Properties

Label 945.2.cs.a.104.4
Level $945$
Weight $2$
Character 945.104
Analytic conductor $7.546$
Analytic rank $0$
Dimension $24$
CM discriminant -35
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(104,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11, 9, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cs (of order \(18\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 104.4
Character \(\chi\) \(=\) 945.104
Dual form 945.2.cs.a.209.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.62968 + 0.586627i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(2.20210 + 0.388289i) q^{5} +(-2.02676 + 1.70066i) q^{7} +(2.31174 + 1.91203i) q^{9} +O(q^{10})\) \(q+(1.62968 + 0.586627i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(2.20210 + 0.388289i) q^{5} +(-2.02676 + 1.70066i) q^{7} +(2.31174 + 1.91203i) q^{9} +(4.46437 - 0.787188i) q^{11} +(-1.74267 - 2.99385i) q^{12} +(-6.63961 + 2.41662i) q^{13} +(3.36094 + 1.92460i) q^{15} +(0.694593 + 3.93923i) q^{16} +(5.90187 + 3.40745i) q^{17} +(-2.87463 - 3.42585i) q^{20} +(-4.30063 + 1.58258i) q^{21} +(4.69846 + 1.71010i) q^{25} +(2.64575 + 4.47214i) q^{27} +5.29150 q^{28} +(1.65407 - 4.54453i) q^{29} +(7.73729 + 1.33605i) q^{33} +(-5.12348 + 2.95804i) q^{35} +(-1.08373 - 5.90132i) q^{36} +(-12.2381 + 0.0433527i) q^{39} +(-7.85180 - 4.53324i) q^{44} +(4.34825 + 5.10810i) q^{45} +(6.27240 + 7.47515i) q^{47} +(-1.17889 + 6.82717i) q^{48} +(1.21554 - 6.89365i) q^{49} +(7.61929 + 9.01526i) q^{51} +(13.2792 + 4.83324i) q^{52} +10.1366 q^{55} +(-2.67504 - 7.26940i) q^{60} +(-7.93705 + 0.0562338i) q^{63} +(4.00000 - 6.92820i) q^{64} +(-15.5594 + 2.74354i) q^{65} +(-4.66166 - 12.8078i) q^{68} +(1.16957 + 0.675254i) q^{71} +(-8.36548 - 14.4894i) q^{73} +(6.65382 + 5.54317i) q^{75} +(-7.70948 + 9.18780i) q^{77} +(-6.58720 - 2.39754i) q^{79} +8.94427i q^{80} +(1.68826 + 8.84024i) q^{81} +(4.73160 - 13.0000i) q^{83} +(8.62348 + 3.10414i) q^{84} +(11.6734 + 9.79516i) q^{85} +(5.36156 - 6.43582i) q^{87} +(9.34707 - 16.1896i) q^{91} +(1.71484 + 9.72536i) q^{97} +(11.8256 + 6.71624i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 6 q^{9} - 18 q^{11} - 24 q^{39} + 96 q^{64} - 180 q^{65} - 12 q^{79} + 102 q^{81} + 84 q^{84} + 60 q^{85} + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1\) \(e\left(\frac{11}{18}\right)\) \(-1\)

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 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(3\) 1.62968 + 0.586627i 0.940898 + 0.338689i
\(4\) −1.53209 1.28558i −0.766044 0.642788i
\(5\) 2.20210 + 0.388289i 0.984808 + 0.173648i
\(6\) 0 0
\(7\) −2.02676 + 1.70066i −0.766044 + 0.642788i
\(8\) 0 0
\(9\) 2.31174 + 1.91203i 0.770579 + 0.637344i
\(10\) 0 0
\(11\) 4.46437 0.787188i 1.34606 0.237346i 0.546259 0.837616i \(-0.316051\pi\)
0.799798 + 0.600270i \(0.204940\pi\)
\(12\) −1.74267 2.99385i −0.503065 0.864249i
\(13\) −6.63961 + 2.41662i −1.84150 + 0.670250i −0.852416 + 0.522864i \(0.824864\pi\)
−0.989079 + 0.147386i \(0.952914\pi\)
\(14\) 0 0
\(15\) 3.36094 + 1.92460i 0.867791 + 0.496929i
\(16\) 0.694593 + 3.93923i 0.173648 + 0.984808i
\(17\) 5.90187 + 3.40745i 1.43141 + 0.826428i 0.997229 0.0743959i \(-0.0237028\pi\)
0.434186 + 0.900823i \(0.357036\pi\)
\(18\) 0 0
\(19\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) −2.87463 3.42585i −0.642788 0.766044i
\(21\) −4.30063 + 1.58258i −0.938475 + 0.345347i
\(22\) 0 0
\(23\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(24\) 0 0
\(25\) 4.69846 + 1.71010i 0.939693 + 0.342020i
\(26\) 0 0
\(27\) 2.64575 + 4.47214i 0.509175 + 0.860663i
\(28\) 5.29150 1.00000
\(29\) 1.65407 4.54453i 0.307154 0.843898i −0.686055 0.727550i \(-0.740659\pi\)
0.993208 0.116348i \(-0.0371189\pi\)
\(30\) 0 0
\(31\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(32\) 0 0
\(33\) 7.73729 + 1.33605i 1.34689 + 0.232576i
\(34\) 0 0
\(35\) −5.12348 + 2.95804i −0.866025 + 0.500000i
\(36\) −1.08373 5.90132i −0.180621 0.983553i
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) −12.2381 + 0.0433527i −1.95967 + 0.00694200i
\(40\) 0 0
\(41\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(42\) 0 0
\(43\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(44\) −7.85180 4.53324i −1.18370 0.683411i
\(45\) 4.34825 + 5.10810i 0.648199 + 0.761471i
\(46\) 0 0
