Properties

Label 1960.2.q.f.961.1
Level $1960$
Weight $2$
Character 1960.961
Analytic conductor $15.651$
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] = [1960,2,Mod(361,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, 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("1960.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.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: \(15.6506787962\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1960.961
Dual form 1960.2.q.f.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000 q^{13} -1.00000 q^{15} +(-2.50000 - 4.33013i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(-0.500000 - 0.866025i) q^{25} -5.00000 q^{27} -5.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(2.00000 - 3.46410i) q^{37} +(0.500000 + 0.866025i) q^{39} +2.00000 q^{41} +10.0000 q^{43} +(-1.00000 - 1.73205i) q^{45} +(-4.50000 + 7.79423i) q^{47} +(-2.50000 + 4.33013i) q^{51} +(-3.00000 - 5.19615i) q^{53} -3.00000 q^{55} +6.00000 q^{57} +(-3.00000 - 5.19615i) q^{59} +(-6.00000 + 10.3923i) q^{61} +(-0.500000 + 0.866025i) q^{65} +(1.00000 + 1.73205i) q^{67} +(-7.00000 - 12.1244i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} -5.00000 q^{85} +(2.50000 + 4.33013i) q^{87} +(1.00000 - 1.73205i) q^{93} +(3.00000 + 5.19615i) q^{95} +9.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{5} + 2 q^{9} - 3 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 6 q^{19} - q^{25} - 10 q^{27} - 10 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{37} + q^{39} + 4 q^{41} + 20 q^{43} - 2 q^{45} - 9 q^{47} - 5 q^{51} - 6 q^{53} - 6 q^{55} + 12 q^{57} - 6 q^{59} - 12 q^{61} - q^{65} + 2 q^{67} - 14 q^{73} - q^{75} - q^{79} - q^{81} - 24 q^{83} - 10 q^{85} + 5 q^{87} + 2 q^{93} + 6 q^{95} + 18 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(17\) −2.50000 4.33013i −0.606339 1.05021i −0.991838 0.127502i \(-0.959304\pi\)
0.385499 0.922708i \(-0.374029\pi\)
\(18\) 0 0
\(19\) −3.00000 + 5.19615i −0.688247 + 1.19208i 0.284157 + 0.958778i \(0.408286\pi\)
−0.972404 + 0.233301i \(0.925047\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 1.00000 + 1.73205i 0.179605 + 0.311086i 0.941745 0.336327i \(-0.109185\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) 0 0
\(33\) −1.50000 + 2.59808i −0.261116 + 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 3.46410i 0.328798 0.569495i −0.653476 0.756948i \(-0.726690\pi\)
0.982274 + 0.187453i \(0.0600231\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.0800641 + 0.138675i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) −1.00000 1.73205i −0.149071 0.258199i
\(46\) 0 0
\(47\) −4.50000 + 7.79423i −0.656392 + 1.13691i 0.325150 + 0.945662i \(0.394585\pi\)
−0.981543 + 0.191243i \(0.938748\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.50000 + 4.33013i −0.350070 + 0.606339i
\(52\) 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
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 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
\(61\) −6.00000 + 10.3923i −0.768221 + 1.33060i 0.170305 + 0.985391i \(0.445525\pi\)
−0.938527 + 0.345207i \(0.887809\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.500000 + 0.866025i −0.0620174 + 0.107417i
\(66\) 0 0
\(67\) 1.00000 + 1.73205i 0.122169 + 0.211604i 0.920623 0.390453i \(-0.127682\pi\)
−0.798454 + 0.602056i \(0.794348\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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
\(79\) −0.500000 + 0.866025i −0.0562544 + 0.0974355i −0.892781 0.450490i \(-0.851249\pi\)
0.836527 + 0.547926i \(0.184582\pi\)
\(80\) 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
\(85\) −5.00000 −0.542326
\(86\) 0 0
\(87\) 2.50000 + 4.33013i 0.268028 + 0.464238i
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.00000 1.73205i 0.103695 0.179605i
\(94\) 0 0
