Properties

Label 1620.2.i.g.541.1
Level $1620$
Weight $2$
Character 1620.541
Analytic conductor $12.936$
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] = [1620,2,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.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: \(12.9357651274\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.2.i.g.1081.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(-1.50000 + 2.59808i) q^{11} +(2.00000 + 3.46410i) q^{13} +6.00000 q^{17} -7.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-1.50000 + 2.59808i) q^{29} +(-2.50000 - 4.33013i) q^{31} -2.00000 q^{35} -4.00000 q^{37} +(-1.50000 - 2.59808i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(1.50000 + 2.59808i) q^{49} -6.00000 q^{53} -3.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(-7.00000 + 12.1244i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-1.00000 - 1.73205i) q^{67} +15.0000 q^{71} -10.0000 q^{73} +(-3.00000 - 5.19615i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(3.00000 + 5.19615i) q^{85} +15.0000 q^{89} -8.00000 q^{91} +(-3.50000 - 6.06218i) q^{95} +(-4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} - 2 q^{7} - 3 q^{11} + 4 q^{13} + 12 q^{17} - 14 q^{19} - 6 q^{23} - q^{25} - 3 q^{29} - 5 q^{31} - 4 q^{35} - 8 q^{37} - 3 q^{41} - 8 q^{43} + 3 q^{49} - 12 q^{53} - 6 q^{55} + 3 q^{59} - 14 q^{61} - 4 q^{65} - 2 q^{67} + 30 q^{71} - 20 q^{73} - 6 q^{77} - 8 q^{79} + 6 q^{85} + 30 q^{89} - 16 q^{91} - 7 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 0
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) −1.00000 + 1.73205i −0.377964 + 0.654654i −0.990766 0.135583i \(-0.956709\pi\)
0.612801 + 0.790237i \(0.290043\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 0.238987i \(-0.923185\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(30\) 0 0
\(31\) −2.50000 4.33013i −0.449013 0.777714i 0.549309 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.50000 2.59808i −0.234261 0.405751i 0.724797 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) −7.00000 + 12.1244i −0.896258 + 1.55236i −0.0640184 + 0.997949i \(0.520392\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 + 3.46410i −0.248069 + 0.429669i
\(66\) 0 0
\(67\) −1.00000 1.73205i −0.122169 0.211604i 0.798454 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 5.19615i −0.341882 0.592157i
\(78\) 0 0
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 3.00000 + 5.19615i 0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.50000 6.06218i −0.359092 0.621966i
\(96\) 0 0
\(97\) −4.00000 + 6.92820i −0.406138 + 0.703452i −0.994453 0.105180i \(-0.966458\pi\)
0.588315 + 0.808632i \(0.299792\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.50000 2.59808i 0.149256 0.258518i −0.781697 0.623658i \(-0.785646\pi\)
0.930953 + 0.365140i \(0.118979\pi\)
\(102\) 0 0
\(103\) −1.00000 1.73205i −0.0985329 0.170664i 0.812545 0.582899i \(-0.198082\pi\)
−0.911078 + 0.412235i \(0.864748\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.00000 15.5885i −0.846649 1.46644i −0.884182 0.467143i \(-0.845283\pi\)
0.0375328 0.999295i \(-0.488050\pi\)
\(114\) 0 0
\(115\) 3.00000 5.19615i 0.279751 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 + 10.3923i −0.550019 + 0.952661i
\(120\) 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
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.50000 + 12.9904i 0.655278 + 1.13497i 0.981824 + 0.189794i \(0.0607819\pi\)
−0.326546 + 0.945181i \(0.605885\pi\)
\(132\) 0 0
