Properties

Label 980.2.i.d.961.1
Level $980$
Weight $2$
Character 980.961
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
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] = [980,2,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("980.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (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: \(7.82533939809\)
Analytic rank: \(0\)
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 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 980.961
Dual form 980.2.i.d.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +1.00000 q^{15} +(1.50000 + 2.59808i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -5.00000 q^{27} -9.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(5.00000 - 8.66025i) q^{37} +(0.500000 + 0.866025i) q^{39} +2.00000 q^{43} +(1.00000 + 1.73205i) q^{45} +(1.50000 - 2.59808i) q^{47} +(1.50000 - 2.59808i) q^{51} +3.00000 q^{55} +2.00000 q^{57} +(-6.00000 - 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(0.500000 - 0.866025i) q^{65} +(-4.00000 - 6.92820i) q^{67} -6.00000 q^{69} +(-7.00000 - 12.1244i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-2.50000 + 4.33013i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} -3.00000 q^{85} +(4.50000 + 7.79423i) q^{87} +(-6.00000 + 10.3923i) q^{89} +(-4.00000 + 6.92820i) q^{93} +(-1.00000 - 1.73205i) q^{95} +17.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} + 2 q^{9} - 3 q^{11} - 2 q^{13} + 2 q^{15} + 3 q^{17} - 2 q^{19} + 6 q^{23} - q^{25} - 10 q^{27} - 18 q^{29} - 8 q^{31} - 3 q^{33} + 10 q^{37} + q^{39} + 4 q^{43} + 2 q^{45} + 3 q^{47} + 3 q^{51} + 6 q^{55} + 4 q^{57} - 12 q^{59} - 8 q^{61} + q^{65} - 8 q^{67} - 12 q^{69} - 14 q^{73} - q^{75} - 5 q^{79} - q^{81} - 24 q^{83} - 6 q^{85} + 9 q^{87} - 12 q^{89} - 8 q^{93} - 2 q^{95} + 34 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −0.500000 0.866025i −0.288675 0.500000i 0.684819 0.728714i \(-0.259881\pi\)
−0.973494 + 0.228714i \(0.926548\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 1.73205i 0.333333 0.577350i
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −4.00000 6.92820i −0.718421 1.24434i −0.961625 0.274367i \(-0.911532\pi\)
0.243204 0.969975i \(-0.421802\pi\)
\(32\) 0 0
\(33\) −1.50000 + 2.59808i −0.261116 + 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 8.66025i 0.821995 1.42374i −0.0821995 0.996616i \(-0.526194\pi\)
0.904194 0.427121i \(-0.140472\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.0800641 + 0.138675i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 1.00000 + 1.73205i 0.149071 + 0.258199i
\(46\) 0 0
\(47\) 1.50000 2.59808i 0.218797 0.378968i −0.735643 0.677369i \(-0.763120\pi\)
0.954441 + 0.298401i \(0.0964533\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.50000 2.59808i 0.210042 0.363803i
\(52\) 0 0
\(53\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −6.00000 10.3923i −0.781133 1.35296i −0.931282 0.364299i \(-0.881308\pi\)
0.150148 0.988663i \(-0.452025\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.500000 0.866025i 0.0620174 0.107417i
\(66\) 0 0
\(67\) −4.00000 6.92820i −0.488678 0.846415i 0.511237 0.859440i \(-0.329187\pi\)
−0.999915 + 0.0130248i \(0.995854\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.00000 12.1244i −0.819288 1.41905i −0.906208 0.422833i \(-0.861036\pi\)
0.0869195 0.996215i \(-0.472298\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.50000 + 4.33013i −0.281272 + 0.487177i −0.971698 0.236225i \(-0.924090\pi\)
0.690426 + 0.723403i \(0.257423\pi\)
\(80\) 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
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 4.50000 + 7.79423i 0.482451 + 0.835629i
\(88\) 0 0
\(89\) −6.00000 + 10.3923i −0.635999 + 1.10158i 0.350304 + 0.936636i \(0.386078\pi\)
−0.986303 + 0.164946i \(0.947255\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 + 6.92820i −0.414781 + 0.718421i
