Properties

Label 1764.2.j.b.589.1
Level $1764$
Weight $2$
Character 1764.589
Analytic conductor $14.086$
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] = [1764,2,Mod(589,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1764.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.j (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0856109166\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 589.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.589
Dual form 1764.2.j.b.1177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +(1.50000 - 2.59808i) q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +(1.50000 - 2.59808i) q^{5} -3.00000 q^{9} +(-1.50000 - 2.59808i) q^{11} +(-0.500000 + 0.866025i) q^{13} +(4.50000 + 2.59808i) q^{15} -6.00000 q^{17} +4.00000 q^{19} +(1.50000 - 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} -5.19615i q^{27} +(-1.50000 - 2.59808i) q^{29} +(2.50000 - 4.33013i) q^{31} +(4.50000 - 2.59808i) q^{33} +2.00000 q^{37} +(-1.50000 - 0.866025i) q^{39} +(1.50000 - 2.59808i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-4.50000 + 7.79423i) q^{45} +(-4.50000 - 7.79423i) q^{47} -10.3923i q^{51} -6.00000 q^{53} -9.00000 q^{55} +6.92820i q^{57} +(-1.50000 + 2.59808i) q^{59} +(-6.50000 - 11.2583i) q^{61} +(1.50000 + 2.59808i) q^{65} +(3.50000 - 6.06218i) q^{67} +(4.50000 + 2.59808i) q^{69} -12.0000 q^{71} +10.0000 q^{73} +(6.00000 - 3.46410i) q^{75} +(-5.50000 - 9.52628i) q^{79} +9.00000 q^{81} +(-4.50000 - 7.79423i) q^{83} +(-9.00000 + 15.5885i) q^{85} +(4.50000 - 2.59808i) q^{87} -6.00000 q^{89} +(7.50000 + 4.33013i) q^{93} +(6.00000 - 10.3923i) q^{95} +(5.50000 + 9.52628i) q^{97} +(4.50000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - 6 q^{9} - 3 q^{11} - q^{13} + 9 q^{15} - 12 q^{17} + 8 q^{19} + 3 q^{23} - 4 q^{25} - 3 q^{29} + 5 q^{31} + 9 q^{33} + 4 q^{37} - 3 q^{39} + 3 q^{41} + q^{43} - 9 q^{45} - 9 q^{47} - 12 q^{53} - 18 q^{55} - 3 q^{59} - 13 q^{61} + 3 q^{65} + 7 q^{67} + 9 q^{69} - 24 q^{71} + 20 q^{73} + 12 q^{75} - 11 q^{79} + 18 q^{81} - 9 q^{83} - 18 q^{85} + 9 q^{87} - 12 q^{89} + 15 q^{93} + 12 q^{95} + 11 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) 1.50000 2.59808i 0.670820 1.16190i −0.306851 0.951757i \(-0.599275\pi\)
0.977672 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 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
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 4.50000 + 2.59808i 1.16190 + 0.670820i
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 2.59808i 0.312772 0.541736i −0.666190 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −1.50000 2.59808i −0.278543 0.482451i 0.692480 0.721437i \(-0.256518\pi\)
−0.971023 + 0.238987i \(0.923185\pi\)
\(30\) 0 0
\(31\) 2.50000 4.33013i 0.449013 0.777714i −0.549309 0.835619i \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 0 0
\(33\) 4.50000 2.59808i 0.783349 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −1.50000 0.866025i −0.240192 0.138675i
\(40\) 0 0
\(41\) 1.50000 2.59808i 0.234261 0.405751i −0.724797 0.688963i \(-0.758066\pi\)
0.959058 + 0.283211i \(0.0913998\pi\)
\(42\) 0 0
\(43\) 0.500000 + 0.866025i 0.0762493 + 0.132068i 0.901629 0.432511i \(-0.142372\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) 0 0
\(45\) −4.50000 + 7.79423i −0.670820 + 1.16190i
\(46\) 0 0
\(47\) −4.50000 7.79423i −0.656392 1.13691i −0.981543 0.191243i \(-0.938748\pi\)
0.325150 0.945662i \(-0.394585\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 10.3923i 1.45521i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) 0 0
\(59\) −1.50000 + 2.59808i −0.195283 + 0.338241i −0.946993 0.321253i \(-0.895896\pi\)
0.751710 + 0.659494i \(0.229229\pi\)
\(60\) 0 0
\(61\) −6.50000 11.2583i −0.832240 1.44148i −0.896258 0.443533i \(-0.853725\pi\)
0.0640184 0.997949i \(-0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.50000 + 2.59808i 0.186052 + 0.322252i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 4.50000 + 2.59808i 0.541736 + 0.312772i
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 6.00000 3.46410i 0.692820 0.400000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50000 9.52628i −0.618798 1.07179i −0.989705 0.143120i \(-0.954286\pi\)
0.370907 0.928670i \(-0.379047\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −4.50000 7.79423i −0.493939 0.855528i 0.506036 0.862512i \(-0.331110\pi\)
−0.999976 + 0.00698436i \(0.997777\pi\)
\(84\) 0 0