\(133\) 7.00000 12.1244i 0.606977 1.05131i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) 0.500000 + 0.866025i 0.0424094 + 0.0734553i 0.886451 0.462822i \(-0.153163\pi\)
−0.844042 + 0.536278i \(0.819830\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 5.19615i −0.245770 0.425685i 0.716578 0.697507i \(-0.245707\pi\)
−0.962348 + 0.271821i \(0.912374\pi\)
\(150\) 0 0
\(151\) −2.50000 + 4.33013i −0.203447 + 0.352381i −0.949637 0.313353i \(-0.898548\pi\)
0.746190 + 0.665733i \(0.231881\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.50000 4.33013i 0.200805 0.347804i
\(156\) 0 0
\(157\) 2.00000 + 3.46410i 0.159617 + 0.276465i 0.934731 0.355357i \(-0.115641\pi\)
−0.775113 + 0.631822i \(0.782307\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 + 15.5885i 0.696441 + 1.20627i 0.969693 + 0.244328i \(0.0785675\pi\)
−0.273252 + 0.961943i \(0.588099\pi\)
\(168\) 0 0
\(169\) −1.50000 + 2.59808i −0.115385 + 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.00000 + 15.5885i −0.684257 + 1.18517i 0.289412 + 0.957205i \(0.406540\pi\)
−0.973670 + 0.227964i \(0.926793\pi\)
\(174\) 0 0
\(175\) −1.00000 1.73205i −0.0755929 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 23.0000 1.70958 0.854788 0.518977i \(-0.173687\pi\)
0.854788 + 0.518977i \(0.173687\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 3.46410i −0.147043 0.254686i
\(186\) 0 0
\(187\) −9.00000 + 15.5885i −0.658145 + 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.5000 18.1865i 0.759753 1.31593i −0.183223 0.983071i \(-0.558653\pi\)
0.942976 0.332860i \(-0.108014\pi\)
\(192\) 0 0
\(193\) 8.00000 + 13.8564i 0.575853 + 0.997406i 0.995948 + 0.0899262i \(0.0286631\pi\)
−0.420096 + 0.907480i \(0.638004\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 5.19615i −0.210559 0.364698i
\(204\) 0 0
\(205\) 1.50000 2.59808i 0.104765 0.181458i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.5000 18.1865i 0.726300 1.25799i
\(210\) 0 0
\(211\) −2.50000 4.33013i −0.172107 0.298098i 0.767049 0.641588i \(-0.221724\pi\)
−0.939156 + 0.343490i \(0.888391\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 + 20.7846i 0.807207 + 1.39812i
\(222\) 0 0
\(223\) 14.0000 24.2487i 0.937509 1.62381i 0.167412 0.985887i \(-0.446459\pi\)
0.770097 0.637927i \(-0.220208\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.00000 15.5885i 0.597351 1.03464i −0.395860 0.918311i \(-0.629553\pi\)
0.993210 0.116331i \(-0.0371134\pi\)
\(228\) 0 0
\(229\) −7.00000 12.1244i −0.462573 0.801200i 0.536515 0.843891i \(-0.319740\pi\)
−0.999088 + 0.0426906i \(0.986407\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 3.50000 6.06218i 0.225455 0.390499i −0.731001 0.682376i \(-0.760947\pi\)
0.956456 + 0.291877i \(0.0942799\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.50000 + 2.59808i −0.0958315 + 0.165985i
\(246\) 0 0
\(247\) −14.0000 24.2487i −0.890799 1.54291i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 + 15.5885i 0.561405 + 0.972381i 0.997374 + 0.0724199i \(0.0230722\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(258\) 0 0
\(259\) 4.00000 6.92820i 0.248548 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.00000 + 5.19615i −0.184988 + 0.320408i −0.943572 0.331166i \(-0.892558\pi\)
0.758585 + 0.651575i \(0.225891\pi\)
\(264\) 0 0
\(265\) −3.00000 5.19615i −0.184289 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.50000 2.59808i −0.0904534 0.156670i
\(276\) 0 0
\(277\) 14.0000 24.2487i 0.841178 1.45696i −0.0477206 0.998861i \(-0.515196\pi\)
0.888899 0.458103i \(-0.151471\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −15.0000 + 25.9808i −0.894825 + 1.54988i −0.0608039 + 0.998150i \(0.519366\pi\)
