Properties

Label 1260.2.bm.b.289.1
Level $1260$
Weight $2$
Character 1260.289
Analytic conductor $10.061$
Analytic rank $0$
Dimension $4$
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] = [1260,2,Mod(109,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.bm (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 289.1
Root \(-1.63746 - 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 1260.289
Dual form 1260.2.bm.b.109.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 2.17945i) q^{5} +(2.63746 - 0.209313i) q^{7} +O(q^{10})\) \(q+(0.500000 - 2.17945i) q^{5} +(2.63746 - 0.209313i) q^{7} +(1.13746 + 1.97014i) q^{11} -6.09095i q^{13} +(4.13746 - 2.38876i) q^{17} +(-2.13746 + 3.70219i) q^{19} +(0.774917 + 0.447399i) q^{23} +(-4.50000 - 2.17945i) q^{25} -3.27492 q^{29} +(2.13746 + 3.70219i) q^{31} +(0.862541 - 5.85286i) q^{35} +(-4.86254 - 2.80739i) q^{37} +11.2749 q^{41} -6.50958i q^{43} +(-1.86254 - 1.07534i) q^{47} +(6.91238 - 1.10411i) q^{49} +(-6.41238 + 3.70219i) q^{53} +(4.86254 - 1.49397i) q^{55} +(-2.13746 - 3.70219i) q^{59} +(-0.774917 + 1.34220i) q^{61} +(-13.2749 - 3.04547i) q^{65} +(12.0498 - 6.95698i) q^{67} -10.5498 q^{71} +(1.86254 - 1.07534i) q^{73} +(3.41238 + 4.95807i) q^{77} +(-0.137459 + 0.238085i) q^{79} -5.67232i q^{83} +(-3.13746 - 10.2118i) q^{85} +(3.50000 - 6.06218i) q^{89} +(-1.27492 - 16.0646i) q^{91} +(7.00000 + 6.50958i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 3 q^{7} - 3 q^{11} + 9 q^{17} - q^{19} - 12 q^{23} - 18 q^{25} + 2 q^{29} + q^{31} + 11 q^{35} - 27 q^{37} + 30 q^{41} - 15 q^{47} + 5 q^{49} - 3 q^{53} + 27 q^{55} - q^{59} + 12 q^{61} - 38 q^{65} + 18 q^{67} - 12 q^{71} + 15 q^{73} - 9 q^{77} + 7 q^{79} - 5 q^{85} + 14 q^{89} + 10 q^{91} + 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.500000 2.17945i 0.223607 0.974679i
\(6\) 0 0
\(7\) 2.63746 0.209313i 0.996866 0.0791130i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.13746 + 1.97014i 0.342957 + 0.594018i 0.984980 0.172666i \(-0.0552383\pi\)
−0.642024 + 0.766685i \(0.721905\pi\)
\(12\) 0 0
\(13\) 6.09095i 1.68933i −0.535299 0.844663i \(-0.679801\pi\)
0.535299 0.844663i \(-0.320199\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.13746 2.38876i 1.00348 0.579360i 0.0942047 0.995553i \(-0.469969\pi\)
0.909276 + 0.416193i \(0.136636\pi\)
\(18\) 0 0
\(19\) −2.13746 + 3.70219i −0.490367 + 0.849340i −0.999939 0.0110882i \(-0.996470\pi\)
0.509572 + 0.860428i \(0.329804\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.774917 + 0.447399i 0.161581 + 0.0932891i 0.578610 0.815604i \(-0.303595\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.50000 2.17945i −0.900000 0.435890i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.27492 −0.608137 −0.304068 0.952650i \(-0.598345\pi\)
−0.304068 + 0.952650i \(0.598345\pi\)
\(30\) 0 0
\(31\) 2.13746 + 3.70219i 0.383899 + 0.664932i 0.991616 0.129221i \(-0.0412478\pi\)
−0.607717 + 0.794154i \(0.707914\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.862541 5.85286i 0.145796 0.989315i
\(36\) 0 0
\(37\) −4.86254 2.80739i −0.799397 0.461532i 0.0438633 0.999038i \(-0.486033\pi\)
−0.843260 + 0.537506i \(0.819367\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.2749 1.76085 0.880423 0.474189i \(-0.157259\pi\)
0.880423 + 0.474189i \(0.157259\pi\)
\(42\) 0 0
\(43\) 6.50958i 0.992701i −0.868122 0.496351i \(-0.834673\pi\)
0.868122 0.496351i \(-0.165327\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.86254 1.07534i −0.271680 0.156854i 0.357971 0.933733i \(-0.383469\pi\)
−0.629651 + 0.776878i \(0.716802\pi\)
\(48\) 0 0
\(49\) 6.91238 1.10411i 0.987482 0.157730i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.41238 + 3.70219i −0.880808 + 0.508534i −0.870925 0.491417i \(-0.836479\pi\)
−0.00988297 + 0.999951i \(0.503146\pi\)
\(54\) 0 0
\(55\) 4.86254 1.49397i 0.655665 0.201446i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.13746 3.70219i −0.278273 0.481984i 0.692682 0.721243i \(-0.256429\pi\)
−0.970956 + 0.239259i \(0.923095\pi\)
\(60\) 0 0
\(61\) −0.774917 + 1.34220i −0.0992180 + 0.171851i −0.911361 0.411608i \(-0.864967\pi\)
0.812143 + 0.583458i \(0.198301\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −13.2749 3.04547i −1.64655 0.377745i
\(66\) 0 0
\(67\) 12.0498 6.95698i 1.47212 0.849930i 0.472613 0.881270i \(-0.343311\pi\)
0.999509 + 0.0313404i \(0.00997759\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.5498 −1.25204 −0.626018 0.779809i \(-0.715316\pi\)
−0.626018 + 0.779809i \(0.715316\pi\)
\(72\) 0 0
\(73\) 1.86254 1.07534i 0.217994 0.125859i −0.387027 0.922068i \(-0.626498\pi\)
0.605021 + 0.796209i \(0.293165\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.41238 + 4.95807i 0.388876 + 0.565024i
\(78\) 0 0
