Properties

Label 525.2.bc.a.82.2
Level $525$
Weight $2$
Character 525.82
Analytic conductor $4.192$
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] = [525,2,Mod(82,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.bc (of order \(12\), degree \(4\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 82.2
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 525.82
Dual form 525.2.bc.a.493.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-0.189469 + 2.63896i) q^{7} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\) \(q+(0.258819 - 0.965926i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-0.189469 + 2.63896i) q^{7} +(-0.866025 - 0.500000i) q^{9} +(-3.00000 - 5.19615i) q^{11} +(0.517638 + 1.93185i) q^{12} +(-1.41421 - 1.41421i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-5.79555 - 1.55291i) q^{17} +(1.73205 - 3.00000i) q^{19} +(2.50000 + 0.866025i) q^{21} +(-0.707107 + 0.707107i) q^{27} +(-2.31079 - 4.76028i) q^{28} +(-4.50000 + 2.59808i) q^{31} +(-5.79555 + 1.55291i) q^{33} +2.00000 q^{36} +(-8.36516 + 2.24144i) q^{37} +(-1.73205 + 1.00000i) q^{39} -10.3923i q^{41} +(-3.67423 + 3.67423i) q^{43} +(10.3923 + 6.00000i) q^{44} +(1.55291 + 5.79555i) q^{47} +(-2.82843 - 2.82843i) q^{48} +(-6.92820 - 1.00000i) q^{49} +(-3.00000 + 5.19615i) q^{51} +(3.86370 + 1.03528i) q^{52} +(10.0382 + 2.68973i) q^{53} +(-2.44949 - 2.44949i) q^{57} +(-5.19615 - 9.00000i) q^{59} +(7.50000 + 4.33013i) q^{61} +(1.48356 - 2.19067i) q^{63} +8.00000i q^{64} +(0.896575 - 3.34607i) q^{67} +(11.5911 - 3.10583i) q^{68} -6.00000 q^{71} +(-0.258819 + 0.965926i) q^{73} +6.92820i q^{76} +(14.2808 - 6.93237i) q^{77} +(4.33013 + 2.50000i) q^{79} +(0.500000 + 0.866025i) q^{81} +(4.24264 + 4.24264i) q^{83} +(-5.19615 + 1.00000i) q^{84} +(5.19615 - 9.00000i) q^{89} +(4.00000 - 3.46410i) q^{91} +(1.34486 + 5.01910i) q^{93} +(9.19239 - 9.19239i) q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{11} + 16 q^{16} + 20 q^{21} - 36 q^{31} + 16 q^{36} - 24 q^{51} + 60 q^{61} - 48 q^{71} + 4 q^{81} + 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(3\) 0.258819 0.965926i 0.149429 0.557678i
\(4\) −1.73205 + 1.00000i −0.866025 + 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.189469 + 2.63896i −0.0716124 + 0.997433i
\(8\) 0 0
\(9\) −0.866025 0.500000i −0.288675 0.166667i
\(10\) 0 0
\(11\) −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i \(-0.806886\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0.517638 + 1.93185i 0.149429 + 0.557678i
\(13\) −1.41421 1.41421i −0.392232 0.392232i 0.483250 0.875482i \(-0.339456\pi\)
−0.875482 + 0.483250i \(0.839456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 3.46410i 0.500000 0.866025i
\(17\) −5.79555 1.55291i −1.40563 0.376637i −0.525266 0.850938i \(-0.676034\pi\)
−0.880363 + 0.474301i \(0.842701\pi\)
\(18\) 0 0
\(19\) 1.73205 3.00000i 0.397360 0.688247i −0.596040 0.802955i \(-0.703260\pi\)
0.993399 + 0.114708i \(0.0365932\pi\)
\(20\) 0 0
\(21\) 2.50000 + 0.866025i 0.545545 + 0.188982i
\(22\) 0 0
\(23\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) −2.31079 4.76028i −0.436698 0.899608i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.50000 + 2.59808i −0.808224 + 0.466628i −0.846339 0.532645i \(-0.821198\pi\)
0.0381148 + 0.999273i \(0.487865\pi\)
\(32\) 0 0
\(33\) −5.79555 + 1.55291i −1.00888 + 0.270328i
\(34\) 0 0
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −8.36516 + 2.24144i −1.37522 + 0.368490i −0.869384 0.494137i \(-0.835484\pi\)
−0.505840 + 0.862627i \(0.668817\pi\)
\(38\) 0 0
\(39\) −1.73205 + 1.00000i −0.277350 + 0.160128i
\(40\) 0 0
\(41\) 10.3923i 1.62301i −0.584349 0.811503i \(-0.698650\pi\)
0.584349 0.811503i \(-0.301350\pi\)
\(42\) 0 0
\(43\) −3.67423 + 3.67423i −0.560316 + 0.560316i −0.929397 0.369082i \(-0.879672\pi\)
0.369082 + 0.929397i \(0.379672\pi\)
\(44\) 10.3923 + 6.00000i 1.56670 + 0.904534i
\(45\) 0 0
\(46\) 0 0
\(47\) 1.55291 + 5.79555i 0.226516 + 0.845369i 0.981792 + 0.189961i \(0.0608363\pi\)
−0.755276 + 0.655407i \(0.772497\pi\)
\(48\) −2.82843 2.82843i −0.408248 0.408248i
\(49\) −6.92820 1.00000i −0.989743 0.142857i
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 3.86370 + 1.03528i 0.535799 + 0.143567i
\(53\) 10.0382 + 2.68973i 1.37885 + 0.369462i 0.870704 0.491807i \(-0.163664\pi\)
0.508148 + 0.861270i \(0.330330\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.44949 2.44949i −0.324443 0.324443i
\(58\) 0 0
\(59\) −5.19615 9.00000i −0.676481 1.17170i −0.976034 0.217620i \(-0.930171\pi\)
0.299552 0.954080i \(-0.403163\pi\)
\(60\) 0 0
\(61\) 7.50000 + 4.33013i 0.960277 + 0.554416i 0.896258 0.443533i \(-0.146275\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 1.48356 2.19067i 0.186911 0.275999i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.896575 3.34607i 0.109534 0.408787i −0.889286 0.457352i \(-0.848798\pi\)
0.998820 + 0.0485648i \(0.0154647\pi\)
\(68\) 11.5911 3.10583i 1.40563 0.376637i
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −0.258819 + 0.965926i −0.0302925 + 0.113053i −0.979417 0.201849i \(-0.935305\pi\)
0.949124 + 0.314902i \(0.101972\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 6.92820i 0.794719i
