Properties

Label 1232.2.q.g.529.1
Level $1232$
Weight $2$
Character 1232.529
Analytic conductor $9.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,2,Mod(177,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(-1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 1232.529
Dual form 1232.2.q.g.177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.32288 + 2.29129i) q^{3} +(-1.82288 - 3.15731i) q^{5} +(-1.32288 - 2.29129i) q^{7} +(-2.00000 - 3.46410i) q^{9} +O(q^{10})\) \(q+(-1.32288 + 2.29129i) q^{3} +(-1.82288 - 3.15731i) q^{5} +(-1.32288 - 2.29129i) q^{7} +(-2.00000 - 3.46410i) q^{9} +(0.500000 - 0.866025i) q^{11} +5.00000 q^{13} +9.64575 q^{15} +(-3.00000 + 5.19615i) q^{17} +(-0.177124 - 0.306788i) q^{19} +7.00000 q^{21} +(-1.82288 - 3.15731i) q^{23} +(-4.14575 + 7.18065i) q^{25} +2.64575 q^{27} -4.29150 q^{29} +(-2.00000 + 3.46410i) q^{31} +(1.32288 + 2.29129i) q^{33} +(-4.82288 + 8.35347i) q^{35} +(0.822876 + 1.42526i) q^{37} +(-6.61438 + 11.4564i) q^{39} -4.93725 q^{41} +4.00000 q^{43} +(-7.29150 + 12.6293i) q^{45} +(6.64575 + 11.5108i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(-7.93725 - 13.7477i) q^{51} +(1.82288 - 3.15731i) q^{53} -3.64575 q^{55} +0.937254 q^{57} +(-0.322876 + 0.559237i) q^{59} +(-1.85425 - 3.21165i) q^{61} +(-5.29150 + 9.16515i) q^{63} +(-9.11438 - 15.7866i) q^{65} +(1.96863 - 3.40976i) q^{67} +9.64575 q^{69} -9.64575 q^{71} +(-2.82288 + 4.88936i) q^{73} +(-10.9686 - 18.9982i) q^{75} -2.64575 q^{77} +(1.32288 + 2.29129i) q^{79} +(2.50000 - 4.33013i) q^{81} -13.2915 q^{83} +21.8745 q^{85} +(5.67712 - 9.83307i) q^{87} +(7.29150 + 12.6293i) q^{89} +(-6.61438 - 11.4564i) q^{91} +(-5.29150 - 9.16515i) q^{93} +(-0.645751 + 1.11847i) q^{95} -5.70850 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 8 q^{9} + 2 q^{11} + 20 q^{13} + 28 q^{15} - 12 q^{17} - 6 q^{19} + 28 q^{21} - 2 q^{23} - 6 q^{25} + 4 q^{29} - 8 q^{31} - 14 q^{35} - 2 q^{37} + 12 q^{41} + 16 q^{43} - 8 q^{45} + 16 q^{47} - 14 q^{49} + 2 q^{53} - 4 q^{55} - 28 q^{57} + 4 q^{59} - 18 q^{61} - 10 q^{65} - 8 q^{67} + 28 q^{69} - 28 q^{71} - 6 q^{73} - 28 q^{75} + 10 q^{81} - 32 q^{83} + 24 q^{85} + 28 q^{87} + 8 q^{89} + 8 q^{95} - 44 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.32288 + 2.29129i −0.763763 + 1.32288i 0.177136 + 0.984186i \(0.443317\pi\)
−0.940898 + 0.338689i \(0.890016\pi\)
\(4\) 0 0
\(5\) −1.82288 3.15731i −0.815215 1.41199i −0.909174 0.416417i \(-0.863286\pi\)
0.0939588 0.995576i \(-0.470048\pi\)
\(6\) 0 0
\(7\) −1.32288 2.29129i −0.500000 0.866025i
\(8\) 0 0
\(9\) −2.00000 3.46410i −0.666667 1.15470i
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 9.64575 2.49052
\(16\) 0 0
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) −0.177124 0.306788i −0.0406351 0.0703821i 0.844993 0.534778i \(-0.179605\pi\)
−0.885628 + 0.464396i \(0.846271\pi\)
\(20\) 0 0
\(21\) 7.00000 1.52753
\(22\) 0 0
\(23\) −1.82288 3.15731i −0.380096 0.658345i 0.610980 0.791646i \(-0.290776\pi\)
−0.991076 + 0.133301i \(0.957442\pi\)
\(24\) 0 0
\(25\) −4.14575 + 7.18065i −0.829150 + 1.43613i
\(26\) 0 0
\(27\) 2.64575 0.509175
\(28\) 0 0
\(29\) −4.29150 −0.796912 −0.398456 0.917187i \(-0.630454\pi\)
−0.398456 + 0.917187i \(0.630454\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 1.32288 + 2.29129i 0.230283 + 0.398862i
\(34\) 0 0
\(35\) −4.82288 + 8.35347i −0.815215 + 1.41199i
\(36\) 0 0
\(37\) 0.822876 + 1.42526i 0.135280 + 0.234312i 0.925704 0.378248i \(-0.123473\pi\)
−0.790424 + 0.612560i \(0.790140\pi\)
\(38\) 0 0
\(39\) −6.61438 + 11.4564i −1.05915 + 1.83450i
\(40\) 0 0
\(41\) −4.93725 −0.771070 −0.385535 0.922693i \(-0.625983\pi\)
−0.385535 + 0.922693i \(0.625983\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −7.29150 + 12.6293i −1.08695 + 1.88266i
\(46\) 0 0
\(47\) 6.64575 + 11.5108i 0.969382 + 1.67902i 0.697349 + 0.716732i \(0.254363\pi\)
0.272034 + 0.962288i \(0.412304\pi\)
\(48\) 0 0
\(49\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) −7.93725 13.7477i −1.11144 1.92507i
\(52\) 0 0
\(53\) 1.82288 3.15731i 0.250391 0.433690i −0.713242 0.700918i \(-0.752774\pi\)
0.963634 + 0.267227i \(0.0861074\pi\)
\(54\) 0 0
\(55\) −3.64575 −0.491593
\(56\) 0 0
\(57\) 0.937254 0.124142
\(58\) 0 0
\(59\) −0.322876 + 0.559237i −0.0420348 + 0.0728065i −0.886277 0.463155i \(-0.846717\pi\)
0.844243 + 0.535961i \(0.180051\pi\)
\(60\) 0 0
\(61\) −1.85425 3.21165i −0.237412 0.411210i 0.722559 0.691310i \(-0.242966\pi\)
−0.959971 + 0.280099i \(0.909633\pi\)
\(62\) 0 0
\(63\) −5.29150 + 9.16515i −0.666667 + 1.15470i
\(64\) 0 0
\(65\) −9.11438 15.7866i −1.13050 1.95808i
\(66\) 0 0
\(67\) 1.96863 3.40976i 0.240506 0.416569i −0.720352 0.693608i \(-0.756020\pi\)
0.960859 + 0.277039i \(0.0893533\pi\)
\(68\) 0 0
\(69\) 9.64575 1.16121
\(70\) 0 0
\(71\) −9.64575 −1.14474 −0.572370 0.819995i \(-0.693976\pi\)
−0.572370 + 0.819995i \(0.693976\pi\)
\(72\) 0 0
\(73\) −2.82288 + 4.88936i −0.330393 + 0.572257i −0.982589 0.185793i \(-0.940514\pi\)
0.652196 + 0.758050i \(0.273848\pi\)
\(74\) 0 0
\(75\) −10.9686 18.9982i −1.26655 2.19373i
\(76\) 0 0