\(79\) −0.137459 + 0.238085i −0.0154653 + 0.0267867i −0.873654 0.486547i \(-0.838256\pi\)
0.858189 + 0.513334i \(0.171590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.67232i 0.622618i −0.950309 0.311309i \(-0.899233\pi\)
0.950309 0.311309i \(-0.100767\pi\)
\(84\) 0 0
\(85\) −3.13746 10.2118i −0.340305 1.10762i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.50000 6.06218i 0.370999 0.642590i −0.618720 0.785611i \(-0.712349\pi\)
0.989720 + 0.143022i \(0.0456819\pi\)
\(90\) 0 0
\(91\) −1.27492 16.0646i −0.133648 1.68403i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.00000 + 6.50958i 0.718185 + 0.667868i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.774917 + 1.34220i 0.0771071 + 0.133553i 0.902001 0.431735i \(-0.142098\pi\)
−0.824894 + 0.565288i \(0.808765\pi\)
\(102\) 0 0
\(103\) −2.22508 1.28465i −0.219244 0.126581i 0.386356 0.922350i \(-0.373734\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0498 + 6.95698i 1.16490 + 0.672556i 0.952474 0.304621i \(-0.0985297\pi\)
0.212428 + 0.977177i \(0.431863\pi\)
\(108\) 0 0
\(109\) 1.77492 + 3.07425i 0.170006 + 0.294459i 0.938422 0.345492i \(-0.112288\pi\)
−0.768416 + 0.639951i \(0.778955\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.0192i 1.22474i −0.790572 0.612369i \(-0.790217\pi\)
0.790572 0.612369i \(-0.209783\pi\)
\(114\) 0 0
\(115\) 1.36254 1.46519i 0.127058 0.136630i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.4124 7.16629i 0.954501 0.656933i
\(120\) 0 0
\(121\) 2.91238 5.04438i 0.264761 0.458580i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.00000 + 8.71780i −0.626099 + 0.779744i
\(126\) 0 0
\(127\) 1.78959i 0.158801i −0.996843 0.0794004i \(-0.974699\pi\)
0.996843 0.0794004i \(-0.0253006\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.13746 + 15.8265i −0.798343 + 1.38277i 0.122351 + 0.992487i \(0.460957\pi\)
−0.920694 + 0.390285i \(0.872377\pi\)
\(132\) 0 0
\(133\) −4.86254 + 10.2118i −0.421636 + 0.885472i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.96221 4.59698i 0.680258 0.392747i −0.119695 0.992811i \(-0.538192\pi\)
0.799952 + 0.600064i \(0.204858\pi\)
\(138\) 0 0
\(139\) 17.0997 1.45037 0.725187 0.688551i \(-0.241753\pi\)
0.725187 + 0.688551i \(0.241753\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 6.92820i 1.00349 0.579365i
\(144\) 0 0
\(145\) −1.63746 + 7.13752i −0.135984 + 0.592738i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.77492 + 6.53835i −0.309253 + 0.535642i −0.978199 0.207669i \(-0.933412\pi\)
0.668946 + 0.743311i \(0.266746\pi\)
\(150\) 0 0
\(151\) 10.1375 + 17.5586i 0.824975 + 1.42890i 0.901939 + 0.431864i \(0.142144\pi\)
−0.0769640 + 0.997034i \(0.524523\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.13746 2.80739i 0.733938 0.225495i
\(156\) 0 0
\(157\) −9.41238 + 5.43424i −0.751189 + 0.433699i −0.826123 0.563489i \(-0.809459\pi\)
0.0749341 + 0.997188i \(0.476125\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.13746 + 1.01779i 0.168455 + 0.0802135i
\(162\) 0 0
\(163\) 6.41238 + 3.70219i 0.502256 + 0.289978i 0.729645 0.683826i \(-0.239685\pi\)
−0.227389 + 0.973804i \(0.573019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.6005i 0.975058i 0.873107 + 0.487529i \(0.162102\pi\)
−0.873107 + 0.487529i \(0.837898\pi\)
\(168\) 0 0
\(169\) −24.0997 −1.85382
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −16.2371 9.37451i −1.23449 0.712731i −0.266524 0.963828i \(-0.585875\pi\)
−0.967962 + 0.251097i \(0.919209\pi\)
\(174\) 0 0
\(175\) −12.3248 4.80630i −0.931664 0.363322i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −0.137459 0.238085i −0.0102741 0.0177953i 0.860843 0.508871i \(-0.169937\pi\)
−0.871117 + 0.491076i \(0.836604\pi\)
\(180\) 0 0
\(181\) −16.7251 −1.24317 −0.621583 0.783348i \(-0.713510\pi\)
−0.621583 + 0.783348i \(0.713510\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.54983 + 9.19397i −0.628596 + 0.675954i
\(186\) 0 0
\(187\) 9.41238 + 5.43424i 0.688301 + 0.397391i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.4124 19.7668i 0.825771 1.43028i −0.0755585 0.997141i \(-0.524074\pi\)
0.901329 0.433135i \(-0.142593\pi\)
\(192\) 0 0
\(193\) −7.96221 + 4.59698i −0.573132 + 0.330898i −0.758399 0.651790i \(-0.774018\pi\)
0.185267 + 0.982688i \(0.440685\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.0383i 1.85515i 0.373634 + 0.927576i \(0.378112\pi\)
−0.373634 + 0.927576i \(0.621888\pi\)
\(198\) 0 0
\(199\) 4.86254 + 8.42217i 0.344696 + 0.597032i 0.985299 0.170841i \(-0.0546485\pi\)
−0.640602 + 0.767873i \(0.721315\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.63746 + 0.685484i −0.606231 + 0.0481115i
\(204\) 0 0
\(205\) 5.63746 24.5731i 0.393737 1.71626i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.72508 −0.672698
\(210\) 0 0
\(211\) −19.6495 −1.35273 −0.676364 0.736568i \(-0.736445\pi\)
