Properties

Label 3150.2.bf.d.1601.2
Level $3150$
Weight $2$
Character 3150.1601
Analytic conductor $25.153$
Analytic rank $0$
Dimension $24$
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] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1601.2
Character \(\chi\) \(=\) 3150.1601
Dual form 3150.2.bf.d.1151.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(2.34325 - 1.22849i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(2.34325 - 1.22849i) q^{7} -1.00000i q^{8} +(-2.03986 + 1.17771i) q^{11} +4.64698i q^{13} +(-2.64356 - 0.107718i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.28109 + 3.95097i) q^{17} +(0.491268 + 0.283634i) q^{19} +2.35542 q^{22} +(-5.04588 - 2.91324i) q^{23} +(2.32349 - 4.02440i) q^{26} +(2.23553 + 1.41507i) q^{28} -2.55339i q^{29} +(-1.89659 + 1.09500i) q^{31} +(0.866025 - 0.500000i) q^{32} -4.56218i q^{34} +(-4.63355 + 8.02554i) q^{37} +(-0.283634 - 0.491268i) q^{38} -8.68451 q^{41} +6.57695 q^{43} +(-2.03986 - 1.17771i) q^{44} +(2.91324 + 5.04588i) q^{46} +(-3.15616 + 5.46663i) q^{47} +(3.98161 - 5.75732i) q^{49} +(-4.02440 + 2.32349i) q^{52} +(-10.5228 + 6.07533i) q^{53} +(-1.22849 - 2.34325i) q^{56} +(-1.27670 + 2.21130i) q^{58} +(-1.67739 - 2.90532i) q^{59} +(-6.85523 - 3.95787i) q^{61} +2.18999 q^{62} -1.00000 q^{64} +(-2.00143 - 3.46657i) q^{67} +(-2.28109 + 3.95097i) q^{68} -2.02720i q^{71} +(-7.11528 + 4.10801i) q^{73} +(8.02554 - 4.63355i) q^{74} +0.567267i q^{76} +(-3.33308 + 5.26562i) q^{77} +(4.13212 - 7.15704i) q^{79} +(7.52101 + 4.34226i) q^{82} +0.171637 q^{83} +(-5.69581 - 3.28848i) q^{86} +(1.17771 + 2.03986i) q^{88} +(2.72938 - 4.72742i) q^{89} +(5.70878 + 10.8890i) q^{91} -5.82648i q^{92} +(5.46663 - 3.15616i) q^{94} +10.8564i q^{97} +(-6.32684 + 2.99518i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 12 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 12 q^{4} - 4 q^{7} - 12 q^{16} + 12 q^{19} + 4 q^{28} + 28 q^{37} + 96 q^{43} - 8 q^{46} - 52 q^{49} - 12 q^{52} + 8 q^{58} - 12 q^{61} - 24 q^{64} - 4 q^{67} - 12 q^{73} + 4 q^{79} + 68 q^{91} - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 0.500000i −0.612372 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.34325 1.22849i 0.885664 0.464326i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −2.03986 + 1.17771i −0.615040 + 0.355093i −0.774935 0.632041i \(-0.782218\pi\)
0.159896 + 0.987134i \(0.448884\pi\)
\(12\) 0 0
\(13\) 4.64698i 1.28884i 0.764672 + 0.644420i \(0.222901\pi\)
−0.764672 + 0.644420i \(0.777099\pi\)
\(14\) −2.64356 0.107718i −0.706520 0.0287888i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 2.28109 + 3.95097i 0.553246 + 0.958250i 0.998038 + 0.0626158i \(0.0199443\pi\)
−0.444792 + 0.895634i \(0.646722\pi\)
\(18\) 0 0
\(19\) 0.491268 + 0.283634i 0.112705 + 0.0650700i 0.555293 0.831655i \(-0.312606\pi\)
−0.442588 + 0.896725i \(0.645940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.35542 0.502178
\(23\) −5.04588 2.91324i −1.05214 0.607453i −0.128891 0.991659i \(-0.541142\pi\)
−0.923247 + 0.384206i \(0.874475\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.32349 4.02440i 0.455674 0.789250i
\(27\) 0 0
\(28\) 2.23553 + 1.41507i 0.422475 + 0.267422i
\(29\) 2.55339i 0.474153i −0.971491 0.237077i \(-0.923811\pi\)
0.971491 0.237077i \(-0.0761893\pi\)
\(30\) 0 0
\(31\) −1.89659 + 1.09500i −0.340638 + 0.196667i −0.660554 0.750778i \(-0.729679\pi\)
0.319916 + 0.947446i \(0.396345\pi\)
\(32\) 0.866025 0.500000i 0.153093 0.0883883i
\(33\) 0 0
\(34\) 4.56218i 0.782408i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.63355 + 8.02554i −0.761750 + 1.31939i 0.180197 + 0.983630i \(0.442326\pi\)
−0.941948 + 0.335760i \(0.891007\pi\)
\(38\) −0.283634 0.491268i −0.0460114 0.0796942i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.68451 −1.35629 −0.678147 0.734927i \(-0.737217\pi\)
−0.678147 + 0.734927i \(0.737217\pi\)
\(42\) 0 0
\(43\) 6.57695 1.00298 0.501488 0.865165i \(-0.332786\pi\)
0.501488 + 0.865165i \(0.332786\pi\)
\(44\) −2.03986 1.17771i −0.307520 0.177547i
\(45\) 0 0
\(46\) 2.91324 + 5.04588i 0.429534 + 0.743975i
\(47\) −3.15616 + 5.46663i −0.460374 + 0.797391i −0.998979 0.0451673i \(-0.985618\pi\)
0.538606 + 0.842558i \(0.318951\pi\)
\(48\) 0 0
\(49\) 3.98161 5.75732i 0.568802 0.822475i
\(50\) 0 0
\(51\) 0 0
\(52\) −4.02440 + 2.32349i −0.558084 + 0.322210i
\(53\) −10.5228 + 6.07533i −1.44542 + 0.834511i −0.998203 0.0599168i \(-0.980916\pi\)
−0.447212 + 0.894428i \(0.647583\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.22849 2.34325i −0.164164 0.313130i
\(57\) 0 0
\(58\) −1.27670 + 2.21130i −0.167639 + 0.290359i
\(59\) −1.67739 2.90532i −0.218377 0.378241i 0.735935 0.677053i \(-0.236743\pi\)
−0.954312 + 0.298812i \(0.903410\pi\)
\(60\) 0 0
\(61\) −6.85523 3.95787i −0.877722 0.506753i −0.00781543 0.999969i \(-0.502488\pi\)
−0.869907 + 0.493216i \(0.835821\pi\)
\(62\) 2.18999 0.278130
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00143 3.46657i −0.244513 0.423509i 0.717481 0.696578i \(-0.245295\pi\)
−0.961995 + 0.273068i \(0.911961\pi\)
\(68\) −2.28109 + 3.95097i −0.276623 + 0.479125i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.02720i 0.240585i −0.992738 0.120292i \(-0.961617\pi\)
0.992738 0.120292i \(-0.0383832\pi\)
\(72\) 0 0
\(73\) −7.11528 + 4.10801i −0.832780 + 0.480806i −0.854804 0.518952i \(-0.826322\pi\)
0.0220235 + 0.999757i \(0.492989\pi\)
\(74\) 8.02554 4.63355i 0.932950 0.538639i
\(75\) 0 0
\(76\) 0.567267i 0.0650700i
\(77\) −3.33308 + 5.26562i −0.379839 + 0.600073i
\(78\) 0 0
