Properties

Label 525.2.r.b.499.1
Level $525$
Weight $2$
Character 525.499
Analytic conductor $4.192$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(424,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.r (of order \(6\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 499.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 525.499
Dual form 525.2.r.b.424.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.866025 + 2.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.866025 + 2.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +(1.73205 + 1.00000i) q^{12} +1.00000i q^{13} +(-2.00000 + 3.46410i) q^{16} +(5.19615 - 3.00000i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-2.00000 - 1.73205i) q^{21} +(5.19615 + 3.00000i) q^{23} +1.00000i q^{27} +(3.46410 - 4.00000i) q^{28} +6.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} -2.00000 q^{36} +(6.06218 + 3.50000i) q^{37} +(-0.500000 - 0.866025i) q^{39} +12.0000 q^{41} +1.00000i q^{43} +(-5.19615 - 3.00000i) q^{47} -4.00000i q^{48} +(-5.50000 + 4.33013i) q^{49} +(-3.00000 + 5.19615i) q^{51} +(1.73205 - 1.00000i) q^{52} +5.00000i q^{57} +(-3.00000 - 5.19615i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(2.59808 + 0.500000i) q^{63} +8.00000 q^{64} +(-6.06218 + 3.50000i) q^{67} +(-10.3923 - 6.00000i) q^{68} -6.00000 q^{69} +12.0000 q^{71} +(-9.52628 + 5.50000i) q^{73} -10.0000 q^{76} +(-6.50000 + 11.2583i) q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000i q^{83} +(-1.00000 + 5.19615i) q^{84} +(-5.19615 + 3.00000i) q^{87} +(3.00000 - 5.19615i) q^{89} +(-2.50000 + 0.866025i) q^{91} -12.0000i q^{92} +(4.33013 + 2.50000i) q^{93} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 2 q^{9} - 8 q^{16} + 10 q^{19} - 8 q^{21} + 24 q^{29} - 10 q^{31} - 8 q^{36} - 2 q^{39} + 48 q^{41} - 22 q^{49} - 12 q^{51} - 12 q^{59} - 4 q^{61} + 32 q^{64} - 24 q^{69} + 48 q^{71} - 40 q^{76} - 26 q^{79} - 2 q^{81} - 4 q^{84} + 12 q^{89} - 10 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) 0.866025 + 2.50000i 0.327327 + 0.944911i
\(8\) 0 0
\(9\) 0.500000 0.866025i 0.166667 0.288675i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 1.73205 + 1.00000i 0.500000 + 0.288675i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 5.19615 3.00000i 1.26025 0.727607i 0.287129 0.957892i \(-0.407299\pi\)
0.973123 + 0.230285i \(0.0739659\pi\)
\(18\) 0 0
\(19\) 2.50000 4.33013i 0.573539 0.993399i −0.422659 0.906289i \(-0.638903\pi\)
0.996199 0.0871106i \(-0.0277634\pi\)
\(20\) 0 0
\(21\) −2.00000 1.73205i −0.436436 0.377964i
\(22\) 0 0
\(23\) 5.19615 + 3.00000i 1.08347 + 0.625543i 0.931831 0.362892i \(-0.118211\pi\)
0.151642 + 0.988436i \(0.451544\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 3.46410 4.00000i 0.654654 0.755929i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 6.06218 + 3.50000i 0.996616 + 0.575396i 0.907245 0.420602i \(-0.138181\pi\)
0.0893706 + 0.995998i \(0.471514\pi\)
\(38\) 0 0
\(39\) −0.500000 0.866025i −0.0800641 0.138675i
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.19615 3.00000i −0.757937 0.437595i 0.0706177 0.997503i \(-0.477503\pi\)
−0.828554 + 0.559908i \(0.810836\pi\)
\(48\) 4.00000i 0.577350i
\(49\) −5.50000 + 4.33013i −0.785714 + 0.618590i
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 1.73205 1.00000i 0.240192 0.138675i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000i 0.662266i
\(58\) 0 0
\(59\) −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) 0 0
\(63\) 2.59808 + 0.500000i 0.327327 + 0.0629941i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.06218 + 3.50000i −0.740613 + 0.427593i −0.822292 0.569066i \(-0.807305\pi\)
0.0816792 + 0.996659i \(0.473972\pi\)
\(68\) −10.3923 6.00000i −1.26025 0.727607i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −9.52628 + 5.50000i −1.11497 + 0.643726i −0.940111 0.340868i \(-0.889279\pi\)
−0.174855 + 0.984594i \(0.555946\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) 0 0
\(78\) 0 0
\(79\) −6.50000 + 11.2583i −0.731307 + 1.26666i 0.225018 + 0.974355i \(0.427756\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) −1.00000 + 5.19615i −0.109109 + 0.566947i
\(85\) 0 0
\(86\) 0 0
\(87\) −5.19615 + 3.00000i −0.557086 + 0.321634i
\(88\) 0 0
\(89\) 3.00000 5.19615i 0.317999 0.550791i −0.662071 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) −2.50000 + 0.866025i −0.262071 + 0.0907841i
\(92\) 12.0000i 1.25109i