−0.676364 + 0.736568i \(0.736445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −14.1873 3.25479i −0.967565 0.221975i
\(216\) 0 0
\(217\) 6.41238 + 9.31697i 0.435300 + 0.632477i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −14.5498 25.2011i −0.978728 1.69521i
\(222\) 0 0
\(223\) 8.71780i 0.583787i 0.956451 + 0.291893i \(0.0942853\pi\)
−0.956451 + 0.291893i \(0.905715\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.58762 + 3.22602i −0.370864 + 0.214118i −0.673836 0.738881i \(-0.735354\pi\)
0.302972 + 0.952999i \(0.402021\pi\)
\(228\) 0 0
\(229\) −2.13746 + 3.70219i −0.141247 + 0.244647i −0.927967 0.372663i \(-0.878445\pi\)
0.786719 + 0.617311i \(0.211778\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.1375 + 9.31697i 1.05720 + 0.610375i 0.924656 0.380802i \(-0.124352\pi\)
0.132544 + 0.991177i \(0.457686\pi\)
\(234\) 0 0
\(235\) −3.27492 + 3.52165i −0.213632 + 0.229727i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.5498 −0.941151 −0.470575 0.882360i \(-0.655954\pi\)
−0.470575 + 0.882360i \(0.655954\pi\)
\(240\) 0 0
\(241\) 6.41238 + 11.1066i 0.413057 + 0.715436i 0.995222 0.0976343i \(-0.0311275\pi\)
−0.582165 + 0.813071i \(0.697794\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.04983 15.6172i 0.0670715 0.997748i
\(246\) 0 0
\(247\) 22.5498 + 13.0192i 1.43481 + 0.828389i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.45017 0.344011 0.172006 0.985096i \(-0.444975\pi\)
0.172006 + 0.985096i \(0.444975\pi\)
\(252\) 0 0
\(253\) 2.03559i 0.127976i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.4124 + 12.3624i 1.33567 + 0.771148i 0.986162 0.165786i \(-0.0530162\pi\)
0.349506 + 0.936934i \(0.386350\pi\)
\(258\) 0 0
\(259\) −13.4124 6.38658i −0.833404 0.396843i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −23.3248 + 13.4666i −1.43827 + 0.830383i −0.997729 0.0673516i \(-0.978545\pi\)
−0.440536 + 0.897735i \(0.645212\pi\)
\(264\) 0 0
\(265\) 4.86254 + 15.8265i 0.298704 + 0.972217i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.7749 + 25.5909i 0.900843 + 1.56031i 0.826403 + 0.563080i \(0.190384\pi\)
0.0744400 + 0.997225i \(0.476283\pi\)
\(270\) 0 0
\(271\) 6.41238 11.1066i 0.389524 0.674676i −0.602861 0.797846i \(-0.705973\pi\)
0.992386 + 0.123170i \(0.0393062\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.824752 11.3446i −0.0497344 0.684108i
\(276\) 0 0
\(277\) −16.1375 + 9.31697i −0.969606 + 0.559802i −0.899116 0.437710i \(-0.855790\pi\)
−0.0704898 + 0.997512i \(0.522456\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −16.9622 + 9.79314i −1.00830 + 0.582142i −0.910693 0.413084i \(-0.864452\pi\)
−0.0976056 + 0.995225i \(0.531118\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 29.7371 2.35999i 1.75533 0.139306i
\(288\) 0 0
\(289\) 2.91238 5.04438i 0.171316 0.296728i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.92820i 0.404750i −0.979308 0.202375i \(-0.935134\pi\)
0.979308 0.202375i \(-0.0648660\pi\)
\(294\) 0 0
\(295\) −9.13746 + 2.80739i −0.532003 + 0.163453i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.72508 4.71998i 0.157596 0.272964i
\(300\) 0 0
\(301\) −1.36254 17.1687i −0.0785356 0.989590i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.53779 + 2.35999i 0.145313 + 0.135133i
\(306\) 0 0
\(307\) 3.99782i 0.228167i 0.993471 + 0.114084i \(0.0363932\pi\)
−0.993471 + 0.114084i \(0.963607\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.41238 + 11.1066i 0.363612 + 0.629795i 0.988552 0.150878i \(-0.0482099\pi\)
−0.624940 + 0.780673i \(0.714877\pi\)
\(312\) 0 0
\(313\) −12.5120 7.22383i −0.707223 0.408315i 0.102809 0.994701i \(-0.467217\pi\)
−0.810032 + 0.586386i \(0.800550\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.31271 1.91259i −0.186060 0.107422i 0.404077 0.914725i \(-0.367593\pi\)
−0.590137 + 0.807303i \(0.700926\pi\)
\(318\) 0 0
\(319\) −3.72508 6.45203i −0.208565 0.361244i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.4235i 1.13640i
\(324\) 0 0
\(325\) −13.2749 + 27.4093i −0.736360 + 1.52039i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.13746 2.44631i −0.283237 0.134869i
\(330\) 0 0
\(331\) 2.41238 4.17836i 0.132596 0.229663i −0.792080 0.610417i \(-0.791002\pi\)
0.924677 + 0.380753i \(0.124335\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.13746 29.7405i −0.499233 1.62490i
\(336\) 0 0
\(337\) 13.0192i 0.709198i 0.935018 + 0.354599i \(0.115383\pi\)
−0.935018 + 0.354599i \(0.884617\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.86254 + 8.42217i −0.263321 + 0.456086i
\(342\) 0 0
\(343\) 18.0000 4.35890i 0.971909 0.235358i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.5000 6.06218i 0.563670 0.325435i −0.190947 0.981600i \(-0.561156\pi\)
0.754617 + 0.656165i \(0.227823\pi\)
\(348\) 0 0
\(349\) 11.2749 0.603532 0.301766 0.953382i \(-0.402424\pi\)