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) 9.00000 0.913812 0.456906 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −9.00000 15.5885i −0.895533 1.55111i −0.833143 0.553058i \(-0.813461\pi\)
−0.0623905 0.998052i \(-0.519872\pi\)
\(102\) 0 0
\(103\) −0.500000 + 0.866025i −0.0492665 + 0.0853320i −0.889607 0.456727i \(-0.849022\pi\)
0.840341 + 0.542059i \(0.182355\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.00000 + 1.73205i −0.0966736 + 0.167444i −0.910306 0.413936i \(-0.864154\pi\)
0.813632 + 0.581380i \(0.197487\pi\)
\(108\) 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
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(123\) −1.00000 1.73205i −0.0901670 0.156174i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) −5.00000 8.66025i −0.440225 0.762493i
\(130\) 0 0
\(131\) −8.00000 + 13.8564i −0.698963 + 1.21064i 0.269863 + 0.962899i \(0.413022\pi\)
−0.968826 + 0.247741i \(0.920312\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.50000 + 4.33013i −0.215166 + 0.372678i
\(136\) 0 0
\(137\) −2.00000 3.46410i −0.170872 0.295958i 0.767853 0.640626i \(-0.221325\pi\)
−0.938725 + 0.344668i \(0.887992\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 1.50000 + 2.59808i 0.125436 + 0.217262i
\(144\) 0 0
\(145\) −2.50000 + 4.33013i −0.207614 + 0.359597i
\(146\) 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
\(151\) −4.50000 7.79423i −0.366205 0.634285i 0.622764 0.782410i \(-0.286010\pi\)
−0.988969 + 0.148124i \(0.952676\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −1.00000 1.73205i −0.0798087 0.138233i 0.823359 0.567521i \(-0.192098\pi\)
−0.903167 + 0.429289i \(0.858764\pi\)
\(158\) 0 0
\(159\) −3.00000 + 5.19615i −0.237915 + 0.412082i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 6.92820i 0.313304 0.542659i −0.665771 0.746156i \(-0.731897\pi\)
0.979076 + 0.203497i \(0.0652307\pi\)
\(164\) 0 0
\(165\) 1.50000 + 2.59808i 0.116775 + 0.202260i
\(166\) 0 0
\(167\) 5.00000 0.386912 0.193456 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 6.00000 + 10.3923i 0.458831 + 0.794719i
\(172\) 0 0
\(173\) 7.50000 12.9904i 0.570214 0.987640i −0.426329 0.904568i \(-0.640193\pi\)
0.996544 0.0830722i \(-0.0264732\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.00000 + 5.19615i −0.225494 + 0.390567i
\(178\) 0 0
\(179\) 10.0000 + 17.3205i 0.747435 + 1.29460i 0.949048 + 0.315130i \(0.102048\pi\)
−0.201613 + 0.979465i \(0.564618\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) −2.00000 3.46410i −0.147043 0.254686i
\(186\) 0 0
\(187\) −7.50000 + 12.9904i −0.548454 + 0.949951i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.5000 21.6506i 0.904468 1.56658i 0.0828388 0.996563i \(-0.473601\pi\)
0.821629 0.570022i \(-0.193065\pi\)
\(192\) 0 0
\(193\) −12.0000 20.7846i −0.863779 1.49611i −0.868255 0.496119i \(-0.834758\pi\)
0.00447566 0.999990i \(-0.498575\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 11.0000 + 19.0526i 0.779769 + 1.35060i 0.932075 + 0.362267i \(0.117997\pi\)
−0.152305 + 0.988334i \(0.548670\pi\)
\(200\) 0 0
\(201\) 1.00000 1.73205i 0.0705346 0.122169i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.00000 1.73205i 0.0698430 0.120972i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 18.0000 1.24509
\(210\) 0 0
\(211\) −11.0000 −0.757271 −0.378636 0.925546i \(-0.623607\pi\)
−0.378636 + 0.925546i \(0.623607\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.00000 8.66025i 0.340997 0.590624i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −7.00000 + 12.1244i −0.473016 + 0.819288i
\(220\) 0 0
\(221\) 2.50000 + 4.33013i 0.168168 + 0.291276i
\(222\) 0 0
\(223\) −11.0000 −0.736614 −0.368307 0.929704i \(-0.620063\pi\)
−0.368307 + 0.929704i \(0.620063\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −8.50000 14.7224i −0.564165 0.977162i −0.997127 0.0757500i \(-0.975865\pi\)