\(85\) −9.00000 + 15.5885i −0.976187 + 1.69081i
\(86\) 0 0
\(87\) 4.50000 2.59808i 0.482451 0.278543i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.50000 + 4.33013i 0.777714 + 0.449013i
\(94\) 0 0
\(95\) 6.00000 10.3923i 0.615587 1.06623i
\(96\) 0 0
\(97\) 5.50000 + 9.52628i 0.558440 + 0.967247i 0.997627 + 0.0688512i \(0.0219334\pi\)
−0.439187 + 0.898396i \(0.644733\pi\)
\(98\) 0 0
\(99\) 4.50000 + 7.79423i 0.452267 + 0.783349i
\(100\) 0 0
\(101\) 7.50000 + 12.9904i 0.746278 + 1.29259i 0.949595 + 0.313478i \(0.101494\pi\)
−0.203317 + 0.979113i \(0.565172\pi\)
\(102\) 0 0
\(103\) −3.50000 + 6.06218i −0.344865 + 0.597324i −0.985329 0.170664i \(-0.945409\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 3.46410i 0.328798i
\(112\) 0 0
\(113\) 4.50000 7.79423i 0.423324 0.733219i −0.572938 0.819599i \(-0.694196\pi\)
0.996262 + 0.0863794i \(0.0275297\pi\)
\(114\) 0 0
\(115\) −4.50000 7.79423i −0.419627 0.726816i
\(116\) 0 0
\(117\) 1.50000 2.59808i 0.138675 0.240192i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 4.50000 + 2.59808i 0.405751 + 0.234261i
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −1.50000 + 0.866025i −0.132068 + 0.0762493i
\(130\) 0 0
\(131\) 10.5000 18.1865i 0.917389 1.58896i 0.114024 0.993478i \(-0.463626\pi\)
0.803365 0.595487i \(-0.203041\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −13.5000 7.79423i −1.16190 0.670820i
\(136\) 0 0
\(137\) −1.50000 2.59808i −0.128154 0.221969i 0.794808 0.606861i \(-0.207572\pi\)
−0.922961 + 0.384893i \(0.874238\pi\)
\(138\) 0 0
\(139\) 2.50000 4.33013i 0.212047 0.367277i −0.740308 0.672268i \(-0.765320\pi\)
0.952355 + 0.304991i \(0.0986536\pi\)
\(140\) 0 0
\(141\) 13.5000 7.79423i 1.13691 0.656392i
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.50000 + 12.9904i −0.614424 + 1.06421i 0.376061 + 0.926595i \(0.377278\pi\)
−0.990485 + 0.137619i \(0.956055\pi\)
\(150\) 0 0
\(151\) 6.50000 + 11.2583i 0.528962 + 0.916190i 0.999430 + 0.0337724i \(0.0107521\pi\)
−0.470467 + 0.882418i \(0.655915\pi\)
\(152\) 0 0
\(153\) 18.0000 1.45521
\(154\) 0 0
\(155\) −7.50000 12.9904i −0.602414 1.04341i
\(156\) 0 0
\(157\) −6.50000 + 11.2583i −0.518756 + 0.898513i 0.481006 + 0.876717i \(0.340272\pi\)
−0.999762 + 0.0217953i \(0.993062\pi\)
\(158\) 0 0
\(159\) 10.3923i 0.824163i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 15.5885i 1.21356i
\(166\) 0 0
\(167\) 4.50000 7.79423i 0.348220 0.603136i −0.637713 0.770274i \(-0.720119\pi\)
0.985933 + 0.167139i \(0.0534527\pi\)
\(168\) 0 0
\(169\) 6.00000 + 10.3923i 0.461538 + 0.799408i
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) 0 0
\(173\) −4.50000 7.79423i −0.342129 0.592584i 0.642699 0.766119i \(-0.277815\pi\)
−0.984828 + 0.173534i \(0.944481\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.50000 2.59808i −0.338241 0.195283i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 19.5000 11.2583i 1.44148 0.832240i
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) 9.00000 + 15.5885i 0.658145 + 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.50000 12.9904i −0.542681 0.939951i −0.998749 0.0500060i \(-0.984076\pi\)
0.456068 0.889945i \(-0.349257\pi\)
\(192\) 0 0
\(193\) −5.50000 + 9.52628i −0.395899 + 0.685717i −0.993215 0.116289i \(-0.962900\pi\)
0.597317 + 0.802005i \(0.296234\pi\)
\(194\) 0 0
\(195\) −4.50000 + 2.59808i −0.322252 + 0.186052i
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 10.5000 + 6.06218i 0.740613 + 0.427593i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.50000 7.79423i −0.314294 0.544373i
\(206\) 0 0
\(207\) −4.50000 + 7.79423i −0.312772 + 0.541736i
\(208\) 0 0
\(209\) −6.00000 10.3923i −0.415029 0.718851i
\(210\) 0 0
\(211\) −8.50000 + 14.7224i −0.585164 + 1.01353i 0.409691 + 0.912224i \(0.365637\pi\)
−0.994855 + 0.101310i \(0.967697\pi\)
\(212\) 0 0
\(213\) 20.7846i 1.42414i
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 17.3205i 1.17041i
\(220\) 0 0
\(221\) 3.00000 5.19615i 0.201802 0.349531i
\(222\) 0 0
\(223\) −0.500000 0.866025i −0.0334825 0.0579934i 0.848799 0.528716i \(-0.177326\pi\)
−0.882281 + 0.470723i \(0.843993\pi\)
\(224\) 0 0
\(225\) 6.00000 + 10.3923i 0.400000 + 0.692820i
\(226\) 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
\(229\) −6.50000 + 11.2583i −0.429532 + 0.743971i −0.996832 0.0795401i \(-0.974655\pi\)
0.567300 + 0.823511i \(0.307988\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −27.0000 −1.76129
\(236\) 0 0
