\(449\) 34.0000i 1.60456i 0.596948 + 0.802280i \(0.296380\pi\)
−0.596948 + 0.802280i \(0.703620\pi\)
\(450\) 0 0
\(451\) 21.1660i 0.996669i
\(452\) 0 0
\(453\) 11.3137 11.3137i 0.531564 0.531564i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.48331 + 7.48331i −0.350055 + 0.350055i −0.860130 0.510075i \(-0.829618\pi\)
0.510075 + 0.860130i \(0.329618\pi\)
\(458\) 0 0
\(459\) 4.00000i 0.186704i
\(460\) 0 0
\(461\) 21.1660i 0.985799i −0.870086 0.492900i \(-0.835937\pi\)
0.870086 0.492900i \(-0.164063\pi\)
\(462\) 0 0
\(463\) −3.74166 3.74166i −0.173890 0.173890i 0.614796 0.788686i \(-0.289238\pi\)
−0.788686 + 0.614796i \(0.789238\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.65685 + 5.65685i 0.261768 + 0.261768i 0.825772 0.564004i \(-0.190740\pi\)
−0.564004 + 0.825772i \(0.690740\pi\)
\(468\) 0 0
\(469\) 14.0000i 0.646460i
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −7.48331 7.48331i −0.344083 0.344083i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.74166 + 3.74166i −0.171319 + 0.171319i
\(478\) 0 0
\(479\) 31.7490 1.45065 0.725325 0.688407i \(-0.241690\pi\)
0.725325 + 0.688407i \(0.241690\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 9.89949 9.89949i 0.450443 0.450443i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 26.1916 26.1916i 1.18685 1.18685i 0.208923 0.977932i \(-0.433004\pi\)
0.977932 0.208923i \(-0.0669957\pi\)
\(488\) 0 0
\(489\) 15.8745 0.717870
\(490\) 0 0
\(491\) 34.0000 1.53440 0.767199 0.641409i \(-0.221650\pi\)
0.767199 + 0.641409i \(0.221650\pi\)
\(492\) 0 0
\(493\) −16.9706 + 16.9706i −0.764316 + 0.764316i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.74166 3.74166i 0.167836 0.167836i
\(498\) 0 0
\(499\) 4.00000i 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −8.48528 + 8.48528i −0.378340 + 0.378340i −0.870503 0.492163i \(-0.836206\pi\)
0.492163 + 0.870503i \(0.336206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.36396 + 6.36396i 0.282633 + 0.282633i
\(508\) 0 0
\(509\) 31.7490 1.40725 0.703625 0.710571i \(-0.251563\pi\)
0.703625 + 0.710571i \(0.251563\pi\)
\(510\) 0 0
\(511\) 37.0405i 1.63858i
\(512\) 0 0
\(513\) 3.74166 + 3.74166i 0.165198 + 0.165198i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 11.3137 + 11.3137i 0.497576 + 0.497576i
\(518\) 0 0
\(519\) 8.00000i 0.351161i
\(520\) 0 0
\(521\) 10.5830i 0.463650i 0.972758 + 0.231825i \(0.0744697\pi\)
−0.972758 + 0.231825i \(0.925530\pi\)
\(522\) 0 0
\(523\) 19.7990 19.7990i 0.865749 0.865749i −0.126249 0.991999i \(-0.540294\pi\)
0.991999 + 0.126249i \(0.0402939\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.9666 + 14.9666i −0.651957 + 0.651957i
\(528\) 0 0
\(529\) 5.00000i 0.217391i
\(530\) 0 0
\(531\) 10.5830i 0.459263i
\(532\) 0 0
\(533\) 14.9666 + 14.9666i 0.648277 + 0.648277i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.7279 12.7279i −0.549250 0.549250i
\(538\) 0 0
\(539\) 14.0000i 0.603023i
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) 7.48331 + 7.48331i 0.321140 + 0.321140i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.74166 + 3.74166i −0.159982 + 0.159982i −0.782559 0.622577i \(-0.786086\pi\)
0.622577 + 0.782559i \(0.286086\pi\)
\(548\) 0 0
\(549\) 10.5830 0.451672
\(550\) 0 0
\(551\) 31.7490i 1.35255i
\(552\) 0 0
\(553\) −7.48331 7.48331i −0.318223 0.318223i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.2250 11.2250i 0.475617 0.475617i −0.428110 0.903727i \(-0.640820\pi\)
0.903727 + 0.428110i \(0.140820\pi\)
