Properties

Label 260.2.x.a.121.1
Level $260$
Weight $2$
Character 260.121
Analytic conductor $2.076$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,2,Mod(101,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.x (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: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 121.1
Root \(1.20036 - 0.747754i\) of defining polynomial
Character \(\chi\) \(=\) 260.121
Dual form 260.2.x.a.101.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.41342 + 2.44811i) q^{3} +1.00000i q^{5} +(-1.81414 + 1.04739i) q^{7} +(-2.49551 - 4.32235i) q^{9} +O(q^{10})\) \(q+(-1.41342 + 2.44811i) q^{3} +1.00000i q^{5} +(-1.81414 + 1.04739i) q^{7} +(-2.49551 - 4.32235i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-3.59030 - 0.331331i) q^{13} +(-2.44811 - 1.41342i) q^{15} +(-1.81414 - 3.14218i) q^{17} +(0.926118 - 0.534695i) q^{19} -5.92163i q^{21} +(-3.90893 + 6.77046i) q^{23} -1.00000 q^{25} +5.62828 q^{27} +(0.263457 - 0.456321i) q^{29} +5.84325i q^{31} +(-4.24026 + 2.44811i) q^{33} +(-1.04739 - 1.81414i) q^{35} +(8.44242 + 4.87423i) q^{37} +(5.88573 - 8.32114i) q^{39} +(-3.69615 - 2.13397i) q^{41} +(4.67238 + 8.09281i) q^{43} +(4.32235 - 2.49551i) q^{45} -3.46410i q^{47} +(-1.30593 + 2.26194i) q^{49} +10.2566 q^{51} +12.5939 q^{53} +(-0.866025 + 1.50000i) q^{55} +3.02299i q^{57} +(-1.21564 + 0.701848i) q^{59} +(5.55440 + 9.62050i) q^{61} +(9.05440 + 5.22756i) q^{63} +(0.331331 - 3.59030i) q^{65} +(-9.38201 - 5.41671i) q^{67} +(-11.0499 - 19.1390i) q^{69} +(-12.2709 + 7.08460i) q^{71} +2.64469i q^{73} +(1.41342 - 2.44811i) q^{75} -3.62828 q^{77} +13.5729 q^{79} +(-0.468594 + 0.811629i) q^{81} -15.7925i q^{83} +(3.14218 - 1.81414i) q^{85} +(0.744750 + 1.28994i) q^{87} +(-4.78436 - 2.76225i) q^{89} +(6.86033 - 3.15937i) q^{91} +(-14.3049 - 8.25896i) q^{93} +(0.534695 + 0.926118i) q^{95} +(-13.1589 + 7.59730i) q^{97} -8.64469i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9} + 12 q^{11} - 8 q^{13} - 6 q^{15} + 6 q^{17} - 6 q^{23} - 8 q^{25} + 4 q^{27} - 6 q^{33} - 6 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} + 10 q^{43} - 4 q^{49} + 24 q^{53} - 24 q^{59} - 4 q^{61} + 24 q^{63} - 54 q^{67} - 24 q^{69} - 36 q^{71} + 2 q^{75} + 12 q^{77} - 16 q^{79} + 8 q^{81} + 18 q^{85} - 6 q^{87} - 24 q^{89} + 24 q^{93} - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.41342 + 2.44811i −0.816038 + 1.41342i 0.0925423 + 0.995709i \(0.470501\pi\)
−0.908580 + 0.417710i \(0.862833\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −1.81414 + 1.04739i −0.685680 + 0.395878i −0.801992 0.597335i \(-0.796226\pi\)
0.116312 + 0.993213i \(0.462893\pi\)
\(8\) 0 0
\(9\) −2.49551 4.32235i −0.831836 1.44078i
\(10\) 0 0
\(11\) 1.50000 + 0.866025i 0.452267 + 0.261116i 0.708787 0.705422i \(-0.249243\pi\)
−0.256520 + 0.966539i \(0.582576\pi\)
\(12\) 0 0
\(13\) −3.59030 0.331331i −0.995769 0.0918946i
\(14\) 0 0
\(15\) −2.44811 1.41342i −0.632100 0.364943i
\(16\) 0 0
\(17\) −1.81414 3.14218i −0.439993 0.762091i 0.557695 0.830046i \(-0.311686\pi\)
−0.997688 + 0.0679550i \(0.978353\pi\)
\(18\) 0 0
\(19\) 0.926118 0.534695i 0.212466 0.122667i −0.389991 0.920819i \(-0.627522\pi\)
0.602457 + 0.798151i \(0.294189\pi\)
\(20\) 0 0
\(21\) 5.92163i 1.29220i
\(22\) 0 0
\(23\) −3.90893 + 6.77046i −0.815068 + 1.41174i 0.0942118 + 0.995552i \(0.469967\pi\)
−0.909279 + 0.416186i \(0.863366\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.62828 1.08316
\(28\) 0 0
\(29\) 0.263457 0.456321i 0.0489227 0.0847366i −0.840527 0.541770i \(-0.817755\pi\)
0.889450 + 0.457033i \(0.151088\pi\)
\(30\) 0 0
\(31\) 5.84325i 1.04948i 0.851263 + 0.524740i \(0.175837\pi\)
−0.851263 + 0.524740i \(0.824163\pi\)
\(32\) 0 0
\(33\) −4.24026 + 2.44811i −0.738134 + 0.426162i
\(34\) 0 0
\(35\) −1.04739 1.81414i −0.177042 0.306646i
\(36\) 0 0
\(37\) 8.44242 + 4.87423i 1.38792 + 0.801319i 0.993081 0.117429i \(-0.0374654\pi\)
0.394844 + 0.918748i \(0.370799\pi\)
\(38\) 0 0
\(39\) 5.88573 8.32114i 0.942471 1.33245i
\(40\) 0 0
\(41\) −3.69615 2.13397i −0.577242 0.333271i 0.182795 0.983151i \(-0.441486\pi\)
−0.760037 + 0.649880i \(0.774819\pi\)
\(42\) 0 0
\(43\) 4.67238 + 8.09281i 0.712532 + 1.23414i 0.963904 + 0.266251i \(0.0857849\pi\)
−0.251372 + 0.967891i \(0.580882\pi\)
\(44\) 0 0
\(45\) 4.32235 2.49551i 0.644337 0.372008i
\(46\) 0 0
\(47\) 3.46410i 0.505291i −0.967559 0.252646i \(-0.918699\pi\)
0.967559 0.252646i \(-0.0813007\pi\)
\(48\) 0 0
\(49\) −1.30593 + 2.26194i −0.186562 + 0.323134i
\(50\) 0 0
\(51\) 10.2566 1.43621
\(52\) 0 0
\(53\) 12.5939 1.72990 0.864952 0.501854i \(-0.167349\pi\)
0.864952 + 0.501854i \(0.167349\pi\)
\(54\) 0 0
\(55\) −0.866025 + 1.50000i −0.116775 + 0.202260i
\(56\) 0 0
\(57\) 3.02299i 0.400405i
\(58\) 0 0
\(59\) −1.21564 + 0.701848i −0.158262 + 0.0913729i −0.577040 0.816716i \(-0.695792\pi\)
0.418777 + 0.908089i \(0.362459\pi\)
\(60\) 0 0
\(61\) 5.55440 + 9.62050i 0.711168 + 1.23178i 0.964419 + 0.264378i \(0.0851667\pi\)
−0.253251 + 0.967400i \(0.581500\pi\)
\(62\) 0 0
\(63\) 9.05440 + 5.22756i 1.14075 + 0.658610i
\(64\) 0 0
\(65\) 0.331331 3.59030i 0.0410965 0.445321i
\(66\) 0 0
\(67\) −9.38201 5.41671i −1.14620 0.661756i −0.198238 0.980154i \(-0.563522\pi\)
−0.947957 + 0.318398i \(0.896855\pi\)
\(68\) 0 0