\(94\) 0 0
\(95\) −1.00000 1.73205i −0.102598 0.177705i
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 3.00000 + 5.19615i 0.298511 + 0.517036i 0.975796 0.218685i \(-0.0701767\pi\)
−0.677284 + 0.735721i \(0.736843\pi\)
\(102\) 0 0
\(103\) 3.50000 6.06218i 0.344865 0.597324i −0.640464 0.767988i \(-0.721258\pi\)
0.985329 + 0.170664i \(0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 5.19615i 0.290021 0.502331i −0.683793 0.729676i \(-0.739671\pi\)
0.973814 + 0.227345i \(0.0730044\pi\)
\(108\) 0 0
\(109\) 9.50000 + 16.4545i 0.909935 + 1.57605i 0.814152 + 0.580651i \(0.197202\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 3.00000 + 5.19615i 0.279751 + 0.484544i
\(116\) 0 0
\(117\) −1.00000 + 1.73205i −0.0924500 + 0.160128i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) −1.00000 1.73205i −0.0880451 0.152499i
\(130\) 0 0
\(131\) 9.00000 15.5885i 0.786334 1.36197i −0.141865 0.989886i \(-0.545310\pi\)
0.928199 0.372084i \(-0.121357\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.50000 4.33013i 0.215166 0.372678i
\(136\) 0 0
\(137\) −6.00000 10.3923i −0.512615 0.887875i −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) 1.50000 + 2.59808i 0.125436 + 0.217262i
\(144\) 0 0
\(145\) 4.50000 7.79423i 0.373705 0.647275i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 + 15.5885i −0.737309 + 1.27706i 0.216394 + 0.976306i \(0.430570\pi\)
−0.953703 + 0.300750i \(0.902763\pi\)
\(150\) 0 0
\(151\) 9.50000 + 16.4545i 0.773099 + 1.33905i 0.935857 + 0.352381i \(0.114628\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 11.0000 + 19.0526i 0.877896 + 1.52056i 0.853646 + 0.520854i \(0.174386\pi\)
0.0242497 + 0.999706i \(0.492280\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 + 1.73205i −0.0783260 + 0.135665i −0.902528 0.430632i \(-0.858291\pi\)
0.824202 + 0.566296i \(0.191624\pi\)
\(164\) 0 0
\(165\) −1.50000 2.59808i −0.116775 0.202260i
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 2.00000 + 3.46410i 0.152944 + 0.264906i
\(172\) 0 0
\(173\) 1.50000 2.59808i 0.114043 0.197528i −0.803354 0.595502i \(-0.796953\pi\)
0.917397 + 0.397974i \(0.130287\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 + 10.3923i −0.450988 + 0.781133i
\(178\) 0 0
\(179\) −6.00000 10.3923i −0.448461 0.776757i 0.549825 0.835280i \(-0.314694\pi\)
−0.998286 + 0.0585225i \(0.981361\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 5.00000 + 8.66025i 0.367607 + 0.636715i
\(186\) 0 0
\(187\) 4.50000 7.79423i 0.329073 0.569970i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.50000 + 2.59808i −0.108536 + 0.187990i −0.915177 0.403051i \(-0.867950\pi\)
0.806641 + 0.591041i \(0.201283\pi\)
\(192\) 0 0
\(193\) 2.00000 + 3.46410i 0.143963 + 0.249351i 0.928986 0.370116i \(-0.120682\pi\)
−0.785022 + 0.619467i \(0.787349\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 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
\(201\) −4.00000 + 6.92820i −0.282138 + 0.488678i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 10.3923i −0.417029 0.722315i
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 + 1.73205i −0.0681994 + 0.118125i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −7.00000 + 12.1244i −0.473016 + 0.819288i
\(220\) 0 0
\(221\) −1.50000 2.59808i −0.100901 0.174766i
\(222\) 0 0
\(223\) 17.0000 1.13840 0.569202 0.822198i \(-0.307252\pi\)
0.569202 + 0.822198i \(0.307252\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 13.5000 + 23.3827i 0.896026 + 1.55196i 0.832529 + 0.553981i \(0.186892\pi\)
0.0634974 + 0.997982i \(0.479775\pi\)
\(228\) 0 0
\(229\) 8.00000 13.8564i 0.528655 0.915657i −0.470787 0.882247i \(-0.656030\pi\)
0.999442 0.0334101i \(-0.0106368\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 + 20.7846i −0.786146 + 1.36165i 0.142166 + 0.989843i \(0.454593\pi\)
−0.928312 + 0.371802i \(0.878740\pi\)