\(77\) 14.2808 6.93237i 1.62745 0.790017i
\(78\) 0 0
\(79\) 4.33013 + 2.50000i 0.487177 + 0.281272i 0.723403 0.690426i \(-0.242577\pi\)
−0.236225 + 0.971698i \(0.575910\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 4.24264 + 4.24264i 0.465690 + 0.465690i 0.900515 0.434825i \(-0.143190\pi\)
−0.434825 + 0.900515i \(0.643190\pi\)
\(84\) −5.19615 + 1.00000i −0.566947 + 0.109109i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 9.00000i 0.550791 0.953998i −0.447427 0.894321i \(-0.647659\pi\)
0.998218 0.0596775i \(-0.0190072\pi\)
\(90\) 0 0
\(91\) 4.00000 3.46410i 0.419314 0.363137i
\(92\) 0 0
\(93\) 1.34486 + 5.01910i 0.139456 + 0.520456i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.19239 9.19239i 0.933346 0.933346i −0.0645677 0.997913i \(-0.520567\pi\)
0.997913 + 0.0645677i \(0.0205669\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
\(100\) 0 0
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) −6.76148 + 1.81173i −0.666228 + 0.178515i −0.576055 0.817411i \(-0.695409\pi\)
−0.0901732 + 0.995926i \(0.528742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(108\) 0.517638 1.93185i 0.0498097 0.185893i
\(109\) −8.66025 + 5.00000i −0.829502 + 0.478913i −0.853682 0.520794i \(-0.825636\pi\)
0.0241802 + 0.999708i \(0.492302\pi\)
\(110\) 0 0
\(111\) 8.66025i 0.821995i
\(112\) 8.76268 + 5.93426i 0.827996 + 0.560734i
\(113\) −7.34847 + 7.34847i −0.691286 + 0.691286i −0.962515 0.271229i \(-0.912570\pi\)
0.271229 + 0.962515i \(0.412570\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.517638 + 1.93185i 0.0478557 + 0.178600i
\(118\) 0 0
\(119\) 5.19615 15.0000i 0.476331 1.37505i
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 0 0
\(123\) −10.0382 2.68973i −0.905114 0.242524i
\(124\) 5.19615 9.00000i 0.466628 0.808224i
\(125\) 0 0
\(126\) 0 0
\(127\) −8.57321 8.57321i −0.760750 0.760750i 0.215708 0.976458i \(-0.430794\pi\)
−0.976458 + 0.215708i \(0.930794\pi\)
\(128\) 0 0
\(129\) 2.59808 + 4.50000i 0.228748 + 0.396203i
\(130\) 0 0
\(131\) 9.00000 + 5.19615i 0.786334 + 0.453990i 0.838670 0.544640i \(-0.183334\pi\)
−0.0523366 + 0.998630i \(0.516667\pi\)
\(132\) 8.48528 8.48528i 0.738549 0.738549i
\(133\) 7.58871 + 5.13922i 0.658024 + 0.445627i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.37945 20.0764i 0.459598 1.71524i −0.214610 0.976700i \(-0.568848\pi\)
0.674207 0.738542i \(-0.264485\pi\)
\(138\) 0 0
\(139\) 1.73205 0.146911 0.0734553 0.997299i \(-0.476597\pi\)
0.0734553 + 0.997299i \(0.476597\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) −3.10583 + 11.5911i −0.259722 + 0.969297i
\(144\) −3.46410 + 2.00000i −0.288675 + 0.166667i
\(145\) 0 0
\(146\) 0 0
\(147\) −2.75908 + 6.43331i −0.227565 + 0.530611i
\(148\) 12.2474 12.2474i 1.00673 1.00673i
\(149\) 10.3923 + 6.00000i 0.851371 + 0.491539i 0.861113 0.508413i \(-0.169768\pi\)
−0.00974235 + 0.999953i \(0.503101\pi\)
\(150\) 0 0
\(151\) −2.50000 4.33013i −0.203447 0.352381i 0.746190 0.665733i \(-0.231881\pi\)
−0.949637 + 0.313353i \(0.898548\pi\)
\(152\) 0 0
\(153\) 4.24264 + 4.24264i 0.342997 + 0.342997i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 3.46410i 0.160128 0.277350i
\(157\) −6.76148 1.81173i −0.539625 0.144592i −0.0212957 0.999773i \(-0.506779\pi\)
−0.518329 + 0.855181i \(0.673446\pi\)
\(158\) 0 0
\(159\) 5.19615 9.00000i 0.412082 0.713746i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.24144 + 8.36516i 0.175563 + 0.655210i 0.996455 + 0.0841267i \(0.0268101\pi\)
−0.820892 + 0.571083i \(0.806523\pi\)
\(164\) 10.3923 + 18.0000i 0.811503 + 1.40556i
\(165\) 0 0
\(166\) 0 0
\(167\) −8.48528 + 8.48528i −0.656611 + 0.656611i −0.954577 0.297966i \(-0.903692\pi\)
0.297966 + 0.954577i \(0.403692\pi\)
\(168\) 0 0
\(169\) 9.00000i 0.692308i
\(170\) 0 0
\(171\) −3.00000 + 1.73205i −0.229416 + 0.132453i
\(172\) 2.68973 10.0382i 0.205090 0.765405i
\(173\) 17.3867 4.65874i 1.32188 0.354198i 0.472200 0.881491i \(-0.343460\pi\)
0.849683 + 0.527294i \(0.176793\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −24.0000 −1.80907
\(177\) −10.0382 + 2.68973i −0.754517 + 0.202172i
\(178\) 0 0
\(179\) −20.7846 + 12.0000i −1.55351 + 0.896922i −0.555663 + 0.831408i \(0.687536\pi\)
−0.997852 + 0.0655145i \(0.979131\pi\)
\(180\) 0 0
\(181\) 5.19615i 0.386227i −0.981176 0.193113i \(-0.938141\pi\)
0.981176 0.193113i \(-0.0618586\pi\)
\(182\) 0 0
\(183\) 6.12372 6.12372i 0.452679 0.452679i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.31749 + 34.7733i 0.681362 + 2.54288i
\(188\) −8.48528 8.48528i −0.618853 0.618853i
\(189\) −1.73205 2.00000i −0.125988 0.145479i
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 7.72741 + 2.07055i 0.557678 + 0.149429i
\(193\) −15.0573 4.03459i −1.08385 0.290416i −0.327677 0.944790i \(-0.606266\pi\)
−0.756171 + 0.654374i \(0.772932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 13.0000 5.19615i 0.928571 0.371154i
\(197\) 7.34847 + 7.34847i 0.523557 + 0.523557i 0.918644 0.395087i \(-0.129286\pi\)
−0.395087 + 0.918644i \(0.629286\pi\)
\(198\) 0 0
\(199\) 9.52628 + 16.5000i 0.675300 + 1.16965i 0.976381 + 0.216055i \(0.0693192\pi\)