\(93\) 4.33013 + 2.50000i 0.449013 + 0.259238i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −0.866025 0.500000i −0.0853320 0.0492665i 0.456727 0.889607i \(-0.349022\pi\)
−0.542059 + 0.840341i \(0.682355\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.5885 9.00000i −1.50699 0.870063i −0.999967 0.00813215i \(-0.997411\pi\)
−0.507026 0.861931i \(-0.669255\pi\)
\(108\) 1.73205 1.00000i 0.166667 0.0962250i
\(109\) −3.50000 6.06218i −0.335239 0.580651i 0.648292 0.761392i \(-0.275484\pi\)
−0.983531 + 0.180741i \(0.942150\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −10.3923 2.00000i −0.981981 0.188982i
\(113\) 18.0000i 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 10.3923i −0.557086 0.964901i
\(117\) 0.866025 + 0.500000i 0.0800641 + 0.0462250i
\(118\) 0 0
\(119\) 12.0000 + 10.3923i 1.10004 + 0.952661i
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) −10.3923 + 6.00000i −0.937043 + 0.541002i
\(124\) −5.00000 + 8.66025i −0.449013 + 0.777714i
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000i 0.976092i 0.872818 + 0.488046i \(0.162290\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) −0.500000 0.866025i −0.0440225 0.0762493i
\(130\) 0 0
\(131\) 3.00000 5.19615i 0.262111 0.453990i −0.704692 0.709514i \(-0.748915\pi\)
0.966803 + 0.255524i \(0.0822479\pi\)
\(132\) 0 0
\(133\) 12.9904 + 2.50000i 1.12641 + 0.216777i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.19615 3.00000i 0.443937 0.256307i −0.261329 0.965250i \(-0.584161\pi\)
0.705266 + 0.708942i \(0.250827\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 2.00000 + 3.46410i 0.166667 + 0.288675i
\(145\) 0 0
\(146\) 0 0
\(147\) 2.59808 6.50000i 0.214286 0.536111i
\(148\) 14.0000i 1.15079i
\(149\) −12.0000 + 20.7846i −0.983078 + 1.70274i −0.332896 + 0.942964i \(0.608026\pi\)
−0.650183 + 0.759778i \(0.725308\pi\)
\(150\) 0 0
\(151\) 8.00000 + 13.8564i 0.651031 + 1.12762i 0.982873 + 0.184284i \(0.0589965\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 + 1.73205i −0.0800641 + 0.138675i
\(157\) −8.66025 + 5.00000i −0.691164 + 0.399043i −0.804048 0.594565i \(-0.797324\pi\)
0.112884 + 0.993608i \(0.463991\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 + 15.5885i −0.236433 + 1.22854i
\(162\) 0 0
\(163\) −3.46410 2.00000i −0.271329 0.156652i 0.358162 0.933659i \(-0.383403\pi\)
−0.629492 + 0.777007i \(0.716737\pi\)
\(164\) −12.0000 20.7846i −0.937043 1.62301i
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −2.50000 4.33013i −0.191180 0.331133i
\(172\) 1.73205 1.00000i 0.132068 0.0762493i
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.19615 + 3.00000i 0.390567 + 0.225494i
\(178\) 0 0
\(179\) 9.00000 + 15.5885i 0.672692 + 1.16514i 0.977138 + 0.212607i \(0.0681952\pi\)
−0.304446 + 0.952529i \(0.598471\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000i 0.875190i
\(189\) −2.50000 + 0.866025i −0.181848 + 0.0629941i
\(190\) 0 0
\(191\) 3.00000 5.19615i 0.217072 0.375980i −0.736839 0.676068i \(-0.763683\pi\)
0.953912 + 0.300088i \(0.0970159\pi\)
\(192\) −6.92820 + 4.00000i −0.500000 + 0.288675i
\(193\) −4.33013 + 2.50000i −0.311689 + 0.179954i −0.647682 0.761911i \(-0.724262\pi\)
0.335993 + 0.941865i \(0.390928\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 13.0000 + 5.19615i 0.928571 + 0.371154i
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −8.00000 13.8564i −0.567105 0.982255i −0.996850 0.0793045i \(-0.974730\pi\)
0.429745 0.902950i \(-0.358603\pi\)
\(200\) 0 0
\(201\) 3.50000 6.06218i 0.246871 0.427593i
\(202\) 0 0
\(203\) 5.19615 + 15.0000i 0.364698 + 1.05279i
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 0 0
\(207\) 5.19615 3.00000i 0.361158 0.208514i
\(208\) −3.46410 2.00000i −0.240192 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −10.3923 + 6.00000i −0.712069 + 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.66025 10.0000i 0.587896 0.678844i
\(218\) 0 0
\(219\) 5.50000 9.52628i 0.371656 0.643726i
\(220\) 0 0
\(221\) 3.00000 + 5.19615i 0.201802 + 0.349531i
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.3923 6.00000i 0.689761 0.398234i −0.113761 0.993508i \(-0.536290\pi\)
0.803523 + 0.595274i \(0.202957\pi\)
\(228\) 8.66025 5.00000i 0.573539 0.331133i
\(229\) 14.5000 25.1147i 0.958187 1.65963i 0.231287 0.972886i \(-0.425707\pi\)
0.726900 0.686743i \(-0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.19615 + 3.00000i 0.340411 + 0.196537i 0.660454 0.750867i \(-0.270364\pi\)
−0.320043 + 0.947403i \(0.603697\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 + 10.3923i −0.390567 + 0.676481i