\(234\) 0 0
\(235\) 1.50000 + 2.59808i 0.0978492 + 0.169480i
\(236\) 0 0
\(237\) 5.00000 0.324785
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 5.00000 + 8.66025i 0.322078 + 0.557856i 0.980917 0.194429i \(-0.0622852\pi\)
−0.658838 + 0.752285i \(0.728952\pi\)
\(242\) 0 0
\(243\) −8.00000 + 13.8564i −0.513200 + 0.888889i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 1.73205i 0.0636285 0.110208i
\(248\) 0 0
\(249\) 6.00000 + 10.3923i 0.380235 + 0.658586i
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) 1.50000 + 2.59808i 0.0939336 + 0.162698i
\(256\) 0 0
\(257\) 3.00000 5.19615i 0.187135 0.324127i −0.757159 0.653231i \(-0.773413\pi\)
0.944294 + 0.329104i \(0.106747\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −9.00000 + 15.5885i −0.557086 + 0.964901i
\(262\) 0 0
\(263\) 15.0000 + 25.9808i 0.924940 + 1.60204i 0.791658 + 0.610964i \(0.209218\pi\)
0.133281 + 0.991078i \(0.457449\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −9.00000 15.5885i −0.548740 0.950445i −0.998361 0.0572259i \(-0.981774\pi\)
0.449622 0.893219i \(-0.351559\pi\)
\(270\) 0 0
\(271\) 8.00000 13.8564i 0.485965 0.841717i −0.513905 0.857847i \(-0.671801\pi\)
0.999870 + 0.0161307i \(0.00513477\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.50000 + 2.59808i −0.0904534 + 0.156670i
\(276\) 0 0
\(277\) 11.0000 + 19.0526i 0.660926 + 1.14476i 0.980373 + 0.197153i \(0.0631696\pi\)
−0.319447 + 0.947604i \(0.603497\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −21.0000 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 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
\(285\) −1.00000 + 1.73205i −0.0592349 + 0.102598i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.00000 6.92820i 0.235294 0.407541i
\(290\) 0 0
\(291\) −8.50000 14.7224i −0.498279 0.863044i
\(292\) 0 0
\(293\) −15.0000 −0.876309 −0.438155 0.898900i \(-0.644368\pi\)
−0.438155 + 0.898900i \(0.644368\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 7.50000 + 12.9904i 0.435194 + 0.753778i
\(298\) 0 0
\(299\) −3.00000 + 5.19615i −0.173494 + 0.300501i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.00000 5.19615i 0.172345 0.298511i
\(304\) 0 0
\(305\) −4.00000 6.92820i −0.229039 0.396708i
\(306\) 0 0
\(307\) −13.0000 −0.741949 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −9.00000 15.5885i −0.510343 0.883940i −0.999928 0.0119847i \(-0.996185\pi\)
0.489585 0.871956i \(-0.337148\pi\)
\(312\) 0 0
\(313\) 6.50000 11.2583i 0.367402 0.636358i −0.621757 0.783210i \(-0.713581\pi\)
0.989158 + 0.146852i \(0.0469141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 15.5885i 0.505490 0.875535i −0.494489 0.869184i \(-0.664645\pi\)
0.999980 0.00635137i \(-0.00202172\pi\)
\(318\) 0 0
\(319\) 13.5000 + 23.3827i 0.755855 + 1.30918i
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 0.500000 + 0.866025i 0.0277350 + 0.0480384i
\(326\) 0 0
\(327\) 9.50000 16.4545i 0.525351 0.909935i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 + 17.3205i −0.549650 + 0.952021i 0.448649 + 0.893708i \(0.351905\pi\)
−0.998298 + 0.0583130i \(0.981428\pi\)
\(332\) 0 0
\(333\) −10.0000 17.3205i −0.547997 0.949158i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 3.00000 + 5.19615i 0.162938 + 0.282216i
\(340\) 0 0
\(341\) −12.0000 + 20.7846i −0.649836 + 1.12555i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3.00000 5.19615i 0.161515 0.279751i
\(346\) 0 0
\(347\) −15.0000 25.9808i −0.805242 1.39472i −0.916127 0.400887i \(-0.868702\pi\)
0.110885 0.993833i \(-0.464631\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 7.50000 + 12.9904i 0.399185 + 0.691408i 0.993626 0.112731i \(-0.0359599\pi\)
−0.594441 + 0.804139i \(0.702627\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 7.50000 + 12.9904i 0.394737 + 0.683704i
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −2.50000 4.33013i −0.130499 0.226031i 0.793370 0.608740i \(-0.208325\pi\)