\(69\) −11.0499 19.1390i −1.33025 2.30406i
\(70\) 0 0
\(71\) −12.2709 + 7.08460i −1.45629 + 0.840787i −0.998826 0.0484428i \(-0.984574\pi\)
−0.457460 + 0.889230i \(0.651241\pi\)
\(72\) 0 0
\(73\) 2.64469i 0.309538i 0.987951 + 0.154769i \(0.0494633\pi\)
−0.987951 + 0.154769i \(0.950537\pi\)
\(74\) 0 0
\(75\) 1.41342 2.44811i 0.163208 0.282684i
\(76\) 0 0
\(77\) −3.62828 −0.413481
\(78\) 0 0
\(79\) 13.5729 1.52707 0.763535 0.645766i \(-0.223462\pi\)
0.763535 + 0.645766i \(0.223462\pi\)
\(80\) 0 0
\(81\) −0.468594 + 0.811629i −0.0520660 + 0.0901809i
\(82\) 0 0
\(83\) 15.7925i 1.73345i −0.498789 0.866724i \(-0.666222\pi\)
0.498789 0.866724i \(-0.333778\pi\)
\(84\) 0 0
\(85\) 3.14218 1.81414i 0.340817 0.196771i
\(86\) 0 0
\(87\) 0.744750 + 1.28994i 0.0798456 + 0.138297i
\(88\) 0 0
\(89\) −4.78436 2.76225i −0.507141 0.292798i 0.224516 0.974470i \(-0.427920\pi\)
−0.731658 + 0.681672i \(0.761253\pi\)
\(90\) 0 0
\(91\) 6.86033 3.15937i 0.719158 0.331192i
\(92\) 0 0
\(93\) −14.3049 8.25896i −1.48335 0.856415i
\(94\) 0 0
\(95\) 0.534695 + 0.926118i 0.0548585 + 0.0950177i
\(96\) 0 0
\(97\) −13.1589 + 7.59730i −1.33608 + 0.771389i −0.986224 0.165413i \(-0.947104\pi\)
−0.349860 + 0.936802i \(0.613771\pi\)
\(98\) 0 0
\(99\) 8.64469i 0.868824i
\(100\) 0 0
\(101\) −1.83133 + 3.17196i −0.182224 + 0.315622i −0.942638 0.333818i \(-0.891663\pi\)
0.760413 + 0.649439i \(0.224996\pi\)
\(102\) 0 0
\(103\) 13.7804 1.35783 0.678914 0.734218i \(-0.262451\pi\)
0.678914 + 0.734218i \(0.262451\pi\)
\(104\) 0 0
\(105\) 5.92163 0.577892
\(106\) 0 0
\(107\) −1.61856 + 2.80342i −0.156472 + 0.271017i −0.933594 0.358333i \(-0.883345\pi\)
0.777122 + 0.629350i \(0.216679\pi\)
\(108\) 0 0
\(109\) 9.12979i 0.874476i 0.899346 + 0.437238i \(0.144043\pi\)
−0.899346 + 0.437238i \(0.855957\pi\)
\(110\) 0 0
\(111\) −23.8654 + 13.7787i −2.26520 + 1.30781i
\(112\) 0 0
\(113\) 5.47680 + 9.48610i 0.515214 + 0.892377i 0.999844 + 0.0176577i \(0.00562092\pi\)
−0.484630 + 0.874719i \(0.661046\pi\)
\(114\) 0 0
\(115\) −6.77046 3.90893i −0.631349 0.364509i
\(116\) 0 0
\(117\) 7.52748 + 16.3453i 0.695916 + 1.51113i
\(118\) 0 0
\(119\) 6.58220 + 3.80024i 0.603390 + 0.348367i
\(120\) 0 0
\(121\) −4.00000 6.92820i −0.363636 0.629837i
\(122\) 0 0
\(123\) 10.4484 6.03240i 0.942103 0.543923i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 0.453810 0.786022i 0.0402691 0.0697482i −0.845188 0.534469i \(-0.820512\pi\)
0.885458 + 0.464720i \(0.153845\pi\)
\(128\) 0 0
\(129\) −26.4161 −2.32581
\(130\) 0 0
\(131\) −13.1626 −1.15002 −0.575012 0.818145i \(-0.695003\pi\)
−0.575012 + 0.818145i \(0.695003\pi\)
\(132\) 0 0
\(133\) −1.12007 + 1.94002i −0.0971225 + 0.168221i
\(134\) 0 0
\(135\) 5.62828i 0.484405i
\(136\) 0 0
\(137\) 10.4484 6.03240i 0.892669 0.515383i 0.0178546 0.999841i \(-0.494316\pi\)
0.874815 + 0.484458i \(0.160983\pi\)
\(138\) 0 0
\(139\) 2.80593 + 4.86002i 0.237996 + 0.412221i 0.960139 0.279522i \(-0.0901761\pi\)
−0.722143 + 0.691744i \(0.756843\pi\)
\(140\) 0 0
\(141\) 8.48052 + 4.89623i 0.714188 + 0.412337i
\(142\) 0 0
\(143\) −5.09850 3.60628i −0.426358 0.301573i
\(144\) 0 0
\(145\) 0.456321 + 0.263457i 0.0378954 + 0.0218789i
\(146\) 0 0
\(147\) −3.69166 6.39414i −0.304483 0.527380i
\(148\) 0 0
\(149\) 18.1767 10.4943i 1.48909 0.859727i 0.489168 0.872189i \(-0.337300\pi\)
0.999922 + 0.0124625i \(0.00396705\pi\)
\(150\) 0 0
\(151\) 6.99102i 0.568921i −0.958688 0.284460i \(-0.908186\pi\)
0.958688 0.284460i \(-0.0918144\pi\)
\(152\) 0 0
\(153\) −9.05440 + 15.6827i −0.732005 + 1.26787i
\(154\) 0 0
\(155\) −5.84325 −0.469341
\(156\) 0 0
\(157\) 3.74761 0.299092 0.149546 0.988755i \(-0.452219\pi\)
0.149546 + 0.988755i \(0.452219\pi\)
\(158\) 0 0
\(159\) −17.8005 + 30.8313i −1.41167 + 2.44508i
\(160\) 0 0
\(161\) 16.3767i 1.29067i
\(162\) 0 0
\(163\) −5.52377 + 3.18915i −0.432655 + 0.249793i −0.700477 0.713675i \(-0.747029\pi\)
0.267822 + 0.963468i \(0.413696\pi\)
\(164\) 0 0
\(165\) −2.44811 4.24026i −0.190585 0.330104i
\(166\) 0 0
\(167\) −13.4484 7.76445i −1.04067 0.600831i −0.120647 0.992695i \(-0.538497\pi\)
−0.920023 + 0.391864i \(0.871830\pi\)
\(168\) 0 0
\(169\) 12.7804 + 2.37915i 0.983111 + 0.183012i
\(170\) 0 0
\(171\) −4.62227 2.66867i −0.353474 0.204078i
\(172\) 0 0
\(173\) −2.38802 4.13617i −0.181558 0.314467i 0.760853 0.648924i \(-0.224781\pi\)
−0.942411 + 0.334456i \(0.891447\pi\)
\(174\) 0 0
\(175\) 1.81414 1.04739i 0.137136 0.0791755i
\(176\) 0 0
\(177\) 3.96802i 0.298255i
\(178\) 0 0
\(179\) 7.85134 13.5989i 0.586837 1.01643i −0.407807 0.913068i \(-0.633706\pi\)
0.994644 0.103363i \(-0.0329604\pi\)
\(180\) 0 0
\(181\) 10.8851 0.809080 0.404540 0.914520i \(-0.367432\pi\)
0.404540 + 0.914520i \(0.367432\pi\)
\(182\) 0 0
\(183\) −31.4028 −2.32136
\(184\) 0 0
\(185\) −4.87423 + 8.44242i −0.358361 + 0.620699i
\(186\) 0 0
\(187\) 6.28436i 0.459558i
\(188\) 0 0
\(189\) −10.2105 + 5.89502i −0.742703 + 0.428800i
\(190\) 0 0
\(191\) 2.97909 + 5.15994i 0.215560 + 0.373360i 0.953446 0.301565i \(-0.0975091\pi\)
−0.737886 + 0.674925i \(0.764176\pi\)
\(192\) 0 0
\(193\) 11.0587 + 6.38473i 0.796021 + 0.459583i 0.842078 0.539356i \(-0.181332\pi\)
−0.0460568 + 0.998939i \(0.514666\pi\)
\(194\) 0 0
\(195\) 8.32114 + 5.88573i 0.595889 + 0.421486i
\(196\) 0 0
\(197\) −13.9499 8.05397i −0.993888 0.573822i −0.0874540 0.996169i \(-0.527873\pi\)