0.301766 + 0.953382i \(0.402424\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.4124 + 10.6304i −0.979992 + 0.565799i −0.902268 0.431176i \(-0.858099\pi\)
−0.0777242 + 0.996975i \(0.524765\pi\)
\(354\) 0 0
\(355\) −5.27492 + 22.9928i −0.279964 + 1.22033i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.687293 1.19043i 0.0362739 0.0628283i −0.847318 0.531085i \(-0.821784\pi\)
0.883592 + 0.468257i \(0.155118\pi\)
\(360\) 0 0
\(361\) 0.362541 + 0.627940i 0.0190811 + 0.0330495i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.41238 4.59698i −0.0739271 0.240617i
\(366\) 0 0
\(367\) 12.7749 7.37560i 0.666845 0.385003i −0.128035 0.991770i \(-0.540867\pi\)
0.794880 + 0.606766i \(0.207534\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −16.1375 + 11.1066i −0.837815 + 0.576624i
\(372\) 0 0
\(373\) −4.86254 2.80739i −0.251773 0.145361i 0.368803 0.929508i \(-0.379768\pi\)
−0.620576 + 0.784146i \(0.713101\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.9474i 1.02734i
\(378\) 0 0
\(379\) −23.6495 −1.21479 −0.607397 0.794399i \(-0.707786\pi\)
−0.607397 + 0.794399i \(0.707786\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.3248 + 10.0025i 0.885253 + 0.511101i 0.872387 0.488816i \(-0.162571\pi\)
0.0128665 + 0.999917i \(0.495904\pi\)
\(384\) 0 0
\(385\) 12.5120 4.95807i 0.637673 0.252686i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.68729 + 4.65453i 0.136251 + 0.235994i 0.926075 0.377340i \(-0.123161\pi\)
−0.789824 + 0.613334i \(0.789828\pi\)
\(390\) 0 0
\(391\) 4.27492 0.216192
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.450166 + 0.418627i 0.0226503 + 0.0210634i
\(396\) 0 0
\(397\) 13.1375 + 7.58492i 0.659350 + 0.380676i 0.792029 0.610483i \(-0.209025\pi\)
−0.132679 + 0.991159i \(0.542358\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 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
\(403\) 22.5498 13.0192i 1.12329 0.648530i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.7732i 0.633142i
\(408\) 0 0
\(409\) 5.04983 + 8.74657i 0.249698 + 0.432490i 0.963442 0.267917i \(-0.0863352\pi\)
−0.713744 + 0.700407i \(0.753002\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.41238 9.31697i −0.315532 0.458458i
\(414\) 0 0
\(415\) −12.3625 2.83616i −0.606853 0.139222i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.0997 0.835373 0.417687 0.908591i \(-0.362841\pi\)
0.417687 + 0.908591i \(0.362841\pi\)
\(420\) 0 0
\(421\) 3.27492 0.159610 0.0798048 0.996811i \(-0.474570\pi\)
0.0798048 + 0.996811i \(0.474570\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −23.8248 + 1.73205i −1.15567 + 0.0840168i
\(426\) 0 0
\(427\) −1.76287 + 3.70219i −0.0853114 + 0.179161i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.68729 + 16.7789i 0.466620 + 0.808210i 0.999273 0.0381236i \(-0.0121381\pi\)
−0.532653 + 0.846334i \(0.678805\pi\)
\(432\) 0 0
\(433\) 26.8756i 1.29156i −0.763525 0.645778i \(-0.776533\pi\)
0.763525 0.645778i \(-0.223467\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.31271 + 1.91259i −0.158468 + 0.0914917i
\(438\) 0 0
\(439\) 0.587624 1.01779i 0.0280458 0.0485767i −0.851662 0.524092i \(-0.824405\pi\)
0.879708 + 0.475515i \(0.157738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5000 6.06218i −0.498870 0.288023i 0.229377 0.973338i \(-0.426331\pi\)
−0.728247 + 0.685315i \(0.759665\pi\)
\(444\) 0 0
\(445\) −11.4622 10.6592i −0.543361 0.505293i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −25.8248 −1.21875 −0.609373 0.792884i \(-0.708579\pi\)
−0.609373 + 0.792884i \(0.708579\pi\)
\(450\) 0 0
\(451\) 12.8248 + 22.2131i 0.603894 + 1.04598i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −35.6495 5.25370i −1.67127 0.246297i
\(456\) 0 0
\(457\) 17.6873 + 10.2118i 0.827377 + 0.477686i 0.852954 0.521987i \(-0.174809\pi\)
−0.0255769 + 0.999673i \(0.508142\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 6.50958i 0.302526i 0.988494 + 0.151263i \(0.0483340\pi\)
−0.988494 + 0.151263i \(0.951666\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.5997 + 9.58382i 0.768141 + 0.443486i 0.832211 0.554459i \(-0.187075\pi\)
−0.0640700 + 0.997945i \(0.520408\pi\)
\(468\) 0 0
\(469\) 30.3248 20.8709i 1.40027 0.963730i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.8248 7.40437i 0.589683 0.340453i
\(474\) 0 0
\(475\) 17.6873 12.0014i 0.811549 0.550660i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.41238 + 11.1066i 0.292989 + 0.507472i 0.974515 0.224322i \(-0.0720168\pi\)
−0.681526 + 0.731794i \(0.738683\pi\)
\(480\) 0 0
\(481\) −17.0997 + 29.6175i −0.779678 + 1.35044i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.0997 + 3.46410i 0.685641 + 0.157297i
\(486\) 0 0
\(487\) −28.9622 + 16.7213i −1.31240 + 0.757716i −0.982494 0.186296i \(-0.940352\pi\)
