Properties

Label 672.2.bi.b.593.1
Level $672$
Weight $2$
Character 672.593
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

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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 593.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 672.593
Dual form 672.2.bi.b.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} +(-0.621320 - 0.358719i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} +(-0.621320 - 0.358719i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.50000 - 2.59808i) q^{9} +(2.91421 + 5.04757i) q^{11} -1.24264 q^{15} +(3.62132 - 2.80821i) q^{21} +(-2.24264 - 3.88437i) q^{25} -5.19615i q^{27} -7.58579 q^{29} +(9.62132 - 5.55487i) q^{31} +(8.74264 + 5.04757i) q^{33} +(-1.75736 - 0.717439i) q^{35} +(-1.86396 + 1.07616i) q^{45} +(6.74264 - 1.88064i) q^{49} +(2.03553 + 3.52565i) q^{53} -4.18154i q^{55} +(-12.9853 + 7.49706i) q^{59} +(3.00000 - 7.34847i) q^{63} +(-8.48528 + 4.89898i) q^{73} +(-6.72792 - 3.88437i) q^{75} +(9.44975 + 12.1859i) q^{77} +(-8.86396 + 15.3528i) q^{79} +(-4.50000 - 7.79423i) q^{81} -13.5592i q^{83} +(-11.3787 + 6.56948i) q^{87} +(9.62132 - 16.6646i) q^{93} -8.06591i q^{97} +17.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) −0.621320 0.358719i −0.277863 0.160424i 0.354593 0.935021i \(-0.384620\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 2.62132 0.358719i 0.990766 0.135583i
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 2.91421 + 5.04757i 0.878668 + 1.52190i 0.852803 + 0.522233i \(0.174901\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.24264 −0.320848
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 3.62132 2.80821i 0.790237 0.612801i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −2.24264 3.88437i −0.448528 0.776874i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −7.58579 −1.40865 −0.704323 0.709880i \(-0.748749\pi\)
−0.704323 + 0.709880i \(0.748749\pi\)
\(30\) 0 0
\(31\) 9.62132 5.55487i 1.72804 0.997684i 0.830014 0.557743i \(-0.188333\pi\)
0.898027 0.439941i \(-0.145001\pi\)
\(32\) 0 0
\(33\) 8.74264 + 5.04757i 1.52190 + 0.878668i
\(34\) 0 0
\(35\) −1.75736 0.717439i −0.297048 0.121269i
\(36\) 0 0
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 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
\(45\) −1.86396 + 1.07616i −0.277863 + 0.160424i
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 6.74264 1.88064i 0.963234 0.268662i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.03553 + 3.52565i 0.279602 + 0.484285i 0.971286 0.237915i \(-0.0764641\pi\)
−0.691684 + 0.722200i \(0.743131\pi\)
\(54\) 0 0
\(55\) 4.18154i 0.563839i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.9853 + 7.49706i −1.69054 + 0.976034i −0.736460 + 0.676481i \(0.763504\pi\)
−0.954080 + 0.299552i \(0.903163\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 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 \(-0.861036\pi\)
−0.0869195 + 0.996215i \(0.527702\pi\)
\(74\) 0 0
\(75\) −6.72792 3.88437i −0.776874 0.448528i
\(76\) 0 0
\(77\) 9.44975 + 12.1859i 1.07690 + 1.38871i
\(78\) 0 0
\(79\) −8.86396 + 15.3528i −0.997274 + 1.72733i −0.434730 + 0.900561i \(0.643156\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 13.5592i 1.48832i −0.668002 0.744160i \(-0.732850\pi\)
0.668002 0.744160i \(-0.267150\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −11.3787 + 6.56948i −1.21992 + 0.704323i
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.62132 16.6646i 0.997684 1.72804i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.06591i 0.818969i −0.912317 0.409484i \(-0.865709\pi\)
0.912317 0.409484i \(-0.134291\pi\)
\(98\) 0 0
\(99\) 17.4853 1.75734
\(100\) 0 0