\(47\) 6.27240 + 7.47515i 0.914923 + 1.09036i 0.995608 + 0.0936230i \(0.0298448\pi\)
−0.0806848 + 0.996740i \(0.525711\pi\)
\(48\) −1.17889 + 6.82717i −0.170158 + 0.985417i
\(49\) 1.21554 6.89365i 0.173648 0.984808i
\(50\) 0 0
\(51\) 7.61929 + 9.01526i 1.06691 + 1.26239i
\(52\) 13.2792 + 4.83324i 1.84150 + 0.670250i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 10.1366 1.36682
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(60\) −2.67504 7.26940i −0.345347 0.938475i
\(61\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(62\) 0 0
\(63\) −7.93705 + 0.0562338i −0.999975 + 0.00708479i
\(64\) 4.00000 6.92820i 0.500000 0.866025i
\(65\) −15.5594 + 2.74354i −1.92991 + 0.340295i
\(66\) 0 0
\(67\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(68\) −4.66166 12.8078i −0.565310 1.55318i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.16957 + 0.675254i 0.138803 + 0.0801380i 0.567793 0.823171i \(-0.307797\pi\)
−0.428990 + 0.903309i \(0.641131\pi\)
\(72\) 0 0
\(73\) −8.36548 14.4894i −0.979105 1.69586i −0.665664 0.746252i \(-0.731851\pi\)
−0.313441 0.949608i \(-0.601482\pi\)
\(74\) 0 0
\(75\) 6.65382 + 5.54317i 0.768317 + 0.640070i
\(76\) 0 0
\(77\) −7.70948 + 9.18780i −0.878576 + 1.04705i
\(78\) 0 0
\(79\) −6.58720 2.39754i −0.741117 0.269745i −0.0562544 0.998416i \(-0.517916\pi\)
−0.684863 + 0.728672i \(0.740138\pi\)
\(80\) 8.94427i 1.00000i
\(81\) 1.68826 + 8.84024i 0.187585 + 0.982248i
\(82\) 0 0
\(83\) 4.73160 13.0000i 0.519360 1.42693i −0.351866 0.936050i \(-0.614453\pi\)
0.871227 0.490881i \(-0.163325\pi\)
\(84\) 8.62348 + 3.10414i 0.940898 + 0.338689i
\(85\) 11.6734 + 9.79516i 1.26616 + 1.06243i
\(86\) 0 0
\(87\) 5.36156 6.43582i 0.574820 0.689993i
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 9.34707 16.1896i 0.979839 1.69713i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.71484 + 9.72536i 0.174116 + 0.987461i 0.939159 + 0.343482i \(0.111606\pi\)
−0.765043 + 0.643979i \(0.777282\pi\)
\(98\) 0 0
\(99\) 11.8256 + 6.71624i 1.18851 + 0.675008i
\(100\) −5.00000 8.66025i −0.500000 0.866025i
\(101\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(102\) 0 0
\(103\) −3.25613 + 18.4664i −0.320836 + 1.81955i 0.216616 + 0.976257i \(0.430498\pi\)
−0.537452 + 0.843294i \(0.680613\pi\)
\(104\) 0 0
\(105\) −10.0849 + 1.81510i −0.984186 + 0.177136i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.69574 10.2530i 0.163173 0.986598i
\(109\) −15.5685 −1.49119 −0.745595 0.666400i \(-0.767834\pi\)
−0.745595 + 0.666400i \(0.767834\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.10705 6.80262i −0.766044 0.642788i
\(113\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.37652 + 4.83619i −0.777741 + 0.449029i
\(117\) −19.9697 7.10855i −1.84620 0.657186i
\(118\) 0 0
\(119\) −17.7566 + 3.13097i −1.62774 + 0.287015i
\(120\) 0 0
\(121\) 8.97428 3.26637i 0.815844 0.296943i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.68246 + 5.59017i 0.866025 + 0.500000i
\(126\) 0 0
\(127\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(132\) −10.1366 11.9938i −0.882280 1.04393i
\(133\) 0 0
\(134\) 0 0
\(135\) 4.08972 + 10.8754i 0.351987 + 0.936005i
\(136\) 0 0
\(137\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(138\) 0 0
\(139\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(140\) 11.6524 + 2.05463i 0.984808 + 0.173648i
\(141\) 5.83690 + 15.8617i 0.491555 + 1.33579i
\(142\) 0 0
\(143\) −27.7393 + 16.0153i −2.31968 + 1.33927i
\(144\) −5.92622 + 10.4346i −0.493852 + 0.869546i
\(145\) 5.40702 9.36524i 0.449029 0.777741i
\(146\) 0 0
\(147\) 6.02494 10.5214i 0.496929 0.867791i
\(148\) 0 0
\(149\) −3.42640 9.41395i −0.280701 0.771221i −0.997279 0.0737137i \(-0.976515\pi\)
0.716578 0.697507i \(-0.245707\pi\)
\(150\) 0 0
\(151\) −1.71990 9.75405i −0.139964 0.793773i −0.971274 0.237964i \(-0.923520\pi\)
0.831310 0.555809i \(-0.187591\pi\)
\(152\) 0 0
\(153\) 7.12843 + 19.1617i 0.576299 + 1.54913i
\(154\) 0 0
\(155\) 0 0
\(156\) 18.8056 + 15.6666i 1.50565 + 1.25433i
\(157\) 4.25060 24.1063i 0.339235 1.92390i −0.0413387 0.999145i \(-0.513162\pi\)
0.380573 0.924751i \(-0.375727\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 16.5195 + 5.94642i 1.28604 + 0.462928i
\(166\) 0 0
\(167\) −25.4357 4.48500i −1.96827 0.347060i −0.990546 0.137179i \(-0.956196\pi\)
−0.977727 0.209881i \(-0.932692\pi\)
