Properties

Label 672.2.bi.b.17.2
Level $672$
Weight $2$
Character 672.17
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(17,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bi (of order \(6\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 17.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 672.17
Dual form 672.2.bi.b.593.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(3.62132 - 2.09077i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 + 0.866025i) q^{3} +(3.62132 - 2.09077i) q^{5} +(-1.62132 - 2.09077i) q^{7} +(1.50000 + 2.59808i) q^{9} +(0.0857864 - 0.148586i) q^{11} +7.24264 q^{15} +(-0.621320 - 4.54026i) q^{21} +(6.24264 - 10.8126i) q^{25} +5.19615i q^{27} -10.4142 q^{29} +(5.37868 + 3.10538i) q^{31} +(0.257359 - 0.148586i) q^{33} +(-10.2426 - 4.18154i) q^{35} +(10.8640 + 6.27231i) q^{45} +(-1.74264 + 6.77962i) q^{49} +(-5.03553 + 8.72180i) q^{53} -0.717439i q^{55} +(3.98528 + 2.30090i) q^{59} +(3.00000 - 7.34847i) q^{63} +(8.48528 + 4.89898i) q^{73} +(18.7279 - 10.8126i) q^{75} +(-0.449747 + 0.0615465i) q^{77} +(3.86396 + 6.69258i) q^{79} +(-4.50000 + 7.79423i) q^{81} +3.76127i q^{83} +(-15.6213 - 9.01897i) q^{87} +(5.37868 + 9.31615i) q^{93} -11.5300i q^{97} +0.514719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 6 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} + 12 q^{15} + 6 q^{21} + 8 q^{25} - 36 q^{29} + 30 q^{31} + 18 q^{33} - 24 q^{35} + 18 q^{45} + 10 q^{49} - 6 q^{53} - 18 q^{59} + 12 q^{63} + 24 q^{75} + 18 q^{77} - 10 q^{79} - 18 q^{81} - 54 q^{87} + 30 q^{93} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) 3.62132 2.09077i 1.61950 0.935021i 0.632456 0.774597i \(-0.282047\pi\)
0.987048 0.160424i \(-0.0512862\pi\)
\(6\) 0 0
\(7\) −1.62132 2.09077i −0.612801 0.790237i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0.0857864 0.148586i 0.0258656 0.0448005i −0.852803 0.522233i \(-0.825099\pi\)
0.878668 + 0.477432i \(0.158432\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 7.24264 1.87004
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) −0.621320 4.54026i −0.135583 0.990766i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 6.24264 10.8126i 1.24853 2.16251i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −10.4142 −1.93387 −0.966935 0.255021i \(-0.917918\pi\)
−0.966935 + 0.255021i \(0.917918\pi\)
\(30\) 0 0
\(31\) 5.37868 + 3.10538i 0.966039 + 0.557743i 0.898027 0.439941i \(-0.145001\pi\)
0.0680129 + 0.997684i \(0.478334\pi\)
\(32\) 0 0
\(33\) 0.257359 0.148586i 0.0448005 0.0258656i
\(34\) 0 0
\(35\) −10.2426 4.18154i −1.73132 0.706809i
\(36\) 0 0
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 10.8640 + 6.27231i 1.61950 + 0.935021i
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.03553 + 8.72180i −0.691684 + 1.19803i 0.279602 + 0.960116i \(0.409797\pi\)
−0.971286 + 0.237915i \(0.923536\pi\)
\(54\) 0 0
\(55\) 0.717439i 0.0967394i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.98528 + 2.30090i 0.518839 + 0.299552i 0.736460 0.676481i \(-0.236496\pi\)
−0.217620 + 0.976034i \(0.569829\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 3.00000 7.34847i 0.377964 0.925820i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 8.48528 + 4.89898i 0.993127 + 0.573382i 0.906208 0.422833i \(-0.138964\pi\)
0.0869195 + 0.996215i \(0.472298\pi\)
\(74\) 0 0
\(75\) 18.7279 10.8126i 2.16251 1.24853i
\(76\) 0 0
\(77\) −0.449747 + 0.0615465i −0.0512535 + 0.00701388i
\(78\) 0 0
\(79\) 3.86396 + 6.69258i 0.434730 + 0.752974i 0.997274 0.0737937i \(-0.0235106\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 3.76127i 0.412854i 0.978462 + 0.206427i \(0.0661835\pi\)
−0.978462 + 0.206427i \(0.933816\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −15.6213 9.01897i −1.67478 0.966935i
