Properties

Label 2100.2.f.b.1049.2
Level $2100$
Weight $2$
Character 2100.1049
Analytic conductor $16.769$
Analytic rank $1$
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] = [2100,2,Mod(1049,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1049");
 
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.f (of order \(2\), degree \(1\), 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: \(16.7685844245\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1049.2
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1049
Dual form 2100.2.f.b.1049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.61803 + 0.618034i) q^{3} +(-0.381966 - 2.61803i) q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})\) \(q+(-1.61803 + 0.618034i) q^{3} +(-0.381966 - 2.61803i) q^{7} +(2.23607 - 2.00000i) q^{9} +5.23607i q^{11} -3.23607 q^{13} -4.47214i q^{17} -2.76393i q^{19} +(2.23607 + 4.00000i) q^{21} +5.70820 q^{23} +(-2.38197 + 4.61803i) q^{27} +4.00000i q^{29} +1.23607i q^{31} +(-3.23607 - 8.47214i) q^{33} -4.47214i q^{37} +(5.23607 - 2.00000i) q^{39} -12.4721 q^{41} +2.76393i q^{43} +1.23607i q^{47} +(-6.70820 + 2.00000i) q^{49} +(2.76393 + 7.23607i) q^{51} -4.76393 q^{53} +(1.70820 + 4.47214i) q^{57} +8.94427 q^{59} +12.9443i q^{61} +(-6.09017 - 5.09017i) q^{63} -3.70820i q^{67} +(-9.23607 + 3.52786i) q^{69} -10.1803i q^{71} -11.2361 q^{73} +(13.7082 - 2.00000i) q^{77} -1.52786 q^{79} +(1.00000 - 8.94427i) q^{81} +2.76393i q^{83} +(-2.47214 - 6.47214i) q^{87} +3.52786 q^{89} +(1.23607 + 8.47214i) q^{91} +(-0.763932 - 2.00000i) q^{93} -8.18034 q^{97} +(10.4721 + 11.7082i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 6 q^{7} - 4 q^{13} - 4 q^{23} - 14 q^{27} - 4 q^{33} + 12 q^{39} - 32 q^{41} + 20 q^{51} - 28 q^{53} - 20 q^{57} - 2 q^{63} - 28 q^{69} - 36 q^{73} + 28 q^{77} - 24 q^{79} + 4 q^{81} + 8 q^{87} + 32 q^{89} - 4 q^{91} - 12 q^{93} + 12 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.61803 + 0.618034i −0.934172 + 0.356822i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.381966 2.61803i −0.144370 0.989524i
\(8\) 0 0
\(9\) 2.23607 2.00000i 0.745356 0.666667i
\(10\) 0 0
\(11\) 5.23607i 1.57873i 0.613922 + 0.789367i \(0.289591\pi\)
−0.613922 + 0.789367i \(0.710409\pi\)
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214i 1.08465i −0.840168 0.542326i \(-0.817544\pi\)
0.840168 0.542326i \(-0.182456\pi\)
\(18\) 0 0
\(19\) 2.76393i 0.634089i −0.948411 0.317045i \(-0.897309\pi\)
0.948411 0.317045i \(-0.102691\pi\)
\(20\) 0 0
\(21\) 2.23607 + 4.00000i 0.487950 + 0.872872i
\(22\) 0 0
\(23\) 5.70820 1.19024 0.595121 0.803636i \(-0.297104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.38197 + 4.61803i −0.458410 + 0.888741i
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 1.23607i 0.222004i 0.993820 + 0.111002i \(0.0354061\pi\)
−0.993820 + 0.111002i \(0.964594\pi\)
\(32\) 0 0
\(33\) −3.23607 8.47214i −0.563327 1.47481i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.47214i 0.735215i −0.929981 0.367607i \(-0.880177\pi\)
0.929981 0.367607i \(-0.119823\pi\)
\(38\) 0 0
\(39\) 5.23607 2.00000i 0.838442 0.320256i
\(40\) 0 0
\(41\) −12.4721 −1.94782 −0.973910 0.226934i \(-0.927130\pi\)
−0.973910 + 0.226934i \(0.927130\pi\)
\(42\) 0 0
\(43\) 2.76393i 0.421496i 0.977540 + 0.210748i \(0.0675899\pi\)
−0.977540 + 0.210748i \(0.932410\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.23607i 0.180299i 0.995928 + 0.0901495i \(0.0287345\pi\)
−0.995928 + 0.0901495i \(0.971266\pi\)
\(48\) 0 0
\(49\) −6.70820 + 2.00000i −0.958315 + 0.285714i
\(50\) 0 0
\(51\) 2.76393 + 7.23607i 0.387028 + 1.01325i
\(52\) 0 0
\(53\) −4.76393 −0.654376 −0.327188 0.944959i \(-0.606101\pi\)
−0.327188 + 0.944959i \(0.606101\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.70820 + 4.47214i 0.226257 + 0.592349i
\(58\) 0 0
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 12.9443i 1.65734i 0.559734 + 0.828672i \(0.310903\pi\)
−0.559734 + 0.828672i \(0.689097\pi\)
\(62\) 0 0
\(63\) −6.09017 5.09017i −0.767289 0.641301i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.70820i 0.453029i −0.974008 0.226515i \(-0.927267\pi\)
0.974008 0.226515i \(-0.0727331\pi\)
\(68\) 0 0
\(69\) −9.23607 + 3.52786i −1.11189 + 0.424705i
\(70\) 0 0
\(71\) 10.1803i 1.20818i −0.796915 0.604092i \(-0.793536\pi\)
0.796915 0.604092i \(-0.206464\pi\)
\(72\) 0 0
\(73\) −11.2361 −1.31508 −0.657541 0.753419i \(-0.728403\pi\)
−0.657541 + 0.753419i \(0.728403\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.7082 2.00000i 1.56219 0.227921i