\(79\) 4.13212 7.15704i 0.464900 0.805230i −0.534297 0.845297i \(-0.679424\pi\)
0.999197 + 0.0400666i \(0.0127570\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 7.52101 + 4.34226i 0.830556 + 0.479522i
\(83\) 0.171637 0.0188396 0.00941978 0.999956i \(-0.497002\pi\)
0.00941978 + 0.999956i \(0.497002\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.69581 3.28848i −0.614195 0.354606i
\(87\) 0 0
\(88\) 1.17771 + 2.03986i 0.125544 + 0.217449i
\(89\) 2.72938 4.72742i 0.289313 0.501105i −0.684333 0.729170i \(-0.739906\pi\)
0.973646 + 0.228065i \(0.0732397\pi\)
\(90\) 0 0
\(91\) 5.70878 + 10.8890i 0.598443 + 1.14148i
\(92\) 5.82648i 0.607453i
\(93\) 0 0
\(94\) 5.46663 3.15616i 0.563840 0.325533i
\(95\) 0 0
\(96\) 0 0
\(97\) 10.8564i 1.10230i 0.834406 + 0.551151i \(0.185811\pi\)
−0.834406 + 0.551151i \(0.814189\pi\)
\(98\) −6.32684 + 2.99518i −0.639107 + 0.302559i
\(99\) 0 0
\(100\) 0 0
\(101\) 5.74827 + 9.95630i 0.571975 + 0.990689i 0.996363 + 0.0852090i \(0.0271558\pi\)
−0.424388 + 0.905480i \(0.639511\pi\)
\(102\) 0 0
\(103\) 16.7782 + 9.68690i 1.65320 + 0.954478i 0.975743 + 0.218920i \(0.0702535\pi\)
0.677462 + 0.735558i \(0.263080\pi\)
\(104\) 4.64698 0.455674
\(105\) 0 0
\(106\) 12.1507 1.18018
\(107\) −9.28430 5.36029i −0.897547 0.518199i −0.0211436 0.999776i \(-0.506731\pi\)
−0.876404 + 0.481577i \(0.840064\pi\)
\(108\) 0 0
\(109\) −5.41186 9.37362i −0.518363 0.897830i −0.999772 0.0213347i \(-0.993208\pi\)
0.481410 0.876496i \(-0.340125\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.107718 + 2.64356i −0.0101784 + 0.249793i
\(113\) 18.4343i 1.73415i −0.498176 0.867076i \(-0.665997\pi\)
0.498176 0.867076i \(-0.334003\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.21130 1.27670i 0.205314 0.118538i
\(117\) 0 0
\(118\) 3.35478i 0.308832i
\(119\) 10.1989 + 6.45578i 0.934931 + 0.591801i
\(120\) 0 0
\(121\) −2.72599 + 4.72156i −0.247817 + 0.429232i
\(122\) 3.95787 + 6.85523i 0.358329 + 0.620643i
\(123\) 0 0
\(124\) −1.89659 1.09500i −0.170319 0.0983337i
\(125\) 0 0
\(126\) 0 0
\(127\) −4.04880 −0.359273 −0.179637 0.983733i \(-0.557492\pi\)
−0.179637 + 0.983733i \(0.557492\pi\)
\(128\) 0.866025 + 0.500000i 0.0765466 + 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) −7.14129 + 12.3691i −0.623937 + 1.08069i 0.364808 + 0.931083i \(0.381134\pi\)
−0.988745 + 0.149608i \(0.952199\pi\)
\(132\) 0 0
\(133\) 1.49960 + 0.0611049i 0.130032 + 0.00529846i
\(134\) 4.00285i 0.345794i
\(135\) 0 0
\(136\) 3.95097 2.28109i 0.338792 0.195602i
\(137\) 4.52794 2.61421i 0.386848 0.223347i −0.293945 0.955822i \(-0.594968\pi\)
0.680794 + 0.732475i \(0.261635\pi\)
\(138\) 0 0
\(139\) 19.4726i 1.65164i −0.563933 0.825820i \(-0.690712\pi\)
0.563933 0.825820i \(-0.309288\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.01360 + 1.75561i −0.0850596 + 0.147328i
\(143\) −5.47280 9.47917i −0.457659 0.792688i
\(144\) 0 0
\(145\) 0 0
\(146\) 8.21601 0.679962
\(147\) 0 0
\(148\) −9.26709 −0.761750
\(149\) −4.79814 2.77021i −0.393079 0.226944i 0.290414 0.956901i \(-0.406207\pi\)
−0.683493 + 0.729957i \(0.739540\pi\)
\(150\) 0 0
\(151\) 4.23984 + 7.34362i 0.345033 + 0.597615i 0.985360 0.170488i \(-0.0545343\pi\)
−0.640327 + 0.768103i \(0.721201\pi\)
\(152\) 0.283634 0.491268i 0.0230057 0.0398471i
\(153\) 0 0
\(154\) 5.51934 2.89362i 0.444761 0.233174i
\(155\) 0 0
\(156\) 0 0
\(157\) 0.970763 0.560470i 0.0774753 0.0447304i −0.460762 0.887524i \(-0.652424\pi\)
0.538237 + 0.842793i \(0.319090\pi\)
\(158\) −7.15704 + 4.13212i −0.569384 + 0.328734i
\(159\) 0 0
\(160\) 0 0
\(161\) −15.4026 0.627617i −1.21390 0.0494631i
\(162\) 0 0
\(163\) −3.35749 + 5.81534i −0.262979 + 0.455493i −0.967032 0.254654i \(-0.918038\pi\)
0.704053 + 0.710147i \(0.251372\pi\)
\(164\) −4.34226 7.52101i −0.339073 0.587292i
\(165\) 0 0
\(166\) −0.148642 0.0858183i −0.0115368 0.00666079i
\(167\) −2.80110 −0.216756 −0.108378 0.994110i \(-0.534566\pi\)
−0.108378 + 0.994110i \(0.534566\pi\)
\(168\) 0 0
\(169\) −8.59442 −0.661109
\(170\) 0 0
\(171\) 0 0
\(172\) 3.28848 + 5.69581i 0.250744 + 0.434301i
\(173\) −6.90018 + 11.9515i −0.524611 + 0.908653i 0.474978 + 0.879998i \(0.342456\pi\)
−0.999589 + 0.0286558i \(0.990877\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.35542i 0.177547i
\(177\) 0 0
\(178\) −4.72742 + 2.72938i −0.354335 + 0.204575i
\(179\) −2.31980 + 1.33933i −0.173390 + 0.100107i −0.584183 0.811622i \(-0.698585\pi\)
0.410794 + 0.911728i \(0.365252\pi\)
\(180\) 0 0
\(181\) 6.88082i 0.511447i 0.966750 + 0.255724i \(0.0823137\pi\)
−0.966750 + 0.255724i \(0.917686\pi\)
\(182\) 0.500563 12.2846i 0.0371042 0.910592i
\(183\) 0 0
\(184\) −2.91324 + 5.04588i −0.214767 + 0.371987i
\(185\) 0 0
\(186\) 0 0
\(187\) −9.30619 5.37293i −0.680536 0.392908i
\(188\) −6.31233 −0.460374
\(189\) 0 0
\(190\) 0 0
\(191\) −8.65356 4.99614i −0.626150 0.361508i 0.153110 0.988209i \(-0.451071\pi\)
−0.779260 + 0.626701i \(0.784405\pi\)
\(192\) 0 0
\(193\) 12.5643 + 21.7620i 0.904398 + 1.56646i 0.821723 + 0.569887i \(0.193013\pi\)
0.0826753 + 0.996577i \(0.473654\pi\)
\(194\) 5.42821 9.40193i 0.389723 0.675019i
\(195\) 0 0
\(196\) 6.97679 + 0.569517i 0.498342 + 0.0406798i
\(197\) 20.0811i 1.43072i 0.698757 + 0.715359i \(0.253737\pi\)
−0.698757 + 0.715359i \(0.746263\pi\)
\(198\) 0 0
\(199\) −10.3028 + 5.94834i −0.730348 + 0.421667i −0.818550 0.574436i \(-0.805221\pi\)
0.0882014 + 0.996103i \(0.471888\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 11.4965i 0.808894i
\(203\) −3.13683 5.98323i −0.220162 0.419941i
\(204\) 0 0