\(168\) 0 0
\(169\) 28.2857 23.7346i 2.17583 1.82574i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 25.8135 4.55162i 1.96257 0.346054i 0.966607 0.256265i \(-0.0824918\pi\)
0.995961 0.0897890i \(-0.0286193\pi\)
\(174\) 0 0
\(175\) −12.4310 + 4.52450i −0.939693 + 0.342020i
\(176\) 6.20183 + 17.0394i 0.467481 + 1.28439i
\(177\) 0 0
\(178\) 0 0
\(179\) −18.6895 10.7904i −1.39692 0.806511i −0.402849 0.915267i \(-0.631980\pi\)
−0.994068 + 0.108756i \(0.965313\pi\)
\(180\) −0.0950524 13.4161i −0.00708479 0.999975i
\(181\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 29.0304 + 10.5662i 2.12291 + 0.772678i
\(188\) 19.5162i 1.42337i
\(189\) −12.9679 4.56445i −0.943274 0.332015i
\(190\) 0 0
\(191\) 2.02342 5.55930i 0.146409 0.402257i −0.844711 0.535222i \(-0.820228\pi\)
0.991121 + 0.132966i \(0.0424500\pi\)
\(192\) 10.5830 8.94427i 0.763763 0.645497i
\(193\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(194\) 0 0
\(195\) −26.9663 4.65646i −1.93110 0.333456i
\(196\) −10.7246 + 8.99903i −0.766044 + 0.642788i
\(197\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0 0
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.37627 + 12.0237i 0.307154 + 0.843898i
\(204\) −0.0836275 23.6073i −0.00585510 1.65284i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −14.1314 24.4764i −0.979839 1.69713i
\(209\) 0 0
\(210\) 0 0
\(211\) 5.04055 28.5864i 0.347006 1.96797i 0.133226 0.991086i \(-0.457467\pi\)
0.213780 0.976882i \(-0.431422\pi\)
\(212\) 0 0
\(213\) 1.50991 + 1.78656i 0.103458 + 0.122413i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.13319 28.5206i −0.346869 1.92724i
\(220\) −15.5302 13.0314i −1.04705 0.878576i
\(221\) −47.4206 8.36153i −3.18986 0.562458i
\(222\) 0 0
\(223\) −20.3816 + 17.1022i −1.36486 + 1.14525i −0.390406 + 0.920643i \(0.627665\pi\)
−0.974449 + 0.224607i \(0.927890\pi\)
\(224\) 0 0
\(225\) 7.59185 + 12.9369i 0.506123 + 0.862461i
\(226\) 0 0
\(227\) −2.72684 + 0.480816i −0.180987 + 0.0319129i −0.263407 0.964685i \(-0.584846\pi\)
0.0824200 + 0.996598i \(0.473735\pi\)
\(228\) 0 0
\(229\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(230\) 0 0
\(231\) −17.9538 + 10.4506i −1.18127 + 0.687600i
\(232\) 0 0
\(233\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 0 0
\(235\) 10.9099 + 18.8965i 0.711684 + 1.23267i
\(236\) 0 0
\(237\) −9.32858 7.77146i −0.605957 0.504811i
\(238\) 0 0
\(239\) −14.5170 + 17.3007i −0.939026 + 1.11909i 0.0536837 + 0.998558i \(0.482904\pi\)
−0.992710 + 0.120530i \(0.961541\pi\)
\(240\) −5.24695 + 14.5763i −0.338689 + 0.940898i
\(241\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(242\) 0 0
\(243\) −2.43459 + 15.3972i −0.156179 + 0.987729i
\(244\) 0 0
\(245\) 5.35346 14.7085i 0.342020 0.939693i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 15.3371 18.4101i 0.971951 1.16670i
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 12.2326 + 10.1175i 0.770579 + 0.637344i
\(253\) 0 0
\(254\) 0 0
\(255\) 13.2779 + 22.8110i 0.831493 + 1.42848i
\(256\) −15.0351 + 5.47232i −0.939693 + 0.342020i
\(257\) −10.9594 30.1108i −0.683631 1.87826i −0.374317 0.927301i \(-0.622123\pi\)
−0.309314 0.950960i \(-0.600099\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 27.3654 + 15.7994i 1.69713 + 0.979839i
\(261\) 12.5131 7.34312i 0.774540 0.454528i
\(262\) 0 0
\(263\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −9.32333 + 25.6156i −0.565310 + 1.55318i
\(273\) 24.7300 20.9007i 1.49673 1.26497i
\(274\) 0 0
\(275\) 22.3218 + 3.93594i 1.34606 + 0.237346i
\(276\) 0 0
\(277\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.4266 2.89645i 0.979929 0.172788i 0.339333 0.940666i \(-0.389799\pi\)
0.640596 + 0.767878i \(0.278687\pi\)
\(282\) 0 0
\(283\) −22.1641 + 8.06706i −1.31752 + 0.479537i −0.902662 0.430350i \(-0.858390\pi\)
−0.414855 + 0.909887i \(0.636168\pi\)
\(284\) −0.923802 2.53813i −0.0548176 0.150610i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 14.7214 + 25.4982i 0.865965 + 1.49990i
\(290\) 0 0
\(291\) −2.91051 + 16.8552i −0.170617 + 0.988071i
\(292\) −5.81060 + 32.9535i −0.340040 + 1.92846i
\(293\) 21.1605 25.2182i 1.23621 1.47326i 0.407868 0.913041i \(-0.366272\pi\)
0.828344 0.560220i \(-0.189283\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.3320 + 17.8825i 0.889654 + 1.03765i
\(298\) 0 0
\(299\) 0 0