\(77\) −2.64575 −0.301511
\(78\) 0 0
\(79\) 1.32288 + 2.29129i 0.148835 + 0.257790i 0.930797 0.365536i \(-0.119114\pi\)
−0.781962 + 0.623326i \(0.785781\pi\)
\(80\) 0 0
\(81\) 2.50000 4.33013i 0.277778 0.481125i
\(82\) 0 0
\(83\) −13.2915 −1.45893 −0.729466 0.684017i \(-0.760231\pi\)
−0.729466 + 0.684017i \(0.760231\pi\)
\(84\) 0 0
\(85\) 21.8745 2.37262
\(86\) 0 0
\(87\) 5.67712 9.83307i 0.608652 1.05422i
\(88\) 0 0
\(89\) 7.29150 + 12.6293i 0.772898 + 1.33870i 0.935968 + 0.352084i \(0.114527\pi\)
−0.163071 + 0.986614i \(0.552140\pi\)
\(90\) 0 0
\(91\) −6.61438 11.4564i −0.693375 1.20096i
\(92\) 0 0
\(93\) −5.29150 9.16515i −0.548703 0.950382i
\(94\) 0 0
\(95\) −0.645751 + 1.11847i −0.0662527 + 0.114753i
\(96\) 0 0
\(97\) −5.70850 −0.579610 −0.289805 0.957086i \(-0.593590\pi\)
−0.289805 + 0.957086i \(0.593590\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −1.50000 + 2.59808i −0.149256 + 0.258518i −0.930953 0.365140i \(-0.881021\pi\)
0.781697 + 0.623658i \(0.214354\pi\)
\(102\) 0 0
\(103\) 6.46863 + 11.2040i 0.637373 + 1.10396i 0.986007 + 0.166703i \(0.0533122\pi\)
−0.348634 + 0.937259i \(0.613354\pi\)
\(104\) 0 0
\(105\) −12.7601 22.1012i −1.24526 2.15686i
\(106\) 0 0
\(107\) 2.46863 + 4.27579i 0.238651 + 0.413356i 0.960327 0.278875i \(-0.0899614\pi\)
−0.721676 + 0.692231i \(0.756628\pi\)
\(108\) 0 0
\(109\) −5.29150 + 9.16515i −0.506834 + 0.877862i 0.493135 + 0.869953i \(0.335851\pi\)
−0.999969 + 0.00790932i \(0.997482\pi\)
\(110\) 0 0
\(111\) −4.35425 −0.413287
\(112\) 0 0
\(113\) −7.70850 −0.725154 −0.362577 0.931954i \(-0.618103\pi\)
−0.362577 + 0.931954i \(0.618103\pi\)
\(114\) 0 0
\(115\) −6.64575 + 11.5108i −0.619720 + 1.07339i
\(116\) 0 0
\(117\) −10.0000 17.3205i −0.924500 1.60128i
\(118\) 0 0
\(119\) 15.8745 1.45521
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.0454545 0.0787296i
\(122\) 0 0
\(123\) 6.53137 11.3127i 0.588914 1.02003i
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −0.0627461 −0.00556781 −0.00278391 0.999996i \(-0.500886\pi\)
−0.00278391 + 0.999996i \(0.500886\pi\)
\(128\) 0 0
\(129\) −5.29150 + 9.16515i −0.465891 + 0.806947i
\(130\) 0 0
\(131\) 7.82288 + 13.5496i 0.683488 + 1.18384i 0.973909 + 0.226937i \(0.0728711\pi\)
−0.290422 + 0.956899i \(0.593796\pi\)
\(132\) 0 0
\(133\) −0.468627 + 0.811686i −0.0406351 + 0.0703821i
\(134\) 0 0
\(135\) −4.82288 8.35347i −0.415087 0.718952i
\(136\) 0 0
\(137\) −9.43725 + 16.3458i −0.806279 + 1.39652i 0.109145 + 0.994026i \(0.465189\pi\)
−0.915424 + 0.402490i \(0.868145\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −35.1660 −2.96151
\(142\) 0 0
\(143\) 2.50000 4.33013i 0.209061 0.362103i
\(144\) 0 0
\(145\) 7.82288 + 13.5496i 0.649654 + 1.12523i
\(146\) 0 0
\(147\) −9.26013 16.0390i −0.763763 1.32288i
\(148\) 0 0
\(149\) 2.35425 + 4.07768i 0.192868 + 0.334056i 0.946199 0.323584i \(-0.104888\pi\)
−0.753332 + 0.657641i \(0.771555\pi\)
\(150\) 0 0
\(151\) −1.67712 + 2.90486i −0.136482 + 0.236395i −0.926163 0.377124i \(-0.876913\pi\)
0.789680 + 0.613519i \(0.210246\pi\)
\(152\) 0 0
\(153\) 24.0000 1.94029
\(154\) 0 0
\(155\) 14.5830 1.17134
\(156\) 0 0
\(157\) 10.5830 18.3303i 0.844616 1.46292i −0.0413387 0.999145i \(-0.513162\pi\)
0.885954 0.463772i \(-0.153504\pi\)
\(158\) 0 0
\(159\) 4.82288 + 8.35347i 0.382479 + 0.662473i
\(160\) 0 0
\(161\) −4.82288 + 8.35347i −0.380096 + 0.658345i
\(162\) 0 0
\(163\) −2.32288 4.02334i −0.181942 0.315132i 0.760600 0.649221i \(-0.224905\pi\)
−0.942542 + 0.334089i \(0.891572\pi\)
\(164\) 0 0
\(165\) 4.82288 8.35347i 0.375460 0.650316i
\(166\) 0 0
\(167\) −15.2288 −1.17844 −0.589218 0.807974i \(-0.700564\pi\)
−0.589218 + 0.807974i \(0.700564\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −0.708497 + 1.22715i −0.0541801 + 0.0938428i
\(172\) 0 0
\(173\) −5.14575 8.91270i −0.391224 0.677620i 0.601387 0.798958i \(-0.294615\pi\)
−0.992611 + 0.121338i \(0.961282\pi\)
\(174\) 0 0
\(175\) 21.9373 1.65830
\(176\) 0 0
\(177\) −0.854249 1.47960i −0.0642093 0.111214i
\(178\) 0 0
\(179\) −2.03137 + 3.51844i −0.151832 + 0.262981i −0.931901 0.362713i \(-0.881851\pi\)
0.780069 + 0.625693i \(0.215184\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 9.81176 0.725306
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) 3.00000 + 5.19615i 0.219382 + 0.379980i
\(188\) 0 0
\(189\) −3.50000 6.06218i −0.254588 0.440959i
\(190\) 0 0
\(191\) −6.64575 11.5108i −0.480870 0.832891i 0.518889 0.854841i \(-0.326346\pi\)
−0.999759 + 0.0219507i \(0.993012\pi\)
\(192\) 0 0
\(193\) 5.76013 9.97684i 0.414623 0.718148i −0.580766 0.814071i \(-0.697247\pi\)
0.995389 + 0.0959224i \(0.0305801\pi\)
\(194\) 0 0
\(195\) 48.2288 3.45373
\(196\) 0 0
\(197\) 18.8745 1.34475 0.672377 0.740209i \(-0.265274\pi\)
0.672377 + 0.740209i \(0.265274\pi\)
\(198\) 0 0
\(199\) −11.1144 + 19.2507i −0.787877 + 1.36464i 0.139388 + 0.990238i \(0.455487\pi\)
−0.927265 + 0.374406i \(0.877847\pi\)
\(200\) 0 0
\(201\) 5.20850 + 9.02138i 0.367379 + 0.636319i
\(202\) 0 0
\(203\) 5.67712 + 9.83307i 0.398456 + 0.690146i
\(204\) 0 0
\(205\) 9.00000 + 15.5885i 0.628587 + 1.08875i
\(206\) 0 0