\(205\) 0 0
\(206\) −9.68690 16.7782i −0.674918 1.16899i
\(207\) 0 0
\(208\) −4.02440 2.32349i −0.279042 0.161105i
\(209\) −1.33615 −0.0924237
\(210\) 0 0
\(211\) −6.78640 −0.467195 −0.233597 0.972333i \(-0.575050\pi\)
−0.233597 + 0.972333i \(0.575050\pi\)
\(212\) −10.5228 6.07533i −0.722708 0.417256i
\(213\) 0 0
\(214\) 5.36029 + 9.28430i 0.366422 + 0.634662i
\(215\) 0 0
\(216\) 0 0
\(217\) −3.09899 + 4.89580i −0.210373 + 0.332348i
\(218\) 10.8237i 0.733075i
\(219\) 0 0
\(220\) 0 0
\(221\) −18.3601 + 10.6002i −1.23503 + 0.713045i
\(222\) 0 0
\(223\) 28.7684i 1.92648i 0.268646 + 0.963239i \(0.413424\pi\)
−0.268646 + 0.963239i \(0.586576\pi\)
\(224\) 1.41507 2.23553i 0.0945480 0.149368i
\(225\) 0 0
\(226\) −9.21714 + 15.9646i −0.613115 + 1.06195i
\(227\) −1.05185 1.82186i −0.0698140 0.120921i 0.829005 0.559241i \(-0.188907\pi\)
−0.898819 + 0.438319i \(0.855574\pi\)
\(228\) 0 0
\(229\) −14.0269 8.09841i −0.926920 0.535158i −0.0410842 0.999156i \(-0.513081\pi\)
−0.885836 + 0.463998i \(0.846415\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.55339 −0.167639
\(233\) 6.93805 + 4.00569i 0.454527 + 0.262421i 0.709740 0.704464i \(-0.248812\pi\)
−0.255213 + 0.966885i \(0.582146\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.67739 2.90532i 0.109189 0.189120i
\(237\) 0 0
\(238\) −5.60460 10.6903i −0.363293 0.692950i
\(239\) 15.6233i 1.01059i 0.862947 + 0.505294i \(0.168616\pi\)
−0.862947 + 0.505294i \(0.831384\pi\)
\(240\) 0 0
\(241\) −3.54491 + 2.04665i −0.228348 + 0.131837i −0.609809 0.792548i \(-0.708754\pi\)
0.381462 + 0.924385i \(0.375421\pi\)
\(242\) 4.72156 2.72599i 0.303513 0.175233i
\(243\) 0 0
\(244\) 7.91574i 0.506753i
\(245\) 0 0
\(246\) 0 0
\(247\) −1.31804 + 2.28291i −0.0838648 + 0.145258i
\(248\) 1.09500 + 1.89659i 0.0695324 + 0.120434i
\(249\) 0 0
\(250\) 0 0
\(251\) 19.9413 1.25869 0.629343 0.777128i \(-0.283324\pi\)
0.629343 + 0.777128i \(0.283324\pi\)
\(252\) 0 0
\(253\) 13.7238 0.862810
\(254\) 3.50637 + 2.02440i 0.220009 + 0.127022i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.0312500 0.0541266i
\(257\) 10.1722 17.6188i 0.634527 1.09903i −0.352088 0.935967i \(-0.614528\pi\)
0.986615 0.163067i \(-0.0521385\pi\)
\(258\) 0 0
\(259\) −0.998233 + 24.4981i −0.0620272 + 1.52224i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.3691 7.14129i 0.764164 0.441190i
\(263\) 3.13156 1.80801i 0.193100 0.111487i −0.400333 0.916370i \(-0.631105\pi\)
0.593433 + 0.804883i \(0.297772\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.26814 0.802720i −0.0777548 0.0492179i
\(267\) 0 0
\(268\) 2.00143 3.46657i 0.122257 0.211755i
\(269\) 10.6172 + 18.3896i 0.647345 + 1.12123i 0.983755 + 0.179518i \(0.0574539\pi\)
−0.336410 + 0.941716i \(0.609213\pi\)
\(270\) 0 0
\(271\) −18.9957 10.9672i −1.15390 0.666207i −0.204069 0.978956i \(-0.565417\pi\)
−0.949836 + 0.312749i \(0.898750\pi\)
\(272\) −4.56218 −0.276623
\(273\) 0 0
\(274\) −5.22842 −0.315860
\(275\) 0 0
\(276\) 0 0
\(277\) −4.96452 8.59880i −0.298289 0.516652i 0.677456 0.735564i \(-0.263083\pi\)
−0.975745 + 0.218912i \(0.929749\pi\)
\(278\) −9.73628 + 16.8637i −0.583943 + 1.01142i
\(279\) 0 0
\(280\) 0 0
\(281\) 11.0696i 0.660358i 0.943918 + 0.330179i \(0.107109\pi\)
−0.943918 + 0.330179i \(0.892891\pi\)
\(282\) 0 0
\(283\) −19.6392 + 11.3387i −1.16743 + 0.674014i −0.953072 0.302742i \(-0.902098\pi\)
−0.214354 + 0.976756i \(0.568765\pi\)
\(284\) 1.75561 1.01360i 0.104176 0.0601462i
\(285\) 0 0
\(286\) 10.9456i 0.647227i
\(287\) −20.3500 + 10.6689i −1.20122 + 0.629763i
\(288\) 0 0
\(289\) −1.90675 + 3.30259i −0.112162 + 0.194270i
\(290\) 0 0
\(291\) 0 0
\(292\) −7.11528 4.10801i −0.416390 0.240403i
\(293\) 12.3248 0.720020 0.360010 0.932949i \(-0.382773\pi\)
0.360010 + 0.932949i \(0.382773\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.02554 + 4.63355i 0.466475 + 0.269319i
\(297\) 0 0
\(298\) 2.77021 + 4.79814i 0.160474 + 0.277949i
\(299\) 13.5378 23.4481i 0.782909 1.35604i
\(300\) 0 0
\(301\) 15.4114 8.07974i 0.888300 0.465708i
\(302\) 8.47968i 0.487951i
\(303\) 0 0
\(304\) −0.491268 + 0.283634i −0.0281761 + 0.0162675i
\(305\) 0 0
\(306\) 0 0
\(307\) 8.06274i 0.460165i 0.973171 + 0.230082i \(0.0738996\pi\)
−0.973171 + 0.230082i \(0.926100\pi\)
\(308\) −6.22670 0.253721i −0.354799 0.0144571i
\(309\) 0 0
\(310\) 0 0
\(311\) 13.2215 + 22.9003i 0.749721 + 1.29855i 0.947956 + 0.318400i \(0.103146\pi\)
−0.198236 + 0.980154i \(0.563521\pi\)
\(312\) 0 0
\(313\) −14.3180 8.26650i −0.809301 0.467250i 0.0374122 0.999300i \(-0.488089\pi\)
−0.846713 + 0.532050i \(0.821422\pi\)
\(314\) −1.12094 −0.0632583
\(315\) 0 0
\(316\) 8.26424 0.464900
\(317\) 10.6181 + 6.13038i 0.596374 + 0.344317i 0.767614 0.640913i \(-0.221444\pi\)
−0.171240 + 0.985229i \(0.554777\pi\)
\(318\) 0 0
\(319\) 3.00716 + 5.20856i 0.168369 + 0.291623i
\(320\) 0 0
\(321\) 0 0
\(322\) 13.0253 + 8.24485i 0.725870 + 0.459468i
\(323\) 2.58798i 0.143999i
\(324\) 0 0
\(325\) 0 0
\(326\) 5.81534 3.35749i 0.322082 0.185954i
\(327\) 0 0
\(328\) 8.68451i 0.479522i
\(329\) −0.679951 + 16.6870i −0.0374869 + 0.919984i
\(330\) 0 0
\(331\) −17.1942 + 29.7812i −0.945077 + 1.63692i −0.189479 + 0.981885i \(0.560680\pi\)
−0.755598 + 0.655036i \(0.772654\pi\)
\(332\) 0.0858183 + 0.148642i 0.00470989 + 0.00815777i
\(333\) 0 0
\(334\) 2.42583 + 1.40055i 0.132735 + 0.0766348i
\(335\) 0 0
\(336\) 0 0
\(337\) 23.9536 1.30484 0.652418 0.757860i \(-0.273755\pi\)
0.652418 + 0.757860i \(0.273755\pi\)
\(338\) 7.44298 + 4.29721i 0.404845 + 0.233737i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.57918 4.46727i 0.139671 0.241916i