\(300\) −3.06808 17.0466i −0.177136 0.984186i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.3074 29.9773i 0.987787 1.71090i 0.358957 0.933354i \(-0.383132\pi\)
0.628830 0.777543i \(-0.283534\pi\)
\(308\) 23.6232 4.16541i 1.34606 0.237346i
\(309\) −16.1394 + 28.1843i −0.918136 + 1.60335i
\(310\) 0 0
\(311\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(312\) 0 0
\(313\) −0.499550 2.83309i −0.0282362 0.160136i 0.967429 0.253141i \(-0.0814637\pi\)
−0.995666 + 0.0930055i \(0.970353\pi\)
\(314\) 0 0
\(315\) −17.5000 2.95804i −0.986013 0.166667i
\(316\) 7.00995 + 12.1416i 0.394340 + 0.683018i
\(317\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(318\) 0 0
\(319\) 3.80699 21.5905i 0.213151 1.20884i
\(320\) 11.4985 13.7034i 0.642788 0.766044i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 8.77822 15.7144i 0.487679 0.873023i
\(325\) −35.3286 −1.95968
\(326\) 0 0
\(327\) −25.3717 9.13289i −1.40306 0.505050i
\(328\) 0 0
\(329\) −25.4253 4.48317i −1.40174 0.247165i
\(330\) 0 0
\(331\) −12.7840 + 10.7271i −0.702673 + 0.589612i −0.922533 0.385919i \(-0.873884\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −23.9617 + 13.8343i −1.31507 + 0.759254i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −9.22133 15.8419i −0.503065 0.864249i
\(337\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −5.29230 30.0141i −0.287015 1.62774i
\(341\) 0 0
\(342\) 0 0
\(343\) 9.26013 + 16.0390i 0.500000 + 0.866025i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(348\) −16.4881 + 2.96756i −0.883856 + 0.159078i
\(349\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(350\) 0 0
\(351\) −28.3742 23.2994i −1.51450 1.24363i
\(352\) 0 0
\(353\) −2.08193 + 5.72005i −0.110810 + 0.304447i −0.982686 0.185279i \(-0.940681\pi\)
0.871876 + 0.489726i \(0.162903\pi\)
\(354\) 0 0
\(355\) 2.31332 + 1.94111i 0.122778 + 0.103023i
\(356\) 0 0
\(357\) −30.7743 5.31401i −1.62875 0.281247i
\(358\) 0 0
\(359\) 32.1235 18.5465i 1.69541 0.978847i 0.745409 0.666608i \(-0.232254\pi\)
0.950004 0.312239i \(-0.101079\pi\)
\(360\) 0 0
\(361\) −9.50000 + 16.4545i −0.500000 + 0.866025i
\(362\) 0 0
\(363\) 16.5414 0.0585968i 0.868197 0.00307553i
\(364\) −35.1335 + 12.7875i −1.84150 + 0.670250i
\(365\) −12.7955 35.1554i −0.669747 1.84012i
\(366\) 0 0
\(367\) −1.03002 5.84153i −0.0537666 0.304925i 0.946051 0.324017i \(-0.105034\pi\)
−0.999818 + 0.0190919i \(0.993923\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(374\) 0 0
\(375\) 12.5000 + 14.7902i 0.645497 + 0.763763i
\(376\) 0 0
\(377\) 34.1712i 1.75990i
\(378\) 0 0
\(379\) 2.89457 0.148684 0.0743421 0.997233i \(-0.476314\pi\)
0.0743421 + 0.997233i \(0.476314\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.9416 2.98725i −0.865673 0.152641i −0.276859 0.960910i \(-0.589294\pi\)
−0.588813 + 0.808269i \(0.700405\pi\)
\(384\) 0 0
\(385\) −20.5445 + 17.2389i −1.04705 + 0.878576i
\(386\) 0 0
\(387\) 0 0
\(388\) 9.87539 17.1047i 0.501347 0.868359i
\(389\) 36.8123 6.49100i 1.86646 0.329107i 0.877768 0.479085i \(-0.159032\pi\)
0.988689 + 0.149979i \(0.0479205\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.5747 7.83736i −0.683018 0.394340i
\(396\) −9.48360 25.4925i −0.476569 1.28105i
\(397\) 17.3129 + 29.9868i 0.868910 + 1.50500i 0.863112 + 0.505013i \(0.168512\pi\)
0.00579782 + 0.999983i \(0.498154\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.47296 + 19.6962i −0.173648 + 0.984808i
\(401\) 19.0139 22.6599i 0.949510 1.13158i −0.0416801 0.999131i \(-0.513271\pi\)
0.991190 0.132450i \(-0.0422845\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.285150 + 20.1226i 0.0141692 + 0.999900i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 28.7287 24.1062i 1.41536 1.18763i
\(413\) 0 0
\(414\) 0 0
\(415\) 15.4672 26.7899i 0.759254 1.31507i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(420\) 17.7844 + 10.1840i 0.867791 + 0.496929i
\(421\) 7.09160 + 40.2184i 0.345623 + 1.96013i 0.269360 + 0.963039i \(0.413188\pi\)
0.0762630 + 0.997088i \(0.475701\pi\)
\(422\) 0 0
\(423\) 0.207402 + 29.2736i 0.0100843 + 1.42333i
\(424\) 0 0
\(425\) 21.9027 + 26.1026i 1.06243 + 1.26616i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −54.6013 + 9.82724i −2.63617 + 0.474464i
\(430\) 0 0
\(431\) 4.23679i 0.204079i 0.994780 + 0.102040i \(0.0325368\pi\)
−0.994780 + 0.102040i \(0.967463\pi\)
