Properties

Label 2646.2.f.c.1765.1
Level $2646$
Weight $2$
Character 2646.1765
Analytic conductor $21.128$
Analytic rank $0$
Dimension $2$
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] = [2646,2,Mod(883,2646)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2646, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2646.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1765.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2646.1765
Dual form 2646.2.f.c.883.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.50000 - 2.59808i) q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.50000 - 2.59808i) q^{5} +1.00000 q^{8} -3.00000 q^{10} +(-3.00000 - 5.19615i) q^{11} +(1.00000 - 1.73205i) q^{13} +(-0.500000 - 0.866025i) q^{16} +6.00000 q^{17} +7.00000 q^{19} +(1.50000 + 2.59808i) q^{20} +(-3.00000 + 5.19615i) q^{22} +(1.50000 - 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} -2.00000 q^{26} +(3.00000 + 5.19615i) q^{29} +(1.00000 - 1.73205i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-3.00000 - 5.19615i) q^{34} +2.00000 q^{37} +(-3.50000 - 6.06218i) q^{38} +(1.50000 - 2.59808i) q^{40} +(-1.00000 - 1.73205i) q^{43} +6.00000 q^{44} -3.00000 q^{46} +(-2.00000 + 3.46410i) q^{50} +(1.00000 + 1.73205i) q^{52} -6.00000 q^{53} -18.0000 q^{55} +(3.00000 - 5.19615i) q^{58} +(2.50000 + 4.33013i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-3.00000 - 5.19615i) q^{65} +(-4.00000 + 6.92820i) q^{67} +(-3.00000 + 5.19615i) q^{68} -3.00000 q^{71} -2.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(-3.50000 + 6.06218i) q^{76} +(-2.50000 - 4.33013i) q^{79} -3.00000 q^{80} +(-6.00000 - 10.3923i) q^{83} +(9.00000 - 15.5885i) q^{85} +(-1.00000 + 1.73205i) q^{86} +(-3.00000 - 5.19615i) q^{88} +(1.50000 + 2.59808i) q^{92} +(10.5000 - 18.1865i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{8} - 6 q^{10} - 6 q^{11} + 2 q^{13} - q^{16} + 12 q^{17} + 14 q^{19} + 3 q^{20} - 6 q^{22} + 3 q^{23} - 4 q^{25} - 4 q^{26} + 6 q^{29} + 2 q^{31} - q^{32} - 6 q^{34} + 4 q^{37} - 7 q^{38} + 3 q^{40} - 2 q^{43} + 12 q^{44} - 6 q^{46} - 4 q^{50} + 2 q^{52} - 12 q^{53} - 36 q^{55} + 6 q^{58} + 5 q^{61} - 4 q^{62} + 2 q^{64} - 6 q^{65} - 8 q^{67} - 6 q^{68} - 6 q^{71} - 4 q^{73} - 2 q^{74} - 7 q^{76} - 5 q^{79} - 6 q^{80} - 12 q^{83} + 18 q^{85} - 2 q^{86} - 6 q^{88} + 3 q^{92} + 21 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 0.866025i −0.353553 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 1.50000 2.59808i 0.670820 1.16190i −0.306851 0.951757i \(-0.599275\pi\)
0.977672 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i \(-0.806886\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) 1.00000 1.73205i 0.277350 0.480384i −0.693375 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 1.50000 + 2.59808i 0.335410 + 0.580948i
\(21\) 0 0
\(22\) −3.00000 + 5.19615i −0.639602 + 1.10782i
\(23\) 1.50000 2.59808i 0.312772 0.541736i −0.666190 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) 1.00000 1.73205i 0.179605 0.311086i −0.762140 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −3.00000 5.19615i −0.514496 0.891133i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −3.50000 6.06218i −0.567775 0.983415i
\(39\) 0 0
\(40\) 1.50000 2.59808i 0.237171 0.410792i
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −1.00000 1.73205i −0.152499 0.264135i 0.779647 0.626219i \(-0.215399\pi\)
−0.932145 + 0.362084i \(0.882065\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.00000 + 3.46410i −0.282843 + 0.489898i
\(51\) 0 0
\(52\) 1.00000 + 1.73205i 0.138675 + 0.240192i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) 3.00000 5.19615i 0.393919 0.682288i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 5.19615i −0.372104 0.644503i
\(66\) 0 0
\(67\) −4.00000 + 6.92820i −0.488678 + 0.846415i −0.999915 0.0130248i \(-0.995854\pi\)
0.511237 + 0.859440i \(0.329187\pi\)
\(68\) −3.00000 + 5.19615i −0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.00000 1.73205i −0.116248 0.201347i
\(75\) 0 0
\(76\) −3.50000 + 6.06218i −0.401478 + 0.695379i
\(77\) 0 0
\(78\) 0 0
\(79\) −2.50000 4.33013i −0.281272 0.487177i 0.690426 0.723403i \(-0.257423\pi\)
−0.971698 + 0.236225i \(0.924090\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 10.3923i −0.658586 1.14070i −0.980982 0.194099i \(-0.937822\pi\)
0.322396 0.946605i \(-0.395512\pi\)
\(84\) 0 0
\(85\) 9.00000 15.5885i 0.976187 1.69081i
\(86\) −1.00000 + 1.73205i −0.107833 + 0.186772i
\(87\) 0 0
\(88\) −3.00000 5.19615i −0.319801 0.553912i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.50000 + 2.59808i 0.156386 + 0.270868i