\(101\) −3.00000 + 1.73205i −0.298511 + 0.172345i −0.641774 0.766894i \(-0.721801\pi\)
0.343263 + 0.939239i \(0.388468\pi\)
\(102\) 0 0
\(103\) 12.7279 + 7.34847i 1.25412 + 0.724066i 0.971925 0.235291i \(-0.0756043\pi\)
0.282194 + 0.959357i \(0.408938\pi\)
\(104\) 0 0
\(105\) −3.25736 + 0.445759i −0.317886 + 0.0435017i
\(106\) 0 0
\(107\) −4.67157 + 8.09140i −0.451618 + 0.782225i −0.998487 0.0549930i \(-0.982486\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 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
\(121\) −11.4853 + 19.8931i −1.04412 + 1.80846i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.80511i 0.608668i
\(126\) 0 0
\(127\) −15.2426 −1.35257 −0.676283 0.736642i \(-0.736410\pi\)
−0.676283 + 0.736642i \(0.736410\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.4706 + 8.93193i 1.35167 + 0.780387i 0.988483 0.151330i \(-0.0483556\pi\)
0.363186 + 0.931717i \(0.381689\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.86396 + 3.22848i −0.160424 + 0.277863i
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 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
\(145\) 4.71320 + 2.72117i 0.391410 + 0.225981i
\(146\) 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 \(-0.870295\pi\)
0.802265 + 0.596968i \(0.203628\pi\)
\(150\) 0 0
\(151\) −10.1066 17.5051i −0.822464 1.42455i −0.903842 0.427865i \(-0.859266\pi\)
0.0813788 0.996683i \(-0.474068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.97056 −0.640211
\(156\) 0 0
\(157\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) 0 0
\(159\) 6.10660 + 3.52565i 0.484285 + 0.279602i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) −3.62132 6.27231i −0.281919 0.488299i
\(166\) 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 −0.193892 0.981023i \(-0.562111\pi\)
−0.946537 + 0.322596i \(0.895445\pi\)
\(174\) 0 0
\(175\) −7.27208 9.37769i −0.549717 0.708887i
\(176\) 0 0
\(177\) −12.9853 + 22.4912i −0.976034 + 1.69054i
\(178\) 0 0
\(179\) 5.65685 + 9.79796i 0.422813 + 0.732334i 0.996213 0.0869415i \(-0.0277093\pi\)
−0.573400 + 0.819275i \(0.694376\pi\)
\(180\) 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
\(189\) −1.86396 13.6208i −0.135583 0.990766i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 2.25736 + 3.90986i 0.162488 + 0.281438i 0.935760 0.352636i \(-0.114715\pi\)
−0.773272 + 0.634074i \(0.781381\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.1421 −1.00759 −0.503793 0.863825i \(-0.668062\pi\)
−0.503793 + 0.863825i \(0.668062\pi\)
\(198\) 0 0
\(199\) −21.2132 + 12.2474i −1.50376 + 0.868199i −0.503774 + 0.863836i \(0.668055\pi\)
−0.999990 + 0.00436292i \(0.998611\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −19.8848 + 2.72117i −1.39564 + 0.190989i
\(204\) 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
\(217\) 23.2279 18.0125i 1.57681 1.22277i
\(218\) 0 0
\(219\) −8.48528 + 14.6969i −0.573382 + 0.993127i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 15.1682i 1.01574i 0.861435 + 0.507869i \(0.169566\pi\)
−0.861435 + 0.507869i \(0.830434\pi\)
\(224\) 0 0
\(225\) −13.4558 −0.897056
\(226\) 0 0
\(227\) 16.7132 9.64937i 1.10929 0.640451i 0.170648 0.985332i \(-0.445414\pi\)
0.938647 + 0.344881i \(0.112081\pi\)
\(228\) 0 0
\(229\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) 24.7279 + 10.0951i 1.62698 + 0.664211i
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 30.7057i 1.99455i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 5.22792 3.01834i 0.336760 0.194429i −0.322078 0.946713i \(-0.604381\pi\)
0.658838 + 0.752285i \(0.271048\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) 0 0