−0.329909 + 0.944013i \(0.607018\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.4502 0.606997 0.303499 0.952832i \(-0.401845\pi\)
0.303499 + 0.952832i \(0.401845\pi\)
\(492\) 0 0
\(493\) −13.5498 + 7.82300i −0.610254 + 0.352330i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −27.8248 + 2.20822i −1.24811 + 0.0990523i
\(498\) 0 0
\(499\) 19.6873 34.0994i 0.881324 1.52650i 0.0314548 0.999505i \(-0.489986\pi\)
0.849869 0.526993i \(-0.176681\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.1797i 0.721418i 0.932678 + 0.360709i \(0.117465\pi\)
−0.932678 + 0.360709i \(0.882535\pi\)
\(504\) 0 0
\(505\) 3.31271 1.01779i 0.147414 0.0452913i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.7749 25.5909i 0.654887 1.13430i −0.327036 0.945012i \(-0.606050\pi\)
0.981922 0.189285i \(-0.0606170\pi\)
\(510\) 0 0
\(511\) 4.68729 3.22602i 0.207354 0.142711i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.91238 + 4.20713i −0.172400 + 0.185388i
\(516\) 0 0
\(517\) 4.89261i 0.215177i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.41238 11.1066i −0.280931 0.486587i 0.690683 0.723158i \(-0.257310\pi\)
−0.971614 + 0.236570i \(0.923977\pi\)
\(522\) 0 0
\(523\) −10.1375 5.85286i −0.443280 0.255928i 0.261708 0.965147i \(-0.415714\pi\)
−0.704988 + 0.709219i \(0.749048\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.6873 + 10.2118i 0.770471 + 0.444831i
\(528\) 0 0
\(529\) −11.0997 19.2252i −0.482594 0.835878i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 68.6750i 2.97464i
\(534\) 0 0
\(535\) 21.1873 22.7835i 0.916007 0.985017i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0378 + 12.3624i 0.432358 + 0.532488i
\(540\) 0 0
\(541\) 1.22508 2.12191i 0.0526704 0.0912278i −0.838488 0.544920i \(-0.816560\pi\)
0.891159 + 0.453692i \(0.149893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.58762 2.33122i 0.325018 0.0998585i
\(546\) 0 0
\(547\) 36.1271i 1.54468i −0.635208 0.772341i \(-0.719086\pi\)
0.635208 0.772341i \(-0.280914\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.00000 12.1244i 0.298210 0.516515i
\(552\) 0 0
\(553\) −0.312707 + 0.656712i −0.0132977 + 0.0279262i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.86254 2.80739i 0.206032 0.118953i −0.393434 0.919353i \(-0.628713\pi\)
0.599466 + 0.800400i \(0.295380\pi\)
\(558\) 0 0
\(559\) −39.6495 −1.67700
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.5997 + 6.11972i −0.446723 + 0.257916i −0.706445 0.707768i \(-0.749702\pi\)
0.259722 + 0.965683i \(0.416369\pi\)
\(564\) 0 0
\(565\) −28.3746 6.50958i −1.19373 0.273860i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.6873 25.4391i 0.615723 1.06646i −0.374534 0.927213i \(-0.622197\pi\)
0.990257 0.139251i \(-0.0444694\pi\)
\(570\) 0 0
\(571\) 0.137459 + 0.238085i 0.00575246 + 0.00996356i 0.868887 0.495010i \(-0.164836\pi\)
−0.863135 + 0.504974i \(0.831502\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.51204 3.70219i −0.104760 0.154392i
\(576\) 0 0
\(577\) 7.13746 4.12081i 0.297136 0.171552i −0.344019 0.938963i \(-0.611789\pi\)
0.641156 + 0.767411i \(0.278455\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.18729 14.9605i −0.0492572 0.620667i
\(582\) 0 0
\(583\) −14.5876 8.42217i −0.604158 0.348811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.9715i 0.576665i −0.957530 0.288333i \(-0.906899\pi\)
0.957530 0.288333i \(-0.0931009\pi\)
\(588\) 0 0
\(589\) −18.2749 −0.753005
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.5876 + 10.1542i 0.722237 + 0.416984i 0.815576 0.578651i \(-0.196421\pi\)
−0.0933384 + 0.995634i \(0.529754\pi\)
\(594\) 0 0
\(595\) −10.4124 26.2764i −0.426866 1.07723i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.13746 + 1.97014i 0.0464753 + 0.0804976i 0.888327 0.459211i \(-0.151868\pi\)
−0.841852 + 0.539709i \(0.818534\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.53779 8.86957i −0.387766 0.360599i
\(606\) 0 0
\(607\) −27.8746 16.0934i −1.13139 0.653211i −0.187109 0.982339i \(-0.559912\pi\)
−0.944285 + 0.329128i \(0.893245\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.54983 + 11.3446i −0.264978 + 0.458955i
\(612\) 0 0
\(613\) 32.0619 18.5109i 1.29497 0.747650i 0.315437 0.948947i \(-0.397849\pi\)
0.979530 + 0.201297i \(0.0645156\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.57919i 0.144093i 0.997401 + 0.0720464i \(0.0229530\pi\)
−0.997401 + 0.0720464i \(0.977047\pi\)
\(618\) 0 0
\(619\) 21.9622 + 38.0397i 0.882736 + 1.52894i 0.848287 + 0.529537i \(0.177634\pi\)
0.0344487 + 0.999406i \(0.489032\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.96221 16.7213i 0.318999 0.669926i
\(624\) 0 0
\(625\) 15.5000 + 19.6150i 0.620000 + 0.784602i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −26.8248 −1.06957
\(630\) 0 0
\(631\) 2.90033 0.115460 0.0577302 0.998332i \(-0.481614\pi\)