\(78\) 0 0
\(79\) −1.52786 −0.171898 −0.0859491 0.996300i \(-0.527392\pi\)
−0.0859491 + 0.996300i \(0.527392\pi\)
\(80\) 0 0
\(81\) 1.00000 8.94427i 0.111111 0.993808i
\(82\) 0 0
\(83\) 2.76393i 0.303381i 0.988428 + 0.151690i \(0.0484717\pi\)
−0.988428 + 0.151690i \(0.951528\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.47214 6.47214i −0.265041 0.693886i
\(88\) 0 0
\(89\) 3.52786 0.373953 0.186976 0.982364i \(-0.440131\pi\)
0.186976 + 0.982364i \(0.440131\pi\)
\(90\) 0 0
\(91\) 1.23607 + 8.47214i 0.129575 + 0.888121i
\(92\) 0 0
\(93\) −0.763932 2.00000i −0.0792161 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.18034 −0.830588 −0.415294 0.909687i \(-0.636321\pi\)
−0.415294 + 0.909687i \(0.636321\pi\)
\(98\) 0 0
\(99\) 10.4721 + 11.7082i 1.05249 + 1.17672i
\(100\) 0 0
\(101\) 10.9443 1.08900 0.544498 0.838762i \(-0.316720\pi\)
0.544498 + 0.838762i \(0.316720\pi\)
\(102\) 0 0
\(103\) −15.2361 −1.50125 −0.750627 0.660726i \(-0.770249\pi\)
−0.750627 + 0.660726i \(0.770249\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.7082 −1.32522 −0.662611 0.748964i \(-0.730552\pi\)
−0.662611 + 0.748964i \(0.730552\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) 2.76393 + 7.23607i 0.262341 + 0.686817i
\(112\) 0 0
\(113\) −12.7639 −1.20073 −0.600365 0.799726i \(-0.704978\pi\)
−0.600365 + 0.799726i \(0.704978\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −7.23607 + 6.47214i −0.668975 + 0.598349i
\(118\) 0 0
\(119\) −11.7082 + 1.70820i −1.07329 + 0.156591i
\(120\) 0 0
\(121\) −16.4164 −1.49240
\(122\) 0 0
\(123\) 20.1803 7.70820i 1.81960 0.695025i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 19.1246i 1.69703i 0.529167 + 0.848517i \(0.322504\pi\)
−0.529167 + 0.848517i \(0.677496\pi\)
\(128\) 0 0
\(129\) −1.70820 4.47214i −0.150399 0.393750i
\(130\) 0 0
\(131\) −11.4164 −0.997456 −0.498728 0.866758i \(-0.666199\pi\)
−0.498728 + 0.866758i \(0.666199\pi\)
\(132\) 0 0
\(133\) −7.23607 + 1.05573i −0.627447 + 0.0915432i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.23607 0.276476 0.138238 0.990399i \(-0.455856\pi\)
0.138238 + 0.990399i \(0.455856\pi\)
\(138\) 0 0
\(139\) 10.7639i 0.912985i −0.889727 0.456492i \(-0.849106\pi\)
0.889727 0.456492i \(-0.150894\pi\)
\(140\) 0 0
\(141\) −0.763932 2.00000i −0.0643347 0.168430i
\(142\) 0 0
\(143\) 16.9443i 1.41695i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.61803 7.38197i 0.793282 0.608854i
\(148\) 0 0
\(149\) 11.4164i 0.935269i −0.883922 0.467634i \(-0.845106\pi\)
0.883922 0.467634i \(-0.154894\pi\)
\(150\) 0 0
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) 0 0
\(153\) −8.94427 10.0000i −0.723102 0.808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −24.1803 −1.92980 −0.964901 0.262615i \(-0.915415\pi\)
−0.964901 + 0.262615i \(0.915415\pi\)
\(158\) 0 0
\(159\) 7.70820 2.94427i 0.611300 0.233496i
\(160\) 0 0
\(161\) −2.18034 14.9443i −0.171835 1.17777i
\(162\) 0 0
\(163\) 12.6525i 0.991018i 0.868603 + 0.495509i \(0.165019\pi\)
−0.868603 + 0.495509i \(0.834981\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.2361i 1.02424i −0.858915 0.512119i \(-0.828861\pi\)
0.858915 0.512119i \(-0.171139\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) −5.52786 6.18034i −0.422726 0.472622i
\(172\) 0 0
\(173\) 2.00000i 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −14.4721 + 5.52786i −1.08779 + 0.415500i
\(178\) 0 0
\(179\) 17.2361i 1.28828i −0.764906 0.644142i \(-0.777215\pi\)
0.764906 0.644142i \(-0.222785\pi\)
\(180\) 0 0
\(181\) 2.47214i 0.183752i −0.995770 0.0918762i \(-0.970714\pi\)
0.995770 0.0918762i \(-0.0292864\pi\)
\(182\) 0 0
\(183\) −8.00000 20.9443i −0.591377 1.54825i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 23.4164 1.71238
\(188\) 0 0
\(189\) 13.0000 + 4.47214i 0.945611 + 0.325300i
\(190\) 0 0
\(191\) 2.76393i 0.199991i 0.994988 + 0.0999956i \(0.0318829\pi\)
−0.994988 + 0.0999956i \(0.968117\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.7639 −1.47937 −0.739684 0.672954i \(-0.765025\pi\)
−0.739684 + 0.672954i \(0.765025\pi\)
\(198\) 0 0
\(199\) 13.2361i 0.938280i −0.883124 0.469140i \(-0.844564\pi\)
0.883124 0.469140i \(-0.155436\pi\)
\(200\) 0 0
\(201\) 2.29180 + 6.00000i 0.161651 + 0.423207i
\(202\) 0 0
\(203\) 10.4721 1.52786i 0.735000 0.107235i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.7639 11.4164i 0.887155 0.793495i
\(208\) 0 0
\(209\) 14.4721 1.00106
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 6.29180 + 16.4721i 0.431107 + 1.12865i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.23607 0.472136i 0.219679 0.0320507i