−0.906434 + 0.422347i \(0.861206\pi\)
\(198\) 0 0
\(199\) −4.26403 7.38551i −0.302269 0.523545i 0.674381 0.738384i \(-0.264411\pi\)
−0.976650 + 0.214839i \(0.931077\pi\)
\(200\) 0 0
\(201\) 26.5214 15.3122i 1.87068 1.08004i
\(202\) 0 0
\(203\) 1.10377i 0.0774696i
\(204\) 0 0
\(205\) 2.13397 3.69615i 0.149043 0.258150i
\(206\) 0 0
\(207\) 39.0190 2.71201
\(208\) 0 0
\(209\) 1.85224 0.128122
\(210\) 0 0
\(211\) 2.09030 3.62050i 0.143902 0.249245i −0.785061 0.619419i \(-0.787368\pi\)
0.928963 + 0.370173i \(0.120702\pi\)
\(212\) 0 0
\(213\) 40.0540i 2.74446i
\(214\) 0 0
\(215\) −8.09281 + 4.67238i −0.551925 + 0.318654i
\(216\) 0 0
\(217\) −6.12019 10.6005i −0.415465 0.719607i
\(218\) 0 0
\(219\) −6.47451 3.73806i −0.437507 0.252595i
\(220\) 0 0
\(221\) 5.47219 + 11.8824i 0.368100 + 0.799299i
\(222\) 0 0
\(223\) −9.09249 5.24955i −0.608878 0.351536i 0.163648 0.986519i \(-0.447674\pi\)
−0.772526 + 0.634983i \(0.781007\pi\)
\(224\) 0 0
\(225\) 2.49551 + 4.32235i 0.166367 + 0.288156i
\(226\) 0 0
\(227\) 13.4977 7.79288i 0.895872 0.517232i 0.0200131 0.999800i \(-0.493629\pi\)
0.875858 + 0.482568i \(0.160296\pi\)
\(228\) 0 0
\(229\) 19.2714i 1.27349i 0.771074 + 0.636745i \(0.219720\pi\)
−0.771074 + 0.636745i \(0.780280\pi\)
\(230\) 0 0
\(231\) 5.12828 8.88244i 0.337416 0.584422i
\(232\) 0 0
\(233\) 2.48794 0.162991 0.0814953 0.996674i \(-0.474030\pi\)
0.0814953 + 0.996674i \(0.474030\pi\)
\(234\) 0 0
\(235\) 3.46410 0.225973
\(236\) 0 0
\(237\) −19.1842 + 33.2280i −1.24615 + 2.15839i
\(238\) 0 0
\(239\) 16.7775i 1.08525i 0.839976 + 0.542624i \(0.182569\pi\)
−0.839976 + 0.542624i \(0.817431\pi\)
\(240\) 0 0
\(241\) 25.1835 14.5397i 1.62221 0.936585i 0.635887 0.771782i \(-0.280634\pi\)
0.986326 0.164803i \(-0.0526990\pi\)
\(242\) 0 0
\(243\) 7.11778 + 12.3284i 0.456606 + 0.790864i
\(244\) 0 0
\(245\) −2.26194 1.30593i −0.144510 0.0834330i
\(246\) 0 0
\(247\) −3.50220 + 1.61286i −0.222840 + 0.102624i
\(248\) 0 0
\(249\) 38.6617 + 22.3214i 2.45009 + 1.41456i
\(250\) 0 0
\(251\) 9.56040 + 16.5591i 0.603447 + 1.04520i 0.992295 + 0.123899i \(0.0395400\pi\)
−0.388847 + 0.921302i \(0.627127\pi\)
\(252\) 0 0
\(253\) −11.7268 + 6.77046i −0.737256 + 0.425655i
\(254\) 0 0
\(255\) 10.2566i 0.642291i
\(256\) 0 0
\(257\) 6.85453 11.8724i 0.427574 0.740580i −0.569083 0.822280i \(-0.692702\pi\)
0.996657 + 0.0817004i \(0.0260351\pi\)
\(258\) 0 0
\(259\) −20.4210 −1.26890
\(260\) 0 0
\(261\) −2.62983 −0.162783
\(262\) 0 0
\(263\) 5.95675 10.3174i 0.367309 0.636197i −0.621835 0.783148i \(-0.713613\pi\)
0.989144 + 0.146951i \(0.0469459\pi\)
\(264\) 0 0
\(265\) 12.5939i 0.773637i
\(266\) 0 0
\(267\) 13.5246 7.80844i 0.827693 0.477869i
\(268\) 0 0
\(269\) 10.4656 + 18.1270i 0.638100 + 1.10522i 0.985849 + 0.167634i \(0.0536127\pi\)
−0.347749 + 0.937588i \(0.613054\pi\)
\(270\) 0 0
\(271\) 1.69014 + 0.975805i 0.102669 + 0.0592760i 0.550455 0.834865i \(-0.314454\pi\)
−0.447786 + 0.894141i \(0.647787\pi\)
\(272\) 0 0
\(273\) −1.96202 + 21.2604i −0.118747 + 1.28674i
\(274\) 0 0
\(275\) −1.50000 0.866025i −0.0904534 0.0522233i
\(276\) 0 0
\(277\) 6.24026 + 10.8084i 0.374941 + 0.649416i 0.990318 0.138816i \(-0.0443297\pi\)
−0.615377 + 0.788233i \(0.710996\pi\)
\(278\) 0 0
\(279\) 25.2566 14.5819i 1.51207 0.872994i
\(280\) 0 0
\(281\) 2.29553i 0.136940i −0.997653 0.0684698i \(-0.978188\pi\)
0.997653 0.0684698i \(-0.0218117\pi\)
\(282\) 0 0
\(283\) 7.46484 12.9295i 0.443739 0.768578i −0.554225 0.832367i \(-0.686985\pi\)
0.997963 + 0.0637892i \(0.0203185\pi\)
\(284\) 0 0
\(285\) −3.02299 −0.179067
\(286\) 0 0
\(287\) 8.94045 0.527738
\(288\) 0 0
\(289\) 1.91780 3.32172i 0.112812 0.195395i
\(290\) 0 0
\(291\) 42.9527i 2.51793i
\(292\) 0 0
\(293\) 1.24026 0.716063i 0.0724566 0.0418329i −0.463334 0.886184i \(-0.653347\pi\)
0.535791 + 0.844351i \(0.320014\pi\)
\(294\) 0 0
\(295\) −0.701848 1.21564i −0.0408632 0.0707771i
\(296\) 0 0
\(297\) 8.44242 + 4.87423i 0.489879 + 0.282832i
\(298\) 0 0
\(299\) 16.2775 23.0128i 0.941350 1.33086i
\(300\) 0 0
\(301\) −16.9527 9.78765i −0.977138 0.564151i
\(302\) 0 0
\(303\) −5.17688 8.96661i −0.297404 0.515118i
\(304\) 0 0
\(305\) −9.62050 + 5.55440i −0.550868 + 0.318044i
\(306\) 0 0
\(307\) 17.3833i 0.992118i 0.868289 + 0.496059i \(0.165220\pi\)
−0.868289 + 0.496059i \(0.834780\pi\)
\(308\) 0 0
\(309\) −19.4775 + 33.7361i −1.10804 + 1.91918i
\(310\) 0 0
\(311\) −20.2164 −1.14637 −0.573185 0.819426i \(-0.694292\pi\)
−0.573185 + 0.819426i \(0.694292\pi\)
\(312\) 0 0
\(313\) −4.86425 −0.274944 −0.137472 0.990506i \(-0.543898\pi\)
−0.137472 + 0.990506i \(0.543898\pi\)
\(314\) 0 0
\(315\) −5.22756 + 9.05440i −0.294540 + 0.510157i
\(316\) 0 0
\(317\) 23.6177i 1.32650i −0.748396 0.663252i \(-0.769176\pi\)
0.748396 0.663252i \(-0.230824\pi\)
\(318\) 0 0
\(319\) 0.790371 0.456321i 0.0442523 0.0255491i
\(320\) 0 0
\(321\) −4.57540 7.92482i −0.255374 0.442320i
\(322\) 0 0
\(323\) −3.36022 1.94002i −0.186967 0.107946i
\(324\) 0 0
\(325\) 3.59030 + 0.331331i 0.199154 + 0.0183789i
\(326\) 0 0
\(327\) −22.3508 12.9042i −1.23600 0.713605i
\(328\) 0 0
\(329\) 3.62828 + 6.28436i 0.200033 + 0.346468i
\(330\) 0 0
\(331\) −12.8863 + 7.43991i −0.708295 + 0.408934i −0.810429 0.585836i \(-0.800766\pi\)
0.102134 + 0.994771i \(0.467433\pi\)