\(237\) 16.5000 9.52628i 1.07179 0.618798i
\(238\) 0 0
\(239\) 13.5000 23.3827i 0.873242 1.51250i 0.0146191 0.999893i \(-0.495346\pi\)
0.858623 0.512607i \(-0.171320\pi\)
\(240\) 0 0
\(241\) −0.500000 0.866025i −0.0322078 0.0557856i 0.849472 0.527633i \(-0.176921\pi\)
−0.881680 + 0.471848i \(0.843587\pi\)
\(242\) 0 0
\(243\) 15.5885i 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 + 3.46410i −0.127257 + 0.220416i
\(248\) 0 0
\(249\) 13.5000 7.79423i 0.855528 0.493939i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) −27.0000 15.5885i −1.69081 0.976187i
\(256\) 0 0
\(257\) −4.50000 + 7.79423i −0.280702 + 0.486191i −0.971558 0.236802i \(-0.923901\pi\)
0.690856 + 0.722993i \(0.257234\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 4.50000 + 7.79423i 0.278543 + 0.482451i
\(262\) 0 0
\(263\) 10.5000 + 18.1865i 0.647458 + 1.12143i 0.983728 + 0.179664i \(0.0575011\pi\)
−0.336270 + 0.941766i \(0.609166\pi\)
\(264\) 0 0
\(265\) −9.00000 + 15.5885i −0.552866 + 0.957591i
\(266\) 0 0
\(267\) 10.3923i 0.635999i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 + 10.3923i −0.361814 + 0.626680i
\(276\) 0 0
\(277\) 0.500000 + 0.866025i 0.0300421 + 0.0520344i 0.880656 0.473757i \(-0.157103\pi\)
−0.850613 + 0.525792i \(0.823769\pi\)
\(278\) 0 0
\(279\) −7.50000 + 12.9904i −0.449013 + 0.777714i
\(280\) 0 0
\(281\) −1.50000 2.59808i −0.0894825 0.154988i 0.817810 0.575488i \(-0.195188\pi\)
−0.907293 + 0.420500i \(0.861855\pi\)
\(282\) 0 0
\(283\) 2.50000 4.33013i 0.148610 0.257399i −0.782104 0.623148i \(-0.785854\pi\)
0.930714 + 0.365748i \(0.119187\pi\)
\(284\) 0 0
\(285\) 18.0000 + 10.3923i 1.06623 + 0.615587i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.5000 + 9.52628i −0.967247 + 0.558440i
\(292\) 0 0
\(293\) −10.5000 + 18.1865i −0.613417 + 1.06247i 0.377244 + 0.926114i \(0.376872\pi\)
−0.990660 + 0.136355i \(0.956461\pi\)
\(294\) 0 0
\(295\) 4.50000 + 7.79423i 0.262000 + 0.453798i
\(296\) 0 0
\(297\) −13.5000 + 7.79423i −0.783349 + 0.452267i
\(298\) 0 0
\(299\) 1.50000 + 2.59808i 0.0867472 + 0.150251i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −22.5000 + 12.9904i −1.29259 + 0.746278i
\(304\) 0 0
\(305\) −39.0000 −2.23313
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −10.5000 6.06218i −0.597324 0.344865i
\(310\) 0 0
\(311\) 10.5000 18.1865i 0.595400 1.03126i −0.398090 0.917346i \(-0.630327\pi\)
0.993490 0.113917i \(-0.0363399\pi\)
\(312\) 0 0
\(313\) −0.500000 0.866025i −0.0282617 0.0489506i 0.851549 0.524276i \(-0.175664\pi\)
−0.879810 + 0.475325i \(0.842331\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.5000 + 18.1865i 0.589739 + 1.02146i 0.994266 + 0.106932i \(0.0341026\pi\)
−0.404528 + 0.914526i \(0.632564\pi\)
\(318\) 0 0
\(319\) −4.50000 + 7.79423i −0.251952 + 0.436393i
\(320\) 0 0
\(321\) 20.7846i 1.16008i
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 3.46410i 0.191565i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.50000 9.52628i −0.302307 0.523612i 0.674351 0.738411i \(-0.264424\pi\)
−0.976658 + 0.214799i \(0.931090\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) −10.5000 18.1865i −0.573676 0.993636i
\(336\) 0 0
\(337\) −11.5000 + 19.9186i −0.626445 + 1.08503i 0.361815 + 0.932250i \(0.382157\pi\)
−0.988260 + 0.152784i \(0.951176\pi\)
\(338\) 0 0
\(339\) 13.5000 + 7.79423i 0.733219 + 0.423324i
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 13.5000 7.79423i 0.726816 0.419627i
\(346\) 0 0
\(347\) −4.50000 + 7.79423i −0.241573 + 0.418416i −0.961162 0.275983i \(-0.910997\pi\)
0.719590 + 0.694399i \(0.244330\pi\)
\(348\) 0 0
\(349\) −0.500000 0.866025i −0.0267644 0.0463573i 0.852333 0.523000i \(-0.175187\pi\)
−0.879097 + 0.476642i \(0.841854\pi\)
\(350\) 0 0
\(351\) 4.50000 + 2.59808i 0.240192 + 0.138675i
\(352\) 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
\(355\) −18.0000 + 31.1769i −0.955341 + 1.65470i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 3.00000 + 1.73205i 0.157459 + 0.0909091i
\(364\) 0 0
\(365\) 15.0000 25.9808i 0.785136 1.35990i
\(366\) 0 0
\(367\) −6.50000 11.2583i −0.339297 0.587680i 0.645003 0.764180i \(-0.276856\pi\)
−0.984301 + 0.176500i \(0.943523\pi\)
\(368\) 0 0
\(369\) −4.50000 + 7.79423i −0.234261 + 0.405751i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.500000 0.866025i 0.0258890 0.0448411i −0.852791 0.522253i \(-0.825092\pi\)
0.878680 + 0.477412i \(0.158425\pi\)
\(374\) 0 0
\(375\) 5.19615i 0.268328i
