Properties

Label 2520.2.bi.f.361.1
Level $2520$
Weight $2$
Character 2520.361
Analytic conductor $20.122$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2520,2,Mod(361,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("2520.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (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: \(20.1223013094\)
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 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2520.361
Dual form 2520.2.bi.f.1801.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(2.50000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(2.50000 + 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{11} +5.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(-1.50000 - 2.59808i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} -4.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(-0.500000 - 2.59808i) q^{35} +(0.500000 + 0.866025i) q^{37} -6.00000 q^{41} -1.00000 q^{43} +(5.00000 + 8.66025i) q^{47} +(5.50000 + 4.33013i) q^{49} -2.00000 q^{55} +(3.00000 + 5.19615i) q^{61} +(-2.50000 - 4.33013i) q^{65} +(7.50000 - 12.9904i) q^{67} +6.00000 q^{71} +(1.50000 - 2.59808i) q^{73} +(4.00000 - 3.46410i) q^{77} +(-4.50000 - 7.79423i) q^{79} -6.00000 q^{83} -4.00000 q^{85} +(12.5000 + 4.33013i) q^{91} +(-1.50000 + 2.59808i) q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 5 q^{7} + 2 q^{11} + 10 q^{13} + 4 q^{17} - 3 q^{19} - 4 q^{23} - q^{25} - 8 q^{29} + q^{31} - q^{35} + q^{37} - 12 q^{41} - 2 q^{43} + 10 q^{47} + 11 q^{49} - 4 q^{55} + 6 q^{61} - 5 q^{65} + 15 q^{67} + 12 q^{71} + 3 q^{73} + 8 q^{77} - 9 q^{79} - 12 q^{83} - 8 q^{85} + 25 q^{91} - 3 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 2.50000 + 0.866025i 0.944911 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 3.46410i 0.485071 0.840168i −0.514782 0.857321i \(-0.672127\pi\)
0.999853 + 0.0171533i \(0.00546033\pi\)
\(18\) 0 0
\(19\) −1.50000 2.59808i −0.344124 0.596040i 0.641071 0.767482i \(-0.278491\pi\)
−0.985194 + 0.171442i \(0.945157\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 3.46410i −0.417029 0.722315i 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0.500000 0.866025i 0.0898027 0.155543i −0.817625 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.500000 2.59808i −0.0845154 0.439155i
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.00000 + 8.66025i 0.729325 + 1.26323i 0.957169 + 0.289530i \(0.0934991\pi\)
−0.227844 + 0.973698i \(0.573168\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 3.00000 + 5.19615i 0.384111 + 0.665299i 0.991645 0.128994i \(-0.0411748\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.50000 4.33013i −0.310087 0.537086i
\(66\) 0 0
\(67\) 7.50000 12.9904i 0.916271 1.58703i 0.111241 0.993793i \(-0.464517\pi\)
0.805030 0.593234i \(-0.202149\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 1.50000 2.59808i 0.175562 0.304082i −0.764794 0.644275i \(-0.777159\pi\)
0.940356 + 0.340193i \(0.110493\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 3.46410i 0.455842 0.394771i
\(78\) 0 0
\(79\) −4.50000 7.79423i −0.506290 0.876919i −0.999974 0.00727784i \(-0.997683\pi\)
0.493684 0.869641i \(-0.335650\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 12.5000 + 4.33013i 1.31036 + 0.453921i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.50000 + 2.59808i −0.153897 + 0.266557i
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 15.5885i 0.895533 1.55111i 0.0623905 0.998052i \(-0.480128\pi\)