\(207\) −7.29150 + 12.6293i −0.506794 + 0.877794i
\(208\) 0 0
\(209\) −0.354249 −0.0245039
\(210\) 0 0
\(211\) 14.9373 1.02832 0.514161 0.857693i \(-0.328103\pi\)
0.514161 + 0.857693i \(0.328103\pi\)
\(212\) 0 0
\(213\) 12.7601 22.1012i 0.874310 1.51435i
\(214\) 0 0
\(215\) −7.29150 12.6293i −0.497276 0.861308i
\(216\) 0 0
\(217\) 10.5830 0.718421
\(218\) 0 0
\(219\) −7.46863 12.9360i −0.504683 0.874137i
\(220\) 0 0
\(221\) −15.0000 + 25.9808i −1.00901 + 1.74766i
\(222\) 0 0
\(223\) 12.3542 0.827302 0.413651 0.910436i \(-0.364253\pi\)
0.413651 + 0.910436i \(0.364253\pi\)
\(224\) 0 0
\(225\) 33.1660 2.21107
\(226\) 0 0
\(227\) −6.64575 + 11.5108i −0.441094 + 0.763997i −0.997771 0.0667318i \(-0.978743\pi\)
0.556677 + 0.830729i \(0.312076\pi\)
\(228\) 0 0
\(229\) 8.00000 + 13.8564i 0.528655 + 0.915657i 0.999442 + 0.0334101i \(0.0106368\pi\)
−0.470787 + 0.882247i \(0.656030\pi\)
\(230\) 0 0
\(231\) 3.50000 6.06218i 0.230283 0.398862i
\(232\) 0 0
\(233\) −8.46863 14.6681i −0.554798 0.960939i −0.997919 0.0644769i \(-0.979462\pi\)
0.443121 0.896462i \(-0.353871\pi\)
\(234\) 0 0
\(235\) 24.2288 41.9654i 1.58051 2.73752i
\(236\) 0 0
\(237\) −7.00000 −0.454699
\(238\) 0 0
\(239\) −9.22876 −0.596959 −0.298479 0.954416i \(-0.596479\pi\)
−0.298479 + 0.954416i \(0.596479\pi\)
\(240\) 0 0
\(241\) −11.4059 + 19.7556i −0.734717 + 1.27257i 0.220130 + 0.975471i \(0.429352\pi\)
−0.954847 + 0.297097i \(0.903981\pi\)
\(242\) 0 0
\(243\) 10.5830 + 18.3303i 0.678900 + 1.17589i
\(244\) 0 0
\(245\) 25.5203 1.63043
\(246\) 0 0
\(247\) −0.885622 1.53394i −0.0563508 0.0976024i
\(248\) 0 0
\(249\) 17.5830 30.4547i 1.11428 1.92999i
\(250\) 0 0
\(251\) −7.29150 −0.460236 −0.230118 0.973163i \(-0.573911\pi\)
−0.230118 + 0.973163i \(0.573911\pi\)
\(252\) 0 0
\(253\) −3.64575 −0.229206
\(254\) 0 0
\(255\) −28.9373 + 50.1208i −1.81212 + 3.13869i
\(256\) 0 0
\(257\) 0.208497 + 0.361128i 0.0130057 + 0.0225265i 0.872455 0.488694i \(-0.162527\pi\)
−0.859449 + 0.511221i \(0.829193\pi\)
\(258\) 0 0
\(259\) 2.17712 3.77089i 0.135280 0.234312i
\(260\) 0 0
\(261\) 8.58301 + 14.8662i 0.531275 + 0.920195i
\(262\) 0 0
\(263\) 2.03137 3.51844i 0.125260 0.216956i −0.796575 0.604540i \(-0.793357\pi\)
0.921834 + 0.387584i \(0.126690\pi\)
\(264\) 0 0
\(265\) −13.2915 −0.816491
\(266\) 0 0
\(267\) −38.5830 −2.36124
\(268\) 0 0
\(269\) −13.2915 + 23.0216i −0.810397 + 1.40365i 0.102189 + 0.994765i \(0.467415\pi\)
−0.912586 + 0.408884i \(0.865918\pi\)
\(270\) 0 0
\(271\) −8.96863 15.5341i −0.544805 0.943630i −0.998619 0.0525339i \(-0.983270\pi\)
0.453814 0.891097i \(-0.350063\pi\)
\(272\) 0 0
\(273\) 35.0000 2.11830
\(274\) 0 0
\(275\) 4.14575 + 7.18065i 0.249998 + 0.433010i
\(276\) 0 0
\(277\) 5.85425 10.1399i 0.351748 0.609245i −0.634808 0.772670i \(-0.718921\pi\)
0.986556 + 0.163425i \(0.0522542\pi\)
\(278\) 0 0
\(279\) 16.0000 0.957895
\(280\) 0 0
\(281\) −7.06275 −0.421328 −0.210664 0.977559i \(-0.567563\pi\)
−0.210664 + 0.977559i \(0.567563\pi\)
\(282\) 0 0
\(283\) −3.82288 + 6.62141i −0.227246 + 0.393602i −0.956991 0.290118i \(-0.906306\pi\)
0.729745 + 0.683720i \(0.239639\pi\)
\(284\) 0 0
\(285\) −1.70850 2.95920i −0.101203 0.175288i
\(286\) 0 0
\(287\) 6.53137 + 11.3127i 0.385535 + 0.667766i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 7.55163 13.0798i 0.442685 0.766752i
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 2.35425 0.137070
\(296\) 0 0
\(297\) 1.32288 2.29129i 0.0767610 0.132954i
\(298\) 0 0
\(299\) −9.11438 15.7866i −0.527098 0.912961i
\(300\) 0 0
\(301\) −5.29150 9.16515i −0.304997 0.528271i
\(302\) 0 0
\(303\) −3.96863 6.87386i −0.227992 0.394893i
\(304\) 0 0
\(305\) −6.76013 + 11.7089i −0.387084 + 0.670449i
\(306\) 0 0
\(307\) 4.22876 0.241348 0.120674 0.992692i \(-0.461494\pi\)
0.120674 + 0.992692i \(0.461494\pi\)
\(308\) 0 0
\(309\) −34.2288 −1.94721
\(310\) 0 0
\(311\) 8.46863 14.6681i 0.480212 0.831751i −0.519531 0.854452i \(-0.673893\pi\)
0.999742 + 0.0227007i \(0.00722647\pi\)
\(312\) 0 0
\(313\) −1.20850 2.09318i −0.0683083 0.118313i 0.829848 0.557989i \(-0.188427\pi\)
−0.898157 + 0.439675i \(0.855093\pi\)
\(314\) 0 0
\(315\) 38.5830 2.17391
\(316\) 0 0
\(317\) −6.00000 10.3923i −0.336994 0.583690i 0.646872 0.762598i \(-0.276077\pi\)
−0.983866 + 0.178908i \(0.942743\pi\)
\(318\) 0 0
\(319\) −2.14575 + 3.71655i −0.120139 + 0.208087i
\(320\) 0 0
\(321\) −13.0627 −0.729091
\(322\) 0 0
\(323\) 2.12549 0.118266
\(324\) 0 0
\(325\) −20.7288 + 35.9033i −1.14982 + 1.99155i
\(326\) 0 0
\(327\) −14.0000 24.2487i −0.774202 1.34096i
\(328\) 0 0
\(329\) 17.5830 30.4547i 0.969382 1.67902i
\(330\) 0 0
\(331\) −7.67712 13.2972i −0.421973 0.730879i 0.574159 0.818743i \(-0.305329\pi\)
−0.996132 + 0.0878650i \(0.971996\pi\)
\(332\) 0 0
\(333\) 3.29150 5.70105i 0.180373 0.312416i
\(334\) 0 0
\(335\) −14.3542 −0.784256
\(336\) 0 0
\(337\) 24.9373 1.35842 0.679209 0.733945i \(-0.262323\pi\)
0.679209 + 0.733945i \(0.262323\pi\)
\(338\) 0 0
\(339\) 10.1974 17.6624i 0.553846 0.959289i
\(340\) 0 0
\(341\) 2.00000 + 3.46410i 0.108306 + 0.187592i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) −17.5830 30.4547i −0.946637 1.63962i