\(218\) 0 0
\(219\) 18.1803 6.94427i 1.22851 0.469250i
\(220\) 0 0
\(221\) 14.4721i 0.973501i
\(222\) 0 0
\(223\) 1.70820 0.114390 0.0571949 0.998363i \(-0.481784\pi\)
0.0571949 + 0.998363i \(0.481784\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.7639i 1.24541i 0.782458 + 0.622703i \(0.213965\pi\)
−0.782458 + 0.622703i \(0.786035\pi\)
\(228\) 0 0
\(229\) 21.8885i 1.44644i 0.690620 + 0.723218i \(0.257338\pi\)
−0.690620 + 0.723218i \(0.742662\pi\)
\(230\) 0 0
\(231\) −20.9443 + 11.7082i −1.37803 + 0.770343i
\(232\) 0 0
\(233\) 24.1803 1.58411 0.792053 0.610452i \(-0.209012\pi\)
0.792053 + 0.610452i \(0.209012\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.47214 0.944272i 0.160582 0.0613371i
\(238\) 0 0
\(239\) 22.1803i 1.43473i 0.696699 + 0.717363i \(0.254651\pi\)
−0.696699 + 0.717363i \(0.745349\pi\)
\(240\) 0 0
\(241\) 28.3607i 1.82687i −0.406982 0.913436i \(-0.633419\pi\)
0.406982 0.913436i \(-0.366581\pi\)
\(242\) 0 0
\(243\) 3.90983 + 15.0902i 0.250816 + 0.968035i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94427i 0.569110i
\(248\) 0 0
\(249\) −1.70820 4.47214i −0.108253 0.283410i
\(250\) 0 0
\(251\) 24.3607 1.53763 0.768816 0.639470i \(-0.220846\pi\)
0.768816 + 0.639470i \(0.220846\pi\)
\(252\) 0 0
\(253\) 29.8885i 1.87908i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.52786i 0.469575i −0.972047 0.234788i \(-0.924561\pi\)
0.972047 0.234788i \(-0.0754395\pi\)
\(258\) 0 0
\(259\) −11.7082 + 1.70820i −0.727512 + 0.106143i
\(260\) 0 0
\(261\) 8.00000 + 8.94427i 0.495188 + 0.553637i
\(262\) 0 0
\(263\) −0.180340 −0.0111202 −0.00556012 0.999985i \(-0.501770\pi\)
−0.00556012 + 0.999985i \(0.501770\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.70820 + 2.18034i −0.349336 + 0.133435i
\(268\) 0 0
\(269\) −28.4721 −1.73598 −0.867988 0.496584i \(-0.834587\pi\)
−0.867988 + 0.496584i \(0.834587\pi\)
\(270\) 0 0
\(271\) 19.1246i 1.16174i 0.813997 + 0.580869i \(0.197287\pi\)
−0.813997 + 0.580869i \(0.802713\pi\)
\(272\) 0 0
\(273\) −7.23607 12.9443i −0.437947 0.783423i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 25.4164i 1.52712i −0.645735 0.763562i \(-0.723449\pi\)
0.645735 0.763562i \(-0.276551\pi\)
\(278\) 0 0
\(279\) 2.47214 + 2.76393i 0.148003 + 0.165472i
\(280\) 0 0
\(281\) 15.4164i 0.919666i 0.888005 + 0.459833i \(0.152091\pi\)
−0.888005 + 0.459833i \(0.847909\pi\)
\(282\) 0 0
\(283\) 22.6525 1.34655 0.673275 0.739392i \(-0.264887\pi\)
0.673275 + 0.739392i \(0.264887\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.76393 + 32.6525i 0.281206 + 1.92741i
\(288\) 0 0
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 13.2361 5.05573i 0.775912 0.296372i
\(292\) 0 0
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −24.1803 12.4721i −1.40309 0.723707i
\(298\) 0 0
\(299\) −18.4721 −1.06827
\(300\) 0 0
\(301\) 7.23607 1.05573i 0.417080 0.0608512i
\(302\) 0 0
\(303\) −17.7082 + 6.76393i −1.01731 + 0.388578i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.18034 0.238585 0.119292 0.992859i \(-0.461937\pi\)
0.119292 + 0.992859i \(0.461937\pi\)
\(308\) 0 0
\(309\) 24.6525 9.41641i 1.40243 0.535681i
\(310\) 0 0
\(311\) 12.3607 0.700910 0.350455 0.936580i \(-0.386027\pi\)
0.350455 + 0.936580i \(0.386027\pi\)
\(312\) 0 0
\(313\) 17.7082 1.00093 0.500463 0.865758i \(-0.333163\pi\)
0.500463 + 0.865758i \(0.333163\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.1803 0.908778 0.454389 0.890803i \(-0.349858\pi\)
0.454389 + 0.890803i \(0.349858\pi\)
\(318\) 0 0
\(319\) −20.9443 −1.17265
\(320\) 0 0
\(321\) 22.1803 8.47214i 1.23799 0.472869i
\(322\) 0 0
\(323\) −12.3607 −0.687767
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 24.1803 9.23607i 1.33718 0.510756i
\(328\) 0 0
\(329\) 3.23607 0.472136i 0.178410 0.0260297i
\(330\) 0 0
\(331\) −3.05573 −0.167958 −0.0839790 0.996468i \(-0.526763\pi\)
−0.0839790 + 0.996468i \(0.526763\pi\)
\(332\) 0 0
\(333\) −8.94427 10.0000i −0.490143 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 0 0
\(339\) 20.6525 7.88854i 1.12169 0.428447i
\(340\) 0 0
\(341\) −6.47214 −0.350486
\(342\) 0 0
\(343\) 7.79837 + 16.7984i 0.421073 + 0.907027i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.34752 0.0723389 0.0361694 0.999346i \(-0.488484\pi\)
0.0361694 + 0.999346i \(0.488484\pi\)
\(348\) 0 0
\(349\) 8.94427i 0.478776i 0.970924 + 0.239388i \(0.0769468\pi\)
−0.970924 + 0.239388i \(0.923053\pi\)
\(350\) 0 0
\(351\) 7.70820 14.9443i 0.411433 0.797666i
\(352\) 0 0
