Properties

Label 380.2.i.c.121.4
Level $380$
Weight $2$
Character 380.121
Analytic conductor $3.034$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,2,Mod(121,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.i (of order \(3\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.4
Root \(-1.26041 - 2.18309i\) of defining polynomial
Character \(\chi\) \(=\) 380.121
Dual form 380.2.i.c.201.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.26041 + 2.18309i) q^{3} +(-0.500000 - 0.866025i) q^{5} +2.72743 q^{7} +(-1.67727 + 2.90511i) q^{9} +O(q^{10})\) \(q+(1.26041 + 2.18309i) q^{3} +(-0.500000 - 0.866025i) q^{5} +2.72743 q^{7} +(-1.67727 + 2.90511i) q^{9} +3.31421 q^{11} +(-1.62412 + 2.81306i) q^{13} +(1.26041 - 2.18309i) q^{15} +(1.17727 + 2.03909i) q^{17} +(-3.11494 - 3.04912i) q^{19} +(3.43768 + 5.95423i) q^{21} +(-1.07396 + 1.86016i) q^{23} +(-0.500000 + 0.866025i) q^{25} -0.893714 q^{27} +(1.96702 - 3.40697i) q^{29} -10.1896 q^{31} +(4.17727 + 7.23524i) q^{33} +(-1.36371 - 2.36202i) q^{35} -3.68579 q^{37} -8.18825 q^{39} +(0.363714 + 0.629971i) q^{41} +(1.18645 + 2.05499i) q^{43} +3.35453 q^{45} +(5.51164 - 9.54644i) q^{47} +0.438860 q^{49} +(-2.96768 + 5.14017i) q^{51} +(4.49148 - 7.77947i) q^{53} +(-1.65711 - 2.87019i) q^{55} +(2.73041 - 10.6434i) q^{57} +(5.48784 + 9.50521i) q^{59} +(4.22743 - 7.32212i) q^{61} +(-4.57462 + 7.92348i) q^{63} +3.24825 q^{65} +(4.87535 - 8.44436i) q^{67} -5.41453 q^{69} +(-3.45850 - 5.99029i) q^{71} +(1.24025 + 2.14818i) q^{73} -2.52082 q^{75} +9.03927 q^{77} +(-5.99948 - 10.3914i) q^{79} +(3.90535 + 6.76427i) q^{81} +4.68711 q^{83} +(1.17727 - 2.03909i) q^{85} +9.91699 q^{87} +(-4.27205 + 7.39941i) q^{89} +(-4.42968 + 7.67243i) q^{91} +(-12.8430 - 22.2448i) q^{93} +(-1.08314 + 4.22218i) q^{95} +(-3.61494 - 6.26127i) q^{97} +(-5.55882 + 9.62816i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} - 4 q^{5} - 5 q^{9} + 4 q^{11} + 9 q^{13} - q^{15} + q^{17} + 3 q^{19} + 8 q^{21} - 4 q^{25} + 20 q^{27} + 5 q^{29} - 20 q^{31} + 25 q^{33} - 52 q^{37} - 54 q^{39} - 8 q^{41} + 7 q^{43} + 10 q^{45} + 16 q^{47} + 20 q^{49} + 12 q^{51} + 5 q^{53} - 2 q^{55} + 27 q^{57} + 11 q^{59} + 12 q^{61} - 3 q^{63} - 18 q^{65} + 6 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 44 q^{77} + 13 q^{79} - 24 q^{81} + 10 q^{83} + q^{85} - 4 q^{87} + 5 q^{89} - 46 q^{91} - 28 q^{93} - 6 q^{95} - q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.26041 + 2.18309i 0.727698 + 1.26041i 0.957854 + 0.287256i \(0.0927431\pi\)
−0.230156 + 0.973154i \(0.573924\pi\)
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 2.72743 1.03087 0.515435 0.856928i \(-0.327630\pi\)
0.515435 + 0.856928i \(0.327630\pi\)
\(8\) 0 0
\(9\) −1.67727 + 2.90511i −0.559089 + 0.968370i
\(10\) 0 0
\(11\) 3.31421 0.999273 0.499636 0.866235i \(-0.333467\pi\)
0.499636 + 0.866235i \(0.333467\pi\)
\(12\) 0 0
\(13\) −1.62412 + 2.81306i −0.450451 + 0.780204i −0.998414 0.0562987i \(-0.982070\pi\)
0.547963 + 0.836502i \(0.315403\pi\)
\(14\) 0 0
\(15\) 1.26041 2.18309i 0.325436 0.563672i
\(16\) 0 0
\(17\) 1.17727 + 2.03909i 0.285529 + 0.494551i 0.972737 0.231910i \(-0.0744974\pi\)
−0.687208 + 0.726460i \(0.741164\pi\)
\(18\) 0 0
\(19\) −3.11494 3.04912i −0.714617 0.699516i
\(20\) 0 0
\(21\) 3.43768 + 5.95423i 0.750163 + 1.29932i
\(22\) 0 0
\(23\) −1.07396 + 1.86016i −0.223937 + 0.387870i −0.956000 0.293367i \(-0.905224\pi\)
0.732063 + 0.681237i \(0.238558\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) −0.893714 −0.171995
\(28\) 0 0
\(29\) 1.96702 3.40697i 0.365266 0.632659i −0.623553 0.781781i \(-0.714311\pi\)
0.988819 + 0.149122i \(0.0476447\pi\)
\(30\) 0 0
\(31\) −10.1896 −1.83010 −0.915050 0.403340i \(-0.867849\pi\)
−0.915050 + 0.403340i \(0.867849\pi\)
\(32\) 0 0
\(33\) 4.17727 + 7.23524i 0.727169 + 1.25949i
\(34\) 0 0
\(35\) −1.36371 2.36202i −0.230510 0.399254i
\(36\) 0 0
\(37\) −3.68579 −0.605940 −0.302970 0.953000i \(-0.597978\pi\)
−0.302970 + 0.953000i \(0.597978\pi\)
\(38\) 0 0
\(39\) −8.18825 −1.31117
\(40\) 0 0
\(41\) 0.363714 + 0.629971i 0.0568025 + 0.0983849i 0.893028 0.450000i \(-0.148576\pi\)
−0.836226 + 0.548385i \(0.815243\pi\)
\(42\) 0 0
\(43\) 1.18645 + 2.05499i 0.180931 + 0.313383i 0.942198 0.335057i \(-0.108755\pi\)
−0.761267 + 0.648439i \(0.775422\pi\)
\(44\) 0 0
\(45\) 3.35453 0.500064
\(46\) 0 0
\(47\) 5.51164 9.54644i 0.803955 1.39249i −0.113039 0.993591i \(-0.536058\pi\)
0.916994 0.398901i \(-0.130608\pi\)
\(48\) 0 0
\(49\) 0.438860 0.0626942
\(50\) 0 0
\(51\) −2.96768 + 5.14017i −0.415558 + 0.719767i
\(52\) 0 0
\(53\) 4.49148 7.77947i 0.616952 1.06859i −0.373087 0.927797i \(-0.621701\pi\)
0.990039 0.140796i \(-0.0449661\pi\)
\(54\) 0 0
\(55\) −1.65711 2.87019i −0.223444 0.387017i
\(56\) 0 0
\(57\) 2.73041 10.6434i 0.361652 1.40975i
\(58\) 0 0
\(59\) 5.48784 + 9.50521i 0.714456 + 1.23747i 0.963169 + 0.268896i \(0.0866589\pi\)
−0.248714 + 0.968577i \(0.580008\pi\)
\(60\) 0 0
\(61\) 4.22743 7.32212i 0.541267 0.937501i −0.457565 0.889176i \(-0.651278\pi\)
0.998832 0.0483251i \(-0.0153884\pi\)
\(62\) 0 0
\(63\) −4.57462 + 7.92348i −0.576348 + 0.998264i
\(64\) 0 0
\(65\) 3.24825 0.402895
\(66\) 0 0
\(67\) 4.87535 8.44436i 0.595619 1.03164i −0.397840 0.917455i \(-0.630240\pi\)