0.833143 0.553058i \(-0.186539\pi\)
\(102\) 0 0
\(103\) −3.50000 6.06218i −0.344865 0.597324i 0.640464 0.767988i \(-0.278742\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 + 3.46410i 0.193347 + 0.334887i 0.946357 0.323122i \(-0.104732\pi\)
−0.753010 + 0.658009i \(0.771399\pi\)
\(108\) 0 0
\(109\) 5.50000 9.52628i 0.526804 0.912452i −0.472708 0.881219i \(-0.656723\pi\)
0.999512 0.0312328i \(-0.00994332\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −2.00000 + 3.46410i −0.186501 + 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.00000 6.92820i 0.733359 0.635107i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.00000 15.5885i −0.786334 1.36197i −0.928199 0.372084i \(-0.878643\pi\)
0.141865 0.989886i \(-0.454690\pi\)
\(132\) 0 0
\(133\) −1.50000 7.79423i −0.130066 0.675845i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 10.3923i 0.512615 0.887875i −0.487278 0.873247i \(-0.662010\pi\)
0.999893 0.0146279i \(-0.00465636\pi\)
\(138\) 0 0
\(139\) −17.0000 −1.44192 −0.720961 0.692976i \(-0.756299\pi\)
−0.720961 + 0.692976i \(0.756299\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.00000 8.66025i 0.418121 0.724207i
\(144\) 0 0
\(145\) 2.00000 + 3.46410i 0.166091 + 0.287678i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 + 17.3205i 0.819232 + 1.41895i 0.906249 + 0.422744i \(0.138933\pi\)
−0.0870170 + 0.996207i \(0.527733\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −1.00000 + 1.73205i −0.0798087 + 0.138233i −0.903167 0.429289i \(-0.858764\pi\)
0.823359 + 0.567521i \(0.192098\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 10.3923i −0.157622 0.819028i
\(162\) 0 0
\(163\) −10.0000 17.3205i −0.783260 1.35665i −0.930033 0.367477i \(-0.880222\pi\)
0.146772 0.989170i \(-0.453112\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.00000 3.46410i −0.152057 0.263371i 0.779926 0.625871i \(-0.215256\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(174\) 0 0
\(175\) −2.00000 + 1.73205i −0.151186 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.00000 1.73205i 0.0747435 0.129460i −0.826231 0.563331i \(-0.809520\pi\)
0.900975 + 0.433872i \(0.142853\pi\)
\(180\) 0 0
\(181\) 25.0000 1.85824 0.929118 0.369784i \(-0.120568\pi\)
0.929118 + 0.369784i \(0.120568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.500000 0.866025i 0.0367607 0.0636715i
\(186\) 0 0
\(187\) −4.00000 6.92820i −0.292509 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.00000 + 15.5885i 0.651217 + 1.12794i 0.982828 + 0.184525i \(0.0590746\pi\)
−0.331611 + 0.943416i \(0.607592\pi\)
\(192\) 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
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −8.00000 + 13.8564i −0.567105 + 0.982255i 0.429745 + 0.902950i \(0.358603\pi\)
−0.996850 + 0.0793045i \(0.974730\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.0000 3.46410i −0.701862 0.243132i
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.500000 + 0.866025i 0.0340997 + 0.0590624i
\(216\) 0 0
\(217\) 2.00000 1.73205i 0.135769 0.117579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0000 17.3205i 0.672673 1.16510i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.00000 + 8.66025i −0.331862 + 0.574801i −0.982877 0.184263i \(-0.941010\pi\)
0.651015 + 0.759065i \(0.274343\pi\)
\(228\) 0 0
\(229\) −12.5000 21.6506i −0.826023 1.43071i −0.901135 0.433539i \(-0.857265\pi\)
0.0751115 0.997175i \(-0.476069\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 + 15.5885i 0.589610 + 1.02123i 0.994283 + 0.106773i \(0.0340517\pi\)
