Properties

Label 882.3.n.c.19.1
Level $882$
Weight $3$
Character 882.19
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,3,Mod(19,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.n (of order \(6\), 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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 19.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.19
Dual form 882.3.n.c.325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-4.24264 - 2.44949i) q^{5} +2.82843 q^{8} +O(q^{10})\) \(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-4.24264 - 2.44949i) q^{5} +2.82843 q^{8} +(6.00000 - 3.46410i) q^{10} +(8.48528 + 14.6969i) q^{11} -1.73205i q^{13} +(-2.00000 + 3.46410i) q^{16} +(4.24264 - 2.44949i) q^{17} +(-25.5000 - 14.7224i) q^{19} +9.79796i q^{20} -24.0000 q^{22} +(4.24264 - 7.34847i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.12132 + 1.22474i) q^{26} -33.9411 q^{29} +(10.5000 - 6.06218i) q^{31} +(-2.82843 - 4.89898i) q^{32} +6.92820i q^{34} +(23.5000 - 40.7032i) q^{37} +(36.0624 - 20.8207i) q^{38} +(-12.0000 - 6.92820i) q^{40} +68.5857i q^{41} +31.0000 q^{43} +(16.9706 - 29.3939i) q^{44} +(6.00000 + 10.3923i) q^{46} +(72.1249 + 41.6413i) q^{47} +1.41421 q^{50} +(-3.00000 + 1.73205i) q^{52} +(38.1838 + 66.1362i) q^{53} -83.1384i q^{55} +(24.0000 - 41.5692i) q^{58} +(72.1249 - 41.6413i) q^{59} +(72.0000 + 41.5692i) q^{61} +17.1464i q^{62} +8.00000 q^{64} +(-4.24264 + 7.34847i) q^{65} +(15.5000 + 26.8468i) q^{67} +(-8.48528 - 4.89898i) q^{68} +59.3970 q^{71} +(70.5000 - 40.7032i) q^{73} +(33.2340 + 57.5630i) q^{74} +58.8897i q^{76} +(-20.5000 + 35.5070i) q^{79} +(16.9706 - 9.79796i) q^{80} +(-84.0000 - 48.4974i) q^{82} -4.89898i q^{83} -24.0000 q^{85} +(-21.9203 + 37.9671i) q^{86} +(24.0000 + 41.5692i) q^{88} +(-50.9117 - 29.3939i) q^{89} -16.9706 q^{92} +(-102.000 + 58.8897i) q^{94} +(72.1249 + 124.924i) q^{95} +41.5692i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 24 q^{10} - 8 q^{16} - 102 q^{19} - 96 q^{22} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 48 q^{40} + 124 q^{43} + 24 q^{46} - 12 q^{52} + 96 q^{58} + 288 q^{61} + 32 q^{64} + 62 q^{67} + 282 q^{73} - 82 q^{79} - 336 q^{82} - 96 q^{85} + 96 q^{88} - 408 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 1.22474i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −1.00000 1.73205i −0.250000 0.433013i
\(5\) −4.24264 2.44949i −0.848528 0.489898i 0.0116258 0.999932i \(-0.496299\pi\)
−0.860154 + 0.510034i \(0.829633\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) 6.00000 3.46410i 0.600000 0.346410i
\(11\) 8.48528 + 14.6969i 0.771389 + 1.33609i 0.936802 + 0.349861i \(0.113771\pi\)
−0.165412 + 0.986224i \(0.552896\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.133235i −0.997779 0.0666173i \(-0.978779\pi\)
0.997779 0.0666173i \(-0.0212207\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.125000 + 0.216506i
\(17\) 4.24264 2.44949i 0.249567 0.144088i −0.369999 0.929032i \(-0.620642\pi\)
0.619566 + 0.784945i \(0.287309\pi\)
\(18\) 0 0
\(19\) −25.5000 14.7224i −1.34211 0.774865i −0.354989 0.934870i \(-0.615515\pi\)
−0.987116 + 0.160006i \(0.948849\pi\)
\(20\) 9.79796i 0.489898i
\(21\) 0 0
\(22\) −24.0000 −1.09091
\(23\) 4.24264 7.34847i 0.184463 0.319499i −0.758933 0.651169i \(-0.774279\pi\)
0.943395 + 0.331670i \(0.107612\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.0200000 0.0346410i
\(26\) 2.12132 + 1.22474i 0.0815892 + 0.0471056i
\(27\) 0 0
\(28\) 0 0
\(29\) −33.9411 −1.17038 −0.585192 0.810895i \(-0.698981\pi\)
−0.585192 + 0.810895i \(0.698981\pi\)
\(30\) 0 0
\(31\) 10.5000 6.06218i 0.338710 0.195554i −0.320992 0.947082i \(-0.604016\pi\)
0.659701 + 0.751528i \(0.270683\pi\)
\(32\) −2.82843 4.89898i −0.0883883 0.153093i
\(33\) 0 0
\(34\) 6.92820i 0.203771i
\(35\) 0 0
\(36\) 0 0
\(37\) 23.5000 40.7032i 0.635135 1.10009i −0.351351 0.936244i \(-0.614278\pi\)
0.986486 0.163843i \(-0.0523889\pi\)
\(38\) 36.0624 20.8207i 0.949012 0.547912i
\(39\) 0 0
\(40\) −12.0000 6.92820i −0.300000 0.173205i
\(41\) 68.5857i 1.67282i 0.548103 + 0.836411i \(0.315350\pi\)
−0.548103 + 0.836411i \(0.684650\pi\)
\(42\) 0 0
\(43\) 31.0000 0.720930 0.360465 0.932773i \(-0.382618\pi\)
0.360465 + 0.932773i \(0.382618\pi\)
\(44\) 16.9706 29.3939i 0.385695 0.668043i
\(45\) 0 0
\(46\) 6.00000 + 10.3923i 0.130435 + 0.225920i
\(47\) 72.1249 + 41.6413i 1.53457 + 0.885986i 0.999142 + 0.0414059i \(0.0131837\pi\)
0.535430 + 0.844580i \(0.320150\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.41421 0.0282843
\(51\) 0 0
\(52\) −3.00000 + 1.73205i −0.0576923 + 0.0333087i
\(53\) 38.1838 + 66.1362i 0.720448 + 1.24785i 0.960820 + 0.277172i \(0.0893973\pi\)
−0.240372 + 0.970681i \(0.577269\pi\)
\(54\) 0 0
\(55\) 83.1384i 1.51161i
\(56\) 0 0
\(57\) 0 0
\(58\) 24.0000 41.5692i 0.413793 0.716711i
\(59\) 72.1249 41.6413i 1.22246 0.705785i 0.257015 0.966407i \(-0.417261\pi\)
0.965441 + 0.260622i \(0.0839277\pi\)
\(60\) 0 0
\(61\) 72.0000 + 41.5692i 1.18033 + 0.681463i 0.956090 0.293072i \(-0.0946775\pi\)
0.224237 + 0.974535i \(0.428011\pi\)
\(62\) 17.1464i 0.276555i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) −4.24264 + 7.34847i −0.0652714 + 0.113053i
\(66\) 0 0
\(67\) 15.5000 + 26.8468i 0.231343 + 0.400698i 0.958204 0.286087i \(-0.0923546\pi\)
−0.726860 + 0.686785i \(0.759021\pi\)
\(68\) −8.48528 4.89898i −0.124784 0.0720438i
\(69\) 0 0
\(70\) 0 0
\(71\) 59.3970 0.836577 0.418289 0.908314i \(-0.362630\pi\)
0.418289 + 0.908314i \(0.362630\pi\)
\(72\) 0 0
\(73\) 70.5000 40.7032i 0.965753 0.557578i 0.0678144 0.997698i \(-0.478397\pi\)