−0.301081 + 0.953599i \(0.597347\pi\)
\(200\) 0 0
\(201\) −3.00000 1.73205i −0.211604 0.122169i
\(202\) 0 0
\(203\) 0 0
\(204\) 12.0000i 0.840168i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −7.72741 + 2.07055i −0.535799 + 0.143567i
\(209\) −20.7846 −1.43770
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) −20.0764 + 5.37945i −1.37885 + 0.369462i
\(213\) −1.55291 + 5.79555i −0.106404 + 0.397105i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.00361 12.3676i −0.407551 0.839565i
\(218\) 0 0
\(219\) 0.866025 + 0.500000i 0.0585206 + 0.0337869i
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) 3.53553 + 3.53553i 0.236757 + 0.236757i 0.815506 0.578749i \(-0.196459\pi\)
−0.578749 + 0.815506i \(0.696459\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.79555 1.55291i −0.384664 0.103071i 0.0613041 0.998119i \(-0.480474\pi\)
−0.445969 + 0.895049i \(0.647141\pi\)
\(228\) 6.69213 + 1.79315i 0.443197 + 0.118754i
\(229\) 3.46410 6.00000i 0.228914 0.396491i −0.728572 0.684969i \(-0.759816\pi\)
0.957487 + 0.288478i \(0.0931491\pi\)
\(230\) 0 0
\(231\) −3.00000 15.5885i −0.197386 1.02565i
\(232\) 0 0
\(233\) −5.37945 20.0764i −0.352420 1.31525i −0.883701 0.468052i \(-0.844956\pi\)
0.531281 0.847196i \(-0.321711\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 18.0000 + 10.3923i 1.17170 + 0.676481i
\(237\) 3.53553 3.53553i 0.229658 0.229658i
\(238\) 0 0
\(239\) 18.0000i 1.16432i 0.813073 + 0.582162i \(0.197793\pi\)
−0.813073 + 0.582162i \(0.802207\pi\)
\(240\) 0 0
\(241\) 16.5000 9.52628i 1.06286 0.613642i 0.136637 0.990621i \(-0.456371\pi\)
0.926222 + 0.376980i \(0.123037\pi\)
\(242\) 0 0
\(243\) 0.965926 0.258819i 0.0619642 0.0166032i
\(244\) −17.3205 −1.10883
\(245\) 0 0
\(246\) 0 0
\(247\) −6.69213 + 1.79315i −0.425810 + 0.114095i
\(248\) 0 0
\(249\) 5.19615 3.00000i 0.329293 0.190117i
\(250\) 0 0
\(251\) 20.7846i 1.31191i −0.754799 0.655956i \(-0.772265\pi\)
0.754799 0.655956i \(-0.227735\pi\)
\(252\) −0.378937 + 5.27792i −0.0238708 + 0.332478i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) −1.55291 5.79555i −0.0968681 0.361517i 0.900428 0.435005i \(-0.143253\pi\)
−0.997296 + 0.0734884i \(0.976587\pi\)
\(258\) 0 0
\(259\) −4.33013 22.5000i −0.269061 1.39808i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −30.1146 8.06918i −1.85694 0.497567i −0.857100 0.515151i \(-0.827736\pi\)
−0.999845 + 0.0175838i \(0.994403\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −7.34847 7.34847i −0.449719 0.449719i
\(268\) 1.79315 + 6.69213i 0.109534 + 0.408787i
\(269\) −10.3923 18.0000i −0.633630 1.09748i −0.986804 0.161922i \(-0.948231\pi\)
0.353174 0.935558i \(-0.385102\pi\)
\(270\) 0 0
\(271\) −27.0000 15.5885i −1.64013 0.946931i −0.980785 0.195094i \(-0.937499\pi\)
−0.659349 0.751837i \(-0.729168\pi\)
\(272\) −16.9706 + 16.9706i −1.02899 + 1.02899i
\(273\) −2.31079 4.76028i −0.139855 0.288105i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.58630 + 13.3843i −0.215480 + 0.804182i 0.770517 + 0.637419i \(0.219998\pi\)
−0.985997 + 0.166763i \(0.946669\pi\)
\(278\) 0 0
\(279\) 5.19615 0.311086
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.14110 15.4548i 0.246163 0.918693i −0.726632 0.687027i \(-0.758916\pi\)
0.972795 0.231667i \(-0.0744178\pi\)
\(284\) 10.3923 6.00000i 0.616670 0.356034i
\(285\) 0 0
\(286\) 0 0
\(287\) 27.4249 + 1.96902i 1.61884 + 0.116227i
\(288\) 0 0
\(289\) 16.4545 + 9.50000i 0.967911 + 0.558824i
\(290\) 0 0
\(291\) −6.50000 11.2583i −0.381037 0.659975i
\(292\) −0.517638 1.93185i −0.0302925 0.113053i
\(293\) −12.7279 12.7279i −0.743573 0.743573i 0.229691 0.973264i \(-0.426229\pi\)
−0.973264 + 0.229691i \(0.926229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.79555 + 1.55291i 0.336292 + 0.0901092i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −9.00000 10.3923i −0.518751 0.599002i
\(302\) 0 0
\(303\) 0 0
\(304\) −6.92820 12.0000i −0.397360 0.688247i
\(305\) 0 0
\(306\) 0 0
\(307\) −4.94975 + 4.94975i −0.282497 + 0.282497i −0.834104 0.551607i \(-0.814015\pi\)
0.551607 + 0.834104i \(0.314015\pi\)
\(308\) −17.8028 + 26.2880i −1.01441 + 1.49790i
\(309\) 7.00000i 0.398216i
\(310\) 0 0
\(311\) 18.0000 10.3923i 1.02069 0.589294i 0.106384 0.994325i \(-0.466073\pi\)
0.914303 + 0.405032i \(0.132739\pi\)
\(312\) 0 0
\(313\) 18.3526 4.91756i 1.03735 0.277957i 0.300335 0.953834i \(-0.402902\pi\)
0.737015 + 0.675877i \(0.236235\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −10.0382 + 2.68973i −0.563801 + 0.151070i −0.529452 0.848340i \(-0.677602\pi\)
−0.0343491 + 0.999410i \(0.510936\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14.6969 + 14.6969i −0.817760 + 0.817760i
\(324\) −1.73205 1.00000i −0.0962250 0.0555556i
\(325\) 0 0
\(326\) 0 0
\(327\) 2.58819 + 9.65926i 0.143127 + 0.534158i
\(328\) 0 0
\(329\) −15.5885 + 3.00000i −0.859419 + 0.165395i
\(330\) 0 0
\(331\) 2.50000 4.33013i 0.137412 0.238005i −0.789104 0.614260i \(-0.789455\pi\)
0.926516 + 0.376254i \(0.122788\pi\)
\(332\) −11.5911 3.10583i −0.636145 0.170454i
\(333\) 8.36516 + 2.24144i 0.458408 + 0.122830i
\(334\) 0 0
\(335\) 0 0
\(336\) 8.00000 6.92820i 0.436436 0.377964i