−0.404674 + 0.914461i \(0.632615\pi\)
\(234\) 0 0
\(235\) 5.00000 8.66025i 0.326164 0.564933i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.0000 0.905585 0.452792 0.891616i \(-0.350428\pi\)
0.452792 + 0.891616i \(0.350428\pi\)
\(240\) 0 0
\(241\) 1.00000 1.73205i 0.0644157 0.111571i −0.832019 0.554747i \(-0.812815\pi\)
0.896435 + 0.443176i \(0.146148\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 6.92820i 0.0638877 0.442627i
\(246\) 0 0
\(247\) −7.50000 12.9904i −0.477214 0.826558i
\(248\) 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
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.0000 + 25.9808i 0.935674 + 1.62064i 0.773427 + 0.633885i \(0.218541\pi\)
0.162247 + 0.986750i \(0.448126\pi\)
\(258\) 0 0
\(259\) 0.500000 + 2.59808i 0.0310685 + 0.161437i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.00000 3.46410i 0.123325 0.213606i −0.797752 0.602986i \(-0.793977\pi\)
0.921077 + 0.389380i \(0.127311\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.00000 12.1244i 0.426798 0.739235i −0.569789 0.821791i \(-0.692975\pi\)
0.996586 + 0.0825561i \(0.0263084\pi\)
\(270\) 0 0
\(271\) 4.00000 + 6.92820i 0.242983 + 0.420858i 0.961563 0.274586i \(-0.0885408\pi\)
−0.718580 + 0.695444i \(0.755208\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 + 1.73205i 0.0603023 + 0.104447i
\(276\) 0 0
\(277\) 11.5000 19.9186i 0.690968 1.19679i −0.280553 0.959839i \(-0.590518\pi\)
0.971521 0.236953i \(-0.0761488\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −15.5000 + 26.8468i −0.921379 + 1.59588i −0.124096 + 0.992270i \(0.539603\pi\)
−0.797283 + 0.603606i \(0.793730\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −15.0000 5.19615i −0.885422 0.306719i
\(288\) 0 0
\(289\) 0.500000 + 0.866025i 0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.0000 −1.86946 −0.934730 0.355359i \(-0.884359\pi\)
−0.934730 + 0.355359i \(0.884359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.0000 17.3205i −0.578315 1.00167i
\(300\) 0 0
\(301\) −2.50000 0.866025i −0.144098 0.0499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.00000 5.19615i 0.171780 0.297531i
\(306\) 0 0
\(307\) 21.0000 1.19853 0.599267 0.800549i \(-0.295459\pi\)
0.599267 + 0.800549i \(0.295459\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 5.19615i 0.170114 0.294647i −0.768345 0.640036i \(-0.778920\pi\)
0.938460 + 0.345389i \(0.112253\pi\)
\(312\) 0 0
\(313\) 4.50000 + 7.79423i 0.254355 + 0.440556i 0.964720 0.263278i \(-0.0848035\pi\)
−0.710365 + 0.703833i \(0.751470\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 + 20.7846i 0.673987 + 1.16738i 0.976764 + 0.214318i \(0.0687530\pi\)
−0.302777 + 0.953062i \(0.597914\pi\)
\(318\) 0 0
\(319\) −4.00000 + 6.92820i −0.223957 + 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −2.50000 + 4.33013i −0.138675 + 0.240192i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.00000 + 25.9808i 0.275659 + 1.43237i
\(330\) 0 0
\(331\) 17.5000 + 30.3109i 0.961887 + 1.66604i 0.717756 + 0.696295i \(0.245169\pi\)
0.244131 + 0.969742i \(0.421497\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.00000 1.73205i −0.0541530 0.0937958i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 + 10.3923i −0.322097 + 0.557888i −0.980921 0.194409i \(-0.937721\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.0000 + 22.5167i −0.691920 + 1.19844i 0.279288 + 0.960207i \(0.409902\pi\)
−0.971208 + 0.238233i \(0.923432\pi\)
\(354\) 0 0