0.897939 + 0.440120i \(0.145064\pi\)
\(74\) 33.2340 + 57.5630i 0.449108 + 0.777878i
\(75\) 0 0
\(76\) 58.8897i 0.774865i
\(77\) 0 0
\(78\) 0 0
\(79\) −20.5000 + 35.5070i −0.259494 + 0.449456i −0.966106 0.258144i \(-0.916889\pi\)
0.706613 + 0.707601i \(0.250222\pi\)
\(80\) 16.9706 9.79796i 0.212132 0.122474i
\(81\) 0 0
\(82\) −84.0000 48.4974i −1.02439 0.591432i
\(83\) 4.89898i 0.0590238i −0.999564 0.0295119i \(-0.990605\pi\)
0.999564 0.0295119i \(-0.00939530\pi\)
\(84\) 0 0
\(85\) −24.0000 −0.282353
\(86\) −21.9203 + 37.9671i −0.254887 + 0.441478i
\(87\) 0 0
\(88\) 24.0000 + 41.5692i 0.272727 + 0.472377i
\(89\) −50.9117 29.3939i −0.572041 0.330268i 0.185923 0.982564i \(-0.440473\pi\)
−0.757964 + 0.652296i \(0.773806\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −16.9706 −0.184463
\(93\) 0 0
\(94\) −102.000 + 58.8897i −1.08511 + 0.626486i
\(95\) 72.1249 + 124.924i 0.759209 + 1.31499i
\(96\) 0 0
\(97\) 41.5692i 0.428549i 0.976774 + 0.214274i \(0.0687387\pi\)
−0.976774 + 0.214274i \(0.931261\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 + 1.73205i −0.0100000 + 0.0173205i
\(101\) 152.735 88.1816i 1.51223 0.873085i 0.512331 0.858788i \(-0.328782\pi\)
0.999898 0.0142971i \(-0.00455107\pi\)
\(102\) 0 0
\(103\) 25.5000 + 14.7224i 0.247573 + 0.142936i 0.618652 0.785665i \(-0.287679\pi\)
−0.371080 + 0.928601i \(0.621012\pi\)
\(104\) 4.89898i 0.0471056i
\(105\) 0 0
\(106\) −108.000 −1.01887
\(107\) 72.1249 124.924i 0.674064 1.16751i −0.302677 0.953093i \(-0.597880\pi\)
0.976741 0.214421i \(-0.0687863\pi\)
\(108\) 0 0
\(109\) −84.5000 146.358i −0.775229 1.34274i −0.934665 0.355528i \(-0.884301\pi\)
0.159436 0.987208i \(-0.449032\pi\)
\(110\) 101.823 + 58.7878i 0.925667 + 0.534434i
\(111\) 0 0
\(112\) 0 0
\(113\) −59.3970 −0.525637 −0.262818 0.964845i \(-0.584652\pi\)
−0.262818 + 0.964845i \(0.584652\pi\)
\(114\) 0 0
\(115\) −36.0000 + 20.7846i −0.313043 + 0.180736i
\(116\) 33.9411 + 58.7878i 0.292596 + 0.506791i
\(117\) 0 0
\(118\) 117.779i 0.998131i
\(119\) 0 0
\(120\) 0 0
\(121\) −83.5000 + 144.626i −0.690083 + 1.19526i
\(122\) −101.823 + 58.7878i −0.834618 + 0.481867i
\(123\) 0 0
\(124\) −21.0000 12.1244i −0.169355 0.0977771i
\(125\) 127.373i 1.01899i
\(126\) 0 0
\(127\) 209.000 1.64567 0.822835 0.568281i \(-0.192391\pi\)
0.822835 + 0.568281i \(0.192391\pi\)
\(128\) −5.65685 + 9.79796i −0.0441942 + 0.0765466i
\(129\) 0 0
\(130\) −6.00000 10.3923i −0.0461538 0.0799408i
\(131\) 50.9117 + 29.3939i 0.388639 + 0.224381i 0.681570 0.731753i \(-0.261297\pi\)
−0.292931 + 0.956133i \(0.594631\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −43.8406 −0.327169
\(135\) 0 0
\(136\) 12.0000 6.92820i 0.0882353 0.0509427i
\(137\) −76.3675 132.272i −0.557427 0.965492i −0.997710 0.0676333i \(-0.978455\pi\)
0.440283 0.897859i \(-0.354878\pi\)
\(138\) 0 0
\(139\) 195.722i 1.40807i −0.710165 0.704035i \(-0.751380\pi\)
0.710165 0.704035i \(-0.248620\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −42.0000 + 72.7461i −0.295775 + 0.512297i
\(143\) 25.4558 14.6969i 0.178013 0.102776i
\(144\) 0 0
\(145\) 144.000 + 83.1384i 0.993103 + 0.573369i
\(146\) 115.126i 0.788534i
\(147\) 0 0
\(148\) −94.0000 −0.635135
\(149\) −25.4558 + 44.0908i −0.170845 + 0.295912i −0.938715 0.344693i \(-0.887983\pi\)
0.767871 + 0.640605i \(0.221316\pi\)
\(150\) 0 0
\(151\) −5.00000 8.66025i −0.0331126 0.0573527i 0.848994 0.528402i \(-0.177209\pi\)
−0.882107 + 0.471049i \(0.843875\pi\)
\(152\) −72.1249 41.6413i −0.474506 0.273956i
\(153\) 0 0
\(154\) 0 0
\(155\) −59.3970 −0.383206
\(156\) 0 0
\(157\) −36.0000 + 20.7846i −0.229299 + 0.132386i −0.610249 0.792210i \(-0.708931\pi\)
0.380949 + 0.924596i \(0.375597\pi\)
\(158\) −28.9914 50.2145i −0.183490 0.317814i
\(159\) 0 0
\(160\) 27.7128i 0.173205i
\(161\) 0 0
\(162\) 0 0
\(163\) −43.0000 + 74.4782i −0.263804 + 0.456921i −0.967250 0.253828i \(-0.918310\pi\)
0.703446 + 0.710749i \(0.251644\pi\)
\(164\) 118.794 68.5857i 0.724353 0.418206i
\(165\) 0 0
\(166\) 6.00000 + 3.46410i 0.0361446 + 0.0208681i
\(167\) 181.262i 1.08540i 0.839926 + 0.542701i \(0.182598\pi\)
−0.839926 + 0.542701i \(0.817402\pi\)
\(168\) 0 0
\(169\) 166.000 0.982249
\(170\) 16.9706 29.3939i 0.0998268 0.172905i
\(171\) 0 0
\(172\) −31.0000 53.6936i −0.180233 0.312172i
\(173\) 38.1838 + 22.0454i 0.220715 + 0.127430i 0.606281 0.795250i \(-0.292660\pi\)
−0.385566 + 0.922680i \(0.625994\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −67.8823 −0.385695
\(177\) 0 0
\(178\) 72.0000 41.5692i 0.404494 0.233535i
\(179\) 4.24264 + 7.34847i 0.0237019 + 0.0410529i 0.877633 0.479333i \(-0.159121\pi\)
−0.853931 + 0.520386i \(0.825788\pi\)
\(180\) 0 0
\(181\) 43.3013i 0.239234i 0.992820 + 0.119617i \(0.0381666\pi\)
−0.992820 + 0.119617i \(0.961833\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.0000 20.7846i 0.0652174 0.112960i
\(185\) −199.404 + 115.126i −1.07786 + 0.622303i
\(186\) 0 0
\(187\) 72.0000 + 41.5692i 0.385027 + 0.222295i
\(188\) 166.565i 0.885986i
\(189\) 0 0
\(190\) −204.000 −1.07368
\(191\) −38.1838 + 66.1362i −0.199915 + 0.346263i −0.948501 0.316775i \(-0.897400\pi\)
0.748586 + 0.663038i \(0.230733\pi\)
\(192\) 0 0
\(193\) 143.500 + 248.549i 0.743523 + 1.28782i 0.950882 + 0.309555i \(0.100180\pi\)
−0.207358 + 0.978265i \(0.566487\pi\)
\(194\) −50.9117 29.3939i −0.262431 0.151515i
\(195\) 0 0
\(196\) 0 0
\(197\) 127.279 0.646087 0.323044 0.946384i \(-0.395294\pi\)
0.323044 + 0.946384i \(0.395294\pi\)
\(198\) 0 0
\(199\) −180.000 + 103.923i −0.904523 + 0.522226i −0.878665 0.477439i \(-0.841565\pi\)