0.432962 0.901412i \(-0.357468\pi\)
\(228\) 0 0
\(229\) 1.00000 1.73205i 0.0660819 0.114457i −0.831092 0.556136i \(-0.812283\pi\)
0.897173 + 0.441679i \(0.145617\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 20.7846i 0.786146 1.36165i −0.142166 0.989843i \(-0.545407\pi\)
0.928312 0.371802i \(-0.121260\pi\)
\(234\) 0 0
\(235\) 4.50000 + 7.79423i 0.293548 + 0.508439i
\(236\) 0 0
\(237\) 1.00000 0.0649570
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 2.00000 + 3.46410i 0.128831 + 0.223142i 0.923224 0.384262i \(-0.125544\pi\)
−0.794393 + 0.607404i \(0.792211\pi\)
\(242\) 0 0
\(243\) −8.00000 + 13.8564i −0.513200 + 0.888889i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.00000 5.19615i 0.190885 0.330623i
\(248\) 0 0
\(249\) 6.00000 + 10.3923i 0.380235 + 0.658586i
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2.50000 + 4.33013i 0.156556 + 0.271163i
\(256\) 0 0
\(257\) 7.00000 12.1244i 0.436648 0.756297i −0.560781 0.827964i \(-0.689499\pi\)
0.997429 + 0.0716680i \(0.0228322\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −5.00000 + 8.66025i −0.309492 + 0.536056i
\(262\) 0 0
\(263\) −13.0000 22.5167i −0.801614 1.38844i −0.918553 0.395298i \(-0.870641\pi\)
0.116939 0.993139i \(-0.462692\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.00000 13.8564i −0.487769 0.844840i 0.512132 0.858906i \(-0.328856\pi\)
−0.999901 + 0.0140665i \(0.995522\pi\)
\(270\) 0 0
\(271\) 14.0000 24.2487i 0.850439 1.47300i −0.0303728 0.999539i \(-0.509669\pi\)
0.880812 0.473466i \(-0.156997\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.50000 + 2.59808i −0.0904534 + 0.156670i
\(276\) 0 0
\(277\) −2.00000 3.46410i −0.120168 0.208138i 0.799666 0.600446i \(-0.205010\pi\)
−0.919834 + 0.392308i \(0.871677\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 0 0
\(283\) 0.500000 + 0.866025i 0.0297219 + 0.0514799i 0.880504 0.474039i \(-0.157204\pi\)
−0.850782 + 0.525519i \(0.823871\pi\)
\(284\) 0 0
\(285\) 3.00000 5.19615i 0.177705 0.307794i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.00000 + 6.92820i −0.235294 + 0.407541i
\(290\) 0 0
\(291\) −4.50000 7.79423i −0.263795 0.456906i
\(292\) 0 0
\(293\) 29.0000 1.69420 0.847099 0.531435i \(-0.178347\pi\)
0.847099 + 0.531435i \(0.178347\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 7.50000 + 12.9904i 0.435194 + 0.753778i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −9.00000 + 15.5885i −0.517036 + 0.895533i
\(304\) 0 0
\(305\) 6.00000 + 10.3923i 0.343559 + 0.595062i
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) 11.0000 + 19.0526i 0.623753 + 1.08037i 0.988781 + 0.149375i \(0.0477261\pi\)
−0.365028 + 0.930997i \(0.618941\pi\)
\(312\) 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
\(317\) −5.00000 + 8.66025i −0.280828 + 0.486408i −0.971589 0.236675i \(-0.923942\pi\)
0.690761 + 0.723083i \(0.257276\pi\)
\(318\) 0 0
\(319\) 7.50000 + 12.9904i 0.419919 + 0.727322i
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) 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 \(-0.648095\pi\)
0.998298 0.0583130i \(-0.0185721\pi\)
\(332\) 0 0
\(333\) −4.00000 6.92820i −0.219199 0.379663i
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 0 0
\(339\) 2.00000 + 3.46410i 0.108625 + 0.188144i
\(340\) 0 0
\(341\) 3.00000 5.19615i 0.162459 0.281387i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.0000 31.1769i −0.966291 1.67366i −0.706107 0.708105i \(-0.749550\pi\)
−0.260184 0.965559i \(-0.583783\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 1.50000 + 2.59808i 0.0798369 + 0.138282i 0.903179 0.429263i \(-0.141227\pi\)
−0.823343 + 0.567545i \(0.807893\pi\)
\(354\) 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
\(361\) −8.50000 14.7224i −0.447368 0.774865i
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 11.5000 + 19.9186i 0.600295 + 1.03974i 0.992776 + 0.119982i \(0.0382835\pi\)
