Properties

Label 1344.2.q.a.961.1
Level $1344$
Weight $2$
Character 1344.961
Analytic conductor $10.732$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(193,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1344.961
Dual form 1344.2.q.a.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.00000 + 5.19615i) q^{11} -5.00000 q^{13} +4.00000 q^{15} +(-1.00000 - 1.73205i) q^{17} +(0.500000 - 0.866025i) q^{19} +(2.00000 + 1.73205i) q^{21} +(3.00000 - 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +1.00000 q^{27} +(1.50000 + 2.59808i) q^{31} +(3.00000 - 5.19615i) q^{33} +(2.00000 - 10.3923i) q^{35} +(1.50000 - 2.59808i) q^{37} +(2.50000 + 4.33013i) q^{39} -6.00000 q^{41} -5.00000 q^{43} +(-2.00000 - 3.46410i) q^{45} +(2.00000 - 3.46410i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-3.00000 - 5.19615i) q^{53} -24.0000 q^{55} -1.00000 q^{57} +(-3.00000 - 5.19615i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(0.500000 - 2.59808i) q^{63} +(10.0000 - 17.3205i) q^{65} +(3.50000 + 6.06218i) q^{67} -6.00000 q^{69} +16.0000 q^{71} +(1.50000 + 2.59808i) q^{73} +(-5.50000 + 9.52628i) q^{75} +(-12.0000 - 10.3923i) q^{77} +(-5.50000 + 9.52628i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} +8.00000 q^{85} +(-2.00000 + 3.46410i) q^{89} +(12.5000 - 4.33013i) q^{91} +(1.50000 - 2.59808i) q^{93} +(2.00000 + 3.46410i) q^{95} -6.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 4 q^{5} - 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 4 q^{5} - 5 q^{7} - q^{9} + 6 q^{11} - 10 q^{13} + 8 q^{15} - 2 q^{17} + q^{19} + 4 q^{21} + 6 q^{23} - 11 q^{25} + 2 q^{27} + 3 q^{31} + 6 q^{33} + 4 q^{35} + 3 q^{37} + 5 q^{39} - 12 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{47} + 11 q^{49} - 2 q^{51} - 6 q^{53} - 48 q^{55} - 2 q^{57} - 6 q^{59} - 2 q^{61} + q^{63} + 20 q^{65} + 7 q^{67} - 12 q^{69} + 32 q^{71} + 3 q^{73} - 11 q^{75} - 24 q^{77} - 11 q^{79} - q^{81} - 24 q^{83} + 16 q^{85} - 4 q^{89} + 25 q^{91} + 3 q^{93} + 4 q^{95} - 12 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 0.866025i −0.288675 0.500000i
\(4\) 0 0
\(5\) −2.00000 + 3.46410i −0.894427 + 1.54919i −0.0599153 + 0.998203i \(0.519083\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) −2.50000 + 0.866025i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 3.00000 + 5.19615i 0.904534 + 1.56670i 0.821541 + 0.570149i \(0.193114\pi\)
0.0829925 + 0.996550i \(0.473552\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 0.500000 0.866025i 0.114708 0.198680i −0.802955 0.596040i \(-0.796740\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) −5.50000 9.52628i −1.10000 1.90526i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.50000 + 2.59808i 0.269408 + 0.466628i 0.968709 0.248199i \(-0.0798387\pi\)
−0.699301 + 0.714827i \(0.746505\pi\)
\(32\) 0 0
\(33\) 3.00000 5.19615i 0.522233 0.904534i
\(34\) 0 0
\(35\) 2.00000 10.3923i 0.338062 1.75662i
\(36\) 0 0
\(37\) 1.50000 2.59808i 0.246598 0.427121i −0.715981 0.698119i \(-0.754020\pi\)
0.962580 + 0.270998i \(0.0873538\pi\)
\(38\) 0 0
\(39\) 2.50000 + 4.33013i 0.400320 + 0.693375i
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) −2.00000 3.46410i −0.298142 0.516398i
\(46\) 0 0
\(47\) 2.00000 3.46410i 0.291730 0.505291i −0.682489 0.730896i \(-0.739102\pi\)
0.974219 + 0.225605i \(0.0724358\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) −1.00000 + 1.73205i −0.140028 + 0.242536i
\(52\) 0 0
\(53\) −3.00000 5.19615i −0.412082 0.713746i 0.583036 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 0 0
\(55\) −24.0000 −3.23616
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) 0 0
\(63\) 0.500000 2.59808i 0.0629941 0.327327i
\(64\) 0 0
\(65\) 10.0000 17.3205i 1.24035 2.14834i
\(66\) 0 0
\(67\) 3.50000 + 6.06218i 0.427593 + 0.740613i 0.996659 0.0816792i \(-0.0260283\pi\)
−0.569066 + 0.822292i \(0.692695\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 1.50000 + 2.59808i 0.175562 + 0.304082i 0.940356 0.340193i \(-0.110493\pi\)
−0.764794 + 0.644275i \(0.777159\pi\)
\(74\) 0 0
\(75\) −5.50000 + 9.52628i −0.635085 + 1.10000i
\(76\) 0 0
\(77\) −12.0000 10.3923i −1.36753 1.18431i
\(78\) 0 0
\(79\) −5.50000 + 9.52628i −0.618798 + 1.07179i 0.370907 + 0.928670i \(0.379047\pi\)
−0.989705 + 0.143120i \(0.954286\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 + 3.46410i −0.212000 + 0.367194i −0.952340 0.305038i \(-0.901331\pi\)