\(355\) −3.00000 5.19615i −0.159223 0.275783i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 + 10.3923i 0.316668 + 0.548485i 0.979791 0.200026i \(-0.0641026\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(360\) 0 0
\(361\) 5.00000 8.66025i 0.263158 0.455803i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 −0.157027
\(366\) 0 0
\(367\) 11.5000 19.9186i 0.600295 1.03974i −0.392481 0.919760i \(-0.628383\pi\)
0.992776 0.119982i \(-0.0382835\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.5000 23.3827i −0.699004 1.21071i −0.968812 0.247796i \(-0.920294\pi\)
0.269809 0.962914i \(-0.413039\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 20.7846i −0.613171 1.06204i −0.990702 0.136047i \(-0.956560\pi\)
0.377531 0.925997i \(-0.376773\pi\)
\(384\) 0 0
\(385\) −5.00000 1.73205i −0.254824 0.0882735i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.00000 5.19615i 0.152106 0.263455i −0.779895 0.625910i \(-0.784728\pi\)
0.932002 + 0.362454i \(0.118061\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.50000 + 7.79423i −0.226420 + 0.392170i
\(396\) 0 0
\(397\) 14.5000 + 25.1147i 0.727734 + 1.26047i 0.957839 + 0.287307i \(0.0927599\pi\)
−0.230105 + 0.973166i \(0.573907\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.00000 + 6.92820i 0.199750 + 0.345978i 0.948447 0.316934i \(-0.102654\pi\)
−0.748697 + 0.662912i \(0.769320\pi\)
\(402\) 0 0
\(403\) 2.50000 4.33013i 0.124534 0.215699i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) −14.5000 + 25.1147i −0.716979 + 1.24184i 0.245212 + 0.969469i \(0.421142\pi\)
−0.962191 + 0.272374i \(0.912191\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3.00000 + 5.19615i 0.147264 + 0.255069i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 + 3.46410i 0.0970143 + 0.168034i
\(426\) 0 0
\(427\) 3.00000 + 15.5885i 0.145180 + 0.754378i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.0000 + 22.5167i −0.626188 + 1.08459i 0.362122 + 0.932131i \(0.382052\pi\)
−0.988310 + 0.152459i \(0.951281\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 + 10.3923i −0.287019 + 0.497131i
\(438\) 0 0
\(439\) 4.00000 + 6.92820i 0.190910 + 0.330665i 0.945552 0.325471i \(-0.105523\pi\)
−0.754642 + 0.656136i \(0.772190\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 + 20.7846i 0.570137 + 0.987507i 0.996551 + 0.0829786i \(0.0264433\pi\)
−0.426414 + 0.904528i \(0.640223\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) −6.00000 + 10.3923i −0.282529 + 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.50000 12.9904i −0.117202 0.608998i
\(456\) 0 0
\(457\) 13.5000 + 23.3827i 0.631503 + 1.09380i 0.987245 + 0.159211i \(0.0508951\pi\)
−0.355741 + 0.934585i \(0.615772\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.0000 36.3731i −0.971764 1.68314i −0.690225 0.723595i \(-0.742488\pi\)
−0.281539 0.959550i \(-0.590845\pi\)
\(468\) 0 0
\(469\) 30.0000 25.9808i 1.38527 1.19968i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.00000 + 1.73205i −0.0459800 + 0.0796398i
\(474\) 0 0
\(475\) 3.00000 0.137649
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 20.7846i 0.548294 0.949673i −0.450098 0.892979i \(-0.648611\pi\)
0.998392 0.0566937i \(-0.0180558\pi\)
\(480\) 0 0
\(481\) 2.50000 + 4.33013i 0.113990 + 0.197437i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.00000 1.73205i −0.0454077 0.0786484i
\(486\) 0 0
\(487\) 11.5000 19.9186i 0.521115 0.902597i −0.478584 0.878042i \(-0.658850\pi\)