\(342\) 0 0
\(343\) 2.25708 18.3822i 0.121871 0.992546i
\(344\) 6.57695i 0.354606i
\(345\) 0 0
\(346\) 11.9515 6.90018i 0.642515 0.370956i
\(347\) −24.8552 + 14.3501i −1.33429 + 0.770355i −0.985955 0.167013i \(-0.946588\pi\)
−0.348340 + 0.937368i \(0.613254\pi\)
\(348\) 0 0
\(349\) 3.57176i 0.191192i −0.995420 0.0955960i \(-0.969524\pi\)
0.995420 0.0955960i \(-0.0304757\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.17771 + 2.03986i −0.0627722 + 0.108725i
\(353\) −6.26984 10.8597i −0.333710 0.578003i 0.649526 0.760339i \(-0.274967\pi\)
−0.983236 + 0.182337i \(0.941634\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.45875 0.289313
\(357\) 0 0
\(358\) 2.67867 0.141572
\(359\) 13.6940 + 7.90624i 0.722742 + 0.417275i 0.815761 0.578389i \(-0.196318\pi\)
−0.0930190 + 0.995664i \(0.529652\pi\)
\(360\) 0 0
\(361\) −9.33910 16.1758i −0.491532 0.851358i
\(362\) 3.44041 5.95896i 0.180824 0.313196i
\(363\) 0 0
\(364\) −6.57578 + 10.3885i −0.344664 + 0.544503i
\(365\) 0 0
\(366\) 0 0
\(367\) 30.8855 17.8317i 1.61221 0.930809i 0.623351 0.781942i \(-0.285771\pi\)
0.988857 0.148867i \(-0.0475627\pi\)
\(368\) 5.04588 2.91324i 0.263035 0.151863i
\(369\) 0 0
\(370\) 0 0
\(371\) −17.1940 + 27.1632i −0.892667 + 1.41024i
\(372\) 0 0
\(373\) 11.5306 19.9717i 0.597034 1.03409i −0.396222 0.918155i \(-0.629679\pi\)
0.993256 0.115939i \(-0.0369876\pi\)
\(374\) 5.37293 + 9.30619i 0.277828 + 0.481212i
\(375\) 0 0
\(376\) 5.46663 + 3.15616i 0.281920 + 0.162767i
\(377\) 11.8656 0.611108
\(378\) 0 0
\(379\) 8.20110 0.421262 0.210631 0.977566i \(-0.432448\pi\)
0.210631 + 0.977566i \(0.432448\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.99614 + 8.65356i 0.255625 + 0.442755i
\(383\) 2.31637 4.01207i 0.118361 0.205007i −0.800757 0.598989i \(-0.795569\pi\)
0.919118 + 0.393982i \(0.128903\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 25.1286i 1.27901i
\(387\) 0 0
\(388\) −9.40193 + 5.42821i −0.477311 + 0.275575i
\(389\) −29.0194 + 16.7544i −1.47134 + 0.849480i −0.999482 0.0321938i \(-0.989751\pi\)
−0.471860 + 0.881673i \(0.656417\pi\)
\(390\) 0 0
\(391\) 26.5815i 1.34428i
\(392\) −5.75732 3.98161i −0.290789 0.201102i
\(393\) 0 0
\(394\) 10.0405 17.3907i 0.505835 0.876132i
\(395\) 0 0
\(396\) 0 0
\(397\) −16.0748 9.28081i −0.806772 0.465790i 0.0390613 0.999237i \(-0.487563\pi\)
−0.845834 + 0.533447i \(0.820897\pi\)
\(398\) 11.8967 0.596327
\(399\) 0 0
\(400\) 0 0
\(401\) −12.1377 7.00770i −0.606128 0.349948i 0.165321 0.986240i \(-0.447134\pi\)
−0.771448 + 0.636292i \(0.780467\pi\)
\(402\) 0 0
\(403\) −5.08843 8.81342i −0.253473 0.439028i
\(404\) −5.74827 + 9.95630i −0.285987 + 0.495345i
\(405\) 0 0
\(406\) −0.275047 + 6.75005i −0.0136503 + 0.334999i
\(407\) 21.8279i 1.08197i
\(408\) 0 0
\(409\) 20.6162 11.9028i 1.01941 0.588555i 0.105474 0.994422i \(-0.466364\pi\)
0.913932 + 0.405868i \(0.133031\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 19.3738i 0.954478i
\(413\) −7.49970 4.74723i −0.369036 0.233596i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.32349 + 4.02440i 0.113918 + 0.197313i
\(417\) 0 0
\(418\) 1.15714 + 0.668077i 0.0565977 + 0.0326767i
\(419\) 18.1374 0.886068 0.443034 0.896505i \(-0.353902\pi\)
0.443034 + 0.896505i \(0.353902\pi\)
\(420\) 0 0
\(421\) 7.98092 0.388966 0.194483 0.980906i \(-0.437697\pi\)
0.194483 + 0.980906i \(0.437697\pi\)
\(422\) 5.87719 + 3.39320i 0.286097 + 0.165178i
\(423\) 0 0
\(424\) 6.07533 + 10.5228i 0.295044 + 0.511032i
\(425\) 0 0
\(426\) 0 0
\(427\) −20.9257 0.852667i −1.01267 0.0412635i
\(428\) 10.7206i 0.518199i
\(429\) 0 0
\(430\) 0 0
\(431\) 27.1353 15.6666i 1.30706 0.754632i 0.325456 0.945557i \(-0.394482\pi\)
0.981605 + 0.190925i \(0.0611488\pi\)
\(432\) 0 0
\(433\) 5.21564i 0.250648i −0.992116 0.125324i \(-0.960003\pi\)
0.992116 0.125324i \(-0.0399970\pi\)
\(434\) 5.13170 2.69039i 0.246329 0.129143i
\(435\) 0 0
\(436\) 5.41186 9.37362i 0.259181 0.448915i
\(437\) −1.65259 2.86236i −0.0790539 0.136925i
\(438\) 0 0
\(439\) −8.91887 5.14931i −0.425675 0.245763i 0.271828 0.962346i \(-0.412372\pi\)
−0.697502 + 0.716583i \(0.745705\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 21.2004 1.00840
\(443\) 35.9968 + 20.7828i 1.71026 + 0.987419i 0.934193 + 0.356768i \(0.116121\pi\)
0.776067 + 0.630651i \(0.217212\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.3842 24.9142i 0.681113 1.17972i
\(447\) 0 0
\(448\) −2.34325 + 1.22849i −0.110708 + 0.0580408i
\(449\) 23.7838i 1.12242i 0.827672 + 0.561212i \(0.189665\pi\)
−0.827672 + 0.561212i \(0.810335\pi\)
\(450\) 0 0
\(451\) 17.7152 10.2279i 0.834174 0.481611i
\(452\) 15.9646 9.21714i 0.750910 0.433538i
\(453\) 0 0
\(454\) 2.10371i 0.0987319i
\(455\) 0 0
\(456\) 0 0
\(457\) −9.38745 + 16.2595i −0.439126 + 0.760589i −0.997622 0.0689182i \(-0.978045\pi\)
0.558496 + 0.829507i \(0.311379\pi\)
\(458\) 8.09841 + 14.0269i 0.378414 + 0.655432i
\(459\) 0 0
\(460\) 0 0
\(461\) 17.4662 0.813484 0.406742 0.913543i \(-0.366665\pi\)
0.406742 + 0.913543i \(0.366665\pi\)
\(462\) 0 0
\(463\) 2.99345 0.139117 0.0695586 0.997578i \(-0.477841\pi\)
0.0695586 + 0.997578i \(0.477841\pi\)
\(464\) 2.21130 + 1.27670i 0.102657 + 0.0592692i
\(465\) 0 0
\(466\) −4.00569 6.93805i −0.185560 0.321399i
\(467\) −12.7050 + 22.0058i −0.587920 + 1.01831i 0.406585 + 0.913613i \(0.366720\pi\)
−0.994505 + 0.104694i \(0.966614\pi\)
\(468\) 0 0
\(469\) −8.94849 5.66430i −0.413203 0.261553i
\(470\) 0 0
\(471\) 0 0
\(472\) −2.90532 + 1.67739i −0.133728 + 0.0772081i