−0.923869 + 0.382709i \(0.874991\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 3.46410i 0.103556 0.179364i −0.809591 0.586994i \(-0.800311\pi\)
0.913147 + 0.407630i \(0.133645\pi\)
\(374\) 0 0
\(375\) −0.500000 0.866025i −0.0258199 0.0447214i
\(376\) 0 0
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −10.0000 17.3205i −0.512316 0.887357i
\(382\) 0 0
\(383\) −18.0000 + 31.1769i −0.919757 + 1.59307i −0.119974 + 0.992777i \(0.538281\pi\)
−0.799783 + 0.600289i \(0.795052\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 3.46410i 0.101666 0.176090i
\(388\) 0 0
\(389\) −4.50000 7.79423i −0.228159 0.395183i 0.729103 0.684403i \(-0.239937\pi\)
−0.957263 + 0.289220i \(0.906604\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) −2.50000 4.33013i −0.125789 0.217872i
\(396\) 0 0
\(397\) −2.50000 + 4.33013i −0.125471 + 0.217323i −0.921917 0.387387i \(-0.873378\pi\)
0.796446 + 0.604710i \(0.206711\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.50000 12.9904i 0.374532 0.648709i −0.615725 0.787961i \(-0.711137\pi\)
0.990257 + 0.139253i \(0.0444700\pi\)
\(402\) 0 0
\(403\) 4.00000 + 6.92820i 0.199254 + 0.345118i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −13.0000 22.5167i −0.642809 1.11338i −0.984803 0.173675i \(-0.944436\pi\)
0.341994 0.939702i \(-0.388898\pi\)
\(410\) 0 0
\(411\) −6.00000 + 10.3923i −0.295958 + 0.512615i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.00000 10.3923i 0.294528 0.510138i
\(416\) 0 0
\(417\) −1.00000 1.73205i −0.0489702 0.0848189i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 29.0000 1.41337 0.706687 0.707527i \(-0.250189\pi\)
0.706687 + 0.707527i \(0.250189\pi\)
\(422\) 0 0
\(423\) −3.00000 5.19615i −0.145865 0.252646i
\(424\) 0 0
\(425\) 1.50000 2.59808i 0.0727607 0.126025i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.50000 2.59808i 0.0724207 0.125436i
\(430\) 0 0
\(431\) −7.50000 12.9904i −0.361262 0.625725i 0.626907 0.779094i \(-0.284321\pi\)
−0.988169 + 0.153370i \(0.950987\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 6.00000 + 10.3923i 0.287019 + 0.497131i
\(438\) 0 0
\(439\) 5.00000 8.66025i 0.238637 0.413331i −0.721686 0.692220i \(-0.756633\pi\)
0.960323 + 0.278889i \(0.0899661\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.00000 5.19615i 0.142534 0.246877i −0.785916 0.618333i \(-0.787808\pi\)
0.928450 + 0.371457i \(0.121142\pi\)
\(444\) 0 0
\(445\) −6.00000 10.3923i −0.284427 0.492642i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 9.50000 16.4545i 0.446349 0.773099i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 24.2487i 0.654892 1.13431i −0.327028 0.945015i \(-0.606047\pi\)
0.981921 0.189292i \(-0.0606194\pi\)
\(458\) 0 0
\(459\) −7.50000 12.9904i −0.350070 0.606339i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) −4.00000 6.92820i −0.185496 0.321288i
\(466\) 0 0
\(467\) −7.50000 + 12.9904i −0.347059 + 0.601123i −0.985726 0.168360i \(-0.946153\pi\)
0.638667 + 0.769483i \(0.279486\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 11.0000 19.0526i 0.506853 0.877896i
\(472\) 0 0
\(473\) −3.00000 5.19615i −0.137940 0.238919i
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.00000 5.19615i −0.137073 0.237418i 0.789314 0.613990i \(-0.210436\pi\)
−0.926388 + 0.376571i \(0.877103\pi\)
\(480\) 0 0
\(481\) −5.00000 + 8.66025i −0.227980 + 0.394874i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.50000 + 14.7224i −0.385965 + 0.668511i
\(486\) 0 0
\(487\) 5.00000 + 8.66025i 0.226572 + 0.392434i 0.956790 0.290780i \(-0.0939149\pi\)
−0.730218 + 0.683214i \(0.760582\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) −13.5000 23.3827i −0.608009 1.05310i
\(494\) 0 0
\(495\) 3.00000 5.19615i 0.134840 0.233550i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.5000 21.6506i 0.559577 0.969216i −0.437955 0.898997i \(-0.644297\pi\)