\(346\) 0 0
\(347\) 13.4059 23.2197i 0.719665 1.24650i −0.241467 0.970409i \(-0.577629\pi\)
0.961132 0.276088i \(-0.0890381\pi\)
\(348\) 0 0
\(349\) 29.8745 1.59915 0.799573 0.600569i \(-0.205059\pi\)
0.799573 + 0.600569i \(0.205059\pi\)
\(350\) 0 0
\(351\) 13.2288 0.706099
\(352\) 0 0
\(353\) 17.5830 30.4547i 0.935849 1.62094i 0.162736 0.986670i \(-0.447968\pi\)
0.773113 0.634268i \(-0.218699\pi\)
\(354\) 0 0
\(355\) 17.5830 + 30.4547i 0.933209 + 1.61637i
\(356\) 0 0
\(357\) −21.0000 + 36.3731i −1.11144 + 1.92507i
\(358\) 0 0
\(359\) −5.03137 8.71459i −0.265546 0.459939i 0.702161 0.712018i \(-0.252219\pi\)
−0.967706 + 0.252080i \(0.918885\pi\)
\(360\) 0 0
\(361\) 9.43725 16.3458i 0.496698 0.860305i
\(362\) 0 0
\(363\) 2.64575 0.138866
\(364\) 0 0
\(365\) 20.5830 1.07736
\(366\) 0 0
\(367\) −4.88562 + 8.46215i −0.255027 + 0.441720i −0.964903 0.262607i \(-0.915418\pi\)
0.709876 + 0.704327i \(0.248751\pi\)
\(368\) 0 0
\(369\) 9.87451 + 17.1031i 0.514046 + 0.890354i
\(370\) 0 0
\(371\) −9.64575 −0.500782
\(372\) 0 0
\(373\) 5.43725 + 9.41760i 0.281530 + 0.487625i 0.971762 0.235964i \(-0.0758247\pi\)
−0.690232 + 0.723589i \(0.742491\pi\)
\(374\) 0 0
\(375\) −15.8745 + 27.4955i −0.819756 + 1.41986i
\(376\) 0 0
\(377\) −21.4575 −1.10512
\(378\) 0 0
\(379\) −21.9373 −1.12684 −0.563421 0.826170i \(-0.690515\pi\)
−0.563421 + 0.826170i \(0.690515\pi\)
\(380\) 0 0
\(381\) 0.0830052 0.143769i 0.00425249 0.00736552i
\(382\) 0 0
\(383\) −17.0516 29.5343i −0.871298 1.50913i −0.860655 0.509189i \(-0.829946\pi\)
−0.0106427 0.999943i \(-0.503388\pi\)
\(384\) 0 0
\(385\) 4.82288 + 8.35347i 0.245797 + 0.425732i
\(386\) 0 0
\(387\) −8.00000 13.8564i −0.406663 0.704361i
\(388\) 0 0
\(389\) 10.4059 18.0235i 0.527599 0.913828i −0.471883 0.881661i \(-0.656426\pi\)
0.999482 0.0321675i \(-0.0102410\pi\)
\(390\) 0 0
\(391\) 21.8745 1.10624
\(392\) 0 0
\(393\) −41.3948 −2.08809
\(394\) 0 0
\(395\) 4.82288 8.35347i 0.242665 0.420308i
\(396\) 0 0
\(397\) −15.5830 26.9906i −0.782089 1.35462i −0.930723 0.365725i \(-0.880821\pi\)
0.148634 0.988892i \(-0.452512\pi\)
\(398\) 0 0
\(399\) −1.23987 2.14752i −0.0620712 0.107510i
\(400\) 0 0
\(401\) −0.208497 0.361128i −0.0104119 0.0180339i 0.860773 0.508990i \(-0.169981\pi\)
−0.871184 + 0.490956i \(0.836648\pi\)
\(402\) 0 0
\(403\) −10.0000 + 17.3205i −0.498135 + 0.862796i
\(404\) 0 0
\(405\) −18.2288 −0.905794
\(406\) 0 0
\(407\) 1.64575 0.0815769
\(408\) 0 0
\(409\) −9.46863 + 16.4001i −0.468193 + 0.810935i −0.999339 0.0363456i \(-0.988428\pi\)
0.531146 + 0.847280i \(0.321762\pi\)
\(410\) 0 0
\(411\) −24.9686 43.2469i −1.23161 2.13321i
\(412\) 0 0
\(413\) 1.70850 0.0840697
\(414\) 0 0
\(415\) 24.2288 + 41.9654i 1.18934 + 2.06000i
\(416\) 0 0
\(417\) −5.29150 + 9.16515i −0.259126 + 0.448819i
\(418\) 0 0
\(419\) −21.8745 −1.06864 −0.534320 0.845282i \(-0.679432\pi\)
−0.534320 + 0.845282i \(0.679432\pi\)
\(420\) 0 0
\(421\) −33.1660 −1.61641 −0.808206 0.588900i \(-0.799561\pi\)
−0.808206 + 0.588900i \(0.799561\pi\)
\(422\) 0 0
\(423\) 26.5830 46.0431i 1.29251 2.23869i
\(424\) 0 0
\(425\) −24.8745 43.0839i −1.20659 2.08988i
\(426\) 0 0
\(427\) −4.90588 + 8.49723i −0.237412 + 0.411210i
\(428\) 0 0
\(429\) 6.61438 + 11.4564i 0.319345 + 0.553122i
\(430\) 0 0
\(431\) 1.38562 2.39997i 0.0667430 0.115602i −0.830723 0.556686i \(-0.812073\pi\)
0.897466 + 0.441084i \(0.145406\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) −41.3948 −1.98473
\(436\) 0 0
\(437\) −0.645751 + 1.11847i −0.0308905 + 0.0535039i
\(438\) 0 0
\(439\) −5.96863 10.3380i −0.284867 0.493404i 0.687710 0.725986i \(-0.258616\pi\)
−0.972577 + 0.232581i \(0.925283\pi\)
\(440\) 0 0
\(441\) 28.0000 1.33333
\(442\) 0 0
\(443\) −9.22876 15.9847i −0.438471 0.759455i 0.559100 0.829100i \(-0.311147\pi\)
−0.997572 + 0.0696451i \(0.977813\pi\)
\(444\) 0 0
\(445\) 26.5830 46.0431i 1.26016 2.18265i
\(446\) 0 0
\(447\) −12.4575 −0.589220
\(448\) 0 0
\(449\) 9.87451 0.466007 0.233003 0.972476i \(-0.425145\pi\)
0.233003 + 0.972476i \(0.425145\pi\)
\(450\) 0 0
\(451\) −2.46863 + 4.27579i −0.116243 + 0.201339i
\(452\) 0 0
\(453\) −4.43725 7.68555i −0.208480 0.361099i
\(454\) 0 0
\(455\) −24.1144 + 41.7673i −1.13050 + 1.95808i
\(456\) 0 0
\(457\) 19.5830 + 33.9188i 0.916054 + 1.58665i 0.805350 + 0.592799i \(0.201977\pi\)
0.110704 + 0.993853i \(0.464689\pi\)
\(458\) 0 0
\(459\) −7.93725 + 13.7477i −0.370479 + 0.641689i
\(460\) 0 0
\(461\) −32.1660 −1.49812 −0.749060 0.662502i \(-0.769495\pi\)
−0.749060 + 0.662502i \(0.769495\pi\)
\(462\) 0 0
\(463\) 22.4575 1.04369 0.521845 0.853041i \(-0.325244\pi\)
0.521845 + 0.853041i \(0.325244\pi\)
\(464\) 0 0
\(465\) −19.2915 + 33.4139i −0.894622 + 1.54953i
\(466\) 0 0
\(467\) 5.35425 + 9.27383i 0.247765 + 0.429142i 0.962905 0.269839i \(-0.0869706\pi\)
−0.715140 + 0.698981i \(0.753637\pi\)
\(468\) 0 0
\(469\) −10.4170 −0.481012
\(470\) 0 0
\(471\) 28.0000 + 48.4974i 1.29017 + 2.23464i
\(472\) 0 0
\(473\) 2.00000 3.46410i 0.0919601 0.159280i
\(474\) 0 0
\(475\) 2.93725 0.134770
\(476\) 0 0
\(477\) −14.5830 −0.667710
\(478\) 0 0