0.993459 0.114188i \(-0.0364266\pi\)
\(68\) 0 0
\(69\) −5.41453 −0.651833
\(70\) 0 0
\(71\) −3.45850 5.99029i −0.410448 0.710917i 0.584491 0.811400i \(-0.301294\pi\)
−0.994939 + 0.100484i \(0.967961\pi\)
\(72\) 0 0
\(73\) 1.24025 + 2.14818i 0.145160 + 0.251425i 0.929433 0.368992i \(-0.120297\pi\)
−0.784272 + 0.620417i \(0.786964\pi\)
\(74\) 0 0
\(75\) −2.52082 −0.291079
\(76\) 0 0
\(77\) 9.03927 1.03012
\(78\) 0 0
\(79\) −5.99948 10.3914i −0.674994 1.16912i −0.976471 0.215650i \(-0.930813\pi\)
0.301477 0.953474i \(-0.402520\pi\)
\(80\) 0 0
\(81\) 3.90535 + 6.76427i 0.433928 + 0.751586i
\(82\) 0 0
\(83\) 4.68711 0.514477 0.257238 0.966348i \(-0.417187\pi\)
0.257238 + 0.966348i \(0.417187\pi\)
\(84\) 0 0
\(85\) 1.17727 2.03909i 0.127692 0.221170i
\(86\) 0 0
\(87\) 9.91699 1.06321
\(88\) 0 0
\(89\) −4.27205 + 7.39941i −0.452836 + 0.784336i −0.998561 0.0536291i \(-0.982921\pi\)
0.545725 + 0.837965i \(0.316254\pi\)
\(90\) 0 0
\(91\) −4.42968 + 7.67243i −0.464357 + 0.804289i
\(92\) 0 0
\(93\) −12.8430 22.2448i −1.33176 2.30668i
\(94\) 0 0
\(95\) −1.08314 + 4.22218i −0.111128 + 0.433186i
\(96\) 0 0
\(97\) −3.61494 6.26127i −0.367042 0.635735i 0.622060 0.782970i \(-0.286296\pi\)
−0.989102 + 0.147235i \(0.952963\pi\)
\(98\) 0 0
\(99\) −5.55882 + 9.62816i −0.558682 + 0.967666i
\(100\) 0 0
\(101\) 2.88818 5.00247i 0.287384 0.497764i −0.685800 0.727790i \(-0.740548\pi\)
0.973185 + 0.230026i \(0.0738810\pi\)
\(102\) 0 0
\(103\) −15.0576 −1.48367 −0.741836 0.670581i \(-0.766045\pi\)
−0.741836 + 0.670581i \(0.766045\pi\)
\(104\) 0 0
\(105\) 3.43768 5.95423i 0.335483 0.581073i
\(106\) 0 0
\(107\) −15.1896 −1.46843 −0.734215 0.678917i \(-0.762450\pi\)
−0.734215 + 0.678917i \(0.762450\pi\)
\(108\) 0 0
\(109\) 6.31057 + 10.9302i 0.604443 + 1.04693i 0.992139 + 0.125139i \(0.0399376\pi\)
−0.387696 + 0.921787i \(0.626729\pi\)
\(110\) 0 0
\(111\) −4.64560 8.04642i −0.440941 0.763732i
\(112\) 0 0
\(113\) −6.10761 −0.574555 −0.287278 0.957847i \(-0.592750\pi\)
−0.287278 + 0.957847i \(0.592750\pi\)
\(114\) 0 0
\(115\) 2.14793 0.200295
\(116\) 0 0
\(117\) −5.44818 9.43652i −0.503684 0.872406i
\(118\) 0 0
\(119\) 3.21091 + 5.56146i 0.294344 + 0.509818i
\(120\) 0 0
\(121\) −0.0159950 −0.00145409
\(122\) 0 0
\(123\) −0.916857 + 1.58804i −0.0826702 + 0.143189i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.56114 7.90013i 0.404736 0.701023i −0.589555 0.807728i \(-0.700697\pi\)
0.994291 + 0.106705i \(0.0340302\pi\)
\(128\) 0 0
\(129\) −2.99082 + 5.18025i −0.263327 + 0.456096i
\(130\) 0 0
\(131\) −11.1328 19.2825i −0.972676 1.68472i −0.687402 0.726277i \(-0.741249\pi\)
−0.285274 0.958446i \(-0.592085\pi\)
\(132\) 0 0
\(133\) −8.49578 8.31625i −0.736678 0.721110i
\(134\) 0 0
\(135\) 0.446857 + 0.773979i 0.0384593 + 0.0666135i
\(136\) 0 0
\(137\) 3.81355 6.60527i 0.325814 0.564326i −0.655863 0.754880i \(-0.727695\pi\)
0.981677 + 0.190554i \(0.0610284\pi\)
\(138\) 0 0
\(139\) −8.45916 + 14.6517i −0.717496 + 1.24274i 0.244493 + 0.969651i \(0.421379\pi\)
−0.961989 + 0.273089i \(0.911955\pi\)
\(140\) 0 0
\(141\) 27.7877 2.34015
\(142\) 0 0
\(143\) −5.38269 + 9.32309i −0.450123 + 0.779636i
\(144\) 0 0
\(145\) −3.93403 −0.326704
\(146\) 0 0
\(147\) 0.553143 + 0.958072i 0.0456225 + 0.0790204i
\(148\) 0 0
\(149\) 6.19809 + 10.7354i 0.507767 + 0.879478i 0.999960 + 0.00899193i \(0.00286226\pi\)
−0.492193 + 0.870486i \(0.663804\pi\)
\(150\) 0 0
\(151\) −14.4549 −1.17632 −0.588160 0.808745i \(-0.700147\pi\)
−0.588160 + 0.808745i \(0.700147\pi\)
\(152\) 0 0
\(153\) −7.89836 −0.638544
\(154\) 0 0
\(155\) 5.09478 + 8.82442i 0.409223 + 0.708795i
\(156\) 0 0
\(157\) 9.10642 + 15.7728i 0.726772 + 1.25881i 0.958241 + 0.285963i \(0.0923135\pi\)
−0.231469 + 0.972842i \(0.574353\pi\)
\(158\) 0 0
\(159\) 22.6444 1.79582
\(160\) 0 0
\(161\) −2.92916 + 5.07345i −0.230850 + 0.399844i
\(162\) 0 0
\(163\) 19.2642 1.50889 0.754446 0.656362i \(-0.227906\pi\)
0.754446 + 0.656362i \(0.227906\pi\)
\(164\) 0 0
\(165\) 4.17727 7.23524i 0.325200 0.563263i
\(166\) 0 0
\(167\) −2.84289 + 4.92404i −0.219990 + 0.381033i −0.954805 0.297234i \(-0.903936\pi\)
0.734815 + 0.678268i \(0.237269\pi\)
\(168\) 0 0
\(169\) 1.22445 + 2.12080i 0.0941881 + 0.163139i
\(170\) 0 0
\(171\) 14.0826 3.93507i 1.07692 0.300922i
\(172\) 0 0
\(173\) 4.57760 + 7.92864i 0.348029 + 0.602804i 0.985899 0.167340i \(-0.0535178\pi\)
−0.637870 + 0.770144i \(0.720184\pi\)
\(174\) 0 0
\(175\) −1.36371 + 2.36202i −0.103087 + 0.178552i
\(176\) 0 0
\(177\) −13.8338 + 23.9609i −1.03982 + 1.80101i
\(178\) 0 0
\(179\) 12.1810 0.910448 0.455224 0.890377i \(-0.349559\pi\)
0.455224 + 0.890377i \(0.349559\pi\)
\(180\) 0 0
\(181\) −7.70608 + 13.3473i −0.572789 + 0.992099i 0.423489 + 0.905901i \(0.360805\pi\)
−0.996278 + 0.0861980i \(0.972528\pi\)
\(182\) 0 0
\(183\) 21.3132 1.57551
\(184\) 0 0
\(185\) 1.84289 + 3.19199i 0.135492 + 0.234679i
\(186\) 0 0
\(187\) 3.90171 + 6.75796i 0.285321 + 0.494191i
\(188\) 0 0
\(189\) −2.43754 −0.177305
\(190\) 0 0
\(191\) −16.2482 −1.17568 −0.587841 0.808977i \(-0.700022\pi\)
−0.587841 + 0.808977i \(0.700022\pi\)
\(192\) 0 0
\(193\) −0.425514 0.737011i −0.0306292 0.0530512i 0.850305 0.526291i \(-0.176418\pi\)
−0.880934 + 0.473240i \(0.843084\pi\)
\(194\) 0 0