\(432\) −15.7791 + 13.5285i −0.759170 + 0.650892i
\(433\) 29.5653 1.42082 0.710410 0.703788i \(-0.248510\pi\)
0.710410 + 0.703788i \(0.248510\pi\)
\(434\) 0 0
\(435\) 14.3056 12.0905i 0.685903 0.579694i
\(436\) 23.8523 + 20.0144i 1.14232 + 0.958518i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(440\) 0 0
\(441\) 15.9909 13.6122i 0.761471 0.648199i
\(442\) 0 0
\(443\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.0614676 17.3518i −0.00290732 0.820711i
\(448\) 3.67544 + 20.8445i 0.173648 + 0.984808i
\(449\) 11.1822 + 6.45603i 0.527719 + 0.304679i 0.740087 0.672511i \(-0.234784\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.91909 16.9050i 0.137151 0.794264i
\(454\) 0 0
\(455\) 26.8694 32.0217i 1.25966 1.50120i
\(456\) 0 0
\(457\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(458\) 0 0
\(459\) 0.376318 + 35.4092i 0.0175650 + 1.65276i
\(460\) 0 0
\(461\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(462\) 0 0
\(463\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(464\) 19.0509 + 3.35918i 0.884414 + 0.155946i
\(465\) 0 0
\(466\) 0 0
\(467\) −32.6038 + 18.8238i −1.50872 + 0.871061i −0.508774 + 0.860900i \(0.669901\pi\)
−0.999948 + 0.0101613i \(0.996765\pi\)
\(468\) 21.4567 + 36.5635i 0.991838 + 1.69015i
\(469\) 0 0
\(470\) 0 0
\(471\) 21.0686 36.7922i 0.970788 1.69530i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 31.2298 + 18.0305i 1.43141 + 0.826428i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −17.9486 6.53274i −0.815844 0.296943i
\(485\) 22.0820i 1.00269i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.9464 + 4.75137i 1.21607 + 0.214427i 0.744635 0.667472i \(-0.232624\pi\)
0.471439 + 0.881899i \(0.343735\pi\)
\(492\) 0 0
\(493\) 25.2474 21.1851i 1.13709 0.954128i
\(494\) 0 0
\(495\) 23.4332 + 19.3816i 1.05324 + 0.871136i
\(496\) 0 0
\(497\) −3.51883 + 0.620464i −0.157841 + 0.0278316i
\(498\) 0 0
\(499\) −40.2340 + 14.6440i −1.80112 + 0.655555i −0.802890 + 0.596127i \(0.796706\pi\)
−0.998233 + 0.0594285i \(0.981072\pi\)
\(500\) −7.64780 21.0122i −0.342020 0.939693i
\(501\) −38.8211 22.2304i −1.73440 0.993181i
\(502\) 0 0
\(503\) 34.3190 + 19.8141i 1.53021 + 0.883466i 0.999352 + 0.0360049i \(0.0114632\pi\)
0.530857 + 0.847461i \(0.321870\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 60.0201 22.0866i 2.66559 0.980902i
\(508\) 0 0
\(509\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(510\) 0 0
\(511\) 41.5964 + 15.1398i 1.84012 + 0.669747i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −14.3406 + 39.4005i −0.631923 + 1.73620i
\(516\) 0 0
\(517\) 33.8866 + 28.4342i 1.49033 + 1.25054i
\(518\) 0 0
\(519\) 44.7380 + 7.72521i 1.96378 + 0.339099i
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) −9.99657 + 17.3146i −0.437120 + 0.757113i −0.997466 0.0711450i \(-0.977335\pi\)
0.560346 + 0.828258i \(0.310668\pi\)
\(524\) 0 0
\(525\) −22.9127 + 0.0811669i −0.999994 + 0.00354242i
\(526\) 0 0
\(527\) 0 0
\(528\) 0.111257 + 31.4070i 0.00484185 + 1.36681i
\(529\) −3.99391 22.6506i −0.173648 0.984808i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −24.1280 28.5487i −1.04120 1.23197i
\(538\) 0 0
\(539\) 31.7327i 1.36682i
\(540\) 7.71532 21.9197i 0.332015 0.943274i
\(541\) −24.9455 −1.07249 −0.536245 0.844062i \(-0.680158\pi\)
−0.536245 + 0.844062i \(0.680158\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −34.2833 6.04507i −1.46853 0.258942i
\(546\) 0 0
\(547\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 17.4281 6.34330i 0.741117 0.269745i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −15.2111 18.1279i −0.642788 0.766044i
\(561\) 41.1120 + 34.2496i 1.73575 + 1.44602i
\(562\) 0 0
\(563\) −28.5799 + 34.0602i −1.20450 + 1.43547i −0.334515 + 0.942390i \(0.608573\pi\)
−0.869985 + 0.493078i \(0.835872\pi\)
\(564\) 11.4487 31.8053i 0.482079 1.33924i
\(565\) 0 0
\(566\) 0 0
\(567\) −18.4559 15.0459i −0.775075 0.631869i
\(568\) 0 0
\(569\) −13.7193 + 37.6935i −0.575143 + 1.58019i 0.221123 + 0.975246i \(0.429028\pi\)
−0.796266 + 0.604947i \(0.793194\pi\)
\(570\) 0 0
\(571\) 31.0355 + 26.0419i 1.29879 + 1.08982i 0.990352 + 0.138575i \(0.0442523\pi\)
0.308443 + 0.951243i \(0.400192\pi\)
\(572\) 63.0879 + 11.1241i 2.63784 + 0.465122i
\(573\) 6.55877 7.87290i 0.273996 0.328895i
\(574\) 0 0
\(575\) 0 0
\(576\) 22.4939 8.36806i 0.937246 0.348669i