−0.0258579 + 0.999666i \(0.508232\pi\)
\(200\) −1.41421 2.44949i −0.00707107 0.0122474i
\(201\) 0 0
\(202\) 249.415i 1.23473i
\(203\) 0 0
\(204\) 0 0
\(205\) 168.000 290.985i 0.819512 1.41944i
\(206\) −36.0624 + 20.8207i −0.175060 + 0.101071i
\(207\) 0 0
\(208\) 6.00000 + 3.46410i 0.0288462 + 0.0166543i
\(209\) 499.696i 2.39089i
\(210\) 0 0
\(211\) 82.0000 0.388626 0.194313 0.980940i \(-0.437752\pi\)
0.194313 + 0.980940i \(0.437752\pi\)
\(212\) 76.3675 132.272i 0.360224 0.623927i
\(213\) 0 0
\(214\) 102.000 + 176.669i 0.476636 + 0.825557i
\(215\) −131.522 75.9342i −0.611730 0.353182i
\(216\) 0 0
\(217\) 0 0
\(218\) 239.002 1.09634
\(219\) 0 0
\(220\) −144.000 + 83.1384i −0.654545 + 0.377902i
\(221\) −4.24264 7.34847i −0.0191975 0.0332510i
\(222\) 0 0
\(223\) 41.5692i 0.186409i −0.995647 0.0932045i \(-0.970289\pi\)
0.995647 0.0932045i \(-0.0297110\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 42.0000 72.7461i 0.185841 0.321886i
\(227\) −330.926 + 191.060i −1.45782 + 0.841675i −0.998904 0.0468029i \(-0.985097\pi\)
−0.458920 + 0.888478i \(0.651763\pi\)
\(228\) 0 0
\(229\) 70.5000 + 40.7032i 0.307860 + 0.177743i 0.645969 0.763364i \(-0.276454\pi\)
−0.338108 + 0.941107i \(0.609787\pi\)
\(230\) 58.7878i 0.255599i
\(231\) 0 0
\(232\) −96.0000 −0.413793
\(233\) −114.551 + 198.409i −0.491636 + 0.851539i −0.999954 0.00963059i \(-0.996934\pi\)
0.508317 + 0.861170i \(0.330268\pi\)
\(234\) 0 0
\(235\) −204.000 353.338i −0.868085 1.50357i
\(236\) −144.250 83.2827i −0.611228 0.352893i
\(237\) 0 0
\(238\) 0 0
\(239\) −67.8823 −0.284026 −0.142013 0.989865i \(-0.545358\pi\)
−0.142013 + 0.989865i \(0.545358\pi\)
\(240\) 0 0
\(241\) 396.000 228.631i 1.64315 0.948675i 0.663451 0.748220i \(-0.269091\pi\)
0.979703 0.200455i \(-0.0642421\pi\)
\(242\) −118.087 204.532i −0.487962 0.845175i
\(243\) 0 0
\(244\) 166.277i 0.681463i
\(245\) 0 0
\(246\) 0 0
\(247\) −25.5000 + 44.1673i −0.103239 + 0.178815i
\(248\) 29.6985 17.1464i 0.119752 0.0691388i
\(249\) 0 0
\(250\) −156.000 90.0666i −0.624000 0.360267i
\(251\) 347.828i 1.38577i −0.721050 0.692884i \(-0.756340\pi\)
0.721050 0.692884i \(-0.243660\pi\)
\(252\) 0 0
\(253\) 144.000 0.569170
\(254\) −147.785 + 255.972i −0.581832 + 1.00776i
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.0312500 0.0541266i
\(257\) −140.007 80.8332i −0.544775 0.314526i 0.202237 0.979337i \(-0.435179\pi\)
−0.747012 + 0.664811i \(0.768512\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 16.9706 0.0652714
\(261\) 0 0
\(262\) −72.0000 + 41.5692i −0.274809 + 0.158661i
\(263\) −127.279 220.454i −0.483951 0.838228i 0.515879 0.856662i \(-0.327466\pi\)
−0.999830 + 0.0184332i \(0.994132\pi\)
\(264\) 0 0
\(265\) 374.123i 1.41178i
\(266\) 0 0
\(267\) 0 0
\(268\) 31.0000 53.6936i 0.115672 0.200349i
\(269\) −16.9706 + 9.79796i −0.0630876 + 0.0364236i −0.531212 0.847239i \(-0.678263\pi\)
0.468124 + 0.883663i \(0.344930\pi\)
\(270\) 0 0
\(271\) −36.0000 20.7846i −0.132841 0.0766960i 0.432107 0.901823i \(-0.357770\pi\)
−0.564948 + 0.825127i \(0.691104\pi\)
\(272\) 19.5959i 0.0720438i
\(273\) 0 0
\(274\) 216.000 0.788321
\(275\) 8.48528 14.6969i 0.0308556 0.0534434i
\(276\) 0 0
\(277\) 168.500 + 291.851i 0.608303 + 1.05361i 0.991520 + 0.129954i \(0.0414829\pi\)
−0.383217 + 0.923658i \(0.625184\pi\)
\(278\) 239.709 + 138.396i 0.862263 + 0.497828i
\(279\) 0 0
\(280\) 0 0
\(281\) 246.073 0.875705 0.437853 0.899047i \(-0.355739\pi\)
0.437853 + 0.899047i \(0.355739\pi\)
\(282\) 0 0
\(283\) 169.500 97.8609i 0.598940 0.345798i −0.169685 0.985498i \(-0.554275\pi\)
0.768624 + 0.639700i \(0.220942\pi\)
\(284\) −59.3970 102.879i −0.209144 0.362248i
\(285\) 0 0
\(286\) 41.5692i 0.145347i
\(287\) 0 0
\(288\) 0 0
\(289\) −132.500 + 229.497i −0.458478 + 0.794106i
\(290\) −203.647 + 117.576i −0.702230 + 0.405433i
\(291\) 0 0
\(292\) −141.000 81.4064i −0.482877 0.278789i
\(293\) 97.9796i 0.334401i 0.985923 + 0.167201i \(0.0534728\pi\)
−0.985923 + 0.167201i \(0.946527\pi\)
\(294\) 0 0
\(295\) −408.000 −1.38305
\(296\) 66.4680 115.126i 0.224554 0.388939i
\(297\) 0 0
\(298\) −36.0000 62.3538i −0.120805 0.209241i
\(299\) −12.7279 7.34847i −0.0425683 0.0245768i
\(300\) 0 0
\(301\) 0 0
\(302\) 14.1421 0.0468283
\(303\) 0 0
\(304\) 102.000 58.8897i 0.335526 0.193716i
\(305\) −203.647 352.727i −0.667694 1.15648i
\(306\) 0 0
\(307\) 71.0141i 0.231316i −0.993289 0.115658i \(-0.963102\pi\)
0.993289 0.115658i \(-0.0368977\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 42.0000 72.7461i 0.135484 0.234665i
\(311\) −186.676 + 107.778i −0.600245 + 0.346552i −0.769138 0.639083i \(-0.779314\pi\)
0.168893 + 0.985634i \(0.445981\pi\)
\(312\) 0 0
\(313\) −253.500 146.358i −0.809904 0.467598i 0.0370184 0.999315i \(-0.488214\pi\)
−0.846923 + 0.531716i \(0.821547\pi\)
\(314\) 58.7878i 0.187222i
\(315\) 0 0
\(316\) 82.0000 0.259494
\(317\) 118.794 205.757i 0.374744 0.649076i −0.615544 0.788102i \(-0.711064\pi\)
0.990289 + 0.139026i \(0.0443972\pi\)
\(318\) 0 0
\(319\) −288.000 498.831i −0.902821 1.56373i
\(320\) −33.9411 19.5959i −0.106066 0.0612372i
\(321\) 0 0
\(322\) 0 0
\(323\) −144.250 −0.446594
\(324\) 0 0
\(325\) −1.50000 + 0.866025i −0.00461538 + 0.00266469i
\(326\) −60.8112 105.328i −0.186537 0.323092i
\(327\) 0 0
\(328\) 193.990i 0.591432i
\(329\) 0 0
\(330\) 0 0
\(331\) 92.5000 160.215i 0.279456 0.484032i −0.691794 0.722095i \(-0.743179\pi\)
0.971250 + 0.238063i \(0.0765125\pi\)
\(332\) −8.48528 + 4.89898i −0.0255581 + 0.0147560i
\(333\) 0 0
\(334\) −222.000 128.172i −0.664671 0.383748i
\(335\) 151.868i 0.453338i
\(336\) 0 0