\(237\) 13.0000i 0.844441i
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −1.00000 1.73205i −0.0644157 0.111571i 0.832019 0.554747i \(-0.187185\pi\)
−0.896435 + 0.443176i \(0.853852\pi\)
\(242\) 0 0
\(243\) 0.866025 + 0.500000i 0.0555556 + 0.0320750i
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 4.33013 + 2.50000i 0.275519 + 0.159071i
\(248\) 0 0
\(249\) −6.00000 10.3923i −0.380235 0.658586i
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) −1.73205 5.00000i −0.109109 0.314970i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 10.3923 + 6.00000i 0.648254 + 0.374270i 0.787787 0.615948i \(-0.211227\pi\)
−0.139533 + 0.990217i \(0.544560\pi\)
\(258\) 0 0
\(259\) −3.50000 + 18.1865i −0.217479 + 1.13006i
\(260\) 0 0
\(261\) 3.00000 5.19615i 0.185695 0.321634i
\(262\) 0 0
\(263\) 10.3923 6.00000i 0.640817 0.369976i −0.144112 0.989561i \(-0.546033\pi\)
0.784929 + 0.619586i \(0.212699\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 12.1244 + 7.00000i 0.740613 + 0.427593i
\(269\) −9.00000 15.5885i −0.548740 0.950445i −0.998361 0.0572259i \(-0.981774\pi\)
0.449622 0.893219i \(-0.351559\pi\)
\(270\) 0 0
\(271\) 8.00000 13.8564i 0.485965 0.841717i −0.513905 0.857847i \(-0.671801\pi\)
0.999870 + 0.0161307i \(0.00513477\pi\)
\(272\) 24.0000i 1.45521i
\(273\) 1.73205 2.00000i 0.104828 0.121046i
\(274\) 0 0
\(275\) 0 0
\(276\) 6.00000 + 10.3923i 0.361158 + 0.625543i
\(277\) 19.9186 11.5000i 1.19679 0.690968i 0.236953 0.971521i \(-0.423851\pi\)
0.959839 + 0.280553i \(0.0905179\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) −9.52628 + 5.50000i −0.566279 + 0.326941i −0.755662 0.654962i \(-0.772685\pi\)
0.189383 + 0.981903i \(0.439351\pi\)
\(284\) −12.0000 20.7846i −0.712069 1.23334i
\(285\) 0 0
\(286\) 0 0
\(287\) 10.3923 + 30.0000i 0.613438 + 1.77084i
\(288\) 0 0
\(289\) 9.50000 16.4545i 0.558824 0.967911i
\(290\) 0 0
\(291\) 5.00000 + 8.66025i 0.293105 + 0.507673i
\(292\) 19.0526 + 11.0000i 1.11497 + 0.643726i
\(293\) 12.0000i 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 + 5.19615i −0.173494 + 0.300501i
\(300\) 0 0
\(301\) −2.50000 + 0.866025i −0.144098 + 0.0499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 10.0000 + 17.3205i 0.573539 + 0.993399i
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000i 0.399511i −0.979846 0.199756i \(-0.935985\pi\)
0.979846 0.199756i \(-0.0640148\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) −6.00000 10.3923i −0.340229 0.589294i 0.644246 0.764818i \(-0.277171\pi\)
−0.984475 + 0.175525i \(0.943838\pi\)
\(312\) 0 0
\(313\) 14.7224 + 8.50000i 0.832161 + 0.480448i 0.854592 0.519300i \(-0.173807\pi\)
−0.0224310 + 0.999748i \(0.507141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 26.0000 1.46261
\(317\) −5.19615 3.00000i −0.291845 0.168497i 0.346929 0.937892i \(-0.387225\pi\)
−0.638774 + 0.769395i \(0.720558\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 30.0000i 1.66924i
\(324\) −1.00000 + 1.73205i −0.0555556 + 0.0962250i
\(325\) 0 0
\(326\) 0 0
\(327\) 6.06218 + 3.50000i 0.335239 + 0.193550i
\(328\) 0 0
\(329\) 3.00000 15.5885i 0.165395 0.859419i
\(330\) 0 0
\(331\) −11.5000 + 19.9186i −0.632097 + 1.09482i 0.355025 + 0.934857i \(0.384472\pi\)
−0.987122 + 0.159968i \(0.948861\pi\)
\(332\) 20.7846 12.0000i 1.14070 0.658586i
\(333\) 6.06218 3.50000i 0.332205 0.191799i
\(334\) 0 0
\(335\) 0 0
\(336\) 10.0000 3.46410i 0.545545 0.188982i
\(337\) 13.0000i 0.708155i −0.935216 0.354078i \(-0.884795\pi\)
0.935216 0.354078i \(-0.115205\pi\)
\(338\) 0 0
\(339\) 9.00000 + 15.5885i 0.488813 + 0.846649i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.5885 10.0000i −0.841698 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.7846 + 12.0000i −1.11578 + 0.644194i −0.940319 0.340293i \(-0.889474\pi\)
−0.175457 + 0.984487i \(0.556140\pi\)
\(348\) 10.3923 + 6.00000i 0.557086 + 0.321634i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) −15.5885 3.00000i −0.825029 0.158777i
\(358\) 0 0
\(359\) −3.00000 + 5.19615i −0.158334 + 0.274242i −0.934268 0.356572i \(-0.883946\pi\)
0.775934 + 0.630814i \(0.217279\pi\)
\(360\) 0 0
\(361\) −3.00000 5.19615i −0.157895 0.273482i
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 4.00000 + 3.46410i 0.209657 + 0.181568i
\(365\) 0 0
\(366\) 0 0
\(367\) −16.4545 + 9.50000i −0.858917 + 0.495896i −0.863649 0.504093i \(-0.831827\pi\)
0.00473247 + 0.999989i \(0.498494\pi\)
\(368\) −20.7846 + 12.0000i −1.08347 + 0.625543i