−0.392481 + 0.919760i \(0.628383\pi\)
\(368\) 0 0
\(369\) 2.00000 3.46410i 0.104116 0.180334i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.0000 20.7846i 0.621336 1.07619i −0.367901 0.929865i \(-0.619923\pi\)
0.989237 0.146321i \(-0.0467433\pi\)
\(374\) 0 0
\(375\) 0.500000 + 0.866025i 0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) 3.00000 + 5.19615i 0.153695 + 0.266207i
\(382\) 0 0
\(383\) 16.0000 27.7128i 0.817562 1.41606i −0.0899119 0.995950i \(-0.528659\pi\)
0.907474 0.420109i \(-0.138008\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0000 17.3205i 0.508329 0.880451i
\(388\) 0 0
\(389\) 7.50000 + 12.9904i 0.380265 + 0.658638i 0.991100 0.133120i \(-0.0424994\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) 0.500000 + 0.866025i 0.0251577 + 0.0435745i
\(396\) 0 0
\(397\) 5.50000 9.52628i 0.276037 0.478110i −0.694359 0.719629i \(-0.744312\pi\)
0.970396 + 0.241518i \(0.0776454\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.50000 + 11.2583i −0.324595 + 0.562214i −0.981430 0.191820i \(-0.938561\pi\)
0.656836 + 0.754034i \(0.271895\pi\)
\(402\) 0 0
\(403\) −1.00000 1.73205i −0.0498135 0.0862796i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) 17.0000 + 29.4449i 0.840596 + 1.45595i 0.889392 + 0.457146i \(0.151128\pi\)
−0.0487958 + 0.998809i \(0.515538\pi\)
\(410\) 0 0
\(411\) −2.00000 + 3.46410i −0.0986527 + 0.170872i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.00000 + 10.3923i −0.294528 + 0.510138i
\(416\) 0 0
\(417\) 8.00000 + 13.8564i 0.391762 + 0.678551i
\(418\) 0 0
\(419\) 22.0000 1.07477 0.537385 0.843337i \(-0.319412\pi\)
0.537385 + 0.843337i \(0.319412\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 0 0
\(423\) 9.00000 + 15.5885i 0.437595 + 0.757937i
\(424\) 0 0
\(425\) −2.50000 + 4.33013i −0.121268 + 0.210042i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.50000 2.59808i 0.0724207 0.125436i
\(430\) 0 0
\(431\) −1.50000 2.59808i −0.0722525 0.125145i 0.827636 0.561266i \(-0.189685\pi\)
−0.899888 + 0.436121i \(0.856352\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 5.00000 0.239732
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 14.0000 24.2487i 0.668184 1.15733i −0.310228 0.950662i \(-0.600405\pi\)
0.978412 0.206666i \(-0.0662612\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.00000 + 15.5885i −0.427603 + 0.740630i −0.996660 0.0816684i \(-0.973975\pi\)
0.569057 + 0.822298i \(0.307309\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) −3.00000 5.19615i −0.141264 0.244677i
\(452\) 0 0
\(453\) −4.50000 + 7.79423i −0.211428 + 0.366205i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.00000 + 15.5885i −0.421002 + 0.729197i −0.996038 0.0889312i \(-0.971655\pi\)
0.575036 + 0.818128i \(0.304988\pi\)
\(458\) 0 0
\(459\) 12.5000 + 21.6506i 0.583450 + 1.01057i
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) −1.00000 1.73205i −0.0463739 0.0803219i
\(466\) 0 0
\(467\) −1.50000 + 2.59808i −0.0694117 + 0.120225i −0.898642 0.438682i \(-0.855446\pi\)
0.829231 + 0.558906i \(0.188779\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1.00000 + 1.73205i −0.0460776 + 0.0798087i
\(472\) 0 0
\(473\) −15.0000 25.9808i −0.689701 1.19460i
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) −2.00000 3.46410i −0.0913823 0.158279i 0.816711 0.577047i \(-0.195795\pi\)
−0.908093 + 0.418769i \(0.862462\pi\)
\(480\) 0 0
\(481\) −2.00000 + 3.46410i −0.0911922 + 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.50000 7.79423i 0.204334 0.353918i
\(486\) 0 0
\(487\) 14.0000 + 24.2487i 0.634401 + 1.09881i 0.986642 + 0.162905i \(0.0520863\pi\)
−0.352241 + 0.935909i \(0.614580\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) 12.5000 + 21.6506i 0.562972 + 0.975096i
\(494\) 0 0