\(473\) −13.4160 + 7.74575i −0.616870 + 0.356150i
\(474\) 0 0
\(475\) 0 0
\(476\) −0.491429 + 12.0604i −0.0225246 + 0.552787i
\(477\) 0 0
\(478\) 7.81165 13.5302i 0.357297 0.618856i
\(479\) −11.8516 20.5276i −0.541514 0.937929i −0.998817 0.0486188i \(-0.984518\pi\)
0.457304 0.889311i \(-0.348815\pi\)
\(480\) 0 0
\(481\) −37.2945 21.5320i −1.70048 0.981774i
\(482\) 4.09331 0.186445
\(483\) 0 0
\(484\) −5.45198 −0.247817
\(485\) 0 0
\(486\) 0 0
\(487\) −5.10772 8.84683i −0.231453 0.400888i 0.726783 0.686867i \(-0.241015\pi\)
−0.958236 + 0.285979i \(0.907681\pi\)
\(488\) −3.95787 + 6.85523i −0.179164 + 0.310322i
\(489\) 0 0
\(490\) 0 0
\(491\) 2.25910i 0.101952i 0.998700 + 0.0509758i \(0.0162331\pi\)
−0.998700 + 0.0509758i \(0.983767\pi\)
\(492\) 0 0
\(493\) 10.0884 5.82452i 0.454357 0.262323i
\(494\) 2.28291 1.31804i 0.102713 0.0593014i
\(495\) 0 0
\(496\) 2.18999i 0.0983337i
\(497\) −2.49041 4.75024i −0.111710 0.213077i
\(498\) 0 0
\(499\) −17.6811 + 30.6246i −0.791517 + 1.37095i 0.133511 + 0.991047i \(0.457375\pi\)
−0.925028 + 0.379900i \(0.875958\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −17.2697 9.97066i −0.770784 0.445012i
\(503\) 30.7297 1.37017 0.685084 0.728464i \(-0.259766\pi\)
0.685084 + 0.728464i \(0.259766\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −11.8852 6.86191i −0.528361 0.305049i
\(507\) 0 0
\(508\) −2.02440 3.50637i −0.0898183 0.155570i
\(509\) 1.03925 1.80003i 0.0460637 0.0797847i −0.842074 0.539362i \(-0.818666\pi\)
0.888138 + 0.459577i \(0.151999\pi\)
\(510\) 0 0
\(511\) −11.6262 + 18.3671i −0.514313 + 0.812514i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −17.6188 + 10.1722i −0.777134 + 0.448679i
\(515\) 0 0
\(516\) 0 0
\(517\) 14.8682i 0.653903i
\(518\) 13.1135 20.7169i 0.576176 0.910246i
\(519\) 0 0
\(520\) 0 0
\(521\) 4.86182 + 8.42093i 0.213000 + 0.368927i 0.952652 0.304062i \(-0.0983431\pi\)
−0.739652 + 0.672990i \(0.765010\pi\)
\(522\) 0 0
\(523\) −1.20378 0.695001i −0.0526375 0.0303903i 0.473450 0.880821i \(-0.343008\pi\)
−0.526088 + 0.850430i \(0.676342\pi\)
\(524\) −14.2826 −0.623937
\(525\) 0 0
\(526\) −3.61602 −0.157666
\(527\) −8.65259 4.99558i −0.376913 0.217611i
\(528\) 0 0
\(529\) 5.47394 + 9.48114i 0.237997 + 0.412224i
\(530\) 0 0
\(531\) 0 0
\(532\) 0.696883 + 1.32925i 0.0302137 + 0.0576302i
\(533\) 40.3568i 1.74805i
\(534\) 0 0
\(535\) 0 0
\(536\) −3.46657 + 2.00143i −0.149733 + 0.0864485i
\(537\) 0 0
\(538\) 21.2345i 0.915484i
\(539\) −1.34145 + 16.4333i −0.0577805 + 0.707832i
\(540\) 0 0
\(541\) −22.5510 + 39.0594i −0.969541 + 1.67930i −0.272658 + 0.962111i \(0.587903\pi\)
−0.696884 + 0.717184i \(0.745431\pi\)
\(542\) 10.9672 + 18.9957i 0.471080 + 0.815934i
\(543\) 0 0
\(544\) 3.95097 + 2.28109i 0.169396 + 0.0978010i
\(545\) 0 0
\(546\) 0 0
\(547\) 11.1372 0.476193 0.238096 0.971242i \(-0.423477\pi\)
0.238096 + 0.971242i \(0.423477\pi\)
\(548\) 4.52794 + 2.61421i 0.193424 + 0.111673i
\(549\) 0 0
\(550\) 0 0
\(551\) 0.724228 1.25440i 0.0308532 0.0534393i
\(552\) 0 0
\(553\) 0.890207 21.8470i 0.0378555 0.929029i
\(554\) 9.92903i 0.421844i
\(555\) 0 0
\(556\) 16.8637 9.73628i 0.715181 0.412910i
\(557\) −23.7662 + 13.7214i −1.00701 + 0.581395i −0.910314 0.413919i \(-0.864160\pi\)
−0.0966925 + 0.995314i \(0.530826\pi\)
\(558\) 0 0
\(559\) 30.5630i 1.29268i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.53481 9.58656i 0.233472 0.404385i
\(563\) 16.6414 + 28.8238i 0.701352 + 1.21478i 0.967992 + 0.250981i \(0.0807531\pi\)
−0.266640 + 0.963796i \(0.585914\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22.6773 0.953200
\(567\) 0 0
\(568\) −2.02720 −0.0850596
\(569\) −24.6215 14.2152i −1.03219 0.595932i −0.114575 0.993415i \(-0.536551\pi\)
−0.917610 + 0.397482i \(0.869884\pi\)
\(570\) 0 0
\(571\) −14.0784 24.3845i −0.589162 1.02046i −0.994343 0.106221i \(-0.966125\pi\)
0.405181 0.914237i \(-0.367209\pi\)
\(572\) 5.47280 9.47917i 0.228829 0.396344i
\(573\) 0 0
\(574\) 22.9580 + 0.935478i 0.958249 + 0.0390461i
\(575\) 0 0
\(576\) 0 0
\(577\) 1.77604 1.02540i 0.0739377 0.0426879i −0.462575 0.886580i \(-0.653075\pi\)
0.536513 + 0.843892i \(0.319741\pi\)
\(578\) 3.30259 1.90675i 0.137370 0.0793103i
\(579\) 0 0
\(580\) 0 0
\(581\) 0.402187 0.210854i 0.0166855 0.00874771i
\(582\) 0 0
\(583\) 14.3100 24.7856i 0.592659 1.02652i
\(584\) 4.10801 + 7.11528i 0.169991 + 0.294432i
\(585\) 0 0
\(586\) −10.6735 6.16238i −0.440920 0.254565i
\(587\) 36.2336 1.49552 0.747761 0.663968i \(-0.231129\pi\)
0.747761 + 0.663968i \(0.231129\pi\)
\(588\) 0 0
\(589\) −1.24231 −0.0511886
\(590\) 0 0
\(591\) 0 0
\(592\) −4.63355 8.02554i −0.190438 0.329848i
\(593\) −17.3920 + 30.1239i −0.714205 + 1.23704i 0.249061 + 0.968488i \(0.419878\pi\)
−0.963265 + 0.268551i \(0.913455\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 5.54041i 0.226944i
\(597\) 0 0
\(598\) −23.4481 + 13.5378i −0.958864 + 0.553601i
\(599\) 31.6459 18.2708i 1.29302 0.746524i 0.313829 0.949479i \(-0.398388\pi\)
0.979188 + 0.202956i \(0.0650547\pi\)
\(600\) 0 0
\(601\) 45.5137i 1.85654i −0.371902 0.928272i \(-0.621294\pi\)
0.371902 0.928272i \(-0.378706\pi\)
\(602\) −17.3866 0.708456i −0.708623 0.0288745i
\(603\) 0 0
\(604\) −4.23984 + 7.34362i −0.172517 + 0.298807i
\(605\) 0 0
\(606\) 0 0
\(607\) −31.8414 18.3836i −1.29240 0.746169i −0.313323 0.949647i \(-0.601442\pi\)
−0.979080 + 0.203478i \(0.934776\pi\)
\(608\) 0.567267 0.0230057
\(609\) 0 0
\(610\) 0 0
\(611\) −25.4033 14.6666i −1.02771 0.593348i
\(612\) 0 0
\(613\) −16.3557 28.3288i −0.660599 1.14419i −0.980459 0.196726i \(-0.936969\pi\)