−0.834021 + 0.551733i \(0.813967\pi\)
\(282\) 0 0
\(283\) 11.0000 + 19.0526i 0.653882 + 1.13256i 0.982173 + 0.187980i \(0.0601941\pi\)
−0.328291 + 0.944577i \(0.606473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −12.0000 20.7846i −0.701047 1.21425i −0.968099 0.250568i \(-0.919383\pi\)
0.267052 0.963682i \(-0.413951\pi\)
\(294\) 0 0
\(295\) −1.50000 + 2.59808i −0.0873334 + 0.151266i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 20.7846i 0.693978 1.20201i
\(300\) 0 0
\(301\) −8.00000 13.8564i −0.461112 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.50000 + 12.9904i 0.425286 + 0.736617i 0.996447 0.0842210i \(-0.0268402\pi\)
−0.571161 + 0.820838i \(0.693507\pi\)
\(312\) 0 0
\(313\) 8.00000 13.8564i 0.452187 0.783210i −0.546335 0.837567i \(-0.683977\pi\)
0.998522 + 0.0543564i \(0.0173107\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 10.3923i 0.336994 0.583690i −0.646872 0.762598i \(-0.723923\pi\)
0.983866 + 0.178908i \(0.0572566\pi\)
\(318\) 0 0
\(319\) −4.50000 7.79423i −0.251952 0.436393i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −42.0000 −2.33694
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.50000 + 14.7224i −0.467202 + 0.809218i −0.999298 0.0374662i \(-0.988071\pi\)
0.532096 + 0.846684i \(0.321405\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 1.73205i 0.0546358 0.0946320i
\(336\) 0 0
\(337\) 14.0000 + 24.2487i 0.762629 + 1.32091i 0.941491 + 0.337037i \(0.109425\pi\)
−0.178863 + 0.983874i \(0.557242\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 15.5885i −0.483145 0.836832i 0.516667 0.856186i \(-0.327172\pi\)
−0.999813 + 0.0193540i \(0.993839\pi\)
\(348\) 0 0
\(349\) 3.50000 6.06218i 0.187351 0.324501i −0.757015 0.653397i \(-0.773343\pi\)
0.944366 + 0.328896i \(0.106677\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 20.7846i 0.638696 1.10625i −0.347024 0.937856i \(-0.612808\pi\)
0.985719 0.168397i \(-0.0538590\pi\)
\(354\) 0 0
\(355\) 7.50000 + 12.9904i 0.398059 + 0.689458i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.0000 1.42501 0.712503 0.701669i \(-0.247562\pi\)
0.712503 + 0.701669i \(0.247562\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.00000 8.66025i −0.261712 0.453298i
\(366\) 0 0
\(367\) 11.0000 19.0526i 0.574195 0.994535i −0.421933 0.906627i \(-0.638648\pi\)
0.996129 0.0879086i \(-0.0280183\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 10.3923i 0.311504 0.539542i
\(372\) 0 0
\(373\) 11.0000 + 19.0526i 0.569558 + 0.986504i 0.996610 + 0.0822766i \(0.0262191\pi\)
−0.427051 + 0.904227i \(0.640448\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 32.0000 1.64373 0.821865 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 + 20.7846i 0.613171 + 1.06204i 0.990702 + 0.136047i \(0.0434398\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(384\) 0 0
\(385\) 3.00000 5.19615i 0.152894 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.00000 + 5.19615i −0.152106 + 0.263455i −0.932002 0.362454i \(-0.881939\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(390\) 0 0
\(391\) −18.0000 31.1769i −0.910299 1.57668i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 + 5.19615i 0.149813 + 0.259483i 0.931158 0.364615i \(-0.118800\pi\)
−0.781345 + 0.624099i \(0.785466\pi\)
\(402\) 0 0
\(403\) 10.0000 17.3205i 0.498135 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 10.3923i 0.297409 0.515127i
\(408\) 0 0
\(409\) 5.00000 + 8.66025i 0.247234 + 0.428222i 0.962757 0.270367i \(-0.0871450\pi\)