\(337\) −359.000 −1.06528 −0.532641 0.846341i \(-0.678800\pi\)
−0.532641 + 0.846341i \(0.678800\pi\)
\(338\) −117.380 + 203.308i −0.347277 + 0.601502i
\(339\) 0 0
\(340\) 24.0000 + 41.5692i 0.0705882 + 0.122262i
\(341\) 178.191 + 102.879i 0.522554 + 0.301697i
\(342\) 0 0
\(343\) 0 0
\(344\) 87.6812 0.254887
\(345\) 0 0
\(346\) −54.0000 + 31.1769i −0.156069 + 0.0901067i
\(347\) −233.345 404.166i −0.672465 1.16474i −0.977203 0.212307i \(-0.931902\pi\)
0.304738 0.952436i \(-0.401431\pi\)
\(348\) 0 0
\(349\) 581.969i 1.66753i 0.552117 + 0.833767i \(0.313820\pi\)
−0.552117 + 0.833767i \(0.686180\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 48.0000 83.1384i 0.136364 0.236189i
\(353\) −250.316 + 144.520i −0.709110 + 0.409405i −0.810731 0.585418i \(-0.800930\pi\)
0.101621 + 0.994823i \(0.467597\pi\)
\(354\) 0 0
\(355\) −252.000 145.492i −0.709859 0.409837i
\(356\) 117.576i 0.330268i
\(357\) 0 0
\(358\) −12.0000 −0.0335196
\(359\) −169.706 + 293.939i −0.472718 + 0.818771i −0.999512 0.0312215i \(-0.990060\pi\)
0.526795 + 0.849992i \(0.323394\pi\)
\(360\) 0 0
\(361\) 253.000 + 438.209i 0.700831 + 1.21387i
\(362\) −53.0330 30.6186i −0.146500 0.0845818i
\(363\) 0 0
\(364\) 0 0
\(365\) −398.808 −1.09263
\(366\) 0 0
\(367\) −133.500 + 77.0763i −0.363760 + 0.210017i −0.670729 0.741703i \(-0.734019\pi\)
0.306969 + 0.951720i \(0.400685\pi\)
\(368\) 16.9706 + 29.3939i 0.0461157 + 0.0798747i
\(369\) 0 0
\(370\) 325.626i 0.880069i
\(371\) 0 0
\(372\) 0 0
\(373\) 144.500 250.281i 0.387399 0.670996i −0.604699 0.796454i \(-0.706707\pi\)
0.992099 + 0.125458i \(0.0400401\pi\)
\(374\) −101.823 + 58.7878i −0.272255 + 0.157187i
\(375\) 0 0
\(376\) 204.000 + 117.779i 0.542553 + 0.313243i
\(377\) 58.7878i 0.155936i
\(378\) 0 0
\(379\) 7.00000 0.0184697 0.00923483 0.999957i \(-0.497060\pi\)
0.00923483 + 0.999957i \(0.497060\pi\)
\(380\) 144.250 249.848i 0.379605 0.657495i
\(381\) 0 0
\(382\) −54.0000 93.5307i −0.141361 0.244845i
\(383\) −428.507 247.398i −1.11882 0.645949i −0.177718 0.984081i \(-0.556871\pi\)
−0.941099 + 0.338132i \(0.890205\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −405.879 −1.05150
\(387\) 0 0
\(388\) 72.0000 41.5692i 0.185567 0.107137i
\(389\) 114.551 + 198.409i 0.294476 + 0.510048i 0.974863 0.222805i \(-0.0715214\pi\)
−0.680387 + 0.732853i \(0.738188\pi\)
\(390\) 0 0
\(391\) 41.5692i 0.106315i
\(392\) 0 0
\(393\) 0 0
\(394\) −90.0000 + 155.885i −0.228426 + 0.395646i
\(395\) 173.948 100.429i 0.440375 0.254251i
\(396\) 0 0
\(397\) −70.5000 40.7032i −0.177582 0.102527i 0.408574 0.912725i \(-0.366026\pi\)
−0.586156 + 0.810198i \(0.699359\pi\)
\(398\) 293.939i 0.738540i
\(399\) 0 0
\(400\) 4.00000 0.0100000
\(401\) 46.6690 80.8332i 0.116382 0.201579i −0.801950 0.597392i \(-0.796204\pi\)
0.918331 + 0.395813i \(0.129537\pi\)
\(402\) 0 0
\(403\) −10.5000 18.1865i −0.0260546 0.0451279i
\(404\) −305.470 176.363i −0.756114 0.436543i
\(405\) 0 0
\(406\) 0 0
\(407\) 797.616 1.95975
\(408\) 0 0
\(409\) 361.500 208.712i 0.883863 0.510299i 0.0119329 0.999929i \(-0.496202\pi\)
0.871930 + 0.489630i \(0.162868\pi\)
\(410\) 237.588 + 411.514i 0.579483 + 1.00369i
\(411\) 0 0
\(412\) 58.8897i 0.142936i
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 + 20.7846i −0.0289157 + 0.0500834i
\(416\) −8.48528 + 4.89898i −0.0203973 + 0.0117764i
\(417\) 0 0
\(418\) 612.000 + 353.338i 1.46411 + 0.845307i
\(419\) 19.5959i 0.0467683i 0.999727 + 0.0233842i \(0.00744408\pi\)
−0.999727 + 0.0233842i \(0.992556\pi\)
\(420\) 0 0
\(421\) 407.000 0.966746 0.483373 0.875415i \(-0.339412\pi\)
0.483373 + 0.875415i \(0.339412\pi\)
\(422\) −57.9828 + 100.429i −0.137400 + 0.237984i
\(423\) 0 0
\(424\) 108.000 + 187.061i 0.254717 + 0.441183i
\(425\) −4.24264 2.44949i −0.00998268 0.00576351i
\(426\) 0 0
\(427\) 0 0
\(428\) −288.500 −0.674064
\(429\) 0 0
\(430\) 186.000 107.387i 0.432558 0.249738i
\(431\) 80.6102 + 139.621i 0.187031 + 0.323946i 0.944259 0.329204i \(-0.106780\pi\)
−0.757228 + 0.653150i \(0.773447\pi\)
\(432\) 0 0
\(433\) 168.009i 0.388011i −0.981000 0.194006i \(-0.937852\pi\)
0.981000 0.194006i \(-0.0621480\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −169.000 + 292.717i −0.387615 + 0.671368i
\(437\) −216.375 + 124.924i −0.495137 + 0.285867i
\(438\) 0 0
\(439\) 468.000 + 270.200i 1.06606 + 0.615490i 0.927102 0.374809i \(-0.122292\pi\)
0.138957 + 0.990298i \(0.455625\pi\)
\(440\) 235.151i 0.534434i
\(441\) 0 0
\(442\) 12.0000 0.0271493
\(443\) −63.6396 + 110.227i −0.143656 + 0.248819i −0.928871 0.370404i \(-0.879219\pi\)
0.785215 + 0.619224i \(0.212553\pi\)
\(444\) 0 0
\(445\) 144.000 + 249.415i 0.323596 + 0.560484i
\(446\) 50.9117 + 29.3939i 0.114152 + 0.0659056i
\(447\) 0 0
\(448\) 0 0
\(449\) 110.309 0.245676 0.122838 0.992427i \(-0.460800\pi\)
0.122838 + 0.992427i \(0.460800\pi\)
\(450\) 0 0
\(451\) −1008.00 + 581.969i −2.23503 + 1.29040i
\(452\) 59.3970 + 102.879i 0.131409 + 0.227607i
\(453\) 0 0
\(454\) 540.400i 1.19031i
\(455\) 0 0
\(456\) 0 0
\(457\) −12.5000 + 21.6506i −0.0273523 + 0.0473756i −0.879378 0.476125i \(-0.842041\pi\)
0.852025 + 0.523501i \(0.175374\pi\)
\(458\) −99.7021 + 57.5630i −0.217690 + 0.125683i
\(459\) 0 0
\(460\) 72.0000 + 41.5692i 0.156522 + 0.0903679i
\(461\) 78.3837i 0.170030i −0.996380 0.0850148i \(-0.972906\pi\)
0.996380 0.0850148i \(-0.0270938\pi\)
\(462\) 0 0
\(463\) 521.000 1.12527 0.562635 0.826705i \(-0.309788\pi\)
0.562635 + 0.826705i \(0.309788\pi\)
\(464\) 67.8823 117.576i 0.146298 0.253395i
\(465\) 0 0
\(466\) −162.000 280.592i −0.347639 0.602129i