\(195\) 4.09412 + 7.09123i 0.293186 + 0.507813i
\(196\) 0 0
\(197\) 19.9843 1.42382 0.711910 0.702270i \(-0.247830\pi\)
0.711910 + 0.702270i \(0.247830\pi\)
\(198\) 0 0
\(199\) 3.22861 5.59212i 0.228870 0.396415i −0.728603 0.684936i \(-0.759830\pi\)
0.957474 + 0.288521i \(0.0931636\pi\)
\(200\) 0 0
\(201\) 24.5798 1.73372
\(202\) 0 0
\(203\) 5.36490 9.29227i 0.376542 0.652190i
\(204\) 0 0
\(205\) 0.363714 0.629971i 0.0254029 0.0439990i
\(206\) 0 0
\(207\) −3.60264 6.23996i −0.250401 0.433707i
\(208\) 0 0
\(209\) −10.3236 10.1054i −0.714097 0.699007i
\(210\) 0 0
\(211\) −5.27205 9.13146i −0.362943 0.628635i 0.625501 0.780223i \(-0.284895\pi\)
−0.988444 + 0.151588i \(0.951561\pi\)
\(212\) 0 0
\(213\) 8.71825 15.1004i 0.597364 1.03467i
\(214\) 0 0
\(215\) 1.18645 2.05499i 0.0809150 0.140149i
\(216\) 0 0
\(217\) −27.7913 −1.88660
\(218\) 0 0
\(219\) −3.12645 + 5.41516i −0.211266 + 0.365923i
\(220\) 0 0
\(221\) −7.64811 −0.514467
\(222\) 0 0
\(223\) −4.86319 8.42329i −0.325663 0.564065i 0.655983 0.754776i \(-0.272254\pi\)
−0.981646 + 0.190710i \(0.938921\pi\)
\(224\) 0 0
\(225\) −1.67727 2.90511i −0.111818 0.193674i
\(226\) 0 0
\(227\) 22.5798 1.49867 0.749336 0.662190i \(-0.230373\pi\)
0.749336 + 0.662190i \(0.230373\pi\)
\(228\) 0 0
\(229\) −16.2239 −1.07211 −0.536053 0.844184i \(-0.680085\pi\)
−0.536053 + 0.844184i \(0.680085\pi\)
\(230\) 0 0
\(231\) 11.3932 + 19.7336i 0.749617 + 1.29837i
\(232\) 0 0
\(233\) 9.33139 + 16.1624i 0.611320 + 1.05884i 0.991018 + 0.133727i \(0.0426944\pi\)
−0.379699 + 0.925110i \(0.623972\pi\)
\(234\) 0 0
\(235\) −11.0233 −0.719079
\(236\) 0 0
\(237\) 15.1236 26.1948i 0.982383 1.70154i
\(238\) 0 0
\(239\) 0.602780 0.0389906 0.0194953 0.999810i \(-0.493794\pi\)
0.0194953 + 0.999810i \(0.493794\pi\)
\(240\) 0 0
\(241\) −10.9408 + 18.9500i −0.704759 + 1.22068i 0.262020 + 0.965062i \(0.415611\pi\)
−0.966779 + 0.255615i \(0.917722\pi\)
\(242\) 0 0
\(243\) −11.1853 + 19.3734i −0.717535 + 1.24281i
\(244\) 0 0
\(245\) −0.219430 0.380064i −0.0140189 0.0242814i
\(246\) 0 0
\(247\) 13.6364 3.81039i 0.867665 0.242449i
\(248\) 0 0
\(249\) 5.90768 + 10.2324i 0.374384 + 0.648452i
\(250\) 0 0
\(251\) −4.13629 + 7.16426i −0.261080 + 0.452204i −0.966529 0.256557i \(-0.917412\pi\)
0.705449 + 0.708761i \(0.250745\pi\)
\(252\) 0 0
\(253\) −3.55934 + 6.16496i −0.223774 + 0.387588i
\(254\) 0 0
\(255\) 5.93535 0.371686
\(256\) 0 0
\(257\) −7.09544 + 12.2897i −0.442602 + 0.766608i −0.997882 0.0650550i \(-0.979278\pi\)
0.555280 + 0.831663i \(0.312611\pi\)
\(258\) 0 0
\(259\) −10.0527 −0.624645
\(260\) 0 0
\(261\) 6.59842 + 11.4288i 0.408432 + 0.707425i
\(262\) 0 0
\(263\) 9.27153 + 16.0588i 0.571707 + 0.990225i 0.996391 + 0.0848836i \(0.0270518\pi\)
−0.424684 + 0.905342i \(0.639615\pi\)
\(264\) 0 0
\(265\) −8.98296 −0.551819
\(266\) 0 0
\(267\) −21.5381 −1.31811
\(268\) 0 0
\(269\) −3.33371 5.77416i −0.203260 0.352057i 0.746317 0.665591i \(-0.231820\pi\)
−0.949577 + 0.313534i \(0.898487\pi\)
\(270\) 0 0
\(271\) −11.0331 19.1099i −0.670214 1.16085i −0.977843 0.209339i \(-0.932869\pi\)
0.307629 0.951506i \(-0.400464\pi\)
\(272\) 0 0
\(273\) −22.3328 −1.35165
\(274\) 0 0
\(275\) −1.65711 + 2.87019i −0.0999273 + 0.173079i
\(276\) 0 0
\(277\) −24.4624 −1.46980 −0.734902 0.678173i \(-0.762772\pi\)
−0.734902 + 0.678173i \(0.762772\pi\)
\(278\) 0 0
\(279\) 17.0906 29.6018i 1.02319 1.77221i
\(280\) 0 0
\(281\) −9.33139 + 16.1624i −0.556664 + 0.964170i 0.441108 + 0.897454i \(0.354586\pi\)
−0.997772 + 0.0667164i \(0.978748\pi\)
\(282\) 0 0
\(283\) −2.24811 3.89384i −0.133636 0.231465i 0.791439 0.611248i \(-0.209332\pi\)
−0.925076 + 0.379783i \(0.875999\pi\)
\(284\) 0 0
\(285\) −10.5826 + 2.95707i −0.626860 + 0.175162i
\(286\) 0 0
\(287\) 0.992002 + 1.71820i 0.0585561 + 0.101422i
\(288\) 0 0
\(289\) 5.72809 9.92134i 0.336946 0.583608i
\(290\) 0 0
\(291\) 9.11262 15.7835i 0.534191 0.925246i
\(292\) 0 0
\(293\) 1.27021 0.0742063 0.0371031 0.999311i \(-0.488187\pi\)
0.0371031 + 0.999311i \(0.488187\pi\)
\(294\) 0 0
\(295\) 5.48784 9.50521i 0.319514 0.553415i
\(296\) 0 0
\(297\) −2.96196 −0.171870
\(298\) 0 0
\(299\) −3.48850 6.04225i −0.201745 0.349433i
\(300\) 0 0
\(301\) 3.23595 + 5.60483i 0.186517 + 0.323057i
\(302\) 0 0
\(303\) 14.5611 0.836516
\(304\) 0 0
\(305\) −8.45485 −0.484124
\(306\) 0 0
\(307\) 4.32638 + 7.49350i 0.246919 + 0.427677i 0.962669 0.270680i \(-0.0872484\pi\)
−0.715750 + 0.698356i \(0.753915\pi\)
\(308\) 0 0
\(309\) −18.9788 32.8722i −1.07967 1.87004i
\(310\) 0 0
\(311\) 23.8497 1.35239 0.676196 0.736721i \(-0.263627\pi\)
0.676196 + 0.736721i \(0.263627\pi\)
\(312\) 0 0
\(313\) 0.388175 0.672340i 0.0219410 0.0380029i −0.854846 0.518881i \(-0.826349\pi\)
0.876787 + 0.480878i \(0.159682\pi\)
\(314\) 0 0
\(315\) 9.14925 0.515502
\(316\) 0 0
\(317\) 6.92116 11.9878i 0.388731 0.673302i −0.603548 0.797327i \(-0.706247\pi\)
0.992279 + 0.124025i \(0.0395802\pi\)
\(318\) 0 0
\(319\) 6.51911 11.2914i 0.365000 0.632199i
\(320\) 0 0
\(321\) −19.1451 33.1603i −1.06857 1.85082i
\(322\) 0 0
\(323\) 2.55030 9.94126i 0.141902 0.553147i
\(324\) 0 0
\(325\) −1.62412 2.81306i −0.0900902 0.156041i
\(326\) 0 0
\(327\) −15.9078 + 27.5531i −0.879704 + 1.52369i
\(328\) 0 0
\(329\) 15.0326 26.0372i 0.828774 1.43548i