\(332\) 0 0
\(333\) 48.6547i 2.66626i
\(334\) 0 0
\(335\) 5.41671 9.38201i 0.295946 0.512594i
\(336\) 0 0
\(337\) −29.1906 −1.59012 −0.795058 0.606534i \(-0.792559\pi\)
−0.795058 + 0.606534i \(0.792559\pi\)
\(338\) 0 0
\(339\) −30.9641 −1.68174
\(340\) 0 0
\(341\) −5.06040 + 8.76488i −0.274036 + 0.474645i
\(342\) 0 0
\(343\) 20.1348i 1.08718i
\(344\) 0 0
\(345\) 19.1390 11.0499i 1.03041 0.594907i
\(346\) 0 0
\(347\) 12.1125 + 20.9795i 0.650234 + 1.12624i 0.983066 + 0.183252i \(0.0586623\pi\)
−0.332833 + 0.942986i \(0.608004\pi\)
\(348\) 0 0
\(349\) −11.1557 6.44076i −0.597152 0.344766i 0.170768 0.985311i \(-0.445375\pi\)
−0.767920 + 0.640545i \(0.778708\pi\)
\(350\) 0 0
\(351\) −20.2072 1.86482i −1.07858 0.0995368i
\(352\) 0 0
\(353\) 14.0441 + 8.10837i 0.747492 + 0.431565i 0.824787 0.565443i \(-0.191295\pi\)
−0.0772948 + 0.997008i \(0.524628\pi\)
\(354\) 0 0
\(355\) −7.08460 12.2709i −0.376011 0.651271i
\(356\) 0 0
\(357\) −18.6068 + 10.7427i −0.984777 + 0.568562i
\(358\) 0 0
\(359\) 9.19261i 0.485167i 0.970131 + 0.242584i \(0.0779949\pi\)
−0.970131 + 0.242584i \(0.922005\pi\)
\(360\) 0 0
\(361\) −8.92820 + 15.4641i −0.469905 + 0.813900i
\(362\) 0 0
\(363\) 22.6147 1.18696
\(364\) 0 0
\(365\) −2.64469 −0.138430
\(366\) 0 0
\(367\) 2.30066 3.98486i 0.120094 0.208008i −0.799711 0.600385i \(-0.795014\pi\)
0.919804 + 0.392377i \(0.128347\pi\)
\(368\) 0 0
\(369\) 21.3014i 1.10891i
\(370\) 0 0
\(371\) −22.8471 + 13.1908i −1.18616 + 0.684831i
\(372\) 0 0
\(373\) 10.2463 + 17.7471i 0.530532 + 0.918908i 0.999365 + 0.0356212i \(0.0113410\pi\)
−0.468834 + 0.883286i \(0.655326\pi\)
\(374\) 0 0
\(375\) 2.44811 + 1.41342i 0.126420 + 0.0729887i
\(376\) 0 0
\(377\) −1.09708 + 1.55103i −0.0565025 + 0.0798823i
\(378\) 0 0
\(379\) 6.95307 + 4.01436i 0.357155 + 0.206204i 0.667832 0.744312i \(-0.267222\pi\)
−0.310677 + 0.950516i \(0.600556\pi\)
\(380\) 0 0
\(381\) 1.28285 + 2.22196i 0.0657223 + 0.113834i
\(382\) 0 0
\(383\) −26.1041 + 15.0712i −1.33386 + 0.770104i −0.985889 0.167401i \(-0.946462\pi\)
−0.347971 + 0.937505i \(0.613129\pi\)
\(384\) 0 0
\(385\) 3.62828i 0.184914i
\(386\) 0 0
\(387\) 23.3199 40.3913i 1.18542 2.05321i
\(388\) 0 0
\(389\) −35.8264 −1.81647 −0.908236 0.418459i \(-0.862570\pi\)
−0.908236 + 0.418459i \(0.862570\pi\)
\(390\) 0 0
\(391\) 28.3654 1.43450
\(392\) 0 0
\(393\) 18.6043 32.2236i 0.938463 1.62547i
\(394\) 0 0
\(395\) 13.5729i 0.682926i
\(396\) 0 0
\(397\) −13.3221 + 7.69152i −0.668617 + 0.386026i −0.795552 0.605885i \(-0.792819\pi\)
0.126935 + 0.991911i \(0.459486\pi\)
\(398\) 0 0
\(399\) −3.16626 5.48413i −0.158511 0.274550i
\(400\) 0 0
\(401\) −8.46704 4.88845i −0.422824 0.244117i 0.273461 0.961883i \(-0.411832\pi\)
−0.696285 + 0.717766i \(0.745165\pi\)
\(402\) 0 0
\(403\) 1.93605 20.9790i 0.0964415 1.04504i
\(404\) 0 0
\(405\) −0.811629 0.468594i −0.0403301 0.0232846i
\(406\) 0 0
\(407\) 8.44242 + 14.6227i 0.418475 + 0.724820i
\(408\) 0 0
\(409\) 0.871721 0.503289i 0.0431039 0.0248860i −0.478293 0.878200i \(-0.658744\pi\)
0.521397 + 0.853314i \(0.325411\pi\)
\(410\) 0 0
\(411\) 34.1052i 1.68229i
\(412\) 0 0
\(413\) 1.47022 2.54650i 0.0723450 0.125305i
\(414\) 0 0
\(415\) 15.7925 0.775221
\(416\) 0 0
\(417\) −15.8638 −0.776855
\(418\) 0 0
\(419\) −4.85134 + 8.40278i −0.237004 + 0.410502i −0.959853 0.280503i \(-0.909499\pi\)
0.722849 + 0.691006i \(0.242832\pi\)
\(420\) 0 0
\(421\) 14.2955i 0.696721i 0.937361 + 0.348361i \(0.113262\pi\)
−0.937361 + 0.348361i \(0.886738\pi\)
\(422\) 0 0
\(423\) −14.9730 + 8.64469i −0.728014 + 0.420319i
\(424\) 0 0
\(425\) 1.81414 + 3.14218i 0.0879987 + 0.152418i
\(426\) 0 0
\(427\) −20.1529 11.6353i −0.975267 0.563071i
\(428\) 0 0
\(429\) 16.0349 7.38452i 0.774173 0.356528i
\(430\) 0 0
\(431\) 10.5197 + 6.07357i 0.506718 + 0.292554i 0.731483 0.681859i \(-0.238828\pi\)
−0.224766 + 0.974413i \(0.572162\pi\)
\(432\) 0 0
\(433\) −0.104510 0.181016i −0.00502242 0.00869909i 0.863503 0.504343i \(-0.168265\pi\)
−0.868526 + 0.495644i \(0.834932\pi\)
\(434\) 0 0
\(435\) −1.28994 + 0.744750i −0.0618481 + 0.0357080i
\(436\) 0 0
\(437\) 8.36033i 0.399929i
\(438\) 0 0
\(439\) −13.8603 + 24.0068i −0.661517 + 1.14578i 0.318700 + 0.947856i \(0.396754\pi\)
−0.980217 + 0.197926i \(0.936579\pi\)
\(440\) 0 0
\(441\) 13.0359 0.620755
\(442\) 0 0
\(443\) −7.98798 −0.379521 −0.189760 0.981830i \(-0.560771\pi\)
−0.189760 + 0.981830i \(0.560771\pi\)
\(444\) 0 0
\(445\) 2.76225 4.78436i 0.130943 0.226801i
\(446\) 0 0
\(447\) 59.3314i 2.80628i
\(448\) 0 0
\(449\) 6.17667 3.56610i 0.291495 0.168295i −0.347121 0.937820i \(-0.612841\pi\)
0.638616 + 0.769526i \(0.279507\pi\)
\(450\) 0 0
\(451\) −3.69615 6.40192i −0.174045 0.301455i
\(452\) 0 0
\(453\) 17.1148 + 9.88124i 0.804124 + 0.464261i
\(454\) 0 0
\(455\) 3.15937 + 6.86033i 0.148114 + 0.321617i
\(456\) 0 0
\(457\) −27.4364 15.8404i −1.28342 0.740984i −0.305949 0.952048i \(-0.598974\pi\)
−0.977472 + 0.211064i \(0.932307\pi\)
\(458\) 0 0
\(459\) −10.2105 17.6851i −0.476584 0.825468i
\(460\) 0 0
\(461\) −10.2649 + 5.92643i −0.478083 + 0.276021i −0.719617 0.694371i \(-0.755683\pi\)
0.241534 + 0.970392i \(0.422349\pi\)
\(462\) 0 0
\(463\) 12.7655i 0.593263i −0.954992 0.296632i \(-0.904137\pi\)
0.954992 0.296632i \(-0.0958633\pi\)
\(464\) 0 0