\(467\) 190.919 + 110.227i 0.408820 + 0.236032i 0.690283 0.723540i \(-0.257486\pi\)
−0.281463 + 0.959572i \(0.590820\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 576.999 1.22766
\(471\) 0 0
\(472\) 204.000 117.779i 0.432203 0.249533i
\(473\) 263.044 + 455.605i 0.556118 + 0.963224i
\(474\) 0 0
\(475\) 29.4449i 0.0619892i
\(476\) 0 0
\(477\) 0 0
\(478\) 48.0000 83.1384i 0.100418 0.173930i
\(479\) 759.433 438.459i 1.58545 0.915363i 0.591412 0.806370i \(-0.298571\pi\)
0.994043 0.108993i \(-0.0347626\pi\)
\(480\) 0 0
\(481\) −70.5000 40.7032i −0.146570 0.0846220i
\(482\) 646.665i 1.34163i
\(483\) 0 0
\(484\) 334.000 0.690083
\(485\) 101.823 176.363i 0.209945 0.363636i
\(486\) 0 0
\(487\) −63.5000 109.985i −0.130390 0.225842i 0.793437 0.608653i \(-0.208290\pi\)
−0.923827 + 0.382810i \(0.874956\pi\)
\(488\) 203.647 + 117.576i 0.417309 + 0.240933i
\(489\) 0 0
\(490\) 0 0
\(491\) −627.911 −1.27884 −0.639420 0.768857i \(-0.720826\pi\)
−0.639420 + 0.768857i \(0.720826\pi\)
\(492\) 0 0
\(493\) −144.000 + 83.1384i −0.292089 + 0.168638i
\(494\) −36.0624 62.4620i −0.0730009 0.126441i
\(495\) 0 0
\(496\) 48.4974i 0.0977771i
\(497\) 0 0
\(498\) 0 0
\(499\) 116.500 201.784i 0.233467 0.404377i −0.725359 0.688371i \(-0.758326\pi\)
0.958826 + 0.283994i \(0.0916596\pi\)
\(500\) 220.617 127.373i 0.441235 0.254747i
\(501\) 0 0
\(502\) 426.000 + 245.951i 0.848606 + 0.489943i
\(503\) 538.888i 1.07135i 0.844425 + 0.535674i \(0.179942\pi\)
−0.844425 + 0.535674i \(0.820058\pi\)
\(504\) 0 0
\(505\) −864.000 −1.71089
\(506\) −101.823 + 176.363i −0.201232 + 0.348544i
\(507\) 0 0
\(508\) −209.000 361.999i −0.411417 0.712596i
\(509\) −275.772 159.217i −0.541791 0.312803i 0.204013 0.978968i \(-0.434601\pi\)
−0.745805 + 0.666165i \(0.767935\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) 198.000 114.315i 0.385214 0.222403i
\(515\) −72.1249 124.924i −0.140048 0.242571i
\(516\) 0 0
\(517\) 1413.35i 2.73376i
\(518\) 0 0
\(519\) 0 0
\(520\) −12.0000 + 20.7846i −0.0230769 + 0.0399704i
\(521\) 492.146 284.141i 0.944619 0.545376i 0.0532135 0.998583i \(-0.483054\pi\)
0.891405 + 0.453207i \(0.149720\pi\)
\(522\) 0 0
\(523\) −457.500 264.138i −0.874761 0.505043i −0.00583355 0.999983i \(-0.501857\pi\)
−0.868927 + 0.494939i \(0.835190\pi\)
\(524\) 117.576i 0.224381i
\(525\) 0 0
\(526\) 360.000 0.684411
\(527\) 29.6985 51.4393i 0.0563539 0.0976078i
\(528\) 0 0
\(529\) 228.500 + 395.774i 0.431947 + 0.748154i
\(530\) 458.205 + 264.545i 0.864538 + 0.499141i
\(531\) 0 0
\(532\) 0 0
\(533\) 118.794 0.222878
\(534\) 0 0
\(535\) −612.000 + 353.338i −1.14393 + 0.660446i
\(536\) 43.8406 + 75.9342i 0.0817922 + 0.141668i
\(537\) 0 0
\(538\) 27.7128i 0.0515108i
\(539\) 0 0
\(540\) 0 0
\(541\) −167.500 + 290.119i −0.309612 + 0.536263i −0.978277 0.207300i \(-0.933532\pi\)
0.668666 + 0.743563i \(0.266866\pi\)
\(542\) 50.9117 29.3939i 0.0939330 0.0542322i
\(543\) 0 0
\(544\) −24.0000 13.8564i −0.0441176 0.0254713i
\(545\) 827.928i 1.51913i
\(546\) 0 0
\(547\) −658.000 −1.20293 −0.601463 0.798901i \(-0.705415\pi\)
−0.601463 + 0.798901i \(0.705415\pi\)
\(548\) −152.735 + 264.545i −0.278714 + 0.482746i
\(549\) 0 0
\(550\) 12.0000 + 20.7846i 0.0218182 + 0.0377902i
\(551\) 865.499 + 499.696i 1.57078 + 0.906889i
\(552\) 0 0
\(553\) 0 0
\(554\) −476.590 −0.860271
\(555\) 0 0
\(556\) −339.000 + 195.722i −0.609712 + 0.352018i
\(557\) 135.765 + 235.151i 0.243742 + 0.422174i 0.961777 0.273833i \(-0.0882915\pi\)
−0.718035 + 0.696007i \(0.754958\pi\)
\(558\) 0 0
\(559\) 53.6936i 0.0960529i
\(560\) 0 0
\(561\) 0 0
\(562\) −174.000 + 301.377i −0.309609 + 0.536258i
\(563\) 12.7279 7.34847i 0.0226073 0.0130523i −0.488654 0.872478i \(-0.662512\pi\)
0.511261 + 0.859425i \(0.329179\pi\)
\(564\) 0 0
\(565\) 252.000 + 145.492i 0.446018 + 0.257508i
\(566\) 276.792i 0.489032i
\(567\) 0 0
\(568\) 168.000 0.295775
\(569\) 424.264 734.847i 0.745631 1.29147i −0.204268 0.978915i \(-0.565481\pi\)
0.949899 0.312556i \(-0.101185\pi\)
\(570\) 0 0
\(571\) −224.500 388.845i −0.393170 0.680990i 0.599696 0.800228i \(-0.295288\pi\)
−0.992866 + 0.119238i \(0.961955\pi\)
\(572\) −50.9117 29.3939i −0.0890064 0.0513879i
\(573\) 0 0
\(574\) 0 0
\(575\) −8.48528 −0.0147570
\(576\) 0 0
\(577\) 253.500 146.358i 0.439341 0.253654i −0.263977 0.964529i \(-0.585034\pi\)
0.703318 + 0.710875i \(0.251701\pi\)
\(578\) −187.383 324.557i −0.324193 0.561518i
\(579\) 0 0
\(580\) 332.554i 0.573369i
\(581\) 0 0
\(582\) 0 0
\(583\) −648.000 + 1122.37i −1.11149 + 1.92516i
\(584\) 199.404 115.126i 0.341445 0.197134i
\(585\) 0 0
\(586\) −120.000 69.2820i −0.204778 0.118229i
\(587\) 529.090i 0.901345i −0.892689 0.450673i \(-0.851184\pi\)
0.892689 0.450673i \(-0.148816\pi\)
\(588\) 0 0
\(589\) −357.000 −0.606112
\(590\) 288.500 499.696i 0.488982 0.846942i
\(591\) 0 0
\(592\) 94.0000 + 162.813i 0.158784 + 0.275022i
\(593\) 907.925 + 524.191i 1.53107 + 0.883964i 0.999313 + 0.0370681i \(0.0118018\pi\)
0.531758 + 0.846896i \(0.321531\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 101.823 0.170845
\(597\) 0 0
\(598\) 18.0000 10.3923i 0.0301003 0.0173784i
\(599\) 322.441 + 558.484i 0.538298 + 0.932360i 0.998996 + 0.0448028i \(0.0142660\pi\)
−0.460698 + 0.887557i \(0.652401\pi\)
\(600\) 0 0
\(601\) 458.993i 0.763716i 0.924221 + 0.381858i \(0.124716\pi\)
−0.924221 + 0.381858i \(0.875284\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.0000 + 17.3205i −0.0165563 + 0.0286763i
\(605\) 708.521 409.065i 1.17111 0.676140i
\(606\) 0 0
\(607\) −910.500 525.677i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(608\) 166.565i 0.273956i
\(609\) 0 0