\(495\) −3.00000 + 5.19615i −0.134840 + 0.233550i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3.50000 + 6.06218i −0.156682 + 0.271380i −0.933670 0.358134i \(-0.883413\pi\)
0.776989 + 0.629515i \(0.216746\pi\)
\(500\) 0 0
\(501\) −2.50000 4.33013i −0.111692 0.193456i
\(502\) 0 0
\(503\) 39.0000 1.73892 0.869462 0.494000i \(-0.164466\pi\)
0.869462 + 0.494000i \(0.164466\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 6.00000 + 10.3923i 0.266469 + 0.461538i
\(508\) 0 0
\(509\) 20.0000 34.6410i 0.886484 1.53544i 0.0424816 0.999097i \(-0.486474\pi\)
0.844003 0.536339i \(-0.180193\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 15.0000 25.9808i 0.662266 1.14708i
\(514\) 0 0
\(515\) 0.500000 + 0.866025i 0.0220326 + 0.0381616i
\(516\) 0 0
\(517\) 27.0000 1.18746
\(518\) 0 0
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) −13.0000 22.5167i −0.569540 0.986473i −0.996611 0.0822547i \(-0.973788\pi\)
0.427071 0.904218i \(-0.359545\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.00000 8.66025i 0.217803 0.377247i
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 1.00000 + 1.73205i 0.0432338 + 0.0748831i
\(536\) 0 0
\(537\) 10.0000 17.3205i 0.431532 0.747435i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.50000 + 12.9904i −0.322450 + 0.558500i −0.980993 0.194043i \(-0.937840\pi\)
0.658543 + 0.752543i \(0.271173\pi\)
\(542\) 0 0
\(543\) −5.00000 8.66025i −0.214571 0.371647i
\(544\) 0 0
\(545\) 19.0000 0.813871
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) 12.0000 + 20.7846i 0.512148 + 0.887066i
\(550\) 0 0
\(551\) 15.0000 25.9808i 0.639021 1.10682i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.00000 + 3.46410i −0.0848953 + 0.147043i
\(556\) 0 0
\(557\) 6.00000 + 10.3923i 0.254228 + 0.440336i 0.964686 0.263404i \(-0.0848453\pi\)
−0.710457 + 0.703740i \(0.751512\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 0 0
\(563\) −10.0000 17.3205i −0.421450 0.729972i 0.574632 0.818412i \(-0.305145\pi\)
−0.996082 + 0.0884397i \(0.971812\pi\)
\(564\) 0 0
\(565\) −2.00000 + 3.46410i −0.0841406 + 0.145736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.00000 + 1.73205i −0.0419222 + 0.0726113i −0.886225 0.463255i \(-0.846681\pi\)
0.844303 + 0.535866i \(0.180015\pi\)
\(570\) 0 0
\(571\) −18.0000 31.1769i −0.753277 1.30471i −0.946227 0.323505i \(-0.895139\pi\)
0.192950 0.981209i \(-0.438194\pi\)
\(572\) 0 0
\(573\) −25.0000 −1.04439
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.50000 + 7.79423i 0.187337 + 0.324478i 0.944362 0.328909i \(-0.106681\pi\)
−0.757024 + 0.653387i \(0.773348\pi\)
\(578\) 0 0
\(579\) −12.0000 + 20.7846i −0.498703 + 0.863779i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −9.00000 + 15.5885i −0.372742 + 0.645608i
\(584\) 0 0
\(585\) 1.00000 + 1.73205i 0.0413449 + 0.0716115i
\(586\) 0 0
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 3.00000 + 5.19615i 0.123404 + 0.213741i
\(592\) 0 0
\(593\) −1.50000 + 2.59808i −0.0615976 + 0.106690i −0.895180 0.445705i \(-0.852953\pi\)
0.833582 + 0.552396i \(0.186286\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0000 19.0526i 0.450200 0.779769i
\(598\) 0 0
\(599\) 8.50000 + 14.7224i 0.347301 + 0.601542i 0.985769 0.168106i \(-0.0537650\pi\)
−0.638468 + 0.769648i \(0.720432\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −1.00000 1.73205i −0.0406558 0.0704179i
\(606\) 0 0
\(607\) 6.50000 11.2583i 0.263827 0.456962i −0.703429 0.710766i \(-0.748349\pi\)
0.967256 + 0.253804i \(0.0816819\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.50000 7.79423i 0.182051 0.315321i
\(612\) 0 0
\(613\) −11.0000 19.0526i −0.444286 0.769526i 0.553716 0.832705i \(-0.313209\pi\)
−0.998002 + 0.0631797i \(0.979876\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −22.0000 38.1051i −0.884255 1.53157i −0.846566 0.532284i \(-0.821334\pi\)
−0.0376891 0.999290i \(-0.512000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −9.00000 15.5885i −0.359425 0.622543i