−0.715523 + 0.698589i \(0.753812\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 + 31.1769i 0.879358 + 1.52309i 0.852047 + 0.523465i \(0.175361\pi\)
0.0273103 + 0.999627i \(0.491306\pi\)
\(420\) 0 0
\(421\) 3.50000 6.06218i 0.170580 0.295452i −0.768043 0.640398i \(-0.778769\pi\)
0.938623 + 0.344946i \(0.112103\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 + 5.19615i −0.145521 + 0.252050i
\(426\) 0 0
\(427\) −14.0000 24.2487i −0.677507 1.17348i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.0000 + 36.3731i 1.00457 + 1.73996i
\(438\) 0 0
\(439\) 9.50000 16.4545i 0.453410 0.785330i −0.545185 0.838316i \(-0.683541\pi\)
0.998595 + 0.0529862i \(0.0168739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 7.50000 + 12.9904i 0.355534 + 0.615803i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.00000 6.92820i −0.187523 0.324799i
\(456\) 0 0
\(457\) −4.00000 + 6.92820i −0.187112 + 0.324088i −0.944286 0.329125i \(-0.893246\pi\)
0.757174 + 0.653213i \(0.226579\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.50000 12.9904i 0.349310 0.605022i −0.636817 0.771015i \(-0.719749\pi\)
0.986127 + 0.165992i \(0.0530827\pi\)
\(462\) 0 0
\(463\) −7.00000 12.1244i −0.325318 0.563467i 0.656259 0.754536i \(-0.272138\pi\)
−0.981577 + 0.191069i \(0.938805\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 20.7846i −0.551761 0.955677i
\(474\) 0 0
\(475\) 3.50000 6.06218i 0.160591 0.278152i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13.5000 + 23.3827i −0.616831 + 1.06838i 0.373230 + 0.927739i \(0.378250\pi\)
−0.990060 + 0.140643i \(0.955083\pi\)
\(480\) 0 0
\(481\) −8.00000 13.8564i −0.364769 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.5000 + 33.7750i 0.880023 + 1.52424i 0.851314 + 0.524656i \(0.175806\pi\)
0.0287085 + 0.999588i \(0.490861\pi\)
\(492\) 0 0
\(493\) −9.00000 + 15.5885i −0.405340 + 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.0000 + 25.9808i −0.672842 + 1.16540i
\(498\) 0 0
\(499\) −2.50000 4.33013i −0.111915 0.193843i 0.804627 0.593780i \(-0.202365\pi\)
−0.916542 + 0.399937i \(0.869032\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.0000 25.9808i −0.664863 1.15158i −0.979322 0.202306i \(-0.935156\pi\)
0.314459 0.949271i \(-0.398177\pi\)
\(510\) 0 0
\(511\) 10.0000 17.3205i 0.442374 0.766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.00000 1.73205i 0.0440653 0.0763233i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.0000 25.9808i −0.653410 1.13174i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 10.3923i 0.259889 0.450141i
\(534\) 0 0
\(535\) −3.00000 5.19615i −0.129701 0.224649i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.50000 + 9.52628i 0.235594 + 0.408061i
\(546\) 0 0
\(547\) −16.0000 + 27.7128i −0.684111 + 1.18491i 0.289605 + 0.957146i \(0.406476\pi\)
−0.973715 + 0.227768i \(0.926857\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.5000 18.1865i 0.447315 0.774772i
\(552\) 0 0
\(553\) −8.00000 13.8564i −0.340195 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0000 31.1769i −0.758610 1.31395i −0.943560 0.331202i \(-0.892546\pi\)
0.184950 0.982748i \(-0.440788\pi\)
\(564\) 0 0
\(565\) 9.00000 15.5885i 0.378633 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.50000 2.59808i 0.0628833 0.108917i −0.832870 0.553469i \(-0.813304\pi\)
0.895753 + 0.444552i \(0.146637\pi\)
\(570\) 0 0
\(571\) 18.5000 + 32.0429i 0.774201 + 1.34096i 0.935243 + 0.354008i \(0.115181\pi\)
−0.161042 + 0.986948i \(0.551485\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 15.5885i 0.372742 0.645608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 10.3923i 0.247647 0.428936i −0.715226 0.698893i \(-0.753676\pi\)