\(330\) 0 0
\(331\) 28.6993 1.57746 0.788728 0.614742i \(-0.210740\pi\)
0.788728 + 0.614742i \(0.210740\pi\)
\(332\) 0 0
\(333\) 6.18205 10.7076i 0.338774 0.586774i
\(334\) 0 0
\(335\) −9.75071 −0.532738
\(336\) 0 0
\(337\) 17.6628 + 30.5928i 0.962153 + 1.66650i 0.717077 + 0.696994i \(0.245480\pi\)
0.245076 + 0.969504i \(0.421187\pi\)
\(338\) 0 0
\(339\) −7.69809 13.3335i −0.418103 0.724175i
\(340\) 0 0
\(341\) −33.7704 −1.82877
\(342\) 0 0
\(343\) −17.8950 −0.966241
\(344\) 0 0
\(345\) 2.70727 + 4.68912i 0.145754 + 0.252454i
\(346\) 0 0
\(347\) −13.0031 22.5221i −0.698044 1.20905i −0.969144 0.246496i \(-0.920721\pi\)
0.271100 0.962551i \(-0.412613\pi\)
\(348\) 0 0
\(349\) 2.44757 0.131015 0.0655077 0.997852i \(-0.479133\pi\)
0.0655077 + 0.997852i \(0.479133\pi\)
\(350\) 0 0
\(351\) 1.45150 2.51408i 0.0774755 0.134191i
\(352\) 0 0
\(353\) −4.85103 −0.258194 −0.129097 0.991632i \(-0.541208\pi\)
−0.129097 + 0.991632i \(0.541208\pi\)
\(354\) 0 0
\(355\) −3.45850 + 5.99029i −0.183558 + 0.317932i
\(356\) 0 0
\(357\) −8.09412 + 14.0194i −0.428386 + 0.741987i
\(358\) 0 0
\(359\) 3.67376 + 6.36314i 0.193894 + 0.335834i 0.946537 0.322594i \(-0.104555\pi\)
−0.752644 + 0.658428i \(0.771222\pi\)
\(360\) 0 0
\(361\) 0.405741 + 18.9957i 0.0213548 + 0.999772i
\(362\) 0 0
\(363\) −0.0201603 0.0349186i −0.00105814 0.00183275i
\(364\) 0 0
\(365\) 1.24025 2.14818i 0.0649176 0.112441i
\(366\) 0 0
\(367\) −6.51784 + 11.2892i −0.340228 + 0.589293i −0.984475 0.175525i \(-0.943838\pi\)
0.644247 + 0.764818i \(0.277171\pi\)
\(368\) 0 0
\(369\) −2.44018 −0.127031
\(370\) 0 0
\(371\) 12.2502 21.2179i 0.635998 1.10158i
\(372\) 0 0
\(373\) −28.5514 −1.47833 −0.739167 0.673522i \(-0.764781\pi\)
−0.739167 + 0.673522i \(0.764781\pi\)
\(374\) 0 0
\(375\) 1.26041 + 2.18309i 0.0650873 + 0.112734i
\(376\) 0 0
\(377\) 6.38936 + 11.0667i 0.329069 + 0.569964i
\(378\) 0 0
\(379\) 13.2128 0.678698 0.339349 0.940661i \(-0.389793\pi\)
0.339349 + 0.940661i \(0.389793\pi\)
\(380\) 0 0
\(381\) 22.9956 1.17810
\(382\) 0 0
\(383\) 18.8309 + 32.6160i 0.962212 + 1.66660i 0.716925 + 0.697150i \(0.245549\pi\)
0.245287 + 0.969450i \(0.421118\pi\)
\(384\) 0 0
\(385\) −4.51964 7.82824i −0.230342 0.398964i
\(386\) 0 0
\(387\) −7.95995 −0.404627
\(388\) 0 0
\(389\) 12.8223 22.2090i 0.650119 1.12604i −0.332975 0.942936i \(-0.608052\pi\)
0.983094 0.183103i \(-0.0586142\pi\)
\(390\) 0 0
\(391\) −5.05736 −0.255762
\(392\) 0 0
\(393\) 28.0637 48.6078i 1.41563 2.45194i
\(394\) 0 0
\(395\) −5.99948 + 10.3914i −0.301866 + 0.522848i
\(396\) 0 0
\(397\) 16.6366 + 28.8154i 0.834965 + 1.44620i 0.894059 + 0.447949i \(0.147845\pi\)
−0.0590940 + 0.998252i \(0.518821\pi\)
\(398\) 0 0
\(399\) 7.44699 29.0290i 0.372816 1.45327i
\(400\) 0 0
\(401\) −4.70674 8.15232i −0.235044 0.407107i 0.724242 0.689546i \(-0.242190\pi\)
−0.959285 + 0.282439i \(0.908857\pi\)
\(402\) 0 0
\(403\) 16.5491 28.6639i 0.824370 1.42785i
\(404\) 0 0
\(405\) 3.90535 6.76427i 0.194059 0.336119i
\(406\) 0 0
\(407\) −12.2155 −0.605499
\(408\) 0 0
\(409\) 1.25545 2.17450i 0.0620779 0.107522i −0.833316 0.552797i \(-0.813561\pi\)
0.895394 + 0.445275i \(0.146894\pi\)
\(410\) 0 0
\(411\) 19.2266 0.948376
\(412\) 0 0
\(413\) 14.9677 + 25.9248i 0.736511 + 1.27567i
\(414\) 0 0
\(415\) −2.34355 4.05915i −0.115041 0.199256i
\(416\) 0 0
\(417\) −42.6480 −2.08848
\(418\) 0 0
\(419\) −39.5638 −1.93282 −0.966409 0.257011i \(-0.917262\pi\)
−0.966409 + 0.257011i \(0.917262\pi\)
\(420\) 0 0
\(421\) 7.07892 + 12.2611i 0.345006 + 0.597567i 0.985355 0.170517i \(-0.0545437\pi\)
−0.640349 + 0.768084i \(0.721210\pi\)
\(422\) 0 0
\(423\) 18.4890 + 32.0238i 0.898965 + 1.55705i
\(424\) 0 0
\(425\) −2.35453 −0.114212
\(426\) 0 0
\(427\) 11.5300 19.9705i 0.557976 0.966442i
\(428\) 0 0
\(429\) −27.1376 −1.31022
\(430\) 0 0
\(431\) 3.26287 5.65145i 0.157167 0.272221i −0.776679 0.629897i \(-0.783097\pi\)
0.933846 + 0.357676i \(0.116431\pi\)
\(432\) 0 0
\(433\) 4.30073 7.44908i 0.206680 0.357980i −0.743987 0.668194i \(-0.767067\pi\)
0.950667 + 0.310214i \(0.100401\pi\)
\(434\) 0 0
\(435\) −4.95850 8.58837i −0.237742 0.411781i
\(436\) 0 0
\(437\) 9.01718 2.51965i 0.431350 0.120531i
\(438\) 0 0
\(439\) 5.77939 + 10.0102i 0.275835 + 0.477760i 0.970345 0.241722i \(-0.0777123\pi\)
−0.694510 + 0.719483i \(0.744379\pi\)
\(440\) 0 0
\(441\) −0.736084 + 1.27494i −0.0350516 + 0.0607112i
\(442\) 0 0
\(443\) 5.69629 9.86626i 0.270639 0.468760i −0.698387 0.715721i \(-0.746098\pi\)
0.969026 + 0.246960i \(0.0794318\pi\)
\(444\) 0 0
\(445\) 8.54410 0.405029
\(446\) 0 0
\(447\) −15.6243 + 27.0620i −0.739002 + 1.27999i
\(448\) 0 0
\(449\) 17.2252 0.812909 0.406455 0.913671i \(-0.366765\pi\)
0.406455 + 0.913671i \(0.366765\pi\)
\(450\) 0 0
\(451\) 1.20542 + 2.08786i 0.0567612 + 0.0983133i
\(452\) 0 0
\(453\) −18.2190 31.5563i −0.856005 1.48264i
\(454\) 0 0
\(455\) 8.85936 0.415333
\(456\) 0 0
\(457\) −26.1885 −1.22505 −0.612524 0.790452i \(-0.709846\pi\)
−0.612524 + 0.790452i \(0.709846\pi\)
\(458\) 0 0
\(459\) −1.05214 1.82236i −0.0491097 0.0850605i
\(460\) 0 0
\(461\) −20.2166 35.0162i −0.941580 1.63086i −0.762458 0.647038i \(-0.776008\pi\)
−0.179122 0.983827i \(-0.557326\pi\)
\(462\) 0 0
\(463\) 9.48542 0.440825 0.220412 0.975407i \(-0.429260\pi\)