0.319860 0.947465i \(-0.396364\pi\)
\(614\) 4.03137 6.98254i 0.162693 0.281792i
\(615\) 0 0
\(616\) 5.26562 + 3.33308i 0.212158 + 0.134294i
\(617\) 20.5530i 0.827435i −0.910405 0.413717i \(-0.864230\pi\)
0.910405 0.413717i \(-0.135770\pi\)
\(618\) 0 0
\(619\) 14.9593 8.63675i 0.601265 0.347140i −0.168274 0.985740i \(-0.553819\pi\)
0.769539 + 0.638600i \(0.220486\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 26.4429i 1.06027i
\(623\) 0.588006 14.4305i 0.0235580 0.578147i
\(624\) 0 0
\(625\) 0 0
\(626\) 8.26650 + 14.3180i 0.330396 + 0.572262i
\(627\) 0 0
\(628\) 0.970763 + 0.560470i 0.0387377 + 0.0223652i
\(629\) −42.2782 −1.68574
\(630\) 0 0
\(631\) 32.7501 1.30376 0.651880 0.758322i \(-0.273981\pi\)
0.651880 + 0.758322i \(0.273981\pi\)
\(632\) −7.15704 4.13212i −0.284692 0.164367i
\(633\) 0 0
\(634\) −6.13038 10.6181i −0.243469 0.421700i
\(635\) 0 0
\(636\) 0 0
\(637\) 26.7542 + 18.5025i 1.06004 + 0.733095i
\(638\) 6.01432i 0.238109i
\(639\) 0 0
\(640\) 0 0
\(641\) −18.9300 + 10.9292i −0.747688 + 0.431678i −0.824858 0.565340i \(-0.808745\pi\)
0.0771698 + 0.997018i \(0.475412\pi\)
\(642\) 0 0
\(643\) 4.13643i 0.163125i 0.996668 + 0.0815624i \(0.0259910\pi\)
−0.996668 + 0.0815624i \(0.974009\pi\)
\(644\) −7.15779 13.6529i −0.282056 0.537999i
\(645\) 0 0
\(646\) 1.29399 2.24125i 0.0509113 0.0881809i
\(647\) 15.4046 + 26.6816i 0.605618 + 1.04896i 0.991953 + 0.126603i \(0.0404075\pi\)
−0.386335 + 0.922358i \(0.626259\pi\)
\(648\) 0 0
\(649\) 6.84327 + 3.95096i 0.268622 + 0.155089i
\(650\) 0 0
\(651\) 0 0
\(652\) −6.71498 −0.262979
\(653\) −5.97747 3.45110i −0.233917 0.135052i 0.378461 0.925617i \(-0.376453\pi\)
−0.612378 + 0.790565i \(0.709787\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.34226 7.52101i 0.169537 0.293646i
\(657\) 0 0
\(658\) 8.93235 14.1114i 0.348219 0.550119i
\(659\) 11.0895i 0.431985i −0.976395 0.215992i \(-0.930701\pi\)
0.976395 0.215992i \(-0.0692986\pi\)
\(660\) 0 0
\(661\) 13.7077 7.91416i 0.533169 0.307825i −0.209137 0.977886i \(-0.567065\pi\)
0.742306 + 0.670061i \(0.233732\pi\)
\(662\) 29.7812 17.1942i 1.15748 0.668270i
\(663\) 0 0
\(664\) 0.171637i 0.00666079i
\(665\) 0 0
\(666\) 0 0
\(667\) −7.43865 + 12.8841i −0.288026 + 0.498875i
\(668\) −1.40055 2.42583i −0.0541890 0.0938581i
\(669\) 0 0
\(670\) 0 0
\(671\) 18.6449 0.719779
\(672\) 0 0
\(673\) −27.8980 −1.07539 −0.537695 0.843139i \(-0.680705\pi\)
−0.537695 + 0.843139i \(0.680705\pi\)
\(674\) −20.7444 11.9768i −0.799045 0.461329i
\(675\) 0 0
\(676\) −4.29721 7.44298i −0.165277 0.286269i
\(677\) 15.5739 26.9749i 0.598555 1.03673i −0.394479 0.918905i \(-0.629075\pi\)
0.993035 0.117823i \(-0.0375916\pi\)
\(678\) 0 0
\(679\) 13.3370 + 25.4393i 0.511828 + 0.976269i
\(680\) 0 0
\(681\) 0 0
\(682\) −4.46727 + 2.57918i −0.171061 + 0.0987620i
\(683\) 21.8168 12.5960i 0.834798 0.481971i −0.0206948 0.999786i \(-0.506588\pi\)
0.855493 + 0.517815i \(0.173254\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −11.1458 + 14.7909i −0.425548 + 0.564720i
\(687\) 0 0
\(688\) −3.28848 + 5.69581i −0.125372 + 0.217151i
\(689\) −28.2319 48.8992i −1.07555 1.86291i
\(690\) 0 0
\(691\) −27.2549 15.7356i −1.03683 0.598612i −0.117894 0.993026i \(-0.537614\pi\)
−0.918933 + 0.394414i \(0.870948\pi\)
\(692\) −13.8004 −0.524611
\(693\) 0 0
\(694\) 28.7003 1.08945
\(695\) 0 0
\(696\) 0 0
\(697\) −19.8102 34.3122i −0.750363 1.29967i
\(698\) −1.78588 + 3.09324i −0.0675966 + 0.117081i
\(699\) 0 0
\(700\) 0 0
\(701\) 5.19395i 0.196173i 0.995178 + 0.0980864i \(0.0312721\pi\)
−0.995178 + 0.0980864i \(0.968728\pi\)
\(702\) 0 0
\(703\) −4.55262 + 2.62846i −0.171705 + 0.0991342i
\(704\) 2.03986 1.17771i 0.0768800 0.0443867i
\(705\) 0 0
\(706\) 12.5397i 0.471937i
\(707\) 25.7009 + 16.2684i 0.966581 + 0.611835i
\(708\) 0 0
\(709\) −17.9440 + 31.0800i −0.673903 + 1.16723i 0.302885 + 0.953027i \(0.402050\pi\)
−0.976788 + 0.214207i \(0.931283\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.72742 2.72938i −0.177167 0.102288i
\(713\) 12.7600 0.477864
\(714\) 0 0
\(715\) 0 0
\(716\) −2.31980 1.33933i −0.0866948 0.0500533i
\(717\) 0 0
\(718\) −7.90624 13.6940i −0.295058 0.511056i
\(719\) −13.0548 + 22.6115i −0.486861 + 0.843268i −0.999886 0.0151058i \(-0.995191\pi\)
0.513025 + 0.858374i \(0.328525\pi\)
\(720\) 0 0
\(721\) 51.2157 + 2.08691i 1.90737 + 0.0777204i
\(722\) 18.6782i 0.695131i
\(723\) 0 0
\(724\) −5.95896 + 3.44041i −0.221463 + 0.127862i
\(725\) 0 0
\(726\) 0 0
\(727\) 19.0193i 0.705386i −0.935739 0.352693i \(-0.885266\pi\)
0.935739 0.352693i \(-0.114734\pi\)
\(728\) 10.8890 5.70878i 0.403574 0.211581i
\(729\) 0 0
\(730\) 0 0
\(731\) 15.0026 + 25.9853i 0.554892 + 0.961101i
\(732\) 0 0
\(733\) 7.65553 + 4.41992i 0.282763 + 0.163254i 0.634674 0.772780i \(-0.281134\pi\)
−0.351910 + 0.936034i \(0.614468\pi\)
\(734\) −35.6635 −1.31636
\(735\) 0 0
\(736\) −5.82648 −0.214767
\(737\) 8.16524 + 4.71421i 0.300771 + 0.173650i
\(738\) 0 0
\(739\) 8.20546 + 14.2123i 0.301843 + 0.522807i 0.976553 0.215276i \(-0.0690650\pi\)
−0.674711 + 0.738082i \(0.735732\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 28.4720 14.9270i 1.04524 0.547987i
\(743\) 47.4829i 1.74198i −0.491302 0.870989i \(-0.663479\pi\)
0.491302 0.870989i \(-0.336521\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −19.9717 + 11.5306i −0.731214 + 0.422167i
\(747\) 0 0
\(748\) 10.7459i 0.392908i
\(749\) −28.3405 1.15480i −1.03554 0.0421955i
\(750\) 0 0
\(751\) 17.4048 30.1459i 0.635109 1.10004i −0.351383 0.936232i \(-0.614289\pi\)