\(610\) 576.000 0.944262
\(611\) 72.1249 124.924i 0.118044 0.204458i
\(612\) 0 0
\(613\) −145.000 251.147i −0.236542 0.409702i 0.723178 0.690662i \(-0.242681\pi\)
−0.959720 + 0.280960i \(0.909347\pi\)
\(614\) 86.9741 + 50.2145i 0.141652 + 0.0817826i
\(615\) 0 0
\(616\) 0 0
\(617\) −729.734 −1.18271 −0.591357 0.806410i \(-0.701407\pi\)
−0.591357 + 0.806410i \(0.701407\pi\)
\(618\) 0 0
\(619\) 709.500 409.630i 1.14620 0.661761i 0.198244 0.980153i \(-0.436476\pi\)
0.947959 + 0.318392i \(0.103143\pi\)
\(620\) 59.3970 + 102.879i 0.0958016 + 0.165933i
\(621\) 0 0
\(622\) 304.841i 0.490098i
\(623\) 0 0
\(624\) 0 0
\(625\) 299.500 518.749i 0.479200 0.829999i
\(626\) 358.503 206.982i 0.572689 0.330642i
\(627\) 0 0
\(628\) 72.0000 + 41.5692i 0.114650 + 0.0661930i
\(629\) 230.252i 0.366060i
\(630\) 0 0
\(631\) −58.0000 −0.0919176 −0.0459588 0.998943i \(-0.514634\pi\)
−0.0459588 + 0.998943i \(0.514634\pi\)
\(632\) −57.9828 + 100.429i −0.0917449 + 0.158907i
\(633\) 0 0
\(634\) 168.000 + 290.985i 0.264984 + 0.458966i
\(635\) −886.712 511.943i −1.39640 0.806210i
\(636\) 0 0
\(637\) 0 0
\(638\) 814.587 1.27678
\(639\) 0 0
\(640\) 48.0000 27.7128i 0.0750000 0.0433013i
\(641\) 479.418 + 830.377i 0.747923 + 1.29544i 0.948817 + 0.315827i \(0.102282\pi\)
−0.200894 + 0.979613i \(0.564385\pi\)
\(642\) 0 0
\(643\) 760.370i 1.18254i −0.806475 0.591268i \(-0.798628\pi\)
0.806475 0.591268i \(-0.201372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 102.000 176.669i 0.157895 0.273482i
\(647\) −305.470 + 176.363i −0.472133 + 0.272586i −0.717132 0.696937i \(-0.754546\pi\)
0.244999 + 0.969523i \(0.421212\pi\)
\(648\) 0 0
\(649\) 1224.00 + 706.677i 1.88598 + 1.08887i
\(650\) 2.44949i 0.00376845i
\(651\) 0 0
\(652\) 172.000 0.263804
\(653\) −220.617 + 382.120i −0.337852 + 0.585177i −0.984028 0.178011i \(-0.943034\pi\)
0.646177 + 0.763188i \(0.276367\pi\)
\(654\) 0 0
\(655\) −144.000 249.415i −0.219847 0.380787i
\(656\) −237.588 137.171i −0.362177 0.209103i
\(657\) 0 0
\(658\) 0 0
\(659\) −161.220 −0.244644 −0.122322 0.992490i \(-0.539034\pi\)
−0.122322 + 0.992490i \(0.539034\pi\)
\(660\) 0 0
\(661\) 721.500 416.558i 1.09153 0.630194i 0.157545 0.987512i \(-0.449642\pi\)
0.933983 + 0.357318i \(0.116309\pi\)
\(662\) 130.815 + 226.578i 0.197605 + 0.342263i
\(663\) 0 0
\(664\) 13.8564i 0.0208681i
\(665\) 0 0
\(666\) 0 0
\(667\) −144.000 + 249.415i −0.215892 + 0.373936i
\(668\) 313.955 181.262i 0.469993 0.271351i
\(669\) 0 0
\(670\) 186.000 + 107.387i 0.277612 + 0.160279i
\(671\) 1410.91i 2.10269i
\(672\) 0 0
\(673\) −263.000 −0.390788 −0.195394 0.980725i \(-0.562598\pi\)
−0.195394 + 0.980725i \(0.562598\pi\)
\(674\) 253.851 439.683i 0.376634 0.652349i
\(675\) 0 0
\(676\) −166.000 287.520i −0.245562 0.425326i
\(677\) 432.749 + 249.848i 0.639216 + 0.369052i 0.784313 0.620366i \(-0.213016\pi\)
−0.145096 + 0.989418i \(0.546349\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −67.8823 −0.0998268
\(681\) 0 0
\(682\) −252.000 + 145.492i −0.369501 + 0.213332i
\(683\) 479.418 + 830.377i 0.701930 + 1.21578i 0.967788 + 0.251767i \(0.0810115\pi\)
−0.265858 + 0.964012i \(0.585655\pi\)
\(684\) 0 0
\(685\) 748.246i 1.09233i
\(686\) 0 0
\(687\) 0 0
\(688\) −62.0000 + 107.387i −0.0901163 + 0.156086i
\(689\) 114.551 66.1362i 0.166257 0.0959887i
\(690\) 0 0
\(691\) −1069.50 617.476i −1.54776 0.893598i −0.998313 0.0580674i \(-0.981506\pi\)
−0.549444 0.835530i \(-0.685161\pi\)
\(692\) 88.1816i 0.127430i
\(693\) 0 0
\(694\) 660.000 0.951009
\(695\) −479.418 + 830.377i −0.689811 + 1.19479i
\(696\) 0 0
\(697\) 168.000 + 290.985i 0.241033 + 0.417481i
\(698\) −712.764 411.514i −1.02115 0.589562i
\(699\) 0 0
\(700\) 0 0
\(701\) 975.807 1.39202 0.696011 0.718031i \(-0.254956\pi\)
0.696011 + 0.718031i \(0.254956\pi\)
\(702\) 0 0
\(703\) −1198.50 + 691.954i −1.70484 + 0.984288i
\(704\) 67.8823 + 117.576i 0.0964237 + 0.167011i
\(705\) 0 0
\(706\) 408.764i 0.578986i
\(707\) 0 0
\(708\) 0 0
\(709\) 553.000 957.824i 0.779972 1.35095i −0.151986 0.988383i \(-0.548567\pi\)
0.931957 0.362568i \(-0.118100\pi\)
\(710\) 356.382 205.757i 0.501946 0.289799i
\(711\) 0 0
\(712\) −144.000 83.1384i −0.202247 0.116767i
\(713\) 102.879i 0.144290i
\(714\) 0 0
\(715\) −144.000 −0.201399
\(716\) 8.48528 14.6969i 0.0118510 0.0205265i
\(717\) 0 0
\(718\) −240.000 415.692i −0.334262 0.578958i
\(719\) −593.970 342.929i −0.826105 0.476952i 0.0264120 0.999651i \(-0.491592\pi\)
−0.852517 + 0.522699i \(0.824925\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −715.592 −0.991125
\(723\) 0 0
\(724\) 75.0000 43.3013i 0.103591 0.0598084i
\(725\) 16.9706 + 29.3939i 0.0234077 + 0.0405433i
\(726\) 0 0
\(727\) 427.817i 0.588468i −0.955733 0.294234i \(-0.904935\pi\)
0.955733 0.294234i \(-0.0950646\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 282.000 488.438i 0.386301 0.669094i
\(731\) 131.522 75.9342i 0.179920 0.103877i
\(732\) 0 0
\(733\) −34.5000 19.9186i −0.0470668 0.0271741i 0.476282 0.879293i \(-0.341984\pi\)
−0.523349 + 0.852119i \(0.675318\pi\)
\(734\) 218.005i 0.297009i
\(735\) 0 0
\(736\) −48.0000 −0.0652174
\(737\) −263.044 + 455.605i −0.356911 + 0.618189i
\(738\) 0 0
\(739\) 243.500 + 421.754i 0.329499 + 0.570710i 0.982413 0.186723i \(-0.0597867\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(740\) 398.808 + 230.252i 0.538930 + 0.311151i
\(741\) 0 0
\(742\) 0 0
\(743\) −509.117 −0.685218 −0.342609 0.939478i \(-0.611311\pi\)
−0.342609 + 0.939478i \(0.611311\pi\)
\(744\) 0 0
\(745\) 216.000 124.708i 0.289933 0.167393i
\(746\) 204.354 + 353.951i 0.273933 + 0.474466i
\(747\) 0 0