\(353\) 1.41641i 0.0753878i 0.999289 + 0.0376939i \(0.0120012\pi\)
−0.999289 + 0.0376939i \(0.987999\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 17.8885 10.0000i 0.946762 0.529256i
\(358\) 0 0
\(359\) 24.0689i 1.27031i 0.772386 + 0.635154i \(0.219063\pi\)
−0.772386 + 0.635154i \(0.780937\pi\)
\(360\) 0 0
\(361\) 11.3607 0.597931
\(362\) 0 0
\(363\) 26.5623 10.1459i 1.39416 0.532522i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.65248 −0.347256 −0.173628 0.984811i \(-0.555549\pi\)
−0.173628 + 0.984811i \(0.555549\pi\)
\(368\) 0 0
\(369\) −27.8885 + 24.9443i −1.45182 + 1.29855i
\(370\) 0 0
\(371\) 1.81966 + 12.4721i 0.0944720 + 0.647521i
\(372\) 0 0
\(373\) 16.4721i 0.852895i −0.904512 0.426447i \(-0.859765\pi\)
0.904512 0.426447i \(-0.140235\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.9443i 0.666664i
\(378\) 0 0
\(379\) −10.4721 −0.537917 −0.268959 0.963152i \(-0.586680\pi\)
−0.268959 + 0.963152i \(0.586680\pi\)
\(380\) 0 0
\(381\) −11.8197 30.9443i −0.605540 1.58532i
\(382\) 0 0
\(383\) 10.7639i 0.550011i 0.961443 + 0.275006i \(0.0886797\pi\)
−0.961443 + 0.275006i \(0.911320\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.52786 + 6.18034i 0.280997 + 0.314164i
\(388\) 0 0
\(389\) 6.47214i 0.328150i 0.986448 + 0.164075i \(0.0524640\pi\)
−0.986448 + 0.164075i \(0.947536\pi\)
\(390\) 0 0
\(391\) 25.5279i 1.29100i
\(392\) 0 0
\(393\) 18.4721 7.05573i 0.931796 0.355914i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.70820 0.487241 0.243620 0.969871i \(-0.421665\pi\)
0.243620 + 0.969871i \(0.421665\pi\)
\(398\) 0 0
\(399\) 11.0557 6.18034i 0.553479 0.309404i
\(400\) 0 0
\(401\) 11.0557i 0.552097i −0.961144 0.276048i \(-0.910975\pi\)
0.961144 0.276048i \(-0.0890250\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.4164 1.16071
\(408\) 0 0
\(409\) 12.0000i 0.593362i −0.954977 0.296681i \(-0.904120\pi\)
0.954977 0.296681i \(-0.0958798\pi\)
\(410\) 0 0
\(411\) −5.23607 + 2.00000i −0.258276 + 0.0986527i
\(412\) 0 0
\(413\) −3.41641 23.4164i −0.168110 1.15225i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.65248 + 17.4164i 0.325773 + 0.852885i
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 23.5279 1.14668 0.573339 0.819318i \(-0.305648\pi\)
0.573339 + 0.819318i \(0.305648\pi\)
\(422\) 0 0
\(423\) 2.47214 + 2.76393i 0.120199 + 0.134387i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 33.8885 4.94427i 1.63998 0.239270i
\(428\) 0 0
\(429\) 10.4721 + 27.4164i 0.505599 + 1.32368i
\(430\) 0 0
\(431\) 5.23607i 0.252213i −0.992017 0.126106i \(-0.959752\pi\)
0.992017 0.126106i \(-0.0402480\pi\)
\(432\) 0 0
\(433\) −1.34752 −0.0647579 −0.0323789 0.999476i \(-0.510308\pi\)
−0.0323789 + 0.999476i \(0.510308\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.7771i 0.754720i
\(438\) 0 0
\(439\) 18.1803i 0.867700i −0.900985 0.433850i \(-0.857155\pi\)
0.900985 0.433850i \(-0.142845\pi\)
\(440\) 0 0
\(441\) −11.0000 + 17.8885i −0.523810 + 0.851835i
\(442\) 0 0
\(443\) −34.0689 −1.61866 −0.809331 0.587353i \(-0.800170\pi\)
−0.809331 + 0.587353i \(0.800170\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7.05573 + 18.4721i 0.333724 + 0.873702i
\(448\) 0 0
\(449\) 5.88854i 0.277898i −0.990300 0.138949i \(-0.955628\pi\)
0.990300 0.138949i \(-0.0443723\pi\)
\(450\) 0 0
\(451\) 65.3050i 3.07509i
\(452\) 0 0
\(453\) 14.4721 5.52786i 0.679960 0.259722i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.9443i 1.26040i −0.776433 0.630200i \(-0.782973\pi\)
0.776433 0.630200i \(-0.217027\pi\)
\(458\) 0 0
\(459\) 20.6525 + 10.6525i 0.963975 + 0.497215i
\(460\) 0 0
\(461\) −15.5279 −0.723205 −0.361602 0.932332i \(-0.617770\pi\)
−0.361602 + 0.932332i \(0.617770\pi\)
\(462\) 0 0
\(463\) 25.2361i 1.17282i 0.810015 + 0.586410i \(0.199459\pi\)
−0.810015 + 0.586410i \(0.800541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.5410i 1.41327i 0.707578 + 0.706635i \(0.249788\pi\)
−0.707578 + 0.706635i \(0.750212\pi\)
\(468\) 0 0
\(469\) −9.70820 + 1.41641i −0.448283 + 0.0654036i
\(470\) 0 0
\(471\) 39.1246 14.9443i 1.80277 0.688596i
\(472\) 0 0
\(473\) −14.4721 −0.665430
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.6525 + 9.52786i −0.487743 + 0.436251i
\(478\) 0 0
\(479\) 4.94427 0.225910 0.112955 0.993600i \(-0.463968\pi\)
0.112955 + 0.993600i \(0.463968\pi\)
\(480\) 0 0
\(481\) 14.4721i 0.659873i
\(482\) 0 0
\(483\) 12.7639 + 22.8328i 0.580779 + 1.03893i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.23607i 0.0560116i 0.999608 + 0.0280058i \(0.00891569\pi\)
−0.999608 + 0.0280058i \(0.991084\pi\)