\(558\) 0 0
\(559\) 10.5830 0.447613
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) −19.7990 + 19.7990i −0.834428 + 0.834428i −0.988119 0.153691i \(-0.950884\pi\)
0.153691 + 0.988119i \(0.450884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.87083 1.87083i 0.0785674 0.0785674i
\(568\) 0 0
\(569\) 34.0000i 1.42535i −0.701492 0.712677i \(-0.747483\pi\)
0.701492 0.712677i \(-0.252517\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 0 0
\(573\) −7.07107 + 7.07107i −0.295398 + 0.295398i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 32.5269 + 32.5269i 1.35411 + 1.35411i 0.881004 + 0.473109i \(0.156868\pi\)
0.473109 + 0.881004i \(0.343132\pi\)
\(578\) 0 0
\(579\) 10.5830 0.439815
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7.48331 + 7.48331i 0.309927 + 0.309927i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.7990 19.7990i −0.817192 0.817192i 0.168508 0.985700i \(-0.446105\pi\)
−0.985700 + 0.168508i \(0.946105\pi\)
\(588\) 0 0
\(589\) 28.0000i 1.15372i
\(590\) 0 0
\(591\) 15.8745i 0.652990i
\(592\) 0 0
\(593\) −5.65685 + 5.65685i −0.232299 + 0.232299i −0.813652 0.581353i \(-0.802524\pi\)
0.581353 + 0.813652i \(0.302524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.74166 + 3.74166i −0.153136 + 0.153136i
\(598\) 0 0
\(599\) 26.0000i 1.06233i 0.847268 + 0.531166i \(0.178246\pi\)
−0.847268 + 0.531166i \(0.821754\pi\)
\(600\) 0 0
\(601\) 21.1660i 0.863380i −0.902022 0.431690i \(-0.857918\pi\)
0.902022 0.431690i \(-0.142082\pi\)
\(602\) 0 0
\(603\) 3.74166 + 3.74166i 0.152372 + 0.152372i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −33.9411 33.9411i −1.37763 1.37763i −0.848612 0.529016i \(-0.822561\pi\)
−0.529016 0.848612i \(-0.677439\pi\)
\(608\) 0 0
\(609\) −15.8745 −0.643268
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 7.48331 + 7.48331i 0.302248 + 0.302248i 0.841893 0.539645i \(-0.181441\pi\)
−0.539645 + 0.841893i \(0.681441\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.1916 26.1916i 1.05443 1.05443i 0.0560036 0.998431i \(-0.482164\pi\)
0.998431 0.0560036i \(-0.0178358\pi\)
\(618\) 0 0
\(619\) −26.4575 −1.06342 −0.531709 0.846927i \(-0.678450\pi\)
−0.531709 + 0.846927i \(0.678450\pi\)
\(620\) 0 0
\(621\) 5.29150i 0.212341i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.48331 7.48331i 0.298855 0.298855i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) −14.1421 + 14.1421i −0.562099 + 0.562099i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.89949 9.89949i −0.392232 0.392232i
\(638\) 0 0
\(639\) 2.00000i 0.0791188i
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) −8.48528 + 8.48528i −0.334627 + 0.334627i −0.854341 0.519714i \(-0.826039\pi\)
0.519714 + 0.854341i \(0.326039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.7990 + 19.7990i 0.778379 + 0.778379i 0.979555 0.201176i \(-0.0644765\pi\)
−0.201176 + 0.979555i \(0.564476\pi\)
\(648\) 0 0
\(649\) 21.1660 0.830839
\(650\) 0 0
\(651\) −14.0000 −0.548703
\(652\) 0 0
\(653\) 33.6749 + 33.6749i 1.31780 + 1.31780i 0.915513 + 0.402288i \(0.131785\pi\)
0.402288 + 0.915513i \(0.368215\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.89949 + 9.89949i 0.386216 + 0.386216i
\(658\) 0 0
\(659\) 6.00000i 0.233727i −0.993148 0.116863i \(-0.962716\pi\)
0.993148 0.116863i \(-0.0372840\pi\)
\(660\) 0 0
\(661\) 10.5830i 0.411631i −0.978591 0.205816i \(-0.934015\pi\)
0.978591 0.205816i \(-0.0659847\pi\)
\(662\) 0 0