0.740341 + 0.672232i \(0.234664\pi\)
\(90\) 0 0
\(91\) 12.5000 4.33013i 1.31036 0.453921i
\(92\) 0 0
\(93\) 1.50000 2.59808i 0.155543 0.269408i
\(94\) 0 0
\(95\) 2.00000 + 3.46410i 0.205196 + 0.355409i
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 1.00000 + 1.73205i 0.0995037 + 0.172345i 0.911479 0.411346i \(-0.134941\pi\)
−0.811976 + 0.583691i \(0.801608\pi\)
\(102\) 0 0
\(103\) 5.50000 9.52628i 0.541931 0.938652i −0.456862 0.889538i \(-0.651027\pi\)
0.998793 0.0491146i \(-0.0156400\pi\)
\(104\) 0 0
\(105\) −10.0000 + 3.46410i −0.975900 + 0.338062i
\(106\) 0 0
\(107\) 5.00000 8.66025i 0.483368 0.837218i −0.516449 0.856318i \(-0.672747\pi\)
0.999818 + 0.0190994i \(0.00607989\pi\)
\(108\) 0 0
\(109\) −7.50000 12.9904i −0.718370 1.24425i −0.961645 0.274296i \(-0.911555\pi\)
0.243276 0.969957i \(-0.421778\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 12.0000 + 20.7846i 1.11901 + 1.93817i
\(116\) 0 0
\(117\) 2.50000 4.33013i 0.231125 0.400320i
\(118\) 0 0
\(119\) 4.00000 + 3.46410i 0.366679 + 0.317554i
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 0 0
\(123\) 3.00000 + 5.19615i 0.270501 + 0.468521i
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 2.50000 + 4.33013i 0.220113 + 0.381246i
\(130\) 0 0
\(131\) 3.00000 5.19615i 0.262111 0.453990i −0.704692 0.709514i \(-0.748915\pi\)
0.966803 + 0.255524i \(0.0822479\pi\)
\(132\) 0 0
\(133\) −0.500000 + 2.59808i −0.0433555 + 0.225282i
\(134\) 0 0
\(135\) −2.00000 + 3.46410i −0.172133 + 0.298142i
\(136\) 0 0
\(137\) 6.00000 + 10.3923i 0.512615 + 0.887875i 0.999893 + 0.0146279i \(0.00465636\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) −15.0000 25.9808i −1.25436 2.17262i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.50000 2.59808i −0.536111 0.214286i
\(148\) 0 0
\(149\) −2.00000 + 3.46410i −0.163846 + 0.283790i −0.936245 0.351348i \(-0.885723\pi\)
0.772399 + 0.635138i \(0.219057\pi\)
\(150\) 0 0
\(151\) 4.00000 + 6.92820i 0.325515 + 0.563809i 0.981617 0.190864i \(-0.0611289\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) 5.00000 + 8.66025i 0.399043 + 0.691164i 0.993608 0.112884i \(-0.0360089\pi\)
−0.594565 + 0.804048i \(0.702676\pi\)
\(158\) 0 0
\(159\) −3.00000 + 5.19615i −0.237915 + 0.412082i
\(160\) 0 0
\(161\) −3.00000 + 15.5885i −0.236433 + 1.22854i
\(162\) 0 0
\(163\) 10.0000 17.3205i 0.783260 1.35665i −0.146772 0.989170i \(-0.546888\pi\)
0.930033 0.367477i \(-0.119778\pi\)
\(164\) 0 0
\(165\) 12.0000 + 20.7846i 0.934199 + 1.61808i
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0.500000 + 0.866025i 0.0382360 + 0.0662266i
\(172\) 0 0
\(173\) 11.0000 19.0526i 0.836315 1.44854i −0.0566411 0.998395i \(-0.518039\pi\)
0.892956 0.450145i \(-0.148628\pi\)
\(174\) 0 0
\(175\) 22.0000 + 19.0526i 1.66304 + 1.44024i
\(176\) 0 0
\(177\) −3.00000 + 5.19615i −0.225494 + 0.390567i
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 6.00000 + 10.3923i 0.441129 + 0.764057i
\(186\) 0 0
\(187\) 6.00000 10.3923i 0.438763 0.759961i
\(188\) 0 0
\(189\) −2.50000 + 0.866025i −0.181848 + 0.0629941i
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 0.500000 + 0.866025i 0.0359908 + 0.0623379i 0.883460 0.468507i \(-0.155208\pi\)
−0.847469 + 0.530845i \(0.821875\pi\)
\(194\) 0 0
\(195\) −20.0000 −1.43223
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −10.0000 17.3205i −0.708881 1.22782i −0.965272 0.261245i \(-0.915867\pi\)
0.256391 0.966573i \(-0.417466\pi\)
\(200\) 0 0
\(201\) 3.50000 6.06218i 0.246871 0.427593i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000 20.7846i 0.838116 1.45166i
\(206\) 0 0
\(207\) 3.00000 + 5.19615i 0.208514 + 0.361158i
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) −8.00000 13.8564i −0.548151 0.949425i
\(214\) 0 0
\(215\) 10.0000 17.3205i 0.681994 1.18125i
\(216\) 0 0
\(217\) −6.00000 5.19615i −0.407307 0.352738i
\(218\) 0 0
\(219\) 1.50000 2.59808i 0.101361 0.175562i
\(220\) 0 0
\(221\) 5.00000 + 8.66025i 0.336336 + 0.582552i
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 11.0000 + 19.0526i 0.730096 + 1.26456i 0.956842 + 0.290609i \(0.0938578\pi\)
−0.226746 + 0.973954i \(0.572809\pi\)
\(228\) 0 0
\(229\) −5.50000 + 9.52628i −0.363450 + 0.629514i −0.988526 0.151050i \(-0.951735\pi\)
0.625076 + 0.780564i \(0.285068\pi\)
\(230\) 0 0
\(231\) −3.00000 + 15.5885i −0.197386 + 1.02565i
\(232\) 0 0
\(233\) 4.00000 6.92820i 0.262049 0.453882i −0.704737 0.709468i \(-0.748935\pi\)