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) 0 0
\(633\) 5.50000 + 9.52628i 0.218605 + 0.378636i
\(634\) 0 0
\(635\) −3.00000 + 5.19615i −0.119051 + 0.206203i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.00000 + 8.66025i 0.197488 + 0.342059i 0.947713 0.319123i \(-0.103388\pi\)
−0.750225 + 0.661182i \(0.770055\pi\)
\(642\) 0 0
\(643\) 9.00000 0.354925 0.177463 0.984128i \(-0.443211\pi\)
0.177463 + 0.984128i \(0.443211\pi\)
\(644\) 0 0
\(645\) −10.0000 −0.393750
\(646\) 0 0
\(647\) −10.0000 17.3205i −0.393141 0.680939i 0.599721 0.800209i \(-0.295278\pi\)
−0.992862 + 0.119269i \(0.961945\pi\)
\(648\) 0 0
\(649\) −9.00000 + 15.5885i −0.353281 + 0.611900i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.00000 12.1244i 0.273931 0.474463i −0.695934 0.718106i \(-0.745009\pi\)
0.969865 + 0.243643i \(0.0783426\pi\)
\(654\) 0 0
\(655\) 8.00000 + 13.8564i 0.312586 + 0.541415i
\(656\) 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
\(661\) 22.0000 + 38.1051i 0.855701 + 1.48212i 0.875993 + 0.482323i \(0.160207\pi\)
−0.0202925 + 0.999794i \(0.506460\pi\)
\(662\) 0 0
\(663\) 2.50000 4.33013i 0.0970920 0.168168i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 5.50000 + 9.52628i 0.212642 + 0.368307i
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) 2.50000 + 4.33013i 0.0962250 + 0.166667i
\(676\) 0 0
\(677\) −1.50000 + 2.59808i −0.0576497 + 0.0998522i −0.893410 0.449242i \(-0.851694\pi\)
0.835760 + 0.549095i \(0.185027\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8.50000 + 14.7224i −0.325721 + 0.564165i
\(682\) 0 0
\(683\) 18.0000 + 31.1769i 0.688751 + 1.19295i 0.972242 + 0.233977i \(0.0751739\pi\)
−0.283491 + 0.958975i \(0.591493\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) 3.00000 + 5.19615i 0.114291 + 0.197958i
\(690\) 0 0
\(691\) 6.00000 10.3923i 0.228251 0.395342i −0.729039 0.684472i \(-0.760033\pi\)
0.957290 + 0.289130i \(0.0933661\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.00000 + 13.8564i −0.303457 + 0.525603i
\(696\) 0 0
\(697\) −5.00000 8.66025i −0.189389 0.328031i
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) 12.0000 + 20.7846i 0.452589 + 0.783906i
\(704\) 0 0
\(705\) 4.50000 7.79423i 0.169480 0.293548i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −13.5000 + 23.3827i −0.507003 + 0.878155i 0.492964 + 0.870050i \(0.335913\pi\)
−0.999967 + 0.00810550i \(0.997420\pi\)
\(710\) 0 0
\(711\) 1.00000 + 1.73205i 0.0375029 + 0.0649570i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) 7.50000 + 12.9904i 0.280093 + 0.485135i
\(718\) 0 0
\(719\) 12.0000 20.7846i 0.447524 0.775135i −0.550700 0.834703i \(-0.685639\pi\)
0.998224 + 0.0595683i \(0.0189724\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.00000 3.46410i 0.0743808 0.128831i
\(724\) 0 0
\(725\) 2.50000 + 4.33013i 0.0928477 + 0.160817i
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −25.0000 43.3013i −0.924658 1.60156i
\(732\) 0 0
\(733\) −25.5000 + 44.1673i −0.941864 + 1.63136i −0.179952 + 0.983675i \(0.557594\pi\)
−0.761912 + 0.647681i \(0.775739\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.00000 5.19615i 0.110506 0.191403i
\(738\) 0 0
\(739\) 16.5000 + 28.5788i 0.606962 + 1.05129i 0.991738 + 0.128279i \(0.0409454\pi\)
−0.384776 + 0.923010i \(0.625721\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) 54.0000 1.98107 0.990534 0.137268i \(-0.0438322\pi\)
0.990534 + 0.137268i \(0.0438322\pi\)
\(744\) 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
\(751\) 15.5000 26.8468i 0.565603 0.979653i −0.431390 0.902165i \(-0.641977\pi\)
0.996993 0.0774878i \(-0.0246899\pi\)
\(752\) 0 0
\(753\) −6.00000 10.3923i −0.218652 0.378717i
\(754\) 0 0
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) 24.0000 0.872295 0.436147 0.899875i \(-0.356343\pi\)