\(663\) 5.65685 5.65685i 0.219694 0.219694i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22.4499 22.4499i 0.869265 0.869265i
\(668\) 0 0
\(669\) 16.0000i 0.618596i
\(670\) 0 0
\(671\) 21.1660i 0.817105i
\(672\) 0 0
\(673\) −7.48331 7.48331i −0.288461 0.288461i 0.548011 0.836471i \(-0.315385\pi\)
−0.836471 + 0.548011i \(0.815385\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25.4558 25.4558i −0.978348 0.978348i 0.0214229 0.999771i \(-0.493180\pi\)
−0.999771 + 0.0214229i \(0.993180\pi\)
\(678\) 0 0
\(679\) 5.29150 0.203069
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 3.74166 + 3.74166i 0.143171 + 0.143171i 0.775059 0.631889i \(-0.217720\pi\)
−0.631889 + 0.775059i \(0.717720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.48331 7.48331i 0.285506 0.285506i
\(688\) 0 0
\(689\) −10.5830 −0.403180
\(690\) 0 0
\(691\) 37.0405i 1.40909i 0.709660 + 0.704544i \(0.248848\pi\)
−0.709660 + 0.704544i \(0.751152\pi\)
\(692\) 0 0
\(693\) −3.74166 3.74166i −0.142134 0.142134i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −29.9333 + 29.9333i −1.13380 + 1.13380i
\(698\) 0 0
\(699\) −15.8745 −0.600429
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −19.7990 19.7990i −0.744618 0.744618i
\(708\) 0 0
\(709\) 46.0000i 1.72757i 0.503864 + 0.863783i \(0.331911\pi\)
−0.503864 + 0.863783i \(0.668089\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 19.7990 19.7990i 0.741478 0.741478i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.24264 + 4.24264i 0.158444 + 0.158444i
\(718\) 0 0
\(719\) −10.5830 −0.394679 −0.197340 0.980335i \(-0.563230\pi\)
−0.197340 + 0.980335i \(0.563230\pi\)
\(720\) 0 0
\(721\) 21.1660i 0.788263i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 21.1660i 0.782853i
\(732\) 0 0
\(733\) −24.0416 + 24.0416i −0.887998 + 0.887998i −0.994331 0.106333i \(-0.966089\pi\)
0.106333 + 0.994331i \(0.466089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.48331 7.48331i 0.275651 0.275651i
\(738\) 0 0
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) 10.5830i 0.388776i
\(742\) 0 0
\(743\) −18.7083 18.7083i −0.686340 0.686340i 0.275081 0.961421i \(-0.411295\pi\)
−0.961421 + 0.275081i \(0.911295\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 14.0000i 0.511549i
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) −7.48331 7.48331i −0.272707 0.272707i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 29.9333 29.9333i 1.08794 1.08794i 0.0922025 0.995740i \(-0.470609\pi\)
0.995740 0.0922025i \(-0.0293907\pi\)
\(758\) 0 0
\(759\) 10.5830 0.384139
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −3.74166 3.74166i −0.135457 0.135457i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.9666 + 14.9666i −0.540414 + 0.540414i
\(768\) 0 0
\(769\) −21.1660 −0.763266 −0.381633 0.924314i \(-0.624638\pi\)
−0.381633 + 0.924314i \(0.624638\pi\)
\(770\) 0 0
\(771\) −28.0000 −1.00840
\(772\) 0 0
\(773\) −16.9706 + 16.9706i −0.610389 + 0.610389i −0.943047 0.332659i \(-0.892054\pi\)
0.332659 + 0.943047i \(0.392054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 56.0000i 2.00641i
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 4.24264 4.24264i 0.151620 0.151620i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.7696 + 36.7696i 1.31069 + 1.31069i 0.920904 + 0.389789i \(0.127452\pi\)
0.389789 + 0.920904i \(0.372548\pi\)
\(788\) 0 0
\(789\) −26.4575 −0.941912
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) 14.9666 + 14.9666i 0.531481 + 0.531481i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.3137 + 11.3137i 0.400752 + 0.400752i 0.878498 0.477746i \(-0.158546\pi\)