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.37868 + 9.31615i 0.557743 + 0.966039i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.5300i 1.17070i −0.810782 0.585348i \(-0.800958\pi\)
0.810782 0.585348i \(-0.199042\pi\)
\(98\) 0 0
\(99\) 0.514719 0.0517312
\(100\) 0 0
\(101\) −3.00000 1.73205i −0.298511 0.172345i 0.343263 0.939239i \(-0.388468\pi\)
−0.641774 + 0.766894i \(0.721801\pi\)
\(102\) 0 0
\(103\) −12.7279 + 7.34847i −1.25412 + 0.724066i −0.971925 0.235291i \(-0.924396\pi\)
−0.282194 + 0.959357i \(0.591062\pi\)
\(104\) 0 0
\(105\) −11.7426 15.1427i −1.14596 1.47778i
\(106\) 0 0
\(107\) −10.3284 17.8894i −0.998487 1.72943i −0.546869 0.837218i \(-0.684180\pi\)
−0.451618 0.892211i \(-0.649153\pi\)
\(108\) 0 0
\(109\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.48528 + 9.50079i 0.498662 + 0.863708i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 31.3000i 2.79956i
\(126\) 0 0
\(127\) −6.75736 −0.599619 −0.299809 0.953999i \(-0.596923\pi\)
−0.299809 + 0.953999i \(0.596923\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −18.4706 + 10.6640i −1.61378 + 0.931717i −0.625297 + 0.780387i \(0.715022\pi\)
−0.988483 + 0.151330i \(0.951644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 10.8640 + 18.8169i 0.935021 + 1.61950i
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −37.7132 + 21.7737i −3.13191 + 1.80821i
\(146\) 0 0
\(147\) −8.48528 + 8.66025i −0.699854 + 0.714286i
\(148\) 0 0
\(149\) 1.41421 + 2.44949i 0.115857 + 0.200670i 0.918122 0.396298i \(-0.129705\pi\)
−0.802265 + 0.596968i \(0.796372\pi\)
\(150\) 0 0
\(151\) 11.1066 19.2372i 0.903842 1.56550i 0.0813788 0.996683i \(-0.474068\pi\)
0.822464 0.568818i \(-0.192599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 25.9706 2.08601
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 0 0
\(159\) −15.1066 + 8.72180i −1.19803 + 0.691684i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) 0 0
\(165\) 0.621320 1.07616i 0.0483697 0.0837788i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.0000 + 8.66025i −1.14043 + 0.658427i −0.946537 0.322596i \(-0.895445\pi\)
−0.193892 + 0.981023i \(0.562111\pi\)
\(174\) 0 0
\(175\) −32.7279 + 4.47871i −2.47400 + 0.338559i
\(176\) 0 0
\(177\) 3.98528 + 6.90271i 0.299552 + 0.518839i
\(178\) 0 0
\(179\) −5.65685 + 9.79796i −0.422813 + 0.732334i −0.996213 0.0869415i \(-0.972291\pi\)
0.573400 + 0.819275i \(0.305624\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.8640 8.42463i 0.790237 0.612801i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 10.7426 18.6068i 0.773272 1.33935i −0.162488 0.986710i \(-0.551952\pi\)
0.935760 0.352636i \(-0.114715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.1421 1.00759 0.503793 0.863825i \(-0.331938\pi\)
0.503793 + 0.863825i \(0.331938\pi\)
\(198\) 0 0
\(199\) 21.2132 + 12.2474i 1.50376 + 0.868199i 0.999990 + 0.00436292i \(0.00138876\pi\)
0.503774 + 0.863836i \(0.331945\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.8848 + 21.7737i 1.18508 + 1.52822i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.22792 16.2804i −0.151241 1.10519i
\(218\) 0 0
\(219\) 8.48528 + 14.6969i 0.573382 + 0.993127i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 29.8651i 1.99992i −0.00910984 0.999959i \(-0.502900\pi\)
0.00910984 0.999959i \(-0.497100\pi\)
\(224\) 0 0
\(225\) 37.4558 2.49706
\(226\) 0 0
\(227\) −25.7132 14.8455i −1.70665 0.985332i −0.938647 0.344881i \(-0.887919\pi\)
−0.767999 0.640451i \(-0.778747\pi\)
\(228\) 0 0
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) −0.727922 0.297173i −0.0478938 0.0195525i
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.3852i 0.869459i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −20.2279 11.6786i −1.30300 0.752285i −0.322078 0.946713i \(-0.604381\pi\)
−0.980917 + 0.194429i \(0.937715\pi\)
\(242\) 0 0
\(243\) −13.5000 + 7.79423i −0.866025 + 0.500000i