0.0577302 + 0.998332i \(0.481614\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.90033 0.894797i −0.154780 0.0355089i
\(636\) 0 0
\(637\) −6.72508 42.1029i −0.266457 1.66818i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.0498 + 24.3350i 0.554935 + 0.961176i 0.997909 + 0.0646411i \(0.0205902\pi\)
−0.442973 + 0.896535i \(0.646076\pi\)
\(642\) 0 0
\(643\) 38.3353i 1.51180i 0.654689 + 0.755898i \(0.272800\pi\)
−0.654689 + 0.755898i \(0.727200\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.675248 0.389855i 0.0265468 0.0153268i −0.486668 0.873587i \(-0.661788\pi\)
0.513215 + 0.858260i \(0.328454\pi\)
\(648\) 0 0
\(649\) 4.86254 8.42217i 0.190871 0.330599i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −32.0619 18.5109i −1.25468 0.724389i −0.282643 0.959225i \(-0.591211\pi\)
−0.972035 + 0.234836i \(0.924545\pi\)
\(654\) 0 0
\(655\) 29.9244 + 27.8279i 1.16924 + 1.08733i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −25.4502 −0.991398 −0.495699 0.868494i \(-0.665088\pi\)
−0.495699 + 0.868494i \(0.665088\pi\)
\(660\) 0 0
\(661\) 7.77492 + 13.4666i 0.302409 + 0.523788i 0.976681 0.214695i \(-0.0688758\pi\)
−0.674272 + 0.738483i \(0.735542\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 19.8248 + 15.7035i 0.768771 + 0.608957i
\(666\) 0 0
\(667\) −2.53779 1.46519i −0.0982636 0.0567325i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.52575 −0.136110
\(672\) 0 0
\(673\) 3.57919i 0.137968i 0.997618 + 0.0689838i \(0.0219757\pi\)
−0.997618 + 0.0689838i \(0.978024\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.3127 12.3049i −0.819114 0.472916i 0.0309969 0.999519i \(-0.490132\pi\)
−0.850111 + 0.526604i \(0.823465\pi\)
\(678\) 0 0
\(679\) 1.45017 + 18.2728i 0.0556522 + 0.701248i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 13.5997 7.85177i 0.520377 0.300440i −0.216712 0.976236i \(-0.569533\pi\)
0.737089 + 0.675796i \(0.236200\pi\)
\(684\) 0 0
\(685\) −6.03779 19.6517i −0.230692 0.750854i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 22.5498 + 39.0575i 0.859080 + 1.48797i
\(690\) 0 0
\(691\) 3.68729 6.38658i 0.140271 0.242957i −0.787327 0.616535i \(-0.788536\pi\)
0.927599 + 0.373578i \(0.121869\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.54983 37.2679i 0.324314 1.41365i
\(696\) 0 0
\(697\) 46.6495 26.9331i 1.76698 1.02016i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.8248 0.522154 0.261077 0.965318i \(-0.415922\pi\)
0.261077 + 0.965318i \(0.415922\pi\)
\(702\) 0 0
\(703\) 20.7870 12.0014i 0.783995 0.452640i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.32475 + 3.37779i 0.0874313 + 0.127035i
\(708\) 0 0
\(709\) −12.7749 + 22.1268i −0.479772 + 0.830990i −0.999731 0.0232018i \(-0.992614\pi\)
0.519959 + 0.854191i \(0.325947\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.82518i 0.143254i
\(714\) 0 0
\(715\) −9.09967 29.6175i −0.340308 1.10763i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.68729 + 6.38658i −0.137513 + 0.238179i −0.926555 0.376160i \(-0.877244\pi\)
0.789042 + 0.614340i \(0.210577\pi\)
\(720\) 0 0
\(721\) −6.13746 2.92248i −0.228571 0.108839i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.7371 + 7.13752i 0.547323 + 0.265081i
\(726\) 0 0
\(727\) 18.6915i 0.693228i 0.938008 + 0.346614i \(0.112669\pi\)
−0.938008 + 0.346614i \(0.887331\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15.5498 26.9331i −0.575131 0.996157i
\(732\) 0 0
\(733\) −28.8625 16.6638i −1.06606 0.615491i −0.138959 0.990298i \(-0.544376\pi\)
−0.927103 + 0.374807i \(0.877709\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.4124 + 15.8265i 1.00975 + 0.582978i
\(738\) 0 0
\(739\) 15.9622 + 27.6474i 0.587179 + 1.01702i 0.994600 + 0.103784i \(0.0330950\pi\)
−0.407420 + 0.913241i \(0.633572\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.5287i 0.716440i 0.933637 + 0.358220i \(0.116616\pi\)
−0.933637 + 0.358220i \(0.883384\pi\)
\(744\) 0 0
\(745\) 12.3625 + 11.4964i 0.452928 + 0.421196i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 33.2371 + 15.8265i 1.21446 + 0.578289i
\(750\) 0 0
\(751\) 11.1375 19.2906i 0.406412 0.703926i −0.588073 0.808808i \(-0.700113\pi\)
0.994485 + 0.104882i \(0.0334466\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 43.3368 13.3148i 1.57719 0.484575i
\(756\) 0 0
\(757\) 9.43996i 0.343101i −0.985175 0.171551i \(-0.945122\pi\)
0.985175 0.171551i \(-0.0548777\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.9622 + 25.9153i −0.542380 + 0.939429i 0.456387 + 0.889781i \(0.349143\pi\)
−0.998767 + 0.0496479i \(0.984190\pi\)
\(762\) 0 0
\(763\) 5.32475 + 7.73668i 0.192769 + 0.280087i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.5498 + 13.0192i −0.814227 + 0.470094i