−0.477746 + 0.878498i \(0.658546\pi\)
\(798\) 0 0
\(799\) 32.0000i 1.13208i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.7990 19.7990i 0.698691 0.698691i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 7.48331 7.48331i 0.263425 0.263425i
\(808\) 0 0
\(809\) 30.0000i 1.05474i −0.849635 0.527372i \(-0.823177\pi\)
0.849635 0.527372i \(-0.176823\pi\)
\(810\) 0 0
\(811\) 37.0405i 1.30067i 0.759648 + 0.650334i \(0.225371\pi\)
−0.759648 + 0.650334i \(0.774629\pi\)
\(812\) 0 0
\(813\) −18.7083 18.7083i −0.656128 0.656128i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −19.7990 19.7990i −0.692679 0.692679i
\(818\) 0 0
\(819\) 5.29150 0.184900
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 0 0
\(823\) −18.7083 18.7083i −0.652130 0.652130i 0.301376 0.953506i \(-0.402554\pi\)
−0.953506 + 0.301376i \(0.902554\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −33.6749 + 33.6749i −1.17099 + 1.17099i −0.189018 + 0.981974i \(0.560530\pi\)
−0.981974 + 0.189018i \(0.939470\pi\)
\(828\) 0 0
\(829\) 10.5830 0.367563 0.183781 0.982967i \(-0.441166\pi\)
0.183781 + 0.982967i \(0.441166\pi\)
\(830\) 0 0
\(831\) 10.5830i 0.367120i
\(832\) 0 0
\(833\) 19.7990 19.7990i 0.685994 0.685994i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.74166 3.74166i 0.129331 0.129331i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 4.24264 4.24264i 0.146124 0.146124i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 13.0958 13.0958i 0.449977 0.449977i
\(848\) 0 0
\(849\) 4.00000i 0.137280i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 9.89949 9.89949i 0.338952 0.338952i −0.517021 0.855973i \(-0.672959\pi\)
0.855973 + 0.517021i \(0.172959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.5980 + 39.5980i 1.35264 + 1.35264i 0.882695 + 0.469946i \(0.155727\pi\)
0.469946 + 0.882695i \(0.344273\pi\)
\(858\) 0 0
\(859\) −15.8745 −0.541631 −0.270816 0.962631i \(-0.587293\pi\)
−0.270816 + 0.962631i \(0.587293\pi\)
\(860\) 0 0
\(861\) −28.0000 −0.954237
\(862\) 0 0
\(863\) 3.74166 + 3.74166i 0.127367 + 0.127367i 0.767917 0.640549i \(-0.221293\pi\)
−0.640549 + 0.767917i \(0.721293\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.707107 0.707107i −0.0240146 0.0240146i
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) 10.5830i 0.358591i
\(872\) 0 0
\(873\) −1.41421 + 1.41421i −0.0478639 + 0.0478639i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.4499 + 22.4499i −0.758081 + 0.758081i −0.975973 0.217892i \(-0.930082\pi\)
0.217892 + 0.975973i \(0.430082\pi\)
\(878\) 0 0
\(879\) 12.0000i 0.404750i
\(880\) 0 0
\(881\) 42.3320i 1.42620i 0.701061 + 0.713101i \(0.252710\pi\)
−0.701061 + 0.713101i \(0.747290\pi\)
\(882\) 0 0
\(883\) 18.7083 + 18.7083i 0.629584 + 0.629584i 0.947963 0.318379i \(-0.103139\pi\)
−0.318379 + 0.947963i \(0.603139\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(888\) 0 0
\(889\) 14.0000i 0.469545i
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 29.9333 + 29.9333i 1.00168 + 1.00168i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.48331 + 7.48331i −0.249861 + 0.249861i
\(898\) 0 0
\(899\) −31.7490 −1.05889
\(900\) 0 0
\(901\) 21.1660i 0.705142i
\(902\) 0 0
\(903\) −9.89949 + 9.89949i −0.329435 + 0.329435i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.74166 + 3.74166i −0.124240 + 0.124240i −0.766493 0.642253i \(-0.778000\pi\)