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 27.7128i 1.41977i
\(382\) 0 0
\(383\) −7.50000 + 12.9904i −0.383232 + 0.663777i −0.991522 0.129937i \(-0.958522\pi\)
0.608290 + 0.793715i \(0.291856\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.50000 2.59808i −0.0762493 0.132068i
\(388\) 0 0
\(389\) −7.50000 12.9904i −0.380265 0.658638i 0.610835 0.791758i \(-0.290834\pi\)
−0.991100 + 0.133120i \(0.957501\pi\)
\(390\) 0 0
\(391\) −9.00000 + 15.5885i −0.455150 + 0.788342i
\(392\) 0 0
\(393\) 31.5000 + 18.1865i 1.58896 + 0.917389i
\(394\) 0 0
\(395\) −33.0000 −1.66041
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.50000 + 2.59808i −0.0749064 + 0.129742i −0.901046 0.433724i \(-0.857199\pi\)
0.826139 + 0.563466i \(0.190532\pi\)
\(402\) 0 0
\(403\) 2.50000 + 4.33013i 0.124534 + 0.215699i
\(404\) 0 0
\(405\) 13.5000 23.3827i 0.670820 1.16190i
\(406\) 0 0
\(407\) −3.00000 5.19615i −0.148704 0.257564i
\(408\) 0 0
\(409\) 11.5000 19.9186i 0.568638 0.984911i −0.428063 0.903749i \(-0.640804\pi\)
0.996701 0.0811615i \(-0.0258630\pi\)
\(410\) 0 0
\(411\) 4.50000 2.59808i 0.221969 0.128154i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −27.0000 −1.32538
\(416\) 0 0
\(417\) 7.50000 + 4.33013i 0.367277 + 0.212047i
\(418\) 0 0
\(419\) 4.50000 7.79423i 0.219839 0.380773i −0.734919 0.678155i \(-0.762780\pi\)
0.954759 + 0.297382i \(0.0961133\pi\)
\(420\) 0 0
\(421\) −17.5000 30.3109i −0.852898 1.47726i −0.878582 0.477592i \(-0.841510\pi\)
0.0256838 0.999670i \(-0.491824\pi\)
\(422\) 0 0
\(423\) 13.5000 + 23.3827i 0.656392 + 1.13691i
\(424\) 0 0
\(425\) 12.0000 + 20.7846i 0.582086 + 1.00820i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5.19615i 0.250873i
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 15.5885i 0.747409i
\(436\) 0 0
\(437\) 6.00000 10.3923i 0.287019 0.497131i
\(438\) 0 0
\(439\) 17.5000 + 30.3109i 0.835229 + 1.44666i 0.893843 + 0.448379i \(0.147999\pi\)
−0.0586141 + 0.998281i \(0.518668\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.50000 + 7.79423i 0.213801 + 0.370315i 0.952901 0.303281i \(-0.0980821\pi\)
−0.739100 + 0.673596i \(0.764749\pi\)
\(444\) 0 0
\(445\) −9.00000 + 15.5885i −0.426641 + 0.738964i
\(446\) 0 0
\(447\) −22.5000 12.9904i −1.06421 0.614424i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) −19.5000 + 11.2583i −0.916190 + 0.528962i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.5000 + 32.0429i 0.865393 + 1.49891i 0.866656 + 0.498906i \(0.166265\pi\)
−0.00126243 + 0.999999i \(0.500402\pi\)
\(458\) 0 0
\(459\) 31.1769i 1.45521i
\(460\) 0 0
\(461\) 1.50000 + 2.59808i 0.0698620 + 0.121004i 0.898840 0.438276i \(-0.144411\pi\)
−0.828978 + 0.559281i \(0.811077\pi\)
\(462\) 0 0
\(463\) 9.50000 16.4545i 0.441502 0.764705i −0.556299 0.830982i \(-0.687779\pi\)
0.997801 + 0.0662777i \(0.0211123\pi\)
\(464\) 0 0
\(465\) 22.5000 12.9904i 1.04341 0.602414i
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −19.5000 11.2583i −0.898513 0.518756i
\(472\) 0 0
\(473\) 1.50000 2.59808i 0.0689701 0.119460i
\(474\) 0 0
\(475\) −8.00000 13.8564i −0.367065 0.635776i
\(476\) 0 0
\(477\) 18.0000 0.824163
\(478\) 0 0
\(479\) 13.5000 + 23.3827i 0.616831 + 1.06838i 0.990060 + 0.140643i \(0.0449170\pi\)
−0.373230 + 0.927739i \(0.621750\pi\)
\(480\) 0 0
\(481\) −1.00000 + 1.73205i −0.0455961 + 0.0789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 33.0000 1.49845
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 34.6410i 1.56652i
\(490\) 0 0
\(491\) 1.50000 2.59808i 0.0676941 0.117250i −0.830192 0.557478i \(-0.811769\pi\)
0.897886 + 0.440228i \(0.145102\pi\)
\(492\) 0 0
\(493\) 9.00000 + 15.5885i 0.405340 + 0.702069i
\(494\) 0 0
\(495\) 27.0000 1.21356
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.50000 + 4.33013i −0.111915 + 0.193843i −0.916542 0.399937i \(-0.869032\pi\)
0.804627 + 0.593780i \(0.202365\pi\)
\(500\) 0 0
\(501\) 13.5000 + 7.79423i 0.603136 + 0.348220i
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) −18.0000 + 10.3923i −0.799408 + 0.461538i
\(508\) 0 0
\(509\) 19.5000 33.7750i 0.864322 1.49705i −0.00339621 0.999994i \(-0.501081\pi\)
0.867719 0.497056i \(-0.165586\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 20.7846i 0.917663i
\(514\) 0 0
\(515\) 10.5000 + 18.1865i 0.462685 + 0.801394i
\(516\) 0 0
\(517\) −13.5000 + 23.3827i −0.593729 + 1.02837i
\(518\) 0 0
\(519\) 13.5000 7.79423i 0.592584 0.342129i