\(93\) 0 0
\(94\) 0 0
\(95\) 10.5000 18.1865i 1.07728 1.86590i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −4.50000 7.79423i −0.447767 0.775555i 0.550474 0.834853i \(-0.314447\pi\)
−0.998240 + 0.0592978i \(0.981114\pi\)
\(102\) 0 0
\(103\) −5.00000 + 8.66025i −0.492665 + 0.853320i −0.999964 0.00844953i \(-0.997310\pi\)
0.507300 + 0.861770i \(0.330644\pi\)
\(104\) 1.00000 1.73205i 0.0980581 0.169842i
\(105\) 0 0
\(106\) 3.00000 + 5.19615i 0.291386 + 0.504695i
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 9.00000 + 15.5885i 0.858116 + 1.48630i
\(111\) 0 0
\(112\) 0 0
\(113\) 7.50000 12.9904i 0.705541 1.22203i −0.260955 0.965351i \(-0.584038\pi\)
0.966496 0.256681i \(-0.0826291\pi\)
\(114\) 0 0
\(115\) −4.50000 7.79423i −0.419627 0.726816i
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 2.50000 4.33013i 0.226339 0.392031i
\(123\) 0 0
\(124\) 1.00000 + 1.73205i 0.0898027 + 0.155543i
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) −0.500000 0.866025i −0.0441942 0.0765466i
\(129\) 0 0
\(130\) −3.00000 + 5.19615i −0.263117 + 0.455733i
\(131\) 4.50000 7.79423i 0.393167 0.680985i −0.599699 0.800226i \(-0.704713\pi\)
0.992865 + 0.119241i \(0.0380462\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 3.00000 + 5.19615i 0.256307 + 0.443937i 0.965250 0.261329i \(-0.0841608\pi\)
−0.708942 + 0.705266i \(0.750827\pi\)
\(138\) 0 0
\(139\) 2.50000 4.33013i 0.212047 0.367277i −0.740308 0.672268i \(-0.765320\pi\)
0.952355 + 0.304991i \(0.0986536\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.50000 + 2.59808i 0.125877 + 0.218026i
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) 1.00000 + 1.73205i 0.0827606 + 0.143346i
\(147\) 0 0
\(148\) −1.00000 + 1.73205i −0.0821995 + 0.142374i
\(149\) −3.00000 + 5.19615i −0.245770 + 0.425685i −0.962348 0.271821i \(-0.912374\pi\)
0.716578 + 0.697507i \(0.245707\pi\)
\(150\) 0 0
\(151\) −11.5000 19.9186i −0.935857 1.62095i −0.773099 0.634285i \(-0.781294\pi\)
−0.162758 0.986666i \(-0.552039\pi\)
\(152\) 7.00000 0.567775
\(153\) 0 0
\(154\) 0 0
\(155\) −3.00000 5.19615i −0.240966 0.417365i
\(156\) 0 0
\(157\) −6.50000 + 11.2583i −0.518756 + 0.898513i 0.481006 + 0.876717i \(0.340272\pi\)
−0.999762 + 0.0217953i \(0.993062\pi\)
\(158\) −2.50000 + 4.33013i −0.198889 + 0.344486i
\(159\) 0 0
\(160\) 1.50000 + 2.59808i 0.118585 + 0.205396i
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −6.00000 + 10.3923i −0.465690 + 0.806599i
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) 4.50000 + 7.79423i 0.346154 + 0.599556i
\(170\) −18.0000 −1.38054
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 3.00000 + 5.19615i 0.228086 + 0.395056i 0.957241 0.289292i \(-0.0934200\pi\)
−0.729155 + 0.684349i \(0.760087\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 + 5.19615i −0.226134 + 0.391675i
\(177\) 0 0
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 25.0000 1.85824 0.929118 0.369784i \(-0.120568\pi\)
0.929118 + 0.369784i \(0.120568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.50000 2.59808i 0.110581 0.191533i
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) −18.0000 31.1769i −1.31629 2.27988i
\(188\) 0 0
\(189\) 0 0
\(190\) −21.0000 −1.52350
\(191\) −4.50000 7.79423i −0.325609 0.563971i 0.656027 0.754738i \(-0.272236\pi\)
−0.981635 + 0.190767i \(0.938902\pi\)
\(192\) 0 0
\(193\) −8.50000 + 14.7224i −0.611843 + 1.05974i 0.379086 + 0.925361i \(0.376238\pi\)
−0.990930 + 0.134382i \(0.957095\pi\)
\(194\) 1.00000 1.73205i 0.0717958 0.124354i
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −2.00000 3.46410i −0.141421 0.244949i
\(201\) 0 0
\(202\) −4.50000 + 7.79423i −0.316619 + 0.548400i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 10.0000 0.696733
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −21.0000 36.3731i −1.45260 2.51598i
\(210\) 0 0
\(211\) −4.00000 + 6.92820i −0.275371 + 0.476957i −0.970229 0.242190i \(-0.922134\pi\)
0.694857 + 0.719148i \(0.255467\pi\)
\(212\) 3.00000 5.19615i 0.206041 0.356873i
\(213\) 0 0
\(214\) −6.00000 10.3923i −0.410152 0.710403i
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 5.00000 + 8.66025i 0.338643 + 0.586546i
\(219\) 0 0
\(220\) 9.00000 15.5885i 0.606780 1.05097i
\(221\) 6.00000 10.3923i 0.403604 0.699062i
\(222\) 0 0
\(223\) −14.0000 24.2487i −0.937509 1.62381i −0.770097 0.637927i \(-0.779792\pi\)
−0.167412 0.985887i \(-0.553541\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 7.50000 + 12.9904i 0.497792 + 0.862202i 0.999997 0.00254715i \(-0.000810783\pi\)
−0.502204 + 0.864749i \(0.667477\pi\)
\(228\) 0 0