\(337\) −18.3712 18.3712i −1.00074 1.00074i −1.00000 0.000741840i \(-0.999764\pi\)
−0.000741840 1.00000i \(-0.500236\pi\)
\(338\) 0 0
\(339\) 5.19615 + 9.00000i 0.282216 + 0.488813i
\(340\) 0 0
\(341\) 27.0000 + 15.5885i 1.46213 + 0.844162i
\(342\) 0 0
\(343\) 3.95164 18.0938i 0.213368 0.976972i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.68973 + 10.0382i −0.144392 + 0.538879i 0.855390 + 0.517985i \(0.173318\pi\)
−0.999782 + 0.0208935i \(0.993349\pi\)
\(348\) 0 0
\(349\) 20.7846 1.11257 0.556287 0.830990i \(-0.312225\pi\)
0.556287 + 0.830990i \(0.312225\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 9.31749 34.7733i 0.495920 1.85080i −0.0288971 0.999582i \(-0.509200\pi\)
0.524817 0.851215i \(-0.324134\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 20.7846i 1.10158i
\(357\) −13.1440 8.90138i −0.695656 0.471111i
\(358\) 0 0
\(359\) −15.5885 9.00000i −0.822727 0.475002i 0.0286287 0.999590i \(-0.490886\pi\)
−0.851356 + 0.524588i \(0.824219\pi\)
\(360\) 0 0
\(361\) 3.50000 + 6.06218i 0.184211 + 0.319062i
\(362\) 0 0
\(363\) 17.6777 + 17.6777i 0.927837 + 0.927837i
\(364\) −3.46410 + 10.0000i −0.181568 + 0.524142i
\(365\) 0 0
\(366\) 0 0
\(367\) 16.4207 + 4.39992i 0.857156 + 0.229674i 0.660526 0.750804i \(-0.270334\pi\)
0.196630 + 0.980478i \(0.437000\pi\)
\(368\) 0 0
\(369\) −5.19615 + 9.00000i −0.270501 + 0.468521i
\(370\) 0 0
\(371\) −9.00000 + 25.9808i −0.467257 + 1.34885i
\(372\) −7.34847 7.34847i −0.381000 0.381000i
\(373\) 1.79315 + 6.69213i 0.0928458 + 0.346505i 0.996684 0.0813690i \(-0.0259292\pi\)
−0.903838 + 0.427874i \(0.859263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) 0 0
\(381\) −10.5000 + 6.06218i −0.537931 + 0.310575i
\(382\) 0 0
\(383\) −23.1822 + 6.21166i −1.18456 + 0.317401i −0.796732 0.604333i \(-0.793440\pi\)
−0.387824 + 0.921733i \(0.626773\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.01910 1.34486i 0.255135 0.0683632i
\(388\) −6.72930 + 25.1141i −0.341628 + 1.27497i
\(389\) 5.19615 3.00000i 0.263455 0.152106i −0.362454 0.932002i \(-0.618061\pi\)
0.625910 + 0.779895i \(0.284728\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 7.34847 7.34847i 0.370681 0.370681i
\(394\) 0 0
\(395\) 0 0
\(396\) −6.00000 10.3923i −0.301511 0.522233i
\(397\) −1.81173 6.76148i −0.0909283 0.339349i 0.905442 0.424469i \(-0.139539\pi\)
−0.996371 + 0.0851201i \(0.972873\pi\)
\(398\) 0 0
\(399\) 6.92820 6.00000i 0.346844 0.300376i
\(400\) 0 0
\(401\) −9.00000 + 15.5885i −0.449439 + 0.778450i −0.998350 0.0574304i \(-0.981709\pi\)
0.548911 + 0.835881i \(0.315043\pi\)
\(402\) 0 0
\(403\) 10.0382 + 2.68973i 0.500038 + 0.133985i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.7423 + 36.7423i 1.82125 + 1.82125i
\(408\) 0 0
\(409\) 3.46410 + 6.00000i 0.171289 + 0.296681i 0.938871 0.344270i \(-0.111874\pi\)
−0.767582 + 0.640951i \(0.778540\pi\)
\(410\) 0 0
\(411\) −18.0000 10.3923i −0.887875 0.512615i
\(412\) 9.89949 9.89949i 0.487713 0.487713i
\(413\) 24.7351 12.0072i 1.21714 0.590836i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.448288 1.67303i 0.0219527 0.0819288i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 31.0000 1.51085 0.755424 0.655237i \(-0.227431\pi\)
0.755424 + 0.655237i \(0.227431\pi\)
\(422\) 0 0
\(423\) 1.55291 5.79555i 0.0755053 0.281790i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −12.8480 + 18.9718i −0.621760 + 0.918108i
\(428\) 0 0
\(429\) 10.3923 + 6.00000i 0.501745 + 0.289683i
\(430\) 0 0
\(431\) 6.00000 + 10.3923i 0.289010 + 0.500580i 0.973574 0.228373i \(-0.0733406\pi\)
−0.684564 + 0.728953i \(0.740007\pi\)
\(432\) 1.03528 + 3.86370i 0.0498097 + 0.185893i
\(433\) 20.5061 + 20.5061i 0.985460 + 0.985460i 0.999896 0.0144357i \(-0.00459518\pi\)
−0.0144357 + 0.999896i \(0.504595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 17.3205i 0.478913 0.829502i
\(437\) 0 0
\(438\) 0 0
\(439\) −4.33013 + 7.50000i −0.206666 + 0.357955i −0.950662 0.310228i \(-0.899595\pi\)
0.743996 + 0.668184i \(0.232928\pi\)
\(440\) 0 0
\(441\) 5.50000 + 4.33013i 0.261905 + 0.206197i
\(442\) 0 0
\(443\) 2.68973 + 10.0382i 0.127793 + 0.476929i 0.999924 0.0123433i \(-0.00392908\pi\)
−0.872131 + 0.489272i \(0.837262\pi\)
\(444\) −8.66025 15.0000i −0.410997 0.711868i
\(445\) 0 0
\(446\) 0 0
\(447\) 8.48528 8.48528i 0.401340 0.401340i
\(448\) −21.1117 1.51575i −0.997433 0.0716124i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) −54.0000 + 31.1769i −2.54276 + 1.46806i
\(452\) 5.37945 20.0764i 0.253028 0.944314i
\(453\) −4.82963 + 1.29410i −0.226916 + 0.0608019i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.4034 4.93117i 0.860873 0.230670i 0.198736 0.980053i \(-0.436316\pi\)
0.662137 + 0.749383i \(0.269650\pi\)
\(458\) 0 0
\(459\) 5.19615 3.00000i 0.242536 0.140028i
\(460\) 0 0
\(461\) 31.1769i 1.45205i −0.687666 0.726027i \(-0.741365\pi\)
0.687666 0.726027i \(-0.258635\pi\)
\(462\) 0 0
\(463\) −1.22474 + 1.22474i −0.0569187 + 0.0569187i −0.734993 0.678074i \(-0.762815\pi\)
0.678074 + 0.734993i \(0.262815\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.76457 28.9778i −0.359302 1.34093i −0.874984 0.484152i \(-0.839128\pi\)