0.997532 0.0702185i \(-0.0223697\pi\)
\(500\) 0 0
\(501\) −4.50000 7.79423i −0.201045 0.348220i
\(502\) 0 0
\(503\) −33.0000 −1.47140 −0.735699 0.677309i \(-0.763146\pi\)
−0.735699 + 0.677309i \(0.763146\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 6.00000 + 10.3923i 0.266469 + 0.461538i
\(508\) 0 0
\(509\) −9.00000 + 15.5885i −0.398918 + 0.690946i −0.993593 0.113020i \(-0.963948\pi\)
0.594675 + 0.803966i \(0.297281\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.00000 8.66025i 0.220755 0.382360i
\(514\) 0 0
\(515\) 3.50000 + 6.06218i 0.154228 + 0.267131i
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) 15.0000 + 25.9808i 0.657162 + 1.13824i 0.981347 + 0.192244i \(0.0615766\pi\)
−0.324185 + 0.945994i \(0.605090\pi\)
\(522\) 0 0
\(523\) −10.0000 + 17.3205i −0.437269 + 0.757373i −0.997478 0.0709788i \(-0.977388\pi\)
0.560208 + 0.828352i \(0.310721\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 20.7846i 0.522728 0.905392i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 3.00000 + 5.19615i 0.129701 + 0.224649i
\(536\) 0 0
\(537\) −6.00000 + 10.3923i −0.258919 + 0.448461i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.5000 21.6506i 0.537417 0.930834i −0.461625 0.887075i \(-0.652733\pi\)
0.999042 0.0437584i \(-0.0139332\pi\)
\(542\) 0 0
\(543\) −4.00000 6.92820i −0.171656 0.297318i
\(544\) 0 0
\(545\) −19.0000 −0.813871
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 8.00000 + 13.8564i 0.341432 + 0.591377i
\(550\) 0 0
\(551\) 9.00000 15.5885i 0.383413 0.664091i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5.00000 8.66025i 0.212238 0.367607i
\(556\) 0 0
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) 6.00000 + 10.3923i 0.252870 + 0.437983i 0.964315 0.264758i \(-0.0852922\pi\)
−0.711445 + 0.702742i \(0.751959\pi\)
\(564\) 0 0
\(565\) 3.00000 5.19615i 0.126211 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i \(-0.793194\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 0 0
\(571\) 2.00000 + 3.46410i 0.0836974 + 0.144968i 0.904835 0.425762i \(-0.139994\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −23.5000 40.7032i −0.978318 1.69450i −0.668521 0.743693i \(-0.733072\pi\)
−0.309797 0.950803i \(-0.600261\pi\)
\(578\) 0 0
\(579\) 2.00000 3.46410i 0.0831172 0.143963i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.00000 1.73205i −0.0413449 0.0716115i
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) −6.00000 10.3923i −0.246807 0.427482i
\(592\) 0 0
\(593\) 16.5000 28.5788i 0.677574 1.17359i −0.298136 0.954524i \(-0.596365\pi\)
0.975709 0.219069i \(-0.0703019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10.0000 + 17.3205i −0.409273 + 0.708881i
\(598\) 0 0
\(599\) 4.50000 + 7.79423i 0.183865 + 0.318464i 0.943193 0.332244i \(-0.107806\pi\)
−0.759328 + 0.650708i \(0.774472\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −16.0000 −0.651570
\(604\) 0 0
\(605\) 1.00000 + 1.73205i 0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) 0.500000 0.866025i 0.0202944 0.0351509i −0.855700 0.517472i \(-0.826873\pi\)
0.875994 + 0.482322i \(0.160206\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.50000 + 2.59808i −0.0606835 + 0.105107i
\(612\) 0 0
\(613\) −1.00000 1.73205i −0.0403896 0.0699569i 0.845124 0.534570i \(-0.179527\pi\)
−0.885514 + 0.464614i \(0.846193\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 5.00000 + 8.66025i 0.200967 + 0.348085i 0.948840 0.315757i \(-0.102258\pi\)
−0.747873 + 0.663842i \(0.768925\pi\)
\(620\) 0 0
\(621\) −15.0000 + 25.9808i −0.601929 + 1.04257i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −3.00000 5.19615i −0.119808 0.207514i
\(628\) 0 0
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) −2.50000 4.33013i −0.0993661 0.172107i
\(634\) 0 0
\(635\) −10.0000 + 17.3205i −0.396838 + 0.687343i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.0000 + 25.9808i 0.592464 + 1.02618i 0.993899 + 0.110291i \(0.0351782\pi\)