\(479\) −12.9686 + 22.4623i −0.592552 + 1.02633i 0.401336 + 0.915931i \(0.368546\pi\)
−0.993887 + 0.110399i \(0.964787\pi\)
\(480\) 0 0
\(481\) 4.11438 + 7.12631i 0.187600 + 0.324932i
\(482\) 0 0
\(483\) −12.7601 22.1012i −0.580606 1.00564i
\(484\) 0 0
\(485\) 10.4059 + 18.0235i 0.472507 + 0.818406i
\(486\) 0 0
\(487\) −15.2915 + 26.4857i −0.692924 + 1.20018i 0.277951 + 0.960595i \(0.410345\pi\)
−0.970875 + 0.239585i \(0.922989\pi\)
\(488\) 0 0
\(489\) 12.2915 0.555841
\(490\) 0 0
\(491\) −10.7085 −0.483268 −0.241634 0.970367i \(-0.577683\pi\)
−0.241634 + 0.970367i \(0.577683\pi\)
\(492\) 0 0
\(493\) 12.8745 22.2993i 0.579839 1.00431i
\(494\) 0 0
\(495\) 7.29150 + 12.6293i 0.327729 + 0.567643i
\(496\) 0 0
\(497\) 12.7601 + 22.1012i 0.572370 + 0.991374i
\(498\) 0 0
\(499\) 8.93725 + 15.4798i 0.400086 + 0.692970i 0.993736 0.111754i \(-0.0356468\pi\)
−0.593650 + 0.804724i \(0.702313\pi\)
\(500\) 0 0
\(501\) 20.1458 34.8935i 0.900046 1.55893i
\(502\) 0 0
\(503\) 19.9373 0.888958 0.444479 0.895789i \(-0.353389\pi\)
0.444479 + 0.895789i \(0.353389\pi\)
\(504\) 0 0
\(505\) 10.9373 0.486701
\(506\) 0 0
\(507\) −15.8745 + 27.4955i −0.705012 + 1.22112i
\(508\) 0 0
\(509\) −10.2915 17.8254i −0.456163 0.790097i 0.542591 0.839997i \(-0.317443\pi\)
−0.998754 + 0.0498996i \(0.984110\pi\)
\(510\) 0 0
\(511\) 14.9373 0.660785
\(512\) 0 0
\(513\) −0.468627 0.811686i −0.0206904 0.0358368i
\(514\) 0 0
\(515\) 23.5830 40.8470i 1.03919 1.79993i
\(516\) 0 0
\(517\) 13.2915 0.584560
\(518\) 0 0
\(519\) 27.2288 1.19521
\(520\) 0 0
\(521\) 1.06275 1.84073i 0.0465598 0.0806439i −0.841806 0.539780i \(-0.818508\pi\)
0.888366 + 0.459136i \(0.151841\pi\)
\(522\) 0 0
\(523\) 7.76013 + 13.4409i 0.339327 + 0.587731i 0.984306 0.176469i \(-0.0564674\pi\)
−0.644979 + 0.764200i \(0.723134\pi\)
\(524\) 0 0
\(525\) −29.0203 + 50.2646i −1.26655 + 2.19373i
\(526\) 0 0
\(527\) −12.0000 20.7846i −0.522728 0.905392i
\(528\) 0 0
\(529\) 4.85425 8.40781i 0.211054 0.365557i
\(530\) 0 0
\(531\) 2.58301 0.112093
\(532\) 0 0
\(533\) −24.6863 −1.06928
\(534\) 0 0
\(535\) 9.00000 15.5885i 0.389104 0.673948i
\(536\) 0 0
\(537\) −5.37451 9.30892i −0.231927 0.401710i
\(538\) 0 0
\(539\) 3.50000 + 6.06218i 0.150756 + 0.261116i
\(540\) 0 0
\(541\) 4.14575 + 7.18065i 0.178240 + 0.308720i 0.941278 0.337633i \(-0.109626\pi\)
−0.763038 + 0.646354i \(0.776293\pi\)
\(542\) 0 0
\(543\) 13.2288 22.9129i 0.567700 0.983286i
\(544\) 0 0
\(545\) 38.5830 1.65271
\(546\) 0 0
\(547\) −27.5203 −1.17668 −0.588341 0.808613i \(-0.700219\pi\)
−0.588341 + 0.808613i \(0.700219\pi\)
\(548\) 0 0
\(549\) −7.41699 + 12.8466i −0.316550 + 0.548280i
\(550\) 0 0
\(551\) 0.760130 + 1.31658i 0.0323826 + 0.0560883i
\(552\) 0 0
\(553\) 3.50000 6.06218i 0.148835 0.257790i
\(554\) 0 0
\(555\) 7.93725 + 13.7477i 0.336918 + 0.583559i
\(556\) 0 0
\(557\) −15.8745 + 27.4955i −0.672624 + 1.16502i 0.304533 + 0.952502i \(0.401500\pi\)
−0.977157 + 0.212518i \(0.931834\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) −15.8745 −0.670222
\(562\) 0 0
\(563\) 6.53137 11.3127i 0.275265 0.476772i −0.694937 0.719070i \(-0.744568\pi\)
0.970202 + 0.242298i \(0.0779012\pi\)
\(564\) 0 0
\(565\) 14.0516 + 24.3381i 0.591157 + 1.02391i
\(566\) 0 0
\(567\) −13.2288 −0.555556
\(568\) 0 0
\(569\) 20.5830 + 35.6508i 0.862884 + 1.49456i 0.869133 + 0.494579i \(0.164678\pi\)
−0.00624806 + 0.999980i \(0.501989\pi\)
\(570\) 0 0
\(571\) −22.4686 + 38.9168i −0.940283 + 1.62862i −0.175351 + 0.984506i \(0.556106\pi\)
−0.764932 + 0.644112i \(0.777227\pi\)
\(572\) 0 0
\(573\) 35.1660 1.46908
\(574\) 0 0
\(575\) 30.2288 1.26063
\(576\) 0 0
\(577\) 12.7288 22.0469i 0.529905 0.917823i −0.469486 0.882940i \(-0.655561\pi\)
0.999391 0.0348828i \(-0.0111058\pi\)
\(578\) 0 0
\(579\) 15.2399 + 26.3962i 0.633347 + 1.09699i
\(580\) 0 0
\(581\) 17.5830 + 30.4547i 0.729466 + 1.26347i
\(582\) 0 0
\(583\) −1.82288 3.15731i −0.0754958 0.130763i
\(584\) 0 0
\(585\) −36.4575 + 63.1463i −1.50733 + 2.61078i
\(586\) 0 0
\(587\) 7.93725 0.327606 0.163803 0.986493i \(-0.447624\pi\)
0.163803 + 0.986493i \(0.447624\pi\)
\(588\) 0 0
\(589\) 1.41699 0.0583863
\(590\) 0 0
\(591\) −24.9686 + 43.2469i −1.02707 + 1.77894i
\(592\) 0 0
\(593\) −11.4686 19.8642i −0.470960 0.815727i 0.528488 0.848941i \(-0.322759\pi\)
−0.999448 + 0.0332139i \(0.989426\pi\)
\(594\) 0 0
\(595\) −28.9373 50.1208i −1.18631 2.05475i
\(596\) 0 0
\(597\) −29.4059 50.9325i −1.20350 2.08453i
\(598\) 0 0
\(599\) −9.87451 + 17.1031i −0.403461 + 0.698816i −0.994141 0.108090i \(-0.965526\pi\)
0.590680 + 0.806906i \(0.298860\pi\)
\(600\) 0 0
\(601\) −24.5830 −1.00276 −0.501381 0.865227i \(-0.667174\pi\)
−0.501381 + 0.865227i \(0.667174\pi\)
\(602\) 0 0
\(603\) −15.7490 −0.641350
\(604\) 0 0
\(605\) −1.82288 + 3.15731i −0.0741104 + 0.128363i
\(606\) 0 0
\(607\) 10.6458 + 18.4390i 0.432098 + 0.748415i 0.997054 0.0767058i \(-0.0244402\pi\)
−0.564956 + 0.825121i \(0.691107\pi\)
\(608\) 0 0
\(609\) −30.0405 −1.21730
\(610\) 0 0
\(611\) 33.2288 + 57.5539i 1.34429 + 2.32838i
\(612\) 0 0
\(613\) −1.00000 + 1.73205i −0.0403896 + 0.0699569i −0.885514 0.464614i \(-0.846193\pi\)