\(577\) 13.5294 23.4337i 0.563238 0.975556i −0.433974 0.900926i \(-0.642889\pi\)
0.997211 0.0746307i \(-0.0237778\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −20.3238 + 7.39724i −0.843898 + 0.307154i
\(581\) 12.5186 + 34.3947i 0.519360 + 1.42693i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −41.2150 23.4077i −1.70403 0.967791i
\(586\) 0 0
\(587\) 29.3852 + 35.0200i 1.21286 + 1.44543i 0.860414 + 0.509596i \(0.170205\pi\)
0.352445 + 0.935833i \(0.385350\pi\)
\(588\) −22.7568 + 8.37421i −0.938475 + 0.345347i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.621055i 0.0255037i 0.999919 + 0.0127518i \(0.00405915\pi\)
−0.999919 + 0.0127518i \(0.995941\pi\)
\(594\) 0 0
\(595\) −40.3175 −1.65286
\(596\) −6.85279 + 18.8279i −0.280701 + 0.771221i
\(597\) 0 0
\(598\) 0 0
\(599\) 40.5103 + 7.14306i 1.65521 + 0.291858i 0.921723 0.387849i \(-0.126782\pi\)
0.733484 + 0.679706i \(0.237893\pi\)
\(600\) 0 0
\(601\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.90452 + 17.1551i −0.403009 + 0.698032i
\(605\) 21.0305 3.70825i 0.855013 0.150762i
\(606\) 0 0
\(607\) −4.75528 + 1.73078i −0.193011 + 0.0702502i −0.436717 0.899599i \(-0.643859\pi\)
0.243706 + 0.969849i \(0.421637\pi\)
\(608\) 0 0
\(609\) 0.0785077 + 22.1621i 0.00318129 + 0.898052i
\(610\) 0 0
\(611\) −59.7108 34.4741i −2.41564 1.39467i
\(612\) 13.7124 38.5216i 0.554292 1.55714i
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(618\) 0 0
\(619\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −8.67128 48.1786i −0.347129 1.92869i
\(625\) 19.1511 + 16.0697i 0.766044 + 0.642788i
\(626\) 0 0
\(627\) 0 0
\(628\) −37.5028 + 31.4686i −1.49653 + 1.25573i
\(629\) 0 0
\(630\) 0 0
\(631\) 4.65857 8.06888i 0.185455 0.321217i −0.758275 0.651935i \(-0.773958\pi\)
0.943730 + 0.330718i \(0.107291\pi\)
\(632\) 0 0
\(633\) 24.9840 43.6298i 0.993027 1.73413i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.58865 + 48.7086i 0.340295 + 1.92991i
\(638\) 0 0
\(639\) 1.41264 + 3.79728i 0.0558833 + 0.150218i
\(640\) 0 0
\(641\) 5.19106 + 6.18647i 0.205035 + 0.244351i 0.858756 0.512384i \(-0.171238\pi\)
−0.653722 + 0.756735i \(0.726793\pi\)
\(642\) 0 0
\(643\) 2.66297 15.1025i 0.105017 0.595584i −0.886196 0.463311i \(-0.846661\pi\)
0.991213 0.132273i \(-0.0422275\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.2989i 1.26980i 0.772594 + 0.634901i \(0.218959\pi\)
−0.772594 + 0.634901i \(0.781041\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.36548 49.4908i 0.326368 1.93082i
\(658\) 0 0
\(659\) 22.6529 3.99431i 0.882430 0.155596i 0.285970 0.958239i \(-0.407684\pi\)
0.596461 + 0.802642i \(0.296573\pi\)
\(660\) −17.6648 30.3475i −0.687600 1.18127i
\(661\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(662\) 0 0
\(663\) −72.3755 41.4449i −2.81083 1.60959i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 33.2039 + 39.5709i 1.28470 + 1.53104i
\(669\) −43.2483 + 15.9148i −1.67207 + 0.615302i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(674\) 0 0
\(675\) 4.78316 + 25.5367i 0.184104 + 0.982907i
\(676\) −73.8488 −2.84034
\(677\) 5.72792 15.7373i 0.220142 0.604835i −0.779629 0.626242i \(-0.784592\pi\)
0.999771 + 0.0214069i \(0.00681454\pi\)
\(678\) 0 0
\(679\) −20.0151 16.7946i −0.768108 0.644519i
\(680\) 0 0
\(681\) −4.72595 0.816062i −0.181099 0.0312716i
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(692\) −45.4001 26.2118i −1.72585 0.996422i
\(693\) −35.3897 + 6.49900i −1.34434 + 0.246877i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 24.8619 + 9.04900i 0.939693 + 0.342020i
\(701\) 50.2017i 1.89609i −0.318131 0.948047i \(-0.603055\pi\)
0.318131 0.948047i \(-0.396945\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 12.4037 34.0788i 0.467481 1.28439i
\(705\) 6.69449 + 37.1954i 0.252129 + 1.40086i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 35.7063 29.9612i 1.34098 1.12521i 0.359605 0.933105i \(-0.382911\pi\)
0.981373 0.192110i \(-0.0615331\pi\)
\(710\) 0 0
\(711\) −10.6437 18.1374i −0.399170 0.680207i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −67.3032 + 24.4964i −2.51700 + 0.916112i
\(716\) 14.7621 + 40.5585i 0.551686 + 1.51574i
\(717\) −33.8071 + 19.6786i −1.26255 + 0.734910i
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) −17.1017 + 20.6768i −0.637344 + 0.770579i