0.436147 + 0.899875i \(0.356343\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12.0000 20.7846i 0.435000 0.753442i −0.562296 0.826936i \(-0.690082\pi\)
0.997296 + 0.0734946i \(0.0234152\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5.00000 + 8.66025i −0.180775 + 0.313112i
\(766\) 0 0
\(767\) 3.00000 + 5.19615i 0.108324 + 0.187622i
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) −11.5000 19.9186i −0.413626 0.716422i 0.581657 0.813434i \(-0.302405\pi\)
−0.995283 + 0.0970125i \(0.969071\pi\)
\(774\) 0 0
\(775\) 1.00000 1.73205i 0.0359211 0.0622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 + 10.3923i −0.214972 + 0.372343i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 25.0000 0.893427
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) −22.5000 38.9711i −0.802038 1.38917i −0.918272 0.395949i \(-0.870416\pi\)
0.116234 0.993222i \(-0.462918\pi\)
\(788\) 0 0
\(789\) −13.0000 + 22.5167i −0.462812 + 0.801614i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.00000 10.3923i 0.213066 0.369042i
\(794\) 0 0
\(795\) 3.00000 + 5.19615i 0.106399 + 0.184289i
\(796\) 0 0
\(797\) −39.0000 −1.38145 −0.690725 0.723117i \(-0.742709\pi\)
−0.690725 + 0.723117i \(0.742709\pi\)
\(798\) 0 0
\(799\) 45.0000 1.59199
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −21.0000 + 36.3731i −0.741074 + 1.28358i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −8.00000 + 13.8564i −0.281613 + 0.487769i
\(808\) 0 0
\(809\) 1.50000 + 2.59808i 0.0527372 + 0.0913435i 0.891189 0.453632i \(-0.149872\pi\)
−0.838452 + 0.544976i \(0.816539\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) −4.00000 6.92820i −0.140114 0.242684i
\(816\) 0 0
\(817\) −30.0000 + 51.9615i −1.04957 + 1.81790i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.50000 + 4.33013i −0.0872506 + 0.151122i −0.906348 0.422532i \(-0.861141\pi\)
0.819097 + 0.573654i \(0.194475\pi\)
\(822\) 0 0
\(823\) −13.0000 22.5167i −0.453152 0.784881i 0.545428 0.838157i \(-0.316367\pi\)
−0.998580 + 0.0532760i \(0.983034\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 0 0
\(829\) −5.00000 8.66025i −0.173657 0.300783i 0.766039 0.642795i \(-0.222225\pi\)
−0.939696 + 0.342012i \(0.888892\pi\)
\(830\) 0 0
\(831\) −2.00000 + 3.46410i −0.0693792 + 0.120168i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 2.50000 4.33013i 0.0865161 0.149850i
\(836\) 0 0
\(837\) −5.00000 8.66025i −0.172825 0.299342i
\(838\) 0 0
\(839\) 52.0000 1.79524 0.897620 0.440771i \(-0.145295\pi\)
0.897620 + 0.440771i \(0.145295\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −13.5000 23.3827i −0.464965 0.805342i
\(844\) 0 0
\(845\) −6.00000 + 10.3923i −0.206406 + 0.357506i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.500000 0.866025i 0.0171600 0.0297219i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −19.0000 32.9090i −0.649028 1.12415i −0.983355 0.181692i \(-0.941843\pi\)
0.334328 0.942457i \(-0.391491\pi\)
\(858\) 0 0
\(859\) −10.0000 + 17.3205i −0.341196 + 0.590968i −0.984655 0.174512i \(-0.944165\pi\)
0.643459 + 0.765480i \(0.277499\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.00000 15.5885i 0.306364 0.530637i −0.671200 0.741276i \(-0.734221\pi\)
0.977564 + 0.210639i \(0.0675543\pi\)
\(864\) 0 0
\(865\) −7.50000 12.9904i −0.255008 0.441686i
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −1.00000 1.73205i −0.0338837 0.0586883i
\(872\) 0 0
\(873\) 9.00000 15.5885i 0.304604 0.527589i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −15.0000 + 25.9808i −0.506514 + 0.877308i 0.493458 + 0.869770i \(0.335733\pi\)
−0.999972 + 0.00753813i \(0.997601\pi\)
\(878\) 0 0
\(879\) −14.5000 25.1147i −0.489073 0.847099i
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 3.00000 + 5.19615i 0.100844 + 0.174667i
\(886\) 0 0
\(887\) 16.0000 27.7128i 0.537227 0.930505i −0.461825 0.886971i \(-0.652805\pi\)