0.986492 0.163809i \(-0.0523781\pi\)
\(752\) −3.15616 5.46663i −0.115093 0.199348i
\(753\) 0 0
\(754\) −10.2759 5.93279i −0.374226 0.216059i
\(755\) 0 0
\(756\) 0 0
\(757\) −9.68581 −0.352037 −0.176018 0.984387i \(-0.556322\pi\)
−0.176018 + 0.984387i \(0.556322\pi\)
\(758\) −7.10236 4.10055i −0.257969 0.148939i
\(759\) 0 0
\(760\) 0 0
\(761\) 22.2236 38.4924i 0.805605 1.39535i −0.110277 0.993901i \(-0.535174\pi\)
0.915882 0.401448i \(-0.131493\pi\)
\(762\) 0 0
\(763\) −24.1968 15.3163i −0.875982 0.554487i
\(764\) 9.99228i 0.361508i
\(765\) 0 0
\(766\) −4.01207 + 2.31637i −0.144962 + 0.0836939i
\(767\) 13.5010 7.79479i 0.487492 0.281454i
\(768\) 0 0
\(769\) 41.0290i 1.47954i −0.672858 0.739771i \(-0.734934\pi\)
0.672858 0.739771i \(-0.265066\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.5643 + 21.7620i −0.452199 + 0.783232i
\(773\) −17.4971 30.3059i −0.629327 1.09003i −0.987687 0.156443i \(-0.949997\pi\)
0.358360 0.933583i \(-0.383336\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.8564 0.389723
\(777\) 0 0
\(778\) 33.5087 1.20135
\(779\) −4.26642 2.46322i −0.152860 0.0882540i
\(780\) 0 0
\(781\) 2.38746 + 4.13521i 0.0854301 + 0.147969i
\(782\) −13.2907 + 23.0202i −0.475276 + 0.823201i
\(783\) 0 0
\(784\) 2.99518 + 6.32684i 0.106971 + 0.225959i
\(785\) 0 0
\(786\) 0 0
\(787\) 19.3139 11.1509i 0.688465 0.397485i −0.114572 0.993415i \(-0.536550\pi\)
0.803037 + 0.595930i \(0.203216\pi\)
\(788\) −17.3907 + 10.0405i −0.619519 + 0.357679i
\(789\) 0 0
\(790\) 0 0
\(791\) −22.6464 43.1961i −0.805212 1.53588i
\(792\) 0 0
\(793\) 18.3921 31.8561i 0.653124 1.13124i
\(794\) 9.28081 + 16.0748i 0.329363 + 0.570474i
\(795\) 0 0
\(796\) −10.3028 5.94834i −0.365174 0.210833i
\(797\) 49.1382 1.74057 0.870283 0.492553i \(-0.163936\pi\)
0.870283 + 0.492553i \(0.163936\pi\)
\(798\) 0 0
\(799\) −28.7980 −1.01880
\(800\) 0 0
\(801\) 0 0
\(802\) 7.00770 + 12.1377i 0.247451 + 0.428597i
\(803\) 9.67609 16.7595i 0.341462 0.591429i
\(804\) 0 0
\(805\) 0 0
\(806\) 10.1769i 0.358465i
\(807\) 0 0
\(808\) 9.95630 5.74827i 0.350262 0.202224i
\(809\) −32.5217 + 18.7764i −1.14340 + 0.660144i −0.947271 0.320434i \(-0.896171\pi\)
−0.196132 + 0.980578i \(0.562838\pi\)
\(810\) 0 0
\(811\) 32.3972i 1.13762i 0.822469 + 0.568810i \(0.192596\pi\)
−0.822469 + 0.568810i \(0.807404\pi\)
\(812\) 3.61322 5.70819i 0.126799 0.200318i
\(813\) 0 0
\(814\) −10.9140 + 18.9035i −0.382534 + 0.662569i
\(815\) 0 0
\(816\) 0 0
\(817\) 3.23104 + 1.86544i 0.113040 + 0.0652636i
\(818\) −23.8056 −0.832342
\(819\) 0 0
\(820\) 0 0
\(821\) 16.8756 + 9.74316i 0.588964 + 0.340039i 0.764688 0.644401i \(-0.222893\pi\)
−0.175724 + 0.984440i \(0.556227\pi\)
\(822\) 0 0
\(823\) −14.3974 24.9370i −0.501861 0.869250i −0.999998 0.00215079i \(-0.999315\pi\)
0.498136 0.867099i \(-0.334018\pi\)
\(824\) 9.68690 16.7782i 0.337459 0.584496i
\(825\) 0 0
\(826\) 4.12132 + 7.86107i 0.143399 + 0.273522i
\(827\) 32.2720i 1.12221i 0.827745 + 0.561104i \(0.189623\pi\)
−0.827745 + 0.561104i \(0.810377\pi\)
\(828\) 0 0
\(829\) 47.2675 27.2899i 1.64167 0.947817i 0.661426 0.750011i \(-0.269952\pi\)
0.980241 0.197806i \(-0.0633816\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.64698i 0.161105i
\(833\) 31.8294 + 2.59824i 1.10282 + 0.0900237i
\(834\) 0 0
\(835\) 0 0
\(836\) −0.668077 1.15714i −0.0231059 0.0400206i
\(837\) 0 0
\(838\) −15.7074 9.06868i −0.542604 0.313272i
\(839\) 47.8295 1.65126 0.825629 0.564213i \(-0.190820\pi\)
0.825629 + 0.564213i \(0.190820\pi\)
\(840\) 0 0
\(841\) 22.4802 0.775179
\(842\) −6.91168 3.99046i −0.238192 0.137520i
\(843\) 0 0
\(844\) −3.39320 5.87719i −0.116799 0.202301i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.587277 + 14.4126i −0.0201791 + 0.495224i
\(848\) 12.1507i 0.417256i
\(849\) 0 0
\(850\) 0 0
\(851\) 46.7606 26.9973i 1.60293 0.925455i
\(852\) 0 0
\(853\) 10.9274i 0.374148i −0.982346 0.187074i \(-0.940100\pi\)
0.982346 0.187074i \(-0.0599004\pi\)
\(854\) 17.6959 + 11.2013i 0.605540 + 0.383300i
\(855\) 0 0
\(856\) −5.36029 + 9.28430i −0.183211 + 0.317331i
\(857\) −11.1878 19.3778i −0.382167 0.661932i 0.609205 0.793013i \(-0.291489\pi\)
−0.991372 + 0.131081i \(0.958155\pi\)
\(858\) 0 0
\(859\) 28.2119 + 16.2881i 0.962577 + 0.555744i 0.896965 0.442101i \(-0.145767\pi\)
0.0656116 + 0.997845i \(0.479100\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −31.3331 −1.06721
\(863\) 36.2871 + 20.9504i 1.23523 + 0.713160i 0.968115 0.250506i \(-0.0805969\pi\)
0.267113 + 0.963665i \(0.413930\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.60782 + 4.51688i −0.0886174 + 0.153490i
\(867\) 0 0
\(868\) −5.78938 0.235902i −0.196504 0.00800703i
\(869\) 19.4658i 0.660331i
\(870\) 0 0
\(871\) 16.1091 9.30059i 0.545836 0.315138i
\(872\) −9.37362 + 5.41186i −0.317431 + 0.183269i
\(873\) 0 0
\(874\) 3.30517i 0.111799i
\(875\) 0 0
\(876\) 0 0
\(877\) −29.3084 + 50.7637i −0.989675 + 1.71417i −0.370710 + 0.928749i \(0.620886\pi\)
−0.618965 + 0.785419i \(0.712448\pi\)
\(878\) 5.14931 + 8.91887i 0.173781 + 0.300997i
\(879\) 0 0
\(880\) 0 0
\(881\) −4.97791 −0.167710 −0.0838550 0.996478i \(-0.526723\pi\)
−0.0838550 + 0.996478i \(0.526723\pi\)
\(882\) 0 0
\(883\) 20.8600 0.701995 0.350998 0.936376i \(-0.385843\pi\)
0.350998 + 0.936376i \(0.385843\pi\)
\(884\) −18.3601 10.6002i −0.617515 0.356523i
\(885\) 0 0
\(886\) −20.7828 35.9968i −0.698211 1.20934i
\(887\) 24.9109 43.1470i 0.836427 1.44873i −0.0564371 0.998406i \(-0.517974\pi\)
0.892864 0.450327i \(-0.148693\pi\)
\(888\) 0 0
\(889\) −9.48735 + 4.97392i −0.318195 + 0.166820i