\(465\) 8.25896 14.3049i 0.383000 0.663376i
\(466\) 0 0
\(467\) 31.3402 1.45025 0.725125 0.688617i \(-0.241782\pi\)
0.725125 + 0.688617i \(0.241782\pi\)
\(468\) 0 0
\(469\) 22.6937 1.04790
\(470\) 0 0
\(471\) −5.29695 + 9.17458i −0.244070 + 0.422742i
\(472\) 0 0
\(473\) 16.1856i 0.744215i
\(474\) 0 0
\(475\) −0.926118 + 0.534695i −0.0424932 + 0.0245335i
\(476\) 0 0
\(477\) −31.4282 54.4352i −1.43900 2.49242i
\(478\) 0 0
\(479\) 13.9656 + 8.06303i 0.638103 + 0.368409i 0.783884 0.620908i \(-0.213236\pi\)
−0.145780 + 0.989317i \(0.546569\pi\)
\(480\) 0 0
\(481\) −28.6958 20.2972i −1.30842 0.925471i
\(482\) 0 0
\(483\) 40.0921 + 23.1472i 1.82426 + 1.05323i
\(484\) 0 0
\(485\) −7.59730 13.1589i −0.344976 0.597515i
\(486\) 0 0
\(487\) −4.84109 + 2.79501i −0.219371 + 0.126654i −0.605659 0.795724i \(-0.707090\pi\)
0.386288 + 0.922378i \(0.373757\pi\)
\(488\) 0 0
\(489\) 18.0304i 0.815364i
\(490\) 0 0
\(491\) −9.14772 + 15.8443i −0.412831 + 0.715044i −0.995198 0.0978817i \(-0.968793\pi\)
0.582367 + 0.812926i \(0.302127\pi\)
\(492\) 0 0
\(493\) −1.91179 −0.0861027
\(494\) 0 0
\(495\) 8.64469 0.388550
\(496\) 0 0
\(497\) 14.8407 25.7049i 0.665698 1.15302i
\(498\) 0 0
\(499\) 0.553868i 0.0247945i 0.999923 + 0.0123973i \(0.00394627\pi\)
−0.999923 + 0.0123973i \(0.996054\pi\)
\(500\) 0 0
\(501\) 38.0165 21.9489i 1.69845 0.980602i
\(502\) 0 0
\(503\) −7.34162 12.7161i −0.327347 0.566981i 0.654638 0.755943i \(-0.272821\pi\)
−0.981984 + 0.188962i \(0.939488\pi\)
\(504\) 0 0
\(505\) −3.17196 1.83133i −0.141150 0.0814931i
\(506\) 0 0
\(507\) −23.8886 + 27.9252i −1.06093 + 1.24020i
\(508\) 0 0
\(509\) 5.99545 + 3.46148i 0.265744 + 0.153427i 0.626952 0.779058i \(-0.284302\pi\)
−0.361208 + 0.932485i \(0.617636\pi\)
\(510\) 0 0
\(511\) −2.77003 4.79784i −0.122539 0.212244i
\(512\) 0 0
\(513\) 5.21245 3.00941i 0.230135 0.132869i
\(514\) 0 0
\(515\) 13.7804i 0.607239i
\(516\) 0 0
\(517\) 3.00000 5.19615i 0.131940 0.228527i
\(518\) 0 0
\(519\) 13.5011 0.592632
\(520\) 0 0
\(521\) −18.3551 −0.804152 −0.402076 0.915606i \(-0.631711\pi\)
−0.402076 + 0.915606i \(0.631711\pi\)
\(522\) 0 0
\(523\) 9.13563 15.8234i 0.399473 0.691908i −0.594188 0.804326i \(-0.702526\pi\)
0.993661 + 0.112418i \(0.0358597\pi\)
\(524\) 0 0
\(525\) 5.92163i 0.258441i
\(526\) 0 0
\(527\) 18.3606 10.6005i 0.799798 0.461764i
\(528\) 0 0
\(529\) −19.0594 33.0119i −0.828670 1.43530i
\(530\) 0 0
\(531\) 6.06726 + 3.50294i 0.263297 + 0.152014i
\(532\) 0 0
\(533\) 12.5632 + 8.88625i 0.544174 + 0.384906i
\(534\) 0 0
\(535\) −2.80342 1.61856i −0.121202 0.0699763i
\(536\) 0 0
\(537\) 22.1945 + 38.4420i 0.957763 + 1.65889i
\(538\) 0 0
\(539\) −3.91780 + 2.26194i −0.168751 + 0.0974287i
\(540\) 0 0
\(541\) 31.8881i 1.37098i −0.728084 0.685488i \(-0.759589\pi\)
0.728084 0.685488i \(-0.240411\pi\)
\(542\) 0 0
\(543\) −15.3852 + 26.6479i −0.660240 + 1.14357i
\(544\) 0 0
\(545\) −9.12979 −0.391077
\(546\) 0 0
\(547\) 44.7966 1.91537 0.957683 0.287826i \(-0.0929325\pi\)
0.957683 + 0.287826i \(0.0929325\pi\)
\(548\) 0 0
\(549\) 27.7221 48.0161i 1.18315 2.04928i
\(550\) 0 0
\(551\) 0.563476i 0.0240049i
\(552\) 0 0
\(553\) −24.6231 + 14.2162i −1.04708 + 0.604533i
\(554\) 0 0
\(555\) −13.7787 23.8654i −0.584872 1.01303i
\(556\) 0 0
\(557\) 22.7820 + 13.1532i 0.965306 + 0.557319i 0.897802 0.440400i \(-0.145163\pi\)
0.0675037 + 0.997719i \(0.478497\pi\)
\(558\) 0 0
\(559\) −14.0938 30.6037i −0.596106 1.29440i
\(560\) 0 0
\(561\) 15.3848 + 8.88244i 0.649548 + 0.375017i
\(562\) 0 0
\(563\) −2.89492 5.01415i −0.122006 0.211321i 0.798552 0.601925i \(-0.205599\pi\)
−0.920559 + 0.390604i \(0.872266\pi\)
\(564\) 0 0
\(565\) −9.48610 + 5.47680i −0.399083 + 0.230411i
\(566\) 0 0
\(567\) 1.96321i 0.0824471i
\(568\) 0 0
\(569\) 11.8184 20.4700i 0.495452 0.858149i −0.504534 0.863392i \(-0.668336\pi\)
0.999986 + 0.00524320i \(0.00166897\pi\)
\(570\) 0 0
\(571\) −11.4641 −0.479758 −0.239879 0.970803i \(-0.577108\pi\)
−0.239879 + 0.970803i \(0.577108\pi\)
\(572\) 0 0
\(573\) −16.8428 −0.703620
\(574\) 0 0
\(575\) 3.90893 6.77046i 0.163014 0.282348i
\(576\) 0 0
\(577\) 34.2415i 1.42549i 0.701422 + 0.712746i \(0.252549\pi\)
−0.701422 + 0.712746i \(0.747451\pi\)
\(578\) 0 0
\(579\) −31.2611 + 18.0486i −1.29917 + 0.750074i
\(580\) 0 0
\(581\) 16.5409 + 28.6497i 0.686233 + 1.18859i
\(582\) 0 0
\(583\) 18.8908 + 10.9066i 0.782379 + 0.451707i
\(584\) 0 0
\(585\) −16.3453 + 7.52748i −0.675797 + 0.311223i
\(586\) 0 0
\(587\) −4.88091 2.81800i −0.201457 0.116311i 0.395878 0.918303i \(-0.370440\pi\)
−0.597335 + 0.801992i \(0.703774\pi\)
\(588\) 0 0
\(589\) 3.12436 + 5.41154i 0.128737 + 0.222979i
\(590\) 0 0
\(591\) 39.4341 22.7673i 1.62210 0.936521i
\(592\) 0 0
\(593\) 15.3014i 0.628353i 0.949365 + 0.314177i \(0.101728\pi\)
−0.949365 + 0.314177i \(0.898272\pi\)
\(594\) 0 0
\(595\) −3.80024 + 6.58220i −0.155795 + 0.269844i
\(596\) 0 0
\(597\) 24.1074 0.986651
\(598\) 0 0
\(599\) 9.82414 0.401404 0.200702 0.979652i \(-0.435678\pi\)
0.200702 + 0.979652i \(0.435678\pi\)
\(600\) 0 0
\(601\) 23.0945 40.0008i 0.942043 1.63167i 0.180476 0.983579i \(-0.442236\pi\)
0.761566 0.648087i \(-0.224431\pi\)
\(602\) 0 0
\(603\) 54.0697i 2.20189i
\(604\) 0 0
\(605\) 6.92820 4.00000i 0.281672 0.162623i
\(606\) 0 0
\(607\) −12.3482 21.3877i −0.501198 0.868100i −0.999999 0.00138384i \(-0.999560\pi\)