\(748\) 166.277i 0.222295i
\(749\) 0 0
\(750\) 0 0
\(751\) 272.500 471.984i 0.362850 0.628474i −0.625579 0.780161i \(-0.715137\pi\)
0.988429 + 0.151687i \(0.0484706\pi\)
\(752\) −288.500 + 166.565i −0.383643 + 0.221496i
\(753\) 0 0
\(754\) −72.0000 41.5692i −0.0954907 0.0551316i
\(755\) 48.9898i 0.0648871i
\(756\) 0 0
\(757\) −770.000 −1.01717 −0.508587 0.861011i \(-0.669832\pi\)
−0.508587 + 0.861011i \(0.669832\pi\)
\(758\) −4.94975 + 8.57321i −0.00653001 + 0.0113103i
\(759\) 0 0
\(760\) 204.000 + 353.338i 0.268421 + 0.464919i
\(761\) −148.492 85.7321i −0.195128 0.112657i 0.399253 0.916841i \(-0.369270\pi\)
−0.594381 + 0.804184i \(0.702603\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 152.735 0.199915
\(765\) 0 0
\(766\) 606.000 349.874i 0.791123 0.456755i
\(767\) −72.1249 124.924i −0.0940351 0.162874i
\(768\) 0 0
\(769\) 704.945i 0.916703i 0.888771 + 0.458352i \(0.151560\pi\)
−0.888771 + 0.458352i \(0.848440\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 287.000 497.099i 0.371762 0.643910i
\(773\) −797.616 + 460.504i −1.03185 + 0.595736i −0.917513 0.397706i \(-0.869806\pi\)
−0.114333 + 0.993443i \(0.536473\pi\)
\(774\) 0 0
\(775\) −10.5000 6.06218i −0.0135484 0.00782216i
\(776\) 117.576i 0.151515i
\(777\) 0 0
\(778\) −324.000 −0.416452
\(779\) 1009.75 1748.94i 1.29621 2.24510i
\(780\) 0 0
\(781\) 504.000 + 872.954i 0.645327 + 1.11774i
\(782\) 50.9117 + 29.3939i 0.0651045 + 0.0375881i
\(783\) 0 0
\(784\) 0 0
\(785\) 203.647 0.259423
\(786\) 0 0
\(787\) −396.000 + 228.631i −0.503177 + 0.290509i −0.730024 0.683421i \(-0.760491\pi\)
0.226848 + 0.973930i \(0.427158\pi\)
\(788\) −127.279 220.454i −0.161522 0.279764i
\(789\) 0 0
\(790\) 284.056i 0.359565i
\(791\) 0 0
\(792\) 0 0
\(793\) 72.0000 124.708i 0.0907945 0.157261i
\(794\) 99.7021 57.5630i 0.125569 0.0724975i
\(795\) 0 0
\(796\) 360.000 + 207.846i 0.452261 + 0.261113i
\(797\) 14.6969i 0.0184403i −0.999957 0.00922016i \(-0.997065\pi\)
0.999957 0.00922016i \(-0.00293491\pi\)
\(798\) 0 0
\(799\) 408.000 0.510638
\(800\) −2.82843 + 4.89898i −0.00353553 + 0.00612372i
\(801\) 0 0
\(802\) 66.0000 + 114.315i 0.0822943 + 0.142538i
\(803\) 1196.42 + 690.756i 1.48994 + 0.860219i
\(804\) 0 0
\(805\) 0 0
\(806\) 29.6985 0.0368468
\(807\) 0 0
\(808\) 432.000 249.415i 0.534653 0.308682i
\(809\) −470.933 815.680i −0.582118 1.00826i −0.995228 0.0975763i \(-0.968891\pi\)
0.413110 0.910681i \(-0.364442\pi\)
\(810\) 0 0
\(811\) 498.831i 0.615081i −0.951535 0.307540i \(-0.900494\pi\)
0.951535 0.307540i \(-0.0995059\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −564.000 + 976.877i −0.692875 + 1.20009i
\(815\) 364.867 210.656i 0.447690 0.258474i
\(816\) 0 0
\(817\) −790.500 456.395i −0.967564 0.558623i
\(818\) 590.327i 0.721671i
\(819\) 0 0
\(820\) −672.000 −0.819512
\(821\) −301.227 + 521.741i −0.366903 + 0.635495i −0.989080 0.147382i \(-0.952915\pi\)
0.622176 + 0.782877i \(0.286249\pi\)
\(822\) 0 0
\(823\) −19.0000 32.9090i −0.0230863 0.0399866i 0.854251 0.519860i \(-0.174016\pi\)
−0.877338 + 0.479873i \(0.840683\pi\)
\(824\) 72.1249 + 41.6413i 0.0875302 + 0.0505356i
\(825\) 0 0
\(826\) 0 0
\(827\) −687.308 −0.831086 −0.415543 0.909574i \(-0.636408\pi\)
−0.415543 + 0.909574i \(0.636408\pi\)
\(828\) 0 0
\(829\) 721.500 416.558i 0.870326 0.502483i 0.00286924 0.999996i \(-0.499087\pi\)
0.867456 + 0.497513i \(0.165753\pi\)
\(830\) −16.9706 29.3939i −0.0204465 0.0354143i
\(831\) 0 0
\(832\) 13.8564i 0.0166543i
\(833\) 0 0
\(834\) 0 0
\(835\) 444.000 769.031i 0.531737 0.920995i
\(836\) −865.499 + 499.696i −1.03529 + 0.597722i
\(837\) 0 0
\(838\) −24.0000 13.8564i −0.0286396 0.0165351i
\(839\) 244.949i 0.291953i −0.989288 0.145977i \(-0.953368\pi\)
0.989288 0.145977i \(-0.0466325\pi\)
\(840\) 0 0
\(841\) 311.000 0.369798
\(842\) −287.792 + 498.471i −0.341796 + 0.592009i
\(843\) 0 0
\(844\) −82.0000 142.028i −0.0971564 0.168280i
\(845\) −704.278 406.615i −0.833466 0.481202i
\(846\) 0 0
\(847\) 0 0
\(848\) −305.470 −0.360224
\(849\) 0 0
\(850\) 6.00000 3.46410i 0.00705882 0.00407541i
\(851\) −199.404 345.378i −0.234317 0.405850i
\(852\) 0 0
\(853\) 1245.34i 1.45996i −0.683469 0.729979i \(-0.739530\pi\)
0.683469 0.729979i \(-0.260470\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 204.000 353.338i 0.238318 0.412778i
\(857\) 1060.66 612.372i 1.23764 0.714554i 0.269031 0.963131i \(-0.413297\pi\)
0.968612 + 0.248578i \(0.0799632\pi\)
\(858\) 0 0
\(859\) −216.000 124.708i −0.251455 0.145178i 0.368975 0.929439i \(-0.379709\pi\)
−0.620430 + 0.784262i \(0.713042\pi\)
\(860\) 303.737i 0.353182i
\(861\) 0 0
\(862\) −228.000 −0.264501
\(863\) 330.926 573.181i 0.383460 0.664172i −0.608094 0.793865i \(-0.708066\pi\)
0.991554 + 0.129693i \(0.0413991\pi\)
\(864\) 0 0
\(865\) −108.000 187.061i −0.124855 0.216256i
\(866\) 205.768 + 118.800i 0.237607 + 0.137183i
\(867\) 0 0
\(868\) 0 0
\(869\) −695.793 −0.800682
\(870\) 0 0
\(871\) 46.5000 26.8468i 0.0533869 0.0308229i
\(872\) −239.002 413.964i −0.274085 0.474729i
\(873\) 0 0
\(874\) 353.338i 0.404277i
\(875\) 0 0
\(876\) 0 0
\(877\) 287.000 497.099i 0.327252 0.566817i −0.654714 0.755877i \(-0.727211\pi\)
0.981966 + 0.189060i \(0.0605441\pi\)
\(878\) −661.852 + 382.120i −0.753818 + 0.435217i
\(879\) 0 0
\(880\) 288.000 + 166.277i 0.327273 + 0.188951i
\(881\) 161.666i 0.183503i −0.995782 0.0917516i \(-0.970753\pi\)
0.995782 0.0917516i \(-0.0292466\pi\)
\(882\) 0 0
\(883\) 1735.00 1.96489 0.982446 0.186546i \(-0.0597294\pi\)
0.982446 + 0.186546i \(0.0597294\pi\)
\(884\) −8.48528 + 14.6969i −0.00959873 + 0.0166255i
\(885\) 0 0