0.642253 + 0.766493i \(0.278000\pi\)
\(908\) 0 0
\(909\) 10.5830 0.351016
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −19.7990 19.7990i −0.653820 0.653820i
\(918\) 0 0
\(919\) 24.0000i 0.791687i 0.918318 + 0.395843i \(0.129548\pi\)
−0.918318 + 0.395843i \(0.870452\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −2.82843 + 2.82843i −0.0930988 + 0.0930988i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −5.65685 5.65685i −0.185795 0.185795i
\(928\) 0 0
\(929\) −21.1660 −0.694434 −0.347217 0.937785i \(-0.612873\pi\)
−0.347217 + 0.937785i \(0.612873\pi\)
\(930\) 0 0
\(931\) 37.0405i 1.21395i
\(932\) 0 0
\(933\) −22.4499 22.4499i −0.734978 0.734978i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.41421 + 1.41421i 0.0462003 + 0.0462003i 0.729830 0.683629i \(-0.239599\pi\)
−0.683629 + 0.729830i \(0.739599\pi\)
\(938\) 0 0
\(939\) 6.00000i 0.195803i
\(940\) 0 0
\(941\) 21.1660i 0.689992i 0.938604 + 0.344996i \(0.112120\pi\)
−0.938604 + 0.344996i \(0.887880\pi\)
\(942\) 0 0
\(943\) 39.5980 39.5980i 1.28949 1.28949i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.7083 + 18.7083i −0.607938 + 0.607938i −0.942407 0.334469i \(-0.891443\pi\)
0.334469 + 0.942407i \(0.391443\pi\)
\(948\) 0 0
\(949\) 28.0000i 0.908918i
\(950\) 0 0
\(951\) 26.4575i 0.857944i
\(952\) 0 0
\(953\) 26.1916 + 26.1916i 0.848429 + 0.848429i 0.989937 0.141508i \(-0.0451951\pi\)
−0.141508 + 0.989937i \(0.545195\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.48528 8.48528i −0.274290 0.274290i
\(958\) 0 0
\(959\) 42.0000i 1.35625i
\(960\) 0 0
\(961\) 3.00000 0.0967742
\(962\) 0 0
\(963\) −3.74166 3.74166i −0.120573 0.120573i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.74166 3.74166i 0.120324 0.120324i −0.644381 0.764705i \(-0.722885\pi\)
0.764705 + 0.644381i \(0.222885\pi\)
\(968\) 0 0
\(969\) −21.1660 −0.679950
\(970\) 0 0
\(971\) 42.3320i 1.35850i 0.733907 + 0.679250i \(0.237695\pi\)
−0.733907 + 0.679250i \(0.762305\pi\)
\(972\) 0 0
\(973\) 29.6985 29.6985i 0.952090 0.952090i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.2250 11.2250i 0.359119 0.359119i −0.504369 0.863488i \(-0.668275\pi\)
0.863488 + 0.504369i \(0.168275\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −22.6274 + 22.6274i −0.721703 + 0.721703i −0.968952 0.247249i \(-0.920473\pi\)
0.247249 + 0.968952i \(0.420473\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 14.9666 14.9666i 0.476393 0.476393i
\(988\) 0 0
\(989\) 28.0000i 0.890348i
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 14.1421 14.1421i 0.448787 0.448787i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −12.7279 12.7279i −0.403097 0.403097i 0.476226 0.879323i \(-0.342005\pi\)
−0.879323 + 0.476226i \(0.842005\pi\)
\(998\) 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 2100.2.x.b.1693.3 yes 8
5.2 odd 4 inner 2100.2.x.b.1357.1 8
5.3 odd 4 inner 2100.2.x.b.1357.4 yes 8
5.4 even 2 inner 2100.2.x.b.1693.2 yes 8
7.6 odd 2 inner 2100.2.x.b.1693.1 yes 8
35.13 even 4 inner 2100.2.x.b.1357.2 yes 8
35.27 even 4 inner 2100.2.x.b.1357.3 yes 8
35.34 odd 2 inner 2100.2.x.b.1693.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.x.b.1357.1 8 5.2 odd 4 inner
2100.2.x.b.1357.2 yes 8 35.13 even 4 inner
2100.2.x.b.1357.3 yes 8 35.27 even 4 inner
2100.2.x.b.1357.4 yes 8 5.3 odd 4 inner
2100.2.x.b.1693.1 yes 8 7.6 odd 2 inner
2100.2.x.b.1693.2 yes 8 5.4 even 2 inner
2100.2.x.b.1693.3 yes 8 1.1 even 1 trivial
2100.2.x.b.1693.4 yes 8 35.34 odd 2 inner