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.0000 + 25.9808i −0.653410 + 1.13174i
\(528\) 0 0
\(529\) 7.00000 + 12.1244i 0.304348 + 0.527146i
\(530\) 0 0
\(531\) 4.50000 7.79423i 0.195283 0.338241i
\(532\) 0 0
\(533\) 1.50000 + 2.59808i 0.0649722 + 0.112535i
\(534\) 0 0
\(535\) 18.0000 31.1769i 0.778208 1.34790i
\(536\) 0 0
\(537\) 20.7846i 0.896922i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 3.46410i 0.148659i
\(544\) 0 0
\(545\) 3.00000 5.19615i 0.128506 0.222579i
\(546\) 0 0
\(547\) 6.50000 + 11.2583i 0.277920 + 0.481371i 0.970868 0.239616i \(-0.0770217\pi\)
−0.692948 + 0.720988i \(0.743688\pi\)
\(548\) 0 0
\(549\) 19.5000 + 33.7750i 0.832240 + 1.44148i
\(550\) 0 0
\(551\) −6.00000 10.3923i −0.255609 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 9.00000 + 5.19615i 0.382029 + 0.220564i
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −27.0000 + 15.5885i −1.13994 + 0.658145i
\(562\) 0 0
\(563\) 4.50000 7.79423i 0.189652 0.328488i −0.755482 0.655169i \(-0.772597\pi\)
0.945134 + 0.326682i \(0.105931\pi\)
\(564\) 0 0
\(565\) −13.5000 23.3827i −0.567949 0.983717i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.50000 12.9904i −0.314416 0.544585i 0.664897 0.746935i \(-0.268475\pi\)
−0.979313 + 0.202350i \(0.935142\pi\)
\(570\) 0 0
\(571\) 15.5000 26.8468i 0.648655 1.12350i −0.334790 0.942293i \(-0.608665\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) 22.5000 12.9904i 0.939951 0.542681i
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) −16.5000 9.52628i −0.685717 0.395899i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 + 15.5885i 0.372742 + 0.645608i
\(584\) 0 0
\(585\) −4.50000 7.79423i −0.186052 0.322252i
\(586\) 0 0
\(587\) 7.50000 + 12.9904i 0.309558 + 0.536170i 0.978266 0.207355i \(-0.0664855\pi\)
−0.668708 + 0.743525i \(0.733152\pi\)
\(588\) 0 0
\(589\) 10.0000 17.3205i 0.412043 0.713679i
\(590\) 0 0
\(591\) 10.3923i 0.427482i
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.92820i 0.283552i
\(598\) 0 0
\(599\) 19.5000 33.7750i 0.796748 1.38001i −0.124975 0.992160i \(-0.539885\pi\)
0.921723 0.387849i \(-0.126782\pi\)
\(600\) 0 0
\(601\) 17.5000 + 30.3109i 0.713840 + 1.23641i 0.963405 + 0.268049i \(0.0863789\pi\)
−0.249565 + 0.968358i \(0.580288\pi\)
\(602\) 0 0
\(603\) −10.5000 + 18.1865i −0.427593 + 0.740613i
\(604\) 0 0
\(605\) −3.00000 5.19615i −0.121967 0.211254i
\(606\) 0 0
\(607\) 20.5000 35.5070i 0.832069 1.44119i −0.0643251 0.997929i \(-0.520489\pi\)
0.896394 0.443257i \(-0.146177\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) 13.5000 7.79423i 0.544373 0.314294i
\(616\) 0 0
\(617\) −1.50000 + 2.59808i −0.0603877 + 0.104595i −0.894639 0.446790i \(-0.852567\pi\)
0.834251 + 0.551385i \(0.185900\pi\)
\(618\) 0 0
\(619\) −6.50000 11.2583i −0.261257 0.452510i 0.705319 0.708890i \(-0.250804\pi\)
−0.966576 + 0.256379i \(0.917470\pi\)
\(620\) 0 0
\(621\) −13.5000 7.79423i −0.541736 0.312772i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 18.0000 10.3923i 0.718851 0.415029i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −25.5000 14.7224i −1.01353 0.585164i
\(634\) 0 0
\(635\) −24.0000 + 41.5692i −0.952411 + 1.64962i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 36.0000 1.42414
\(640\) 0 0
\(641\) 16.5000 + 28.5788i 0.651711 + 1.12880i 0.982708 + 0.185164i \(0.0592817\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(642\) 0 0
\(643\) 20.5000 35.5070i 0.808441 1.40026i −0.105502 0.994419i \(-0.533645\pi\)
0.913943 0.405842i \(-0.133022\pi\)
\(644\) 0 0
\(645\) 5.19615i 0.204598i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.5000 18.1865i 0.410897 0.711694i −0.584091 0.811688i \(-0.698549\pi\)
0.994988 + 0.0999939i \(0.0318823\pi\)
\(654\) 0 0
\(655\) −31.5000 54.5596i −1.23081 2.13182i
\(656\) 0 0
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) 10.5000 + 18.1865i 0.409022 + 0.708447i 0.994780 0.102039i \(-0.0325366\pi\)
−0.585758 + 0.810486i \(0.699203\pi\)
\(660\) 0 0
\(661\) 5.50000 9.52628i 0.213925 0.370529i −0.739014 0.673690i \(-0.764708\pi\)
0.952940 + 0.303160i \(0.0980418\pi\)
\(662\) 0 0
\(663\) 9.00000 + 5.19615i 0.349531 + 0.201802i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) 1.50000 0.866025i 0.0579934 0.0334825i
\(670\) 0 0
\(671\) −19.5000 + 33.7750i −0.752789 + 1.30387i
\(672\) 0 0
\(673\) −5.50000 9.52628i −0.212009 0.367211i 0.740334 0.672239i \(-0.234667\pi\)
−0.952343 + 0.305028i \(0.901334\pi\)
\(674\) 0 0