0.845124 + 0.534570i \(0.179527\pi\)
\(614\) 0 0
\(615\) −47.6235 −1.92037
\(616\) 0 0
\(617\) −16.2915 −0.655871 −0.327936 0.944700i \(-0.606353\pi\)
−0.327936 + 0.944700i \(0.606353\pi\)
\(618\) 0 0
\(619\) 17.2915 29.9498i 0.695004 1.20378i −0.275175 0.961394i \(-0.588736\pi\)
0.970179 0.242388i \(-0.0779308\pi\)
\(620\) 0 0
\(621\) −4.82288 8.35347i −0.193535 0.335213i
\(622\) 0 0
\(623\) 19.2915 33.4139i 0.772898 1.33870i
\(624\) 0 0
\(625\) −1.14575 1.98450i −0.0458301 0.0793800i
\(626\) 0 0
\(627\) 0.468627 0.811686i 0.0187152 0.0324156i
\(628\) 0 0
\(629\) −9.87451 −0.393722
\(630\) 0 0
\(631\) −34.8118 −1.38583 −0.692917 0.721017i \(-0.743675\pi\)
−0.692917 + 0.721017i \(0.743675\pi\)
\(632\) 0 0
\(633\) −19.7601 + 34.2255i −0.785395 + 1.36034i
\(634\) 0 0
\(635\) 0.114378 + 0.198109i 0.00453896 + 0.00786172i
\(636\) 0 0
\(637\) −17.5000 + 30.3109i −0.693375 + 1.20096i
\(638\) 0 0
\(639\) 19.2915 + 33.4139i 0.763160 + 1.32183i
\(640\) 0 0
\(641\) 9.43725 16.3458i 0.372749 0.645620i −0.617238 0.786776i \(-0.711749\pi\)
0.989987 + 0.141156i \(0.0450819\pi\)
\(642\) 0 0
\(643\) −6.52026 −0.257134 −0.128567 0.991701i \(-0.541038\pi\)
−0.128567 + 0.991701i \(0.541038\pi\)
\(644\) 0 0
\(645\) 38.5830 1.51920
\(646\) 0 0
\(647\) 19.4059 33.6120i 0.762924 1.32142i −0.178413 0.983956i \(-0.557096\pi\)
0.941337 0.337467i \(-0.109570\pi\)
\(648\) 0 0
\(649\) 0.322876 + 0.559237i 0.0126740 + 0.0219520i
\(650\) 0 0
\(651\) −14.0000 + 24.2487i −0.548703 + 0.950382i
\(652\) 0 0
\(653\) −1.17712 2.03884i −0.0460644 0.0797859i 0.842074 0.539362i \(-0.181335\pi\)
−0.888138 + 0.459576i \(0.848001\pi\)
\(654\) 0 0
\(655\) 28.5203 49.3985i 1.11438 1.93016i
\(656\) 0 0
\(657\) 22.5830 0.881047
\(658\) 0 0
\(659\) −14.5830 −0.568073 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(660\) 0 0
\(661\) −14.2915 + 24.7536i −0.555875 + 0.962804i 0.441960 + 0.897035i \(0.354283\pi\)
−0.997835 + 0.0657690i \(0.979050\pi\)
\(662\) 0 0
\(663\) −39.6863 68.7386i −1.54129 2.66959i
\(664\) 0 0
\(665\) 3.41699 0.132505
\(666\) 0 0
\(667\) 7.82288 + 13.5496i 0.302903 + 0.524643i
\(668\) 0 0
\(669\) −16.3431 + 28.3071i −0.631862 + 1.09442i
\(670\) 0 0
\(671\) −3.70850 −0.143165
\(672\) 0 0
\(673\) −14.9373 −0.575789 −0.287894 0.957662i \(-0.592955\pi\)
−0.287894 + 0.957662i \(0.592955\pi\)
\(674\) 0 0
\(675\) −10.9686 + 18.9982i −0.422183 + 0.731242i
\(676\) 0 0
\(677\) 1.06275 + 1.84073i 0.0408446 + 0.0707450i 0.885725 0.464210i \(-0.153662\pi\)
−0.844880 + 0.534955i \(0.820328\pi\)
\(678\) 0 0
\(679\) 7.55163 + 13.0798i 0.289805 + 0.501957i
\(680\) 0 0
\(681\) −17.5830 30.4547i −0.673782 1.16703i
\(682\) 0 0
\(683\) −6.96863 + 12.0700i −0.266647 + 0.461846i −0.967994 0.250974i \(-0.919249\pi\)
0.701347 + 0.712820i \(0.252582\pi\)
\(684\) 0 0
\(685\) 68.8118 2.62916
\(686\) 0 0
\(687\) −42.3320 −1.61507
\(688\) 0 0
\(689\) 9.11438 15.7866i 0.347230 0.601420i
\(690\) 0 0
\(691\) −9.38562 16.2564i −0.357046 0.618422i 0.630420 0.776254i \(-0.282883\pi\)
−0.987466 + 0.157833i \(0.949549\pi\)
\(692\) 0 0
\(693\) 5.29150 + 9.16515i 0.201008 + 0.348155i
\(694\) 0 0
\(695\) −7.29150 12.6293i −0.276582 0.479055i
\(696\) 0 0
\(697\) 14.8118 25.6547i 0.561035 0.971742i
\(698\) 0 0
\(699\) 44.8118 1.69494
\(700\) 0 0
\(701\) −6.87451 −0.259647 −0.129823 0.991537i \(-0.541441\pi\)
−0.129823 + 0.991537i \(0.541441\pi\)
\(702\) 0 0
\(703\) 0.291503 0.504897i 0.0109942 0.0190426i
\(704\) 0 0
\(705\) 64.1033 + 111.030i 2.41427 + 4.18164i
\(706\) 0 0
\(707\) 7.93725 0.298511
\(708\) 0 0
\(709\) 3.40588 + 5.89916i 0.127911 + 0.221548i 0.922867 0.385119i \(-0.125840\pi\)
−0.794956 + 0.606667i \(0.792506\pi\)
\(710\) 0 0
\(711\) 5.29150 9.16515i 0.198447 0.343720i
\(712\) 0 0
\(713\) 14.5830 0.546138
\(714\) 0 0
\(715\) −18.2288 −0.681717
\(716\) 0 0
\(717\) 12.2085 21.1457i 0.455935 0.789702i
\(718\) 0 0
\(719\) −1.93725 3.35542i −0.0722474 0.125136i 0.827639 0.561261i \(-0.189684\pi\)
−0.899886 + 0.436125i \(0.856350\pi\)
\(720\) 0 0
\(721\) 17.1144 29.6430i 0.637373 1.10396i
\(722\) 0 0
\(723\) −30.1771 52.2683i −1.12230 1.94388i
\(724\) 0 0
\(725\) 17.7915 30.8158i 0.660760 1.14447i
\(726\) 0 0
\(727\) 17.2915 0.641306 0.320653 0.947197i \(-0.396098\pi\)
0.320653 + 0.947197i \(0.396098\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) −12.0000 + 20.7846i −0.443836 + 0.768747i
\(732\) 0 0
\(733\) −20.7288 35.9033i −0.765634 1.32612i −0.939911 0.341420i \(-0.889092\pi\)
0.174277 0.984697i \(-0.444241\pi\)
\(734\) 0 0
\(735\) −33.7601 + 58.4743i −1.24526 + 2.15686i
\(736\) 0 0
\(737\) −1.96863 3.40976i −0.0725153 0.125600i
\(738\) 0 0
\(739\) −3.93725 + 6.81952i −0.144834 + 0.250860i −0.929311 0.369298i \(-0.879598\pi\)
0.784477 + 0.620158i \(0.212932\pi\)
\(740\) 0 0
\(741\) 4.68627 0.172154
\(742\) 0 0
\(743\) 34.7085 1.27333 0.636666 0.771140i \(-0.280313\pi\)
0.636666 + 0.771140i \(0.280313\pi\)
\(744\) 0 0
\(745\) 8.58301 14.8662i 0.314457 0.544655i
\(746\) 0 0
\(747\) 26.5830 + 46.0431i 0.972621 + 1.68463i
\(748\) 0 0
\(749\) 6.53137 11.3127i 0.238651 0.413356i