0.220412 + 0.975407i \(0.429260\pi\)
\(464\) 0 0
\(465\) −12.8430 + 22.2448i −0.595581 + 1.03158i
\(466\) 0 0
\(467\) −39.2462 −1.81610 −0.908048 0.418867i \(-0.862427\pi\)
−0.908048 + 0.418867i \(0.862427\pi\)
\(468\) 0 0
\(469\) 13.2972 23.0314i 0.614006 1.06349i
\(470\) 0 0
\(471\) −22.9557 + 39.7604i −1.05774 + 1.83206i
\(472\) 0 0
\(473\) 3.93214 + 6.81066i 0.180800 + 0.313155i
\(474\) 0 0
\(475\) 4.19809 1.17306i 0.192621 0.0538237i
\(476\) 0 0
\(477\) 15.0668 + 26.0965i 0.689862 + 1.19488i
\(478\) 0 0
\(479\) −2.72677 + 4.72290i −0.124589 + 0.215795i −0.921572 0.388207i \(-0.873095\pi\)
0.796983 + 0.604002i \(0.206428\pi\)
\(480\) 0 0
\(481\) 5.98617 10.3684i 0.272946 0.472756i
\(482\) 0 0
\(483\) −14.7677 −0.671956
\(484\) 0 0
\(485\) −3.61494 + 6.26127i −0.164146 + 0.284309i
\(486\) 0 0
\(487\) 25.0662 1.13586 0.567930 0.823077i \(-0.307744\pi\)
0.567930 + 0.823077i \(0.307744\pi\)
\(488\) 0 0
\(489\) 24.2808 + 42.0557i 1.09802 + 1.90182i
\(490\) 0 0
\(491\) −14.8412 25.7057i −0.669773 1.16008i −0.977967 0.208758i \(-0.933058\pi\)
0.308194 0.951324i \(-0.400275\pi\)
\(492\) 0 0
\(493\) 9.26281 0.417176
\(494\) 0 0
\(495\) 11.1176 0.499701
\(496\) 0 0
\(497\) −9.43280 16.3381i −0.423119 0.732863i
\(498\) 0 0
\(499\) 6.85519 + 11.8735i 0.306881 + 0.531533i 0.977678 0.210108i \(-0.0673814\pi\)
−0.670798 + 0.741640i \(0.734048\pi\)
\(500\) 0 0
\(501\) −14.3328 −0.640344
\(502\) 0 0
\(503\) −18.3253 + 31.7404i −0.817086 + 1.41523i 0.0907343 + 0.995875i \(0.471079\pi\)
−0.907820 + 0.419359i \(0.862255\pi\)
\(504\) 0 0
\(505\) −5.77635 −0.257044
\(506\) 0 0
\(507\) −3.08661 + 5.34616i −0.137081 + 0.237431i
\(508\) 0 0
\(509\) 6.32519 10.9556i 0.280359 0.485596i −0.691114 0.722746i \(-0.742880\pi\)
0.971473 + 0.237149i \(0.0762131\pi\)
\(510\) 0 0
\(511\) 3.38269 + 5.85899i 0.149641 + 0.259187i
\(512\) 0 0
\(513\) 2.78387 + 2.72504i 0.122911 + 0.120314i
\(514\) 0 0
\(515\) 7.52882 + 13.0403i 0.331759 + 0.574624i
\(516\) 0 0
\(517\) 18.2667 31.6389i 0.803371 1.39148i
\(518\) 0 0
\(519\) −11.5393 + 19.9867i −0.506520 + 0.877318i
\(520\) 0 0
\(521\) 16.3339 0.715601 0.357800 0.933798i \(-0.383527\pi\)
0.357800 + 0.933798i \(0.383527\pi\)
\(522\) 0 0
\(523\) 4.56734 7.91086i 0.199716 0.345918i −0.748720 0.662886i \(-0.769331\pi\)
0.948436 + 0.316968i \(0.102665\pi\)
\(524\) 0 0
\(525\) −6.87535 −0.300065
\(526\) 0 0
\(527\) −11.9958 20.7774i −0.522547 0.905078i
\(528\) 0 0
\(529\) 9.19321 + 15.9231i 0.399705 + 0.692309i
\(530\) 0 0
\(531\) −36.8183 −1.59778
\(532\) 0 0
\(533\) −2.36286 −0.102347
\(534\) 0 0
\(535\) 7.59478 + 13.1545i 0.328351 + 0.568721i
\(536\) 0 0
\(537\) 15.3530 + 26.5922i 0.662531 + 1.14754i
\(538\) 0 0
\(539\) 1.45447 0.0626486
\(540\) 0 0
\(541\) 1.45954 2.52800i 0.0627507 0.108687i −0.832943 0.553358i \(-0.813346\pi\)
0.895694 + 0.444671i \(0.146679\pi\)
\(542\) 0 0
\(543\) −38.8513 −1.66727
\(544\) 0 0
\(545\) 6.31057 10.9302i 0.270315 0.468200i
\(546\) 0 0
\(547\) 1.73973 3.01329i 0.0743853 0.128839i −0.826433 0.563034i \(-0.809634\pi\)
0.900819 + 0.434195i \(0.142967\pi\)
\(548\) 0 0
\(549\) 14.1810 + 24.5623i 0.605232 + 1.04829i
\(550\) 0 0
\(551\) −16.5154 + 4.61486i −0.703580 + 0.196600i
\(552\) 0 0
\(553\) −16.3631 28.3418i −0.695831 1.20522i
\(554\) 0 0
\(555\) −4.64560 + 8.04642i −0.197195 + 0.341552i
\(556\) 0 0
\(557\) −7.96280 + 13.7920i −0.337395 + 0.584385i −0.983942 0.178489i \(-0.942879\pi\)
0.646547 + 0.762874i \(0.276212\pi\)
\(558\) 0 0
\(559\) −7.70775 −0.326003
\(560\) 0 0
\(561\) −9.83551 + 17.0356i −0.415256 + 0.719244i
\(562\) 0 0
\(563\) 19.6274 0.827195 0.413598 0.910460i \(-0.364272\pi\)
0.413598 + 0.910460i \(0.364272\pi\)
\(564\) 0 0
\(565\) 3.05380 + 5.28934i 0.128474 + 0.222524i
\(566\) 0 0
\(567\) 10.6516 + 18.4491i 0.447324 + 0.774788i
\(568\) 0 0
\(569\) −11.7544 −0.492770 −0.246385 0.969172i \(-0.579243\pi\)
−0.246385 + 0.969172i \(0.579243\pi\)
\(570\) 0 0
\(571\) 32.4868 1.35953 0.679766 0.733429i \(-0.262081\pi\)
0.679766 + 0.733429i \(0.262081\pi\)
\(572\) 0 0
\(573\) −20.4795 35.4715i −0.855541 1.48184i
\(574\) 0 0
\(575\) −1.07396 1.86016i −0.0447873 0.0775740i
\(576\) 0 0
\(577\) 25.7287 1.07110 0.535551 0.844503i \(-0.320104\pi\)
0.535551 + 0.844503i \(0.320104\pi\)
\(578\) 0 0
\(579\) 1.07264 1.85787i 0.0445775 0.0772106i
\(580\) 0 0
\(581\) 12.7837 0.530359
\(582\) 0 0
\(583\) 14.8857 25.7828i 0.616503 1.06782i
\(584\) 0 0
\(585\) −5.44818 + 9.43652i −0.225254 + 0.390152i
\(586\) 0 0
\(587\) 20.8279 + 36.0750i 0.859659 + 1.48897i 0.872255 + 0.489052i \(0.162657\pi\)
−0.0125958 + 0.999921i \(0.504009\pi\)
\(588\) 0 0
\(589\) 31.7399 + 31.0692i 1.30782 + 1.28018i
\(590\) 0 0
\(591\) 25.1884 + 43.6276i 1.03611 + 1.79460i
\(592\) 0 0
\(593\) −8.83267 + 15.2986i −0.362714 + 0.628239i −0.988407 0.151831i \(-0.951483\pi\)
0.625692 + 0.780070i \(0.284817\pi\)
\(594\) 0 0
\(595\) 3.21091 5.56146i 0.131634 0.227998i
\(596\) 0 0
\(597\) 16.2775 0.666193
\(598\) 0 0
\(599\) 13.1925 22.8502i 0.539033 0.933632i −0.459924 0.887959i \(-0.652123\pi\)
0.998956 0.0456738i \(-0.0145435\pi\)
\(600\) 0 0
\(601\) 9.46838 0.386223 0.193112 0.981177i \(-0.438142\pi\)
0.193112 + 0.981177i \(0.438142\pi\)
\(602\) 0 0
\(603\) 16.3545 + 28.3269i 0.666008 + 1.15356i