0.966786 + 0.255586i \(0.0822686\pi\)
\(234\) 0 0
\(235\) 8.00000 + 13.8564i 0.521862 + 0.903892i
\(236\) 0 0
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) −5.00000 8.66025i −0.322078 0.557856i 0.658838 0.752285i \(-0.271048\pi\)
−0.980917 + 0.194429i \(0.937715\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 4.00000 + 27.7128i 0.255551 + 1.77051i
\(246\) 0 0
\(247\) −2.50000 + 4.33013i −0.159071 + 0.275519i
\(248\) 0 0
\(249\) 6.00000 + 10.3923i 0.380235 + 0.658586i
\(250\) 0 0
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) −4.00000 6.92820i −0.250490 0.433861i
\(256\) 0 0
\(257\) 6.00000 10.3923i 0.374270 0.648254i −0.615948 0.787787i \(-0.711227\pi\)
0.990217 + 0.139533i \(0.0445601\pi\)
\(258\) 0 0
\(259\) −1.50000 + 7.79423i −0.0932055 + 0.484310i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 0 0
\(269\) −1.00000 1.73205i −0.0609711 0.105605i 0.833929 0.551872i \(-0.186086\pi\)
−0.894900 + 0.446267i \(0.852753\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) −10.0000 8.66025i −0.605228 0.524142i
\(274\) 0 0
\(275\) 33.0000 57.1577i 1.98997 3.44674i
\(276\) 0 0
\(277\) 0.500000 + 0.866025i 0.0300421 + 0.0520344i 0.880656 0.473757i \(-0.157103\pi\)
−0.850613 + 0.525792i \(0.823769\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −5.50000 9.52628i −0.326941 0.566279i 0.654962 0.755662i \(-0.272685\pi\)
−0.981903 + 0.189383i \(0.939351\pi\)
\(284\) 0 0
\(285\) 2.00000 3.46410i 0.118470 0.205196i
\(286\) 0 0
\(287\) 15.0000 5.19615i 0.885422 0.306719i
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 3.00000 + 5.19615i 0.175863 + 0.304604i
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 3.00000 + 5.19615i 0.174078 + 0.301511i
\(298\) 0 0
\(299\) −15.0000 + 25.9808i −0.867472 + 1.50251i
\(300\) 0 0
\(301\) 12.5000 4.33013i 0.720488 0.249584i
\(302\) 0 0
\(303\) 1.00000 1.73205i 0.0574485 0.0995037i
\(304\) 0 0
\(305\) −4.00000 6.92820i −0.229039 0.396708i
\(306\) 0 0
\(307\) −11.0000 −0.627803 −0.313902 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) 0 0
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) 1.00000 + 1.73205i 0.0567048 + 0.0982156i 0.892984 0.450088i \(-0.148607\pi\)
−0.836280 + 0.548303i \(0.815274\pi\)
\(312\) 0 0
\(313\) −15.5000 + 26.8468i −0.876112 + 1.51747i −0.0205381 + 0.999789i \(0.506538\pi\)
−0.855574 + 0.517681i \(0.826795\pi\)
\(314\) 0 0
\(315\) 8.00000 + 6.92820i 0.450749 + 0.390360i
\(316\) 0 0
\(317\) −10.0000 + 17.3205i −0.561656 + 0.972817i 0.435696 + 0.900094i \(0.356502\pi\)
−0.997352 + 0.0727229i \(0.976831\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) 27.5000 + 47.6314i 1.52543 + 2.64211i
\(326\) 0 0
\(327\) −7.50000 + 12.9904i −0.414751 + 0.718370i
\(328\) 0 0
\(329\) −2.00000 + 10.3923i −0.110264 + 0.572946i
\(330\) 0 0
\(331\) −2.50000 + 4.33013i −0.137412 + 0.238005i −0.926516 0.376254i \(-0.877212\pi\)
0.789104 + 0.614260i \(0.210545\pi\)
\(332\) 0 0
\(333\) 1.50000 + 2.59808i 0.0821995 + 0.142374i
\(334\) 0 0
\(335\) −28.0000 −1.52980
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 0 0
\(339\) 8.00000 + 13.8564i 0.434500 + 0.752577i
\(340\) 0 0
\(341\) −9.00000 + 15.5885i −0.487377 + 0.844162i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 12.0000 20.7846i 0.646058 1.11901i
\(346\) 0 0
\(347\) −11.0000 19.0526i −0.590511 1.02279i −0.994164 0.107883i \(-0.965593\pi\)
0.403653 0.914912i \(-0.367740\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) 2.00000 + 3.46410i 0.106449 + 0.184376i 0.914329 0.404971i \(-0.132718\pi\)
−0.807880 + 0.589347i \(0.799385\pi\)
\(354\) 0 0
\(355\) −32.0000 + 55.4256i −1.69838 + 2.94169i
\(356\) 0 0
\(357\) 1.00000 5.19615i 0.0529256 0.275010i
\(358\) 0 0
\(359\) 12.0000 20.7846i 0.633336 1.09697i −0.353529 0.935423i \(-0.615019\pi\)
0.986865 0.161546i \(-0.0516481\pi\)
\(360\) 0 0
\(361\) 9.00000 + 15.5885i 0.473684 + 0.820445i
\(362\) 0 0
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −13.5000 23.3827i −0.704694 1.22057i −0.966802 0.255528i \(-0.917751\pi\)
0.262108 0.965039i \(-0.415582\pi\)
\(368\) 0 0
\(369\) 3.00000 5.19615i 0.156174 0.270501i
\(370\) 0 0
\(371\) 12.0000 + 10.3923i 0.623009 + 0.539542i
\(372\) 0 0
\(373\) −14.5000 + 25.1147i −0.750782 + 1.30039i 0.196663 + 0.980471i \(0.436990\pi\)
−0.947444 + 0.319921i \(0.896344\pi\)
\(374\) 0 0