\(229\) −0.500000 + 0.866025i −0.0330409 + 0.0572286i −0.882073 0.471113i \(-0.843853\pi\)
0.849032 + 0.528341i \(0.177186\pi\)
\(230\) −4.50000 + 7.79423i −0.296721 + 0.513936i
\(231\) 0 0
\(232\) 3.00000 + 5.19615i 0.196960 + 0.341144i
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.50000 + 12.9904i −0.485135 + 0.840278i −0.999854 0.0170808i \(-0.994563\pi\)
0.514719 + 0.857359i \(0.327896\pi\)
\(240\) 0 0
\(241\) 4.00000 + 6.92820i 0.257663 + 0.446285i 0.965615 0.259975i \(-0.0837143\pi\)
−0.707953 + 0.706260i \(0.750381\pi\)
\(242\) 25.0000 1.60706
\(243\) 0 0
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) 7.00000 12.1244i 0.445399 0.771454i
\(248\) 1.00000 1.73205i 0.0635001 0.109985i
\(249\) 0 0
\(250\) −1.50000 2.59808i −0.0948683 0.164317i
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) −8.50000 14.7224i −0.533337 0.923768i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) −9.00000 + 15.5885i −0.561405 + 0.972381i 0.435970 + 0.899961i \(0.356405\pi\)
−0.997374 + 0.0724199i \(0.976928\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −9.00000 −0.556022
\(263\) −10.5000 18.1865i −0.647458 1.12143i −0.983728 0.179664i \(-0.942499\pi\)
0.336270 0.941766i \(-0.390834\pi\)
\(264\) 0 0
\(265\) −9.00000 + 15.5885i −0.552866 + 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 6.92820i −0.244339 0.423207i
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −3.00000 5.19615i −0.181902 0.315063i
\(273\) 0 0
\(274\) 3.00000 5.19615i 0.181237 0.313911i
\(275\) −12.0000 + 20.7846i −0.723627 + 1.25336i
\(276\) 0 0
\(277\) 8.00000 + 13.8564i 0.480673 + 0.832551i 0.999754 0.0221745i \(-0.00705893\pi\)
−0.519081 + 0.854725i \(0.673726\pi\)
\(278\) −5.00000 −0.299880
\(279\) 0 0
\(280\) 0 0
\(281\) −13.5000 23.3827i −0.805342 1.39489i −0.916060 0.401042i \(-0.868648\pi\)
0.110717 0.993852i \(-0.464685\pi\)
\(282\) 0 0
\(283\) −9.50000 + 16.4545i −0.564716 + 0.978117i 0.432360 + 0.901701i \(0.357681\pi\)
−0.997076 + 0.0764162i \(0.975652\pi\)
\(284\) 1.50000 2.59808i 0.0890086 0.154167i
\(285\) 0 0
\(286\) 6.00000 + 10.3923i 0.354787 + 0.614510i
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −9.00000 15.5885i −0.528498 0.915386i
\(291\) 0 0
\(292\) 1.00000 1.73205i 0.0585206 0.101361i
\(293\) 1.50000 2.59808i 0.0876309 0.151781i −0.818878 0.573967i \(-0.805404\pi\)
0.906509 + 0.422186i \(0.138737\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −3.00000 5.19615i −0.173494 0.300501i
\(300\) 0 0
\(301\) 0 0
\(302\) −11.5000 + 19.9186i −0.661751 + 1.14619i
\(303\) 0 0
\(304\) −3.50000 6.06218i −0.200739 0.347690i
\(305\) 15.0000 0.858898
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.00000 + 5.19615i −0.170389 + 0.295122i
\(311\) −6.00000 + 10.3923i −0.340229 + 0.589294i −0.984475 0.175525i \(-0.943838\pi\)
0.644246 + 0.764818i \(0.277171\pi\)
\(312\) 0 0
\(313\) −5.00000 8.66025i −0.282617 0.489506i 0.689412 0.724370i \(-0.257869\pi\)
−0.972028 + 0.234863i \(0.924536\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 9.00000 + 15.5885i 0.505490 + 0.875535i 0.999980 + 0.00635137i \(0.00202172\pi\)
−0.494489 + 0.869184i \(0.664645\pi\)
\(318\) 0 0
\(319\) 18.0000 31.1769i 1.00781 1.74557i
\(320\) 1.50000 2.59808i 0.0838525 0.145237i
\(321\) 0 0
\(322\) 0 0
\(323\) 42.0000 2.33694
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) −1.00000 1.73205i −0.0553849 0.0959294i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 22.5167i −0.714545 1.23763i −0.963135 0.269019i \(-0.913301\pi\)
0.248590 0.968609i \(-0.420033\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 + 20.7846i 0.655630 + 1.13558i
\(336\) 0 0
\(337\) 11.0000 19.0526i 0.599208 1.03786i −0.393730 0.919226i \(-0.628816\pi\)
0.992938 0.118633i \(-0.0378512\pi\)
\(338\) 4.50000 7.79423i 0.244768 0.423950i
\(339\) 0 0
\(340\) 9.00000 + 15.5885i 0.488094 + 0.845403i
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) −1.00000 1.73205i −0.0539164 0.0933859i
\(345\) 0 0
\(346\) 3.00000 5.19615i 0.161281 0.279347i
\(347\) −12.0000 + 20.7846i −0.644194 + 1.11578i 0.340293 + 0.940319i \(0.389474\pi\)
−0.984487 + 0.175457i \(0.943860\pi\)
\(348\) 0 0
\(349\) 13.0000 + 22.5167i 0.695874 + 1.20529i 0.969885 + 0.243563i \(0.0783162\pi\)
−0.274011 + 0.961727i \(0.588351\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) 9.00000 + 15.5885i 0.479022 + 0.829690i 0.999711 0.0240566i \(-0.00765819\pi\)
−0.520689 + 0.853746i \(0.674325\pi\)
\(354\) 0 0
\(355\) −4.50000 + 7.79423i −0.238835 + 0.413675i
\(356\) 0 0
\(357\) 0 0
\(358\) 9.00000 + 15.5885i 0.475665 + 0.823876i
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −12.5000 21.6506i −0.656985 1.13793i