0.999052 0.0435339i \(-0.0138616\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.50000 + 2.59808i −0.0502519 + 0.0870388i
\(892\) 0 0
\(893\) −27.0000 46.7654i −0.903521 1.56494i
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.00000 8.66025i −0.166759 0.288836i
\(900\) 0 0
\(901\) −15.0000 + 25.9808i −0.499722 + 0.865545i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 8.66025i 0.166206 0.287877i
\(906\) 0 0
\(907\) −10.0000 17.3205i −0.332045 0.575118i 0.650868 0.759191i \(-0.274405\pi\)
−0.982913 + 0.184073i \(0.941072\pi\)
\(908\) 0 0
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 18.0000 + 31.1769i 0.595713 + 1.03181i
\(914\) 0 0
\(915\) 6.00000 10.3923i 0.198354 0.343559i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.50000 + 14.7224i −0.280389 + 0.485648i −0.971481 0.237119i \(-0.923797\pi\)
0.691091 + 0.722767i \(0.257130\pi\)
\(920\) 0 0
\(921\) 12.5000 + 21.6506i 0.411889 + 0.713413i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 1.00000 + 1.73205i 0.0328443 + 0.0568880i
\(928\) 0 0
\(929\) −21.0000 + 36.3731i −0.688988 + 1.19336i 0.283178 + 0.959067i \(0.408611\pi\)
−0.972166 + 0.234294i \(0.924722\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 11.0000 19.0526i 0.360124 0.623753i
\(934\) 0 0
\(935\) 7.50000 + 12.9904i 0.245276 + 0.424831i
\(936\) 0 0
\(937\) −43.0000 −1.40475 −0.702374 0.711808i \(-0.747877\pi\)
−0.702374 + 0.711808i \(0.747877\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) −6.00000 10.3923i −0.195594 0.338779i 0.751501 0.659732i \(-0.229330\pi\)
−0.947095 + 0.320953i \(0.895997\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.0000 22.5167i 0.422443 0.731693i −0.573735 0.819041i \(-0.694506\pi\)
0.996178 + 0.0873481i \(0.0278392\pi\)
\(948\) 0 0
\(949\) 7.00000 + 12.1244i 0.227230 + 0.393573i
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −12.5000 21.6506i −0.404491 0.700598i
\(956\) 0 0
\(957\) 7.50000 12.9904i 0.242441 0.419919i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 13.5000 23.3827i 0.435484 0.754280i
\(962\) 0 0
\(963\) 2.00000 + 3.46410i 0.0644491 + 0.111629i
\(964\) 0 0
\(965\) −24.0000 −0.772587
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 0 0
\(969\) −15.0000 25.9808i −0.481869 0.834622i
\(970\) 0 0
\(971\) −7.00000 + 12.1244i −0.224641 + 0.389089i −0.956212 0.292676i \(-0.905454\pi\)
0.731571 + 0.681765i \(0.238788\pi\)
\(972\) 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
\(979\) 0 0
\(980\) 0 0
\(981\) 38.0000 1.21325
\(982\) 0 0
\(983\) −17.5000 30.3109i −0.558163 0.966767i −0.997650 0.0685181i \(-0.978173\pi\)
0.439487 0.898249i \(-0.355160\pi\)
\(984\) 0 0
\(985\) −3.00000 + 5.19615i −0.0955879 + 0.165563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −18.0000 31.1769i −0.571789 0.990367i −0.996382 0.0849833i \(-0.972916\pi\)
0.424594 0.905384i \(-0.360417\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 22.0000 0.697447
\(996\) 0 0
\(997\) 7.50000 + 12.9904i 0.237527 + 0.411409i 0.960004 0.279986i \(-0.0903297\pi\)
−0.722477 + 0.691395i \(0.756996\pi\)
\(998\) 0 0
\(999\) −10.0000 + 17.3205i −0.316386 + 0.547997i
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 1960.2.q.f.961.1 2
7.2 even 3 1960.2.a.j.1.1 yes 1
7.3 odd 6 1960.2.q.j.361.1 2
7.4 even 3 inner 1960.2.q.f.361.1 2
7.5 odd 6 1960.2.a.f.1.1 1
7.6 odd 2 1960.2.q.j.961.1 2
28.19 even 6 3920.2.a.x.1.1 1
28.23 odd 6 3920.2.a.l.1.1 1
35.9 even 6 9800.2.a.t.1.1 1
35.19 odd 6 9800.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.a.f.1.1 1 7.5 odd 6
1960.2.a.j.1.1 yes 1 7.2 even 3
1960.2.q.f.361.1 2 7.4 even 3 inner
1960.2.q.f.961.1 2 1.1 even 1 trivial
1960.2.q.j.361.1 2 7.3 odd 6
1960.2.q.j.961.1 2 7.6 odd 2
3920.2.a.l.1.1 1 28.23 odd 6
3920.2.a.x.1.1 1 28.19 even 6
9800.2.a.t.1.1 1 35.9 even 6
9800.2.a.bd.1.1 1 35.19 odd 6