0.515683 0.856780i \(-0.327538\pi\)
\(468\) −2.82843 2.82843i −0.130744 0.130744i
\(469\) 8.66025 + 3.00000i 0.399893 + 0.138527i
\(470\) 0 0
\(471\) −3.50000 + 6.06218i −0.161271 + 0.279330i
\(472\) 0 0
\(473\) 30.1146 + 8.06918i 1.38467 + 0.371021i
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 + 31.1769i 0.275010 + 1.42899i
\(477\) −7.34847 7.34847i −0.336463 0.336463i
\(478\) 0 0
\(479\) 10.3923 + 18.0000i 0.474837 + 0.822441i 0.999585 0.0288165i \(-0.00917385\pi\)
−0.524748 + 0.851258i \(0.675841\pi\)
\(480\) 0 0
\(481\) 15.0000 + 8.66025i 0.683941 + 0.394874i
\(482\) 0 0
\(483\) 0 0
\(484\) 50.0000i 2.27273i
\(485\) 0 0
\(486\) 0 0
\(487\) 9.41404 35.1337i 0.426591 1.59206i −0.333833 0.942632i \(-0.608342\pi\)
0.760424 0.649427i \(-0.224991\pi\)
\(488\) 0 0
\(489\) 8.66025 0.391630
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 20.0764 5.37945i 0.905114 0.242524i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 20.7846i 0.933257i
\(497\) 1.13681 15.8338i 0.0509930 0.710241i
\(498\) 0 0
\(499\) −17.3205 10.0000i −0.775372 0.447661i 0.0594153 0.998233i \(-0.481076\pi\)
−0.834788 + 0.550572i \(0.814410\pi\)
\(500\) 0 0
\(501\) 6.00000 + 10.3923i 0.268060 + 0.464294i
\(502\) 0 0
\(503\) −25.4558 25.4558i −1.13502 1.13502i −0.989330 0.145690i \(-0.953460\pi\)
−0.145690 0.989330i \(-0.546540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −8.69333 2.32937i −0.386084 0.103451i
\(508\) 23.4225 + 6.27603i 1.03920 + 0.278454i
\(509\) 5.19615 9.00000i 0.230315 0.398918i −0.727586 0.686017i \(-0.759358\pi\)
0.957901 + 0.287099i \(0.0926909\pi\)
\(510\) 0 0
\(511\) −2.50000 0.866025i −0.110593 0.0383107i
\(512\) 0 0
\(513\) 0.896575 + 3.34607i 0.0395848 + 0.147732i
\(514\) 0 0
\(515\) 0 0
\(516\) −9.00000 5.19615i −0.396203 0.228748i
\(517\) 25.4558 25.4558i 1.11955 1.11955i
\(518\) 0 0
\(519\) 18.0000i 0.790112i
\(520\) 0 0
\(521\) −9.00000 + 5.19615i −0.394297 + 0.227648i −0.684020 0.729463i \(-0.739770\pi\)
0.289723 + 0.957110i \(0.406437\pi\)
\(522\) 0 0
\(523\) −6.76148 + 1.81173i −0.295659 + 0.0792216i −0.403599 0.914936i \(-0.632241\pi\)
0.107941 + 0.994157i \(0.465574\pi\)
\(524\) −20.7846 −0.907980
\(525\) 0 0
\(526\) 0 0
\(527\) 30.1146 8.06918i 1.31181 0.351499i
\(528\) −6.21166 + 23.1822i −0.270328 + 1.00888i
\(529\) 19.9186 11.5000i 0.866025 0.500000i
\(530\) 0 0
\(531\) 10.3923i 0.450988i
\(532\) −18.2832 1.31268i −0.792679 0.0569118i
\(533\) −14.6969 + 14.6969i −0.636595 + 0.636595i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.21166 + 23.1822i 0.268053 + 1.00039i
\(538\) 0 0
\(539\) 15.5885 + 39.0000i 0.671442 + 1.67985i
\(540\) 0 0
\(541\) 5.00000 8.66025i 0.214967 0.372333i −0.738296 0.674477i \(-0.764369\pi\)
0.953262 + 0.302144i \(0.0977023\pi\)
\(542\) 0 0
\(543\) −5.01910 1.34486i −0.215390 0.0577136i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 22.0454 + 22.0454i 0.942594 + 0.942594i 0.998439 0.0558458i \(-0.0177855\pi\)
−0.0558458 + 0.998439i \(0.517786\pi\)
\(548\) 10.7589 + 40.1528i 0.459598 + 1.71524i
\(549\) −4.33013 7.50000i −0.184805 0.320092i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −7.41782 + 10.9534i −0.315438 + 0.465784i
\(554\) 0 0
\(555\) 0 0
\(556\) −3.00000 + 1.73205i −0.127228 + 0.0734553i
\(557\) −2.68973 + 10.0382i −0.113967 + 0.425332i −0.999208 0.0398021i \(-0.987327\pi\)
0.885240 + 0.465134i \(0.153994\pi\)
\(558\) 0 0
\(559\) 10.3923 0.439548
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 0 0
\(563\) 1.55291 5.79555i 0.0654475 0.244254i −0.925450 0.378869i \(-0.876313\pi\)
0.990898 + 0.134615i \(0.0429798\pi\)
\(564\) −10.3923 + 6.00000i −0.437595 + 0.252646i
\(565\) 0 0
\(566\) 0 0
\(567\) −2.38014 + 1.15539i −0.0999565 + 0.0485220i
\(568\) 0 0
\(569\) −20.7846 12.0000i −0.871336 0.503066i −0.00354413 0.999994i \(-0.501128\pi\)
−0.867792 + 0.496928i \(0.834461\pi\)
\(570\) 0 0
\(571\) −10.0000 17.3205i −0.418487 0.724841i 0.577301 0.816532i \(-0.304106\pi\)
−0.995788 + 0.0916910i \(0.970773\pi\)
\(572\) −6.21166 23.1822i −0.259722 0.969297i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 4.00000 6.92820i 0.166667 0.288675i
\(577\) 6.76148 + 1.81173i 0.281484 + 0.0754234i 0.396799 0.917906i \(-0.370121\pi\)
−0.115315 + 0.993329i \(0.536788\pi\)
\(578\) 0 0
\(579\) −7.79423 + 13.5000i −0.323917 + 0.561041i
\(580\) 0 0
\(581\) −12.0000 + 10.3923i −0.497844 + 0.431145i
\(582\) 0 0
\(583\) −16.1384 60.2292i −0.668383 2.49444i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.7279 + 12.7279i −0.525338 + 0.525338i −0.919179 0.393841i \(-0.871146\pi\)
0.393841 + 0.919179i \(0.371146\pi\)
\(588\) −1.65445 13.9019i −0.0682284 0.573305i
\(589\) 18.0000i 0.741677i
\(590\) 0 0
\(591\) 9.00000 5.19615i 0.370211 0.213741i
\(592\) −8.96575 + 33.4607i −0.368490 + 1.37522i
\(593\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24.0000 −0.983078
\(597\) 18.4034 4.93117i 0.753199 0.201819i
\(598\) 0 0
\(599\) −10.3923 + 6.00000i −0.424618 + 0.245153i −0.697051 0.717021i \(-0.745505\pi\)
0.272433 + 0.962175i \(0.412172\pi\)
\(600\) 0 0
\(601\) 34.6410i 1.41304i 0.707695 + 0.706518i \(0.249735\pi\)
−0.707695 + 0.706518i \(0.750265\pi\)