\(375\) −12.0000 20.7846i −0.619677 1.07331i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −27.0000 −1.38690 −0.693448 0.720506i \(-0.743909\pi\)
−0.693448 + 0.720506i \(0.743909\pi\)
\(380\) 0 0
\(381\) 3.50000 + 6.06218i 0.179310 + 0.310575i
\(382\) 0 0
\(383\) −13.0000 + 22.5167i −0.664269 + 1.15055i 0.315214 + 0.949021i \(0.397924\pi\)
−0.979483 + 0.201527i \(0.935410\pi\)
\(384\) 0 0
\(385\) 60.0000 20.7846i 3.05788 1.05928i
\(386\) 0 0
\(387\) 2.50000 4.33013i 0.127082 0.220113i
\(388\) 0 0
\(389\) −2.00000 3.46410i −0.101404 0.175637i 0.810859 0.585241i \(-0.199000\pi\)
−0.912263 + 0.409604i \(0.865667\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) −22.0000 38.1051i −1.10694 1.91728i
\(396\) 0 0
\(397\) −10.5000 + 18.1865i −0.526980 + 0.912756i 0.472526 + 0.881317i \(0.343342\pi\)
−0.999506 + 0.0314391i \(0.989991\pi\)
\(398\) 0 0
\(399\) 2.50000 0.866025i 0.125157 0.0433555i
\(400\) 0 0
\(401\) 1.00000 1.73205i 0.0499376 0.0864945i −0.839976 0.542623i \(-0.817431\pi\)
0.889914 + 0.456129i \(0.150764\pi\)
\(402\) 0 0
\(403\) −7.50000 12.9904i −0.373602 0.647097i
\(404\) 0 0
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) 18.0000 0.892227
\(408\) 0 0
\(409\) −6.50000 11.2583i −0.321404 0.556689i 0.659374 0.751815i \(-0.270822\pi\)
−0.980778 + 0.195127i \(0.937488\pi\)
\(410\) 0 0
\(411\) 6.00000 10.3923i 0.295958 0.512615i
\(412\) 0 0
\(413\) 12.0000 + 10.3923i 0.590481 + 0.511372i
\(414\) 0 0
\(415\) 24.0000 41.5692i 1.17811 2.04055i
\(416\) 0 0
\(417\) 2.50000 + 4.33013i 0.122426 + 0.212047i
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 0 0
\(423\) 2.00000 + 3.46410i 0.0972433 + 0.168430i
\(424\) 0 0
\(425\) −11.0000 + 19.0526i −0.533578 + 0.924185i
\(426\) 0 0
\(427\) 1.00000 5.19615i 0.0483934 0.251459i
\(428\) 0 0
\(429\) −15.0000 + 25.9808i −0.724207 + 1.25436i
\(430\) 0 0
\(431\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.00000 5.19615i −0.143509 0.248566i
\(438\) 0 0
\(439\) −12.0000 + 20.7846i −0.572729 + 0.991995i 0.423556 + 0.905870i \(0.360782\pi\)
−0.996284 + 0.0861252i \(0.972552\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) −13.0000 + 22.5167i −0.617649 + 1.06980i 0.372265 + 0.928126i \(0.378581\pi\)
−0.989914 + 0.141672i \(0.954752\pi\)
\(444\) 0 0
\(445\) −8.00000 13.8564i −0.379236 0.656857i
\(446\) 0 0
\(447\) 4.00000 0.189194
\(448\) 0 0
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −18.0000 31.1769i −0.847587 1.46806i
\(452\) 0 0
\(453\) 4.00000 6.92820i 0.187936 0.325515i
\(454\) 0 0
\(455\) −10.0000 + 51.9615i −0.468807 + 2.43599i
\(456\) 0 0
\(457\) −12.5000 + 21.6506i −0.584725 + 1.01277i 0.410184 + 0.912003i \(0.365464\pi\)
−0.994910 + 0.100771i \(0.967869\pi\)
\(458\) 0 0
\(459\) −1.00000 1.73205i −0.0466760 0.0808452i
\(460\) 0 0
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) −5.00000 −0.232370 −0.116185 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) 0 0
\(465\) 6.00000 + 10.3923i 0.278243 + 0.481932i
\(466\) 0 0
\(467\) 4.00000 6.92820i 0.185098 0.320599i −0.758512 0.651660i \(-0.774073\pi\)
0.943610 + 0.331061i \(0.107406\pi\)
\(468\) 0 0
\(469\) −14.0000 12.1244i −0.646460 0.559851i
\(470\) 0 0
\(471\) 5.00000 8.66025i 0.230388 0.399043i
\(472\) 0 0
\(473\) −15.0000 25.9808i −0.689701 1.19460i
\(474\) 0 0
\(475\) −11.0000 −0.504715
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −1.00000 1.73205i −0.0456912 0.0791394i 0.842275 0.539048i \(-0.181216\pi\)
−0.887967 + 0.459908i \(0.847882\pi\)
\(480\) 0 0
\(481\) −7.50000 + 12.9904i −0.341971 + 0.592310i
\(482\) 0 0
\(483\) 15.0000 5.19615i 0.682524 0.236433i
\(484\) 0 0
\(485\) 12.0000 20.7846i 0.544892 0.943781i
\(486\) 0 0
\(487\) 0.500000 + 0.866025i 0.0226572 + 0.0392434i 0.877132 0.480250i \(-0.159454\pi\)
−0.854475 + 0.519493i \(0.826121\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 12.0000 20.7846i 0.539360 0.934199i
\(496\) 0 0
\(497\) −40.0000 + 13.8564i −1.79425 + 0.621545i
\(498\) 0 0
\(499\) 14.5000 25.1147i 0.649109 1.12429i −0.334227 0.942493i \(-0.608475\pi\)
0.983336 0.181797i \(-0.0581915\pi\)
\(500\) 0 0
\(501\) 4.00000 + 6.92820i 0.178707 + 0.309529i
\(502\) 0 0
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) −6.00000 10.3923i −0.266469 0.461538i
\(508\) 0 0
\(509\) −12.0000 + 20.7846i −0.531891 + 0.921262i 0.467416 + 0.884037i \(0.345185\pi\)
−0.999307 + 0.0372243i \(0.988148\pi\)
\(510\) 0 0
\(511\) −6.00000 5.19615i −0.265424 0.229864i
\(512\) 0 0