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 + 5.19615i −0.157027 + 0.271979i
\(366\) 0 0
\(367\) 4.00000 + 6.92820i 0.208798 + 0.361649i 0.951336 0.308155i \(-0.0997115\pi\)
−0.742538 + 0.669804i \(0.766378\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) 0 0
\(372\) 0 0
\(373\) −7.00000 + 12.1244i −0.362446 + 0.627775i −0.988363 0.152115i \(-0.951392\pi\)
0.625917 + 0.779890i \(0.284725\pi\)
\(374\) −18.0000 + 31.1769i −0.930758 + 1.61212i
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 10.5000 + 18.1865i 0.538639 + 0.932949i
\(381\) 0 0
\(382\) −4.50000 + 7.79423i −0.230240 + 0.398787i
\(383\) 9.00000 15.5885i 0.459879 0.796533i −0.539076 0.842257i \(-0.681226\pi\)
0.998954 + 0.0457244i \(0.0145596\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.0000 0.865277
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 12.0000 + 20.7846i 0.608424 + 1.05382i 0.991500 + 0.130105i \(0.0415314\pi\)
−0.383076 + 0.923717i \(0.625135\pi\)
\(390\) 0 0
\(391\) 9.00000 15.5885i 0.455150 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 9.00000 + 15.5885i 0.453413 + 0.785335i
\(395\) −15.0000 −0.754732
\(396\) 0 0
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 7.00000 + 12.1244i 0.350878 + 0.607739i
\(399\) 0 0
\(400\) −2.00000 + 3.46410i −0.100000 + 0.173205i
\(401\) 1.50000 2.59808i 0.0749064 0.129742i −0.826139 0.563466i \(-0.809468\pi\)
0.901046 + 0.433724i \(0.142801\pi\)
\(402\) 0 0
\(403\) −2.00000 3.46410i −0.0996271 0.172559i
\(404\) 9.00000 0.447767
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 10.3923i −0.297409 0.515127i
\(408\) 0 0
\(409\) 16.0000 27.7128i 0.791149 1.37031i −0.134107 0.990967i \(-0.542817\pi\)
0.925256 0.379344i \(-0.123850\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −5.00000 8.66025i −0.246332 0.426660i
\(413\) 0 0
\(414\) 0 0
\(415\) −36.0000 −1.76717
\(416\) 1.00000 + 1.73205i 0.0490290 + 0.0849208i
\(417\) 0 0
\(418\) −21.0000 + 36.3731i −1.02714 + 1.77906i
\(419\) −7.50000 + 12.9904i −0.366399 + 0.634622i −0.989000 0.147918i \(-0.952743\pi\)
0.622601 + 0.782540i \(0.286076\pi\)
\(420\) 0 0
\(421\) 5.00000 + 8.66025i 0.243685 + 0.422075i 0.961761 0.273890i \(-0.0883103\pi\)
−0.718076 + 0.695965i \(0.754977\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −12.0000 20.7846i −0.582086 1.00820i
\(426\) 0 0
\(427\) 0 0
\(428\) −6.00000 + 10.3923i −0.290021 + 0.502331i
\(429\) 0 0
\(430\) 3.00000 + 5.19615i 0.144673 + 0.250581i
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5.00000 8.66025i 0.239457 0.414751i
\(437\) 10.5000 18.1865i 0.502283 0.869980i
\(438\) 0 0
\(439\) 4.00000 + 6.92820i 0.190910 + 0.330665i 0.945552 0.325471i \(-0.105523\pi\)
−0.754642 + 0.656136i \(0.772190\pi\)
\(440\) −18.0000 −0.858116
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 9.00000 + 15.5885i 0.427603 + 0.740630i 0.996660 0.0816684i \(-0.0260248\pi\)
−0.569057 + 0.822298i \(0.692691\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −14.0000 + 24.2487i −0.662919 + 1.14821i
\(447\) 0 0
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.50000 + 12.9904i 0.352770 + 0.611016i
\(453\) 0 0
\(454\) 7.50000 12.9904i 0.351992 0.609669i
\(455\) 0 0
\(456\) 0 0
\(457\) −14.5000 25.1147i −0.678281 1.17482i −0.975498 0.220008i \(-0.929392\pi\)
0.297217 0.954810i \(-0.403942\pi\)
\(458\) 1.00000 0.0467269
\(459\) 0 0
\(460\) 9.00000 0.419627
\(461\) 16.5000 + 28.5788i 0.768482 + 1.33105i 0.938386 + 0.345589i \(0.112321\pi\)
−0.169904 + 0.985461i \(0.554346\pi\)
\(462\) 0 0
\(463\) 6.50000 11.2583i 0.302081 0.523219i −0.674526 0.738251i \(-0.735652\pi\)
0.976607 + 0.215032i \(0.0689855\pi\)
\(464\) 3.00000 5.19615i 0.139272 0.241225i
\(465\) 0 0
\(466\) 4.50000 + 7.79423i 0.208458 + 0.361061i
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 + 10.3923i −0.275880 + 0.477839i
\(474\) 0 0
\(475\) −14.0000 24.2487i −0.642364 1.11261i
\(476\) 0 0
\(477\) 0 0
\(478\) 15.0000 0.686084
\(479\) 3.00000 + 5.19615i 0.137073 + 0.237418i 0.926388 0.376571i \(-0.122897\pi\)
−0.789314 + 0.613990i \(0.789564\pi\)
\(480\) 0 0
\(481\) 2.00000 3.46410i 0.0911922 0.157949i
\(482\) 4.00000 6.92820i 0.182195 0.315571i
\(483\) 0 0
\(484\) −12.5000 21.6506i −0.568182 0.984120i
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 29.0000 1.31412 0.657058 0.753840i \(-0.271801\pi\)
0.657058 + 0.753840i \(0.271801\pi\)
\(488\) 2.50000 + 4.33013i 0.113170 + 0.196016i
\(489\) 0 0
\(490\) 0 0
\(491\) 9.00000 15.5885i 0.406164 0.703497i −0.588292 0.808649i \(-0.700199\pi\)
0.994456 + 0.105151i \(0.0335327\pi\)
\(492\) 0 0