\(886\) −90.0000 155.885i −0.101580 0.175942i
\(887\) −169.706 97.9796i −0.191325 0.110462i 0.401277 0.915957i \(-0.368566\pi\)
−0.592603 + 0.805495i \(0.701900\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −407.294 −0.457633
\(891\) 0 0
\(892\) −72.0000 + 41.5692i −0.0807175 + 0.0466023i
\(893\) −1226.12 2123.71i −1.37304 2.37817i
\(894\) 0 0
\(895\) 41.5692i 0.0464461i
\(896\) 0 0
\(897\) 0 0
\(898\) −78.0000 + 135.100i −0.0868597 + 0.150445i
\(899\) −356.382 + 205.757i −0.396420 + 0.228873i
\(900\) 0 0
\(901\) 324.000 + 187.061i 0.359600 + 0.207615i
\(902\) 1646.06i 1.82490i
\(903\) 0 0
\(904\) −168.000 −0.185841
\(905\) 106.066 183.712i 0.117200 0.202996i
\(906\) 0 0
\(907\) −375.500 650.385i −0.414002 0.717073i 0.581321 0.813674i \(-0.302536\pi\)
−0.995323 + 0.0966015i \(0.969203\pi\)
\(908\) 661.852 + 382.120i 0.728912 + 0.420837i
\(909\) 0 0
\(910\) 0 0
\(911\) 1247.34 1.36919 0.684597 0.728921i \(-0.259978\pi\)
0.684597 + 0.728921i \(0.259978\pi\)
\(912\) 0 0
\(913\) 72.0000 41.5692i 0.0788609 0.0455304i
\(914\) −17.6777 30.6186i −0.0193410 0.0334996i
\(915\) 0 0
\(916\) 162.813i 0.177743i
\(917\) 0 0
\(918\) 0 0
\(919\) −507.500 + 879.016i −0.552231 + 0.956492i 0.445883 + 0.895091i \(0.352890\pi\)
−0.998113 + 0.0614001i \(0.980443\pi\)
\(920\) −101.823 + 58.7878i −0.110678 + 0.0638997i
\(921\) 0 0
\(922\) 96.0000 + 55.4256i 0.104121 + 0.0601146i
\(923\) 102.879i 0.111461i
\(924\) 0 0
\(925\) −47.0000 −0.0508108
\(926\) −368.403 + 638.092i −0.397843 + 0.689084i
\(927\) 0 0
\(928\) 96.0000 + 166.277i 0.103448 + 0.179178i
\(929\) −946.109 546.236i −1.01842 0.587983i −0.104772 0.994496i \(-0.533411\pi\)
−0.913645 + 0.406513i \(0.866744\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 458.205 0.491636
\(933\) 0 0
\(934\) −270.000 + 155.885i −0.289079 + 0.166900i
\(935\) −203.647 352.727i −0.217804 0.377248i
\(936\) 0 0
\(937\) 1747.64i 1.86514i 0.360985 + 0.932572i \(0.382441\pi\)
−0.360985 + 0.932572i \(0.617559\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −408.000 + 706.677i −0.434043 + 0.751784i
\(941\) −1073.39 + 619.721i −1.14069 + 0.658577i −0.946601 0.322407i \(-0.895508\pi\)
−0.194088 + 0.980984i \(0.562175\pi\)
\(942\) 0 0
\(943\) 504.000 + 290.985i 0.534464 + 0.308573i
\(944\) 333.131i 0.352893i
\(945\) 0 0
\(946\) −744.000 −0.786469
\(947\) −602.455 + 1043.48i −0.636172 + 1.10188i 0.350093 + 0.936715i \(0.386150\pi\)
−0.986266 + 0.165168i \(0.947183\pi\)
\(948\) 0 0
\(949\) −70.5000 122.110i −0.0742887 0.128672i
\(950\) −36.0624 20.8207i −0.0379605 0.0219165i
\(951\) 0 0
\(952\) 0 0
\(953\) 1026.72 1.07735 0.538677 0.842512i \(-0.318924\pi\)
0.538677 + 0.842512i \(0.318924\pi\)
\(954\) 0 0
\(955\) 324.000 187.061i 0.339267 0.195876i
\(956\) 67.8823 + 117.576i 0.0710065 + 0.122987i
\(957\) 0 0
\(958\) 1240.15i 1.29452i
\(959\) 0 0
\(960\) 0 0
\(961\) −407.000 + 704.945i −0.423517 + 0.733553i
\(962\) 99.7021 57.5630i 0.103640 0.0598368i
\(963\) 0 0
\(964\) −792.000 457.261i −0.821577 0.474338i
\(965\) 1406.01i 1.45700i
\(966\) 0 0
\(967\) −895.000 −0.925543 −0.462771 0.886478i \(-0.653145\pi\)
−0.462771 + 0.886478i \(0.653145\pi\)
\(968\) −236.174 + 409.065i −0.243981 + 0.422588i
\(969\) 0 0
\(970\) 144.000 + 249.415i 0.148454 + 0.257129i
\(971\) 182.434 + 105.328i 0.187882 + 0.108474i 0.590991 0.806678i \(-0.298737\pi\)
−0.403109 + 0.915152i \(0.632070\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 179.605 0.184400
\(975\) 0 0
\(976\) −288.000 + 166.277i −0.295082 + 0.170366i
\(977\) 627.911 + 1087.57i 0.642693 + 1.11318i 0.984829 + 0.173526i \(0.0555161\pi\)
−0.342136 + 0.939650i \(0.611151\pi\)
\(978\) 0 0
\(979\) 997.661i 1.01906i
\(980\) 0 0
\(981\) 0 0
\(982\) 444.000 769.031i 0.452138 0.783127i
\(983\) −1022.48 + 590.327i −1.04016 + 0.600536i −0.919878 0.392204i \(-0.871713\pi\)
−0.120281 + 0.992740i \(0.538379\pi\)
\(984\) 0 0
\(985\) −540.000 311.769i −0.548223 0.316517i
\(986\) 235.151i 0.238490i
\(987\) 0 0
\(988\) 102.000 0.103239
\(989\) 131.522 227.803i 0.132985 0.230336i
\(990\) 0 0
\(991\) −327.500 567.247i −0.330474 0.572398i 0.652131 0.758107i \(-0.273875\pi\)
−0.982605 + 0.185708i \(0.940542\pi\)
\(992\) −59.3970 34.2929i −0.0598760 0.0345694i
\(993\) 0 0
\(994\) 0 0
\(995\) 1018.23 1.02335
\(996\) 0 0
\(997\) −397.500 + 229.497i −0.398696 + 0.230187i −0.685921 0.727676i \(-0.740601\pi\)
0.287225 + 0.957863i \(0.407267\pi\)
\(998\) 164.756 + 285.366i 0.165086 + 0.285937i
\(999\) 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 882.3.n.c.19.1 4
3.2 odd 2 inner 882.3.n.c.19.2 4
7.2 even 3 882.3.c.c.685.4 4
7.3 odd 6 inner 882.3.n.c.325.1 4
7.4 even 3 126.3.n.b.73.1 yes 4
7.5 odd 6 882.3.c.c.685.3 4
7.6 odd 2 126.3.n.b.19.1 4
21.2 odd 6 882.3.c.c.685.1 4
21.5 even 6 882.3.c.c.685.2 4
21.11 odd 6 126.3.n.b.73.2 yes 4
21.17 even 6 inner 882.3.n.c.325.2 4
21.20 even 2 126.3.n.b.19.2 yes 4
28.11 odd 6 1008.3.cg.i.577.2 4
28.27 even 2 1008.3.cg.i.145.2 4
84.11 even 6 1008.3.cg.i.577.1 4
84.83 odd 2 1008.3.cg.i.145.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.n.b.19.1 4 7.6 odd 2
126.3.n.b.19.2 yes 4 21.20 even 2
126.3.n.b.73.1 yes 4 7.4 even 3
126.3.n.b.73.2 yes 4 21.11 odd 6
882.3.c.c.685.1 4 21.2 odd 6
882.3.c.c.685.2 4 21.5 even 6
882.3.c.c.685.3 4 7.5 odd 6
882.3.c.c.685.4 4 7.2 even 3
882.3.n.c.19.1 4 1.1 even 1 trivial
882.3.n.c.19.2 4 3.2 odd 2 inner
882.3.n.c.325.1 4 7.3 odd 6 inner
882.3.n.c.325.2 4 21.17 even 6 inner
1008.3.cg.i.145.1 4 84.83 odd 2
1008.3.cg.i.145.2 4 28.27 even 2
1008.3.cg.i.577.1 4 84.11 even 6
1008.3.cg.i.577.2 4 28.11 odd 6