\(604\) 0 0
\(605\) 0.00799750 + 0.0138521i 0.000325145 + 0.000563167i
\(606\) 0 0
\(607\) −8.26529 −0.335478 −0.167739 0.985831i \(-0.553647\pi\)
−0.167739 + 0.985831i \(0.553647\pi\)
\(608\) 0 0
\(609\) 27.0479 1.09604
\(610\) 0 0
\(611\) 17.9032 + 31.0092i 0.724285 + 1.25450i
\(612\) 0 0
\(613\) −3.40417 5.89620i −0.137493 0.238145i 0.789054 0.614324i \(-0.210571\pi\)
−0.926547 + 0.376179i \(0.877238\pi\)
\(614\) 0 0
\(615\) 1.83371 0.0739425
\(616\) 0 0
\(617\) −3.00668 + 5.20772i −0.121044 + 0.209655i −0.920180 0.391496i \(-0.871958\pi\)
0.799135 + 0.601151i \(0.205291\pi\)
\(618\) 0 0
\(619\) 13.2299 0.531754 0.265877 0.964007i \(-0.414338\pi\)
0.265877 + 0.964007i \(0.414338\pi\)
\(620\) 0 0
\(621\) 0.959816 1.66245i 0.0385161 0.0667118i
\(622\) 0 0
\(623\) −11.6517 + 20.1813i −0.466816 + 0.808548i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 9.04916 35.2743i 0.361389 1.40872i
\(628\) 0 0
\(629\) −4.33915 7.51564i −0.173013 0.299668i
\(630\) 0 0
\(631\) 11.8932 20.5996i 0.473460 0.820058i −0.526078 0.850436i \(-0.676338\pi\)
0.999538 + 0.0303788i \(0.00967135\pi\)
\(632\) 0 0
\(633\) 13.2899 23.0188i 0.528226 0.914914i
\(634\) 0 0
\(635\) −9.12228 −0.362007
\(636\) 0 0
\(637\) −0.712762 + 1.23454i −0.0282407 + 0.0489143i
\(638\) 0 0
\(639\) 23.2033 0.917908
\(640\) 0 0
\(641\) −6.14239 10.6389i −0.242610 0.420212i 0.718847 0.695168i \(-0.244670\pi\)
−0.961457 + 0.274956i \(0.911337\pi\)
\(642\) 0 0
\(643\) −14.5776 25.2492i −0.574885 0.995729i −0.996054 0.0887474i \(-0.971714\pi\)
0.421170 0.906982i \(-0.361620\pi\)
\(644\) 0 0
\(645\) 5.98164 0.235527
\(646\) 0 0
\(647\) −36.7499 −1.44479 −0.722394 0.691481i \(-0.756958\pi\)
−0.722394 + 0.691481i \(0.756958\pi\)
\(648\) 0 0
\(649\) 18.1879 + 31.5023i 0.713936 + 1.23657i
\(650\) 0 0
\(651\) −35.0284 60.6710i −1.37287 2.37788i
\(652\) 0 0
\(653\) 24.7707 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(654\) 0 0
\(655\) −11.1328 + 19.2825i −0.434994 + 0.753431i
\(656\) 0 0
\(657\) −8.32092 −0.324630
\(658\) 0 0
\(659\) −11.1156 + 19.2528i −0.433002 + 0.749982i −0.997130 0.0757053i \(-0.975879\pi\)
0.564128 + 0.825687i \(0.309213\pi\)
\(660\) 0 0
\(661\) −20.4684 + 35.4524i −0.796130 + 1.37894i 0.125990 + 0.992032i \(0.459789\pi\)
−0.922119 + 0.386905i \(0.873544\pi\)
\(662\) 0 0
\(663\) −9.63975 16.6965i −0.374377 0.648440i
\(664\) 0 0
\(665\) −2.95419 + 11.5157i −0.114559 + 0.446559i
\(666\) 0 0
\(667\) 4.22501 + 7.31793i 0.163593 + 0.283351i
\(668\) 0 0
\(669\) 12.2592 21.2336i 0.473969 0.820939i
\(670\) 0 0
\(671\) 14.0106 24.2671i 0.540873 0.936819i
\(672\) 0 0
\(673\) −33.5992 −1.29515 −0.647577 0.762000i \(-0.724217\pi\)
−0.647577 + 0.762000i \(0.724217\pi\)
\(674\) 0 0
\(675\) 0.446857 0.773979i 0.0171995 0.0297905i
\(676\) 0 0
\(677\) 29.5883 1.13717 0.568585 0.822624i \(-0.307491\pi\)
0.568585 + 0.822624i \(0.307491\pi\)
\(678\) 0 0
\(679\) −9.85949 17.0771i −0.378373 0.655361i
\(680\) 0 0
\(681\) 28.4598 + 49.2938i 1.09058 + 1.88894i
\(682\) 0 0
\(683\) 41.0567 1.57099 0.785495 0.618868i \(-0.212408\pi\)
0.785495 + 0.618868i \(0.212408\pi\)
\(684\) 0 0
\(685\) −7.62711 −0.291417
\(686\) 0 0
\(687\) −20.4488 35.4183i −0.780170 1.35129i
\(688\) 0 0
\(689\) 14.5894 + 25.2696i 0.555813 + 0.962697i
\(690\) 0 0
\(691\) 1.22497 0.0466000 0.0233000 0.999729i \(-0.492583\pi\)
0.0233000 + 0.999729i \(0.492583\pi\)
\(692\) 0 0
\(693\) −15.1613 + 26.2601i −0.575929 + 0.997538i
\(694\) 0 0
\(695\) 16.9183 0.641748
\(696\) 0 0
\(697\) −0.856376 + 1.48329i −0.0324375 + 0.0561835i
\(698\) 0 0
\(699\) −23.5228 + 40.7426i −0.889712 + 1.54103i
\(700\) 0 0
\(701\) −22.0101 38.1226i −0.831309 1.43987i −0.897001 0.442029i \(-0.854259\pi\)
0.0656918 0.997840i \(-0.479075\pi\)
\(702\) 0 0
\(703\) 11.4810 + 11.2384i 0.433015 + 0.423865i
\(704\) 0 0
\(705\) −13.8939 24.0649i −0.523273 0.906335i
\(706\) 0 0
\(707\) 7.87729 13.6439i 0.296256 0.513130i
\(708\) 0 0
\(709\) −8.16255 + 14.1379i −0.306551 + 0.530962i −0.977605 0.210446i \(-0.932508\pi\)
0.671055 + 0.741408i \(0.265842\pi\)
\(710\) 0 0
\(711\) 40.2509 1.50953
\(712\) 0 0
\(713\) 10.9432 18.9542i 0.409827 0.709841i
\(714\) 0 0
\(715\) 10.7654 0.402602
\(716\) 0 0
\(717\) 0.759750 + 1.31593i 0.0283734 + 0.0491442i
\(718\) 0 0
\(719\) −5.08574 8.80876i −0.189666 0.328511i 0.755473 0.655180i \(-0.227407\pi\)
−0.945139 + 0.326669i \(0.894074\pi\)
\(720\) 0 0
\(721\) −41.0686 −1.52947
\(722\) 0 0
\(723\) −55.1595 −2.05141
\(724\) 0 0
\(725\) 1.96702 + 3.40697i 0.0730532 + 0.126532i
\(726\) 0 0
\(727\) −4.02148 6.96541i −0.149148 0.258333i 0.781765 0.623574i \(-0.214320\pi\)
−0.930913 + 0.365241i \(0.880987\pi\)
\(728\) 0 0
\(729\) −32.9600 −1.22074
\(730\) 0 0
\(731\) −2.79353 + 4.83853i −0.103322 + 0.178960i
\(732\) 0 0
\(733\) 41.2325 1.52296 0.761479 0.648190i \(-0.224474\pi\)
0.761479 + 0.648190i \(0.224474\pi\)
\(734\) 0 0
\(735\) 0.553143 0.958072i 0.0204030 0.0353390i
\(736\) 0 0
\(737\) 16.1580 27.9864i 0.595186 1.03089i
\(738\) 0 0
\(739\) −16.5253 28.6227i −0.607893 1.05290i −0.991587 0.129442i \(-0.958681\pi\)
0.383694 0.923460i \(-0.374652\pi\)
\(740\) 0 0
\(741\) 25.5059 + 24.9669i 0.936983 + 0.917184i
\(742\) 0 0