\(244\) 0 0
\(245\) 7.86396 + 28.1946i 0.502410 + 1.80129i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −3.25736 + 5.64191i −0.206427 + 0.357542i
\(250\) 0 0
\(251\) 20.4874i 1.29316i −0.762848 0.646578i \(-0.776200\pi\)
0.762848 0.646578i \(-0.223800\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −15.6213 27.0569i −0.966935 1.67478i
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 42.1126i 2.58696i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.8345 10.8741i −1.14836 0.663007i −0.199874 0.979822i \(-0.564053\pi\)
−0.948487 + 0.316815i \(0.897387\pi\)
\(270\) 0 0
\(271\) 5.89340 3.40256i 0.357998 0.206691i −0.310204 0.950670i \(-0.600397\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.07107 1.85514i −0.0645878 0.111869i
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) 18.6323i 1.11549i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 9.98528 17.2950i 0.585348 1.01385i
\(292\) 0 0
\(293\) 3.34101i 0.195184i 0.995227 + 0.0975919i \(0.0311140\pi\)
−0.995227 + 0.0975919i \(0.968886\pi\)
\(294\) 0 0
\(295\) 19.2426 1.12035
\(296\) 0 0
\(297\) 0.772078 + 0.445759i 0.0448005 + 0.0258656i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.00000 5.19615i −0.172345 0.298511i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −25.4558 −1.44813
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 21.2574 12.2729i 1.20154 0.693708i 0.240640 0.970614i \(-0.422643\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 0 0
\(315\) −4.50000 32.8835i −0.253546 1.85277i
\(316\) 0 0
\(317\) 15.2782 + 26.4626i 0.858108 + 1.48629i 0.873732 + 0.486408i \(0.161693\pi\)
−0.0156238 + 0.999878i \(0.504973\pi\)
\(318\) 0 0
\(319\) −0.893398 + 1.54741i −0.0500207 + 0.0866384i
\(320\) 0 0
\(321\) 35.7787i 1.99697i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 36.4558 1.98588 0.992938 0.118633i \(-0.0378512\pi\)
0.992938 + 0.118633i \(0.0378512\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.922836 0.532799i 0.0499743 0.0288527i
\(342\) 0 0
\(343\) 17.0000 7.34847i 0.917914 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.1421 24.4949i 0.759190 1.31495i −0.184075 0.982912i \(-0.558929\pi\)
0.943264 0.332043i \(-0.107738\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 9.50000 16.4545i 0.500000 0.866025i
\(362\) 0 0
\(363\) 19.0016i 0.997324i
\(364\) 0 0
\(365\) 40.9706 2.14450
\(366\) 0 0
\(367\) 26.3787 + 15.2297i 1.37696 + 0.794986i 0.991792 0.127862i \(-0.0408116\pi\)
0.385164 + 0.922848i \(0.374145\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 26.3995 3.61269i 1.37059 0.187561i
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 27.1066 46.9500i 1.39978 2.42449i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −10.1360 5.85204i −0.519285 0.299809i
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) −1.50000 + 1.16320i −0.0764471 + 0.0592821i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.5563 26.9444i 0.788738 1.36613i −0.138002 0.990432i \(-0.544068\pi\)
0.926740 0.375703i \(-0.122599\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −36.9411 −1.86343
\(394\) 0 0
\(395\) 27.9853 + 16.1573i 1.40809 + 0.812962i
\(396\) 0 0
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 37.6339i 1.87004i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 9.47056 + 5.46783i 0.468289 + 0.270367i 0.715523 0.698589i \(-0.246188\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.65076 12.0628i −0.0812285 0.593572i
\(414\) 0 0
\(415\) 7.86396 + 13.6208i 0.386027 + 0.668618i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923i 0.507697i 0.967244 + 0.253849i \(0.0816965\pi\)
−0.967244 + 0.253849i \(0.918303\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 39.1918i 1.88344i −0.336399 0.941720i \(-0.609209\pi\)
0.336399 0.941720i \(-0.390791\pi\)