\(369\) 6.00000 10.3923i 0.312348 0.541002i
\(370\) 0 0
\(371\) 0 0
\(372\) 10.0000i 0.518476i
\(373\) 14.7224 + 8.50000i 0.762299 + 0.440113i 0.830120 0.557584i \(-0.188272\pi\)
−0.0678218 + 0.997697i \(0.521605\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −5.50000 9.52628i −0.281774 0.488046i
\(382\) 0 0
\(383\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.866025 + 0.500000i 0.0440225 + 0.0254164i
\(388\) −17.3205 + 10.0000i −0.879316 + 0.507673i
\(389\) 6.00000 + 10.3923i 0.304212 + 0.526911i 0.977086 0.212847i \(-0.0682735\pi\)
−0.672874 + 0.739758i \(0.734940\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −25.1147 14.5000i −1.26047 0.727734i −0.287307 0.957839i \(-0.592760\pi\)
−0.973166 + 0.230105i \(0.926093\pi\)
\(398\) 0 0
\(399\) −12.5000 + 4.33013i −0.625783 + 0.216777i
\(400\) 0 0
\(401\) −12.0000 + 20.7846i −0.599251 + 1.03793i 0.393680 + 0.919247i \(0.371202\pi\)
−0.992932 + 0.118686i \(0.962132\pi\)
\(402\) 0 0
\(403\) 4.33013 2.50000i 0.215699 0.124534i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.50000 + 4.33013i 0.123617 + 0.214111i 0.921192 0.389109i \(-0.127217\pi\)
−0.797574 + 0.603220i \(0.793884\pi\)
\(410\) 0 0
\(411\) −3.00000 + 5.19615i −0.147979 + 0.256307i
\(412\) 2.00000i 0.0985329i
\(413\) 10.3923 12.0000i 0.511372 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.33013 2.50000i 0.212047 0.122426i
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 0 0
\(423\) −5.19615 + 3.00000i −0.252646 + 0.145865i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.19615 1.00000i −0.251459 0.0483934i
\(428\) 36.0000i 1.74013i
\(429\) 0 0
\(430\) 0 0
\(431\) −3.00000 5.19615i −0.144505 0.250290i 0.784683 0.619897i \(-0.212826\pi\)
−0.929188 + 0.369607i \(0.879492\pi\)
\(432\) −3.46410 2.00000i −0.166667 0.0962250i
\(433\) 29.0000i 1.39365i −0.717241 0.696826i \(-0.754595\pi\)
0.717241 0.696826i \(-0.245405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7.00000 + 12.1244i −0.335239 + 0.580651i
\(437\) 25.9808 15.0000i 1.24283 0.717547i
\(438\) 0 0
\(439\) −14.0000 + 24.2487i −0.668184 + 1.15733i 0.310228 + 0.950662i \(0.399595\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) −31.1769 18.0000i −1.48126 0.855206i −0.481486 0.876454i \(-0.659903\pi\)
−0.999774 + 0.0212481i \(0.993236\pi\)
\(444\) 7.00000 + 12.1244i 0.332205 + 0.575396i
\(445\) 0 0
\(446\) 0 0
\(447\) 24.0000i 1.13516i
\(448\) 6.92820 + 20.0000i 0.327327 + 0.944911i
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −31.1769 + 18.0000i −1.46644 + 0.846649i
\(453\) −13.8564 8.00000i −0.651031 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.52628 5.50000i −0.445621 0.257279i 0.260358 0.965512i \(-0.416159\pi\)
−0.705979 + 0.708233i \(0.749493\pi\)
\(458\) 0 0
\(459\) 3.00000 + 5.19615i 0.140028 + 0.242536i
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 7.00000i 0.325318i 0.986682 + 0.162659i \(0.0520070\pi\)
−0.986682 + 0.162659i \(0.947993\pi\)
\(464\) −12.0000 + 20.7846i −0.557086 + 0.964901i
\(465\) 0 0
\(466\) 0 0
\(467\) −25.9808 15.0000i −1.20225 0.694117i −0.241192 0.970477i \(-0.577538\pi\)
−0.961054 + 0.276360i \(0.910872\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −14.0000 12.1244i −0.646460 0.559851i
\(470\) 0 0
\(471\) 5.00000 8.66025i 0.230388 0.399043i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 31.1769i 0.275010 1.42899i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) −3.50000 + 6.06218i −0.159586 + 0.276412i
\(482\) 0 0
\(483\) −5.19615 15.0000i −0.236433 0.682524i
\(484\) −22.0000 −1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −6.06218 + 3.50000i −0.274703 + 0.158600i −0.631023 0.775764i \(-0.717365\pi\)
0.356320 + 0.934364i \(0.384031\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 20.7846 + 12.0000i 0.937043 + 0.541002i
\(493\) 31.1769 18.0000i 1.40414 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) 10.3923 + 30.0000i 0.466159 + 1.34568i
\(498\) 0 0
\(499\) 8.50000 14.7224i 0.380512 0.659067i −0.610623 0.791921i \(-0.709081\pi\)
0.991136 + 0.132855i \(0.0424144\pi\)
\(500\) 0 0
\(501\) −6.00000 10.3923i −0.268060 0.464294i
\(502\) 0 0
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −10.3923 + 6.00000i −0.461538 + 0.266469i
\(508\) 19.0526 11.0000i 0.845321 0.488046i
\(509\) −9.00000 + 15.5885i −0.398918 + 0.690946i −0.993593 0.113020i \(-0.963948\pi\)
0.594675 + 0.803966i \(0.297281\pi\)
\(510\) 0 0
\(511\) −22.0000 19.0526i −0.973223 0.842836i