\(488\) 0 0
\(489\) −7.81966 20.4721i −0.353617 0.925782i
\(490\) 0 0
\(491\) 0.875388i 0.0395057i −0.999805 0.0197529i \(-0.993712\pi\)
0.999805 0.0197529i \(-0.00628794\pi\)
\(492\) 0 0
\(493\) 17.8885 0.805659
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −26.6525 + 3.88854i −1.19553 + 0.174425i
\(498\) 0 0
\(499\) −28.3607 −1.26960 −0.634799 0.772677i \(-0.718917\pi\)
−0.634799 + 0.772677i \(0.718917\pi\)
\(500\) 0 0
\(501\) 8.18034 + 21.4164i 0.365471 + 0.956815i
\(502\) 0 0
\(503\) 12.6525i 0.564146i 0.959393 + 0.282073i \(0.0910220\pi\)
−0.959393 + 0.282073i \(0.908978\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.09017 1.56231i 0.181651 0.0693844i
\(508\) 0 0
\(509\) 5.05573 0.224091 0.112046 0.993703i \(-0.464260\pi\)
0.112046 + 0.993703i \(0.464260\pi\)
\(510\) 0 0
\(511\) 4.29180 + 29.4164i 0.189858 + 1.30131i
\(512\) 0 0
\(513\) 12.7639 + 6.58359i 0.563541 + 0.290673i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.47214 −0.284644
\(518\) 0 0
\(519\) 1.23607 + 3.23607i 0.0542574 + 0.142048i
\(520\) 0 0
\(521\) 27.8885 1.22182 0.610910 0.791700i \(-0.290804\pi\)
0.610910 + 0.791700i \(0.290804\pi\)
\(522\) 0 0
\(523\) 21.7082 0.949233 0.474617 0.880193i \(-0.342587\pi\)
0.474617 + 0.880193i \(0.342587\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.52786 0.240798
\(528\) 0 0
\(529\) 9.58359 0.416678
\(530\) 0 0
\(531\) 20.0000 17.8885i 0.867926 0.776297i
\(532\) 0 0
\(533\) 40.3607 1.74822
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.6525 + 27.8885i 0.459688 + 1.20348i
\(538\) 0 0
\(539\) −10.4721 35.1246i −0.451067 1.51292i
\(540\) 0 0
\(541\) −8.11146 −0.348739 −0.174369 0.984680i \(-0.555789\pi\)
−0.174369 + 0.984680i \(0.555789\pi\)
\(542\) 0 0
\(543\) 1.52786 + 4.00000i 0.0655669 + 0.171656i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.29180i 0.183504i 0.995782 + 0.0917520i \(0.0292467\pi\)
−0.995782 + 0.0917520i \(0.970753\pi\)
\(548\) 0 0
\(549\) 25.8885 + 28.9443i 1.10490 + 1.23531i
\(550\) 0 0
\(551\) 11.0557 0.470990
\(552\) 0 0
\(553\) 0.583592 + 4.00000i 0.0248169 + 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.18034 0.346612 0.173306 0.984868i \(-0.444555\pi\)
0.173306 + 0.984868i \(0.444555\pi\)
\(558\) 0 0
\(559\) 8.94427i 0.378302i
\(560\) 0 0
\(561\) −37.8885 + 14.4721i −1.59966 + 0.611014i
\(562\) 0 0
\(563\) 40.0689i 1.68870i 0.535790 + 0.844351i \(0.320014\pi\)
−0.535790 + 0.844351i \(0.679986\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −23.7984 + 0.798374i −0.999438 + 0.0335286i
\(568\) 0 0
\(569\) 13.8885i 0.582238i −0.956687 0.291119i \(-0.905972\pi\)
0.956687 0.291119i \(-0.0940276\pi\)
\(570\) 0 0
\(571\) −46.8328 −1.95989 −0.979946 0.199262i \(-0.936145\pi\)
−0.979946 + 0.199262i \(0.936145\pi\)
\(572\) 0 0
\(573\) −1.70820 4.47214i −0.0713612 0.186826i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −26.0689 −1.08526 −0.542631 0.839971i \(-0.682572\pi\)
−0.542631 + 0.839971i \(0.682572\pi\)
\(578\) 0 0
\(579\) −3.70820 9.70820i −0.154108 0.403459i
\(580\) 0 0
\(581\) 7.23607 1.05573i 0.300203 0.0437990i
\(582\) 0 0
\(583\) 24.9443i 1.03309i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.7082i 0.813445i −0.913552 0.406722i \(-0.866672\pi\)
0.913552 0.406722i \(-0.133328\pi\)
\(588\) 0 0
\(589\) 3.41641 0.140771
\(590\) 0 0
\(591\) 33.5967 12.8328i 1.38199 0.527872i
\(592\) 0 0
\(593\) 28.4721i 1.16921i 0.811318 + 0.584605i \(0.198751\pi\)
−0.811318 + 0.584605i \(0.801249\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.18034 + 21.4164i 0.334799 + 0.876515i
\(598\) 0 0
\(599\) 0.652476i 0.0266594i −0.999911 0.0133297i \(-0.995757\pi\)
0.999911 0.0133297i \(-0.00424311\pi\)
\(600\) 0 0
\(601\) 17.3050i 0.705884i −0.935645 0.352942i \(-0.885181\pi\)
0.935645 0.352942i \(-0.114819\pi\)
\(602\) 0 0
\(603\) −7.41641 8.29180i −0.302019 0.337668i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.0689 0.733393 0.366697 0.930341i \(-0.380489\pi\)
0.366697 + 0.930341i \(0.380489\pi\)
\(608\) 0 0
\(609\) −16.0000 + 8.94427i −0.648353 + 0.362440i
\(610\) 0 0
\(611\) 4.00000i 0.161823i
\(612\) 0 0
\(613\) 22.3607i 0.903139i 0.892236 + 0.451570i \(0.149136\pi\)
−0.892236 + 0.451570i \(0.850864\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −40.5410 −1.63212 −0.816060 0.577967i \(-0.803846\pi\)
−0.816060 + 0.577967i \(0.803846\pi\)
\(618\) 0 0
\(619\) 7.70820i 0.309819i −0.987929 0.154909i \(-0.950491\pi\)
0.987929 0.154909i \(-0.0495086\pi\)
\(620\) 0 0
\(621\) −13.5967 + 26.3607i −0.545619 + 1.05782i