\(721\) −24.8056 42.9646i −0.923810 1.60009i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.5432 18.5237i 0.577260 0.687952i
\(726\) 0 0
\(727\) −28.6062 10.4118i −1.06095 0.386152i −0.248162 0.968719i \(-0.579826\pi\)
−0.812783 + 0.582566i \(0.802049\pi\)
\(728\) 0 0
\(729\) −13.0000 + 23.6643i −0.481481 + 0.876456i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −32.6052 27.3590i −1.20430 1.01053i −0.999497 0.0317250i \(-0.989900\pi\)
−0.204804 0.978803i \(-0.565656\pi\)
\(734\) 0 0
\(735\) 17.3529 20.8297i 0.640070 0.768317i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.3056 + 35.1704i −0.746955 + 1.29376i 0.202321 + 0.979319i \(0.435152\pi\)
−0.949276 + 0.314445i \(0.898182\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(744\) 0 0
\(745\) −3.88992 22.0609i −0.142516 0.808247i
\(746\) 0 0
\(747\) 35.7946 21.0055i 1.30965 0.768552i
\(748\) −30.8935 53.5092i −1.12958 1.95649i
\(749\) 0 0
\(750\) 0 0
\(751\) 9.13585 51.8120i 0.333372 1.89065i −0.109376 0.994000i \(-0.534885\pi\)
0.442748 0.896646i \(-0.354004\pi\)
\(752\) −25.0896 + 29.9006i −0.914923 + 1.09036i
\(753\) 0 0
\(754\) 0 0
\(755\) 22.1472i 0.806018i
\(756\) 14.0000 + 23.6643i 0.509175 + 0.860663i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(762\) 0 0
\(763\) 31.5536 26.4766i 1.14232 0.958518i
\(764\) −10.2470 + 5.91608i −0.370722 + 0.214036i
\(765\) 8.25722 + 44.9638i 0.298540 + 1.62567i
\(766\) 0 0
\(767\) 0 0
\(768\) −27.7126 + 0.0981703i −0.999994 + 0.00354242i
\(769\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(770\) 0 0
\(771\) −0.196606 55.5002i −0.00708059 1.99879i
\(772\) 0 0
\(773\) −47.0498 27.1642i −1.69226 0.977028i −0.952686 0.303956i \(-0.901693\pi\)
−0.739576 0.673073i \(-0.764974\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 35.3286 + 41.8014i 1.26497 + 1.49673i
\(781\) 5.75296 + 2.09391i 0.205857 + 0.0749259i
\(782\) 0 0
\(783\) 24.7000 4.62645i 0.882707 0.165336i
\(784\) 28.0000 1.00000
\(785\) 18.7205 51.4341i 0.668162 1.83576i
\(786\) 0 0
\(787\) −9.87831 8.28889i −0.352124 0.295467i 0.449518 0.893271i \(-0.351596\pi\)
−0.801642 + 0.597804i \(0.796040\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −17.0380 46.8115i −0.603516 1.65815i −0.744092 0.668077i \(-0.767118\pi\)
0.140576 0.990070i \(-0.455105\pi\)
\(798\) 0 0
\(799\) 11.5477 + 65.4902i 0.408528 + 2.31688i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −48.7525 58.1009i −1.72044 2.05034i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.7926i 1.99672i 0.0572419 + 0.998360i \(0.481769\pi\)
−0.0572419 + 0.998360i \(0.518231\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 8.75254 24.0474i 0.307154 0.843898i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −30.2209 + 36.2761i −1.05794 + 1.26992i
\(817\) 0 0
\(818\) 0 0
\(819\) 52.5630 19.5542i 1.83670 0.683279i
\(820\) 0 0
\(821\) −13.7241 + 2.41994i −0.478976 + 0.0844563i −0.407923 0.913016i \(-0.633747\pi\)
−0.0710524 + 0.997473i \(0.522636\pi\)
\(822\) 0 0
\(823\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(824\) 0 0
\(825\) 34.0686 + 19.5089i 1.18612 + 0.679214i
\(826\) 0 0
\(827\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(828\) 0 0
\(829\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −9.81560 + 55.6670i −0.340295 + 1.92991i
\(833\) 30.6637 36.5436i 1.06243 1.26616i
\(834\) 0 0
\(835\) −54.2704 19.7528i −1.87810 0.683574i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(840\) 0 0
\(841\) 4.29849 + 3.60686i 0.148224 + 0.124374i
\(842\) 0 0
\(843\) 28.4693 + 4.91599i 0.980535 + 0.169316i
\(844\) −44.4725 + 37.3169i −1.53081 + 1.28450i
\(845\) 71.5038 41.2828i 2.45981 1.42017i
\(846\) 0 0
\(847\) −12.6338 + 21.8823i −0.434101 + 0.751886i
\(848\) 0 0
\(849\) −40.8528 + 0.144718i −1.40206 + 0.00496672i
\(850\) 0 0
\(851\) 0 0
\(852\) −0.0165725 4.67827i −0.000567764 0.160275i
\(853\) 2.37739 + 13.4828i 0.0814003 + 0.461644i 0.998075 + 0.0620108i \(0.0197513\pi\)
−0.916675 + 0.399633i \(0.869138\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −37.3410 44.5013i −1.27554 1.52013i −0.733281 0.679925i \(-0.762012\pi\)
−0.542263 0.840209i \(-0.682432\pi\)
\(858\) 0 0
\(859\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 58.6113 1.99284
\(866\) 0 0
\(867\) 9.03329 + 50.1900i 0.306787 + 1.70454i
\(868\) 0 0