0.498801 0.866716i \(-0.333774\pi\)
\(608\) 0 0
\(609\) −2.70216 1.56009i −0.109497 0.0632182i
\(610\) 0 0
\(611\) −1.14776 + 12.4371i −0.0464335 + 0.503153i
\(612\) 0 0
\(613\) −14.4155 8.32277i −0.582235 0.336154i 0.179786 0.983706i \(-0.442459\pi\)
−0.762021 + 0.647552i \(0.775793\pi\)
\(614\) 0 0
\(615\) 6.03240 + 10.4484i 0.243250 + 0.421321i
\(616\) 0 0
\(617\) −1.39918 + 0.807820i −0.0563291 + 0.0325216i −0.527900 0.849307i \(-0.677020\pi\)
0.471571 + 0.881828i \(0.343687\pi\)
\(618\) 0 0
\(619\) 23.3922i 0.940213i −0.882610 0.470106i \(-0.844216\pi\)
0.882610 0.470106i \(-0.155784\pi\)
\(620\) 0 0
\(621\) −22.0005 + 38.1060i −0.882851 + 1.52914i
\(622\) 0 0
\(623\) 11.5727 0.463649
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.61799 + 4.53449i −0.104552 + 0.181090i
\(628\) 0 0
\(629\) 35.3701i 1.41030i
\(630\) 0 0
\(631\) 2.77458 1.60190i 0.110454 0.0637708i −0.443755 0.896148i \(-0.646354\pi\)
0.554209 + 0.832377i \(0.313021\pi\)
\(632\) 0 0
\(633\) 5.90893 + 10.2346i 0.234859 + 0.406787i
\(634\) 0 0
\(635\) 0.786022 + 0.453810i 0.0311923 + 0.0180089i
\(636\) 0 0
\(637\) 5.43813 7.68834i 0.215467 0.304623i
\(638\) 0 0
\(639\) 61.2442 + 35.3593i 2.42278 + 1.39879i
\(640\) 0 0
\(641\) −9.98794 17.2996i −0.394500 0.683294i 0.598537 0.801095i \(-0.295749\pi\)
−0.993037 + 0.117801i \(0.962416\pi\)
\(642\) 0 0
\(643\) −24.7136 + 14.2684i −0.974607 + 0.562690i −0.900638 0.434571i \(-0.856900\pi\)
−0.0739696 + 0.997260i \(0.523567\pi\)
\(644\) 0 0
\(645\) 26.4161i 1.04013i
\(646\) 0 0
\(647\) 3.04325 5.27107i 0.119643 0.207227i −0.799983 0.600022i \(-0.795158\pi\)
0.919626 + 0.392795i \(0.128492\pi\)
\(648\) 0 0
\(649\) −2.43127 −0.0954359
\(650\) 0 0
\(651\) 34.6016 1.35614
\(652\) 0 0
\(653\) −9.81499 + 17.0001i −0.384090 + 0.665264i −0.991643 0.129016i \(-0.958818\pi\)
0.607552 + 0.794280i \(0.292152\pi\)
\(654\) 0 0
\(655\) 13.1626i 0.514306i
\(656\) 0 0
\(657\) 11.4313 6.59985i 0.445976 0.257485i
\(658\) 0 0
\(659\) 9.79752 + 16.9698i 0.381657 + 0.661049i 0.991299 0.131627i \(-0.0420202\pi\)
−0.609642 + 0.792677i \(0.708687\pi\)
\(660\) 0 0
\(661\) 30.0735 + 17.3630i 1.16972 + 0.675341i 0.953615 0.301029i \(-0.0973300\pi\)
0.216109 + 0.976369i \(0.430663\pi\)
\(662\) 0 0
\(663\) −36.8241 3.39831i −1.43013 0.131980i
\(664\) 0 0
\(665\) −1.94002 1.12007i −0.0752308 0.0434345i
\(666\) 0 0
\(667\) 2.05967 + 3.56745i 0.0797506 + 0.138132i
\(668\) 0 0
\(669\) 25.7030 14.8396i 0.993736 0.573734i
\(670\) 0 0
\(671\) 19.2410i 0.742790i
\(672\) 0 0
\(673\) −17.0051 + 29.4538i −0.655500 + 1.13536i 0.326268 + 0.945277i \(0.394209\pi\)
−0.981768 + 0.190082i \(0.939125\pi\)
\(674\) 0 0
\(675\) −5.62828 −0.216633
\(676\) 0 0
\(677\) 29.9209 1.14995 0.574977 0.818169i \(-0.305011\pi\)
0.574977 + 0.818169i \(0.305011\pi\)
\(678\) 0 0
\(679\) 15.9147 27.5651i 0.610751 1.05785i
\(680\) 0 0
\(681\) 44.0584i 1.68832i
\(682\) 0 0
\(683\) 24.1251 13.9286i 0.923121 0.532964i 0.0384916 0.999259i \(-0.487745\pi\)
0.884629 + 0.466295i \(0.154411\pi\)
\(684\) 0 0
\(685\) 6.03240 + 10.4484i 0.230486 + 0.399214i
\(686\) 0 0
\(687\) −47.1786 27.2386i −1.79998 1.03922i
\(688\) 0 0
\(689\) −45.2158 4.17274i −1.72258 0.158969i
\(690\) 0 0
\(691\) 31.4550 + 18.1606i 1.19661 + 0.690860i 0.959797 0.280696i \(-0.0905652\pi\)
0.236809 + 0.971556i \(0.423899\pi\)
\(692\) 0 0
\(693\) 9.05440 + 15.6827i 0.343948 + 0.595736i
\(694\) 0 0
\(695\) −4.86002 + 2.80593i −0.184351 + 0.106435i
\(696\) 0 0
\(697\) 15.4853i 0.586548i
\(698\) 0 0
\(699\) −3.51651 + 6.09077i −0.133007 + 0.230374i
\(700\) 0 0
\(701\) 2.16156 0.0816411 0.0408206 0.999166i \(-0.487003\pi\)
0.0408206 + 0.999166i \(0.487003\pi\)
\(702\) 0 0
\(703\) 10.4249 0.393183
\(704\) 0 0
\(705\) −4.89623 + 8.48052i −0.184403 + 0.319395i
\(706\) 0 0
\(707\) 7.67250i 0.288554i
\(708\) 0 0
\(709\) −41.2371 + 23.8082i −1.54869 + 0.894137i −0.550449 + 0.834869i \(0.685543\pi\)
−0.998242 + 0.0592680i \(0.981123\pi\)
\(710\) 0 0
\(711\) −33.8713 58.6668i −1.27027 2.20018i
\(712\) 0 0
\(713\) −39.5615 22.8408i −1.48159 0.855396i
\(714\) 0 0
\(715\) 3.60628 5.09850i 0.134867 0.190673i
\(716\) 0 0
\(717\) −41.0733 23.7137i −1.53391 0.885603i
\(718\) 0 0
\(719\) −10.1469 17.5749i −0.378414 0.655433i 0.612417 0.790535i \(-0.290197\pi\)
−0.990832 + 0.135102i \(0.956864\pi\)
\(720\) 0 0
\(721\) −24.9996 + 14.4335i −0.931035 + 0.537533i
\(722\) 0 0
\(723\) 82.2029i 3.05716i
\(724\) 0 0
\(725\) −0.263457 + 0.456321i −0.00978454 + 0.0169473i
\(726\) 0 0
\(727\) 11.0681 0.410494 0.205247 0.978710i \(-0.434200\pi\)
0.205247 + 0.978710i \(0.434200\pi\)
\(728\) 0 0
\(729\) −43.0532 −1.59456
\(730\) 0 0
\(731\) 16.9527 29.3630i 0.627019 1.08603i
\(732\) 0 0
\(733\) 23.4002i 0.864304i 0.901801 + 0.432152i \(0.142246\pi\)
−0.901801 + 0.432152i \(0.857754\pi\)
\(734\) 0 0
\(735\) 6.39414 3.69166i 0.235852 0.136169i
\(736\) 0 0
\(737\) −9.38201 16.2501i −0.345591 0.598581i
\(738\) 0 0
\(739\) −22.1617 12.7951i −0.815232 0.470675i 0.0335372 0.999437i \(-0.489323\pi\)
−0.848770 + 0.528763i \(0.822656\pi\)
\(740\) 0 0
\(741\) 1.00161 10.8534i 0.0367951 0.398711i
\(742\) 0 0
\(743\) 7.28694 + 4.20712i 0.267332 + 0.154344i 0.627675 0.778476i \(-0.284007\pi\)
−0.360343 + 0.932820i \(0.617340\pi\)
\(744\) 0 0