0.999698 0.0245553i \(-0.00781698\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) −8.00000 + 13.8564i −0.360302 + 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.0000 + 5.19615i 0.672842 + 0.233079i
\(498\) 0 0
\(499\) −3.50000 6.06218i −0.156682 0.271380i 0.776989 0.629515i \(-0.216746\pi\)
−0.933670 + 0.358134i \(0.883413\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.0000 + 25.9808i 0.664863 + 1.15158i 0.979322 + 0.202306i \(0.0648436\pi\)
−0.314459 + 0.949271i \(0.601823\pi\)
\(510\) 0 0
\(511\) 6.00000 5.19615i 0.265424 0.229864i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.50000 + 6.06218i −0.154228 + 0.267131i
\(516\) 0 0
\(517\) 20.0000 0.879599
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 + 10.3923i −0.262865 + 0.455295i −0.967002 0.254769i \(-0.918001\pi\)
0.704137 + 0.710064i \(0.251334\pi\)
\(522\) 0 0
\(523\) −6.50000 11.2583i −0.284225 0.492292i 0.688196 0.725525i \(-0.258403\pi\)
−0.972421 + 0.233233i \(0.925070\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.00000 3.46410i −0.0871214 0.150899i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) 2.00000 3.46410i 0.0864675 0.149766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.0000 5.19615i 0.559950 0.223814i
\(540\) 0 0
\(541\) −8.50000 14.7224i −0.365444 0.632967i 0.623404 0.781900i \(-0.285749\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −11.0000 −0.471188
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 + 10.3923i 0.255609 + 0.442727i
\(552\) 0 0
\(553\) −4.50000 23.3827i −0.191359 0.994333i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 25.9808i 0.635570 1.10084i −0.350824 0.936442i \(-0.614098\pi\)
0.986394 0.164399i \(-0.0525683\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.00000 + 8.66025i −0.210725 + 0.364986i −0.951942 0.306280i \(-0.900916\pi\)
0.741217 + 0.671266i \(0.234249\pi\)
\(564\) 0 0
\(565\) −9.00000 15.5885i −0.378633 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 + 5.19615i 0.125767 + 0.217834i 0.922032 0.387113i \(-0.126528\pi\)
−0.796266 + 0.604947i \(0.793194\pi\)
\(570\) 0 0
\(571\) 0.500000 0.866025i 0.0209243 0.0362420i −0.855374 0.518012i \(-0.826672\pi\)
0.876298 + 0.481770i \(0.160006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −23.5000 + 40.7032i −0.978318 + 1.69450i −0.309797 + 0.950803i \(0.600261\pi\)
−0.668521 + 0.743693i \(0.733072\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0000 5.19615i −0.622305 0.215573i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11.0000 19.0526i −0.451716 0.782395i 0.546777 0.837278i \(-0.315855\pi\)
−0.998493 + 0.0548835i \(0.982521\pi\)
\(594\) 0 0
\(595\) −10.0000 3.46410i −0.409960 0.142014i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.00000 3.46410i 0.0817178 0.141539i −0.822270 0.569097i \(-0.807293\pi\)
0.903988 + 0.427558i \(0.140626\pi\)
\(600\) 0 0
\(601\) 15.0000 0.611863 0.305931 0.952054i \(-0.401032\pi\)
0.305931 + 0.952054i \(0.401032\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.50000 6.06218i 0.142295 0.246463i
\(606\) 0 0
\(607\) −8.50000 14.7224i −0.345004 0.597565i 0.640350 0.768083i \(-0.278789\pi\)
−0.985355 + 0.170518i \(0.945456\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 25.0000 + 43.3013i 1.01139 + 1.75178i
\(612\) 0 0
\(613\) −5.00000 + 8.66025i −0.201948 + 0.349784i −0.949156 0.314806i \(-0.898061\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −0.500000 + 0.866025i −0.0200967 + 0.0348085i −0.875899 0.482495i \(-0.839731\pi\)
0.855802 + 0.517303i \(0.173064\pi\)
\(620\) 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