\(513\) 0.500000 0.866025i 0.0220755 0.0382360i
\(514\) 0 0
\(515\) 22.0000 + 38.1051i 0.969436 + 1.67911i
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −12.0000 20.7846i −0.525730 0.910590i −0.999551 0.0299693i \(-0.990459\pi\)
0.473821 0.880621i \(-0.342874\pi\)
\(522\) 0 0
\(523\) −8.50000 + 14.7224i −0.371679 + 0.643767i −0.989824 0.142297i \(-0.954551\pi\)
0.618145 + 0.786064i \(0.287884\pi\)
\(524\) 0 0
\(525\) 5.50000 28.5788i 0.240040 1.24728i
\(526\) 0 0
\(527\) 3.00000 5.19615i 0.130682 0.226348i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 0 0
\(535\) 20.0000 + 34.6410i 0.864675 + 1.49766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 39.0000 + 15.5885i 1.67985 + 0.671442i
\(540\) 0 0
\(541\) 18.5000 32.0429i 0.795377 1.37763i −0.127222 0.991874i \(-0.540606\pi\)
0.922599 0.385759i \(-0.126061\pi\)
\(542\) 0 0
\(543\) 12.5000 + 21.6506i 0.536426 + 0.929118i
\(544\) 0 0
\(545\) 60.0000 2.57012
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −1.00000 1.73205i −0.0426790 0.0739221i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.50000 28.5788i 0.233884 1.21530i
\(554\) 0 0
\(555\) 6.00000 10.3923i 0.254686 0.441129i
\(556\) 0 0
\(557\) −3.00000 5.19615i −0.127114 0.220168i 0.795443 0.606028i \(-0.207238\pi\)
−0.922557 + 0.385860i \(0.873905\pi\)
\(558\) 0 0
\(559\) 25.0000 1.05739
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) 12.0000 + 20.7846i 0.505740 + 0.875967i 0.999978 + 0.00664037i \(0.00211371\pi\)
−0.494238 + 0.869326i \(0.664553\pi\)
\(564\) 0 0
\(565\) 32.0000 55.4256i 1.34625 2.33177i
\(566\) 0 0
\(567\) 2.00000 + 1.73205i 0.0839921 + 0.0727393i
\(568\) 0 0
\(569\) −9.00000 + 15.5885i −0.377300 + 0.653502i −0.990668 0.136295i \(-0.956481\pi\)
0.613369 + 0.789797i \(0.289814\pi\)
\(570\) 0 0
\(571\) 14.5000 + 25.1147i 0.606806 + 1.05102i 0.991763 + 0.128085i \(0.0408829\pi\)
−0.384957 + 0.922934i \(0.625784\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −66.0000 −2.75239
\(576\) 0 0
\(577\) −11.5000 19.9186i −0.478751 0.829222i 0.520952 0.853586i \(-0.325577\pi\)
−0.999703 + 0.0243645i \(0.992244\pi\)
\(578\) 0 0
\(579\) 0.500000 0.866025i 0.0207793 0.0359908i
\(580\) 0 0
\(581\) 30.0000 10.3923i 1.24461 0.431145i
\(582\) 0 0
\(583\) 18.0000 31.1769i 0.745484 1.29122i
\(584\) 0 0
\(585\) 10.0000 + 17.3205i 0.413449 + 0.716115i
\(586\) 0 0
\(587\) −40.0000 −1.65098 −0.825488 0.564419i \(-0.809100\pi\)
−0.825488 + 0.564419i \(0.809100\pi\)
\(588\) 0 0
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) 9.00000 + 15.5885i 0.370211 + 0.641223i
\(592\) 0 0
\(593\) 13.0000 22.5167i 0.533846 0.924648i −0.465372 0.885115i \(-0.654080\pi\)
0.999218 0.0395334i \(-0.0125871\pi\)
\(594\) 0 0
\(595\) −20.0000 + 6.92820i −0.819920 + 0.284029i
\(596\) 0 0
\(597\) −10.0000 + 17.3205i −0.409273 + 0.708881i
\(598\) 0 0
\(599\) −10.0000 17.3205i −0.408589 0.707697i 0.586143 0.810208i \(-0.300646\pi\)
−0.994732 + 0.102511i \(0.967312\pi\)
\(600\) 0 0
\(601\) −21.0000 −0.856608 −0.428304 0.903635i \(-0.640889\pi\)
−0.428304 + 0.903635i \(0.640889\pi\)
\(602\) 0 0
\(603\) −7.00000 −0.285062
\(604\) 0 0
\(605\) −50.0000 86.6025i −2.03279 3.52089i
\(606\) 0 0
\(607\) 2.50000 4.33013i 0.101472 0.175754i −0.810819 0.585296i \(-0.800978\pi\)
0.912291 + 0.409542i \(0.134311\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.0000 + 17.3205i −0.404557 + 0.700713i
\(612\) 0 0
\(613\) −23.0000 39.8372i −0.928961 1.60901i −0.785063 0.619416i \(-0.787370\pi\)
−0.143898 0.989593i \(-0.545964\pi\)
\(614\) 0 0
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −12.5000 21.6506i −0.502417 0.870212i −0.999996 0.00279365i \(-0.999111\pi\)
0.497579 0.867419i \(-0.334223\pi\)
\(620\) 0 0
\(621\) 3.00000 5.19615i 0.120386 0.208514i
\(622\) 0 0
\(623\) 2.00000 10.3923i 0.0801283 0.416359i
\(624\) 0 0
\(625\) −20.5000 + 35.5070i −0.820000 + 1.42028i
\(626\) 0 0
\(627\) −3.00000 5.19615i −0.119808 0.207514i
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 0 0
\(633\) 6.00000 + 10.3923i 0.238479 + 0.413057i
\(634\) 0 0
\(635\) 14.0000 24.2487i 0.555573 0.962281i
\(636\) 0 0
\(637\) −27.5000 + 21.6506i −1.08959 + 0.857829i
\(638\) 0 0
\(639\) −8.00000 + 13.8564i −0.316475 + 0.548151i
\(640\) 0 0
\(641\) −19.0000 32.9090i −0.750455 1.29983i −0.947602 0.319452i \(-0.896501\pi\)
0.197148 0.980374i \(-0.436832\pi\)
\(642\) 0 0