\(245\) −4.86396 1.25024i −0.310747 0.0798748i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −11.7426 20.3389i −0.744160 1.28892i
\(250\) 0 0
\(251\) 10.6895i 0.674714i 0.941377 + 0.337357i \(0.109533\pi\)
−0.941377 + 0.337357i \(0.890467\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −11.3787 + 19.7085i −0.704323 + 1.21992i
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 2.92074i 0.179420i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.8345 16.0703i 1.69710 0.979822i 0.748614 0.663007i \(-0.230720\pi\)
0.948487 0.316815i \(-0.102613\pi\)
\(270\) 0 0
\(271\) 27.1066 + 15.6500i 1.64661 + 0.950670i 0.978406 + 0.206691i \(0.0662693\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.0711 22.6398i 0.788215 1.36523i
\(276\) 0 0
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 33.3292i 1.99537i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 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
\(291\) −6.98528 12.0989i −0.409484 0.709248i
\(292\) 0 0
\(293\) 27.8359i 1.62619i −0.582130 0.813095i \(-0.697781\pi\)
0.582130 0.813095i \(-0.302219\pi\)
\(294\) 0 0
\(295\) 10.7574 0.626318
\(296\) 0 0
\(297\) 26.2279 15.1427i 1.52190 0.878668i
\(298\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 29.7426 + 17.1719i 1.68115 + 0.970614i 0.960897 + 0.276907i \(0.0893093\pi\)
0.720257 + 0.693708i \(0.244024\pi\)
\(314\) 0 0
\(315\) −4.50000 + 3.48960i −0.253546 + 0.196616i
\(316\) 0 0
\(317\) −0.278175 + 0.481813i −0.0156238 + 0.0270613i −0.873732 0.486408i \(-0.838307\pi\)
0.858108 + 0.513470i \(0.171640\pi\)
\(318\) 0 0
\(319\) −22.1066 38.2898i −1.23773 2.14381i
\(320\) 0 0
\(321\) 16.1828i 0.903236i
\(322\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −14.4558 −0.787460 −0.393730 0.919226i \(-0.628816\pi\)
−0.393730 + 0.919226i \(0.628816\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 56.0772 + 32.3762i 3.03675 + 1.75327i
\(342\) 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 −0.943264 0.332043i \(-0.892262\pi\)
0.184075 0.982912i \(-0.441071\pi\)
\(348\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 9.50000 + 16.4545i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 39.7862i 2.08823i
\(364\) 0 0
\(365\) 7.02944 0.367938
\(366\) 0 0
\(367\) 30.6213 17.6792i 1.59842 0.922848i 0.606628 0.794986i \(-0.292522\pi\)
0.991792 0.127862i \(-0.0408116\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.60051 + 8.51167i 0.342681 + 0.441904i
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 5.89340 + 10.2077i 0.304334 + 0.527122i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −22.8640 + 13.2005i −1.17136 + 0.676283i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) −1.50000 10.9612i −0.0764471 0.558632i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.5563 26.9444i −0.788738 1.36613i −0.926740 0.375703i \(-0.877401\pi\)
0.138002 0.990432i \(-0.455932\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 30.9411 1.56077
\(394\) 0 0
\(395\) 11.0147 6.35935i 0.554211 0.319974i
\(396\) 0 0
\(397\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 6.45695i 0.320848i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −24.4706 + 14.1281i −1.20999 + 0.698589i −0.962757 0.270367i \(-0.912855\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −31.3492 + 24.3103i −1.54260 + 1.19623i
\(414\) 0 0
\(415\) −4.86396 + 8.42463i −0.238762 + 0.413549i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923i 0.507697i −0.967244 0.253849i \(-0.918303\pi\)
0.967244 0.253849i \(-0.0816965\pi\)
\(420\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0 0
\(433\) 39.1918i 1.88344i −0.336399 0.941720i \(-0.609209\pi\)
0.336399 0.941720i \(-0.390791\pi\)
\(434\) 0 0
\(435\) 9.42641 0.451962