\(627\) 0 0
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.50000 + 14.7224i 0.337312 + 0.584242i
\(636\) 0 0
\(637\) 27.5000 + 21.6506i 1.08959 + 0.857829i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 + 3.46410i −0.0789953 + 0.136824i −0.902817 0.430026i \(-0.858505\pi\)
0.823821 + 0.566849i \(0.191838\pi\)
\(642\) 0 0
\(643\) −49.0000 −1.93237 −0.966186 0.257847i \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.0000 + 36.3731i −0.825595 + 1.42997i 0.0758684 + 0.997118i \(0.475827\pi\)
−0.901464 + 0.432855i \(0.857506\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.00000 12.1244i −0.273931 0.474463i 0.695934 0.718106i \(-0.254991\pi\)
−0.969865 + 0.243643i \(0.921657\pi\)
\(654\) 0 0
\(655\) −9.00000 + 15.5885i −0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) 6.50000 11.2583i 0.252821 0.437898i −0.711481 0.702706i \(-0.751975\pi\)
0.964301 + 0.264807i \(0.0853084\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.00000 + 5.19615i −0.232670 + 0.201498i
\(666\) 0 0
\(667\) 8.00000 + 13.8564i 0.309761 + 0.536522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 + 31.1769i 0.691796 + 1.19823i 0.971249 + 0.238067i \(0.0765137\pi\)
−0.279453 + 0.960159i \(0.590153\pi\)
\(678\) 0 0
\(679\) 5.00000 + 1.73205i 0.191882 + 0.0664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.00000 + 13.8564i −0.306111 + 0.530201i −0.977508 0.210898i \(-0.932361\pi\)
0.671397 + 0.741098i \(0.265695\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 7.50000 + 12.9904i 0.285313 + 0.494177i 0.972685 0.232128i \(-0.0745690\pi\)
−0.687372 + 0.726306i \(0.741236\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.50000 + 14.7224i 0.322423 + 0.558454i
\(696\) 0 0
\(697\) −12.0000 + 20.7846i −0.454532 + 0.787273i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 1.50000 2.59808i 0.0565736 0.0979883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000 31.1769i 1.35392 1.17253i
\(708\) 0 0
\(709\) 17.0000 + 29.4449i 0.638448 + 1.10583i 0.985773 + 0.168080i \(0.0537568\pi\)
−0.347325 + 0.937745i \(0.612910\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) −10.0000 −0.373979
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.00000 + 15.5885i 0.335643 + 0.581351i 0.983608 0.180319i \(-0.0577130\pi\)
−0.647965 + 0.761670i \(0.724380\pi\)
\(720\) 0 0
\(721\) −3.50000 18.1865i −0.130347 0.677302i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000 3.46410i 0.0742781 0.128654i
\(726\) 0 0
\(727\) −11.0000 −0.407967 −0.203984 0.978974i \(-0.565389\pi\)
−0.203984 + 0.978974i \(0.565389\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.00000 + 3.46410i −0.0739727 + 0.128124i
\(732\) 0 0
\(733\) −14.5000 25.1147i −0.535570 0.927634i −0.999136 0.0415715i \(-0.986764\pi\)
0.463566 0.886062i \(-0.346570\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.0000 25.9808i −0.552532 0.957014i
\(738\) 0 0
\(739\) −17.5000 + 30.3109i −0.643748 + 1.11500i 0.340841 + 0.940121i \(0.389288\pi\)
−0.984589 + 0.174883i \(0.944045\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22.0000 −0.807102 −0.403551 0.914957i \(-0.632224\pi\)
−0.403551 + 0.914957i \(0.632224\pi\)
\(744\) 0 0
\(745\) 10.0000 17.3205i 0.366372 0.634574i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.00000 + 10.3923i 0.0730784 + 0.379727i
\(750\) 0 0
\(751\) 6.50000 + 11.2583i 0.237188 + 0.410822i 0.959906 0.280321i \(-0.0904408\pi\)
−0.722718 + 0.691143i \(0.757107\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.00000 + 3.46410i 0.0724999 + 0.125574i 0.899996 0.435897i \(-0.143569\pi\)