\(869\) −31.2950 5.51815i −1.06161 0.187190i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −14.6309 + 25.7613i −0.495182 + 0.871889i
\(874\) 0 0
\(875\) −29.1310 + 5.13658i −0.984808 + 0.173648i
\(876\) −28.8009 + 50.2952i −0.973091 + 1.69932i
\(877\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(878\) 0 0
\(879\) 49.2786 28.6843i 1.66213 0.967496i
\(880\) 7.04083 + 39.9305i 0.237346 + 1.34606i
\(881\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(882\) 0 0
\(883\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(884\) 61.9032 + 73.7734i 2.08203 + 2.48127i
\(885\) 0 0
\(886\) 0 0
\(887\) −38.0091 + 45.2975i −1.27622 + 1.52094i −0.545873 + 0.837868i \(0.683802\pi\)
−0.730349 + 0.683074i \(0.760642\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 14.4960 + 38.1371i 0.485633 + 1.27764i
\(892\) 53.2127 1.78169
\(893\) 0 0
\(894\) 0 0
\(895\) −36.9663 31.0184i −1.23565 1.03683i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 5.00000 29.5804i 0.166667 0.986013i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(908\) 4.79589 + 2.76891i 0.159157 + 0.0918895i
\(909\) 0 0
\(910\) 0 0
\(911\) −0.992804 1.18318i −0.0328931 0.0392004i 0.749347 0.662177i \(-0.230367\pi\)
−0.782240 + 0.622977i \(0.785923\pi\)
\(912\) 0 0
\(913\) 10.8902 61.7612i 0.360412 2.04400i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 31.6113 1.04276 0.521380 0.853325i \(-0.325417\pi\)
0.521380 + 0.853325i \(0.325417\pi\)
\(920\) 0 0
\(921\) 45.7911 38.7006i 1.50887 1.27523i
\(922\) 0 0
\(923\) −9.39735 1.65701i −0.309318 0.0545410i
\(924\) 40.9419 + 7.06971i 1.34689 + 0.232576i
\(925\) 0 0
\(926\) 0 0
\(927\) −42.8357 + 36.4637i −1.40691 + 1.19763i
\(928\) 0 0
\(929\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 59.8251 + 34.5400i 1.95649 + 1.12958i
\(936\) 0 0
\(937\) −27.4440 47.5345i −0.896558 1.55288i −0.831864 0.554979i \(-0.812726\pi\)
−0.0646935 0.997905i \(-0.520607\pi\)
\(938\) 0 0
\(939\) 0.847857 4.91009i 0.0276688 0.160235i
\(940\) 7.57794 42.9766i 0.247165 1.40174i
\(941\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −26.7842 15.0866i −0.871290 0.490768i
\(946\) 0 0
\(947\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(948\) 4.30142 + 23.8992i 0.139704 + 0.776209i
\(949\) 90.5589 + 75.9879i 2.93967 + 2.46667i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) 6.61438 11.4564i 0.214036 0.370722i
\(956\) 44.4826 7.84349i 1.43867 0.253677i
\(957\) 18.8698 32.9524i 0.609973 1.06520i
\(958\) 0 0
\(959\) 0 0
\(960\) 26.7778 15.5869i 0.864249 0.503065i
\(961\) 5.38309 + 30.5290i 0.173648 + 0.984808i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 23.5242 20.4600i 0.754540 0.656254i
\(973\) 0 0
\(974\) 0 0
\(975\) −57.5745 20.7247i −1.84386 0.663722i
\(976\) 0 0
\(977\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −27.1109 + 15.6525i −0.866025 + 0.500000i
\(981\) −35.9902 29.7674i −1.14908 0.950401i
\(982\) 0 0
\(983\) −8.18662 + 1.44352i −0.261113 + 0.0460412i −0.302672 0.953095i \(-0.597879\pi\)
0.0415592 + 0.999136i \(0.486767\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −38.8053 22.2213i −1.23519 0.707313i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −18.3618 31.8035i −0.583280 1.01027i −0.995087 0.0990001i \(-0.968436\pi\)
0.411807 0.911271i \(-0.364898\pi\)
\(992\) 0 0
\(993\) −27.1267 + 9.98227i −0.860839 + 0.316778i
\(994\) 0 0
\(995\) 0 0
\(996\) −47.1655 + 8.48893i −1.49450 + 0.268982i
\(997\) −17.4034 6.33430i −0.551170 0.200609i 0.0513964 0.998678i \(-0.483633\pi\)
−0.602566 + 0.798069i \(0.705855\pi\)
\(998\) 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 945.2.cs.a.104.4 yes 24
5.4 even 2 inner 945.2.cs.a.104.1 24
7.6 odd 2 inner 945.2.cs.a.104.1 24
27.20 odd 18 inner 945.2.cs.a.209.4 yes 24
35.34 odd 2 CM 945.2.cs.a.104.4 yes 24
135.74 odd 18 inner 945.2.cs.a.209.1 yes 24
189.20 even 18 inner 945.2.cs.a.209.1 yes 24
945.209 even 18 inner 945.2.cs.a.209.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
945.2.cs.a.104.1 24 5.4 even 2 inner
945.2.cs.a.104.1 24 7.6 odd 2 inner
945.2.cs.a.104.4 yes 24 1.1 even 1 trivial
945.2.cs.a.104.4 yes 24 35.34 odd 2 CM
945.2.cs.a.209.1 yes 24 135.74 odd 18 inner
945.2.cs.a.209.1 yes 24 189.20 even 18 inner
945.2.cs.a.209.4 yes 24 27.20 odd 18 inner
945.2.cs.a.209.4 yes 24 945.209 even 18 inner