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −14.8934 8.59871i −0.710823 0.410394i 0.100543 0.994933i \(-0.467942\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 0 0
\(441\) 5.22792 20.3389i 0.248949 0.968517i
\(442\) 0 0
\(443\) 6.42893 11.1352i 0.305448 0.529051i −0.671913 0.740630i \(-0.734527\pi\)
0.977361 + 0.211579i \(0.0678605\pi\)
\(444\) 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
\(453\) −30.3198 17.5051i −1.42455 0.822464i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.01472 1.75754i 0.0474665 0.0822145i −0.841316 0.540544i \(-0.818219\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.1051i 1.77473i −0.461065 0.887366i \(-0.652533\pi\)
0.461065 0.887366i \(-0.347467\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 0 0
\(465\) −11.9558 + 6.90271i −0.554439 + 0.320106i
\(466\) 0 0
\(467\) 15.0000 + 8.66025i 0.694117 + 0.400749i 0.805153 0.593068i \(-0.202083\pi\)
−0.111035 + 0.993816i \(0.535417\pi\)
\(468\) 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
\(477\) 12.2132 0.559204
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.89340 + 5.01151i −0.131382 + 0.227561i
\(486\) 0 0
\(487\) 19.5919 + 33.9341i 0.887793 + 1.53770i 0.842479 + 0.538730i \(0.181096\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −44.3137 −1.99985 −0.999925 0.0122607i \(-0.996097\pi\)
−0.999925 + 0.0122607i \(0.996097\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −10.8640 6.27231i −0.488299 0.281919i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 2.48528 0.110594
\(506\) 0 0
\(507\) −19.5000 + 11.2583i −0.866025 + 0.500000i
\(508\) 0 0
\(509\) −21.6213 12.4831i −0.958348 0.553303i −0.0626839 0.998033i \(-0.519966\pi\)
−0.895664 + 0.444731i \(0.853299\pi\)
\(510\) 0 0
\(511\) −20.4853 + 15.8856i −0.906215 + 0.702739i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.27208 9.13151i −0.232316 0.402382i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −30.0000 −1.31685
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) −19.0294 7.76874i −0.830513 0.339055i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11.5000 19.9186i −0.500000 0.866025i
\(530\) 0 0
\(531\) 44.9823i 1.95207i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.80509 3.35157i 0.250976 0.144901i
\(536\) 0 0
\(537\) 16.9706 + 9.79796i 0.732334 + 0.422813i
\(538\) 0 0
\(539\) 29.1421 + 28.5533i 1.25524 + 1.22988i
\(540\) 0 0
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 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
\(553\) −17.7279 + 43.4244i −0.753868 + 1.84659i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.0355 + 39.8987i 0.976047 + 1.69056i 0.676436 + 0.736501i \(0.263523\pi\)
0.299611 + 0.954062i \(0.403143\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 37.7132 21.7737i 1.58942 0.917653i 0.596020 0.802970i \(-0.296748\pi\)
0.993402 0.114684i \(-0.0365854\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −14.5919 18.8169i −0.612801 0.790237i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −15.7721 + 9.10601i −0.656600 + 0.379088i −0.790980 0.611842i \(-0.790429\pi\)
0.134380 + 0.990930i \(0.457096\pi\)
\(578\) 0 0
\(579\) 6.77208 + 3.90986i 0.281438 + 0.162488i
\(580\) 0 0
\(581\) −4.86396 35.5431i −0.201791 1.47458i
\(582\) 0 0
\(583\) −11.8640 + 20.5490i −0.491355 + 0.851052i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 47.8521i 1.97507i −0.157409 0.987534i \(-0.550314\pi\)
0.157409 0.987534i \(-0.449686\pi\)
\(588\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 26.2269i 1.06982i 0.844909 + 0.534910i \(0.179654\pi\)
−0.844909 + 0.534910i \(0.820346\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.2721 8.23999i 0.580242 0.335003i
\(606\) 0 0