−0.401435 + 0.915888i \(0.631488\pi\)
\(642\) 0 0
\(643\) 41.0000 1.61688 0.808441 0.588577i \(-0.200312\pi\)
0.808441 + 0.588577i \(0.200312\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −24.0000 41.5692i −0.943537 1.63425i −0.758654 0.651494i \(-0.774142\pi\)
−0.184884 0.982760i \(-0.559191\pi\)
\(648\) 0 0
\(649\) −18.0000 + 31.1769i −0.706562 + 1.22380i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 5.19615i 0.117399 0.203341i −0.801337 0.598213i \(-0.795878\pi\)
0.918736 + 0.394872i \(0.129211\pi\)
\(654\) 0 0
\(655\) 9.00000 + 15.5885i 0.351659 + 0.609091i
\(656\) 0 0
\(657\) −28.0000 −1.09238
\(658\) 0 0
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) −4.00000 6.92820i −0.155582 0.269476i 0.777689 0.628649i \(-0.216392\pi\)
−0.933271 + 0.359174i \(0.883059\pi\)
\(662\) 0 0
\(663\) −1.50000 + 2.59808i −0.0582552 + 0.100901i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −27.0000 + 46.7654i −1.04544 + 1.81076i
\(668\) 0 0
\(669\) −8.50000 14.7224i −0.328629 0.569202i
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 2.50000 + 4.33013i 0.0962250 + 0.166667i
\(676\) 0 0
\(677\) −7.50000 + 12.9904i −0.288248 + 0.499261i −0.973392 0.229147i \(-0.926406\pi\)
0.685143 + 0.728408i \(0.259740\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 13.5000 23.3827i 0.517321 0.896026i
\(682\) 0 0
\(683\) −12.0000 20.7846i −0.459167 0.795301i 0.539750 0.841825i \(-0.318519\pi\)
−0.998917 + 0.0465244i \(0.985185\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 14.0000 24.2487i 0.532585 0.922464i −0.466691 0.884420i \(-0.654554\pi\)
0.999276 0.0380440i \(-0.0121127\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.00000 + 1.73205i −0.0379322 + 0.0657004i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 0 0
\(703\) 10.0000 + 17.3205i 0.377157 + 0.653255i
\(704\) 0 0
\(705\) 1.50000 2.59808i 0.0564933 0.0978492i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.500000 0.866025i 0.0187779 0.0325243i −0.856484 0.516174i \(-0.827356\pi\)
0.875262 + 0.483650i \(0.160689\pi\)
\(710\) 0 0
\(711\) 5.00000 + 8.66025i 0.187515 + 0.324785i
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 0 0
\(717\) −4.50000 7.79423i −0.168056 0.291081i
\(718\) 0 0
\(719\) 9.00000 15.5885i 0.335643 0.581351i −0.647965 0.761670i \(-0.724380\pi\)
0.983608 + 0.180319i \(0.0577130\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.00000 8.66025i 0.185952 0.322078i
\(724\) 0 0
\(725\) 4.50000 + 7.79423i 0.167126 + 0.289470i
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 3.00000 + 5.19615i 0.110959 + 0.192187i
\(732\) 0 0
\(733\) 6.50000 11.2583i 0.240083 0.415836i −0.720655 0.693294i \(-0.756159\pi\)
0.960738 + 0.277458i \(0.0894920\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 + 20.7846i −0.442026 + 0.765611i
\(738\) 0 0
\(739\) −5.50000 9.52628i −0.202321 0.350430i 0.746955 0.664875i \(-0.231515\pi\)
−0.949276 + 0.314445i \(0.898182\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) −9.00000 15.5885i −0.329734 0.571117i
\(746\) 0 0
\(747\) −12.0000 + 20.7846i −0.439057 + 0.760469i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.5000 + 45.8993i −0.966999 + 1.67489i −0.262852 + 0.964836i \(0.584663\pi\)
−0.704146 + 0.710055i \(0.748670\pi\)
\(752\) 0 0
\(753\) −3.00000 5.19615i −0.109326 0.189358i
\(754\) 0 0
\(755\) −19.0000 −0.691481
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 9.00000 + 15.5885i 0.326679 + 0.565825i
\(760\) 0 0
\(761\) 21.0000 36.3731i 0.761249 1.31852i −0.180957 0.983491i \(-0.557920\pi\)
0.942207 0.335032i \(-0.108747\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.00000 + 5.19615i −0.108465 + 0.187867i
\(766\) 0 0
\(767\) 6.00000 + 10.3923i 0.216647 + 0.375244i
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) −7.50000 12.9904i −0.269756 0.467232i 0.699043 0.715080i \(-0.253610\pi\)