0.962872 + 0.269957i \(0.0870095\pi\)
\(588\) 0 0
\(589\) 17.5000 + 30.3109i 0.721075 + 1.24894i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.5000 + 33.7750i 0.796748 + 1.38001i 0.921723 + 0.387849i \(0.126782\pi\)
−0.124975 + 0.992160i \(0.539885\pi\)
\(600\) 0 0
\(601\) −8.50000 + 14.7224i −0.346722 + 0.600541i −0.985665 0.168714i \(-0.946039\pi\)
0.638943 + 0.769254i \(0.279372\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 + 1.73205i −0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) −4.00000 6.92820i −0.162355 0.281207i 0.773358 0.633970i \(-0.218576\pi\)
−0.935713 + 0.352763i \(0.885242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 + 20.7846i 0.483102 + 0.836757i 0.999812 0.0194037i \(-0.00617676\pi\)
−0.516710 + 0.856161i \(0.672843\pi\)
\(618\) 0 0
\(619\) 2.00000 3.46410i 0.0803868 0.139234i −0.823029 0.567999i \(-0.807718\pi\)
0.903416 + 0.428765i \(0.141051\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15.0000 + 25.9808i −0.600962 + 1.04090i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00000 + 1.73205i 0.0396838 + 0.0687343i
\(636\) 0 0
\(637\) −6.00000 + 10.3923i −0.237729 + 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5000 28.5788i 0.651711 1.12880i −0.330997 0.943632i \(-0.607385\pi\)
0.982708 0.185164i \(-0.0592817\pi\)
\(642\) 0 0
\(643\) 20.0000 + 34.6410i 0.788723 + 1.36611i 0.926750 + 0.375680i \(0.122591\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12.0000 20.7846i −0.469596 0.813365i 0.529799 0.848123i \(-0.322267\pi\)
−0.999396 + 0.0347583i \(0.988934\pi\)
\(654\) 0 0
\(655\) −7.50000 + 12.9904i −0.293049 + 0.507576i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 + 20.7846i −0.467454 + 0.809653i −0.999309 0.0371821i \(-0.988162\pi\)
0.531855 + 0.846836i \(0.321495\pi\)
\(660\) 0 0
\(661\) 3.50000 + 6.06218i 0.136134 + 0.235791i 0.926030 0.377450i \(-0.123199\pi\)
−0.789896 + 0.613241i \(0.789865\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.0000 0.542897
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −21.0000 36.3731i −0.810696 1.40417i
\(672\) 0 0
\(673\) 5.00000 8.66025i 0.192736 0.333828i −0.753420 0.657539i \(-0.771597\pi\)
0.946156 + 0.323711i \(0.104931\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.00000 + 5.19615i −0.115299 + 0.199704i −0.917899 0.396813i \(-0.870116\pi\)
0.802600 + 0.596518i \(0.203449\pi\)
\(678\) 0 0
\(679\) −8.00000 13.8564i −0.307012 0.531760i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 20.7846i −0.457164 0.791831i
\(690\) 0 0
\(691\) −22.0000 + 38.1051i −0.836919 + 1.44959i 0.0555386 + 0.998457i \(0.482312\pi\)
−0.892458 + 0.451130i \(0.851021\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.500000 + 0.866025i −0.0189661 + 0.0328502i
\(696\) 0 0
\(697\) −9.00000 15.5885i −0.340899 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.00000 + 5.19615i 0.112827 + 0.195421i
\(708\) 0 0
\(709\) −13.0000 + 22.5167i −0.488225 + 0.845631i −0.999908 0.0135434i \(-0.995689\pi\)
0.511683 + 0.859174i \(0.329022\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.0000 + 25.9808i −0.561754 + 0.972987i
\(714\) 0 0
\(715\) −6.00000 10.3923i −0.224387 0.388650i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.00000 0.335643 0.167822 0.985817i \(-0.446327\pi\)
0.167822 + 0.985817i \(0.446327\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.50000 2.59808i −0.0557086 0.0964901i
\(726\) 0 0
\(727\) −22.0000 + 38.1051i −0.815935 + 1.41324i 0.0927199 + 0.995692i \(0.470444\pi\)