\(890\) 0 0
\(891\) 0 0
\(892\) −24.9142 + 14.3842i −0.834189 + 0.481619i
\(893\) −3.10104 + 1.79039i −0.103772 + 0.0599130i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.64356 + 0.107718i 0.0883151 + 0.00359861i
\(897\) 0 0
\(898\) 11.8919 20.5973i 0.396837 0.687342i
\(899\) 2.79596 + 4.84275i 0.0932505 + 0.161515i
\(900\) 0 0
\(901\) −48.0069 27.7168i −1.59934 0.923379i
\(902\) −20.4557 −0.681100
\(903\) 0 0
\(904\) −18.4343 −0.613115
\(905\) 0 0
\(906\) 0 0
\(907\) −18.0797 31.3149i −0.600326 1.03980i −0.992771 0.120020i \(-0.961704\pi\)
0.392445 0.919775i \(-0.371629\pi\)
\(908\) 1.05185 1.82186i 0.0349070 0.0604607i
\(909\) 0 0
\(910\) 0 0
\(911\) 2.05205i 0.0679873i −0.999422 0.0339937i \(-0.989177\pi\)
0.999422 0.0339937i \(-0.0108226\pi\)
\(912\) 0 0
\(913\) −0.350114 + 0.202138i −0.0115871 + 0.00668980i
\(914\) 16.2595 9.38745i 0.537818 0.310509i
\(915\) 0 0
\(916\) 16.1968i 0.535158i
\(917\) −1.53849 + 37.7568i −0.0508054 + 1.24684i
\(918\) 0 0
\(919\) −0.696500 + 1.20637i −0.0229754 + 0.0397946i −0.877285 0.479971i \(-0.840647\pi\)
0.854309 + 0.519765i \(0.173981\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −15.1262 8.73312i −0.498155 0.287610i
\(923\) 9.42038 0.310076
\(924\) 0 0
\(925\) 0 0
\(926\) −2.59240 1.49672i −0.0851916 0.0491854i
\(927\) 0 0
\(928\) −1.27670 2.21130i −0.0419096 0.0725896i
\(929\) 16.5276 28.6266i 0.542252 0.939208i −0.456522 0.889712i \(-0.650905\pi\)
0.998774 0.0494960i \(-0.0157615\pi\)
\(930\) 0 0
\(931\) 3.58901 1.69907i 0.117625 0.0556847i
\(932\) 8.01137i 0.262421i
\(933\) 0 0
\(934\) 22.0058 12.7050i 0.720051 0.415722i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.53551i 0.0828314i 0.999142 + 0.0414157i \(0.0131868\pi\)
−0.999142 + 0.0414157i \(0.986813\pi\)
\(938\) 4.91747 + 9.37967i 0.160561 + 0.306257i
\(939\) 0 0
\(940\) 0 0
\(941\) 18.8858 + 32.7112i 0.615659 + 1.06635i 0.990268 + 0.139171i \(0.0444436\pi\)
−0.374609 + 0.927183i \(0.622223\pi\)
\(942\) 0 0
\(943\) 43.8210 + 25.3001i 1.42701 + 0.823884i
\(944\) 3.35478 0.109189
\(945\) 0 0
\(946\) 15.4915 0.503672
\(947\) −49.0138 28.2981i −1.59274 0.919566i −0.992835 0.119491i \(-0.961874\pi\)
−0.599900 0.800075i \(-0.704793\pi\)
\(948\) 0 0
\(949\) −19.0898 33.0645i −0.619682 1.07332i
\(950\) 0 0
\(951\) 0 0
\(952\) 6.45578 10.1989i 0.209233 0.330548i
\(953\) 34.3234i 1.11184i 0.831234 + 0.555922i \(0.187635\pi\)
−0.831234 + 0.555922i \(0.812365\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −13.5302 + 7.81165i −0.437597 + 0.252647i
\(957\) 0 0
\(958\) 23.7032i 0.765816i
\(959\) 7.39855 11.6883i 0.238912 0.377434i
\(960\) 0 0
\(961\) −13.1020 + 22.6933i −0.422644 + 0.732041i
\(962\) 21.5320 + 37.2945i 0.694219 + 1.20242i
\(963\) 0 0
\(964\) −3.54491 2.04665i −0.114174 0.0659183i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.903990 −0.0290703 −0.0145352 0.999894i \(-0.504627\pi\)
−0.0145352 + 0.999894i \(0.504627\pi\)
\(968\) 4.72156 + 2.72599i 0.151757 + 0.0876167i
\(969\) 0 0
\(970\) 0 0
\(971\) 21.2293 36.7703i 0.681282 1.18001i −0.293308 0.956018i \(-0.594756\pi\)
0.974590 0.223997i \(-0.0719105\pi\)
\(972\) 0 0
\(973\) −23.9219 45.6290i −0.766901 1.46280i
\(974\) 10.2154i 0.327324i
\(975\) 0 0
\(976\) 6.85523 3.95787i 0.219431 0.126688i
\(977\) 5.32079 3.07196i 0.170227 0.0982807i −0.412466 0.910973i \(-0.635332\pi\)
0.582693 + 0.812692i \(0.301999\pi\)
\(978\) 0 0
\(979\) 12.8577i 0.410933i
\(980\) 0 0
\(981\) 0 0
\(982\) 1.12955 1.95643i 0.0360453 0.0624323i
\(983\) −19.7933 34.2830i −0.631309 1.09346i −0.987284 0.158964i \(-0.949185\pi\)
0.355976 0.934495i \(-0.384149\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −11.6490 −0.370981
\(987\) 0 0
\(988\) −2.63608 −0.0838648
\(989\) −33.1865 19.1602i −1.05527 0.609260i
\(990\) 0 0
\(991\) 19.4602 + 33.7060i 0.618172 + 1.07071i 0.989819 + 0.142331i \(0.0454599\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(992\) −1.09500 + 1.89659i −0.0347662 + 0.0602168i
\(993\) 0 0
\(994\) −0.218366 + 5.35903i −0.00692616 + 0.169978i
\(995\) 0 0
\(996\) 0 0
\(997\) 18.5522 10.7111i 0.587555 0.339225i −0.176575 0.984287i \(-0.556502\pi\)
0.764130 + 0.645062i \(0.223168\pi\)
\(998\) 30.6246 17.6811i 0.969406 0.559687i
\(999\) 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 3150.2.bf.d.1601.2 yes 24
3.2 odd 2 inner 3150.2.bf.d.1601.11 yes 24
5.2 odd 4 3150.2.bp.h.1349.8 24
5.3 odd 4 3150.2.bp.g.1349.5 24
5.4 even 2 3150.2.bf.e.1601.11 yes 24
7.3 odd 6 inner 3150.2.bf.d.1151.11 yes 24
15.2 even 4 3150.2.bp.g.1349.8 24
15.8 even 4 3150.2.bp.h.1349.5 24
15.14 odd 2 3150.2.bf.e.1601.2 yes 24
21.17 even 6 inner 3150.2.bf.d.1151.2 24
35.3 even 12 3150.2.bp.g.899.8 24
35.17 even 12 3150.2.bp.h.899.5 24
35.24 odd 6 3150.2.bf.e.1151.2 yes 24
105.17 odd 12 3150.2.bp.g.899.5 24
105.38 odd 12 3150.2.bp.h.899.8 24
105.59 even 6 3150.2.bf.e.1151.11 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3150.2.bf.d.1151.2 24 21.17 even 6 inner
3150.2.bf.d.1151.11 yes 24 7.3 odd 6 inner
3150.2.bf.d.1601.2 yes 24 1.1 even 1 trivial
3150.2.bf.d.1601.11 yes 24 3.2 odd 2 inner
3150.2.bf.e.1151.2 yes 24 35.24 odd 6
3150.2.bf.e.1151.11 yes 24 105.59 even 6
3150.2.bf.e.1601.2 yes 24 15.14 odd 2
3150.2.bf.e.1601.11 yes 24 5.4 even 2
3150.2.bp.g.899.5 24 105.17 odd 12
3150.2.bp.g.899.8 24 35.3 even 12
3150.2.bp.g.1349.5 24 5.3 odd 4
3150.2.bp.g.1349.8 24 15.2 even 4
3150.2.bp.h.899.5 24 35.17 even 12
3150.2.bp.h.899.8 24 105.38 odd 12
3150.2.bp.h.1349.5 24 15.8 even 4
3150.2.bp.h.1349.8 24 5.2 odd 4