\(643\) 23.0000 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) 0 0
\(645\) −20.0000 −0.787499
\(646\) 0 0
\(647\) −5.00000 8.66025i −0.196570 0.340470i 0.750844 0.660480i \(-0.229647\pi\)
−0.947414 + 0.320010i \(0.896314\pi\)
\(648\) 0 0
\(649\) 18.0000 31.1769i 0.706562 1.22380i
\(650\) 0 0
\(651\) −1.50000 + 7.79423i −0.0587896 + 0.305480i
\(652\) 0 0
\(653\) 17.0000 29.4449i 0.665261 1.15227i −0.313953 0.949439i \(-0.601653\pi\)
0.979214 0.202828i \(-0.0650132\pi\)
\(654\) 0 0
\(655\) 12.0000 + 20.7846i 0.468879 + 0.812122i
\(656\) 0 0
\(657\) −3.00000 −0.117041
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 1.50000 + 2.59808i 0.0583432 + 0.101053i 0.893722 0.448622i \(-0.148085\pi\)
−0.835379 + 0.549675i \(0.814752\pi\)
\(662\) 0 0
\(663\) 5.00000 8.66025i 0.194184 0.336336i
\(664\) 0 0
\(665\) −8.00000 6.92820i −0.310227 0.268664i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −8.00000 13.8564i −0.309298 0.535720i
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 0 0
\(675\) −5.50000 9.52628i −0.211695 0.366667i
\(676\) 0 0
\(677\) −6.00000 + 10.3923i −0.230599 + 0.399409i −0.957984 0.286820i \(-0.907402\pi\)
0.727386 + 0.686229i \(0.240735\pi\)
\(678\) 0 0
\(679\) 15.0000 5.19615i 0.575647 0.199410i
\(680\) 0 0
\(681\) 11.0000 19.0526i 0.421521 0.730096i
\(682\) 0 0
\(683\) 9.00000 + 15.5885i 0.344375 + 0.596476i 0.985240 0.171178i \(-0.0547574\pi\)
−0.640865 + 0.767654i \(0.721424\pi\)
\(684\) 0 0
\(685\) −48.0000 −1.83399
\(686\) 0 0
\(687\) 11.0000 0.419676
\(688\) 0 0
\(689\) 15.0000 + 25.9808i 0.571454 + 0.989788i
\(690\) 0 0
\(691\) 11.5000 19.9186i 0.437481 0.757739i −0.560014 0.828483i \(-0.689204\pi\)
0.997494 + 0.0707446i \(0.0225375\pi\)
\(692\) 0 0
\(693\) 15.0000 5.19615i 0.569803 0.197386i
\(694\) 0 0
\(695\) 10.0000 17.3205i 0.379322 0.657004i
\(696\) 0 0
\(697\) 6.00000 + 10.3923i 0.227266 + 0.393637i
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −1.50000 2.59808i −0.0565736 0.0979883i
\(704\) 0 0
\(705\) 8.00000 13.8564i 0.301297 0.521862i
\(706\) 0 0
\(707\) −4.00000 3.46410i −0.150435 0.130281i
\(708\) 0 0
\(709\) −3.00000 + 5.19615i −0.112667 + 0.195146i −0.916845 0.399244i \(-0.869273\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(710\) 0 0
\(711\) −5.50000 9.52628i −0.206266 0.357263i
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) 120.000 4.48775
\(716\) 0 0
\(717\) −1.00000 1.73205i −0.0373457 0.0646846i
\(718\) 0 0
\(719\) −15.0000 + 25.9808i −0.559406 + 0.968919i 0.438141 + 0.898906i \(0.355637\pi\)
−0.997546 + 0.0700124i \(0.977696\pi\)
\(720\) 0 0
\(721\) −5.50000 + 28.5788i −0.204831 + 1.06433i
\(722\) 0 0
\(723\) −5.00000 + 8.66025i −0.185952 + 0.322078i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −41.0000 −1.52061 −0.760303 0.649569i \(-0.774949\pi\)
−0.760303 + 0.649569i \(0.774949\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.00000 + 8.66025i 0.184932 + 0.320311i
\(732\) 0 0
\(733\) 10.5000 18.1865i 0.387826 0.671735i −0.604331 0.796734i \(-0.706559\pi\)
0.992157 + 0.124999i \(0.0398927\pi\)
\(734\) 0 0
\(735\) 22.0000 17.3205i 0.811482 0.638877i
\(736\) 0 0
\(737\) −21.0000 + 36.3731i −0.773545 + 1.33982i
\(738\) 0 0
\(739\) −25.5000 44.1673i −0.938033 1.62472i −0.769135 0.639087i \(-0.779313\pi\)
−0.168898 0.985634i \(-0.554021\pi\)
\(740\) 0 0
\(741\) 5.00000 0.183680
\(742\) 0 0
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 0 0
\(745\) −8.00000 13.8564i −0.293097 0.507659i
\(746\) 0 0
\(747\) 6.00000 10.3923i 0.219529 0.380235i
\(748\) 0 0
\(749\) −5.00000 + 25.9808i −0.182696 + 0.949316i
\(750\) 0 0
\(751\) 3.50000 6.06218i 0.127717 0.221212i −0.795075 0.606511i \(-0.792568\pi\)
0.922792 + 0.385299i \(0.125902\pi\)
\(752\) 0 0
\(753\) −7.00000 12.1244i −0.255094 0.441836i
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) −18.0000 31.1769i −0.653359 1.13165i
\(760\) 0 0
\(761\) −25.0000 + 43.3013i −0.906249 + 1.56967i −0.0870179 + 0.996207i \(0.527734\pi\)
−0.819231 + 0.573463i \(0.805600\pi\)
\(762\) 0 0
\(763\) 30.0000 + 25.9808i 1.08607 + 0.940567i
\(764\) 0 0
\(765\) −4.00000 + 6.92820i −0.144620 + 0.250490i
\(766\) 0 0
\(767\) 15.0000 + 25.9808i 0.541619 + 0.938111i
\(768\) 0 0
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) −11.0000 19.0526i −0.395643 0.685273i 0.597540 0.801839i \(-0.296145\pi\)
−0.993183 + 0.116566i \(0.962811\pi\)
\(774\) 0 0
\(775\) 16.5000 28.5788i 0.592697 1.02658i