\(493\) 18.0000 + 31.1769i 0.810679 + 1.40414i
\(494\) −14.0000 −0.629890
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 + 27.7128i −0.716258 + 1.24060i 0.246214 + 0.969216i \(0.420813\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) −1.50000 + 2.59808i −0.0670820 + 0.116190i
\(501\) 0 0
\(502\) −1.50000 2.59808i −0.0669483 0.115958i
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −27.0000 −1.20148
\(506\) 9.00000 + 15.5885i 0.400099 + 0.692991i
\(507\) 0 0
\(508\) −8.50000 + 14.7224i −0.377127 + 0.653202i
\(509\) −15.0000 + 25.9808i −0.664863 + 1.15158i 0.314459 + 0.949271i \(0.398177\pi\)
−0.979322 + 0.202306i \(0.935156\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 15.0000 + 25.9808i 0.660979 + 1.14485i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −3.00000 5.19615i −0.131559 0.227866i
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 0 0
\(523\) 13.0000 0.568450 0.284225 0.958758i \(-0.408264\pi\)
0.284225 + 0.958758i \(0.408264\pi\)
\(524\) 4.50000 + 7.79423i 0.196583 + 0.340492i
\(525\) 0 0
\(526\) −10.5000 + 18.1865i −0.457822 + 0.792971i
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) 7.00000 + 12.1244i 0.304348 + 0.527146i
\(530\) 18.0000 0.781870
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 18.0000 31.1769i 0.778208 1.34790i
\(536\) −4.00000 + 6.92820i −0.172774 + 0.299253i
\(537\) 0 0
\(538\) 4.50000 + 7.79423i 0.194009 + 0.336033i
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −14.0000 24.2487i −0.601351 1.04157i
\(543\) 0 0
\(544\) −3.00000 + 5.19615i −0.128624 + 0.222783i
\(545\) −15.0000 + 25.9808i −0.642529 + 1.11289i
\(546\) 0 0
\(547\) −16.0000 27.7128i −0.684111 1.18491i −0.973715 0.227768i \(-0.926857\pi\)
0.289605 0.957146i \(-0.406476\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 24.0000 1.02336
\(551\) 21.0000 + 36.3731i 0.894630 + 1.54954i
\(552\) 0 0
\(553\) 0 0
\(554\) 8.00000 13.8564i 0.339887 0.588702i
\(555\) 0 0
\(556\) 2.50000 + 4.33013i 0.106024 + 0.183638i
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −13.5000 + 23.3827i −0.569463 + 0.986339i
\(563\) 16.5000 28.5788i 0.695392 1.20445i −0.274656 0.961542i \(-0.588564\pi\)
0.970048 0.242912i \(-0.0781026\pi\)
\(564\) 0 0
\(565\) −22.5000 38.9711i −0.946582 1.63953i
\(566\) 19.0000 0.798630
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) −9.00000 15.5885i −0.377300 0.653502i 0.613369 0.789797i \(-0.289814\pi\)
−0.990668 + 0.136295i \(0.956481\pi\)
\(570\) 0 0
\(571\) −16.0000 + 27.7128i −0.669579 + 1.15975i 0.308443 + 0.951243i \(0.400192\pi\)
−0.978022 + 0.208502i \(0.933141\pi\)
\(572\) 6.00000 10.3923i 0.250873 0.434524i
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −9.50000 16.4545i −0.395148 0.684416i
\(579\) 0 0
\(580\) −9.00000 + 15.5885i −0.373705 + 0.647275i
\(581\) 0 0
\(582\) 0 0
\(583\) 18.0000 + 31.1769i 0.745484 + 1.29122i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −3.00000 −0.123929
\(587\) 1.50000 + 2.59808i 0.0619116 + 0.107234i 0.895320 0.445424i \(-0.146947\pi\)
−0.833408 + 0.552658i \(0.813614\pi\)
\(588\) 0 0
\(589\) 7.00000 12.1244i 0.288430 0.499575i
\(590\) 0 0
\(591\) 0 0
\(592\) −1.00000 1.73205i −0.0410997 0.0711868i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.00000 5.19615i −0.122885 0.212843i
\(597\) 0 0
\(598\) −3.00000 + 5.19615i −0.122679 + 0.212486i
\(599\) −12.0000 + 20.7846i −0.490307 + 0.849236i −0.999938 0.0111569i \(-0.996449\pi\)
0.509631 + 0.860393i \(0.329782\pi\)
\(600\) 0 0
\(601\) 7.00000 + 12.1244i 0.285536 + 0.494563i 0.972739 0.231903i \(-0.0744951\pi\)
−0.687203 + 0.726465i \(0.741162\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 23.0000 0.935857
\(605\) 37.5000 + 64.9519i 1.52459 + 2.64067i
\(606\) 0 0
\(607\) −11.0000 + 19.0526i −0.446476 + 0.773320i −0.998154 0.0607380i \(-0.980655\pi\)
0.551678 + 0.834058i \(0.313988\pi\)
\(608\) −3.50000 + 6.06218i −0.141944 + 0.245854i
\(609\) 0 0
\(610\) −7.50000 12.9904i −0.303666 0.525965i
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −12.5000 21.6506i −0.504459 0.873749i
\(615\) 0 0
\(616\) 0 0
\(617\) 21.0000 36.3731i 0.845428 1.46432i −0.0398207 0.999207i \(-0.512679\pi\)
0.885249 0.465118i \(-0.153988\pi\)
\(618\) 0 0
\(619\) −3.50000 6.06218i −0.140677 0.243659i 0.787075 0.616858i \(-0.211595\pi\)
−0.927752 + 0.373198i \(0.878261\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) −5.00000 + 8.66025i −0.199840 + 0.346133i
\(627\) 0 0
\(628\) −6.50000 11.2583i −0.259378 0.449256i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −2.50000 4.33013i −0.0994447 0.172243i
\(633\) 0 0
\(634\) 9.00000 15.5885i 0.357436 0.619097i