\(745\) 10.4943 + 18.1767i 0.384482 + 0.665942i
\(746\) 0 0
\(747\) −68.2605 + 39.4102i −2.49752 + 1.44194i
\(748\) 0 0
\(749\) 6.78106i 0.247775i
\(750\) 0 0
\(751\) 20.2595 35.0905i 0.739279 1.28047i −0.213541 0.976934i \(-0.568500\pi\)
0.952820 0.303535i \(-0.0981671\pi\)
\(752\) 0 0
\(753\) −54.0514 −1.96974
\(754\) 0 0
\(755\) 6.99102 0.254429
\(756\) 0 0
\(757\) 5.67593 9.83100i 0.206295 0.357314i −0.744249 0.667902i \(-0.767193\pi\)
0.950545 + 0.310588i \(0.100526\pi\)
\(758\) 0 0
\(759\) 38.2780i 1.38940i
\(760\) 0 0
\(761\) −5.86364 + 3.38538i −0.212557 + 0.122720i −0.602499 0.798120i \(-0.705828\pi\)
0.389942 + 0.920839i \(0.372495\pi\)
\(762\) 0 0
\(763\) −9.56249 16.5627i −0.346185 0.599611i
\(764\) 0 0
\(765\) −15.6827 9.05440i −0.567008 0.327362i
\(766\) 0 0
\(767\) 4.59704 2.11707i 0.165990 0.0764428i
\(768\) 0 0
\(769\) −20.0563 11.5795i −0.723250 0.417569i 0.0926975 0.995694i \(-0.470451\pi\)
−0.815948 + 0.578126i \(0.803784\pi\)
\(770\) 0 0
\(771\) 19.3766 + 33.5613i 0.697833 + 1.20868i
\(772\) 0 0
\(773\) 36.3333 20.9770i 1.30682 0.754491i 0.325253 0.945627i \(-0.394551\pi\)
0.981564 + 0.191136i \(0.0612172\pi\)
\(774\) 0 0
\(775\) 5.84325i 0.209896i
\(776\) 0 0
\(777\) 28.8634 49.9928i 1.03547 1.79348i
\(778\) 0 0
\(779\) −4.56410 −0.163526
\(780\) 0 0
\(781\) −24.5418 −0.878174
\(782\) 0 0
\(783\) 1.48281 2.56830i 0.0529913 0.0917835i
\(784\) 0 0
\(785\) 3.74761i 0.133758i
\(786\) 0 0
\(787\) 32.4026 18.7076i 1.15503 0.666856i 0.204920 0.978779i \(-0.434306\pi\)
0.950107 + 0.311923i \(0.100973\pi\)
\(788\) 0 0
\(789\) 16.8388 + 29.1656i 0.599476 + 1.03832i
\(790\) 0 0
\(791\) −19.8714 11.4727i −0.706544 0.407923i
\(792\) 0 0
\(793\) −16.7544 36.3808i −0.594965 1.29192i
\(794\) 0 0
\(795\) −30.8313 17.8005i −1.09347 0.631317i
\(796\) 0 0
\(797\) 23.6200 + 40.9110i 0.836663 + 1.44914i 0.892670 + 0.450712i \(0.148830\pi\)
−0.0560071 + 0.998430i \(0.517837\pi\)
\(798\) 0 0
\(799\) −10.8848 + 6.28436i −0.385078 + 0.222325i
\(800\) 0 0
\(801\) 27.5729i 0.974240i
\(802\) 0 0
\(803\) −2.29037 + 3.96704i −0.0808254 + 0.139994i
\(804\) 0 0
\(805\) 16.3767 0.577204
\(806\) 0 0
\(807\) −59.1692 −2.08286
\(808\) 0 0
\(809\) 23.2371 40.2478i 0.816972 1.41504i −0.0909313 0.995857i \(-0.528984\pi\)
0.907903 0.419180i \(-0.137682\pi\)
\(810\) 0 0
\(811\) 11.4041i 0.400453i −0.979750 0.200227i \(-0.935832\pi\)
0.979750 0.200227i \(-0.0641678\pi\)
\(812\) 0 0
\(813\) −4.77777 + 2.75844i −0.167564 + 0.0967429i
\(814\) 0 0
\(815\) −3.18915 5.52377i −0.111711 0.193489i
\(816\) 0 0
\(817\) 8.65436 + 4.99660i 0.302778 + 0.174809i
\(818\) 0 0
\(819\) −30.7759 21.7685i −1.07540 0.760652i
\(820\) 0 0
\(821\) −21.2709 12.2808i −0.742359 0.428601i 0.0805674 0.996749i \(-0.474327\pi\)
−0.822926 + 0.568148i \(0.807660\pi\)
\(822\) 0 0
\(823\) 1.48224 + 2.56731i 0.0516676 + 0.0894909i 0.890702 0.454587i \(-0.150213\pi\)
−0.839035 + 0.544078i \(0.816880\pi\)
\(824\) 0 0
\(825\) 4.24026 2.44811i 0.147627 0.0852324i
\(826\) 0 0
\(827\) 17.7265i 0.616412i −0.951320 0.308206i \(-0.900271\pi\)
0.951320 0.308206i \(-0.0997286\pi\)
\(828\) 0 0
\(829\) −5.78791 + 10.0250i −0.201022 + 0.348181i −0.948858 0.315703i \(-0.897760\pi\)
0.747836 + 0.663884i \(0.231093\pi\)
\(830\) 0 0
\(831\) −35.2804 −1.22386
\(832\) 0 0
\(833\) 9.47657 0.328344
\(834\) 0 0
\(835\) 7.76445 13.4484i 0.268700 0.465402i
\(836\) 0 0
\(837\) 32.8874i 1.13676i
\(838\) 0 0
\(839\) 37.1348 21.4398i 1.28203 0.740183i 0.304815 0.952412i \(-0.401405\pi\)
0.977220 + 0.212229i \(0.0680721\pi\)
\(840\) 0 0
\(841\) 14.3612 + 24.8743i 0.495213 + 0.857734i
\(842\) 0 0
\(843\) 5.61971 + 3.24454i 0.193553 + 0.111748i
\(844\) 0 0
\(845\) −2.37915 + 12.7804i −0.0818453 + 0.439660i
\(846\) 0 0
\(847\) 14.5131 + 8.37915i 0.498677 + 0.287911i
\(848\) 0 0
\(849\) 21.1019 + 36.5496i 0.724215 + 1.25438i
\(850\) 0 0
\(851\) −66.0016 + 38.1060i −2.26251 + 1.30626i
\(852\) 0 0
\(853\) 13.4599i 0.460857i 0.973089 + 0.230428i \(0.0740127\pi\)
−0.973089 + 0.230428i \(0.925987\pi\)
\(854\) 0 0
\(855\) 2.66867 4.62227i 0.0912666 0.158078i
\(856\) 0 0
\(857\) −10.5950 −0.361919 −0.180960 0.983491i \(-0.557920\pi\)
−0.180960 + 0.983491i \(0.557920\pi\)
\(858\) 0 0
\(859\) −8.75716 −0.298791 −0.149395 0.988778i \(-0.547733\pi\)
−0.149395 + 0.988778i \(0.547733\pi\)
\(860\) 0 0
\(861\) −12.6366 + 21.8872i −0.430654 + 0.745915i
\(862\) 0 0
\(863\) 30.8640i 1.05062i −0.850910 0.525311i \(-0.823949\pi\)
0.850910 0.525311i \(-0.176051\pi\)
\(864\) 0 0
\(865\) 4.13617 2.38802i 0.140634 0.0811951i
\(866\) 0 0
\(867\) 5.42130 + 9.38997i 0.184117 + 0.318900i
\(868\) 0 0
\(869\) 20.3593 + 11.7545i 0.690643 + 0.398743i
\(870\) 0 0
\(871\) 31.8895 + 22.5561i 1.08053 + 0.764285i
\(872\) 0 0
\(873\) 65.6763 + 37.9182i 2.22281 + 1.28334i
\(874\) 0 0
\(875\) 1.04739 + 1.81414i 0.0354084 + 0.0613291i
\(876\) 0 0
\(877\) 17.4103 10.0518i 0.587904 0.339427i −0.176364 0.984325i \(-0.556434\pi\)
0.764268 + 0.644898i \(0.223100\pi\)
\(878\) 0 0
\(879\) 4.04839i 0.136549i
\(880\) 0 0
\(881\) 1.56698 2.71409i 0.0527930 0.0914401i −0.838421 0.545023i \(-0.816521\pi\)
0.891214 + 0.453583i \(0.149854\pi\)
\(882\) 0 0
\(883\) −34.0429 −1.14563 −0.572817 0.819683i \(-0.694149\pi\)
−0.572817 + 0.819683i \(0.694149\pi\)
\(884\) 0 0
\(885\) 3.96802 0.133384