−0.968799 + 0.247849i \(0.920276\pi\)
\(774\) 0 0
\(775\) −4.00000 + 6.92820i −0.143684 + 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 45.0000 1.60817
\(784\) 0 0
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) −2.50000 4.33013i −0.0891154 0.154352i 0.818022 0.575187i \(-0.195071\pi\)
−0.907137 + 0.420834i \(0.861737\pi\)
\(788\) 0 0
\(789\) 15.0000 25.9808i 0.534014 0.924940i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.00000 6.92820i 0.142044 0.246028i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 12.0000 + 20.7846i 0.423999 + 0.734388i
\(802\) 0 0
\(803\) −21.0000 + 36.3731i −0.741074 + 1.28358i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.00000 + 15.5885i −0.316815 + 0.548740i
\(808\) 0 0
\(809\) −4.50000 7.79423i −0.158212 0.274030i 0.776012 0.630718i \(-0.217239\pi\)
−0.934224 + 0.356687i \(0.883906\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −1.00000 1.73205i −0.0350285 0.0606711i
\(816\) 0 0
\(817\) −2.00000 + 3.46410i −0.0699711 + 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.50000 12.9904i 0.261752 0.453367i −0.704956 0.709251i \(-0.749033\pi\)
0.966708 + 0.255884i \(0.0823665\pi\)
\(822\) 0 0
\(823\) 20.0000 + 34.6410i 0.697156 + 1.20751i 0.969448 + 0.245295i \(0.0788849\pi\)
−0.272292 + 0.962215i \(0.587782\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) −42.0000 −1.46048 −0.730242 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 0 0
\(829\) 20.0000 + 34.6410i 0.694629 + 1.20313i 0.970306 + 0.241882i \(0.0777647\pi\)
−0.275677 + 0.961250i \(0.588902\pi\)
\(830\) 0 0
\(831\) 11.0000 19.0526i 0.381586 0.660926i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −4.50000 + 7.79423i −0.155729 + 0.269730i
\(836\) 0 0
\(837\) 20.0000 + 34.6410i 0.691301 + 1.19737i
\(838\) 0 0
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 10.5000 + 18.1865i 0.361639 + 0.626377i
\(844\) 0 0
\(845\) 6.00000 10.3923i 0.206406 0.357506i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −5.50000 + 9.52628i −0.188760 + 0.326941i
\(850\) 0 0
\(851\) −30.0000 51.9615i −1.02839 1.78122i
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −3.00000 5.19615i −0.102478 0.177497i 0.810227 0.586116i \(-0.199344\pi\)
−0.912705 + 0.408619i \(0.866010\pi\)
\(858\) 0 0
\(859\) 2.00000 3.46410i 0.0682391 0.118194i −0.829887 0.557931i \(-0.811595\pi\)
0.898126 + 0.439738i \(0.144929\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 + 10.3923i −0.204242 + 0.353758i −0.949891 0.312581i \(-0.898806\pi\)
0.745649 + 0.666339i \(0.232140\pi\)
\(864\) 0 0
\(865\) 1.50000 + 2.59808i 0.0510015 + 0.0883372i
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 4.00000 + 6.92820i 0.135535 + 0.234753i
\(872\) 0 0
\(873\) 17.0000 29.4449i 0.575363 0.996558i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.0000 19.0526i 0.371444 0.643359i −0.618344 0.785907i \(-0.712196\pi\)
0.989788 + 0.142548i \(0.0455296\pi\)
\(878\) 0 0
\(879\) 7.50000 + 12.9904i 0.252969 + 0.438155i
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) −6.00000 10.3923i −0.201688 0.349334i
\(886\) 0 0
\(887\) 24.0000 41.5692i 0.805841 1.39576i −0.109881 0.993945i \(-0.535047\pi\)
0.915722 0.401813i \(-0.131620\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.50000 + 2.59808i −0.0502519 + 0.0870388i
\(892\) 0 0
\(893\) 3.00000 + 5.19615i 0.100391 + 0.173883i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 0 0
\(899\) 36.0000 + 62.3538i 1.20067 + 2.07962i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 + 6.92820i −0.132964 + 0.230301i
\(906\) 0 0
\(907\) 11.0000 + 19.0526i 0.365249 + 0.632630i 0.988816 0.149140i \(-0.0476505\pi\)
−0.623567 + 0.781770i \(0.714317\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 18.0000 + 31.1769i 0.595713 + 1.03181i
\(914\) 0 0