\(622\) 0 0
\(623\) −1.34752 9.23607i −0.0539874 0.370035i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −23.4164 + 8.94427i −0.935161 + 0.357200i
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −8.94427 −0.356066 −0.178033 0.984025i \(-0.556973\pi\)
−0.178033 + 0.984025i \(0.556973\pi\)
\(632\) 0 0
\(633\) 12.9443 4.94427i 0.514489 0.196517i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 21.7082 6.47214i 0.860110 0.256435i
\(638\) 0 0
\(639\) −20.3607 22.7639i −0.805456 0.900527i
\(640\) 0 0
\(641\) 39.4164i 1.55685i −0.627735 0.778427i \(-0.716018\pi\)
0.627735 0.778427i \(-0.283982\pi\)
\(642\) 0 0
\(643\) 37.7082 1.48707 0.743533 0.668699i \(-0.233149\pi\)
0.743533 + 0.668699i \(0.233149\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 31.1246i 1.22363i −0.790999 0.611817i \(-0.790439\pi\)
0.790999 0.611817i \(-0.209561\pi\)
\(648\) 0 0
\(649\) 46.8328i 1.83835i
\(650\) 0 0
\(651\) −4.94427 + 2.76393i −0.193781 + 0.108327i
\(652\) 0 0
\(653\) −27.5967 −1.07994 −0.539972 0.841683i \(-0.681565\pi\)
−0.539972 + 0.841683i \(0.681565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −25.1246 + 22.4721i −0.980204 + 0.876722i
\(658\) 0 0
\(659\) 41.2361i 1.60633i −0.595757 0.803165i \(-0.703148\pi\)
0.595757 0.803165i \(-0.296852\pi\)
\(660\) 0 0
\(661\) 30.8328i 1.19926i −0.800278 0.599629i \(-0.795315\pi\)
0.800278 0.599629i \(-0.204685\pi\)
\(662\) 0 0
\(663\) −8.94427 23.4164i −0.347367 0.909418i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22.8328i 0.884090i
\(668\) 0 0
\(669\) −2.76393 + 1.05573i −0.106860 + 0.0408168i
\(670\) 0 0
\(671\) −67.7771 −2.61651
\(672\) 0 0
\(673\) 2.94427i 0.113493i 0.998389 + 0.0567467i \(0.0180727\pi\)
−0.998389 + 0.0567467i \(0.981927\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000i 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 0 0
\(679\) 3.12461 + 21.4164i 0.119912 + 0.821886i
\(680\) 0 0
\(681\) −11.5967 30.3607i −0.444388 1.16342i
\(682\) 0 0
\(683\) 33.7082 1.28981 0.644904 0.764263i \(-0.276897\pi\)
0.644904 + 0.764263i \(0.276897\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13.5279 35.4164i −0.516120 1.35122i
\(688\) 0 0
\(689\) 15.4164 0.587318
\(690\) 0 0
\(691\) 14.1803i 0.539446i −0.962938 0.269723i \(-0.913068\pi\)
0.962938 0.269723i \(-0.0869321\pi\)
\(692\) 0 0
\(693\) 26.6525 31.8885i 1.01244 1.21135i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 55.7771i 2.11271i
\(698\) 0 0
\(699\) −39.1246 + 14.9443i −1.47983 + 0.565244i
\(700\) 0 0
\(701\) 20.9443i 0.791054i 0.918454 + 0.395527i \(0.129438\pi\)
−0.918454 + 0.395527i \(0.870562\pi\)
\(702\) 0 0
\(703\) −12.3607 −0.466192
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.18034 28.6525i −0.157218 1.07759i
\(708\) 0 0
\(709\) 5.05573 0.189872 0.0949359 0.995483i \(-0.469735\pi\)
0.0949359 + 0.995483i \(0.469735\pi\)
\(710\) 0 0
\(711\) −3.41641 + 3.05573i −0.128125 + 0.114599i
\(712\) 0 0
\(713\) 7.05573i 0.264239i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −13.7082 35.8885i −0.511942 1.34028i
\(718\) 0 0
\(719\) 41.8885 1.56218 0.781090 0.624419i \(-0.214664\pi\)
0.781090 + 0.624419i \(0.214664\pi\)
\(720\) 0 0
\(721\) 5.81966 + 39.8885i 0.216735 + 1.48553i
\(722\) 0 0
\(723\) 17.5279 + 45.8885i 0.651868 + 1.70661i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −9.70820 −0.360057 −0.180029 0.983661i \(-0.557619\pi\)
−0.180029 + 0.983661i \(0.557619\pi\)
\(728\) 0 0
\(729\) −15.6525 22.0000i −0.579721 0.814815i
\(730\) 0 0
\(731\) 12.3607 0.457176
\(732\) 0 0
\(733\) −17.3475 −0.640745 −0.320373 0.947292i \(-0.603808\pi\)
−0.320373 + 0.947292i \(0.603808\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.4164 0.715213
\(738\) 0 0
\(739\) 4.36068 0.160410 0.0802051 0.996778i \(-0.474442\pi\)
0.0802051 + 0.996778i \(0.474442\pi\)
\(740\) 0 0
\(741\) −5.52786 14.4721i −0.203071 0.531647i
\(742\) 0 0
\(743\) 6.65248 0.244056 0.122028 0.992527i \(-0.461060\pi\)
0.122028 + 0.992527i \(0.461060\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.52786 + 6.18034i 0.202254 + 0.226127i
\(748\) 0 0
\(749\) 5.23607 + 35.8885i 0.191322 + 1.31134i
\(750\) 0 0
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 0 0
\(753\) −39.4164 + 15.0557i −1.43641 + 0.548661i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 27.3050i 0.992415i −0.868204 0.496208i \(-0.834725\pi\)
0.868204 0.496208i \(-0.165275\pi\)
\(758\) 0 0
\(759\) −18.4721 48.3607i −0.670496 1.75538i
\(760\) 0 0
\(761\) −31.3050 −1.13480 −0.567402 0.823441i \(-0.692051\pi\)