\(750\) 0 0
\(751\) −8.00000 13.8564i −0.291924 0.505627i 0.682341 0.731034i \(-0.260962\pi\)
−0.974265 + 0.225407i \(0.927629\pi\)
\(752\) 0 0
\(753\) 9.64575 16.7069i 0.351511 0.608834i
\(754\) 0 0
\(755\) 12.2288 0.445050
\(756\) 0 0
\(757\) 19.1660 0.696600 0.348300 0.937383i \(-0.386759\pi\)
0.348300 + 0.937383i \(0.386759\pi\)
\(758\) 0 0
\(759\) 4.82288 8.35347i 0.175059 0.303212i
\(760\) 0 0
\(761\) 3.00000 + 5.19615i 0.108750 + 0.188360i 0.915264 0.402854i \(-0.131982\pi\)
−0.806514 + 0.591215i \(0.798649\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) 0 0
\(765\) −43.7490 75.7755i −1.58175 2.73967i
\(766\) 0 0
\(767\) −1.61438 + 2.79619i −0.0582918 + 0.100964i
\(768\) 0 0
\(769\) −15.1660 −0.546900 −0.273450 0.961886i \(-0.588165\pi\)
−0.273450 + 0.961886i \(0.588165\pi\)
\(770\) 0 0
\(771\) −1.10326 −0.0397331
\(772\) 0 0
\(773\) −1.29150 + 2.23695i −0.0464521 + 0.0804574i −0.888317 0.459232i \(-0.848125\pi\)
0.841864 + 0.539689i \(0.181458\pi\)
\(774\) 0 0
\(775\) −16.5830 28.7226i −0.595679 1.03175i
\(776\) 0 0
\(777\) 5.76013 + 9.97684i 0.206643 + 0.357917i
\(778\) 0 0
\(779\) 0.874508 + 1.51469i 0.0313325 + 0.0542695i
\(780\) 0 0
\(781\) −4.82288 + 8.35347i −0.172576 + 0.298911i
\(782\) 0 0
\(783\) −11.3542 −0.405768
\(784\) 0 0
\(785\) −77.1660 −2.75417
\(786\) 0 0
\(787\) −0.405881 + 0.703006i −0.0144681 + 0.0250595i −0.873169 0.487418i \(-0.837939\pi\)
0.858701 + 0.512477i \(0.171272\pi\)
\(788\) 0 0
\(789\) 5.37451 + 9.30892i 0.191338 + 0.331406i
\(790\) 0 0
\(791\) 10.1974 + 17.6624i 0.362577 + 0.628002i
\(792\) 0 0
\(793\) −9.27124 16.0583i −0.329232 0.570246i
\(794\) 0 0
\(795\) 17.5830 30.4547i 0.623605 1.08012i
\(796\) 0 0
\(797\) −35.1660 −1.24564 −0.622822 0.782364i \(-0.714014\pi\)
−0.622822 + 0.782364i \(0.714014\pi\)
\(798\) 0 0
\(799\) −79.7490 −2.82132
\(800\) 0 0
\(801\) 29.1660 50.5170i 1.03053 1.78493i
\(802\) 0 0
\(803\) 2.82288 + 4.88936i 0.0996171 + 0.172542i
\(804\) 0 0
\(805\) 35.1660 1.23944
\(806\) 0 0
\(807\) −35.1660 60.9093i −1.23790 2.14411i
\(808\) 0 0
\(809\) −25.2915 + 43.8062i −0.889202 + 1.54014i −0.0483813 + 0.998829i \(0.515406\pi\)
−0.840821 + 0.541314i \(0.817927\pi\)
\(810\) 0 0
\(811\) −27.7490 −0.974400 −0.487200 0.873290i \(-0.661982\pi\)
−0.487200 + 0.873290i \(0.661982\pi\)
\(812\) 0 0
\(813\) 47.4575 1.66441
\(814\) 0 0
\(815\) −8.46863 + 14.6681i −0.296643 + 0.513801i
\(816\) 0 0
\(817\) −0.708497 1.22715i −0.0247872 0.0429327i
\(818\) 0 0
\(819\) −26.4575 + 45.8258i −0.924500 + 1.60128i
\(820\) 0 0
\(821\) 3.85425 + 6.67575i 0.134514 + 0.232985i 0.925412 0.378963i \(-0.123719\pi\)
−0.790898 + 0.611949i \(0.790386\pi\)
\(822\) 0 0
\(823\) −21.9373 + 37.9964i −0.764685 + 1.32447i 0.175729 + 0.984439i \(0.443772\pi\)
−0.940413 + 0.340034i \(0.889561\pi\)
\(824\) 0 0
\(825\) −21.9373 −0.763757
\(826\) 0 0
\(827\) 35.3948 1.23080 0.615398 0.788216i \(-0.288995\pi\)
0.615398 + 0.788216i \(0.288995\pi\)
\(828\) 0 0
\(829\) −21.6974 + 37.5810i −0.753581 + 1.30524i 0.192495 + 0.981298i \(0.438342\pi\)
−0.946077 + 0.323943i \(0.894991\pi\)
\(830\) 0 0
\(831\) 15.4889 + 26.8275i 0.537304 + 0.930637i
\(832\) 0 0
\(833\) −21.0000 36.3731i −0.727607 1.26025i
\(834\) 0 0
\(835\) 27.7601 + 48.0820i 0.960679 + 1.66394i
\(836\) 0 0
\(837\) −5.29150 + 9.16515i −0.182901 + 0.316794i
\(838\) 0 0
\(839\) −27.8745 −0.962335 −0.481167 0.876629i \(-0.659787\pi\)
−0.481167 + 0.876629i \(0.659787\pi\)
\(840\) 0 0
\(841\) −10.5830 −0.364931
\(842\) 0 0
\(843\) 9.34313 16.1828i 0.321795 0.557365i
\(844\) 0 0
\(845\) −21.8745 37.8878i −0.752506 1.30338i
\(846\) 0 0
\(847\) −1.32288 + 2.29129i −0.0454545 + 0.0787296i
\(848\) 0 0
\(849\) −10.1144 17.5186i −0.347125 0.601237i
\(850\) 0 0
\(851\) 3.00000 5.19615i 0.102839 0.178122i
\(852\) 0 0
\(853\) −3.16601 −0.108402 −0.0542011 0.998530i \(-0.517261\pi\)
−0.0542011 + 0.998530i \(0.517261\pi\)
\(854\) 0 0
\(855\) 5.16601 0.176674
\(856\) 0 0
\(857\) −18.0000 + 31.1769i −0.614868 + 1.06498i 0.375539 + 0.926806i \(0.377458\pi\)
−0.990408 + 0.138177i \(0.955876\pi\)
\(858\) 0 0
\(859\) 18.9059 + 32.7459i 0.645060 + 1.11728i 0.984288 + 0.176573i \(0.0565011\pi\)
−0.339227 + 0.940704i \(0.610166\pi\)
\(860\) 0 0
\(861\) −34.5608 −1.17783
\(862\) 0 0
\(863\) −24.7601 42.8858i −0.842845 1.45985i −0.887480 0.460846i \(-0.847546\pi\)
0.0446353 0.999003i \(-0.485787\pi\)
\(864\) 0 0
\(865\) −18.7601 + 32.4935i −0.637864 + 1.10481i
\(866\) 0 0
\(867\) 50.2693 1.70723
\(868\) 0 0
\(869\) 2.64575 0.0897510
\(870\) 0 0
\(871\) 9.84313 17.0488i 0.333522 0.577677i
\(872\) 0 0
\(873\) 11.4170 + 19.7748i 0.386407 + 0.669276i
\(874\) 0 0
\(875\) −15.8745 27.4955i −0.536656 0.929516i
\(876\) 0 0
\(877\) −4.43725 7.68555i −0.149835 0.259523i 0.781331 0.624117i \(-0.214541\pi\)
−0.931167 + 0.364594i \(0.881208\pi\)
\(878\) 0 0
\(879\) −15.8745 + 27.4955i −0.535434 + 0.927399i
\(880\) 0 0
\(881\) 6.87451 0.231608 0.115804 0.993272i \(-0.463056\pi\)
0.115804 + 0.993272i \(0.463056\pi\)
\(882\) 0 0
\(883\) −6.06275 −0.204028 −0.102014 0.994783i \(-0.532529\pi\)
−0.102014 + 0.994783i \(0.532529\pi\)