\(776\) 0 0
\(777\) 7.50000 2.59808i 0.269061 0.0932055i
\(778\) 0 0
\(779\) −3.00000 + 5.19615i −0.107486 + 0.186171i
\(780\) 0 0
\(781\) 48.0000 + 83.1384i 1.71758 + 2.97493i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −40.0000 −1.42766
\(786\) 0 0
\(787\) 16.0000 + 27.7128i 0.570338 + 0.987855i 0.996531 + 0.0832226i \(0.0265213\pi\)
−0.426193 + 0.904632i \(0.640145\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 40.0000 13.8564i 1.42224 0.492677i
\(792\) 0 0
\(793\) 5.00000 8.66025i 0.177555 0.307535i
\(794\) 0 0
\(795\) −12.0000 20.7846i −0.425596 0.737154i
\(796\) 0 0
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) −2.00000 3.46410i −0.0706665 0.122398i
\(802\) 0 0
\(803\) −9.00000 + 15.5885i −0.317603 + 0.550105i
\(804\) 0 0
\(805\) −48.0000 41.5692i −1.69178 1.46512i
\(806\) 0 0
\(807\) −1.00000 + 1.73205i −0.0352017 + 0.0609711i
\(808\) 0 0
\(809\) −4.00000 6.92820i −0.140633 0.243583i 0.787102 0.616822i \(-0.211580\pi\)
−0.927735 + 0.373240i \(0.878247\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 40.0000 + 69.2820i 1.40114 + 2.42684i
\(816\) 0 0
\(817\) −2.50000 + 4.33013i −0.0874639 + 0.151492i
\(818\) 0 0
\(819\) −2.50000 + 12.9904i −0.0873571 + 0.453921i
\(820\) 0 0
\(821\) 2.00000 3.46410i 0.0698005 0.120898i −0.829013 0.559229i \(-0.811097\pi\)
0.898813 + 0.438331i \(0.144430\pi\)
\(822\) 0 0
\(823\) 12.0000 + 20.7846i 0.418294 + 0.724506i 0.995768 0.0919029i \(-0.0292950\pi\)
−0.577474 + 0.816409i \(0.695962\pi\)
\(824\) 0 0
\(825\) −66.0000 −2.29783
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 2.50000 + 4.33013i 0.0868286 + 0.150392i 0.906169 0.422916i \(-0.138993\pi\)
−0.819340 + 0.573307i \(0.805660\pi\)
\(830\) 0 0
\(831\) 0.500000 0.866025i 0.0173448 0.0300421i
\(832\) 0 0
\(833\) −13.0000 5.19615i −0.450423 0.180036i
\(834\) 0 0
\(835\) 16.0000 27.7128i 0.553703 0.959041i
\(836\) 0 0
\(837\) 1.50000 + 2.59808i 0.0518476 + 0.0898027i
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −3.00000 5.19615i −0.103325 0.178965i
\(844\) 0 0
\(845\) −24.0000 + 41.5692i −0.825625 + 1.43002i
\(846\) 0 0
\(847\) 12.5000 64.9519i 0.429505 2.23177i
\(848\) 0 0
\(849\) −5.50000 + 9.52628i −0.188760 + 0.326941i
\(850\) 0 0
\(851\) −9.00000 15.5885i −0.308516 0.534365i
\(852\) 0 0
\(853\) 33.0000 1.12990 0.564949 0.825126i \(-0.308896\pi\)
0.564949 + 0.825126i \(0.308896\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −12.0000 20.7846i −0.409912 0.709989i 0.584967 0.811057i \(-0.301107\pi\)
−0.994880 + 0.101068i \(0.967774\pi\)
\(858\) 0 0
\(859\) 26.0000 45.0333i 0.887109 1.53652i 0.0438309 0.999039i \(-0.486044\pi\)
0.843278 0.537478i \(-0.180623\pi\)
\(860\) 0 0
\(861\) −12.0000 10.3923i −0.408959 0.354169i
\(862\) 0 0
\(863\) −27.0000 + 46.7654i −0.919091 + 1.59191i −0.118291 + 0.992979i \(0.537742\pi\)
−0.800799 + 0.598933i \(0.795592\pi\)
\(864\) 0 0
\(865\) 44.0000 + 76.2102i 1.49604 + 2.59123i
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −66.0000 −2.23890
\(870\) 0 0
\(871\) −17.5000 30.3109i −0.592965 1.02705i
\(872\) 0 0
\(873\) 3.00000 5.19615i 0.101535 0.175863i
\(874\) 0 0
\(875\) −60.0000 + 20.7846i −2.02837 + 0.702648i
\(876\) 0 0
\(877\) 1.00000 1.73205i 0.0337676 0.0584872i −0.848648 0.528958i \(-0.822583\pi\)
0.882415 + 0.470471i \(0.155916\pi\)
\(878\) 0 0
\(879\) −6.00000 10.3923i −0.202375 0.350524i
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −7.00000 −0.235569 −0.117784 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 0 0
\(885\) −12.0000 20.7846i −0.403376 0.698667i
\(886\) 0 0
\(887\) −17.0000 + 29.4449i −0.570804 + 0.988662i 0.425679 + 0.904874i \(0.360035\pi\)
−0.996484 + 0.0837878i \(0.973298\pi\)
\(888\) 0 0
\(889\) 17.5000 6.06218i 0.586931 0.203319i
\(890\) 0 0
\(891\) 3.00000 5.19615i 0.100504 0.174078i
\(892\) 0 0
\(893\) −2.00000 3.46410i −0.0669274 0.115922i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 30.0000 1.00167
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −6.00000 + 10.3923i −0.199889 + 0.346218i
\(902\) 0 0
\(903\) −10.0000 8.66025i −0.332779 0.288195i
\(904\) 0 0
\(905\) 50.0000 86.6025i 1.66206 2.87877i
\(906\) 0 0
\(907\) 21.5000 + 37.2391i 0.713896 + 1.23650i 0.963384 + 0.268126i \(0.0864043\pi\)
−0.249488 + 0.968378i \(0.580262\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −26.0000 −0.861418 −0.430709 0.902491i \(-0.641737\pi\)
−0.430709 + 0.902491i \(0.641737\pi\)