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 23.5876 13.6183i 0.848388 0.489817i −0.0117187 0.999931i \(-0.503730\pi\)
0.860107 + 0.510114i \(0.170397\pi\)
\(774\) 0 0
\(775\) −1.54983 21.3183i −0.0556717 0.765777i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0997 + 41.7419i −0.863460 + 1.49556i
\(780\) 0 0
\(781\) −12.0000 20.7846i −0.429394 0.743732i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.13746 + 23.2309i 0.254747 + 0.829147i
\(786\) 0 0
\(787\) 1.50000 0.866025i 0.0534692 0.0308705i −0.473027 0.881048i \(-0.656839\pi\)
0.526496 + 0.850177i \(0.323505\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.72508 34.3375i −0.0968928 1.22090i
\(792\) 0 0
\(793\) 8.17525 + 4.71998i 0.290312 + 0.167611i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.3873i 1.04095i −0.853876 0.520476i \(-0.825754\pi\)
0.853876 0.520476i \(-0.174246\pi\)
\(798\) 0 0
\(799\) −10.2749 −0.363500
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.23713 + 2.44631i 0.149525 + 0.0863283i
\(804\) 0 0
\(805\) 3.28696 4.14959i 0.115850 0.146254i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21.5997 37.4117i −0.759404 1.31533i −0.943155 0.332353i \(-0.892157\pi\)
0.183751 0.982973i \(-0.441176\pi\)
\(810\) 0 0
\(811\) 22.5498 0.791832 0.395916 0.918287i \(-0.370427\pi\)
0.395916 + 0.918287i \(0.370427\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11.2749 12.1244i 0.394943 0.424698i
\(816\) 0 0
\(817\) 24.0997 + 13.9140i 0.843141 + 0.486788i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.68729 15.0468i 0.303189 0.525138i −0.673668 0.739034i \(-0.735282\pi\)
0.976856 + 0.213896i \(0.0686154\pi\)
\(822\) 0 0
\(823\) −27.9743 + 16.1509i −0.975121 + 0.562987i −0.900794 0.434247i \(-0.857014\pi\)
−0.0743276 + 0.997234i \(0.523681\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.5670i 1.58452i −0.610183 0.792261i \(-0.708904\pi\)
0.610183 0.792261i \(-0.291096\pi\)
\(828\) 0 0
\(829\) 0.962210 + 1.66660i 0.0334189 + 0.0578833i 0.882251 0.470779i \(-0.156027\pi\)
−0.848832 + 0.528662i \(0.822694\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25.9622 21.0802i 0.899537 0.730387i
\(834\) 0 0
\(835\) 27.4622 + 6.30026i 0.950369 + 0.218030i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.9003 0.376321 0.188161 0.982138i \(-0.439747\pi\)
0.188161 + 0.982138i \(0.439747\pi\)
\(840\) 0 0
\(841\) −18.2749 −0.630170
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.0498 + 52.5240i −0.414527 + 1.80688i
\(846\) 0 0
\(847\) 6.62541 13.9140i 0.227652 0.478089i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.51204 4.35099i −0.0861118 0.149150i
\(852\) 0 0
\(853\) 30.4547i 1.04275i −0.853327 0.521375i \(-0.825419\pi\)
0.853327 0.521375i \(-0.174581\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.8625 16.6638i 0.985926 0.569224i 0.0818717 0.996643i \(-0.473910\pi\)
0.904054 + 0.427418i \(0.140577\pi\)
\(858\) 0 0
\(859\) 17.6873 30.6353i 0.603483 1.04526i −0.388807 0.921319i \(-0.627113\pi\)
0.992289 0.123943i \(-0.0395541\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21.7749 12.5718i −0.741227 0.427947i 0.0812884 0.996691i \(-0.474097\pi\)
−0.822515 + 0.568743i \(0.807430\pi\)
\(864\) 0 0
\(865\) −28.5498 + 30.7007i −0.970723 + 1.04386i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.625414 −0.0212157
\(870\) 0 0
\(871\) −42.3746 73.3949i −1.43581 2.48689i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −16.6375 + 24.4580i −0.562449 + 0.826832i
\(876\) 0 0
\(877\) −38.6873 22.3361i −1.30638 0.754237i −0.324887 0.945753i \(-0.605326\pi\)
−0.981490 + 0.191516i \(0.938660\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40.0241 −1.34845 −0.674223 0.738528i \(-0.735521\pi\)
−0.674223 + 0.738528i \(0.735521\pi\)
\(882\) 0 0
\(883\) 20.6695i 0.695585i 0.937572 + 0.347792i \(0.113069\pi\)
−0.937572 + 0.347792i \(0.886931\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.9743 19.6150i −1.14074 0.658609i −0.194129 0.980976i \(-0.562188\pi\)
−0.946615 + 0.322367i \(0.895521\pi\)
\(888\) 0 0
\(889\) −0.374586 4.71998i −0.0125632 0.158303i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.96221 4.59698i 0.266445 0.153832i
\(894\) 0 0
\(895\) −0.587624 + 0.180541i −0.0196421 + 0.00603483i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.00000 12.1244i −0.233463 0.404370i
\(900\) 0 0
\(901\) −17.6873 + 30.6353i −0.589249 + 1.02061i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.36254 + 36.4515i −0.277980 + 1.21169i
\(906\) 0 0
\(907\) −39.2492 + 22.6605i −1.30325 + 0.752430i −0.980960 0.194212i \(-0.937785\pi\)
−0.322288 + 0.946642i \(0.604452\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.09967 −0.168960 −0.0844798 0.996425i \(-0.526923\pi\)
−0.0844798 + 0.996425i \(0.526923\pi\)
\(912\) 0 0