\(512\) 0 0
\(513\) 4.33013 + 2.50000i 0.191180 + 0.110378i
\(514\) 0 0
\(515\) 0 0
\(516\) −1.00000 + 1.73205i −0.0440225 + 0.0762493i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.0000 + 20.7846i 0.525730 + 0.910590i 0.999551 + 0.0299693i \(0.00954094\pi\)
−0.473821 + 0.880621i \(0.657126\pi\)
\(522\) 0 0
\(523\) −37.2391 21.5000i −1.62835 0.940129i −0.984585 0.174908i \(-0.944037\pi\)
−0.643767 0.765222i \(-0.722629\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) −25.9808 15.0000i −1.13174 0.653410i
\(528\) 0 0
\(529\) 6.50000 + 11.2583i 0.282609 + 0.489493i
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −8.66025 25.0000i −0.375470 1.08389i
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −15.5885 9.00000i −0.672692 0.388379i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.50000 16.4545i 0.408437 0.707433i −0.586278 0.810110i \(-0.699407\pi\)
0.994715 + 0.102677i \(0.0327407\pi\)
\(542\) 0 0
\(543\) 6.06218 3.50000i 0.260153 0.150199i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) −10.3923 6.00000i −0.443937 0.256307i
\(549\) 1.00000 + 1.73205i 0.0426790 + 0.0739221i
\(550\) 0 0
\(551\) 15.0000 25.9808i 0.639021 1.10682i
\(552\) 0 0
\(553\) −33.7750 6.50000i −1.43626 0.276408i
\(554\) 0 0
\(555\) 0 0
\(556\) 5.00000 + 8.66025i 0.212047 + 0.367277i
\(557\) 15.5885 9.00000i 0.660504 0.381342i −0.131965 0.991254i \(-0.542129\pi\)
0.792469 + 0.609912i \(0.208795\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.5885 9.00000i 0.656975 0.379305i −0.134148 0.990961i \(-0.542830\pi\)
0.791123 + 0.611656i \(0.209497\pi\)
\(564\) −6.00000 10.3923i −0.252646 0.437595i
\(565\) 0 0
\(566\) 0 0
\(567\) 1.73205 2.00000i 0.0727393 0.0839921i
\(568\) 0 0
\(569\) −12.0000 + 20.7846i −0.503066 + 0.871336i 0.496928 + 0.867792i \(0.334461\pi\)
−0.999994 + 0.00354413i \(0.998872\pi\)
\(570\) 0 0
\(571\) 3.50000 + 6.06218i 0.146470 + 0.253694i 0.929921 0.367760i \(-0.119875\pi\)
−0.783450 + 0.621455i \(0.786542\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 4.00000 6.92820i 0.166667 0.288675i
\(577\) −6.06218 + 3.50000i −0.252372 + 0.145707i −0.620850 0.783930i \(-0.713212\pi\)
0.368478 + 0.929636i \(0.379879\pi\)
\(578\) 0 0
\(579\) 2.50000 4.33013i 0.103896 0.179954i
\(580\) 0 0
\(581\) −30.0000 + 10.3923i −1.24461 + 0.431145i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000i 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) −13.8564 + 2.00000i −0.571429 + 0.0824786i
\(589\) −25.0000 −1.03011
\(590\) 0 0
\(591\) −3.00000 5.19615i −0.123404 0.213741i
\(592\) −24.2487 + 14.0000i −0.996616 + 0.575396i
\(593\) 31.1769 + 18.0000i 1.28028 + 0.739171i 0.976900 0.213697i \(-0.0685507\pi\)
0.303383 + 0.952869i \(0.401884\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 48.0000 1.96616
\(597\) 13.8564 + 8.00000i 0.567105 + 0.327418i
\(598\) 0 0
\(599\) 6.00000 + 10.3923i 0.245153 + 0.424618i 0.962175 0.272433i \(-0.0878284\pi\)
−0.717021 + 0.697051i \(0.754495\pi\)
\(600\) 0 0
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 0 0
\(603\) 7.00000i 0.285062i
\(604\) 16.0000 27.7128i 0.651031 1.12762i
\(605\) 0 0
\(606\) 0 0
\(607\) 16.4545 + 9.50000i 0.667867 + 0.385593i 0.795268 0.606258i \(-0.207330\pi\)
−0.127401 + 0.991851i \(0.540664\pi\)
\(608\) 0 0
\(609\) −12.0000 10.3923i −0.486265 0.421117i
\(610\) 0 0
\(611\) 3.00000 5.19615i 0.121367 0.210214i
\(612\) −10.3923 + 6.00000i −0.420084 + 0.242536i
\(613\) −32.9090 + 19.0000i −1.32918 + 0.767403i −0.985173 0.171564i \(-0.945118\pi\)
−0.344008 + 0.938967i \(0.611785\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) 17.5000 + 30.3109i 0.703384 + 1.21830i 0.967271 + 0.253744i \(0.0816620\pi\)
−0.263887 + 0.964554i \(0.585005\pi\)
\(620\) 0 0
\(621\) −3.00000 + 5.19615i −0.120386 + 0.208514i
\(622\) 0 0
\(623\) 15.5885 + 3.00000i 0.624538 + 0.120192i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 17.3205 + 10.0000i 0.691164 + 0.399043i
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 3.46410 2.00000i 0.137686 0.0794929i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.33013 5.50000i −0.171566 0.217918i
\(638\) 0 0
\(639\) 6.00000 10.3923i 0.237356 0.411113i
\(640\) 0 0
\(641\) 15.0000 + 25.9808i 0.592464 + 1.02618i 0.993899 + 0.110291i \(0.0351782\pi\)
−0.401435 + 0.915888i \(0.631488\pi\)
\(642\) 0 0
\(643\) 25.0000i 0.985904i 0.870057 + 0.492952i \(0.164082\pi\)
−0.870057 + 0.492952i \(0.835918\pi\)
\(644\) 30.0000 10.3923i 1.18217 0.409514i