−0.908655 + 0.417548i \(0.862889\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.0000 + 41.5692i −0.887672 + 1.53749i
\(732\) 0 0
\(733\) −7.00000 12.1244i −0.258551 0.447823i 0.707303 0.706910i \(-0.249912\pi\)
−0.965854 + 0.259087i \(0.916578\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) −25.0000 −0.919640 −0.459820 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −27.0000 46.7654i −0.990534 1.71566i −0.614145 0.789193i \(-0.710499\pi\)
−0.376389 0.926462i \(-0.622834\pi\)
\(744\) 0 0
\(745\) 3.00000 5.19615i 0.109911 0.190372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.00000 10.3923i 0.219235 0.379727i
\(750\) 0 0
\(751\) −4.00000 6.92820i −0.145962 0.252814i 0.783769 0.621052i \(-0.213294\pi\)
−0.929731 + 0.368238i \(0.879961\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.50000 7.79423i −0.163125 0.282541i 0.772863 0.634573i \(-0.218824\pi\)
−0.935988 + 0.352032i \(0.885491\pi\)
\(762\) 0 0
\(763\) −11.0000 + 19.0526i −0.398227 + 0.689749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.00000 + 10.3923i −0.216647 + 0.375244i
\(768\) 0 0
\(769\) −20.5000 35.5070i −0.739249 1.28042i −0.952834 0.303492i \(-0.901847\pi\)
0.213585 0.976924i \(-0.431486\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.5000 + 18.1865i 0.376202 + 0.651600i
\(780\) 0 0
\(781\) −22.5000 + 38.9711i −0.805113 + 1.39450i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 + 3.46410i −0.0713831 + 0.123639i
\(786\) 0 0
\(787\) 17.0000 + 29.4449i 0.605985 + 1.04960i 0.991895 + 0.127060i \(0.0405540\pi\)
−0.385911 + 0.922536i \(0.626113\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) −56.0000 −1.98862
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.0000 25.9808i 0.529339 0.916841i
\(804\) 0 0
\(805\) 6.00000 + 10.3923i 0.211472 + 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.0000 19.0526i −0.385313 0.667382i
\(816\) 0 0
\(817\) 28.0000 48.4974i 0.979596 1.69671i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.50000 + 12.9904i −0.261752 + 0.453367i −0.966708 0.255884i \(-0.917634\pi\)
0.704956 + 0.709251i \(0.250967\pi\)
\(822\) 0 0
\(823\) −7.00000 12.1244i −0.244005 0.422628i 0.717847 0.696201i \(-0.245128\pi\)
−0.961851 + 0.273573i \(0.911795\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.00000 + 15.5885i 0.311832 + 0.540108i
\(834\) 0 0
\(835\) −9.00000 + 15.5885i −0.311458 + 0.539461i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.50000 7.79423i 0.155357 0.269087i −0.777832 0.628473i \(-0.783680\pi\)
0.933189 + 0.359386i \(0.117014\pi\)
\(840\) 0 0
\(841\) 10.0000 + 17.3205i 0.344828 + 0.597259i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 + 20.7846i 0.411355 + 0.712487i
\(852\) 0 0
\(853\) −7.00000 + 12.1244i −0.239675 + 0.415130i −0.960621 0.277862i \(-0.910374\pi\)
0.720946 + 0.692992i \(0.243708\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) 27.5000 + 47.6314i 0.938288 + 1.62516i 0.768663 + 0.639654i \(0.220922\pi\)
0.169625 + 0.985509i \(0.445745\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.0000 20.7846i −0.407072 0.705070i
\(870\) 0 0
\(871\) 4.00000 6.92820i 0.135535 0.234753i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 1.73205i 0.0338062 0.0585540i
\(876\) 0 0
\(877\) 14.0000 + 24.2487i 0.472746 + 0.818821i 0.999514 0.0311889i \(-0.00992933\pi\)
−0.526767 + 0.850010i \(0.676596\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 41.5692i −0.805841 1.39576i −0.915722 0.401813i \(-0.868380\pi\)
0.109881 0.993945i \(-0.464953\pi\)
\(888\) 0 0