\(913\) 11.1752 6.45203i 0.369847 0.213531i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −20.7870 + 43.6544i −0.686446 + 1.44160i
\(918\) 0 0
\(919\) 2.96221 5.13070i 0.0977143 0.169246i −0.813024 0.582230i \(-0.802180\pi\)
0.910738 + 0.412984i \(0.135514\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 64.2585i 2.11509i
\(924\) 0 0
\(925\) 15.7629 + 23.2309i 0.518280 + 0.763828i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.95017 15.5021i 0.293645 0.508609i −0.681023 0.732262i \(-0.738465\pi\)
0.974669 + 0.223653i \(0.0717982\pi\)
\(930\) 0 0
\(931\) −10.6873 + 27.9509i −0.350262 + 0.916054i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16.5498 17.7967i 0.541237 0.582014i
\(936\) 0 0
\(937\) 10.5074i 0.343262i 0.985161 + 0.171631i \(0.0549036\pi\)
−0.985161 + 0.171631i \(0.945096\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.13746 + 3.70219i 0.0696792 + 0.120688i 0.898760 0.438441i \(-0.144469\pi\)
−0.829081 + 0.559129i \(0.811136\pi\)
\(942\) 0 0
\(943\) 8.73713 + 5.04438i 0.284520 + 0.164268i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.1495 20.8709i −1.17470 0.678214i −0.219918 0.975518i \(-0.570579\pi\)
−0.954783 + 0.297304i \(0.903912\pi\)
\(948\) 0 0
\(949\) −6.54983 11.3446i −0.212617 0.368263i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29.6175i 0.959405i 0.877431 + 0.479702i \(0.159255\pi\)
−0.877431 + 0.479702i \(0.840745\pi\)
\(954\) 0 0
\(955\) −37.3746 34.7561i −1.20941 1.12468i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.0378 13.7910i 0.647054 0.445333i
\(960\) 0 0
\(961\) 6.36254 11.0202i 0.205243 0.355492i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.03779 + 19.6517i 0.194363 + 0.632611i
\(966\) 0 0
\(967\) 6.50958i 0.209334i 0.994507 + 0.104667i \(0.0333777\pi\)
−0.994507 + 0.104667i \(0.966622\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.96221 13.7910i 0.255519 0.442573i −0.709517 0.704688i \(-0.751087\pi\)
0.965036 + 0.262116i \(0.0844202\pi\)
\(972\) 0 0
\(973\) 45.0997 3.57919i 1.44583 0.114744i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.5876 + 8.42217i −0.466699 + 0.269449i −0.714857 0.699271i \(-0.753508\pi\)
0.248158 + 0.968720i \(0.420175\pi\)
\(978\) 0 0
\(979\) 15.9244 0.508947
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −32.2251 + 18.6052i −1.02782 + 0.593412i −0.916360 0.400356i \(-0.868886\pi\)
−0.111461 + 0.993769i \(0.535553\pi\)
\(984\) 0 0
\(985\) 56.7492 + 13.0192i 1.80818 + 0.414825i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.91238 5.04438i 0.0926082 0.160402i
\(990\) 0 0
\(991\) 17.2371 + 29.8556i 0.547555 + 0.948394i 0.998441 + 0.0558122i \(0.0177748\pi\)
−0.450886 + 0.892582i \(0.648892\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.7870 6.38658i 0.658991 0.202468i
\(996\) 0 0
\(997\) −4.13746 + 2.38876i −0.131035 + 0.0756529i −0.564084 0.825717i \(-0.690771\pi\)
0.433050 + 0.901370i \(0.357437\pi\)
\(998\) 0 0
\(999\) 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 1260.2.bm.b.289.1 4
3.2 odd 2 140.2.q.b.9.2 yes 4
5.4 even 2 1260.2.bm.a.289.2 4
7.4 even 3 1260.2.bm.a.109.2 4
12.11 even 2 560.2.bw.a.289.2 4
15.2 even 4 700.2.i.f.401.1 8
15.8 even 4 700.2.i.f.401.4 8
15.14 odd 2 140.2.q.a.9.1 4
21.2 odd 6 980.2.e.f.589.4 4
21.5 even 6 980.2.e.c.589.1 4
21.11 odd 6 140.2.q.a.109.1 yes 4
21.17 even 6 980.2.q.g.949.2 4
21.20 even 2 980.2.q.b.569.1 4
35.4 even 6 inner 1260.2.bm.b.109.2 4
60.59 even 2 560.2.bw.e.289.1 4
84.11 even 6 560.2.bw.e.529.1 4
105.2 even 12 4900.2.a.be.1.4 4
105.23 even 12 4900.2.a.be.1.2 4
105.32 even 12 700.2.i.f.501.1 8
105.44 odd 6 980.2.e.f.589.2 4
105.47 odd 12 4900.2.a.bf.1.2 4
105.53 even 12 700.2.i.f.501.4 8
105.59 even 6 980.2.q.b.949.2 4
105.68 odd 12 4900.2.a.bf.1.4 4
105.74 odd 6 140.2.q.b.109.1 yes 4
105.89 even 6 980.2.e.c.589.3 4
105.104 even 2 980.2.q.g.569.2 4
420.179 even 6 560.2.bw.a.529.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.q.a.9.1 4 15.14 odd 2
140.2.q.a.109.1 yes 4 21.11 odd 6
140.2.q.b.9.2 yes 4 3.2 odd 2
140.2.q.b.109.1 yes 4 105.74 odd 6
560.2.bw.a.289.2 4 12.11 even 2
560.2.bw.a.529.1 4 420.179 even 6
560.2.bw.e.289.1 4 60.59 even 2
560.2.bw.e.529.1 4 84.11 even 6
700.2.i.f.401.1 8 15.2 even 4
700.2.i.f.401.4 8 15.8 even 4
700.2.i.f.501.1 8 105.32 even 12
700.2.i.f.501.4 8 105.53 even 12
980.2.e.c.589.1 4 21.5 even 6
980.2.e.c.589.3 4 105.89 even 6
980.2.e.f.589.2 4 105.44 odd 6
980.2.e.f.589.4 4 21.2 odd 6
980.2.q.b.569.1 4 21.20 even 2
980.2.q.b.949.2 4 105.59 even 6
980.2.q.g.569.2 4 105.104 even 2
980.2.q.g.949.2 4 21.17 even 6
1260.2.bm.a.109.2 4 7.4 even 3
1260.2.bm.a.289.2 4 5.4 even 2
1260.2.bm.b.109.2 4 35.4 even 6 inner
1260.2.bm.b.289.1 4 1.1 even 1 trivial
4900.2.a.be.1.2 4 105.23 even 12
4900.2.a.be.1.4 4 105.2 even 12
4900.2.a.bf.1.2 4 105.47 odd 12
4900.2.a.bf.1.4 4 105.68 odd 12