\(675\) −18.0000 + 10.3923i −0.692820 + 0.400000i
\(676\) 0 0
\(677\) 7.50000 + 12.9904i 0.288248 + 0.499261i 0.973392 0.229147i \(-0.0735938\pi\)
−0.685143 + 0.728408i \(0.740260\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −40.5000 + 23.3827i −1.55196 + 0.896026i
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) −19.5000 11.2583i −0.743971 0.429532i
\(688\) 0 0
\(689\) 3.00000 5.19615i 0.114291 0.197958i
\(690\) 0 0
\(691\) −0.500000 0.866025i −0.0190209 0.0329452i 0.856358 0.516382i \(-0.172722\pi\)
−0.875379 + 0.483437i \(0.839388\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.50000 12.9904i −0.284491 0.492753i
\(696\) 0 0
\(697\) −9.00000 + 15.5885i −0.340899 + 0.590455i
\(698\) 0 0
\(699\) 10.3923i 0.393073i
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 46.7654i 1.76129i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12.5000 + 21.6506i 0.469447 + 0.813107i 0.999390 0.0349269i \(-0.0111198\pi\)
−0.529943 + 0.848034i \(0.677787\pi\)
\(710\) 0 0
\(711\) 16.5000 + 28.5788i 0.618798 + 1.07179i
\(712\) 0 0
\(713\) −7.50000 12.9904i −0.280877 0.486494i
\(714\) 0 0
\(715\) 4.50000 7.79423i 0.168290 0.291488i
\(716\) 0 0
\(717\) 40.5000 + 23.3827i 1.51250 + 0.873242i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.50000 0.866025i 0.0557856 0.0322078i
\(724\) 0 0
\(725\) −6.00000 + 10.3923i −0.222834 + 0.385961i
\(726\) 0 0
\(727\) −18.5000 32.0429i −0.686127 1.18841i −0.973081 0.230463i \(-0.925976\pi\)
0.286954 0.957944i \(-0.407357\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −3.00000 5.19615i −0.110959 0.192187i
\(732\) 0 0
\(733\) 11.5000 19.9186i 0.424762 0.735710i −0.571636 0.820507i \(-0.693691\pi\)
0.996398 + 0.0847976i \(0.0270244\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.0000 −0.773545
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) −6.00000 3.46410i −0.220416 0.127257i
\(742\) 0 0
\(743\) −4.50000 + 7.79423i −0.165089 + 0.285943i −0.936687 0.350168i \(-0.886124\pi\)
0.771598 + 0.636111i \(0.219458\pi\)
\(744\) 0 0
\(745\) 22.5000 + 38.9711i 0.824336 + 1.42779i
\(746\) 0 0
\(747\) 13.5000 + 23.3827i 0.493939 + 0.855528i
\(748\) 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
\(753\) 20.7846i 0.757433i
\(754\) 0 0
\(755\) 39.0000 1.41936
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 15.5885i 0.565825i
\(760\) 0 0
\(761\) 13.5000 23.3827i 0.489375 0.847622i −0.510551 0.859848i \(-0.670558\pi\)
0.999925 + 0.0122260i \(0.00389175\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 27.0000 46.7654i 0.976187 1.69081i
\(766\) 0 0
\(767\) −1.50000 2.59808i −0.0541619 0.0938111i
\(768\) 0 0
\(769\) −0.500000 + 0.866025i −0.0180305 + 0.0312297i −0.874900 0.484304i \(-0.839073\pi\)
0.856869 + 0.515534i \(0.172406\pi\)
\(770\) 0 0
\(771\) −13.5000 7.79423i −0.486191 0.280702i
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 10.3923i 0.214972 0.372343i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) −13.5000 + 7.79423i −0.482451 + 0.278543i
\(784\) 0 0
\(785\) 19.5000 + 33.7750i 0.695985 + 1.20548i
\(786\) 0 0
\(787\) −21.5000 + 37.2391i −0.766392 + 1.32743i 0.173115 + 0.984902i \(0.444617\pi\)
−0.939507 + 0.342529i \(0.888717\pi\)
\(788\) 0 0
\(789\) −31.5000 + 18.1865i −1.12143 + 0.647458i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13.0000 0.461644
\(794\) 0 0
\(795\) −27.0000 15.5885i −0.957591 0.552866i
\(796\) 0 0
\(797\) −4.50000 + 7.79423i −0.159398 + 0.276086i −0.934652 0.355564i \(-0.884289\pi\)
0.775254 + 0.631650i \(0.217622\pi\)
\(798\) 0 0
\(799\) 27.0000 + 46.7654i 0.955191 + 1.65444i
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) −15.0000 25.9808i −0.529339 0.916841i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.3923i 0.365826i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 13.8564i 0.485965i
\(814\) 0 0
\(815\) 30.0000 51.9615i 1.05085 1.82013i
\(816\) 0 0
\(817\) 2.00000 + 3.46410i 0.0699711 + 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.5000 44.1673i −0.889956 1.54145i −0.839926 0.542702i \(-0.817401\pi\)
−0.0500305 0.998748i \(-0.515932\pi\)
\(822\) 0 0
\(823\) 9.50000 16.4545i 0.331149 0.573567i −0.651588 0.758573i \(-0.725897\pi\)
0.982737 + 0.185006i \(0.0592303\pi\)
\(824\) 0 0
\(825\) −18.0000 10.3923i −0.626680 0.361814i
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) −1.50000 + 0.866025i −0.0520344 + 0.0300421i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −13.5000 23.3827i −0.467187 0.809191i