\(886\) 0 0
\(887\) −25.4034 + 44.0001i −0.852964 + 1.47738i 0.0255565 + 0.999673i \(0.491864\pi\)
−0.878521 + 0.477704i \(0.841469\pi\)
\(888\) 0 0
\(889\) 1.90127i 0.0637666i
\(890\) 0 0
\(891\) −1.40578 + 0.811629i −0.0470955 + 0.0271906i
\(892\) 0 0
\(893\) −1.85224 3.20817i −0.0619827 0.107357i
\(894\) 0 0
\(895\) 13.5989 + 7.85134i 0.454562 + 0.262442i
\(896\) 0 0
\(897\) 33.3311 + 72.3758i 1.11289 + 2.41656i
\(898\) 0 0
\(899\) 2.66640 + 1.53944i 0.0889293 + 0.0513434i
\(900\) 0 0
\(901\) −22.8471 39.5723i −0.761147 1.31834i
\(902\) 0 0
\(903\) 47.9226 27.6681i 1.59476 0.920737i
\(904\) 0 0
\(905\) 10.8851i 0.361832i
\(906\) 0 0
\(907\) −25.4803 + 44.1331i −0.846058 + 1.46542i 0.0386406 + 0.999253i \(0.487697\pi\)
−0.884699 + 0.466163i \(0.845636\pi\)
\(908\) 0 0
\(909\) 18.2804 0.606323
\(910\) 0 0
\(911\) −23.7176 −0.785800 −0.392900 0.919581i \(-0.628528\pi\)
−0.392900 + 0.919581i \(0.628528\pi\)
\(912\) 0 0
\(913\) 13.6767 23.6887i 0.452632 0.783981i
\(914\) 0 0
\(915\) 31.4028i 1.03814i
\(916\) 0 0
\(917\) 23.8788 13.7864i 0.788548 0.455269i
\(918\) 0 0
\(919\) 21.1516 + 36.6356i 0.697725 + 1.20850i 0.969253 + 0.246065i \(0.0791377\pi\)
−0.271528 + 0.962431i \(0.587529\pi\)
\(920\) 0 0
\(921\) −42.5563 24.5699i −1.40228 0.809606i
\(922\) 0 0
\(923\) 46.4034 21.3701i 1.52739 0.703405i
\(924\) 0 0
\(925\) −8.44242 4.87423i −0.277585 0.160264i
\(926\) 0 0
\(927\) −34.3892 59.5638i −1.12949 1.95633i
\(928\) 0 0
\(929\) −11.5432 + 6.66449i −0.378721 + 0.218655i −0.677262 0.735742i \(-0.736834\pi\)
0.298541 + 0.954397i \(0.403500\pi\)
\(930\) 0 0
\(931\) 2.79310i 0.0915402i
\(932\) 0 0
\(933\) 28.5743 49.4922i 0.935481 1.62030i
\(934\) 0 0
\(935\) 6.28436 0.205521
\(936\) 0 0
\(937\) 14.0848 0.460129 0.230065 0.973175i \(-0.426106\pi\)
0.230065 + 0.973175i \(0.426106\pi\)
\(938\) 0 0
\(939\) 6.87523 11.9082i 0.224365 0.388611i
\(940\) 0 0
\(941\) 49.0399i 1.59866i 0.600895 + 0.799328i \(0.294811\pi\)
−0.600895 + 0.799328i \(0.705189\pi\)
\(942\) 0 0
\(943\) 28.8960 16.6831i 0.940983 0.543277i
\(944\) 0 0
\(945\) −5.89502 10.2105i −0.191765 0.332147i
\(946\) 0 0
\(947\) −15.5323 8.96760i −0.504733 0.291408i 0.225933 0.974143i \(-0.427457\pi\)
−0.730666 + 0.682735i \(0.760790\pi\)
\(948\) 0 0
\(949\) 0.876268 9.49523i 0.0284449 0.308228i
\(950\) 0 0
\(951\) 57.8189 + 33.3818i 1.87491 + 1.08248i
\(952\) 0 0
\(953\) 22.1915 + 38.4368i 0.718853 + 1.24509i 0.961454 + 0.274964i \(0.0886661\pi\)
−0.242601 + 0.970126i \(0.578001\pi\)
\(954\) 0 0
\(955\) −5.15994 + 2.97909i −0.166972 + 0.0964012i
\(956\) 0 0
\(957\) 2.57989i 0.0833960i
\(958\) 0 0
\(959\) −12.6366 + 21.8872i −0.408057 + 0.706776i
\(960\) 0 0
\(961\) −3.14359 −0.101406
\(962\) 0 0
\(963\) 16.1565 0.520635
\(964\) 0 0
\(965\) −6.38473 + 11.0587i −0.205532 + 0.355992i
\(966\) 0 0
\(967\) 24.3595i 0.783348i 0.920104 + 0.391674i \(0.128104\pi\)
−0.920104 + 0.391674i \(0.871896\pi\)
\(968\) 0 0
\(969\) 9.49879 5.48413i 0.305145 0.176176i
\(970\) 0 0
\(971\) 0.587359 + 1.01734i 0.0188492 + 0.0326478i 0.875296 0.483587i \(-0.160666\pi\)
−0.856447 + 0.516235i \(0.827333\pi\)
\(972\) 0 0
\(973\) −10.1807 5.87783i −0.326378 0.188435i
\(974\) 0 0
\(975\) −5.88573 + 8.32114i −0.188494 + 0.266490i
\(976\) 0 0
\(977\) 10.6609 + 6.15505i 0.341071 + 0.196917i 0.660746 0.750610i \(-0.270240\pi\)
−0.319675 + 0.947527i \(0.603574\pi\)
\(978\) 0 0
\(979\) −4.78436 8.28676i −0.152909 0.264846i
\(980\) 0 0
\(981\) 39.4621 22.7835i 1.25993 0.727420i
\(982\) 0 0
\(983\) 2.29060i 0.0730589i 0.999333 + 0.0365295i \(0.0116303\pi\)
−0.999333 + 0.0365295i \(0.988370\pi\)
\(984\) 0 0
\(985\) 8.05397 13.9499i 0.256621 0.444480i
\(986\) 0 0
\(987\) −20.5131 −0.652940
\(988\) 0 0
\(989\) −73.0560 −2.32305
\(990\) 0 0
\(991\) −10.8499 + 18.7926i −0.344659 + 0.596967i −0.985292 0.170880i \(-0.945339\pi\)
0.640633 + 0.767848i \(0.278672\pi\)
\(992\) 0 0
\(993\) 42.0628i 1.33482i
\(994\) 0 0
\(995\) 7.38551 4.26403i 0.234136 0.135179i
\(996\) 0 0
\(997\) 9.53125 + 16.5086i 0.301858 + 0.522833i 0.976557 0.215260i \(-0.0690599\pi\)
−0.674699 + 0.738093i \(0.735727\pi\)
\(998\) 0 0
\(999\) 47.5163 + 27.4335i 1.50335 + 0.867959i
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 260.2.x.a.121.1 yes 8
3.2 odd 2 2340.2.dj.d.901.1 8
4.3 odd 2 1040.2.da.c.641.4 8
5.2 odd 4 1300.2.ba.c.849.4 8
5.3 odd 4 1300.2.ba.b.849.1 8
5.4 even 2 1300.2.y.b.901.4 8
13.4 even 6 3380.2.f.i.3041.7 8
13.6 odd 12 3380.2.a.q.1.4 4
13.7 odd 12 3380.2.a.p.1.4 4
13.9 even 3 3380.2.f.i.3041.8 8
13.10 even 6 inner 260.2.x.a.101.1 8
39.23 odd 6 2340.2.dj.d.361.3 8
52.23 odd 6 1040.2.da.c.881.4 8
65.23 odd 12 1300.2.ba.c.49.4 8
65.49 even 6 1300.2.y.b.101.4 8
65.62 odd 12 1300.2.ba.b.49.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.x.a.101.1 8 13.10 even 6 inner
260.2.x.a.121.1 yes 8 1.1 even 1 trivial
1040.2.da.c.641.4 8 4.3 odd 2
1040.2.da.c.881.4 8 52.23 odd 6
1300.2.y.b.101.4 8 65.49 even 6
1300.2.y.b.901.4 8 5.4 even 2
1300.2.ba.b.49.1 8 65.62 odd 12
1300.2.ba.b.849.1 8 5.3 odd 4
1300.2.ba.c.49.4 8 65.23 odd 12
1300.2.ba.c.849.4 8 5.2 odd 4
2340.2.dj.d.361.3 8 39.23 odd 6
2340.2.dj.d.901.1 8 3.2 odd 2
3380.2.a.p.1.4 4 13.7 odd 12
3380.2.a.q.1.4 4 13.6 odd 12
3380.2.f.i.3041.7 8 13.4 even 6
3380.2.f.i.3041.8 8 13.9 even 3