−0.827496 + 0.561471i \(0.810236\pi\)
\(762\) 0 0
\(763\) 22.0000 19.0526i 0.796453 0.689749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −41.0000 −1.47850 −0.739249 0.673432i \(-0.764819\pi\)
−0.739249 + 0.673432i \(0.764819\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.00000 + 8.66025i −0.179838 + 0.311488i −0.941825 0.336104i \(-0.890891\pi\)
0.761987 + 0.647592i \(0.224224\pi\)
\(774\) 0 0
\(775\) 0.500000 + 0.866025i 0.0179605 + 0.0311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.00000 + 15.5885i 0.322458 + 0.558514i
\(780\) 0 0
\(781\) 6.00000 10.3923i 0.214697 0.371866i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 45.0000 + 15.5885i 1.60002 + 0.554262i
\(792\) 0 0
\(793\) 15.0000 + 25.9808i 0.532666 + 0.922604i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.0000 −0.991811 −0.495905 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.00000 5.19615i −0.105868 0.183368i
\(804\) 0 0
\(805\) −8.00000 + 6.92820i −0.281963 + 0.244187i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −11.0000 + 19.0526i −0.386739 + 0.669852i −0.992009 0.126168i \(-0.959732\pi\)
0.605269 + 0.796021i \(0.293065\pi\)
\(810\) 0 0
\(811\) −24.0000 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10.0000 + 17.3205i −0.350285 + 0.606711i
\(816\) 0 0
\(817\) 1.50000 + 2.59808i 0.0524784 + 0.0908952i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 + 25.9808i 0.523504 + 0.906735i 0.999626 + 0.0273557i \(0.00870868\pi\)
−0.476122 + 0.879379i \(0.657958\pi\)
\(822\) 0 0
\(823\) −12.0000 + 20.7846i −0.418294 + 0.724506i −0.995768 0.0919029i \(-0.970705\pi\)
0.577474 + 0.816409i \(0.304038\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 0 0
\(829\) −18.5000 + 32.0429i −0.642532 + 1.11290i 0.342334 + 0.939578i \(0.388783\pi\)
−0.984866 + 0.173319i \(0.944551\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26.0000 10.3923i 0.900847 0.360072i
\(834\) 0 0
\(835\) 1.00000 + 1.73205i 0.0346064 + 0.0599401i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.00000 10.3923i −0.206406 0.357506i
\(846\) 0 0
\(847\) 3.50000 + 18.1865i 0.120261 + 0.624897i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.00000 3.46410i 0.0685591 0.118748i
\(852\) 0 0
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20.0000 + 34.6410i −0.683187 + 1.18331i 0.290816 + 0.956779i \(0.406073\pi\)
−0.974003 + 0.226536i \(0.927260\pi\)
\(858\) 0 0
\(859\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.00000 + 12.1244i 0.238283 + 0.412718i 0.960222 0.279239i \(-0.0900822\pi\)
−0.721939 + 0.691957i \(0.756749\pi\)
\(864\) 0 0
\(865\) −2.00000 + 3.46410i −0.0680020 + 0.117783i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.0000 −0.610608
\(870\) 0 0
\(871\) 37.5000 64.9519i 1.27064 2.20081i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.50000 + 0.866025i 0.0845154 + 0.0292770i
\(876\) 0 0
\(877\) 7.00000 + 12.1244i 0.236373 + 0.409410i 0.959671 0.281126i \(-0.0907079\pi\)
−0.723298 + 0.690536i \(0.757375\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 0 0
\(883\) −15.0000 −0.504790 −0.252395 0.967624i \(-0.581218\pi\)
−0.252395 + 0.967624i \(0.581218\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15.0000 25.9808i −0.503651 0.872349i −0.999991 0.00422062i \(-0.998657\pi\)
0.496340 0.868128i \(-0.334677\pi\)
\(888\) 0 0
\(889\) −42.5000 14.7224i −1.42540 0.493775i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.0000 25.9808i 0.501956 0.869413i