\(602\) 0 0
\(603\) −2.44949 + 2.44949i −0.0997509 + 0.0997509i
\(604\) 8.66025 + 5.00000i 0.352381 + 0.203447i
\(605\) 0 0
\(606\) 0 0
\(607\) −6.47048 24.1481i −0.262629 0.980143i −0.963686 0.267038i \(-0.913955\pi\)
0.701057 0.713105i \(-0.252712\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.00000 10.3923i 0.242734 0.420428i
\(612\) −11.5911 3.10583i −0.468543 0.125546i
\(613\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.34847 + 7.34847i 0.295838 + 0.295838i 0.839381 0.543543i \(-0.182918\pi\)
−0.543543 + 0.839381i \(0.682918\pi\)
\(618\) 0 0
\(619\) −23.3827 40.5000i −0.939829 1.62783i −0.765787 0.643094i \(-0.777650\pi\)
−0.174042 0.984738i \(-0.555683\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 22.7661 + 15.4176i 0.912105 + 0.617695i
\(624\) 8.00000i 0.320256i
\(625\) 0 0
\(626\) 0 0
\(627\) −5.37945 + 20.0764i −0.214835 + 0.801774i
\(628\) 13.5230 3.62347i 0.539625 0.144592i
\(629\) 51.9615 2.07184
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) 1.81173 6.76148i 0.0720099 0.268745i
\(634\) 0 0
\(635\) 0 0
\(636\) 20.7846i 0.824163i
\(637\) 8.38375 + 11.2122i 0.332176 + 0.444242i
\(638\) 0 0
\(639\) 5.19615 + 3.00000i 0.205557 + 0.118678i
\(640\) 0 0
\(641\) 9.00000 + 15.5885i 0.355479 + 0.615707i 0.987200 0.159489i \(-0.0509845\pi\)
−0.631721 + 0.775196i \(0.717651\pi\)
\(642\) 0 0
\(643\) 20.5061 + 20.5061i 0.808682 + 0.808682i 0.984434 0.175753i \(-0.0562359\pi\)
−0.175753 + 0.984434i \(0.556236\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.3867 4.65874i −0.683540 0.183154i −0.0996938 0.995018i \(-0.531786\pi\)
−0.583846 + 0.811864i \(0.698453\pi\)
\(648\) 0 0
\(649\) −31.1769 + 54.0000i −1.22380 + 2.11969i
\(650\) 0 0
\(651\) −13.5000 + 2.59808i −0.529107 + 0.101827i
\(652\) −12.2474 12.2474i −0.479647 0.479647i
\(653\) −2.68973 10.0382i −0.105257 0.392825i 0.893117 0.449824i \(-0.148513\pi\)
−0.998374 + 0.0569993i \(0.981847\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36.0000 20.7846i −1.40556 0.811503i
\(657\) 0.707107 0.707107i 0.0275869 0.0275869i
\(658\) 0 0
\(659\) 6.00000i 0.233727i 0.993148 + 0.116863i \(0.0372840\pi\)
−0.993148 + 0.116863i \(0.962716\pi\)
\(660\) 0 0
\(661\) 37.5000 21.6506i 1.45858 0.842112i 0.459639 0.888106i \(-0.347979\pi\)
0.998942 + 0.0459936i \(0.0146454\pi\)
\(662\) 0 0
\(663\) 11.5911 3.10583i 0.450161 0.120620i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 6.21166 23.1822i 0.240336 0.896947i
\(669\) 4.33013 2.50000i 0.167412 0.0966556i
\(670\) 0 0
\(671\) 51.9615i 2.00595i
\(672\) 0 0
\(673\) 8.57321 8.57321i 0.330473 0.330473i −0.522293 0.852766i \(-0.674923\pi\)
0.852766 + 0.522293i \(0.174923\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 9.00000 + 15.5885i 0.346154 + 0.599556i
\(677\) −12.4233 46.3644i −0.477467 1.78193i −0.611820 0.790997i \(-0.709562\pi\)
0.134354 0.990933i \(-0.457104\pi\)
\(678\) 0 0
\(679\) 22.5167 + 26.0000i 0.864110 + 0.997788i
\(680\) 0 0
\(681\) −3.00000 + 5.19615i −0.114960 + 0.199117i
\(682\) 0 0
\(683\) 30.1146 + 8.06918i 1.15230 + 0.308759i 0.783888 0.620903i \(-0.213234\pi\)
0.368415 + 0.929661i \(0.379901\pi\)
\(684\) 3.46410 6.00000i 0.132453 0.229416i
\(685\) 0 0
\(686\) 0 0
\(687\) −4.89898 4.89898i −0.186908 0.186908i
\(688\) 5.37945 + 20.0764i 0.205090 + 0.765405i
\(689\) −10.3923 18.0000i −0.395915 0.685745i
\(690\) 0 0
\(691\) 10.5000 + 6.06218i 0.399439 + 0.230616i 0.686242 0.727373i \(-0.259259\pi\)
−0.286803 + 0.957990i \(0.592593\pi\)
\(692\) −25.4558 + 25.4558i −0.967686 + 0.967686i
\(693\) −15.8338 1.13681i −0.601474 0.0431839i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −16.1384 + 60.2292i −0.611284 + 2.28134i
\(698\) 0 0
\(699\) −20.7846 −0.786146
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −7.76457 + 28.9778i −0.292846 + 1.09292i
\(704\) 41.5692 24.0000i 1.56670 0.904534i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 14.6969 14.6969i 0.552345 0.552345i
\(709\) −26.8468 15.5000i −1.00825 0.582115i −0.0975728 0.995228i \(-0.531108\pi\)
−0.910679 + 0.413114i \(0.864441\pi\)
\(710\) 0 0
\(711\) −2.50000 4.33013i −0.0937573 0.162392i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 41.5692i 0.896922 1.55351i
\(717\) 17.3867 + 4.65874i 0.649317 + 0.173984i
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) −3.50000 18.1865i −0.130347 0.677302i
\(722\) 0 0
\(723\) −4.93117 18.4034i −0.183392 0.684428i
\(724\) 5.19615 + 9.00000i 0.193113 + 0.334482i
\(725\) 0 0
\(726\) 0 0
\(727\) −28.2843 + 28.2843i −1.04901 + 1.04901i −0.0502699 + 0.998736i \(0.516008\pi\)
−0.998736 + 0.0502699i \(0.983992\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 27.0000 15.5885i 0.998631 0.576560i
\(732\) −4.48288 + 16.7303i −0.165692 + 0.618371i
\(733\) −45.3985 + 12.1645i −1.67683 + 0.449306i −0.966940 0.255004i \(-0.917923\pi\)
−0.709892 + 0.704310i \(0.751256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −20.0764 + 5.37945i −0.739523 + 0.198155i
\(738\) 0 0
\(739\) −0.866025 + 0.500000i −0.0318573 + 0.0183928i −0.515844 0.856683i \(-0.672522\pi\)
0.483987 + 0.875075i \(0.339188\pi\)
\(740\) 0 0
\(741\) 6.92820i 0.254514i