\(836\) 0 0
\(837\) −22.5000 12.9904i −0.777714 0.449013i
\(838\) 0 0
\(839\) −4.50000 7.79423i −0.155357 0.269087i 0.777832 0.628473i \(-0.216320\pi\)
−0.933189 + 0.359386i \(0.882986\pi\)
\(840\) 0 0
\(841\) 10.0000 17.3205i 0.344828 0.597259i
\(842\) 0 0
\(843\) 4.50000 2.59808i 0.154988 0.0894825i
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.50000 + 4.33013i 0.257399 + 0.148610i
\(850\) 0 0
\(851\) 3.00000 5.19615i 0.102839 0.178122i
\(852\) 0 0
\(853\) −6.50000 11.2583i −0.222556 0.385478i 0.733028 0.680199i \(-0.238107\pi\)
−0.955583 + 0.294721i \(0.904773\pi\)
\(854\) 0 0
\(855\) −18.0000 + 31.1769i −0.615587 + 1.06623i
\(856\) 0 0
\(857\) 13.5000 + 23.3827i 0.461151 + 0.798737i 0.999019 0.0442921i \(-0.0141032\pi\)
−0.537867 + 0.843029i \(0.680770\pi\)
\(858\) 0 0
\(859\) 20.5000 35.5070i 0.699451 1.21148i −0.269206 0.963083i \(-0.586761\pi\)
0.968657 0.248402i \(-0.0799054\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) 32.9090i 1.11765i
\(868\) 0 0
\(869\) −16.5000 + 28.5788i −0.559724 + 0.969471i
\(870\) 0 0
\(871\) 3.50000 + 6.06218i 0.118593 + 0.205409i
\(872\) 0 0
\(873\) −16.5000 28.5788i −0.558440 0.967247i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −11.5000 + 19.9186i −0.388327 + 0.672603i −0.992225 0.124459i \(-0.960280\pi\)
0.603897 + 0.797062i \(0.293614\pi\)
\(878\) 0 0
\(879\) −31.5000 18.1865i −1.06247 0.613417i
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) −13.5000 + 7.79423i −0.453798 + 0.262000i
\(886\) 0 0
\(887\) 10.5000 18.1865i 0.352555 0.610644i −0.634141 0.773217i \(-0.718646\pi\)
0.986696 + 0.162573i \(0.0519794\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −13.5000 23.3827i −0.452267 0.783349i
\(892\) 0 0
\(893\) −18.0000 31.1769i −0.602347 1.04330i
\(894\) 0 0
\(895\) 18.0000 31.1769i 0.601674 1.04213i
\(896\) 0 0
\(897\) −4.50000 + 2.59808i −0.150251 + 0.0867472i
\(898\) 0 0
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.00000 + 5.19615i −0.0997234 + 0.172726i
\(906\) 0 0
\(907\) −23.5000 40.7032i −0.780305 1.35153i −0.931764 0.363064i \(-0.881731\pi\)
0.151460 0.988463i \(-0.451603\pi\)
\(908\) 0 0
\(909\) −22.5000 38.9711i −0.746278 1.29259i
\(910\) 0 0
\(911\) 22.5000 + 38.9711i 0.745458 + 1.29117i 0.949980 + 0.312310i \(0.101103\pi\)
−0.204522 + 0.978862i \(0.565564\pi\)
\(912\) 0 0
\(913\) −13.5000 + 23.3827i −0.446785 + 0.773854i
\(914\) 0 0
\(915\) 67.5500i 2.23313i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 34.6410i 1.14146i
\(922\) 0 0
\(923\) 6.00000 10.3923i 0.197492 0.342067i
\(924\) 0 0
\(925\) −4.00000 6.92820i −0.131519 0.227798i
\(926\) 0 0
\(927\) 10.5000 18.1865i 0.344865 0.597324i
\(928\) 0 0
\(929\) 13.5000 + 23.3827i 0.442921 + 0.767161i 0.997905 0.0646999i \(-0.0206090\pi\)
−0.554984 + 0.831861i \(0.687276\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 31.5000 + 18.1865i 1.03126 + 0.595400i
\(934\) 0 0
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) 1.50000 0.866025i 0.0489506 0.0282617i
\(940\) 0 0
\(941\) −10.5000 + 18.1865i −0.342290 + 0.592864i −0.984858 0.173365i \(-0.944536\pi\)
0.642567 + 0.766229i \(0.277869\pi\)
\(942\) 0 0
\(943\) −4.50000 7.79423i −0.146540 0.253815i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.5000 23.3827i −0.438691 0.759835i 0.558898 0.829237i \(-0.311224\pi\)
−0.997589 + 0.0694014i \(0.977891\pi\)
\(948\) 0 0
\(949\) −5.00000 + 8.66025i −0.162307 + 0.281124i
\(950\) 0 0
\(951\) −31.5000 + 18.1865i −1.02146 + 0.589739i
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) −45.0000 −1.45617
\(956\) 0 0
\(957\) −13.5000 7.79423i −0.436393 0.251952i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 3.00000 + 5.19615i 0.0967742 + 0.167618i
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 0 0
\(965\) 16.5000 + 28.5788i 0.531154 + 0.919985i
\(966\) 0 0
\(967\) 21.5000 37.2391i 0.691393 1.19753i −0.279988 0.960003i \(-0.590331\pi\)
0.971381 0.237525i \(-0.0763362\pi\)
\(968\) 0 0
\(969\) 41.5692i 1.33540i
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 6.92820i 0.221880i
\(976\) 0 0
\(977\) 28.5000 49.3634i 0.911796 1.57928i 0.100270 0.994960i \(-0.468029\pi\)
0.811526 0.584316i \(-0.198637\pi\)
\(978\) 0 0
\(979\) 9.00000 + 15.5885i 0.287641 + 0.498209i
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 25.5000 + 44.1673i 0.813324 + 1.40872i 0.910525 + 0.413453i \(0.135677\pi\)
−0.0972017 + 0.995265i \(0.530989\pi\)