−0.567402 + 0.823441i \(0.692051\pi\)
\(762\) 0 0
\(763\) 5.70820 + 39.1246i 0.206651 + 1.41641i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.9443 −1.04512
\(768\) 0 0
\(769\) 4.58359i 0.165289i 0.996579 + 0.0826443i \(0.0263365\pi\)
−0.996579 + 0.0826443i \(0.973663\pi\)
\(770\) 0 0
\(771\) 4.65248 + 12.1803i 0.167555 + 0.438664i
\(772\) 0 0
\(773\) 18.0000i 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 17.8885 10.0000i 0.641748 0.358748i
\(778\) 0 0
\(779\) 34.4721i 1.23509i
\(780\) 0 0
\(781\) 53.3050 1.90740
\(782\) 0 0
\(783\) −18.4721 9.52786i −0.660140 0.340498i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −29.7082 −1.05898 −0.529492 0.848315i \(-0.677617\pi\)
−0.529492 + 0.848315i \(0.677617\pi\)
\(788\) 0 0
\(789\) 0.291796 0.111456i 0.0103882 0.00396795i
\(790\) 0 0
\(791\) 4.87539 + 33.4164i 0.173349 + 1.18815i
\(792\) 0 0
\(793\) 41.8885i 1.48751i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.8328i 0.879623i 0.898090 + 0.439812i \(0.144955\pi\)
−0.898090 + 0.439812i \(0.855045\pi\)
\(798\) 0 0
\(799\) 5.52786 0.195562
\(800\) 0 0
\(801\) 7.88854 7.05573i 0.278728 0.249302i
\(802\) 0 0
\(803\) 58.8328i 2.07616i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 46.0689 17.5967i 1.62170 0.619435i
\(808\) 0 0
\(809\) 32.3607i 1.13774i 0.822427 + 0.568870i \(0.192619\pi\)
−0.822427 + 0.568870i \(0.807381\pi\)
\(810\) 0 0
\(811\) 11.7082i 0.411131i 0.978643 + 0.205565i \(0.0659033\pi\)
−0.978643 + 0.205565i \(0.934097\pi\)
\(812\) 0 0
\(813\) −11.8197 30.9443i −0.414534 1.08526i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.63932 0.267266
\(818\) 0 0
\(819\) 19.7082 + 16.4721i 0.688660 + 0.575583i
\(820\) 0 0
\(821\) 6.83282i 0.238467i −0.992866 0.119233i \(-0.961956\pi\)
0.992866 0.119233i \(-0.0380437\pi\)
\(822\) 0 0
\(823\) 37.5967i 1.31054i −0.755395 0.655270i \(-0.772555\pi\)
0.755395 0.655270i \(-0.227445\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.2918 0.914255 0.457128 0.889401i \(-0.348878\pi\)
0.457128 + 0.889401i \(0.348878\pi\)
\(828\) 0 0
\(829\) 24.3607i 0.846081i −0.906111 0.423041i \(-0.860963\pi\)
0.906111 0.423041i \(-0.139037\pi\)
\(830\) 0 0
\(831\) 15.7082 + 41.1246i 0.544912 + 1.42660i
\(832\) 0 0
\(833\) 8.94427 + 30.0000i 0.309901 + 1.03944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.70820 2.94427i −0.197304 0.101769i
\(838\) 0 0
\(839\) −4.94427 −0.170695 −0.0853476 0.996351i \(-0.527200\pi\)
−0.0853476 + 0.996351i \(0.527200\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) −9.52786 24.9443i −0.328157 0.859126i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.27051 + 42.9787i 0.215457 + 1.47677i
\(848\) 0 0
\(849\) −36.6525 + 14.0000i −1.25791 + 0.480479i
\(850\) 0 0
\(851\) 25.5279i 0.875084i
\(852\) 0 0
\(853\) −21.1246 −0.723293 −0.361646 0.932315i \(-0.617785\pi\)
−0.361646 + 0.932315i \(0.617785\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.5279i 0.667059i 0.942740 + 0.333530i \(0.108240\pi\)
−0.942740 + 0.333530i \(0.891760\pi\)
\(858\) 0 0
\(859\) 44.6525i 1.52352i −0.647858 0.761761i \(-0.724335\pi\)
0.647858 0.761761i \(-0.275665\pi\)
\(860\) 0 0
\(861\) −27.8885 49.8885i −0.950439 1.70020i
\(862\) 0 0
\(863\) −21.3475 −0.726678 −0.363339 0.931657i \(-0.618363\pi\)
−0.363339 + 0.931657i \(0.618363\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.85410 1.85410i 0.164854 0.0629686i
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) 12.0000i 0.406604i
\(872\) 0 0
\(873\) −18.2918 + 16.3607i −0.619083 + 0.553725i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23.5279i 0.794480i −0.917715 0.397240i \(-0.869968\pi\)
0.917715 0.397240i \(-0.130032\pi\)
\(878\) 0 0
\(879\) −8.65248 22.6525i −0.291841 0.764049i
\(880\) 0 0
\(881\) 36.8328 1.24093 0.620465 0.784234i \(-0.286944\pi\)
0.620465 + 0.784234i \(0.286944\pi\)
\(882\) 0 0
\(883\) 13.5967i 0.457567i −0.973477 0.228783i \(-0.926525\pi\)
0.973477 0.228783i \(-0.0734748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.18034i 0.207516i 0.994603 + 0.103758i \(0.0330867\pi\)
−0.994603 + 0.103758i \(0.966913\pi\)
\(888\) 0 0
\(889\) 50.0689 7.30495i 1.67926 0.245000i
\(890\) 0 0
\(891\) 46.8328 + 5.23607i 1.56896 + 0.175415i
\(892\) 0 0
\(893\) 3.41641 0.114326
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 29.8885 11.4164i 0.997949 0.381183i
\(898\) 0 0
\(899\) −4.94427 −0.164901
\(900\) 0 0
\(901\) 21.3050i 0.709771i
\(902\) 0 0
\(903\) −11.0557 + 6.18034i −0.367912 + 0.205669i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.7639i 0.755864i −0.925833 0.377932i \(-0.876635\pi\)