\(645\) 0 0
\(646\) 0 0
\(647\) −20.7846 + 12.0000i −0.817127 + 0.471769i −0.849425 0.527710i \(-0.823051\pi\)
0.0322975 + 0.999478i \(0.489718\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.50000 + 12.9904i −0.0979827 + 0.509133i
\(652\) 8.00000i 0.313304i
\(653\) −20.7846 12.0000i −0.813365 0.469596i 0.0347583 0.999396i \(-0.488934\pi\)
−0.848123 + 0.529799i \(0.822267\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −24.0000 + 41.5692i −0.937043 + 1.62301i
\(657\) 11.0000i 0.429151i
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 6.50000 + 11.2583i 0.252821 + 0.437898i 0.964301 0.264807i \(-0.0853084\pi\)
−0.711481 + 0.702706i \(0.751975\pi\)
\(662\) 0 0
\(663\) −5.19615 3.00000i −0.201802 0.116510i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 31.1769 + 18.0000i 1.20717 + 0.696963i
\(668\) 20.7846 12.0000i 0.804181 0.464294i
\(669\) −2.00000 3.46410i −0.0773245 0.133930i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 31.0000i 1.19496i 0.801883 + 0.597481i \(0.203832\pi\)
−0.801883 + 0.597481i \(0.796168\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −12.0000 20.7846i −0.461538 0.799408i
\(677\) 15.5885 + 9.00000i 0.599113 + 0.345898i 0.768693 0.639618i \(-0.220908\pi\)
−0.169580 + 0.985517i \(0.554241\pi\)
\(678\) 0 0
\(679\) 25.0000 8.66025i 0.959412 0.332350i
\(680\) 0 0
\(681\) −6.00000 + 10.3923i −0.229920 + 0.398234i
\(682\) 0 0
\(683\) −15.5885 + 9.00000i −0.596476 + 0.344375i −0.767654 0.640865i \(-0.778576\pi\)
0.171178 + 0.985240i \(0.445243\pi\)
\(684\) −5.00000 + 8.66025i −0.191180 + 0.331133i
\(685\) 0 0
\(686\) 0 0
\(687\) 29.0000i 1.10642i
\(688\) −3.46410 2.00000i −0.132068 0.0762493i
\(689\) 0 0
\(690\) 0 0
\(691\) −5.50000 + 9.52628i −0.209230 + 0.362397i −0.951472 0.307735i \(-0.900429\pi\)
0.742242 + 0.670132i \(0.233762\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 62.3538 36.0000i 2.36182 1.36360i
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 30.3109 17.5000i 1.14320 0.660025i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) −5.00000 + 8.66025i −0.187779 + 0.325243i −0.944509 0.328484i \(-0.893462\pi\)
0.756730 + 0.653727i \(0.226796\pi\)
\(710\) 0 0
\(711\) 6.50000 + 11.2583i 0.243769 + 0.422220i
\(712\) 0 0
\(713\) 30.0000i 1.12351i
\(714\) 0 0
\(715\) 0 0
\(716\) 18.0000 31.1769i 0.672692 1.16514i
\(717\) −5.19615 + 3.00000i −0.194054 + 0.112037i
\(718\) 0 0
\(719\) 3.00000 5.19615i 0.111881 0.193784i −0.804648 0.593753i \(-0.797646\pi\)
0.916529 + 0.399969i \(0.130979\pi\)
\(720\) 0 0
\(721\) 0.500000 2.59808i 0.0186210 0.0967574i
\(722\) 0 0
\(723\) 1.73205 + 1.00000i 0.0644157 + 0.0371904i
\(724\) 7.00000 + 12.1244i 0.260153 + 0.450598i
\(725\) 0 0
\(726\) 0 0
\(727\) 7.00000i 0.259616i −0.991539 0.129808i \(-0.958564\pi\)
0.991539 0.129808i \(-0.0414360\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 3.00000 + 5.19615i 0.110959 + 0.192187i
\(732\) −3.46410 + 2.00000i −0.128037 + 0.0739221i
\(733\) −11.2583 6.50000i −0.415836 0.240083i 0.277458 0.960738i \(-0.410508\pi\)
−0.693294 + 0.720655i \(0.743841\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3.50000 6.06218i −0.128750 0.223001i 0.794443 0.607339i \(-0.207763\pi\)
−0.923192 + 0.384338i \(0.874430\pi\)
\(740\) 0 0
\(741\) −5.00000 −0.183680
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 10.3923 + 6.00000i 0.380235 + 0.219529i
\(748\) 0 0
\(749\) 9.00000 46.7654i 0.328853 1.70877i
\(750\) 0 0
\(751\) 9.50000 16.4545i 0.346660 0.600433i −0.638994 0.769212i \(-0.720649\pi\)
0.985654 + 0.168779i \(0.0539825\pi\)
\(752\) 20.7846 12.0000i 0.757937 0.437595i
\(753\) 15.5885 9.00000i 0.568075 0.327978i
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 + 3.46410i 0.145479 + 0.125988i
\(757\) 50.0000i 1.81728i 0.417579 + 0.908640i \(0.362879\pi\)
−0.417579 + 0.908640i \(0.637121\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 36.3731i 0.761249 1.31852i −0.180957 0.983491i \(-0.557920\pi\)
0.942207 0.335032i \(-0.108747\pi\)
\(762\) 0 0
\(763\) 12.1244 14.0000i 0.438931 0.506834i
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 0 0
\(767\) 5.19615 3.00000i 0.187622 0.108324i
\(768\) 13.8564 + 8.00000i 0.500000 + 0.288675i
\(769\) 13.0000 0.468792 0.234396 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 8.66025 + 5.00000i 0.311689 + 0.179954i
\(773\) 20.7846 12.0000i 0.747570 0.431610i −0.0772449 0.997012i \(-0.524612\pi\)
0.824815 + 0.565402i \(0.191279\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.06218 17.5000i −0.217479 0.627809i