\(434\) 0 0
\(435\) −75.4264 −3.61642
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −36.1066 + 20.8462i −1.72327 + 0.994933i −0.811366 + 0.584539i \(0.801275\pi\)
−0.911908 + 0.410394i \(0.865391\pi\)
\(440\) 0 0
\(441\) −20.2279 + 5.64191i −0.963234 + 0.268662i
\(442\) 0 0
\(443\) 20.5711 + 35.6301i 0.977361 + 1.69284i 0.671913 + 0.740630i \(0.265473\pi\)
0.305448 + 0.952209i \(0.401194\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.89898i 0.231714i
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 33.3198 19.2372i 1.56550 0.903842i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.9853 + 31.1514i 0.841316 + 1.45720i 0.888783 + 0.458329i \(0.151552\pi\)
−0.0474665 + 0.998873i \(0.515115\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.1051i 1.77473i 0.461065 + 0.887366i \(0.347467\pi\)
−0.461065 + 0.887366i \(0.652533\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 0 0
\(465\) 38.9558 + 22.4912i 1.80653 + 1.04300i
\(466\) 0 0
\(467\) 15.0000 8.66025i 0.694117 0.400749i −0.111035 0.993816i \(-0.535417\pi\)
0.805153 + 0.593068i \(0.202083\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −30.2132 −1.38337
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −24.1066 41.7539i −1.09462 1.89595i
\(486\) 0 0
\(487\) −18.5919 + 32.2021i −0.842479 + 1.45922i 0.0453143 + 0.998973i \(0.485571\pi\)
−0.887793 + 0.460243i \(0.847762\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −21.6863 −0.978689 −0.489344 0.872091i \(-0.662764\pi\)
−0.489344 + 0.872091i \(0.662764\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.86396 1.07616i 0.0837788 0.0483697i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −14.4853 −0.644587
\(506\) 0 0
\(507\) −19.5000 11.2583i −0.866025 0.500000i
\(508\) 0 0
\(509\) −17.3787 + 10.0336i −0.770296 + 0.444731i −0.832980 0.553303i \(-0.813367\pi\)
0.0626839 + 0.998033i \(0.480034\pi\)
\(510\) 0 0
\(511\) −3.51472 25.6836i −0.155482 1.13618i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −30.7279 + 53.2223i −1.35403 + 2.34526i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −30.0000 −1.31685
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) −52.9706 21.6251i −2.31182 0.943799i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11.5000 + 19.9186i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 13.8054i 0.599104i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −74.8051 43.1887i −3.23411 1.86721i
\(536\) 0 0
\(537\) −16.9706 + 9.79796i −0.732334 + 0.422813i
\(538\) 0 0
\(539\) 0.857864 + 0.840532i 0.0369508 + 0.0362043i
\(540\) 0 0
\(541\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 7.72792 18.9295i 0.328625 0.804963i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.9645 27.6513i 0.676436 1.17162i −0.299611 0.954062i \(-0.596857\pi\)
0.976047 0.217560i \(-0.0698099\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.71320 2.72117i −0.198638 0.114684i 0.397382 0.917653i \(-0.369919\pi\)
−0.596020 + 0.802970i \(0.703252\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 23.5919 3.22848i 0.990766 0.135583i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −41.2279 23.8030i −1.71634 0.990930i −0.925361 0.379088i \(-0.876238\pi\)
−0.790980 0.611842i \(-0.790429\pi\)
\(578\) 0 0
\(579\) 32.2279 18.6068i 1.33935 0.773272i
\(580\) 0 0
\(581\) 7.86396 6.09823i 0.326252 0.252997i
\(582\) 0 0
\(583\) 0.863961 + 1.49642i 0.0357816 + 0.0619756i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.5316i 1.26017i −0.776525 0.630087i \(-0.783019\pi\)
0.776525 0.630087i \(-0.216981\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 21.2132 + 12.2474i 0.872595 + 0.503793i
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21.2132 + 36.7423i 0.868199 + 1.50376i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 22.7628i 0.928516i 0.885700 + 0.464258i \(0.153679\pi\)