\(635\) 25.5000 44.1673i 1.01194 1.75273i
\(636\) 0 0
\(637\) 0 0
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 13.5000 + 23.3827i 0.533218 + 0.923561i 0.999247 + 0.0387913i \(0.0123508\pi\)
−0.466029 + 0.884769i \(0.654316\pi\)
\(642\) 0 0
\(643\) −2.00000 + 3.46410i −0.0788723 + 0.136611i −0.902764 0.430137i \(-0.858465\pi\)
0.823891 + 0.566748i \(0.191799\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.0000 36.3731i −0.826234 1.43108i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.00000 + 6.92820i 0.156893 + 0.271746i
\(651\) 0 0
\(652\) −1.00000 + 1.73205i −0.0391630 + 0.0678323i
\(653\) −18.0000 + 31.1769i −0.704394 + 1.22005i 0.262515 + 0.964928i \(0.415448\pi\)
−0.966910 + 0.255119i \(0.917885\pi\)
\(654\) 0 0
\(655\) −13.5000 23.3827i −0.527489 0.913637i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.0000 + 36.3731i 0.818044 + 1.41689i 0.907122 + 0.420869i \(0.138275\pi\)
−0.0890776 + 0.996025i \(0.528392\pi\)
\(660\) 0 0
\(661\) 2.50000 4.33013i 0.0972387 0.168422i −0.813302 0.581842i \(-0.802332\pi\)
0.910541 + 0.413419i \(0.135666\pi\)
\(662\) −13.0000 + 22.5167i −0.505259 + 0.875135i
\(663\) 0 0
\(664\) −6.00000 10.3923i −0.232845 0.403300i
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 0 0
\(670\) 12.0000 20.7846i 0.463600 0.802980i
\(671\) 15.0000 25.9808i 0.579069 1.00298i
\(672\) 0 0
\(673\) 18.5000 + 32.0429i 0.713123 + 1.23516i 0.963679 + 0.267063i \(0.0860531\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −21.0000 36.3731i −0.807096 1.39793i −0.914867 0.403755i \(-0.867705\pi\)
0.107772 0.994176i \(-0.465628\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 9.00000 15.5885i 0.345134 0.597790i
\(681\) 0 0
\(682\) 6.00000 + 10.3923i 0.229752 + 0.397942i
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) −1.00000 + 1.73205i −0.0381246 + 0.0660338i
\(689\) −6.00000 + 10.3923i −0.228582 + 0.395915i
\(690\) 0 0
\(691\) 23.5000 + 40.7032i 0.893982 + 1.54842i 0.835059 + 0.550160i \(0.185433\pi\)
0.0589228 + 0.998263i \(0.481233\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) −7.50000 12.9904i −0.284491 0.492753i
\(696\) 0 0
\(697\) 0 0
\(698\) 13.0000 22.5167i 0.492057 0.852268i
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) −3.00000 5.19615i −0.113067 0.195837i
\(705\) 0 0
\(706\) 9.00000 15.5885i 0.338719 0.586679i
\(707\) 0 0
\(708\) 0 0
\(709\) 26.0000 + 45.0333i 0.976450 + 1.69126i 0.675063 + 0.737760i \(0.264116\pi\)
0.301388 + 0.953502i \(0.402550\pi\)
\(710\) 9.00000 0.337764
\(711\) 0 0
\(712\) 0 0
\(713\) −3.00000 5.19615i −0.112351 0.194597i
\(714\) 0 0
\(715\) −18.0000 + 31.1769i −0.673162 + 1.16595i
\(716\) 9.00000 15.5885i 0.336346 0.582568i
\(717\) 0 0
\(718\) −1.50000 2.59808i −0.0559795 0.0969593i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15.0000 25.9808i −0.558242 0.966904i
\(723\) 0 0
\(724\) −12.5000 + 21.6506i −0.464559 + 0.804640i
\(725\) 12.0000 20.7846i 0.445669 0.771921i
\(726\) 0 0
\(727\) 4.00000 + 6.92820i 0.148352 + 0.256953i 0.930618 0.365991i \(-0.119270\pi\)
−0.782267 + 0.622944i \(0.785937\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) −6.00000 10.3923i −0.221918 0.384373i
\(732\) 0 0
\(733\) 14.5000 25.1147i 0.535570 0.927634i −0.463566 0.886062i \(-0.653430\pi\)
0.999136 0.0415715i \(-0.0132364\pi\)
\(734\) 4.00000 6.92820i 0.147643 0.255725i
\(735\) 0 0
\(736\) 1.50000 + 2.59808i 0.0552907 + 0.0957664i
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 3.00000 + 5.19615i 0.110282 + 0.191014i
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0000 31.1769i 0.660356 1.14377i −0.320166 0.947361i \(-0.603739\pi\)
0.980522 0.196409i \(-0.0629279\pi\)
\(744\) 0 0
\(745\) 9.00000 + 15.5885i 0.329734 + 0.571117i
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) 0 0
\(750\) 0 0
\(751\) 15.5000 26.8468i 0.565603 0.979653i −0.431390 0.902165i \(-0.641977\pi\)
0.996993 0.0774878i \(-0.0246899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 10.3923i −0.218507 0.378465i
\(755\) −69.0000 −2.51117
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −1.00000 1.73205i −0.0363216 0.0629109i
\(759\) 0 0
\(760\) 10.5000 18.1865i 0.380875 0.659695i
\(761\) 21.0000 36.3731i 0.761249 1.31852i −0.180957 0.983491i \(-0.557920\pi\)
0.942207 0.335032i \(-0.108747\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 0 0
\(768\) 0 0
\(769\) 7.00000 12.1244i 0.252426 0.437215i −0.711767 0.702416i \(-0.752105\pi\)
0.964193 + 0.265200i \(0.0854381\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.50000 14.7224i −0.305922 0.529872i
\(773\) −51.0000 −1.83434 −0.917171 0.398493i \(-0.869533\pi\)