\(778\) 0 0
\(779\) 30.0000 51.9615i 1.07486 1.86171i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) −4.00000 27.7128i −0.142857 0.989743i
\(785\) 0 0
\(786\) 0 0
\(787\) −24.2487 + 14.0000i −0.864373 + 0.499046i −0.865474 0.500953i \(-0.832983\pi\)
0.00110111 + 0.999999i \(0.499650\pi\)
\(788\) 10.3923 6.00000i 0.370211 0.213741i
\(789\) −6.00000 + 10.3923i −0.213606 + 0.369976i
\(790\) 0 0
\(791\) 45.0000 15.5885i 1.60002 0.554262i
\(792\) 0 0
\(793\) −1.73205 1.00000i −0.0615069 0.0355110i
\(794\) 0 0
\(795\) 0 0
\(796\) −16.0000 + 27.7128i −0.567105 + 0.982255i
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) −3.00000 5.19615i −0.106000 0.183597i
\(802\) 0 0
\(803\) 0 0
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) 0 0
\(807\) 15.5885 + 9.00000i 0.548740 + 0.316815i
\(808\) 0 0
\(809\) 9.00000 + 15.5885i 0.316423 + 0.548061i 0.979739 0.200279i \(-0.0641847\pi\)
−0.663316 + 0.748340i \(0.730851\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 20.7846 24.0000i 0.729397 0.842235i
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) −12.0000 20.7846i −0.420084 0.727607i
\(817\) 4.33013 + 2.50000i 0.151492 + 0.0874639i
\(818\) 0 0
\(819\) −0.500000 + 2.59808i −0.0174714 + 0.0907841i
\(820\) 0 0
\(821\) 12.0000 20.7846i 0.418803 0.725388i −0.577016 0.816733i \(-0.695783\pi\)
0.995819 + 0.0913446i \(0.0291165\pi\)
\(822\) 0 0
\(823\) 34.6410 20.0000i 1.20751 0.697156i 0.245295 0.969448i \(-0.421115\pi\)
0.962215 + 0.272292i \(0.0877817\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000i 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) −10.3923 6.00000i −0.361158 0.208514i
\(829\) −9.50000 16.4545i −0.329949 0.571488i 0.652553 0.757743i \(-0.273698\pi\)
−0.982501 + 0.186256i \(0.940365\pi\)
\(830\) 0 0
\(831\) −11.5000 + 19.9186i −0.398931 + 0.690968i
\(832\) 8.00000i 0.277350i
\(833\) −15.5885 + 39.0000i −0.540108 + 1.35127i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.33013 2.50000i 0.149671 0.0864126i
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 20.7846 12.0000i 0.715860 0.413302i
\(844\) 4.00000 + 6.92820i 0.137686 + 0.238479i
\(845\) 0 0
\(846\) 0 0
\(847\) 28.5788 + 5.50000i 0.981981 + 0.188982i
\(848\) 0 0
\(849\) 5.50000 9.52628i 0.188760 0.326941i
\(850\) 0 0
\(851\) 21.0000 + 36.3731i 0.719871 + 1.24685i
\(852\) 20.7846 + 12.0000i 0.712069 + 0.411113i
\(853\) 17.0000i 0.582069i −0.956713 0.291034i \(-0.906001\pi\)
0.956713 0.291034i \(-0.0939994\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 36.3731 21.0000i 1.24248 0.717346i 0.272882 0.962048i \(-0.412023\pi\)
0.969599 + 0.244701i \(0.0786899\pi\)
\(858\) 0 0
\(859\) −2.00000 + 3.46410i −0.0682391 + 0.118194i −0.898126 0.439738i \(-0.855071\pi\)
0.829887 + 0.557931i \(0.188405\pi\)
\(860\) 0 0
\(861\) −24.0000 20.7846i −0.817918 0.708338i
\(862\) 0 0
\(863\) −5.19615 3.00000i −0.176879 0.102121i 0.408946 0.912558i \(-0.365896\pi\)
−0.585826 + 0.810437i \(0.699230\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) −25.9808 5.00000i −0.881845 0.169711i
\(869\) 0 0
\(870\) 0 0
\(871\) −3.50000 6.06218i −0.118593 0.205409i
\(872\) 0 0
\(873\) −8.66025 5.00000i −0.293105 0.169224i
\(874\) 0 0
\(875\) 0 0
\(876\) −22.0000 −0.743311
\(877\) −1.73205 1.00000i −0.0584872 0.0337676i 0.470471 0.882415i \(-0.344084\pi\)
−0.528958 + 0.848648i \(0.677417\pi\)
\(878\) 0 0
\(879\) 6.00000 + 10.3923i 0.202375 + 0.350524i
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) 31.0000i 1.04323i 0.853180 + 0.521617i \(0.174671\pi\)
−0.853180 + 0.521617i \(0.825329\pi\)
\(884\) 6.00000 10.3923i 0.201802 0.349531i
\(885\) 0 0
\(886\) 0 0
\(887\) −10.3923 6.00000i −0.348939 0.201460i 0.315279 0.948999i \(-0.397902\pi\)
−0.664218 + 0.747539i \(0.731235\pi\)
\(888\) 0 0
\(889\) −27.5000 + 9.52628i −0.922320 + 0.319501i
\(890\) 0 0
\(891\) 0 0
\(892\) 6.92820 4.00000i 0.231973 0.133930i
\(893\) −25.9808 + 15.0000i −0.869413 + 0.501956i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000i 0.200334i
\(898\) 0 0
\(899\) −15.0000 25.9808i −0.500278 0.866507i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.73205 2.00000i 0.0576390 0.0665558i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −26.8468 + 15.5000i −0.891433 + 0.514669i −0.874411 0.485186i \(-0.838752\pi\)
−0.0170220 + 0.999855i \(0.505419\pi\)
\(908\) −20.7846 12.0000i −0.689761 0.398234i
\(909\) 0 0