\(743\) −17.3933 30.1261i −0.638099 1.10522i −0.985849 0.167633i \(-0.946388\pi\)
0.347750 0.937587i \(-0.386946\pi\)
\(744\) 0 0
\(745\) 6.19809 10.7354i 0.227080 0.393315i
\(746\) 0 0
\(747\) −7.86153 + 13.6166i −0.287638 + 0.498204i
\(748\) 0 0
\(749\) −41.4284 −1.51376
\(750\) 0 0
\(751\) 21.7350 37.6462i 0.793123 1.37373i −0.130902 0.991395i \(-0.541787\pi\)
0.924024 0.382334i \(-0.124879\pi\)
\(752\) 0 0
\(753\) −20.8537 −0.759950
\(754\) 0 0
\(755\) 7.22743 + 12.5183i 0.263033 + 0.455587i
\(756\) 0 0
\(757\) 3.08494 + 5.34328i 0.112124 + 0.194205i 0.916626 0.399745i \(-0.130901\pi\)
−0.804502 + 0.593949i \(0.797568\pi\)
\(758\) 0 0
\(759\) −17.9449 −0.651359
\(760\) 0 0
\(761\) −9.81318 −0.355728 −0.177864 0.984055i \(-0.556919\pi\)
−0.177864 + 0.984055i \(0.556919\pi\)
\(762\) 0 0
\(763\) 17.2116 + 29.8114i 0.623103 + 1.07925i
\(764\) 0 0
\(765\) 3.94918 + 6.84018i 0.142783 + 0.247307i
\(766\) 0 0
\(767\) −35.6517 −1.28731
\(768\) 0 0
\(769\) 20.7452 35.9317i 0.748090 1.29573i −0.200647 0.979664i \(-0.564305\pi\)
0.948737 0.316066i \(-0.102362\pi\)
\(770\) 0 0
\(771\) −35.7727 −1.28832
\(772\) 0 0
\(773\) −20.8584 + 36.1279i −0.750226 + 1.29943i 0.197487 + 0.980306i \(0.436722\pi\)
−0.947713 + 0.319124i \(0.896611\pi\)
\(774\) 0 0
\(775\) 5.09478 8.82442i 0.183010 0.316983i
\(776\) 0 0
\(777\) −12.6705 21.9460i −0.454553 0.787309i
\(778\) 0 0
\(779\) 0.787908 3.07133i 0.0282297 0.110042i
\(780\) 0 0
\(781\) −11.4622 19.8531i −0.410149 0.710400i
\(782\) 0 0
\(783\) −1.75795 + 3.04486i −0.0628240 + 0.108814i
\(784\) 0 0
\(785\) 9.10642 15.7728i 0.325022 0.562955i
\(786\) 0 0
\(787\) −13.0342 −0.464621 −0.232310 0.972642i \(-0.574629\pi\)
−0.232310 + 0.972642i \(0.574629\pi\)
\(788\) 0 0
\(789\) −23.3718 + 40.4812i −0.832060 + 1.44117i
\(790\) 0 0
\(791\) −16.6580 −0.592292
\(792\) 0 0
\(793\) 13.7317 + 23.7841i 0.487628 + 0.844596i
\(794\) 0 0
\(795\) −11.3222 19.6106i −0.401557 0.695518i
\(796\) 0 0
\(797\) 33.1911 1.17569 0.587844 0.808974i \(-0.299977\pi\)
0.587844 + 0.808974i \(0.299977\pi\)
\(798\) 0 0
\(799\) 25.9547 0.918210
\(800\) 0 0
\(801\) −14.3307 24.8216i −0.506351 0.877026i
\(802\) 0 0
\(803\) 4.11045 + 7.11951i 0.145055 + 0.251242i
\(804\) 0 0
\(805\) 5.85831 0.206478
\(806\) 0 0
\(807\) 8.40369 14.5556i 0.295824 0.512382i
\(808\) 0 0
\(809\) −29.5271 −1.03812 −0.519058 0.854739i \(-0.673717\pi\)
−0.519058 + 0.854739i \(0.673717\pi\)
\(810\) 0 0
\(811\) −14.4097 + 24.9583i −0.505993 + 0.876406i 0.493983 + 0.869472i \(0.335541\pi\)
−0.999976 + 0.00693438i \(0.997793\pi\)
\(812\) 0 0
\(813\) 27.8125 48.1727i 0.975427 1.68949i
\(814\) 0 0
\(815\) −9.63212 16.6833i −0.337398 0.584391i
\(816\) 0 0
\(817\) 2.57018 10.0188i 0.0899194 0.350513i
\(818\) 0 0
\(819\) −14.8595 25.7374i −0.519233 0.899338i
\(820\) 0 0
\(821\) 9.40280 16.2861i 0.328160 0.568390i −0.653987 0.756506i \(-0.726905\pi\)
0.982147 + 0.188116i \(0.0602382\pi\)
\(822\) 0 0
\(823\) −3.78479 + 6.55545i −0.131929 + 0.228509i −0.924420 0.381375i \(-0.875451\pi\)
0.792491 + 0.609884i \(0.208784\pi\)
\(824\) 0 0
\(825\) −8.35453 −0.290868
\(826\) 0 0
\(827\) −17.4683 + 30.2560i −0.607434 + 1.05211i 0.384228 + 0.923238i \(0.374468\pi\)
−0.991662 + 0.128868i \(0.958866\pi\)
\(828\) 0 0
\(829\) −45.8376 −1.59201 −0.796003 0.605293i \(-0.793056\pi\)
−0.796003 + 0.605293i \(0.793056\pi\)
\(830\) 0 0
\(831\) −30.8327 53.4037i −1.06957 1.85256i
\(832\) 0 0
\(833\) 0.516655 + 0.894872i 0.0179010 + 0.0310055i
\(834\) 0 0
\(835\) 5.68579 0.196765
\(836\) 0 0
\(837\) 9.10656 0.314769
\(838\) 0 0
\(839\) −7.00502 12.1330i −0.241840 0.418879i 0.719398 0.694598i \(-0.244418\pi\)
−0.961238 + 0.275719i \(0.911084\pi\)
\(840\) 0 0
\(841\) 6.76169 + 11.7116i 0.233162 + 0.403848i
\(842\) 0 0
\(843\) −47.0455 −1.62033
\(844\) 0 0
\(845\) 1.22445 2.12080i 0.0421222 0.0729578i
\(846\) 0 0
\(847\) −0.0436252 −0.00149898
\(848\) 0 0
\(849\) 5.66708 9.81568i 0.194494 0.336873i
\(850\) 0 0
\(851\) 3.95840 6.85615i 0.135692 0.235026i
\(852\) 0 0
\(853\) 18.7757 + 32.5205i 0.642869 + 1.11348i 0.984789 + 0.173753i \(0.0555895\pi\)
−0.341920 + 0.939729i \(0.611077\pi\)
\(854\) 0 0
\(855\) −10.4492 10.2284i −0.357354 0.349803i
\(856\) 0 0
\(857\) −2.03378 3.52261i −0.0694726 0.120330i 0.829197 0.558957i \(-0.188798\pi\)
−0.898669 + 0.438627i \(0.855465\pi\)
\(858\) 0 0
\(859\) −6.31785 + 10.9428i −0.215562 + 0.373365i −0.953446 0.301563i \(-0.902492\pi\)
0.737884 + 0.674928i \(0.235825\pi\)
\(860\) 0 0
\(861\) −2.50066 + 4.33127i −0.0852223 + 0.147609i
\(862\) 0 0
\(863\) −14.5147 −0.494085 −0.247042 0.969005i \(-0.579459\pi\)
−0.247042 + 0.969005i \(0.579459\pi\)
\(864\) 0 0
\(865\) 4.57760 7.92864i 0.155643 0.269582i
\(866\) 0 0
\(867\) 28.8790 0.980781
\(868\) 0 0
\(869\) −19.8835 34.4393i −0.674503 1.16827i
\(870\) 0 0
\(871\) 15.8364 + 27.4294i 0.536594 + 0.929409i
\(872\) 0 0
\(873\) 24.2529 0.820836
\(874\) 0 0
\(875\) 2.72743 0.0922039
\(876\) 0 0
\(877\) −0.967541 1.67583i −0.0326715 0.0565887i 0.849227 0.528027i \(-0.177068\pi\)
−0.881899 + 0.471439i \(0.843735\pi\)
\(878\) 0 0
\(879\) 1.60098 + 2.77298i 0.0539998 + 0.0935303i
\(880\) 0 0
\(881\) 15.9037 0.535811 0.267905 0.963445i \(-0.413669\pi\)
0.267905 + 0.963445i \(0.413669\pi\)
\(882\) 0 0