−0.917171 + 0.398493i \(0.869533\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 1.00000 + 1.73205i 0.0358979 + 0.0621770i
\(777\) 0 0
\(778\) 12.0000 20.7846i 0.430221 0.745164i
\(779\) 0 0
\(780\) 0 0
\(781\) 9.00000 + 15.5885i 0.322045 + 0.557799i
\(782\) −18.0000 −0.643679
\(783\) 0 0
\(784\) 0 0
\(785\) 19.5000 + 33.7750i 0.695985 + 1.20548i
\(786\) 0 0
\(787\) 10.0000 17.3205i 0.356462 0.617409i −0.630905 0.775860i \(-0.717316\pi\)
0.987367 + 0.158450i \(0.0506498\pi\)
\(788\) 9.00000 15.5885i 0.320612 0.555316i
\(789\) 0 0
\(790\) 7.50000 + 12.9904i 0.266838 + 0.462177i
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 13.0000 + 22.5167i 0.461353 + 0.799086i
\(795\) 0 0
\(796\) 7.00000 12.1244i 0.248108 0.429736i
\(797\) −1.50000 + 2.59808i −0.0531327 + 0.0920286i −0.891368 0.453279i \(-0.850254\pi\)
0.838236 + 0.545308i \(0.183587\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −3.00000 −0.105934
\(803\) 6.00000 + 10.3923i 0.211735 + 0.366736i
\(804\) 0 0
\(805\) 0 0
\(806\) −2.00000 + 3.46410i −0.0704470 + 0.122018i
\(807\) 0 0
\(808\) −4.50000 7.79423i −0.158309 0.274200i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.00000 + 10.3923i −0.210300 + 0.364250i
\(815\) 3.00000 5.19615i 0.105085 0.182013i
\(816\) 0 0
\(817\) −7.00000 12.1244i −0.244899 0.424178i
\(818\) −32.0000 −1.11885
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 20.7846i −0.418803 0.725388i 0.577016 0.816733i \(-0.304217\pi\)
−0.995819 + 0.0913446i \(0.970884\pi\)
\(822\) 0 0
\(823\) −4.00000 + 6.92820i −0.139431 + 0.241502i −0.927281 0.374365i \(-0.877861\pi\)
0.787850 + 0.615867i \(0.211194\pi\)
\(824\) −5.00000 + 8.66025i −0.174183 + 0.301694i
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 18.0000 + 31.1769i 0.624789 + 1.08217i
\(831\) 0 0
\(832\) 1.00000 1.73205i 0.0346688 0.0600481i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 42.0000 1.45260
\(837\) 0 0
\(838\) 15.0000 0.518166
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 5.00000 8.66025i 0.172311 0.298452i
\(843\) 0 0
\(844\) −4.00000 6.92820i −0.137686 0.238479i
\(845\) 27.0000 0.928828
\(846\) 0 0
\(847\) 0 0
\(848\) 3.00000 + 5.19615i 0.103020 + 0.178437i
\(849\) 0 0
\(850\) −12.0000 + 20.7846i −0.411597 + 0.712906i
\(851\) 3.00000 5.19615i 0.102839 0.178122i
\(852\) 0 0
\(853\) 17.5000 + 30.3109i 0.599189 + 1.03783i 0.992941 + 0.118609i \(0.0378434\pi\)
−0.393753 + 0.919216i \(0.628823\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −27.0000 46.7654i −0.922302 1.59747i −0.795843 0.605503i \(-0.792972\pi\)
−0.126459 0.991972i \(-0.540361\pi\)
\(858\) 0 0
\(859\) −2.00000 + 3.46410i −0.0682391 + 0.118194i −0.898126 0.439738i \(-0.855071\pi\)
0.829887 + 0.557931i \(0.188405\pi\)
\(860\) 3.00000 5.19615i 0.102299 0.177187i
\(861\) 0 0
\(862\) 6.00000 + 10.3923i 0.204361 + 0.353963i
\(863\) −9.00000 −0.306364 −0.153182 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 7.00000 + 12.1244i 0.237870 + 0.412002i
\(867\) 0 0
\(868\) 0 0
\(869\) −15.0000 + 25.9808i −0.508840 + 0.881337i
\(870\) 0 0
\(871\) 8.00000 + 13.8564i 0.271070 + 0.469506i
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) −21.0000 −0.710336
\(875\) 0 0
\(876\) 0 0
\(877\) 11.0000 19.0526i 0.371444 0.643359i −0.618344 0.785907i \(-0.712196\pi\)
0.989788 + 0.142548i \(0.0455296\pi\)
\(878\) 4.00000 6.92820i 0.134993 0.233816i
\(879\) 0 0
\(880\) 9.00000 + 15.5885i 0.303390 + 0.525487i
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 6.00000 + 10.3923i 0.201802 + 0.349531i
\(885\) 0 0
\(886\) 9.00000 15.5885i 0.302361 0.523704i
\(887\) −12.0000 + 20.7846i −0.402921 + 0.697879i −0.994077 0.108678i \(-0.965338\pi\)
0.591156 + 0.806557i \(0.298672\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 0 0
\(895\) −27.0000 + 46.7654i −0.902510 + 1.56319i
\(896\) 0 0
\(897\) 0 0
\(898\) 16.5000 + 28.5788i 0.550612 + 0.953688i
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 7.50000 12.9904i 0.249446 0.432054i
\(905\) 37.5000 64.9519i 1.24654 2.15907i
\(906\) 0 0
\(907\) −16.0000 27.7128i −0.531271 0.920189i −0.999334 0.0364935i \(-0.988381\pi\)
0.468063 0.883695i \(-0.344952\pi\)
\(908\) −15.0000 −0.497792
\(909\) 0 0
\(910\) 0 0
\(911\) 7.50000 + 12.9904i 0.248486 + 0.430391i 0.963106 0.269122i \(-0.0867336\pi\)
−0.714620 + 0.699513i \(0.753400\pi\)
\(912\) 0 0
\(913\) −36.0000 + 62.3538i −1.19143 + 2.06361i
\(914\) −14.5000 + 25.1147i −0.479617 + 0.830722i
\(915\) 0 0
\(916\) −0.500000 0.866025i −0.0165205 0.0286143i
\(917\) 0 0
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) −4.50000 7.79423i −0.148361 0.256968i