\(984\) 0 0
\(985\) 9.00000 15.5885i 0.286764 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 16.5000 9.52628i 0.523612 0.302307i
\(994\) 0 0
\(995\) 6.00000 10.3923i 0.190213 0.329458i
\(996\) 0 0
\(997\) −0.500000 0.866025i −0.0158352 0.0274273i 0.857999 0.513651i \(-0.171707\pi\)
−0.873834 + 0.486224i \(0.838374\pi\)
\(998\) 0 0
\(999\) 10.3923i 0.328798i
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 1764.2.j.b.589.1 2
3.2 odd 2 5292.2.j.a.1765.1 2
7.2 even 3 1764.2.l.a.949.1 2
7.3 odd 6 1764.2.i.a.373.1 2
7.4 even 3 1764.2.i.c.373.1 2
7.5 odd 6 1764.2.l.c.949.1 2
7.6 odd 2 36.2.e.a.13.1 2
9.2 odd 6 5292.2.j.a.3529.1 2
9.7 even 3 inner 1764.2.j.b.1177.1 2
21.2 odd 6 5292.2.l.c.361.1 2
21.5 even 6 5292.2.l.a.361.1 2
21.11 odd 6 5292.2.i.a.1549.1 2
21.17 even 6 5292.2.i.c.1549.1 2
21.20 even 2 108.2.e.a.37.1 2
28.27 even 2 144.2.i.a.49.1 2
35.13 even 4 900.2.s.b.49.2 4
35.27 even 4 900.2.s.b.49.1 4
35.34 odd 2 900.2.i.b.301.1 2
56.13 odd 2 576.2.i.f.193.1 2
56.27 even 2 576.2.i.e.193.1 2
63.2 odd 6 5292.2.i.a.2125.1 2
63.11 odd 6 5292.2.l.c.3313.1 2
63.13 odd 6 324.2.a.c.1.1 1
63.16 even 3 1764.2.i.c.1537.1 2
63.20 even 6 108.2.e.a.73.1 2
63.25 even 3 1764.2.l.a.961.1 2
63.34 odd 6 36.2.e.a.25.1 yes 2
63.38 even 6 5292.2.l.a.3313.1 2
63.41 even 6 324.2.a.a.1.1 1
63.47 even 6 5292.2.i.c.2125.1 2
63.52 odd 6 1764.2.l.c.961.1 2
63.61 odd 6 1764.2.i.a.1537.1 2
84.83 odd 2 432.2.i.c.145.1 2
105.62 odd 4 2700.2.s.b.1549.1 4
105.83 odd 4 2700.2.s.b.1549.2 4
105.104 even 2 2700.2.i.b.901.1 2
168.83 odd 2 1728.2.i.c.577.1 2
168.125 even 2 1728.2.i.d.577.1 2
252.83 odd 6 432.2.i.c.289.1 2
252.139 even 6 1296.2.a.k.1.1 1
252.167 odd 6 1296.2.a.b.1.1 1
252.223 even 6 144.2.i.a.97.1 2
315.13 even 12 8100.2.d.h.649.2 2
315.34 odd 6 900.2.i.b.601.1 2
315.83 odd 12 2700.2.s.b.2449.1 4
315.97 even 12 900.2.s.b.349.2 4
315.104 even 6 8100.2.a.g.1.1 1
315.139 odd 6 8100.2.a.j.1.1 1
315.167 odd 12 8100.2.d.c.649.1 2
315.202 even 12 8100.2.d.h.649.1 2
315.209 even 6 2700.2.i.b.1801.1 2
315.223 even 12 900.2.s.b.349.1 4
315.272 odd 12 2700.2.s.b.2449.2 4
315.293 odd 12 8100.2.d.c.649.2 2
504.13 odd 6 5184.2.a.e.1.1 1
504.83 odd 6 1728.2.i.c.1153.1 2
504.139 even 6 5184.2.a.f.1.1 1
504.293 even 6 5184.2.a.ba.1.1 1
504.349 odd 6 576.2.i.f.385.1 2
504.419 odd 6 5184.2.a.bb.1.1 1
504.461 even 6 1728.2.i.d.1153.1 2
504.475 even 6 576.2.i.e.385.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.2.e.a.13.1 2 7.6 odd 2
36.2.e.a.25.1 yes 2 63.34 odd 6
108.2.e.a.37.1 2 21.20 even 2
108.2.e.a.73.1 2 63.20 even 6
144.2.i.a.49.1 2 28.27 even 2
144.2.i.a.97.1 2 252.223 even 6
324.2.a.a.1.1 1 63.41 even 6
324.2.a.c.1.1 1 63.13 odd 6
432.2.i.c.145.1 2 84.83 odd 2
432.2.i.c.289.1 2 252.83 odd 6
576.2.i.e.193.1 2 56.27 even 2
576.2.i.e.385.1 2 504.475 even 6
576.2.i.f.193.1 2 56.13 odd 2
576.2.i.f.385.1 2 504.349 odd 6
900.2.i.b.301.1 2 35.34 odd 2
900.2.i.b.601.1 2 315.34 odd 6
900.2.s.b.49.1 4 35.27 even 4
900.2.s.b.49.2 4 35.13 even 4
900.2.s.b.349.1 4 315.223 even 12
900.2.s.b.349.2 4 315.97 even 12
1296.2.a.b.1.1 1 252.167 odd 6
1296.2.a.k.1.1 1 252.139 even 6
1728.2.i.c.577.1 2 168.83 odd 2
1728.2.i.c.1153.1 2 504.83 odd 6
1728.2.i.d.577.1 2 168.125 even 2
1728.2.i.d.1153.1 2 504.461 even 6
1764.2.i.a.373.1 2 7.3 odd 6
1764.2.i.a.1537.1 2 63.61 odd 6
1764.2.i.c.373.1 2 7.4 even 3
1764.2.i.c.1537.1 2 63.16 even 3
1764.2.j.b.589.1 2 1.1 even 1 trivial
1764.2.j.b.1177.1 2 9.7 even 3 inner
1764.2.l.a.949.1 2 7.2 even 3
1764.2.l.a.961.1 2 63.25 even 3
1764.2.l.c.949.1 2 7.5 odd 6
1764.2.l.c.961.1 2 63.52 odd 6
2700.2.i.b.901.1 2 105.104 even 2
2700.2.i.b.1801.1 2 315.209 even 6
2700.2.s.b.1549.1 4 105.62 odd 4
2700.2.s.b.1549.2 4 105.83 odd 4
2700.2.s.b.2449.1 4 315.83 odd 12
2700.2.s.b.2449.2 4 315.272 odd 12
5184.2.a.e.1.1 1 504.13 odd 6
5184.2.a.f.1.1 1 504.139 even 6
5184.2.a.ba.1.1 1 504.293 even 6
5184.2.a.bb.1.1 1 504.419 odd 6
5292.2.i.a.1549.1 2 21.11 odd 6
5292.2.i.a.2125.1 2 63.2 odd 6
5292.2.i.c.1549.1 2 21.17 even 6
5292.2.i.c.2125.1 2 63.47 even 6
5292.2.j.a.1765.1 2 3.2 odd 2
5292.2.j.a.3529.1 2 9.2 odd 6
5292.2.l.a.361.1 2 21.5 even 6
5292.2.l.a.3313.1 2 63.38 even 6
5292.2.l.c.361.1 2 21.2 odd 6
5292.2.l.c.3313.1 2 63.11 odd 6
8100.2.a.g.1.1 1 315.104 even 6
8100.2.a.j.1.1 1 315.139 odd 6
8100.2.d.c.649.1 2 315.167 odd 12
8100.2.d.c.649.2 2 315.293 odd 12
8100.2.d.h.649.1 2 315.202 even 12
8100.2.d.h.649.2 2 315.13 even 12