0.925833 0.377932i \(-0.123365\pi\)
\(908\) 0 0
\(909\) 24.4721 21.8885i 0.811690 0.725997i
\(910\) 0 0
\(911\) 12.0689i 0.399860i −0.979810 0.199930i \(-0.935929\pi\)
0.979810 0.199930i \(-0.0640715\pi\)
\(912\) 0 0
\(913\) −14.4721 −0.478958
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.36068 + 29.8885i 0.144002 + 0.987007i
\(918\) 0 0
\(919\) 25.5279 0.842087 0.421043 0.907041i \(-0.361664\pi\)
0.421043 + 0.907041i \(0.361664\pi\)
\(920\) 0 0
\(921\) −6.76393 + 2.58359i −0.222879 + 0.0851323i
\(922\) 0 0
\(923\) 32.9443i 1.08437i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −34.0689 + 30.4721i −1.11897 + 1.00084i
\(928\) 0 0
\(929\) −15.8885 −0.521286 −0.260643 0.965435i \(-0.583935\pi\)
−0.260643 + 0.965435i \(0.583935\pi\)
\(930\) 0 0
\(931\) 5.52786 + 18.5410i 0.181168 + 0.607657i
\(932\) 0 0
\(933\) −20.0000 + 7.63932i −0.654771 + 0.250100i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.2361 −0.628415 −0.314207 0.949354i \(-0.601739\pi\)
−0.314207 + 0.949354i \(0.601739\pi\)
\(938\) 0 0
\(939\) −28.6525 + 10.9443i −0.935038 + 0.357153i
\(940\) 0 0
\(941\) 36.2492 1.18169 0.590845 0.806785i \(-0.298794\pi\)
0.590845 + 0.806785i \(0.298794\pi\)
\(942\) 0 0
\(943\) −71.1935 −2.31838
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.7639 −0.674737 −0.337369 0.941373i \(-0.609537\pi\)
−0.337369 + 0.941373i \(0.609537\pi\)
\(948\) 0 0
\(949\) 36.3607 1.18032
\(950\) 0 0
\(951\) −26.1803 + 10.0000i −0.848956 + 0.324272i
\(952\) 0 0
\(953\) −9.70820 −0.314480 −0.157240 0.987560i \(-0.550260\pi\)
−0.157240 + 0.987560i \(0.550260\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 33.8885 12.9443i 1.09546 0.418429i
\(958\) 0 0
\(959\) −1.23607 8.47214i −0.0399147 0.273580i
\(960\) 0 0
\(961\) 29.4721 0.950714
\(962\) 0 0
\(963\) −30.6525 + 27.4164i −0.987762 + 0.883481i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 21.8197i 0.701673i 0.936437 + 0.350836i \(0.114103\pi\)
−0.936437 + 0.350836i \(0.885897\pi\)
\(968\) 0 0
\(969\) 20.0000 7.63932i 0.642493 0.245410i
\(970\) 0 0
\(971\) −27.4164 −0.879834 −0.439917 0.898038i \(-0.644992\pi\)
−0.439917 + 0.898038i \(0.644992\pi\)
\(972\) 0 0
\(973\) −28.1803 + 4.11146i −0.903420 + 0.131807i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.9574 1.40632 0.703161 0.711030i \(-0.251771\pi\)
0.703161 + 0.711030i \(0.251771\pi\)
\(978\) 0 0
\(979\) 18.4721i 0.590372i
\(980\) 0 0
\(981\) −33.4164 + 29.8885i −1.06690 + 0.954268i
\(982\) 0 0
\(983\) 26.5410i 0.846527i −0.906007 0.423264i \(-0.860884\pi\)
0.906007 0.423264i \(-0.139116\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.94427 + 2.76393i −0.157378 + 0.0879769i
\(988\) 0 0
\(989\) 15.7771i 0.501682i
\(990\) 0 0
\(991\) −31.7771 −1.00943 −0.504716 0.863285i \(-0.668403\pi\)
−0.504716 + 0.863285i \(0.668403\pi\)
\(992\) 0 0
\(993\) 4.94427 1.88854i 0.156902 0.0599311i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.0689 −1.33233 −0.666167 0.745802i \(-0.732066\pi\)
−0.666167 + 0.745802i \(0.732066\pi\)
\(998\) 0 0
\(999\) 20.6525 + 10.6525i 0.653415 + 0.337029i
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.f.b.1049.2 4
3.2 odd 2 2100.2.f.a.1049.1 4
5.2 odd 4 2100.2.d.g.1301.4 4
5.3 odd 4 420.2.d.d.41.1 yes 4
5.4 even 2 2100.2.f.h.1049.3 4
7.6 odd 2 2100.2.f.g.1049.3 4
15.2 even 4 2100.2.d.h.1301.2 4
15.8 even 4 420.2.d.c.41.3 4
15.14 odd 2 2100.2.f.g.1049.4 4
20.3 even 4 1680.2.f.f.881.4 4
21.20 even 2 2100.2.f.h.1049.4 4
35.13 even 4 420.2.d.c.41.4 yes 4
35.27 even 4 2100.2.d.h.1301.1 4
35.34 odd 2 2100.2.f.a.1049.2 4
60.23 odd 4 1680.2.f.j.881.2 4
105.62 odd 4 2100.2.d.g.1301.3 4
105.83 odd 4 420.2.d.d.41.2 yes 4
105.104 even 2 inner 2100.2.f.b.1049.1 4
140.83 odd 4 1680.2.f.j.881.1 4
420.83 even 4 1680.2.f.f.881.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.d.c.41.3 4 15.8 even 4
420.2.d.c.41.4 yes 4 35.13 even 4
420.2.d.d.41.1 yes 4 5.3 odd 4
420.2.d.d.41.2 yes 4 105.83 odd 4
1680.2.f.f.881.3 4 420.83 even 4
1680.2.f.f.881.4 4 20.3 even 4
1680.2.f.j.881.1 4 140.83 odd 4
1680.2.f.j.881.2 4 60.23 odd 4
2100.2.d.g.1301.3 4 105.62 odd 4
2100.2.d.g.1301.4 4 5.2 odd 4
2100.2.d.h.1301.1 4 35.27 even 4
2100.2.d.h.1301.2 4 15.2 even 4
2100.2.f.a.1049.1 4 3.2 odd 2
2100.2.f.a.1049.2 4 35.34 odd 2
2100.2.f.b.1049.1 4 105.104 even 2 inner
2100.2.f.b.1049.2 4 1.1 even 1 trivial
2100.2.f.g.1049.3 4 7.6 odd 2
2100.2.f.g.1049.4 4 15.14 odd 2
2100.2.f.h.1049.3 4 5.4 even 2
2100.2.f.h.1049.4 4 21.20 even 2