\(921\) 0 0
\(922\) 16.5000 28.5788i 0.543399 0.941194i
\(923\) −3.00000 + 5.19615i −0.0987462 + 0.171033i
\(924\) 0 0
\(925\) −4.00000 6.92820i −0.131519 0.227798i
\(926\) −13.0000 −0.427207
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.50000 7.79423i 0.147402 0.255308i
\(933\) 0 0
\(934\) −6.00000 10.3923i −0.196326 0.340047i
\(935\) −108.000 −3.53198
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10.5000 + 18.1865i −0.342290 + 0.592864i −0.984858 0.173365i \(-0.944536\pi\)
0.642567 + 0.766229i \(0.277869\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −12.0000 20.7846i −0.389948 0.675409i 0.602494 0.798123i \(-0.294174\pi\)
−0.992442 + 0.122714i \(0.960840\pi\)
\(948\) 0 0
\(949\) −2.00000 + 3.46410i −0.0649227 + 0.112449i
\(950\) −14.0000 + 24.2487i −0.454220 + 0.786732i
\(951\) 0 0
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) −27.0000 −0.873699
\(956\) −7.50000 12.9904i −0.242567 0.420139i
\(957\) 0 0
\(958\) 3.00000 5.19615i 0.0969256 0.167880i
\(959\) 0 0
\(960\) 0 0
\(961\) 13.5000 + 23.3827i 0.435484 + 0.754280i
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) −8.00000 −0.257663
\(965\) 25.5000 + 44.1673i 0.820874 + 1.42180i
\(966\) 0 0
\(967\) −8.50000 + 14.7224i −0.273342 + 0.473441i −0.969715 0.244238i \(-0.921462\pi\)
0.696374 + 0.717679i \(0.254796\pi\)
\(968\) −12.5000 + 21.6506i −0.401765 + 0.695878i
\(969\) 0 0
\(970\) −3.00000 5.19615i −0.0963242 0.166838i
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −14.5000 25.1147i −0.464610 0.804728i
\(975\) 0 0
\(976\) 2.50000 4.33013i 0.0800230 0.138604i
\(977\) 3.00000 5.19615i 0.0959785 0.166240i −0.814038 0.580812i \(-0.802735\pi\)
0.910017 + 0.414572i \(0.136069\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −18.0000 −0.574403
\(983\) −9.00000 15.5885i −0.287055 0.497195i 0.686050 0.727554i \(-0.259343\pi\)
−0.973106 + 0.230360i \(0.926010\pi\)
\(984\) 0 0
\(985\) −27.0000 + 46.7654i −0.860292 + 1.49007i
\(986\) 18.0000 31.1769i 0.573237 0.992875i
\(987\) 0 0
\(988\) 7.00000 + 12.1244i 0.222700 + 0.385727i
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 1.00000 + 1.73205i 0.0317500 + 0.0549927i
\(993\) 0 0
\(994\) 0 0
\(995\) −21.0000 + 36.3731i −0.665745 + 1.15310i
\(996\) 0 0
\(997\) −27.5000 47.6314i −0.870934 1.50850i −0.861032 0.508551i \(-0.830182\pi\)
−0.00990158 0.999951i \(-0.503152\pi\)
\(998\) 32.0000 1.01294
\(999\) 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 2646.2.f.c.1765.1 2
3.2 odd 2 882.2.f.h.589.1 2
7.2 even 3 2646.2.h.a.361.1 2
7.3 odd 6 2646.2.e.f.1549.1 2
7.4 even 3 2646.2.e.j.1549.1 2
7.5 odd 6 2646.2.h.e.361.1 2
7.6 odd 2 378.2.f.a.253.1 2
9.2 odd 6 882.2.f.h.295.1 2
9.4 even 3 7938.2.a.u.1.1 1
9.5 odd 6 7938.2.a.l.1.1 1
9.7 even 3 inner 2646.2.f.c.883.1 2
21.2 odd 6 882.2.h.f.67.1 2
21.5 even 6 882.2.h.j.67.1 2
21.11 odd 6 882.2.e.d.373.1 2
21.17 even 6 882.2.e.b.373.1 2
21.20 even 2 126.2.f.a.85.1 yes 2
28.27 even 2 3024.2.r.a.1009.1 2
63.2 odd 6 882.2.e.d.655.1 2
63.11 odd 6 882.2.h.f.79.1 2
63.13 odd 6 1134.2.a.h.1.1 1
63.16 even 3 2646.2.e.j.2125.1 2
63.20 even 6 126.2.f.a.43.1 2
63.25 even 3 2646.2.h.a.667.1 2
63.34 odd 6 378.2.f.a.127.1 2
63.38 even 6 882.2.h.j.79.1 2
63.41 even 6 1134.2.a.a.1.1 1
63.47 even 6 882.2.e.b.655.1 2
63.52 odd 6 2646.2.h.e.667.1 2
63.61 odd 6 2646.2.e.f.2125.1 2
84.83 odd 2 1008.2.r.d.337.1 2
252.83 odd 6 1008.2.r.d.673.1 2
252.139 even 6 9072.2.a.w.1.1 1
252.167 odd 6 9072.2.a.c.1.1 1
252.223 even 6 3024.2.r.a.2017.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.a.43.1 2 63.20 even 6
126.2.f.a.85.1 yes 2 21.20 even 2
378.2.f.a.127.1 2 63.34 odd 6
378.2.f.a.253.1 2 7.6 odd 2
882.2.e.b.373.1 2 21.17 even 6
882.2.e.b.655.1 2 63.47 even 6
882.2.e.d.373.1 2 21.11 odd 6
882.2.e.d.655.1 2 63.2 odd 6
882.2.f.h.295.1 2 9.2 odd 6
882.2.f.h.589.1 2 3.2 odd 2
882.2.h.f.67.1 2 21.2 odd 6
882.2.h.f.79.1 2 63.11 odd 6
882.2.h.j.67.1 2 21.5 even 6
882.2.h.j.79.1 2 63.38 even 6
1008.2.r.d.337.1 2 84.83 odd 2
1008.2.r.d.673.1 2 252.83 odd 6
1134.2.a.a.1.1 1 63.41 even 6
1134.2.a.h.1.1 1 63.13 odd 6
2646.2.e.f.1549.1 2 7.3 odd 6
2646.2.e.f.2125.1 2 63.61 odd 6
2646.2.e.j.1549.1 2 7.4 even 3
2646.2.e.j.2125.1 2 63.16 even 3
2646.2.f.c.883.1 2 9.7 even 3 inner
2646.2.f.c.1765.1 2 1.1 even 1 trivial
2646.2.h.a.361.1 2 7.2 even 3
2646.2.h.a.667.1 2 63.25 even 3
2646.2.h.e.361.1 2 7.5 odd 6
2646.2.h.e.667.1 2 63.52 odd 6
3024.2.r.a.1009.1 2 28.27 even 2
3024.2.r.a.2017.1 2 252.223 even 6
7938.2.a.l.1.1 1 9.5 odd 6
7938.2.a.u.1.1 1 9.4 even 3
9072.2.a.c.1.1 1 252.167 odd 6
9072.2.a.w.1.1 1 252.139 even 6