\(915\) −4.00000 + 6.92820i −0.132236 + 0.229039i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −20.5000 + 35.5070i −0.676233 + 1.17127i 0.299874 + 0.953979i \(0.403055\pi\)
−0.976107 + 0.217291i \(0.930278\pi\)
\(920\) 0 0
\(921\) 6.50000 + 11.2583i 0.214182 + 0.370975i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) −7.00000 12.1244i −0.229910 0.398216i
\(928\) 0 0
\(929\) 12.0000 20.7846i 0.393707 0.681921i −0.599228 0.800578i \(-0.704526\pi\)
0.992935 + 0.118657i \(0.0378590\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −9.00000 + 15.5885i −0.294647 + 0.510343i
\(934\) 0 0
\(935\) 4.50000 + 7.79423i 0.147166 + 0.254899i
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.0000 + 31.1769i −0.584921 + 1.01311i 0.409964 + 0.912102i \(0.365541\pi\)
−0.994885 + 0.101012i \(0.967792\pi\)
\(948\) 0 0
\(949\) 7.00000 + 12.1244i 0.227230 + 0.393573i
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) −1.50000 2.59808i −0.0485389 0.0840718i
\(956\) 0 0
\(957\) 13.5000 23.3827i 0.436393 0.755855i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) −6.00000 10.3923i −0.193347 0.334887i
\(964\) 0 0
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) 3.00000 + 5.19615i 0.0963739 + 0.166924i
\(970\) 0 0
\(971\) −18.0000 + 31.1769i −0.577647 + 1.00051i 0.418101 + 0.908401i \(0.362696\pi\)
−0.995748 + 0.0921142i \(0.970638\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0.500000 0.866025i 0.0160128 0.0277350i
\(976\) 0 0
\(977\) 9.00000 + 15.5885i 0.287936 + 0.498719i 0.973317 0.229465i \(-0.0736978\pi\)
−0.685381 + 0.728184i \(0.740364\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 38.0000 1.21325
\(982\) 0 0
\(983\) 16.5000 + 28.5788i 0.526268 + 0.911523i 0.999532 + 0.0306024i \(0.00974257\pi\)
−0.473263 + 0.880921i \(0.656924\pi\)
\(984\) 0 0
\(985\) −6.00000 + 10.3923i −0.191176 + 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.00000 10.3923i 0.190789 0.330456i
\(990\) 0 0
\(991\) 8.00000 + 13.8564i 0.254128 + 0.440163i 0.964658 0.263504i \(-0.0848781\pi\)
−0.710530 + 0.703667i \(0.751545\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) −8.50000 14.7224i −0.269198 0.466264i 0.699457 0.714675i \(-0.253425\pi\)
−0.968655 + 0.248410i \(0.920092\pi\)
\(998\) 0 0
\(999\) −25.0000 + 43.3013i −0.790965 + 1.36999i
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 980.2.i.d.961.1 2
7.2 even 3 140.2.a.a.1.1 1
7.3 odd 6 980.2.i.h.361.1 2
7.4 even 3 inner 980.2.i.d.361.1 2
7.5 odd 6 980.2.a.c.1.1 1
7.6 odd 2 980.2.i.h.961.1 2
21.2 odd 6 1260.2.a.c.1.1 1
21.5 even 6 8820.2.a.r.1.1 1
28.19 even 6 3920.2.a.u.1.1 1
28.23 odd 6 560.2.a.c.1.1 1
35.2 odd 12 700.2.e.c.449.1 2
35.9 even 6 700.2.a.d.1.1 1
35.12 even 12 4900.2.e.l.2549.2 2
35.19 odd 6 4900.2.a.p.1.1 1
35.23 odd 12 700.2.e.c.449.2 2
35.33 even 12 4900.2.e.l.2549.1 2
56.37 even 6 2240.2.a.g.1.1 1
56.51 odd 6 2240.2.a.r.1.1 1
84.23 even 6 5040.2.a.h.1.1 1
105.2 even 12 6300.2.k.c.6049.2 2
105.23 even 12 6300.2.k.c.6049.1 2
105.44 odd 6 6300.2.a.d.1.1 1
140.23 even 12 2800.2.g.j.449.1 2
140.79 odd 6 2800.2.a.y.1.1 1
140.107 even 12 2800.2.g.j.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.a.a.1.1 1 7.2 even 3
560.2.a.c.1.1 1 28.23 odd 6
700.2.a.d.1.1 1 35.9 even 6
700.2.e.c.449.1 2 35.2 odd 12
700.2.e.c.449.2 2 35.23 odd 12
980.2.a.c.1.1 1 7.5 odd 6
980.2.i.d.361.1 2 7.4 even 3 inner
980.2.i.d.961.1 2 1.1 even 1 trivial
980.2.i.h.361.1 2 7.3 odd 6
980.2.i.h.961.1 2 7.6 odd 2
1260.2.a.c.1.1 1 21.2 odd 6
2240.2.a.g.1.1 1 56.37 even 6
2240.2.a.r.1.1 1 56.51 odd 6
2800.2.a.y.1.1 1 140.79 odd 6
2800.2.g.j.449.1 2 140.23 even 12
2800.2.g.j.449.2 2 140.107 even 12
3920.2.a.u.1.1 1 28.19 even 6
4900.2.a.p.1.1 1 35.19 odd 6
4900.2.e.l.2549.1 2 35.33 even 12
4900.2.e.l.2549.2 2 35.12 even 12
5040.2.a.h.1.1 1 84.23 even 6
6300.2.a.d.1.1 1 105.44 odd 6
6300.2.k.c.6049.1 2 105.23 even 12
6300.2.k.c.6049.2 2 105.2 even 12
8820.2.a.r.1.1 1 21.5 even 6