−0.885700 + 0.464258i \(0.846321\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 39.7279 + 22.9369i 1.61517 + 0.932519i
\(606\) 0 0
\(607\) 35.5919 20.5490i 1.44463 0.834058i 0.446476 0.894795i \(-0.352679\pi\)
0.998154 + 0.0607380i \(0.0193454\pi\)
\(608\) 0 0
\(609\) 6.47056 + 47.2832i 0.262200 + 1.91601i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −34.2279 59.2845i −1.36912 2.37138i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −20.7574 −0.826337 −0.413169 0.910654i \(-0.635578\pi\)
−0.413169 + 0.910654i \(0.635578\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.4706 + 14.1281i −0.971085 + 0.560656i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0.683766 0.394773i 0.0268402 0.0154962i
\(650\) 0 0
\(651\) 10.7574 26.3500i 0.421614 1.03274i
\(652\) 0 0
\(653\) −4.52082 7.83028i −0.176913 0.306423i 0.763909 0.645325i \(-0.223278\pi\)
−0.940822 + 0.338902i \(0.889945\pi\)
\(654\) 0 0
\(655\) −44.5919 + 77.2354i −1.74235 + 3.01784i
\(656\) 0 0
\(657\) 29.3939i 1.14676i
\(658\) 0 0
\(659\) 45.2548 1.76288 0.881439 0.472298i \(-0.156575\pi\)
0.881439 + 0.472298i \(0.156575\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 25.8640 44.7977i 0.999959 1.73198i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −50.9411 −1.96364 −0.981818 0.189824i \(-0.939208\pi\)
−0.981818 + 0.189824i \(0.939208\pi\)
\(674\) 0 0
\(675\) 56.1838 + 32.4377i 2.16251 + 1.24853i
\(676\) 0 0
\(677\) 24.6213 14.2151i 0.946274 0.546332i 0.0543526 0.998522i \(-0.482690\pi\)
0.891922 + 0.452190i \(0.149357\pi\)
\(678\) 0 0
\(679\) −24.1066 + 18.6938i −0.925126 + 0.717404i
\(680\) 0 0
\(681\) −25.7132 44.5366i −0.985332 1.70665i
\(682\) 0 0
\(683\) 21.0858 36.5217i 0.806825 1.39746i −0.108227 0.994126i \(-0.534517\pi\)
0.915052 0.403336i \(-0.132149\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) −0.834524 1.07616i −0.0317009 0.0408799i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 51.3848 1.94078 0.970388 0.241551i \(-0.0776561\pi\)
0.970388 + 0.241551i \(0.0776561\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.24264 + 9.08052i 0.0467343 + 0.341508i
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) −11.5919 + 20.0777i −0.434730 + 0.752974i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 36.0000 + 14.6969i 1.34071 + 0.547343i
\(722\) 0 0
\(723\) −20.2279 35.0358i −0.752285 1.30300i
\(724\) 0 0
\(725\) −65.0122 + 112.604i −2.41449 + 4.18202i
\(726\) 0 0
\(727\) 28.6764i 1.06355i 0.846886 + 0.531775i \(0.178475\pi\)
−0.846886 + 0.531775i \(0.821525\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) −12.6213 + 49.1023i −0.465544 + 1.81117i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 10.2426 + 5.91359i 0.375261 + 0.216657i
\(746\) 0 0
\(747\) −9.77208 + 5.64191i −0.357542 + 0.206427i
\(748\) 0 0
\(749\) −20.6569 + 50.5988i −0.754785 + 1.84884i
\(750\) 0 0
\(751\) −25.8345 44.7467i −0.942715 1.63283i −0.760263 0.649616i \(-0.774930\pi\)
−0.182453 0.983215i \(-0.558404\pi\)
\(752\) 0 0
\(753\) 17.7426 30.7312i 0.646578 1.11991i
\(754\) 0 0
\(755\) 92.8854i 3.38045i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 47.0116i 1.69528i 0.530572 + 0.847640i \(0.321977\pi\)
−0.530572 + 0.847640i \(0.678023\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −45.0000 25.9808i −1.61854 0.934463i −0.987299 0.158874i \(-0.949213\pi\)
−0.631239 0.775589i \(-0.717453\pi\)
\(774\) 0 0
\(775\) 67.1543 38.7716i 2.41225 1.39272i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 54.1138i 1.93387i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −36.4706 + 63.1689i −1.29348 + 2.24037i
\(796\) 0 0
\(797\) 55.2006i 1.95530i −0.210230 0.977652i \(-0.567421\pi\)
0.210230 0.977652i \(-0.432579\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0