\(883\) 17.5216 30.3483i 0.589648 1.02130i −0.404631 0.914480i \(-0.632600\pi\)
0.994278 0.106820i \(-0.0340668\pi\)
\(884\) 0 0
\(885\) 27.6677 0.930040
\(886\) 0 0
\(887\) 8.08806 14.0089i 0.271571 0.470374i −0.697694 0.716396i \(-0.745790\pi\)
0.969264 + 0.246022i \(0.0791237\pi\)
\(888\) 0 0
\(889\) 12.4402 21.5470i 0.417230 0.722664i
\(890\) 0 0
\(891\) 12.9432 + 22.4182i 0.433613 + 0.751039i
\(892\) 0 0
\(893\) −46.2767 + 12.9310i −1.54859 + 0.432718i
\(894\) 0 0
\(895\) −6.09048 10.5490i −0.203582 0.352615i
\(896\) 0 0
\(897\) 8.79387 15.2314i 0.293619 0.508563i
\(898\) 0 0
\(899\) −20.0431 + 34.7156i −0.668473 + 1.15783i
\(900\) 0 0
\(901\) 21.1507 0.704631
\(902\) 0 0
\(903\) −8.15724 + 14.1288i −0.271456 + 0.470176i
\(904\) 0 0
\(905\) 15.4122 0.512318
\(906\) 0 0
\(907\) −11.4975 19.9143i −0.381770 0.661244i 0.609546 0.792751i \(-0.291352\pi\)
−0.991315 + 0.131507i \(0.958019\pi\)
\(908\) 0 0
\(909\) 9.68848 + 16.7809i 0.321347 + 0.556589i
\(910\) 0 0
\(911\) −23.0833 −0.764783 −0.382392 0.924000i \(-0.624899\pi\)
−0.382392 + 0.924000i \(0.624899\pi\)
\(912\) 0 0
\(913\) 15.5341 0.514103
\(914\) 0 0
\(915\) −10.6566 18.4577i −0.352296 0.610194i
\(916\) 0 0
\(917\) −30.3638 52.5917i −1.00270 1.73673i
\(918\) 0 0
\(919\) 12.5454 0.413835 0.206918 0.978358i \(-0.433657\pi\)
0.206918 + 0.978358i \(0.433657\pi\)
\(920\) 0 0
\(921\) −10.9060 + 18.8898i −0.359365 + 0.622439i
\(922\) 0 0
\(923\) 22.4681 0.739547
\(924\) 0 0
\(925\) 1.84289 3.19199i 0.0605940 0.104952i
\(926\) 0 0
\(927\) 25.2557 43.7441i 0.829505 1.43674i
\(928\) 0 0
\(929\) 26.6494 + 46.1581i 0.874338 + 1.51440i 0.857466 + 0.514540i \(0.172037\pi\)
0.0168714 + 0.999858i \(0.494629\pi\)
\(930\) 0 0
\(931\) −1.36702 1.33814i −0.0448024 0.0438556i
\(932\) 0 0
\(933\) 30.0604 + 52.0662i 0.984134 + 1.70457i
\(934\) 0 0
\(935\) 3.90171 6.75796i 0.127600 0.221009i
\(936\) 0 0
\(937\) 17.5602 30.4152i 0.573668 0.993622i −0.422517 0.906355i \(-0.638853\pi\)
0.996185 0.0872667i \(-0.0278132\pi\)
\(938\) 0 0
\(939\) 1.95704 0.0638656
\(940\) 0 0
\(941\) −6.57898 + 11.3951i −0.214469 + 0.371470i −0.953108 0.302630i \(-0.902135\pi\)
0.738639 + 0.674101i \(0.235469\pi\)
\(942\) 0 0
\(943\) −1.56246 −0.0508807
\(944\) 0 0
\(945\) 1.21877 + 2.11097i 0.0396466 + 0.0686699i
\(946\) 0 0
\(947\) 5.22510 + 9.05015i 0.169793 + 0.294090i 0.938347 0.345695i \(-0.112357\pi\)
−0.768554 + 0.639785i \(0.779023\pi\)
\(948\) 0 0
\(949\) −8.05728 −0.261550
\(950\) 0 0
\(951\) 34.8940 1.13152
\(952\) 0 0
\(953\) −2.17117 3.76057i −0.0703310 0.121817i 0.828715 0.559670i \(-0.189072\pi\)
−0.899046 + 0.437853i \(0.855739\pi\)
\(954\) 0 0
\(955\) 8.12412 + 14.0714i 0.262890 + 0.455340i
\(956\) 0 0
\(957\) 32.8670 1.06244
\(958\) 0 0
\(959\) 10.4012 18.0154i 0.335872 0.581747i
\(960\) 0 0
\(961\) 72.8272 2.34927
\(962\) 0 0
\(963\) 25.4770 44.1274i 0.820983 1.42198i
\(964\) 0 0
\(965\) −0.425514 + 0.737011i −0.0136978 + 0.0237252i
\(966\) 0 0
\(967\) 7.84100 + 13.5810i 0.252149 + 0.436736i 0.964117 0.265476i \(-0.0855292\pi\)
−0.711968 + 0.702212i \(0.752196\pi\)
\(968\) 0 0
\(969\) 24.9171 6.96253i 0.800454 0.223669i
\(970\) 0 0
\(971\) −1.33867 2.31865i −0.0429601 0.0744091i 0.843746 0.536743i \(-0.180346\pi\)
−0.886706 + 0.462334i \(0.847012\pi\)
\(972\) 0 0
\(973\) −23.0717 + 39.9614i −0.739646 + 1.28110i
\(974\) 0 0
\(975\) 4.09412 7.09123i 0.131117 0.227101i
\(976\) 0 0
\(977\) 2.87156 0.0918693 0.0459347 0.998944i \(-0.485373\pi\)
0.0459347 + 0.998944i \(0.485373\pi\)
\(978\) 0 0
\(979\) −14.1585 + 24.5232i −0.452507 + 0.783765i
\(980\) 0 0
\(981\) −42.3380 −1.35175
\(982\) 0 0
\(983\) 16.9545 + 29.3660i 0.540764 + 0.936630i 0.998860 + 0.0477276i \(0.0151979\pi\)
−0.458097 + 0.888902i \(0.651469\pi\)
\(984\) 0 0
\(985\) −9.99214 17.3069i −0.318376 0.551443i
\(986\) 0 0
\(987\) 75.7889 2.41239
\(988\) 0 0
\(989\) −5.09680 −0.162069
\(990\) 0 0
\(991\) 20.0649 + 34.7535i 0.637383 + 1.10398i 0.986005 + 0.166716i \(0.0533165\pi\)
−0.348622 + 0.937264i \(0.613350\pi\)
\(992\) 0 0
\(993\) 36.1729 + 62.6533i 1.14791 + 1.98824i
\(994\) 0 0
\(995\) −6.45722 −0.204708
\(996\) 0 0
\(997\) −1.75369 + 3.03748i −0.0555399 + 0.0961979i −0.892459 0.451129i \(-0.851021\pi\)
0.836919 + 0.547327i \(0.184355\pi\)
\(998\) 0 0
\(999\) 3.29404 0.104219
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 380.2.i.c.121.4 8
3.2 odd 2 3420.2.t.w.1261.3 8
4.3 odd 2 1520.2.q.m.881.1 8
5.2 odd 4 1900.2.s.d.349.7 16
5.3 odd 4 1900.2.s.d.349.2 16
5.4 even 2 1900.2.i.d.501.1 8
19.7 even 3 7220.2.a.r.1.1 4
19.11 even 3 inner 380.2.i.c.201.4 yes 8
19.12 odd 6 7220.2.a.p.1.4 4
57.11 odd 6 3420.2.t.w.3241.3 8
76.11 odd 6 1520.2.q.m.961.1 8
95.49 even 6 1900.2.i.d.201.1 8
95.68 odd 12 1900.2.s.d.49.7 16
95.87 odd 12 1900.2.s.d.49.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.i.c.121.4 8 1.1 even 1 trivial
380.2.i.c.201.4 yes 8 19.11 even 3 inner
1520.2.q.m.881.1 8 4.3 odd 2
1520.2.q.m.961.1 8 76.11 odd 6
1900.2.i.d.201.1 8 95.49 even 6
1900.2.i.d.501.1 8 5.4 even 2
1900.2.s.d.49.2 16 95.87 odd 12
1900.2.s.d.49.7 16 95.68 odd 12
1900.2.s.d.349.2 16 5.3 odd 4
1900.2.s.d.349.7 16 5.2 odd 4
3420.2.t.w.1261.3 8 3.2 odd 2
3420.2.t.w.3241.3 8 57.11 odd 6
7220.2.a.p.1.4 4 19.12 odd 6
7220.2.a.r.1.1 4 19.7 even 3