\(912\) 0 0
\(913\) −36.0000 62.3538i −1.19143 2.06361i
\(914\) 0 0
\(915\) −4.00000 + 6.92820i −0.132236 + 0.229039i
\(916\) 0 0
\(917\) −3.00000 + 15.5885i −0.0990687 + 0.514776i
\(918\) 0 0
\(919\) −21.5000 + 37.2391i −0.709220 + 1.22840i 0.255927 + 0.966696i \(0.417619\pi\)
−0.965147 + 0.261708i \(0.915714\pi\)
\(920\) 0 0
\(921\) 5.50000 + 9.52628i 0.181231 + 0.313902i
\(922\) 0 0
\(923\) −80.0000 −2.63323
\(924\) 0 0
\(925\) −33.0000 −1.08503
\(926\) 0 0
\(927\) 5.50000 + 9.52628i 0.180644 + 0.312884i
\(928\) 0 0
\(929\) 18.0000 31.1769i 0.590561 1.02288i −0.403596 0.914937i \(-0.632240\pi\)
0.994157 0.107944i \(-0.0344268\pi\)
\(930\) 0 0
\(931\) −1.00000 6.92820i −0.0327737 0.227063i
\(932\) 0 0
\(933\) 1.00000 1.73205i 0.0327385 0.0567048i
\(934\) 0 0
\(935\) 24.0000 + 41.5692i 0.784884 + 1.35946i
\(936\) 0 0
\(937\) 51.0000 1.66610 0.833049 0.553200i \(-0.186593\pi\)
0.833049 + 0.553200i \(0.186593\pi\)
\(938\) 0 0
\(939\) 31.0000 1.01165
\(940\) 0 0
\(941\) 27.0000 + 46.7654i 0.880175 + 1.52451i 0.851146 + 0.524929i \(0.175908\pi\)
0.0290288 + 0.999579i \(0.490759\pi\)
\(942\) 0 0
\(943\) −18.0000 + 31.1769i −0.586161 + 1.01526i
\(944\) 0 0
\(945\) 2.00000 10.3923i 0.0650600 0.338062i
\(946\) 0 0
\(947\) −15.0000 + 25.9808i −0.487435 + 0.844261i −0.999896 0.0144491i \(-0.995401\pi\)
0.512461 + 0.858710i \(0.328734\pi\)
\(948\) 0 0
\(949\) −7.50000 12.9904i −0.243460 0.421686i
\(950\) 0 0
\(951\) 20.0000 0.648544
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −24.0000 20.7846i −0.775000 0.671170i
\(960\) 0 0
\(961\) 11.0000 19.0526i 0.354839 0.614599i
\(962\) 0 0
\(963\) 5.00000 + 8.66025i 0.161123 + 0.279073i
\(964\) 0 0
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) −53.0000 −1.70437 −0.852183 0.523245i \(-0.824721\pi\)
−0.852183 + 0.523245i \(0.824721\pi\)
\(968\) 0 0
\(969\) 1.00000 + 1.73205i 0.0321246 + 0.0556415i
\(970\) 0 0
\(971\) −13.0000 + 22.5167i −0.417190 + 0.722594i −0.995656 0.0931127i \(-0.970318\pi\)
0.578466 + 0.815707i \(0.303652\pi\)
\(972\) 0 0
\(973\) 12.5000 4.33013i 0.400732 0.138817i
\(974\) 0 0
\(975\) 27.5000 47.6314i 0.880705 1.52543i
\(976\) 0 0
\(977\) −15.0000 25.9808i −0.479893 0.831198i 0.519841 0.854263i \(-0.325991\pi\)
−0.999734 + 0.0230645i \(0.992658\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 15.0000 0.478913
\(982\) 0 0
\(983\) −13.0000 22.5167i −0.414636 0.718170i 0.580755 0.814079i \(-0.302758\pi\)
−0.995390 + 0.0959088i \(0.969424\pi\)
\(984\) 0 0
\(985\) 36.0000 62.3538i 1.14706 1.98676i
\(986\) 0 0
\(987\) 10.0000 3.46410i 0.318304 0.110264i
\(988\) 0 0
\(989\) −15.0000 + 25.9808i −0.476972 + 0.826140i
\(990\) 0 0
\(991\) −12.5000 21.6506i −0.397076 0.687755i 0.596288 0.802771i \(-0.296642\pi\)
−0.993364 + 0.115015i \(0.963308\pi\)
\(992\) 0 0
\(993\) 5.00000 0.158670
\(994\) 0 0
\(995\) 80.0000 2.53617
\(996\) 0 0
\(997\) −2.50000 4.33013i −0.0791758 0.137136i 0.823719 0.566999i \(-0.191896\pi\)
−0.902895 + 0.429862i \(0.858562\pi\)
\(998\) 0 0
\(999\) 1.50000 2.59808i 0.0474579 0.0821995i
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 1344.2.q.a.961.1 2
4.3 odd 2 1344.2.q.l.961.1 2
7.2 even 3 9408.2.a.dd.1.1 1
7.4 even 3 inner 1344.2.q.a.193.1 2
7.5 odd 6 9408.2.a.a.1.1 1
8.3 odd 2 672.2.q.e.289.1 yes 2
8.5 even 2 672.2.q.j.289.1 yes 2
24.5 odd 2 2016.2.s.a.289.1 2
24.11 even 2 2016.2.s.b.289.1 2
28.11 odd 6 1344.2.q.l.193.1 2
28.19 even 6 9408.2.a.bs.1.1 1
28.23 odd 6 9408.2.a.bp.1.1 1
56.5 odd 6 4704.2.a.bh.1.1 1
56.11 odd 6 672.2.q.e.193.1 2
56.19 even 6 4704.2.a.p.1.1 1
56.37 even 6 4704.2.a.a.1.1 1
56.51 odd 6 4704.2.a.r.1.1 1
56.53 even 6 672.2.q.j.193.1 yes 2
168.11 even 6 2016.2.s.b.865.1 2
168.53 odd 6 2016.2.s.a.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.e.193.1 2 56.11 odd 6
672.2.q.e.289.1 yes 2 8.3 odd 2
672.2.q.j.193.1 yes 2 56.53 even 6
672.2.q.j.289.1 yes 2 8.5 even 2
1344.2.q.a.193.1 2 7.4 even 3 inner
1344.2.q.a.961.1 2 1.1 even 1 trivial
1344.2.q.l.193.1 2 28.11 odd 6
1344.2.q.l.961.1 2 4.3 odd 2
2016.2.s.a.289.1 2 24.5 odd 2
2016.2.s.a.865.1 2 168.53 odd 6
2016.2.s.b.289.1 2 24.11 even 2
2016.2.s.b.865.1 2 168.11 even 6
4704.2.a.a.1.1 1 56.37 even 6
4704.2.a.p.1.1 1 56.19 even 6
4704.2.a.r.1.1 1 56.51 odd 6
4704.2.a.bh.1.1 1 56.5 odd 6
9408.2.a.a.1.1 1 7.5 odd 6
9408.2.a.bp.1.1 1 28.23 odd 6
9408.2.a.bs.1.1 1 28.19 even 6
9408.2.a.dd.1.1 1 7.2 even 3