\(889\) −2.00000 + 3.46410i −0.0670778 + 0.116182i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −7.50000 12.9904i −0.250697 0.434221i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.5000 + 19.9186i 0.382273 + 0.662116i
\(906\) 0 0
\(907\) −13.0000 + 22.5167i −0.431658 + 0.747653i −0.997016 0.0771920i \(-0.975405\pi\)
0.565358 + 0.824845i \(0.308738\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −22.5000 + 38.9711i −0.745458 + 1.29117i 0.204522 + 0.978862i \(0.434436\pi\)
−0.949980 + 0.312310i \(0.898897\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −30.0000 −0.990687
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.0000 + 51.9615i 0.987462 + 1.71033i
\(924\) 0 0
\(925\) 2.00000 3.46410i 0.0657596 0.113899i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.50000 + 7.79423i −0.147640 + 0.255720i −0.930355 0.366660i \(-0.880501\pi\)
0.782715 + 0.622381i \(0.213834\pi\)
\(930\) 0 0
\(931\) −10.5000 18.1865i −0.344124 0.596040i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) −58.0000 −1.89478 −0.947389 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.00000 5.19615i −0.0977972 0.169390i 0.812975 0.582298i \(-0.197846\pi\)
−0.910773 + 0.412908i \(0.864513\pi\)
\(942\) 0 0
\(943\) −9.00000 + 15.5885i −0.293080 + 0.507630i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000 46.7654i 0.877382 1.51967i 0.0231788 0.999731i \(-0.492621\pi\)
0.854203 0.519939i \(-0.174045\pi\)
\(948\) 0 0
\(949\) −20.0000 34.6410i −0.649227 1.12449i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 21.0000 0.679544
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 20.7846i −0.387500 0.671170i
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.00000 + 13.8564i −0.257529 + 0.446054i
\(966\) 0 0
\(967\) −1.00000 1.73205i −0.0321578 0.0556990i 0.849499 0.527591i \(-0.176905\pi\)
−0.881656 + 0.471892i \(0.843571\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 0 0
\(973\) −2.00000 −0.0641171
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0000 20.7846i −0.383914 0.664959i 0.607704 0.794164i \(-0.292091\pi\)
−0.991618 + 0.129205i \(0.958757\pi\)
\(978\) 0 0
\(979\) −22.5000 + 38.9711i −0.719103 + 1.24552i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21.0000 + 36.3731i −0.669796 + 1.16012i 0.308165 + 0.951333i \(0.400285\pi\)
−0.977961 + 0.208788i \(0.933048\pi\)
\(984\) 0 0
\(985\) 12.0000 + 20.7846i 0.382352 + 0.662253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.00000 13.8564i −0.253617 0.439278i
\(996\) 0 0
\(997\) −13.0000 + 22.5167i −0.411714 + 0.713110i −0.995077 0.0991016i \(-0.968403\pi\)
0.583363 + 0.812211i \(0.301736\pi\)
\(998\) 0 0
\(999\) 0 0
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 1620.2.i.g.541.1 2
3.2 odd 2 1620.2.i.c.541.1 2
9.2 odd 6 1620.2.a.f.1.1 yes 1
9.4 even 3 inner 1620.2.i.g.1081.1 2
9.5 odd 6 1620.2.i.c.1081.1 2
9.7 even 3 1620.2.a.c.1.1 1
36.7 odd 6 6480.2.a.b.1.1 1
36.11 even 6 6480.2.a.p.1.1 1
45.2 even 12 8100.2.d.d.649.2 2
45.7 odd 12 8100.2.d.i.649.2 2
45.29 odd 6 8100.2.a.b.1.1 1
45.34 even 6 8100.2.a.e.1.1 1
45.38 even 12 8100.2.d.d.649.1 2
45.43 odd 12 8100.2.d.i.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.c.1.1 1 9.7 even 3
1620.2.a.f.1.1 yes 1 9.2 odd 6
1620.2.i.c.541.1 2 3.2 odd 2
1620.2.i.c.1081.1 2 9.5 odd 6
1620.2.i.g.541.1 2 1.1 even 1 trivial
1620.2.i.g.1081.1 2 9.4 even 3 inner
6480.2.a.b.1.1 1 36.7 odd 6
6480.2.a.p.1.1 1 36.11 even 6
8100.2.a.b.1.1 1 45.29 odd 6
8100.2.a.e.1.1 1 45.34 even 6
8100.2.d.d.649.1 2 45.38 even 12
8100.2.d.d.649.2 2 45.2 even 12
8100.2.d.i.649.1 2 45.43 odd 12
8100.2.d.i.649.2 2 45.7 odd 12