\(894\) 0 0
\(895\) −2.00000 −0.0668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.00000 + 3.46410i −0.0667037 + 0.115534i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −12.5000 21.6506i −0.415514 0.719691i
\(906\) 0 0
\(907\) −8.50000 + 14.7224i −0.282238 + 0.488850i −0.971936 0.235247i \(-0.924410\pi\)
0.689698 + 0.724097i \(0.257743\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −6.00000 + 10.3923i −0.198571 + 0.343935i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.00000 46.7654i −0.297206 1.54433i
\(918\) 0 0
\(919\) −28.5000 49.3634i −0.940128 1.62835i −0.765224 0.643765i \(-0.777372\pi\)
−0.174905 0.984585i \(-0.555962\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.0000 32.9090i −0.623370 1.07971i −0.988854 0.148890i \(-0.952430\pi\)
0.365484 0.930818i \(-0.380904\pi\)
\(930\) 0 0
\(931\) 3.00000 20.7846i 0.0983210 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.00000 + 6.92820i −0.130814 + 0.226576i
\(936\) 0 0
\(937\) 39.0000 1.27407 0.637037 0.770833i \(-0.280160\pi\)
0.637037 + 0.770833i \(0.280160\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(942\) 0 0
\(943\) 12.0000 + 20.7846i 0.390774 + 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.0000 25.9808i −0.487435 0.844261i 0.512461 0.858710i \(-0.328734\pi\)
−0.999896 + 0.0144491i \(0.995401\pi\)
\(948\) 0 0
\(949\) 7.50000 12.9904i 0.243460 0.421686i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −48.0000 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(954\) 0 0
\(955\) 9.00000 15.5885i 0.291233 0.504431i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 20.7846i 0.775000 0.671170i
\(960\) 0 0
\(961\) 15.0000 + 25.9808i 0.483871 + 0.838089i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.0000 0.354103
\(966\) 0 0
\(967\) 5.00000 0.160789 0.0803946 0.996763i \(-0.474382\pi\)
0.0803946 + 0.996763i \(0.474382\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 16.0000 + 27.7128i 0.513464 + 0.889346i 0.999878 + 0.0156178i \(0.00497150\pi\)
−0.486414 + 0.873729i \(0.661695\pi\)
\(972\) 0 0
\(973\) −42.5000 14.7224i −1.36249 0.471979i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.0000 19.0526i 0.351921 0.609545i −0.634665 0.772787i \(-0.718862\pi\)
0.986586 + 0.163242i \(0.0521952\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 41.5692i 0.765481 1.32585i −0.174511 0.984655i \(-0.555834\pi\)
0.939992 0.341197i \(-0.110832\pi\)
\(984\) 0 0
\(985\) −6.00000 10.3923i −0.191176 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.00000 + 3.46410i 0.0635963 + 0.110152i
\(990\) 0 0
\(991\) 12.5000 21.6506i 0.397076 0.687755i −0.596288 0.802771i \(-0.703358\pi\)
0.993364 + 0.115015i \(0.0366917\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 12.5000 21.6506i 0.395879 0.685682i −0.597334 0.801993i \(-0.703773\pi\)
0.993213 + 0.116310i \(0.0371066\pi\)
\(998\) 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 2520.2.bi.f.361.1 2
3.2 odd 2 840.2.bg.f.361.1 yes 2
7.2 even 3 inner 2520.2.bi.f.1801.1 2
12.11 even 2 1680.2.bg.f.1201.1 2
21.2 odd 6 840.2.bg.f.121.1 2
21.11 odd 6 5880.2.a.e.1.1 1
21.17 even 6 5880.2.a.bg.1.1 1
84.23 even 6 1680.2.bg.f.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bg.f.121.1 2 21.2 odd 6
840.2.bg.f.361.1 yes 2 3.2 odd 2
1680.2.bg.f.961.1 2 84.23 even 6
1680.2.bg.f.1201.1 2 12.11 even 2
2520.2.bi.f.361.1 2 1.1 even 1 trivial
2520.2.bi.f.1801.1 2 7.2 even 3 inner
5880.2.a.e.1.1 1 21.11 odd 6
5880.2.a.bg.1.1 1 21.17 even 6