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.55291 5.79555i −0.0568182 0.212048i
\(748\) −50.9117 50.9117i −1.86152 1.86152i
\(749\) 0 0
\(750\) 0 0
\(751\) −5.50000 + 9.52628i −0.200698 + 0.347619i −0.948753 0.316017i \(-0.897654\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 23.1822 + 6.21166i 0.845369 + 0.226516i
\(753\) −20.0764 5.37945i −0.731624 0.196038i
\(754\) 0 0
\(755\) 0 0
\(756\) 5.00000 + 1.73205i 0.181848 + 0.0629941i
\(757\) −20.8207 20.8207i −0.756740 0.756740i 0.218988 0.975728i \(-0.429725\pi\)
−0.975728 + 0.218988i \(0.929725\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 10.3923i −0.652499 0.376721i 0.136914 0.990583i \(-0.456282\pi\)
−0.789413 + 0.613862i \(0.789615\pi\)
\(762\) 0 0
\(763\) −11.5539 23.8014i −0.418281 0.861668i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.37945 + 20.0764i −0.194241 + 0.724916i
\(768\) −15.4548 + 4.14110i −0.557678 + 0.149429i
\(769\) 5.19615 0.187378 0.0936890 0.995602i \(-0.470134\pi\)
0.0936890 + 0.995602i \(0.470134\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 30.1146 8.06918i 1.08385 0.290416i
\(773\) −6.21166 + 23.1822i −0.223418 + 0.833806i 0.759614 + 0.650374i \(0.225388\pi\)
−0.983032 + 0.183433i \(0.941279\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −22.8541 1.64085i −0.819884 0.0588651i
\(778\) 0 0
\(779\) −31.1769 18.0000i −1.11703 0.644917i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) −17.3205 + 22.0000i −0.618590 + 0.785714i
\(785\) 0 0
\(786\) 0 0
\(787\) −10.6252 2.84701i −0.378747 0.101485i 0.0644227 0.997923i \(-0.479479\pi\)
−0.443170 + 0.896438i \(0.646146\pi\)
\(788\) −20.0764 5.37945i −0.715192 0.191635i
\(789\) −15.5885 + 27.0000i −0.554964 + 0.961225i
\(790\) 0 0
\(791\) −18.0000 20.7846i −0.640006 0.739016i
\(792\) 0 0
\(793\) −4.48288 16.7303i −0.159192 0.594111i
\(794\) 0 0
\(795\) 0 0
\(796\) −33.0000 19.0526i −1.16965 0.675300i
\(797\) −8.48528 + 8.48528i −0.300564 + 0.300564i −0.841235 0.540670i \(-0.818171\pi\)
0.540670 + 0.841235i \(0.318171\pi\)
\(798\) 0 0
\(799\) 36.0000i 1.27359i
\(800\) 0 0
\(801\) −9.00000 + 5.19615i −0.317999 + 0.183597i
\(802\) 0 0
\(803\) 5.79555 1.55291i 0.204521 0.0548012i
\(804\) 6.92820 0.244339
\(805\) 0 0
\(806\) 0 0
\(807\) −20.0764 + 5.37945i −0.706722 + 0.189366i
\(808\) 0 0
\(809\) 20.7846 12.0000i 0.730748 0.421898i −0.0879478 0.996125i \(-0.528031\pi\)
0.818696 + 0.574228i \(0.194698\pi\)
\(810\) 0 0
\(811\) 39.8372i 1.39887i −0.714695 0.699436i \(-0.753435\pi\)
0.714695 0.699436i \(-0.246565\pi\)
\(812\) 0 0
\(813\) −22.0454 + 22.0454i −0.773166 + 0.773166i
\(814\) 0 0
\(815\) 0 0
\(816\) 12.0000 + 20.7846i 0.420084 + 0.727607i
\(817\) 4.65874 + 17.3867i 0.162989 + 0.608282i
\(818\) 0 0
\(819\) −5.19615 + 1.00000i −0.181568 + 0.0349428i
\(820\) 0 0
\(821\) 12.0000 20.7846i 0.418803 0.725388i −0.577016 0.816733i \(-0.695783\pi\)
0.995819 + 0.0913446i \(0.0291165\pi\)
\(822\) 0 0
\(823\) −38.4797 10.3106i −1.34132 0.359406i −0.484396 0.874849i \(-0.660961\pi\)
−0.856924 + 0.515443i \(0.827627\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0454 + 22.0454i 0.766594 + 0.766594i 0.977505 0.210911i \(-0.0676431\pi\)
−0.210911 + 0.977505i \(0.567643\pi\)
\(828\) 0 0
\(829\) 4.33013 + 7.50000i 0.150392 + 0.260486i 0.931371 0.364070i \(-0.118613\pi\)
−0.780980 + 0.624556i \(0.785280\pi\)
\(830\) 0 0
\(831\) 12.0000 + 6.92820i 0.416275 + 0.240337i
\(832\) 11.3137 11.3137i 0.392232 0.392232i
\(833\) 38.5999 + 16.5545i 1.33741 + 0.573578i
\(834\) 0 0
\(835\) 0 0
\(836\) 36.0000 20.7846i 1.24509 0.718851i
\(837\) 1.34486 5.01910i 0.0464853 0.173485i
\(838\) 0 0
\(839\) 31.1769 1.07635 0.538173 0.842834i \(-0.319115\pi\)
0.538173 + 0.842834i \(0.319115\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) −1.55291 + 5.79555i −0.0534852 + 0.199610i
\(844\) −12.1244 + 7.00000i −0.417338 + 0.240950i
\(845\) 0 0
\(846\) 0 0
\(847\) −54.7668 37.0891i −1.88181 1.27440i
\(848\) 29.3939 29.3939i 1.00939 1.00939i
\(849\) −13.8564 8.00000i −0.475551 0.274559i
\(850\) 0 0
\(851\) 0 0
\(852\) −3.10583 11.5911i −0.106404 0.397105i
\(853\) 13.4350 + 13.4350i 0.460007 + 0.460007i 0.898658 0.438651i \(-0.144544\pi\)
−0.438651 + 0.898658i \(0.644544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.1822 + 6.21166i 0.791890 + 0.212186i 0.632020 0.774952i \(-0.282226\pi\)
0.159869 + 0.987138i \(0.448893\pi\)
\(858\) 0 0
\(859\) 5.19615 9.00000i 0.177290 0.307076i −0.763661 0.645617i \(-0.776600\pi\)
0.940952 + 0.338541i \(0.109933\pi\)
\(860\) 0 0
\(861\) 9.00000 25.9808i 0.306719 0.885422i
\(862\) 0 0
\(863\) −8.06918 30.1146i −0.274678 1.02511i −0.956057 0.293181i \(-0.905286\pi\)
0.681379 0.731931i \(-0.261381\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.4350 13.4350i 0.456278 0.456278i
\(868\) 22.7661 + 15.4176i 0.772732 + 0.523309i
\(869\) 30.0000i 1.01768i
\(870\) 0 0
\(871\) −6.00000 + 3.46410i −0.203302 + 0.117377i
\(872\) 0 0
\(873\) −12.5570 + 3.36465i −0.424991 + 0.113876i
\(874\) 0 0
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 21.7494 5.82774i 0.734426 0.196789i 0.127827 0.991797i \(-0.459200\pi\)