\(607\) −2.59188 1.49642i −0.105201 0.0607380i 0.446476 0.894795i \(-0.352679\pi\)
−0.551678 + 0.834058i \(0.686012\pi\)
\(608\) 0 0
\(609\) −27.4706 + 21.3025i −1.11316 + 0.863220i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −8.77208 + 15.1937i −0.350883 + 0.607747i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −29.2426 −1.16413 −0.582066 0.813142i \(-0.697755\pi\)
−0.582066 + 0.813142i \(0.697755\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.47056 + 5.46783i 0.375828 + 0.216984i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) −75.6838 43.6960i −2.97085 1.71522i
\(650\) 0 0
\(651\) 19.2426 47.1347i 0.754179 1.84735i
\(652\) 0 0
\(653\) 19.5208 33.8110i 0.763909 1.32313i −0.176913 0.984226i \(-0.556611\pi\)
0.940822 0.338902i \(-0.110055\pi\)
\(654\) 0 0
\(655\) −6.40812 11.0992i −0.250386 0.433681i
\(656\) 0 0
\(657\) 29.3939i 1.14676i
\(658\) 0 0
\(659\) −45.2548 −1.76288 −0.881439 0.472298i \(-0.843425\pi\)
−0.881439 + 0.472298i \(0.843425\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 13.1360 + 22.7523i 0.507869 + 0.879654i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 16.9411 0.653032 0.326516 0.945192i \(-0.394125\pi\)
0.326516 + 0.945192i \(0.394125\pi\)
\(674\) 0 0
\(675\) −20.1838 + 11.6531i −0.776874 + 0.448528i
\(676\) 0 0
\(677\) 20.3787 + 11.7656i 0.783216 + 0.452190i 0.837569 0.546332i \(-0.183976\pi\)
−0.0543526 + 0.998522i \(0.517310\pi\)
\(678\) 0 0
\(679\) −2.89340 21.1433i −0.111038 0.811407i
\(680\) 0 0
\(681\) 16.7132 28.9481i 0.640451 1.10929i
\(682\) 0 0
\(683\) 23.9142 + 41.4206i 0.915052 + 1.58492i 0.806825 + 0.590790i \(0.201184\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(684\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 45.8345 6.27231i 1.74111 0.238265i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.6152 0.552009 0.276005 0.961156i \(-0.410989\pi\)
0.276005 + 0.961156i \(0.410989\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.24264 + 5.61642i −0.272388 + 0.211227i
\(708\) 0 0
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 26.5919 + 46.0585i 0.997274 + 1.72733i
\(712\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 36.0000 + 14.6969i 1.34071 + 0.547343i
\(722\) 0 0
\(723\) 5.22792 9.05503i 0.194429 0.336760i
\(724\) 0 0
\(725\) 17.0122 + 29.4660i 0.631817 + 1.09434i
\(726\) 0 0
\(727\) 25.2123i 0.935074i 0.883974 + 0.467537i \(0.154858\pi\)
−0.883974 + 0.467537i \(0.845142\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) −8.37868 + 2.33696i −0.309052 + 0.0861999i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1.75736 1.01461i 0.0643847 0.0371725i
\(746\) 0 0
\(747\) −35.2279 20.3389i −1.28892 0.744160i
\(748\) 0 0
\(749\) −9.34315 + 22.8859i −0.341391 + 0.836234i
\(750\) 0 0
\(751\) 20.8345 36.0865i 0.760263 1.31681i −0.182453 0.983215i \(-0.558404\pi\)
0.942715 0.333599i \(-0.108263\pi\)
\(752\) 0 0
\(753\) 9.25736 + 16.0342i 0.337357 + 0.584319i
\(754\) 0 0
\(755\) 14.5017i 0.527772i
\(756\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.97824i 0.0713370i 0.999364 + 0.0356685i \(0.0113561\pi\)
−0.999364 + 0.0356685i \(0.988644\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −45.0000 + 25.9808i −1.61854 + 0.934463i −0.631239 + 0.775589i \(0.717453\pi\)
−0.987299 + 0.158874i \(0.949213\pi\)
\(774\) 0 0
\(775\) −43.1543 24.9152i −1.55015 0.894979i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 39.4169i 1.40865i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −2.52944 4.38111i −0.0897099 0.155382i
\(796\) 0 0
\(797\) 37.8801i 1.34178i −0.741557 0.670890i \(-0.765912\pi\)
0.741557 0.670890i \(-0.234088\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0