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) −17.3205 10.0000i −0.573539 0.331133i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −58.0000 −1.91637
\(917\) 15.5885 + 3.00000i 0.514776 + 0.0990687i
\(918\) 0 0
\(919\) 23.5000 40.7032i 0.775193 1.34267i −0.159492 0.987199i \(-0.550986\pi\)
0.934686 0.355475i \(-0.115681\pi\)
\(920\) 0 0
\(921\) 3.50000 + 6.06218i 0.115329 + 0.199756i
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −0.866025 + 0.500000i −0.0284440 + 0.0164222i
\(928\) 0 0
\(929\) −6.00000 + 10.3923i −0.196854 + 0.340960i −0.947507 0.319736i \(-0.896406\pi\)
0.750653 + 0.660697i \(0.229739\pi\)
\(930\) 0 0
\(931\) 5.00000 + 34.6410i 0.163868 + 1.13531i
\(932\) 12.0000i 0.393073i
\(933\) 10.3923 + 6.00000i 0.340229 + 0.196431i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 29.0000i 0.947389i 0.880689 + 0.473694i \(0.157080\pi\)
−0.880689 + 0.473694i \(0.842920\pi\)
\(938\) 0 0
\(939\) −17.0000 −0.554774
\(940\) 0 0
\(941\) 15.0000 + 25.9808i 0.488986 + 0.846949i 0.999920 0.0126715i \(-0.00403357\pi\)
−0.510934 + 0.859620i \(0.670700\pi\)
\(942\) 0 0
\(943\) 62.3538 + 36.0000i 2.03052 + 1.17232i
\(944\) 24.0000 0.781133
\(945\) 0 0
\(946\) 0 0
\(947\) −41.5692 24.0000i −1.35082 0.779895i −0.362454 0.932002i \(-0.618061\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(948\) −22.5167 + 13.0000i −0.731307 + 0.422220i
\(949\) −5.50000 9.52628i −0.178538 0.309236i
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.00000 10.3923i −0.194054 0.336111i
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 + 10.3923i 0.387500 + 0.335585i
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) −15.5885 + 9.00000i −0.502331 + 0.290021i
\(964\) −2.00000 + 3.46410i −0.0644157 + 0.111571i
\(965\) 0 0
\(966\) 0 0
\(967\) 55.0000i 1.76868i −0.466843 0.884340i \(-0.654609\pi\)
0.466843 0.884340i \(-0.345391\pi\)
\(968\) 0 0
\(969\) 15.0000 + 25.9808i 0.481869 + 0.834622i
\(970\) 0 0
\(971\) 12.0000 20.7846i 0.385098 0.667010i −0.606685 0.794943i \(-0.707501\pi\)
0.991783 + 0.127933i \(0.0408342\pi\)
\(972\) 2.00000i 0.0641500i
\(973\) −4.33013 12.5000i −0.138817 0.400732i
\(974\) 0 0
\(975\) 0 0
\(976\) −4.00000 6.92820i −0.128037 0.221766i
\(977\) −10.3923 + 6.00000i −0.332479 + 0.191957i −0.656941 0.753942i \(-0.728150\pi\)
0.324462 + 0.945899i \(0.394817\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 0 0
\(983\) −15.5885 + 9.00000i −0.497195 + 0.287055i −0.727554 0.686050i \(-0.759343\pi\)
0.230360 + 0.973106i \(0.426010\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.19615 + 15.0000i 0.165395 + 0.477455i
\(988\) 10.0000i 0.318142i
\(989\) −3.00000 + 5.19615i −0.0953945 + 0.165228i
\(990\) 0 0
\(991\) 21.5000 + 37.2391i 0.682970 + 1.18294i 0.974070 + 0.226246i \(0.0726454\pi\)
−0.291100 + 0.956693i \(0.594021\pi\)
\(992\) 0 0
\(993\) 23.0000i 0.729883i
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 + 20.7846i −0.380235 + 0.658586i
\(997\) −26.8468 + 15.5000i −0.850246 + 0.490890i −0.860734 0.509055i \(-0.829995\pi\)
0.0104877 + 0.999945i \(0.496662\pi\)
\(998\) 0 0
\(999\) −3.50000 + 6.06218i −0.110735 + 0.191799i
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 525.2.r.b.499.1 4
5.2 odd 4 525.2.i.c.226.1 2
5.3 odd 4 105.2.i.a.16.1 2
5.4 even 2 inner 525.2.r.b.499.2 4
7.4 even 3 inner 525.2.r.b.424.2 4
15.8 even 4 315.2.j.b.226.1 2
20.3 even 4 1680.2.bg.m.961.1 2
35.2 odd 12 3675.2.a.h.1.1 1
35.3 even 12 735.2.i.c.361.1 2
35.4 even 6 inner 525.2.r.b.424.1 4
35.12 even 12 3675.2.a.i.1.1 1
35.13 even 4 735.2.i.c.226.1 2
35.18 odd 12 105.2.i.a.46.1 yes 2
35.23 odd 12 735.2.a.e.1.1 1
35.32 odd 12 525.2.i.c.151.1 2
35.33 even 12 735.2.a.d.1.1 1
105.23 even 12 2205.2.a.f.1.1 1
105.53 even 12 315.2.j.b.46.1 2
105.68 odd 12 2205.2.a.d.1.1 1
140.123 even 12 1680.2.bg.m.1201.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.a.16.1 2 5.3 odd 4
105.2.i.a.46.1 yes 2 35.18 odd 12
315.2.j.b.46.1 2 105.53 even 12
315.2.j.b.226.1 2 15.8 even 4
525.2.i.c.151.1 2 35.32 odd 12
525.2.i.c.226.1 2 5.2 odd 4
525.2.r.b.424.1 4 35.4 even 6 inner
525.2.r.b.424.2 4 7.4 even 3 inner
525.2.r.b.499.1 4 1.1 even 1 trivial
525.2.r.b.499.2 4 5.4 even 2 inner
735.2.a.d.1.1 1 35.33 even 12
735.2.a.e.1.1 1 35.23 odd 12
735.2.i.c.226.1 2 35.13 even 4
735.2.i.c.361.1 2 35.3 even 12
1680.2.bg.m.961.1 2 20.3 even 4
1680.2.bg.m.1201.1 2 140.123 even 12
2205.2.a.d.1.1 1 105.68 odd 12
2205.2.a.f.1.1 1 105.23 even 12
3675.2.a.h.1.1 1 35.2 odd 12
3675.2.a.i.1.1 1 35.12 even 12