0.606599 + 0.795008i \(0.292533\pi\)
\(878\) 0 0
\(879\) −15.5885 + 9.00000i −0.525786 + 0.303562i
\(880\) 0 0
\(881\) 10.3923i 0.350126i −0.984557 0.175063i \(-0.943987\pi\)
0.984557 0.175063i \(-0.0560129\pi\)
\(882\) 0 0
\(883\) 2.44949 2.44949i 0.0824319 0.0824319i −0.664689 0.747121i \(-0.731436\pi\)
0.747121 + 0.664689i \(0.231436\pi\)
\(884\) −20.7846 12.0000i −0.699062 0.403604i
\(885\) 0 0
\(886\) 0 0
\(887\) −4.65874 17.3867i −0.156425 0.583787i −0.998979 0.0451749i \(-0.985615\pi\)
0.842554 0.538612i \(-0.181051\pi\)
\(888\) 0 0
\(889\) 24.2487 21.0000i 0.813276 0.704317i
\(890\) 0 0
\(891\) 3.00000 5.19615i 0.100504 0.174078i
\(892\) −9.65926 2.58819i −0.323416 0.0866590i
\(893\) 20.0764 + 5.37945i 0.671831 + 0.180017i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −54.0000 31.1769i −1.79900 1.03865i
\(902\) 0 0
\(903\) −12.3676 + 6.00361i −0.411567 + 0.199787i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.448288 + 1.67303i −0.0148851 + 0.0555521i −0.972969 0.230936i \(-0.925821\pi\)
0.958084 + 0.286488i \(0.0924879\pi\)
\(908\) 11.5911 3.10583i 0.384664 0.103071i
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −13.3843 + 3.58630i −0.443197 + 0.118754i
\(913\) 9.31749 34.7733i 0.308364 1.15083i
\(914\) 0 0
\(915\) 0 0
\(916\) 13.8564i 0.457829i
\(917\) −15.4176 + 22.7661i −0.509136 + 0.751803i
\(918\) 0 0
\(919\) −21.6506 12.5000i −0.714189 0.412337i 0.0984214 0.995145i \(-0.468621\pi\)
−0.812610 + 0.582808i \(0.801954\pi\)
\(920\) 0 0
\(921\) 3.50000 + 6.06218i 0.115329 + 0.199756i
\(922\) 0 0
\(923\) 8.48528 + 8.48528i 0.279296 + 0.279296i
\(924\) 20.7846 + 24.0000i 0.683763 + 0.789542i
\(925\) 0 0
\(926\) 0 0
\(927\) 6.76148 + 1.81173i 0.222076 + 0.0595051i
\(928\) 0 0
\(929\) −5.19615 + 9.00000i −0.170480 + 0.295280i −0.938588 0.345040i \(-0.887865\pi\)
0.768108 + 0.640321i \(0.221199\pi\)
\(930\) 0 0
\(931\) −15.0000 + 19.0526i −0.491605 + 0.624422i
\(932\) 29.3939 + 29.3939i 0.962828 + 0.962828i
\(933\) −5.37945 20.0764i −0.176115 0.657272i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −20.5061 + 20.5061i −0.669905 + 0.669905i −0.957694 0.287789i \(-0.907080\pi\)
0.287789 + 0.957694i \(0.407080\pi\)
\(938\) 0 0
\(939\) 19.0000i 0.620042i
\(940\) 0 0
\(941\) −27.0000 + 15.5885i −0.880175 + 0.508169i −0.870716 0.491786i \(-0.836344\pi\)
−0.00945879 + 0.999955i \(0.503011\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −41.5692 −1.35296
\(945\) 0 0
\(946\) 0 0
\(947\) 10.0382 2.68973i 0.326198 0.0874044i −0.0920040 0.995759i \(-0.529327\pi\)
0.418202 + 0.908354i \(0.362661\pi\)
\(948\) −2.58819 + 9.65926i −0.0840605 + 0.313718i
\(949\) 1.73205 1.00000i 0.0562247 0.0324614i
\(950\) 0 0
\(951\) 10.3923i 0.336994i
\(952\) 0 0
\(953\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −18.0000 31.1769i −0.582162 1.00833i
\(957\) 0 0
\(958\) 0 0
\(959\) 51.9615 + 18.0000i 1.67793 + 0.581250i
\(960\) 0 0
\(961\) −2.00000 + 3.46410i −0.0645161 + 0.111745i
\(962\) 0 0
\(963\) 0 0
\(964\) −19.0526 + 33.0000i −0.613642 + 1.06286i
\(965\) 0 0
\(966\) 0 0
\(967\) 30.6186 + 30.6186i 0.984628 + 0.984628i 0.999884 0.0152551i \(-0.00485605\pi\)
−0.0152551 + 0.999884i \(0.504856\pi\)
\(968\) 0 0
\(969\) 10.3923 + 18.0000i 0.333849 + 0.578243i
\(970\) 0 0
\(971\) 18.0000 + 10.3923i 0.577647 + 0.333505i 0.760198 0.649692i \(-0.225102\pi\)
−0.182550 + 0.983196i \(0.558435\pi\)
\(972\) −1.41421 + 1.41421i −0.0453609 + 0.0453609i
\(973\) −0.328169 + 4.57081i −0.0105206 + 0.146533i
\(974\) 0 0
\(975\) 0 0
\(976\) 30.0000 17.3205i 0.960277 0.554416i
\(977\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(978\) 0 0
\(979\) −62.3538 −1.99284
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) −10.8704 + 40.5689i −0.346712 + 1.29395i 0.543888 + 0.839158i \(0.316952\pi\)
−0.890600 + 0.454788i \(0.849715\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.13681 + 15.8338i −0.0361851 + 0.503994i
\(988\) 9.79796 9.79796i 0.311715 0.311715i
\(989\) 0 0
\(990\) 0 0
\(991\) 14.5000 + 25.1147i 0.460608 + 0.797796i 0.998991 0.0449040i \(-0.0142982\pi\)
−0.538384 + 0.842700i \(0.680965\pi\)
\(992\) 0 0
\(993\) −3.53553 3.53553i −0.112197 0.112197i
\(994\) 0 0
\(995\) 0 0
\(996\) −6.00000 + 10.3923i −0.190117 + 0.329293i
\(997\) 4.82963 + 1.29410i 0.152956 + 0.0409844i 0.334484 0.942401i \(-0.391438\pi\)
−0.181529 + 0.983386i \(0.558104\pi\)
\(998\) 0 0
\(999\) 4.33013 7.50000i 0.136999 0.237289i
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 525.2.bc.a.82.2 yes 8
5.2 odd 4 inner 525.2.bc.a.418.1 yes 8
5.3 odd 4 inner 525.2.bc.a.418.2 yes 8
5.4 even 2 inner 525.2.bc.a.82.1 8
7.3 odd 6 inner 525.2.bc.a.157.2 yes 8
35.3 even 12 inner 525.2.bc.a.493.2 yes 8
35.17 even 12 inner 525.2.bc.a.493.1 yes 8
35.24 odd 6 inner 525.2.bc.a.157.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.2.bc.a.82.1 8 5.4 even 2 inner
525.2.bc.a.82.2 yes 8 1.1 even 1 trivial
525.2.bc.a.157.1 yes 8 35.24 odd 6 inner
525.2.bc.a.157.2 yes 8 7.3 odd 6 inner
525.2.bc.a.418.1 yes 8 5.2 odd 4 inner
525.2.bc.a.418.2 yes 8 5.3 odd 4 inner
525.2.bc.a.493.1 yes 8 35.17 even 12 inner
525.2.bc.a.493.2 yes 8 35.3 even 12 inner