\(884\) 0 0
\(885\) −3.11438 + 5.39426i −0.104689 + 0.181326i
\(886\) 0 0
\(887\) −6.55163 11.3478i −0.219982 0.381020i 0.734820 0.678262i \(-0.237267\pi\)
−0.954802 + 0.297242i \(0.903933\pi\)
\(888\) 0 0
\(889\) 0.0830052 + 0.143769i 0.00278391 + 0.00482187i
\(890\) 0 0
\(891\) −2.50000 4.33013i −0.0837532 0.145065i
\(892\) 0 0
\(893\) 2.35425 4.07768i 0.0787819 0.136454i
\(894\) 0 0
\(895\) 14.8118 0.495103
\(896\) 0 0
\(897\) 48.2288 1.61031
\(898\) 0 0
\(899\) 8.58301 14.8662i 0.286259 0.495816i
\(900\) 0 0
\(901\) 10.9373 + 18.9439i 0.364373 + 0.631112i
\(902\) 0 0
\(903\) 28.0000 0.931782
\(904\) 0 0
\(905\) 18.2288 + 31.5731i 0.605944 + 1.04953i
\(906\) 0 0
\(907\) −5.22876 + 9.05647i −0.173618 + 0.300715i −0.939682 0.342049i \(-0.888879\pi\)
0.766064 + 0.642764i \(0.222212\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 1.29150 0.0427894 0.0213947 0.999771i \(-0.493189\pi\)
0.0213947 + 0.999771i \(0.493189\pi\)
\(912\) 0 0
\(913\) −6.64575 + 11.5108i −0.219942 + 0.380951i
\(914\) 0 0
\(915\) −17.8856 30.9788i −0.591280 1.02413i
\(916\) 0 0
\(917\) 20.6974 35.8489i 0.683488 1.18384i
\(918\) 0 0
\(919\) −17.6458 30.5633i −0.582080 1.00819i −0.995233 0.0975298i \(-0.968906\pi\)
0.413153 0.910662i \(-0.364427\pi\)
\(920\) 0 0
\(921\) −5.59412 + 9.68930i −0.184332 + 0.319273i
\(922\) 0 0
\(923\) −48.2288 −1.58747
\(924\) 0 0
\(925\) −13.6458 −0.448670
\(926\) 0 0
\(927\) 25.8745 44.8160i 0.849830 1.47195i
\(928\) 0 0
\(929\) −17.7915 30.8158i −0.583720 1.01103i −0.995034 0.0995392i \(-0.968263\pi\)
0.411313 0.911494i \(-0.365070\pi\)
\(930\) 0 0
\(931\) 2.47974 0.0812702
\(932\) 0 0
\(933\) 22.4059 + 38.8081i 0.733536 + 1.27052i
\(934\) 0 0
\(935\) 10.9373 18.9439i 0.357686 0.619531i
\(936\) 0 0
\(937\) −46.6863 −1.52517 −0.762587 0.646886i \(-0.776071\pi\)
−0.762587 + 0.646886i \(0.776071\pi\)
\(938\) 0 0
\(939\) 6.39477 0.208685
\(940\) 0 0
\(941\) 22.3118 38.6451i 0.727343 1.25979i −0.230660 0.973034i \(-0.574088\pi\)
0.958002 0.286760i \(-0.0925782\pi\)
\(942\) 0 0
\(943\) 9.00000 + 15.5885i 0.293080 + 0.507630i
\(944\) 0 0
\(945\) −12.7601 + 22.1012i −0.415087 + 0.718952i
\(946\) 0 0
\(947\) −16.9373 29.3362i −0.550387 0.953298i −0.998246 0.0591941i \(-0.981147\pi\)
0.447860 0.894104i \(-0.352186\pi\)
\(948\) 0 0
\(949\) −14.1144 + 24.4468i −0.458172 + 0.793577i
\(950\) 0 0
\(951\) 31.7490 1.02953
\(952\) 0 0
\(953\) −25.5203 −0.826682 −0.413341 0.910576i \(-0.635638\pi\)
−0.413341 + 0.910576i \(0.635638\pi\)
\(954\) 0 0
\(955\) −24.2288 + 41.9654i −0.784024 + 1.35797i
\(956\) 0 0
\(957\) −5.67712 9.83307i −0.183515 0.317858i
\(958\) 0 0
\(959\) 49.9373 1.61256
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) 9.87451 17.1031i 0.318202 0.551141i
\(964\) 0 0
\(965\) −42.0000 −1.35203
\(966\) 0 0
\(967\) 26.3320 0.846781 0.423390 0.905947i \(-0.360840\pi\)
0.423390 + 0.905947i \(0.360840\pi\)
\(968\) 0 0
\(969\) −2.81176 + 4.87011i −0.0903268 + 0.156451i
\(970\) 0 0
\(971\) −12.9686 22.4623i −0.416183 0.720850i 0.579369 0.815066i \(-0.303299\pi\)
−0.995552 + 0.0942153i \(0.969966\pi\)
\(972\) 0 0
\(973\) −5.29150 9.16515i −0.169638 0.293821i
\(974\) 0 0
\(975\) −54.8431 94.9911i −1.75639 3.04215i
\(976\) 0 0
\(977\) −18.2288 + 31.5731i −0.583190 + 1.01011i 0.411909 + 0.911225i \(0.364862\pi\)
−0.995098 + 0.0988890i \(0.968471\pi\)
\(978\) 0 0
\(979\) 14.5830 0.466075
\(980\) 0 0
\(981\) 42.3320 1.35156
\(982\) 0 0
\(983\) −18.6458 + 32.2954i −0.594707 + 1.03006i 0.398881 + 0.917003i \(0.369399\pi\)
−0.993588 + 0.113060i \(0.963935\pi\)
\(984\) 0 0
\(985\) −34.4059 59.5927i −1.09626 1.89878i
\(986\) 0 0
\(987\) 46.5203 + 80.5755i 1.48076 + 2.56474i
\(988\) 0 0
\(989\) −7.29150 12.6293i −0.231856 0.401587i
\(990\) 0 0
\(991\) 28.3431 49.0917i 0.900349 1.55945i 0.0733083 0.997309i \(-0.476644\pi\)
0.827041 0.562141i \(-0.190022\pi\)
\(992\) 0 0
\(993\) 40.6235 1.28915
\(994\) 0 0
\(995\) 81.0405 2.56916
\(996\) 0 0
\(997\) 21.2915 36.8780i 0.674309 1.16794i −0.302362 0.953193i \(-0.597775\pi\)
0.976670 0.214744i \(-0.0688916\pi\)
\(998\) 0 0
\(999\) 2.17712 + 3.77089i 0.0688812 + 0.119306i
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 1232.2.q.g.529.1 4
4.3 odd 2 154.2.e.f.67.2 yes 4
7.2 even 3 inner 1232.2.q.g.177.1 4
7.3 odd 6 8624.2.a.bk.1.1 2
7.4 even 3 8624.2.a.ca.1.2 2
12.11 even 2 1386.2.k.s.991.2 4
28.3 even 6 1078.2.a.n.1.2 2
28.11 odd 6 1078.2.a.s.1.1 2
28.19 even 6 1078.2.e.v.177.1 4
28.23 odd 6 154.2.e.f.23.2 4
28.27 even 2 1078.2.e.v.67.1 4
84.11 even 6 9702.2.a.cz.1.1 2
84.23 even 6 1386.2.k.s.793.2 4
84.59 odd 6 9702.2.a.dr.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.f.23.2 4 28.23 odd 6
154.2.e.f.67.2 yes 4 4.3 odd 2
1078.2.a.n.1.2 2 28.3 even 6
1078.2.a.s.1.1 2 28.11 odd 6
1078.2.e.v.67.1 4 28.27 even 2
1078.2.e.v.177.1 4 28.19 even 6
1232.2.q.g.177.1 4 7.2 even 3 inner
1232.2.q.g.529.1 4 1.1 even 1 trivial
1386.2.k.s.793.2 4 84.23 even 6
1386.2.k.s.991.2 4 12.11 even 2
8624.2.a.bk.1.1 2 7.3 odd 6
8624.2.a.ca.1.2 2 7.4 even 3
9702.2.a.cz.1.1 2 84.11 even 6
9702.2.a.dr.1.2 2 84.59 odd 6